LMFDB - Embedded newform 8128.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8128,2,Mod(1,8128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8128, 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("8128.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8128 = 2^{6} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8128.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:	$64.9024067629$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 508)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	8128.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.593579 q^{3} -3.46050 q^{5} +4.46050 q^{7} -2.64766 q^{9} +O(q^{10})$	$q+0.593579 q^{3} -3.46050 q^{5} +4.46050 q^{7} -2.64766 q^{9} -1.59358 q^{11} +1.40642 q^{13} -2.05408 q^{15} -1.86693 q^{17} +1.27335 q^{19} +2.64766 q^{21} -0.866926 q^{23} +6.97509 q^{25} -3.35234 q^{27} -4.46050 q^{29} +0.539495 q^{31} -0.945916 q^{33} -15.4356 q^{35} +11.9502 q^{37} +0.834822 q^{39} -7.10817 q^{43} +9.16225 q^{45} +11.1082 q^{47} +12.8961 q^{49} -1.10817 q^{51} +7.62276 q^{53} +5.51459 q^{55} +0.755832 q^{57} -1.48541 q^{59} +2.07899 q^{61} -11.8099 q^{63} -4.86693 q^{65} -11.3815 q^{67} -0.514589 q^{69} -14.8420 q^{71} -16.5687 q^{73} +4.14027 q^{75} -7.10817 q^{77} -3.50739 q^{79} +5.95311 q^{81} +14.7309 q^{83} +6.46050 q^{85} -2.64766 q^{87} -3.24124 q^{89} +6.27335 q^{91} +0.320233 q^{93} -4.40642 q^{95} -10.6008 q^{97} +4.21926 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 4 q^{5} + 7 q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q - q^3 - 4 * q^5 + 7 * q^7 + 4 * q^9	$ 3 q - q^{3} - 4 q^{5} + 7 q^{7} + 4 q^{9} - 2 q^{11} + 7 q^{13} + 3 q^{15} - 2 q^{17} + 3 q^{19} - 4 q^{21} + q^{23} - q^{25} - 22 q^{27} - 7 q^{29} + 8 q^{31} - 12 q^{33} - 18 q^{35} - 8 q^{37} - 15 q^{39} - 3 q^{43} + 15 q^{47} + 4 q^{49} + 15 q^{51} - 11 q^{53} + q^{55} - 28 q^{57} - 20 q^{59} + 19 q^{61} + 4 q^{63} - 11 q^{65} - 15 q^{67} + 14 q^{69} - 19 q^{71} - 25 q^{73} + 8 q^{75} - 3 q^{77} - 3 q^{79} + 19 q^{81} - 8 q^{83} + 13 q^{85} + 4 q^{87} + 5 q^{89} + 18 q^{91} - q^{93} - 16 q^{95} - 21 q^{97} + 21 q^{99}+O(q^{100}) $ 3 * q - q^3 - 4 * q^5 + 7 * q^7 + 4 * q^9 - 2 * q^11 + 7 * q^13 + 3 * q^15 - 2 * q^17 + 3 * q^19 - 4 * q^21 + q^23 - q^25 - 22 * q^27 - 7 * q^29 + 8 * q^31 - 12 * q^33 - 18 * q^35 - 8 * q^37 - 15 * q^39 - 3 * q^43 + 15 * q^47 + 4 * q^49 + 15 * q^51 - 11 * q^53 + q^55 - 28 * q^57 - 20 * q^59 + 19 * q^61 + 4 * q^63 - 11 * q^65 - 15 * q^67 + 14 * q^69 - 19 * q^71 - 25 * q^73 + 8 * q^75 - 3 * q^77 - 3 * q^79 + 19 * q^81 - 8 * q^83 + 13 * q^85 + 4 * q^87 + 5 * q^89 + 18 * q^91 - q^93 - 16 * q^95 - 21 * q^97 + 21 * 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
$3$	0.593579	0.342703	0.171352	−	0.985210i	$-0.445187\pi$
$3$	0.593579	0.342703	0.171352	+	0.985210i	$0.445187\pi$
$4$	0	0
$5$	−3.46050	−1.54758	−0.773792	−	0.633439i	$-0.781643\pi$
$5$	−3.46050	−1.54758	−0.773792	+	0.633439i	$0.781643\pi$
$6$	0	0
$7$	4.46050	1.68591	0.842956	−	0.537982i	$-0.180813\pi$
$7$	4.46050	1.68591	0.842956	+	0.537982i	$0.180813\pi$
$8$	0	0
$9$	−2.64766	−0.882555
$10$	0	0
$11$	−1.59358	−0.480482	−0.240241	−	0.970713i	$-0.577227\pi$
$11$	−1.59358	−0.480482	−0.240241	+	0.970713i	$0.577227\pi$
$12$	0	0
$13$	1.40642	0.390071	0.195035	−	0.980796i	$-0.437518\pi$
$13$	1.40642	0.390071	0.195035	+	0.980796i	$0.437518\pi$
$14$	0	0
$15$	−2.05408	−0.530362
$16$	0	0
$17$	−1.86693	−0.452796	−0.226398	−	0.974035i	$-0.572695\pi$
$17$	−1.86693	−0.452796	−0.226398	+	0.974035i	$0.572695\pi$
$18$	0	0
$19$	1.27335	0.292126	0.146063	−	0.989275i	$-0.453340\pi$
$19$	1.27335	0.292126	0.146063	+	0.989275i	$0.453340\pi$
$20$	0	0
$21$	2.64766	0.577768
$22$	0	0
$23$	−0.866926	−0.180766	−0.0903832	−	0.995907i	$-0.528809\pi$
$23$	−0.866926	−0.180766	−0.0903832	+	0.995907i	$0.528809\pi$
$24$	0	0
$25$	6.97509	1.39502
$26$	0	0
$27$	−3.35234	−0.645157
$28$	0	0
$29$	−4.46050	−0.828295	−0.414147	−	0.910210i	$-0.635920\pi$
$29$	−4.46050	−0.828295	−0.414147	+	0.910210i	$0.635920\pi$
$30$	0	0
$31$	0.539495	0.0968962	0.0484481	−	0.998826i	$-0.484572\pi$
$31$	0.539495	0.0968962	0.0484481	+	0.998826i	$0.484572\pi$
$32$	0	0
$33$	−0.945916	−0.164663
$34$	0	0
$35$	−15.4356	−2.60909
$36$	0	0
$37$	11.9502	1.96460	0.982299	−	0.187318i	$-0.0599794\pi$
$37$	11.9502	1.96460	0.982299	+	0.187318i	$0.0599794\pi$
$38$	0	0
$39$	0.834822	0.133679
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−7.10817	−1.08399	−0.541993	−	0.840383i	$-0.682330\pi$
$43$	−7.10817	−1.08399	−0.541993	+	0.840383i	$0.682330\pi$
$44$	0	0
$45$	9.16225	1.36583
$46$	0	0
$47$	11.1082	1.62029	0.810146	−	0.586228i	$-0.199388\pi$
$47$	11.1082	1.62029	0.810146	+	0.586228i	$0.199388\pi$
$48$	0	0
$49$	12.8961	1.84230
$50$	0	0
$51$	−1.10817	−0.155175
$52$	0	0
$53$	7.62276	1.04707	0.523533	−	0.852005i	$-0.324614\pi$
$53$	7.62276	1.04707	0.523533	+	0.852005i	$0.324614\pi$
$54$	0	0
$55$	5.51459	0.743587
$56$	0	0
$57$	0.755832	0.100112
$58$	0	0
$59$	−1.48541	−0.193384	−0.0966920	−	0.995314i	$-0.530826\pi$
$59$	−1.48541	−0.193384	−0.0966920	+	0.995314i	$0.530826\pi$
$60$	0	0
$61$	2.07899	0.266187	0.133094	−	0.991103i	$-0.457509\pi$
$61$	2.07899	0.266187	0.133094	+	0.991103i	$0.457509\pi$
$62$	0	0
$63$	−11.8099	−1.48791
$64$	0	0
$65$	−4.86693	−0.603668
$66$	0	0
$67$	−11.3815	−1.39047	−0.695237	−	0.718781i	$-0.744700\pi$
$67$	−11.3815	−1.39047	−0.695237	+	0.718781i	$0.744700\pi$
$68$	0	0
$69$	−0.514589	−0.0619492
$70$	0	0
$71$	−14.8420	−1.76142	−0.880712	−	0.473652i	$-0.842935\pi$
$71$	−14.8420	−1.76142	−0.880712	+	0.473652i	$0.842935\pi$
$72$	0	0
$73$	−16.5687	−1.93922	−0.969608	−	0.244663i	$-0.921323\pi$
$73$	−16.5687	−1.93922	−0.969608	+	0.244663i	$0.921323\pi$
$74$	0	0
$75$	4.14027	0.478077
$76$	0	0
$77$	−7.10817	−0.810051
$78$	0	0
$79$	−3.50739	−0.394612	−0.197306	−	0.980342i	$-0.563219\pi$
$79$	−3.50739	−0.394612	−0.197306	+	0.980342i	$0.563219\pi$
$80$	0	0
$81$	5.95311	0.661457
$82$	0	0
$83$	14.7309	1.61693	0.808465	−	0.588545i	$-0.200299\pi$
$83$	14.7309	1.61693	0.808465	+	0.588545i	$0.200299\pi$
$84$	0	0
$85$	6.46050	0.700740
$86$	0	0
$87$	−2.64766	−0.283859
$88$	0	0
$89$	−3.24124	−0.343571	−0.171786	−	0.985134i	$-0.554954\pi$
$89$	−3.24124	−0.343571	−0.171786	+	0.985134i	$0.554954\pi$
$90$	0	0
$91$	6.27335	0.657625
$92$	0	0
$93$	0.320233	0.0332066
$94$	0	0
$95$	−4.40642	−0.452089
$96$	0	0
$97$	−10.6008	−1.07635	−0.538173	−	0.842834i	$-0.680885\pi$
$97$	−10.6008	−1.07635	−0.538173	+	0.842834i	$0.680885\pi$
$98$	0	0
$99$	4.21926	0.424052
$100$	0	0
$101$	−12.5218	−1.24596	−0.622982	−	0.782236i	$-0.714079\pi$
$101$	−12.5218	−1.24596	−0.622982	+	0.782236i	$0.714079\pi$
$102$	0	0
$103$	−4.42840	−0.436343	−0.218172	−	0.975910i	$-0.570009\pi$
$103$	−4.42840	−0.436343	−0.218172	+	0.975910i	$0.570009\pi$
$104$	0	0
$105$	−9.16225	−0.894144
$106$	0	0
$107$	−13.5438	−1.30933	−0.654663	−	0.755921i	$-0.727189\pi$
$107$	−13.5438	−1.30933	−0.654663	+	0.755921i	$0.727189\pi$
$108$	0	0
$109$	9.65486	0.924768	0.462384	−	0.886680i	$-0.346994\pi$
$109$	9.65486	0.924768	0.462384	+	0.886680i	$0.346994\pi$
$110$	0	0
$111$	7.09338	0.673274
$112$	0	0
$113$	8.16225	0.767840	0.383920	−	0.923366i	$-0.374574\pi$
$113$	8.16225	0.767840	0.383920	+	0.923366i	$0.374574\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	−3.72373	−0.344259
$118$	0	0
$119$	−8.32743	−0.763374
$120$	0	0
$121$	−8.46050	−0.769137
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.83482	−0.611325
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	0	0
$129$	−4.21926	−0.371485
$130$	0	0
$131$	1.60078	0.139860	0.0699302	−	0.997552i	$-0.477722\pi$
$131$	1.60078	0.139860	0.0699302	+	0.997552i	$0.477722\pi$
$132$	0	0
$133$	5.67977	0.492498
$134$	0	0
$135$	11.6008	0.998436
$136$	0	0
$137$	7.48249	0.639272	0.319636	−	0.947540i	$-0.396439\pi$
$137$	7.48249	0.639272	0.319636	+	0.947540i	$0.396439\pi$
$138$	0	0
$139$	−5.81284	−0.493039	−0.246519	−	0.969138i	$-0.579287\pi$
$139$	−5.81284	−0.493039	−0.246519	+	0.969138i	$0.579287\pi$
$140$	0	0
$141$	6.59358	0.555280
$142$	0	0
$143$	−2.24124	−0.187422
$144$	0	0
$145$	15.4356	1.28186
$146$	0	0
$147$	7.65486	0.631362
$148$	0	0
$149$	−11.6228	−0.952173	−0.476087	−	0.879398i	$-0.657945\pi$
$149$	−11.6228	−0.952173	−0.476087	+	0.879398i	$0.657945\pi$
$150$	0	0
$151$	−8.73385	−0.710751	−0.355375	−	0.934724i	$-0.615647\pi$
$151$	−8.73385	−0.710751	−0.355375	+	0.934724i	$0.615647\pi$
$152$	0	0
$153$	4.94299	0.399617
$154$	0	0
$155$	−1.86693	−0.149955
$156$	0	0
$157$	−3.72665	−0.297419	−0.148710	−	0.988881i	$-0.547512\pi$
$157$	−3.72665	−0.297419	−0.148710	+	0.988881i	$0.547512\pi$
$158$	0	0
$159$	4.52471	0.358833
$160$	0	0
$161$	−3.86693	−0.304756
$162$	0	0
$163$	21.2953	1.66798	0.833989	−	0.551781i	$-0.186051\pi$
$163$	21.2953	1.66798	0.833989	+	0.551781i	$0.186051\pi$
$164$	0	0
$165$	3.27335	0.254830
$166$	0	0
$167$	−17.9794	−1.39129	−0.695643	−	0.718388i	$-0.744880\pi$
$167$	−17.9794	−1.39129	−0.695643	+	0.718388i	$0.744880\pi$
$168$	0	0
$169$	−11.0220	−0.847845
$170$	0	0
$171$	−3.37139	−0.257817
$172$	0	0
$173$	−13.7339	−1.04417	−0.522083	−	0.852895i	$-0.674845\pi$
$173$	−13.7339	−1.04417	−0.522083	+	0.852895i	$0.674845\pi$
$174$	0	0
$175$	31.1124	2.35188
$176$	0	0
$177$	−0.881709	−0.0662733
$178$	0	0
$179$	2.16225	0.161614	0.0808072	−	0.996730i	$-0.474250\pi$
$179$	2.16225	0.161614	0.0808072	+	0.996730i	$0.474250\pi$
$180$	0	0
$181$	22.6300	1.68207	0.841036	−	0.540980i	$-0.181946\pi$
$181$	22.6300	1.68207	0.841036	+	0.540980i	$0.181946\pi$
$182$	0	0
$183$	1.23405	0.0912233
$184$	0	0
$185$	−41.3537	−3.04038
$186$	0	0
$187$	2.97509	0.217560
$188$	0	0
$189$	−14.9531	−1.08768
$190$	0	0
$191$	5.08619	0.368024	0.184012	−	0.982924i	$-0.441092\pi$
$191$	5.08619	0.368024	0.184012	+	0.982924i	$0.441092\pi$
$192$	0	0
$193$	26.8391	1.93192	0.965960	−	0.258691i	$-0.0832911\pi$
$193$	26.8391	1.93192	0.965960	+	0.258691i	$0.0832911\pi$
$194$	0	0
$195$	−2.88891	−0.206879
$196$	0	0
$197$	−7.69455	−0.548214	−0.274107	−	0.961699i	$-0.588382\pi$
$197$	−7.69455	−0.548214	−0.274107	+	0.961699i	$0.588382\pi$
$198$	0	0
$199$	9.21634	0.653329	0.326664	−	0.945140i	$-0.394075\pi$
$199$	9.21634	0.653329	0.326664	+	0.945140i	$0.394075\pi$
$200$	0	0
$201$	−6.75583	−0.476520
$202$	0	0
$203$	−19.8961	−1.39643
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.29533	0.159536
$208$	0	0
$209$	−2.02918	−0.140361
$210$	0	0
$211$	8.83482	0.608214	0.304107	−	0.952638i	$-0.401642\pi$
$211$	8.83482	0.608214	0.304107	+	0.952638i	$0.401642\pi$
$212$	0	0
$213$	−8.80992	−0.603646
$214$	0	0
$215$	24.5979	1.67756
$216$	0	0
$217$	2.40642	0.163358
$218$	0	0
$219$	−9.83482	−0.664576
$220$	0	0
$221$	−2.62568	−0.176623
$222$	0	0
$223$	−2.02491	−0.135598	−0.0677989	−	0.997699i	$-0.521598\pi$
$223$	−2.02491	−0.135598	−0.0677989	+	0.997699i	$0.521598\pi$
$224$	0	0
$225$	−18.4677	−1.23118
$226$	0	0
$227$	−12.9679	−0.860710	−0.430355	−	0.902660i	$-0.641612\pi$
$227$	−12.9679	−0.860710	−0.430355	+	0.902660i	$0.641612\pi$
$228$	0	0
$229$	11.6300	0.768529	0.384265	−	0.923223i	$-0.374455\pi$
$229$	11.6300	0.768529	0.384265	+	0.923223i	$0.374455\pi$
$230$	0	0
$231$	−4.21926	−0.277607
$232$	0	0
$233$	−25.0512	−1.64116	−0.820578	−	0.571535i	$-0.806348\pi$
$233$	−25.0512	−1.64116	−0.820578	+	0.571535i	$0.806348\pi$
$234$	0	0
$235$	−38.4399	−2.50754
$236$	0	0
$237$	−2.08192	−0.135235
$238$	0	0
$239$	−19.3566	−1.25207	−0.626037	−	0.779793i	$-0.715324\pi$
$239$	−19.3566	−1.25207	−0.626037	+	0.779793i	$0.715324\pi$
$240$	0	0
$241$	−15.7850	−1.01680	−0.508401	−	0.861120i	$-0.669763\pi$
$241$	−15.7850	−1.01680	−0.508401	+	0.861120i	$0.669763\pi$
$242$	0	0
$243$	13.5907	0.871841
$244$	0	0
$245$	−44.6270	−2.85112
$246$	0	0
$247$	1.79086	0.113950
$248$	0	0
$249$	8.74397	0.554127
$250$	0	0
$251$	−27.0364	−1.70652	−0.853260	−	0.521485i	$-0.825378\pi$
$251$	−27.0364	−1.70652	−0.853260	+	0.521485i	$0.825378\pi$
$252$	0	0
$253$	1.38151	0.0868551
$254$	0	0
$255$	3.83482	0.240146
$256$	0	0
$257$	−5.14027	−0.320641	−0.160321	−	0.987065i	$-0.551253\pi$
$257$	−5.14027	−0.320641	−0.160321	+	0.987065i	$0.551253\pi$
$258$	0	0
$259$	53.3039	3.31214
$260$	0	0
$261$	11.8099	0.731015
$262$	0	0
$263$	26.4428	1.63053	0.815266	−	0.579086i	$-0.196591\pi$
$263$	26.4428	1.63053	0.815266	+	0.579086i	$0.196591\pi$
$264$	0	0
$265$	−26.3786	−1.62042
$266$	0	0
$267$	−1.92393	−0.117743
$268$	0	0
$269$	−21.2307	−1.29446	−0.647230	−	0.762295i	$-0.724073\pi$
$269$	−21.2307	−1.29446	−0.647230	+	0.762295i	$0.724073\pi$
$270$	0	0
$271$	−15.8741	−0.964284	−0.482142	−	0.876093i	$-0.660141\pi$
$271$	−15.8741	−0.964284	−0.482142	+	0.876093i	$0.660141\pi$
$272$	0	0
$273$	3.72373	0.225370
$274$	0	0
$275$	−11.1154	−0.670282
$276$	0	0
$277$	−20.9181	−1.25685	−0.628423	−	0.777872i	$-0.716299\pi$
$277$	−20.9181	−1.25685	−0.628423	+	0.777872i	$0.716299\pi$
$278$	0	0
$279$	−1.42840	−0.0855162
$280$	0	0
$281$	10.0115	0.597234	0.298617	−	0.954373i	$-0.403475\pi$
$281$	10.0115	0.597234	0.298617	+	0.954373i	$0.403475\pi$
$282$	0	0
$283$	10.5438	0.626762	0.313381	−	0.949627i	$-0.398538\pi$
$283$	10.5438	0.626762	0.313381	+	0.949627i	$0.398538\pi$
$284$	0	0
$285$	−2.61556	−0.154932
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.5146	−0.794976
$290$	0	0
$291$	−6.29240	−0.368867
$292$	0	0
$293$	−3.88463	−0.226943	−0.113471	−	0.993541i	$-0.536197\pi$
$293$	−3.88463	−0.226943	−0.113471	+	0.993541i	$0.536197\pi$
$294$	0	0
$295$	5.14027	0.299278
$296$	0	0
$297$	5.34221	0.309987
$298$	0	0
$299$	−1.21926	−0.0705117
$300$	0	0
$301$	−31.7060	−1.82750
$302$	0	0
$303$	−7.43267	−0.426996
$304$	0	0
$305$	−7.19436	−0.411948
$306$	0	0
$307$	−5.54377	−0.316400	−0.158200	−	0.987407i	$-0.550569\pi$
$307$	−5.54377	−0.316400	−0.158200	+	0.987407i	$0.550569\pi$
$308$	0	0
$309$	−2.62861	−0.149536
$310$	0	0
$311$	−10.9574	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$311$	−10.9574	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$312$	0	0
$313$	−9.63288	−0.544483	−0.272241	−	0.962229i	$-0.587765\pi$
$313$	−9.63288	−0.544483	−0.272241	+	0.962229i	$0.587765\pi$
$314$	0	0
$315$	40.8683	2.30267
$316$	0	0
$317$	−6.17237	−0.346675	−0.173338	−	0.984862i	$-0.555455\pi$
$317$	−6.17237	−0.346675	−0.173338	+	0.984862i	$0.555455\pi$
$318$	0	0
$319$	7.10817	0.397981
$320$	0	0
$321$	−8.03930	−0.448710
$322$	0	0
$323$	−2.37724	−0.132273
$324$	0	0
$325$	9.80992	0.544156
$326$	0	0
$327$	5.73093	0.316921
$328$	0	0
$329$	49.5480	2.73167
$330$	0	0
$331$	−17.6270	−0.968869	−0.484435	−	0.874827i	$-0.660975\pi$
$331$	−17.6270	−0.968869	−0.484435	+	0.874827i	$0.660975\pi$
$332$	0	0
$333$	−31.6401	−1.73387
$334$	0	0
$335$	39.3858	2.15188
$336$	0	0
$337$	9.66206	0.526326	0.263163	−	0.964751i	$-0.415234\pi$
$337$	9.66206	0.526326	0.263163	+	0.964751i	$0.415234\pi$
$338$	0	0
$339$	4.84494	0.263141
$340$	0	0
$341$	−0.859728	−0.0465569
$342$	0	0
$343$	26.2996	1.42004
$344$	0	0
$345$	1.78074	0.0958717
$346$	0	0
$347$	−0.125877	−0.00675744	−0.00337872	−	0.999994i	$-0.501075\pi$
$347$	−0.125877	−0.00675744	−0.00337872	+	0.999994i	$0.501075\pi$
$348$	0	0
$349$	−9.78794	−0.523936	−0.261968	−	0.965077i	$-0.584372\pi$
$349$	−9.78794	−0.523936	−0.261968	+	0.965077i	$0.584372\pi$
$350$	0	0
$351$	−4.71480	−0.251657
$352$	0	0
$353$	26.3173	1.40073	0.700364	−	0.713785i	$-0.253021\pi$
$353$	26.3173	1.40073	0.700364	+	0.713785i	$0.253021\pi$
$354$	0	0
$355$	51.3609	2.72595
$356$	0	0
$357$	−4.94299	−0.261611
$358$	0	0
$359$	−1.42840	−0.0753882	−0.0376941	−	0.999289i	$-0.512001\pi$
$359$	−1.42840	−0.0753882	−0.0376941	+	0.999289i	$0.512001\pi$
$360$	0	0
$361$	−17.3786	−0.914663
$362$	0	0
$363$	−5.02198	−0.263586
$364$	0	0
$365$	57.3360	3.00110
$366$	0	0
$367$	17.5615	0.916702	0.458351	−	0.888771i	$-0.348440\pi$
$367$	17.5615	0.916702	0.458351	+	0.888771i	$0.348440\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	34.0013	1.76526
$372$	0	0
$373$	−0.165178	−0.00855258	−0.00427629	−	0.999991i	$-0.501361\pi$
$373$	−0.165178	−0.00855258	−0.00427629	+	0.999991i	$0.501361\pi$
$374$	0	0
$375$	−4.05701	−0.209503
$376$	0	0
$377$	−6.27335	−0.323094
$378$	0	0
$379$	0.694551	0.0356767	0.0178383	−	0.999841i	$-0.494322\pi$
$379$	0.694551	0.0356767	0.0178383	+	0.999841i	$0.494322\pi$
$380$	0	0
$381$	−0.593579	−0.0304100
$382$	0	0
$383$	3.62276	0.185114	0.0925571	−	0.995707i	$-0.470496\pi$
$383$	3.62276	0.185114	0.0925571	+	0.995707i	$0.470496\pi$
$384$	0	0
$385$	24.5979	1.25362
$386$	0	0
$387$	18.8200	0.956676
$388$	0	0
$389$	−17.9679	−0.911009	−0.455504	−	0.890234i	$-0.650541\pi$
$389$	−17.9679	−0.911009	−0.455504	+	0.890234i	$0.650541\pi$
$390$	0	0
$391$	1.61849	0.0818503
$392$	0	0
$393$	0.950188	0.0479306
$394$	0	0
$395$	12.1373	0.610696
$396$	0	0
$397$	−31.2527	−1.56853	−0.784264	−	0.620427i	$-0.786959\pi$
$397$	−31.2527	−1.56853	−0.784264	+	0.620427i	$0.786959\pi$
$398$	0	0
$399$	3.37139	0.168781
$400$	0	0
$401$	−28.6726	−1.43184	−0.715920	−	0.698182i	$-0.753992\pi$
$401$	−28.6726	−1.43184	−0.715920	+	0.698182i	$0.753992\pi$
$402$	0	0
$403$	0.758757	0.0377964
$404$	0	0
$405$	−20.6008	−1.02366
$406$	0	0
$407$	−19.0436	−0.943955
$408$	0	0
$409$	−19.1373	−0.946281	−0.473140	−	0.880987i	$-0.656880\pi$
$409$	−19.1373	−0.946281	−0.473140	+	0.880987i	$0.656880\pi$
$410$	0	0
$411$	4.44145	0.219081
$412$	0	0
$413$	−6.62568	−0.326029
$414$	0	0
$415$	−50.9764	−2.50234
$416$	0	0
$417$	−3.45038	−0.168966
$418$	0	0
$419$	23.4825	1.14719	0.573597	−	0.819137i	$-0.305547\pi$
$419$	23.4825	1.14719	0.573597	+	0.819137i	$0.305547\pi$
$420$	0	0
$421$	24.2412	1.18145	0.590723	−	0.806874i	$-0.298843\pi$
$421$	24.2412	1.18145	0.590723	+	0.806874i	$0.298843\pi$
$422$	0	0
$423$	−29.4107	−1.43000
$424$	0	0
$425$	−13.0220	−0.631659
$426$	0	0
$427$	9.27335	0.448769
$428$	0	0
$429$	−1.33036	−0.0642302
$430$	0	0
$431$	−23.0263	−1.10914	−0.554568	−	0.832139i	$-0.687116\pi$
$431$	−23.0263	−1.10914	−0.554568	+	0.832139i	$0.687116\pi$
$432$	0	0
$433$	25.2556	1.21371	0.606854	−	0.794813i	$-0.292431\pi$
$433$	25.2556	1.21371	0.606854	+	0.794813i	$0.292431\pi$
$434$	0	0
$435$	9.16225	0.439296
$436$	0	0
$437$	−1.10390	−0.0528065
$438$	0	0
$439$	−19.1010	−0.911640	−0.455820	−	0.890072i	$-0.650654\pi$
$439$	−19.1010	−0.911640	−0.455820	+	0.890072i	$0.650654\pi$
$440$	0	0
$441$	−34.1445	−1.62593
$442$	0	0
$443$	−28.5012	−1.35413	−0.677065	−	0.735923i	$-0.736749\pi$
$443$	−28.5012	−1.35413	−0.677065	+	0.735923i	$0.736749\pi$
$444$	0	0
$445$	11.2163	0.531705
$446$	0	0
$447$	−6.89903	−0.326313
$448$	0	0
$449$	−26.5261	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$449$	−26.5261	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−5.18423	−0.243577
$454$	0	0
$455$	−21.7089	−1.01773
$456$	0	0
$457$	5.59358	0.261657	0.130828	−	0.991405i	$-0.458236\pi$
$457$	5.59358	0.261657	0.130828	+	0.991405i	$0.458236\pi$
$458$	0	0
$459$	6.25856	0.292125
$460$	0	0
$461$	−2.67684	−0.124673	−0.0623365	−	0.998055i	$-0.519855\pi$
$461$	−2.67684	−0.124673	−0.0623365	+	0.998055i	$0.519855\pi$
$462$	0	0
$463$	14.3097	0.665029	0.332515	−	0.943098i	$-0.392103\pi$
$463$	14.3097	0.665029	0.332515	+	0.943098i	$0.392103\pi$
$464$	0	0
$465$	−1.10817	−0.0513901
$466$	0	0
$467$	−34.3782	−1.59083	−0.795417	−	0.606063i	$-0.792748\pi$
$467$	−34.3782	−1.59083	−0.795417	+	0.606063i	$0.792748\pi$
$468$	0	0
$469$	−50.7673	−2.34422
$470$	0	0
$471$	−2.21206	−0.101927
$472$	0	0
$473$	11.3274	0.520836
$474$	0	0
$475$	8.88171	0.407521
$476$	0	0
$477$	−20.1825	−0.924093
$478$	0	0
$479$	1.86400	0.0851684	0.0425842	−	0.999093i	$-0.486441\pi$
$479$	1.86400	0.0851684	0.0425842	+	0.999093i	$0.486441\pi$
$480$	0	0
$481$	16.8070	0.766333
$482$	0	0
$483$	−2.29533	−0.104441
$484$	0	0
$485$	36.6840	1.66574
$486$	0	0
$487$	−21.6372	−0.980473	−0.490236	−	0.871589i	$-0.663090\pi$
$487$	−21.6372	−0.980473	−0.490236	+	0.871589i	$0.663090\pi$
$488$	0	0
$489$	12.6405	0.571622
$490$	0	0
$491$	−25.6021	−1.15541	−0.577704	−	0.816246i	$-0.696051\pi$
$491$	−25.6021	−1.15541	−0.577704	+	0.816246i	$0.696051\pi$
$492$	0	0
$493$	8.32743	0.375049
$494$	0	0
$495$	−14.6008	−0.656256
$496$	0	0
$497$	−66.2029	−2.96961
$498$	0	0
$499$	−30.2134	−1.35254	−0.676269	−	0.736655i	$-0.736404\pi$
$499$	−30.2134	−1.35254	−0.676269	+	0.736655i	$0.736404\pi$
$500$	0	0
$501$	−10.6722	−0.476798
$502$	0	0
$503$	28.4792	1.26982	0.634912	−	0.772585i	$-0.281036\pi$
$503$	28.4792	1.26982	0.634912	+	0.772585i	$0.281036\pi$
$504$	0	0
$505$	43.3317	1.92824
$506$	0	0
$507$	−6.54242	−0.290559
$508$	0	0
$509$	−43.0335	−1.90742	−0.953712	−	0.300721i	$-0.902773\pi$
$509$	−43.0335	−1.90742	−0.953712	+	0.300721i	$0.902773\pi$
$510$	0	0
$511$	−73.9046	−3.26935
$512$	0	0
$513$	−4.26868	−0.188467
$514$	0	0
$515$	15.3245	0.675278
$516$	0	0
$517$	−17.7017	−0.778522
$518$	0	0
$519$	−8.15213	−0.357839
$520$	0	0
$521$	14.0085	0.613725	0.306863	−	0.951754i	$-0.400721\pi$
$521$	14.0085	0.613725	0.306863	+	0.951754i	$0.400721\pi$
$522$	0	0
$523$	37.7749	1.65178	0.825890	−	0.563831i	$-0.190673\pi$
$523$	37.7749	1.65178	0.825890	+	0.563831i	$0.190673\pi$
$524$	0	0
$525$	18.4677	0.805997
$526$	0	0
$527$	−1.00720	−0.0438742
$528$	0	0
$529$	−22.2484	−0.967323
$530$	0	0
$531$	3.93287	0.170672
$532$	0	0
$533$	0	0
$534$	0	0
$535$	46.8683	2.02629
$536$	0	0
$537$	1.28347	0.0553858
$538$	0	0
$539$	−20.5510	−0.885193
$540$	0	0
$541$	18.9109	0.813043	0.406521	−	0.913641i	$-0.366742\pi$
$541$	18.9109	0.813043	0.406521	+	0.913641i	$0.366742\pi$
$542$	0	0
$543$	13.4327	0.576451
$544$	0	0
$545$	−33.4107	−1.43116
$546$	0	0
$547$	−43.9118	−1.87753	−0.938767	−	0.344551i	$-0.888031\pi$
$547$	−43.9118	−1.87753	−0.938767	+	0.344551i	$0.888031\pi$
$548$	0	0
$549$	−5.50447	−0.234925
$550$	0	0
$551$	−5.67977	−0.241966
$552$	0	0
$553$	−15.6447	−0.665282
$554$	0	0
$555$	−24.5467	−1.04195
$556$	0	0
$557$	0.0714038	0.00302548	0.00151274	−	0.999999i	$-0.499518\pi$
$557$	0.0714038	0.00302548	0.00151274	+	0.999999i	$0.499518\pi$
$558$	0	0
$559$	−9.99707	−0.422831
$560$	0	0
$561$	1.76595	0.0745586
$562$	0	0
$563$	−27.7850	−1.17100	−0.585499	−	0.810673i	$-0.699101\pi$
$563$	−27.7850	−1.17100	−0.585499	+	0.810673i	$0.699101\pi$
$564$	0	0
$565$	−28.2455	−1.18830
$566$	0	0
$567$	26.5539	1.11516
$568$	0	0
$569$	−14.9636	−0.627308	−0.313654	−	0.949537i	$-0.601553\pi$
$569$	−14.9636	−0.627308	−0.313654	+	0.949537i	$0.601553\pi$
$570$	0	0
$571$	15.5438	0.650486	0.325243	−	0.945630i	$-0.394554\pi$
$571$	15.5438	0.650486	0.325243	+	0.945630i	$0.394554\pi$
$572$	0	0
$573$	3.01906	0.126123
$574$	0	0
$575$	−6.04689	−0.252173
$576$	0	0
$577$	45.1944	1.88147	0.940733	−	0.339149i	$-0.110139\pi$
$577$	45.1944	1.88147	0.940733	+	0.339149i	$0.110139\pi$
$578$	0	0
$579$	15.9311	0.662075
$580$	0	0
$581$	65.7074	2.72600
$582$	0	0
$583$	−12.1475	−0.503097
$584$	0	0
$585$	12.8860	0.532770
$586$	0	0
$587$	29.4150	1.21409	0.607043	−	0.794669i	$-0.292356\pi$
$587$	29.4150	1.21409	0.607043	+	0.794669i	$0.292356\pi$
$588$	0	0
$589$	0.686964	0.0283059
$590$	0	0
$591$	−4.56733	−0.187875
$592$	0	0
$593$	−24.1944	−0.993543	−0.496772	−	0.867881i	$-0.665481\pi$
$593$	−24.1944	−0.993543	−0.496772	+	0.867881i	$0.665481\pi$
$594$	0	0
$595$	28.8171	1.18139
$596$	0	0
$597$	5.47063	0.223898
$598$	0	0
$599$	−15.6519	−0.639521	−0.319760	−	0.947498i	$-0.603602\pi$
$599$	−15.6519	−0.639521	−0.319760	+	0.947498i	$0.603602\pi$
$600$	0	0
$601$	20.8348	0.849870	0.424935	−	0.905224i	$-0.360297\pi$
$601$	20.8348	0.849870	0.424935	+	0.905224i	$0.360297\pi$
$602$	0	0
$603$	30.1344	1.22717
$604$	0	0
$605$	29.2776	1.19030
$606$	0	0
$607$	27.1301	1.10118	0.550589	−	0.834776i	$-0.314403\pi$
$607$	27.1301	1.10118	0.550589	+	0.834776i	$0.314403\pi$
$608$	0	0
$609$	−11.8099	−0.478562
$610$	0	0
$611$	15.6228	0.632029
$612$	0	0
$613$	−14.5543	−0.587842	−0.293921	−	0.955830i	$-0.594960\pi$
$613$	−14.5543	−0.587842	−0.293921	+	0.955830i	$0.594960\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.2134	−1.21635	−0.608173	−	0.793804i	$-0.708097\pi$
$617$	−30.2134	−1.21635	−0.608173	+	0.793804i	$0.708097\pi$
$618$	0	0
$619$	−19.1766	−0.770775	−0.385387	−	0.922755i	$-0.625932\pi$
$619$	−19.1766	−0.770775	−0.385387	+	0.922755i	$0.625932\pi$
$620$	0	0
$621$	2.90623	0.116623
$622$	0	0
$623$	−14.4576	−0.579231
$624$	0	0
$625$	−11.2235	−0.448941
$626$	0	0
$627$	−1.20448	−0.0481022
$628$	0	0
$629$	−22.3101	−0.889562
$630$	0	0
$631$	−0.654861	−0.0260696	−0.0130348	−	0.999915i	$-0.504149\pi$
$631$	−0.654861	−0.0260696	−0.0130348	+	0.999915i	$0.504149\pi$
$632$	0	0
$633$	5.24417	0.208437
$634$	0	0
$635$	3.46050	0.137326
$636$	0	0
$637$	18.1373	0.718628
$638$	0	0
$639$	39.2967	1.55455
$640$	0	0
$641$	6.18716	0.244378	0.122189	−	0.992507i	$-0.461009\pi$
$641$	6.18716	0.244378	0.122189	+	0.992507i	$0.461009\pi$
$642$	0	0
$643$	−22.6549	−0.893420	−0.446710	−	0.894679i	$-0.647405\pi$
$643$	−22.6549	−0.893420	−0.446710	+	0.894679i	$0.647405\pi$
$644$	0	0
$645$	14.6008	0.574905
$646$	0	0
$647$	20.0368	0.787727	0.393863	−	0.919169i	$-0.371138\pi$
$647$	20.0368	0.787727	0.393863	+	0.919169i	$0.371138\pi$
$648$	0	0
$649$	2.36712	0.0929176
$650$	0	0
$651$	1.42840	0.0559835
$652$	0	0
$653$	19.1694	0.750159	0.375079	−	0.926993i	$-0.377615\pi$
$653$	19.1694	0.750159	0.375079	+	0.926993i	$0.377615\pi$
$654$	0	0
$655$	−5.53950	−0.216446
$656$	0	0
$657$	43.8683	1.71146
$658$	0	0
$659$	3.49981	0.136333	0.0681665	−	0.997674i	$-0.478285\pi$
$659$	3.49981	0.136333	0.0681665	+	0.997674i	$0.478285\pi$
$660$	0	0
$661$	−26.1196	−1.01594	−0.507968	−	0.861376i	$-0.669603\pi$
$661$	−26.1196	−1.01594	−0.507968	+	0.861376i	$0.669603\pi$
$662$	0	0
$663$	−1.55855	−0.0605291
$664$	0	0
$665$	−19.6549	−0.762183
$666$	0	0
$667$	3.86693	0.149728
$668$	0	0
$669$	−1.20194	−0.0464698
$670$	0	0
$671$	−3.31304	−0.127898
$672$	0	0
$673$	35.4792	1.36762	0.683811	−	0.729659i	$-0.260321\pi$
$673$	35.4792	1.36762	0.683811	+	0.729659i	$0.260321\pi$
$674$	0	0
$675$	−23.3829	−0.900007
$676$	0	0
$677$	45.5189	1.74943	0.874716	−	0.484636i	$-0.161048\pi$
$677$	45.5189	1.74943	0.874716	+	0.484636i	$0.161048\pi$
$678$	0	0
$679$	−47.2848	−1.81462
$680$	0	0
$681$	−7.69748	−0.294968
$682$	0	0
$683$	−36.1521	−1.38332	−0.691661	−	0.722222i	$-0.743121\pi$
$683$	−36.1521	−1.38332	−0.691661	+	0.722222i	$0.743121\pi$
$684$	0	0
$685$	−25.8932	−0.989327
$686$	0	0
$687$	6.90330	0.263377
$688$	0	0
$689$	10.7208	0.408430
$690$	0	0
$691$	−27.8319	−1.05878	−0.529388	−	0.848380i	$-0.677578\pi$
$691$	−27.8319	−1.05878	−0.529388	+	0.848380i	$0.677578\pi$
$692$	0	0
$693$	18.8200	0.714914
$694$	0	0
$695$	20.1154	0.763019
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−14.8699	−0.562429
$700$	0	0
$701$	5.67977	0.214522	0.107261	−	0.994231i	$-0.465792\pi$
$701$	5.67977	0.214522	0.107261	+	0.994231i	$0.465792\pi$
$702$	0	0
$703$	15.2167	0.573910
$704$	0	0
$705$	−22.8171	−0.859342
$706$	0	0
$707$	−55.8535	−2.10059
$708$	0	0
$709$	−23.3173	−0.875700	−0.437850	−	0.899048i	$-0.644260\pi$
$709$	−23.3173	−0.875700	−0.437850	+	0.899048i	$0.644260\pi$
$710$	0	0
$711$	9.28639	0.348267
$712$	0	0
$713$	−0.467702	−0.0175156
$714$	0	0
$715$	7.75583	0.290052
$716$	0	0
$717$	−11.4897	−0.429090
$718$	0	0
$719$	7.53230	0.280907	0.140454	−	0.990087i	$-0.455144\pi$
$719$	7.53230	0.280907	0.140454	+	0.990087i	$0.455144\pi$
$720$	0	0
$721$	−19.7529	−0.735637
$722$	0	0
$723$	−9.36966	−0.348461
$724$	0	0
$725$	−31.1124	−1.15549
$726$	0	0
$727$	25.2881	0.937885	0.468942	−	0.883229i	$-0.344635\pi$
$727$	25.2881	0.937885	0.468942	+	0.883229i	$0.344635\pi$
$728$	0	0
$729$	−9.79221	−0.362674
$730$	0	0
$731$	13.2704	0.490824
$732$	0	0
$733$	36.3068	1.34102	0.670511	−	0.741900i	$-0.266075\pi$
$733$	36.3068	1.34102	0.670511	+	0.741900i	$0.266075\pi$
$734$	0	0
$735$	−26.4897	−0.977087
$736$	0	0
$737$	18.1373	0.668098
$738$	0	0
$739$	−1.45623	−0.0535684	−0.0267842	−	0.999641i	$-0.508527\pi$
$739$	−1.45623	−0.0535684	−0.0267842	+	0.999641i	$0.508527\pi$
$740$	0	0
$741$	1.06302	0.0390509
$742$	0	0
$743$	−24.4150	−0.895698	−0.447849	−	0.894109i	$-0.647810\pi$
$743$	−24.4150	−0.895698	−0.447849	+	0.894109i	$0.647810\pi$
$744$	0	0
$745$	40.2206	1.47357
$746$	0	0
$747$	−39.0025	−1.42703
$748$	0	0
$749$	−60.4120	−2.20741
$750$	0	0
$751$	49.2393	1.79677	0.898383	−	0.439212i	$-0.144742\pi$
$751$	49.2393	1.79677	0.898383	+	0.439212i	$0.144742\pi$
$752$	0	0
$753$	−16.0482	−0.584830
$754$	0	0
$755$	30.2235	1.09995
$756$	0	0
$757$	2.53657	0.0921932	0.0460966	−	0.998937i	$-0.485322\pi$
$757$	2.53657	0.0921932	0.0460966	+	0.998937i	$0.485322\pi$
$758$	0	0
$759$	0.820039	0.0297655
$760$	0	0
$761$	49.7424	1.80316	0.901580	−	0.432612i	$-0.142408\pi$
$761$	49.7424	1.80316	0.901580	+	0.432612i	$0.142408\pi$
$762$	0	0
$763$	43.0656	1.55908
$764$	0	0
$765$	−17.1052	−0.618441
$766$	0	0
$767$	−2.08911	−0.0754335
$768$	0	0
$769$	28.5514	1.02959	0.514794	−	0.857314i	$-0.327868\pi$
$769$	28.5514	1.02959	0.514794	+	0.857314i	$0.327868\pi$
$770$	0	0
$771$	−3.05116	−0.109885
$772$	0	0
$773$	−18.9718	−0.682367	−0.341184	−	0.939997i	$-0.610828\pi$
$773$	−18.9718	−0.682367	−0.341184	+	0.939997i	$0.610828\pi$
$774$	0	0
$775$	3.76303	0.135172
$776$	0	0
$777$	31.6401	1.13508
$778$	0	0
$779$	0	0
$780$	0	0
$781$	23.6519	0.846333
$782$	0	0
$783$	14.9531	0.534381
$784$	0	0
$785$	12.8961	0.460282
$786$	0	0
$787$	−28.6414	−1.02096	−0.510478	−	0.859891i	$-0.670532\pi$
$787$	−28.6414	−1.02096	−0.510478	+	0.859891i	$0.670532\pi$
$788$	0	0
$789$	15.6959	0.558789
$790$	0	0
$791$	36.4078	1.29451
$792$	0	0
$793$	2.92393	0.103832
$794$	0	0
$795$	−15.6578	−0.555324
$796$	0	0
$797$	−0.999611	−0.0354080	−0.0177040	−	0.999843i	$-0.505636\pi$
$797$	−0.999611	−0.0354080	−0.0177040	+	0.999843i	$0.505636\pi$
$798$	0	0
$799$	−20.7381	−0.733662
$800$	0	0
$801$	8.58172	0.303220
$802$	0	0
$803$	26.4035	0.931759
$804$	0	0
$805$	13.3815	0.471636
$806$	0	0
$807$	−12.6021	−0.443616
$808$	0	0
$809$	24.8243	0.872776	0.436388	−	0.899759i	$-0.356257\pi$
$809$	24.8243	0.872776	0.436388	+	0.899759i	$0.356257\pi$
$810$	0	0
$811$	−31.3274	−1.10005	−0.550027	−	0.835147i	$-0.685383\pi$
$811$	−31.3274	−1.10005	−0.550027	+	0.835147i	$0.685383\pi$
$812$	0	0
$813$	−9.42255	−0.330463
$814$	0	0
$815$	−73.6926	−2.58134
$816$	0	0
$817$	−9.05116	−0.316660
$818$	0	0
$819$	−16.6097	−0.580390
$820$	0	0
$821$	−8.52510	−0.297528	−0.148764	−	0.988873i	$-0.547529\pi$
$821$	−8.52510	−0.297528	−0.148764	+	0.988873i	$0.547529\pi$
$822$	0	0
$823$	−17.2704	−0.602009	−0.301005	−	0.953623i	$-0.597322\pi$
$823$	−17.2704	−0.602009	−0.301005	+	0.953623i	$0.597322\pi$
$824$	0	0
$825$	−6.59785	−0.229708
$826$	0	0
$827$	27.7850	0.966179	0.483090	−	0.875571i	$-0.339514\pi$
$827$	27.7850	0.966179	0.483090	+	0.875571i	$0.339514\pi$
$828$	0	0
$829$	−39.2852	−1.36443	−0.682216	−	0.731151i	$-0.738984\pi$
$829$	−39.2852	−1.36443	−0.682216	+	0.731151i	$0.738984\pi$
$830$	0	0
$831$	−12.4165	−0.430725
$832$	0	0
$833$	−24.0761	−0.834186
$834$	0	0
$835$	62.2177	2.15313
$836$	0	0
$837$	−1.80857	−0.0625133
$838$	0	0
$839$	25.7807	0.890050	0.445025	−	0.895518i	$-0.353195\pi$
$839$	25.7807	0.890050	0.445025	+	0.895518i	$0.353195\pi$
$840$	0	0
$841$	−9.10390	−0.313927
$842$	0	0
$843$	5.94260	0.204674
$844$	0	0
$845$	38.1416	1.31211
$846$	0	0
$847$	−37.7381	−1.29670
$848$	0	0
$849$	6.25856	0.214793
$850$	0	0
$851$	−10.3599	−0.355134
$852$	0	0
$853$	36.4576	1.24828	0.624142	−	0.781311i	$-0.285449\pi$
$853$	36.4576	1.24828	0.624142	+	0.781311i	$0.285449\pi$
$854$	0	0
$855$	11.6667	0.398993
$856$	0	0
$857$	5.07568	0.173382	0.0866909	−	0.996235i	$-0.472371\pi$
$857$	5.07568	0.173382	0.0866909	+	0.996235i	$0.472371\pi$
$858$	0	0
$859$	11.4399	0.390323	0.195162	−	0.980771i	$-0.437477\pi$
$859$	11.4399	0.390323	0.195162	+	0.980771i	$0.437477\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	35.2848	1.20111	0.600555	−	0.799584i	$-0.294947\pi$
$863$	35.2848	1.20111	0.600555	+	0.799584i	$0.294947\pi$
$864$	0	0
$865$	47.5261	1.61593
$866$	0	0
$867$	−8.02198	−0.272441
$868$	0	0
$869$	5.58931	0.189604
$870$	0	0
$871$	−16.0072	−0.542383
$872$	0	0
$873$	28.0673	0.949934
$874$	0	0
$875$	−30.4868	−1.03064
$876$	0	0
$877$	37.2235	1.25695	0.628475	−	0.777830i	$-0.283680\pi$
$877$	37.2235	1.25695	0.628475	+	0.777830i	$0.283680\pi$
$878$	0	0
$879$	−2.30584	−0.0777740
$880$	0	0
$881$	−1.38017	−0.0464990	−0.0232495	−	0.999730i	$-0.507401\pi$
$881$	−1.38017	−0.0464990	−0.0232495	+	0.999730i	$0.507401\pi$
$882$	0	0
$883$	37.6883	1.26831	0.634156	−	0.773205i	$-0.281348\pi$
$883$	37.6883	1.26831	0.634156	+	0.773205i	$0.281348\pi$
$884$	0	0
$885$	3.05116	0.102564
$886$	0	0
$887$	37.3360	1.25362	0.626810	−	0.779172i	$-0.284360\pi$
$887$	37.3360	1.25362	0.626810	+	0.779172i	$0.284360\pi$
$888$	0	0
$889$	−4.46050	−0.149601
$890$	0	0
$891$	−9.48676	−0.317818
$892$	0	0
$893$	14.1445	0.473329
$894$	0	0
$895$	−7.48249	−0.250112
$896$	0	0
$897$	−0.723729	−0.0241646
$898$	0	0
$899$	−2.40642	−0.0802586
$900$	0	0
$901$	−14.2311	−0.474107
$902$	0	0
$903$	−18.8200	−0.626292
$904$	0	0
$905$	−78.3111	−2.60315
$906$	0	0
$907$	40.5552	1.34661	0.673307	−	0.739363i	$-0.264873\pi$
$907$	40.5552	1.34661	0.673307	+	0.739363i	$0.264873\pi$
$908$	0	0
$909$	33.1535	1.09963
$910$	0	0
$911$	47.4661	1.57262	0.786311	−	0.617830i	$-0.211988\pi$
$911$	47.4661	1.57262	0.786311	+	0.617830i	$0.211988\pi$
$912$	0	0
$913$	−23.4749	−0.776906
$914$	0	0
$915$	−4.27042	−0.141176
$916$	0	0
$917$	7.14027	0.235793
$918$	0	0
$919$	26.4504	0.872518	0.436259	−	0.899821i	$-0.356303\pi$
$919$	26.4504	0.872518	0.436259	+	0.899821i	$0.356303\pi$
$920$	0	0
$921$	−3.29067	−0.108431
$922$	0	0
$923$	−20.8741	−0.687080
$924$	0	0
$925$	83.3537	2.74065
$926$	0	0
$927$	11.7249	0.385097
$928$	0	0
$929$	−21.2062	−0.695753	−0.347877	−	0.937540i	$-0.613097\pi$
$929$	−21.2062	−0.695753	−0.347877	+	0.937540i	$0.613097\pi$
$930$	0	0
$931$	16.4212	0.538183
$932$	0	0
$933$	−6.50408	−0.212934
$934$	0	0
$935$	−10.2953	−0.336693
$936$	0	0
$937$	28.5539	0.932815	0.466407	−	0.884570i	$-0.345548\pi$
$937$	28.5539	0.932815	0.466407	+	0.884570i	$0.345548\pi$
$938$	0	0
$939$	−5.71788	−0.186596
$940$	0	0
$941$	48.0263	1.56561	0.782806	−	0.622266i	$-0.213788\pi$
$941$	48.0263	1.56561	0.782806	+	0.622266i	$0.213788\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	51.7453	1.68328
$946$	0	0
$947$	56.8142	1.84621	0.923107	−	0.384544i	$-0.125641\pi$
$947$	56.8142	1.84621	0.923107	+	0.384544i	$0.125641\pi$
$948$	0	0
$949$	−23.3025	−0.756432
$950$	0	0
$951$	−3.66379	−0.118807
$952$	0	0
$953$	−33.5700	−1.08744	−0.543720	−	0.839267i	$-0.682985\pi$
$953$	−33.5700	−1.08744	−0.543720	+	0.839267i	$0.682985\pi$
$954$	0	0
$955$	−17.6008	−0.569548
$956$	0	0
$957$	4.21926	0.136389
$958$	0	0
$959$	33.3757	1.07776
$960$	0	0
$961$	−30.7089	−0.990611
$962$	0	0
$963$	35.8593	1.15555
$964$	0	0
$965$	−92.8768	−2.98981
$966$	0	0
$967$	1.14786	0.0369126	0.0184563	−	0.999830i	$-0.494125\pi$
$967$	1.14786	0.0369126	0.0184563	+	0.999830i	$0.494125\pi$
$968$	0	0
$969$	−1.41108	−0.0453305
$970$	0	0
$971$	−3.20060	−0.102712	−0.0513560	−	0.998680i	$-0.516354\pi$
$971$	−3.20060	−0.102712	−0.0513560	+	0.998680i	$0.516354\pi$
$972$	0	0
$973$	−25.9282	−0.831220
$974$	0	0
$975$	5.82296	0.186484
$976$	0	0
$977$	−33.1652	−1.06105	−0.530524	−	0.847670i	$-0.678005\pi$
$977$	−33.1652	−1.06105	−0.530524	+	0.847670i	$0.678005\pi$
$978$	0	0
$979$	5.16518	0.165080
$980$	0	0
$981$	−25.5628	−0.816158
$982$	0	0
$983$	35.3346	1.12700	0.563500	−	0.826116i	$-0.309455\pi$
$983$	35.3346	1.12700	0.563500	+	0.826116i	$0.309455\pi$
$984$	0	0
$985$	26.6270	0.848408
$986$	0	0
$987$	29.4107	0.936153
$988$	0	0
$989$	6.16225	0.195948
$990$	0	0
$991$	−30.8037	−0.978511	−0.489256	−	0.872140i	$-0.662731\pi$
$991$	−30.8037	−0.978511	−0.489256	+	0.872140i	$0.662731\pi$
$992$	0	0
$993$	−10.4630	−0.332035
$994$	0	0
$995$	−31.8932	−1.01108
$996$	0	0
$997$	−44.5041	−1.40946	−0.704729	−	0.709476i	$-0.748931\pi$
$997$	−44.5041	−1.40946	−0.704729	+	0.709476i	$0.748931\pi$
$998$	0	0
$999$	−40.0611	−1.26748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8128.2.a.x.1.2		3
4.3	odd	2		8128.2.a.ba.1.2		3
8.3	odd	2		2032.2.a.i.1.2		3
8.5	even	2		508.2.a.d.1.2	✓	3
24.5	odd	2		4572.2.a.n.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
508.2.a.d.1.2	✓	3	8.5	even	2
2032.2.a.i.1.2		3	8.3	odd	2
4572.2.a.n.1.1		3	24.5	odd	2
8128.2.a.x.1.2		3	1.1	even	1	trivial
8128.2.a.ba.1.2		3	4.3	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:	\( N \)	\(=\)	\( 8128 = 2^{6} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8128.a (trivial)

\(f(q)\)	\(=\)	\(q+0.593579 q^{3} -3.46050 q^{5} +4.46050 q^{7} -2.64766 q^{9} +O(q^{10})\)	\(q+0.593579 q^{3} -3.46050 q^{5} +4.46050 q^{7} -2.64766 q^{9} -1.59358 q^{11} +1.40642 q^{13} -2.05408 q^{15} -1.86693 q^{17} +1.27335 q^{19} +2.64766 q^{21} -0.866926 q^{23} +6.97509 q^{25} -3.35234 q^{27} -4.46050 q^{29} +0.539495 q^{31} -0.945916 q^{33} -15.4356 q^{35} +11.9502 q^{37} +0.834822 q^{39} -7.10817 q^{43} +9.16225 q^{45} +11.1082 q^{47} +12.8961 q^{49} -1.10817 q^{51} +7.62276 q^{53} +5.51459 q^{55} +0.755832 q^{57} -1.48541 q^{59} +2.07899 q^{61} -11.8099 q^{63} -4.86693 q^{65} -11.3815 q^{67} -0.514589 q^{69} -14.8420 q^{71} -16.5687 q^{73} +4.14027 q^{75} -7.10817 q^{77} -3.50739 q^{79} +5.95311 q^{81} +14.7309 q^{83} +6.46050 q^{85} -2.64766 q^{87} -3.24124 q^{89} +6.27335 q^{91} +0.320233 q^{93} -4.40642 q^{95} -10.6008 q^{97} +4.21926 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 4 q^{5} + 7 q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q - q^3 - 4 * q^5 + 7 * q^7 + 4 * q^9	\( 3 q - q^{3} - 4 q^{5} + 7 q^{7} + 4 q^{9} - 2 q^{11} + 7 q^{13} + 3 q^{15} - 2 q^{17} + 3 q^{19} - 4 q^{21} + q^{23} - q^{25} - 22 q^{27} - 7 q^{29} + 8 q^{31} - 12 q^{33} - 18 q^{35} - 8 q^{37} - 15 q^{39} - 3 q^{43} + 15 q^{47} + 4 q^{49} + 15 q^{51} - 11 q^{53} + q^{55} - 28 q^{57} - 20 q^{59} + 19 q^{61} + 4 q^{63} - 11 q^{65} - 15 q^{67} + 14 q^{69} - 19 q^{71} - 25 q^{73} + 8 q^{75} - 3 q^{77} - 3 q^{79} + 19 q^{81} - 8 q^{83} + 13 q^{85} + 4 q^{87} + 5 q^{89} + 18 q^{91} - q^{93} - 16 q^{95} - 21 q^{97} + 21 q^{99}+O(q^{100}) \) 3 * q - q^3 - 4 * q^5 + 7 * q^7 + 4 * q^9 - 2 * q^11 + 7 * q^13 + 3 * q^15 - 2 * q^17 + 3 * q^19 - 4 * q^21 + q^23 - q^25 - 22 * q^27 - 7 * q^29 + 8 * q^31 - 12 * q^33 - 18 * q^35 - 8 * q^37 - 15 * q^39 - 3 * q^43 + 15 * q^47 + 4 * q^49 + 15 * q^51 - 11 * q^53 + q^55 - 28 * q^57 - 20 * q^59 + 19 * q^61 + 4 * q^63 - 11 * q^65 - 15 * q^67 + 14 * q^69 - 19 * q^71 - 25 * q^73 + 8 * q^75 - 3 * q^77 - 3 * q^79 + 19 * q^81 - 8 * q^83 + 13 * q^85 + 4 * q^87 + 5 * q^89 + 18 * q^91 - q^93 - 16 * q^95 - 21 * q^97 + 21 * q^99

Introduction

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