LMFDB - Embedded newform 786.4.a.g.1.8

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			1.8
Root			$-19.2844$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +16.1323 q^{5} +6.00000 q^{6} +13.6210 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +16.1323 q^{5} +6.00000 q^{6} +13.6210 q^{7} -8.00000 q^{8} +9.00000 q^{9} -32.2646 q^{10} -29.4592 q^{11} -12.0000 q^{12} +78.0092 q^{13} -27.2421 q^{14} -48.3969 q^{15} +16.0000 q^{16} -74.8903 q^{17} -18.0000 q^{18} -142.121 q^{19} +64.5292 q^{20} -40.8631 q^{21} +58.9184 q^{22} -212.644 q^{23} +24.0000 q^{24} +135.251 q^{25} -156.018 q^{26} -27.0000 q^{27} +54.4842 q^{28} -101.881 q^{29} +96.7937 q^{30} -139.203 q^{31} -32.0000 q^{32} +88.3777 q^{33} +149.781 q^{34} +219.739 q^{35} +36.0000 q^{36} -118.064 q^{37} +284.243 q^{38} -234.028 q^{39} -129.058 q^{40} +327.735 q^{41} +81.7263 q^{42} -371.164 q^{43} -117.837 q^{44} +145.191 q^{45} +425.289 q^{46} -304.569 q^{47} -48.0000 q^{48} -157.467 q^{49} -270.502 q^{50} +224.671 q^{51} +312.037 q^{52} -124.090 q^{53} +54.0000 q^{54} -475.245 q^{55} -108.968 q^{56} +426.364 q^{57} +203.762 q^{58} +546.327 q^{59} -193.587 q^{60} +62.3673 q^{61} +278.406 q^{62} +122.589 q^{63} +64.0000 q^{64} +1258.47 q^{65} -176.755 q^{66} +374.558 q^{67} -299.561 q^{68} +637.933 q^{69} -439.477 q^{70} -810.618 q^{71} -72.0000 q^{72} +61.6896 q^{73} +236.128 q^{74} -405.752 q^{75} -568.486 q^{76} -401.265 q^{77} +468.055 q^{78} +1385.13 q^{79} +258.117 q^{80} +81.0000 q^{81} -655.471 q^{82} -106.765 q^{83} -163.453 q^{84} -1208.15 q^{85} +742.328 q^{86} +305.642 q^{87} +235.674 q^{88} -291.112 q^{89} -290.381 q^{90} +1062.57 q^{91} -850.577 q^{92} +417.610 q^{93} +609.138 q^{94} -2292.74 q^{95} +96.0000 q^{96} +1461.47 q^{97} +314.934 q^{98} -265.133 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	16.1323	1.44292	0.721458	−	0.692458i	$-0.243472\pi$
$5$	16.1323	1.44292	0.721458	+	0.692458i	$0.243472\pi$
$6$	6.00000	0.408248
$7$	13.6210	0.735467	0.367734	−	0.929931i	$-0.380134\pi$
$7$	13.6210	0.735467	0.367734	+	0.929931i	$0.380134\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−32.2646	−1.02030
$11$	−29.4592	−0.807481	−0.403740	−	0.914874i	$-0.632290\pi$
$11$	−29.4592	−0.807481	−0.403740	+	0.914874i	$0.632290\pi$
$12$	−12.0000	−0.288675
$13$	78.0092	1.66430	0.832149	−	0.554553i	$-0.187111\pi$
$13$	78.0092	1.66430	0.832149	+	0.554553i	$0.187111\pi$
$14$	−27.2421	−0.520054
$15$	−48.3969	−0.833068
$16$	16.0000	0.250000
$17$	−74.8903	−1.06845	−0.534223	−	0.845344i	$-0.679396\pi$
$17$	−74.8903	−1.06845	−0.534223	+	0.845344i	$0.679396\pi$
$18$	−18.0000	−0.235702
$19$	−142.121	−1.71605	−0.858023	−	0.513611i	$-0.828307\pi$
$19$	−142.121	−1.71605	−0.858023	+	0.513611i	$0.828307\pi$
$20$	64.5292	0.721458
$21$	−40.8631	−0.424622
$22$	58.9184	0.570975
$23$	−212.644	−1.92780	−0.963900	−	0.266263i	$-0.914211\pi$
$23$	−212.644	−1.92780	−0.963900	+	0.266263i	$0.914211\pi$
$24$	24.0000	0.204124
$25$	135.251	1.08201
$26$	−156.018	−1.17684
$27$	−27.0000	−0.192450
$28$	54.4842	0.367734
$29$	−101.881	−0.652372	−0.326186	−	0.945306i	$-0.605764\pi$
$29$	−101.881	−0.652372	−0.326186	+	0.945306i	$0.605764\pi$
$30$	96.7937	0.589068
$31$	−139.203	−0.806505	−0.403252	−	0.915089i	$-0.632120\pi$
$31$	−139.203	−0.806505	−0.403252	+	0.915089i	$0.632120\pi$
$32$	−32.0000	−0.176777
$33$	88.3777	0.466199
$34$	149.781	0.755505
$35$	219.739	1.06122
$36$	36.0000	0.166667
$37$	−118.064	−0.524585	−0.262292	−	0.964988i	$-0.584478\pi$
$37$	−118.064	−0.524585	−0.262292	+	0.964988i	$0.584478\pi$
$38$	284.243	1.21343
$39$	−234.028	−0.960883
$40$	−129.058	−0.510148
$41$	327.735	1.24838	0.624191	−	0.781272i	$-0.285429\pi$
$41$	327.735	1.24838	0.624191	+	0.781272i	$0.285429\pi$
$42$	81.7263	0.300253
$43$	−371.164	−1.31633	−0.658163	−	0.752876i	$-0.728666\pi$
$43$	−371.164	−1.31633	−0.658163	+	0.752876i	$0.728666\pi$
$44$	−117.837	−0.403740
$45$	145.191	0.480972
$46$	425.289	1.36316
$47$	−304.569	−0.945233	−0.472617	−	0.881268i	$-0.656690\pi$
$47$	−304.569	−0.945233	−0.472617	+	0.881268i	$0.656690\pi$
$48$	−48.0000	−0.144338
$49$	−157.467	−0.459088
$50$	−270.502	−0.765094
$51$	224.671	0.616867
$52$	312.037	0.832149
$53$	−124.090	−0.321606	−0.160803	−	0.986987i	$-0.551408\pi$
$53$	−124.090	−0.321606	−0.160803	+	0.986987i	$0.551408\pi$
$54$	54.0000	0.136083
$55$	−475.245	−1.16513
$56$	−108.968	−0.260027
$57$	426.364	0.990760
$58$	203.762	0.461297
$59$	546.327	1.20552	0.602760	−	0.797922i	$-0.294067\pi$
$59$	546.327	1.20552	0.602760	+	0.797922i	$0.294067\pi$
$60$	−193.587	−0.416534
$61$	62.3673	0.130907	0.0654534	−	0.997856i	$-0.479151\pi$
$61$	62.3673	0.130907	0.0654534	+	0.997856i	$0.479151\pi$
$62$	278.406	0.570285
$63$	122.589	0.245156
$64$	64.0000	0.125000
$65$	1258.47	2.40144
$66$	−176.755	−0.329653
$67$	374.558	0.682978	0.341489	−	0.939886i	$-0.389069\pi$
$67$	374.558	0.682978	0.341489	+	0.939886i	$0.389069\pi$
$68$	−299.561	−0.534223
$69$	637.933	1.11302
$70$	−439.477	−0.750394
$71$	−810.618	−1.35497	−0.677484	−	0.735538i	$-0.736930\pi$
$71$	−810.618	−1.35497	−0.677484	+	0.735538i	$0.736930\pi$
$72$	−72.0000	−0.117851
$73$	61.6896	0.0989071	0.0494535	−	0.998776i	$-0.484252\pi$
$73$	61.6896	0.0989071	0.0494535	+	0.998776i	$0.484252\pi$
$74$	236.128	0.370937
$75$	−405.752	−0.624697
$76$	−568.486	−0.858023
$77$	−401.265	−0.593876
$78$	468.055	0.679447
$79$	1385.13	1.97265	0.986323	−	0.164824i	$-0.0527058\pi$
$79$	1385.13	1.97265	0.986323	+	0.164824i	$0.0527058\pi$
$80$	258.117	0.360729
$81$	81.0000	0.111111
$82$	−655.471	−0.882739
$83$	−106.765	−0.141193	−0.0705965	−	0.997505i	$-0.522490\pi$
$83$	−106.765	−0.141193	−0.0705965	+	0.997505i	$0.522490\pi$
$84$	−163.453	−0.212311
$85$	−1208.15	−1.54168
$86$	742.328	0.930783
$87$	305.642	0.376647
$88$	235.674	0.285488
$89$	−291.112	−0.346717	−0.173358	−	0.984859i	$-0.555462\pi$
$89$	−291.112	−0.346717	−0.173358	+	0.984859i	$0.555462\pi$
$90$	−290.381	−0.340099
$91$	1062.57	1.22404
$92$	−850.577	−0.963900
$93$	417.610	0.465636
$94$	609.138	0.668381
$95$	−2292.74	−2.47611
$96$	96.0000	0.102062
$97$	1461.47	1.52979	0.764895	−	0.644155i	$-0.222791\pi$
$97$	1461.47	1.52979	0.764895	+	0.644155i	$0.222791\pi$
$98$	314.934	0.324624
$99$	−265.133	−0.269160
$100$	541.003	0.541003
$101$	−409.845	−0.403773	−0.201887	−	0.979409i	$-0.564707\pi$
$101$	−409.845	−0.403773	−0.201887	+	0.979409i	$0.564707\pi$
$102$	−449.342	−0.436191
$103$	−1059.35	−1.01341	−0.506704	−	0.862120i	$-0.669136\pi$
$103$	−1059.35	−1.01341	−0.506704	+	0.862120i	$0.669136\pi$
$104$	−624.074	−0.588418
$105$	−659.216	−0.612694
$106$	248.181	0.227410
$107$	−982.568	−0.887743	−0.443871	−	0.896091i	$-0.646395\pi$
$107$	−982.568	−0.887743	−0.443871	+	0.896091i	$0.646395\pi$
$108$	−108.000	−0.0962250
$109$	−1404.27	−1.23399	−0.616995	−	0.786967i	$-0.711650\pi$
$109$	−1404.27	−1.23399	−0.616995	+	0.786967i	$0.711650\pi$
$110$	950.489	0.823869
$111$	354.192	0.302869
$112$	217.937	0.183867
$113$	421.956	0.351277	0.175638	−	0.984455i	$-0.443801\pi$
$113$	421.956	0.351277	0.175638	+	0.984455i	$0.443801\pi$
$114$	−852.728	−0.700573
$115$	−3430.44	−2.78165
$116$	−407.523	−0.326186
$117$	702.083	0.554766
$118$	−1092.65	−0.852432
$119$	−1020.08	−0.785807
$120$	387.175	0.294534
$121$	−463.155	−0.347975
$122$	−124.735	−0.0925651
$123$	−983.206	−0.720754
$124$	−556.813	−0.403252
$125$	165.369	0.118329
$126$	−245.179	−0.173351
$127$	506.054	0.353583	0.176791	−	0.984248i	$-0.443428\pi$
$127$	506.054	0.353583	0.176791	+	0.984248i	$0.443428\pi$
$128$	−128.000	−0.0883883
$129$	1113.49	0.759981
$130$	−2516.94	−1.69808
$131$	131.000	0.0873704
$131$	131.000	0.0873704
$132$	353.511	0.233100
$133$	−1935.84	−1.26210
$134$	−749.116	−0.482939
$135$	−435.572	−0.277689
$136$	599.123	0.377752
$137$	−2720.88	−1.69679	−0.848395	−	0.529364i	$-0.822431\pi$
$137$	−2720.88	−1.69679	−0.848395	+	0.529364i	$0.822431\pi$
$138$	−1275.87	−0.787021
$139$	−1117.69	−0.682022	−0.341011	−	0.940059i	$-0.610769\pi$
$139$	−1117.69	−0.682022	−0.341011	+	0.940059i	$0.610769\pi$
$140$	878.955	0.530609
$141$	913.707	0.545731
$142$	1621.24	0.958107
$143$	−2298.09	−1.34389
$144$	144.000	0.0833333
$145$	−1643.57	−0.941318
$146$	−123.379	−0.0699379
$147$	472.401	0.265054
$148$	−472.257	−0.262292
$149$	−1433.78	−0.788320	−0.394160	−	0.919042i	$-0.628964\pi$
$149$	−1433.78	−0.788320	−0.394160	+	0.919042i	$0.628964\pi$
$150$	811.505	0.441727
$151$	3418.13	1.84214	0.921071	−	0.389395i	$-0.127316\pi$
$151$	3418.13	1.84214	0.921071	+	0.389395i	$0.127316\pi$
$152$	1136.97	0.606714
$153$	−674.013	−0.356148
$154$	802.531	0.419934
$155$	−2245.67	−1.16372
$156$	−936.111	−0.480441
$157$	−1430.40	−0.727121	−0.363561	−	0.931571i	$-0.618439\pi$
$157$	−1430.40	−0.727121	−0.363561	+	0.931571i	$0.618439\pi$
$158$	−2770.26	−1.39487
$159$	372.271	0.185679
$160$	−516.233	−0.255074
$161$	−2896.44	−1.41783
$162$	−162.000	−0.0785674
$163$	2819.07	1.35464	0.677321	−	0.735688i	$-0.263141\pi$
$163$	2819.07	1.35464	0.677321	+	0.735688i	$0.263141\pi$
$164$	1310.94	0.624191
$165$	1425.73	0.672686
$166$	213.531	0.0998385
$167$	1441.91	0.668132	0.334066	−	0.942550i	$-0.391579\pi$
$167$	1441.91	0.668132	0.334066	+	0.942550i	$0.391579\pi$
$168$	326.905	0.150127
$169$	3888.44	1.76989
$170$	2416.30	1.09013
$171$	−1279.09	−0.572015
$172$	−1484.66	−0.658163
$173$	1395.98	0.613493	0.306746	−	0.951791i	$-0.400760\pi$
$173$	1395.98	0.613493	0.306746	+	0.951791i	$0.400760\pi$
$174$	−611.285	−0.266330
$175$	1842.26	0.795781
$176$	−471.347	−0.201870
$177$	−1638.98	−0.696008
$178$	582.224	0.245166
$179$	−374.555	−0.156400	−0.0781998	−	0.996938i	$-0.524917\pi$
$179$	−374.555	−0.156400	−0.0781998	+	0.996938i	$0.524917\pi$
$180$	580.762	0.240486
$181$	3231.17	1.32691	0.663456	−	0.748215i	$-0.269089\pi$
$181$	3231.17	1.32691	0.663456	+	0.748215i	$0.269089\pi$
$182$	−2125.13	−0.865524
$183$	−187.102	−0.0755791
$184$	1701.15	0.681580
$185$	−1904.65	−0.756931
$186$	−835.219	−0.329254
$187$	2206.21	0.862749
$188$	−1218.28	−0.472617
$189$	−367.768	−0.141541
$190$	4585.49	1.75087
$191$	−3451.90	−1.30770	−0.653850	−	0.756624i	$-0.726847\pi$
$191$	−3451.90	−1.30770	−0.653850	+	0.756624i	$0.726847\pi$
$192$	−192.000	−0.0721688
$193$	−2049.47	−0.764374	−0.382187	−	0.924085i	$-0.624829\pi$
$193$	−2049.47	−0.764374	−0.382187	+	0.924085i	$0.624829\pi$
$194$	−2922.94	−1.08173
$195$	−3775.40	−1.38647
$196$	−629.868	−0.229544
$197$	−2254.88	−0.815502	−0.407751	−	0.913093i	$-0.633687\pi$
$197$	−2254.88	−0.815502	−0.407751	+	0.913093i	$0.633687\pi$
$198$	530.266	0.190325
$199$	1617.32	0.576124	0.288062	−	0.957612i	$-0.406989\pi$
$199$	1617.32	0.576124	0.288062	+	0.957612i	$0.406989\pi$
$200$	−1082.01	−0.382547
$201$	−1123.67	−0.394318
$202$	819.690	0.285511
$203$	−1387.72	−0.479799
$204$	898.684	0.308434
$205$	5287.12	1.80131
$206$	2118.70	0.716588
$207$	−1913.80	−0.642600
$208$	1248.15	0.416074
$209$	4186.78	1.38567
$210$	1318.43	0.433240
$211$	−5809.87	−1.89558	−0.947791	−	0.318892i	$-0.896689\pi$
$211$	−5809.87	−1.89558	−0.947791	+	0.318892i	$0.896689\pi$
$212$	−496.361	−0.160803
$213$	2431.86	0.782291
$214$	1965.14	0.627729
$215$	−5987.73	−1.89935
$216$	216.000	0.0680414
$217$	−1896.09	−0.593158
$218$	2808.54	0.872563
$219$	−185.069	−0.0571040
$220$	−1900.98	−0.582563
$221$	−5842.14	−1.77821
$222$	−708.385	−0.214161
$223$	1715.97	0.515290	0.257645	−	0.966240i	$-0.417053\pi$
$223$	1715.97	0.515290	0.257645	+	0.966240i	$0.417053\pi$
$224$	−435.873	−0.130013
$225$	1217.26	0.360669
$226$	−843.912	−0.248390
$227$	4640.94	1.35696	0.678480	−	0.734618i	$-0.262639\pi$
$227$	4640.94	1.35696	0.678480	+	0.734618i	$0.262639\pi$
$228$	1705.46	0.495380
$229$	2734.85	0.789189	0.394594	−	0.918855i	$-0.370885\pi$
$229$	2734.85	0.789189	0.394594	+	0.918855i	$0.370885\pi$
$230$	6860.88	1.96693
$231$	1203.80	0.342874
$232$	815.047	0.230648
$233$	−2033.86	−0.571856	−0.285928	−	0.958251i	$-0.592302\pi$
$233$	−2033.86	−0.571856	−0.285928	+	0.958251i	$0.592302\pi$
$234$	−1404.17	−0.392279
$235$	−4913.40	−1.36389
$236$	2185.31	0.602760
$237$	−4155.38	−1.13891
$238$	2040.17	0.555649
$239$	4909.88	1.32884	0.664421	−	0.747358i	$-0.268678\pi$
$239$	4909.88	1.32884	0.664421	+	0.747358i	$0.268678\pi$
$240$	−774.350	−0.208267
$241$	−1040.14	−0.278014	−0.139007	−	0.990291i	$-0.544391\pi$
$241$	−1040.14	−0.278014	−0.139007	+	0.990291i	$0.544391\pi$
$242$	926.309	0.246055
$243$	−243.000	−0.0641500
$244$	249.469	0.0654534
$245$	−2540.31	−0.662425
$246$	1966.41	0.509650
$247$	−11086.8	−2.85601
$248$	1113.63	0.285142
$249$	320.296	0.0815178
$250$	−330.739	−0.0836710
$251$	1570.52	0.394941	0.197471	−	0.980309i	$-0.436727\pi$
$251$	1570.52	0.394941	0.197471	+	0.980309i	$0.436727\pi$
$252$	490.358	0.122578
$253$	6264.34	1.55666
$254$	−1012.11	−0.250021
$255$	3624.46	0.890087
$256$	256.000	0.0625000
$257$	1627.33	0.394980	0.197490	−	0.980305i	$-0.436721\pi$
$257$	1627.33	0.394980	0.197490	+	0.980305i	$0.436721\pi$
$258$	−2226.99	−0.537388
$259$	−1608.16	−0.385815
$260$	5033.87	1.20072
$261$	−916.927	−0.217457
$262$	−262.000	−0.0617802
$263$	304.221	0.0713272	0.0356636	−	0.999364i	$-0.488646\pi$
$263$	304.221	0.0713272	0.0356636	+	0.999364i	$0.488646\pi$
$264$	−707.021	−0.164826
$265$	−2001.86	−0.464050
$266$	3871.68	0.892437
$267$	873.336	0.200177
$268$	1498.23	0.341489
$269$	−1875.71	−0.425146	−0.212573	−	0.977145i	$-0.568184\pi$
$269$	−1875.71	−0.425146	−0.212573	+	0.977145i	$0.568184\pi$
$270$	871.144	0.196356
$271$	4360.96	0.977526	0.488763	−	0.872417i	$-0.337448\pi$
$271$	4360.96	0.977526	0.488763	+	0.872417i	$0.337448\pi$
$272$	−1198.25	−0.267111
$273$	−3187.70	−0.706698
$274$	5441.75	1.19981
$275$	−3984.38	−0.873699
$276$	2551.73	0.556508
$277$	6296.86	1.36585	0.682927	−	0.730487i	$-0.260707\pi$
$277$	6296.86	1.36585	0.682927	+	0.730487i	$0.260707\pi$
$278$	2235.38	0.482262
$279$	−1252.83	−0.268835
$280$	−1757.91	−0.375197
$281$	4516.18	0.958763	0.479382	−	0.877607i	$-0.340861\pi$
$281$	4516.18	0.958763	0.479382	+	0.877607i	$0.340861\pi$
$282$	−1827.41	−0.385890
$283$	1544.25	0.324367	0.162184	−	0.986761i	$-0.448146\pi$
$283$	1544.25	0.324367	0.162184	+	0.986761i	$0.448146\pi$
$284$	−3242.47	−0.677484
$285$	6878.23	1.42958
$286$	4596.18	0.950272
$287$	4464.10	0.918144
$288$	−288.000	−0.0589256
$289$	695.559	0.141575
$290$	3287.14	0.665613
$291$	−4384.41	−0.883225
$292$	246.758	0.0494535
$293$	−97.2916	−0.0193988	−0.00969938	−	0.999953i	$-0.503087\pi$
$293$	−97.2916	−0.0193988	−0.00969938	+	0.999953i	$0.503087\pi$
$294$	−944.803	−0.187422
$295$	8813.51	1.73947
$296$	944.513	0.185469
$297$	795.399	0.155400
$298$	2867.56	0.557427
$299$	−16588.2	−3.20843
$300$	−1623.01	−0.312348
$301$	−5055.64	−0.968115
$302$	−6836.26	−1.30259
$303$	1229.53	0.233118
$304$	−2273.94	−0.429012
$305$	1006.13	0.188888
$306$	1348.03	0.251835
$307$	8828.60	1.64129	0.820643	−	0.571441i	$-0.193615\pi$
$307$	8828.60	1.64129	0.820643	+	0.571441i	$0.193615\pi$
$308$	−1605.06	−0.296938
$309$	3178.06	0.585092
$310$	4491.33	0.822873
$311$	−4656.55	−0.849030	−0.424515	−	0.905421i	$-0.639555\pi$
$311$	−4656.55	−0.849030	−0.424515	+	0.905421i	$0.639555\pi$
$312$	1872.22	0.339723
$313$	−3121.65	−0.563725	−0.281863	−	0.959455i	$-0.590952\pi$
$313$	−3121.65	−0.563725	−0.281863	+	0.959455i	$0.590952\pi$
$314$	2860.79	0.514152
$315$	1977.65	0.353739
$316$	5540.51	0.986323
$317$	5418.56	0.960053	0.480026	−	0.877254i	$-0.340627\pi$
$317$	5418.56	0.960053	0.480026	+	0.877254i	$0.340627\pi$
$318$	−744.542	−0.131295
$319$	3001.33	0.526778
$320$	1032.47	0.180365
$321$	2947.70	0.512538
$322$	5792.88	1.00256
$323$	10643.5	1.83350
$324$	324.000	0.0555556
$325$	10550.8	1.80078
$326$	−5638.14	−0.957876
$327$	4212.82	0.712444
$328$	−2621.88	−0.441370
$329$	−4148.55	−0.695188
$330$	−2851.47	−0.475661
$331$	−10945.1	−1.81751	−0.908755	−	0.417330i	$-0.862966\pi$
$331$	−10945.1	−1.81751	−0.908755	+	0.417330i	$0.862966\pi$
$332$	−427.061	−0.0705965
$333$	−1062.58	−0.174862
$334$	−2883.81	−0.472441
$335$	6042.48	0.985480
$336$	−653.810	−0.106156
$337$	−10113.5	−1.63477	−0.817387	−	0.576088i	$-0.804578\pi$
$337$	−10113.5	−1.63477	−0.817387	+	0.576088i	$0.804578\pi$
$338$	−7776.88	−1.25150
$339$	−1265.87	−0.202810
$340$	−4832.61	−0.770838
$341$	4100.82	0.651237
$342$	2558.18	0.404476
$343$	−6816.89	−1.07311
$344$	2969.31	0.465391
$345$	10291.3	1.60599
$346$	−2791.96	−0.433805
$347$	−7112.54	−1.10035	−0.550175	−	0.835049i	$-0.685439\pi$
$347$	−7112.54	−1.10035	−0.550175	+	0.835049i	$0.685439\pi$
$348$	1222.57	0.188324
$349$	1723.47	0.264341	0.132171	−	0.991227i	$-0.457805\pi$
$349$	1723.47	0.264341	0.132171	+	0.991227i	$0.457805\pi$
$350$	−3684.52	−0.562702
$351$	−2106.25	−0.320294
$352$	942.695	0.142744
$353$	−1651.85	−0.249062	−0.124531	−	0.992216i	$-0.539743\pi$
$353$	−1651.85	−0.249062	−0.124531	+	0.992216i	$0.539743\pi$
$354$	3277.96	0.492152
$355$	−13077.1	−1.95510
$356$	−1164.45	−0.173358
$357$	3060.25	0.453686
$358$	749.109	0.110591
$359$	−4494.00	−0.660680	−0.330340	−	0.943862i	$-0.607163\pi$
$359$	−4494.00	−0.660680	−0.330340	+	0.943862i	$0.607163\pi$
$360$	−1161.52	−0.170049
$361$	13339.5	1.94482
$362$	−6462.34	−0.938269
$363$	1389.46	0.200903
$364$	4250.27	0.612018
$365$	995.194	0.142715
$366$	374.204	0.0534425
$367$	3034.93	0.431668	0.215834	−	0.976430i	$-0.430753\pi$
$367$	3034.93	0.431668	0.215834	+	0.976430i	$0.430753\pi$
$368$	−3402.31	−0.481950
$369$	2949.62	0.416127
$370$	3809.29	0.535231
$371$	−1690.24	−0.236531
$372$	1670.44	0.232818
$373$	4584.37	0.636380	0.318190	−	0.948027i	$-0.396925\pi$
$373$	4584.37	0.636380	0.318190	+	0.948027i	$0.396925\pi$
$374$	−4412.42	−0.610056
$375$	−496.108	−0.0683171
$376$	2436.55	0.334190
$377$	−7947.64	−1.08574
$378$	735.537	0.100084
$379$	−11327.1	−1.53518	−0.767590	−	0.640941i	$-0.778544\pi$
$379$	−11327.1	−1.53518	−0.767590	+	0.640941i	$0.778544\pi$
$380$	−9170.97	−1.23806
$381$	−1518.16	−0.204141
$382$	6903.80	0.924683
$383$	−2932.23	−0.391201	−0.195601	−	0.980684i	$-0.562666\pi$
$383$	−2932.23	−0.391201	−0.195601	+	0.980684i	$0.562666\pi$
$384$	384.000	0.0510310
$385$	−6473.33	−0.856913
$386$	4098.94	0.540494
$387$	−3340.48	−0.438775
$388$	5845.87	0.764895
$389$	−6566.44	−0.855866	−0.427933	−	0.903810i	$-0.640758\pi$
$389$	−6566.44	−0.855866	−0.427933	+	0.903810i	$0.640758\pi$
$390$	7550.81	0.980384
$391$	15925.0	2.05975
$392$	1259.74	0.162312
$393$	−393.000	−0.0504433
$394$	4509.77	0.576647
$395$	22345.3	2.84636
$396$	−1060.53	−0.134580
$397$	−460.720	−0.0582440	−0.0291220	−	0.999576i	$-0.509271\pi$
$397$	−460.720	−0.0582440	−0.0291220	+	0.999576i	$0.509271\pi$
$398$	−3234.64	−0.407381
$399$	5807.53	0.728672
$400$	2164.01	0.270502
$401$	729.741	0.0908766	0.0454383	−	0.998967i	$-0.485532\pi$
$401$	729.741	0.0908766	0.0454383	+	0.998967i	$0.485532\pi$
$402$	2247.35	0.278825
$403$	−10859.1	−1.34226
$404$	−1639.38	−0.201887
$405$	1306.72	0.160324
$406$	2775.45	0.339269
$407$	3478.08	0.423592
$408$	−1797.37	−0.218095
$409$	6741.89	0.815074	0.407537	−	0.913189i	$-0.366388\pi$
$409$	6741.89	0.815074	0.407537	+	0.913189i	$0.366388\pi$
$410$	−10574.2	−1.27372
$411$	8162.63	0.979642
$412$	−4237.41	−0.506704
$413$	7441.55	0.886621
$414$	3827.60	0.454387
$415$	−1722.37	−0.203730
$416$	−2496.30	−0.294209
$417$	3353.06	0.393766
$418$	−8373.57	−0.979820
$419$	−3324.43	−0.387611	−0.193806	−	0.981040i	$-0.562083\pi$
$419$	−3324.43	−0.387611	−0.193806	+	0.981040i	$0.562083\pi$
$420$	−2636.86	−0.306347
$421$	−7366.27	−0.852755	−0.426377	−	0.904545i	$-0.640210\pi$
$421$	−7366.27	−0.852755	−0.426377	+	0.904545i	$0.640210\pi$
$422$	11619.7	1.34038
$423$	−2741.12	−0.315078
$424$	992.723	0.113705
$425$	−10129.0	−1.15606
$426$	−4863.71	−0.553163
$427$	849.508	0.0962777
$428$	−3930.27	−0.443871
$429$	6894.27	0.775894
$430$	11975.5	1.34304
$431$	887.960	0.0992379	0.0496189	−	0.998768i	$-0.484199\pi$
$431$	887.960	0.0992379	0.0496189	+	0.998768i	$0.484199\pi$
$432$	−432.000	−0.0481125
$433$	−1329.43	−0.147548	−0.0737740	−	0.997275i	$-0.523504\pi$
$433$	−1329.43	−0.147548	−0.0737740	+	0.997275i	$0.523504\pi$
$434$	3792.19	0.419426
$435$	4930.71	0.543470
$436$	−5617.09	−0.616995
$437$	30221.3	3.30820
$438$	370.137	0.0403786
$439$	−16611.2	−1.80595	−0.902975	−	0.429694i	$-0.858621\pi$
$439$	−16611.2	−1.80595	−0.902975	+	0.429694i	$0.858621\pi$
$440$	3801.96	0.411935
$441$	−1417.20	−0.153029
$442$	11684.3	1.25738
$443$	9949.34	1.06706	0.533530	−	0.845781i	$-0.320865\pi$
$443$	9949.34	1.06706	0.533530	+	0.845781i	$0.320865\pi$
$444$	1416.77	0.151435
$445$	−4696.30	−0.500283
$446$	−3431.93	−0.364365
$447$	4301.34	0.455137
$448$	871.747	0.0919334
$449$	−7836.84	−0.823705	−0.411852	−	0.911251i	$-0.635118\pi$
$449$	−7836.84	−0.823705	−0.411852	+	0.911251i	$0.635118\pi$
$450$	−2434.51	−0.255031
$451$	−9654.83	−1.00804
$452$	1687.82	0.175638
$453$	−10254.4	−1.06356
$454$	−9281.88	−0.959516
$455$	17141.6	1.76618
$456$	−3410.91	−0.350287
$457$	13204.9	1.35164	0.675821	−	0.737066i	$-0.263789\pi$
$457$	13204.9	1.35164	0.675821	+	0.737066i	$0.263789\pi$
$458$	−5469.71	−0.558041
$459$	2022.04	0.205622
$460$	−13721.8	−1.39083
$461$	−15408.8	−1.55674	−0.778372	−	0.627803i	$-0.783954\pi$
$461$	−15408.8	−1.55674	−0.778372	+	0.627803i	$0.783954\pi$
$462$	−2407.59	−0.242449
$463$	18680.7	1.87509	0.937546	−	0.347860i	$-0.113092\pi$
$463$	18680.7	1.87509	0.937546	+	0.347860i	$0.113092\pi$
$464$	−1630.09	−0.163093
$465$	6737.00	0.671873
$466$	4067.72	0.404363
$467$	4131.32	0.409367	0.204684	−	0.978828i	$-0.434383\pi$
$467$	4131.32	0.409367	0.204684	+	0.978828i	$0.434383\pi$
$468$	2808.33	0.277383
$469$	5101.87	0.502308
$470$	9826.79	0.964417
$471$	4291.19	0.419804
$472$	−4370.62	−0.426216
$473$	10934.2	1.06291
$474$	8310.77	0.805329
$475$	−19222.0	−1.85677
$476$	−4080.34	−0.392903
$477$	−1116.81	−0.107202
$478$	−9819.75	−0.939634
$479$	−5951.42	−0.567698	−0.283849	−	0.958869i	$-0.591611\pi$
$479$	−5951.42	−0.567698	−0.283849	+	0.958869i	$0.591611\pi$
$480$	1548.70	0.147267
$481$	−9210.09	−0.873065
$482$	2080.29	0.196586
$483$	8689.32	0.818587
$484$	−1852.62	−0.173987
$485$	23576.8	2.20736
$486$	486.000	0.0453609
$487$	−9284.63	−0.863916	−0.431958	−	0.901894i	$-0.642177\pi$
$487$	−9284.63	−0.863916	−0.431958	+	0.901894i	$0.642177\pi$
$488$	−498.939	−0.0462826
$489$	−8457.21	−0.782103
$490$	5080.61	0.468405
$491$	−6345.79	−0.583262	−0.291631	−	0.956531i	$-0.594198\pi$
$491$	−6345.79	−0.583262	−0.291631	+	0.956531i	$0.594198\pi$
$492$	−3932.82	−0.360377
$493$	7629.89	0.697024
$494$	22173.6	2.01951
$495$	−4277.20	−0.388376
$496$	−2227.25	−0.201626
$497$	−11041.5	−0.996535
$498$	−640.592	−0.0576418
$499$	10655.4	0.955913	0.477957	−	0.878383i	$-0.341378\pi$
$499$	10655.4	0.955913	0.477957	+	0.878383i	$0.341378\pi$
$500$	661.477	0.0591643
$501$	−4325.72	−0.385746
$502$	−3141.04	−0.279266
$503$	15095.8	1.33815	0.669074	−	0.743196i	$-0.266691\pi$
$503$	15095.8	1.33815	0.669074	+	0.743196i	$0.266691\pi$
$504$	−980.715	−0.0866757
$505$	−6611.74	−0.582611
$506$	−12528.7	−1.10073
$507$	−11665.3	−1.02184
$508$	2024.21	0.176791
$509$	−11119.4	−0.968290	−0.484145	−	0.874988i	$-0.660869\pi$
$509$	−11119.4	−0.968290	−0.484145	+	0.874988i	$0.660869\pi$
$510$	−7248.91	−0.629387
$511$	840.276	0.0727429
$512$	−512.000	−0.0441942
$513$	3837.28	0.330253
$514$	−3254.65	−0.279293
$515$	−17089.8	−1.46226
$516$	4453.97	0.379991
$517$	8972.37	0.763258
$518$	3216.31	0.272812
$519$	−4187.93	−0.354200
$520$	−10067.7	−0.849038
$521$	15166.2	1.27533	0.637663	−	0.770316i	$-0.279901\pi$
$521$	15166.2	1.27533	0.637663	+	0.770316i	$0.279901\pi$
$522$	1833.85	0.153766
$523$	−11145.4	−0.931842	−0.465921	−	0.884826i	$-0.654277\pi$
$523$	−11145.4	−0.931842	−0.465921	+	0.884826i	$0.654277\pi$
$524$	524.000	0.0436852
$525$	−5526.77	−0.459444
$526$	−608.442	−0.0504360
$527$	10425.0	0.861706
$528$	1414.04	0.116550
$529$	33050.6	2.71642
$530$	4003.72	0.328133
$531$	4916.94	0.401840
$532$	−7743.37	−0.631048
$533$	25566.4	2.07768
$534$	−1746.67	−0.141547
$535$	−15851.1	−1.28094
$536$	−2996.46	−0.241469
$537$	1123.66	0.0902974
$538$	3751.43	0.300624
$539$	4638.86	0.370705
$540$	−1742.29	−0.138845
$541$	−7156.33	−0.568715	−0.284358	−	0.958718i	$-0.591780\pi$
$541$	−7156.33	−0.568715	−0.284358	+	0.958718i	$0.591780\pi$
$542$	−8721.93	−0.691215
$543$	−9693.52	−0.766093
$544$	2396.49	0.188876
$545$	−22654.1	−1.78054
$546$	6375.40	0.499711
$547$	−20865.3	−1.63096	−0.815482	−	0.578783i	$-0.803528\pi$
$547$	−20865.3	−1.63096	−0.815482	+	0.578783i	$0.803528\pi$
$548$	−10883.5	−0.848395
$549$	561.306	0.0436356
$550$	7968.77	0.617799
$551$	14479.4	1.11950
$552$	−5103.46	−0.393511
$553$	18866.9	1.45082
$554$	−12593.7	−0.965804
$555$	5713.94	0.437015
$556$	−4470.75	−0.341011
$557$	−9879.57	−0.751545	−0.375773	−	0.926712i	$-0.622623\pi$
$557$	−9879.57	−0.751545	−0.375773	+	0.926712i	$0.622623\pi$
$558$	2505.66	0.190095
$559$	−28954.2	−2.19076
$560$	3515.82	0.265304
$561$	−6618.63	−0.498108
$562$	−9032.35	−0.677948
$563$	−3657.57	−0.273798	−0.136899	−	0.990585i	$-0.543714\pi$
$563$	−3657.57	−0.273798	−0.136899	+	0.990585i	$0.543714\pi$
$564$	3654.83	0.272865
$565$	6807.11	0.506863
$566$	−3088.49	−0.229362
$567$	1103.30	0.0817186
$568$	6484.95	0.479053
$569$	−17977.0	−1.32449	−0.662246	−	0.749287i	$-0.730396\pi$
$569$	−17977.0	−1.32449	−0.662246	+	0.749287i	$0.730396\pi$
$570$	−13756.5	−1.01087
$571$	−2480.65	−0.181807	−0.0909035	−	0.995860i	$-0.528975\pi$
$571$	−2480.65	−0.181807	−0.0909035	+	0.995860i	$0.528975\pi$
$572$	−9192.36	−0.671944
$573$	10355.7	0.755001
$574$	−8928.20	−0.649226
$575$	−28760.3	−2.08589
$576$	576.000	0.0416667
$577$	−18108.5	−1.30653	−0.653263	−	0.757131i	$-0.726600\pi$
$577$	−18108.5	−1.30653	−0.653263	+	0.757131i	$0.726600\pi$
$578$	−1391.12	−0.100109
$579$	6148.41	0.441312
$580$	−6574.29	−0.470659
$581$	−1454.25	−0.103843
$582$	8768.81	0.624534
$583$	3655.60	0.259691
$584$	−493.517	−0.0349689
$585$	11326.2	0.800480
$586$	194.583	0.0137170
$587$	8887.99	0.624952	0.312476	−	0.949926i	$-0.398842\pi$
$587$	8887.99	0.624952	0.312476	+	0.949926i	$0.398842\pi$
$588$	1889.61	0.132527
$589$	19783.8	1.38400
$590$	−17627.0	−1.22999
$591$	6764.65	0.470830
$592$	−1889.03	−0.131146
$593$	8747.45	0.605758	0.302879	−	0.953029i	$-0.402052\pi$
$593$	8747.45	0.605758	0.302879	+	0.953029i	$0.402052\pi$
$594$	−1590.80	−0.109884
$595$	−16456.3	−1.13385
$596$	−5735.12	−0.394160
$597$	−4851.95	−0.332625
$598$	33176.4	2.26871
$599$	−12632.7	−0.861701	−0.430850	−	0.902423i	$-0.641786\pi$
$599$	−12632.7	−0.861701	−0.430850	+	0.902423i	$0.641786\pi$
$600$	3246.02	0.220864
$601$	25171.6	1.70844	0.854218	−	0.519915i	$-0.174036\pi$
$601$	25171.6	1.70844	0.854218	+	0.519915i	$0.174036\pi$
$602$	10111.3	0.684560
$603$	3371.02	0.227659
$604$	13672.5	0.921071
$605$	−7471.74	−0.502098
$606$	−2459.07	−0.164840
$607$	7979.19	0.533551	0.266775	−	0.963759i	$-0.414042\pi$
$607$	7979.19	0.533551	0.266775	+	0.963759i	$0.414042\pi$
$608$	4547.88	0.303357
$609$	4163.17	0.277012
$610$	−2012.26	−0.133564
$611$	−23759.2	−1.57315
$612$	−2696.05	−0.178074
$613$	27.6902	0.00182447	0.000912233	−	1.00000i	$-0.499710\pi$
$613$	27.6902	0.00182447	0.000912233		1.00000i	$0.499710\pi$
$614$	−17657.2	−1.16056
$615$	−15861.4	−1.03999
$616$	3210.12	0.209967
$617$	−18630.4	−1.21561	−0.607806	−	0.794086i	$-0.707950\pi$
$617$	−18630.4	−1.21561	−0.607806	+	0.794086i	$0.707950\pi$
$618$	−6356.11	−0.413722
$619$	27026.7	1.75492	0.877459	−	0.479652i	$-0.159237\pi$
$619$	27026.7	1.75492	0.877459	+	0.479652i	$0.159237\pi$
$620$	−8982.67	−0.581859
$621$	5741.40	0.371005
$622$	9313.09	0.600355
$623$	−3965.25	−0.254999
$624$	−3744.44	−0.240221
$625$	−14238.6	−0.911268
$626$	6243.30	0.398614
$627$	−12560.4	−0.800020
$628$	−5721.58	−0.363561
$629$	8841.86	0.560490
$630$	−3955.30	−0.250131
$631$	21759.4	1.37278	0.686392	−	0.727232i	$-0.259194\pi$
$631$	21759.4	1.37278	0.686392	+	0.727232i	$0.259194\pi$
$632$	−11081.0	−0.697436
$633$	17429.6	1.09441
$634$	−10837.1	−0.678860
$635$	8163.80	0.510190
$636$	1489.08	0.0928397
$637$	−12283.9	−0.764059
$638$	−6002.66	−0.372488
$639$	−7295.57	−0.451656
$640$	−2064.93	−0.127537
$641$	5007.99	0.308586	0.154293	−	0.988025i	$-0.450690\pi$
$641$	5007.99	0.308586	0.154293	+	0.988025i	$0.450690\pi$
$642$	−5895.41	−0.362419
$643$	4023.42	0.246762	0.123381	−	0.992359i	$-0.460626\pi$
$643$	4023.42	0.246762	0.123381	+	0.992359i	$0.460626\pi$
$644$	−11585.8	−0.708917
$645$	17963.2	1.09659
$646$	−21287.0	−1.29648
$647$	12954.7	0.787174	0.393587	−	0.919287i	$-0.371234\pi$
$647$	12954.7	0.787174	0.393587	+	0.919287i	$0.371234\pi$
$648$	−648.000	−0.0392837
$649$	−16094.4	−0.973435
$650$	−21101.6	−1.27334
$651$	5688.28	0.342460
$652$	11276.3	0.677321
$653$	−6282.21	−0.376481	−0.188240	−	0.982123i	$-0.560278\pi$
$653$	−6282.21	−0.376481	−0.188240	+	0.982123i	$0.560278\pi$
$654$	−8425.63	−0.503774
$655$	2113.33	0.126068
$656$	5243.77	0.312096
$657$	555.206	0.0329690
$658$	8297.10	0.491572
$659$	11367.3	0.671938	0.335969	−	0.941873i	$-0.390936\pi$
$659$	11367.3	0.671938	0.335969	+	0.941873i	$0.390936\pi$
$660$	5702.94	0.336343
$661$	−6808.80	−0.400653	−0.200326	−	0.979729i	$-0.564200\pi$
$661$	−6808.80	−0.400653	−0.200326	+	0.979729i	$0.564200\pi$
$662$	21890.2	1.28517
$663$	17526.4	1.02665
$664$	854.122	0.0499192
$665$	−31229.6	−1.82110
$666$	2125.15	0.123646
$667$	21664.4	1.25764
$668$	5767.62	0.334066
$669$	−5147.90	−0.297503
$670$	−12085.0	−0.696840
$671$	−1837.29	−0.105705
$672$	1307.62	0.0750633
$673$	−15821.8	−0.906218	−0.453109	−	0.891455i	$-0.649685\pi$
$673$	−15821.8	−0.906218	−0.453109	+	0.891455i	$0.649685\pi$
$674$	20227.1	1.15596
$675$	−3651.77	−0.208232
$676$	15553.8	0.884943
$677$	−9484.07	−0.538408	−0.269204	−	0.963083i	$-0.586761\pi$
$677$	−9484.07	−0.538408	−0.269204	+	0.963083i	$0.586761\pi$
$678$	2531.73	0.143408
$679$	19906.7	1.12511
$680$	9665.22	0.545065
$681$	−13922.8	−0.783442
$682$	−8201.64	−0.460494
$683$	15489.9	0.867794	0.433897	−	0.900963i	$-0.357138\pi$
$683$	15489.9	0.867794	0.433897	+	0.900963i	$0.357138\pi$
$684$	−5116.37	−0.286008
$685$	−43894.0	−2.44833
$686$	13633.8	0.758804
$687$	−8204.56	−0.455638
$688$	−5938.63	−0.329081
$689$	−9680.19	−0.535248
$690$	−20582.6	−1.13561
$691$	14796.4	0.814590	0.407295	−	0.913297i	$-0.366472\pi$
$691$	14796.4	0.814590	0.407295	+	0.913297i	$0.366472\pi$
$692$	5583.91	0.306746
$693$	−3611.39	−0.197959
$694$	14225.1	0.778065
$695$	−18030.9	−0.984100
$696$	−2445.14	−0.133165
$697$	−24544.2	−1.33383
$698$	−3446.94	−0.186918
$699$	6101.58	0.330161
$700$	7369.03	0.397890
$701$	−14196.8	−0.764913	−0.382457	−	0.923973i	$-0.624922\pi$
$701$	−14196.8	−0.764913	−0.382457	+	0.923973i	$0.624922\pi$
$702$	4212.50	0.226482
$703$	16779.4	0.900211
$704$	−1885.39	−0.100935
$705$	14740.2	0.787444
$706$	3303.70	0.176114
$707$	−5582.51	−0.296962
$708$	−6555.93	−0.348004
$709$	−21142.5	−1.11992	−0.559959	−	0.828520i	$-0.689183\pi$
$709$	−21142.5	−1.11992	−0.559959	+	0.828520i	$0.689183\pi$
$710$	26154.3	1.38247
$711$	12466.1	0.657549
$712$	2328.90	0.122583
$713$	29600.8	1.55478
$714$	−6120.51	−0.320804
$715$	−37073.5	−1.93912
$716$	−1498.22	−0.0781998
$717$	−14729.6	−0.767208
$718$	8987.99	0.467171
$719$	−30327.9	−1.57307	−0.786536	−	0.617544i	$-0.788128\pi$
$719$	−30327.9	−1.57307	−0.786536	+	0.617544i	$0.788128\pi$
$720$	2323.05	0.120243
$721$	−14429.5	−0.745329
$722$	−26679.0	−1.37519
$723$	3120.43	0.160512
$724$	12924.7	0.663456
$725$	−13779.5	−0.705871
$726$	−2778.93	−0.142060
$727$	−26202.1	−1.33670	−0.668351	−	0.743846i	$-0.733000\pi$
$727$	−26202.1	−1.33670	−0.668351	+	0.743846i	$0.733000\pi$
$728$	−8500.54	−0.432762
$729$	729.000	0.0370370
$730$	−1990.39	−0.100914
$731$	27796.6	1.40642
$732$	−748.408	−0.0377896
$733$	6147.17	0.309755	0.154878	−	0.987934i	$-0.450502\pi$
$733$	6147.17	0.309755	0.154878	+	0.987934i	$0.450502\pi$
$734$	−6069.87	−0.305236
$735$	7620.92	0.382451
$736$	6804.62	0.340790
$737$	−11034.2	−0.551492
$738$	−5899.24	−0.294246
$739$	−36313.1	−1.80757	−0.903787	−	0.427983i	$-0.859224\pi$
$739$	−36313.1	−1.80757	−0.903787	+	0.427983i	$0.859224\pi$
$740$	−7618.58	−0.378466
$741$	33260.3	1.64892
$742$	3380.48	0.167252
$743$	−28751.9	−1.41966	−0.709830	−	0.704373i	$-0.751228\pi$
$743$	−28751.9	−1.41966	−0.709830	+	0.704373i	$0.751228\pi$
$744$	−3340.88	−0.164627
$745$	−23130.1	−1.13748
$746$	−9168.74	−0.449988
$747$	−960.888	−0.0470643
$748$	8824.84	0.431374
$749$	−13383.6	−0.652906
$750$	992.216	0.0483075
$751$	−21642.9	−1.05161	−0.525806	−	0.850605i	$-0.676236\pi$
$751$	−21642.9	−1.05161	−0.525806	+	0.850605i	$0.676236\pi$
$752$	−4873.10	−0.236308
$753$	−4711.55	−0.228019
$754$	15895.3	0.767735
$755$	55142.3	2.65806
$756$	−1471.07	−0.0707704
$757$	24250.8	1.16435	0.582174	−	0.813064i	$-0.302202\pi$
$757$	24250.8	1.16435	0.582174	+	0.813064i	$0.302202\pi$
$758$	22654.2	1.08554
$759$	−18793.0	−0.898739
$760$	18341.9	0.875437
$761$	−11254.6	−0.536110	−0.268055	−	0.963404i	$-0.586381\pi$
$761$	−11254.6	−0.536110	−0.268055	+	0.963404i	$0.586381\pi$
$762$	3036.32	0.144349
$763$	−19127.7	−0.907559
$764$	−13807.6	−0.653850
$765$	−10873.4	−0.513892
$766$	5864.47	0.276621
$767$	42618.6	2.00635
$768$	−768.000	−0.0360844
$769$	−14015.7	−0.657242	−0.328621	−	0.944462i	$-0.606584\pi$
$769$	−14015.7	−0.657242	−0.328621	+	0.944462i	$0.606584\pi$
$770$	12946.7	0.605929
$771$	−4881.98	−0.228042
$772$	−8197.89	−0.382187
$773$	3900.91	0.181508	0.0907542	−	0.995873i	$-0.471072\pi$
$773$	3900.91	0.181508	0.0907542	+	0.995873i	$0.471072\pi$
$774$	6680.96	0.310261
$775$	−18827.4	−0.872643
$776$	−11691.7	−0.540863
$777$	4824.47	0.222750
$778$	13132.9	0.605189
$779$	−46578.2	−2.14228
$780$	−15101.6	−0.693236
$781$	23880.2	1.09411
$782$	−31850.0	−1.45646
$783$	2750.78	0.125549
$784$	−2519.47	−0.114772
$785$	−23075.6	−1.04917
$786$	786.000	0.0356688
$787$	32955.4	1.49267	0.746336	−	0.665569i	$-0.231811\pi$
$787$	32955.4	1.49267	0.746336	+	0.665569i	$0.231811\pi$
$788$	−9019.54	−0.407751
$789$	−912.663	−0.0411808
$790$	−44690.6	−2.01268
$791$	5747.48	0.258353
$792$	2121.06	0.0951625
$793$	4865.23	0.217868
$794$	921.439	0.0411847
$795$	6005.58	0.267920
$796$	6469.27	0.288062
$797$	43827.6	1.94787	0.973936	−	0.226824i	$-0.0728342\pi$
$797$	43827.6	1.94787	0.973936	+	0.226824i	$0.0728342\pi$
$798$	−11615.1	−0.515249
$799$	22809.3	1.00993
$800$	−4328.03	−0.191274
$801$	−2620.01	−0.115572
$802$	−1459.48	−0.0642595
$803$	−1817.33	−0.0798656
$804$	−4494.70	−0.197159
$805$	−46726.2	−2.04582
$806$	21718.3	0.949124
$807$	5627.14	0.245458
$808$	3278.76	0.142755
$809$	−27593.8	−1.19919	−0.599595	−	0.800304i	$-0.704672\pi$
$809$	−27593.8	−1.19919	−0.599595	+	0.800304i	$0.704672\pi$
$810$	−2613.43	−0.113366
$811$	−18400.8	−0.796721	−0.398360	−	0.917229i	$-0.630421\pi$
$811$	−18400.8	−0.796721	−0.398360	+	0.917229i	$0.630421\pi$
$812$	−5550.89	−0.239899
$813$	−13082.9	−0.564375
$814$	−6956.16	−0.299525
$815$	45478.1	1.95463
$816$	3594.74	0.154217
$817$	52750.4	2.25888
$818$	−13483.8	−0.576344
$819$	9563.11	0.408012
$820$	21148.5	0.900655
$821$	10404.7	0.442299	0.221149	−	0.975240i	$-0.429019\pi$
$821$	10404.7	0.442299	0.221149	+	0.975240i	$0.429019\pi$
$822$	−16325.3	−0.692712
$823$	8244.21	0.349180	0.174590	−	0.984641i	$-0.444140\pi$
$823$	8244.21	0.349180	0.174590	+	0.984641i	$0.444140\pi$
$824$	8474.82	0.358294
$825$	11953.2	0.504431
$826$	−14883.1	−0.626936
$827$	33159.7	1.39429	0.697144	−	0.716931i	$-0.254454\pi$
$827$	33159.7	1.39429	0.697144	+	0.716931i	$0.254454\pi$
$828$	−7655.20	−0.321300
$829$	23267.8	0.974817	0.487409	−	0.873174i	$-0.337942\pi$
$829$	23267.8	0.974817	0.487409	+	0.873174i	$0.337942\pi$
$830$	3444.74	0.144059
$831$	−18890.6	−0.788576
$832$	4992.59	0.208037
$833$	11792.8	0.490510
$834$	−6706.13	−0.278434
$835$	23261.2	0.964058
$836$	16747.1	0.692837
$837$	3758.49	0.155212
$838$	6648.86	0.274082
$839$	−21256.9	−0.874698	−0.437349	−	0.899292i	$-0.644082\pi$
$839$	−21256.9	−0.874698	−0.437349	+	0.899292i	$0.644082\pi$
$840$	5273.73	0.216620
$841$	−14009.3	−0.574410
$842$	14732.5	0.602989
$843$	−13548.5	−0.553542
$844$	−23239.5	−0.947791
$845$	62729.4	2.55380
$846$	5482.24	0.222794
$847$	−6308.65	−0.255924
$848$	−1985.45	−0.0804015
$849$	−4632.74	−0.187273
$850$	20258.0	0.817461
$851$	25105.7	1.01129
$852$	9727.42	0.391145
$853$	39196.8	1.57336	0.786678	−	0.617363i	$-0.211799\pi$
$853$	39196.8	1.57336	0.786678	+	0.617363i	$0.211799\pi$
$854$	−1699.02	−0.0680786
$855$	−20634.7	−0.825370
$856$	7860.55	0.313864
$857$	34413.6	1.37170	0.685850	−	0.727743i	$-0.259431\pi$
$857$	34413.6	1.37170	0.685850	+	0.727743i	$0.259431\pi$
$858$	−13788.5	−0.548640
$859$	30884.1	1.22672	0.613360	−	0.789804i	$-0.289818\pi$
$859$	30884.1	1.22672	0.613360	+	0.789804i	$0.289818\pi$
$860$	−23950.9	−0.949674
$861$	−13392.3	−0.530091
$862$	−1775.92	−0.0701718
$863$	−45852.5	−1.80862	−0.904309	−	0.426878i	$-0.859613\pi$
$863$	−45852.5	−1.80862	−0.904309	+	0.426878i	$0.859613\pi$
$864$	864.000	0.0340207
$865$	22520.3	0.885218
$866$	2658.86	0.104332
$867$	−2086.68	−0.0817385
$868$	−7584.38	−0.296579
$869$	−40804.8	−1.59287
$870$	−9861.43	−0.384292
$871$	29219.0	1.13668
$872$	11234.2	0.436281
$873$	13153.2	0.509930
$874$	−60442.6	−2.33925
$875$	2252.50	0.0870268
$876$	−740.275	−0.0285520
$877$	−11101.9	−0.427463	−0.213732	−	0.976892i	$-0.568562\pi$
$877$	−11101.9	−0.427463	−0.213732	+	0.976892i	$0.568562\pi$
$878$	33222.5	1.27700
$879$	291.875	0.0111999
$880$	−7603.91	−0.291282
$881$	32752.0	1.25249	0.626245	−	0.779626i	$-0.284591\pi$
$881$	32752.0	1.25249	0.626245	+	0.779626i	$0.284591\pi$
$882$	2834.41	0.108208
$883$	−964.546	−0.0367606	−0.0183803	−	0.999831i	$-0.505851\pi$
$883$	−964.546	−0.0367606	−0.0183803	+	0.999831i	$0.505851\pi$
$884$	−23368.5	−0.889105
$885$	−26440.5	−1.00428
$886$	−19898.7	−0.754525
$887$	46989.3	1.77874	0.889372	−	0.457183i	$-0.151142\pi$
$887$	46989.3	1.77874	0.889372	+	0.457183i	$0.151142\pi$
$888$	−2833.54	−0.107080
$889$	6892.98	0.260048
$890$	9392.61	0.353754
$891$	−2386.20	−0.0897201
$892$	6863.87	0.257645
$893$	43285.8	1.62206
$894$	−8602.67	−0.321830
$895$	−6042.43	−0.225672
$896$	−1743.49	−0.0650067
$897$	49764.7	1.85239
$898$	15673.7	0.582447
$899$	14182.1	0.526141
$900$	4869.03	0.180334
$901$	9293.16	0.343618
$902$	19309.7	0.712795
$903$	15166.9	0.558941
$904$	−3375.65	−0.124195
$905$	52126.2	1.91462
$906$	20508.8	0.752051
$907$	22680.7	0.830319	0.415159	−	0.909749i	$-0.363726\pi$
$907$	22680.7	0.830319	0.415159	+	0.909749i	$0.363726\pi$
$908$	18563.8	0.678480
$909$	−3688.60	−0.134591
$910$	−34283.3	−1.24888
$911$	−2257.77	−0.0821111	−0.0410556	−	0.999157i	$-0.513072\pi$
$911$	−2257.77	−0.0821111	−0.0410556	+	0.999157i	$0.513072\pi$
$912$	6821.83	0.247690
$913$	3145.22	0.114011
$914$	−26409.8	−0.955755
$915$	−3018.38	−0.109054
$916$	10939.4	0.394594
$917$	1784.36	0.0642581
$918$	−4044.08	−0.145397
$919$	−12950.7	−0.464859	−0.232430	−	0.972613i	$-0.574668\pi$
$919$	−12950.7	−0.464859	−0.232430	+	0.972613i	$0.574668\pi$
$920$	27443.5	0.983463
$921$	−26485.8	−0.947597
$922$	30817.6	1.10078
$923$	−63235.7	−2.25507
$924$	4815.18	0.171437
$925$	−15968.3	−0.567604
$926$	−37361.5	−1.32589
$927$	−9534.17	−0.337803
$928$	3260.19	0.115324
$929$	−21148.3	−0.746881	−0.373440	−	0.927654i	$-0.621822\pi$
$929$	−21148.3	−0.746881	−0.373440	+	0.927654i	$0.621822\pi$
$930$	−13474.0	−0.475086
$931$	22379.4	0.787816
$932$	−8135.43	−0.285928
$933$	13969.6	0.490188
$934$	−8262.63	−0.289466
$935$	35591.2	1.24487
$936$	−5616.66	−0.196139
$937$	−16255.3	−0.566741	−0.283370	−	0.959011i	$-0.591453\pi$
$937$	−16255.3	−0.566741	−0.283370	+	0.959011i	$0.591453\pi$
$938$	−10203.7	−0.355186
$939$	9364.95	0.325467
$940$	−19653.6	−0.681946
$941$	−44929.7	−1.55650	−0.778250	−	0.627954i	$-0.783892\pi$
$941$	−44929.7	−1.55650	−0.778250	+	0.627954i	$0.783892\pi$
$942$	−8582.37	−0.296846
$943$	−69691.1	−2.40663
$944$	8741.23	0.301380
$945$	−5932.94	−0.204231
$946$	−21868.4	−0.751589
$947$	−18757.1	−0.643637	−0.321819	−	0.946801i	$-0.604294\pi$
$947$	−18757.1	−0.643637	−0.321819	+	0.946801i	$0.604294\pi$
$948$	−16621.5	−0.569454
$949$	4812.35	0.164611
$950$	38444.1	1.31294
$951$	−16255.7	−0.554287
$952$	8160.68	0.277825
$953$	26719.1	0.908203	0.454102	−	0.890950i	$-0.349960\pi$
$953$	26719.1	0.908203	0.454102	+	0.890950i	$0.349960\pi$
$954$	2233.63	0.0758033
$955$	−55687.0	−1.88690
$956$	19639.5	0.664421
$957$	−9003.99	−0.304135
$958$	11902.8	0.401423
$959$	−37061.2	−1.24793
$960$	−3097.40	−0.104133
$961$	−10413.5	−0.349550
$962$	18420.2	0.617350
$963$	−8843.11	−0.295914
$964$	−4160.57	−0.139007
$965$	−33062.7	−1.10293
$966$	−17378.6	−0.578829
$967$	39761.5	1.32228	0.661139	−	0.750263i	$-0.270073\pi$
$967$	39761.5	1.32228	0.661139	+	0.750263i	$0.270073\pi$
$968$	3705.24	0.123028
$969$	−31930.5	−1.05857
$970$	−47153.7	−1.56084
$971$	−13031.6	−0.430694	−0.215347	−	0.976538i	$-0.569088\pi$
$971$	−13031.6	−0.430694	−0.215347	+	0.976538i	$0.569088\pi$
$972$	−972.000	−0.0320750
$973$	−15224.1	−0.501605
$974$	18569.3	0.610881
$975$	−31652.4	−1.03968
$976$	997.877	0.0327267
$977$	17883.0	0.585596	0.292798	−	0.956174i	$-0.405414\pi$
$977$	17883.0	0.585596	0.292798	+	0.956174i	$0.405414\pi$
$978$	16914.4	0.553030
$979$	8575.93	0.279967
$980$	−10161.2	−0.331213
$981$	−12638.5	−0.411330
$982$	12691.6	0.412428
$983$	48249.6	1.56554	0.782769	−	0.622312i	$-0.213807\pi$
$983$	48249.6	1.56554	0.782769	+	0.622312i	$0.213807\pi$
$984$	7865.65	0.254825
$985$	−36376.5	−1.17670
$986$	−15259.8	−0.492870
$987$	12445.6	0.401367
$988$	−44347.1	−1.42801
$989$	78926.0	2.53761
$990$	8554.40	0.274623
$991$	−40785.7	−1.30737	−0.653684	−	0.756768i	$-0.726777\pi$
$991$	−40785.7	−1.30737	−0.653684	+	0.756768i	$0.726777\pi$
$992$	4454.50	0.142571
$993$	32835.2	1.04934
$994$	22082.9	0.704656
$995$	26091.0	0.831298
$996$	1281.18	0.0407589
$997$	−33666.7	−1.06944	−0.534722	−	0.845028i	$-0.679584\pi$
$997$	−33666.7	−1.06944	−0.534722	+	0.845028i	$0.679584\pi$
$998$	−21310.8	−0.675933
$999$	3187.73	0.100956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	786.4.a.g.1.8	✓	8
3.2	odd	2		2358.4.a.i.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.4.a.g.1.8	✓	8	1.1	even	1	trivial
2358.4.a.i.1.1		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +16.1323 q^{5} +6.00000 q^{6} +13.6210 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +16.1323 q^{5} +6.00000 q^{6} +13.6210 q^{7} -8.00000 q^{8} +9.00000 q^{9} -32.2646 q^{10} -29.4592 q^{11} -12.0000 q^{12} +78.0092 q^{13} -27.2421 q^{14} -48.3969 q^{15} +16.0000 q^{16} -74.8903 q^{17} -18.0000 q^{18} -142.121 q^{19} +64.5292 q^{20} -40.8631 q^{21} +58.9184 q^{22} -212.644 q^{23} +24.0000 q^{24} +135.251 q^{25} -156.018 q^{26} -27.0000 q^{27} +54.4842 q^{28} -101.881 q^{29} +96.7937 q^{30} -139.203 q^{31} -32.0000 q^{32} +88.3777 q^{33} +149.781 q^{34} +219.739 q^{35} +36.0000 q^{36} -118.064 q^{37} +284.243 q^{38} -234.028 q^{39} -129.058 q^{40} +327.735 q^{41} +81.7263 q^{42} -371.164 q^{43} -117.837 q^{44} +145.191 q^{45} +425.289 q^{46} -304.569 q^{47} -48.0000 q^{48} -157.467 q^{49} -270.502 q^{50} +224.671 q^{51} +312.037 q^{52} -124.090 q^{53} +54.0000 q^{54} -475.245 q^{55} -108.968 q^{56} +426.364 q^{57} +203.762 q^{58} +546.327 q^{59} -193.587 q^{60} +62.3673 q^{61} +278.406 q^{62} +122.589 q^{63} +64.0000 q^{64} +1258.47 q^{65} -176.755 q^{66} +374.558 q^{67} -299.561 q^{68} +637.933 q^{69} -439.477 q^{70} -810.618 q^{71} -72.0000 q^{72} +61.6896 q^{73} +236.128 q^{74} -405.752 q^{75} -568.486 q^{76} -401.265 q^{77} +468.055 q^{78} +1385.13 q^{79} +258.117 q^{80} +81.0000 q^{81} -655.471 q^{82} -106.765 q^{83} -163.453 q^{84} -1208.15 q^{85} +742.328 q^{86} +305.642 q^{87} +235.674 q^{88} -291.112 q^{89} -290.381 q^{90} +1062.57 q^{91} -850.577 q^{92} +417.610 q^{93} +609.138 q^{94} -2292.74 q^{95} +96.0000 q^{96} +1461.47 q^{97} +314.934 q^{98} -265.133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) \) 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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