LMFDB - Embedded newform 4096.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4096,2,Mod(1,4096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4096, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4096.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4096 = 2^{12} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4096.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:	$32.7067246679$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{48})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 1 $ x^8 - 8x^6 + 20x^4 - 16*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.98289$ of defining polynomial
Character	$\chi$	$=$	4096.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.51764 q^{3} +3.56960 q^{5} +1.45158 q^{7} -0.696775 q^{9} +O(q^{10})$	$q-1.51764 q^{3} +3.56960 q^{5} +1.45158 q^{7} -0.696775 q^{9} -5.38134 q^{11} -4.09170 q^{13} -5.41736 q^{15} +2.35311 q^{17} +3.02823 q^{19} -2.20297 q^{21} +1.50557 q^{23} +7.74202 q^{25} +5.61037 q^{27} +4.18388 q^{29} -10.5829 q^{31} +8.16693 q^{33} +5.18154 q^{35} -4.57823 q^{37} +6.20972 q^{39} +8.87832 q^{41} -9.31079 q^{43} -2.48720 q^{45} -3.06910 q^{47} -4.89293 q^{49} -3.57117 q^{51} -0.415948 q^{53} -19.2092 q^{55} -4.59575 q^{57} -8.42178 q^{59} -2.79305 q^{61} -1.01142 q^{63} -14.6057 q^{65} -5.18859 q^{67} -2.28492 q^{69} +4.02652 q^{71} +11.9236 q^{73} -11.7496 q^{75} -7.81143 q^{77} -4.59983 q^{79} -6.42418 q^{81} -8.22902 q^{83} +8.39967 q^{85} -6.34962 q^{87} +1.36773 q^{89} -5.93942 q^{91} +16.0610 q^{93} +10.8096 q^{95} -11.2672 q^{97} +3.74958 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3}+O(q^{10}) $ 8 * q - 8 * q^3	$ 8 q - 8 q^{3} - 8 q^{11} + 8 q^{17} - 8 q^{25} - 8 q^{27} + 40 q^{33} + 8 q^{35} + 32 q^{41} - 56 q^{43} + 8 q^{49} - 48 q^{51} + 8 q^{57} - 32 q^{59} - 16 q^{65} - 24 q^{67} - 8 q^{73} + 16 q^{81} - 48 q^{83} - 8 q^{89} - 8 q^{91} + 8 q^{97} - 64 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 - 8 * q^11 + 8 * q^17 - 8 * q^25 - 8 * q^27 + 40 * q^33 + 8 * q^35 + 32 * q^41 - 56 * q^43 + 8 * q^49 - 48 * q^51 + 8 * q^57 - 32 * q^59 - 16 * q^65 - 24 * q^67 - 8 * q^73 + 16 * q^81 - 48 * q^83 - 8 * q^89 - 8 * q^91 + 8 * q^97 - 64 * 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
$3$	−1.51764	−0.876209	−0.438104	−	0.898924i	$-0.644350\pi$
$3$	−1.51764	−0.876209	−0.438104	+	0.898924i	$0.644350\pi$
$4$	0	0
$5$	3.56960	1.59637	0.798186	−	0.602411i	$-0.205793\pi$
$5$	3.56960	1.59637	0.798186	+	0.602411i	$0.205793\pi$
$6$	0	0
$7$	1.45158	0.548644	0.274322	−	0.961638i	$-0.411547\pi$
$7$	1.45158	0.548644	0.274322	+	0.961638i	$0.411547\pi$
$8$	0	0
$9$	−0.696775	−0.232258
$10$	0	0
$11$	−5.38134	−1.62254	−0.811268	−	0.584675i	$-0.801222\pi$
$11$	−5.38134	−1.62254	−0.811268	+	0.584675i	$0.801222\pi$
$12$	0	0
$13$	−4.09170	−1.13483	−0.567417	−	0.823431i	$-0.692057\pi$
$13$	−4.09170	−1.13483	−0.567417	+	0.823431i	$0.692057\pi$
$14$	0	0
$15$	−5.41736	−1.39876
$16$	0	0
$17$	2.35311	0.570714	0.285357	−	0.958421i	$-0.407888\pi$
$17$	2.35311	0.570714	0.285357	+	0.958421i	$0.407888\pi$
$18$	0	0
$19$	3.02823	0.694723	0.347362	−	0.937731i	$-0.387078\pi$
$19$	3.02823	0.694723	0.347362	+	0.937731i	$0.387078\pi$
$20$	0	0
$21$	−2.20297	−0.480727
$22$	0	0
$23$	1.50557	0.313934	0.156967	−	0.987604i	$-0.449828\pi$
$23$	1.50557	0.313934	0.156967	+	0.987604i	$0.449828\pi$
$24$	0	0
$25$	7.74202	1.54840
$26$	0	0
$27$	5.61037	1.07972
$28$	0	0
$29$	4.18388	0.776927	0.388464	−	0.921464i	$-0.373006\pi$
$29$	4.18388	0.776927	0.388464	+	0.921464i	$0.373006\pi$
$30$	0	0
$31$	−10.5829	−1.90074	−0.950370	−	0.311123i	$-0.899295\pi$
$31$	−10.5829	−1.90074	−0.950370	+	0.311123i	$0.899295\pi$
$32$	0	0
$33$	8.16693	1.42168
$34$	0	0
$35$	5.18154	0.875840
$36$	0	0
$37$	−4.57823	−0.752656	−0.376328	−	0.926487i	$-0.622813\pi$
$37$	−4.57823	−0.752656	−0.376328	+	0.926487i	$0.622813\pi$
$38$	0	0
$39$	6.20972	0.994351
$40$	0	0
$41$	8.87832	1.38656	0.693280	−	0.720668i	$-0.256165\pi$
$41$	8.87832	1.38656	0.693280	+	0.720668i	$0.256165\pi$
$42$	0	0
$43$	−9.31079	−1.41988	−0.709941	−	0.704261i	$-0.751278\pi$
$43$	−9.31079	−1.41988	−0.709941	+	0.704261i	$0.751278\pi$
$44$	0	0
$45$	−2.48720	−0.370771
$46$	0	0
$47$	−3.06910	−0.447674	−0.223837	−	0.974627i	$-0.571858\pi$
$47$	−3.06910	−0.447674	−0.223837	+	0.974627i	$0.571858\pi$
$48$	0	0
$49$	−4.89293	−0.698990
$50$	0	0
$51$	−3.57117	−0.500064
$52$	0	0
$53$	−0.415948	−0.0571348	−0.0285674	−	0.999592i	$-0.509095\pi$
$53$	−0.415948	−0.0571348	−0.0285674	+	0.999592i	$0.509095\pi$
$54$	0	0
$55$	−19.2092	−2.59017
$56$	0	0
$57$	−4.59575	−0.608723
$58$	0	0
$59$	−8.42178	−1.09642	−0.548211	−	0.836340i	$-0.684691\pi$
$59$	−8.42178	−1.09642	−0.548211	+	0.836340i	$0.684691\pi$
$60$	0	0
$61$	−2.79305	−0.357613	−0.178806	−	0.983884i	$-0.557224\pi$
$61$	−2.79305	−0.357613	−0.178806	+	0.983884i	$0.557224\pi$
$62$	0	0
$63$	−1.01142	−0.127427
$64$	0	0
$65$	−14.6057	−1.81162
$66$	0	0
$67$	−5.18859	−0.633887	−0.316944	−	0.948444i	$-0.602657\pi$
$67$	−5.18859	−0.633887	−0.316944	+	0.948444i	$0.602657\pi$
$68$	0	0
$69$	−2.28492	−0.275072
$70$	0	0
$71$	4.02652	0.477860	0.238930	−	0.971037i	$-0.423203\pi$
$71$	4.02652	0.477860	0.238930	+	0.971037i	$0.423203\pi$
$72$	0	0
$73$	11.9236	1.39555	0.697774	−	0.716318i	$-0.254174\pi$
$73$	11.9236	1.39555	0.697774	+	0.716318i	$0.254174\pi$
$74$	0	0
$75$	−11.7496	−1.35672
$76$	0	0
$77$	−7.81143	−0.890195
$78$	0	0
$79$	−4.59983	−0.517521	−0.258761	−	0.965941i	$-0.583314\pi$
$79$	−4.59983	−0.517521	−0.258761	+	0.965941i	$0.583314\pi$
$80$	0	0
$81$	−6.42418	−0.713798
$82$	0	0
$83$	−8.22902	−0.903253	−0.451626	−	0.892207i	$-0.649156\pi$
$83$	−8.22902	−0.903253	−0.451626	+	0.892207i	$0.649156\pi$
$84$	0	0
$85$	8.39967	0.911072
$86$	0	0
$87$	−6.34962	−0.680751
$88$	0	0
$89$	1.36773	0.144979	0.0724893	−	0.997369i	$-0.476906\pi$
$89$	1.36773	0.144979	0.0724893	+	0.997369i	$0.476906\pi$
$90$	0	0
$91$	−5.93942	−0.622620
$92$	0	0
$93$	16.0610	1.66544
$94$	0	0
$95$	10.8096	1.10904
$96$	0	0
$97$	−11.2672	−1.14401	−0.572006	−	0.820249i	$-0.693835\pi$
$97$	−11.2672	−1.14401	−0.572006	+	0.820249i	$0.693835\pi$
$98$	0	0
$99$	3.74958	0.376847
$100$	0	0
$101$	−9.37195	−0.932544	−0.466272	−	0.884641i	$-0.654403\pi$
$101$	−9.37195	−0.932544	−0.466272	+	0.884641i	$0.654403\pi$
$102$	0	0
$103$	1.70342	0.167843	0.0839215	−	0.996472i	$-0.473256\pi$
$103$	1.70342	0.167843	0.0839215	+	0.996472i	$0.473256\pi$
$104$	0	0
$105$	−7.86370	−0.767419
$106$	0	0
$107$	−2.91119	−0.281435	−0.140718	−	0.990050i	$-0.544941\pi$
$107$	−2.91119	−0.281435	−0.140718	+	0.990050i	$0.544941\pi$
$108$	0	0
$109$	3.82144	0.366028	0.183014	−	0.983110i	$-0.441415\pi$
$109$	3.82144	0.366028	0.183014	+	0.983110i	$0.441415\pi$
$110$	0	0
$111$	6.94809	0.659483
$112$	0	0
$113$	2.13630	0.200966	0.100483	−	0.994939i	$-0.467961\pi$
$113$	2.13630	0.200966	0.100483	+	0.994939i	$0.467961\pi$
$114$	0	0
$115$	5.37429	0.501155
$116$	0	0
$117$	2.85099	0.263574
$118$	0	0
$119$	3.41572	0.313119
$120$	0	0
$121$	17.9588	1.63262
$122$	0	0
$123$	−13.4741	−1.21492
$124$	0	0
$125$	9.78790	0.875456
$126$	0	0
$127$	1.09821	0.0974502	0.0487251	−	0.998812i	$-0.484484\pi$
$127$	1.09821	0.0974502	0.0487251	+	0.998812i	$0.484484\pi$
$128$	0	0
$129$	14.1304	1.24411
$130$	0	0
$131$	0.709146	0.0619584	0.0309792	−	0.999520i	$-0.490137\pi$
$131$	0.709146	0.0619584	0.0309792	+	0.999520i	$0.490137\pi$
$132$	0	0
$133$	4.39570	0.381156
$134$	0	0
$135$	20.0267	1.72363
$136$	0	0
$137$	−16.2474	−1.38811	−0.694057	−	0.719920i	$-0.744178\pi$
$137$	−16.2474	−1.38811	−0.694057	+	0.719920i	$0.744178\pi$
$138$	0	0
$139$	0.845443	0.0717095	0.0358548	−	0.999357i	$-0.488585\pi$
$139$	0.845443	0.0717095	0.0358548	+	0.999357i	$0.488585\pi$
$140$	0	0
$141$	4.65778	0.392256
$142$	0	0
$143$	22.0188	1.84131
$144$	0	0
$145$	14.9348	1.24027
$146$	0	0
$147$	7.42569	0.612461
$148$	0	0
$149$	−7.06099	−0.578459	−0.289229	−	0.957260i	$-0.593399\pi$
$149$	−7.06099	−0.578459	−0.289229	+	0.957260i	$0.593399\pi$
$150$	0	0
$151$	−13.9645	−1.13642	−0.568209	−	0.822884i	$-0.692364\pi$
$151$	−13.9645	−1.13642	−0.568209	+	0.822884i	$0.692364\pi$
$152$	0	0
$153$	−1.63959	−0.132553
$154$	0	0
$155$	−37.7766	−3.03429
$156$	0	0
$157$	−24.2106	−1.93222	−0.966109	−	0.258134i	$-0.916892\pi$
$157$	−24.2106	−1.93222	−0.966109	+	0.258134i	$0.916892\pi$
$158$	0	0
$159$	0.631258	0.0500620
$160$	0	0
$161$	2.18546	0.172238
$162$	0	0
$163$	−17.5103	−1.37152	−0.685758	−	0.727830i	$-0.740529\pi$
$163$	−17.5103	−1.37152	−0.685758	+	0.727830i	$0.740529\pi$
$164$	0	0
$165$	29.1526	2.26953
$166$	0	0
$167$	22.5149	1.74226	0.871128	−	0.491057i	$-0.163389\pi$
$167$	22.5149	1.74226	0.871128	+	0.491057i	$0.163389\pi$
$168$	0	0
$169$	3.74202	0.287848
$170$	0	0
$171$	−2.10999	−0.161355
$172$	0	0
$173$	−9.05532	−0.688463	−0.344232	−	0.938885i	$-0.611861\pi$
$173$	−9.05532	−0.688463	−0.344232	+	0.938885i	$0.611861\pi$
$174$	0	0
$175$	11.2381	0.849523
$176$	0	0
$177$	12.7812	0.960695
$178$	0	0
$179$	−0.143345	−0.0107141	−0.00535705	−	0.999986i	$-0.501705\pi$
$179$	−0.143345	−0.0107141	−0.00535705	+	0.999986i	$0.501705\pi$
$180$	0	0
$181$	2.57310	0.191257	0.0956286	−	0.995417i	$-0.469514\pi$
$181$	2.57310	0.191257	0.0956286	+	0.995417i	$0.469514\pi$
$182$	0	0
$183$	4.23883	0.313344
$184$	0	0
$185$	−16.3424	−1.20152
$186$	0	0
$187$	−12.6629	−0.926003
$188$	0	0
$189$	8.14387	0.592380
$190$	0	0
$191$	16.4603	1.19103	0.595514	−	0.803345i	$-0.296948\pi$
$191$	16.4603	1.19103	0.595514	+	0.803345i	$0.296948\pi$
$192$	0	0
$193$	9.84445	0.708619	0.354309	−	0.935128i	$-0.384716\pi$
$193$	9.84445	0.708619	0.354309	+	0.935128i	$0.384716\pi$
$194$	0	0
$195$	22.1662	1.58735
$196$	0	0
$197$	−10.2186	−0.728044	−0.364022	−	0.931390i	$-0.618597\pi$
$197$	−10.2186	−0.728044	−0.364022	+	0.931390i	$0.618597\pi$
$198$	0	0
$199$	−3.41852	−0.242332	−0.121166	−	0.992632i	$-0.538663\pi$
$199$	−3.41852	−0.242332	−0.121166	+	0.992632i	$0.538663\pi$
$200$	0	0
$201$	7.87440	0.555417
$202$	0	0
$203$	6.07322	0.426257
$204$	0	0
$205$	31.6920	2.21347
$206$	0	0
$207$	−1.04905	−0.0729137
$208$	0	0
$209$	−16.2959	−1.12721
$210$	0	0
$211$	15.8494	1.09111	0.545557	−	0.838073i	$-0.316318\pi$
$211$	15.8494	1.09111	0.545557	+	0.838073i	$0.316318\pi$
$212$	0	0
$213$	−6.11080	−0.418705
$214$	0	0
$215$	−33.2358	−2.26666
$216$	0	0
$217$	−15.3618	−1.04283
$218$	0	0
$219$	−18.0956	−1.22279
$220$	0	0
$221$	−9.62824	−0.647665
$222$	0	0
$223$	20.9031	1.39978	0.699888	−	0.714253i	$-0.253233\pi$
$223$	20.9031	1.39978	0.699888	+	0.714253i	$0.253233\pi$
$224$	0	0
$225$	−5.39444	−0.359629
$226$	0	0
$227$	−13.5517	−0.899456	−0.449728	−	0.893166i	$-0.648479\pi$
$227$	−13.5517	−0.899456	−0.449728	+	0.893166i	$0.648479\pi$
$228$	0	0
$229$	5.79917	0.383220	0.191610	−	0.981471i	$-0.438629\pi$
$229$	5.79917	0.383220	0.191610	+	0.981471i	$0.438629\pi$
$230$	0	0
$231$	11.8549	0.779996
$232$	0	0
$233$	0.0379288	0.00248480	0.00124240	−	0.999999i	$-0.499605\pi$
$233$	0.0379288	0.00248480	0.00124240	+	0.999999i	$0.499605\pi$
$234$	0	0
$235$	−10.9554	−0.714654
$236$	0	0
$237$	6.98088	0.453457
$238$	0	0
$239$	−25.6128	−1.65676	−0.828378	−	0.560169i	$-0.810736\pi$
$239$	−25.6128	−1.65676	−0.828378	+	0.560169i	$0.810736\pi$
$240$	0	0
$241$	1.28132	0.0825369	0.0412684	−	0.999148i	$-0.486860\pi$
$241$	1.28132	0.0825369	0.0412684	+	0.999148i	$0.486860\pi$
$242$	0	0
$243$	−7.08152	−0.454279
$244$	0	0
$245$	−17.4658	−1.11585
$246$	0	0
$247$	−12.3906	−0.788395
$248$	0	0
$249$	12.4887	0.791438
$250$	0	0
$251$	10.4850	0.661809	0.330905	−	0.943664i	$-0.392646\pi$
$251$	10.4850	0.661809	0.330905	+	0.943664i	$0.392646\pi$
$252$	0	0
$253$	−8.10201	−0.509369
$254$	0	0
$255$	−12.7477	−0.798289
$256$	0	0
$257$	8.01513	0.499970	0.249985	−	0.968250i	$-0.419574\pi$
$257$	8.01513	0.499970	0.249985	+	0.968250i	$0.419574\pi$
$258$	0	0
$259$	−6.64564	−0.412940
$260$	0	0
$261$	−2.91522	−0.180448
$262$	0	0
$263$	19.6098	1.20919	0.604597	−	0.796531i	$-0.293334\pi$
$263$	19.6098	1.20919	0.604597	+	0.796531i	$0.293334\pi$
$264$	0	0
$265$	−1.48477	−0.0912084
$266$	0	0
$267$	−2.07571	−0.127032
$268$	0	0
$269$	−17.1059	−1.04297	−0.521484	−	0.853261i	$-0.674621\pi$
$269$	−17.1059	−1.04297	−0.521484	+	0.853261i	$0.674621\pi$
$270$	0	0
$271$	15.6152	0.948557	0.474279	−	0.880375i	$-0.342709\pi$
$271$	15.6152	0.948557	0.474279	+	0.880375i	$0.342709\pi$
$272$	0	0
$273$	9.01388	0.545545
$274$	0	0
$275$	−41.6624	−2.51234
$276$	0	0
$277$	−13.9076	−0.835624	−0.417812	−	0.908534i	$-0.637203\pi$
$277$	−13.9076	−0.835624	−0.417812	+	0.908534i	$0.637203\pi$
$278$	0	0
$279$	7.37387	0.441462
$280$	0	0
$281$	−26.4867	−1.58006	−0.790032	−	0.613066i	$-0.789936\pi$
$281$	−26.4867	−1.58006	−0.790032	+	0.613066i	$0.789936\pi$
$282$	0	0
$283$	1.07155	0.0636969	0.0318485	−	0.999493i	$-0.489861\pi$
$283$	1.07155	0.0636969	0.0318485	+	0.999493i	$0.489861\pi$
$284$	0	0
$285$	−16.4050	−0.971748
$286$	0	0
$287$	12.8875	0.760728
$288$	0	0
$289$	−11.4629	−0.674286
$290$	0	0
$291$	17.0996	1.00239
$292$	0	0
$293$	−17.7977	−1.03975	−0.519875	−	0.854242i	$-0.674022\pi$
$293$	−17.7977	−1.03975	−0.519875	+	0.854242i	$0.674022\pi$
$294$	0	0
$295$	−30.0623	−1.75030
$296$	0	0
$297$	−30.1913	−1.75188
$298$	0	0
$299$	−6.16036	−0.356263
$300$	0	0
$301$	−13.5153	−0.779010
$302$	0	0
$303$	14.2232	0.817103
$304$	0	0
$305$	−9.97005	−0.570883
$306$	0	0
$307$	20.6193	1.17681	0.588404	−	0.808567i	$-0.299757\pi$
$307$	20.6193	1.17681	0.588404	+	0.808567i	$0.299757\pi$
$308$	0	0
$309$	−2.58517	−0.147065
$310$	0	0
$311$	2.01815	0.114439	0.0572193	−	0.998362i	$-0.481777\pi$
$311$	2.01815	0.114439	0.0572193	+	0.998362i	$0.481777\pi$
$312$	0	0
$313$	−1.79869	−0.101668	−0.0508339	−	0.998707i	$-0.516188\pi$
$313$	−1.79869	−0.101668	−0.0508339	+	0.998707i	$0.516188\pi$
$314$	0	0
$315$	−3.61037	−0.203421
$316$	0	0
$317$	−24.7983	−1.39281	−0.696406	−	0.717648i	$-0.745219\pi$
$317$	−24.7983	−1.39281	−0.696406	+	0.717648i	$0.745219\pi$
$318$	0	0
$319$	−22.5149	−1.26059
$320$	0	0
$321$	4.41813	0.246596
$322$	0	0
$323$	7.12576	0.396488
$324$	0	0
$325$	−31.6780	−1.75718
$326$	0	0
$327$	−5.79956	−0.320717
$328$	0	0
$329$	−4.45503	−0.245614
$330$	0	0
$331$	−22.9238	−1.26001	−0.630004	−	0.776592i	$-0.716947\pi$
$331$	−22.9238	−1.26001	−0.630004	+	0.776592i	$0.716947\pi$
$332$	0	0
$333$	3.18999	0.174810
$334$	0	0
$335$	−18.5212	−1.01192
$336$	0	0
$337$	−27.4961	−1.49781	−0.748905	−	0.662677i	$-0.769420\pi$
$337$	−27.4961	−1.49781	−0.748905	+	0.662677i	$0.769420\pi$
$338$	0	0
$339$	−3.24213	−0.176088
$340$	0	0
$341$	56.9500	3.08402
$342$	0	0
$343$	−17.2635	−0.932141
$344$	0	0
$345$	−8.15623	−0.439117
$346$	0	0
$347$	4.90706	0.263425	0.131712	−	0.991288i	$-0.457952\pi$
$347$	4.90706	0.263425	0.131712	+	0.991288i	$0.457952\pi$
$348$	0	0
$349$	21.6452	1.15864	0.579321	−	0.815100i	$-0.303318\pi$
$349$	21.6452	1.15864	0.579321	+	0.815100i	$0.303318\pi$
$350$	0	0
$351$	−22.9559	−1.22530
$352$	0	0
$353$	11.5498	0.614733	0.307366	−	0.951591i	$-0.400552\pi$
$353$	11.5498	0.614733	0.307366	+	0.951591i	$0.400552\pi$
$354$	0	0
$355$	14.3730	0.762842
$356$	0	0
$357$	−5.18383	−0.274357
$358$	0	0
$359$	26.9748	1.42368	0.711839	−	0.702343i	$-0.247863\pi$
$359$	26.9748	1.42368	0.711839	+	0.702343i	$0.247863\pi$
$360$	0	0
$361$	−9.82984	−0.517360
$362$	0	0
$363$	−27.2550	−1.43052
$364$	0	0
$365$	42.5623	2.22781
$366$	0	0
$367$	−12.4379	−0.649251	−0.324626	−	0.945843i	$-0.605238\pi$
$367$	−12.4379	−0.649251	−0.324626	+	0.945843i	$0.605238\pi$
$368$	0	0
$369$	−6.18618	−0.322040
$370$	0	0
$371$	−0.603780	−0.0313467
$372$	0	0
$373$	33.6882	1.74431	0.872154	−	0.489232i	$-0.162723\pi$
$373$	33.6882	1.74431	0.872154	+	0.489232i	$0.162723\pi$
$374$	0	0
$375$	−14.8545	−0.767083
$376$	0	0
$377$	−17.1192	−0.881683
$378$	0	0
$379$	−14.1784	−0.728296	−0.364148	−	0.931341i	$-0.618640\pi$
$379$	−14.1784	−0.728296	−0.364148	+	0.931341i	$0.618640\pi$
$380$	0	0
$381$	−1.66668	−0.0853867
$382$	0	0
$383$	−7.14287	−0.364984	−0.182492	−	0.983207i	$-0.558416\pi$
$383$	−7.14287	−0.364984	−0.182492	+	0.983207i	$0.558416\pi$
$384$	0	0
$385$	−27.8836	−1.42108
$386$	0	0
$387$	6.48752	0.329779
$388$	0	0
$389$	28.3729	1.43856	0.719282	−	0.694719i	$-0.244471\pi$
$389$	28.3729	1.43856	0.719282	+	0.694719i	$0.244471\pi$
$390$	0	0
$391$	3.54279	0.179166
$392$	0	0
$393$	−1.07623	−0.0542885
$394$	0	0
$395$	−16.4195	−0.826157
$396$	0	0
$397$	9.28605	0.466054	0.233027	−	0.972470i	$-0.425137\pi$
$397$	9.28605	0.466054	0.233027	+	0.972470i	$0.425137\pi$
$398$	0	0
$399$	−6.67109	−0.333972
$400$	0	0
$401$	−9.43274	−0.471049	−0.235524	−	0.971868i	$-0.575681\pi$
$401$	−9.43274	−0.471049	−0.235524	+	0.971868i	$0.575681\pi$
$402$	0	0
$403$	43.3019	2.15702
$404$	0	0
$405$	−22.9317	−1.13949
$406$	0	0
$407$	24.6370	1.22121
$408$	0	0
$409$	21.8489	1.08036	0.540179	−	0.841550i	$-0.318357\pi$
$409$	21.8489	1.08036	0.540179	+	0.841550i	$0.318357\pi$
$410$	0	0
$411$	24.6577	1.21628
$412$	0	0
$413$	−12.2248	−0.601546
$414$	0	0
$415$	−29.3743	−1.44193
$416$	0	0
$417$	−1.28308	−0.0628325
$418$	0	0
$419$	−21.3872	−1.04483	−0.522416	−	0.852691i	$-0.674969\pi$
$419$	−21.3872	−1.04483	−0.522416	+	0.852691i	$0.674969\pi$
$420$	0	0
$421$	26.5243	1.29271	0.646357	−	0.763035i	$-0.276292\pi$
$421$	26.5243	1.29271	0.646357	+	0.763035i	$0.276292\pi$
$422$	0	0
$423$	2.13847	0.103976
$424$	0	0
$425$	18.2178	0.883695
$426$	0	0
$427$	−4.05432	−0.196202
$428$	0	0
$429$	−33.4166	−1.61337
$430$	0	0
$431$	−24.7162	−1.19054	−0.595268	−	0.803528i	$-0.702954\pi$
$431$	−24.7162	−1.19054	−0.595268	+	0.803528i	$0.702954\pi$
$432$	0	0
$433$	−9.69501	−0.465913	−0.232956	−	0.972487i	$-0.574840\pi$
$433$	−9.69501	−0.465913	−0.232956	+	0.972487i	$0.574840\pi$
$434$	0	0
$435$	−22.6656	−1.08673
$436$	0	0
$437$	4.55922	0.218097
$438$	0	0
$439$	−11.9562	−0.570637	−0.285319	−	0.958433i	$-0.592099\pi$
$439$	−11.9562	−0.570637	−0.285319	+	0.958433i	$0.592099\pi$
$440$	0	0
$441$	3.40927	0.162346
$442$	0	0
$443$	37.9379	1.80248	0.901241	−	0.433318i	$-0.142657\pi$
$443$	37.9379	1.80248	0.901241	+	0.433318i	$0.142657\pi$
$444$	0	0
$445$	4.88223	0.231440
$446$	0	0
$447$	10.7160	0.506851
$448$	0	0
$449$	−22.3365	−1.05413	−0.527063	−	0.849826i	$-0.676707\pi$
$449$	−22.3365	−1.05413	−0.527063	+	0.849826i	$0.676707\pi$
$450$	0	0
$451$	−47.7772	−2.24974
$452$	0	0
$453$	21.1931	0.995740
$454$	0	0
$455$	−21.2013	−0.993933
$456$	0	0
$457$	−31.6853	−1.48218	−0.741089	−	0.671407i	$-0.765690\pi$
$457$	−31.6853	−1.48218	−0.741089	+	0.671407i	$0.765690\pi$
$458$	0	0
$459$	13.2018	0.616209
$460$	0	0
$461$	3.16134	0.147238	0.0736192	−	0.997286i	$-0.476545\pi$
$461$	3.16134	0.147238	0.0736192	+	0.997286i	$0.476545\pi$
$462$	0	0
$463$	−26.3825	−1.22610	−0.613048	−	0.790045i	$-0.710057\pi$
$463$	−26.3825	−1.22610	−0.613048	+	0.790045i	$0.710057\pi$
$464$	0	0
$465$	57.3312	2.65867
$466$	0	0
$467$	−10.2631	−0.474918	−0.237459	−	0.971398i	$-0.576314\pi$
$467$	−10.2631	−0.474918	−0.237459	+	0.971398i	$0.576314\pi$
$468$	0	0
$469$	−7.53163	−0.347778
$470$	0	0
$471$	36.7430	1.69303
$472$	0	0
$473$	50.1045	2.30381
$474$	0	0
$475$	23.4446	1.07571
$476$	0	0
$477$	0.289822	0.0132700
$478$	0	0
$479$	−26.4855	−1.21015	−0.605077	−	0.796167i	$-0.706858\pi$
$479$	−26.4855	−1.21015	−0.605077	+	0.796167i	$0.706858\pi$
$480$	0	0
$481$	18.7327	0.854139
$482$	0	0
$483$	−3.31673	−0.150916
$484$	0	0
$485$	−40.2194	−1.82627
$486$	0	0
$487$	−7.61404	−0.345025	−0.172513	−	0.985007i	$-0.555189\pi$
$487$	−7.61404	−0.345025	−0.172513	+	0.985007i	$0.555189\pi$
$488$	0	0
$489$	26.5744	1.20173
$490$	0	0
$491$	17.3063	0.781023	0.390511	−	0.920598i	$-0.372298\pi$
$491$	17.3063	0.781023	0.390511	+	0.920598i	$0.372298\pi$
$492$	0	0
$493$	9.84515	0.443403
$494$	0	0
$495$	13.3845	0.601588
$496$	0	0
$497$	5.84480	0.262175
$498$	0	0
$499$	−4.40768	−0.197315	−0.0986574	−	0.995121i	$-0.531455\pi$
$499$	−4.40768	−0.197315	−0.0986574	+	0.995121i	$0.531455\pi$
$500$	0	0
$501$	−34.1695	−1.52658
$502$	0	0
$503$	31.2728	1.39438	0.697192	−	0.716885i	$-0.254433\pi$
$503$	31.2728	1.39438	0.697192	+	0.716885i	$0.254433\pi$
$504$	0	0
$505$	−33.4541	−1.48869
$506$	0	0
$507$	−5.67903	−0.252215
$508$	0	0
$509$	13.6952	0.607031	0.303515	−	0.952827i	$-0.401840\pi$
$509$	13.6952	0.607031	0.303515	+	0.952827i	$0.401840\pi$
$510$	0	0
$511$	17.3080	0.765659
$512$	0	0
$513$	16.9895	0.750103
$514$	0	0
$515$	6.08052	0.267940
$516$	0	0
$517$	16.5159	0.726366
$518$	0	0
$519$	13.7427	0.603238
$520$	0	0
$521$	11.1689	0.489316	0.244658	−	0.969609i	$-0.421324\pi$
$521$	11.1689	0.489316	0.244658	+	0.969609i	$0.421324\pi$
$522$	0	0
$523$	−12.1884	−0.532960	−0.266480	−	0.963840i	$-0.585861\pi$
$523$	−12.1884	−0.532960	−0.266480	+	0.963840i	$0.585861\pi$
$524$	0	0
$525$	−17.0554	−0.744359
$526$	0	0
$527$	−24.9027	−1.08478
$528$	0	0
$529$	−20.7332	−0.901445
$530$	0	0
$531$	5.86808	0.254653
$532$	0	0
$533$	−36.3274	−1.57351
$534$	0	0
$535$	−10.3918	−0.449275
$536$	0	0
$537$	0.217546	0.00938779
$538$	0	0
$539$	26.3305	1.13414
$540$	0	0
$541$	12.7571	0.548471	0.274236	−	0.961663i	$-0.411575\pi$
$541$	12.7571	0.548471	0.274236	+	0.961663i	$0.411575\pi$
$542$	0	0
$543$	−3.90504	−0.167581
$544$	0	0
$545$	13.6410	0.584316
$546$	0	0
$547$	28.2882	1.20951	0.604757	−	0.796410i	$-0.293270\pi$
$547$	28.2882	1.20951	0.604757	+	0.796410i	$0.293270\pi$
$548$	0	0
$549$	1.94612	0.0830585
$550$	0	0
$551$	12.6697	0.539749
$552$	0	0
$553$	−6.67700	−0.283935
$554$	0	0
$555$	24.8019	1.05278
$556$	0	0
$557$	35.5121	1.50470	0.752348	−	0.658766i	$-0.228921\pi$
$557$	35.5121	1.50470	0.752348	+	0.658766i	$0.228921\pi$
$558$	0	0
$559$	38.0970	1.61133
$560$	0	0
$561$	19.2177	0.811372
$562$	0	0
$563$	−11.7262	−0.494200	−0.247100	−	0.968990i	$-0.579478\pi$
$563$	−11.7262	−0.494200	−0.247100	+	0.968990i	$0.579478\pi$
$564$	0	0
$565$	7.62572	0.320816
$566$	0	0
$567$	−9.32519	−0.391621
$568$	0	0
$569$	26.5410	1.11266	0.556328	−	0.830962i	$-0.312210\pi$
$569$	26.5410	1.11266	0.556328	+	0.830962i	$0.312210\pi$
$570$	0	0
$571$	−13.1527	−0.550425	−0.275213	−	0.961383i	$-0.588748\pi$
$571$	−13.1527	−0.550425	−0.275213	+	0.961383i	$0.588748\pi$
$572$	0	0
$573$	−24.9808	−1.04359
$574$	0	0
$575$	11.6562	0.486097
$576$	0	0
$577$	25.2275	1.05024	0.525118	−	0.851029i	$-0.324021\pi$
$577$	25.2275	1.05024	0.525118	+	0.851029i	$0.324021\pi$
$578$	0	0
$579$	−14.9403	−0.620898
$580$	0	0
$581$	−11.9451	−0.495564
$582$	0	0
$583$	2.23836	0.0927033
$584$	0	0
$585$	10.1769	0.420763
$586$	0	0
$587$	−17.8563	−0.737009	−0.368504	−	0.929626i	$-0.620130\pi$
$587$	−17.8563	−0.737009	−0.368504	+	0.929626i	$0.620130\pi$
$588$	0	0
$589$	−32.0473	−1.32049
$590$	0	0
$591$	15.5081	0.637919
$592$	0	0
$593$	41.3009	1.69602	0.848012	−	0.529976i	$-0.177799\pi$
$593$	41.3009	1.69602	0.848012	+	0.529976i	$0.177799\pi$
$594$	0	0
$595$	12.1928	0.499854
$596$	0	0
$597$	5.18807	0.212334
$598$	0	0
$599$	9.53066	0.389412	0.194706	−	0.980862i	$-0.437625\pi$
$599$	9.53066	0.389412	0.194706	+	0.980862i	$0.437625\pi$
$600$	0	0
$601$	−32.8742	−1.34097	−0.670483	−	0.741925i	$-0.733913\pi$
$601$	−32.8742	−1.34097	−0.670483	+	0.741925i	$0.733913\pi$
$602$	0	0
$603$	3.61528	0.147225
$604$	0	0
$605$	64.1058	2.60627
$606$	0	0
$607$	18.7402	0.760642	0.380321	−	0.924855i	$-0.375813\pi$
$607$	18.7402	0.760642	0.380321	+	0.924855i	$0.375813\pi$
$608$	0	0
$609$	−9.21695	−0.373490
$610$	0	0
$611$	12.5578	0.508035
$612$	0	0
$613$	−37.2655	−1.50514	−0.752569	−	0.658513i	$-0.771186\pi$
$613$	−37.2655	−1.50514	−0.752569	+	0.658513i	$0.771186\pi$
$614$	0	0
$615$	−48.0970	−1.93946
$616$	0	0
$617$	16.3300	0.657419	0.328710	−	0.944431i	$-0.393386\pi$
$617$	16.3300	0.657419	0.328710	+	0.944431i	$0.393386\pi$
$618$	0	0
$619$	43.1365	1.73380	0.866901	−	0.498480i	$-0.166108\pi$
$619$	43.1365	1.73380	0.866901	+	0.498480i	$0.166108\pi$
$620$	0	0
$621$	8.44682	0.338959
$622$	0	0
$623$	1.98536	0.0795417
$624$	0	0
$625$	−3.77124	−0.150850
$626$	0	0
$627$	24.7313	0.987674
$628$	0	0
$629$	−10.7731	−0.429551
$630$	0	0
$631$	−6.34403	−0.252552	−0.126276	−	0.991995i	$-0.540302\pi$
$631$	−6.34403	−0.252552	−0.126276	+	0.991995i	$0.540302\pi$
$632$	0	0
$633$	−24.0536	−0.956044
$634$	0	0
$635$	3.92016	0.155567
$636$	0	0
$637$	20.0204	0.793237
$638$	0	0
$639$	−2.80558	−0.110987
$640$	0	0
$641$	−39.5996	−1.56409	−0.782045	−	0.623221i	$-0.785824\pi$
$641$	−39.5996	−1.56409	−0.782045	+	0.623221i	$0.785824\pi$
$642$	0	0
$643$	24.4683	0.964934	0.482467	−	0.875914i	$-0.339741\pi$
$643$	24.4683	0.964934	0.482467	+	0.875914i	$0.339741\pi$
$644$	0	0
$645$	50.4399	1.98607
$646$	0	0
$647$	−35.9659	−1.41396	−0.706982	−	0.707232i	$-0.749944\pi$
$647$	−35.9659	−1.41396	−0.706982	+	0.707232i	$0.749944\pi$
$648$	0	0
$649$	45.3205	1.77898
$650$	0	0
$651$	23.3137	0.913736
$652$	0	0
$653$	−24.4970	−0.958641	−0.479320	−	0.877640i	$-0.659117\pi$
$653$	−24.4970	−0.958641	−0.479320	+	0.877640i	$0.659117\pi$
$654$	0	0
$655$	2.53137	0.0989087
$656$	0	0
$657$	−8.30803	−0.324127
$658$	0	0
$659$	−27.0706	−1.05452	−0.527260	−	0.849704i	$-0.676781\pi$
$659$	−27.0706	−1.05452	−0.527260	+	0.849704i	$0.676781\pi$
$660$	0	0
$661$	12.6336	0.491390	0.245695	−	0.969347i	$-0.420984\pi$
$661$	12.6336	0.491390	0.245695	+	0.969347i	$0.420984\pi$
$662$	0	0
$663$	14.6122	0.567490
$664$	0	0
$665$	15.6909	0.608466
$666$	0	0
$667$	6.29915	0.243904
$668$	0	0
$669$	−31.7233	−1.22650
$670$	0	0
$671$	15.0303	0.580240
$672$	0	0
$673$	−9.29926	−0.358460	−0.179230	−	0.983807i	$-0.557361\pi$
$673$	−9.29926	−0.358460	−0.179230	+	0.983807i	$0.557361\pi$
$674$	0	0
$675$	43.4356	1.67184
$676$	0	0
$677$	22.4650	0.863399	0.431700	−	0.902017i	$-0.357914\pi$
$677$	22.4650	0.863399	0.431700	+	0.902017i	$0.357914\pi$
$678$	0	0
$679$	−16.3552	−0.627656
$680$	0	0
$681$	20.5665	0.788111
$682$	0	0
$683$	−5.57986	−0.213508	−0.106754	−	0.994285i	$-0.534046\pi$
$683$	−5.57986	−0.213508	−0.106754	+	0.994285i	$0.534046\pi$
$684$	0	0
$685$	−57.9968	−2.21595
$686$	0	0
$687$	−8.80104	−0.335781
$688$	0	0
$689$	1.70193	0.0648385
$690$	0	0
$691$	24.8340	0.944731	0.472365	−	0.881403i	$-0.343400\pi$
$691$	24.8340	0.944731	0.472365	+	0.881403i	$0.343400\pi$
$692$	0	0
$693$	5.44280	0.206755
$694$	0	0
$695$	3.01789	0.114475
$696$	0	0
$697$	20.8917	0.791329
$698$	0	0
$699$	−0.0575622	−0.00217720
$700$	0	0
$701$	19.8755	0.750686	0.375343	−	0.926886i	$-0.377525\pi$
$701$	19.8755	0.750686	0.375343	+	0.926886i	$0.377525\pi$
$702$	0	0
$703$	−13.8639	−0.522887
$704$	0	0
$705$	16.6264	0.626186
$706$	0	0
$707$	−13.6041	−0.511635
$708$	0	0
$709$	−26.3575	−0.989879	−0.494939	−	0.868928i	$-0.664810\pi$
$709$	−26.3575	−0.989879	−0.494939	+	0.868928i	$0.664810\pi$
$710$	0	0
$711$	3.20504	0.120199
$712$	0	0
$713$	−15.9333	−0.596707
$714$	0	0
$715$	78.5984	2.93941
$716$	0	0
$717$	38.8710	1.45166
$718$	0	0
$719$	34.6091	1.29070	0.645350	−	0.763887i	$-0.276711\pi$
$719$	34.6091	1.29070	0.645350	+	0.763887i	$0.276711\pi$
$720$	0	0
$721$	2.47264	0.0920860
$722$	0	0
$723$	−1.94457	−0.0723195
$724$	0	0
$725$	32.3917	1.20300
$726$	0	0
$727$	43.5223	1.61415	0.807077	−	0.590447i	$-0.201048\pi$
$727$	43.5223	1.61415	0.807077	+	0.590447i	$0.201048\pi$
$728$	0	0
$729$	30.0197	1.11184
$730$	0	0
$731$	−21.9093	−0.810346
$732$	0	0
$733$	45.5242	1.68147	0.840737	−	0.541444i	$-0.182122\pi$
$733$	45.5242	1.68147	0.840737	+	0.541444i	$0.182122\pi$
$734$	0	0
$735$	26.5067	0.977715
$736$	0	0
$737$	27.9216	1.02850
$738$	0	0
$739$	−27.9863	−1.02949	−0.514747	−	0.857342i	$-0.672114\pi$
$739$	−27.9863	−1.02949	−0.514747	+	0.857342i	$0.672114\pi$
$740$	0	0
$741$	18.8045	0.690799
$742$	0	0
$743$	−37.1907	−1.36440	−0.682198	−	0.731167i	$-0.738976\pi$
$743$	−37.1907	−1.36440	−0.682198	+	0.731167i	$0.738976\pi$
$744$	0	0
$745$	−25.2049	−0.923435
$746$	0	0
$747$	5.73378	0.209788
$748$	0	0
$749$	−4.22581	−0.154408
$750$	0	0
$751$	37.4098	1.36510	0.682552	−	0.730837i	$-0.260870\pi$
$751$	37.4098	1.36510	0.682552	+	0.730837i	$0.260870\pi$
$752$	0	0
$753$	−15.9125	−0.579883
$754$	0	0
$755$	−49.8478	−1.81415
$756$	0	0
$757$	−8.15361	−0.296348	−0.148174	−	0.988961i	$-0.547340\pi$
$757$	−8.15361	−0.296348	−0.148174	+	0.988961i	$0.547340\pi$
$758$	0	0
$759$	12.2959	0.446314
$760$	0	0
$761$	11.4341	0.414488	0.207244	−	0.978289i	$-0.433551\pi$
$761$	11.4341	0.414488	0.207244	+	0.978289i	$0.433551\pi$
$762$	0	0
$763$	5.54711	0.200819
$764$	0	0
$765$	−5.85267	−0.211604
$766$	0	0
$767$	34.4594	1.24426
$768$	0	0
$769$	−30.6572	−1.10553	−0.552764	−	0.833338i	$-0.686427\pi$
$769$	−30.6572	−1.10553	−0.552764	+	0.833338i	$0.686427\pi$
$770$	0	0
$771$	−12.1641	−0.438078
$772$	0	0
$773$	−31.8650	−1.14610	−0.573052	−	0.819519i	$-0.694241\pi$
$773$	−31.8650	−1.14610	−0.573052	+	0.819519i	$0.694241\pi$
$774$	0	0
$775$	−81.9328	−2.94311
$776$	0	0
$777$	10.0857	0.361822
$778$	0	0
$779$	26.8856	0.963275
$780$	0	0
$781$	−21.6681	−0.775345
$782$	0	0
$783$	23.4731	0.838861
$784$	0	0
$785$	−86.4222	−3.08454
$786$	0	0
$787$	0.547427	0.0195137	0.00975683	−	0.999952i	$-0.496894\pi$
$787$	0.547427	0.0195137	0.00975683	+	0.999952i	$0.496894\pi$
$788$	0	0
$789$	−29.7606	−1.05951
$790$	0	0
$791$	3.10100	0.110259
$792$	0	0
$793$	11.4283	0.405831
$794$	0	0
$795$	2.25334	0.0799176
$796$	0	0
$797$	−31.3517	−1.11053	−0.555267	−	0.831672i	$-0.687384\pi$
$797$	−31.3517	−1.11053	−0.555267	+	0.831672i	$0.687384\pi$
$798$	0	0
$799$	−7.22193	−0.255494
$800$	0	0
$801$	−0.952996	−0.0336725
$802$	0	0
$803$	−64.1647	−2.26432
$804$	0	0
$805$	7.80120	0.274956
$806$	0	0
$807$	25.9606	0.913857
$808$	0	0
$809$	−7.93121	−0.278846	−0.139423	−	0.990233i	$-0.544525\pi$
$809$	−7.93121	−0.278846	−0.139423	+	0.990233i	$0.544525\pi$
$810$	0	0
$811$	41.1458	1.44482	0.722412	−	0.691463i	$-0.243033\pi$
$811$	41.1458	1.44482	0.722412	+	0.691463i	$0.243033\pi$
$812$	0	0
$813$	−23.6983	−0.831134
$814$	0	0
$815$	−62.5049	−2.18945
$816$	0	0
$817$	−28.1952	−0.986425
$818$	0	0
$819$	4.13843	0.144609
$820$	0	0
$821$	26.6623	0.930521	0.465261	−	0.885174i	$-0.345961\pi$
$821$	26.6623	0.930521	0.465261	+	0.885174i	$0.345961\pi$
$822$	0	0
$823$	37.3733	1.30275	0.651376	−	0.758755i	$-0.274192\pi$
$823$	37.3733	1.30275	0.651376	+	0.758755i	$0.274192\pi$
$824$	0	0
$825$	63.2285	2.20133
$826$	0	0
$827$	−4.97031	−0.172835	−0.0864174	−	0.996259i	$-0.527542\pi$
$827$	−4.97031	−0.172835	−0.0864174	+	0.996259i	$0.527542\pi$
$828$	0	0
$829$	−0.0401144	−0.00139323	−0.000696615	−	1.00000i	$-0.500222\pi$
$829$	−0.0401144	−0.00139323	−0.000696615		1.00000i	$0.500222\pi$
$830$	0	0
$831$	21.1066	0.732181
$832$	0	0
$833$	−11.5136	−0.398923
$834$	0	0
$835$	80.3691	2.78129
$836$	0	0
$837$	−59.3738	−2.05226
$838$	0	0
$839$	5.34593	0.184562	0.0922809	−	0.995733i	$-0.470584\pi$
$839$	5.34593	0.184562	0.0922809	+	0.995733i	$0.470584\pi$
$840$	0	0
$841$	−11.4951	−0.396384
$842$	0	0
$843$	40.1972	1.38447
$844$	0	0
$845$	13.3575	0.459512
$846$	0	0
$847$	26.0686	0.895728
$848$	0	0
$849$	−1.62622	−0.0558118
$850$	0	0
$851$	−6.89286	−0.236284
$852$	0	0
$853$	46.3126	1.58571	0.792856	−	0.609409i	$-0.208593\pi$
$853$	46.3126	1.58571	0.792856	+	0.609409i	$0.208593\pi$
$854$	0	0
$855$	−7.53182	−0.257583
$856$	0	0
$857$	1.78178	0.0608643	0.0304321	−	0.999537i	$-0.490312\pi$
$857$	1.78178	0.0608643	0.0304321	+	0.999537i	$0.490312\pi$
$858$	0	0
$859$	12.1905	0.415935	0.207967	−	0.978136i	$-0.433315\pi$
$859$	12.1905	0.415935	0.207967	+	0.978136i	$0.433315\pi$
$860$	0	0
$861$	−19.5586	−0.666557
$862$	0	0
$863$	55.7303	1.89708	0.948541	−	0.316653i	$-0.102559\pi$
$863$	55.7303	1.89708	0.948541	+	0.316653i	$0.102559\pi$
$864$	0	0
$865$	−32.3238	−1.09904
$866$	0	0
$867$	17.3965	0.590815
$868$	0	0
$869$	24.7533	0.839697
$870$	0	0
$871$	21.2302	0.719356
$872$	0	0
$873$	7.85071	0.265706
$874$	0	0
$875$	14.2079	0.480314
$876$	0	0
$877$	11.1729	0.377284	0.188642	−	0.982046i	$-0.439591\pi$
$877$	11.1729	0.377284	0.188642	+	0.982046i	$0.439591\pi$
$878$	0	0
$879$	27.0104	0.911039
$880$	0	0
$881$	13.1185	0.441972	0.220986	−	0.975277i	$-0.429072\pi$
$881$	13.1185	0.441972	0.220986	+	0.975277i	$0.429072\pi$
$882$	0	0
$883$	11.4079	0.383906	0.191953	−	0.981404i	$-0.438518\pi$
$883$	11.4079	0.383906	0.191953	+	0.981404i	$0.438518\pi$
$884$	0	0
$885$	45.6238	1.53363
$886$	0	0
$887$	45.4677	1.52666	0.763328	−	0.646012i	$-0.223564\pi$
$887$	45.4677	1.52666	0.763328	+	0.646012i	$0.223564\pi$
$888$	0	0
$889$	1.59413	0.0534655
$890$	0	0
$891$	34.5707	1.15816
$892$	0	0
$893$	−9.29392	−0.311009
$894$	0	0
$895$	−0.511683	−0.0171037
$896$	0	0
$897$	9.34920	0.312161
$898$	0	0
$899$	−44.2775	−1.47674
$900$	0	0
$901$	−0.978772	−0.0326076
$902$	0	0
$903$	20.5114	0.682575
$904$	0	0
$905$	9.18494	0.305318
$906$	0	0
$907$	31.6525	1.05100	0.525502	−	0.850793i	$-0.323878\pi$
$907$	31.6525	1.05100	0.525502	+	0.850793i	$0.323878\pi$
$908$	0	0
$909$	6.53014	0.216591
$910$	0	0
$911$	−41.1501	−1.36336	−0.681681	−	0.731649i	$-0.738751\pi$
$911$	−41.1501	−1.36336	−0.681681	+	0.731649i	$0.738751\pi$
$912$	0	0
$913$	44.2832	1.46556
$914$	0	0
$915$	15.1309	0.500213
$916$	0	0
$917$	1.02938	0.0339931
$918$	0	0
$919$	1.42185	0.0469025	0.0234512	−	0.999725i	$-0.492535\pi$
$919$	1.42185	0.0469025	0.0234512	+	0.999725i	$0.492535\pi$
$920$	0	0
$921$	−31.2927	−1.03113
$922$	0	0
$923$	−16.4753	−0.542292
$924$	0	0
$925$	−35.4447	−1.16541
$926$	0	0
$927$	−1.18690	−0.0389829
$928$	0	0
$929$	−52.8710	−1.73464	−0.867320	−	0.497751i	$-0.834159\pi$
$929$	−52.8710	−1.73464	−0.867320	+	0.497751i	$0.834159\pi$
$930$	0	0
$931$	−14.8169	−0.485604
$932$	0	0
$933$	−3.06281	−0.100272
$934$	0	0
$935$	−45.2015	−1.47825
$936$	0	0
$937$	−5.44757	−0.177964	−0.0889821	−	0.996033i	$-0.528361\pi$
$937$	−5.44757	−0.177964	−0.0889821	+	0.996033i	$0.528361\pi$
$938$	0	0
$939$	2.72976	0.0890823
$940$	0	0
$941$	−42.4414	−1.38355	−0.691775	−	0.722113i	$-0.743171\pi$
$941$	−42.4414	−1.38355	−0.691775	+	0.722113i	$0.743171\pi$
$942$	0	0
$943$	13.3670	0.435288
$944$	0	0
$945$	29.0703	0.945658
$946$	0	0
$947$	35.0305	1.13834	0.569170	−	0.822220i	$-0.307265\pi$
$947$	35.0305	1.13834	0.569170	+	0.822220i	$0.307265\pi$
$948$	0	0
$949$	−48.7876	−1.58371
$950$	0	0
$951$	37.6349	1.22039
$952$	0	0
$953$	−19.6541	−0.636660	−0.318330	−	0.947980i	$-0.603122\pi$
$953$	−19.6541	−0.636660	−0.318330	+	0.947980i	$0.603122\pi$
$954$	0	0
$955$	58.7567	1.90132
$956$	0	0
$957$	34.1695	1.10454
$958$	0	0
$959$	−23.5844	−0.761580
$960$	0	0
$961$	80.9971	2.61281
$962$	0	0
$963$	2.02844	0.0653656
$964$	0	0
$965$	35.1407	1.13122
$966$	0	0
$967$	52.1177	1.67599	0.837995	−	0.545677i	$-0.183728\pi$
$967$	52.1177	1.67599	0.837995	+	0.545677i	$0.183728\pi$
$968$	0	0
$969$	−10.8143	−0.347406
$970$	0	0
$971$	−40.6668	−1.30506	−0.652530	−	0.757763i	$-0.726292\pi$
$971$	−40.6668	−1.30506	−0.652530	+	0.757763i	$0.726292\pi$
$972$	0	0
$973$	1.22722	0.0393430
$974$	0	0
$975$	48.0758	1.53966
$976$	0	0
$977$	2.89004	0.0924607	0.0462303	−	0.998931i	$-0.485279\pi$
$977$	2.89004	0.0924607	0.0462303	+	0.998931i	$0.485279\pi$
$978$	0	0
$979$	−7.36020	−0.235233
$980$	0	0
$981$	−2.66268	−0.0850129
$982$	0	0
$983$	−14.3720	−0.458396	−0.229198	−	0.973380i	$-0.573610\pi$
$983$	−14.3720	−0.458396	−0.229198	+	0.973380i	$0.573610\pi$
$984$	0	0
$985$	−36.4762	−1.16223
$986$	0	0
$987$	6.76112	0.215209
$988$	0	0
$989$	−14.0181	−0.445749
$990$	0	0
$991$	−51.7294	−1.64324	−0.821619	−	0.570038i	$-0.806929\pi$
$991$	−51.7294	−1.64324	−0.821619	+	0.570038i	$0.806929\pi$
$992$	0	0
$993$	34.7901	1.10403
$994$	0	0
$995$	−12.2027	−0.386852
$996$	0	0
$997$	−19.5134	−0.617996	−0.308998	−	0.951063i	$-0.599994\pi$
$997$	−19.5134	−0.617996	−0.308998	+	0.951063i	$0.599994\pi$
$998$	0	0
$999$	−25.6855	−0.812654

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4096.2.a.i.1.4		8
4.3	odd	2		4096.2.a.s.1.6		8
8.3	odd	2	inner	4096.2.a.i.1.3		8
8.5	even	2		4096.2.a.s.1.5		8
64.3	odd	16		1024.2.g.f.641.1	yes	16
64.5	even	16		1024.2.g.g.897.1	yes	16
64.11	odd	16		1024.2.g.a.385.4	yes	16
64.13	even	16		1024.2.g.g.129.1	yes	16
64.19	odd	16		1024.2.g.d.129.1	yes	16
64.21	even	16		1024.2.g.f.385.4	yes	16
64.27	odd	16		1024.2.g.d.897.1	yes	16
64.29	even	16		1024.2.g.a.641.1	yes	16
64.35	odd	16		1024.2.g.a.641.4	yes	16
64.37	even	16		1024.2.g.d.897.4	yes	16
64.43	odd	16		1024.2.g.f.385.1	yes	16
64.45	even	16		1024.2.g.d.129.4	yes	16
64.51	odd	16		1024.2.g.g.129.4	yes	16
64.53	even	16		1024.2.g.a.385.1	✓	16
64.59	odd	16		1024.2.g.g.897.4	yes	16
64.61	even	16		1024.2.g.f.641.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1024.2.g.a.385.1	✓	16	64.53	even	16
1024.2.g.a.385.4	yes	16	64.11	odd	16
1024.2.g.a.641.1	yes	16	64.29	even	16
1024.2.g.a.641.4	yes	16	64.35	odd	16
1024.2.g.d.129.1	yes	16	64.19	odd	16
1024.2.g.d.129.4	yes	16	64.45	even	16
1024.2.g.d.897.1	yes	16	64.27	odd	16
1024.2.g.d.897.4	yes	16	64.37	even	16
1024.2.g.f.385.1	yes	16	64.43	odd	16
1024.2.g.f.385.4	yes	16	64.21	even	16
1024.2.g.f.641.1	yes	16	64.3	odd	16
1024.2.g.f.641.4	yes	16	64.61	even	16
1024.2.g.g.129.1	yes	16	64.13	even	16
1024.2.g.g.129.4	yes	16	64.51	odd	16
1024.2.g.g.897.1	yes	16	64.5	even	16
1024.2.g.g.897.4	yes	16	64.59	odd	16
4096.2.a.i.1.3		8	8.3	odd	2	inner
4096.2.a.i.1.4		8	1.1	even	1	trivial
4096.2.a.s.1.5		8	8.5	even	2
4096.2.a.s.1.6		8	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:	\( N \)	\(=\)	\( 4096 = 2^{12} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4096.a (trivial)

\(f(q)\)	\(=\)	\(q-1.51764 q^{3} +3.56960 q^{5} +1.45158 q^{7} -0.696775 q^{9} +O(q^{10})\)	\(q-1.51764 q^{3} +3.56960 q^{5} +1.45158 q^{7} -0.696775 q^{9} -5.38134 q^{11} -4.09170 q^{13} -5.41736 q^{15} +2.35311 q^{17} +3.02823 q^{19} -2.20297 q^{21} +1.50557 q^{23} +7.74202 q^{25} +5.61037 q^{27} +4.18388 q^{29} -10.5829 q^{31} +8.16693 q^{33} +5.18154 q^{35} -4.57823 q^{37} +6.20972 q^{39} +8.87832 q^{41} -9.31079 q^{43} -2.48720 q^{45} -3.06910 q^{47} -4.89293 q^{49} -3.57117 q^{51} -0.415948 q^{53} -19.2092 q^{55} -4.59575 q^{57} -8.42178 q^{59} -2.79305 q^{61} -1.01142 q^{63} -14.6057 q^{65} -5.18859 q^{67} -2.28492 q^{69} +4.02652 q^{71} +11.9236 q^{73} -11.7496 q^{75} -7.81143 q^{77} -4.59983 q^{79} -6.42418 q^{81} -8.22902 q^{83} +8.39967 q^{85} -6.34962 q^{87} +1.36773 q^{89} -5.93942 q^{91} +16.0610 q^{93} +10.8096 q^{95} -11.2672 q^{97} +3.74958 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3}+O(q^{10}) \) 8 * q - 8 * q^3	\( 8 q - 8 q^{3} - 8 q^{11} + 8 q^{17} - 8 q^{25} - 8 q^{27} + 40 q^{33} + 8 q^{35} + 32 q^{41} - 56 q^{43} + 8 q^{49} - 48 q^{51} + 8 q^{57} - 32 q^{59} - 16 q^{65} - 24 q^{67} - 8 q^{73} + 16 q^{81} - 48 q^{83} - 8 q^{89} - 8 q^{91} + 8 q^{97} - 64 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 - 8 * q^11 + 8 * q^17 - 8 * q^25 - 8 * q^27 + 40 * q^33 + 8 * q^35 + 32 * q^41 - 56 * q^43 + 8 * q^49 - 48 * q^51 + 8 * q^57 - 32 * q^59 - 16 * q^65 - 24 * q^67 - 8 * q^73 + 16 * q^81 - 48 * q^83 - 8 * q^89 - 8 * q^91 + 8 * q^97 - 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more