LMFDB - Embedded newform 1792.2.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(1,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1792.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.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:	$14.3091920422$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.9248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 2 $ x^4 - 5*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.662153$ of defining polynomial
Character	$\chi$	$=$	1792.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.02045 q^{3} +1.69614 q^{5} +1.00000 q^{7} +6.12311 q^{9} +O(q^{10})$	$q+3.02045 q^{3} +1.69614 q^{5} +1.00000 q^{7} +6.12311 q^{9} +1.32431 q^{11} +1.69614 q^{13} +5.12311 q^{15} +2.00000 q^{17} +3.02045 q^{19} +3.02045 q^{21} -5.12311 q^{23} -2.12311 q^{25} +9.43318 q^{27} -6.04090 q^{29} -10.2462 q^{31} +4.00000 q^{33} +1.69614 q^{35} -6.04090 q^{37} +5.12311 q^{39} -4.24621 q^{41} +1.32431 q^{43} +10.3857 q^{45} +1.00000 q^{49} +6.04090 q^{51} +2.64861 q^{53} +2.24621 q^{55} +9.12311 q^{57} +0.371834 q^{59} +1.69614 q^{61} +6.12311 q^{63} +2.87689 q^{65} -11.5012 q^{67} -15.4741 q^{69} +8.00000 q^{71} +6.00000 q^{73} -6.41273 q^{75} +1.32431 q^{77} +10.1231 q^{81} -5.66906 q^{83} +3.39228 q^{85} -18.2462 q^{87} +16.2462 q^{89} +1.69614 q^{91} -30.9481 q^{93} +5.12311 q^{95} +12.2462 q^{97} +8.10887 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^7 + 8 * q^9	$ 4 q + 4 q^{7} + 8 q^{9} + 4 q^{15} + 8 q^{17} - 4 q^{23} + 8 q^{25} - 8 q^{31} + 16 q^{33} + 4 q^{39} + 16 q^{41} + 4 q^{49} - 24 q^{55} + 20 q^{57} + 8 q^{63} + 28 q^{65} + 32 q^{71} + 24 q^{73} + 24 q^{81} - 40 q^{87} + 32 q^{89} + 4 q^{95} + 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 8 * q^9 + 4 * q^15 + 8 * q^17 - 4 * q^23 + 8 * q^25 - 8 * q^31 + 16 * q^33 + 4 * q^39 + 16 * q^41 + 4 * q^49 - 24 * q^55 + 20 * q^57 + 8 * q^63 + 28 * q^65 + 32 * q^71 + 24 * q^73 + 24 * q^81 - 40 * q^87 + 32 * q^89 + 4 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.02045	1.74386	0.871928	−	0.489634i	$-0.162870\pi$
$3$	3.02045	1.74386	0.871928	+	0.489634i	$0.162870\pi$
$4$	0	0
$5$	1.69614	0.758537	0.379269	−	0.925287i	$-0.376176\pi$
$5$	1.69614	0.758537	0.379269	+	0.925287i	$0.376176\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	6.12311	2.04104
$10$	0	0
$11$	1.32431	0.399294	0.199647	−	0.979868i	$-0.436021\pi$
$11$	1.32431	0.399294	0.199647	+	0.979868i	$0.436021\pi$
$12$	0	0
$13$	1.69614	0.470425	0.235212	−	0.971944i	$-0.424421\pi$
$13$	1.69614	0.470425	0.235212	+	0.971944i	$0.424421\pi$
$14$	0	0
$15$	5.12311	1.32278
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	3.02045	0.692938	0.346469	−	0.938061i	$-0.387381\pi$
$19$	3.02045	0.692938	0.346469	+	0.938061i	$0.387381\pi$
$20$	0	0
$21$	3.02045	0.659116
$22$	0	0
$23$	−5.12311	−1.06824	−0.534121	−	0.845408i	$-0.679357\pi$
$23$	−5.12311	−1.06824	−0.534121	+	0.845408i	$0.679357\pi$
$24$	0	0
$25$	−2.12311	−0.424621
$26$	0	0
$27$	9.43318	1.81542
$28$	0	0
$29$	−6.04090	−1.12177	−0.560883	−	0.827895i	$-0.689538\pi$
$29$	−6.04090	−1.12177	−0.560883	+	0.827895i	$0.689538\pi$
$30$	0	0
$31$	−10.2462	−1.84027	−0.920137	−	0.391597i	$-0.871923\pi$
$31$	−10.2462	−1.84027	−0.920137	+	0.391597i	$0.871923\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	1.69614	0.286700
$36$	0	0
$37$	−6.04090	−0.993117	−0.496559	−	0.868003i	$-0.665403\pi$
$37$	−6.04090	−0.993117	−0.496559	+	0.868003i	$0.665403\pi$
$38$	0	0
$39$	5.12311	0.820353
$40$	0	0
$41$	−4.24621	−0.663147	−0.331573	−	0.943429i	$-0.607579\pi$
$41$	−4.24621	−0.663147	−0.331573	+	0.943429i	$0.607579\pi$
$42$	0	0
$43$	1.32431	0.201955	0.100977	−	0.994889i	$-0.467803\pi$
$43$	1.32431	0.201955	0.100977	+	0.994889i	$0.467803\pi$
$44$	0	0
$45$	10.3857	1.54820
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.04090	0.845895
$52$	0	0
$53$	2.64861	0.363815	0.181908	−	0.983316i	$-0.441773\pi$
$53$	2.64861	0.363815	0.181908	+	0.983316i	$0.441773\pi$
$54$	0	0
$55$	2.24621	0.302879
$56$	0	0
$57$	9.12311	1.20838
$58$	0	0
$59$	0.371834	0.0484087	0.0242043	−	0.999707i	$-0.492295\pi$
$59$	0.371834	0.0484087	0.0242043	+	0.999707i	$0.492295\pi$
$60$	0	0
$61$	1.69614	0.217169	0.108584	−	0.994087i	$-0.465368\pi$
$61$	1.69614	0.217169	0.108584	+	0.994087i	$0.465368\pi$
$62$	0	0
$63$	6.12311	0.771439
$64$	0	0
$65$	2.87689	0.356835
$66$	0	0
$67$	−11.5012	−1.40509	−0.702545	−	0.711640i	$-0.747953\pi$
$67$	−11.5012	−1.40509	−0.702545	+	0.711640i	$0.747953\pi$
$68$	0	0
$69$	−15.4741	−1.86286
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	−6.41273	−0.740478
$76$	0	0
$77$	1.32431	0.150919
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	10.1231	1.12479
$82$	0	0
$83$	−5.66906	−0.622260	−0.311130	−	0.950367i	$-0.600708\pi$
$83$	−5.66906	−0.622260	−0.311130	+	0.950367i	$0.600708\pi$
$84$	0	0
$85$	3.39228	0.367945
$86$	0	0
$87$	−18.2462	−1.95620
$88$	0	0
$89$	16.2462	1.72209	0.861047	−	0.508525i	$-0.169809\pi$
$89$	16.2462	1.72209	0.861047	+	0.508525i	$0.169809\pi$
$90$	0	0
$91$	1.69614	0.177804
$92$	0	0
$93$	−30.9481	−3.20917
$94$	0	0
$95$	5.12311	0.525620
$96$	0	0
$97$	12.2462	1.24341	0.621707	−	0.783250i	$-0.286439\pi$
$97$	12.2462	1.24341	0.621707	+	0.783250i	$0.286439\pi$
$98$	0	0
$99$	8.10887	0.814972
$100$	0	0
$101$	10.3857	1.03341	0.516705	−	0.856163i	$-0.327158\pi$
$101$	10.3857	1.03341	0.516705	+	0.856163i	$0.327158\pi$
$102$	0	0
$103$	−2.24621	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$104$	0	0
$105$	5.12311	0.499964
$106$	0	0
$107$	−14.1498	−1.36791	−0.683955	−	0.729524i	$-0.739741\pi$
$107$	−14.1498	−1.36791	−0.683955	+	0.729524i	$0.739741\pi$
$108$	0	0
$109$	18.1227	1.73584	0.867919	−	0.496705i	$-0.165457\pi$
$109$	18.1227	1.73584	0.867919	+	0.496705i	$0.165457\pi$
$110$	0	0
$111$	−18.2462	−1.73185
$112$	0	0
$113$	4.87689	0.458780	0.229390	−	0.973335i	$-0.426327\pi$
$113$	4.87689	0.458780	0.229390	+	0.973335i	$0.426327\pi$
$114$	0	0
$115$	−8.68951	−0.810301
$116$	0	0
$117$	10.3857	0.960154
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−9.24621	−0.840565
$122$	0	0
$123$	−12.8255	−1.15643
$124$	0	0
$125$	−12.0818	−1.08063
$126$	0	0
$127$	13.1231	1.16449	0.582244	−	0.813014i	$-0.302175\pi$
$127$	13.1231	1.16449	0.582244	+	0.813014i	$0.302175\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−12.4536	−1.08808	−0.544039	−	0.839060i	$-0.683106\pi$
$131$	−12.4536	−1.08808	−0.544039	+	0.839060i	$0.683106\pi$
$132$	0	0
$133$	3.02045	0.261906
$134$	0	0
$135$	16.0000	1.37706
$136$	0	0
$137$	16.2462	1.38801	0.694004	−	0.719971i	$-0.255845\pi$
$137$	16.2462	1.38801	0.694004	+	0.719971i	$0.255845\pi$
$138$	0	0
$139$	−8.31768	−0.705496	−0.352748	−	0.935718i	$-0.614753\pi$
$139$	−8.31768	−0.705496	−0.352748	+	0.935718i	$0.614753\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.24621	0.187838
$144$	0	0
$145$	−10.2462	−0.850902
$146$	0	0
$147$	3.02045	0.249122
$148$	0	0
$149$	−14.7304	−1.20676	−0.603381	−	0.797453i	$-0.706180\pi$
$149$	−14.7304	−1.20676	−0.603381	+	0.797453i	$0.706180\pi$
$150$	0	0
$151$	−10.8769	−0.885149	−0.442575	−	0.896732i	$-0.645935\pi$
$151$	−10.8769	−0.885149	−0.442575	+	0.896732i	$0.645935\pi$
$152$	0	0
$153$	12.2462	0.990048
$154$	0	0
$155$	−17.3790	−1.39592
$156$	0	0
$157$	8.48071	0.676834	0.338417	−	0.940996i	$-0.390109\pi$
$157$	8.48071	0.676834	0.338417	+	0.940996i	$0.390109\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	−5.12311	−0.403757
$162$	0	0
$163$	−13.4061	−1.05005	−0.525023	−	0.851088i	$-0.675943\pi$
$163$	−13.4061	−1.05005	−0.525023	+	0.851088i	$0.675943\pi$
$164$	0	0
$165$	6.78456	0.528178
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−10.1231	−0.778700
$170$	0	0
$171$	18.4945	1.41431
$172$	0	0
$173$	19.0752	1.45026	0.725129	−	0.688613i	$-0.241780\pi$
$173$	19.0752	1.45026	0.725129	+	0.688613i	$0.241780\pi$
$174$	0	0
$175$	−2.12311	−0.160492
$176$	0	0
$177$	1.12311	0.0844178
$178$	0	0
$179$	−4.71659	−0.352534	−0.176267	−	0.984342i	$-0.556402\pi$
$179$	−4.71659	−0.352534	−0.176267	+	0.984342i	$0.556402\pi$
$180$	0	0
$181$	−6.99337	−0.519813	−0.259906	−	0.965634i	$-0.583692\pi$
$181$	−6.99337	−0.519813	−0.259906	+	0.965634i	$0.583692\pi$
$182$	0	0
$183$	5.12311	0.378711
$184$	0	0
$185$	−10.2462	−0.753316
$186$	0	0
$187$	2.64861	0.193686
$188$	0	0
$189$	9.43318	0.686163
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	−11.1231	−0.800659	−0.400329	−	0.916371i	$-0.631104\pi$
$193$	−11.1231	−0.800659	−0.400329	+	0.916371i	$0.631104\pi$
$194$	0	0
$195$	8.68951	0.622269
$196$	0	0
$197$	−21.5150	−1.53288	−0.766439	−	0.642317i	$-0.777973\pi$
$197$	−21.5150	−1.53288	−0.766439	+	0.642317i	$0.777973\pi$
$198$	0	0
$199$	18.2462	1.29344	0.646720	−	0.762728i	$-0.276140\pi$
$199$	18.2462	1.29344	0.646720	+	0.762728i	$0.276140\pi$
$200$	0	0
$201$	−34.7386	−2.45027
$202$	0	0
$203$	−6.04090	−0.423988
$204$	0	0
$205$	−7.20217	−0.503022
$206$	0	0
$207$	−31.3693	−2.18032
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−4.71659	−0.324703	−0.162352	−	0.986733i	$-0.551908\pi$
$211$	−4.71659	−0.324703	−0.162352	+	0.986733i	$0.551908\pi$
$212$	0	0
$213$	24.1636	1.65566
$214$	0	0
$215$	2.24621	0.153190
$216$	0	0
$217$	−10.2462	−0.695558
$218$	0	0
$219$	18.1227	1.22462
$220$	0	0
$221$	3.39228	0.228190
$222$	0	0
$223$	−5.75379	−0.385302	−0.192651	−	0.981267i	$-0.561709\pi$
$223$	−5.75379	−0.385302	−0.192651	+	0.981267i	$0.561709\pi$
$224$	0	0
$225$	−13.0000	−0.866667
$226$	0	0
$227$	9.80501	0.650782	0.325391	−	0.945580i	$-0.394504\pi$
$227$	9.80501	0.650782	0.325391	+	0.945580i	$0.394504\pi$
$228$	0	0
$229$	25.8597	1.70886	0.854429	−	0.519568i	$-0.173907\pi$
$229$	25.8597	1.70886	0.854429	+	0.519568i	$0.173907\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	16.2462	1.06432	0.532162	−	0.846642i	$-0.321380\pi$
$233$	16.2462	1.06432	0.532162	+	0.846642i	$0.321380\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−17.6155	−1.13945	−0.569727	−	0.821834i	$-0.692951\pi$
$239$	−17.6155	−1.13945	−0.569727	+	0.821834i	$0.692951\pi$
$240$	0	0
$241$	−3.75379	−0.241803	−0.120901	−	0.992665i	$-0.538578\pi$
$241$	−3.75379	−0.241803	−0.120901	+	0.992665i	$0.538578\pi$
$242$	0	0
$243$	2.27678	0.146055
$244$	0	0
$245$	1.69614	0.108362
$246$	0	0
$247$	5.12311	0.325975
$248$	0	0
$249$	−17.1231	−1.08513
$250$	0	0
$251$	10.9663	0.692186	0.346093	−	0.938200i	$-0.387508\pi$
$251$	10.9663	0.692186	0.346093	+	0.938200i	$0.387508\pi$
$252$	0	0
$253$	−6.78456	−0.426542
$254$	0	0
$255$	10.2462	0.641643
$256$	0	0
$257$	22.4924	1.40304	0.701519	−	0.712650i	$-0.252505\pi$
$257$	22.4924	1.40304	0.701519	+	0.712650i	$0.252505\pi$
$258$	0	0
$259$	−6.04090	−0.375363
$260$	0	0
$261$	−36.9890	−2.28956
$262$	0	0
$263$	−12.4924	−0.770316	−0.385158	−	0.922851i	$-0.625853\pi$
$263$	−12.4924	−0.770316	−0.385158	+	0.922851i	$0.625853\pi$
$264$	0	0
$265$	4.49242	0.275967
$266$	0	0
$267$	49.0708	3.00309
$268$	0	0
$269$	−11.8730	−0.723909	−0.361954	−	0.932196i	$-0.617890\pi$
$269$	−11.8730	−0.723909	−0.361954	+	0.932196i	$0.617890\pi$
$270$	0	0
$271$	10.2462	0.622413	0.311207	−	0.950342i	$-0.399267\pi$
$271$	10.2462	0.622413	0.311207	+	0.950342i	$0.399267\pi$
$272$	0	0
$273$	5.12311	0.310064
$274$	0	0
$275$	−2.81164	−0.169548
$276$	0	0
$277$	2.64861	0.159140	0.0795699	−	0.996829i	$-0.474645\pi$
$277$	2.64861	0.159140	0.0795699	+	0.996829i	$0.474645\pi$
$278$	0	0
$279$	−62.7386	−3.75606
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−21.8868	−1.30104	−0.650518	−	0.759491i	$-0.725448\pi$
$283$	−21.8868	−1.30104	−0.650518	+	0.759491i	$0.725448\pi$
$284$	0	0
$285$	15.4741	0.916605
$286$	0	0
$287$	−4.24621	−0.250646
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	36.9890	2.16834
$292$	0	0
$293$	10.3857	0.606736	0.303368	−	0.952873i	$-0.401889\pi$
$293$	10.3857	0.606736	0.303368	+	0.952873i	$0.401889\pi$
$294$	0	0
$295$	0.630683	0.0367198
$296$	0	0
$297$	12.4924	0.724884
$298$	0	0
$299$	−8.68951	−0.502527
$300$	0	0
$301$	1.32431	0.0763318
$302$	0	0
$303$	31.3693	1.80212
$304$	0	0
$305$	2.87689	0.164730
$306$	0	0
$307$	25.2791	1.44275	0.721377	−	0.692543i	$-0.243510\pi$
$307$	25.2791	1.44275	0.721377	+	0.692543i	$0.243510\pi$
$308$	0	0
$309$	−6.78456	−0.385960
$310$	0	0
$311$	12.4924	0.708380	0.354190	−	0.935173i	$-0.384757\pi$
$311$	12.4924	0.708380	0.354190	+	0.935173i	$0.384757\pi$
$312$	0	0
$313$	20.7386	1.17222	0.586108	−	0.810233i	$-0.300659\pi$
$313$	20.7386	1.17222	0.586108	+	0.810233i	$0.300659\pi$
$314$	0	0
$315$	10.3857	0.585165
$316$	0	0
$317$	−4.13595	−0.232298	−0.116149	−	0.993232i	$-0.537055\pi$
$317$	−4.13595	−0.232298	−0.116149	+	0.993232i	$0.537055\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	−42.7386	−2.38544
$322$	0	0
$323$	6.04090	0.336124
$324$	0	0
$325$	−3.60109	−0.199752
$326$	0	0
$327$	54.7386	3.02705
$328$	0	0
$329$	0	0
$330$	0	0
$331$	14.8934	0.818617	0.409309	−	0.912396i	$-0.365770\pi$
$331$	14.8934	0.818617	0.409309	+	0.912396i	$0.365770\pi$
$332$	0	0
$333$	−36.9890	−2.02699
$334$	0	0
$335$	−19.5076	−1.06581
$336$	0	0
$337$	−0.876894	−0.0477675	−0.0238837	−	0.999715i	$-0.507603\pi$
$337$	−0.876894	−0.0477675	−0.0238837	+	0.999715i	$0.507603\pi$
$338$	0	0
$339$	14.7304	0.800046
$340$	0	0
$341$	−13.5691	−0.734809
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−26.2462	−1.41305
$346$	0	0
$347$	8.10887	0.435307	0.217654	−	0.976026i	$-0.430160\pi$
$347$	8.10887	0.435307	0.217654	+	0.976026i	$0.430160\pi$
$348$	0	0
$349$	−27.3471	−1.46385	−0.731927	−	0.681383i	$-0.761379\pi$
$349$	−27.3471	−1.46385	−0.731927	+	0.681383i	$0.761379\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	7.75379	0.412693	0.206346	−	0.978479i	$-0.433843\pi$
$353$	7.75379	0.412693	0.206346	+	0.978479i	$0.433843\pi$
$354$	0	0
$355$	13.5691	0.720175
$356$	0	0
$357$	6.04090	0.319718
$358$	0	0
$359$	31.3693	1.65561	0.827805	−	0.561017i	$-0.189590\pi$
$359$	31.3693	1.65561	0.827805	+	0.561017i	$0.189590\pi$
$360$	0	0
$361$	−9.87689	−0.519837
$362$	0	0
$363$	−27.9277	−1.46582
$364$	0	0
$365$	10.1768	0.532680
$366$	0	0
$367$	10.2462	0.534848	0.267424	−	0.963579i	$-0.413828\pi$
$367$	10.2462	0.534848	0.267424	+	0.963579i	$0.413828\pi$
$368$	0	0
$369$	−26.0000	−1.35351
$370$	0	0
$371$	2.64861	0.137509
$372$	0	0
$373$	−14.7304	−0.762711	−0.381356	−	0.924428i	$-0.624543\pi$
$373$	−14.7304	−0.762711	−0.381356	+	0.924428i	$0.624543\pi$
$374$	0	0
$375$	−36.4924	−1.88446
$376$	0	0
$377$	−10.2462	−0.527707
$378$	0	0
$379$	−36.4084	−1.87017	−0.935087	−	0.354418i	$-0.884679\pi$
$379$	−36.4084	−1.87017	−0.935087	+	0.354418i	$0.884679\pi$
$380$	0	0
$381$	39.6377	2.03070
$382$	0	0
$383$	−4.49242	−0.229552	−0.114776	−	0.993391i	$-0.536615\pi$
$383$	−4.49242	−0.229552	−0.114776	+	0.993391i	$0.536615\pi$
$384$	0	0
$385$	2.24621	0.114478
$386$	0	0
$387$	8.10887	0.412197
$388$	0	0
$389$	24.9073	1.26285	0.631424	−	0.775438i	$-0.282471\pi$
$389$	24.9073	1.26285	0.631424	+	0.775438i	$0.282471\pi$
$390$	0	0
$391$	−10.2462	−0.518173
$392$	0	0
$393$	−37.6155	−1.89745
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−20.5625	−1.03200	−0.516001	−	0.856588i	$-0.672580\pi$
$397$	−20.5625	−1.03200	−0.516001	+	0.856588i	$0.672580\pi$
$398$	0	0
$399$	9.12311	0.456727
$400$	0	0
$401$	−0.876894	−0.0437900	−0.0218950	−	0.999760i	$-0.506970\pi$
$401$	−0.876894	−0.0437900	−0.0218950	+	0.999760i	$0.506970\pi$
$402$	0	0
$403$	−17.3790	−0.865711
$404$	0	0
$405$	17.1702	0.853195
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	49.0708	2.42049
$412$	0	0
$413$	0.371834	0.0182968
$414$	0	0
$415$	−9.61553	−0.472008
$416$	0	0
$417$	−25.1231	−1.23028
$418$	0	0
$419$	−27.9277	−1.36436	−0.682179	−	0.731185i	$-0.738967\pi$
$419$	−27.9277	−1.36436	−0.682179	+	0.731185i	$0.738967\pi$
$420$	0	0
$421$	26.8122	1.30675	0.653373	−	0.757036i	$-0.273353\pi$
$421$	26.8122	1.30675	0.653373	+	0.757036i	$0.273353\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.24621	−0.205971
$426$	0	0
$427$	1.69614	0.0820820
$428$	0	0
$429$	6.78456	0.327562
$430$	0	0
$431$	17.6155	0.848510	0.424255	−	0.905543i	$-0.360536\pi$
$431$	17.6155	0.848510	0.424255	+	0.905543i	$0.360536\pi$
$432$	0	0
$433$	−18.4924	−0.888689	−0.444345	−	0.895856i	$-0.646563\pi$
$433$	−18.4924	−0.888689	−0.444345	+	0.895856i	$0.646563\pi$
$434$	0	0
$435$	−30.9481	−1.48385
$436$	0	0
$437$	−15.4741	−0.740225
$438$	0	0
$439$	−22.7386	−1.08526	−0.542628	−	0.839973i	$-0.682571\pi$
$439$	−22.7386	−1.08526	−0.542628	+	0.839973i	$0.682571\pi$
$440$	0	0
$441$	6.12311	0.291576
$442$	0	0
$443$	−7.36520	−0.349931	−0.174966	−	0.984575i	$-0.555981\pi$
$443$	−7.36520	−0.349931	−0.174966	+	0.984575i	$0.555981\pi$
$444$	0	0
$445$	27.5559	1.30627
$446$	0	0
$447$	−44.4924	−2.10442
$448$	0	0
$449$	16.7386	0.789945	0.394972	−	0.918693i	$-0.370754\pi$
$449$	16.7386	0.789945	0.394972	+	0.918693i	$0.370754\pi$
$450$	0	0
$451$	−5.62329	−0.264790
$452$	0	0
$453$	−32.8531	−1.54357
$454$	0	0
$455$	2.87689	0.134871
$456$	0	0
$457$	−17.3693	−0.812502	−0.406251	−	0.913761i	$-0.633164\pi$
$457$	−17.3693	−0.812502	−0.406251	+	0.913761i	$0.633164\pi$
$458$	0	0
$459$	18.8664	0.880606
$460$	0	0
$461$	−24.3724	−1.13514	−0.567568	−	0.823327i	$-0.692115\pi$
$461$	−24.3724	−1.13514	−0.567568	+	0.823327i	$0.692115\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	−52.4924	−2.43428
$466$	0	0
$467$	3.02045	0.139770	0.0698848	−	0.997555i	$-0.477737\pi$
$467$	3.02045	0.139770	0.0698848	+	0.997555i	$0.477737\pi$
$468$	0	0
$469$	−11.5012	−0.531074
$470$	0	0
$471$	25.6155	1.18030
$472$	0	0
$473$	1.75379	0.0806393
$474$	0	0
$475$	−6.41273	−0.294236
$476$	0	0
$477$	16.2177	0.742559
$478$	0	0
$479$	10.2462	0.468161	0.234081	−	0.972217i	$-0.424792\pi$
$479$	10.2462	0.468161	0.234081	+	0.972217i	$0.424792\pi$
$480$	0	0
$481$	−10.2462	−0.467187
$482$	0	0
$483$	−15.4741	−0.704095
$484$	0	0
$485$	20.7713	0.943176
$486$	0	0
$487$	−0.630683	−0.0285790	−0.0142895	−	0.999898i	$-0.504549\pi$
$487$	−0.630683	−0.0285790	−0.0142895	+	0.999898i	$0.504549\pi$
$488$	0	0
$489$	−40.4924	−1.83113
$490$	0	0
$491$	34.1774	1.54240	0.771202	−	0.636590i	$-0.219656\pi$
$491$	34.1774	1.54240	0.771202	+	0.636590i	$0.219656\pi$
$492$	0	0
$493$	−12.0818	−0.544137
$494$	0	0
$495$	13.7538	0.618187
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	8.85254	0.396294	0.198147	−	0.980172i	$-0.436508\pi$
$499$	8.85254	0.396294	0.198147	+	0.980172i	$0.436508\pi$
$500$	0	0
$501$	−24.1636	−1.07955
$502$	0	0
$503$	−13.7538	−0.613251	−0.306626	−	0.951830i	$-0.599200\pi$
$503$	−13.7538	−0.613251	−0.306626	+	0.951830i	$0.599200\pi$
$504$	0	0
$505$	17.6155	0.783881
$506$	0	0
$507$	−30.5763	−1.35794
$508$	0	0
$509$	30.7393	1.36250	0.681249	−	0.732052i	$-0.261437\pi$
$509$	30.7393	1.36250	0.681249	+	0.732052i	$0.261437\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	28.4924	1.25797
$514$	0	0
$515$	−3.80989	−0.167884
$516$	0	0
$517$	0	0
$518$	0	0
$519$	57.6155	2.52904
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	−41.1708	−1.80027	−0.900136	−	0.435609i	$-0.856533\pi$
$523$	−41.1708	−1.80027	−0.900136	+	0.435609i	$0.856533\pi$
$524$	0	0
$525$	−6.41273	−0.279874
$526$	0	0
$527$	−20.4924	−0.892664
$528$	0	0
$529$	3.24621	0.141140
$530$	0	0
$531$	2.27678	0.0988038
$532$	0	0
$533$	−7.20217	−0.311961
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	−14.2462	−0.614769
$538$	0	0
$539$	1.32431	0.0570419
$540$	0	0
$541$	13.2431	0.569364	0.284682	−	0.958622i	$-0.408112\pi$
$541$	13.2431	0.569364	0.284682	+	0.958622i	$0.408112\pi$
$542$	0	0
$543$	−21.1231	−0.906479
$544$	0	0
$545$	30.7386	1.31670
$546$	0	0
$547$	−9.59621	−0.410304	−0.205152	−	0.978730i	$-0.565769\pi$
$547$	−9.59621	−0.410304	−0.205152	+	0.978730i	$0.565769\pi$
$548$	0	0
$549$	10.3857	0.443249
$550$	0	0
$551$	−18.2462	−0.777315
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−30.9481	−1.31368
$556$	0	0
$557$	2.64861	0.112225	0.0561127	−	0.998424i	$-0.482129\pi$
$557$	2.64861	0.112225	0.0561127	+	0.998424i	$0.482129\pi$
$558$	0	0
$559$	2.24621	0.0950046
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	−29.8326	−1.25730	−0.628648	−	0.777690i	$-0.716391\pi$
$563$	−29.8326	−1.25730	−0.628648	+	0.777690i	$0.716391\pi$
$564$	0	0
$565$	8.27190	0.348001
$566$	0	0
$567$	10.1231	0.425130
$568$	0	0
$569$	13.3693	0.560471	0.280235	−	0.959931i	$-0.409587\pi$
$569$	13.3693	0.560471	0.280235	+	0.959931i	$0.409587\pi$
$570$	0	0
$571$	−9.27015	−0.387944	−0.193972	−	0.981007i	$-0.562137\pi$
$571$	−9.27015	−0.387944	−0.193972	+	0.981007i	$0.562137\pi$
$572$	0	0
$573$	48.3272	2.01890
$574$	0	0
$575$	10.8769	0.453598
$576$	0	0
$577$	7.75379	0.322794	0.161397	−	0.986890i	$-0.448400\pi$
$577$	7.75379	0.322794	0.161397	+	0.986890i	$0.448400\pi$
$578$	0	0
$579$	−33.5968	−1.39623
$580$	0	0
$581$	−5.66906	−0.235192
$582$	0	0
$583$	3.50758	0.145269
$584$	0	0
$585$	17.6155	0.728312
$586$	0	0
$587$	−21.8868	−0.903365	−0.451683	−	0.892179i	$-0.649176\pi$
$587$	−21.8868	−0.903365	−0.451683	+	0.892179i	$0.649176\pi$
$588$	0	0
$589$	−30.9481	−1.27520
$590$	0	0
$591$	−64.9848	−2.67312
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	3.39228	0.139070
$596$	0	0
$597$	55.1117	2.25557
$598$	0	0
$599$	12.4924	0.510427	0.255213	−	0.966885i	$-0.417854\pi$
$599$	12.4924	0.510427	0.255213	+	0.966885i	$0.417854\pi$
$600$	0	0
$601$	16.2462	0.662697	0.331348	−	0.943508i	$-0.392496\pi$
$601$	16.2462	0.662697	0.331348	+	0.943508i	$0.392496\pi$
$602$	0	0
$603$	−70.4228	−2.86784
$604$	0	0
$605$	−15.6829	−0.637600
$606$	0	0
$607$	40.9848	1.66352	0.831762	−	0.555133i	$-0.187333\pi$
$607$	40.9848	1.66352	0.831762	+	0.555133i	$0.187333\pi$
$608$	0	0
$609$	−18.2462	−0.739374
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−2.23100	−0.0901094	−0.0450547	−	0.998985i	$-0.514346\pi$
$613$	−2.23100	−0.0901094	−0.0450547	+	0.998985i	$0.514346\pi$
$614$	0	0
$615$	−21.7538	−0.877197
$616$	0	0
$617$	29.3693	1.18236	0.591182	−	0.806538i	$-0.298661\pi$
$617$	29.3693	1.18236	0.591182	+	0.806538i	$0.298661\pi$
$618$	0	0
$619$	19.6558	0.790033	0.395017	−	0.918674i	$-0.370739\pi$
$619$	19.6558	0.790033	0.395017	+	0.918674i	$0.370739\pi$
$620$	0	0
$621$	−48.3272	−1.93930
$622$	0	0
$623$	16.2462	0.650891
$624$	0	0
$625$	−9.87689	−0.395076
$626$	0	0
$627$	12.0818	0.482500
$628$	0	0
$629$	−12.0818	−0.481733
$630$	0	0
$631$	3.50758	0.139634	0.0698172	−	0.997560i	$-0.477758\pi$
$631$	3.50758	0.139634	0.0698172	+	0.997560i	$0.477758\pi$
$632$	0	0
$633$	−14.2462	−0.566236
$634$	0	0
$635$	22.2586	0.883307
$636$	0	0
$637$	1.69614	0.0672036
$638$	0	0
$639$	48.9848	1.93781
$640$	0	0
$641$	−5.36932	−0.212075	−0.106038	−	0.994362i	$-0.533816\pi$
$641$	−5.36932	−0.212075	−0.106038	+	0.994362i	$0.533816\pi$
$642$	0	0
$643$	−5.66906	−0.223566	−0.111783	−	0.993733i	$-0.535656\pi$
$643$	−5.66906	−0.223566	−0.111783	+	0.993733i	$0.535656\pi$
$644$	0	0
$645$	6.78456	0.267142
$646$	0	0
$647$	−43.2311	−1.69959	−0.849794	−	0.527115i	$-0.823274\pi$
$647$	−43.2311	−1.69959	−0.849794	+	0.527115i	$0.823274\pi$
$648$	0	0
$649$	0.492423	0.0193293
$650$	0	0
$651$	−30.9481	−1.21295
$652$	0	0
$653$	31.6918	1.24020	0.620098	−	0.784524i	$-0.287093\pi$
$653$	31.6918	1.24020	0.620098	+	0.784524i	$0.287093\pi$
$654$	0	0
$655$	−21.1231	−0.825348
$656$	0	0
$657$	36.7386	1.43331
$658$	0	0
$659$	−6.62153	−0.257938	−0.128969	−	0.991649i	$-0.541167\pi$
$659$	−6.62153	−0.257938	−0.128969	+	0.991649i	$0.541167\pi$
$660$	0	0
$661$	−44.7261	−1.73964	−0.869821	−	0.493367i	$-0.835766\pi$
$661$	−44.7261	−1.73964	−0.869821	+	0.493367i	$0.835766\pi$
$662$	0	0
$663$	10.2462	0.397930
$664$	0	0
$665$	5.12311	0.198666
$666$	0	0
$667$	30.9481	1.19832
$668$	0	0
$669$	−17.3790	−0.671912
$670$	0	0
$671$	2.24621	0.0867140
$672$	0	0
$673$	−38.9848	−1.50276	−0.751378	−	0.659872i	$-0.770610\pi$
$673$	−38.9848	−1.50276	−0.751378	+	0.659872i	$0.770610\pi$
$674$	0	0
$675$	−20.0276	−0.770864
$676$	0	0
$677$	17.1702	0.659905	0.329952	−	0.943998i	$-0.392967\pi$
$677$	17.1702	0.659905	0.329952	+	0.943998i	$0.392967\pi$
$678$	0	0
$679$	12.2462	0.469966
$680$	0	0
$681$	29.6155	1.13487
$682$	0	0
$683$	−0.580639	−0.0222175	−0.0111088	−	0.999938i	$-0.503536\pi$
$683$	−0.580639	−0.0222175	−0.0111088	+	0.999938i	$0.503536\pi$
$684$	0	0
$685$	27.5559	1.05286
$686$	0	0
$687$	78.1080	2.98000
$688$	0	0
$689$	4.49242	0.171148
$690$	0	0
$691$	38.8482	1.47786	0.738928	−	0.673785i	$-0.235332\pi$
$691$	38.8482	1.47786	0.738928	+	0.673785i	$0.235332\pi$
$692$	0	0
$693$	8.10887	0.308031
$694$	0	0
$695$	−14.1080	−0.535145
$696$	0	0
$697$	−8.49242	−0.321673
$698$	0	0
$699$	49.0708	1.85603
$700$	0	0
$701$	−2.23100	−0.0842639	−0.0421319	−	0.999112i	$-0.513415\pi$
$701$	−2.23100	−0.0842639	−0.0421319	+	0.999112i	$0.513415\pi$
$702$	0	0
$703$	−18.2462	−0.688169
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.3857	0.390593
$708$	0	0
$709$	28.7171	1.07849	0.539247	−	0.842147i	$-0.318709\pi$
$709$	28.7171	1.07849	0.539247	+	0.842147i	$0.318709\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	52.4924	1.96586
$714$	0	0
$715$	3.80989	0.142482
$716$	0	0
$717$	−53.2068	−1.98704
$718$	0	0
$719$	−52.4924	−1.95764	−0.978819	−	0.204730i	$-0.934368\pi$
$719$	−52.4924	−1.95764	−0.978819	+	0.204730i	$0.934368\pi$
$720$	0	0
$721$	−2.24621	−0.0836533
$722$	0	0
$723$	−11.3381	−0.421669
$724$	0	0
$725$	12.8255	0.476326
$726$	0	0
$727$	16.9848	0.629933	0.314967	−	0.949103i	$-0.398007\pi$
$727$	16.9848	0.629933	0.314967	+	0.949103i	$0.398007\pi$
$728$	0	0
$729$	−23.4924	−0.870090
$730$	0	0
$731$	2.64861	0.0979625
$732$	0	0
$733$	−29.2520	−1.08045	−0.540224	−	0.841521i	$-0.681660\pi$
$733$	−29.2520	−1.08045	−0.540224	+	0.841521i	$0.681660\pi$
$734$	0	0
$735$	5.12311	0.188969
$736$	0	0
$737$	−15.2311	−0.561043
$738$	0	0
$739$	21.3519	0.785444	0.392722	−	0.919657i	$-0.371533\pi$
$739$	21.3519	0.785444	0.392722	+	0.919657i	$0.371533\pi$
$740$	0	0
$741$	15.4741	0.568454
$742$	0	0
$743$	−0.630683	−0.0231375	−0.0115688	−	0.999933i	$-0.503683\pi$
$743$	−0.630683	−0.0231375	−0.0115688	+	0.999933i	$0.503683\pi$
$744$	0	0
$745$	−24.9848	−0.915374
$746$	0	0
$747$	−34.7123	−1.27006
$748$	0	0
$749$	−14.1498	−0.517021
$750$	0	0
$751$	8.63068	0.314938	0.157469	−	0.987524i	$-0.449667\pi$
$751$	8.63068	0.314938	0.157469	+	0.987524i	$0.449667\pi$
$752$	0	0
$753$	33.1231	1.20707
$754$	0	0
$755$	−18.4487	−0.671419
$756$	0	0
$757$	−26.3946	−0.959328	−0.479664	−	0.877452i	$-0.659241\pi$
$757$	−26.3946	−0.959328	−0.479664	+	0.877452i	$0.659241\pi$
$758$	0	0
$759$	−20.4924	−0.743828
$760$	0	0
$761$	−8.73863	−0.316775	−0.158388	−	0.987377i	$-0.550630\pi$
$761$	−8.73863	−0.316775	−0.158388	+	0.987377i	$0.550630\pi$
$762$	0	0
$763$	18.1227	0.656085
$764$	0	0
$765$	20.7713	0.750988
$766$	0	0
$767$	0.630683	0.0227726
$768$	0	0
$769$	−40.2462	−1.45132	−0.725658	−	0.688056i	$-0.758464\pi$
$769$	−40.2462	−1.45132	−0.725658	+	0.688056i	$0.758464\pi$
$770$	0	0
$771$	67.9372	2.44670
$772$	0	0
$773$	1.69614	0.0610060	0.0305030	−	0.999535i	$-0.490289\pi$
$773$	1.69614	0.0610060	0.0305030	+	0.999535i	$0.490289\pi$
$774$	0	0
$775$	21.7538	0.781419
$776$	0	0
$777$	−18.2462	−0.654579
$778$	0	0
$779$	−12.8255	−0.459520
$780$	0	0
$781$	10.5945	0.379099
$782$	0	0
$783$	−56.9848	−2.03647
$784$	0	0
$785$	14.3845	0.513404
$786$	0	0
$787$	−10.5487	−0.376020	−0.188010	−	0.982167i	$-0.560204\pi$
$787$	−10.5487	−0.376020	−0.188010	+	0.982167i	$0.560204\pi$
$788$	0	0
$789$	−37.7327	−1.34332
$790$	0	0
$791$	4.87689	0.173402
$792$	0	0
$793$	2.87689	0.102162
$794$	0	0
$795$	13.5691	0.481247
$796$	0	0
$797$	25.8597	0.915998	0.457999	−	0.888953i	$-0.348566\pi$
$797$	25.8597	0.915998	0.457999	+	0.888953i	$0.348566\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	99.4773	3.51486
$802$	0	0
$803$	7.94584	0.280403
$804$	0	0
$805$	−8.68951	−0.306265
$806$	0	0
$807$	−35.8617	−1.26239
$808$	0	0
$809$	−37.8617	−1.33115	−0.665574	−	0.746332i	$-0.731813\pi$
$809$	−37.8617	−1.33115	−0.665574	+	0.746332i	$0.731813\pi$
$810$	0	0
$811$	15.8459	0.556425	0.278213	−	0.960520i	$-0.410258\pi$
$811$	15.8459	0.556425	0.278213	+	0.960520i	$0.410258\pi$
$812$	0	0
$813$	30.9481	1.08540
$814$	0	0
$815$	−22.7386	−0.796500
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	10.3857	0.362904
$820$	0	0
$821$	33.5968	1.17254	0.586268	−	0.810118i	$-0.300597\pi$
$821$	33.5968	1.17254	0.586268	+	0.810118i	$0.300597\pi$
$822$	0	0
$823$	32.9848	1.14978	0.574890	−	0.818231i	$-0.305045\pi$
$823$	32.9848	1.14978	0.574890	+	0.818231i	$0.305045\pi$
$824$	0	0
$825$	−8.49242	−0.295668
$826$	0	0
$827$	6.20393	0.215732	0.107866	−	0.994165i	$-0.465598\pi$
$827$	6.20393	0.215732	0.107866	+	0.994165i	$0.465598\pi$
$828$	0	0
$829$	−46.6310	−1.61956	−0.809781	−	0.586732i	$-0.800414\pi$
$829$	−46.6310	−1.61956	−0.809781	+	0.586732i	$0.800414\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−13.5691	−0.469579
$836$	0	0
$837$	−96.6543	−3.34086
$838$	0	0
$839$	6.73863	0.232643	0.116322	−	0.993212i	$-0.462890\pi$
$839$	6.73863	0.232643	0.116322	+	0.993212i	$0.462890\pi$
$840$	0	0
$841$	7.49242	0.258359
$842$	0	0
$843$	18.1227	0.624179
$844$	0	0
$845$	−17.1702	−0.590673
$846$	0	0
$847$	−9.24621	−0.317704
$848$	0	0
$849$	−66.1080	−2.26882
$850$	0	0
$851$	30.9481	1.06089
$852$	0	0
$853$	−25.4421	−0.871121	−0.435561	−	0.900159i	$-0.643450\pi$
$853$	−25.4421	−0.871121	−0.435561	+	0.900159i	$0.643450\pi$
$854$	0	0
$855$	31.3693	1.07281
$856$	0	0
$857$	46.9848	1.60497	0.802486	−	0.596671i	$-0.203510\pi$
$857$	46.9848	1.60497	0.802486	+	0.596671i	$0.203510\pi$
$858$	0	0
$859$	28.3453	0.967129	0.483565	−	0.875309i	$-0.339342\pi$
$859$	28.3453	0.967129	0.483565	+	0.875309i	$0.339342\pi$
$860$	0	0
$861$	−12.8255	−0.437091
$862$	0	0
$863$	36.4924	1.24222	0.621108	−	0.783725i	$-0.286683\pi$
$863$	36.4924	1.24222	0.621108	+	0.783725i	$0.286683\pi$
$864$	0	0
$865$	32.3542	1.10007
$866$	0	0
$867$	−39.2658	−1.33354
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−19.5076	−0.660989
$872$	0	0
$873$	74.9848	2.53785
$874$	0	0
$875$	−12.0818	−0.408439
$876$	0	0
$877$	35.5017	1.19881	0.599404	−	0.800447i	$-0.295404\pi$
$877$	35.5017	1.19881	0.599404	+	0.800447i	$0.295404\pi$
$878$	0	0
$879$	31.3693	1.05806
$880$	0	0
$881$	44.2462	1.49069	0.745346	−	0.666677i	$-0.232284\pi$
$881$	44.2462	1.49069	0.745346	+	0.666677i	$0.232284\pi$
$882$	0	0
$883$	−4.71659	−0.158726	−0.0793629	−	0.996846i	$-0.525289\pi$
$883$	−4.71659	−0.158726	−0.0793629	+	0.996846i	$0.525289\pi$
$884$	0	0
$885$	1.90495	0.0640340
$886$	0	0
$887$	28.4924	0.956682	0.478341	−	0.878174i	$-0.341238\pi$
$887$	28.4924	0.956682	0.478341	+	0.878174i	$0.341238\pi$
$888$	0	0
$889$	13.1231	0.440135
$890$	0	0
$891$	13.4061	0.449121
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	−26.2462	−0.876335
$898$	0	0
$899$	61.8963	2.06436
$900$	0	0
$901$	5.29723	0.176476
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−11.8617	−0.394298
$906$	0	0
$907$	51.5564	1.71190	0.855951	−	0.517056i	$-0.172972\pi$
$907$	51.5564	1.71190	0.855951	+	0.517056i	$0.172972\pi$
$908$	0	0
$909$	63.5924	2.10923
$910$	0	0
$911$	11.8617	0.392997	0.196498	−	0.980504i	$-0.437043\pi$
$911$	11.8617	0.392997	0.196498	+	0.980504i	$0.437043\pi$
$912$	0	0
$913$	−7.50758	−0.248465
$914$	0	0
$915$	8.68951	0.287266
$916$	0	0
$917$	−12.4536	−0.411255
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	76.3542	2.51596
$922$	0	0
$923$	13.5691	0.446633
$924$	0	0
$925$	12.8255	0.421699
$926$	0	0
$927$	−13.7538	−0.451734
$928$	0	0
$929$	−2.49242	−0.0817737	−0.0408869	−	0.999164i	$-0.513018\pi$
$929$	−2.49242	−0.0817737	−0.0408869	+	0.999164i	$0.513018\pi$
$930$	0	0
$931$	3.02045	0.0989912
$932$	0	0
$933$	37.7327	1.23531
$934$	0	0
$935$	4.49242	0.146918
$936$	0	0
$937$	−34.9848	−1.14291	−0.571453	−	0.820635i	$-0.693620\pi$
$937$	−34.9848	−1.14291	−0.571453	+	0.820635i	$0.693620\pi$
$938$	0	0
$939$	62.6400	2.04418
$940$	0	0
$941$	30.7393	1.00207	0.501037	−	0.865426i	$-0.332952\pi$
$941$	30.7393	1.00207	0.501037	+	0.865426i	$0.332952\pi$
$942$	0	0
$943$	21.7538	0.708401
$944$	0	0
$945$	16.0000	0.520480
$946$	0	0
$947$	41.7056	1.35525	0.677625	−	0.735407i	$-0.263009\pi$
$947$	41.7056	1.35525	0.677625	+	0.735407i	$0.263009\pi$
$948$	0	0
$949$	10.1768	0.330354
$950$	0	0
$951$	−12.4924	−0.405095
$952$	0	0
$953$	17.5076	0.567126	0.283563	−	0.958954i	$-0.408483\pi$
$953$	17.5076	0.567126	0.283563	+	0.958954i	$0.408483\pi$
$954$	0	0
$955$	27.1383	0.878173
$956$	0	0
$957$	−24.1636	−0.781098
$958$	0	0
$959$	16.2462	0.524618
$960$	0	0
$961$	73.9848	2.38661
$962$	0	0
$963$	−86.6405	−2.79195
$964$	0	0
$965$	−18.8664	−0.607329
$966$	0	0
$967$	10.8769	0.349777	0.174889	−	0.984588i	$-0.444043\pi$
$967$	10.8769	0.349777	0.174889	+	0.984588i	$0.444043\pi$
$968$	0	0
$969$	18.2462	0.586153
$970$	0	0
$971$	10.9663	0.351925	0.175962	−	0.984397i	$-0.443696\pi$
$971$	10.9663	0.351925	0.175962	+	0.984397i	$0.443696\pi$
$972$	0	0
$973$	−8.31768	−0.266652
$974$	0	0
$975$	−10.8769	−0.348339
$976$	0	0
$977$	23.7538	0.759951	0.379976	−	0.924997i	$-0.375932\pi$
$977$	23.7538	0.759951	0.379976	+	0.924997i	$0.375932\pi$
$978$	0	0
$979$	21.5150	0.687621
$980$	0	0
$981$	110.967	3.54291
$982$	0	0
$983$	34.2462	1.09228	0.546142	−	0.837692i	$-0.316096\pi$
$983$	34.2462	1.09228	0.546142	+	0.837692i	$0.316096\pi$
$984$	0	0
$985$	−36.4924	−1.16275
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.78456	−0.215737
$990$	0	0
$991$	−36.4924	−1.15922	−0.579610	−	0.814894i	$-0.696795\pi$
$991$	−36.4924	−1.15922	−0.579610	+	0.814894i	$0.696795\pi$
$992$	0	0
$993$	44.9848	1.42755
$994$	0	0
$995$	30.9481	0.981122
$996$	0	0
$997$	−33.0619	−1.04708	−0.523540	−	0.852001i	$-0.675389\pi$
$997$	−33.0619	−1.04708	−0.523540	+	0.852001i	$0.675389\pi$
$998$	0	0
$999$	−56.9848	−1.80292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.a.x.1.4		4
4.3	odd	2		1792.2.a.v.1.1		4
8.3	odd	2		1792.2.a.v.1.4		4
8.5	even	2	inner	1792.2.a.x.1.1		4
16.3	odd	4		224.2.b.b.113.1		4
16.5	even	4		56.2.b.b.29.4	yes	4
16.11	odd	4		224.2.b.b.113.4		4
16.13	even	4		56.2.b.b.29.3	✓	4
48.5	odd	4		504.2.c.d.253.1		4
48.11	even	4		2016.2.c.c.1009.2		4
48.29	odd	4		504.2.c.d.253.2		4
48.35	even	4		2016.2.c.c.1009.3		4
112.3	even	12		1568.2.t.e.177.4		8
112.5	odd	12		392.2.p.e.165.3		8
112.11	odd	12		1568.2.t.d.177.4		8
112.13	odd	4		392.2.b.c.197.3		4
112.19	even	12		1568.2.t.e.753.1		8
112.27	even	4		1568.2.b.d.785.1		4
112.37	even	12		392.2.p.f.165.3		8
112.45	odd	12		392.2.p.e.373.3		8
112.51	odd	12		1568.2.t.d.753.4		8
112.53	even	12		392.2.p.f.373.1		8
112.59	even	12		1568.2.t.e.177.1		8
112.61	odd	12		392.2.p.e.165.1		8
112.67	odd	12		1568.2.t.d.177.1		8
112.69	odd	4		392.2.b.c.197.4		4
112.75	even	12		1568.2.t.e.753.4		8
112.83	even	4		1568.2.b.d.785.4		4
112.93	even	12		392.2.p.f.165.1		8
112.101	odd	12		392.2.p.e.373.1		8
112.107	odd	12		1568.2.t.d.753.1		8
112.109	even	12		392.2.p.f.373.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.b.b.29.3	✓	4	16.13	even	4
56.2.b.b.29.4	yes	4	16.5	even	4
224.2.b.b.113.1		4	16.3	odd	4
224.2.b.b.113.4		4	16.11	odd	4
392.2.b.c.197.3		4	112.13	odd	4
392.2.b.c.197.4		4	112.69	odd	4
392.2.p.e.165.1		8	112.61	odd	12
392.2.p.e.165.3		8	112.5	odd	12
392.2.p.e.373.1		8	112.101	odd	12
392.2.p.e.373.3		8	112.45	odd	12
392.2.p.f.165.1		8	112.93	even	12
392.2.p.f.165.3		8	112.37	even	12
392.2.p.f.373.1		8	112.53	even	12
392.2.p.f.373.3		8	112.109	even	12
504.2.c.d.253.1		4	48.5	odd	4
504.2.c.d.253.2		4	48.29	odd	4
1568.2.b.d.785.1		4	112.27	even	4
1568.2.b.d.785.4		4	112.83	even	4
1568.2.t.d.177.1		8	112.67	odd	12
1568.2.t.d.177.4		8	112.11	odd	12
1568.2.t.d.753.1		8	112.107	odd	12
1568.2.t.d.753.4		8	112.51	odd	12
1568.2.t.e.177.1		8	112.59	even	12
1568.2.t.e.177.4		8	112.3	even	12
1568.2.t.e.753.1		8	112.19	even	12
1568.2.t.e.753.4		8	112.75	even	12
1792.2.a.v.1.1		4	4.3	odd	2
1792.2.a.v.1.4		4	8.3	odd	2
1792.2.a.x.1.1		4	8.5	even	2	inner
1792.2.a.x.1.4		4	1.1	even	1	trivial
2016.2.c.c.1009.2		4	48.11	even	4
2016.2.c.c.1009.3		4	48.35	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.a (trivial)

\(f(q)\)	\(=\)	\(q+3.02045 q^{3} +1.69614 q^{5} +1.00000 q^{7} +6.12311 q^{9} +O(q^{10})\)	\(q+3.02045 q^{3} +1.69614 q^{5} +1.00000 q^{7} +6.12311 q^{9} +1.32431 q^{11} +1.69614 q^{13} +5.12311 q^{15} +2.00000 q^{17} +3.02045 q^{19} +3.02045 q^{21} -5.12311 q^{23} -2.12311 q^{25} +9.43318 q^{27} -6.04090 q^{29} -10.2462 q^{31} +4.00000 q^{33} +1.69614 q^{35} -6.04090 q^{37} +5.12311 q^{39} -4.24621 q^{41} +1.32431 q^{43} +10.3857 q^{45} +1.00000 q^{49} +6.04090 q^{51} +2.64861 q^{53} +2.24621 q^{55} +9.12311 q^{57} +0.371834 q^{59} +1.69614 q^{61} +6.12311 q^{63} +2.87689 q^{65} -11.5012 q^{67} -15.4741 q^{69} +8.00000 q^{71} +6.00000 q^{73} -6.41273 q^{75} +1.32431 q^{77} +10.1231 q^{81} -5.66906 q^{83} +3.39228 q^{85} -18.2462 q^{87} +16.2462 q^{89} +1.69614 q^{91} -30.9481 q^{93} +5.12311 q^{95} +12.2462 q^{97} +8.10887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^7 + 8 * q^9	\( 4 q + 4 q^{7} + 8 q^{9} + 4 q^{15} + 8 q^{17} - 4 q^{23} + 8 q^{25} - 8 q^{31} + 16 q^{33} + 4 q^{39} + 16 q^{41} + 4 q^{49} - 24 q^{55} + 20 q^{57} + 8 q^{63} + 28 q^{65} + 32 q^{71} + 24 q^{73} + 24 q^{81} - 40 q^{87} + 32 q^{89} + 4 q^{95} + 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 8 * q^9 + 4 * q^15 + 8 * q^17 - 4 * q^23 + 8 * q^25 - 8 * q^31 + 16 * q^33 + 4 * q^39 + 16 * q^41 + 4 * q^49 - 24 * q^55 + 20 * q^57 + 8 * q^63 + 28 * q^65 + 32 * q^71 + 24 * q^73 + 24 * q^81 - 40 * q^87 + 32 * q^89 + 4 * q^95 + 16 * q^97

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