LMFDB - Embedded newform 2352.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.27492 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.27492 q^{5} +1.00000 q^{9} +4.27492 q^{11} +1.27492 q^{13} -4.27492 q^{15} +4.00000 q^{17} +1.27492 q^{19} -4.00000 q^{23} +13.2749 q^{25} -1.00000 q^{27} -2.27492 q^{29} -1.00000 q^{31} -4.27492 q^{33} +5.27492 q^{37} -1.27492 q^{39} -10.5498 q^{41} +7.27492 q^{43} +4.27492 q^{45} -6.00000 q^{47} -4.00000 q^{51} +1.72508 q^{53} +18.2749 q^{55} -1.27492 q^{57} +6.27492 q^{59} -10.0000 q^{61} +5.45017 q^{65} -7.27492 q^{67} +4.00000 q^{69} -2.00000 q^{71} -3.27492 q^{73} -13.2749 q^{75} +3.54983 q^{79} +1.00000 q^{81} -0.274917 q^{83} +17.0997 q^{85} +2.27492 q^{87} -4.54983 q^{89} +1.00000 q^{93} +5.45017 q^{95} -16.2749 q^{97} +4.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + q^5 + 2 * q^9	$ 2 q - 2 q^{3} + q^{5} + 2 q^{9} + q^{11} - 5 q^{13} - q^{15} + 8 q^{17} - 5 q^{19} - 8 q^{23} + 19 q^{25} - 2 q^{27} + 3 q^{29} - 2 q^{31} - q^{33} + 3 q^{37} + 5 q^{39} - 6 q^{41} + 7 q^{43} + q^{45} - 12 q^{47} - 8 q^{51} + 11 q^{53} + 29 q^{55} + 5 q^{57} + 5 q^{59} - 20 q^{61} + 26 q^{65} - 7 q^{67} + 8 q^{69} - 4 q^{71} + q^{73} - 19 q^{75} - 8 q^{79} + 2 q^{81} + 7 q^{83} + 4 q^{85} - 3 q^{87} + 6 q^{89} + 2 q^{93} + 26 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + q^5 + 2 * q^9 + q^11 - 5 * q^13 - q^15 + 8 * q^17 - 5 * q^19 - 8 * q^23 + 19 * q^25 - 2 * q^27 + 3 * q^29 - 2 * q^31 - q^33 + 3 * q^37 + 5 * q^39 - 6 * q^41 + 7 * q^43 + q^45 - 12 * q^47 - 8 * q^51 + 11 * q^53 + 29 * q^55 + 5 * q^57 + 5 * q^59 - 20 * q^61 + 26 * q^65 - 7 * q^67 + 8 * q^69 - 4 * q^71 + q^73 - 19 * q^75 - 8 * q^79 + 2 * q^81 + 7 * q^83 + 4 * q^85 - 3 * q^87 + 6 * q^89 + 2 * q^93 + 26 * q^95 - 25 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	4.27492	1.91180	0.955901	−	0.293691i	$-0.0948835\pi$
$5$	4.27492	1.91180	0.955901	+	0.293691i	$0.0948835\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.27492	1.28894	0.644468	−	0.764631i	$-0.277079\pi$
$11$	4.27492	1.28894	0.644468	+	0.764631i	$0.277079\pi$
$12$	0	0
$13$	1.27492	0.353598	0.176799	−	0.984247i	$-0.443426\pi$
$13$	1.27492	0.353598	0.176799	+	0.984247i	$0.443426\pi$
$14$	0	0
$15$	−4.27492	−1.10378
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	1.27492	0.292486	0.146243	−	0.989249i	$-0.453282\pi$
$19$	1.27492	0.292486	0.146243	+	0.989249i	$0.453282\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	13.2749	2.65498
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.27492	−0.422442	−0.211221	−	0.977438i	$-0.567744\pi$
$29$	−2.27492	−0.422442	−0.211221	+	0.977438i	$0.567744\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	−4.27492	−0.744168
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.27492	0.867191	0.433596	−	0.901108i	$-0.357245\pi$
$37$	5.27492	0.867191	0.433596	+	0.901108i	$0.357245\pi$
$38$	0	0
$39$	−1.27492	−0.204150
$40$	0	0
$41$	−10.5498	−1.64761	−0.823804	−	0.566875i	$-0.808152\pi$
$41$	−10.5498	−1.64761	−0.823804	+	0.566875i	$0.808152\pi$
$42$	0	0
$43$	7.27492	1.10941	0.554707	−	0.832046i	$-0.312830\pi$
$43$	7.27492	1.10941	0.554707	+	0.832046i	$0.312830\pi$
$44$	0	0
$45$	4.27492	0.637267
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	1.72508	0.236958	0.118479	−	0.992957i	$-0.462198\pi$
$53$	1.72508	0.236958	0.118479	+	0.992957i	$0.462198\pi$
$54$	0	0
$55$	18.2749	2.46419
$56$	0	0
$57$	−1.27492	−0.168867
$58$	0	0
$59$	6.27492	0.816925	0.408462	−	0.912775i	$-0.366065\pi$
$59$	6.27492	0.816925	0.408462	+	0.912775i	$0.366065\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.45017	0.676010
$66$	0	0
$67$	−7.27492	−0.888773	−0.444386	−	0.895835i	$-0.646578\pi$
$67$	−7.27492	−0.888773	−0.444386	+	0.895835i	$0.646578\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−3.27492	−0.383300	−0.191650	−	0.981463i	$-0.561384\pi$
$73$	−3.27492	−0.383300	−0.191650	+	0.981463i	$0.561384\pi$
$74$	0	0
$75$	−13.2749	−1.53286
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.54983	0.399388	0.199694	−	0.979858i	$-0.436005\pi$
$79$	3.54983	0.399388	0.199694	+	0.979858i	$0.436005\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.274917	−0.0301761	−0.0150880	−	0.999886i	$-0.504803\pi$
$83$	−0.274917	−0.0301761	−0.0150880	+	0.999886i	$0.504803\pi$
$84$	0	0
$85$	17.0997	1.85472
$86$	0	0
$87$	2.27492	0.243897
$88$	0	0
$89$	−4.54983	−0.482281	−0.241141	−	0.970490i	$-0.577522\pi$
$89$	−4.54983	−0.482281	−0.241141	+	0.970490i	$0.577522\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	5.45017	0.559175
$96$	0	0
$97$	−16.2749	−1.65247	−0.826234	−	0.563327i	$-0.809521\pi$
$97$	−16.2749	−1.65247	−0.826234	+	0.563327i	$0.809521\pi$
$98$	0	0
$99$	4.27492	0.429645
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−11.8248	−1.16513	−0.582564	−	0.812785i	$-0.697950\pi$
$103$	−11.8248	−1.16513	−0.582564	+	0.812785i	$0.697950\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.82475	−0.659774	−0.329887	−	0.944020i	$-0.607011\pi$
$107$	−6.82475	−0.659774	−0.329887	+	0.944020i	$0.607011\pi$
$108$	0	0
$109$	−5.82475	−0.557910	−0.278955	−	0.960304i	$-0.589988\pi$
$109$	−5.82475	−0.557910	−0.278955	+	0.960304i	$0.589988\pi$
$110$	0	0
$111$	−5.27492	−0.500673
$112$	0	0
$113$	10.5498	0.992445	0.496222	−	0.868195i	$-0.334720\pi$
$113$	10.5498	0.992445	0.496222	+	0.868195i	$0.334720\pi$
$114$	0	0
$115$	−17.0997	−1.59455
$116$	0	0
$117$	1.27492	0.117866
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.27492	0.661356
$122$	0	0
$123$	10.5498	0.951247
$124$	0	0
$125$	35.3746	3.16400
$126$	0	0
$127$	21.5498	1.91224	0.956119	−	0.292978i	$-0.0946462\pi$
$127$	21.5498	1.91224	0.956119	+	0.292978i	$0.0946462\pi$
$128$	0	0
$129$	−7.27492	−0.640521
$130$	0	0
$131$	−0.274917	−0.0240196	−0.0120098	−	0.999928i	$-0.503823\pi$
$131$	−0.274917	−0.0240196	−0.0120098	+	0.999928i	$0.503823\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.27492	−0.367926
$136$	0	0
$137$	16.5498	1.41395	0.706974	−	0.707240i	$-0.250060\pi$
$137$	16.5498	1.41395	0.706974	+	0.707240i	$0.250060\pi$
$138$	0	0
$139$	0.725083	0.0615007	0.0307504	−	0.999527i	$-0.490210\pi$
$139$	0.725083	0.0615007	0.0307504	+	0.999527i	$0.490210\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	5.45017	0.455766
$144$	0	0
$145$	−9.72508	−0.807624
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.549834	0.0450442	0.0225221	−	0.999746i	$-0.492830\pi$
$149$	0.549834	0.0450442	0.0225221	+	0.999746i	$0.492830\pi$
$150$	0	0
$151$	15.3746	1.25117	0.625583	−	0.780158i	$-0.284861\pi$
$151$	15.3746	1.25117	0.625583	+	0.780158i	$0.284861\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−4.27492	−0.343370
$156$	0	0
$157$	−14.5498	−1.16120	−0.580602	−	0.814188i	$-0.697183\pi$
$157$	−14.5498	−1.16120	−0.580602	+	0.814188i	$0.697183\pi$
$158$	0	0
$159$	−1.72508	−0.136808
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	−18.2749	−1.42270
$166$	0	0
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	−11.3746	−0.874968
$170$	0	0
$171$	1.27492	0.0974954
$172$	0	0
$173$	7.45017	0.566426	0.283213	−	0.959057i	$-0.408600\pi$
$173$	7.45017	0.566426	0.283213	+	0.959057i	$0.408600\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.27492	−0.471652
$178$	0	0
$179$	1.45017	0.108390	0.0541952	−	0.998530i	$-0.482741\pi$
$179$	1.45017	0.108390	0.0541952	+	0.998530i	$0.482741\pi$
$180$	0	0
$181$	−3.82475	−0.284292	−0.142146	−	0.989846i	$-0.545400\pi$
$181$	−3.82475	−0.284292	−0.142146	+	0.989846i	$0.545400\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	22.5498	1.65790
$186$	0	0
$187$	17.0997	1.25045
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.5498	0.763359	0.381680	−	0.924295i	$-0.375346\pi$
$191$	10.5498	0.763359	0.381680	+	0.924295i	$0.375346\pi$
$192$	0	0
$193$	15.5498	1.11930	0.559651	−	0.828729i	$-0.310935\pi$
$193$	15.5498	1.11930	0.559651	+	0.828729i	$0.310935\pi$
$194$	0	0
$195$	−5.45017	−0.390294
$196$	0	0
$197$	−16.5498	−1.17913	−0.589563	−	0.807722i	$-0.700700\pi$
$197$	−16.5498	−1.17913	−0.589563	+	0.807722i	$0.700700\pi$
$198$	0	0
$199$	25.0997	1.77927	0.889634	−	0.456674i	$-0.150959\pi$
$199$	25.0997	1.77927	0.889634	+	0.456674i	$0.150959\pi$
$200$	0	0
$201$	7.27492	0.513133
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−45.0997	−3.14990
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	5.45017	0.376996
$210$	0	0
$211$	−17.6495	−1.21504	−0.607521	−	0.794304i	$-0.707836\pi$
$211$	−17.6495	−1.21504	−0.607521	+	0.794304i	$0.707836\pi$
$212$	0	0
$213$	2.00000	0.137038
$214$	0	0
$215$	31.0997	2.12098
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.27492	0.221298
$220$	0	0
$221$	5.09967	0.343041
$222$	0	0
$223$	6.27492	0.420200	0.210100	−	0.977680i	$-0.432621\pi$
$223$	6.27492	0.420200	0.210100	+	0.977680i	$0.432621\pi$
$224$	0	0
$225$	13.2749	0.884994
$226$	0	0
$227$	−3.72508	−0.247242	−0.123621	−	0.992329i	$-0.539451\pi$
$227$	−3.72508	−0.247242	−0.123621	+	0.992329i	$0.539451\pi$
$228$	0	0
$229$	−10.7251	−0.708733	−0.354367	−	0.935107i	$-0.615304\pi$
$229$	−10.7251	−0.708733	−0.354367	+	0.935107i	$0.615304\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.5498	0.953191	0.476596	−	0.879123i	$-0.341871\pi$
$233$	14.5498	0.953191	0.476596	+	0.879123i	$0.341871\pi$
$234$	0	0
$235$	−25.6495	−1.67319
$236$	0	0
$237$	−3.54983	−0.230587
$238$	0	0
$239$	−30.5498	−1.97610	−0.988052	−	0.154119i	$-0.950746\pi$
$239$	−30.5498	−1.97610	−0.988052	+	0.154119i	$0.950746\pi$
$240$	0	0
$241$	12.8248	0.826115	0.413057	−	0.910705i	$-0.364461\pi$
$241$	12.8248	0.826115	0.413057	+	0.910705i	$0.364461\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.62541	0.103423
$248$	0	0
$249$	0.274917	0.0174222
$250$	0	0
$251$	19.3746	1.22291	0.611457	−	0.791278i	$-0.290584\pi$
$251$	19.3746	1.22291	0.611457	+	0.791278i	$0.290584\pi$
$252$	0	0
$253$	−17.0997	−1.07505
$254$	0	0
$255$	−17.0997	−1.07082
$256$	0	0
$257$	19.0997	1.19140	0.595702	−	0.803205i	$-0.296874\pi$
$257$	19.0997	1.19140	0.595702	+	0.803205i	$0.296874\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.27492	−0.140814
$262$	0	0
$263$	24.5498	1.51381	0.756904	−	0.653526i	$-0.226711\pi$
$263$	24.5498	1.51381	0.756904	+	0.653526i	$0.226711\pi$
$264$	0	0
$265$	7.37459	0.453017
$266$	0	0
$267$	4.54983	0.278445
$268$	0	0
$269$	−28.2749	−1.72395	−0.861976	−	0.506949i	$-0.830773\pi$
$269$	−28.2749	−1.72395	−0.861976	+	0.506949i	$0.830773\pi$
$270$	0	0
$271$	−6.27492	−0.381174	−0.190587	−	0.981670i	$-0.561039\pi$
$271$	−6.27492	−0.381174	−0.190587	+	0.981670i	$0.561039\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	56.7492	3.42210
$276$	0	0
$277$	−4.17525	−0.250866	−0.125433	−	0.992102i	$-0.540032\pi$
$277$	−4.17525	−0.250866	−0.125433	+	0.992102i	$0.540032\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−11.4502	−0.683060	−0.341530	−	0.939871i	$-0.610945\pi$
$281$	−11.4502	−0.683060	−0.341530	+	0.939871i	$0.610945\pi$
$282$	0	0
$283$	26.9244	1.60049	0.800245	−	0.599673i	$-0.204703\pi$
$283$	26.9244	1.60049	0.800245	+	0.599673i	$0.204703\pi$
$284$	0	0
$285$	−5.45017	−0.322840
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	16.2749	0.954053
$292$	0	0
$293$	5.17525	0.302341	0.151171	−	0.988508i	$-0.451696\pi$
$293$	5.17525	0.302341	0.151171	+	0.988508i	$0.451696\pi$
$294$	0	0
$295$	26.8248	1.56180
$296$	0	0
$297$	−4.27492	−0.248056
$298$	0	0
$299$	−5.09967	−0.294921
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−42.7492	−2.44781
$306$	0	0
$307$	−26.3746	−1.50528	−0.752639	−	0.658434i	$-0.771219\pi$
$307$	−26.3746	−1.50528	−0.752639	+	0.658434i	$0.771219\pi$
$308$	0	0
$309$	11.8248	0.672687
$310$	0	0
$311$	10.5498	0.598226	0.299113	−	0.954218i	$-0.403309\pi$
$311$	10.5498	0.598226	0.299113	+	0.954218i	$0.403309\pi$
$312$	0	0
$313$	4.45017	0.251538	0.125769	−	0.992060i	$-0.459860\pi$
$313$	4.45017	0.251538	0.125769	+	0.992060i	$0.459860\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.17525	−0.290671	−0.145335	−	0.989382i	$-0.546426\pi$
$317$	−5.17525	−0.290671	−0.145335	+	0.989382i	$0.546426\pi$
$318$	0	0
$319$	−9.72508	−0.544500
$320$	0	0
$321$	6.82475	0.380920
$322$	0	0
$323$	5.09967	0.283753
$324$	0	0
$325$	16.9244	0.938798
$326$	0	0
$327$	5.82475	0.322110
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−23.8248	−1.30953	−0.654763	−	0.755834i	$-0.727232\pi$
$331$	−23.8248	−1.30953	−0.654763	+	0.755834i	$0.727232\pi$
$332$	0	0
$333$	5.27492	0.289064
$334$	0	0
$335$	−31.0997	−1.69916
$336$	0	0
$337$	6.09967	0.332270	0.166135	−	0.986103i	$-0.446871\pi$
$337$	6.09967	0.332270	0.166135	+	0.986103i	$0.446871\pi$
$338$	0	0
$339$	−10.5498	−0.572988
$340$	0	0
$341$	−4.27492	−0.231500
$342$	0	0
$343$	0	0
$344$	0	0
$345$	17.0997	0.920615
$346$	0	0
$347$	30.1993	1.62119	0.810593	−	0.585610i	$-0.199145\pi$
$347$	30.1993	1.62119	0.810593	+	0.585610i	$0.199145\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	−1.27492	−0.0680500
$352$	0	0
$353$	5.45017	0.290083	0.145042	−	0.989426i	$-0.453668\pi$
$353$	5.45017	0.290083	0.145042	+	0.989426i	$0.453668\pi$
$354$	0	0
$355$	−8.54983	−0.453778
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.6495	−1.35373	−0.676865	−	0.736108i	$-0.736662\pi$
$359$	−25.6495	−1.35373	−0.676865	+	0.736108i	$0.736662\pi$
$360$	0	0
$361$	−17.3746	−0.914452
$362$	0	0
$363$	−7.27492	−0.381834
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	−8.09967	−0.422799	−0.211400	−	0.977400i	$-0.567802\pi$
$367$	−8.09967	−0.422799	−0.211400	+	0.977400i	$0.567802\pi$
$368$	0	0
$369$	−10.5498	−0.549202
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.27492	0.0660127	0.0330064	−	0.999455i	$-0.489492\pi$
$373$	1.27492	0.0660127	0.0330064	+	0.999455i	$0.489492\pi$
$374$	0	0
$375$	−35.3746	−1.82674
$376$	0	0
$377$	−2.90033	−0.149375
$378$	0	0
$379$	−35.8248	−1.84019	−0.920097	−	0.391691i	$-0.871890\pi$
$379$	−35.8248	−1.84019	−0.920097	+	0.391691i	$0.871890\pi$
$380$	0	0
$381$	−21.5498	−1.10403
$382$	0	0
$383$	−4.54983	−0.232486	−0.116243	−	0.993221i	$-0.537085\pi$
$383$	−4.54983	−0.232486	−0.116243	+	0.993221i	$0.537085\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.27492	0.369805
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	0.274917	0.0138677
$394$	0	0
$395$	15.1752	0.763550
$396$	0	0
$397$	−34.3746	−1.72521	−0.862606	−	0.505877i	$-0.831169\pi$
$397$	−34.3746	−1.72521	−0.862606	+	0.505877i	$0.831169\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	−1.27492	−0.0635082
$404$	0	0
$405$	4.27492	0.212422
$406$	0	0
$407$	22.5498	1.11775
$408$	0	0
$409$	−25.5498	−1.26336	−0.631679	−	0.775230i	$-0.717634\pi$
$409$	−25.5498	−1.26336	−0.631679	+	0.775230i	$0.717634\pi$
$410$	0	0
$411$	−16.5498	−0.816343
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.17525	−0.0576907
$416$	0	0
$417$	−0.725083	−0.0355075
$418$	0	0
$419$	13.4502	0.657084	0.328542	−	0.944489i	$-0.393443\pi$
$419$	13.4502	0.657084	0.328542	+	0.944489i	$0.393443\pi$
$420$	0	0
$421$	−13.8248	−0.673777	−0.336889	−	0.941545i	$-0.609375\pi$
$421$	−13.8248	−0.673777	−0.336889	+	0.941545i	$0.609375\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	53.0997	2.57571
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.45017	−0.263136
$430$	0	0
$431$	−27.6495	−1.33183	−0.665915	−	0.746028i	$-0.731959\pi$
$431$	−27.6495	−1.33183	−0.665915	+	0.746028i	$0.731959\pi$
$432$	0	0
$433$	−25.8248	−1.24106	−0.620529	−	0.784183i	$-0.713082\pi$
$433$	−25.8248	−1.24106	−0.620529	+	0.784183i	$0.713082\pi$
$434$	0	0
$435$	9.72508	0.466282
$436$	0	0
$437$	−5.09967	−0.243950
$438$	0	0
$439$	9.72508	0.464153	0.232076	−	0.972698i	$-0.425448\pi$
$439$	9.72508	0.464153	0.232076	+	0.972698i	$0.425448\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−31.3746	−1.49065	−0.745326	−	0.666700i	$-0.767706\pi$
$443$	−31.3746	−1.49065	−0.745326	+	0.666700i	$0.767706\pi$
$444$	0	0
$445$	−19.4502	−0.922026
$446$	0	0
$447$	−0.549834	−0.0260063
$448$	0	0
$449$	5.45017	0.257209	0.128605	−	0.991696i	$-0.458950\pi$
$449$	5.45017	0.257209	0.128605	+	0.991696i	$0.458950\pi$
$450$	0	0
$451$	−45.0997	−2.12366
$452$	0	0
$453$	−15.3746	−0.722361
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.64950	−0.404607	−0.202303	−	0.979323i	$-0.564843\pi$
$457$	−8.64950	−0.404607	−0.202303	+	0.979323i	$0.564843\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−41.6495	−1.93981	−0.969905	−	0.243482i	$-0.921710\pi$
$461$	−41.6495	−1.93981	−0.969905	+	0.243482i	$0.921710\pi$
$462$	0	0
$463$	−35.8248	−1.66492	−0.832459	−	0.554087i	$-0.813067\pi$
$463$	−35.8248	−1.66492	−0.832459	+	0.554087i	$0.813067\pi$
$464$	0	0
$465$	4.27492	0.198245
$466$	0	0
$467$	25.4502	1.17769	0.588847	−	0.808245i	$-0.299582\pi$
$467$	25.4502	1.17769	0.588847	+	0.808245i	$0.299582\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.5498	0.670421
$472$	0	0
$473$	31.0997	1.42996
$474$	0	0
$475$	16.9244	0.776546
$476$	0	0
$477$	1.72508	0.0789861
$478$	0	0
$479$	−20.5498	−0.938946	−0.469473	−	0.882947i	$-0.655556\pi$
$479$	−20.5498	−0.938946	−0.469473	+	0.882947i	$0.655556\pi$
$480$	0	0
$481$	6.72508	0.306637
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−69.5739	−3.15919
$486$	0	0
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	15.9244	0.718659	0.359330	−	0.933211i	$-0.383005\pi$
$491$	15.9244	0.718659	0.359330	+	0.933211i	$0.383005\pi$
$492$	0	0
$493$	−9.09967	−0.409828
$494$	0	0
$495$	18.2749	0.821396
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−24.7251	−1.10685	−0.553423	−	0.832900i	$-0.686679\pi$
$499$	−24.7251	−1.10685	−0.553423	+	0.832900i	$0.686679\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	0	0
$503$	7.64950	0.341074	0.170537	−	0.985351i	$-0.445450\pi$
$503$	7.64950	0.341074	0.170537	+	0.985351i	$0.445450\pi$
$504$	0	0
$505$	25.6495	1.14139
$506$	0	0
$507$	11.3746	0.505163
$508$	0	0
$509$	3.72508	0.165111	0.0825557	−	0.996586i	$-0.473692\pi$
$509$	3.72508	0.165111	0.0825557	+	0.996586i	$0.473692\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.27492	−0.0562890
$514$	0	0
$515$	−50.5498	−2.22749
$516$	0	0
$517$	−25.6495	−1.12806
$518$	0	0
$519$	−7.45017	−0.327026
$520$	0	0
$521$	−0.549834	−0.0240887	−0.0120443	−	0.999927i	$-0.503834\pi$
$521$	−0.549834	−0.0240887	−0.0120443	+	0.999927i	$0.503834\pi$
$522$	0	0
$523$	−25.2749	−1.10519	−0.552597	−	0.833448i	$-0.686363\pi$
$523$	−25.2749	−1.10519	−0.552597	+	0.833448i	$0.686363\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.27492	0.272308
$532$	0	0
$533$	−13.4502	−0.582591
$534$	0	0
$535$	−29.1752	−1.26136
$536$	0	0
$537$	−1.45017	−0.0625793
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.725083	−0.0311737	−0.0155869	−	0.999879i	$-0.504962\pi$
$541$	−0.725083	−0.0311737	−0.0155869	+	0.999879i	$0.504962\pi$
$542$	0	0
$543$	3.82475	0.164136
$544$	0	0
$545$	−24.9003	−1.06661
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−2.90033	−0.123558
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−22.5498	−0.957187
$556$	0	0
$557$	−7.17525	−0.304025	−0.152013	−	0.988379i	$-0.548575\pi$
$557$	−7.17525	−0.304025	−0.152013	+	0.988379i	$0.548575\pi$
$558$	0	0
$559$	9.27492	0.392287
$560$	0	0
$561$	−17.0997	−0.721949
$562$	0	0
$563$	7.72508	0.325573	0.162787	−	0.986661i	$-0.447952\pi$
$563$	7.72508	0.325573	0.162787	+	0.986661i	$0.447952\pi$
$564$	0	0
$565$	45.0997	1.89736
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.5498	1.11303	0.556513	−	0.830839i	$-0.312139\pi$
$569$	26.5498	1.11303	0.556513	+	0.830839i	$0.312139\pi$
$570$	0	0
$571$	−0.725083	−0.0303438	−0.0151719	−	0.999885i	$-0.504830\pi$
$571$	−0.725083	−0.0303438	−0.0151719	+	0.999885i	$0.504830\pi$
$572$	0	0
$573$	−10.5498	−0.440726
$574$	0	0
$575$	−53.0997	−2.21441
$576$	0	0
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	0	0
$579$	−15.5498	−0.646229
$580$	0	0
$581$	0	0
$582$	0	0
$583$	7.37459	0.305424
$584$	0	0
$585$	5.45017	0.225337
$586$	0	0
$587$	1.72508	0.0712018	0.0356009	−	0.999366i	$-0.488665\pi$
$587$	1.72508	0.0712018	0.0356009	+	0.999366i	$0.488665\pi$
$588$	0	0
$589$	−1.27492	−0.0525320
$590$	0	0
$591$	16.5498	0.680769
$592$	0	0
$593$	−14.5498	−0.597490	−0.298745	−	0.954333i	$-0.596568\pi$
$593$	−14.5498	−0.597490	−0.298745	+	0.954333i	$0.596568\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−25.0997	−1.02726
$598$	0	0
$599$	−7.45017	−0.304406	−0.152203	−	0.988349i	$-0.548637\pi$
$599$	−7.45017	−0.304406	−0.152203	+	0.988349i	$0.548637\pi$
$600$	0	0
$601$	26.0997	1.06463	0.532314	−	0.846547i	$-0.321323\pi$
$601$	26.0997	1.06463	0.532314	+	0.846547i	$0.321323\pi$
$602$	0	0
$603$	−7.27492	−0.296258
$604$	0	0
$605$	31.0997	1.26438
$606$	0	0
$607$	−7.00000	−0.284121	−0.142061	−	0.989858i	$-0.545373\pi$
$607$	−7.00000	−0.284121	−0.142061	+	0.989858i	$0.545373\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.64950	−0.309466
$612$	0	0
$613$	6.54983	0.264545	0.132273	−	0.991213i	$-0.457773\pi$
$613$	6.54983	0.264545	0.132273	+	0.991213i	$0.457773\pi$
$614$	0	0
$615$	45.0997	1.81859
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	6.17525	0.248204	0.124102	−	0.992269i	$-0.460395\pi$
$619$	6.17525	0.248204	0.124102	+	0.992269i	$0.460395\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	84.8488	3.39395
$626$	0	0
$627$	−5.45017	−0.217659
$628$	0	0
$629$	21.0997	0.841299
$630$	0	0
$631$	2.82475	0.112452	0.0562258	−	0.998418i	$-0.482093\pi$
$631$	2.82475	0.112452	0.0562258	+	0.998418i	$0.482093\pi$
$632$	0	0
$633$	17.6495	0.701505
$634$	0	0
$635$	92.1238	3.65582
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	41.6495	1.64506	0.822528	−	0.568724i	$-0.192563\pi$
$641$	41.6495	1.64506	0.822528	+	0.568724i	$0.192563\pi$
$642$	0	0
$643$	−32.3746	−1.27673	−0.638365	−	0.769734i	$-0.720389\pi$
$643$	−32.3746	−1.27673	−0.638365	+	0.769734i	$0.720389\pi$
$644$	0	0
$645$	−31.0997	−1.22455
$646$	0	0
$647$	−34.0000	−1.33668	−0.668339	−	0.743857i	$-0.732994\pi$
$647$	−34.0000	−1.33668	−0.668339	+	0.743857i	$0.732994\pi$
$648$	0	0
$649$	26.8248	1.05296
$650$	0	0
$651$	0	0
$652$	0	0
$653$	45.9244	1.79716	0.898581	−	0.438808i	$-0.144599\pi$
$653$	45.9244	1.79716	0.898581	+	0.438808i	$0.144599\pi$
$654$	0	0
$655$	−1.17525	−0.0459208
$656$	0	0
$657$	−3.27492	−0.127767
$658$	0	0
$659$	−18.1993	−0.708946	−0.354473	−	0.935066i	$-0.615340\pi$
$659$	−18.1993	−0.708946	−0.354473	+	0.935066i	$0.615340\pi$
$660$	0	0
$661$	−29.8248	−1.16005	−0.580024	−	0.814599i	$-0.696957\pi$
$661$	−29.8248	−1.16005	−0.580024	+	0.814599i	$0.696957\pi$
$662$	0	0
$663$	−5.09967	−0.198055
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.09967	0.352341
$668$	0	0
$669$	−6.27492	−0.242602
$670$	0	0
$671$	−42.7492	−1.65031
$672$	0	0
$673$	26.4502	1.01958	0.509789	−	0.860299i	$-0.329723\pi$
$673$	26.4502	1.01958	0.509789	+	0.860299i	$0.329723\pi$
$674$	0	0
$675$	−13.2749	−0.510952
$676$	0	0
$677$	5.72508	0.220033	0.110016	−	0.993930i	$-0.464910\pi$
$677$	5.72508	0.220033	0.110016	+	0.993930i	$0.464910\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.72508	0.142745
$682$	0	0
$683$	−34.8248	−1.33253	−0.666266	−	0.745714i	$-0.732108\pi$
$683$	−34.8248	−1.33253	−0.666266	+	0.745714i	$0.732108\pi$
$684$	0	0
$685$	70.7492	2.70319
$686$	0	0
$687$	10.7251	0.409187
$688$	0	0
$689$	2.19934	0.0837881
$690$	0	0
$691$	11.8248	0.449835	0.224917	−	0.974378i	$-0.427789\pi$
$691$	11.8248	0.449835	0.224917	+	0.974378i	$0.427789\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.09967	0.117577
$696$	0	0
$697$	−42.1993	−1.59841
$698$	0	0
$699$	−14.5498	−0.550325
$700$	0	0
$701$	39.9244	1.50792	0.753962	−	0.656918i	$-0.228140\pi$
$701$	39.9244	1.50792	0.753962	+	0.656918i	$0.228140\pi$
$702$	0	0
$703$	6.72508	0.253641
$704$	0	0
$705$	25.6495	0.966016
$706$	0	0
$707$	0	0
$708$	0	0
$709$	32.1993	1.20927	0.604636	−	0.796502i	$-0.293319\pi$
$709$	32.1993	1.20927	0.604636	+	0.796502i	$0.293319\pi$
$710$	0	0
$711$	3.54983	0.133129
$712$	0	0
$713$	4.00000	0.149801
$714$	0	0
$715$	23.2990	0.871333
$716$	0	0
$717$	30.5498	1.14090
$718$	0	0
$719$	−32.1993	−1.20083	−0.600416	−	0.799688i	$-0.704998\pi$
$719$	−32.1993	−1.20083	−0.600416	+	0.799688i	$0.704998\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−12.8248	−0.476958
$724$	0	0
$725$	−30.1993	−1.12158
$726$	0	0
$727$	16.4502	0.610103	0.305051	−	0.952336i	$-0.401326\pi$
$727$	16.4502	0.610103	0.305051	+	0.952336i	$0.401326\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	29.0997	1.07629
$732$	0	0
$733$	46.9244	1.73319	0.866597	−	0.499010i	$-0.166303\pi$
$733$	46.9244	1.73319	0.866597	+	0.499010i	$0.166303\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.0997	−1.14557
$738$	0	0
$739$	36.3746	1.33806	0.669030	−	0.743235i	$-0.266710\pi$
$739$	36.3746	1.33806	0.669030	+	0.743235i	$0.266710\pi$
$740$	0	0
$741$	−1.62541	−0.0597111
$742$	0	0
$743$	−16.1993	−0.594296	−0.297148	−	0.954831i	$-0.596035\pi$
$743$	−16.1993	−0.594296	−0.297148	+	0.954831i	$0.596035\pi$
$744$	0	0
$745$	2.35050	0.0861155
$746$	0	0
$747$	−0.274917	−0.0100587
$748$	0	0
$749$	0	0
$750$	0	0
$751$	25.5498	0.932327	0.466163	−	0.884699i	$-0.345636\pi$
$751$	25.5498	0.932327	0.466163	+	0.884699i	$0.345636\pi$
$752$	0	0
$753$	−19.3746	−0.706049
$754$	0	0
$755$	65.7251	2.39198
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	17.0997	0.620679
$760$	0	0
$761$	−5.09967	−0.184863	−0.0924314	−	0.995719i	$-0.529464\pi$
$761$	−5.09967	−0.184863	−0.0924314	+	0.995719i	$0.529464\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	17.0997	0.618240
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−12.6495	−0.456153	−0.228076	−	0.973643i	$-0.573244\pi$
$769$	−12.6495	−0.456153	−0.228076	+	0.973643i	$0.573244\pi$
$770$	0	0
$771$	−19.0997	−0.687858
$772$	0	0
$773$	27.0997	0.974707	0.487354	−	0.873205i	$-0.337962\pi$
$773$	27.0997	0.974707	0.487354	+	0.873205i	$0.337962\pi$
$774$	0	0
$775$	−13.2749	−0.476849
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.4502	−0.481902
$780$	0	0
$781$	−8.54983	−0.305937
$782$	0	0
$783$	2.27492	0.0812989
$784$	0	0
$785$	−62.1993	−2.21999
$786$	0	0
$787$	−12.5498	−0.447353	−0.223677	−	0.974663i	$-0.571806\pi$
$787$	−12.5498	−0.447353	−0.223677	+	0.974663i	$0.571806\pi$
$788$	0	0
$789$	−24.5498	−0.873997
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12.7492	−0.452736
$794$	0	0
$795$	−7.37459	−0.261550
$796$	0	0
$797$	−36.4743	−1.29198	−0.645992	−	0.763344i	$-0.723556\pi$
$797$	−36.4743	−1.29198	−0.645992	+	0.763344i	$0.723556\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	−4.54983	−0.160760
$802$	0	0
$803$	−14.0000	−0.494049
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.2749	0.995324
$808$	0	0
$809$	15.6495	0.550207	0.275104	−	0.961415i	$-0.411288\pi$
$809$	15.6495	0.550207	0.275104	+	0.961415i	$0.411288\pi$
$810$	0	0
$811$	−27.4502	−0.963906	−0.481953	−	0.876197i	$-0.660072\pi$
$811$	−27.4502	−0.963906	−0.481953	+	0.876197i	$0.660072\pi$
$812$	0	0
$813$	6.27492	0.220071
$814$	0	0
$815$	51.2990	1.79693
$816$	0	0
$817$	9.27492	0.324488
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.2749	0.707599	0.353800	−	0.935321i	$-0.384889\pi$
$821$	20.2749	0.707599	0.353800	+	0.935321i	$0.384889\pi$
$822$	0	0
$823$	−26.1993	−0.913252	−0.456626	−	0.889659i	$-0.650942\pi$
$823$	−26.1993	−0.913252	−0.456626	+	0.889659i	$0.650942\pi$
$824$	0	0
$825$	−56.7492	−1.97575
$826$	0	0
$827$	−39.0241	−1.35700	−0.678500	−	0.734600i	$-0.737370\pi$
$827$	−39.0241	−1.35700	−0.678500	+	0.734600i	$0.737370\pi$
$828$	0	0
$829$	−22.7251	−0.789275	−0.394637	−	0.918837i	$-0.629130\pi$
$829$	−22.7251	−0.789275	−0.394637	+	0.918837i	$0.629130\pi$
$830$	0	0
$831$	4.17525	0.144838
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−25.6495	−0.887638
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	30.1993	1.04260	0.521298	−	0.853374i	$-0.325448\pi$
$839$	30.1993	1.04260	0.521298	+	0.853374i	$0.325448\pi$
$840$	0	0
$841$	−23.8248	−0.821543
$842$	0	0
$843$	11.4502	0.394365
$844$	0	0
$845$	−48.6254	−1.67277
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−26.9244	−0.924044
$850$	0	0
$851$	−21.0997	−0.723287
$852$	0	0
$853$	24.3746	0.834570	0.417285	−	0.908776i	$-0.362982\pi$
$853$	24.3746	0.834570	0.417285	+	0.908776i	$0.362982\pi$
$854$	0	0
$855$	5.45017	0.186392
$856$	0	0
$857$	28.5498	0.975244	0.487622	−	0.873055i	$-0.337864\pi$
$857$	28.5498	0.975244	0.487622	+	0.873055i	$0.337864\pi$
$858$	0	0
$859$	−10.3505	−0.353154	−0.176577	−	0.984287i	$-0.556503\pi$
$859$	−10.3505	−0.353154	−0.176577	+	0.984287i	$0.556503\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.0997	0.377837	0.188919	−	0.981993i	$-0.439502\pi$
$863$	11.0997	0.377837	0.188919	+	0.981993i	$0.439502\pi$
$864$	0	0
$865$	31.8488	1.08289
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	15.1752	0.514785
$870$	0	0
$871$	−9.27492	−0.314269
$872$	0	0
$873$	−16.2749	−0.550822
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.7492	−0.970791	−0.485395	−	0.874295i	$-0.661324\pi$
$877$	−28.7492	−0.970791	−0.485395	+	0.874295i	$0.661324\pi$
$878$	0	0
$879$	−5.17525	−0.174557
$880$	0	0
$881$	−24.5498	−0.827105	−0.413552	−	0.910480i	$-0.635712\pi$
$881$	−24.5498	−0.827105	−0.413552	+	0.910480i	$0.635712\pi$
$882$	0	0
$883$	30.3746	1.02219	0.511093	−	0.859525i	$-0.329241\pi$
$883$	30.3746	1.02219	0.511093	+	0.859525i	$0.329241\pi$
$884$	0	0
$885$	−26.8248	−0.901704
$886$	0	0
$887$	−11.6495	−0.391152	−0.195576	−	0.980689i	$-0.562658\pi$
$887$	−11.6495	−0.391152	−0.195576	+	0.980689i	$0.562658\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.27492	0.143215
$892$	0	0
$893$	−7.64950	−0.255981
$894$	0	0
$895$	6.19934	0.207221
$896$	0	0
$897$	5.09967	0.170273
$898$	0	0
$899$	2.27492	0.0758727
$900$	0	0
$901$	6.90033	0.229883
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.3505	−0.543509
$906$	0	0
$907$	37.2749	1.23769	0.618847	−	0.785512i	$-0.287600\pi$
$907$	37.2749	1.23769	0.618847	+	0.785512i	$0.287600\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	51.8488	1.71783	0.858914	−	0.512119i	$-0.171139\pi$
$911$	51.8488	1.71783	0.858914	+	0.512119i	$0.171139\pi$
$912$	0	0
$913$	−1.17525	−0.0388950
$914$	0	0
$915$	42.7492	1.41324
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−19.8248	−0.653958	−0.326979	−	0.945032i	$-0.606031\pi$
$919$	−19.8248	−0.653958	−0.326979	+	0.945032i	$0.606031\pi$
$920$	0	0
$921$	26.3746	0.869072
$922$	0	0
$923$	−2.54983	−0.0839288
$924$	0	0
$925$	70.0241	2.30238
$926$	0	0
$927$	−11.8248	−0.388376
$928$	0	0
$929$	−20.1993	−0.662719	−0.331359	−	0.943505i	$-0.607507\pi$
$929$	−20.1993	−0.662719	−0.331359	+	0.943505i	$0.607507\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−10.5498	−0.345386
$934$	0	0
$935$	73.0997	2.39061
$936$	0	0
$937$	−24.0997	−0.787302	−0.393651	−	0.919260i	$-0.628788\pi$
$937$	−24.0997	−0.787302	−0.393651	+	0.919260i	$0.628788\pi$
$938$	0	0
$939$	−4.45017	−0.145226
$940$	0	0
$941$	−29.1752	−0.951086	−0.475543	−	0.879693i	$-0.657748\pi$
$941$	−29.1752	−0.951086	−0.475543	+	0.879693i	$0.657748\pi$
$942$	0	0
$943$	42.1993	1.37420
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.5498	1.12272	0.561359	−	0.827572i	$-0.310279\pi$
$947$	34.5498	1.12272	0.561359	+	0.827572i	$0.310279\pi$
$948$	0	0
$949$	−4.17525	−0.135534
$950$	0	0
$951$	5.17525	0.167819
$952$	0	0
$953$	10.3505	0.335285	0.167643	−	0.985848i	$-0.446384\pi$
$953$	10.3505	0.335285	0.167643	+	0.985848i	$0.446384\pi$
$954$	0	0
$955$	45.0997	1.45939
$956$	0	0
$957$	9.72508	0.314367
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−6.82475	−0.219925
$964$	0	0
$965$	66.4743	2.13988
$966$	0	0
$967$	38.4502	1.23647	0.618237	−	0.785992i	$-0.287847\pi$
$967$	38.4502	1.23647	0.618237	+	0.785992i	$0.287847\pi$
$968$	0	0
$969$	−5.09967	−0.163825
$970$	0	0
$971$	−41.5739	−1.33417	−0.667085	−	0.744981i	$-0.732458\pi$
$971$	−41.5739	−1.33417	−0.667085	+	0.744981i	$0.732458\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−16.9244	−0.542015
$976$	0	0
$977$	39.0997	1.25091	0.625455	−	0.780261i	$-0.284914\pi$
$977$	39.0997	1.25091	0.625455	+	0.780261i	$0.284914\pi$
$978$	0	0
$979$	−19.4502	−0.621630
$980$	0	0
$981$	−5.82475	−0.185970
$982$	0	0
$983$	−51.2990	−1.63618	−0.818092	−	0.575087i	$-0.804968\pi$
$983$	−51.2990	−1.63618	−0.818092	+	0.575087i	$0.804968\pi$
$984$	0	0
$985$	−70.7492	−2.25426
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−29.0997	−0.925316
$990$	0	0
$991$	−16.0997	−0.511423	−0.255711	−	0.966753i	$-0.582310\pi$
$991$	−16.0997	−0.511423	−0.255711	+	0.966753i	$0.582310\pi$
$992$	0	0
$993$	23.8248	0.756056
$994$	0	0
$995$	107.299	3.40161
$996$	0	0
$997$	22.1752	0.702297	0.351149	−	0.936320i	$-0.385791\pi$
$997$	22.1752	0.702297	0.351149	+	0.936320i	$0.385791\pi$
$998$	0	0
$999$	−5.27492	−0.166891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.a.ba.1.2		2
3.2	odd	2		7056.2.a.ch.1.1		2
4.3	odd	2		1176.2.a.n.1.2		2
7.2	even	3		2352.2.q.bf.1537.1		4
7.3	odd	6		336.2.q.g.289.2		4
7.4	even	3		2352.2.q.bf.961.1		4
7.5	odd	6		336.2.q.g.193.2		4
7.6	odd	2		2352.2.a.bf.1.1		2
8.3	odd	2		9408.2.a.dj.1.1		2
8.5	even	2		9408.2.a.dw.1.1		2
12.11	even	2		3528.2.a.bd.1.1		2
21.5	even	6		1008.2.s.r.865.1		4
21.17	even	6		1008.2.s.r.289.1		4
21.20	even	2		7056.2.a.cu.1.2		2
28.3	even	6		168.2.q.c.121.2	yes	4
28.11	odd	6		1176.2.q.l.961.1		4
28.19	even	6		168.2.q.c.25.2	✓	4
28.23	odd	6		1176.2.q.l.361.1		4
28.27	even	2		1176.2.a.k.1.1		2
56.3	even	6		1344.2.q.w.961.1		4
56.5	odd	6		1344.2.q.x.193.1		4
56.13	odd	2		9408.2.a.dp.1.2		2
56.19	even	6		1344.2.q.w.193.1		4
56.27	even	2		9408.2.a.ec.1.2		2
56.45	odd	6		1344.2.q.x.961.1		4
84.11	even	6		3528.2.s.bk.3313.2		4
84.23	even	6		3528.2.s.bk.361.2		4
84.47	odd	6		504.2.s.i.361.1		4
84.59	odd	6		504.2.s.i.289.1		4
84.83	odd	2		3528.2.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.q.c.25.2	✓	4	28.19	even	6
168.2.q.c.121.2	yes	4	28.3	even	6
336.2.q.g.193.2		4	7.5	odd	6
336.2.q.g.289.2		4	7.3	odd	6
504.2.s.i.289.1		4	84.59	odd	6
504.2.s.i.361.1		4	84.47	odd	6
1008.2.s.r.289.1		4	21.17	even	6
1008.2.s.r.865.1		4	21.5	even	6
1176.2.a.k.1.1		2	28.27	even	2
1176.2.a.n.1.2		2	4.3	odd	2
1176.2.q.l.361.1		4	28.23	odd	6
1176.2.q.l.961.1		4	28.11	odd	6
1344.2.q.w.193.1		4	56.19	even	6
1344.2.q.w.961.1		4	56.3	even	6
1344.2.q.x.193.1		4	56.5	odd	6
1344.2.q.x.961.1		4	56.45	odd	6
2352.2.a.ba.1.2		2	1.1	even	1	trivial
2352.2.a.bf.1.1		2	7.6	odd	2
2352.2.q.bf.961.1		4	7.4	even	3
2352.2.q.bf.1537.1		4	7.2	even	3
3528.2.a.bd.1.1		2	12.11	even	2
3528.2.a.bk.1.2		2	84.83	odd	2
3528.2.s.bk.361.2		4	84.23	even	6
3528.2.s.bk.3313.2		4	84.11	even	6
7056.2.a.ch.1.1		2	3.2	odd	2
7056.2.a.cu.1.2		2	21.20	even	2
9408.2.a.dj.1.1		2	8.3	odd	2
9408.2.a.dp.1.2		2	56.13	odd	2
9408.2.a.dw.1.1		2	8.5	even	2
9408.2.a.ec.1.2		2	56.27	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.27492 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.27492 q^{5} +1.00000 q^{9} +4.27492 q^{11} +1.27492 q^{13} -4.27492 q^{15} +4.00000 q^{17} +1.27492 q^{19} -4.00000 q^{23} +13.2749 q^{25} -1.00000 q^{27} -2.27492 q^{29} -1.00000 q^{31} -4.27492 q^{33} +5.27492 q^{37} -1.27492 q^{39} -10.5498 q^{41} +7.27492 q^{43} +4.27492 q^{45} -6.00000 q^{47} -4.00000 q^{51} +1.72508 q^{53} +18.2749 q^{55} -1.27492 q^{57} +6.27492 q^{59} -10.0000 q^{61} +5.45017 q^{65} -7.27492 q^{67} +4.00000 q^{69} -2.00000 q^{71} -3.27492 q^{73} -13.2749 q^{75} +3.54983 q^{79} +1.00000 q^{81} -0.274917 q^{83} +17.0997 q^{85} +2.27492 q^{87} -4.54983 q^{89} +1.00000 q^{93} +5.45017 q^{95} -16.2749 q^{97} +4.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + q^5 + 2 * q^9	\( 2 q - 2 q^{3} + q^{5} + 2 q^{9} + q^{11} - 5 q^{13} - q^{15} + 8 q^{17} - 5 q^{19} - 8 q^{23} + 19 q^{25} - 2 q^{27} + 3 q^{29} - 2 q^{31} - q^{33} + 3 q^{37} + 5 q^{39} - 6 q^{41} + 7 q^{43} + q^{45} - 12 q^{47} - 8 q^{51} + 11 q^{53} + 29 q^{55} + 5 q^{57} + 5 q^{59} - 20 q^{61} + 26 q^{65} - 7 q^{67} + 8 q^{69} - 4 q^{71} + q^{73} - 19 q^{75} - 8 q^{79} + 2 q^{81} + 7 q^{83} + 4 q^{85} - 3 q^{87} + 6 q^{89} + 2 q^{93} + 26 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + q^5 + 2 * q^9 + q^11 - 5 * q^13 - q^15 + 8 * q^17 - 5 * q^19 - 8 * q^23 + 19 * q^25 - 2 * q^27 + 3 * q^29 - 2 * q^31 - q^33 + 3 * q^37 + 5 * q^39 - 6 * q^41 + 7 * q^43 + q^45 - 12 * q^47 - 8 * q^51 + 11 * q^53 + 29 * q^55 + 5 * q^57 + 5 * q^59 - 20 * q^61 + 26 * q^65 - 7 * q^67 + 8 * q^69 - 4 * q^71 + q^73 - 19 * q^75 - 8 * q^79 + 2 * q^81 + 7 * q^83 + 4 * q^85 - 3 * q^87 + 6 * q^89 + 2 * q^93 + 26 * q^95 - 25 * q^97 + q^99

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