LMFDB - Embedded newform 9464.2.a.t.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-4,0,3,0,5,0,1,0,0,0,-7,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69202 q^{3} -1.44504 q^{5} +1.00000 q^{7} -0.137063 q^{9} +4.85086 q^{11} -2.44504 q^{15} -0.149145 q^{17} +1.08815 q^{19} +1.69202 q^{21} +9.15883 q^{23} -2.91185 q^{25} -5.30798 q^{27} +3.15883 q^{29} -5.93900 q^{31} +8.20775 q^{33} -1.44504 q^{35} -4.15883 q^{37} +8.76271 q^{41} -3.08815 q^{43} +0.198062 q^{45} +11.4426 q^{47} +1.00000 q^{49} -0.252356 q^{51} -0.131687 q^{53} -7.00969 q^{55} +1.84117 q^{57} -0.454731 q^{59} +2.53319 q^{61} -0.137063 q^{63} +2.81402 q^{67} +15.4969 q^{69} -8.74094 q^{71} +6.96615 q^{73} -4.92692 q^{75} +4.85086 q^{77} -16.3817 q^{79} -8.57002 q^{81} +4.00538 q^{83} +0.215521 q^{85} +5.34481 q^{87} +6.95108 q^{89} -10.0489 q^{93} -1.57242 q^{95} +14.3666 q^{97} -0.664874 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{5} + 3 q^{7} + 5 q^{9} + q^{11} - 7 q^{15} - 14 q^{17} + 7 q^{19} + 19 q^{23} - 5 q^{25} - 21 q^{27} + q^{29} - 8 q^{31} + 7 q^{33} - 4 q^{35} - 4 q^{37} + 9 q^{41} - 13 q^{43} + 5 q^{45} - 7 q^{47}+ \cdots - 3 q^{99}+O(q^{100}) $ 3 * q - 4 * q^5 + 3 * q^7 + 5 * q^9 + q^11 - 7 * q^15 - 14 * q^17 + 7 * q^19 + 19 * q^23 - 5 * q^25 - 21 * q^27 + q^29 - 8 * q^31 + 7 * q^33 - 4 * q^35 - 4 * q^37 + 9 * q^41 - 13 * q^43 + 5 * q^45 - 7 * q^47 + 3 * q^49 + 7 * q^51 + 2 * q^53 + q^55 + 14 * q^57 + 21 * q^59 + 11 * q^61 + 5 * q^63 + 23 * q^67 - 7 * q^69 - 12 * q^71 + 5 * q^73 + 14 * q^75 + q^77 + 2 * q^79 - q^81 + 9 * q^83 + 21 * q^85 - 7 * q^87 + 30 * q^89 - 21 * q^93 - 21 * q^95 + 17 * q^97 - 3 * 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
$3$	1.69202	0.976889	0.488445	−	0.872595i	$-0.337564\pi$
$3$	1.69202	0.976889	0.488445	+	0.872595i	$0.337564\pi$
$4$	0	0
$5$	−1.44504	−0.646242	−0.323121	−	0.946358i	$-0.604732\pi$
$5$	−1.44504	−0.646242	−0.323121	+	0.946358i	$0.604732\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.137063	−0.0456878
$10$	0	0
$11$	4.85086	1.46259	0.731294	−	0.682062i	$-0.238917\pi$
$11$	4.85086	1.46259	0.731294	+	0.682062i	$0.238917\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.44504	−0.631307
$16$	0	0
$17$	−0.149145	−0.0361730	−0.0180865	−	0.999836i	$-0.505757\pi$
$17$	−0.149145	−0.0361730	−0.0180865	+	0.999836i	$0.505757\pi$
$18$	0	0
$19$	1.08815	0.249638	0.124819	−	0.992180i	$-0.460165\pi$
$19$	1.08815	0.249638	0.124819	+	0.992180i	$0.460165\pi$
$20$	0	0
$21$	1.69202	0.369229
$22$	0	0
$23$	9.15883	1.90975	0.954874	−	0.297010i	$-0.0959894\pi$
$23$	9.15883	1.90975	0.954874	+	0.297010i	$0.0959894\pi$
$24$	0	0
$25$	−2.91185	−0.582371
$26$	0	0
$27$	−5.30798	−1.02152
$28$	0	0
$29$	3.15883	0.586581	0.293290	−	0.956023i	$-0.405250\pi$
$29$	3.15883	0.586581	0.293290	+	0.956023i	$0.405250\pi$
$30$	0	0
$31$	−5.93900	−1.06668	−0.533338	−	0.845902i	$-0.679063\pi$
$31$	−5.93900	−1.06668	−0.533338	+	0.845902i	$0.679063\pi$
$32$	0	0
$33$	8.20775	1.42879
$34$	0	0
$35$	−1.44504	−0.244257
$36$	0	0
$37$	−4.15883	−0.683708	−0.341854	−	0.939753i	$-0.611055\pi$
$37$	−4.15883	−0.683708	−0.341854	+	0.939753i	$0.611055\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.76271	1.36851	0.684253	−	0.729245i	$-0.260129\pi$
$41$	8.76271	1.36851	0.684253	+	0.729245i	$0.260129\pi$
$42$	0	0
$43$	−3.08815	−0.470938	−0.235469	−	0.971882i	$-0.575663\pi$
$43$	−3.08815	−0.470938	−0.235469	+	0.971882i	$0.575663\pi$
$44$	0	0
$45$	0.198062	0.0295254
$46$	0	0
$47$	11.4426	1.66908	0.834541	−	0.550946i	$-0.185733\pi$
$47$	11.4426	1.66908	0.834541	+	0.550946i	$0.185733\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.252356	−0.0353370
$52$	0	0
$53$	−0.131687	−0.0180885	−0.00904427	−	0.999959i	$-0.502879\pi$
$53$	−0.131687	−0.0180885	−0.00904427	+	0.999959i	$0.502879\pi$
$54$	0	0
$55$	−7.00969	−0.945186
$56$	0	0
$57$	1.84117	0.243868
$58$	0	0
$59$	−0.454731	−0.0592009	−0.0296004	−	0.999562i	$-0.509423\pi$
$59$	−0.454731	−0.0592009	−0.0296004	+	0.999562i	$0.509423\pi$
$60$	0	0
$61$	2.53319	0.324341	0.162171	−	0.986763i	$-0.448150\pi$
$61$	2.53319	0.324341	0.162171	+	0.986763i	$0.448150\pi$
$62$	0	0
$63$	−0.137063	−0.0172684
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.81402	0.343787	0.171894	−	0.985116i	$-0.445011\pi$
$67$	2.81402	0.343787	0.171894	+	0.985116i	$0.445011\pi$
$68$	0	0
$69$	15.4969	1.86561
$70$	0	0
$71$	−8.74094	−1.03736	−0.518679	−	0.854969i	$-0.673576\pi$
$71$	−8.74094	−1.03736	−0.518679	+	0.854969i	$0.673576\pi$
$72$	0	0
$73$	6.96615	0.815326	0.407663	−	0.913132i	$-0.366344\pi$
$73$	6.96615	0.815326	0.407663	+	0.913132i	$0.366344\pi$
$74$	0	0
$75$	−4.92692	−0.568912
$76$	0	0
$77$	4.85086	0.552806
$78$	0	0
$79$	−16.3817	−1.84308	−0.921540	−	0.388284i	$-0.873068\pi$
$79$	−16.3817	−1.84308	−0.921540	+	0.388284i	$0.873068\pi$
$80$	0	0
$81$	−8.57002	−0.952225
$82$	0	0
$83$	4.00538	0.439647	0.219824	−	0.975540i	$-0.429452\pi$
$83$	4.00538	0.439647	0.219824	+	0.975540i	$0.429452\pi$
$84$	0	0
$85$	0.215521	0.0233765
$86$	0	0
$87$	5.34481	0.573024
$88$	0	0
$89$	6.95108	0.736813	0.368407	−	0.929665i	$-0.379903\pi$
$89$	6.95108	0.736813	0.368407	+	0.929665i	$0.379903\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.0489	−1.04202
$94$	0	0
$95$	−1.57242	−0.161327
$96$	0	0
$97$	14.3666	1.45871	0.729353	−	0.684138i	$-0.239821\pi$
$97$	14.3666	1.45871	0.729353	+	0.684138i	$0.239821\pi$
$98$	0	0
$99$	−0.664874	−0.0668224
$100$	0	0
$101$	0.872625	0.0868295	0.0434147	−	0.999057i	$-0.486176\pi$
$101$	0.872625	0.0868295	0.0434147	+	0.999057i	$0.486176\pi$
$102$	0	0
$103$	13.0097	1.28188	0.640941	−	0.767590i	$-0.278544\pi$
$103$	13.0097	1.28188	0.640941	+	0.767590i	$0.278544\pi$
$104$	0	0
$105$	−2.44504	−0.238612
$106$	0	0
$107$	19.3884	1.87434	0.937172	−	0.348869i	$-0.113434\pi$
$107$	19.3884	1.87434	0.937172	+	0.348869i	$0.113434\pi$
$108$	0	0
$109$	1.60925	0.154138	0.0770692	−	0.997026i	$-0.475444\pi$
$109$	1.60925	0.154138	0.0770692	+	0.997026i	$0.475444\pi$
$110$	0	0
$111$	−7.03684	−0.667907
$112$	0	0
$113$	−9.86054	−0.927602	−0.463801	−	0.885939i	$-0.653515\pi$
$113$	−9.86054	−0.927602	−0.463801	+	0.885939i	$0.653515\pi$
$114$	0	0
$115$	−13.2349	−1.23416
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.149145	−0.0136721
$120$	0	0
$121$	12.5308	1.13916
$122$	0	0
$123$	14.8267	1.33688
$124$	0	0
$125$	11.4330	1.02260
$126$	0	0
$127$	7.13169	0.632835	0.316417	−	0.948620i	$-0.397520\pi$
$127$	7.13169	0.632835	0.316417	+	0.948620i	$0.397520\pi$
$128$	0	0
$129$	−5.22521	−0.460054
$130$	0	0
$131$	−21.8509	−1.90912	−0.954559	−	0.298022i	$-0.903673\pi$
$131$	−21.8509	−1.90912	−0.954559	+	0.298022i	$0.903673\pi$
$132$	0	0
$133$	1.08815	0.0943542
$134$	0	0
$135$	7.67025	0.660150
$136$	0	0
$137$	8.03146	0.686174	0.343087	−	0.939304i	$-0.388527\pi$
$137$	8.03146	0.686174	0.343087	+	0.939304i	$0.388527\pi$
$138$	0	0
$139$	−19.2078	−1.62918	−0.814590	−	0.580037i	$-0.803038\pi$
$139$	−19.2078	−1.62918	−0.814590	+	0.580037i	$0.803038\pi$
$140$	0	0
$141$	19.3612	1.63051
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.56465	−0.379073
$146$	0	0
$147$	1.69202	0.139556
$148$	0	0
$149$	12.8509	1.05278	0.526392	−	0.850242i	$-0.323545\pi$
$149$	12.8509	1.05278	0.526392	+	0.850242i	$0.323545\pi$
$150$	0	0
$151$	−6.74094	−0.548570	−0.274285	−	0.961648i	$-0.588441\pi$
$151$	−6.74094	−0.548570	−0.274285	+	0.961648i	$0.588441\pi$
$152$	0	0
$153$	0.0204423	0.00165266
$154$	0	0
$155$	8.58211	0.689331
$156$	0	0
$157$	−12.7627	−1.01858	−0.509288	−	0.860596i	$-0.670091\pi$
$157$	−12.7627	−1.01858	−0.509288	+	0.860596i	$0.670091\pi$
$158$	0	0
$159$	−0.222816	−0.0176705
$160$	0	0
$161$	9.15883	0.721817
$162$	0	0
$163$	10.0368	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$163$	10.0368	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$164$	0	0
$165$	−11.8605	−0.923342
$166$	0	0
$167$	−8.57673	−0.663687	−0.331844	−	0.943334i	$-0.607671\pi$
$167$	−8.57673	−0.663687	−0.331844	+	0.943334i	$0.607671\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−0.149145	−0.0114054
$172$	0	0
$173$	−19.9051	−1.51336	−0.756680	−	0.653785i	$-0.773180\pi$
$173$	−19.9051	−1.51336	−0.756680	+	0.653785i	$0.773180\pi$
$174$	0	0
$175$	−2.91185	−0.220115
$176$	0	0
$177$	−0.769414	−0.0578327
$178$	0	0
$179$	4.16421	0.311248	0.155624	−	0.987816i	$-0.450261\pi$
$179$	4.16421	0.311248	0.155624	+	0.987816i	$0.450261\pi$
$180$	0	0
$181$	−2.37196	−0.176306	−0.0881532	−	0.996107i	$-0.528097\pi$
$181$	−2.37196	−0.176306	−0.0881532	+	0.996107i	$0.528097\pi$
$182$	0	0
$183$	4.28621	0.316846
$184$	0	0
$185$	6.00969	0.441841
$186$	0	0
$187$	−0.723480	−0.0529061
$188$	0	0
$189$	−5.30798	−0.386099
$190$	0	0
$191$	17.1588	1.24157	0.620785	−	0.783981i	$-0.286814\pi$
$191$	17.1588	1.24157	0.620785	+	0.783981i	$0.286814\pi$
$192$	0	0
$193$	10.1914	0.733590	0.366795	−	0.930302i	$-0.380455\pi$
$193$	10.1914	0.733590	0.366795	+	0.930302i	$0.380455\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.41657	0.670903	0.335451	−	0.942058i	$-0.391111\pi$
$197$	9.41657	0.670903	0.335451	+	0.942058i	$0.391111\pi$
$198$	0	0
$199$	−8.52542	−0.604351	−0.302175	−	0.953252i	$-0.597713\pi$
$199$	−8.52542	−0.604351	−0.302175	+	0.953252i	$0.597713\pi$
$200$	0	0
$201$	4.76138	0.335842
$202$	0	0
$203$	3.15883	0.221707
$204$	0	0
$205$	−12.6625	−0.884386
$206$	0	0
$207$	−1.25534	−0.0872522
$208$	0	0
$209$	5.27844	0.365117
$210$	0	0
$211$	−1.75302	−0.120683	−0.0603415	−	0.998178i	$-0.519219\pi$
$211$	−1.75302	−0.120683	−0.0603415	+	0.998178i	$0.519219\pi$
$212$	0	0
$213$	−14.7899	−1.01338
$214$	0	0
$215$	4.46250	0.304340
$216$	0	0
$217$	−5.93900	−0.403166
$218$	0	0
$219$	11.7869	0.796483
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.88769	0.126409	0.0632046	−	0.998001i	$-0.479868\pi$
$223$	1.88769	0.126409	0.0632046	+	0.998001i	$0.479868\pi$
$224$	0	0
$225$	0.399108	0.0266072
$226$	0	0
$227$	27.9463	1.85486	0.927430	−	0.373996i	$-0.122013\pi$
$227$	27.9463	1.85486	0.927430	+	0.373996i	$0.122013\pi$
$228$	0	0
$229$	11.1860	0.739190	0.369595	−	0.929193i	$-0.379496\pi$
$229$	11.1860	0.739190	0.369595	+	0.929193i	$0.379496\pi$
$230$	0	0
$231$	8.20775	0.540030
$232$	0	0
$233$	18.1129	1.18662	0.593308	−	0.804976i	$-0.297822\pi$
$233$	18.1129	1.18662	0.593308	+	0.804976i	$0.297822\pi$
$234$	0	0
$235$	−16.5351	−1.07863
$236$	0	0
$237$	−27.7181	−1.80048
$238$	0	0
$239$	16.9312	1.09519	0.547595	−	0.836743i	$-0.315543\pi$
$239$	16.9312	1.09519	0.547595	+	0.836743i	$0.315543\pi$
$240$	0	0
$241$	−0.283815	−0.0182822	−0.00914108	−	0.999958i	$-0.502910\pi$
$241$	−0.283815	−0.0182822	−0.00914108	+	0.999958i	$0.502910\pi$
$242$	0	0
$243$	1.42327	0.0913029
$244$	0	0
$245$	−1.44504	−0.0923203
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.77718	0.429487
$250$	0	0
$251$	−11.6528	−0.735518	−0.367759	−	0.929921i	$-0.619875\pi$
$251$	−11.6528	−0.735518	−0.367759	+	0.929921i	$0.619875\pi$
$252$	0	0
$253$	44.4282	2.79318
$254$	0	0
$255$	0.364666	0.0228362
$256$	0	0
$257$	−1.46921	−0.0916465	−0.0458232	−	0.998950i	$-0.514591\pi$
$257$	−1.46921	−0.0916465	−0.0458232	+	0.998950i	$0.514591\pi$
$258$	0	0
$259$	−4.15883	−0.258417
$260$	0	0
$261$	−0.432960	−0.0267996
$262$	0	0
$263$	−21.5381	−1.32810	−0.664048	−	0.747690i	$-0.731163\pi$
$263$	−21.5381	−1.32810	−0.664048	+	0.747690i	$0.731163\pi$
$264$	0	0
$265$	0.190293	0.0116896
$266$	0	0
$267$	11.7614	0.719785
$268$	0	0
$269$	7.04221	0.429371	0.214686	−	0.976683i	$-0.431127\pi$
$269$	7.04221	0.429371	0.214686	+	0.976683i	$0.431127\pi$
$270$	0	0
$271$	−20.0344	−1.21701	−0.608503	−	0.793552i	$-0.708230\pi$
$271$	−20.0344	−1.21701	−0.608503	+	0.793552i	$0.708230\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−14.1250	−0.851768
$276$	0	0
$277$	29.5719	1.77681	0.888403	−	0.459065i	$-0.151815\pi$
$277$	29.5719	1.77681	0.888403	+	0.459065i	$0.151815\pi$
$278$	0	0
$279$	0.814019	0.0487341
$280$	0	0
$281$	13.1793	0.786210	0.393105	−	0.919494i	$-0.371401\pi$
$281$	13.1793	0.786210	0.393105	+	0.919494i	$0.371401\pi$
$282$	0	0
$283$	15.8073	0.939647	0.469824	−	0.882760i	$-0.344317\pi$
$283$	15.8073	0.939647	0.469824	+	0.882760i	$0.344317\pi$
$284$	0	0
$285$	−2.66056	−0.157598
$286$	0	0
$287$	8.76271	0.517246
$288$	0	0
$289$	−16.9778	−0.998692
$290$	0	0
$291$	24.3086	1.42499
$292$	0	0
$293$	−5.76032	−0.336521	−0.168261	−	0.985743i	$-0.553815\pi$
$293$	−5.76032	−0.336521	−0.168261	+	0.985743i	$0.553815\pi$
$294$	0	0
$295$	0.657105	0.0382581
$296$	0	0
$297$	−25.7482	−1.49406
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−3.08815	−0.177998
$302$	0	0
$303$	1.47650	0.0848228
$304$	0	0
$305$	−3.66056	−0.209603
$306$	0	0
$307$	11.5036	0.656548	0.328274	−	0.944583i	$-0.393533\pi$
$307$	11.5036	0.656548	0.328274	+	0.944583i	$0.393533\pi$
$308$	0	0
$309$	22.0127	1.25226
$310$	0	0
$311$	1.24400	0.0705405	0.0352703	−	0.999378i	$-0.488771\pi$
$311$	1.24400	0.0705405	0.0352703	+	0.999378i	$0.488771\pi$
$312$	0	0
$313$	10.7356	0.606810	0.303405	−	0.952862i	$-0.401877\pi$
$313$	10.7356	0.606810	0.303405	+	0.952862i	$0.401877\pi$
$314$	0	0
$315$	0.198062	0.0111595
$316$	0	0
$317$	−11.3556	−0.637792	−0.318896	−	0.947790i	$-0.603312\pi$
$317$	−11.3556	−0.637792	−0.318896	+	0.947790i	$0.603312\pi$
$318$	0	0
$319$	15.3230	0.857926
$320$	0	0
$321$	32.8055	1.83103
$322$	0	0
$323$	−0.162291	−0.00903014
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.72289	0.150576
$328$	0	0
$329$	11.4426	0.630854
$330$	0	0
$331$	−8.70410	−0.478421	−0.239210	−	0.970968i	$-0.576889\pi$
$331$	−8.70410	−0.478421	−0.239210	+	0.970968i	$0.576889\pi$
$332$	0	0
$333$	0.570024	0.0312371
$334$	0	0
$335$	−4.06638	−0.222170
$336$	0	0
$337$	9.10560	0.496014	0.248007	−	0.968758i	$-0.420224\pi$
$337$	9.10560	0.496014	0.248007	+	0.968758i	$0.420224\pi$
$338$	0	0
$339$	−16.6843	−0.906164
$340$	0	0
$341$	−28.8092	−1.56011
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−22.3937	−1.20564
$346$	0	0
$347$	16.8823	0.906290	0.453145	−	0.891437i	$-0.350302\pi$
$347$	16.8823	0.906290	0.453145	+	0.891437i	$0.350302\pi$
$348$	0	0
$349$	−1.08038	−0.0578312	−0.0289156	−	0.999582i	$-0.509205\pi$
$349$	−1.08038	−0.0578312	−0.0289156	+	0.999582i	$0.509205\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.6450	−0.992374	−0.496187	−	0.868216i	$-0.665267\pi$
$353$	−18.6450	−0.992374	−0.496187	+	0.868216i	$0.665267\pi$
$354$	0	0
$355$	12.6310	0.670385
$356$	0	0
$357$	−0.252356	−0.0133561
$358$	0	0
$359$	23.3666	1.23324	0.616621	−	0.787260i	$-0.288501\pi$
$359$	23.3666	1.23324	0.616621	+	0.787260i	$0.288501\pi$
$360$	0	0
$361$	−17.8159	−0.937681
$362$	0	0
$363$	21.2024	1.11284
$364$	0	0
$365$	−10.0664	−0.526898
$366$	0	0
$367$	22.1497	1.15621	0.578103	−	0.815964i	$-0.303793\pi$
$367$	22.1497	1.15621	0.578103	+	0.815964i	$0.303793\pi$
$368$	0	0
$369$	−1.20105	−0.0625240
$370$	0	0
$371$	−0.131687	−0.00683682
$372$	0	0
$373$	2.81940	0.145983	0.0729914	−	0.997333i	$-0.476745\pi$
$373$	2.81940	0.145983	0.0729914	+	0.997333i	$0.476745\pi$
$374$	0	0
$375$	19.3448	0.998962
$376$	0	0
$377$	0	0
$378$	0	0
$379$	13.7651	0.707066	0.353533	−	0.935422i	$-0.384980\pi$
$379$	13.7651	0.707066	0.353533	+	0.935422i	$0.384980\pi$
$380$	0	0
$381$	12.0670	0.618209
$382$	0	0
$383$	−33.2433	−1.69865	−0.849326	−	0.527869i	$-0.822991\pi$
$383$	−33.2433	−1.69865	−0.849326	+	0.527869i	$0.822991\pi$
$384$	0	0
$385$	−7.00969	−0.357247
$386$	0	0
$387$	0.423272	0.0215161
$388$	0	0
$389$	15.2881	0.775139	0.387569	−	0.921841i	$-0.373315\pi$
$389$	15.2881	0.775139	0.387569	+	0.921841i	$0.373315\pi$
$390$	0	0
$391$	−1.36599	−0.0690813
$392$	0	0
$393$	−36.9721	−1.86500
$394$	0	0
$395$	23.6722	1.19108
$396$	0	0
$397$	−4.49635	−0.225665	−0.112833	−	0.993614i	$-0.535992\pi$
$397$	−4.49635	−0.225665	−0.112833	+	0.993614i	$0.535992\pi$
$398$	0	0
$399$	1.84117	0.0921736
$400$	0	0
$401$	−22.2610	−1.11166	−0.555830	−	0.831296i	$-0.687600\pi$
$401$	−22.2610	−1.11166	−0.555830	+	0.831296i	$0.687600\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	12.3840	0.615368
$406$	0	0
$407$	−20.1739	−0.999983
$408$	0	0
$409$	37.7453	1.86638	0.933191	−	0.359380i	$-0.117012\pi$
$409$	37.7453	1.86638	0.933191	+	0.359380i	$0.117012\pi$
$410$	0	0
$411$	13.5894	0.670316
$412$	0	0
$413$	−0.454731	−0.0223758
$414$	0	0
$415$	−5.78794	−0.284119
$416$	0	0
$417$	−32.4999	−1.59153
$418$	0	0
$419$	35.9571	1.75662	0.878308	−	0.478094i	$-0.158672\pi$
$419$	35.9571	1.75662	0.878308	+	0.478094i	$0.158672\pi$
$420$	0	0
$421$	6.46921	0.315290	0.157645	−	0.987496i	$-0.449610\pi$
$421$	6.46921	0.315290	0.157645	+	0.987496i	$0.449610\pi$
$422$	0	0
$423$	−1.56837	−0.0762566
$424$	0	0
$425$	0.434288	0.0210661
$426$	0	0
$427$	2.53319	0.122590
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−35.9711	−1.73266	−0.866332	−	0.499468i	$-0.833529\pi$
$431$	−35.9711	−1.73266	−0.866332	+	0.499468i	$0.833529\pi$
$432$	0	0
$433$	5.47352	0.263040	0.131520	−	0.991313i	$-0.458014\pi$
$433$	5.47352	0.263040	0.131520	+	0.991313i	$0.458014\pi$
$434$	0	0
$435$	−7.72348	−0.370313
$436$	0	0
$437$	9.96615	0.476746
$438$	0	0
$439$	−2.35557	−0.112425	−0.0562126	−	0.998419i	$-0.517902\pi$
$439$	−2.35557	−0.112425	−0.0562126	+	0.998419i	$0.517902\pi$
$440$	0	0
$441$	−0.137063	−0.00652683
$442$	0	0
$443$	−25.7845	−1.22506	−0.612529	−	0.790448i	$-0.709848\pi$
$443$	−25.7845	−1.22506	−0.612529	+	0.790448i	$0.709848\pi$
$444$	0	0
$445$	−10.0446	−0.476160
$446$	0	0
$447$	21.7439	1.02845
$448$	0	0
$449$	22.7573	1.07398	0.536992	−	0.843587i	$-0.319560\pi$
$449$	22.7573	1.07398	0.536992	+	0.843587i	$0.319560\pi$
$450$	0	0
$451$	42.5066	2.00156
$452$	0	0
$453$	−11.4058	−0.535892
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.7573	1.25166	0.625828	−	0.779961i	$-0.284761\pi$
$457$	26.7573	1.25166	0.625828	+	0.779961i	$0.284761\pi$
$458$	0	0
$459$	0.791658	0.0369514
$460$	0	0
$461$	−20.8224	−0.969795	−0.484897	−	0.874571i	$-0.661143\pi$
$461$	−20.8224	−0.969795	−0.484897	+	0.874571i	$0.661143\pi$
$462$	0	0
$463$	40.2097	1.86870	0.934351	−	0.356354i	$-0.115980\pi$
$463$	40.2097	1.86870	0.934351	+	0.356354i	$0.115980\pi$
$464$	0	0
$465$	14.5211	0.673400
$466$	0	0
$467$	2.71678	0.125717	0.0628587	−	0.998022i	$-0.479978\pi$
$467$	2.71678	0.125717	0.0628587	+	0.998022i	$0.479978\pi$
$468$	0	0
$469$	2.81402	0.129939
$470$	0	0
$471$	−21.5948	−0.995035
$472$	0	0
$473$	−14.9801	−0.688788
$474$	0	0
$475$	−3.16852	−0.145382
$476$	0	0
$477$	0.0180494	0.000826425 0
$478$	0	0
$479$	5.41550	0.247441	0.123720	−	0.992317i	$-0.460517\pi$
$479$	5.41550	0.247441	0.123720	+	0.992317i	$0.460517\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	15.4969	0.705135
$484$	0	0
$485$	−20.7603	−0.942677
$486$	0	0
$487$	−20.6340	−0.935016	−0.467508	−	0.883989i	$-0.654848\pi$
$487$	−20.6340	−0.935016	−0.467508	+	0.883989i	$0.654848\pi$
$488$	0	0
$489$	16.9825	0.767977
$490$	0	0
$491$	35.4276	1.59882	0.799412	−	0.600783i	$-0.205144\pi$
$491$	35.4276	1.59882	0.799412	+	0.600783i	$0.205144\pi$
$492$	0	0
$493$	−0.471124	−0.0212184
$494$	0	0
$495$	0.960771	0.0431835
$496$	0	0
$497$	−8.74094	−0.392085
$498$	0	0
$499$	−4.07367	−0.182363	−0.0911813	−	0.995834i	$-0.529064\pi$
$499$	−4.07367	−0.182363	−0.0911813	+	0.995834i	$0.529064\pi$
$500$	0	0
$501$	−14.5120	−0.648349
$502$	0	0
$503$	−20.2271	−0.901883	−0.450942	−	0.892553i	$-0.648912\pi$
$503$	−20.2271	−0.901883	−0.450942	+	0.892553i	$0.648912\pi$
$504$	0	0
$505$	−1.26098	−0.0561129
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.0368	−0.755144	−0.377572	−	0.925980i	$-0.623241\pi$
$509$	−17.0368	−0.755144	−0.377572	+	0.925980i	$0.623241\pi$
$510$	0	0
$511$	6.96615	0.308164
$512$	0	0
$513$	−5.77586	−0.255010
$514$	0	0
$515$	−18.7995	−0.828407
$516$	0	0
$517$	55.5066	2.44118
$518$	0	0
$519$	−33.6799	−1.47839
$520$	0	0
$521$	25.6504	1.12376	0.561882	−	0.827217i	$-0.310077\pi$
$521$	25.6504	1.12376	0.561882	+	0.827217i	$0.310077\pi$
$522$	0	0
$523$	−18.8974	−0.826325	−0.413162	−	0.910657i	$-0.635576\pi$
$523$	−18.8974	−0.826325	−0.413162	+	0.910657i	$0.635576\pi$
$524$	0	0
$525$	−4.92692	−0.215028
$526$	0	0
$527$	0.885772	0.0385848
$528$	0	0
$529$	60.8842	2.64714
$530$	0	0
$531$	0.0623269	0.00270476
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−28.0170	−1.21128
$536$	0	0
$537$	7.04593	0.304054
$538$	0	0
$539$	4.85086	0.208941
$540$	0	0
$541$	−23.9691	−1.03051	−0.515257	−	0.857036i	$-0.672303\pi$
$541$	−23.9691	−1.03051	−0.515257	+	0.857036i	$0.672303\pi$
$542$	0	0
$543$	−4.01341	−0.172232
$544$	0	0
$545$	−2.32544	−0.0996108
$546$	0	0
$547$	28.1618	1.20411	0.602056	−	0.798454i	$-0.294348\pi$
$547$	28.1618	1.20411	0.602056	+	0.798454i	$0.294348\pi$
$548$	0	0
$549$	−0.347207	−0.0148184
$550$	0	0
$551$	3.43727	0.146433
$552$	0	0
$553$	−16.3817	−0.696619
$554$	0	0
$555$	10.1685	0.431630
$556$	0	0
$557$	35.7851	1.51626	0.758131	−	0.652102i	$-0.226113\pi$
$557$	35.7851	1.51626	0.758131	+	0.652102i	$0.226113\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−1.22414	−0.0516834
$562$	0	0
$563$	−2.02177	−0.0852075	−0.0426037	−	0.999092i	$-0.513565\pi$
$563$	−2.02177	−0.0852075	−0.0426037	+	0.999092i	$0.513565\pi$
$564$	0	0
$565$	14.2489	0.599456
$566$	0	0
$567$	−8.57002	−0.359907
$568$	0	0
$569$	−28.7590	−1.20564	−0.602820	−	0.797877i	$-0.705956\pi$
$569$	−28.7590	−1.20564	−0.602820	+	0.797877i	$0.705956\pi$
$570$	0	0
$571$	24.3967	1.02097	0.510485	−	0.859886i	$-0.329466\pi$
$571$	24.3967	1.02097	0.510485	+	0.859886i	$0.329466\pi$
$572$	0	0
$573$	29.0331	1.21288
$574$	0	0
$575$	−26.6692	−1.11218
$576$	0	0
$577$	−9.56225	−0.398082	−0.199041	−	0.979991i	$-0.563783\pi$
$577$	−9.56225	−0.398082	−0.199041	+	0.979991i	$0.563783\pi$
$578$	0	0
$579$	17.2440	0.716636
$580$	0	0
$581$	4.00538	0.166171
$582$	0	0
$583$	−0.638792	−0.0264561
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.3672	−0.881918	−0.440959	−	0.897527i	$-0.645362\pi$
$587$	−21.3672	−0.881918	−0.440959	+	0.897527i	$0.645362\pi$
$588$	0	0
$589$	−6.46250	−0.266283
$590$	0	0
$591$	15.9330	0.655397
$592$	0	0
$593$	−0.804331	−0.0330299	−0.0165150	−	0.999864i	$-0.505257\pi$
$593$	−0.804331	−0.0330299	−0.0165150	+	0.999864i	$0.505257\pi$
$594$	0	0
$595$	0.215521	0.00883549
$596$	0	0
$597$	−14.4252	−0.590384
$598$	0	0
$599$	−3.23490	−0.132174	−0.0660872	−	0.997814i	$-0.521052\pi$
$599$	−3.23490	−0.132174	−0.0660872	+	0.997814i	$0.521052\pi$
$600$	0	0
$601$	13.8398	0.564539	0.282269	−	0.959335i	$-0.408913\pi$
$601$	13.8398	0.564539	0.282269	+	0.959335i	$0.408913\pi$
$602$	0	0
$603$	−0.385699	−0.0157069
$604$	0	0
$605$	−18.1075	−0.736176
$606$	0	0
$607$	3.92394	0.159268	0.0796338	−	0.996824i	$-0.474625\pi$
$607$	3.92394	0.159268	0.0796338	+	0.996824i	$0.474625\pi$
$608$	0	0
$609$	5.34481	0.216583
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−11.1629	−0.450865	−0.225432	−	0.974259i	$-0.572379\pi$
$613$	−11.1629	−0.450865	−0.225432	+	0.974259i	$0.572379\pi$
$614$	0	0
$615$	−21.4252	−0.863947
$616$	0	0
$617$	36.8713	1.48438	0.742191	−	0.670188i	$-0.233787\pi$
$617$	36.8713	1.48438	0.742191	+	0.670188i	$0.233787\pi$
$618$	0	0
$619$	−7.24219	−0.291088	−0.145544	−	0.989352i	$-0.546493\pi$
$619$	−7.24219	−0.291088	−0.145544	+	0.989352i	$0.546493\pi$
$620$	0	0
$621$	−48.6149	−1.95085
$622$	0	0
$623$	6.95108	0.278489
$624$	0	0
$625$	−1.96184	−0.0784735
$626$	0	0
$627$	8.93123	0.356679
$628$	0	0
$629$	0.620269	0.0247317
$630$	0	0
$631$	−26.2664	−1.04565	−0.522824	−	0.852441i	$-0.675121\pi$
$631$	−26.2664	−1.04565	−0.522824	+	0.852441i	$0.675121\pi$
$632$	0	0
$633$	−2.96615	−0.117894
$634$	0	0
$635$	−10.3056	−0.408965
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.19806	0.0473946
$640$	0	0
$641$	27.0248	1.06741	0.533707	−	0.845670i	$-0.320799\pi$
$641$	27.0248	1.06741	0.533707	+	0.845670i	$0.320799\pi$
$642$	0	0
$643$	8.40044	0.331281	0.165640	−	0.986186i	$-0.447031\pi$
$643$	8.40044	0.331281	0.165640	+	0.986186i	$0.447031\pi$
$644$	0	0
$645$	7.55065	0.297306
$646$	0	0
$647$	21.5657	0.847836	0.423918	−	0.905701i	$-0.360655\pi$
$647$	21.5657	0.847836	0.423918	+	0.905701i	$0.360655\pi$
$648$	0	0
$649$	−2.20583	−0.0865865
$650$	0	0
$651$	−10.0489	−0.393848
$652$	0	0
$653$	−12.7216	−0.497833	−0.248917	−	0.968525i	$-0.580075\pi$
$653$	−12.7216	−0.497833	−0.248917	+	0.968525i	$0.580075\pi$
$654$	0	0
$655$	31.5754	1.23375
$656$	0	0
$657$	−0.954804	−0.0372504
$658$	0	0
$659$	46.9764	1.82994	0.914971	−	0.403520i	$-0.132213\pi$
$659$	46.9764	1.82994	0.914971	+	0.403520i	$0.132213\pi$
$660$	0	0
$661$	32.2573	1.25466	0.627331	−	0.778753i	$-0.284147\pi$
$661$	32.2573	1.25466	0.627331	+	0.778753i	$0.284147\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.57242	−0.0609757
$666$	0	0
$667$	28.9312	1.12022
$668$	0	0
$669$	3.19401	0.123488
$670$	0	0
$671$	12.2881	0.474378
$672$	0	0
$673$	−12.8528	−0.495438	−0.247719	−	0.968832i	$-0.579681\pi$
$673$	−12.8528	−0.495438	−0.247719	+	0.968832i	$0.579681\pi$
$674$	0	0
$675$	15.4561	0.594904
$676$	0	0
$677$	−28.2118	−1.08427	−0.542134	−	0.840292i	$-0.682383\pi$
$677$	−28.2118	−1.08427	−0.542134	+	0.840292i	$0.682383\pi$
$678$	0	0
$679$	14.3666	0.551339
$680$	0	0
$681$	47.2857	1.81199
$682$	0	0
$683$	1.12929	0.0432112	0.0216056	−	0.999767i	$-0.493122\pi$
$683$	1.12929	0.0432112	0.0216056	+	0.999767i	$0.493122\pi$
$684$	0	0
$685$	−11.6058	−0.443435
$686$	0	0
$687$	18.9269	0.722107
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−15.8810	−0.604141	−0.302071	−	0.953286i	$-0.597678\pi$
$691$	−15.8810	−0.604141	−0.302071	+	0.953286i	$0.597678\pi$
$692$	0	0
$693$	−0.664874	−0.0252565
$694$	0	0
$695$	27.7560	1.05285
$696$	0	0
$697$	−1.30691	−0.0495029
$698$	0	0
$699$	30.6474	1.15919
$700$	0	0
$701$	−30.0810	−1.13614	−0.568071	−	0.822979i	$-0.692310\pi$
$701$	−30.0810	−1.13614	−0.568071	+	0.822979i	$0.692310\pi$
$702$	0	0
$703$	−4.52542	−0.170679
$704$	0	0
$705$	−27.9778	−1.05370
$706$	0	0
$707$	0.872625	0.0328185
$708$	0	0
$709$	0.473517	0.0177833	0.00889165	−	0.999960i	$-0.497170\pi$
$709$	0.473517	0.0177833	0.00889165	+	0.999960i	$0.497170\pi$
$710$	0	0
$711$	2.24532	0.0842062
$712$	0	0
$713$	−54.3943	−2.03708
$714$	0	0
$715$	0	0
$716$	0	0
$717$	28.6480	1.06988
$718$	0	0
$719$	−37.8562	−1.41180	−0.705900	−	0.708312i	$-0.749457\pi$
$719$	−37.8562	−1.41180	−0.705900	+	0.708312i	$0.749457\pi$
$720$	0	0
$721$	13.0097	0.484506
$722$	0	0
$723$	−0.480222	−0.0178596
$724$	0	0
$725$	−9.19806	−0.341607
$726$	0	0
$727$	37.8877	1.40518	0.702588	−	0.711597i	$-0.252028\pi$
$727$	37.8877	1.40518	0.702588	+	0.711597i	$0.252028\pi$
$728$	0	0
$729$	28.1183	1.04142
$730$	0	0
$731$	0.460581	0.0170352
$732$	0	0
$733$	−37.2161	−1.37461	−0.687304	−	0.726370i	$-0.741206\pi$
$733$	−37.2161	−1.37461	−0.687304	+	0.726370i	$0.741206\pi$
$734$	0	0
$735$	−2.44504	−0.0901867
$736$	0	0
$737$	13.6504	0.502819
$738$	0	0
$739$	23.0019	0.846139	0.423070	−	0.906097i	$-0.360953\pi$
$739$	23.0019	0.846139	0.423070	+	0.906097i	$0.360953\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−45.7284	−1.67761	−0.838806	−	0.544431i	$-0.816746\pi$
$743$	−45.7284	−1.67761	−0.838806	+	0.544431i	$0.816746\pi$
$744$	0	0
$745$	−18.5700	−0.680353
$746$	0	0
$747$	−0.548990	−0.0200865
$748$	0	0
$749$	19.3884	0.708435
$750$	0	0
$751$	6.28680	0.229409	0.114704	−	0.993400i	$-0.463408\pi$
$751$	6.28680	0.229409	0.114704	+	0.993400i	$0.463408\pi$
$752$	0	0
$753$	−19.7168	−0.718519
$754$	0	0
$755$	9.74094	0.354509
$756$	0	0
$757$	9.35019	0.339838	0.169919	−	0.985458i	$-0.445649\pi$
$757$	9.35019	0.339838	0.169919	+	0.985458i	$0.445649\pi$
$758$	0	0
$759$	75.1734	2.72862
$760$	0	0
$761$	−17.6649	−0.640351	−0.320176	−	0.947358i	$-0.603742\pi$
$761$	−17.6649	−0.640351	−0.320176	+	0.947358i	$0.603742\pi$
$762$	0	0
$763$	1.60925	0.0582588
$764$	0	0
$765$	−0.0295400	−0.00106802
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−47.2325	−1.70325	−0.851624	−	0.524153i	$-0.824382\pi$
$769$	−47.2325	−1.70325	−0.851624	+	0.524153i	$0.824382\pi$
$770$	0	0
$771$	−2.48593	−0.0895285
$772$	0	0
$773$	−24.9463	−0.897256	−0.448628	−	0.893719i	$-0.648087\pi$
$773$	−24.9463	−0.897256	−0.448628	+	0.893719i	$0.648087\pi$
$774$	0	0
$775$	17.2935	0.621201
$776$	0	0
$777$	−7.03684	−0.252445
$778$	0	0
$779$	9.53511	0.341631
$780$	0	0
$781$	−42.4010	−1.51723
$782$	0	0
$783$	−16.7670	−0.599204
$784$	0	0
$785$	18.4426	0.658246
$786$	0	0
$787$	−51.7864	−1.84599	−0.922993	−	0.384817i	$-0.874265\pi$
$787$	−51.7864	−1.84599	−0.922993	+	0.384817i	$0.874265\pi$
$788$	0	0
$789$	−36.4429	−1.29740
$790$	0	0
$791$	−9.86054	−0.350601
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.321979	0.0114194
$796$	0	0
$797$	−17.1395	−0.607111	−0.303555	−	0.952814i	$-0.598174\pi$
$797$	−17.1395	−0.607111	−0.303555	+	0.952814i	$0.598174\pi$
$798$	0	0
$799$	−1.70661	−0.0603756
$800$	0	0
$801$	−0.952739	−0.0336634
$802$	0	0
$803$	33.7918	1.19249
$804$	0	0
$805$	−13.2349	−0.466469
$806$	0	0
$807$	11.9156	0.419448
$808$	0	0
$809$	−14.6437	−0.514845	−0.257422	−	0.966299i	$-0.582873\pi$
$809$	−14.6437	−0.514845	−0.257422	+	0.966299i	$0.582873\pi$
$810$	0	0
$811$	17.3002	0.607492	0.303746	−	0.952753i	$-0.401762\pi$
$811$	17.3002	0.607492	0.303746	+	0.952753i	$0.401762\pi$
$812$	0	0
$813$	−33.8987	−1.18888
$814$	0	0
$815$	−14.5036	−0.508041
$816$	0	0
$817$	−3.36035	−0.117564
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44.4965	−1.55294	−0.776469	−	0.630156i	$-0.782991\pi$
$821$	−44.4965	−1.55294	−0.776469	+	0.630156i	$0.782991\pi$
$822$	0	0
$823$	−48.9566	−1.70652	−0.853259	−	0.521487i	$-0.825378\pi$
$823$	−48.9566	−1.70652	−0.853259	+	0.521487i	$0.825378\pi$
$824$	0	0
$825$	−23.8998	−0.832083
$826$	0	0
$827$	−14.1390	−0.491661	−0.245830	−	0.969313i	$-0.579061\pi$
$827$	−14.1390	−0.491661	−0.245830	+	0.969313i	$0.579061\pi$
$828$	0	0
$829$	−35.0157	−1.21614	−0.608072	−	0.793882i	$-0.708057\pi$
$829$	−35.0157	−1.21614	−0.608072	+	0.793882i	$0.708057\pi$
$830$	0	0
$831$	50.0364	1.73574
$832$	0	0
$833$	−0.149145	−0.00516757
$834$	0	0
$835$	12.3937	0.428903
$836$	0	0
$837$	31.5241	1.08963
$838$	0	0
$839$	18.8146	0.649552	0.324776	−	0.945791i	$-0.394711\pi$
$839$	18.8146	0.649552	0.324776	+	0.945791i	$0.394711\pi$
$840$	0	0
$841$	−19.0218	−0.655923
$842$	0	0
$843$	22.2996	0.768040
$844$	0	0
$845$	0	0
$846$	0	0
$847$	12.5308	0.430563
$848$	0	0
$849$	26.7463	0.917931
$850$	0	0
$851$	−38.0901	−1.30571
$852$	0	0
$853$	49.6058	1.69847	0.849235	−	0.528015i	$-0.177064\pi$
$853$	49.6058	1.69847	0.849235	+	0.528015i	$0.177064\pi$
$854$	0	0
$855$	0.215521	0.00737065
$856$	0	0
$857$	−34.6426	−1.18337	−0.591685	−	0.806169i	$-0.701537\pi$
$857$	−34.6426	−1.18337	−0.591685	+	0.806169i	$0.701537\pi$
$858$	0	0
$859$	−13.7345	−0.468615	−0.234307	−	0.972163i	$-0.575282\pi$
$859$	−13.7345	−0.468615	−0.234307	+	0.972163i	$0.575282\pi$
$860$	0	0
$861$	14.8267	0.505292
$862$	0	0
$863$	−29.7103	−1.01135	−0.505676	−	0.862724i	$-0.668757\pi$
$863$	−29.7103	−1.01135	−0.505676	+	0.862724i	$0.668757\pi$
$864$	0	0
$865$	28.7638	0.977998
$866$	0	0
$867$	−28.7267	−0.975611
$868$	0	0
$869$	−79.4650	−2.69567
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1.96913	−0.0666450
$874$	0	0
$875$	11.4330	0.386505
$876$	0	0
$877$	20.4513	0.690590	0.345295	−	0.938494i	$-0.387779\pi$
$877$	20.4513	0.690590	0.345295	+	0.938494i	$0.387779\pi$
$878$	0	0
$879$	−9.74658	−0.328744
$880$	0	0
$881$	−14.0291	−0.472651	−0.236326	−	0.971674i	$-0.575943\pi$
$881$	−14.0291	−0.472651	−0.236326	+	0.971674i	$0.575943\pi$
$882$	0	0
$883$	−22.0218	−0.741092	−0.370546	−	0.928814i	$-0.620829\pi$
$883$	−22.0218	−0.741092	−0.370546	+	0.928814i	$0.620829\pi$
$884$	0	0
$885$	1.11184	0.0373739
$886$	0	0
$887$	−40.5056	−1.36004	−0.680022	−	0.733192i	$-0.738030\pi$
$887$	−40.5056	−1.36004	−0.680022	+	0.733192i	$0.738030\pi$
$888$	0	0
$889$	7.13169	0.239189
$890$	0	0
$891$	−41.5719	−1.39271
$892$	0	0
$893$	12.4513	0.416666
$894$	0	0
$895$	−6.01746	−0.201141
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−18.7603	−0.625692
$900$	0	0
$901$	0.0196404	0.000654316 0
$902$	0	0
$903$	−5.22521	−0.173884
$904$	0	0
$905$	3.42758	0.113937
$906$	0	0
$907$	−58.3967	−1.93903	−0.969516	−	0.245030i	$-0.921202\pi$
$907$	−58.3967	−1.93903	−0.969516	+	0.245030i	$0.921202\pi$
$908$	0	0
$909$	−0.119605	−0.00396705
$910$	0	0
$911$	30.1728	0.999671	0.499835	−	0.866120i	$-0.333394\pi$
$911$	30.1728	0.999671	0.499835	+	0.866120i	$0.333394\pi$
$912$	0	0
$913$	19.4295	0.643023
$914$	0	0
$915$	−6.19375	−0.204759
$916$	0	0
$917$	−21.8509	−0.721579
$918$	0	0
$919$	−27.8015	−0.917086	−0.458543	−	0.888672i	$-0.651628\pi$
$919$	−27.8015	−0.917086	−0.458543	+	0.888672i	$0.651628\pi$
$920$	0	0
$921$	19.4644	0.641374
$922$	0	0
$923$	0	0
$924$	0	0
$925$	12.1099	0.398172
$926$	0	0
$927$	−1.78315	−0.0585664
$928$	0	0
$929$	−38.9627	−1.27832	−0.639162	−	0.769072i	$-0.720719\pi$
$929$	−38.9627	−1.27832	−0.639162	+	0.769072i	$0.720719\pi$
$930$	0	0
$931$	1.08815	0.0356625
$932$	0	0
$933$	2.10487	0.0689103
$934$	0	0
$935$	1.04546	0.0341902
$936$	0	0
$937$	−31.5687	−1.03130	−0.515652	−	0.856798i	$-0.672450\pi$
$937$	−31.5687	−1.03130	−0.515652	+	0.856798i	$0.672450\pi$
$938$	0	0
$939$	18.1648	0.592786
$940$	0	0
$941$	−9.85756	−0.321347	−0.160674	−	0.987008i	$-0.551367\pi$
$941$	−9.85756	−0.321347	−0.160674	+	0.987008i	$0.551367\pi$
$942$	0	0
$943$	80.2562	2.61350
$944$	0	0
$945$	7.67025	0.249513
$946$	0	0
$947$	50.5435	1.64244	0.821221	−	0.570610i	$-0.193293\pi$
$947$	50.5435	1.64244	0.821221	+	0.570610i	$0.193293\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−19.2139	−0.623052
$952$	0	0
$953$	−24.0218	−0.778141	−0.389071	−	0.921208i	$-0.627204\pi$
$953$	−24.0218	−0.778141	−0.389071	+	0.921208i	$0.627204\pi$
$954$	0	0
$955$	−24.7952	−0.802355
$956$	0	0
$957$	25.9269	0.838098
$958$	0	0
$959$	8.03146	0.259349
$960$	0	0
$961$	4.27173	0.137798
$962$	0	0
$963$	−2.65743	−0.0856346
$964$	0	0
$965$	−14.7269	−0.474077
$966$	0	0
$967$	−52.4969	−1.68819	−0.844094	−	0.536195i	$-0.819861\pi$
$967$	−52.4969	−1.68819	−0.844094	+	0.536195i	$0.819861\pi$
$968$	0	0
$969$	−0.274601	−0.00882144
$970$	0	0
$971$	−7.21957	−0.231687	−0.115844	−	0.993267i	$-0.536957\pi$
$971$	−7.21957	−0.231687	−0.115844	+	0.993267i	$0.536957\pi$
$972$	0	0
$973$	−19.2078	−0.615772
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.43237	0.0778184	0.0389092	−	0.999243i	$-0.487612\pi$
$977$	2.43237	0.0778184	0.0389092	+	0.999243i	$0.487612\pi$
$978$	0	0
$979$	33.7187	1.07765
$980$	0	0
$981$	−0.220569	−0.00704224
$982$	0	0
$983$	35.9452	1.14647	0.573237	−	0.819389i	$-0.305687\pi$
$983$	35.9452	1.14647	0.573237	+	0.819389i	$0.305687\pi$
$984$	0	0
$985$	−13.6073	−0.433566
$986$	0	0
$987$	19.3612	0.616274
$988$	0	0
$989$	−28.2838	−0.899373
$990$	0	0
$991$	48.5695	1.54286	0.771431	−	0.636313i	$-0.219541\pi$
$991$	48.5695	1.54286	0.771431	+	0.636313i	$0.219541\pi$
$992$	0	0
$993$	−14.7275	−0.467364
$994$	0	0
$995$	12.3196	0.390557
$996$	0	0
$997$	−24.6179	−0.779656	−0.389828	−	0.920888i	$-0.627466\pi$
$997$	−24.6179	−0.779656	−0.389828	+	0.920888i	$0.627466\pi$
$998$	0	0
$999$	22.0750	0.698422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.t.1.3	✓	3
13.12	even	2		9464.2.a.u.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.t.1.3	✓	3	1.1	even	1	trivial
9464.2.a.u.1.3	yes	3	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q+1.69202 q^{3} -1.44504 q^{5} +1.00000 q^{7} -0.137063 q^{9} +4.85086 q^{11} -2.44504 q^{15} -0.149145 q^{17} +1.08815 q^{19} +1.69202 q^{21} +9.15883 q^{23} -2.91185 q^{25} -5.30798 q^{27} +3.15883 q^{29} -5.93900 q^{31} +8.20775 q^{33} -1.44504 q^{35} -4.15883 q^{37} +8.76271 q^{41} -3.08815 q^{43} +0.198062 q^{45} +11.4426 q^{47} +1.00000 q^{49} -0.252356 q^{51} -0.131687 q^{53} -7.00969 q^{55} +1.84117 q^{57} -0.454731 q^{59} +2.53319 q^{61} -0.137063 q^{63} +2.81402 q^{67} +15.4969 q^{69} -8.74094 q^{71} +6.96615 q^{73} -4.92692 q^{75} +4.85086 q^{77} -16.3817 q^{79} -8.57002 q^{81} +4.00538 q^{83} +0.215521 q^{85} +5.34481 q^{87} +6.95108 q^{89} -10.0489 q^{93} -1.57242 q^{95} +14.3666 q^{97} -0.664874 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{5} + 3 q^{7} + 5 q^{9} + q^{11} - 7 q^{15} - 14 q^{17} + 7 q^{19} + 19 q^{23} - 5 q^{25} - 21 q^{27} + q^{29} - 8 q^{31} + 7 q^{33} - 4 q^{35} - 4 q^{37} + 9 q^{41} - 13 q^{43} + 5 q^{45} - 7 q^{47}+ \cdots - 3 q^{99}+O(q^{100}) \) 3 * q - 4 * q^5 + 3 * q^7 + 5 * q^9 + q^11 - 7 * q^15 - 14 * q^17 + 7 * q^19 + 19 * q^23 - 5 * q^25 - 21 * q^27 + q^29 - 8 * q^31 + 7 * q^33 - 4 * q^35 - 4 * q^37 + 9 * q^41 - 13 * q^43 + 5 * q^45 - 7 * q^47 + 3 * q^49 + 7 * q^51 + 2 * q^53 + q^55 + 14 * q^57 + 21 * q^59 + 11 * q^61 + 5 * q^63 + 23 * q^67 - 7 * q^69 - 12 * q^71 + 5 * q^73 + 14 * q^75 + q^77 + 2 * q^79 - q^81 + 9 * q^83 + 21 * q^85 - 7 * q^87 + 30 * q^89 - 21 * q^93 - 21 * q^95 + 17 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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