LMFDB - Embedded newform 6012.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 26x^{8} + 82x^{7} + 211x^{6} - 340x^{5} - 593x^{4} + 192x^{3} + 423x^{2} + 126x + 9 $ x^10 - 4x^9 - 26x^8 + 82x^7 + 211x^6 - 340x^5 - 593x^4 + 192x^3 + 423x^2 + 126*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.108196$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.17375 q^{5} -0.108196 q^{7} +O(q^{10})$	$q-3.17375 q^{5} -0.108196 q^{7} +1.62861 q^{11} -5.77598 q^{13} -4.70432 q^{17} -5.48341 q^{19} +0.0951583 q^{23} +5.07268 q^{25} +4.83785 q^{29} -8.21996 q^{31} +0.343388 q^{35} +5.57843 q^{37} -3.02591 q^{41} +6.54435 q^{43} +3.97831 q^{47} -6.98829 q^{49} -5.77802 q^{53} -5.16880 q^{55} -3.26570 q^{59} -2.91122 q^{61} +18.3315 q^{65} -11.7958 q^{67} +7.35576 q^{71} -4.72675 q^{73} -0.176210 q^{77} -7.99913 q^{79} +9.73353 q^{83} +14.9303 q^{85} +9.42181 q^{89} +0.624940 q^{91} +17.4030 q^{95} +12.2956 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{5} + 4 q^{7}+O(q^{10}) $ 10 * q + 6 * q^5 + 4 * q^7	$ 10 q + 6 q^{5} + 4 q^{7} + 8 q^{11} - 2 q^{13} + 6 q^{17} + 20 q^{23} + 24 q^{25} + 8 q^{29} - 4 q^{31} - 4 q^{37} - 14 q^{41} + 20 q^{43} + 48 q^{47} - 2 q^{49} + 22 q^{53} - 6 q^{55} + 2 q^{59} - 8 q^{61} + 28 q^{65} - 6 q^{67} + 20 q^{71} + 20 q^{73} + 24 q^{77} - 4 q^{79} + 46 q^{83} - 18 q^{85} - 8 q^{89} + 28 q^{91} + 36 q^{95} - 34 q^{97}+O(q^{100}) $ 10 * q + 6 * q^5 + 4 * q^7 + 8 * q^11 - 2 * q^13 + 6 * q^17 + 20 * q^23 + 24 * q^25 + 8 * q^29 - 4 * q^31 - 4 * q^37 - 14 * q^41 + 20 * q^43 + 48 * q^47 - 2 * q^49 + 22 * q^53 - 6 * q^55 + 2 * q^59 - 8 * q^61 + 28 * q^65 - 6 * q^67 + 20 * q^71 + 20 * q^73 + 24 * q^77 - 4 * q^79 + 46 * q^83 - 18 * q^85 - 8 * q^89 + 28 * q^91 + 36 * q^95 - 34 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.17375	−1.41934	−0.709672	−	0.704533i	$-0.751157\pi$
$5$	−3.17375	−1.41934	−0.709672	+	0.704533i	$0.751157\pi$
$6$	0	0
$7$	−0.108196	−0.0408944	−0.0204472	−	0.999791i	$-0.506509\pi$
$7$	−0.108196	−0.0408944	−0.0204472	+	0.999791i	$0.506509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.62861	0.491045	0.245522	−	0.969391i	$-0.421041\pi$
$11$	1.62861	0.491045	0.245522	+	0.969391i	$0.421041\pi$
$12$	0	0
$13$	−5.77598	−1.60197	−0.800984	−	0.598685i	$-0.795690\pi$
$13$	−5.77598	−1.60197	−0.800984	+	0.598685i	$0.795690\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.70432	−1.14097	−0.570483	−	0.821309i	$-0.693244\pi$
$17$	−4.70432	−1.14097	−0.570483	+	0.821309i	$0.693244\pi$
$18$	0	0
$19$	−5.48341	−1.25798	−0.628990	−	0.777413i	$-0.716531\pi$
$19$	−5.48341	−1.25798	−0.628990	+	0.777413i	$0.716531\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.0951583	0.0198419	0.00992094	−	0.999951i	$-0.496842\pi$
$23$	0.0951583	0.0198419	0.00992094	+	0.999951i	$0.496842\pi$
$24$	0	0
$25$	5.07268	1.01454
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.83785	0.898367	0.449183	−	0.893440i	$-0.351715\pi$
$29$	4.83785	0.898367	0.449183	+	0.893440i	$0.351715\pi$
$30$	0	0
$31$	−8.21996	−1.47635	−0.738174	−	0.674610i	$-0.764312\pi$
$31$	−8.21996	−1.47635	−0.738174	+	0.674610i	$0.764312\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.343388	0.0580432
$36$	0	0
$37$	5.57843	0.917088	0.458544	−	0.888672i	$-0.348371\pi$
$37$	5.57843	0.917088	0.458544	+	0.888672i	$0.348371\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.02591	−0.472568	−0.236284	−	0.971684i	$-0.575930\pi$
$41$	−3.02591	−0.472568	−0.236284	+	0.971684i	$0.575930\pi$
$42$	0	0
$43$	6.54435	0.998004	0.499002	−	0.866601i	$-0.333700\pi$
$43$	6.54435	0.998004	0.499002	+	0.866601i	$0.333700\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.97831	0.580297	0.290148	−	0.956982i	$-0.406295\pi$
$47$	3.97831	0.580297	0.290148	+	0.956982i	$0.406295\pi$
$48$	0	0
$49$	−6.98829	−0.998328
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.77802	−0.793672	−0.396836	−	0.917889i	$-0.629892\pi$
$53$	−5.77802	−0.793672	−0.396836	+	0.917889i	$0.629892\pi$
$54$	0	0
$55$	−5.16880	−0.696961
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.26570	−0.425158	−0.212579	−	0.977144i	$-0.568186\pi$
$59$	−3.26570	−0.425158	−0.212579	+	0.977144i	$0.568186\pi$
$60$	0	0
$61$	−2.91122	−0.372743	−0.186372	−	0.982479i	$-0.559673\pi$
$61$	−2.91122	−0.372743	−0.186372	+	0.982479i	$0.559673\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.3315	2.27374
$66$	0	0
$67$	−11.7958	−1.44109	−0.720545	−	0.693408i	$-0.756108\pi$
$67$	−11.7958	−1.44109	−0.720545	+	0.693408i	$0.756108\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.35576	0.872968	0.436484	−	0.899712i	$-0.356223\pi$
$71$	7.35576	0.872968	0.436484	+	0.899712i	$0.356223\pi$
$72$	0	0
$73$	−4.72675	−0.553224	−0.276612	−	0.960982i	$-0.589212\pi$
$73$	−4.72675	−0.553224	−0.276612	+	0.960982i	$0.589212\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.176210	−0.0200810
$78$	0	0
$79$	−7.99913	−0.899972	−0.449986	−	0.893035i	$-0.648571\pi$
$79$	−7.99913	−0.899972	−0.449986	+	0.893035i	$0.648571\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.73353	1.06839	0.534197	−	0.845360i	$-0.320614\pi$
$83$	9.73353	1.06839	0.534197	+	0.845360i	$0.320614\pi$
$84$	0	0
$85$	14.9303	1.61942
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.42181	0.998709	0.499355	−	0.866398i	$-0.333570\pi$
$89$	9.42181	0.998709	0.499355	+	0.866398i	$0.333570\pi$
$90$	0	0
$91$	0.624940	0.0655115
$92$	0	0
$93$	0	0
$94$	0	0
$95$	17.4030	1.78551
$96$	0	0
$97$	12.2956	1.24843	0.624214	−	0.781253i	$-0.285419\pi$
$97$	12.2956	1.24843	0.624214	+	0.781253i	$0.285419\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.0701	1.49953	0.749766	−	0.661703i	$-0.230166\pi$
$101$	15.0701	1.49953	0.749766	+	0.661703i	$0.230166\pi$
$102$	0	0
$103$	−0.762117	−0.0750936	−0.0375468	−	0.999295i	$-0.511954\pi$
$103$	−0.762117	−0.0750936	−0.0375468	+	0.999295i	$0.511954\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.07185	0.103620	0.0518099	−	0.998657i	$-0.483501\pi$
$107$	1.07185	0.103620	0.0518099	+	0.998657i	$0.483501\pi$
$108$	0	0
$109$	8.75387	0.838469	0.419234	−	0.907878i	$-0.362299\pi$
$109$	8.75387	0.838469	0.419234	+	0.907878i	$0.362299\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.46678	0.326127	0.163064	−	0.986616i	$-0.447862\pi$
$113$	3.46678	0.326127	0.163064	+	0.986616i	$0.447862\pi$
$114$	0	0
$115$	−0.302008	−0.0281624
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.508991	0.0466591
$120$	0	0
$121$	−8.34763	−0.758875
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−0.230656	−0.0206305
$126$	0	0
$127$	2.76258	0.245140	0.122570	−	0.992460i	$-0.460886\pi$
$127$	2.76258	0.245140	0.122570	+	0.992460i	$0.460886\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.1508	−1.58584	−0.792922	−	0.609323i	$-0.791441\pi$
$131$	−18.1508	−1.58584	−0.792922	+	0.609323i	$0.791441\pi$
$132$	0	0
$133$	0.593285	0.0514443
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.7440	−1.60141	−0.800704	−	0.599060i	$-0.795541\pi$
$137$	−18.7440	−1.60141	−0.800704	+	0.599060i	$0.795541\pi$
$138$	0	0
$139$	−6.48580	−0.550118	−0.275059	−	0.961427i	$-0.588697\pi$
$139$	−6.48580	−0.550118	−0.275059	+	0.961427i	$0.588697\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.40682	−0.786638
$144$	0	0
$145$	−15.3541	−1.27509
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.2953	1.33496	0.667480	−	0.744628i	$-0.267373\pi$
$149$	16.2953	1.33496	0.667480	+	0.744628i	$0.267373\pi$
$150$	0	0
$151$	−5.15815	−0.419764	−0.209882	−	0.977727i	$-0.567308\pi$
$151$	−5.15815	−0.419764	−0.209882	+	0.977727i	$0.567308\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	26.0881	2.09545
$156$	0	0
$157$	−5.42441	−0.432915	−0.216458	−	0.976292i	$-0.569450\pi$
$157$	−5.42441	−0.432915	−0.216458	+	0.976292i	$0.569450\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.0102958	−0.000811421 0
$162$	0	0
$163$	4.50881	0.353157	0.176579	−	0.984287i	$-0.443497\pi$
$163$	4.50881	0.353157	0.176579	+	0.984287i	$0.443497\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	20.3619	1.56630
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.67627	0.507588	0.253794	−	0.967258i	$-0.418322\pi$
$173$	6.67627	0.507588	0.253794	+	0.967258i	$0.418322\pi$
$174$	0	0
$175$	−0.548845	−0.0414888
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.46839	0.408726	0.204363	−	0.978895i	$-0.434488\pi$
$179$	5.46839	0.408726	0.204363	+	0.978895i	$0.434488\pi$
$180$	0	0
$181$	1.80001	0.133794	0.0668970	−	0.997760i	$-0.478690\pi$
$181$	1.80001	0.133794	0.0668970	+	0.997760i	$0.478690\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−17.7045	−1.30166
$186$	0	0
$187$	−7.66151	−0.560265
$188$	0	0
$189$	0	0
$190$	0	0
$191$	26.6018	1.92484	0.962418	−	0.271573i	$-0.0875437\pi$
$191$	26.6018	1.92484	0.962418	+	0.271573i	$0.0875437\pi$
$192$	0	0
$193$	11.3523	0.817155	0.408578	−	0.912724i	$-0.366025\pi$
$193$	11.3523	0.817155	0.408578	+	0.912724i	$0.366025\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.43812	0.672438	0.336219	−	0.941784i	$-0.390852\pi$
$197$	9.43812	0.672438	0.336219	+	0.941784i	$0.390852\pi$
$198$	0	0
$199$	2.43867	0.172872	0.0864362	−	0.996257i	$-0.472452\pi$
$199$	2.43867	0.172872	0.0864362	+	0.996257i	$0.472452\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.523438	−0.0367382
$204$	0	0
$205$	9.60348	0.670736
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.93033	−0.617724
$210$	0	0
$211$	−10.5804	−0.728386	−0.364193	−	0.931324i	$-0.618655\pi$
$211$	−10.5804	−0.728386	−0.364193	+	0.931324i	$0.618655\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−20.7701	−1.41651
$216$	0	0
$217$	0.889370	0.0603744
$218$	0	0
$219$	0	0
$220$	0	0
$221$	27.1721	1.82779
$222$	0	0
$223$	−16.7046	−1.11862	−0.559312	−	0.828957i	$-0.688935\pi$
$223$	−16.7046	−1.11862	−0.559312	+	0.828957i	$0.688935\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.1621	−0.939969	−0.469984	−	0.882675i	$-0.655740\pi$
$227$	−14.1621	−0.939969	−0.469984	+	0.882675i	$0.655740\pi$
$228$	0	0
$229$	−10.4964	−0.693624	−0.346812	−	0.937935i	$-0.612736\pi$
$229$	−10.4964	−0.693624	−0.346812	+	0.937935i	$0.612736\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.1845	0.798233	0.399116	−	0.916900i	$-0.369317\pi$
$233$	12.1845	0.798233	0.399116	+	0.916900i	$0.369317\pi$
$234$	0	0
$235$	−12.6262	−0.823640
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.6029	−0.685845	−0.342923	−	0.939364i	$-0.611417\pi$
$239$	−10.6029	−0.685845	−0.342923	+	0.939364i	$0.611417\pi$
$240$	0	0
$241$	26.5490	1.71017	0.855085	−	0.518487i	$-0.173505\pi$
$241$	26.5490	1.71017	0.855085	+	0.518487i	$0.173505\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	22.1791	1.41697
$246$	0	0
$247$	31.6721	2.01524
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.73360	0.361902	0.180951	−	0.983492i	$-0.442082\pi$
$251$	5.73360	0.361902	0.180951	+	0.983492i	$0.442082\pi$
$252$	0	0
$253$	0.154976	0.00974324
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.49757	0.280551	0.140275	−	0.990113i	$-0.455201\pi$
$257$	4.49757	0.280551	0.140275	+	0.990113i	$0.455201\pi$
$258$	0	0
$259$	−0.603566	−0.0375038
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.0370	−0.742230	−0.371115	−	0.928587i	$-0.621025\pi$
$263$	−12.0370	−0.742230	−0.371115	+	0.928587i	$0.621025\pi$
$264$	0	0
$265$	18.3380	1.12649
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.71493	−0.470388	−0.235194	−	0.971948i	$-0.575572\pi$
$269$	−7.71493	−0.470388	−0.235194	+	0.971948i	$0.575572\pi$
$270$	0	0
$271$	24.1647	1.46790	0.733952	−	0.679202i	$-0.237674\pi$
$271$	24.1647	1.46790	0.733952	+	0.679202i	$0.237674\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.26141	0.498182
$276$	0	0
$277$	1.66146	0.0998273	0.0499137	−	0.998754i	$-0.484105\pi$
$277$	1.66146	0.0998273	0.0499137	+	0.998754i	$0.484105\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−29.3601	−1.75148	−0.875738	−	0.482786i	$-0.839625\pi$
$281$	−29.3601	−1.75148	−0.875738	+	0.482786i	$0.839625\pi$
$282$	0	0
$283$	16.7275	0.994349	0.497174	−	0.867651i	$-0.334371\pi$
$283$	16.7275	0.994349	0.497174	+	0.867651i	$0.334371\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.327393	0.0193254
$288$	0	0
$289$	5.13065	0.301803
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.3390	1.65558	0.827792	−	0.561035i	$-0.189597\pi$
$293$	28.3390	1.65558	0.827792	+	0.561035i	$0.189597\pi$
$294$	0	0
$295$	10.3645	0.603446
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.549632	−0.0317861
$300$	0	0
$301$	−0.708075	−0.0408128
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.23947	0.529051
$306$	0	0
$307$	27.1960	1.55216	0.776079	−	0.630636i	$-0.217206\pi$
$307$	27.1960	1.55216	0.776079	+	0.630636i	$0.217206\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.86096	0.502459	0.251230	−	0.967928i	$-0.419165\pi$
$311$	8.86096	0.502459	0.251230	+	0.967928i	$0.419165\pi$
$312$	0	0
$313$	6.48572	0.366595	0.183297	−	0.983058i	$-0.441323\pi$
$313$	6.48572	0.366595	0.183297	+	0.983058i	$0.441323\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.5839	−1.54926	−0.774632	−	0.632412i	$-0.782065\pi$
$317$	−27.5839	−1.54926	−0.774632	+	0.632412i	$0.782065\pi$
$318$	0	0
$319$	7.87898	0.441138
$320$	0	0
$321$	0	0
$322$	0	0
$323$	25.7957	1.43531
$324$	0	0
$325$	−29.2997	−1.62525
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.430439	−0.0237309
$330$	0	0
$331$	−4.82514	−0.265214	−0.132607	−	0.991169i	$-0.542335\pi$
$331$	−4.82514	−0.265214	−0.132607	+	0.991169i	$0.542335\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	37.4370	2.04540
$336$	0	0
$337$	2.40824	0.131185	0.0655926	−	0.997846i	$-0.479106\pi$
$337$	2.40824	0.131185	0.0655926	+	0.997846i	$0.479106\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−13.3871	−0.724953
$342$	0	0
$343$	1.51348	0.0817204
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.0749	1.23873	0.619364	−	0.785104i	$-0.287391\pi$
$347$	23.0749	1.23873	0.619364	+	0.785104i	$0.287391\pi$
$348$	0	0
$349$	22.0733	1.18156	0.590779	−	0.806833i	$-0.298821\pi$
$349$	22.0733	1.18156	0.590779	+	0.806833i	$0.298821\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.49823	−0.186192	−0.0930959	−	0.995657i	$-0.529676\pi$
$353$	−3.49823	−0.186192	−0.0930959	+	0.995657i	$0.529676\pi$
$354$	0	0
$355$	−23.3453	−1.23904
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.5094	0.924110	0.462055	−	0.886851i	$-0.347112\pi$
$359$	17.5094	0.924110	0.462055	+	0.886851i	$0.347112\pi$
$360$	0	0
$361$	11.0678	0.582514
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.0015	0.785215
$366$	0	0
$367$	−9.86088	−0.514734	−0.257367	−	0.966314i	$-0.582855\pi$
$367$	−9.86088	−0.514734	−0.257367	+	0.966314i	$0.582855\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.625161	0.0324567
$372$	0	0
$373$	9.68648	0.501547	0.250774	−	0.968046i	$-0.419315\pi$
$373$	9.68648	0.501547	0.250774	+	0.968046i	$0.419315\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−27.9433	−1.43916
$378$	0	0
$379$	−34.0415	−1.74859	−0.874297	−	0.485391i	$-0.838677\pi$
$379$	−34.0415	−1.74859	−0.874297	+	0.485391i	$0.838677\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.51666	0.486279	0.243139	−	0.969991i	$-0.421823\pi$
$383$	9.51666	0.486279	0.243139	+	0.969991i	$0.421823\pi$
$384$	0	0
$385$	0.559245	0.0285018
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.9355	1.21358	0.606790	−	0.794862i	$-0.292457\pi$
$389$	23.9355	1.21358	0.606790	+	0.794862i	$0.292457\pi$
$390$	0	0
$391$	−0.447655	−0.0226389
$392$	0	0
$393$	0	0
$394$	0	0
$395$	25.3872	1.27737
$396$	0	0
$397$	7.62148	0.382511	0.191256	−	0.981540i	$-0.438744\pi$
$397$	7.62148	0.382511	0.191256	+	0.981540i	$0.438744\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.219424	−0.0109575	−0.00547875	−	0.999985i	$-0.501744\pi$
$401$	−0.219424	−0.0109575	−0.00547875	+	0.999985i	$0.501744\pi$
$402$	0	0
$403$	47.4783	2.36506
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.08509	0.450331
$408$	0	0
$409$	−11.5659	−0.571896	−0.285948	−	0.958245i	$-0.592309\pi$
$409$	−11.5659	−0.571896	−0.285948	+	0.958245i	$0.592309\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.353337	0.0173866
$414$	0	0
$415$	−30.8918	−1.51642
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.228250	0.0111508	0.00557538	−	0.999984i	$-0.498225\pi$
$419$	0.228250	0.0111508	0.00557538	+	0.999984i	$0.498225\pi$
$420$	0	0
$421$	24.1596	1.17747	0.588733	−	0.808327i	$-0.299627\pi$
$421$	24.1596	1.17747	0.588733	+	0.808327i	$0.299627\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−23.8635	−1.15755
$426$	0	0
$427$	0.314983	0.0152431
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.0430	1.44712	0.723561	−	0.690261i	$-0.242504\pi$
$431$	30.0430	1.44712	0.723561	+	0.690261i	$0.242504\pi$
$432$	0	0
$433$	−16.8721	−0.810823	−0.405412	−	0.914134i	$-0.632872\pi$
$433$	−16.8721	−0.810823	−0.405412	+	0.914134i	$0.632872\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.521792	−0.0249607
$438$	0	0
$439$	27.9456	1.33377	0.666885	−	0.745161i	$-0.267627\pi$
$439$	27.9456	1.33377	0.666885	+	0.745161i	$0.267627\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.9769	1.75682	0.878412	−	0.477904i	$-0.158603\pi$
$443$	36.9769	1.75682	0.878412	+	0.477904i	$0.158603\pi$
$444$	0	0
$445$	−29.9024	−1.41751
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.31444	−0.0620322	−0.0310161	−	0.999519i	$-0.509874\pi$
$449$	−1.31444	−0.0620322	−0.0310161	+	0.999519i	$0.509874\pi$
$450$	0	0
$451$	−4.92803	−0.232052
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.98340	−0.0929833
$456$	0	0
$457$	4.49596	0.210312	0.105156	−	0.994456i	$-0.466466\pi$
$457$	4.49596	0.210312	0.105156	+	0.994456i	$0.466466\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.5045	−1.04814	−0.524070	−	0.851675i	$-0.675587\pi$
$461$	−22.5045	−1.04814	−0.524070	+	0.851675i	$0.675587\pi$
$462$	0	0
$463$	−6.37850	−0.296434	−0.148217	−	0.988955i	$-0.547353\pi$
$463$	−6.37850	−0.296434	−0.148217	+	0.988955i	$0.547353\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.6179	1.50938	0.754688	−	0.656084i	$-0.227788\pi$
$467$	32.6179	1.50938	0.754688	+	0.656084i	$0.227788\pi$
$468$	0	0
$469$	1.27627	0.0589325
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.6582	0.490065
$474$	0	0
$475$	−27.8156	−1.27627
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.36146	−0.0622067	−0.0311034	−	0.999516i	$-0.509902\pi$
$479$	−1.36146	−0.0622067	−0.0311034	+	0.999516i	$0.509902\pi$
$480$	0	0
$481$	−32.2209	−1.46915
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−39.0231	−1.77195
$486$	0	0
$487$	3.85683	0.174770	0.0873849	−	0.996175i	$-0.472149\pi$
$487$	3.85683	0.174770	0.0873849	+	0.996175i	$0.472149\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	32.0239	1.44522	0.722610	−	0.691256i	$-0.242942\pi$
$491$	32.0239	1.44522	0.722610	+	0.691256i	$0.242942\pi$
$492$	0	0
$493$	−22.7588	−1.02501
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.795867	−0.0356995
$498$	0	0
$499$	−7.39903	−0.331226	−0.165613	−	0.986191i	$-0.552960\pi$
$499$	−7.39903	−0.331226	−0.165613	+	0.986191i	$0.552960\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−29.5283	−1.31660	−0.658301	−	0.752755i	$-0.728725\pi$
$503$	−29.5283	−1.31660	−0.658301	+	0.752755i	$0.728725\pi$
$504$	0	0
$505$	−47.8287	−2.12835
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.9724	1.01823	0.509116	−	0.860698i	$-0.329972\pi$
$509$	22.9724	1.01823	0.509116	+	0.860698i	$0.329972\pi$
$510$	0	0
$511$	0.511417	0.0226238
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.41877	0.106584
$516$	0	0
$517$	6.47912	0.284951
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−29.8260	−1.30670	−0.653350	−	0.757056i	$-0.726637\pi$
$521$	−29.8260	−1.30670	−0.653350	+	0.757056i	$0.726637\pi$
$522$	0	0
$523$	26.9295	1.17754	0.588772	−	0.808299i	$-0.299611\pi$
$523$	26.9295	1.17754	0.588772	+	0.808299i	$0.299611\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	38.6694	1.68446
$528$	0	0
$529$	−22.9909	−0.999606
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.4776	0.757039
$534$	0	0
$535$	−3.40178	−0.147072
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−11.3812	−0.490223
$540$	0	0
$541$	−7.68550	−0.330426	−0.165213	−	0.986258i	$-0.552831\pi$
$541$	−7.68550	−0.330426	−0.165213	+	0.986258i	$0.552831\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−27.7826	−1.19008
$546$	0	0
$547$	27.8176	1.18940	0.594698	−	0.803949i	$-0.297272\pi$
$547$	27.8176	1.18940	0.594698	+	0.803949i	$0.297272\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−26.5279	−1.13013
$552$	0	0
$553$	0.865477	0.0368038
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.4563	−0.909134	−0.454567	−	0.890713i	$-0.650206\pi$
$557$	−21.4563	−0.909134	−0.454567	+	0.890713i	$0.650206\pi$
$558$	0	0
$559$	−37.8000	−1.59877
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.31031	−0.265948	−0.132974	−	0.991120i	$-0.542453\pi$
$563$	−6.31031	−0.265948	−0.132974	+	0.991120i	$0.542453\pi$
$564$	0	0
$565$	−11.0027	−0.462887
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−32.1557	−1.34804	−0.674019	−	0.738714i	$-0.735433\pi$
$569$	−32.1557	−1.34804	−0.674019	+	0.738714i	$0.735433\pi$
$570$	0	0
$571$	−28.6570	−1.19926	−0.599630	−	0.800278i	$-0.704685\pi$
$571$	−28.6570	−1.19926	−0.599630	+	0.800278i	$0.704685\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.482707	0.0201303
$576$	0	0
$577$	−17.0145	−0.708323	−0.354162	−	0.935184i	$-0.615234\pi$
$577$	−17.0145	−0.708323	−0.354162	+	0.935184i	$0.615234\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.05313	−0.0436913
$582$	0	0
$583$	−9.41015	−0.389728
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.850957	0.0351228	0.0175614	−	0.999846i	$-0.494410\pi$
$587$	0.850957	0.0351228	0.0175614	+	0.999846i	$0.494410\pi$
$588$	0	0
$589$	45.0734	1.85722
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12.3839	−0.508546	−0.254273	−	0.967132i	$-0.581836\pi$
$593$	−12.3839	−0.508546	−0.254273	+	0.967132i	$0.581836\pi$
$594$	0	0
$595$	−1.61541	−0.0662253
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−32.4250	−1.32485	−0.662424	−	0.749129i	$-0.730472\pi$
$599$	−32.4250	−1.32485	−0.662424	+	0.749129i	$0.730472\pi$
$600$	0	0
$601$	−37.8140	−1.54247	−0.771233	−	0.636553i	$-0.780360\pi$
$601$	−37.8140	−1.54247	−0.771233	+	0.636553i	$0.780360\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	26.4933	1.07710
$606$	0	0
$607$	31.5756	1.28161	0.640807	−	0.767702i	$-0.278600\pi$
$607$	31.5756	1.28161	0.640807	+	0.767702i	$0.278600\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22.9787	−0.929617
$612$	0	0
$613$	−37.3450	−1.50835	−0.754176	−	0.656672i	$-0.771963\pi$
$613$	−37.3450	−1.50835	−0.754176	+	0.656672i	$0.771963\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.4927	−1.06656	−0.533278	−	0.845940i	$-0.679040\pi$
$617$	−26.4927	−1.06656	−0.533278	+	0.845940i	$0.679040\pi$
$618$	0	0
$619$	−33.0123	−1.32688	−0.663438	−	0.748231i	$-0.730903\pi$
$619$	−33.0123	−1.32688	−0.663438	+	0.748231i	$0.730903\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.01941	−0.0408416
$624$	0	0
$625$	−24.6313	−0.985253
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−26.2427	−1.04637
$630$	0	0
$631$	−9.17407	−0.365214	−0.182607	−	0.983186i	$-0.558454\pi$
$631$	−9.17407	−0.365214	−0.182607	+	0.983186i	$0.558454\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.76775	−0.347937
$636$	0	0
$637$	40.3642	1.59929
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.3355	0.842700	0.421350	−	0.906898i	$-0.361556\pi$
$641$	21.3355	0.842700	0.421350	+	0.906898i	$0.361556\pi$
$642$	0	0
$643$	−24.6077	−0.970432	−0.485216	−	0.874394i	$-0.661259\pi$
$643$	−24.6077	−0.970432	−0.485216	+	0.874394i	$0.661259\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.7432	0.894127	0.447063	−	0.894502i	$-0.352470\pi$
$647$	22.7432	0.894127	0.447063	+	0.894502i	$0.352470\pi$
$648$	0	0
$649$	−5.31856	−0.208772
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.9801	1.52541	0.762704	−	0.646748i	$-0.223872\pi$
$653$	38.9801	1.52541	0.762704	+	0.646748i	$0.223872\pi$
$654$	0	0
$655$	57.6061	2.25086
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.4227	1.06824	0.534118	−	0.845410i	$-0.320644\pi$
$659$	27.4227	1.06824	0.534118	+	0.845410i	$0.320644\pi$
$660$	0	0
$661$	30.7998	1.19797	0.598987	−	0.800759i	$-0.295570\pi$
$661$	30.7998	1.19797	0.598987	+	0.800759i	$0.295570\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.88294	−0.0730172
$666$	0	0
$667$	0.460362	0.0178253
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.74124	−0.183034
$672$	0	0
$673$	−2.37961	−0.0917271	−0.0458636	−	0.998948i	$-0.514604\pi$
$673$	−2.37961	−0.0917271	−0.0458636	+	0.998948i	$0.514604\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−37.7926	−1.45249	−0.726243	−	0.687438i	$-0.758735\pi$
$677$	−37.7926	−1.45249	−0.726243	+	0.687438i	$0.758735\pi$
$678$	0	0
$679$	−1.33034	−0.0510537
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.9130	1.25938	0.629690	−	0.776847i	$-0.283182\pi$
$683$	32.9130	1.25938	0.629690	+	0.776847i	$0.283182\pi$
$684$	0	0
$685$	59.4887	2.27295
$686$	0	0
$687$	0	0
$688$	0	0
$689$	33.3737	1.27144
$690$	0	0
$691$	28.3508	1.07851	0.539257	−	0.842141i	$-0.318705\pi$
$691$	28.3508	1.07851	0.539257	+	0.842141i	$0.318705\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.5843	0.780807
$696$	0	0
$697$	14.2349	0.539184
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−31.4669	−1.18849	−0.594245	−	0.804284i	$-0.702549\pi$
$701$	−31.4669	−1.18849	−0.594245	+	0.804284i	$0.702549\pi$
$702$	0	0
$703$	−30.5888	−1.15368
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.63053	−0.0613224
$708$	0	0
$709$	20.5920	0.773348	0.386674	−	0.922216i	$-0.373624\pi$
$709$	20.5920	0.773348	0.386674	+	0.922216i	$0.373624\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.782198	−0.0292935
$714$	0	0
$715$	29.8549	1.11651
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.1994	−1.35001	−0.675004	−	0.737814i	$-0.735858\pi$
$719$	−36.1994	−1.35001	−0.675004	+	0.737814i	$0.735858\pi$
$720$	0	0
$721$	0.0824583	0.00307091
$722$	0	0
$723$	0	0
$724$	0	0
$725$	24.5409	0.911425
$726$	0	0
$727$	−34.8040	−1.29081	−0.645405	−	0.763840i	$-0.723311\pi$
$727$	−34.8040	−1.29081	−0.645405	+	0.763840i	$0.723311\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−30.7867	−1.13869
$732$	0	0
$733$	−33.6718	−1.24370	−0.621848	−	0.783138i	$-0.713618\pi$
$733$	−33.6718	−1.24370	−0.621848	+	0.783138i	$0.713618\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.2108	−0.707640
$738$	0	0
$739$	−3.46595	−0.127497	−0.0637485	−	0.997966i	$-0.520306\pi$
$739$	−3.46595	−0.127497	−0.0637485	+	0.997966i	$0.520306\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.0548	0.442247	0.221124	−	0.975246i	$-0.429028\pi$
$743$	12.0548	0.442247	0.221124	+	0.975246i	$0.429028\pi$
$744$	0	0
$745$	−51.7170	−1.89477
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−0.115970	−0.00423747
$750$	0	0
$751$	−13.4001	−0.488976	−0.244488	−	0.969652i	$-0.578620\pi$
$751$	−13.4001	−0.488976	−0.244488	+	0.969652i	$0.578620\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.3707	0.595790
$756$	0	0
$757$	3.39860	0.123524	0.0617621	−	0.998091i	$-0.480328\pi$
$757$	3.39860	0.123524	0.0617621	+	0.998091i	$0.480328\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.91634	0.141967	0.0709837	−	0.997477i	$-0.477386\pi$
$761$	3.91634	0.141967	0.0709837	+	0.997477i	$0.477386\pi$
$762$	0	0
$763$	−0.947137	−0.0342887
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.8626	0.681090
$768$	0	0
$769$	12.9339	0.466408	0.233204	−	0.972428i	$-0.425079\pi$
$769$	12.9339	0.466408	0.233204	+	0.972428i	$0.425079\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	42.4154	1.52558	0.762788	−	0.646649i	$-0.223830\pi$
$773$	42.4154	1.52558	0.762788	+	0.646649i	$0.223830\pi$
$774$	0	0
$775$	−41.6972	−1.49781
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.5923	0.594481
$780$	0	0
$781$	11.9797	0.428666
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17.2157	0.614455
$786$	0	0
$787$	−13.0863	−0.466476	−0.233238	−	0.972420i	$-0.574932\pi$
$787$	−13.0863	−0.466476	−0.233238	+	0.972420i	$0.574932\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.375093	−0.0133368
$792$	0	0
$793$	16.8151	0.597123
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.99116	−0.283061	−0.141531	−	0.989934i	$-0.545202\pi$
$797$	−7.99116	−0.283061	−0.141531	+	0.989934i	$0.545202\pi$
$798$	0	0
$799$	−18.7153	−0.662099
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.69803	−0.271658
$804$	0	0
$805$	0.0326762	0.00115169
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−12.2797	−0.431732	−0.215866	−	0.976423i	$-0.569257\pi$
$809$	−12.2797	−0.431732	−0.215866	+	0.976423i	$0.569257\pi$
$810$	0	0
$811$	2.28647	0.0802888	0.0401444	−	0.999194i	$-0.487218\pi$
$811$	2.28647	0.0802888	0.0401444	+	0.999194i	$0.487218\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14.3098	−0.501251
$816$	0	0
$817$	−35.8854	−1.25547
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37.6177	1.31287	0.656434	−	0.754383i	$-0.272064\pi$
$821$	37.6177	1.31287	0.656434	+	0.754383i	$0.272064\pi$
$822$	0	0
$823$	31.0226	1.08138	0.540691	−	0.841221i	$-0.318163\pi$
$823$	31.0226	1.08138	0.540691	+	0.841221i	$0.318163\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−55.5761	−1.93257	−0.966285	−	0.257476i	$-0.917109\pi$
$827$	−55.5761	−1.93257	−0.966285	+	0.257476i	$0.917109\pi$
$828$	0	0
$829$	−22.1109	−0.767944	−0.383972	−	0.923345i	$-0.625444\pi$
$829$	−22.1109	−0.767944	−0.383972	+	0.923345i	$0.625444\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	32.8752	1.13906
$834$	0	0
$835$	3.17375	0.109832
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42.2423	−1.45836	−0.729182	−	0.684319i	$-0.760099\pi$
$839$	−42.2423	−1.45836	−0.729182	+	0.684319i	$0.760099\pi$
$840$	0	0
$841$	−5.59518	−0.192937
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−64.6237	−2.22312
$846$	0	0
$847$	0.903183	0.0310337
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.530834	0.0181968
$852$	0	0
$853$	30.4244	1.04171	0.520855	−	0.853645i	$-0.325613\pi$
$853$	30.4244	1.04171	0.520855	+	0.853645i	$0.325613\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.8726	−0.644675	−0.322337	−	0.946625i	$-0.604469\pi$
$857$	−18.8726	−0.644675	−0.322337	+	0.946625i	$0.604469\pi$
$858$	0	0
$859$	−26.0106	−0.887469	−0.443734	−	0.896158i	$-0.646347\pi$
$859$	−26.0106	−0.887469	−0.443734	+	0.896158i	$0.646347\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13.3710	0.455153	0.227577	−	0.973760i	$-0.426920\pi$
$863$	13.3710	0.455153	0.227577	+	0.973760i	$0.426920\pi$
$864$	0	0
$865$	−21.1888	−0.720441
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.0275	−0.441927
$870$	0	0
$871$	68.1325	2.30858
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.0249561	0.000843671 0
$876$	0	0
$877$	55.9378	1.88889	0.944443	−	0.328676i	$-0.106602\pi$
$877$	55.9378	1.88889	0.944443	+	0.328676i	$0.106602\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−31.3596	−1.05653	−0.528266	−	0.849079i	$-0.677158\pi$
$881$	−31.3596	−1.05653	−0.528266	+	0.849079i	$0.677158\pi$
$882$	0	0
$883$	43.1397	1.45177	0.725884	−	0.687817i	$-0.241431\pi$
$883$	43.1397	1.45177	0.725884	+	0.687817i	$0.241431\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.56766	−0.0862136	−0.0431068	−	0.999070i	$-0.513726\pi$
$887$	−2.56766	−0.0862136	−0.0431068	+	0.999070i	$0.513726\pi$
$888$	0	0
$889$	−0.298902	−0.0100248
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−21.8147	−0.730001
$894$	0	0
$895$	−17.3553	−0.580123
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−39.7670	−1.32630
$900$	0	0
$901$	27.1817	0.905553
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.71279	−0.189900
$906$	0	0
$907$	−22.4393	−0.745085	−0.372542	−	0.928015i	$-0.621514\pi$
$907$	−22.4393	−0.745085	−0.372542	+	0.928015i	$0.621514\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.0910	−0.831303	−0.415651	−	0.909524i	$-0.636446\pi$
$911$	−25.0910	−0.831303	−0.415651	+	0.909524i	$0.636446\pi$
$912$	0	0
$913$	15.8521	0.524629
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.96385	0.0648521
$918$	0	0
$919$	13.0442	0.430288	0.215144	−	0.976582i	$-0.430978\pi$
$919$	13.0442	0.430288	0.215144	+	0.976582i	$0.430978\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−42.4867	−1.39847
$924$	0	0
$925$	28.2976	0.930418
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−49.9403	−1.63849	−0.819244	−	0.573445i	$-0.805607\pi$
$929$	−49.9403	−1.63849	−0.819244	+	0.573445i	$0.805607\pi$
$930$	0	0
$931$	38.3197	1.25588
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.3157	0.795208
$936$	0	0
$937$	−22.4946	−0.734865	−0.367433	−	0.930050i	$-0.619763\pi$
$937$	−22.4946	−0.734865	−0.367433	+	0.930050i	$0.619763\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−0.902077	−0.0294069	−0.0147034	−	0.999892i	$-0.504680\pi$
$941$	−0.902077	−0.0294069	−0.0147034	+	0.999892i	$0.504680\pi$
$942$	0	0
$943$	−0.287940	−0.00937663
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.966532	0.0314081	0.0157040	−	0.999877i	$-0.495001\pi$
$947$	0.966532	0.0314081	0.0157040	+	0.999877i	$0.495001\pi$
$948$	0	0
$949$	27.3016	0.886248
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.8557	0.740369	0.370184	−	0.928958i	$-0.379295\pi$
$953$	22.8557	0.740369	0.370184	+	0.928958i	$0.379295\pi$
$954$	0	0
$955$	−84.4273	−2.73200
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.02803	0.0654886
$960$	0	0
$961$	36.5678	1.17961
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−36.0293	−1.15982
$966$	0	0
$967$	2.06744	0.0664844	0.0332422	−	0.999447i	$-0.489417\pi$
$967$	2.06744	0.0664844	0.0332422	+	0.999447i	$0.489417\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15.1407	0.485887	0.242944	−	0.970040i	$-0.421887\pi$
$971$	15.1407	0.485887	0.242944	+	0.970040i	$0.421887\pi$
$972$	0	0
$973$	0.701740	0.0224968
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.709890	0.0227114	0.0113557	−	0.999936i	$-0.496385\pi$
$977$	0.709890	0.0227114	0.0113557	+	0.999936i	$0.496385\pi$
$978$	0	0
$979$	15.3444	0.490411
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.5656	0.911102	0.455551	−	0.890210i	$-0.349442\pi$
$983$	28.5656	0.911102	0.455551	+	0.890210i	$0.349442\pi$
$984$	0	0
$985$	−29.9542	−0.954420
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.622749	0.0198023
$990$	0	0
$991$	15.5630	0.494374	0.247187	−	0.968968i	$-0.420494\pi$
$991$	15.5630	0.494374	0.247187	+	0.968968i	$0.420494\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7.73971	−0.245365
$996$	0	0
$997$	8.54160	0.270515	0.135258	−	0.990810i	$-0.456814\pi$
$997$	8.54160	0.270515	0.135258	+	0.990810i	$0.456814\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.k.1.2	yes	10
3.2	odd	2		6012.2.a.j.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.a.j.1.9	✓	10	3.2	odd	2
6012.2.a.k.1.2	yes	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-3.17375 q^{5} -0.108196 q^{7} +O(q^{10})\)	\(q-3.17375 q^{5} -0.108196 q^{7} +1.62861 q^{11} -5.77598 q^{13} -4.70432 q^{17} -5.48341 q^{19} +0.0951583 q^{23} +5.07268 q^{25} +4.83785 q^{29} -8.21996 q^{31} +0.343388 q^{35} +5.57843 q^{37} -3.02591 q^{41} +6.54435 q^{43} +3.97831 q^{47} -6.98829 q^{49} -5.77802 q^{53} -5.16880 q^{55} -3.26570 q^{59} -2.91122 q^{61} +18.3315 q^{65} -11.7958 q^{67} +7.35576 q^{71} -4.72675 q^{73} -0.176210 q^{77} -7.99913 q^{79} +9.73353 q^{83} +14.9303 q^{85} +9.42181 q^{89} +0.624940 q^{91} +17.4030 q^{95} +12.2956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{5} + 4 q^{7}+O(q^{10}) \) 10 * q + 6 * q^5 + 4 * q^7	\( 10 q + 6 q^{5} + 4 q^{7} + 8 q^{11} - 2 q^{13} + 6 q^{17} + 20 q^{23} + 24 q^{25} + 8 q^{29} - 4 q^{31} - 4 q^{37} - 14 q^{41} + 20 q^{43} + 48 q^{47} - 2 q^{49} + 22 q^{53} - 6 q^{55} + 2 q^{59} - 8 q^{61} + 28 q^{65} - 6 q^{67} + 20 q^{71} + 20 q^{73} + 24 q^{77} - 4 q^{79} + 46 q^{83} - 18 q^{85} - 8 q^{89} + 28 q^{91} + 36 q^{95} - 34 q^{97}+O(q^{100}) \) 10 * q + 6 * q^5 + 4 * q^7 + 8 * q^11 - 2 * q^13 + 6 * q^17 + 20 * q^23 + 24 * q^25 + 8 * q^29 - 4 * q^31 - 4 * q^37 - 14 * q^41 + 20 * q^43 + 48 * q^47 - 2 * q^49 + 22 * q^53 - 6 * q^55 + 2 * q^59 - 8 * q^61 + 28 * q^65 - 6 * q^67 + 20 * q^71 + 20 * q^73 + 24 * q^77 - 4 * q^79 + 46 * q^83 - 18 * q^85 - 8 * q^89 + 28 * q^91 + 36 * q^95 - 34 * q^97

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