LMFDB - Embedded newform 5712.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5712,2,Mod(1,5712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5712, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5712.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:	$45.6105496346$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.4764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x - 6 $ x^3 - x^2 - 12*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1428)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.19852$ of defining polynomial
Character	$\chi$	$=$	5712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.19852 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.23055 q^{11} -3.19852 q^{13} -3.19852 q^{15} +1.00000 q^{17} +3.19852 q^{19} -1.00000 q^{21} -2.23055 q^{23} +5.23055 q^{25} +1.00000 q^{27} -7.42907 q^{31} +2.23055 q^{33} +3.19852 q^{35} +4.96797 q^{37} -3.19852 q^{39} -3.19852 q^{41} -4.16650 q^{43} -3.19852 q^{45} +7.42907 q^{47} +1.00000 q^{49} +1.00000 q^{51} -1.42907 q^{53} -7.13447 q^{55} +3.19852 q^{57} -3.42907 q^{59} +5.03203 q^{61} -1.00000 q^{63} +10.2306 q^{65} -14.3970 q^{67} -2.23055 q^{69} -6.39705 q^{71} +14.3970 q^{73} +5.23055 q^{75} -2.23055 q^{77} -5.03203 q^{79} +1.00000 q^{81} +16.3330 q^{83} -3.19852 q^{85} +8.39705 q^{89} +3.19852 q^{91} -7.42907 q^{93} -10.2306 q^{95} +8.00000 q^{97} +2.23055 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} + 3 q^{17} - 2 q^{19} - 3 q^{21} - 2 q^{23} + 11 q^{25} + 3 q^{27} - 6 q^{31} + 2 q^{33} - 2 q^{35} + 8 q^{37} + 2 q^{39} + 2 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 + 3 * q^17 - 2 * q^19 - 3 * q^21 - 2 * q^23 + 11 * q^25 + 3 * q^27 - 6 * q^31 + 2 * q^33 - 2 * q^35 + 8 * q^37 + 2 * q^39 + 2 * q^41 + 6 * q^43 + 2 * q^45 + 6 * q^47 + 3 * q^49 + 3 * q^51 + 12 * q^53 + 4 * q^55 - 2 * q^57 + 6 * q^59 + 22 * q^61 - 3 * q^63 + 26 * q^65 - 20 * q^67 - 2 * q^69 + 4 * q^71 + 20 * q^73 + 11 * q^75 - 2 * q^77 - 22 * q^79 + 3 * q^81 + 12 * q^83 + 2 * q^85 + 2 * q^89 - 2 * q^91 - 6 * q^93 - 26 * q^95 + 24 * q^97 + 2 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.19852	−1.43042	−0.715212	−	0.698908i	$-0.753670\pi$
$5$	−3.19852	−1.43042	−0.715212	+	0.698908i	$0.753670\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.23055	0.672536	0.336268	−	0.941766i	$-0.390835\pi$
$11$	2.23055	0.672536	0.336268	+	0.941766i	$0.390835\pi$
$12$	0	0
$13$	−3.19852	−0.887111	−0.443555	−	0.896247i	$-0.646283\pi$
$13$	−3.19852	−0.887111	−0.443555	+	0.896247i	$0.646283\pi$
$14$	0	0
$15$	−3.19852	−0.825855
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.19852	0.733792	0.366896	−	0.930262i	$-0.380421\pi$
$19$	3.19852	0.733792	0.366896	+	0.930262i	$0.380421\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−2.23055	−0.465102	−0.232551	−	0.972584i	$-0.574707\pi$
$23$	−2.23055	−0.465102	−0.232551	+	0.972584i	$0.574707\pi$
$24$	0	0
$25$	5.23055	1.04611
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−7.42907	−1.33430	−0.667151	−	0.744923i	$-0.732486\pi$
$31$	−7.42907	−1.33430	−0.667151	+	0.744923i	$0.732486\pi$
$32$	0	0
$33$	2.23055	0.388289
$34$	0	0
$35$	3.19852	0.540649
$36$	0	0
$37$	4.96797	0.816730	0.408365	−	0.912819i	$-0.366099\pi$
$37$	4.96797	0.816730	0.408365	+	0.912819i	$0.366099\pi$
$38$	0	0
$39$	−3.19852	−0.512174
$40$	0	0
$41$	−3.19852	−0.499525	−0.249763	−	0.968307i	$-0.580353\pi$
$41$	−3.19852	−0.499525	−0.249763	+	0.968307i	$0.580353\pi$
$42$	0	0
$43$	−4.16650	−0.635385	−0.317692	−	0.948194i	$-0.602908\pi$
$43$	−4.16650	−0.635385	−0.317692	+	0.948194i	$0.602908\pi$
$44$	0	0
$45$	−3.19852	−0.476808
$46$	0	0
$47$	7.42907	1.08364	0.541821	−	0.840494i	$-0.317735\pi$
$47$	7.42907	1.08364	0.541821	+	0.840494i	$0.317735\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	−1.42907	−0.196298	−0.0981492	−	0.995172i	$-0.531292\pi$
$53$	−1.42907	−0.196298	−0.0981492	+	0.995172i	$0.531292\pi$
$54$	0	0
$55$	−7.13447	−0.962011
$56$	0	0
$57$	3.19852	0.423655
$58$	0	0
$59$	−3.42907	−0.446427	−0.223214	−	0.974770i	$-0.571655\pi$
$59$	−3.42907	−0.446427	−0.223214	+	0.974770i	$0.571655\pi$
$60$	0	0
$61$	5.03203	0.644285	0.322143	−	0.946691i	$-0.395597\pi$
$61$	5.03203	0.644285	0.322143	+	0.946691i	$0.395597\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	10.2306	1.26894
$66$	0	0
$67$	−14.3970	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$67$	−14.3970	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$68$	0	0
$69$	−2.23055	−0.268527
$70$	0	0
$71$	−6.39705	−0.759190	−0.379595	−	0.925153i	$-0.623937\pi$
$71$	−6.39705	−0.759190	−0.379595	+	0.925153i	$0.623937\pi$
$72$	0	0
$73$	14.3970	1.68505	0.842523	−	0.538660i	$-0.181069\pi$
$73$	14.3970	1.68505	0.842523	+	0.538660i	$0.181069\pi$
$74$	0	0
$75$	5.23055	0.603972
$76$	0	0
$77$	−2.23055	−0.254195
$78$	0	0
$79$	−5.03203	−0.566147	−0.283074	−	0.959098i	$-0.591354\pi$
$79$	−5.03203	−0.566147	−0.283074	+	0.959098i	$0.591354\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.3330	1.79278	0.896389	−	0.443268i	$-0.146181\pi$
$83$	16.3330	1.79278	0.896389	+	0.443268i	$0.146181\pi$
$84$	0	0
$85$	−3.19852	−0.346929
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.39705	0.890085	0.445043	−	0.895509i	$-0.353189\pi$
$89$	8.39705	0.890085	0.445043	+	0.895509i	$0.353189\pi$
$90$	0	0
$91$	3.19852	0.335296
$92$	0	0
$93$	−7.42907	−0.770359
$94$	0	0
$95$	−10.2306	−1.04963
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	2.23055	0.224179
$100$	0	0
$101$	11.3650	1.13086	0.565431	−	0.824796i	$-0.308710\pi$
$101$	11.3650	1.13086	0.565431	+	0.824796i	$0.308710\pi$
$102$	0	0
$103$	−0.801477	−0.0789719	−0.0394859	−	0.999220i	$-0.512572\pi$
$103$	−0.801477	−0.0789719	−0.0394859	+	0.999220i	$0.512572\pi$
$104$	0	0
$105$	3.19852	0.312144
$106$	0	0
$107$	−6.23055	−0.602330	−0.301165	−	0.953572i	$-0.597376\pi$
$107$	−6.23055	−0.602330	−0.301165	+	0.953572i	$0.597376\pi$
$108$	0	0
$109$	8.96797	0.858976	0.429488	−	0.903073i	$-0.358694\pi$
$109$	8.96797	0.858976	0.429488	+	0.903073i	$0.358694\pi$
$110$	0	0
$111$	4.96797	0.471539
$112$	0	0
$113$	8.62760	0.811616	0.405808	−	0.913958i	$-0.366990\pi$
$113$	8.62760	0.811616	0.405808	+	0.913958i	$0.366990\pi$
$114$	0	0
$115$	7.13447	0.665293
$116$	0	0
$117$	−3.19852	−0.295704
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−6.02464	−0.547695
$122$	0	0
$123$	−3.19852	−0.288401
$124$	0	0
$125$	−0.737422	−0.0659570
$126$	0	0
$127$	−0.166496	−0.0147741	−0.00738705	−	0.999973i	$-0.502351\pi$
$127$	−0.166496	−0.0147741	−0.00738705	+	0.999973i	$0.502351\pi$
$128$	0	0
$129$	−4.16650	−0.366839
$130$	0	0
$131$	13.1985	1.15316	0.576580	−	0.817041i	$-0.304387\pi$
$131$	13.1985	1.15316	0.576580	+	0.817041i	$0.304387\pi$
$132$	0	0
$133$	−3.19852	−0.277347
$134$	0	0
$135$	−3.19852	−0.275285
$136$	0	0
$137$	6.57093	0.561392	0.280696	−	0.959797i	$-0.409435\pi$
$137$	6.57093	0.561392	0.280696	+	0.959797i	$0.409435\pi$
$138$	0	0
$139$	14.3970	1.22114	0.610571	−	0.791962i	$-0.290940\pi$
$139$	14.3970	1.22114	0.610571	+	0.791962i	$0.290940\pi$
$140$	0	0
$141$	7.42907	0.625641
$142$	0	0
$143$	−7.13447	−0.596614
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−0.0640550	−0.00524759	−0.00262380	−	0.999997i	$-0.500835\pi$
$149$	−0.0640550	−0.00524759	−0.00262380	+	0.999997i	$0.500835\pi$
$150$	0	0
$151$	−8.46110	−0.688555	−0.344277	−	0.938868i	$-0.611876\pi$
$151$	−8.46110	−0.688555	−0.344277	+	0.938868i	$0.611876\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	23.7621	1.90862
$156$	0	0
$157$	−3.65962	−0.292070	−0.146035	−	0.989279i	$-0.546651\pi$
$157$	−3.65962	−0.292070	−0.146035	+	0.989279i	$0.546651\pi$
$158$	0	0
$159$	−1.42907	−0.113333
$160$	0	0
$161$	2.23055	0.175792
$162$	0	0
$163$	13.2552	1.03823	0.519113	−	0.854705i	$-0.326262\pi$
$163$	13.2552	1.03823	0.519113	+	0.854705i	$0.326262\pi$
$164$	0	0
$165$	−7.13447	−0.555418
$166$	0	0
$167$	3.26258	0.252466	0.126233	−	0.992001i	$-0.459711\pi$
$167$	3.26258	0.252466	0.126233	+	0.992001i	$0.459711\pi$
$168$	0	0
$169$	−2.76945	−0.213035
$170$	0	0
$171$	3.19852	0.244597
$172$	0	0
$173$	−2.40443	−0.182805	−0.0914027	−	0.995814i	$-0.529135\pi$
$173$	−2.40443	−0.182805	−0.0914027	+	0.995814i	$0.529135\pi$
$174$	0	0
$175$	−5.23055	−0.395392
$176$	0	0
$177$	−3.42907	−0.257745
$178$	0	0
$179$	7.93594	0.593160	0.296580	−	0.955008i	$-0.404154\pi$
$179$	7.93594	0.593160	0.296580	+	0.955008i	$0.404154\pi$
$180$	0	0
$181$	19.3192	1.43599	0.717994	−	0.696049i	$-0.245060\pi$
$181$	19.3192	1.43599	0.717994	+	0.696049i	$0.245060\pi$
$182$	0	0
$183$	5.03203	0.371978
$184$	0	0
$185$	−15.8902	−1.16827
$186$	0	0
$187$	2.23055	0.163114
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	−12.2872	−0.884454	−0.442227	−	0.896903i	$-0.645811\pi$
$193$	−12.2872	−0.884454	−0.442227	+	0.896903i	$0.645811\pi$
$194$	0	0
$195$	10.2306	0.732625
$196$	0	0
$197$	14.5635	1.03761	0.518805	−	0.854893i	$-0.326377\pi$
$197$	14.5635	1.03761	0.518805	+	0.854893i	$0.326377\pi$
$198$	0	0
$199$	15.8902	1.12642	0.563212	−	0.826312i	$-0.309565\pi$
$199$	15.8902	1.12642	0.563212	+	0.826312i	$0.309565\pi$
$200$	0	0
$201$	−14.3970	−1.01549
$202$	0	0
$203$	0	0
$204$	0	0
$205$	10.2306	0.714533
$206$	0	0
$207$	−2.23055	−0.155034
$208$	0	0
$209$	7.13447	0.493501
$210$	0	0
$211$	−7.89018	−0.543182	−0.271591	−	0.962413i	$-0.587550\pi$
$211$	−7.89018	−0.543182	−0.271591	+	0.962413i	$0.587550\pi$
$212$	0	0
$213$	−6.39705	−0.438318
$214$	0	0
$215$	13.3266	0.908869
$216$	0	0
$217$	7.42907	0.504318
$218$	0	0
$219$	14.3970	0.972862
$220$	0	0
$221$	−3.19852	−0.215156
$222$	0	0
$223$	13.5956	0.910427	0.455213	−	0.890382i	$-0.349563\pi$
$223$	13.5956	0.910427	0.455213	+	0.890382i	$0.349563\pi$
$224$	0	0
$225$	5.23055	0.348703
$226$	0	0
$227$	−9.99262	−0.663233	−0.331617	−	0.943414i	$-0.607594\pi$
$227$	−9.99262	−0.663233	−0.331617	+	0.943414i	$0.607594\pi$
$228$	0	0
$229$	−22.3330	−1.47581	−0.737903	−	0.674907i	$-0.764184\pi$
$229$	−22.3330	−1.47581	−0.737903	+	0.674907i	$0.764184\pi$
$230$	0	0
$231$	−2.23055	−0.146759
$232$	0	0
$233$	10.6917	0.700433	0.350217	−	0.936669i	$-0.386108\pi$
$233$	10.6917	0.700433	0.350217	+	0.936669i	$0.386108\pi$
$234$	0	0
$235$	−23.7621	−1.55007
$236$	0	0
$237$	−5.03203	−0.326865
$238$	0	0
$239$	25.7621	1.66641	0.833205	−	0.552965i	$-0.186504\pi$
$239$	25.7621	1.66641	0.833205	+	0.552965i	$0.186504\pi$
$240$	0	0
$241$	4.79409	0.308815	0.154407	−	0.988007i	$-0.450653\pi$
$241$	4.79409	0.308815	0.154407	+	0.988007i	$0.450653\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.19852	−0.204346
$246$	0	0
$247$	−10.2306	−0.650954
$248$	0	0
$249$	16.3330	1.03506
$250$	0	0
$251$	25.8261	1.63013	0.815065	−	0.579369i	$-0.196701\pi$
$251$	25.8261	1.63013	0.815065	+	0.579369i	$0.196701\pi$
$252$	0	0
$253$	−4.97536	−0.312798
$254$	0	0
$255$	−3.19852	−0.200299
$256$	0	0
$257$	−2.57093	−0.160370	−0.0801850	−	0.996780i	$-0.525551\pi$
$257$	−2.57093	−0.160370	−0.0801850	+	0.996780i	$0.525551\pi$
$258$	0	0
$259$	−4.96797	−0.308695
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.1911	−1.06005	−0.530026	−	0.847982i	$-0.677818\pi$
$263$	−17.1911	−1.06005	−0.530026	+	0.847982i	$0.677818\pi$
$264$	0	0
$265$	4.57093	0.280790
$266$	0	0
$267$	8.39705	0.513891
$268$	0	0
$269$	30.3897	1.85289	0.926445	−	0.376430i	$-0.122848\pi$
$269$	30.3897	1.85289	0.926445	+	0.376430i	$0.122848\pi$
$270$	0	0
$271$	−26.3897	−1.60306	−0.801529	−	0.597956i	$-0.795980\pi$
$271$	−26.3897	−1.60306	−0.801529	+	0.597956i	$0.795980\pi$
$272$	0	0
$273$	3.19852	0.193583
$274$	0	0
$275$	11.6670	0.703547
$276$	0	0
$277$	24.8581	1.49358	0.746791	−	0.665059i	$-0.231594\pi$
$277$	24.8581	1.49358	0.746791	+	0.665059i	$0.231594\pi$
$278$	0	0
$279$	−7.42907	−0.444767
$280$	0	0
$281$	−12.3970	−0.739546	−0.369773	−	0.929122i	$-0.620565\pi$
$281$	−12.3970	−0.739546	−0.369773	+	0.929122i	$0.620565\pi$
$282$	0	0
$283$	16.2232	0.964367	0.482184	−	0.876070i	$-0.339844\pi$
$283$	16.2232	0.964367	0.482184	+	0.876070i	$0.339844\pi$
$284$	0	0
$285$	−10.2306	−0.606006
$286$	0	0
$287$	3.19852	0.188803
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	11.2552	0.657535	0.328768	−	0.944411i	$-0.393367\pi$
$293$	11.2552	0.657535	0.328768	+	0.944411i	$0.393367\pi$
$294$	0	0
$295$	10.9680	0.638580
$296$	0	0
$297$	2.23055	0.129430
$298$	0	0
$299$	7.13447	0.412597
$300$	0	0
$301$	4.16650	0.240153
$302$	0	0
$303$	11.3650	0.652903
$304$	0	0
$305$	−16.0951	−0.921600
$306$	0	0
$307$	−19.2552	−1.09895	−0.549476	−	0.835510i	$-0.685173\pi$
$307$	−19.2552	−1.09895	−0.549476	+	0.835510i	$0.685173\pi$
$308$	0	0
$309$	−0.801477	−0.0455944
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−9.03203	−0.510520	−0.255260	−	0.966872i	$-0.582161\pi$
$313$	−9.03203	−0.510520	−0.255260	+	0.966872i	$0.582161\pi$
$314$	0	0
$315$	3.19852	0.180216
$316$	0	0
$317$	34.1774	1.91959	0.959797	−	0.280695i	$-0.0905650\pi$
$317$	34.1774	1.91959	0.959797	+	0.280695i	$0.0905650\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−6.23055	−0.347755
$322$	0	0
$323$	3.19852	0.177971
$324$	0	0
$325$	−16.7300	−0.928016
$326$	0	0
$327$	8.96797	0.495930
$328$	0	0
$329$	−7.42907	−0.409578
$330$	0	0
$331$	−6.10244	−0.335420	−0.167710	−	0.985836i	$-0.553637\pi$
$331$	−6.10244	−0.335420	−0.167710	+	0.985836i	$0.553637\pi$
$332$	0	0
$333$	4.96797	0.272243
$334$	0	0
$335$	46.0493	2.51594
$336$	0	0
$337$	−26.2232	−1.42847	−0.714233	−	0.699908i	$-0.753225\pi$
$337$	−26.2232	−1.42847	−0.714233	+	0.699908i	$0.753225\pi$
$338$	0	0
$339$	8.62760	0.468587
$340$	0	0
$341$	−16.5709	−0.897366
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.13447	0.384107
$346$	0	0
$347$	−8.46110	−0.454216	−0.227108	−	0.973870i	$-0.572927\pi$
$347$	−8.46110	−0.454216	−0.227108	+	0.973870i	$0.572927\pi$
$348$	0	0
$349$	8.67337	0.464275	0.232137	−	0.972683i	$-0.425428\pi$
$349$	8.67337	0.464275	0.232137	+	0.972683i	$0.425428\pi$
$350$	0	0
$351$	−3.19852	−0.170725
$352$	0	0
$353$	34.6843	1.84606	0.923029	−	0.384731i	$-0.125706\pi$
$353$	34.6843	1.84606	0.923029	+	0.384731i	$0.125706\pi$
$354$	0	0
$355$	20.4611	1.08596
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	−2.90392	−0.153263	−0.0766315	−	0.997059i	$-0.524416\pi$
$359$	−2.90392	−0.153263	−0.0766315	+	0.997059i	$0.524416\pi$
$360$	0	0
$361$	−8.76945	−0.461550
$362$	0	0
$363$	−6.02464	−0.316212
$364$	0	0
$365$	−46.0493	−2.41033
$366$	0	0
$367$	17.5882	0.918096	0.459048	−	0.888412i	$-0.348191\pi$
$367$	17.5882	0.918096	0.459048	+	0.888412i	$0.348191\pi$
$368$	0	0
$369$	−3.19852	−0.166508
$370$	0	0
$371$	1.42907	0.0741938
$372$	0	0
$373$	16.2689	0.842374	0.421187	−	0.906974i	$-0.361614\pi$
$373$	16.2689	0.842374	0.421187	+	0.906974i	$0.361614\pi$
$374$	0	0
$375$	−0.737422	−0.0380803
$376$	0	0
$377$	0	0
$378$	0	0
$379$	4.92220	0.252837	0.126418	−	0.991977i	$-0.459652\pi$
$379$	4.92220	0.252837	0.126418	+	0.991977i	$0.459652\pi$
$380$	0	0
$381$	−0.166496	−0.00852983
$382$	0	0
$383$	25.6980	1.31311	0.656553	−	0.754279i	$-0.272014\pi$
$383$	25.6980	1.31311	0.656553	+	0.754279i	$0.272014\pi$
$384$	0	0
$385$	7.13447	0.363606
$386$	0	0
$387$	−4.16650	−0.211795
$388$	0	0
$389$	−17.3192	−0.878121	−0.439060	−	0.898458i	$-0.644689\pi$
$389$	−17.3192	−0.878121	−0.439060	+	0.898458i	$0.644689\pi$
$390$	0	0
$391$	−2.23055	−0.112804
$392$	0	0
$393$	13.1985	0.665777
$394$	0	0
$395$	16.0951	0.809830
$396$	0	0
$397$	−20.6843	−1.03811	−0.519057	−	0.854740i	$-0.673717\pi$
$397$	−20.6843	−1.03811	−0.519057	+	0.854740i	$0.673717\pi$
$398$	0	0
$399$	−3.19852	−0.160126
$400$	0	0
$401$	5.30835	0.265086	0.132543	−	0.991177i	$-0.457686\pi$
$401$	5.30835	0.265086	0.132543	+	0.991177i	$0.457686\pi$
$402$	0	0
$403$	23.7621	1.18367
$404$	0	0
$405$	−3.19852	−0.158936
$406$	0	0
$407$	11.0813	0.549280
$408$	0	0
$409$	−0.673367	−0.0332958	−0.0166479	−	0.999861i	$-0.505299\pi$
$409$	−0.673367	−0.0332958	−0.0166479	+	0.999861i	$0.505299\pi$
$410$	0	0
$411$	6.57093	0.324120
$412$	0	0
$413$	3.42907	0.168734
$414$	0	0
$415$	−52.2415	−2.56443
$416$	0	0
$417$	14.3970	0.705026
$418$	0	0
$419$	17.6030	0.859961	0.429980	−	0.902838i	$-0.358521\pi$
$419$	17.6030	0.859961	0.429980	+	0.902838i	$0.358521\pi$
$420$	0	0
$421$	−30.6276	−1.49270	−0.746349	−	0.665555i	$-0.768195\pi$
$421$	−30.6276	−1.49270	−0.746349	+	0.665555i	$0.768195\pi$
$422$	0	0
$423$	7.42907	0.361214
$424$	0	0
$425$	5.23055	0.253719
$426$	0	0
$427$	−5.03203	−0.243517
$428$	0	0
$429$	−7.13447	−0.344455
$430$	0	0
$431$	−3.87189	−0.186502	−0.0932512	−	0.995643i	$-0.529726\pi$
$431$	−3.87189	−0.186502	−0.0932512	+	0.995643i	$0.529726\pi$
$432$	0	0
$433$	20.7867	0.998945	0.499473	−	0.866330i	$-0.333527\pi$
$433$	20.7867	0.998945	0.499473	+	0.866330i	$0.333527\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.13447	−0.341288
$438$	0	0
$439$	−32.6843	−1.55994	−0.779968	−	0.625820i	$-0.784764\pi$
$439$	−32.6843	−1.55994	−0.779968	+	0.625820i	$0.784764\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−2.44282	−0.116062	−0.0580308	−	0.998315i	$-0.518482\pi$
$443$	−2.44282	−0.116062	−0.0580308	+	0.998315i	$0.518482\pi$
$444$	0	0
$445$	−26.8581	−1.27320
$446$	0	0
$447$	−0.0640550	−0.00302970
$448$	0	0
$449$	−2.85815	−0.134884	−0.0674422	−	0.997723i	$-0.521484\pi$
$449$	−2.85815	−0.134884	−0.0674422	+	0.997723i	$0.521484\pi$
$450$	0	0
$451$	−7.13447	−0.335949
$452$	0	0
$453$	−8.46110	−0.397537
$454$	0	0
$455$	−10.2306	−0.479616
$456$	0	0
$457$	2.75571	0.128907	0.0644533	−	0.997921i	$-0.479470\pi$
$457$	2.75571	0.128907	0.0644533	+	0.997921i	$0.479470\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	34.6843	1.61541	0.807704	−	0.589589i	$-0.200710\pi$
$461$	34.6843	1.61541	0.807704	+	0.589589i	$0.200710\pi$
$462$	0	0
$463$	−17.2552	−0.801917	−0.400958	−	0.916096i	$-0.631323\pi$
$463$	−17.2552	−0.801917	−0.400958	+	0.916096i	$0.631323\pi$
$464$	0	0
$465$	23.7621	1.10194
$466$	0	0
$467$	−7.19114	−0.332766	−0.166383	−	0.986061i	$-0.553209\pi$
$467$	−7.19114	−0.332766	−0.166383	+	0.986061i	$0.553209\pi$
$468$	0	0
$469$	14.3970	0.664794
$470$	0	0
$471$	−3.65962	−0.168627
$472$	0	0
$473$	−9.29358	−0.427319
$474$	0	0
$475$	16.7300	0.767627
$476$	0	0
$477$	−1.42907	−0.0654328
$478$	0	0
$479$	−22.5818	−1.03179	−0.515895	−	0.856652i	$-0.672541\pi$
$479$	−22.5818	−1.03179	−0.515895	+	0.856652i	$0.672541\pi$
$480$	0	0
$481$	−15.8902	−0.724530
$482$	0	0
$483$	2.23055	0.101494
$484$	0	0
$485$	−25.5882	−1.16190
$486$	0	0
$487$	−11.5389	−0.522877	−0.261439	−	0.965220i	$-0.584197\pi$
$487$	−11.5389	−0.522877	−0.261439	+	0.965220i	$0.584197\pi$
$488$	0	0
$489$	13.2552	0.599421
$490$	0	0
$491$	30.7941	1.38972	0.694859	−	0.719146i	$-0.255467\pi$
$491$	30.7941	1.38972	0.694859	+	0.719146i	$0.255467\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−7.13447	−0.320670
$496$	0	0
$497$	6.39705	0.286947
$498$	0	0
$499$	−9.95423	−0.445613	−0.222806	−	0.974863i	$-0.571522\pi$
$499$	−9.95423	−0.445613	−0.222806	+	0.974863i	$0.571522\pi$
$500$	0	0
$501$	3.26258	0.145761
$502$	0	0
$503$	−27.5956	−1.23043	−0.615213	−	0.788361i	$-0.710930\pi$
$503$	−27.5956	−1.23043	−0.615213	+	0.788361i	$0.710930\pi$
$504$	0	0
$505$	−36.3513	−1.61761
$506$	0	0
$507$	−2.76945	−0.122996
$508$	0	0
$509$	20.0493	0.888669	0.444335	−	0.895861i	$-0.353440\pi$
$509$	20.0493	0.888669	0.444335	+	0.895861i	$0.353440\pi$
$510$	0	0
$511$	−14.3970	−0.636888
$512$	0	0
$513$	3.19852	0.141218
$514$	0	0
$515$	2.56354	0.112963
$516$	0	0
$517$	16.5709	0.728788
$518$	0	0
$519$	−2.40443	−0.105543
$520$	0	0
$521$	−30.9789	−1.35721	−0.678605	−	0.734504i	$-0.737415\pi$
$521$	−30.9789	−1.35721	−0.678605	+	0.734504i	$0.737415\pi$
$522$	0	0
$523$	−5.31925	−0.232595	−0.116297	−	0.993214i	$-0.537103\pi$
$523$	−5.31925	−0.232595	−0.116297	+	0.993214i	$0.537103\pi$
$524$	0	0
$525$	−5.23055	−0.228280
$526$	0	0
$527$	−7.42907	−0.323616
$528$	0	0
$529$	−18.0246	−0.783680
$530$	0	0
$531$	−3.42907	−0.148809
$532$	0	0
$533$	10.2306	0.443134
$534$	0	0
$535$	19.9286	0.861587
$536$	0	0
$537$	7.93594	0.342461
$538$	0	0
$539$	2.23055	0.0960766
$540$	0	0
$541$	17.1911	0.739105	0.369552	−	0.929210i	$-0.379511\pi$
$541$	17.1911	0.739105	0.369552	+	0.929210i	$0.379511\pi$
$542$	0	0
$543$	19.3192	0.829068
$544$	0	0
$545$	−28.6843	−1.22870
$546$	0	0
$547$	17.3650	0.742475	0.371237	−	0.928538i	$-0.378934\pi$
$547$	17.3650	0.742475	0.371237	+	0.928538i	$0.378934\pi$
$548$	0	0
$549$	5.03203	0.214762
$550$	0	0
$551$	0	0
$552$	0	0
$553$	5.03203	0.213984
$554$	0	0
$555$	−15.8902	−0.674500
$556$	0	0
$557$	14.5709	0.617390	0.308695	−	0.951161i	$-0.400108\pi$
$557$	14.5709	0.617390	0.308695	+	0.951161i	$0.400108\pi$
$558$	0	0
$559$	13.3266	0.563657
$560$	0	0
$561$	2.23055	0.0941739
$562$	0	0
$563$	−23.9852	−1.01086	−0.505429	−	0.862868i	$-0.668666\pi$
$563$	−23.9852	−1.01086	−0.505429	+	0.862868i	$0.668666\pi$
$564$	0	0
$565$	−27.5956	−1.16095
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−38.9074	−1.63108	−0.815542	−	0.578698i	$-0.803561\pi$
$569$	−38.9074	−1.63108	−0.815542	+	0.578698i	$0.803561\pi$
$570$	0	0
$571$	−15.8719	−0.664218	−0.332109	−	0.943241i	$-0.607760\pi$
$571$	−15.8719	−0.664218	−0.332109	+	0.943241i	$0.607760\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	0	0
$575$	−11.6670	−0.486548
$576$	0	0
$577$	41.7089	1.73636	0.868182	−	0.496245i	$-0.165288\pi$
$577$	41.7089	1.73636	0.868182	+	0.496245i	$0.165288\pi$
$578$	0	0
$579$	−12.2872	−0.510640
$580$	0	0
$581$	−16.3330	−0.677607
$582$	0	0
$583$	−3.18762	−0.132018
$584$	0	0
$585$	10.2306	0.422981
$586$	0	0
$587$	−23.7621	−0.980765	−0.490383	−	0.871507i	$-0.663143\pi$
$587$	−23.7621	−0.980765	−0.490383	+	0.871507i	$0.663143\pi$
$588$	0	0
$589$	−23.7621	−0.979099
$590$	0	0
$591$	14.5635	0.599064
$592$	0	0
$593$	−20.9680	−0.861051	−0.430526	−	0.902578i	$-0.641672\pi$
$593$	−20.9680	−0.861051	−0.430526	+	0.902578i	$0.641672\pi$
$594$	0	0
$595$	3.19852	0.131127
$596$	0	0
$597$	15.8902	0.650342
$598$	0	0
$599$	−23.5882	−0.963787	−0.481894	−	0.876230i	$-0.660051\pi$
$599$	−23.5882	−0.963787	−0.481894	+	0.876230i	$0.660051\pi$
$600$	0	0
$601$	−6.28722	−0.256461	−0.128231	−	0.991744i	$-0.540930\pi$
$601$	−6.28722	−0.256461	−0.128231	+	0.991744i	$0.540930\pi$
$602$	0	0
$603$	−14.3970	−0.586293
$604$	0	0
$605$	19.2700	0.783435
$606$	0	0
$607$	−44.0951	−1.78976	−0.894882	−	0.446304i	$-0.852740\pi$
$607$	−44.0951	−1.78976	−0.894882	+	0.446304i	$0.852740\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−23.7621	−0.961310
$612$	0	0
$613$	5.61385	0.226741	0.113371	−	0.993553i	$-0.463835\pi$
$613$	5.61385	0.226741	0.113371	+	0.993553i	$0.463835\pi$
$614$	0	0
$615$	10.2306	0.412536
$616$	0	0
$617$	36.7941	1.48127	0.740637	−	0.671905i	$-0.234524\pi$
$617$	36.7941	1.48127	0.740637	+	0.671905i	$0.234524\pi$
$618$	0	0
$619$	−27.3010	−1.09732	−0.548659	−	0.836046i	$-0.684862\pi$
$619$	−27.3010	−1.09732	−0.548659	+	0.836046i	$0.684862\pi$
$620$	0	0
$621$	−2.23055	−0.0895089
$622$	0	0
$623$	−8.39705	−0.336421
$624$	0	0
$625$	−23.7941	−0.951764
$626$	0	0
$627$	7.13447	0.284923
$628$	0	0
$629$	4.96797	0.198086
$630$	0	0
$631$	30.1024	1.19836	0.599180	−	0.800615i	$-0.295494\pi$
$631$	30.1024	1.19836	0.599180	+	0.800615i	$0.295494\pi$
$632$	0	0
$633$	−7.89018	−0.313606
$634$	0	0
$635$	0.532540	0.0211332
$636$	0	0
$637$	−3.19852	−0.126730
$638$	0	0
$639$	−6.39705	−0.253063
$640$	0	0
$641$	−1.89756	−0.0749491	−0.0374745	−	0.999298i	$-0.511931\pi$
$641$	−1.89756	−0.0749491	−0.0374745	+	0.999298i	$0.511931\pi$
$642$	0	0
$643$	−17.7163	−0.698662	−0.349331	−	0.936999i	$-0.613591\pi$
$643$	−17.7163	−0.698662	−0.349331	+	0.936999i	$0.613591\pi$
$644$	0	0
$645$	13.3266	0.524736
$646$	0	0
$647$	−9.60295	−0.377531	−0.188766	−	0.982022i	$-0.560449\pi$
$647$	−9.60295	−0.377531	−0.188766	+	0.982022i	$0.560449\pi$
$648$	0	0
$649$	−7.64872	−0.300239
$650$	0	0
$651$	7.42907	0.291168
$652$	0	0
$653$	8.16650	0.319580	0.159790	−	0.987151i	$-0.448918\pi$
$653$	8.16650	0.319580	0.159790	+	0.987151i	$0.448918\pi$
$654$	0	0
$655$	−42.2158	−1.64951
$656$	0	0
$657$	14.3970	0.561682
$658$	0	0
$659$	−43.1454	−1.68070	−0.840352	−	0.542040i	$-0.817652\pi$
$659$	−43.1454	−1.68070	−0.840352	+	0.542040i	$0.817652\pi$
$660$	0	0
$661$	−21.2626	−0.827018	−0.413509	−	0.910500i	$-0.635697\pi$
$661$	−21.2626	−0.827018	−0.413509	+	0.910500i	$0.635697\pi$
$662$	0	0
$663$	−3.19852	−0.124220
$664$	0	0
$665$	10.2306	0.396724
$666$	0	0
$667$	0	0
$668$	0	0
$669$	13.5956	0.525635
$670$	0	0
$671$	11.2242	0.433305
$672$	0	0
$673$	24.5104	0.944806	0.472403	−	0.881383i	$-0.343387\pi$
$673$	24.5104	0.944806	0.472403	+	0.881383i	$0.343387\pi$
$674$	0	0
$675$	5.23055	0.201324
$676$	0	0
$677$	−22.9789	−0.883150	−0.441575	−	0.897224i	$-0.645580\pi$
$677$	−22.9789	−0.883150	−0.441575	+	0.897224i	$0.645580\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−9.99262	−0.382918
$682$	0	0
$683$	9.76945	0.373818	0.186909	−	0.982377i	$-0.440153\pi$
$683$	9.76945	0.373818	0.186909	+	0.982377i	$0.440153\pi$
$684$	0	0
$685$	−21.0173	−0.803028
$686$	0	0
$687$	−22.3330	−0.852057
$688$	0	0
$689$	4.57093	0.174138
$690$	0	0
$691$	32.6843	1.24337	0.621684	−	0.783268i	$-0.286449\pi$
$691$	32.6843	1.24337	0.621684	+	0.783268i	$0.286449\pi$
$692$	0	0
$693$	−2.23055	−0.0847316
$694$	0	0
$695$	−46.0493	−1.74675
$696$	0	0
$697$	−3.19852	−0.121153
$698$	0	0
$699$	10.6917	0.404395
$700$	0	0
$701$	20.2689	0.765547	0.382774	−	0.923842i	$-0.374969\pi$
$701$	20.2689	0.765547	0.382774	+	0.923842i	$0.374969\pi$
$702$	0	0
$703$	15.8902	0.599309
$704$	0	0
$705$	−23.7621	−0.894931
$706$	0	0
$707$	−11.3650	−0.427426
$708$	0	0
$709$	15.5882	0.585427	0.292713	−	0.956200i	$-0.405442\pi$
$709$	15.5882	0.585427	0.292713	+	0.956200i	$0.405442\pi$
$710$	0	0
$711$	−5.03203	−0.188716
$712$	0	0
$713$	16.5709	0.620586
$714$	0	0
$715$	22.8198	0.853411
$716$	0	0
$717$	25.7621	0.962102
$718$	0	0
$719$	−30.5818	−1.14051	−0.570255	−	0.821468i	$-0.693156\pi$
$719$	−30.5818	−1.14051	−0.570255	+	0.821468i	$0.693156\pi$
$720$	0	0
$721$	0.801477	0.0298486
$722$	0	0
$723$	4.79409	0.178294
$724$	0	0
$725$	0	0
$726$	0	0
$727$	34.9074	1.29465	0.647323	−	0.762216i	$-0.275889\pi$
$727$	34.9074	1.29465	0.647323	+	0.762216i	$0.275889\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.16650	−0.154103
$732$	0	0
$733$	8.06406	0.297853	0.148926	−	0.988848i	$-0.452418\pi$
$733$	8.06406	0.297853	0.148926	+	0.988848i	$0.452418\pi$
$734$	0	0
$735$	−3.19852	−0.117979
$736$	0	0
$737$	−32.1133	−1.18291
$738$	0	0
$739$	8.16650	0.300409	0.150205	−	0.988655i	$-0.452007\pi$
$739$	8.16650	0.300409	0.150205	+	0.988655i	$0.452007\pi$
$740$	0	0
$741$	−10.2306	−0.375829
$742$	0	0
$743$	2.61670	0.0959973	0.0479986	−	0.998847i	$-0.484716\pi$
$743$	2.61670	0.0959973	0.0479986	+	0.998847i	$0.484716\pi$
$744$	0	0
$745$	0.204881	0.00750627
$746$	0	0
$747$	16.3330	0.597593
$748$	0	0
$749$	6.23055	0.227659
$750$	0	0
$751$	43.4143	1.58421	0.792105	−	0.610385i	$-0.208985\pi$
$751$	43.4143	1.58421	0.792105	+	0.610385i	$0.208985\pi$
$752$	0	0
$753$	25.8261	0.941156
$754$	0	0
$755$	27.0630	0.984924
$756$	0	0
$757$	32.3439	1.17556	0.587779	−	0.809021i	$-0.300002\pi$
$757$	32.3439	1.17556	0.587779	+	0.809021i	$0.300002\pi$
$758$	0	0
$759$	−4.97536	−0.180594
$760$	0	0
$761$	−5.07780	−0.184070	−0.0920350	−	0.995756i	$-0.529337\pi$
$761$	−5.07780	−0.184070	−0.0920350	+	0.995756i	$0.529337\pi$
$762$	0	0
$763$	−8.96797	−0.324662
$764$	0	0
$765$	−3.19852	−0.115643
$766$	0	0
$767$	10.9680	0.396031
$768$	0	0
$769$	17.0429	0.614584	0.307292	−	0.951615i	$-0.400577\pi$
$769$	17.0429	0.614584	0.307292	+	0.951615i	$0.400577\pi$
$770$	0	0
$771$	−2.57093	−0.0925896
$772$	0	0
$773$	−28.1591	−1.01281	−0.506406	−	0.862295i	$-0.669026\pi$
$773$	−28.1591	−1.01281	−0.506406	+	0.862295i	$0.669026\pi$
$774$	0	0
$775$	−38.8581	−1.39583
$776$	0	0
$777$	−4.96797	−0.178225
$778$	0	0
$779$	−10.2306	−0.366548
$780$	0	0
$781$	−14.2689	−0.510583
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.7054	0.417783
$786$	0	0
$787$	6.83986	0.243815	0.121907	−	0.992541i	$-0.461099\pi$
$787$	6.83986	0.243815	0.121907	+	0.992541i	$0.461099\pi$
$788$	0	0
$789$	−17.1911	−0.612021
$790$	0	0
$791$	−8.62760	−0.306762
$792$	0	0
$793$	−16.0951	−0.571552
$794$	0	0
$795$	4.57093	0.162114
$796$	0	0
$797$	28.1774	0.998095	0.499047	−	0.866575i	$-0.333683\pi$
$797$	28.1774	0.998095	0.499047	+	0.866575i	$0.333683\pi$
$798$	0	0
$799$	7.42907	0.262822
$800$	0	0
$801$	8.39705	0.296695
$802$	0	0
$803$	32.1133	1.13326
$804$	0	0
$805$	−7.13447	−0.251457
$806$	0	0
$807$	30.3897	1.06977
$808$	0	0
$809$	49.0887	1.72587	0.862933	−	0.505318i	$-0.168625\pi$
$809$	49.0887	1.72587	0.862933	+	0.505318i	$0.168625\pi$
$810$	0	0
$811$	−9.38330	−0.329492	−0.164746	−	0.986336i	$-0.552681\pi$
$811$	−9.38330	−0.329492	−0.164746	+	0.986336i	$0.552681\pi$
$812$	0	0
$813$	−26.3897	−0.925526
$814$	0	0
$815$	−42.3970	−1.48510
$816$	0	0
$817$	−13.3266	−0.466240
$818$	0	0
$819$	3.19852	0.111765
$820$	0	0
$821$	12.1665	0.424614	0.212307	−	0.977203i	$-0.431902\pi$
$821$	12.1665	0.424614	0.212307	+	0.977203i	$0.431902\pi$
$822$	0	0
$823$	33.9395	1.18306	0.591528	−	0.806285i	$-0.298525\pi$
$823$	33.9395	1.18306	0.591528	+	0.806285i	$0.298525\pi$
$824$	0	0
$825$	11.6670	0.406193
$826$	0	0
$827$	−29.0887	−1.01151	−0.505757	−	0.862676i	$-0.668787\pi$
$827$	−29.0887	−1.01151	−0.505757	+	0.862676i	$0.668787\pi$
$828$	0	0
$829$	−6.92220	−0.240418	−0.120209	−	0.992749i	$-0.538356\pi$
$829$	−6.92220	−0.240418	−0.120209	+	0.992749i	$0.538356\pi$
$830$	0	0
$831$	24.8581	0.862320
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−10.4354	−0.361133
$836$	0	0
$837$	−7.42907	−0.256786
$838$	0	0
$839$	−4.18478	−0.144475	−0.0722373	−	0.997387i	$-0.523014\pi$
$839$	−4.18478	−0.144475	−0.0722373	+	0.997387i	$0.523014\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−12.3970	−0.426977
$844$	0	0
$845$	8.85815	0.304730
$846$	0	0
$847$	6.02464	0.207009
$848$	0	0
$849$	16.2232	0.556778
$850$	0	0
$851$	−11.0813	−0.379863
$852$	0	0
$853$	−49.9395	−1.70989	−0.854947	−	0.518715i	$-0.826411\pi$
$853$	−49.9395	−1.70989	−0.854947	+	0.518715i	$0.826411\pi$
$854$	0	0
$855$	−10.2306	−0.349877
$856$	0	0
$857$	44.8581	1.53233	0.766163	−	0.642647i	$-0.222164\pi$
$857$	44.8581	1.53233	0.766163	+	0.642647i	$0.222164\pi$
$858$	0	0
$859$	−48.3970	−1.65129	−0.825643	−	0.564193i	$-0.809187\pi$
$859$	−48.3970	−1.65129	−0.825643	+	0.564193i	$0.809187\pi$
$860$	0	0
$861$	3.19852	0.109005
$862$	0	0
$863$	−3.37979	−0.115049	−0.0575246	−	0.998344i	$-0.518321\pi$
$863$	−3.37979	−0.115049	−0.0575246	+	0.998344i	$0.518321\pi$
$864$	0	0
$865$	7.69063	0.261489
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−11.2242	−0.380755
$870$	0	0
$871$	46.0493	1.56032
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	0.737422	0.0249294
$876$	0	0
$877$	20.6350	0.696794	0.348397	−	0.937347i	$-0.386726\pi$
$877$	20.6350	0.696794	0.348397	+	0.937347i	$0.386726\pi$
$878$	0	0
$879$	11.2552	0.379628
$880$	0	0
$881$	−8.73004	−0.294122	−0.147061	−	0.989127i	$-0.546981\pi$
$881$	−8.73004	−0.294122	−0.147061	+	0.989127i	$0.546981\pi$
$882$	0	0
$883$	55.9321	1.88226	0.941132	−	0.338039i	$-0.109764\pi$
$883$	55.9321	1.88226	0.941132	+	0.338039i	$0.109764\pi$
$884$	0	0
$885$	10.9680	0.368684
$886$	0	0
$887$	46.1060	1.54809	0.774043	−	0.633133i	$-0.218231\pi$
$887$	46.1060	1.54809	0.774043	+	0.633133i	$0.218231\pi$
$888$	0	0
$889$	0.166496	0.00558409
$890$	0	0
$891$	2.23055	0.0747263
$892$	0	0
$893$	23.7621	0.795167
$894$	0	0
$895$	−25.3833	−0.848470
$896$	0	0
$897$	7.13447	0.238213
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−1.42907	−0.0476094
$902$	0	0
$903$	4.16650	0.138652
$904$	0	0
$905$	−61.7931	−2.05407
$906$	0	0
$907$	39.0630	1.29707	0.648533	−	0.761186i	$-0.275383\pi$
$907$	39.0630	1.29707	0.648533	+	0.761186i	$0.275383\pi$
$908$	0	0
$909$	11.3650	0.376954
$910$	0	0
$911$	−36.4995	−1.20928	−0.604641	−	0.796498i	$-0.706683\pi$
$911$	−36.4995	−1.20928	−0.604641	+	0.796498i	$0.706683\pi$
$912$	0	0
$913$	36.4316	1.20571
$914$	0	0
$915$	−16.0951	−0.532086
$916$	0	0
$917$	−13.1985	−0.435854
$918$	0	0
$919$	9.88279	0.326003	0.163002	−	0.986626i	$-0.447882\pi$
$919$	9.88279	0.326003	0.163002	+	0.986626i	$0.447882\pi$
$920$	0	0
$921$	−19.2552	−0.634480
$922$	0	0
$923$	20.4611	0.673485
$924$	0	0
$925$	25.9852	0.854389
$926$	0	0
$927$	−0.801477	−0.0263240
$928$	0	0
$929$	−34.3897	−1.12829	−0.564144	−	0.825676i	$-0.690794\pi$
$929$	−34.3897	−1.12829	−0.564144	+	0.825676i	$0.690794\pi$
$930$	0	0
$931$	3.19852	0.104827
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.13447	−0.233322
$936$	0	0
$937$	33.5389	1.09567	0.547834	−	0.836587i	$-0.315453\pi$
$937$	33.5389	1.09567	0.547834	+	0.836587i	$0.315453\pi$
$938$	0	0
$939$	−9.03203	−0.294749
$940$	0	0
$941$	−1.07780	−0.0351352	−0.0175676	−	0.999846i	$-0.505592\pi$
$941$	−1.07780	−0.0351352	−0.0175676	+	0.999846i	$0.505592\pi$
$942$	0	0
$943$	7.13447	0.232330
$944$	0	0
$945$	3.19852	0.104048
$946$	0	0
$947$	−5.12708	−0.166608	−0.0833039	−	0.996524i	$-0.526547\pi$
$947$	−5.12708	−0.166608	−0.0833039	+	0.996524i	$0.526547\pi$
$948$	0	0
$949$	−46.0493	−1.49482
$950$	0	0
$951$	34.1774	1.10828
$952$	0	0
$953$	29.5424	0.956973	0.478486	−	0.878095i	$-0.341186\pi$
$953$	29.5424	0.956973	0.478486	+	0.878095i	$0.341186\pi$
$954$	0	0
$955$	−19.1911	−0.621011
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.57093	−0.212186
$960$	0	0
$961$	24.1911	0.780359
$962$	0	0
$963$	−6.23055	−0.200777
$964$	0	0
$965$	39.3010	1.26514
$966$	0	0
$967$	4.51426	0.145169	0.0725843	−	0.997362i	$-0.476875\pi$
$967$	4.51426	0.145169	0.0725843	+	0.997362i	$0.476875\pi$
$968$	0	0
$969$	3.19852	0.102751
$970$	0	0
$971$	44.3330	1.42271	0.711357	−	0.702831i	$-0.248081\pi$
$971$	44.3330	1.42271	0.711357	+	0.702831i	$0.248081\pi$
$972$	0	0
$973$	−14.3970	−0.461548
$974$	0	0
$975$	−16.7300	−0.535790
$976$	0	0
$977$	33.6522	1.07663	0.538315	−	0.842744i	$-0.319061\pi$
$977$	33.6522	1.07663	0.538315	+	0.842744i	$0.319061\pi$
$978$	0	0
$979$	18.7300	0.598615
$980$	0	0
$981$	8.96797	0.286325
$982$	0	0
$983$	−57.8645	−1.84559	−0.922796	−	0.385290i	$-0.874101\pi$
$983$	−57.8645	−1.84559	−0.922796	+	0.385290i	$0.874101\pi$
$984$	0	0
$985$	−46.5818	−1.48422
$986$	0	0
$987$	−7.42907	−0.236470
$988$	0	0
$989$	9.29358	0.295519
$990$	0	0
$991$	12.7941	0.406418	0.203209	−	0.979135i	$-0.434863\pi$
$991$	12.7941	0.406418	0.203209	+	0.979135i	$0.434863\pi$
$992$	0	0
$993$	−6.10244	−0.193655
$994$	0	0
$995$	−50.8251	−1.61126
$996$	0	0
$997$	1.95071	0.0617797	0.0308898	−	0.999523i	$-0.490166\pi$
$997$	1.95071	0.0617797	0.0308898	+	0.999523i	$0.490166\pi$
$998$	0	0
$999$	4.96797	0.157180

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5712.2.a.bw.1.1		3
4.3	odd	2		1428.2.a.j.1.1	✓	3
12.11	even	2		4284.2.a.s.1.3		3
28.27	even	2		9996.2.a.x.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1428.2.a.j.1.1	✓	3	4.3	odd	2
4284.2.a.s.1.3		3	12.11	even	2
5712.2.a.bw.1.1		3	1.1	even	1	trivial
9996.2.a.x.1.3		3	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5712.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.19852 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.23055 q^{11} -3.19852 q^{13} -3.19852 q^{15} +1.00000 q^{17} +3.19852 q^{19} -1.00000 q^{21} -2.23055 q^{23} +5.23055 q^{25} +1.00000 q^{27} -7.42907 q^{31} +2.23055 q^{33} +3.19852 q^{35} +4.96797 q^{37} -3.19852 q^{39} -3.19852 q^{41} -4.16650 q^{43} -3.19852 q^{45} +7.42907 q^{47} +1.00000 q^{49} +1.00000 q^{51} -1.42907 q^{53} -7.13447 q^{55} +3.19852 q^{57} -3.42907 q^{59} +5.03203 q^{61} -1.00000 q^{63} +10.2306 q^{65} -14.3970 q^{67} -2.23055 q^{69} -6.39705 q^{71} +14.3970 q^{73} +5.23055 q^{75} -2.23055 q^{77} -5.03203 q^{79} +1.00000 q^{81} +16.3330 q^{83} -3.19852 q^{85} +8.39705 q^{89} +3.19852 q^{91} -7.42907 q^{93} -10.2306 q^{95} +8.00000 q^{97} +2.23055 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} + 3 q^{17} - 2 q^{19} - 3 q^{21} - 2 q^{23} + 11 q^{25} + 3 q^{27} - 6 q^{31} + 2 q^{33} - 2 q^{35} + 8 q^{37} + 2 q^{39} + 2 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 + 3 * q^17 - 2 * q^19 - 3 * q^21 - 2 * q^23 + 11 * q^25 + 3 * q^27 - 6 * q^31 + 2 * q^33 - 2 * q^35 + 8 * q^37 + 2 * q^39 + 2 * q^41 + 6 * q^43 + 2 * q^45 + 6 * q^47 + 3 * q^49 + 3 * q^51 + 12 * q^53 + 4 * q^55 - 2 * q^57 + 6 * q^59 + 22 * q^61 - 3 * q^63 + 26 * q^65 - 20 * q^67 - 2 * q^69 + 4 * q^71 + 20 * q^73 + 11 * q^75 - 2 * q^77 - 22 * q^79 + 3 * q^81 + 12 * q^83 + 2 * q^85 + 2 * q^89 - 2 * q^91 - 6 * q^93 - 26 * q^95 + 24 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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