LMFDB - Embedded newform 1792.3.d.j.1023.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,3,Mod(1023,1792)]

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([1, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1792.1023");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1792.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.8284633734$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 5x^{14} + 48x^{12} + 180x^{10} + 1056x^{8} + 2880x^{6} + 12288x^{4} + 20480x^{2} + 65536 $ x^16 + 5x^14 + 48x^12 + 180x^10 + 1056x^8 + 2880x^6 + 12288x^4 + 20480*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{38} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1023.2
Root			$1.45617 + 1.37098i$ of defining polynomial
Character	$\chi$	$=$	1792.1023
Dual form			1792.3.d.j.1023.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.22363i q^{3} +6.26788 q^{5} +2.64575i q^{7} -18.2863 q^{9} +O(q^{10})$	$q-5.22363i q^{3} +6.26788 q^{5} +2.64575i q^{7} -18.2863 q^{9} -9.80688i q^{11} -2.41653 q^{13} -32.7411i q^{15} +6.89452 q^{17} +2.77637i q^{19} +13.8204 q^{21} -42.8332i q^{23} +14.2863 q^{25} +48.5083i q^{27} +37.3505 q^{29} -7.16835i q^{31} -51.2275 q^{33} +16.5833i q^{35} -0.202653 q^{37} +12.6231i q^{39} -63.5494 q^{41} +35.3384i q^{43} -114.616 q^{45} -37.9129i q^{47} -7.00000 q^{49} -36.0144i q^{51} -54.6651 q^{53} -61.4684i q^{55} +14.5027 q^{57} -104.795i q^{59} -43.7668 q^{61} -48.3810i q^{63} -15.1465 q^{65} +31.1021i q^{67} -223.745 q^{69} -23.1294i q^{71} +69.2275 q^{73} -74.6264i q^{75} +25.9466 q^{77} +19.9328i q^{79} +88.8125 q^{81} -5.11617i q^{83} +43.2140 q^{85} -195.105i q^{87} +17.9889 q^{89} -6.39353i q^{91} -37.4448 q^{93} +17.4019i q^{95} +12.4864 q^{97} +179.332i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 96 q^{9}+O(q^{10}) $ 16 * q - 96 * q^9	$ 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) $ 16 * q - 96 * q^9 - 160 * q^17 + 32 * q^25 + 64 * q^33 - 256 * q^41 - 112 * q^49 - 112 * q^57 - 144 * q^65 + 224 * q^73 + 96 * q^81 + 1024 * q^89 + 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 5.22363i	− 1.74121i	−0.491982	−	0.870605i	$-0.663728\pi$
$3$	− 5.22363i	− 1.74121i	0.491982	−	0.870605i	$-0.336272\pi$
$4$	0	0
$5$	6.26788	1.25358	0.626788	−	0.779190i	$-0.284369\pi$
$5$	6.26788	1.25358	0.626788	+	0.779190i	$0.284369\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−18.2863	−2.03181
$10$	0	0
$11$	− 9.80688i	− 0.891535i	−0.895149	−	0.445767i	$-0.852931\pi$
$11$	− 9.80688i	− 0.891535i	0.895149	−	0.445767i	$-0.147069\pi$
$12$	0	0
$13$	−2.41653	−0.185887	−0.0929434	−	0.995671i	$-0.529628\pi$
$13$	−2.41653	−0.185887	−0.0929434	+	0.995671i	$0.529628\pi$
$14$	0	0
$15$	− 32.7411i	− 2.18274i
$16$	0	0
$17$	6.89452	0.405560	0.202780	−	0.979224i	$-0.435002\pi$
$17$	6.89452	0.405560	0.202780	+	0.979224i	$0.435002\pi$
$18$	0	0
$19$	2.77637i	0.146125i	0.997327	+	0.0730624i	$0.0232772\pi$
$19$	2.77637i	0.146125i	−0.997327	+	0.0730624i	$0.976723\pi$
$20$	0	0
$21$	13.8204	0.658116
$22$	0	0
$23$	− 42.8332i	− 1.86231i	−0.364617	−	0.931157i	$-0.618800\pi$
$23$	− 42.8332i	− 1.86231i	0.364617	−	0.931157i	$-0.381200\pi$
$24$	0	0
$25$	14.2863	0.571453
$26$	0	0
$27$	48.5083i	1.79660i
$28$	0	0
$29$	37.3505	1.28795	0.643974	−	0.765048i	$-0.277285\pi$
$29$	37.3505	1.28795	0.643974	+	0.765048i	$0.277285\pi$
$30$	0	0
$31$	− 7.16835i	− 0.231237i	−0.993294	−	0.115619i	$-0.963115\pi$
$31$	− 7.16835i	− 0.231237i	0.993294	−	0.115619i	$-0.0368850\pi$
$32$	0	0
$33$	−51.2275	−1.55235
$34$	0	0
$35$	16.5833i	0.473807i
$36$	0	0
$37$	−0.202653	−0.00547709	−0.00273855	−	0.999996i	$-0.500872\pi$
$37$	−0.202653	−0.00547709	−0.00273855	+	0.999996i	$0.500872\pi$
$38$	0	0
$39$	12.6231i	0.323668i
$40$	0	0
$41$	−63.5494	−1.54999	−0.774993	−	0.631970i	$-0.782247\pi$
$41$	−63.5494	−1.54999	−0.774993	+	0.631970i	$0.782247\pi$
$42$	0	0
$43$	35.3384i	0.821823i	0.911675	+	0.410911i	$0.134789\pi$
$43$	35.3384i	0.821823i	−0.911675	+	0.410911i	$0.865211\pi$
$44$	0	0
$45$	−114.616	−2.54703
$46$	0	0
$47$	− 37.9129i	− 0.806657i	−0.915055	−	0.403329i	$-0.867853\pi$
$47$	− 37.9129i	− 0.806657i	0.915055	−	0.403329i	$-0.132147\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 36.0144i	− 0.706165i
$52$	0	0
$53$	−54.6651	−1.03142	−0.515709	−	0.856764i	$-0.672471\pi$
$53$	−54.6651	−1.03142	−0.515709	+	0.856764i	$0.672471\pi$
$54$	0	0
$55$	− 61.4684i	− 1.11761i
$56$	0	0
$57$	14.5027	0.254434
$58$	0	0
$59$	− 104.795i	− 1.77619i	−0.459665	−	0.888093i	$-0.652030\pi$
$59$	− 104.795i	− 1.77619i	0.459665	−	0.888093i	$-0.347970\pi$
$60$	0	0
$61$	−43.7668	−0.717489	−0.358745	−	0.933436i	$-0.616795\pi$
$61$	−43.7668	−0.717489	−0.358745	+	0.933436i	$0.616795\pi$
$62$	0	0
$63$	− 48.3810i	− 0.767953i
$64$	0	0
$65$	−15.1465	−0.233023
$66$	0	0
$67$	31.1021i	0.464210i	0.972691	+	0.232105i	$0.0745613\pi$
$67$	31.1021i	0.464210i	−0.972691	+	0.232105i	$0.925439\pi$
$68$	0	0
$69$	−223.745	−3.24268
$70$	0	0
$71$	− 23.1294i	− 0.325766i	−0.986645	−	0.162883i	$-0.947921\pi$
$71$	− 23.1294i	− 0.325766i	0.986645	−	0.162883i	$-0.0520794\pi$
$72$	0	0
$73$	69.2275	0.948322	0.474161	−	0.880438i	$-0.342751\pi$
$73$	69.2275	0.948322	0.474161	+	0.880438i	$0.342751\pi$
$74$	0	0
$75$	− 74.6264i	− 0.995019i
$76$	0	0
$77$	25.9466	0.336968
$78$	0	0
$79$	19.9328i	0.252315i	0.992010	+	0.126157i	$0.0402644\pi$
$79$	19.9328i	0.252315i	−0.992010	+	0.126157i	$0.959736\pi$
$80$	0	0
$81$	88.8125	1.09645
$82$	0	0
$83$	− 5.11617i	− 0.0616406i	−0.999525	−	0.0308203i	$-0.990188\pi$
$83$	− 5.11617i	− 0.0616406i	0.999525	−	0.0308203i	$-0.00981197\pi$
$84$	0	0
$85$	43.2140	0.508400
$86$	0	0
$87$	− 195.105i	− 2.24259i
$88$	0	0
$89$	17.9889	0.202122	0.101061	−	0.994880i	$-0.467776\pi$
$89$	17.9889	0.202122	0.101061	+	0.994880i	$0.467776\pi$
$90$	0	0
$91$	− 6.39353i	− 0.0702586i
$92$	0	0
$93$	−37.4448	−0.402632
$94$	0	0
$95$	17.4019i	0.183178i
$96$	0	0
$97$	12.4864	0.128726	0.0643629	−	0.997927i	$-0.479498\pi$
$97$	12.4864	0.128726	0.0643629	+	0.997927i	$0.479498\pi$
$98$	0	0
$99$	179.332i	1.81143i
$100$	0	0
$101$	−68.0753	−0.674013	−0.337006	−	0.941502i	$-0.609414\pi$
$101$	−68.0753	−0.674013	−0.337006	+	0.941502i	$0.609414\pi$
$102$	0	0
$103$	58.2931i	0.565952i	0.959127	+	0.282976i	$0.0913217\pi$
$103$	58.2931i	0.565952i	−0.959127	+	0.282976i	$0.908678\pi$
$104$	0	0
$105$	86.6248	0.824998
$106$	0	0
$107$	− 135.868i	− 1.26979i	−0.772597	−	0.634897i	$-0.781043\pi$
$107$	− 135.868i	− 1.26979i	0.772597	−	0.634897i	$-0.218957\pi$
$108$	0	0
$109$	−44.4981	−0.408239	−0.204120	−	0.978946i	$-0.565433\pi$
$109$	−44.4981	−0.408239	−0.204120	+	0.978946i	$0.565433\pi$
$110$	0	0
$111$	1.05858i	0.00953677i
$112$	0	0
$113$	−133.391	−1.18045	−0.590224	−	0.807240i	$-0.700961\pi$
$113$	−133.391	−1.18045	−0.590224	+	0.807240i	$0.700961\pi$
$114$	0	0
$115$	− 268.474i	− 2.33455i
$116$	0	0
$117$	44.1894	0.377687
$118$	0	0
$119$	18.2412i	0.153287i
$120$	0	0
$121$	24.8251	0.205166
$122$	0	0
$123$	331.959i	2.69885i
$124$	0	0
$125$	−67.1521	−0.537217
$126$	0	0
$127$	− 130.977i	− 1.03131i	−0.856795	−	0.515657i	$-0.827548\pi$
$127$	− 130.977i	− 1.03131i	0.856795	−	0.515657i	$-0.172452\pi$
$128$	0	0
$129$	184.595	1.43097
$130$	0	0
$131$	− 53.3311i	− 0.407108i	−0.979064	−	0.203554i	$-0.934751\pi$
$131$	− 53.3311i	− 0.407108i	0.979064	−	0.203554i	$-0.0652492\pi$
$132$	0	0
$133$	−7.34558	−0.0552299
$134$	0	0
$135$	304.044i	2.25218i
$136$	0	0
$137$	57.7179	0.421299	0.210649	−	0.977562i	$-0.432442\pi$
$137$	57.7179	0.421299	0.210649	+	0.977562i	$0.432442\pi$
$138$	0	0
$139$	172.422i	1.24045i	0.784425	+	0.620224i	$0.212958\pi$
$139$	172.422i	1.24045i	−0.784425	+	0.620224i	$0.787042\pi$
$140$	0	0
$141$	−198.043	−1.40456
$142$	0	0
$143$	23.6986i	0.165725i
$144$	0	0
$145$	234.108	1.61454
$146$	0	0
$147$	36.5654i	0.248744i
$148$	0	0
$149$	219.529	1.47335	0.736673	−	0.676249i	$-0.236396\pi$
$149$	219.529	1.47335	0.736673	+	0.676249i	$0.236396\pi$
$150$	0	0
$151$	185.668i	1.22959i	0.788687	+	0.614795i	$0.210761\pi$
$151$	185.668i	1.22959i	−0.788687	+	0.614795i	$0.789239\pi$
$152$	0	0
$153$	−126.075	−0.824022
$154$	0	0
$155$	− 44.9304i	− 0.289873i
$156$	0	0
$157$	188.182	1.19861	0.599305	−	0.800521i	$-0.295444\pi$
$157$	188.182	1.19861	0.599305	+	0.800521i	$0.295444\pi$
$158$	0	0
$159$	285.550i	1.79591i
$160$	0	0
$161$	113.326	0.703889
$162$	0	0
$163$	54.5154i	0.334450i	0.985919	+	0.167225i	$0.0534806\pi$
$163$	54.5154i	0.334450i	−0.985919	+	0.167225i	$0.946519\pi$
$164$	0	0
$165$	−321.088	−1.94599
$166$	0	0
$167$	− 266.435i	− 1.59542i	−0.603042	−	0.797709i	$-0.706045\pi$
$167$	− 266.435i	− 1.59542i	0.603042	−	0.797709i	$-0.293955\pi$
$168$	0	0
$169$	−163.160	−0.965446
$170$	0	0
$171$	− 50.7696i	− 0.296898i
$172$	0	0
$173$	−114.835	−0.663786	−0.331893	−	0.943317i	$-0.607687\pi$
$173$	−114.835	−0.663786	−0.331893	+	0.943317i	$0.607687\pi$
$174$	0	0
$175$	37.7980i	0.215989i
$176$	0	0
$177$	−547.410	−3.09271
$178$	0	0
$179$	112.849i	0.630439i	0.949019	+	0.315220i	$0.102078\pi$
$179$	112.849i	0.630439i	−0.949019	+	0.315220i	$0.897922\pi$
$180$	0	0
$181$	60.9470	0.336724	0.168362	−	0.985725i	$-0.446152\pi$
$181$	60.9470	0.336724	0.168362	+	0.985725i	$0.446152\pi$
$182$	0	0
$183$	228.622i	1.24930i
$184$	0	0
$185$	−1.27020	−0.00686595
$186$	0	0
$187$	− 67.6138i	− 0.361571i
$188$	0	0
$189$	−128.341	−0.679052
$190$	0	0
$191$	− 178.459i	− 0.934342i	−0.884167	−	0.467171i	$-0.845273\pi$
$191$	− 178.459i	− 0.934342i	0.884167	−	0.467171i	$-0.154727\pi$
$192$	0	0
$193$	221.588	1.14812	0.574062	−	0.818812i	$-0.305367\pi$
$193$	221.588	1.14812	0.574062	+	0.818812i	$0.305367\pi$
$194$	0	0
$195$	79.1198i	0.405742i
$196$	0	0
$197$	−242.298	−1.22994	−0.614970	−	0.788550i	$-0.710832\pi$
$197$	−242.298	−1.22994	−0.614970	+	0.788550i	$0.710832\pi$
$198$	0	0
$199$	297.047i	1.49270i	0.665555	+	0.746349i	$0.268195\pi$
$199$	297.047i	1.49270i	−0.665555	+	0.746349i	$0.731805\pi$
$200$	0	0
$201$	162.466	0.808287
$202$	0	0
$203$	98.8201i	0.486798i
$204$	0	0
$205$	−398.320	−1.94302
$206$	0	0
$207$	783.262i	3.78388i
$208$	0	0
$209$	27.2275	0.130275
$210$	0	0
$211$	− 141.020i	− 0.668341i	−0.942513	−	0.334171i	$-0.891544\pi$
$211$	− 141.020i	− 0.668341i	0.942513	−	0.334171i	$-0.108456\pi$
$212$	0	0
$213$	−120.820	−0.567228
$214$	0	0
$215$	221.497i	1.03022i
$216$	0	0
$217$	18.9657	0.0873994
$218$	0	0
$219$	− 361.619i	− 1.65123i
$220$	0	0
$221$	−16.6608	−0.0753883
$222$	0	0
$223$	− 40.8267i	− 0.183079i	−0.995801	−	0.0915396i	$-0.970821\pi$
$223$	− 40.8267i	− 0.183079i	0.995801	−	0.0915396i	$-0.0291788\pi$
$224$	0	0
$225$	−261.244	−1.16108
$226$	0	0
$227$	8.18598i	0.0360616i	0.999837	+	0.0180308i	$0.00573969\pi$
$227$	8.18598i	0.0360616i	−0.999837	+	0.0180308i	$0.994260\pi$
$228$	0	0
$229$	−332.252	−1.45088	−0.725440	−	0.688285i	$-0.758364\pi$
$229$	−332.252	−1.45088	−0.725440	+	0.688285i	$0.758364\pi$
$230$	0	0
$231$	− 135.535i	− 0.586733i
$232$	0	0
$233$	−329.260	−1.41314	−0.706568	−	0.707646i	$-0.749757\pi$
$233$	−329.260	−1.41314	−0.706568	+	0.707646i	$0.749757\pi$
$234$	0	0
$235$	− 237.633i	− 1.01121i
$236$	0	0
$237$	104.122	0.439333
$238$	0	0
$239$	137.719i	0.576230i	0.957596	+	0.288115i	$0.0930286\pi$
$239$	137.719i	0.576230i	−0.957596	+	0.288115i	$0.906971\pi$
$240$	0	0
$241$	201.854	0.837567	0.418783	−	0.908086i	$-0.362457\pi$
$241$	201.854	0.837567	0.418783	+	0.908086i	$0.362457\pi$
$242$	0	0
$243$	− 27.3492i	− 0.112548i
$244$	0	0
$245$	−43.8752	−0.179082
$246$	0	0
$247$	− 6.70917i	− 0.0271627i
$248$	0	0
$249$	−26.7250	−0.107329
$250$	0	0
$251$	269.203i	1.07252i	0.844052	+	0.536261i	$0.180164\pi$
$251$	269.203i	1.07252i	−0.844052	+	0.536261i	$0.819836\pi$
$252$	0	0
$253$	−420.061	−1.66032
$254$	0	0
$255$	− 225.734i	− 0.885232i
$256$	0	0
$257$	−242.359	−0.943032	−0.471516	−	0.881858i	$-0.656293\pi$
$257$	−242.359	−0.943032	−0.471516	+	0.881858i	$0.656293\pi$
$258$	0	0
$259$	− 0.536168i	− 0.00207015i
$260$	0	0
$261$	−683.002	−2.61687
$262$	0	0
$263$	33.8470i	0.128696i	0.997928	+	0.0643479i	$0.0204967\pi$
$263$	33.8470i	0.128696i	−0.997928	+	0.0643479i	$0.979503\pi$
$264$	0	0
$265$	−342.634	−1.29296
$266$	0	0
$267$	− 93.9673i	− 0.351937i
$268$	0	0
$269$	165.598	0.615606	0.307803	−	0.951450i	$-0.400406\pi$
$269$	165.598	0.615606	0.307803	+	0.951450i	$0.400406\pi$
$270$	0	0
$271$	− 148.308i	− 0.547263i	−0.961835	−	0.273632i	$-0.911775\pi$
$271$	− 148.308i	− 0.547263i	0.961835	−	0.273632i	$-0.0882249\pi$
$272$	0	0
$273$	−33.3975	−0.122335
$274$	0	0
$275$	− 140.104i	− 0.509470i
$276$	0	0
$277$	478.358	1.72693	0.863463	−	0.504413i	$-0.168291\pi$
$277$	478.358	1.72693	0.863463	+	0.504413i	$0.168291\pi$
$278$	0	0
$279$	131.083i	0.469830i
$280$	0	0
$281$	226.066	0.804506	0.402253	−	0.915528i	$-0.368227\pi$
$281$	226.066	0.804506	0.402253	+	0.915528i	$0.368227\pi$
$282$	0	0
$283$	254.628i	0.899745i	0.893093	+	0.449873i	$0.148531\pi$
$283$	254.628i	0.899745i	−0.893093	+	0.449873i	$0.851469\pi$
$284$	0	0
$285$	90.9014	0.318952
$286$	0	0
$287$	− 168.136i	− 0.585840i
$288$	0	0
$289$	−241.466	−0.835521
$290$	0	0
$291$	− 65.2244i	− 0.224139i
$292$	0	0
$293$	149.558	0.510437	0.255218	−	0.966883i	$-0.417853\pi$
$293$	149.558	0.510437	0.255218	+	0.966883i	$0.417853\pi$
$294$	0	0
$295$	− 656.842i	− 2.22658i
$296$	0	0
$297$	475.715	1.60173
$298$	0	0
$299$	103.508i	0.346180i
$300$	0	0
$301$	−93.4966	−0.310620
$302$	0	0
$303$	355.600i	1.17360i
$304$	0	0
$305$	−274.325	−0.899427
$306$	0	0
$307$	271.779i	0.885272i	0.896701	+	0.442636i	$0.145957\pi$
$307$	271.779i	0.885272i	−0.896701	+	0.442636i	$0.854043\pi$
$308$	0	0
$309$	304.502	0.985442
$310$	0	0
$311$	− 534.180i	− 1.71762i	−0.512293	−	0.858811i	$-0.671204\pi$
$311$	− 534.180i	− 1.71762i	0.512293	−	0.858811i	$-0.328796\pi$
$312$	0	0
$313$	556.232	1.77710	0.888550	−	0.458780i	$-0.151713\pi$
$313$	556.232	1.77710	0.888550	+	0.458780i	$0.151713\pi$
$314$	0	0
$315$	− 303.247i	− 0.962688i
$316$	0	0
$317$	387.459	1.22227	0.611134	−	0.791527i	$-0.290714\pi$
$317$	387.459	1.22227	0.611134	+	0.791527i	$0.290714\pi$
$318$	0	0
$319$	− 366.292i	− 1.14825i
$320$	0	0
$321$	−709.724	−2.21098
$322$	0	0
$323$	19.1417i	0.0592624i
$324$	0	0
$325$	−34.5233	−0.106225
$326$	0	0
$327$	232.442i	0.710830i
$328$	0	0
$329$	100.308	0.304888
$330$	0	0
$331$	383.205i	1.15772i	0.815427	+	0.578860i	$0.196502\pi$
$331$	383.205i	1.15772i	−0.815427	+	0.578860i	$0.803498\pi$
$332$	0	0
$333$	3.70577	0.0111284
$334$	0	0
$335$	194.944i	0.581923i
$336$	0	0
$337$	563.726	1.67278	0.836388	−	0.548138i	$-0.184663\pi$
$337$	563.726	1.67278	0.836388	+	0.548138i	$0.184663\pi$
$338$	0	0
$339$	696.783i	2.05541i
$340$	0	0
$341$	−70.2992	−0.206156
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−1402.41	−4.06495
$346$	0	0
$347$	− 51.5890i	− 0.148671i	−0.997233	−	0.0743357i	$-0.976316\pi$
$347$	− 51.5890i	− 0.148671i	0.997233	−	0.0743357i	$-0.0236836\pi$
$348$	0	0
$349$	586.383	1.68018	0.840090	−	0.542447i	$-0.182502\pi$
$349$	586.383	1.68018	0.840090	+	0.542447i	$0.182502\pi$
$350$	0	0
$351$	− 117.222i	− 0.333965i
$352$	0	0
$353$	303.844	0.860746	0.430373	−	0.902651i	$-0.358382\pi$
$353$	303.844	0.860746	0.430373	+	0.902651i	$0.358382\pi$
$354$	0	0
$355$	− 144.972i	− 0.408373i
$356$	0	0
$357$	95.2852	0.266905
$358$	0	0
$359$	− 116.130i	− 0.323481i	−0.986833	−	0.161741i	$-0.948289\pi$
$359$	− 116.130i	− 0.323481i	0.986833	−	0.161741i	$-0.0517108\pi$
$360$	0	0
$361$	353.292	0.978648
$362$	0	0
$363$	− 129.677i	− 0.357237i
$364$	0	0
$365$	433.910	1.18879
$366$	0	0
$367$	− 476.288i	− 1.29779i	−0.760879	−	0.648894i	$-0.775232\pi$
$367$	− 476.288i	− 1.29779i	0.760879	−	0.648894i	$-0.224768\pi$
$368$	0	0
$369$	1162.08	3.14928
$370$	0	0
$371$	− 144.630i	− 0.389839i
$372$	0	0
$373$	49.2857	0.132133	0.0660666	−	0.997815i	$-0.478955\pi$
$373$	49.2857	0.132133	0.0660666	+	0.997815i	$0.478955\pi$
$374$	0	0
$375$	350.778i	0.935407i
$376$	0	0
$377$	−90.2585	−0.239412
$378$	0	0
$379$	− 167.511i	− 0.441983i	−0.975276	−	0.220991i	$-0.929071\pi$
$379$	− 167.511i	− 0.441983i	0.975276	−	0.220991i	$-0.0709292\pi$
$380$	0	0
$381$	−684.174	−1.79573
$382$	0	0
$383$	− 513.207i	− 1.33997i	−0.742376	−	0.669983i	$-0.766301\pi$
$383$	− 513.207i	− 1.33997i	0.742376	−	0.669983i	$-0.233699\pi$
$384$	0	0
$385$	162.630	0.422416
$386$	0	0
$387$	− 646.209i	− 1.66979i
$388$	0	0
$389$	709.398	1.82365	0.911823	−	0.410584i	$-0.134675\pi$
$389$	709.398	1.82365	0.911823	+	0.410584i	$0.134675\pi$
$390$	0	0
$391$	− 295.315i	− 0.755281i
$392$	0	0
$393$	−278.582	−0.708860
$394$	0	0
$395$	124.937i	0.316295i
$396$	0	0
$397$	−309.404	−0.779355	−0.389677	−	0.920951i	$-0.627413\pi$
$397$	−309.404	−0.779355	−0.389677	+	0.920951i	$0.627413\pi$
$398$	0	0
$399$	38.3706i	0.0961669i
$400$	0	0
$401$	−528.073	−1.31689	−0.658445	−	0.752629i	$-0.728785\pi$
$401$	−528.073	−1.31689	−0.658445	+	0.752629i	$0.728785\pi$
$402$	0	0
$403$	17.3225i	0.0429839i
$404$	0	0
$405$	556.666	1.37448
$406$	0	0
$407$	1.98739i	0.00488302i
$408$	0	0
$409$	612.830	1.49836	0.749181	−	0.662366i	$-0.230448\pi$
$409$	612.830	1.49836	0.749181	+	0.662366i	$0.230448\pi$
$410$	0	0
$411$	− 301.497i	− 0.733569i
$412$	0	0
$413$	277.261	0.671335
$414$	0	0
$415$	− 32.0676i	− 0.0772712i
$416$	0	0
$417$	900.670	2.15988
$418$	0	0
$419$	− 237.642i	− 0.567165i	−0.958948	−	0.283583i	$-0.908477\pi$
$419$	− 237.642i	− 0.567165i	0.958948	−	0.283583i	$-0.0915230\pi$
$420$	0	0
$421$	−394.516	−0.937092	−0.468546	−	0.883439i	$-0.655222\pi$
$421$	−394.516	−0.937092	−0.468546	+	0.883439i	$0.655222\pi$
$422$	0	0
$423$	693.287i	1.63898i
$424$	0	0
$425$	98.4973	0.231758
$426$	0	0
$427$	− 115.796i	− 0.271185i
$428$	0	0
$429$	123.793	0.288561
$430$	0	0
$431$	− 363.359i	− 0.843059i	−0.906815	−	0.421530i	$-0.861493\pi$
$431$	− 363.359i	− 0.843059i	0.906815	−	0.421530i	$-0.138507\pi$
$432$	0	0
$433$	119.733	0.276520	0.138260	−	0.990396i	$-0.455849\pi$
$433$	119.733	0.276520	0.138260	+	0.990396i	$0.455849\pi$
$434$	0	0
$435$	− 1222.89i	− 2.81125i
$436$	0	0
$437$	118.921	0.272130
$438$	0	0
$439$	− 871.477i	− 1.98514i	−0.121672	−	0.992570i	$-0.538826\pi$
$439$	− 871.477i	− 1.98514i	0.121672	−	0.992570i	$-0.461174\pi$
$440$	0	0
$441$	128.004	0.290259
$442$	0	0
$443$	343.956i	0.776424i	0.921570	+	0.388212i	$0.126907\pi$
$443$	343.956i	0.776424i	−0.921570	+	0.388212i	$0.873093\pi$
$444$	0	0
$445$	112.752	0.253376
$446$	0	0
$447$	− 1146.74i	− 2.56541i
$448$	0	0
$449$	−242.849	−0.540866	−0.270433	−	0.962739i	$-0.587167\pi$
$449$	−242.849	−0.540866	−0.270433	+	0.962739i	$0.587167\pi$
$450$	0	0
$451$	623.222i	1.38187i
$452$	0	0
$453$	969.861	2.14097
$454$	0	0
$455$	− 40.0739i	− 0.0880745i
$456$	0	0
$457$	42.2571	0.0924662	0.0462331	−	0.998931i	$-0.485278\pi$
$457$	42.2571	0.0924662	0.0462331	+	0.998931i	$0.485278\pi$
$458$	0	0
$459$	334.441i	0.728631i
$460$	0	0
$461$	−816.370	−1.77087	−0.885434	−	0.464766i	$-0.846139\pi$
$461$	−816.370	−1.77087	−0.885434	+	0.464766i	$0.846139\pi$
$462$	0	0
$463$	115.161i	0.248727i	0.992237	+	0.124363i	$0.0396889\pi$
$463$	115.161i	0.248727i	−0.992237	+	0.124363i	$0.960311\pi$
$464$	0	0
$465$	−234.700	−0.504730
$466$	0	0
$467$	603.424i	1.29213i	0.763283	+	0.646064i	$0.223586\pi$
$467$	603.424i	1.29213i	−0.763283	+	0.646064i	$0.776414\pi$
$468$	0	0
$469$	−82.2884	−0.175455
$470$	0	0
$471$	− 982.992i	− 2.08703i
$472$	0	0
$473$	346.559	0.732684
$474$	0	0
$475$	39.6641i	0.0835034i
$476$	0	0
$477$	999.624	2.09565
$478$	0	0
$479$	158.595i	0.331097i	0.986202	+	0.165548i	$0.0529394\pi$
$479$	158.595i	0.331097i	−0.986202	+	0.165548i	$0.947061\pi$
$480$	0	0
$481$	0.489715	0.00101812
$482$	0	0
$483$	− 591.974i	− 1.22562i
$484$	0	0
$485$	78.2633	0.161368
$486$	0	0
$487$	106.987i	0.219687i	0.993949	+	0.109843i	$0.0350349\pi$
$487$	106.987i	0.219687i	−0.993949	+	0.109843i	$0.964965\pi$
$488$	0	0
$489$	284.768	0.582348
$490$	0	0
$491$	− 616.591i	− 1.25579i	−0.778299	−	0.627893i	$-0.783917\pi$
$491$	− 616.591i	− 1.25579i	0.778299	−	0.627893i	$-0.216083\pi$
$492$	0	0
$493$	257.514	0.522340
$494$	0	0
$495$	1124.03i	2.27077i
$496$	0	0
$497$	61.1947	0.123128
$498$	0	0
$499$	554.090i	1.11040i	0.831717	+	0.555200i	$0.187358\pi$
$499$	554.090i	1.11040i	−0.831717	+	0.555200i	$0.812642\pi$
$500$	0	0
$501$	−1391.76	−2.77796
$502$	0	0
$503$	− 148.158i	− 0.294548i	−0.989096	−	0.147274i	$-0.952950\pi$
$503$	− 148.158i	− 0.294548i	0.989096	−	0.147274i	$-0.0470499\pi$
$504$	0	0
$505$	−426.688	−0.844926
$506$	0	0
$507$	852.290i	1.68104i
$508$	0	0
$509$	182.889	0.359310	0.179655	−	0.983730i	$-0.442502\pi$
$509$	182.889	0.359310	0.179655	+	0.983730i	$0.442502\pi$
$510$	0	0
$511$	183.159i	0.358432i
$512$	0	0
$513$	−134.677	−0.262528
$514$	0	0
$515$	365.374i	0.709464i
$516$	0	0
$517$	−371.807	−0.719163
$518$	0	0
$519$	599.856i	1.15579i
$520$	0	0
$521$	−98.2461	−0.188572	−0.0942861	−	0.995545i	$-0.530057\pi$
$521$	−98.2461	−0.188572	−0.0942861	+	0.995545i	$0.530057\pi$
$522$	0	0
$523$	574.764i	1.09898i	0.835502	+	0.549488i	$0.185177\pi$
$523$	574.764i	1.09898i	−0.835502	+	0.549488i	$0.814823\pi$
$524$	0	0
$525$	197.443	0.376082
$526$	0	0
$527$	− 49.4223i	− 0.0937805i
$528$	0	0
$529$	−1305.69	−2.46822
$530$	0	0
$531$	1916.31i	3.60888i
$532$	0	0
$533$	153.569	0.288122
$534$	0	0
$535$	− 851.604i	− 1.59178i
$536$	0	0
$537$	589.480	1.09773
$538$	0	0
$539$	68.6482i	0.127362i
$540$	0	0
$541$	370.654	0.685128	0.342564	−	0.939495i	$-0.388705\pi$
$541$	370.654	0.685128	0.342564	+	0.939495i	$0.388705\pi$
$542$	0	0
$543$	− 318.365i	− 0.586307i
$544$	0	0
$545$	−278.909	−0.511759
$546$	0	0
$547$	− 56.5966i	− 0.103467i	−0.998661	−	0.0517336i	$-0.983525\pi$
$547$	− 56.5966i	− 0.103467i	0.998661	−	0.0517336i	$-0.0164747\pi$
$548$	0	0
$549$	800.334	1.45780
$550$	0	0
$551$	103.699i	0.188201i
$552$	0	0
$553$	−52.7374	−0.0953659
$554$	0	0
$555$	6.63506i	0.0119551i
$556$	0	0
$557$	151.525	0.272037	0.136019	−	0.990706i	$-0.456569\pi$
$557$	151.525	0.272037	0.136019	+	0.990706i	$0.456569\pi$
$558$	0	0
$559$	− 85.3962i	− 0.152766i
$560$	0	0
$561$	−353.189	−0.629571
$562$	0	0
$563$	− 318.048i	− 0.564917i	−0.959280	−	0.282458i	$-0.908850\pi$
$563$	− 318.048i	− 0.564917i	0.959280	−	0.282458i	$-0.0911499\pi$
$564$	0	0
$565$	−836.076	−1.47978
$566$	0	0
$567$	234.976i	0.414419i
$568$	0	0
$569$	−356.654	−0.626808	−0.313404	−	0.949620i	$-0.601469\pi$
$569$	−356.654	−0.626808	−0.313404	+	0.949620i	$0.601469\pi$
$570$	0	0
$571$	− 831.014i	− 1.45537i	−0.685914	−	0.727683i	$-0.740597\pi$
$571$	− 831.014i	− 1.45537i	0.685914	−	0.727683i	$-0.259403\pi$
$572$	0	0
$573$	−932.205	−1.62689
$574$	0	0
$575$	− 611.929i	− 1.06422i
$576$	0	0
$577$	771.483	1.33706	0.668530	−	0.743685i	$-0.266924\pi$
$577$	771.483	1.33706	0.668530	+	0.743685i	$0.266924\pi$
$578$	0	0
$579$	− 1157.49i	− 1.99912i
$580$	0	0
$581$	13.5361	0.0232980
$582$	0	0
$583$	536.094i	0.919544i
$584$	0	0
$585$	276.974	0.473460
$586$	0	0
$587$	− 144.376i	− 0.245956i	−0.992409	−	0.122978i	$-0.960756\pi$
$587$	− 144.376i	− 0.245956i	0.992409	−	0.122978i	$-0.0392445\pi$
$588$	0	0
$589$	19.9020	0.0337894
$590$	0	0
$591$	1265.68i	2.14158i
$592$	0	0
$593$	838.926	1.41471	0.707357	−	0.706856i	$-0.249887\pi$
$593$	838.926	1.41471	0.707357	+	0.706856i	$0.249887\pi$
$594$	0	0
$595$	114.334i	0.192157i
$596$	0	0
$597$	1551.66	2.59910
$598$	0	0
$599$	− 711.341i	− 1.18755i	−0.804632	−	0.593774i	$-0.797637\pi$
$599$	− 711.341i	− 1.18755i	0.804632	−	0.593774i	$-0.202363\pi$
$600$	0	0
$601$	356.394	0.593002	0.296501	−	0.955033i	$-0.404180\pi$
$601$	356.394	0.593002	0.296501	+	0.955033i	$0.404180\pi$
$602$	0	0
$603$	− 568.742i	− 0.943188i
$604$	0	0
$605$	155.601	0.257191
$606$	0	0
$607$	60.3719i	0.0994595i	0.998763	+	0.0497298i	$0.0158360\pi$
$607$	60.3719i	0.0994595i	−0.998763	+	0.0497298i	$0.984164\pi$
$608$	0	0
$609$	516.199	0.847618
$610$	0	0
$611$	91.6176i	0.149947i
$612$	0	0
$613$	482.989	0.787911	0.393955	−	0.919130i	$-0.371106\pi$
$613$	482.989	0.787911	0.393955	+	0.919130i	$0.371106\pi$
$614$	0	0
$615$	2080.68i	3.38321i
$616$	0	0
$617$	−712.490	−1.15476	−0.577382	−	0.816474i	$-0.695926\pi$
$617$	−712.490	−1.15476	−0.577382	+	0.816474i	$0.695926\pi$
$618$	0	0
$619$	93.0817i	0.150374i	0.997169	+	0.0751872i	$0.0239554\pi$
$619$	93.0817i	0.150374i	−0.997169	+	0.0751872i	$0.976045\pi$
$620$	0	0
$621$	2077.77	3.34584
$622$	0	0
$623$	47.5941i	0.0763951i
$624$	0	0
$625$	−778.059	−1.24489
$626$	0	0
$627$	− 142.227i	− 0.226837i
$628$	0	0
$629$	−1.39719	−0.00222129
$630$	0	0
$631$	− 610.573i	− 0.967628i	−0.875171	−	0.483814i	$-0.839251\pi$
$631$	− 610.573i	− 0.967628i	0.875171	−	0.483814i	$-0.160749\pi$
$632$	0	0
$633$	−736.636	−1.16372
$634$	0	0
$635$	− 820.947i	− 1.29283i
$636$	0	0
$637$	16.9157	0.0265553
$638$	0	0
$639$	422.952i	0.661897i
$640$	0	0
$641$	590.153	0.920676	0.460338	−	0.887744i	$-0.347728\pi$
$641$	590.153	0.920676	0.460338	+	0.887744i	$0.347728\pi$
$642$	0	0
$643$	− 257.971i	− 0.401199i	−0.979673	−	0.200600i	$-0.935711\pi$
$643$	− 257.971i	− 0.401199i	0.979673	−	0.200600i	$-0.0642890\pi$
$644$	0	0
$645$	1157.02	1.79383
$646$	0	0
$647$	− 379.964i	− 0.587271i	−0.955918	−	0.293635i	$-0.905135\pi$
$647$	− 379.964i	− 0.587271i	0.955918	−	0.293635i	$-0.0948651\pi$
$648$	0	0
$649$	−1027.71	−1.58353
$650$	0	0
$651$	− 99.0696i	− 0.152181i
$652$	0	0
$653$	952.773	1.45907	0.729535	−	0.683943i	$-0.239736\pi$
$653$	952.773	1.45907	0.729535	+	0.683943i	$0.239736\pi$
$654$	0	0
$655$	− 334.273i	− 0.510340i
$656$	0	0
$657$	−1265.92	−1.92681
$658$	0	0
$659$	− 963.119i	− 1.46149i	−0.682653	−	0.730743i	$-0.739174\pi$
$659$	− 963.119i	− 1.46149i	0.682653	−	0.730743i	$-0.260826\pi$
$660$	0	0
$661$	71.6817	0.108444	0.0542222	−	0.998529i	$-0.482732\pi$
$661$	71.6817	0.108444	0.0542222	+	0.998529i	$0.482732\pi$
$662$	0	0
$663$	87.0299i	0.131267i
$664$	0	0
$665$	−46.0412	−0.0692349
$666$	0	0
$667$	− 1599.84i	− 2.39856i
$668$	0	0
$669$	−213.263	−0.318779
$670$	0	0
$671$	429.216i	0.639667i
$672$	0	0
$673$	−712.783	−1.05911	−0.529556	−	0.848275i	$-0.677642\pi$
$673$	−712.783	−1.05911	−0.529556	+	0.848275i	$0.677642\pi$
$674$	0	0
$675$	693.005i	1.02667i
$676$	0	0
$677$	767.527	1.13372	0.566859	−	0.823815i	$-0.308159\pi$
$677$	767.527	1.13372	0.566859	+	0.823815i	$0.308159\pi$
$678$	0	0
$679$	33.0359i	0.0486538i
$680$	0	0
$681$	42.7605	0.0627908
$682$	0	0
$683$	484.354i	0.709156i	0.935026	+	0.354578i	$0.115375\pi$
$683$	484.354i	0.709156i	−0.935026	+	0.354578i	$0.884625\pi$
$684$	0	0
$685$	361.769	0.528130
$686$	0	0
$687$	1735.56i	2.52629i
$688$	0	0
$689$	132.100	0.191727
$690$	0	0
$691$	− 574.851i	− 0.831912i	−0.909385	−	0.415956i	$-0.863447\pi$
$691$	− 574.851i	− 0.831912i	0.909385	−	0.415956i	$-0.136553\pi$
$692$	0	0
$693$	−474.467	−0.684657
$694$	0	0
$695$	1080.72i	1.55500i
$696$	0	0
$697$	−438.143	−0.628612
$698$	0	0
$699$	1719.94i	2.46057i
$700$	0	0
$701$	143.138	0.204191	0.102096	−	0.994775i	$-0.467445\pi$
$701$	143.138	0.204191	0.102096	+	0.994775i	$0.467445\pi$
$702$	0	0
$703$	− 0.562638i	0 0.000800339i
$704$	0	0
$705$	−1241.31	−1.76072
$706$	0	0
$707$	− 180.110i	− 0.254753i
$708$	0	0
$709$	255.311	0.360101	0.180050	−	0.983657i	$-0.442374\pi$
$709$	255.311	0.360101	0.180050	+	0.983657i	$0.442374\pi$
$710$	0	0
$711$	− 364.498i	− 0.512656i
$712$	0	0
$713$	−307.044	−0.430636
$714$	0	0
$715$	148.540i	0.207748i
$716$	0	0
$717$	719.393	1.00334
$718$	0	0
$719$	− 415.630i	− 0.578067i	−0.957319	−	0.289034i	$-0.906666\pi$
$719$	− 415.630i	− 0.578067i	0.957319	−	0.289034i	$-0.0933340\pi$
$720$	0	0
$721$	−154.229	−0.213910
$722$	0	0
$723$	− 1054.41i	− 1.45838i
$724$	0	0
$725$	533.601	0.736001
$726$	0	0
$727$	896.838i	1.23361i	0.787114	+	0.616807i	$0.211574\pi$
$727$	896.838i	1.23361i	−0.787114	+	0.616807i	$0.788426\pi$
$728$	0	0
$729$	656.451	0.900481
$730$	0	0
$731$	243.641i	0.333299i
$732$	0	0
$733$	−509.059	−0.694487	−0.347244	−	0.937775i	$-0.612882\pi$
$733$	−509.059	−0.694487	−0.347244	+	0.937775i	$0.612882\pi$
$734$	0	0
$735$	229.188i	0.311820i
$736$	0	0
$737$	305.014	0.413859
$738$	0	0
$739$	741.427i	1.00328i	0.865075	+	0.501642i	$0.167271\pi$
$739$	741.427i	1.00328i	−0.865075	+	0.501642i	$0.832729\pi$
$740$	0	0
$741$	−35.0463	−0.0472959
$742$	0	0
$743$	1344.98i	1.81021i	0.425191	+	0.905104i	$0.360207\pi$
$743$	1344.98i	1.81021i	−0.425191	+	0.905104i	$0.639793\pi$
$744$	0	0
$745$	1375.98	1.84695
$746$	0	0
$747$	93.5560i	0.125242i
$748$	0	0
$749$	359.473	0.479937
$750$	0	0
$751$	− 27.6931i	− 0.0368749i	−0.999830	−	0.0184375i	$-0.994131\pi$
$751$	− 27.6931i	− 0.0368749i	0.999830	−	0.0184375i	$-0.00586916\pi$
$752$	0	0
$753$	1406.22	1.86749
$754$	0	0
$755$	1163.74i	1.54138i
$756$	0	0
$757$	−1341.69	−1.77238	−0.886192	−	0.463318i	$-0.846659\pi$
$757$	−1341.69	−1.77238	−0.886192	+	0.463318i	$0.846659\pi$
$758$	0	0
$759$	2194.24i	2.89096i
$760$	0	0
$761$	112.001	0.147176	0.0735881	−	0.997289i	$-0.476555\pi$
$761$	112.001	0.147176	0.0735881	+	0.997289i	$0.476555\pi$
$762$	0	0
$763$	− 117.731i	− 0.154300i
$764$	0	0
$765$	−790.226	−1.03297
$766$	0	0
$767$	253.240i	0.330169i
$768$	0	0
$769$	140.749	0.183028	0.0915142	−	0.995804i	$-0.470829\pi$
$769$	140.749	0.183028	0.0915142	+	0.995804i	$0.470829\pi$
$770$	0	0
$771$	1265.99i	1.64202i
$772$	0	0
$773$	1325.14	1.71428	0.857138	−	0.515087i	$-0.172240\pi$
$773$	1325.14	1.71428	0.857138	+	0.515087i	$0.172240\pi$
$774$	0	0
$775$	− 102.409i	− 0.132141i
$776$	0	0
$777$	−2.80074	−0.00360456
$778$	0	0
$779$	− 176.437i	− 0.226491i
$780$	0	0
$781$	−226.827	−0.290432
$782$	0	0
$783$	1811.81i	2.31393i
$784$	0	0
$785$	1179.50	1.50255
$786$	0	0
$787$	− 327.801i	− 0.416519i	−0.978074	−	0.208260i	$-0.933220\pi$
$787$	− 327.801i	− 0.416519i	0.978074	−	0.208260i	$-0.0667799\pi$
$788$	0	0
$789$	176.804	0.224086
$790$	0	0
$791$	− 352.918i	− 0.446167i
$792$	0	0
$793$	105.764	0.133372
$794$	0	0
$795$	1789.80i	2.25132i
$796$	0	0
$797$	393.650	0.493915	0.246958	−	0.969026i	$-0.420569\pi$
$797$	393.650	0.493915	0.246958	+	0.969026i	$0.420569\pi$
$798$	0	0
$799$	− 261.391i	− 0.327148i
$800$	0	0
$801$	−328.950	−0.410675
$802$	0	0
$803$	− 678.906i	− 0.845462i
$804$	0	0
$805$	710.314	0.882378
$806$	0	0
$807$	− 865.023i	− 1.07190i
$808$	0	0
$809$	−416.641	−0.515008	−0.257504	−	0.966277i	$-0.582900\pi$
$809$	−416.641	−0.515008	−0.257504	+	0.966277i	$0.582900\pi$
$810$	0	0
$811$	748.707i	0.923190i	0.887091	+	0.461595i	$0.152723\pi$
$811$	748.707i	0.923190i	−0.887091	+	0.461595i	$0.847277\pi$
$812$	0	0
$813$	−774.708	−0.952901
$814$	0	0
$815$	341.696i	0.419259i
$816$	0	0
$817$	−98.1124	−0.120089
$818$	0	0
$819$	116.914i	0.142752i
$820$	0	0
$821$	−554.169	−0.674993	−0.337496	−	0.941327i	$-0.609580\pi$
$821$	−554.169	−0.674993	−0.337496	+	0.941327i	$0.609580\pi$
$822$	0	0
$823$	121.452i	0.147572i	0.997274	+	0.0737861i	$0.0235082\pi$
$823$	121.452i	0.147572i	−0.997274	+	0.0737861i	$0.976492\pi$
$824$	0	0
$825$	−731.853	−0.887094
$826$	0	0
$827$	1516.61i	1.83386i	0.399043	+	0.916932i	$0.369343\pi$
$827$	1516.61i	1.83386i	−0.399043	+	0.916932i	$0.630657\pi$
$828$	0	0
$829$	−325.042	−0.392089	−0.196044	−	0.980595i	$-0.562810\pi$
$829$	−325.042	−0.392089	−0.196044	+	0.980595i	$0.562810\pi$
$830$	0	0
$831$	− 2498.77i	− 3.00694i
$832$	0	0
$833$	−48.2617	−0.0579372
$834$	0	0
$835$	− 1669.98i	− 1.99998i
$836$	0	0
$837$	347.724	0.415441
$838$	0	0
$839$	− 1165.70i	− 1.38939i	−0.719303	−	0.694696i	$-0.755539\pi$
$839$	− 1165.70i	− 1.38939i	0.719303	−	0.694696i	$-0.244461\pi$
$840$	0	0
$841$	554.058	0.658808
$842$	0	0
$843$	− 1180.89i	− 1.40081i
$844$	0	0
$845$	−1022.67	−1.21026
$846$	0	0
$847$	65.6810i	0.0775454i
$848$	0	0
$849$	1330.08	1.56665
$850$	0	0
$851$	8.68026i	0.0102001i
$852$	0	0
$853$	151.949	0.178134	0.0890672	−	0.996026i	$-0.471611\pi$
$853$	151.949	0.178134	0.0890672	+	0.996026i	$0.471611\pi$
$854$	0	0
$855$	− 318.218i	− 0.372184i
$856$	0	0
$857$	−412.018	−0.480768	−0.240384	−	0.970678i	$-0.577273\pi$
$857$	−412.018	−0.480768	−0.240384	+	0.970678i	$0.577273\pi$
$858$	0	0
$859$	− 159.993i	− 0.186255i	−0.995654	−	0.0931274i	$-0.970314\pi$
$859$	− 159.993i	− 0.186255i	0.995654	−	0.0931274i	$-0.0296864\pi$
$860$	0	0
$861$	−878.280	−1.02007
$862$	0	0
$863$	− 992.910i	− 1.15053i	−0.817966	−	0.575266i	$-0.804898\pi$
$863$	− 992.910i	− 1.15053i	0.817966	−	0.575266i	$-0.195102\pi$
$864$	0	0
$865$	−719.772	−0.832107
$866$	0	0
$867$	1261.33i	1.45482i
$868$	0	0
$869$	195.479	0.224947
$870$	0	0
$871$	− 75.1590i	− 0.0862905i
$872$	0	0
$873$	−228.330	−0.261547
$874$	0	0
$875$	− 177.668i	− 0.203049i
$876$	0	0
$877$	−825.096	−0.940817	−0.470408	−	0.882449i	$-0.655893\pi$
$877$	−825.096	−0.940817	−0.470408	+	0.882449i	$0.655893\pi$
$878$	0	0
$879$	− 781.236i	− 0.888778i
$880$	0	0
$881$	1352.83	1.53556	0.767780	−	0.640714i	$-0.221361\pi$
$881$	1352.83	1.53556	0.767780	+	0.640714i	$0.221361\pi$
$882$	0	0
$883$	− 1013.40i	− 1.14768i	−0.818969	−	0.573838i	$-0.805454\pi$
$883$	− 1013.40i	− 1.14768i	0.818969	−	0.573838i	$-0.194546\pi$
$884$	0	0
$885$	−3431.10	−3.87695
$886$	0	0
$887$	− 233.760i	− 0.263540i	−0.991280	−	0.131770i	$-0.957934\pi$
$887$	− 233.760i	− 0.263540i	0.991280	−	0.131770i	$-0.0420661\pi$
$888$	0	0
$889$	346.532	0.389800
$890$	0	0
$891$	− 870.974i	− 0.977524i
$892$	0	0
$893$	105.260	0.117873
$894$	0	0
$895$	707.322i	0.790304i
$896$	0	0
$897$	540.686	0.602772
$898$	0	0
$899$	− 267.741i	− 0.297821i
$900$	0	0
$901$	−376.890	−0.418302
$902$	0	0
$903$	488.392i	0.540854i
$904$	0	0
$905$	382.008	0.422109
$906$	0	0
$907$	203.590i	0.224465i	0.993682	+	0.112233i	$0.0358002\pi$
$907$	203.590i	0.224465i	−0.993682	+	0.112233i	$0.964200\pi$
$908$	0	0
$909$	1244.85	1.36947
$910$	0	0
$911$	− 712.022i	− 0.781583i	−0.920479	−	0.390792i	$-0.872201\pi$
$911$	− 712.022i	− 0.781583i	0.920479	−	0.390792i	$-0.127799\pi$
$912$	0	0
$913$	−50.1737	−0.0549548
$914$	0	0
$915$	1432.97i	1.56609i
$916$	0	0
$917$	141.101	0.153872
$918$	0	0
$919$	− 786.783i	− 0.856129i	−0.903748	−	0.428065i	$-0.859196\pi$
$919$	− 786.783i	− 0.856129i	0.903748	−	0.428065i	$-0.140804\pi$
$920$	0	0
$921$	1419.67	1.54145
$922$	0	0
$923$	55.8929i	0.0605557i
$924$	0	0
$925$	−2.89516	−0.00312990
$926$	0	0
$927$	− 1065.97i	− 1.14991i
$928$	0	0
$929$	−1657.99	−1.78471	−0.892353	−	0.451338i	$-0.850947\pi$
$929$	−1657.99	−1.78471	−0.892353	+	0.451338i	$0.850947\pi$
$930$	0	0
$931$	− 19.4346i	− 0.0208750i
$932$	0	0
$933$	−2790.36	−2.99074
$934$	0	0
$935$	− 423.795i	− 0.453257i
$936$	0	0
$937$	−333.736	−0.356175	−0.178088	−	0.984015i	$-0.556991\pi$
$937$	−333.736	−0.356175	−0.178088	+	0.984015i	$0.556991\pi$
$938$	0	0
$939$	− 2905.55i	− 3.09430i
$940$	0	0
$941$	543.324	0.577390	0.288695	−	0.957421i	$-0.406779\pi$
$941$	543.324	0.577390	0.288695	+	0.957421i	$0.406779\pi$
$942$	0	0
$943$	2722.03i	2.88656i
$944$	0	0
$945$	−804.425	−0.851243
$946$	0	0
$947$	− 359.728i	− 0.379861i	−0.981798	−	0.189930i	$-0.939174\pi$
$947$	− 359.728i	− 0.379861i	0.981798	−	0.189930i	$-0.0608262\pi$
$948$	0	0
$949$	−167.290	−0.176281
$950$	0	0
$951$	− 2023.94i	− 2.12823i
$952$	0	0
$953$	904.225	0.948820	0.474410	−	0.880304i	$-0.342661\pi$
$953$	904.225	0.948820	0.474410	+	0.880304i	$0.342661\pi$
$954$	0	0
$955$	− 1118.56i	− 1.17127i
$956$	0	0
$957$	−1913.37	−1.99934
$958$	0	0
$959$	152.707i	0.159236i
$960$	0	0
$961$	909.615	0.946529
$962$	0	0
$963$	2484.52i	2.57998i
$964$	0	0
$965$	1388.89	1.43926
$966$	0	0
$967$	− 920.961i	− 0.952390i	−0.879340	−	0.476195i	$-0.842016\pi$
$967$	− 920.961i	− 0.952390i	0.879340	−	0.476195i	$-0.157984\pi$
$968$	0	0
$969$	99.9894	0.103188
$970$	0	0
$971$	− 1327.31i	− 1.36695i	−0.729973	−	0.683476i	$-0.760468\pi$
$971$	− 1327.31i	− 1.36695i	0.729973	−	0.683476i	$-0.239532\pi$
$972$	0	0
$973$	−456.186	−0.468845
$974$	0	0
$975$	180.337i	0.184961i
$976$	0	0
$977$	−688.490	−0.704698	−0.352349	−	0.935869i	$-0.614617\pi$
$977$	−688.490	−0.704698	−0.352349	+	0.935869i	$0.614617\pi$
$978$	0	0
$979$	− 176.415i	− 0.180199i
$980$	0	0
$981$	813.706	0.829466
$982$	0	0
$983$	1687.08i	1.71625i	0.513438	+	0.858127i	$0.328372\pi$
$983$	1687.08i	1.71625i	−0.513438	+	0.858127i	$0.671628\pi$
$984$	0	0
$985$	−1518.70	−1.54182
$986$	0	0
$987$	− 523.972i	− 0.530874i
$988$	0	0
$989$	1513.66	1.53049
$990$	0	0
$991$	− 1139.22i	− 1.14957i	−0.818304	−	0.574785i	$-0.805086\pi$
$991$	− 1139.22i	− 1.14957i	0.818304	−	0.574785i	$-0.194914\pi$
$992$	0	0
$993$	2001.72	2.01583
$994$	0	0
$995$	1861.85i	1.87121i
$996$	0	0
$997$	206.085	0.206705	0.103353	−	0.994645i	$-0.467043\pi$
$997$	206.085	0.206705	0.103353	+	0.994645i	$0.467043\pi$
$998$	0	0
$999$	− 9.83033i	− 0.00984017i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.d.j.1023.2		16
4.3	odd	2	inner	1792.3.d.j.1023.16		16
8.3	odd	2	inner	1792.3.d.j.1023.1		16
8.5	even	2	inner	1792.3.d.j.1023.15		16
16.3	odd	4		56.3.g.b.43.5	✓	8
16.5	even	4		56.3.g.b.43.6	yes	8
16.11	odd	4		224.3.g.b.15.8		8
16.13	even	4		224.3.g.b.15.7		8
48.5	odd	4		504.3.g.b.379.3		8
48.11	even	4		2016.3.g.b.1135.1		8
48.29	odd	4		2016.3.g.b.1135.8		8
48.35	even	4		504.3.g.b.379.4		8
112.3	even	12		392.3.k.n.275.6		16
112.5	odd	12		392.3.k.n.67.6		16
112.13	odd	4		1568.3.g.m.687.2		8
112.19	even	12		392.3.k.n.67.1		16
112.27	even	4		1568.3.g.m.687.1		8
112.37	even	12		392.3.k.o.67.6		16
112.51	odd	12		392.3.k.o.67.1		16
112.53	even	12		392.3.k.o.275.1		16
112.67	odd	12		392.3.k.o.275.6		16
112.69	odd	4		392.3.g.m.99.6		8
112.83	even	4		392.3.g.m.99.5		8
112.101	odd	12		392.3.k.n.275.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.b.43.5	✓	8	16.3	odd	4
56.3.g.b.43.6	yes	8	16.5	even	4
224.3.g.b.15.7		8	16.13	even	4
224.3.g.b.15.8		8	16.11	odd	4
392.3.g.m.99.5		8	112.83	even	4
392.3.g.m.99.6		8	112.69	odd	4
392.3.k.n.67.1		16	112.19	even	12
392.3.k.n.67.6		16	112.5	odd	12
392.3.k.n.275.1		16	112.101	odd	12
392.3.k.n.275.6		16	112.3	even	12
392.3.k.o.67.1		16	112.51	odd	12
392.3.k.o.67.6		16	112.37	even	12
392.3.k.o.275.1		16	112.53	even	12
392.3.k.o.275.6		16	112.67	odd	12
504.3.g.b.379.3		8	48.5	odd	4
504.3.g.b.379.4		8	48.35	even	4
1568.3.g.m.687.1		8	112.27	even	4
1568.3.g.m.687.2		8	112.13	odd	4
1792.3.d.j.1023.1		16	8.3	odd	2	inner
1792.3.d.j.1023.2		16	1.1	even	1	trivial
1792.3.d.j.1023.15		16	8.5	even	2	inner
1792.3.d.j.1023.16		16	4.3	odd	2	inner
2016.3.g.b.1135.1		8	48.11	even	4
2016.3.g.b.1135.8		8	48.29	odd	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1792.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.22363i q^{3} +6.26788 q^{5} +2.64575i q^{7} -18.2863 q^{9} +O(q^{10})\)	\(q-5.22363i q^{3} +6.26788 q^{5} +2.64575i q^{7} -18.2863 q^{9} -9.80688i q^{11} -2.41653 q^{13} -32.7411i q^{15} +6.89452 q^{17} +2.77637i q^{19} +13.8204 q^{21} -42.8332i q^{23} +14.2863 q^{25} +48.5083i q^{27} +37.3505 q^{29} -7.16835i q^{31} -51.2275 q^{33} +16.5833i q^{35} -0.202653 q^{37} +12.6231i q^{39} -63.5494 q^{41} +35.3384i q^{43} -114.616 q^{45} -37.9129i q^{47} -7.00000 q^{49} -36.0144i q^{51} -54.6651 q^{53} -61.4684i q^{55} +14.5027 q^{57} -104.795i q^{59} -43.7668 q^{61} -48.3810i q^{63} -15.1465 q^{65} +31.1021i q^{67} -223.745 q^{69} -23.1294i q^{71} +69.2275 q^{73} -74.6264i q^{75} +25.9466 q^{77} +19.9328i q^{79} +88.8125 q^{81} -5.11617i q^{83} +43.2140 q^{85} -195.105i q^{87} +17.9889 q^{89} -6.39353i q^{91} -37.4448 q^{93} +17.4019i q^{95} +12.4864 q^{97} +179.332i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 96 q^{9}+O(q^{10}) \) 16 * q - 96 * q^9	\( 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) \) 16 * q - 96 * q^9 - 160 * q^17 + 32 * q^25 + 64 * q^33 - 256 * q^41 - 112 * q^49 - 112 * q^57 - 144 * q^65 + 224 * q^73 + 96 * q^81 + 1024 * q^89 + 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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