LMFDB - Embedded newform 9072.2.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.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:	$72.4402847137$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 252)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46050 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-2.46050 q^{5} +1.00000 q^{7} +4.64766 q^{11} +7.10817 q^{13} +4.51459 q^{17} -4.32743 q^{19} +5.86693 q^{23} +1.05408 q^{25} +6.97509 q^{29} +7.38151 q^{31} -2.46050 q^{35} -0.726654 q^{37} -0.273346 q^{41} +4.83482 q^{43} +3.67257 q^{47} +1.00000 q^{49} -5.05408 q^{53} -11.4356 q^{55} +9.13307 q^{59} -13.8171 q^{61} -17.4897 q^{65} +1.32743 q^{67} +13.5218 q^{71} -4.32743 q^{73} +4.64766 q^{77} -6.43560 q^{79} +1.48541 q^{83} -11.1082 q^{85} -9.83482 q^{89} +7.10817 q^{91} +10.6477 q^{95} -0.492608 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q - q^5 + 3 * q^7	$ 3 q - q^{5} + 3 q^{7} + 2 q^{11} + 3 q^{13} - 2 q^{17} - 3 q^{19} + 14 q^{23} - 6 q^{25} - q^{29} + 3 q^{31} - q^{35} - 3 q^{37} - 3 q^{43} + 21 q^{47} + 3 q^{49} - 6 q^{53} - 6 q^{55} + 31 q^{59} + 6 q^{61} - 15 q^{65} - 6 q^{67} + 17 q^{71} - 3 q^{73} + 2 q^{77} + 9 q^{79} + 20 q^{83} - 15 q^{85} - 12 q^{89} + 3 q^{91} + 20 q^{95} - 9 q^{97}+O(q^{100}) $ 3 * q - q^5 + 3 * q^7 + 2 * q^11 + 3 * q^13 - 2 * q^17 - 3 * q^19 + 14 * q^23 - 6 * q^25 - q^29 + 3 * q^31 - q^35 - 3 * q^37 - 3 * q^43 + 21 * q^47 + 3 * q^49 - 6 * q^53 - 6 * q^55 + 31 * q^59 + 6 * q^61 - 15 * q^65 - 6 * q^67 + 17 * q^71 - 3 * q^73 + 2 * q^77 + 9 * q^79 + 20 * q^83 - 15 * q^85 - 12 * q^89 + 3 * q^91 + 20 * q^95 - 9 * 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
$5$	−2.46050	−1.10037	−0.550186	−	0.835042i	$-0.685443\pi$
$5$	−2.46050	−1.10037	−0.550186	+	0.835042i	$0.685443\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.64766	1.40132	0.700662	−	0.713494i	$-0.252888\pi$
$11$	4.64766	1.40132	0.700662	+	0.713494i	$0.252888\pi$
$12$	0	0
$13$	7.10817	1.97145	0.985726	−	0.168360i	$-0.0538471\pi$
$13$	7.10817	1.97145	0.985726	+	0.168360i	$0.0538471\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.51459	1.09495	0.547474	−	0.836822i	$-0.315589\pi$
$17$	4.51459	1.09495	0.547474	+	0.836822i	$0.315589\pi$
$18$	0	0
$19$	−4.32743	−0.992781	−0.496390	−	0.868099i	$-0.665342\pi$
$19$	−4.32743	−0.992781	−0.496390	+	0.868099i	$0.665342\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.86693	1.22334	0.611669	−	0.791114i	$-0.290498\pi$
$23$	5.86693	1.22334	0.611669	+	0.791114i	$0.290498\pi$
$24$	0	0
$25$	1.05408	0.210817
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.97509	1.29524	0.647621	−	0.761962i	$-0.275764\pi$
$29$	6.97509	1.29524	0.647621	+	0.761962i	$0.275764\pi$
$30$	0	0
$31$	7.38151	1.32576	0.662880	−	0.748726i	$-0.269334\pi$
$31$	7.38151	1.32576	0.662880	+	0.748726i	$0.269334\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.46050	−0.415901
$36$	0	0
$37$	−0.726654	−0.119461	−0.0597306	−	0.998215i	$-0.519024\pi$
$37$	−0.726654	−0.119461	−0.0597306	+	0.998215i	$0.519024\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.273346	−0.0426895	−0.0213448	−	0.999772i	$-0.506795\pi$
$41$	−0.273346	−0.0426895	−0.0213448	+	0.999772i	$0.506795\pi$
$42$	0	0
$43$	4.83482	0.737303	0.368652	−	0.929568i	$-0.379819\pi$
$43$	4.83482	0.737303	0.368652	+	0.929568i	$0.379819\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.67257	0.535699	0.267850	−	0.963461i	$-0.413687\pi$
$47$	3.67257	0.535699	0.267850	+	0.963461i	$0.413687\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.05408	−0.694232	−0.347116	−	0.937822i	$-0.612839\pi$
$53$	−5.05408	−0.694232	−0.347116	+	0.937822i	$0.612839\pi$
$54$	0	0
$55$	−11.4356	−1.54198
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.13307	1.18903	0.594513	−	0.804086i	$-0.297345\pi$
$59$	9.13307	1.18903	0.594513	+	0.804086i	$0.297345\pi$
$60$	0	0
$61$	−13.8171	−1.76910	−0.884550	−	0.466445i	$-0.845534\pi$
$61$	−13.8171	−1.76910	−0.884550	+	0.466445i	$0.845534\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−17.4897	−2.16933
$66$	0	0
$67$	1.32743	0.162171	0.0810857	−	0.996707i	$-0.474161\pi$
$67$	1.32743	0.162171	0.0810857	+	0.996707i	$0.474161\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.5218	1.60474	0.802370	−	0.596826i	$-0.203572\pi$
$71$	13.5218	1.60474	0.802370	+	0.596826i	$0.203572\pi$
$72$	0	0
$73$	−4.32743	−0.506487	−0.253244	−	0.967403i	$-0.581497\pi$
$73$	−4.32743	−0.506487	−0.253244	+	0.967403i	$0.581497\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.64766	0.529650
$78$	0	0
$79$	−6.43560	−0.724061	−0.362031	−	0.932166i	$-0.617916\pi$
$79$	−6.43560	−0.724061	−0.362031	+	0.932166i	$0.617916\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.48541	0.163045	0.0815225	−	0.996672i	$-0.474022\pi$
$83$	1.48541	0.163045	0.0815225	+	0.996672i	$0.474022\pi$
$84$	0	0
$85$	−11.1082	−1.20485
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.83482	−1.04249	−0.521245	−	0.853407i	$-0.674532\pi$
$89$	−9.83482	−1.04249	−0.521245	+	0.853407i	$0.674532\pi$
$90$	0	0
$91$	7.10817	0.745139
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.6477	1.09243
$96$	0	0
$97$	−0.492608	−0.0500168	−0.0250084	−	0.999687i	$-0.507961\pi$
$97$	−0.492608	−0.0500168	−0.0250084	+	0.999687i	$0.507961\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.40642	−0.338952	−0.169476	−	0.985534i	$-0.554207\pi$
$101$	−3.40642	−0.338952	−0.169476	+	0.985534i	$0.554207\pi$
$102$	0	0
$103$	−5.16225	−0.508652	−0.254326	−	0.967119i	$-0.581854\pi$
$103$	−5.16225	−0.508652	−0.254326	+	0.967119i	$0.581854\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.76303	−0.557133	−0.278567	−	0.960417i	$-0.589859\pi$
$107$	−5.76303	−0.557133	−0.278567	+	0.960417i	$0.589859\pi$
$108$	0	0
$109$	−8.98229	−0.860347	−0.430174	−	0.902746i	$-0.641548\pi$
$109$	−8.98229	−0.860347	−0.430174	+	0.902746i	$0.641548\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.35953	0.127894	0.0639471	−	0.997953i	$-0.479631\pi$
$113$	1.35953	0.127894	0.0639471	+	0.997953i	$0.479631\pi$
$114$	0	0
$115$	−14.4356	−1.34613
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.51459	0.413852
$120$	0	0
$121$	10.6008	0.963707
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.70895	0.868394
$126$	0	0
$127$	0.820039	0.0727667	0.0363833	−	0.999338i	$-0.488416\pi$
$127$	0.820039	0.0727667	0.0363833	+	0.999338i	$0.488416\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.78794	0.680435	0.340218	−	0.940347i	$-0.389499\pi$
$131$	7.78794	0.680435	0.340218	+	0.940347i	$0.389499\pi$
$132$	0	0
$133$	−4.32743	−0.375236
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.99280	0.255692	0.127846	−	0.991794i	$-0.459194\pi$
$137$	2.99280	0.255692	0.127846	+	0.991794i	$0.459194\pi$
$138$	0	0
$139$	−6.32743	−0.536686	−0.268343	−	0.963323i	$-0.586476\pi$
$139$	−6.32743	−0.536686	−0.268343	+	0.963323i	$0.586476\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	33.0364	2.76264
$144$	0	0
$145$	−17.1623	−1.42525
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.38151	0.358948	0.179474	−	0.983763i	$-0.442560\pi$
$149$	4.38151	0.358948	0.179474	+	0.983763i	$0.442560\pi$
$150$	0	0
$151$	−6.60078	−0.537164	−0.268582	−	0.963257i	$-0.586555\pi$
$151$	−6.60078	−0.537164	−0.268582	+	0.963257i	$0.586555\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−18.1623	−1.45883
$156$	0	0
$157$	−5.78074	−0.461353	−0.230677	−	0.973030i	$-0.574094\pi$
$157$	−5.78074	−0.461353	−0.230677	+	0.973030i	$0.574094\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.86693	0.462379
$162$	0	0
$163$	−7.32743	−0.573929	−0.286964	−	0.957941i	$-0.592646\pi$
$163$	−7.32743	−0.573929	−0.286964	+	0.957941i	$0.592646\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0292	−0.930846	−0.465423	−	0.885088i	$-0.654098\pi$
$167$	−12.0292	−0.930846	−0.465423	+	0.885088i	$0.654098\pi$
$168$	0	0
$169$	37.5261	2.88662
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.89903	−0.372466	−0.186233	−	0.982506i	$-0.559628\pi$
$173$	−4.89903	−0.372466	−0.186233	+	0.982506i	$0.559628\pi$
$174$	0	0
$175$	1.05408	0.0796813
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.78074	−0.133099	−0.0665493	−	0.997783i	$-0.521199\pi$
$179$	−1.78074	−0.133099	−0.0665493	+	0.997783i	$0.521199\pi$
$180$	0	0
$181$	−16.9430	−1.25936	−0.629681	−	0.776854i	$-0.716815\pi$
$181$	−16.9430	−1.25936	−0.629681	+	0.776854i	$0.716815\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.78794	0.131452
$186$	0	0
$187$	20.9823	1.53438
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.48968	−0.397220	−0.198610	−	0.980079i	$-0.563643\pi$
$191$	−5.48968	−0.397220	−0.198610	+	0.980079i	$0.563643\pi$
$192$	0	0
$193$	−5.50739	−0.396431	−0.198215	−	0.980158i	$-0.563515\pi$
$193$	−5.50739	−0.396431	−0.198215	+	0.980158i	$0.563515\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.6300	−0.828600	−0.414300	−	0.910140i	$-0.635974\pi$
$197$	−11.6300	−0.828600	−0.414300	+	0.910140i	$0.635974\pi$
$198$	0	0
$199$	4.14747	0.294006	0.147003	−	0.989136i	$-0.453037\pi$
$199$	4.14747	0.294006	0.147003	+	0.989136i	$0.453037\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.97509	0.489556
$204$	0	0
$205$	0.672570	0.0469743
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−20.1124	−1.39121
$210$	0	0
$211$	27.2163	1.87365	0.936825	−	0.349799i	$-0.113750\pi$
$211$	27.2163	1.87365	0.936825	+	0.349799i	$0.113750\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−11.8961	−0.811308
$216$	0	0
$217$	7.38151	0.501090
$218$	0	0
$219$	0	0
$220$	0	0
$221$	32.0905	2.15864
$222$	0	0
$223$	3.21634	0.215382	0.107691	−	0.994184i	$-0.465654\pi$
$223$	3.21634	0.215382	0.107691	+	0.994184i	$0.465654\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.9459	1.05837	0.529184	−	0.848507i	$-0.322498\pi$
$227$	15.9459	1.05837	0.529184	+	0.848507i	$0.322498\pi$
$228$	0	0
$229$	−1.21634	−0.0803778	−0.0401889	−	0.999192i	$-0.512796\pi$
$229$	−1.21634	−0.0803778	−0.0401889	+	0.999192i	$0.512796\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.9722	−1.30842	−0.654210	−	0.756313i	$-0.726999\pi$
$233$	−19.9722	−1.30842	−0.654210	+	0.756313i	$0.726999\pi$
$234$	0	0
$235$	−9.03638	−0.589468
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.01478	0.389064	0.194532	−	0.980896i	$-0.437681\pi$
$239$	6.01478	0.389064	0.194532	+	0.980896i	$0.437681\pi$
$240$	0	0
$241$	−18.6156	−1.19913	−0.599567	−	0.800325i	$-0.704660\pi$
$241$	−18.6156	−1.19913	−0.599567	+	0.800325i	$0.704660\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.46050	−0.157196
$246$	0	0
$247$	−30.7601	−1.95722
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.99707	0.441651	0.220826	−	0.975313i	$-0.429125\pi$
$251$	6.99707	0.441651	0.220826	+	0.975313i	$0.429125\pi$
$252$	0	0
$253$	27.2675	1.71429
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.7778	1.10895	0.554475	−	0.832201i	$-0.312919\pi$
$257$	17.7778	1.10895	0.554475	+	0.832201i	$0.312919\pi$
$258$	0	0
$259$	−0.726654	−0.0451521
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.1986	−1.67714	−0.838570	−	0.544794i	$-0.816608\pi$
$263$	−27.1986	−1.67714	−0.838570	+	0.544794i	$0.816608\pi$
$264$	0	0
$265$	12.4356	0.763913
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−23.8961	−1.45697	−0.728486	−	0.685061i	$-0.759775\pi$
$269$	−23.8961	−1.45697	−0.728486	+	0.685061i	$0.759775\pi$
$270$	0	0
$271$	−12.2733	−0.745553	−0.372776	−	0.927921i	$-0.621594\pi$
$271$	−12.2733	−0.745553	−0.372776	+	0.927921i	$0.621594\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.89903	0.295423
$276$	0	0
$277$	12.7807	0.767920	0.383960	−	0.923350i	$-0.374560\pi$
$277$	12.7807	0.767920	0.383960	+	0.923350i	$0.374560\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	28.5146	1.70104	0.850519	−	0.525945i	$-0.176288\pi$
$281$	28.5146	1.70104	0.850519	+	0.525945i	$0.176288\pi$
$282$	0	0
$283$	−0.726654	−0.0431951	−0.0215975	−	0.999767i	$-0.506875\pi$
$283$	−0.726654	−0.0431951	−0.0215975	+	0.999767i	$0.506875\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.273346	−0.0161351
$288$	0	0
$289$	3.38151	0.198913
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−25.5801	−1.49441	−0.747204	−	0.664595i	$-0.768604\pi$
$293$	−25.5801	−1.49441	−0.747204	+	0.664595i	$0.768604\pi$
$294$	0	0
$295$	−22.4720	−1.30837
$296$	0	0
$297$	0	0
$298$	0	0
$299$	41.7031	2.41175
$300$	0	0
$301$	4.83482	0.278675
$302$	0	0
$303$	0	0
$304$	0	0
$305$	33.9971	1.94667
$306$	0	0
$307$	6.23405	0.355796	0.177898	−	0.984049i	$-0.443070\pi$
$307$	6.23405	0.355796	0.177898	+	0.984049i	$0.443070\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	29.2383	1.65795	0.828976	−	0.559284i	$-0.188924\pi$
$311$	29.2383	1.65795	0.828976	+	0.559284i	$0.188924\pi$
$312$	0	0
$313$	28.4868	1.61017	0.805083	−	0.593162i	$-0.202121\pi$
$313$	28.4868	1.61017	0.805083	+	0.593162i	$0.202121\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.61849	−0.0909032	−0.0454516	−	0.998967i	$-0.514473\pi$
$317$	−1.61849	−0.0909032	−0.0454516	+	0.998967i	$0.514473\pi$
$318$	0	0
$319$	32.4179	1.81505
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−19.5366	−1.08704
$324$	0	0
$325$	7.49261	0.415615
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.67257	0.202475
$330$	0	0
$331$	13.9823	0.768536	0.384268	−	0.923222i	$-0.374454\pi$
$331$	13.9823	0.768536	0.384268	+	0.923222i	$0.374454\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.26615	−0.178449
$336$	0	0
$337$	27.7237	1.51021	0.755104	−	0.655605i	$-0.227586\pi$
$337$	27.7237	1.51021	0.755104	+	0.655605i	$0.227586\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	34.3068	1.85782
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.52898	0.404177	0.202089	−	0.979367i	$-0.435227\pi$
$347$	7.52898	0.404177	0.202089	+	0.979367i	$0.435227\pi$
$348$	0	0
$349$	−30.1082	−1.61165	−0.805827	−	0.592152i	$-0.798279\pi$
$349$	−30.1082	−1.61165	−0.805827	+	0.592152i	$0.798279\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.3638	1.08386	0.541928	−	0.840425i	$-0.317695\pi$
$353$	20.3638	1.08386	0.541928	+	0.840425i	$0.317695\pi$
$354$	0	0
$355$	−33.2704	−1.76581
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.0263	−0.845833	−0.422917	−	0.906169i	$-0.638994\pi$
$359$	−16.0263	−0.845833	−0.422917	+	0.906169i	$0.638994\pi$
$360$	0	0
$361$	−0.273346	−0.0143866
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.6477	0.557324
$366$	0	0
$367$	13.5979	0.709802	0.354901	−	0.934904i	$-0.384515\pi$
$367$	13.5979	0.709802	0.354901	+	0.934904i	$0.384515\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.05408	−0.262395
$372$	0	0
$373$	−21.9282	−1.13540	−0.567700	−	0.823236i	$-0.692167\pi$
$373$	−21.9282	−1.13540	−0.567700	+	0.823236i	$0.692167\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	49.5801	2.55351
$378$	0	0
$379$	29.7965	1.53054	0.765271	−	0.643708i	$-0.222605\pi$
$379$	29.7965	1.53054	0.765271	+	0.643708i	$0.222605\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.0219809	−0.00112317	−0.000561587	−	1.00000i	$-0.500179\pi$
$383$	−0.0219809	−0.00112317	−0.000561587		1.00000i	$0.500179\pi$
$384$	0	0
$385$	−11.4356	−0.582812
$386$	0	0
$387$	0	0
$388$	0	0
$389$	35.3566	1.79265	0.896326	−	0.443396i	$-0.146227\pi$
$389$	35.3566	1.79265	0.896326	+	0.443396i	$0.146227\pi$
$390$	0	0
$391$	26.4868	1.33949
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.8348	0.796736
$396$	0	0
$397$	−16.9430	−0.850344	−0.425172	−	0.905112i	$-0.639786\pi$
$397$	−16.9430	−0.850344	−0.425172	+	0.905112i	$0.639786\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.96362	0.147996	0.0739982	−	0.997258i	$-0.476424\pi$
$401$	2.96362	0.147996	0.0739982	+	0.997258i	$0.476424\pi$
$402$	0	0
$403$	52.4690	2.61367
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.37724	−0.167404
$408$	0	0
$409$	−14.6549	−0.724636	−0.362318	−	0.932054i	$-0.618015\pi$
$409$	−14.6549	−0.724636	−0.362318	+	0.932054i	$0.618015\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.13307	0.449409
$414$	0	0
$415$	−3.65486	−0.179410
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−25.2704	−1.23454	−0.617270	−	0.786751i	$-0.711762\pi$
$419$	−25.2704	−1.23454	−0.617270	+	0.786751i	$0.711762\pi$
$420$	0	0
$421$	−15.9971	−0.779650	−0.389825	−	0.920889i	$-0.627464\pi$
$421$	−15.9971	−0.779650	−0.389825	+	0.920889i	$0.627464\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.75876	0.230834
$426$	0	0
$427$	−13.8171	−0.668657
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.0335	0.627799	0.313900	−	0.949456i	$-0.398364\pi$
$431$	13.0335	0.627799	0.313900	+	0.949456i	$0.398364\pi$
$432$	0	0
$433$	23.5467	1.13158	0.565791	−	0.824549i	$-0.308571\pi$
$433$	23.5467	1.13158	0.565791	+	0.824549i	$0.308571\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−25.3887	−1.21451
$438$	0	0
$439$	−6.70895	−0.320200	−0.160100	−	0.987101i	$-0.551182\pi$
$439$	−6.70895	−0.320200	−0.160100	+	0.987101i	$0.551182\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.2455	1.67456	0.837282	−	0.546771i	$-0.184143\pi$
$443$	35.2455	1.67456	0.837282	+	0.546771i	$0.184143\pi$
$444$	0	0
$445$	24.1986	1.14712
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.9387	0.610616	0.305308	−	0.952254i	$-0.401241\pi$
$449$	12.9387	0.610616	0.305308	+	0.952254i	$0.401241\pi$
$450$	0	0
$451$	−1.27042	−0.0598218
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.4897	−0.819929
$456$	0	0
$457$	−29.1986	−1.36585	−0.682927	−	0.730487i	$-0.739293\pi$
$457$	−29.1986	−1.36585	−0.682927	+	0.730487i	$0.739293\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.6870	0.870339	0.435169	−	0.900349i	$-0.356689\pi$
$461$	18.6870	0.870339	0.435169	+	0.900349i	$0.356689\pi$
$462$	0	0
$463$	38.2498	1.77762	0.888809	−	0.458277i	$-0.151533\pi$
$463$	38.2498	1.77762	0.888809	+	0.458277i	$0.151533\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.2877	−0.707432	−0.353716	−	0.935353i	$-0.615082\pi$
$467$	−15.2877	−0.707432	−0.353716	+	0.935353i	$0.615082\pi$
$468$	0	0
$469$	1.32743	0.0612950
$470$	0	0
$471$	0	0
$472$	0	0
$473$	22.4706	1.03320
$474$	0	0
$475$	−4.56148	−0.209295
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11.0321	0.504070	0.252035	−	0.967718i	$-0.418900\pi$
$479$	11.0321	0.504070	0.252035	+	0.967718i	$0.418900\pi$
$480$	0	0
$481$	−5.16518	−0.235512
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.21206	0.0550370
$486$	0	0
$487$	−16.6008	−0.752253	−0.376126	−	0.926568i	$-0.622744\pi$
$487$	−16.6008	−0.752253	−0.376126	+	0.926568i	$0.622744\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.7267	−1.20616	−0.603079	−	0.797682i	$-0.706059\pi$
$491$	−26.7267	−1.20616	−0.603079	+	0.797682i	$0.706059\pi$
$492$	0	0
$493$	31.4897	1.41822
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.5218	0.606535
$498$	0	0
$499$	−1.23697	−0.0553744	−0.0276872	−	0.999617i	$-0.508814\pi$
$499$	−1.23697	−0.0553744	−0.0276872	+	0.999617i	$0.508814\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.07179	−0.0477889	−0.0238944	−	0.999714i	$-0.507607\pi$
$503$	−1.07179	−0.0477889	−0.0238944	+	0.999714i	$0.507607\pi$
$504$	0	0
$505$	8.38151	0.372973
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.0689	−0.889537	−0.444768	−	0.895646i	$-0.646714\pi$
$509$	−20.0689	−0.889537	−0.444768	+	0.895646i	$0.646714\pi$
$510$	0	0
$511$	−4.32743	−0.191434
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.7017	0.559706
$516$	0	0
$517$	17.0689	0.750688
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.8860	1.35314	0.676570	−	0.736379i	$-0.263466\pi$
$521$	30.8860	1.35314	0.676570	+	0.736379i	$0.263466\pi$
$522$	0	0
$523$	7.39922	0.323545	0.161773	−	0.986828i	$-0.448279\pi$
$523$	7.39922	0.323545	0.161773	+	0.986828i	$0.448279\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	33.3245	1.45164
$528$	0	0
$529$	11.4208	0.496557
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.94299	−0.0841603
$534$	0	0
$535$	14.1800	0.613053
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.64766	0.200189
$540$	0	0
$541$	22.6696	0.974644	0.487322	−	0.873222i	$-0.337974\pi$
$541$	22.6696	0.974644	0.487322	+	0.873222i	$0.337974\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.1010	0.946702
$546$	0	0
$547$	6.14747	0.262847	0.131423	−	0.991326i	$-0.458045\pi$
$547$	6.14747	0.262847	0.131423	+	0.991326i	$0.458045\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−30.1842	−1.28589
$552$	0	0
$553$	−6.43560	−0.273669
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.6739	−1.25732	−0.628662	−	0.777679i	$-0.716397\pi$
$557$	−29.6739	−1.25732	−0.628662	+	0.777679i	$0.716397\pi$
$558$	0	0
$559$	34.3667	1.45356
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−29.3111	−1.23531	−0.617657	−	0.786447i	$-0.711918\pi$
$563$	−29.3111	−1.23531	−0.617657	+	0.786447i	$0.711918\pi$
$564$	0	0
$565$	−3.34514	−0.140731
$566$	0	0
$567$	0	0
$568$	0	0
$569$	36.8860	1.54634	0.773170	−	0.634198i	$-0.218670\pi$
$569$	36.8860	1.54634	0.773170	+	0.634198i	$0.218670\pi$
$570$	0	0
$571$	−32.3786	−1.35500	−0.677501	−	0.735522i	$-0.736937\pi$
$571$	−32.3786	−1.35500	−0.677501	+	0.735522i	$0.736937\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.18423	0.257900
$576$	0	0
$577$	23.0187	0.958280	0.479140	−	0.877739i	$-0.340949\pi$
$577$	23.0187	0.958280	0.479140	+	0.877739i	$0.340949\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.48541	0.0616252
$582$	0	0
$583$	−23.4897	−0.972843
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.74105	0.236958	0.118479	−	0.992957i	$-0.462198\pi$
$587$	5.74105	0.236958	0.118479	+	0.992957i	$0.462198\pi$
$588$	0	0
$589$	−31.9430	−1.31619
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−27.7453	−1.13936	−0.569682	−	0.821865i	$-0.692934\pi$
$593$	−27.7453	−1.13936	−0.569682	+	0.821865i	$0.692934\pi$
$594$	0	0
$595$	−11.1082	−0.455391
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.10817	0.167855	0.0839276	−	0.996472i	$-0.473254\pi$
$599$	4.10817	0.167855	0.0839276	+	0.996472i	$0.473254\pi$
$600$	0	0
$601$	15.6185	0.637091	0.318546	−	0.947908i	$-0.396806\pi$
$601$	15.6185	0.637091	0.318546	+	0.947908i	$0.396806\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−26.0833	−1.06044
$606$	0	0
$607$	0.561476	0.0227896	0.0113948	−	0.999935i	$-0.496373\pi$
$607$	0.561476	0.0227896	0.0113948	+	0.999935i	$0.496373\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26.1052	1.05611
$612$	0	0
$613$	−20.2016	−0.815933	−0.407967	−	0.912997i	$-0.633762\pi$
$613$	−20.2016	−0.815933	−0.407967	+	0.912997i	$0.633762\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.9138	0.922475	0.461238	−	0.887277i	$-0.347406\pi$
$617$	22.9138	0.922475	0.461238	+	0.887277i	$0.347406\pi$
$618$	0	0
$619$	−39.7031	−1.59580	−0.797901	−	0.602788i	$-0.794056\pi$
$619$	−39.7031	−1.59580	−0.797901	+	0.602788i	$0.794056\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−9.83482	−0.394024
$624$	0	0
$625$	−29.1593	−1.16637
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.28054	−0.130804
$630$	0	0
$631$	31.0364	1.23554	0.617769	−	0.786359i	$-0.288037\pi$
$631$	31.0364	1.23554	0.617769	+	0.786359i	$0.288037\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.01771	−0.0800703
$636$	0	0
$637$	7.10817	0.281636
$638$	0	0
$639$	0	0
$640$	0	0
$641$	29.5864	1.16859	0.584296	−	0.811541i	$-0.301371\pi$
$641$	29.5864	1.16859	0.584296	+	0.811541i	$0.301371\pi$
$642$	0	0
$643$	−25.6883	−1.01305	−0.506524	−	0.862226i	$-0.669070\pi$
$643$	−25.6883	−1.01305	−0.506524	+	0.862226i	$0.669070\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.0177	0.669035	0.334518	−	0.942390i	$-0.391427\pi$
$647$	17.0177	0.669035	0.334518	+	0.942390i	$0.391427\pi$
$648$	0	0
$649$	42.4475	1.66621
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.47102	−0.0575653	−0.0287827	−	0.999586i	$-0.509163\pi$
$653$	−1.47102	−0.0575653	−0.0287827	+	0.999586i	$0.509163\pi$
$654$	0	0
$655$	−19.1623	−0.748731
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−41.4006	−1.61274	−0.806369	−	0.591413i	$-0.798570\pi$
$659$	−41.4006	−1.61274	−0.806369	+	0.591413i	$0.798570\pi$
$660$	0	0
$661$	38.2704	1.48855	0.744273	−	0.667875i	$-0.232796\pi$
$661$	38.2704	1.48855	0.744273	+	0.667875i	$0.232796\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.6477	0.412899
$666$	0	0
$667$	40.9224	1.58452
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−64.2173	−2.47908
$672$	0	0
$673$	−30.4897	−1.17529	−0.587645	−	0.809119i	$-0.699945\pi$
$673$	−30.4897	−1.17529	−0.587645	+	0.809119i	$0.699945\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−44.9253	−1.72662	−0.863309	−	0.504675i	$-0.831612\pi$
$677$	−44.9253	−1.72662	−0.863309	+	0.504675i	$0.831612\pi$
$678$	0	0
$679$	−0.492608	−0.0189046
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−48.3973	−1.85187	−0.925935	−	0.377683i	$-0.876721\pi$
$683$	−48.3973	−1.85187	−0.925935	+	0.377683i	$0.876721\pi$
$684$	0	0
$685$	−7.36381	−0.281357
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−35.9253	−1.36864
$690$	0	0
$691$	−18.3815	−0.699266	−0.349633	−	0.936887i	$-0.613694\pi$
$691$	−18.3815	−0.699266	−0.349633	+	0.936887i	$0.613694\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.5687	0.590553
$696$	0	0
$697$	−1.23405	−0.0467428
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27.0292	1.02088	0.510439	−	0.859914i	$-0.329483\pi$
$701$	27.0292	1.02088	0.510439	+	0.859914i	$0.329483\pi$
$702$	0	0
$703$	3.14454	0.118599
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.40642	−0.128112
$708$	0	0
$709$	4.98522	0.187224	0.0936119	−	0.995609i	$-0.470159\pi$
$709$	4.98522	0.187224	0.0936119	+	0.995609i	$0.470159\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	43.3068	1.62185
$714$	0	0
$715$	−81.2862	−3.03993
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15.6942	0.585293	0.292647	−	0.956221i	$-0.405464\pi$
$719$	15.6942	0.585293	0.292647	+	0.956221i	$0.405464\pi$
$720$	0	0
$721$	−5.16225	−0.192252
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.35234	0.273059
$726$	0	0
$727$	−21.8142	−0.809043	−0.404522	−	0.914528i	$-0.632562\pi$
$727$	−21.8142	−0.809043	−0.404522	+	0.914528i	$0.632562\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	21.8272	0.807309
$732$	0	0
$733$	−24.0148	−0.887006	−0.443503	−	0.896273i	$-0.646264\pi$
$733$	−24.0148	−0.887006	−0.443503	+	0.896273i	$0.646264\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.16945	0.227255
$738$	0	0
$739$	−18.7089	−0.688220	−0.344110	−	0.938929i	$-0.611819\pi$
$739$	−18.7089	−0.688220	−0.344110	+	0.938929i	$0.611819\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−40.3068	−1.47871	−0.739356	−	0.673314i	$-0.764870\pi$
$743$	−40.3068	−1.47871	−0.739356	+	0.673314i	$0.764870\pi$
$744$	0	0
$745$	−10.7807	−0.394976
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.76303	−0.210577
$750$	0	0
$751$	21.1259	0.770894	0.385447	−	0.922730i	$-0.374047\pi$
$751$	21.1259	0.770894	0.385447	+	0.922730i	$0.374047\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.2412	0.591079
$756$	0	0
$757$	8.85934	0.321998	0.160999	−	0.986955i	$-0.448528\pi$
$757$	8.85934	0.321998	0.160999	+	0.986955i	$0.448528\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.38910	−0.0503549	−0.0251774	−	0.999683i	$-0.508015\pi$
$761$	−1.38910	−0.0503549	−0.0251774	+	0.999683i	$0.508015\pi$
$762$	0	0
$763$	−8.98229	−0.325181
$764$	0	0
$765$	0	0
$766$	0	0
$767$	64.9194	2.34410
$768$	0	0
$769$	−37.9253	−1.36762	−0.683810	−	0.729660i	$-0.739678\pi$
$769$	−37.9253	−1.36762	−0.683810	+	0.729660i	$0.739678\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.31596	−0.0473318	−0.0236659	−	0.999720i	$-0.507534\pi$
$773$	−1.31596	−0.0473318	−0.0236659	+	0.999720i	$0.507534\pi$
$774$	0	0
$775$	7.78074	0.279492
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.18289	0.0423813
$780$	0	0
$781$	62.8447	2.24876
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.2235	0.507660
$786$	0	0
$787$	−12.2586	−0.436971	−0.218485	−	0.975840i	$-0.570112\pi$
$787$	−12.2586	−0.436971	−0.218485	+	0.975840i	$0.570112\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.35953	0.0483395
$792$	0	0
$793$	−98.2144	−3.48769
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.4356	−0.759288	−0.379644	−	0.925133i	$-0.623953\pi$
$797$	−21.4356	−0.759288	−0.379644	+	0.925133i	$0.623953\pi$
$798$	0	0
$799$	16.5801	0.586563
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.1124	−0.709753
$804$	0	0
$805$	−14.4356	−0.508788
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.6955	−0.938564	−0.469282	−	0.883048i	$-0.655487\pi$
$809$	−26.6955	−0.938564	−0.469282	+	0.883048i	$0.655487\pi$
$810$	0	0
$811$	−38.2852	−1.34438	−0.672188	−	0.740381i	$-0.734645\pi$
$811$	−38.2852	−1.34438	−0.672188	+	0.740381i	$0.734645\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.0292	0.631535
$816$	0	0
$817$	−20.9224	−0.731981
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.4998	0.366446	0.183223	−	0.983071i	$-0.441347\pi$
$821$	10.4998	0.366446	0.183223	+	0.983071i	$0.441347\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.7817	−1.69631	−0.848153	−	0.529752i	$-0.822285\pi$
$827$	−48.7817	−1.69631	−0.848153	+	0.529752i	$0.822285\pi$
$828$	0	0
$829$	−6.21926	−0.216004	−0.108002	−	0.994151i	$-0.534445\pi$
$829$	−6.21926	−0.216004	−0.108002	+	0.994151i	$0.534445\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.51459	0.156421
$834$	0	0
$835$	29.5979	1.02428
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42.0731	−1.45253	−0.726263	−	0.687417i	$-0.758745\pi$
$839$	−42.0731	−1.45253	−0.726263	+	0.687417i	$0.758745\pi$
$840$	0	0
$841$	19.6519	0.677653
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−92.3330	−3.17635
$846$	0	0
$847$	10.6008	0.364247
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.26322	−0.146141
$852$	0	0
$853$	−13.4504	−0.460532	−0.230266	−	0.973128i	$-0.573960\pi$
$853$	−13.4504	−0.460532	−0.230266	+	0.973128i	$0.573960\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.3786	−1.41347	−0.706733	−	0.707481i	$-0.749832\pi$
$857$	−41.3786	−1.41347	−0.706733	+	0.707481i	$0.749832\pi$
$858$	0	0
$859$	39.7630	1.35670	0.678349	−	0.734740i	$-0.262696\pi$
$859$	39.7630	1.35670	0.678349	+	0.734740i	$0.262696\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	53.3858	1.81727	0.908637	−	0.417588i	$-0.137124\pi$
$863$	53.3858	1.81727	0.908637	+	0.417588i	$0.137124\pi$
$864$	0	0
$865$	12.0541	0.409851
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−29.9105	−1.01464
$870$	0	0
$871$	9.43560	0.319713
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.70895	0.328222
$876$	0	0
$877$	−6.85349	−0.231426	−0.115713	−	0.993283i	$-0.536915\pi$
$877$	−6.85349	−0.231426	−0.115713	+	0.993283i	$0.536915\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.5103	0.421483	0.210742	−	0.977542i	$-0.432412\pi$
$881$	12.5103	0.421483	0.210742	+	0.977542i	$0.432412\pi$
$882$	0	0
$883$	−6.69124	−0.225178	−0.112589	−	0.993642i	$-0.535914\pi$
$883$	−6.69124	−0.225178	−0.112589	+	0.993642i	$0.535914\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.1416	1.07921	0.539605	−	0.841918i	$-0.318574\pi$
$887$	32.1416	1.07921	0.539605	+	0.841918i	$0.318574\pi$
$888$	0	0
$889$	0.820039	0.0275032
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−15.8928	−0.531832
$894$	0	0
$895$	4.38151	0.146458
$896$	0	0
$897$	0	0
$898$	0	0
$899$	51.4868	1.71718
$900$	0	0
$901$	−22.8171	−0.760148
$902$	0	0
$903$	0	0
$904$	0	0
$905$	41.6883	1.38577
$906$	0	0
$907$	31.4031	1.04272	0.521362	−	0.853336i	$-0.325424\pi$
$907$	31.4031	1.04272	0.521362	+	0.853336i	$0.325424\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.7965	1.51731	0.758653	−	0.651495i	$-0.225858\pi$
$911$	45.7965	1.51731	0.758653	+	0.651495i	$0.225858\pi$
$912$	0	0
$913$	6.90369	0.228479
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.78794	0.257180
$918$	0	0
$919$	−27.1800	−0.896584	−0.448292	−	0.893887i	$-0.647968\pi$
$919$	−27.1800	−0.896584	−0.448292	+	0.893887i	$0.647968\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	96.1151	3.16367
$924$	0	0
$925$	−0.765954	−0.0251844
$926$	0	0
$927$	0	0
$928$	0	0
$929$	40.6677	1.33426	0.667132	−	0.744940i	$-0.267522\pi$
$929$	40.6677	1.33426	0.667132	+	0.744940i	$0.267522\pi$
$930$	0	0
$931$	−4.32743	−0.141826
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−51.6270	−1.68838
$936$	0	0
$937$	16.4150	0.536254	0.268127	−	0.963384i	$-0.413595\pi$
$937$	16.4150	0.536254	0.268127	+	0.963384i	$0.413595\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−7.33755	−0.239197	−0.119599	−	0.992822i	$-0.538161\pi$
$941$	−7.33755	−0.239197	−0.119599	+	0.992822i	$0.538161\pi$
$942$	0	0
$943$	−1.60370	−0.0522237
$944$	0	0
$945$	0	0
$946$	0	0
$947$	59.1124	1.92090	0.960448	−	0.278459i	$-0.0898236\pi$
$947$	59.1124	1.92090	0.960448	+	0.278459i	$0.0898236\pi$
$948$	0	0
$949$	−30.7601	−0.998515
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.9354	0.548592	0.274296	−	0.961645i	$-0.411555\pi$
$953$	16.9354	0.548592	0.274296	+	0.961645i	$0.411555\pi$
$954$	0	0
$955$	13.5074	0.437089
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.99280	0.0966426
$960$	0	0
$961$	23.4868	0.757637
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.5510	0.436221
$966$	0	0
$967$	7.11109	0.228677	0.114339	−	0.993442i	$-0.463525\pi$
$967$	7.11109	0.228677	0.114339	+	0.993442i	$0.463525\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.47102	−0.0472072	−0.0236036	−	0.999721i	$-0.507514\pi$
$971$	−1.47102	−0.0472072	−0.0236036	+	0.999721i	$0.507514\pi$
$972$	0	0
$973$	−6.32743	−0.202848
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19.4327	0.621706	0.310853	−	0.950458i	$-0.399385\pi$
$977$	19.4327	0.621706	0.310853	+	0.950458i	$0.399385\pi$
$978$	0	0
$979$	−45.7089	−1.46086
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.74436	0.247007	0.123503	−	0.992344i	$-0.460587\pi$
$983$	7.74436	0.247007	0.123503	+	0.992344i	$0.460587\pi$
$984$	0	0
$985$	28.6156	0.911768
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28.3655	0.901972
$990$	0	0
$991$	14.4710	0.459687	0.229843	−	0.973228i	$-0.426179\pi$
$991$	14.4710	0.459687	0.229843	+	0.973228i	$0.426179\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.2049	−0.323516
$996$	0	0
$997$	−55.3097	−1.75168	−0.875838	−	0.482605i	$-0.839691\pi$
$997$	−55.3097	−1.75168	−0.875838	+	0.482605i	$0.839691\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.bv.1.1		3
3.2	odd	2		9072.2.a.by.1.3		3
4.3	odd	2		2268.2.a.h.1.1		3
9.2	odd	6		1008.2.r.j.337.2		6
9.4	even	3		3024.2.r.j.2017.3		6
9.5	odd	6		1008.2.r.j.673.2		6
9.7	even	3		3024.2.r.j.1009.3		6
12.11	even	2		2268.2.a.i.1.3		3
36.7	odd	6		756.2.j.b.253.3		6
36.11	even	6		252.2.j.a.85.2	✓	6
36.23	even	6		252.2.j.a.169.2	yes	6
36.31	odd	6		756.2.j.b.505.3		6
252.11	even	6		1764.2.i.g.373.3		6
252.23	even	6		1764.2.i.g.1537.3		6
252.31	even	6		5292.2.l.f.3313.3		6
252.47	odd	6		1764.2.l.f.949.2		6
252.59	odd	6		1764.2.l.f.961.2		6
252.67	odd	6		5292.2.l.e.3313.1		6
252.79	odd	6		5292.2.l.e.361.1		6
252.83	odd	6		1764.2.j.e.589.2		6
252.95	even	6		1764.2.l.e.961.2		6
252.103	even	6		5292.2.i.e.2125.1		6
252.115	even	6		5292.2.i.e.1549.1		6
252.131	odd	6		1764.2.i.d.1537.1		6
252.139	even	6		5292.2.j.d.3529.1		6
252.151	odd	6		5292.2.i.f.1549.3		6
252.167	odd	6		1764.2.j.e.1177.2		6
252.187	even	6		5292.2.l.f.361.3		6
252.191	even	6		1764.2.l.e.949.2		6
252.223	even	6		5292.2.j.d.1765.1		6
252.227	odd	6		1764.2.i.d.373.1		6
252.247	odd	6		5292.2.i.f.2125.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.a.85.2	✓	6	36.11	even	6
252.2.j.a.169.2	yes	6	36.23	even	6
756.2.j.b.253.3		6	36.7	odd	6
756.2.j.b.505.3		6	36.31	odd	6
1008.2.r.j.337.2		6	9.2	odd	6
1008.2.r.j.673.2		6	9.5	odd	6
1764.2.i.d.373.1		6	252.227	odd	6
1764.2.i.d.1537.1		6	252.131	odd	6
1764.2.i.g.373.3		6	252.11	even	6
1764.2.i.g.1537.3		6	252.23	even	6
1764.2.j.e.589.2		6	252.83	odd	6
1764.2.j.e.1177.2		6	252.167	odd	6
1764.2.l.e.949.2		6	252.191	even	6
1764.2.l.e.961.2		6	252.95	even	6
1764.2.l.f.949.2		6	252.47	odd	6
1764.2.l.f.961.2		6	252.59	odd	6
2268.2.a.h.1.1		3	4.3	odd	2
2268.2.a.i.1.3		3	12.11	even	2
3024.2.r.j.1009.3		6	9.7	even	3
3024.2.r.j.2017.3		6	9.4	even	3
5292.2.i.e.1549.1		6	252.115	even	6
5292.2.i.e.2125.1		6	252.103	even	6
5292.2.i.f.1549.3		6	252.151	odd	6
5292.2.i.f.2125.3		6	252.247	odd	6
5292.2.j.d.1765.1		6	252.223	even	6
5292.2.j.d.3529.1		6	252.139	even	6
5292.2.l.e.361.1		6	252.79	odd	6
5292.2.l.e.3313.1		6	252.67	odd	6
5292.2.l.f.361.3		6	252.187	even	6
5292.2.l.f.3313.3		6	252.31	even	6
9072.2.a.bv.1.1		3	1.1	even	1	trivial
9072.2.a.by.1.3		3	3.2	odd	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:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46050 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-2.46050 q^{5} +1.00000 q^{7} +4.64766 q^{11} +7.10817 q^{13} +4.51459 q^{17} -4.32743 q^{19} +5.86693 q^{23} +1.05408 q^{25} +6.97509 q^{29} +7.38151 q^{31} -2.46050 q^{35} -0.726654 q^{37} -0.273346 q^{41} +4.83482 q^{43} +3.67257 q^{47} +1.00000 q^{49} -5.05408 q^{53} -11.4356 q^{55} +9.13307 q^{59} -13.8171 q^{61} -17.4897 q^{65} +1.32743 q^{67} +13.5218 q^{71} -4.32743 q^{73} +4.64766 q^{77} -6.43560 q^{79} +1.48541 q^{83} -11.1082 q^{85} -9.83482 q^{89} +7.10817 q^{91} +10.6477 q^{95} -0.492608 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q - q^5 + 3 * q^7	\( 3 q - q^{5} + 3 q^{7} + 2 q^{11} + 3 q^{13} - 2 q^{17} - 3 q^{19} + 14 q^{23} - 6 q^{25} - q^{29} + 3 q^{31} - q^{35} - 3 q^{37} - 3 q^{43} + 21 q^{47} + 3 q^{49} - 6 q^{53} - 6 q^{55} + 31 q^{59} + 6 q^{61} - 15 q^{65} - 6 q^{67} + 17 q^{71} - 3 q^{73} + 2 q^{77} + 9 q^{79} + 20 q^{83} - 15 q^{85} - 12 q^{89} + 3 q^{91} + 20 q^{95} - 9 q^{97}+O(q^{100}) \) 3 * q - q^5 + 3 * q^7 + 2 * q^11 + 3 * q^13 - 2 * q^17 - 3 * q^19 + 14 * q^23 - 6 * q^25 - q^29 + 3 * q^31 - q^35 - 3 * q^37 - 3 * q^43 + 21 * q^47 + 3 * q^49 - 6 * q^53 - 6 * q^55 + 31 * q^59 + 6 * q^61 - 15 * q^65 - 6 * q^67 + 17 * q^71 - 3 * q^73 + 2 * q^77 + 9 * q^79 + 20 * q^83 - 15 * q^85 - 12 * q^89 + 3 * q^91 + 20 * q^95 - 9 * q^97

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