LMFDB - Embedded newform 768.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.13251087523$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.82843 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.82843 q^{5} +2.82843 q^{7} +1.00000 q^{9} -4.00000 q^{11} +5.65685 q^{13} +2.82843 q^{15} -2.00000 q^{17} -4.00000 q^{19} -2.82843 q^{21} -5.65685 q^{23} +3.00000 q^{25} -1.00000 q^{27} +2.82843 q^{29} -8.48528 q^{31} +4.00000 q^{33} -8.00000 q^{35} -5.65685 q^{39} -10.0000 q^{41} -12.0000 q^{43} -2.82843 q^{45} +5.65685 q^{47} +1.00000 q^{49} +2.00000 q^{51} +2.82843 q^{53} +11.3137 q^{55} +4.00000 q^{57} +4.00000 q^{59} -11.3137 q^{61} +2.82843 q^{63} -16.0000 q^{65} -4.00000 q^{67} +5.65685 q^{69} +5.65685 q^{71} +2.00000 q^{73} -3.00000 q^{75} -11.3137 q^{77} -8.48528 q^{79} +1.00000 q^{81} +4.00000 q^{83} +5.65685 q^{85} -2.82843 q^{87} -6.00000 q^{89} +16.0000 q^{91} +8.48528 q^{93} +11.3137 q^{95} +14.0000 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} - 8 q^{11} - 4 q^{17} - 8 q^{19} + 6 q^{25} - 2 q^{27} + 8 q^{33} - 16 q^{35} - 20 q^{41} - 24 q^{43} + 2 q^{49} + 4 q^{51} + 8 q^{57} + 8 q^{59} - 32 q^{65} - 8 q^{67} + 4 q^{73} - 6 q^{75} + 2 q^{81} + 8 q^{83} - 12 q^{89} + 32 q^{91} + 28 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 8 * q^11 - 4 * q^17 - 8 * q^19 + 6 * q^25 - 2 * q^27 + 8 * q^33 - 16 * q^35 - 20 * q^41 - 24 * q^43 + 2 * q^49 + 4 * q^51 + 8 * q^57 + 8 * q^59 - 32 * q^65 - 8 * q^67 + 4 * q^73 - 6 * q^75 + 2 * q^81 + 8 * q^83 - 12 * q^89 + 32 * q^91 + 28 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	0	0
$15$	2.82843	0.730297
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.82843	0.525226	0.262613	−	0.964901i	$-0.415416\pi$
$29$	2.82843	0.525226	0.262613	+	0.964901i	$0.415416\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	−8.00000	−1.35225
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−5.65685	−0.905822
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	5.65685	0.825137	0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685	0.825137	0.412568	+	0.910927i	$0.364632\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	2.82843	0.388514	0.194257	−	0.980951i	$-0.437770\pi$
$53$	2.82843	0.388514	0.194257	+	0.980951i	$0.437770\pi$
$54$	0	0
$55$	11.3137	1.52554
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−11.3137	−1.44857	−0.724286	−	0.689500i	$-0.757830\pi$
$61$	−11.3137	−1.44857	−0.724286	+	0.689500i	$0.757830\pi$
$62$	0	0
$63$	2.82843	0.356348
$64$	0	0
$65$	−16.0000	−1.98456
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	5.65685	0.681005
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	−11.3137	−1.28932
$78$	0	0
$79$	−8.48528	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	5.65685	0.613572
$86$	0	0
$87$	−2.82843	−0.303239
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	0	0
$93$	8.48528	0.879883
$94$	0	0
$95$	11.3137	1.16076
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	14.1421	1.40720	0.703598	−	0.710599i	$-0.251576\pi$
$101$	14.1421	1.40720	0.703598	+	0.710599i	$0.251576\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−5.65685	−0.541828	−0.270914	−	0.962604i	$-0.587326\pi$
$109$	−5.65685	−0.541828	−0.270914	+	0.962604i	$0.587326\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	16.0000	1.49201
$116$	0	0
$117$	5.65685	0.522976
$118$	0	0
$119$	−5.65685	−0.518563
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	10.0000	0.901670
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	8.48528	0.752947	0.376473	−	0.926427i	$-0.377137\pi$
$127$	8.48528	0.752947	0.376473	+	0.926427i	$0.377137\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−11.3137	−0.981023
$134$	0	0
$135$	2.82843	0.243432
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−5.65685	−0.476393
$142$	0	0
$143$	−22.6274	−1.89220
$144$	0	0
$145$	−8.00000	−0.664364
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−14.1421	−1.15857	−0.579284	−	0.815125i	$-0.696668\pi$
$149$	−14.1421	−1.15857	−0.579284	+	0.815125i	$0.696668\pi$
$150$	0	0
$151$	−8.48528	−0.690522	−0.345261	−	0.938507i	$-0.612210\pi$
$151$	−8.48528	−0.690522	−0.345261	+	0.938507i	$0.612210\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	24.0000	1.92773
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−2.82843	−0.224309
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	−11.3137	−0.880771
$166$	0	0
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	19.7990	1.50529	0.752645	−	0.658427i	$-0.228778\pi$
$173$	19.7990	1.50529	0.752645	+	0.658427i	$0.228778\pi$
$174$	0	0
$175$	8.48528	0.641427
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−5.65685	−0.420471	−0.210235	−	0.977651i	$-0.567423\pi$
$181$	−5.65685	−0.420471	−0.210235	+	0.977651i	$0.567423\pi$
$182$	0	0
$183$	11.3137	0.836333
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−2.82843	−0.205738
$190$	0	0
$191$	−11.3137	−0.818631	−0.409316	−	0.912393i	$-0.634232\pi$
$191$	−11.3137	−0.818631	−0.409316	+	0.912393i	$0.634232\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	16.0000	1.14578
$196$	0	0
$197$	−8.48528	−0.604551	−0.302276	−	0.953221i	$-0.597746\pi$
$197$	−8.48528	−0.604551	−0.302276	+	0.953221i	$0.597746\pi$
$198$	0	0
$199$	−2.82843	−0.200502	−0.100251	−	0.994962i	$-0.531965\pi$
$199$	−2.82843	−0.200502	−0.100251	+	0.994962i	$0.531965\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	28.2843	1.97546
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	−5.65685	−0.387601
$214$	0	0
$215$	33.9411	2.31477
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−11.3137	−0.761042
$222$	0	0
$223$	19.7990	1.32584	0.662919	−	0.748691i	$-0.269317\pi$
$223$	19.7990	1.32584	0.662919	+	0.748691i	$0.269317\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	5.65685	0.373815	0.186908	−	0.982377i	$-0.440153\pi$
$229$	5.65685	0.373815	0.186908	+	0.982377i	$0.440153\pi$
$230$	0	0
$231$	11.3137	0.744387
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	0	0
$237$	8.48528	0.551178
$238$	0	0
$239$	22.6274	1.46365	0.731823	−	0.681495i	$-0.238670\pi$
$239$	22.6274	1.46365	0.731823	+	0.681495i	$0.238670\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.82843	−0.180702
$246$	0	0
$247$	−22.6274	−1.43975
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	22.6274	1.42257
$254$	0	0
$255$	−5.65685	−0.354246
$256$	0	0
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.82843	0.175075
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	−8.48528	−0.517357	−0.258678	−	0.965964i	$-0.583287\pi$
$269$	−8.48528	−0.517357	−0.258678	+	0.965964i	$0.583287\pi$
$270$	0	0
$271$	19.7990	1.20270	0.601351	−	0.798985i	$-0.294629\pi$
$271$	19.7990	1.20270	0.601351	+	0.798985i	$0.294629\pi$
$272$	0	0
$273$	−16.0000	−0.968364
$274$	0	0
$275$	−12.0000	−0.723627
$276$	0	0
$277$	16.9706	1.01966	0.509831	−	0.860274i	$-0.329708\pi$
$277$	16.9706	1.01966	0.509831	+	0.860274i	$0.329708\pi$
$278$	0	0
$279$	−8.48528	−0.508001
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−11.3137	−0.670166
$286$	0	0
$287$	−28.2843	−1.66957
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	−19.7990	−1.15667	−0.578335	−	0.815800i	$-0.696297\pi$
$293$	−19.7990	−1.15667	−0.578335	+	0.815800i	$0.696297\pi$
$294$	0	0
$295$	−11.3137	−0.658710
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	−32.0000	−1.85061
$300$	0	0
$301$	−33.9411	−1.95633
$302$	0	0
$303$	−14.1421	−0.812444
$304$	0	0
$305$	32.0000	1.83231
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	2.82843	0.160904
$310$	0	0
$311$	−11.3137	−0.641542	−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137	−0.641542	−0.320771	+	0.947157i	$0.603942\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	−8.00000	−0.450749
$316$	0	0
$317$	−8.48528	−0.476581	−0.238290	−	0.971194i	$-0.576587\pi$
$317$	−8.48528	−0.476581	−0.238290	+	0.971194i	$0.576587\pi$
$318$	0	0
$319$	−11.3137	−0.633446
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	16.9706	0.941357
$326$	0	0
$327$	5.65685	0.312825
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.3137	0.618134
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	33.9411	1.83801
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−16.0000	−0.861411
$346$	0	0
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	0	0
$349$	−11.3137	−0.605609	−0.302804	−	0.953053i	$-0.597923\pi$
$349$	−11.3137	−0.605609	−0.302804	+	0.953053i	$0.597923\pi$
$350$	0	0
$351$	−5.65685	−0.301941
$352$	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
$355$	−16.0000	−0.849192
$356$	0	0
$357$	5.65685	0.299392
$358$	0	0
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	−5.65685	−0.296093
$366$	0	0
$367$	−8.48528	−0.442928	−0.221464	−	0.975169i	$-0.571084\pi$
$367$	−8.48528	−0.442928	−0.221464	+	0.975169i	$0.571084\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	−33.9411	−1.75740	−0.878702	−	0.477370i	$-0.841590\pi$
$373$	−33.9411	−1.75740	−0.878702	+	0.477370i	$0.841590\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	16.0000	0.824042
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	−8.48528	−0.434714
$382$	0	0
$383$	11.3137	0.578103	0.289052	−	0.957313i	$-0.406660\pi$
$383$	11.3137	0.578103	0.289052	+	0.957313i	$0.406660\pi$
$384$	0	0
$385$	32.0000	1.63087
$386$	0	0
$387$	−12.0000	−0.609994
$388$	0	0
$389$	31.1127	1.57748	0.788738	−	0.614729i	$-0.210735\pi$
$389$	31.1127	1.57748	0.788738	+	0.614729i	$0.210735\pi$
$390$	0	0
$391$	11.3137	0.572159
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	24.0000	1.20757
$396$	0	0
$397$	−33.9411	−1.70346	−0.851728	−	0.523984i	$-0.824445\pi$
$397$	−33.9411	−1.70346	−0.851728	+	0.523984i	$0.824445\pi$
$398$	0	0
$399$	11.3137	0.566394
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	−48.0000	−2.39105
$404$	0	0
$405$	−2.82843	−0.140546
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	0	0
$413$	11.3137	0.556711
$414$	0	0
$415$	−11.3137	−0.555368
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	16.9706	0.827095	0.413547	−	0.910483i	$-0.364290\pi$
$421$	16.9706	0.827095	0.413547	+	0.910483i	$0.364290\pi$
$422$	0	0
$423$	5.65685	0.275046
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	−32.0000	−1.54859
$428$	0	0
$429$	22.6274	1.09246
$430$	0	0
$431$	−28.2843	−1.36241	−0.681203	−	0.732095i	$-0.738543\pi$
$431$	−28.2843	−1.36241	−0.681203	+	0.732095i	$0.738543\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	8.00000	0.383571
$436$	0	0
$437$	22.6274	1.08242
$438$	0	0
$439$	31.1127	1.48493	0.742464	−	0.669886i	$-0.233657\pi$
$439$	31.1127	1.48493	0.742464	+	0.669886i	$0.233657\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	16.9706	0.804482
$446$	0	0
$447$	14.1421	0.668900
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	0	0
$453$	8.48528	0.398673
$454$	0	0
$455$	−45.2548	−2.12158
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	8.48528	0.395199	0.197599	−	0.980283i	$-0.436685\pi$
$461$	8.48528	0.395199	0.197599	+	0.980283i	$0.436685\pi$
$462$	0	0
$463$	19.7990	0.920137	0.460069	−	0.887883i	$-0.347825\pi$
$463$	19.7990	0.920137	0.460069	+	0.887883i	$0.347825\pi$
$464$	0	0
$465$	−24.0000	−1.11297
$466$	0	0
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	−11.3137	−0.522419
$470$	0	0
$471$	0	0
$472$	0	0
$473$	48.0000	2.20704
$474$	0	0
$475$	−12.0000	−0.550598
$476$	0	0
$477$	2.82843	0.129505
$478$	0	0
$479$	28.2843	1.29234	0.646171	−	0.763193i	$-0.276369\pi$
$479$	28.2843	1.29234	0.646171	+	0.763193i	$0.276369\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	16.0000	0.728025
$484$	0	0
$485$	−39.5980	−1.79805
$486$	0	0
$487$	14.1421	0.640841	0.320421	−	0.947275i	$-0.396176\pi$
$487$	14.1421	0.640841	0.320421	+	0.947275i	$0.396176\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−5.65685	−0.254772
$494$	0	0
$495$	11.3137	0.508513
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	11.3137	0.505459
$502$	0	0
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	−40.0000	−1.77998
$506$	0	0
$507$	−19.0000	−0.843820
$508$	0	0
$509$	2.82843	0.125368	0.0626839	−	0.998033i	$-0.480034\pi$
$509$	2.82843	0.125368	0.0626839	+	0.998033i	$0.480034\pi$
$510$	0	0
$511$	5.65685	0.250244
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−22.6274	−0.995153
$518$	0	0
$519$	−19.7990	−0.869079
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	−8.48528	−0.370328
$526$	0	0
$527$	16.9706	0.739249
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−56.5685	−2.45026
$534$	0	0
$535$	33.9411	1.46740
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−4.00000	−0.172292
$540$	0	0
$541$	16.9706	0.729621	0.364811	−	0.931082i	$-0.381134\pi$
$541$	16.9706	0.729621	0.364811	+	0.931082i	$0.381134\pi$
$542$	0	0
$543$	5.65685	0.242759
$544$	0	0
$545$	16.0000	0.685365
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	−11.3137	−0.482857
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.48528	0.359533	0.179766	−	0.983709i	$-0.442466\pi$
$557$	8.48528	0.359533	0.179766	+	0.983709i	$0.442466\pi$
$558$	0	0
$559$	−67.8823	−2.87111
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−5.65685	−0.237986
$566$	0	0
$567$	2.82843	0.118783
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	11.3137	0.472637
$574$	0	0
$575$	−16.9706	−0.707721
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	11.3137	0.469372
$582$	0	0
$583$	−11.3137	−0.468566
$584$	0	0
$585$	−16.0000	−0.661519
$586$	0	0
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	33.9411	1.39852
$590$	0	0
$591$	8.48528	0.349038
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	0	0
$597$	2.82843	0.115760
$598$	0	0
$599$	−39.5980	−1.61793	−0.808965	−	0.587857i	$-0.799972\pi$
$599$	−39.5980	−1.61793	−0.808965	+	0.587857i	$0.799972\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−14.1421	−0.574960
$606$	0	0
$607$	−14.1421	−0.574012	−0.287006	−	0.957929i	$-0.592660\pi$
$607$	−14.1421	−0.574012	−0.287006	+	0.957929i	$0.592660\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	32.0000	1.29458
$612$	0	0
$613$	−11.3137	−0.456956	−0.228478	−	0.973549i	$-0.573375\pi$
$613$	−11.3137	−0.456956	−0.228478	+	0.973549i	$0.573375\pi$
$614$	0	0
$615$	−28.2843	−1.14053
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	5.65685	0.227002
$622$	0	0
$623$	−16.9706	−0.679911
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	−16.0000	−0.638978
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−8.48528	−0.337794	−0.168897	−	0.985634i	$-0.554020\pi$
$631$	−8.48528	−0.337794	−0.168897	+	0.985634i	$0.554020\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	−24.0000	−0.952411
$636$	0	0
$637$	5.65685	0.224133
$638$	0	0
$639$	5.65685	0.223782
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	−33.9411	−1.33643
$646$	0	0
$647$	39.5980	1.55676	0.778379	−	0.627795i	$-0.216042\pi$
$647$	39.5980	1.55676	0.778379	+	0.627795i	$0.216042\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	24.0000	0.940634
$652$	0	0
$653$	42.4264	1.66027	0.830137	−	0.557560i	$-0.188262\pi$
$653$	42.4264	1.66027	0.830137	+	0.557560i	$0.188262\pi$
$654$	0	0
$655$	−33.9411	−1.32619
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	45.2548	1.76021	0.880105	−	0.474780i	$-0.157472\pi$
$661$	45.2548	1.76021	0.880105	+	0.474780i	$0.157472\pi$
$662$	0	0
$663$	11.3137	0.439388
$664$	0	0
$665$	32.0000	1.24091
$666$	0	0
$667$	−16.0000	−0.619522
$668$	0	0
$669$	−19.7990	−0.765473
$670$	0	0
$671$	45.2548	1.74704
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	−3.00000	−0.115470
$676$	0	0
$677$	19.7990	0.760937	0.380468	−	0.924794i	$-0.375763\pi$
$677$	19.7990	0.760937	0.380468	+	0.924794i	$0.375763\pi$
$678$	0	0
$679$	39.5980	1.51963
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	28.2843	1.08069
$686$	0	0
$687$	−5.65685	−0.215822
$688$	0	0
$689$	16.0000	0.609551
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	−11.3137	−0.429772
$694$	0	0
$695$	−11.3137	−0.429153
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	−19.7990	−0.747798	−0.373899	−	0.927470i	$-0.621979\pi$
$701$	−19.7990	−0.747798	−0.373899	+	0.927470i	$0.621979\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	40.0000	1.50435
$708$	0	0
$709$	−16.9706	−0.637343	−0.318671	−	0.947865i	$-0.603237\pi$
$709$	−16.9706	−0.637343	−0.318671	+	0.947865i	$0.603237\pi$
$710$	0	0
$711$	−8.48528	−0.318223
$712$	0	0
$713$	48.0000	1.79761
$714$	0	0
$715$	64.0000	2.39346
$716$	0	0
$717$	−22.6274	−0.845036
$718$	0	0
$719$	16.9706	0.632895	0.316448	−	0.948610i	$-0.397510\pi$
$719$	16.9706	0.632895	0.316448	+	0.948610i	$0.397510\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−6.00000	−0.223142
$724$	0	0
$725$	8.48528	0.315135
$726$	0	0
$727$	25.4558	0.944105	0.472052	−	0.881570i	$-0.343513\pi$
$727$	25.4558	0.944105	0.472052	+	0.881570i	$0.343513\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	−16.9706	−0.626822	−0.313411	−	0.949618i	$-0.601472\pi$
$733$	−16.9706	−0.626822	−0.313411	+	0.949618i	$0.601472\pi$
$734$	0	0
$735$	2.82843	0.104328
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	0	0
$741$	22.6274	0.831239
$742$	0	0
$743$	22.6274	0.830119	0.415060	−	0.909794i	$-0.363761\pi$
$743$	22.6274	0.830119	0.415060	+	0.909794i	$0.363761\pi$
$744$	0	0
$745$	40.0000	1.46549
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	−33.9411	−1.24018
$750$	0	0
$751$	48.0833	1.75458	0.877292	−	0.479958i	$-0.159348\pi$
$751$	48.0833	1.75458	0.877292	+	0.479958i	$0.159348\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	24.0000	0.873449
$756$	0	0
$757$	28.2843	1.02801	0.514005	−	0.857787i	$-0.328161\pi$
$757$	28.2843	1.02801	0.514005	+	0.857787i	$0.328161\pi$
$758$	0	0
$759$	−22.6274	−0.821323
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	0	0
$765$	5.65685	0.204524
$766$	0	0
$767$	22.6274	0.817029
$768$	0	0
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	0	0
$773$	2.82843	0.101731	0.0508657	−	0.998706i	$-0.483802\pi$
$773$	2.82843	0.101731	0.0508657	+	0.998706i	$0.483802\pi$
$774$	0	0
$775$	−25.4558	−0.914401
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40.0000	1.43315
$780$	0	0
$781$	−22.6274	−0.809673
$782$	0	0
$783$	−2.82843	−0.101080
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.65685	0.201135
$792$	0	0
$793$	−64.0000	−2.27271
$794$	0	0
$795$	8.00000	0.283731
$796$	0	0
$797$	19.7990	0.701316	0.350658	−	0.936504i	$-0.385958\pi$
$797$	19.7990	0.701316	0.350658	+	0.936504i	$0.385958\pi$
$798$	0	0
$799$	−11.3137	−0.400250
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−8.00000	−0.282314
$804$	0	0
$805$	45.2548	1.59502
$806$	0	0
$807$	8.48528	0.298696
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	−19.7990	−0.694381
$814$	0	0
$815$	11.3137	0.396302
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	16.0000	0.559085
$820$	0	0
$821$	−31.1127	−1.08584	−0.542920	−	0.839784i	$-0.682681\pi$
$821$	−31.1127	−1.08584	−0.542920	+	0.839784i	$0.682681\pi$
$822$	0	0
$823$	−19.7990	−0.690149	−0.345075	−	0.938575i	$-0.612146\pi$
$823$	−19.7990	−0.690149	−0.345075	+	0.938575i	$0.612146\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−50.9117	−1.76824	−0.884118	−	0.467264i	$-0.845240\pi$
$829$	−50.9117	−1.76824	−0.884118	+	0.467264i	$0.845240\pi$
$830$	0	0
$831$	−16.9706	−0.588702
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	32.0000	1.10741
$836$	0	0
$837$	8.48528	0.293294
$838$	0	0
$839$	−5.65685	−0.195296	−0.0976481	−	0.995221i	$-0.531132\pi$
$839$	−5.65685	−0.195296	−0.0976481	+	0.995221i	$0.531132\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	−10.0000	−0.344418
$844$	0	0
$845$	−53.7401	−1.84872
$846$	0	0
$847$	14.1421	0.485930
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−22.6274	−0.774748	−0.387374	−	0.921923i	$-0.626618\pi$
$853$	−22.6274	−0.774748	−0.387374	+	0.921923i	$0.626618\pi$
$854$	0	0
$855$	11.3137	0.386921
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	28.2843	0.963925
$862$	0	0
$863$	−56.5685	−1.92562	−0.962808	−	0.270187i	$-0.912914\pi$
$863$	−56.5685	−1.92562	−0.962808	+	0.270187i	$0.912914\pi$
$864$	0	0
$865$	−56.0000	−1.90406
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	33.9411	1.15137
$870$	0	0
$871$	−22.6274	−0.766701
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	16.0000	0.540899
$876$	0	0
$877$	11.3137	0.382037	0.191018	−	0.981586i	$-0.438821\pi$
$877$	11.3137	0.382037	0.191018	+	0.981586i	$0.438821\pi$
$878$	0	0
$879$	19.7990	0.667803
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	0	0
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	0	0
$885$	11.3137	0.380306
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	−22.6274	−0.757198
$894$	0	0
$895$	−33.9411	−1.13453
$896$	0	0
$897$	32.0000	1.06845
$898$	0	0
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−5.65685	−0.188457
$902$	0	0
$903$	33.9411	1.12949
$904$	0	0
$905$	16.0000	0.531858
$906$	0	0
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	0	0
$909$	14.1421	0.469065
$910$	0	0
$911$	−22.6274	−0.749680	−0.374840	−	0.927090i	$-0.622302\pi$
$911$	−22.6274	−0.749680	−0.374840	+	0.927090i	$0.622302\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	−32.0000	−1.05789
$916$	0	0
$917$	33.9411	1.12083
$918$	0	0
$919$	2.82843	0.0933012	0.0466506	−	0.998911i	$-0.485145\pi$
$919$	2.82843	0.0933012	0.0466506	+	0.998911i	$0.485145\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	32.0000	1.05329
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.82843	−0.0928977
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	11.3137	0.370394
$934$	0	0
$935$	−22.6274	−0.739996
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	−14.1421	−0.461020	−0.230510	−	0.973070i	$-0.574040\pi$
$941$	−14.1421	−0.461020	−0.230510	+	0.973070i	$0.574040\pi$
$942$	0	0
$943$	56.5685	1.84213
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	11.3137	0.367259
$950$	0	0
$951$	8.48528	0.275154
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	32.0000	1.03550
$956$	0	0
$957$	11.3137	0.365720
$958$	0	0
$959$	−28.2843	−0.913347
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	−16.9706	−0.546302
$966$	0	0
$967$	−14.1421	−0.454780	−0.227390	−	0.973804i	$-0.573019\pi$
$967$	−14.1421	−0.454780	−0.227390	+	0.973804i	$0.573019\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	11.3137	0.362701
$974$	0	0
$975$	−16.9706	−0.543493
$976$	0	0
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	−5.65685	−0.180609
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	0	0
$987$	−16.0000	−0.509286
$988$	0	0
$989$	67.8823	2.15853
$990$	0	0
$991$	25.4558	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	−33.9411	−1.07493	−0.537463	−	0.843287i	$-0.680617\pi$
$997$	−33.9411	−1.07493	−0.537463	+	0.843287i	$0.680617\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.a.i.1.1		2
3.2	odd	2		2304.2.a.x.1.2		2
4.3	odd	2		768.2.a.l.1.1		2
8.3	odd	2	inner	768.2.a.i.1.2		2
8.5	even	2		768.2.a.l.1.2		2
12.11	even	2		2304.2.a.r.1.2		2
16.3	odd	4		384.2.d.c.193.4	yes	4
16.5	even	4		384.2.d.c.193.3	yes	4
16.11	odd	4		384.2.d.c.193.1	✓	4
16.13	even	4		384.2.d.c.193.2	yes	4
24.5	odd	2		2304.2.a.r.1.1		2
24.11	even	2		2304.2.a.x.1.1		2
48.5	odd	4		1152.2.d.h.577.3		4
48.11	even	4		1152.2.d.h.577.4		4
48.29	odd	4		1152.2.d.h.577.1		4
48.35	even	4		1152.2.d.h.577.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.d.c.193.1	✓	4	16.11	odd	4
384.2.d.c.193.2	yes	4	16.13	even	4
384.2.d.c.193.3	yes	4	16.5	even	4
384.2.d.c.193.4	yes	4	16.3	odd	4
768.2.a.i.1.1		2	1.1	even	1	trivial
768.2.a.i.1.2		2	8.3	odd	2	inner
768.2.a.l.1.1		2	4.3	odd	2
768.2.a.l.1.2		2	8.5	even	2
1152.2.d.h.577.1		4	48.29	odd	4
1152.2.d.h.577.2		4	48.35	even	4
1152.2.d.h.577.3		4	48.5	odd	4
1152.2.d.h.577.4		4	48.11	even	4
2304.2.a.r.1.1		2	24.5	odd	2
2304.2.a.r.1.2		2	12.11	even	2
2304.2.a.x.1.1		2	24.11	even	2
2304.2.a.x.1.2		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.82843 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.82843 q^{5} +2.82843 q^{7} +1.00000 q^{9} -4.00000 q^{11} +5.65685 q^{13} +2.82843 q^{15} -2.00000 q^{17} -4.00000 q^{19} -2.82843 q^{21} -5.65685 q^{23} +3.00000 q^{25} -1.00000 q^{27} +2.82843 q^{29} -8.48528 q^{31} +4.00000 q^{33} -8.00000 q^{35} -5.65685 q^{39} -10.0000 q^{41} -12.0000 q^{43} -2.82843 q^{45} +5.65685 q^{47} +1.00000 q^{49} +2.00000 q^{51} +2.82843 q^{53} +11.3137 q^{55} +4.00000 q^{57} +4.00000 q^{59} -11.3137 q^{61} +2.82843 q^{63} -16.0000 q^{65} -4.00000 q^{67} +5.65685 q^{69} +5.65685 q^{71} +2.00000 q^{73} -3.00000 q^{75} -11.3137 q^{77} -8.48528 q^{79} +1.00000 q^{81} +4.00000 q^{83} +5.65685 q^{85} -2.82843 q^{87} -6.00000 q^{89} +16.0000 q^{91} +8.48528 q^{93} +11.3137 q^{95} +14.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} - 8 q^{11} - 4 q^{17} - 8 q^{19} + 6 q^{25} - 2 q^{27} + 8 q^{33} - 16 q^{35} - 20 q^{41} - 24 q^{43} + 2 q^{49} + 4 q^{51} + 8 q^{57} + 8 q^{59} - 32 q^{65} - 8 q^{67} + 4 q^{73} - 6 q^{75} + 2 q^{81} + 8 q^{83} - 12 q^{89} + 32 q^{91} + 28 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 8 * q^11 - 4 * q^17 - 8 * q^19 + 6 * q^25 - 2 * q^27 + 8 * q^33 - 16 * q^35 - 20 * q^41 - 24 * q^43 + 2 * q^49 + 4 * q^51 + 8 * q^57 + 8 * q^59 - 32 * q^65 - 8 * q^67 + 4 * q^73 - 6 * q^75 + 2 * q^81 + 8 * q^83 - 12 * q^89 + 32 * q^91 + 28 * q^97 - 8 * q^99

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