LMFDB - Embedded newform 8128.2.a.x.1.3

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			1.3
Root			$-1.69963$ 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)

$f(q)$	$=$	$q+1.58836 q^{3} +0.699628 q^{5} +0.300372 q^{7} -0.477100 q^{9} +O(q^{10})$	$q+1.58836 q^{3} +0.699628 q^{5} +0.300372 q^{7} -0.477100 q^{9} -2.58836 q^{11} +0.411636 q^{13} +1.11126 q^{15} +3.28799 q^{17} -4.87636 q^{19} +0.477100 q^{21} +4.28799 q^{23} -4.51052 q^{25} -5.52290 q^{27} -0.300372 q^{29} +4.69963 q^{31} -4.11126 q^{33} +0.210149 q^{35} -11.0210 q^{37} +0.653828 q^{39} -0.777472 q^{43} -0.333792 q^{45} +4.77747 q^{47} -6.90978 q^{49} +5.22253 q^{51} -6.03342 q^{53} -1.81089 q^{55} -7.74543 q^{57} -8.81089 q^{59} +10.3993 q^{61} -0.143307 q^{63} +0.287992 q^{65} +1.09888 q^{67} +6.81089 q^{69} +1.79851 q^{71} -6.07784 q^{73} -7.16435 q^{75} -0.777472 q^{77} -12.6414 q^{79} -7.34108 q^{81} -5.25595 q^{83} +2.30037 q^{85} -0.477100 q^{87} -2.06546 q^{89} +0.123644 q^{91} +7.46472 q^{93} -3.41164 q^{95} +4.86398 q^{97} +1.23491 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.58836	0.917042	0.458521	−	0.888683i	$-0.348379\pi$
$3$	1.58836	0.917042	0.458521	+	0.888683i	$0.348379\pi$
$4$	0	0
$5$	0.699628	0.312883	0.156442	−	0.987687i	$-0.449998\pi$
$5$	0.699628	0.312883	0.156442	+	0.987687i	$0.449998\pi$
$6$	0	0
$7$	0.300372	0.113530	0.0567649	−	0.998388i	$-0.481921\pi$
$7$	0.300372	0.113530	0.0567649	+	0.998388i	$0.481921\pi$
$8$	0	0
$9$	−0.477100	−0.159033
$10$	0	0
$11$	−2.58836	−0.780421	−0.390211	−	0.920726i	$-0.627598\pi$
$11$	−2.58836	−0.780421	−0.390211	+	0.920726i	$0.627598\pi$
$12$	0	0
$13$	0.411636	0.114167	0.0570836	−	0.998369i	$-0.481820\pi$
$13$	0.411636	0.114167	0.0570836	+	0.998369i	$0.481820\pi$
$14$	0	0
$15$	1.11126	0.286927
$16$	0	0
$17$	3.28799	0.797455	0.398728	−	0.917069i	$-0.369452\pi$
$17$	3.28799	0.797455	0.398728	+	0.917069i	$0.369452\pi$
$18$	0	0
$19$	−4.87636	−1.11871	−0.559356	−	0.828927i	$-0.688952\pi$
$19$	−4.87636	−1.11871	−0.559356	+	0.828927i	$0.688952\pi$
$20$	0	0
$21$	0.477100	0.104112
$22$	0	0
$23$	4.28799	0.894108	0.447054	−	0.894507i	$-0.352473\pi$
$23$	4.28799	0.894108	0.447054	+	0.894507i	$0.352473\pi$
$24$	0	0
$25$	−4.51052	−0.902104
$26$	0	0
$27$	−5.52290	−1.06288
$28$	0	0
$29$	−0.300372	−0.0557777	−0.0278888	−	0.999611i	$-0.508878\pi$
$29$	−0.300372	−0.0557777	−0.0278888	+	0.999611i	$0.508878\pi$
$30$	0	0
$31$	4.69963	0.844078	0.422039	−	0.906578i	$-0.361315\pi$
$31$	4.69963	0.844078	0.422039	+	0.906578i	$0.361315\pi$
$32$	0	0
$33$	−4.11126	−0.715679
$34$	0	0
$35$	0.210149	0.0355216
$36$	0	0
$37$	−11.0210	−1.81185	−0.905924	−	0.423440i	$-0.860822\pi$
$37$	−11.0210	−1.81185	−0.905924	+	0.423440i	$0.860822\pi$
$38$	0	0
$39$	0.653828	0.104696
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−0.777472	−0.118563	−0.0592817	−	0.998241i	$-0.518881\pi$
$43$	−0.777472	−0.118563	−0.0592817	+	0.998241i	$0.518881\pi$
$44$	0	0
$45$	−0.333792	−0.0497588
$46$	0	0
$47$	4.77747	0.696866	0.348433	−	0.937334i	$-0.386714\pi$
$47$	4.77747	0.696866	0.348433	+	0.937334i	$0.386714\pi$
$48$	0	0
$49$	−6.90978	−0.987111
$50$	0	0
$51$	5.22253	0.731300
$52$	0	0
$53$	−6.03342	−0.828754	−0.414377	−	0.910105i	$-0.636001\pi$
$53$	−6.03342	−0.828754	−0.414377	+	0.910105i	$0.636001\pi$
$54$	0	0
$55$	−1.81089	−0.244181
$56$	0	0
$57$	−7.74543	−1.02591
$58$	0	0
$59$	−8.81089	−1.14708	−0.573540	−	0.819177i	$-0.694430\pi$
$59$	−8.81089	−1.14708	−0.573540	+	0.819177i	$0.694430\pi$
$60$	0	0
$61$	10.3993	1.33149	0.665744	−	0.746180i	$-0.268114\pi$
$61$	10.3993	1.33149	0.665744	+	0.746180i	$0.268114\pi$
$62$	0	0
$63$	−0.143307	−0.0180550
$64$	0	0
$65$	0.287992	0.0357210
$66$	0	0
$67$	1.09888	0.134250	0.0671251	−	0.997745i	$-0.478617\pi$
$67$	1.09888	0.134250	0.0671251	+	0.997745i	$0.478617\pi$
$68$	0	0
$69$	6.81089	0.819935
$70$	0	0
$71$	1.79851	0.213444	0.106722	−	0.994289i	$-0.465964\pi$
$71$	1.79851	0.213444	0.106722	+	0.994289i	$0.465964\pi$
$72$	0	0
$73$	−6.07784	−0.711358	−0.355679	−	0.934608i	$-0.615750\pi$
$73$	−6.07784	−0.711358	−0.355679	+	0.934608i	$0.615750\pi$
$74$	0	0
$75$	−7.16435	−0.827268
$76$	0	0
$77$	−0.777472	−0.0886011
$78$	0	0
$79$	−12.6414	−1.42227	−0.711137	−	0.703053i	$-0.751819\pi$
$79$	−12.6414	−1.42227	−0.711137	+	0.703053i	$0.751819\pi$
$80$	0	0
$81$	−7.34108	−0.815675
$82$	0	0
$83$	−5.25595	−0.576915	−0.288458	−	0.957493i	$-0.593142\pi$
$83$	−5.25595	−0.576915	−0.288458	+	0.957493i	$0.593142\pi$
$84$	0	0
$85$	2.30037	0.249510
$86$	0	0
$87$	−0.477100	−0.0511505
$88$	0	0
$89$	−2.06546	−0.218939	−0.109469	−	0.993990i	$-0.534915\pi$
$89$	−2.06546	−0.218939	−0.109469	+	0.993990i	$0.534915\pi$
$90$	0	0
$91$	0.123644	0.0129614
$92$	0	0
$93$	7.46472	0.774055
$94$	0	0
$95$	−3.41164	−0.350026
$96$	0	0
$97$	4.86398	0.493862	0.246931	−	0.969033i	$-0.420578\pi$
$97$	4.86398	0.493862	0.246931	+	0.969033i	$0.420578\pi$
$98$	0	0
$99$	1.23491	0.124113
$100$	0	0
$101$	11.2632	1.12073	0.560367	−	0.828245i	$-0.310660\pi$
$101$	11.2632	1.12073	0.560367	+	0.828245i	$0.310660\pi$
$102$	0	0
$103$	−5.24219	−0.516529	−0.258264	−	0.966074i	$-0.583151\pi$
$103$	−5.24219	−0.516529	−0.258264	+	0.966074i	$0.583151\pi$
$104$	0	0
$105$	0.333792	0.0325748
$106$	0	0
$107$	8.43268	0.815218	0.407609	−	0.913157i	$-0.366363\pi$
$107$	8.43268	0.815218	0.407609	+	0.913157i	$0.366363\pi$
$108$	0	0
$109$	−8.97524	−0.859672	−0.429836	−	0.902907i	$-0.641429\pi$
$109$	−8.97524	−0.859672	−0.429836	+	0.902907i	$0.641429\pi$
$110$	0	0
$111$	−17.5054	−1.66154
$112$	0	0
$113$	−1.33379	−0.125473	−0.0627363	−	0.998030i	$-0.519983\pi$
$113$	−1.33379	−0.125473	−0.0627363	+	0.998030i	$0.519983\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	−0.196391	−0.0181564
$118$	0	0
$119$	0.987620	0.0905350
$120$	0	0
$121$	−4.30037	−0.390943
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.65383	−0.595136
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	0	0
$129$	−1.23491	−0.108728
$130$	0	0
$131$	−13.8640	−1.21130	−0.605651	−	0.795731i	$-0.707087\pi$
$131$	−13.8640	−1.21130	−0.605651	+	0.795731i	$0.707087\pi$
$132$	0	0
$133$	−1.46472	−0.127007
$134$	0	0
$135$	−3.86398	−0.332558
$136$	0	0
$137$	5.13093	0.438365	0.219182	−	0.975684i	$-0.429661\pi$
$137$	5.13093	0.438365	0.219182	+	0.975684i	$0.429661\pi$
$138$	0	0
$139$	−3.82327	−0.324286	−0.162143	−	0.986767i	$-0.551841\pi$
$139$	−3.82327	−0.324286	−0.162143	+	0.986767i	$0.551841\pi$
$140$	0	0
$141$	7.58836	0.639056
$142$	0	0
$143$	−1.06546	−0.0890986
$144$	0	0
$145$	−0.210149	−0.0174519
$146$	0	0
$147$	−10.9752	−0.905223
$148$	0	0
$149$	2.03342	0.166584	0.0832922	−	0.996525i	$-0.473457\pi$
$149$	2.03342	0.166584	0.0832922	+	0.996525i	$0.473457\pi$
$150$	0	0
$151$	1.57598	0.128252	0.0641259	−	0.997942i	$-0.479574\pi$
$151$	1.57598	0.128252	0.0641259	+	0.997942i	$0.479574\pi$
$152$	0	0
$153$	−1.56870	−0.126822
$154$	0	0
$155$	3.28799	0.264098
$156$	0	0
$157$	−9.87636	−0.788219	−0.394110	−	0.919063i	$-0.628947\pi$
$157$	−9.87636	−0.788219	−0.394110	+	0.919063i	$0.628947\pi$
$158$	0	0
$159$	−9.58327	−0.760003
$160$	0	0
$161$	1.28799	0.101508
$162$	0	0
$163$	16.9542	1.32796	0.663978	−	0.747752i	$-0.268867\pi$
$163$	16.9542	1.32796	0.663978	+	0.747752i	$0.268867\pi$
$164$	0	0
$165$	−2.87636	−0.223924
$166$	0	0
$167$	19.6428	1.52001	0.760004	−	0.649919i	$-0.225197\pi$
$167$	19.6428	1.52001	0.760004	+	0.649919i	$0.225197\pi$
$168$	0	0
$169$	−12.8306	−0.986966
$170$	0	0
$171$	2.32651	0.177913
$172$	0	0
$173$	−3.42402	−0.260323	−0.130162	−	0.991493i	$-0.541550\pi$
$173$	−3.42402	−0.260323	−0.130162	+	0.991493i	$0.541550\pi$
$174$	0	0
$175$	−1.35483	−0.102416
$176$	0	0
$177$	−13.9949	−1.05192
$178$	0	0
$179$	−7.33379	−0.548153	−0.274077	−	0.961708i	$-0.588372\pi$
$179$	−7.33379	−0.548153	−0.274077	+	0.961708i	$0.588372\pi$
$180$	0	0
$181$	−7.48576	−0.556412	−0.278206	−	0.960521i	$-0.589740\pi$
$181$	−7.48576	−0.556412	−0.278206	+	0.960521i	$0.589740\pi$
$182$	0	0
$183$	16.5178	1.22103
$184$	0	0
$185$	−7.71063	−0.566897
$186$	0	0
$187$	−8.51052	−0.622351
$188$	0	0
$189$	−1.65892	−0.120669
$190$	0	0
$191$	−3.05308	−0.220913	−0.110457	−	0.993881i	$-0.535231\pi$
$191$	−3.05308	−0.220913	−0.110457	+	0.993881i	$0.535231\pi$
$192$	0	0
$193$	0.521523	0.0375400	0.0187700	−	0.999824i	$-0.494025\pi$
$193$	0.521523	0.0375400	0.0187700	+	0.999824i	$0.494025\pi$
$194$	0	0
$195$	0.457436	0.0327577
$196$	0	0
$197$	−18.8182	−1.34074	−0.670370	−	0.742027i	$-0.733864\pi$
$197$	−18.8182	−1.34074	−0.670370	+	0.742027i	$0.733864\pi$
$198$	0	0
$199$	−3.44506	−0.244214	−0.122107	−	0.992517i	$-0.538965\pi$
$199$	−3.44506	−0.244214	−0.122107	+	0.992517i	$0.538965\pi$
$200$	0	0
$201$	1.74543	0.123113
$202$	0	0
$203$	−0.0902232	−0.00633243
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.04580	−0.142193
$208$	0	0
$209$	12.6218	0.873067
$210$	0	0
$211$	8.65383	0.595754	0.297877	−	0.954604i	$-0.403721\pi$
$211$	8.65383	0.595754	0.297877	+	0.954604i	$0.403721\pi$
$212$	0	0
$213$	2.85669	0.195737
$214$	0	0
$215$	−0.543941	−0.0370965
$216$	0	0
$217$	1.41164	0.0958281
$218$	0	0
$219$	−9.65383	−0.652345
$220$	0	0
$221$	1.35346	0.0910433
$222$	0	0
$223$	−13.5105	−0.904731	−0.452366	−	0.891833i	$-0.649420\pi$
$223$	−13.5105	−0.904731	−0.452366	+	0.891833i	$0.649420\pi$
$224$	0	0
$225$	2.15197	0.143465
$226$	0	0
$227$	−17.9418	−1.19084	−0.595420	−	0.803414i	$-0.703014\pi$
$227$	−17.9418	−1.19084	−0.595420	+	0.803414i	$0.703014\pi$
$228$	0	0
$229$	−18.4858	−1.22157	−0.610787	−	0.791795i	$-0.709147\pi$
$229$	−18.4858	−1.22157	−0.610787	+	0.791795i	$0.709147\pi$
$230$	0	0
$231$	−1.23491	−0.0812510
$232$	0	0
$233$	−12.2088	−0.799823	−0.399912	−	0.916554i	$-0.630959\pi$
$233$	−12.2088	−0.799823	−0.399912	+	0.916554i	$0.630959\pi$
$234$	0	0
$235$	3.34245	0.218038
$236$	0	0
$237$	−20.0792	−1.30429
$238$	0	0
$239$	4.60940	0.298158	0.149079	−	0.988825i	$-0.452369\pi$
$239$	4.60940	0.298158	0.149079	+	0.988825i	$0.452369\pi$
$240$	0	0
$241$	7.36721	0.474564	0.237282	−	0.971441i	$-0.423743\pi$
$241$	7.36721	0.474564	0.237282	+	0.971441i	$0.423743\pi$
$242$	0	0
$243$	4.90840	0.314874
$244$	0	0
$245$	−4.83427	−0.308850
$246$	0	0
$247$	−2.00728	−0.127720
$248$	0	0
$249$	−8.34836	−0.529056
$250$	0	0
$251$	4.07413	0.257156	0.128578	−	0.991699i	$-0.458959\pi$
$251$	4.07413	0.257156	0.128578	+	0.991699i	$0.458959\pi$
$252$	0	0
$253$	−11.0989	−0.697781
$254$	0	0
$255$	3.65383	0.228812
$256$	0	0
$257$	6.16435	0.384521	0.192261	−	0.981344i	$-0.438418\pi$
$257$	6.16435	0.384521	0.192261	+	0.981344i	$0.438418\pi$
$258$	0	0
$259$	−3.31041	−0.205699
$260$	0	0
$261$	0.143307	0.00887050
$262$	0	0
$263$	−5.66249	−0.349164	−0.174582	−	0.984643i	$-0.555857\pi$
$263$	−5.66249	−0.349164	−0.174582	+	0.984643i	$0.555857\pi$
$264$	0	0
$265$	−4.22115	−0.259303
$266$	0	0
$267$	−3.28071	−0.200776
$268$	0	0
$269$	24.3497	1.48463	0.742315	−	0.670051i	$-0.233728\pi$
$269$	24.3497	1.48463	0.742315	+	0.670051i	$0.233728\pi$
$270$	0	0
$271$	5.74033	0.348700	0.174350	−	0.984684i	$-0.444218\pi$
$271$	5.74033	0.348700	0.174350	+	0.984684i	$0.444218\pi$
$272$	0	0
$273$	0.196391	0.0118862
$274$	0	0
$275$	11.6749	0.704021
$276$	0	0
$277$	−2.92078	−0.175493	−0.0877463	−	0.996143i	$-0.527966\pi$
$277$	−2.92078	−0.175493	−0.0877463	+	0.996143i	$0.527966\pi$
$278$	0	0
$279$	−2.24219	−0.134237
$280$	0	0
$281$	−32.5846	−1.94384	−0.971918	−	0.235318i	$-0.924387\pi$
$281$	−32.5846	−1.94384	−0.971918	+	0.235318i	$0.924387\pi$
$282$	0	0
$283$	−11.4327	−0.679602	−0.339801	−	0.940497i	$-0.610360\pi$
$283$	−11.4327	−0.679602	−0.339801	+	0.940497i	$0.610360\pi$
$284$	0	0
$285$	−5.41892	−0.320989
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−6.18911	−0.364065
$290$	0	0
$291$	7.72576	0.452892
$292$	0	0
$293$	−26.6749	−1.55836	−0.779181	−	0.626799i	$-0.784365\pi$
$293$	−26.6749	−1.55836	−0.779181	+	0.626799i	$0.784365\pi$
$294$	0	0
$295$	−6.16435	−0.358902
$296$	0	0
$297$	14.2953	0.829496
$298$	0	0
$299$	1.76509	0.102078
$300$	0	0
$301$	−0.233531	−0.0134605
$302$	0	0
$303$	17.8901	1.02776
$304$	0	0
$305$	7.27561	0.416600
$306$	0	0
$307$	16.4327	0.937862	0.468931	−	0.883235i	$-0.344639\pi$
$307$	16.4327	0.937862	0.468931	+	0.883235i	$0.344639\pi$
$308$	0	0
$309$	−8.32651	−0.473679
$310$	0	0
$311$	28.4734	1.61458	0.807289	−	0.590157i	$-0.200934\pi$
$311$	28.4734	1.61458	0.807289	+	0.590157i	$0.200934\pi$
$312$	0	0
$313$	10.8058	0.610780	0.305390	−	0.952227i	$-0.401213\pi$
$313$	10.8058	0.610780	0.305390	+	0.952227i	$0.401213\pi$
$314$	0	0
$315$	−0.100262	−0.00564912
$316$	0	0
$317$	10.1062	0.567619	0.283809	−	0.958881i	$-0.408402\pi$
$317$	10.1062	0.567619	0.283809	+	0.958881i	$0.408402\pi$
$318$	0	0
$319$	0.777472	0.0435301
$320$	0	0
$321$	13.3942	0.747589
$322$	0	0
$323$	−16.0334	−0.892123
$324$	0	0
$325$	−1.85669	−0.102991
$326$	0	0
$327$	−14.2559	−0.788356
$328$	0	0
$329$	1.43502	0.0791151
$330$	0	0
$331$	22.1657	1.21834	0.609169	−	0.793040i	$-0.291503\pi$
$331$	22.1657	1.21834	0.609169	+	0.793040i	$0.291503\pi$
$332$	0	0
$333$	5.25814	0.288144
$334$	0	0
$335$	0.768810	0.0420046
$336$	0	0
$337$	−25.4276	−1.38513	−0.692564	−	0.721356i	$-0.743519\pi$
$337$	−25.4276	−1.38513	−0.692564	+	0.721356i	$0.743519\pi$
$338$	0	0
$339$	−2.11855	−0.115064
$340$	0	0
$341$	−12.1643	−0.658736
$342$	0	0
$343$	−4.17811	−0.225596
$344$	0	0
$345$	4.76509	0.256544
$346$	0	0
$347$	−21.7403	−1.16708	−0.583541	−	0.812084i	$-0.698333\pi$
$347$	−21.7403	−1.16708	−0.583541	+	0.812084i	$0.698333\pi$
$348$	0	0
$349$	3.68725	0.197374	0.0986869	−	0.995119i	$-0.468536\pi$
$349$	3.68725	0.197374	0.0986869	+	0.995119i	$0.468536\pi$
$350$	0	0
$351$	−2.27342	−0.121346
$352$	0	0
$353$	23.7848	1.26593	0.632967	−	0.774178i	$-0.281837\pi$
$353$	23.7848	1.26593	0.632967	+	0.774178i	$0.281837\pi$
$354$	0	0
$355$	1.25829	0.0667831
$356$	0	0
$357$	1.56870	0.0830244
$358$	0	0
$359$	−2.24219	−0.118338	−0.0591692	−	0.998248i	$-0.518845\pi$
$359$	−2.24219	−0.118338	−0.0591692	+	0.998248i	$0.518845\pi$
$360$	0	0
$361$	4.77885	0.251518
$362$	0	0
$363$	−6.83056	−0.358511
$364$	0	0
$365$	−4.25223	−0.222572
$366$	0	0
$367$	23.5302	1.22827	0.614133	−	0.789203i	$-0.289506\pi$
$367$	23.5302	1.22827	0.614133	+	0.789203i	$0.289506\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.81227	−0.0940884
$372$	0	0
$373$	−0.346172	−0.0179241	−0.00896206	−	0.999960i	$-0.502853\pi$
$373$	−0.346172	−0.0179241	−0.00896206	+	0.999960i	$0.502853\pi$
$374$	0	0
$375$	−10.5687	−0.545765
$376$	0	0
$377$	−0.123644	−0.00636798
$378$	0	0
$379$	11.8182	0.607059	0.303529	−	0.952822i	$-0.401835\pi$
$379$	11.8182	0.607059	0.303529	+	0.952822i	$0.401835\pi$
$380$	0	0
$381$	−1.58836	−0.0813744
$382$	0	0
$383$	−10.0334	−0.512684	−0.256342	−	0.966586i	$-0.582517\pi$
$383$	−10.0334	−0.512684	−0.256342	+	0.966586i	$0.582517\pi$
$384$	0	0
$385$	−0.543941	−0.0277218
$386$	0	0
$387$	0.370932	0.0188555
$388$	0	0
$389$	−22.9418	−1.16320	−0.581598	−	0.813476i	$-0.697572\pi$
$389$	−22.9418	−1.16320	−0.581598	+	0.813476i	$0.697572\pi$
$390$	0	0
$391$	14.0989	0.713011
$392$	0	0
$393$	−22.0210	−1.11081
$394$	0	0
$395$	−8.84431	−0.445006
$396$	0	0
$397$	12.5192	0.628320	0.314160	−	0.949370i	$-0.398277\pi$
$397$	12.5192	0.628320	0.314160	+	0.949370i	$0.398277\pi$
$398$	0	0
$399$	−2.32651	−0.116471
$400$	0	0
$401$	−37.9876	−1.89701	−0.948506	−	0.316760i	$-0.897405\pi$
$401$	−37.9876	−1.89701	−0.948506	+	0.316760i	$0.897405\pi$
$402$	0	0
$403$	1.93454	0.0963661
$404$	0	0
$405$	−5.13602	−0.255211
$406$	0	0
$407$	28.5265	1.41400
$408$	0	0
$409$	1.84431	0.0911954	0.0455977	−	0.998960i	$-0.485481\pi$
$409$	1.84431	0.0911954	0.0455977	+	0.998960i	$0.485481\pi$
$410$	0	0
$411$	8.14978	0.401999
$412$	0	0
$413$	−2.64654	−0.130228
$414$	0	0
$415$	−3.67721	−0.180507
$416$	0	0
$417$	−6.07275	−0.297384
$418$	0	0
$419$	21.1309	1.03231	0.516157	−	0.856494i	$-0.327362\pi$
$419$	21.1309	1.03231	0.516157	+	0.856494i	$0.327362\pi$
$420$	0	0
$421$	23.0655	1.12414	0.562071	−	0.827089i	$-0.310005\pi$
$421$	23.0655	1.12414	0.562071	+	0.827089i	$0.310005\pi$
$422$	0	0
$423$	−2.27933	−0.110825
$424$	0	0
$425$	−14.8306	−0.719388
$426$	0	0
$427$	3.12364	0.151164
$428$	0	0
$429$	−1.69234	−0.0817072
$430$	0	0
$431$	1.30175	0.0627031	0.0313515	−	0.999508i	$-0.490019\pi$
$431$	1.30175	0.0627031	0.0313515	+	0.999508i	$0.490019\pi$
$432$	0	0
$433$	−8.83922	−0.424786	−0.212393	−	0.977184i	$-0.568126\pi$
$433$	−8.83922	−0.424786	−0.212393	+	0.977184i	$0.568126\pi$
$434$	0	0
$435$	−0.333792	−0.0160041
$436$	0	0
$437$	−20.9098	−1.00025
$438$	0	0
$439$	−29.2298	−1.39506	−0.697531	−	0.716554i	$-0.745718\pi$
$439$	−29.2298	−1.39506	−0.697531	+	0.716554i	$0.745718\pi$
$440$	0	0
$441$	3.29665	0.156983
$442$	0	0
$443$	32.9061	1.56341	0.781707	−	0.623646i	$-0.214349\pi$
$443$	32.9061	1.56341	0.781707	+	0.623646i	$0.214349\pi$
$444$	0	0
$445$	−1.44506	−0.0685023
$446$	0	0
$447$	3.22981	0.152765
$448$	0	0
$449$	23.3955	1.10410	0.552052	−	0.833810i	$-0.313845\pi$
$449$	23.3955	1.10410	0.552052	+	0.833810i	$0.313845\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.50324	0.117612
$454$	0	0
$455$	0.0865047	0.00405540
$456$	0	0
$457$	6.58836	0.308191	0.154095	−	0.988056i	$-0.450754\pi$
$457$	6.58836	0.308191	0.154095	+	0.988056i	$0.450754\pi$
$458$	0	0
$459$	−18.1593	−0.847601
$460$	0	0
$461$	14.1447	0.658784	0.329392	−	0.944193i	$-0.393156\pi$
$461$	14.1447	0.658784	0.329392	+	0.944193i	$0.393156\pi$
$462$	0	0
$463$	−22.9505	−1.06660	−0.533300	−	0.845926i	$-0.679048\pi$
$463$	−22.9505	−1.06660	−0.533300	+	0.845926i	$0.679048\pi$
$464$	0	0
$465$	5.22253	0.242189
$466$	0	0
$467$	38.9664	1.80315	0.901576	−	0.432622i	$-0.142411\pi$
$467$	38.9664	1.80315	0.901576	+	0.432622i	$0.142411\pi$
$468$	0	0
$469$	0.330074	0.0152414
$470$	0	0
$471$	−15.6872	−0.722830
$472$	0	0
$473$	2.01238	0.0925293
$474$	0	0
$475$	21.9949	1.00920
$476$	0	0
$477$	2.87854	0.131799
$478$	0	0
$479$	−12.9680	−0.592521	−0.296261	−	0.955107i	$-0.595740\pi$
$479$	−12.9680	−0.592521	−0.296261	+	0.955107i	$0.595740\pi$
$480$	0	0
$481$	−4.53666	−0.206854
$482$	0	0
$483$	2.04580	0.0930871
$484$	0	0
$485$	3.40297	0.154521
$486$	0	0
$487$	24.9381	1.13005	0.565027	−	0.825073i	$-0.308866\pi$
$487$	24.9381	1.13005	0.565027	+	0.825073i	$0.308866\pi$
$488$	0	0
$489$	26.9294	1.21779
$490$	0	0
$491$	25.6762	1.15875	0.579376	−	0.815060i	$-0.303296\pi$
$491$	25.6762	1.15875	0.579376	+	0.815060i	$0.303296\pi$
$492$	0	0
$493$	−0.987620	−0.0444802
$494$	0	0
$495$	0.863976	0.0388329
$496$	0	0
$497$	0.540223	0.0242323
$498$	0	0
$499$	−7.87498	−0.352532	−0.176266	−	0.984343i	$-0.556402\pi$
$499$	−7.87498	−0.352532	−0.176266	+	0.984343i	$0.556402\pi$
$500$	0	0
$501$	31.2000	1.39391
$502$	0	0
$503$	−34.7366	−1.54883	−0.774415	−	0.632679i	$-0.781955\pi$
$503$	−34.7366	−1.54883	−0.774415	+	0.632679i	$0.781955\pi$
$504$	0	0
$505$	7.88007	0.350659
$506$	0	0
$507$	−20.3796	−0.905089
$508$	0	0
$509$	−2.24591	−0.0995482	−0.0497741	−	0.998761i	$-0.515850\pi$
$509$	−2.24591	−0.0995482	−0.0497741	+	0.998761i	$0.515850\pi$
$510$	0	0
$511$	−1.82561	−0.0807604
$512$	0	0
$513$	26.9316	1.18906
$514$	0	0
$515$	−3.66758	−0.161613
$516$	0	0
$517$	−12.3658	−0.543849
$518$	0	0
$519$	−5.43858	−0.238727
$520$	0	0
$521$	−38.2646	−1.67640	−0.838202	−	0.545361i	$-0.816393\pi$
$521$	−38.2646	−1.67640	−0.838202	+	0.545361i	$0.816393\pi$
$522$	0	0
$523$	21.4052	0.935982	0.467991	−	0.883733i	$-0.344978\pi$
$523$	21.4052	0.935982	0.467991	+	0.883733i	$0.344978\pi$
$524$	0	0
$525$	−2.15197	−0.0939196
$526$	0	0
$527$	15.4523	0.673115
$528$	0	0
$529$	−4.61312	−0.200571
$530$	0	0
$531$	4.20368	0.182424
$532$	0	0
$533$	0	0
$534$	0	0
$535$	5.89974	0.255068
$536$	0	0
$537$	−11.6487	−0.502680
$538$	0	0
$539$	17.8850	0.770362
$540$	0	0
$541$	17.3731	0.746929	0.373464	−	0.927645i	$-0.378170\pi$
$541$	17.3731	0.746929	0.373464	+	0.927645i	$0.378170\pi$
$542$	0	0
$543$	−11.8901	−0.510254
$544$	0	0
$545$	−6.27933	−0.268977
$546$	0	0
$547$	44.6267	1.90810	0.954051	−	0.299646i	$-0.0968685\pi$
$547$	44.6267	1.90810	0.954051	+	0.299646i	$0.0968685\pi$
$548$	0	0
$549$	−4.96148	−0.211751
$550$	0	0
$551$	1.46472	0.0623992
$552$	0	0
$553$	−3.79714	−0.161471
$554$	0	0
$555$	−12.2473	−0.519868
$556$	0	0
$557$	−26.3360	−1.11589	−0.557946	−	0.829877i	$-0.688410\pi$
$557$	−26.3360	−1.11589	−0.557946	+	0.829877i	$0.688410\pi$
$558$	0	0
$559$	−0.320035	−0.0135361
$560$	0	0
$561$	−13.5178	−0.570722
$562$	0	0
$563$	−4.63279	−0.195249	−0.0976243	−	0.995223i	$-0.531124\pi$
$563$	−4.63279	−0.195249	−0.0976243	+	0.995223i	$0.531124\pi$
$564$	0	0
$565$	−0.933159	−0.0392583
$566$	0	0
$567$	−2.20505	−0.0926035
$568$	0	0
$569$	−46.0741	−1.93153	−0.965764	−	0.259423i	$-0.916468\pi$
$569$	−46.0741	−1.93153	−0.965764	+	0.259423i	$0.916468\pi$
$570$	0	0
$571$	−6.43268	−0.269199	−0.134600	−	0.990900i	$-0.542975\pi$
$571$	−6.43268	−0.269199	−0.134600	+	0.990900i	$0.542975\pi$
$572$	0	0
$573$	−4.84941	−0.202587
$574$	0	0
$575$	−19.3411	−0.806579
$576$	0	0
$577$	30.7244	1.27907	0.639536	−	0.768761i	$-0.279126\pi$
$577$	30.7244	1.27907	0.639536	+	0.768761i	$0.279126\pi$
$578$	0	0
$579$	0.828368	0.0344258
$580$	0	0
$581$	−1.57874	−0.0654971
$582$	0	0
$583$	15.6167	0.646777
$584$	0	0
$585$	−0.137401	−0.00568083
$586$	0	0
$587$	−23.8530	−0.984518	−0.492259	−	0.870449i	$-0.663829\pi$
$587$	−23.8530	−0.984518	−0.492259	+	0.870449i	$0.663829\pi$
$588$	0	0
$589$	−22.9171	−0.944281
$590$	0	0
$591$	−29.8901	−1.22951
$592$	0	0
$593$	−9.72439	−0.399333	−0.199666	−	0.979864i	$-0.563986\pi$
$593$	−9.72439	−0.399333	−0.199666	+	0.979864i	$0.563986\pi$
$594$	0	0
$595$	0.690967	0.0283269
$596$	0	0
$597$	−5.47200	−0.223954
$598$	0	0
$599$	12.6552	0.517078	0.258539	−	0.966001i	$-0.416759\pi$
$599$	12.6552	0.517078	0.258539	+	0.966001i	$0.416759\pi$
$600$	0	0
$601$	20.6538	0.842487	0.421244	−	0.906948i	$-0.361594\pi$
$601$	20.6538	0.842487	0.421244	+	0.906948i	$0.361594\pi$
$602$	0	0
$603$	−0.524278	−0.0213502
$604$	0	0
$605$	−3.00866	−0.122319
$606$	0	0
$607$	22.6080	0.917632	0.458816	−	0.888531i	$-0.348274\pi$
$607$	22.6080	0.917632	0.458816	+	0.888531i	$0.348274\pi$
$608$	0	0
$609$	−0.143307	−0.00580711
$610$	0	0
$611$	1.96658	0.0795593
$612$	0	0
$613$	−36.9825	−1.49371	−0.746855	−	0.664987i	$-0.768437\pi$
$613$	−36.9825	−1.49371	−0.746855	+	0.664987i	$0.768437\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.87498	−0.317035	−0.158517	−	0.987356i	$-0.550671\pi$
$617$	−7.87498	−0.317035	−0.158517	+	0.987356i	$0.550671\pi$
$618$	0	0
$619$	23.2385	0.934033	0.467017	−	0.884249i	$-0.345329\pi$
$619$	23.2385	0.934033	0.467017	+	0.884249i	$0.345329\pi$
$620$	0	0
$621$	−23.6822	−0.950332
$622$	0	0
$623$	−0.620407	−0.0248561
$624$	0	0
$625$	17.8974	0.715896
$626$	0	0
$627$	20.0480	0.800640
$628$	0	0
$629$	−36.2371	−1.44487
$630$	0	0
$631$	17.9752	0.715583	0.357792	−	0.933801i	$-0.383530\pi$
$631$	17.9752	0.715583	0.357792	+	0.933801i	$0.383530\pi$
$632$	0	0
$633$	13.7454	0.546332
$634$	0	0
$635$	−0.699628	−0.0277639
$636$	0	0
$637$	−2.84431	−0.112696
$638$	0	0
$639$	−0.858070	−0.0339447
$640$	0	0
$641$	8.17673	0.322961	0.161481	−	0.986876i	$-0.448373\pi$
$641$	8.17673	0.322961	0.161481	+	0.986876i	$0.448373\pi$
$642$	0	0
$643$	−4.02476	−0.158721	−0.0793605	−	0.996846i	$-0.525288\pi$
$643$	−4.02476	−0.158721	−0.0793605	+	0.996846i	$0.525288\pi$
$644$	0	0
$645$	−0.863976	−0.0340190
$646$	0	0
$647$	40.1135	1.57702	0.788511	−	0.615020i	$-0.210852\pi$
$647$	40.1135	1.57702	0.788511	+	0.615020i	$0.210852\pi$
$648$	0	0
$649$	22.8058	0.895206
$650$	0	0
$651$	2.24219	0.0878784
$652$	0	0
$653$	−6.78613	−0.265562	−0.132781	−	0.991145i	$-0.542391\pi$
$653$	−6.78613	−0.265562	−0.132781	+	0.991145i	$0.542391\pi$
$654$	0	0
$655$	−9.69963	−0.378996
$656$	0	0
$657$	2.89974	0.113130
$658$	0	0
$659$	−22.0938	−0.860652	−0.430326	−	0.902674i	$-0.641601\pi$
$659$	−22.0938	−0.860652	−0.430326	+	0.902674i	$0.641601\pi$
$660$	0	0
$661$	22.8072	0.887096	0.443548	−	0.896251i	$-0.353720\pi$
$661$	22.8072	0.887096	0.443548	+	0.896251i	$0.353720\pi$
$662$	0	0
$663$	2.14978	0.0834906
$664$	0	0
$665$	−1.02476	−0.0397385
$666$	0	0
$667$	−1.28799	−0.0498713
$668$	0	0
$669$	−21.4596	−0.829677
$670$	0	0
$671$	−26.9171	−1.03912
$672$	0	0
$673$	−27.7366	−1.06917	−0.534584	−	0.845115i	$-0.679532\pi$
$673$	−27.7366	−1.06917	−0.534584	+	0.845115i	$0.679532\pi$
$674$	0	0
$675$	24.9112	0.958831
$676$	0	0
$677$	12.0568	0.463381	0.231690	−	0.972790i	$-0.425574\pi$
$677$	12.0568	0.463381	0.231690	+	0.972790i	$0.425574\pi$
$678$	0	0
$679$	1.46100	0.0560681
$680$	0	0
$681$	−28.4981	−1.09205
$682$	0	0
$683$	−33.4386	−1.27949	−0.639746	−	0.768586i	$-0.720960\pi$
$683$	−33.4386	−1.27949	−0.639746	+	0.768586i	$0.720960\pi$
$684$	0	0
$685$	3.58974	0.137157
$686$	0	0
$687$	−29.3621	−1.12023
$688$	0	0
$689$	−2.48357	−0.0946166
$690$	0	0
$691$	−17.9739	−0.683758	−0.341879	−	0.939744i	$-0.611063\pi$
$691$	−17.9739	−0.683758	−0.341879	+	0.939744i	$0.611063\pi$
$692$	0	0
$693$	0.370932	0.0140905
$694$	0	0
$695$	−2.67487	−0.101464
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−19.3920	−0.733472
$700$	0	0
$701$	−1.46472	−0.0553217	−0.0276609	−	0.999617i	$-0.508806\pi$
$701$	−1.46472	−0.0553217	−0.0276609	+	0.999617i	$0.508806\pi$
$702$	0	0
$703$	53.7425	2.02694
$704$	0	0
$705$	5.30903	0.199950
$706$	0	0
$707$	3.38316	0.127237
$708$	0	0
$709$	−20.7848	−0.780588	−0.390294	−	0.920690i	$-0.627627\pi$
$709$	−20.7848	−0.780588	−0.390294	+	0.920690i	$0.627627\pi$
$710$	0	0
$711$	6.03123	0.226189
$712$	0	0
$713$	20.1520	0.754697
$714$	0	0
$715$	−0.745428	−0.0278774
$716$	0	0
$717$	7.32141	0.273423
$718$	0	0
$719$	28.1520	1.04989	0.524946	−	0.851136i	$-0.324086\pi$
$719$	28.1520	1.04989	0.524946	+	0.851136i	$0.324086\pi$
$720$	0	0
$721$	−1.57461	−0.0586414
$722$	0	0
$723$	11.7018	0.435195
$724$	0	0
$725$	1.35483	0.0503172
$726$	0	0
$727$	37.4065	1.38733	0.693666	−	0.720297i	$-0.255994\pi$
$727$	37.4065	1.38733	0.693666	+	0.720297i	$0.255994\pi$
$728$	0	0
$729$	29.8196	1.10443
$730$	0	0
$731$	−2.55632	−0.0945489
$732$	0	0
$733$	−10.6304	−0.392644	−0.196322	−	0.980539i	$-0.562900\pi$
$733$	−10.6304	−0.392644	−0.196322	+	0.980539i	$0.562900\pi$
$734$	0	0
$735$	−7.67859	−0.283229
$736$	0	0
$737$	−2.84431	−0.104772
$738$	0	0
$739$	−23.4327	−0.861985	−0.430992	−	0.902356i	$-0.641836\pi$
$739$	−23.4327	−0.861985	−0.430992	+	0.902356i	$0.641836\pi$
$740$	0	0
$741$	−3.18830	−0.117125
$742$	0	0
$743$	28.8530	1.05851	0.529256	−	0.848462i	$-0.322471\pi$
$743$	28.8530	1.05851	0.529256	+	0.848462i	$0.322471\pi$
$744$	0	0
$745$	1.42264	0.0521214
$746$	0	0
$747$	2.50761	0.0917487
$748$	0	0
$749$	2.53294	0.0925516
$750$	0	0
$751$	−48.6143	−1.77396	−0.886981	−	0.461805i	$-0.847202\pi$
$751$	−48.6143	−1.77396	−0.886981	+	0.461805i	$0.847202\pi$
$752$	0	0
$753$	6.47119	0.235823
$754$	0	0
$755$	1.10260	0.0401278
$756$	0	0
$757$	−2.98034	−0.108322	−0.0541611	−	0.998532i	$-0.517248\pi$
$757$	−2.98034	−0.108322	−0.0541611	+	0.998532i	$0.517248\pi$
$758$	0	0
$759$	−17.6291	−0.639895
$760$	0	0
$761$	−12.8406	−0.465471	−0.232736	−	0.972540i	$-0.574768\pi$
$761$	−12.8406	−0.465471	−0.232736	+	0.972540i	$0.574768\pi$
$762$	0	0
$763$	−2.69591	−0.0975985
$764$	0	0
$765$	−1.09751	−0.0396805
$766$	0	0
$767$	−3.62688	−0.130959
$768$	0	0
$769$	41.3026	1.48941	0.744704	−	0.667395i	$-0.232591\pi$
$769$	41.3026	1.48941	0.744704	+	0.667395i	$0.232591\pi$
$770$	0	0
$771$	9.79123	0.352622
$772$	0	0
$773$	53.3781	1.91988	0.959938	−	0.280213i	$-0.0904052\pi$
$773$	53.3781	1.91988	0.959938	+	0.280213i	$0.0904052\pi$
$774$	0	0
$775$	−21.1978	−0.761446
$776$	0	0
$777$	−5.25814	−0.188635
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−4.65521	−0.166576
$782$	0	0
$783$	1.65892	0.0592851
$784$	0	0
$785$	−6.90978	−0.246621
$786$	0	0
$787$	44.0704	1.57094	0.785470	−	0.618900i	$-0.212421\pi$
$787$	44.0704	1.57094	0.785470	+	0.618900i	$0.212421\pi$
$788$	0	0
$789$	−8.99409	−0.320198
$790$	0	0
$791$	−0.400634	−0.0142449
$792$	0	0
$793$	4.28071	0.152012
$794$	0	0
$795$	−6.70472	−0.237792
$796$	0	0
$797$	50.1876	1.77774	0.888868	−	0.458164i	$-0.151493\pi$
$797$	50.1876	1.77774	0.888868	+	0.458164i	$0.151493\pi$
$798$	0	0
$799$	15.7083	0.555719
$800$	0	0
$801$	0.985432	0.0348185
$802$	0	0
$803$	15.7317	0.555159
$804$	0	0
$805$	0.901116	0.0317602
$806$	0	0
$807$	38.6762	1.36147
$808$	0	0
$809$	−19.7614	−0.694773	−0.347386	−	0.937722i	$-0.612931\pi$
$809$	−19.7614	−0.694773	−0.347386	+	0.937722i	$0.612931\pi$
$810$	0	0
$811$	−22.0124	−0.772959	−0.386480	−	0.922298i	$-0.626309\pi$
$811$	−22.0124	−0.772959	−0.386480	+	0.922298i	$0.626309\pi$
$812$	0	0
$813$	9.11774	0.319773
$814$	0	0
$815$	11.8616	0.415495
$816$	0	0
$817$	3.79123	0.132638
$818$	0	0
$819$	−0.0589905	−0.00206129
$820$	0	0
$821$	−45.6043	−1.59160	−0.795801	−	0.605559i	$-0.792950\pi$
$821$	−45.6043	−1.59160	−0.795801	+	0.605559i	$0.792950\pi$
$822$	0	0
$823$	−1.44368	−0.0503235	−0.0251617	−	0.999683i	$-0.508010\pi$
$823$	−1.44368	−0.0503235	−0.0251617	+	0.999683i	$0.508010\pi$
$824$	0	0
$825$	18.5439	0.645617
$826$	0	0
$827$	4.63279	0.161098	0.0805489	−	0.996751i	$-0.474333\pi$
$827$	4.63279	0.161098	0.0805489	+	0.996751i	$0.474333\pi$
$828$	0	0
$829$	−41.7266	−1.44922	−0.724612	−	0.689157i	$-0.757981\pi$
$829$	−41.7266	−1.44922	−0.724612	+	0.689157i	$0.757981\pi$
$830$	0	0
$831$	−4.63926	−0.160934
$832$	0	0
$833$	−22.7193	−0.787177
$834$	0	0
$835$	13.7427	0.475585
$836$	0	0
$837$	−25.9556	−0.897156
$838$	0	0
$839$	28.7651	0.993081	0.496541	−	0.868013i	$-0.334603\pi$
$839$	28.7651	0.993081	0.496541	+	0.868013i	$0.334603\pi$
$840$	0	0
$841$	−28.9098	−0.996889
$842$	0	0
$843$	−51.7563	−1.78258
$844$	0	0
$845$	−8.97662	−0.308805
$846$	0	0
$847$	−1.29171	−0.0443837
$848$	0	0
$849$	−18.1593	−0.623224
$850$	0	0
$851$	−47.2581	−1.61999
$852$	0	0
$853$	22.6204	0.774508	0.387254	−	0.921973i	$-0.373424\pi$
$853$	22.6204	0.774508	0.387254	+	0.921973i	$0.373424\pi$
$854$	0	0
$855$	1.62769	0.0556659
$856$	0	0
$857$	−47.4683	−1.62149	−0.810743	−	0.585402i	$-0.800937\pi$
$857$	−47.4683	−1.62149	−0.810743	+	0.585402i	$0.800937\pi$
$858$	0	0
$859$	−30.3425	−1.03527	−0.517636	−	0.855601i	$-0.673188\pi$
$859$	−30.3425	−1.03527	−0.517636	+	0.855601i	$0.673188\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.4610	−0.458218	−0.229109	−	0.973401i	$-0.573581\pi$
$863$	−13.4610	−0.458218	−0.229109	+	0.973401i	$0.573581\pi$
$864$	0	0
$865$	−2.39554	−0.0814507
$866$	0	0
$867$	−9.83056	−0.333863
$868$	0	0
$869$	32.7207	1.10997
$870$	0	0
$871$	0.452340	0.0153270
$872$	0	0
$873$	−2.32060	−0.0785405
$874$	0	0
$875$	−1.99862	−0.0675658
$876$	0	0
$877$	8.10260	0.273605	0.136803	−	0.990598i	$-0.456317\pi$
$877$	8.10260	0.273605	0.136803	+	0.990598i	$0.456317\pi$
$878$	0	0
$879$	−42.3694	−1.42908
$880$	0	0
$881$	−24.7134	−0.832615	−0.416308	−	0.909224i	$-0.636676\pi$
$881$	−24.7134	−0.832615	−0.416308	+	0.909224i	$0.636676\pi$
$882$	0	0
$883$	−21.7293	−0.731250	−0.365625	−	0.930762i	$-0.619145\pi$
$883$	−21.7293	−0.731250	−0.365625	+	0.930762i	$0.619145\pi$
$884$	0	0
$885$	−9.79123	−0.329129
$886$	0	0
$887$	−24.2522	−0.814310	−0.407155	−	0.913359i	$-0.633479\pi$
$887$	−24.2522	−0.814310	−0.407155	+	0.913359i	$0.633479\pi$
$888$	0	0
$889$	−0.300372	−0.0100741
$890$	0	0
$891$	19.0014	0.636570
$892$	0	0
$893$	−23.2967	−0.779593
$894$	0	0
$895$	−5.13093	−0.171508
$896$	0	0
$897$	2.80361	0.0936098
$898$	0	0
$899$	−1.41164	−0.0470807
$900$	0	0
$901$	−19.8378	−0.660894
$902$	0	0
$903$	−0.370932	−0.0123438
$904$	0	0
$905$	−5.23725	−0.174092
$906$	0	0
$907$	−24.0173	−0.797482	−0.398741	−	0.917064i	$-0.630553\pi$
$907$	−24.0173	−0.797482	−0.398741	+	0.917064i	$0.630553\pi$
$908$	0	0
$909$	−5.37369	−0.178234
$910$	0	0
$911$	−18.6442	−0.617710	−0.308855	−	0.951109i	$-0.599946\pi$
$911$	−18.6442	−0.617710	−0.308855	+	0.951109i	$0.599946\pi$
$912$	0	0
$913$	13.6043	0.450237
$914$	0	0
$915$	11.5563	0.382040
$916$	0	0
$917$	−4.16435	−0.137519
$918$	0	0
$919$	29.0727	0.959021	0.479511	−	0.877536i	$-0.340814\pi$
$919$	29.0727	0.959021	0.479511	+	0.877536i	$0.340814\pi$
$920$	0	0
$921$	26.1011	0.860060
$922$	0	0
$923$	0.740333	0.0243683
$924$	0	0
$925$	49.7106	1.63448
$926$	0	0
$927$	2.50105	0.0821452
$928$	0	0
$929$	−15.3273	−0.502873	−0.251437	−	0.967874i	$-0.580903\pi$
$929$	−15.3273	−0.502873	−0.251437	+	0.967874i	$0.580903\pi$
$930$	0	0
$931$	33.6945	1.10429
$932$	0	0
$933$	45.2261	1.48064
$934$	0	0
$935$	−5.95420	−0.194723
$936$	0	0
$937$	−0.205053	−0.00669878	−0.00334939	−	0.999994i	$-0.501066\pi$
$937$	−0.205053	−0.00669878	−0.00334939	+	0.999994i	$0.501066\pi$
$938$	0	0
$939$	17.1635	0.560111
$940$	0	0
$941$	23.6983	0.772541	0.386270	−	0.922386i	$-0.373763\pi$
$941$	23.6983	0.772541	0.386270	+	0.922386i	$0.373763\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.16063	−0.0377553
$946$	0	0
$947$	19.0110	0.617775	0.308887	−	0.951099i	$-0.400043\pi$
$947$	19.0110	0.617775	0.308887	+	0.951099i	$0.400043\pi$
$948$	0	0
$949$	−2.50186	−0.0812138
$950$	0	0
$951$	16.0523	0.520531
$952$	0	0
$953$	12.7344	0.412509	0.206254	−	0.978498i	$-0.433873\pi$
$953$	12.7344	0.412509	0.206254	+	0.978498i	$0.433873\pi$
$954$	0	0
$955$	−2.13602	−0.0691201
$956$	0	0
$957$	1.23491	0.0399189
$958$	0	0
$959$	1.54119	0.0497675
$960$	0	0
$961$	−8.91350	−0.287532
$962$	0	0
$963$	−4.02323	−0.129647
$964$	0	0
$965$	0.364872	0.0117456
$966$	0	0
$967$	24.5709	0.790147	0.395073	−	0.918650i	$-0.370719\pi$
$967$	24.5709	0.790147	0.395073	+	0.918650i	$0.370719\pi$
$968$	0	0
$969$	−25.4669	−0.818115
$970$	0	0
$971$	−59.2719	−1.90213	−0.951063	−	0.308998i	$-0.900006\pi$
$971$	−59.2719	−1.90213	−0.951063	+	0.308998i	$0.900006\pi$
$972$	0	0
$973$	−1.14840	−0.0368161
$974$	0	0
$975$	−2.94910	−0.0944469
$976$	0	0
$977$	−33.3462	−1.06684	−0.533419	−	0.845851i	$-0.679093\pi$
$977$	−33.3462	−1.06684	−0.533419	+	0.845851i	$0.679093\pi$
$978$	0	0
$979$	5.34617	0.170864
$980$	0	0
$981$	4.28209	0.136716
$982$	0	0
$983$	9.56004	0.304918	0.152459	−	0.988310i	$-0.451281\pi$
$983$	9.56004	0.304918	0.152459	+	0.988310i	$0.451281\pi$
$984$	0	0
$985$	−13.1657	−0.419495
$986$	0	0
$987$	2.27933	0.0725519
$988$	0	0
$989$	−3.33379	−0.106008
$990$	0	0
$991$	51.4042	1.63291	0.816454	−	0.577410i	$-0.195937\pi$
$991$	51.4042	1.63291	0.816454	+	0.577410i	$0.195937\pi$
$992$	0	0
$993$	35.2072	1.11727
$994$	0	0
$995$	−2.41026	−0.0764103
$996$	0	0
$997$	7.22609	0.228853	0.114426	−	0.993432i	$-0.463497\pi$
$997$	7.22609	0.228853	0.114426	+	0.993432i	$0.463497\pi$
$998$	0	0
$999$	60.8681	1.92578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8128.2.a.x.1.3		3
4.3	odd	2		8128.2.a.ba.1.1		3
8.3	odd	2		2032.2.a.i.1.3		3
8.5	even	2		508.2.a.d.1.1	✓	3
24.5	odd	2		4572.2.a.n.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
508.2.a.d.1.1	✓	3	8.5	even	2
2032.2.a.i.1.3		3	8.3	odd	2
4572.2.a.n.1.3		3	24.5	odd	2
8128.2.a.x.1.3		3	1.1	even	1	trivial
8128.2.a.ba.1.1		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)

\(f(q)\)	\(=\)	\(q+1.58836 q^{3} +0.699628 q^{5} +0.300372 q^{7} -0.477100 q^{9} +O(q^{10})\)	\(q+1.58836 q^{3} +0.699628 q^{5} +0.300372 q^{7} -0.477100 q^{9} -2.58836 q^{11} +0.411636 q^{13} +1.11126 q^{15} +3.28799 q^{17} -4.87636 q^{19} +0.477100 q^{21} +4.28799 q^{23} -4.51052 q^{25} -5.52290 q^{27} -0.300372 q^{29} +4.69963 q^{31} -4.11126 q^{33} +0.210149 q^{35} -11.0210 q^{37} +0.653828 q^{39} -0.777472 q^{43} -0.333792 q^{45} +4.77747 q^{47} -6.90978 q^{49} +5.22253 q^{51} -6.03342 q^{53} -1.81089 q^{55} -7.74543 q^{57} -8.81089 q^{59} +10.3993 q^{61} -0.143307 q^{63} +0.287992 q^{65} +1.09888 q^{67} +6.81089 q^{69} +1.79851 q^{71} -6.07784 q^{73} -7.16435 q^{75} -0.777472 q^{77} -12.6414 q^{79} -7.34108 q^{81} -5.25595 q^{83} +2.30037 q^{85} -0.477100 q^{87} -2.06546 q^{89} +0.123644 q^{91} +7.46472 q^{93} -3.41164 q^{95} +4.86398 q^{97} +1.23491 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 4 q^{5} + 7 q^{7} + 4 q^{9}+O(q^{10}) \) 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

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