LMFDB - Embedded newform 4368.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 4x + 2$ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.28917 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.20555 q^{11} -1.00000 q^{13} -1.28917 q^{15} +1.62721 q^{17} +6.33804 q^{19} +1.00000 q^{21} +2.71083 q^{23} -3.33804 q^{25} +1.00000 q^{27} -4.33804 q^{29} +7.49472 q^{31} -4.20555 q^{33} -1.28917 q^{35} +3.42166 q^{37} -1.00000 q^{39} -7.62721 q^{41} +4.91638 q^{43} -1.28917 q^{45} +1.08362 q^{47} +1.00000 q^{49} +1.62721 q^{51} +1.75971 q^{53} +5.42166 q^{55} +6.33804 q^{57} +4.57834 q^{59} -5.25443 q^{61} +1.00000 q^{63} +1.28917 q^{65} -8.67609 q^{67} +2.71083 q^{69} +6.78389 q^{71} +4.07306 q^{73} -3.33804 q^{75} -4.20555 q^{77} +8.91638 q^{79} +1.00000 q^{81} +4.33804 q^{83} -2.09775 q^{85} -4.33804 q^{87} +4.54359 q^{89} -1.00000 q^{91} +7.49472 q^{93} -8.17081 q^{95} +11.3275 q^{97} -4.20555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 3 q^{15} - 8 q^{17} + 7 q^{19} + 3 q^{21} + 9 q^{23} + 2 q^{25} + 3 q^{27} - q^{29} + 7 q^{31} + 2 q^{33} - 3 q^{35} + 12 q^{37} - 3 q^{39}+ \cdots + 2 q^{99}+O(q^{100})$ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 3 * q^15 - 8 * q^17 + 7 * q^19 + 3 * q^21 + 9 * q^23 + 2 * q^25 + 3 * q^27 - q^29 + 7 * q^31 + 2 * q^33 - 3 * q^35 + 12 * q^37 - 3 * q^39 - 10 * q^41 + q^43 - 3 * q^45 + 17 * q^47 + 3 * q^49 - 8 * q^51 - 5 * q^53 + 18 * q^55 + 7 * q^57 + 12 * q^59 + 10 * q^61 + 3 * q^63 + 3 * q^65 - 2 * q^67 + 9 * q^69 + 4 * q^71 - 5 * q^73 + 2 * q^75 + 2 * q^77 + 13 * q^79 + 3 * q^81 + q^83 + 16 * q^85 - q^87 - 13 * q^89 - 3 * q^91 + 7 * q^93 + 15 * q^95 - 9 * 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
$5$	−1.28917	−0.576534	−0.288267	−	0.957550i	$-0.593079\pi$
$5$	−1.28917	−0.576534	−0.288267	+	0.957550i	$0.593079\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.20555	−1.26802	−0.634011	−	0.773324i	$-0.718592\pi$
$11$	−4.20555	−1.26802	−0.634011	+	0.773324i	$0.718592\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.28917	−0.332862
$16$	0	0
$17$	1.62721	0.394657	0.197329	−	0.980337i	$-0.436773\pi$
$17$	1.62721	0.394657	0.197329	+	0.980337i	$0.436773\pi$
$18$	0	0
$19$	6.33804	1.45405	0.727024	−	0.686613i	$-0.240903\pi$
$19$	6.33804	1.45405	0.727024	+	0.686613i	$0.240903\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	2.71083	0.565247	0.282624	−	0.959231i	$-0.408795\pi$
$23$	2.71083	0.565247	0.282624	+	0.959231i	$0.408795\pi$
$24$	0	0
$25$	−3.33804	−0.667609
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.33804	−0.805555	−0.402777	−	0.915298i	$-0.631955\pi$
$29$	−4.33804	−0.805555	−0.402777	+	0.915298i	$0.631955\pi$
$30$	0	0
$31$	7.49472	1.34609	0.673046	−	0.739601i	$-0.264986\pi$
$31$	7.49472	1.34609	0.673046	+	0.739601i	$0.264986\pi$
$32$	0	0
$33$	−4.20555	−0.732092
$34$	0	0
$35$	−1.28917	−0.217909
$36$	0	0
$37$	3.42166	0.562518	0.281259	−	0.959632i	$-0.409248\pi$
$37$	3.42166	0.562518	0.281259	+	0.959632i	$0.409248\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−7.62721	−1.19117	−0.595585	−	0.803292i	$-0.703080\pi$
$41$	−7.62721	−1.19117	−0.595585	+	0.803292i	$0.703080\pi$
$42$	0	0
$43$	4.91638	0.749741	0.374871	−	0.927077i	$-0.377687\pi$
$43$	4.91638	0.749741	0.374871	+	0.927077i	$0.377687\pi$
$44$	0	0
$45$	−1.28917	−0.192178
$46$	0	0
$47$	1.08362	0.158062	0.0790310	−	0.996872i	$-0.474817\pi$
$47$	1.08362	0.158062	0.0790310	+	0.996872i	$0.474817\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.62721	0.227855
$52$	0	0
$53$	1.75971	0.241714	0.120857	−	0.992670i	$-0.461436\pi$
$53$	1.75971	0.241714	0.120857	+	0.992670i	$0.461436\pi$
$54$	0	0
$55$	5.42166	0.731057
$56$	0	0
$57$	6.33804	0.839494
$58$	0	0
$59$	4.57834	0.596049	0.298024	−	0.954558i	$-0.403672\pi$
$59$	4.57834	0.596049	0.298024	+	0.954558i	$0.403672\pi$
$60$	0	0
$61$	−5.25443	−0.672760	−0.336380	−	0.941726i	$-0.609203\pi$
$61$	−5.25443	−0.672760	−0.336380	+	0.941726i	$0.609203\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	1.28917	0.159902
$66$	0	0
$67$	−8.67609	−1.05995	−0.529976	−	0.848012i	$-0.677799\pi$
$67$	−8.67609	−1.05995	−0.529976	+	0.848012i	$0.677799\pi$
$68$	0	0
$69$	2.71083	0.326346
$70$	0	0
$71$	6.78389	0.805099	0.402550	−	0.915398i	$-0.368124\pi$
$71$	6.78389	0.805099	0.402550	+	0.915398i	$0.368124\pi$
$72$	0	0
$73$	4.07306	0.476715	0.238358	−	0.971177i	$-0.423391\pi$
$73$	4.07306	0.476715	0.238358	+	0.971177i	$0.423391\pi$
$74$	0	0
$75$	−3.33804	−0.385444
$76$	0	0
$77$	−4.20555	−0.479267
$78$	0	0
$79$	8.91638	1.00317	0.501586	−	0.865108i	$-0.332750\pi$
$79$	8.91638	1.00317	0.501586	+	0.865108i	$0.332750\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.33804	0.476162	0.238081	−	0.971245i	$-0.423482\pi$
$83$	4.33804	0.476162	0.238081	+	0.971245i	$0.423482\pi$
$84$	0	0
$85$	−2.09775	−0.227533
$86$	0	0
$87$	−4.33804	−0.465087
$88$	0	0
$89$	4.54359	0.481620	0.240810	−	0.970572i	$-0.422587\pi$
$89$	4.54359	0.481620	0.240810	+	0.970572i	$0.422587\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	7.49472	0.777166
$94$	0	0
$95$	−8.17081	−0.838307
$96$	0	0
$97$	11.3275	1.15013	0.575066	−	0.818107i	$-0.304976\pi$
$97$	11.3275	1.15013	0.575066	+	0.818107i	$0.304976\pi$
$98$	0	0
$99$	−4.20555	−0.422674
$100$	0	0
$101$	14.3033	1.42323	0.711616	−	0.702569i	$-0.247964\pi$
$101$	14.3033	1.42323	0.711616	+	0.702569i	$0.247964\pi$
$102$	0	0
$103$	19.6655	1.93770	0.968851	−	0.247645i	$-0.0796565\pi$
$103$	19.6655	1.93770	0.968851	+	0.247645i	$0.0796565\pi$
$104$	0	0
$105$	−1.28917	−0.125810
$106$	0	0
$107$	−2.20555	−0.213219	−0.106609	−	0.994301i	$-0.533999\pi$
$107$	−2.20555	−0.213219	−0.106609	+	0.994301i	$0.533999\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	3.42166	0.324770
$112$	0	0
$113$	−2.17081	−0.204212	−0.102106	−	0.994774i	$-0.532558\pi$
$113$	−2.17081	−0.204212	−0.102106	+	0.994774i	$0.532558\pi$
$114$	0	0
$115$	−3.49472	−0.325884
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	1.62721	0.149166
$120$	0	0
$121$	6.68665	0.607877
$122$	0	0
$123$	−7.62721	−0.687723
$124$	0	0
$125$	10.7491	0.961433
$126$	0	0
$127$	0.745574	0.0661590	0.0330795	−	0.999453i	$-0.489469\pi$
$127$	0.745574	0.0661590	0.0330795	+	0.999453i	$0.489469\pi$
$128$	0	0
$129$	4.91638	0.432863
$130$	0	0
$131$	14.5089	1.26764	0.633822	−	0.773479i	$-0.281485\pi$
$131$	14.5089	1.26764	0.633822	+	0.773479i	$0.281485\pi$
$132$	0	0
$133$	6.33804	0.549578
$134$	0	0
$135$	−1.28917	−0.110954
$136$	0	0
$137$	4.41110	0.376866	0.188433	−	0.982086i	$-0.439659\pi$
$137$	4.41110	0.376866	0.188433	+	0.982086i	$0.439659\pi$
$138$	0	0
$139$	−18.9894	−1.61066	−0.805332	−	0.592825i	$-0.798013\pi$
$139$	−18.9894	−1.61066	−0.805332	+	0.592825i	$0.798013\pi$
$140$	0	0
$141$	1.08362	0.0912571
$142$	0	0
$143$	4.20555	0.351686
$144$	0	0
$145$	5.59247	0.464429
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−9.42166	−0.771853	−0.385926	−	0.922530i	$-0.626118\pi$
$149$	−9.42166	−0.771853	−0.385926	+	0.922530i	$0.626118\pi$
$150$	0	0
$151$	16.4111	1.33552	0.667758	−	0.744378i	$-0.267254\pi$
$151$	16.4111	1.33552	0.667758	+	0.744378i	$0.267254\pi$
$152$	0	0
$153$	1.62721	0.131552
$154$	0	0
$155$	−9.66196	−0.776067
$156$	0	0
$157$	−9.51941	−0.759732	−0.379866	−	0.925042i	$-0.624030\pi$
$157$	−9.51941	−0.759732	−0.379866	+	0.925042i	$0.624030\pi$
$158$	0	0
$159$	1.75971	0.139554
$160$	0	0
$161$	2.71083	0.213643
$162$	0	0
$163$	0.745574	0.0583979	0.0291989	−	0.999574i	$-0.490704\pi$
$163$	0.745574	0.0583979	0.0291989	+	0.999574i	$0.490704\pi$
$164$	0	0
$165$	5.42166	0.422076
$166$	0	0
$167$	−3.59247	−0.277994	−0.138997	−	0.990293i	$-0.544388\pi$
$167$	−3.59247	−0.277994	−0.138997	+	0.990293i	$0.544388\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.33804	0.484682
$172$	0	0
$173$	−15.7250	−1.19555	−0.597773	−	0.801665i	$-0.703948\pi$
$173$	−15.7250	−1.19555	−0.597773	+	0.801665i	$0.703948\pi$
$174$	0	0
$175$	−3.33804	−0.252332
$176$	0	0
$177$	4.57834	0.344129
$178$	0	0
$179$	−24.9547	−1.86520	−0.932601	−	0.360910i	$-0.882466\pi$
$179$	−24.9547	−1.86520	−0.932601	+	0.360910i	$0.882466\pi$
$180$	0	0
$181$	10.9411	0.813244	0.406622	−	0.913597i	$-0.366707\pi$
$181$	10.9411	0.813244	0.406622	+	0.913597i	$0.366707\pi$
$182$	0	0
$183$	−5.25443	−0.388418
$184$	0	0
$185$	−4.41110	−0.324311
$186$	0	0
$187$	−6.84333	−0.500434
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−5.04888	−0.365324	−0.182662	−	0.983176i	$-0.558471\pi$
$191$	−5.04888	−0.365324	−0.182662	+	0.983176i	$0.558471\pi$
$192$	0	0
$193$	2.74557	0.197631	0.0988154	−	0.995106i	$-0.468495\pi$
$193$	2.74557	0.197631	0.0988154	+	0.995106i	$0.468495\pi$
$194$	0	0
$195$	1.28917	0.0923193
$196$	0	0
$197$	5.42166	0.386277	0.193139	−	0.981171i	$-0.438133\pi$
$197$	5.42166	0.386277	0.193139	+	0.981171i	$0.438133\pi$
$198$	0	0
$199$	20.4111	1.44690	0.723452	−	0.690374i	$-0.242554\pi$
$199$	20.4111	1.44690	0.723452	+	0.690374i	$0.242554\pi$
$200$	0	0
$201$	−8.67609	−0.611964
$202$	0	0
$203$	−4.33804	−0.304471
$204$	0	0
$205$	9.83276	0.686750
$206$	0	0
$207$	2.71083	0.188416
$208$	0	0
$209$	−26.6550	−1.84376
$210$	0	0
$211$	14.4842	0.997130	0.498565	−	0.866852i	$-0.333860\pi$
$211$	14.4842	0.997130	0.498565	+	0.866852i	$0.333860\pi$
$212$	0	0
$213$	6.78389	0.464824
$214$	0	0
$215$	−6.33804	−0.432251
$216$	0	0
$217$	7.49472	0.508775
$218$	0	0
$219$	4.07306	0.275232
$220$	0	0
$221$	−1.62721	−0.109458
$222$	0	0
$223$	−16.8469	−1.12815	−0.564076	−	0.825723i	$-0.690767\pi$
$223$	−16.8469	−1.12815	−0.564076	+	0.825723i	$0.690767\pi$
$224$	0	0
$225$	−3.33804	−0.222536
$226$	0	0
$227$	21.9305	1.45558	0.727790	−	0.685800i	$-0.240548\pi$
$227$	21.9305	1.45558	0.727790	+	0.685800i	$0.240548\pi$
$228$	0	0
$229$	−5.25443	−0.347222	−0.173611	−	0.984814i	$-0.555544\pi$
$229$	−5.25443	−0.347222	−0.173611	+	0.984814i	$0.555544\pi$
$230$	0	0
$231$	−4.20555	−0.276705
$232$	0	0
$233$	4.07306	0.266835	0.133417	−	0.991060i	$-0.457405\pi$
$233$	4.07306	0.266835	0.133417	+	0.991060i	$0.457405\pi$
$234$	0	0
$235$	−1.39697	−0.0911281
$236$	0	0
$237$	8.91638	0.579181
$238$	0	0
$239$	2.37279	0.153483	0.0767414	−	0.997051i	$-0.475548\pi$
$239$	2.37279	0.153483	0.0767414	+	0.997051i	$0.475548\pi$
$240$	0	0
$241$	14.9164	0.960849	0.480424	−	0.877036i	$-0.340483\pi$
$241$	14.9164	0.960849	0.480424	+	0.877036i	$0.340483\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.28917	−0.0823620
$246$	0	0
$247$	−6.33804	−0.403280
$248$	0	0
$249$	4.33804	0.274912
$250$	0	0
$251$	1.08719	0.0686228	0.0343114	−	0.999411i	$-0.489076\pi$
$251$	1.08719	0.0686228	0.0343114	+	0.999411i	$0.489076\pi$
$252$	0	0
$253$	−11.4005	−0.716746
$254$	0	0
$255$	−2.09775	−0.131366
$256$	0	0
$257$	−16.6167	−1.03652	−0.518259	−	0.855224i	$-0.673420\pi$
$257$	−16.6167	−1.03652	−0.518259	+	0.855224i	$0.673420\pi$
$258$	0	0
$259$	3.42166	0.212612
$260$	0	0
$261$	−4.33804	−0.268518
$262$	0	0
$263$	27.0524	1.66813	0.834063	−	0.551670i	$-0.186009\pi$
$263$	27.0524	1.66813	0.834063	+	0.551670i	$0.186009\pi$
$264$	0	0
$265$	−2.26856	−0.139356
$266$	0	0
$267$	4.54359	0.278063
$268$	0	0
$269$	7.52946	0.459079	0.229540	−	0.973299i	$-0.426278\pi$
$269$	7.52946	0.459079	0.229540	+	0.973299i	$0.426278\pi$
$270$	0	0
$271$	−16.8222	−1.02188	−0.510938	−	0.859618i	$-0.670702\pi$
$271$	−16.8222	−1.02188	−0.510938	+	0.859618i	$0.670702\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	14.0383	0.846542
$276$	0	0
$277$	17.4947	1.05116	0.525578	−	0.850745i	$-0.323849\pi$
$277$	17.4947	1.05116	0.525578	+	0.850745i	$0.323849\pi$
$278$	0	0
$279$	7.49472	0.448697
$280$	0	0
$281$	27.4005	1.63458	0.817290	−	0.576227i	$-0.195476\pi$
$281$	27.4005	1.63458	0.817290	+	0.576227i	$0.195476\pi$
$282$	0	0
$283$	−20.0766	−1.19343	−0.596716	−	0.802453i	$-0.703528\pi$
$283$	−20.0766	−1.19343	−0.596716	+	0.802453i	$0.703528\pi$
$284$	0	0
$285$	−8.17081	−0.483997
$286$	0	0
$287$	−7.62721	−0.450220
$288$	0	0
$289$	−14.3522	−0.844246
$290$	0	0
$291$	11.3275	0.664029
$292$	0	0
$293$	−27.0524	−1.58042	−0.790210	−	0.612836i	$-0.790029\pi$
$293$	−27.0524	−1.58042	−0.790210	+	0.612836i	$0.790029\pi$
$294$	0	0
$295$	−5.90225	−0.343642
$296$	0	0
$297$	−4.20555	−0.244031
$298$	0	0
$299$	−2.71083	−0.156771
$300$	0	0
$301$	4.91638	0.283376
$302$	0	0
$303$	14.3033	0.821703
$304$	0	0
$305$	6.77384	0.387869
$306$	0	0
$307$	20.1708	1.15121	0.575604	−	0.817728i	$-0.304767\pi$
$307$	20.1708	1.15121	0.575604	+	0.817728i	$0.304767\pi$
$308$	0	0
$309$	19.6655	1.11873
$310$	0	0
$311$	6.57834	0.373023	0.186512	−	0.982453i	$-0.440282\pi$
$311$	6.57834	0.373023	0.186512	+	0.982453i	$0.440282\pi$
$312$	0	0
$313$	−10.7456	−0.607376	−0.303688	−	0.952772i	$-0.598218\pi$
$313$	−10.7456	−0.607376	−0.303688	+	0.952772i	$0.598218\pi$
$314$	0	0
$315$	−1.28917	−0.0726364
$316$	0	0
$317$	24.6066	1.38204	0.691022	−	0.722833i	$-0.257161\pi$
$317$	24.6066	1.38204	0.691022	+	0.722833i	$0.257161\pi$
$318$	0	0
$319$	18.2439	1.02146
$320$	0	0
$321$	−2.20555	−0.123102
$322$	0	0
$323$	10.3133	0.573850
$324$	0	0
$325$	3.33804	0.185161
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	1.08362	0.0597418
$330$	0	0
$331$	−30.6550	−1.68495	−0.842475	−	0.538736i	$-0.818902\pi$
$331$	−30.6550	−1.68495	−0.842475	+	0.538736i	$0.818902\pi$
$332$	0	0
$333$	3.42166	0.187506
$334$	0	0
$335$	11.1849	0.611099
$336$	0	0
$337$	−6.50528	−0.354365	−0.177183	−	0.984178i	$-0.556698\pi$
$337$	−6.50528	−0.354365	−0.177183	+	0.984178i	$0.556698\pi$
$338$	0	0
$339$	−2.17081	−0.117902
$340$	0	0
$341$	−31.5194	−1.70687
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.49472	−0.188149
$346$	0	0
$347$	−24.3033	−1.30467	−0.652335	−	0.757931i	$-0.726210\pi$
$347$	−24.3033	−1.30467	−0.652335	+	0.757931i	$0.726210\pi$
$348$	0	0
$349$	17.7597	0.950655	0.475328	−	0.879809i	$-0.342330\pi$
$349$	17.7597	0.950655	0.475328	+	0.879809i	$0.342330\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	1.94056	0.103286	0.0516428	−	0.998666i	$-0.483554\pi$
$353$	1.94056	0.103286	0.0516428	+	0.998666i	$0.483554\pi$
$354$	0	0
$355$	−8.74557	−0.464167
$356$	0	0
$357$	1.62721	0.0861212
$358$	0	0
$359$	33.6272	1.77478	0.887388	−	0.461023i	$-0.152517\pi$
$359$	33.6272	1.77478	0.887388	+	0.461023i	$0.152517\pi$
$360$	0	0
$361$	21.1708	1.11425
$362$	0	0
$363$	6.68665	0.350958
$364$	0	0
$365$	−5.25086	−0.274842
$366$	0	0
$367$	10.5783	0.552185	0.276092	−	0.961131i	$-0.410960\pi$
$367$	10.5783	0.552185	0.276092	+	0.961131i	$0.410960\pi$
$368$	0	0
$369$	−7.62721	−0.397057
$370$	0	0
$371$	1.75971	0.0913595
$372$	0	0
$373$	−5.47002	−0.283227	−0.141614	−	0.989922i	$-0.545229\pi$
$373$	−5.47002	−0.283227	−0.141614	+	0.989922i	$0.545229\pi$
$374$	0	0
$375$	10.7491	0.555083
$376$	0	0
$377$	4.33804	0.223421
$378$	0	0
$379$	5.42166	0.278492	0.139246	−	0.990258i	$-0.455532\pi$
$379$	5.42166	0.278492	0.139246	+	0.990258i	$0.455532\pi$
$380$	0	0
$381$	0.745574	0.0381969
$382$	0	0
$383$	−14.1461	−0.722833	−0.361416	−	0.932405i	$-0.617707\pi$
$383$	−14.1461	−0.722833	−0.361416	+	0.932405i	$0.617707\pi$
$384$	0	0
$385$	5.42166	0.276314
$386$	0	0
$387$	4.91638	0.249914
$388$	0	0
$389$	0.313348	0.0158874	0.00794370	−	0.999968i	$-0.497471\pi$
$389$	0.313348	0.0158874	0.00794370	+	0.999968i	$0.497471\pi$
$390$	0	0
$391$	4.41110	0.223079
$392$	0	0
$393$	14.5089	0.731875
$394$	0	0
$395$	−11.4947	−0.578362
$396$	0	0
$397$	20.6797	1.03788	0.518941	−	0.854810i	$-0.326326\pi$
$397$	20.6797	1.03788	0.518941	+	0.854810i	$0.326326\pi$
$398$	0	0
$399$	6.33804	0.317299
$400$	0	0
$401$	−12.7456	−0.636484	−0.318242	−	0.948010i	$-0.603092\pi$
$401$	−12.7456	−0.636484	−0.318242	+	0.948010i	$0.603092\pi$
$402$	0	0
$403$	−7.49472	−0.373339
$404$	0	0
$405$	−1.28917	−0.0640593
$406$	0	0
$407$	−14.3900	−0.713285
$408$	0	0
$409$	−18.6514	−0.922252	−0.461126	−	0.887335i	$-0.652554\pi$
$409$	−18.6514	−0.922252	−0.461126	+	0.887335i	$0.652554\pi$
$410$	0	0
$411$	4.41110	0.217584
$412$	0	0
$413$	4.57834	0.225285
$414$	0	0
$415$	−5.59247	−0.274524
$416$	0	0
$417$	−18.9894	−0.929917
$418$	0	0
$419$	−3.10831	−0.151851	−0.0759256	−	0.997113i	$-0.524191\pi$
$419$	−3.10831	−0.151851	−0.0759256	+	0.997113i	$0.524191\pi$
$420$	0	0
$421$	13.4005	0.653102	0.326551	−	0.945180i	$-0.394113\pi$
$421$	13.4005	0.653102	0.326551	+	0.945180i	$0.394113\pi$
$422$	0	0
$423$	1.08362	0.0526873
$424$	0	0
$425$	−5.43171	−0.263477
$426$	0	0
$427$	−5.25443	−0.254279
$428$	0	0
$429$	4.20555	0.203046
$430$	0	0
$431$	−9.89220	−0.476491	−0.238245	−	0.971205i	$-0.576572\pi$
$431$	−9.89220	−0.476491	−0.238245	+	0.971205i	$0.576572\pi$
$432$	0	0
$433$	0.578337	0.0277931	0.0138966	−	0.999903i	$-0.495576\pi$
$433$	0.578337	0.0277931	0.0138966	+	0.999903i	$0.495576\pi$
$434$	0	0
$435$	5.59247	0.268138
$436$	0	0
$437$	17.1814	0.821896
$438$	0	0
$439$	−23.7350	−1.13281	−0.566405	−	0.824127i	$-0.691666\pi$
$439$	−23.7350	−1.13281	−0.566405	+	0.824127i	$0.691666\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	3.86751	0.183751	0.0918754	−	0.995771i	$-0.470714\pi$
$443$	3.86751	0.183751	0.0918754	+	0.995771i	$0.470714\pi$
$444$	0	0
$445$	−5.85746	−0.277670
$446$	0	0
$447$	−9.42166	−0.445629
$448$	0	0
$449$	1.68665	0.0795980	0.0397990	−	0.999208i	$-0.487328\pi$
$449$	1.68665	0.0795980	0.0397990	+	0.999208i	$0.487328\pi$
$450$	0	0
$451$	32.0766	1.51043
$452$	0	0
$453$	16.4111	0.771061
$454$	0	0
$455$	1.28917	0.0604372
$456$	0	0
$457$	−3.83276	−0.179289	−0.0896445	−	0.995974i	$-0.528573\pi$
$457$	−3.83276	−0.179289	−0.0896445	+	0.995974i	$0.528573\pi$
$458$	0	0
$459$	1.62721	0.0759518
$460$	0	0
$461$	−11.2161	−0.522386	−0.261193	−	0.965287i	$-0.584116\pi$
$461$	−11.2161	−0.522386	−0.261193	+	0.965287i	$0.584116\pi$
$462$	0	0
$463$	25.8328	1.20055	0.600275	−	0.799794i	$-0.295058\pi$
$463$	25.8328	1.20055	0.600275	+	0.799794i	$0.295058\pi$
$464$	0	0
$465$	−9.66196	−0.448062
$466$	0	0
$467$	−41.0872	−1.90129	−0.950644	−	0.310283i	$-0.899576\pi$
$467$	−41.0872	−1.90129	−0.950644	+	0.310283i	$0.899576\pi$
$468$	0	0
$469$	−8.67609	−0.400625
$470$	0	0
$471$	−9.51941	−0.438631
$472$	0	0
$473$	−20.6761	−0.950688
$474$	0	0
$475$	−21.1567	−0.970735
$476$	0	0
$477$	1.75971	0.0805715
$478$	0	0
$479$	−2.43580	−0.111294	−0.0556472	−	0.998450i	$-0.517722\pi$
$479$	−2.43580	−0.111294	−0.0556472	+	0.998450i	$0.517722\pi$
$480$	0	0
$481$	−3.42166	−0.156014
$482$	0	0
$483$	2.71083	0.123347
$484$	0	0
$485$	−14.6030	−0.663090
$486$	0	0
$487$	6.57834	0.298093	0.149046	−	0.988830i	$-0.452380\pi$
$487$	6.57834	0.298093	0.149046	+	0.988830i	$0.452380\pi$
$488$	0	0
$489$	0.745574	0.0337160
$490$	0	0
$491$	−28.3033	−1.27731	−0.638655	−	0.769493i	$-0.720509\pi$
$491$	−28.3033	−1.27731	−0.638655	+	0.769493i	$0.720509\pi$
$492$	0	0
$493$	−7.05892	−0.317918
$494$	0	0
$495$	5.42166	0.243686
$496$	0	0
$497$	6.78389	0.304299
$498$	0	0
$499$	−28.8222	−1.29026	−0.645129	−	0.764073i	$-0.723197\pi$
$499$	−28.8222	−1.29026	−0.645129	+	0.764073i	$0.723197\pi$
$500$	0	0
$501$	−3.59247	−0.160500
$502$	0	0
$503$	−8.67609	−0.386848	−0.193424	−	0.981115i	$-0.561959\pi$
$503$	−8.67609	−0.386848	−0.193424	+	0.981115i	$0.561959\pi$
$504$	0	0
$505$	−18.4394	−0.820541
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−9.28917	−0.411735	−0.205868	−	0.978580i	$-0.566002\pi$
$509$	−9.28917	−0.411735	−0.205868	+	0.978580i	$0.566002\pi$
$510$	0	0
$511$	4.07306	0.180181
$512$	0	0
$513$	6.33804	0.279831
$514$	0	0
$515$	−25.3522	−1.11715
$516$	0	0
$517$	−4.55721	−0.200426
$518$	0	0
$519$	−15.7250	−0.690249
$520$	0	0
$521$	−11.7250	−0.513680	−0.256840	−	0.966454i	$-0.582681\pi$
$521$	−11.7250	−0.513680	−0.256840	+	0.966454i	$0.582681\pi$
$522$	0	0
$523$	−0.676089	−0.0295633	−0.0147817	−	0.999891i	$-0.504705\pi$
$523$	−0.676089	−0.0295633	−0.0147817	+	0.999891i	$0.504705\pi$
$524$	0	0
$525$	−3.33804	−0.145684
$526$	0	0
$527$	12.1955	0.531244
$528$	0	0
$529$	−15.6514	−0.680495
$530$	0	0
$531$	4.57834	0.198683
$532$	0	0
$533$	7.62721	0.330371
$534$	0	0
$535$	2.84333	0.122928
$536$	0	0
$537$	−24.9547	−1.07687
$538$	0	0
$539$	−4.20555	−0.181146
$540$	0	0
$541$	−0.578337	−0.0248647	−0.0124323	−	0.999923i	$-0.503957\pi$
$541$	−0.578337	−0.0248647	−0.0124323	+	0.999923i	$0.503957\pi$
$542$	0	0
$543$	10.9411	0.469527
$544$	0	0
$545$	−12.8917	−0.552219
$546$	0	0
$547$	0.985867	0.0421526	0.0210763	−	0.999778i	$-0.493291\pi$
$547$	0.985867	0.0421526	0.0210763	+	0.999778i	$0.493291\pi$
$548$	0	0
$549$	−5.25443	−0.224253
$550$	0	0
$551$	−27.4947	−1.17131
$552$	0	0
$553$	8.91638	0.379163
$554$	0	0
$555$	−4.41110	−0.187241
$556$	0	0
$557$	41.2333	1.74711	0.873556	−	0.486725i	$-0.161808\pi$
$557$	41.2333	1.74711	0.873556	+	0.486725i	$0.161808\pi$
$558$	0	0
$559$	−4.91638	−0.207941
$560$	0	0
$561$	−6.84333	−0.288925
$562$	0	0
$563$	−6.91995	−0.291641	−0.145821	−	0.989311i	$-0.546582\pi$
$563$	−6.91995	−0.291641	−0.145821	+	0.989311i	$0.546582\pi$
$564$	0	0
$565$	2.79854	0.117735
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−29.5713	−1.23970	−0.619848	−	0.784722i	$-0.712806\pi$
$569$	−29.5713	−1.23970	−0.619848	+	0.784722i	$0.712806\pi$
$570$	0	0
$571$	19.4252	0.812921	0.406460	−	0.913668i	$-0.366763\pi$
$571$	19.4252	0.812921	0.406460	+	0.913668i	$0.366763\pi$
$572$	0	0
$573$	−5.04888	−0.210920
$574$	0	0
$575$	−9.04888	−0.377364
$576$	0	0
$577$	−38.8222	−1.61619	−0.808095	−	0.589053i	$-0.799501\pi$
$577$	−38.8222	−1.61619	−0.808095	+	0.589053i	$0.799501\pi$
$578$	0	0
$579$	2.74557	0.114102
$580$	0	0
$581$	4.33804	0.179972
$582$	0	0
$583$	−7.40054	−0.306499
$584$	0	0
$585$	1.28917	0.0533006
$586$	0	0
$587$	−24.0731	−0.993601	−0.496801	−	0.867865i	$-0.665492\pi$
$587$	−24.0731	−0.993601	−0.496801	+	0.867865i	$0.665492\pi$
$588$	0	0
$589$	47.5019	1.95728
$590$	0	0
$591$	5.42166	0.223017
$592$	0	0
$593$	40.4741	1.66207	0.831036	−	0.556218i	$-0.187748\pi$
$593$	40.4741	1.66207	0.831036	+	0.556218i	$0.187748\pi$
$594$	0	0
$595$	−2.09775	−0.0859994
$596$	0	0
$597$	20.4111	0.835371
$598$	0	0
$599$	2.49523	0.101953	0.0509763	−	0.998700i	$-0.483767\pi$
$599$	2.49523	0.101953	0.0509763	+	0.998700i	$0.483767\pi$
$600$	0	0
$601$	35.2333	1.43720	0.718598	−	0.695426i	$-0.244784\pi$
$601$	35.2333	1.43720	0.718598	+	0.695426i	$0.244784\pi$
$602$	0	0
$603$	−8.67609	−0.353318
$604$	0	0
$605$	−8.62022	−0.350462
$606$	0	0
$607$	−4.89169	−0.198547	−0.0992737	−	0.995060i	$-0.531652\pi$
$607$	−4.89169	−0.198547	−0.0992737	+	0.995060i	$0.531652\pi$
$608$	0	0
$609$	−4.33804	−0.175786
$610$	0	0
$611$	−1.08362	−0.0438385
$612$	0	0
$613$	7.83276	0.316362	0.158181	−	0.987410i	$-0.449437\pi$
$613$	7.83276	0.316362	0.158181	+	0.987410i	$0.449437\pi$
$614$	0	0
$615$	9.83276	0.396495
$616$	0	0
$617$	15.7350	0.633468	0.316734	−	0.948514i	$-0.397414\pi$
$617$	15.7350	0.633468	0.316734	+	0.948514i	$0.397414\pi$
$618$	0	0
$619$	41.0177	1.64864	0.824320	−	0.566124i	$-0.191558\pi$
$619$	41.0177	1.64864	0.824320	+	0.566124i	$0.191558\pi$
$620$	0	0
$621$	2.71083	0.108782
$622$	0	0
$623$	4.54359	0.182035
$624$	0	0
$625$	2.83276	0.113311
$626$	0	0
$627$	−26.6550	−1.06450
$628$	0	0
$629$	5.56777	0.222002
$630$	0	0
$631$	19.5960	0.780106	0.390053	−	0.920792i	$-0.372457\pi$
$631$	19.5960	0.780106	0.390053	+	0.920792i	$0.372457\pi$
$632$	0	0
$633$	14.4842	0.575694
$634$	0	0
$635$	−0.961171	−0.0381429
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	6.78389	0.268366
$640$	0	0
$641$	−20.8953	−0.825313	−0.412656	−	0.910887i	$-0.635399\pi$
$641$	−20.8953	−0.825313	−0.412656	+	0.910887i	$0.635399\pi$
$642$	0	0
$643$	9.15667	0.361104	0.180552	−	0.983565i	$-0.442212\pi$
$643$	9.15667	0.361104	0.180552	+	0.983565i	$0.442212\pi$
$644$	0	0
$645$	−6.33804	−0.249560
$646$	0	0
$647$	6.24386	0.245472	0.122736	−	0.992439i	$-0.460833\pi$
$647$	6.24386	0.245472	0.122736	+	0.992439i	$0.460833\pi$
$648$	0	0
$649$	−19.2544	−0.755802
$650$	0	0
$651$	7.49472	0.293741
$652$	0	0
$653$	−38.8222	−1.51923	−0.759615	−	0.650373i	$-0.774613\pi$
$653$	−38.8222	−1.51923	−0.759615	+	0.650373i	$0.774613\pi$
$654$	0	0
$655$	−18.7044	−0.730840
$656$	0	0
$657$	4.07306	0.158905
$658$	0	0
$659$	0.954695	0.0371896	0.0185948	−	0.999827i	$-0.494081\pi$
$659$	0.954695	0.0371896	0.0185948	+	0.999827i	$0.494081\pi$
$660$	0	0
$661$	2.28968	0.0890584	0.0445292	−	0.999008i	$-0.485821\pi$
$661$	2.28968	0.0890584	0.0445292	+	0.999008i	$0.485821\pi$
$662$	0	0
$663$	−1.62721	−0.0631957
$664$	0	0
$665$	−8.17081	−0.316850
$666$	0	0
$667$	−11.7597	−0.455338
$668$	0	0
$669$	−16.8469	−0.651339
$670$	0	0
$671$	22.0978	0.853074
$672$	0	0
$673$	−44.6797	−1.72227	−0.861137	−	0.508373i	$-0.830247\pi$
$673$	−44.6797	−1.72227	−0.861137	+	0.508373i	$0.830247\pi$
$674$	0	0
$675$	−3.33804	−0.128481
$676$	0	0
$677$	−37.0278	−1.42309	−0.711546	−	0.702639i	$-0.752005\pi$
$677$	−37.0278	−1.42309	−0.711546	+	0.702639i	$0.752005\pi$
$678$	0	0
$679$	11.3275	0.434709
$680$	0	0
$681$	21.9305	0.840379
$682$	0	0
$683$	26.8605	1.02779	0.513894	−	0.857853i	$-0.328202\pi$
$683$	26.8605	1.02779	0.513894	+	0.857853i	$0.328202\pi$
$684$	0	0
$685$	−5.68665	−0.217276
$686$	0	0
$687$	−5.25443	−0.200469
$688$	0	0
$689$	−1.75971	−0.0670395
$690$	0	0
$691$	−2.86802	−0.109105	−0.0545523	−	0.998511i	$-0.517373\pi$
$691$	−2.86802	−0.109105	−0.0545523	+	0.998511i	$0.517373\pi$
$692$	0	0
$693$	−4.20555	−0.159756
$694$	0	0
$695$	24.4806	0.928602
$696$	0	0
$697$	−12.4111	−0.470104
$698$	0	0
$699$	4.07306	0.154057
$700$	0	0
$701$	−42.9930	−1.62382	−0.811912	−	0.583780i	$-0.801573\pi$
$701$	−42.9930	−1.62382	−0.811912	+	0.583780i	$0.801573\pi$
$702$	0	0
$703$	21.6867	0.817928
$704$	0	0
$705$	−1.39697	−0.0526128
$706$	0	0
$707$	14.3033	0.537931
$708$	0	0
$709$	3.49115	0.131113	0.0655564	−	0.997849i	$-0.479118\pi$
$709$	3.49115	0.131113	0.0655564	+	0.997849i	$0.479118\pi$
$710$	0	0
$711$	8.91638	0.334390
$712$	0	0
$713$	20.3169	0.760875
$714$	0	0
$715$	−5.42166	−0.202759
$716$	0	0
$717$	2.37279	0.0886134
$718$	0	0
$719$	−27.5194	−1.02630	−0.513150	−	0.858299i	$-0.671522\pi$
$719$	−27.5194	−1.02630	−0.513150	+	0.858299i	$0.671522\pi$
$720$	0	0
$721$	19.6655	0.732382
$722$	0	0
$723$	14.9164	0.554746
$724$	0	0
$725$	14.4806	0.537795
$726$	0	0
$727$	−13.4983	−0.500624	−0.250312	−	0.968165i	$-0.580533\pi$
$727$	−13.4983	−0.500624	−0.250312	+	0.968165i	$0.580533\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−5.08362	−0.187768	−0.0938839	−	0.995583i	$-0.529928\pi$
$733$	−5.08362	−0.187768	−0.0938839	+	0.995583i	$0.529928\pi$
$734$	0	0
$735$	−1.28917	−0.0475517
$736$	0	0
$737$	36.4877	1.34404
$738$	0	0
$739$	53.1638	1.95566	0.977831	−	0.209394i	$-0.0671492\pi$
$739$	53.1638	1.95566	0.977831	+	0.209394i	$0.0671492\pi$
$740$	0	0
$741$	−6.33804	−0.232834
$742$	0	0
$743$	46.1149	1.69179	0.845897	−	0.533347i	$-0.179066\pi$
$743$	46.1149	1.69179	0.845897	+	0.533347i	$0.179066\pi$
$744$	0	0
$745$	12.1461	0.444999
$746$	0	0
$747$	4.33804	0.158721
$748$	0	0
$749$	−2.20555	−0.0805890
$750$	0	0
$751$	32.7774	1.19606	0.598032	−	0.801472i	$-0.295949\pi$
$751$	32.7774	1.19606	0.598032	+	0.801472i	$0.295949\pi$
$752$	0	0
$753$	1.08719	0.0396194
$754$	0	0
$755$	−21.1567	−0.769970
$756$	0	0
$757$	−30.1708	−1.09658	−0.548288	−	0.836289i	$-0.684720\pi$
$757$	−30.1708	−1.09658	−0.548288	+	0.836289i	$0.684720\pi$
$758$	0	0
$759$	−11.4005	−0.413813
$760$	0	0
$761$	−1.81915	−0.0659440	−0.0329720	−	0.999456i	$-0.510497\pi$
$761$	−1.81915	−0.0659440	−0.0329720	+	0.999456i	$0.510497\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	0	0
$765$	−2.09775	−0.0758444
$766$	0	0
$767$	−4.57834	−0.165314
$768$	0	0
$769$	−0.937507	−0.0338074	−0.0169037	−	0.999857i	$-0.505381\pi$
$769$	−0.937507	−0.0338074	−0.0169037	+	0.999857i	$0.505381\pi$
$770$	0	0
$771$	−16.6167	−0.598434
$772$	0	0
$773$	−8.03831	−0.289118	−0.144559	−	0.989496i	$-0.546176\pi$
$773$	−8.03831	−0.289118	−0.144559	+	0.989496i	$0.546176\pi$
$774$	0	0
$775$	−25.0177	−0.898662
$776$	0	0
$777$	3.42166	0.122751
$778$	0	0
$779$	−48.3416	−1.73202
$780$	0	0
$781$	−28.5300	−1.02088
$782$	0	0
$783$	−4.33804	−0.155029
$784$	0	0
$785$	12.2721	0.438011
$786$	0	0
$787$	−25.3275	−0.902827	−0.451414	−	0.892315i	$-0.649080\pi$
$787$	−25.3275	−0.902827	−0.451414	+	0.892315i	$0.649080\pi$
$788$	0	0
$789$	27.0524	0.963093
$790$	0	0
$791$	−2.17081	−0.0771850
$792$	0	0
$793$	5.25443	0.186590
$794$	0	0
$795$	−2.26856	−0.0804575
$796$	0	0
$797$	40.8122	1.44564	0.722820	−	0.691036i	$-0.242845\pi$
$797$	40.8122	1.44564	0.722820	+	0.691036i	$0.242845\pi$
$798$	0	0
$799$	1.76328	0.0623803
$800$	0	0
$801$	4.54359	0.160540
$802$	0	0
$803$	−17.1294	−0.604485
$804$	0	0
$805$	−3.49472	−0.123173
$806$	0	0
$807$	7.52946	0.265050
$808$	0	0
$809$	25.4947	0.896347	0.448173	−	0.893947i	$-0.352075\pi$
$809$	25.4947	0.896347	0.448173	+	0.893947i	$0.352075\pi$
$810$	0	0
$811$	13.1567	0.461993	0.230997	−	0.972955i	$-0.425801\pi$
$811$	13.1567	0.461993	0.230997	+	0.972955i	$0.425801\pi$
$812$	0	0
$813$	−16.8222	−0.589980
$814$	0	0
$815$	−0.961171	−0.0336683
$816$	0	0
$817$	31.1602	1.09016
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−23.1083	−0.806486	−0.403243	−	0.915093i	$-0.632117\pi$
$821$	−23.1083	−0.806486	−0.403243	+	0.915093i	$0.632117\pi$
$822$	0	0
$823$	17.4911	0.609703	0.304852	−	0.952400i	$-0.401393\pi$
$823$	17.4911	0.609703	0.304852	+	0.952400i	$0.401393\pi$
$824$	0	0
$825$	14.0383	0.488751
$826$	0	0
$827$	−39.3905	−1.36974	−0.684871	−	0.728665i	$-0.740141\pi$
$827$	−39.3905	−1.36974	−0.684871	+	0.728665i	$0.740141\pi$
$828$	0	0
$829$	10.8222	0.375871	0.187935	−	0.982181i	$-0.439820\pi$
$829$	10.8222	0.375871	0.187935	+	0.982181i	$0.439820\pi$
$830$	0	0
$831$	17.4947	0.606885
$832$	0	0
$833$	1.62721	0.0563796
$834$	0	0
$835$	4.63130	0.160273
$836$	0	0
$837$	7.49472	0.259055
$838$	0	0
$839$	−4.57834	−0.158062	−0.0790309	−	0.996872i	$-0.525183\pi$
$839$	−4.57834	−0.158062	−0.0790309	+	0.996872i	$0.525183\pi$
$840$	0	0
$841$	−10.1814	−0.351082
$842$	0	0
$843$	27.4005	0.943725
$844$	0	0
$845$	−1.28917	−0.0443487
$846$	0	0
$847$	6.68665	0.229756
$848$	0	0
$849$	−20.0766	−0.689028
$850$	0	0
$851$	9.27555	0.317962
$852$	0	0
$853$	−54.6585	−1.87147	−0.935736	−	0.352700i	$-0.885263\pi$
$853$	−54.6585	−1.87147	−0.935736	+	0.352700i	$0.885263\pi$
$854$	0	0
$855$	−8.17081	−0.279436
$856$	0	0
$857$	8.20555	0.280296	0.140148	−	0.990131i	$-0.455242\pi$
$857$	8.20555	0.280296	0.140148	+	0.990131i	$0.455242\pi$
$858$	0	0
$859$	3.20503	0.109354	0.0546772	−	0.998504i	$-0.482587\pi$
$859$	3.20503	0.109354	0.0546772	+	0.998504i	$0.482587\pi$
$860$	0	0
$861$	−7.62721	−0.259935
$862$	0	0
$863$	21.7038	0.738807	0.369404	−	0.929269i	$-0.379562\pi$
$863$	21.7038	0.738807	0.369404	+	0.929269i	$0.379562\pi$
$864$	0	0
$865$	20.2721	0.689273
$866$	0	0
$867$	−14.3522	−0.487426
$868$	0	0
$869$	−37.4983	−1.27204
$870$	0	0
$871$	8.67609	0.293978
$872$	0	0
$873$	11.3275	0.383377
$874$	0	0
$875$	10.7491	0.363387
$876$	0	0
$877$	−38.5371	−1.30131	−0.650653	−	0.759375i	$-0.725505\pi$
$877$	−38.5371	−1.30131	−0.650653	+	0.759375i	$0.725505\pi$
$878$	0	0
$879$	−27.0524	−0.912456
$880$	0	0
$881$	−20.2056	−0.680742	−0.340371	−	0.940291i	$-0.610553\pi$
$881$	−20.2056	−0.680742	−0.340371	+	0.940291i	$0.610553\pi$
$882$	0	0
$883$	−5.49115	−0.184792	−0.0923959	−	0.995722i	$-0.529453\pi$
$883$	−5.49115	−0.184792	−0.0923959	+	0.995722i	$0.529453\pi$
$884$	0	0
$885$	−5.90225	−0.198402
$886$	0	0
$887$	−45.7633	−1.53658	−0.768290	−	0.640102i	$-0.778892\pi$
$887$	−45.7633	−1.53658	−0.768290	+	0.640102i	$0.778892\pi$
$888$	0	0
$889$	0.745574	0.0250057
$890$	0	0
$891$	−4.20555	−0.140891
$892$	0	0
$893$	6.86802	0.229830
$894$	0	0
$895$	32.1708	1.07535
$896$	0	0
$897$	−2.71083	−0.0905120
$898$	0	0
$899$	−32.5124	−1.08435
$900$	0	0
$901$	2.86342	0.0953943
$902$	0	0
$903$	4.91638	0.163607
$904$	0	0
$905$	−14.1049	−0.468863
$906$	0	0
$907$	−6.67252	−0.221557	−0.110779	−	0.993845i	$-0.535334\pi$
$907$	−6.67252	−0.221557	−0.110779	+	0.993845i	$0.535334\pi$
$908$	0	0
$909$	14.3033	0.474411
$910$	0	0
$911$	−23.5330	−0.779684	−0.389842	−	0.920882i	$-0.627470\pi$
$911$	−23.5330	−0.779684	−0.389842	+	0.920882i	$0.627470\pi$
$912$	0	0
$913$	−18.2439	−0.603784
$914$	0	0
$915$	6.77384	0.223936
$916$	0	0
$917$	14.5089	0.479125
$918$	0	0
$919$	28.1955	0.930084	0.465042	−	0.885289i	$-0.346039\pi$
$919$	28.1955	0.930084	0.465042	+	0.885289i	$0.346039\pi$
$920$	0	0
$921$	20.1708	0.664651
$922$	0	0
$923$	−6.78389	−0.223294
$924$	0	0
$925$	−11.4217	−0.375542
$926$	0	0
$927$	19.6655	0.645901
$928$	0	0
$929$	−6.97582	−0.228869	−0.114435	−	0.993431i	$-0.536506\pi$
$929$	−6.97582	−0.228869	−0.114435	+	0.993431i	$0.536506\pi$
$930$	0	0
$931$	6.33804	0.207721
$932$	0	0
$933$	6.57834	0.215365
$934$	0	0
$935$	8.82220	0.288517
$936$	0	0
$937$	45.5960	1.48956	0.744779	−	0.667311i	$-0.232555\pi$
$937$	45.5960	1.48956	0.744779	+	0.667311i	$0.232555\pi$
$938$	0	0
$939$	−10.7456	−0.350669
$940$	0	0
$941$	−52.8086	−1.72151	−0.860755	−	0.509019i	$-0.830008\pi$
$941$	−52.8086	−1.72151	−0.860755	+	0.509019i	$0.830008\pi$
$942$	0	0
$943$	−20.6761	−0.673306
$944$	0	0
$945$	−1.28917	−0.0419367
$946$	0	0
$947$	−8.73553	−0.283867	−0.141933	−	0.989876i	$-0.545332\pi$
$947$	−8.73553	−0.283867	−0.141933	+	0.989876i	$0.545332\pi$
$948$	0	0
$949$	−4.07306	−0.132217
$950$	0	0
$951$	24.6066	0.797924
$952$	0	0
$953$	56.0036	1.81413	0.907067	−	0.420987i	$-0.138316\pi$
$953$	56.0036	1.81413	0.907067	+	0.420987i	$0.138316\pi$
$954$	0	0
$955$	6.50885	0.210622
$956$	0	0
$957$	18.2439	0.589740
$958$	0	0
$959$	4.41110	0.142442
$960$	0	0
$961$	25.1708	0.811962
$962$	0	0
$963$	−2.20555	−0.0710729
$964$	0	0
$965$	−3.53951	−0.113941
$966$	0	0
$967$	−29.9094	−0.961821	−0.480911	−	0.876770i	$-0.659694\pi$
$967$	−29.9094	−0.961821	−0.480911	+	0.876770i	$0.659694\pi$
$968$	0	0
$969$	10.3133	0.331312
$970$	0	0
$971$	−11.5194	−0.369676	−0.184838	−	0.982769i	$-0.559176\pi$
$971$	−11.5194	−0.369676	−0.184838	+	0.982769i	$0.559176\pi$
$972$	0	0
$973$	−18.9894	−0.608773
$974$	0	0
$975$	3.33804	0.106903
$976$	0	0
$977$	−36.4182	−1.16512	−0.582561	−	0.812787i	$-0.697949\pi$
$977$	−36.4182	−1.16512	−0.582561	+	0.812787i	$0.697949\pi$
$978$	0	0
$979$	−19.1083	−0.610704
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−17.6902	−0.564230	−0.282115	−	0.959381i	$-0.591036\pi$
$983$	−17.6902	−0.564230	−0.282115	+	0.959381i	$0.591036\pi$
$984$	0	0
$985$	−6.98944	−0.222702
$986$	0	0
$987$	1.08362	0.0344920
$988$	0	0
$989$	13.3275	0.423789
$990$	0	0
$991$	11.0589	0.351298	0.175649	−	0.984453i	$-0.443798\pi$
$991$	11.0589	0.351298	0.175649	+	0.984453i	$0.443798\pi$
$992$	0	0
$993$	−30.6550	−0.972806
$994$	0	0
$995$	−26.3133	−0.834189
$996$	0	0
$997$	−49.6727	−1.57315	−0.786575	−	0.617495i	$-0.788147\pi$
$997$	−49.6727	−1.57315	−0.786575	+	0.617495i	$0.788147\pi$
$998$	0	0
$999$	3.42166	0.108257

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bq.1.2		3
4.3	odd	2		273.2.a.d.1.1	✓	3
12.11	even	2		819.2.a.j.1.3		3
20.19	odd	2		6825.2.a.bd.1.3		3
28.27	even	2		1911.2.a.n.1.1		3
52.51	odd	2		3549.2.a.t.1.3		3
84.83	odd	2		5733.2.a.bc.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.d.1.1	✓	3	4.3	odd	2
819.2.a.j.1.3		3	12.11	even	2
1911.2.a.n.1.1		3	28.27	even	2
3549.2.a.t.1.3		3	52.51	odd	2
4368.2.a.bq.1.2		3	1.1	even	1	trivial
5733.2.a.bc.1.3		3	84.83	odd	2
6825.2.a.bd.1.3		3	20.19	odd	2

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

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Embedding invariants

$q$ -expansion

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4368.2.a.bq.1.2

Level	$4368$
Weight	$2$
Character	4368.1
Self dual	yes
Analytic conductor	$34.879$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$