LMFDB - Embedded newform 495.6.a.e.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:	3.3.34253.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 52x + 48 $ x^3 - x^2 - 52*x + 48
Coefficient ring:	$\Z[a_1, a_2]$
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			1.2
Root			$0.921799$ 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)

$f(q)$	$=$	$q+2.92180 q^{2} -23.4631 q^{4} -25.0000 q^{5} -85.0105 q^{7} -162.052 q^{8} +O(q^{10})$	\(q+2.92180 q^{2} -23.4631 q^{4} -25.0000 q^{5} -85.0105 q^{7} -162.052 q^{8} -73.0450 q^{10} +121.000 q^{11} +724.085 q^{13} -248.384 q^{14} +277.335 q^{16} +2098.75 q^{17} -6.40223 q^{19} +586.577 q^{20} +353.538 q^{22} -1569.80 q^{23} +625.000 q^{25} +2115.63 q^{26} +1994.61 q^{28} +5145.94 q^{29} -1031.88 q^{31} +5995.98 q^{32} +6132.14 q^{34} +2125.26 q^{35} -12641.6 q^{37} -18.7060 q^{38} +4051.30 q^{40} -13808.7 q^{41} -17012.0 q^{43} -2839.03 q^{44} -4586.64 q^{46} -8078.06 q^{47} -9580.22 q^{49} +1826.12 q^{50} -16989.3 q^{52} +22110.8 q^{53} -3025.00 q^{55} +13776.1 q^{56} +15035.4 q^{58} +16890.6 q^{59} -34398.6 q^{61} -3014.94 q^{62} +8644.32 q^{64} -18102.1 q^{65} -37306.5 q^{67} -49243.3 q^{68} +6209.59 q^{70} +56607.5 q^{71} -27777.9 q^{73} -36936.2 q^{74} +150.216 q^{76} -10286.3 q^{77} -12759.9 q^{79} -6933.39 q^{80} -40346.2 q^{82} +69258.6 q^{83} -52468.9 q^{85} -49705.6 q^{86} -19608.3 q^{88} -59029.2 q^{89} -61554.8 q^{91} +36832.4 q^{92} -23602.5 q^{94} +160.056 q^{95} +104905. q^{97} -27991.5 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8}+O(q^{10}) $ 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8	$ 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8} - 175 q^{10} + 363 q^{11} - 654 q^{13} + 728 q^{14} - 415 q^{16} + 2366 q^{17} - 2872 q^{19} - 625 q^{20} + 847 q^{22} - 2272 q^{23} + 1875 q^{25} - 3422 q^{26} + 4592 q^{28} + 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 2506 q^{34} + 4300 q^{35} - 9126 q^{37} - 13076 q^{38} - 5775 q^{40} + 8758 q^{41} - 14672 q^{43} + 3025 q^{44} - 28768 q^{46} + 19392 q^{47} - 26629 q^{49} + 4375 q^{50} - 61506 q^{52} + 4598 q^{53} - 9075 q^{55} - 2688 q^{56} + 8550 q^{58} + 9348 q^{59} - 60078 q^{61} + 14096 q^{62} - 7087 q^{64} + 16350 q^{65} - 38468 q^{67} - 59778 q^{68} - 18200 q^{70} + 74032 q^{71} - 44442 q^{73} - 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 108116 q^{79} + 10375 q^{80} - 92230 q^{82} + 81892 q^{83} - 59150 q^{85} - 126412 q^{86} + 27951 q^{88} - 167342 q^{89} - 31832 q^{91} - 72960 q^{92} + 12728 q^{94} + 71800 q^{95} + 159702 q^{97} - 163121 q^{98}+O(q^{100}) $ 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8 - 175 * q^10 + 363 * q^11 - 654 * q^13 + 728 * q^14 - 415 * q^16 + 2366 * q^17 - 2872 * q^19 - 625 * q^20 + 847 * q^22 - 2272 * q^23 + 1875 * q^25 - 3422 * q^26 + 4592 * q^28 + 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 2506 * q^34 + 4300 * q^35 - 9126 * q^37 - 13076 * q^38 - 5775 * q^40 + 8758 * q^41 - 14672 * q^43 + 3025 * q^44 - 28768 * q^46 + 19392 * q^47 - 26629 * q^49 + 4375 * q^50 - 61506 * q^52 + 4598 * q^53 - 9075 * q^55 - 2688 * q^56 + 8550 * q^58 + 9348 * q^59 - 60078 * q^61 + 14096 * q^62 - 7087 * q^64 + 16350 * q^65 - 38468 * q^67 - 59778 * q^68 - 18200 * q^70 + 74032 * q^71 - 44442 * q^73 - 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 108116 * q^79 + 10375 * q^80 - 92230 * q^82 + 81892 * q^83 - 59150 * q^85 - 126412 * q^86 + 27951 * q^88 - 167342 * q^89 - 31832 * q^91 - 72960 * q^92 + 12728 * q^94 + 71800 * q^95 + 159702 * q^97 - 163121 * 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 $

$2$	2.92180	0.516506	0.258253	−	0.966077i	$-0.416853\pi$
$2$	2.92180	0.516506	0.258253	+	0.966077i	$0.416853\pi$
$3$	0	0
$3$	0	0
$4$	−23.4631	−0.733222
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−85.0105	−0.655734	−0.327867	−	0.944724i	$-0.606330\pi$
$7$	−85.0105	−0.655734	−0.327867	+	0.944724i	$0.606330\pi$
$8$	−162.052	−0.895219
$9$	0	0
$10$	−73.0450	−0.230988
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	724.085	1.18831	0.594157	−	0.804349i	$-0.297486\pi$
$13$	724.085	1.18831	0.594157	+	0.804349i	$0.297486\pi$
$14$	−248.384	−0.338690
$15$	0	0
$16$	277.335	0.270835
$17$	2098.75	1.76132	0.880662	−	0.473745i	$-0.157098\pi$
$17$	2098.75	1.76132	0.880662	+	0.473745i	$0.157098\pi$
$18$	0	0
$19$	−6.40223	−0.00406862	−0.00203431	−	0.999998i	$-0.500648\pi$
$19$	−6.40223	−0.00406862	−0.00203431	+	0.999998i	$0.500648\pi$
$20$	586.577	0.327907
$21$	0	0
$22$	353.538	0.155732
$23$	−1569.80	−0.618764	−0.309382	−	0.950938i	$-0.600122\pi$
$23$	−1569.80	−0.618764	−0.309382	+	0.950938i	$0.600122\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	2115.63	0.613771
$27$	0	0
$28$	1994.61	0.480798
$29$	5145.94	1.13624	0.568119	−	0.822946i	$-0.307671\pi$
$29$	5145.94	1.13624	0.568119	+	0.822946i	$0.307671\pi$
$30$	0	0
$31$	−1031.88	−0.192852	−0.0964259	−	0.995340i	$-0.530741\pi$
$31$	−1031.88	−0.192852	−0.0964259	+	0.995340i	$0.530741\pi$
$32$	5995.98	1.03511
$33$	0	0
$34$	6132.14	0.909735
$35$	2125.26	0.293253
$36$	0	0
$37$	−12641.6	−1.51809	−0.759045	−	0.651039i	$-0.774334\pi$
$37$	−12641.6	−1.51809	−0.759045	+	0.651039i	$0.774334\pi$
$38$	−18.7060	−0.00210147
$39$	0	0
$40$	4051.30	0.400354
$41$	−13808.7	−1.28290	−0.641450	−	0.767165i	$-0.721667\pi$
$41$	−13808.7	−1.28290	−0.641450	+	0.767165i	$0.721667\pi$
$42$	0	0
$43$	−17012.0	−1.40308	−0.701542	−	0.712628i	$-0.747505\pi$
$43$	−17012.0	−1.40308	−0.701542	+	0.712628i	$0.747505\pi$
$44$	−2839.03	−0.221075
$45$	0	0
$46$	−4586.64	−0.319595
$47$	−8078.06	−0.533412	−0.266706	−	0.963778i	$-0.585935\pi$
$47$	−8078.06	−0.533412	−0.266706	+	0.963778i	$0.585935\pi$
$48$	0	0
$49$	−9580.22	−0.570013
$50$	1826.12	0.103301
$51$	0	0
$52$	−16989.3	−0.871297
$53$	22110.8	1.08122	0.540612	−	0.841272i	$-0.318193\pi$
$53$	22110.8	1.08122	0.540612	+	0.841272i	$0.318193\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	13776.1	0.587025
$57$	0	0
$58$	15035.4	0.586874
$59$	16890.6	0.631706	0.315853	−	0.948808i	$-0.397709\pi$
$59$	16890.6	0.631706	0.315853	+	0.948808i	$0.397709\pi$
$60$	0	0
$61$	−34398.6	−1.18363	−0.591816	−	0.806073i	$-0.701589\pi$
$61$	−34398.6	−1.18363	−0.591816	+	0.806073i	$0.701589\pi$
$62$	−3014.94	−0.0996091
$63$	0	0
$64$	8644.32	0.263804
$65$	−18102.1	−0.531430
$66$	0	0
$67$	−37306.5	−1.01531	−0.507654	−	0.861561i	$-0.669487\pi$
$67$	−37306.5	−1.01531	−0.507654	+	0.861561i	$0.669487\pi$
$68$	−49243.3	−1.29144
$69$	0	0
$70$	6209.59	0.151467
$71$	56607.5	1.33269	0.666344	−	0.745645i	$-0.267858\pi$
$71$	56607.5	1.33269	0.666344	+	0.745645i	$0.267858\pi$
$72$	0	0
$73$	−27777.9	−0.610088	−0.305044	−	0.952338i	$-0.598671\pi$
$73$	−27777.9	−0.610088	−0.305044	+	0.952338i	$0.598671\pi$
$74$	−36936.2	−0.784102
$75$	0	0
$76$	150.216	0.00298320
$77$	−10286.3	−0.197711
$78$	0	0
$79$	−12759.9	−0.230028	−0.115014	−	0.993364i	$-0.536691\pi$
$79$	−12759.9	−0.230028	−0.115014	+	0.993364i	$0.536691\pi$
$80$	−6933.39	−0.121121
$81$	0	0
$82$	−40346.2	−0.662625
$83$	69258.6	1.10352	0.551758	−	0.834004i	$-0.313957\pi$
$83$	69258.6	1.10352	0.551758	+	0.834004i	$0.313957\pi$
$84$	0	0
$85$	−52468.9	−0.787688
$86$	−49705.6	−0.724702
$87$	0	0
$88$	−19608.3	−0.269919
$89$	−59029.2	−0.789936	−0.394968	−	0.918695i	$-0.629244\pi$
$89$	−59029.2	−0.789936	−0.394968	+	0.918695i	$0.629244\pi$
$90$	0	0
$91$	−61554.8	−0.779217
$92$	36832.4	0.453691
$93$	0	0
$94$	−23602.5	−0.275510
$95$	160.056	0.00181954
$96$	0	0
$97$	104905.	1.13205	0.566027	−	0.824387i	$-0.308480\pi$
$97$	104905.	1.13205	0.566027	+	0.824387i	$0.308480\pi$
$98$	−27991.5	−0.294415
$99$	0	0
$100$	−14664.4	−0.146644
$101$	−135417.	−1.32090	−0.660451	−	0.750869i	$-0.729635\pi$
$101$	−135417.	−1.32090	−0.660451	+	0.750869i	$0.729635\pi$
$102$	0	0
$103$	167505.	1.55573	0.777867	−	0.628428i	$-0.216302\pi$
$103$	167505.	1.55573	0.777867	+	0.628428i	$0.216302\pi$
$104$	−117339.	−1.06380
$105$	0	0
$106$	64603.4	0.558458
$107$	−57989.6	−0.489656	−0.244828	−	0.969567i	$-0.578731\pi$
$107$	−57989.6	−0.489656	−0.244828	+	0.969567i	$0.578731\pi$
$108$	0	0
$109$	−14956.1	−0.120574	−0.0602869	−	0.998181i	$-0.519202\pi$
$109$	−14956.1	−0.120574	−0.0602869	+	0.998181i	$0.519202\pi$
$110$	−8838.44	−0.0696457
$111$	0	0
$112$	−23576.4	−0.177596
$113$	−198323.	−1.46109	−0.730544	−	0.682866i	$-0.760733\pi$
$113$	−198323.	−1.46109	−0.730544	+	0.682866i	$0.760733\pi$
$114$	0	0
$115$	39245.0	0.276720
$116$	−120740.	−0.833115
$117$	0	0
$118$	49351.0	0.326280
$119$	−178416.	−1.15496
$120$	0	0
$121$	14641.0	0.0909091
$122$	−100506.	−0.611353
$123$	0	0
$124$	24211.0	0.141403
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−40986.5	−0.225492	−0.112746	−	0.993624i	$-0.535965\pi$
$127$	−40986.5	−0.225492	−0.112746	+	0.993624i	$0.535965\pi$
$128$	−166614.	−0.898851
$129$	0	0
$130$	−52890.8	−0.274487
$131$	238406.	1.21378	0.606889	−	0.794787i	$-0.292417\pi$
$131$	238406.	1.21378	0.606889	+	0.794787i	$0.292417\pi$
$132$	0	0
$133$	544.257	0.00266793
$134$	−109002.	−0.524413
$135$	0	0
$136$	−340107.	−1.57677
$137$	−149534.	−0.680673	−0.340337	−	0.940304i	$-0.610541\pi$
$137$	−149534.	−0.680673	−0.340337	+	0.940304i	$0.610541\pi$
$138$	0	0
$139$	167700.	0.736202	0.368101	−	0.929786i	$-0.380008\pi$
$139$	167700.	0.736202	0.368101	+	0.929786i	$0.380008\pi$
$140$	−49865.2	−0.215019
$141$	0	0
$142$	165396.	0.688341
$143$	87614.3	0.358290
$144$	0	0
$145$	−128648.	−0.508142
$146$	−81161.5	−0.315114
$147$	0	0
$148$	296611.	1.11310
$149$	272698.	1.00627	0.503136	−	0.864207i	$-0.332179\pi$
$149$	272698.	1.00627	0.503136	+	0.864207i	$0.332179\pi$
$150$	0	0
$151$	−448320.	−1.60009	−0.800047	−	0.599937i	$-0.795192\pi$
$151$	−448320.	−1.60009	−0.800047	+	0.599937i	$0.795192\pi$
$152$	1037.49	0.00364231
$153$	0	0
$154$	−30054.4	−0.102119
$155$	25796.9	0.0862459
$156$	0	0
$157$	−366634.	−1.18709	−0.593545	−	0.804801i	$-0.702272\pi$
$157$	−366634.	−1.18709	−0.593545	+	0.804801i	$0.702272\pi$
$158$	−37281.9	−0.118811
$159$	0	0
$160$	−149900.	−0.462914
$161$	133450.	0.405744
$162$	0	0
$163$	501806.	1.47934	0.739668	−	0.672972i	$-0.234982\pi$
$163$	501806.	1.47934	0.739668	+	0.672972i	$0.234982\pi$
$164$	323994.	0.940650
$165$	0	0
$166$	202360.	0.569973
$167$	195231.	0.541700	0.270850	−	0.962622i	$-0.412695\pi$
$167$	195231.	0.541700	0.270850	+	0.962622i	$0.412695\pi$
$168$	0	0
$169$	153006.	0.412089
$170$	−153303.	−0.406846
$171$	0	0
$172$	399154.	1.02877
$173$	−618180.	−1.57036	−0.785181	−	0.619266i	$-0.787430\pi$
$173$	−618180.	−1.57036	−0.785181	+	0.619266i	$0.787430\pi$
$174$	0	0
$175$	−53131.6	−0.131147
$176$	33557.6	0.0816600
$177$	0	0
$178$	−172471.	−0.408007
$179$	88898.6	0.207378	0.103689	−	0.994610i	$-0.466935\pi$
$179$	88898.6	0.207378	0.103689	+	0.994610i	$0.466935\pi$
$180$	0	0
$181$	−378128.	−0.857911	−0.428956	−	0.903326i	$-0.641118\pi$
$181$	−378128.	−0.857911	−0.428956	+	0.903326i	$0.641118\pi$
$182$	−179851.	−0.402470
$183$	0	0
$184$	254389.	0.553930
$185$	316040.	0.678910
$186$	0	0
$187$	253949.	0.531059
$188$	189536.	0.391109
$189$	0	0
$190$	467.651	0.000939805 0
$191$	−125442.	−0.248806	−0.124403	−	0.992232i	$-0.539701\pi$
$191$	−125442.	−0.248806	−0.124403	+	0.992232i	$0.539701\pi$
$192$	0	0
$193$	−928398.	−1.79408	−0.897038	−	0.441953i	$-0.854286\pi$
$193$	−928398.	−1.79408	−0.897038	+	0.441953i	$0.854286\pi$
$194$	306512.	0.584713
$195$	0	0
$196$	224781.	0.417946
$197$	−1.03233e6	−1.89519	−0.947595	−	0.319473i	$-0.896494\pi$
$197$	−1.03233e6	−1.89519	−0.947595	+	0.319473i	$0.896494\pi$
$198$	0	0
$199$	−155017.	−0.277489	−0.138744	−	0.990328i	$-0.544307\pi$
$199$	−155017.	−0.277489	−0.138744	+	0.990328i	$0.544307\pi$
$200$	−101283.	−0.179044
$201$	0	0
$202$	−395662.	−0.682254
$203$	−437459.	−0.745070
$204$	0	0
$205$	345217.	0.573730
$206$	489417.	0.803546
$207$	0	0
$208$	200814.	0.321837
$209$	−774.670	−0.00122674
$210$	0	0
$211$	−364263.	−0.563259	−0.281630	−	0.959523i	$-0.590875\pi$
$211$	−364263.	−0.563259	−0.281630	+	0.959523i	$0.590875\pi$
$212$	−518789.	−0.792777
$213$	0	0
$214$	−169434.	−0.252910
$215$	425300.	0.627478
$216$	0	0
$217$	87720.4	0.126459
$218$	−43698.8	−0.0622771
$219$	0	0
$220$	70975.8	0.0988676
$221$	1.51968e6	2.09301
$222$	0	0
$223$	−74806.0	−0.100734	−0.0503668	−	0.998731i	$-0.516039\pi$
$223$	−74806.0	−0.100734	−0.0503668	+	0.998731i	$0.516039\pi$
$224$	−509721.	−0.678755
$225$	0	0
$226$	−579459.	−0.754661
$227$	−1.22677e6	−1.58015	−0.790075	−	0.613010i	$-0.789958\pi$
$227$	−1.22677e6	−1.58015	−0.790075	+	0.613010i	$0.789958\pi$
$228$	0	0
$229$	−440852.	−0.555526	−0.277763	−	0.960650i	$-0.589593\pi$
$229$	−440852.	−0.555526	−0.277763	+	0.960650i	$0.589593\pi$
$230$	114666.	0.142927
$231$	0	0
$232$	−833910.	−1.01718
$233$	514549.	0.620922	0.310461	−	0.950586i	$-0.399517\pi$
$233$	514549.	0.620922	0.310461	+	0.950586i	$0.399517\pi$
$234$	0	0
$235$	201952.	0.238549
$236$	−396306.	−0.463181
$237$	0	0
$238$	−521296.	−0.596544
$239$	984315.	1.11465	0.557326	−	0.830294i	$-0.311827\pi$
$239$	984315.	1.11465	0.557326	+	0.830294i	$0.311827\pi$
$240$	0	0
$241$	284794.	0.315855	0.157927	−	0.987451i	$-0.449519\pi$
$241$	284794.	0.315855	0.157927	+	0.987451i	$0.449519\pi$
$242$	42778.1	0.0469551
$243$	0	0
$244$	807098.	0.867865
$245$	239505.	0.254918
$246$	0	0
$247$	−4635.76	−0.00483480
$248$	167218.	0.172645
$249$	0	0
$250$	−45653.1	−0.0461977
$251$	−134529.	−0.134782	−0.0673911	−	0.997727i	$-0.521468\pi$
$251$	−134529.	−0.134782	−0.0673911	+	0.997727i	$0.521468\pi$
$252$	0	0
$253$	−189946.	−0.186564
$254$	−119754.	−0.116468
$255$	0	0
$256$	−763432.	−0.728066
$257$	−2.06732e6	−1.95242	−0.976212	−	0.216817i	$-0.930433\pi$
$257$	−2.06732e6	−1.95242	−0.976212	+	0.216817i	$0.930433\pi$
$258$	0	0
$259$	1.07467e6	0.995462
$260$	424732.	0.389656
$261$	0	0
$262$	696575.	0.626924
$263$	661739.	0.589926	0.294963	−	0.955509i	$-0.404693\pi$
$263$	661739.	0.589926	0.294963	+	0.955509i	$0.404693\pi$
$264$	0	0
$265$	−552771.	−0.483538
$266$	1590.21	0.00137800
$267$	0	0
$268$	875327.	0.744446
$269$	−703008.	−0.592351	−0.296176	−	0.955133i	$-0.595711\pi$
$269$	−703008.	−0.592351	−0.296176	+	0.955133i	$0.595711\pi$
$270$	0	0
$271$	−1.33847e6	−1.10710	−0.553549	−	0.832817i	$-0.686727\pi$
$271$	−1.33847e6	−1.10710	−0.553549	+	0.832817i	$0.686727\pi$
$272$	582059.	0.477029
$273$	0	0
$274$	−436908.	−0.351572
$275$	75625.0	0.0603023
$276$	0	0
$277$	−33171.1	−0.0259753	−0.0129877	−	0.999916i	$-0.504134\pi$
$277$	−33171.1	−0.0259753	−0.0129877	+	0.999916i	$0.504134\pi$
$278$	489987.	0.380253
$279$	0	0
$280$	−344403.	−0.262526
$281$	321114.	0.242601	0.121301	−	0.992616i	$-0.461293\pi$
$281$	321114.	0.242601	0.121301	+	0.992616i	$0.461293\pi$
$282$	0	0
$283$	−1.90591e6	−1.41461	−0.707305	−	0.706908i	$-0.750089\pi$
$283$	−1.90591e6	−1.41461	−0.707305	+	0.706908i	$0.750089\pi$
$284$	−1.32819e6	−0.977155
$285$	0	0
$286$	255991.	0.185059
$287$	1.17388e6	0.841240
$288$	0	0
$289$	2.98491e6	2.10226
$290$	−375885.	−0.262458
$291$	0	0
$292$	651756.	0.447330
$293$	−272957.	−0.185748	−0.0928742	−	0.995678i	$-0.529605\pi$
$293$	−272957.	−0.185748	−0.0928742	+	0.995678i	$0.529605\pi$
$294$	0	0
$295$	−422265.	−0.282508
$296$	2.04859e6	1.35902
$297$	0	0
$298$	796768.	0.519746
$299$	−1.13667e6	−0.735286
$300$	0	0
$301$	1.44620e6	0.920050
$302$	−1.30990e6	−0.826459
$303$	0	0
$304$	−1775.57	−0.00110193
$305$	859966.	0.529336
$306$	0	0
$307$	−843806.	−0.510971	−0.255486	−	0.966813i	$-0.582235\pi$
$307$	−843806.	−0.510971	−0.255486	+	0.966813i	$0.582235\pi$
$308$	241348.	0.144966
$309$	0	0
$310$	75373.4	0.0445465
$311$	−2.06909e6	−1.21305	−0.606523	−	0.795066i	$-0.707436\pi$
$311$	−2.06909e6	−1.21305	−0.606523	+	0.795066i	$0.707436\pi$
$312$	0	0
$313$	603113.	0.347967	0.173983	−	0.984749i	$-0.444336\pi$
$313$	603113.	0.347967	0.173983	+	0.984749i	$0.444336\pi$
$314$	−1.07123e6	−0.613139
$315$	0	0
$316$	299387.	0.168661
$317$	1.67334e6	0.935268	0.467634	−	0.883922i	$-0.345107\pi$
$317$	1.67334e6	0.935268	0.467634	+	0.883922i	$0.345107\pi$
$318$	0	0
$319$	622659.	0.342589
$320$	−216108.	−0.117977
$321$	0	0
$322$	389913.	0.209569
$323$	−13436.7	−0.00716616
$324$	0	0
$325$	452553.	0.237663
$326$	1.46618e6	0.764086
$327$	0	0
$328$	2.23772e6	1.14848
$329$	686720.	0.349776
$330$	0	0
$331$	−2.20421e6	−1.10582	−0.552909	−	0.833242i	$-0.686482\pi$
$331$	−2.20421e6	−1.10582	−0.552909	+	0.833242i	$0.686482\pi$
$332$	−1.62502e6	−0.809122
$333$	0	0
$334$	570427.	0.279791
$335$	932663.	0.454060
$336$	0	0
$337$	1.07377e6	0.515036	0.257518	−	0.966273i	$-0.417095\pi$
$337$	1.07377e6	0.515036	0.257518	+	0.966273i	$0.417095\pi$
$338$	447052.	0.212847
$339$	0	0
$340$	1.23108e6	0.577550
$341$	−124857.	−0.0581470
$342$	0	0
$343$	2.24319e6	1.02951
$344$	2.75683e6	1.25607
$345$	0	0
$346$	−1.80620e6	−0.811101
$347$	1.39783e6	0.623203	0.311601	−	0.950213i	$-0.399135\pi$
$347$	1.39783e6	0.623203	0.311601	+	0.950213i	$0.399135\pi$
$348$	0	0
$349$	−2.66674e6	−1.17197	−0.585985	−	0.810322i	$-0.699292\pi$
$349$	−2.66674e6	−1.17197	−0.585985	+	0.810322i	$0.699292\pi$
$350$	−155240.	−0.0677381
$351$	0	0
$352$	725514.	0.312097
$353$	594048.	0.253738	0.126869	−	0.991920i	$-0.459507\pi$
$353$	594048.	0.253738	0.126869	+	0.991920i	$0.459507\pi$
$354$	0	0
$355$	−1.41519e6	−0.595996
$356$	1.38501e6	0.579198
$357$	0	0
$358$	259744.	0.107112
$359$	3.35774e6	1.37503	0.687513	−	0.726172i	$-0.258702\pi$
$359$	3.35774e6	1.37503	0.687513	+	0.726172i	$0.258702\pi$
$360$	0	0
$361$	−2.47606e6	−0.999983
$362$	−1.10481e6	−0.443116
$363$	0	0
$364$	1.44427e6	0.571339
$365$	694448.	0.272840
$366$	0	0
$367$	2.58775e6	1.00290	0.501450	−	0.865187i	$-0.332800\pi$
$367$	2.58775e6	1.00290	0.501450	+	0.865187i	$0.332800\pi$
$368$	−435362.	−0.167583
$369$	0	0
$370$	923404.	0.350661
$371$	−1.87965e6	−0.708995
$372$	0	0
$373$	−1.07376e6	−0.399608	−0.199804	−	0.979836i	$-0.564031\pi$
$373$	−1.07376e6	−0.399608	−0.199804	+	0.979836i	$0.564031\pi$
$374$	741989.	0.274295
$375$	0	0
$376$	1.30907e6	0.477520
$377$	3.72610e6	1.35021
$378$	0	0
$379$	5.16745e6	1.84790	0.923949	−	0.382516i	$-0.124942\pi$
$379$	5.16745e6	1.84790	0.923949	+	0.382516i	$0.124942\pi$
$380$	−3755.40	−0.00133413
$381$	0	0
$382$	−366517.	−0.128510
$383$	−2.42092e6	−0.843302	−0.421651	−	0.906758i	$-0.638549\pi$
$383$	−2.42092e6	−0.843302	−0.421651	+	0.906758i	$0.638549\pi$
$384$	0	0
$385$	257157.	0.0884191
$386$	−2.71259e6	−0.926651
$387$	0	0
$388$	−2.46140e6	−0.830047
$389$	−1.92154e6	−0.643835	−0.321918	−	0.946768i	$-0.604327\pi$
$389$	−1.92154e6	−0.643835	−0.321918	+	0.946768i	$0.604327\pi$
$390$	0	0
$391$	−3.29463e6	−1.08984
$392$	1.55249e6	0.510287
$393$	0	0
$394$	−3.01626e6	−0.978877
$395$	318998.	0.102871
$396$	0	0
$397$	4.09572e6	1.30423	0.652114	−	0.758121i	$-0.273882\pi$
$397$	4.09572e6	1.30423	0.652114	+	0.758121i	$0.273882\pi$
$398$	−452928.	−0.143325
$399$	0	0
$400$	173335.	0.0541671
$401$	−5.97525e6	−1.85565	−0.927823	−	0.373022i	$-0.878322\pi$
$401$	−5.97525e6	−1.85565	−0.927823	+	0.373022i	$0.878322\pi$
$402$	0	0
$403$	−747167.	−0.229168
$404$	3.17731e6	0.968514
$405$	0	0
$406$	−1.27817e6	−0.384833
$407$	−1.52963e6	−0.457721
$408$	0	0
$409$	−1.92665e6	−0.569500	−0.284750	−	0.958602i	$-0.591911\pi$
$409$	−1.92665e6	−0.569500	−0.284750	+	0.958602i	$0.591911\pi$
$410$	1.00865e6	0.296335
$411$	0	0
$412$	−3.93019e6	−1.14070
$413$	−1.43588e6	−0.414231
$414$	0	0
$415$	−1.73147e6	−0.493507
$416$	4.34160e6	1.23003
$417$	0	0
$418$	−2263.43	−0.000633616 0
$419$	1.47857e6	0.411441	0.205720	−	0.978611i	$-0.434046\pi$
$419$	1.47857e6	0.411441	0.205720	+	0.978611i	$0.434046\pi$
$420$	0	0
$421$	4.82644e6	1.32715	0.663577	−	0.748108i	$-0.269037\pi$
$421$	4.82644e6	1.32715	0.663577	+	0.748108i	$0.269037\pi$
$422$	−1.06430e6	−0.290927
$423$	0	0
$424$	−3.58311e6	−0.967932
$425$	1.31172e6	0.352265
$426$	0	0
$427$	2.92425e6	0.776147
$428$	1.36062e6	0.359026
$429$	0	0
$430$	1.24264e6	0.324096
$431$	−2.89229e6	−0.749978	−0.374989	−	0.927029i	$-0.622353\pi$
$431$	−2.89229e6	−0.749978	−0.374989	+	0.927029i	$0.622353\pi$
$432$	0	0
$433$	2.36700e6	0.606706	0.303353	−	0.952878i	$-0.401894\pi$
$433$	2.36700e6	0.606706	0.303353	+	0.952878i	$0.401894\pi$
$434$	256301.	0.0653170
$435$	0	0
$436$	350917.	0.0884073
$437$	10050.2	0.00251752
$438$	0	0
$439$	3.22037e6	0.797525	0.398762	−	0.917054i	$-0.369440\pi$
$439$	3.22037e6	0.797525	0.398762	+	0.917054i	$0.369440\pi$
$440$	490207.	0.120711
$441$	0	0
$442$	4.44019e6	1.08105
$443$	3.80335e6	0.920782	0.460391	−	0.887716i	$-0.347709\pi$
$443$	3.80335e6	0.920782	0.460391	+	0.887716i	$0.347709\pi$
$444$	0	0
$445$	1.47573e6	0.353270
$446$	−218568.	−0.0520295
$447$	0	0
$448$	−734858.	−0.172985
$449$	6.55303e6	1.53400	0.767001	−	0.641646i	$-0.221748\pi$
$449$	6.55303e6	1.53400	0.767001	+	0.641646i	$0.221748\pi$
$450$	0	0
$451$	−1.67085e6	−0.386809
$452$	4.65326e6	1.07130
$453$	0	0
$454$	−3.58437e6	−0.816157
$455$	1.53887e6	0.348477
$456$	0	0
$457$	−7.41470e6	−1.66074	−0.830372	−	0.557209i	$-0.811872\pi$
$457$	−7.41470e6	−1.66074	−0.830372	+	0.557209i	$0.811872\pi$
$458$	−1.28808e6	−0.286932
$459$	0	0
$460$	−920810.	−0.202897
$461$	−409525.	−0.0897487	−0.0448743	−	0.998993i	$-0.514289\pi$
$461$	−409525.	−0.0897487	−0.0448743	+	0.998993i	$0.514289\pi$
$462$	0	0
$463$	7.07192e6	1.53315	0.766575	−	0.642154i	$-0.221959\pi$
$463$	7.07192e6	1.53315	0.766575	+	0.642154i	$0.221959\pi$
$464$	1.42715e6	0.307734
$465$	0	0
$466$	1.50341e6	0.320710
$467$	6.35423e6	1.34825	0.674125	−	0.738617i	$-0.264521\pi$
$467$	6.35423e6	1.34825	0.674125	+	0.738617i	$0.264521\pi$
$468$	0	0
$469$	3.17145e6	0.665772
$470$	590062.	0.123212
$471$	0	0
$472$	−2.73716e6	−0.565516
$473$	−2.05845e6	−0.423046
$474$	0	0
$475$	−4001.39	−0.000813724 0
$476$	4.18619e6	0.846841
$477$	0	0
$478$	2.87597e6	0.575725
$479$	−6.36751e6	−1.26803	−0.634017	−	0.773319i	$-0.718595\pi$
$479$	−6.36751e6	−1.26803	−0.634017	+	0.773319i	$0.718595\pi$
$480$	0	0
$481$	−9.15358e6	−1.80397
$482$	832109.	0.163141
$483$	0	0
$484$	−343523.	−0.0666565
$485$	−2.62263e6	−0.506270
$486$	0	0
$487$	−2.94280e6	−0.562261	−0.281130	−	0.959670i	$-0.590709\pi$
$487$	−2.94280e6	−0.562261	−0.281130	+	0.959670i	$0.590709\pi$
$488$	5.57437e6	1.05961
$489$	0	0
$490$	699787.	0.131667
$491$	501628.	0.0939027	0.0469513	−	0.998897i	$-0.485049\pi$
$491$	501628.	0.0939027	0.0469513	+	0.998897i	$0.485049\pi$
$492$	0	0
$493$	1.08001e7	2.00129
$494$	−13544.8	−0.00249720
$495$	0	0
$496$	−286176.	−0.0522311
$497$	−4.81224e6	−0.873888
$498$	0	0
$499$	−9.19784e6	−1.65362	−0.826808	−	0.562484i	$-0.809846\pi$
$499$	−9.19784e6	−1.65362	−0.826808	+	0.562484i	$0.809846\pi$
$500$	366611.	0.0655813
$501$	0	0
$502$	−393068.	−0.0696158
$503$	6.53811e6	1.15221	0.576106	−	0.817375i	$-0.304572\pi$
$503$	6.53811e6	1.15221	0.576106	+	0.817375i	$0.304572\pi$
$504$	0	0
$505$	3.38543e6	0.590726
$506$	−554984.	−0.0963616
$507$	0	0
$508$	961669.	0.165336
$509$	1.49017e6	0.254942	0.127471	−	0.991842i	$-0.459314\pi$
$509$	1.49017e6	0.254942	0.127471	+	0.991842i	$0.459314\pi$
$510$	0	0
$511$	2.36141e6	0.400055
$512$	3.10107e6	0.522801
$513$	0	0
$514$	−6.04029e6	−1.00844
$515$	−4.18763e6	−0.695746
$516$	0	0
$517$	−977445.	−0.160830
$518$	3.13996e6	0.514162
$519$	0	0
$520$	2.93349e6	0.475746
$521$	−3.82163e6	−0.616814	−0.308407	−	0.951255i	$-0.599796\pi$
$521$	−3.82163e6	−0.616814	−0.308407	+	0.951255i	$0.599796\pi$
$522$	0	0
$523$	−4.18273e6	−0.668660	−0.334330	−	0.942456i	$-0.608510\pi$
$523$	−4.18273e6	−0.668660	−0.334330	+	0.942456i	$0.608510\pi$
$524$	−5.59375e6	−0.889968
$525$	0	0
$526$	1.93347e6	0.304700
$527$	−2.16566e6	−0.339675
$528$	0	0
$529$	−3.97207e6	−0.617131
$530$	−1.61509e6	−0.249750
$531$	0	0
$532$	−12769.9	−0.00195619
$533$	−9.99866e6	−1.52449
$534$	0	0
$535$	1.44974e6	0.218981
$536$	6.04560e6	0.908923
$537$	0	0
$538$	−2.05405e6	−0.305953
$539$	−1.15921e6	−0.171866
$540$	0	0
$541$	2.80424e6	0.411928	0.205964	−	0.978560i	$-0.433967\pi$
$541$	2.80424e6	0.411928	0.205964	+	0.978560i	$0.433967\pi$
$542$	−3.91075e6	−0.571823
$543$	0	0
$544$	1.25841e7	1.82316
$545$	373903.	0.0539222
$546$	0	0
$547$	−8.42403e6	−1.20379	−0.601896	−	0.798574i	$-0.705588\pi$
$547$	−8.42403e6	−1.20379	−0.601896	+	0.798574i	$0.705588\pi$
$548$	3.50853e6	0.499084
$549$	0	0
$550$	220961.	0.0311465
$551$	−32945.5	−0.00462293
$552$	0	0
$553$	1.08473e6	0.150837
$554$	−96919.4	−0.0134164
$555$	0	0
$556$	−3.93477e6	−0.539799
$557$	3.56702e6	0.487155	0.243577	−	0.969881i	$-0.421679\pi$
$557$	3.56702e6	0.487155	0.243577	+	0.969881i	$0.421679\pi$
$558$	0	0
$559$	−1.23181e7	−1.66730
$560$	589411.	0.0794233
$561$	0	0
$562$	938231.	0.125305
$563$	−4.70552e6	−0.625658	−0.312829	−	0.949810i	$-0.601277\pi$
$563$	−4.70552e6	−0.625658	−0.312829	+	0.949810i	$0.601277\pi$
$564$	0	0
$565$	4.95807e6	0.653418
$566$	−5.56869e6	−0.730655
$567$	0	0
$568$	−9.17337e6	−1.19305
$569$	−2.60879e6	−0.337799	−0.168900	−	0.985633i	$-0.554021\pi$
$569$	−2.60879e6	−0.337799	−0.168900	+	0.985633i	$0.554021\pi$
$570$	0	0
$571$	1.13036e6	0.145086	0.0725431	−	0.997365i	$-0.476889\pi$
$571$	1.13036e6	0.145086	0.0725431	+	0.997365i	$0.476889\pi$
$572$	−2.05570e6	−0.262706
$573$	0	0
$574$	3.42985e6	0.434506
$575$	−981126.	−0.123753
$576$	0	0
$577$	1.12075e7	1.40142	0.700710	−	0.713446i	$-0.252867\pi$
$577$	1.12075e7	1.40142	0.700710	+	0.713446i	$0.252867\pi$
$578$	8.72132e6	1.08583
$579$	0	0
$580$	3.01849e6	0.372580
$581$	−5.88771e6	−0.723613
$582$	0	0
$583$	2.67541e6	0.326001
$584$	4.50147e6	0.546162
$585$	0	0
$586$	−797526.	−0.0959402
$587$	−9.05798e6	−1.08502	−0.542508	−	0.840051i	$-0.682525\pi$
$587$	−9.05798e6	−1.08502	−0.542508	+	0.840051i	$0.682525\pi$
$588$	0	0
$589$	6606.31	0.000784641 0
$590$	−1.23377e6	−0.145917
$591$	0	0
$592$	−3.50596e6	−0.411152
$593$	−1.00907e7	−1.17837	−0.589186	−	0.807997i	$-0.700552\pi$
$593$	−1.00907e7	−1.17837	−0.589186	+	0.807997i	$0.700552\pi$
$594$	0	0
$595$	4.46040e6	0.516514
$596$	−6.39833e6	−0.737821
$597$	0	0
$598$	−3.32112e6	−0.379780
$599$	1.29126e7	1.47044	0.735222	−	0.677827i	$-0.237078\pi$
$599$	1.29126e7	1.47044	0.735222	+	0.677827i	$0.237078\pi$
$600$	0	0
$601$	1.82591e6	0.206202	0.103101	−	0.994671i	$-0.467124\pi$
$601$	1.82591e6	0.206202	0.103101	+	0.994671i	$0.467124\pi$
$602$	4.22550e6	0.475211
$603$	0	0
$604$	1.05190e7	1.17322
$605$	−366025.	−0.0406558
$606$	0	0
$607$	−1.26409e6	−0.139253	−0.0696267	−	0.997573i	$-0.522181\pi$
$607$	−1.26409e6	−0.139253	−0.0696267	+	0.997573i	$0.522181\pi$
$608$	−38387.7	−0.00421146
$609$	0	0
$610$	2.51265e6	0.273405
$611$	−5.84920e6	−0.633860
$612$	0	0
$613$	1.59125e7	1.71036	0.855180	−	0.518332i	$-0.173447\pi$
$613$	1.59125e7	1.71036	0.855180	+	0.518332i	$0.173447\pi$
$614$	−2.46543e6	−0.263920
$615$	0	0
$616$	1.66691e6	0.176995
$617$	−85225.6	−0.00901274	−0.00450637	−	0.999990i	$-0.501434\pi$
$617$	−85225.6	−0.00901274	−0.00450637	+	0.999990i	$0.501434\pi$
$618$	0	0
$619$	−1.20387e7	−1.26285	−0.631425	−	0.775437i	$-0.717530\pi$
$619$	−1.20387e7	−1.26285	−0.631425	+	0.775437i	$0.717530\pi$
$620$	−605276.	−0.0632374
$621$	0	0
$622$	−6.04545e6	−0.626546
$623$	5.01810e6	0.517987
$624$	0	0
$625$	390625.	0.0400000
$626$	1.76218e6	0.179727
$627$	0	0
$628$	8.60236e6	0.870400
$629$	−2.65316e7	−2.67385
$630$	0	0
$631$	121417.	0.0121396	0.00606981	−	0.999982i	$-0.498068\pi$
$631$	121417.	0.0121396	0.00606981	+	0.999982i	$0.498068\pi$
$632$	2.06777e6	0.205925
$633$	0	0
$634$	4.88916e6	0.483071
$635$	1.02466e6	0.100843
$636$	0	0
$637$	−6.93689e6	−0.677355
$638$	1.81928e6	0.176949
$639$	0	0
$640$	4.16536e6	0.401978
$641$	−1.25328e7	−1.20477	−0.602385	−	0.798206i	$-0.705783\pi$
$641$	−1.25328e7	−1.20477	−0.602385	+	0.798206i	$0.705783\pi$
$642$	0	0
$643$	−3.45380e6	−0.329435	−0.164717	−	0.986341i	$-0.552671\pi$
$643$	−3.45380e6	−0.329435	−0.164717	+	0.986341i	$0.552671\pi$
$644$	−3.13114e6	−0.297501
$645$	0	0
$646$	−39259.4	−0.00370137
$647$	−1.71115e7	−1.60704	−0.803521	−	0.595276i	$-0.797043\pi$
$647$	−1.71115e7	−1.60704	−0.803521	+	0.595276i	$0.797043\pi$
$648$	0	0
$649$	2.04376e6	0.190467
$650$	1.32227e6	0.122754
$651$	0	0
$652$	−1.17739e7	−1.08468
$653$	−1.61622e7	−1.48326	−0.741630	−	0.670809i	$-0.765947\pi$
$653$	−1.61622e7	−1.48326	−0.741630	+	0.670809i	$0.765947\pi$
$654$	0	0
$655$	−5.96015e6	−0.542818
$656$	−3.82964e6	−0.347455
$657$	0	0
$658$	2.00646e6	0.180661
$659$	−9.54805e6	−0.856449	−0.428224	−	0.903672i	$-0.640861\pi$
$659$	−9.54805e6	−0.856449	−0.428224	+	0.903672i	$0.640861\pi$
$660$	0	0
$661$	−1.24374e7	−1.10720	−0.553599	−	0.832784i	$-0.686746\pi$
$661$	−1.24374e7	−1.10720	−0.553599	+	0.832784i	$0.686746\pi$
$662$	−6.44027e6	−0.571162
$663$	0	0
$664$	−1.12235e7	−0.987889
$665$	−13606.4	−0.00119314
$666$	0	0
$667$	−8.07810e6	−0.703064
$668$	−4.58073e6	−0.397186
$669$	0	0
$670$	2.72505e6	0.234525
$671$	−4.16224e6	−0.356878
$672$	0	0
$673$	1.37772e7	1.17253	0.586266	−	0.810119i	$-0.300597\pi$
$673$	1.37772e7	1.17253	0.586266	+	0.810119i	$0.300597\pi$
$674$	3.13735e6	0.266019
$675$	0	0
$676$	−3.58999e6	−0.302153
$677$	1.06456e6	0.0892683	0.0446341	−	0.999003i	$-0.485788\pi$
$677$	1.06456e6	0.0892683	0.0446341	+	0.999003i	$0.485788\pi$
$678$	0	0
$679$	−8.91804e6	−0.742326
$680$	8.50268e6	0.705154
$681$	0	0
$682$	−364807.	−0.0300333
$683$	1.48070e7	1.21455	0.607275	−	0.794492i	$-0.292263\pi$
$683$	1.48070e7	1.21455	0.607275	+	0.794492i	$0.292263\pi$
$684$	0	0
$685$	3.73835e6	0.304406
$686$	6.55415e6	0.531748
$687$	0	0
$688$	−4.71803e6	−0.380005
$689$	1.60101e7	1.28483
$690$	0	0
$691$	1.56554e7	1.24729	0.623646	−	0.781707i	$-0.285651\pi$
$691$	1.56554e7	1.24729	0.623646	+	0.781707i	$0.285651\pi$
$692$	1.45044e7	1.15142
$693$	0	0
$694$	4.08417e6	0.321888
$695$	−4.19251e6	−0.329240
$696$	0	0
$697$	−2.89810e7	−2.25960
$698$	−7.79167e6	−0.605330
$699$	0	0
$700$	1.24663e6	0.0961596
$701$	−1.56474e7	−1.20267	−0.601334	−	0.798998i	$-0.705364\pi$
$701$	−1.56474e7	−1.20267	−0.601334	+	0.798998i	$0.705364\pi$
$702$	0	0
$703$	80934.4	0.00617653
$704$	1.04596e6	0.0795398
$705$	0	0
$706$	1.73569e6	0.131057
$707$	1.15119e7	0.866160
$708$	0	0
$709$	−635875.	−0.0475068	−0.0237534	−	0.999718i	$-0.507562\pi$
$709$	−635875.	−0.0475068	−0.0237534	+	0.999718i	$0.507562\pi$
$710$	−4.13490e6	−0.307836
$711$	0	0
$712$	9.56580e6	0.707166
$713$	1.61984e6	0.119330
$714$	0	0
$715$	−2.19036e6	−0.160232
$716$	−2.08584e6	−0.152054
$717$	0	0
$718$	9.81064e6	0.710209
$719$	2.50359e6	0.180610	0.0903048	−	0.995914i	$-0.471216\pi$
$719$	2.50359e6	0.180610	0.0903048	+	0.995914i	$0.471216\pi$
$720$	0	0
$721$	−1.42397e7	−1.02015
$722$	−7.23454e6	−0.516497
$723$	0	0
$724$	8.87205e6	0.629039
$725$	3.21621e6	0.227248
$726$	0	0
$727$	8.82888e6	0.619540	0.309770	−	0.950811i	$-0.399748\pi$
$727$	8.82888e6	0.619540	0.309770	+	0.950811i	$0.399748\pi$
$728$	9.97508e6	0.697570
$729$	0	0
$730$	2.02904e6	0.140923
$731$	−3.57040e7	−2.47129
$732$	0	0
$733$	−2.93960e6	−0.202082	−0.101041	−	0.994882i	$-0.532217\pi$
$733$	−2.93960e6	−0.202082	−0.101041	+	0.994882i	$0.532217\pi$
$734$	7.56089e6	0.518004
$735$	0	0
$736$	−9.41250e6	−0.640487
$737$	−4.51409e6	−0.306127
$738$	0	0
$739$	−1.51762e6	−0.102224	−0.0511120	−	0.998693i	$-0.516277\pi$
$739$	−1.51762e6	−0.102224	−0.0511120	+	0.998693i	$0.516277\pi$
$740$	−7.41527e6	−0.497792
$741$	0	0
$742$	−5.49197e6	−0.366200
$743$	1.07165e7	0.712168	0.356084	−	0.934454i	$-0.384112\pi$
$743$	1.07165e7	0.712168	0.356084	+	0.934454i	$0.384112\pi$
$744$	0	0
$745$	−6.81744e6	−0.450019
$746$	−3.13731e6	−0.206400
$747$	0	0
$748$	−5.95844e6	−0.389384
$749$	4.92973e6	0.321084
$750$	0	0
$751$	−2.69694e7	−1.74490	−0.872452	−	0.488700i	$-0.837471\pi$
$751$	−2.69694e7	−1.74490	−0.872452	+	0.488700i	$0.837471\pi$
$752$	−2.24033e6	−0.144467
$753$	0	0
$754$	1.08869e7	0.697391
$755$	1.12080e7	0.715584
$756$	0	0
$757$	−2.81995e7	−1.78855	−0.894275	−	0.447518i	$-0.852308\pi$
$757$	−2.81995e7	−1.78855	−0.894275	+	0.447518i	$0.852308\pi$
$758$	1.50982e7	0.954450
$759$	0	0
$760$	−25937.4	−0.00162889
$761$	−5.29643e6	−0.331529	−0.165764	−	0.986165i	$-0.553009\pi$
$761$	−5.29643e6	−0.331529	−0.165764	+	0.986165i	$0.553009\pi$
$762$	0	0
$763$	1.27143e6	0.0790643
$764$	2.94326e6	0.182430
$765$	0	0
$766$	−7.07344e6	−0.435571
$767$	1.22302e7	0.750665
$768$	0	0
$769$	6.09571e6	0.371714	0.185857	−	0.982577i	$-0.440494\pi$
$769$	6.09571e6	0.371714	0.185857	+	0.982577i	$0.440494\pi$
$770$	751360.	0.0456690
$771$	0	0
$772$	2.17831e7	1.31546
$773$	−1.02486e7	−0.616901	−0.308451	−	0.951240i	$-0.599810\pi$
$773$	−1.02486e7	−0.616901	−0.308451	+	0.951240i	$0.599810\pi$
$774$	0	0
$775$	−644923.	−0.0385704
$776$	−1.70001e7	−1.01344
$777$	0	0
$778$	−5.61434e6	−0.332545
$779$	88406.4	0.00521963
$780$	0	0
$781$	6.84951e6	0.401820
$782$	−9.62624e6	−0.562911
$783$	0	0
$784$	−2.65693e6	−0.154380
$785$	9.16585e6	0.530883
$786$	0	0
$787$	−1.62333e7	−0.934267	−0.467133	−	0.884187i	$-0.654713\pi$
$787$	−1.62333e7	−0.934267	−0.467133	+	0.884187i	$0.654713\pi$
$788$	2.42217e7	1.38959
$789$	0	0
$790$	932048.	0.0531337
$791$	1.68595e7	0.958084
$792$	0	0
$793$	−2.49075e7	−1.40653
$794$	1.19669e7	0.673642
$795$	0	0
$796$	3.63717e6	0.203461
$797$	−1.29153e7	−0.720211	−0.360105	−	0.932912i	$-0.617259\pi$
$797$	−1.29153e7	−0.720211	−0.360105	+	0.932912i	$0.617259\pi$
$798$	0	0
$799$	−1.69539e7	−0.939511
$800$	3.74749e6	0.207021
$801$	0	0
$802$	−1.74585e7	−0.958452
$803$	−3.36113e6	−0.183948
$804$	0	0
$805$	−3.33624e6	−0.181454
$806$	−2.18307e6	−0.118367
$807$	0	0
$808$	2.19447e7	1.18250
$809$	−1.40361e6	−0.0754007	−0.0377004	−	0.999289i	$-0.512003\pi$
$809$	−1.40361e6	−0.0754007	−0.0377004	+	0.999289i	$0.512003\pi$
$810$	0	0
$811$	−1.27909e7	−0.682887	−0.341444	−	0.939902i	$-0.610916\pi$
$811$	−1.27909e7	−0.682887	−0.341444	+	0.939902i	$0.610916\pi$
$812$	1.02641e7	0.546301
$813$	0	0
$814$	−4.46928e6	−0.236416
$815$	−1.25452e7	−0.661580
$816$	0	0
$817$	108915.	0.00570862
$818$	−5.62928e6	−0.294150
$819$	0	0
$820$	−8.09986e6	−0.420671
$821$	5.20603e6	0.269556	0.134778	−	0.990876i	$-0.456968\pi$
$821$	5.20603e6	0.269556	0.134778	+	0.990876i	$0.456968\pi$
$822$	0	0
$823$	3.77702e7	1.94379	0.971896	−	0.235409i	$-0.0756429\pi$
$823$	3.77702e7	1.94379	0.971896	+	0.235409i	$0.0756429\pi$
$824$	−2.71446e7	−1.39272
$825$	0	0
$826$	−4.19535e6	−0.213953
$827$	2.54890e7	1.29595	0.647976	−	0.761661i	$-0.275616\pi$
$827$	2.54890e7	1.29595	0.647976	+	0.761661i	$0.275616\pi$
$828$	0	0
$829$	1.71410e7	0.866262	0.433131	−	0.901331i	$-0.357409\pi$
$829$	1.71410e7	0.866262	0.433131	+	0.901331i	$0.357409\pi$
$830$	−5.05900e6	−0.254900
$831$	0	0
$832$	6.25922e6	0.313482
$833$	−2.01065e7	−1.00398
$834$	0	0
$835$	−4.88079e6	−0.242255
$836$	18176.1	0.000899469 0
$837$	0	0
$838$	4.32009e6	0.212512
$839$	1.01094e7	0.495818	0.247909	−	0.968783i	$-0.420257\pi$
$839$	1.01094e7	0.495818	0.247909	+	0.968783i	$0.420257\pi$
$840$	0	0
$841$	5.96954e6	0.291039
$842$	1.41019e7	0.685483
$843$	0	0
$844$	8.54672e6	0.412994
$845$	−3.82515e6	−0.184292
$846$	0	0
$847$	−1.24464e6	−0.0596121
$848$	6.13212e6	0.292834
$849$	0	0
$850$	3.83259e6	0.181947
$851$	1.98448e7	0.939339
$852$	0	0
$853$	3.29141e7	1.54885	0.774425	−	0.632666i	$-0.218039\pi$
$853$	3.29141e7	1.54885	0.774425	+	0.632666i	$0.218039\pi$
$854$	8.54406e6	0.400885
$855$	0	0
$856$	9.39734e6	0.438349
$857$	−6.94241e6	−0.322893	−0.161446	−	0.986881i	$-0.551616\pi$
$857$	−6.94241e6	−0.322893	−0.161446	+	0.986881i	$0.551616\pi$
$858$	0	0
$859$	−1.84275e7	−0.852085	−0.426042	−	0.904703i	$-0.640093\pi$
$859$	−1.84275e7	−0.852085	−0.426042	+	0.904703i	$0.640093\pi$
$860$	−9.97884e6	−0.460081
$861$	0	0
$862$	−8.45068e6	−0.387368
$863$	2.60648e7	1.19131	0.595657	−	0.803239i	$-0.296892\pi$
$863$	2.60648e7	1.19131	0.595657	+	0.803239i	$0.296892\pi$
$864$	0	0
$865$	1.54545e7	0.702287
$866$	6.91589e6	0.313367
$867$	0	0
$868$	−2.05819e6	−0.0927228
$869$	−1.54395e6	−0.0693559
$870$	0	0
$871$	−2.70131e7	−1.20650
$872$	2.42367e6	0.107940
$873$	0	0
$874$	29364.8	0.00130031
$875$	1.32829e6	0.0586506
$876$	0	0
$877$	1.86288e6	0.0817875	0.0408937	−	0.999164i	$-0.486979\pi$
$877$	1.86288e6	0.0817875	0.0408937	+	0.999164i	$0.486979\pi$
$878$	9.40927e6	0.411926
$879$	0	0
$880$	−838940.	−0.0365194
$881$	−1.01838e7	−0.442051	−0.221025	−	0.975268i	$-0.570940\pi$
$881$	−1.01838e7	−0.442051	−0.221025	+	0.975268i	$0.570940\pi$
$882$	0	0
$883$	6.25954e6	0.270172	0.135086	−	0.990834i	$-0.456869\pi$
$883$	6.25954e6	0.270172	0.135086	+	0.990834i	$0.456869\pi$
$884$	−3.56563e7	−1.53464
$885$	0	0
$886$	1.11126e7	0.475589
$887$	−1.80319e7	−0.769540	−0.384770	−	0.923012i	$-0.625719\pi$
$887$	−1.80319e7	−0.769540	−0.384770	+	0.923012i	$0.625719\pi$
$888$	0	0
$889$	3.48428e6	0.147863
$890$	4.31179e6	0.182466
$891$	0	0
$892$	1.75518e6	0.0738600
$893$	51717.6	0.00217025
$894$	0	0
$895$	−2.22247e6	−0.0927422
$896$	1.41640e7	0.589407
$897$	0	0
$898$	1.91466e7	0.792321
$899$	−5.30998e6	−0.219126
$900$	0	0
$901$	4.64052e7	1.90439
$902$	−4.88189e6	−0.199789
$903$	0	0
$904$	3.21386e7	1.30799
$905$	9.45320e6	0.383670
$906$	0	0
$907$	2.69733e7	1.08872	0.544360	−	0.838852i	$-0.316772\pi$
$907$	2.69733e7	1.08872	0.544360	+	0.838852i	$0.316772\pi$
$908$	2.87838e7	1.15860
$909$	0	0
$910$	4.49627e6	0.179990
$911$	4.31660e7	1.72324	0.861621	−	0.507553i	$-0.169450\pi$
$911$	4.31660e7	1.72324	0.861621	+	0.507553i	$0.169450\pi$
$912$	0	0
$913$	8.38029e6	0.332723
$914$	−2.16643e7	−0.857784
$915$	0	0
$916$	1.03438e7	0.407323
$917$	−2.02670e7	−0.795915
$918$	0	0
$919$	−3.69944e7	−1.44493	−0.722465	−	0.691407i	$-0.756991\pi$
$919$	−3.69944e7	−1.44493	−0.722465	+	0.691407i	$0.756991\pi$
$920$	−6.35974e6	−0.247725
$921$	0	0
$922$	−1.19655e6	−0.0463557
$923$	4.09887e7	1.58365
$924$	0	0
$925$	−7.90099e6	−0.303618
$926$	2.06627e7	0.791882
$927$	0	0
$928$	3.08550e7	1.17613
$929$	−1.63853e6	−0.0622895	−0.0311447	−	0.999515i	$-0.509915\pi$
$929$	−1.63853e6	−0.0622895	−0.0311447	+	0.999515i	$0.509915\pi$
$930$	0	0
$931$	61334.7	0.00231917
$932$	−1.20729e7	−0.455273
$933$	0	0
$934$	1.85658e7	0.696380
$935$	−6.34873e6	−0.237497
$936$	0	0
$937$	−2.02831e7	−0.754719	−0.377360	−	0.926067i	$-0.623168\pi$
$937$	−2.02831e7	−0.754719	−0.377360	+	0.926067i	$0.623168\pi$
$938$	9.26633e6	0.343875
$939$	0	0
$940$	−4.73841e6	−0.174909
$941$	1.83342e7	0.674975	0.337487	−	0.941330i	$-0.390423\pi$
$941$	1.83342e7	0.674975	0.337487	+	0.941330i	$0.390423\pi$
$942$	0	0
$943$	2.16769e7	0.793812
$944$	4.68436e6	0.171088
$945$	0	0
$946$	−6.01438e6	−0.218506
$947$	6.09095e6	0.220704	0.110352	−	0.993893i	$-0.464802\pi$
$947$	6.09095e6	0.220704	0.110352	+	0.993893i	$0.464802\pi$
$948$	0	0
$949$	−2.01136e7	−0.724976
$950$	−11691.3	−0.000420293 0
$951$	0	0
$952$	2.89127e7	1.03394
$953$	5.31211e7	1.89467	0.947337	−	0.320238i	$-0.103763\pi$
$953$	5.31211e7	1.89467	0.947337	+	0.320238i	$0.103763\pi$
$954$	0	0
$955$	3.13605e6	0.111269
$956$	−2.30951e7	−0.817287
$957$	0	0
$958$	−1.86046e7	−0.654947
$959$	1.27120e7	0.446340
$960$	0	0
$961$	−2.75644e7	−0.962808
$962$	−2.67449e7	−0.931759
$963$	0	0
$964$	−6.68214e6	−0.231592
$965$	2.32099e7	0.802335
$966$	0	0
$967$	−2.17670e6	−0.0748570	−0.0374285	−	0.999299i	$-0.511917\pi$
$967$	−2.17670e6	−0.0748570	−0.0374285	+	0.999299i	$0.511917\pi$
$968$	−2.37260e6	−0.0813836
$969$	0	0
$970$	−7.66279e6	−0.261492
$971$	−4.83305e7	−1.64503	−0.822514	−	0.568745i	$-0.807429\pi$
$971$	−4.83305e7	−1.64503	−0.822514	+	0.568745i	$0.807429\pi$
$972$	0	0
$973$	−1.42563e7	−0.482753
$974$	−8.59826e6	−0.290411
$975$	0	0
$976$	−9.53996e6	−0.320569
$977$	3.84816e7	1.28978	0.644892	−	0.764274i	$-0.276902\pi$
$977$	3.84816e7	1.28978	0.644892	+	0.764274i	$0.276902\pi$
$978$	0	0
$979$	−7.14253e6	−0.238175
$980$	−5.61954e6	−0.186911
$981$	0	0
$982$	1.46566e6	0.0485013
$983$	1.53385e7	0.506288	0.253144	−	0.967429i	$-0.418535\pi$
$983$	1.53385e7	0.506288	0.253144	+	0.967429i	$0.418535\pi$
$984$	0	0
$985$	2.58082e7	0.847555
$986$	3.15556e7	1.03368
$987$	0	0
$988$	108769.	0.00354498
$989$	2.67054e7	0.868178
$990$	0	0
$991$	−4.29143e7	−1.38809	−0.694046	−	0.719931i	$-0.744174\pi$
$991$	−4.29143e7	−1.38809	−0.694046	+	0.719931i	$0.744174\pi$
$992$	−6.18712e6	−0.199622
$993$	0	0
$994$	−1.40604e7	−0.451368
$995$	3.87542e6	0.124097
$996$	0	0
$997$	6.10887e7	1.94636	0.973180	−	0.230044i	$-0.0738869\pi$
$997$	6.10887e7	1.94636	0.973180	+	0.230044i	$0.0738869\pi$
$998$	−2.68742e7	−0.854102
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.92180 q^{2} -23.4631 q^{4} -25.0000 q^{5} -85.0105 q^{7} -162.052 q^{8} +O(q^{10})\)	\(q+2.92180 q^{2} -23.4631 q^{4} -25.0000 q^{5} -85.0105 q^{7} -162.052 q^{8} -73.0450 q^{10} +121.000 q^{11} +724.085 q^{13} -248.384 q^{14} +277.335 q^{16} +2098.75 q^{17} -6.40223 q^{19} +586.577 q^{20} +353.538 q^{22} -1569.80 q^{23} +625.000 q^{25} +2115.63 q^{26} +1994.61 q^{28} +5145.94 q^{29} -1031.88 q^{31} +5995.98 q^{32} +6132.14 q^{34} +2125.26 q^{35} -12641.6 q^{37} -18.7060 q^{38} +4051.30 q^{40} -13808.7 q^{41} -17012.0 q^{43} -2839.03 q^{44} -4586.64 q^{46} -8078.06 q^{47} -9580.22 q^{49} +1826.12 q^{50} -16989.3 q^{52} +22110.8 q^{53} -3025.00 q^{55} +13776.1 q^{56} +15035.4 q^{58} +16890.6 q^{59} -34398.6 q^{61} -3014.94 q^{62} +8644.32 q^{64} -18102.1 q^{65} -37306.5 q^{67} -49243.3 q^{68} +6209.59 q^{70} +56607.5 q^{71} -27777.9 q^{73} -36936.2 q^{74} +150.216 q^{76} -10286.3 q^{77} -12759.9 q^{79} -6933.39 q^{80} -40346.2 q^{82} +69258.6 q^{83} -52468.9 q^{85} -49705.6 q^{86} -19608.3 q^{88} -59029.2 q^{89} -61554.8 q^{91} +36832.4 q^{92} -23602.5 q^{94} +160.056 q^{95} +104905. q^{97} -27991.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8}+O(q^{10}) \) 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8	\( 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8} - 175 q^{10} + 363 q^{11} - 654 q^{13} + 728 q^{14} - 415 q^{16} + 2366 q^{17} - 2872 q^{19} - 625 q^{20} + 847 q^{22} - 2272 q^{23} + 1875 q^{25} - 3422 q^{26} + 4592 q^{28} + 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 2506 q^{34} + 4300 q^{35} - 9126 q^{37} - 13076 q^{38} - 5775 q^{40} + 8758 q^{41} - 14672 q^{43} + 3025 q^{44} - 28768 q^{46} + 19392 q^{47} - 26629 q^{49} + 4375 q^{50} - 61506 q^{52} + 4598 q^{53} - 9075 q^{55} - 2688 q^{56} + 8550 q^{58} + 9348 q^{59} - 60078 q^{61} + 14096 q^{62} - 7087 q^{64} + 16350 q^{65} - 38468 q^{67} - 59778 q^{68} - 18200 q^{70} + 74032 q^{71} - 44442 q^{73} - 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 108116 q^{79} + 10375 q^{80} - 92230 q^{82} + 81892 q^{83} - 59150 q^{85} - 126412 q^{86} + 27951 q^{88} - 167342 q^{89} - 31832 q^{91} - 72960 q^{92} + 12728 q^{94} + 71800 q^{95} + 159702 q^{97} - 163121 q^{98}+O(q^{100}) \) 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8 - 175 * q^10 + 363 * q^11 - 654 * q^13 + 728 * q^14 - 415 * q^16 + 2366 * q^17 - 2872 * q^19 - 625 * q^20 + 847 * q^22 - 2272 * q^23 + 1875 * q^25 - 3422 * q^26 + 4592 * q^28 + 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 2506 * q^34 + 4300 * q^35 - 9126 * q^37 - 13076 * q^38 - 5775 * q^40 + 8758 * q^41 - 14672 * q^43 + 3025 * q^44 - 28768 * q^46 + 19392 * q^47 - 26629 * q^49 + 4375 * q^50 - 61506 * q^52 + 4598 * q^53 - 9075 * q^55 - 2688 * q^56 + 8550 * q^58 + 9348 * q^59 - 60078 * q^61 + 14096 * q^62 - 7087 * q^64 + 16350 * q^65 - 38468 * q^67 - 59778 * q^68 - 18200 * q^70 + 74032 * q^71 - 44442 * q^73 - 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 108116 * q^79 + 10375 * q^80 - 92230 * q^82 + 81892 * q^83 - 59150 * q^85 - 126412 * q^86 + 27951 * q^88 - 167342 * q^89 - 31832 * q^91 - 72960 * q^92 + 12728 * q^94 + 71800 * q^95 + 159702 * q^97 - 163121 * q^98

Introduction

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