LMFDB - Embedded newform 4056.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(1,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.36147$ of defining polynomial
Character	$\chi$	$=$	4056.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.93800 q^{5} +1.78493 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.93800 q^{5} +1.78493 q^{7} +1.00000 q^{9} +4.72294 q^{11} +3.93800 q^{15} +0.784934 q^{17} +5.15307 q^{19} -1.78493 q^{21} +4.72294 q^{23} +10.5079 q^{25} -1.00000 q^{27} -7.93800 q^{29} -2.93800 q^{31} -4.72294 q^{33} -7.02908 q^{35} +1.21507 q^{37} -8.66094 q^{41} -6.21507 q^{43} -3.93800 q^{45} +0.722938 q^{47} -3.81401 q^{49} -0.784934 q^{51} -13.8140 q^{53} -18.5989 q^{55} -5.15307 q^{57} +0.430132 q^{59} +4.15307 q^{61} +1.78493 q^{63} +9.78493 q^{67} -4.72294 q^{69} +8.72294 q^{71} +4.87601 q^{73} -10.5079 q^{75} +8.43013 q^{77} +16.3839 q^{79} +1.00000 q^{81} +6.59894 q^{83} -3.09107 q^{85} +7.93800 q^{87} +15.8760 q^{89} +2.93800 q^{93} -20.2928 q^{95} +3.06200 q^{97} +4.72294 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 6 q^{19} - 3 q^{21} + 15 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} + 12 q^{35} + 6 q^{37} - 21 q^{43} - 12 q^{47} + 24 q^{49} - 6 q^{53} - 18 q^{55} - 6 q^{57} + 6 q^{59} + 3 q^{61} + 3 q^{63} + 27 q^{67} + 12 q^{71} - 9 q^{73} - 15 q^{75} + 30 q^{77} + 9 q^{79} + 3 q^{81} - 18 q^{83} + 12 q^{85} + 12 q^{87} + 24 q^{89} - 3 q^{93} - 42 q^{95} + 21 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 6 * q^19 - 3 * q^21 + 15 * q^25 - 3 * q^27 - 12 * q^29 + 3 * q^31 + 12 * q^35 + 6 * q^37 - 21 * q^43 - 12 * q^47 + 24 * q^49 - 6 * q^53 - 18 * q^55 - 6 * q^57 + 6 * q^59 + 3 * q^61 + 3 * q^63 + 27 * q^67 + 12 * q^71 - 9 * q^73 - 15 * q^75 + 30 * q^77 + 9 * q^79 + 3 * q^81 - 18 * q^83 + 12 * q^85 + 12 * q^87 + 24 * q^89 - 3 * q^93 - 42 * q^95 + 21 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.93800	−1.76113	−0.880564	−	0.473927i	$-0.842836\pi$
$5$	−3.93800	−1.76113	−0.880564	+	0.473927i	$0.842836\pi$
$6$	0	0
$7$	1.78493	0.674642	0.337321	−	0.941390i	$-0.390479\pi$
$7$	1.78493	0.674642	0.337321	+	0.941390i	$0.390479\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.72294	1.42402	0.712010	−	0.702170i	$-0.247785\pi$
$11$	4.72294	1.42402	0.712010	+	0.702170i	$0.247785\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.93800	1.01679
$16$	0	0
$17$	0.784934	0.190374	0.0951872	−	0.995459i	$-0.469655\pi$
$17$	0.784934	0.190374	0.0951872	+	0.995459i	$0.469655\pi$
$18$	0	0
$19$	5.15307	1.18220	0.591098	−	0.806600i	$-0.298695\pi$
$19$	5.15307	1.18220	0.591098	+	0.806600i	$0.298695\pi$
$20$	0	0
$21$	−1.78493	−0.389505
$22$	0	0
$23$	4.72294	0.984801	0.492400	−	0.870369i	$-0.336120\pi$
$23$	4.72294	0.984801	0.492400	+	0.870369i	$0.336120\pi$
$24$	0	0
$25$	10.5079	2.10157
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.93800	−1.47405	−0.737025	−	0.675865i	$-0.763770\pi$
$29$	−7.93800	−1.47405	−0.737025	+	0.675865i	$0.763770\pi$
$30$	0	0
$31$	−2.93800	−0.527681	−0.263841	−	0.964566i	$-0.584989\pi$
$31$	−2.93800	−0.527681	−0.263841	+	0.964566i	$0.584989\pi$
$32$	0	0
$33$	−4.72294	−0.822158
$34$	0	0
$35$	−7.02908	−1.18813
$36$	0	0
$37$	1.21507	0.199756	0.0998778	−	0.995000i	$-0.468155\pi$
$37$	1.21507	0.199756	0.0998778	+	0.995000i	$0.468155\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.66094	−1.35261	−0.676306	−	0.736621i	$-0.736420\pi$
$41$	−8.66094	−1.35261	−0.676306	+	0.736621i	$0.736420\pi$
$42$	0	0
$43$	−6.21507	−0.947789	−0.473894	−	0.880582i	$-0.657152\pi$
$43$	−6.21507	−0.947789	−0.473894	+	0.880582i	$0.657152\pi$
$44$	0	0
$45$	−3.93800	−0.587043
$46$	0	0
$47$	0.722938	0.105451	0.0527256	−	0.998609i	$-0.483209\pi$
$47$	0.722938	0.105451	0.0527256	+	0.998609i	$0.483209\pi$
$48$	0	0
$49$	−3.81401	−0.544859
$50$	0	0
$51$	−0.784934	−0.109913
$52$	0	0
$53$	−13.8140	−1.89750	−0.948750	−	0.316027i	$-0.897651\pi$
$53$	−13.8140	−1.89750	−0.948750	+	0.316027i	$0.897651\pi$
$54$	0	0
$55$	−18.5989	−2.50788
$56$	0	0
$57$	−5.15307	−0.682541
$58$	0	0
$59$	0.430132	0.0559984	0.0279992	−	0.999608i	$-0.491086\pi$
$59$	0.430132	0.0559984	0.0279992	+	0.999608i	$0.491086\pi$
$60$	0	0
$61$	4.15307	0.531746	0.265873	−	0.964008i	$-0.414340\pi$
$61$	4.15307	0.531746	0.265873	+	0.964008i	$0.414340\pi$
$62$	0	0
$63$	1.78493	0.224881
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.78493	1.19542	0.597710	−	0.801712i	$-0.296077\pi$
$67$	9.78493	1.19542	0.597710	+	0.801712i	$0.296077\pi$
$68$	0	0
$69$	−4.72294	−0.568575
$70$	0	0
$71$	8.72294	1.03522	0.517611	−	0.855616i	$-0.326821\pi$
$71$	8.72294	1.03522	0.517611	+	0.855616i	$0.326821\pi$
$72$	0	0
$73$	4.87601	0.570693	0.285347	−	0.958424i	$-0.407891\pi$
$73$	4.87601	0.570693	0.285347	+	0.958424i	$0.407891\pi$
$74$	0	0
$75$	−10.5079	−1.21334
$76$	0	0
$77$	8.43013	0.960703
$78$	0	0
$79$	16.3839	1.84333	0.921665	−	0.387986i	$-0.126829\pi$
$79$	16.3839	1.84333	0.921665	+	0.387986i	$0.126829\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.59894	0.724328	0.362164	−	0.932114i	$-0.382038\pi$
$83$	6.59894	0.724328	0.362164	+	0.932114i	$0.382038\pi$
$84$	0	0
$85$	−3.09107	−0.335274
$86$	0	0
$87$	7.93800	0.851043
$88$	0	0
$89$	15.8760	1.68285	0.841427	−	0.540371i	$-0.181716\pi$
$89$	15.8760	1.68285	0.841427	+	0.540371i	$0.181716\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.93800	0.304657
$94$	0	0
$95$	−20.2928	−2.08200
$96$	0	0
$97$	3.06200	0.310899	0.155449	−	0.987844i	$-0.450317\pi$
$97$	3.06200	0.310899	0.155449	+	0.987844i	$0.450317\pi$
$98$	0	0
$99$	4.72294	0.474673
$100$	0	0
$101$	−1.63186	−0.162377	−0.0811883	−	0.996699i	$-0.525872\pi$
$101$	−1.63186	−0.162377	−0.0811883	+	0.996699i	$0.525872\pi$
$102$	0	0
$103$	−7.66094	−0.754855	−0.377427	−	0.926039i	$-0.623191\pi$
$103$	−7.66094	−0.754855	−0.377427	+	0.926039i	$0.623191\pi$
$104$	0	0
$105$	7.02908	0.685968
$106$	0	0
$107$	0.722938	0.0698890	0.0349445	−	0.999389i	$-0.488875\pi$
$107$	0.722938	0.0698890	0.0349445	+	0.999389i	$0.488875\pi$
$108$	0	0
$109$	5.36814	0.514174	0.257087	−	0.966388i	$-0.417237\pi$
$109$	5.36814	0.514174	0.257087	+	0.966388i	$0.417237\pi$
$110$	0	0
$111$	−1.21507	−0.115329
$112$	0	0
$113$	13.2151	1.24317	0.621584	−	0.783347i	$-0.286489\pi$
$113$	13.2151	1.24317	0.621584	+	0.783347i	$0.286489\pi$
$114$	0	0
$115$	−18.5989	−1.73436
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.40106	0.128435
$120$	0	0
$121$	11.3061	1.02783
$122$	0	0
$123$	8.66094	0.780931
$124$	0	0
$125$	−21.6900	−1.94001
$126$	0	0
$127$	16.3839	1.45383	0.726917	−	0.686725i	$-0.240952\pi$
$127$	16.3839	1.45383	0.726917	+	0.686725i	$0.240952\pi$
$128$	0	0
$129$	6.21507	0.547206
$130$	0	0
$131$	−21.0157	−1.83615	−0.918077	−	0.396402i	$-0.870259\pi$
$131$	−21.0157	−1.83615	−0.918077	+	0.396402i	$0.870259\pi$
$132$	0	0
$133$	9.19789	0.797558
$134$	0	0
$135$	3.93800	0.338929
$136$	0	0
$137$	12.5369	1.07110	0.535552	−	0.844502i	$-0.320104\pi$
$137$	12.5369	1.07110	0.535552	+	0.844502i	$0.320104\pi$
$138$	0	0
$139$	8.81401	0.747595	0.373797	−	0.927510i	$-0.378056\pi$
$139$	8.81401	0.747595	0.373797	+	0.927510i	$0.378056\pi$
$140$	0	0
$141$	−0.722938	−0.0608823
$142$	0	0
$143$	0	0
$144$	0	0
$145$	31.2599	2.59599
$146$	0	0
$147$	3.81401	0.314574
$148$	0	0
$149$	3.63186	0.297534	0.148767	−	0.988872i	$-0.452470\pi$
$149$	3.63186	0.297534	0.148767	+	0.988872i	$0.452470\pi$
$150$	0	0
$151$	−14.5989	−1.18805	−0.594023	−	0.804448i	$-0.702461\pi$
$151$	−14.5989	−1.18805	−0.594023	+	0.804448i	$0.702461\pi$
$152$	0	0
$153$	0.784934	0.0634582
$154$	0	0
$155$	11.5699	0.929314
$156$	0	0
$157$	−15.5989	−1.24493	−0.622466	−	0.782647i	$-0.713869\pi$
$157$	−15.5989	−1.24493	−0.622466	+	0.782647i	$0.713869\pi$
$158$	0	0
$159$	13.8140	1.09552
$160$	0	0
$161$	8.43013	0.664387
$162$	0	0
$163$	10.9380	0.856731	0.428365	−	0.903606i	$-0.359090\pi$
$163$	10.9380	0.856731	0.428365	+	0.903606i	$0.359090\pi$
$164$	0	0
$165$	18.5989	1.44793
$166$	0	0
$167$	−20.5856	−1.59296	−0.796481	−	0.604663i	$-0.793308\pi$
$167$	−20.5856	−1.59296	−0.796481	+	0.604663i	$0.793308\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	5.15307	0.394065
$172$	0	0
$173$	−0.306139	−0.0232753	−0.0116377	−	0.999932i	$-0.503704\pi$
$173$	−0.306139	−0.0232753	−0.0116377	+	0.999932i	$0.503704\pi$
$174$	0	0
$175$	18.7559	1.41781
$176$	0	0
$177$	−0.430132	−0.0323307
$178$	0	0
$179$	−2.84693	−0.212790	−0.106395	−	0.994324i	$-0.533931\pi$
$179$	−2.84693	−0.212790	−0.106395	+	0.994324i	$0.533931\pi$
$180$	0	0
$181$	13.2151	0.982268	0.491134	−	0.871084i	$-0.336583\pi$
$181$	13.2151	0.982268	0.491134	+	0.871084i	$0.336583\pi$
$182$	0	0
$183$	−4.15307	−0.307004
$184$	0	0
$185$	−4.78493	−0.351795
$186$	0	0
$187$	3.70719	0.271097
$188$	0	0
$189$	−1.78493	−0.129835
$190$	0	0
$191$	−3.56987	−0.258307	−0.129153	−	0.991625i	$-0.541226\pi$
$191$	−3.56987	−0.258307	−0.129153	+	0.991625i	$0.541226\pi$
$192$	0	0
$193$	12.4459	0.895874	0.447937	−	0.894065i	$-0.352159\pi$
$193$	12.4459	0.895874	0.447937	+	0.894065i	$0.352159\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.73627	−0.622434	−0.311217	−	0.950339i	$-0.600737\pi$
$197$	−8.73627	−0.622434	−0.311217	+	0.950339i	$0.600737\pi$
$198$	0	0
$199$	−16.0911	−1.14067	−0.570333	−	0.821414i	$-0.693186\pi$
$199$	−16.0911	−1.14067	−0.570333	+	0.821414i	$0.693186\pi$
$200$	0	0
$201$	−9.78493	−0.690176
$202$	0	0
$203$	−14.1688	−0.994456
$204$	0	0
$205$	34.1068	2.38212
$206$	0	0
$207$	4.72294	0.328267
$208$	0	0
$209$	24.3376	1.68347
$210$	0	0
$211$	4.38388	0.301799	0.150899	−	0.988549i	$-0.451783\pi$
$211$	4.38388	0.301799	0.150899	+	0.988549i	$0.451783\pi$
$212$	0	0
$213$	−8.72294	−0.597686
$214$	0	0
$215$	24.4750	1.66918
$216$	0	0
$217$	−5.24414	−0.355996
$218$	0	0
$219$	−4.87601	−0.329490
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	10.5079	0.700525
$226$	0	0
$227$	−17.7387	−1.17736	−0.588679	−	0.808367i	$-0.700352\pi$
$227$	−17.7387	−1.17736	−0.588679	+	0.808367i	$0.700352\pi$
$228$	0	0
$229$	11.4459	0.756365	0.378182	−	0.925731i	$-0.376549\pi$
$229$	11.4459	0.756365	0.378182	+	0.925731i	$0.376549\pi$
$230$	0	0
$231$	−8.43013	−0.554662
$232$	0	0
$233$	−9.75201	−0.638876	−0.319438	−	0.947607i	$-0.603494\pi$
$233$	−9.75201	−0.638876	−0.319438	+	0.947607i	$0.603494\pi$
$234$	0	0
$235$	−2.84693	−0.185713
$236$	0	0
$237$	−16.3839	−1.06425
$238$	0	0
$239$	6.59894	0.426850	0.213425	−	0.976959i	$-0.431538\pi$
$239$	6.59894	0.426850	0.213425	+	0.976959i	$0.431538\pi$
$240$	0	0
$241$	23.8140	1.53400	0.766998	−	0.641650i	$-0.221750\pi$
$241$	23.8140	1.53400	0.766998	+	0.641650i	$0.221750\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	15.0196	0.959566
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.59894	−0.418191
$250$	0	0
$251$	9.44588	0.596218	0.298109	−	0.954532i	$-0.403644\pi$
$251$	9.44588	0.596218	0.298109	+	0.954532i	$0.403644\pi$
$252$	0	0
$253$	22.3061	1.40237
$254$	0	0
$255$	3.09107	0.193570
$256$	0	0
$257$	−12.6609	−0.789768	−0.394884	−	0.918731i	$-0.629215\pi$
$257$	−12.6609	−0.789768	−0.394884	+	0.918731i	$0.629215\pi$
$258$	0	0
$259$	2.16881	0.134763
$260$	0	0
$261$	−7.93800	−0.491350
$262$	0	0
$263$	−0.292806	−0.0180552	−0.00902758	−	0.999959i	$-0.502874\pi$
$263$	−0.292806	−0.0180552	−0.00902758	+	0.999959i	$0.502874\pi$
$264$	0	0
$265$	54.3996	3.34174
$266$	0	0
$267$	−15.8760	−0.971596
$268$	0	0
$269$	5.56987	0.339601	0.169800	−	0.985478i	$-0.445688\pi$
$269$	5.56987	0.339601	0.169800	+	0.985478i	$0.445688\pi$
$270$	0	0
$271$	11.9537	0.726138	0.363069	−	0.931762i	$-0.381729\pi$
$271$	11.9537	0.726138	0.363069	+	0.931762i	$0.381729\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	49.6280	2.99268
$276$	0	0
$277$	18.6609	1.12123	0.560614	−	0.828078i	$-0.310565\pi$
$277$	18.6609	1.12123	0.560614	+	0.828078i	$0.310565\pi$
$278$	0	0
$279$	−2.93800	−0.175894
$280$	0	0
$281$	4.96708	0.296311	0.148156	−	0.988964i	$-0.452666\pi$
$281$	4.96708	0.296311	0.148156	+	0.988964i	$0.452666\pi$
$282$	0	0
$283$	−10.8007	−0.642034	−0.321017	−	0.947073i	$-0.604025\pi$
$283$	−10.8007	−0.642034	−0.321017	+	0.947073i	$0.604025\pi$
$284$	0	0
$285$	20.2928	1.20204
$286$	0	0
$287$	−15.4592	−0.912528
$288$	0	0
$289$	−16.3839	−0.963758
$290$	0	0
$291$	−3.06200	−0.179497
$292$	0	0
$293$	−26.2441	−1.53320	−0.766600	−	0.642125i	$-0.778053\pi$
$293$	−26.2441	−1.53320	−0.766600	+	0.642125i	$0.778053\pi$
$294$	0	0
$295$	−1.69386	−0.0986204
$296$	0	0
$297$	−4.72294	−0.274053
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−11.0935	−0.639418
$302$	0	0
$303$	1.63186	0.0937482
$304$	0	0
$305$	−16.3548	−0.936473
$306$	0	0
$307$	21.5369	1.22918	0.614589	−	0.788847i	$-0.289322\pi$
$307$	21.5369	1.22918	0.614589	+	0.788847i	$0.289322\pi$
$308$	0	0
$309$	7.66094	0.435816
$310$	0	0
$311$	21.1531	1.19948	0.599740	−	0.800195i	$-0.295271\pi$
$311$	21.1531	1.19948	0.599740	+	0.800195i	$0.295271\pi$
$312$	0	0
$313$	9.79827	0.553831	0.276915	−	0.960894i	$-0.410688\pi$
$313$	9.79827	0.553831	0.276915	+	0.960894i	$0.410688\pi$
$314$	0	0
$315$	−7.02908	−0.396044
$316$	0	0
$317$	22.5236	1.26505	0.632526	−	0.774539i	$-0.282018\pi$
$317$	22.5236	1.26505	0.632526	+	0.774539i	$0.282018\pi$
$318$	0	0
$319$	−37.4907	−2.09908
$320$	0	0
$321$	−0.722938	−0.0403504
$322$	0	0
$323$	4.04482	0.225060
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.36814	−0.296859
$328$	0	0
$329$	1.29040	0.0711418
$330$	0	0
$331$	21.0620	1.15767	0.578836	−	0.815444i	$-0.303507\pi$
$331$	21.0620	1.15767	0.578836	+	0.815444i	$0.303507\pi$
$332$	0	0
$333$	1.21507	0.0665852
$334$	0	0
$335$	−38.5331	−2.10529
$336$	0	0
$337$	−2.01574	−0.109805	−0.0549023	−	0.998492i	$-0.517485\pi$
$337$	−2.01574	−0.109805	−0.0549023	+	0.998492i	$0.517485\pi$
$338$	0	0
$339$	−13.2151	−0.717744
$340$	0	0
$341$	−13.8760	−0.751428
$342$	0	0
$343$	−19.3023	−1.04223
$344$	0	0
$345$	18.5989	1.00133
$346$	0	0
$347$	22.5989	1.21317	0.606587	−	0.795017i	$-0.292538\pi$
$347$	22.5989	1.21317	0.606587	+	0.795017i	$0.292538\pi$
$348$	0	0
$349$	−4.25989	−0.228026	−0.114013	−	0.993479i	$-0.536371\pi$
$349$	−4.25989	−0.228026	−0.114013	+	0.993479i	$0.536371\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.52120	0.187415	0.0937074	−	0.995600i	$-0.470128\pi$
$353$	3.52120	0.187415	0.0937074	+	0.995600i	$0.470128\pi$
$354$	0	0
$355$	−34.3510	−1.82316
$356$	0	0
$357$	−1.40106	−0.0741517
$358$	0	0
$359$	9.15307	0.483081	0.241540	−	0.970391i	$-0.422347\pi$
$359$	9.15307	0.483081	0.241540	+	0.970391i	$0.422347\pi$
$360$	0	0
$361$	7.55412	0.397586
$362$	0	0
$363$	−11.3061	−0.593418
$364$	0	0
$365$	−19.2017	−1.00506
$366$	0	0
$367$	2.21507	0.115626	0.0578128	−	0.998327i	$-0.481587\pi$
$367$	2.21507	0.115626	0.0578128	+	0.998327i	$0.481587\pi$
$368$	0	0
$369$	−8.66094	−0.450871
$370$	0	0
$371$	−24.6571	−1.28013
$372$	0	0
$373$	−28.4592	−1.47356	−0.736781	−	0.676131i	$-0.763655\pi$
$373$	−28.4592	−1.47356	−0.736781	+	0.676131i	$0.763655\pi$
$374$	0	0
$375$	21.6900	1.12007
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−34.0119	−1.74707	−0.873537	−	0.486758i	$-0.838179\pi$
$379$	−34.0119	−1.74707	−0.873537	+	0.486758i	$0.838179\pi$
$380$	0	0
$381$	−16.3839	−0.839372
$382$	0	0
$383$	26.8918	1.37410	0.687052	−	0.726608i	$-0.258904\pi$
$383$	26.8918	1.37410	0.687052	+	0.726608i	$0.258904\pi$
$384$	0	0
$385$	−33.1979	−1.69192
$386$	0	0
$387$	−6.21507	−0.315930
$388$	0	0
$389$	34.2756	1.73784	0.868922	−	0.494950i	$-0.164813\pi$
$389$	34.2756	1.73784	0.868922	+	0.494950i	$0.164813\pi$
$390$	0	0
$391$	3.70719	0.187481
$392$	0	0
$393$	21.0157	1.06010
$394$	0	0
$395$	−64.5198	−3.24634
$396$	0	0
$397$	−27.3996	−1.37515	−0.687574	−	0.726115i	$-0.741324\pi$
$397$	−27.3996	−1.37515	−0.687574	+	0.726115i	$0.741324\pi$
$398$	0	0
$399$	−9.19789	−0.460470
$400$	0	0
$401$	17.3972	0.868775	0.434388	−	0.900726i	$-0.356965\pi$
$401$	17.3972	0.868775	0.434388	+	0.900726i	$0.356965\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−3.93800	−0.195681
$406$	0	0
$407$	5.73868	0.284456
$408$	0	0
$409$	35.5198	1.75634	0.878170	−	0.478349i	$-0.158765\pi$
$409$	35.5198	1.75634	0.878170	+	0.478349i	$0.158765\pi$
$410$	0	0
$411$	−12.5369	−0.618402
$412$	0	0
$413$	0.767757	0.0377789
$414$	0	0
$415$	−25.9867	−1.27564
$416$	0	0
$417$	−8.81401	−0.431624
$418$	0	0
$419$	7.13974	0.348799	0.174399	−	0.984675i	$-0.444202\pi$
$419$	7.13974	0.348799	0.174399	+	0.984675i	$0.444202\pi$
$420$	0	0
$421$	18.0291	0.878683	0.439342	−	0.898320i	$-0.355212\pi$
$421$	18.0291	0.878683	0.439342	+	0.898320i	$0.355212\pi$
$422$	0	0
$423$	0.722938	0.0351504
$424$	0	0
$425$	8.24799	0.400086
$426$	0	0
$427$	7.41296	0.358738
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.40106	−0.260160	−0.130080	−	0.991504i	$-0.541523\pi$
$431$	−5.40106	−0.260160	−0.130080	+	0.991504i	$0.541523\pi$
$432$	0	0
$433$	33.8918	1.62873	0.814367	−	0.580351i	$-0.197084\pi$
$433$	33.8918	1.62873	0.814367	+	0.580351i	$0.197084\pi$
$434$	0	0
$435$	−31.2599	−1.49880
$436$	0	0
$437$	24.3376	1.16423
$438$	0	0
$439$	6.64520	0.317158	0.158579	−	0.987346i	$-0.449309\pi$
$439$	6.64520	0.317158	0.158579	+	0.987346i	$0.449309\pi$
$440$	0	0
$441$	−3.81401	−0.181620
$442$	0	0
$443$	9.44588	0.448787	0.224394	−	0.974499i	$-0.427960\pi$
$443$	9.44588	0.448787	0.224394	+	0.974499i	$0.427960\pi$
$444$	0	0
$445$	−62.5198	−2.96372
$446$	0	0
$447$	−3.63186	−0.171781
$448$	0	0
$449$	−9.75201	−0.460226	−0.230113	−	0.973164i	$-0.573910\pi$
$449$	−9.75201	−0.460226	−0.230113	+	0.973164i	$0.573910\pi$
$450$	0	0
$451$	−40.9051	−1.92615
$452$	0	0
$453$	14.5989	0.685918
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.4616	0.816820	0.408410	−	0.912799i	$-0.366083\pi$
$457$	17.4616	0.816820	0.408410	+	0.912799i	$0.366083\pi$
$458$	0	0
$459$	−0.784934	−0.0366376
$460$	0	0
$461$	0.244142	0.0113708	0.00568542	−	0.999984i	$-0.498190\pi$
$461$	0.244142	0.0113708	0.00568542	+	0.999984i	$0.498190\pi$
$462$	0	0
$463$	−0.0910730	−0.00423252	−0.00211626	−	0.999998i	$-0.500674\pi$
$463$	−0.0910730	−0.00423252	−0.00211626	+	0.999998i	$0.500674\pi$
$464$	0	0
$465$	−11.5699	−0.536540
$466$	0	0
$467$	−14.1688	−0.655654	−0.327827	−	0.944738i	$-0.606316\pi$
$467$	−14.1688	−0.655654	−0.327827	+	0.944738i	$0.606316\pi$
$468$	0	0
$469$	17.4655	0.806480
$470$	0	0
$471$	15.5989	0.718761
$472$	0	0
$473$	−29.3534	−1.34967
$474$	0	0
$475$	54.1478	2.48447
$476$	0	0
$477$	−13.8140	−0.632500
$478$	0	0
$479$	1.69386	0.0773945	0.0386972	−	0.999251i	$-0.487679\pi$
$479$	1.69386	0.0773945	0.0386972	+	0.999251i	$0.487679\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−8.43013	−0.383584
$484$	0	0
$485$	−12.0582	−0.547533
$486$	0	0
$487$	18.8469	0.854036	0.427018	−	0.904243i	$-0.359564\pi$
$487$	18.8469	0.854036	0.427018	+	0.904243i	$0.359564\pi$
$488$	0	0
$489$	−10.9380	−0.494634
$490$	0	0
$491$	21.9208	0.989273	0.494637	−	0.869100i	$-0.335301\pi$
$491$	21.9208	0.989273	0.494637	+	0.869100i	$0.335301\pi$
$492$	0	0
$493$	−6.23081	−0.280622
$494$	0	0
$495$	−18.5989	−0.835960
$496$	0	0
$497$	15.5699	0.698404
$498$	0	0
$499$	13.4459	0.601920	0.300960	−	0.953637i	$-0.402693\pi$
$499$	13.4459	0.601920	0.300960	+	0.953637i	$0.402693\pi$
$500$	0	0
$501$	20.5856	0.919697
$502$	0	0
$503$	9.92083	0.442348	0.221174	−	0.975234i	$-0.429011\pi$
$503$	9.92083	0.442348	0.221174	+	0.975234i	$0.429011\pi$
$504$	0	0
$505$	6.42629	0.285966
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.3996	0.904197	0.452099	−	0.891968i	$-0.350675\pi$
$509$	20.3996	0.904197	0.452099	+	0.891968i	$0.350675\pi$
$510$	0	0
$511$	8.70335	0.385014
$512$	0	0
$513$	−5.15307	−0.227514
$514$	0	0
$515$	30.1688	1.32940
$516$	0	0
$517$	3.41439	0.150165
$518$	0	0
$519$	0.306139	0.0134380
$520$	0	0
$521$	−1.49454	−0.0654769	−0.0327385	−	0.999464i	$-0.510423\pi$
$521$	−1.49454	−0.0654769	−0.0327385	+	0.999464i	$0.510423\pi$
$522$	0	0
$523$	−32.0448	−1.40122	−0.700611	−	0.713543i	$-0.747089\pi$
$523$	−32.0448	−1.40122	−0.700611	+	0.713543i	$0.747089\pi$
$524$	0	0
$525$	−18.7559	−0.818573
$526$	0	0
$527$	−2.30614	−0.100457
$528$	0	0
$529$	−0.693861	−0.0301679
$530$	0	0
$531$	0.430132	0.0186661
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−2.84693	−0.123084
$536$	0	0
$537$	2.84693	0.122854
$538$	0	0
$539$	−18.0133	−0.775889
$540$	0	0
$541$	1.44347	0.0620594	0.0310297	−	0.999518i	$-0.490121\pi$
$541$	1.44347	0.0620594	0.0310297	+	0.999518i	$0.490121\pi$
$542$	0	0
$543$	−13.2151	−0.567113
$544$	0	0
$545$	−21.1397	−0.905527
$546$	0	0
$547$	−18.3972	−0.786608	−0.393304	−	0.919408i	$-0.628668\pi$
$547$	−18.3972	−0.786608	−0.393304	+	0.919408i	$0.628668\pi$
$548$	0	0
$549$	4.15307	0.177249
$550$	0	0
$551$	−40.9051	−1.74262
$552$	0	0
$553$	29.2441	1.24359
$554$	0	0
$555$	4.78493	0.203109
$556$	0	0
$557$	−16.0935	−0.681903	−0.340951	−	0.940081i	$-0.610749\pi$
$557$	−16.0935	−0.681903	−0.340951	+	0.940081i	$0.610749\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3.70719	−0.156518
$562$	0	0
$563$	13.0424	0.549672	0.274836	−	0.961491i	$-0.411376\pi$
$563$	13.0424	0.549672	0.274836	+	0.961491i	$0.411376\pi$
$564$	0	0
$565$	−52.0410	−2.18938
$566$	0	0
$567$	1.78493	0.0749602
$568$	0	0
$569$	6.58561	0.276083	0.138042	−	0.990426i	$-0.455919\pi$
$569$	6.58561	0.276083	0.138042	+	0.990426i	$0.455919\pi$
$570$	0	0
$571$	−26.8469	−1.12351	−0.561755	−	0.827304i	$-0.689873\pi$
$571$	−26.8469	−1.12351	−0.561755	+	0.827304i	$0.689873\pi$
$572$	0	0
$573$	3.56987	0.149133
$574$	0	0
$575$	49.6280	2.06963
$576$	0	0
$577$	−22.5818	−0.940091	−0.470046	−	0.882642i	$-0.655763\pi$
$577$	−22.5818	−0.940091	−0.470046	+	0.882642i	$0.655763\pi$
$578$	0	0
$579$	−12.4459	−0.517233
$580$	0	0
$581$	11.7787	0.488662
$582$	0	0
$583$	−65.2427	−2.70208
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.87601	−0.242529	−0.121264	−	0.992620i	$-0.538695\pi$
$587$	−5.87601	−0.242529	−0.121264	+	0.992620i	$0.538695\pi$
$588$	0	0
$589$	−15.1397	−0.623822
$590$	0	0
$591$	8.73627	0.359362
$592$	0	0
$593$	−12.4788	−0.512443	−0.256221	−	0.966618i	$-0.582478\pi$
$593$	−12.4788	−0.512443	−0.256221	+	0.966618i	$0.582478\pi$
$594$	0	0
$595$	−5.51736	−0.226190
$596$	0	0
$597$	16.0911	0.658564
$598$	0	0
$599$	41.6280	1.70087	0.850437	−	0.526076i	$-0.176337\pi$
$599$	41.6280	1.70087	0.850437	+	0.526076i	$0.176337\pi$
$600$	0	0
$601$	29.2599	1.19354	0.596768	−	0.802414i	$-0.296451\pi$
$601$	29.2599	1.19354	0.596768	+	0.802414i	$0.296451\pi$
$602$	0	0
$603$	9.78493	0.398473
$604$	0	0
$605$	−44.5236	−1.81014
$606$	0	0
$607$	−18.0582	−0.732958	−0.366479	−	0.930426i	$-0.619437\pi$
$607$	−18.0582	−0.732958	−0.366479	+	0.930426i	$0.619437\pi$
$608$	0	0
$609$	14.1688	0.574149
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−17.9051	−0.723180	−0.361590	−	0.932337i	$-0.617766\pi$
$613$	−17.9051	−0.723180	−0.361590	+	0.932337i	$0.617766\pi$
$614$	0	0
$615$	−34.1068	−1.37532
$616$	0	0
$617$	−23.3972	−0.941936	−0.470968	−	0.882150i	$-0.656095\pi$
$617$	−23.3972	−0.941936	−0.470968	+	0.882150i	$0.656095\pi$
$618$	0	0
$619$	43.9537	1.76665	0.883325	−	0.468761i	$-0.155299\pi$
$619$	43.9537	1.76665	0.883325	+	0.468761i	$0.155299\pi$
$620$	0	0
$621$	−4.72294	−0.189525
$622$	0	0
$623$	28.3376	1.13532
$624$	0	0
$625$	32.8760	1.31504
$626$	0	0
$627$	−24.3376	−0.971951
$628$	0	0
$629$	0.953747	0.0380284
$630$	0	0
$631$	−9.67427	−0.385127	−0.192563	−	0.981285i	$-0.561680\pi$
$631$	−9.67427	−0.385127	−0.192563	+	0.981285i	$0.561680\pi$
$632$	0	0
$633$	−4.38388	−0.174244
$634$	0	0
$635$	−64.5198	−2.56039
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.72294	0.345074
$640$	0	0
$641$	14.4130	0.569277	0.284639	−	0.958635i	$-0.408126\pi$
$641$	14.4130	0.569277	0.284639	+	0.958635i	$0.408126\pi$
$642$	0	0
$643$	18.6900	0.737062	0.368531	−	0.929615i	$-0.379861\pi$
$643$	18.6900	0.737062	0.368531	+	0.929615i	$0.379861\pi$
$644$	0	0
$645$	−24.4750	−0.963700
$646$	0	0
$647$	22.7363	0.893855	0.446928	−	0.894570i	$-0.352518\pi$
$647$	22.7363	0.893855	0.446928	+	0.894570i	$0.352518\pi$
$648$	0	0
$649$	2.03149	0.0797428
$650$	0	0
$651$	5.24414	0.205534
$652$	0	0
$653$	20.2136	0.791021	0.395510	−	0.918462i	$-0.370568\pi$
$653$	20.2136	0.791021	0.395510	+	0.918462i	$0.370568\pi$
$654$	0	0
$655$	82.7601	3.23370
$656$	0	0
$657$	4.87601	0.190231
$658$	0	0
$659$	19.7520	0.769429	0.384715	−	0.923036i	$-0.374300\pi$
$659$	19.7520	0.769429	0.384715	+	0.923036i	$0.374300\pi$
$660$	0	0
$661$	−10.7653	−0.418723	−0.209362	−	0.977838i	$-0.567139\pi$
$661$	−10.7653	−0.418723	−0.209362	+	0.977838i	$0.567139\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−36.2213	−1.40460
$666$	0	0
$667$	−37.4907	−1.45165
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	19.6147	0.757217
$672$	0	0
$673$	46.5040	1.79260	0.896299	−	0.443450i	$-0.146246\pi$
$673$	46.5040	1.79260	0.896299	+	0.443450i	$0.146246\pi$
$674$	0	0
$675$	−10.5079	−0.404448
$676$	0	0
$677$	2.18215	0.0838667	0.0419333	−	0.999120i	$-0.486648\pi$
$677$	2.18215	0.0838667	0.0419333	+	0.999120i	$0.486648\pi$
$678$	0	0
$679$	5.46546	0.209745
$680$	0	0
$681$	17.7387	0.679748
$682$	0	0
$683$	−5.87601	−0.224839	−0.112420	−	0.993661i	$-0.535860\pi$
$683$	−5.87601	−0.224839	−0.112420	+	0.993661i	$0.535860\pi$
$684$	0	0
$685$	−49.3705	−1.88635
$686$	0	0
$687$	−11.4459	−0.436687
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−11.4788	−0.436674	−0.218337	−	0.975873i	$-0.570063\pi$
$691$	−11.4788	−0.436674	−0.218337	+	0.975873i	$0.570063\pi$
$692$	0	0
$693$	8.43013	0.320234
$694$	0	0
$695$	−34.7096	−1.31661
$696$	0	0
$697$	−6.79827	−0.257503
$698$	0	0
$699$	9.75201	0.368855
$700$	0	0
$701$	−26.4301	−0.998252	−0.499126	−	0.866529i	$-0.666346\pi$
$701$	−26.4301	−0.998252	−0.499126	+	0.866529i	$0.666346\pi$
$702$	0	0
$703$	6.26132	0.236150
$704$	0	0
$705$	2.84693	0.107222
$706$	0	0
$707$	−2.91277	−0.109546
$708$	0	0
$709$	52.0606	1.95518	0.977588	−	0.210528i	$-0.0675185\pi$
$709$	52.0606	1.95518	0.977588	+	0.210528i	$0.0675185\pi$
$710$	0	0
$711$	16.3839	0.614443
$712$	0	0
$713$	−13.8760	−0.519661
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.59894	−0.246442
$718$	0	0
$719$	34.4616	1.28520	0.642601	−	0.766201i	$-0.277855\pi$
$719$	34.4616	1.28520	0.642601	+	0.766201i	$0.277855\pi$
$720$	0	0
$721$	−13.6743	−0.509257
$722$	0	0
$723$	−23.8140	−0.885653
$724$	0	0
$725$	−83.4115	−3.09783
$726$	0	0
$727$	−29.1335	−1.08050	−0.540251	−	0.841504i	$-0.681671\pi$
$727$	−29.1335	−1.08050	−0.540251	+	0.841504i	$0.681671\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.87842	−0.180435
$732$	0	0
$733$	−37.7229	−1.39333	−0.696664	−	0.717397i	$-0.745333\pi$
$733$	−37.7229	−1.39333	−0.696664	+	0.717397i	$0.745333\pi$
$734$	0	0
$735$	−15.0196	−0.554006
$736$	0	0
$737$	46.2136	1.70230
$738$	0	0
$739$	27.7520	1.02087	0.510437	−	0.859915i	$-0.329484\pi$
$739$	27.7520	1.02087	0.510437	+	0.859915i	$0.329484\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24.1821	−0.887157	−0.443578	−	0.896236i	$-0.646291\pi$
$743$	−24.1821	−0.887157	−0.443578	+	0.896236i	$0.646291\pi$
$744$	0	0
$745$	−14.3023	−0.523996
$746$	0	0
$747$	6.59894	0.241443
$748$	0	0
$749$	1.29040	0.0471500
$750$	0	0
$751$	43.7968	1.59817	0.799085	−	0.601219i	$-0.205318\pi$
$751$	43.7968	1.59817	0.799085	+	0.601219i	$0.205318\pi$
$752$	0	0
$753$	−9.44588	−0.344227
$754$	0	0
$755$	57.4907	2.09230
$756$	0	0
$757$	16.8918	0.613941	0.306971	−	0.951719i	$-0.400685\pi$
$757$	16.8918	0.613941	0.306971	+	0.951719i	$0.400685\pi$
$758$	0	0
$759$	−22.3061	−0.809662
$760$	0	0
$761$	−0.123993	−0.00449474	−0.00224737	−	0.999997i	$-0.500715\pi$
$761$	−0.123993	−0.00449474	−0.00224737	+	0.999997i	$0.500715\pi$
$762$	0	0
$763$	9.58177	0.346883
$764$	0	0
$765$	−3.09107	−0.111758
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−43.8102	−1.57984	−0.789918	−	0.613213i	$-0.789877\pi$
$769$	−43.8102	−1.57984	−0.789918	+	0.613213i	$0.789877\pi$
$770$	0	0
$771$	12.6609	0.455973
$772$	0	0
$773$	−4.55412	−0.163800	−0.0819002	−	0.996641i	$-0.526099\pi$
$773$	−4.55412	−0.163800	−0.0819002	+	0.996641i	$0.526099\pi$
$774$	0	0
$775$	−30.8722	−1.10896
$776$	0	0
$777$	−2.16881	−0.0778057
$778$	0	0
$779$	−44.6304	−1.59905
$780$	0	0
$781$	41.1979	1.47418
$782$	0	0
$783$	7.93800	0.283681
$784$	0	0
$785$	61.4287	2.19248
$786$	0	0
$787$	0.0462534	0.00164875	0.000824377	−	1.00000i	$-0.499738\pi$
$787$	0.0462534	0.00164875	0.000824377		1.00000i	$0.499738\pi$
$788$	0	0
$789$	0.292806	0.0104242
$790$	0	0
$791$	23.5880	0.838693
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−54.3996	−1.92936
$796$	0	0
$797$	18.5198	0.656004	0.328002	−	0.944677i	$-0.393625\pi$
$797$	18.5198	0.656004	0.328002	+	0.944677i	$0.393625\pi$
$798$	0	0
$799$	0.567458	0.0200752
$800$	0	0
$801$	15.8760	0.560951
$802$	0	0
$803$	23.0291	0.812678
$804$	0	0
$805$	−33.1979	−1.17007
$806$	0	0
$807$	−5.56987	−0.196069
$808$	0	0
$809$	43.7034	1.53653	0.768264	−	0.640133i	$-0.221121\pi$
$809$	43.7034	1.53653	0.768264	+	0.640133i	$0.221121\pi$
$810$	0	0
$811$	12.2017	0.428461	0.214230	−	0.976783i	$-0.431276\pi$
$811$	12.2017	0.428461	0.214230	+	0.976783i	$0.431276\pi$
$812$	0	0
$813$	−11.9537	−0.419236
$814$	0	0
$815$	−43.0739	−1.50881
$816$	0	0
$817$	−32.0267	−1.12047
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.5856	−1.34665	−0.673324	−	0.739348i	$-0.735134\pi$
$821$	−38.5856	−1.34665	−0.673324	+	0.739348i	$0.735134\pi$
$822$	0	0
$823$	48.3376	1.68494	0.842472	−	0.538740i	$-0.181100\pi$
$823$	48.3376	1.68494	0.842472	+	0.538740i	$0.181100\pi$
$824$	0	0
$825$	−49.6280	−1.72783
$826$	0	0
$827$	−1.19789	−0.0416547	−0.0208273	−	0.999783i	$-0.506630\pi$
$827$	−1.19789	−0.0416547	−0.0208273	+	0.999783i	$0.506630\pi$
$828$	0	0
$829$	−18.3352	−0.636808	−0.318404	−	0.947955i	$-0.603147\pi$
$829$	−18.3352	−0.636808	−0.318404	+	0.947955i	$0.603147\pi$
$830$	0	0
$831$	−18.6609	−0.647341
$832$	0	0
$833$	−2.99375	−0.103727
$834$	0	0
$835$	81.0662	2.80541
$836$	0	0
$837$	2.93800	0.101552
$838$	0	0
$839$	46.4883	1.60495	0.802477	−	0.596683i	$-0.203515\pi$
$839$	46.4883	1.60495	0.802477	+	0.596683i	$0.203515\pi$
$840$	0	0
$841$	34.0119	1.17282
$842$	0	0
$843$	−4.96708	−0.171075
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.1807	0.693417
$848$	0	0
$849$	10.8007	0.370678
$850$	0	0
$851$	5.73868	0.196719
$852$	0	0
$853$	−10.7653	−0.368598	−0.184299	−	0.982870i	$-0.559002\pi$
$853$	−10.7653	−0.368598	−0.184299	+	0.982870i	$0.559002\pi$
$854$	0	0
$855$	−20.2928	−0.693999
$856$	0	0
$857$	−23.3972	−0.799234	−0.399617	−	0.916682i	$-0.630857\pi$
$857$	−23.3972	−0.799234	−0.399617	+	0.916682i	$0.630857\pi$
$858$	0	0
$859$	0.339059	0.0115685	0.00578427	−	0.999983i	$-0.498159\pi$
$859$	0.339059	0.0115685	0.00578427	+	0.999983i	$0.498159\pi$
$860$	0	0
$861$	15.4592	0.526848
$862$	0	0
$863$	13.9208	0.473870	0.236935	−	0.971525i	$-0.423857\pi$
$863$	13.9208	0.473870	0.236935	+	0.971525i	$0.423857\pi$
$864$	0	0
$865$	1.20558	0.0409908
$866$	0	0
$867$	16.3839	0.556426
$868$	0	0
$869$	77.3800	2.62494
$870$	0	0
$871$	0	0
$872$	0	0
$873$	3.06200	0.103633
$874$	0	0
$875$	−38.7153	−1.30881
$876$	0	0
$877$	−7.37055	−0.248886	−0.124443	−	0.992227i	$-0.539714\pi$
$877$	−7.37055	−0.248886	−0.124443	+	0.992227i	$0.539714\pi$
$878$	0	0
$879$	26.2441	0.885193
$880$	0	0
$881$	−30.9671	−1.04331	−0.521654	−	0.853157i	$-0.674685\pi$
$881$	−30.9671	−1.04331	−0.521654	+	0.853157i	$0.674685\pi$
$882$	0	0
$883$	24.5660	0.826713	0.413356	−	0.910569i	$-0.364356\pi$
$883$	24.5660	0.826713	0.413356	+	0.910569i	$0.364356\pi$
$884$	0	0
$885$	1.69386	0.0569385
$886$	0	0
$887$	−20.1821	−0.677650	−0.338825	−	0.940849i	$-0.610029\pi$
$887$	−20.1821	−0.677650	−0.338825	+	0.940849i	$0.610029\pi$
$888$	0	0
$889$	29.2441	0.980817
$890$	0	0
$891$	4.72294	0.158224
$892$	0	0
$893$	3.72535	0.124664
$894$	0	0
$895$	11.2112	0.374750
$896$	0	0
$897$	0	0
$898$	0	0
$899$	23.3219	0.777828
$900$	0	0
$901$	−10.8431	−0.361236
$902$	0	0
$903$	11.0935	0.369168
$904$	0	0
$905$	−52.0410	−1.72990
$906$	0	0
$907$	47.2294	1.56823	0.784113	−	0.620618i	$-0.213118\pi$
$907$	47.2294	1.56823	0.784113	+	0.620618i	$0.213118\pi$
$908$	0	0
$909$	−1.63186	−0.0541255
$910$	0	0
$911$	−48.3376	−1.60150	−0.800748	−	0.599001i	$-0.795565\pi$
$911$	−48.3376	−1.60150	−0.800748	+	0.599001i	$0.795565\pi$
$912$	0	0
$913$	31.1664	1.03146
$914$	0	0
$915$	16.3548	0.540673
$916$	0	0
$917$	−37.5117	−1.23875
$918$	0	0
$919$	−10.0582	−0.331788	−0.165894	−	0.986144i	$-0.553051\pi$
$919$	−10.0582	−0.331788	−0.165894	+	0.986144i	$0.553051\pi$
$920$	0	0
$921$	−21.5369	−0.709667
$922$	0	0
$923$	0	0
$924$	0	0
$925$	12.7678	0.419801
$926$	0	0
$927$	−7.66094	−0.251618
$928$	0	0
$929$	−12.0487	−0.395304	−0.197652	−	0.980272i	$-0.563332\pi$
$929$	−12.0487	−0.395304	−0.197652	+	0.980272i	$0.563332\pi$
$930$	0	0
$931$	−19.6539	−0.644129
$932$	0	0
$933$	−21.1531	−0.692520
$934$	0	0
$935$	−14.5989	−0.477437
$936$	0	0
$937$	−22.8298	−0.745816	−0.372908	−	0.927868i	$-0.621639\pi$
$937$	−22.8298	−0.745816	−0.372908	+	0.927868i	$0.621639\pi$
$938$	0	0
$939$	−9.79827	−0.319754
$940$	0	0
$941$	−39.8102	−1.29777	−0.648887	−	0.760885i	$-0.724765\pi$
$941$	−39.8102	−1.29777	−0.648887	+	0.760885i	$0.724765\pi$
$942$	0	0
$943$	−40.9051	−1.33205
$944$	0	0
$945$	7.02908	0.228656
$946$	0	0
$947$	−30.3061	−0.984817	−0.492409	−	0.870364i	$-0.663883\pi$
$947$	−30.3061	−0.984817	−0.492409	+	0.870364i	$0.663883\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−22.5236	−0.730378
$952$	0	0
$953$	10.0267	0.324796	0.162398	−	0.986725i	$-0.448077\pi$
$953$	10.0267	0.324796	0.162398	+	0.986725i	$0.448077\pi$
$954$	0	0
$955$	14.0582	0.454911
$956$	0	0
$957$	37.4907	1.21190
$958$	0	0
$959$	22.3776	0.722611
$960$	0	0
$961$	−22.3681	−0.721553
$962$	0	0
$963$	0.722938	0.0232963
$964$	0	0
$965$	−49.0119	−1.57775
$966$	0	0
$967$	−43.7702	−1.40755	−0.703777	−	0.710421i	$-0.748505\pi$
$967$	−43.7702	−1.40755	−0.703777	+	0.710421i	$0.748505\pi$
$968$	0	0
$969$	−4.04482	−0.129938
$970$	0	0
$971$	−48.9232	−1.57002	−0.785011	−	0.619482i	$-0.787343\pi$
$971$	−48.9232	−1.57002	−0.785011	+	0.619482i	$0.787343\pi$
$972$	0	0
$973$	15.7324	0.504358
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.5794	−0.882342	−0.441171	−	0.897423i	$-0.645437\pi$
$977$	−27.5794	−0.882342	−0.441171	+	0.897423i	$0.645437\pi$
$978$	0	0
$979$	74.9814	2.39642
$980$	0	0
$981$	5.36814	0.171391
$982$	0	0
$983$	−34.7096	−1.10706	−0.553532	−	0.832828i	$-0.686720\pi$
$983$	−34.7096	−1.10706	−0.553532	+	0.832828i	$0.686720\pi$
$984$	0	0
$985$	34.4035	1.09619
$986$	0	0
$987$	−1.29040	−0.0410738
$988$	0	0
$989$	−29.3534	−0.933383
$990$	0	0
$991$	−32.6571	−1.03739	−0.518693	−	0.854960i	$-0.673581\pi$
$991$	−32.6571	−1.03739	−0.518693	+	0.854960i	$0.673581\pi$
$992$	0	0
$993$	−21.0620	−0.668382
$994$	0	0
$995$	63.3667	2.00886
$996$	0	0
$997$	5.59894	0.177320	0.0886602	−	0.996062i	$-0.471741\pi$
$997$	5.59894	0.177320	0.0886602	+	0.996062i	$0.471741\pi$
$998$	0	0
$999$	−1.21507	−0.0384430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.z.1.1		3
4.3	odd	2		8112.2.a.ck.1.1		3
13.3	even	3		312.2.q.e.217.1	✓	6
13.5	odd	4		4056.2.c.m.337.6		6
13.8	odd	4		4056.2.c.m.337.1		6
13.9	even	3		312.2.q.e.289.1	yes	6
13.12	even	2		4056.2.a.y.1.3		3
39.29	odd	6		936.2.t.h.217.3		6
39.35	odd	6		936.2.t.h.289.3		6
52.3	odd	6		624.2.q.j.529.1		6
52.35	odd	6		624.2.q.j.289.1		6
52.51	odd	2		8112.2.a.cl.1.3		3
156.35	even	6		1872.2.t.u.289.3		6
156.107	even	6		1872.2.t.u.1153.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.e.217.1	✓	6	13.3	even	3
312.2.q.e.289.1	yes	6	13.9	even	3
624.2.q.j.289.1		6	52.35	odd	6
624.2.q.j.529.1		6	52.3	odd	6
936.2.t.h.217.3		6	39.29	odd	6
936.2.t.h.289.3		6	39.35	odd	6
1872.2.t.u.289.3		6	156.35	even	6
1872.2.t.u.1153.3		6	156.107	even	6
4056.2.a.y.1.3		3	13.12	even	2
4056.2.a.z.1.1		3	1.1	even	1	trivial
4056.2.c.m.337.1		6	13.8	odd	4
4056.2.c.m.337.6		6	13.5	odd	4
8112.2.a.ck.1.1		3	4.3	odd	2
8112.2.a.cl.1.3		3	52.51	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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.93800 q^{5} +1.78493 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.93800 q^{5} +1.78493 q^{7} +1.00000 q^{9} +4.72294 q^{11} +3.93800 q^{15} +0.784934 q^{17} +5.15307 q^{19} -1.78493 q^{21} +4.72294 q^{23} +10.5079 q^{25} -1.00000 q^{27} -7.93800 q^{29} -2.93800 q^{31} -4.72294 q^{33} -7.02908 q^{35} +1.21507 q^{37} -8.66094 q^{41} -6.21507 q^{43} -3.93800 q^{45} +0.722938 q^{47} -3.81401 q^{49} -0.784934 q^{51} -13.8140 q^{53} -18.5989 q^{55} -5.15307 q^{57} +0.430132 q^{59} +4.15307 q^{61} +1.78493 q^{63} +9.78493 q^{67} -4.72294 q^{69} +8.72294 q^{71} +4.87601 q^{73} -10.5079 q^{75} +8.43013 q^{77} +16.3839 q^{79} +1.00000 q^{81} +6.59894 q^{83} -3.09107 q^{85} +7.93800 q^{87} +15.8760 q^{89} +2.93800 q^{93} -20.2928 q^{95} +3.06200 q^{97} +4.72294 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 6 q^{19} - 3 q^{21} + 15 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} + 12 q^{35} + 6 q^{37} - 21 q^{43} - 12 q^{47} + 24 q^{49} - 6 q^{53} - 18 q^{55} - 6 q^{57} + 6 q^{59} + 3 q^{61} + 3 q^{63} + 27 q^{67} + 12 q^{71} - 9 q^{73} - 15 q^{75} + 30 q^{77} + 9 q^{79} + 3 q^{81} - 18 q^{83} + 12 q^{85} + 12 q^{87} + 24 q^{89} - 3 q^{93} - 42 q^{95} + 21 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 6 * q^19 - 3 * q^21 + 15 * q^25 - 3 * q^27 - 12 * q^29 + 3 * q^31 + 12 * q^35 + 6 * q^37 - 21 * q^43 - 12 * q^47 + 24 * q^49 - 6 * q^53 - 18 * q^55 - 6 * q^57 + 6 * q^59 + 3 * q^61 + 3 * q^63 + 27 * q^67 + 12 * q^71 - 9 * q^73 - 15 * q^75 + 30 * q^77 + 9 * q^79 + 3 * q^81 - 18 * q^83 + 12 * q^85 + 12 * q^87 + 24 * q^89 - 3 * q^93 - 42 * q^95 + 21 * q^97

Introduction

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