LMFDB - Embedded newform 786.4.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [786,4,Mod(1,786)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(786, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("786.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 786 = 2 \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	786.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:	$46.3755012645$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 436x^{6} + 1403x^{5} + 41156x^{4} - 104947x^{3} - 993314x^{2} + 1535040x + 1863168 $ x^8 - x^7 - 436x^6 + 1403x^5 + 41156x^4 - 104947x^3 - 993314x^2 + 1535040x + 1863168
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$14.7762$ of defining polynomial
Character	$\chi$	$=$	786.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +4.74723 q^{5} +6.00000 q^{6} +27.5663 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +4.74723 q^{5} +6.00000 q^{6} +27.5663 q^{7} -8.00000 q^{8} +9.00000 q^{9} -9.49446 q^{10} +35.9318 q^{11} -12.0000 q^{12} -64.3215 q^{13} -55.1327 q^{14} -14.2417 q^{15} +16.0000 q^{16} -105.772 q^{17} -18.0000 q^{18} -6.65620 q^{19} +18.9889 q^{20} -82.6990 q^{21} -71.8636 q^{22} +6.46461 q^{23} +24.0000 q^{24} -102.464 q^{25} +128.643 q^{26} -27.0000 q^{27} +110.265 q^{28} -182.493 q^{29} +28.4834 q^{30} +15.2217 q^{31} -32.0000 q^{32} -107.795 q^{33} +211.545 q^{34} +130.864 q^{35} +36.0000 q^{36} -166.716 q^{37} +13.3124 q^{38} +192.964 q^{39} -37.9778 q^{40} -241.404 q^{41} +165.398 q^{42} +500.596 q^{43} +143.727 q^{44} +42.7251 q^{45} -12.9292 q^{46} -6.09555 q^{47} -48.0000 q^{48} +416.903 q^{49} +204.928 q^{50} +317.317 q^{51} -257.286 q^{52} -8.02587 q^{53} +54.0000 q^{54} +170.577 q^{55} -220.531 q^{56} +19.9686 q^{57} +364.987 q^{58} +44.5960 q^{59} -56.9668 q^{60} -401.549 q^{61} -30.4433 q^{62} +248.097 q^{63} +64.0000 q^{64} -305.349 q^{65} +215.591 q^{66} -258.326 q^{67} -423.089 q^{68} -19.3938 q^{69} -261.728 q^{70} -1058.36 q^{71} -72.0000 q^{72} +547.136 q^{73} +333.433 q^{74} +307.391 q^{75} -26.6248 q^{76} +990.509 q^{77} -385.929 q^{78} +179.688 q^{79} +75.9557 q^{80} +81.0000 q^{81} +482.807 q^{82} -300.532 q^{83} -330.796 q^{84} -502.126 q^{85} -1001.19 q^{86} +547.480 q^{87} -287.455 q^{88} -1144.79 q^{89} -85.4502 q^{90} -1773.11 q^{91} +25.8584 q^{92} -45.6650 q^{93} +12.1911 q^{94} -31.5985 q^{95} +96.0000 q^{96} +1335.12 q^{97} -833.806 q^{98} +323.386 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9	\( 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9} - 2 q^{10} - 77 q^{11} - 96 q^{12} + 29 q^{13} - 44 q^{14} - 3 q^{15} + 128 q^{16} + 59 q^{17} - 144 q^{18} - 150 q^{19} + 4 q^{20} - 66 q^{21} + 154 q^{22} - 269 q^{23} + 192 q^{24} + 273 q^{25} - 58 q^{26} - 216 q^{27} + 88 q^{28} - 123 q^{29} + 6 q^{30} - 86 q^{31} - 256 q^{32} + 231 q^{33} - 118 q^{34} - 292 q^{35} + 288 q^{36} + 412 q^{37} + 300 q^{38} - 87 q^{39} - 8 q^{40} - 114 q^{41} + 132 q^{42} + 1087 q^{43} - 308 q^{44} + 9 q^{45} + 538 q^{46} - 442 q^{47} - 384 q^{48} + 1292 q^{49} - 546 q^{50} - 177 q^{51} + 116 q^{52} + 5 q^{53} + 432 q^{54} + 456 q^{55} - 176 q^{56} + 450 q^{57} + 246 q^{58} + 252 q^{59} - 12 q^{60} + 1482 q^{61} + 172 q^{62} + 198 q^{63} + 512 q^{64} + 475 q^{65} - 462 q^{66} + 330 q^{67} + 236 q^{68} + 807 q^{69} + 584 q^{70} - 2946 q^{71} - 576 q^{72} - 214 q^{73} - 824 q^{74} - 819 q^{75} - 600 q^{76} - 960 q^{77} + 174 q^{78} - 64 q^{79} + 16 q^{80} + 648 q^{81} + 228 q^{82} - 276 q^{83} - 264 q^{84} + 80 q^{85} - 2174 q^{86} + 369 q^{87} + 616 q^{88} - 3177 q^{89} - 18 q^{90} - 781 q^{91} - 1076 q^{92} + 258 q^{93} + 884 q^{94} - 2700 q^{95} + 768 q^{96} + 200 q^{97} - 2584 q^{98} - 693 q^{99}+O(q^{100}) \) 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9 - 2 * q^10 - 77 * q^11 - 96 * q^12 + 29 * q^13 - 44 * q^14 - 3 * q^15 + 128 * q^16 + 59 * q^17 - 144 * q^18 - 150 * q^19 + 4 * q^20 - 66 * q^21 + 154 * q^22 - 269 * q^23 + 192 * q^24 + 273 * q^25 - 58 * q^26 - 216 * q^27 + 88 * q^28 - 123 * q^29 + 6 * q^30 - 86 * q^31 - 256 * q^32 + 231 * q^33 - 118 * q^34 - 292 * q^35 + 288 * q^36 + 412 * q^37 + 300 * q^38 - 87 * q^39 - 8 * q^40 - 114 * q^41 + 132 * q^42 + 1087 * q^43 - 308 * q^44 + 9 * q^45 + 538 * q^46 - 442 * q^47 - 384 * q^48 + 1292 * q^49 - 546 * q^50 - 177 * q^51 + 116 * q^52 + 5 * q^53 + 432 * q^54 + 456 * q^55 - 176 * q^56 + 450 * q^57 + 246 * q^58 + 252 * q^59 - 12 * q^60 + 1482 * q^61 + 172 * q^62 + 198 * q^63 + 512 * q^64 + 475 * q^65 - 462 * q^66 + 330 * q^67 + 236 * q^68 + 807 * q^69 + 584 * q^70 - 2946 * q^71 - 576 * q^72 - 214 * q^73 - 824 * q^74 - 819 * q^75 - 600 * q^76 - 960 * q^77 + 174 * q^78 - 64 * q^79 + 16 * q^80 + 648 * q^81 + 228 * q^82 - 276 * q^83 - 264 * q^84 + 80 * q^85 - 2174 * q^86 + 369 * q^87 + 616 * q^88 - 3177 * q^89 - 18 * q^90 - 781 * q^91 - 1076 * q^92 + 258 * q^93 + 884 * q^94 - 2700 * q^95 + 768 * q^96 + 200 * q^97 - 2584 * q^98 - 693 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	4.74723	0.424605	0.212303	−	0.977204i	$-0.431904\pi$
$5$	4.74723	0.424605	0.212303	+	0.977204i	$0.431904\pi$
$6$	6.00000	0.408248
$7$	27.5663	1.48844	0.744221	−	0.667933i	$-0.232821\pi$
$7$	27.5663	1.48844	0.744221	+	0.667933i	$0.232821\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−9.49446	−0.300241
$11$	35.9318	0.984896	0.492448	−	0.870342i	$-0.336102\pi$
$11$	35.9318	0.984896	0.492448	+	0.870342i	$0.336102\pi$
$12$	−12.0000	−0.288675
$13$	−64.3215	−1.37227	−0.686137	−	0.727472i	$-0.740695\pi$
$13$	−64.3215	−1.37227	−0.686137	+	0.727472i	$0.740695\pi$
$14$	−55.1327	−1.05249
$15$	−14.2417	−0.245146
$16$	16.0000	0.250000
$17$	−105.772	−1.50903	−0.754517	−	0.656281i	$-0.772129\pi$
$17$	−105.772	−1.50903	−0.754517	+	0.656281i	$0.772129\pi$
$18$	−18.0000	−0.235702
$19$	−6.65620	−0.0803704	−0.0401852	−	0.999192i	$-0.512795\pi$
$19$	−6.65620	−0.0803704	−0.0401852	+	0.999192i	$0.512795\pi$
$20$	18.9889	0.212303
$21$	−82.6990	−0.859353
$22$	−71.8636	−0.696426
$23$	6.46461	0.0586072	0.0293036	−	0.999571i	$-0.490671\pi$
$23$	6.46461	0.0586072	0.0293036	+	0.999571i	$0.490671\pi$
$24$	24.0000	0.204124
$25$	−102.464	−0.819710
$26$	128.643	0.970345
$27$	−27.0000	−0.192450
$28$	110.265	0.744221
$29$	−182.493	−1.16856	−0.584279	−	0.811553i	$-0.698623\pi$
$29$	−182.493	−1.16856	−0.584279	+	0.811553i	$0.698623\pi$
$30$	28.4834	0.173344
$31$	15.2217	0.0881900	0.0440950	−	0.999027i	$-0.485960\pi$
$31$	15.2217	0.0881900	0.0440950	+	0.999027i	$0.485960\pi$
$32$	−32.0000	−0.176777
$33$	−107.795	−0.568630
$34$	211.545	1.06705
$35$	130.864	0.632000
$36$	36.0000	0.166667
$37$	−166.716	−0.740757	−0.370378	−	0.928881i	$-0.620772\pi$
$37$	−166.716	−0.740757	−0.370378	+	0.928881i	$0.620772\pi$
$38$	13.3124	0.0568305
$39$	192.964	0.792283
$40$	−37.9778	−0.150121
$41$	−241.404	−0.919535	−0.459767	−	0.888039i	$-0.652067\pi$
$41$	−241.404	−0.919535	−0.459767	+	0.888039i	$0.652067\pi$
$42$	165.398	0.607654
$43$	500.596	1.77535	0.887677	−	0.460467i	$-0.152318\pi$
$43$	500.596	1.77535	0.887677	+	0.460467i	$0.152318\pi$
$44$	143.727	0.492448
$45$	42.7251	0.141535
$46$	−12.9292	−0.0414415
$47$	−6.09555	−0.0189176	−0.00945881	−	0.999955i	$-0.503011\pi$
$47$	−6.09555	−0.0189176	−0.00945881	+	0.999955i	$0.503011\pi$
$48$	−48.0000	−0.144338
$49$	416.903	1.21546
$50$	204.928	0.579623
$51$	317.317	0.871241
$52$	−257.286	−0.686137
$53$	−8.02587	−0.0208007	−0.0104004	−	0.999946i	$-0.503311\pi$
$53$	−8.02587	−0.0208007	−0.0104004	+	0.999946i	$0.503311\pi$
$54$	54.0000	0.136083
$55$	170.577	0.418192
$56$	−220.531	−0.526244
$57$	19.9686	0.0464019
$58$	364.987	0.826296
$59$	44.5960	0.0984052	0.0492026	−	0.998789i	$-0.484332\pi$
$59$	44.5960	0.0984052	0.0492026	+	0.998789i	$0.484332\pi$
$60$	−56.9668	−0.122573
$61$	−401.549	−0.842837	−0.421419	−	0.906866i	$-0.638468\pi$
$61$	−401.549	−0.842837	−0.421419	+	0.906866i	$0.638468\pi$
$62$	−30.4433	−0.0623598
$63$	248.097	0.496147
$64$	64.0000	0.125000
$65$	−305.349	−0.582675
$66$	215.591	0.402082
$67$	−258.326	−0.471037	−0.235519	−	0.971870i	$-0.575679\pi$
$67$	−258.326	−0.471037	−0.235519	+	0.971870i	$0.575679\pi$
$68$	−423.089	−0.754517
$69$	−19.3938	−0.0338369
$70$	−261.728	−0.446892
$71$	−1058.36	−1.76907	−0.884534	−	0.466476i	$-0.845524\pi$
$71$	−1058.36	−1.76907	−0.884534	+	0.466476i	$0.845524\pi$
$72$	−72.0000	−0.117851
$73$	547.136	0.877224	0.438612	−	0.898676i	$-0.355470\pi$
$73$	547.136	0.877224	0.438612	+	0.898676i	$0.355470\pi$
$74$	333.433	0.523794
$75$	307.391	0.473260
$76$	−26.6248	−0.0401852
$77$	990.509	1.46596
$78$	−385.929	−0.560229
$79$	179.688	0.255905	0.127952	−	0.991780i	$-0.459160\pi$
$79$	179.688	0.255905	0.127952	+	0.991780i	$0.459160\pi$
$80$	75.9557	0.106151
$81$	81.0000	0.111111
$82$	482.807	0.650209
$83$	−300.532	−0.397442	−0.198721	−	0.980056i	$-0.563679\pi$
$83$	−300.532	−0.397442	−0.198721	+	0.980056i	$0.563679\pi$
$84$	−330.796	−0.429676
$85$	−502.126	−0.640744
$86$	−1001.19	−1.25536
$87$	547.480	0.674667
$88$	−287.455	−0.348213
$89$	−1144.79	−1.36345	−0.681727	−	0.731606i	$-0.738771\pi$
$89$	−1144.79	−1.36345	−0.681727	+	0.731606i	$0.738771\pi$
$90$	−85.4502	−0.100080
$91$	−1773.11	−2.04255
$92$	25.8584	0.0293036
$93$	−45.6650	−0.0509165
$94$	12.1911	0.0133768
$95$	−31.5985	−0.0341257
$96$	96.0000	0.102062
$97$	1335.12	1.39753	0.698765	−	0.715351i	$-0.253733\pi$
$97$	1335.12	1.39753	0.698765	+	0.715351i	$0.253733\pi$
$98$	−833.806	−0.859460
$99$	323.386	0.328299
$100$	−409.855	−0.409855
$101$	−132.956	−0.130986	−0.0654930	−	0.997853i	$-0.520862\pi$
$101$	−132.956	−0.130986	−0.0654930	+	0.997853i	$0.520862\pi$
$102$	−634.634	−0.616060
$103$	1118.54	1.07003	0.535014	−	0.844843i	$-0.320306\pi$
$103$	1118.54	1.07003	0.535014	+	0.844843i	$0.320306\pi$
$104$	514.572	0.485172
$105$	−392.591	−0.364886
$106$	16.0517	0.0147083
$107$	762.248	0.688685	0.344342	−	0.938844i	$-0.388102\pi$
$107$	762.248	0.688685	0.344342	+	0.938844i	$0.388102\pi$
$108$	−108.000	−0.0962250
$109$	−1414.79	−1.24324	−0.621618	−	0.783321i	$-0.713524\pi$
$109$	−1414.79	−1.24324	−0.621618	+	0.783321i	$0.713524\pi$
$110$	−341.153	−0.295706
$111$	500.149	0.427676
$112$	441.061	0.372111
$113$	−1165.60	−0.970357	−0.485178	−	0.874415i	$-0.661245\pi$
$113$	−1165.60	−0.970357	−0.485178	+	0.874415i	$0.661245\pi$
$114$	−39.9372	−0.0328111
$115$	30.6890	0.0248849
$116$	−729.974	−0.584279
$117$	−578.893	−0.457425
$118$	−89.1920	−0.0695830
$119$	−2915.76	−2.24611
$120$	113.934	0.0866722
$121$	−39.9041	−0.0299805
$122$	803.098	0.595976
$123$	724.211	0.530894
$124$	60.8866	0.0440950
$125$	−1079.82	−0.772659
$126$	−496.194	−0.350829
$127$	−384.018	−0.268316	−0.134158	−	0.990960i	$-0.542833\pi$
$127$	−384.018	−0.268316	−0.134158	+	0.990960i	$0.542833\pi$
$128$	−128.000	−0.0883883
$129$	−1501.79	−1.02500
$130$	610.698	0.412014
$131$	131.000	0.0873704
$131$	131.000	0.0873704
$132$	−431.182	−0.284315
$133$	−183.487	−0.119627
$134$	516.651	0.333074
$135$	−128.175	−0.0817153
$136$	846.179	0.533524
$137$	783.707	0.488734	0.244367	−	0.969683i	$-0.421420\pi$
$137$	783.707	0.488734	0.244367	+	0.969683i	$0.421420\pi$
$138$	38.7877	0.0239263
$139$	−167.685	−0.102323	−0.0511614	−	0.998690i	$-0.516292\pi$
$139$	−167.685	−0.102323	−0.0511614	+	0.998690i	$0.516292\pi$
$140$	523.455	0.316000
$141$	18.2867	0.0109221
$142$	2116.71	1.25092
$143$	−2311.19	−1.35155
$144$	144.000	0.0833333
$145$	−866.339	−0.496176
$146$	−1094.27	−0.620291
$147$	−1250.71	−0.701746
$148$	−666.865	−0.370378
$149$	−584.887	−0.321583	−0.160791	−	0.986988i	$-0.551405\pi$
$149$	−584.887	−0.321583	−0.160791	+	0.986988i	$0.551405\pi$
$150$	−614.783	−0.334645
$151$	157.698	0.0849885	0.0424943	−	0.999097i	$-0.486470\pi$
$151$	157.698	0.0849885	0.0424943	+	0.999097i	$0.486470\pi$
$152$	53.2496	0.0284152
$153$	−951.951	−0.503011
$154$	−1981.02	−1.03659
$155$	72.2607	0.0374460
$156$	771.858	0.396142
$157$	−913.409	−0.464318	−0.232159	−	0.972678i	$-0.574579\pi$
$157$	−913.409	−0.464318	−0.232159	+	0.972678i	$0.574579\pi$
$158$	−359.376	−0.180952
$159$	24.0776	0.0120093
$160$	−151.911	−0.0750603
$161$	178.206	0.0872334
$162$	−162.000	−0.0785674
$163$	−2232.94	−1.07299	−0.536495	−	0.843903i	$-0.680252\pi$
$163$	−2232.94	−1.07299	−0.536495	+	0.843903i	$0.680252\pi$
$164$	−965.615	−0.459767
$165$	−511.730	−0.241443
$166$	601.064	0.281034
$167$	−2783.04	−1.28957	−0.644784	−	0.764365i	$-0.723053\pi$
$167$	−2783.04	−1.28957	−0.644784	+	0.764365i	$0.723053\pi$
$168$	661.592	0.303827
$169$	1940.25	0.883138
$170$	1004.25	0.453074
$171$	−59.9058	−0.0267901
$172$	2002.38	0.887677
$173$	−85.6841	−0.0376557	−0.0188279	−	0.999823i	$-0.505993\pi$
$173$	−85.6841	−0.0376557	−0.0188279	+	0.999823i	$0.505993\pi$
$174$	−1094.96	−0.477062
$175$	−2824.55	−1.22009
$176$	574.909	0.246224
$177$	−133.788	−0.0568143
$178$	2289.58	0.964108
$179$	−327.175	−0.136616	−0.0683078	−	0.997664i	$-0.521760\pi$
$179$	−327.175	−0.136616	−0.0683078	+	0.997664i	$0.521760\pi$
$180$	170.900	0.0707675
$181$	−1052.23	−0.432110	−0.216055	−	0.976381i	$-0.569319\pi$
$181$	−1052.23	−0.432110	−0.216055	+	0.976381i	$0.569319\pi$
$182$	3546.22	1.44430
$183$	1204.65	0.486612
$184$	−51.7169	−0.0207208
$185$	−791.441	−0.314529
$186$	91.3300	0.0360034
$187$	−3800.59	−1.48624
$188$	−24.3822	−0.00945881
$189$	−744.291	−0.286451
$190$	63.1971	0.0241305
$191$	2194.36	0.831301	0.415651	−	0.909524i	$-0.363554\pi$
$191$	2194.36	0.831301	0.415651	+	0.909524i	$0.363554\pi$
$192$	−192.000	−0.0721688
$193$	463.834	0.172992	0.0864961	−	0.996252i	$-0.472433\pi$
$193$	463.834	0.172992	0.0864961	+	0.996252i	$0.472433\pi$
$194$	−2670.23	−0.988203
$195$	916.047	0.336408
$196$	1667.61	0.607730
$197$	357.100	0.129149	0.0645744	−	0.997913i	$-0.479431\pi$
$197$	357.100	0.129149	0.0645744	+	0.997913i	$0.479431\pi$
$198$	−646.773	−0.232142
$199$	−2492.30	−0.887812	−0.443906	−	0.896073i	$-0.646408\pi$
$199$	−2492.30	−0.887812	−0.443906	+	0.896073i	$0.646408\pi$
$200$	819.710	0.289811
$201$	774.977	0.271954
$202$	265.911	0.0926211
$203$	−5030.68	−1.73933
$204$	1269.27	0.435620
$205$	−1146.00	−0.390439
$206$	−2237.08	−0.756625
$207$	58.1815	0.0195357
$208$	−1029.14	−0.343069
$209$	−239.170	−0.0791565
$210$	785.183	0.258013
$211$	3939.98	1.28549	0.642747	−	0.766078i	$-0.277795\pi$
$211$	3939.98	1.28549	0.642747	+	0.766078i	$0.277795\pi$
$212$	−32.1035	−0.0104004
$213$	3175.07	1.02137
$214$	−1524.50	−0.486974
$215$	2376.45	0.753824
$216$	216.000	0.0680414
$217$	419.605	0.131266
$218$	2829.59	0.879101
$219$	−1641.41	−0.506466
$220$	682.307	0.209096
$221$	6803.44	2.07081
$222$	−1000.30	−0.302413
$223$	298.984	0.0897822	0.0448911	−	0.998992i	$-0.485706\pi$
$223$	298.984	0.0897822	0.0448911	+	0.998992i	$0.485706\pi$
$224$	−882.123	−0.263122
$225$	−922.174	−0.273237
$226$	2331.20	0.686146
$227$	3830.16	1.11990	0.559948	−	0.828527i	$-0.310821\pi$
$227$	3830.16	1.11990	0.559948	+	0.828527i	$0.310821\pi$
$228$	79.8744	0.0232009
$229$	510.984	0.147453	0.0737265	−	0.997278i	$-0.476511\pi$
$229$	510.984	0.147453	0.0737265	+	0.997278i	$0.476511\pi$
$230$	−61.3780	−0.0175963
$231$	−2971.53	−0.846373
$232$	1459.95	0.413148
$233$	−4801.92	−1.35015	−0.675074	−	0.737750i	$-0.735888\pi$
$233$	−4801.92	−1.35015	−0.675074	+	0.737750i	$0.735888\pi$
$234$	1157.79	0.323448
$235$	−28.9370	−0.00803252
$236$	178.384	0.0492026
$237$	−539.063	−0.147747
$238$	5831.51	1.58824
$239$	−3414.16	−0.924032	−0.462016	−	0.886872i	$-0.652874\pi$
$239$	−3414.16	−0.924032	−0.462016	+	0.886872i	$0.652874\pi$
$240$	−227.867	−0.0612865
$241$	1884.35	0.503658	0.251829	−	0.967772i	$-0.418968\pi$
$241$	1884.35	0.503658	0.251829	+	0.967772i	$0.418968\pi$
$242$	79.8081	0.0211994
$243$	−243.000	−0.0641500
$244$	−1606.20	−0.421419
$245$	1979.13	0.516091
$246$	−1448.42	−0.375398
$247$	428.137	0.110290
$248$	−121.773	−0.0311799
$249$	901.596	0.229463
$250$	2159.65	0.546352
$251$	−6511.82	−1.63754	−0.818770	−	0.574121i	$-0.805344\pi$
$251$	−6511.82	−1.63754	−0.818770	+	0.574121i	$0.805344\pi$
$252$	992.388	0.248074
$253$	232.285	0.0577219
$254$	768.037	0.189728
$255$	1506.38	0.369933
$256$	256.000	0.0625000
$257$	−4587.42	−1.11344	−0.556722	−	0.830699i	$-0.687941\pi$
$257$	−4587.42	−1.11344	−0.556722	+	0.830699i	$0.687941\pi$
$258$	3003.58	0.724785
$259$	−4595.76	−1.10257
$260$	−1221.40	−0.291338
$261$	−1642.44	−0.389519
$262$	−262.000	−0.0617802
$263$	1645.67	0.385841	0.192921	−	0.981214i	$-0.438204\pi$
$263$	1645.67	0.385841	0.192921	+	0.981214i	$0.438204\pi$
$264$	862.364	0.201041
$265$	−38.1007	−0.00883209
$266$	366.974	0.0845889
$267$	3434.37	0.787191
$268$	−1033.30	−0.235519
$269$	3070.88	0.696041	0.348021	−	0.937487i	$-0.386854\pi$
$269$	3070.88	0.696041	0.348021	+	0.937487i	$0.386854\pi$
$270$	256.350	0.0577815
$271$	−2542.68	−0.569950	−0.284975	−	0.958535i	$-0.591985\pi$
$271$	−2542.68	−0.569950	−0.284975	+	0.958535i	$0.591985\pi$
$272$	−1692.36	−0.377258
$273$	5319.32	1.17927
$274$	−1567.41	−0.345587
$275$	−3681.71	−0.807329
$276$	−77.5753	−0.0169184
$277$	8159.97	1.76998	0.884991	−	0.465607i	$-0.154164\pi$
$277$	8159.97	1.76998	0.884991	+	0.465607i	$0.154164\pi$
$278$	335.370	0.0723531
$279$	136.995	0.0293967
$280$	−1046.91	−0.223446
$281$	−7283.85	−1.54633	−0.773164	−	0.634206i	$-0.781327\pi$
$281$	−7283.85	−1.54633	−0.773164	+	0.634206i	$0.781327\pi$
$282$	−36.5733	−0.00772308
$283$	5915.63	1.24257	0.621286	−	0.783584i	$-0.286611\pi$
$283$	5915.63	1.24257	0.621286	+	0.783584i	$0.286611\pi$
$284$	−4233.43	−0.884534
$285$	94.7956	0.0197025
$286$	4622.38	0.955688
$287$	−6654.61	−1.36867
$288$	−288.000	−0.0589256
$289$	6274.79	1.27718
$290$	1732.68	0.350849
$291$	−4005.35	−0.806864
$292$	2188.54	0.438612
$293$	−8978.01	−1.79011	−0.895053	−	0.445960i	$-0.852862\pi$
$293$	−8978.01	−1.79011	−0.895053	+	0.445960i	$0.852862\pi$
$294$	2501.42	0.496210
$295$	211.708	0.0417834
$296$	1333.73	0.261897
$297$	−970.159	−0.189543
$298$	1169.77	0.227393
$299$	−415.814	−0.0804251
$300$	1229.57	0.236630
$301$	13799.6	2.64251
$302$	−315.396	−0.0600960
$303$	398.867	0.0756248
$304$	−106.499	−0.0200926
$305$	−1906.25	−0.357873
$306$	1903.90	0.355683
$307$	843.729	0.156854	0.0784270	−	0.996920i	$-0.475010\pi$
$307$	843.729	0.156854	0.0784270	+	0.996920i	$0.475010\pi$
$308$	3962.04	0.732980
$309$	−3355.62	−0.617782
$310$	−144.521	−0.0264783
$311$	2256.30	0.411392	0.205696	−	0.978616i	$-0.434054\pi$
$311$	2256.30	0.411392	0.205696	+	0.978616i	$0.434054\pi$
$312$	−1543.72	−0.280114
$313$	4962.47	0.896151	0.448075	−	0.893996i	$-0.352110\pi$
$313$	4962.47	0.896151	0.448075	+	0.893996i	$0.352110\pi$
$314$	1826.82	0.328323
$315$	1177.77	0.210667
$316$	718.751	0.127952
$317$	801.022	0.141924	0.0709620	−	0.997479i	$-0.477393\pi$
$317$	801.022	0.141924	0.0709620	+	0.997479i	$0.477393\pi$
$318$	−48.1552	−0.00849186
$319$	−6557.32	−1.15091
$320$	303.823	0.0530757
$321$	−2286.74	−0.397612
$322$	−356.411	−0.0616833
$323$	704.042	0.121282
$324$	324.000	0.0555556
$325$	6590.62	1.12487
$326$	4465.88	0.758719
$327$	4244.38	0.717783
$328$	1931.23	0.325105
$329$	−168.032	−0.0281578
$330$	1023.46	0.170726
$331$	11740.9	1.94966	0.974828	−	0.222956i	$-0.0715707\pi$
$331$	11740.9	1.94966	0.974828	+	0.222956i	$0.0715707\pi$
$332$	−1202.13	−0.198721
$333$	−1500.45	−0.246919
$334$	5566.07	0.911862
$335$	−1226.33	−0.200005
$336$	−1323.18	−0.214838
$337$	11905.6	1.92445	0.962226	−	0.272254i	$-0.0877690\pi$
$337$	11905.6	1.92445	0.962226	+	0.272254i	$0.0877690\pi$
$338$	−3880.51	−0.624473
$339$	3496.80	0.560236
$340$	−2008.50	−0.320372
$341$	546.942	0.0868580
$342$	119.812	0.0189435
$343$	2037.23	0.320701
$344$	−4004.77	−0.627682
$345$	−92.0670	−0.0143673
$346$	171.368	0.0266266
$347$	−1407.86	−0.217804	−0.108902	−	0.994052i	$-0.534733\pi$
$347$	−1407.86	−0.217804	−0.108902	+	0.994052i	$0.534733\pi$
$348$	2189.92	0.337334
$349$	−3817.86	−0.585575	−0.292787	−	0.956178i	$-0.594583\pi$
$349$	−3817.86	−0.585575	−0.292787	+	0.956178i	$0.594583\pi$
$350$	5649.10	0.862735
$351$	1736.68	0.264094
$352$	−1149.82	−0.174107
$353$	3863.95	0.582599	0.291299	−	0.956632i	$-0.405912\pi$
$353$	3863.95	0.582599	0.291299	+	0.956632i	$0.405912\pi$
$354$	267.576	0.0401737
$355$	−5024.26	−0.751156
$356$	−4579.16	−0.681727
$357$	8747.27	1.29679
$358$	654.350	0.0966018
$359$	342.395	0.0503368	0.0251684	−	0.999683i	$-0.491988\pi$
$359$	342.395	0.0503368	0.0251684	+	0.999683i	$0.491988\pi$
$360$	−341.801	−0.0500402
$361$	−6814.69	−0.993541
$362$	2104.46	0.305548
$363$	119.712	0.0173093
$364$	−7092.43	−1.02128
$365$	2597.38	0.372474
$366$	−2409.29	−0.344087
$367$	−696.319	−0.0990397	−0.0495199	−	0.998773i	$-0.515769\pi$
$367$	−696.319	−0.0990397	−0.0495199	+	0.998773i	$0.515769\pi$
$368$	103.434	0.0146518
$369$	−2172.63	−0.306512
$370$	1582.88	0.222406
$371$	−221.244	−0.0309607
$372$	−182.660	−0.0254583
$373$	−9513.94	−1.32068	−0.660339	−	0.750968i	$-0.729587\pi$
$373$	−9513.94	−1.32068	−0.660339	+	0.750968i	$0.729587\pi$
$374$	7601.19	1.05093
$375$	3239.47	0.446095
$376$	48.7644	0.00668839
$377$	11738.3	1.60358
$378$	1488.58	0.202551
$379$	−1426.08	−0.193280	−0.0966398	−	0.995319i	$-0.530809\pi$
$379$	−1426.08	−0.193280	−0.0966398	+	0.995319i	$0.530809\pi$
$380$	−126.394	−0.0170629
$381$	1152.06	0.154912
$382$	−4388.73	−0.587819
$383$	8087.12	1.07894	0.539468	−	0.842006i	$-0.318625\pi$
$383$	8087.12	1.07894	0.539468	+	0.842006i	$0.318625\pi$
$384$	384.000	0.0510310
$385$	4702.17	0.622454
$386$	−927.668	−0.122324
$387$	4505.37	0.591784
$388$	5340.46	0.698765
$389$	−3201.57	−0.417290	−0.208645	−	0.977991i	$-0.566905\pi$
$389$	−3201.57	−0.417290	−0.208645	+	0.977991i	$0.566905\pi$
$390$	−1832.09	−0.237876
$391$	−683.777	−0.0884402
$392$	−3335.22	−0.429730
$393$	−393.000	−0.0504433
$394$	−714.199	−0.0913219
$395$	853.020	0.108658
$396$	1293.55	0.164149
$397$	−5512.83	−0.696930	−0.348465	−	0.937322i	$-0.613297\pi$
$397$	−5512.83	−0.696930	−0.348465	+	0.937322i	$0.613297\pi$
$398$	4984.60	0.627778
$399$	550.462	0.0690665
$400$	−1639.42	−0.204928
$401$	3319.19	0.413348	0.206674	−	0.978410i	$-0.433736\pi$
$401$	3319.19	0.413348	0.206674	+	0.978410i	$0.433736\pi$
$402$	−1549.95	−0.192300
$403$	−979.080	−0.121021
$404$	−531.823	−0.0654930
$405$	384.526	0.0471784
$406$	10061.4	1.22989
$407$	−5990.42	−0.729568
$408$	−2538.54	−0.308030
$409$	−10474.2	−1.26630	−0.633151	−	0.774028i	$-0.718239\pi$
$409$	−10474.2	−1.26630	−0.633151	+	0.774028i	$0.718239\pi$
$410$	2292.00	0.276082
$411$	−2351.12	−0.282171
$412$	4474.16	0.535014
$413$	1229.35	0.146470
$414$	−116.363	−0.0138138
$415$	−1426.69	−0.168756
$416$	2058.29	0.242586
$417$	503.055	0.0590761
$418$	478.339	0.0559721
$419$	3731.03	0.435019	0.217510	−	0.976058i	$-0.430207\pi$
$419$	3731.03	0.435019	0.217510	+	0.976058i	$0.430207\pi$
$420$	−1570.37	−0.182443
$421$	14106.5	1.63304	0.816519	−	0.577319i	$-0.195901\pi$
$421$	14106.5	1.63304	0.816519	+	0.577319i	$0.195901\pi$
$422$	−7879.96	−0.908982
$423$	−54.8600	−0.00630587
$424$	64.2070	0.00735416
$425$	10837.8	1.23697
$426$	−6350.14	−0.722219
$427$	−11069.2	−1.25451
$428$	3048.99	0.344342
$429$	6933.57	0.780316
$430$	−4752.89	−0.533034
$431$	−648.073	−0.0724283	−0.0362141	−	0.999344i	$-0.511530\pi$
$431$	−648.073	−0.0724283	−0.0362141	+	0.999344i	$0.511530\pi$
$432$	−432.000	−0.0481125
$433$	−12721.4	−1.41189	−0.705947	−	0.708265i	$-0.749478\pi$
$433$	−12721.4	−1.41189	−0.705947	+	0.708265i	$0.749478\pi$
$434$	−839.211	−0.0928189
$435$	2599.02	0.286467
$436$	−5659.18	−0.621618
$437$	−43.0298	−0.00471028
$438$	3282.81	0.358125
$439$	−7181.74	−0.780788	−0.390394	−	0.920648i	$-0.627661\pi$
$439$	−7181.74	−0.780788	−0.390394	+	0.920648i	$0.627661\pi$
$440$	−1364.61	−0.147853
$441$	3752.13	0.405154
$442$	−13606.9	−1.46428
$443$	6831.15	0.732636	0.366318	−	0.930490i	$-0.380618\pi$
$443$	6831.15	0.732636	0.366318	+	0.930490i	$0.380618\pi$
$444$	2000.60	0.213838
$445$	−5434.58	−0.578930
$446$	−597.968	−0.0634856
$447$	1754.66	0.185666
$448$	1764.25	0.186055
$449$	−2958.87	−0.310997	−0.155499	−	0.987836i	$-0.549698\pi$
$449$	−2958.87	−0.310997	−0.155499	+	0.987836i	$0.549698\pi$
$450$	1844.35	0.193208
$451$	−8674.07	−0.905646
$452$	−4662.40	−0.485178
$453$	−473.093	−0.0490681
$454$	−7660.32	−0.791887
$455$	−8417.35	−0.867278
$456$	−159.749	−0.0164055
$457$	6808.35	0.696896	0.348448	−	0.937328i	$-0.386709\pi$
$457$	6808.35	0.696896	0.348448	+	0.937328i	$0.386709\pi$
$458$	−1021.97	−0.104265
$459$	2855.85	0.290414
$460$	122.756	0.0124425
$461$	−18404.2	−1.85937	−0.929684	−	0.368359i	$-0.879920\pi$
$461$	−18404.2	−1.85937	−0.929684	+	0.368359i	$0.879920\pi$
$462$	5943.05	0.598476
$463$	−9405.88	−0.944121	−0.472061	−	0.881566i	$-0.656490\pi$
$463$	−9405.88	−0.944121	−0.472061	+	0.881566i	$0.656490\pi$
$464$	−2919.90	−0.292140
$465$	−216.782	−0.0216194
$466$	9603.84	0.954698
$467$	−8945.27	−0.886377	−0.443188	−	0.896428i	$-0.646153\pi$
$467$	−8945.27	−0.886377	−0.443188	+	0.896428i	$0.646153\pi$
$468$	−2315.57	−0.228712
$469$	−7121.09	−0.701112
$470$	57.8740	0.00567985
$471$	2740.23	0.268074
$472$	−356.768	−0.0347915
$473$	17987.3	1.74854
$474$	1078.13	0.104473
$475$	682.020	0.0658805
$476$	−11663.0	−1.12305
$477$	−72.2328	−0.00693357
$478$	6828.32	0.653390
$479$	−1295.96	−0.123619	−0.0618097	−	0.998088i	$-0.519687\pi$
$479$	−1295.96	−0.123619	−0.0618097	+	0.998088i	$0.519687\pi$
$480$	455.734	0.0433361
$481$	10723.4	1.01652
$482$	−3768.70	−0.356140
$483$	−534.617	−0.0503642
$484$	−159.616	−0.0149903
$485$	6338.10	0.593399
$486$	486.000	0.0453609
$487$	−12379.2	−1.15186	−0.575928	−	0.817500i	$-0.695359\pi$
$487$	−12379.2	−1.15186	−0.575928	+	0.817500i	$0.695359\pi$
$488$	3212.39	0.297988
$489$	6698.82	0.619491
$490$	−3958.27	−0.364931
$491$	12584.5	1.15668	0.578341	−	0.815795i	$-0.303700\pi$
$491$	12584.5	1.15668	0.578341	+	0.815795i	$0.303700\pi$
$492$	2896.84	0.265447
$493$	19302.8	1.76339
$494$	−856.274	−0.0779870
$495$	1535.19	0.139397
$496$	243.547	0.0220475
$497$	−29175.0	−2.63316
$498$	−1803.19	−0.162255
$499$	5269.08	0.472698	0.236349	−	0.971668i	$-0.424049\pi$
$499$	5269.08	0.472698	0.236349	+	0.971668i	$0.424049\pi$
$500$	−4319.29	−0.386329
$501$	8349.11	0.744532
$502$	13023.6	1.15792
$503$	−4054.03	−0.359364	−0.179682	−	0.983725i	$-0.557507\pi$
$503$	−4054.03	−0.359364	−0.179682	+	0.983725i	$0.557507\pi$
$504$	−1984.78	−0.175415
$505$	−631.171	−0.0556173
$506$	−464.571	−0.0408156
$507$	−5820.76	−0.509880
$508$	−1536.07	−0.134158
$509$	−5252.98	−0.457435	−0.228717	−	0.973493i	$-0.573453\pi$
$509$	−5252.98	−0.457435	−0.228717	+	0.973493i	$0.573453\pi$
$510$	−3012.76	−0.261582
$511$	15082.5	1.30570
$512$	−512.000	−0.0441942
$513$	179.718	0.0154673
$514$	9174.84	0.787324
$515$	5309.97	0.454340
$516$	−6007.15	−0.512500
$517$	−219.024	−0.0186319
$518$	9191.52	0.779637
$519$	257.052	0.0217405
$520$	2442.79	0.206007
$521$	16954.8	1.42573	0.712863	−	0.701303i	$-0.247398\pi$
$521$	16954.8	1.42573	0.712863	+	0.701303i	$0.247398\pi$
$522$	3284.88	0.275432
$523$	−14829.1	−1.23983	−0.619917	−	0.784668i	$-0.712834\pi$
$523$	−14829.1	−1.23983	−0.619917	+	0.784668i	$0.712834\pi$
$524$	524.000	0.0436852
$525$	8473.65	0.704420
$526$	−3291.34	−0.272831
$527$	−1610.03	−0.133082
$528$	−1724.73	−0.142157
$529$	−12125.2	−0.996565
$530$	76.2013	0.00624523
$531$	401.364	0.0328017
$532$	−733.949	−0.0598134
$533$	15527.4	1.26185
$534$	−6868.74	−0.556628
$535$	3618.57	0.292419
$536$	2066.61	0.166537
$537$	981.525	0.0788751
$538$	−6141.77	−0.492175
$539$	14980.1	1.19710
$540$	−512.701	−0.0408577
$541$	18747.5	1.48987	0.744934	−	0.667138i	$-0.232481\pi$
$541$	18747.5	1.48987	0.744934	+	0.667138i	$0.232481\pi$
$542$	5085.35	0.403016
$543$	3156.70	0.249479
$544$	3384.72	0.266762
$545$	−6716.36	−0.527885
$546$	−10638.6	−0.833868
$547$	7024.39	0.549070	0.274535	−	0.961577i	$-0.411476\pi$
$547$	7024.39	0.549070	0.274535	+	0.961577i	$0.411476\pi$
$548$	3134.83	0.244367
$549$	−3613.94	−0.280946
$550$	7363.42	0.570868
$551$	1214.71	0.0939175
$552$	155.151	0.0119631
$553$	4953.34	0.380899
$554$	−16319.9	−1.25157
$555$	2374.32	0.181594
$556$	−670.740	−0.0511614
$557$	25715.4	1.95619	0.978093	−	0.208169i	$-0.0667504\pi$
$557$	25715.4	1.95619	0.978093	+	0.208169i	$0.0667504\pi$
$558$	−273.990	−0.0207866
$559$	−32199.1	−2.43627
$560$	2093.82	0.158000
$561$	11401.8	0.858081
$562$	14567.7	1.09342
$563$	−9142.06	−0.684355	−0.342177	−	0.939635i	$-0.611164\pi$
$563$	−9142.06	−0.684355	−0.342177	+	0.939635i	$0.611164\pi$
$564$	73.1467	0.00546105
$565$	−5533.37	−0.412019
$566$	−11831.3	−0.878630
$567$	2232.87	0.165382
$568$	8466.85	0.625460
$569$	−20135.5	−1.48352	−0.741762	−	0.670663i	$-0.766010\pi$
$569$	−20135.5	−1.48352	−0.741762	+	0.670663i	$0.766010\pi$
$570$	−189.591	−0.0139318
$571$	−10419.1	−0.763617	−0.381809	−	0.924241i	$-0.624699\pi$
$571$	−10419.1	−0.763617	−0.381809	+	0.924241i	$0.624699\pi$
$572$	−9244.75	−0.675774
$573$	−6583.09	−0.479952
$574$	13309.2	0.967799
$575$	−662.389	−0.0480409
$576$	576.000	0.0416667
$577$	−14692.8	−1.06009	−0.530043	−	0.847971i	$-0.677824\pi$
$577$	−14692.8	−1.06009	−0.530043	+	0.847971i	$0.677824\pi$
$578$	−12549.6	−0.903104
$579$	−1391.50	−0.0998771
$580$	−3465.36	−0.248088
$581$	−8284.57	−0.591569
$582$	8010.69	0.570539
$583$	−288.384	−0.0204865
$584$	−4377.08	−0.310146
$585$	−2748.14	−0.194225
$586$	17956.0	1.26580
$587$	19592.4	1.37762	0.688811	−	0.724941i	$-0.258133\pi$
$587$	19592.4	1.37762	0.688811	+	0.724941i	$0.258133\pi$
$588$	−5002.84	−0.350873
$589$	−101.318	−0.00708787
$590$	−423.415	−0.0295453
$591$	−1071.30	−0.0745640
$592$	−2667.46	−0.185189
$593$	−2472.54	−0.171223	−0.0856113	−	0.996329i	$-0.527284\pi$
$593$	−2472.54	−0.171223	−0.0856113	+	0.996329i	$0.527284\pi$
$594$	1940.32	0.134027
$595$	−13841.8	−0.953710
$596$	−2339.55	−0.160791
$597$	7476.90	0.512578
$598$	831.627	0.0568692
$599$	20997.2	1.43226	0.716128	−	0.697969i	$-0.245913\pi$
$599$	20997.2	1.43226	0.716128	+	0.697969i	$0.245913\pi$
$600$	−2459.13	−0.167323
$601$	−23915.8	−1.62320	−0.811602	−	0.584211i	$-0.801404\pi$
$601$	−23915.8	−1.62320	−0.811602	+	0.584211i	$0.801404\pi$
$602$	−27599.2	−1.86854
$603$	−2324.93	−0.157012
$604$	630.791	0.0424943
$605$	−189.434	−0.0127299
$606$	−797.734	−0.0534748
$607$	2343.68	0.156717	0.0783584	−	0.996925i	$-0.475032\pi$
$607$	2343.68	0.156717	0.0783584	+	0.996925i	$0.475032\pi$
$608$	212.999	0.0142076
$609$	15092.0	1.00420
$610$	3812.49	0.253055
$611$	392.075	0.0259602
$612$	−3807.81	−0.251506
$613$	6412.29	0.422496	0.211248	−	0.977433i	$-0.432247\pi$
$613$	6412.29	0.422496	0.211248	+	0.977433i	$0.432247\pi$
$614$	−1687.46	−0.110912
$615$	3438.00	0.225420
$616$	−7924.07	−0.518295
$617$	983.512	0.0641729	0.0320865	−	0.999485i	$-0.489785\pi$
$617$	983.512	0.0641729	0.0320865	+	0.999485i	$0.489785\pi$
$618$	6711.24	0.436838
$619$	−19031.0	−1.23574	−0.617868	−	0.786282i	$-0.712003\pi$
$619$	−19031.0	−1.23574	−0.617868	+	0.786282i	$0.712003\pi$
$620$	289.043	0.0187230
$621$	−174.545	−0.0112790
$622$	−4512.60	−0.290898
$623$	−31557.7	−2.02942
$624$	3087.43	0.198071
$625$	7681.80	0.491635
$626$	−9924.93	−0.633674
$627$	717.509	0.0457010
$628$	−3653.63	−0.232159
$629$	17634.0	1.11783
$630$	−2355.55	−0.148964
$631$	19726.9	1.24456	0.622280	−	0.782795i	$-0.286207\pi$
$631$	19726.9	1.24456	0.622280	+	0.782795i	$0.286207\pi$
$632$	−1437.50	−0.0904759
$633$	−11819.9	−0.742181
$634$	−1602.04	−0.100355
$635$	−1823.02	−0.113928
$636$	96.3104	0.00600465
$637$	−26815.8	−1.66795
$638$	13114.6	0.813815
$639$	−9525.21	−0.589689
$640$	−607.646	−0.0375302
$641$	−30783.8	−1.89686	−0.948429	−	0.316990i	$-0.897328\pi$
$641$	−30783.8	−1.89686	−0.948429	+	0.316990i	$0.897328\pi$
$642$	4573.49	0.281154
$643$	−1917.17	−0.117583	−0.0587916	−	0.998270i	$-0.518725\pi$
$643$	−1917.17	−0.117583	−0.0587916	+	0.998270i	$0.518725\pi$
$644$	712.823	0.0436167
$645$	−7129.34	−0.435221
$646$	−1408.08	−0.0857591
$647$	11843.0	0.719624	0.359812	−	0.933025i	$-0.382841\pi$
$647$	11843.0	0.719624	0.359812	+	0.933025i	$0.382841\pi$
$648$	−648.000	−0.0392837
$649$	1602.42	0.0969188
$650$	−13181.2	−0.795402
$651$	−1258.82	−0.0757863
$652$	−8931.76	−0.536495
$653$	8292.16	0.496933	0.248466	−	0.968640i	$-0.420073\pi$
$653$	8292.16	0.496933	0.248466	+	0.968640i	$0.420073\pi$
$654$	−8488.77	−0.507549
$655$	621.887	0.0370979
$656$	−3862.46	−0.229884
$657$	4924.22	0.292408
$658$	336.064	0.0199106
$659$	2458.49	0.145325	0.0726624	−	0.997357i	$-0.476850\pi$
$659$	2458.49	0.145325	0.0726624	+	0.997357i	$0.476850\pi$
$660$	−2046.92	−0.120722
$661$	−2135.22	−0.125643	−0.0628217	−	0.998025i	$-0.520010\pi$
$661$	−2135.22	−0.125643	−0.0628217	+	0.998025i	$0.520010\pi$
$662$	−23481.7	−1.37862
$663$	−20410.3	−1.19558
$664$	2404.26	0.140517
$665$	−871.056	−0.0507941
$666$	3000.89	0.174598
$667$	−1179.75	−0.0684859
$668$	−11132.1	−0.644784
$669$	−896.951	−0.0518358
$670$	2452.66	0.141425
$671$	−14428.4	−0.830107
$672$	2646.37	0.151914
$673$	−24368.1	−1.39572	−0.697861	−	0.716234i	$-0.745865\pi$
$673$	−24368.1	−1.39572	−0.697861	+	0.716234i	$0.745865\pi$
$674$	−23811.2	−1.36079
$675$	2766.52	0.157753
$676$	7761.02	0.441569
$677$	−31148.8	−1.76831	−0.884156	−	0.467192i	$-0.845266\pi$
$677$	−31148.8	−1.76831	−0.884156	+	0.467192i	$0.845266\pi$
$678$	−6993.59	−0.396146
$679$	36804.2	2.08014
$680$	4017.01	0.226537
$681$	−11490.5	−0.646573
$682$	−1093.88	−0.0614179
$683$	12312.0	0.689760	0.344880	−	0.938647i	$-0.387920\pi$
$683$	12312.0	0.689760	0.344880	+	0.938647i	$0.387920\pi$
$684$	−239.623	−0.0133951
$685$	3720.44	0.207519
$686$	−4074.47	−0.226770
$687$	−1532.95	−0.0851320
$688$	8009.54	0.443838
$689$	516.236	0.0285443
$690$	184.134	0.0101592
$691$	−3262.56	−0.179614	−0.0898071	−	0.995959i	$-0.528625\pi$
$691$	−3262.56	−0.179614	−0.0898071	+	0.995959i	$0.528625\pi$
$692$	−342.736	−0.0188279
$693$	8914.58	0.488653
$694$	2815.72	0.154011
$695$	−796.040	−0.0434468
$696$	−4379.84	−0.238531
$697$	25533.8	1.38761
$698$	7635.73	0.414064
$699$	14405.8	0.779508
$700$	−11298.2	−0.610046
$701$	22364.4	1.20498	0.602490	−	0.798126i	$-0.294175\pi$
$701$	22364.4	1.20498	0.602490	+	0.798126i	$0.294175\pi$
$702$	−3473.36	−0.186743
$703$	1109.70	0.0595349
$704$	2299.64	0.123112
$705$	86.8110	0.00463758
$706$	−7727.90	−0.411959
$707$	−3665.10	−0.194965
$708$	−535.152	−0.0284071
$709$	2009.31	0.106434	0.0532168	−	0.998583i	$-0.483053\pi$
$709$	2009.31	0.106434	0.0532168	+	0.998583i	$0.483053\pi$
$710$	10048.5	0.531147
$711$	1617.19	0.0853015
$712$	9158.32	0.482054
$713$	98.4021	0.00516857
$714$	−17494.5	−0.916970
$715$	−10971.7	−0.573874
$716$	−1308.70	−0.0683078
$717$	10242.5	0.533490
$718$	−684.789	−0.0355935
$719$	4508.59	0.233855	0.116928	−	0.993140i	$-0.462695\pi$
$719$	4508.59	0.233855	0.116928	+	0.993140i	$0.462695\pi$
$720$	683.601	0.0353838
$721$	30834.0	1.59268
$722$	13629.4	0.702539
$723$	−5653.05	−0.290787
$724$	−4208.93	−0.216055
$725$	18699.0	0.957879
$726$	−239.424	−0.0122395
$727$	24137.6	1.23138	0.615690	−	0.787989i	$-0.288878\pi$
$727$	24137.6	1.23138	0.615690	+	0.787989i	$0.288878\pi$
$728$	14184.9	0.722151
$729$	729.000	0.0370370
$730$	−5194.76	−0.263379
$731$	−52949.2	−2.67907
$732$	4818.59	0.243306
$733$	−27168.3	−1.36901	−0.684505	−	0.729008i	$-0.739982\pi$
$733$	−27168.3	−1.36901	−0.684505	+	0.729008i	$0.739982\pi$
$734$	1392.64	0.0700317
$735$	−5937.40	−0.297965
$736$	−206.868	−0.0103604
$737$	−9282.11	−0.463923
$738$	4345.27	0.216736
$739$	−15532.6	−0.773173	−0.386586	−	0.922253i	$-0.626346\pi$
$739$	−15532.6	−0.773173	−0.386586	+	0.922253i	$0.626346\pi$
$740$	−3165.76	−0.157265
$741$	−1284.41	−0.0636761
$742$	442.488	0.0218925
$743$	10291.2	0.508137	0.254068	−	0.967186i	$-0.418231\pi$
$743$	10291.2	0.508137	0.254068	+	0.967186i	$0.418231\pi$
$744$	365.320	0.0180017
$745$	−2776.60	−0.136546
$746$	19027.9	0.933860
$747$	−2704.79	−0.132481
$748$	−15202.4	−0.743120
$749$	21012.4	1.02507
$750$	−6478.94	−0.315437
$751$	−21547.7	−1.04699	−0.523494	−	0.852029i	$-0.675372\pi$
$751$	−21547.7	−1.04699	−0.523494	+	0.852029i	$0.675372\pi$
$752$	−97.5289	−0.00472940
$753$	19535.5	0.945435
$754$	−23476.5	−1.13390
$755$	748.628	0.0360866
$756$	−2977.16	−0.143225
$757$	22502.1	1.08039	0.540194	−	0.841540i	$-0.318351\pi$
$757$	22502.1	1.08039	0.540194	+	0.841540i	$0.318351\pi$
$758$	2852.17	0.136669
$759$	−696.856	−0.0333258
$760$	252.788	0.0120653
$761$	5908.54	0.281451	0.140726	−	0.990049i	$-0.455056\pi$
$761$	5908.54	0.281451	0.140726	+	0.990049i	$0.455056\pi$
$762$	−2304.11	−0.109540
$763$	−39000.7	−1.85049
$764$	8777.45	0.415651
$765$	−4519.13	−0.213581
$766$	−16174.2	−0.762923
$767$	−2868.48	−0.135039
$768$	−768.000	−0.0360844
$769$	−37198.6	−1.74436	−0.872181	−	0.489183i	$-0.837295\pi$
$769$	−37198.6	−1.74436	−0.872181	+	0.489183i	$0.837295\pi$
$770$	−9404.35	−0.440142
$771$	13762.3	0.642848
$772$	1855.34	0.0864961
$773$	9536.04	0.443709	0.221855	−	0.975080i	$-0.428789\pi$
$773$	9536.04	0.443709	0.221855	+	0.975080i	$0.428789\pi$
$774$	−9010.73	−0.418455
$775$	−1559.67	−0.0722903
$776$	−10680.9	−0.494102
$777$	13787.3	0.636571
$778$	6403.13	0.295069
$779$	1606.83	0.0739034
$780$	3664.19	0.168204
$781$	−38028.7	−1.74235
$782$	1367.55	0.0625366
$783$	4927.32	0.224889
$784$	6670.45	0.303865
$785$	−4336.16	−0.197152
$786$	786.000	0.0356688
$787$	5803.36	0.262856	0.131428	−	0.991326i	$-0.458044\pi$
$787$	5803.36	0.262856	0.131428	+	0.991326i	$0.458044\pi$
$788$	1428.40	0.0645744
$789$	−4937.01	−0.222766
$790$	−1706.04	−0.0768331
$791$	−32131.3	−1.44432
$792$	−2587.09	−0.116071
$793$	25828.2	1.15660
$794$	11025.7	0.492804
$795$	114.302	0.00509921
$796$	−9969.20	−0.443906
$797$	26394.5	1.17307	0.586537	−	0.809922i	$-0.300491\pi$
$797$	26394.5	1.17307	0.586537	+	0.809922i	$0.300491\pi$
$798$	−1100.92	−0.0488374
$799$	644.741	0.0285473
$800$	3278.84	0.144906
$801$	−10303.1	−0.454485
$802$	−6638.39	−0.292281
$803$	19659.6	0.863974
$804$	3099.91	0.135977
$805$	845.984	0.0370398
$806$	1958.16	0.0855747
$807$	−9212.65	−0.401860
$808$	1063.65	0.0463105
$809$	17.9118	0.000778424 0	0.000389212	−	1.00000i	$-0.499876\pi$
$809$	17.9118	0.000778424 0	0.000389212		1.00000i	$0.499876\pi$
$810$	−769.051	−0.0333601
$811$	28719.3	1.24349	0.621745	−	0.783219i	$-0.286424\pi$
$811$	28719.3	1.24349	0.621745	+	0.783219i	$0.286424\pi$
$812$	−20122.7	−0.869666
$813$	7628.03	0.329061
$814$	11980.8	0.515883
$815$	−10600.3	−0.455597
$816$	5077.07	0.217810
$817$	−3332.07	−0.142686
$818$	20948.5	0.895411
$819$	−15958.0	−0.680851
$820$	−4584.00	−0.195220
$821$	22548.1	0.958505	0.479253	−	0.877677i	$-0.340908\pi$
$821$	22548.1	0.958505	0.479253	+	0.877677i	$0.340908\pi$
$822$	4702.24	0.199525
$823$	−11962.4	−0.506663	−0.253331	−	0.967380i	$-0.581526\pi$
$823$	−11962.4	−0.506663	−0.253331	+	0.967380i	$0.581526\pi$
$824$	−8948.32	−0.378312
$825$	11045.1	0.466112
$826$	−2458.70	−0.103570
$827$	25303.6	1.06396	0.531979	−	0.846758i	$-0.321449\pi$
$827$	25303.6	1.06396	0.531979	+	0.846758i	$0.321449\pi$
$828$	232.726	0.00976786
$829$	10265.9	0.430096	0.215048	−	0.976604i	$-0.431009\pi$
$829$	10265.9	0.430096	0.215048	+	0.976604i	$0.431009\pi$
$830$	2853.39	0.119328
$831$	−24479.9	−1.02190
$832$	−4116.58	−0.171534
$833$	−44096.8	−1.83417
$834$	−1006.11	−0.0417731
$835$	−13211.7	−0.547557
$836$	−956.678	−0.0395782
$837$	−410.985	−0.0169722
$838$	−7462.07	−0.307605
$839$	−29647.4	−1.21995	−0.609976	−	0.792420i	$-0.708821\pi$
$839$	−29647.4	−1.21995	−0.609976	+	0.792420i	$0.708821\pi$
$840$	3140.73	0.129007
$841$	8914.88	0.365529
$842$	−28213.0	−1.15473
$843$	21851.6	0.892773
$844$	15759.9	0.642747
$845$	9210.84	0.374985
$846$	109.720	0.00445893
$847$	−1100.01	−0.0446243
$848$	−128.414	−0.00520018
$849$	−17746.9	−0.717399
$850$	−21675.7	−0.874670
$851$	−1077.76	−0.0434137
$852$	12700.3	0.510686
$853$	9226.08	0.370334	0.185167	−	0.982707i	$-0.440717\pi$
$853$	9226.08	0.370334	0.185167	+	0.982707i	$0.440717\pi$
$854$	22138.5	0.887076
$855$	−284.387	−0.0113752
$856$	−6097.98	−0.243487
$857$	−48259.9	−1.92360	−0.961800	−	0.273753i	$-0.911735\pi$
$857$	−48259.9	−1.92360	−0.961800	+	0.273753i	$0.911735\pi$
$858$	−13867.1	−0.551767
$859$	37115.5	1.47423	0.737115	−	0.675767i	$-0.236187\pi$
$859$	37115.5	1.47423	0.737115	+	0.675767i	$0.236187\pi$
$860$	9505.78	0.376912
$861$	19963.8	0.790204
$862$	1296.15	0.0512145
$863$	32430.6	1.27920	0.639600	−	0.768708i	$-0.279100\pi$
$863$	32430.6	1.27920	0.639600	+	0.768708i	$0.279100\pi$
$864$	864.000	0.0340207
$865$	−406.762	−0.0159888
$866$	25442.7	0.998360
$867$	−18824.4	−0.737381
$868$	1678.42	0.0656329
$869$	6456.51	0.252039
$870$	−5198.03	−0.202563
$871$	16615.9	0.646393
$872$	11318.4	0.439550
$873$	12016.0	0.465843
$874$	86.0596	0.00333067
$875$	−29766.8	−1.15006
$876$	−6565.63	−0.253233
$877$	−35449.1	−1.36491	−0.682457	−	0.730925i	$-0.739089\pi$
$877$	−35449.1	−1.36491	−0.682457	+	0.730925i	$0.739089\pi$
$878$	14363.5	0.552100
$879$	26934.0	1.03352
$880$	2729.23	0.104548
$881$	2298.43	0.0878958	0.0439479	−	0.999034i	$-0.486006\pi$
$881$	2298.43	0.0878958	0.0439479	+	0.999034i	$0.486006\pi$
$882$	−7504.25	−0.286487
$883$	17689.0	0.674158	0.337079	−	0.941476i	$-0.390561\pi$
$883$	17689.0	0.674158	0.337079	+	0.941476i	$0.390561\pi$
$884$	27213.7	1.03540
$885$	−635.123	−0.0241236
$886$	−13662.3	−0.518052
$887$	30941.2	1.17125	0.585627	−	0.810581i	$-0.300848\pi$
$887$	30941.2	1.17125	0.585627	+	0.810581i	$0.300848\pi$
$888$	−4001.19	−0.151206
$889$	−10586.0	−0.399373
$890$	10869.2	0.409365
$891$	2910.48	0.109433
$892$	1195.94	0.0448911
$893$	40.5733	0.00152042
$894$	−3509.32	−0.131286
$895$	−1553.17	−0.0580077
$896$	−3528.49	−0.131561
$897$	1247.44	0.0464335
$898$	5917.74	0.219908
$899$	−2777.85	−0.103055
$900$	−3688.70	−0.136618
$901$	848.915	0.0313890
$902$	17348.1	0.640388
$903$	−41398.8	−1.52565
$904$	9324.79	0.343073
$905$	−4995.19	−0.183476
$906$	946.187	0.0346964
$907$	−20790.3	−0.761113	−0.380557	−	0.924758i	$-0.624268\pi$
$907$	−20790.3	−0.761113	−0.380557	+	0.924758i	$0.624268\pi$
$908$	15320.6	0.559948
$909$	−1196.60	−0.0436620
$910$	16834.7	0.613258
$911$	12884.2	0.468576	0.234288	−	0.972167i	$-0.424724\pi$
$911$	12884.2	0.468576	0.234288	+	0.972167i	$0.424724\pi$
$912$	319.498	0.0116005
$913$	−10798.7	−0.391439
$914$	−13616.7	−0.492780
$915$	5718.74	0.206618
$916$	2043.93	0.0737265
$917$	3611.19	0.130046
$918$	−5711.71	−0.205353
$919$	40556.4	1.45575	0.727873	−	0.685712i	$-0.240509\pi$
$919$	40556.4	1.45575	0.727873	+	0.685712i	$0.240509\pi$
$920$	−245.512	−0.00879814
$921$	−2531.19	−0.0905597
$922$	36808.4	1.31477
$923$	68075.1	2.42765
$924$	−11886.1	−0.423186
$925$	17082.4	0.607206
$926$	18811.8	0.667595
$927$	10066.9	0.356676
$928$	5839.79	0.206574
$929$	48125.8	1.69963	0.849814	−	0.527082i	$-0.176714\pi$
$929$	48125.8	1.69963	0.849814	+	0.527082i	$0.176714\pi$
$930$	433.564	0.0152872
$931$	−2774.99	−0.0976871
$932$	−19207.7	−0.675074
$933$	−6768.90	−0.237518
$934$	17890.5	0.626763
$935$	−18042.3	−0.631066
$936$	4631.15	0.161724
$937$	12743.9	0.444316	0.222158	−	0.975011i	$-0.428690\pi$
$937$	12743.9	0.444316	0.222158	+	0.975011i	$0.428690\pi$
$938$	14242.2	0.495761
$939$	−14887.4	−0.517393
$940$	−115.748	−0.00401626
$941$	28791.5	0.997425	0.498712	−	0.866767i	$-0.333806\pi$
$941$	28791.5	0.997425	0.498712	+	0.866767i	$0.333806\pi$
$942$	−5480.45	−0.189557
$943$	−1560.58	−0.0538913
$944$	713.536	0.0246013
$945$	−3533.32	−0.121629
$946$	−35974.7	−1.23640
$947$	52867.5	1.81411	0.907055	−	0.421012i	$-0.138325\pi$
$947$	52867.5	1.81411	0.907055	+	0.421012i	$0.138325\pi$
$948$	−2156.25	−0.0738733
$949$	−35192.6	−1.20379
$950$	−1364.04	−0.0465845
$951$	−2403.07	−0.0819398
$952$	23326.1	0.794120
$953$	28127.1	0.956059	0.478030	−	0.878344i	$-0.341351\pi$
$953$	28127.1	0.956059	0.478030	+	0.878344i	$0.341351\pi$
$954$	144.466	0.00490278
$955$	10417.1	0.352975
$956$	−13656.6	−0.462016
$957$	19672.0	0.664477
$958$	2591.91	0.0874121
$959$	21603.9	0.727453
$960$	−911.468	−0.0306432
$961$	−29559.3	−0.992223
$962$	−21446.9	−0.718790
$963$	6860.23	0.229562
$964$	7537.40	0.251829
$965$	2201.93	0.0734534
$966$	1069.23	0.0356129
$967$	35420.7	1.17793	0.588963	−	0.808160i	$-0.299536\pi$
$967$	35420.7	1.17793	0.588963	+	0.808160i	$0.299536\pi$
$968$	319.232	0.0105997
$969$	−2112.13	−0.0700220
$970$	−12676.2	−0.419596
$971$	−18794.9	−0.621173	−0.310586	−	0.950545i	$-0.600525\pi$
$971$	−18794.9	−0.621173	−0.310586	+	0.950545i	$0.600525\pi$
$972$	−972.000	−0.0320750
$973$	−4622.46	−0.152302
$974$	24758.3	0.814485
$975$	−19771.9	−0.649443
$976$	−6424.78	−0.210709
$977$	23453.7	0.768013	0.384007	−	0.923330i	$-0.374544\pi$
$977$	23453.7	0.768013	0.384007	+	0.923330i	$0.374544\pi$
$978$	−13397.6	−0.438046
$979$	−41134.4	−1.34286
$980$	7916.54	0.258045
$981$	−12733.1	−0.414412
$982$	−25169.0	−0.817898
$983$	−10432.9	−0.338513	−0.169256	−	0.985572i	$-0.554137\pi$
$983$	−10432.9	−0.338513	−0.169256	+	0.985572i	$0.554137\pi$
$984$	−5793.69	−0.187699
$985$	1695.23	0.0548372
$986$	−38605.5	−1.24691
$987$	504.096	0.0162569
$988$	1712.55	0.0551452
$989$	3236.16	0.104048
$990$	−3070.38	−0.0985688
$991$	22691.5	0.727366	0.363683	−	0.931523i	$-0.381519\pi$
$991$	22691.5	0.727366	0.363683	+	0.931523i	$0.381519\pi$
$992$	−487.093	−0.0155899
$993$	−35222.6	−1.12563
$994$	58350.0	1.86192
$995$	−11831.5	−0.376969
$996$	3606.38	0.114732
$997$	26889.0	0.854146	0.427073	−	0.904217i	$-0.359545\pi$
$997$	26889.0	0.854146	0.427073	+	0.904217i	$0.359545\pi$
$998$	−10538.2	−0.334248
$999$	4501.34	0.142559

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	786.4.a.g.1.5	✓	8
3.2	odd	2		2358.4.a.i.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.4.a.g.1.5	✓	8	1.1	even	1	trivial
2358.4.a.i.1.4		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 786 = 2 \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	786.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +4.74723 q^{5} +6.00000 q^{6} +27.5663 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +4.74723 q^{5} +6.00000 q^{6} +27.5663 q^{7} -8.00000 q^{8} +9.00000 q^{9} -9.49446 q^{10} +35.9318 q^{11} -12.0000 q^{12} -64.3215 q^{13} -55.1327 q^{14} -14.2417 q^{15} +16.0000 q^{16} -105.772 q^{17} -18.0000 q^{18} -6.65620 q^{19} +18.9889 q^{20} -82.6990 q^{21} -71.8636 q^{22} +6.46461 q^{23} +24.0000 q^{24} -102.464 q^{25} +128.643 q^{26} -27.0000 q^{27} +110.265 q^{28} -182.493 q^{29} +28.4834 q^{30} +15.2217 q^{31} -32.0000 q^{32} -107.795 q^{33} +211.545 q^{34} +130.864 q^{35} +36.0000 q^{36} -166.716 q^{37} +13.3124 q^{38} +192.964 q^{39} -37.9778 q^{40} -241.404 q^{41} +165.398 q^{42} +500.596 q^{43} +143.727 q^{44} +42.7251 q^{45} -12.9292 q^{46} -6.09555 q^{47} -48.0000 q^{48} +416.903 q^{49} +204.928 q^{50} +317.317 q^{51} -257.286 q^{52} -8.02587 q^{53} +54.0000 q^{54} +170.577 q^{55} -220.531 q^{56} +19.9686 q^{57} +364.987 q^{58} +44.5960 q^{59} -56.9668 q^{60} -401.549 q^{61} -30.4433 q^{62} +248.097 q^{63} +64.0000 q^{64} -305.349 q^{65} +215.591 q^{66} -258.326 q^{67} -423.089 q^{68} -19.3938 q^{69} -261.728 q^{70} -1058.36 q^{71} -72.0000 q^{72} +547.136 q^{73} +333.433 q^{74} +307.391 q^{75} -26.6248 q^{76} +990.509 q^{77} -385.929 q^{78} +179.688 q^{79} +75.9557 q^{80} +81.0000 q^{81} +482.807 q^{82} -300.532 q^{83} -330.796 q^{84} -502.126 q^{85} -1001.19 q^{86} +547.480 q^{87} -287.455 q^{88} -1144.79 q^{89} -85.4502 q^{90} -1773.11 q^{91} +25.8584 q^{92} -45.6650 q^{93} +12.1911 q^{94} -31.5985 q^{95} +96.0000 q^{96} +1335.12 q^{97} -833.806 q^{98} +323.386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9	\( 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9} - 2 q^{10} - 77 q^{11} - 96 q^{12} + 29 q^{13} - 44 q^{14} - 3 q^{15} + 128 q^{16} + 59 q^{17} - 144 q^{18} - 150 q^{19} + 4 q^{20} - 66 q^{21} + 154 q^{22} - 269 q^{23} + 192 q^{24} + 273 q^{25} - 58 q^{26} - 216 q^{27} + 88 q^{28} - 123 q^{29} + 6 q^{30} - 86 q^{31} - 256 q^{32} + 231 q^{33} - 118 q^{34} - 292 q^{35} + 288 q^{36} + 412 q^{37} + 300 q^{38} - 87 q^{39} - 8 q^{40} - 114 q^{41} + 132 q^{42} + 1087 q^{43} - 308 q^{44} + 9 q^{45} + 538 q^{46} - 442 q^{47} - 384 q^{48} + 1292 q^{49} - 546 q^{50} - 177 q^{51} + 116 q^{52} + 5 q^{53} + 432 q^{54} + 456 q^{55} - 176 q^{56} + 450 q^{57} + 246 q^{58} + 252 q^{59} - 12 q^{60} + 1482 q^{61} + 172 q^{62} + 198 q^{63} + 512 q^{64} + 475 q^{65} - 462 q^{66} + 330 q^{67} + 236 q^{68} + 807 q^{69} + 584 q^{70} - 2946 q^{71} - 576 q^{72} - 214 q^{73} - 824 q^{74} - 819 q^{75} - 600 q^{76} - 960 q^{77} + 174 q^{78} - 64 q^{79} + 16 q^{80} + 648 q^{81} + 228 q^{82} - 276 q^{83} - 264 q^{84} + 80 q^{85} - 2174 q^{86} + 369 q^{87} + 616 q^{88} - 3177 q^{89} - 18 q^{90} - 781 q^{91} - 1076 q^{92} + 258 q^{93} + 884 q^{94} - 2700 q^{95} + 768 q^{96} + 200 q^{97} - 2584 q^{98} - 693 q^{99}+O(q^{100}) \) 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9 - 2 * q^10 - 77 * q^11 - 96 * q^12 + 29 * q^13 - 44 * q^14 - 3 * q^15 + 128 * q^16 + 59 * q^17 - 144 * q^18 - 150 * q^19 + 4 * q^20 - 66 * q^21 + 154 * q^22 - 269 * q^23 + 192 * q^24 + 273 * q^25 - 58 * q^26 - 216 * q^27 + 88 * q^28 - 123 * q^29 + 6 * q^30 - 86 * q^31 - 256 * q^32 + 231 * q^33 - 118 * q^34 - 292 * q^35 + 288 * q^36 + 412 * q^37 + 300 * q^38 - 87 * q^39 - 8 * q^40 - 114 * q^41 + 132 * q^42 + 1087 * q^43 - 308 * q^44 + 9 * q^45 + 538 * q^46 - 442 * q^47 - 384 * q^48 + 1292 * q^49 - 546 * q^50 - 177 * q^51 + 116 * q^52 + 5 * q^53 + 432 * q^54 + 456 * q^55 - 176 * q^56 + 450 * q^57 + 246 * q^58 + 252 * q^59 - 12 * q^60 + 1482 * q^61 + 172 * q^62 + 198 * q^63 + 512 * q^64 + 475 * q^65 - 462 * q^66 + 330 * q^67 + 236 * q^68 + 807 * q^69 + 584 * q^70 - 2946 * q^71 - 576 * q^72 - 214 * q^73 - 824 * q^74 - 819 * q^75 - 600 * q^76 - 960 * q^77 + 174 * q^78 - 64 * q^79 + 16 * q^80 + 648 * q^81 + 228 * q^82 - 276 * q^83 - 264 * q^84 + 80 * q^85 - 2174 * q^86 + 369 * q^87 + 616 * q^88 - 3177 * q^89 - 18 * q^90 - 781 * q^91 - 1076 * q^92 + 258 * q^93 + 884 * q^94 - 2700 * q^95 + 768 * q^96 + 200 * q^97 - 2584 * q^98 - 693 * q^99

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