LMFDB - Embedded newform 1728.3.g.l.703.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(703,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, 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("1728.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.207360000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 $ x^8 + 6x^6 + 32x^4 + 24*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.7
Root			$-1.14412 - 1.98168i$ of defining polynomial
Character	$\chi$	$=$	1728.703
Dual form			1728.3.g.l.703.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.40492 q^{5} -9.47802i q^{7} +O(q^{10})$	$q+7.40492 q^{5} -9.47802i q^{7} -6.77022i q^{11} +14.4164 q^{13} -17.8933 q^{17} -5.19615i q^{19} -24.9366i q^{23} +29.8328 q^{25} +29.6197 q^{29} +17.1275i q^{31} -70.1839i q^{35} -6.41641 q^{37} -8.64290 q^{41} -50.1329i q^{43} -52.0175i q^{47} -40.8328 q^{49} -82.6921 q^{53} -50.1329i q^{55} +83.7244i q^{59} +8.41641 q^{61} +106.752 q^{65} +29.0588i q^{67} +113.287i q^{71} +2.16718 q^{73} -64.1683 q^{77} -149.485i q^{79} +13.5404i q^{83} -132.498 q^{85} -17.8933 q^{89} -136.639i q^{91} -38.4771i q^{95} -138.331 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{13} + 24 q^{25} + 56 q^{37} - 112 q^{49} - 40 q^{61} + 232 q^{73} - 416 q^{85} - 248 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 + 24 * q^25 + 56 * q^37 - 112 * q^49 - 40 * q^61 + 232 * q^73 - 416 * q^85 - 248 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	7.40492	1.48098	0.740492	−	0.672065i	$-0.234593\pi$
$5$	7.40492	1.48098	0.740492	+	0.672065i	$0.234593\pi$
$6$	0	0
$7$	− 9.47802i	− 1.35400i	−0.735982	−	0.677001i	$-0.763279\pi$
$7$	− 9.47802i	− 1.35400i	0.735982	−	0.677001i	$-0.236721\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 6.77022i	− 0.615475i	−0.951471	−	0.307737i	$-0.900428\pi$
$11$	− 6.77022i	− 0.615475i	0.951471	−	0.307737i	$-0.0995718\pi$
$12$	0	0
$13$	14.4164	1.10895	0.554477	−	0.832199i	$-0.312918\pi$
$13$	14.4164	1.10895	0.554477	+	0.832199i	$0.312918\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−17.8933	−1.05255	−0.526274	−	0.850315i	$-0.676411\pi$
$17$	−17.8933	−1.05255	−0.526274	+	0.850315i	$0.676411\pi$
$18$	0	0
$19$	− 5.19615i	− 0.273482i	−0.990607	−	0.136741i	$-0.956337\pi$
$19$	− 5.19615i	− 0.273482i	0.990607	−	0.136741i	$-0.0436628\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 24.9366i	− 1.08420i	−0.840313	−	0.542101i	$-0.817629\pi$
$23$	− 24.9366i	− 1.08420i	0.840313	−	0.542101i	$-0.182371\pi$
$24$	0	0
$25$	29.8328	1.19331
$26$	0	0
$27$	0	0
$28$	0	0
$29$	29.6197	1.02137	0.510684	−	0.859768i	$-0.329392\pi$
$29$	29.6197	1.02137	0.510684	+	0.859768i	$0.329392\pi$
$30$	0	0
$31$	17.1275i	0.552499i	0.961086	+	0.276249i	$0.0890916\pi$
$31$	17.1275i	0.552499i	−0.961086	+	0.276249i	$0.910908\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 70.1839i	− 2.00526i
$36$	0	0
$37$	−6.41641	−0.173416	−0.0867082	−	0.996234i	$-0.527635\pi$
$37$	−6.41641	−0.173416	−0.0867082	+	0.996234i	$0.527635\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.64290	−0.210803	−0.105401	−	0.994430i	$-0.533613\pi$
$41$	−8.64290	−0.210803	−0.105401	+	0.994430i	$0.533613\pi$
$42$	0	0
$43$	− 50.1329i	− 1.16588i	−0.812514	−	0.582941i	$-0.801902\pi$
$43$	− 50.1329i	− 1.16588i	0.812514	−	0.582941i	$-0.198098\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 52.0175i	− 1.10676i	−0.832930	−	0.553378i	$-0.813339\pi$
$47$	− 52.0175i	− 1.10676i	0.832930	−	0.553378i	$-0.186661\pi$
$48$	0	0
$49$	−40.8328	−0.833323
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−82.6921	−1.56023	−0.780114	−	0.625637i	$-0.784839\pi$
$53$	−82.6921	−1.56023	−0.780114	+	0.625637i	$0.784839\pi$
$54$	0	0
$55$	− 50.1329i	− 0.911508i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	83.7244i	1.41906i	0.704676	+	0.709529i	$0.251092\pi$
$59$	83.7244i	1.41906i	−0.704676	+	0.709529i	$0.748908\pi$
$60$	0	0
$61$	8.41641	0.137974	0.0689869	−	0.997618i	$-0.478023\pi$
$61$	8.41641	0.137974	0.0689869	+	0.997618i	$0.478023\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	106.752	1.64234
$66$	0	0
$67$	29.0588i	0.433713i	0.976203	+	0.216856i	$0.0695804\pi$
$67$	29.0588i	0.433713i	−0.976203	+	0.216856i	$0.930420\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	113.287i	1.59559i	0.602928	+	0.797796i	$0.294001\pi$
$71$	113.287i	1.59559i	−0.602928	+	0.797796i	$0.705999\pi$
$72$	0	0
$73$	2.16718	0.0296875	0.0148437	−	0.999890i	$-0.495275\pi$
$73$	2.16718	0.0296875	0.0148437	+	0.999890i	$0.495275\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−64.1683	−0.833354
$78$	0	0
$79$	− 149.485i	− 1.89221i	−0.323861	−	0.946105i	$-0.604981\pi$
$79$	− 149.485i	− 1.89221i	0.323861	−	0.946105i	$-0.395019\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.5404i	0.163138i	0.996668	+	0.0815690i	$0.0259931\pi$
$83$	13.5404i	0.163138i	−0.996668	+	0.0815690i	$0.974007\pi$
$84$	0	0
$85$	−132.498	−1.55881
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.8933	−0.201048	−0.100524	−	0.994935i	$-0.532052\pi$
$89$	−17.8933	−0.201048	−0.100524	+	0.994935i	$0.532052\pi$
$90$	0	0
$91$	− 136.639i	− 1.50153i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 38.4771i	− 0.405022i
$96$	0	0
$97$	−138.331	−1.42610	−0.713048	−	0.701115i	$-0.752686\pi$
$97$	−138.331	−1.42610	−0.713048	+	0.701115i	$0.752686\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	97.5019	0.965366	0.482683	−	0.875795i	$-0.339662\pi$
$101$	97.5019	0.965366	0.482683	+	0.875795i	$0.339662\pi$
$102$	0	0
$103$	− 42.4835i	− 0.412461i	−0.978503	−	0.206231i	$-0.933880\pi$
$103$	− 42.4835i	− 0.412461i	0.978503	−	0.206231i	$-0.0661197\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	97.2648i	0.909017i	0.890742	+	0.454509i	$0.150185\pi$
$107$	97.2648i	0.909017i	−0.890742	+	0.454509i	$0.849815\pi$
$108$	0	0
$109$	96.8328	0.888374	0.444187	−	0.895934i	$-0.353493\pi$
$109$	96.8328	0.888374	0.444187	+	0.895934i	$0.353493\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−38.8701	−0.343983	−0.171991	−	0.985098i	$-0.555020\pi$
$113$	−38.8701	−0.343983	−0.171991	+	0.985098i	$0.555020\pi$
$114$	0	0
$115$	− 184.654i	− 1.60568i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	169.593i	1.42515i
$120$	0	0
$121$	75.1641	0.621191
$122$	0	0
$123$	0	0
$124$	0	0
$125$	35.7866	0.286293
$126$	0	0
$127$	− 99.0165i	− 0.779657i	−0.920887	−	0.389829i	$-0.872534\pi$
$127$	− 99.0165i	− 0.779657i	0.920887	−	0.389829i	$-0.127466\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 54.1618i	− 0.413449i	−0.978399	−	0.206724i	$-0.933720\pi$
$131$	− 54.1618i	− 0.413449i	0.978399	−	0.206724i	$-0.0662803\pi$
$132$	0	0
$133$	−49.2492	−0.370295
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.55944	0.0405798	0.0202899	−	0.999794i	$-0.493541\pi$
$137$	5.55944	0.0405798	0.0202899	+	0.999794i	$0.493541\pi$
$138$	0	0
$139$	− 152.517i	− 1.09724i	−0.836070	−	0.548622i	$-0.815153\pi$
$139$	− 152.517i	− 1.09724i	0.836070	−	0.548622i	$-0.184847\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 97.6023i	− 0.682534i
$144$	0	0
$145$	219.331	1.51263
$146$	0	0
$147$	0	0
$148$	0	0
$149$	136.979	0.919325	0.459663	−	0.888094i	$-0.347970\pi$
$149$	136.979	0.919325	0.459663	+	0.888094i	$0.347970\pi$
$150$	0	0
$151$	77.9879i	0.516476i	0.966081	+	0.258238i	$0.0831419\pi$
$151$	77.9879i	0.516476i	−0.966081	+	0.258238i	$0.916858\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	126.827i	0.818242i
$156$	0	0
$157$	72.1641	0.459644	0.229822	−	0.973233i	$-0.426186\pi$
$157$	72.1641	0.459644	0.229822	+	0.973233i	$0.426186\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−236.350	−1.46801
$162$	0	0
$163$	56.5785i	0.347108i	0.984824	+	0.173554i	$0.0555250\pi$
$163$	56.5785i	0.347108i	−0.984824	+	0.173554i	$0.944475\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	201.637i	1.20741i	0.797208	+	0.603705i	$0.206309\pi$
$167$	201.637i	1.20741i	−0.797208	+	0.603705i	$0.793691\pi$
$168$	0	0
$169$	38.8328	0.229780
$170$	0	0
$171$	0	0
$172$	0	0
$173$	92.5500	0.534971	0.267485	−	0.963562i	$-0.413807\pi$
$173$	92.5500	0.534971	0.267485	+	0.963562i	$0.413807\pi$
$174$	0	0
$175$	− 282.756i	− 1.61575i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4.28851i	− 0.0239581i	−0.999928	−	0.0119791i	$-0.996187\pi$
$179$	− 4.28851i	− 0.0239581i	0.999928	−	0.0119791i	$-0.00381315\pi$
$180$	0	0
$181$	289.748	1.60082	0.800408	−	0.599456i	$-0.204616\pi$
$181$	289.748	1.60082	0.800408	+	0.599456i	$0.204616\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−47.5130	−0.256827
$186$	0	0
$187$	121.142i	0.647816i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 92.6389i	− 0.485020i	−0.970149	−	0.242510i	$-0.922029\pi$
$191$	− 92.6389i	− 0.485020i	0.970149	−	0.242510i	$-0.0779708\pi$
$192$	0	0
$193$	−243.164	−1.25992	−0.629959	−	0.776629i	$-0.716928\pi$
$193$	−243.164	−1.25992	−0.629959	+	0.776629i	$0.716928\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−139.432	−0.707779	−0.353890	−	0.935287i	$-0.615141\pi$
$197$	−139.432	−0.707779	−0.353890	+	0.935287i	$0.615141\pi$
$198$	0	0
$199$	− 110.993i	− 0.557756i	−0.960327	−	0.278878i	$-0.910038\pi$
$199$	− 110.993i	− 0.557756i	0.960327	−	0.278878i	$-0.0899624\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 280.736i	− 1.38293i
$204$	0	0
$205$	−64.0000	−0.312195
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−35.1791	−0.168321
$210$	0	0
$211$	− 265.095i	− 1.25637i	−0.778062	−	0.628187i	$-0.783797\pi$
$211$	− 265.095i	− 1.25637i	0.778062	−	0.628187i	$-0.216203\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 371.230i	− 1.72665i
$216$	0	0
$217$	162.334	0.748085
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−257.957	−1.16723
$222$	0	0
$223$	− 317.925i	− 1.42567i	−0.701330	−	0.712837i	$-0.747410\pi$
$223$	− 317.925i	− 1.42567i	0.701330	−	0.712837i	$-0.252590\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 249.366i	− 1.09853i	−0.835648	−	0.549265i	$-0.814908\pi$
$227$	− 249.366i	− 1.09853i	0.835648	−	0.549265i	$-0.185092\pi$
$228$	0	0
$229$	−152.334	−0.665216	−0.332608	−	0.943065i	$-0.607929\pi$
$229$	−152.334	−0.665216	−0.332608	+	0.943065i	$0.607929\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	346.793	1.48838	0.744191	−	0.667966i	$-0.232835\pi$
$233$	346.793	1.48838	0.744191	+	0.667966i	$0.232835\pi$
$234$	0	0
$235$	− 385.186i	− 1.63909i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 104.035i	− 0.435293i	−0.976028	−	0.217647i	$-0.930162\pi$
$239$	− 104.035i	− 0.435293i	0.976028	−	0.217647i	$-0.0698380\pi$
$240$	0	0
$241$	281.827	1.16940	0.584702	−	0.811248i	$-0.301211\pi$
$241$	281.827	1.16940	0.584702	+	0.811248i	$0.301211\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−302.364	−1.23414
$246$	0	0
$247$	− 74.9099i	− 0.303279i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	271.484i	1.08161i	0.841148	+	0.540804i	$0.181880\pi$
$251$	271.484i	1.08161i	−0.841148	+	0.540804i	$0.818120\pi$
$252$	0	0
$253$	−168.827	−0.667299
$254$	0	0
$255$	0	0
$256$	0	0
$257$	459.105	1.78640	0.893200	−	0.449659i	$-0.148455\pi$
$257$	459.105	1.78640	0.893200	+	0.449659i	$0.148455\pi$
$258$	0	0
$259$	60.8148i	0.234806i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 203.782i	− 0.774835i	−0.921904	−	0.387418i	$-0.873367\pi$
$263$	− 203.782i	− 0.774835i	0.921904	−	0.387418i	$-0.126633\pi$
$264$	0	0
$265$	−612.328	−2.31067
$266$	0	0
$267$	0	0
$268$	0	0
$269$	323.363	1.20209	0.601047	−	0.799213i	$-0.294750\pi$
$269$	323.363	1.20209	0.601047	+	0.799213i	$0.294750\pi$
$270$	0	0
$271$	104.928i	0.387190i	0.981082	+	0.193595i	$0.0620148\pi$
$271$	104.928i	0.387190i	−0.981082	+	0.193595i	$0.937985\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 201.975i	− 0.734454i
$276$	0	0
$277$	48.8328	0.176292	0.0881459	−	0.996108i	$-0.471906\pi$
$277$	48.8328	0.176292	0.0881459	+	0.996108i	$0.471906\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−29.6197	−0.105408	−0.0527040	−	0.998610i	$-0.516784\pi$
$281$	−29.6197	−0.105408	−0.0527040	+	0.998610i	$0.516784\pi$
$282$	0	0
$283$	− 385.186i	− 1.36108i	−0.732711	−	0.680540i	$-0.761745\pi$
$283$	− 385.186i	− 1.36108i	0.732711	−	0.680540i	$-0.238255\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	81.9176i	0.285427i
$288$	0	0
$289$	31.1703	0.107856
$290$	0	0
$291$	0	0
$292$	0	0
$293$	281.410	0.960443	0.480222	−	0.877147i	$-0.340556\pi$
$293$	281.410	0.960443	0.480222	+	0.877147i	$0.340556\pi$
$294$	0	0
$295$	619.972i	2.10160i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 359.497i	− 1.20233i
$300$	0	0
$301$	−475.161	−1.57861
$302$	0	0
$303$	0	0
$304$	0	0
$305$	62.3228	0.204337
$306$	0	0
$307$	553.961i	1.80443i	0.431282	+	0.902217i	$0.358061\pi$
$307$	553.961i	1.80443i	−0.431282	+	0.902217i	$0.641939\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.6847i	0.0504331i	0.999682	+	0.0252166i	$0.00802753\pi$
$311$	15.6847i	0.0504331i	−0.999682	+	0.0252166i	$0.991972\pi$
$312$	0	0
$313$	−308.161	−0.984540	−0.492270	−	0.870443i	$-0.663833\pi$
$313$	−308.161	−0.984540	−0.492270	+	0.870443i	$0.663833\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−496.107	−1.56500	−0.782502	−	0.622648i	$-0.786057\pi$
$317$	−496.107	−1.56500	−0.782502	+	0.622648i	$0.786057\pi$
$318$	0	0
$319$	− 200.532i	− 0.628626i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	92.9763i	0.287852i
$324$	0	0
$325$	430.082	1.32333
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−493.023	−1.49855
$330$	0	0
$331$	− 86.5060i	− 0.261347i	−0.991425	−	0.130674i	$-0.958286\pi$
$331$	− 86.5060i	− 0.261347i	0.991425	−	0.130674i	$-0.0417140\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	215.178i	0.642322i
$336$	0	0
$337$	320.659	0.951512	0.475756	−	0.879577i	$-0.342175\pi$
$337$	320.659	0.951512	0.475756	+	0.879577i	$0.342175\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	115.957	0.340049
$342$	0	0
$343$	− 77.4087i	− 0.225681i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	674.759i	1.94455i	0.233842	+	0.972275i	$0.424870\pi$
$347$	674.759i	1.94455i	−0.233842	+	0.972275i	$0.575130\pi$
$348$	0	0
$349$	−153.918	−0.441026	−0.220513	−	0.975384i	$-0.570773\pi$
$349$	−153.918	−0.441026	−0.220513	+	0.975384i	$0.570773\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−623.228	−1.76552	−0.882759	−	0.469825i	$-0.844317\pi$
$353$	−623.228	−1.76552	−0.882759	+	0.469825i	$0.844317\pi$
$354$	0	0
$355$	838.881i	2.36305i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	160.341i	0.446633i	0.974746	+	0.223316i	$0.0716883\pi$
$359$	160.341i	0.446633i	−0.974746	+	0.223316i	$0.928312\pi$
$360$	0	0
$361$	334.000	0.925208
$362$	0	0
$363$	0	0
$364$	0	0
$365$	16.0478	0.0439666
$366$	0	0
$367$	− 284.585i	− 0.775435i	−0.921778	−	0.387717i	$-0.873264\pi$
$367$	− 284.585i	− 0.775435i	0.921778	−	0.387717i	$-0.126736\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	783.757i	2.11255i
$372$	0	0
$373$	−301.420	−0.808095	−0.404048	−	0.914738i	$-0.632397\pi$
$373$	−301.420	−0.808095	−0.404048	+	0.914738i	$0.632397\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	427.009	1.13265
$378$	0	0
$379$	− 248.547i	− 0.655796i	−0.944713	−	0.327898i	$-0.893660\pi$
$379$	− 248.547i	− 0.655796i	0.944713	−	0.327898i	$-0.106340\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	488.131i	1.27449i	0.770660	+	0.637247i	$0.219927\pi$
$383$	488.131i	1.27449i	−0.770660	+	0.637247i	$0.780073\pi$
$384$	0	0
$385$	−475.161	−1.23418
$386$	0	0
$387$	0	0
$388$	0	0
$389$	415.867	1.06907	0.534534	−	0.845147i	$-0.320487\pi$
$389$	415.867	1.06907	0.534534	+	0.845147i	$0.320487\pi$
$390$	0	0
$391$	446.199i	1.14117i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1106.92i	− 2.80233i
$396$	0	0
$397$	670.827	1.68974	0.844870	−	0.534972i	$-0.179678\pi$
$397$	670.827	1.68974	0.844870	+	0.534972i	$0.179678\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	114.742	0.286139	0.143070	−	0.989713i	$-0.454303\pi$
$401$	114.742	0.286139	0.143070	+	0.989713i	$0.454303\pi$
$402$	0	0
$403$	246.916i	0.612696i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	43.4405i	0.106733i
$408$	0	0
$409$	−146.672	−0.358611	−0.179305	−	0.983793i	$-0.557385\pi$
$409$	−146.672	−0.358611	−0.179305	+	0.983793i	$0.557385\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	793.541	1.92141
$414$	0	0
$415$	100.266i	0.241605i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	536.872i	1.28132i	0.767825	+	0.640659i	$0.221339\pi$
$419$	536.872i	1.28132i	−0.767825	+	0.640659i	$0.778661\pi$
$420$	0	0
$421$	367.413	0.872716	0.436358	−	0.899773i	$-0.356268\pi$
$421$	367.413	0.872716	0.436358	+	0.899773i	$0.356268\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−533.808	−1.25602
$426$	0	0
$427$	− 79.7709i	− 0.186817i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	735.353i	1.70616i	0.521784	+	0.853078i	$0.325267\pi$
$431$	735.353i	1.70616i	−0.521784	+	0.853078i	$0.674733\pi$
$432$	0	0
$433$	765.161	1.76712	0.883558	−	0.468322i	$-0.155141\pi$
$433$	765.161	1.76712	0.883558	+	0.468322i	$0.155141\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−129.575	−0.296509
$438$	0	0
$439$	46.3847i	0.105660i	0.998604	+	0.0528299i	$0.0168241\pi$
$439$	46.3847i	0.105660i	−0.998604	+	0.0528299i	$0.983176\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	139.693i	0.315334i	0.987492	+	0.157667i	$0.0503973\pi$
$443$	139.693i	0.315334i	−0.987492	+	0.157667i	$0.949603\pi$
$444$	0	0
$445$	−132.498	−0.297749
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−267.139	−0.594963	−0.297482	−	0.954728i	$-0.596147\pi$
$449$	−267.139	−0.594963	−0.297482	+	0.954728i	$0.596147\pi$
$450$	0	0
$451$	58.5144i	0.129744i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1011.80i	− 2.22374i
$456$	0	0
$457$	237.830	0.520415	0.260208	−	0.965553i	$-0.416209\pi$
$457$	237.830	0.520415	0.260208	+	0.965553i	$0.416209\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−298.650	−0.647830	−0.323915	−	0.946086i	$-0.604999\pi$
$461$	−298.650	−0.647830	−0.323915	+	0.946086i	$0.604999\pi$
$462$	0	0
$463$	489.535i	1.05731i	0.848837	+	0.528655i	$0.177304\pi$
$463$	489.535i	1.05731i	−0.848837	+	0.528655i	$0.822696\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	454.280i	0.972762i	0.873747	+	0.486381i	$0.161683\pi$
$467$	454.280i	0.972762i	−0.873747	+	0.486381i	$0.838317\pi$
$468$	0	0
$469$	275.420	0.587248
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−339.411	−0.717571
$474$	0	0
$475$	− 155.016i	− 0.326349i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.61359i	0.00754403i	0.999993	+	0.00377201i	$0.00120067\pi$
$479$	3.61359i	0.00754403i	−0.999993	+	0.00377201i	$0.998799\pi$
$480$	0	0
$481$	−92.5016	−0.192311
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1024.33	−2.11202
$486$	0	0
$487$	874.538i	1.79577i	0.440234	+	0.897883i	$0.354896\pi$
$487$	874.538i	1.79577i	−0.440234	+	0.897883i	$0.645104\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 645.196i	− 1.31404i	−0.753871	−	0.657022i	$-0.771816\pi$
$491$	− 645.196i	− 1.31404i	0.753871	−	0.657022i	$-0.228184\pi$
$492$	0	0
$493$	−529.994	−1.07504
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1073.74	2.16043
$498$	0	0
$499$	564.842i	1.13195i	0.824423	+	0.565974i	$0.191500\pi$
$499$	564.842i	1.13195i	−0.824423	+	0.565974i	$0.808500\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	483.048i	0.960334i	0.877177	+	0.480167i	$0.159424\pi$
$503$	483.048i	0.960334i	−0.877177	+	0.480167i	$0.840576\pi$
$504$	0	0
$505$	721.994	1.42969
$506$	0	0
$507$	0	0
$508$	0	0
$509$	216.004	0.424369	0.212184	−	0.977230i	$-0.431942\pi$
$509$	216.004	0.424369	0.212184	+	0.977230i	$0.431942\pi$
$510$	0	0
$511$	− 20.5406i	− 0.0401969i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 314.587i	− 0.610848i
$516$	0	0
$517$	−352.170	−0.681180
$518$	0	0
$519$	0	0
$520$	0	0
$521$	454.806	0.872949	0.436475	−	0.899717i	$-0.356227\pi$
$521$	454.806	0.872949	0.436475	+	0.899717i	$0.356227\pi$
$522$	0	0
$523$	325.041i	0.621493i	0.950493	+	0.310747i	$0.100579\pi$
$523$	325.041i	0.621493i	−0.950493	+	0.310747i	$0.899421\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 306.467i	− 0.581531i
$528$	0	0
$529$	−92.8359	−0.175493
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−124.600	−0.233770
$534$	0	0
$535$	720.238i	1.34624i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	276.447i	0.512889i
$540$	0	0
$541$	962.574	1.77925	0.889625	−	0.456692i	$-0.150966\pi$
$541$	962.574	1.77925	0.889625	+	0.456692i	$0.150966\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	717.039	1.31567
$546$	0	0
$547$	133.470i	0.244003i	0.992530	+	0.122002i	$0.0389313\pi$
$547$	133.470i	0.244003i	−0.992530	+	0.122002i	$0.961069\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 153.908i	− 0.279325i
$552$	0	0
$553$	−1416.82	−2.56206
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−413.437	−0.742258	−0.371129	−	0.928581i	$-0.621029\pi$
$557$	−413.437	−0.742258	−0.371129	+	0.928581i	$0.621029\pi$
$558$	0	0
$559$	− 722.737i	− 1.29291i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	516.562i	0.917516i	0.888561	+	0.458758i	$0.151706\pi$
$563$	516.562i	0.917516i	−0.888561	+	0.458758i	$0.848294\pi$
$564$	0	0
$565$	−287.830	−0.509433
$566$	0	0
$567$	0	0
$568$	0	0
$569$	468.355	0.823120	0.411560	−	0.911383i	$-0.364984\pi$
$569$	468.355	0.823120	0.411560	+	0.911383i	$0.364984\pi$
$570$	0	0
$571$	675.972i	1.18384i	0.805997	+	0.591919i	$0.201630\pi$
$571$	675.972i	1.18384i	−0.805997	+	0.591919i	$0.798370\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 743.930i	− 1.29379i
$576$	0	0
$577$	−354.823	−0.614945	−0.307473	−	0.951557i	$-0.599483\pi$
$577$	−354.823	−0.614945	−0.307473	+	0.951557i	$0.599483\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	128.337	0.220889
$582$	0	0
$583$	559.844i	0.960281i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 220.479i	− 0.375603i	−0.982207	−	0.187801i	$-0.939864\pi$
$587$	− 220.479i	− 0.375603i	0.982207	−	0.187801i	$-0.0601361\pi$
$588$	0	0
$589$	88.9969	0.151098
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−993.520	−1.67541	−0.837707	−	0.546121i	$-0.816104\pi$
$593$	−993.520	−1.67541	−0.837707	+	0.546121i	$0.816104\pi$
$594$	0	0
$595$	1255.82i	2.11063i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	280.736i	0.468674i	0.972155	+	0.234337i	$0.0752919\pi$
$599$	280.736i	0.468674i	−0.972155	+	0.234337i	$0.924708\pi$
$600$	0	0
$601$	561.830	0.934825	0.467412	−	0.884039i	$-0.345186\pi$
$601$	561.830	0.934825	0.467412	+	0.884039i	$0.345186\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	556.584	0.919973
$606$	0	0
$607$	− 84.0527i	− 0.138472i	−0.997600	−	0.0692362i	$-0.977944\pi$
$607$	− 84.0527i	− 0.138472i	0.997600	−	0.0692362i	$-0.0220562\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 749.906i	− 1.22734i
$612$	0	0
$613$	−730.234	−1.19125	−0.595623	−	0.803264i	$-0.703095\pi$
$613$	−730.234	−1.19125	−0.595623	+	0.803264i	$0.703095\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	549.786	0.891064	0.445532	−	0.895266i	$-0.353015\pi$
$617$	549.786	0.891064	0.445532	+	0.895266i	$0.353015\pi$
$618$	0	0
$619$	73.1269i	0.118137i	0.998254	+	0.0590685i	$0.0188131\pi$
$619$	73.1269i	0.118137i	−0.998254	+	0.0590685i	$0.981187\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	169.593i	0.272220i
$624$	0	0
$625$	−480.823	−0.769318
$626$	0	0
$627$	0	0
$628$	0	0
$629$	114.811	0.182529
$630$	0	0
$631$	779.849i	1.23589i	0.786220	+	0.617947i	$0.212035\pi$
$631$	779.849i	1.23589i	−0.786220	+	0.617947i	$0.787965\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 733.209i	− 1.15466i
$636$	0	0
$637$	−588.663	−0.924117
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−444.341	−0.693200	−0.346600	−	0.938013i	$-0.612664\pi$
$641$	−444.341	−0.693200	−0.346600	+	0.938013i	$0.612664\pi$
$642$	0	0
$643$	446.199i	0.693933i	0.937878	+	0.346966i	$0.112788\pi$
$643$	446.199i	0.693933i	−0.937878	+	0.346966i	$0.887212\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 861.386i	− 1.33135i	−0.746240	−	0.665677i	$-0.768143\pi$
$647$	− 861.386i	− 1.33135i	0.746240	−	0.665677i	$-0.231857\pi$
$648$	0	0
$649$	566.833	0.873394
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−876.302	−1.34196	−0.670982	−	0.741474i	$-0.734127\pi$
$653$	−876.302	−1.34196	−0.670982	+	0.741474i	$0.734127\pi$
$654$	0	0
$655$	− 401.064i	− 0.612311i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 565.760i	− 0.858513i	−0.903183	−	0.429257i	$-0.858776\pi$
$659$	− 565.760i	− 0.858513i	0.903183	−	0.429257i	$-0.141224\pi$
$660$	0	0
$661$	−632.580	−0.957005	−0.478503	−	0.878086i	$-0.658820\pi$
$661$	−632.580	−0.957005	−0.478503	+	0.878086i	$0.658820\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−364.686	−0.548401
$666$	0	0
$667$	− 738.615i	− 1.10737i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 56.9810i	− 0.0849195i
$672$	0	0
$673$	−1072.66	−1.59385	−0.796924	−	0.604080i	$-0.793541\pi$
$673$	−1072.66	−1.59385	−0.796924	+	0.604080i	$0.793541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	478.798	0.707234	0.353617	−	0.935390i	$-0.384952\pi$
$677$	478.798	0.707234	0.353617	+	0.935390i	$0.384952\pi$
$678$	0	0
$679$	1311.11i	1.93094i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	318.418i	0.466206i	0.972452	+	0.233103i	$0.0748879\pi$
$683$	318.418i	0.466206i	−0.972452	+	0.233103i	$0.925112\pi$
$684$	0	0
$685$	41.1672	0.0600981
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1192.12	−1.73022
$690$	0	0
$691$	1121.30i	1.62272i	0.584545	+	0.811362i	$0.301273\pi$
$691$	1121.30i	1.62272i	−0.584545	+	0.811362i	$0.698727\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1129.38i	− 1.62500i
$696$	0	0
$697$	154.650	0.221880
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−413.437	−0.589782	−0.294891	−	0.955531i	$-0.595283\pi$
$701$	−413.437	−0.589782	−0.294891	+	0.955531i	$0.595283\pi$
$702$	0	0
$703$	33.3406i	0.0474262i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 924.125i	− 1.30711i
$708$	0	0
$709$	682.915	0.963209	0.481604	−	0.876389i	$-0.340054\pi$
$709$	682.915	0.963209	0.481604	+	0.876389i	$0.340054\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	427.101	0.599020
$714$	0	0
$715$	− 722.737i	− 1.01082i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 286.374i	− 0.398295i	−0.979970	−	0.199148i	$-0.936183\pi$
$719$	− 286.374i	− 0.398295i	0.979970	−	0.199148i	$-0.0638173\pi$
$720$	0	0
$721$	−402.659	−0.558474
$722$	0	0
$723$	0	0
$724$	0	0
$725$	883.638	1.21881
$726$	0	0
$727$	322.923i	0.444186i	0.975026	+	0.222093i	$0.0712888\pi$
$727$	322.923i	0.444186i	−0.975026	+	0.222093i	$0.928711\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	897.044i	1.22715i
$732$	0	0
$733$	521.830	0.711910	0.355955	−	0.934503i	$-0.384156\pi$
$733$	521.830	0.711910	0.355955	+	0.934503i	$0.384156\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	196.734	0.266939
$738$	0	0
$739$	− 965.020i	− 1.30585i	−0.757424	−	0.652923i	$-0.773542\pi$
$739$	− 965.020i	− 1.30585i	0.757424	−	0.652923i	$-0.226458\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1097.33i	1.47689i	0.674312	+	0.738447i	$0.264440\pi$
$743$	1097.33i	1.47689i	−0.674312	+	0.738447i	$0.735560\pi$
$744$	0	0
$745$	1014.32	1.36151
$746$	0	0
$747$	0	0
$748$	0	0
$749$	921.878	1.23081
$750$	0	0
$751$	− 1381.05i	− 1.83894i	−0.393154	−	0.919472i	$-0.628616\pi$
$751$	− 1381.05i	− 1.83894i	0.393154	−	0.919472i	$-0.371384\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	577.494i	0.764892i
$756$	0	0
$757$	516.252	0.681971	0.340986	−	0.940068i	$-0.389239\pi$
$757$	516.252	0.681971	0.340986	+	0.940068i	$0.389239\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1216.18	−1.59814	−0.799069	−	0.601239i	$-0.794674\pi$
$761$	−1216.18	−1.59814	−0.799069	+	0.601239i	$0.794674\pi$
$762$	0	0
$763$	− 917.783i	− 1.20286i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1207.00i	1.57367i
$768$	0	0
$769$	1004.33	1.30601	0.653007	−	0.757352i	$-0.273507\pi$
$769$	1004.33	1.30601	0.653007	+	0.757352i	$0.273507\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	382.580	0.494929	0.247464	−	0.968897i	$-0.420403\pi$
$773$	382.580	0.494929	0.247464	+	0.968897i	$0.420403\pi$
$774$	0	0
$775$	510.960i	0.659304i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	44.9098i	0.0576506i
$780$	0	0
$781$	766.978	0.982046
$782$	0	0
$783$	0	0
$784$	0	0
$785$	534.369	0.680725
$786$	0	0
$787$	− 477.268i	− 0.606440i	−0.952921	−	0.303220i	$-0.901938\pi$
$787$	− 477.268i	− 0.606440i	0.952921	−	0.303220i	$-0.0980617\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	368.411i	0.465754i
$792$	0	0
$793$	121.334	0.153007
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−514.607	−0.645680	−0.322840	−	0.946453i	$-0.604638\pi$
$797$	−514.607	−0.645680	−0.322840	+	0.946453i	$0.604638\pi$
$798$	0	0
$799$	930.765i	1.16491i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 14.6723i	− 0.0182719i
$804$	0	0
$805$	−1750.15	−2.17410
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1463.74	−1.80932	−0.904662	−	0.426129i	$-0.859877\pi$
$809$	−1463.74	−1.80932	−0.904662	+	0.426129i	$0.859877\pi$
$810$	0	0
$811$	1163.05i	1.43410i	0.697023	+	0.717049i	$0.254508\pi$
$811$	1163.05i	1.43410i	−0.697023	+	0.717049i	$0.745492\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	418.959i	0.514061i
$816$	0	0
$817$	−260.498	−0.318848
$818$	0	0
$819$	0	0
$820$	0	0
$821$	392.553	0.478140	0.239070	−	0.971002i	$-0.423158\pi$
$821$	392.553	0.478140	0.239070	+	0.971002i	$0.423158\pi$
$822$	0	0
$823$	− 1248.84i	− 1.51743i	−0.651424	−	0.758714i	$-0.725828\pi$
$823$	− 1248.84i	− 1.51743i	0.651424	−	0.758714i	$-0.274172\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1369.15i	1.65557i	0.561049	+	0.827783i	$0.310398\pi$
$827$	1369.15i	1.65557i	−0.561049	+	0.827783i	$0.689602\pi$
$828$	0	0
$829$	73.7477	0.0889598	0.0444799	−	0.999010i	$-0.485837\pi$
$829$	73.7477	0.0889598	0.0444799	+	0.999010i	$0.485837\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	730.634	0.877112
$834$	0	0
$835$	1493.11i	1.78815i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1200.57i	1.43096i	0.698635	+	0.715478i	$0.253791\pi$
$839$	1200.57i	1.43096i	−0.698635	+	0.715478i	$0.746209\pi$
$840$	0	0
$841$	36.3251	0.0431927
$842$	0	0
$843$	0	0
$844$	0	0
$845$	287.554	0.340300
$846$	0	0
$847$	− 712.406i	− 0.841094i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	160.004i	0.188018i
$852$	0	0
$853$	439.079	0.514747	0.257373	−	0.966312i	$-0.417143\pi$
$853$	439.079	0.514747	0.257373	+	0.966312i	$0.417143\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	510.962	0.596222	0.298111	−	0.954531i	$-0.403643\pi$
$857$	510.962	0.596222	0.298111	+	0.954531i	$0.403643\pi$
$858$	0	0
$859$	− 176.776i	− 0.205793i	−0.994692	−	0.102897i	$-0.967189\pi$
$859$	− 176.776i	− 0.205793i	0.994692	−	0.102897i	$-0.0328111\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1028.28i	− 1.19152i	−0.803163	−	0.595759i	$-0.796851\pi$
$863$	− 1028.28i	− 1.19152i	0.803163	−	0.595759i	$-0.203149\pi$
$864$	0	0
$865$	685.325	0.792283
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1012.04	−1.16461
$870$	0	0
$871$	418.923i	0.480968i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 339.186i	− 0.387641i
$876$	0	0
$877$	940.915	1.07288	0.536439	−	0.843939i	$-0.319769\pi$
$877$	940.915	1.07288	0.536439	+	0.843939i	$0.319769\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1065.75	1.20970	0.604851	−	0.796339i	$-0.293233\pi$
$881$	1065.75	1.20970	0.604851	+	0.796339i	$0.293233\pi$
$882$	0	0
$883$	1001.97i	1.13474i	0.823464	+	0.567368i	$0.192038\pi$
$883$	1001.97i	1.13474i	−0.823464	+	0.567368i	$0.807962\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	674.084i	0.759959i	0.924995	+	0.379980i	$0.124069\pi$
$887$	674.084i	0.759959i	−0.924995	+	0.379980i	$0.875931\pi$
$888$	0	0
$889$	−938.480	−1.05566
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−270.291	−0.302678
$894$	0	0
$895$	− 31.7561i	− 0.0354816i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	507.310i	0.564305i
$900$	0	0
$901$	1479.63	1.64221
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2145.56	2.37078
$906$	0	0
$907$	− 1181.45i	− 1.30259i	−0.758826	−	0.651293i	$-0.774227\pi$
$907$	− 1181.45i	− 1.30259i	0.758826	−	0.651293i	$-0.225773\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1394.43i	− 1.53066i	−0.643641	−	0.765328i	$-0.722577\pi$
$911$	− 1394.43i	− 1.53066i	0.643641	−	0.765328i	$-0.277423\pi$
$912$	0	0
$913$	91.6718	0.100407
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−513.346	−0.559811
$918$	0	0
$919$	− 210.163i	− 0.228686i	−0.993441	−	0.114343i	$-0.963524\pi$
$919$	− 210.163i	− 0.228686i	0.993441	−	0.114343i	$-0.0364763\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1633.19i	1.76944i
$924$	0	0
$925$	−191.420	−0.206940
$926$	0	0
$927$	0	0
$928$	0	0
$929$	166.599	0.179332	0.0896659	−	0.995972i	$-0.471420\pi$
$929$	166.599	0.179332	0.0896659	+	0.995972i	$0.471420\pi$
$930$	0	0
$931$	212.174i	0.227899i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	897.044i	0.959405i
$936$	0	0
$937$	251.158	0.268045	0.134022	−	0.990978i	$-0.457211\pi$
$937$	251.158	0.268045	0.134022	+	0.990978i	$0.457211\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	814.564	0.865637	0.432818	−	0.901481i	$-0.357519\pi$
$941$	814.564	0.865637	0.432818	+	0.901481i	$0.357519\pi$
$942$	0	0
$943$	215.525i	0.228552i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 592.384i	− 0.625537i	−0.949829	−	0.312769i	$-0.898744\pi$
$947$	− 592.384i	− 0.625537i	0.949829	−	0.312769i	$-0.101256\pi$
$948$	0	0
$949$	31.2430	0.0329220
$950$	0	0
$951$	0	0
$952$	0	0
$953$	844.768	0.886430	0.443215	−	0.896415i	$-0.353838\pi$
$953$	844.768	0.886430	0.443215	+	0.896415i	$0.353838\pi$
$954$	0	0
$955$	− 685.983i	− 0.718307i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 52.6925i	− 0.0549452i
$960$	0	0
$961$	667.650	0.694745
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1800.61	−1.86592
$966$	0	0
$967$	− 180.173i	− 0.186322i	−0.995651	−	0.0931611i	$-0.970303\pi$
$967$	− 180.173i	− 0.186322i	0.995651	−	0.0931611i	$-0.0296971\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 411.395i	− 0.423682i	−0.977304	−	0.211841i	$-0.932054\pi$
$971$	− 411.395i	− 0.423682i	0.977304	−	0.211841i	$-0.0679458\pi$
$972$	0	0
$973$	−1445.56	−1.48567
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1756.25	−1.79759	−0.898797	−	0.438365i	$-0.855558\pi$
$977$	−1756.25	−1.79759	−0.898797	+	0.438365i	$0.855558\pi$
$978$	0	0
$979$	121.142i	0.123740i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	803.055i	0.816943i	0.912771	+	0.408472i	$0.133938\pi$
$983$	803.055i	0.816943i	−0.912771	+	0.408472i	$0.866062\pi$
$984$	0	0
$985$	−1032.49	−1.04821
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1250.15	−1.26405
$990$	0	0
$991$	− 338.466i	− 0.341540i	−0.985311	−	0.170770i	$-0.945375\pi$
$991$	− 338.466i	− 0.341540i	0.985311	−	0.170770i	$-0.0546254\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 821.897i	− 0.826027i
$996$	0	0
$997$	−591.811	−0.593592	−0.296796	−	0.954941i	$-0.595918\pi$
$997$	−591.811	−0.593592	−0.296796	+	0.954941i	$0.595918\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.g.l.703.7		8
3.2	odd	2	inner	1728.3.g.l.703.1		8
4.3	odd	2	inner	1728.3.g.l.703.8		8
8.3	odd	2		108.3.d.d.55.3	✓	8
8.5	even	2		108.3.d.d.55.4	yes	8
12.11	even	2	inner	1728.3.g.l.703.2		8
24.5	odd	2		108.3.d.d.55.5	yes	8
24.11	even	2		108.3.d.d.55.6	yes	8
72.5	odd	6		324.3.f.p.55.4		8
72.11	even	6		324.3.f.p.271.3		8
72.13	even	6		324.3.f.p.55.1		8
72.29	odd	6		324.3.f.o.271.1		8
72.43	odd	6		324.3.f.p.271.2		8
72.59	even	6		324.3.f.o.55.1		8
72.61	even	6		324.3.f.o.271.4		8
72.67	odd	6		324.3.f.o.55.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.d.d.55.3	✓	8	8.3	odd	2
108.3.d.d.55.4	yes	8	8.5	even	2
108.3.d.d.55.5	yes	8	24.5	odd	2
108.3.d.d.55.6	yes	8	24.11	even	2
324.3.f.o.55.1		8	72.59	even	6
324.3.f.o.55.4		8	72.67	odd	6
324.3.f.o.271.1		8	72.29	odd	6
324.3.f.o.271.4		8	72.61	even	6
324.3.f.p.55.1		8	72.13	even	6
324.3.f.p.55.4		8	72.5	odd	6
324.3.f.p.271.2		8	72.43	odd	6
324.3.f.p.271.3		8	72.11	even	6
1728.3.g.l.703.1		8	3.2	odd	2	inner
1728.3.g.l.703.2		8	12.11	even	2	inner
1728.3.g.l.703.7		8	1.1	even	1	trivial
1728.3.g.l.703.8		8	4.3	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.40492 q^{5} -9.47802i q^{7} +O(q^{10})\)	\(q+7.40492 q^{5} -9.47802i q^{7} -6.77022i q^{11} +14.4164 q^{13} -17.8933 q^{17} -5.19615i q^{19} -24.9366i q^{23} +29.8328 q^{25} +29.6197 q^{29} +17.1275i q^{31} -70.1839i q^{35} -6.41641 q^{37} -8.64290 q^{41} -50.1329i q^{43} -52.0175i q^{47} -40.8328 q^{49} -82.6921 q^{53} -50.1329i q^{55} +83.7244i q^{59} +8.41641 q^{61} +106.752 q^{65} +29.0588i q^{67} +113.287i q^{71} +2.16718 q^{73} -64.1683 q^{77} -149.485i q^{79} +13.5404i q^{83} -132.498 q^{85} -17.8933 q^{89} -136.639i q^{91} -38.4771i q^{95} -138.331 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{13} + 24 q^{25} + 56 q^{37} - 112 q^{49} - 40 q^{61} + 232 q^{73} - 416 q^{85} - 248 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 + 24 * q^25 + 56 * q^37 - 112 * q^49 - 40 * q^61 + 232 * q^73 - 416 * q^85 - 248 * q^97

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