LMFDB - Embedded newform 45.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,6,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, 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("45.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	45.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:	$7.21727189158$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{145}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 36 $ x^2 - x - 36
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.52080$ of defining polynomial
Character	$\chi$	$=$	45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.52080 q^{2} -19.6040 q^{4} +25.0000 q^{5} -80.4159 q^{7} +181.687 q^{8} +O(q^{10})$	\(q-3.52080 q^{2} -19.6040 q^{4} +25.0000 q^{5} -80.4159 q^{7} +181.687 q^{8} -88.0199 q^{10} +520.416 q^{11} +421.664 q^{13} +283.128 q^{14} -12.3561 q^{16} +1162.67 q^{17} +1170.66 q^{19} -490.100 q^{20} -1832.28 q^{22} +2374.48 q^{23} +625.000 q^{25} -1484.59 q^{26} +1576.47 q^{28} -7056.62 q^{29} -5561.15 q^{31} -5770.49 q^{32} -4093.51 q^{34} -2010.40 q^{35} +2389.95 q^{37} -4121.64 q^{38} +4542.18 q^{40} +8388.28 q^{41} +20546.6 q^{43} -10202.2 q^{44} -8360.07 q^{46} +8352.17 q^{47} -10340.3 q^{49} -2200.50 q^{50} -8266.29 q^{52} +28390.6 q^{53} +13010.4 q^{55} -14610.5 q^{56} +24844.9 q^{58} +37500.4 q^{59} -57311.1 q^{61} +19579.7 q^{62} +20712.1 q^{64} +10541.6 q^{65} -45848.9 q^{67} -22792.9 q^{68} +7078.21 q^{70} +41510.8 q^{71} +76806.5 q^{73} -8414.52 q^{74} -22949.5 q^{76} -41849.7 q^{77} -7008.26 q^{79} -308.904 q^{80} -29533.4 q^{82} -93271.1 q^{83} +29066.6 q^{85} -72340.5 q^{86} +94552.9 q^{88} +7074.70 q^{89} -33908.5 q^{91} -46549.3 q^{92} -29406.3 q^{94} +29266.4 q^{95} +22429.9 q^{97} +36406.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} + 21 q^{4} + 50 q^{5} + 80 q^{7} + 255 q^{8}+O(q^{10}) $ 2 * q + 5 * q^2 + 21 * q^4 + 50 * q^5 + 80 * q^7 + 255 * q^8	$ 2 q + 5 q^{2} + 21 q^{4} + 50 q^{5} + 80 q^{7} + 255 q^{8} + 125 q^{10} + 800 q^{11} - 120 q^{13} + 1650 q^{14} - 687 q^{16} + 1940 q^{17} - 1512 q^{19} + 525 q^{20} + 550 q^{22} - 1320 q^{23} + 1250 q^{25} - 6100 q^{26} + 8090 q^{28} + 1300 q^{29} - 5824 q^{31} - 13865 q^{32} + 2530 q^{34} + 2000 q^{35} - 12560 q^{37} - 26980 q^{38} + 6375 q^{40} + 400 q^{41} + 25680 q^{43} + 1150 q^{44} - 39840 q^{46} + 18920 q^{47} - 1414 q^{49} + 3125 q^{50} - 30260 q^{52} + 49460 q^{53} + 20000 q^{55} - 2850 q^{56} + 96050 q^{58} + 63200 q^{59} - 49116 q^{61} + 17340 q^{62} - 26671 q^{64} - 3000 q^{65} + 6080 q^{67} + 8770 q^{68} + 41250 q^{70} + 65200 q^{71} + 97740 q^{73} - 135800 q^{74} - 131876 q^{76} + 3000 q^{77} - 46288 q^{79} - 17175 q^{80} - 97600 q^{82} - 57360 q^{83} + 48500 q^{85} - 28600 q^{86} + 115050 q^{88} - 87000 q^{89} - 120800 q^{91} - 196560 q^{92} + 60640 q^{94} - 37800 q^{95} + 10180 q^{97} + 112465 q^{98}+O(q^{100}) $ 2 * q + 5 * q^2 + 21 * q^4 + 50 * q^5 + 80 * q^7 + 255 * q^8 + 125 * q^10 + 800 * q^11 - 120 * q^13 + 1650 * q^14 - 687 * q^16 + 1940 * q^17 - 1512 * q^19 + 525 * q^20 + 550 * q^22 - 1320 * q^23 + 1250 * q^25 - 6100 * q^26 + 8090 * q^28 + 1300 * q^29 - 5824 * q^31 - 13865 * q^32 + 2530 * q^34 + 2000 * q^35 - 12560 * q^37 - 26980 * q^38 + 6375 * q^40 + 400 * q^41 + 25680 * q^43 + 1150 * q^44 - 39840 * q^46 + 18920 * q^47 - 1414 * q^49 + 3125 * q^50 - 30260 * q^52 + 49460 * q^53 + 20000 * q^55 - 2850 * q^56 + 96050 * q^58 + 63200 * q^59 - 49116 * q^61 + 17340 * q^62 - 26671 * q^64 - 3000 * q^65 + 6080 * q^67 + 8770 * q^68 + 41250 * q^70 + 65200 * q^71 + 97740 * q^73 - 135800 * q^74 - 131876 * q^76 + 3000 * q^77 - 46288 * q^79 - 17175 * q^80 - 97600 * q^82 - 57360 * q^83 + 48500 * q^85 - 28600 * q^86 + 115050 * q^88 - 87000 * q^89 - 120800 * q^91 - 196560 * q^92 + 60640 * q^94 - 37800 * q^95 + 10180 * q^97 + 112465 * q^98

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 $

$2$	−3.52080	−0.622395	−0.311197	−	0.950345i	$-0.600730\pi$
$2$	−3.52080	−0.622395	−0.311197	+	0.950345i	$0.600730\pi$
$3$	0	0
$3$	0	0
$4$	−19.6040	−0.612625
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−80.4159	−0.620293	−0.310147	−	0.950689i	$-0.600378\pi$
$7$	−80.4159	−0.620293	−0.310147	+	0.950689i	$0.600378\pi$
$8$	181.687	1.00369
$9$	0	0
$10$	−88.0199	−0.278343
$11$	520.416	1.29679	0.648394	−	0.761305i	$-0.275441\pi$
$11$	520.416	1.29679	0.648394	+	0.761305i	$0.275441\pi$
$12$	0	0
$13$	421.664	0.692003	0.346001	−	0.938234i	$-0.387539\pi$
$13$	421.664	0.692003	0.346001	+	0.938234i	$0.387539\pi$
$14$	283.128	0.386067
$15$	0	0
$16$	−12.3561	−0.0120666
$17$	1162.67	0.975736	0.487868	−	0.872917i	$-0.337775\pi$
$17$	1162.67	0.975736	0.487868	+	0.872917i	$0.337775\pi$
$18$	0	0
$19$	1170.66	0.743952	0.371976	−	0.928242i	$-0.378680\pi$
$19$	1170.66	0.743952	0.371976	+	0.928242i	$0.378680\pi$
$20$	−490.100	−0.273974
$21$	0	0
$22$	−1832.28	−0.807114
$23$	2374.48	0.935943	0.467971	−	0.883744i	$-0.344985\pi$
$23$	2374.48	0.935943	0.467971	+	0.883744i	$0.344985\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−1484.59	−0.430699
$27$	0	0
$28$	1576.47	0.380007
$29$	−7056.62	−1.55812	−0.779062	−	0.626947i	$-0.784304\pi$
$29$	−7056.62	−1.55812	−0.779062	+	0.626947i	$0.784304\pi$
$30$	0	0
$31$	−5561.15	−1.03935	−0.519673	−	0.854365i	$-0.673946\pi$
$31$	−5561.15	−1.03935	−0.519673	+	0.854365i	$0.673946\pi$
$32$	−5770.49	−0.996179
$33$	0	0
$34$	−4093.51	−0.607293
$35$	−2010.40	−0.277404
$36$	0	0
$37$	2389.95	0.287001	0.143501	−	0.989650i	$-0.454164\pi$
$37$	2389.95	0.287001	0.143501	+	0.989650i	$0.454164\pi$
$38$	−4121.64	−0.463032
$39$	0	0
$40$	4542.18	0.448864
$41$	8388.28	0.779316	0.389658	−	0.920960i	$-0.372593\pi$
$41$	8388.28	0.779316	0.389658	+	0.920960i	$0.372593\pi$
$42$	0	0
$43$	20546.6	1.69461	0.847304	−	0.531108i	$-0.178224\pi$
$43$	20546.6	1.69461	0.847304	+	0.531108i	$0.178224\pi$
$44$	−10202.2	−0.794444
$45$	0	0
$46$	−8360.07	−0.582526
$47$	8352.17	0.551512	0.275756	−	0.961228i	$-0.411072\pi$
$47$	8352.17	0.551512	0.275756	+	0.961228i	$0.411072\pi$
$48$	0	0
$49$	−10340.3	−0.615236
$50$	−2200.50	−0.124479
$51$	0	0
$52$	−8266.29	−0.423938
$53$	28390.6	1.38831	0.694154	−	0.719827i	$-0.255779\pi$
$53$	28390.6	1.38831	0.694154	+	0.719827i	$0.255779\pi$
$54$	0	0
$55$	13010.4	0.579941
$56$	−14610.5	−0.622582
$57$	0	0
$58$	24844.9	0.969768
$59$	37500.4	1.40251	0.701254	−	0.712911i	$-0.252624\pi$
$59$	37500.4	1.40251	0.701254	+	0.712911i	$0.252624\pi$
$60$	0	0
$61$	−57311.1	−1.97203	−0.986017	−	0.166644i	$-0.946707\pi$
$61$	−57311.1	−1.97203	−0.986017	+	0.166644i	$0.946707\pi$
$62$	19579.7	0.646884
$63$	0	0
$64$	20712.1	0.632083
$65$	10541.6	0.309473
$66$	0	0
$67$	−45848.9	−1.24779	−0.623895	−	0.781508i	$-0.714451\pi$
$67$	−45848.9	−1.24779	−0.623895	+	0.781508i	$0.714451\pi$
$68$	−22792.9	−0.597760
$69$	0	0
$70$	7078.21	0.172655
$71$	41510.8	0.977271	0.488636	−	0.872488i	$-0.337495\pi$
$71$	41510.8	0.977271	0.488636	+	0.872488i	$0.337495\pi$
$72$	0	0
$73$	76806.5	1.68691	0.843453	−	0.537203i	$-0.180519\pi$
$73$	76806.5	1.68691	0.843453	+	0.537203i	$0.180519\pi$
$74$	−8414.52	−0.178628
$75$	0	0
$76$	−22949.5	−0.455763
$77$	−41849.7	−0.804389
$78$	0	0
$79$	−7008.26	−0.126341	−0.0631703	−	0.998003i	$-0.520121\pi$
$79$	−7008.26	−0.126341	−0.0631703	+	0.998003i	$0.520121\pi$
$80$	−308.904	−0.00539633
$81$	0	0
$82$	−29533.4	−0.485042
$83$	−93271.1	−1.48611	−0.743057	−	0.669229i	$-0.766625\pi$
$83$	−93271.1	−1.48611	−0.743057	+	0.669229i	$0.766625\pi$
$84$	0	0
$85$	29066.6	0.436363
$86$	−72340.5	−1.05472
$87$	0	0
$88$	94552.9	1.30157
$89$	7074.70	0.0946745	0.0473372	−	0.998879i	$-0.484926\pi$
$89$	7074.70	0.0946745	0.0473372	+	0.998879i	$0.484926\pi$
$90$	0	0
$91$	−33908.5	−0.429245
$92$	−46549.3	−0.573382
$93$	0	0
$94$	−29406.3	−0.343258
$95$	29266.4	0.332705
$96$	0	0
$97$	22429.9	0.242046	0.121023	−	0.992650i	$-0.461383\pi$
$97$	22429.9	0.242046	0.121023	+	0.992650i	$0.461383\pi$
$98$	36406.0	0.382920
$99$	0	0
$100$	−12252.5	−0.122525
$101$	−107340.	−1.04703	−0.523515	−	0.852017i	$-0.675379\pi$
$101$	−107340.	−1.04703	−0.523515	+	0.852017i	$0.675379\pi$
$102$	0	0
$103$	−14473.5	−0.134425	−0.0672125	−	0.997739i	$-0.521411\pi$
$103$	−14473.5	−0.134425	−0.0672125	+	0.997739i	$0.521411\pi$
$104$	76610.9	0.694556
$105$	0	0
$106$	−99957.7	−0.864075
$107$	47956.9	0.404941	0.202470	−	0.979288i	$-0.435103\pi$
$107$	47956.9	0.404941	0.202470	+	0.979288i	$0.435103\pi$
$108$	0	0
$109$	−67023.6	−0.540333	−0.270167	−	0.962814i	$-0.587079\pi$
$109$	−67023.6	−0.540333	−0.270167	+	0.962814i	$0.587079\pi$
$110$	−45807.0	−0.360952
$111$	0	0
$112$	993.631	0.00748480
$113$	−7799.34	−0.0574595	−0.0287297	−	0.999587i	$-0.509146\pi$
$113$	−7799.34	−0.0574595	−0.0287297	+	0.999587i	$0.509146\pi$
$114$	0	0
$115$	59362.0	0.418566
$116$	138338.	0.954545
$117$	0	0
$118$	−132031.	−0.872914
$119$	−93496.8	−0.605243
$120$	0	0
$121$	109782.	0.681658
$122$	201781.	1.22738
$123$	0	0
$124$	109021.	0.636729
$125$	15625.0	0.0894427
$126$	0	0
$127$	157992.	0.869214	0.434607	−	0.900620i	$-0.356887\pi$
$127$	157992.	0.869214	0.434607	+	0.900620i	$0.356887\pi$
$128$	111732.	0.602774
$129$	0	0
$130$	−37114.8	−0.192614
$131$	328644.	1.67320	0.836600	−	0.547814i	$-0.184540\pi$
$131$	328644.	1.67320	0.836600	+	0.547814i	$0.184540\pi$
$132$	0	0
$133$	−94139.3	−0.461468
$134$	161425.	0.776618
$135$	0	0
$136$	211241.	0.979336
$137$	−297430.	−1.35389	−0.676944	−	0.736034i	$-0.736696\pi$
$137$	−297430.	−1.35389	−0.676944	+	0.736034i	$0.736696\pi$
$138$	0	0
$139$	−16801.8	−0.0737594	−0.0368797	−	0.999320i	$-0.511742\pi$
$139$	−16801.8	−0.0737594	−0.0368797	+	0.999320i	$0.511742\pi$
$140$	39411.8	0.169944
$141$	0	0
$142$	−146151.	−0.608249
$143$	219441.	0.897381
$144$	0	0
$145$	−176416.	−0.696814
$146$	−270420.	−1.04992
$147$	0	0
$148$	−46852.5	−0.175824
$149$	−490724.	−1.81081	−0.905403	−	0.424553i	$-0.860431\pi$
$149$	−490724.	−1.81081	−0.905403	+	0.424553i	$0.860431\pi$
$150$	0	0
$151$	132845.	0.474137	0.237069	−	0.971493i	$-0.423813\pi$
$151$	132845.	0.474137	0.237069	+	0.971493i	$0.423813\pi$
$152$	212693.	0.746697
$153$	0	0
$154$	147344.	0.500647
$155$	−139029.	−0.464810
$156$	0	0
$157$	300751.	0.973772	0.486886	−	0.873465i	$-0.338133\pi$
$157$	300751.	0.973772	0.486886	+	0.873465i	$0.338133\pi$
$158$	24674.7	0.0786337
$159$	0	0
$160$	−144262.	−0.445505
$161$	−190946.	−0.580559
$162$	0	0
$163$	−663660.	−1.95649	−0.978243	−	0.207464i	$-0.933479\pi$
$163$	−663660.	−1.95649	−0.978243	+	0.207464i	$0.933479\pi$
$164$	−164444.	−0.477428
$165$	0	0
$166$	328389.	0.924949
$167$	−198828.	−0.551678	−0.275839	−	0.961204i	$-0.588956\pi$
$167$	−198828.	−0.551678	−0.275839	+	0.961204i	$0.588956\pi$
$168$	0	0
$169$	−193493.	−0.521132
$170$	−102338.	−0.271590
$171$	0	0
$172$	−402796.	−1.03816
$173$	−425570.	−1.08108	−0.540538	−	0.841320i	$-0.681779\pi$
$173$	−425570.	−1.08108	−0.540538	+	0.841320i	$0.681779\pi$
$174$	0	0
$175$	−50260.0	−0.124059
$176$	−6430.34	−0.0156478
$177$	0	0
$178$	−24908.6	−0.0589249
$179$	271380.	0.633061	0.316530	−	0.948582i	$-0.397482\pi$
$179$	271380.	0.633061	0.316530	+	0.948582i	$0.397482\pi$
$180$	0	0
$181$	390577.	0.886156	0.443078	−	0.896483i	$-0.353886\pi$
$181$	390577.	0.886156	0.443078	+	0.896483i	$0.353886\pi$
$182$	119385.	0.267160
$183$	0	0
$184$	431413.	0.939396
$185$	59748.7	0.128351
$186$	0	0
$187$	605070.	1.26532
$188$	−163736.	−0.337870
$189$	0	0
$190$	−103041.	−0.207074
$191$	185922.	0.368764	0.184382	−	0.982855i	$-0.440972\pi$
$191$	185922.	0.368764	0.184382	+	0.982855i	$0.440972\pi$
$192$	0	0
$193$	−774298.	−1.49629	−0.748143	−	0.663537i	$-0.769054\pi$
$193$	−774298.	−1.49629	−0.748143	+	0.663537i	$0.769054\pi$
$194$	−78971.1	−0.150648
$195$	0	0
$196$	202711.	0.376909
$197$	−39823.2	−0.0731090	−0.0365545	−	0.999332i	$-0.511638\pi$
$197$	−39823.2	−0.0731090	−0.0365545	+	0.999332i	$0.511638\pi$
$198$	0	0
$199$	−20033.4	−0.0358610	−0.0179305	−	0.999839i	$-0.505708\pi$
$199$	−20033.4	−0.0358610	−0.0179305	+	0.999839i	$0.505708\pi$
$200$	113554.	0.200738
$201$	0	0
$202$	377923.	0.651666
$203$	567465.	0.966493
$204$	0	0
$205$	209707.	0.348521
$206$	50958.2	0.0836654
$207$	0	0
$208$	−5210.14	−0.00835009
$209$	609228.	0.964748
$210$	0	0
$211$	−231466.	−0.357917	−0.178958	−	0.983857i	$-0.557273\pi$
$211$	−231466.	−0.357917	−0.178958	+	0.983857i	$0.557273\pi$
$212$	−556570.	−0.850511
$213$	0	0
$214$	−168846.	−0.252033
$215$	513666.	0.757852
$216$	0	0
$217$	447205.	0.644700
$218$	235977.	0.336301
$219$	0	0
$220$	−255056.	−0.355286
$221$	490254.	0.675212
$222$	0	0
$223$	134460.	0.181063	0.0905315	−	0.995894i	$-0.471143\pi$
$223$	134460.	0.181063	0.0905315	+	0.995894i	$0.471143\pi$
$224$	464039.	0.617923
$225$	0	0
$226$	27459.9	0.0357625
$227$	1.48703e6	1.91538	0.957690	−	0.287803i	$-0.0929248\pi$
$227$	1.48703e6	1.91538	0.957690	+	0.287803i	$0.0929248\pi$
$228$	0	0
$229$	−1.23314e6	−1.55390	−0.776950	−	0.629563i	$-0.783234\pi$
$229$	−1.23314e6	−1.55390	−0.776950	+	0.629563i	$0.783234\pi$
$230$	−209002.	−0.260514
$231$	0	0
$232$	−1.28210e6	−1.56387
$233$	−941066.	−1.13561	−0.567806	−	0.823162i	$-0.692208\pi$
$233$	−941066.	−1.13561	−0.567806	+	0.823162i	$0.692208\pi$
$234$	0	0
$235$	208804.	0.246644
$236$	−735157.	−0.859211
$237$	0	0
$238$	329183.	0.376700
$239$	835483.	0.946113	0.473056	−	0.881032i	$-0.343151\pi$
$239$	835483.	0.946113	0.473056	+	0.881032i	$0.343151\pi$
$240$	0	0
$241$	32145.1	0.0356511	0.0178255	−	0.999841i	$-0.494326\pi$
$241$	32145.1	0.0356511	0.0178255	+	0.999841i	$0.494326\pi$
$242$	−386519.	−0.424261
$243$	0	0
$244$	1.12353e6	1.20812
$245$	−258507.	−0.275142
$246$	0	0
$247$	493623.	0.514817
$248$	−1.01039e6	−1.04318
$249$	0	0
$250$	−55012.5	−0.0556687
$251$	−680645.	−0.681924	−0.340962	−	0.940077i	$-0.610753\pi$
$251$	−680645.	−0.681924	−0.340962	+	0.940077i	$0.610753\pi$
$252$	0	0
$253$	1.23572e6	1.21372
$254$	−556259.	−0.540995
$255$	0	0
$256$	−1.05617e6	−1.00725
$257$	−959946.	−0.906596	−0.453298	−	0.891359i	$-0.649753\pi$
$257$	−959946.	−0.906596	−0.453298	+	0.891359i	$0.649753\pi$
$258$	0	0
$259$	−192190.	−0.178025
$260$	−206657.	−0.189591
$261$	0	0
$262$	−1.15709e6	−1.04139
$263$	828601.	0.738680	0.369340	−	0.929294i	$-0.379584\pi$
$263$	828601.	0.738680	0.369340	+	0.929294i	$0.379584\pi$
$264$	0	0
$265$	709766.	0.620870
$266$	331446.	0.287216
$267$	0	0
$268$	898821.	0.764427
$269$	−1.47945e6	−1.24658	−0.623290	−	0.781991i	$-0.714204\pi$
$269$	−1.47945e6	−1.24658	−0.623290	+	0.781991i	$0.714204\pi$
$270$	0	0
$271$	−812554.	−0.672093	−0.336046	−	0.941845i	$-0.609090\pi$
$271$	−812554.	−0.672093	−0.336046	+	0.941845i	$0.609090\pi$
$272$	−14366.1	−0.0117738
$273$	0	0
$274$	1.04719e6	0.842653
$275$	325260.	0.259358
$276$	0	0
$277$	275667.	0.215866	0.107933	−	0.994158i	$-0.465577\pi$
$277$	275667.	0.215866	0.107933	+	0.994158i	$0.465577\pi$
$278$	59155.6	0.0459075
$279$	0	0
$280$	−365264.	−0.278427
$281$	−509442.	−0.384883	−0.192442	−	0.981308i	$-0.561641\pi$
$281$	−509442.	−0.384883	−0.192442	+	0.981308i	$0.561641\pi$
$282$	0	0
$283$	986519.	0.732216	0.366108	−	0.930572i	$-0.380690\pi$
$283$	986519.	0.732216	0.366108	+	0.930572i	$0.380690\pi$
$284$	−813777.	−0.598700
$285$	0	0
$286$	−772606.	−0.558525
$287$	−674552.	−0.483404
$288$	0	0
$289$	−68065.9	−0.0479386
$290$	621123.	0.433693
$291$	0	0
$292$	−1.50571e6	−1.03344
$293$	201092.	0.136844	0.0684219	−	0.997656i	$-0.478204\pi$
$293$	201092.	0.136844	0.0684219	+	0.997656i	$0.478204\pi$
$294$	0	0
$295$	937510.	0.627221
$296$	434223.	0.288060
$297$	0	0
$298$	1.72774e6	1.12704
$299$	1.00123e6	0.647675
$300$	0	0
$301$	−1.65228e6	−1.05115
$302$	−467722.	−0.295101
$303$	0	0
$304$	−14464.8	−0.00897694
$305$	−1.43278e6	−0.881920
$306$	0	0
$307$	−982845.	−0.595167	−0.297584	−	0.954696i	$-0.596181\pi$
$307$	−982845.	−0.595167	−0.297584	+	0.954696i	$0.596181\pi$
$308$	820422.	0.492788
$309$	0	0
$310$	489492.	0.289295
$311$	1.70689e6	1.00070	0.500351	−	0.865823i	$-0.333204\pi$
$311$	1.70689e6	1.00070	0.500351	+	0.865823i	$0.333204\pi$
$312$	0	0
$313$	−511933.	−0.295360	−0.147680	−	0.989035i	$-0.547181\pi$
$313$	−511933.	−0.295360	−0.147680	+	0.989035i	$0.547181\pi$
$314$	−1.05888e6	−0.606071
$315$	0	0
$316$	137390.	0.0773993
$317$	2.51314e6	1.40465	0.702325	−	0.711857i	$-0.252145\pi$
$317$	2.51314e6	1.40465	0.702325	+	0.711857i	$0.252145\pi$
$318$	0	0
$319$	−3.67238e6	−2.02055
$320$	517803.	0.282676
$321$	0	0
$322$	672283.	0.361337
$323$	1.36108e6	0.725901
$324$	0	0
$325$	263540.	0.138401
$326$	2.33661e6	1.21771
$327$	0	0
$328$	1.52404e6	0.782191
$329$	−671648.	−0.342099
$330$	0	0
$331$	−1.89334e6	−0.949857	−0.474928	−	0.880024i	$-0.657526\pi$
$331$	−1.89334e6	−0.949857	−0.474928	+	0.880024i	$0.657526\pi$
$332$	1.82849e6	0.910430
$333$	0	0
$334$	700032.	0.343361
$335$	−1.14622e6	−0.558029
$336$	0	0
$337$	3.29119e6	1.57862	0.789310	−	0.613994i	$-0.210438\pi$
$337$	3.29119e6	1.57862	0.789310	+	0.613994i	$0.210438\pi$
$338$	681248.	0.324350
$339$	0	0
$340$	−569822.	−0.267326
$341$	−2.89411e6	−1.34781
$342$	0	0
$343$	2.18307e6	1.00192
$344$	3.73306e6	1.70086
$345$	0	0
$346$	1.49835e6	0.672856
$347$	−182036.	−0.0811584	−0.0405792	−	0.999176i	$-0.512920\pi$
$347$	−182036.	−0.0811584	−0.0405792	+	0.999176i	$0.512920\pi$
$348$	0	0
$349$	796176.	0.349901	0.174951	−	0.984577i	$-0.444023\pi$
$349$	796176.	0.349901	0.174951	+	0.984577i	$0.444023\pi$
$350$	176955.	0.0772135
$351$	0	0
$352$	−3.00305e6	−1.29183
$353$	727614.	0.310788	0.155394	−	0.987853i	$-0.450335\pi$
$353$	727614.	0.310788	0.155394	+	0.987853i	$0.450335\pi$
$354$	0	0
$355$	1.03777e6	0.437049
$356$	−138692.	−0.0579999
$357$	0	0
$358$	−955475.	−0.394014
$359$	−2.55650e6	−1.04691	−0.523454	−	0.852054i	$-0.675357\pi$
$359$	−2.55650e6	−1.04691	−0.523454	+	0.852054i	$0.675357\pi$
$360$	0	0
$361$	−1.10567e6	−0.446535
$362$	−1.37514e6	−0.551539
$363$	0	0
$364$	664742.	0.262966
$365$	1.92016e6	0.754407
$366$	0	0
$367$	2.15447e6	0.834977	0.417489	−	0.908682i	$-0.362910\pi$
$367$	2.15447e6	0.834977	0.417489	+	0.908682i	$0.362910\pi$
$368$	−29339.5	−0.0112936
$369$	0	0
$370$	−210363.	−0.0798850
$371$	−2.28306e6	−0.861158
$372$	0	0
$373$	−3.61293e6	−1.34458	−0.672291	−	0.740287i	$-0.734689\pi$
$373$	−3.61293e6	−1.34458	−0.672291	+	0.740287i	$0.734689\pi$
$374$	−2.13033e6	−0.787531
$375$	0	0
$376$	1.51748e6	0.553547
$377$	−2.97552e6	−1.07823
$378$	0	0
$379$	660411.	0.236165	0.118083	−	0.993004i	$-0.462325\pi$
$379$	660411.	0.236165	0.118083	+	0.993004i	$0.462325\pi$
$380$	−573738.	−0.203824
$381$	0	0
$382$	−654595.	−0.229517
$383$	−5.10573e6	−1.77853	−0.889264	−	0.457395i	$-0.848783\pi$
$383$	−5.10573e6	−1.77853	−0.889264	+	0.457395i	$0.848783\pi$
$384$	0	0
$385$	−1.04624e6	−0.359734
$386$	2.72615e6	0.931281
$387$	0	0
$388$	−439715.	−0.148283
$389$	492425.	0.164993	0.0824966	−	0.996591i	$-0.473711\pi$
$389$	492425.	0.164993	0.0824966	+	0.996591i	$0.473711\pi$
$390$	0	0
$391$	2.76073e6	0.913233
$392$	−1.87870e6	−0.617506
$393$	0	0
$394$	140210.	0.0455027
$395$	−175207.	−0.0565012
$396$	0	0
$397$	−2.72078e6	−0.866397	−0.433199	−	0.901298i	$-0.642615\pi$
$397$	−2.72078e6	−0.866397	−0.433199	+	0.901298i	$0.642615\pi$
$398$	70533.6	0.0223197
$399$	0	0
$400$	−7722.59	−0.00241331
$401$	204131.	0.0633941	0.0316970	−	0.999498i	$-0.489909\pi$
$401$	204131.	0.0633941	0.0316970	+	0.999498i	$0.489909\pi$
$402$	0	0
$403$	−2.34494e6	−0.719231
$404$	2.10429e6	0.641436
$405$	0	0
$406$	−1.99793e6	−0.601541
$407$	1.24377e6	0.372180
$408$	0	0
$409$	−468364.	−0.138444	−0.0692221	−	0.997601i	$-0.522052\pi$
$409$	−468364.	−0.138444	−0.0692221	+	0.997601i	$0.522052\pi$
$410$	−738336.	−0.216917
$411$	0	0
$412$	283738.	0.0823521
$413$	−3.01563e6	−0.869967
$414$	0	0
$415$	−2.33178e6	−0.664610
$416$	−2.43321e6	−0.689359
$417$	0	0
$418$	−2.14497e6	−0.600454
$419$	2.45205e6	0.682328	0.341164	−	0.940004i	$-0.389179\pi$
$419$	2.45205e6	0.682328	0.341164	+	0.940004i	$0.389179\pi$
$420$	0	0
$421$	5.51677e6	1.51698	0.758489	−	0.651686i	$-0.225938\pi$
$421$	5.51677e6	1.51698	0.758489	+	0.651686i	$0.225938\pi$
$422$	814946.	0.222765
$423$	0	0
$424$	5.15822e6	1.39343
$425$	726666.	0.195147
$426$	0	0
$427$	4.60873e6	1.22324
$428$	−940146.	−0.248077
$429$	0	0
$430$	−1.80851e6	−0.471683
$431$	−2.25875e6	−0.585701	−0.292850	−	0.956158i	$-0.594604\pi$
$431$	−2.25875e6	−0.585701	−0.292850	+	0.956158i	$0.594604\pi$
$432$	0	0
$433$	−2.02853e6	−0.519951	−0.259976	−	0.965615i	$-0.583715\pi$
$433$	−2.02853e6	−0.519951	−0.259976	+	0.965615i	$0.583715\pi$
$434$	−1.57452e6	−0.401258
$435$	0	0
$436$	1.31393e6	0.331021
$437$	2.77970e6	0.696297
$438$	0	0
$439$	−1.74449e6	−0.432024	−0.216012	−	0.976391i	$-0.569305\pi$
$439$	−1.74449e6	−0.432024	−0.216012	+	0.976391i	$0.569305\pi$
$440$	2.36382e6	0.582081
$441$	0	0
$442$	−1.72608e6	−0.420249
$443$	987200.	0.238999	0.119499	−	0.992834i	$-0.461871\pi$
$443$	987200.	0.238999	0.119499	+	0.992834i	$0.461871\pi$
$444$	0	0
$445$	176867.	0.0423397
$446$	−473405.	−0.112693
$447$	0	0
$448$	−1.66558e6	−0.392077
$449$	7.78246e6	1.82180	0.910901	−	0.412624i	$-0.135388\pi$
$449$	7.78246e6	1.82180	0.910901	+	0.412624i	$0.135388\pi$
$450$	0	0
$451$	4.36540e6	1.01061
$452$	152898.	0.0352011
$453$	0	0
$454$	−5.23553e6	−1.19212
$455$	−847712.	−0.191964
$456$	0	0
$457$	4.01729e6	0.899793	0.449896	−	0.893081i	$-0.351461\pi$
$457$	4.01729e6	0.899793	0.449896	+	0.893081i	$0.351461\pi$
$458$	4.34163e6	0.967139
$459$	0	0
$460$	−1.16373e6	−0.256424
$461$	1.42822e6	0.313000	0.156500	−	0.987678i	$-0.449979\pi$
$461$	1.42822e6	0.313000	0.156500	+	0.987678i	$0.449979\pi$
$462$	0	0
$463$	−2.56420e6	−0.555904	−0.277952	−	0.960595i	$-0.589656\pi$
$463$	−2.56420e6	−0.555904	−0.277952	+	0.960595i	$0.589656\pi$
$464$	87192.7	0.0188012
$465$	0	0
$466$	3.31330e6	0.706800
$467$	2.07027e6	0.439272	0.219636	−	0.975582i	$-0.429513\pi$
$467$	2.07027e6	0.439272	0.219636	+	0.975582i	$0.429513\pi$
$468$	0	0
$469$	3.68698e6	0.773996
$470$	−735158.	−0.153510
$471$	0	0
$472$	6.81334e6	1.40768
$473$	1.06928e7	2.19755
$474$	0	0
$475$	731659.	0.148790
$476$	1.83291e6	0.370787
$477$	0	0
$478$	−2.94157e6	−0.588856
$479$	−3.78107e6	−0.752967	−0.376484	−	0.926423i	$-0.622867\pi$
$479$	−3.78107e6	−0.752967	−0.376484	+	0.926423i	$0.622867\pi$
$480$	0	0
$481$	1.00775e6	0.198606
$482$	−113176.	−0.0221890
$483$	0	0
$484$	−2.15216e6	−0.417601
$485$	560747.	0.108246
$486$	0	0
$487$	−7.73232e6	−1.47736	−0.738682	−	0.674054i	$-0.764551\pi$
$487$	−7.73232e6	−1.47736	−0.738682	+	0.674054i	$0.764551\pi$
$488$	−1.04127e7	−1.97931
$489$	0	0
$490$	910150.	0.171247
$491$	8.20848e6	1.53659	0.768296	−	0.640094i	$-0.221105\pi$
$491$	8.20848e6	1.53659	0.768296	+	0.640094i	$0.221105\pi$
$492$	0	0
$493$	−8.20449e6	−1.52032
$494$	−1.73795e6	−0.320419
$495$	0	0
$496$	68714.4	0.0125413
$497$	−3.33813e6	−0.606195
$498$	0	0
$499$	58456.0	0.0105094	0.00525470	−	0.999986i	$-0.498327\pi$
$499$	58456.0	0.0105094	0.00525470	+	0.999986i	$0.498327\pi$
$500$	−306312.	−0.0547948
$501$	0	0
$502$	2.39641e6	0.424426
$503$	−4.83775e6	−0.852557	−0.426278	−	0.904592i	$-0.640176\pi$
$503$	−4.83775e6	−0.852557	−0.426278	+	0.904592i	$0.640176\pi$
$504$	0	0
$505$	−2.68350e6	−0.468246
$506$	−4.35071e6	−0.755413
$507$	0	0
$508$	−3.09728e6	−0.532502
$509$	−1.42306e6	−0.243460	−0.121730	−	0.992563i	$-0.538844\pi$
$509$	−1.42306e6	−0.243460	−0.121730	+	0.992563i	$0.538844\pi$
$510$	0	0
$511$	−6.17647e6	−1.04638
$512$	143140.	0.0241315
$513$	0	0
$514$	3.37977e6	0.564261
$515$	−361837.	−0.0601167
$516$	0	0
$517$	4.34660e6	0.715194
$518$	676662.	0.110802
$519$	0	0
$520$	1.91527e6	0.310615
$521$	5.26166e6	0.849236	0.424618	−	0.905373i	$-0.360408\pi$
$521$	5.26166e6	0.849236	0.424618	+	0.905373i	$0.360408\pi$
$522$	0	0
$523$	6.53462e6	1.04464	0.522319	−	0.852750i	$-0.325067\pi$
$523$	6.53462e6	1.04464	0.522319	+	0.852750i	$0.325067\pi$
$524$	−6.44274e6	−1.02504
$525$	0	0
$526$	−2.91734e6	−0.459751
$527$	−6.46576e6	−1.01413
$528$	0	0
$529$	−798179.	−0.124011
$530$	−2.49894e6	−0.386426
$531$	0	0
$532$	1.84551e6	0.282707
$533$	3.53704e6	0.539289
$534$	0	0
$535$	1.19892e6	0.181095
$536$	−8.33015e6	−1.25239
$537$	0	0
$538$	5.20885e6	0.775865
$539$	−5.38124e6	−0.797831
$540$	0	0
$541$	9.38001e6	1.37788	0.688938	−	0.724820i	$-0.258077\pi$
$541$	9.38001e6	1.37788	0.688938	+	0.724820i	$0.258077\pi$
$542$	2.86084e6	0.418307
$543$	0	0
$544$	−6.70915e6	−0.972008
$545$	−1.67559e6	−0.241644
$546$	0	0
$547$	−7.75854e6	−1.10869	−0.554347	−	0.832286i	$-0.687032\pi$
$547$	−7.75854e6	−1.10869	−0.554347	+	0.832286i	$0.687032\pi$
$548$	5.83081e6	0.829425
$549$	0	0
$550$	−1.14517e6	−0.161423
$551$	−8.26087e6	−1.15917
$552$	0	0
$553$	563576.	0.0783682
$554$	−970567.	−0.134354
$555$	0	0
$556$	329381.	0.0451868
$557$	7.61675e6	1.04024	0.520118	−	0.854094i	$-0.325888\pi$
$557$	7.61675e6	1.04024	0.520118	+	0.854094i	$0.325888\pi$
$558$	0	0
$559$	8.66377e6	1.17267
$560$	24840.8	0.00334730
$561$	0	0
$562$	1.79364e6	0.239549
$563$	2.87806e6	0.382674	0.191337	−	0.981524i	$-0.438718\pi$
$563$	2.87806e6	0.382674	0.191337	+	0.981524i	$0.438718\pi$
$564$	0	0
$565$	−194983.	−0.0256967
$566$	−3.47333e6	−0.455728
$567$	0	0
$568$	7.54198e6	0.980876
$569$	−6.23789e6	−0.807713	−0.403856	−	0.914822i	$-0.632330\pi$
$569$	−6.23789e6	−0.807713	−0.403856	+	0.914822i	$0.632330\pi$
$570$	0	0
$571$	−1.14465e7	−1.46920	−0.734601	−	0.678499i	$-0.762631\pi$
$571$	−1.14465e7	−1.46920	−0.734601	+	0.678499i	$0.762631\pi$
$572$	−4.30191e6	−0.549758
$573$	0	0
$574$	2.37496e6	0.300868
$575$	1.48405e6	0.187189
$576$	0	0
$577$	−1.48097e7	−1.85186	−0.925928	−	0.377700i	$-0.876715\pi$
$577$	−1.48097e7	−1.85186	−0.925928	+	0.377700i	$0.876715\pi$
$578$	239646.	0.0298367
$579$	0	0
$580$	3.45845e6	0.426885
$581$	7.50048e6	0.921826
$582$	0	0
$583$	1.47749e7	1.80034
$584$	1.39548e7	1.69313
$585$	0	0
$586$	−708003.	−0.0851708
$587$	−7.19551e6	−0.861919	−0.430959	−	0.902371i	$-0.641825\pi$
$587$	−7.19551e6	−0.861919	−0.430959	+	0.902371i	$0.641825\pi$
$588$	0	0
$589$	−6.51019e6	−0.773224
$590$	−3.30078e6	−0.390379
$591$	0	0
$592$	−29530.6	−0.00346312
$593$	−2.20920e6	−0.257987	−0.128994	−	0.991645i	$-0.541175\pi$
$593$	−2.20920e6	−0.257987	−0.128994	+	0.991645i	$0.541175\pi$
$594$	0	0
$595$	−2.33742e6	−0.270673
$596$	9.62015e6	1.10934
$597$	0	0
$598$	−3.52514e6	−0.403110
$599$	−309553.	−0.0352508	−0.0176254	−	0.999845i	$-0.505611\pi$
$599$	−309553.	−0.0352508	−0.0176254	+	0.999845i	$0.505611\pi$
$600$	0	0
$601$	−5.93532e6	−0.670283	−0.335141	−	0.942168i	$-0.608784\pi$
$601$	−5.93532e6	−0.670283	−0.335141	+	0.942168i	$0.608784\pi$
$602$	5.81733e6	0.654233
$603$	0	0
$604$	−2.60430e6	−0.290468
$605$	2.74454e6	0.304847
$606$	0	0
$607$	4.30961e6	0.474751	0.237376	−	0.971418i	$-0.423713\pi$
$607$	4.30961e6	0.474751	0.237376	+	0.971418i	$0.423713\pi$
$608$	−6.75525e6	−0.741110
$609$	0	0
$610$	5.04452e6	0.548903
$611$	3.52181e6	0.381648
$612$	0	0
$613$	1.06508e7	1.14480	0.572402	−	0.819973i	$-0.306012\pi$
$613$	1.06508e7	1.14480	0.572402	+	0.819973i	$0.306012\pi$
$614$	3.46040e6	0.370429
$615$	0	0
$616$	−7.60356e6	−0.807356
$617$	1.57352e7	1.66403	0.832014	−	0.554754i	$-0.187188\pi$
$617$	1.57352e7	1.66403	0.832014	+	0.554754i	$0.187188\pi$
$618$	0	0
$619$	3.90140e6	0.409255	0.204627	−	0.978840i	$-0.434402\pi$
$619$	3.90140e6	0.409255	0.204627	+	0.978840i	$0.434402\pi$
$620$	2.72552e6	0.284754
$621$	0	0
$622$	−6.00962e6	−0.622832
$623$	−568918.	−0.0587259
$624$	0	0
$625$	390625.	0.0400000
$626$	1.80241e6	0.183831
$627$	0	0
$628$	−5.89591e6	−0.596557
$629$	2.77871e6	0.280038
$630$	0	0
$631$	−1.35074e7	−1.35051	−0.675254	−	0.737586i	$-0.735966\pi$
$631$	−1.35074e7	−1.35051	−0.675254	+	0.737586i	$0.735966\pi$
$632$	−1.27331e6	−0.126807
$633$	0	0
$634$	−8.84825e6	−0.874247
$635$	3.94981e6	0.388724
$636$	0	0
$637$	−4.36012e6	−0.425745
$638$	1.29297e7	1.25758
$639$	0	0
$640$	2.79331e6	0.269569
$641$	7.66035e6	0.736382	0.368191	−	0.929750i	$-0.379977\pi$
$641$	7.66035e6	0.736382	0.368191	+	0.929750i	$0.379977\pi$
$642$	0	0
$643$	1.12191e7	1.07012	0.535060	−	0.844814i	$-0.320289\pi$
$643$	1.12191e7	1.07012	0.535060	+	0.844814i	$0.320289\pi$
$644$	3.74331e6	0.355665
$645$	0	0
$646$	−4.79209e6	−0.451797
$647$	−1.67988e7	−1.57767	−0.788835	−	0.614604i	$-0.789316\pi$
$647$	−1.67988e7	−1.57767	−0.788835	+	0.614604i	$0.789316\pi$
$648$	0	0
$649$	1.95158e7	1.81876
$650$	−927870.	−0.0861398
$651$	0	0
$652$	1.30104e7	1.19859
$653$	−9.92579e6	−0.910924	−0.455462	−	0.890255i	$-0.650526\pi$
$653$	−9.92579e6	−0.910924	−0.455462	+	0.890255i	$0.650526\pi$
$654$	0	0
$655$	8.21611e6	0.748278
$656$	−103647.	−0.00940365
$657$	0	0
$658$	2.36474e6	0.212921
$659$	7.26028e6	0.651238	0.325619	−	0.945501i	$-0.394427\pi$
$659$	7.26028e6	0.651238	0.325619	+	0.945501i	$0.394427\pi$
$660$	0	0
$661$	4.89224e6	0.435516	0.217758	−	0.976003i	$-0.430126\pi$
$661$	4.89224e6	0.435516	0.217758	+	0.976003i	$0.430126\pi$
$662$	6.66606e6	0.591186
$663$	0	0
$664$	−1.69462e7	−1.49160
$665$	−2.35348e6	−0.206375
$666$	0	0
$667$	−1.67558e7	−1.45831
$668$	3.89781e6	0.337971
$669$	0	0
$670$	4.03561e6	0.347314
$671$	−2.98256e7	−2.55731
$672$	0	0
$673$	8.33639e6	0.709481	0.354740	−	0.934965i	$-0.384569\pi$
$673$	8.33639e6	0.709481	0.354740	+	0.934965i	$0.384569\pi$
$674$	−1.15876e7	−0.982526
$675$	0	0
$676$	3.79323e6	0.319258
$677$	−1.56824e7	−1.31505	−0.657525	−	0.753433i	$-0.728396\pi$
$677$	−1.56824e7	−1.31505	−0.657525	+	0.753433i	$0.728396\pi$
$678$	0	0
$679$	−1.80372e6	−0.150140
$680$	5.28104e6	0.437972
$681$	0	0
$682$	1.01896e7	0.838871
$683$	−1.39324e7	−1.14281	−0.571407	−	0.820667i	$-0.693602\pi$
$683$	−1.39324e7	−1.14281	−0.571407	+	0.820667i	$0.693602\pi$
$684$	0	0
$685$	−7.43574e6	−0.605477
$686$	−7.68616e6	−0.623590
$687$	0	0
$688$	−253877.	−0.0204481
$689$	1.19713e7	0.960713
$690$	0	0
$691$	−3.60504e6	−0.287220	−0.143610	−	0.989634i	$-0.545871\pi$
$691$	−3.60504e6	−0.287220	−0.143610	+	0.989634i	$0.545871\pi$
$692$	8.34288e6	0.662294
$693$	0	0
$694$	640912.	0.0505126
$695$	−420044.	−0.0329862
$696$	0	0
$697$	9.75277e6	0.760407
$698$	−2.80318e6	−0.217777
$699$	0	0
$700$	985296.	0.0760014
$701$	−5.10596e6	−0.392448	−0.196224	−	0.980559i	$-0.562868\pi$
$701$	−5.10596e6	−0.392448	−0.196224	+	0.980559i	$0.562868\pi$
$702$	0	0
$703$	2.79781e6	0.213515
$704$	1.07789e7	0.819678
$705$	0	0
$706$	−2.56178e6	−0.193433
$707$	8.63186e6	0.649465
$708$	0	0
$709$	−8.89610e6	−0.664637	−0.332318	−	0.943167i	$-0.607831\pi$
$709$	−8.89610e6	−0.664637	−0.332318	+	0.943167i	$0.607831\pi$
$710$	−3.65378e6	−0.272017
$711$	0	0
$712$	1.28538e6	0.0950237
$713$	−1.32049e7	−0.972769
$714$	0	0
$715$	5.48601e6	0.401321
$716$	−5.32013e6	−0.387829
$717$	0	0
$718$	9.00090e6	0.651591
$719$	−1.71802e7	−1.23938	−0.619691	−	0.784846i	$-0.712742\pi$
$719$	−1.71802e7	−1.23938	−0.619691	+	0.784846i	$0.712742\pi$
$720$	0	0
$721$	1.16390e6	0.0833829
$722$	3.89282e6	0.277921
$723$	0	0
$724$	−7.65687e6	−0.542881
$725$	−4.41039e6	−0.311625
$726$	0	0
$727$	−1.48569e7	−1.04254	−0.521270	−	0.853392i	$-0.674542\pi$
$727$	−1.48569e7	−1.04254	−0.521270	+	0.853392i	$0.674542\pi$
$728$	−6.16074e6	−0.430828
$729$	0	0
$730$	−6.76050e6	−0.469539
$731$	2.38888e7	1.65349
$732$	0	0
$733$	−1.97878e7	−1.36031	−0.680154	−	0.733069i	$-0.738087\pi$
$733$	−1.97878e7	−1.36031	−0.680154	+	0.733069i	$0.738087\pi$
$734$	−7.58544e6	−0.519686
$735$	0	0
$736$	−1.37019e7	−0.932367
$737$	−2.38605e7	−1.61812
$738$	0	0
$739$	1.26011e7	0.848783	0.424392	−	0.905479i	$-0.360488\pi$
$739$	1.26011e7	0.848783	0.424392	+	0.905479i	$0.360488\pi$
$740$	−1.17131e6	−0.0786310
$741$	0	0
$742$	8.03819e6	0.535980
$743$	1.19632e7	0.795014	0.397507	−	0.917599i	$-0.369875\pi$
$743$	1.19632e7	0.795014	0.397507	+	0.917599i	$0.369875\pi$
$744$	0	0
$745$	−1.22681e7	−0.809817
$746$	1.27204e7	0.836860
$747$	0	0
$748$	−1.18618e7	−0.775168
$749$	−3.85650e6	−0.251182
$750$	0	0
$751$	1.44504e7	0.934933	0.467466	−	0.884011i	$-0.345167\pi$
$751$	1.44504e7	0.934933	0.467466	+	0.884011i	$0.345167\pi$
$752$	−103201.	−0.00665485
$753$	0	0
$754$	1.04762e7	0.671082
$755$	3.32113e6	0.212041
$756$	0	0
$757$	−2.08448e7	−1.32208	−0.661040	−	0.750351i	$-0.729885\pi$
$757$	−2.08448e7	−1.32208	−0.661040	+	0.750351i	$0.729885\pi$
$758$	−2.32517e6	−0.146988
$759$	0	0
$760$	5.31733e6	0.333933
$761$	−6.27907e6	−0.393037	−0.196518	−	0.980500i	$-0.562964\pi$
$761$	−6.27907e6	−0.393037	−0.196518	+	0.980500i	$0.562964\pi$
$762$	0	0
$763$	5.38977e6	0.335165
$764$	−3.64482e6	−0.225914
$765$	0	0
$766$	1.79762e7	1.10695
$767$	1.58126e7	0.970540
$768$	0	0
$769$	6.28302e6	0.383136	0.191568	−	0.981479i	$-0.438643\pi$
$769$	6.28302e6	0.383136	0.191568	+	0.981479i	$0.438643\pi$
$770$	3.68361e6	0.223896
$771$	0	0
$772$	1.51793e7	0.916662
$773$	2.10517e7	1.26718	0.633591	−	0.773668i	$-0.281580\pi$
$773$	2.10517e7	1.26718	0.633591	+	0.773668i	$0.281580\pi$
$774$	0	0
$775$	−3.47572e6	−0.207869
$776$	4.07522e6	0.242939
$777$	0	0
$778$	−1.73373e6	−0.102691
$779$	9.81979e6	0.579774
$780$	0	0
$781$	2.16029e7	1.26731
$782$	−9.71996e6	−0.568392
$783$	0	0
$784$	127766.	0.00742378
$785$	7.51877e6	0.435484
$786$	0	0
$787$	−1.95429e6	−0.112474	−0.0562370	−	0.998417i	$-0.517910\pi$
$787$	−1.95429e6	−0.112474	−0.0562370	+	0.998417i	$0.517910\pi$
$788$	780694.	0.0447884
$789$	0	0
$790$	616867.	0.0351661
$791$	627191.	0.0356417
$792$	0	0
$793$	−2.41660e7	−1.36465
$794$	9.57931e6	0.539241
$795$	0	0
$796$	392735.	0.0219693
$797$	1.56292e7	0.871547	0.435774	−	0.900056i	$-0.356475\pi$
$797$	1.56292e7	0.871547	0.435774	+	0.900056i	$0.356475\pi$
$798$	0	0
$799$	9.71078e6	0.538130
$800$	−3.60655e6	−0.199236
$801$	0	0
$802$	−718705.	−0.0394562
$803$	3.99713e7	2.18756
$804$	0	0
$805$	−4.77366e6	−0.259634
$806$	8.25604e6	0.447646
$807$	0	0
$808$	−1.95023e7	−1.05089
$809$	−8.57518e6	−0.460651	−0.230326	−	0.973114i	$-0.573979\pi$
$809$	−8.57518e6	−0.460651	−0.230326	+	0.973114i	$0.573979\pi$
$810$	0	0
$811$	−5.81206e6	−0.310297	−0.155149	−	0.987891i	$-0.549586\pi$
$811$	−5.81206e6	−0.310297	−0.155149	+	0.987891i	$0.549586\pi$
$812$	−1.11246e7	−0.592098
$813$	0	0
$814$	−4.37905e6	−0.231643
$815$	−1.65915e7	−0.874967
$816$	0	0
$817$	2.40530e7	1.26071
$818$	1.64901e6	0.0861670
$819$	0	0
$820$	−4.11110e6	−0.213512
$821$	8.43637e6	0.436815	0.218408	−	0.975858i	$-0.429914\pi$
$821$	8.43637e6	0.436815	0.218408	+	0.975858i	$0.429914\pi$
$822$	0	0
$823$	−3.48078e6	−0.179133	−0.0895667	−	0.995981i	$-0.528548\pi$
$823$	−3.48078e6	−0.179133	−0.0895667	+	0.995981i	$0.528548\pi$
$824$	−2.62965e6	−0.134921
$825$	0	0
$826$	1.06174e7	0.541463
$827$	3.76000e7	1.91172	0.955861	−	0.293820i	$-0.0949268\pi$
$827$	3.76000e7	1.91172	0.955861	+	0.293820i	$0.0949268\pi$
$828$	0	0
$829$	1.78630e7	0.902752	0.451376	−	0.892334i	$-0.350933\pi$
$829$	1.78630e7	0.902752	0.451376	+	0.892334i	$0.350933\pi$
$830$	8.20972e6	0.413650
$831$	0	0
$832$	8.73355e6	0.437404
$833$	−1.20223e7	−0.600308
$834$	0	0
$835$	−4.97069e6	−0.246718
$836$	−1.19433e7	−0.591028
$837$	0	0
$838$	−8.63315e6	−0.424678
$839$	−1.01511e7	−0.497861	−0.248931	−	0.968521i	$-0.580079\pi$
$839$	−1.01511e7	−0.497861	−0.248931	+	0.968521i	$0.580079\pi$
$840$	0	0
$841$	2.92847e7	1.42775
$842$	−1.94234e7	−0.944160
$843$	0	0
$844$	4.53766e6	0.219269
$845$	−4.83732e6	−0.233057
$846$	0	0
$847$	−8.82820e6	−0.422828
$848$	−350799.	−0.0167521
$849$	0	0
$850$	−2.55844e6	−0.121459
$851$	5.67489e6	0.268617
$852$	0	0
$853$	−2.57022e7	−1.20948	−0.604738	−	0.796425i	$-0.706722\pi$
$853$	−2.57022e7	−1.20948	−0.604738	+	0.796425i	$0.706722\pi$
$854$	−1.62264e7	−0.761338
$855$	0	0
$856$	8.71315e6	0.406435
$857$	2.47250e7	1.14996	0.574982	−	0.818166i	$-0.305009\pi$
$857$	2.47250e7	1.14996	0.574982	+	0.818166i	$0.305009\pi$
$858$	0	0
$859$	−4.01446e7	−1.85628	−0.928141	−	0.372228i	$-0.878594\pi$
$859$	−4.01446e7	−1.85628	−0.928141	+	0.372228i	$0.878594\pi$
$860$	−1.00699e7	−0.464279
$861$	0	0
$862$	7.95261e6	0.364537
$863$	−3.30616e7	−1.51111	−0.755556	−	0.655084i	$-0.772633\pi$
$863$	−3.30616e7	−1.51111	−0.755556	+	0.655084i	$0.772633\pi$
$864$	0	0
$865$	−1.06393e7	−0.483472
$866$	7.14206e6	0.323615
$867$	0	0
$868$	−8.76700e6	−0.394959
$869$	−3.64721e6	−0.163837
$870$	0	0
$871$	−1.93328e7	−0.863475
$872$	−1.21773e7	−0.542327
$873$	0	0
$874$	−9.78676e6	−0.433371
$875$	−1.25650e6	−0.0554807
$876$	0	0
$877$	2.39879e7	1.05316	0.526579	−	0.850126i	$-0.323475\pi$
$877$	2.39879e7	1.05316	0.526579	+	0.850126i	$0.323475\pi$
$878$	6.14200e6	0.268889
$879$	0	0
$880$	−160758.	−0.00699789
$881$	−3.84976e7	−1.67107	−0.835534	−	0.549439i	$-0.814842\pi$
$881$	−3.84976e7	−1.67107	−0.835534	+	0.549439i	$0.814842\pi$
$882$	0	0
$883$	2.51833e7	1.08695	0.543476	−	0.839425i	$-0.317108\pi$
$883$	2.51833e7	1.08695	0.543476	+	0.839425i	$0.317108\pi$
$884$	−9.61093e6	−0.413652
$885$	0	0
$886$	−3.47573e6	−0.148752
$887$	−3.20905e7	−1.36952	−0.684759	−	0.728770i	$-0.740092\pi$
$887$	−3.20905e7	−1.36952	−0.684759	+	0.728770i	$0.740092\pi$
$888$	0	0
$889$	−1.27051e7	−0.539168
$890$	−622714.	−0.0263520
$891$	0	0
$892$	−2.63594e6	−0.110924
$893$	9.77751e6	0.410298
$894$	0	0
$895$	6.78450e6	0.283113
$896$	−8.98507e6	−0.373896
$897$	0	0
$898$	−2.74005e7	−1.13388
$899$	3.92429e7	1.61943
$900$	0	0
$901$	3.30088e7	1.35462
$902$	−1.53697e7	−0.628997
$903$	0	0
$904$	−1.41704e6	−0.0576715
$905$	9.76443e6	0.396301
$906$	0	0
$907$	−463821.	−0.0187211	−0.00936057	−	0.999956i	$-0.502980\pi$
$907$	−463821.	−0.0187211	−0.00936057	+	0.999956i	$0.502980\pi$
$908$	−2.91517e7	−1.17341
$909$	0	0
$910$	2.98462e6	0.119477
$911$	2.98906e7	1.19327	0.596634	−	0.802513i	$-0.296504\pi$
$911$	2.98906e7	1.19327	0.596634	+	0.802513i	$0.296504\pi$
$912$	0	0
$913$	−4.85398e7	−1.92717
$914$	−1.41441e7	−0.560026
$915$	0	0
$916$	2.41744e7	0.951957
$917$	−2.64282e7	−1.03787
$918$	0	0
$919$	3.02895e7	1.18305	0.591524	−	0.806287i	$-0.298526\pi$
$919$	3.02895e7	1.18305	0.591524	+	0.806287i	$0.298526\pi$
$920$	1.07853e7	0.420111
$921$	0	0
$922$	−5.02849e6	−0.194809
$923$	1.75036e7	0.676274
$924$	0	0
$925$	1.49372e6	0.0574003
$926$	9.02803e6	0.345992
$927$	0	0
$928$	4.07201e7	1.55217
$929$	−2.64394e7	−1.00511	−0.502554	−	0.864546i	$-0.667606\pi$
$929$	−2.64394e7	−1.00511	−0.502554	+	0.864546i	$0.667606\pi$
$930$	0	0
$931$	−1.21049e7	−0.457706
$932$	1.84486e7	0.695704
$933$	0	0
$934$	−7.28899e6	−0.273401
$935$	1.51267e7	0.565870
$936$	0	0
$937$	1.87274e7	0.696833	0.348416	−	0.937340i	$-0.386720\pi$
$937$	1.87274e7	0.696833	0.348416	+	0.937340i	$0.386720\pi$
$938$	−1.29811e7	−0.481731
$939$	0	0
$940$	−4.09340e6	−0.151100
$941$	−2.91313e7	−1.07247	−0.536236	−	0.844068i	$-0.680154\pi$
$941$	−2.91313e7	−1.07247	−0.536236	+	0.844068i	$0.680154\pi$
$942$	0	0
$943$	1.99178e7	0.729395
$944$	−463360.	−0.0169234
$945$	0	0
$946$	−3.76471e7	−1.36774
$947$	9.75601e6	0.353506	0.176753	−	0.984255i	$-0.443441\pi$
$947$	9.75601e6	0.353506	0.176753	+	0.984255i	$0.443441\pi$
$948$	0	0
$949$	3.23865e7	1.16734
$950$	−2.57602e6	−0.0926064
$951$	0	0
$952$	−1.69872e7	−0.607476
$953$	4.75019e6	0.169425	0.0847127	−	0.996405i	$-0.473003\pi$
$953$	4.75019e6	0.169425	0.0847127	+	0.996405i	$0.473003\pi$
$954$	0	0
$955$	4.64806e6	0.164916
$956$	−1.63788e7	−0.579612
$957$	0	0
$958$	1.33124e7	0.468643
$959$	2.39181e7	0.839808
$960$	0	0
$961$	2.29725e6	0.0802415
$962$	−3.54810e6	−0.123611
$963$	0	0
$964$	−630173.	−0.0218407
$965$	−1.93574e7	−0.669160
$966$	0	0
$967$	−208056.	−0.00715508	−0.00357754	−	0.999994i	$-0.501139\pi$
$967$	−208056.	−0.00715508	−0.00357754	+	0.999994i	$0.501139\pi$
$968$	1.99459e7	0.684173
$969$	0	0
$970$	−1.97428e6	−0.0673719
$971$	1.65816e7	0.564388	0.282194	−	0.959357i	$-0.408938\pi$
$971$	1.65816e7	0.564388	0.282194	+	0.959357i	$0.408938\pi$
$972$	0	0
$973$	1.35113e6	0.0457525
$974$	2.72239e7	0.919503
$975$	0	0
$976$	708145.	0.0237957
$977$	1.41891e7	0.475575	0.237787	−	0.971317i	$-0.423578\pi$
$977$	1.41891e7	0.475575	0.237787	+	0.971317i	$0.423578\pi$
$978$	0	0
$979$	3.68179e6	0.122773
$980$	5.06777e6	0.168559
$981$	0	0
$982$	−2.89004e7	−0.956367
$983$	−4.24448e7	−1.40101	−0.700504	−	0.713649i	$-0.747041\pi$
$983$	−4.24448e7	−1.40101	−0.700504	+	0.713649i	$0.747041\pi$
$984$	0	0
$985$	−995581.	−0.0326953
$986$	2.88863e7	0.946238
$987$	0	0
$988$	−9.67698e6	−0.315390
$989$	4.87876e7	1.58606
$990$	0	0
$991$	−3.61915e7	−1.17064	−0.585320	−	0.810803i	$-0.699031\pi$
$991$	−3.61915e7	−1.17064	−0.585320	+	0.810803i	$0.699031\pi$
$992$	3.20905e7	1.03538
$993$	0	0
$994$	1.17529e7	0.377292
$995$	−500836.	−0.0160375
$996$	0	0
$997$	−9.68344e6	−0.308526	−0.154263	−	0.988030i	$-0.549300\pi$
$997$	−9.68344e6	−0.308526	−0.154263	+	0.988030i	$0.549300\pi$
$998$	−205812.	−0.00654100
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.6.a.f.1.1	yes	2
3.2	odd	2		45.6.a.d.1.2	✓	2
4.3	odd	2		720.6.a.be.1.2		2
5.2	odd	4		225.6.b.k.199.2		4
5.3	odd	4		225.6.b.k.199.3		4
5.4	even	2		225.6.a.k.1.2		2
12.11	even	2		720.6.a.y.1.2		2
15.2	even	4		225.6.b.j.199.3		4
15.8	even	4		225.6.b.j.199.2		4
15.14	odd	2		225.6.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.6.a.d.1.2	✓	2	3.2	odd	2
45.6.a.f.1.1	yes	2	1.1	even	1	trivial
225.6.a.k.1.2		2	5.4	even	2
225.6.a.r.1.1		2	15.14	odd	2
225.6.b.j.199.2		4	15.8	even	4
225.6.b.j.199.3		4	15.2	even	4
225.6.b.k.199.2		4	5.2	odd	4
225.6.b.k.199.3		4	5.3	odd	4
720.6.a.y.1.2		2	12.11	even	2
720.6.a.be.1.2		2	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q-3.52080 q^{2} -19.6040 q^{4} +25.0000 q^{5} -80.4159 q^{7} +181.687 q^{8} +O(q^{10})\)	\(q-3.52080 q^{2} -19.6040 q^{4} +25.0000 q^{5} -80.4159 q^{7} +181.687 q^{8} -88.0199 q^{10} +520.416 q^{11} +421.664 q^{13} +283.128 q^{14} -12.3561 q^{16} +1162.67 q^{17} +1170.66 q^{19} -490.100 q^{20} -1832.28 q^{22} +2374.48 q^{23} +625.000 q^{25} -1484.59 q^{26} +1576.47 q^{28} -7056.62 q^{29} -5561.15 q^{31} -5770.49 q^{32} -4093.51 q^{34} -2010.40 q^{35} +2389.95 q^{37} -4121.64 q^{38} +4542.18 q^{40} +8388.28 q^{41} +20546.6 q^{43} -10202.2 q^{44} -8360.07 q^{46} +8352.17 q^{47} -10340.3 q^{49} -2200.50 q^{50} -8266.29 q^{52} +28390.6 q^{53} +13010.4 q^{55} -14610.5 q^{56} +24844.9 q^{58} +37500.4 q^{59} -57311.1 q^{61} +19579.7 q^{62} +20712.1 q^{64} +10541.6 q^{65} -45848.9 q^{67} -22792.9 q^{68} +7078.21 q^{70} +41510.8 q^{71} +76806.5 q^{73} -8414.52 q^{74} -22949.5 q^{76} -41849.7 q^{77} -7008.26 q^{79} -308.904 q^{80} -29533.4 q^{82} -93271.1 q^{83} +29066.6 q^{85} -72340.5 q^{86} +94552.9 q^{88} +7074.70 q^{89} -33908.5 q^{91} -46549.3 q^{92} -29406.3 q^{94} +29266.4 q^{95} +22429.9 q^{97} +36406.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} + 21 q^{4} + 50 q^{5} + 80 q^{7} + 255 q^{8}+O(q^{10}) \) 2 * q + 5 * q^2 + 21 * q^4 + 50 * q^5 + 80 * q^7 + 255 * q^8	\( 2 q + 5 q^{2} + 21 q^{4} + 50 q^{5} + 80 q^{7} + 255 q^{8} + 125 q^{10} + 800 q^{11} - 120 q^{13} + 1650 q^{14} - 687 q^{16} + 1940 q^{17} - 1512 q^{19} + 525 q^{20} + 550 q^{22} - 1320 q^{23} + 1250 q^{25} - 6100 q^{26} + 8090 q^{28} + 1300 q^{29} - 5824 q^{31} - 13865 q^{32} + 2530 q^{34} + 2000 q^{35} - 12560 q^{37} - 26980 q^{38} + 6375 q^{40} + 400 q^{41} + 25680 q^{43} + 1150 q^{44} - 39840 q^{46} + 18920 q^{47} - 1414 q^{49} + 3125 q^{50} - 30260 q^{52} + 49460 q^{53} + 20000 q^{55} - 2850 q^{56} + 96050 q^{58} + 63200 q^{59} - 49116 q^{61} + 17340 q^{62} - 26671 q^{64} - 3000 q^{65} + 6080 q^{67} + 8770 q^{68} + 41250 q^{70} + 65200 q^{71} + 97740 q^{73} - 135800 q^{74} - 131876 q^{76} + 3000 q^{77} - 46288 q^{79} - 17175 q^{80} - 97600 q^{82} - 57360 q^{83} + 48500 q^{85} - 28600 q^{86} + 115050 q^{88} - 87000 q^{89} - 120800 q^{91} - 196560 q^{92} + 60640 q^{94} - 37800 q^{95} + 10180 q^{97} + 112465 q^{98}+O(q^{100}) \) 2 * q + 5 * q^2 + 21 * q^4 + 50 * q^5 + 80 * q^7 + 255 * q^8 + 125 * q^10 + 800 * q^11 - 120 * q^13 + 1650 * q^14 - 687 * q^16 + 1940 * q^17 - 1512 * q^19 + 525 * q^20 + 550 * q^22 - 1320 * q^23 + 1250 * q^25 - 6100 * q^26 + 8090 * q^28 + 1300 * q^29 - 5824 * q^31 - 13865 * q^32 + 2530 * q^34 + 2000 * q^35 - 12560 * q^37 - 26980 * q^38 + 6375 * q^40 + 400 * q^41 + 25680 * q^43 + 1150 * q^44 - 39840 * q^46 + 18920 * q^47 - 1414 * q^49 + 3125 * q^50 - 30260 * q^52 + 49460 * q^53 + 20000 * q^55 - 2850 * q^56 + 96050 * q^58 + 63200 * q^59 - 49116 * q^61 + 17340 * q^62 - 26671 * q^64 - 3000 * q^65 + 6080 * q^67 + 8770 * q^68 + 41250 * q^70 + 65200 * q^71 + 97740 * q^73 - 135800 * q^74 - 131876 * q^76 + 3000 * q^77 - 46288 * q^79 - 17175 * q^80 - 97600 * q^82 - 57360 * q^83 + 48500 * q^85 - 28600 * q^86 + 115050 * q^88 - 87000 * q^89 - 120800 * q^91 - 196560 * q^92 + 60640 * q^94 - 37800 * q^95 + 10180 * q^97 + 112465 * q^98

Introduction

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