LMFDB - Embedded newform 867.6.a.u.1.14

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			1.14
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)

$f(q)$	$=$	$q+0.614853 q^{2} +9.00000 q^{3} -31.6220 q^{4} +24.2891 q^{5} +5.53368 q^{6} +30.9562 q^{7} -39.1182 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+0.614853 q^{2} +9.00000 q^{3} -31.6220 q^{4} +24.2891 q^{5} +5.53368 q^{6} +30.9562 q^{7} -39.1182 q^{8} +81.0000 q^{9} +14.9342 q^{10} -419.282 q^{11} -284.598 q^{12} -58.8613 q^{13} +19.0335 q^{14} +218.602 q^{15} +987.851 q^{16} +49.8031 q^{18} +1303.64 q^{19} -768.068 q^{20} +278.606 q^{21} -257.797 q^{22} -263.649 q^{23} -352.064 q^{24} -2535.04 q^{25} -36.1910 q^{26} +729.000 q^{27} -978.896 q^{28} +5854.78 q^{29} +134.408 q^{30} -2748.95 q^{31} +1859.16 q^{32} -3773.54 q^{33} +751.897 q^{35} -2561.38 q^{36} +13325.4 q^{37} +801.546 q^{38} -529.751 q^{39} -950.144 q^{40} -5427.52 q^{41} +171.302 q^{42} -9213.25 q^{43} +13258.5 q^{44} +1967.41 q^{45} -162.105 q^{46} -5141.34 q^{47} +8890.66 q^{48} -15848.7 q^{49} -1558.68 q^{50} +1861.31 q^{52} +12943.9 q^{53} +448.228 q^{54} -10184.0 q^{55} -1210.95 q^{56} +11732.7 q^{57} +3599.83 q^{58} -28866.2 q^{59} -6912.61 q^{60} -36643.4 q^{61} -1690.20 q^{62} +2507.45 q^{63} -30468.1 q^{64} -1429.68 q^{65} -2320.17 q^{66} +25220.6 q^{67} -2372.84 q^{69} +462.307 q^{70} +38755.3 q^{71} -3168.57 q^{72} -11438.5 q^{73} +8193.18 q^{74} -22815.4 q^{75} -41223.6 q^{76} -12979.4 q^{77} -325.719 q^{78} -21240.0 q^{79} +23994.0 q^{80} +6561.00 q^{81} -3337.13 q^{82} +12937.0 q^{83} -8810.06 q^{84} -5664.80 q^{86} +52693.0 q^{87} +16401.6 q^{88} -48987.0 q^{89} +1209.67 q^{90} -1822.12 q^{91} +8337.09 q^{92} -24740.6 q^{93} -3161.17 q^{94} +31664.1 q^{95} +16732.5 q^{96} +102511. q^{97} -9744.64 q^{98} -33961.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.614853	0.108692	0.0543459	−	0.998522i	$-0.482693\pi$
$2$	0.614853	0.108692	0.0543459	+	0.998522i	$0.482693\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−31.6220	−0.988186
$5$	24.2891	0.434496	0.217248	−	0.976116i	$-0.430292\pi$
$5$	24.2891	0.434496	0.217248	+	0.976116i	$0.430292\pi$
$6$	5.53368	0.0627532
$7$	30.9562	0.238783	0.119391	−	0.992847i	$-0.461906\pi$
$7$	30.9562	0.238783	0.119391	+	0.992847i	$0.461906\pi$
$8$	−39.1182	−0.216099
$9$	81.0000	0.333333
$10$	14.9342	0.0472261
$11$	−419.282	−1.04478	−0.522390	−	0.852707i	$-0.674959\pi$
$11$	−419.282	−1.04478	−0.522390	+	0.852707i	$0.674959\pi$
$12$	−284.598	−0.570530
$13$	−58.8613	−0.0965987	−0.0482993	−	0.998833i	$-0.515380\pi$
$13$	−58.8613	−0.0965987	−0.0482993	+	0.998833i	$0.515380\pi$
$14$	19.0335	0.0259537
$15$	218.602	0.250856
$16$	987.851	0.964698
$17$	0	0
$17$	0	0
$18$	49.8031	0.0362306
$19$	1303.64	0.828463	0.414231	−	0.910172i	$-0.364050\pi$
$19$	1303.64	0.828463	0.414231	+	0.910172i	$0.364050\pi$
$20$	−768.068	−0.429363
$21$	278.606	0.137861
$22$	−257.797	−0.113559
$23$	−263.649	−0.103922	−0.0519609	−	0.998649i	$-0.516547\pi$
$23$	−263.649	−0.103922	−0.0519609	+	0.998649i	$0.516547\pi$
$24$	−352.064	−0.124765
$25$	−2535.04	−0.811213
$26$	−36.1910	−0.0104995
$27$	729.000	0.192450
$28$	−978.896	−0.235962
$29$	5854.78	1.29275	0.646376	−	0.763019i	$-0.276284\pi$
$29$	5854.78	1.29275	0.646376	+	0.763019i	$0.276284\pi$
$30$	134.408	0.0272660
$31$	−2748.95	−0.513763	−0.256882	−	0.966443i	$-0.582695\pi$
$31$	−2748.95	−0.513763	−0.256882	+	0.966443i	$0.582695\pi$
$32$	1859.16	0.320954
$33$	−3773.54	−0.603204
$34$	0	0
$35$	751.897	0.103750
$36$	−2561.38	−0.329395
$37$	13325.4	1.60021	0.800104	−	0.599861i	$-0.204778\pi$
$37$	13325.4	1.60021	0.800104	+	0.599861i	$0.204778\pi$
$38$	801.546	0.0900471
$39$	−529.751	−0.0557713
$40$	−950.144	−0.0938943
$41$	−5427.52	−0.504245	−0.252123	−	0.967695i	$-0.581129\pi$
$41$	−5427.52	−0.504245	−0.252123	+	0.967695i	$0.581129\pi$
$42$	171.302	0.0149844
$43$	−9213.25	−0.759874	−0.379937	−	0.925012i	$-0.624054\pi$
$43$	−9213.25	−0.759874	−0.379937	+	0.925012i	$0.624054\pi$
$44$	13258.5	1.03244
$45$	1967.41	0.144832
$46$	−162.105	−0.0112954
$47$	−5141.34	−0.339494	−0.169747	−	0.985488i	$-0.554295\pi$
$47$	−5141.34	−0.339494	−0.169747	+	0.985488i	$0.554295\pi$
$48$	8890.66	0.556969
$49$	−15848.7	−0.942983
$50$	−1558.68	−0.0881722
$51$	0	0
$52$	1861.31	0.0954575
$53$	12943.9	0.632960	0.316480	−	0.948599i	$-0.397499\pi$
$53$	12943.9	0.632960	0.316480	+	0.948599i	$0.397499\pi$
$54$	448.228	0.0209177
$55$	−10184.0	−0.453953
$56$	−1210.95	−0.0516008
$57$	11732.7	0.478313
$58$	3599.83	0.140512
$59$	−28866.2	−1.07959	−0.539796	−	0.841796i	$-0.681499\pi$
$59$	−28866.2	−1.07959	−0.539796	+	0.841796i	$0.681499\pi$
$60$	−6912.61	−0.247893
$61$	−36643.4	−1.26087	−0.630436	−	0.776241i	$-0.717124\pi$
$61$	−36643.4	−1.26087	−0.630436	+	0.776241i	$0.717124\pi$
$62$	−1690.20	−0.0558418
$63$	2507.45	0.0795942
$64$	−30468.1	−0.929813
$65$	−1429.68	−0.0419717
$66$	−2320.17	−0.0655633
$67$	25220.6	0.686387	0.343193	−	0.939265i	$-0.388491\pi$
$67$	25220.6	0.686387	0.343193	+	0.939265i	$0.388491\pi$
$68$	0	0
$69$	−2372.84	−0.0599992
$70$	462.307	0.0112768
$71$	38755.3	0.912399	0.456199	−	0.889878i	$-0.349210\pi$
$71$	38755.3	0.912399	0.456199	+	0.889878i	$0.349210\pi$
$72$	−3168.57	−0.0720331
$73$	−11438.5	−0.251225	−0.125613	−	0.992079i	$-0.540090\pi$
$73$	−11438.5	−0.251225	−0.125613	+	0.992079i	$0.540090\pi$
$74$	8193.18	0.173929
$75$	−22815.4	−0.468354
$76$	−41223.6	−0.818675
$77$	−12979.4	−0.249475
$78$	−325.719	−0.00606188
$79$	−21240.0	−0.382902	−0.191451	−	0.981502i	$-0.561319\pi$
$79$	−21240.0	−0.382902	−0.191451	+	0.981502i	$0.561319\pi$
$80$	23994.0	0.419157
$81$	6561.00	0.111111
$82$	−3337.13	−0.0548073
$83$	12937.0	0.206128	0.103064	−	0.994675i	$-0.467135\pi$
$83$	12937.0	0.206128	0.103064	+	0.994675i	$0.467135\pi$
$84$	−8810.06	−0.136233
$85$	0	0
$86$	−5664.80	−0.0825921
$87$	52693.0	0.746371
$88$	16401.6	0.225776
$89$	−48987.0	−0.655550	−0.327775	−	0.944756i	$-0.606299\pi$
$89$	−48987.0	−0.655550	−0.327775	+	0.944756i	$0.606299\pi$
$90$	1209.67	0.0157420
$91$	−1822.12	−0.0230661
$92$	8337.09	0.102694
$93$	−24740.6	−0.296621
$94$	−3161.17	−0.0369002
$95$	31664.1	0.359964
$96$	16732.5	0.185303
$97$	102511.	1.10622	0.553109	−	0.833109i	$-0.313441\pi$
$97$	102511.	1.10622	0.553109	+	0.833109i	$0.313441\pi$
$98$	−9744.64	−0.102494
$99$	−33961.9	−0.348260
$100$	80163.0	0.801630
$101$	−21708.2	−0.211749	−0.105874	−	0.994380i	$-0.533764\pi$
$101$	−21708.2	−0.211749	−0.105874	+	0.994380i	$0.533764\pi$
$102$	0	0
$103$	120026.	1.11476	0.557382	−	0.830256i	$-0.311806\pi$
$103$	120026.	1.11476	0.557382	+	0.830256i	$0.311806\pi$
$104$	2302.55	0.0208749
$105$	6767.08	0.0599001
$106$	7958.62	0.0687976
$107$	44440.0	0.375244	0.187622	−	0.982241i	$-0.439922\pi$
$107$	44440.0	0.375244	0.187622	+	0.982241i	$0.439922\pi$
$108$	−23052.4	−0.190177
$109$	−151039.	−1.21765	−0.608826	−	0.793304i	$-0.708359\pi$
$109$	−151039.	−1.21765	−0.608826	+	0.793304i	$0.708359\pi$
$110$	−6261.65	−0.0493409
$111$	119929.	0.923881
$112$	30580.1	0.230353
$113$	24162.3	0.178009	0.0890045	−	0.996031i	$-0.471631\pi$
$113$	24162.3	0.178009	0.0890045	+	0.996031i	$0.471631\pi$
$114$	7213.92	0.0519887
$115$	−6403.78	−0.0451536
$116$	−185140.	−1.27748
$117$	−4767.76	−0.0321996
$118$	−17748.5	−0.117343
$119$	0	0
$120$	−8551.30	−0.0542099
$121$	14746.6	0.0915645
$122$	−22530.3	−0.137046
$123$	−48847.7	−0.291126
$124$	86927.3	0.507694
$125$	−137477.	−0.786965
$126$	1541.72	0.00865123
$127$	−110007.	−0.605217	−0.302608	−	0.953115i	$-0.597857\pi$
$127$	−110007.	−0.605217	−0.302608	+	0.953115i	$0.597857\pi$
$128$	−78226.7	−0.422017
$129$	−82919.3	−0.438714
$130$	−879.047	−0.00456198
$131$	−346593.	−1.76458	−0.882291	−	0.470704i	$-0.844000\pi$
$131$	−346593.	−1.76458	−0.882291	+	0.470704i	$0.844000\pi$
$132$	119327.	0.596078
$133$	40355.7	0.197822
$134$	15507.0	0.0746046
$135$	17706.7	0.0836188
$136$	0	0
$137$	−116944.	−0.532324	−0.266162	−	0.963928i	$-0.585756\pi$
$137$	−116944.	−0.532324	−0.266162	+	0.963928i	$0.585756\pi$
$138$	−1458.95	−0.00652142
$139$	−128259.	−0.563054	−0.281527	−	0.959553i	$-0.590841\pi$
$139$	−128259.	−0.563054	−0.281527	+	0.959553i	$0.590841\pi$
$140$	−23776.5	−0.102524
$141$	−46272.1	−0.196007
$142$	23828.8	0.0991702
$143$	24679.5	0.100924
$144$	80015.9	0.321566
$145$	142207.	0.561696
$146$	−7033.03	−0.0273061
$147$	−142638.	−0.544431
$148$	−421376.	−1.58130
$149$	−487374.	−1.79844	−0.899222	−	0.437492i	$-0.855867\pi$
$149$	−487374.	−1.79844	−0.899222	+	0.437492i	$0.855867\pi$
$150$	−14028.1	−0.0509062
$151$	225435.	0.804598	0.402299	−	0.915508i	$-0.368211\pi$
$151$	225435.	0.804598	0.402299	+	0.915508i	$0.368211\pi$
$152$	−50995.9	−0.179030
$153$	0	0
$154$	−7980.42	−0.0271159
$155$	−66769.5	−0.223228
$156$	16751.8	0.0551124
$157$	−240705.	−0.779357	−0.389678	−	0.920951i	$-0.627414\pi$
$157$	−240705.	−0.779357	−0.389678	+	0.920951i	$0.627414\pi$
$158$	−13059.5	−0.0416183
$159$	116495.	0.365440
$160$	45157.4	0.139453
$161$	−8161.57	−0.0248147
$162$	4034.05	0.0120769
$163$	−166722.	−0.491499	−0.245750	−	0.969333i	$-0.579034\pi$
$163$	−166722.	−0.491499	−0.245750	+	0.969333i	$0.579034\pi$
$164$	171629.	0.498288
$165$	−91655.7	−0.262090
$166$	7954.35	0.0224045
$167$	−500194.	−1.38787	−0.693933	−	0.720040i	$-0.744124\pi$
$167$	−500194.	−1.38787	−0.693933	+	0.720040i	$0.744124\pi$
$168$	−10898.6	−0.0297917
$169$	−367828.	−0.990669
$170$	0	0
$171$	105595.	0.276154
$172$	291341.	0.750897
$173$	−238308.	−0.605374	−0.302687	−	0.953090i	$-0.597884\pi$
$173$	−238308.	−0.605374	−0.302687	+	0.953090i	$0.597884\pi$
$174$	32398.5	0.0811244
$175$	−78475.3	−0.193704
$176$	−414188.	−1.00790
$177$	−259796.	−0.623303
$178$	−30119.8	−0.0712528
$179$	28577.2	0.0666633	0.0333317	−	0.999444i	$-0.489388\pi$
$179$	28577.2	0.0666633	0.0333317	+	0.999444i	$0.489388\pi$
$180$	−62213.5	−0.143121
$181$	728176.	1.65211	0.826057	−	0.563587i	$-0.190579\pi$
$181$	728176.	1.65211	0.826057	+	0.563587i	$0.190579\pi$
$182$	−1120.34	−0.00250709
$183$	−329791.	−0.727965
$184$	10313.5	0.0224574
$185$	323662.	0.695284
$186$	−15211.8	−0.0322403
$187$	0	0
$188$	162579.	0.335483
$189$	22567.1	0.0459537
$190$	19468.8	0.0391251
$191$	493169.	0.978165	0.489082	−	0.872238i	$-0.337332\pi$
$191$	493169.	0.978165	0.489082	+	0.872238i	$0.337332\pi$
$192$	−274213.	−0.536828
$193$	−610100.	−1.17898	−0.589492	−	0.807774i	$-0.700672\pi$
$193$	−610100.	−1.17898	−0.589492	+	0.807774i	$0.700672\pi$
$194$	63029.2	0.120237
$195$	−12867.2	−0.0242324
$196$	501167.	0.931843
$197$	924896.	1.69796	0.848980	−	0.528425i	$-0.177217\pi$
$197$	924896.	1.69796	0.848980	+	0.528425i	$0.177217\pi$
$198$	−20881.6	−0.0378530
$199$	−712382.	−1.27521	−0.637603	−	0.770365i	$-0.720074\pi$
$199$	−712382.	−1.27521	−0.637603	+	0.770365i	$0.720074\pi$
$200$	99166.2	0.175303
$201$	226986.	0.396286
$202$	−13347.4	−0.0230153
$203$	181242.	0.308687
$204$	0	0
$205$	−131829.	−0.219092
$206$	73798.5	0.121166
$207$	−21355.6	−0.0346406
$208$	−58146.1	−0.0931885
$209$	−546592.	−0.865561
$210$	4160.76	0.00651065
$211$	−557679.	−0.862340	−0.431170	−	0.902271i	$-0.641899\pi$
$211$	−557679.	−0.862340	−0.431170	+	0.902271i	$0.641899\pi$
$212$	−409312.	−0.625483
$213$	348797.	0.526774
$214$	27324.1	0.0407860
$215$	−223781.	−0.330162
$216$	−28517.2	−0.0415884
$217$	−85097.2	−0.122678
$218$	−92866.9	−0.132349
$219$	−102947.	−0.145045
$220$	322037.	0.448590
$221$	0	0
$222$	73738.6	0.100418
$223$	1.35745e6	1.82794	0.913972	−	0.405777i	$-0.132999\pi$
$223$	1.35745e6	1.82794	0.913972	+	0.405777i	$0.132999\pi$
$224$	57552.7	0.0766383
$225$	−205338.	−0.270404
$226$	14856.3	0.0193481
$227$	−1.27434e6	−1.64143	−0.820713	−	0.571341i	$-0.806423\pi$
$227$	−1.27434e6	−1.64143	−0.820713	+	0.571341i	$0.806423\pi$
$228$	−371012.	−0.472662
$229$	−952725.	−1.20055	−0.600273	−	0.799795i	$-0.704941\pi$
$229$	−952725.	−1.20055	−0.600273	+	0.799795i	$0.704941\pi$
$230$	−3937.39	−0.00490782
$231$	−116815.	−0.144035
$232$	−229028.	−0.279363
$233$	−944541.	−1.13981	−0.569903	−	0.821712i	$-0.693019\pi$
$233$	−944541.	−1.13981	−0.569903	+	0.821712i	$0.693019\pi$
$234$	−2931.47	−0.00349983
$235$	−124878.	−0.147509
$236$	912806.	1.06684
$237$	−191160.	−0.221068
$238$	0	0
$239$	−975205.	−1.10434	−0.552168	−	0.833733i	$-0.686199\pi$
$239$	−975205.	−1.10434	−0.552168	+	0.833733i	$0.686199\pi$
$240$	215946.	0.242001
$241$	824453.	0.914373	0.457186	−	0.889371i	$-0.348857\pi$
$241$	824453.	0.914373	0.457186	+	0.889371i	$0.348857\pi$
$242$	9066.97	0.00995231
$243$	59049.0	0.0641500
$244$	1.15874e6	1.24598
$245$	−384950.	−0.409722
$246$	−30034.2	−0.0316430
$247$	−76733.8	−0.0800284
$248$	107534.	0.111024
$249$	116433.	0.119008
$250$	−84528.3	−0.0855366
$251$	−1.68333e6	−1.68649	−0.843246	−	0.537528i	$-0.819358\pi$
$251$	−1.68333e6	−1.68649	−0.843246	+	0.537528i	$0.819358\pi$
$252$	−79290.6	−0.0786539
$253$	110543.	0.108575
$254$	−67638.2	−0.0657821
$255$	0	0
$256$	926881.	0.883943
$257$	1.85825e6	1.75498	0.877488	−	0.479599i	$-0.159218\pi$
$257$	1.85825e6	1.75498	0.877488	+	0.479599i	$0.159218\pi$
$258$	−50983.2	−0.0476846
$259$	412504.	0.382102
$260$	45209.4	0.0414759
$261$	474237.	0.430918
$262$	−213104.	−0.191796
$263$	−1.10150e6	−0.981967	−0.490983	−	0.871169i	$-0.663362\pi$
$263$	−1.10150e6	−0.981967	−0.490983	+	0.871169i	$0.663362\pi$
$264$	147614.	0.130352
$265$	314396.	0.275019
$266$	24812.8	0.0215017
$267$	−440883.	−0.378482
$268$	−797526.	−0.678278
$269$	−746569.	−0.629056	−0.314528	−	0.949248i	$-0.601846\pi$
$269$	−746569.	−0.629056	−0.314528	+	0.949248i	$0.601846\pi$
$270$	10887.0	0.00908867
$271$	229380.	0.189729	0.0948643	−	0.995490i	$-0.469758\pi$
$271$	229380.	0.189729	0.0948643	+	0.995490i	$0.469758\pi$
$272$	0	0
$273$	−16399.1	−0.0133172
$274$	−71903.3	−0.0578592
$275$	1.06290e6	0.847539
$276$	75033.8	0.0592904
$277$	−2.10079e6	−1.64507	−0.822534	−	0.568716i	$-0.807440\pi$
$277$	−2.10079e6	−1.64507	−0.822534	+	0.568716i	$0.807440\pi$
$278$	−78860.4	−0.0611994
$279$	−222665.	−0.171254
$280$	−29412.9	−0.0224203
$281$	−1.63076e6	−1.23204	−0.616020	−	0.787730i	$-0.711256\pi$
$281$	−1.63076e6	−1.23204	−0.616020	+	0.787730i	$0.711256\pi$
$282$	−28450.5	−0.0213043
$283$	117536.	0.0872378	0.0436189	−	0.999048i	$-0.486111\pi$
$283$	117536.	0.0872378	0.0436189	+	0.999048i	$0.486111\pi$
$284$	−1.22552e6	−0.901620
$285$	284977.	0.207825
$286$	15174.3	0.0109696
$287$	−168015.	−0.120405
$288$	150592.	0.106985
$289$	0	0
$290$	87436.5	0.0610517
$291$	922598.	0.638675
$292$	361709.	0.248257
$293$	29078.0	0.0197877	0.00989384	−	0.999951i	$-0.496851\pi$
$293$	29078.0	0.0197877	0.00989384	+	0.999951i	$0.496851\pi$
$294$	−87701.7	−0.0591752
$295$	−701133.	−0.469079
$296$	−521266.	−0.345804
$297$	−305657.	−0.201068
$298$	−299664.	−0.195476
$299$	15518.7	0.0100387
$300$	721467.	0.462821
$301$	−285207.	−0.181445
$302$	138610.	0.0874532
$303$	−195374.	−0.122253
$304$	1.28780e6	0.799216
$305$	−890034.	−0.547844
$306$	0	0
$307$	668518.	0.404825	0.202413	−	0.979300i	$-0.435122\pi$
$307$	668518.	0.404825	0.202413	+	0.979300i	$0.435122\pi$
$308$	410434.	0.246528
$309$	1.08024e6	0.643609
$310$	−41053.4	−0.0242630
$311$	1.48060e6	0.868037	0.434018	−	0.900904i	$-0.357095\pi$
$311$	1.48060e6	0.868037	0.434018	+	0.900904i	$0.357095\pi$
$312$	20722.9	0.0120521
$313$	−2.41305e6	−1.39222	−0.696108	−	0.717938i	$-0.745086\pi$
$313$	−2.41305e6	−1.39222	−0.696108	+	0.717938i	$0.745086\pi$
$314$	−147998.	−0.0847097
$315$	60903.7	0.0345834
$316$	671651.	0.378378
$317$	1.66443e6	0.930290	0.465145	−	0.885235i	$-0.346002\pi$
$317$	1.66443e6	0.930290	0.465145	+	0.885235i	$0.346002\pi$
$318$	71627.6	0.0397203
$319$	−2.45480e6	−1.35064
$320$	−740042.	−0.404000
$321$	399960.	0.216647
$322$	−5018.17	−0.00269715
$323$	0	0
$324$	−207472.	−0.109798
$325$	149216.	0.0783621
$326$	−102509.	−0.0534219
$327$	−1.35935e6	−0.703011
$328$	212315.	0.108967
$329$	−159156.	−0.0810652
$330$	−56354.8	−0.0284870
$331$	−2.04528e6	−1.02608	−0.513041	−	0.858364i	$-0.671481\pi$
$331$	−2.04528e6	−1.02608	−0.513041	+	0.858364i	$0.671481\pi$
$332$	−409093.	−0.203693
$333$	1.07936e6	0.533403
$334$	−307546.	−0.150850
$335$	612585.	0.298232
$336$	275221.	0.132994
$337$	−3.09905e6	−1.48646	−0.743231	−	0.669035i	$-0.766708\pi$
$337$	−3.09905e6	−1.48646	−0.743231	+	0.669035i	$0.766708\pi$
$338$	−226161.	−0.107678
$339$	217461.	0.102774
$340$	0	0
$341$	1.15259e6	0.536769
$342$	64925.2	0.0300157
$343$	−1.01090e6	−0.463951
$344$	360406.	0.164208
$345$	−57634.1	−0.0260694
$346$	−146525.	−0.0657992
$347$	1.58670e6	0.707412	0.353706	−	0.935357i	$-0.384921\pi$
$347$	1.58670e6	0.707412	0.353706	+	0.935357i	$0.384921\pi$
$348$	−1.66626e6	−0.737554
$349$	2.90941e6	1.27862	0.639310	−	0.768949i	$-0.279220\pi$
$349$	2.90941e6	1.27862	0.639310	+	0.768949i	$0.279220\pi$
$350$	−48250.8	−0.0210540
$351$	−42909.9	−0.0185904
$352$	−779515.	−0.335326
$353$	−1.15158e6	−0.491876	−0.245938	−	0.969286i	$-0.579096\pi$
$353$	−1.15158e6	−0.491876	−0.245938	+	0.969286i	$0.579096\pi$
$354$	−159736.	−0.0677479
$355$	941329.	0.396434
$356$	1.54906e6	0.647805
$357$	0	0
$358$	17570.8	0.00724575
$359$	2.83289e6	1.16010	0.580048	−	0.814582i	$-0.303034\pi$
$359$	2.83289e6	1.16010	0.580048	+	0.814582i	$0.303034\pi$
$360$	−76961.7	−0.0312981
$361$	−776628.	−0.313650
$362$	447721.	0.179571
$363$	132719.	0.0528648
$364$	57619.0	0.0227936
$365$	−277831.	−0.109156
$366$	−202773.	−0.0791238
$367$	−459547.	−0.178100	−0.0890501	−	0.996027i	$-0.528383\pi$
$367$	−459547.	−0.178100	−0.0890501	+	0.996027i	$0.528383\pi$
$368$	−260446.	−0.100253
$369$	−439629.	−0.168082
$370$	199005.	0.0755716
$371$	400695.	0.151140
$372$	782345.	0.293117
$373$	68887.5	0.0256371	0.0128185	−	0.999918i	$-0.495920\pi$
$373$	68887.5	0.0256371	0.0128185	+	0.999918i	$0.495920\pi$
$374$	0	0
$375$	−1.23729e6	−0.454354
$376$	201120.	0.0733644
$377$	−344620.	−0.124878
$378$	13875.4	0.00499479
$379$	195048.	0.0697499	0.0348749	−	0.999392i	$-0.488897\pi$
$379$	195048.	0.0697499	0.0348749	+	0.999392i	$0.488897\pi$
$380$	−1.00128e6	−0.355711
$381$	−990063.	−0.349422
$382$	303226.	0.106318
$383$	4.47316e6	1.55818	0.779090	−	0.626912i	$-0.215681\pi$
$383$	4.47316e6	1.55818	0.779090	+	0.626912i	$0.215681\pi$
$384$	−704040.	−0.243652
$385$	−315257.	−0.108396
$386$	−375122.	−0.128146
$387$	−746273.	−0.253291
$388$	−3.24159e6	−1.09315
$389$	5.38626e6	1.80474	0.902368	−	0.430967i	$-0.141828\pi$
$389$	5.38626e6	1.80474	0.902368	+	0.430967i	$0.141828\pi$
$390$	−7911.42	−0.00263386
$391$	0	0
$392$	619973.	0.203778
$393$	−3.11934e6	−1.01878
$394$	568676.	0.184554
$395$	−515901.	−0.166369
$396$	1.07394e6	0.344146
$397$	101712.	0.0323887	0.0161944	−	0.999869i	$-0.494845\pi$
$397$	101712.	0.0323887	0.0161944	+	0.999869i	$0.494845\pi$
$398$	−438011.	−0.138604
$399$	363201.	0.114213
$400$	−2.50424e6	−0.782576
$401$	−4.09892e6	−1.27294	−0.636471	−	0.771300i	$-0.719607\pi$
$401$	−4.09892e6	−1.27294	−0.636471	+	0.771300i	$0.719607\pi$
$402$	139563.	0.0430730
$403$	161807.	0.0496288
$404$	686456.	0.209247
$405$	159361.	0.0482773
$406$	111437.	0.0335517
$407$	−5.58711e6	−1.67187
$408$	0	0
$409$	−5.93976e6	−1.75574	−0.877870	−	0.478898i	$-0.841036\pi$
$409$	−5.93976e6	−1.75574	−0.877870	+	0.478898i	$0.841036\pi$
$410$	−81055.7	−0.0238135
$411$	−1.05249e6	−0.307337
$412$	−3.79546e6	−1.10159
$413$	−893589.	−0.257788
$414$	−13130.5	−0.00376514
$415$	314227.	0.0895620
$416$	−109433.	−0.0310037
$417$	−1.15433e6	−0.325080
$418$	−336074.	−0.0940793
$419$	−4.86832e6	−1.35470	−0.677352	−	0.735659i	$-0.736873\pi$
$419$	−4.86832e6	−1.35470	−0.677352	+	0.735659i	$0.736873\pi$
$420$	−213988.	−0.0591925
$421$	547620.	0.150582	0.0752912	−	0.997162i	$-0.476011\pi$
$421$	547620.	0.150582	0.0752912	+	0.997162i	$0.476011\pi$
$422$	−342891.	−0.0937292
$423$	−416449.	−0.113165
$424$	−506343.	−0.136782
$425$	0	0
$426$	214459.	0.0572560
$427$	−1.13434e6	−0.301075
$428$	−1.40528e6	−0.370811
$429$	222115.	0.0582687
$430$	−137593.	−0.0358859
$431$	−3.36096e6	−0.871505	−0.435752	−	0.900067i	$-0.643518\pi$
$431$	−3.36096e6	−0.871505	−0.435752	+	0.900067i	$0.643518\pi$
$432$	720143.	0.185656
$433$	−97700.7	−0.0250425	−0.0125213	−	0.999922i	$-0.503986\pi$
$433$	−97700.7	−0.0250425	−0.0125213	+	0.999922i	$0.503986\pi$
$434$	−52322.3	−0.0133341
$435$	1.27986e6	0.324295
$436$	4.77615e6	1.20327
$437$	−343703.	−0.0860953
$438$	−63297.2	−0.0157652
$439$	1.72944e6	0.428297	0.214148	−	0.976801i	$-0.431302\pi$
$439$	1.72944e6	0.428297	0.214148	+	0.976801i	$0.431302\pi$
$440$	398378.	0.0980989
$441$	−1.28375e6	−0.314328
$442$	0	0
$443$	6.76458e6	1.63769	0.818845	−	0.574015i	$-0.194615\pi$
$443$	6.76458e6	1.63769	0.818845	+	0.574015i	$0.194615\pi$
$444$	−3.79238e6	−0.912966
$445$	−1.18985e6	−0.284834
$446$	834635.	0.198682
$447$	−4.38637e6	−1.03833
$448$	−943177.	−0.222023
$449$	−1.41418e6	−0.331046	−0.165523	−	0.986206i	$-0.552931\pi$
$449$	−1.41418e6	−0.331046	−0.165523	+	0.986206i	$0.552931\pi$
$450$	−126253.	−0.0293907
$451$	2.27566e6	0.526825
$452$	−764059.	−0.175906
$453$	2.02892e6	0.464535
$454$	−783533.	−0.178409
$455$	−44257.6	−0.0100221
$456$	−458963.	−0.103363
$457$	−7.97893e6	−1.78712	−0.893561	−	0.448942i	$-0.851801\pi$
$457$	−7.97893e6	−1.78712	−0.893561	+	0.448942i	$0.851801\pi$
$458$	−585786.	−0.130489
$459$	0	0
$460$	202500.	0.0446201
$461$	641495.	0.140586	0.0702928	−	0.997526i	$-0.477607\pi$
$461$	641495.	0.140586	0.0702928	+	0.997526i	$0.477607\pi$
$462$	−71823.8	−0.0156554
$463$	2.52897e6	0.548267	0.274133	−	0.961692i	$-0.411609\pi$
$463$	2.52897e6	0.548267	0.274133	+	0.961692i	$0.411609\pi$
$464$	5.78365e6	1.24712
$465$	−600925.	−0.128881
$466$	−580754.	−0.123888
$467$	8.32818e6	1.76709	0.883544	−	0.468348i	$-0.155151\pi$
$467$	8.32818e6	1.76709	0.883544	+	0.468348i	$0.155151\pi$
$468$	150766.	0.0318192
$469$	780735.	0.163897
$470$	−76781.9	−0.0160330
$471$	−2.16635e6	−0.449962
$472$	1.12919e6	0.233299
$473$	3.86295e6	0.793901
$474$	−117536.	−0.0240283
$475$	−3.30478e6	−0.672060
$476$	0	0
$477$	1.04846e6	0.210987
$478$	−599608.	−0.120032
$479$	2.62475e6	0.522696	0.261348	−	0.965245i	$-0.415833\pi$
$479$	2.62475e6	0.522696	0.261348	+	0.965245i	$0.415833\pi$
$480$	406416.	0.0805134
$481$	−784351.	−0.154578
$482$	506918.	0.0993848
$483$	−73454.1	−0.0143268
$484$	−466315.	−0.0904828
$485$	2.48989e6	0.480647
$486$	36306.5	0.00697258
$487$	2.67906e6	0.511871	0.255935	−	0.966694i	$-0.417617\pi$
$487$	2.67906e6	0.511871	0.255935	+	0.966694i	$0.417617\pi$
$488$	1.43342e6	0.272474
$489$	−1.50049e6	−0.283767
$490$	−236688.	−0.0445334
$491$	−5.40127e6	−1.01110	−0.505548	−	0.862799i	$-0.668710\pi$
$491$	−5.40127e6	−1.01110	−0.505548	+	0.862799i	$0.668710\pi$
$492$	1.54466e6	0.287687
$493$	0	0
$494$	−47180.0	−0.00869843
$495$	−824902.	−0.151318
$496$	−2.71555e6	−0.495626
$497$	1.19972e6	0.217865
$498$	71589.2	0.0129352
$499$	3.57523e6	0.642765	0.321382	−	0.946949i	$-0.395853\pi$
$499$	3.57523e6	0.642765	0.321382	+	0.946949i	$0.395853\pi$
$500$	4.34729e6	0.777668
$501$	−4.50175e6	−0.801285
$502$	−1.03500e6	−0.183308
$503$	−3.43708e6	−0.605717	−0.302859	−	0.953035i	$-0.597941\pi$
$503$	−3.43708e6	−0.605717	−0.302859	+	0.953035i	$0.597941\pi$
$504$	−98087.0	−0.0172003
$505$	−527272.	−0.0920039
$506$	67967.9	0.0118012
$507$	−3.31046e6	−0.571963
$508$	3.47864e6	0.598067
$509$	2.17199e6	0.371590	0.185795	−	0.982589i	$-0.440514\pi$
$509$	2.17199e6	0.371590	0.185795	+	0.982589i	$0.440514\pi$
$510$	0	0
$511$	−354094.	−0.0599883
$512$	3.07315e6	0.518094
$513$	950352.	0.159438
$514$	1.14255e6	0.190751
$515$	2.91532e6	0.484361
$516$	2.62207e6	0.433531
$517$	2.15567e6	0.354696
$518$	253630.	0.0415313
$519$	−2.14478e6	−0.349513
$520$	55926.7	0.00907007
$521$	−1.01721e7	−1.64178	−0.820891	−	0.571085i	$-0.806523\pi$
$521$	−1.01721e7	−1.64178	−0.820891	+	0.571085i	$0.806523\pi$
$522$	291586.	0.0468372
$523$	3.18936e6	0.509858	0.254929	−	0.966960i	$-0.417948\pi$
$523$	3.18936e6	0.509858	0.254929	+	0.966960i	$0.417948\pi$
$524$	1.09600e7	1.74374
$525$	−706278.	−0.111835
$526$	−677264.	−0.106732
$527$	0	0
$528$	−3.72769e6	−0.581909
$529$	−6.36683e6	−0.989200
$530$	193307.	0.0298923
$531$	−2.33816e6	−0.359864
$532$	−1.27613e6	−0.195485
$533$	319471.	0.0487094
$534$	−271078.	−0.0411378
$535$	1.07941e6	0.163042
$536$	−986585.	−0.148328
$537$	257195.	0.0384881
$538$	−459030.	−0.0683732
$539$	6.64508e6	0.985209
$540$	−559921.	−0.0826309
$541$	157115.	0.0230794	0.0115397	−	0.999933i	$-0.496327\pi$
$541$	157115.	0.0230794	0.0115397	+	0.999933i	$0.496327\pi$
$542$	141035.	0.0206219
$543$	6.55358e6	0.953848
$544$	0	0
$545$	−3.66860e6	−0.529065
$546$	−10083.0	−0.00144747
$547$	−2.67809e6	−0.382698	−0.191349	−	0.981522i	$-0.561286\pi$
$547$	−2.67809e6	−0.382698	−0.191349	+	0.981522i	$0.561286\pi$
$548$	3.69799e6	0.526035
$549$	−2.96812e6	−0.420291
$550$	653526.	0.0921205
$551$	7.63251e6	1.07100
$552$	92821.2	0.0129658
$553$	−657511.	−0.0914303
$554$	−1.29168e6	−0.178805
$555$	2.91296e6	0.401422
$556$	4.05580e6	0.556403
$557$	6.73612e6	0.919966	0.459983	−	0.887928i	$-0.347856\pi$
$557$	6.73612e6	0.919966	0.459983	+	0.887928i	$0.347856\pi$
$558$	−136906.	−0.0186139
$559$	542304.	0.0734029
$560$	742762.	0.100087
$561$	0	0
$562$	−1.00268e6	−0.133913
$563$	1.17720e7	1.56523	0.782614	−	0.622507i	$-0.213886\pi$
$563$	1.17720e7	1.56523	0.782614	+	0.622507i	$0.213886\pi$
$564$	1.46321e6	0.193691
$565$	586880.	0.0773442
$566$	72267.4	0.00948203
$567$	203104.	0.0265314
$568$	−1.51604e6	−0.197169
$569$	1.21420e7	1.57221	0.786105	−	0.618093i	$-0.212094\pi$
$569$	1.21420e7	1.57221	0.786105	+	0.618093i	$0.212094\pi$
$570$	175219.	0.0225889
$571$	1.28339e7	1.64729	0.823645	−	0.567106i	$-0.191937\pi$
$571$	1.28339e7	1.64729	0.823645	+	0.567106i	$0.191937\pi$
$572$	−780413.	−0.0997320
$573$	4.43852e6	0.564744
$574$	−103305.	−0.0130870
$575$	668361.	0.0843027
$576$	−2.46792e6	−0.309938
$577$	1.34450e7	1.68121	0.840606	−	0.541647i	$-0.182199\pi$
$577$	1.34450e7	1.68121	0.840606	+	0.541647i	$0.182199\pi$
$578$	0	0
$579$	−5.49090e6	−0.680686
$580$	−4.49687e6	−0.555060
$581$	400480.	0.0492199
$582$	567262.	0.0694187
$583$	−5.42716e6	−0.661304
$584$	447455.	0.0542897
$585$	−115804.	−0.0139906
$586$	17878.7	0.00215076
$587$	−8.62340e6	−1.03296	−0.516480	−	0.856299i	$-0.672758\pi$
$587$	−8.62340e6	−1.03296	−0.516480	+	0.856299i	$0.672758\pi$
$588$	4.51051e6	0.538000
$589$	−3.58364e6	−0.425634
$590$	−431094.	−0.0509850
$591$	8.32407e6	0.980318
$592$	1.31635e7	1.54372
$593$	−6.92497e6	−0.808688	−0.404344	−	0.914607i	$-0.632500\pi$
$593$	−6.92497e6	−0.808688	−0.404344	+	0.914607i	$0.632500\pi$
$594$	−187934.	−0.0218544
$595$	0	0
$596$	1.54117e7	1.77720
$597$	−6.41144e6	−0.736241
$598$	9541.73	0.00109112
$599$	6.45992e6	0.735631	0.367816	−	0.929899i	$-0.380106\pi$
$599$	6.45992e6	0.735631	0.367816	+	0.929899i	$0.380106\pi$
$600$	892496.	0.101211
$601$	5.26407e6	0.594477	0.297239	−	0.954803i	$-0.403934\pi$
$601$	5.26407e6	0.594477	0.297239	+	0.954803i	$0.403934\pi$
$602$	−175361.	−0.0197216
$603$	2.04287e6	0.228796
$604$	−7.12870e6	−0.795093
$605$	358180.	0.0397844
$606$	−120126.	−0.0132879
$607$	1.08456e7	1.19476	0.597382	−	0.801957i	$-0.296208\pi$
$607$	1.08456e7	1.19476	0.597382	+	0.801957i	$0.296208\pi$
$608$	2.42368e6	0.265899
$609$	1.63118e6	0.178220
$610$	−547241.	−0.0595461
$611$	302626.	0.0327946
$612$	0	0
$613$	1.97089e6	0.211842	0.105921	−	0.994375i	$-0.466221\pi$
$613$	1.97089e6	0.211842	0.105921	+	0.994375i	$0.466221\pi$
$614$	411041.	0.0440011
$615$	−1.18646e6	−0.126493
$616$	507730.	0.0539115
$617$	3.65891e6	0.386935	0.193468	−	0.981107i	$-0.438027\pi$
$617$	3.65891e6	0.386935	0.193468	+	0.981107i	$0.438027\pi$
$618$	664187.	0.0699550
$619$	369356.	0.0387452	0.0193726	−	0.999812i	$-0.493833\pi$
$619$	369356.	0.0387452	0.0193726	+	0.999812i	$0.493833\pi$
$620$	2.11138e6	0.220591
$621$	−192200.	−0.0199997
$622$	910355.	0.0943484
$623$	−1.51645e6	−0.156534
$624$	−523315.	−0.0538024
$625$	4.58281e6	0.469280
$626$	−1.48367e6	−0.151322
$627$	−4.91933e6	−0.499732
$628$	7.61157e6	0.770150
$629$	0	0
$630$	37446.8	0.00375893
$631$	−1.58925e7	−1.58898	−0.794491	−	0.607276i	$-0.792262\pi$
$631$	−1.58925e7	−1.58898	−0.794491	+	0.607276i	$0.792262\pi$
$632$	830871.	0.0827449
$633$	−5.01911e6	−0.497872
$634$	1.02338e6	0.101115
$635$	−2.67197e6	−0.262964
$636$	−3.68381e6	−0.361123
$637$	932875.	0.0910909
$638$	−1.50934e6	−0.146804
$639$	3.13918e6	0.304133
$640$	−1.90005e6	−0.183365
$641$	4.87938e6	0.469051	0.234525	−	0.972110i	$-0.424646\pi$
$641$	4.87938e6	0.469051	0.234525	+	0.972110i	$0.424646\pi$
$642$	245917.	0.0235478
$643$	−1.33841e6	−0.127662	−0.0638311	−	0.997961i	$-0.520332\pi$
$643$	−1.33841e6	−0.127662	−0.0638311	+	0.997961i	$0.520332\pi$
$644$	258085.	0.0245215
$645$	−2.01403e6	−0.190619
$646$	0	0
$647$	−5.64993e6	−0.530618	−0.265309	−	0.964163i	$-0.585474\pi$
$647$	−5.64993e6	−0.530618	−0.265309	+	0.964163i	$0.585474\pi$
$648$	−256654.	−0.0240110
$649$	1.21031e7	1.12794
$650$	91745.8	0.00851732
$651$	−765874.	−0.0708280
$652$	5.27206e6	0.485693
$653$	1.43413e7	1.31615	0.658074	−	0.752953i	$-0.271371\pi$
$653$	1.43413e7	1.31615	0.658074	+	0.752953i	$0.271371\pi$
$654$	−835802.	−0.0764115
$655$	−8.41842e6	−0.766704
$656$	−5.36158e6	−0.486444
$657$	−926522.	−0.0837418
$658$	−97857.9	−0.00881112
$659$	4.41609e6	0.396117	0.198059	−	0.980190i	$-0.436536\pi$
$659$	4.41609e6	0.396117	0.198059	+	0.980190i	$0.436536\pi$
$660$	2.89833e6	0.258993
$661$	−1.54572e7	−1.37603	−0.688013	−	0.725699i	$-0.741517\pi$
$661$	−1.54572e7	−1.37603	−0.688013	+	0.725699i	$0.741517\pi$
$662$	−1.25754e6	−0.111527
$663$	0	0
$664$	−506071.	−0.0445442
$665$	980202.	0.0859531
$666$	663647.	0.0579765
$667$	−1.54361e6	−0.134345
$668$	1.58171e7	1.37147
$669$	1.22171e7	1.05536
$670$	376650.	0.0324154
$671$	1.53639e7	1.31733
$672$	517974.	0.0442471
$673$	−7.65216e6	−0.651248	−0.325624	−	0.945499i	$-0.605574\pi$
$673$	−7.65216e6	−0.651248	−0.325624	+	0.945499i	$0.605574\pi$
$674$	−1.90546e6	−0.161566
$675$	−1.84805e6	−0.156118
$676$	1.16315e7	0.978965
$677$	1.25672e7	1.05382	0.526911	−	0.849921i	$-0.323350\pi$
$677$	1.25672e7	1.05382	0.526911	+	0.849921i	$0.323350\pi$
$678$	133706.	0.0111706
$679$	3.17335e6	0.264146
$680$	0	0
$681$	−1.14691e7	−0.947677
$682$	708672.	0.0583424
$683$	−1.70300e7	−1.39689	−0.698447	−	0.715662i	$-0.746125\pi$
$683$	−1.70300e7	−1.39689	−0.698447	+	0.715662i	$0.746125\pi$
$684$	−3.33911e6	−0.272892
$685$	−2.84046e6	−0.231293
$686$	−621554.	−0.0504276
$687$	−8.57452e6	−0.693136
$688$	−9.10132e6	−0.733049
$689$	−761896.	−0.0611431
$690$	−35436.5	−0.00283353
$691$	4.44462e6	0.354111	0.177056	−	0.984201i	$-0.443343\pi$
$691$	4.44462e6	0.354111	0.177056	+	0.984201i	$0.443343\pi$
$692$	7.53578e6	0.598223
$693$	−1.05133e6	−0.0831584
$694$	975591.	0.0768898
$695$	−3.11529e6	−0.244645
$696$	−2.06125e6	−0.161290
$697$	0	0
$698$	1.78886e6	0.138975
$699$	−8.50087e6	−0.658067
$700$	2.48154e6	0.191415
$701$	2.48267e7	1.90820	0.954102	−	0.299482i	$-0.0968138\pi$
$701$	2.48267e7	1.90820	0.954102	+	0.299482i	$0.0968138\pi$
$702$	−26383.3	−0.00202063
$703$	1.73715e7	1.32571
$704$	1.27747e7	0.971449
$705$	−1.12391e6	−0.0851642
$706$	−708051.	−0.0534629
$707$	−672004.	−0.0505619
$708$	8.21526e6	0.615939
$709$	−3.69397e6	−0.275980	−0.137990	−	0.990434i	$-0.544064\pi$
$709$	−3.69397e6	−0.275980	−0.137990	+	0.990434i	$0.544064\pi$
$710$	578779.	0.0430891
$711$	−1.72044e6	−0.127634
$712$	1.91628e6	0.141664
$713$	724758.	0.0533912
$714$	0	0
$715$	599441.	0.0438512
$716$	−903667.	−0.0658758
$717$	−8.77684e6	−0.637589
$718$	1.74181e6	0.126093
$719$	1.04250e7	0.752060	0.376030	−	0.926608i	$-0.377289\pi$
$719$	1.04250e7	0.752060	0.376030	+	0.926608i	$0.377289\pi$
$720$	1.94351e6	0.139719
$721$	3.71556e6	0.266186
$722$	−477512.	−0.0340911
$723$	7.42008e6	0.527913
$724$	−2.30263e7	−1.63260
$725$	−1.48421e7	−1.04870
$726$	81602.7	0.00574597
$727$	−512050.	−0.0359316	−0.0179658	−	0.999839i	$-0.505719\pi$
$727$	−512050.	−0.0359316	−0.0179658	+	0.999839i	$0.505719\pi$
$728$	71278.1	0.00498457
$729$	531441.	0.0370370
$730$	−170826.	−0.0118644
$731$	0	0
$732$	1.04286e7	0.719365
$733$	−9.14250e6	−0.628500	−0.314250	−	0.949340i	$-0.601753\pi$
$733$	−9.14250e6	−0.628500	−0.314250	+	0.949340i	$0.601753\pi$
$734$	−282554.	−0.0193580
$735$	−3.46455e6	−0.236553
$736$	−490167.	−0.0333541
$737$	−1.05746e7	−0.717123
$738$	−270307.	−0.0182691
$739$	1.18421e7	0.797659	0.398829	−	0.917025i	$-0.369417\pi$
$739$	1.18421e7	0.797659	0.398829	+	0.917025i	$0.369417\pi$
$740$	−1.02348e7	−0.687070
$741$	−690604.	−0.0462044
$742$	246369.	0.0164277
$743$	2.20626e7	1.46617	0.733087	−	0.680135i	$-0.238079\pi$
$743$	2.20626e7	1.46617	0.733087	+	0.680135i	$0.238079\pi$
$744$	967806.	0.0640997
$745$	−1.18379e7	−0.781417
$746$	42355.7	0.00278654
$747$	1.04790e6	0.0687095
$748$	0	0
$749$	1.37569e6	0.0896018
$750$	−760754.	−0.0493846
$751$	−8.35950e6	−0.540855	−0.270427	−	0.962740i	$-0.587165\pi$
$751$	−8.35950e6	−0.540855	−0.270427	+	0.962740i	$0.587165\pi$
$752$	−5.07888e6	−0.327509
$753$	−1.51499e7	−0.973696
$754$	−211891.	−0.0135732
$755$	5.47561e6	0.349595
$756$	−713615.	−0.0454108
$757$	2.15816e7	1.36881	0.684406	−	0.729101i	$-0.260061\pi$
$757$	2.15816e7	1.36881	0.684406	+	0.729101i	$0.260061\pi$
$758$	119926.	0.00758123
$759$	994889.	0.0626860
$760$	−1.23864e6	−0.0777879
$761$	−361606.	−0.0226347	−0.0113173	−	0.999936i	$-0.503602\pi$
$761$	−361606.	−0.0226347	−0.0113173	+	0.999936i	$0.503602\pi$
$762$	−608744.	−0.0379793
$763$	−4.67560e6	−0.290754
$764$	−1.55950e7	−0.966609
$765$	0	0
$766$	2.75034e6	0.169361
$767$	1.69910e6	0.104287
$768$	8.34193e6	0.510345
$769$	−3.22981e7	−1.96952	−0.984762	−	0.173910i	$-0.944360\pi$
$769$	−3.22981e7	−1.96952	−0.984762	+	0.173910i	$0.944360\pi$
$770$	−193837.	−0.0117818
$771$	1.67242e7	1.01324
$772$	1.92925e7	1.16505
$773$	2.32135e7	1.39731	0.698653	−	0.715461i	$-0.253783\pi$
$773$	2.32135e7	1.39731	0.698653	+	0.715461i	$0.253783\pi$
$774$	−458849.	−0.0275307
$775$	6.96871e6	0.416772
$776$	−4.01004e6	−0.239053
$777$	3.71254e6	0.220607
$778$	3.31176e6	0.196160
$779$	−7.07552e6	−0.417748
$780$	406885.	0.0239461
$781$	−1.62494e7	−0.953256
$782$	0	0
$783$	4.26813e6	0.248790
$784$	−1.56562e7	−0.909694
$785$	−5.84650e6	−0.338627
$786$	−1.91794e6	−0.110733
$787$	5.20593e6	0.299614	0.149807	−	0.988715i	$-0.452135\pi$
$787$	5.20593e6	0.299614	0.149807	+	0.988715i	$0.452135\pi$
$788$	−2.92470e7	−1.67790
$789$	−9.91354e6	−0.566939
$790$	−317203.	−0.0180830
$791$	747973.	0.0425055
$792$	1.32853e6	0.0752588
$793$	2.15688e6	0.121799
$794$	62537.7	0.00352039
$795$	2.82956e6	0.158782
$796$	2.25269e7	1.26014
$797$	2.55456e7	1.42453	0.712264	−	0.701912i	$-0.247670\pi$
$797$	2.55456e7	1.42453	0.712264	+	0.701912i	$0.247670\pi$
$798$	223316.	0.0124140
$799$	0	0
$800$	−4.71306e6	−0.260362
$801$	−3.96794e6	−0.218517
$802$	−2.52024e6	−0.138358
$803$	4.79598e6	0.262475
$804$	−7.17773e6	−0.391604
$805$	−198237.	−0.0107819
$806$	99487.5	0.00539425
$807$	−6.71912e6	−0.363185
$808$	849186.	0.0457588
$809$	1.65742e6	0.0890351	0.0445175	−	0.999009i	$-0.485825\pi$
$809$	1.65742e6	0.0890351	0.0445175	+	0.999009i	$0.485825\pi$
$810$	97983.4	0.00524735
$811$	1.83266e7	0.978432	0.489216	−	0.872163i	$-0.337283\pi$
$811$	1.83266e7	0.978432	0.489216	+	0.872163i	$0.337283\pi$
$812$	−5.73122e6	−0.305040
$813$	2.06442e6	0.109540
$814$	−3.43525e6	−0.181718
$815$	−4.04951e6	−0.213554
$816$	0	0
$817$	−1.20107e7	−0.629528
$818$	−3.65208e6	−0.190835
$819$	−147592.	−0.00768870
$820$	4.16870e6	0.216504
$821$	2.51447e7	1.30193	0.650966	−	0.759107i	$-0.274364\pi$
$821$	2.51447e7	1.30193	0.650966	+	0.759107i	$0.274364\pi$
$822$	−647130.	−0.0334050
$823$	−3.12807e7	−1.60982	−0.804909	−	0.593398i	$-0.797786\pi$
$823$	−3.12807e7	−1.60982	−0.804909	+	0.593398i	$0.797786\pi$
$824$	−4.69521e6	−0.240900
$825$	9.56608e6	0.489327
$826$	−549426.	−0.0280194
$827$	−2.34297e7	−1.19125	−0.595625	−	0.803262i	$-0.703096\pi$
$827$	−2.34297e7	−1.19125	−0.595625	+	0.803262i	$0.703096\pi$
$828$	675304.	0.0342313
$829$	1.85109e7	0.935494	0.467747	−	0.883862i	$-0.345066\pi$
$829$	1.85109e7	0.935494	0.467747	+	0.883862i	$0.345066\pi$
$830$	193204.	0.00973465
$831$	−1.89071e7	−0.949780
$832$	1.79339e6	0.0898187
$833$	0	0
$834$	−709744.	−0.0353335
$835$	−1.21492e7	−0.603022
$836$	1.72843e7	0.855335
$837$	−2.00399e6	−0.0988738
$838$	−2.99331e6	−0.147245
$839$	−3.39530e7	−1.66523	−0.832614	−	0.553854i	$-0.813156\pi$
$839$	−3.39530e7	−1.66523	−0.832614	+	0.553854i	$0.813156\pi$
$840$	−264716.	−0.0129444
$841$	1.37673e7	0.671209
$842$	336706.	0.0163671
$843$	−1.46769e7	−0.711319
$844$	1.76349e7	0.852152
$845$	−8.93421e6	−0.430442
$846$	−256055.	−0.0123001
$847$	456498.	0.0218640
$848$	1.27867e7	0.610616
$849$	1.05782e6	0.0503668
$850$	0	0
$851$	−3.51323e6	−0.166296
$852$	−1.10297e7	−0.520551
$853$	−4.93753e6	−0.232347	−0.116174	−	0.993229i	$-0.537063\pi$
$853$	−4.93753e6	−0.232347	−0.116174	+	0.993229i	$0.537063\pi$
$854$	−697454.	−0.0327243
$855$	2.56480e6	0.119988
$856$	−1.73841e6	−0.0810901
$857$	3.52560e7	1.63976	0.819882	−	0.572533i	$-0.194039\pi$
$857$	3.52560e7	1.63976	0.819882	+	0.572533i	$0.194039\pi$
$858$	136568.	0.00633333
$859$	1.60234e7	0.740922	0.370461	−	0.928848i	$-0.379200\pi$
$859$	1.60234e7	0.740922	0.370461	+	0.928848i	$0.379200\pi$
$860$	7.07640e6	0.326262
$861$	−1.51214e6	−0.0695158
$862$	−2.06650e6	−0.0947254
$863$	−1.68268e7	−0.769083	−0.384542	−	0.923108i	$-0.625640\pi$
$863$	−1.68268e7	−0.769083	−0.384542	+	0.923108i	$0.625640\pi$
$864$	1.35533e6	0.0617677
$865$	−5.78829e6	−0.263033
$866$	−60071.6	−0.00272192
$867$	0	0
$868$	2.69094e6	0.121228
$869$	8.90557e6	0.400048
$870$	786929.	0.0352482
$871$	−1.48452e6	−0.0663040
$872$	5.90837e6	0.263134
$873$	8.30338e6	0.368739
$874$	−211327.	−0.00935784
$875$	−4.25577e6	−0.187914
$876$	3.25538e6	0.143332
$877$	2.77920e6	0.122017	0.0610085	−	0.998137i	$-0.480568\pi$
$877$	2.77920e6	0.122017	0.0610085	+	0.998137i	$0.480568\pi$
$878$	1.06335e6	0.0465523
$879$	261702.	0.0114244
$880$	−1.00602e7	−0.437927
$881$	−3.79557e7	−1.64754	−0.823772	−	0.566922i	$-0.808134\pi$
$881$	−3.79557e7	−1.64754	−0.823772	+	0.566922i	$0.808134\pi$
$882$	−789315.	−0.0341648
$883$	−2.36153e7	−1.01927	−0.509637	−	0.860389i	$-0.670220\pi$
$883$	−2.36153e7	−1.01927	−0.509637	+	0.860389i	$0.670220\pi$
$884$	0	0
$885$	−6.31020e6	−0.270823
$886$	4.15922e6	0.178003
$887$	−1.16857e7	−0.498708	−0.249354	−	0.968412i	$-0.580218\pi$
$887$	−1.16857e7	−0.498708	−0.249354	+	0.968412i	$0.580218\pi$
$888$	−4.69139e6	−0.199650
$889$	−3.40540e6	−0.144515
$890$	−731582.	−0.0309591
$891$	−2.75091e6	−0.116087
$892$	−4.29253e7	−1.80635
$893$	−6.70245e6	−0.281258
$894$	−2.69697e6	−0.112858
$895$	694113.	0.0289649
$896$	−2.42160e6	−0.100770
$897$	139668.	0.00579585
$898$	−869512.	−0.0359819
$899$	−1.60945e7	−0.664169
$900$	6.49320e6	0.267210
$901$	0	0
$902$	1.39920e6	0.0572615
$903$	−2.56687e6	−0.104757
$904$	−945185.	−0.0384677
$905$	1.76867e7	0.717836
$906$	1.24749e6	0.0504911
$907$	4.12437e7	1.66471	0.832357	−	0.554240i	$-0.186991\pi$
$907$	4.12437e7	1.66471	0.832357	+	0.554240i	$0.186991\pi$
$908$	4.02972e7	1.62203
$909$	−1.75837e6	−0.0705829
$910$	−27212.0	−0.00108932
$911$	−3.21158e7	−1.28210	−0.641051	−	0.767498i	$-0.721501\pi$
$911$	−3.21158e7	−1.28210	−0.641051	+	0.767498i	$0.721501\pi$
$912$	1.15902e7	0.461428
$913$	−5.42425e6	−0.215359
$914$	−4.90587e6	−0.194245
$915$	−8.01031e6	−0.316298
$916$	3.01270e7	1.18636
$917$	−1.07292e7	−0.421352
$918$	0	0
$919$	−2.13657e7	−0.834505	−0.417252	−	0.908791i	$-0.637007\pi$
$919$	−2.13657e7	−0.834505	−0.417252	+	0.908791i	$0.637007\pi$
$920$	250504.	0.00975766
$921$	6.01667e6	0.233726
$922$	394425.	0.0152805
$923$	−2.28118e6	−0.0881365
$924$	3.69390e6	0.142333
$925$	−3.37805e7	−1.29811
$926$	1.55495e6	0.0595921
$927$	9.72212e6	0.371588
$928$	1.08850e7	0.414914
$929$	4.28274e6	0.162811	0.0814053	−	0.996681i	$-0.474059\pi$
$929$	4.28274e6	0.162811	0.0814053	+	0.996681i	$0.474059\pi$
$930$	−369481.	−0.0140083
$931$	−2.06610e7	−0.781226
$932$	2.98682e7	1.12634
$933$	1.33254e7	0.501161
$934$	5.12061e6	0.192068
$935$	0	0
$936$	186506.	0.00695831
$937$	2.41941e7	0.900245	0.450123	−	0.892967i	$-0.351380\pi$
$937$	2.41941e7	0.900245	0.450123	+	0.892967i	$0.351380\pi$
$938$	480038.	0.0178143
$939$	−2.17175e7	−0.803796
$940$	3.94890e6	0.145766
$941$	−8.57284e6	−0.315610	−0.157805	−	0.987470i	$-0.550442\pi$
$941$	−8.57284e6	−0.315610	−0.157805	+	0.987470i	$0.550442\pi$
$942$	−1.33199e6	−0.0489071
$943$	1.43096e6	0.0524020
$944$	−2.85155e7	−1.04148
$945$	548133.	0.0199667
$946$	2.37515e6	0.0862905
$947$	−2.57839e7	−0.934271	−0.467135	−	0.884186i	$-0.654714\pi$
$947$	−2.57839e7	−0.934271	−0.467135	+	0.884186i	$0.654714\pi$
$948$	6.04486e6	0.218457
$949$	673287.	0.0242680
$950$	−2.03195e6	−0.0730474
$951$	1.49799e7	0.537103
$952$	0	0
$953$	1.64177e7	0.585572	0.292786	−	0.956178i	$-0.405418\pi$
$953$	1.64177e7	0.585572	0.292786	+	0.956178i	$0.405418\pi$
$954$	644648.	0.0229325
$955$	1.19786e7	0.425009
$956$	3.08379e7	1.09129
$957$	−2.20932e7	−0.779793
$958$	1.61384e6	0.0568127
$959$	−3.62014e6	−0.127110
$960$	−6.66038e6	−0.233249
$961$	−2.10724e7	−0.736047
$962$	−482261.	−0.0168014
$963$	3.59964e6	0.125081
$964$	−2.60708e7	−0.903571
$965$	−1.48188e7	−0.512263
$966$	−45163.5	−0.00155720
$967$	−1.26667e7	−0.435610	−0.217805	−	0.975992i	$-0.569890\pi$
$967$	−1.26667e7	−0.435610	−0.217805	+	0.975992i	$0.569890\pi$
$968$	−576858.	−0.0197870
$969$	0	0
$970$	1.53092e6	0.0522424
$971$	5.08971e7	1.73239	0.866193	−	0.499710i	$-0.166560\pi$
$971$	5.08971e7	1.73239	0.866193	+	0.499710i	$0.166560\pi$
$972$	−1.86724e6	−0.0633922
$973$	−3.97041e6	−0.134448
$974$	1.64723e6	0.0556361
$975$	1.34294e6	0.0452424
$976$	−3.61982e7	−1.21636
$977$	4.18736e7	1.40347	0.701736	−	0.712437i	$-0.252408\pi$
$977$	4.18736e7	1.40347	0.701736	+	0.712437i	$0.252408\pi$
$978$	−922584.	−0.0308431
$979$	2.05394e7	0.684905
$980$	1.21729e7	0.404882
$981$	−1.22342e7	−0.405884
$982$	−3.32099e6	−0.109898
$983$	−2.63945e7	−0.871224	−0.435612	−	0.900134i	$-0.643468\pi$
$983$	−2.63945e7	−0.871224	−0.435612	+	0.900134i	$0.643468\pi$
$984$	1.91083e6	0.0629122
$985$	2.24649e7	0.737757
$986$	0	0
$987$	−1.43241e6	−0.0468030
$988$	2.42647e6	0.0790829
$989$	2.42906e6	0.0789675
$990$	−507194.	−0.0164470
$991$	−4.72736e7	−1.52910	−0.764548	−	0.644567i	$-0.777038\pi$
$991$	−4.72736e7	−1.52910	−0.764548	+	0.644567i	$0.777038\pi$
$992$	−5.11076e6	−0.164894
$993$	−1.84075e7	−0.592408
$994$	737650.	0.0236801
$995$	−1.73031e7	−0.554072
$996$	−3.68184e6	−0.117602
$997$	−1.12181e7	−0.357421	−0.178711	−	0.983902i	$-0.557193\pi$
$997$	−1.12181e7	−0.357421	−0.178711	+	0.983902i	$0.557193\pi$
$998$	2.19824e6	0.0698632
$999$	9.71423e6	0.307960

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.6.a.u.1.14		28
17.5	odd	16		51.6.h.a.25.8	✓	56
17.7	odd	16		51.6.h.a.49.8	yes	56
17.16	even	2		867.6.a.t.1.14		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.6.h.a.25.8	✓	56	17.5	odd	16
51.6.h.a.49.8	yes	56	17.7	odd	16
867.6.a.t.1.14		28	17.16	even	2
867.6.a.u.1.14		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)

\(f(q)\)	\(=\)	\(q+0.614853 q^{2} +9.00000 q^{3} -31.6220 q^{4} +24.2891 q^{5} +5.53368 q^{6} +30.9562 q^{7} -39.1182 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+0.614853 q^{2} +9.00000 q^{3} -31.6220 q^{4} +24.2891 q^{5} +5.53368 q^{6} +30.9562 q^{7} -39.1182 q^{8} +81.0000 q^{9} +14.9342 q^{10} -419.282 q^{11} -284.598 q^{12} -58.8613 q^{13} +19.0335 q^{14} +218.602 q^{15} +987.851 q^{16} +49.8031 q^{18} +1303.64 q^{19} -768.068 q^{20} +278.606 q^{21} -257.797 q^{22} -263.649 q^{23} -352.064 q^{24} -2535.04 q^{25} -36.1910 q^{26} +729.000 q^{27} -978.896 q^{28} +5854.78 q^{29} +134.408 q^{30} -2748.95 q^{31} +1859.16 q^{32} -3773.54 q^{33} +751.897 q^{35} -2561.38 q^{36} +13325.4 q^{37} +801.546 q^{38} -529.751 q^{39} -950.144 q^{40} -5427.52 q^{41} +171.302 q^{42} -9213.25 q^{43} +13258.5 q^{44} +1967.41 q^{45} -162.105 q^{46} -5141.34 q^{47} +8890.66 q^{48} -15848.7 q^{49} -1558.68 q^{50} +1861.31 q^{52} +12943.9 q^{53} +448.228 q^{54} -10184.0 q^{55} -1210.95 q^{56} +11732.7 q^{57} +3599.83 q^{58} -28866.2 q^{59} -6912.61 q^{60} -36643.4 q^{61} -1690.20 q^{62} +2507.45 q^{63} -30468.1 q^{64} -1429.68 q^{65} -2320.17 q^{66} +25220.6 q^{67} -2372.84 q^{69} +462.307 q^{70} +38755.3 q^{71} -3168.57 q^{72} -11438.5 q^{73} +8193.18 q^{74} -22815.4 q^{75} -41223.6 q^{76} -12979.4 q^{77} -325.719 q^{78} -21240.0 q^{79} +23994.0 q^{80} +6561.00 q^{81} -3337.13 q^{82} +12937.0 q^{83} -8810.06 q^{84} -5664.80 q^{86} +52693.0 q^{87} +16401.6 q^{88} -48987.0 q^{89} +1209.67 q^{90} -1822.12 q^{91} +8337.09 q^{92} -24740.6 q^{93} -3161.17 q^{94} +31664.1 q^{95} +16732.5 q^{96} +102511. q^{97} -9744.64 q^{98} -33961.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9}+O(q^{10}) \) 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

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