LMFDB - Embedded newform 1024.2.b.g.513.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,2,Mod(513,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1024.513");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1024.b (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:	$8.17668116698$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			513.2
Root			$0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1024.513
Dual form			1024.2.b.g.513.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61313i q^{3} +3.41421i q^{5} -1.53073 q^{7} -3.82843 q^{9} +O(q^{10})$	$q-2.61313i q^{3} +3.41421i q^{5} -1.53073 q^{7} -3.82843 q^{9} -4.77791i q^{11} -0.585786i q^{13} +8.92177 q^{15} -2.82843 q^{17} +0.448342i q^{19} +4.00000i q^{21} -5.86030 q^{23} -6.65685 q^{25} +2.16478i q^{27} -4.58579i q^{29} -7.39104 q^{31} -12.4853 q^{33} -5.22625i q^{35} -5.07107i q^{37} -1.53073 q^{39} +4.00000 q^{41} -2.61313i q^{43} -13.0711i q^{45} -7.39104 q^{47} -4.65685 q^{49} +7.39104i q^{51} +7.41421i q^{53} +16.3128 q^{55} +1.17157 q^{57} +2.61313i q^{59} -13.0711i q^{61} +5.86030 q^{63} +2.00000 q^{65} +10.0042i q^{67} +15.3137i q^{69} +11.9832 q^{71} -10.4853 q^{73} +17.3952i q^{75} +7.31371i q^{77} -6.12293 q^{79} -5.82843 q^{81} -3.50981i q^{83} -9.65685i q^{85} -11.9832 q^{87} +0.828427 q^{89} +0.896683i q^{91} +19.3137i q^{93} -1.53073 q^{95} +10.8284 q^{97} +18.2919i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 8 q^{25} - 32 q^{33} + 32 q^{41} + 8 q^{49} + 32 q^{57} + 16 q^{65} - 16 q^{73} - 24 q^{81} - 16 q^{89} + 64 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^25 - 32 * q^33 + 32 * q^41 + 8 * q^49 + 32 * q^57 + 16 * q^65 - 16 * q^73 - 24 * q^81 - 16 * q^89 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$-1$	$1$

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
$3$	− 2.61313i	− 1.50869i	−0.656479	−	0.754344i	$-0.727955\pi$
$3$	− 2.61313i	− 1.50869i	0.656479	−	0.754344i	$-0.272045\pi$
$4$	0	0
$5$	3.41421i	1.52688i	0.645877	+	0.763441i	$0.276492\pi$
$5$	3.41421i	1.52688i	−0.645877	+	0.763441i	$0.723508\pi$
$6$	0	0
$7$	−1.53073	−0.578563	−0.289281	−	0.957244i	$-0.593416\pi$
$7$	−1.53073	−0.578563	−0.289281	+	0.957244i	$0.593416\pi$
$8$	0	0
$9$	−3.82843	−1.27614
$10$	0	0
$11$	− 4.77791i	− 1.44059i	−0.693666	−	0.720297i	$-0.744005\pi$
$11$	− 4.77791i	− 1.44059i	0.693666	−	0.720297i	$-0.255995\pi$
$12$	0	0
$13$	− 0.585786i	− 0.162468i	−0.996695	−	0.0812340i	$-0.974114\pi$
$13$	− 0.585786i	− 0.162468i	0.996695	−	0.0812340i	$-0.0258861\pi$
$14$	0	0
$15$	8.92177	2.30359
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	0.448342i	0.102857i	0.998677	+	0.0514283i	$0.0163774\pi$
$19$	0.448342i	0.102857i	−0.998677	+	0.0514283i	$0.983623\pi$
$20$	0	0
$21$	4.00000i	0.872872i
$22$	0	0
$23$	−5.86030	−1.22196	−0.610979	−	0.791647i	$-0.709224\pi$
$23$	−5.86030	−1.22196	−0.610979	+	0.791647i	$0.709224\pi$
$24$	0	0
$25$	−6.65685	−1.33137
$26$	0	0
$27$	2.16478i	0.416613i
$28$	0	0
$29$	− 4.58579i	− 0.851559i	−0.904827	−	0.425780i	$-0.860000\pi$
$29$	− 4.58579i	− 0.851559i	0.904827	−	0.425780i	$-0.140000\pi$
$30$	0	0
$31$	−7.39104	−1.32747	−0.663735	−	0.747968i	$-0.731030\pi$
$31$	−7.39104	−1.32747	−0.663735	+	0.747968i	$0.731030\pi$
$32$	0	0
$33$	−12.4853	−2.17341
$34$	0	0
$35$	− 5.22625i	− 0.883398i
$36$	0	0
$37$	− 5.07107i	− 0.833678i	−0.908980	−	0.416839i	$-0.863138\pi$
$37$	− 5.07107i	− 0.833678i	0.908980	−	0.416839i	$-0.136862\pi$
$38$	0	0
$39$	−1.53073	−0.245114
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	− 2.61313i	− 0.398498i	−0.979949	−	0.199249i	$-0.936150\pi$
$43$	− 2.61313i	− 0.398498i	0.979949	−	0.199249i	$-0.0638503\pi$
$44$	0	0
$45$	− 13.0711i	− 1.94852i
$46$	0	0
$47$	−7.39104	−1.07809	−0.539047	−	0.842276i	$-0.681215\pi$
$47$	−7.39104	−1.07809	−0.539047	+	0.842276i	$0.681215\pi$
$48$	0	0
$49$	−4.65685	−0.665265
$50$	0	0
$51$	7.39104i	1.03495i
$52$	0	0
$53$	7.41421i	1.01842i	0.860642	+	0.509210i	$0.170062\pi$
$53$	7.41421i	1.01842i	−0.860642	+	0.509210i	$0.829938\pi$
$54$	0	0
$55$	16.3128	2.19962
$56$	0	0
$57$	1.17157	0.155179
$58$	0	0
$59$	2.61313i	0.340200i	0.985427	+	0.170100i	$0.0544091\pi$
$59$	2.61313i	0.340200i	−0.985427	+	0.170100i	$0.945591\pi$
$60$	0	0
$61$	− 13.0711i	− 1.67358i	−0.547525	−	0.836789i	$-0.684430\pi$
$61$	− 13.0711i	− 1.67358i	0.547525	−	0.836789i	$-0.315570\pi$
$62$	0	0
$63$	5.86030	0.738329
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	10.0042i	1.22220i	0.791552	+	0.611101i	$0.209273\pi$
$67$	10.0042i	1.22220i	−0.791552	+	0.611101i	$0.790727\pi$
$68$	0	0
$69$	15.3137i	1.84355i
$70$	0	0
$71$	11.9832	1.42215	0.711074	−	0.703117i	$-0.248209\pi$
$71$	11.9832	1.42215	0.711074	+	0.703117i	$0.248209\pi$
$72$	0	0
$73$	−10.4853	−1.22721	−0.613605	−	0.789613i	$-0.710281\pi$
$73$	−10.4853	−1.22721	−0.613605	+	0.789613i	$0.710281\pi$
$74$	0	0
$75$	17.3952i	2.00862i
$76$	0	0
$77$	7.31371i	0.833474i
$78$	0	0
$79$	−6.12293	−0.688884	−0.344442	−	0.938808i	$-0.611932\pi$
$79$	−6.12293	−0.688884	−0.344442	+	0.938808i	$0.611932\pi$
$80$	0	0
$81$	−5.82843	−0.647603
$82$	0	0
$83$	− 3.50981i	− 0.385252i	−0.981272	−	0.192626i	$-0.938300\pi$
$83$	− 3.50981i	− 0.385252i	0.981272	−	0.192626i	$-0.0617003\pi$
$84$	0	0
$85$	− 9.65685i	− 1.04743i
$86$	0	0
$87$	−11.9832	−1.28474
$88$	0	0
$89$	0.828427	0.0878131	0.0439065	−	0.999036i	$-0.486020\pi$
$89$	0.828427	0.0878131	0.0439065	+	0.999036i	$0.486020\pi$
$90$	0	0
$91$	0.896683i	0.0939979i
$92$	0	0
$93$	19.3137i	2.00274i
$94$	0	0
$95$	−1.53073	−0.157050
$96$	0	0
$97$	10.8284	1.09946	0.549730	−	0.835342i	$-0.314731\pi$
$97$	10.8284	1.09946	0.549730	+	0.835342i	$0.314731\pi$
$98$	0	0
$99$	18.2919i	1.83840i
$100$	0	0
$101$	2.92893i	0.291440i	0.989326	+	0.145720i	$0.0465498\pi$
$101$	2.92893i	0.291440i	−0.989326	+	0.145720i	$0.953450\pi$
$102$	0	0
$103$	−2.79884	−0.275777	−0.137889	−	0.990448i	$-0.544032\pi$
$103$	−2.79884	−0.275777	−0.137889	+	0.990448i	$0.544032\pi$
$104$	0	0
$105$	−13.6569	−1.33277
$106$	0	0
$107$	10.0042i	0.967139i	0.875306	+	0.483569i	$0.160660\pi$
$107$	10.0042i	0.967139i	−0.875306	+	0.483569i	$0.839340\pi$
$108$	0	0
$109$	− 1.07107i	− 0.102590i	−0.998684	−	0.0512948i	$-0.983665\pi$
$109$	− 1.07107i	− 0.102590i	0.998684	−	0.0512948i	$-0.0163348\pi$
$110$	0	0
$111$	−13.2513	−1.25776
$112$	0	0
$113$	3.65685	0.344008	0.172004	−	0.985096i	$-0.444976\pi$
$113$	3.65685	0.344008	0.172004	+	0.985096i	$0.444976\pi$
$114$	0	0
$115$	− 20.0083i	− 1.86579i
$116$	0	0
$117$	2.24264i	0.207332i
$118$	0	0
$119$	4.32957	0.396891
$120$	0	0
$121$	−11.8284	−1.07531
$122$	0	0
$123$	− 10.4525i	− 0.942471i
$124$	0	0
$125$	− 5.65685i	− 0.505964i
$126$	0	0
$127$	13.5140	1.19917	0.599586	−	0.800311i	$-0.295332\pi$
$127$	13.5140	1.19917	0.599586	+	0.800311i	$0.295332\pi$
$128$	0	0
$129$	−6.82843	−0.601209
$130$	0	0
$131$	− 9.10748i	− 0.795724i	−0.917445	−	0.397862i	$-0.869752\pi$
$131$	− 9.10748i	− 0.795724i	0.917445	−	0.397862i	$-0.130248\pi$
$132$	0	0
$133$	− 0.686292i	− 0.0595090i
$134$	0	0
$135$	−7.39104	−0.636119
$136$	0	0
$137$	20.9706	1.79164	0.895818	−	0.444421i	$-0.146591\pi$
$137$	20.9706	1.79164	0.895818	+	0.444421i	$0.146591\pi$
$138$	0	0
$139$	− 19.1886i	− 1.62755i	−0.581178	−	0.813776i	$-0.697408\pi$
$139$	− 19.1886i	− 1.62755i	0.581178	−	0.813776i	$-0.302592\pi$
$140$	0	0
$141$	19.3137i	1.62651i
$142$	0	0
$143$	−2.79884	−0.234050
$144$	0	0
$145$	15.6569	1.30023
$146$	0	0
$147$	12.1689i	1.00368i
$148$	0	0
$149$	10.2426i	0.839110i	0.907730	+	0.419555i	$0.137814\pi$
$149$	10.2426i	0.839110i	−0.907730	+	0.419555i	$0.862186\pi$
$150$	0	0
$151$	8.92177	0.726043	0.363022	−	0.931781i	$-0.381745\pi$
$151$	8.92177	0.726043	0.363022	+	0.931781i	$0.381745\pi$
$152$	0	0
$153$	10.8284	0.875426
$154$	0	0
$155$	− 25.2346i	− 2.02689i
$156$	0	0
$157$	6.24264i	0.498217i	0.968476	+	0.249108i	$0.0801376\pi$
$157$	6.24264i	0.498217i	−0.968476	+	0.249108i	$0.919862\pi$
$158$	0	0
$159$	19.3743	1.53648
$160$	0	0
$161$	8.97056	0.706979
$162$	0	0
$163$	− 13.0656i	− 1.02338i	−0.859170	−	0.511690i	$-0.829020\pi$
$163$	− 13.0656i	− 1.02338i	0.859170	−	0.511690i	$-0.170980\pi$
$164$	0	0
$165$	− 42.6274i	− 3.31854i
$166$	0	0
$167$	−1.53073	−0.118452	−0.0592259	−	0.998245i	$-0.518863\pi$
$167$	−1.53073	−0.118452	−0.0592259	+	0.998245i	$0.518863\pi$
$168$	0	0
$169$	12.6569	0.973604
$170$	0	0
$171$	− 1.71644i	− 0.131260i
$172$	0	0
$173$	− 23.2132i	− 1.76487i	−0.470437	−	0.882434i	$-0.655904\pi$
$173$	− 23.2132i	− 1.76487i	0.470437	−	0.882434i	$-0.344096\pi$
$174$	0	0
$175$	10.1899	0.770282
$176$	0	0
$177$	6.82843	0.513256
$178$	0	0
$179$	− 1.71644i	− 0.128293i	−0.997940	−	0.0641465i	$-0.979568\pi$
$179$	− 1.71644i	− 0.128293i	0.997940	−	0.0641465i	$-0.0204325\pi$
$180$	0	0
$181$	− 5.75736i	− 0.427941i	−0.976840	−	0.213971i	$-0.931360\pi$
$181$	− 5.75736i	− 0.427941i	0.976840	−	0.213971i	$-0.0686397\pi$
$182$	0	0
$183$	−34.1563	−2.52491
$184$	0	0
$185$	17.3137	1.27293
$186$	0	0
$187$	13.5140i	0.988239i
$188$	0	0
$189$	− 3.31371i	− 0.241037i
$190$	0	0
$191$	−13.5140	−0.977837	−0.488918	−	0.872330i	$-0.662608\pi$
$191$	−13.5140	−0.977837	−0.488918	+	0.872330i	$0.662608\pi$
$192$	0	0
$193$	−16.4853	−1.18664	−0.593318	−	0.804968i	$-0.702182\pi$
$193$	−16.4853	−1.18664	−0.593318	+	0.804968i	$0.702182\pi$
$194$	0	0
$195$	− 5.22625i	− 0.374260i
$196$	0	0
$197$	14.7279i	1.04932i	0.851312	+	0.524660i	$0.175808\pi$
$197$	14.7279i	1.04932i	−0.851312	+	0.524660i	$0.824192\pi$
$198$	0	0
$199$	−22.4357	−1.59043	−0.795214	−	0.606329i	$-0.792641\pi$
$199$	−22.4357	−1.59043	−0.795214	+	0.606329i	$0.792641\pi$
$200$	0	0
$201$	26.1421	1.84392
$202$	0	0
$203$	7.01962i	0.492681i
$204$	0	0
$205$	13.6569i	0.953836i
$206$	0	0
$207$	22.4357	1.55939
$208$	0	0
$209$	2.14214	0.148175
$210$	0	0
$211$	− 26.9510i	− 1.85538i	−0.373346	−	0.927692i	$-0.621789\pi$
$211$	− 26.9510i	− 1.85538i	0.373346	−	0.927692i	$-0.378211\pi$
$212$	0	0
$213$	− 31.3137i	− 2.14558i
$214$	0	0
$215$	8.92177	0.608460
$216$	0	0
$217$	11.3137	0.768025
$218$	0	0
$219$	27.3994i	1.85148i
$220$	0	0
$221$	1.65685i	0.111452i
$222$	0	0
$223$	−7.39104	−0.494940	−0.247470	−	0.968896i	$-0.579599\pi$
$223$	−7.39104	−0.494940	−0.247470	+	0.968896i	$0.579599\pi$
$224$	0	0
$225$	25.4853	1.69902
$226$	0	0
$227$	− 2.61313i	− 0.173439i	−0.996233	−	0.0867196i	$-0.972362\pi$
$227$	− 2.61313i	− 0.173439i	0.996233	−	0.0867196i	$-0.0276384\pi$
$228$	0	0
$229$	− 15.8995i	− 1.05067i	−0.850896	−	0.525334i	$-0.823940\pi$
$229$	− 15.8995i	− 1.05067i	0.850896	−	0.525334i	$-0.176060\pi$
$230$	0	0
$231$	19.1116	1.25745
$232$	0	0
$233$	−15.1716	−0.993923	−0.496961	−	0.867773i	$-0.665551\pi$
$233$	−15.1716	−0.993923	−0.496961	+	0.867773i	$0.665551\pi$
$234$	0	0
$235$	− 25.2346i	− 1.64612i
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	29.5641	1.91235	0.956173	−	0.292803i	$-0.0945880\pi$
$239$	29.5641	1.91235	0.956173	+	0.292803i	$0.0945880\pi$
$240$	0	0
$241$	−18.8284	−1.21285	−0.606423	−	0.795142i	$-0.707396\pi$
$241$	−18.8284	−1.21285	−0.606423	+	0.795142i	$0.707396\pi$
$242$	0	0
$243$	21.7248i	1.39364i
$244$	0	0
$245$	− 15.8995i	− 1.01578i
$246$	0	0
$247$	0.262632	0.0167109
$248$	0	0
$249$	−9.17157	−0.581225
$250$	0	0
$251$	− 13.9623i	− 0.881293i	−0.897681	−	0.440647i	$-0.854749\pi$
$251$	− 13.9623i	− 0.881293i	0.897681	−	0.440647i	$-0.145251\pi$
$252$	0	0
$253$	28.0000i	1.76034i
$254$	0	0
$255$	−25.2346	−1.58025
$256$	0	0
$257$	18.9706	1.18335	0.591676	−	0.806176i	$-0.298467\pi$
$257$	18.9706	1.18335	0.591676	+	0.806176i	$0.298467\pi$
$258$	0	0
$259$	7.76245i	0.482335i
$260$	0	0
$261$	17.5563i	1.08671i
$262$	0	0
$263$	20.6424	1.27286	0.636432	−	0.771333i	$-0.280410\pi$
$263$	20.6424	1.27286	0.636432	+	0.771333i	$0.280410\pi$
$264$	0	0
$265$	−25.3137	−1.55501
$266$	0	0
$267$	− 2.16478i	− 0.132483i
$268$	0	0
$269$	5.55635i	0.338777i	0.985549	+	0.169388i	$0.0541792\pi$
$269$	5.55635i	0.338777i	−0.985549	+	0.169388i	$0.945821\pi$
$270$	0	0
$271$	−28.2960	−1.71886	−0.859431	−	0.511252i	$-0.829182\pi$
$271$	−28.2960	−1.71886	−0.859431	+	0.511252i	$0.829182\pi$
$272$	0	0
$273$	2.34315	0.141814
$274$	0	0
$275$	31.8059i	1.91797i
$276$	0	0
$277$	− 0.585786i	− 0.0351965i	−0.999845	−	0.0175982i	$-0.994398\pi$
$277$	− 0.585786i	− 0.0351965i	0.999845	−	0.0175982i	$-0.00560199\pi$
$278$	0	0
$279$	28.2960	1.69404
$280$	0	0
$281$	21.7990	1.30042	0.650209	−	0.759755i	$-0.274681\pi$
$281$	21.7990	1.30042	0.650209	+	0.759755i	$0.274681\pi$
$282$	0	0
$283$	9.10748i	0.541383i	0.962666	+	0.270692i	$0.0872524\pi$
$283$	9.10748i	0.541383i	−0.962666	+	0.270692i	$0.912748\pi$
$284$	0	0
$285$	4.00000i	0.236940i
$286$	0	0
$287$	−6.12293	−0.361425
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	− 28.2960i	− 1.65874i
$292$	0	0
$293$	3.41421i	0.199460i	0.995015	+	0.0997302i	$0.0317980\pi$
$293$	3.41421i	0.199460i	−0.995015	+	0.0997302i	$0.968202\pi$
$294$	0	0
$295$	−8.92177	−0.519446
$296$	0	0
$297$	10.3431	0.600170
$298$	0	0
$299$	3.43289i	0.198529i
$300$	0	0
$301$	4.00000i	0.230556i
$302$	0	0
$303$	7.65367	0.439692
$304$	0	0
$305$	44.6274	2.55536
$306$	0	0
$307$	25.6829i	1.46580i	0.680336	+	0.732901i	$0.261834\pi$
$307$	25.6829i	1.46580i	−0.680336	+	0.732901i	$0.738166\pi$
$308$	0	0
$309$	7.31371i	0.416062i
$310$	0	0
$311$	−7.12840	−0.404215	−0.202107	−	0.979363i	$-0.564779\pi$
$311$	−7.12840	−0.404215	−0.202107	+	0.979363i	$0.564779\pi$
$312$	0	0
$313$	8.68629	0.490978	0.245489	−	0.969399i	$-0.421051\pi$
$313$	8.68629	0.490978	0.245489	+	0.969399i	$0.421051\pi$
$314$	0	0
$315$	20.0083i	1.12734i
$316$	0	0
$317$	− 12.5858i	− 0.706888i	−0.935456	−	0.353444i	$-0.885010\pi$
$317$	− 12.5858i	− 0.706888i	0.935456	−	0.353444i	$-0.114990\pi$
$318$	0	0
$319$	−21.9105	−1.22675
$320$	0	0
$321$	26.1421	1.45911
$322$	0	0
$323$	− 1.26810i	− 0.0705590i
$324$	0	0
$325$	3.89949i	0.216305i
$326$	0	0
$327$	−2.79884	−0.154776
$328$	0	0
$329$	11.3137	0.623745
$330$	0	0
$331$	5.67459i	0.311904i	0.987765	+	0.155952i	$0.0498445\pi$
$331$	5.67459i	0.311904i	−0.987765	+	0.155952i	$0.950156\pi$
$332$	0	0
$333$	19.4142i	1.06389i
$334$	0	0
$335$	−34.1563	−1.86616
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	− 9.55582i	− 0.519001i
$340$	0	0
$341$	35.3137i	1.91234i
$342$	0	0
$343$	17.8435	0.963461
$344$	0	0
$345$	−52.2843	−2.81489
$346$	0	0
$347$	21.7248i	1.16625i	0.812384	+	0.583123i	$0.198170\pi$
$347$	21.7248i	1.16625i	−0.812384	+	0.583123i	$0.801830\pi$
$348$	0	0
$349$	22.7279i	1.21660i	0.793708	+	0.608299i	$0.208148\pi$
$349$	22.7279i	1.21660i	−0.793708	+	0.608299i	$0.791852\pi$
$350$	0	0
$351$	1.26810	0.0676862
$352$	0	0
$353$	−10.9706	−0.583904	−0.291952	−	0.956433i	$-0.594305\pi$
$353$	−10.9706	−0.583904	−0.291952	+	0.956433i	$0.594305\pi$
$354$	0	0
$355$	40.9133i	2.17145i
$356$	0	0
$357$	− 11.3137i	− 0.598785i
$358$	0	0
$359$	−8.92177	−0.470873	−0.235437	−	0.971890i	$-0.575652\pi$
$359$	−8.92177	−0.470873	−0.235437	+	0.971890i	$0.575652\pi$
$360$	0	0
$361$	18.7990	0.989421
$362$	0	0
$363$	30.9092i	1.62231i
$364$	0	0
$365$	− 35.7990i	− 1.87380i
$366$	0	0
$367$	22.1731	1.15743	0.578713	−	0.815531i	$-0.303555\pi$
$367$	22.1731	1.15743	0.578713	+	0.815531i	$0.303555\pi$
$368$	0	0
$369$	−15.3137	−0.797200
$370$	0	0
$371$	− 11.3492i	− 0.589220i
$372$	0	0
$373$	18.7279i	0.969695i	0.874599	+	0.484848i	$0.161125\pi$
$373$	18.7279i	0.969695i	−0.874599	+	0.484848i	$0.838875\pi$
$374$	0	0
$375$	−14.7821	−0.763343
$376$	0	0
$377$	−2.68629	−0.138351
$378$	0	0
$379$	4.40649i	0.226346i	0.993575	+	0.113173i	$0.0361015\pi$
$379$	4.40649i	0.226346i	−0.993575	+	0.113173i	$0.963899\pi$
$380$	0	0
$381$	− 35.3137i	− 1.80918i
$382$	0	0
$383$	8.65914	0.442461	0.221231	−	0.975222i	$-0.428993\pi$
$383$	8.65914	0.442461	0.221231	+	0.975222i	$0.428993\pi$
$384$	0	0
$385$	−24.9706	−1.27262
$386$	0	0
$387$	10.0042i	0.508540i
$388$	0	0
$389$	− 16.3848i	− 0.830741i	−0.909652	−	0.415371i	$-0.863652\pi$
$389$	− 16.3848i	− 0.830741i	0.909652	−	0.415371i	$-0.136348\pi$
$390$	0	0
$391$	16.5754	0.838256
$392$	0	0
$393$	−23.7990	−1.20050
$394$	0	0
$395$	− 20.9050i	− 1.05185i
$396$	0	0
$397$	− 9.07107i	− 0.455264i	−0.973747	−	0.227632i	$-0.926902\pi$
$397$	− 9.07107i	− 0.455264i	0.973747	−	0.227632i	$-0.0730983\pi$
$398$	0	0
$399$	−1.79337	−0.0897806
$400$	0	0
$401$	24.4853	1.22274	0.611368	−	0.791346i	$-0.290619\pi$
$401$	24.4853	1.22274	0.611368	+	0.791346i	$0.290619\pi$
$402$	0	0
$403$	4.32957i	0.215671i
$404$	0	0
$405$	− 19.8995i	− 0.988814i
$406$	0	0
$407$	−24.2291	−1.20099
$408$	0	0
$409$	8.68629	0.429509	0.214755	−	0.976668i	$-0.431105\pi$
$409$	8.68629	0.429509	0.214755	+	0.976668i	$0.431105\pi$
$410$	0	0
$411$	− 54.7987i	− 2.70302i
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	11.9832	0.588234
$416$	0	0
$417$	−50.1421	−2.45547
$418$	0	0
$419$	14.3337i	0.700249i	0.936703	+	0.350124i	$0.113861\pi$
$419$	14.3337i	0.700249i	−0.936703	+	0.350124i	$0.886139\pi$
$420$	0	0
$421$	11.4142i	0.556295i	0.960538	+	0.278147i	$0.0897204\pi$
$421$	11.4142i	0.556295i	−0.960538	+	0.278147i	$0.910280\pi$
$422$	0	0
$423$	28.2960	1.37580
$424$	0	0
$425$	18.8284	0.913313
$426$	0	0
$427$	20.0083i	0.968271i
$428$	0	0
$429$	7.31371i	0.353109i
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−25.4558	−1.22333	−0.611665	−	0.791117i	$-0.709500\pi$
$433$	−25.4558	−1.22333	−0.611665	+	0.791117i	$0.709500\pi$
$434$	0	0
$435$	− 40.9133i	− 1.96164i
$436$	0	0
$437$	− 2.62742i	− 0.125686i
$438$	0	0
$439$	18.8490	0.899614	0.449807	−	0.893126i	$-0.351493\pi$
$439$	18.8490	0.899614	0.449807	+	0.893126i	$0.351493\pi$
$440$	0	0
$441$	17.8284	0.848973
$442$	0	0
$443$	− 10.0042i	− 0.475312i	−0.971349	−	0.237656i	$-0.923621\pi$
$443$	− 10.0042i	− 0.475312i	0.971349	−	0.237656i	$-0.0763791\pi$
$444$	0	0
$445$	2.82843i	0.134080i
$446$	0	0
$447$	26.7653	1.26596
$448$	0	0
$449$	22.1421	1.04495	0.522476	−	0.852654i	$-0.325008\pi$
$449$	22.1421	1.04495	0.522476	+	0.852654i	$0.325008\pi$
$450$	0	0
$451$	− 19.1116i	− 0.899932i
$452$	0	0
$453$	− 23.3137i	− 1.09537i
$454$	0	0
$455$	−3.06147	−0.143524
$456$	0	0
$457$	−18.6274	−0.871354	−0.435677	−	0.900103i	$-0.643491\pi$
$457$	−18.6274	−0.871354	−0.435677	+	0.900103i	$0.643491\pi$
$458$	0	0
$459$	− 6.12293i	− 0.285794i
$460$	0	0
$461$	− 29.7574i	− 1.38594i	−0.720967	−	0.692969i	$-0.756302\pi$
$461$	− 29.7574i	− 1.38594i	0.720967	−	0.692969i	$-0.243698\pi$
$462$	0	0
$463$	−2.53620	−0.117867	−0.0589337	−	0.998262i	$-0.518770\pi$
$463$	−2.53620	−0.117867	−0.0589337	+	0.998262i	$0.518770\pi$
$464$	0	0
$465$	−65.9411	−3.05795
$466$	0	0
$467$	− 17.9205i	− 0.829260i	−0.909990	−	0.414630i	$-0.863911\pi$
$467$	− 17.9205i	− 0.829260i	0.909990	−	0.414630i	$-0.136089\pi$
$468$	0	0
$469$	− 15.3137i	− 0.707121i
$470$	0	0
$471$	16.3128	0.751654
$472$	0	0
$473$	−12.4853	−0.574074
$474$	0	0
$475$	− 2.98454i	− 0.136940i
$476$	0	0
$477$	− 28.3848i	− 1.29965i
$478$	0	0
$479$	40.5419	1.85241	0.926204	−	0.377024i	$-0.123052\pi$
$479$	40.5419	1.85241	0.926204	+	0.377024i	$0.123052\pi$
$480$	0	0
$481$	−2.97056	−0.135446
$482$	0	0
$483$	− 23.4412i	− 1.06661i
$484$	0	0
$485$	36.9706i	1.67875i
$486$	0	0
$487$	−37.2178	−1.68650	−0.843250	−	0.537521i	$-0.819361\pi$
$487$	−37.2178	−1.68650	−0.843250	+	0.537521i	$0.819361\pi$
$488$	0	0
$489$	−34.1421	−1.54396
$490$	0	0
$491$	− 17.3952i	− 0.785034i	−0.919745	−	0.392517i	$-0.871604\pi$
$491$	− 17.3952i	− 0.785034i	0.919745	−	0.392517i	$-0.128396\pi$
$492$	0	0
$493$	12.9706i	0.584165i
$494$	0	0
$495$	−62.4524	−2.80703
$496$	0	0
$497$	−18.3431	−0.822803
$498$	0	0
$499$	− 16.4985i	− 0.738575i	−0.929315	−	0.369287i	$-0.879602\pi$
$499$	− 16.4985i	− 0.738575i	0.929315	−	0.369287i	$-0.120398\pi$
$500$	0	0
$501$	4.00000i	0.178707i
$502$	0	0
$503$	−18.1062	−0.807314	−0.403657	−	0.914910i	$-0.632261\pi$
$503$	−18.1062	−0.807314	−0.403657	+	0.914910i	$0.632261\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	− 33.0740i	− 1.46887i
$508$	0	0
$509$	− 17.7574i	− 0.787081i	−0.919307	−	0.393541i	$-0.871250\pi$
$509$	− 17.7574i	− 0.787081i	0.919307	−	0.393541i	$-0.128750\pi$
$510$	0	0
$511$	16.0502	0.710018
$512$	0	0
$513$	−0.970563	−0.0428514
$514$	0	0
$515$	− 9.55582i	− 0.421080i
$516$	0	0
$517$	35.3137i	1.55310i
$518$	0	0
$519$	−60.6590	−2.66264
$520$	0	0
$521$	−17.6569	−0.773561	−0.386780	−	0.922172i	$-0.626413\pi$
$521$	−17.6569	−0.773561	−0.386780	+	0.922172i	$0.626413\pi$
$522$	0	0
$523$	39.5683i	1.73020i	0.501598	+	0.865101i	$0.332746\pi$
$523$	39.5683i	1.73020i	−0.501598	+	0.865101i	$0.667254\pi$
$524$	0	0
$525$	− 26.6274i	− 1.16212i
$526$	0	0
$527$	20.9050	0.910636
$528$	0	0
$529$	11.3431	0.493180
$530$	0	0
$531$	− 10.0042i	− 0.434144i
$532$	0	0
$533$	− 2.34315i	− 0.101493i
$534$	0	0
$535$	−34.1563	−1.47671
$536$	0	0
$537$	−4.48528	−0.193554
$538$	0	0
$539$	22.2500i	0.958377i
$540$	0	0
$541$	− 1.75736i	− 0.0755548i	−0.999286	−	0.0377774i	$-0.987972\pi$
$541$	− 1.75736i	− 0.0755548i	0.999286	−	0.0377774i	$-0.0120278\pi$
$542$	0	0
$543$	−15.0447	−0.645630
$544$	0	0
$545$	3.65685	0.156642
$546$	0	0
$547$	1.34502i	0.0575091i	0.999587	+	0.0287545i	$0.00915412\pi$
$547$	1.34502i	0.0575091i	−0.999587	+	0.0287545i	$0.990846\pi$
$548$	0	0
$549$	50.0416i	2.13572i
$550$	0	0
$551$	2.05600	0.0875885
$552$	0	0
$553$	9.37258	0.398563
$554$	0	0
$555$	− 45.2429i	− 1.92045i
$556$	0	0
$557$	− 8.58579i	− 0.363791i	−0.983318	−	0.181896i	$-0.941777\pi$
$557$	− 8.58579i	− 0.363791i	0.983318	−	0.181896i	$-0.0582233\pi$
$558$	0	0
$559$	−1.53073	−0.0647431
$560$	0	0
$561$	35.3137	1.49095
$562$	0	0
$563$	21.3533i	0.899936i	0.893045	+	0.449968i	$0.148565\pi$
$563$	21.3533i	0.899936i	−0.893045	+	0.449968i	$0.851435\pi$
$564$	0	0
$565$	12.4853i	0.525260i
$566$	0	0
$567$	8.92177	0.374679
$568$	0	0
$569$	14.3431	0.601296	0.300648	−	0.953735i	$-0.402797\pi$
$569$	14.3431	0.601296	0.300648	+	0.953735i	$0.402797\pi$
$570$	0	0
$571$	− 22.2500i	− 0.931135i	−0.885012	−	0.465567i	$-0.845850\pi$
$571$	− 22.2500i	− 0.931135i	0.885012	−	0.465567i	$-0.154150\pi$
$572$	0	0
$573$	35.3137i	1.47525i
$574$	0	0
$575$	39.0112	1.62688
$576$	0	0
$577$	−5.31371	−0.221213	−0.110606	−	0.993864i	$-0.535279\pi$
$577$	−5.31371	−0.221213	−0.110606	+	0.993864i	$0.535279\pi$
$578$	0	0
$579$	43.0781i	1.79027i
$580$	0	0
$581$	5.37258i	0.222892i
$582$	0	0
$583$	35.4244	1.46713
$584$	0	0
$585$	−7.65685	−0.316572
$586$	0	0
$587$	29.1158i	1.20174i	0.799348	+	0.600869i	$0.205179\pi$
$587$	29.1158i	1.20174i	−0.799348	+	0.600869i	$0.794821\pi$
$588$	0	0
$589$	− 3.31371i	− 0.136539i
$590$	0	0
$591$	38.4859	1.58310
$592$	0	0
$593$	−13.3137	−0.546728	−0.273364	−	0.961911i	$-0.588136\pi$
$593$	−13.3137	−0.546728	−0.273364	+	0.961911i	$0.588136\pi$
$594$	0	0
$595$	14.7821i	0.606006i
$596$	0	0
$597$	58.6274i	2.39946i
$598$	0	0
$599$	−15.7875	−0.645061	−0.322531	−	0.946559i	$-0.604534\pi$
$599$	−15.7875	−0.645061	−0.322531	+	0.946559i	$0.604534\pi$
$600$	0	0
$601$	−15.1716	−0.618861	−0.309431	−	0.950922i	$-0.600138\pi$
$601$	−15.1716	−0.618861	−0.309431	+	0.950922i	$0.600138\pi$
$602$	0	0
$603$	− 38.3002i	− 1.55970i
$604$	0	0
$605$	− 40.3848i	− 1.64187i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	18.3431	0.743302
$610$	0	0
$611$	4.32957i	0.175156i
$612$	0	0
$613$	− 36.5858i	− 1.47769i	−0.673878	−	0.738843i	$-0.735373\pi$
$613$	− 36.5858i	− 1.47769i	0.673878	−	0.738843i	$-0.264627\pi$
$614$	0	0
$615$	35.6871	1.43904
$616$	0	0
$617$	−21.5147	−0.866150	−0.433075	−	0.901358i	$-0.642571\pi$
$617$	−21.5147	−0.866150	−0.433075	+	0.901358i	$0.642571\pi$
$618$	0	0
$619$	− 4.77791i	− 0.192040i	−0.995379	−	0.0960202i	$-0.969389\pi$
$619$	− 4.77791i	− 0.192040i	0.995379	−	0.0960202i	$-0.0306113\pi$
$620$	0	0
$621$	− 12.6863i	− 0.509083i
$622$	0	0
$623$	−1.26810	−0.0508054
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	− 5.59767i	− 0.223549i
$628$	0	0
$629$	14.3431i	0.571899i
$630$	0	0
$631$	2.79884	0.111420	0.0557099	−	0.998447i	$-0.482258\pi$
$631$	2.79884	0.111420	0.0557099	+	0.998447i	$0.482258\pi$
$632$	0	0
$633$	−70.4264	−2.79920
$634$	0	0
$635$	46.1396i	1.83099i
$636$	0	0
$637$	2.72792i	0.108084i
$638$	0	0
$639$	−45.8770	−1.81486
$640$	0	0
$641$	−9.85786	−0.389362	−0.194681	−	0.980867i	$-0.562367\pi$
$641$	−9.85786	−0.389362	−0.194681	+	0.980867i	$0.562367\pi$
$642$	0	0
$643$	1.34502i	0.0530426i	0.999648	+	0.0265213i	$0.00844298\pi$
$643$	1.34502i	0.0530426i	−0.999648	+	0.0265213i	$0.991557\pi$
$644$	0	0
$645$	− 23.3137i	− 0.917976i
$646$	0	0
$647$	−0.262632	−0.0103251	−0.00516257	−	0.999987i	$-0.501643\pi$
$647$	−0.262632	−0.0103251	−0.00516257	+	0.999987i	$0.501643\pi$
$648$	0	0
$649$	12.4853	0.490090
$650$	0	0
$651$	− 29.5641i	− 1.15871i
$652$	0	0
$653$	37.5563i	1.46969i	0.678233	+	0.734847i	$0.262746\pi$
$653$	37.5563i	1.46969i	−0.678233	+	0.734847i	$0.737254\pi$
$654$	0	0
$655$	31.0949	1.21498
$656$	0	0
$657$	40.1421	1.56609
$658$	0	0
$659$	− 33.0740i	− 1.28838i	−0.764866	−	0.644189i	$-0.777195\pi$
$659$	− 33.0740i	− 1.28838i	0.764866	−	0.644189i	$-0.222805\pi$
$660$	0	0
$661$	14.9289i	0.580668i	0.956925	+	0.290334i	$0.0937664\pi$
$661$	14.9289i	0.580668i	−0.956925	+	0.290334i	$0.906234\pi$
$662$	0	0
$663$	4.32957	0.168147
$664$	0	0
$665$	2.34315	0.0908633
$666$	0	0
$667$	26.8741i	1.04057i
$668$	0	0
$669$	19.3137i	0.746711i
$670$	0	0
$671$	−62.4524	−2.41095
$672$	0	0
$673$	−21.1716	−0.816104	−0.408052	−	0.912959i	$-0.633792\pi$
$673$	−21.1716	−0.816104	−0.408052	+	0.912959i	$0.633792\pi$
$674$	0	0
$675$	− 14.4107i	− 0.554666i
$676$	0	0
$677$	17.5563i	0.674745i	0.941371	+	0.337373i	$0.109538\pi$
$677$	17.5563i	0.674745i	−0.941371	+	0.337373i	$0.890462\pi$
$678$	0	0
$679$	−16.5754	−0.636107
$680$	0	0
$681$	−6.82843	−0.261666
$682$	0	0
$683$	20.0852i	0.768541i	0.923221	+	0.384270i	$0.125547\pi$
$683$	20.0852i	0.768541i	−0.923221	+	0.384270i	$0.874453\pi$
$684$	0	0
$685$	71.5980i	2.73562i
$686$	0	0
$687$	−41.5474	−1.58513
$688$	0	0
$689$	4.34315	0.165461
$690$	0	0
$691$	19.9314i	0.758226i	0.925350	+	0.379113i	$0.123771\pi$
$691$	19.9314i	0.758226i	−0.925350	+	0.379113i	$0.876229\pi$
$692$	0	0
$693$	− 28.0000i	− 1.06363i
$694$	0	0
$695$	65.5139	2.48508
$696$	0	0
$697$	−11.3137	−0.428537
$698$	0	0
$699$	39.6452i	1.49952i
$700$	0	0
$701$	28.8701i	1.09041i	0.838304	+	0.545204i	$0.183548\pi$
$701$	28.8701i	1.09041i	−0.838304	+	0.545204i	$0.816452\pi$
$702$	0	0
$703$	2.27357	0.0857493
$704$	0	0
$705$	−65.9411	−2.48349
$706$	0	0
$707$	− 4.48342i	− 0.168616i
$708$	0	0
$709$	− 35.6985i	− 1.34068i	−0.742052	−	0.670342i	$-0.766147\pi$
$709$	− 35.6985i	− 1.34068i	0.742052	−	0.670342i	$-0.233853\pi$
$710$	0	0
$711$	23.4412	0.879114
$712$	0	0
$713$	43.3137	1.62211
$714$	0	0
$715$	− 9.55582i	− 0.357367i
$716$	0	0
$717$	− 77.2548i	− 2.88513i
$718$	0	0
$719$	−12.2459	−0.456694	−0.228347	−	0.973580i	$-0.573332\pi$
$719$	−12.2459	−0.456694	−0.228347	+	0.973580i	$0.573332\pi$
$720$	0	0
$721$	4.28427	0.159555
$722$	0	0
$723$	49.2011i	1.82981i
$724$	0	0
$725$	30.5269i	1.13374i
$726$	0	0
$727$	−42.8155	−1.58794	−0.793969	−	0.607958i	$-0.791989\pi$
$727$	−42.8155	−1.58794	−0.793969	+	0.607958i	$0.791989\pi$
$728$	0	0
$729$	39.2843	1.45497
$730$	0	0
$731$	7.39104i	0.273367i
$732$	0	0
$733$	− 46.5269i	− 1.71851i	−0.511547	−	0.859255i	$-0.670927\pi$
$733$	− 46.5269i	− 1.71851i	0.511547	−	0.859255i	$-0.329073\pi$
$734$	0	0
$735$	−41.5474	−1.53250
$736$	0	0
$737$	47.7990	1.76070
$738$	0	0
$739$	− 31.4344i	− 1.15633i	−0.815918	−	0.578167i	$-0.803768\pi$
$739$	− 31.4344i	− 1.15633i	0.815918	−	0.578167i	$-0.196232\pi$
$740$	0	0
$741$	− 0.686292i	− 0.0252115i
$742$	0	0
$743$	−28.5587	−1.04772	−0.523858	−	0.851806i	$-0.675508\pi$
$743$	−28.5587	−1.04772	−0.523858	+	0.851806i	$0.675508\pi$
$744$	0	0
$745$	−34.9706	−1.28122
$746$	0	0
$747$	13.4370i	0.491636i
$748$	0	0
$749$	− 15.3137i	− 0.559551i
$750$	0	0
$751$	6.12293	0.223429	0.111715	−	0.993740i	$-0.464366\pi$
$751$	6.12293	0.223429	0.111715	+	0.993740i	$0.464366\pi$
$752$	0	0
$753$	−36.4853	−1.32960
$754$	0	0
$755$	30.4608i	1.10858i
$756$	0	0
$757$	− 33.0711i	− 1.20199i	−0.799253	−	0.600994i	$-0.794771\pi$
$757$	− 33.0711i	− 1.20199i	0.799253	−	0.600994i	$-0.205229\pi$
$758$	0	0
$759$	73.1675	2.65581
$760$	0	0
$761$	10.6274	0.385244	0.192622	−	0.981273i	$-0.438301\pi$
$761$	10.6274	0.385244	0.192622	+	0.981273i	$0.438301\pi$
$762$	0	0
$763$	1.63952i	0.0593546i
$764$	0	0
$765$	36.9706i	1.33667i
$766$	0	0
$767$	1.53073	0.0552716
$768$	0	0
$769$	10.8284	0.390483	0.195242	−	0.980755i	$-0.437451\pi$
$769$	10.8284	0.390483	0.195242	+	0.980755i	$0.437451\pi$
$770$	0	0
$771$	− 49.5725i	− 1.78531i
$772$	0	0
$773$	− 19.6985i	− 0.708505i	−0.935150	−	0.354253i	$-0.884735\pi$
$773$	− 19.6985i	− 0.708505i	0.935150	−	0.354253i	$-0.115265\pi$
$774$	0	0
$775$	49.2011	1.76735
$776$	0	0
$777$	20.2843	0.727694
$778$	0	0
$779$	1.79337i	0.0642540i
$780$	0	0
$781$	− 57.2548i	− 2.04874i
$782$	0	0
$783$	9.92724	0.354771
$784$	0	0
$785$	−21.3137	−0.760719
$786$	0	0
$787$	48.7527i	1.73785i	0.494947	+	0.868923i	$0.335187\pi$
$787$	48.7527i	1.73785i	−0.494947	+	0.868923i	$0.664813\pi$
$788$	0	0
$789$	− 53.9411i	− 1.92035i
$790$	0	0
$791$	−5.59767	−0.199030
$792$	0	0
$793$	−7.65685	−0.271903
$794$	0	0
$795$	66.1479i	2.34602i
$796$	0	0
$797$	− 19.6985i	− 0.697756i	−0.937168	−	0.348878i	$-0.886563\pi$
$797$	− 19.6985i	− 0.697756i	0.937168	−	0.348878i	$-0.113437\pi$
$798$	0	0
$799$	20.9050	0.739566
$800$	0	0
$801$	−3.17157	−0.112062
$802$	0	0
$803$	50.0977i	1.76791i
$804$	0	0
$805$	30.6274i	1.07947i
$806$	0	0
$807$	14.5194	0.511108
$808$	0	0
$809$	47.3137	1.66346	0.831731	−	0.555179i	$-0.187350\pi$
$809$	47.3137	1.66346	0.831731	+	0.555179i	$0.187350\pi$
$810$	0	0
$811$	20.4567i	0.718331i	0.933274	+	0.359165i	$0.116939\pi$
$811$	20.4567i	0.718331i	−0.933274	+	0.359165i	$0.883061\pi$
$812$	0	0
$813$	73.9411i	2.59323i
$814$	0	0
$815$	44.6088	1.56258
$816$	0	0
$817$	1.17157	0.0409881
$818$	0	0
$819$	− 3.43289i	− 0.119955i
$820$	0	0
$821$	− 26.5269i	− 0.925796i	−0.886412	−	0.462898i	$-0.846810\pi$
$821$	− 26.5269i	− 0.925796i	0.886412	−	0.462898i	$-0.153190\pi$
$822$	0	0
$823$	−42.8155	−1.49245	−0.746227	−	0.665692i	$-0.768137\pi$
$823$	−42.8155	−1.49245	−0.746227	+	0.665692i	$0.768137\pi$
$824$	0	0
$825$	83.1127	2.89361
$826$	0	0
$827$	− 29.1158i	− 1.01246i	−0.862400	−	0.506228i	$-0.831039\pi$
$827$	− 29.1158i	− 1.01246i	0.862400	−	0.506228i	$-0.168961\pi$
$828$	0	0
$829$	− 24.3848i	− 0.846918i	−0.905915	−	0.423459i	$-0.860816\pi$
$829$	− 24.3848i	− 0.846918i	0.905915	−	0.423459i	$-0.139184\pi$
$830$	0	0
$831$	−1.53073	−0.0531006
$832$	0	0
$833$	13.1716	0.456368
$834$	0	0
$835$	− 5.22625i	− 0.180862i
$836$	0	0
$837$	− 16.0000i	− 0.553041i
$838$	0	0
$839$	−39.7540	−1.37246	−0.686231	−	0.727384i	$-0.740736\pi$
$839$	−39.7540	−1.37246	−0.686231	+	0.727384i	$0.740736\pi$
$840$	0	0
$841$	7.97056	0.274847
$842$	0	0
$843$	− 56.9635i	− 1.96193i
$844$	0	0
$845$	43.2132i	1.48658i
$846$	0	0
$847$	18.1062	0.622135
$848$	0	0
$849$	23.7990	0.816779
$850$	0	0
$851$	29.7180i	1.01872i
$852$	0	0
$853$	− 51.8995i	− 1.77700i	−0.458872	−	0.888502i	$-0.651746\pi$
$853$	− 51.8995i	− 1.77700i	0.458872	−	0.888502i	$-0.348254\pi$
$854$	0	0
$855$	5.86030	0.200418
$856$	0	0
$857$	32.2843	1.10281	0.551405	−	0.834238i	$-0.314092\pi$
$857$	32.2843	1.10281	0.551405	+	0.834238i	$0.314092\pi$
$858$	0	0
$859$	− 28.7444i	− 0.980746i	−0.871513	−	0.490373i	$-0.836861\pi$
$859$	− 28.7444i	− 0.980746i	0.871513	−	0.490373i	$-0.163139\pi$
$860$	0	0
$861$	16.0000i	0.545279i
$862$	0	0
$863$	−43.0781	−1.46640	−0.733198	−	0.680015i	$-0.761973\pi$
$863$	−43.0781	−1.46640	−0.733198	+	0.680015i	$0.761973\pi$
$864$	0	0
$865$	79.2548	2.69475
$866$	0	0
$867$	23.5181i	0.798718i
$868$	0	0
$869$	29.2548i	0.992402i
$870$	0	0
$871$	5.86030	0.198569
$872$	0	0
$873$	−41.4558	−1.40307
$874$	0	0
$875$	8.65914i	0.292732i
$876$	0	0
$877$	− 18.4437i	− 0.622798i	−0.950279	−	0.311399i	$-0.899202\pi$
$877$	− 18.4437i	− 0.622798i	0.950279	−	0.311399i	$-0.100798\pi$
$878$	0	0
$879$	8.92177	0.300924
$880$	0	0
$881$	27.9411	0.941360	0.470680	−	0.882304i	$-0.344009\pi$
$881$	27.9411	0.941360	0.470680	+	0.882304i	$0.344009\pi$
$882$	0	0
$883$	2.98454i	0.100438i	0.998738	+	0.0502190i	$0.0159919\pi$
$883$	2.98454i	0.100438i	−0.998738	+	0.0502190i	$0.984008\pi$
$884$	0	0
$885$	23.3137i	0.783682i
$886$	0	0
$887$	10.1899	0.342142	0.171071	−	0.985259i	$-0.445277\pi$
$887$	10.1899	0.342142	0.171071	+	0.985259i	$0.445277\pi$
$888$	0	0
$889$	−20.6863	−0.693796
$890$	0	0
$891$	27.8477i	0.932933i
$892$	0	0
$893$	− 3.31371i	− 0.110889i
$894$	0	0
$895$	5.86030	0.195888
$896$	0	0
$897$	8.97056	0.299518
$898$	0	0
$899$	33.8937i	1.13042i
$900$	0	0
$901$	− 20.9706i	− 0.698631i
$902$	0	0
$903$	10.4525	0.347838
$904$	0	0
$905$	19.6569	0.653416
$906$	0	0
$907$	− 9.47890i	− 0.314742i	−0.987540	−	0.157371i	$-0.949698\pi$
$907$	− 9.47890i	− 0.314742i	0.987540	−	0.157371i	$-0.0503018\pi$
$908$	0	0
$909$	− 11.2132i	− 0.371918i
$910$	0	0
$911$	−16.0502	−0.531766	−0.265883	−	0.964005i	$-0.585663\pi$
$911$	−16.0502	−0.531766	−0.265883	+	0.964005i	$0.585663\pi$
$912$	0	0
$913$	−16.7696	−0.554991
$914$	0	0
$915$	− 116.617i	− 3.85524i
$916$	0	0
$917$	13.9411i	0.460377i
$918$	0	0
$919$	−35.4244	−1.16854	−0.584272	−	0.811558i	$-0.698620\pi$
$919$	−35.4244	−1.16854	−0.584272	+	0.811558i	$0.698620\pi$
$920$	0	0
$921$	67.1127	2.21144
$922$	0	0
$923$	− 7.01962i	− 0.231054i
$924$	0	0
$925$	33.7574i	1.10994i
$926$	0	0
$927$	10.7151	0.351931
$928$	0	0
$929$	−37.1716	−1.21956	−0.609780	−	0.792571i	$-0.708742\pi$
$929$	−37.1716	−1.21956	−0.609780	+	0.792571i	$0.708742\pi$
$930$	0	0
$931$	− 2.08786i	− 0.0684269i
$932$	0	0
$933$	18.6274i	0.609834i
$934$	0	0
$935$	−46.1396	−1.50893
$936$	0	0
$937$	7.45584	0.243572	0.121786	−	0.992556i	$-0.461138\pi$
$937$	7.45584	0.243572	0.121786	+	0.992556i	$0.461138\pi$
$938$	0	0
$939$	− 22.6984i	− 0.740733i
$940$	0	0
$941$	− 17.0711i	− 0.556501i	−0.960509	−	0.278250i	$-0.910245\pi$
$941$	− 17.0711i	− 0.556501i	0.960509	−	0.278250i	$-0.0897545\pi$
$942$	0	0
$943$	−23.4412	−0.763351
$944$	0	0
$945$	11.3137	0.368035
$946$	0	0
$947$	− 37.0321i	− 1.20338i	−0.798729	−	0.601691i	$-0.794494\pi$
$947$	− 37.0321i	− 1.20338i	0.798729	−	0.601691i	$-0.205506\pi$
$948$	0	0
$949$	6.14214i	0.199382i
$950$	0	0
$951$	−32.8882	−1.06647
$952$	0	0
$953$	−44.9706	−1.45674	−0.728370	−	0.685184i	$-0.759722\pi$
$953$	−44.9706	−1.45674	−0.728370	+	0.685184i	$0.759722\pi$
$954$	0	0
$955$	− 46.1396i	− 1.49304i
$956$	0	0
$957$	57.2548i	1.85079i
$958$	0	0
$959$	−32.1003	−1.03657
$960$	0	0
$961$	23.6274	0.762175
$962$	0	0
$963$	− 38.3002i	− 1.23421i
$964$	0	0
$965$	− 56.2843i	− 1.81185i
$966$	0	0
$967$	4.59220	0.147675	0.0738376	−	0.997270i	$-0.476475\pi$
$967$	4.59220	0.147675	0.0738376	+	0.997270i	$0.476475\pi$
$968$	0	0
$969$	−3.31371	−0.106452
$970$	0	0
$971$	8.58221i	0.275416i	0.990473	+	0.137708i	$0.0439736\pi$
$971$	8.58221i	0.275416i	−0.990473	+	0.137708i	$0.956026\pi$
$972$	0	0
$973$	29.3726i	0.941642i
$974$	0	0
$975$	10.1899	0.326337
$976$	0	0
$977$	15.1127	0.483498	0.241749	−	0.970339i	$-0.422279\pi$
$977$	15.1127	0.483498	0.241749	+	0.970339i	$0.422279\pi$
$978$	0	0
$979$	− 3.95815i	− 0.126503i
$980$	0	0
$981$	4.10051i	0.130919i
$982$	0	0
$983$	−2.05600	−0.0655762	−0.0327881	−	0.999462i	$-0.510439\pi$
$983$	−2.05600	−0.0655762	−0.0327881	+	0.999462i	$0.510439\pi$
$984$	0	0
$985$	−50.2843	−1.60219
$986$	0	0
$987$	− 29.5641i	− 0.941037i
$988$	0	0
$989$	15.3137i	0.486948i
$990$	0	0
$991$	23.4412	0.744635	0.372317	−	0.928106i	$-0.378563\pi$
$991$	23.4412	0.744635	0.372317	+	0.928106i	$0.378563\pi$
$992$	0	0
$993$	14.8284	0.470566
$994$	0	0
$995$	− 76.6004i	− 2.42840i
$996$	0	0
$997$	− 8.38478i	− 0.265549i	−0.991146	−	0.132774i	$-0.957611\pi$
$997$	− 8.38478i	− 0.265549i	0.991146	−	0.132774i	$-0.0423885\pi$
$998$	0	0
$999$	10.9778	0.347321

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.b.g.513.2		8
4.3	odd	2	inner	1024.2.b.g.513.8		8
8.3	odd	2	inner	1024.2.b.g.513.1		8
8.5	even	2	inner	1024.2.b.g.513.7		8
16.3	odd	4		1024.2.a.i.1.1		4
16.5	even	4		1024.2.a.h.1.1		4
16.11	odd	4		1024.2.a.h.1.4		4
16.13	even	4		1024.2.a.i.1.4		4
32.3	odd	8		512.2.e.i.385.1	yes	8
32.5	even	8		512.2.e.i.129.4	yes	8
32.11	odd	8		512.2.e.j.129.4	yes	8
32.13	even	8		512.2.e.j.385.1	yes	8
32.19	odd	8		512.2.e.j.385.4	yes	8
32.21	even	8		512.2.e.j.129.1	yes	8
32.27	odd	8		512.2.e.i.129.1	✓	8
32.29	even	8		512.2.e.i.385.4	yes	8
48.5	odd	4		9216.2.a.bp.1.4		4
48.11	even	4		9216.2.a.bp.1.3		4
48.29	odd	4		9216.2.a.w.1.2		4
48.35	even	4		9216.2.a.w.1.1		4
96.5	odd	8		4608.2.k.bi.1153.3		8
96.11	even	8		4608.2.k.bd.1153.2		8
96.29	odd	8		4608.2.k.bi.3457.4		8
96.35	even	8		4608.2.k.bi.3457.3		8
96.53	odd	8		4608.2.k.bd.1153.1		8
96.59	even	8		4608.2.k.bi.1153.4		8
96.77	odd	8		4608.2.k.bd.3457.2		8
96.83	even	8		4608.2.k.bd.3457.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.i.129.1	✓	8	32.27	odd	8
512.2.e.i.129.4	yes	8	32.5	even	8
512.2.e.i.385.1	yes	8	32.3	odd	8
512.2.e.i.385.4	yes	8	32.29	even	8
512.2.e.j.129.1	yes	8	32.21	even	8
512.2.e.j.129.4	yes	8	32.11	odd	8
512.2.e.j.385.1	yes	8	32.13	even	8
512.2.e.j.385.4	yes	8	32.19	odd	8
1024.2.a.h.1.1		4	16.5	even	4
1024.2.a.h.1.4		4	16.11	odd	4
1024.2.a.i.1.1		4	16.3	odd	4
1024.2.a.i.1.4		4	16.13	even	4
1024.2.b.g.513.1		8	8.3	odd	2	inner
1024.2.b.g.513.2		8	1.1	even	1	trivial
1024.2.b.g.513.7		8	8.5	even	2	inner
1024.2.b.g.513.8		8	4.3	odd	2	inner
4608.2.k.bd.1153.1		8	96.53	odd	8
4608.2.k.bd.1153.2		8	96.11	even	8
4608.2.k.bd.3457.1		8	96.83	even	8
4608.2.k.bd.3457.2		8	96.77	odd	8
4608.2.k.bi.1153.3		8	96.5	odd	8
4608.2.k.bi.1153.4		8	96.59	even	8
4608.2.k.bi.3457.3		8	96.35	even	8
4608.2.k.bi.3457.4		8	96.29	odd	8
9216.2.a.w.1.1		4	48.35	even	4
9216.2.a.w.1.2		4	48.29	odd	4
9216.2.a.bp.1.3		4	48.11	even	4
9216.2.a.bp.1.4		4	48.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.61313i q^{3} +3.41421i q^{5} -1.53073 q^{7} -3.82843 q^{9} +O(q^{10})\)	\(q-2.61313i q^{3} +3.41421i q^{5} -1.53073 q^{7} -3.82843 q^{9} -4.77791i q^{11} -0.585786i q^{13} +8.92177 q^{15} -2.82843 q^{17} +0.448342i q^{19} +4.00000i q^{21} -5.86030 q^{23} -6.65685 q^{25} +2.16478i q^{27} -4.58579i q^{29} -7.39104 q^{31} -12.4853 q^{33} -5.22625i q^{35} -5.07107i q^{37} -1.53073 q^{39} +4.00000 q^{41} -2.61313i q^{43} -13.0711i q^{45} -7.39104 q^{47} -4.65685 q^{49} +7.39104i q^{51} +7.41421i q^{53} +16.3128 q^{55} +1.17157 q^{57} +2.61313i q^{59} -13.0711i q^{61} +5.86030 q^{63} +2.00000 q^{65} +10.0042i q^{67} +15.3137i q^{69} +11.9832 q^{71} -10.4853 q^{73} +17.3952i q^{75} +7.31371i q^{77} -6.12293 q^{79} -5.82843 q^{81} -3.50981i q^{83} -9.65685i q^{85} -11.9832 q^{87} +0.828427 q^{89} +0.896683i q^{91} +19.3137i q^{93} -1.53073 q^{95} +10.8284 q^{97} +18.2919i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 8 q^{25} - 32 q^{33} + 32 q^{41} + 8 q^{49} + 32 q^{57} + 16 q^{65} - 16 q^{73} - 24 q^{81} - 16 q^{89} + 64 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^25 - 32 * q^33 + 32 * q^41 + 8 * q^49 + 32 * q^57 + 16 * q^65 - 16 * q^73 - 24 * q^81 - 16 * q^89 + 64 * q^97

Introduction

L-functions

Modular forms

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Representations

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