LMFDB - Embedded newform 966.4.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,4,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("966.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	966.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:	$56.9958450655$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 356x^{2} + 245x + 16751 $ x^4 - x^3 - 356x^2 + 245x + 16751
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.09048$ of defining polynomial
Character	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.09048 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.09048 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -12.1810 q^{10} -17.9649 q^{11} -12.0000 q^{12} +17.6855 q^{13} +14.0000 q^{14} -18.2714 q^{15} +16.0000 q^{16} +6.89339 q^{17} -18.0000 q^{18} +26.1458 q^{19} +24.3619 q^{20} +21.0000 q^{21} +35.9298 q^{22} +23.0000 q^{23} +24.0000 q^{24} -87.9060 q^{25} -35.3711 q^{26} -27.0000 q^{27} -28.0000 q^{28} +30.1942 q^{29} +36.5429 q^{30} +61.0604 q^{31} -32.0000 q^{32} +53.8946 q^{33} -13.7868 q^{34} -42.6334 q^{35} +36.0000 q^{36} -299.610 q^{37} -52.2917 q^{38} -53.0566 q^{39} -48.7239 q^{40} -118.877 q^{41} -42.0000 q^{42} +412.909 q^{43} -71.8595 q^{44} +54.8143 q^{45} -46.0000 q^{46} -463.381 q^{47} -48.0000 q^{48} +49.0000 q^{49} +175.812 q^{50} -20.6802 q^{51} +70.7421 q^{52} +423.877 q^{53} +54.0000 q^{54} -109.415 q^{55} +56.0000 q^{56} -78.4375 q^{57} -60.3884 q^{58} +218.896 q^{59} -73.0858 q^{60} -85.7890 q^{61} -122.121 q^{62} -63.0000 q^{63} +64.0000 q^{64} +107.713 q^{65} -107.789 q^{66} +1049.52 q^{67} +27.5736 q^{68} -69.0000 q^{69} +85.2667 q^{70} -1000.28 q^{71} -72.0000 q^{72} -917.587 q^{73} +599.221 q^{74} +263.718 q^{75} +104.583 q^{76} +125.754 q^{77} +106.113 q^{78} +474.244 q^{79} +97.4477 q^{80} +81.0000 q^{81} +237.755 q^{82} +865.832 q^{83} +84.0000 q^{84} +41.9841 q^{85} -825.818 q^{86} -90.5826 q^{87} +143.719 q^{88} +465.302 q^{89} -109.629 q^{90} -123.799 q^{91} +92.0000 q^{92} -183.181 q^{93} +926.763 q^{94} +159.241 q^{95} +96.0000 q^{96} +49.5554 q^{97} -98.0000 q^{98} -161.684 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 5 q^{5} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 5 * q^5 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 5 q^{5} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} + 10 q^{10} - 41 q^{11} - 48 q^{12} + 23 q^{13} + 56 q^{14} + 15 q^{15} + 64 q^{16} + 18 q^{17} - 72 q^{18} + 15 q^{19} - 20 q^{20} + 84 q^{21} + 82 q^{22} + 92 q^{23} + 96 q^{24} + 219 q^{25} - 46 q^{26} - 108 q^{27} - 112 q^{28} - 46 q^{29} - 30 q^{30} + 142 q^{31} - 128 q^{32} + 123 q^{33} - 36 q^{34} + 35 q^{35} + 144 q^{36} - 142 q^{37} - 30 q^{38} - 69 q^{39} + 40 q^{40} - 621 q^{41} - 168 q^{42} + 185 q^{43} - 164 q^{44} - 45 q^{45} - 184 q^{46} + 669 q^{47} - 192 q^{48} + 196 q^{49} - 438 q^{50} - 54 q^{51} + 92 q^{52} + 422 q^{53} + 216 q^{54} + 1435 q^{55} + 224 q^{56} - 45 q^{57} + 92 q^{58} + 270 q^{59} + 60 q^{60} + 272 q^{61} - 284 q^{62} - 252 q^{63} + 256 q^{64} - 1992 q^{65} - 246 q^{66} - 67 q^{67} + 72 q^{68} - 276 q^{69} - 70 q^{70} + 611 q^{71} - 288 q^{72} - 236 q^{73} + 284 q^{74} - 657 q^{75} + 60 q^{76} + 287 q^{77} + 138 q^{78} + 558 q^{79} - 80 q^{80} + 324 q^{81} + 1242 q^{82} + 468 q^{83} + 336 q^{84} + 1045 q^{85} - 370 q^{86} + 138 q^{87} + 328 q^{88} - 1519 q^{89} + 90 q^{90} - 161 q^{91} + 368 q^{92} - 426 q^{93} - 1338 q^{94} + 23 q^{95} + 384 q^{96} + 600 q^{97} - 392 q^{98} - 369 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 5 * q^5 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9 + 10 * q^10 - 41 * q^11 - 48 * q^12 + 23 * q^13 + 56 * q^14 + 15 * q^15 + 64 * q^16 + 18 * q^17 - 72 * q^18 + 15 * q^19 - 20 * q^20 + 84 * q^21 + 82 * q^22 + 92 * q^23 + 96 * q^24 + 219 * q^25 - 46 * q^26 - 108 * q^27 - 112 * q^28 - 46 * q^29 - 30 * q^30 + 142 * q^31 - 128 * q^32 + 123 * q^33 - 36 * q^34 + 35 * q^35 + 144 * q^36 - 142 * q^37 - 30 * q^38 - 69 * q^39 + 40 * q^40 - 621 * q^41 - 168 * q^42 + 185 * q^43 - 164 * q^44 - 45 * q^45 - 184 * q^46 + 669 * q^47 - 192 * q^48 + 196 * q^49 - 438 * q^50 - 54 * q^51 + 92 * q^52 + 422 * q^53 + 216 * q^54 + 1435 * q^55 + 224 * q^56 - 45 * q^57 + 92 * q^58 + 270 * q^59 + 60 * q^60 + 272 * q^61 - 284 * q^62 - 252 * q^63 + 256 * q^64 - 1992 * q^65 - 246 * q^66 - 67 * q^67 + 72 * q^68 - 276 * q^69 - 70 * q^70 + 611 * q^71 - 288 * q^72 - 236 * q^73 + 284 * q^74 - 657 * q^75 + 60 * q^76 + 287 * q^77 + 138 * q^78 + 558 * q^79 - 80 * q^80 + 324 * q^81 + 1242 * q^82 + 468 * q^83 + 336 * q^84 + 1045 * q^85 - 370 * q^86 + 138 * q^87 + 328 * q^88 - 1519 * q^89 + 90 * q^90 - 161 * q^91 + 368 * q^92 - 426 * q^93 - 1338 * q^94 + 23 * q^95 + 384 * q^96 + 600 * q^97 - 392 * q^98 - 369 * 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
$5$	6.09048	0.544749	0.272375	−	0.962191i	$-0.412191\pi$
$5$	6.09048	0.544749	0.272375	+	0.962191i	$0.412191\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−12.1810	−0.385196
$11$	−17.9649	−0.492419	−0.246210	−	0.969217i	$-0.579185\pi$
$11$	−17.9649	−0.492419	−0.246210	+	0.969217i	$0.579185\pi$
$12$	−12.0000	−0.288675
$13$	17.6855	0.377314	0.188657	−	0.982043i	$-0.439587\pi$
$13$	17.6855	0.377314	0.188657	+	0.982043i	$0.439587\pi$
$14$	14.0000	0.267261
$15$	−18.2714	−0.314511
$16$	16.0000	0.250000
$17$	6.89339	0.0983467	0.0491733	−	0.998790i	$-0.484341\pi$
$17$	6.89339	0.0983467	0.0491733	+	0.998790i	$0.484341\pi$
$18$	−18.0000	−0.235702
$19$	26.1458	0.315698	0.157849	−	0.987463i	$-0.449544\pi$
$19$	26.1458	0.315698	0.157849	+	0.987463i	$0.449544\pi$
$20$	24.3619	0.272375
$21$	21.0000	0.218218
$22$	35.9298	0.348193
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	24.0000	0.204124
$25$	−87.9060	−0.703248
$26$	−35.3711	−0.266801
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	30.1942	0.193342	0.0966711	−	0.995316i	$-0.469181\pi$
$29$	30.1942	0.193342	0.0966711	+	0.995316i	$0.469181\pi$
$30$	36.5429	0.222393
$31$	61.0604	0.353767	0.176883	−	0.984232i	$-0.443398\pi$
$31$	61.0604	0.353767	0.176883	+	0.984232i	$0.443398\pi$
$32$	−32.0000	−0.176777
$33$	53.8946	0.284298
$34$	−13.7868	−0.0695416
$35$	−42.6334	−0.205896
$36$	36.0000	0.166667
$37$	−299.610	−1.33123	−0.665617	−	0.746294i	$-0.731831\pi$
$37$	−299.610	−1.33123	−0.665617	+	0.746294i	$0.731831\pi$
$38$	−52.2917	−0.223232
$39$	−53.0566	−0.217842
$40$	−48.7239	−0.192598
$41$	−118.877	−0.452817	−0.226409	−	0.974032i	$-0.572698\pi$
$41$	−118.877	−0.452817	−0.226409	+	0.974032i	$0.572698\pi$
$42$	−42.0000	−0.154303
$43$	412.909	1.46437	0.732187	−	0.681104i	$-0.238500\pi$
$43$	412.909	1.46437	0.732187	+	0.681104i	$0.238500\pi$
$44$	−71.8595	−0.246210
$45$	54.8143	0.181583
$46$	−46.0000	−0.147442
$47$	−463.381	−1.43811	−0.719055	−	0.694954i	$-0.755425\pi$
$47$	−463.381	−1.43811	−0.719055	+	0.694954i	$0.755425\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	175.812	0.497272
$51$	−20.6802	−0.0567805
$52$	70.7421	0.188657
$53$	423.877	1.09857	0.549283	−	0.835636i	$-0.314901\pi$
$53$	423.877	1.09857	0.549283	+	0.835636i	$0.314901\pi$
$54$	54.0000	0.136083
$55$	−109.415	−0.268245
$56$	56.0000	0.133631
$57$	−78.4375	−0.182268
$58$	−60.3884	−0.136714
$59$	218.896	0.483013	0.241507	−	0.970399i	$-0.422358\pi$
$59$	218.896	0.483013	0.241507	+	0.970399i	$0.422358\pi$
$60$	−73.0858	−0.157256
$61$	−85.7890	−0.180068	−0.0900340	−	0.995939i	$-0.528698\pi$
$61$	−85.7890	−0.180068	−0.0900340	+	0.995939i	$0.528698\pi$
$62$	−122.121	−0.250151
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	107.713	0.205542
$66$	−107.789	−0.201029
$67$	1049.52	1.91372	0.956860	−	0.290549i	$-0.0938380\pi$
$67$	1049.52	1.91372	0.956860	+	0.290549i	$0.0938380\pi$
$68$	27.5736	0.0491733
$69$	−69.0000	−0.120386
$70$	85.2667	0.145590
$71$	−1000.28	−1.67199	−0.835997	−	0.548734i	$-0.815110\pi$
$71$	−1000.28	−1.67199	−0.835997	+	0.548734i	$0.815110\pi$
$72$	−72.0000	−0.117851
$73$	−917.587	−1.47117	−0.735585	−	0.677432i	$-0.763093\pi$
$73$	−917.587	−1.47117	−0.735585	+	0.677432i	$0.763093\pi$
$74$	599.221	0.941324
$75$	263.718	0.406021
$76$	104.583	0.157849
$77$	125.754	0.186117
$78$	106.113	0.154038
$79$	474.244	0.675400	0.337700	−	0.941254i	$-0.390351\pi$
$79$	474.244	0.675400	0.337700	+	0.941254i	$0.390351\pi$
$80$	97.4477	0.136187
$81$	81.0000	0.111111
$82$	237.755	0.320190
$83$	865.832	1.14503	0.572515	−	0.819895i	$-0.305968\pi$
$83$	865.832	1.14503	0.572515	+	0.819895i	$0.305968\pi$
$84$	84.0000	0.109109
$85$	41.9841	0.0535743
$86$	−825.818	−1.03547
$87$	−90.5826	−0.111626
$88$	143.719	0.174097
$89$	465.302	0.554179	0.277089	−	0.960844i	$-0.410630\pi$
$89$	465.302	0.554179	0.277089	+	0.960844i	$0.410630\pi$
$90$	−109.629	−0.128399
$91$	−123.799	−0.142611
$92$	92.0000	0.104257
$93$	−183.181	−0.204247
$94$	926.763	1.01690
$95$	159.241	0.171976
$96$	96.0000	0.102062
$97$	49.5554	0.0518721	0.0259360	−	0.999664i	$-0.491743\pi$
$97$	49.5554	0.0518721	0.0259360	+	0.999664i	$0.491743\pi$
$98$	−98.0000	−0.101015
$99$	−161.684	−0.164140
$100$	−351.624	−0.351624
$101$	−852.356	−0.839729	−0.419864	−	0.907587i	$-0.637922\pi$
$101$	−852.356	−0.839729	−0.419864	+	0.907587i	$0.637922\pi$
$102$	41.3604	0.0401499
$103$	−579.828	−0.554681	−0.277341	−	0.960772i	$-0.589453\pi$
$103$	−579.828	−0.554681	−0.277341	+	0.960772i	$0.589453\pi$
$104$	−141.484	−0.133401
$105$	127.900	0.118874
$106$	−847.754	−0.776803
$107$	327.101	0.295533	0.147767	−	0.989022i	$-0.452792\pi$
$107$	327.101	0.295533	0.147767	+	0.989022i	$0.452792\pi$
$108$	−108.000	−0.0962250
$109$	−906.894	−0.796924	−0.398462	−	0.917185i	$-0.630456\pi$
$109$	−906.894	−0.796924	−0.398462	+	0.917185i	$0.630456\pi$
$110$	218.829	0.189678
$111$	898.831	0.768588
$112$	−112.000	−0.0944911
$113$	−487.194	−0.405587	−0.202793	−	0.979222i	$-0.565002\pi$
$113$	−487.194	−0.405587	−0.202793	+	0.979222i	$0.565002\pi$
$114$	156.875	0.128883
$115$	140.081	0.113588
$116$	120.777	0.0966711
$117$	159.170	0.125771
$118$	−437.791	−0.341542
$119$	−48.2538	−0.0371716
$120$	146.172	0.111196
$121$	−1008.26	−0.757523
$122$	171.578	0.127327
$123$	356.632	0.261434
$124$	244.241	0.176883
$125$	−1296.70	−0.927843
$126$	126.000	0.0890871
$127$	1411.70	0.986365	0.493183	−	0.869926i	$-0.335833\pi$
$127$	1411.70	0.986365	0.493183	+	0.869926i	$0.335833\pi$
$128$	−128.000	−0.0883883
$129$	−1238.73	−0.845456
$130$	−215.427	−0.145340
$131$	−2303.68	−1.53644	−0.768221	−	0.640185i	$-0.778858\pi$
$131$	−2303.68	−1.53644	−0.768221	+	0.640185i	$0.778858\pi$
$132$	215.579	0.142149
$133$	−183.021	−0.119323
$134$	−2099.04	−1.35320
$135$	−164.443	−0.104837
$136$	−55.1472	−0.0347708
$137$	−2639.94	−1.64632	−0.823160	−	0.567810i	$-0.807791\pi$
$137$	−2639.94	−1.64632	−0.823160	+	0.567810i	$0.807791\pi$
$138$	138.000	0.0851257
$139$	608.541	0.371336	0.185668	−	0.982613i	$-0.440555\pi$
$139$	608.541	0.371336	0.185668	+	0.982613i	$0.440555\pi$
$140$	−170.533	−0.102948
$141$	1390.14	0.830293
$142$	2000.56	1.18228
$143$	−317.718	−0.185797
$144$	144.000	0.0833333
$145$	183.897	0.105323
$146$	1835.17	1.04027
$147$	−147.000	−0.0824786
$148$	−1198.44	−0.665617
$149$	−1239.18	−0.681325	−0.340662	−	0.940186i	$-0.610651\pi$
$149$	−1239.18	−0.681325	−0.340662	+	0.940186i	$0.610651\pi$
$150$	−527.436	−0.287100
$151$	−1045.86	−0.563650	−0.281825	−	0.959466i	$-0.590940\pi$
$151$	−1045.86	−0.563650	−0.281825	+	0.959466i	$0.590940\pi$
$152$	−209.167	−0.111616
$153$	62.0405	0.0327822
$154$	−251.508	−0.131605
$155$	371.887	0.192714
$156$	−212.226	−0.108921
$157$	−688.801	−0.350142	−0.175071	−	0.984556i	$-0.556016\pi$
$157$	−688.801	−0.350142	−0.175071	+	0.984556i	$0.556016\pi$
$158$	−948.487	−0.477580
$159$	−1271.63	−0.634257
$160$	−194.895	−0.0962990
$161$	−161.000	−0.0788110
$162$	−162.000	−0.0785674
$163$	35.5164	0.0170666	0.00853330	−	0.999964i	$-0.497284\pi$
$163$	35.5164	0.0170666	0.00853330	+	0.999964i	$0.497284\pi$
$164$	−475.509	−0.226409
$165$	328.244	0.154871
$166$	−1731.66	−0.809658
$167$	−936.021	−0.433722	−0.216861	−	0.976203i	$-0.569582\pi$
$167$	−936.021	−0.433722	−0.216861	+	0.976203i	$0.569582\pi$
$168$	−168.000	−0.0771517
$169$	−1884.22	−0.857634
$170$	−83.9682	−0.0378827
$171$	235.313	0.105233
$172$	1651.64	0.732187
$173$	−3729.53	−1.63902	−0.819511	−	0.573063i	$-0.805755\pi$
$173$	−3729.53	−1.63902	−0.819511	+	0.573063i	$0.805755\pi$
$174$	181.165	0.0789316
$175$	615.342	0.265803
$176$	−287.438	−0.123105
$177$	−656.687	−0.278868
$178$	−930.604	−0.391864
$179$	−1349.85	−0.563644	−0.281822	−	0.959467i	$-0.590939\pi$
$179$	−1349.85	−0.563644	−0.281822	+	0.959467i	$0.590939\pi$
$180$	219.257	0.0907915
$181$	482.526	0.198154	0.0990770	−	0.995080i	$-0.468411\pi$
$181$	482.526	0.198154	0.0990770	+	0.995080i	$0.468411\pi$
$182$	247.597	0.100841
$183$	257.367	0.103962
$184$	−184.000	−0.0737210
$185$	−1824.77	−0.725188
$186$	366.362	0.144425
$187$	−123.839	−0.0484278
$188$	−1853.53	−0.719055
$189$	189.000	0.0727393
$190$	−318.481	−0.121606
$191$	−2106.01	−0.797831	−0.398916	−	0.916988i	$-0.630613\pi$
$191$	−2106.01	−0.797831	−0.398916	+	0.916988i	$0.630613\pi$
$192$	−192.000	−0.0721688
$193$	1140.93	0.425524	0.212762	−	0.977104i	$-0.431754\pi$
$193$	1140.93	0.425524	0.212762	+	0.977104i	$0.431754\pi$
$194$	−99.1108	−0.0366791
$195$	−323.140	−0.118669
$196$	196.000	0.0714286
$197$	−4371.44	−1.58098	−0.790488	−	0.612477i	$-0.790173\pi$
$197$	−4371.44	−1.58098	−0.790488	+	0.612477i	$0.790173\pi$
$198$	323.368	0.116064
$199$	−1174.27	−0.418299	−0.209149	−	0.977884i	$-0.567069\pi$
$199$	−1174.27	−0.418299	−0.209149	+	0.977884i	$0.567069\pi$
$200$	703.248	0.248636
$201$	−3148.56	−1.10489
$202$	1704.71	0.593778
$203$	−211.359	−0.0730765
$204$	−82.7207	−0.0283902
$205$	−724.020	−0.246672
$206$	1159.66	0.392219
$207$	207.000	0.0695048
$208$	282.968	0.0943285
$209$	−469.707	−0.155456
$210$	−255.800	−0.0840566
$211$	−4787.23	−1.56193	−0.780963	−	0.624577i	$-0.785271\pi$
$211$	−4787.23	−1.56193	−0.780963	+	0.624577i	$0.785271\pi$
$212$	1695.51	0.549283
$213$	3000.84	0.965326
$214$	−654.202	−0.208974
$215$	2514.82	0.797716
$216$	216.000	0.0680414
$217$	−427.423	−0.133711
$218$	1813.79	0.563510
$219$	2752.76	0.849381
$220$	−437.659	−0.134123
$221$	121.913	0.0371076
$222$	−1797.66	−0.543474
$223$	4812.21	1.44506	0.722532	−	0.691338i	$-0.242978\pi$
$223$	4812.21	1.44506	0.722532	+	0.691338i	$0.242978\pi$
$224$	224.000	0.0668153
$225$	−791.154	−0.234416
$226$	974.387	0.286793
$227$	−6415.27	−1.87575	−0.937877	−	0.346968i	$-0.887211\pi$
$227$	−6415.27	−1.87575	−0.937877	+	0.346968i	$0.887211\pi$
$228$	−313.750	−0.0911342
$229$	2262.29	0.652823	0.326412	−	0.945228i	$-0.394160\pi$
$229$	2262.29	0.652823	0.326412	+	0.945228i	$0.394160\pi$
$230$	−280.162	−0.0803189
$231$	−377.262	−0.107455
$232$	−241.554	−0.0683568
$233$	331.393	0.0931773	0.0465886	−	0.998914i	$-0.485165\pi$
$233$	331.393	0.0931773	0.0465886	+	0.998914i	$0.485165\pi$
$234$	−318.340	−0.0889338
$235$	−2822.22	−0.783409
$236$	875.582	0.241507
$237$	−1422.73	−0.389942
$238$	96.5075	0.0262843
$239$	−98.7038	−0.0267139	−0.0133569	−	0.999911i	$-0.504252\pi$
$239$	−98.7038	−0.0267139	−0.0133569	+	0.999911i	$0.504252\pi$
$240$	−292.343	−0.0786278
$241$	6773.24	1.81038	0.905192	−	0.425003i	$-0.139727\pi$
$241$	6773.24	1.81038	0.905192	+	0.425003i	$0.139727\pi$
$242$	2016.53	0.535650
$243$	−243.000	−0.0641500
$244$	−343.156	−0.0900340
$245$	298.434	0.0778213
$246$	−713.264	−0.184862
$247$	462.403	0.119117
$248$	−488.483	−0.125075
$249$	−2597.50	−0.661083
$250$	2593.40	0.656084
$251$	−6221.89	−1.56463	−0.782315	−	0.622883i	$-0.785961\pi$
$251$	−6221.89	−1.56463	−0.782315	+	0.622883i	$0.785961\pi$
$252$	−252.000	−0.0629941
$253$	−413.192	−0.102677
$254$	−2823.41	−0.697465
$255$	−125.952	−0.0309311
$256$	256.000	0.0625000
$257$	−4867.49	−1.18142	−0.590711	−	0.806883i	$-0.701153\pi$
$257$	−4867.49	−1.18142	−0.590711	+	0.806883i	$0.701153\pi$
$258$	2477.45	0.597828
$259$	2097.27	0.503159
$260$	430.854	0.102771
$261$	271.748	0.0644474
$262$	4607.37	1.08643
$263$	3120.48	0.731623	0.365811	−	0.930689i	$-0.380792\pi$
$263$	3120.48	0.731623	0.365811	+	0.930689i	$0.380792\pi$
$264$	−431.157	−0.100515
$265$	2581.62	0.598443
$266$	366.042	0.0843739
$267$	−1395.91	−0.319955
$268$	4198.08	0.956860
$269$	−6411.00	−1.45311	−0.726553	−	0.687110i	$-0.758879\pi$
$269$	−6411.00	−1.45311	−0.726553	+	0.687110i	$0.758879\pi$
$270$	328.886	0.0741310
$271$	−711.002	−0.159374	−0.0796869	−	0.996820i	$-0.525392\pi$
$271$	−711.002	−0.159374	−0.0796869	+	0.996820i	$0.525392\pi$
$272$	110.294	0.0245867
$273$	371.396	0.0823367
$274$	5279.89	1.16412
$275$	1579.22	0.346293
$276$	−276.000	−0.0601929
$277$	−6042.46	−1.31067	−0.655336	−	0.755337i	$-0.727473\pi$
$277$	−6042.46	−1.31067	−0.655336	+	0.755337i	$0.727473\pi$
$278$	−1217.08	−0.262574
$279$	549.543	0.117922
$280$	341.067	0.0727952
$281$	1329.48	0.282243	0.141122	−	0.989992i	$-0.454929\pi$
$281$	1329.48	0.282243	0.141122	+	0.989992i	$0.454929\pi$
$282$	−2780.29	−0.587106
$283$	−1486.75	−0.312291	−0.156145	−	0.987734i	$-0.549907\pi$
$283$	−1486.75	−0.312291	−0.156145	+	0.987734i	$0.549907\pi$
$284$	−4001.13	−0.835997
$285$	−477.722	−0.0992906
$286$	635.437	0.131378
$287$	832.141	0.171149
$288$	−288.000	−0.0589256
$289$	−4865.48	−0.990328
$290$	−367.794	−0.0744746
$291$	−148.666	−0.0299483
$292$	−3670.35	−0.735585
$293$	4344.49	0.866239	0.433119	−	0.901337i	$-0.357413\pi$
$293$	4344.49	0.866239	0.433119	+	0.901337i	$0.357413\pi$
$294$	294.000	0.0583212
$295$	1333.18	0.263121
$296$	2396.88	0.470662
$297$	485.052	0.0947662
$298$	2478.35	0.481769
$299$	406.767	0.0786754
$300$	1054.87	0.203010
$301$	−2890.36	−0.553481
$302$	2091.73	0.398561
$303$	2557.07	0.484818
$304$	418.333	0.0789246
$305$	−522.496	−0.0980920
$306$	−124.081	−0.0231805
$307$	8622.91	1.60305	0.801523	−	0.597963i	$-0.204023\pi$
$307$	8622.91	1.60305	0.801523	+	0.597963i	$0.204023\pi$
$308$	503.017	0.0930585
$309$	1739.48	0.320245
$310$	−743.774	−0.136269
$311$	4709.34	0.858656	0.429328	−	0.903149i	$-0.358750\pi$
$311$	4709.34	0.858656	0.429328	+	0.903149i	$0.358750\pi$
$312$	424.453	0.0770189
$313$	1731.53	0.312690	0.156345	−	0.987703i	$-0.450029\pi$
$313$	1731.53	0.312690	0.156345	+	0.987703i	$0.450029\pi$
$314$	1377.60	0.247588
$315$	−383.700	−0.0686319
$316$	1896.97	0.337700
$317$	5640.48	0.999372	0.499686	−	0.866206i	$-0.333449\pi$
$317$	5640.48	0.999372	0.499686	+	0.866206i	$0.333449\pi$
$318$	2543.26	0.448488
$319$	−542.435	−0.0952054
$320$	389.791	0.0680937
$321$	−981.303	−0.170626
$322$	322.000	0.0557278
$323$	180.234	0.0310479
$324$	324.000	0.0555556
$325$	−1554.66	−0.265345
$326$	−71.0327	−0.0120679
$327$	2720.68	0.460104
$328$	951.018	0.160095
$329$	3243.67	0.543554
$330$	−656.488	−0.109511
$331$	10335.2	1.71624	0.858120	−	0.513450i	$-0.171633\pi$
$331$	10335.2	1.71624	0.858120	+	0.513450i	$0.171633\pi$
$332$	3463.33	0.572515
$333$	−2696.49	−0.443744
$334$	1872.04	0.306687
$335$	6392.08	1.04250
$336$	336.000	0.0545545
$337$	−1822.57	−0.294604	−0.147302	−	0.989092i	$-0.547059\pi$
$337$	−1822.57	−0.294604	−0.147302	+	0.989092i	$0.547059\pi$
$338$	3768.44	0.606439
$339$	1461.58	0.234166
$340$	167.936	0.0267871
$341$	−1096.94	−0.174202
$342$	−470.625	−0.0744108
$343$	−343.000	−0.0539949
$344$	−3303.27	−0.517734
$345$	−420.243	−0.0655801
$346$	7459.06	1.15896
$347$	−609.110	−0.0942326	−0.0471163	−	0.998889i	$-0.515003\pi$
$347$	−609.110	−0.0942326	−0.0471163	+	0.998889i	$0.515003\pi$
$348$	−362.330	−0.0558131
$349$	8746.35	1.34149	0.670747	−	0.741686i	$-0.265974\pi$
$349$	8746.35	1.34149	0.670747	+	0.741686i	$0.265974\pi$
$350$	−1230.68	−0.187951
$351$	−477.509	−0.0726141
$352$	574.876	0.0870483
$353$	−5303.10	−0.799591	−0.399795	−	0.916604i	$-0.630919\pi$
$353$	−5303.10	−0.799591	−0.399795	+	0.916604i	$0.630919\pi$
$354$	1313.37	0.197189
$355$	−6092.20	−0.910817
$356$	1861.21	0.277089
$357$	144.761	0.0214610
$358$	2699.69	0.398556
$359$	−11963.3	−1.75877	−0.879385	−	0.476112i	$-0.842046\pi$
$359$	−11963.3	−1.75877	−0.879385	+	0.476112i	$0.842046\pi$
$360$	−438.515	−0.0641993
$361$	−6175.40	−0.900335
$362$	−965.052	−0.140116
$363$	3024.79	0.437356
$364$	−495.195	−0.0713057
$365$	−5588.55	−0.801419
$366$	−514.734	−0.0735125
$367$	−10160.1	−1.44510	−0.722551	−	0.691318i	$-0.757031\pi$
$367$	−10160.1	−1.44510	−0.722551	+	0.691318i	$0.757031\pi$
$368$	368.000	0.0521286
$369$	−1069.90	−0.150939
$370$	3649.54	0.512786
$371$	−2967.14	−0.415219
$372$	−732.724	−0.102124
$373$	2691.70	0.373649	0.186825	−	0.982393i	$-0.440180\pi$
$373$	2691.70	0.373649	0.186825	+	0.982393i	$0.440180\pi$
$374$	247.678	0.0342436
$375$	3890.10	0.535691
$376$	3707.05	0.508448
$377$	534.000	0.0729507
$378$	−378.000	−0.0514344
$379$	349.535	0.0473731	0.0236866	−	0.999719i	$-0.492460\pi$
$379$	349.535	0.0473731	0.0236866	+	0.999719i	$0.492460\pi$
$380$	636.963	0.0859882
$381$	−4235.11	−0.569478
$382$	4212.03	0.564152
$383$	−9336.92	−1.24568	−0.622839	−	0.782350i	$-0.714021\pi$
$383$	−9336.92	−1.24568	−0.622839	+	0.782350i	$0.714021\pi$
$384$	384.000	0.0510310
$385$	765.903	0.101387
$386$	−2281.87	−0.300891
$387$	3716.18	0.488124
$388$	198.222	0.0259360
$389$	6328.26	0.824821	0.412410	−	0.910998i	$-0.364687\pi$
$389$	6328.26	0.824821	0.412410	+	0.910998i	$0.364687\pi$
$390$	646.280	0.0839120
$391$	158.548	0.0205067
$392$	−392.000	−0.0505076
$393$	6911.05	0.887065
$394$	8742.89	1.11792
$395$	2888.37	0.367923
$396$	−646.736	−0.0820699
$397$	6442.53	0.814461	0.407231	−	0.913325i	$-0.366495\pi$
$397$	6442.53	0.814461	0.407231	+	0.913325i	$0.366495\pi$
$398$	2348.53	0.295782
$399$	549.063	0.0688910
$400$	−1406.50	−0.175812
$401$	−4846.22	−0.603513	−0.301756	−	0.953385i	$-0.597573\pi$
$401$	−4846.22	−0.603513	−0.301756	+	0.953385i	$0.597573\pi$
$402$	6297.12	0.781273
$403$	1079.89	0.133481
$404$	−3409.42	−0.419864
$405$	493.329	0.0605277
$406$	422.719	0.0516729
$407$	5382.46	0.655525
$408$	165.441	0.0200749
$409$	6301.68	0.761853	0.380926	−	0.924605i	$-0.375605\pi$
$409$	6301.68	0.761853	0.380926	+	0.924605i	$0.375605\pi$
$410$	1448.04	0.174423
$411$	7919.83	0.950503
$412$	−2319.31	−0.277341
$413$	−1532.27	−0.182562
$414$	−414.000	−0.0491473
$415$	5273.33	0.623754
$416$	−565.937	−0.0667003
$417$	−1825.62	−0.214391
$418$	939.413	0.109924
$419$	1848.26	0.215497	0.107749	−	0.994178i	$-0.465636\pi$
$419$	1848.26	0.215497	0.107749	+	0.994178i	$0.465636\pi$
$420$	511.600	0.0594370
$421$	−8088.72	−0.936390	−0.468195	−	0.883625i	$-0.655096\pi$
$421$	−8088.72	−0.936390	−0.468195	+	0.883625i	$0.655096\pi$
$422$	9574.46	1.10445
$423$	−4170.43	−0.479370
$424$	−3391.02	−0.388402
$425$	−605.971	−0.0691621
$426$	−6001.69	−0.682589
$427$	600.523	0.0680593
$428$	1308.40	0.147767
$429$	953.155	0.107270
$430$	−5029.63	−0.564071
$431$	−9876.71	−1.10382	−0.551908	−	0.833905i	$-0.686100\pi$
$431$	−9876.71	−1.10382	−0.551908	+	0.833905i	$0.686100\pi$
$432$	−432.000	−0.0481125
$433$	−3824.73	−0.424491	−0.212246	−	0.977216i	$-0.568078\pi$
$433$	−3824.73	−0.424491	−0.212246	+	0.977216i	$0.568078\pi$
$434$	854.845	0.0945481
$435$	−551.692	−0.0608083
$436$	−3627.58	−0.398462
$437$	601.354	0.0658276
$438$	−5505.52	−0.600603
$439$	−12862.8	−1.39842	−0.699212	−	0.714914i	$-0.746466\pi$
$439$	−12862.8	−1.39842	−0.699212	+	0.714914i	$0.746466\pi$
$440$	875.318	0.0948390
$441$	441.000	0.0476190
$442$	−243.827	−0.0262390
$443$	−447.340	−0.0479768	−0.0239884	−	0.999712i	$-0.507636\pi$
$443$	−447.340	−0.0479768	−0.0239884	+	0.999712i	$0.507636\pi$
$444$	3595.32	0.384294
$445$	2833.91	0.301888
$446$	−9624.41	−1.02181
$447$	3717.53	0.393363
$448$	−448.000	−0.0472456
$449$	5081.59	0.534109	0.267055	−	0.963681i	$-0.413950\pi$
$449$	5081.59	0.534109	0.267055	+	0.963681i	$0.413950\pi$
$450$	1582.31	0.165757
$451$	2135.62	0.222976
$452$	−1948.77	−0.202793
$453$	3137.59	0.325423
$454$	12830.5	1.32636
$455$	−753.994	−0.0776874
$456$	627.500	0.0644416
$457$	2206.72	0.225878	0.112939	−	0.993602i	$-0.463974\pi$
$457$	2206.72	0.225878	0.112939	+	0.993602i	$0.463974\pi$
$458$	−4524.59	−0.461616
$459$	−186.122	−0.0189268
$460$	560.324	0.0567940
$461$	14866.2	1.50193	0.750963	−	0.660344i	$-0.229589\pi$
$461$	14866.2	1.50193	0.750963	+	0.660344i	$0.229589\pi$
$462$	754.525	0.0759820
$463$	9427.32	0.946274	0.473137	−	0.880989i	$-0.343122\pi$
$463$	9427.32	0.946274	0.473137	+	0.880989i	$0.343122\pi$
$464$	483.107	0.0483355
$465$	−1115.66	−0.111264
$466$	−662.787	−0.0658863
$467$	10300.3	1.02065	0.510324	−	0.859982i	$-0.329526\pi$
$467$	10300.3	1.02065	0.510324	+	0.859982i	$0.329526\pi$
$468$	636.679	0.0628857
$469$	−7346.64	−0.723318
$470$	5644.43	0.553954
$471$	2066.40	0.202155
$472$	−1751.16	−0.170771
$473$	−7417.86	−0.721086
$474$	2845.46	0.275731
$475$	−2298.38	−0.222014
$476$	−193.015	−0.0185858
$477$	3814.89	0.366189
$478$	197.408	0.0188896
$479$	18496.3	1.76434	0.882169	−	0.470933i	$-0.156083\pi$
$479$	18496.3	1.76434	0.882169	+	0.470933i	$0.156083\pi$
$480$	584.686	0.0555982
$481$	−5298.77	−0.502293
$482$	−13546.5	−1.28013
$483$	483.000	0.0455016
$484$	−4033.05	−0.378762
$485$	301.816	0.0282573
$486$	486.000	0.0453609
$487$	−7324.26	−0.681507	−0.340753	−	0.940153i	$-0.610682\pi$
$487$	−7324.26	−0.681507	−0.340753	+	0.940153i	$0.610682\pi$
$488$	686.312	0.0636637
$489$	−106.549	−0.00985341
$490$	−596.867	−0.0550280
$491$	8680.11	0.797816	0.398908	−	0.916991i	$-0.369389\pi$
$491$	8680.11	0.797816	0.398908	+	0.916991i	$0.369389\pi$
$492$	1426.53	0.130717
$493$	208.141	0.0190146
$494$	−924.806	−0.0842287
$495$	−984.733	−0.0894150
$496$	976.966	0.0884417
$497$	7001.97	0.631954
$498$	5194.99	0.467456
$499$	−7679.57	−0.688948	−0.344474	−	0.938796i	$-0.611943\pi$
$499$	−7679.57	−0.688948	−0.344474	+	0.938796i	$0.611943\pi$
$500$	−5186.80	−0.463922
$501$	2808.06	0.250409
$502$	12443.8	1.10636
$503$	12518.0	1.10964	0.554821	−	0.831970i	$-0.312787\pi$
$503$	12518.0	1.10964	0.554821	+	0.831970i	$0.312787\pi$
$504$	504.000	0.0445435
$505$	−5191.26	−0.457442
$506$	826.384	0.0726033
$507$	5652.67	0.495155
$508$	5646.81	0.493183
$509$	−7981.60	−0.695046	−0.347523	−	0.937672i	$-0.612977\pi$
$509$	−7981.60	−0.695046	−0.347523	+	0.937672i	$0.612977\pi$
$510$	251.905	0.0218716
$511$	6423.11	0.556050
$512$	−512.000	−0.0441942
$513$	−705.938	−0.0607562
$514$	9734.98	0.835392
$515$	−3531.43	−0.302162
$516$	−4954.91	−0.422728
$517$	8324.59	0.708153
$518$	−4194.54	−0.355787
$519$	11188.6	0.946290
$520$	−861.707	−0.0726699
$521$	−8340.61	−0.701360	−0.350680	−	0.936495i	$-0.614050\pi$
$521$	−8340.61	−0.701360	−0.350680	+	0.936495i	$0.614050\pi$
$522$	−543.496	−0.0455712
$523$	16430.2	1.37370	0.686848	−	0.726801i	$-0.258994\pi$
$523$	16430.2	1.37370	0.686848	+	0.726801i	$0.258994\pi$
$524$	−9214.74	−0.768221
$525$	−1846.03	−0.153461
$526$	−6240.95	−0.517335
$527$	420.913	0.0347918
$528$	862.314	0.0710746
$529$	529.000	0.0434783
$530$	−5163.23	−0.423163
$531$	1970.06	0.161004
$532$	−732.083	−0.0596614
$533$	−2102.41	−0.170854
$534$	2791.81	0.226243
$535$	1992.20	0.160991
$536$	−8396.16	−0.676602
$537$	4049.54	0.325420
$538$	12822.0	1.02750
$539$	−880.279	−0.0703456
$540$	−657.772	−0.0524185
$541$	−3675.89	−0.292124	−0.146062	−	0.989275i	$-0.546660\pi$
$541$	−3675.89	−0.292124	−0.146062	+	0.989275i	$0.546660\pi$
$542$	1422.00	0.112694
$543$	−1447.58	−0.114404
$544$	−220.589	−0.0173854
$545$	−5523.42	−0.434124
$546$	−742.792	−0.0582208
$547$	1949.46	0.152382	0.0761911	−	0.997093i	$-0.475724\pi$
$547$	1949.46	0.152382	0.0761911	+	0.997093i	$0.475724\pi$
$548$	−10559.8	−0.823160
$549$	−772.101	−0.0600227
$550$	−3158.44	−0.244866
$551$	789.453	0.0610378
$552$	552.000	0.0425628
$553$	−3319.70	−0.255277
$554$	12084.9	0.926785
$555$	5474.31	0.418688
$556$	2434.16	0.185668
$557$	5917.07	0.450115	0.225058	−	0.974345i	$-0.427743\pi$
$557$	5917.07	0.450115	0.225058	+	0.974345i	$0.427743\pi$
$558$	−1099.09	−0.0833836
$559$	7302.52	0.552529
$560$	−682.134	−0.0514740
$561$	371.517	0.0279598
$562$	−2658.97	−0.199576
$563$	−10321.4	−0.772640	−0.386320	−	0.922365i	$-0.626254\pi$
$563$	−10321.4	−0.772640	−0.386320	+	0.922365i	$0.626254\pi$
$564$	5560.58	0.415146
$565$	−2967.24	−0.220943
$566$	2973.50	0.220823
$567$	−567.000	−0.0419961
$568$	8002.25	0.591139
$569$	−10995.4	−0.810106	−0.405053	−	0.914293i	$-0.632747\pi$
$569$	−10995.4	−0.810106	−0.405053	+	0.914293i	$0.632747\pi$
$570$	955.444	0.0702091
$571$	−16788.6	−1.23044	−0.615219	−	0.788356i	$-0.710933\pi$
$571$	−16788.6	−1.23044	−0.615219	+	0.788356i	$0.710933\pi$
$572$	−1270.87	−0.0928984
$573$	6318.04	0.460628
$574$	−1664.28	−0.121021
$575$	−2021.84	−0.146637
$576$	576.000	0.0416667
$577$	10882.4	0.785165	0.392582	−	0.919717i	$-0.371582\pi$
$577$	10882.4	0.785165	0.392582	+	0.919717i	$0.371582\pi$
$578$	9730.96	0.700268
$579$	−3422.80	−0.245677
$580$	735.589	0.0526615
$581$	−6060.83	−0.432780
$582$	297.332	0.0211767
$583$	−7614.90	−0.540955
$584$	7340.70	0.520137
$585$	969.420	0.0685139
$586$	−8688.99	−0.612523
$587$	13026.0	0.915915	0.457958	−	0.888974i	$-0.348581\pi$
$587$	13026.0	0.915915	0.457958	+	0.888974i	$0.348581\pi$
$588$	−588.000	−0.0412393
$589$	1596.47	0.111684
$590$	−2666.36	−0.186055
$591$	13114.3	0.912777
$592$	−4793.76	−0.332808
$593$	−4356.00	−0.301652	−0.150826	−	0.988560i	$-0.548193\pi$
$593$	−4356.00	−0.301652	−0.150826	+	0.988560i	$0.548193\pi$
$594$	−970.103	−0.0670098
$595$	−293.889	−0.0202492
$596$	−4956.71	−0.340662
$597$	3522.80	0.241505
$598$	−813.534	−0.0556319
$599$	−4894.82	−0.333885	−0.166942	−	0.985967i	$-0.553389\pi$
$599$	−4894.82	−0.333885	−0.166942	+	0.985967i	$0.553389\pi$
$600$	−2109.74	−0.143550
$601$	4142.57	0.281163	0.140581	−	0.990069i	$-0.455103\pi$
$601$	4142.57	0.281163	0.140581	+	0.990069i	$0.455103\pi$
$602$	5780.73	0.391370
$603$	9445.68	0.637907
$604$	−4183.45	−0.281825
$605$	−6140.81	−0.412660
$606$	−5114.14	−0.342818
$607$	27409.6	1.83282	0.916410	−	0.400241i	$-0.131074\pi$
$607$	27409.6	1.83282	0.916410	+	0.400241i	$0.131074\pi$
$608$	−836.667	−0.0558081
$609$	634.078	0.0421907
$610$	1044.99	0.0693615
$611$	−8195.15	−0.542619
$612$	248.162	0.0163911
$613$	−29413.9	−1.93803	−0.969017	−	0.246994i	$-0.920557\pi$
$613$	−29413.9	−1.93803	−0.969017	+	0.246994i	$0.920557\pi$
$614$	−17245.8	−1.13353
$615$	2172.06	0.142416
$616$	−1006.03	−0.0658023
$617$	3164.17	0.206458	0.103229	−	0.994658i	$-0.467082\pi$
$617$	3164.17	0.206458	0.103229	+	0.994658i	$0.467082\pi$
$618$	−3478.97	−0.226448
$619$	6616.16	0.429606	0.214803	−	0.976657i	$-0.431089\pi$
$619$	6616.16	0.429606	0.214803	+	0.976657i	$0.431089\pi$
$620$	1487.55	0.0963571
$621$	−621.000	−0.0401286
$622$	−9418.68	−0.607162
$623$	−3257.11	−0.209460
$624$	−848.905	−0.0544606
$625$	3090.73	0.197806
$626$	−3463.06	−0.221105
$627$	1409.12	0.0897525
$628$	−2755.20	−0.175071
$629$	−2065.33	−0.130922
$630$	767.401	0.0485301
$631$	3450.57	0.217694	0.108847	−	0.994059i	$-0.465284\pi$
$631$	3450.57	0.217694	0.108847	+	0.994059i	$0.465284\pi$
$632$	−3793.95	−0.238790
$633$	14361.7	0.901779
$634$	−11281.0	−0.706663
$635$	8597.95	0.537322
$636$	−5086.52	−0.317129
$637$	866.591	0.0539020
$638$	1084.87	0.0673204
$639$	−9002.53	−0.557331
$640$	−779.582	−0.0481495
$641$	1853.06	0.114183	0.0570917	−	0.998369i	$-0.481817\pi$
$641$	1853.06	0.114183	0.0570917	+	0.998369i	$0.481817\pi$
$642$	1962.61	0.120651
$643$	28241.3	1.73208	0.866042	−	0.499972i	$-0.166656\pi$
$643$	28241.3	1.73208	0.866042	+	0.499972i	$0.166656\pi$
$644$	−644.000	−0.0394055
$645$	−7544.45	−0.460562
$646$	−360.467	−0.0219542
$647$	931.775	0.0566180	0.0283090	−	0.999599i	$-0.490988\pi$
$647$	931.775	0.0566180	0.0283090	+	0.999599i	$0.490988\pi$
$648$	−648.000	−0.0392837
$649$	−3932.43	−0.237845
$650$	3109.33	0.187628
$651$	1282.27	0.0771982
$652$	142.065	0.00853330
$653$	29760.7	1.78350	0.891752	−	0.452525i	$-0.149477\pi$
$653$	29760.7	1.78350	0.891752	+	0.452525i	$0.149477\pi$
$654$	−5441.36	−0.325343
$655$	−14030.5	−0.836975
$656$	−1902.04	−0.113204
$657$	−8258.29	−0.490390
$658$	−6487.34	−0.384351
$659$	−2342.40	−0.138463	−0.0692313	−	0.997601i	$-0.522055\pi$
$659$	−2342.40	−0.138463	−0.0692313	+	0.997601i	$0.522055\pi$
$660$	1312.98	0.0774357
$661$	−23474.1	−1.38130	−0.690648	−	0.723191i	$-0.742674\pi$
$661$	−23474.1	−1.38130	−0.690648	+	0.723191i	$0.742674\pi$
$662$	−20670.4	−1.21356
$663$	−365.740	−0.0214241
$664$	−6926.66	−0.404829
$665$	−1114.69	−0.0650010
$666$	5392.99	0.313775
$667$	694.467	0.0403146
$668$	−3744.08	−0.216861
$669$	−14436.6	−0.834308
$670$	−12784.2	−0.737157
$671$	1541.19	0.0886690
$672$	−672.000	−0.0385758
$673$	7369.05	0.422075	0.211037	−	0.977478i	$-0.432316\pi$
$673$	7369.05	0.422075	0.211037	+	0.977478i	$0.432316\pi$
$674$	3645.13	0.208316
$675$	2373.46	0.135340
$676$	−7536.89	−0.428817
$677$	−19746.1	−1.12098	−0.560492	−	0.828160i	$-0.689388\pi$
$677$	−19746.1	−1.12098	−0.560492	+	0.828160i	$0.689388\pi$
$678$	−2923.16	−0.165580
$679$	−346.888	−0.0196058
$680$	−335.873	−0.0189414
$681$	19245.8	1.08297
$682$	2193.88	0.123179
$683$	−14886.1	−0.833969	−0.416984	−	0.908914i	$-0.636913\pi$
$683$	−14886.1	−0.833969	−0.416984	+	0.908914i	$0.636913\pi$
$684$	941.250	0.0526164
$685$	−16078.5	−0.896831
$686$	686.000	0.0381802
$687$	−6786.88	−0.376908
$688$	6606.55	0.366093
$689$	7496.49	0.414504
$690$	840.486	0.0463721
$691$	−29417.2	−1.61951	−0.809756	−	0.586767i	$-0.800400\pi$
$691$	−29417.2	−1.61951	−0.809756	+	0.586767i	$0.800400\pi$
$692$	−14918.1	−0.819511
$693$	1131.79	0.0620390
$694$	1218.22	0.0666325
$695$	3706.31	0.202285
$696$	724.661	0.0394658
$697$	−819.468	−0.0445331
$698$	−17492.7	−0.948579
$699$	−994.180	−0.0537959
$700$	2461.37	0.132901
$701$	16916.6	0.911459	0.455729	−	0.890118i	$-0.349378\pi$
$701$	16916.6	0.911459	0.455729	+	0.890118i	$0.349378\pi$
$702$	955.019	0.0513459
$703$	−7833.56	−0.420268
$704$	−1149.75	−0.0615524
$705$	8466.65	0.452301
$706$	10606.2	0.565396
$707$	5966.49	0.317388
$708$	−2626.75	−0.139434
$709$	2348.39	0.124394	0.0621972	−	0.998064i	$-0.480189\pi$
$709$	2348.39	0.124394	0.0621972	+	0.998064i	$0.480189\pi$
$710$	12184.4	0.644045
$711$	4268.19	0.225133
$712$	−3722.42	−0.195932
$713$	1404.39	0.0737655
$714$	−289.523	−0.0151752
$715$	−1935.06	−0.101213
$716$	−5399.39	−0.281822
$717$	296.111	0.0154233
$718$	23926.6	1.24364
$719$	−22922.1	−1.18894	−0.594472	−	0.804116i	$-0.702639\pi$
$719$	−22922.1	−1.18894	−0.594472	+	0.804116i	$0.702639\pi$
$720$	877.029	0.0453958
$721$	4058.80	0.209650
$722$	12350.8	0.636633
$723$	−20319.7	−1.04523
$724$	1930.10	0.0990770
$725$	−2654.25	−0.135968
$726$	−6049.58	−0.309258
$727$	4142.50	0.211330	0.105665	−	0.994402i	$-0.466303\pi$
$727$	4142.50	0.211330	0.105665	+	0.994402i	$0.466303\pi$
$728$	990.390	0.0504207
$729$	729.000	0.0370370
$730$	11177.1	0.566689
$731$	2846.35	0.144016
$732$	1029.47	0.0519812
$733$	−7694.58	−0.387730	−0.193865	−	0.981028i	$-0.562102\pi$
$733$	−7694.58	−0.387730	−0.193865	+	0.981028i	$0.562102\pi$
$734$	20320.2	1.02184
$735$	−895.301	−0.0449302
$736$	−736.000	−0.0368605
$737$	−18854.5	−0.942353
$738$	2139.79	0.106730
$739$	10499.4	0.522635	0.261317	−	0.965253i	$-0.415843\pi$
$739$	10499.4	0.522635	0.261317	+	0.965253i	$0.415843\pi$
$740$	−7299.08	−0.362594
$741$	−1387.21	−0.0687725
$742$	5934.28	0.293604
$743$	21082.3	1.04096	0.520481	−	0.853873i	$-0.325753\pi$
$743$	21082.3	1.04096	0.520481	+	0.853873i	$0.325753\pi$
$744$	1465.45	0.0722123
$745$	−7547.19	−0.371151
$746$	−5383.41	−0.264210
$747$	7792.49	0.381676
$748$	−495.356	−0.0242139
$749$	−2289.71	−0.111701
$750$	−7780.20	−0.378790
$751$	−14861.0	−0.722083	−0.361041	−	0.932550i	$-0.617579\pi$
$751$	−14861.0	−0.722083	−0.361041	+	0.932550i	$0.617579\pi$
$752$	−7414.10	−0.359527
$753$	18665.7	0.903340
$754$	−1068.00	−0.0515840
$755$	−6369.81	−0.307048
$756$	756.000	0.0363696
$757$	−6521.83	−0.313131	−0.156565	−	0.987668i	$-0.550042\pi$
$757$	−6521.83	−0.313131	−0.156565	+	0.987668i	$0.550042\pi$
$758$	−699.070	−0.0334979
$759$	1239.58	0.0592803
$760$	−1273.93	−0.0608028
$761$	27425.8	1.30642	0.653210	−	0.757177i	$-0.273422\pi$
$761$	27425.8	1.30642	0.653210	+	0.757177i	$0.273422\pi$
$762$	8470.22	0.402682
$763$	6348.26	0.301209
$764$	−8424.05	−0.398916
$765$	377.857	0.0178581
$766$	18673.8	0.880827
$767$	3871.28	0.182248
$768$	−768.000	−0.0360844
$769$	−6930.68	−0.325002	−0.162501	−	0.986708i	$-0.551956\pi$
$769$	−6930.68	−0.325002	−0.162501	+	0.986708i	$0.551956\pi$
$770$	−1531.81	−0.0716915
$771$	14602.5	0.682095
$772$	4563.73	0.212762
$773$	150.252	0.00699118	0.00349559	−	0.999994i	$-0.498887\pi$
$773$	150.252	0.00699118	0.00349559	+	0.999994i	$0.498887\pi$
$774$	−7432.36	−0.345156
$775$	−5367.58	−0.248786
$776$	−396.443	−0.0183395
$777$	−6291.82	−0.290499
$778$	−12656.5	−0.583237
$779$	−3108.15	−0.142954
$780$	−1292.56	−0.0593347
$781$	17969.9	0.823322
$782$	−317.096	−0.0145004
$783$	−815.243	−0.0372087
$784$	784.000	0.0357143
$785$	−4195.13	−0.190740
$786$	−13822.1	−0.627250
$787$	−18876.0	−0.854962	−0.427481	−	0.904024i	$-0.640599\pi$
$787$	−18876.0	−0.854962	−0.427481	+	0.904024i	$0.640599\pi$
$788$	−17485.8	−0.790488
$789$	−9361.43	−0.422403
$790$	−5776.74	−0.260161
$791$	3410.35	0.153297
$792$	1293.47	0.0580322
$793$	−1517.22	−0.0679422
$794$	−12885.1	−0.575911
$795$	−7744.85	−0.345511
$796$	−4697.06	−0.209149
$797$	7532.56	0.334777	0.167388	−	0.985891i	$-0.446467\pi$
$797$	7532.56	0.334777	0.167388	+	0.985891i	$0.446467\pi$
$798$	−1098.13	−0.0487133
$799$	−3194.27	−0.141433
$800$	2812.99	0.124318
$801$	4187.72	0.184726
$802$	9692.44	0.426748
$803$	16484.3	0.724433
$804$	−12594.2	−0.552443
$805$	−980.568	−0.0429323
$806$	−2159.77	−0.0943854
$807$	19233.0	0.838951
$808$	6818.85	0.296889
$809$	−42648.2	−1.85344	−0.926718	−	0.375757i	$-0.877383\pi$
$809$	−42648.2	−1.85344	−0.926718	+	0.375757i	$0.877383\pi$
$810$	−986.658	−0.0427995
$811$	−14705.2	−0.636705	−0.318352	−	0.947972i	$-0.603130\pi$
$811$	−14705.2	−0.636705	−0.318352	+	0.947972i	$0.603130\pi$
$812$	−845.438	−0.0365382
$813$	2133.01	0.0920145
$814$	−10764.9	−0.463526
$815$	216.312	0.00929702
$816$	−330.883	−0.0141951
$817$	10795.9	0.462300
$818$	−12603.4	−0.538711
$819$	−1114.19	−0.0475371
$820$	−2896.08	−0.123336
$821$	−1041.15	−0.0442587	−0.0221293	−	0.999755i	$-0.507045\pi$
$821$	−1041.15	−0.0442587	−0.0221293	+	0.999755i	$0.507045\pi$
$822$	−15839.7	−0.672107
$823$	4625.66	0.195918	0.0979589	−	0.995190i	$-0.468769\pi$
$823$	4625.66	0.195918	0.0979589	+	0.995190i	$0.468769\pi$
$824$	4638.62	0.196109
$825$	−4737.66	−0.199932
$826$	3064.54	0.129091
$827$	−17983.1	−0.756145	−0.378073	−	0.925776i	$-0.623413\pi$
$827$	−17983.1	−0.756145	−0.378073	+	0.925776i	$0.623413\pi$
$828$	828.000	0.0347524
$829$	−9058.90	−0.379528	−0.189764	−	0.981830i	$-0.560772\pi$
$829$	−9058.90	−0.379528	−0.189764	+	0.981830i	$0.560772\pi$
$830$	−10546.7	−0.441061
$831$	18127.4	0.756717
$832$	1131.87	0.0471643
$833$	337.776	0.0140495
$834$	3651.24	0.151597
$835$	−5700.82	−0.236269
$836$	−1878.83	−0.0777280
$837$	−1648.63	−0.0680824
$838$	−3696.52	−0.152380
$839$	−22389.0	−0.921280	−0.460640	−	0.887587i	$-0.652380\pi$
$839$	−22389.0	−0.921280	−0.460640	+	0.887587i	$0.652380\pi$
$840$	−1023.20	−0.0420283
$841$	−23477.3	−0.962619
$842$	16177.4	0.662128
$843$	−3988.45	−0.162953
$844$	−19148.9	−0.780963
$845$	−11475.8	−0.467195
$846$	8340.87	0.338966
$847$	7057.84	0.286317
$848$	6782.03	0.274641
$849$	4460.26	0.180301
$850$	1211.94	0.0489050
$851$	−6891.04	−0.277581
$852$	12003.4	0.482663
$853$	48544.6	1.94858	0.974289	−	0.225302i	$-0.0723368\pi$
$853$	48544.6	1.94858	0.974289	+	0.225302i	$0.0723368\pi$
$854$	−1201.05	−0.0481252
$855$	1433.17	0.0573255
$856$	−2616.81	−0.104487
$857$	5133.70	0.204625	0.102313	−	0.994752i	$-0.467376\pi$
$857$	5133.70	0.204625	0.102313	+	0.994752i	$0.467376\pi$
$858$	−1906.31	−0.0758512
$859$	−2566.23	−0.101931	−0.0509654	−	0.998700i	$-0.516230\pi$
$859$	−2566.23	−0.101931	−0.0509654	+	0.998700i	$0.516230\pi$
$860$	10059.3	0.398858
$861$	−2496.42	−0.0988128
$862$	19753.4	0.780516
$863$	29389.7	1.15926	0.579628	−	0.814881i	$-0.303198\pi$
$863$	29389.7	1.15926	0.579628	+	0.814881i	$0.303198\pi$
$864$	864.000	0.0340207
$865$	−22714.6	−0.892856
$866$	7649.46	0.300161
$867$	14596.4	0.571766
$868$	−1709.69	−0.0668556
$869$	−8519.73	−0.332580
$870$	1103.38	0.0429979
$871$	18561.3	0.722074
$872$	7255.15	0.281755
$873$	445.999	0.0172907
$874$	−1202.71	−0.0465472
$875$	9076.90	0.350692
$876$	11011.0	0.424690
$877$	−40295.4	−1.55152	−0.775759	−	0.631030i	$-0.782633\pi$
$877$	−40295.4	−1.55152	−0.775759	+	0.631030i	$0.782633\pi$
$878$	25725.6	0.988836
$879$	−13033.5	−0.500123
$880$	−1750.64	−0.0670613
$881$	−25843.7	−0.988306	−0.494153	−	0.869375i	$-0.664522\pi$
$881$	−25843.7	−0.988306	−0.494153	+	0.869375i	$0.664522\pi$
$882$	−882.000	−0.0336718
$883$	−36656.2	−1.39703	−0.698517	−	0.715594i	$-0.746156\pi$
$883$	−36656.2	−1.39703	−0.698517	+	0.715594i	$0.746156\pi$
$884$	487.653	0.0185538
$885$	−3999.54	−0.151913
$886$	894.679	0.0339248
$887$	−38953.1	−1.47454	−0.737270	−	0.675598i	$-0.763885\pi$
$887$	−38953.1	−1.47454	−0.737270	+	0.675598i	$0.763885\pi$
$888$	−7190.65	−0.271737
$889$	−9881.92	−0.372811
$890$	−5667.83	−0.213467
$891$	−1455.15	−0.0547133
$892$	19248.8	0.722532
$893$	−12115.5	−0.454009
$894$	−7435.06	−0.278150
$895$	−8221.21	−0.307045
$896$	896.000	0.0334077
$897$	−1220.30	−0.0454233
$898$	−10163.2	−0.377672
$899$	1843.67	0.0683980
$900$	−3164.62	−0.117208
$901$	2921.95	0.108040
$902$	−4271.23	−0.157668
$903$	8671.09	0.319552
$904$	3897.55	0.143397
$905$	2938.81	0.107944
$906$	−6275.18	−0.230109
$907$	−12747.4	−0.466671	−0.233335	−	0.972396i	$-0.574964\pi$
$907$	−12747.4	−0.466671	−0.233335	+	0.972396i	$0.574964\pi$
$908$	−25661.1	−0.937877
$909$	−7671.20	−0.279910
$910$	1507.99	0.0549333
$911$	14200.2	0.516437	0.258218	−	0.966087i	$-0.416865\pi$
$911$	14200.2	0.516437	0.258218	+	0.966087i	$0.416865\pi$
$912$	−1255.00	−0.0455671
$913$	−15554.6	−0.563835
$914$	−4413.45	−0.159720
$915$	1567.49	0.0566334
$916$	9049.17	0.326412
$917$	16125.8	0.580720
$918$	372.243	0.0133833
$919$	−1539.71	−0.0552671	−0.0276335	−	0.999618i	$-0.508797\pi$
$919$	−1539.71	−0.0552671	−0.0276335	+	0.999618i	$0.508797\pi$
$920$	−1120.65	−0.0401594
$921$	−25868.7	−0.925520
$922$	−29732.4	−1.06202
$923$	−17690.5	−0.630867
$924$	−1509.05	−0.0537274
$925$	26337.6	0.936187
$926$	−18854.6	−0.669116
$927$	−5218.45	−0.184894
$928$	−966.215	−0.0341784
$929$	−13275.6	−0.468846	−0.234423	−	0.972135i	$-0.575320\pi$
$929$	−13275.6	−0.468846	−0.234423	+	0.972135i	$0.575320\pi$
$930$	2231.32	0.0786752
$931$	1281.15	0.0450997
$932$	1325.57	0.0465886
$933$	−14128.0	−0.495745
$934$	−20600.7	−0.721707
$935$	−754.239	−0.0263810
$936$	−1273.36	−0.0444669
$937$	11738.6	0.409269	0.204634	−	0.978838i	$-0.434399\pi$
$937$	11738.6	0.409269	0.204634	+	0.978838i	$0.434399\pi$
$938$	14693.3	0.511463
$939$	−5194.60	−0.180532
$940$	−11288.9	−0.391704
$941$	−35829.7	−1.24125	−0.620624	−	0.784108i	$-0.713121\pi$
$941$	−35829.7	−1.24125	−0.620624	+	0.784108i	$0.713121\pi$
$942$	−4132.81	−0.142945
$943$	−2734.18	−0.0944189
$944$	3502.33	0.120753
$945$	1151.10	0.0396247
$946$	14835.7	0.509885
$947$	41724.1	1.43173	0.715867	−	0.698236i	$-0.246032\pi$
$947$	41724.1	1.43173	0.715867	+	0.698236i	$0.246032\pi$
$948$	−5690.92	−0.194971
$949$	−16228.0	−0.555094
$950$	4596.75	0.156988
$951$	−16921.4	−0.576988
$952$	386.030	0.0131421
$953$	30972.6	1.05278	0.526391	−	0.850243i	$-0.323545\pi$
$953$	30972.6	1.05278	0.526391	+	0.850243i	$0.323545\pi$
$954$	−7629.79	−0.258934
$955$	−12826.6	−0.434618
$956$	−394.815	−0.0133569
$957$	1627.31	0.0549669
$958$	−36992.6	−1.24758
$959$	18479.6	0.622250
$960$	−1169.37	−0.0393139
$961$	−26062.6	−0.874849
$962$	10597.5	0.355175
$963$	2943.91	0.0985111
$964$	27093.0	0.905192
$965$	6948.83	0.231804
$966$	−966.000	−0.0321745
$967$	42865.6	1.42551	0.712754	−	0.701415i	$-0.247448\pi$
$967$	42865.6	1.42551	0.712754	+	0.701415i	$0.247448\pi$
$968$	8066.11	0.267825
$969$	−540.701	−0.0179255
$970$	−603.633	−0.0199809
$971$	48742.8	1.61095	0.805474	−	0.592632i	$-0.201911\pi$
$971$	48742.8	1.61095	0.805474	+	0.592632i	$0.201911\pi$
$972$	−972.000	−0.0320750
$973$	−4259.78	−0.140352
$974$	14648.5	0.481898
$975$	4663.99	0.153197
$976$	−1372.62	−0.0450170
$977$	−24010.7	−0.786253	−0.393127	−	0.919484i	$-0.628607\pi$
$977$	−24010.7	−0.786253	−0.393127	+	0.919484i	$0.628607\pi$
$978$	213.098	0.00696741
$979$	−8359.09	−0.272888
$980$	1193.73	0.0389107
$981$	−8162.05	−0.265641
$982$	−17360.2	−0.564141
$983$	53743.0	1.74378	0.871890	−	0.489702i	$-0.162894\pi$
$983$	53743.0	1.74378	0.871890	+	0.489702i	$0.162894\pi$
$984$	−2853.05	−0.0924310
$985$	−26624.2	−0.861236
$986$	−416.281	−0.0134453
$987$	−9731.01	−0.313821
$988$	1849.61	0.0595587
$989$	9496.91	0.305343
$990$	1969.47	0.0632260
$991$	−12616.6	−0.404420	−0.202210	−	0.979342i	$-0.564812\pi$
$991$	−12616.6	−0.404420	−0.202210	+	0.979342i	$0.564812\pi$
$992$	−1953.93	−0.0625377
$993$	−31005.7	−0.990871
$994$	−14003.9	−0.446859
$995$	−7151.84	−0.227868
$996$	−10390.0	−0.330541
$997$	41624.0	1.32221	0.661105	−	0.750293i	$-0.270088\pi$
$997$	41624.0	1.32221	0.661105	+	0.750293i	$0.270088\pi$
$998$	15359.1	0.487160
$999$	8089.48	0.256196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.4.a.h.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.4.a.h.1.3	✓	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.09048 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.09048 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -12.1810 q^{10} -17.9649 q^{11} -12.0000 q^{12} +17.6855 q^{13} +14.0000 q^{14} -18.2714 q^{15} +16.0000 q^{16} +6.89339 q^{17} -18.0000 q^{18} +26.1458 q^{19} +24.3619 q^{20} +21.0000 q^{21} +35.9298 q^{22} +23.0000 q^{23} +24.0000 q^{24} -87.9060 q^{25} -35.3711 q^{26} -27.0000 q^{27} -28.0000 q^{28} +30.1942 q^{29} +36.5429 q^{30} +61.0604 q^{31} -32.0000 q^{32} +53.8946 q^{33} -13.7868 q^{34} -42.6334 q^{35} +36.0000 q^{36} -299.610 q^{37} -52.2917 q^{38} -53.0566 q^{39} -48.7239 q^{40} -118.877 q^{41} -42.0000 q^{42} +412.909 q^{43} -71.8595 q^{44} +54.8143 q^{45} -46.0000 q^{46} -463.381 q^{47} -48.0000 q^{48} +49.0000 q^{49} +175.812 q^{50} -20.6802 q^{51} +70.7421 q^{52} +423.877 q^{53} +54.0000 q^{54} -109.415 q^{55} +56.0000 q^{56} -78.4375 q^{57} -60.3884 q^{58} +218.896 q^{59} -73.0858 q^{60} -85.7890 q^{61} -122.121 q^{62} -63.0000 q^{63} +64.0000 q^{64} +107.713 q^{65} -107.789 q^{66} +1049.52 q^{67} +27.5736 q^{68} -69.0000 q^{69} +85.2667 q^{70} -1000.28 q^{71} -72.0000 q^{72} -917.587 q^{73} +599.221 q^{74} +263.718 q^{75} +104.583 q^{76} +125.754 q^{77} +106.113 q^{78} +474.244 q^{79} +97.4477 q^{80} +81.0000 q^{81} +237.755 q^{82} +865.832 q^{83} +84.0000 q^{84} +41.9841 q^{85} -825.818 q^{86} -90.5826 q^{87} +143.719 q^{88} +465.302 q^{89} -109.629 q^{90} -123.799 q^{91} +92.0000 q^{92} -183.181 q^{93} +926.763 q^{94} +159.241 q^{95} +96.0000 q^{96} +49.5554 q^{97} -98.0000 q^{98} -161.684 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 5 q^{5} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 5 * q^5 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 5 q^{5} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} + 10 q^{10} - 41 q^{11} - 48 q^{12} + 23 q^{13} + 56 q^{14} + 15 q^{15} + 64 q^{16} + 18 q^{17} - 72 q^{18} + 15 q^{19} - 20 q^{20} + 84 q^{21} + 82 q^{22} + 92 q^{23} + 96 q^{24} + 219 q^{25} - 46 q^{26} - 108 q^{27} - 112 q^{28} - 46 q^{29} - 30 q^{30} + 142 q^{31} - 128 q^{32} + 123 q^{33} - 36 q^{34} + 35 q^{35} + 144 q^{36} - 142 q^{37} - 30 q^{38} - 69 q^{39} + 40 q^{40} - 621 q^{41} - 168 q^{42} + 185 q^{43} - 164 q^{44} - 45 q^{45} - 184 q^{46} + 669 q^{47} - 192 q^{48} + 196 q^{49} - 438 q^{50} - 54 q^{51} + 92 q^{52} + 422 q^{53} + 216 q^{54} + 1435 q^{55} + 224 q^{56} - 45 q^{57} + 92 q^{58} + 270 q^{59} + 60 q^{60} + 272 q^{61} - 284 q^{62} - 252 q^{63} + 256 q^{64} - 1992 q^{65} - 246 q^{66} - 67 q^{67} + 72 q^{68} - 276 q^{69} - 70 q^{70} + 611 q^{71} - 288 q^{72} - 236 q^{73} + 284 q^{74} - 657 q^{75} + 60 q^{76} + 287 q^{77} + 138 q^{78} + 558 q^{79} - 80 q^{80} + 324 q^{81} + 1242 q^{82} + 468 q^{83} + 336 q^{84} + 1045 q^{85} - 370 q^{86} + 138 q^{87} + 328 q^{88} - 1519 q^{89} + 90 q^{90} - 161 q^{91} + 368 q^{92} - 426 q^{93} - 1338 q^{94} + 23 q^{95} + 384 q^{96} + 600 q^{97} - 392 q^{98} - 369 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 5 * q^5 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9 + 10 * q^10 - 41 * q^11 - 48 * q^12 + 23 * q^13 + 56 * q^14 + 15 * q^15 + 64 * q^16 + 18 * q^17 - 72 * q^18 + 15 * q^19 - 20 * q^20 + 84 * q^21 + 82 * q^22 + 92 * q^23 + 96 * q^24 + 219 * q^25 - 46 * q^26 - 108 * q^27 - 112 * q^28 - 46 * q^29 - 30 * q^30 + 142 * q^31 - 128 * q^32 + 123 * q^33 - 36 * q^34 + 35 * q^35 + 144 * q^36 - 142 * q^37 - 30 * q^38 - 69 * q^39 + 40 * q^40 - 621 * q^41 - 168 * q^42 + 185 * q^43 - 164 * q^44 - 45 * q^45 - 184 * q^46 + 669 * q^47 - 192 * q^48 + 196 * q^49 - 438 * q^50 - 54 * q^51 + 92 * q^52 + 422 * q^53 + 216 * q^54 + 1435 * q^55 + 224 * q^56 - 45 * q^57 + 92 * q^58 + 270 * q^59 + 60 * q^60 + 272 * q^61 - 284 * q^62 - 252 * q^63 + 256 * q^64 - 1992 * q^65 - 246 * q^66 - 67 * q^67 + 72 * q^68 - 276 * q^69 - 70 * q^70 + 611 * q^71 - 288 * q^72 - 236 * q^73 + 284 * q^74 - 657 * q^75 + 60 * q^76 + 287 * q^77 + 138 * q^78 + 558 * q^79 - 80 * q^80 + 324 * q^81 + 1242 * q^82 + 468 * q^83 + 336 * q^84 + 1045 * q^85 - 370 * q^86 + 138 * q^87 + 328 * q^88 - 1519 * q^89 + 90 * q^90 - 161 * q^91 + 368 * q^92 - 426 * q^93 - 1338 * q^94 + 23 * q^95 + 384 * q^96 + 600 * q^97 - 392 * q^98 - 369 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more