LMFDB - Embedded newform 495.6.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
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} - 119x^{3} + 206x^{2} + 1428x - 1320 $ x^5 - 2x^4 - 119x^3 + 206x^2 + 1428x - 1320
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-10.1900$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.1900 q^{2} +71.8359 q^{4} +25.0000 q^{5} +134.644 q^{7} -405.928 q^{8} +O(q^{10})$	\(q-10.1900 q^{2} +71.8359 q^{4} +25.0000 q^{5} +134.644 q^{7} -405.928 q^{8} -254.750 q^{10} +121.000 q^{11} +328.709 q^{13} -1372.02 q^{14} +1837.65 q^{16} -636.868 q^{17} -975.686 q^{19} +1795.90 q^{20} -1232.99 q^{22} +1141.85 q^{23} +625.000 q^{25} -3349.55 q^{26} +9672.26 q^{28} +1070.39 q^{29} +8566.05 q^{31} -5735.96 q^{32} +6489.68 q^{34} +3366.09 q^{35} -9884.23 q^{37} +9942.23 q^{38} -10148.2 q^{40} +9045.27 q^{41} +14934.4 q^{43} +8692.15 q^{44} -11635.4 q^{46} +25313.8 q^{47} +1321.95 q^{49} -6368.74 q^{50} +23613.1 q^{52} -5137.53 q^{53} +3025.00 q^{55} -54655.7 q^{56} -10907.3 q^{58} +23659.7 q^{59} -25728.0 q^{61} -87288.0 q^{62} -355.443 q^{64} +8217.73 q^{65} +21264.1 q^{67} -45750.0 q^{68} -34300.5 q^{70} -42804.2 q^{71} -52136.5 q^{73} +100720. q^{74} -70089.3 q^{76} +16291.9 q^{77} -72664.1 q^{79} +45941.3 q^{80} -92171.3 q^{82} +70100.2 q^{83} -15921.7 q^{85} -152181. q^{86} -49117.3 q^{88} -46714.1 q^{89} +44258.7 q^{91} +82025.8 q^{92} -257948. q^{94} -24392.1 q^{95} +180979. q^{97} -13470.7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 82 q^{4} + 125 q^{5} + 184 q^{7} - 24 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 82 * q^4 + 125 * q^5 + 184 * q^7 - 24 * q^8	$ 5 q + 2 q^{2} + 82 q^{4} + 125 q^{5} + 184 q^{7} - 24 q^{8} + 50 q^{10} + 605 q^{11} + 1082 q^{13} - 432 q^{14} + 4770 q^{16} - 2174 q^{17} + 1632 q^{19} + 2050 q^{20} + 242 q^{22} - 1212 q^{23} + 3125 q^{25} - 5600 q^{26} + 16508 q^{28} - 82 q^{29} + 12120 q^{31} + 4864 q^{32} - 4524 q^{34} + 4600 q^{35} - 6530 q^{37} + 15132 q^{38} - 600 q^{40} - 6782 q^{41} + 46184 q^{43} + 9922 q^{44} + 12048 q^{46} + 11692 q^{47} + 34445 q^{49} + 1250 q^{50} + 50020 q^{52} - 10314 q^{53} + 15125 q^{55} - 54928 q^{56} + 75048 q^{58} - 92892 q^{59} + 106 q^{61} - 97160 q^{62} + 44550 q^{64} + 27050 q^{65} + 100476 q^{67} - 119928 q^{68} - 10800 q^{70} + 13772 q^{71} + 94154 q^{73} + 47924 q^{74} - 51524 q^{76} + 22264 q^{77} + 178744 q^{79} + 119250 q^{80} - 299848 q^{82} + 100116 q^{83} - 54350 q^{85} + 167704 q^{86} - 2904 q^{88} - 119410 q^{89} + 47536 q^{91} + 404560 q^{92} - 310288 q^{94} + 40800 q^{95} + 100682 q^{97} + 16434 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 82 * q^4 + 125 * q^5 + 184 * q^7 - 24 * q^8 + 50 * q^10 + 605 * q^11 + 1082 * q^13 - 432 * q^14 + 4770 * q^16 - 2174 * q^17 + 1632 * q^19 + 2050 * q^20 + 242 * q^22 - 1212 * q^23 + 3125 * q^25 - 5600 * q^26 + 16508 * q^28 - 82 * q^29 + 12120 * q^31 + 4864 * q^32 - 4524 * q^34 + 4600 * q^35 - 6530 * q^37 + 15132 * q^38 - 600 * q^40 - 6782 * q^41 + 46184 * q^43 + 9922 * q^44 + 12048 * q^46 + 11692 * q^47 + 34445 * q^49 + 1250 * q^50 + 50020 * q^52 - 10314 * q^53 + 15125 * q^55 - 54928 * q^56 + 75048 * q^58 - 92892 * q^59 + 106 * q^61 - 97160 * q^62 + 44550 * q^64 + 27050 * q^65 + 100476 * q^67 - 119928 * q^68 - 10800 * q^70 + 13772 * q^71 + 94154 * q^73 + 47924 * q^74 - 51524 * q^76 + 22264 * q^77 + 178744 * q^79 + 119250 * q^80 - 299848 * q^82 + 100116 * q^83 - 54350 * q^85 + 167704 * q^86 - 2904 * q^88 - 119410 * q^89 + 47536 * q^91 + 404560 * q^92 - 310288 * q^94 + 40800 * q^95 + 100682 * q^97 + 16434 * 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 $

$2$	−10.1900	−1.80135	−0.900677	−	0.434490i	$-0.856929\pi$
$2$	−10.1900	−1.80135	−0.900677	+	0.434490i	$0.856929\pi$
$3$	0	0
$3$	0	0
$4$	71.8359	2.24487
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	134.644	1.03858	0.519292	−	0.854597i	$-0.326196\pi$
$7$	134.644	1.03858	0.519292	+	0.854597i	$0.326196\pi$
$8$	−405.928	−2.24246
$9$	0	0
$10$	−254.750	−0.805590
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	328.709	0.539453	0.269727	−	0.962937i	$-0.413067\pi$
$13$	328.709	0.539453	0.269727	+	0.962937i	$0.413067\pi$
$14$	−1372.02	−1.87085
$15$	0	0
$16$	1837.65	1.79458
$17$	−636.868	−0.534475	−0.267238	−	0.963631i	$-0.586111\pi$
$17$	−636.868	−0.534475	−0.267238	+	0.963631i	$0.586111\pi$
$18$	0	0
$19$	−975.686	−0.620049	−0.310024	−	0.950729i	$-0.600337\pi$
$19$	−975.686	−0.620049	−0.310024	+	0.950729i	$0.600337\pi$
$20$	1795.90	1.00394
$21$	0	0
$22$	−1232.99	−0.543128
$23$	1141.85	0.450079	0.225040	−	0.974350i	$-0.427749\pi$
$23$	1141.85	0.450079	0.225040	+	0.974350i	$0.427749\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−3349.55	−0.971746
$27$	0	0
$28$	9672.26	2.33149
$29$	1070.39	0.236345	0.118173	−	0.992993i	$-0.462296\pi$
$29$	1070.39	0.236345	0.118173	+	0.992993i	$0.462296\pi$
$30$	0	0
$31$	8566.05	1.60095	0.800473	−	0.599369i	$-0.204582\pi$
$31$	8566.05	1.60095	0.800473	+	0.599369i	$0.204582\pi$
$32$	−5735.96	−0.990219
$33$	0	0
$34$	6489.68	0.962778
$35$	3366.09	0.464468
$36$	0	0
$37$	−9884.23	−1.18697	−0.593483	−	0.804846i	$-0.702248\pi$
$37$	−9884.23	−1.18697	−0.593483	+	0.804846i	$0.702248\pi$
$38$	9942.23	1.11693
$39$	0	0
$40$	−10148.2	−1.00286
$41$	9045.27	0.840354	0.420177	−	0.907442i	$-0.361968\pi$
$41$	9045.27	0.840354	0.420177	+	0.907442i	$0.361968\pi$
$42$	0	0
$43$	14934.4	1.23173	0.615866	−	0.787851i	$-0.288806\pi$
$43$	14934.4	1.23173	0.615866	+	0.787851i	$0.288806\pi$
$44$	8692.15	0.676855
$45$	0	0
$46$	−11635.4	−0.810752
$47$	25313.8	1.67153	0.835763	−	0.549091i	$-0.185026\pi$
$47$	25313.8	1.67153	0.835763	+	0.549091i	$0.185026\pi$
$48$	0	0
$49$	1321.95	0.0786547
$50$	−6368.74	−0.360271
$51$	0	0
$52$	23613.1	1.21100
$53$	−5137.53	−0.251226	−0.125613	−	0.992079i	$-0.540090\pi$
$53$	−5137.53	−0.251226	−0.125613	+	0.992079i	$0.540090\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	−54655.7	−2.32898
$57$	0	0
$58$	−10907.3	−0.425742
$59$	23659.7	0.884870	0.442435	−	0.896801i	$-0.354115\pi$
$59$	23659.7	0.884870	0.442435	+	0.896801i	$0.354115\pi$
$60$	0	0
$61$	−25728.0	−0.885280	−0.442640	−	0.896699i	$-0.645958\pi$
$61$	−25728.0	−0.885280	−0.442640	+	0.896699i	$0.645958\pi$
$62$	−87288.0	−2.88387
$63$	0	0
$64$	−355.443	−0.0108473
$65$	8217.73	0.241251
$66$	0	0
$67$	21264.1	0.578707	0.289354	−	0.957222i	$-0.406560\pi$
$67$	21264.1	0.578707	0.289354	+	0.957222i	$0.406560\pi$
$68$	−45750.0	−1.19983
$69$	0	0
$70$	−34300.5	−0.836672
$71$	−42804.2	−1.00772	−0.503861	−	0.863785i	$-0.668088\pi$
$71$	−42804.2	−1.00772	−0.503861	+	0.863785i	$0.668088\pi$
$72$	0	0
$73$	−52136.5	−1.14508	−0.572539	−	0.819878i	$-0.694041\pi$
$73$	−52136.5	−1.14508	−0.572539	+	0.819878i	$0.694041\pi$
$74$	100720.	2.13815
$75$	0	0
$76$	−70089.3	−1.39193
$77$	16291.9	0.313145
$78$	0	0
$79$	−72664.1	−1.30994	−0.654971	−	0.755654i	$-0.727319\pi$
$79$	−72664.1	−1.30994	−0.654971	+	0.755654i	$0.727319\pi$
$80$	45941.3	0.802561
$81$	0	0
$82$	−92171.3	−1.51377
$83$	70100.2	1.11692	0.558462	−	0.829530i	$-0.311392\pi$
$83$	70100.2	1.11692	0.558462	+	0.829530i	$0.311392\pi$
$84$	0	0
$85$	−15921.7	−0.239024
$86$	−152181.	−2.21879
$87$	0	0
$88$	−49117.3	−0.676126
$89$	−46714.1	−0.625134	−0.312567	−	0.949896i	$-0.601189\pi$
$89$	−46714.1	−0.625134	−0.312567	+	0.949896i	$0.601189\pi$
$90$	0	0
$91$	44258.7	0.560267
$92$	82025.8	1.01037
$93$	0	0
$94$	−257948.	−3.01101
$95$	−24392.1	−0.277294
$96$	0	0
$97$	180979.	1.95298	0.976492	−	0.215555i	$-0.0691559\pi$
$97$	180979.	1.95298	0.976492	+	0.215555i	$0.0691559\pi$
$98$	−13470.7	−0.141685
$99$	0	0
$100$	44897.5	0.448975
$101$	−176753.	−1.72411	−0.862053	−	0.506819i	$-0.830821\pi$
$101$	−176753.	−1.72411	−0.862053	+	0.506819i	$0.830821\pi$
$102$	0	0
$103$	−124306.	−1.15451	−0.577256	−	0.816563i	$-0.695877\pi$
$103$	−124306.	−1.15451	−0.577256	+	0.816563i	$0.695877\pi$
$104$	−133432.	−1.20970
$105$	0	0
$106$	52351.4	0.452547
$107$	128872.	1.08818	0.544088	−	0.839028i	$-0.316876\pi$
$107$	128872.	1.08818	0.544088	+	0.839028i	$0.316876\pi$
$108$	0	0
$109$	89843.2	0.724300	0.362150	−	0.932120i	$-0.382043\pi$
$109$	89843.2	0.724300	0.362150	+	0.932120i	$0.382043\pi$
$110$	−30824.7	−0.242894
$111$	0	0
$112$	247428.	1.86382
$113$	55260.0	0.407113	0.203556	−	0.979063i	$-0.434750\pi$
$113$	55260.0	0.407113	0.203556	+	0.979063i	$0.434750\pi$
$114$	0	0
$115$	28546.2	0.201282
$116$	76892.5	0.530565
$117$	0	0
$118$	−241092.	−1.59396
$119$	−85750.4	−0.555097
$120$	0	0
$121$	14641.0	0.0909091
$122$	262168.	1.59470
$123$	0	0
$124$	615350.	3.59392
$125$	15625.0	0.0894427
$126$	0	0
$127$	−50763.1	−0.279280	−0.139640	−	0.990202i	$-0.544594\pi$
$127$	−50763.1	−0.279280	−0.139640	+	0.990202i	$0.544594\pi$
$128$	187173.	1.00976
$129$	0	0
$130$	−83738.6	−0.434578
$131$	−28827.5	−0.146767	−0.0733836	−	0.997304i	$-0.523380\pi$
$131$	−28827.5	−0.146767	−0.0733836	+	0.997304i	$0.523380\pi$
$132$	0	0
$133$	−131370.	−0.643972
$134$	−216681.	−1.04246
$135$	0	0
$136$	258523.	1.19854
$137$	295351.	1.34443	0.672214	−	0.740357i	$-0.265344\pi$
$137$	295351.	1.34443	0.672214	+	0.740357i	$0.265344\pi$
$138$	0	0
$139$	59066.7	0.259302	0.129651	−	0.991560i	$-0.458614\pi$
$139$	59066.7	0.259302	0.129651	+	0.991560i	$0.458614\pi$
$140$	241807.	1.04267
$141$	0	0
$142$	436175.	1.81526
$143$	39773.8	0.162651
$144$	0	0
$145$	26759.8	0.105697
$146$	531271.	2.06269
$147$	0	0
$148$	−710043.	−2.66459
$149$	−248102.	−0.915511	−0.457756	−	0.889078i	$-0.651347\pi$
$149$	−248102.	−0.915511	−0.457756	+	0.889078i	$0.651347\pi$
$150$	0	0
$151$	−156472.	−0.558462	−0.279231	−	0.960224i	$-0.590080\pi$
$151$	−156472.	−0.558462	−0.279231	+	0.960224i	$0.590080\pi$
$152$	396058.	1.39043
$153$	0	0
$154$	−166014.	−0.564084
$155$	214151.	0.715964
$156$	0	0
$157$	391275.	1.26687	0.633436	−	0.773795i	$-0.281644\pi$
$157$	391275.	1.26687	0.633436	+	0.773795i	$0.281644\pi$
$158$	740446.	2.35967
$159$	0	0
$160$	−143399.	−0.442839
$161$	153743.	0.467445
$162$	0	0
$163$	−78318.8	−0.230886	−0.115443	−	0.993314i	$-0.536829\pi$
$163$	−78318.8	−0.230886	−0.115443	+	0.993314i	$0.536829\pi$
$164$	649776.	1.88649
$165$	0	0
$166$	−714320.	−2.01198
$167$	−1114.53	−0.00309242	−0.00154621	−	0.999999i	$-0.500492\pi$
$167$	−1114.53	−0.00309242	−0.00154621	+	0.999999i	$0.500492\pi$
$168$	0	0
$169$	−263243.	−0.708990
$170$	162242.	0.430567
$171$	0	0
$172$	1.07283e6	2.76508
$173$	347712.	0.883293	0.441647	−	0.897189i	$-0.354395\pi$
$173$	347712.	0.883293	0.441647	+	0.897189i	$0.354395\pi$
$174$	0	0
$175$	84152.4	0.207717
$176$	222356.	0.541087
$177$	0	0
$178$	476017.	1.12609
$179$	−226477.	−0.528312	−0.264156	−	0.964480i	$-0.585093\pi$
$179$	−226477.	−0.528312	−0.264156	+	0.964480i	$0.585093\pi$
$180$	0	0
$181$	472699.	1.07248	0.536239	−	0.844066i	$-0.319845\pi$
$181$	472699.	1.07248	0.536239	+	0.844066i	$0.319845\pi$
$182$	−450996.	−1.00924
$183$	0	0
$184$	−463508.	−1.00928
$185$	−247106.	−0.530828
$186$	0	0
$187$	−77061.1	−0.161150
$188$	1.81844e6	3.75236
$189$	0	0
$190$	248556.	0.499505
$191$	360223.	0.714477	0.357239	−	0.934013i	$-0.383718\pi$
$191$	360223.	0.714477	0.357239	+	0.934013i	$0.383718\pi$
$192$	0	0
$193$	244438.	0.472363	0.236181	−	0.971709i	$-0.424104\pi$
$193$	244438.	0.472363	0.236181	+	0.971709i	$0.424104\pi$
$194$	−1.84417e6	−3.51801
$195$	0	0
$196$	94963.5	0.176570
$197$	−541174.	−0.993508	−0.496754	−	0.867891i	$-0.665475\pi$
$197$	−541174.	−0.993508	−0.496754	+	0.867891i	$0.665475\pi$
$198$	0	0
$199$	999402.	1.78899	0.894494	−	0.447079i	$-0.147536\pi$
$199$	999402.	1.78899	0.894494	+	0.447079i	$0.147536\pi$
$200$	−253705.	−0.448491
$201$	0	0
$202$	1.80111e6	3.10572
$203$	144121.	0.245464
$204$	0	0
$205$	226132.	0.375818
$206$	1.26668e6	2.07969
$207$	0	0
$208$	604053.	0.968093
$209$	−118058.	−0.186952
$210$	0	0
$211$	463100.	0.716092	0.358046	−	0.933704i	$-0.383443\pi$
$211$	463100.	0.716092	0.358046	+	0.933704i	$0.383443\pi$
$212$	−369060.	−0.563971
$213$	0	0
$214$	−1.31321e6	−1.96019
$215$	373360.	0.550848
$216$	0	0
$217$	1.15337e6	1.66271
$218$	−915501.	−1.30472
$219$	0	0
$220$	217304.	0.302699
$221$	−209345.	−0.288324
$222$	0	0
$223$	1.34479e6	1.81089	0.905446	−	0.424462i	$-0.139537\pi$
$223$	1.34479e6	1.81089	0.905446	+	0.424462i	$0.139537\pi$
$224$	−772312.	−1.02842
$225$	0	0
$226$	−563099.	−0.733354
$227$	−1.28112e6	−1.65016	−0.825078	−	0.565019i	$-0.808869\pi$
$227$	−1.28112e6	−1.65016	−0.825078	+	0.565019i	$0.808869\pi$
$228$	0	0
$229$	293472.	0.369810	0.184905	−	0.982756i	$-0.440802\pi$
$229$	293472.	0.369810	0.184905	+	0.982756i	$0.440802\pi$
$230$	−290886.	−0.362579
$231$	0	0
$232$	−434501.	−0.529994
$233$	1.01992e6	1.23077	0.615385	−	0.788227i	$-0.289001\pi$
$233$	1.01992e6	1.23077	0.615385	+	0.788227i	$0.289001\pi$
$234$	0	0
$235$	632845.	0.747529
$236$	1.69962e6	1.98642
$237$	0	0
$238$	873796.	0.999925
$239$	705249.	0.798634	0.399317	−	0.916813i	$-0.369247\pi$
$239$	705249.	0.798634	0.399317	+	0.916813i	$0.369247\pi$
$240$	0	0
$241$	255105.	0.282928	0.141464	−	0.989943i	$-0.454819\pi$
$241$	255105.	0.282928	0.141464	+	0.989943i	$0.454819\pi$
$242$	−149192.	−0.163759
$243$	0	0
$244$	−1.84819e6	−1.98734
$245$	33048.7	0.0351754
$246$	0	0
$247$	−320717.	−0.334487
$248$	−3.47720e6	−3.59005
$249$	0	0
$250$	−159219.	−0.161118
$251$	−321566.	−0.322171	−0.161085	−	0.986940i	$-0.551499\pi$
$251$	−321566.	−0.322171	−0.161085	+	0.986940i	$0.551499\pi$
$252$	0	0
$253$	138164.	0.135704
$254$	517276.	0.503081
$255$	0	0
$256$	−1.89591e6	−1.80808
$257$	−1.37206e6	−1.29580	−0.647902	−	0.761723i	$-0.724354\pi$
$257$	−1.37206e6	−1.29580	−0.647902	+	0.761723i	$0.724354\pi$
$258$	0	0
$259$	−1.33085e6	−1.23276
$260$	590329.	0.541577
$261$	0	0
$262$	293752.	0.264380
$263$	−1.69197e6	−1.50835	−0.754177	−	0.656671i	$-0.771964\pi$
$263$	−1.69197e6	−1.50835	−0.754177	+	0.656671i	$0.771964\pi$
$264$	0	0
$265$	−128438.	−0.112352
$266$	1.33866e6	1.16002
$267$	0	0
$268$	1.52752e6	1.29912
$269$	−1.63565e6	−1.37819	−0.689097	−	0.724669i	$-0.741993\pi$
$269$	−1.63565e6	−1.37819	−0.689097	+	0.724669i	$0.741993\pi$
$270$	0	0
$271$	1.37446e6	1.13687	0.568434	−	0.822729i	$-0.307550\pi$
$271$	1.37446e6	1.13687	0.568434	+	0.822729i	$0.307550\pi$
$272$	−1.17034e6	−0.959159
$273$	0	0
$274$	−3.00963e6	−2.42179
$275$	75625.0	0.0603023
$276$	0	0
$277$	−1.06340e6	−0.832714	−0.416357	−	0.909201i	$-0.636693\pi$
$277$	−1.06340e6	−0.832714	−0.416357	+	0.909201i	$0.636693\pi$
$278$	−601889.	−0.467094
$279$	0	0
$280$	−1.36639e6	−1.04155
$281$	−1.83504e6	−1.38637	−0.693186	−	0.720759i	$-0.743793\pi$
$281$	−1.83504e6	−1.38637	−0.693186	+	0.720759i	$0.743793\pi$
$282$	0	0
$283$	−202576.	−0.150357	−0.0751783	−	0.997170i	$-0.523953\pi$
$283$	−202576.	−0.150357	−0.0751783	+	0.997170i	$0.523953\pi$
$284$	−3.07488e6	−2.26221
$285$	0	0
$286$	−405295.	−0.292992
$287$	1.21789e6	0.872777
$288$	0	0
$289$	−1.01426e6	−0.714336
$290$	−272682.	−0.190397
$291$	0	0
$292$	−3.74528e6	−2.57055
$293$	205193.	0.139634	0.0698172	−	0.997560i	$-0.477758\pi$
$293$	205193.	0.139634	0.0698172	+	0.997560i	$0.477758\pi$
$294$	0	0
$295$	591493.	0.395726
$296$	4.01228e6	2.66172
$297$	0	0
$298$	2.52815e6	1.64916
$299$	375337.	0.242797
$300$	0	0
$301$	2.01082e6	1.27926
$302$	1.59445e6	1.00599
$303$	0	0
$304$	−1.79297e6	−1.11273
$305$	−643199.	−0.395909
$306$	0	0
$307$	1.71849e6	1.04064	0.520322	−	0.853970i	$-0.325812\pi$
$307$	1.71849e6	1.04064	0.520322	+	0.853970i	$0.325812\pi$
$308$	1.17034e6	0.702970
$309$	0	0
$310$	−2.18220e6	−1.28970
$311$	1.87637e6	1.10006	0.550031	−	0.835144i	$-0.314616\pi$
$311$	1.87637e6	1.10006	0.550031	+	0.835144i	$0.314616\pi$
$312$	0	0
$313$	1.84176e6	1.06261	0.531304	−	0.847181i	$-0.321702\pi$
$313$	1.84176e6	1.06261	0.531304	+	0.847181i	$0.321702\pi$
$314$	−3.98709e6	−2.28208
$315$	0	0
$316$	−5.21989e6	−2.94065
$317$	2.25359e6	1.25958	0.629791	−	0.776764i	$-0.283140\pi$
$317$	2.25359e6	1.25958	0.629791	+	0.776764i	$0.283140\pi$
$318$	0	0
$319$	129517.	0.0712608
$320$	−8886.07	−0.00485104
$321$	0	0
$322$	−1.56664e6	−0.842033
$323$	621383.	0.331401
$324$	0	0
$325$	205443.	0.107891
$326$	798068.	0.415907
$327$	0	0
$328$	−3.67173e6	−1.88446
$329$	3.40835e6	1.73602
$330$	0	0
$331$	−372999.	−0.187128	−0.0935638	−	0.995613i	$-0.529826\pi$
$331$	−372999.	−0.187128	−0.0935638	+	0.995613i	$0.529826\pi$
$332$	5.03571e6	2.50735
$333$	0	0
$334$	11357.0	0.00557055
$335$	531601.	0.258806
$336$	0	0
$337$	1.52472e6	0.731332	0.365666	−	0.930746i	$-0.380841\pi$
$337$	1.52472e6	0.731332	0.365666	+	0.930746i	$0.380841\pi$
$338$	2.68245e6	1.27714
$339$	0	0
$340$	−1.14375e6	−0.536580
$341$	1.03649e6	0.482703
$342$	0	0
$343$	−2.08497e6	−0.956894
$344$	−6.06229e6	−2.76211
$345$	0	0
$346$	−3.54319e6	−1.59112
$347$	−1.27128e6	−0.566785	−0.283392	−	0.959004i	$-0.591460\pi$
$347$	−1.27128e6	−0.566785	−0.283392	+	0.959004i	$0.591460\pi$
$348$	0	0
$349$	−2.76246e6	−1.21404	−0.607019	−	0.794687i	$-0.707635\pi$
$349$	−2.76246e6	−1.21404	−0.607019	+	0.794687i	$0.707635\pi$
$350$	−857512.	−0.374171
$351$	0	0
$352$	−694051.	−0.298562
$353$	2.44445e6	1.04411	0.522053	−	0.852913i	$-0.325166\pi$
$353$	2.44445e6	1.04411	0.522053	+	0.852913i	$0.325166\pi$
$354$	0	0
$355$	−1.07011e6	−0.450667
$356$	−3.35575e6	−1.40335
$357$	0	0
$358$	2.30779e6	0.951677
$359$	1.86633e6	0.764278	0.382139	−	0.924105i	$-0.375188\pi$
$359$	1.86633e6	0.764278	0.382139	+	0.924105i	$0.375188\pi$
$360$	0	0
$361$	−1.52414e6	−0.615539
$362$	−4.81680e6	−1.93191
$363$	0	0
$364$	3.17936e6	1.25773
$365$	−1.30341e6	−0.512094
$366$	0	0
$367$	4.79188e6	1.85713	0.928563	−	0.371176i	$-0.121045\pi$
$367$	4.79188e6	1.85713	0.928563	+	0.371176i	$0.121045\pi$
$368$	2.09832e6	0.807704
$369$	0	0
$370$	2.51801e6	0.956208
$371$	−691737.	−0.260919
$372$	0	0
$373$	−2.82793e6	−1.05244	−0.526218	−	0.850350i	$-0.676391\pi$
$373$	−2.82793e6	−1.05244	−0.526218	+	0.850350i	$0.676391\pi$
$374$	785252.	0.290289
$375$	0	0
$376$	−1.02756e7	−3.74832
$377$	351847.	0.127497
$378$	0	0
$379$	3.30279e6	1.18109	0.590545	−	0.807005i	$-0.298913\pi$
$379$	3.30279e6	1.18109	0.590545	+	0.807005i	$0.298913\pi$
$380$	−1.75223e6	−0.622491
$381$	0	0
$382$	−3.67067e6	−1.28703
$383$	5.55972e6	1.93667	0.968336	−	0.249652i	$-0.0803161\pi$
$383$	5.55972e6	1.93667	0.968336	+	0.249652i	$0.0803161\pi$
$384$	0	0
$385$	407297.	0.140043
$386$	−2.49082e6	−0.850892
$387$	0	0
$388$	1.30008e7	4.38420
$389$	−3.97769e6	−1.33278	−0.666388	−	0.745605i	$-0.732161\pi$
$389$	−3.97769e6	−1.33278	−0.666388	+	0.745605i	$0.732161\pi$
$390$	0	0
$391$	−727208.	−0.240556
$392$	−536616.	−0.176380
$393$	0	0
$394$	5.51456e6	1.78966
$395$	−1.81660e6	−0.585824
$396$	0	0
$397$	3.97422e6	1.26554	0.632770	−	0.774340i	$-0.281918\pi$
$397$	3.97422e6	1.26554	0.632770	+	0.774340i	$0.281918\pi$
$398$	−1.01839e7	−3.22260
$399$	0	0
$400$	1.14853e6	0.358916
$401$	5.52342e6	1.71533	0.857663	−	0.514212i	$-0.171916\pi$
$401$	5.52342e6	1.71533	0.857663	+	0.514212i	$0.171916\pi$
$402$	0	0
$403$	2.81574e6	0.863635
$404$	−1.26972e7	−3.87040
$405$	0	0
$406$	−1.46860e6	−0.442168
$407$	−1.19599e6	−0.357884
$408$	0	0
$409$	−3.15672e6	−0.933099	−0.466549	−	0.884495i	$-0.654503\pi$
$409$	−3.15672e6	−0.933099	−0.466549	+	0.884495i	$0.654503\pi$
$410$	−2.30428e6	−0.676980
$411$	0	0
$412$	−8.92963e6	−2.59173
$413$	3.18563e6	0.919011
$414$	0	0
$415$	1.75250e6	0.499504
$416$	−1.88546e6	−0.534177
$417$	0	0
$418$	1.20301e6	0.336766
$419$	5.38025e6	1.49716	0.748579	−	0.663045i	$-0.230736\pi$
$419$	5.38025e6	1.49716	0.748579	+	0.663045i	$0.230736\pi$
$420$	0	0
$421$	6.14869e6	1.69074	0.845370	−	0.534181i	$-0.179380\pi$
$421$	6.14869e6	1.69074	0.845370	+	0.534181i	$0.179380\pi$
$422$	−4.71898e6	−1.28993
$423$	0	0
$424$	2.08547e6	0.563364
$425$	−398043.	−0.106895
$426$	0	0
$427$	−3.46411e6	−0.919437
$428$	9.25764e6	2.44282
$429$	0	0
$430$	−3.80453e6	−0.992271
$431$	1.77150e6	0.459356	0.229678	−	0.973267i	$-0.426233\pi$
$431$	1.77150e6	0.459356	0.229678	+	0.973267i	$0.426233\pi$
$432$	0	0
$433$	−639769.	−0.163985	−0.0819924	−	0.996633i	$-0.526128\pi$
$433$	−639769.	−0.163985	−0.0819924	+	0.996633i	$0.526128\pi$
$434$	−1.17528e7	−2.99514
$435$	0	0
$436$	6.45397e6	1.62596
$437$	−1.11409e6	−0.279071
$438$	0	0
$439$	−3.37312e6	−0.835353	−0.417677	−	0.908596i	$-0.637155\pi$
$439$	−3.37312e6	−0.835353	−0.417677	+	0.908596i	$0.637155\pi$
$440$	−1.22793e6	−0.302373
$441$	0	0
$442$	2.13322e6	0.519374
$443$	−2.61585e6	−0.633291	−0.316645	−	0.948544i	$-0.602557\pi$
$443$	−2.61585e6	−0.633291	−0.316645	+	0.948544i	$0.602557\pi$
$444$	0	0
$445$	−1.16785e6	−0.279569
$446$	−1.37034e7	−3.26205
$447$	0	0
$448$	−47858.2	−0.0112658
$449$	6.10127e6	1.42825	0.714125	−	0.700018i	$-0.246825\pi$
$449$	6.10127e6	1.42825	0.714125	+	0.700018i	$0.246825\pi$
$450$	0	0
$451$	1.09448e6	0.253376
$452$	3.96965e6	0.913916
$453$	0	0
$454$	1.30546e7	2.97251
$455$	1.10647e6	0.250559
$456$	0	0
$457$	−3.44204e6	−0.770950	−0.385475	−	0.922718i	$-0.625962\pi$
$457$	−3.44204e6	−0.770950	−0.385475	+	0.922718i	$0.625962\pi$
$458$	−2.99048e6	−0.666158
$459$	0	0
$460$	2.05065e6	0.451852
$461$	1.07647e6	0.235912	0.117956	−	0.993019i	$-0.462366\pi$
$461$	1.07647e6	0.235912	0.117956	+	0.993019i	$0.462366\pi$
$462$	0	0
$463$	5.81934e6	1.26160	0.630799	−	0.775946i	$-0.282727\pi$
$463$	5.81934e6	1.26160	0.630799	+	0.775946i	$0.282727\pi$
$464$	1.96700e6	0.424141
$465$	0	0
$466$	−1.03930e7	−2.21705
$467$	−869141.	−0.184416	−0.0922078	−	0.995740i	$-0.529392\pi$
$467$	−869141.	−0.184416	−0.0922078	+	0.995740i	$0.529392\pi$
$468$	0	0
$469$	2.86307e6	0.601036
$470$	−6.44869e6	−1.34656
$471$	0	0
$472$	−9.60414e6	−1.98428
$473$	1.80706e6	0.371381
$474$	0	0
$475$	−609804.	−0.124010
$476$	−6.15996e6	−1.24612
$477$	0	0
$478$	−7.18648e6	−1.43862
$479$	2.82970e6	0.563509	0.281755	−	0.959487i	$-0.409084\pi$
$479$	2.82970e6	0.563509	0.281755	+	0.959487i	$0.409084\pi$
$480$	0	0
$481$	−3.24904e6	−0.640313
$482$	−2.59951e6	−0.509653
$483$	0	0
$484$	1.05175e6	0.204079
$485$	4.52447e6	0.873401
$486$	0	0
$487$	−9.69730e6	−1.85280	−0.926400	−	0.376541i	$-0.877114\pi$
$487$	−9.69730e6	−1.85280	−0.926400	+	0.376541i	$0.877114\pi$
$488$	1.04437e7	1.98520
$489$	0	0
$490$	−336766.	−0.0633634
$491$	5.04634e6	0.944654	0.472327	−	0.881423i	$-0.343414\pi$
$491$	5.04634e6	0.944654	0.472327	+	0.881423i	$0.343414\pi$
$492$	0	0
$493$	−681698.	−0.126321
$494$	3.26810e6	0.602530
$495$	0	0
$496$	1.57414e7	2.87303
$497$	−5.76333e6	−1.04660
$498$	0	0
$499$	−8.28396e6	−1.48932	−0.744658	−	0.667446i	$-0.767387\pi$
$499$	−8.28396e6	−1.48932	−0.744658	+	0.667446i	$0.767387\pi$
$500$	1.12244e6	0.200788
$501$	0	0
$502$	3.27676e6	0.580344
$503$	4.33940e6	0.764734	0.382367	−	0.924011i	$-0.375109\pi$
$503$	4.33940e6	0.764734	0.382367	+	0.924011i	$0.375109\pi$
$504$	0	0
$505$	−4.41883e6	−0.771043
$506$	−1.40789e6	−0.244451
$507$	0	0
$508$	−3.64662e6	−0.626947
$509$	1.97982e6	0.338712	0.169356	−	0.985555i	$-0.445831\pi$
$509$	1.97982e6	0.338712	0.169356	+	0.985555i	$0.445831\pi$
$510$	0	0
$511$	−7.01986e6	−1.18926
$512$	1.33298e7	2.24724
$513$	0	0
$514$	1.39813e7	2.33420
$515$	−3.10765e6	−0.516314
$516$	0	0
$517$	3.06297e6	0.503984
$518$	1.35614e7	2.22064
$519$	0	0
$520$	−3.33581e6	−0.540994
$521$	897869.	0.144917	0.0724584	−	0.997371i	$-0.476916\pi$
$521$	897869.	0.144917	0.0724584	+	0.997371i	$0.476916\pi$
$522$	0	0
$523$	5.03726e6	0.805267	0.402633	−	0.915361i	$-0.368095\pi$
$523$	5.03726e6	0.805267	0.402633	+	0.915361i	$0.368095\pi$
$524$	−2.07085e6	−0.329474
$525$	0	0
$526$	1.72412e7	2.71708
$527$	−5.45545e6	−0.855665
$528$	0	0
$529$	−5.13252e6	−0.797428
$530$	1.30879e6	0.202385
$531$	0	0
$532$	−9.43709e6	−1.44564
$533$	2.97327e6	0.453331
$534$	0	0
$535$	3.22180e6	0.486647
$536$	−8.63167e6	−1.29773
$537$	0	0
$538$	1.66673e7	2.48262
$539$	159956.	0.0237153
$540$	0	0
$541$	1.34111e7	1.97003	0.985013	−	0.172483i	$-0.0551788\pi$
$541$	1.34111e7	1.97003	0.985013	+	0.172483i	$0.0551788\pi$
$542$	−1.40058e7	−2.04790
$543$	0	0
$544$	3.65305e6	0.529247
$545$	2.24608e6	0.323917
$546$	0	0
$547$	1.17349e7	1.67691	0.838456	−	0.544969i	$-0.183459\pi$
$547$	1.17349e7	1.67691	0.838456	+	0.544969i	$0.183459\pi$
$548$	2.12168e7	3.01807
$549$	0	0
$550$	−770618.	−0.108626
$551$	−1.04436e6	−0.146546
$552$	0	0
$553$	−9.78377e6	−1.36048
$554$	1.08360e7	1.50001
$555$	0	0
$556$	4.24311e6	0.582099
$557$	1.15265e7	1.57420	0.787101	−	0.616824i	$-0.211581\pi$
$557$	1.15265e7	1.57420	0.787101	+	0.616824i	$0.211581\pi$
$558$	0	0
$559$	4.90908e6	0.664462
$560$	6.18571e6	0.833526
$561$	0	0
$562$	1.86990e7	2.49734
$563$	−7.10264e6	−0.944384	−0.472192	−	0.881496i	$-0.656537\pi$
$563$	−7.10264e6	−0.944384	−0.472192	+	0.881496i	$0.656537\pi$
$564$	0	0
$565$	1.38150e6	0.182066
$566$	2.06425e6	0.270845
$567$	0	0
$568$	1.73754e7	2.25977
$569$	−1.29610e6	−0.167826	−0.0839128	−	0.996473i	$-0.526742\pi$
$569$	−1.29610e6	−0.167826	−0.0839128	+	0.996473i	$0.526742\pi$
$570$	0	0
$571$	1.13637e7	1.45858	0.729290	−	0.684205i	$-0.239851\pi$
$571$	1.13637e7	1.45858	0.729290	+	0.684205i	$0.239851\pi$
$572$	2.85719e6	0.365131
$573$	0	0
$574$	−1.24103e7	−1.57218
$575$	713656.	0.0900159
$576$	0	0
$577$	2.63730e6	0.329777	0.164888	−	0.986312i	$-0.447274\pi$
$577$	2.63730e6	0.329777	0.164888	+	0.986312i	$0.447274\pi$
$578$	1.03353e7	1.28677
$579$	0	0
$580$	1.92231e6	0.237276
$581$	9.43855e6	1.16002
$582$	0	0
$583$	−621642.	−0.0757476
$584$	2.11637e7	2.56779
$585$	0	0
$586$	−2.09091e6	−0.251531
$587$	−9.19302e6	−1.10119	−0.550596	−	0.834772i	$-0.685599\pi$
$587$	−9.19302e6	−1.10119	−0.550596	+	0.834772i	$0.685599\pi$
$588$	0	0
$589$	−8.35777e6	−0.992664
$590$	−6.02731e6	−0.712842
$591$	0	0
$592$	−1.81638e7	−2.13011
$593$	−1.36988e7	−1.59973	−0.799863	−	0.600183i	$-0.795095\pi$
$593$	−1.36988e7	−1.59973	−0.799863	+	0.600183i	$0.795095\pi$
$594$	0	0
$595$	−2.14376e6	−0.248247
$596$	−1.78226e7	−2.05521
$597$	0	0
$598$	−3.82468e6	−0.437363
$599$	4.07920e6	0.464524	0.232262	−	0.972653i	$-0.425387\pi$
$599$	4.07920e6	0.464524	0.232262	+	0.972653i	$0.425387\pi$
$600$	0	0
$601$	1.12151e7	1.26654	0.633269	−	0.773932i	$-0.281713\pi$
$601$	1.12151e7	1.26654	0.633269	+	0.773932i	$0.281713\pi$
$602$	−2.04903e7	−2.30439
$603$	0	0
$604$	−1.12403e7	−1.25368
$605$	366025.	0.0406558
$606$	0	0
$607$	−5.17748e6	−0.570357	−0.285179	−	0.958474i	$-0.592053\pi$
$607$	−5.17748e6	−0.570357	−0.285179	+	0.958474i	$0.592053\pi$
$608$	5.59650e6	0.613984
$609$	0	0
$610$	6.55419e6	0.713173
$611$	8.32089e6	0.901710
$612$	0	0
$613$	−1.47716e7	−1.58773	−0.793866	−	0.608093i	$-0.791935\pi$
$613$	−1.47716e7	−1.58773	−0.793866	+	0.608093i	$0.791935\pi$
$614$	−1.75114e7	−1.87457
$615$	0	0
$616$	−6.61333e6	−0.702213
$617$	−5.40990e6	−0.572105	−0.286053	−	0.958214i	$-0.592343\pi$
$617$	−5.40990e6	−0.572105	−0.286053	+	0.958214i	$0.592343\pi$
$618$	0	0
$619$	−1.06530e7	−1.11749	−0.558746	−	0.829339i	$-0.688717\pi$
$619$	−1.06530e7	−1.11749	−0.558746	+	0.829339i	$0.688717\pi$
$620$	1.53838e7	1.60725
$621$	0	0
$622$	−1.91202e7	−1.98160
$623$	−6.28977e6	−0.649254
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.87676e7	−1.91413
$627$	0	0
$628$	2.81076e7	2.84397
$629$	6.29495e6	0.634404
$630$	0	0
$631$	−4.13999e6	−0.413929	−0.206964	−	0.978348i	$-0.566358\pi$
$631$	−4.13999e6	−0.413929	−0.206964	+	0.978348i	$0.566358\pi$
$632$	2.94964e7	2.93749
$633$	0	0
$634$	−2.29641e7	−2.26895
$635$	−1.26908e6	−0.124898
$636$	0	0
$637$	434537.	0.0424305
$638$	−1.31978e6	−0.128366
$639$	0	0
$640$	4.67932e6	0.451578
$641$	8.19469e6	0.787748	0.393874	−	0.919164i	$-0.371135\pi$
$641$	8.19469e6	0.787748	0.393874	+	0.919164i	$0.371135\pi$
$642$	0	0
$643$	−4.42289e6	−0.421870	−0.210935	−	0.977500i	$-0.567651\pi$
$643$	−4.42289e6	−0.421870	−0.210935	+	0.977500i	$0.567651\pi$
$644$	1.10443e7	1.04935
$645$	0	0
$646$	−6.33189e6	−0.596970
$647$	3.31323e6	0.311165	0.155583	−	0.987823i	$-0.450275\pi$
$647$	3.31323e6	0.311165	0.155583	+	0.987823i	$0.450275\pi$
$648$	0	0
$649$	2.86283e6	0.266798
$650$	−2.09347e6	−0.194349
$651$	0	0
$652$	−5.62610e6	−0.518309
$653$	7.52954e6	0.691011	0.345506	−	0.938417i	$-0.387707\pi$
$653$	7.52954e6	0.691011	0.345506	+	0.938417i	$0.387707\pi$
$654$	0	0
$655$	−720688.	−0.0656363
$656$	1.66221e7	1.50808
$657$	0	0
$658$	−3.47310e7	−3.12718
$659$	−654806.	−0.0587353	−0.0293676	−	0.999569i	$-0.509349\pi$
$659$	−654806.	−0.0587353	−0.0293676	+	0.999569i	$0.509349\pi$
$660$	0	0
$661$	−268318.	−0.0238861	−0.0119431	−	0.999929i	$-0.503802\pi$
$661$	−268318.	−0.0238861	−0.0119431	+	0.999929i	$0.503802\pi$
$662$	3.80086e6	0.337083
$663$	0	0
$664$	−2.84556e7	−2.50465
$665$	−3.28425e6	−0.287993
$666$	0	0
$667$	1.22222e6	0.106374
$668$	−80063.0	−0.00694210
$669$	0	0
$670$	−5.41701e6	−0.466201
$671$	−3.11308e6	−0.266922
$672$	0	0
$673$	1.09013e7	0.927773	0.463887	−	0.885895i	$-0.346455\pi$
$673$	1.09013e7	0.927773	0.463887	+	0.885895i	$0.346455\pi$
$674$	−1.55369e7	−1.31739
$675$	0	0
$676$	−1.89103e7	−1.59159
$677$	−1.46402e7	−1.22766	−0.613828	−	0.789440i	$-0.710371\pi$
$677$	−1.46402e7	−1.22766	−0.613828	+	0.789440i	$0.710371\pi$
$678$	0	0
$679$	2.43677e7	2.02834
$680$	6.46307e6	0.536002
$681$	0	0
$682$	−1.05618e7	−0.869519
$683$	−3.65651e6	−0.299927	−0.149963	−	0.988692i	$-0.547916\pi$
$683$	−3.65651e6	−0.299927	−0.149963	+	0.988692i	$0.547916\pi$
$684$	0	0
$685$	7.38378e6	0.601246
$686$	2.12458e7	1.72370
$687$	0	0
$688$	2.74442e7	2.21044
$689$	−1.68876e6	−0.135525
$690$	0	0
$691$	1.57863e7	1.25772	0.628862	−	0.777517i	$-0.283521\pi$
$691$	1.57863e7	1.25772	0.628862	+	0.777517i	$0.283521\pi$
$692$	2.49782e7	1.98288
$693$	0	0
$694$	1.29543e7	1.02098
$695$	1.47667e6	0.115963
$696$	0	0
$697$	−5.76065e6	−0.449148
$698$	2.81495e7	2.18691
$699$	0	0
$700$	6.04516e6	0.466297
$701$	−2.47966e6	−0.190589	−0.0952945	−	0.995449i	$-0.530379\pi$
$701$	−2.47966e6	−0.190589	−0.0952945	+	0.995449i	$0.530379\pi$
$702$	0	0
$703$	9.64390e6	0.735977
$704$	−43008.6	−0.00327057
$705$	0	0
$706$	−2.49090e7	−1.88080
$707$	−2.37987e7	−1.79063
$708$	0	0
$709$	−1.95693e7	−1.46204	−0.731021	−	0.682355i	$-0.760956\pi$
$709$	−1.95693e7	−1.46204	−0.731021	+	0.682355i	$0.760956\pi$
$710$	1.09044e7	0.811811
$711$	0	0
$712$	1.89626e7	1.40184
$713$	9.78114e6	0.720553
$714$	0	0
$715$	994346.	0.0727398
$716$	−1.62692e7	−1.18599
$717$	0	0
$718$	−1.90178e7	−1.37673
$719$	−2.23668e7	−1.61355	−0.806775	−	0.590859i	$-0.798789\pi$
$719$	−2.23668e7	−1.61355	−0.806775	+	0.590859i	$0.798789\pi$
$720$	0	0
$721$	−1.67370e7	−1.19906
$722$	1.55309e7	1.10880
$723$	0	0
$724$	3.39568e7	2.40758
$725$	668994.	0.0472691
$726$	0	0
$727$	1.15759e6	0.0812301	0.0406151	−	0.999175i	$-0.487068\pi$
$727$	1.15759e6	0.0812301	0.0406151	+	0.999175i	$0.487068\pi$
$728$	−1.79658e7	−1.25637
$729$	0	0
$730$	1.32818e7	0.922463
$731$	−9.51125e6	−0.658330
$732$	0	0
$733$	−690471.	−0.0474663	−0.0237332	−	0.999718i	$-0.507555\pi$
$733$	−690471.	−0.0474663	−0.0237332	+	0.999718i	$0.507555\pi$
$734$	−4.88293e7	−3.34534
$735$	0	0
$736$	−6.54960e6	−0.445677
$737$	2.57295e6	0.174487
$738$	0	0
$739$	−1.97471e7	−1.33013	−0.665064	−	0.746787i	$-0.731596\pi$
$739$	−1.97471e7	−1.33013	−0.665064	+	0.746787i	$0.731596\pi$
$740$	−1.77511e7	−1.19164
$741$	0	0
$742$	7.04880e6	0.470008
$743$	−2.32892e7	−1.54768	−0.773841	−	0.633380i	$-0.781667\pi$
$743$	−2.32892e7	−1.54768	−0.773841	+	0.633380i	$0.781667\pi$
$744$	0	0
$745$	−6.20254e6	−0.409429
$746$	2.88165e7	1.89581
$747$	0	0
$748$	−5.53575e6	−0.361762
$749$	1.73518e7	1.13016
$750$	0	0
$751$	−9.35160e6	−0.605043	−0.302521	−	0.953143i	$-0.597828\pi$
$751$	−9.35160e6	−0.605043	−0.302521	+	0.953143i	$0.597828\pi$
$752$	4.65180e7	2.99969
$753$	0	0
$754$	−3.58532e6	−0.229668
$755$	−3.91179e6	−0.249752
$756$	0	0
$757$	−5.83342e6	−0.369985	−0.184992	−	0.982740i	$-0.559226\pi$
$757$	−5.83342e6	−0.369985	−0.184992	+	0.982740i	$0.559226\pi$
$758$	−3.36554e7	−2.12756
$759$	0	0
$760$	9.90145e6	0.621820
$761$	−1.44516e7	−0.904596	−0.452298	−	0.891867i	$-0.649396\pi$
$761$	−1.44516e7	−0.904596	−0.452298	+	0.891867i	$0.649396\pi$
$762$	0	0
$763$	1.20968e7	0.752246
$764$	2.58770e7	1.60391
$765$	0	0
$766$	−5.66535e7	−3.48863
$767$	7.77717e6	0.477346
$768$	0	0
$769$	5.82446e6	0.355173	0.177586	−	0.984105i	$-0.443171\pi$
$769$	5.82446e6	0.355173	0.177586	+	0.984105i	$0.443171\pi$
$770$	−4.15036e6	−0.252266
$771$	0	0
$772$	1.75594e7	1.06039
$773$	1.10046e7	0.662407	0.331204	−	0.943559i	$-0.392545\pi$
$773$	1.10046e7	0.662407	0.331204	+	0.943559i	$0.392545\pi$
$774$	0	0
$775$	5.35378e6	0.320189
$776$	−7.34644e7	−4.37948
$777$	0	0
$778$	4.05326e7	2.40080
$779$	−8.82535e6	−0.521060
$780$	0	0
$781$	−5.17931e6	−0.303840
$782$	7.41024e6	0.433327
$783$	0	0
$784$	2.42928e6	0.141152
$785$	9.78187e6	0.566562
$786$	0	0
$787$	2.08102e7	1.19768	0.598839	−	0.800869i	$-0.295629\pi$
$787$	2.08102e7	1.19768	0.598839	+	0.800869i	$0.295629\pi$
$788$	−3.88757e7	−2.23030
$789$	0	0
$790$	1.85112e7	1.05528
$791$	7.44041e6	0.422820
$792$	0	0
$793$	−8.45702e6	−0.477567
$794$	−4.04973e7	−2.27969
$795$	0	0
$796$	7.17930e7	4.01605
$797$	3.43533e7	1.91568	0.957840	−	0.287301i	$-0.0927581\pi$
$797$	3.43533e7	1.91568	0.957840	+	0.287301i	$0.0927581\pi$
$798$	0	0
$799$	−1.61216e7	−0.893389
$800$	−3.58498e6	−0.198044
$801$	0	0
$802$	−5.62836e7	−3.08991
$803$	−6.30852e6	−0.345254
$804$	0	0
$805$	3.84357e6	0.209048
$806$	−2.86924e7	−1.55571
$807$	0	0
$808$	7.17490e7	3.86623
$809$	1.11150e7	0.597086	0.298543	−	0.954396i	$-0.403499\pi$
$809$	1.11150e7	0.597086	0.298543	+	0.954396i	$0.403499\pi$
$810$	0	0
$811$	−2.67488e7	−1.42808	−0.714039	−	0.700106i	$-0.753136\pi$
$811$	−2.67488e7	−1.42808	−0.714039	+	0.700106i	$0.753136\pi$
$812$	1.03531e7	0.551036
$813$	0	0
$814$	1.21871e7	0.644675
$815$	−1.95797e6	−0.103255
$816$	0	0
$817$	−1.45713e7	−0.763735
$818$	3.21669e7	1.68084
$819$	0	0
$820$	1.62444e7	0.843663
$821$	1.05297e7	0.545202	0.272601	−	0.962127i	$-0.412116\pi$
$821$	1.05297e7	0.545202	0.272601	+	0.962127i	$0.412116\pi$
$822$	0	0
$823$	−2.49743e7	−1.28527	−0.642633	−	0.766174i	$-0.722158\pi$
$823$	−2.49743e7	−1.28527	−0.642633	+	0.766174i	$0.722158\pi$
$824$	5.04592e7	2.58894
$825$	0	0
$826$	−3.24616e7	−1.65546
$827$	−1.13990e7	−0.579566	−0.289783	−	0.957092i	$-0.593583\pi$
$827$	−1.13990e7	−0.579566	−0.289783	+	0.957092i	$0.593583\pi$
$828$	0	0
$829$	−8.48674e6	−0.428899	−0.214449	−	0.976735i	$-0.568796\pi$
$829$	−8.48674e6	−0.428899	−0.214449	+	0.976735i	$0.568796\pi$
$830$	−1.78580e7	−0.899783
$831$	0	0
$832$	−116837.	−0.00585158
$833$	−841908.	−0.0420390
$834$	0	0
$835$	−27863.1	−0.00138297
$836$	−8.48081e6	−0.419683
$837$	0	0
$838$	−5.48247e7	−2.69691
$839$	1.94174e7	0.952329	0.476165	−	0.879356i	$-0.342027\pi$
$839$	1.94174e7	0.952329	0.476165	+	0.879356i	$0.342027\pi$
$840$	0	0
$841$	−1.93654e7	−0.944141
$842$	−6.26551e7	−3.04562
$843$	0	0
$844$	3.32672e7	1.60753
$845$	−6.58108e6	−0.317070
$846$	0	0
$847$	1.97132e6	0.0944166
$848$	−9.44100e6	−0.450846
$849$	0	0
$850$	4.05605e6	0.192556
$851$	−1.12863e7	−0.534229
$852$	0	0
$853$	7.23170e6	0.340304	0.170152	−	0.985418i	$-0.445574\pi$
$853$	7.23170e6	0.340304	0.170152	+	0.985418i	$0.445574\pi$
$854$	3.52993e7	1.65623
$855$	0	0
$856$	−5.23127e7	−2.44019
$857$	2.59073e7	1.20495	0.602477	−	0.798136i	$-0.294180\pi$
$857$	2.59073e7	1.20495	0.602477	+	0.798136i	$0.294180\pi$
$858$	0	0
$859$	6.83977e6	0.316270	0.158135	−	0.987417i	$-0.449452\pi$
$859$	6.83977e6	0.316270	0.158135	+	0.987417i	$0.449452\pi$
$860$	2.68207e7	1.23658
$861$	0	0
$862$	−1.80516e7	−0.827462
$863$	1.68524e7	0.770255	0.385127	−	0.922863i	$-0.374157\pi$
$863$	1.68524e7	0.770255	0.385127	+	0.922863i	$0.374157\pi$
$864$	0	0
$865$	8.69281e6	0.395021
$866$	6.51925e6	0.295395
$867$	0	0
$868$	8.28531e7	3.73258
$869$	−8.79235e6	−0.394962
$870$	0	0
$871$	6.98969e6	0.312186
$872$	−3.64698e7	−1.62421
$873$	0	0
$874$	1.13525e7	0.502706
$875$	2.10381e6	0.0928937
$876$	0	0
$877$	2.73229e7	1.19958	0.599788	−	0.800159i	$-0.295252\pi$
$877$	2.73229e7	1.19958	0.599788	+	0.800159i	$0.295252\pi$
$878$	3.43720e7	1.50477
$879$	0	0
$880$	5.55890e6	0.241981
$881$	7.68266e6	0.333482	0.166741	−	0.986001i	$-0.446676\pi$
$881$	7.68266e6	0.333482	0.166741	+	0.986001i	$0.446676\pi$
$882$	0	0
$883$	−4.15161e6	−0.179190	−0.0895951	−	0.995978i	$-0.528557\pi$
$883$	−4.15161e6	−0.179190	−0.0895951	+	0.995978i	$0.528557\pi$
$884$	−1.50385e7	−0.647251
$885$	0	0
$886$	2.66555e7	1.14078
$887$	−2.13978e7	−0.913190	−0.456595	−	0.889675i	$-0.650931\pi$
$887$	−2.13978e7	−0.913190	−0.456595	+	0.889675i	$0.650931\pi$
$888$	0	0
$889$	−6.83494e6	−0.290055
$890$	1.19004e7	0.503602
$891$	0	0
$892$	9.66043e7	4.06522
$893$	−2.46983e7	−1.03643
$894$	0	0
$895$	−5.66191e6	−0.236268
$896$	2.52016e7	1.04872
$897$	0	0
$898$	−6.21719e7	−2.57278
$899$	9.16902e6	0.378376
$900$	0	0
$901$	3.27193e6	0.134274
$902$	−1.11527e7	−0.456420
$903$	0	0
$904$	−2.24316e7	−0.912932
$905$	1.18175e7	0.479627
$906$	0	0
$907$	2.98758e7	1.20587	0.602936	−	0.797789i	$-0.293997\pi$
$907$	2.98758e7	1.20587	0.602936	+	0.797789i	$0.293997\pi$
$908$	−9.20304e7	−3.70439
$909$	0	0
$910$	−1.12749e7	−0.451345
$911$	−1.11801e7	−0.446323	−0.223162	−	0.974781i	$-0.571638\pi$
$911$	−1.11801e7	−0.446323	−0.223162	+	0.974781i	$0.571638\pi$
$912$	0	0
$913$	8.48212e6	0.336765
$914$	3.50744e7	1.38875
$915$	0	0
$916$	2.10818e7	0.830175
$917$	−3.88145e6	−0.152430
$918$	0	0
$919$	6.09049e6	0.237883	0.118941	−	0.992901i	$-0.462050\pi$
$919$	6.09049e6	0.237883	0.118941	+	0.992901i	$0.462050\pi$
$920$	−1.15877e7	−0.451365
$921$	0	0
$922$	−1.09692e7	−0.424961
$923$	−1.40702e7	−0.543619
$924$	0	0
$925$	−6.17764e6	−0.237393
$926$	−5.92990e7	−2.27258
$927$	0	0
$928$	−6.13972e6	−0.234034
$929$	1.73381e7	0.659118	0.329559	−	0.944135i	$-0.393100\pi$
$929$	1.73381e7	0.659118	0.329559	+	0.944135i	$0.393100\pi$
$930$	0	0
$931$	−1.28981e6	−0.0487698
$932$	7.32670e7	2.76292
$933$	0	0
$934$	8.85654e6	0.332198
$935$	−1.92653e6	−0.0720686
$936$	0	0
$937$	6.37937e6	0.237372	0.118686	−	0.992932i	$-0.462132\pi$
$937$	6.37937e6	0.237372	0.118686	+	0.992932i	$0.462132\pi$
$938$	−2.91747e7	−1.08268
$939$	0	0
$940$	4.54610e7	1.67811
$941$	2.57786e7	0.949041	0.474521	−	0.880244i	$-0.342621\pi$
$941$	2.57786e7	0.949041	0.474521	+	0.880244i	$0.342621\pi$
$942$	0	0
$943$	1.03283e7	0.378226
$944$	4.34783e7	1.58797
$945$	0	0
$946$	−1.84139e7	−0.668989
$947$	−1.57500e7	−0.570696	−0.285348	−	0.958424i	$-0.592109\pi$
$947$	−1.57500e7	−0.570696	−0.285348	+	0.958424i	$0.592109\pi$
$948$	0	0
$949$	−1.71378e7	−0.617716
$950$	6.21389e6	0.223385
$951$	0	0
$952$	3.48085e7	1.24478
$953$	1.94812e6	0.0694837	0.0347419	−	0.999396i	$-0.488939\pi$
$953$	1.94812e6	0.0694837	0.0347419	+	0.999396i	$0.488939\pi$
$954$	0	0
$955$	9.00558e6	0.319524
$956$	5.06622e7	1.79283
$957$	0	0
$958$	−2.88346e7	−1.01508
$959$	3.97672e7	1.39630
$960$	0	0
$961$	4.47481e7	1.56303
$962$	3.31077e7	1.15343
$963$	0	0
$964$	1.83257e7	0.635137
$965$	6.11095e6	0.211247
$966$	0	0
$967$	4.61582e7	1.58739	0.793693	−	0.608319i	$-0.208156\pi$
$967$	4.61582e7	1.58739	0.793693	+	0.608319i	$0.208156\pi$
$968$	−5.94319e6	−0.203860
$969$	0	0
$970$	−4.61043e7	−1.57330
$971$	3.05303e7	1.03916	0.519580	−	0.854422i	$-0.326088\pi$
$971$	3.05303e7	1.03916	0.519580	+	0.854422i	$0.326088\pi$
$972$	0	0
$973$	7.95296e6	0.269306
$974$	9.88154e7	3.33755
$975$	0	0
$976$	−4.72790e7	−1.58871
$977$	−8.71629e6	−0.292143	−0.146072	−	0.989274i	$-0.546663\pi$
$977$	−8.71629e6	−0.292143	−0.146072	+	0.989274i	$0.546663\pi$
$978$	0	0
$979$	−5.65241e6	−0.188485
$980$	2.37409e6	0.0789644
$981$	0	0
$982$	−5.14222e7	−1.70166
$983$	−5.94838e7	−1.96343	−0.981714	−	0.190360i	$-0.939034\pi$
$983$	−5.94838e7	−1.96343	−0.981714	+	0.190360i	$0.939034\pi$
$984$	0	0
$985$	−1.35294e7	−0.444310
$986$	6.94650e6	0.227548
$987$	0	0
$988$	−2.30390e7	−0.750882
$989$	1.70528e7	0.554378
$990$	0	0
$991$	−2.48497e7	−0.803779	−0.401889	−	0.915688i	$-0.631646\pi$
$991$	−2.48497e7	−0.803779	−0.401889	+	0.915688i	$0.631646\pi$
$992$	−4.91345e7	−1.58529
$993$	0	0
$994$	5.87282e7	1.88530
$995$	2.49851e7	0.800060
$996$	0	0
$997$	−5.50375e7	−1.75356	−0.876781	−	0.480890i	$-0.840314\pi$
$997$	−5.50375e7	−1.75356	−0.876781	+	0.480890i	$0.840314\pi$
$998$	8.44135e7	2.68278
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.j.1.1		5
3.2	odd	2		165.6.a.f.1.5	✓	5
15.14	odd	2		825.6.a.l.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.f.1.5	✓	5	3.2	odd	2
495.6.a.j.1.1		5	1.1	even	1	trivial
825.6.a.l.1.1		5	15.14	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-10.1900 q^{2} +71.8359 q^{4} +25.0000 q^{5} +134.644 q^{7} -405.928 q^{8} +O(q^{10})\)	\(q-10.1900 q^{2} +71.8359 q^{4} +25.0000 q^{5} +134.644 q^{7} -405.928 q^{8} -254.750 q^{10} +121.000 q^{11} +328.709 q^{13} -1372.02 q^{14} +1837.65 q^{16} -636.868 q^{17} -975.686 q^{19} +1795.90 q^{20} -1232.99 q^{22} +1141.85 q^{23} +625.000 q^{25} -3349.55 q^{26} +9672.26 q^{28} +1070.39 q^{29} +8566.05 q^{31} -5735.96 q^{32} +6489.68 q^{34} +3366.09 q^{35} -9884.23 q^{37} +9942.23 q^{38} -10148.2 q^{40} +9045.27 q^{41} +14934.4 q^{43} +8692.15 q^{44} -11635.4 q^{46} +25313.8 q^{47} +1321.95 q^{49} -6368.74 q^{50} +23613.1 q^{52} -5137.53 q^{53} +3025.00 q^{55} -54655.7 q^{56} -10907.3 q^{58} +23659.7 q^{59} -25728.0 q^{61} -87288.0 q^{62} -355.443 q^{64} +8217.73 q^{65} +21264.1 q^{67} -45750.0 q^{68} -34300.5 q^{70} -42804.2 q^{71} -52136.5 q^{73} +100720. q^{74} -70089.3 q^{76} +16291.9 q^{77} -72664.1 q^{79} +45941.3 q^{80} -92171.3 q^{82} +70100.2 q^{83} -15921.7 q^{85} -152181. q^{86} -49117.3 q^{88} -46714.1 q^{89} +44258.7 q^{91} +82025.8 q^{92} -257948. q^{94} -24392.1 q^{95} +180979. q^{97} -13470.7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 82 q^{4} + 125 q^{5} + 184 q^{7} - 24 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 82 * q^4 + 125 * q^5 + 184 * q^7 - 24 * q^8	\( 5 q + 2 q^{2} + 82 q^{4} + 125 q^{5} + 184 q^{7} - 24 q^{8} + 50 q^{10} + 605 q^{11} + 1082 q^{13} - 432 q^{14} + 4770 q^{16} - 2174 q^{17} + 1632 q^{19} + 2050 q^{20} + 242 q^{22} - 1212 q^{23} + 3125 q^{25} - 5600 q^{26} + 16508 q^{28} - 82 q^{29} + 12120 q^{31} + 4864 q^{32} - 4524 q^{34} + 4600 q^{35} - 6530 q^{37} + 15132 q^{38} - 600 q^{40} - 6782 q^{41} + 46184 q^{43} + 9922 q^{44} + 12048 q^{46} + 11692 q^{47} + 34445 q^{49} + 1250 q^{50} + 50020 q^{52} - 10314 q^{53} + 15125 q^{55} - 54928 q^{56} + 75048 q^{58} - 92892 q^{59} + 106 q^{61} - 97160 q^{62} + 44550 q^{64} + 27050 q^{65} + 100476 q^{67} - 119928 q^{68} - 10800 q^{70} + 13772 q^{71} + 94154 q^{73} + 47924 q^{74} - 51524 q^{76} + 22264 q^{77} + 178744 q^{79} + 119250 q^{80} - 299848 q^{82} + 100116 q^{83} - 54350 q^{85} + 167704 q^{86} - 2904 q^{88} - 119410 q^{89} + 47536 q^{91} + 404560 q^{92} - 310288 q^{94} + 40800 q^{95} + 100682 q^{97} + 16434 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 82 * q^4 + 125 * q^5 + 184 * q^7 - 24 * q^8 + 50 * q^10 + 605 * q^11 + 1082 * q^13 - 432 * q^14 + 4770 * q^16 - 2174 * q^17 + 1632 * q^19 + 2050 * q^20 + 242 * q^22 - 1212 * q^23 + 3125 * q^25 - 5600 * q^26 + 16508 * q^28 - 82 * q^29 + 12120 * q^31 + 4864 * q^32 - 4524 * q^34 + 4600 * q^35 - 6530 * q^37 + 15132 * q^38 - 600 * q^40 - 6782 * q^41 + 46184 * q^43 + 9922 * q^44 + 12048 * q^46 + 11692 * q^47 + 34445 * q^49 + 1250 * q^50 + 50020 * q^52 - 10314 * q^53 + 15125 * q^55 - 54928 * q^56 + 75048 * q^58 - 92892 * q^59 + 106 * q^61 - 97160 * q^62 + 44550 * q^64 + 27050 * q^65 + 100476 * q^67 - 119928 * q^68 - 10800 * q^70 + 13772 * q^71 + 94154 * q^73 + 47924 * q^74 - 51524 * q^76 + 22264 * q^77 + 178744 * q^79 + 119250 * q^80 - 299848 * q^82 + 100116 * q^83 - 54350 * q^85 + 167704 * q^86 - 2904 * q^88 - 119410 * q^89 + 47536 * q^91 + 404560 * q^92 - 310288 * q^94 + 40800 * q^95 + 100682 * q^97 + 16434 * q^98

Introduction

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Database

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