LMFDB - Embedded newform 867.6.a.u.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,6,Mod(1,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("867.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	867.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:	$139.052771778$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68843 q^{2} +9.00000 q^{3} -24.7723 q^{4} +41.7848 q^{5} -24.1959 q^{6} +18.3667 q^{7} +152.629 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-2.68843 q^{2} +9.00000 q^{3} -24.7723 q^{4} +41.7848 q^{5} -24.1959 q^{6} +18.3667 q^{7} +152.629 q^{8} +81.0000 q^{9} -112.336 q^{10} +159.103 q^{11} -222.951 q^{12} +128.048 q^{13} -49.3775 q^{14} +376.063 q^{15} +382.382 q^{16} -217.763 q^{18} -755.418 q^{19} -1035.11 q^{20} +165.300 q^{21} -427.738 q^{22} -2118.60 q^{23} +1373.66 q^{24} -1379.03 q^{25} -344.249 q^{26} +729.000 q^{27} -454.985 q^{28} +185.978 q^{29} -1011.02 q^{30} -9632.85 q^{31} -5912.13 q^{32} +1431.93 q^{33} +767.447 q^{35} -2006.56 q^{36} +7142.77 q^{37} +2030.89 q^{38} +1152.43 q^{39} +6377.56 q^{40} +3847.59 q^{41} -444.398 q^{42} +6664.88 q^{43} -3941.35 q^{44} +3384.57 q^{45} +5695.72 q^{46} +4537.14 q^{47} +3441.44 q^{48} -16469.7 q^{49} +3707.43 q^{50} -3172.05 q^{52} -25734.9 q^{53} -1959.87 q^{54} +6648.09 q^{55} +2803.28 q^{56} -6798.76 q^{57} -499.988 q^{58} -21619.5 q^{59} -9315.96 q^{60} +246.249 q^{61} +25897.3 q^{62} +1487.70 q^{63} +3658.14 q^{64} +5350.47 q^{65} -3849.64 q^{66} -16736.8 q^{67} -19067.4 q^{69} -2063.23 q^{70} +14396.2 q^{71} +12362.9 q^{72} +76437.5 q^{73} -19202.9 q^{74} -12411.3 q^{75} +18713.4 q^{76} +2922.19 q^{77} -3098.24 q^{78} +30955.8 q^{79} +15977.8 q^{80} +6561.00 q^{81} -10344.0 q^{82} -9646.03 q^{83} -4094.86 q^{84} -17918.1 q^{86} +1673.80 q^{87} +24283.7 q^{88} +4276.73 q^{89} -9099.19 q^{90} +2351.82 q^{91} +52482.7 q^{92} -86695.7 q^{93} -12197.8 q^{94} -31565.0 q^{95} -53209.1 q^{96} +15464.2 q^{97} +44277.6 q^{98} +12887.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9}+O(q^{10}) $ 28 * q + 252 * q^3 + 384 * q^4 - 244 * q^5 - 392 * q^7 + 2268 * q^9	\( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9} - 800 q^{10} - 1132 q^{11} + 3456 q^{12} - 828 q^{13} - 1568 q^{14} - 2196 q^{15} + 4096 q^{16} - 7532 q^{19} - 9216 q^{20} - 3528 q^{21} - 664 q^{22} - 4548 q^{23} + 9904 q^{25} + 20412 q^{27} - 36936 q^{28} - 36680 q^{29} - 7200 q^{30} - 7688 q^{31} - 10188 q^{33} + 13136 q^{35} + 31104 q^{36} - 42360 q^{37} - 1984 q^{38} - 7452 q^{39} - 81000 q^{40} - 6060 q^{41} - 14112 q^{42} - 50044 q^{43} - 56960 q^{44} - 19764 q^{45} - 51488 q^{46} + 16552 q^{47} + 36864 q^{48} + 38084 q^{49} - 48064 q^{50} - 31360 q^{52} - 4328 q^{53} - 71740 q^{55} + 9248 q^{56} - 67788 q^{57} - 95256 q^{58} - 89832 q^{59} - 82944 q^{60} - 154104 q^{61} - 106624 q^{62} - 31752 q^{63} + 93720 q^{64} - 13260 q^{65} - 5976 q^{66} - 182136 q^{67} - 40932 q^{69} - 145464 q^{70} + 24504 q^{71} - 58184 q^{73} - 308992 q^{74} + 89136 q^{75} - 602544 q^{76} - 240576 q^{77} - 136080 q^{79} - 317440 q^{80} + 183708 q^{81} - 46648 q^{82} - 203576 q^{83} - 332424 q^{84} - 421280 q^{86} - 330120 q^{87} - 145152 q^{88} - 120880 q^{89} - 64800 q^{90} - 290152 q^{91} - 491616 q^{92} - 69192 q^{93} - 799168 q^{94} - 543084 q^{95} - 409328 q^{97} + 193072 q^{98} - 91692 q^{99}+O(q^{100}) \) 28 * q + 252 * q^3 + 384 * q^4 - 244 * q^5 - 392 * q^7 + 2268 * q^9 - 800 * q^10 - 1132 * q^11 + 3456 * q^12 - 828 * q^13 - 1568 * q^14 - 2196 * q^15 + 4096 * q^16 - 7532 * q^19 - 9216 * q^20 - 3528 * q^21 - 664 * q^22 - 4548 * q^23 + 9904 * q^25 + 20412 * q^27 - 36936 * q^28 - 36680 * q^29 - 7200 * q^30 - 7688 * q^31 - 10188 * q^33 + 13136 * q^35 + 31104 * q^36 - 42360 * q^37 - 1984 * q^38 - 7452 * q^39 - 81000 * q^40 - 6060 * q^41 - 14112 * q^42 - 50044 * q^43 - 56960 * q^44 - 19764 * q^45 - 51488 * q^46 + 16552 * q^47 + 36864 * q^48 + 38084 * q^49 - 48064 * q^50 - 31360 * q^52 - 4328 * q^53 - 71740 * q^55 + 9248 * q^56 - 67788 * q^57 - 95256 * q^58 - 89832 * q^59 - 82944 * q^60 - 154104 * q^61 - 106624 * q^62 - 31752 * q^63 + 93720 * q^64 - 13260 * q^65 - 5976 * q^66 - 182136 * q^67 - 40932 * q^69 - 145464 * q^70 + 24504 * q^71 - 58184 * q^73 - 308992 * q^74 + 89136 * q^75 - 602544 * q^76 - 240576 * q^77 - 136080 * q^79 - 317440 * q^80 + 183708 * q^81 - 46648 * q^82 - 203576 * q^83 - 332424 * q^84 - 421280 * q^86 - 330120 * q^87 - 145152 * q^88 - 120880 * q^89 - 64800 * q^90 - 290152 * q^91 - 491616 * q^92 - 69192 * q^93 - 799168 * q^94 - 543084 * q^95 - 409328 * q^97 + 193072 * q^98 - 91692 * q^99

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 $

$2$	−2.68843	−0.475253	−0.237626	−	0.971357i	$-0.576369\pi$
$2$	−2.68843	−0.475253	−0.237626	+	0.971357i	$0.576369\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−24.7723	−0.774135
$5$	41.7848	0.747469	0.373735	−	0.927536i	$-0.378077\pi$
$5$	41.7848	0.747469	0.373735	+	0.927536i	$0.378077\pi$
$6$	−24.1959	−0.274387
$7$	18.3667	0.141672	0.0708361	−	0.997488i	$-0.477433\pi$
$7$	18.3667	0.141672	0.0708361	+	0.997488i	$0.477433\pi$
$8$	152.629	0.843162
$9$	81.0000	0.333333
$10$	−112.336	−0.355237
$11$	159.103	0.396458	0.198229	−	0.980156i	$-0.436481\pi$
$11$	159.103	0.396458	0.198229	+	0.980156i	$0.436481\pi$
$12$	−222.951	−0.446947
$13$	128.048	0.210143	0.105072	−	0.994465i	$-0.466493\pi$
$13$	128.048	0.210143	0.105072	+	0.994465i	$0.466493\pi$
$14$	−49.3775	−0.0673301
$15$	376.063	0.431552
$16$	382.382	0.373420
$17$	0	0
$17$	0	0
$18$	−217.763	−0.158418
$19$	−755.418	−0.480068	−0.240034	−	0.970764i	$-0.577159\pi$
$19$	−755.418	−0.480068	−0.240034	+	0.970764i	$0.577159\pi$
$20$	−1035.11	−0.578642
$21$	165.300	0.0817945
$22$	−427.738	−0.188417
$23$	−2118.60	−0.835083	−0.417542	−	0.908658i	$-0.637108\pi$
$23$	−2118.60	−0.835083	−0.417542	+	0.908658i	$0.637108\pi$
$24$	1373.66	0.486800
$25$	−1379.03	−0.441290
$26$	−344.249	−0.0998711
$27$	729.000	0.192450
$28$	−454.985	−0.109673
$29$	185.978	0.0410644	0.0205322	−	0.999789i	$-0.493464\pi$
$29$	185.978	0.0410644	0.0205322	+	0.999789i	$0.493464\pi$
$30$	−1011.02	−0.205096
$31$	−9632.85	−1.80032	−0.900162	−	0.435556i	$-0.856552\pi$
$31$	−9632.85	−1.80032	−0.900162	+	0.435556i	$0.856552\pi$
$32$	−5912.13	−1.02063
$33$	1431.93	0.228895
$34$	0	0
$35$	767.447	0.105896
$36$	−2006.56	−0.258045
$37$	7142.77	0.857753	0.428876	−	0.903363i	$-0.358910\pi$
$37$	7142.77	0.857753	0.428876	+	0.903363i	$0.358910\pi$
$38$	2030.89	0.228154
$39$	1152.43	0.121326
$40$	6377.56	0.630238
$41$	3847.59	0.357461	0.178731	−	0.983898i	$-0.442801\pi$
$41$	3847.59	0.357461	0.178731	+	0.983898i	$0.442801\pi$
$42$	−444.398	−0.0388731
$43$	6664.88	0.549694	0.274847	−	0.961488i	$-0.411373\pi$
$43$	6664.88	0.549694	0.274847	+	0.961488i	$0.411373\pi$
$44$	−3941.35	−0.306912
$45$	3384.57	0.249156
$46$	5695.72	0.396875
$47$	4537.14	0.299597	0.149799	−	0.988717i	$-0.452137\pi$
$47$	4537.14	0.299597	0.149799	+	0.988717i	$0.452137\pi$
$48$	3441.44	0.215594
$49$	−16469.7	−0.979929
$50$	3707.43	0.209724
$51$	0	0
$52$	−3172.05	−0.162679
$53$	−25734.9	−1.25844	−0.629221	−	0.777226i	$-0.716626\pi$
$53$	−25734.9	−1.25844	−0.629221	+	0.777226i	$0.716626\pi$
$54$	−1959.87	−0.0914624
$55$	6648.09	0.296340
$56$	2803.28	0.119453
$57$	−6798.76	−0.277168
$58$	−499.988	−0.0195160
$59$	−21619.5	−0.808567	−0.404284	−	0.914634i	$-0.632479\pi$
$59$	−21619.5	−0.808567	−0.404284	+	0.914634i	$0.632479\pi$
$60$	−9315.96	−0.334079
$61$	246.249	0.00847323	0.00423662	−	0.999991i	$-0.498651\pi$
$61$	246.249	0.00847323	0.00423662	+	0.999991i	$0.498651\pi$
$62$	25897.3	0.855608
$63$	1487.70	0.0472241
$64$	3658.14	0.111637
$65$	5350.47	0.157076
$66$	−3849.64	−0.108783
$67$	−16736.8	−0.455496	−0.227748	−	0.973720i	$-0.573136\pi$
$67$	−16736.8	−0.455496	−0.227748	+	0.973720i	$0.573136\pi$
$68$	0	0
$69$	−19067.4	−0.482136
$70$	−2063.23	−0.0503272
$71$	14396.2	0.338923	0.169461	−	0.985537i	$-0.445797\pi$
$71$	14396.2	0.338923	0.169461	+	0.985537i	$0.445797\pi$
$72$	12362.9	0.281054
$73$	76437.5	1.67880	0.839401	−	0.543513i	$-0.182906\pi$
$73$	76437.5	1.67880	0.839401	+	0.543513i	$0.182906\pi$
$74$	−19202.9	−0.407649
$75$	−12411.3	−0.254779
$76$	18713.4	0.371638
$77$	2922.19	0.0561670
$78$	−3098.24	−0.0576606
$79$	30955.8	0.558051	0.279026	−	0.960284i	$-0.409989\pi$
$79$	30955.8	0.558051	0.279026	+	0.960284i	$0.409989\pi$
$80$	15977.8	0.279120
$81$	6561.00	0.111111
$82$	−10344.0	−0.169884
$83$	−9646.03	−0.153693	−0.0768464	−	0.997043i	$-0.524485\pi$
$83$	−9646.03	−0.153693	−0.0768464	+	0.997043i	$0.524485\pi$
$84$	−4094.86	−0.0633200
$85$	0	0
$86$	−17918.1	−0.261243
$87$	1673.80	0.0237085
$88$	24283.7	0.334278
$89$	4276.73	0.0572317	0.0286159	−	0.999590i	$-0.490890\pi$
$89$	4276.73	0.0572317	0.0286159	+	0.999590i	$0.490890\pi$
$90$	−9099.19	−0.118412
$91$	2351.82	0.0297715
$92$	52482.7	0.646467
$93$	−86695.7	−1.03942
$94$	−12197.8	−0.142384
$95$	−31565.0	−0.358836
$96$	−53209.1	−0.589262
$97$	15464.2	0.166878	0.0834389	−	0.996513i	$-0.473410\pi$
$97$	15464.2	0.166878	0.0834389	+	0.996513i	$0.473410\pi$
$98$	44277.6	0.465714
$99$	12887.3	0.132153
$100$	34161.8	0.341618
$101$	−169175.	−1.65018	−0.825092	−	0.564999i	$-0.808877\pi$
$101$	−169175.	−1.65018	−0.825092	+	0.564999i	$0.808877\pi$
$102$	0	0
$103$	−1529.18	−0.0142025	−0.00710127	−	0.999975i	$-0.502260\pi$
$103$	−1529.18	−0.0142025	−0.00710127	+	0.999975i	$0.502260\pi$
$104$	19543.8	0.177185
$105$	6907.02	0.0611389
$106$	69186.7	0.598078
$107$	9953.32	0.0840444	0.0420222	−	0.999117i	$-0.486620\pi$
$107$	9953.32	0.0840444	0.0420222	+	0.999117i	$0.486620\pi$
$108$	−18059.0	−0.148982
$109$	92625.3	0.746729	0.373365	−	0.927685i	$-0.378204\pi$
$109$	92625.3	0.746729	0.373365	+	0.927685i	$0.378204\pi$
$110$	−17872.9	−0.140836
$111$	64284.9	0.495224
$112$	7023.08	0.0529033
$113$	−169936.	−1.25195	−0.625977	−	0.779842i	$-0.715300\pi$
$113$	−169936.	−1.25195	−0.625977	+	0.779842i	$0.715300\pi$
$114$	18278.0	0.131725
$115$	−88525.4	−0.624199
$116$	−4607.09	−0.0317894
$117$	10371.9	0.0700477
$118$	58122.7	0.384274
$119$	0	0
$120$	57398.0	0.363868
$121$	−135737.	−0.842821
$122$	−662.023	−0.00402692
$123$	34628.3	0.206380
$124$	238628.	1.39369
$125$	−188200.	−1.07732
$126$	−3999.58	−0.0224434
$127$	179541.	0.987766	0.493883	−	0.869528i	$-0.335577\pi$
$127$	179541.	0.987766	0.493883	+	0.869528i	$0.335577\pi$
$128$	179353.	0.967575
$129$	59983.9	0.317366
$130$	−14384.4	−0.0746505
$131$	−33354.2	−0.169813	−0.0849067	−	0.996389i	$-0.527059\pi$
$131$	−33354.2	−0.169813	−0.0849067	+	0.996389i	$0.527059\pi$
$132$	−35472.2	−0.177196
$133$	−13874.5	−0.0680124
$134$	44995.7	0.216476
$135$	30461.1	0.143851
$136$	0	0
$137$	−96521.5	−0.439362	−0.219681	−	0.975572i	$-0.570502\pi$
$137$	−96521.5	−0.439362	−0.219681	+	0.975572i	$0.570502\pi$
$138$	51261.5	0.229136
$139$	−312549.	−1.37208	−0.686042	−	0.727562i	$-0.740653\pi$
$139$	−312549.	−1.37208	−0.686042	+	0.727562i	$0.740653\pi$
$140$	−19011.4	−0.0819775
$141$	40834.3	0.172973
$142$	−38703.1	−0.161074
$143$	20372.9	0.0833128
$144$	30973.0	0.124473
$145$	7771.03	0.0306944
$146$	−205497.	−0.797855
$147$	−148227.	−0.565762
$148$	−176943.	−0.664016
$149$	−252388.	−0.931329	−0.465665	−	0.884961i	$-0.654185\pi$
$149$	−252388.	−0.931329	−0.465665	+	0.884961i	$0.654185\pi$
$150$	33366.9	0.121084
$151$	214705.	0.766303	0.383152	−	0.923686i	$-0.374839\pi$
$151$	214705.	0.766303	0.383152	+	0.923686i	$0.374839\pi$
$152$	−115298.	−0.404776
$153$	0	0
$154$	−7856.11	−0.0266935
$155$	−402507.	−1.34569
$156$	−28548.5	−0.0939228
$157$	397578.	1.28728	0.643641	−	0.765328i	$-0.277423\pi$
$157$	397578.	1.28728	0.643641	+	0.765328i	$0.277423\pi$
$158$	−83222.6	−0.265215
$159$	−231614.	−0.726562
$160$	−247037.	−0.762890
$161$	−38911.6	−0.118308
$162$	−17638.8	−0.0528058
$163$	−227680.	−0.671206	−0.335603	−	0.942004i	$-0.608940\pi$
$163$	−227680.	−0.671206	−0.335603	+	0.942004i	$0.608940\pi$
$164$	−95313.6	−0.276723
$165$	59832.8	0.171092
$166$	25932.7	0.0730429
$167$	241861.	0.671081	0.335540	−	0.942026i	$-0.391081\pi$
$167$	241861.	0.671081	0.335540	+	0.942026i	$0.391081\pi$
$168$	25229.5	0.0689661
$169$	−354897.	−0.955840
$170$	0	0
$171$	−61188.8	−0.160023
$172$	−165104.	−0.425537
$173$	361681.	0.918778	0.459389	−	0.888235i	$-0.348068\pi$
$173$	361681.	0.918778	0.459389	+	0.888235i	$0.348068\pi$
$174$	−4499.90	−0.0112675
$175$	−25328.2	−0.0625185
$176$	60838.1	0.148045
$177$	−194576.	−0.466827
$178$	−11497.7	−0.0271995
$179$	564802.	1.31754	0.658770	−	0.752344i	$-0.271077\pi$
$179$	564802.	1.31754	0.658770	+	0.752344i	$0.271077\pi$
$180$	−83843.6	−0.192881
$181$	−736469.	−1.67093	−0.835464	−	0.549545i	$-0.814801\pi$
$181$	−736469.	−1.67093	−0.835464	+	0.549545i	$0.814801\pi$
$182$	−6322.71	−0.0141490
$183$	2216.24	0.00489202
$184$	−323359.	−0.704111
$185$	298459.	0.641144
$186$	233076.	0.493986
$187$	0	0
$188$	−112396.	−0.231929
$189$	13389.3	0.0272648
$190$	84860.4	0.170538
$191$	−102844.	−0.203984	−0.101992	−	0.994785i	$-0.532522\pi$
$191$	−102844.	−0.203984	−0.101992	+	0.994785i	$0.532522\pi$
$192$	32923.2	0.0644539
$193$	590572.	1.14125	0.570624	−	0.821212i	$-0.306701\pi$
$193$	590572.	1.14125	0.570624	+	0.821212i	$0.306701\pi$
$194$	−41574.5	−0.0793091
$195$	48154.2	0.0906876
$196$	407992.	0.758597
$197$	−949032.	−1.74227	−0.871134	−	0.491045i	$-0.836615\pi$
$197$	−949032.	−1.74227	−0.871134	+	0.491045i	$0.836615\pi$
$198$	−34646.8	−0.0628058
$199$	1.10144e6	1.97165	0.985825	−	0.167774i	$-0.0536580\pi$
$199$	1.10144e6	1.97165	0.985825	+	0.167774i	$0.0536580\pi$
$200$	−210480.	−0.372079
$201$	−150631.	−0.262981
$202$	454815.	0.784254
$203$	3415.78	0.00581769
$204$	0	0
$205$	160771.	0.267191
$206$	4111.11	0.00674980
$207$	−171607.	−0.278361
$208$	48963.3	0.0784716
$209$	−120189.	−0.190327
$210$	−18569.1	−0.0290564
$211$	−670510.	−1.03681	−0.518405	−	0.855135i	$-0.673474\pi$
$211$	−670510.	−1.03681	−0.518405	+	0.855135i	$0.673474\pi$
$212$	637514.	0.974204
$213$	129565.	0.195677
$214$	−26758.9	−0.0399423
$215$	278490.	0.410879
$216$	111266.	0.162267
$217$	−176923.	−0.255056
$218$	−249017.	−0.354885
$219$	687938.	0.969256
$220$	−164689.	−0.229407
$221$	0	0
$222$	−172826.	−0.235356
$223$	−1.34693e6	−1.81377	−0.906886	−	0.421377i	$-0.861547\pi$
$223$	−1.34693e6	−1.81377	−0.906886	+	0.421377i	$0.861547\pi$
$224$	−108586.	−0.144595
$225$	−111701.	−0.147097
$226$	456861.	0.594994
$227$	−487782.	−0.628292	−0.314146	−	0.949375i	$-0.601718\pi$
$227$	−487782.	−0.628292	−0.314146	+	0.949375i	$0.601718\pi$
$228$	168421.	0.214565
$229$	−1.15185e6	−1.45147	−0.725736	−	0.687974i	$-0.758500\pi$
$229$	−1.15185e6	−1.45147	−0.725736	+	0.687974i	$0.758500\pi$
$230$	237995.	0.296652
$231$	26299.7	0.0324281
$232$	28385.5	0.0346239
$233$	777721.	0.938499	0.469250	−	0.883066i	$-0.344524\pi$
$233$	777721.	0.938499	0.469250	+	0.883066i	$0.344524\pi$
$234$	−27884.2	−0.0332904
$235$	189584.	0.223940
$236$	535566.	0.625940
$237$	278602.	0.322191
$238$	0	0
$239$	−1.18464e6	−1.34150	−0.670751	−	0.741683i	$-0.734028\pi$
$239$	−1.18464e6	−1.34150	−0.670751	+	0.741683i	$0.734028\pi$
$240$	143800.	0.161150
$241$	−1.00161e6	−1.11085	−0.555424	−	0.831567i	$-0.687444\pi$
$241$	−1.00161e6	−1.11085	−0.555424	+	0.831567i	$0.687444\pi$
$242$	364921.	0.400553
$243$	59049.0	0.0641500
$244$	−6100.15	−0.00655942
$245$	−688182.	−0.732467
$246$	−93095.9	−0.0980827
$247$	−96729.9	−0.100883
$248$	−1.47025e6	−1.51796
$249$	−86814.3	−0.0887346
$250$	505963.	0.511999
$251$	1.18322e6	1.18545	0.592725	−	0.805405i	$-0.298052\pi$
$251$	1.18322e6	1.18545	0.592725	+	0.805405i	$0.298052\pi$
$252$	−36853.8	−0.0365578
$253$	−337076.	−0.331075
$254$	−482684.	−0.469438
$255$	0	0
$256$	−599240.	−0.571480
$257$	−1.28106e6	−1.20987	−0.604934	−	0.796276i	$-0.706800\pi$
$257$	−1.28106e6	−1.20987	−0.604934	+	0.796276i	$0.706800\pi$
$258$	−161263.	−0.150829
$259$	131189.	0.121520
$260$	−132544.	−0.121598
$261$	15064.2	0.0136881
$262$	89670.5	0.0807043
$263$	−442254.	−0.394259	−0.197130	−	0.980377i	$-0.563162\pi$
$263$	−442254.	−0.394259	−0.197130	+	0.980377i	$0.563162\pi$
$264$	218553.	0.192995
$265$	−1.07533e6	−0.940647
$266$	37300.7	0.0323231
$267$	38490.6	0.0330428
$268$	414609.	0.352616
$269$	−2.08175e6	−1.75408	−0.877038	−	0.480421i	$-0.840484\pi$
$269$	−2.08175e6	−1.75408	−0.877038	+	0.480421i	$0.840484\pi$
$270$	−81892.7	−0.0683653
$271$	791590.	0.654752	0.327376	−	0.944894i	$-0.393836\pi$
$271$	791590.	0.654752	0.327376	+	0.944894i	$0.393836\pi$
$272$	0	0
$273$	21166.4	0.0171886
$274$	259492.	0.208808
$275$	−219408.	−0.174953
$276$	472344.	0.373238
$277$	−198548.	−0.155477	−0.0777385	−	0.996974i	$-0.524770\pi$
$277$	−198548.	−0.155477	−0.0777385	+	0.996974i	$0.524770\pi$
$278$	840267.	0.652086
$279$	−780261.	−0.600108
$280$	117134.	0.0892872
$281$	2.46387e6	1.86145	0.930725	−	0.365720i	$-0.119177\pi$
$281$	2.46387e6	1.86145	0.930725	+	0.365720i	$0.119177\pi$
$282$	−109780.	−0.0822057
$283$	−112266.	−0.0833262	−0.0416631	−	0.999132i	$-0.513266\pi$
$283$	−112266.	−0.0833262	−0.0416631	+	0.999132i	$0.513266\pi$
$284$	−356626.	−0.262372
$285$	−284085.	−0.207174
$286$	−54771.1	−0.0395946
$287$	70667.3	0.0506423
$288$	−478882.	−0.340210
$289$	0	0
$290$	−20891.9	−0.0145876
$291$	139178.	0.0963469
$292$	−1.89353e6	−1.29962
$293$	1.05318e6	0.716691	0.358346	−	0.933589i	$-0.383341\pi$
$293$	1.05318e6	0.716691	0.358346	+	0.933589i	$0.383341\pi$
$294$	398499.	0.268880
$295$	−903368.	−0.604379
$296$	1.09019e6	0.723225
$297$	115986.	0.0762983
$298$	678529.	0.442617
$299$	−271283.	−0.175487
$300$	307456.	0.197233
$301$	122411.	0.0778764
$302$	−577221.	−0.364187
$303$	−1.52257e6	−0.952734
$304$	−288858.	−0.179267
$305$	10289.4	0.00633348
$306$	0	0
$307$	−2.94983e6	−1.78628	−0.893142	−	0.449775i	$-0.851504\pi$
$307$	−2.94983e6	−1.78628	−0.893142	+	0.449775i	$0.851504\pi$
$308$	−72389.4	−0.0434809
$309$	−13762.6	−0.00819984
$310$	1.08211e6	0.639541
$311$	−1.27401e6	−0.746917	−0.373459	−	0.927647i	$-0.621828\pi$
$311$	−1.27401e6	−0.746917	−0.373459	+	0.927647i	$0.621828\pi$
$312$	175894.	0.102298
$313$	−929760.	−0.536427	−0.268213	−	0.963360i	$-0.586433\pi$
$313$	−929760.	−0.536427	−0.268213	+	0.963360i	$0.586433\pi$
$314$	−1.06886e6	−0.611784
$315$	62163.2	0.0352986
$316$	−766847.	−0.432007
$317$	−1.89555e6	−1.05947	−0.529733	−	0.848165i	$-0.677708\pi$
$317$	−1.89555e6	−1.05947	−0.529733	+	0.848165i	$0.677708\pi$
$318$	622680.	0.345300
$319$	29589.6	0.0162803
$320$	152855.	0.0834456
$321$	89579.9	0.0485230
$322$	104611.	0.0562263
$323$	0	0
$324$	−162531.	−0.0860150
$325$	−176582.	−0.0927340
$326$	612102.	0.318992
$327$	833627.	0.431124
$328$	587252.	0.301398
$329$	83332.1	0.0424446
$330$	−160857.	−0.0813118
$331$	−2.15384e6	−1.08055	−0.540274	−	0.841489i	$-0.681680\pi$
$331$	−2.15384e6	−1.08055	−0.540274	+	0.841489i	$0.681680\pi$
$332$	238955.	0.118979
$333$	578564.	0.285918
$334$	−650228.	−0.318933
$335$	−699343.	−0.340469
$336$	63207.7	0.0305437
$337$	331452.	0.158981	0.0794906	−	0.996836i	$-0.474671\pi$
$337$	331452.	0.158981	0.0794906	+	0.996836i	$0.474671\pi$
$338$	954116.	0.454265
$339$	−1.52942e6	−0.722816
$340$	0	0
$341$	−1.53262e6	−0.713752
$342$	164502.	0.0760512
$343$	−611181.	−0.280501
$344$	1.01725e6	0.463481
$345$	−796728.	−0.360382
$346$	−972356.	−0.436652
$347$	1.20897e6	0.539006	0.269503	−	0.963000i	$-0.413141\pi$
$347$	1.20897e6	0.539006	0.269503	+	0.963000i	$0.413141\pi$
$348$	−41463.9	−0.0183536
$349$	820344.	0.360522	0.180261	−	0.983619i	$-0.442306\pi$
$349$	820344.	0.360522	0.180261	+	0.983619i	$0.442306\pi$
$350$	68093.1	0.0297121
$351$	93347.2	0.0404421
$352$	−940637.	−0.404637
$353$	2.92723e6	1.25032	0.625159	−	0.780497i	$-0.285034\pi$
$353$	2.92723e6	1.25032	0.625159	+	0.780497i	$0.285034\pi$
$354$	523104.	0.221861
$355$	601540.	0.253334
$356$	−105945.	−0.0443051
$357$	0	0
$358$	−1.51843e6	−0.626164
$359$	−972924.	−0.398421	−0.199211	−	0.979957i	$-0.563838\pi$
$359$	−972924.	−0.398421	−0.199211	+	0.979957i	$0.563838\pi$
$360$	516582.	0.210079
$361$	−1.90544e6	−0.769534
$362$	1.97995e6	0.794113
$363$	−1.22164e6	−0.486603
$364$	−58260.0	−0.0230471
$365$	3.19393e6	1.25485
$366$	−5958.21	−0.00232495
$367$	−4.59535e6	−1.78096	−0.890478	−	0.455026i	$-0.849630\pi$
$367$	−4.59535e6	−1.78096	−0.890478	+	0.455026i	$0.849630\pi$
$368$	−810116.	−0.311837
$369$	311654.	0.119154
$370$	−802388.	−0.304705
$371$	−472664.	−0.178286
$372$	2.14765e6	0.804649
$373$	517418.	0.192562	0.0962808	−	0.995354i	$-0.469305\pi$
$373$	517418.	0.192562	0.0962808	+	0.995354i	$0.469305\pi$
$374$	0	0
$375$	−1.69380e6	−0.621991
$376$	692498.	0.252609
$377$	23814.1	0.00862940
$378$	−35996.2	−0.0129577
$379$	1.50042e6	0.536555	0.268278	−	0.963342i	$-0.413546\pi$
$379$	1.50042e6	0.536555	0.268278	+	0.963342i	$0.413546\pi$
$380$	781938.	0.277788
$381$	1.61587e6	0.570287
$382$	276489.	0.0969437
$383$	−3.94372e6	−1.37376	−0.686878	−	0.726773i	$-0.741019\pi$
$383$	−3.94372e6	−1.37376	−0.686878	+	0.726773i	$0.741019\pi$
$384$	1.61418e6	0.558630
$385$	122103.	0.0419831
$386$	−1.58772e6	−0.542381
$387$	539855.	0.183231
$388$	−383084.	−0.129186
$389$	310454.	0.104022	0.0520108	−	0.998647i	$-0.483437\pi$
$389$	310454.	0.104022	0.0520108	+	0.998647i	$0.483437\pi$
$390$	−129459.	−0.0430995
$391$	0	0
$392$	−2.51374e6	−0.826239
$393$	−300188.	−0.0980418
$394$	2.55141e6	0.828018
$395$	1.29348e6	0.417126
$396$	−319249.	−0.102304
$397$	−695416.	−0.221446	−0.110723	−	0.993851i	$-0.535317\pi$
$397$	−695416.	−0.221446	−0.110723	+	0.993851i	$0.535317\pi$
$398$	−2.96116e6	−0.937032
$399$	−124870.	−0.0392670
$400$	−527317.	−0.164786
$401$	−4.73941e6	−1.47185	−0.735924	−	0.677064i	$-0.763252\pi$
$401$	−4.73941e6	−1.47185	−0.735924	+	0.677064i	$0.763252\pi$
$402$	404962.	0.124982
$403$	−1.23347e6	−0.378326
$404$	4.19085e6	1.27747
$405$	274150.	0.0830521
$406$	−9183.11	−0.00276487
$407$	1.13644e6	0.340063
$408$	0	0
$409$	5.35820e6	1.58384	0.791918	−	0.610627i	$-0.209083\pi$
$409$	5.35820e6	1.58384	0.791918	+	0.610627i	$0.209083\pi$
$410$	−432221.	−0.126983
$411$	−868694.	−0.253666
$412$	37881.4	0.0109947
$413$	−397078.	−0.114552
$414$	461354.	0.132292
$415$	−403058.	−0.114881
$416$	−757037.	−0.214479
$417$	−2.81294e6	−0.792173
$418$	323121.	0.0904533
$419$	−4.81531e6	−1.33995	−0.669976	−	0.742383i	$-0.733696\pi$
$419$	−4.81531e6	−1.33995	−0.669976	+	0.742383i	$0.733696\pi$
$420$	−171103.	−0.0473298
$421$	−3.60145e6	−0.990312	−0.495156	−	0.868804i	$-0.664889\pi$
$421$	−3.60145e6	−0.990312	−0.495156	+	0.868804i	$0.664889\pi$
$422$	1.80262e6	0.492747
$423$	367509.	0.0998658
$424$	−3.92789e6	−1.06107
$425$	0	0
$426$	−348328.	−0.0929960
$427$	4522.76	0.00120042
$428$	−246567.	−0.0650617
$429$	183356.	0.0481007
$430$	−748703.	−0.195271
$431$	1.29522e6	0.335855	0.167928	−	0.985799i	$-0.446293\pi$
$431$	1.29522e6	0.335855	0.167928	+	0.985799i	$0.446293\pi$
$432$	278757.	0.0718647
$433$	−3.63297e6	−0.931198	−0.465599	−	0.884996i	$-0.654161\pi$
$433$	−3.63297e6	−0.931198	−0.465599	+	0.884996i	$0.654161\pi$
$434$	475646.	0.121216
$435$	69939.3	0.0177214
$436$	−2.29454e6	−0.578069
$437$	1.60043e6	0.400897
$438$	−1.84947e6	−0.460642
$439$	−1.48503e6	−0.367769	−0.183884	−	0.982948i	$-0.558867\pi$
$439$	−1.48503e6	−0.367769	−0.183884	+	0.982948i	$0.558867\pi$
$440$	1.01469e6	0.249863
$441$	−1.33404e6	−0.326643
$442$	0	0
$443$	5.66139e6	1.37061	0.685305	−	0.728256i	$-0.259669\pi$
$443$	5.66139e6	1.37061	0.685305	+	0.728256i	$0.259669\pi$
$444$	−1.59249e6	−0.383370
$445$	178702.	0.0427790
$446$	3.62113e6	0.861999
$447$	−2.27149e6	−0.537703
$448$	67187.7	0.0158159
$449$	−2.22865e6	−0.521705	−0.260853	−	0.965379i	$-0.584004\pi$
$449$	−2.22865e6	−0.521705	−0.260853	+	0.965379i	$0.584004\pi$
$450$	300302.	0.0699080
$451$	612163.	0.141718
$452$	4.20970e6	0.969181
$453$	1.93235e6	0.442425
$454$	1.31137e6	0.298597
$455$	98270.2	0.0222532
$456$	−1.03769e6	−0.233697
$457$	−3.13715e6	−0.702660	−0.351330	−	0.936252i	$-0.614270\pi$
$457$	−3.13715e6	−0.702660	−0.351330	+	0.936252i	$0.614270\pi$
$458$	3.09668e6	0.689815
$459$	0	0
$460$	2.19298e6	0.483214
$461$	−2.99266e6	−0.655850	−0.327925	−	0.944704i	$-0.606349\pi$
$461$	−2.99266e6	−0.655850	−0.327925	+	0.944704i	$0.606349\pi$
$462$	−70705.0	−0.0154115
$463$	−7.61132e6	−1.65009	−0.825044	−	0.565068i	$-0.808850\pi$
$463$	−7.61132e6	−1.65009	−0.825044	+	0.565068i	$0.808850\pi$
$464$	71114.5	0.0153343
$465$	−3.62256e6	−0.776933
$466$	−2.09085e6	−0.446024
$467$	3.18537e6	0.675878	0.337939	−	0.941168i	$-0.390270\pi$
$467$	3.18537e6	0.675878	0.337939	+	0.941168i	$0.390270\pi$
$468$	−256936.	−0.0542264
$469$	−307399.	−0.0645312
$470$	−509683.	−0.106428
$471$	3.57820e6	0.743212
$472$	−3.29976e6	−0.681753
$473$	1.06040e6	0.217930
$474$	−749003.	−0.153122
$475$	1.04174e6	0.211849
$476$	0	0
$477$	−2.08453e6	−0.419481
$478$	3.18482e6	0.637552
$479$	7.92998e6	1.57919	0.789593	−	0.613630i	$-0.210292\pi$
$479$	7.92998e6	1.57919	0.789593	+	0.613630i	$0.210292\pi$
$480$	−2.22333e6	−0.440455
$481$	914619.	0.180251
$482$	2.69275e6	0.527933
$483$	−350205.	−0.0683052
$484$	3.36253e6	0.652458
$485$	646169.	0.124736
$486$	−158749.	−0.0304875
$487$	6.23621e6	1.19151	0.595756	−	0.803165i	$-0.296852\pi$
$487$	6.23621e6	1.19151	0.595756	+	0.803165i	$0.296852\pi$
$488$	37584.6	0.00714431
$489$	−2.04912e6	−0.387521
$490$	1.85013e6	0.348107
$491$	1.50441e6	0.281618	0.140809	−	0.990037i	$-0.455030\pi$
$491$	1.50441e6	0.281618	0.140809	+	0.990037i	$0.455030\pi$
$492$	−857823.	−0.159766
$493$	0	0
$494$	260052.	0.0479449
$495$	538495.	0.0987799
$496$	−3.68343e6	−0.672277
$497$	264409.	0.0480159
$498$	233395.	0.0421713
$499$	−2.76877e6	−0.497778	−0.248889	−	0.968532i	$-0.580065\pi$
$499$	−2.76877e6	−0.497778	−0.248889	+	0.968532i	$0.580065\pi$
$500$	4.66215e6	0.833991
$501$	2.17675e6	0.387449
$502$	−3.18102e6	−0.563388
$503$	4.74015e6	0.835356	0.417678	−	0.908595i	$-0.362844\pi$
$503$	4.74015e6	0.835356	0.417678	+	0.908595i	$0.362844\pi$
$504$	227065.	0.0398176
$505$	−7.06894e6	−1.23346
$506$	906207.	0.157344
$507$	−3.19407e6	−0.551854
$508$	−4.44764e6	−0.764664
$509$	9.23651e6	1.58021	0.790103	−	0.612974i	$-0.210027\pi$
$509$	9.23651e6	1.58021	0.790103	+	0.612974i	$0.210027\pi$
$510$	0	0
$511$	1.40390e6	0.237840
$512$	−4.12829e6	−0.695978
$513$	−550699.	−0.0923892
$514$	3.44405e6	0.574992
$515$	−63896.6	−0.0106160
$516$	−1.48594e6	−0.245684
$517$	721873.	0.118778
$518$	−352692.	−0.0577526
$519$	3.25513e6	0.530457
$520$	816635.	0.132440
$521$	−1.83570e6	−0.296284	−0.148142	−	0.988966i	$-0.547329\pi$
$521$	−1.83570e6	−0.296284	−0.148142	+	0.988966i	$0.547329\pi$
$522$	−40499.1	−0.00650532
$523$	−4.99319e6	−0.798223	−0.399111	−	0.916902i	$-0.630681\pi$
$523$	−4.99319e6	−0.798223	−0.399111	+	0.916902i	$0.630681\pi$
$524$	826261.	0.131459
$525$	−227954.	−0.0360951
$526$	1.18897e6	0.187373
$527$	0	0
$528$	547543.	0.0854739
$529$	−1.94787e6	−0.302636
$530$	2.89095e6	0.447045
$531$	−1.75118e6	−0.269522
$532$	343703.	0.0526508
$533$	492677.	0.0751180
$534$	−103479.	−0.0157037
$535$	415898.	0.0628206
$536$	−2.55451e6	−0.384057
$537$	5.08322e6	0.760682
$538$	5.59666e6	0.833629
$539$	−2.62037e6	−0.388500
$540$	−754593.	−0.111360
$541$	921698.	0.135393	0.0676964	−	0.997706i	$-0.478435\pi$
$541$	921698.	0.135393	0.0676964	+	0.997706i	$0.478435\pi$
$542$	−2.12814e6	−0.311173
$543$	−6.62822e6	−0.964711
$544$	0	0
$545$	3.87033e6	0.558157
$546$	−56904.4	−0.00816891
$547$	−6.77714e6	−0.968453	−0.484226	−	0.874943i	$-0.660899\pi$
$547$	−6.77714e6	−0.968453	−0.484226	+	0.874943i	$0.660899\pi$
$548$	2.39106e6	0.340126
$549$	19946.1	0.00282441
$550$	589864.	0.0831467
$551$	−140491.	−0.0197137
$552$	−2.91023e6	−0.406518
$553$	568554.	0.0790604
$554$	533784.	0.0738909
$555$	2.68613e6	0.370165
$556$	7.74256e6	1.06218
$557$	5.14680e6	0.702909	0.351455	−	0.936205i	$-0.385687\pi$
$557$	5.14680e6	0.702909	0.351455	+	0.936205i	$0.385687\pi$
$558$	2.09768e6	0.285203
$559$	853425.	0.115514
$560$	293458.	0.0395436
$561$	0	0
$562$	−6.62394e6	−0.884659
$563$	1.28370e7	1.70684	0.853421	−	0.521222i	$-0.174524\pi$
$563$	1.28370e7	1.70684	0.853421	+	0.521222i	$0.174524\pi$
$564$	−1.01156e6	−0.133904
$565$	−7.10073e6	−0.935797
$566$	301819.	0.0396010
$567$	120504.	0.0157414
$568$	2.19727e6	0.285767
$569$	424479.	0.0549636	0.0274818	−	0.999622i	$-0.491251\pi$
$569$	424479.	0.0549636	0.0274818	+	0.999622i	$0.491251\pi$
$570$	763743.	0.0984601
$571$	1.38150e7	1.77321	0.886607	−	0.462524i	$-0.153056\pi$
$571$	1.38150e7	1.77321	0.886607	+	0.462524i	$0.153056\pi$
$572$	−504683.	−0.0644954
$573$	−925595.	−0.117770
$574$	−189984.	−0.0240679
$575$	2.92162e6	0.368514
$576$	296309.	0.0372125
$577$	−1.09950e6	−0.137486	−0.0687428	−	0.997634i	$-0.521899\pi$
$577$	−1.09950e6	−0.137486	−0.0687428	+	0.997634i	$0.521899\pi$
$578$	0	0
$579$	5.31515e6	0.658900
$580$	−192507.	−0.0237616
$581$	−177165.	−0.0217740
$582$	−374171.	−0.0457891
$583$	−4.09450e6	−0.498919
$584$	1.16666e7	1.41550
$585$	433388.	0.0523585
$586$	−2.83140e6	−0.340609
$587$	1.19579e7	1.43238	0.716191	−	0.697905i	$-0.245884\pi$
$587$	1.19579e7	1.43238	0.716191	+	0.697905i	$0.245884\pi$
$588$	3.67193e6	0.437976
$589$	7.27683e6	0.864279
$590$	2.42864e6	0.287233
$591$	−8.54129e6	−1.00590
$592$	2.73127e6	0.320302
$593$	−9.26740e6	−1.08223	−0.541117	−	0.840947i	$-0.681998\pi$
$593$	−9.26740e6	−1.08223	−0.541117	+	0.840947i	$0.681998\pi$
$594$	−311821.	−0.0362610
$595$	0	0
$596$	6.25224e6	0.720975
$597$	9.91300e6	1.13833
$598$	729327.	0.0834007
$599$	−4.05157e6	−0.461378	−0.230689	−	0.973028i	$-0.574098\pi$
$599$	−4.05157e6	−0.461378	−0.230689	+	0.973028i	$0.574098\pi$
$600$	−1.89432e6	−0.214820
$601$	710778.	0.0802690	0.0401345	−	0.999194i	$-0.487221\pi$
$601$	710778.	0.0802690	0.0401345	+	0.999194i	$0.487221\pi$
$602$	−329095.	−0.0370110
$603$	−1.35568e6	−0.151832
$604$	−5.31875e6	−0.593222
$605$	−5.67175e6	−0.629983
$606$	4.09334e6	0.452789
$607$	7.75127e6	0.853888	0.426944	−	0.904278i	$-0.359590\pi$
$607$	7.75127e6	0.853888	0.426944	+	0.904278i	$0.359590\pi$
$608$	4.46612e6	0.489973
$609$	30742.1	0.00335884
$610$	−27662.5	−0.00301000
$611$	580973.	0.0629583
$612$	0	0
$613$	1.10280e7	1.18535	0.592674	−	0.805443i	$-0.298072\pi$
$613$	1.10280e7	1.18535	0.592674	+	0.805443i	$0.298072\pi$
$614$	7.93041e6	0.848936
$615$	1.44694e6	0.154263
$616$	446010.	0.0473579
$617$	1.07836e7	1.14039	0.570193	−	0.821511i	$-0.306868\pi$
$617$	1.07836e7	1.14039	0.570193	+	0.821511i	$0.306868\pi$
$618$	37000.0	0.00389700
$619$	9.18874e6	0.963895	0.481947	−	0.876200i	$-0.339930\pi$
$619$	9.18874e6	0.963895	0.481947	+	0.876200i	$0.339930\pi$
$620$	9.97103e6	1.04174
$621$	−1.54446e6	−0.160712
$622$	3.42510e6	0.354974
$623$	78549.2	0.00810815
$624$	440670.	0.0453056
$625$	−3.55443e6	−0.363974
$626$	2.49960e6	0.254938
$627$	−1.08170e6	−0.109885
$628$	−9.84894e6	−0.996530
$629$	0	0
$630$	−167122.	−0.0167757
$631$	2.53270e6	0.253227	0.126614	−	0.991952i	$-0.459589\pi$
$631$	2.53270e6	0.253227	0.126614	+	0.991952i	$0.459589\pi$
$632$	4.72474e6	0.470528
$633$	−6.03459e6	−0.598603
$634$	5.09606e6	0.503514
$635$	7.50208e6	0.738325
$636$	5.73762e6	0.562457
$637$	−2.10891e6	−0.205925
$638$	−79549.7	−0.00773725
$639$	1.16609e6	0.112974
$640$	7.49425e6	0.723233
$641$	−1.68624e7	−1.62097	−0.810484	−	0.585761i	$-0.800796\pi$
$641$	−1.68624e7	−1.62097	−0.810484	+	0.585761i	$0.800796\pi$
$642$	−240830.	−0.0230607
$643$	1.65867e7	1.58210	0.791050	−	0.611752i	$-0.209535\pi$
$643$	1.65867e7	1.58210	0.791050	+	0.611752i	$0.209535\pi$
$644$	963931.	0.0915865
$645$	2.50641e6	0.237221
$646$	0	0
$647$	1.41124e7	1.32538	0.662691	−	0.748893i	$-0.269414\pi$
$647$	1.41124e7	1.32538	0.662691	+	0.748893i	$0.269414\pi$
$648$	1.00140e6	0.0936847
$649$	−3.43973e6	−0.320563
$650$	474730.	0.0440721
$651$	−1.59231e6	−0.147257
$652$	5.64016e6	0.519604
$653$	1.28463e7	1.17895	0.589474	−	0.807787i	$-0.299335\pi$
$653$	1.28463e7	1.17895	0.589474	+	0.807787i	$0.299335\pi$
$654$	−2.24115e6	−0.204893
$655$	−1.39370e6	−0.126930
$656$	1.47125e6	0.133483
$657$	6.19144e6	0.559600
$658$	−224033.	−0.0201719
$659$	1.98714e7	1.78244	0.891222	−	0.453568i	$-0.149849\pi$
$659$	1.98714e7	1.78244	0.891222	+	0.453568i	$0.149849\pi$
$660$	−1.48220e6	−0.132448
$661$	−6.96341e6	−0.619895	−0.309948	−	0.950754i	$-0.600312\pi$
$661$	−6.96341e6	−0.619895	−0.309948	+	0.950754i	$0.600312\pi$
$662$	5.79047e6	0.513534
$663$	0	0
$664$	−1.47226e6	−0.129588
$665$	−579743.	−0.0508372
$666$	−1.55543e6	−0.135883
$667$	−394012.	−0.0342922
$668$	−5.99146e6	−0.519507
$669$	−1.21224e7	−1.04718
$670$	1.88014e6	0.161809
$671$	39178.9	0.00335928
$672$	−977274.	−0.0834820
$673$	1.72849e7	1.47105	0.735527	−	0.677496i	$-0.236935\pi$
$673$	1.72849e7	1.47105	0.735527	+	0.677496i	$0.236935\pi$
$674$	−891087.	−0.0755562
$675$	−1.00531e6	−0.0849263
$676$	8.79161e6	0.739949
$677$	1.57798e7	1.32322	0.661609	−	0.749849i	$-0.269874\pi$
$677$	1.57798e7	1.32322	0.661609	+	0.749849i	$0.269874\pi$
$678$	4.11175e6	0.343520
$679$	284026.	0.0236420
$680$	0	0
$681$	−4.39004e6	−0.362744
$682$	4.12034e6	0.339212
$683$	7.77722e6	0.637929	0.318965	−	0.947767i	$-0.396665\pi$
$683$	7.77722e6	0.637929	0.318965	+	0.947767i	$0.396665\pi$
$684$	1.51579e6	0.123879
$685$	−4.03313e6	−0.328410
$686$	1.64312e6	0.133309
$687$	−1.03667e7	−0.838007
$688$	2.54853e6	0.205267
$689$	−3.29531e6	−0.264453
$690$	2.14195e6	0.171272
$691$	−6.90076e6	−0.549796	−0.274898	−	0.961473i	$-0.588644\pi$
$691$	−6.90076e6	−0.549796	−0.274898	+	0.961473i	$0.588644\pi$
$692$	−8.95968e6	−0.711258
$693$	236697.	0.0187223
$694$	−3.25025e6	−0.256164
$695$	−1.30598e7	−1.02559
$696$	255470.	0.0199901
$697$	0	0
$698$	−2.20544e6	−0.171339
$699$	6.99949e6	0.541843
$700$	627438.	0.0483978
$701$	−9.42864e6	−0.724693	−0.362347	−	0.932043i	$-0.618024\pi$
$701$	−9.42864e6	−0.724693	−0.362347	+	0.932043i	$0.618024\pi$
$702$	−250958.	−0.0192202
$703$	−5.39577e6	−0.411780
$704$	582021.	0.0442595
$705$	1.70625e6	0.129292
$706$	−7.86968e6	−0.594217
$707$	−3.10718e6	−0.233785
$708$	4.82009e6	0.361387
$709$	4.09959e6	0.306284	0.153142	−	0.988204i	$-0.451061\pi$
$709$	4.09959e6	0.306284	0.153142	+	0.988204i	$0.451061\pi$
$710$	−1.61720e6	−0.120398
$711$	2.50742e6	0.186017
$712$	652752.	0.0482556
$713$	2.04082e7	1.50342
$714$	0	0
$715$	851276.	0.0622738
$716$	−1.39915e7	−1.01995
$717$	−1.06617e7	−0.774516
$718$	2.61564e6	0.189351
$719$	−1.56754e7	−1.13083	−0.565413	−	0.824808i	$-0.691283\pi$
$719$	−1.56754e7	−1.13083	−0.565413	+	0.824808i	$0.691283\pi$
$720$	1.29420e6	0.0930400
$721$	−28086.0	−0.00201211
$722$	5.12266e6	0.365723
$723$	−9.01446e6	−0.641349
$724$	1.82440e7	1.29352
$725$	−256469.	−0.0181213
$726$	3.28429e6	0.231259
$727$	−8.73453e6	−0.612920	−0.306460	−	0.951884i	$-0.599145\pi$
$727$	−8.73453e6	−0.612920	−0.306460	+	0.951884i	$0.599145\pi$
$728$	358955.	0.0251022
$729$	531441.	0.0370370
$730$	−8.58666e6	−0.596372
$731$	0	0
$732$	−54901.3	−0.00378709
$733$	2.10160e7	1.44474	0.722372	−	0.691505i	$-0.243052\pi$
$733$	2.10160e7	1.44474	0.722372	+	0.691505i	$0.243052\pi$
$734$	1.23543e7	0.846404
$735$	−6.19364e6	−0.422890
$736$	1.25254e7	0.852312
$737$	−2.66287e6	−0.180585
$738$	−837863.	−0.0566281
$739$	5.08712e6	0.342658	0.171329	−	0.985214i	$-0.445194\pi$
$739$	5.08712e6	0.342658	0.171329	+	0.985214i	$0.445194\pi$
$740$	−7.39352e6	−0.496332
$741$	−870569.	−0.0582449
$742$	1.27073e6	0.0847310
$743$	−2.24865e7	−1.49434	−0.747171	−	0.664632i	$-0.768588\pi$
$743$	−2.24865e7	−1.49434	−0.747171	+	0.664632i	$0.768588\pi$
$744$	−1.32322e7	−0.876397
$745$	−1.05460e7	−0.696140
$746$	−1.39104e6	−0.0915154
$747$	−781329.	−0.0512309
$748$	0	0
$749$	182809.	0.0119068
$750$	4.55367e6	0.295603
$751$	4.49261e6	0.290669	0.145335	−	0.989383i	$-0.453574\pi$
$751$	4.49261e6	0.290669	0.145335	+	0.989383i	$0.453574\pi$
$752$	1.73492e6	0.111876
$753$	1.06490e7	0.684419
$754$	−64022.6	−0.00410114
$755$	8.97142e6	0.572788
$756$	−331684.	−0.0211067
$757$	−1.06659e7	−0.676486	−0.338243	−	0.941059i	$-0.609833\pi$
$757$	−1.06659e7	−0.676486	−0.338243	+	0.941059i	$0.609833\pi$
$758$	−4.03378e6	−0.254999
$759$	−3.03368e6	−0.191146
$760$	−4.81772e6	−0.302557
$761$	8.53092e6	0.533992	0.266996	−	0.963698i	$-0.413969\pi$
$761$	8.53092e6	0.533992	0.266996	+	0.963698i	$0.413969\pi$
$762$	−4.34415e6	−0.271030
$763$	1.70122e6	0.105791
$764$	2.54768e6	0.157911
$765$	0	0
$766$	1.06024e7	0.652881
$767$	−2.76834e6	−0.169915
$768$	−5.39316e6	−0.329944
$769$	−4.58683e6	−0.279703	−0.139851	−	0.990172i	$-0.544663\pi$
$769$	−4.58683e6	−0.279703	−0.139851	+	0.990172i	$0.544663\pi$
$770$	−328266.	−0.0199526
$771$	−1.15296e7	−0.698517
$772$	−1.46298e7	−0.883480
$773$	1.33452e7	0.803295	0.401647	−	0.915794i	$-0.368438\pi$
$773$	1.33452e7	0.803295	0.401647	+	0.915794i	$0.368438\pi$
$774$	−1.45136e6	−0.0870811
$775$	1.32840e7	0.794464
$776$	2.36028e6	0.140705
$777$	1.18070e6	0.0701595
$778$	−834635.	−0.0494365
$779$	−2.90653e6	−0.171606
$780$	−1.19289e6	−0.0702044
$781$	2.29047e6	0.134368
$782$	0	0
$783$	135578.	0.00790285
$784$	−6.29771e6	−0.365925
$785$	1.66127e7	0.962203
$786$	807035.	0.0465946
$787$	−3.00879e7	−1.73163	−0.865816	−	0.500363i	$-0.833200\pi$
$787$	−3.00879e7	−1.73163	−0.865816	+	0.500363i	$0.833200\pi$
$788$	2.35097e7	1.34875
$789$	−3.98028e6	−0.227626
$790$	−3.47744e6	−0.198240
$791$	−3.12115e6	−0.177367
$792$	1.96698e6	0.111426
$793$	31531.7	0.00178059
$794$	1.86958e6	0.105243
$795$	−9.67796e6	−0.543083
$796$	−2.72853e7	−1.52632
$797$	−1.53800e7	−0.857654	−0.428827	−	0.903387i	$-0.641073\pi$
$797$	−1.53800e7	−0.857654	−0.428827	+	0.903387i	$0.641073\pi$
$798$	335706.	0.0186617
$799$	0	0
$800$	8.15300e6	0.450394
$801$	346415.	0.0190772
$802$	1.27416e7	0.699499
$803$	1.21614e7	0.665573
$804$	3.73148e6	0.203583
$805$	−1.62591e6	−0.0884317
$806$	3.31610e6	0.179800
$807$	−1.87358e7	−1.01272
$808$	−2.58209e7	−1.39137
$809$	−2.36710e7	−1.27158	−0.635791	−	0.771861i	$-0.719326\pi$
$809$	−2.36710e7	−1.27158	−0.635791	+	0.771861i	$0.719326\pi$
$810$	−737034.	−0.0394707
$811$	−3.61567e6	−0.193035	−0.0965176	−	0.995331i	$-0.530770\pi$
$811$	−3.61567e6	−0.193035	−0.0965176	+	0.995331i	$0.530770\pi$
$812$	−84616.9	−0.00450368
$813$	7.12431e6	0.378021
$814$	−3.05523e6	−0.161616
$815$	−9.51356e6	−0.501706
$816$	0	0
$817$	−5.03476e6	−0.263891
$818$	−1.44052e7	−0.752722
$819$	190497.	0.00992382
$820$	−3.98266e6	−0.206842
$821$	−1.82213e7	−0.943457	−0.471729	−	0.881744i	$-0.656370\pi$
$821$	−1.82213e7	−0.943457	−0.471729	+	0.881744i	$0.656370\pi$
$822$	2.33543e6	0.120555
$823$	2.97111e6	0.152904	0.0764522	−	0.997073i	$-0.475641\pi$
$823$	2.97111e6	0.152904	0.0764522	+	0.997073i	$0.475641\pi$
$824$	−233397.	−0.0119751
$825$	−1.97467e6	−0.101009
$826$	1.06752e6	0.0544409
$827$	−3.24683e7	−1.65081	−0.825403	−	0.564544i	$-0.809052\pi$
$827$	−3.24683e7	−1.65081	−0.825403	+	0.564544i	$0.809052\pi$
$828$	4.25110e6	0.215489
$829$	−2.88453e7	−1.45777	−0.728885	−	0.684636i	$-0.759961\pi$
$829$	−2.88453e7	−1.45777	−0.728885	+	0.684636i	$0.759961\pi$
$830$	1.08359e6	0.0545973
$831$	−1.78693e6	−0.0897647
$832$	468418.	0.0234598
$833$	0	0
$834$	7.56240e6	0.376482
$835$	1.01061e7	0.501612
$836$	2.97737e6	0.147339
$837$	−7.02235e6	−0.346472
$838$	1.29457e7	0.636816
$839$	−1.43021e7	−0.701446	−0.350723	−	0.936479i	$-0.614064\pi$
$839$	−1.43021e7	−0.701446	−0.350723	+	0.936479i	$0.614064\pi$
$840$	1.05421e6	0.0515500
$841$	−2.04766e7	−0.998314
$842$	9.68226e6	0.470648
$843$	2.21748e7	1.07471
$844$	1.66101e7	0.802631
$845$	−1.48293e7	−0.714461
$846$	−988023.	−0.0474615
$847$	−2.49304e6	−0.119404
$848$	−9.84057e6	−0.469927
$849$	−1.01039e6	−0.0481084
$850$	0	0
$851$	−1.51327e7	−0.716295
$852$	−3.20963e6	−0.151480
$853$	−6.56414e6	−0.308891	−0.154445	−	0.988001i	$-0.549359\pi$
$853$	−6.56414e6	−0.308891	−0.154445	+	0.988001i	$0.549359\pi$
$854$	−12159.1	−0.000570504 0
$855$	−2.55676e6	−0.119612
$856$	1.51916e6	0.0708630
$857$	−1.44754e7	−0.673256	−0.336628	−	0.941638i	$-0.609286\pi$
$857$	−1.44754e7	−0.673256	−0.336628	+	0.941638i	$0.609286\pi$
$858$	−492940.	−0.0228600
$859$	−2.01584e7	−0.932122	−0.466061	−	0.884753i	$-0.654327\pi$
$859$	−2.01584e7	−0.932122	−0.466061	+	0.884753i	$0.654327\pi$
$860$	−6.89886e6	−0.318076
$861$	636005.	0.0292384
$862$	−3.48213e6	−0.159616
$863$	3.07900e7	1.40729	0.703643	−	0.710553i	$-0.251555\pi$
$863$	3.07900e7	1.40729	0.703643	+	0.710553i	$0.251555\pi$
$864$	−4.30994e6	−0.196421
$865$	1.51128e7	0.686758
$866$	9.76700e6	0.442554
$867$	0	0
$868$	4.38280e6	0.197448
$869$	4.92516e6	0.221244
$870$	−188027.	−0.00842214
$871$	−2.14311e6	−0.0957194
$872$	1.41373e7	0.629614
$873$	1.25260e6	0.0556259
$874$	−4.30265e6	−0.190527
$875$	−3.45660e6	−0.152626
$876$	−1.70418e7	−0.750335
$877$	−1.67793e7	−0.736673	−0.368337	−	0.929693i	$-0.620073\pi$
$877$	−1.67793e7	−0.736673	−0.368337	+	0.929693i	$0.620073\pi$
$878$	3.99241e6	0.174783
$879$	9.47859e6	0.413782
$880$	2.54211e6	0.110659
$881$	7.61395e6	0.330499	0.165250	−	0.986252i	$-0.447157\pi$
$881$	7.61395e6	0.330499	0.165250	+	0.986252i	$0.447157\pi$
$882$	3.58649e6	0.155238
$883$	3.00177e7	1.29561	0.647806	−	0.761805i	$-0.275687\pi$
$883$	3.00177e7	1.29561	0.647806	+	0.761805i	$0.275687\pi$
$884$	0	0
$885$	−8.13031e6	−0.348938
$886$	−1.52203e7	−0.651386
$887$	3.73087e7	1.59221	0.796107	−	0.605157i	$-0.206889\pi$
$887$	3.73087e7	1.59221	0.796107	+	0.605157i	$0.206889\pi$
$888$	9.81172e6	0.417554
$889$	3.29756e6	0.139939
$890$	−480430.	−0.0203308
$891$	1.04387e6	0.0440508
$892$	3.33666e7	1.40410
$893$	−3.42744e6	−0.143827
$894$	6.10676e6	0.255545
$895$	2.36002e7	0.984821
$896$	3.29412e6	0.137079
$897$	−2.44155e6	−0.101317
$898$	5.99157e6	0.247942
$899$	−1.79149e6	−0.0739292
$900$	2.76710e6	0.113873
$901$	0	0
$902$	−1.64576e6	−0.0673519
$903$	1.10170e6	0.0449620
$904$	−2.59370e7	−1.05560
$905$	−3.07732e7	−1.24897
$906$	−5.19499e6	−0.210264
$907$	−3.76377e7	−1.51917	−0.759583	−	0.650410i	$-0.774597\pi$
$907$	−3.76377e7	−1.51917	−0.759583	+	0.650410i	$0.774597\pi$
$908$	1.20835e7	0.486382
$909$	−1.37032e7	−0.550061
$910$	−264193.	−0.0105759
$911$	−3.72829e7	−1.48838	−0.744189	−	0.667969i	$-0.767164\pi$
$911$	−3.72829e7	−1.48838	−0.744189	+	0.667969i	$0.767164\pi$
$912$	−2.59972e6	−0.103500
$913$	−1.53471e6	−0.0609327
$914$	8.43403e6	0.333941
$915$	92605.0	0.00365664
$916$	2.85341e7	1.12363
$917$	−612605.	−0.0240579
$918$	0	0
$919$	−4.15385e7	−1.62242	−0.811208	−	0.584758i	$-0.801189\pi$
$919$	−4.15385e7	−1.62242	−0.811208	+	0.584758i	$0.801189\pi$
$920$	−1.35115e7	−0.526301
$921$	−2.65484e7	−1.03131
$922$	8.04556e6	0.311694
$923$	1.84340e6	0.0712223
$924$	−651505.	−0.0251037
$925$	−9.85009e6	−0.378518
$926$	2.04625e7	0.784209
$927$	−123864.	−0.00473418
$928$	−1.09952e6	−0.0419116
$929$	2.56479e6	0.0975016	0.0487508	−	0.998811i	$-0.484476\pi$
$929$	2.56479e6	0.0975016	0.0487508	+	0.998811i	$0.484476\pi$
$930$	9.73902e6	0.369239
$931$	1.24415e7	0.470433
$932$	−1.92659e7	−0.726525
$933$	−1.14661e7	−0.431233
$934$	−8.56366e6	−0.321213
$935$	0	0
$936$	1.58305e6	0.0590616
$937$	3.65092e7	1.35848	0.679239	−	0.733917i	$-0.262310\pi$
$937$	3.65092e7	1.35848	0.679239	+	0.733917i	$0.262310\pi$
$938$	826421.	0.0306686
$939$	−8.36784e6	−0.309706
$940$	−4.69643e6	−0.173360
$941$	3.96136e7	1.45838	0.729189	−	0.684313i	$-0.239898\pi$
$941$	3.96136e7	1.45838	0.729189	+	0.684313i	$0.239898\pi$
$942$	−9.61977e6	−0.353214
$943$	−8.15150e6	−0.298510
$944$	−8.26692e6	−0.301935
$945$	559469.	0.0203796
$946$	−2.85082e6	−0.103572
$947$	−1.37680e6	−0.0498879	−0.0249439	−	0.999689i	$-0.507941\pi$
$947$	−1.37680e6	−0.0498879	−0.0249439	+	0.999689i	$0.507941\pi$
$948$	−6.90162e6	−0.249419
$949$	9.78769e6	0.352789
$950$	−2.80066e6	−0.100682
$951$	−1.70599e7	−0.611683
$952$	0	0
$953$	2.27494e7	0.811405	0.405703	−	0.914005i	$-0.367027\pi$
$953$	2.27494e7	0.811405	0.405703	+	0.914005i	$0.367027\pi$
$954$	5.60412e6	0.199359
$955$	−4.29731e6	−0.152471
$956$	2.93463e7	1.03850
$957$	266306.	0.00939943
$958$	−2.13192e7	−0.750513
$959$	−1.77278e6	−0.0622455
$960$	1.37569e6	0.0481773
$961$	6.41627e7	2.24117
$962$	−2.45889e6	−0.0856647
$963$	806219.	0.0280148
$964$	2.48121e7	0.859947
$965$	2.46769e7	0.853047
$966$	941502.	0.0324622
$967$	3.71032e7	1.27598	0.637991	−	0.770044i	$-0.279766\pi$
$967$	3.71032e7	1.27598	0.637991	+	0.770044i	$0.279766\pi$
$968$	−2.07174e7	−0.710635
$969$	0	0
$970$	−1.73718e6	−0.0592811
$971$	−2.84211e7	−0.967369	−0.483685	−	0.875242i	$-0.660702\pi$
$971$	−2.84211e7	−0.967369	−0.483685	+	0.875242i	$0.660702\pi$
$972$	−1.46278e6	−0.0496608
$973$	−5.74047e6	−0.194386
$974$	−1.67657e7	−0.566269
$975$	−1.58924e6	−0.0535400
$976$	94161.0	0.00316407
$977$	−4.85304e7	−1.62659	−0.813294	−	0.581853i	$-0.802328\pi$
$977$	−4.85304e7	−1.62659	−0.813294	+	0.581853i	$0.802328\pi$
$978$	5.50892e6	0.184170
$979$	680441.	0.0226900
$980$	1.70479e7	0.567028
$981$	7.50265e6	0.248910
$982$	−4.04449e6	−0.133840
$983$	3.61937e7	1.19467	0.597336	−	0.801991i	$-0.296226\pi$
$983$	3.61937e7	1.19467	0.597336	+	0.801991i	$0.296226\pi$
$984$	5.28527e6	0.174012
$985$	−3.96551e7	−1.30229
$986$	0	0
$987$	749989.	0.0245054
$988$	2.39622e6	0.0780971
$989$	−1.41202e7	−0.459040
$990$	−1.44771e6	−0.0469454
$991$	−7.76459e6	−0.251151	−0.125575	−	0.992084i	$-0.540078\pi$
$991$	−7.76459e6	−0.251151	−0.125575	+	0.992084i	$0.540078\pi$
$992$	5.69506e7	1.83747
$993$	−1.93846e7	−0.623855
$994$	−710847.	−0.0228197
$995$	4.60236e7	1.47375
$996$	2.15059e6	0.0686926
$997$	−3.52774e7	−1.12398	−0.561989	−	0.827144i	$-0.689964\pi$
$997$	−3.52774e7	−1.12398	−0.561989	+	0.827144i	$0.689964\pi$
$998$	7.44366e6	0.236570
$999$	5.20708e6	0.165075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.6.a.u.1.12		28
17.3	odd	16		51.6.h.a.43.6	yes	56
17.6	odd	16		51.6.h.a.19.6	✓	56
17.16	even	2		867.6.a.t.1.12		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.6.h.a.19.6	✓	56	17.6	odd	16
51.6.h.a.43.6	yes	56	17.3	odd	16
867.6.a.t.1.12		28	17.16	even	2
867.6.a.u.1.12		28	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68843 q^{2} +9.00000 q^{3} -24.7723 q^{4} +41.7848 q^{5} -24.1959 q^{6} +18.3667 q^{7} +152.629 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-2.68843 q^{2} +9.00000 q^{3} -24.7723 q^{4} +41.7848 q^{5} -24.1959 q^{6} +18.3667 q^{7} +152.629 q^{8} +81.0000 q^{9} -112.336 q^{10} +159.103 q^{11} -222.951 q^{12} +128.048 q^{13} -49.3775 q^{14} +376.063 q^{15} +382.382 q^{16} -217.763 q^{18} -755.418 q^{19} -1035.11 q^{20} +165.300 q^{21} -427.738 q^{22} -2118.60 q^{23} +1373.66 q^{24} -1379.03 q^{25} -344.249 q^{26} +729.000 q^{27} -454.985 q^{28} +185.978 q^{29} -1011.02 q^{30} -9632.85 q^{31} -5912.13 q^{32} +1431.93 q^{33} +767.447 q^{35} -2006.56 q^{36} +7142.77 q^{37} +2030.89 q^{38} +1152.43 q^{39} +6377.56 q^{40} +3847.59 q^{41} -444.398 q^{42} +6664.88 q^{43} -3941.35 q^{44} +3384.57 q^{45} +5695.72 q^{46} +4537.14 q^{47} +3441.44 q^{48} -16469.7 q^{49} +3707.43 q^{50} -3172.05 q^{52} -25734.9 q^{53} -1959.87 q^{54} +6648.09 q^{55} +2803.28 q^{56} -6798.76 q^{57} -499.988 q^{58} -21619.5 q^{59} -9315.96 q^{60} +246.249 q^{61} +25897.3 q^{62} +1487.70 q^{63} +3658.14 q^{64} +5350.47 q^{65} -3849.64 q^{66} -16736.8 q^{67} -19067.4 q^{69} -2063.23 q^{70} +14396.2 q^{71} +12362.9 q^{72} +76437.5 q^{73} -19202.9 q^{74} -12411.3 q^{75} +18713.4 q^{76} +2922.19 q^{77} -3098.24 q^{78} +30955.8 q^{79} +15977.8 q^{80} +6561.00 q^{81} -10344.0 q^{82} -9646.03 q^{83} -4094.86 q^{84} -17918.1 q^{86} +1673.80 q^{87} +24283.7 q^{88} +4276.73 q^{89} -9099.19 q^{90} +2351.82 q^{91} +52482.7 q^{92} -86695.7 q^{93} -12197.8 q^{94} -31565.0 q^{95} -53209.1 q^{96} +15464.2 q^{97} +44277.6 q^{98} +12887.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9}+O(q^{10}) \) 28 * q + 252 * q^3 + 384 * q^4 - 244 * q^5 - 392 * q^7 + 2268 * q^9	\( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9} - 800 q^{10} - 1132 q^{11} + 3456 q^{12} - 828 q^{13} - 1568 q^{14} - 2196 q^{15} + 4096 q^{16} - 7532 q^{19} - 9216 q^{20} - 3528 q^{21} - 664 q^{22} - 4548 q^{23} + 9904 q^{25} + 20412 q^{27} - 36936 q^{28} - 36680 q^{29} - 7200 q^{30} - 7688 q^{31} - 10188 q^{33} + 13136 q^{35} + 31104 q^{36} - 42360 q^{37} - 1984 q^{38} - 7452 q^{39} - 81000 q^{40} - 6060 q^{41} - 14112 q^{42} - 50044 q^{43} - 56960 q^{44} - 19764 q^{45} - 51488 q^{46} + 16552 q^{47} + 36864 q^{48} + 38084 q^{49} - 48064 q^{50} - 31360 q^{52} - 4328 q^{53} - 71740 q^{55} + 9248 q^{56} - 67788 q^{57} - 95256 q^{58} - 89832 q^{59} - 82944 q^{60} - 154104 q^{61} - 106624 q^{62} - 31752 q^{63} + 93720 q^{64} - 13260 q^{65} - 5976 q^{66} - 182136 q^{67} - 40932 q^{69} - 145464 q^{70} + 24504 q^{71} - 58184 q^{73} - 308992 q^{74} + 89136 q^{75} - 602544 q^{76} - 240576 q^{77} - 136080 q^{79} - 317440 q^{80} + 183708 q^{81} - 46648 q^{82} - 203576 q^{83} - 332424 q^{84} - 421280 q^{86} - 330120 q^{87} - 145152 q^{88} - 120880 q^{89} - 64800 q^{90} - 290152 q^{91} - 491616 q^{92} - 69192 q^{93} - 799168 q^{94} - 543084 q^{95} - 409328 q^{97} + 193072 q^{98} - 91692 q^{99}+O(q^{100}) \) 28 * q + 252 * q^3 + 384 * q^4 - 244 * q^5 - 392 * q^7 + 2268 * q^9 - 800 * q^10 - 1132 * q^11 + 3456 * q^12 - 828 * q^13 - 1568 * q^14 - 2196 * q^15 + 4096 * q^16 - 7532 * q^19 - 9216 * q^20 - 3528 * q^21 - 664 * q^22 - 4548 * q^23 + 9904 * q^25 + 20412 * q^27 - 36936 * q^28 - 36680 * q^29 - 7200 * q^30 - 7688 * q^31 - 10188 * q^33 + 13136 * q^35 + 31104 * q^36 - 42360 * q^37 - 1984 * q^38 - 7452 * q^39 - 81000 * q^40 - 6060 * q^41 - 14112 * q^42 - 50044 * q^43 - 56960 * q^44 - 19764 * q^45 - 51488 * q^46 + 16552 * q^47 + 36864 * q^48 + 38084 * q^49 - 48064 * q^50 - 31360 * q^52 - 4328 * q^53 - 71740 * q^55 + 9248 * q^56 - 67788 * q^57 - 95256 * q^58 - 89832 * q^59 - 82944 * q^60 - 154104 * q^61 - 106624 * q^62 - 31752 * q^63 + 93720 * q^64 - 13260 * q^65 - 5976 * q^66 - 182136 * q^67 - 40932 * q^69 - 145464 * q^70 + 24504 * q^71 - 58184 * q^73 - 308992 * q^74 + 89136 * q^75 - 602544 * q^76 - 240576 * q^77 - 136080 * q^79 - 317440 * q^80 + 183708 * q^81 - 46648 * q^82 - 203576 * q^83 - 332424 * q^84 - 421280 * q^86 - 330120 * q^87 - 145152 * q^88 - 120880 * q^89 - 64800 * q^90 - 290152 * q^91 - 491616 * q^92 - 69192 * q^93 - 799168 * q^94 - 543084 * q^95 - 409328 * q^97 + 193072 * q^98 - 91692 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more