LMFDB - Embedded newform 495.6.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$79.3899908074$
Analytic rank:	$1$
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:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30119$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.30119 q^{2} -13.4998 q^{4} +25.0000 q^{5} -148.216 q^{7} +195.703 q^{8} +O(q^{10})$	\(q-4.30119 q^{2} -13.4998 q^{4} +25.0000 q^{5} -148.216 q^{7} +195.703 q^{8} -107.530 q^{10} -121.000 q^{11} +234.137 q^{13} +637.504 q^{14} -409.764 q^{16} -1031.76 q^{17} +1266.29 q^{19} -337.494 q^{20} +520.444 q^{22} -384.710 q^{23} +625.000 q^{25} -1007.07 q^{26} +2000.88 q^{28} +4484.85 q^{29} -997.235 q^{31} -4500.03 q^{32} +4437.78 q^{34} -3705.39 q^{35} +5168.11 q^{37} -5446.56 q^{38} +4892.58 q^{40} -2259.17 q^{41} +15818.9 q^{43} +1633.47 q^{44} +1654.71 q^{46} -12033.2 q^{47} +5160.90 q^{49} -2688.24 q^{50} -3160.79 q^{52} +3851.67 q^{53} -3025.00 q^{55} -29006.3 q^{56} -19290.2 q^{58} +20261.0 q^{59} +2006.88 q^{61} +4289.30 q^{62} +32467.9 q^{64} +5853.41 q^{65} +40945.6 q^{67} +13928.5 q^{68} +15937.6 q^{70} -16970.8 q^{71} -56640.5 q^{73} -22229.0 q^{74} -17094.6 q^{76} +17934.1 q^{77} +58507.9 q^{79} -10244.1 q^{80} +9717.12 q^{82} +52243.8 q^{83} -25793.9 q^{85} -68040.2 q^{86} -23680.1 q^{88} -55114.4 q^{89} -34702.7 q^{91} +5193.50 q^{92} +51757.0 q^{94} +31657.3 q^{95} +99383.8 q^{97} -22198.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8}+O(q^{10}) $ 3 * q - 7 * q^2 + 73 * q^4 + 75 * q^5 + 92 * q^7 - 231 * q^8	$ 3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8} - 175 q^{10} - 363 q^{11} - 90 q^{13} + 784 q^{14} - 415 q^{16} - 1934 q^{17} + 2084 q^{19} + 1825 q^{20} + 847 q^{22} - 1220 q^{23} + 1875 q^{25} - 17062 q^{26} + 11120 q^{28} - 4402 q^{29} - 10688 q^{31} - 12439 q^{32} - 4094 q^{34} + 2300 q^{35} - 8190 q^{37} - 13792 q^{38} - 5775 q^{40} - 5974 q^{41} + 18868 q^{43} - 8833 q^{44} + 46220 q^{46} - 55500 q^{47} + 1907 q^{49} - 4375 q^{50} + 27330 q^{52} - 9206 q^{53} - 9075 q^{55} - 73248 q^{56} + 15366 q^{58} + 59196 q^{59} + 79902 q^{61} - 64616 q^{62} + 2129 q^{64} - 2250 q^{65} + 4468 q^{67} + 1218 q^{68} + 19600 q^{70} + 75164 q^{71} - 61290 q^{73} + 56766 q^{74} + 37816 q^{76} - 11132 q^{77} - 83564 q^{79} - 10375 q^{80} + 147410 q^{82} - 74764 q^{83} - 48350 q^{85} + 253432 q^{86} + 27951 q^{88} - 37342 q^{89} - 126488 q^{91} - 148164 q^{92} + 59252 q^{94} + 52100 q^{95} + 33486 q^{97} + 95249 q^{98}+O(q^{100}) $ 3 * q - 7 * q^2 + 73 * q^4 + 75 * q^5 + 92 * q^7 - 231 * q^8 - 175 * q^10 - 363 * q^11 - 90 * q^13 + 784 * q^14 - 415 * q^16 - 1934 * q^17 + 2084 * q^19 + 1825 * q^20 + 847 * q^22 - 1220 * q^23 + 1875 * q^25 - 17062 * q^26 + 11120 * q^28 - 4402 * q^29 - 10688 * q^31 - 12439 * q^32 - 4094 * q^34 + 2300 * q^35 - 8190 * q^37 - 13792 * q^38 - 5775 * q^40 - 5974 * q^41 + 18868 * q^43 - 8833 * q^44 + 46220 * q^46 - 55500 * q^47 + 1907 * q^49 - 4375 * q^50 + 27330 * q^52 - 9206 * q^53 - 9075 * q^55 - 73248 * q^56 + 15366 * q^58 + 59196 * q^59 + 79902 * q^61 - 64616 * q^62 + 2129 * q^64 - 2250 * q^65 + 4468 * q^67 + 1218 * q^68 + 19600 * q^70 + 75164 * q^71 - 61290 * q^73 + 56766 * q^74 + 37816 * q^76 - 11132 * q^77 - 83564 * q^79 - 10375 * q^80 + 147410 * q^82 - 74764 * q^83 - 48350 * q^85 + 253432 * q^86 + 27951 * q^88 - 37342 * q^89 - 126488 * q^91 - 148164 * q^92 + 59252 * q^94 + 52100 * q^95 + 33486 * q^97 + 95249 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.30119	−0.760350	−0.380175	−	0.924915i	$-0.624136\pi$
$2$	−4.30119	−0.760350	−0.380175	+	0.924915i	$0.624136\pi$
$3$	0	0
$3$	0	0
$4$	−13.4998	−0.421868
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−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
$9$	0	0
$10$	−107.530	−0.340039
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	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
$15$	0	0
$16$	−409.764	−0.400160
$17$	−1031.76	−0.865875	−0.432937	−	0.901424i	$-0.642523\pi$
$17$	−1031.76	−0.865875	−0.432937	+	0.901424i	$0.642523\pi$
$18$	0	0
$19$	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
$21$	0	0
$22$	520.444	0.229254
$23$	−384.710	−0.151640	−0.0758201	−	0.997122i	$-0.524157\pi$
$23$	−384.710	−0.151640	−0.0758201	+	0.997122i	$0.524157\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−1007.07	−0.292163
$27$	0	0
$28$	2000.88	0.482309
$29$	4484.85	0.990268	0.495134	−	0.868817i	$-0.335119\pi$
$29$	4484.85	0.990268	0.495134	+	0.868817i	$0.335119\pi$
$30$	0	0
$31$	−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
$33$	0	0
$34$	4437.78	0.658368
$35$	−3705.39	−0.511286
$36$	0	0
$37$	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
$39$	0	0
$40$	4892.58	0.483490
$41$	−2259.17	−0.209889	−0.104944	−	0.994478i	$-0.533466\pi$
$41$	−2259.17	−0.209889	−0.104944	+	0.994478i	$0.533466\pi$
$42$	0	0
$43$	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
$45$	0	0
$46$	1654.71	0.115300
$47$	−12033.2	−0.794577	−0.397289	−	0.917694i	$-0.630049\pi$
$47$	−12033.2	−0.794577	−0.397289	+	0.917694i	$0.630049\pi$
$48$	0	0
$49$	5160.90	0.307069
$50$	−2688.24	−0.152070
$51$	0	0
$52$	−3160.79	−0.162102
$53$	3851.67	0.188347	0.0941737	−	0.995556i	$-0.469979\pi$
$53$	3851.67	0.188347	0.0941737	+	0.995556i	$0.469979\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	−29006.3	−1.23601
$57$	0	0
$58$	−19290.2	−0.752950
$59$	20261.0	0.757760	0.378880	−	0.925446i	$-0.376309\pi$
$59$	20261.0	0.757760	0.378880	+	0.925446i	$0.376309\pi$
$60$	0	0
$61$	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
$63$	0	0
$64$	32467.9	0.990842
$65$	5853.41	0.171841
$66$	0	0
$67$	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
$69$	0	0
$70$	15937.6	0.388757
$71$	−16970.8	−0.399537	−0.199769	−	0.979843i	$-0.564019\pi$
$71$	−16970.8	−0.399537	−0.199769	+	0.979843i	$0.564019\pi$
$72$	0	0
$73$	−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
$75$	0	0
$76$	−17094.6	−0.339489
$77$	17934.1	0.344709
$78$	0	0
$79$	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
$81$	0	0
$82$	9717.12	0.159589
$83$	52243.8	0.832415	0.416207	−	0.909270i	$-0.363359\pi$
$83$	52243.8	0.832415	0.416207	+	0.909270i	$0.363359\pi$
$84$	0	0
$85$	−25793.9	−0.387231
$86$	−68040.2	−0.992018
$87$	0	0
$88$	−23680.1	−0.325969
$89$	−55114.4	−0.737548	−0.368774	−	0.929519i	$-0.620222\pi$
$89$	−55114.4	−0.737548	−0.368774	+	0.929519i	$0.620222\pi$
$90$	0	0
$91$	−34702.7	−0.439299
$92$	5193.50	0.0639721
$93$	0	0
$94$	51757.0	0.604157
$95$	31657.3	0.359886
$96$	0	0
$97$	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
$99$	0	0
$100$	−8437.35	−0.0843735
$101$	25334.3	0.247118	0.123559	−	0.992337i	$-0.460569\pi$
$101$	25334.3	0.247118	0.123559	+	0.992337i	$0.460569\pi$
$102$	0	0
$103$	−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
$105$	0	0
$106$	−16566.8	−0.143210
$107$	−205830.	−1.73799	−0.868997	−	0.494817i	$-0.835235\pi$
$107$	−205830.	−1.73799	−0.868997	+	0.494817i	$0.835235\pi$
$108$	0	0
$109$	−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
$111$	0	0
$112$	60733.4	0.457491
$113$	−40304.5	−0.296932	−0.148466	−	0.988917i	$-0.547434\pi$
$113$	−40304.5	−0.296932	−0.148466	+	0.988917i	$0.547434\pi$
$114$	0	0
$115$	−9617.75	−0.0678155
$116$	−60544.4	−0.417762
$117$	0	0
$118$	−87146.6	−0.576163
$119$	152923.	0.989929
$120$	0	0
$121$	14641.0	0.0909091
$122$	−8631.97	−0.0525062
$123$	0	0
$124$	13462.4	0.0786266
$125$	15625.0	0.0894427
$126$	0	0
$127$	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
$129$	0	0
$130$	−25176.6	−0.130659
$131$	−266474.	−1.35668	−0.678338	−	0.734750i	$-0.737299\pi$
$131$	−266474.	−1.35668	−0.678338	+	0.734750i	$0.737299\pi$
$132$	0	0
$133$	−187684.	−0.920023
$134$	−176115.	−0.847292
$135$	0	0
$136$	−201918.	−0.936112
$137$	−17176.5	−0.0781866	−0.0390933	−	0.999236i	$-0.512447\pi$
$137$	−17176.5	−0.0781866	−0.0390933	+	0.999236i	$0.512447\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	72994.7	0.303788
$143$	−28330.5	−0.115855
$144$	0	0
$145$	112121.	0.442861
$146$	243621.	0.945874
$147$	0	0
$148$	−69768.3	−0.261820
$149$	37407.9	0.138038	0.0690188	−	0.997615i	$-0.478013\pi$
$149$	37407.9	0.138038	0.0690188	+	0.997615i	$0.478013\pi$
$150$	0	0
$151$	−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
$153$	0	0
$154$	−77138.0	−0.262100
$155$	−24930.9	−0.0833505
$156$	0	0
$157$	−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
$159$	0	0
$160$	−112501.	−0.347420
$161$	57020.1	0.173366
$162$	0	0
$163$	−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
$165$	0	0
$166$	−224711.	−0.632927
$167$	−416990.	−1.15700	−0.578502	−	0.815681i	$-0.696363\pi$
$167$	−416990.	−1.15700	−0.578502	+	0.815681i	$0.696363\pi$
$168$	0	0
$169$	−316473.	−0.852354
$170$	110945.	0.294431
$171$	0	0
$172$	−213552.	−0.550405
$173$	−608651.	−1.54615	−0.773077	−	0.634312i	$-0.781284\pi$
$173$	−608651.	−1.54615	−0.773077	+	0.634312i	$0.781284\pi$
$174$	0	0
$175$	−92634.8	−0.228654
$176$	49581.4	0.120653
$177$	0	0
$178$	237058.	0.560795
$179$	−511566.	−1.19335	−0.596676	−	0.802482i	$-0.703512\pi$
$179$	−511566.	−1.19335	−0.596676	+	0.802482i	$0.703512\pi$
$180$	0	0
$181$	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
$183$	0	0
$184$	−75289.0	−0.163941
$185$	129203.	0.277551
$186$	0	0
$187$	124843.	0.261071
$188$	162445.	0.335206
$189$	0	0
$190$	−136164.	−0.273639
$191$	−45986.9	−0.0912117	−0.0456059	−	0.998960i	$-0.514522\pi$
$191$	−45986.9	−0.0912117	−0.0456059	+	0.998960i	$0.514522\pi$
$192$	0	0
$193$	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
$195$	0	0
$196$	−69671.0	−0.129542
$197$	−978922.	−1.79714	−0.898571	−	0.438828i	$-0.855394\pi$
$197$	−978922.	−1.79714	−0.898571	+	0.438828i	$0.855394\pi$
$198$	0	0
$199$	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
$201$	0	0
$202$	−108968.	−0.187897
$203$	−664725.	−1.13214
$204$	0	0
$205$	−56479.3	−0.0938652
$206$	400942.	0.658285
$207$	0	0
$208$	−95940.7	−0.153760
$209$	−153221.	−0.242635
$210$	0	0
$211$	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
$213$	0	0
$214$	885312.	1.32148
$215$	395473.	0.583473
$216$	0	0
$217$	147806.	0.213080
$218$	670558.	0.955642
$219$	0	0
$220$	40836.8	0.0568846
$221$	−241572.	−0.332710
$222$	0	0
$223$	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
$225$	0	0
$226$	173357.	0.225773
$227$	−606704.	−0.781470	−0.390735	−	0.920503i	$-0.627779\pi$
$227$	−606704.	−0.781470	−0.390735	+	0.920503i	$0.627779\pi$
$228$	0	0
$229$	−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
$231$	0	0
$232$	877699.	1.07060
$233$	−1.49288e6	−1.80151	−0.900755	−	0.434328i	$-0.856986\pi$
$233$	−1.49288e6	−1.80151	−0.900755	+	0.434328i	$0.856986\pi$
$234$	0	0
$235$	−300830.	−0.355346
$236$	−273519.	−0.319675
$237$	0	0
$238$	−657749.	−0.752693
$239$	−388141.	−0.439536	−0.219768	−	0.975552i	$-0.570530\pi$
$239$	−388141.	−0.439536	−0.219768	+	0.975552i	$0.570530\pi$
$240$	0	0
$241$	−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
$243$	0	0
$244$	−27092.4	−0.0291322
$245$	129023.	0.137325
$246$	0	0
$247$	296485.	0.309215
$248$	−195162.	−0.201496
$249$	0	0
$250$	−67206.1	−0.0680078
$251$	346975.	0.347627	0.173814	−	0.984779i	$-0.444391\pi$
$251$	346975.	0.347627	0.173814	+	0.984779i	$0.444391\pi$
$252$	0	0
$253$	46549.9	0.0457212
$254$	−519691.	−0.505430
$255$	0	0
$256$	−1.05768e6	−1.00869
$257$	−1.97267e6	−1.86304	−0.931518	−	0.363696i	$-0.881515\pi$
$257$	−1.97267e6	−1.86304	−0.931518	+	0.363696i	$0.881515\pi$
$258$	0	0
$259$	−765995.	−0.709539
$260$	−79019.7	−0.0724940
$261$	0	0
$262$	1.14615e6	1.03155
$263$	190986.	0.170260	0.0851300	−	0.996370i	$-0.472869\pi$
$263$	190986.	0.170260	0.0851300	+	0.996370i	$0.472869\pi$
$264$	0	0
$265$	96291.8	0.0842316
$266$	807266.	0.699540
$267$	0	0
$268$	−552755.	−0.470106
$269$	−933912.	−0.786911	−0.393455	−	0.919344i	$-0.628720\pi$
$269$	−933912.	−0.786911	−0.393455	+	0.919344i	$0.628720\pi$
$270$	0	0
$271$	−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
$273$	0	0
$274$	73879.3	0.0594492
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	−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
$279$	0	0
$280$	−725157.	−0.552760
$281$	2.05374e6	1.55160	0.775798	−	0.630981i	$-0.217348\pi$
$281$	2.05374e6	1.55160	0.775798	+	0.630981i	$0.217348\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	121855.	0.0880903
$287$	334845.	0.239960
$288$	0	0
$289$	−355335.	−0.250261
$290$	−482255.	−0.336730
$291$	0	0
$292$	764633.	0.524803
$293$	1.19434e6	0.812751	0.406376	−	0.913706i	$-0.366792\pi$
$293$	1.19434e6	0.812751	0.406376	+	0.913706i	$0.366792\pi$
$294$	0	0
$295$	506526.	0.338881
$296$	1.01141e6	0.670965
$297$	0	0
$298$	−160898.	−0.104957
$299$	−90074.7	−0.0582673
$300$	0	0
$301$	−2.34461e6	−1.49161
$302$	971438.	0.612912
$303$	0	0
$304$	−518880.	−0.322020
$305$	50172.0	0.0308825
$306$	0	0
$307$	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
$309$	0	0
$310$	107232.	0.0633755
$311$	2.60616e6	1.52792	0.763960	−	0.645263i	$-0.223252\pi$
$311$	2.60616e6	1.52792	0.763960	+	0.645263i	$0.223252\pi$
$312$	0	0
$313$	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
$315$	0	0
$316$	−789843.	−0.444962
$317$	700043.	0.391270	0.195635	−	0.980677i	$-0.437323\pi$
$317$	700043.	0.391270	0.195635	+	0.980677i	$0.437323\pi$
$318$	0	0
$319$	−542667.	−0.298577
$320$	811698.	0.443118
$321$	0	0
$322$	−245254.	−0.131819
$323$	−1.30650e6	−0.696794
$324$	0	0
$325$	146335.	0.0768495
$326$	2.75017e6	1.43323
$327$	0	0
$328$	−442127.	−0.226914
$329$	1.78351e6	0.908417
$330$	0	0
$331$	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
$333$	0	0
$334$	1.79355e6	0.879728
$335$	1.02364e6	0.498350
$336$	0	0
$337$	−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
$339$	0	0
$340$	348212.	0.163360
$341$	120665.	0.0561949
$342$	0	0
$343$	1.72613e6	0.792208
$344$	3.09581e6	1.41052
$345$	0	0
$346$	2.61792e6	1.17562
$347$	3.80533e6	1.69656	0.848278	−	0.529551i	$-0.177639\pi$
$347$	3.80533e6	1.69656	0.848278	+	0.529551i	$0.177639\pi$
$348$	0	0
$349$	−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
$351$	0	0
$352$	544503.	0.234231
$353$	−2.48978e6	−1.06347	−0.531733	−	0.846912i	$-0.678459\pi$
$353$	−2.48978e6	−1.06347	−0.531733	+	0.846912i	$0.678459\pi$
$354$	0	0
$355$	−424271.	−0.178678
$356$	744032.	0.311148
$357$	0	0
$358$	2.20034e6	0.907366
$359$	2.15148e6	0.881050	0.440525	−	0.897740i	$-0.354792\pi$
$359$	2.15148e6	0.881050	0.440525	+	0.897740i	$0.354792\pi$
$360$	0	0
$361$	−872605.	−0.352411
$362$	−242454.	−0.0972427
$363$	0	0
$364$	468479.	0.185326
$365$	−1.41601e6	−0.556333
$366$	0	0
$367$	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
$369$	0	0
$370$	−555725.	−0.211036
$371$	−570879.	−0.215332
$372$	0	0
$373$	−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
$375$	0	0
$376$	−2.35493e6	−0.859031
$377$	1.05007e6	0.380508
$378$	0	0
$379$	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
$381$	0	0
$382$	197798.	0.0693528
$383$	57968.7	0.0201928	0.0100964	−	0.999949i	$-0.496786\pi$
$383$	57968.7	0.0201928	0.0100964	+	0.999949i	$0.496786\pi$
$384$	0	0
$385$	448353.	0.154159
$386$	−1.55225e6	−0.530267
$387$	0	0
$388$	−1.34166e6	−0.452442
$389$	4.53325e6	1.51892	0.759461	−	0.650553i	$-0.225463\pi$
$389$	4.53325e6	1.51892	0.759461	+	0.650553i	$0.225463\pi$
$390$	0	0
$391$	396927.	0.131301
$392$	1.01000e6	0.331977
$393$	0	0
$394$	4.21053e6	1.36646
$395$	1.46270e6	0.471695
$396$	0	0
$397$	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
$399$	0	0
$400$	−256102.	−0.0800320
$401$	−954326.	−0.296371	−0.148186	−	0.988960i	$-0.547343\pi$
$401$	−954326.	−0.296371	−0.148186	+	0.988960i	$0.547343\pi$
$402$	0	0
$403$	−233489.	−0.0716150
$404$	−342007.	−0.104251
$405$	0	0
$406$	2.85911e6	0.860826
$407$	−625341.	−0.187125
$408$	0	0
$409$	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
$411$	0	0
$412$	1.25840e6	0.365239
$413$	−3.00301e6	−0.866325
$414$	0	0
$415$	1.30610e6	0.372267
$416$	−1.05362e6	−0.298505
$417$	0	0
$418$	659034.	0.184487
$419$	6.60122e6	1.83691	0.918457	−	0.395520i	$-0.129436\pi$
$419$	6.60122e6	1.83691	0.918457	+	0.395520i	$0.129436\pi$
$420$	0	0
$421$	−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
$423$	0	0
$424$	753785.	0.203626
$425$	−644848.	−0.173175
$426$	0	0
$427$	−297451.	−0.0789489
$428$	2.77865e6	0.733204
$429$	0	0
$430$	−1.70100e6	−0.443644
$431$	−4.72300e6	−1.22469	−0.612344	−	0.790592i	$-0.709773\pi$
$431$	−4.72300e6	−1.22469	−0.612344	+	0.790592i	$0.709773\pi$
$432$	0	0
$433$	−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
$435$	0	0
$436$	2.10462e6	0.530222
$437$	−487155.	−0.122029
$438$	0	0
$439$	−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
$441$	0	0
$442$	1.03905e6	0.252976
$443$	−3.40381e6	−0.824055	−0.412028	−	0.911171i	$-0.635179\pi$
$443$	−3.40381e6	−0.824055	−0.412028	+	0.911171i	$0.635179\pi$
$444$	0	0
$445$	−1.37786e6	−0.329841
$446$	−2.84460e6	−0.677149
$447$	0	0
$448$	−4.81226e6	−1.13280
$449$	605757.	0.141802	0.0709010	−	0.997483i	$-0.477413\pi$
$449$	605757.	0.141802	0.0709010	+	0.997483i	$0.477413\pi$
$450$	0	0
$451$	273360.	0.0632839
$452$	544102.	0.125266
$453$	0	0
$454$	2.60955e6	0.594191
$455$	−867568.	−0.196460
$456$	0	0
$457$	−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
$459$	0	0
$460$	129837.	0.0286092
$461$	2.84223e6	0.622883	0.311441	−	0.950265i	$-0.399188\pi$
$461$	2.84223e6	0.622883	0.311441	+	0.950265i	$0.399188\pi$
$462$	0	0
$463$	−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
$465$	0	0
$466$	6.42118e6	1.36978
$467$	−7.00140e6	−1.48557	−0.742784	−	0.669531i	$-0.766495\pi$
$467$	−7.00140e6	−1.48557	−0.742784	+	0.669531i	$0.766495\pi$
$468$	0	0
$469$	−6.06877e6	−1.27400
$470$	1.29393e6	0.270187
$471$	0	0
$472$	3.96515e6	0.819228
$473$	−1.91409e6	−0.393377
$474$	0	0
$475$	791432.	0.160946
$476$	−2.06442e6	−0.417619
$477$	0	0
$478$	1.66947e6	0.334201
$479$	−1.06826e6	−0.212734	−0.106367	−	0.994327i	$-0.533922\pi$
$479$	−1.06826e6	−0.212734	−0.106367	+	0.994327i	$0.533922\pi$
$480$	0	0
$481$	1.21004e6	0.238472
$482$	324040.	0.0635303
$483$	0	0
$484$	−197650.	−0.0383516
$485$	2.48460e6	0.479625
$486$	0	0
$487$	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
$489$	0	0
$490$	−554950.	−0.104415
$491$	−9.23719e6	−1.72916	−0.864582	−	0.502491i	$-0.832417\pi$
$491$	−9.23719e6	−1.72916	−0.864582	+	0.502491i	$0.832417\pi$
$492$	0	0
$493$	−4.62727e6	−0.857448
$494$	−1.27524e6	−0.235112
$495$	0	0
$496$	408631.	0.0745807
$497$	2.51534e6	0.456779
$498$	0	0
$499$	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
$501$	0	0
$502$	−1.49240e6	−0.264318
$503$	−3.28052e6	−0.578127	−0.289064	−	0.957310i	$-0.593344\pi$
$503$	−3.28052e6	−0.578127	−0.289064	+	0.957310i	$0.593344\pi$
$504$	0	0
$505$	633357.	0.110515
$506$	−200220.	−0.0347641
$507$	0	0
$508$	−1.63111e6	−0.280430
$509$	2.50882e6	0.429215	0.214608	−	0.976700i	$-0.431153\pi$
$509$	2.50882e6	0.429215	0.214608	+	0.976700i	$0.431153\pi$
$510$	0	0
$511$	8.39501e6	1.42223
$512$	4.41009e6	0.743486
$513$	0	0
$514$	8.48482e6	1.41656
$515$	−2.33042e6	−0.387182
$516$	0	0
$517$	1.45602e6	0.239574
$518$	3.29469e6	0.539498
$519$	0	0
$520$	1.14553e6	0.185780
$521$	3.63347e6	0.586444	0.293222	−	0.956044i	$-0.405272\pi$
$521$	3.63347e6	0.586444	0.293222	+	0.956044i	$0.405272\pi$
$522$	0	0
$523$	−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
$525$	0	0
$526$	−821468.	−0.129457
$527$	1.02890e6	0.161379
$528$	0	0
$529$	−6.28834e6	−0.977005
$530$	−414170.	−0.0640455
$531$	0	0
$532$	2.53369e6	0.388128
$533$	−528954.	−0.0806492
$534$	0	0
$535$	−5.14574e6	−0.777255
$536$	8.01317e6	1.20474
$537$	0	0
$538$	4.01693e6	0.598328
$539$	−624469.	−0.0925847
$540$	0	0
$541$	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
$543$	0	0
$544$	4.64293e6	0.672660
$545$	−3.89752e6	−0.562078
$546$	0	0
$547$	−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
$549$	0	0
$550$	325277.	0.0458508
$551$	5.67912e6	0.796897
$552$	0	0
$553$	−8.67179e6	−1.20586
$554$	2.03236e6	0.281337
$555$	0	0
$556$	67930.6	0.00931920
$557$	−8.45756e6	−1.15507	−0.577534	−	0.816367i	$-0.695985\pi$
$557$	−8.45756e6	−1.15507	−0.577534	+	0.816367i	$0.695985\pi$
$558$	0	0
$559$	3.70379e6	0.501322
$560$	1.51834e6	0.204596
$561$	0	0
$562$	−8.83350e6	−1.17976
$563$	−9.96372e6	−1.32480	−0.662401	−	0.749150i	$-0.730462\pi$
$563$	−9.96372e6	−1.32480	−0.662401	+	0.749150i	$0.730462\pi$
$564$	0	0
$565$	−1.00761e6	−0.132792
$566$	−2.00254e6	−0.262748
$567$	0	0
$568$	−3.32124e6	−0.431946
$569$	1.05157e7	1.36163	0.680813	−	0.732458i	$-0.261627\pi$
$569$	1.05157e7	1.36163	0.680813	+	0.732458i	$0.261627\pi$
$570$	0	0
$571$	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
$573$	0	0
$574$	−1.44023e6	−0.182453
$575$	−240444.	−0.0303280
$576$	0	0
$577$	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
$579$	0	0
$580$	−1.51361e6	−0.186829
$581$	−7.74336e6	−0.951676
$582$	0	0
$583$	−466053.	−0.0567889
$584$	−1.10847e7	−1.34491
$585$	0	0
$586$	−5.13707e6	−0.617975
$587$	−2.20946e6	−0.264661	−0.132331	−	0.991206i	$-0.542246\pi$
$587$	−2.20946e6	−0.264661	−0.132331	+	0.991206i	$0.542246\pi$
$588$	0	0
$589$	−1.26279e6	−0.149983
$590$	−2.17866e6	−0.257668
$591$	0	0
$592$	−2.11770e6	−0.248348
$593$	261524.	0.0305404	0.0152702	−	0.999883i	$-0.495139\pi$
$593$	261524.	0.0305404	0.0152702	+	0.999883i	$0.495139\pi$
$594$	0	0
$595$	3.82306e6	0.442710
$596$	−504998.	−0.0582336
$597$	0	0
$598$	387429.	0.0443036
$599$	−7.09631e6	−0.808101	−0.404050	−	0.914737i	$-0.632398\pi$
$599$	−7.09631e6	−0.808101	−0.404050	+	0.914737i	$0.632398\pi$
$600$	0	0
$601$	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
$603$	0	0
$604$	3.04897e6	0.340064
$605$	366025.	0.0406558
$606$	0	0
$607$	−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
$609$	0	0
$610$	−215799.	−0.0234815
$611$	−2.81741e6	−0.305314
$612$	0	0
$613$	−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
$615$	0	0
$616$	3.50976e6	0.372671
$617$	6.18055e6	0.653604	0.326802	−	0.945093i	$-0.394029\pi$
$617$	6.18055e6	0.653604	0.326802	+	0.945093i	$0.394029\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	−1.12096e7	−1.16175
$623$	8.16882e6	0.843217
$624$	0	0
$625$	390625.	0.0400000
$626$	−5.26085e6	−0.536562
$627$	0	0
$628$	2.77470e6	0.280748
$629$	−5.33223e6	−0.537381
$630$	0	0
$631$	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
$633$	0	0
$634$	−3.01102e6	−0.297502
$635$	3.02062e6	0.297278
$636$	0	0
$637$	1.20836e6	0.117990
$638$	2.33411e6	0.227023
$639$	0	0
$640$	108756.	0.0104955
$641$	−6.80878e6	−0.654522	−0.327261	−	0.944934i	$-0.606126\pi$
$641$	−6.80878e6	−0.654522	−0.327261	+	0.944934i	$0.606126\pi$
$642$	0	0
$643$	−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
$645$	0	0
$646$	5.61952e6	0.529808
$647$	−2.61990e6	−0.246050	−0.123025	−	0.992404i	$-0.539260\pi$
$647$	−2.61990e6	−0.246050	−0.123025	+	0.992404i	$0.539260\pi$
$648$	0	0
$649$	−2.45159e6	−0.228473
$650$	−629416.	−0.0584325
$651$	0	0
$652$	8.63173e6	0.795204
$653$	−1.48492e7	−1.36276	−0.681379	−	0.731931i	$-0.738619\pi$
$653$	−1.48492e7	−1.36276	−0.681379	+	0.731931i	$0.738619\pi$
$654$	0	0
$655$	−6.66184e6	−0.606724
$656$	925726.	0.0839891
$657$	0	0
$658$	−7.67120e6	−0.690715
$659$	2.47470e6	0.221977	0.110989	−	0.993822i	$-0.464598\pi$
$659$	2.47470e6	0.221977	0.110989	+	0.993822i	$0.464598\pi$
$660$	0	0
$661$	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
$663$	0	0
$664$	1.02243e7	0.899938
$665$	−4.69211e6	−0.411447
$666$	0	0
$667$	−1.72537e6	−0.150164
$668$	5.62927e6	0.488103
$669$	0	0
$670$	−4.40286e6	−0.378921
$671$	−242833.	−0.0208210
$672$	0	0
$673$	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
$675$	0	0
$676$	4.27231e6	0.359581
$677$	−228067.	−0.0191245	−0.00956227	−	0.999954i	$-0.503044\pi$
$677$	−228067.	−0.0191245	−0.00956227	+	0.999954i	$0.503044\pi$
$678$	0	0
$679$	−1.47302e7	−1.22613
$680$	−5.04795e6	−0.418642
$681$	0	0
$682$	−519005.	−0.0427278
$683$	1.58370e6	0.129903	0.0649517	−	0.997888i	$-0.479311\pi$
$683$	1.58370e6	0.129903	0.0649517	+	0.997888i	$0.479311\pi$
$684$	0	0
$685$	−429412.	−0.0349661
$686$	−7.42443e6	−0.602356
$687$	0	0
$688$	−6.48202e6	−0.522083
$689$	901818.	0.0723720
$690$	0	0
$691$	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
$693$	0	0
$694$	−1.63674e7	−1.28998
$695$	−125800.	−0.00987910
$696$	0	0
$697$	2.33091e6	0.181737
$698$	3.43238e6	0.266660
$699$	0	0
$700$	1.25055e6	0.0964618
$701$	1.54594e7	1.18822	0.594109	−	0.804384i	$-0.297505\pi$
$701$	1.54594e7	1.18822	0.594109	+	0.804384i	$0.297505\pi$
$702$	0	0
$703$	6.54433e6	0.499433
$704$	−3.92862e6	−0.298750
$705$	0	0
$706$	1.07090e7	0.808606
$707$	−3.75494e6	−0.282523
$708$	0	0
$709$	−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
$711$	0	0
$712$	−1.07861e7	−0.797376
$713$	383646.	0.0282623
$714$	0	0
$715$	−708263.	−0.0518119
$716$	6.90602e6	0.503437
$717$	0	0
$718$	−9.25391e6	−0.669906
$719$	−2.29858e6	−0.165820	−0.0829099	−	0.996557i	$-0.526421\pi$
$719$	−2.29858e6	−0.165820	−0.0829099	+	0.996557i	$0.526421\pi$
$720$	0	0
$721$	1.38162e7	0.989805
$722$	3.75324e6	0.267956
$723$	0	0
$724$	−760968.	−0.0539535
$725$	2.80303e6	0.198054
$726$	0	0
$727$	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
$729$	0	0
$730$	6.09054e6	0.423008
$731$	−1.63213e7	−1.12969
$732$	0	0
$733$	−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
$735$	0	0
$736$	1.73121e6	0.117802
$737$	−4.95441e6	−0.335988
$738$	0	0
$739$	−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
$741$	0	0
$742$	2.45546e6	0.163728
$743$	7.93013e6	0.526997	0.263499	−	0.964660i	$-0.415124\pi$
$743$	7.93013e6	0.526997	0.263499	+	0.964660i	$0.415124\pi$
$744$	0	0
$745$	935197.	0.0617323
$746$	2.02711e7	1.33361
$747$	0	0
$748$	−1.68535e6	−0.110137
$749$	3.05072e7	1.98700
$750$	0	0
$751$	−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
$753$	0	0
$754$	−4.51654e6	−0.289319
$755$	−5.64633e6	−0.360495
$756$	0	0
$757$	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
$759$	0	0
$760$	6.19543e6	0.389079
$761$	−2.68159e7	−1.67853	−0.839267	−	0.543719i	$-0.817016\pi$
$761$	−2.68159e7	−1.67853	−0.839267	+	0.543719i	$0.817016\pi$
$762$	0	0
$763$	2.31069e7	1.43691
$764$	620812.	0.0384793
$765$	0	0
$766$	−249334.	−0.0153536
$767$	4.74385e6	0.291167
$768$	0	0
$769$	−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
$771$	0	0
$772$	−4.87192e6	−0.294210
$773$	−2.41553e7	−1.45400	−0.727000	−	0.686637i	$-0.759086\pi$
$773$	−2.41553e7	−1.45400	−0.727000	+	0.686637i	$0.759086\pi$
$774$	0	0
$775$	−623272.	−0.0372755
$776$	1.94497e7	1.15947
$777$	0	0
$778$	−1.94983e7	−1.15491
$779$	−2.86077e6	−0.168904
$780$	0	0
$781$	2.05347e6	0.120465
$782$	−1.70726e6	−0.0998350
$783$	0	0
$784$	−2.11475e6	−0.122877
$785$	−5.13842e6	−0.297616
$786$	0	0
$787$	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
$789$	0	0
$790$	−6.29134e6	−0.358654
$791$	5.97376e6	0.339474
$792$	0	0
$793$	469884.	0.0265343
$794$	−2.67351e6	−0.150498
$795$	0	0
$796$	−1.21812e7	−0.681409
$797$	−2.41989e6	−0.134943	−0.0674714	−	0.997721i	$-0.521493\pi$
$797$	−2.41989e6	−0.134943	−0.0674714	+	0.997721i	$0.521493\pi$
$798$	0	0
$799$	1.24153e7	0.688004
$800$	−2.81252e6	−0.155371
$801$	0	0
$802$	4.10474e6	0.225346
$803$	6.85350e6	0.375080
$804$	0	0
$805$	1.42550e6	0.0775315
$806$	1.00428e6	0.0544525
$807$	0	0
$808$	4.95800e6	0.267164
$809$	−2.06594e7	−1.10981	−0.554903	−	0.831915i	$-0.687245\pi$
$809$	−2.06594e7	−1.10981	−0.554903	+	0.831915i	$0.687245\pi$
$810$	0	0
$811$	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
$813$	0	0
$814$	2.68971e6	0.142280
$815$	−1.59850e7	−0.842980
$816$	0	0
$817$	2.00314e7	1.04992
$818$	−1.37580e7	−0.718904
$819$	0	0
$820$	762457.	0.0395987
$821$	−5.26859e6	−0.272795	−0.136398	−	0.990654i	$-0.543552\pi$
$821$	−5.26859e6	−0.272795	−0.136398	+	0.990654i	$0.543552\pi$
$822$	0	0
$823$	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
$825$	0	0
$826$	1.29165e7	0.658710
$827$	−3.01917e7	−1.53505	−0.767527	−	0.641017i	$-0.778513\pi$
$827$	−3.01917e7	−1.53505	−0.767527	+	0.641017i	$0.778513\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	7.60193e6	0.380728
$833$	−5.32479e6	−0.265883
$834$	0	0
$835$	−1.04248e7	−0.517428
$836$	2.06845e6	0.102360
$837$	0	0
$838$	−2.83931e7	−1.39670
$839$	−1.57845e7	−0.774150	−0.387075	−	0.922048i	$-0.626515\pi$
$839$	−1.57845e7	−0.774150	−0.387075	+	0.922048i	$0.626515\pi$
$840$	0	0
$841$	−397286.	−0.0193692
$842$	1.81730e7	0.883376
$843$	0	0
$844$	−4.29822e6	−0.207698
$845$	−7.91183e6	−0.381184
$846$	0	0
$847$	−2.17003e6	−0.103934
$848$	−1.57828e6	−0.0753691
$849$	0	0
$850$	2.77361e6	0.131674
$851$	−1.98822e6	−0.0941112
$852$	0	0
$853$	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
$855$	0	0
$856$	−4.02815e7	−1.87898
$857$	2.59115e7	1.20515	0.602574	−	0.798063i	$-0.294142\pi$
$857$	2.59115e7	1.20515	0.602574	+	0.798063i	$0.294142\pi$
$858$	0	0
$859$	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
$861$	0	0
$862$	2.03145e7	0.931191
$863$	−2.28033e7	−1.04225	−0.521125	−	0.853481i	$-0.674487\pi$
$863$	−2.28033e7	−1.04225	−0.521125	+	0.853481i	$0.674487\pi$
$864$	0	0
$865$	−1.52163e7	−0.691461
$866$	3.79420e6	0.171920
$867$	0	0
$868$	−1.99534e6	−0.0898915
$869$	−7.07945e6	−0.318017
$870$	0	0
$871$	9.58685e6	0.428184
$872$	−3.05102e7	−1.35880
$873$	0	0
$874$	2.09535e6	0.0927849
$875$	−2.31587e6	−0.102257
$876$	0	0
$877$	−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
$879$	0	0
$880$	1.23954e6	0.0539575
$881$	3.45208e7	1.49844	0.749222	−	0.662319i	$-0.230427\pi$
$881$	3.45208e7	1.49844	0.749222	+	0.662319i	$0.230427\pi$
$882$	0	0
$883$	−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
$885$	0	0
$886$	1.46404e7	0.626571
$887$	2.11009e7	0.900519	0.450260	−	0.892898i	$-0.351331\pi$
$887$	2.11009e7	0.900519	0.450260	+	0.892898i	$0.351331\pi$
$888$	0	0
$889$	−1.79082e7	−0.759970
$890$	5.92644e6	0.250795
$891$	0	0
$892$	−8.92810e6	−0.375705
$893$	−1.52375e7	−0.639419
$894$	0	0
$895$	−1.27891e7	−0.533684
$896$	−644775.	−0.0268311
$897$	0	0
$898$	−2.60547e6	−0.107819
$899$	−4.47245e6	−0.184564
$900$	0	0
$901$	−3.97399e6	−0.163085
$902$	−1.17577e6	−0.0481179
$903$	0	0
$904$	−7.88772e6	−0.321019
$905$	1.40922e6	0.0571951
$906$	0	0
$907$	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
$909$	0	0
$910$	3.73157e6	0.149379
$911$	4.09652e7	1.63538	0.817690	−	0.575658i	$-0.195254\pi$
$911$	4.09652e7	1.63538	0.817690	+	0.575658i	$0.195254\pi$
$912$	0	0
$913$	−6.32150e6	−0.250982
$914$	2.34987e7	0.930417
$915$	0	0
$916$	4.75700e6	0.187325
$917$	3.94956e7	1.55105
$918$	0	0
$919$	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
$921$	0	0
$922$	−1.22250e7	−0.473609
$923$	−3.97349e6	−0.153521
$924$	0	0
$925$	3.23007e6	0.124124
$926$	3.66312e7	1.40386
$927$	0	0
$928$	−2.01819e7	−0.769295
$929$	4.15645e7	1.58010	0.790048	−	0.613045i	$-0.210056\pi$
$929$	4.15645e7	1.58010	0.790048	+	0.613045i	$0.210056\pi$
$930$	0	0
$931$	6.53520e6	0.247107
$932$	2.01536e7	0.759999
$933$	0	0
$934$	3.01144e7	1.12955
$935$	3.12106e6	0.116755
$936$	0	0
$937$	−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
$939$	0	0
$940$	4.06113e6	0.149909
$941$	−1.33538e7	−0.491622	−0.245811	−	0.969318i	$-0.579054\pi$
$941$	−1.33538e7	−0.491622	−0.245811	+	0.969318i	$0.579054\pi$
$942$	0	0
$943$	869126.	0.0318276
$944$	−8.30224e6	−0.303225
$945$	0	0
$946$	8.23286e6	0.299105
$947$	−1.36642e7	−0.495117	−0.247559	−	0.968873i	$-0.579628\pi$
$947$	−1.36642e7	−0.495117	−0.247559	+	0.968873i	$0.579628\pi$
$948$	0	0
$949$	−1.32616e7	−0.478003
$950$	−3.40410e6	−0.122375
$951$	0	0
$952$	2.99274e7	1.07023
$953$	2.92858e7	1.04454	0.522270	−	0.852780i	$-0.325085\pi$
$953$	2.92858e7	1.04454	0.522270	+	0.852780i	$0.325085\pi$
$954$	0	0
$955$	−1.14967e6	−0.0407911
$956$	5.23981e6	0.185426
$957$	0	0
$958$	4.59478e6	0.161752
$959$	2.54582e6	0.0893885
$960$	0	0
$961$	−2.76347e7	−0.965263
$962$	−5.20462e6	−0.181323
$963$	0	0
$964$	1.01704e6	0.0352487
$965$	9.02223e6	0.311886
$966$	0	0
$967$	−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
$969$	0	0
$970$	−1.06867e7	−0.364683
$971$	−2.28334e7	−0.777183	−0.388592	−	0.921410i	$-0.627038\pi$
$971$	−2.28334e7	−0.777183	−0.388592	+	0.921410i	$0.627038\pi$
$972$	0	0
$973$	745819.	0.0252552
$974$	−4.31444e6	−0.145723
$975$	0	0
$976$	−822347.	−0.0276332
$977$	5.35594e7	1.79515	0.897573	−	0.440866i	$-0.145329\pi$
$977$	5.35594e7	1.79515	0.897573	+	0.440866i	$0.145329\pi$
$978$	0	0
$979$	6.66884e6	0.222379
$980$	−1.74177e6	−0.0579331
$981$	0	0
$982$	3.97309e7	1.31477
$983$	2.25516e7	0.744378	0.372189	−	0.928157i	$-0.378607\pi$
$983$	2.25516e7	0.744378	0.372189	+	0.928157i	$0.378607\pi$
$984$	0	0
$985$	−2.44730e7	−0.803706
$986$	1.99028e7	0.651961
$987$	0	0
$988$	−4.00248e6	−0.130448
$989$	−6.08570e6	−0.197843
$990$	0	0
$991$	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
$993$	0	0
$994$	−1.08190e7	−0.347312
$995$	2.25582e7	0.722348
$996$	0	0
$997$	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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.a.1.2		3
3.2	odd	2		165.6.a.e.1.2	✓	3
15.14	odd	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	3.2	odd	2
495.6.a.a.1.2		3	1.1	even	1	trivial
825.6.a.f.1.2		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-4.30119 q^{2} -13.4998 q^{4} +25.0000 q^{5} -148.216 q^{7} +195.703 q^{8} +O(q^{10})\)	\(q-4.30119 q^{2} -13.4998 q^{4} +25.0000 q^{5} -148.216 q^{7} +195.703 q^{8} -107.530 q^{10} -121.000 q^{11} +234.137 q^{13} +637.504 q^{14} -409.764 q^{16} -1031.76 q^{17} +1266.29 q^{19} -337.494 q^{20} +520.444 q^{22} -384.710 q^{23} +625.000 q^{25} -1007.07 q^{26} +2000.88 q^{28} +4484.85 q^{29} -997.235 q^{31} -4500.03 q^{32} +4437.78 q^{34} -3705.39 q^{35} +5168.11 q^{37} -5446.56 q^{38} +4892.58 q^{40} -2259.17 q^{41} +15818.9 q^{43} +1633.47 q^{44} +1654.71 q^{46} -12033.2 q^{47} +5160.90 q^{49} -2688.24 q^{50} -3160.79 q^{52} +3851.67 q^{53} -3025.00 q^{55} -29006.3 q^{56} -19290.2 q^{58} +20261.0 q^{59} +2006.88 q^{61} +4289.30 q^{62} +32467.9 q^{64} +5853.41 q^{65} +40945.6 q^{67} +13928.5 q^{68} +15937.6 q^{70} -16970.8 q^{71} -56640.5 q^{73} -22229.0 q^{74} -17094.6 q^{76} +17934.1 q^{77} +58507.9 q^{79} -10244.1 q^{80} +9717.12 q^{82} +52243.8 q^{83} -25793.9 q^{85} -68040.2 q^{86} -23680.1 q^{88} -55114.4 q^{89} -34702.7 q^{91} +5193.50 q^{92} +51757.0 q^{94} +31657.3 q^{95} +99383.8 q^{97} -22198.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8}+O(q^{10}) \) 3 * q - 7 * q^2 + 73 * q^4 + 75 * q^5 + 92 * q^7 - 231 * q^8	\( 3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8} - 175 q^{10} - 363 q^{11} - 90 q^{13} + 784 q^{14} - 415 q^{16} - 1934 q^{17} + 2084 q^{19} + 1825 q^{20} + 847 q^{22} - 1220 q^{23} + 1875 q^{25} - 17062 q^{26} + 11120 q^{28} - 4402 q^{29} - 10688 q^{31} - 12439 q^{32} - 4094 q^{34} + 2300 q^{35} - 8190 q^{37} - 13792 q^{38} - 5775 q^{40} - 5974 q^{41} + 18868 q^{43} - 8833 q^{44} + 46220 q^{46} - 55500 q^{47} + 1907 q^{49} - 4375 q^{50} + 27330 q^{52} - 9206 q^{53} - 9075 q^{55} - 73248 q^{56} + 15366 q^{58} + 59196 q^{59} + 79902 q^{61} - 64616 q^{62} + 2129 q^{64} - 2250 q^{65} + 4468 q^{67} + 1218 q^{68} + 19600 q^{70} + 75164 q^{71} - 61290 q^{73} + 56766 q^{74} + 37816 q^{76} - 11132 q^{77} - 83564 q^{79} - 10375 q^{80} + 147410 q^{82} - 74764 q^{83} - 48350 q^{85} + 253432 q^{86} + 27951 q^{88} - 37342 q^{89} - 126488 q^{91} - 148164 q^{92} + 59252 q^{94} + 52100 q^{95} + 33486 q^{97} + 95249 q^{98}+O(q^{100}) \) 3 * q - 7 * q^2 + 73 * q^4 + 75 * q^5 + 92 * q^7 - 231 * q^8 - 175 * q^10 - 363 * q^11 - 90 * q^13 + 784 * q^14 - 415 * q^16 - 1934 * q^17 + 2084 * q^19 + 1825 * q^20 + 847 * q^22 - 1220 * q^23 + 1875 * q^25 - 17062 * q^26 + 11120 * q^28 - 4402 * q^29 - 10688 * q^31 - 12439 * q^32 - 4094 * q^34 + 2300 * q^35 - 8190 * q^37 - 13792 * q^38 - 5775 * q^40 - 5974 * q^41 + 18868 * q^43 - 8833 * q^44 + 46220 * q^46 - 55500 * q^47 + 1907 * q^49 - 4375 * q^50 + 27330 * q^52 - 9206 * q^53 - 9075 * q^55 - 73248 * q^56 + 15366 * q^58 + 59196 * q^59 + 79902 * q^61 - 64616 * q^62 + 2129 * q^64 - 2250 * q^65 + 4468 * q^67 + 1218 * q^68 + 19600 * q^70 + 75164 * q^71 - 61290 * q^73 + 56766 * q^74 + 37816 * q^76 - 11132 * q^77 - 83564 * q^79 - 10375 * q^80 + 147410 * q^82 - 74764 * q^83 - 48350 * q^85 + 253432 * q^86 + 27951 * q^88 - 37342 * q^89 - 126488 * q^91 - 148164 * q^92 + 59252 * q^94 + 52100 * q^95 + 33486 * q^97 + 95249 * q^98

Introduction

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