LMFDB - Embedded newform 912.6.a.v.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [912,6,Mod(1,912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("912.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	912.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-45,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$146.270043669$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7184x^{3} - 76134x^{2} + 12883743x + 275533272 $ x^5 - 2x^4 - 7184x^3 - 76134x^2 + 12883743x + 275533272
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 228)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$57.9047$ of defining polynomial
Character	$\chi$	$=$	912.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +68.7054 q^{5} +92.5541 q^{7} +81.0000 q^{9} -455.495 q^{11} -138.319 q^{13} -618.349 q^{15} -25.2508 q^{17} -361.000 q^{19} -832.987 q^{21} -4594.45 q^{23} +1595.44 q^{25} -729.000 q^{27} +5839.25 q^{29} +8095.84 q^{31} +4099.46 q^{33} +6358.97 q^{35} -5483.74 q^{37} +1244.87 q^{39} +13069.3 q^{41} -9161.20 q^{43} +5565.14 q^{45} +25131.8 q^{47} -8240.74 q^{49} +227.257 q^{51} +31247.6 q^{53} -31295.0 q^{55} +3249.00 q^{57} -26035.2 q^{59} +23473.6 q^{61} +7496.88 q^{63} -9503.24 q^{65} -1346.30 q^{67} +41350.0 q^{69} -62531.0 q^{71} +60494.1 q^{73} -14358.9 q^{75} -42158.0 q^{77} +56411.2 q^{79} +6561.00 q^{81} +79425.3 q^{83} -1734.87 q^{85} -52553.3 q^{87} +7121.69 q^{89} -12802.0 q^{91} -72862.6 q^{93} -24802.7 q^{95} +115588. q^{97} -36895.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 45 q^{3} - 6 q^{5} - 54 q^{7} + 405 q^{9} - 272 q^{11} + 440 q^{13} + 54 q^{15} + 1940 q^{17} - 1805 q^{19} + 486 q^{21} - 3224 q^{23} + 7313 q^{25} - 3645 q^{27} + 7524 q^{29} - 11774 q^{31} + 2448 q^{33}+ \cdots - 22032 q^{99}+O(q^{100}) $ 5 * q - 45 * q^3 - 6 * q^5 - 54 * q^7 + 405 * q^9 - 272 * q^11 + 440 * q^13 + 54 * q^15 + 1940 * q^17 - 1805 * q^19 + 486 * q^21 - 3224 * q^23 + 7313 * q^25 - 3645 * q^27 + 7524 * q^29 - 11774 * q^31 + 2448 * q^33 - 12834 * q^35 + 17760 * q^37 - 3960 * q^39 + 10736 * q^41 - 21914 * q^43 - 486 * q^45 - 10190 * q^47 + 1503 * q^49 - 17460 * q^51 - 1676 * q^53 - 36474 * q^55 + 16245 * q^57 - 54920 * q^59 + 39268 * q^61 - 4374 * q^63 + 39684 * q^65 - 104168 * q^67 + 29016 * q^69 - 50620 * q^71 + 98616 * q^73 - 65817 * q^75 - 9418 * q^77 - 115782 * q^79 + 32805 * q^81 - 143266 * q^83 + 113334 * q^85 - 67716 * q^87 + 88112 * q^89 - 247372 * q^91 + 105966 * q^93 + 2166 * q^95 + 186490 * q^97 - 22032 * q^99

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	68.7054	1.22904	0.614520	−	0.788901i	$-0.289350\pi$
$5$	68.7054	1.22904	0.614520	+	0.788901i	$0.289350\pi$
$6$	0	0
$7$	92.5541	0.713922	0.356961	−	0.934119i	$-0.383813\pi$
$7$	92.5541	0.713922	0.356961	+	0.934119i	$0.383813\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−455.495	−1.13502	−0.567508	−	0.823368i	$-0.692093\pi$
$11$	−455.495	−1.13502	−0.567508	+	0.823368i	$0.692093\pi$
$12$	0	0
$13$	−138.319	−0.226998	−0.113499	−	0.993538i	$-0.536206\pi$
$13$	−138.319	−0.226998	−0.113499	+	0.993538i	$0.536206\pi$
$14$	0	0
$15$	−618.349	−0.709587
$16$	0	0
$17$	−25.2508	−0.0211910	−0.0105955	−	0.999944i	$-0.503373\pi$
$17$	−25.2508	−0.0211910	−0.0105955	+	0.999944i	$0.503373\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	−832.987	−0.412183
$22$	0	0
$23$	−4594.45	−1.81098	−0.905490	−	0.424368i	$-0.860496\pi$
$23$	−4594.45	−1.81098	−0.905490	+	0.424368i	$0.860496\pi$
$24$	0	0
$25$	1595.44	0.510540
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	5839.25	1.28932	0.644662	−	0.764467i	$-0.276998\pi$
$29$	5839.25	1.28932	0.644662	+	0.764467i	$0.276998\pi$
$30$	0	0
$31$	8095.84	1.51307	0.756533	−	0.653956i	$-0.226892\pi$
$31$	8095.84	1.51307	0.756533	+	0.653956i	$0.226892\pi$
$32$	0	0
$33$	4099.46	0.655302
$34$	0	0
$35$	6358.97	0.877438
$36$	0	0
$37$	−5483.74	−0.658525	−0.329263	−	0.944238i	$-0.606800\pi$
$37$	−5483.74	−0.658525	−0.329263	+	0.944238i	$0.606800\pi$
$38$	0	0
$39$	1244.87	0.131057
$40$	0	0
$41$	13069.3	1.21420	0.607102	−	0.794624i	$-0.292332\pi$
$41$	13069.3	1.21420	0.607102	+	0.794624i	$0.292332\pi$
$42$	0	0
$43$	−9161.20	−0.755581	−0.377791	−	0.925891i	$-0.623316\pi$
$43$	−9161.20	−0.755581	−0.377791	+	0.925891i	$0.623316\pi$
$44$	0	0
$45$	5565.14	0.409680
$46$	0	0
$47$	25131.8	1.65951	0.829754	−	0.558130i	$-0.188481\pi$
$47$	25131.8	1.65951	0.829754	+	0.558130i	$0.188481\pi$
$48$	0	0
$49$	−8240.74	−0.490316
$50$	0	0
$51$	227.257	0.0122347
$52$	0	0
$53$	31247.6	1.52801	0.764006	−	0.645209i	$-0.223230\pi$
$53$	31247.6	1.52801	0.764006	+	0.645209i	$0.223230\pi$
$54$	0	0
$55$	−31295.0	−1.39498
$56$	0	0
$57$	3249.00	0.132453
$58$	0	0
$59$	−26035.2	−0.973711	−0.486856	−	0.873482i	$-0.661856\pi$
$59$	−26035.2	−0.973711	−0.486856	+	0.873482i	$0.661856\pi$
$60$	0	0
$61$	23473.6	0.807708	0.403854	−	0.914823i	$-0.367670\pi$
$61$	23473.6	0.807708	0.403854	+	0.914823i	$0.367670\pi$
$62$	0	0
$63$	7496.88	0.237974
$64$	0	0
$65$	−9503.24	−0.278990
$66$	0	0
$67$	−1346.30	−0.0366401	−0.0183200	−	0.999832i	$-0.505832\pi$
$67$	−1346.30	−0.0366401	−0.0183200	+	0.999832i	$0.505832\pi$
$68$	0	0
$69$	41350.0	1.04557
$70$	0	0
$71$	−62531.0	−1.47214	−0.736071	−	0.676904i	$-0.763321\pi$
$71$	−62531.0	−1.47214	−0.736071	+	0.676904i	$0.763321\pi$
$72$	0	0
$73$	60494.1	1.32864	0.664318	−	0.747451i	$-0.268722\pi$
$73$	60494.1	1.32864	0.664318	+	0.747451i	$0.268722\pi$
$74$	0	0
$75$	−14358.9	−0.294760
$76$	0	0
$77$	−42158.0	−0.810313
$78$	0	0
$79$	56411.2	1.01694	0.508472	−	0.861078i	$-0.330210\pi$
$79$	56411.2	1.01694	0.508472	+	0.861078i	$0.330210\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	79425.3	1.26550	0.632752	−	0.774354i	$-0.281925\pi$
$83$	79425.3	1.26550	0.632752	+	0.774354i	$0.281925\pi$
$84$	0	0
$85$	−1734.87	−0.0260446
$86$	0	0
$87$	−52553.3	−0.744392
$88$	0	0
$89$	7121.69	0.0953033	0.0476516	−	0.998864i	$-0.484826\pi$
$89$	7121.69	0.0953033	0.0476516	+	0.998864i	$0.484826\pi$
$90$	0	0
$91$	−12802.0	−0.162059
$92$	0	0
$93$	−72862.6	−0.873569
$94$	0	0
$95$	−24802.7	−0.281961
$96$	0	0
$97$	115588.	1.24733	0.623666	−	0.781691i	$-0.285642\pi$
$97$	115588.	1.24733	0.623666	+	0.781691i	$0.285642\pi$
$98$	0	0
$99$	−36895.1	−0.378339
$100$	0	0
$101$	−32109.9	−0.313210	−0.156605	−	0.987661i	$-0.550055\pi$
$101$	−32109.9	−0.313210	−0.156605	+	0.987661i	$0.550055\pi$
$102$	0	0
$103$	−195110.	−1.81212	−0.906061	−	0.423148i	$-0.860925\pi$
$103$	−195110.	−1.81212	−0.906061	+	0.423148i	$0.860925\pi$
$104$	0	0
$105$	−57230.7	−0.506589
$106$	0	0
$107$	−20216.6	−0.170706	−0.0853531	−	0.996351i	$-0.527202\pi$
$107$	−20216.6	−0.170706	−0.0853531	+	0.996351i	$0.527202\pi$
$108$	0	0
$109$	−94942.4	−0.765410	−0.382705	−	0.923871i	$-0.625007\pi$
$109$	−94942.4	−0.765410	−0.382705	+	0.923871i	$0.625007\pi$
$110$	0	0
$111$	49353.7	0.380200
$112$	0	0
$113$	−40546.3	−0.298713	−0.149357	−	0.988783i	$-0.547720\pi$
$113$	−40546.3	−0.298713	−0.149357	+	0.988783i	$0.547720\pi$
$114$	0	0
$115$	−315663.	−2.22577
$116$	0	0
$117$	−11203.8	−0.0756661
$118$	0	0
$119$	−2337.06	−0.0151287
$120$	0	0
$121$	46425.0	0.288262
$122$	0	0
$123$	−117623.	−0.701021
$124$	0	0
$125$	−105089.	−0.601566
$126$	0	0
$127$	179240.	0.986112	0.493056	−	0.869998i	$-0.335880\pi$
$127$	179240.	0.986112	0.493056	+	0.869998i	$0.335880\pi$
$128$	0	0
$129$	82450.8	0.436235
$130$	0	0
$131$	264193.	1.34506	0.672532	−	0.740068i	$-0.265207\pi$
$131$	264193.	1.34506	0.672532	+	0.740068i	$0.265207\pi$
$132$	0	0
$133$	−33412.0	−0.163785
$134$	0	0
$135$	−50086.3	−0.236529
$136$	0	0
$137$	180893.	0.823419	0.411710	−	0.911315i	$-0.364932\pi$
$137$	180893.	0.823419	0.411710	+	0.911315i	$0.364932\pi$
$138$	0	0
$139$	40272.0	0.176794	0.0883968	−	0.996085i	$-0.471826\pi$
$139$	40272.0	0.176794	0.0883968	+	0.996085i	$0.471826\pi$
$140$	0	0
$141$	−226186.	−0.958117
$142$	0	0
$143$	63003.5	0.257647
$144$	0	0
$145$	401188.	1.58463
$146$	0	0
$147$	74166.7	0.283084
$148$	0	0
$149$	−106352.	−0.392447	−0.196224	−	0.980559i	$-0.562868\pi$
$149$	−106352.	−0.392447	−0.196224	+	0.980559i	$0.562868\pi$
$150$	0	0
$151$	473107.	1.68856	0.844280	−	0.535902i	$-0.180028\pi$
$151$	473107.	1.68856	0.844280	+	0.535902i	$0.180028\pi$
$152$	0	0
$153$	−2045.31	−0.00706368
$154$	0	0
$155$	556228.	1.85962
$156$	0	0
$157$	−221657.	−0.717681	−0.358841	−	0.933399i	$-0.616828\pi$
$157$	−221657.	−0.717681	−0.358841	+	0.933399i	$0.616828\pi$
$158$	0	0
$159$	−281228.	−0.882199
$160$	0	0
$161$	−425235.	−1.29290
$162$	0	0
$163$	632179.	1.86368	0.931840	−	0.362870i	$-0.118203\pi$
$163$	632179.	1.86368	0.931840	+	0.362870i	$0.118203\pi$
$164$	0	0
$165$	281655.	0.805393
$166$	0	0
$167$	−219742.	−0.609707	−0.304853	−	0.952399i	$-0.598607\pi$
$167$	−219742.	−0.609707	−0.304853	+	0.952399i	$0.598607\pi$
$168$	0	0
$169$	−352161.	−0.948472
$170$	0	0
$171$	−29241.0	−0.0764719
$172$	0	0
$173$	−185721.	−0.471786	−0.235893	−	0.971779i	$-0.575802\pi$
$173$	−185721.	−0.471786	−0.235893	+	0.971779i	$0.575802\pi$
$174$	0	0
$175$	147664.	0.364485
$176$	0	0
$177$	234316.	0.562173
$178$	0	0
$179$	452972.	1.05667	0.528335	−	0.849036i	$-0.322817\pi$
$179$	452972.	1.05667	0.528335	+	0.849036i	$0.322817\pi$
$180$	0	0
$181$	−246811.	−0.559975	−0.279988	−	0.960004i	$-0.590330\pi$
$181$	−246811.	−0.559975	−0.279988	+	0.960004i	$0.590330\pi$
$182$	0	0
$183$	−211262.	−0.466330
$184$	0	0
$185$	−376763.	−0.809354
$186$	0	0
$187$	11501.6	0.0240522
$188$	0	0
$189$	−67471.9	−0.137394
$190$	0	0
$191$	569883.	1.13032	0.565161	−	0.824981i	$-0.308814\pi$
$191$	569883.	1.13032	0.565161	+	0.824981i	$0.308814\pi$
$192$	0	0
$193$	832118.	1.60802	0.804010	−	0.594616i	$-0.202696\pi$
$193$	832118.	1.60802	0.804010	+	0.594616i	$0.202696\pi$
$194$	0	0
$195$	85529.2	0.161075
$196$	0	0
$197$	−38624.6	−0.0709086	−0.0354543	−	0.999371i	$-0.511288\pi$
$197$	−38624.6	−0.0709086	−0.0354543	+	0.999371i	$0.511288\pi$
$198$	0	0
$199$	63345.6	0.113392	0.0566962	−	0.998391i	$-0.481943\pi$
$199$	63345.6	0.113392	0.0566962	+	0.998391i	$0.481943\pi$
$200$	0	0
$201$	12116.7	0.0211542
$202$	0	0
$203$	540447.	0.920477
$204$	0	0
$205$	897930.	1.49231
$206$	0	0
$207$	−372150.	−0.603660
$208$	0	0
$209$	164434.	0.260391
$210$	0	0
$211$	123375.	0.190774	0.0953871	−	0.995440i	$-0.469591\pi$
$211$	123375.	0.190774	0.0953871	+	0.995440i	$0.469591\pi$
$212$	0	0
$213$	562779.	0.849942
$214$	0	0
$215$	−629424.	−0.928640
$216$	0	0
$217$	749303.	1.08021
$218$	0	0
$219$	−544447.	−0.767088
$220$	0	0
$221$	3492.65	0.00481033
$222$	0	0
$223$	−72181.4	−0.0971993	−0.0485996	−	0.998818i	$-0.515476\pi$
$223$	−72181.4	−0.0971993	−0.0485996	+	0.998818i	$0.515476\pi$
$224$	0	0
$225$	129230.	0.170180
$226$	0	0
$227$	386692.	0.498082	0.249041	−	0.968493i	$-0.419885\pi$
$227$	386692.	0.498082	0.249041	+	0.968493i	$0.419885\pi$
$228$	0	0
$229$	423629.	0.533822	0.266911	−	0.963721i	$-0.413997\pi$
$229$	423629.	0.533822	0.266911	+	0.963721i	$0.413997\pi$
$230$	0	0
$231$	379422.	0.467834
$232$	0	0
$233$	436055.	0.526201	0.263101	−	0.964768i	$-0.415255\pi$
$233$	436055.	0.526201	0.263101	+	0.964768i	$0.415255\pi$
$234$	0	0
$235$	1.72669e6	2.03960
$236$	0	0
$237$	−507701.	−0.587133
$238$	0	0
$239$	1.30409e6	1.47677	0.738384	−	0.674380i	$-0.235589\pi$
$239$	1.30409e6	1.47677	0.738384	+	0.674380i	$0.235589\pi$
$240$	0	0
$241$	148667.	0.164882	0.0824409	−	0.996596i	$-0.473728\pi$
$241$	148667.	0.164882	0.0824409	+	0.996596i	$0.473728\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−566184.	−0.602618
$246$	0	0
$247$	49933.0	0.0520770
$248$	0	0
$249$	−714828.	−0.730640
$250$	0	0
$251$	−362686.	−0.363367	−0.181684	−	0.983357i	$-0.558155\pi$
$251$	−362686.	−0.363367	−0.181684	+	0.983357i	$0.558155\pi$
$252$	0	0
$253$	2.09275e6	2.05549
$254$	0	0
$255$	15613.8	0.0150369
$256$	0	0
$257$	−582935.	−0.550538	−0.275269	−	0.961367i	$-0.588767\pi$
$257$	−582935.	−0.550538	−0.275269	+	0.961367i	$0.588767\pi$
$258$	0	0
$259$	−507543.	−0.470136
$260$	0	0
$261$	472980.	0.429775
$262$	0	0
$263$	1.66782e6	1.48682	0.743412	−	0.668833i	$-0.233206\pi$
$263$	1.66782e6	1.48682	0.743412	+	0.668833i	$0.233206\pi$
$264$	0	0
$265$	2.14688e6	1.87799
$266$	0	0
$267$	−64095.2	−0.0550234
$268$	0	0
$269$	316816.	0.266948	0.133474	−	0.991052i	$-0.457387\pi$
$269$	316816.	0.266948	0.133474	+	0.991052i	$0.457387\pi$
$270$	0	0
$271$	−2.05650e6	−1.70101	−0.850503	−	0.525970i	$-0.823703\pi$
$271$	−2.05650e6	−1.70101	−0.850503	+	0.525970i	$0.823703\pi$
$272$	0	0
$273$	115218.	0.0935648
$274$	0	0
$275$	−726714.	−0.579471
$276$	0	0
$277$	−422934.	−0.331187	−0.165594	−	0.986194i	$-0.552954\pi$
$277$	−422934.	−0.331187	−0.165594	+	0.986194i	$0.552954\pi$
$278$	0	0
$279$	655763.	0.504355
$280$	0	0
$281$	−98335.2	−0.0742922	−0.0371461	−	0.999310i	$-0.511827\pi$
$281$	−98335.2	−0.0742922	−0.0371461	+	0.999310i	$0.511827\pi$
$282$	0	0
$283$	1.43796e6	1.06728	0.533642	−	0.845710i	$-0.320823\pi$
$283$	1.43796e6	1.06728	0.533642	+	0.845710i	$0.320823\pi$
$284$	0	0
$285$	223224.	0.162790
$286$	0	0
$287$	1.20961e6	0.866847
$288$	0	0
$289$	−1.41922e6	−0.999551
$290$	0	0
$291$	−1.04029e6	−0.720148
$292$	0	0
$293$	2.05305e6	1.39711	0.698555	−	0.715557i	$-0.253827\pi$
$293$	2.05305e6	1.39711	0.698555	+	0.715557i	$0.253827\pi$
$294$	0	0
$295$	−1.78876e6	−1.19673
$296$	0	0
$297$	332056.	0.218434
$298$	0	0
$299$	635498.	0.411089
$300$	0	0
$301$	−847907.	−0.539426
$302$	0	0
$303$	288989.	0.180832
$304$	0	0
$305$	1.61276e6	0.992706
$306$	0	0
$307$	2.78894e6	1.68886	0.844429	−	0.535668i	$-0.179940\pi$
$307$	2.78894e6	1.68886	0.844429	+	0.535668i	$0.179940\pi$
$308$	0	0
$309$	1.75599e6	1.04623
$310$	0	0
$311$	−881911.	−0.517040	−0.258520	−	0.966006i	$-0.583235\pi$
$311$	−881911.	−0.517040	−0.258520	+	0.966006i	$0.583235\pi$
$312$	0	0
$313$	1.26094e6	0.727499	0.363749	−	0.931497i	$-0.381496\pi$
$313$	1.26094e6	0.727499	0.363749	+	0.931497i	$0.381496\pi$
$314$	0	0
$315$	515076.	0.292479
$316$	0	0
$317$	445866.	0.249205	0.124602	−	0.992207i	$-0.460234\pi$
$317$	445866.	0.249205	0.124602	+	0.992207i	$0.460234\pi$
$318$	0	0
$319$	−2.65975e6	−1.46340
$320$	0	0
$321$	181950.	0.0985572
$322$	0	0
$323$	9115.53	0.00486156
$324$	0	0
$325$	−220679.	−0.115892
$326$	0	0
$327$	854481.	0.441909
$328$	0	0
$329$	2.32605e6	1.18476
$330$	0	0
$331$	−199749.	−0.100211	−0.0501055	−	0.998744i	$-0.515956\pi$
$331$	−199749.	−0.100211	−0.0501055	+	0.998744i	$0.515956\pi$
$332$	0	0
$333$	−444183.	−0.219508
$334$	0	0
$335$	−92498.4	−0.0450321
$336$	0	0
$337$	−487801.	−0.233974	−0.116987	−	0.993133i	$-0.537324\pi$
$337$	−487801.	−0.233974	−0.116987	+	0.993133i	$0.537324\pi$
$338$	0	0
$339$	364916.	0.172462
$340$	0	0
$341$	−3.68762e6	−1.71735
$342$	0	0
$343$	−2.31827e6	−1.06397
$344$	0	0
$345$	2.84097e6	1.28505
$346$	0	0
$347$	−340706.	−0.151900	−0.0759498	−	0.997112i	$-0.524199\pi$
$347$	−340706.	−0.151900	−0.0759498	+	0.997112i	$0.524199\pi$
$348$	0	0
$349$	1.95453e6	0.858969	0.429485	−	0.903074i	$-0.358695\pi$
$349$	1.95453e6	0.858969	0.429485	+	0.903074i	$0.358695\pi$
$350$	0	0
$351$	100834.	0.0436858
$352$	0	0
$353$	−189240.	−0.0808307	−0.0404153	−	0.999183i	$-0.512868\pi$
$353$	−189240.	−0.0808307	−0.0404153	+	0.999183i	$0.512868\pi$
$354$	0	0
$355$	−4.29622e6	−1.80932
$356$	0	0
$357$	21033.6	0.00873459
$358$	0	0
$359$	1.38450e6	0.566967	0.283484	−	0.958977i	$-0.408510\pi$
$359$	1.38450e6	0.566967	0.283484	+	0.958977i	$0.408510\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	−417825.	−0.166428
$364$	0	0
$365$	4.15627e6	1.63295
$366$	0	0
$367$	2.01107e6	0.779403	0.389701	−	0.920941i	$-0.372578\pi$
$367$	2.01107e6	0.779403	0.389701	+	0.920941i	$0.372578\pi$
$368$	0	0
$369$	1.05861e6	0.404735
$370$	0	0
$371$	2.89209e6	1.09088
$372$	0	0
$373$	−3.33720e6	−1.24197	−0.620983	−	0.783824i	$-0.713266\pi$
$373$	−3.33720e6	−1.24197	−0.620983	+	0.783824i	$0.713266\pi$
$374$	0	0
$375$	945804.	0.347315
$376$	0	0
$377$	−807678.	−0.292674
$378$	0	0
$379$	320740.	0.114698	0.0573490	−	0.998354i	$-0.481735\pi$
$379$	320740.	0.114698	0.0573490	+	0.998354i	$0.481735\pi$
$380$	0	0
$381$	−1.61316e6	−0.569332
$382$	0	0
$383$	−5.60942e6	−1.95398	−0.976992	−	0.213278i	$-0.931586\pi$
$383$	−5.60942e6	−1.95398	−0.976992	+	0.213278i	$0.931586\pi$
$384$	0	0
$385$	−2.89648e6	−0.995907
$386$	0	0
$387$	−742057.	−0.251860
$388$	0	0
$389$	1.26415e6	0.423571	0.211785	−	0.977316i	$-0.432072\pi$
$389$	1.26415e6	0.423571	0.211785	+	0.977316i	$0.432072\pi$
$390$	0	0
$391$	116013.	0.0383766
$392$	0	0
$393$	−2.37774e6	−0.776573
$394$	0	0
$395$	3.87575e6	1.24987
$396$	0	0
$397$	−3.69231e6	−1.17577	−0.587884	−	0.808945i	$-0.700039\pi$
$397$	−3.69231e6	−1.17577	−0.587884	+	0.808945i	$0.700039\pi$
$398$	0	0
$399$	300708.	0.0945612
$400$	0	0
$401$	2.16581e6	0.672605	0.336303	−	0.941754i	$-0.390823\pi$
$401$	2.16581e6	0.672605	0.336303	+	0.941754i	$0.390823\pi$
$402$	0	0
$403$	−1.11981e6	−0.343463
$404$	0	0
$405$	450776.	0.136560
$406$	0	0
$407$	2.49782e6	0.747437
$408$	0	0
$409$	−293772.	−0.0868364	−0.0434182	−	0.999057i	$-0.513825\pi$
$409$	−293772.	−0.0868364	−0.0434182	+	0.999057i	$0.513825\pi$
$410$	0	0
$411$	−1.62804e6	−0.475401
$412$	0	0
$413$	−2.40966e6	−0.695154
$414$	0	0
$415$	5.45695e6	1.55536
$416$	0	0
$417$	−362448.	−0.102072
$418$	0	0
$419$	−3.02446e6	−0.841615	−0.420807	−	0.907150i	$-0.638253\pi$
$419$	−3.02446e6	−0.841615	−0.420807	+	0.907150i	$0.638253\pi$
$420$	0	0
$421$	−5.71044e6	−1.57023	−0.785117	−	0.619348i	$-0.787397\pi$
$421$	−5.71044e6	−1.57023	−0.785117	+	0.619348i	$0.787397\pi$
$422$	0	0
$423$	2.03568e6	0.553169
$424$	0	0
$425$	−40286.0	−0.0108189
$426$	0	0
$427$	2.17257e6	0.576640
$428$	0	0
$429$	−567031.	−0.148752
$430$	0	0
$431$	−855091.	−0.221727	−0.110864	−	0.993836i	$-0.535362\pi$
$431$	−855091.	−0.221727	−0.110864	+	0.993836i	$0.535362\pi$
$432$	0	0
$433$	−3.21472e6	−0.823992	−0.411996	−	0.911186i	$-0.635168\pi$
$433$	−3.21472e6	−0.823992	−0.411996	+	0.911186i	$0.635168\pi$
$434$	0	0
$435$	−3.61070e6	−0.914888
$436$	0	0
$437$	1.65859e6	0.415467
$438$	0	0
$439$	6.88316e6	1.70462	0.852308	−	0.523040i	$-0.175202\pi$
$439$	6.88316e6	1.70462	0.852308	+	0.523040i	$0.175202\pi$
$440$	0	0
$441$	−667500.	−0.163439
$442$	0	0
$443$	1.98623e6	0.480863	0.240431	−	0.970666i	$-0.422711\pi$
$443$	1.98623e6	0.480863	0.240431	+	0.970666i	$0.422711\pi$
$444$	0	0
$445$	489299.	0.117132
$446$	0	0
$447$	957170.	0.226579
$448$	0	0
$449$	−2.18473e6	−0.511425	−0.255712	−	0.966753i	$-0.582310\pi$
$449$	−2.18473e6	−0.511425	−0.255712	+	0.966753i	$0.582310\pi$
$450$	0	0
$451$	−5.95299e6	−1.37814
$452$	0	0
$453$	−4.25796e6	−0.974891
$454$	0	0
$455$	−879564.	−0.199177
$456$	0	0
$457$	3.89170e6	0.871663	0.435831	−	0.900028i	$-0.356454\pi$
$457$	3.89170e6	0.871663	0.435831	+	0.900028i	$0.356454\pi$
$458$	0	0
$459$	18407.8	0.00407822
$460$	0	0
$461$	2.81521e6	0.616962	0.308481	−	0.951231i	$-0.400179\pi$
$461$	2.81521e6	0.616962	0.308481	+	0.951231i	$0.400179\pi$
$462$	0	0
$463$	334859.	0.0725955	0.0362978	−	0.999341i	$-0.488444\pi$
$463$	334859.	0.0725955	0.0362978	+	0.999341i	$0.488444\pi$
$464$	0	0
$465$	−5.00606e6	−1.07365
$466$	0	0
$467$	−4.92891e6	−1.04582	−0.522912	−	0.852386i	$-0.675155\pi$
$467$	−4.92891e6	−1.04582	−0.522912	+	0.852386i	$0.675155\pi$
$468$	0	0
$469$	−124606.	−0.0261581
$470$	0	0
$471$	1.99491e6	0.414354
$472$	0	0
$473$	4.17288e6	0.857597
$474$	0	0
$475$	−575952.	−0.117126
$476$	0	0
$477$	2.53106e6	0.509338
$478$	0	0
$479$	8.65727e6	1.72402	0.862009	−	0.506892i	$-0.169206\pi$
$479$	8.65727e6	1.72402	0.862009	+	0.506892i	$0.169206\pi$
$480$	0	0
$481$	758504.	0.149484
$482$	0	0
$483$	3.82711e6	0.746455
$484$	0	0
$485$	7.94150e6	1.53302
$486$	0	0
$487$	687262.	0.131311	0.0656553	−	0.997842i	$-0.479086\pi$
$487$	687262.	0.131311	0.0656553	+	0.997842i	$0.479086\pi$
$488$	0	0
$489$	−5.68961e6	−1.07600
$490$	0	0
$491$	5.24841e6	0.982481	0.491240	−	0.871024i	$-0.336544\pi$
$491$	5.24841e6	0.982481	0.491240	+	0.871024i	$0.336544\pi$
$492$	0	0
$493$	−147446.	−0.0273221
$494$	0	0
$495$	−2.53490e6	−0.464994
$496$	0	0
$497$	−5.78750e6	−1.05099
$498$	0	0
$499$	1.73505e6	0.311932	0.155966	−	0.987762i	$-0.450151\pi$
$499$	1.73505e6	0.311932	0.155966	+	0.987762i	$0.450151\pi$
$500$	0	0
$501$	1.97767e6	0.352014
$502$	0	0
$503$	7.46626e6	1.31578	0.657889	−	0.753115i	$-0.271449\pi$
$503$	7.46626e6	1.31578	0.657889	+	0.753115i	$0.271449\pi$
$504$	0	0
$505$	−2.20613e6	−0.384948
$506$	0	0
$507$	3.16945e6	0.547600
$508$	0	0
$509$	7.25555e6	1.24130	0.620648	−	0.784089i	$-0.286869\pi$
$509$	7.25555e6	1.24130	0.620648	+	0.784089i	$0.286869\pi$
$510$	0	0
$511$	5.59897e6	0.948541
$512$	0	0
$513$	263169.	0.0441511
$514$	0	0
$515$	−1.34051e7	−2.22717
$516$	0	0
$517$	−1.14474e7	−1.88357
$518$	0	0
$519$	1.67149e6	0.272386
$520$	0	0
$521$	1.04532e7	1.68716	0.843580	−	0.537003i	$-0.180444\pi$
$521$	1.04532e7	1.68716	0.843580	+	0.537003i	$0.180444\pi$
$522$	0	0
$523$	−5.76718e6	−0.921955	−0.460977	−	0.887412i	$-0.652501\pi$
$523$	−5.76718e6	−0.921955	−0.460977	+	0.887412i	$0.652501\pi$
$524$	0	0
$525$	−1.32898e6	−0.210436
$526$	0	0
$527$	−204426.	−0.0320634
$528$	0	0
$529$	1.46726e7	2.27965
$530$	0	0
$531$	−2.10885e6	−0.324570
$532$	0	0
$533$	−1.80772e6	−0.275622
$534$	0	0
$535$	−1.38899e6	−0.209805
$536$	0	0
$537$	−4.07675e6	−0.610068
$538$	0	0
$539$	3.75362e6	0.556517
$540$	0	0
$541$	−8.23533e6	−1.20973	−0.604864	−	0.796329i	$-0.706773\pi$
$541$	−8.23533e6	−1.20973	−0.604864	+	0.796329i	$0.706773\pi$
$542$	0	0
$543$	2.22130e6	0.323302
$544$	0	0
$545$	−6.52306e6	−0.940719
$546$	0	0
$547$	−4.43504e6	−0.633767	−0.316883	−	0.948465i	$-0.602636\pi$
$547$	−4.43504e6	−0.633767	−0.316883	+	0.948465i	$0.602636\pi$
$548$	0	0
$549$	1.90136e6	0.269236
$550$	0	0
$551$	−2.10797e6	−0.295791
$552$	0	0
$553$	5.22108e6	0.726019
$554$	0	0
$555$	3.39086e6	0.467281
$556$	0	0
$557$	−9.40423e6	−1.28436	−0.642178	−	0.766555i	$-0.721969\pi$
$557$	−9.40423e6	−1.28436	−0.642178	+	0.766555i	$0.721969\pi$
$558$	0	0
$559$	1.26717e6	0.171516
$560$	0	0
$561$	−103514.	−0.0138865
$562$	0	0
$563$	8.42988e6	1.12086	0.560429	−	0.828203i	$-0.310636\pi$
$563$	8.42988e6	1.12086	0.560429	+	0.828203i	$0.310636\pi$
$564$	0	0
$565$	−2.78575e6	−0.367131
$566$	0	0
$567$	607247.	0.0793246
$568$	0	0
$569$	8.14622e6	1.05481	0.527407	−	0.849613i	$-0.323164\pi$
$569$	8.14622e6	1.05481	0.527407	+	0.849613i	$0.323164\pi$
$570$	0	0
$571$	1.44633e6	0.185643	0.0928215	−	0.995683i	$-0.470411\pi$
$571$	1.44633e6	0.185643	0.0928215	+	0.995683i	$0.470411\pi$
$572$	0	0
$573$	−5.12895e6	−0.652592
$574$	0	0
$575$	−7.33015e6	−0.924577
$576$	0	0
$577$	7.86543e6	0.983519	0.491760	−	0.870731i	$-0.336354\pi$
$577$	7.86543e6	0.983519	0.491760	+	0.870731i	$0.336354\pi$
$578$	0	0
$579$	−7.48906e6	−0.928391
$580$	0	0
$581$	7.35114e6	0.903471
$582$	0	0
$583$	−1.42331e7	−1.73432
$584$	0	0
$585$	−769763.	−0.0929966
$586$	0	0
$587$	−1.49205e7	−1.78726	−0.893631	−	0.448802i	$-0.851851\pi$
$587$	−1.49205e7	−1.78726	−0.893631	+	0.448802i	$0.851851\pi$
$588$	0	0
$589$	−2.92260e6	−0.347121
$590$	0	0
$591$	347622.	0.0409391
$592$	0	0
$593$	1.14093e7	1.33236	0.666179	−	0.745792i	$-0.267929\pi$
$593$	1.14093e7	1.33236	0.666179	+	0.745792i	$0.267929\pi$
$594$	0	0
$595$	−160569.	−0.0185938
$596$	0	0
$597$	−570111.	−0.0654671
$598$	0	0
$599$	7.64606e6	0.870705	0.435352	−	0.900260i	$-0.356624\pi$
$599$	7.64606e6	0.870705	0.435352	+	0.900260i	$0.356624\pi$
$600$	0	0
$601$	−7.31161e6	−0.825708	−0.412854	−	0.910797i	$-0.635468\pi$
$601$	−7.31161e6	−0.825708	−0.412854	+	0.910797i	$0.635468\pi$
$602$	0	0
$603$	−109051.	−0.0122134
$604$	0	0
$605$	3.18965e6	0.354286
$606$	0	0
$607$	−4.14398e6	−0.456506	−0.228253	−	0.973602i	$-0.573301\pi$
$607$	−4.14398e6	−0.456506	−0.228253	+	0.973602i	$0.573301\pi$
$608$	0	0
$609$	−4.86402e6	−0.531438
$610$	0	0
$611$	−3.47620e6	−0.376705
$612$	0	0
$613$	−8.51734e6	−0.915488	−0.457744	−	0.889084i	$-0.651342\pi$
$613$	−8.51734e6	−0.915488	−0.457744	+	0.889084i	$0.651342\pi$
$614$	0	0
$615$	−8.08137e6	−0.861583
$616$	0	0
$617$	4.77541e6	0.505008	0.252504	−	0.967596i	$-0.418746\pi$
$617$	4.77541e6	0.505008	0.252504	+	0.967596i	$0.418746\pi$
$618$	0	0
$619$	−1.38607e7	−1.45398	−0.726989	−	0.686649i	$-0.759081\pi$
$619$	−1.38607e7	−1.45398	−0.726989	+	0.686649i	$0.759081\pi$
$620$	0	0
$621$	3.34935e6	0.348523
$622$	0	0
$623$	659141.	0.0680391
$624$	0	0
$625$	−1.22059e7	−1.24989
$626$	0	0
$627$	−1.47990e6	−0.150337
$628$	0	0
$629$	138469.	0.0139548
$630$	0	0
$631$	1.49814e7	1.49789	0.748946	−	0.662631i	$-0.230560\pi$
$631$	1.49814e7	1.49789	0.748946	+	0.662631i	$0.230560\pi$
$632$	0	0
$633$	−1.11037e6	−0.110143
$634$	0	0
$635$	1.23148e7	1.21197
$636$	0	0
$637$	1.13985e6	0.111301
$638$	0	0
$639$	−5.06501e6	−0.490714
$640$	0	0
$641$	3.32362e6	0.319497	0.159748	−	0.987158i	$-0.448932\pi$
$641$	3.32362e6	0.319497	0.159748	+	0.987158i	$0.448932\pi$
$642$	0	0
$643$	−6.63132e6	−0.632518	−0.316259	−	0.948673i	$-0.602427\pi$
$643$	−6.63132e6	−0.632518	−0.316259	+	0.948673i	$0.602427\pi$
$644$	0	0
$645$	5.66482e6	0.536150
$646$	0	0
$647$	−9.63184e6	−0.904583	−0.452292	−	0.891870i	$-0.649393\pi$
$647$	−9.63184e6	−0.904583	−0.452292	+	0.891870i	$0.649393\pi$
$648$	0	0
$649$	1.18589e7	1.10518
$650$	0	0
$651$	−6.74373e6	−0.623660
$652$	0	0
$653$	1.92143e7	1.76336	0.881680	−	0.471848i	$-0.156413\pi$
$653$	1.92143e7	1.76336	0.881680	+	0.471848i	$0.156413\pi$
$654$	0	0
$655$	1.81515e7	1.65314
$656$	0	0
$657$	4.90002e6	0.442878
$658$	0	0
$659$	−1.36828e7	−1.22733	−0.613664	−	0.789568i	$-0.710305\pi$
$659$	−1.36828e7	−1.22733	−0.613664	+	0.789568i	$0.710305\pi$
$660$	0	0
$661$	−1.42269e7	−1.26651	−0.633254	−	0.773944i	$-0.718281\pi$
$661$	−1.42269e7	−1.26651	−0.633254	+	0.773944i	$0.718281\pi$
$662$	0	0
$663$	−31433.9	−0.00277724
$664$	0	0
$665$	−2.29559e6	−0.201298
$666$	0	0
$667$	−2.68281e7	−2.33494
$668$	0	0
$669$	649633.	0.0561180
$670$	0	0
$671$	−1.06921e7	−0.916762
$672$	0	0
$673$	−1.34068e7	−1.14100	−0.570501	−	0.821297i	$-0.693251\pi$
$673$	−1.34068e7	−1.14100	−0.570501	+	0.821297i	$0.693251\pi$
$674$	0	0
$675$	−1.16307e6	−0.0982534
$676$	0	0
$677$	−9.06473e6	−0.760122	−0.380061	−	0.924961i	$-0.624097\pi$
$677$	−9.06473e6	−0.760122	−0.380061	+	0.924961i	$0.624097\pi$
$678$	0	0
$679$	1.06981e7	0.890498
$680$	0	0
$681$	−3.48023e6	−0.287568
$682$	0	0
$683$	−1.14492e7	−0.939123	−0.469562	−	0.882900i	$-0.655588\pi$
$683$	−1.14492e7	−0.939123	−0.469562	+	0.882900i	$0.655588\pi$
$684$	0	0
$685$	1.24283e7	1.01202
$686$	0	0
$687$	−3.81266e6	−0.308202
$688$	0	0
$689$	−4.32213e6	−0.346856
$690$	0	0
$691$	9.17204e6	0.730753	0.365377	−	0.930860i	$-0.380940\pi$
$691$	9.17204e6	0.730753	0.365377	+	0.930860i	$0.380940\pi$
$692$	0	0
$693$	−3.41479e6	−0.270104
$694$	0	0
$695$	2.76691e6	0.217286
$696$	0	0
$697$	−330009.	−0.0257303
$698$	0	0
$699$	−3.92450e6	−0.303802
$700$	0	0
$701$	1.57928e6	0.121385	0.0606924	−	0.998157i	$-0.480669\pi$
$701$	1.57928e6	0.121385	0.0606924	+	0.998157i	$0.480669\pi$
$702$	0	0
$703$	1.97963e6	0.151076
$704$	0	0
$705$	−1.55402e7	−1.17756
$706$	0	0
$707$	−2.97190e6	−0.223607
$708$	0	0
$709$	2.08665e7	1.55896	0.779480	−	0.626428i	$-0.215484\pi$
$709$	2.08665e7	1.55896	0.779480	+	0.626428i	$0.215484\pi$
$710$	0	0
$711$	4.56930e6	0.338982
$712$	0	0
$713$	−3.71959e7	−2.74013
$714$	0	0
$715$	4.32868e6	0.316658
$716$	0	0
$717$	−1.17368e7	−0.852613
$718$	0	0
$719$	1.24652e6	0.0899241	0.0449620	−	0.998989i	$-0.485683\pi$
$719$	1.24652e6	0.0899241	0.0449620	+	0.998989i	$0.485683\pi$
$720$	0	0
$721$	−1.80583e7	−1.29371
$722$	0	0
$723$	−1.33800e6	−0.0951945
$724$	0	0
$725$	9.31616e6	0.658251
$726$	0	0
$727$	8.16134e6	0.572698	0.286349	−	0.958125i	$-0.407558\pi$
$727$	8.16134e6	0.572698	0.286349	+	0.958125i	$0.407558\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	231327.	0.0160116
$732$	0	0
$733$	4.20287e6	0.288926	0.144463	−	0.989510i	$-0.453855\pi$
$733$	4.20287e6	0.288926	0.144463	+	0.989510i	$0.453855\pi$
$734$	0	0
$735$	5.09565e6	0.347922
$736$	0	0
$737$	613235.	0.0415871
$738$	0	0
$739$	−1.97042e7	−1.32724	−0.663619	−	0.748071i	$-0.730980\pi$
$739$	−1.97042e7	−1.32724	−0.663619	+	0.748071i	$0.730980\pi$
$740$	0	0
$741$	−449397.	−0.0300666
$742$	0	0
$743$	1.76027e6	0.116979	0.0584895	−	0.998288i	$-0.481372\pi$
$743$	1.76027e6	0.116979	0.0584895	+	0.998288i	$0.481372\pi$
$744$	0	0
$745$	−7.30698e6	−0.482333
$746$	0	0
$747$	6.43345e6	0.421835
$748$	0	0
$749$	−1.87113e6	−0.121871
$750$	0	0
$751$	−1.93990e7	−1.25510	−0.627551	−	0.778575i	$-0.715943\pi$
$751$	−1.93990e7	−1.25510	−0.627551	+	0.778575i	$0.715943\pi$
$752$	0	0
$753$	3.26417e6	0.209790
$754$	0	0
$755$	3.25050e7	2.07531
$756$	0	0
$757$	−3.66152e6	−0.232232	−0.116116	−	0.993236i	$-0.537044\pi$
$757$	−3.66152e6	−0.232232	−0.116116	+	0.993236i	$0.537044\pi$
$758$	0	0
$759$	−1.88347e7	−1.18674
$760$	0	0
$761$	−2.09526e7	−1.31152	−0.655762	−	0.754967i	$-0.727653\pi$
$761$	−2.09526e7	−1.31152	−0.655762	+	0.754967i	$0.727653\pi$
$762$	0	0
$763$	−8.78731e6	−0.546442
$764$	0	0
$765$	−140524.	−0.00868155
$766$	0	0
$767$	3.60115e6	0.221031
$768$	0	0
$769$	1.71458e7	1.04555	0.522773	−	0.852472i	$-0.324898\pi$
$769$	1.71458e7	1.04555	0.522773	+	0.852472i	$0.324898\pi$
$770$	0	0
$771$	5.24641e6	0.317853
$772$	0	0
$773$	1.42888e7	0.860099	0.430049	−	0.902805i	$-0.358496\pi$
$773$	1.42888e7	0.860099	0.430049	+	0.902805i	$0.358496\pi$
$774$	0	0
$775$	1.29164e7	0.772480
$776$	0	0
$777$	4.56788e6	0.271433
$778$	0	0
$779$	−4.71801e6	−0.278558
$780$	0	0
$781$	2.84826e7	1.67091
$782$	0	0
$783$	−4.25682e6	−0.248131
$784$	0	0
$785$	−1.52290e7	−0.882059
$786$	0	0
$787$	1.68977e6	0.0972502	0.0486251	−	0.998817i	$-0.484516\pi$
$787$	1.68977e6	0.0972502	0.0486251	+	0.998817i	$0.484516\pi$
$788$	0	0
$789$	−1.50104e7	−0.858419
$790$	0	0
$791$	−3.75272e6	−0.213258
$792$	0	0
$793$	−3.24683e6	−0.183348
$794$	0	0
$795$	−1.93219e7	−1.08426
$796$	0	0
$797$	−8.07902e6	−0.450519	−0.225259	−	0.974299i	$-0.572323\pi$
$797$	−8.07902e6	−0.450519	−0.225259	+	0.974299i	$0.572323\pi$
$798$	0	0
$799$	−634598.	−0.0351667
$800$	0	0
$801$	576857.	0.0317678
$802$	0	0
$803$	−2.75548e7	−1.50802
$804$	0	0
$805$	−2.92159e7	−1.58902
$806$	0	0
$807$	−2.85134e6	−0.154122
$808$	0	0
$809$	3.20293e7	1.72059	0.860293	−	0.509799i	$-0.170280\pi$
$809$	3.20293e7	1.72059	0.860293	+	0.509799i	$0.170280\pi$
$810$	0	0
$811$	−2.79665e7	−1.49309	−0.746545	−	0.665334i	$-0.768289\pi$
$811$	−2.79665e7	−1.49309	−0.746545	+	0.665334i	$0.768289\pi$
$812$	0	0
$813$	1.85085e7	0.982077
$814$	0	0
$815$	4.34342e7	2.29054
$816$	0	0
$817$	3.30719e6	0.173342
$818$	0	0
$819$	−1.03696e6	−0.0540196
$820$	0	0
$821$	−1.93844e7	−1.00368	−0.501838	−	0.864961i	$-0.667343\pi$
$821$	−1.93844e7	−1.00368	−0.501838	+	0.864961i	$0.667343\pi$
$822$	0	0
$823$	−3.64416e7	−1.87542	−0.937709	−	0.347421i	$-0.887058\pi$
$823$	−3.64416e7	−1.87542	−0.937709	+	0.347421i	$0.887058\pi$
$824$	0	0
$825$	6.54042e6	0.334558
$826$	0	0
$827$	1.28526e7	0.653473	0.326736	−	0.945116i	$-0.394051\pi$
$827$	1.28526e7	0.653473	0.326736	+	0.945116i	$0.394051\pi$
$828$	0	0
$829$	−3.23505e7	−1.63491	−0.817457	−	0.575989i	$-0.804617\pi$
$829$	−3.23505e7	−1.63491	−0.817457	+	0.575989i	$0.804617\pi$
$830$	0	0
$831$	3.80641e6	0.191211
$832$	0	0
$833$	208085.	0.0103903
$834$	0	0
$835$	−1.50974e7	−0.749354
$836$	0	0
$837$	−5.90187e6	−0.291190
$838$	0	0
$839$	1.00791e7	0.494330	0.247165	−	0.968973i	$-0.420501\pi$
$839$	1.00791e7	0.494330	0.247165	+	0.968973i	$0.420501\pi$
$840$	0	0
$841$	1.35857e7	0.662358
$842$	0	0
$843$	885017.	0.0428926
$844$	0	0
$845$	−2.41954e7	−1.16571
$846$	0	0
$847$	4.29682e6	0.205797
$848$	0	0
$849$	−1.29416e7	−0.616197
$850$	0	0
$851$	2.51947e7	1.19258
$852$	0	0
$853$	−2.85150e7	−1.34184	−0.670919	−	0.741530i	$-0.734100\pi$
$853$	−2.85150e7	−1.34184	−0.670919	+	0.741530i	$0.734100\pi$
$854$	0	0
$855$	−2.00902e6	−0.0939870
$856$	0	0
$857$	1.42267e7	0.661685	0.330843	−	0.943686i	$-0.392667\pi$
$857$	1.42267e7	0.661685	0.330843	+	0.943686i	$0.392667\pi$
$858$	0	0
$859$	8.83950e6	0.408738	0.204369	−	0.978894i	$-0.434486\pi$
$859$	8.83950e6	0.408738	0.204369	+	0.978894i	$0.434486\pi$
$860$	0	0
$861$	−1.08865e7	−0.500474
$862$	0	0
$863$	2.21209e7	1.01106	0.505528	−	0.862810i	$-0.331298\pi$
$863$	2.21209e7	1.01106	0.505528	+	0.862810i	$0.331298\pi$
$864$	0	0
$865$	−1.27600e7	−0.579845
$866$	0	0
$867$	1.27730e7	0.577091
$868$	0	0
$869$	−2.56950e7	−1.15425
$870$	0	0
$871$	186219.	0.00831723
$872$	0	0
$873$	9.36260e6	0.415778
$874$	0	0
$875$	−9.72645e6	−0.429471
$876$	0	0
$877$	1.81374e7	0.796300	0.398150	−	0.917320i	$-0.369652\pi$
$877$	1.81374e7	0.796300	0.398150	+	0.917320i	$0.369652\pi$
$878$	0	0
$879$	−1.84774e7	−0.806621
$880$	0	0
$881$	−2.33506e7	−1.01358	−0.506790	−	0.862069i	$-0.669168\pi$
$881$	−2.33506e7	−1.01358	−0.506790	+	0.862069i	$0.669168\pi$
$882$	0	0
$883$	3.19662e7	1.37971	0.689857	−	0.723945i	$-0.257673\pi$
$883$	3.19662e7	1.37971	0.689857	+	0.723945i	$0.257673\pi$
$884$	0	0
$885$	1.60988e7	0.690933
$886$	0	0
$887$	3.72115e7	1.58806	0.794031	−	0.607877i	$-0.207979\pi$
$887$	3.72115e7	1.58806	0.794031	+	0.607877i	$0.207979\pi$
$888$	0	0
$889$	1.65894e7	0.704007
$890$	0	0
$891$	−2.98850e6	−0.126113
$892$	0	0
$893$	−9.07258e6	−0.380717
$894$	0	0
$895$	3.11217e7	1.29869
$896$	0	0
$897$	−5.71948e6	−0.237342
$898$	0	0
$899$	4.72737e7	1.95083
$900$	0	0
$901$	−789026.	−0.0323802
$902$	0	0
$903$	7.63116e6	0.311438
$904$	0	0
$905$	−1.69573e7	−0.688232
$906$	0	0
$907$	−3.16967e7	−1.27937	−0.639685	−	0.768637i	$-0.720935\pi$
$907$	−3.16967e7	−1.27937	−0.639685	+	0.768637i	$0.720935\pi$
$908$	0	0
$909$	−2.60090e6	−0.104403
$910$	0	0
$911$	−3.01177e7	−1.20234	−0.601168	−	0.799123i	$-0.705298\pi$
$911$	−3.01177e7	−1.20234	−0.601168	+	0.799123i	$0.705298\pi$
$912$	0	0
$913$	−3.61779e7	−1.43637
$914$	0	0
$915$	−1.45148e7	−0.573139
$916$	0	0
$917$	2.44521e7	0.960271
$918$	0	0
$919$	3.19659e7	1.24853	0.624263	−	0.781214i	$-0.285399\pi$
$919$	3.19659e7	1.24853	0.624263	+	0.781214i	$0.285399\pi$
$920$	0	0
$921$	−2.51004e7	−0.975062
$922$	0	0
$923$	8.64921e6	0.334174
$924$	0	0
$925$	−8.74896e6	−0.336203
$926$	0	0
$927$	−1.58039e7	−0.604041
$928$	0	0
$929$	−2.68114e7	−1.01925	−0.509624	−	0.860398i	$-0.670215\pi$
$929$	−2.68114e7	−1.01925	−0.509624	+	0.860398i	$0.670215\pi$
$930$	0	0
$931$	2.97491e6	0.112486
$932$	0	0
$933$	7.93720e6	0.298513
$934$	0	0
$935$	790223.	0.0295611
$936$	0	0
$937$	3.98690e7	1.48350	0.741748	−	0.670678i	$-0.233997\pi$
$937$	3.98690e7	1.48350	0.741748	+	0.670678i	$0.233997\pi$
$938$	0	0
$939$	−1.13484e7	−0.420022
$940$	0	0
$941$	−2.21442e7	−0.815242	−0.407621	−	0.913151i	$-0.633642\pi$
$941$	−2.21442e7	−0.815242	−0.407621	+	0.913151i	$0.633642\pi$
$942$	0	0
$943$	−6.00461e7	−2.19890
$944$	0	0
$945$	−4.63569e6	−0.168863
$946$	0	0
$947$	−183531.	−0.00665021	−0.00332511	−	0.999994i	$-0.501058\pi$
$947$	−183531.	−0.00665021	−0.00332511	+	0.999994i	$0.501058\pi$
$948$	0	0
$949$	−8.36746e6	−0.301598
$950$	0	0
$951$	−4.01279e6	−0.143878
$952$	0	0
$953$	3.95151e7	1.40939	0.704694	−	0.709511i	$-0.251084\pi$
$953$	3.95151e7	1.40939	0.704694	+	0.709511i	$0.251084\pi$
$954$	0	0
$955$	3.91540e7	1.38921
$956$	0	0
$957$	2.39378e7	0.844897
$958$	0	0
$959$	1.67424e7	0.587857
$960$	0	0
$961$	3.69135e7	1.28937
$962$	0	0
$963$	−1.63755e6	−0.0569020
$964$	0	0
$965$	5.71710e7	1.97632
$966$	0	0
$967$	2.88127e7	0.990872	0.495436	−	0.868645i	$-0.335008\pi$
$967$	2.88127e7	0.990872	0.495436	+	0.868645i	$0.335008\pi$
$968$	0	0
$969$	−82039.8	−0.00280682
$970$	0	0
$971$	3.63561e7	1.23745	0.618727	−	0.785606i	$-0.287649\pi$
$971$	3.63561e7	1.23745	0.618727	+	0.785606i	$0.287649\pi$
$972$	0	0
$973$	3.72734e6	0.126217
$974$	0	0
$975$	1.98611e6	0.0669100
$976$	0	0
$977$	1.64450e7	0.551185	0.275592	−	0.961275i	$-0.411126\pi$
$977$	1.64450e7	0.551185	0.275592	+	0.961275i	$0.411126\pi$
$978$	0	0
$979$	−3.24389e6	−0.108171
$980$	0	0
$981$	−7.69033e6	−0.255137
$982$	0	0
$983$	−678596.	−0.0223989	−0.0111995	−	0.999937i	$-0.503565\pi$
$983$	−678596.	−0.0223989	−0.0111995	+	0.999937i	$0.503565\pi$
$984$	0	0
$985$	−2.65372e6	−0.0871495
$986$	0	0
$987$	−2.09345e7	−0.684020
$988$	0	0
$989$	4.20906e7	1.36834
$990$	0	0
$991$	−3.57109e7	−1.15509	−0.577546	−	0.816358i	$-0.695989\pi$
$991$	−3.57109e7	−1.15509	−0.577546	+	0.816358i	$0.695989\pi$
$992$	0	0
$993$	1.79774e6	0.0578569
$994$	0	0
$995$	4.35219e6	0.139364
$996$	0	0
$997$	−5.41050e7	−1.72385	−0.861926	−	0.507035i	$-0.830742\pi$
$997$	−5.41050e7	−1.72385	−0.861926	+	0.507035i	$0.830742\pi$
$998$	0	0
$999$	3.99765e6	0.126733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.6.a.v.1.5		5
4.3	odd	2		228.6.a.d.1.5	✓	5
12.11	even	2		684.6.a.f.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.6.a.d.1.5	✓	5	4.3	odd	2
684.6.a.f.1.1		5	12.11	even	2
912.6.a.v.1.5		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	912.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +68.7054 q^{5} +92.5541 q^{7} +81.0000 q^{9} -455.495 q^{11} -138.319 q^{13} -618.349 q^{15} -25.2508 q^{17} -361.000 q^{19} -832.987 q^{21} -4594.45 q^{23} +1595.44 q^{25} -729.000 q^{27} +5839.25 q^{29} +8095.84 q^{31} +4099.46 q^{33} +6358.97 q^{35} -5483.74 q^{37} +1244.87 q^{39} +13069.3 q^{41} -9161.20 q^{43} +5565.14 q^{45} +25131.8 q^{47} -8240.74 q^{49} +227.257 q^{51} +31247.6 q^{53} -31295.0 q^{55} +3249.00 q^{57} -26035.2 q^{59} +23473.6 q^{61} +7496.88 q^{63} -9503.24 q^{65} -1346.30 q^{67} +41350.0 q^{69} -62531.0 q^{71} +60494.1 q^{73} -14358.9 q^{75} -42158.0 q^{77} +56411.2 q^{79} +6561.00 q^{81} +79425.3 q^{83} -1734.87 q^{85} -52553.3 q^{87} +7121.69 q^{89} -12802.0 q^{91} -72862.6 q^{93} -24802.7 q^{95} +115588. q^{97} -36895.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 45 q^{3} - 6 q^{5} - 54 q^{7} + 405 q^{9} - 272 q^{11} + 440 q^{13} + 54 q^{15} + 1940 q^{17} - 1805 q^{19} + 486 q^{21} - 3224 q^{23} + 7313 q^{25} - 3645 q^{27} + 7524 q^{29} - 11774 q^{31} + 2448 q^{33}+ \cdots - 22032 q^{99}+O(q^{100}) \) 5 * q - 45 * q^3 - 6 * q^5 - 54 * q^7 + 405 * q^9 - 272 * q^11 + 440 * q^13 + 54 * q^15 + 1940 * q^17 - 1805 * q^19 + 486 * q^21 - 3224 * q^23 + 7313 * q^25 - 3645 * q^27 + 7524 * q^29 - 11774 * q^31 + 2448 * q^33 - 12834 * q^35 + 17760 * q^37 - 3960 * q^39 + 10736 * q^41 - 21914 * q^43 - 486 * q^45 - 10190 * q^47 + 1503 * q^49 - 17460 * q^51 - 1676 * q^53 - 36474 * q^55 + 16245 * q^57 - 54920 * q^59 + 39268 * q^61 - 4374 * q^63 + 39684 * q^65 - 104168 * q^67 + 29016 * q^69 - 50620 * q^71 + 98616 * q^73 - 65817 * q^75 - 9418 * q^77 - 115782 * q^79 + 32805 * q^81 - 143266 * q^83 + 113334 * q^85 - 67716 * q^87 + 88112 * q^89 - 247372 * q^91 + 105966 * q^93 + 2166 * q^95 + 186490 * q^97 - 22032 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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