LMFDB - Embedded newform 165.6.a.e.1.2

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:	3.3.307532.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 76x + 168 $ x^3 - x^2 - 76*x + 168
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30119$ 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)

$f(q)$	$=$	\(q+4.30119 q^{2} -9.00000 q^{3} -13.4998 q^{4} -25.0000 q^{5} -38.7107 q^{6} -148.216 q^{7} -195.703 q^{8} +81.0000 q^{9} -107.530 q^{10} +121.000 q^{11} +121.498 q^{12} +234.137 q^{13} -637.504 q^{14} +225.000 q^{15} -409.764 q^{16} +1031.76 q^{17} +348.396 q^{18} +1266.29 q^{19} +337.494 q^{20} +1333.94 q^{21} +520.444 q^{22} +384.710 q^{23} +1761.33 q^{24} +625.000 q^{25} +1007.07 q^{26} -729.000 q^{27} +2000.88 q^{28} -4484.85 q^{29} +967.768 q^{30} -997.235 q^{31} +4500.03 q^{32} -1089.00 q^{33} +4437.78 q^{34} +3705.39 q^{35} -1093.48 q^{36} +5168.11 q^{37} +5446.56 q^{38} -2107.23 q^{39} +4892.58 q^{40} +2259.17 q^{41} +5737.54 q^{42} +15818.9 q^{43} -1633.47 q^{44} -2025.00 q^{45} +1654.71 q^{46} +12033.2 q^{47} +3687.87 q^{48} +5160.90 q^{49} +2688.24 q^{50} -9285.81 q^{51} -3160.79 q^{52} -3851.67 q^{53} -3135.57 q^{54} -3025.00 q^{55} +29006.3 q^{56} -11396.6 q^{57} -19290.2 q^{58} -20261.0 q^{59} -3037.45 q^{60} +2006.88 q^{61} -4289.30 q^{62} -12005.5 q^{63} +32467.9 q^{64} -5853.41 q^{65} -4684.00 q^{66} +40945.6 q^{67} -13928.5 q^{68} -3462.39 q^{69} +15937.6 q^{70} +16970.8 q^{71} -15852.0 q^{72} -56640.5 q^{73} +22229.0 q^{74} -5625.00 q^{75} -17094.6 q^{76} -17934.1 q^{77} -9063.59 q^{78} +58507.9 q^{79} +10244.1 q^{80} +6561.00 q^{81} +9717.12 q^{82} -52243.8 q^{83} -18007.9 q^{84} -25793.9 q^{85} +68040.2 q^{86} +40363.6 q^{87} -23680.1 q^{88} +55114.4 q^{89} -8709.91 q^{90} -34702.7 q^{91} -5193.50 q^{92} +8975.11 q^{93} +51757.0 q^{94} -31657.3 q^{95} -40500.3 q^{96} +99383.8 q^{97} +22198.0 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 7 q^{2} - 27 q^{3} + 73 q^{4} - 75 q^{5} - 63 q^{6} + 92 q^{7} + 231 q^{8} + 243 q^{9} - 175 q^{10} + 363 q^{11} - 657 q^{12} - 90 q^{13} - 784 q^{14} + 675 q^{15} - 415 q^{16} + 1934 q^{17} + 567 q^{18}+ \cdots + 29403 q^{99}+O(q^{100}) $ 3 * q + 7 * q^2 - 27 * q^3 + 73 * q^4 - 75 * q^5 - 63 * q^6 + 92 * q^7 + 231 * q^8 + 243 * q^9 - 175 * q^10 + 363 * q^11 - 657 * q^12 - 90 * q^13 - 784 * q^14 + 675 * q^15 - 415 * q^16 + 1934 * q^17 + 567 * q^18 + 2084 * q^19 - 1825 * q^20 - 828 * q^21 + 847 * q^22 + 1220 * q^23 - 2079 * q^24 + 1875 * q^25 + 17062 * q^26 - 2187 * q^27 + 11120 * q^28 + 4402 * q^29 + 1575 * q^30 - 10688 * q^31 + 12439 * q^32 - 3267 * q^33 - 4094 * q^34 - 2300 * q^35 + 5913 * q^36 - 8190 * q^37 + 13792 * q^38 + 810 * q^39 - 5775 * q^40 + 5974 * q^41 + 7056 * q^42 + 18868 * q^43 + 8833 * q^44 - 6075 * q^45 + 46220 * q^46 + 55500 * q^47 + 3735 * q^48 + 1907 * q^49 + 4375 * q^50 - 17406 * q^51 + 27330 * q^52 + 9206 * q^53 - 5103 * q^54 - 9075 * q^55 + 73248 * q^56 - 18756 * q^57 + 15366 * q^58 - 59196 * q^59 + 16425 * q^60 + 79902 * q^61 + 64616 * q^62 + 7452 * q^63 + 2129 * q^64 + 2250 * q^65 - 7623 * q^66 + 4468 * q^67 - 1218 * q^68 - 10980 * q^69 + 19600 * q^70 - 75164 * q^71 + 18711 * q^72 - 61290 * q^73 - 56766 * q^74 - 16875 * q^75 + 37816 * q^76 + 11132 * q^77 - 153558 * q^78 - 83564 * q^79 + 10375 * q^80 + 19683 * q^81 + 147410 * q^82 + 74764 * q^83 - 100080 * q^84 - 48350 * q^85 - 253432 * q^86 - 39618 * q^87 + 27951 * q^88 + 37342 * q^89 - 14175 * q^90 - 126488 * q^91 + 148164 * q^92 + 96192 * q^93 + 59252 * q^94 - 52100 * q^95 - 111951 * q^96 + 33486 * q^97 - 95249 * 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 $

$2$	4.30119	0.760350	0.380175	−	0.924915i	$-0.375864\pi$
$2$	4.30119	0.760350	0.380175	+	0.924915i	$0.375864\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−13.4998	−0.421868
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−38.7107	−0.438988
$7$	−148.216	−1.14327	−0.571635	−	0.820508i	$-0.693691\pi$
$7$	−148.216	−1.14327	−0.571635	+	0.820508i	$0.693691\pi$
$8$	−195.703	−1.08112
$9$	81.0000	0.333333
$10$	−107.530	−0.340039
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	121.498	0.243565
$13$	234.137	0.384247	0.192124	−	0.981371i	$-0.438463\pi$
$13$	234.137	0.384247	0.192124	+	0.981371i	$0.438463\pi$
$14$	−637.504	−0.869286
$15$	225.000	0.258199
$16$	−409.764	−0.400160
$17$	1031.76	0.865875	0.432937	−	0.901424i	$-0.357477\pi$
$17$	1031.76	0.865875	0.432937	+	0.901424i	$0.357477\pi$
$18$	348.396	0.253450
$19$	1266.29	0.804729	0.402365	−	0.915480i	$-0.368188\pi$
$19$	1266.29	0.804729	0.402365	+	0.915480i	$0.368188\pi$
$20$	337.494	0.188665
$21$	1333.94	0.660068
$22$	520.444	0.229254
$23$	384.710	0.151640	0.0758201	−	0.997122i	$-0.475843\pi$
$23$	384.710	0.151640	0.0758201	+	0.997122i	$0.475843\pi$
$24$	1761.33	0.624183
$25$	625.000	0.200000
$26$	1007.07	0.292163
$27$	−729.000	−0.192450
$28$	2000.88	0.482309
$29$	−4484.85	−0.990268	−0.495134	−	0.868817i	$-0.664881\pi$
$29$	−4484.85	−0.990268	−0.495134	+	0.868817i	$0.664881\pi$
$30$	967.768	0.196322
$31$	−997.235	−0.186377	−0.0931887	−	0.995648i	$-0.529706\pi$
$31$	−997.235	−0.186377	−0.0931887	+	0.995648i	$0.529706\pi$
$32$	4500.03	0.776856
$33$	−1089.00	−0.174078
$34$	4437.78	0.658368
$35$	3705.39	0.511286
$36$	−1093.48	−0.140623
$37$	5168.11	0.620622	0.310311	−	0.950635i	$-0.399567\pi$
$37$	5168.11	0.620622	0.310311	+	0.950635i	$0.399567\pi$
$38$	5446.56	0.611876
$39$	−2107.23	−0.221845
$40$	4892.58	0.483490
$41$	2259.17	0.209889	0.104944	−	0.994478i	$-0.466534\pi$
$41$	2259.17	0.209889	0.104944	+	0.994478i	$0.466534\pi$
$42$	5737.54	0.501883
$43$	15818.9	1.30469	0.652343	−	0.757924i	$-0.273786\pi$
$43$	15818.9	1.30469	0.652343	+	0.757924i	$0.273786\pi$
$44$	−1633.47	−0.127198
$45$	−2025.00	−0.149071
$46$	1654.71	0.115300
$47$	12033.2	0.794577	0.397289	−	0.917694i	$-0.369951\pi$
$47$	12033.2	0.794577	0.397289	+	0.917694i	$0.369951\pi$
$48$	3687.87	0.231032
$49$	5160.90	0.307069
$50$	2688.24	0.152070
$51$	−9285.81	−0.499913
$52$	−3160.79	−0.162102
$53$	−3851.67	−0.188347	−0.0941737	−	0.995556i	$-0.530021\pi$
$53$	−3851.67	−0.188347	−0.0941737	+	0.995556i	$0.530021\pi$
$54$	−3135.57	−0.146329
$55$	−3025.00	−0.134840
$56$	29006.3	1.23601
$57$	−11396.6	−0.464611
$58$	−19290.2	−0.752950
$59$	−20261.0	−0.757760	−0.378880	−	0.925446i	$-0.623691\pi$
$59$	−20261.0	−0.757760	−0.378880	+	0.925446i	$0.623691\pi$
$60$	−3037.45	−0.108926
$61$	2006.88	0.0690553	0.0345276	−	0.999404i	$-0.489007\pi$
$61$	2006.88	0.0690553	0.0345276	+	0.999404i	$0.489007\pi$
$62$	−4289.30	−0.141712
$63$	−12005.5	−0.381090
$64$	32467.9	0.990842
$65$	−5853.41	−0.171841
$66$	−4684.00	−0.132360
$67$	40945.6	1.11435	0.557173	−	0.830397i	$-0.311886\pi$
$67$	40945.6	1.11435	0.557173	+	0.830397i	$0.311886\pi$
$68$	−13928.5	−0.365285
$69$	−3462.39	−0.0875495
$70$	15937.6	0.388757
$71$	16970.8	0.399537	0.199769	−	0.979843i	$-0.435981\pi$
$71$	16970.8	0.399537	0.199769	+	0.979843i	$0.435981\pi$
$72$	−15852.0	−0.360372
$73$	−56640.5	−1.24400	−0.621999	−	0.783018i	$-0.713679\pi$
$73$	−56640.5	−1.24400	−0.621999	+	0.783018i	$0.713679\pi$
$74$	22229.0	0.471890
$75$	−5625.00	−0.115470
$76$	−17094.6	−0.339489
$77$	−17934.1	−0.344709
$78$	−9063.59	−0.168680
$79$	58507.9	1.05474	0.527372	−	0.849635i	$-0.323178\pi$
$79$	58507.9	1.05474	0.527372	+	0.849635i	$0.323178\pi$
$80$	10244.1	0.178957
$81$	6561.00	0.111111
$82$	9717.12	0.159589
$83$	−52243.8	−0.832415	−0.416207	−	0.909270i	$-0.636641\pi$
$83$	−52243.8	−0.832415	−0.416207	+	0.909270i	$0.636641\pi$
$84$	−18007.9	−0.278461
$85$	−25793.9	−0.387231
$86$	68040.2	0.992018
$87$	40363.6	0.571732
$88$	−23680.1	−0.325969
$89$	55114.4	0.737548	0.368774	−	0.929519i	$-0.379778\pi$
$89$	55114.4	0.737548	0.368774	+	0.929519i	$0.379778\pi$
$90$	−8709.91	−0.113346
$91$	−34702.7	−0.439299
$92$	−5193.50	−0.0639721
$93$	8975.11	0.107605
$94$	51757.0	0.604157
$95$	−31657.3	−0.359886
$96$	−40500.3	−0.448518
$97$	99383.8	1.07247	0.536237	−	0.844068i	$-0.319846\pi$
$97$	99383.8	1.07247	0.536237	+	0.844068i	$0.319846\pi$
$98$	22198.0	0.233480
$99$	9801.00	0.100504
$100$	−8437.35	−0.0843735
$101$	−25334.3	−0.247118	−0.123559	−	0.992337i	$-0.539431\pi$
$101$	−25334.3	−0.247118	−0.123559	+	0.992337i	$0.539431\pi$
$102$	−39940.0	−0.380109
$103$	−93216.6	−0.865766	−0.432883	−	0.901450i	$-0.642504\pi$
$103$	−93216.6	−0.865766	−0.432883	+	0.901450i	$0.642504\pi$
$104$	−45821.3	−0.415416
$105$	−33348.5	−0.295191
$106$	−16566.8	−0.143210
$107$	205830.	1.73799	0.868997	−	0.494817i	$-0.164765\pi$
$107$	205830.	1.73799	0.868997	+	0.494817i	$0.164765\pi$
$108$	9841.33	0.0811885
$109$	−155901.	−1.25684	−0.628422	−	0.777872i	$-0.716299\pi$
$109$	−155901.	−1.25684	−0.628422	+	0.777872i	$0.716299\pi$
$110$	−13011.1	−0.102526
$111$	−46513.0	−0.358316
$112$	60733.4	0.457491
$113$	40304.5	0.296932	0.148466	−	0.988917i	$-0.452566\pi$
$113$	40304.5	0.296932	0.148466	+	0.988917i	$0.452566\pi$
$114$	−49019.0	−0.353267
$115$	−9617.75	−0.0678155
$116$	60544.4	0.417762
$117$	18965.1	0.128082
$118$	−87146.6	−0.576163
$119$	−152923.	−0.989929
$120$	−44033.2	−0.279143
$121$	14641.0	0.0909091
$122$	8631.97	0.0525062
$123$	−20332.5	−0.121179
$124$	13462.4	0.0786266
$125$	−15625.0	−0.0894427
$126$	−51637.8	−0.289762
$127$	120825.	0.664733	0.332367	−	0.943150i	$-0.392153\pi$
$127$	120825.	0.664733	0.332367	+	0.943150i	$0.392153\pi$
$128$	−4350.25	−0.0234687
$129$	−142370.	−0.753260
$130$	−25176.6	−0.130659
$131$	266474.	1.35668	0.678338	−	0.734750i	$-0.262701\pi$
$131$	266474.	1.35668	0.678338	+	0.734750i	$0.262701\pi$
$132$	14701.2	0.0734377
$133$	−187684.	−0.920023
$134$	176115.	0.847292
$135$	18225.0	0.0860663
$136$	−201918.	−0.936112
$137$	17176.5	0.0781866	0.0390933	−	0.999236i	$-0.487553\pi$
$137$	17176.5	0.0781866	0.0390933	+	0.999236i	$0.487553\pi$
$138$	−14892.4	−0.0665682
$139$	−5031.99	−0.0220903	−0.0110452	−	0.999939i	$-0.503516\pi$
$139$	−5031.99	−0.0220903	−0.0110452	+	0.999939i	$0.503516\pi$
$140$	−50021.9	−0.215695
$141$	−108299.	−0.458749
$142$	72994.7	0.303788
$143$	28330.5	0.115855
$144$	−33190.9	−0.133387
$145$	112121.	0.442861
$146$	−243621.	−0.945874
$147$	−46448.1	−0.177286
$148$	−69768.3	−0.261820
$149$	−37407.9	−0.138038	−0.0690188	−	0.997615i	$-0.521987\pi$
$149$	−37407.9	−0.138038	−0.0690188	+	0.997615i	$0.521987\pi$
$150$	−24194.2	−0.0877977
$151$	−225853.	−0.806091	−0.403046	−	0.915180i	$-0.632048\pi$
$151$	−225853.	−0.806091	−0.403046	+	0.915180i	$0.632048\pi$
$152$	−247817.	−0.870006
$153$	83572.3	0.288625
$154$	−77138.0	−0.262100
$155$	24930.9	0.0833505
$156$	28447.1	0.0935894
$157$	−205537.	−0.665489	−0.332744	−	0.943017i	$-0.607975\pi$
$157$	−205537.	−0.665489	−0.332744	+	0.943017i	$0.607975\pi$
$158$	251654.	0.801974
$159$	34665.1	0.108742
$160$	−112501.	−0.347420
$161$	−57020.1	−0.173366
$162$	28220.1	0.0844833
$163$	−639398.	−1.88496	−0.942481	−	0.334261i	$-0.891513\pi$
$163$	−639398.	−1.88496	−0.942481	+	0.334261i	$0.891513\pi$
$164$	−30498.3	−0.0885453
$165$	27225.0	0.0778499
$166$	−224711.	−0.632927
$167$	416990.	1.15700	0.578502	−	0.815681i	$-0.303637\pi$
$167$	416990.	1.15700	0.578502	+	0.815681i	$0.303637\pi$
$168$	−261057.	−0.713611
$169$	−316473.	−0.852354
$170$	−110945.	−0.294431
$171$	102570.	0.268243
$172$	−213552.	−0.550405
$173$	608651.	1.54615	0.773077	−	0.634312i	$-0.218716\pi$
$173$	608651.	1.54615	0.773077	+	0.634312i	$0.218716\pi$
$174$	173612.	0.434716
$175$	−92634.8	−0.228654
$176$	−49581.4	−0.120653
$177$	182349.	0.437493
$178$	237058.	0.560795
$179$	511566.	1.19335	0.596676	−	0.802482i	$-0.296488\pi$
$179$	511566.	1.19335	0.596676	+	0.802482i	$0.296488\pi$
$180$	27337.0	0.0628883
$181$	56369.0	0.127892	0.0639460	−	0.997953i	$-0.479631\pi$
$181$	56369.0	0.127892	0.0639460	+	0.997953i	$0.479631\pi$
$182$	−149263.	−0.334021
$183$	−18061.9	−0.0398691
$184$	−75289.0	−0.163941
$185$	−129203.	−0.277551
$186$	38603.7	0.0818175
$187$	124843.	0.261071
$188$	−162445.	−0.335206
$189$	108049.	0.220023
$190$	−136164.	−0.273639
$191$	45986.9	0.0912117	0.0456059	−	0.998960i	$-0.485478\pi$
$191$	45986.9	0.0912117	0.0456059	+	0.998960i	$0.485478\pi$
$192$	−292211.	−0.572063
$193$	360889.	0.697398	0.348699	−	0.937235i	$-0.386624\pi$
$193$	360889.	0.697398	0.348699	+	0.937235i	$0.386624\pi$
$194$	427469.	0.815455
$195$	52680.7	0.0992122
$196$	−69671.0	−0.129542
$197$	978922.	1.79714	0.898571	−	0.438828i	$-0.144606\pi$
$197$	978922.	1.79714	0.898571	+	0.438828i	$0.144606\pi$
$198$	42156.0	0.0764181
$199$	902328.	1.61522	0.807610	−	0.589717i	$-0.200761\pi$
$199$	902328.	1.61522	0.807610	+	0.589717i	$0.200761\pi$
$200$	−122314.	−0.216223
$201$	−368510.	−0.643367
$202$	−108968.	−0.187897
$203$	664725.	1.13214
$204$	125356.	0.210897
$205$	−56479.3	−0.0938652
$206$	−400942.	−0.658285
$207$	31161.5	0.0505467
$208$	−95940.7	−0.153760
$209$	153221.	0.242635
$210$	−143438.	−0.224449
$211$	318392.	0.492330	0.246165	−	0.969228i	$-0.420830\pi$
$211$	318392.	0.492330	0.246165	+	0.969228i	$0.420830\pi$
$212$	51996.7	0.0794577
$213$	−152737.	−0.230673
$214$	885312.	1.32148
$215$	−395473.	−0.583473
$216$	142668.	0.208061
$217$	147806.	0.213080
$218$	−670558.	−0.955642
$219$	509764.	0.718223
$220$	40836.8	0.0568846
$221$	241572.	0.332710
$222$	−200061.	−0.272446
$223$	661352.	0.890575	0.445287	−	0.895388i	$-0.353101\pi$
$223$	661352.	0.890575	0.445287	+	0.895388i	$0.353101\pi$
$224$	−666975.	−0.888157
$225$	50625.0	0.0666667
$226$	173357.	0.225773
$227$	606704.	0.781470	0.390735	−	0.920503i	$-0.372221\pi$
$227$	606704.	0.781470	0.390735	+	0.920503i	$0.372221\pi$
$228$	153852.	0.196004
$229$	−352377.	−0.444036	−0.222018	−	0.975043i	$-0.571264\pi$
$229$	−352377.	−0.444036	−0.222018	+	0.975043i	$0.571264\pi$
$230$	−41367.8	−0.0515635
$231$	161407.	0.199018
$232$	877699.	1.07060
$233$	1.49288e6	1.80151	0.900755	−	0.434328i	$-0.143014\pi$
$233$	1.49288e6	1.80151	0.900755	+	0.434328i	$0.143014\pi$
$234$	81572.3	0.0973875
$235$	−300830.	−0.355346
$236$	273519.	0.319675
$237$	−526571.	−0.608956
$238$	−657749.	−0.752693
$239$	388141.	0.439536	0.219768	−	0.975552i	$-0.429470\pi$
$239$	388141.	0.439536	0.219768	+	0.975552i	$0.429470\pi$
$240$	−92196.8	−0.103321
$241$	−75337.2	−0.0835540	−0.0417770	−	0.999127i	$-0.513302\pi$
$241$	−75337.2	−0.0835540	−0.0417770	+	0.999127i	$0.513302\pi$
$242$	62973.7	0.0691227
$243$	−59049.0	−0.0641500
$244$	−27092.4	−0.0291322
$245$	−129023.	−0.137325
$246$	−87454.1	−0.0921388
$247$	296485.	0.309215
$248$	195162.	0.201496
$249$	470194.	0.480595
$250$	−67206.1	−0.0680078
$251$	−346975.	−0.347627	−0.173814	−	0.984779i	$-0.555609\pi$
$251$	−346975.	−0.347627	−0.173814	+	0.984779i	$0.555609\pi$
$252$	162071.	0.160770
$253$	46549.9	0.0457212
$254$	519691.	0.505430
$255$	232145.	0.223568
$256$	−1.05768e6	−1.00869
$257$	1.97267e6	1.86304	0.931518	−	0.363696i	$-0.118485\pi$
$257$	1.97267e6	1.86304	0.931518	+	0.363696i	$0.118485\pi$
$258$	−612362.	−0.572742
$259$	−765995.	−0.709539
$260$	79019.7	0.0724940
$261$	−363273.	−0.330089
$262$	1.14615e6	1.03155
$263$	−190986.	−0.170260	−0.0851300	−	0.996370i	$-0.527131\pi$
$263$	−190986.	−0.170260	−0.0851300	+	0.996370i	$0.527131\pi$
$264$	213121.	0.188198
$265$	96291.8	0.0842316
$266$	−807266.	−0.699540
$267$	−496030.	−0.425823
$268$	−552755.	−0.470106
$269$	933912.	0.786911	0.393455	−	0.919344i	$-0.371280\pi$
$269$	933912.	0.786911	0.393455	+	0.919344i	$0.371280\pi$
$270$	78389.2	0.0654405
$271$	−1.17171e6	−0.969163	−0.484582	−	0.874746i	$-0.661028\pi$
$271$	−1.17171e6	−0.969163	−0.484582	+	0.874746i	$0.661028\pi$
$272$	−422776.	−0.346488
$273$	312325.	0.253629
$274$	73879.3	0.0594492
$275$	75625.0	0.0603023
$276$	46741.5	0.0369343
$277$	−472512.	−0.370010	−0.185005	−	0.982738i	$-0.559230\pi$
$277$	−472512.	−0.370010	−0.185005	+	0.982738i	$0.559230\pi$
$278$	−21643.5	−0.0167964
$279$	−80776.0	−0.0621258
$280$	−725157.	−0.552760
$281$	−2.05374e6	−1.55160	−0.775798	−	0.630981i	$-0.782652\pi$
$281$	−2.05374e6	−1.55160	−0.775798	+	0.630981i	$0.782652\pi$
$282$	−465813.	−0.348810
$283$	465578.	0.345562	0.172781	−	0.984960i	$-0.444725\pi$
$283$	465578.	0.345562	0.172781	+	0.984960i	$0.444725\pi$
$284$	−229102.	−0.168552
$285$	284916.	0.207780
$286$	121855.	0.0880903
$287$	−334845.	−0.239960
$288$	364502.	0.258952
$289$	−355335.	−0.250261
$290$	482255.	0.336730
$291$	−894454.	−0.619193
$292$	764633.	0.524803
$293$	−1.19434e6	−0.812751	−0.406376	−	0.913706i	$-0.633208\pi$
$293$	−1.19434e6	−0.812751	−0.406376	+	0.913706i	$0.633208\pi$
$294$	−199782.	−0.134800
$295$	506526.	0.338881
$296$	−1.01141e6	−0.670965
$297$	−88209.0	−0.0580259
$298$	−160898.	−0.104957
$299$	90074.7	0.0582673
$300$	75936.2	0.0487131
$301$	−2.34461e6	−1.49161
$302$	−971438.	−0.612912
$303$	228009.	0.142674
$304$	−518880.	−0.322020
$305$	−50172.0	−0.0308825
$306$	359460.	0.219456
$307$	1.72251e6	1.04308	0.521538	−	0.853228i	$-0.325359\pi$
$307$	1.72251e6	1.04308	0.521538	+	0.853228i	$0.325359\pi$
$308$	242106.	0.145422
$309$	838950.	0.499850
$310$	107232.	0.0633755
$311$	−2.60616e6	−1.52792	−0.763960	−	0.645263i	$-0.776748\pi$
$311$	−2.60616e6	−1.52792	−0.763960	+	0.645263i	$0.776748\pi$
$312$	412391.	0.239841
$313$	1.22311e6	0.705677	0.352839	−	0.935684i	$-0.385216\pi$
$313$	1.22311e6	0.705677	0.352839	+	0.935684i	$0.385216\pi$
$314$	−884053.	−0.506005
$315$	300137.	0.170429
$316$	−789843.	−0.444962
$317$	−700043.	−0.391270	−0.195635	−	0.980677i	$-0.562677\pi$
$317$	−700043.	−0.391270	−0.195635	+	0.980677i	$0.562677\pi$
$318$	149101.	0.0826824
$319$	−542667.	−0.298577
$320$	−811698.	−0.443118
$321$	−1.85247e6	−1.00343
$322$	−245254.	−0.131819
$323$	1.30650e6	0.696794
$324$	−88572.0	−0.0468742
$325$	146335.	0.0768495
$326$	−2.75017e6	−1.43323
$327$	1.40311e6	0.725640
$328$	−442127.	−0.226914
$329$	−1.78351e6	−0.908417
$330$	117100.	0.0591932
$331$	2.78738e6	1.39839	0.699193	−	0.714933i	$-0.253543\pi$
$331$	2.78738e6	1.39839	0.699193	+	0.714933i	$0.253543\pi$
$332$	705279.	0.351169
$333$	418617.	0.206874
$334$	1.79355e6	0.879728
$335$	−1.02364e6	−0.498350
$336$	−546601.	−0.264133
$337$	−795333.	−0.381482	−0.190741	−	0.981640i	$-0.561089\pi$
$337$	−795333.	−0.381482	−0.190741	+	0.981640i	$0.561089\pi$
$338$	−1.36121e6	−0.648087
$339$	−362741.	−0.171434
$340$	348212.	0.163360
$341$	−120665.	−0.0561949
$342$	441171.	0.203959
$343$	1.72613e6	0.792208
$344$	−3.09581e6	−1.41052
$345$	86559.8	0.0391533
$346$	2.61792e6	1.17562
$347$	−3.80533e6	−1.69656	−0.848278	−	0.529551i	$-0.822361\pi$
$347$	−3.80533e6	−1.69656	−0.848278	+	0.529551i	$0.822361\pi$
$348$	−544900.	−0.241195
$349$	−798008.	−0.350706	−0.175353	−	0.984506i	$-0.556107\pi$
$349$	−798008.	−0.350706	−0.175353	+	0.984506i	$0.556107\pi$
$350$	−398440.	−0.173857
$351$	−170686.	−0.0739484
$352$	544503.	0.234231
$353$	2.48978e6	1.06347	0.531733	−	0.846912i	$-0.321541\pi$
$353$	2.48978e6	1.06347	0.531733	+	0.846912i	$0.321541\pi$
$354$	784319.	0.332648
$355$	−424271.	−0.178678
$356$	−744032.	−0.311148
$357$	1.37630e6	0.571536
$358$	2.20034e6	0.907366
$359$	−2.15148e6	−0.881050	−0.440525	−	0.897740i	$-0.645208\pi$
$359$	−2.15148e6	−0.881050	−0.440525	+	0.897740i	$0.645208\pi$
$360$	396299.	0.161163
$361$	−872605.	−0.352411
$362$	242454.	0.0972427
$363$	−131769.	−0.0524864
$364$	468479.	0.185326
$365$	1.41601e6	0.556333
$366$	−77687.8	−0.0303145
$367$	1.47515e6	0.571705	0.285853	−	0.958274i	$-0.407723\pi$
$367$	1.47515e6	0.571705	0.285853	+	0.958274i	$0.407723\pi$
$368$	−157640.	−0.0606803
$369$	182993.	0.0699630
$370$	−555725.	−0.211036
$371$	570879.	0.215332
$372$	−121162.	−0.0453951
$373$	−4.71290e6	−1.75394	−0.876972	−	0.480541i	$-0.840440\pi$
$373$	−4.71290e6	−1.75394	−0.876972	+	0.480541i	$0.840440\pi$
$374$	536972.	0.198505
$375$	140625.	0.0516398
$376$	−2.35493e6	−0.859031
$377$	−1.05007e6	−0.380508
$378$	464740.	0.167294
$379$	4.47900e6	1.60171	0.800853	−	0.598861i	$-0.204380\pi$
$379$	4.47900e6	1.60171	0.800853	+	0.598861i	$0.204380\pi$
$380$	427366.	0.151824
$381$	−1.08742e6	−0.383784
$382$	197798.	0.0693528
$383$	−57968.7	−0.0201928	−0.0100964	−	0.999949i	$-0.503214\pi$
$383$	−57968.7	−0.0201928	−0.0100964	+	0.999949i	$0.503214\pi$
$384$	39152.3	0.0135497
$385$	448353.	0.154159
$386$	1.55225e6	0.530267
$387$	1.28133e6	0.434895
$388$	−1.34166e6	−0.452442
$389$	−4.53325e6	−1.51892	−0.759461	−	0.650553i	$-0.774537\pi$
$389$	−4.53325e6	−1.51892	−0.759461	+	0.650553i	$0.774537\pi$
$390$	226590.	0.0754360
$391$	396927.	0.131301
$392$	−1.01000e6	−0.331977
$393$	−2.39826e6	−0.783277
$394$	4.21053e6	1.36646
$395$	−1.46270e6	−0.471695
$396$	−132311.	−0.0423993
$397$	621573.	0.197932	0.0989660	−	0.995091i	$-0.468446\pi$
$397$	621573.	0.197932	0.0989660	+	0.995091i	$0.468446\pi$
$398$	3.88108e6	1.22813
$399$	1.68916e6	0.531176
$400$	−256102.	−0.0800320
$401$	954326.	0.296371	0.148186	−	0.988960i	$-0.452657\pi$
$401$	954326.	0.296371	0.148186	+	0.988960i	$0.452657\pi$
$402$	−1.58503e6	−0.489185
$403$	−233489.	−0.0716150
$404$	342007.	0.104251
$405$	−164025.	−0.0496904
$406$	2.85911e6	0.860826
$407$	625341.	0.187125
$408$	1.81726e6	0.540465
$409$	3.19864e6	0.945491	0.472746	−	0.881199i	$-0.343263\pi$
$409$	3.19864e6	0.945491	0.472746	+	0.881199i	$0.343263\pi$
$410$	−242928.	−0.0713704
$411$	−154588.	−0.0451411
$412$	1.25840e6	0.365239
$413$	3.00301e6	0.866325
$414$	134032.	0.0384332
$415$	1.30610e6	0.372267
$416$	1.05362e6	0.298505
$417$	45287.9	0.0127539
$418$	659034.	0.184487
$419$	−6.60122e6	−1.83691	−0.918457	−	0.395520i	$-0.870564\pi$
$419$	−6.60122e6	−1.83691	−0.918457	+	0.395520i	$0.870564\pi$
$420$	450198.	0.124532
$421$	−4.22510e6	−1.16180	−0.580901	−	0.813974i	$-0.697300\pi$
$421$	−4.22510e6	−1.16180	−0.580901	+	0.813974i	$0.697300\pi$
$422$	1.36946e6	0.374343
$423$	974688.	0.264859
$424$	753785.	0.203626
$425$	644848.	0.173175
$426$	−656953.	−0.175392
$427$	−297451.	−0.0789489
$428$	−2.77865e6	−0.733204
$429$	−254975.	−0.0668889
$430$	−1.70100e6	−0.443644
$431$	4.72300e6	1.22469	0.612344	−	0.790592i	$-0.290227\pi$
$431$	4.72300e6	1.22469	0.612344	+	0.790592i	$0.290227\pi$
$432$	298718.	0.0770108
$433$	−882129.	−0.226106	−0.113053	−	0.993589i	$-0.536063\pi$
$433$	−882129.	−0.226106	−0.113053	+	0.993589i	$0.536063\pi$
$434$	635741.	0.162015
$435$	−1.00909e6	−0.255686
$436$	2.10462e6	0.530222
$437$	487155.	0.122029
$438$	2.19259e6	0.546101
$439$	−3.07514e6	−0.761560	−0.380780	−	0.924666i	$-0.624344\pi$
$439$	−3.07514e6	−0.761560	−0.380780	+	0.924666i	$0.624344\pi$
$440$	592002.	0.145778
$441$	418033.	0.102356
$442$	1.03905e6	0.252976
$443$	3.40381e6	0.824055	0.412028	−	0.911171i	$-0.364821\pi$
$443$	3.40381e6	0.824055	0.412028	+	0.911171i	$0.364821\pi$
$444$	627914.	0.151162
$445$	−1.37786e6	−0.329841
$446$	2.84460e6	0.677149
$447$	336671.	0.0796961
$448$	−4.81226e6	−1.13280
$449$	−605757.	−0.141802	−0.0709010	−	0.997483i	$-0.522587\pi$
$449$	−605757.	−0.141802	−0.0709010	+	0.997483i	$0.522587\pi$
$450$	217748.	0.0506900
$451$	273360.	0.0632839
$452$	−544102.	−0.125266
$453$	2.03268e6	0.465397
$454$	2.60955e6	0.594191
$455$	867568.	0.196460
$456$	2.23035e6	0.502298
$457$	−5.46329e6	−1.22367	−0.611834	−	0.790986i	$-0.709568\pi$
$457$	−5.46329e6	−1.22367	−0.611834	+	0.790986i	$0.709568\pi$
$458$	−1.51564e6	−0.337623
$459$	−752151.	−0.166638
$460$	129837.	0.0286092
$461$	−2.84223e6	−0.622883	−0.311441	−	0.950265i	$-0.600812\pi$
$461$	−2.84223e6	−0.622883	−0.311441	+	0.950265i	$0.600812\pi$
$462$	694242.	0.151323
$463$	−8.51652e6	−1.84633	−0.923166	−	0.384401i	$-0.874408\pi$
$463$	−8.51652e6	−1.84633	−0.923166	+	0.384401i	$0.874408\pi$
$464$	1.83773e6	0.396266
$465$	−224378.	−0.0481224
$466$	6.42118e6	1.36978
$467$	7.00140e6	1.48557	0.742784	−	0.669531i	$-0.233505\pi$
$467$	7.00140e6	1.48557	0.742784	+	0.669531i	$0.233505\pi$
$468$	−256024.	−0.0540339
$469$	−6.06877e6	−1.27400
$470$	−1.29393e6	−0.270187
$471$	1.84983e6	0.384220
$472$	3.96515e6	0.819228
$473$	1.91409e6	0.393377
$474$	−2.26488e6	−0.463020
$475$	791432.	0.160946
$476$	2.06442e6	0.417619
$477$	−311986.	−0.0627825
$478$	1.66947e6	0.334201
$479$	1.06826e6	0.212734	0.106367	−	0.994327i	$-0.466078\pi$
$479$	1.06826e6	0.212734	0.106367	+	0.994327i	$0.466078\pi$
$480$	1.01251e6	0.200583
$481$	1.21004e6	0.238472
$482$	−324040.	−0.0635303
$483$	513181.	0.100093
$484$	−197650.	−0.0383516
$485$	−2.48460e6	−0.479625
$486$	−253981.	−0.0487765
$487$	1.00308e6	0.191652	0.0958260	−	0.995398i	$-0.469451\pi$
$487$	1.00308e6	0.191652	0.0958260	+	0.995398i	$0.469451\pi$
$488$	−392753.	−0.0746569
$489$	5.75458e6	1.08828
$490$	−554950.	−0.104415
$491$	9.23719e6	1.72916	0.864582	−	0.502491i	$-0.167583\pi$
$491$	9.23719e6	1.72916	0.864582	+	0.502491i	$0.167583\pi$
$492$	274484.	0.0511217
$493$	−4.62727e6	−0.857448
$494$	1.27524e6	0.235112
$495$	−245025.	−0.0449467
$496$	408631.	0.0745807
$497$	−2.51534e6	−0.456779
$498$	2.02240e6	0.365420
$499$	4.31532e6	0.775822	0.387911	−	0.921697i	$-0.373197\pi$
$499$	4.31532e6	0.775822	0.387911	+	0.921697i	$0.373197\pi$
$500$	210934.	0.0377330
$501$	−3.75291e6	−0.667997
$502$	−1.49240e6	−0.264318
$503$	3.28052e6	0.578127	0.289064	−	0.957310i	$-0.406656\pi$
$503$	3.28052e6	0.578127	0.289064	+	0.957310i	$0.406656\pi$
$504$	2.34951e6	0.412003
$505$	633357.	0.110515
$506$	200220.	0.0347641
$507$	2.84826e6	0.492107
$508$	−1.63111e6	−0.280430
$509$	−2.50882e6	−0.429215	−0.214608	−	0.976700i	$-0.568847\pi$
$509$	−2.50882e6	−0.429215	−0.214608	+	0.976700i	$0.568847\pi$
$510$	998501.	0.169990
$511$	8.39501e6	1.42223
$512$	−4.41009e6	−0.743486
$513$	−923126.	−0.154870
$514$	8.48482e6	1.41656
$515$	2.33042e6	0.387182
$516$	1.92197e6	0.317776
$517$	1.45602e6	0.239574
$518$	−3.29469e6	−0.539498
$519$	−5.47786e6	−0.892673
$520$	1.14553e6	0.185780
$521$	−3.63347e6	−0.586444	−0.293222	−	0.956044i	$-0.594728\pi$
$521$	−3.63347e6	−0.586444	−0.293222	+	0.956044i	$0.594728\pi$
$522$	−1.56250e6	−0.250983
$523$	−7.95990e6	−1.27249	−0.636243	−	0.771488i	$-0.719513\pi$
$523$	−7.95990e6	−1.27249	−0.636243	+	0.771488i	$0.719513\pi$
$524$	−3.59733e6	−0.572338
$525$	833713.	0.132014
$526$	−821468.	−0.129457
$527$	−1.02890e6	−0.161379
$528$	446233.	0.0696589
$529$	−6.28834e6	−0.977005
$530$	414170.	0.0640455
$531$	−1.64114e6	−0.252587
$532$	2.53369e6	0.388128
$533$	528954.	0.0806492
$534$	−2.13352e6	−0.323775
$535$	−5.14574e6	−0.777255
$536$	−8.01317e6	−1.20474
$537$	−4.60409e6	−0.688983
$538$	4.01693e6	0.598328
$539$	624469.	0.0925847
$540$	−246033.	−0.0363086
$541$	1.20412e6	0.176879	0.0884395	−	0.996082i	$-0.471812\pi$
$541$	1.20412e6	0.176879	0.0884395	+	0.996082i	$0.471812\pi$
$542$	−5.03975e6	−0.736903
$543$	−507321.	−0.0738385
$544$	4.64293e6	0.672660
$545$	3.89752e6	0.562078
$546$	1.34337e6	0.192847
$547$	−1.04446e7	−1.49254	−0.746268	−	0.665646i	$-0.768156\pi$
$547$	−1.04446e7	−1.49254	−0.746268	+	0.665646i	$0.768156\pi$
$548$	−231878.	−0.0329844
$549$	162557.	0.0230184
$550$	325277.	0.0458508
$551$	−5.67912e6	−0.796897
$552$	677601.	0.0946512
$553$	−8.67179e6	−1.20586
$554$	−2.03236e6	−0.281337
$555$	1.16282e6	0.160244
$556$	67930.6	0.00931920
$557$	8.45756e6	1.15507	0.577534	−	0.816367i	$-0.304015\pi$
$557$	8.45756e6	1.15507	0.577534	+	0.816367i	$0.304015\pi$
$558$	−347433.	−0.0472373
$559$	3.70379e6	0.501322
$560$	−1.51834e6	−0.204596
$561$	−1.12358e6	−0.150729
$562$	−8.83350e6	−1.17976
$563$	9.96372e6	1.32480	0.662401	−	0.749150i	$-0.269538\pi$
$563$	9.96372e6	1.32480	0.662401	+	0.749150i	$0.269538\pi$
$564$	1.46201e6	0.193532
$565$	−1.00761e6	−0.132792
$566$	2.00254e6	0.262748
$567$	−972443.	−0.127030
$568$	−3.32124e6	−0.431946
$569$	−1.05157e7	−1.36163	−0.680813	−	0.732458i	$-0.738373\pi$
$569$	−1.05157e7	−1.36163	−0.680813	+	0.732458i	$0.738373\pi$
$570$	1.22548e6	0.157986
$571$	9.02900e6	1.15891	0.579454	−	0.815005i	$-0.303266\pi$
$571$	9.02900e6	1.15891	0.579454	+	0.815005i	$0.303266\pi$
$572$	−382456.	−0.0488755
$573$	−413882.	−0.0526611
$574$	−1.44023e6	−0.182453
$575$	240444.	0.0303280
$576$	2.62990e6	0.330281
$577$	6.65688e6	0.832398	0.416199	−	0.909273i	$-0.363362\pi$
$577$	6.65688e6	0.832398	0.416199	+	0.909273i	$0.363362\pi$
$578$	−1.52836e6	−0.190286
$579$	−3.24800e6	−0.402643
$580$	−1.51361e6	−0.186829
$581$	7.74336e6	0.951676
$582$	−3.84722e6	−0.470803
$583$	−466053.	−0.0567889
$584$	1.10847e7	1.34491
$585$	−474127.	−0.0572802
$586$	−5.13707e6	−0.617975
$587$	2.20946e6	0.264661	0.132331	−	0.991206i	$-0.457754\pi$
$587$	2.20946e6	0.264661	0.132331	+	0.991206i	$0.457754\pi$
$588$	627039.	0.0747913
$589$	−1.26279e6	−0.149983
$590$	2.17866e6	0.257668
$591$	−8.81030e6	−1.03758
$592$	−2.11770e6	−0.248348
$593$	−261524.	−0.0305404	−0.0152702	−	0.999883i	$-0.504861\pi$
$593$	−261524.	−0.0305404	−0.0152702	+	0.999883i	$0.504861\pi$
$594$	−379404.	−0.0441200
$595$	3.82306e6	0.442710
$596$	504998.	0.0582336
$597$	−8.12095e6	−0.932548
$598$	387429.	0.0443036
$599$	7.09631e6	0.808101	0.404050	−	0.914737i	$-0.367602\pi$
$599$	7.09631e6	0.808101	0.404050	+	0.914737i	$0.367602\pi$
$600$	1.10083e6	0.124837
$601$	6.04793e6	0.683000	0.341500	−	0.939882i	$-0.389065\pi$
$601$	6.04793e6	0.683000	0.341500	+	0.939882i	$0.389065\pi$
$602$	−1.00846e7	−1.13414
$603$	3.31659e6	0.371448
$604$	3.04897e6	0.340064
$605$	−366025.	−0.0406558
$606$	980708.	0.108482
$607$	−1.05203e7	−1.15892	−0.579461	−	0.815000i	$-0.696737\pi$
$607$	−1.05203e7	−1.15892	−0.579461	+	0.815000i	$0.696737\pi$
$608$	5.69835e6	0.625158
$609$	−5.98253e6	−0.653644
$610$	−215799.	−0.0234815
$611$	2.81741e6	0.305314
$612$	−1.12821e6	−0.121762
$613$	−8.55961e6	−0.920032	−0.460016	−	0.887911i	$-0.652156\pi$
$613$	−8.55961e6	−0.920032	−0.460016	+	0.887911i	$0.652156\pi$
$614$	7.40884e6	0.793103
$615$	508313.	0.0541931
$616$	3.50976e6	0.372671
$617$	−6.18055e6	−0.653604	−0.326802	−	0.945093i	$-0.605971\pi$
$617$	−6.18055e6	−0.653604	−0.326802	+	0.945093i	$0.605971\pi$
$618$	3.60848e6	0.380061
$619$	−1.12540e7	−1.18053	−0.590267	−	0.807208i	$-0.700978\pi$
$619$	−1.12540e7	−1.18053	−0.590267	+	0.807208i	$0.700978\pi$
$620$	−336561.	−0.0351629
$621$	−280454.	−0.0291832
$622$	−1.12096e7	−1.16175
$623$	−8.16882e6	−0.843217
$624$	863466.	0.0887736
$625$	390625.	0.0400000
$626$	5.26085e6	0.536562
$627$	−1.37899e6	−0.140085
$628$	2.77470e6	0.280748
$629$	5.33223e6	0.537381
$630$	1.29095e6	0.129586
$631$	1.17369e7	1.17349	0.586746	−	0.809771i	$-0.300409\pi$
$631$	1.17369e7	1.17349	0.586746	+	0.809771i	$0.300409\pi$
$632$	−1.14502e7	−1.14030
$633$	−2.86553e6	−0.284247
$634$	−3.01102e6	−0.297502
$635$	−3.02062e6	−0.297278
$636$	−467970.	−0.0458749
$637$	1.20836e6	0.117990
$638$	−2.33411e6	−0.227023
$639$	1.37464e6	0.133179
$640$	108756.	0.0104955
$641$	6.80878e6	0.654522	0.327261	−	0.944934i	$-0.393874\pi$
$641$	6.80878e6	0.654522	0.327261	+	0.944934i	$0.393874\pi$
$642$	−7.96781e6	−0.762959
$643$	−1.16456e7	−1.11080	−0.555400	−	0.831583i	$-0.687435\pi$
$643$	−1.16456e7	−1.11080	−0.555400	+	0.831583i	$0.687435\pi$
$644$	769758.	0.0731374
$645$	3.55926e6	0.336868
$646$	5.61952e6	0.529808
$647$	2.61990e6	0.246050	0.123025	−	0.992404i	$-0.460740\pi$
$647$	2.61990e6	0.246050	0.123025	+	0.992404i	$0.460740\pi$
$648$	−1.28401e6	−0.120124
$649$	−2.45159e6	−0.228473
$650$	629416.	0.0584325
$651$	−1.33025e6	−0.123022
$652$	8.63173e6	0.795204
$653$	1.48492e7	1.36276	0.681379	−	0.731931i	$-0.261381\pi$
$653$	1.48492e7	1.36276	0.681379	+	0.731931i	$0.261381\pi$
$654$	6.03502e6	0.551740
$655$	−6.66184e6	−0.606724
$656$	−925726.	−0.0839891
$657$	−4.58788e6	−0.414666
$658$	−7.67120e6	−0.690715
$659$	−2.47470e6	−0.221977	−0.110989	−	0.993822i	$-0.535402\pi$
$659$	−2.47470e6	−0.221977	−0.110989	+	0.993822i	$0.535402\pi$
$660$	−367531.	−0.0328424
$661$	7.61212e6	0.677645	0.338822	−	0.940850i	$-0.389971\pi$
$661$	7.61212e6	0.677645	0.338822	+	0.940850i	$0.389971\pi$
$662$	1.19891e7	1.06326
$663$	−2.17415e6	−0.192090
$664$	1.02243e7	0.899938
$665$	4.69211e6	0.411447
$666$	1.80055e6	0.157297
$667$	−1.72537e6	−0.150164
$668$	−5.62927e6	−0.488103
$669$	−5.95217e6	−0.514174
$670$	−4.40286e6	−0.378921
$671$	242833.	0.0208210
$672$	6.00278e6	0.512777
$673$	6.62472e6	0.563807	0.281903	−	0.959443i	$-0.409034\pi$
$673$	6.62472e6	0.563807	0.281903	+	0.959443i	$0.409034\pi$
$674$	−3.42088e6	−0.290060
$675$	−455625.	−0.0384900
$676$	4.27231e6	0.359581
$677$	228067.	0.0191245	0.00956227	−	0.999954i	$-0.496956\pi$
$677$	228067.	0.0191245	0.00956227	+	0.999954i	$0.496956\pi$
$678$	−1.56022e6	−0.130350
$679$	−1.47302e7	−1.22613
$680$	5.04795e6	0.418642
$681$	−5.46034e6	−0.451182
$682$	−519005.	−0.0427278
$683$	−1.58370e6	−0.129903	−0.0649517	−	0.997888i	$-0.520689\pi$
$683$	−1.58370e6	−0.129903	−0.0649517	+	0.997888i	$0.520689\pi$
$684$	−1.38467e6	−0.113163
$685$	−429412.	−0.0349661
$686$	7.42443e6	0.602356
$687$	3.17139e6	0.256364
$688$	−6.48202e6	−0.522083
$689$	−901818.	−0.0723720
$690$	372310.	0.0297702
$691$	2.07411e7	1.65248	0.826241	−	0.563317i	$-0.190475\pi$
$691$	2.07411e7	1.65248	0.826241	+	0.563317i	$0.190475\pi$
$692$	−8.21665e6	−0.652273
$693$	−1.45266e6	−0.114903
$694$	−1.63674e7	−1.28998
$695$	125800.	0.00987910
$696$	−7.89929e6	−0.618109
$697$	2.33091e6	0.181737
$698$	−3.43238e6	−0.266660
$699$	−1.34360e7	−1.04010
$700$	1.25055e6	0.0964618
$701$	−1.54594e7	−1.18822	−0.594109	−	0.804384i	$-0.702495\pi$
$701$	−1.54594e7	−1.18822	−0.594109	+	0.804384i	$0.702495\pi$
$702$	−734151.	−0.0562267
$703$	6.54433e6	0.499433
$704$	3.92862e6	0.298750
$705$	2.70747e6	0.205159
$706$	1.07090e7	0.808606
$707$	3.75494e6	0.282523
$708$	−2.46167e6	−0.184564
$709$	−4.71237e6	−0.352066	−0.176033	−	0.984384i	$-0.556327\pi$
$709$	−4.71237e6	−0.352066	−0.176033	+	0.984384i	$0.556327\pi$
$710$	−1.82487e6	−0.135858
$711$	4.73914e6	0.351581
$712$	−1.07861e7	−0.797376
$713$	−383646.	−0.0282623
$714$	5.91974e6	0.434567
$715$	−708263.	−0.0518119
$716$	−6.90602e6	−0.503437
$717$	−3.49327e6	−0.253766
$718$	−9.25391e6	−0.669906
$719$	2.29858e6	0.165820	0.0829099	−	0.996557i	$-0.473579\pi$
$719$	2.29858e6	0.165820	0.0829099	+	0.996557i	$0.473579\pi$
$720$	829772.	0.0596523
$721$	1.38162e7	0.989805
$722$	−3.75324e6	−0.267956
$723$	678035.	0.0482399
$724$	−760968.	−0.0539535
$725$	−2.80303e6	−0.198054
$726$	−566763.	−0.0399080
$727$	2.81705e7	1.97678	0.988391	−	0.151933i	$-0.0485496\pi$
$727$	2.81705e7	1.97678	0.988391	+	0.151933i	$0.0485496\pi$
$728$	6.79143e6	0.474934
$729$	531441.	0.0370370
$730$	6.09054e6	0.423008
$731$	1.63213e7	1.12969
$732$	243832.	0.0168195
$733$	−1.32094e7	−0.908078	−0.454039	−	0.890982i	$-0.650017\pi$
$733$	−1.32094e7	−0.908078	−0.454039	+	0.890982i	$0.650017\pi$
$734$	6.34492e6	0.434696
$735$	1.16120e6	0.0792848
$736$	1.73121e6	0.117802
$737$	4.95441e6	0.335988
$738$	787087.	0.0531963
$739$	−6.63649e6	−0.447021	−0.223510	−	0.974702i	$-0.571752\pi$
$739$	−6.63649e6	−0.447021	−0.223510	+	0.974702i	$0.571752\pi$
$740$	1.74421e6	0.117090
$741$	−2.66837e6	−0.178525
$742$	2.45546e6	0.163728
$743$	−7.93013e6	−0.526997	−0.263499	−	0.964660i	$-0.584876\pi$
$743$	−7.93013e6	−0.526997	−0.263499	+	0.964660i	$0.584876\pi$
$744$	−1.75646e6	−0.116334
$745$	935197.	0.0617323
$746$	−2.02711e7	−1.33361
$747$	−4.23175e6	−0.277472
$748$	−1.68535e6	−0.110137
$749$	−3.05072e7	−1.98700
$750$	604855.	0.0392643
$751$	−1.43446e7	−0.928086	−0.464043	−	0.885813i	$-0.653602\pi$
$751$	−1.43446e7	−0.928086	−0.464043	+	0.885813i	$0.653602\pi$
$752$	−4.93076e6	−0.317958
$753$	3.12277e6	0.200703
$754$	−4.51654e6	−0.289319
$755$	5.64633e6	0.360495
$756$	−1.45864e6	−0.0928204
$757$	8.38496e6	0.531816	0.265908	−	0.963998i	$-0.414328\pi$
$757$	8.38496e6	0.531816	0.265908	+	0.963998i	$0.414328\pi$
$758$	1.92650e7	1.21786
$759$	−418949.	−0.0263972
$760$	6.19543e6	0.389079
$761$	2.68159e7	1.67853	0.839267	−	0.543719i	$-0.182984\pi$
$761$	2.68159e7	1.67853	0.839267	+	0.543719i	$0.182984\pi$
$762$	−4.67722e6	−0.291810
$763$	2.31069e7	1.43691
$764$	−620812.	−0.0384793
$765$	−2.08931e6	−0.129077
$766$	−249334.	−0.0153536
$767$	−4.74385e6	−0.291167
$768$	9.51916e6	0.582365
$769$	−2.82141e7	−1.72049	−0.860243	−	0.509884i	$-0.829688\pi$
$769$	−2.82141e7	−1.72049	−0.860243	+	0.509884i	$0.829688\pi$
$770$	1.92845e6	0.117215
$771$	−1.77540e7	−1.07562
$772$	−4.87192e6	−0.294210
$773$	2.41553e7	1.45400	0.727000	−	0.686637i	$-0.240914\pi$
$773$	2.41553e7	1.45400	0.727000	+	0.686637i	$0.240914\pi$
$774$	5.51125e6	0.330673
$775$	−623272.	−0.0372755
$776$	−1.94497e7	−1.15947
$777$	6.89395e6	0.409653
$778$	−1.94983e7	−1.15491
$779$	2.86077e6	0.168904
$780$	−711178.	−0.0418544
$781$	2.05347e6	0.120465
$782$	1.70726e6	0.0998350
$783$	3.26945e6	0.190577
$784$	−2.11475e6	−0.122877
$785$	5.13842e6	0.297616
$786$	−1.03154e7	−0.595565
$787$	1.96829e7	1.13280	0.566400	−	0.824131i	$-0.308336\pi$
$787$	1.96829e7	1.13280	0.566400	+	0.824131i	$0.308336\pi$
$788$	−1.32152e7	−0.758156
$789$	1.71888e6	0.0982997
$790$	−6.29134e6	−0.358654
$791$	−5.97376e6	−0.339474
$792$	−1.91809e6	−0.108656
$793$	469884.	0.0265343
$794$	2.67351e6	0.150498
$795$	−866627.	−0.0486311
$796$	−1.21812e7	−0.681409
$797$	2.41989e6	0.134943	0.0674714	−	0.997721i	$-0.478507\pi$
$797$	2.41989e6	0.134943	0.0674714	+	0.997721i	$0.478507\pi$
$798$	7.26539e6	0.403880
$799$	1.24153e7	0.688004
$800$	2.81252e6	0.155371
$801$	4.46427e6	0.245849
$802$	4.10474e6	0.225346
$803$	−6.85350e6	−0.375080
$804$	4.97480e6	0.271416
$805$	1.42550e6	0.0775315
$806$	−1.00428e6	−0.0544525
$807$	−8.40521e6	−0.454323
$808$	4.95800e6	0.267164
$809$	2.06594e7	1.10981	0.554903	−	0.831915i	$-0.312755\pi$
$809$	2.06594e7	1.10981	0.554903	+	0.831915i	$0.312755\pi$
$810$	−705503.	−0.0377821
$811$	1.38880e7	0.741461	0.370730	−	0.928741i	$-0.379107\pi$
$811$	1.38880e7	0.741461	0.370730	+	0.928741i	$0.379107\pi$
$812$	−8.97363e6	−0.477615
$813$	1.05454e7	0.559547
$814$	2.68971e6	0.142280
$815$	1.59850e7	0.842980
$816$	3.80499e6	0.200045
$817$	2.00314e7	1.04992
$818$	1.37580e7	0.718904
$819$	−2.81092e6	−0.146433
$820$	762457.	0.0395987
$821$	5.26859e6	0.272795	0.136398	−	0.990654i	$-0.456448\pi$
$821$	5.26859e6	0.272795	0.136398	+	0.990654i	$0.456448\pi$
$822$	−664913.	−0.0343230
$823$	3.61766e7	1.86178	0.930890	−	0.365299i	$-0.119033\pi$
$823$	3.61766e7	1.86178	0.930890	+	0.365299i	$0.119033\pi$
$824$	1.82428e7	0.935994
$825$	−680625.	−0.0348155
$826$	1.29165e7	0.658710
$827$	3.01917e7	1.53505	0.767527	−	0.641017i	$-0.221487\pi$
$827$	3.01917e7	1.53505	0.767527	+	0.641017i	$0.221487\pi$
$828$	−420673.	−0.0213240
$829$	2.93884e7	1.48522	0.742608	−	0.669727i	$-0.233589\pi$
$829$	2.93884e7	1.48522	0.742608	+	0.669727i	$0.233589\pi$
$830$	5.61777e6	0.283053
$831$	4.25261e6	0.213625
$832$	7.60193e6	0.380728
$833$	5.32479e6	0.265883
$834$	194792.	0.00969740
$835$	−1.04248e7	−0.517428
$836$	−2.06845e6	−0.102360
$837$	726984.	0.0358683
$838$	−2.83931e7	−1.39670
$839$	1.57845e7	0.774150	0.387075	−	0.922048i	$-0.373485\pi$
$839$	1.57845e7	0.774150	0.387075	+	0.922048i	$0.373485\pi$
$840$	6.52641e6	0.319136
$841$	−397286.	−0.0193692
$842$	−1.81730e7	−0.883376
$843$	1.84836e7	0.895814
$844$	−4.29822e6	−0.207698
$845$	7.91183e6	0.381184
$846$	4.19232e6	0.201386
$847$	−2.17003e6	−0.103934
$848$	1.57828e6	0.0753691
$849$	−4.19020e6	−0.199510
$850$	2.77361e6	0.131674
$851$	1.98822e6	0.0941112
$852$	2.06192e6	0.0973134
$853$	2.68137e7	1.26178	0.630891	−	0.775871i	$-0.282689\pi$
$853$	2.68137e7	1.26178	0.630891	+	0.775871i	$0.282689\pi$
$854$	−1.27939e6	−0.0600288
$855$	−2.56424e6	−0.119962
$856$	−4.02815e7	−1.87898
$857$	−2.59115e7	−1.20515	−0.602574	−	0.798063i	$-0.705858\pi$
$857$	−2.59115e7	−1.20515	−0.602574	+	0.798063i	$0.705858\pi$
$858$	−1.09669e6	−0.0508590
$859$	1.20430e7	0.556870	0.278435	−	0.960455i	$-0.410184\pi$
$859$	1.20430e7	0.556870	0.278435	+	0.960455i	$0.410184\pi$
$860$	5.33879e6	0.246148
$861$	3.01360e6	0.138541
$862$	2.03145e7	0.931191
$863$	2.28033e7	1.04225	0.521125	−	0.853481i	$-0.325513\pi$
$863$	2.28033e7	1.04225	0.521125	+	0.853481i	$0.325513\pi$
$864$	−3.28052e6	−0.149506
$865$	−1.52163e7	−0.691461
$866$	−3.79420e6	−0.171920
$867$	3.19801e6	0.144488
$868$	−1.99534e6	−0.0898915
$869$	7.07945e6	0.318017
$870$	−4.34029e6	−0.194411
$871$	9.58685e6	0.428184
$872$	3.05102e7	1.35880
$873$	8.05009e6	0.357491
$874$	2.09535e6	0.0927849
$875$	2.31587e6	0.102257
$876$	−6.88170e6	−0.302995
$877$	−3.10668e7	−1.36395	−0.681974	−	0.731376i	$-0.738878\pi$
$877$	−3.10668e7	−1.36395	−0.681974	+	0.731376i	$0.738878\pi$
$878$	−1.32268e7	−0.579052
$879$	1.07490e7	0.469242
$880$	1.23954e6	0.0539575
$881$	−3.45208e7	−1.49844	−0.749222	−	0.662319i	$-0.769573\pi$
$881$	−3.45208e7	−1.49844	−0.749222	+	0.662319i	$0.769573\pi$
$882$	1.79804e6	0.0778265
$883$	−3.29428e7	−1.42186	−0.710932	−	0.703260i	$-0.751727\pi$
$883$	−3.29428e7	−1.42186	−0.710932	+	0.703260i	$0.751727\pi$
$884$	−3.26117e6	−0.140360
$885$	−4.55873e6	−0.195653
$886$	1.46404e7	0.626571
$887$	−2.11009e7	−0.900519	−0.450260	−	0.892898i	$-0.648669\pi$
$887$	−2.11009e7	−0.900519	−0.450260	+	0.892898i	$0.648669\pi$
$888$	9.10273e6	0.387382
$889$	−1.79082e7	−0.759970
$890$	−5.92644e6	−0.250795
$891$	793881.	0.0335013
$892$	−8.92810e6	−0.375705
$893$	1.52375e7	0.639419
$894$	1.44809e6	0.0605969
$895$	−1.27891e7	−0.533684
$896$	644775.	0.0268311
$897$	−810673.	−0.0336406
$898$	−2.60547e6	−0.107819
$899$	4.47245e6	0.184564
$900$	−683426.	−0.0281245
$901$	−3.97399e6	−0.163085
$902$	1.17577e6	0.0481179
$903$	2.11015e7	0.861181
$904$	−7.88772e6	−0.321019
$905$	−1.40922e6	−0.0571951
$906$	8.74294e6	0.353865
$907$	1.30426e7	0.526438	0.263219	−	0.964736i	$-0.415216\pi$
$907$	1.30426e7	0.526438	0.263219	+	0.964736i	$0.415216\pi$
$908$	−8.19036e6	−0.329677
$909$	−2.05208e6	−0.0823728
$910$	3.73157e6	0.149379
$911$	−4.09652e7	−1.63538	−0.817690	−	0.575658i	$-0.804746\pi$
$911$	−4.09652e7	−1.63538	−0.817690	+	0.575658i	$0.804746\pi$
$912$	4.66992e6	0.185918
$913$	−6.32150e6	−0.250982
$914$	−2.34987e7	−0.930417
$915$	451548.	0.0178300
$916$	4.75700e6	0.187325
$917$	−3.94956e7	−1.55105
$918$	−3.23514e6	−0.126703
$919$	5.59367e6	0.218478	0.109239	−	0.994016i	$-0.465159\pi$
$919$	5.59367e6	0.218478	0.109239	+	0.994016i	$0.465159\pi$
$920$	1.88222e6	0.0733165
$921$	−1.55026e7	−0.602220
$922$	−1.22250e7	−0.473609
$923$	3.97349e6	0.153521
$924$	−2.17896e6	−0.0839592
$925$	3.23007e6	0.124124
$926$	−3.66312e7	−1.40386
$927$	−7.55055e6	−0.288589
$928$	−2.01819e7	−0.769295
$929$	−4.15645e7	−1.58010	−0.790048	−	0.613045i	$-0.789944\pi$
$929$	−4.15645e7	−1.58010	−0.790048	+	0.613045i	$0.789944\pi$
$930$	−965091.	−0.0365899
$931$	6.53520e6	0.247107
$932$	−2.01536e7	−0.759999
$933$	2.34555e7	0.882145
$934$	3.01144e7	1.12955
$935$	−3.12106e6	−0.116755
$936$	−3.71152e6	−0.138472
$937$	−1.62578e7	−0.604941	−0.302470	−	0.953159i	$-0.597811\pi$
$937$	−1.62578e7	−0.604941	−0.302470	+	0.953159i	$0.597811\pi$
$938$	−2.61030e7	−0.968685
$939$	−1.10080e7	−0.407423
$940$	4.06113e6	0.149909
$941$	1.33538e7	0.491622	0.245811	−	0.969318i	$-0.420946\pi$
$941$	1.33538e7	0.491622	0.245811	+	0.969318i	$0.420946\pi$
$942$	7.95648e6	0.292142
$943$	869126.	0.0318276
$944$	8.30224e6	0.303225
$945$	−2.70123e6	−0.0983971
$946$	8.23286e6	0.299105
$947$	1.36642e7	0.495117	0.247559	−	0.968873i	$-0.420372\pi$
$947$	1.36642e7	0.495117	0.247559	+	0.968873i	$0.420372\pi$
$948$	7.10859e6	0.256899
$949$	−1.32616e7	−0.478003
$950$	3.40410e6	0.122375
$951$	6.30038e6	0.225900
$952$	2.99274e7	1.07023
$953$	−2.92858e7	−1.04454	−0.522270	−	0.852780i	$-0.674915\pi$
$953$	−2.92858e7	−1.04454	−0.522270	+	0.852780i	$0.674915\pi$
$954$	−1.34191e6	−0.0477367
$955$	−1.14967e6	−0.0407911
$956$	−5.23981e6	−0.185426
$957$	4.88400e6	0.172384
$958$	4.59478e6	0.161752
$959$	−2.54582e6	−0.0893885
$960$	7.30528e6	0.255834
$961$	−2.76347e7	−0.965263
$962$	5.20462e6	0.181323
$963$	1.66722e7	0.579332
$964$	1.01704e6	0.0352487
$965$	−9.02223e6	−0.311886
$966$	2.20729e6	0.0761055
$967$	−4.41255e7	−1.51748	−0.758741	−	0.651393i	$-0.774185\pi$
$967$	−4.41255e7	−1.51748	−0.758741	+	0.651393i	$0.774185\pi$
$968$	−2.86529e6	−0.0982834
$969$	−1.17585e7	−0.402294
$970$	−1.06867e7	−0.364683
$971$	2.28334e7	0.777183	0.388592	−	0.921410i	$-0.372962\pi$
$971$	2.28334e7	0.777183	0.388592	+	0.921410i	$0.372962\pi$
$972$	797148.	0.0270628
$973$	745819.	0.0252552
$974$	4.31444e6	0.145723
$975$	−1.31702e6	−0.0443691
$976$	−822347.	−0.0276332
$977$	−5.35594e7	−1.79515	−0.897573	−	0.440866i	$-0.854671\pi$
$977$	−5.35594e7	−1.79515	−0.897573	+	0.440866i	$0.854671\pi$
$978$	2.47516e7	0.827476
$979$	6.66884e6	0.222379
$980$	1.74177e6	0.0579331
$981$	−1.26279e7	−0.418948
$982$	3.97309e7	1.31477
$983$	−2.25516e7	−0.744378	−0.372189	−	0.928157i	$-0.621393\pi$
$983$	−2.25516e7	−0.744378	−0.372189	+	0.928157i	$0.621393\pi$
$984$	3.97914e6	0.131009
$985$	−2.44730e7	−0.803706
$986$	−1.99028e7	−0.651961
$987$	1.60516e7	0.524475
$988$	−4.00248e6	−0.130448
$989$	6.08570e6	0.197843
$990$	−1.05390e6	−0.0341752
$991$	7.10889e6	0.229942	0.114971	−	0.993369i	$-0.463323\pi$
$991$	7.10889e6	0.229942	0.114971	+	0.993369i	$0.463323\pi$
$992$	−4.48758e6	−0.144788
$993$	−2.50865e7	−0.807358
$994$	−1.08190e7	−0.347312
$995$	−2.25582e7	−0.722348
$996$	−6.34752e6	−0.202747
$997$	3.59868e7	1.14658	0.573291	−	0.819352i	$-0.305666\pi$
$997$	3.59868e7	1.14658	0.573291	+	0.819352i	$0.305666\pi$
$998$	1.85610e7	0.589896
$999$	−3.76755e6	−0.119439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.6.a.e.1.2	✓	3
3.2	odd	2		495.6.a.a.1.2		3
5.4	even	2		825.6.a.f.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.e.1.2	✓	3	1.1	even	1	trivial
495.6.a.a.1.2		3	3.2	odd	2
825.6.a.f.1.2		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)

\(f(q)\)	\(=\)	\(q+4.30119 q^{2} -9.00000 q^{3} -13.4998 q^{4} -25.0000 q^{5} -38.7107 q^{6} -148.216 q^{7} -195.703 q^{8} +81.0000 q^{9} -107.530 q^{10} +121.000 q^{11} +121.498 q^{12} +234.137 q^{13} -637.504 q^{14} +225.000 q^{15} -409.764 q^{16} +1031.76 q^{17} +348.396 q^{18} +1266.29 q^{19} +337.494 q^{20} +1333.94 q^{21} +520.444 q^{22} +384.710 q^{23} +1761.33 q^{24} +625.000 q^{25} +1007.07 q^{26} -729.000 q^{27} +2000.88 q^{28} -4484.85 q^{29} +967.768 q^{30} -997.235 q^{31} +4500.03 q^{32} -1089.00 q^{33} +4437.78 q^{34} +3705.39 q^{35} -1093.48 q^{36} +5168.11 q^{37} +5446.56 q^{38} -2107.23 q^{39} +4892.58 q^{40} +2259.17 q^{41} +5737.54 q^{42} +15818.9 q^{43} -1633.47 q^{44} -2025.00 q^{45} +1654.71 q^{46} +12033.2 q^{47} +3687.87 q^{48} +5160.90 q^{49} +2688.24 q^{50} -9285.81 q^{51} -3160.79 q^{52} -3851.67 q^{53} -3135.57 q^{54} -3025.00 q^{55} +29006.3 q^{56} -11396.6 q^{57} -19290.2 q^{58} -20261.0 q^{59} -3037.45 q^{60} +2006.88 q^{61} -4289.30 q^{62} -12005.5 q^{63} +32467.9 q^{64} -5853.41 q^{65} -4684.00 q^{66} +40945.6 q^{67} -13928.5 q^{68} -3462.39 q^{69} +15937.6 q^{70} +16970.8 q^{71} -15852.0 q^{72} -56640.5 q^{73} +22229.0 q^{74} -5625.00 q^{75} -17094.6 q^{76} -17934.1 q^{77} -9063.59 q^{78} +58507.9 q^{79} +10244.1 q^{80} +6561.00 q^{81} +9717.12 q^{82} -52243.8 q^{83} -18007.9 q^{84} -25793.9 q^{85} +68040.2 q^{86} +40363.6 q^{87} -23680.1 q^{88} +55114.4 q^{89} -8709.91 q^{90} -34702.7 q^{91} -5193.50 q^{92} +8975.11 q^{93} +51757.0 q^{94} -31657.3 q^{95} -40500.3 q^{96} +99383.8 q^{97} +22198.0 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 7 q^{2} - 27 q^{3} + 73 q^{4} - 75 q^{5} - 63 q^{6} + 92 q^{7} + 231 q^{8} + 243 q^{9} - 175 q^{10} + 363 q^{11} - 657 q^{12} - 90 q^{13} - 784 q^{14} + 675 q^{15} - 415 q^{16} + 1934 q^{17} + 567 q^{18}+ \cdots + 29403 q^{99}+O(q^{100}) \) 3 * q + 7 * q^2 - 27 * q^3 + 73 * q^4 - 75 * q^5 - 63 * q^6 + 92 * q^7 + 231 * q^8 + 243 * q^9 - 175 * q^10 + 363 * q^11 - 657 * q^12 - 90 * q^13 - 784 * q^14 + 675 * q^15 - 415 * q^16 + 1934 * q^17 + 567 * q^18 + 2084 * q^19 - 1825 * q^20 - 828 * q^21 + 847 * q^22 + 1220 * q^23 - 2079 * q^24 + 1875 * q^25 + 17062 * q^26 - 2187 * q^27 + 11120 * q^28 + 4402 * q^29 + 1575 * q^30 - 10688 * q^31 + 12439 * q^32 - 3267 * q^33 - 4094 * q^34 - 2300 * q^35 + 5913 * q^36 - 8190 * q^37 + 13792 * q^38 + 810 * q^39 - 5775 * q^40 + 5974 * q^41 + 7056 * q^42 + 18868 * q^43 + 8833 * q^44 - 6075 * q^45 + 46220 * q^46 + 55500 * q^47 + 3735 * q^48 + 1907 * q^49 + 4375 * q^50 - 17406 * q^51 + 27330 * q^52 + 9206 * q^53 - 5103 * q^54 - 9075 * q^55 + 73248 * q^56 - 18756 * q^57 + 15366 * q^58 - 59196 * q^59 + 16425 * q^60 + 79902 * q^61 + 64616 * q^62 + 7452 * q^63 + 2129 * q^64 + 2250 * q^65 - 7623 * q^66 + 4468 * q^67 - 1218 * q^68 - 10980 * q^69 + 19600 * q^70 - 75164 * q^71 + 18711 * q^72 - 61290 * q^73 - 56766 * q^74 - 16875 * q^75 + 37816 * q^76 + 11132 * q^77 - 153558 * q^78 - 83564 * q^79 + 10375 * q^80 + 19683 * q^81 + 147410 * q^82 + 74764 * q^83 - 100080 * q^84 - 48350 * q^85 - 253432 * q^86 - 39618 * q^87 + 27951 * q^88 + 37342 * q^89 - 14175 * q^90 - 126488 * q^91 + 148164 * q^92 + 96192 * q^93 + 59252 * q^94 - 52100 * q^95 - 111951 * q^96 + 33486 * q^97 - 95249 * q^98 + 29403 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more