LMFDB - Embedded newform 165.6.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,6,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, 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("165.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	165.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:	$26.4633302691$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.87740$ of defining polynomial
Character	$\chi$	$=$	165.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+6.55890 q^{2} -9.00000 q^{3} +11.0192 q^{4} +25.0000 q^{5} -59.0301 q^{6} +146.487 q^{7} -137.611 q^{8} +81.0000 q^{9} +163.973 q^{10} -121.000 q^{11} -99.1731 q^{12} +170.238 q^{13} +960.792 q^{14} -225.000 q^{15} -1255.19 q^{16} +1564.81 q^{17} +531.271 q^{18} +569.786 q^{19} +275.481 q^{20} -1318.38 q^{21} -793.627 q^{22} +3153.25 q^{23} +1238.50 q^{24} +625.000 q^{25} +1116.57 q^{26} -729.000 q^{27} +1614.17 q^{28} +3982.58 q^{29} -1475.75 q^{30} +2990.78 q^{31} -3829.14 q^{32} +1089.00 q^{33} +10263.5 q^{34} +3662.17 q^{35} +892.558 q^{36} +7858.92 q^{37} +3737.17 q^{38} -1532.14 q^{39} -3440.27 q^{40} -5206.60 q^{41} -8647.13 q^{42} +13874.3 q^{43} -1333.33 q^{44} +2025.00 q^{45} +20681.9 q^{46} +6852.01 q^{47} +11296.7 q^{48} +4651.35 q^{49} +4099.32 q^{50} -14083.3 q^{51} +1875.89 q^{52} -3834.15 q^{53} -4781.44 q^{54} -3025.00 q^{55} -20158.2 q^{56} -5128.08 q^{57} +26121.4 q^{58} +9649.38 q^{59} -2479.33 q^{60} -21131.8 q^{61} +19616.2 q^{62} +11865.4 q^{63} +15051.2 q^{64} +4255.95 q^{65} +7142.65 q^{66} -43499.9 q^{67} +17243.0 q^{68} -28379.2 q^{69} +24019.8 q^{70} -52607.2 q^{71} -11146.5 q^{72} -64367.9 q^{73} +51545.9 q^{74} -5625.00 q^{75} +6278.61 q^{76} -17724.9 q^{77} -10049.2 q^{78} +28935.7 q^{79} -31379.8 q^{80} +6561.00 q^{81} -34149.6 q^{82} -4648.64 q^{83} -14527.5 q^{84} +39120.3 q^{85} +91000.1 q^{86} -35843.2 q^{87} +16650.9 q^{88} -103458. q^{89} +13281.8 q^{90} +24937.6 q^{91} +34746.4 q^{92} -26917.0 q^{93} +44941.6 q^{94} +14244.7 q^{95} +34462.2 q^{96} +75809.5 q^{97} +30507.8 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 27 q^{3} - 20 q^{4} + 75 q^{5} - 18 q^{6} + 152 q^{7} - 24 q^{8} + 243 q^{9} + 50 q^{10} - 363 q^{11} + 180 q^{12} - 546 q^{13} - 8 q^{14} - 675 q^{15} - 1360 q^{16} - 314 q^{17} + 162 q^{18}+ \cdots - 29403 q^{99}+O(q^{100}) $ 3 * q + 2 * q^2 - 27 * q^3 - 20 * q^4 + 75 * q^5 - 18 * q^6 + 152 * q^7 - 24 * q^8 + 243 * q^9 + 50 * q^10 - 363 * q^11 + 180 * q^12 - 546 * q^13 - 8 * q^14 - 675 * q^15 - 1360 * q^16 - 314 * q^17 + 162 * q^18 + 1808 * q^19 - 500 * q^20 - 1368 * q^21 - 242 * q^22 + 4288 * q^23 + 216 * q^24 + 1875 * q^25 + 812 * q^26 - 2187 * q^27 + 5888 * q^28 + 5582 * q^29 - 450 * q^30 + 6328 * q^31 - 736 * q^32 + 3267 * q^33 + 11596 * q^34 + 3800 * q^35 - 1620 * q^36 + 16866 * q^37 + 9584 * q^38 + 4914 * q^39 - 600 * q^40 + 23282 * q^41 + 72 * q^42 + 20572 * q^43 + 2420 * q^44 + 6075 * q^45 + 16592 * q^46 + 3432 * q^47 + 12240 * q^48 + 11531 * q^49 + 1250 * q^50 + 2826 * q^51 + 21816 * q^52 + 16138 * q^53 - 1458 * q^54 - 9075 * q^55 + 15648 * q^56 - 16272 * q^57 + 17460 * q^58 + 21972 * q^59 + 4500 * q^60 + 8322 * q^61 + 5056 * q^62 + 12312 * q^63 + 22208 * q^64 - 13650 * q^65 + 2178 * q^66 - 84332 * q^67 + 59832 * q^68 - 38592 * q^69 - 200 * q^70 + 50528 * q^71 - 1944 * q^72 - 53838 * q^73 + 79212 * q^74 - 16875 * q^75 - 52448 * q^76 - 18392 * q^77 - 7308 * q^78 + 6364 * q^79 - 34000 * q^80 + 19683 * q^81 - 68020 * q^82 + 96272 * q^83 - 52992 * q^84 - 7850 * q^85 + 143152 * q^86 - 50238 * q^87 + 2904 * q^88 - 38938 * q^89 + 4050 * q^90 + 104968 * q^91 + 24000 * q^92 - 56952 * q^93 + 49088 * q^94 + 45200 * q^95 + 6624 * q^96 - 103242 * q^97 + 9554 * q^98 - 29403 * 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 $

$2$	6.55890	1.15946	0.579731	−	0.814808i	$-0.303158\pi$
$2$	6.55890	1.15946	0.579731	+	0.814808i	$0.303158\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	11.0192	0.344351
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−59.0301	−0.669415
$7$	146.487	1.12993	0.564967	−	0.825113i	$-0.308889\pi$
$7$	146.487	1.12993	0.564967	+	0.825113i	$0.308889\pi$
$8$	−137.611	−0.760200
$9$	81.0000	0.333333
$10$	163.973	0.518527
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−99.1731	−0.198811
$13$	170.238	0.279382	0.139691	−	0.990195i	$-0.455389\pi$
$13$	170.238	0.279382	0.139691	+	0.990195i	$0.455389\pi$
$14$	960.792	1.31012
$15$	−225.000	−0.258199
$16$	−1255.19	−1.22577
$17$	1564.81	1.31323	0.656614	−	0.754227i	$-0.271988\pi$
$17$	1564.81	1.31323	0.656614	+	0.754227i	$0.271988\pi$
$18$	531.271	0.386487
$19$	569.786	0.362100	0.181050	−	0.983474i	$-0.442050\pi$
$19$	569.786	0.362100	0.181050	+	0.983474i	$0.442050\pi$
$20$	275.481	0.153998
$21$	−1318.38	−0.652368
$22$	−793.627	−0.349591
$23$	3153.25	1.24291	0.621454	−	0.783451i	$-0.286542\pi$
$23$	3153.25	1.24291	0.621454	+	0.783451i	$0.286542\pi$
$24$	1238.50	0.438902
$25$	625.000	0.200000
$26$	1116.57	0.323932
$27$	−729.000	−0.192450
$28$	1614.17	0.389094
$29$	3982.58	0.879366	0.439683	−	0.898153i	$-0.355091\pi$
$29$	3982.58	0.879366	0.439683	+	0.898153i	$0.355091\pi$
$30$	−1475.75	−0.299372
$31$	2990.78	0.558959	0.279480	−	0.960152i	$-0.409838\pi$
$31$	2990.78	0.558959	0.279480	+	0.960152i	$0.409838\pi$
$32$	−3829.14	−0.661037
$33$	1089.00	0.174078
$34$	10263.5	1.52264
$35$	3662.17	0.505322
$36$	892.558	0.114784
$37$	7858.92	0.943753	0.471877	−	0.881665i	$-0.343577\pi$
$37$	7858.92	0.943753	0.471877	+	0.881665i	$0.343577\pi$
$38$	3737.17	0.419841
$39$	−1532.14	−0.161301
$40$	−3440.27	−0.339972
$41$	−5206.60	−0.483721	−0.241860	−	0.970311i	$-0.577758\pi$
$41$	−5206.60	−0.483721	−0.241860	+	0.970311i	$0.577758\pi$
$42$	−8647.13	−0.756395
$43$	13874.3	1.14430	0.572149	−	0.820149i	$-0.306110\pi$
$43$	13874.3	1.14430	0.572149	+	0.820149i	$0.306110\pi$
$44$	−1333.33	−0.103826
$45$	2025.00	0.149071
$46$	20681.9	1.44110
$47$	6852.01	0.452453	0.226226	−	0.974075i	$-0.427361\pi$
$47$	6852.01	0.452453	0.226226	+	0.974075i	$0.427361\pi$
$48$	11296.7	0.707701
$49$	4651.35	0.276751
$50$	4099.32	0.231892
$51$	−14083.3	−0.758192
$52$	1875.89	0.0962054
$53$	−3834.15	−0.187490	−0.0937452	−	0.995596i	$-0.529884\pi$
$53$	−3834.15	−0.187490	−0.0937452	+	0.995596i	$0.529884\pi$
$54$	−4781.44	−0.223138
$55$	−3025.00	−0.134840
$56$	−20158.2	−0.858976
$57$	−5128.08	−0.209058
$58$	26121.4	1.01959
$59$	9649.38	0.360885	0.180443	−	0.983586i	$-0.442247\pi$
$59$	9649.38	0.360885	0.180443	+	0.983586i	$0.442247\pi$
$60$	−2479.33	−0.0889110
$61$	−21131.8	−0.727128	−0.363564	−	0.931569i	$-0.618440\pi$
$61$	−21131.8	−0.727128	−0.363564	+	0.931569i	$0.618440\pi$
$62$	19616.2	0.648092
$63$	11865.4	0.376645
$64$	15051.2	0.459326
$65$	4255.95	0.124943
$66$	7142.65	0.201836
$67$	−43499.9	−1.18386	−0.591931	−	0.805989i	$-0.701634\pi$
$67$	−43499.9	−1.18386	−0.591931	+	0.805989i	$0.701634\pi$
$68$	17243.0	0.452211
$69$	−28379.2	−0.717593
$70$	24019.8	0.585901
$71$	−52607.2	−1.23851	−0.619255	−	0.785190i	$-0.712565\pi$
$71$	−52607.2	−1.23851	−0.619255	+	0.785190i	$0.712565\pi$
$72$	−11146.5	−0.253400
$73$	−64367.9	−1.41372	−0.706858	−	0.707355i	$-0.749888\pi$
$73$	−64367.9	−1.41372	−0.706858	+	0.707355i	$0.749888\pi$
$74$	51545.9	1.09425
$75$	−5625.00	−0.115470
$76$	6278.61	0.124689
$77$	−17724.9	−0.340688
$78$	−10049.2	−0.187022
$79$	28935.7	0.521635	0.260817	−	0.965388i	$-0.416008\pi$
$79$	28935.7	0.521635	0.260817	+	0.965388i	$0.416008\pi$
$80$	−31379.8	−0.548183
$81$	6561.00	0.111111
$82$	−34149.6	−0.560855
$83$	−4648.64	−0.0740681	−0.0370340	−	0.999314i	$-0.511791\pi$
$83$	−4648.64	−0.0740681	−0.0370340	+	0.999314i	$0.511791\pi$
$84$	−14527.5	−0.224643
$85$	39120.3	0.587293
$86$	91000.1	1.32677
$87$	−35843.2	−0.507702
$88$	16650.9	0.229209
$89$	−103458.	−1.38448	−0.692242	−	0.721666i	$-0.743377\pi$
$89$	−103458.	−1.38448	−0.692242	+	0.721666i	$0.743377\pi$
$90$	13281.8	0.172842
$91$	24937.6	0.315683
$92$	34746.4	0.427996
$93$	−26917.0	−0.322715
$94$	44941.6	0.524601
$95$	14244.7	0.161936
$96$	34462.2	0.381650
$97$	75809.5	0.818077	0.409039	−	0.912517i	$-0.365864\pi$
$97$	75809.5	0.818077	0.409039	+	0.912517i	$0.365864\pi$
$98$	30507.8	0.320882
$99$	−9801.00	−0.100504
$100$	6887.02	0.0688702
$101$	35023.6	0.341631	0.170816	−	0.985303i	$-0.445360\pi$
$101$	35023.6	0.341631	0.170816	+	0.985303i	$0.445360\pi$
$102$	−92371.1	−0.879095
$103$	16965.7	0.157572	0.0787859	−	0.996892i	$-0.474896\pi$
$103$	16965.7	0.157572	0.0787859	+	0.996892i	$0.474896\pi$
$104$	−23426.6	−0.212386
$105$	−32959.5	−0.291748
$106$	−25147.8	−0.217388
$107$	12190.4	0.102934	0.0514671	−	0.998675i	$-0.483610\pi$
$107$	12190.4	0.102934	0.0514671	+	0.998675i	$0.483610\pi$
$108$	−8033.02	−0.0662704
$109$	12138.7	0.0978606	0.0489303	−	0.998802i	$-0.484419\pi$
$109$	12138.7	0.0978606	0.0489303	+	0.998802i	$0.484419\pi$
$110$	−19840.7	−0.156342
$111$	−70730.3	−0.544876
$112$	−183869.	−1.38504
$113$	204649.	1.50770	0.753848	−	0.657048i	$-0.228195\pi$
$113$	204649.	1.50770	0.753848	+	0.657048i	$0.228195\pi$
$114$	−33634.6	−0.242395
$115$	78831.2	0.555845
$116$	43885.0	0.302810
$117$	13789.3	0.0931273
$118$	63289.3	0.418433
$119$	229224.	1.48386
$120$	30962.4	0.196283
$121$	14641.0	0.0909091
$122$	−138601.	−0.843077
$123$	46859.4	0.279276
$124$	32956.1	0.192478
$125$	15625.0	0.0894427
$126$	77824.2	0.436705
$127$	−112241.	−0.617506	−0.308753	−	0.951142i	$-0.599912\pi$
$127$	−112241.	−0.617506	−0.308753	+	0.951142i	$0.599912\pi$
$128$	221252.	1.19361
$129$	−124869.	−0.660661
$130$	27914.4	0.144867
$131$	−11679.0	−0.0594602	−0.0297301	−	0.999558i	$-0.509465\pi$
$131$	−11679.0	−0.0594602	−0.0297301	+	0.999558i	$0.509465\pi$
$132$	11999.9	0.0599438
$133$	83466.1	0.409149
$134$	−285311.	−1.37264
$135$	−18225.0	−0.0860663
$136$	−215335.	−0.998315
$137$	−384935.	−1.75221	−0.876105	−	0.482121i	$-0.839867\pi$
$137$	−384935.	−1.75221	−0.876105	+	0.482121i	$0.839867\pi$
$138$	−186137.	−0.832021
$139$	−310605.	−1.36355	−0.681775	−	0.731562i	$-0.738792\pi$
$139$	−310605.	−1.36355	−0.681775	+	0.731562i	$0.738792\pi$
$140$	40354.3	0.174008
$141$	−61668.0	−0.261224
$142$	−345046.	−1.43600
$143$	−20598.8	−0.0842368
$144$	−101671.	−0.408591
$145$	99564.5	0.393264
$146$	−422183.	−1.63915
$147$	−41862.2	−0.159782
$148$	86599.2	0.324982
$149$	285203.	1.05242	0.526209	−	0.850355i	$-0.323613\pi$
$149$	285203.	1.05242	0.526209	+	0.850355i	$0.323613\pi$
$150$	−36893.8	−0.133883
$151$	−522357.	−1.86434	−0.932170	−	0.362020i	$-0.882087\pi$
$151$	−522357.	−1.86434	−0.932170	+	0.362020i	$0.882087\pi$
$152$	−78408.8	−0.275268
$153$	126750.	0.437743
$154$	−116256.	−0.395015
$155$	74769.5	0.249974
$156$	−16883.0	−0.0555442
$157$	−2221.66	−0.00719330	−0.00359665	−	0.999994i	$-0.501145\pi$
$157$	−2221.66	−0.00719330	−0.00359665	+	0.999994i	$0.501145\pi$
$158$	189787.	0.604816
$159$	34507.3	0.108248
$160$	−95728.4	−0.295625
$161$	461909.	1.40440
$162$	43033.0	0.128829
$163$	250146.	0.737436	0.368718	−	0.929541i	$-0.379797\pi$
$163$	250146.	0.737436	0.368718	+	0.929541i	$0.379797\pi$
$164$	−57372.7	−0.166570
$165$	27225.0	0.0778499
$166$	−30490.0	−0.0858791
$167$	−42083.4	−0.116767	−0.0583834	−	0.998294i	$-0.518595\pi$
$167$	−42083.4	−0.116767	−0.0583834	+	0.998294i	$0.518595\pi$
$168$	181423.	0.495930
$169$	−342312.	−0.921946
$170$	256586.	0.680944
$171$	46152.7	0.120700
$172$	152884.	0.394040
$173$	−117206.	−0.297737	−0.148869	−	0.988857i	$-0.547563\pi$
$173$	−117206.	−0.297737	−0.148869	+	0.988857i	$0.547563\pi$
$174$	−235092.	−0.588661
$175$	91554.2	0.225987
$176$	151878.	0.369585
$177$	−86844.4	−0.208357
$178$	−678569.	−1.60526
$179$	582985.	1.35996	0.679978	−	0.733233i	$-0.261989\pi$
$179$	582985.	1.35996	0.679978	+	0.733233i	$0.261989\pi$
$180$	22313.9	0.0513328
$181$	235291.	0.533837	0.266919	−	0.963719i	$-0.413994\pi$
$181$	235291.	0.533837	0.266919	+	0.963719i	$0.413994\pi$
$182$	163563.	0.366022
$183$	190186.	0.419808
$184$	−433921.	−0.944858
$185$	196473.	0.422059
$186$	−176546.	−0.374176
$187$	−189342.	−0.395953
$188$	75503.8	0.155802
$189$	−106789.	−0.217456
$190$	93429.4	0.187758
$191$	591788.	1.17377	0.586885	−	0.809670i	$-0.300354\pi$
$191$	591788.	1.17377	0.586885	+	0.809670i	$0.300354\pi$
$192$	−135461.	−0.265192
$193$	−540825.	−1.04511	−0.522557	−	0.852605i	$-0.675022\pi$
$193$	−540825.	−1.04511	−0.522557	+	0.852605i	$0.675022\pi$
$194$	497227.	0.948529
$195$	−38303.5	−0.0721361
$196$	51254.3	0.0952994
$197$	481711.	0.884344	0.442172	−	0.896930i	$-0.354208\pi$
$197$	481711.	0.884344	0.442172	+	0.896930i	$0.354208\pi$
$198$	−64283.8	−0.116530
$199$	−1.01427e6	−1.81560	−0.907801	−	0.419401i	$-0.862240\pi$
$199$	−1.01427e6	−1.81560	−0.907801	+	0.419401i	$0.862240\pi$
$200$	−86006.8	−0.152040
$201$	391499.	0.683503
$202$	229717.	0.396108
$203$	583395.	0.993625
$204$	−155187.	−0.261084
$205$	−130165.	−0.216326
$206$	111276.	0.182699
$207$	255413.	0.414302
$208$	−213681.	−0.342459
$209$	−68944.2	−0.109177
$210$	−216178.	−0.338270
$211$	619718.	0.958270	0.479135	−	0.877741i	$-0.340950\pi$
$211$	619718.	0.958270	0.479135	+	0.877741i	$0.340950\pi$
$212$	−42249.3	−0.0645625
$213$	473465.	0.715054
$214$	79955.8	0.119348
$215$	346857.	0.511746
$216$	100318.	0.146301
$217$	438109.	0.631587
$218$	79616.9	0.113466
$219$	579311.	0.816210
$220$	−33333.2	−0.0464323
$221$	266391.	0.366892
$222$	−463913.	−0.631763
$223$	894252.	1.20420	0.602099	−	0.798422i	$-0.294331\pi$
$223$	894252.	1.20420	0.602099	+	0.798422i	$0.294331\pi$
$224$	−560918.	−0.746928
$225$	50625.0	0.0666667
$226$	1.34227e6	1.74812
$227$	851468.	1.09674	0.548370	−	0.836236i	$-0.315249\pi$
$227$	851468.	1.09674	0.548370	+	0.836236i	$0.315249\pi$
$228$	−56507.5	−0.0719894
$229$	137840.	0.173695	0.0868475	−	0.996222i	$-0.472321\pi$
$229$	137840.	0.173695	0.0868475	+	0.996222i	$0.472321\pi$
$230$	517047.	0.644481
$231$	159524.	0.196696
$232$	−548046.	−0.668494
$233$	−1.31590e6	−1.58793	−0.793965	−	0.607963i	$-0.791987\pi$
$233$	−1.31590e6	−1.58793	−0.793965	+	0.607963i	$0.791987\pi$
$234$	90442.6	0.107977
$235$	171300.	0.202343
$236$	106329.	0.124271
$237$	−260422.	−0.301166
$238$	1.50346e6	1.72048
$239$	1.11778e6	1.26579	0.632896	−	0.774237i	$-0.281866\pi$
$239$	1.11778e6	1.26579	0.632896	+	0.774237i	$0.281866\pi$
$240$	282418.	0.316493
$241$	−471581.	−0.523015	−0.261507	−	0.965201i	$-0.584220\pi$
$241$	−471581.	−0.523015	−0.261507	+	0.965201i	$0.584220\pi$
$242$	96028.9	0.105406
$243$	−59049.0	−0.0641500
$244$	−232856.	−0.250387
$245$	116284.	0.123767
$246$	307346.	0.323810
$247$	96999.3	0.101164
$248$	−411564.	−0.424921
$249$	41837.8	0.0427632
$250$	102483.	0.103705
$251$	1.76541e6	1.76873	0.884363	−	0.466799i	$-0.154593\pi$
$251$	1.76541e6	1.76873	0.884363	+	0.466799i	$0.154593\pi$
$252$	130748.	0.129698
$253$	−381543.	−0.374751
$254$	−736176.	−0.715974
$255$	−352083.	−0.339074
$256$	969531.	0.924617
$257$	−19920.1	−0.0188130	−0.00940652	−	0.999956i	$-0.502994\pi$
$257$	−19920.1	−0.0188130	−0.00940652	+	0.999956i	$0.502994\pi$
$258$	−819001.	−0.766011
$259$	1.15123e6	1.06638
$260$	46897.3	0.0430244
$261$	322589.	0.293122
$262$	−76601.2	−0.0689418
$263$	−2.18348e6	−1.94652	−0.973262	−	0.229697i	$-0.926227\pi$
$263$	−2.18348e6	−1.94652	−0.973262	+	0.229697i	$0.926227\pi$
$264$	−149858.	−0.132334
$265$	−95853.7	−0.0838483
$266$	547446.	0.474392
$267$	931119.	0.799332
$268$	−479335.	−0.407664
$269$	839300.	0.707190	0.353595	−	0.935399i	$-0.384959\pi$
$269$	839300.	0.707190	0.353595	+	0.935399i	$0.384959\pi$
$270$	−119536.	−0.0997906
$271$	1.42641e6	1.17984	0.589918	−	0.807463i	$-0.299160\pi$
$271$	1.42641e6	1.17984	0.589918	+	0.807463i	$0.299160\pi$
$272$	−1.96414e6	−1.60972
$273$	−224438.	−0.182260
$274$	−2.52475e6	−2.03162
$275$	−75625.0	−0.0603023
$276$	−312717.	−0.247104
$277$	−1.30448e6	−1.02150	−0.510749	−	0.859730i	$-0.670632\pi$
$277$	−1.30448e6	−1.02150	−0.510749	+	0.859730i	$0.670632\pi$
$278$	−2.03723e6	−1.58098
$279$	242253.	0.186320
$280$	−503954.	−0.384146
$281$	−580646.	−0.438678	−0.219339	−	0.975649i	$-0.570390\pi$
$281$	−580646.	−0.438678	−0.219339	+	0.975649i	$0.570390\pi$
$282$	−404475.	−0.302879
$283$	−601974.	−0.446799	−0.223399	−	0.974727i	$-0.571715\pi$
$283$	−601974.	−0.446799	−0.223399	+	0.974727i	$0.571715\pi$
$284$	−579691.	−0.426482
$285$	−128202.	−0.0934937
$286$	−135106.	−0.0976693
$287$	−762698.	−0.546572
$288$	−310160.	−0.220346
$289$	1.02878e6	0.724567
$290$	653034.	0.455975
$291$	−682286.	−0.472317
$292$	−709285.	−0.486815
$293$	−2.40993e6	−1.63996	−0.819982	−	0.572389i	$-0.806017\pi$
$293$	−2.40993e6	−1.63996	−0.819982	+	0.572389i	$0.806017\pi$
$294$	−274570.	−0.185261
$295$	241234.	0.161393
$296$	−1.08147e6	−0.717441
$297$	88209.0	0.0580259
$298$	1.87062e6	1.22024
$299$	536803.	0.347246
$300$	−61983.2	−0.0397622
$301$	2.03240e6	1.29298
$302$	−3.42609e6	−2.16163
$303$	−315213.	−0.197241
$304$	−715191.	−0.443852
$305$	−528294.	−0.325182
$306$	831340.	0.507546
$307$	−1.83380e6	−1.11047	−0.555235	−	0.831694i	$-0.687371\pi$
$307$	−1.83380e6	−1.11047	−0.555235	+	0.831694i	$0.687371\pi$
$308$	−195315.	−0.117316
$309$	−152691.	−0.0909742
$310$	490406.	0.289835
$311$	2.27507e6	1.33381	0.666906	−	0.745142i	$-0.267618\pi$
$311$	2.27507e6	1.33381	0.666906	+	0.745142i	$0.267618\pi$
$312$	210839.	0.122621
$313$	159585.	0.0920728	0.0460364	−	0.998940i	$-0.485341\pi$
$313$	159585.	0.0920728	0.0460364	+	0.998940i	$0.485341\pi$
$314$	−14571.6	−0.00834035
$315$	296636.	0.168441
$316$	318849.	0.179625
$317$	−1.24338e6	−0.694955	−0.347477	−	0.937688i	$-0.612962\pi$
$317$	−1.24338e6	−0.694955	−0.347477	+	0.937688i	$0.612962\pi$
$318$	226330.	0.125509
$319$	−481892.	−0.265139
$320$	376280.	0.205417
$321$	−109714.	−0.0594290
$322$	3.02962e6	1.62835
$323$	891609.	0.475519
$324$	72297.2	0.0382612
$325$	106399.	0.0558764
$326$	1.64068e6	0.855029
$327$	−109249.	−0.0564998
$328$	716485.	0.367724
$329$	1.00373e6	0.511242
$330$	178566.	0.0902640
$331$	−958448.	−0.480838	−0.240419	−	0.970669i	$-0.577285\pi$
$331$	−958448.	−0.480838	−0.240419	+	0.970669i	$0.577285\pi$
$332$	−51224.5	−0.0255054
$333$	636573.	0.314584
$334$	−276021.	−0.135387
$335$	−1.08750e6	−0.529439
$336$	1.65482e6	0.799655
$337$	−1.51074e6	−0.724626	−0.362313	−	0.932056i	$-0.618013\pi$
$337$	−1.51074e6	−0.724626	−0.362313	+	0.932056i	$0.618013\pi$
$338$	−2.24519e6	−1.06896
$339$	−1.84184e6	−0.870469
$340$	431076.	0.202235
$341$	−361884.	−0.168533
$342$	302711.	0.139947
$343$	−1.78064e6	−0.817224
$344$	−1.90925e6	−0.869896
$345$	−709481.	−0.320917
$346$	−768741.	−0.345215
$347$	−1.93847e6	−0.864243	−0.432122	−	0.901815i	$-0.642235\pi$
$347$	−1.93847e6	−0.864243	−0.432122	+	0.901815i	$0.642235\pi$
$348$	−394965.	−0.174828
$349$	−1.62462e6	−0.713985	−0.356992	−	0.934107i	$-0.616198\pi$
$349$	−1.62462e6	−0.713985	−0.356992	+	0.934107i	$0.616198\pi$
$350$	600495.	0.262023
$351$	−124103.	−0.0537671
$352$	463325.	0.199310
$353$	−23347.5	−0.00997249	−0.00498624	−	0.999988i	$-0.501587\pi$
$353$	−23347.5	−0.00997249	−0.00498624	+	0.999988i	$0.501587\pi$
$354$	−569604.	−0.241582
$355$	−1.31518e6	−0.553878
$356$	−1.14002e6	−0.476748
$357$	−2.06302e6	−0.856707
$358$	3.82374e6	1.57682
$359$	3.63918e6	1.49028	0.745139	−	0.666909i	$-0.232383\pi$
$359$	3.63918e6	1.49028	0.745139	+	0.666909i	$0.232383\pi$
$360$	−278662.	−0.113324
$361$	−2.15144e6	−0.868884
$362$	1.54325e6	0.618964
$363$	−131769.	−0.0524864
$364$	274793.	0.108706
$365$	−1.60920e6	−0.632233
$366$	1.24741e6	0.486751
$367$	2.24713e6	0.870890	0.435445	−	0.900215i	$-0.356591\pi$
$367$	2.24713e6	0.870890	0.435445	+	0.900215i	$0.356591\pi$
$368$	−3.95793e6	−1.52352
$369$	−421735.	−0.161240
$370$	1.28865e6	0.489362
$371$	−561651.	−0.211852
$372$	−296605.	−0.111127
$373$	3.82745e6	1.42442	0.712208	−	0.701968i	$-0.247695\pi$
$373$	3.82745e6	1.42442	0.712208	+	0.701968i	$0.247695\pi$
$374$	−1.24188e6	−0.459092
$375$	−140625.	−0.0516398
$376$	−942910.	−0.343954
$377$	677986.	0.245679
$378$	−700418.	−0.252132
$379$	−4.26443e6	−1.52498	−0.762489	−	0.647002i	$-0.776023\pi$
$379$	−4.26443e6	−1.52498	−0.762489	+	0.647002i	$0.776023\pi$
$380$	156965.	0.0557628
$381$	1.01017e6	0.356517
$382$	3.88148e6	1.36094
$383$	2.40920e6	0.839221	0.419611	−	0.907704i	$-0.362167\pi$
$383$	2.40920e6	0.839221	0.419611	+	0.907704i	$0.362167\pi$
$384$	−1.99127e6	−0.689130
$385$	−443122.	−0.152360
$386$	−3.54722e6	−1.21177
$387$	1.12382e6	0.381433
$388$	835362.	0.281706
$389$	3.73204e6	1.25047	0.625233	−	0.780438i	$-0.285004\pi$
$389$	3.73204e6	1.25047	0.625233	+	0.780438i	$0.285004\pi$
$390$	−251229.	−0.0836390
$391$	4.93424e6	1.63222
$392$	−640077.	−0.210386
$393$	105111.	0.0343293
$394$	3.15950e6	1.02536
$395$	723393.	0.233282
$396$	−107999.	−0.0346086
$397$	−4.89288e6	−1.55808	−0.779038	−	0.626977i	$-0.784292\pi$
$397$	−4.89288e6	−1.55808	−0.779038	+	0.626977i	$0.784292\pi$
$398$	−6.65250e6	−2.10512
$399$	−751195.	−0.236222
$400$	−784495.	−0.245155
$401$	−3.93524e6	−1.22211	−0.611055	−	0.791588i	$-0.709255\pi$
$401$	−3.93524e6	−1.22211	−0.611055	+	0.791588i	$0.709255\pi$
$402$	2.56780e6	0.792495
$403$	509144.	0.156163
$404$	385933.	0.117641
$405$	164025.	0.0496904
$406$	3.82643e6	1.15207
$407$	−950929.	−0.284552
$408$	1.93802e6	0.576378
$409$	657037.	0.194215	0.0971073	−	0.995274i	$-0.469041\pi$
$409$	657037.	0.194215	0.0971073	+	0.995274i	$0.469041\pi$
$410$	−853740.	−0.250822
$411$	3.46442e6	1.01164
$412$	186949.	0.0542600
$413$	1.41351e6	0.407777
$414$	1.67523e6	0.480368
$415$	−116216.	−0.0331243
$416$	−651864.	−0.184682
$417$	2.79544e6	0.787246
$418$	−452198.	−0.126587
$419$	4.38504e6	1.22022	0.610110	−	0.792317i	$-0.291125\pi$
$419$	4.38504e6	1.22022	0.610110	+	0.792317i	$0.291125\pi$
$420$	−363188.	−0.100464
$421$	2.34058e6	0.643604	0.321802	−	0.946807i	$-0.395711\pi$
$421$	2.34058e6	0.643604	0.321802	+	0.946807i	$0.395711\pi$
$422$	4.06467e6	1.11108
$423$	555012.	0.150818
$424$	527620.	0.142530
$425$	978008.	0.262646
$426$	3.10541e6	0.829078
$427$	−3.09552e6	−0.821607
$428$	134329.	0.0354455
$429$	185389.	0.0486341
$430$	2.27500e6	0.593350
$431$	−2.18215e6	−0.565838	−0.282919	−	0.959144i	$-0.591303\pi$
$431$	−2.18215e6	−0.565838	−0.282919	+	0.959144i	$0.591303\pi$
$432$	915035.	0.235900
$433$	−3.61080e6	−0.925515	−0.462757	−	0.886485i	$-0.653140\pi$
$433$	−3.61080e6	−0.925515	−0.462757	+	0.886485i	$0.653140\pi$
$434$	2.87352e6	0.732301
$435$	−896080.	−0.227051
$436$	133760.	0.0336984
$437$	1.79668e6	0.450056
$438$	3.79965e6	0.946364
$439$	6.48191e6	1.60525	0.802624	−	0.596486i	$-0.203437\pi$
$439$	6.48191e6	1.60525	0.802624	+	0.596486i	$0.203437\pi$
$440$	416273.	0.102505
$441$	376760.	0.0922503
$442$	1.74723e6	0.425397
$443$	−6.81523e6	−1.64995	−0.824976	−	0.565167i	$-0.808812\pi$
$443$	−6.81523e6	−1.64995	−0.824976	+	0.565167i	$0.808812\pi$
$444$	−779393.	−0.187629
$445$	−2.58644e6	−0.619160
$446$	5.86531e6	1.39622
$447$	−2.56683e6	−0.607614
$448$	2.20480e6	0.519008
$449$	−2.98356e6	−0.698423	−0.349211	−	0.937044i	$-0.613551\pi$
$449$	−2.98356e6	−0.698423	−0.349211	+	0.937044i	$0.613551\pi$
$450$	332045.	0.0772974
$451$	629999.	0.145847
$452$	2.25508e6	0.519177
$453$	4.70122e6	1.07638
$454$	5.58469e6	1.27163
$455$	623440.	0.141178
$456$	705679.	0.158926
$457$	−6.11184e6	−1.36893	−0.684466	−	0.729045i	$-0.739964\pi$
$457$	−6.11184e6	−1.36893	−0.684466	+	0.729045i	$0.739964\pi$
$458$	904081.	0.201393
$459$	−1.14075e6	−0.252731
$460$	868659.	0.191406
$461$	−3.41772e6	−0.749003	−0.374502	−	0.927226i	$-0.622186\pi$
$461$	−3.41772e6	−0.749003	−0.374502	+	0.927226i	$0.622186\pi$
$462$	1.04630e6	0.228062
$463$	−4.60011e6	−0.997277	−0.498639	−	0.866810i	$-0.666166\pi$
$463$	−4.60011e6	−0.997277	−0.498639	+	0.866810i	$0.666166\pi$
$464$	−4.99890e6	−1.07790
$465$	−672925.	−0.144323
$466$	−8.63083e6	−1.84114
$467$	5.03617e6	1.06858	0.534291	−	0.845301i	$-0.320579\pi$
$467$	5.03617e6	1.06858	0.534291	+	0.845301i	$0.320579\pi$
$468$	151947.	0.0320685
$469$	−6.37215e6	−1.33769
$470$	1.12354e6	0.234609
$471$	19994.9	0.00415305
$472$	−1.32786e6	−0.274345
$473$	−1.67879e6	−0.345019
$474$	−1.70808e6	−0.349190
$475$	356117.	0.0724199
$476$	2.52587e6	0.510969
$477$	−310566.	−0.0624968
$478$	7.33142e6	1.46764
$479$	−2.82760e6	−0.563091	−0.281545	−	0.959548i	$-0.590847\pi$
$479$	−2.82760e6	−0.563091	−0.281545	+	0.959548i	$0.590847\pi$
$480$	861556.	0.170679
$481$	1.33789e6	0.263668
$482$	−3.09306e6	−0.606416
$483$	−4.15718e6	−0.810833
$484$	161333.	0.0313046
$485$	1.89524e6	0.365855
$486$	−387297.	−0.0743795
$487$	−5.68931e6	−1.08702	−0.543510	−	0.839403i	$-0.682905\pi$
$487$	−5.68931e6	−1.08702	−0.543510	+	0.839403i	$0.682905\pi$
$488$	2.90796e6	0.552763
$489$	−2.25131e6	−0.425759
$490$	762695.	0.143503
$491$	3.23715e6	0.605981	0.302990	−	0.952994i	$-0.402015\pi$
$491$	3.23715e6	0.605981	0.302990	+	0.952994i	$0.402015\pi$
$492$	516355.	0.0961690
$493$	6.23199e6	1.15481
$494$	636209.	0.117296
$495$	−245025.	−0.0449467
$496$	−3.75400e6	−0.685157
$497$	−7.70626e6	−1.39943
$498$	274410.	0.0495823
$499$	−5.08564e6	−0.914312	−0.457156	−	0.889386i	$-0.651132\pi$
$499$	−5.08564e6	−0.914312	−0.457156	+	0.889386i	$0.651132\pi$
$500$	172175.	0.0307997
$501$	378751.	0.0674154
$502$	1.15791e7	2.05077
$503$	9.02252e6	1.59004	0.795019	−	0.606584i	$-0.207461\pi$
$503$	9.02252e6	1.59004	0.795019	+	0.606584i	$0.207461\pi$
$504$	−1.63281e6	−0.286325
$505$	875591.	0.152782
$506$	−2.50251e6	−0.434509
$507$	3.08081e6	0.532286
$508$	−1.23681e6	−0.212639
$509$	8.98909e6	1.53788	0.768938	−	0.639323i	$-0.220785\pi$
$509$	8.98909e6	1.53788	0.768938	+	0.639323i	$0.220785\pi$
$510$	−2.30928e6	−0.393143
$511$	−9.42904e6	−1.59741
$512$	−720997.	−0.121551
$513$	−415374.	−0.0696861
$514$	−130654.	−0.0218130
$515$	424143.	0.0704683
$516$	−1.37596e6	−0.227499
$517$	−829093.	−0.136420
$518$	7.55079e6	1.23643
$519$	1.05485e6	0.171899
$520$	−585665.	−0.0949819
$521$	−1.11219e7	−1.79508	−0.897539	−	0.440936i	$-0.854647\pi$
$521$	−1.11219e7	−1.79508	−0.897539	+	0.440936i	$0.854647\pi$
$522$	2.11583e6	0.339863
$523$	3.67510e6	0.587510	0.293755	−	0.955881i	$-0.405095\pi$
$523$	3.67510e6	0.587510	0.293755	+	0.955881i	$0.405095\pi$
$524$	−128693.	−0.0204752
$525$	−823988.	−0.130474
$526$	−1.43212e7	−2.25692
$527$	4.68001e6	0.734040
$528$	−1.36690e6	−0.213380
$529$	3.50664e6	0.544818
$530$	−628695.	−0.0972188
$531$	781600.	0.120295
$532$	919733.	0.140891
$533$	−886361.	−0.135143
$534$	6.10712e6	0.926795
$535$	304761.	0.0460335
$536$	5.98605e6	0.899971
$537$	−5.24687e6	−0.785171
$538$	5.50489e6	0.819960
$539$	−562814.	−0.0834436
$540$	−200825.	−0.0296370
$541$	−4.31097e6	−0.633260	−0.316630	−	0.948549i	$-0.602551\pi$
$541$	−4.31097e6	−0.633260	−0.316630	+	0.948549i	$0.602551\pi$
$542$	9.35569e6	1.36797
$543$	−2.11762e6	−0.308211
$544$	−5.99188e6	−0.868092
$545$	303469.	0.0437646
$546$	−1.47207e6	−0.211323
$547$	−1.80626e6	−0.258114	−0.129057	−	0.991637i	$-0.541195\pi$
$547$	−1.80626e6	−0.258114	−0.129057	+	0.991637i	$0.541195\pi$
$548$	−4.24169e6	−0.603375
$549$	−1.71167e6	−0.242376
$550$	−496017.	−0.0699182
$551$	2.26922e6	0.318418
$552$	3.90529e6	0.545514
$553$	4.23870e6	0.589413
$554$	−8.55595e6	−1.18439
$555$	−1.76826e6	−0.243676
$556$	−3.42262e6	−0.469540
$557$	8.42670e6	1.15085	0.575426	−	0.817854i	$-0.304836\pi$
$557$	8.42670e6	1.15085	0.575426	+	0.817854i	$0.304836\pi$
$558$	1.58891e6	0.216031
$559$	2.36193e6	0.319696
$560$	−4.59672e6	−0.619410
$561$	1.70408e6	0.228604
$562$	−3.80840e6	−0.508630
$563$	−9.19966e6	−1.22321	−0.611605	−	0.791163i	$-0.709476\pi$
$563$	−9.19966e6	−1.22321	−0.611605	+	0.791163i	$0.709476\pi$
$564$	−679534.	−0.0899526
$565$	5.11623e6	0.674262
$566$	−3.94829e6	−0.518046
$567$	961099.	0.125548
$568$	7.23932e6	0.941515
$569$	5.18644e6	0.671566	0.335783	−	0.941939i	$-0.390999\pi$
$569$	5.18644e6	0.671566	0.335783	+	0.941939i	$0.390999\pi$
$570$	−840864.	−0.108402
$571$	−3.29390e6	−0.422786	−0.211393	−	0.977401i	$-0.567800\pi$
$571$	−3.29390e6	−0.422786	−0.211393	+	0.977401i	$0.567800\pi$
$572$	−226983.	−0.0290070
$573$	−5.32609e6	−0.677676
$574$	−5.00246e6	−0.633730
$575$	1.97078e6	0.248581
$576$	1.21915e6	0.153109
$577$	−1.02043e7	−1.27598	−0.637989	−	0.770046i	$-0.720233\pi$
$577$	−1.02043e7	−1.27598	−0.637989	+	0.770046i	$0.720233\pi$
$578$	6.74768e6	0.840107
$579$	4.86742e6	0.603396
$580$	1.09712e6	0.135421
$581$	−680965.	−0.0836921
$582$	−4.47505e6	−0.547633
$583$	463932.	0.0565305
$584$	8.85773e6	1.07471
$585$	344732.	0.0416478
$586$	−1.58065e7	−1.90148
$587$	−1.33951e7	−1.60454	−0.802272	−	0.596958i	$-0.796376\pi$
$587$	−1.33951e7	−1.60454	−0.802272	+	0.596958i	$0.796376\pi$
$588$	−461289.	−0.0550212
$589$	1.70411e6	0.202399
$590$	1.58223e6	0.187129
$591$	−4.33540e6	−0.510576
$592$	−9.86445e6	−1.15683
$593$	3.76550e6	0.439730	0.219865	−	0.975530i	$-0.429438\pi$
$593$	3.76550e6	0.439730	0.219865	+	0.975530i	$0.429438\pi$
$594$	578554.	0.0672788
$595$	5.73061e6	0.663603
$596$	3.14272e6	0.362401
$597$	9.12843e6	1.04824
$598$	3.52084e6	0.402618
$599$	1.21252e6	0.138077	0.0690384	−	0.997614i	$-0.478007\pi$
$599$	1.21252e6	0.138077	0.0690384	+	0.997614i	$0.478007\pi$
$600$	774061.	0.0877803
$601$	−7.81714e6	−0.882799	−0.441399	−	0.897311i	$-0.645518\pi$
$601$	−7.81714e6	−0.882799	−0.441399	+	0.897311i	$0.645518\pi$
$602$	1.33303e7	1.49916
$603$	−3.52349e6	−0.394620
$604$	−5.75597e6	−0.641987
$605$	366025.	0.0406558
$606$	−2.06745e6	−0.228693
$607$	5.43853e6	0.599114	0.299557	−	0.954078i	$-0.403161\pi$
$607$	5.43853e6	0.599114	0.299557	+	0.954078i	$0.403161\pi$
$608$	−2.18179e6	−0.239361
$609$	−5.25055e6	−0.573670
$610$	−3.46503e6	−0.377036
$611$	1.16647e6	0.126407
$612$	1.39669e6	0.150737
$613$	1.70637e7	1.83409	0.917045	−	0.398783i	$-0.130567\pi$
$613$	1.70637e7	1.83409	0.917045	+	0.398783i	$0.130567\pi$
$614$	−1.20277e7	−1.28755
$615$	1.17149e6	0.124896
$616$	2.43914e6	0.258991
$617$	7.80667e6	0.825568	0.412784	−	0.910829i	$-0.364556\pi$
$617$	7.80667e6	0.825568	0.412784	+	0.910829i	$0.364556\pi$
$618$	−1.00149e6	−0.105481
$619$	7.31658e6	0.767506	0.383753	−	0.923436i	$-0.374631\pi$
$619$	7.31658e6	0.767506	0.383753	+	0.923436i	$0.374631\pi$
$620$	823902.	0.0860788
$621$	−2.29872e6	−0.239198
$622$	1.49220e7	1.54650
$623$	−1.51552e7	−1.56438
$624$	1.92313e6	0.197719
$625$	390625.	0.0400000
$626$	1.04670e6	0.106755
$627$	620497.	0.0630335
$628$	−24481.0	−0.00247702
$629$	1.22977e7	1.23936
$630$	1.94560e6	0.195300
$631$	−1.18729e7	−1.18709	−0.593543	−	0.804802i	$-0.702271\pi$
$631$	−1.18729e7	−1.18709	−0.593543	+	0.804802i	$0.702271\pi$
$632$	−3.98187e6	−0.396547
$633$	−5.57746e6	−0.553257
$634$	−8.15523e6	−0.805773
$635$	−2.80602e6	−0.276157
$636$	380244.	0.0372752
$637$	791837.	0.0773192
$638$	−3.16068e6	−0.307418
$639$	−4.26118e6	−0.412837
$640$	5.53129e6	0.533798
$641$	−1.85963e7	−1.78764	−0.893821	−	0.448424i	$-0.851985\pi$
$641$	−1.85963e7	−1.78764	−0.893821	+	0.448424i	$0.851985\pi$
$642$	−719603.	−0.0689057
$643$	−1.18028e7	−1.12579	−0.562894	−	0.826529i	$-0.690312\pi$
$643$	−1.18028e7	−1.12579	−0.562894	+	0.826529i	$0.690312\pi$
$644$	5.08988e6	0.483608
$645$	−3.12171e6	−0.295457
$646$	5.84798e6	0.551346
$647$	2.05252e7	1.92764	0.963821	−	0.266551i	$-0.0858842\pi$
$647$	2.05252e7	1.92764	0.963821	+	0.266551i	$0.0858842\pi$
$648$	−902865.	−0.0844666
$649$	−1.16757e6	−0.108811
$650$	697859.	0.0647865
$651$	−3.94298e6	−0.364647
$652$	2.75642e6	0.253937
$653$	−5.78511e6	−0.530919	−0.265460	−	0.964122i	$-0.585524\pi$
$653$	−5.78511e6	−0.530919	−0.265460	+	0.964122i	$0.585524\pi$
$654$	−716552.	−0.0655094
$655$	−291974.	−0.0265914
$656$	6.53528e6	0.592932
$657$	−5.21380e6	−0.471239
$658$	6.58335e6	0.592765
$659$	1.88598e7	1.69170	0.845852	−	0.533418i	$-0.179093\pi$
$659$	1.88598e7	1.69170	0.845852	+	0.533418i	$0.179093\pi$
$660$	299999.	0.0268077
$661$	−1.12652e7	−1.00285	−0.501426	−	0.865201i	$-0.667191\pi$
$661$	−1.12652e7	−1.00285	−0.501426	+	0.865201i	$0.667191\pi$
$662$	−6.28637e6	−0.557513
$663$	−2.39751e6	−0.211825
$664$	639704.	0.0563065
$665$	2.08665e6	0.182977
$666$	4.17522e6	0.364749
$667$	1.25581e7	1.09297
$668$	−463727.	−0.0402088
$669$	−8.04827e6	−0.695244
$670$	−7.13279e6	−0.613864
$671$	2.55694e6	0.219237
$672$	5.04826e6	0.431239
$673$	1.51301e7	1.28767	0.643833	−	0.765166i	$-0.277343\pi$
$673$	1.51301e7	1.28767	0.643833	+	0.765166i	$0.277343\pi$
$674$	−9.90878e6	−0.840176
$675$	−455625.	−0.0384900
$676$	−3.77201e6	−0.317473
$677$	−1.12001e7	−0.939187	−0.469593	−	0.882883i	$-0.655599\pi$
$677$	−1.12001e7	−0.939187	−0.469593	+	0.882883i	$0.655599\pi$
$678$	−1.20805e7	−1.00928
$679$	1.11051e7	0.924373
$680$	−5.38338e6	−0.446460
$681$	−7.66321e6	−0.633203
$682$	−2.37356e6	−0.195407
$683$	2.12907e7	1.74638	0.873190	−	0.487379i	$-0.162047\pi$
$683$	2.12907e7	1.74638	0.873190	+	0.487379i	$0.162047\pi$
$684$	508567.	0.0415631
$685$	−9.62338e6	−0.783612
$686$	−1.16791e7	−0.947539
$687$	−1.24056e6	−0.100283
$688$	−1.74149e7	−1.40265
$689$	−652717.	−0.0523814
$690$	−4.65342e6	−0.372091
$691$	1.77648e6	0.141536	0.0707678	−	0.997493i	$-0.477455\pi$
$691$	1.77648e6	0.141536	0.0707678	+	0.997493i	$0.477455\pi$
$692$	−1.29152e6	−0.102526
$693$	−1.43572e6	−0.113563
$694$	−1.27143e7	−1.00206
$695$	−7.76512e6	−0.609798
$696$	4.93242e6	0.385955
$697$	−8.14735e6	−0.635235
$698$	−1.06557e7	−0.827838
$699$	1.18431e7	0.916792
$700$	1.00886e6	0.0778188
$701$	94420.1	0.00725720	0.00362860	−	0.999993i	$-0.498845\pi$
$701$	94420.1	0.00725720	0.00362860	+	0.999993i	$0.498845\pi$
$702$	−813983.	−0.0623408
$703$	4.47791e6	0.341733
$704$	−1.82120e6	−0.138492
$705$	−1.54170e6	−0.116823
$706$	−153134.	−0.0115627
$707$	5.13050e6	0.386021
$708$	−956958.	−0.0717480
$709$	−1.46333e7	−1.09327	−0.546633	−	0.837372i	$-0.684091\pi$
$709$	−1.46333e7	−1.09327	−0.546633	+	0.837372i	$0.684091\pi$
$710$	−8.62614e6	−0.642201
$711$	2.34379e6	0.173878
$712$	1.42369e7	1.05248
$713$	9.43067e6	0.694734
$714$	−1.35311e7	−0.993319
$715$	−514970.	−0.0376718
$716$	6.42405e6	0.468302
$717$	−1.00600e7	−0.730805
$718$	2.38690e7	1.72792
$719$	1.15215e7	0.831161	0.415580	−	0.909556i	$-0.363578\pi$
$719$	1.15215e7	0.831161	0.415580	+	0.909556i	$0.363578\pi$
$720$	−2.54176e6	−0.182728
$721$	2.48525e6	0.178046
$722$	−1.41111e7	−1.00744
$723$	4.24423e6	0.301963
$724$	2.59273e6	0.183827
$725$	2.48911e6	0.175873
$726$	−864260.	−0.0608559
$727$	1.73993e7	1.22095	0.610473	−	0.792037i	$-0.290979\pi$
$727$	1.73993e7	1.22095	0.610473	+	0.792037i	$0.290979\pi$
$728$	−3.43168e6	−0.239982
$729$	531441.	0.0370370
$730$	−1.05546e7	−0.733050
$731$	2.17107e7	1.50272
$732$	2.09570e6	0.144561
$733$	−1.21356e7	−0.834262	−0.417131	−	0.908846i	$-0.636964\pi$
$733$	−1.21356e7	−0.834262	−0.417131	+	0.908846i	$0.636964\pi$
$734$	1.47387e7	1.00976
$735$	−1.04655e6	−0.0714568
$736$	−1.20742e7	−0.821608
$737$	5.26348e6	0.356948
$738$	−2.76612e6	−0.186952
$739$	1.30600e7	0.879692	0.439846	−	0.898073i	$-0.355033\pi$
$739$	1.30600e7	0.879692	0.439846	+	0.898073i	$0.355033\pi$
$740$	2.16498e6	0.145337
$741$	−872994.	−0.0584071
$742$	−3.68382e6	−0.245634
$743$	−2.36143e6	−0.156929	−0.0784644	−	0.996917i	$-0.525002\pi$
$743$	−2.36143e6	−0.156929	−0.0784644	+	0.996917i	$0.525002\pi$
$744$	3.70407e6	0.245328
$745$	7.13008e6	0.470656
$746$	2.51038e7	1.65156
$747$	−376540.	−0.0246894
$748$	−2.08641e6	−0.136347
$749$	1.78574e6	0.116309
$750$	−922346.	−0.0598743
$751$	4.12172e6	0.266673	0.133336	−	0.991071i	$-0.457431\pi$
$751$	4.12172e6	0.266673	0.133336	+	0.991071i	$0.457431\pi$
$752$	−8.60058e6	−0.554604
$753$	−1.58887e7	−1.02117
$754$	4.44685e6	0.284855
$755$	−1.30589e7	−0.833759
$756$	−1.17673e6	−0.0748811
$757$	4.12364e6	0.261542	0.130771	−	0.991413i	$-0.458255\pi$
$757$	4.12364e6	0.261542	0.130771	+	0.991413i	$0.458255\pi$
$758$	−2.79700e7	−1.76815
$759$	3.43389e6	0.216362
$760$	−1.96022e6	−0.123104
$761$	−2.08230e7	−1.30341	−0.651706	−	0.758472i	$-0.725946\pi$
$761$	−2.08230e7	−1.30341	−0.651706	+	0.758472i	$0.725946\pi$
$762$	6.62558e6	0.413368
$763$	1.77817e6	0.110576
$764$	6.52105e6	0.404189
$765$	3.16875e6	0.195764
$766$	1.58017e7	0.973045
$767$	1.64269e6	0.100825
$768$	−8.72578e6	−0.533828
$769$	1.58156e7	0.964427	0.482213	−	0.876054i	$-0.339833\pi$
$769$	1.58156e7	0.964427	0.482213	+	0.876054i	$0.339833\pi$
$770$	−2.90640e6	−0.176656
$771$	179281.	0.0108617
$772$	−5.95947e6	−0.359886
$773$	−2.58350e6	−0.155510	−0.0777552	−	0.996972i	$-0.524775\pi$
$773$	−2.58350e6	−0.155510	−0.0777552	+	0.996972i	$0.524775\pi$
$774$	7.37101e6	0.442257
$775$	1.86924e6	0.111792
$776$	−1.04322e7	−0.621902
$777$	−1.03610e7	−0.615674
$778$	2.44781e7	1.44987
$779$	−2.96665e6	−0.175155
$780$	−422076.	−0.0248401
$781$	6.36547e6	0.373425
$782$	3.23632e7	1.89250
$783$	−2.90330e6	−0.169234
$784$	−5.83834e6	−0.339234
$785$	−55541.5	−0.00321694
$786$	689411.	0.0398035
$787$	−5.64885e6	−0.325105	−0.162552	−	0.986700i	$-0.551973\pi$
$787$	−5.64885e6	−0.325105	−0.162552	+	0.986700i	$0.551973\pi$
$788$	5.30809e6	0.304525
$789$	1.96513e7	1.12383
$790$	4.74467e6	0.270482
$791$	2.99784e7	1.70360
$792$	1.34872e6	0.0764030
$793$	−3.59743e6	−0.203146
$794$	−3.20920e7	−1.80653
$795$	862683.	0.0484098
$796$	−1.11765e7	−0.625204
$797$	−3.72876e6	−0.207931	−0.103965	−	0.994581i	$-0.533153\pi$
$797$	−3.72876e6	−0.207931	−0.103965	+	0.994581i	$0.533153\pi$
$798$	−4.92702e6	−0.273890
$799$	1.07221e7	0.594173
$800$	−2.39321e6	−0.132207
$801$	−8.38007e6	−0.461495
$802$	−2.58109e7	−1.41699
$803$	7.78852e6	0.426252
$804$	4.31401e6	0.235365
$805$	1.15477e7	0.628068
$806$	3.33943e6	0.181065
$807$	−7.55370e6	−0.408296
$808$	−4.81963e6	−0.259708
$809$	6.35004e6	0.341119	0.170559	−	0.985347i	$-0.445443\pi$
$809$	6.35004e6	0.341119	0.170559	+	0.985347i	$0.445443\pi$
$810$	1.07582e6	0.0576141
$811$	1.34156e7	0.716241	0.358120	−	0.933675i	$-0.383418\pi$
$811$	1.34156e7	0.716241	0.358120	+	0.933675i	$0.383418\pi$
$812$	6.42856e6	0.342156
$813$	−1.28377e7	−0.681178
$814$	−6.23705e6	−0.329928
$815$	6.25365e6	0.329791
$816$	1.76773e7	0.929372
$817$	7.90538e6	0.414350
$818$	4.30945e6	0.225184
$819$	2.01995e6	0.105228
$820$	−1.43432e6	−0.0744922
$821$	2.40201e7	1.24370	0.621851	−	0.783135i	$-0.286381\pi$
$821$	2.40201e7	1.24370	0.621851	+	0.783135i	$0.286381\pi$
$822$	2.27228e7	1.17296
$823$	−1.46523e6	−0.0754058	−0.0377029	−	0.999289i	$-0.512004\pi$
$823$	−1.46523e6	−0.0754058	−0.0377029	+	0.999289i	$0.512004\pi$
$824$	−2.33466e6	−0.119786
$825$	680625.	0.0348155
$826$	9.27105e6	0.472801
$827$	−2.32918e7	−1.18424	−0.592120	−	0.805850i	$-0.701709\pi$
$827$	−2.32918e7	−1.18424	−0.592120	+	0.805850i	$0.701709\pi$
$828$	2.81446e6	0.142665
$829$	−2.02821e6	−0.102501	−0.0512503	−	0.998686i	$-0.516321\pi$
$829$	−2.02821e6	−0.102501	−0.0512503	+	0.998686i	$0.516321\pi$
$830$	−762250.	−0.0384063
$831$	1.17403e7	0.589762
$832$	2.56229e6	0.128327
$833$	7.27850e6	0.363437
$834$	1.83350e7	0.912781
$835$	−1.05208e6	−0.0522197
$836$	−759712.	−0.0375953
$837$	−2.18028e6	−0.107572
$838$	2.87610e7	1.41480
$839$	−3.07518e7	−1.50822	−0.754111	−	0.656747i	$-0.771932\pi$
$839$	−3.07518e7	−1.50822	−0.754111	+	0.656747i	$0.771932\pi$
$840$	4.53559e6	0.221787
$841$	−4.65021e6	−0.226716
$842$	1.53517e7	0.746234
$843$	5.22582e6	0.253271
$844$	6.82881e6	0.329981
$845$	−8.55780e6	−0.412307
$846$	3.64027e6	0.174867
$847$	2.14471e6	0.102721
$848$	4.81259e6	0.229821
$849$	5.41777e6	0.257959
$850$	6.41466e6	0.304527
$851$	2.47811e7	1.17300
$852$	5.21722e6	0.246229
$853$	−2.05598e7	−0.967491	−0.483745	−	0.875209i	$-0.660724\pi$
$853$	−2.05598e7	−0.967491	−0.483745	+	0.875209i	$0.660724\pi$
$854$	−2.03032e7	−0.952621
$855$	1.15382e6	0.0539786
$856$	−1.67753e6	−0.0782505
$857$	3.20463e7	1.49048	0.745240	−	0.666796i	$-0.232335\pi$
$857$	3.20463e7	1.49048	0.745240	+	0.666796i	$0.232335\pi$
$858$	1.21595e6	0.0563894
$859$	2.34894e6	0.108615	0.0543075	−	0.998524i	$-0.482705\pi$
$859$	2.34894e6	0.108615	0.0543075	+	0.998524i	$0.482705\pi$
$860$	3.82210e6	0.176220
$861$	6.86428e6	0.315564
$862$	−1.43125e7	−0.656068
$863$	−1.44887e7	−0.662219	−0.331110	−	0.943592i	$-0.607423\pi$
$863$	−1.44887e7	−0.662219	−0.331110	+	0.943592i	$0.607423\pi$
$864$	2.79144e6	0.127217
$865$	−2.93014e6	−0.133152
$866$	−2.36829e7	−1.07310
$867$	−9.25903e6	−0.418329
$868$	4.82763e6	0.217488
$869$	−3.50122e6	−0.157279
$870$	−5.87731e6	−0.263257
$871$	−7.40533e6	−0.330749
$872$	−1.67042e6	−0.0743936
$873$	6.14057e6	0.272692
$874$	1.17842e7	0.521823
$875$	2.28885e6	0.101064
$876$	6.38356e6	0.281063
$877$	−3.17684e7	−1.39475	−0.697376	−	0.716705i	$-0.745649\pi$
$877$	−3.17684e7	−1.39475	−0.697376	+	0.716705i	$0.745649\pi$
$878$	4.25143e7	1.86122
$879$	2.16893e7	0.946834
$880$	3.79696e6	0.165283
$881$	1.06777e7	0.463486	0.231743	−	0.972777i	$-0.425557\pi$
$881$	1.06777e7	0.463486	0.231743	+	0.972777i	$0.425557\pi$
$882$	2.47113e6	0.106961
$883$	1.77890e7	0.767804	0.383902	−	0.923374i	$-0.374580\pi$
$883$	1.77890e7	0.767804	0.383902	+	0.923374i	$0.374580\pi$
$884$	2.93542e6	0.126340
$885$	−2.17111e6	−0.0931802
$886$	−4.47005e7	−1.91306
$887$	3.13731e7	1.33890	0.669450	−	0.742857i	$-0.266530\pi$
$887$	3.13731e7	1.33890	0.669450	+	0.742857i	$0.266530\pi$
$888$	9.73326e6	0.414215
$889$	−1.64418e7	−0.697741
$890$	−1.69642e7	−0.717892
$891$	−793881.	−0.0335013
$892$	9.85397e6	0.414667
$893$	3.90418e6	0.163833
$894$	−1.68356e7	−0.704505
$895$	1.45746e7	0.608191
$896$	3.24104e7	1.34870
$897$	−4.83123e6	−0.200482
$898$	−1.95689e7	−0.809795
$899$	1.19110e7	0.491529
$900$	557849.	0.0229567
$901$	−5.99972e6	−0.246218
$902$	4.13210e6	0.169104
$903$	−1.82916e7	−0.746504
$904$	−2.81620e7	−1.14615
$905$	5.88228e6	0.238739
$906$	3.08348e7	1.24802
$907$	−8.46355e6	−0.341613	−0.170807	−	0.985305i	$-0.554637\pi$
$907$	−8.46355e6	−0.341613	−0.170807	+	0.985305i	$0.554637\pi$
$908$	9.38252e6	0.377663
$909$	2.83691e6	0.113877
$910$	4.08908e6	0.163690
$911$	−2.18411e7	−0.871923	−0.435961	−	0.899965i	$-0.643592\pi$
$911$	−2.18411e7	−0.871923	−0.435961	+	0.899965i	$0.643592\pi$
$912$	6.43672e6	0.256258
$913$	562486.	0.0223324
$914$	−4.00870e7	−1.58722
$915$	4.75464e6	0.187744
$916$	1.51889e6	0.0598120
$917$	−1.71081e6	−0.0671861
$918$	−7.48206e6	−0.293032
$919$	4.18752e7	1.63557	0.817784	−	0.575525i	$-0.195202\pi$
$919$	4.18752e7	1.63557	0.817784	+	0.575525i	$0.195202\pi$
$920$	−1.08480e7	−0.422553
$921$	1.65042e7	0.641130
$922$	−2.24165e7	−0.868440
$923$	−8.95575e6	−0.346017
$924$	1.75783e6	0.0677325
$925$	4.91183e6	0.188751
$926$	−3.01717e7	−1.15630
$927$	1.37422e6	0.0525240
$928$	−1.52498e7	−0.581293
$929$	8.03740e6	0.305546	0.152773	−	0.988261i	$-0.451180\pi$
$929$	8.03740e6	0.305546	0.152773	+	0.988261i	$0.451180\pi$
$930$	−4.41365e6	−0.167337
$931$	2.65028e6	0.100211
$932$	−1.45002e7	−0.546805
$933$	−2.04757e7	−0.770077
$934$	3.30317e7	1.23898
$935$	−4.73356e6	−0.177076
$936$	−1.89755e6	−0.0707953
$937$	−1.78192e6	−0.0663038	−0.0331519	−	0.999450i	$-0.510555\pi$
$937$	−1.78192e6	−0.0663038	−0.0331519	+	0.999450i	$0.510555\pi$
$938$	−4.17943e7	−1.55099
$939$	−1.43627e6	−0.0531583
$940$	1.88760e6	0.0696770
$941$	2.97245e7	1.09431	0.547154	−	0.837032i	$-0.315711\pi$
$941$	2.97245e7	1.09431	0.547154	+	0.837032i	$0.315711\pi$
$942$	131145.	0.00481531
$943$	−1.64177e7	−0.601220
$944$	−1.21118e7	−0.442364
$945$	−2.66972e6	−0.0972492
$946$	−1.10110e7	−0.400036
$947$	931083.	0.0337375	0.0168688	−	0.999858i	$-0.494630\pi$
$947$	931083.	0.0337375	0.0168688	+	0.999858i	$0.494630\pi$
$948$	−2.86965e6	−0.103707
$949$	−1.09579e7	−0.394967
$950$	2.33573e6	0.0839681
$951$	1.11904e7	0.401232
$952$	−3.15437e7	−1.12803
$953$	−2.74983e7	−0.980784	−0.490392	−	0.871502i	$-0.663146\pi$
$953$	−2.74983e7	−0.980784	−0.490392	+	0.871502i	$0.663146\pi$
$954$	−2.03697e6	−0.0724626
$955$	1.47947e7	0.524926
$956$	1.23171e7	0.435876
$957$	4.33703e6	0.153078
$958$	−1.85459e7	−0.652882
$959$	−5.63879e7	−1.97988
$960$	−3.38652e6	−0.118598
$961$	−1.96844e7	−0.687565
$962$	8.77507e6	0.305712
$963$	987424.	0.0343114
$964$	−5.19646e6	−0.180101
$965$	−1.35206e7	−0.467389
$966$	−2.72666e7	−0.940129
$967$	4.33623e6	0.149123	0.0745617	−	0.997216i	$-0.476244\pi$
$967$	4.33623e6	0.149123	0.0745617	+	0.997216i	$0.476244\pi$
$968$	−2.01476e6	−0.0691091
$969$	−8.02448e6	−0.274541
$970$	1.24307e7	0.424195
$971$	−5.44123e7	−1.85204	−0.926018	−	0.377480i	$-0.876791\pi$
$971$	−5.44123e7	−1.85204	−0.926018	+	0.377480i	$0.876791\pi$
$972$	−650675.	−0.0220901
$973$	−4.54995e7	−1.54072
$974$	−3.73157e7	−1.26036
$975$	−957589.	−0.0322602
$976$	2.65244e7	0.891294
$977$	−3.16213e7	−1.05985	−0.529924	−	0.848045i	$-0.677780\pi$
$977$	−3.16213e7	−1.05985	−0.529924	+	0.848045i	$0.677780\pi$
$978$	−1.47662e7	−0.493651
$979$	1.25184e7	0.417438
$980$	1.28136e6	0.0426192
$981$	983239.	0.0326202
$982$	2.12321e7	0.702611
$983$	1.31368e7	0.433617	0.216808	−	0.976214i	$-0.430435\pi$
$983$	1.31368e7	0.433617	0.216808	+	0.976214i	$0.430435\pi$
$984$	−6.44836e6	−0.212306
$985$	1.20428e7	0.395491
$986$	4.08750e7	1.33895
$987$	−9.03355e6	−0.295165
$988$	1.06886e6	0.0348359
$989$	4.37491e7	1.42226
$990$	−1.60710e6	−0.0521139
$991$	−3.53335e7	−1.14288	−0.571442	−	0.820643i	$-0.693616\pi$
$991$	−3.53335e7	−1.14288	−0.571442	+	0.820643i	$0.693616\pi$
$992$	−1.14521e7	−0.369493
$993$	8.62603e6	0.277612
$994$	−5.05446e7	−1.62259
$995$	−2.53567e7	−0.811962
$996$	461020.	0.0147256
$997$	3.72145e7	1.18570	0.592850	−	0.805313i	$-0.298003\pi$
$997$	3.72145e7	1.18570	0.592850	+	0.805313i	$0.298003\pi$
$998$	−3.33563e7	−1.06011
$999$	−5.72915e6	−0.181625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.6.a.d.1.3	✓	3
3.2	odd	2		495.6.a.c.1.1		3
5.4	even	2		825.6.a.h.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.d.1.3	✓	3	1.1	even	1	trivial
495.6.a.c.1.1		3	3.2	odd	2
825.6.a.h.1.1		3	5.4	even	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:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)

\(f(q)\)	\(=\)	\(q+6.55890 q^{2} -9.00000 q^{3} +11.0192 q^{4} +25.0000 q^{5} -59.0301 q^{6} +146.487 q^{7} -137.611 q^{8} +81.0000 q^{9} +163.973 q^{10} -121.000 q^{11} -99.1731 q^{12} +170.238 q^{13} +960.792 q^{14} -225.000 q^{15} -1255.19 q^{16} +1564.81 q^{17} +531.271 q^{18} +569.786 q^{19} +275.481 q^{20} -1318.38 q^{21} -793.627 q^{22} +3153.25 q^{23} +1238.50 q^{24} +625.000 q^{25} +1116.57 q^{26} -729.000 q^{27} +1614.17 q^{28} +3982.58 q^{29} -1475.75 q^{30} +2990.78 q^{31} -3829.14 q^{32} +1089.00 q^{33} +10263.5 q^{34} +3662.17 q^{35} +892.558 q^{36} +7858.92 q^{37} +3737.17 q^{38} -1532.14 q^{39} -3440.27 q^{40} -5206.60 q^{41} -8647.13 q^{42} +13874.3 q^{43} -1333.33 q^{44} +2025.00 q^{45} +20681.9 q^{46} +6852.01 q^{47} +11296.7 q^{48} +4651.35 q^{49} +4099.32 q^{50} -14083.3 q^{51} +1875.89 q^{52} -3834.15 q^{53} -4781.44 q^{54} -3025.00 q^{55} -20158.2 q^{56} -5128.08 q^{57} +26121.4 q^{58} +9649.38 q^{59} -2479.33 q^{60} -21131.8 q^{61} +19616.2 q^{62} +11865.4 q^{63} +15051.2 q^{64} +4255.95 q^{65} +7142.65 q^{66} -43499.9 q^{67} +17243.0 q^{68} -28379.2 q^{69} +24019.8 q^{70} -52607.2 q^{71} -11146.5 q^{72} -64367.9 q^{73} +51545.9 q^{74} -5625.00 q^{75} +6278.61 q^{76} -17724.9 q^{77} -10049.2 q^{78} +28935.7 q^{79} -31379.8 q^{80} +6561.00 q^{81} -34149.6 q^{82} -4648.64 q^{83} -14527.5 q^{84} +39120.3 q^{85} +91000.1 q^{86} -35843.2 q^{87} +16650.9 q^{88} -103458. q^{89} +13281.8 q^{90} +24937.6 q^{91} +34746.4 q^{92} -26917.0 q^{93} +44941.6 q^{94} +14244.7 q^{95} +34462.2 q^{96} +75809.5 q^{97} +30507.8 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 27 q^{3} - 20 q^{4} + 75 q^{5} - 18 q^{6} + 152 q^{7} - 24 q^{8} + 243 q^{9} + 50 q^{10} - 363 q^{11} + 180 q^{12} - 546 q^{13} - 8 q^{14} - 675 q^{15} - 1360 q^{16} - 314 q^{17} + 162 q^{18}+ \cdots - 29403 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 - 27 * q^3 - 20 * q^4 + 75 * q^5 - 18 * q^6 + 152 * q^7 - 24 * q^8 + 243 * q^9 + 50 * q^10 - 363 * q^11 + 180 * q^12 - 546 * q^13 - 8 * q^14 - 675 * q^15 - 1360 * q^16 - 314 * q^17 + 162 * q^18 + 1808 * q^19 - 500 * q^20 - 1368 * q^21 - 242 * q^22 + 4288 * q^23 + 216 * q^24 + 1875 * q^25 + 812 * q^26 - 2187 * q^27 + 5888 * q^28 + 5582 * q^29 - 450 * q^30 + 6328 * q^31 - 736 * q^32 + 3267 * q^33 + 11596 * q^34 + 3800 * q^35 - 1620 * q^36 + 16866 * q^37 + 9584 * q^38 + 4914 * q^39 - 600 * q^40 + 23282 * q^41 + 72 * q^42 + 20572 * q^43 + 2420 * q^44 + 6075 * q^45 + 16592 * q^46 + 3432 * q^47 + 12240 * q^48 + 11531 * q^49 + 1250 * q^50 + 2826 * q^51 + 21816 * q^52 + 16138 * q^53 - 1458 * q^54 - 9075 * q^55 + 15648 * q^56 - 16272 * q^57 + 17460 * q^58 + 21972 * q^59 + 4500 * q^60 + 8322 * q^61 + 5056 * q^62 + 12312 * q^63 + 22208 * q^64 - 13650 * q^65 + 2178 * q^66 - 84332 * q^67 + 59832 * q^68 - 38592 * q^69 - 200 * q^70 + 50528 * q^71 - 1944 * q^72 - 53838 * q^73 + 79212 * q^74 - 16875 * q^75 - 52448 * q^76 - 18392 * q^77 - 7308 * q^78 + 6364 * q^79 - 34000 * q^80 + 19683 * q^81 - 68020 * q^82 + 96272 * q^83 - 52992 * q^84 - 7850 * q^85 + 143152 * q^86 - 50238 * q^87 + 2904 * q^88 - 38938 * q^89 + 4050 * q^90 + 104968 * q^91 + 24000 * q^92 - 56952 * q^93 + 49088 * q^94 + 45200 * q^95 + 6624 * q^96 - 103242 * q^97 + 9554 * q^98 - 29403 * q^99

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