LMFDB - Embedded newform 2016.3.g.b.1135.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,3,Mod(1135,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2016.1135");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.g (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:	$54.9320212950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.292213762624.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 $ x^8 - x^7 - 2x^6 - 2x^5 + 24x^4 - 8x^3 - 32x^2 - 64x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1135.8
Root			$1.37098 - 1.45617i$ of defining polynomial
Character	$\chi$	$=$	2016.1135
Dual form			2016.3.g.b.1135.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.26788i q^{5} -2.64575i q^{7} +O(q^{10})$	$q+6.26788i q^{5} -2.64575i q^{7} +9.80688 q^{11} -2.41653i q^{13} -6.89452 q^{17} -2.77637 q^{19} -42.8332i q^{23} -14.2863 q^{25} -37.3505i q^{29} -7.16835i q^{31} +16.5833 q^{35} +0.202653i q^{37} -63.5494 q^{41} +35.3384 q^{43} +37.9129i q^{47} -7.00000 q^{49} -54.6651i q^{53} +61.4684i q^{55} +104.795 q^{59} -43.7668i q^{61} +15.1465 q^{65} -31.1021 q^{67} -23.1294i q^{71} -69.2275 q^{73} -25.9466i q^{77} +19.9328i q^{79} -5.11617 q^{83} -43.2140i q^{85} +17.9889 q^{89} -6.39353 q^{91} -17.4019i q^{95} +12.4864 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 32 q^{11} + 80 q^{17} - 56 q^{19} - 16 q^{25} + 56 q^{35} - 128 q^{41} - 56 q^{49} + 104 q^{59} + 72 q^{65} - 304 q^{67} - 112 q^{73} + 72 q^{83} + 512 q^{89} + 56 q^{91} + 64 q^{97}+O(q^{100}) $ 8 * q - 32 * q^11 + 80 * q^17 - 56 * q^19 - 16 * q^25 + 56 * q^35 - 128 * q^41 - 56 * q^49 + 104 * q^59 + 72 * q^65 - 304 * q^67 - 112 * q^73 + 72 * q^83 + 512 * q^89 + 56 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	6.26788i	1.25358i	0.779190	+	0.626788i	$0.215631\pi$
$5$	6.26788i	1.25358i	−0.779190	+	0.626788i	$0.784369\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.80688	0.891535	0.445767	−	0.895149i	$-0.352931\pi$
$11$	9.80688	0.891535	0.445767	+	0.895149i	$0.352931\pi$
$12$	0	0
$13$	− 2.41653i	− 0.185887i	−0.995671	−	0.0929434i	$-0.970372\pi$
$13$	− 2.41653i	− 0.185887i	0.995671	−	0.0929434i	$-0.0296276\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.89452	−0.405560	−0.202780	−	0.979224i	$-0.564998\pi$
$17$	−6.89452	−0.405560	−0.202780	+	0.979224i	$0.564998\pi$
$18$	0	0
$19$	−2.77637	−0.146125	−0.0730624	−	0.997327i	$-0.523277\pi$
$19$	−2.77637	−0.146125	−0.0730624	+	0.997327i	$0.523277\pi$
$20$	0	0
$21$	0	0
$22$	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
$27$	0	0
$28$	0	0
$29$	− 37.3505i	− 1.28795i	−0.765048	−	0.643974i	$-0.777285\pi$
$29$	− 37.3505i	− 1.28795i	0.765048	−	0.643974i	$-0.222715\pi$
$30$	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
$33$	0	0
$34$	0	0
$35$	16.5833	0.473807
$36$	0	0
$37$	0.202653i	0.00547709i	0.999996	+	0.00273855i	$0.000871708\pi$
$37$	0.202653i	0.00547709i	−0.999996	+	0.00273855i	$0.999128\pi$
$38$	0	0
$39$	0	0
$40$	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
$43$	35.3384	0.821823	0.410911	−	0.911675i	$-0.365211\pi$
$43$	35.3384	0.821823	0.410911	+	0.911675i	$0.365211\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	37.9129i	0.806657i	0.915055	+	0.403329i	$0.132147\pi$
$47$	37.9129i	0.806657i	−0.915055	+	0.403329i	$0.867853\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 54.6651i	− 1.03142i	−0.856764	−	0.515709i	$-0.827529\pi$
$53$	− 54.6651i	− 1.03142i	0.856764	−	0.515709i	$-0.172471\pi$
$54$	0	0
$55$	61.4684i	1.11761i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	104.795	1.77619	0.888093	−	0.459665i	$-0.152030\pi$
$59$	104.795	1.77619	0.888093	+	0.459665i	$0.152030\pi$
$60$	0	0
$61$	− 43.7668i	− 0.717489i	−0.933436	−	0.358745i	$-0.883205\pi$
$61$	− 43.7668i	− 0.717489i	0.933436	−	0.358745i	$-0.116795\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	15.1465	0.233023
$66$	0	0
$67$	−31.1021	−0.464210	−0.232105	−	0.972691i	$-0.574561\pi$
$67$	−31.1021	−0.464210	−0.232105	+	0.972691i	$0.574561\pi$
$68$	0	0
$69$	0	0
$70$	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	$-0.657249\pi$
$73$	−69.2275	−0.948322	−0.474161	+	0.880438i	$0.657249\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 25.9466i	− 0.336968i
$78$	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
$81$	0	0
$82$	0	0
$83$	−5.11617	−0.0616406	−0.0308203	−	0.999525i	$-0.509812\pi$
$83$	−5.11617	−0.0616406	−0.0308203	+	0.999525i	$0.509812\pi$
$84$	0	0
$85$	− 43.2140i	− 0.508400i
$86$	0	0
$87$	0	0
$88$	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
$91$	−6.39353	−0.0702586
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	− 68.0753i	− 0.674013i	−0.941502	−	0.337006i	$-0.890586\pi$
$101$	− 68.0753i	− 0.674013i	0.941502	−	0.337006i	$-0.109414\pi$
$102$	0	0
$103$	− 58.2931i	− 0.565952i	−0.959127	−	0.282976i	$-0.908678\pi$
$103$	− 58.2931i	− 0.565952i	0.959127	−	0.282976i	$-0.0913217\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	135.868	1.26979	0.634897	−	0.772597i	$-0.281043\pi$
$107$	135.868	1.26979	0.634897	+	0.772597i	$0.281043\pi$
$108$	0	0
$109$	− 44.4981i	− 0.408239i	−0.978946	−	0.204120i	$-0.934567\pi$
$109$	− 44.4981i	− 0.408239i	0.978946	−	0.204120i	$-0.0654332\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	133.391	1.18045	0.590224	−	0.807240i	$-0.299039\pi$
$113$	133.391	1.18045	0.590224	+	0.807240i	$0.299039\pi$
$114$	0	0
$115$	268.474	2.33455
$116$	0	0
$117$	0	0
$118$	0	0
$119$	18.2412i	0.153287i
$120$	0	0
$121$	−24.8251	−0.205166
$122$	0	0
$123$	0	0
$124$	0	0
$125$	67.1521i	0.537217i
$126$	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
$129$	0	0
$130$	0	0
$131$	−53.3311	−0.407108	−0.203554	−	0.979064i	$-0.565249\pi$
$131$	−53.3311	−0.407108	−0.203554	+	0.979064i	$0.565249\pi$
$132$	0	0
$133$	7.34558i	0.0552299i
$134$	0	0
$135$	0	0
$136$	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
$139$	172.422	1.24045	0.620224	−	0.784425i	$-0.287042\pi$
$139$	172.422	1.24045	0.620224	+	0.784425i	$0.287042\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 23.6986i	− 0.165725i
$144$	0	0
$145$	234.108	1.61454
$146$	0	0
$147$	0	0
$148$	0	0
$149$	219.529i	1.47335i	0.676249	+	0.736673i	$0.263604\pi$
$149$	219.529i	1.47335i	−0.676249	+	0.736673i	$0.736396\pi$
$150$	0	0
$151$	− 185.668i	− 1.22959i	−0.788687	−	0.614795i	$-0.789239\pi$
$151$	− 185.668i	− 1.22959i	0.788687	−	0.614795i	$-0.210761\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	44.9304	0.289873
$156$	0	0
$157$	188.182i	1.19861i	0.800521	+	0.599305i	$0.204556\pi$
$157$	188.182i	1.19861i	−0.800521	+	0.599305i	$0.795444\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−113.326	−0.703889
$162$	0	0
$163$	−54.5154	−0.334450	−0.167225	−	0.985919i	$-0.553481\pi$
$163$	−54.5154	−0.334450	−0.167225	+	0.985919i	$0.553481\pi$
$164$	0	0
$165$	0	0
$166$	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
$171$	0	0
$172$	0	0
$173$	114.835i	0.663786i	0.943317	+	0.331893i	$0.107687\pi$
$173$	114.835i	0.663786i	−0.943317	+	0.331893i	$0.892313\pi$
$174$	0	0
$175$	37.7980i	0.215989i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	112.849	0.630439	0.315220	−	0.949019i	$-0.397922\pi$
$179$	112.849	0.630439	0.315220	+	0.949019i	$0.397922\pi$
$180$	0	0
$181$	− 60.9470i	− 0.336724i	−0.985725	−	0.168362i	$-0.946152\pi$
$181$	− 60.9470i	− 0.336724i	0.985725	−	0.168362i	$-0.0538477\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.27020	−0.00686595
$186$	0	0
$187$	−67.6138	−0.361571
$188$	0	0
$189$	0	0
$190$	0	0
$191$	178.459i	0.934342i	0.884167	+	0.467171i	$0.154727\pi$
$191$	178.459i	0.934342i	−0.884167	+	0.467171i	$0.845273\pi$
$192$	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
$195$	0	0
$196$	0	0
$197$	− 242.298i	− 1.22994i	−0.788550	−	0.614970i	$-0.789168\pi$
$197$	− 242.298i	− 1.22994i	0.788550	−	0.614970i	$-0.210832\pi$
$198$	0	0
$199$	− 297.047i	− 1.49270i	−0.665555	−	0.746349i	$-0.731805\pi$
$199$	− 297.047i	− 1.49270i	0.665555	−	0.746349i	$-0.268195\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−98.8201	−0.486798
$204$	0	0
$205$	− 398.320i	− 1.94302i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−27.2275	−0.130275
$210$	0	0
$211$	141.020	0.668341	0.334171	−	0.942513i	$-0.391544\pi$
$211$	141.020	0.668341	0.334171	+	0.942513i	$0.391544\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	221.497i	1.03022i
$216$	0	0
$217$	−18.9657	−0.0873994
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.6608i	0.0753883i
$222$	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
$225$	0	0
$226$	0	0
$227$	8.18598	0.0360616	0.0180308	−	0.999837i	$-0.494260\pi$
$227$	8.18598	0.0360616	0.0180308	+	0.999837i	$0.494260\pi$
$228$	0	0
$229$	332.252i	1.45088i	0.688285	+	0.725440i	$0.258364\pi$
$229$	332.252i	1.45088i	−0.688285	+	0.725440i	$0.741636\pi$
$230$	0	0
$231$	0	0
$232$	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
$235$	−237.633	−1.01121
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 137.719i	− 0.576230i	−0.957596	−	0.288115i	$-0.906971\pi$
$239$	− 137.719i	− 0.576230i	0.957596	−	0.288115i	$-0.0930286\pi$
$240$	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
$243$	0	0
$244$	0	0
$245$	− 43.8752i	− 0.179082i
$246$	0	0
$247$	6.70917i	0.0271627i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−269.203	−1.07252	−0.536261	−	0.844052i	$-0.680164\pi$
$251$	−269.203	−1.07252	−0.536261	+	0.844052i	$0.680164\pi$
$252$	0	0
$253$	− 420.061i	− 1.66032i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	242.359	0.943032	0.471516	−	0.881858i	$-0.343707\pi$
$257$	242.359	0.943032	0.471516	+	0.881858i	$0.343707\pi$
$258$	0	0
$259$	0.536168	0.00207015
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	0	0
$269$	− 165.598i	− 0.615606i	−0.951450	−	0.307803i	$-0.900406\pi$
$269$	− 165.598i	− 0.615606i	0.951450	−	0.307803i	$-0.0995938\pi$
$270$	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
$273$	0	0
$274$	0	0
$275$	−140.104	−0.509470
$276$	0	0
$277$	− 478.358i	− 1.72693i	−0.504413	−	0.863463i	$-0.668291\pi$
$277$	− 478.358i	− 1.72693i	0.504413	−	0.863463i	$-0.331709\pi$
$278$	0	0
$279$	0	0
$280$	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
$283$	254.628	0.899745	0.449873	−	0.893093i	$-0.351469\pi$
$283$	254.628	0.899745	0.449873	+	0.893093i	$0.351469\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	168.136i	0.585840i
$288$	0	0
$289$	−241.466	−0.835521
$290$	0	0
$291$	0	0
$292$	0	0
$293$	149.558i	0.510437i	0.966883	+	0.255218i	$0.0821474\pi$
$293$	149.558i	0.510437i	−0.966883	+	0.255218i	$0.917853\pi$
$294$	0	0
$295$	656.842i	2.22658i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−103.508	−0.346180
$300$	0	0
$301$	− 93.4966i	− 0.310620i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	274.325	0.899427
$306$	0	0
$307$	−271.779	−0.885272	−0.442636	−	0.896701i	$-0.645957\pi$
$307$	−271.779	−0.885272	−0.442636	+	0.896701i	$0.645957\pi$
$308$	0	0
$309$	0	0
$310$	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	$-0.848287\pi$
$313$	−556.232	−1.77710	−0.888550	+	0.458780i	$0.848287\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 387.459i	− 1.22227i	−0.791527	−	0.611134i	$-0.790714\pi$
$317$	− 387.459i	− 1.22227i	0.791527	−	0.611134i	$-0.209286\pi$
$318$	0	0
$319$	− 366.292i	− 1.14825i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	19.1417	0.0592624
$324$	0	0
$325$	34.5233i	0.106225i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	100.308	0.304888
$330$	0	0
$331$	383.205	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$331$	383.205	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	− 70.2992i	− 0.206156i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	51.5890	0.148671	0.0743357	−	0.997233i	$-0.476316\pi$
$347$	51.5890	0.148671	0.0743357	+	0.997233i	$0.476316\pi$
$348$	0	0
$349$	586.383i	1.68018i	0.542447	+	0.840090i	$0.317498\pi$
$349$	586.383i	1.68018i	−0.542447	+	0.840090i	$0.682502\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−303.844	−0.860746	−0.430373	−	0.902651i	$-0.641618\pi$
$353$	−303.844	−0.860746	−0.430373	+	0.902651i	$0.641618\pi$
$354$	0	0
$355$	144.972	0.408373
$356$	0	0
$357$	0	0
$358$	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
$363$	0	0
$364$	0	0
$365$	− 433.910i	− 1.18879i
$366$	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
$369$	0	0
$370$	0	0
$371$	−144.630	−0.389839
$372$	0	0
$373$	− 49.2857i	− 0.132133i	−0.997815	−	0.0660666i	$-0.978955\pi$
$373$	− 49.2857i	− 0.132133i	0.997815	−	0.0660666i	$-0.0210450\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−90.2585	−0.239412
$378$	0	0
$379$	−167.511	−0.441983	−0.220991	−	0.975276i	$-0.570929\pi$
$379$	−167.511	−0.441983	−0.220991	+	0.975276i	$0.570929\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	513.207i	1.33997i	0.742376	+	0.669983i	$0.233699\pi$
$383$	513.207i	1.33997i	−0.742376	+	0.669983i	$0.766301\pi$
$384$	0	0
$385$	162.630	0.422416
$386$	0	0
$387$	0	0
$388$	0	0
$389$	709.398i	1.82365i	0.410584	+	0.911823i	$0.365325\pi$
$389$	709.398i	1.82365i	−0.410584	+	0.911823i	$0.634675\pi$
$390$	0	0
$391$	295.315i	0.755281i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−124.937	−0.316295
$396$	0	0
$397$	− 309.404i	− 0.779355i	−0.920951	−	0.389677i	$-0.872587\pi$
$397$	− 309.404i	− 0.779355i	0.920951	−	0.389677i	$-0.127413\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	528.073	1.31689	0.658445	−	0.752629i	$-0.271215\pi$
$401$	528.073	1.31689	0.658445	+	0.752629i	$0.271215\pi$
$402$	0	0
$403$	−17.3225	−0.0429839
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.98739i	0.00488302i
$408$	0	0
$409$	−612.830	−1.49836	−0.749181	−	0.662366i	$-0.769552\pi$
$409$	−612.830	−1.49836	−0.749181	+	0.662366i	$0.769552\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 277.261i	− 0.671335i
$414$	0	0
$415$	− 32.0676i	− 0.0772712i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−237.642	−0.567165	−0.283583	−	0.958948i	$-0.591523\pi$
$419$	−237.642	−0.567165	−0.283583	+	0.958948i	$0.591523\pi$
$420$	0	0
$421$	394.516i	0.937092i	0.883439	+	0.468546i	$0.155222\pi$
$421$	394.516i	0.937092i	−0.883439	+	0.468546i	$0.844778\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	98.4973	0.231758
$426$	0	0
$427$	−115.796	−0.271185
$428$	0	0
$429$	0	0
$430$	0	0
$431$	363.359i	0.843059i	0.906815	+	0.421530i	$0.138507\pi$
$431$	363.359i	0.843059i	−0.906815	+	0.421530i	$0.861493\pi$
$432$	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
$435$	0	0
$436$	0	0
$437$	118.921i	0.272130i
$438$	0	0
$439$	871.477i	1.98514i	0.121672	+	0.992570i	$0.461174\pi$
$439$	871.477i	1.98514i	−0.121672	+	0.992570i	$0.538826\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−343.956	−0.776424	−0.388212	−	0.921570i	$-0.626907\pi$
$443$	−343.956	−0.776424	−0.388212	+	0.921570i	$0.626907\pi$
$444$	0	0
$445$	112.752i	0.253376i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	242.849	0.540866	0.270433	−	0.962739i	$-0.412833\pi$
$449$	242.849	0.540866	0.270433	+	0.962739i	$0.412833\pi$
$450$	0	0
$451$	−623.222	−1.38187
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 40.0739i	− 0.0880745i
$456$	0	0
$457$	−42.2571	−0.0924662	−0.0462331	−	0.998931i	$-0.514722\pi$
$457$	−42.2571	−0.0924662	−0.0462331	+	0.998931i	$0.514722\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	816.370i	1.77087i	0.464766	+	0.885434i	$0.346139\pi$
$461$	816.370i	1.77087i	−0.464766	+	0.885434i	$0.653861\pi$
$462$	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
$465$	0	0
$466$	0	0
$467$	603.424	1.29213	0.646064	−	0.763283i	$-0.276414\pi$
$467$	603.424	1.29213	0.646064	+	0.763283i	$0.276414\pi$
$468$	0	0
$469$	82.2884i	0.175455i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	346.559	0.732684
$474$	0	0
$475$	39.6641	0.0835034
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 158.595i	− 0.331097i	−0.986202	−	0.165548i	$-0.947061\pi$
$479$	− 158.595i	− 0.331097i	0.986202	−	0.165548i	$-0.0529394\pi$
$480$	0	0
$481$	0.489715	0.00101812
$482$	0	0
$483$	0	0
$484$	0	0
$485$	78.2633i	0.161368i
$486$	0	0
$487$	− 106.987i	− 0.219687i	−0.993949	−	0.109843i	$-0.964965\pi$
$487$	− 106.987i	− 0.219687i	0.993949	−	0.109843i	$-0.0350349\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	616.591	1.25579	0.627893	−	0.778299i	$-0.283917\pi$
$491$	616.591	1.25579	0.627893	+	0.778299i	$0.283917\pi$
$492$	0	0
$493$	257.514i	0.522340i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−61.1947	−0.123128
$498$	0	0
$499$	−554.090	−1.11040	−0.555200	−	0.831717i	$-0.687358\pi$
$499$	−554.090	−1.11040	−0.555200	+	0.831717i	$0.687358\pi$
$500$	0	0
$501$	0	0
$502$	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
$507$	0	0
$508$	0	0
$509$	− 182.889i	− 0.359310i	−0.983730	−	0.179655i	$-0.942502\pi$
$509$	− 182.889i	− 0.359310i	0.983730	−	0.179655i	$-0.0574981\pi$
$510$	0	0
$511$	183.159i	0.358432i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	365.374	0.709464
$516$	0	0
$517$	371.807i	0.719163i
$518$	0	0
$519$	0	0
$520$	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
$523$	574.764	1.09898	0.549488	−	0.835502i	$-0.314823\pi$
$523$	574.764	1.09898	0.549488	+	0.835502i	$0.314823\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	49.4223i	0.0937805i
$528$	0	0
$529$	−1305.69	−2.46822
$530$	0	0
$531$	0	0
$532$	0	0
$533$	153.569i	0.288122i
$534$	0	0
$535$	851.604i	1.59178i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−68.6482	−0.127362
$540$	0	0
$541$	370.654i	0.685128i	0.939495	+	0.342564i	$0.111295\pi$
$541$	370.654i	0.685128i	−0.939495	+	0.342564i	$0.888705\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	278.909	0.511759
$546$	0	0
$547$	56.5966	0.103467	0.0517336	−	0.998661i	$-0.483525\pi$
$547$	56.5966	0.103467	0.0517336	+	0.998661i	$0.483525\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	103.699i	0.188201i
$552$	0	0
$553$	52.7374	0.0953659
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 151.525i	− 0.272037i	−0.990706	−	0.136019i	$-0.956569\pi$
$557$	− 151.525i	− 0.272037i	0.990706	−	0.136019i	$-0.0434307\pi$
$558$	0	0
$559$	− 85.3962i	− 0.152766i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−318.048	−0.564917	−0.282458	−	0.959280i	$-0.591150\pi$
$563$	−318.048	−0.564917	−0.282458	+	0.959280i	$0.591150\pi$
$564$	0	0
$565$	836.076i	1.47978i
$566$	0	0
$567$	0	0
$568$	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
$571$	−831.014	−1.45537	−0.727683	−	0.685914i	$-0.759403\pi$
$571$	−831.014	−1.45537	−0.727683	+	0.685914i	$0.759403\pi$
$572$	0	0
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	13.5361i	0.0232980i
$582$	0	0
$583$	− 536.094i	− 0.919544i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	144.376	0.245956	0.122978	−	0.992409i	$-0.460756\pi$
$587$	144.376	0.245956	0.122978	+	0.992409i	$0.460756\pi$
$588$	0	0
$589$	19.9020i	0.0337894i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−838.926	−1.41471	−0.707357	−	0.706856i	$-0.750113\pi$
$593$	−838.926	−1.41471	−0.707357	+	0.706856i	$0.750113\pi$
$594$	0	0
$595$	−114.334	−0.192157
$596$	0	0
$597$	0	0
$598$	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	$-0.595820\pi$
$601$	−356.394	−0.593002	−0.296501	+	0.955033i	$0.595820\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 155.601i	− 0.257191i
$606$	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
$609$	0	0
$610$	0	0
$611$	91.6176	0.149947
$612$	0	0
$613$	− 482.989i	− 0.787911i	−0.919130	−	0.393955i	$-0.871106\pi$
$613$	− 482.989i	− 0.787911i	0.919130	−	0.393955i	$-0.128894\pi$
$614$	0	0
$615$	0	0
$616$	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
$619$	93.0817	0.150374	0.0751872	−	0.997169i	$-0.476045\pi$
$619$	93.0817	0.150374	0.0751872	+	0.997169i	$0.476045\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 47.5941i	− 0.0763951i
$624$	0	0
$625$	−778.059	−1.24489
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.39719i	− 0.00222129i
$630$	0	0
$631$	610.573i	0.967628i	0.875171	+	0.483814i	$0.160749\pi$
$631$	610.573i	0.967628i	−0.875171	+	0.483814i	$0.839251\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	820.947	1.29283
$636$	0	0
$637$	16.9157i	0.0265553i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−590.153	−0.920676	−0.460338	−	0.887744i	$-0.652272\pi$
$641$	−590.153	−0.920676	−0.460338	+	0.887744i	$0.652272\pi$
$642$	0	0
$643$	257.971	0.401199	0.200600	−	0.979673i	$-0.435711\pi$
$643$	257.971	0.401199	0.200600	+	0.979673i	$0.435711\pi$
$644$	0	0
$645$	0	0
$646$	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
$651$	0	0
$652$	0	0
$653$	− 952.773i	− 1.45907i	−0.683943	−	0.729535i	$-0.739736\pi$
$653$	− 952.773i	− 1.45907i	0.683943	−	0.729535i	$-0.260264\pi$
$654$	0	0
$655$	− 334.273i	− 0.510340i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−963.119	−1.46149	−0.730743	−	0.682653i	$-0.760826\pi$
$659$	−963.119	−1.46149	−0.730743	+	0.682653i	$0.760826\pi$
$660$	0	0
$661$	− 71.6817i	− 0.108444i	−0.998529	−	0.0542222i	$-0.982732\pi$
$661$	− 71.6817i	− 0.108444i	0.998529	−	0.0542222i	$-0.0172679\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−46.0412	−0.0692349
$666$	0	0
$667$	−1599.84	−2.39856
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	767.527i	1.13372i	0.823815	+	0.566859i	$0.191841\pi$
$677$	767.527i	1.13372i	−0.823815	+	0.566859i	$0.808159\pi$
$678$	0	0
$679$	− 33.0359i	− 0.0486538i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−484.354	−0.709156	−0.354578	−	0.935026i	$-0.615375\pi$
$683$	−484.354	−0.709156	−0.354578	+	0.935026i	$0.615375\pi$
$684$	0	0
$685$	361.769i	0.528130i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−132.100	−0.191727
$690$	0	0
$691$	574.851	0.831912	0.415956	−	0.909385i	$-0.363447\pi$
$691$	574.851	0.831912	0.415956	+	0.909385i	$0.363447\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1080.72i	1.55500i
$696$	0	0
$697$	438.143	0.628612
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 143.138i	− 0.204191i	−0.994775	−	0.102096i	$-0.967445\pi$
$701$	− 143.138i	− 0.204191i	0.994775	−	0.102096i	$-0.0325548\pi$
$702$	0	0
$703$	− 0.562638i	0 0.000800339i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−180.110	−0.254753
$708$	0	0
$709$	− 255.311i	− 0.360101i	−0.983657	−	0.180050i	$-0.942374\pi$
$709$	− 255.311i	− 0.360101i	0.983657	−	0.180050i	$-0.0576261\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−307.044	−0.430636
$714$	0	0
$715$	148.540	0.207748
$716$	0	0
$717$	0	0
$718$	0	0
$719$	415.630i	0.578067i	0.957319	+	0.289034i	$0.0933340\pi$
$719$	415.630i	0.578067i	−0.957319	+	0.289034i	$0.906666\pi$
$720$	0	0
$721$	−154.229	−0.213910
$722$	0	0
$723$	0	0
$724$	0	0
$725$	533.601i	0.736001i
$726$	0	0
$727$	− 896.838i	− 1.23361i	−0.787114	−	0.616807i	$-0.788426\pi$
$727$	− 896.838i	− 1.23361i	0.787114	−	0.616807i	$-0.211574\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−243.641	−0.333299
$732$	0	0
$733$	− 509.059i	− 0.694487i	−0.937775	−	0.347244i	$-0.887118\pi$
$733$	− 509.059i	− 0.694487i	0.937775	−	0.347244i	$-0.112882\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−305.014	−0.413859
$738$	0	0
$739$	−741.427	−1.00328	−0.501642	−	0.865075i	$-0.667271\pi$
$739$	−741.427	−1.00328	−0.501642	+	0.865075i	$0.667271\pi$
$740$	0	0
$741$	0	0
$742$	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
$747$	0	0
$748$	0	0
$749$	− 359.473i	− 0.479937i
$750$	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
$753$	0	0
$754$	0	0
$755$	1163.74	1.54138
$756$	0	0
$757$	1341.69i	1.77238i	0.463318	+	0.886192i	$0.346659\pi$
$757$	1341.69i	1.77238i	−0.463318	+	0.886192i	$0.653341\pi$
$758$	0	0
$759$	0	0
$760$	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
$763$	−117.731	−0.154300
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	1325.14i	1.71428i	0.515087	+	0.857138i	$0.327760\pi$
$773$	1325.14i	1.71428i	−0.515087	+	0.857138i	$0.672240\pi$
$774$	0	0
$775$	102.409i	0.132141i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	176.437	0.226491
$780$	0	0
$781$	− 226.827i	− 0.290432i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1179.50	−1.50255
$786$	0	0
$787$	327.801	0.416519	0.208260	−	0.978074i	$-0.433220\pi$
$787$	327.801	0.416519	0.208260	+	0.978074i	$0.433220\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 352.918i	− 0.446167i
$792$	0	0
$793$	−105.764	−0.133372
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 393.650i	− 0.493915i	−0.969026	−	0.246958i	$-0.920569\pi$
$797$	− 393.650i	− 0.493915i	0.969026	−	0.246958i	$-0.0794308\pi$
$798$	0	0
$799$	− 261.391i	− 0.327148i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−678.906	−0.845462
$804$	0	0
$805$	− 710.314i	− 0.882378i
$806$	0	0
$807$	0	0
$808$	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
$811$	748.707	0.923190	0.461595	−	0.887091i	$-0.347277\pi$
$811$	748.707	0.923190	0.461595	+	0.887091i	$0.347277\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 341.696i	− 0.419259i
$816$	0	0
$817$	−98.1124	−0.120089
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 554.169i	− 0.674993i	−0.941327	−	0.337496i	$-0.890420\pi$
$821$	− 554.169i	− 0.674993i	0.941327	−	0.337496i	$-0.109580\pi$
$822$	0	0
$823$	− 121.452i	− 0.147572i	−0.997274	−	0.0737861i	$-0.976492\pi$
$823$	− 121.452i	− 0.147572i	0.997274	−	0.0737861i	$-0.0235082\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1516.61	−1.83386	−0.916932	−	0.399043i	$-0.869343\pi$
$827$	−1516.61	−1.83386	−0.916932	+	0.399043i	$0.869343\pi$
$828$	0	0
$829$	− 325.042i	− 0.392089i	−0.980595	−	0.196044i	$-0.937190\pi$
$829$	− 325.042i	− 0.392089i	0.980595	−	0.196044i	$-0.0628097\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	48.2617	0.0579372
$834$	0	0
$835$	1669.98	1.99998
$836$	0	0
$837$	0	0
$838$	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
$843$	0	0
$844$	0	0
$845$	1022.67i	1.21026i
$846$	0	0
$847$	65.6810i	0.0775454i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.68026	0.0102001
$852$	0	0
$853$	− 151.949i	− 0.178134i	−0.996026	−	0.0890672i	$-0.971611\pi$
$853$	− 151.949i	− 0.178134i	0.996026	−	0.0890672i	$-0.0283886\pi$
$854$	0	0
$855$	0	0
$856$	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
$859$	−159.993	−0.186255	−0.0931274	−	0.995654i	$-0.529686\pi$
$859$	−159.993	−0.186255	−0.0931274	+	0.995654i	$0.529686\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	992.910i	1.15053i	0.817966	+	0.575266i	$0.195102\pi$
$863$	992.910i	1.15053i	−0.817966	+	0.575266i	$0.804898\pi$
$864$	0	0
$865$	−719.772	−0.832107
$866$	0	0
$867$	0	0
$868$	0	0
$869$	195.479i	0.224947i
$870$	0	0
$871$	75.1590i	0.0862905i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	177.668	0.203049
$876$	0	0
$877$	− 825.096i	− 0.940817i	−0.882449	−	0.470408i	$-0.844107\pi$
$877$	− 825.096i	− 0.940817i	0.882449	−	0.470408i	$-0.155893\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1352.83	−1.53556	−0.767780	−	0.640714i	$-0.778639\pi$
$881$	−1352.83	−1.53556	−0.767780	+	0.640714i	$0.778639\pi$
$882$	0	0
$883$	1013.40	1.14768	0.573838	−	0.818969i	$-0.305454\pi$
$883$	1013.40	1.14768	0.573838	+	0.818969i	$0.305454\pi$
$884$	0	0
$885$	0	0
$886$	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
$891$	0	0
$892$	0	0
$893$	− 105.260i	− 0.117873i
$894$	0	0
$895$	707.322i	0.790304i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−267.741	−0.297821
$900$	0	0
$901$	376.890i	0.418302i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	382.008	0.422109
$906$	0	0
$907$	203.590	0.224465	0.112233	−	0.993682i	$-0.464200\pi$
$907$	203.590	0.224465	0.112233	+	0.993682i	$0.464200\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	712.022i	0.781583i	0.920479	+	0.390792i	$0.127799\pi$
$911$	712.022i	0.781583i	−0.920479	+	0.390792i	$0.872201\pi$
$912$	0	0
$913$	−50.1737	−0.0549548
$914$	0	0
$915$	0	0
$916$	0	0
$917$	141.101i	0.153872i
$918$	0	0
$919$	786.783i	0.856129i	0.903748	+	0.428065i	$0.140804\pi$
$919$	786.783i	0.856129i	−0.903748	+	0.428065i	$0.859196\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−55.8929	−0.0605557
$924$	0	0
$925$	− 2.89516i	− 0.00312990i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1657.99	1.78471	0.892353	−	0.451338i	$-0.149053\pi$
$929$	1657.99	1.78471	0.892353	+	0.451338i	$0.149053\pi$
$930$	0	0
$931$	19.4346	0.0208750
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 423.795i	− 0.453257i
$936$	0	0
$937$	333.736	0.356175	0.178088	−	0.984015i	$-0.443009\pi$
$937$	333.736	0.356175	0.178088	+	0.984015i	$0.443009\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 543.324i	− 0.577390i	−0.957421	−	0.288695i	$-0.906779\pi$
$941$	− 543.324i	− 0.577390i	0.957421	−	0.288695i	$-0.0932214\pi$
$942$	0	0
$943$	2722.03i	2.88656i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−359.728	−0.379861	−0.189930	−	0.981798i	$-0.560826\pi$
$947$	−359.728	−0.379861	−0.189930	+	0.981798i	$0.560826\pi$
$948$	0	0
$949$	167.290i	0.176281i
$950$	0	0
$951$	0	0
$952$	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
$955$	−1118.56	−1.17127
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 152.707i	− 0.159236i
$960$	0	0
$961$	909.615	0.946529
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1388.89i	1.43926i
$966$	0	0
$967$	920.961i	0.952390i	0.879340	+	0.476195i	$0.157984\pi$
$967$	920.961i	0.952390i	−0.879340	+	0.476195i	$0.842016\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1327.31	1.36695	0.683476	−	0.729973i	$-0.260468\pi$
$971$	1327.31	1.36695	0.683476	+	0.729973i	$0.260468\pi$
$972$	0	0
$973$	− 456.186i	− 0.468845i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	688.490	0.704698	0.352349	−	0.935869i	$-0.385383\pi$
$977$	688.490	0.704698	0.352349	+	0.935869i	$0.385383\pi$
$978$	0	0
$979$	176.415	0.180199
$980$	0	0
$981$	0	0
$982$	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
$987$	0	0
$988$	0	0
$989$	− 1513.66i	− 1.53049i
$990$	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
$993$	0	0
$994$	0	0
$995$	1861.85	1.87121
$996$	0	0
$997$	− 206.085i	− 0.206705i	−0.994645	−	0.103353i	$-0.967043\pi$
$997$	− 206.085i	− 0.206705i	0.994645	−	0.103353i	$-0.0329570\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.g.b.1135.8		8
3.2	odd	2		224.3.g.b.15.7		8
4.3	odd	2		504.3.g.b.379.4		8
8.3	odd	2	inner	2016.3.g.b.1135.1		8
8.5	even	2		504.3.g.b.379.3		8
12.11	even	2		56.3.g.b.43.5	✓	8
21.20	even	2		1568.3.g.m.687.2		8
24.5	odd	2		56.3.g.b.43.6	yes	8
24.11	even	2		224.3.g.b.15.8		8
48.5	odd	4		1792.3.d.j.1023.2		16
48.11	even	4		1792.3.d.j.1023.16		16
48.29	odd	4		1792.3.d.j.1023.15		16
48.35	even	4		1792.3.d.j.1023.1		16
84.11	even	6		392.3.k.o.275.6		16
84.23	even	6		392.3.k.o.67.1		16
84.47	odd	6		392.3.k.n.67.1		16
84.59	odd	6		392.3.k.n.275.6		16
84.83	odd	2		392.3.g.m.99.5		8
168.5	even	6		392.3.k.n.67.6		16
168.53	odd	6		392.3.k.o.275.1		16
168.83	odd	2		1568.3.g.m.687.1		8
168.101	even	6		392.3.k.n.275.1		16
168.125	even	2		392.3.g.m.99.6		8
168.149	odd	6		392.3.k.o.67.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.b.43.5	✓	8	12.11	even	2
56.3.g.b.43.6	yes	8	24.5	odd	2
224.3.g.b.15.7		8	3.2	odd	2
224.3.g.b.15.8		8	24.11	even	2
392.3.g.m.99.5		8	84.83	odd	2
392.3.g.m.99.6		8	168.125	even	2
392.3.k.n.67.1		16	84.47	odd	6
392.3.k.n.67.6		16	168.5	even	6
392.3.k.n.275.1		16	168.101	even	6
392.3.k.n.275.6		16	84.59	odd	6
392.3.k.o.67.1		16	84.23	even	6
392.3.k.o.67.6		16	168.149	odd	6
392.3.k.o.275.1		16	168.53	odd	6
392.3.k.o.275.6		16	84.11	even	6
504.3.g.b.379.3		8	8.5	even	2
504.3.g.b.379.4		8	4.3	odd	2
1568.3.g.m.687.1		8	168.83	odd	2
1568.3.g.m.687.2		8	21.20	even	2
1792.3.d.j.1023.1		16	48.35	even	4
1792.3.d.j.1023.2		16	48.5	odd	4
1792.3.d.j.1023.15		16	48.29	odd	4
1792.3.d.j.1023.16		16	48.11	even	4
2016.3.g.b.1135.1		8	8.3	odd	2	inner
2016.3.g.b.1135.8		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2016.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.26788i q^{5} -2.64575i q^{7} +O(q^{10})\)	\(q+6.26788i q^{5} -2.64575i q^{7} +9.80688 q^{11} -2.41653i q^{13} -6.89452 q^{17} -2.77637 q^{19} -42.8332i q^{23} -14.2863 q^{25} -37.3505i q^{29} -7.16835i q^{31} +16.5833 q^{35} +0.202653i q^{37} -63.5494 q^{41} +35.3384 q^{43} +37.9129i q^{47} -7.00000 q^{49} -54.6651i q^{53} +61.4684i q^{55} +104.795 q^{59} -43.7668i q^{61} +15.1465 q^{65} -31.1021 q^{67} -23.1294i q^{71} -69.2275 q^{73} -25.9466i q^{77} +19.9328i q^{79} -5.11617 q^{83} -43.2140i q^{85} +17.9889 q^{89} -6.39353 q^{91} -17.4019i q^{95} +12.4864 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 32 q^{11} + 80 q^{17} - 56 q^{19} - 16 q^{25} + 56 q^{35} - 128 q^{41} - 56 q^{49} + 104 q^{59} + 72 q^{65} - 304 q^{67} - 112 q^{73} + 72 q^{83} + 512 q^{89} + 56 q^{91} + 64 q^{97}+O(q^{100}) \) 8 * q - 32 * q^11 + 80 * q^17 - 56 * q^19 - 16 * q^25 + 56 * q^35 - 128 * q^41 - 56 * q^49 + 104 * q^59 + 72 * q^65 - 304 * q^67 - 112 * q^73 + 72 * q^83 + 512 * q^89 + 56 * q^91 + 64 * q^97

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