LMFDB - Embedded newform 5184.2.d.p.2593.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(2593,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5184.2593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.d (of order $2$, degree $1$, 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:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2593.3
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	5184.2593
Dual form			5184.2.d.p.2593.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.717439i q^{5} -2.44949 q^{7} +O(q^{10})$	$q-0.717439i q^{5} -2.44949 q^{7} -4.24264i q^{11} +4.18154i q^{13} +3.00000 q^{17} -2.24264i q^{19} -5.91359 q^{23} +4.48528 q^{25} -0.717439i q^{29} +1.43488 q^{31} +1.75736i q^{35} +2.74666i q^{37} -8.48528 q^{41} -3.75736i q^{43} +1.43488 q^{47} -1.00000 q^{49} -11.8272i q^{53} -3.04384 q^{55} +9.08052i q^{61} +3.00000 q^{65} -6.24264i q^{67} +16.3059 q^{71} -9.48528 q^{73} +10.3923i q^{77} -1.01461 q^{79} +6.00000i q^{83} -2.15232i q^{85} +0.514719 q^{89} -10.2426i q^{91} -1.60896 q^{95} -16.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 24 q^{17} - 32 q^{25} - 8 q^{49} + 24 q^{65} - 8 q^{73} + 72 q^{89} - 64 q^{97}+O(q^{100}) $ 8 * q + 24 * q^17 - 32 * q^25 - 8 * q^49 + 24 * q^65 - 8 * q^73 + 72 * q^89 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 0.717439i	− 0.320848i	−0.987048	−	0.160424i	$-0.948714\pi$
$5$	− 0.717439i	− 0.320848i	0.987048	−	0.160424i	$-0.0512862\pi$
$6$	0	0
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.24264i	− 1.27920i	−0.768706	−	0.639602i	$-0.779099\pi$
$11$	− 4.24264i	− 1.27920i	0.768706	−	0.639602i	$-0.220901\pi$
$12$	0	0
$13$	4.18154i	1.15975i	0.814705	+	0.579875i	$0.196899\pi$
$13$	4.18154i	1.15975i	−0.814705	+	0.579875i	$0.803101\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	− 2.24264i	− 0.514497i	−0.966345	−	0.257249i	$-0.917184\pi$
$19$	− 2.24264i	− 0.514497i	0.966345	−	0.257249i	$-0.0828159\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.91359	−1.23307	−0.616535	−	0.787328i	$-0.711464\pi$
$23$	−5.91359	−1.23307	−0.616535	+	0.787328i	$0.711464\pi$
$24$	0	0
$25$	4.48528	0.897056
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.717439i	− 0.133225i	−0.997779	−	0.0666125i	$-0.978781\pi$
$29$	− 0.717439i	− 0.133225i	0.997779	−	0.0666125i	$-0.0212191\pi$
$30$	0	0
$31$	1.43488	0.257712	0.128856	−	0.991663i	$-0.458870\pi$
$31$	1.43488	0.257712	0.128856	+	0.991663i	$0.458870\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.75736i	0.297048i
$36$	0	0
$37$	2.74666i	0.451549i	0.974180	+	0.225774i	$0.0724912\pi$
$37$	2.74666i	0.451549i	−0.974180	+	0.225774i	$0.927509\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.48528	−1.32518	−0.662589	−	0.748983i	$-0.730542\pi$
$41$	−8.48528	−1.32518	−0.662589	+	0.748983i	$0.730542\pi$
$42$	0	0
$43$	− 3.75736i	− 0.572992i	−0.958081	−	0.286496i	$-0.907509\pi$
$43$	− 3.75736i	− 0.572992i	0.958081	−	0.286496i	$-0.0924905\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.43488	0.209298	0.104649	−	0.994509i	$-0.466628\pi$
$47$	1.43488	0.209298	0.104649	+	0.994509i	$0.466628\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.8272i	− 1.62459i	−0.583248	−	0.812294i	$-0.698218\pi$
$53$	− 11.8272i	− 1.62459i	0.583248	−	0.812294i	$-0.301782\pi$
$54$	0	0
$55$	−3.04384	−0.410431
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	9.08052i	1.16264i	0.813675	+	0.581321i	$0.197464\pi$
$61$	9.08052i	1.16264i	−0.813675	+	0.581321i	$0.802536\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	− 6.24264i	− 0.762660i	−0.924439	−	0.381330i	$-0.875466\pi$
$67$	− 6.24264i	− 0.762660i	0.924439	−	0.381330i	$-0.124534\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.3059	1.93515	0.967577	−	0.252577i	$-0.0812779\pi$
$71$	16.3059	1.93515	0.967577	+	0.252577i	$0.0812779\pi$
$72$	0	0
$73$	−9.48528	−1.11017	−0.555084	−	0.831794i	$-0.687314\pi$
$73$	−9.48528	−1.11017	−0.555084	+	0.831794i	$0.687314\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.3923i	1.18431i
$78$	0	0
$79$	−1.01461	−0.114153	−0.0570764	−	0.998370i	$-0.518178\pi$
$79$	−1.01461	−0.114153	−0.0570764	+	0.998370i	$0.518178\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	− 2.15232i	− 0.233452i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.514719	0.0545601	0.0272800	−	0.999628i	$-0.491315\pi$
$89$	0.514719	0.0545601	0.0272800	+	0.999628i	$0.491315\pi$
$90$	0	0
$91$	− 10.2426i	− 1.07372i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.60896	−0.165076
$96$	0	0
$97$	−16.4853	−1.67383	−0.836913	−	0.547335i	$-0.815642\pi$
$97$	−16.4853	−1.67383	−0.836913	+	0.547335i	$0.815642\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.2621i	1.31962i	0.751431	+	0.659812i	$0.229364\pi$
$101$	13.2621i	1.31962i	−0.751431	+	0.659812i	$0.770636\pi$
$102$	0	0
$103$	−18.1610	−1.78946	−0.894730	−	0.446607i	$-0.852632\pi$
$103$	−18.1610	−1.78946	−0.894730	+	0.446607i	$0.852632\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	11.1097i	1.06412i	0.846707	+	0.532060i	$0.178582\pi$
$109$	11.1097i	1.06412i	−0.846707	+	0.532060i	$0.821418\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.514719	−0.0484207	−0.0242103	−	0.999707i	$-0.507707\pi$
$113$	−0.514719	−0.0484207	−0.0242103	+	0.999707i	$0.507707\pi$
$114$	0	0
$115$	4.24264i	0.395628i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.34847	−0.673633
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 6.80511i	− 0.608668i
$126$	0	0
$127$	−8.78335	−0.779396	−0.389698	−	0.920943i	$-0.627421\pi$
$127$	−8.78335	−0.779396	−0.389698	+	0.920943i	$0.627421\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 22.2426i	− 1.94335i	−0.236324	−	0.971674i	$-0.575943\pi$
$131$	− 22.2426i	− 1.94335i	0.236324	−	0.971674i	$-0.424057\pi$
$132$	0	0
$133$	5.49333i	0.476332i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.4853	−0.981254	−0.490627	−	0.871370i	$-0.663232\pi$
$137$	−11.4853	−0.981254	−0.490627	+	0.871370i	$0.663232\pi$
$138$	0	0
$139$	− 10.0000i	− 0.848189i	−0.905618	−	0.424094i	$-0.860592\pi$
$139$	− 10.0000i	− 0.848189i	0.905618	−	0.424094i	$-0.139408\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	17.7408	1.48356
$144$	0	0
$145$	−0.514719	−0.0427451
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.9369i	1.87907i	0.342458	+	0.939533i	$0.388741\pi$
$149$	22.9369i	1.87907i	−0.342458	+	0.939533i	$0.611259\pi$
$150$	0	0
$151$	−13.8564	−1.12762	−0.563809	−	0.825905i	$-0.690665\pi$
$151$	−13.8564	−1.12762	−0.563809	+	0.825905i	$0.690665\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.02944i	− 0.0826864i
$156$	0	0
$157$	9.67487i	0.772138i	0.922470	+	0.386069i	$0.126167\pi$
$157$	9.67487i	0.772138i	−0.922470	+	0.386069i	$0.873833\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	14.4853	1.14160
$162$	0	0
$163$	12.9706i	1.01593i	0.861377	+	0.507966i	$0.169603\pi$
$163$	12.9706i	1.01593i	−0.861377	+	0.507966i	$0.830397\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.1757	−1.48386	−0.741928	−	0.670479i	$-0.766089\pi$
$167$	−19.1757	−1.48386	−0.741928	+	0.670479i	$0.766089\pi$
$168$	0	0
$169$	−4.48528	−0.345022
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 15.4144i	− 1.17193i	−0.810335	−	0.585967i	$-0.800715\pi$
$173$	− 15.4144i	− 1.17193i	0.810335	−	0.585967i	$-0.199285\pi$
$174$	0	0
$175$	−10.9867	−0.830513
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.9706i	1.71690i	0.512897	+	0.858450i	$0.328572\pi$
$179$	22.9706i	1.71690i	−0.512897	+	0.858450i	$0.671428\pi$
$180$	0	0
$181$	− 4.89898i	− 0.364138i	−0.983286	−	0.182069i	$-0.941721\pi$
$181$	− 4.89898i	− 0.364138i	0.983286	−	0.182069i	$-0.0582795\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.97056	0.144879
$186$	0	0
$187$	− 12.7279i	− 0.930758i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.47871	−0.324068	−0.162034	−	0.986785i	$-0.551805\pi$
$191$	−4.47871	−0.324068	−0.162034	+	0.986785i	$0.551805\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.9369i	− 1.63419i	−0.576504	−	0.817094i	$-0.695584\pi$
$197$	− 22.9369i	− 1.63419i	0.576504	−	0.817094i	$-0.304416\pi$
$198$	0	0
$199$	−12.4215	−0.880539	−0.440269	−	0.897866i	$-0.645117\pi$
$199$	−12.4215	−0.880539	−0.440269	+	0.897866i	$0.645117\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.75736i	0.123342i
$204$	0	0
$205$	6.08767i	0.425181i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.51472	−0.658147
$210$	0	0
$211$	25.2132i	1.73575i	0.496784	+	0.867874i	$0.334514\pi$
$211$	25.2132i	1.73575i	−0.496784	+	0.867874i	$0.665486\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.69568	−0.183844
$216$	0	0
$217$	−3.51472	−0.238595
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.5446i	0.843843i
$222$	0	0
$223$	−14.8710	−0.995837	−0.497919	−	0.867224i	$-0.665902\pi$
$223$	−14.8710	−0.995837	−0.497919	+	0.867224i	$0.665902\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.75736i	0.116640i	0.998298	+	0.0583200i	$0.0185744\pi$
$227$	1.75736i	0.116640i	−0.998298	+	0.0583200i	$0.981426\pi$
$228$	0	0
$229$	24.3718i	1.61053i	0.592912	+	0.805267i	$0.297978\pi$
$229$	24.3718i	1.61053i	−0.592912	+	0.805267i	$0.702022\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.9706	1.30832	0.654158	−	0.756358i	$-0.273023\pi$
$233$	19.9706	1.30832	0.654158	+	0.756358i	$0.273023\pi$
$234$	0	0
$235$	− 1.02944i	− 0.0671531i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.2195	−1.43726	−0.718630	−	0.695393i	$-0.755230\pi$
$239$	−22.2195	−1.43726	−0.718630	+	0.695393i	$0.755230\pi$
$240$	0	0
$241$	11.0000	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.717439i	0.0458355i
$246$	0	0
$247$	9.37769	0.596688
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.75736i	0.489640i	0.969569	+	0.244820i	$0.0787289\pi$
$251$	7.75736i	0.489640i	−0.969569	+	0.244820i	$0.921271\pi$
$252$	0	0
$253$	25.0892i	1.57735i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.4853	−0.716432	−0.358216	−	0.933639i	$-0.616615\pi$
$257$	−11.4853	−0.716432	−0.358216	+	0.933639i	$0.616615\pi$
$258$	0	0
$259$	− 6.72792i	− 0.418053i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.30463	−0.265435	−0.132718	−	0.991154i	$-0.542370\pi$
$263$	−4.30463	−0.265435	−0.132718	+	0.991154i	$0.542370\pi$
$264$	0	0
$265$	−8.48528	−0.521247
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 22.9369i	− 1.39849i	−0.714883	−	0.699245i	$-0.753520\pi$
$269$	− 22.9369i	− 1.39849i	0.714883	−	0.699245i	$-0.246480\pi$
$270$	0	0
$271$	−25.5095	−1.54959	−0.774796	−	0.632211i	$-0.782148\pi$
$271$	−25.5095	−1.54959	−0.774796	+	0.632211i	$0.782148\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 19.0294i	− 1.14752i
$276$	0	0
$277$	− 2.86976i	− 0.172427i	−0.996277	−	0.0862135i	$-0.972523\pi$
$277$	− 2.86976i	− 0.172427i	0.996277	−	0.0862135i	$-0.0274767\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.4853	0.685154	0.342577	−	0.939490i	$-0.388700\pi$
$281$	11.4853	0.685154	0.342577	+	0.939490i	$0.388700\pi$
$282$	0	0
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.7846	1.22688
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3.58719i	− 0.209566i	−0.994495	−	0.104783i	$-0.966585\pi$
$293$	− 3.58719i	− 0.209566i	0.994495	−	0.104783i	$-0.0334148\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 24.7279i	− 1.43005i
$300$	0	0
$301$	9.20361i	0.530487i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.51472	0.373032
$306$	0	0
$307$	− 18.9706i	− 1.08271i	−0.840795	−	0.541354i	$-0.817912\pi$
$307$	− 18.9706i	− 1.08271i	0.840795	−	0.541354i	$-0.182088\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.86976	0.162729	0.0813645	−	0.996684i	$-0.474072\pi$
$311$	2.86976	0.162729	0.0813645	+	0.996684i	$0.474072\pi$
$312$	0	0
$313$	−27.9706	−1.58099	−0.790495	−	0.612469i	$-0.790177\pi$
$313$	−27.9706	−1.58099	−0.790495	+	0.612469i	$0.790177\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.8493i	0.946348i	0.880969	+	0.473174i	$0.156892\pi$
$317$	16.8493i	0.946348i	−0.880969	+	0.473174i	$0.843108\pi$
$318$	0	0
$319$	−3.04384	−0.170422
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.72792i	− 0.374352i
$324$	0	0
$325$	18.7554i	1.04036i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.51472	−0.193773
$330$	0	0
$331$	− 14.2426i	− 0.782846i	−0.920211	−	0.391423i	$-0.871983\pi$
$331$	− 14.2426i	− 0.782846i	0.920211	−	0.391423i	$-0.128017\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.47871	−0.244698
$336$	0	0
$337$	0.485281	0.0264350	0.0132175	−	0.999913i	$-0.495793\pi$
$337$	0.485281	0.0264350	0.0132175	+	0.999913i	$0.495793\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 6.08767i	− 0.329666i
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.7279i	1.64956i	0.565453	+	0.824781i	$0.308701\pi$
$347$	30.7279i	1.64956i	−0.565453	+	0.824781i	$0.691299\pi$
$348$	0	0
$349$	9.20361i	0.492658i	0.969186	+	0.246329i	$0.0792244\pi$
$349$	9.20361i	0.492658i	−0.969186	+	0.246329i	$0.920776\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	− 11.6985i	− 0.620891i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.6105	1.08778	0.543891	−	0.839156i	$-0.316950\pi$
$359$	20.6105	1.08778	0.543891	+	0.839156i	$0.316950\pi$
$360$	0	0
$361$	13.9706	0.735293
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.80511i	0.356196i
$366$	0	0
$367$	−13.8564	−0.723299	−0.361649	−	0.932314i	$-0.617786\pi$
$367$	−13.8564	−0.723299	−0.361649	+	0.932314i	$0.617786\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	28.9706i	1.50408i
$372$	0	0
$373$	− 12.6677i	− 0.655909i	−0.944693	−	0.327955i	$-0.893641\pi$
$373$	− 12.6677i	− 0.655909i	0.944693	−	0.327955i	$-0.106359\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.00000	0.154508
$378$	0	0
$379$	− 30.9706i	− 1.59085i	−0.606051	−	0.795425i	$-0.707247\pi$
$379$	− 30.9706i	− 1.59085i	0.606051	−	0.795425i	$-0.292753\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.1757	−0.979830	−0.489915	−	0.871770i	$-0.662972\pi$
$383$	−19.1757	−0.979830	−0.489915	+	0.871770i	$0.662972\pi$
$384$	0	0
$385$	7.45584	0.379985
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.95743i	0.454160i	0.973876	+	0.227080i	$0.0729178\pi$
$389$	8.95743i	0.454160i	−0.973876	+	0.227080i	$0.927082\pi$
$390$	0	0
$391$	−17.7408	−0.897190
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.727922i	0.0366257i
$396$	0	0
$397$	− 25.8067i	− 1.29520i	−0.761980	−	0.647600i	$-0.775773\pi$
$397$	− 25.8067i	− 1.29520i	0.761980	−	0.647600i	$-0.224227\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−28.4558	−1.42102	−0.710509	−	0.703689i	$-0.751535\pi$
$401$	−28.4558	−1.42102	−0.710509	+	0.703689i	$0.751535\pi$
$402$	0	0
$403$	6.00000i	0.298881i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.6531	0.577623
$408$	0	0
$409$	15.9706	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$409$	15.9706	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.30463	0.211306
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 28.9706i	− 1.41530i	−0.706561	−	0.707652i	$-0.749754\pi$
$419$	− 28.9706i	− 1.41530i	0.706561	−	0.707652i	$-0.250246\pi$
$420$	0	0
$421$	27.8359i	1.35664i	0.734767	+	0.678320i	$0.237291\pi$
$421$	27.8359i	1.35664i	−0.734767	+	0.678320i	$0.762709\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13.4558	0.652704
$426$	0	0
$427$	− 22.2426i	− 1.07640i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	27.9590	1.34674	0.673369	−	0.739307i	$-0.264847\pi$
$431$	27.9590	1.34674	0.673369	+	0.739307i	$0.264847\pi$
$432$	0	0
$433$	−18.4558	−0.886931	−0.443466	−	0.896291i	$-0.646251\pi$
$433$	−18.4558	−0.886931	−0.443466	+	0.896291i	$0.646251\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.2621i	0.634410i
$438$	0	0
$439$	25.0892	1.19744	0.598722	−	0.800957i	$-0.295675\pi$
$439$	25.0892	1.19744	0.598722	+	0.800957i	$0.295675\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 22.9706i	− 1.09136i	−0.837992	−	0.545682i	$-0.816271\pi$
$443$	− 22.9706i	− 1.09136i	0.837992	−	0.545682i	$-0.183729\pi$
$444$	0	0
$445$	− 0.369279i	− 0.0175055i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−28.9706	−1.36721	−0.683603	−	0.729854i	$-0.739588\pi$
$449$	−28.9706	−1.36721	−0.683603	+	0.729854i	$0.739588\pi$
$450$	0	0
$451$	36.0000i	1.69517i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.34847	−0.344502
$456$	0	0
$457$	7.48528	0.350147	0.175073	−	0.984555i	$-0.443984\pi$
$457$	7.48528	0.350147	0.175073	+	0.984555i	$0.443984\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.7846i	0.968036i	0.875058	+	0.484018i	$0.160823\pi$
$461$	20.7846i	0.968036i	−0.875058	+	0.484018i	$0.839177\pi$
$462$	0	0
$463$	−12.6677	−0.588719	−0.294359	−	0.955695i	$-0.595106\pi$
$463$	−12.6677	−0.588719	−0.294359	+	0.955695i	$0.595106\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.02944i	0.325284i	0.986685	+	0.162642i	$0.0520015\pi$
$467$	7.02944i	0.325284i	−0.986685	+	0.162642i	$0.947999\pi$
$468$	0	0
$469$	15.2913i	0.706086i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−15.9411	−0.732974
$474$	0	0
$475$	− 10.0589i	− 0.461533i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.34847	0.335760	0.167880	−	0.985807i	$-0.446308\pi$
$479$	7.34847	0.335760	0.167880	+	0.985807i	$0.446308\pi$
$480$	0	0
$481$	−11.4853	−0.523684
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.8272i	0.537045i
$486$	0	0
$487$	−8.36308	−0.378967	−0.189484	−	0.981884i	$-0.560681\pi$
$487$	−8.36308	−0.378967	−0.189484	+	0.981884i	$0.560681\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	28.2426i	1.27457i	0.770627	+	0.637286i	$0.219943\pi$
$491$	28.2426i	1.27457i	−0.770627	+	0.637286i	$0.780057\pi$
$492$	0	0
$493$	− 2.15232i	− 0.0969355i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−39.9411	−1.79160
$498$	0	0
$499$	− 0.242641i	− 0.0108621i	−0.999985	−	0.00543104i	$-0.998271\pi$
$499$	− 0.242641i	− 0.0108621i	0.999985	−	0.00543104i	$-0.00172876\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−23.6544	−1.05470	−0.527348	−	0.849649i	$-0.676814\pi$
$503$	−23.6544	−1.05470	−0.527348	+	0.849649i	$0.676814\pi$
$504$	0	0
$505$	9.51472	0.423399
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.86976i	0.127200i	0.997975	+	0.0635998i	$0.0202581\pi$
$509$	2.86976i	0.127200i	−0.997975	+	0.0635998i	$0.979742\pi$
$510$	0	0
$511$	23.2341	1.02782
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.0294i	0.574146i
$516$	0	0
$517$	− 6.08767i	− 0.267735i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.02944	0.307965	0.153983	−	0.988074i	$-0.450790\pi$
$521$	7.02944	0.307965	0.153983	+	0.988074i	$0.450790\pi$
$522$	0	0
$523$	− 12.2426i	− 0.535333i	−0.963512	−	0.267667i	$-0.913747\pi$
$523$	− 12.2426i	− 0.535333i	0.963512	−	0.267667i	$-0.0862526\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.30463	0.187513
$528$	0	0
$529$	11.9706	0.520459
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 35.4815i	− 1.53688i
$534$	0	0
$535$	4.30463	0.186106
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.24264i	0.182743i
$540$	0	0
$541$	− 11.9503i	− 0.513782i	−0.966440	−	0.256891i	$-0.917302\pi$
$541$	− 11.9503i	− 0.513782i	0.966440	−	0.256891i	$-0.0826982\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.97056	0.341421
$546$	0	0
$547$	30.9706i	1.32421i	0.749413	+	0.662103i	$0.230336\pi$
$547$	30.9706i	1.32421i	−0.749413	+	0.662103i	$0.769664\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.60896	−0.0685439
$552$	0	0
$553$	2.48528	0.105685
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6.45695i	− 0.273590i	−0.990599	−	0.136795i	$-0.956320\pi$
$557$	− 6.45695i	− 0.273590i	0.990599	−	0.136795i	$-0.0436801\pi$
$558$	0	0
$559$	15.7116	0.664528
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 18.0000i	− 0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$	− 18.0000i	− 0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$	0	0
$565$	0.369279i	0.0155357i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.4558	−0.438332	−0.219166	−	0.975688i	$-0.570334\pi$
$569$	−10.4558	−0.438332	−0.219166	+	0.975688i	$0.570334\pi$
$570$	0	0
$571$	24.9706i	1.04499i	0.852644	+	0.522493i	$0.174998\pi$
$571$	24.9706i	1.04499i	−0.852644	+	0.522493i	$0.825002\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−26.5241	−1.10613
$576$	0	0
$577$	−15.9706	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$577$	−15.9706	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 14.6969i	− 0.609732i
$582$	0	0
$583$	−50.1785	−2.07818
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 31.7574i	− 1.31077i	−0.755296	−	0.655383i	$-0.772507\pi$
$587$	− 31.7574i	− 1.31077i	0.755296	−	0.655383i	$-0.227493\pi$
$588$	0	0
$589$	− 3.21792i	− 0.132592i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.4853	0.964425	0.482212	−	0.876054i	$-0.339833\pi$
$593$	23.4853	0.964425	0.482212	+	0.876054i	$0.339833\pi$
$594$	0	0
$595$	5.27208i	0.216134i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.8288	1.25963	0.629814	−	0.776746i	$-0.283131\pi$
$599$	30.8288	1.25963	0.629814	+	0.776746i	$0.283131\pi$
$600$	0	0
$601$	−1.00000	−0.0407909	−0.0203954	−	0.999792i	$-0.506493\pi$
$601$	−1.00000	−0.0407909	−0.0203954	+	0.999792i	$0.506493\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.02207i	0.204176i
$606$	0	0
$607$	−2.44949	−0.0994217	−0.0497109	−	0.998764i	$-0.515830\pi$
$607$	−2.44949	−0.0994217	−0.0497109	+	0.998764i	$0.515830\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.00000i	0.242734i
$612$	0	0
$613$	0.840532i	0.0339488i	0.999856	+	0.0169744i	$0.00540337\pi$
$613$	0.840532i	0.0339488i	−0.999856	+	0.0169744i	$0.994597\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.9706	1.28709	0.643543	−	0.765410i	$-0.277464\pi$
$617$	31.9706	1.28709	0.643543	+	0.765410i	$0.277464\pi$
$618$	0	0
$619$	− 4.00000i	− 0.160774i	−0.996764	−	0.0803868i	$-0.974384\pi$
$619$	− 4.00000i	− 0.160774i	0.996764	−	0.0803868i	$-0.0256155\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.26080	−0.0505128
$624$	0	0
$625$	17.5442	0.701766
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.23999i	0.328550i
$630$	0	0
$631$	−19.5959	−0.780101	−0.390051	−	0.920793i	$-0.627542\pi$
$631$	−19.5959	−0.780101	−0.390051	+	0.920793i	$0.627542\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.30152i	0.250068i
$636$	0	0
$637$	− 4.18154i	− 0.165679i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.51472	0.257316	0.128658	−	0.991689i	$-0.458933\pi$
$641$	6.51472	0.257316	0.128658	+	0.991689i	$0.458933\pi$
$642$	0	0
$643$	30.9706i	1.22136i	0.791878	+	0.610680i	$0.209104\pi$
$643$	30.9706i	1.22136i	−0.791878	+	0.610680i	$0.790896\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.2182	0.401720	0.200860	−	0.979620i	$-0.435626\pi$
$647$	10.2182	0.401720	0.200860	+	0.979620i	$0.435626\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 14.6969i	− 0.575136i	−0.957760	−	0.287568i	$-0.907153\pi$
$653$	− 14.6969i	− 0.575136i	0.957760	−	0.287568i	$-0.0928467\pi$
$654$	0	0
$655$	−15.9577	−0.623520
$656$	0	0
$657$	0	0
$658$	0	0
$659$	25.7574i	1.00336i	0.865052	+	0.501682i	$0.167285\pi$
$659$	25.7574i	1.00336i	−0.865052	+	0.501682i	$0.832715\pi$
$660$	0	0
$661$	34.1698i	1.32905i	0.747266	+	0.664525i	$0.231366\pi$
$661$	34.1698i	1.32905i	−0.747266	+	0.664525i	$0.768634\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.94113	0.152830
$666$	0	0
$667$	4.24264i	0.164276i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	38.5254	1.48726
$672$	0	0
$673$	−7.48528	−0.288536	−0.144268	−	0.989539i	$-0.546083\pi$
$673$	−7.48528	−0.288536	−0.144268	+	0.989539i	$0.546083\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 32.2636i	− 1.23999i	−0.784605	−	0.619996i	$-0.787134\pi$
$677$	− 32.2636i	− 1.23999i	0.784605	−	0.619996i	$-0.212866\pi$
$678$	0	0
$679$	40.3805	1.54966
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 39.9411i	− 1.52830i	−0.645036	−	0.764152i	$-0.723158\pi$
$683$	− 39.9411i	− 1.52830i	0.645036	−	0.764152i	$-0.276842\pi$
$684$	0	0
$685$	8.23999i	0.314834i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	49.4558	1.88412
$690$	0	0
$691$	38.7279i	1.47328i	0.676285	+	0.736640i	$0.263589\pi$
$691$	38.7279i	1.47328i	−0.676285	+	0.736640i	$0.736411\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.17439	−0.272140
$696$	0	0
$697$	−25.4558	−0.964209
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.15232i	0.0812919i	0.999174	+	0.0406459i	$0.0129416\pi$
$701$	2.15232i	0.0812919i	−0.999174	+	0.0406459i	$0.987058\pi$
$702$	0	0
$703$	6.15978	0.232320
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 32.4853i	− 1.22173i
$708$	0	0
$709$	− 35.6046i	− 1.33716i	−0.743640	−	0.668580i	$-0.766902\pi$
$709$	− 35.6046i	− 1.33716i	0.743640	−	0.668580i	$-0.233098\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.48528	−0.317776
$714$	0	0
$715$	− 12.7279i	− 0.475997i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−47.1347	−1.75783	−0.878913	−	0.476982i	$-0.841731\pi$
$719$	−47.1347	−1.75783	−0.878913	+	0.476982i	$0.841731\pi$
$720$	0	0
$721$	44.4853	1.65672
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.21792i	− 0.119510i
$726$	0	0
$727$	28.1331	1.04340	0.521699	−	0.853130i	$-0.325298\pi$
$727$	28.1331	1.04340	0.521699	+	0.853130i	$0.325298\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 11.2721i	− 0.416913i
$732$	0	0
$733$	− 41.2211i	− 1.52253i	−0.648438	−	0.761267i	$-0.724578\pi$
$733$	− 41.2211i	− 1.52253i	0.648438	−	0.761267i	$-0.275422\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−26.4853	−0.975598
$738$	0	0
$739$	14.9706i	0.550701i	0.961344	+	0.275351i	$0.0887939\pi$
$739$	14.9706i	0.550701i	−0.961344	+	0.275351i	$0.911206\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−30.8288	−1.13100	−0.565499	−	0.824749i	$-0.691316\pi$
$743$	−30.8288	−1.13100	−0.565499	+	0.824749i	$0.691316\pi$
$744$	0	0
$745$	16.4558	0.602895
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 14.6969i	− 0.537014i
$750$	0	0
$751$	39.1197	1.42750	0.713750	−	0.700401i	$-0.246995\pi$
$751$	39.1197	1.42750	0.713750	+	0.700401i	$0.246995\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9.94113i	0.361795i
$756$	0	0
$757$	− 21.6251i	− 0.785979i	−0.919543	−	0.392990i	$-0.871441\pi$
$757$	− 21.6251i	− 0.785979i	0.919543	−	0.392990i	$-0.128559\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	− 27.2132i	− 0.985184i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−22.5147	−0.811902	−0.405951	−	0.913895i	$-0.633060\pi$
$769$	−22.5147	−0.811902	−0.405951	+	0.913895i	$0.633060\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.8944i	1.14716i	0.819149	+	0.573580i	$0.194446\pi$
$773$	31.8944i	1.14716i	−0.819149	+	0.573580i	$0.805554\pi$
$774$	0	0
$775$	6.43583	0.231182
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.0294i	0.681800i
$780$	0	0
$781$	− 69.1801i	− 2.47546i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.94113	0.247739
$786$	0	0
$787$	49.6985i	1.77156i	0.464106	+	0.885780i	$0.346376\pi$
$787$	49.6985i	1.77156i	−0.464106	+	0.885780i	$0.653624\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.26080	0.0448288
$792$	0	0
$793$	−37.9706	−1.34837
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 31.8944i	− 1.12976i	−0.825174	−	0.564878i	$-0.808923\pi$
$797$	− 31.8944i	− 1.12976i	0.825174	−	0.564878i	$-0.191077\pi$
$798$	0	0
$799$	4.30463	0.152287
$800$	0	0
$801$	0	0
$802$	0	0
$803$	40.2426i	1.42013i
$804$	0	0
$805$	− 10.3923i	− 0.366281i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−51.4264	−1.80806	−0.904028	−	0.427473i	$-0.859404\pi$
$809$	−51.4264	−1.80806	−0.904028	+	0.427473i	$0.859404\pi$
$810$	0	0
$811$	− 20.0000i	− 0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$	− 20.0000i	− 0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.30559	0.325960
$816$	0	0
$817$	−8.42641	−0.294803
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.3718i	0.850582i	0.905057	+	0.425291i	$0.139828\pi$
$821$	24.3718i	0.850582i	−0.905057	+	0.425291i	$0.860172\pi$
$822$	0	0
$823$	−33.2061	−1.15749	−0.578747	−	0.815507i	$-0.696458\pi$
$823$	−33.2061	−1.15749	−0.578747	+	0.815507i	$0.696458\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 31.7574i	− 1.10431i	−0.833741	−	0.552156i	$-0.813805\pi$
$827$	− 31.7574i	− 1.10431i	0.833741	−	0.552156i	$-0.186195\pi$
$828$	0	0
$829$	− 43.5984i	− 1.51424i	−0.653278	−	0.757118i	$-0.726607\pi$
$829$	− 43.5984i	− 1.51424i	0.653278	−	0.757118i	$-0.273393\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.00000	−0.103944
$834$	0	0
$835$	13.7574i	0.476093i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.0013	−0.414330	−0.207165	−	0.978306i	$-0.566424\pi$
$839$	−12.0013	−0.414330	−0.207165	+	0.978306i	$0.566424\pi$
$840$	0	0
$841$	28.4853	0.982251
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.21792i	0.110700i
$846$	0	0
$847$	17.1464	0.589158
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 16.2426i	− 0.556791i
$852$	0	0
$853$	34.6410i	1.18609i	0.805171	+	0.593043i	$0.202074\pi$
$853$	34.6410i	1.18609i	−0.805171	+	0.593043i	$0.797926\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.9706	0.682181	0.341091	−	0.940030i	$-0.389204\pi$
$857$	19.9706	0.682181	0.341091	+	0.940030i	$0.389204\pi$
$858$	0	0
$859$	− 42.9706i	− 1.46614i	−0.680155	−	0.733068i	$-0.738088\pi$
$859$	− 42.9706i	− 1.46614i	0.680155	−	0.733068i	$-0.261912\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.6556	−1.21373	−0.606866	−	0.794804i	$-0.707573\pi$
$863$	−35.6556	−1.21373	−0.606866	+	0.794804i	$0.707573\pi$
$864$	0	0
$865$	−11.0589	−0.376013
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.30463i	0.146025i
$870$	0	0
$871$	26.1039	0.884495
$872$	0	0
$873$	0	0
$874$	0	0
$875$	16.6690i	0.563517i
$876$	0	0
$877$	− 3.34101i	− 0.112818i	−0.998408	−	0.0564089i	$-0.982035\pi$
$877$	− 3.34101i	− 0.112818i	0.998408	−	0.0564089i	$-0.0179651\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.9706	0.571753	0.285876	−	0.958267i	$-0.407715\pi$
$881$	16.9706	0.571753	0.285876	+	0.958267i	$0.407715\pi$
$882$	0	0
$883$	54.9706i	1.84991i	0.380081	+	0.924953i	$0.375896\pi$
$883$	54.9706i	1.84991i	−0.380081	+	0.924953i	$0.624104\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.4377	1.08915	0.544576	−	0.838712i	$-0.316691\pi$
$887$	32.4377	1.08915	0.544576	+	0.838712i	$0.316691\pi$
$888$	0	0
$889$	21.5147	0.721581
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 3.21792i	− 0.107683i
$894$	0	0
$895$	16.4800	0.550865
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.02944i	− 0.0343337i
$900$	0	0
$901$	− 35.4815i	− 1.18206i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.51472	−0.116833
$906$	0	0
$907$	− 41.9411i	− 1.39263i	−0.717735	−	0.696316i	$-0.754821\pi$
$907$	− 41.9411i	− 1.39263i	0.717735	−	0.696316i	$-0.245179\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−20.6105	−0.682857	−0.341429	−	0.939908i	$-0.610911\pi$
$911$	−20.6105	−0.682857	−0.341429	+	0.939908i	$0.610911\pi$
$912$	0	0
$913$	25.4558	0.842465
$914$	0	0
$915$	0	0
$916$	0	0
$917$	54.4831i	1.79919i
$918$	0	0
$919$	23.8284	0.786028	0.393014	−	0.919533i	$-0.371432\pi$
$919$	23.8284	0.786028	0.393014	+	0.919533i	$0.371432\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	68.1838i	2.24430i
$924$	0	0
$925$	12.3196i	0.405064i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	43.9706	1.44263	0.721314	−	0.692609i	$-0.243539\pi$
$929$	43.9706	1.44263	0.721314	+	0.692609i	$0.243539\pi$
$930$	0	0
$931$	2.24264i	0.0734996i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.13151	−0.298632
$936$	0	0
$937$	31.0000	1.01273	0.506363	−	0.862320i	$-0.330990\pi$
$937$	31.0000	1.01273	0.506363	+	0.862320i	$0.330990\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 21.5020i	− 0.700947i	−0.936573	−	0.350473i	$-0.886021\pi$
$941$	− 21.5020i	− 0.700947i	0.936573	−	0.350473i	$-0.113979\pi$
$942$	0	0
$943$	50.1785	1.63404
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.2426i	0.722789i	0.932413	+	0.361394i	$0.117699\pi$
$947$	22.2426i	0.722789i	−0.932413	+	0.361394i	$0.882301\pi$
$948$	0	0
$949$	− 39.6631i	− 1.28752i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−36.5147	−1.18283	−0.591414	−	0.806368i	$-0.701430\pi$
$953$	−36.5147	−1.18283	−0.591414	+	0.806368i	$0.701430\pi$
$954$	0	0
$955$	3.21320i	0.103977i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	28.1331	0.908464
$960$	0	0
$961$	−28.9411	−0.933585
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.02207i	0.161666i
$966$	0	0
$967$	6.50794	0.209281	0.104641	−	0.994510i	$-0.466631\pi$
$967$	6.50794	0.209281	0.104641	+	0.994510i	$0.466631\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.75736i	− 0.0563963i	−0.999602	−	0.0281982i	$-0.991023\pi$
$971$	− 1.75736i	− 0.0563963i	0.999602	−	0.0281982i	$-0.00897695\pi$
$972$	0	0
$973$	24.4949i	0.785270i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.02944	0.224892	0.112446	−	0.993658i	$-0.464132\pi$
$977$	7.02944	0.224892	0.112446	+	0.993658i	$0.464132\pi$
$978$	0	0
$979$	− 2.18377i	− 0.0697935i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	50.1785	1.60045	0.800223	−	0.599703i	$-0.204715\pi$
$983$	50.1785	1.60045	0.800223	+	0.599703i	$0.204715\pi$
$984$	0	0
$985$	−16.4558	−0.524327
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.2195i	0.706539i
$990$	0	0
$991$	55.8459	1.77400	0.887002	−	0.461766i	$-0.152784\pi$
$991$	55.8459	1.77400	0.887002	+	0.461766i	$0.152784\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.91169i	0.282520i
$996$	0	0
$997$	− 33.3292i	− 1.05555i	−0.849385	−	0.527774i	$-0.823027\pi$
$997$	− 33.3292i	− 1.05555i	0.849385	−	0.527774i	$-0.176973\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.d.p.2593.3	yes	8
3.2	odd	2		5184.2.d.o.2593.5	yes	8
4.3	odd	2	inner	5184.2.d.p.2593.4	yes	8
8.3	odd	2	inner	5184.2.d.p.2593.6	yes	8
8.5	even	2	inner	5184.2.d.p.2593.5	yes	8
12.11	even	2		5184.2.d.o.2593.6	yes	8
24.5	odd	2		5184.2.d.o.2593.3	✓	8
24.11	even	2		5184.2.d.o.2593.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5184.2.d.o.2593.3	✓	8	24.5	odd	2
5184.2.d.o.2593.4	yes	8	24.11	even	2
5184.2.d.o.2593.5	yes	8	3.2	odd	2
5184.2.d.o.2593.6	yes	8	12.11	even	2
5184.2.d.p.2593.3	yes	8	1.1	even	1	trivial
5184.2.d.p.2593.4	yes	8	4.3	odd	2	inner
5184.2.d.p.2593.5	yes	8	8.5	even	2	inner
5184.2.d.p.2593.6	yes	8	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.717439i q^{5} -2.44949 q^{7} +O(q^{10})\)	\(q-0.717439i q^{5} -2.44949 q^{7} -4.24264i q^{11} +4.18154i q^{13} +3.00000 q^{17} -2.24264i q^{19} -5.91359 q^{23} +4.48528 q^{25} -0.717439i q^{29} +1.43488 q^{31} +1.75736i q^{35} +2.74666i q^{37} -8.48528 q^{41} -3.75736i q^{43} +1.43488 q^{47} -1.00000 q^{49} -11.8272i q^{53} -3.04384 q^{55} +9.08052i q^{61} +3.00000 q^{65} -6.24264i q^{67} +16.3059 q^{71} -9.48528 q^{73} +10.3923i q^{77} -1.01461 q^{79} +6.00000i q^{83} -2.15232i q^{85} +0.514719 q^{89} -10.2426i q^{91} -1.60896 q^{95} -16.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 24 q^{17} - 32 q^{25} - 8 q^{49} + 24 q^{65} - 8 q^{73} + 72 q^{89} - 64 q^{97}+O(q^{100}) \) 8 * q + 24 * q^17 - 32 * q^25 - 8 * q^49 + 24 * q^65 - 8 * q^73 + 72 * q^89 - 64 * q^97

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