LMFDB - Embedded newform 6040.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} +1.00000 q^{5} -2.56155 q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} +1.00000 q^{5} -2.56155 q^{7} -0.561553 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.56155 q^{15} -2.00000 q^{17} +5.56155 q^{19} -4.00000 q^{21} -4.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} -3.00000 q^{29} +2.12311 q^{31} +1.56155 q^{33} -2.56155 q^{35} +8.24621 q^{37} -1.56155 q^{39} -9.12311 q^{41} -2.00000 q^{43} -0.561553 q^{45} -7.12311 q^{47} -0.438447 q^{49} -3.12311 q^{51} +2.00000 q^{53} +1.00000 q^{55} +8.68466 q^{57} -9.00000 q^{59} +0.876894 q^{61} +1.43845 q^{63} -1.00000 q^{65} -7.24621 q^{67} -6.43845 q^{69} +3.12311 q^{71} +9.24621 q^{73} +1.56155 q^{75} -2.56155 q^{77} -8.00000 q^{79} -7.00000 q^{81} -3.24621 q^{83} -2.00000 q^{85} -4.68466 q^{87} -6.87689 q^{89} +2.56155 q^{91} +3.31534 q^{93} +5.56155 q^{95} -10.2462 q^{97} -0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 + 2 * q^5 - q^7 + 3 * q^9	$ 2 q - q^{3} + 2 q^{5} - q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} - q^{15} - 4 q^{17} + 7 q^{19} - 8 q^{21} + 2 q^{25} - 7 q^{27} - 6 q^{29} - 4 q^{31} - q^{33} - q^{35} + q^{39} - 10 q^{41} - 4 q^{43} + 3 q^{45} - 6 q^{47} - 5 q^{49} + 2 q^{51} + 4 q^{53} + 2 q^{55} + 5 q^{57} - 18 q^{59} + 10 q^{61} + 7 q^{63} - 2 q^{65} + 2 q^{67} - 17 q^{69} - 2 q^{71} + 2 q^{73} - q^{75} - q^{77} - 16 q^{79} - 14 q^{81} + 10 q^{83} - 4 q^{85} + 3 q^{87} - 22 q^{89} + q^{91} + 19 q^{93} + 7 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^5 - q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 - q^15 - 4 * q^17 + 7 * q^19 - 8 * q^21 + 2 * q^25 - 7 * q^27 - 6 * q^29 - 4 * q^31 - q^33 - q^35 + q^39 - 10 * q^41 - 4 * q^43 + 3 * q^45 - 6 * q^47 - 5 * q^49 + 2 * q^51 + 4 * q^53 + 2 * q^55 + 5 * q^57 - 18 * q^59 + 10 * q^61 + 7 * q^63 - 2 * q^65 + 2 * q^67 - 17 * q^69 - 2 * q^71 + 2 * q^73 - q^75 - q^77 - 16 * q^79 - 14 * q^81 + 10 * q^83 - 4 * q^85 + 3 * q^87 - 22 * q^89 + q^91 + 19 * q^93 + 7 * q^95 - 4 * q^97 + 3 * 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.56155	−0.968176	−0.484088	−	0.875019i	$-0.660849\pi$
$7$	−2.56155	−0.968176	−0.484088	+	0.875019i	$0.660849\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	1.56155	0.403191
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	5.56155	1.27591	0.637954	−	0.770075i	$-0.279781\pi$
$19$	5.56155	1.27591	0.637954	+	0.770075i	$0.279781\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−4.12311	−0.859727	−0.429863	−	0.902894i	$-0.641438\pi$
$23$	−4.12311	−0.859727	−0.429863	+	0.902894i	$0.641438\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	2.12311	0.381321	0.190661	−	0.981656i	$-0.438937\pi$
$31$	2.12311	0.381321	0.190661	+	0.981656i	$0.438937\pi$
$32$	0	0
$33$	1.56155	0.271831
$34$	0	0
$35$	−2.56155	−0.432981
$36$	0	0
$37$	8.24621	1.35567	0.677834	−	0.735215i	$-0.262919\pi$
$37$	8.24621	1.35567	0.677834	+	0.735215i	$0.262919\pi$
$38$	0	0
$39$	−1.56155	−0.250049
$40$	0	0
$41$	−9.12311	−1.42479	−0.712395	−	0.701779i	$-0.752389\pi$
$41$	−9.12311	−1.42479	−0.712395	+	0.701779i	$0.752389\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−0.561553	−0.0837114
$46$	0	0
$47$	−7.12311	−1.03901	−0.519506	−	0.854467i	$-0.673884\pi$
$47$	−7.12311	−1.03901	−0.519506	+	0.854467i	$0.673884\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	−3.12311	−0.437322
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	8.68466	1.15031
$58$	0	0
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	0	0
$63$	1.43845	0.181227
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−7.24621	−0.885266	−0.442633	−	0.896703i	$-0.645955\pi$
$67$	−7.24621	−0.885266	−0.442633	+	0.896703i	$0.645955\pi$
$68$	0	0
$69$	−6.43845	−0.775098
$70$	0	0
$71$	3.12311	0.370644	0.185322	−	0.982678i	$-0.440667\pi$
$71$	3.12311	0.370644	0.185322	+	0.982678i	$0.440667\pi$
$72$	0	0
$73$	9.24621	1.08219	0.541094	−	0.840962i	$-0.318010\pi$
$73$	9.24621	1.08219	0.541094	+	0.840962i	$0.318010\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	−2.56155	−0.291916
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−3.24621	−0.356318	−0.178159	−	0.984002i	$-0.557014\pi$
$83$	−3.24621	−0.356318	−0.178159	+	0.984002i	$0.557014\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−4.68466	−0.502248
$88$	0	0
$89$	−6.87689	−0.728949	−0.364475	−	0.931213i	$-0.618751\pi$
$89$	−6.87689	−0.728949	−0.364475	+	0.931213i	$0.618751\pi$
$90$	0	0
$91$	2.56155	0.268524
$92$	0	0
$93$	3.31534	0.343785
$94$	0	0
$95$	5.56155	0.570603
$96$	0	0
$97$	−10.2462	−1.04035	−0.520173	−	0.854061i	$-0.674132\pi$
$97$	−10.2462	−1.04035	−0.520173	+	0.854061i	$0.674132\pi$
$98$	0	0
$99$	−0.561553	−0.0564382
$100$	0	0
$101$	−15.3693	−1.52930	−0.764652	−	0.644443i	$-0.777089\pi$
$101$	−15.3693	−1.52930	−0.764652	+	0.644443i	$0.777089\pi$
$102$	0	0
$103$	20.2462	1.99492	0.997459	−	0.0712395i	$-0.0226955\pi$
$103$	20.2462	1.99492	0.997459	+	0.0712395i	$0.0226955\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	5.87689	0.568141	0.284070	−	0.958803i	$-0.408315\pi$
$107$	5.87689	0.568141	0.284070	+	0.958803i	$0.408315\pi$
$108$	0	0
$109$	−4.87689	−0.467122	−0.233561	−	0.972342i	$-0.575038\pi$
$109$	−4.87689	−0.467122	−0.233561	+	0.972342i	$0.575038\pi$
$110$	0	0
$111$	12.8769	1.22222
$112$	0	0
$113$	−3.43845	−0.323462	−0.161731	−	0.986835i	$-0.551708\pi$
$113$	−3.43845	−0.323462	−0.161731	+	0.986835i	$0.551708\pi$
$114$	0	0
$115$	−4.12311	−0.384482
$116$	0	0
$117$	0.561553	0.0519156
$118$	0	0
$119$	5.12311	0.469634
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−14.2462	−1.28454
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.1231	1.16449	0.582244	−	0.813014i	$-0.302175\pi$
$127$	13.1231	1.16449	0.582244	+	0.813014i	$0.302175\pi$
$128$	0	0
$129$	−3.12311	−0.274974
$130$	0	0
$131$	−13.1231	−1.14657	−0.573286	−	0.819356i	$-0.694331\pi$
$131$	−13.1231	−1.14657	−0.573286	+	0.819356i	$0.694331\pi$
$132$	0	0
$133$	−14.2462	−1.23530
$134$	0	0
$135$	−5.56155	−0.478662
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	0.315342	0.0267469	0.0133735	−	0.999911i	$-0.495743\pi$
$139$	0.315342	0.0267469	0.0133735	+	0.999911i	$0.495743\pi$
$140$	0	0
$141$	−11.1231	−0.936734
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−3.00000	−0.249136
$146$	0	0
$147$	−0.684658	−0.0564697
$148$	0	0
$149$	6.87689	0.563377	0.281689	−	0.959506i	$-0.409106\pi$
$149$	6.87689	0.563377	0.281689	+	0.959506i	$0.409106\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	1.12311	0.0907977
$154$	0	0
$155$	2.12311	0.170532
$156$	0	0
$157$	3.56155	0.284243	0.142121	−	0.989849i	$-0.454608\pi$
$157$	3.56155	0.284243	0.142121	+	0.989849i	$0.454608\pi$
$158$	0	0
$159$	3.12311	0.247678
$160$	0	0
$161$	10.5616	0.832367
$162$	0	0
$163$	−2.12311	−0.166294	−0.0831472	−	0.996537i	$-0.526497\pi$
$163$	−2.12311	−0.166294	−0.0831472	+	0.996537i	$0.526497\pi$
$164$	0	0
$165$	1.56155	0.121567
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−3.12311	−0.238830
$172$	0	0
$173$	15.6155	1.18723	0.593613	−	0.804750i	$-0.297701\pi$
$173$	15.6155	1.18723	0.593613	+	0.804750i	$0.297701\pi$
$174$	0	0
$175$	−2.56155	−0.193635
$176$	0	0
$177$	−14.0540	−1.05636
$178$	0	0
$179$	−16.2462	−1.21430	−0.607149	−	0.794588i	$-0.707687\pi$
$179$	−16.2462	−1.21430	−0.607149	+	0.794588i	$0.707687\pi$
$180$	0	0
$181$	−26.4924	−1.96917	−0.984583	−	0.174917i	$-0.944034\pi$
$181$	−26.4924	−1.96917	−0.984583	+	0.174917i	$0.944034\pi$
$182$	0	0
$183$	1.36932	0.101223
$184$	0	0
$185$	8.24621	0.606274
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	14.2462	1.03626
$190$	0	0
$191$	−19.0540	−1.37870	−0.689349	−	0.724430i	$-0.742103\pi$
$191$	−19.0540	−1.37870	−0.689349	+	0.724430i	$0.742103\pi$
$192$	0	0
$193$	4.24621	0.305649	0.152824	−	0.988253i	$-0.451163\pi$
$193$	4.24621	0.305649	0.152824	+	0.988253i	$0.451163\pi$
$194$	0	0
$195$	−1.56155	−0.111825
$196$	0	0
$197$	−23.9309	−1.70500	−0.852502	−	0.522724i	$-0.824916\pi$
$197$	−23.9309	−1.70500	−0.852502	+	0.522724i	$0.824916\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	−11.3153	−0.798123
$202$	0	0
$203$	7.68466	0.539357
$204$	0	0
$205$	−9.12311	−0.637185
$206$	0	0
$207$	2.31534	0.160927
$208$	0	0
$209$	5.56155	0.384701
$210$	0	0
$211$	6.49242	0.446957	0.223478	−	0.974709i	$-0.428259\pi$
$211$	6.49242	0.446957	0.223478	+	0.974709i	$0.428259\pi$
$212$	0	0
$213$	4.87689	0.334159
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	−5.43845	−0.369186
$218$	0	0
$219$	14.4384	0.975660
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	1.75379	0.117442	0.0587212	−	0.998274i	$-0.481298\pi$
$223$	1.75379	0.117442	0.0587212	+	0.998274i	$0.481298\pi$
$224$	0	0
$225$	−0.561553	−0.0374369
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	25.2462	1.66832	0.834158	−	0.551525i	$-0.185954\pi$
$229$	25.2462	1.66832	0.834158	+	0.551525i	$0.185954\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	0	0
$235$	−7.12311	−0.464660
$236$	0	0
$237$	−12.4924	−0.811470
$238$	0	0
$239$	24.6847	1.59672	0.798359	−	0.602182i	$-0.205702\pi$
$239$	24.6847	1.59672	0.798359	+	0.602182i	$0.205702\pi$
$240$	0	0
$241$	−13.2462	−0.853263	−0.426632	−	0.904425i	$-0.640300\pi$
$241$	−13.2462	−0.853263	−0.426632	+	0.904425i	$0.640300\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	−0.438447	−0.0280114
$246$	0	0
$247$	−5.56155	−0.353873
$248$	0	0
$249$	−5.06913	−0.321243
$250$	0	0
$251$	2.12311	0.134009	0.0670046	−	0.997753i	$-0.478656\pi$
$251$	2.12311	0.134009	0.0670046	+	0.997753i	$0.478656\pi$
$252$	0	0
$253$	−4.12311	−0.259217
$254$	0	0
$255$	−3.12311	−0.195576
$256$	0	0
$257$	17.3153	1.08010	0.540051	−	0.841633i	$-0.318405\pi$
$257$	17.3153	1.08010	0.540051	+	0.841633i	$0.318405\pi$
$258$	0	0
$259$	−21.1231	−1.31253
$260$	0	0
$261$	1.68466	0.104278
$262$	0	0
$263$	18.6155	1.14788	0.573941	−	0.818896i	$-0.305414\pi$
$263$	18.6155	1.14788	0.573941	+	0.818896i	$0.305414\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	−10.7386	−0.657194
$268$	0	0
$269$	−13.3153	−0.811851	−0.405925	−	0.913906i	$-0.633051\pi$
$269$	−13.3153	−0.811851	−0.405925	+	0.913906i	$0.633051\pi$
$270$	0	0
$271$	−25.3693	−1.54108	−0.770538	−	0.637394i	$-0.780012\pi$
$271$	−25.3693	−1.54108	−0.770538	+	0.637394i	$0.780012\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−20.2462	−1.21648	−0.608238	−	0.793754i	$-0.708124\pi$
$277$	−20.2462	−1.21648	−0.608238	+	0.793754i	$0.708124\pi$
$278$	0	0
$279$	−1.19224	−0.0713773
$280$	0	0
$281$	−8.49242	−0.506615	−0.253308	−	0.967386i	$-0.581519\pi$
$281$	−8.49242	−0.506615	−0.253308	+	0.967386i	$0.581519\pi$
$282$	0	0
$283$	−19.8078	−1.17745	−0.588725	−	0.808334i	$-0.700370\pi$
$283$	−19.8078	−1.17745	−0.588725	+	0.808334i	$0.700370\pi$
$284$	0	0
$285$	8.68466	0.514435
$286$	0	0
$287$	23.3693	1.37945
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−16.0000	−0.937937
$292$	0	0
$293$	−14.3153	−0.836311	−0.418156	−	0.908375i	$-0.637323\pi$
$293$	−14.3153	−0.836311	−0.418156	+	0.908375i	$0.637323\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	0	0
$297$	−5.56155	−0.322714
$298$	0	0
$299$	4.12311	0.238445
$300$	0	0
$301$	5.12311	0.295291
$302$	0	0
$303$	−24.0000	−1.37876
$304$	0	0
$305$	0.876894	0.0502108
$306$	0	0
$307$	20.2462	1.15551	0.577756	−	0.816209i	$-0.303928\pi$
$307$	20.2462	1.15551	0.577756	+	0.816209i	$0.303928\pi$
$308$	0	0
$309$	31.6155	1.79854
$310$	0	0
$311$	24.8617	1.40978	0.704890	−	0.709317i	$-0.250996\pi$
$311$	24.8617	1.40978	0.704890	+	0.709317i	$0.250996\pi$
$312$	0	0
$313$	6.87689	0.388705	0.194353	−	0.980932i	$-0.437739\pi$
$313$	6.87689	0.388705	0.194353	+	0.980932i	$0.437739\pi$
$314$	0	0
$315$	1.43845	0.0810473
$316$	0	0
$317$	8.93087	0.501608	0.250804	−	0.968038i	$-0.419305\pi$
$317$	8.93087	0.501608	0.250804	+	0.968038i	$0.419305\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	0	0
$321$	9.17708	0.512215
$322$	0	0
$323$	−11.1231	−0.618906
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−7.61553	−0.421140
$328$	0	0
$329$	18.2462	1.00595
$330$	0	0
$331$	8.49242	0.466786	0.233393	−	0.972383i	$-0.425017\pi$
$331$	8.49242	0.466786	0.233393	+	0.972383i	$0.425017\pi$
$332$	0	0
$333$	−4.63068	−0.253760
$334$	0	0
$335$	−7.24621	−0.395903
$336$	0	0
$337$	−4.43845	−0.241778	−0.120889	−	0.992666i	$-0.538574\pi$
$337$	−4.43845	−0.241778	−0.120889	+	0.992666i	$0.538574\pi$
$338$	0	0
$339$	−5.36932	−0.291621
$340$	0	0
$341$	2.12311	0.114973
$342$	0	0
$343$	19.0540	1.02882
$344$	0	0
$345$	−6.43845	−0.346634
$346$	0	0
$347$	9.36932	0.502971	0.251486	−	0.967861i	$-0.419081\pi$
$347$	9.36932	0.502971	0.251486	+	0.967861i	$0.419081\pi$
$348$	0	0
$349$	28.9309	1.54863	0.774317	−	0.632798i	$-0.218094\pi$
$349$	28.9309	1.54863	0.774317	+	0.632798i	$0.218094\pi$
$350$	0	0
$351$	5.56155	0.296854
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	3.12311	0.165757
$356$	0	0
$357$	8.00000	0.423405
$358$	0	0
$359$	32.7386	1.72788	0.863940	−	0.503596i	$-0.167990\pi$
$359$	32.7386	1.72788	0.863940	+	0.503596i	$0.167990\pi$
$360$	0	0
$361$	11.9309	0.627941
$362$	0	0
$363$	−15.6155	−0.819603
$364$	0	0
$365$	9.24621	0.483969
$366$	0	0
$367$	−6.24621	−0.326050	−0.163025	−	0.986622i	$-0.552125\pi$
$367$	−6.24621	−0.326050	−0.163025	+	0.986622i	$0.552125\pi$
$368$	0	0
$369$	5.12311	0.266698
$370$	0	0
$371$	−5.12311	−0.265978
$372$	0	0
$373$	0.753789	0.0390297	0.0195149	−	0.999810i	$-0.493788\pi$
$373$	0.753789	0.0390297	0.0195149	+	0.999810i	$0.493788\pi$
$374$	0	0
$375$	1.56155	0.0806382
$376$	0	0
$377$	3.00000	0.154508
$378$	0	0
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	20.4924	1.04986
$382$	0	0
$383$	23.8617	1.21928	0.609639	−	0.792679i	$-0.291314\pi$
$383$	23.8617	1.21928	0.609639	+	0.792679i	$0.291314\pi$
$384$	0	0
$385$	−2.56155	−0.130549
$386$	0	0
$387$	1.12311	0.0570907
$388$	0	0
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	8.24621	0.417029
$392$	0	0
$393$	−20.4924	−1.03371
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	23.3693	1.17287	0.586436	−	0.809995i	$-0.300530\pi$
$397$	23.3693	1.17287	0.586436	+	0.809995i	$0.300530\pi$
$398$	0	0
$399$	−22.2462	−1.11370
$400$	0	0
$401$	3.43845	0.171708	0.0858539	−	0.996308i	$-0.472638\pi$
$401$	3.43845	0.171708	0.0858539	+	0.996308i	$0.472638\pi$
$402$	0	0
$403$	−2.12311	−0.105759
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	8.24621	0.408750
$408$	0	0
$409$	24.4924	1.21107	0.605536	−	0.795818i	$-0.292959\pi$
$409$	24.4924	1.21107	0.605536	+	0.795818i	$0.292959\pi$
$410$	0	0
$411$	−3.12311	−0.154051
$412$	0	0
$413$	23.0540	1.13441
$414$	0	0
$415$	−3.24621	−0.159350
$416$	0	0
$417$	0.492423	0.0241140
$418$	0	0
$419$	3.12311	0.152574	0.0762868	−	0.997086i	$-0.475694\pi$
$419$	3.12311	0.152574	0.0762868	+	0.997086i	$0.475694\pi$
$420$	0	0
$421$	−25.3693	−1.23642	−0.618212	−	0.786011i	$-0.712143\pi$
$421$	−25.3693	−1.23642	−0.618212	+	0.786011i	$0.712143\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−2.24621	−0.108702
$428$	0	0
$429$	−1.56155	−0.0753925
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	1.05398	0.0506508	0.0253254	−	0.999679i	$-0.491938\pi$
$433$	1.05398	0.0506508	0.0253254	+	0.999679i	$0.491938\pi$
$434$	0	0
$435$	−4.68466	−0.224612
$436$	0	0
$437$	−22.9309	−1.09693
$438$	0	0
$439$	−14.1231	−0.674059	−0.337030	−	0.941494i	$-0.609422\pi$
$439$	−14.1231	−0.674059	−0.337030	+	0.941494i	$0.609422\pi$
$440$	0	0
$441$	0.246211	0.0117243
$442$	0	0
$443$	23.6847	1.12529	0.562646	−	0.826698i	$-0.309783\pi$
$443$	23.6847	1.12529	0.562646	+	0.826698i	$0.309783\pi$
$444$	0	0
$445$	−6.87689	−0.325996
$446$	0	0
$447$	10.7386	0.507920
$448$	0	0
$449$	26.9848	1.27349	0.636747	−	0.771073i	$-0.280280\pi$
$449$	26.9848	1.27349	0.636747	+	0.771073i	$0.280280\pi$
$450$	0	0
$451$	−9.12311	−0.429590
$452$	0	0
$453$	−1.56155	−0.0733682
$454$	0	0
$455$	2.56155	0.120087
$456$	0	0
$457$	−4.49242	−0.210147	−0.105073	−	0.994464i	$-0.533508\pi$
$457$	−4.49242	−0.210147	−0.105073	+	0.994464i	$0.533508\pi$
$458$	0	0
$459$	11.1231	0.519182
$460$	0	0
$461$	9.05398	0.421686	0.210843	−	0.977520i	$-0.432379\pi$
$461$	9.05398	0.421686	0.210843	+	0.977520i	$0.432379\pi$
$462$	0	0
$463$	26.7386	1.24265	0.621325	−	0.783553i	$-0.286595\pi$
$463$	26.7386	1.24265	0.621325	+	0.783553i	$0.286595\pi$
$464$	0	0
$465$	3.31534	0.153745
$466$	0	0
$467$	−32.4924	−1.50357	−0.751785	−	0.659408i	$-0.770807\pi$
$467$	−32.4924	−1.50357	−0.751785	+	0.659408i	$0.770807\pi$
$468$	0	0
$469$	18.5616	0.857093
$470$	0	0
$471$	5.56155	0.256263
$472$	0	0
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	5.56155	0.255182
$476$	0	0
$477$	−1.12311	−0.0514235
$478$	0	0
$479$	−4.87689	−0.222831	−0.111415	−	0.993774i	$-0.535538\pi$
$479$	−4.87689	−0.222831	−0.111415	+	0.993774i	$0.535538\pi$
$480$	0	0
$481$	−8.24621	−0.375995
$482$	0	0
$483$	16.4924	0.750431
$484$	0	0
$485$	−10.2462	−0.465256
$486$	0	0
$487$	17.3693	0.787079	0.393539	−	0.919308i	$-0.371250\pi$
$487$	17.3693	0.787079	0.393539	+	0.919308i	$0.371250\pi$
$488$	0	0
$489$	−3.31534	−0.149925
$490$	0	0
$491$	7.05398	0.318341	0.159171	−	0.987251i	$-0.449118\pi$
$491$	7.05398	0.318341	0.159171	+	0.987251i	$0.449118\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	0	0
$495$	−0.561553	−0.0252399
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	2.24621	0.100554	0.0502771	−	0.998735i	$-0.483990\pi$
$499$	2.24621	0.100554	0.0502771	+	0.998735i	$0.483990\pi$
$500$	0	0
$501$	−18.7386	−0.837180
$502$	0	0
$503$	−17.3693	−0.774460	−0.387230	−	0.921983i	$-0.626568\pi$
$503$	−17.3693	−0.774460	−0.387230	+	0.921983i	$0.626568\pi$
$504$	0	0
$505$	−15.3693	−0.683926
$506$	0	0
$507$	−18.7386	−0.832212
$508$	0	0
$509$	22.2462	0.986046	0.493023	−	0.870016i	$-0.335892\pi$
$509$	22.2462	0.986046	0.493023	+	0.870016i	$0.335892\pi$
$510$	0	0
$511$	−23.6847	−1.04775
$512$	0	0
$513$	−30.9309	−1.36563
$514$	0	0
$515$	20.2462	0.892155
$516$	0	0
$517$	−7.12311	−0.313274
$518$	0	0
$519$	24.3845	1.07036
$520$	0	0
$521$	11.2462	0.492705	0.246353	−	0.969180i	$-0.420768\pi$
$521$	11.2462	0.492705	0.246353	+	0.969180i	$0.420768\pi$
$522$	0	0
$523$	−15.0000	−0.655904	−0.327952	−	0.944694i	$-0.606358\pi$
$523$	−15.0000	−0.655904	−0.327952	+	0.944694i	$0.606358\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−4.24621	−0.184968
$528$	0	0
$529$	−6.00000	−0.260870
$530$	0	0
$531$	5.05398	0.219324
$532$	0	0
$533$	9.12311	0.395166
$534$	0	0
$535$	5.87689	0.254080
$536$	0	0
$537$	−25.3693	−1.09477
$538$	0	0
$539$	−0.438447	−0.0188853
$540$	0	0
$541$	−41.4233	−1.78093	−0.890463	−	0.455055i	$-0.849620\pi$
$541$	−41.4233	−1.78093	−0.890463	+	0.455055i	$0.849620\pi$
$542$	0	0
$543$	−41.3693	−1.77533
$544$	0	0
$545$	−4.87689	−0.208903
$546$	0	0
$547$	−11.6155	−0.496644	−0.248322	−	0.968678i	$-0.579879\pi$
$547$	−11.6155	−0.496644	−0.248322	+	0.968678i	$0.579879\pi$
$548$	0	0
$549$	−0.492423	−0.0210161
$550$	0	0
$551$	−16.6847	−0.710790
$552$	0	0
$553$	20.4924	0.871426
$554$	0	0
$555$	12.8769	0.546594
$556$	0	0
$557$	−1.00000	−0.0423714	−0.0211857	−	0.999776i	$-0.506744\pi$
$557$	−1.00000	−0.0423714	−0.0211857	+	0.999776i	$0.506744\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	−3.12311	−0.131858
$562$	0	0
$563$	26.2462	1.10615	0.553073	−	0.833133i	$-0.313455\pi$
$563$	26.2462	1.10615	0.553073	+	0.833133i	$0.313455\pi$
$564$	0	0
$565$	−3.43845	−0.144657
$566$	0	0
$567$	17.9309	0.753026
$568$	0	0
$569$	21.6847	0.909068	0.454534	−	0.890729i	$-0.349806\pi$
$569$	21.6847	0.909068	0.454534	+	0.890729i	$0.349806\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	0	0
$573$	−29.7538	−1.24298
$574$	0	0
$575$	−4.12311	−0.171945
$576$	0	0
$577$	−22.2462	−0.926122	−0.463061	−	0.886326i	$-0.653249\pi$
$577$	−22.2462	−0.926122	−0.463061	+	0.886326i	$0.653249\pi$
$578$	0	0
$579$	6.63068	0.275562
$580$	0	0
$581$	8.31534	0.344978
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	0.561553	0.0232174
$586$	0	0
$587$	3.19224	0.131758	0.0658788	−	0.997828i	$-0.479015\pi$
$587$	3.19224	0.131758	0.0658788	+	0.997828i	$0.479015\pi$
$588$	0	0
$589$	11.8078	0.486530
$590$	0	0
$591$	−37.3693	−1.53717
$592$	0	0
$593$	7.56155	0.310516	0.155258	−	0.987874i	$-0.450379\pi$
$593$	7.56155	0.310516	0.155258	+	0.987874i	$0.450379\pi$
$594$	0	0
$595$	5.12311	0.210027
$596$	0	0
$597$	15.6155	0.639101
$598$	0	0
$599$	−13.7538	−0.561965	−0.280982	−	0.959713i	$-0.590660\pi$
$599$	−13.7538	−0.561965	−0.280982	+	0.959713i	$0.590660\pi$
$600$	0	0
$601$	−20.9309	−0.853788	−0.426894	−	0.904302i	$-0.640392\pi$
$601$	−20.9309	−0.853788	−0.426894	+	0.904302i	$0.640392\pi$
$602$	0	0
$603$	4.06913	0.165708
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	40.8078	1.65634	0.828168	−	0.560480i	$-0.189383\pi$
$607$	40.8078	1.65634	0.828168	+	0.560480i	$0.189383\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	7.12311	0.288170
$612$	0	0
$613$	−33.3693	−1.34777	−0.673887	−	0.738834i	$-0.735377\pi$
$613$	−33.3693	−1.34777	−0.673887	+	0.738834i	$0.735377\pi$
$614$	0	0
$615$	−14.2462	−0.574463
$616$	0	0
$617$	35.9848	1.44870	0.724348	−	0.689435i	$-0.242141\pi$
$617$	35.9848	1.44870	0.724348	+	0.689435i	$0.242141\pi$
$618$	0	0
$619$	−16.7386	−0.672782	−0.336391	−	0.941722i	$-0.609206\pi$
$619$	−16.7386	−0.672782	−0.336391	+	0.941722i	$0.609206\pi$
$620$	0	0
$621$	22.9309	0.920184
$622$	0	0
$623$	17.6155	0.705751
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.68466	0.346832
$628$	0	0
$629$	−16.4924	−0.657596
$630$	0	0
$631$	37.3693	1.48765	0.743825	−	0.668375i	$-0.233010\pi$
$631$	37.3693	1.48765	0.743825	+	0.668375i	$0.233010\pi$
$632$	0	0
$633$	10.1383	0.402960
$634$	0	0
$635$	13.1231	0.520775
$636$	0	0
$637$	0.438447	0.0173719
$638$	0	0
$639$	−1.75379	−0.0693788
$640$	0	0
$641$	−31.4924	−1.24388	−0.621938	−	0.783067i	$-0.713654\pi$
$641$	−31.4924	−1.24388	−0.621938	+	0.783067i	$0.713654\pi$
$642$	0	0
$643$	38.7386	1.52770	0.763851	−	0.645392i	$-0.223306\pi$
$643$	38.7386	1.52770	0.763851	+	0.645392i	$0.223306\pi$
$644$	0	0
$645$	−3.12311	−0.122972
$646$	0	0
$647$	−10.0000	−0.393141	−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000	−0.393141	−0.196570	+	0.980490i	$0.562980\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	−8.49242	−0.332844
$652$	0	0
$653$	−16.4924	−0.645398	−0.322699	−	0.946502i	$-0.604590\pi$
$653$	−16.4924	−0.645398	−0.322699	+	0.946502i	$0.604590\pi$
$654$	0	0
$655$	−13.1231	−0.512762
$656$	0	0
$657$	−5.19224	−0.202568
$658$	0	0
$659$	−23.8078	−0.927419	−0.463709	−	0.885987i	$-0.653482\pi$
$659$	−23.8078	−0.927419	−0.463709	+	0.885987i	$0.653482\pi$
$660$	0	0
$661$	−29.1231	−1.13276	−0.566379	−	0.824145i	$-0.691656\pi$
$661$	−29.1231	−1.13276	−0.566379	+	0.824145i	$0.691656\pi$
$662$	0	0
$663$	3.12311	0.121291
$664$	0	0
$665$	−14.2462	−0.552444
$666$	0	0
$667$	12.3693	0.478942
$668$	0	0
$669$	2.73863	0.105882
$670$	0	0
$671$	0.876894	0.0338521
$672$	0	0
$673$	32.2462	1.24300	0.621500	−	0.783414i	$-0.286524\pi$
$673$	32.2462	1.24300	0.621500	+	0.783414i	$0.286524\pi$
$674$	0	0
$675$	−5.56155	−0.214064
$676$	0	0
$677$	−24.4233	−0.938663	−0.469332	−	0.883022i	$-0.655505\pi$
$677$	−24.4233	−0.938663	−0.469332	+	0.883022i	$0.655505\pi$
$678$	0	0
$679$	26.2462	1.00724
$680$	0	0
$681$	−6.24621	−0.239355
$682$	0	0
$683$	14.4384	0.552472	0.276236	−	0.961090i	$-0.410913\pi$
$683$	14.4384	0.552472	0.276236	+	0.961090i	$0.410913\pi$
$684$	0	0
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	39.4233	1.50409
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	49.3693	1.87810	0.939049	−	0.343784i	$-0.111709\pi$
$691$	49.3693	1.87810	0.939049	+	0.343784i	$0.111709\pi$
$692$	0	0
$693$	1.43845	0.0546421
$694$	0	0
$695$	0.315342	0.0119616
$696$	0	0
$697$	18.2462	0.691125
$698$	0	0
$699$	−32.7926	−1.24033
$700$	0	0
$701$	25.6847	0.970096	0.485048	−	0.874487i	$-0.338802\pi$
$701$	25.6847	0.970096	0.485048	+	0.874487i	$0.338802\pi$
$702$	0	0
$703$	45.8617	1.72971
$704$	0	0
$705$	−11.1231	−0.418920
$706$	0	0
$707$	39.3693	1.48064
$708$	0	0
$709$	−20.7538	−0.779425	−0.389712	−	0.920937i	$-0.627426\pi$
$709$	−20.7538	−0.779425	−0.389712	+	0.920937i	$0.627426\pi$
$710$	0	0
$711$	4.49242	0.168479
$712$	0	0
$713$	−8.75379	−0.327832
$714$	0	0
$715$	−1.00000	−0.0373979
$716$	0	0
$717$	38.5464	1.43954
$718$	0	0
$719$	−2.87689	−0.107290	−0.0536450	−	0.998560i	$-0.517084\pi$
$719$	−2.87689	−0.107290	−0.0536450	+	0.998560i	$0.517084\pi$
$720$	0	0
$721$	−51.8617	−1.93143
$722$	0	0
$723$	−20.6847	−0.769271
$724$	0	0
$725$	−3.00000	−0.111417
$726$	0	0
$727$	−6.87689	−0.255050	−0.127525	−	0.991835i	$-0.540703\pi$
$727$	−6.87689	−0.255050	−0.127525	+	0.991835i	$0.540703\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	4.00000	0.147945
$732$	0	0
$733$	−28.0540	−1.03620	−0.518099	−	0.855321i	$-0.673360\pi$
$733$	−28.0540	−1.03620	−0.518099	+	0.855321i	$0.673360\pi$
$734$	0	0
$735$	−0.684658	−0.0252540
$736$	0	0
$737$	−7.24621	−0.266918
$738$	0	0
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	−8.68466	−0.319039
$742$	0	0
$743$	−27.3693	−1.00408	−0.502041	−	0.864844i	$-0.667418\pi$
$743$	−27.3693	−1.00408	−0.502041	+	0.864844i	$0.667418\pi$
$744$	0	0
$745$	6.87689	0.251950
$746$	0	0
$747$	1.82292	0.0666971
$748$	0	0
$749$	−15.0540	−0.550060
$750$	0	0
$751$	30.1080	1.09865	0.549327	−	0.835607i	$-0.314884\pi$
$751$	30.1080	1.09865	0.549327	+	0.835607i	$0.314884\pi$
$752$	0	0
$753$	3.31534	0.120818
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−27.8617	−1.01265	−0.506326	−	0.862342i	$-0.668997\pi$
$757$	−27.8617	−1.01265	−0.506326	+	0.862342i	$0.668997\pi$
$758$	0	0
$759$	−6.43845	−0.233701
$760$	0	0
$761$	−4.87689	−0.176787	−0.0883936	−	0.996086i	$-0.528173\pi$
$761$	−4.87689	−0.176787	−0.0883936	+	0.996086i	$0.528173\pi$
$762$	0	0
$763$	12.4924	0.452256
$764$	0	0
$765$	1.12311	0.0406060
$766$	0	0
$767$	9.00000	0.324971
$768$	0	0
$769$	12.8769	0.464353	0.232176	−	0.972674i	$-0.425415\pi$
$769$	12.8769	0.464353	0.232176	+	0.972674i	$0.425415\pi$
$770$	0	0
$771$	27.0388	0.973779
$772$	0	0
$773$	−39.8617	−1.43373	−0.716864	−	0.697213i	$-0.754423\pi$
$773$	−39.8617	−1.43373	−0.716864	+	0.697213i	$0.754423\pi$
$774$	0	0
$775$	2.12311	0.0762642
$776$	0	0
$777$	−32.9848	−1.18332
$778$	0	0
$779$	−50.7386	−1.81790
$780$	0	0
$781$	3.12311	0.111754
$782$	0	0
$783$	16.6847	0.596261
$784$	0	0
$785$	3.56155	0.127117
$786$	0	0
$787$	−6.24621	−0.222653	−0.111327	−	0.993784i	$-0.535510\pi$
$787$	−6.24621	−0.222653	−0.111327	+	0.993784i	$0.535510\pi$
$788$	0	0
$789$	29.0691	1.03489
$790$	0	0
$791$	8.80776	0.313168
$792$	0	0
$793$	−0.876894	−0.0311394
$794$	0	0
$795$	3.12311	0.110765
$796$	0	0
$797$	22.8769	0.810341	0.405171	−	0.914241i	$-0.367212\pi$
$797$	22.8769	0.810341	0.405171	+	0.914241i	$0.367212\pi$
$798$	0	0
$799$	14.2462	0.503995
$800$	0	0
$801$	3.86174	0.136448
$802$	0	0
$803$	9.24621	0.326292
$804$	0	0
$805$	10.5616	0.372246
$806$	0	0
$807$	−20.7926	−0.731935
$808$	0	0
$809$	−19.8617	−0.698302	−0.349151	−	0.937067i	$-0.613530\pi$
$809$	−19.8617	−0.698302	−0.349151	+	0.937067i	$0.613530\pi$
$810$	0	0
$811$	−23.3693	−0.820608	−0.410304	−	0.911949i	$-0.634577\pi$
$811$	−23.3693	−0.820608	−0.410304	+	0.911949i	$0.634577\pi$
$812$	0	0
$813$	−39.6155	−1.38938
$814$	0	0
$815$	−2.12311	−0.0743691
$816$	0	0
$817$	−11.1231	−0.389148
$818$	0	0
$819$	−1.43845	−0.0502634
$820$	0	0
$821$	−7.61553	−0.265784	−0.132892	−	0.991131i	$-0.542426\pi$
$821$	−7.61553	−0.265784	−0.132892	+	0.991131i	$0.542426\pi$
$822$	0	0
$823$	17.3693	0.605456	0.302728	−	0.953077i	$-0.402103\pi$
$823$	17.3693	0.605456	0.302728	+	0.953077i	$0.402103\pi$
$824$	0	0
$825$	1.56155	0.0543663
$826$	0	0
$827$	−13.3693	−0.464897	−0.232448	−	0.972609i	$-0.574674\pi$
$827$	−13.3693	−0.464897	−0.232448	+	0.972609i	$0.574674\pi$
$828$	0	0
$829$	53.4924	1.85787	0.928934	−	0.370245i	$-0.120726\pi$
$829$	53.4924	1.85787	0.928934	+	0.370245i	$0.120726\pi$
$830$	0	0
$831$	−31.6155	−1.09673
$832$	0	0
$833$	0.876894	0.0303826
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	−11.8078	−0.408136
$838$	0	0
$839$	16.3153	0.563268	0.281634	−	0.959522i	$-0.409124\pi$
$839$	16.3153	0.563268	0.281634	+	0.959522i	$0.409124\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−13.2614	−0.456746
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	25.6155	0.880160
$848$	0	0
$849$	−30.9309	−1.06154
$850$	0	0
$851$	−34.0000	−1.16550
$852$	0	0
$853$	36.2462	1.24105	0.620523	−	0.784188i	$-0.286920\pi$
$853$	36.2462	1.24105	0.620523	+	0.784188i	$0.286920\pi$
$854$	0	0
$855$	−3.12311	−0.106808
$856$	0	0
$857$	−28.9309	−0.988260	−0.494130	−	0.869388i	$-0.664513\pi$
$857$	−28.9309	−0.988260	−0.494130	+	0.869388i	$0.664513\pi$
$858$	0	0
$859$	49.1231	1.67606	0.838029	−	0.545625i	$-0.183708\pi$
$859$	49.1231	1.67606	0.838029	+	0.545625i	$0.183708\pi$
$860$	0	0
$861$	36.4924	1.24366
$862$	0	0
$863$	8.80776	0.299820	0.149910	−	0.988700i	$-0.452102\pi$
$863$	8.80776	0.299820	0.149910	+	0.988700i	$0.452102\pi$
$864$	0	0
$865$	15.6155	0.530944
$866$	0	0
$867$	−20.3002	−0.689430
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	7.24621	0.245529
$872$	0	0
$873$	5.75379	0.194736
$874$	0	0
$875$	−2.56155	−0.0865963
$876$	0	0
$877$	15.0000	0.506514	0.253257	−	0.967399i	$-0.418498\pi$
$877$	15.0000	0.506514	0.253257	+	0.967399i	$0.418498\pi$
$878$	0	0
$879$	−22.3542	−0.753987
$880$	0	0
$881$	−45.6155	−1.53683	−0.768413	−	0.639954i	$-0.778953\pi$
$881$	−45.6155	−1.53683	−0.768413	+	0.639954i	$0.778953\pi$
$882$	0	0
$883$	−38.8769	−1.30831	−0.654156	−	0.756360i	$-0.726976\pi$
$883$	−38.8769	−1.30831	−0.654156	+	0.756360i	$0.726976\pi$
$884$	0	0
$885$	−14.0540	−0.472419
$886$	0	0
$887$	−50.0540	−1.68065	−0.840324	−	0.542084i	$-0.817635\pi$
$887$	−50.0540	−1.68065	−0.840324	+	0.542084i	$0.817635\pi$
$888$	0	0
$889$	−33.6155	−1.12743
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	−39.6155	−1.32568
$894$	0	0
$895$	−16.2462	−0.543051
$896$	0	0
$897$	6.43845	0.214973
$898$	0	0
$899$	−6.36932	−0.212429
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	−26.4924	−0.880638
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	8.63068	0.286262
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	−3.24621	−0.107434
$914$	0	0
$915$	1.36932	0.0452682
$916$	0	0
$917$	33.6155	1.11008
$918$	0	0
$919$	6.00000	0.197922	0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000	0.197922	0.0989609	+	0.995091i	$0.468448\pi$
$920$	0	0
$921$	31.6155	1.04177
$922$	0	0
$923$	−3.12311	−0.102798
$924$	0	0
$925$	8.24621	0.271134
$926$	0	0
$927$	−11.3693	−0.373417
$928$	0	0
$929$	−14.3845	−0.471939	−0.235970	−	0.971760i	$-0.575827\pi$
$929$	−14.3845	−0.471939	−0.235970	+	0.971760i	$0.575827\pi$
$930$	0	0
$931$	−2.43845	−0.0799169
$932$	0	0
$933$	38.8229	1.27101
$934$	0	0
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	1.50758	0.0492504	0.0246252	−	0.999697i	$-0.492161\pi$
$937$	1.50758	0.0492504	0.0246252	+	0.999697i	$0.492161\pi$
$938$	0	0
$939$	10.7386	0.350442
$940$	0	0
$941$	5.61553	0.183061	0.0915305	−	0.995802i	$-0.470824\pi$
$941$	5.61553	0.183061	0.0915305	+	0.995802i	$0.470824\pi$
$942$	0	0
$943$	37.6155	1.22493
$944$	0	0
$945$	14.2462	0.463429
$946$	0	0
$947$	53.0000	1.72227	0.861134	−	0.508378i	$-0.169755\pi$
$947$	53.0000	1.72227	0.861134	+	0.508378i	$0.169755\pi$
$948$	0	0
$949$	−9.24621	−0.300145
$950$	0	0
$951$	13.9460	0.452231
$952$	0	0
$953$	47.6155	1.54242	0.771209	−	0.636582i	$-0.219652\pi$
$953$	47.6155	1.54242	0.771209	+	0.636582i	$0.219652\pi$
$954$	0	0
$955$	−19.0540	−0.616572
$956$	0	0
$957$	−4.68466	−0.151434
$958$	0	0
$959$	5.12311	0.165434
$960$	0	0
$961$	−26.4924	−0.854594
$962$	0	0
$963$	−3.30019	−0.106347
$964$	0	0
$965$	4.24621	0.136690
$966$	0	0
$967$	19.9848	0.642669	0.321335	−	0.946966i	$-0.395869\pi$
$967$	19.9848	0.642669	0.321335	+	0.946966i	$0.395869\pi$
$968$	0	0
$969$	−17.3693	−0.557983
$970$	0	0
$971$	16.0000	0.513464	0.256732	−	0.966483i	$-0.417354\pi$
$971$	16.0000	0.513464	0.256732	+	0.966483i	$0.417354\pi$
$972$	0	0
$973$	−0.807764	−0.0258957
$974$	0	0
$975$	−1.56155	−0.0500097
$976$	0	0
$977$	21.6847	0.693754	0.346877	−	0.937911i	$-0.387242\pi$
$977$	21.6847	0.693754	0.346877	+	0.937911i	$0.387242\pi$
$978$	0	0
$979$	−6.87689	−0.219786
$980$	0	0
$981$	2.73863	0.0874378
$982$	0	0
$983$	42.4233	1.35309	0.676546	−	0.736400i	$-0.263476\pi$
$983$	42.4233	1.35309	0.676546	+	0.736400i	$0.263476\pi$
$984$	0	0
$985$	−23.9309	−0.762501
$986$	0	0
$987$	28.4924	0.906924
$988$	0	0
$989$	8.24621	0.262214
$990$	0	0
$991$	−24.1231	−0.766296	−0.383148	−	0.923687i	$-0.625160\pi$
$991$	−24.1231	−0.766296	−0.383148	+	0.923687i	$0.625160\pi$
$992$	0	0
$993$	13.2614	0.420837
$994$	0	0
$995$	10.0000	0.317021
$996$	0	0
$997$	−40.0000	−1.26681	−0.633406	−	0.773819i	$-0.718344\pi$
$997$	−40.0000	−1.26681	−0.633406	+	0.773819i	$0.718344\pi$
$998$	0	0
$999$	−45.8617	−1.45100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.k.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.k.1.2	✓	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +1.00000 q^{5} -2.56155 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} +1.00000 q^{5} -2.56155 q^{7} -0.561553 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.56155 q^{15} -2.00000 q^{17} +5.56155 q^{19} -4.00000 q^{21} -4.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} -3.00000 q^{29} +2.12311 q^{31} +1.56155 q^{33} -2.56155 q^{35} +8.24621 q^{37} -1.56155 q^{39} -9.12311 q^{41} -2.00000 q^{43} -0.561553 q^{45} -7.12311 q^{47} -0.438447 q^{49} -3.12311 q^{51} +2.00000 q^{53} +1.00000 q^{55} +8.68466 q^{57} -9.00000 q^{59} +0.876894 q^{61} +1.43845 q^{63} -1.00000 q^{65} -7.24621 q^{67} -6.43845 q^{69} +3.12311 q^{71} +9.24621 q^{73} +1.56155 q^{75} -2.56155 q^{77} -8.00000 q^{79} -7.00000 q^{81} -3.24621 q^{83} -2.00000 q^{85} -4.68466 q^{87} -6.87689 q^{89} +2.56155 q^{91} +3.31534 q^{93} +5.56155 q^{95} -10.2462 q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 + 2 * q^5 - q^7 + 3 * q^9	\( 2 q - q^{3} + 2 q^{5} - q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} - q^{15} - 4 q^{17} + 7 q^{19} - 8 q^{21} + 2 q^{25} - 7 q^{27} - 6 q^{29} - 4 q^{31} - q^{33} - q^{35} + q^{39} - 10 q^{41} - 4 q^{43} + 3 q^{45} - 6 q^{47} - 5 q^{49} + 2 q^{51} + 4 q^{53} + 2 q^{55} + 5 q^{57} - 18 q^{59} + 10 q^{61} + 7 q^{63} - 2 q^{65} + 2 q^{67} - 17 q^{69} - 2 q^{71} + 2 q^{73} - q^{75} - q^{77} - 16 q^{79} - 14 q^{81} + 10 q^{83} - 4 q^{85} + 3 q^{87} - 22 q^{89} + q^{91} + 19 q^{93} + 7 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^5 - q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 - q^15 - 4 * q^17 + 7 * q^19 - 8 * q^21 + 2 * q^25 - 7 * q^27 - 6 * q^29 - 4 * q^31 - q^33 - q^35 + q^39 - 10 * q^41 - 4 * q^43 + 3 * q^45 - 6 * q^47 - 5 * q^49 + 2 * q^51 + 4 * q^53 + 2 * q^55 + 5 * q^57 - 18 * q^59 + 10 * q^61 + 7 * q^63 - 2 * q^65 + 2 * q^67 - 17 * q^69 - 2 * q^71 + 2 * q^73 - q^75 - q^77 - 16 * q^79 - 14 * q^81 + 10 * q^83 - 4 * q^85 + 3 * q^87 - 22 * q^89 + q^91 + 19 * q^93 + 7 * q^95 - 4 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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