LMFDB - Embedded newform 3420.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, 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("3420.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.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:	$27.3088374913$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 1 $ x^3 - x^2 - 7*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.27307$ of defining polynomial
Character	$\chi$	$=$	3420.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -0.166860 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -0.166860 q^{7} -4.71301 q^{11} +1.83314 q^{13} +4.54615 q^{17} +1.00000 q^{19} +4.54615 q^{23} +1.00000 q^{25} -6.71301 q^{29} -2.54615 q^{31} +0.166860 q^{35} +2.16686 q^{37} -10.7130 q^{41} +9.25915 q^{43} -0.546146 q^{47} -6.97216 q^{49} -10.5461 q^{53} +4.71301 q^{55} +9.42601 q^{59} +12.8799 q^{61} -1.83314 q^{65} +5.42601 q^{71} +15.0923 q^{73} +0.786413 q^{77} +13.7597 q^{79} +13.9722 q^{83} -4.54615 q^{85} -6.04557 q^{89} -0.305878 q^{91} -1.00000 q^{95} -2.16686 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5}+O(q^{10}) $ 3 * q - 3 * q^5	$ 3 q - 3 q^{5} + 2 q^{11} + 6 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{23} + 3 q^{25} - 4 q^{29} + 8 q^{31} + 6 q^{37} - 16 q^{41} - 4 q^{43} + 14 q^{47} + 27 q^{49} - 16 q^{53} - 2 q^{55} - 4 q^{59} + 22 q^{61} - 6 q^{65} - 16 q^{71} + 14 q^{73} + 20 q^{77} + 8 q^{79} - 6 q^{83} + 2 q^{85} - 4 q^{89} + 48 q^{91} - 3 q^{95} - 6 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^11 + 6 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^23 + 3 * q^25 - 4 * q^29 + 8 * q^31 + 6 * q^37 - 16 * q^41 - 4 * q^43 + 14 * q^47 + 27 * q^49 - 16 * q^53 - 2 * q^55 - 4 * q^59 + 22 * q^61 - 6 * q^65 - 16 * q^71 + 14 * q^73 + 20 * q^77 + 8 * q^79 - 6 * q^83 + 2 * q^85 - 4 * q^89 + 48 * q^91 - 3 * q^95 - 6 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.166860	−0.0630672	−0.0315336	−	0.999503i	$-0.510039\pi$
$7$	−0.166860	−0.0630672	−0.0315336	+	0.999503i	$0.510039\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.71301	−1.42102	−0.710512	−	0.703685i	$-0.751537\pi$
$11$	−4.71301	−1.42102	−0.710512	+	0.703685i	$0.751537\pi$
$12$	0	0
$13$	1.83314	0.508421	0.254211	−	0.967149i	$-0.418184\pi$
$13$	1.83314	0.508421	0.254211	+	0.967149i	$0.418184\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.54615	1.10260	0.551301	−	0.834306i	$-0.314132\pi$
$17$	4.54615	1.10260	0.551301	+	0.834306i	$0.314132\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.54615	0.947937	0.473968	−	0.880542i	$-0.342821\pi$
$23$	4.54615	0.947937	0.473968	+	0.880542i	$0.342821\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.71301	−1.24657	−0.623287	−	0.781993i	$-0.714203\pi$
$29$	−6.71301	−1.24657	−0.623287	+	0.781993i	$0.714203\pi$
$30$	0	0
$31$	−2.54615	−0.457301	−0.228651	−	0.973509i	$-0.573431\pi$
$31$	−2.54615	−0.457301	−0.228651	+	0.973509i	$0.573431\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.166860	0.0282045
$36$	0	0
$37$	2.16686	0.356230	0.178115	−	0.984010i	$-0.443000\pi$
$37$	2.16686	0.356230	0.178115	+	0.984010i	$0.443000\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.7130	−1.67309	−0.836545	−	0.547898i	$-0.815428\pi$
$41$	−10.7130	−1.67309	−0.836545	+	0.547898i	$0.815428\pi$
$42$	0	0
$43$	9.25915	1.41201	0.706004	−	0.708208i	$-0.250496\pi$
$43$	9.25915	1.41201	0.706004	+	0.708208i	$0.250496\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.546146	−0.0796635	−0.0398318	−	0.999206i	$-0.512682\pi$
$47$	−0.546146	−0.0796635	−0.0398318	+	0.999206i	$0.512682\pi$
$48$	0	0
$49$	−6.97216	−0.996023
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.5461	−1.44862	−0.724312	−	0.689472i	$-0.757843\pi$
$53$	−10.5461	−1.44862	−0.724312	+	0.689472i	$0.757843\pi$
$54$	0	0
$55$	4.71301	0.635502
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.42601	1.22716	0.613581	−	0.789632i	$-0.289728\pi$
$59$	9.42601	1.22716	0.613581	+	0.789632i	$0.289728\pi$
$60$	0	0
$61$	12.8799	1.64910	0.824549	−	0.565791i	$-0.191429\pi$
$61$	12.8799	1.64910	0.824549	+	0.565791i	$0.191429\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.83314	−0.227373
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.42601	0.643949	0.321975	−	0.946748i	$-0.395653\pi$
$71$	5.42601	0.643949	0.321975	+	0.946748i	$0.395653\pi$
$72$	0	0
$73$	15.0923	1.76642	0.883210	−	0.468979i	$-0.155378\pi$
$73$	15.0923	1.76642	0.883210	+	0.468979i	$0.155378\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.786413	0.0896201
$78$	0	0
$79$	13.7597	1.54809	0.774045	−	0.633130i	$-0.218230\pi$
$79$	13.7597	1.54809	0.774045	+	0.633130i	$0.218230\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.9722	1.53364	0.766822	−	0.641860i	$-0.221837\pi$
$83$	13.9722	1.53364	0.766822	+	0.641860i	$0.221837\pi$
$84$	0	0
$85$	−4.54615	−0.493099
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.04557	−0.640829	−0.320414	−	0.947278i	$-0.603822\pi$
$89$	−6.04557	−0.640829	−0.320414	+	0.947278i	$0.603822\pi$
$90$	0	0
$91$	−0.305878	−0.0320647
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−2.16686	−0.220011	−0.110006	−	0.993931i	$-0.535087\pi$
$97$	−2.16686	−0.220011	−0.110006	+	0.993931i	$0.535087\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.5183	1.64363	0.821816	−	0.569753i	$-0.192961\pi$
$101$	16.5183	1.64363	0.821816	+	0.569753i	$0.192961\pi$
$102$	0	0
$103$	−5.09229	−0.501758	−0.250879	−	0.968018i	$-0.580720\pi$
$103$	−5.09229	−0.501758	−0.250879	+	0.968018i	$0.580720\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	15.4260	1.47754	0.738772	−	0.673955i	$-0.235406\pi$
$109$	15.4260	1.47754	0.738772	+	0.673955i	$0.235406\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.8799	−1.39978	−0.699890	−	0.714251i	$-0.746768\pi$
$113$	−14.8799	−1.39978	−0.699890	+	0.714251i	$0.746768\pi$
$114$	0	0
$115$	−4.54615	−0.423930
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.758571	−0.0695381
$120$	0	0
$121$	11.2124	1.01931
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.1390	−1.23533	−0.617666	−	0.786441i	$-0.711922\pi$
$131$	−14.1390	−1.23533	−0.617666	+	0.786441i	$0.711922\pi$
$132$	0	0
$133$	−0.166860	−0.0144686
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	9.42601	0.799504	0.399752	−	0.916623i	$-0.369096\pi$
$139$	9.42601	0.799504	0.399752	+	0.916623i	$0.369096\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.63960	−0.722480
$144$	0	0
$145$	6.71301	0.557485
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.0923	1.23641	0.618204	−	0.786017i	$-0.287860\pi$
$149$	15.0923	1.23641	0.618204	+	0.786017i	$0.287860\pi$
$150$	0	0
$151$	11.2136	0.912549	0.456274	−	0.889839i	$-0.349184\pi$
$151$	11.2136	0.912549	0.456274	+	0.889839i	$0.349184\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.54615	0.204511
$156$	0	0
$157$	−14.7586	−1.17786	−0.588931	−	0.808183i	$-0.700451\pi$
$157$	−14.7586	−1.17786	−0.588931	+	0.808183i	$0.700451\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.758571	−0.0597838
$162$	0	0
$163$	−3.49942	−0.274096	−0.137048	−	0.990564i	$-0.543761\pi$
$163$	−3.49942	−0.274096	−0.137048	+	0.990564i	$0.543761\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.21243	−0.480732	−0.240366	−	0.970682i	$-0.577267\pi$
$167$	−6.21243	−0.480732	−0.240366	+	0.970682i	$0.577267\pi$
$168$	0	0
$169$	−9.63960	−0.741508
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.12013	0.0851622	0.0425811	−	0.999093i	$-0.486442\pi$
$173$	1.12013	0.0851622	0.0425811	+	0.999093i	$0.486442\pi$
$174$	0	0
$175$	−0.166860	−0.0126134
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.6663	0.871979	0.435989	−	0.899952i	$-0.356399\pi$
$179$	11.6663	0.871979	0.435989	+	0.899952i	$0.356399\pi$
$180$	0	0
$181$	9.66628	0.718489	0.359244	−	0.933243i	$-0.383034\pi$
$181$	9.66628	0.718489	0.359244	+	0.933243i	$0.383034\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.16686	−0.159311
$186$	0	0
$187$	−21.4260	−1.56683
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.04673	−0.0757385	−0.0378692	−	0.999283i	$-0.512057\pi$
$191$	−1.04673	−0.0757385	−0.0378692	+	0.999283i	$0.512057\pi$
$192$	0	0
$193$	−10.9254	−0.786430	−0.393215	−	0.919447i	$-0.628637\pi$
$193$	−10.9254	−0.786430	−0.393215	+	0.919447i	$0.628637\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.4539	0.816053	0.408027	−	0.912970i	$-0.366217\pi$
$197$	11.4539	0.816053	0.408027	+	0.912970i	$0.366217\pi$
$198$	0	0
$199$	−24.7586	−1.75509	−0.877544	−	0.479496i	$-0.840820\pi$
$199$	−24.7586	−1.75509	−0.877544	+	0.479496i	$0.840820\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.12013	0.0786180
$204$	0	0
$205$	10.7130	0.748229
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.71301	−0.326005
$210$	0	0
$211$	16.4249	1.13073	0.565367	−	0.824840i	$-0.308735\pi$
$211$	16.4249	1.13073	0.565367	+	0.824840i	$0.308735\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.25915	−0.631469
$216$	0	0
$217$	0.424850	0.0288407
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.33372	0.560587
$222$	0	0
$223$	−5.09229	−0.341005	−0.170503	−	0.985357i	$-0.554539\pi$
$223$	−5.09229	−0.341005	−0.170503	+	0.985357i	$0.554539\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.6396	1.63539	0.817694	−	0.575653i	$-0.195252\pi$
$227$	24.6396	1.63539	0.817694	+	0.575653i	$0.195252\pi$
$228$	0	0
$229$	−7.78757	−0.514617	−0.257309	−	0.966329i	$-0.582836\pi$
$229$	−7.78757	−0.514617	−0.257309	+	0.966329i	$0.582836\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.9443	1.69967	0.849834	−	0.527050i	$-0.176702\pi$
$233$	25.9443	1.69967	0.849834	+	0.527050i	$0.176702\pi$
$234$	0	0
$235$	0.546146	0.0356266
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.3793	1.05949	0.529744	−	0.848158i	$-0.322288\pi$
$239$	16.3793	1.05949	0.529744	+	0.848158i	$0.322288\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.97216	0.445435
$246$	0	0
$247$	1.83314	0.116640
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.4716	1.35527	0.677637	−	0.735397i	$-0.263004\pi$
$251$	21.4716	1.35527	0.677637	+	0.735397i	$0.263004\pi$
$252$	0	0
$253$	−21.4260	−1.34704
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.9722	0.996316	0.498158	−	0.867086i	$-0.334010\pi$
$257$	15.9722	0.996316	0.498158	+	0.867086i	$0.334010\pi$
$258$	0	0
$259$	−0.361563	−0.0224664
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.3047	−1.06705	−0.533527	−	0.845783i	$-0.679134\pi$
$263$	−17.3047	−1.06705	−0.533527	+	0.845783i	$0.679134\pi$
$264$	0	0
$265$	10.5461	0.647845
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.04673	0.429646	0.214823	−	0.976653i	$-0.431082\pi$
$269$	7.04673	0.429646	0.214823	+	0.976653i	$0.431082\pi$
$270$	0	0
$271$	11.6663	0.708676	0.354338	−	0.935117i	$-0.384706\pi$
$271$	11.6663	0.708676	0.354338	+	0.935117i	$0.384706\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.71301	−0.284205
$276$	0	0
$277$	29.9443	1.79918	0.899590	−	0.436736i	$-0.143866\pi$
$277$	29.9443	1.79918	0.899590	+	0.436736i	$0.143866\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.1390	−1.44001	−0.720007	−	0.693967i	$-0.755861\pi$
$281$	−24.1390	−1.44001	−0.720007	+	0.693967i	$0.755861\pi$
$282$	0	0
$283$	4.92543	0.292786	0.146393	−	0.989226i	$-0.453234\pi$
$283$	4.92543	0.292786	0.146393	+	0.989226i	$0.453234\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.78757	0.105517
$288$	0	0
$289$	3.66744	0.215732
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.12013	−0.532804	−0.266402	−	0.963862i	$-0.585835\pi$
$293$	−9.12013	−0.532804	−0.266402	+	0.963862i	$0.585835\pi$
$294$	0	0
$295$	−9.42601	−0.548804
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.33372	0.481951
$300$	0	0
$301$	−1.54498	−0.0890514
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.8799	−0.737499
$306$	0	0
$307$	−8.75857	−0.499878	−0.249939	−	0.968262i	$-0.580411\pi$
$307$	−8.75857	−0.499878	−0.249939	+	0.968262i	$0.580411\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.04673	−0.0593544	−0.0296772	−	0.999560i	$-0.509448\pi$
$311$	−1.04673	−0.0593544	−0.0296772	+	0.999560i	$0.509448\pi$
$312$	0	0
$313$	−21.6663	−1.22465	−0.612325	−	0.790606i	$-0.709766\pi$
$313$	−21.6663	−1.22465	−0.612325	+	0.790606i	$0.709766\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.5461	0.816993	0.408496	−	0.912760i	$-0.366053\pi$
$317$	14.5461	0.816993	0.408496	+	0.912760i	$0.366053\pi$
$318$	0	0
$319$	31.6384	1.77141
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.54615	0.252954
$324$	0	0
$325$	1.83314	0.101684
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.0911300	0.00502416
$330$	0	0
$331$	7.21359	0.396495	0.198247	−	0.980152i	$-0.436475\pi$
$331$	7.21359	0.396495	0.198247	+	0.980152i	$0.436475\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.59171	−0.141179	−0.0705897	−	0.997505i	$-0.522488\pi$
$337$	−2.59171	−0.141179	−0.0705897	+	0.997505i	$0.522488\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.0000	0.649836
$342$	0	0
$343$	2.33140	0.125884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.54498	0.190305	0.0951524	−	0.995463i	$-0.469666\pi$
$347$	3.54498	0.190305	0.0951524	+	0.995463i	$0.469666\pi$
$348$	0	0
$349$	−28.8520	−1.54441	−0.772207	−	0.635371i	$-0.780847\pi$
$349$	−28.8520	−1.54441	−0.772207	+	0.635371i	$0.780847\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.8520	−1.10984	−0.554921	−	0.831903i	$-0.687251\pi$
$353$	−20.8520	−1.10984	−0.554921	+	0.831903i	$0.687251\pi$
$354$	0	0
$355$	−5.42601	−0.287983
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.4727	0.552730	0.276365	−	0.961053i	$-0.410870\pi$
$359$	10.4727	0.552730	0.276365	+	0.961053i	$0.410870\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.0923	−0.789967
$366$	0	0
$367$	2.40713	0.125651	0.0628255	−	0.998025i	$-0.479989\pi$
$367$	2.40713	0.125651	0.0628255	+	0.998025i	$0.479989\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.75973	0.0913608
$372$	0	0
$373$	−31.2592	−1.61854	−0.809269	−	0.587439i	$-0.800136\pi$
$373$	−31.2592	−1.61854	−0.809269	+	0.587439i	$0.800136\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.3059	−0.633785
$378$	0	0
$379$	−11.6384	−0.597826	−0.298913	−	0.954280i	$-0.596624\pi$
$379$	−11.6384	−0.597826	−0.298913	+	0.954280i	$0.596624\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−25.0923	−1.28216	−0.641078	−	0.767476i	$-0.721513\pi$
$383$	−25.0923	−1.28216	−0.641078	+	0.767476i	$0.721513\pi$
$384$	0	0
$385$	−0.786413	−0.0400793
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.7597	−0.596242	−0.298121	−	0.954528i	$-0.596360\pi$
$389$	−11.7597	−0.596242	−0.298121	+	0.954528i	$0.596360\pi$
$390$	0	0
$391$	20.6674	1.04520
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.7597	−0.692327
$396$	0	0
$397$	25.5195	1.28079	0.640393	−	0.768048i	$-0.278772\pi$
$397$	25.5195	1.28079	0.640393	+	0.768048i	$0.278772\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.194703	−0.00972298	−0.00486149	−	0.999988i	$-0.501547\pi$
$401$	−0.194703	−0.00972298	−0.00486149	+	0.999988i	$0.501547\pi$
$402$	0	0
$403$	−4.66744	−0.232502
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.2124	−0.506211
$408$	0	0
$409$	29.6106	1.46415	0.732075	−	0.681224i	$-0.238552\pi$
$409$	29.6106	1.46415	0.732075	+	0.681224i	$0.238552\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.57283	−0.0773937
$414$	0	0
$415$	−13.9722	−0.685866
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.95443	0.193187	0.0965934	−	0.995324i	$-0.469205\pi$
$419$	3.95443	0.193187	0.0965934	+	0.995324i	$0.469205\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.54615	0.220520
$426$	0	0
$427$	−2.14914	−0.104004
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.8509	1.43787	0.718933	−	0.695080i	$-0.244631\pi$
$431$	29.8509	1.43787	0.718933	+	0.695080i	$0.244631\pi$
$432$	0	0
$433$	5.07457	0.243868	0.121934	−	0.992538i	$-0.461090\pi$
$433$	5.07457	0.243868	0.121934	+	0.992538i	$0.461090\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.54615	0.217472
$438$	0	0
$439$	−34.8520	−1.66340	−0.831698	−	0.555228i	$-0.812631\pi$
$439$	−34.8520	−1.66340	−0.831698	+	0.555228i	$0.812631\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.8787	−0.754420	−0.377210	−	0.926128i	$-0.623117\pi$
$443$	−15.8787	−0.754420	−0.377210	+	0.926128i	$0.623117\pi$
$444$	0	0
$445$	6.04557	0.286587
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−11.8053	−0.557126	−0.278563	−	0.960418i	$-0.589858\pi$
$449$	−11.8053	−0.557126	−0.278563	+	0.960418i	$0.589858\pi$
$450$	0	0
$451$	50.4905	2.37750
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.305878	0.0143398
$456$	0	0
$457$	−35.4260	−1.65716	−0.828579	−	0.559871i	$-0.810851\pi$
$457$	−35.4260	−1.65716	−0.828579	+	0.559871i	$0.810851\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.6674	0.496832	0.248416	−	0.968653i	$-0.420090\pi$
$461$	10.6674	0.496832	0.248416	+	0.968653i	$0.420090\pi$
$462$	0	0
$463$	13.9266	0.647224	0.323612	−	0.946190i	$-0.395103\pi$
$463$	13.9266	0.647224	0.323612	+	0.946190i	$0.395103\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.30588	−0.106703	−0.0533517	−	0.998576i	$-0.516990\pi$
$467$	−2.30588	−0.106703	−0.0533517	+	0.998576i	$0.516990\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−43.6384	−2.00650
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.13902	−0.0977342	−0.0488671	−	0.998805i	$-0.515561\pi$
$479$	−2.13902	−0.0977342	−0.0488671	+	0.998805i	$0.515561\pi$
$480$	0	0
$481$	3.97216	0.181115
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.16686	0.0983921
$486$	0	0
$487$	−8.09113	−0.366644	−0.183322	−	0.983053i	$-0.558685\pi$
$487$	−8.09113	−0.366644	−0.183322	+	0.983053i	$0.558685\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.37929	−0.197634	−0.0988172	−	0.995106i	$-0.531506\pi$
$491$	−4.37929	−0.197634	−0.0988172	+	0.995106i	$0.531506\pi$
$492$	0	0
$493$	−30.5183	−1.37448
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.905386	−0.0406121
$498$	0	0
$499$	−10.0935	−0.451845	−0.225923	−	0.974145i	$-0.572540\pi$
$499$	−10.0935	−0.451845	−0.225923	+	0.974145i	$0.572540\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.21243	0.366174	0.183087	−	0.983097i	$-0.441391\pi$
$503$	8.21243	0.366174	0.183087	+	0.983097i	$0.441391\pi$
$504$	0	0
$505$	−16.5183	−0.735055
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.8065	−1.45412	−0.727060	−	0.686574i	$-0.759114\pi$
$509$	−32.8065	−1.45412	−0.727060	+	0.686574i	$0.759114\pi$
$510$	0	0
$511$	−2.51830	−0.111403
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.09229	0.224393
$516$	0	0
$517$	2.57399	0.113204
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.62071	0.421491	0.210746	−	0.977541i	$-0.432411\pi$
$521$	9.62071	0.421491	0.210746	+	0.977541i	$0.432411\pi$
$522$	0	0
$523$	−31.6106	−1.38223	−0.691117	−	0.722743i	$-0.742881\pi$
$523$	−31.6106	−1.38223	−0.691117	+	0.722743i	$0.742881\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.5751	−0.504221
$528$	0	0
$529$	−2.33256	−0.101416
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−19.6384	−0.850635
$534$	0	0
$535$	−16.0000	−0.691740
$536$	0	0
$537$	0	0
$538$	0	0
$539$	32.8598	1.41537
$540$	0	0
$541$	−32.1846	−1.38372	−0.691862	−	0.722030i	$-0.743209\pi$
$541$	−32.1846	−1.38372	−0.691862	+	0.722030i	$0.743209\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15.4260	−0.660778
$546$	0	0
$547$	33.4260	1.42919	0.714597	−	0.699537i	$-0.246610\pi$
$547$	33.4260	1.42919	0.714597	+	0.699537i	$0.246610\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.71301	−0.285984
$552$	0	0
$553$	−2.29595	−0.0976338
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.2769	−1.07102	−0.535508	−	0.844530i	$-0.679880\pi$
$557$	−25.2769	−1.07102	−0.535508	+	0.844530i	$0.679880\pi$
$558$	0	0
$559$	16.9733	0.717895
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.6396	1.37560	0.687798	−	0.725903i	$-0.258578\pi$
$563$	32.6396	1.37560	0.687798	+	0.725903i	$0.258578\pi$
$564$	0	0
$565$	14.8799	0.626001
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.6207	−1.07408	−0.537038	−	0.843558i	$-0.680457\pi$
$569$	−25.6207	−1.07408	−0.537038	+	0.843558i	$0.680457\pi$
$570$	0	0
$571$	−27.1857	−1.13769	−0.568844	−	0.822445i	$-0.692609\pi$
$571$	−27.1857	−1.13769	−0.568844	+	0.822445i	$0.692609\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.54615	0.189587
$576$	0	0
$577$	4.51830	0.188099	0.0940497	−	0.995568i	$-0.470019\pi$
$577$	4.51830	0.188099	0.0940497	+	0.995568i	$0.470019\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.33140	−0.0967227
$582$	0	0
$583$	49.7040	2.05853
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.9733	−1.11331	−0.556654	−	0.830744i	$-0.687915\pi$
$587$	−26.9733	−1.11331	−0.556654	+	0.830744i	$0.687915\pi$
$588$	0	0
$589$	−2.54615	−0.104912
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.09229	0.291246	0.145623	−	0.989340i	$-0.453481\pi$
$593$	7.09229	0.291246	0.145623	+	0.989340i	$0.453481\pi$
$594$	0	0
$595$	0.758571	0.0310984
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.85202	0.279966	0.139983	−	0.990154i	$-0.455295\pi$
$599$	6.85202	0.279966	0.139983	+	0.990154i	$0.455295\pi$
$600$	0	0
$601$	−31.8509	−1.29922	−0.649612	−	0.760266i	$-0.725069\pi$
$601$	−31.8509	−1.29922	−0.649612	+	0.760266i	$0.725069\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.2124	−0.455850
$606$	0	0
$607$	41.3703	1.67917	0.839585	−	0.543229i	$-0.182798\pi$
$607$	41.3703	1.67917	0.839585	+	0.543229i	$0.182798\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.00116	−0.0405027
$612$	0	0
$613$	5.66628	0.228859	0.114429	−	0.993431i	$-0.463496\pi$
$613$	5.66628	0.228859	0.114429	+	0.993431i	$0.463496\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.7307	1.23717	0.618586	−	0.785717i	$-0.287706\pi$
$617$	30.7307	1.23717	0.618586	+	0.785717i	$0.287706\pi$
$618$	0	0
$619$	2.99884	0.120533	0.0602667	−	0.998182i	$-0.480805\pi$
$619$	2.99884	0.120533	0.0602667	+	0.998182i	$0.480805\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.00876	0.0404153
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.85086	0.392780
$630$	0	0
$631$	25.5171	1.01582	0.507911	−	0.861410i	$-0.330418\pi$
$631$	25.5171	1.01582	0.507911	+	0.861410i	$0.330418\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	−12.7809	−0.506399
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.13902	0.163481	0.0817407	−	0.996654i	$-0.473952\pi$
$641$	4.13902	0.163481	0.0817407	+	0.996654i	$0.473952\pi$
$642$	0	0
$643$	26.2603	1.03561	0.517803	−	0.855500i	$-0.326750\pi$
$643$	26.2603	1.03561	0.517803	+	0.855500i	$0.326750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.3970	0.880517	0.440259	−	0.897871i	$-0.354887\pi$
$647$	22.3970	0.880517	0.440259	+	0.897871i	$0.354887\pi$
$648$	0	0
$649$	−44.4249	−1.74383
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13.6384	−0.533713	−0.266857	−	0.963736i	$-0.585985\pi$
$653$	−13.6384	−0.533713	−0.266857	+	0.963736i	$0.585985\pi$
$654$	0	0
$655$	14.1390	0.552457
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13.0923	0.510003	0.255002	−	0.966941i	$-0.417924\pi$
$659$	13.0923	0.510003	0.255002	+	0.966941i	$0.417924\pi$
$660$	0	0
$661$	−20.5183	−0.798070	−0.399035	−	0.916936i	$-0.630655\pi$
$661$	−20.5183	−0.798070	−0.399035	+	0.916936i	$0.630655\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.166860	0.00647056
$666$	0	0
$667$	−30.5183	−1.18167
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−60.7029	−2.34341
$672$	0	0
$673$	−2.16686	−0.0835263	−0.0417632	−	0.999128i	$-0.513297\pi$
$673$	−2.16686	−0.0835263	−0.0417632	+	0.999128i	$0.513297\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.3036	0.857195	0.428598	−	0.903495i	$-0.359008\pi$
$677$	22.3036	0.857195	0.428598	+	0.903495i	$0.359008\pi$
$678$	0	0
$679$	0.361563	0.0138755
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.33256	−0.280573	−0.140286	−	0.990111i	$-0.544802\pi$
$683$	−7.33256	−0.280573	−0.140286	+	0.990111i	$0.544802\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−19.3326	−0.736512
$690$	0	0
$691$	0.333720	0.0126953	0.00634766	−	0.999980i	$-0.497979\pi$
$691$	0.333720	0.0126953	0.00634766	+	0.999980i	$0.497979\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.42601	−0.357549
$696$	0	0
$697$	−48.7029	−1.84475
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19.7597	−0.746315	−0.373157	−	0.927768i	$-0.621725\pi$
$701$	−19.7597	−0.746315	−0.373157	+	0.927768i	$0.621725\pi$
$702$	0	0
$703$	2.16686	0.0817247
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.75625	−0.103659
$708$	0	0
$709$	30.3970	1.14158	0.570792	−	0.821095i	$-0.306636\pi$
$709$	30.3970	1.14158	0.570792	+	0.821095i	$0.306636\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−11.5751	−0.433493
$714$	0	0
$715$	8.63960	0.323103
$716$	0	0
$717$	0	0
$718$	0	0
$719$	34.1390	1.27317	0.636585	−	0.771206i	$-0.280346\pi$
$719$	34.1390	1.27317	0.636585	+	0.771206i	$0.280346\pi$
$720$	0	0
$721$	0.849701	0.0316445
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.71301	−0.249315
$726$	0	0
$727$	41.2592	1.53022	0.765109	−	0.643901i	$-0.222685\pi$
$727$	41.2592	1.53022	0.765109	+	0.643901i	$0.222685\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	42.0935	1.55688
$732$	0	0
$733$	3.09229	0.114216	0.0571082	−	0.998368i	$-0.481812\pi$
$733$	3.09229	0.114216	0.0571082	+	0.998368i	$0.481812\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−15.5195	−0.570893	−0.285446	−	0.958395i	$-0.592142\pi$
$739$	−15.5195	−0.570893	−0.285446	+	0.958395i	$0.592142\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.4272	−0.969519	−0.484759	−	0.874647i	$-0.661093\pi$
$743$	−26.4272	−0.969519	−0.484759	+	0.874647i	$0.661093\pi$
$744$	0	0
$745$	−15.0923	−0.552939
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.66976	−0.0975510
$750$	0	0
$751$	40.3059	1.47078	0.735391	−	0.677643i	$-0.236998\pi$
$751$	40.3059	1.47078	0.735391	+	0.677643i	$0.236998\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.2136	−0.408104
$756$	0	0
$757$	0.573988	0.0208620	0.0104310	−	0.999946i	$-0.496680\pi$
$757$	0.573988	0.0208620	0.0104310	+	0.999946i	$0.496680\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	43.7597	1.58629	0.793145	−	0.609033i	$-0.208442\pi$
$761$	43.7597	1.58629	0.793145	+	0.609033i	$0.208442\pi$
$762$	0	0
$763$	−2.57399	−0.0931846
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17.2792	0.623916
$768$	0	0
$769$	13.3326	0.480784	0.240392	−	0.970676i	$-0.422724\pi$
$769$	13.3326	0.480784	0.240392	+	0.970676i	$0.422724\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.2136	1.12267	0.561337	−	0.827587i	$-0.310287\pi$
$773$	31.2136	1.12267	0.561337	+	0.827587i	$0.310287\pi$
$774$	0	0
$775$	−2.54615	−0.0914603
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.7130	−0.383833
$780$	0	0
$781$	−25.5728	−0.915068
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.7586	0.526756
$786$	0	0
$787$	−48.2780	−1.72093	−0.860463	−	0.509513i	$-0.829826\pi$
$787$	−48.2780	−1.72093	−0.860463	+	0.509513i	$0.829826\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.48286	0.0882803
$792$	0	0
$793$	23.6106	0.838437
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.0644	−0.604454	−0.302227	−	0.953236i	$-0.597730\pi$
$797$	−17.0644	−0.604454	−0.302227	+	0.953236i	$0.597730\pi$
$798$	0	0
$799$	−2.48286	−0.0878372
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−71.1301	−2.51013
$804$	0	0
$805$	0.758571	0.0267361
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.75973	0.132185	0.0660926	−	0.997813i	$-0.478947\pi$
$809$	3.75973	0.132185	0.0660926	+	0.997813i	$0.478947\pi$
$810$	0	0
$811$	−5.33488	−0.187333	−0.0936665	−	0.995604i	$-0.529859\pi$
$811$	−5.33488	−0.187333	−0.0936665	+	0.995604i	$0.529859\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.49942	0.122579
$816$	0	0
$817$	9.25915	0.323937
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.42601	−0.259170	−0.129585	−	0.991568i	$-0.541364\pi$
$821$	−7.42601	−0.259170	−0.129585	+	0.991568i	$0.541364\pi$
$822$	0	0
$823$	−30.6852	−1.06962	−0.534809	−	0.844973i	$-0.679616\pi$
$823$	−30.6852	−1.06962	−0.534809	+	0.844973i	$0.679616\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.8242	−1.07186	−0.535931	−	0.844262i	$-0.680039\pi$
$827$	−30.8242	−1.07186	−0.535931	+	0.844262i	$0.680039\pi$
$828$	0	0
$829$	20.5183	0.712630	0.356315	−	0.934366i	$-0.384033\pi$
$829$	20.5183	0.712630	0.356315	+	0.934366i	$0.384033\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−31.6964	−1.09822
$834$	0	0
$835$	6.21243	0.214990
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.9432	−1.06828	−0.534138	−	0.845397i	$-0.679364\pi$
$839$	−30.9432	−1.06828	−0.534138	+	0.845397i	$0.679364\pi$
$840$	0	0
$841$	16.0644	0.553947
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.63960	0.331612
$846$	0	0
$847$	−1.87091	−0.0642852
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.85086	0.337683
$852$	0	0
$853$	−31.4260	−1.07601	−0.538003	−	0.842943i	$-0.680821\pi$
$853$	−31.4260	−1.07601	−0.538003	+	0.842943i	$0.680821\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.6941	0.399464	0.199732	−	0.979851i	$-0.435993\pi$
$857$	11.6941	0.399464	0.199732	+	0.979851i	$0.435993\pi$
$858$	0	0
$859$	−38.1289	−1.30094	−0.650471	−	0.759531i	$-0.725428\pi$
$859$	−38.1289	−1.30094	−0.650471	+	0.759531i	$0.725428\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	0	0
$865$	−1.12013	−0.0380857
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−64.8497	−2.19988
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.166860	0.00564091
$876$	0	0
$877$	22.8697	0.772256	0.386128	−	0.922445i	$-0.373812\pi$
$877$	22.8697	0.772256	0.386128	+	0.922445i	$0.373812\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	58.2780	1.96344	0.981718	−	0.190339i	$-0.0609587\pi$
$881$	58.2780	1.96344	0.981718	+	0.190339i	$0.0609587\pi$
$882$	0	0
$883$	−1.31484	−0.0442478	−0.0221239	−	0.999755i	$-0.507043\pi$
$883$	−1.31484	−0.0442478	−0.0221239	+	0.999755i	$0.507043\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−10.1846	−0.341965	−0.170982	−	0.985274i	$-0.554694\pi$
$887$	−10.1846	−0.341965	−0.170982	+	0.985274i	$0.554694\pi$
$888$	0	0
$889$	−0.667441	−0.0223853
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.546146	−0.0182761
$894$	0	0
$895$	−11.6663	−0.389961
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.0923	0.570060
$900$	0	0
$901$	−47.9443	−1.59726
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.66628	−0.321318
$906$	0	0
$907$	−44.0354	−1.46217	−0.731086	−	0.682285i	$-0.760986\pi$
$907$	−44.0354	−1.46217	−0.731086	+	0.682285i	$0.760986\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36.4249	1.20681	0.603405	−	0.797435i	$-0.293810\pi$
$911$	36.4249	1.20681	0.603405	+	0.797435i	$0.293810\pi$
$912$	0	0
$913$	−65.8509	−2.17935
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.35924	0.0779090
$918$	0	0
$919$	−5.75973	−0.189996	−0.0949980	−	0.995477i	$-0.530284\pi$
$919$	−5.75973	−0.189996	−0.0949980	+	0.995477i	$0.530284\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.94664	0.327398
$924$	0	0
$925$	2.16686	0.0712459
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50.7586	−1.66533	−0.832667	−	0.553774i	$-0.813187\pi$
$929$	−50.7586	−1.66533	−0.832667	+	0.553774i	$0.813187\pi$
$930$	0	0
$931$	−6.97216	−0.228503
$932$	0	0
$933$	0	0
$934$	0	0
$935$	21.4260	0.700706
$936$	0	0
$937$	−38.2780	−1.25049	−0.625244	−	0.780429i	$-0.715001\pi$
$937$	−38.2780	−1.25049	−0.625244	+	0.780429i	$0.715001\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.6573	−0.999400	−0.499700	−	0.866199i	$-0.666556\pi$
$941$	−30.6573	−0.999400	−0.499700	+	0.866199i	$0.666556\pi$
$942$	0	0
$943$	−48.7029	−1.58598
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.1201	−0.361356	−0.180678	−	0.983542i	$-0.557829\pi$
$947$	−11.1201	−0.361356	−0.180678	+	0.983542i	$0.557829\pi$
$948$	0	0
$949$	27.6663	0.898085
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30.6373	0.992439	0.496219	−	0.868197i	$-0.334721\pi$
$953$	30.6373	0.992439	0.496219	+	0.868197i	$0.334721\pi$
$954$	0	0
$955$	1.04673	0.0338713
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.00116	−0.0323292
$960$	0	0
$961$	−24.5171	−0.790876
$962$	0	0
$963$	0	0
$964$	0	0
$965$	10.9254	0.351702
$966$	0	0
$967$	11.6863	0.375807	0.187903	−	0.982188i	$-0.439831\pi$
$967$	11.6863	0.375807	0.187903	+	0.982188i	$0.439831\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.2769	−0.490258	−0.245129	−	0.969490i	$-0.578830\pi$
$971$	−15.2769	−0.490258	−0.245129	+	0.969490i	$0.578830\pi$
$972$	0	0
$973$	−1.57283	−0.0504225
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.97332	0.159111	0.0795553	−	0.996830i	$-0.474650\pi$
$977$	4.97332	0.159111	0.0795553	+	0.996830i	$0.474650\pi$
$978$	0	0
$979$	28.4928	0.910633
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.9722	−0.892173	−0.446087	−	0.894990i	$-0.647183\pi$
$983$	−27.9722	−0.892173	−0.446087	+	0.894990i	$0.647183\pi$
$984$	0	0
$985$	−11.4539	−0.364950
$986$	0	0
$987$	0	0
$988$	0	0
$989$	42.0935	1.33849
$990$	0	0
$991$	−34.8520	−1.10711	−0.553556	−	0.832812i	$-0.686729\pi$
$991$	−34.8520	−1.10711	−0.553556	+	0.832812i	$0.686729\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	24.7586	0.784899
$996$	0	0
$997$	37.6106	1.19114	0.595570	−	0.803304i	$-0.296926\pi$
$997$	37.6106	1.19114	0.595570	+	0.803304i	$0.296926\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.a.k.1.2		3
3.2	odd	2		1140.2.a.f.1.2	✓	3
12.11	even	2		4560.2.a.bu.1.2		3
15.2	even	4		5700.2.f.p.3649.5		6
15.8	even	4		5700.2.f.p.3649.2		6
15.14	odd	2		5700.2.a.z.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.f.1.2	✓	3	3.2	odd	2
3420.2.a.k.1.2		3	1.1	even	1	trivial
4560.2.a.bu.1.2		3	12.11	even	2
5700.2.a.z.1.2		3	15.14	odd	2
5700.2.f.p.3649.2		6	15.8	even	4
5700.2.f.p.3649.5		6	15.2	even	4

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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -0.166860 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -0.166860 q^{7} -4.71301 q^{11} +1.83314 q^{13} +4.54615 q^{17} +1.00000 q^{19} +4.54615 q^{23} +1.00000 q^{25} -6.71301 q^{29} -2.54615 q^{31} +0.166860 q^{35} +2.16686 q^{37} -10.7130 q^{41} +9.25915 q^{43} -0.546146 q^{47} -6.97216 q^{49} -10.5461 q^{53} +4.71301 q^{55} +9.42601 q^{59} +12.8799 q^{61} -1.83314 q^{65} +5.42601 q^{71} +15.0923 q^{73} +0.786413 q^{77} +13.7597 q^{79} +13.9722 q^{83} -4.54615 q^{85} -6.04557 q^{89} -0.305878 q^{91} -1.00000 q^{95} -2.16686 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5}+O(q^{10}) \) 3 * q - 3 * q^5	\( 3 q - 3 q^{5} + 2 q^{11} + 6 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{23} + 3 q^{25} - 4 q^{29} + 8 q^{31} + 6 q^{37} - 16 q^{41} - 4 q^{43} + 14 q^{47} + 27 q^{49} - 16 q^{53} - 2 q^{55} - 4 q^{59} + 22 q^{61} - 6 q^{65} - 16 q^{71} + 14 q^{73} + 20 q^{77} + 8 q^{79} - 6 q^{83} + 2 q^{85} - 4 q^{89} + 48 q^{91} - 3 q^{95} - 6 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^11 + 6 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^23 + 3 * q^25 - 4 * q^29 + 8 * q^31 + 6 * q^37 - 16 * q^41 - 4 * q^43 + 14 * q^47 + 27 * q^49 - 16 * q^53 - 2 * q^55 - 4 * q^59 + 22 * q^61 - 6 * q^65 - 16 * q^71 + 14 * q^73 + 20 * q^77 + 8 * q^79 - 6 * q^83 + 2 * q^85 - 4 * q^89 + 48 * q^91 - 3 * q^95 - 6 * q^97

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