LMFDB - Embedded newform 124.6.a.b.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,6,Mod(1,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("124.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	124.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:	$19.8875936568$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 847x^{4} + 1184x^{3} + 199815x^{2} - 13326x - 12452553 $ x^6 - 2x^5 - 847x^4 + 1184x^3 + 199815x^2 - 13326*x - 12452553
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$21.5036$ of defining polynomial
Character	$\chi$	$=$	124.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+24.5036 q^{3} +52.1028 q^{5} +141.835 q^{7} +357.424 q^{9} +O(q^{10})$	$q+24.5036 q^{3} +52.1028 q^{5} +141.835 q^{7} +357.424 q^{9} -53.4195 q^{11} -287.764 q^{13} +1276.70 q^{15} +338.420 q^{17} -290.332 q^{19} +3475.47 q^{21} -993.316 q^{23} -410.299 q^{25} +2803.81 q^{27} -5503.32 q^{29} -961.000 q^{31} -1308.97 q^{33} +7390.02 q^{35} +9672.81 q^{37} -7051.25 q^{39} +18866.4 q^{41} -1037.64 q^{43} +18622.8 q^{45} -14977.3 q^{47} +3310.29 q^{49} +8292.50 q^{51} +15823.0 q^{53} -2783.30 q^{55} -7114.16 q^{57} +37679.7 q^{59} +17322.4 q^{61} +50695.5 q^{63} -14993.3 q^{65} -19631.1 q^{67} -24339.8 q^{69} -60543.5 q^{71} -54003.4 q^{73} -10053.8 q^{75} -7576.78 q^{77} +39671.6 q^{79} -18150.9 q^{81} -77548.8 q^{83} +17632.6 q^{85} -134851. q^{87} -44216.9 q^{89} -40815.2 q^{91} -23547.9 q^{93} -15127.1 q^{95} -88524.0 q^{97} -19093.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9}+O(q^{10}) $ 6 * q + 20 * q^3 + 25 * q^5 + 39 * q^7 + 306 * q^9	$ 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9} + 280 q^{11} + 1214 q^{13} + 1914 q^{15} + 796 q^{17} + 3147 q^{19} + 1082 q^{21} + 9122 q^{23} + 6481 q^{25} + 7316 q^{27} + 13020 q^{29} - 5766 q^{31} + 24804 q^{33} + 20059 q^{35} + 21678 q^{37} + 30680 q^{39} + 3227 q^{41} + 37882 q^{43} + 26169 q^{45} + 29708 q^{47} + 34849 q^{49} + 24432 q^{51} - 9976 q^{53} + 23758 q^{55} + 17318 q^{57} + 20573 q^{59} + 21610 q^{61} - 17697 q^{63} + 3894 q^{65} - 17024 q^{67} - 83692 q^{69} + 44509 q^{71} - 161864 q^{73} - 49430 q^{75} - 144202 q^{77} - 24420 q^{79} - 181158 q^{81} - 114160 q^{83} - 228882 q^{85} - 56180 q^{87} - 199742 q^{89} - 186774 q^{91} - 19220 q^{93} - 12793 q^{95} - 282951 q^{97} + 34060 q^{99}+O(q^{100}) $ 6 * q + 20 * q^3 + 25 * q^5 + 39 * q^7 + 306 * q^9 + 280 * q^11 + 1214 * q^13 + 1914 * q^15 + 796 * q^17 + 3147 * q^19 + 1082 * q^21 + 9122 * q^23 + 6481 * q^25 + 7316 * q^27 + 13020 * q^29 - 5766 * q^31 + 24804 * q^33 + 20059 * q^35 + 21678 * q^37 + 30680 * q^39 + 3227 * q^41 + 37882 * q^43 + 26169 * q^45 + 29708 * q^47 + 34849 * q^49 + 24432 * q^51 - 9976 * q^53 + 23758 * q^55 + 17318 * q^57 + 20573 * q^59 + 21610 * q^61 - 17697 * q^63 + 3894 * q^65 - 17024 * q^67 - 83692 * q^69 + 44509 * q^71 - 161864 * q^73 - 49430 * q^75 - 144202 * q^77 - 24420 * q^79 - 181158 * q^81 - 114160 * q^83 - 228882 * q^85 - 56180 * q^87 - 199742 * q^89 - 186774 * q^91 - 19220 * q^93 - 12793 * q^95 - 282951 * q^97 + 34060 * 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
$3$	24.5036	1.57190	0.785952	−	0.618287i	$-0.212173\pi$
$3$	24.5036	1.57190	0.785952	+	0.618287i	$0.212173\pi$
$4$	0	0
$5$	52.1028	0.932043	0.466022	−	0.884773i	$-0.345687\pi$
$5$	52.1028	0.932043	0.466022	+	0.884773i	$0.345687\pi$
$6$	0	0
$7$	141.835	1.09406	0.547028	−	0.837114i	$-0.315759\pi$
$7$	141.835	1.09406	0.547028	+	0.837114i	$0.315759\pi$
$8$	0	0
$9$	357.424	1.47088
$10$	0	0
$11$	−53.4195	−0.133112	−0.0665561	−	0.997783i	$-0.521201\pi$
$11$	−53.4195	−0.133112	−0.0665561	+	0.997783i	$0.521201\pi$
$12$	0	0
$13$	−287.764	−0.472257	−0.236128	−	0.971722i	$-0.575879\pi$
$13$	−287.764	−0.472257	−0.236128	+	0.971722i	$0.575879\pi$
$14$	0	0
$15$	1276.70	1.46508
$16$	0	0
$17$	338.420	0.284010	0.142005	−	0.989866i	$-0.454645\pi$
$17$	338.420	0.284010	0.142005	+	0.989866i	$0.454645\pi$
$18$	0	0
$19$	−290.332	−0.184506	−0.0922530	−	0.995736i	$-0.529407\pi$
$19$	−290.332	−0.184506	−0.0922530	+	0.995736i	$0.529407\pi$
$20$	0	0
$21$	3475.47	1.71975
$22$	0	0
$23$	−993.316	−0.391532	−0.195766	−	0.980651i	$-0.562719\pi$
$23$	−993.316	−0.391532	−0.195766	+	0.980651i	$0.562719\pi$
$24$	0	0
$25$	−410.299	−0.131296
$26$	0	0
$27$	2803.81	0.740182
$28$	0	0
$29$	−5503.32	−1.21515	−0.607575	−	0.794263i	$-0.707857\pi$
$29$	−5503.32	−1.21515	−0.607575	+	0.794263i	$0.707857\pi$
$30$	0	0
$31$	−961.000	−0.179605
$31$	−961.000	−0.179605
$32$	0	0
$33$	−1308.97	−0.209240
$34$	0	0
$35$	7390.02	1.01971
$36$	0	0
$37$	9672.81	1.16158	0.580789	−	0.814054i	$-0.302744\pi$
$37$	9672.81	1.16158	0.580789	+	0.814054i	$0.302744\pi$
$38$	0	0
$39$	−7051.25	−0.742343
$40$	0	0
$41$	18866.4	1.75279	0.876395	−	0.481592i	$-0.159941\pi$
$41$	18866.4	1.75279	0.876395	+	0.481592i	$0.159941\pi$
$42$	0	0
$43$	−1037.64	−0.0855804	−0.0427902	−	0.999084i	$-0.513625\pi$
$43$	−1037.64	−0.0855804	−0.0427902	+	0.999084i	$0.513625\pi$
$44$	0	0
$45$	18622.8	1.37093
$46$	0	0
$47$	−14977.3	−0.988981	−0.494490	−	0.869183i	$-0.664645\pi$
$47$	−14977.3	−0.988981	−0.494490	+	0.869183i	$0.664645\pi$
$48$	0	0
$49$	3310.29	0.196959
$50$	0	0
$51$	8292.50	0.446437
$52$	0	0
$53$	15823.0	0.773746	0.386873	−	0.922133i	$-0.373555\pi$
$53$	15823.0	0.773746	0.386873	+	0.922133i	$0.373555\pi$
$54$	0	0
$55$	−2783.30	−0.124066
$56$	0	0
$57$	−7114.16	−0.290026
$58$	0	0
$59$	37679.7	1.40922	0.704608	−	0.709596i	$-0.251123\pi$
$59$	37679.7	1.40922	0.704608	+	0.709596i	$0.251123\pi$
$60$	0	0
$61$	17322.4	0.596052	0.298026	−	0.954558i	$-0.403672\pi$
$61$	17322.4	0.596052	0.298026	+	0.954558i	$0.403672\pi$
$62$	0	0
$63$	50695.5	1.60923
$64$	0	0
$65$	−14993.3	−0.440164
$66$	0	0
$67$	−19631.1	−0.534267	−0.267134	−	0.963659i	$-0.586077\pi$
$67$	−19631.1	−0.534267	−0.267134	+	0.963659i	$0.586077\pi$
$68$	0	0
$69$	−24339.8	−0.615452
$70$	0	0
$71$	−60543.5	−1.42535	−0.712675	−	0.701494i	$-0.752517\pi$
$71$	−60543.5	−1.42535	−0.712675	+	0.701494i	$0.752517\pi$
$72$	0	0
$73$	−54003.4	−1.18608	−0.593040	−	0.805173i	$-0.702072\pi$
$73$	−54003.4	−1.18608	−0.593040	+	0.805173i	$0.702072\pi$
$74$	0	0
$75$	−10053.8	−0.206384
$76$	0	0
$77$	−7576.78	−0.145632
$78$	0	0
$79$	39671.6	0.715174	0.357587	−	0.933880i	$-0.383600\pi$
$79$	39671.6	0.715174	0.357587	+	0.933880i	$0.383600\pi$
$80$	0	0
$81$	−18150.9	−0.307387
$82$	0	0
$83$	−77548.8	−1.23561	−0.617803	−	0.786333i	$-0.711977\pi$
$83$	−77548.8	−1.23561	−0.617803	+	0.786333i	$0.711977\pi$
$84$	0	0
$85$	17632.6	0.264710
$86$	0	0
$87$	−134851.	−1.91010
$88$	0	0
$89$	−44216.9	−0.591716	−0.295858	−	0.955232i	$-0.595605\pi$
$89$	−44216.9	−0.591716	−0.295858	+	0.955232i	$0.595605\pi$
$90$	0	0
$91$	−40815.2	−0.516676
$92$	0	0
$93$	−23547.9	−0.282322
$94$	0	0
$95$	−15127.1	−0.171968
$96$	0	0
$97$	−88524.0	−0.955282	−0.477641	−	0.878555i	$-0.658508\pi$
$97$	−88524.0	−0.955282	−0.477641	+	0.878555i	$0.658508\pi$
$98$	0	0
$99$	−19093.4	−0.195792
$100$	0	0
$101$	−91408.0	−0.891622	−0.445811	−	0.895127i	$-0.647085\pi$
$101$	−91408.0	−0.891622	−0.445811	+	0.895127i	$0.647085\pi$
$102$	0	0
$103$	127971.	1.18855	0.594276	−	0.804261i	$-0.297439\pi$
$103$	127971.	1.18855	0.594276	+	0.804261i	$0.297439\pi$
$104$	0	0
$105$	181082.	1.60288
$106$	0	0
$107$	−35492.2	−0.299691	−0.149846	−	0.988709i	$-0.547878\pi$
$107$	−35492.2	−0.299691	−0.149846	+	0.988709i	$0.547878\pi$
$108$	0	0
$109$	−154760.	−1.24765	−0.623823	−	0.781565i	$-0.714421\pi$
$109$	−154760.	−1.24765	−0.623823	+	0.781565i	$0.714421\pi$
$110$	0	0
$111$	237018.	1.82589
$112$	0	0
$113$	57641.7	0.424659	0.212330	−	0.977198i	$-0.431895\pi$
$113$	57641.7	0.424659	0.212330	+	0.977198i	$0.431895\pi$
$114$	0	0
$115$	−51754.5	−0.364925
$116$	0	0
$117$	−102854.	−0.694635
$118$	0	0
$119$	48000.0	0.310723
$120$	0	0
$121$	−158197.	−0.982281
$122$	0	0
$123$	462295.	2.75522
$124$	0	0
$125$	−184199.	−1.05442
$126$	0	0
$127$	274218.	1.50864	0.754322	−	0.656505i	$-0.227966\pi$
$127$	274218.	1.50864	0.754322	+	0.656505i	$0.227966\pi$
$128$	0	0
$129$	−25425.8	−0.134524
$130$	0	0
$131$	5493.46	0.0279684	0.0139842	−	0.999902i	$-0.495549\pi$
$131$	5493.46	0.0279684	0.0139842	+	0.999902i	$0.495549\pi$
$132$	0	0
$133$	−41179.3	−0.201860
$134$	0	0
$135$	146086.	0.689882
$136$	0	0
$137$	166802.	0.759277	0.379638	−	0.925135i	$-0.376048\pi$
$137$	166802.	0.759277	0.379638	+	0.925135i	$0.376048\pi$
$138$	0	0
$139$	82327.7	0.361417	0.180709	−	0.983537i	$-0.442161\pi$
$139$	82327.7	0.361417	0.180709	+	0.983537i	$0.442161\pi$
$140$	0	0
$141$	−366996.	−1.55458
$142$	0	0
$143$	15372.2	0.0628632
$144$	0	0
$145$	−286738.	−1.13257
$146$	0	0
$147$	81114.0	0.309601
$148$	0	0
$149$	−79655.2	−0.293933	−0.146967	−	0.989141i	$-0.546951\pi$
$149$	−79655.2	−0.293933	−0.146967	+	0.989141i	$0.546951\pi$
$150$	0	0
$151$	124473.	0.444255	0.222127	−	0.975018i	$-0.428700\pi$
$151$	124473.	0.444255	0.222127	+	0.975018i	$0.428700\pi$
$152$	0	0
$153$	120960.	0.417746
$154$	0	0
$155$	−50070.8	−0.167400
$156$	0	0
$157$	−544785.	−1.76391	−0.881954	−	0.471335i	$-0.843772\pi$
$157$	−544785.	−1.76391	−0.881954	+	0.471335i	$0.843772\pi$
$158$	0	0
$159$	387719.	1.21625
$160$	0	0
$161$	−140887.	−0.428359
$162$	0	0
$163$	−223916.	−0.660109	−0.330055	−	0.943962i	$-0.607067\pi$
$163$	−223916.	−0.660109	−0.330055	+	0.943962i	$0.607067\pi$
$164$	0	0
$165$	−68200.9	−0.195020
$166$	0	0
$167$	226961.	0.629737	0.314869	−	0.949135i	$-0.398040\pi$
$167$	226961.	0.629737	0.314869	+	0.949135i	$0.398040\pi$
$168$	0	0
$169$	−288485.	−0.776973
$170$	0	0
$171$	−103772.	−0.271387
$172$	0	0
$173$	256083.	0.650527	0.325264	−	0.945623i	$-0.394547\pi$
$173$	256083.	0.650527	0.325264	+	0.945623i	$0.394547\pi$
$174$	0	0
$175$	−58194.9	−0.143645
$176$	0	0
$177$	923288.	2.21515
$178$	0	0
$179$	461904.	1.07750	0.538752	−	0.842464i	$-0.318896\pi$
$179$	461904.	1.07750	0.538752	+	0.842464i	$0.318896\pi$
$180$	0	0
$181$	604524.	1.37157	0.685783	−	0.727806i	$-0.259460\pi$
$181$	604524.	1.37157	0.685783	+	0.727806i	$0.259460\pi$
$182$	0	0
$183$	424461.	0.936936
$184$	0	0
$185$	503981.	1.08264
$186$	0	0
$187$	−18078.2	−0.0378052
$188$	0	0
$189$	397679.	0.809801
$190$	0	0
$191$	611336.	1.21254	0.606270	−	0.795259i	$-0.292665\pi$
$191$	611336.	1.21254	0.606270	+	0.795259i	$0.292665\pi$
$192$	0	0
$193$	−308093.	−0.595371	−0.297686	−	0.954664i	$-0.596215\pi$
$193$	−308093.	−0.595371	−0.297686	+	0.954664i	$0.596215\pi$
$194$	0	0
$195$	−367390.	−0.691895
$196$	0	0
$197$	872319.	1.60144	0.800718	−	0.599041i	$-0.204451\pi$
$197$	872319.	1.60144	0.800718	+	0.599041i	$0.204451\pi$
$198$	0	0
$199$	−273029.	−0.488739	−0.244369	−	0.969682i	$-0.578581\pi$
$199$	−273029.	−0.488739	−0.244369	+	0.969682i	$0.578581\pi$
$200$	0	0
$201$	−481033.	−0.839817
$202$	0	0
$203$	−780565.	−1.32944
$204$	0	0
$205$	982994.	1.63368
$206$	0	0
$207$	−355035.	−0.575898
$208$	0	0
$209$	15509.4	0.0245600
$210$	0	0
$211$	−910733.	−1.40827	−0.704133	−	0.710068i	$-0.748664\pi$
$211$	−910733.	−1.40827	−0.704133	+	0.710068i	$0.748664\pi$
$212$	0	0
$213$	−1.48353e6	−2.24051
$214$	0	0
$215$	−54063.8	−0.0797646
$216$	0	0
$217$	−136304.	−0.196498
$218$	0	0
$219$	−1.32328e6	−1.86440
$220$	0	0
$221$	−97385.2	−0.134126
$222$	0	0
$223$	546662.	0.736134	0.368067	−	0.929799i	$-0.380020\pi$
$223$	546662.	0.736134	0.368067	+	0.929799i	$0.380020\pi$
$224$	0	0
$225$	−146651.	−0.193121
$226$	0	0
$227$	673173.	0.867085	0.433543	−	0.901133i	$-0.357263\pi$
$227$	673173.	0.867085	0.433543	+	0.901133i	$0.357263\pi$
$228$	0	0
$229$	−459073.	−0.578486	−0.289243	−	0.957256i	$-0.593404\pi$
$229$	−459073.	−0.578486	−0.289243	+	0.957256i	$0.593404\pi$
$230$	0	0
$231$	−185658.	−0.228920
$232$	0	0
$233$	805322.	0.971807	0.485903	−	0.874013i	$-0.338491\pi$
$233$	805322.	0.971807	0.485903	+	0.874013i	$0.338491\pi$
$234$	0	0
$235$	−780357.	−0.921773
$236$	0	0
$237$	972094.	1.12418
$238$	0	0
$239$	690987.	0.782484	0.391242	−	0.920288i	$-0.372046\pi$
$239$	690987.	0.782484	0.391242	+	0.920288i	$0.372046\pi$
$240$	0	0
$241$	−1.62717e6	−1.80464	−0.902321	−	0.431064i	$-0.858138\pi$
$241$	−1.62717e6	−1.80464	−0.902321	+	0.431064i	$0.858138\pi$
$242$	0	0
$243$	−1.12609e6	−1.22337
$244$	0	0
$245$	172475.	0.183574
$246$	0	0
$247$	83547.1	0.0871343
$248$	0	0
$249$	−1.90022e6	−1.94225
$250$	0	0
$251$	−1.22202e6	−1.22432	−0.612161	−	0.790733i	$-0.709700\pi$
$251$	−1.22202e6	−1.22432	−0.612161	+	0.790733i	$0.709700\pi$
$252$	0	0
$253$	53062.4	0.0521178
$254$	0	0
$255$	432062.	0.416098
$256$	0	0
$257$	707021.	0.667728	0.333864	−	0.942621i	$-0.391647\pi$
$257$	707021.	0.667728	0.333864	+	0.942621i	$0.391647\pi$
$258$	0	0
$259$	1.37195e6	1.27083
$260$	0	0
$261$	−1.96702e6	−1.78734
$262$	0	0
$263$	−821447.	−0.732302	−0.366151	−	0.930555i	$-0.619325\pi$
$263$	−821447.	−0.732302	−0.366151	+	0.930555i	$0.619325\pi$
$264$	0	0
$265$	824421.	0.721164
$266$	0	0
$267$	−1.08347e6	−0.930120
$268$	0	0
$269$	646978.	0.545141	0.272570	−	0.962136i	$-0.412126\pi$
$269$	646978.	0.545141	0.272570	+	0.962136i	$0.412126\pi$
$270$	0	0
$271$	1.45638e6	1.20463	0.602313	−	0.798260i	$-0.294246\pi$
$271$	1.45638e6	1.20463	0.602313	+	0.798260i	$0.294246\pi$
$272$	0	0
$273$	−1.00012e6	−0.812165
$274$	0	0
$275$	21918.0	0.0174771
$276$	0	0
$277$	−1.26357e6	−0.989460	−0.494730	−	0.869047i	$-0.664733\pi$
$277$	−1.26357e6	−0.989460	−0.494730	+	0.869047i	$0.664733\pi$
$278$	0	0
$279$	−343485.	−0.264178
$280$	0	0
$281$	−1.37562e6	−1.03928	−0.519639	−	0.854386i	$-0.673933\pi$
$281$	−1.37562e6	−1.03928	−0.519639	+	0.854386i	$0.673933\pi$
$282$	0	0
$283$	404473.	0.300209	0.150104	−	0.988670i	$-0.452039\pi$
$283$	404473.	0.300209	0.150104	+	0.988670i	$0.452039\pi$
$284$	0	0
$285$	−370668.	−0.270317
$286$	0	0
$287$	2.67593e6	1.91765
$288$	0	0
$289$	−1.30533e6	−0.919338
$290$	0	0
$291$	−2.16915e6	−1.50161
$292$	0	0
$293$	2.31185e6	1.57322	0.786612	−	0.617447i	$-0.211833\pi$
$293$	2.31185e6	1.57322	0.786612	+	0.617447i	$0.211833\pi$
$294$	0	0
$295$	1.96322e6	1.31345
$296$	0	0
$297$	−149778.	−0.0985273
$298$	0	0
$299$	285841.	0.184904
$300$	0	0
$301$	−147174.	−0.0936298
$302$	0	0
$303$	−2.23982e6	−1.40154
$304$	0	0
$305$	902546.	0.555546
$306$	0	0
$307$	2.93729e6	1.77869	0.889345	−	0.457237i	$-0.151161\pi$
$307$	2.93729e6	1.77869	0.889345	+	0.457237i	$0.151161\pi$
$308$	0	0
$309$	3.13574e6	1.86829
$310$	0	0
$311$	1.05775e6	0.620129	0.310064	−	0.950716i	$-0.399649\pi$
$311$	1.05775e6	0.620129	0.310064	+	0.950716i	$0.399649\pi$
$312$	0	0
$313$	1.26771e6	0.731409	0.365705	−	0.930731i	$-0.380828\pi$
$313$	1.26771e6	0.731409	0.365705	+	0.930731i	$0.380828\pi$
$314$	0	0
$315$	2.64137e6	1.49987
$316$	0	0
$317$	3.22101e6	1.80030	0.900149	−	0.435582i	$-0.143457\pi$
$317$	3.22101e6	1.80030	0.900149	+	0.435582i	$0.143457\pi$
$318$	0	0
$319$	293984.	0.161751
$320$	0	0
$321$	−869686.	−0.471086
$322$	0	0
$323$	−98254.1	−0.0524016
$324$	0	0
$325$	118069.	0.0620053
$326$	0	0
$327$	−3.79216e6	−1.96118
$328$	0	0
$329$	−2.12431e6	−1.08200
$330$	0	0
$331$	−1.84115e6	−0.923676	−0.461838	−	0.886964i	$-0.652810\pi$
$331$	−1.84115e6	−0.923676	−0.461838	+	0.886964i	$0.652810\pi$
$332$	0	0
$333$	3.45730e6	1.70855
$334$	0	0
$335$	−1.02284e6	−0.497960
$336$	0	0
$337$	105320.	0.0505168	0.0252584	−	0.999681i	$-0.491959\pi$
$337$	105320.	0.0505168	0.0252584	+	0.999681i	$0.491959\pi$
$338$	0	0
$339$	1.41243e6	0.667524
$340$	0	0
$341$	51336.1	0.0239077
$342$	0	0
$343$	−1.91431e6	−0.878572
$344$	0	0
$345$	−1.26817e6	−0.573627
$346$	0	0
$347$	4.08598e6	1.82168	0.910840	−	0.412759i	$-0.135435\pi$
$347$	4.08598e6	1.82168	0.910840	+	0.412759i	$0.135435\pi$
$348$	0	0
$349$	4.21430e6	1.85209	0.926044	−	0.377415i	$-0.123187\pi$
$349$	4.21430e6	1.85209	0.926044	+	0.377415i	$0.123187\pi$
$350$	0	0
$351$	−806835.	−0.349556
$352$	0	0
$353$	−3.13178e6	−1.33769	−0.668844	−	0.743402i	$-0.733211\pi$
$353$	−3.13178e6	−1.33769	−0.668844	+	0.743402i	$0.733211\pi$
$354$	0	0
$355$	−3.15448e6	−1.32849
$356$	0	0
$357$	1.17617e6	0.488427
$358$	0	0
$359$	231786.	0.0949186	0.0474593	−	0.998873i	$-0.484888\pi$
$359$	231786.	0.0949186	0.0474593	+	0.998873i	$0.484888\pi$
$360$	0	0
$361$	−2.39181e6	−0.965958
$362$	0	0
$363$	−3.87640e6	−1.54405
$364$	0	0
$365$	−2.81373e6	−1.10548
$366$	0	0
$367$	1.41407e6	0.548034	0.274017	−	0.961725i	$-0.411648\pi$
$367$	1.41407e6	0.548034	0.274017	+	0.961725i	$0.411648\pi$
$368$	0	0
$369$	6.74332e6	2.57815
$370$	0	0
$371$	2.24426e6	0.846521
$372$	0	0
$373$	−1.99612e6	−0.742872	−0.371436	−	0.928459i	$-0.621134\pi$
$373$	−1.99612e6	−0.742872	−0.371436	+	0.928459i	$0.621134\pi$
$374$	0	0
$375$	−4.51353e6	−1.65744
$376$	0	0
$377$	1.58366e6	0.573863
$378$	0	0
$379$	−325084.	−0.116251	−0.0581256	−	0.998309i	$-0.518512\pi$
$379$	−325084.	−0.116251	−0.0581256	+	0.998309i	$0.518512\pi$
$380$	0	0
$381$	6.71932e6	2.37144
$382$	0	0
$383$	2.03785e6	0.709863	0.354931	−	0.934892i	$-0.384504\pi$
$383$	2.03785e6	0.709863	0.354931	+	0.934892i	$0.384504\pi$
$384$	0	0
$385$	−394771.	−0.135736
$386$	0	0
$387$	−370877.	−0.125879
$388$	0	0
$389$	−222747.	−0.0746343	−0.0373171	−	0.999303i	$-0.511881\pi$
$389$	−222747.	−0.0746343	−0.0373171	+	0.999303i	$0.511881\pi$
$390$	0	0
$391$	−336158.	−0.111199
$392$	0	0
$393$	134609.	0.0439637
$394$	0	0
$395$	2.06700e6	0.666573
$396$	0	0
$397$	−3.81543e6	−1.21497	−0.607487	−	0.794329i	$-0.707823\pi$
$397$	−3.81543e6	−1.21497	−0.607487	+	0.794329i	$0.707823\pi$
$398$	0	0
$399$	−1.00904e6	−0.317305
$400$	0	0
$401$	−1.71517e6	−0.532656	−0.266328	−	0.963882i	$-0.585810\pi$
$401$	−1.71517e6	−0.532656	−0.266328	+	0.963882i	$0.585810\pi$
$402$	0	0
$403$	276541.	0.0848199
$404$	0	0
$405$	−945712.	−0.286498
$406$	0	0
$407$	−516717.	−0.154620
$408$	0	0
$409$	−2.04097e6	−0.603292	−0.301646	−	0.953420i	$-0.597536\pi$
$409$	−2.04097e6	−0.603292	−0.301646	+	0.953420i	$0.597536\pi$
$410$	0	0
$411$	4.08725e6	1.19351
$412$	0	0
$413$	5.34432e6	1.54176
$414$	0	0
$415$	−4.04051e6	−1.15164
$416$	0	0
$417$	2.01732e6	0.568113
$418$	0	0
$419$	5.58376e6	1.55379	0.776893	−	0.629632i	$-0.216794\pi$
$419$	5.58376e6	1.55379	0.776893	+	0.629632i	$0.216794\pi$
$420$	0	0
$421$	−7.12646e6	−1.95961	−0.979803	−	0.199967i	$-0.935916\pi$
$421$	−7.12646e6	−1.95961	−0.979803	+	0.199967i	$0.935916\pi$
$422$	0	0
$423$	−5.35324e6	−1.45467
$424$	0	0
$425$	−138853.	−0.0372893
$426$	0	0
$427$	2.45693e6	0.652114
$428$	0	0
$429$	376674.	0.0988149
$430$	0	0
$431$	2.64014e6	0.684594	0.342297	−	0.939592i	$-0.388795\pi$
$431$	2.64014e6	0.684594	0.342297	+	0.939592i	$0.388795\pi$
$432$	0	0
$433$	5.41452e6	1.38784	0.693921	−	0.720051i	$-0.255882\pi$
$433$	5.41452e6	1.38784	0.693921	+	0.720051i	$0.255882\pi$
$434$	0	0
$435$	−7.02611e6	−1.78029
$436$	0	0
$437$	288391.	0.0722401
$438$	0	0
$439$	3.87837e6	0.960479	0.480239	−	0.877137i	$-0.340550\pi$
$439$	3.87837e6	0.960479	0.480239	+	0.877137i	$0.340550\pi$
$440$	0	0
$441$	1.18318e6	0.289704
$442$	0	0
$443$	−969210.	−0.234644	−0.117322	−	0.993094i	$-0.537431\pi$
$443$	−969210.	−0.234644	−0.117322	+	0.993094i	$0.537431\pi$
$444$	0	0
$445$	−2.30382e6	−0.551505
$446$	0	0
$447$	−1.95184e6	−0.462035
$448$	0	0
$449$	4.23719e6	0.991886	0.495943	−	0.868355i	$-0.334823\pi$
$449$	4.23719e6	0.991886	0.495943	+	0.868355i	$0.334823\pi$
$450$	0	0
$451$	−1.00783e6	−0.233318
$452$	0	0
$453$	3.05003e6	0.698326
$454$	0	0
$455$	−2.12658e6	−0.481564
$456$	0	0
$457$	2.08497e6	0.466992	0.233496	−	0.972358i	$-0.424983\pi$
$457$	2.08497e6	0.466992	0.233496	+	0.972358i	$0.424983\pi$
$458$	0	0
$459$	948865.	0.210219
$460$	0	0
$461$	688956.	0.150987	0.0754934	−	0.997146i	$-0.475947\pi$
$461$	688956.	0.150987	0.0754934	+	0.997146i	$0.475947\pi$
$462$	0	0
$463$	−9.10370e6	−1.97363	−0.986814	−	0.161858i	$-0.948251\pi$
$463$	−9.10370e6	−1.97363	−0.986814	+	0.161858i	$0.948251\pi$
$464$	0	0
$465$	−1.22691e6	−0.263137
$466$	0	0
$467$	−5.91763e6	−1.25561	−0.627806	−	0.778370i	$-0.716047\pi$
$467$	−5.91763e6	−1.25561	−0.627806	+	0.778370i	$0.716047\pi$
$468$	0	0
$469$	−2.78439e6	−0.584519
$470$	0	0
$471$	−1.33492e7	−2.77270
$472$	0	0
$473$	55430.0	0.0113918
$474$	0	0
$475$	119123.	0.0242249
$476$	0	0
$477$	5.65552e6	1.13809
$478$	0	0
$479$	1.00780e6	0.200694	0.100347	−	0.994953i	$-0.468005\pi$
$479$	1.00780e6	0.200694	0.100347	+	0.994953i	$0.468005\pi$
$480$	0	0
$481$	−2.78349e6	−0.548563
$482$	0	0
$483$	−3.45224e6	−0.673339
$484$	0	0
$485$	−4.61235e6	−0.890364
$486$	0	0
$487$	6.38036e6	1.21905	0.609527	−	0.792765i	$-0.291359\pi$
$487$	6.38036e6	1.21905	0.609527	+	0.792765i	$0.291359\pi$
$488$	0	0
$489$	−5.48673e6	−1.03763
$490$	0	0
$491$	5.23949e6	0.980811	0.490405	−	0.871495i	$-0.336849\pi$
$491$	5.23949e6	0.980811	0.490405	+	0.871495i	$0.336849\pi$
$492$	0	0
$493$	−1.86243e6	−0.345115
$494$	0	0
$495$	−994821.	−0.182487
$496$	0	0
$497$	−8.58721e6	−1.55941
$498$	0	0
$499$	−1.01478e7	−1.82441	−0.912203	−	0.409738i	$-0.865620\pi$
$499$	−1.01478e7	−1.82441	−0.912203	+	0.409738i	$0.865620\pi$
$500$	0	0
$501$	5.56134e6	0.989886
$502$	0	0
$503$	−985205.	−0.173623	−0.0868114	−	0.996225i	$-0.527668\pi$
$503$	−985205.	−0.173623	−0.0868114	+	0.996225i	$0.527668\pi$
$504$	0	0
$505$	−4.76261e6	−0.831030
$506$	0	0
$507$	−7.06890e6	−1.22133
$508$	0	0
$509$	5.51699e6	0.943860	0.471930	−	0.881636i	$-0.343557\pi$
$509$	5.51699e6	0.943860	0.471930	+	0.881636i	$0.343557\pi$
$510$	0	0
$511$	−7.65960e6	−1.29764
$512$	0	0
$513$	−814035.	−0.136568
$514$	0	0
$515$	6.66764e6	1.10778
$516$	0	0
$517$	800078.	0.131645
$518$	0	0
$519$	6.27495e6	1.02257
$520$	0	0
$521$	865260.	0.139654	0.0698269	−	0.997559i	$-0.477755\pi$
$521$	865260.	0.139654	0.0698269	+	0.997559i	$0.477755\pi$
$522$	0	0
$523$	−2.87041e6	−0.458871	−0.229435	−	0.973324i	$-0.573688\pi$
$523$	−2.87041e6	−0.458871	−0.229435	+	0.973324i	$0.573688\pi$
$524$	0	0
$525$	−1.42598e6	−0.225796
$526$	0	0
$527$	−325222.	−0.0510097
$528$	0	0
$529$	−5.44967e6	−0.846702
$530$	0	0
$531$	1.34677e7	2.07279
$532$	0	0
$533$	−5.42908e6	−0.827768
$534$	0	0
$535$	−1.84924e6	−0.279325
$536$	0	0
$537$	1.13183e7	1.69373
$538$	0	0
$539$	−176834.	−0.0262177
$540$	0	0
$541$	2.80940e6	0.412687	0.206344	−	0.978480i	$-0.433844\pi$
$541$	2.80940e6	0.412687	0.206344	+	0.978480i	$0.433844\pi$
$542$	0	0
$543$	1.48130e7	2.15597
$544$	0	0
$545$	−8.06341e6	−1.16286
$546$	0	0
$547$	3.01572e6	0.430946	0.215473	−	0.976510i	$-0.430871\pi$
$547$	3.01572e6	0.430946	0.215473	+	0.976510i	$0.430871\pi$
$548$	0	0
$549$	6.19146e6	0.876722
$550$	0	0
$551$	1.59779e6	0.224202
$552$	0	0
$553$	5.62683e6	0.782440
$554$	0	0
$555$	1.23493e7	1.70181
$556$	0	0
$557$	6.91953e6	0.945015	0.472507	−	0.881327i	$-0.343349\pi$
$557$	6.91953e6	0.945015	0.472507	+	0.881327i	$0.343349\pi$
$558$	0	0
$559$	298595.	0.0404159
$560$	0	0
$561$	−442981.	−0.0594262
$562$	0	0
$563$	4.21366e6	0.560259	0.280129	−	0.959962i	$-0.409623\pi$
$563$	4.21366e6	0.560259	0.280129	+	0.959962i	$0.409623\pi$
$564$	0	0
$565$	3.00329e6	0.395801
$566$	0	0
$567$	−2.57444e6	−0.336299
$568$	0	0
$569$	−3.30727e6	−0.428241	−0.214121	−	0.976807i	$-0.568689\pi$
$569$	−3.30727e6	−0.428241	−0.214121	+	0.976807i	$0.568689\pi$
$570$	0	0
$571$	1.36506e7	1.75210	0.876052	−	0.482216i	$-0.160168\pi$
$571$	1.36506e7	1.75210	0.876052	+	0.482216i	$0.160168\pi$
$572$	0	0
$573$	1.49799e7	1.90600
$574$	0	0
$575$	407557.	0.0514065
$576$	0	0
$577$	4.41135e6	0.551609	0.275805	−	0.961214i	$-0.411056\pi$
$577$	4.41135e6	0.551609	0.275805	+	0.961214i	$0.411056\pi$
$578$	0	0
$579$	−7.54936e6	−0.935867
$580$	0	0
$581$	−1.09992e7	−1.35182
$582$	0	0
$583$	−845255.	−0.102995
$584$	0	0
$585$	−5.35898e6	−0.647429
$586$	0	0
$587$	−2.12980e6	−0.255120	−0.127560	−	0.991831i	$-0.540715\pi$
$587$	−2.12980e6	−0.255120	−0.127560	+	0.991831i	$0.540715\pi$
$588$	0	0
$589$	279009.	0.0331383
$590$	0	0
$591$	2.13749e7	2.51730
$592$	0	0
$593$	−5.81278e6	−0.678808	−0.339404	−	0.940641i	$-0.610225\pi$
$593$	−5.81278e6	−0.678808	−0.339404	+	0.940641i	$0.610225\pi$
$594$	0	0
$595$	2.50093e6	0.289607
$596$	0	0
$597$	−6.69019e6	−0.768250
$598$	0	0
$599$	−8.31661e6	−0.947064	−0.473532	−	0.880777i	$-0.657021\pi$
$599$	−8.31661e6	−0.947064	−0.473532	+	0.880777i	$0.657021\pi$
$600$	0	0
$601$	1.37571e7	1.55360	0.776802	−	0.629744i	$-0.216840\pi$
$601$	1.37571e7	1.55360	0.776802	+	0.629744i	$0.216840\pi$
$602$	0	0
$603$	−7.01665e6	−0.785845
$604$	0	0
$605$	−8.24252e6	−0.915528
$606$	0	0
$607$	4.10039e6	0.451703	0.225852	−	0.974162i	$-0.427484\pi$
$607$	4.10039e6	0.451703	0.225852	+	0.974162i	$0.427484\pi$
$608$	0	0
$609$	−1.91266e7	−2.08975
$610$	0	0
$611$	4.30992e6	0.467053
$612$	0	0
$613$	9.73593e6	1.04647	0.523234	−	0.852189i	$-0.324725\pi$
$613$	9.73593e6	1.04647	0.523234	+	0.852189i	$0.324725\pi$
$614$	0	0
$615$	2.40868e7	2.56798
$616$	0	0
$617$	1.90798e6	0.201772	0.100886	−	0.994898i	$-0.467832\pi$
$617$	1.90798e6	0.201772	0.100886	+	0.994898i	$0.467832\pi$
$618$	0	0
$619$	1.45209e7	1.52324	0.761618	−	0.648027i	$-0.224405\pi$
$619$	1.45209e7	1.52324	0.761618	+	0.648027i	$0.224405\pi$
$620$	0	0
$621$	−2.78507e6	−0.289805
$622$	0	0
$623$	−6.27152e6	−0.647370
$624$	0	0
$625$	−8.31510e6	−0.851466
$626$	0	0
$627$	380035.	0.0386060
$628$	0	0
$629$	3.27347e6	0.329900
$630$	0	0
$631$	4.39828e6	0.439754	0.219877	−	0.975528i	$-0.429434\pi$
$631$	4.39828e6	0.439754	0.219877	+	0.975528i	$0.429434\pi$
$632$	0	0
$633$	−2.23162e7	−2.21366
$634$	0	0
$635$	1.42875e7	1.40612
$636$	0	0
$637$	−952584.	−0.0930153
$638$	0	0
$639$	−2.16397e7	−2.09652
$640$	0	0
$641$	−5.87667e6	−0.564919	−0.282460	−	0.959279i	$-0.591150\pi$
$641$	−5.87667e6	−0.564919	−0.282460	+	0.959279i	$0.591150\pi$
$642$	0	0
$643$	−1.20597e7	−1.15029	−0.575147	−	0.818050i	$-0.695055\pi$
$643$	−1.20597e7	−1.15029	−0.575147	+	0.818050i	$0.695055\pi$
$644$	0	0
$645$	−1.32475e6	−0.125382
$646$	0	0
$647$	4.19237e6	0.393730	0.196865	−	0.980431i	$-0.436924\pi$
$647$	4.19237e6	0.393730	0.196865	+	0.980431i	$0.436924\pi$
$648$	0	0
$649$	−2.01283e6	−0.187584
$650$	0	0
$651$	−3.33993e6	−0.308877
$652$	0	0
$653$	−1.09082e7	−1.00108	−0.500541	−	0.865713i	$-0.666866\pi$
$653$	−1.09082e7	−1.00108	−0.500541	+	0.865713i	$0.666866\pi$
$654$	0	0
$655$	286225.	0.0260678
$656$	0	0
$657$	−1.93021e7	−1.74458
$658$	0	0
$659$	1.10808e6	0.0993937	0.0496968	−	0.998764i	$-0.484174\pi$
$659$	1.10808e6	0.0993937	0.0496968	+	0.998764i	$0.484174\pi$
$660$	0	0
$661$	1.31429e7	1.17001	0.585004	−	0.811030i	$-0.301093\pi$
$661$	1.31429e7	1.17001	0.585004	+	0.811030i	$0.301093\pi$
$662$	0	0
$663$	−2.38628e6	−0.210833
$664$	0	0
$665$	−2.14556e6	−0.188142
$666$	0	0
$667$	5.46653e6	0.475770
$668$	0	0
$669$	1.33952e7	1.15713
$670$	0	0
$671$	−925355.	−0.0793418
$672$	0	0
$673$	−1.49242e7	−1.27015	−0.635073	−	0.772452i	$-0.719030\pi$
$673$	−1.49242e7	−1.27015	−0.635073	+	0.772452i	$0.719030\pi$
$674$	0	0
$675$	−1.15040e6	−0.0971827
$676$	0	0
$677$	−1.46851e6	−0.123142	−0.0615708	−	0.998103i	$-0.519611\pi$
$677$	−1.46851e6	−0.123142	−0.0615708	+	0.998103i	$0.519611\pi$
$678$	0	0
$679$	−1.25558e7	−1.04513
$680$	0	0
$681$	1.64951e7	1.36297
$682$	0	0
$683$	1.87397e7	1.53713	0.768566	−	0.639771i	$-0.220971\pi$
$683$	1.87397e7	1.53713	0.768566	+	0.639771i	$0.220971\pi$
$684$	0	0
$685$	8.69086e6	0.707679
$686$	0	0
$687$	−1.12489e7	−0.909324
$688$	0	0
$689$	−4.55328e6	−0.365407
$690$	0	0
$691$	6.78348e6	0.540453	0.270226	−	0.962797i	$-0.412901\pi$
$691$	6.78348e6	0.540453	0.270226	+	0.962797i	$0.412901\pi$
$692$	0	0
$693$	−2.70813e6	−0.214208
$694$	0	0
$695$	4.28950e6	0.336857
$696$	0	0
$697$	6.38478e6	0.497810
$698$	0	0
$699$	1.97333e7	1.52759
$700$	0	0
$701$	−9.49944e6	−0.730135	−0.365067	−	0.930981i	$-0.618954\pi$
$701$	−9.49944e6	−0.730135	−0.365067	+	0.930981i	$0.618954\pi$
$702$	0	0
$703$	−2.80833e6	−0.214318
$704$	0	0
$705$	−1.91215e7	−1.44894
$706$	0	0
$707$	−1.29649e7	−0.975485
$708$	0	0
$709$	4.10250e6	0.306501	0.153251	−	0.988187i	$-0.451026\pi$
$709$	4.10250e6	0.306501	0.153251	+	0.988187i	$0.451026\pi$
$710$	0	0
$711$	1.41796e7	1.05194
$712$	0	0
$713$	954577.	0.0703213
$714$	0	0
$715$	800935.	0.0585912
$716$	0	0
$717$	1.69317e7	1.22999
$718$	0	0
$719$	7.76983e6	0.560518	0.280259	−	0.959924i	$-0.409580\pi$
$719$	7.76983e6	0.560518	0.280259	+	0.959924i	$0.409580\pi$
$720$	0	0
$721$	1.81508e7	1.30034
$722$	0	0
$723$	−3.98715e7	−2.83672
$724$	0	0
$725$	2.25801e6	0.159544
$726$	0	0
$727$	7.04611e6	0.494440	0.247220	−	0.968959i	$-0.420483\pi$
$727$	7.04611e6	0.494440	0.247220	+	0.968959i	$0.420483\pi$
$728$	0	0
$729$	−2.31825e7	−1.61563
$730$	0	0
$731$	−351157.	−0.0243057
$732$	0	0
$733$	−1.14700e6	−0.0788501	−0.0394250	−	0.999223i	$-0.512553\pi$
$733$	−1.14700e6	−0.0788501	−0.0394250	+	0.999223i	$0.512553\pi$
$734$	0	0
$735$	4.22626e6	0.288561
$736$	0	0
$737$	1.04869e6	0.0711175
$738$	0	0
$739$	−1.77682e7	−1.19683	−0.598416	−	0.801185i	$-0.704203\pi$
$739$	−1.77682e7	−1.19683	−0.598416	+	0.801185i	$0.704203\pi$
$740$	0	0
$741$	2.04720e6	0.136967
$742$	0	0
$743$	−7.60693e6	−0.505519	−0.252760	−	0.967529i	$-0.581338\pi$
$743$	−7.60693e6	−0.505519	−0.252760	+	0.967529i	$0.581338\pi$
$744$	0	0
$745$	−4.15026e6	−0.273958
$746$	0	0
$747$	−2.77178e7	−1.81743
$748$	0	0
$749$	−5.03406e6	−0.327879
$750$	0	0
$751$	2.95420e7	1.91135	0.955676	−	0.294421i	$-0.0951268\pi$
$751$	2.95420e7	1.91135	0.955676	+	0.294421i	$0.0951268\pi$
$752$	0	0
$753$	−2.99440e7	−1.92452
$754$	0	0
$755$	6.48538e6	0.414064
$756$	0	0
$757$	−1.66343e7	−1.05503	−0.527514	−	0.849546i	$-0.676876\pi$
$757$	−1.66343e7	−1.05503	−0.527514	+	0.849546i	$0.676876\pi$
$758$	0	0
$759$	1.30022e6	0.0819241
$760$	0	0
$761$	1.31189e7	0.821176	0.410588	−	0.911821i	$-0.365323\pi$
$761$	1.31189e7	0.821176	0.410588	+	0.911821i	$0.365323\pi$
$762$	0	0
$763$	−2.19504e7	−1.36500
$764$	0	0
$765$	6.30233e6	0.389357
$766$	0	0
$767$	−1.08429e7	−0.665512
$768$	0	0
$769$	1.16987e7	0.713384	0.356692	−	0.934222i	$-0.383905\pi$
$769$	1.16987e7	0.713384	0.356692	+	0.934222i	$0.383905\pi$
$770$	0	0
$771$	1.73245e7	1.04960
$772$	0	0
$773$	−1.03272e7	−0.621630	−0.310815	−	0.950470i	$-0.600602\pi$
$773$	−1.03272e7	−0.621630	−0.310815	+	0.950470i	$0.600602\pi$
$774$	0	0
$775$	394297.	0.0235814
$776$	0	0
$777$	3.36176e7	1.99763
$778$	0	0
$779$	−5.47753e6	−0.323401
$780$	0	0
$781$	3.23420e6	0.189732
$782$	0	0
$783$	−1.54302e7	−0.899432
$784$	0	0
$785$	−2.83848e7	−1.64404
$786$	0	0
$787$	1.27310e6	0.0732700	0.0366350	−	0.999329i	$-0.488336\pi$
$787$	1.27310e6	0.0732700	0.0366350	+	0.999329i	$0.488336\pi$
$788$	0	0
$789$	−2.01284e7	−1.15111
$790$	0	0
$791$	8.17564e6	0.464601
$792$	0	0
$793$	−4.98477e6	−0.281490
$794$	0	0
$795$	2.02012e7	1.13360
$796$	0	0
$797$	2.81972e7	1.57239	0.786196	−	0.617977i	$-0.212048\pi$
$797$	2.81972e7	1.57239	0.786196	+	0.617977i	$0.212048\pi$
$798$	0	0
$799$	−5.06861e6	−0.280881
$800$	0	0
$801$	−1.58042e7	−0.870344
$802$	0	0
$803$	2.88483e6	0.157882
$804$	0	0
$805$	−7.34063e6	−0.399249
$806$	0	0
$807$	1.58533e7	0.856909
$808$	0	0
$809$	−1.31398e6	−0.0705858	−0.0352929	−	0.999377i	$-0.511236\pi$
$809$	−1.31398e6	−0.0705858	−0.0352929	+	0.999377i	$0.511236\pi$
$810$	0	0
$811$	−1.58413e7	−0.845744	−0.422872	−	0.906190i	$-0.638978\pi$
$811$	−1.58413e7	−0.845744	−0.422872	+	0.906190i	$0.638978\pi$
$812$	0	0
$813$	3.56865e7	1.89356
$814$	0	0
$815$	−1.16666e7	−0.615250
$816$	0	0
$817$	301259.	0.0157901
$818$	0	0
$819$	−1.45883e7	−0.759969
$820$	0	0
$821$	1.79155e7	0.927622	0.463811	−	0.885934i	$-0.346482\pi$
$821$	1.79155e7	0.927622	0.463811	+	0.885934i	$0.346482\pi$
$822$	0	0
$823$	9.21964e6	0.474476	0.237238	−	0.971452i	$-0.423758\pi$
$823$	9.21964e6	0.474476	0.237238	+	0.971452i	$0.423758\pi$
$824$	0	0
$825$	537068.	0.0274723
$826$	0	0
$827$	1.38118e7	0.702241	0.351120	−	0.936330i	$-0.385801\pi$
$827$	1.38118e7	0.702241	0.351120	+	0.936330i	$0.385801\pi$
$828$	0	0
$829$	−1.77142e7	−0.895234	−0.447617	−	0.894225i	$-0.647727\pi$
$829$	−1.77142e7	−0.895234	−0.447617	+	0.894225i	$0.647727\pi$
$830$	0	0
$831$	−3.09619e7	−1.55534
$832$	0	0
$833$	1.12027e6	0.0559384
$834$	0	0
$835$	1.18253e7	0.586942
$836$	0	0
$837$	−2.69446e6	−0.132941
$838$	0	0
$839$	1.43580e7	0.704191	0.352095	−	0.935964i	$-0.385469\pi$
$839$	1.43580e7	0.704191	0.352095	+	0.935964i	$0.385469\pi$
$840$	0	0
$841$	9.77536e6	0.476587
$842$	0	0
$843$	−3.37075e7	−1.63364
$844$	0	0
$845$	−1.50309e7	−0.724173
$846$	0	0
$847$	−2.24380e7	−1.07467
$848$	0	0
$849$	9.91103e6	0.471900
$850$	0	0
$851$	−9.60816e6	−0.454796
$852$	0	0
$853$	3.36517e7	1.58356	0.791781	−	0.610805i	$-0.209154\pi$
$853$	3.36517e7	1.58356	0.791781	+	0.610805i	$0.209154\pi$
$854$	0	0
$855$	−5.40680e6	−0.252944
$856$	0	0
$857$	−9.01602e6	−0.419337	−0.209668	−	0.977773i	$-0.567238\pi$
$857$	−9.01602e6	−0.419337	−0.209668	+	0.977773i	$0.567238\pi$
$858$	0	0
$859$	9.09097e6	0.420366	0.210183	−	0.977662i	$-0.432594\pi$
$859$	9.09097e6	0.420366	0.210183	+	0.977662i	$0.432594\pi$
$860$	0	0
$861$	6.55698e7	3.01436
$862$	0	0
$863$	−2.42705e7	−1.10931	−0.554653	−	0.832082i	$-0.687149\pi$
$863$	−2.42705e7	−1.10931	−0.554653	+	0.832082i	$0.687149\pi$
$864$	0	0
$865$	1.33426e7	0.606319
$866$	0	0
$867$	−3.19852e7	−1.44511
$868$	0	0
$869$	−2.11923e6	−0.0951983
$870$	0	0
$871$	5.64914e6	0.252311
$872$	0	0
$873$	−3.16406e7	−1.40511
$874$	0	0
$875$	−2.61259e7	−1.15359
$876$	0	0
$877$	−2.70870e7	−1.18922	−0.594610	−	0.804014i	$-0.702694\pi$
$877$	−2.70870e7	−1.18922	−0.594610	+	0.804014i	$0.702694\pi$
$878$	0	0
$879$	5.66486e7	2.47296
$880$	0	0
$881$	−1.87253e7	−0.812809	−0.406404	−	0.913693i	$-0.633218\pi$
$881$	−1.87253e7	−0.812809	−0.406404	+	0.913693i	$0.633218\pi$
$882$	0	0
$883$	1.82581e7	0.788050	0.394025	−	0.919100i	$-0.371082\pi$
$883$	1.82581e7	0.788050	0.394025	+	0.919100i	$0.371082\pi$
$884$	0	0
$885$	4.81059e7	2.06462
$886$	0	0
$887$	−2.16884e7	−0.925589	−0.462795	−	0.886466i	$-0.653153\pi$
$887$	−2.16884e7	−0.925589	−0.462795	+	0.886466i	$0.653153\pi$
$888$	0	0
$889$	3.88938e7	1.65054
$890$	0	0
$891$	969611.	0.0409170
$892$	0	0
$893$	4.34838e6	0.182473
$894$	0	0
$895$	2.40665e7	1.00428
$896$	0	0
$897$	7.00412e6	0.290651
$898$	0	0
$899$	5.28869e6	0.218247
$900$	0	0
$901$	5.35481e6	0.219752
$902$	0	0
$903$	−3.60628e6	−0.147177
$904$	0	0
$905$	3.14974e7	1.27836
$906$	0	0
$907$	−2.93378e7	−1.18416	−0.592078	−	0.805881i	$-0.701692\pi$
$907$	−2.93378e7	−1.18416	−0.592078	+	0.805881i	$0.701692\pi$
$908$	0	0
$909$	−3.26715e7	−1.31147
$910$	0	0
$911$	−373353.	−0.0149047	−0.00745237	−	0.999972i	$-0.502372\pi$
$911$	−373353.	−0.0149047	−0.00745237	+	0.999972i	$0.502372\pi$
$912$	0	0
$913$	4.14262e6	0.164474
$914$	0	0
$915$	2.21156e7	0.873265
$916$	0	0
$917$	779168.	0.0305990
$918$	0	0
$919$	2.96283e7	1.15723	0.578613	−	0.815602i	$-0.303594\pi$
$919$	2.96283e7	1.15723	0.578613	+	0.815602i	$0.303594\pi$
$920$	0	0
$921$	7.19740e7	2.79593
$922$	0	0
$923$	1.74222e7	0.673132
$924$	0	0
$925$	−3.96875e6	−0.152510
$926$	0	0
$927$	4.57399e7	1.74822
$928$	0	0
$929$	−2.80859e7	−1.06770	−0.533849	−	0.845580i	$-0.679255\pi$
$929$	−2.80859e7	−1.06770	−0.533849	+	0.845580i	$0.679255\pi$
$930$	0	0
$931$	−961083.	−0.0363402
$932$	0	0
$933$	2.59186e7	0.974783
$934$	0	0
$935$	−941926.	−0.0352361
$936$	0	0
$937$	−2.04070e6	−0.0759328	−0.0379664	−	0.999279i	$-0.512088\pi$
$937$	−2.04070e6	−0.0759328	−0.0379664	+	0.999279i	$0.512088\pi$
$938$	0	0
$939$	3.10635e7	1.14971
$940$	0	0
$941$	373523.	0.0137513	0.00687564	−	0.999976i	$-0.497811\pi$
$941$	373523.	0.0137513	0.00687564	+	0.999976i	$0.497811\pi$
$942$	0	0
$943$	−1.87403e7	−0.686275
$944$	0	0
$945$	2.07202e7	0.754770
$946$	0	0
$947$	1.37756e7	0.499156	0.249578	−	0.968355i	$-0.419708\pi$
$947$	1.37756e7	0.499156	0.249578	+	0.968355i	$0.419708\pi$
$948$	0	0
$949$	1.55402e7	0.560134
$950$	0	0
$951$	7.89263e7	2.82990
$952$	0	0
$953$	−2.79429e7	−0.996643	−0.498322	−	0.866992i	$-0.666050\pi$
$953$	−2.79429e7	−0.996643	−0.498322	+	0.866992i	$0.666050\pi$
$954$	0	0
$955$	3.18523e7	1.13014
$956$	0	0
$957$	7.20366e6	0.254257
$958$	0	0
$959$	2.36585e7	0.830692
$960$	0	0
$961$	923521.	0.0322581
$962$	0	0
$963$	−1.26858e7	−0.440811
$964$	0	0
$965$	−1.60525e7	−0.554912
$966$	0	0
$967$	−4.86988e7	−1.67476	−0.837378	−	0.546624i	$-0.815913\pi$
$967$	−4.86988e7	−1.67476	−0.837378	+	0.546624i	$0.815913\pi$
$968$	0	0
$969$	−2.40758e6	−0.0823703
$970$	0	0
$971$	−5.54072e7	−1.88590	−0.942948	−	0.332939i	$-0.891960\pi$
$971$	−5.54072e7	−1.88590	−0.942948	+	0.332939i	$0.891960\pi$
$972$	0	0
$973$	1.16770e7	0.395411
$974$	0	0
$975$	2.89312e6	0.0974664
$976$	0	0
$977$	−3.50805e7	−1.17579	−0.587894	−	0.808938i	$-0.700043\pi$
$977$	−3.50805e7	−1.17579	−0.587894	+	0.808938i	$0.700043\pi$
$978$	0	0
$979$	2.36204e6	0.0787646
$980$	0	0
$981$	−5.53149e7	−1.83514
$982$	0	0
$983$	−5.81264e7	−1.91862	−0.959312	−	0.282350i	$-0.908886\pi$
$983$	−5.81264e7	−1.91862	−0.959312	+	0.282350i	$0.908886\pi$
$984$	0	0
$985$	4.54503e7	1.49261
$986$	0	0
$987$	−5.20531e7	−1.70080
$988$	0	0
$989$	1.03070e6	0.0335075
$990$	0	0
$991$	−1.29819e7	−0.419909	−0.209954	−	0.977711i	$-0.567332\pi$
$991$	−1.29819e7	−0.419909	−0.209954	+	0.977711i	$0.567332\pi$
$992$	0	0
$993$	−4.51148e7	−1.45193
$994$	0	0
$995$	−1.42256e7	−0.455526
$996$	0	0
$997$	−4.60185e7	−1.46620	−0.733102	−	0.680119i	$-0.761928\pi$
$997$	−4.60185e7	−1.46620	−0.733102	+	0.680119i	$0.761928\pi$
$998$	0	0
$999$	2.71207e7	0.859780

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.6.a.b.1.6	✓	6
4.3	odd	2		496.6.a.f.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.6.a.b.1.6	✓	6	1.1	even	1	trivial
496.6.a.f.1.1		6	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:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	124.a (trivial)

\(f(q)\)	\(=\)	\(q+24.5036 q^{3} +52.1028 q^{5} +141.835 q^{7} +357.424 q^{9} +O(q^{10})\)	\(q+24.5036 q^{3} +52.1028 q^{5} +141.835 q^{7} +357.424 q^{9} -53.4195 q^{11} -287.764 q^{13} +1276.70 q^{15} +338.420 q^{17} -290.332 q^{19} +3475.47 q^{21} -993.316 q^{23} -410.299 q^{25} +2803.81 q^{27} -5503.32 q^{29} -961.000 q^{31} -1308.97 q^{33} +7390.02 q^{35} +9672.81 q^{37} -7051.25 q^{39} +18866.4 q^{41} -1037.64 q^{43} +18622.8 q^{45} -14977.3 q^{47} +3310.29 q^{49} +8292.50 q^{51} +15823.0 q^{53} -2783.30 q^{55} -7114.16 q^{57} +37679.7 q^{59} +17322.4 q^{61} +50695.5 q^{63} -14993.3 q^{65} -19631.1 q^{67} -24339.8 q^{69} -60543.5 q^{71} -54003.4 q^{73} -10053.8 q^{75} -7576.78 q^{77} +39671.6 q^{79} -18150.9 q^{81} -77548.8 q^{83} +17632.6 q^{85} -134851. q^{87} -44216.9 q^{89} -40815.2 q^{91} -23547.9 q^{93} -15127.1 q^{95} -88524.0 q^{97} -19093.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9}+O(q^{10}) \) 6 * q + 20 * q^3 + 25 * q^5 + 39 * q^7 + 306 * q^9	\( 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9} + 280 q^{11} + 1214 q^{13} + 1914 q^{15} + 796 q^{17} + 3147 q^{19} + 1082 q^{21} + 9122 q^{23} + 6481 q^{25} + 7316 q^{27} + 13020 q^{29} - 5766 q^{31} + 24804 q^{33} + 20059 q^{35} + 21678 q^{37} + 30680 q^{39} + 3227 q^{41} + 37882 q^{43} + 26169 q^{45} + 29708 q^{47} + 34849 q^{49} + 24432 q^{51} - 9976 q^{53} + 23758 q^{55} + 17318 q^{57} + 20573 q^{59} + 21610 q^{61} - 17697 q^{63} + 3894 q^{65} - 17024 q^{67} - 83692 q^{69} + 44509 q^{71} - 161864 q^{73} - 49430 q^{75} - 144202 q^{77} - 24420 q^{79} - 181158 q^{81} - 114160 q^{83} - 228882 q^{85} - 56180 q^{87} - 199742 q^{89} - 186774 q^{91} - 19220 q^{93} - 12793 q^{95} - 282951 q^{97} + 34060 q^{99}+O(q^{100}) \) 6 * q + 20 * q^3 + 25 * q^5 + 39 * q^7 + 306 * q^9 + 280 * q^11 + 1214 * q^13 + 1914 * q^15 + 796 * q^17 + 3147 * q^19 + 1082 * q^21 + 9122 * q^23 + 6481 * q^25 + 7316 * q^27 + 13020 * q^29 - 5766 * q^31 + 24804 * q^33 + 20059 * q^35 + 21678 * q^37 + 30680 * q^39 + 3227 * q^41 + 37882 * q^43 + 26169 * q^45 + 29708 * q^47 + 34849 * q^49 + 24432 * q^51 - 9976 * q^53 + 23758 * q^55 + 17318 * q^57 + 20573 * q^59 + 21610 * q^61 - 17697 * q^63 + 3894 * q^65 - 17024 * q^67 - 83692 * q^69 + 44509 * q^71 - 161864 * q^73 - 49430 * q^75 - 144202 * q^77 - 24420 * q^79 - 181158 * q^81 - 114160 * q^83 - 228882 * q^85 - 56180 * q^87 - 199742 * q^89 - 186774 * q^91 - 19220 * q^93 - 12793 * q^95 - 282951 * q^97 + 34060 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more