LMFDB - Embedded newform 495.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$79.3899908074$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.87740$ 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)

$f(q)$	$=$	$q-6.55890 q^{2} +11.0192 q^{4} -25.0000 q^{5} +146.487 q^{7} +137.611 q^{8} +O(q^{10})$	\(q-6.55890 q^{2} +11.0192 q^{4} -25.0000 q^{5} +146.487 q^{7} +137.611 q^{8} +163.973 q^{10} +121.000 q^{11} +170.238 q^{13} -960.792 q^{14} -1255.19 q^{16} -1564.81 q^{17} +569.786 q^{19} -275.481 q^{20} -793.627 q^{22} -3153.25 q^{23} +625.000 q^{25} -1116.57 q^{26} +1614.17 q^{28} -3982.58 q^{29} +2990.78 q^{31} +3829.14 q^{32} +10263.5 q^{34} -3662.17 q^{35} +7858.92 q^{37} -3737.17 q^{38} -3440.27 q^{40} +5206.60 q^{41} +13874.3 q^{43} +1333.33 q^{44} +20681.9 q^{46} -6852.01 q^{47} +4651.35 q^{49} -4099.32 q^{50} +1875.89 q^{52} +3834.15 q^{53} -3025.00 q^{55} +20158.2 q^{56} +26121.4 q^{58} -9649.38 q^{59} -21131.8 q^{61} -19616.2 q^{62} +15051.2 q^{64} -4255.95 q^{65} -43499.9 q^{67} -17243.0 q^{68} +24019.8 q^{70} +52607.2 q^{71} -64367.9 q^{73} -51545.9 q^{74} +6278.61 q^{76} +17724.9 q^{77} +28935.7 q^{79} +31379.8 q^{80} -34149.6 q^{82} +4648.64 q^{83} +39120.3 q^{85} -91000.1 q^{86} +16650.9 q^{88} +103458. q^{89} +24937.6 q^{91} -34746.4 q^{92} +44941.6 q^{94} -14244.7 q^{95} +75809.5 q^{97} -30507.8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8	$ 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8} + 50 q^{10} + 363 q^{11} - 546 q^{13} + 8 q^{14} - 1360 q^{16} + 314 q^{17} + 1808 q^{19} + 500 q^{20} - 242 q^{22} - 4288 q^{23} + 1875 q^{25} - 812 q^{26} + 5888 q^{28} - 5582 q^{29} + 6328 q^{31} + 736 q^{32} + 11596 q^{34} - 3800 q^{35} + 16866 q^{37} - 9584 q^{38} - 600 q^{40} - 23282 q^{41} + 20572 q^{43} - 2420 q^{44} + 16592 q^{46} - 3432 q^{47} + 11531 q^{49} - 1250 q^{50} + 21816 q^{52} - 16138 q^{53} - 9075 q^{55} - 15648 q^{56} + 17460 q^{58} - 21972 q^{59} + 8322 q^{61} - 5056 q^{62} + 22208 q^{64} + 13650 q^{65} - 84332 q^{67} - 59832 q^{68} - 200 q^{70} - 50528 q^{71} - 53838 q^{73} - 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 6364 q^{79} + 34000 q^{80} - 68020 q^{82} - 96272 q^{83} - 7850 q^{85} - 143152 q^{86} + 2904 q^{88} + 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 49088 q^{94} - 45200 q^{95} - 103242 q^{97} - 9554 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8 + 50 * q^10 + 363 * q^11 - 546 * q^13 + 8 * q^14 - 1360 * q^16 + 314 * q^17 + 1808 * q^19 + 500 * q^20 - 242 * q^22 - 4288 * q^23 + 1875 * q^25 - 812 * q^26 + 5888 * q^28 - 5582 * q^29 + 6328 * q^31 + 736 * q^32 + 11596 * q^34 - 3800 * q^35 + 16866 * q^37 - 9584 * q^38 - 600 * q^40 - 23282 * q^41 + 20572 * q^43 - 2420 * q^44 + 16592 * q^46 - 3432 * q^47 + 11531 * q^49 - 1250 * q^50 + 21816 * q^52 - 16138 * q^53 - 9075 * q^55 - 15648 * q^56 + 17460 * q^58 - 21972 * q^59 + 8322 * q^61 - 5056 * q^62 + 22208 * q^64 + 13650 * q^65 - 84332 * q^67 - 59832 * q^68 - 200 * q^70 - 50528 * q^71 - 53838 * q^73 - 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 6364 * q^79 + 34000 * q^80 - 68020 * q^82 - 96272 * q^83 - 7850 * q^85 - 143152 * q^86 + 2904 * q^88 + 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 49088 * q^94 - 45200 * q^95 - 103242 * q^97 - 9554 * 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 $

$2$	−6.55890	−1.15946	−0.579731	−	0.814808i	$-0.696842\pi$
$2$	−6.55890	−1.15946	−0.579731	+	0.814808i	$0.696842\pi$
$3$	0	0
$3$	0	0
$4$	11.0192	0.344351
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	146.487	1.12993	0.564967	−	0.825113i	$-0.308889\pi$
$7$	146.487	1.12993	0.564967	+	0.825113i	$0.308889\pi$
$8$	137.611	0.760200
$9$	0	0
$10$	163.973	0.518527
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	170.238	0.279382	0.139691	−	0.990195i	$-0.455389\pi$
$13$	170.238	0.279382	0.139691	+	0.990195i	$0.455389\pi$
$14$	−960.792	−1.31012
$15$	0	0
$16$	−1255.19	−1.22577
$17$	−1564.81	−1.31323	−0.656614	−	0.754227i	$-0.728012\pi$
$17$	−1564.81	−1.31323	−0.656614	+	0.754227i	$0.728012\pi$
$18$	0	0
$19$	569.786	0.362100	0.181050	−	0.983474i	$-0.442050\pi$
$19$	569.786	0.362100	0.181050	+	0.983474i	$0.442050\pi$
$20$	−275.481	−0.153998
$21$	0	0
$22$	−793.627	−0.349591
$23$	−3153.25	−1.24291	−0.621454	−	0.783451i	$-0.713458\pi$
$23$	−3153.25	−1.24291	−0.621454	+	0.783451i	$0.713458\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−1116.57	−0.323932
$27$	0	0
$28$	1614.17	0.389094
$29$	−3982.58	−0.879366	−0.439683	−	0.898153i	$-0.644909\pi$
$29$	−3982.58	−0.879366	−0.439683	+	0.898153i	$0.644909\pi$
$30$	0	0
$31$	2990.78	0.558959	0.279480	−	0.960152i	$-0.409838\pi$
$31$	2990.78	0.558959	0.279480	+	0.960152i	$0.409838\pi$
$32$	3829.14	0.661037
$33$	0	0
$34$	10263.5	1.52264
$35$	−3662.17	−0.505322
$36$	0	0
$37$	7858.92	0.943753	0.471877	−	0.881665i	$-0.343577\pi$
$37$	7858.92	0.943753	0.471877	+	0.881665i	$0.343577\pi$
$38$	−3737.17	−0.419841
$39$	0	0
$40$	−3440.27	−0.339972
$41$	5206.60	0.483721	0.241860	−	0.970311i	$-0.422242\pi$
$41$	5206.60	0.483721	0.241860	+	0.970311i	$0.422242\pi$
$42$	0	0
$43$	13874.3	1.14430	0.572149	−	0.820149i	$-0.306110\pi$
$43$	13874.3	1.14430	0.572149	+	0.820149i	$0.306110\pi$
$44$	1333.33	0.103826
$45$	0	0
$46$	20681.9	1.44110
$47$	−6852.01	−0.452453	−0.226226	−	0.974075i	$-0.572639\pi$
$47$	−6852.01	−0.452453	−0.226226	+	0.974075i	$0.572639\pi$
$48$	0	0
$49$	4651.35	0.276751
$50$	−4099.32	−0.231892
$51$	0	0
$52$	1875.89	0.0962054
$53$	3834.15	0.187490	0.0937452	−	0.995596i	$-0.470116\pi$
$53$	3834.15	0.187490	0.0937452	+	0.995596i	$0.470116\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	20158.2	0.858976
$57$	0	0
$58$	26121.4	1.01959
$59$	−9649.38	−0.360885	−0.180443	−	0.983586i	$-0.557753\pi$
$59$	−9649.38	−0.360885	−0.180443	+	0.983586i	$0.557753\pi$
$60$	0	0
$61$	−21131.8	−0.727128	−0.363564	−	0.931569i	$-0.618440\pi$
$61$	−21131.8	−0.727128	−0.363564	+	0.931569i	$0.618440\pi$
$62$	−19616.2	−0.648092
$63$	0	0
$64$	15051.2	0.459326
$65$	−4255.95	−0.124943
$66$	0	0
$67$	−43499.9	−1.18386	−0.591931	−	0.805989i	$-0.701634\pi$
$67$	−43499.9	−1.18386	−0.591931	+	0.805989i	$0.701634\pi$
$68$	−17243.0	−0.452211
$69$	0	0
$70$	24019.8	0.585901
$71$	52607.2	1.23851	0.619255	−	0.785190i	$-0.287435\pi$
$71$	52607.2	1.23851	0.619255	+	0.785190i	$0.287435\pi$
$72$	0	0
$73$	−64367.9	−1.41372	−0.706858	−	0.707355i	$-0.749888\pi$
$73$	−64367.9	−1.41372	−0.706858	+	0.707355i	$0.749888\pi$
$74$	−51545.9	−1.09425
$75$	0	0
$76$	6278.61	0.124689
$77$	17724.9	0.340688
$78$	0	0
$79$	28935.7	0.521635	0.260817	−	0.965388i	$-0.416008\pi$
$79$	28935.7	0.521635	0.260817	+	0.965388i	$0.416008\pi$
$80$	31379.8	0.548183
$81$	0	0
$82$	−34149.6	−0.560855
$83$	4648.64	0.0740681	0.0370340	−	0.999314i	$-0.488209\pi$
$83$	4648.64	0.0740681	0.0370340	+	0.999314i	$0.488209\pi$
$84$	0	0
$85$	39120.3	0.587293
$86$	−91000.1	−1.32677
$87$	0	0
$88$	16650.9	0.229209
$89$	103458.	1.38448	0.692242	−	0.721666i	$-0.256623\pi$
$89$	103458.	1.38448	0.692242	+	0.721666i	$0.256623\pi$
$90$	0	0
$91$	24937.6	0.315683
$92$	−34746.4	−0.427996
$93$	0	0
$94$	44941.6	0.524601
$95$	−14244.7	−0.161936
$96$	0	0
$97$	75809.5	0.818077	0.409039	−	0.912517i	$-0.365864\pi$
$97$	75809.5	0.818077	0.409039	+	0.912517i	$0.365864\pi$
$98$	−30507.8	−0.320882
$99$	0	0
$100$	6887.02	0.0688702
$101$	−35023.6	−0.341631	−0.170816	−	0.985303i	$-0.554640\pi$
$101$	−35023.6	−0.341631	−0.170816	+	0.985303i	$0.554640\pi$
$102$	0	0
$103$	16965.7	0.157572	0.0787859	−	0.996892i	$-0.474896\pi$
$103$	16965.7	0.157572	0.0787859	+	0.996892i	$0.474896\pi$
$104$	23426.6	0.212386
$105$	0	0
$106$	−25147.8	−0.217388
$107$	−12190.4	−0.102934	−0.0514671	−	0.998675i	$-0.516390\pi$
$107$	−12190.4	−0.102934	−0.0514671	+	0.998675i	$0.516390\pi$
$108$	0	0
$109$	12138.7	0.0978606	0.0489303	−	0.998802i	$-0.484419\pi$
$109$	12138.7	0.0978606	0.0489303	+	0.998802i	$0.484419\pi$
$110$	19840.7	0.156342
$111$	0	0
$112$	−183869.	−1.38504
$113$	−204649.	−1.50770	−0.753848	−	0.657048i	$-0.771805\pi$
$113$	−204649.	−1.50770	−0.753848	+	0.657048i	$0.771805\pi$
$114$	0	0
$115$	78831.2	0.555845
$116$	−43885.0	−0.302810
$117$	0	0
$118$	63289.3	0.418433
$119$	−229224.	−1.48386
$120$	0	0
$121$	14641.0	0.0909091
$122$	138601.	0.843077
$123$	0	0
$124$	32956.1	0.192478
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−112241.	−0.617506	−0.308753	−	0.951142i	$-0.599912\pi$
$127$	−112241.	−0.617506	−0.308753	+	0.951142i	$0.599912\pi$
$128$	−221252.	−1.19361
$129$	0	0
$130$	27914.4	0.144867
$131$	11679.0	0.0594602	0.0297301	−	0.999558i	$-0.490535\pi$
$131$	11679.0	0.0594602	0.0297301	+	0.999558i	$0.490535\pi$
$132$	0	0
$133$	83466.1	0.409149
$134$	285311.	1.37264
$135$	0	0
$136$	−215335.	−0.998315
$137$	384935.	1.75221	0.876105	−	0.482121i	$-0.160133\pi$
$137$	384935.	1.75221	0.876105	+	0.482121i	$0.160133\pi$
$138$	0	0
$139$	−310605.	−1.36355	−0.681775	−	0.731562i	$-0.738792\pi$
$139$	−310605.	−1.36355	−0.681775	+	0.731562i	$0.738792\pi$
$140$	−40354.3	−0.174008
$141$	0	0
$142$	−345046.	−1.43600
$143$	20598.8	0.0842368
$144$	0	0
$145$	99564.5	0.393264
$146$	422183.	1.63915
$147$	0	0
$148$	86599.2	0.324982
$149$	−285203.	−1.05242	−0.526209	−	0.850355i	$-0.676387\pi$
$149$	−285203.	−1.05242	−0.526209	+	0.850355i	$0.676387\pi$
$150$	0	0
$151$	−522357.	−1.86434	−0.932170	−	0.362020i	$-0.882087\pi$
$151$	−522357.	−1.86434	−0.932170	+	0.362020i	$0.882087\pi$
$152$	78408.8	0.275268
$153$	0	0
$154$	−116256.	−0.395015
$155$	−74769.5	−0.249974
$156$	0	0
$157$	−2221.66	−0.00719330	−0.00359665	−	0.999994i	$-0.501145\pi$
$157$	−2221.66	−0.00719330	−0.00359665	+	0.999994i	$0.501145\pi$
$158$	−189787.	−0.604816
$159$	0	0
$160$	−95728.4	−0.295625
$161$	−461909.	−1.40440
$162$	0	0
$163$	250146.	0.737436	0.368718	−	0.929541i	$-0.379797\pi$
$163$	250146.	0.737436	0.368718	+	0.929541i	$0.379797\pi$
$164$	57372.7	0.166570
$165$	0	0
$166$	−30490.0	−0.0858791
$167$	42083.4	0.116767	0.0583834	−	0.998294i	$-0.481405\pi$
$167$	42083.4	0.116767	0.0583834	+	0.998294i	$0.481405\pi$
$168$	0	0
$169$	−342312.	−0.921946
$170$	−256586.	−0.680944
$171$	0	0
$172$	152884.	0.394040
$173$	117206.	0.297737	0.148869	−	0.988857i	$-0.452437\pi$
$173$	117206.	0.297737	0.148869	+	0.988857i	$0.452437\pi$
$174$	0	0
$175$	91554.2	0.225987
$176$	−151878.	−0.369585
$177$	0	0
$178$	−678569.	−1.60526
$179$	−582985.	−1.35996	−0.679978	−	0.733233i	$-0.738011\pi$
$179$	−582985.	−1.35996	−0.679978	+	0.733233i	$0.738011\pi$
$180$	0	0
$181$	235291.	0.533837	0.266919	−	0.963719i	$-0.413994\pi$
$181$	235291.	0.533837	0.266919	+	0.963719i	$0.413994\pi$
$182$	−163563.	−0.366022
$183$	0	0
$184$	−433921.	−0.944858
$185$	−196473.	−0.422059
$186$	0	0
$187$	−189342.	−0.395953
$188$	−75503.8	−0.155802
$189$	0	0
$190$	93429.4	0.187758
$191$	−591788.	−1.17377	−0.586885	−	0.809670i	$-0.699646\pi$
$191$	−591788.	−1.17377	−0.586885	+	0.809670i	$0.699646\pi$
$192$	0	0
$193$	−540825.	−1.04511	−0.522557	−	0.852605i	$-0.675022\pi$
$193$	−540825.	−1.04511	−0.522557	+	0.852605i	$0.675022\pi$
$194$	−497227.	−0.948529
$195$	0	0
$196$	51254.3	0.0952994
$197$	−481711.	−0.884344	−0.442172	−	0.896930i	$-0.645792\pi$
$197$	−481711.	−0.884344	−0.442172	+	0.896930i	$0.645792\pi$
$198$	0	0
$199$	−1.01427e6	−1.81560	−0.907801	−	0.419401i	$-0.862240\pi$
$199$	−1.01427e6	−1.81560	−0.907801	+	0.419401i	$0.862240\pi$
$200$	86006.8	0.152040
$201$	0	0
$202$	229717.	0.396108
$203$	−583395.	−0.993625
$204$	0	0
$205$	−130165.	−0.216326
$206$	−111276.	−0.182699
$207$	0	0
$208$	−213681.	−0.342459
$209$	68944.2	0.109177
$210$	0	0
$211$	619718.	0.958270	0.479135	−	0.877741i	$-0.340950\pi$
$211$	619718.	0.958270	0.479135	+	0.877741i	$0.340950\pi$
$212$	42249.3	0.0645625
$213$	0	0
$214$	79955.8	0.119348
$215$	−346857.	−0.511746
$216$	0	0
$217$	438109.	0.631587
$218$	−79616.9	−0.113466
$219$	0	0
$220$	−33333.2	−0.0464323
$221$	−266391.	−0.366892
$222$	0	0
$223$	894252.	1.20420	0.602099	−	0.798422i	$-0.294331\pi$
$223$	894252.	1.20420	0.602099	+	0.798422i	$0.294331\pi$
$224$	560918.	0.746928
$225$	0	0
$226$	1.34227e6	1.74812
$227$	−851468.	−1.09674	−0.548370	−	0.836236i	$-0.684751\pi$
$227$	−851468.	−1.09674	−0.548370	+	0.836236i	$0.684751\pi$
$228$	0	0
$229$	137840.	0.173695	0.0868475	−	0.996222i	$-0.472321\pi$
$229$	137840.	0.173695	0.0868475	+	0.996222i	$0.472321\pi$
$230$	−517047.	−0.644481
$231$	0	0
$232$	−548046.	−0.668494
$233$	1.31590e6	1.58793	0.793965	−	0.607963i	$-0.208013\pi$
$233$	1.31590e6	1.58793	0.793965	+	0.607963i	$0.208013\pi$
$234$	0	0
$235$	171300.	0.202343
$236$	−106329.	−0.124271
$237$	0	0
$238$	1.50346e6	1.72048
$239$	−1.11778e6	−1.26579	−0.632896	−	0.774237i	$-0.718134\pi$
$239$	−1.11778e6	−1.26579	−0.632896	+	0.774237i	$0.718134\pi$
$240$	0	0
$241$	−471581.	−0.523015	−0.261507	−	0.965201i	$-0.584220\pi$
$241$	−471581.	−0.523015	−0.261507	+	0.965201i	$0.584220\pi$
$242$	−96028.9	−0.105406
$243$	0	0
$244$	−232856.	−0.250387
$245$	−116284.	−0.123767
$246$	0	0
$247$	96999.3	0.101164
$248$	411564.	0.424921
$249$	0	0
$250$	102483.	0.103705
$251$	−1.76541e6	−1.76873	−0.884363	−	0.466799i	$-0.845407\pi$
$251$	−1.76541e6	−1.76873	−0.884363	+	0.466799i	$0.845407\pi$
$252$	0	0
$253$	−381543.	−0.374751
$254$	736176.	0.715974
$255$	0	0
$256$	969531.	0.924617
$257$	19920.1	0.0188130	0.00940652	−	0.999956i	$-0.497006\pi$
$257$	19920.1	0.0188130	0.00940652	+	0.999956i	$0.497006\pi$
$258$	0	0
$259$	1.15123e6	1.06638
$260$	−46897.3	−0.0430244
$261$	0	0
$262$	−76601.2	−0.0689418
$263$	2.18348e6	1.94652	0.973262	−	0.229697i	$-0.0737734\pi$
$263$	2.18348e6	1.94652	0.973262	+	0.229697i	$0.0737734\pi$
$264$	0	0
$265$	−95853.7	−0.0838483
$266$	−547446.	−0.474392
$267$	0	0
$268$	−479335.	−0.407664
$269$	−839300.	−0.707190	−0.353595	−	0.935399i	$-0.615041\pi$
$269$	−839300.	−0.707190	−0.353595	+	0.935399i	$0.615041\pi$
$270$	0	0
$271$	1.42641e6	1.17984	0.589918	−	0.807463i	$-0.299160\pi$
$271$	1.42641e6	1.17984	0.589918	+	0.807463i	$0.299160\pi$
$272$	1.96414e6	1.60972
$273$	0	0
$274$	−2.52475e6	−2.03162
$275$	75625.0	0.0603023
$276$	0	0
$277$	−1.30448e6	−1.02150	−0.510749	−	0.859730i	$-0.670632\pi$
$277$	−1.30448e6	−1.02150	−0.510749	+	0.859730i	$0.670632\pi$
$278$	2.03723e6	1.58098
$279$	0	0
$280$	−503954.	−0.384146
$281$	580646.	0.438678	0.219339	−	0.975649i	$-0.429610\pi$
$281$	580646.	0.438678	0.219339	+	0.975649i	$0.429610\pi$
$282$	0	0
$283$	−601974.	−0.446799	−0.223399	−	0.974727i	$-0.571715\pi$
$283$	−601974.	−0.446799	−0.223399	+	0.974727i	$0.571715\pi$
$284$	579691.	0.426482
$285$	0	0
$286$	−135106.	−0.0976693
$287$	762698.	0.546572
$288$	0	0
$289$	1.02878e6	0.724567
$290$	−653034.	−0.455975
$291$	0	0
$292$	−709285.	−0.486815
$293$	2.40993e6	1.63996	0.819982	−	0.572389i	$-0.193983\pi$
$293$	2.40993e6	1.63996	0.819982	+	0.572389i	$0.193983\pi$
$294$	0	0
$295$	241234.	0.161393
$296$	1.08147e6	0.717441
$297$	0	0
$298$	1.87062e6	1.22024
$299$	−536803.	−0.347246
$300$	0	0
$301$	2.03240e6	1.29298
$302$	3.42609e6	2.16163
$303$	0	0
$304$	−715191.	−0.443852
$305$	528294.	0.325182
$306$	0	0
$307$	−1.83380e6	−1.11047	−0.555235	−	0.831694i	$-0.687371\pi$
$307$	−1.83380e6	−1.11047	−0.555235	+	0.831694i	$0.687371\pi$
$308$	195315.	0.117316
$309$	0	0
$310$	490406.	0.289835
$311$	−2.27507e6	−1.33381	−0.666906	−	0.745142i	$-0.732382\pi$
$311$	−2.27507e6	−1.33381	−0.666906	+	0.745142i	$0.732382\pi$
$312$	0	0
$313$	159585.	0.0920728	0.0460364	−	0.998940i	$-0.485341\pi$
$313$	159585.	0.0920728	0.0460364	+	0.998940i	$0.485341\pi$
$314$	14571.6	0.00834035
$315$	0	0
$316$	318849.	0.179625
$317$	1.24338e6	0.694955	0.347477	−	0.937688i	$-0.387038\pi$
$317$	1.24338e6	0.694955	0.347477	+	0.937688i	$0.387038\pi$
$318$	0	0
$319$	−481892.	−0.265139
$320$	−376280.	−0.205417
$321$	0	0
$322$	3.02962e6	1.62835
$323$	−891609.	−0.475519
$324$	0	0
$325$	106399.	0.0558764
$326$	−1.64068e6	−0.855029
$327$	0	0
$328$	716485.	0.367724
$329$	−1.00373e6	−0.511242
$330$	0	0
$331$	−958448.	−0.480838	−0.240419	−	0.970669i	$-0.577285\pi$
$331$	−958448.	−0.480838	−0.240419	+	0.970669i	$0.577285\pi$
$332$	51224.5	0.0255054
$333$	0	0
$334$	−276021.	−0.135387
$335$	1.08750e6	0.529439
$336$	0	0
$337$	−1.51074e6	−0.724626	−0.362313	−	0.932056i	$-0.618013\pi$
$337$	−1.51074e6	−0.724626	−0.362313	+	0.932056i	$0.618013\pi$
$338$	2.24519e6	1.06896
$339$	0	0
$340$	431076.	0.202235
$341$	361884.	0.168533
$342$	0	0
$343$	−1.78064e6	−0.817224
$344$	1.90925e6	0.869896
$345$	0	0
$346$	−768741.	−0.345215
$347$	1.93847e6	0.864243	0.432122	−	0.901815i	$-0.357765\pi$
$347$	1.93847e6	0.864243	0.432122	+	0.901815i	$0.357765\pi$
$348$	0	0
$349$	−1.62462e6	−0.713985	−0.356992	−	0.934107i	$-0.616198\pi$
$349$	−1.62462e6	−0.713985	−0.356992	+	0.934107i	$0.616198\pi$
$350$	−600495.	−0.262023
$351$	0	0
$352$	463325.	0.199310
$353$	23347.5	0.00997249	0.00498624	−	0.999988i	$-0.498413\pi$
$353$	23347.5	0.00997249	0.00498624	+	0.999988i	$0.498413\pi$
$354$	0	0
$355$	−1.31518e6	−0.553878
$356$	1.14002e6	0.476748
$357$	0	0
$358$	3.82374e6	1.57682
$359$	−3.63918e6	−1.49028	−0.745139	−	0.666909i	$-0.767617\pi$
$359$	−3.63918e6	−1.49028	−0.745139	+	0.666909i	$0.767617\pi$
$360$	0	0
$361$	−2.15144e6	−0.868884
$362$	−1.54325e6	−0.618964
$363$	0	0
$364$	274793.	0.108706
$365$	1.60920e6	0.632233
$366$	0	0
$367$	2.24713e6	0.870890	0.435445	−	0.900215i	$-0.356591\pi$
$367$	2.24713e6	0.870890	0.435445	+	0.900215i	$0.356591\pi$
$368$	3.95793e6	1.52352
$369$	0	0
$370$	1.28865e6	0.489362
$371$	561651.	0.211852
$372$	0	0
$373$	3.82745e6	1.42442	0.712208	−	0.701968i	$-0.247695\pi$
$373$	3.82745e6	1.42442	0.712208	+	0.701968i	$0.247695\pi$
$374$	1.24188e6	0.459092
$375$	0	0
$376$	−942910.	−0.343954
$377$	−677986.	−0.245679
$378$	0	0
$379$	−4.26443e6	−1.52498	−0.762489	−	0.647002i	$-0.776023\pi$
$379$	−4.26443e6	−1.52498	−0.762489	+	0.647002i	$0.776023\pi$
$380$	−156965.	−0.0557628
$381$	0	0
$382$	3.88148e6	1.36094
$383$	−2.40920e6	−0.839221	−0.419611	−	0.907704i	$-0.637833\pi$
$383$	−2.40920e6	−0.839221	−0.419611	+	0.907704i	$0.637833\pi$
$384$	0	0
$385$	−443122.	−0.152360
$386$	3.54722e6	1.21177
$387$	0	0
$388$	835362.	0.281706
$389$	−3.73204e6	−1.25047	−0.625233	−	0.780438i	$-0.714996\pi$
$389$	−3.73204e6	−1.25047	−0.625233	+	0.780438i	$0.714996\pi$
$390$	0	0
$391$	4.93424e6	1.63222
$392$	640077.	0.210386
$393$	0	0
$394$	3.15950e6	1.02536
$395$	−723393.	−0.233282
$396$	0	0
$397$	−4.89288e6	−1.55808	−0.779038	−	0.626977i	$-0.784292\pi$
$397$	−4.89288e6	−1.55808	−0.779038	+	0.626977i	$0.784292\pi$
$398$	6.65250e6	2.10512
$399$	0	0
$400$	−784495.	−0.245155
$401$	3.93524e6	1.22211	0.611055	−	0.791588i	$-0.290745\pi$
$401$	3.93524e6	1.22211	0.611055	+	0.791588i	$0.290745\pi$
$402$	0	0
$403$	509144.	0.156163
$404$	−385933.	−0.117641
$405$	0	0
$406$	3.82643e6	1.15207
$407$	950929.	0.284552
$408$	0	0
$409$	657037.	0.194215	0.0971073	−	0.995274i	$-0.469041\pi$
$409$	657037.	0.194215	0.0971073	+	0.995274i	$0.469041\pi$
$410$	853740.	0.250822
$411$	0	0
$412$	186949.	0.0542600
$413$	−1.41351e6	−0.407777
$414$	0	0
$415$	−116216.	−0.0331243
$416$	651864.	0.184682
$417$	0	0
$418$	−452198.	−0.126587
$419$	−4.38504e6	−1.22022	−0.610110	−	0.792317i	$-0.708875\pi$
$419$	−4.38504e6	−1.22022	−0.610110	+	0.792317i	$0.708875\pi$
$420$	0	0
$421$	2.34058e6	0.643604	0.321802	−	0.946807i	$-0.395711\pi$
$421$	2.34058e6	0.643604	0.321802	+	0.946807i	$0.395711\pi$
$422$	−4.06467e6	−1.11108
$423$	0	0
$424$	527620.	0.142530
$425$	−978008.	−0.262646
$426$	0	0
$427$	−3.09552e6	−0.821607
$428$	−134329.	−0.0354455
$429$	0	0
$430$	2.27500e6	0.593350
$431$	2.18215e6	0.565838	0.282919	−	0.959144i	$-0.408697\pi$
$431$	2.18215e6	0.565838	0.282919	+	0.959144i	$0.408697\pi$
$432$	0	0
$433$	−3.61080e6	−0.925515	−0.462757	−	0.886485i	$-0.653140\pi$
$433$	−3.61080e6	−0.925515	−0.462757	+	0.886485i	$0.653140\pi$
$434$	−2.87352e6	−0.732301
$435$	0	0
$436$	133760.	0.0336984
$437$	−1.79668e6	−0.450056
$438$	0	0
$439$	6.48191e6	1.60525	0.802624	−	0.596486i	$-0.203437\pi$
$439$	6.48191e6	1.60525	0.802624	+	0.596486i	$0.203437\pi$
$440$	−416273.	−0.102505
$441$	0	0
$442$	1.74723e6	0.425397
$443$	6.81523e6	1.64995	0.824976	−	0.565167i	$-0.191188\pi$
$443$	6.81523e6	1.64995	0.824976	+	0.565167i	$0.191188\pi$
$444$	0	0
$445$	−2.58644e6	−0.619160
$446$	−5.86531e6	−1.39622
$447$	0	0
$448$	2.20480e6	0.519008
$449$	2.98356e6	0.698423	0.349211	−	0.937044i	$-0.386449\pi$
$449$	2.98356e6	0.698423	0.349211	+	0.937044i	$0.386449\pi$
$450$	0	0
$451$	629999.	0.145847
$452$	−2.25508e6	−0.519177
$453$	0	0
$454$	5.58469e6	1.27163
$455$	−623440.	−0.141178
$456$	0	0
$457$	−6.11184e6	−1.36893	−0.684466	−	0.729045i	$-0.739964\pi$
$457$	−6.11184e6	−1.36893	−0.684466	+	0.729045i	$0.739964\pi$
$458$	−904081.	−0.201393
$459$	0	0
$460$	868659.	0.191406
$461$	3.41772e6	0.749003	0.374502	−	0.927226i	$-0.377814\pi$
$461$	3.41772e6	0.749003	0.374502	+	0.927226i	$0.377814\pi$
$462$	0	0
$463$	−4.60011e6	−0.997277	−0.498639	−	0.866810i	$-0.666166\pi$
$463$	−4.60011e6	−0.997277	−0.498639	+	0.866810i	$0.666166\pi$
$464$	4.99890e6	1.07790
$465$	0	0
$466$	−8.63083e6	−1.84114
$467$	−5.03617e6	−1.06858	−0.534291	−	0.845301i	$-0.679421\pi$
$467$	−5.03617e6	−1.06858	−0.534291	+	0.845301i	$0.679421\pi$
$468$	0	0
$469$	−6.37215e6	−1.33769
$470$	−1.12354e6	−0.234609
$471$	0	0
$472$	−1.32786e6	−0.274345
$473$	1.67879e6	0.345019
$474$	0	0
$475$	356117.	0.0724199
$476$	−2.52587e6	−0.510969
$477$	0	0
$478$	7.33142e6	1.46764
$479$	2.82760e6	0.563091	0.281545	−	0.959548i	$-0.409153\pi$
$479$	2.82760e6	0.563091	0.281545	+	0.959548i	$0.409153\pi$
$480$	0	0
$481$	1.33789e6	0.263668
$482$	3.09306e6	0.606416
$483$	0	0
$484$	161333.	0.0313046
$485$	−1.89524e6	−0.365855
$486$	0	0
$487$	−5.68931e6	−1.08702	−0.543510	−	0.839403i	$-0.682905\pi$
$487$	−5.68931e6	−1.08702	−0.543510	+	0.839403i	$0.682905\pi$
$488$	−2.90796e6	−0.552763
$489$	0	0
$490$	762695.	0.143503
$491$	−3.23715e6	−0.605981	−0.302990	−	0.952994i	$-0.597985\pi$
$491$	−3.23715e6	−0.605981	−0.302990	+	0.952994i	$0.597985\pi$
$492$	0	0
$493$	6.23199e6	1.15481
$494$	−636209.	−0.117296
$495$	0	0
$496$	−3.75400e6	−0.685157
$497$	7.70626e6	1.39943
$498$	0	0
$499$	−5.08564e6	−0.914312	−0.457156	−	0.889386i	$-0.651132\pi$
$499$	−5.08564e6	−0.914312	−0.457156	+	0.889386i	$0.651132\pi$
$500$	−172175.	−0.0307997
$501$	0	0
$502$	1.15791e7	2.05077
$503$	−9.02252e6	−1.59004	−0.795019	−	0.606584i	$-0.792539\pi$
$503$	−9.02252e6	−1.59004	−0.795019	+	0.606584i	$0.792539\pi$
$504$	0	0
$505$	875591.	0.152782
$506$	2.50251e6	0.434509
$507$	0	0
$508$	−1.23681e6	−0.212639
$509$	−8.98909e6	−1.53788	−0.768938	−	0.639323i	$-0.779215\pi$
$509$	−8.98909e6	−1.53788	−0.768938	+	0.639323i	$0.779215\pi$
$510$	0	0
$511$	−9.42904e6	−1.59741
$512$	720997.	0.121551
$513$	0	0
$514$	−130654.	−0.0218130
$515$	−424143.	−0.0704683
$516$	0	0
$517$	−829093.	−0.136420
$518$	−7.55079e6	−1.23643
$519$	0	0
$520$	−585665.	−0.0949819
$521$	1.11219e7	1.79508	0.897539	−	0.440936i	$-0.145353\pi$
$521$	1.11219e7	1.79508	0.897539	+	0.440936i	$0.145353\pi$
$522$	0	0
$523$	3.67510e6	0.587510	0.293755	−	0.955881i	$-0.405095\pi$
$523$	3.67510e6	0.587510	0.293755	+	0.955881i	$0.405095\pi$
$524$	128693.	0.0204752
$525$	0	0
$526$	−1.43212e7	−2.25692
$527$	−4.68001e6	−0.734040
$528$	0	0
$529$	3.50664e6	0.544818
$530$	628695.	0.0972188
$531$	0	0
$532$	919733.	0.140891
$533$	886361.	0.135143
$534$	0	0
$535$	304761.	0.0460335
$536$	−5.98605e6	−0.899971
$537$	0	0
$538$	5.50489e6	0.819960
$539$	562814.	0.0834436
$540$	0	0
$541$	−4.31097e6	−0.633260	−0.316630	−	0.948549i	$-0.602551\pi$
$541$	−4.31097e6	−0.633260	−0.316630	+	0.948549i	$0.602551\pi$
$542$	−9.35569e6	−1.36797
$543$	0	0
$544$	−5.99188e6	−0.868092
$545$	−303469.	−0.0437646
$546$	0	0
$547$	−1.80626e6	−0.258114	−0.129057	−	0.991637i	$-0.541195\pi$
$547$	−1.80626e6	−0.258114	−0.129057	+	0.991637i	$0.541195\pi$
$548$	4.24169e6	0.603375
$549$	0	0
$550$	−496017.	−0.0699182
$551$	−2.26922e6	−0.318418
$552$	0	0
$553$	4.23870e6	0.589413
$554$	8.55595e6	1.18439
$555$	0	0
$556$	−3.42262e6	−0.469540
$557$	−8.42670e6	−1.15085	−0.575426	−	0.817854i	$-0.695164\pi$
$557$	−8.42670e6	−1.15085	−0.575426	+	0.817854i	$0.695164\pi$
$558$	0	0
$559$	2.36193e6	0.319696
$560$	4.59672e6	0.619410
$561$	0	0
$562$	−3.80840e6	−0.508630
$563$	9.19966e6	1.22321	0.611605	−	0.791163i	$-0.290524\pi$
$563$	9.19966e6	1.22321	0.611605	+	0.791163i	$0.290524\pi$
$564$	0	0
$565$	5.11623e6	0.674262
$566$	3.94829e6	0.518046
$567$	0	0
$568$	7.23932e6	0.941515
$569$	−5.18644e6	−0.671566	−0.335783	−	0.941939i	$-0.609001\pi$
$569$	−5.18644e6	−0.671566	−0.335783	+	0.941939i	$0.609001\pi$
$570$	0	0
$571$	−3.29390e6	−0.422786	−0.211393	−	0.977401i	$-0.567800\pi$
$571$	−3.29390e6	−0.422786	−0.211393	+	0.977401i	$0.567800\pi$
$572$	226983.	0.0290070
$573$	0	0
$574$	−5.00246e6	−0.633730
$575$	−1.97078e6	−0.248581
$576$	0	0
$577$	−1.02043e7	−1.27598	−0.637989	−	0.770046i	$-0.720233\pi$
$577$	−1.02043e7	−1.27598	−0.637989	+	0.770046i	$0.720233\pi$
$578$	−6.74768e6	−0.840107
$579$	0	0
$580$	1.09712e6	0.135421
$581$	680965.	0.0836921
$582$	0	0
$583$	463932.	0.0565305
$584$	−8.85773e6	−1.07471
$585$	0	0
$586$	−1.58065e7	−1.90148
$587$	1.33951e7	1.60454	0.802272	−	0.596958i	$-0.203624\pi$
$587$	1.33951e7	1.60454	0.802272	+	0.596958i	$0.203624\pi$
$588$	0	0
$589$	1.70411e6	0.202399
$590$	−1.58223e6	−0.187129
$591$	0	0
$592$	−9.86445e6	−1.15683
$593$	−3.76550e6	−0.439730	−0.219865	−	0.975530i	$-0.570562\pi$
$593$	−3.76550e6	−0.439730	−0.219865	+	0.975530i	$0.570562\pi$
$594$	0	0
$595$	5.73061e6	0.663603
$596$	−3.14272e6	−0.362401
$597$	0	0
$598$	3.52084e6	0.402618
$599$	−1.21252e6	−0.138077	−0.0690384	−	0.997614i	$-0.521993\pi$
$599$	−1.21252e6	−0.138077	−0.0690384	+	0.997614i	$0.521993\pi$
$600$	0	0
$601$	−7.81714e6	−0.882799	−0.441399	−	0.897311i	$-0.645518\pi$
$601$	−7.81714e6	−0.882799	−0.441399	+	0.897311i	$0.645518\pi$
$602$	−1.33303e7	−1.49916
$603$	0	0
$604$	−5.75597e6	−0.641987
$605$	−366025.	−0.0406558
$606$	0	0
$607$	5.43853e6	0.599114	0.299557	−	0.954078i	$-0.403161\pi$
$607$	5.43853e6	0.599114	0.299557	+	0.954078i	$0.403161\pi$
$608$	2.18179e6	0.239361
$609$	0	0
$610$	−3.46503e6	−0.377036
$611$	−1.16647e6	−0.126407
$612$	0	0
$613$	1.70637e7	1.83409	0.917045	−	0.398783i	$-0.130567\pi$
$613$	1.70637e7	1.83409	0.917045	+	0.398783i	$0.130567\pi$
$614$	1.20277e7	1.28755
$615$	0	0
$616$	2.43914e6	0.258991
$617$	−7.80667e6	−0.825568	−0.412784	−	0.910829i	$-0.635444\pi$
$617$	−7.80667e6	−0.825568	−0.412784	+	0.910829i	$0.635444\pi$
$618$	0	0
$619$	7.31658e6	0.767506	0.383753	−	0.923436i	$-0.374631\pi$
$619$	7.31658e6	0.767506	0.383753	+	0.923436i	$0.374631\pi$
$620$	−823902.	−0.0860788
$621$	0	0
$622$	1.49220e7	1.54650
$623$	1.51552e7	1.56438
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.04670e6	−0.106755
$627$	0	0
$628$	−24481.0	−0.00247702
$629$	−1.22977e7	−1.23936
$630$	0	0
$631$	−1.18729e7	−1.18709	−0.593543	−	0.804802i	$-0.702271\pi$
$631$	−1.18729e7	−1.18709	−0.593543	+	0.804802i	$0.702271\pi$
$632$	3.98187e6	0.396547
$633$	0	0
$634$	−8.15523e6	−0.805773
$635$	2.80602e6	0.276157
$636$	0	0
$637$	791837.	0.0773192
$638$	3.16068e6	0.307418
$639$	0	0
$640$	5.53129e6	0.533798
$641$	1.85963e7	1.78764	0.893821	−	0.448424i	$-0.148015\pi$
$641$	1.85963e7	1.78764	0.893821	+	0.448424i	$0.148015\pi$
$642$	0	0
$643$	−1.18028e7	−1.12579	−0.562894	−	0.826529i	$-0.690312\pi$
$643$	−1.18028e7	−1.12579	−0.562894	+	0.826529i	$0.690312\pi$
$644$	−5.08988e6	−0.483608
$645$	0	0
$646$	5.84798e6	0.551346
$647$	−2.05252e7	−1.92764	−0.963821	−	0.266551i	$-0.914116\pi$
$647$	−2.05252e7	−1.92764	−0.963821	+	0.266551i	$0.914116\pi$
$648$	0	0
$649$	−1.16757e6	−0.108811
$650$	−697859.	−0.0647865
$651$	0	0
$652$	2.75642e6	0.253937
$653$	5.78511e6	0.530919	0.265460	−	0.964122i	$-0.414476\pi$
$653$	5.78511e6	0.530919	0.265460	+	0.964122i	$0.414476\pi$
$654$	0	0
$655$	−291974.	−0.0265914
$656$	−6.53528e6	−0.592932
$657$	0	0
$658$	6.58335e6	0.592765
$659$	−1.88598e7	−1.69170	−0.845852	−	0.533418i	$-0.820907\pi$
$659$	−1.88598e7	−1.69170	−0.845852	+	0.533418i	$0.820907\pi$
$660$	0	0
$661$	−1.12652e7	−1.00285	−0.501426	−	0.865201i	$-0.667191\pi$
$661$	−1.12652e7	−1.00285	−0.501426	+	0.865201i	$0.667191\pi$
$662$	6.28637e6	0.557513
$663$	0	0
$664$	639704.	0.0563065
$665$	−2.08665e6	−0.182977
$666$	0	0
$667$	1.25581e7	1.09297
$668$	463727.	0.0402088
$669$	0	0
$670$	−7.13279e6	−0.613864
$671$	−2.55694e6	−0.219237
$672$	0	0
$673$	1.51301e7	1.28767	0.643833	−	0.765166i	$-0.277343\pi$
$673$	1.51301e7	1.28767	0.643833	+	0.765166i	$0.277343\pi$
$674$	9.90878e6	0.840176
$675$	0	0
$676$	−3.77201e6	−0.317473
$677$	1.12001e7	0.939187	0.469593	−	0.882883i	$-0.344401\pi$
$677$	1.12001e7	0.939187	0.469593	+	0.882883i	$0.344401\pi$
$678$	0	0
$679$	1.11051e7	0.924373
$680$	5.38338e6	0.446460
$681$	0	0
$682$	−2.37356e6	−0.195407
$683$	−2.12907e7	−1.74638	−0.873190	−	0.487379i	$-0.837953\pi$
$683$	−2.12907e7	−1.74638	−0.873190	+	0.487379i	$0.837953\pi$
$684$	0	0
$685$	−9.62338e6	−0.783612
$686$	1.16791e7	0.947539
$687$	0	0
$688$	−1.74149e7	−1.40265
$689$	652717.	0.0523814
$690$	0	0
$691$	1.77648e6	0.141536	0.0707678	−	0.997493i	$-0.477455\pi$
$691$	1.77648e6	0.141536	0.0707678	+	0.997493i	$0.477455\pi$
$692$	1.29152e6	0.102526
$693$	0	0
$694$	−1.27143e7	−1.00206
$695$	7.76512e6	0.609798
$696$	0	0
$697$	−8.14735e6	−0.635235
$698$	1.06557e7	0.827838
$699$	0	0
$700$	1.00886e6	0.0778188
$701$	−94420.1	−0.00725720	−0.00362860	−	0.999993i	$-0.501155\pi$
$701$	−94420.1	−0.00725720	−0.00362860	+	0.999993i	$0.501155\pi$
$702$	0	0
$703$	4.47791e6	0.341733
$704$	1.82120e6	0.138492
$705$	0	0
$706$	−153134.	−0.0115627
$707$	−5.13050e6	−0.386021
$708$	0	0
$709$	−1.46333e7	−1.09327	−0.546633	−	0.837372i	$-0.684091\pi$
$709$	−1.46333e7	−1.09327	−0.546633	+	0.837372i	$0.684091\pi$
$710$	8.62614e6	0.642201
$711$	0	0
$712$	1.42369e7	1.05248
$713$	−9.43067e6	−0.694734
$714$	0	0
$715$	−514970.	−0.0376718
$716$	−6.42405e6	−0.468302
$717$	0	0
$718$	2.38690e7	1.72792
$719$	−1.15215e7	−0.831161	−0.415580	−	0.909556i	$-0.636422\pi$
$719$	−1.15215e7	−0.831161	−0.415580	+	0.909556i	$0.636422\pi$
$720$	0	0
$721$	2.48525e6	0.178046
$722$	1.41111e7	1.00744
$723$	0	0
$724$	2.59273e6	0.183827
$725$	−2.48911e6	−0.175873
$726$	0	0
$727$	1.73993e7	1.22095	0.610473	−	0.792037i	$-0.290979\pi$
$727$	1.73993e7	1.22095	0.610473	+	0.792037i	$0.290979\pi$
$728$	3.43168e6	0.239982
$729$	0	0
$730$	−1.05546e7	−0.733050
$731$	−2.17107e7	−1.50272
$732$	0	0
$733$	−1.21356e7	−0.834262	−0.417131	−	0.908846i	$-0.636964\pi$
$733$	−1.21356e7	−0.834262	−0.417131	+	0.908846i	$0.636964\pi$
$734$	−1.47387e7	−1.00976
$735$	0	0
$736$	−1.20742e7	−0.821608
$737$	−5.26348e6	−0.356948
$738$	0	0
$739$	1.30600e7	0.879692	0.439846	−	0.898073i	$-0.355033\pi$
$739$	1.30600e7	0.879692	0.439846	+	0.898073i	$0.355033\pi$
$740$	−2.16498e6	−0.145337
$741$	0	0
$742$	−3.68382e6	−0.245634
$743$	2.36143e6	0.156929	0.0784644	−	0.996917i	$-0.474998\pi$
$743$	2.36143e6	0.156929	0.0784644	+	0.996917i	$0.474998\pi$
$744$	0	0
$745$	7.13008e6	0.470656
$746$	−2.51038e7	−1.65156
$747$	0	0
$748$	−2.08641e6	−0.136347
$749$	−1.78574e6	−0.116309
$750$	0	0
$751$	4.12172e6	0.266673	0.133336	−	0.991071i	$-0.457431\pi$
$751$	4.12172e6	0.266673	0.133336	+	0.991071i	$0.457431\pi$
$752$	8.60058e6	0.554604
$753$	0	0
$754$	4.44685e6	0.284855
$755$	1.30589e7	0.833759
$756$	0	0
$757$	4.12364e6	0.261542	0.130771	−	0.991413i	$-0.458255\pi$
$757$	4.12364e6	0.261542	0.130771	+	0.991413i	$0.458255\pi$
$758$	2.79700e7	1.76815
$759$	0	0
$760$	−1.96022e6	−0.123104
$761$	2.08230e7	1.30341	0.651706	−	0.758472i	$-0.274054\pi$
$761$	2.08230e7	1.30341	0.651706	+	0.758472i	$0.274054\pi$
$762$	0	0
$763$	1.77817e6	0.110576
$764$	−6.52105e6	−0.404189
$765$	0	0
$766$	1.58017e7	0.973045
$767$	−1.64269e6	−0.100825
$768$	0	0
$769$	1.58156e7	0.964427	0.482213	−	0.876054i	$-0.339833\pi$
$769$	1.58156e7	0.964427	0.482213	+	0.876054i	$0.339833\pi$
$770$	2.90640e6	0.176656
$771$	0	0
$772$	−5.95947e6	−0.359886
$773$	2.58350e6	0.155510	0.0777552	−	0.996972i	$-0.475225\pi$
$773$	2.58350e6	0.155510	0.0777552	+	0.996972i	$0.475225\pi$
$774$	0	0
$775$	1.86924e6	0.111792
$776$	1.04322e7	0.621902
$777$	0	0
$778$	2.44781e7	1.44987
$779$	2.96665e6	0.175155
$780$	0	0
$781$	6.36547e6	0.373425
$782$	−3.23632e7	−1.89250
$783$	0	0
$784$	−5.83834e6	−0.339234
$785$	55541.5	0.00321694
$786$	0	0
$787$	−5.64885e6	−0.325105	−0.162552	−	0.986700i	$-0.551973\pi$
$787$	−5.64885e6	−0.325105	−0.162552	+	0.986700i	$0.551973\pi$
$788$	−5.30809e6	−0.304525
$789$	0	0
$790$	4.74467e6	0.270482
$791$	−2.99784e7	−1.70360
$792$	0	0
$793$	−3.59743e6	−0.203146
$794$	3.20920e7	1.80653
$795$	0	0
$796$	−1.11765e7	−0.625204
$797$	3.72876e6	0.207931	0.103965	−	0.994581i	$-0.466847\pi$
$797$	3.72876e6	0.207931	0.103965	+	0.994581i	$0.466847\pi$
$798$	0	0
$799$	1.07221e7	0.594173
$800$	2.39321e6	0.132207
$801$	0	0
$802$	−2.58109e7	−1.41699
$803$	−7.78852e6	−0.426252
$804$	0	0
$805$	1.15477e7	0.628068
$806$	−3.33943e6	−0.181065
$807$	0	0
$808$	−4.81963e6	−0.259708
$809$	−6.35004e6	−0.341119	−0.170559	−	0.985347i	$-0.554557\pi$
$809$	−6.35004e6	−0.341119	−0.170559	+	0.985347i	$0.554557\pi$
$810$	0	0
$811$	1.34156e7	0.716241	0.358120	−	0.933675i	$-0.383418\pi$
$811$	1.34156e7	0.716241	0.358120	+	0.933675i	$0.383418\pi$
$812$	−6.42856e6	−0.342156
$813$	0	0
$814$	−6.23705e6	−0.329928
$815$	−6.25365e6	−0.329791
$816$	0	0
$817$	7.90538e6	0.414350
$818$	−4.30945e6	−0.225184
$819$	0	0
$820$	−1.43432e6	−0.0744922
$821$	−2.40201e7	−1.24370	−0.621851	−	0.783135i	$-0.713619\pi$
$821$	−2.40201e7	−1.24370	−0.621851	+	0.783135i	$0.713619\pi$
$822$	0	0
$823$	−1.46523e6	−0.0754058	−0.0377029	−	0.999289i	$-0.512004\pi$
$823$	−1.46523e6	−0.0754058	−0.0377029	+	0.999289i	$0.512004\pi$
$824$	2.33466e6	0.119786
$825$	0	0
$826$	9.27105e6	0.472801
$827$	2.32918e7	1.18424	0.592120	−	0.805850i	$-0.298291\pi$
$827$	2.32918e7	1.18424	0.592120	+	0.805850i	$0.298291\pi$
$828$	0	0
$829$	−2.02821e6	−0.102501	−0.0512503	−	0.998686i	$-0.516321\pi$
$829$	−2.02821e6	−0.102501	−0.0512503	+	0.998686i	$0.516321\pi$
$830$	762250.	0.0384063
$831$	0	0
$832$	2.56229e6	0.128327
$833$	−7.27850e6	−0.363437
$834$	0	0
$835$	−1.05208e6	−0.0522197
$836$	759712.	0.0375953
$837$	0	0
$838$	2.87610e7	1.41480
$839$	3.07518e7	1.50822	0.754111	−	0.656747i	$-0.228068\pi$
$839$	3.07518e7	1.50822	0.754111	+	0.656747i	$0.228068\pi$
$840$	0	0
$841$	−4.65021e6	−0.226716
$842$	−1.53517e7	−0.746234
$843$	0	0
$844$	6.82881e6	0.329981
$845$	8.55780e6	0.412307
$846$	0	0
$847$	2.14471e6	0.102721
$848$	−4.81259e6	−0.229821
$849$	0	0
$850$	6.41466e6	0.304527
$851$	−2.47811e7	−1.17300
$852$	0	0
$853$	−2.05598e7	−0.967491	−0.483745	−	0.875209i	$-0.660724\pi$
$853$	−2.05598e7	−0.967491	−0.483745	+	0.875209i	$0.660724\pi$
$854$	2.03032e7	0.952621
$855$	0	0
$856$	−1.67753e6	−0.0782505
$857$	−3.20463e7	−1.49048	−0.745240	−	0.666796i	$-0.767665\pi$
$857$	−3.20463e7	−1.49048	−0.745240	+	0.666796i	$0.767665\pi$
$858$	0	0
$859$	2.34894e6	0.108615	0.0543075	−	0.998524i	$-0.482705\pi$
$859$	2.34894e6	0.108615	0.0543075	+	0.998524i	$0.482705\pi$
$860$	−3.82210e6	−0.176220
$861$	0	0
$862$	−1.43125e7	−0.656068
$863$	1.44887e7	0.662219	0.331110	−	0.943592i	$-0.392577\pi$
$863$	1.44887e7	0.662219	0.331110	+	0.943592i	$0.392577\pi$
$864$	0	0
$865$	−2.93014e6	−0.133152
$866$	2.36829e7	1.07310
$867$	0	0
$868$	4.82763e6	0.217488
$869$	3.50122e6	0.157279
$870$	0	0
$871$	−7.40533e6	−0.330749
$872$	1.67042e6	0.0743936
$873$	0	0
$874$	1.17842e7	0.521823
$875$	−2.28885e6	−0.101064
$876$	0	0
$877$	−3.17684e7	−1.39475	−0.697376	−	0.716705i	$-0.745649\pi$
$877$	−3.17684e7	−1.39475	−0.697376	+	0.716705i	$0.745649\pi$
$878$	−4.25143e7	−1.86122
$879$	0	0
$880$	3.79696e6	0.165283
$881$	−1.06777e7	−0.463486	−0.231743	−	0.972777i	$-0.574443\pi$
$881$	−1.06777e7	−0.463486	−0.231743	+	0.972777i	$0.574443\pi$
$882$	0	0
$883$	1.77890e7	0.767804	0.383902	−	0.923374i	$-0.374580\pi$
$883$	1.77890e7	0.767804	0.383902	+	0.923374i	$0.374580\pi$
$884$	−2.93542e6	−0.126340
$885$	0	0
$886$	−4.47005e7	−1.91306
$887$	−3.13731e7	−1.33890	−0.669450	−	0.742857i	$-0.733470\pi$
$887$	−3.13731e7	−1.33890	−0.669450	+	0.742857i	$0.733470\pi$
$888$	0	0
$889$	−1.64418e7	−0.697741
$890$	1.69642e7	0.717892
$891$	0	0
$892$	9.85397e6	0.414667
$893$	−3.90418e6	−0.163833
$894$	0	0
$895$	1.45746e7	0.608191
$896$	−3.24104e7	−1.34870
$897$	0	0
$898$	−1.95689e7	−0.809795
$899$	−1.19110e7	−0.491529
$900$	0	0
$901$	−5.99972e6	−0.246218
$902$	−4.13210e6	−0.169104
$903$	0	0
$904$	−2.81620e7	−1.14615
$905$	−5.88228e6	−0.238739
$906$	0	0
$907$	−8.46355e6	−0.341613	−0.170807	−	0.985305i	$-0.554637\pi$
$907$	−8.46355e6	−0.341613	−0.170807	+	0.985305i	$0.554637\pi$
$908$	−9.38252e6	−0.377663
$909$	0	0
$910$	4.08908e6	0.163690
$911$	2.18411e7	0.871923	0.435961	−	0.899965i	$-0.356408\pi$
$911$	2.18411e7	0.871923	0.435961	+	0.899965i	$0.356408\pi$
$912$	0	0
$913$	562486.	0.0223324
$914$	4.00870e7	1.58722
$915$	0	0
$916$	1.51889e6	0.0598120
$917$	1.71081e6	0.0671861
$918$	0	0
$919$	4.18752e7	1.63557	0.817784	−	0.575525i	$-0.195202\pi$
$919$	4.18752e7	1.63557	0.817784	+	0.575525i	$0.195202\pi$
$920$	1.08480e7	0.422553
$921$	0	0
$922$	−2.24165e7	−0.868440
$923$	8.95575e6	0.346017
$924$	0	0
$925$	4.91183e6	0.188751
$926$	3.01717e7	1.15630
$927$	0	0
$928$	−1.52498e7	−0.581293
$929$	−8.03740e6	−0.305546	−0.152773	−	0.988261i	$-0.548820\pi$
$929$	−8.03740e6	−0.305546	−0.152773	+	0.988261i	$0.548820\pi$
$930$	0	0
$931$	2.65028e6	0.100211
$932$	1.45002e7	0.546805
$933$	0	0
$934$	3.30317e7	1.23898
$935$	4.73356e6	0.177076
$936$	0	0
$937$	−1.78192e6	−0.0663038	−0.0331519	−	0.999450i	$-0.510555\pi$
$937$	−1.78192e6	−0.0663038	−0.0331519	+	0.999450i	$0.510555\pi$
$938$	4.17943e7	1.55099
$939$	0	0
$940$	1.88760e6	0.0696770
$941$	−2.97245e7	−1.09431	−0.547154	−	0.837032i	$-0.684289\pi$
$941$	−2.97245e7	−1.09431	−0.547154	+	0.837032i	$0.684289\pi$
$942$	0	0
$943$	−1.64177e7	−0.601220
$944$	1.21118e7	0.442364
$945$	0	0
$946$	−1.10110e7	−0.400036
$947$	−931083.	−0.0337375	−0.0168688	−	0.999858i	$-0.505370\pi$
$947$	−931083.	−0.0337375	−0.0168688	+	0.999858i	$0.505370\pi$
$948$	0	0
$949$	−1.09579e7	−0.394967
$950$	−2.33573e6	−0.0839681
$951$	0	0
$952$	−3.15437e7	−1.12803
$953$	2.74983e7	0.980784	0.490392	−	0.871502i	$-0.336854\pi$
$953$	2.74983e7	0.980784	0.490392	+	0.871502i	$0.336854\pi$
$954$	0	0
$955$	1.47947e7	0.524926
$956$	−1.23171e7	−0.435876
$957$	0	0
$958$	−1.85459e7	−0.652882
$959$	5.63879e7	1.97988
$960$	0	0
$961$	−1.96844e7	−0.687565
$962$	−8.77507e6	−0.305712
$963$	0	0
$964$	−5.19646e6	−0.180101
$965$	1.35206e7	0.467389
$966$	0	0
$967$	4.33623e6	0.149123	0.0745617	−	0.997216i	$-0.476244\pi$
$967$	4.33623e6	0.149123	0.0745617	+	0.997216i	$0.476244\pi$
$968$	2.01476e6	0.0691091
$969$	0	0
$970$	1.24307e7	0.424195
$971$	5.44123e7	1.85204	0.926018	−	0.377480i	$-0.123209\pi$
$971$	5.44123e7	1.85204	0.926018	+	0.377480i	$0.123209\pi$
$972$	0	0
$973$	−4.54995e7	−1.54072
$974$	3.73157e7	1.26036
$975$	0	0
$976$	2.65244e7	0.891294
$977$	3.16213e7	1.05985	0.529924	−	0.848045i	$-0.322220\pi$
$977$	3.16213e7	1.05985	0.529924	+	0.848045i	$0.322220\pi$
$978$	0	0
$979$	1.25184e7	0.417438
$980$	−1.28136e6	−0.0426192
$981$	0	0
$982$	2.12321e7	0.702611
$983$	−1.31368e7	−0.433617	−0.216808	−	0.976214i	$-0.569565\pi$
$983$	−1.31368e7	−0.433617	−0.216808	+	0.976214i	$0.569565\pi$
$984$	0	0
$985$	1.20428e7	0.395491
$986$	−4.08750e7	−1.33895
$987$	0	0
$988$	1.06886e6	0.0348359
$989$	−4.37491e7	−1.42226
$990$	0	0
$991$	−3.53335e7	−1.14288	−0.571442	−	0.820643i	$-0.693616\pi$
$991$	−3.53335e7	−1.14288	−0.571442	+	0.820643i	$0.693616\pi$
$992$	1.14521e7	0.369493
$993$	0	0
$994$	−5.05446e7	−1.62259
$995$	2.53567e7	0.811962
$996$	0	0
$997$	3.72145e7	1.18570	0.592850	−	0.805313i	$-0.298003\pi$
$997$	3.72145e7	1.18570	0.592850	+	0.805313i	$0.298003\pi$
$998$	3.33563e7	1.06011
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-6.55890 q^{2} +11.0192 q^{4} -25.0000 q^{5} +146.487 q^{7} +137.611 q^{8} +O(q^{10})\)	\(q-6.55890 q^{2} +11.0192 q^{4} -25.0000 q^{5} +146.487 q^{7} +137.611 q^{8} +163.973 q^{10} +121.000 q^{11} +170.238 q^{13} -960.792 q^{14} -1255.19 q^{16} -1564.81 q^{17} +569.786 q^{19} -275.481 q^{20} -793.627 q^{22} -3153.25 q^{23} +625.000 q^{25} -1116.57 q^{26} +1614.17 q^{28} -3982.58 q^{29} +2990.78 q^{31} +3829.14 q^{32} +10263.5 q^{34} -3662.17 q^{35} +7858.92 q^{37} -3737.17 q^{38} -3440.27 q^{40} +5206.60 q^{41} +13874.3 q^{43} +1333.33 q^{44} +20681.9 q^{46} -6852.01 q^{47} +4651.35 q^{49} -4099.32 q^{50} +1875.89 q^{52} +3834.15 q^{53} -3025.00 q^{55} +20158.2 q^{56} +26121.4 q^{58} -9649.38 q^{59} -21131.8 q^{61} -19616.2 q^{62} +15051.2 q^{64} -4255.95 q^{65} -43499.9 q^{67} -17243.0 q^{68} +24019.8 q^{70} +52607.2 q^{71} -64367.9 q^{73} -51545.9 q^{74} +6278.61 q^{76} +17724.9 q^{77} +28935.7 q^{79} +31379.8 q^{80} -34149.6 q^{82} +4648.64 q^{83} +39120.3 q^{85} -91000.1 q^{86} +16650.9 q^{88} +103458. q^{89} +24937.6 q^{91} -34746.4 q^{92} +44941.6 q^{94} -14244.7 q^{95} +75809.5 q^{97} -30507.8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8	\( 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8} + 50 q^{10} + 363 q^{11} - 546 q^{13} + 8 q^{14} - 1360 q^{16} + 314 q^{17} + 1808 q^{19} + 500 q^{20} - 242 q^{22} - 4288 q^{23} + 1875 q^{25} - 812 q^{26} + 5888 q^{28} - 5582 q^{29} + 6328 q^{31} + 736 q^{32} + 11596 q^{34} - 3800 q^{35} + 16866 q^{37} - 9584 q^{38} - 600 q^{40} - 23282 q^{41} + 20572 q^{43} - 2420 q^{44} + 16592 q^{46} - 3432 q^{47} + 11531 q^{49} - 1250 q^{50} + 21816 q^{52} - 16138 q^{53} - 9075 q^{55} - 15648 q^{56} + 17460 q^{58} - 21972 q^{59} + 8322 q^{61} - 5056 q^{62} + 22208 q^{64} + 13650 q^{65} - 84332 q^{67} - 59832 q^{68} - 200 q^{70} - 50528 q^{71} - 53838 q^{73} - 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 6364 q^{79} + 34000 q^{80} - 68020 q^{82} - 96272 q^{83} - 7850 q^{85} - 143152 q^{86} + 2904 q^{88} + 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 49088 q^{94} - 45200 q^{95} - 103242 q^{97} - 9554 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 - 20 * q^4 - 75 * q^5 + 152 * q^7 + 24 * q^8 + 50 * q^10 + 363 * q^11 - 546 * q^13 + 8 * q^14 - 1360 * q^16 + 314 * q^17 + 1808 * q^19 + 500 * q^20 - 242 * q^22 - 4288 * q^23 + 1875 * q^25 - 812 * q^26 + 5888 * q^28 - 5582 * q^29 + 6328 * q^31 + 736 * q^32 + 11596 * q^34 - 3800 * q^35 + 16866 * q^37 - 9584 * q^38 - 600 * q^40 - 23282 * q^41 + 20572 * q^43 - 2420 * q^44 + 16592 * q^46 - 3432 * q^47 + 11531 * q^49 - 1250 * q^50 + 21816 * q^52 - 16138 * q^53 - 9075 * q^55 - 15648 * q^56 + 17460 * q^58 - 21972 * q^59 + 8322 * q^61 - 5056 * q^62 + 22208 * q^64 + 13650 * q^65 - 84332 * q^67 - 59832 * q^68 - 200 * q^70 - 50528 * q^71 - 53838 * q^73 - 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 6364 * q^79 + 34000 * q^80 - 68020 * q^82 - 96272 * q^83 - 7850 * q^85 - 143152 * q^86 + 2904 * q^88 + 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 49088 * q^94 - 45200 * q^95 - 103242 * q^97 - 9554 * q^98

Introduction

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Database

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