LMFDB - Embedded newform 2166.4.a.bi.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, 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("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.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:	$127.798137072$
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} - 2x^{7} - 350x^{6} + 948x^{5} + 37019x^{4} - 115308x^{3} - 1098530x^{2} + 2724222x + 7883581 $ x^8 - 2x^7 - 350x^6 + 948x^5 + 37019x^4 - 115308x^3 - 1098530x^2 + 2724222*x + 7883581
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 19 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-13.2235$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.46644 q^{5} -6.00000 q^{6} -26.1558 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.46644 q^{5} -6.00000 q^{6} -26.1558 q^{7} +8.00000 q^{8} +9.00000 q^{9} -16.9329 q^{10} +38.9688 q^{11} -12.0000 q^{12} -5.45373 q^{13} -52.3116 q^{14} +25.3993 q^{15} +16.0000 q^{16} +6.06565 q^{17} +18.0000 q^{18} -33.8657 q^{20} +78.4674 q^{21} +77.9376 q^{22} +87.9598 q^{23} -24.0000 q^{24} -53.3194 q^{25} -10.9075 q^{26} -27.0000 q^{27} -104.623 q^{28} -73.4834 q^{29} +50.7986 q^{30} -10.2778 q^{31} +32.0000 q^{32} -116.906 q^{33} +12.1313 q^{34} +221.446 q^{35} +36.0000 q^{36} +446.303 q^{37} +16.3612 q^{39} -67.7315 q^{40} +147.386 q^{41} +156.935 q^{42} -88.9771 q^{43} +155.875 q^{44} -76.1979 q^{45} +175.920 q^{46} -175.071 q^{47} -48.0000 q^{48} +341.126 q^{49} -106.639 q^{50} -18.1970 q^{51} -21.8149 q^{52} -462.582 q^{53} -54.0000 q^{54} -329.927 q^{55} -209.246 q^{56} -146.967 q^{58} -94.3063 q^{59} +101.597 q^{60} +784.585 q^{61} -20.5556 q^{62} -235.402 q^{63} +64.0000 q^{64} +46.1737 q^{65} -233.813 q^{66} -109.607 q^{67} +24.2626 q^{68} -263.880 q^{69} +442.893 q^{70} +413.774 q^{71} +72.0000 q^{72} +1123.39 q^{73} +892.606 q^{74} +159.958 q^{75} -1019.26 q^{77} +32.7224 q^{78} -1176.43 q^{79} -135.463 q^{80} +81.0000 q^{81} +294.773 q^{82} +896.711 q^{83} +313.870 q^{84} -51.3545 q^{85} -177.954 q^{86} +220.450 q^{87} +311.750 q^{88} +85.1322 q^{89} -152.396 q^{90} +142.647 q^{91} +351.839 q^{92} +30.8334 q^{93} -350.141 q^{94} -96.0000 q^{96} -1355.02 q^{97} +682.252 q^{98} +350.719 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{2} - 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} + 64 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q + 16 * q^2 - 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 + 64 * q^8 + 72 * q^9	\( 8 q + 16 q^{2} - 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} + 64 q^{8} + 72 q^{9} - 60 q^{10} - 6 q^{11} - 96 q^{12} + 48 q^{13} - 68 q^{14} + 90 q^{15} + 128 q^{16} - 36 q^{17} + 144 q^{18} - 120 q^{20} + 102 q^{21} - 12 q^{22} - 282 q^{23} - 192 q^{24} + 396 q^{25} + 96 q^{26} - 216 q^{27} - 136 q^{28} - 380 q^{29} + 180 q^{30} - 48 q^{31} + 256 q^{32} + 18 q^{33} - 72 q^{34} + 762 q^{35} + 288 q^{36} - 168 q^{37} - 144 q^{39} - 240 q^{40} + 342 q^{41} + 204 q^{42} - 788 q^{43} - 24 q^{44} - 270 q^{45} - 564 q^{46} - 468 q^{47} - 384 q^{48} + 222 q^{49} + 792 q^{50} + 108 q^{51} + 192 q^{52} + 1682 q^{53} - 432 q^{54} - 46 q^{55} - 272 q^{56} - 760 q^{58} + 292 q^{59} + 360 q^{60} + 522 q^{61} - 96 q^{62} - 306 q^{63} + 512 q^{64} - 1120 q^{65} + 36 q^{66} - 2484 q^{67} - 144 q^{68} + 846 q^{69} + 1524 q^{70} + 1182 q^{71} + 576 q^{72} + 182 q^{73} - 336 q^{74} - 1188 q^{75} - 504 q^{77} - 288 q^{78} - 2232 q^{79} - 480 q^{80} + 648 q^{81} + 684 q^{82} - 750 q^{83} + 408 q^{84} - 2238 q^{85} - 1576 q^{86} + 1140 q^{87} - 48 q^{88} + 1304 q^{89} - 540 q^{90} - 624 q^{91} - 1128 q^{92} + 144 q^{93} - 936 q^{94} - 768 q^{96} - 1248 q^{97} + 444 q^{98} - 54 q^{99}+O(q^{100}) \) 8 * q + 16 * q^2 - 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 + 64 * q^8 + 72 * q^9 - 60 * q^10 - 6 * q^11 - 96 * q^12 + 48 * q^13 - 68 * q^14 + 90 * q^15 + 128 * q^16 - 36 * q^17 + 144 * q^18 - 120 * q^20 + 102 * q^21 - 12 * q^22 - 282 * q^23 - 192 * q^24 + 396 * q^25 + 96 * q^26 - 216 * q^27 - 136 * q^28 - 380 * q^29 + 180 * q^30 - 48 * q^31 + 256 * q^32 + 18 * q^33 - 72 * q^34 + 762 * q^35 + 288 * q^36 - 168 * q^37 - 144 * q^39 - 240 * q^40 + 342 * q^41 + 204 * q^42 - 788 * q^43 - 24 * q^44 - 270 * q^45 - 564 * q^46 - 468 * q^47 - 384 * q^48 + 222 * q^49 + 792 * q^50 + 108 * q^51 + 192 * q^52 + 1682 * q^53 - 432 * q^54 - 46 * q^55 - 272 * q^56 - 760 * q^58 + 292 * q^59 + 360 * q^60 + 522 * q^61 - 96 * q^62 - 306 * q^63 + 512 * q^64 - 1120 * q^65 + 36 * q^66 - 2484 * q^67 - 144 * q^68 + 846 * q^69 + 1524 * q^70 + 1182 * q^71 + 576 * q^72 + 182 * q^73 - 336 * q^74 - 1188 * q^75 - 504 * q^77 - 288 * q^78 - 2232 * q^79 - 480 * q^80 + 648 * q^81 + 684 * q^82 - 750 * q^83 + 408 * q^84 - 2238 * q^85 - 1576 * q^86 + 1140 * q^87 - 48 * q^88 + 1304 * q^89 - 540 * q^90 - 624 * q^91 - 1128 * q^92 + 144 * q^93 - 936 * q^94 - 768 * q^96 - 1248 * q^97 + 444 * q^98 - 54 * 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
$5$	−8.46644	−0.757261	−0.378631	−	0.925548i	$-0.623605\pi$
$5$	−8.46644	−0.757261	−0.378631	+	0.925548i	$0.623605\pi$
$6$	−6.00000	−0.408248
$7$	−26.1558	−1.41228	−0.706140	−	0.708072i	$-0.749565\pi$
$7$	−26.1558	−1.41228	−0.706140	+	0.708072i	$0.749565\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−16.9329	−0.535464
$11$	38.9688	1.06814	0.534070	−	0.845441i	$-0.320662\pi$
$11$	38.9688	1.06814	0.534070	+	0.845441i	$0.320662\pi$
$12$	−12.0000	−0.288675
$13$	−5.45373	−0.116353	−0.0581767	−	0.998306i	$-0.518529\pi$
$13$	−5.45373	−0.116353	−0.0581767	+	0.998306i	$0.518529\pi$
$14$	−52.3116	−0.998633
$15$	25.3993	0.437205
$16$	16.0000	0.250000
$17$	6.06565	0.0865375	0.0432687	−	0.999063i	$-0.486223\pi$
$17$	6.06565	0.0865375	0.0432687	+	0.999063i	$0.486223\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	−33.8657	−0.378631
$21$	78.4674	0.815381
$22$	77.9376	0.755288
$23$	87.9598	0.797430	0.398715	−	0.917075i	$-0.369456\pi$
$23$	87.9598	0.797430	0.398715	+	0.917075i	$0.369456\pi$
$24$	−24.0000	−0.204124
$25$	−53.3194	−0.426556
$26$	−10.9075	−0.0822742
$27$	−27.0000	−0.192450
$28$	−104.623	−0.706140
$29$	−73.4834	−0.470535	−0.235268	−	0.971931i	$-0.575597\pi$
$29$	−73.4834	−0.470535	−0.235268	+	0.971931i	$0.575597\pi$
$30$	50.7986	0.309151
$31$	−10.2778	−0.0595467	−0.0297733	−	0.999557i	$-0.509479\pi$
$31$	−10.2778	−0.0595467	−0.0297733	+	0.999557i	$0.509479\pi$
$32$	32.0000	0.176777
$33$	−116.906	−0.616690
$34$	12.1313	0.0611912
$35$	221.446	1.06947
$36$	36.0000	0.166667
$37$	446.303	1.98302	0.991510	−	0.130030i	$-0.0415073\pi$
$37$	446.303	1.98302	0.991510	+	0.130030i	$0.0415073\pi$
$38$	0	0
$39$	16.3612	0.0671766
$40$	−67.7315	−0.267732
$41$	147.386	0.561412	0.280706	−	0.959794i	$-0.409431\pi$
$41$	147.386	0.561412	0.280706	+	0.959794i	$0.409431\pi$
$42$	156.935	0.576561
$43$	−88.9771	−0.315555	−0.157778	−	0.987475i	$-0.550433\pi$
$43$	−88.9771	−0.315555	−0.157778	+	0.987475i	$0.550433\pi$
$44$	155.875	0.534070
$45$	−76.1979	−0.252420
$46$	175.920	0.563868
$47$	−175.071	−0.543334	−0.271667	−	0.962391i	$-0.587575\pi$
$47$	−175.071	−0.543334	−0.271667	+	0.962391i	$0.587575\pi$
$48$	−48.0000	−0.144338
$49$	341.126	0.994537
$50$	−106.639	−0.301620
$51$	−18.1970	−0.0499624
$52$	−21.8149	−0.0581767
$53$	−462.582	−1.19888	−0.599439	−	0.800421i	$-0.704609\pi$
$53$	−462.582	−1.19888	−0.599439	+	0.800421i	$0.704609\pi$
$54$	−54.0000	−0.136083
$55$	−329.927	−0.808860
$56$	−209.246	−0.499317
$57$	0	0
$58$	−146.967	−0.332719
$59$	−94.3063	−0.208096	−0.104048	−	0.994572i	$-0.533179\pi$
$59$	−94.3063	−0.208096	−0.104048	+	0.994572i	$0.533179\pi$
$60$	101.597	0.218602
$61$	784.585	1.64682	0.823409	−	0.567449i	$-0.192070\pi$
$61$	784.585	1.64682	0.823409	+	0.567449i	$0.192070\pi$
$62$	−20.5556	−0.0421059
$63$	−235.402	−0.470760
$64$	64.0000	0.125000
$65$	46.1737	0.0881099
$66$	−233.813	−0.436066
$67$	−109.607	−0.199861	−0.0999303	−	0.994994i	$-0.531862\pi$
$67$	−109.607	−0.199861	−0.0999303	+	0.994994i	$0.531862\pi$
$68$	24.2626	0.0432687
$69$	−263.880	−0.460397
$70$	442.893	0.756226
$71$	413.774	0.691632	0.345816	−	0.938302i	$-0.387602\pi$
$71$	413.774	0.691632	0.345816	+	0.938302i	$0.387602\pi$
$72$	72.0000	0.117851
$73$	1123.39	1.80113	0.900564	−	0.434724i	$-0.143154\pi$
$73$	1123.39	1.80113	0.900564	+	0.434724i	$0.143154\pi$
$74$	892.606	1.40221
$75$	159.958	0.246272
$76$	0	0
$77$	−1019.26	−1.50851
$78$	32.7224	0.0475011
$79$	−1176.43	−1.67542	−0.837712	−	0.546113i	$-0.816107\pi$
$79$	−1176.43	−1.67542	−0.837712	+	0.546113i	$0.816107\pi$
$80$	−135.463	−0.189315
$81$	81.0000	0.111111
$82$	294.773	0.396978
$83$	896.711	1.18587	0.592933	−	0.805252i	$-0.297970\pi$
$83$	896.711	1.18587	0.592933	+	0.805252i	$0.297970\pi$
$84$	313.870	0.407690
$85$	−51.3545	−0.0655315
$86$	−177.954	−0.223131
$87$	220.450	0.271664
$88$	311.750	0.377644
$89$	85.1322	0.101393	0.0506966	−	0.998714i	$-0.483856\pi$
$89$	85.1322	0.101393	0.0506966	+	0.998714i	$0.483856\pi$
$90$	−152.396	−0.178488
$91$	142.647	0.164324
$92$	351.839	0.398715
$93$	30.8334	0.0343793
$94$	−350.141	−0.384195
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−1355.02	−1.41836	−0.709182	−	0.705025i	$-0.750936\pi$
$97$	−1355.02	−1.41836	−0.709182	+	0.705025i	$0.750936\pi$
$98$	682.252	0.703244
$99$	350.719	0.356046
$100$	−213.278	−0.213278
$101$	−1934.93	−1.90627	−0.953133	−	0.302553i	$-0.902161\pi$
$101$	−1934.93	−1.90627	−0.953133	+	0.302553i	$0.902161\pi$
$102$	−36.3939	−0.0353288
$103$	376.924	0.360577	0.180288	−	0.983614i	$-0.442297\pi$
$103$	376.924	0.360577	0.180288	+	0.983614i	$0.442297\pi$
$104$	−43.6299	−0.0411371
$105$	−664.339	−0.617456
$106$	−925.163	−0.847734
$107$	−218.310	−0.197241	−0.0986205	−	0.995125i	$-0.531443\pi$
$107$	−218.310	−0.197241	−0.0986205	+	0.995125i	$0.531443\pi$
$108$	−108.000	−0.0962250
$109$	459.490	0.403772	0.201886	−	0.979409i	$-0.435293\pi$
$109$	459.490	0.403772	0.201886	+	0.979409i	$0.435293\pi$
$110$	−659.853	−0.571951
$111$	−1338.91	−1.14490
$112$	−418.493	−0.353070
$113$	−2071.06	−1.72415	−0.862073	−	0.506783i	$-0.830834\pi$
$113$	−2071.06	−1.72415	−0.862073	+	0.506783i	$0.830834\pi$
$114$	0	0
$115$	−744.706	−0.603863
$116$	−293.933	−0.235268
$117$	−49.0836	−0.0387845
$118$	−188.613	−0.147146
$119$	−158.652	−0.122215
$120$	203.194	0.154575
$121$	187.566	0.140921
$122$	1569.17	1.16448
$123$	−442.159	−0.324131
$124$	−41.1112	−0.0297733
$125$	1509.73	1.08028
$126$	−470.805	−0.332878
$127$	620.284	0.433396	0.216698	−	0.976239i	$-0.430471\pi$
$127$	620.284	0.433396	0.216698	+	0.976239i	$0.430471\pi$
$128$	128.000	0.0883883
$129$	266.931	0.182186
$130$	92.3474	0.0623031
$131$	−2007.46	−1.33888	−0.669438	−	0.742868i	$-0.733465\pi$
$131$	−2007.46	−1.33888	−0.669438	+	0.742868i	$0.733465\pi$
$132$	−467.625	−0.308345
$133$	0	0
$134$	−219.214	−0.141323
$135$	228.594	0.145735
$136$	48.5252	0.0305956
$137$	−2193.09	−1.36765	−0.683827	−	0.729644i	$-0.739686\pi$
$137$	−2193.09	−1.36765	−0.683827	+	0.729644i	$0.739686\pi$
$138$	−527.759	−0.325550
$139$	−2451.89	−1.49616	−0.748081	−	0.663607i	$-0.769025\pi$
$139$	−2451.89	−1.49616	−0.748081	+	0.663607i	$0.769025\pi$
$140$	885.786	0.534733
$141$	525.212	0.313694
$142$	827.547	0.489058
$143$	−212.525	−0.124282
$144$	144.000	0.0833333
$145$	622.142	0.356318
$146$	2246.77	1.27359
$147$	−1023.38	−0.574196
$148$	1785.21	0.991510
$149$	−3070.29	−1.68811	−0.844054	−	0.536258i	$-0.819837\pi$
$149$	−3070.29	−1.68811	−0.844054	+	0.536258i	$0.819837\pi$
$150$	319.917	0.174141
$151$	2407.26	1.29735	0.648676	−	0.761065i	$-0.275323\pi$
$151$	2407.26	1.29735	0.648676	+	0.761065i	$0.275323\pi$
$152$	0	0
$153$	54.5909	0.0288458
$154$	−2038.52	−1.06668
$155$	87.0163	0.0450924
$156$	65.4448	0.0335883
$157$	214.821	0.109201	0.0546006	−	0.998508i	$-0.482611\pi$
$157$	214.821	0.109201	0.0546006	+	0.998508i	$0.482611\pi$
$158$	−2352.86	−1.18470
$159$	1387.75	0.692172
$160$	−270.926	−0.133866
$161$	−2300.66	−1.12620
$162$	162.000	0.0785674
$163$	−1926.98	−0.925969	−0.462984	−	0.886366i	$-0.653221\pi$
$163$	−1926.98	−0.925969	−0.462984	+	0.886366i	$0.653221\pi$
$164$	589.546	0.280706
$165$	989.780	0.466996
$166$	1793.42	0.838534
$167$	3741.50	1.73369	0.866845	−	0.498578i	$-0.166144\pi$
$167$	3741.50	1.73369	0.866845	+	0.498578i	$0.166144\pi$
$168$	627.739	0.288281
$169$	−2167.26	−0.986462
$170$	−102.709	−0.0463378
$171$	0	0
$172$	−355.908	−0.157778
$173$	−1788.17	−0.785851	−0.392925	−	0.919570i	$-0.628537\pi$
$173$	−1788.17	−0.785851	−0.392925	+	0.919570i	$0.628537\pi$
$174$	440.900	0.192095
$175$	1394.61	0.602416
$176$	623.500	0.267035
$177$	282.919	0.120144
$178$	170.264	0.0716958
$179$	−1054.63	−0.440373	−0.220186	−	0.975458i	$-0.570667\pi$
$179$	−1054.63	−0.440373	−0.220186	+	0.975458i	$0.570667\pi$
$180$	−304.792	−0.126210
$181$	271.395	0.111451	0.0557255	−	0.998446i	$-0.482253\pi$
$181$	271.395	0.111451	0.0557255	+	0.998446i	$0.482253\pi$
$182$	285.294	0.116194
$183$	−2353.76	−0.950790
$184$	703.679	0.281934
$185$	−3778.60	−1.50166
$186$	61.6668	0.0243098
$187$	236.371	0.0924341
$188$	−700.283	−0.271667
$189$	706.207	0.271794
$190$	0	0
$191$	−3556.70	−1.34740	−0.673701	−	0.739004i	$-0.735296\pi$
$191$	−3556.70	−1.34740	−0.673701	+	0.739004i	$0.735296\pi$
$192$	−192.000	−0.0721688
$193$	−2825.12	−1.05366	−0.526830	−	0.849970i	$-0.676620\pi$
$193$	−2825.12	−1.05366	−0.526830	+	0.849970i	$0.676620\pi$
$194$	−2710.04	−1.00294
$195$	−138.521	−0.0508703
$196$	1364.50	0.497269
$197$	−2738.77	−0.990505	−0.495253	−	0.868749i	$-0.664924\pi$
$197$	−2738.77	−0.990505	−0.495253	+	0.868749i	$0.664924\pi$
$198$	701.438	0.251763
$199$	−1811.31	−0.645227	−0.322614	−	0.946531i	$-0.604561\pi$
$199$	−1811.31	−0.645227	−0.322614	+	0.946531i	$0.604561\pi$
$200$	−426.556	−0.150810
$201$	328.822	0.115390
$202$	−3869.86	−1.34793
$203$	1922.02	0.664528
$204$	−72.7879	−0.0249812
$205$	−1247.84	−0.425135
$206$	753.848	0.254966
$207$	791.639	0.265810
$208$	−87.2597	−0.0290883
$209$	0	0
$210$	−1328.68	−0.436607
$211$	−4614.52	−1.50558	−0.752788	−	0.658263i	$-0.771291\pi$
$211$	−4614.52	−1.50558	−0.752788	+	0.658263i	$0.771291\pi$
$212$	−1850.33	−0.599439
$213$	−1241.32	−0.399314
$214$	−436.619	−0.139470
$215$	753.319	0.238958
$216$	−216.000	−0.0680414
$217$	268.824	0.0840966
$218$	918.980	0.285510
$219$	−3370.16	−1.03988
$220$	−1319.71	−0.404430
$221$	−33.0805	−0.0100689
$222$	−2677.82	−0.809565
$223$	−1841.76	−0.553065	−0.276533	−	0.961005i	$-0.589185\pi$
$223$	−1841.76	−0.553065	−0.276533	+	0.961005i	$0.589185\pi$
$224$	−836.986	−0.249658
$225$	−479.875	−0.142185
$226$	−4142.11	−1.21916
$227$	1594.09	0.466094	0.233047	−	0.972465i	$-0.425130\pi$
$227$	1594.09	0.466094	0.233047	+	0.972465i	$0.425130\pi$
$228$	0	0
$229$	−4004.53	−1.15558	−0.577788	−	0.816187i	$-0.696084\pi$
$229$	−4004.53	−1.15558	−0.577788	+	0.816187i	$0.696084\pi$
$230$	−1489.41	−0.426996
$231$	3057.78	0.870940
$232$	−587.867	−0.166359
$233$	3154.45	0.886930	0.443465	−	0.896292i	$-0.353749\pi$
$233$	3154.45	0.886930	0.443465	+	0.896292i	$0.353749\pi$
$234$	−98.1672	−0.0274247
$235$	1482.22	0.411445
$236$	−377.225	−0.104048
$237$	3529.28	0.967306
$238$	−317.304	−0.0864192
$239$	213.212	0.0577053	0.0288526	−	0.999584i	$-0.490815\pi$
$239$	213.212	0.0577053	0.0288526	+	0.999584i	$0.490815\pi$
$240$	406.389	0.109301
$241$	4776.93	1.27680	0.638400	−	0.769705i	$-0.279596\pi$
$241$	4776.93	1.27680	0.638400	+	0.769705i	$0.279596\pi$
$242$	375.132	0.0996462
$243$	−243.000	−0.0641500
$244$	3138.34	0.823409
$245$	−2888.12	−0.753124
$246$	−884.318	−0.229195
$247$	0	0
$248$	−82.2224	−0.0210529
$249$	−2690.13	−0.684660
$250$	3019.46	0.763870
$251$	2302.56	0.579029	0.289514	−	0.957174i	$-0.406506\pi$
$251$	2302.56	0.579029	0.289514	+	0.957174i	$0.406506\pi$
$252$	−941.609	−0.235380
$253$	3427.69	0.851766
$254$	1240.57	0.306457
$255$	154.063	0.0378346
$256$	256.000	0.0625000
$257$	5494.60	1.33363	0.666816	−	0.745222i	$-0.267657\pi$
$257$	5494.60	1.33363	0.666816	+	0.745222i	$0.267657\pi$
$258$	533.862	0.128825
$259$	−11673.4	−2.80058
$260$	184.695	0.0440549
$261$	−661.350	−0.156845
$262$	−4014.92	−0.946728
$263$	−3709.79	−0.869793	−0.434896	−	0.900481i	$-0.643215\pi$
$263$	−3709.79	−0.869793	−0.434896	+	0.900481i	$0.643215\pi$
$264$	−935.251	−0.218033
$265$	3916.42	0.907863
$266$	0	0
$267$	−255.397	−0.0585394
$268$	−438.429	−0.0999303
$269$	2032.29	0.460634	0.230317	−	0.973116i	$-0.426024\pi$
$269$	2032.29	0.460634	0.230317	+	0.973116i	$0.426024\pi$
$270$	457.188	0.103050
$271$	−1793.01	−0.401910	−0.200955	−	0.979600i	$-0.564405\pi$
$271$	−1793.01	−0.401910	−0.200955	+	0.979600i	$0.564405\pi$
$272$	97.0505	0.0216344
$273$	−427.940	−0.0948723
$274$	−4386.19	−0.967077
$275$	−2077.79	−0.455621
$276$	−1055.52	−0.230198
$277$	6840.62	1.48380	0.741901	−	0.670510i	$-0.233925\pi$
$277$	6840.62	1.48380	0.741901	+	0.670510i	$0.233925\pi$
$278$	−4903.78	−1.05795
$279$	−92.5002	−0.0198489
$280$	1771.57	0.378113
$281$	7005.00	1.48713	0.743564	−	0.668665i	$-0.233134\pi$
$281$	7005.00	1.48713	0.743564	+	0.668665i	$0.233134\pi$
$282$	1050.42	0.221815
$283$	−350.405	−0.0736022	−0.0368011	−	0.999323i	$-0.511717\pi$
$283$	−350.405	−0.0736022	−0.0368011	+	0.999323i	$0.511717\pi$
$284$	1655.09	0.345816
$285$	0	0
$286$	−425.051	−0.0878803
$287$	−3855.01	−0.792871
$288$	288.000	0.0589256
$289$	−4876.21	−0.992511
$290$	1244.28	0.251955
$291$	4065.06	0.818893
$292$	4493.54	0.900564
$293$	8527.18	1.70021	0.850107	−	0.526609i	$-0.176537\pi$
$293$	8527.18	1.70021	0.850107	+	0.526609i	$0.176537\pi$
$294$	−2046.76	−0.406018
$295$	798.439	0.157583
$296$	3570.42	0.701103
$297$	−1052.16	−0.205563
$298$	−6140.58	−1.19367
$299$	−479.710	−0.0927837
$300$	639.833	0.123136
$301$	2327.27	0.445653
$302$	4814.52	0.917366
$303$	5804.79	1.10058
$304$	0	0
$305$	−6642.64	−1.24707
$306$	109.182	0.0203971
$307$	−1612.51	−0.299775	−0.149887	−	0.988703i	$-0.547891\pi$
$307$	−1612.51	−0.299775	−0.149887	+	0.988703i	$0.547891\pi$
$308$	−4077.04	−0.754256
$309$	−1130.77	−0.208179
$310$	174.033	0.0318851
$311$	4933.39	0.899507	0.449753	−	0.893153i	$-0.351512\pi$
$311$	4933.39	0.899507	0.449753	+	0.893153i	$0.351512\pi$
$312$	130.890	0.0237505
$313$	−3160.93	−0.570818	−0.285409	−	0.958406i	$-0.592129\pi$
$313$	−3160.93	−0.570818	−0.285409	+	0.958406i	$0.592129\pi$
$314$	429.642	0.0772170
$315$	1993.02	0.356488
$316$	−4705.71	−0.837712
$317$	5453.04	0.966162	0.483081	−	0.875576i	$-0.339518\pi$
$317$	5453.04	0.966162	0.483081	+	0.875576i	$0.339518\pi$
$318$	2775.49	0.489439
$319$	−2863.56	−0.502597
$320$	−541.852	−0.0946576
$321$	654.929	0.113877
$322$	−4601.32	−0.796340
$323$	0	0
$324$	324.000	0.0555556
$325$	290.790	0.0496312
$326$	−3853.96	−0.654759
$327$	−1378.47	−0.233118
$328$	1179.09	0.198489
$329$	4579.11	0.767340
$330$	1979.56	0.330216
$331$	5680.41	0.943274	0.471637	−	0.881793i	$-0.343663\pi$
$331$	5680.41	0.943274	0.471637	+	0.881793i	$0.343663\pi$
$332$	3586.84	0.592933
$333$	4016.73	0.661007
$334$	7483.01	1.22590
$335$	927.983	0.151347
$336$	1255.48	0.203845
$337$	−3511.86	−0.567665	−0.283833	−	0.958874i	$-0.591606\pi$
$337$	−3511.86	−0.567665	−0.283833	+	0.958874i	$0.591606\pi$
$338$	−4334.51	−0.697534
$339$	6213.17	0.995437
$340$	−205.418	−0.0327657
$341$	−400.513	−0.0636041
$342$	0	0
$343$	49.0109	0.00771528
$344$	−711.816	−0.111566
$345$	2234.12	0.348640
$346$	−3576.34	−0.555680
$347$	−9775.02	−1.51225	−0.756125	−	0.654428i	$-0.772910\pi$
$347$	−9775.02	−1.51225	−0.756125	+	0.654428i	$0.772910\pi$
$348$	881.800	0.135832
$349$	−11482.5	−1.76116	−0.880580	−	0.473897i	$-0.842847\pi$
$349$	−11482.5	−1.76116	−0.880580	+	0.473897i	$0.842847\pi$
$350$	2789.23	0.425973
$351$	147.251	0.0223922
$352$	1247.00	0.188822
$353$	3824.71	0.576683	0.288341	−	0.957528i	$-0.406896\pi$
$353$	3824.71	0.576683	0.288341	+	0.957528i	$0.406896\pi$
$354$	565.838	0.0849547
$355$	−3503.19	−0.523746
$356$	340.529	0.0506966
$357$	475.956	0.0705610
$358$	−2109.26	−0.311391
$359$	−8176.70	−1.20209	−0.601045	−	0.799216i	$-0.705249\pi$
$359$	−8176.70	−1.20209	−0.601045	+	0.799216i	$0.705249\pi$
$360$	−609.583	−0.0892441
$361$	0	0
$362$	542.790	0.0788077
$363$	−562.698	−0.0813608
$364$	570.587	0.0821618
$365$	−9511.07	−1.36392
$366$	−4707.51	−0.672310
$367$	−5586.79	−0.794627	−0.397314	−	0.917683i	$-0.630057\pi$
$367$	−5586.79	−0.794627	−0.397314	+	0.917683i	$0.630057\pi$
$368$	1407.36	0.199358
$369$	1326.48	0.187137
$370$	−7557.19	−1.06184
$371$	12099.2	1.69315
$372$	123.334	0.0171896
$373$	−9242.17	−1.28295	−0.641476	−	0.767143i	$-0.721678\pi$
$373$	−9242.17	−1.28295	−0.641476	+	0.767143i	$0.721678\pi$
$374$	472.742	0.0653608
$375$	−4529.19	−0.623697
$376$	−1400.57	−0.192097
$377$	400.759	0.0547483
$378$	1412.41	0.192187
$379$	5162.54	0.699688	0.349844	−	0.936808i	$-0.386235\pi$
$379$	5162.54	0.699688	0.349844	+	0.936808i	$0.386235\pi$
$380$	0	0
$381$	−1860.85	−0.250221
$382$	−7113.40	−0.952757
$383$	−9994.42	−1.33340	−0.666699	−	0.745327i	$-0.732293\pi$
$383$	−9994.42	−1.33340	−0.666699	+	0.745327i	$0.732293\pi$
$384$	−384.000	−0.0510310
$385$	8629.50	1.14234
$386$	−5650.24	−0.745051
$387$	−800.794	−0.105185
$388$	−5420.08	−0.709182
$389$	199.512	0.0260043	0.0130022	−	0.999915i	$-0.495861\pi$
$389$	199.512	0.0260043	0.0130022	+	0.999915i	$0.495861\pi$
$390$	−277.042	−0.0359707
$391$	533.534	0.0690076
$392$	2729.01	0.351622
$393$	6022.39	0.773000
$394$	−5477.55	−0.700393
$395$	9960.15	1.26873
$396$	1402.88	0.178023
$397$	1660.51	0.209921	0.104960	−	0.994476i	$-0.466528\pi$
$397$	1660.51	0.209921	0.104960	+	0.994476i	$0.466528\pi$
$398$	−3622.62	−0.456244
$399$	0	0
$400$	−853.111	−0.106639
$401$	−2226.47	−0.277268	−0.138634	−	0.990344i	$-0.544271\pi$
$401$	−2226.47	−0.277268	−0.138634	+	0.990344i	$0.544271\pi$
$402$	657.643	0.0815927
$403$	56.0524	0.00692846
$404$	−7739.72	−0.953133
$405$	−685.781	−0.0841401
$406$	3844.03	0.469892
$407$	17391.9	2.11814
$408$	−145.576	−0.0176644
$409$	1459.66	0.176468	0.0882342	−	0.996100i	$-0.471878\pi$
$409$	1459.66	0.176468	0.0882342	+	0.996100i	$0.471878\pi$
$410$	−2495.68	−0.300616
$411$	6579.28	0.789615
$412$	1507.70	0.180288
$413$	2466.66	0.293889
$414$	1583.28	0.187956
$415$	−7591.95	−0.898010
$416$	−174.519	−0.0205686
$417$	7355.67	0.863810
$418$	0	0
$419$	7574.64	0.883163	0.441581	−	0.897221i	$-0.354418\pi$
$419$	7574.64	0.883163	0.441581	+	0.897221i	$0.354418\pi$
$420$	−2657.36	−0.308728
$421$	−12944.7	−1.49854	−0.749271	−	0.662263i	$-0.769596\pi$
$421$	−12944.7	−1.49854	−0.749271	+	0.662263i	$0.769596\pi$
$422$	−9229.03	−1.06460
$423$	−1575.64	−0.181111
$424$	−3700.65	−0.423867
$425$	−323.417	−0.0369131
$426$	−2482.64	−0.282358
$427$	−20521.5	−2.32577
$428$	−873.238	−0.0986205
$429$	637.576	0.0717540
$430$	1506.64	0.168969
$431$	−8151.65	−0.911023	−0.455512	−	0.890230i	$-0.650544\pi$
$431$	−8151.65	−0.911023	−0.455512	+	0.890230i	$0.650544\pi$
$432$	−432.000	−0.0481125
$433$	1943.44	0.215695	0.107848	−	0.994167i	$-0.465604\pi$
$433$	1943.44	0.215695	0.107848	+	0.994167i	$0.465604\pi$
$434$	537.648	0.0594653
$435$	−1866.43	−0.205720
$436$	1837.96	0.201886
$437$	0	0
$438$	−6740.31	−0.735307
$439$	13646.7	1.48365	0.741823	−	0.670596i	$-0.233961\pi$
$439$	13646.7	1.48365	0.741823	+	0.670596i	$0.233961\pi$
$440$	−2639.41	−0.285975
$441$	3070.14	0.331512
$442$	−66.1609	−0.00711981
$443$	401.445	0.0430546	0.0215273	−	0.999768i	$-0.493147\pi$
$443$	401.445	0.0430546	0.0215273	+	0.999768i	$0.493147\pi$
$444$	−5355.63	−0.572449
$445$	−720.766	−0.0767811
$446$	−3683.52	−0.391076
$447$	9210.87	0.974630
$448$	−1673.97	−0.176535
$449$	3890.88	0.408958	0.204479	−	0.978871i	$-0.434450\pi$
$449$	3890.88	0.408958	0.204479	+	0.978871i	$0.434450\pi$
$450$	−959.750	−0.100540
$451$	5743.47	0.599666
$452$	−8284.23	−0.862073
$453$	−7221.78	−0.749026
$454$	3188.18	0.329578
$455$	−1207.71	−0.124436
$456$	0	0
$457$	5366.73	0.549333	0.274667	−	0.961540i	$-0.411433\pi$
$457$	5366.73	0.549333	0.274667	+	0.961540i	$0.411433\pi$
$458$	−8009.06	−0.817115
$459$	−163.773	−0.0166541
$460$	−2978.83	−0.301931
$461$	4081.15	0.412316	0.206158	−	0.978519i	$-0.433904\pi$
$461$	4081.15	0.412316	0.206158	+	0.978519i	$0.433904\pi$
$462$	6115.56	0.615848
$463$	4127.12	0.414262	0.207131	−	0.978313i	$-0.433587\pi$
$463$	4127.12	0.414262	0.207131	+	0.978313i	$0.433587\pi$
$464$	−1175.73	−0.117634
$465$	−261.049	−0.0260341
$466$	6308.89	0.627154
$467$	−10724.6	−1.06268	−0.531342	−	0.847157i	$-0.678312\pi$
$467$	−10724.6	−1.06268	−0.531342	+	0.847157i	$0.678312\pi$
$468$	−196.334	−0.0193922
$469$	2866.87	0.282259
$470$	2964.45	0.290936
$471$	−644.464	−0.0630474
$472$	−754.451	−0.0735729
$473$	−3467.33	−0.337057
$474$	7058.57	0.683989
$475$	0	0
$476$	−634.608	−0.0611076
$477$	−4163.24	−0.399626
$478$	426.425	0.0408038
$479$	−3248.01	−0.309823	−0.154912	−	0.987928i	$-0.549509\pi$
$479$	−3248.01	−0.309823	−0.154912	+	0.987928i	$0.549509\pi$
$480$	812.778	0.0772876
$481$	−2434.02	−0.230731
$482$	9553.85	0.902834
$483$	6901.98	0.650209
$484$	750.264	0.0704605
$485$	11472.2	1.07407
$486$	−486.000	−0.0453609
$487$	−3777.02	−0.351444	−0.175722	−	0.984440i	$-0.556226\pi$
$487$	−3777.02	−0.351444	−0.175722	+	0.984440i	$0.556226\pi$
$488$	6276.68	0.582238
$489$	5780.95	0.534608
$490$	−5776.25	−0.532539
$491$	−9551.21	−0.877882	−0.438941	−	0.898516i	$-0.644646\pi$
$491$	−9551.21	−0.877882	−0.438941	+	0.898516i	$0.644646\pi$
$492$	−1768.64	−0.162066
$493$	−445.725	−0.0407189
$494$	0	0
$495$	−2969.34	−0.269620
$496$	−164.445	−0.0148867
$497$	−10822.6	−0.976779
$498$	−5380.27	−0.484128
$499$	11025.9	0.989156	0.494578	−	0.869133i	$-0.335323\pi$
$499$	11025.9	0.989156	0.494578	+	0.869133i	$0.335323\pi$
$500$	6038.92	0.540138
$501$	−11224.5	−1.00095
$502$	4605.12	0.409435
$503$	−8862.31	−0.785588	−0.392794	−	0.919626i	$-0.628491\pi$
$503$	−8862.31	−0.785588	−0.392794	+	0.919626i	$0.628491\pi$
$504$	−1883.22	−0.166439
$505$	16382.0	1.44354
$506$	6855.38	0.602290
$507$	6501.77	0.569534
$508$	2481.13	0.216698
$509$	−4177.45	−0.363776	−0.181888	−	0.983319i	$-0.558221\pi$
$509$	−4177.45	−0.363776	−0.181888	+	0.983319i	$0.558221\pi$
$510$	308.127	0.0267531
$511$	−29383.0	−2.54370
$512$	512.000	0.0441942
$513$	0	0
$514$	10989.2	0.943020
$515$	−3191.20	−0.273051
$516$	1067.72	0.0910929
$517$	−6822.29	−0.580356
$518$	−23346.8	−1.98031
$519$	5364.52	0.453711
$520$	369.390	0.0311515
$521$	12176.7	1.02394	0.511968	−	0.859005i	$-0.328917\pi$
$521$	12176.7	1.02394	0.511968	+	0.859005i	$0.328917\pi$
$522$	−1322.70	−0.110906
$523$	−20882.1	−1.74591	−0.872956	−	0.487800i	$-0.837800\pi$
$523$	−20882.1	−1.74591	−0.872956	+	0.487800i	$0.837800\pi$
$524$	−8029.85	−0.669438
$525$	−4183.84	−0.347805
$526$	−7419.58	−0.615036
$527$	−62.3416	−0.00515302
$528$	−1870.50	−0.154173
$529$	−4430.07	−0.364105
$530$	7832.84	0.641956
$531$	−848.757	−0.0693652
$532$	0	0
$533$	−803.806	−0.0653222
$534$	−510.793	−0.0413936
$535$	1848.30	0.149363
$536$	−876.858	−0.0706614
$537$	3163.89	0.254249
$538$	4064.57	0.325718
$539$	13293.3	1.06230
$540$	914.375	0.0728675
$541$	−3677.27	−0.292233	−0.146116	−	0.989267i	$-0.546677\pi$
$541$	−3677.27	−0.292233	−0.146116	+	0.989267i	$0.546677\pi$
$542$	−3586.02	−0.284193
$543$	−814.185	−0.0643462
$544$	194.101	0.0152978
$545$	−3890.24	−0.305761
$546$	−855.881	−0.0670848
$547$	−12677.0	−0.990911	−0.495456	−	0.868633i	$-0.664999\pi$
$547$	−12677.0	−0.990911	−0.495456	+	0.868633i	$0.664999\pi$
$548$	−8772.37	−0.683827
$549$	7061.27	0.548939
$550$	−4155.59	−0.322172
$551$	0	0
$552$	−2111.04	−0.162775
$553$	30770.4	2.36617
$554$	13681.2	1.04921
$555$	11335.8	0.866986
$556$	−9807.56	−0.748081
$557$	−4162.34	−0.316632	−0.158316	−	0.987388i	$-0.550607\pi$
$557$	−4162.34	−0.316632	−0.158316	+	0.987388i	$0.550607\pi$
$558$	−185.000	−0.0140353
$559$	485.257	0.0367159
$560$	3543.14	0.267366
$561$	−709.114	−0.0533668
$562$	14010.0	1.05156
$563$	−7944.92	−0.594739	−0.297370	−	0.954762i	$-0.596109\pi$
$563$	−7944.92	−0.594739	−0.297370	+	0.954762i	$0.596109\pi$
$564$	2100.85	0.156847
$565$	17534.5	1.30563
$566$	−700.810	−0.0520446
$567$	−2118.62	−0.156920
$568$	3310.19	0.244529
$569$	8867.64	0.653340	0.326670	−	0.945138i	$-0.394073\pi$
$569$	8867.64	0.653340	0.326670	+	0.945138i	$0.394073\pi$
$570$	0	0
$571$	−19192.0	−1.40658	−0.703292	−	0.710901i	$-0.748287\pi$
$571$	−19192.0	−1.40658	−0.703292	+	0.710901i	$0.748287\pi$
$572$	−850.101	−0.0621408
$573$	10670.1	0.777922
$574$	−7710.02	−0.560645
$575$	−4689.97	−0.340148
$576$	576.000	0.0416667
$577$	5669.82	0.409077	0.204539	−	0.978859i	$-0.434431\pi$
$577$	5669.82	0.409077	0.204539	+	0.978859i	$0.434431\pi$
$578$	−9752.42	−0.701811
$579$	8475.36	0.608331
$580$	2488.57	0.178159
$581$	−23454.2	−1.67478
$582$	8130.12	0.579045
$583$	−18026.2	−1.28057
$584$	8987.08	0.636795
$585$	415.563	0.0293700
$586$	17054.4	1.20223
$587$	5668.01	0.398542	0.199271	−	0.979944i	$-0.436143\pi$
$587$	5668.01	0.398542	0.199271	+	0.979944i	$0.436143\pi$
$588$	−4093.51	−0.287098
$589$	0	0
$590$	1596.88	0.111428
$591$	8216.32	0.571868
$592$	7140.85	0.495755
$593$	−10764.4	−0.745432	−0.372716	−	0.927946i	$-0.621573\pi$
$593$	−10764.4	−0.745432	−0.372716	+	0.927946i	$0.621573\pi$
$594$	−2104.31	−0.145355
$595$	1343.22	0.0925489
$596$	−12281.2	−0.844054
$597$	5433.92	0.372522
$598$	−959.419	−0.0656080
$599$	−24720.6	−1.68624	−0.843118	−	0.537728i	$-0.819283\pi$
$599$	−24720.6	−1.68624	−0.843118	+	0.537728i	$0.819283\pi$
$600$	1279.67	0.0870703
$601$	−25364.2	−1.72151	−0.860754	−	0.509021i	$-0.830008\pi$
$601$	−25364.2	−1.72151	−0.860754	+	0.509021i	$0.830008\pi$
$602$	4654.53	0.315124
$603$	−986.465	−0.0666202
$604$	9629.04	0.648676
$605$	−1588.01	−0.106714
$606$	11609.6	0.778230
$607$	28334.8	1.89468	0.947341	−	0.320226i	$-0.103759\pi$
$607$	28334.8	1.89468	0.947341	+	0.320226i	$0.103759\pi$
$608$	0	0
$609$	−5766.05	−0.383665
$610$	−13285.3	−0.881812
$611$	954.789	0.0632187
$612$	218.364	0.0144229
$613$	29080.9	1.91610	0.958049	−	0.286604i	$-0.0925263\pi$
$613$	29080.9	1.91610	0.958049	+	0.286604i	$0.0925263\pi$
$614$	−3225.02	−0.211973
$615$	3743.51	0.245452
$616$	−8154.08	−0.533340
$617$	2401.61	0.156702	0.0783510	−	0.996926i	$-0.475035\pi$
$617$	2401.61	0.156702	0.0783510	+	0.996926i	$0.475035\pi$
$618$	−2261.54	−0.147205
$619$	6861.44	0.445532	0.222766	−	0.974872i	$-0.428491\pi$
$619$	6861.44	0.445532	0.222766	+	0.974872i	$0.428491\pi$
$620$	348.065	0.0225462
$621$	−2374.92	−0.153466
$622$	9866.77	0.636047
$623$	−2226.70	−0.143196
$624$	261.779	0.0167942
$625$	−6117.11	−0.391495
$626$	−6321.85	−0.403630
$627$	0	0
$628$	859.285	0.0546006
$629$	2707.12	0.171606
$630$	3986.04	0.252075
$631$	13745.6	0.867200	0.433600	−	0.901105i	$-0.357243\pi$
$631$	13745.6	0.867200	0.433600	+	0.901105i	$0.357243\pi$
$632$	−9411.42	−0.592352
$633$	13843.6	0.869245
$634$	10906.1	0.683180
$635$	−5251.59	−0.328194
$636$	5550.98	0.346086
$637$	−1860.41	−0.115718
$638$	−5727.11	−0.355390
$639$	3723.96	0.230544
$640$	−1083.70	−0.0669331
$641$	−15360.4	−0.946490	−0.473245	−	0.880931i	$-0.656917\pi$
$641$	−15360.4	−0.946490	−0.473245	+	0.880931i	$0.656917\pi$
$642$	1309.86	0.0805233
$643$	3472.77	0.212990	0.106495	−	0.994313i	$-0.466037\pi$
$643$	3472.77	0.212990	0.106495	+	0.994313i	$0.466037\pi$
$644$	−9202.64	−0.563098
$645$	−2259.96	−0.137962
$646$	0	0
$647$	15387.5	0.934999	0.467499	−	0.883993i	$-0.345155\pi$
$647$	15387.5	0.934999	0.467499	+	0.883993i	$0.345155\pi$
$648$	648.000	0.0392837
$649$	−3675.00	−0.222275
$650$	581.580	0.0350945
$651$	−806.472	−0.0485532
$652$	−7707.93	−0.462984
$653$	7443.52	0.446076	0.223038	−	0.974810i	$-0.428403\pi$
$653$	7443.52	0.446076	0.223038	+	0.974810i	$0.428403\pi$
$654$	−2756.94	−0.164839
$655$	16996.0	1.01388
$656$	2358.18	0.140353
$657$	10110.5	0.600376
$658$	9158.23	0.542591
$659$	21288.9	1.25842	0.629210	−	0.777235i	$-0.283378\pi$
$659$	21288.9	1.25842	0.629210	+	0.777235i	$0.283378\pi$
$660$	3959.12	0.233498
$661$	20660.1	1.21571	0.607856	−	0.794047i	$-0.292030\pi$
$661$	20660.1	1.21571	0.607856	+	0.794047i	$0.292030\pi$
$662$	11360.8	0.666996
$663$	99.2414	0.00581330
$664$	7173.69	0.419267
$665$	0	0
$666$	8033.45	0.467402
$667$	−6463.59	−0.375219
$668$	14966.0	0.866845
$669$	5525.29	0.319312
$670$	1855.97	0.107018
$671$	30574.3	1.75903
$672$	2510.96	0.144140
$673$	−18954.0	−1.08562	−0.542810	−	0.839856i	$-0.682639\pi$
$673$	−18954.0	−1.08562	−0.542810	+	0.839856i	$0.682639\pi$
$674$	−7023.72	−0.401400
$675$	1439.63	0.0820907
$676$	−8669.03	−0.493231
$677$	25224.2	1.43197	0.715987	−	0.698114i	$-0.245977\pi$
$677$	25224.2	1.43197	0.715987	+	0.698114i	$0.245977\pi$
$678$	12426.3	0.703880
$679$	35441.6	2.00313
$680$	−410.836	−0.0231689
$681$	−4782.26	−0.269100
$682$	−801.027	−0.0449749
$683$	−6091.07	−0.341242	−0.170621	−	0.985337i	$-0.554577\pi$
$683$	−6091.07	−0.341242	−0.170621	+	0.985337i	$0.554577\pi$
$684$	0	0
$685$	18567.7	1.03567
$686$	98.0218	0.00545552
$687$	12013.6	0.667172
$688$	−1423.63	−0.0788888
$689$	2522.80	0.139493
$690$	4468.24	0.246526
$691$	−30917.7	−1.70212	−0.851059	−	0.525071i	$-0.824039\pi$
$691$	−30917.7	−1.70212	−0.851059	+	0.525071i	$0.824039\pi$
$692$	−7152.69	−0.392925
$693$	−9173.34	−0.502837
$694$	−19550.0	−1.06932
$695$	20758.8	1.13299
$696$	1763.60	0.0960476
$697$	893.995	0.0485832
$698$	−22965.0	−1.24533
$699$	−9463.34	−0.512069
$700$	5578.45	0.301208
$701$	6539.47	0.352343	0.176171	−	0.984359i	$-0.443629\pi$
$701$	6539.47	0.352343	0.176171	+	0.984359i	$0.443629\pi$
$702$	294.502	0.0158337
$703$	0	0
$704$	2494.00	0.133517
$705$	−4446.67	−0.237548
$706$	7649.43	0.407776
$707$	50609.7	2.69218
$708$	1131.68	0.0600720
$709$	−11347.8	−0.601095	−0.300547	−	0.953767i	$-0.597169\pi$
$709$	−11347.8	−0.601095	−0.300547	+	0.953767i	$0.597169\pi$
$710$	−7006.38	−0.370345
$711$	−10587.8	−0.558475
$712$	681.058	0.0358479
$713$	−904.034	−0.0474843
$714$	951.913	0.0498942
$715$	1799.33	0.0941136
$716$	−4218.52	−0.220186
$717$	−639.637	−0.0333162
$718$	−16353.4	−0.850005
$719$	−33580.2	−1.74177	−0.870883	−	0.491490i	$-0.836453\pi$
$719$	−33580.2	−1.74177	−0.870883	+	0.491490i	$0.836453\pi$
$720$	−1219.17	−0.0631051
$721$	−9858.75	−0.509236
$722$	0	0
$723$	−14330.8	−0.737161
$724$	1085.58	0.0557255
$725$	3918.09	0.200709
$726$	−1125.40	−0.0575308
$727$	3710.83	0.189308	0.0946541	−	0.995510i	$-0.469825\pi$
$727$	3710.83	0.189308	0.0946541	+	0.995510i	$0.469825\pi$
$728$	1141.17	0.0580972
$729$	729.000	0.0370370
$730$	−19022.1	−0.964440
$731$	−539.704	−0.0273074
$732$	−9415.02	−0.475395
$733$	−4648.62	−0.234244	−0.117122	−	0.993118i	$-0.537367\pi$
$733$	−4648.62	−0.234244	−0.117122	+	0.993118i	$0.537367\pi$
$734$	−11173.6	−0.561886
$735$	8664.37	0.434816
$736$	2814.72	0.140967
$737$	−4271.26	−0.213479
$738$	2652.95	0.132326
$739$	−35507.0	−1.76745	−0.883724	−	0.468008i	$-0.844972\pi$
$739$	−35507.0	−1.76745	−0.883724	+	0.468008i	$0.844972\pi$
$740$	−15114.4	−0.750832
$741$	0	0
$742$	24198.4	1.19724
$743$	12161.8	0.600503	0.300251	−	0.953860i	$-0.402929\pi$
$743$	12161.8	0.600503	0.300251	+	0.953860i	$0.402929\pi$
$744$	246.667	0.0121549
$745$	25994.4	1.27834
$746$	−18484.3	−0.907185
$747$	8070.40	0.395288
$748$	945.485	0.0462170
$749$	5710.06	0.278560
$750$	−9058.38	−0.441020
$751$	28107.4	1.36572	0.682859	−	0.730551i	$-0.260737\pi$
$751$	28107.4	1.36572	0.682859	+	0.730551i	$0.260737\pi$
$752$	−2801.13	−0.135833
$753$	−6907.68	−0.334302
$754$	801.518	0.0387129
$755$	−20380.9	−0.982434
$756$	2824.83	0.135897
$757$	30876.3	1.48246	0.741228	−	0.671253i	$-0.234244\pi$
$757$	30876.3	1.48246	0.741228	+	0.671253i	$0.234244\pi$
$758$	10325.1	0.494754
$759$	−10283.1	−0.491768
$760$	0	0
$761$	7131.07	0.339686	0.169843	−	0.985471i	$-0.445674\pi$
$761$	7131.07	0.339686	0.169843	+	0.985471i	$0.445674\pi$
$762$	−3721.70	−0.176933
$763$	−12018.3	−0.570240
$764$	−14226.8	−0.673701
$765$	−462.190	−0.0218438
$766$	−19988.8	−0.942854
$767$	514.322	0.0242126
$768$	−768.000	−0.0360844
$769$	31172.0	1.46176	0.730879	−	0.682507i	$-0.239110\pi$
$769$	31172.0	1.46176	0.730879	+	0.682507i	$0.239110\pi$
$770$	17259.0	0.807755
$771$	−16483.8	−0.769973
$772$	−11300.5	−0.526830
$773$	11170.9	0.519779	0.259889	−	0.965638i	$-0.416314\pi$
$773$	11170.9	0.519779	0.259889	+	0.965638i	$0.416314\pi$
$774$	−1601.59	−0.0743771
$775$	548.007	0.0254000
$776$	−10840.2	−0.501468
$777$	35020.2	1.61692
$778$	399.025	0.0183878
$779$	0	0
$780$	−554.084	−0.0254351
$781$	16124.3	0.738759
$782$	1067.07	0.0487958
$783$	1984.05	0.0905545
$784$	5458.02	0.248634
$785$	−1818.77	−0.0826939
$786$	12044.8	0.546594
$787$	−15143.6	−0.685911	−0.342956	−	0.939352i	$-0.611428\pi$
$787$	−15143.6	−0.685911	−0.342956	+	0.939352i	$0.611428\pi$
$788$	−10955.1	−0.495253
$789$	11129.4	0.502175
$790$	19920.3	0.897130
$791$	54170.1	2.43498
$792$	2805.75	0.125881
$793$	−4278.92	−0.191613
$794$	3321.02	0.148437
$795$	−11749.3	−0.524155
$796$	−7245.23	−0.322614
$797$	37804.9	1.68020	0.840100	−	0.542432i	$-0.182496\pi$
$797$	37804.9	1.68020	0.840100	+	0.542432i	$0.182496\pi$
$798$	0	0
$799$	−1061.92	−0.0470187
$800$	−1706.22	−0.0754051
$801$	766.190	0.0337977
$802$	−4452.94	−0.196058
$803$	43777.0	1.92385
$804$	1315.29	0.0576948
$805$	19478.4	0.852824
$806$	112.105	0.00489916
$807$	−6096.86	−0.265947
$808$	−15479.4	−0.673967
$809$	−38365.9	−1.66733	−0.833666	−	0.552268i	$-0.813762\pi$
$809$	−38365.9	−1.66733	−0.833666	+	0.552268i	$0.813762\pi$
$810$	−1371.56	−0.0594961
$811$	−21119.8	−0.914447	−0.457223	−	0.889352i	$-0.651156\pi$
$811$	−21119.8	−0.914447	−0.457223	+	0.889352i	$0.651156\pi$
$812$	7688.07	0.332264
$813$	5379.03	0.232043
$814$	34783.8	1.49775
$815$	16314.7	0.701200
$816$	−291.151	−0.0124906
$817$	0	0
$818$	2919.32	0.124782
$819$	1283.82	0.0547745
$820$	−4991.35	−0.212568
$821$	2117.58	0.0900171	0.0450085	−	0.998987i	$-0.485668\pi$
$821$	2117.58	0.0900171	0.0450085	+	0.998987i	$0.485668\pi$
$822$	13158.6	0.558342
$823$	−30988.0	−1.31248	−0.656242	−	0.754550i	$-0.727855\pi$
$823$	−30988.0	−1.31248	−0.656242	+	0.754550i	$0.727855\pi$
$824$	3015.39	0.127483
$825$	6233.38	0.263053
$826$	4933.32	0.207811
$827$	−252.450	−0.0106149	−0.00530746	−	0.999986i	$-0.501689\pi$
$827$	−252.450	−0.0106149	−0.00530746	+	0.999986i	$0.501689\pi$
$828$	3166.55	0.132905
$829$	27473.7	1.15103	0.575513	−	0.817793i	$-0.304803\pi$
$829$	27473.7	1.15103	0.575513	+	0.817793i	$0.304803\pi$
$830$	−15183.9	−0.634989
$831$	−20521.9	−0.856673
$832$	−349.039	−0.0145442
$833$	2069.15	0.0860647
$834$	14711.3	0.610806
$835$	−31677.2	−1.31286
$836$	0	0
$837$	277.501	0.0114598
$838$	15149.3	0.624490
$839$	−10505.7	−0.432295	−0.216148	−	0.976361i	$-0.569349\pi$
$839$	−10505.7	−0.432295	−0.216148	+	0.976361i	$0.569349\pi$
$840$	−5314.72	−0.218304
$841$	−18989.2	−0.778597
$842$	−25889.4	−1.05963
$843$	−21015.0	−0.858594
$844$	−18458.1	−0.752788
$845$	18348.9	0.747009
$846$	−3151.27	−0.128065
$847$	−4905.94	−0.199020
$848$	−7401.31	−0.299719
$849$	1051.22	0.0424942
$850$	−646.835	−0.0261015
$851$	39256.7	1.58132
$852$	−4965.28	−0.199657
$853$	−12763.7	−0.512335	−0.256167	−	0.966632i	$-0.582460\pi$
$853$	−12763.7	−0.512335	−0.256167	+	0.966632i	$0.582460\pi$
$854$	−41042.9	−1.64457
$855$	0	0
$856$	−1746.48	−0.0697352
$857$	18383.1	0.732736	0.366368	−	0.930470i	$-0.380601\pi$
$857$	18383.1	0.732736	0.366368	+	0.930470i	$0.380601\pi$
$858$	1275.15	0.0507377
$859$	24390.7	0.968799	0.484400	−	0.874847i	$-0.339038\pi$
$859$	24390.7	0.968799	0.484400	+	0.874847i	$0.339038\pi$
$860$	3013.27	0.119479
$861$	11565.0	0.457764
$862$	−16303.3	−0.644191
$863$	21158.2	0.834570	0.417285	−	0.908776i	$-0.362982\pi$
$863$	21158.2	0.834570	0.417285	+	0.908776i	$0.362982\pi$
$864$	−864.000	−0.0340207
$865$	15139.4	0.595094
$866$	3886.89	0.152519
$867$	14628.6	0.573027
$868$	1075.30	0.0420483
$869$	−45844.0	−1.78959
$870$	−3732.85	−0.145466
$871$	597.769	0.0232544
$872$	3675.92	0.142755
$873$	−12195.2	−0.472788
$874$	0	0
$875$	−39488.2	−1.52565
$876$	−13480.6	−0.519941
$877$	−35002.4	−1.34771	−0.673857	−	0.738862i	$-0.735364\pi$
$877$	−35002.4	−1.34771	−0.673857	+	0.738862i	$0.735364\pi$
$878$	27293.4	1.04910
$879$	−25581.5	−0.981620
$880$	−5278.83	−0.202215
$881$	−10355.8	−0.396023	−0.198012	−	0.980200i	$-0.563448\pi$
$881$	−10355.8	−0.396023	−0.198012	+	0.980200i	$0.563448\pi$
$882$	6140.27	0.234415
$883$	30423.2	1.15948	0.579740	−	0.814801i	$-0.303154\pi$
$883$	30423.2	1.15948	0.579740	+	0.814801i	$0.303154\pi$
$884$	−132.322	−0.00503446
$885$	−2395.32	−0.0909804
$886$	802.889	0.0304442
$887$	−19843.0	−0.751142	−0.375571	−	0.926794i	$-0.622553\pi$
$887$	−19843.0	−0.751142	−0.375571	+	0.926794i	$0.622553\pi$
$888$	−10711.3	−0.404782
$889$	−16224.0	−0.612077
$890$	−1441.53	−0.0542925
$891$	3156.47	0.118682
$892$	−7367.05	−0.276533
$893$	0	0
$894$	18421.7	0.689167
$895$	8928.96	0.333477
$896$	−3347.94	−0.124829
$897$	1439.13	0.0535687
$898$	7781.76	0.289177
$899$	755.247	0.0280188
$900$	−1919.50	−0.0710926
$901$	−2805.86	−0.103748
$902$	11486.9	0.424028
$903$	−6981.80	−0.257298
$904$	−16568.5	−0.609578
$905$	−2297.75	−0.0843975
$906$	−14443.6	−0.529642
$907$	−17397.5	−0.636906	−0.318453	−	0.947939i	$-0.603163\pi$
$907$	−17397.5	−0.636906	−0.318453	+	0.947939i	$0.603163\pi$
$908$	6376.35	0.233047
$909$	−17414.4	−0.635422
$910$	−2415.42	−0.0879895
$911$	−49563.7	−1.80255	−0.901273	−	0.433251i	$-0.857366\pi$
$911$	−49563.7	−1.80255	−0.901273	+	0.433251i	$0.857366\pi$
$912$	0	0
$913$	34943.7	1.26667
$914$	10733.5	0.388437
$915$	19927.9	0.719997
$916$	−16018.1	−0.577788
$917$	52506.8	1.89087
$918$	−327.545	−0.0117763
$919$	1412.66	0.0507066	0.0253533	−	0.999679i	$-0.491929\pi$
$919$	1412.66	0.0507066	0.0253533	+	0.999679i	$0.491929\pi$
$920$	−5957.65	−0.213498
$921$	4837.53	0.173075
$922$	8162.29	0.291552
$923$	−2256.61	−0.0804737
$924$	12231.1	0.435470
$925$	−23796.6	−0.845868
$926$	8254.23	0.292928
$927$	3392.32	0.120192
$928$	−2351.47	−0.0831797
$929$	37457.5	1.32286	0.661431	−	0.750006i	$-0.269949\pi$
$929$	37457.5	1.32286	0.661431	+	0.750006i	$0.269949\pi$
$930$	−522.098	−0.0184089
$931$	0	0
$932$	12617.8	0.443465
$933$	−14800.2	−0.519331
$934$	−21449.1	−0.751432
$935$	−2001.22	−0.0699967
$936$	−392.669	−0.0137124
$937$	−49811.4	−1.73668	−0.868339	−	0.495971i	$-0.834812\pi$
$937$	−49811.4	−1.73668	−0.868339	+	0.495971i	$0.834812\pi$
$938$	5733.73	0.199587
$939$	9482.78	0.329562
$940$	5928.90	0.205723
$941$	−38536.6	−1.33503	−0.667513	−	0.744599i	$-0.732641\pi$
$941$	−38536.6	−1.33503	−0.667513	+	0.744599i	$0.732641\pi$
$942$	−1288.93	−0.0445812
$943$	12964.1	0.447687
$944$	−1508.90	−0.0520239
$945$	−5979.05	−0.205819
$946$	−6934.65	−0.238335
$947$	−595.756	−0.0204430	−0.0102215	−	0.999948i	$-0.503254\pi$
$947$	−595.756	−0.0204430	−0.0102215	+	0.999948i	$0.503254\pi$
$948$	14117.1	0.483653
$949$	−6126.64	−0.209567
$950$	0	0
$951$	−16359.1	−0.557814
$952$	−1269.22	−0.0432096
$953$	17705.7	0.601829	0.300915	−	0.953651i	$-0.402708\pi$
$953$	17705.7	0.601829	0.300915	+	0.953651i	$0.402708\pi$
$954$	−8326.47	−0.282578
$955$	30112.6	1.02033
$956$	852.849	0.0288526
$957$	8590.67	0.290175
$958$	−6496.02	−0.219078
$959$	57362.1	1.93151
$960$	1625.56	0.0546506
$961$	−29685.4	−0.996454
$962$	−4868.03	−0.163151
$963$	−1964.79	−0.0657470
$964$	19107.7	0.638400
$965$	23918.7	0.797896
$966$	13804.0	0.459767
$967$	2454.48	0.0816242	0.0408121	−	0.999167i	$-0.487005\pi$
$967$	2454.48	0.0816242	0.0408121	+	0.999167i	$0.487005\pi$
$968$	1500.53	0.0498231
$969$	0	0
$970$	22944.4	0.759484
$971$	−18419.1	−0.608751	−0.304376	−	0.952552i	$-0.598448\pi$
$971$	−18419.1	−0.608751	−0.304376	+	0.952552i	$0.598448\pi$
$972$	−972.000	−0.0320750
$973$	64131.2	2.11300
$974$	−7554.03	−0.248508
$975$	−872.370	−0.0286546
$976$	12553.4	0.411704
$977$	40154.5	1.31490	0.657449	−	0.753499i	$-0.271635\pi$
$977$	40154.5	1.31490	0.657449	+	0.753499i	$0.271635\pi$
$978$	11561.9	0.378025
$979$	3317.50	0.108302
$980$	−11552.5	−0.376562
$981$	4135.41	0.134591
$982$	−19102.4	−0.620757
$983$	21857.5	0.709204	0.354602	−	0.935017i	$-0.384616\pi$
$983$	21857.5	0.709204	0.354602	+	0.935017i	$0.384616\pi$
$984$	−3537.27	−0.114598
$985$	23187.7	0.750071
$986$	−891.450	−0.0287926
$987$	−13737.3	−0.443024
$988$	0	0
$989$	−7826.41	−0.251633
$990$	−5938.68	−0.190650
$991$	21496.5	0.689062	0.344531	−	0.938775i	$-0.388038\pi$
$991$	21496.5	0.689062	0.344531	+	0.938775i	$0.388038\pi$
$992$	−328.890	−0.0105265
$993$	−17041.2	−0.544600
$994$	−21645.2	−0.690687
$995$	15335.3	0.488605
$996$	−10760.5	−0.342330
$997$	42393.3	1.34665	0.673325	−	0.739346i	$-0.264865\pi$
$997$	42393.3	1.34665	0.673325	+	0.739346i	$0.264865\pi$
$998$	22051.9	0.699439
$999$	−12050.2	−0.381632

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.bi.1.3	yes	8
19.18	odd	2		2166.4.a.bh.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.bh.1.3	✓	8	19.18	odd	2
2166.4.a.bi.1.3	yes	8	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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.46644 q^{5} -6.00000 q^{6} -26.1558 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.46644 q^{5} -6.00000 q^{6} -26.1558 q^{7} +8.00000 q^{8} +9.00000 q^{9} -16.9329 q^{10} +38.9688 q^{11} -12.0000 q^{12} -5.45373 q^{13} -52.3116 q^{14} +25.3993 q^{15} +16.0000 q^{16} +6.06565 q^{17} +18.0000 q^{18} -33.8657 q^{20} +78.4674 q^{21} +77.9376 q^{22} +87.9598 q^{23} -24.0000 q^{24} -53.3194 q^{25} -10.9075 q^{26} -27.0000 q^{27} -104.623 q^{28} -73.4834 q^{29} +50.7986 q^{30} -10.2778 q^{31} +32.0000 q^{32} -116.906 q^{33} +12.1313 q^{34} +221.446 q^{35} +36.0000 q^{36} +446.303 q^{37} +16.3612 q^{39} -67.7315 q^{40} +147.386 q^{41} +156.935 q^{42} -88.9771 q^{43} +155.875 q^{44} -76.1979 q^{45} +175.920 q^{46} -175.071 q^{47} -48.0000 q^{48} +341.126 q^{49} -106.639 q^{50} -18.1970 q^{51} -21.8149 q^{52} -462.582 q^{53} -54.0000 q^{54} -329.927 q^{55} -209.246 q^{56} -146.967 q^{58} -94.3063 q^{59} +101.597 q^{60} +784.585 q^{61} -20.5556 q^{62} -235.402 q^{63} +64.0000 q^{64} +46.1737 q^{65} -233.813 q^{66} -109.607 q^{67} +24.2626 q^{68} -263.880 q^{69} +442.893 q^{70} +413.774 q^{71} +72.0000 q^{72} +1123.39 q^{73} +892.606 q^{74} +159.958 q^{75} -1019.26 q^{77} +32.7224 q^{78} -1176.43 q^{79} -135.463 q^{80} +81.0000 q^{81} +294.773 q^{82} +896.711 q^{83} +313.870 q^{84} -51.3545 q^{85} -177.954 q^{86} +220.450 q^{87} +311.750 q^{88} +85.1322 q^{89} -152.396 q^{90} +142.647 q^{91} +351.839 q^{92} +30.8334 q^{93} -350.141 q^{94} -96.0000 q^{96} -1355.02 q^{97} +682.252 q^{98} +350.719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{2} - 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} + 64 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q + 16 * q^2 - 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 + 64 * q^8 + 72 * q^9	\( 8 q + 16 q^{2} - 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} + 64 q^{8} + 72 q^{9} - 60 q^{10} - 6 q^{11} - 96 q^{12} + 48 q^{13} - 68 q^{14} + 90 q^{15} + 128 q^{16} - 36 q^{17} + 144 q^{18} - 120 q^{20} + 102 q^{21} - 12 q^{22} - 282 q^{23} - 192 q^{24} + 396 q^{25} + 96 q^{26} - 216 q^{27} - 136 q^{28} - 380 q^{29} + 180 q^{30} - 48 q^{31} + 256 q^{32} + 18 q^{33} - 72 q^{34} + 762 q^{35} + 288 q^{36} - 168 q^{37} - 144 q^{39} - 240 q^{40} + 342 q^{41} + 204 q^{42} - 788 q^{43} - 24 q^{44} - 270 q^{45} - 564 q^{46} - 468 q^{47} - 384 q^{48} + 222 q^{49} + 792 q^{50} + 108 q^{51} + 192 q^{52} + 1682 q^{53} - 432 q^{54} - 46 q^{55} - 272 q^{56} - 760 q^{58} + 292 q^{59} + 360 q^{60} + 522 q^{61} - 96 q^{62} - 306 q^{63} + 512 q^{64} - 1120 q^{65} + 36 q^{66} - 2484 q^{67} - 144 q^{68} + 846 q^{69} + 1524 q^{70} + 1182 q^{71} + 576 q^{72} + 182 q^{73} - 336 q^{74} - 1188 q^{75} - 504 q^{77} - 288 q^{78} - 2232 q^{79} - 480 q^{80} + 648 q^{81} + 684 q^{82} - 750 q^{83} + 408 q^{84} - 2238 q^{85} - 1576 q^{86} + 1140 q^{87} - 48 q^{88} + 1304 q^{89} - 540 q^{90} - 624 q^{91} - 1128 q^{92} + 144 q^{93} - 936 q^{94} - 768 q^{96} - 1248 q^{97} + 444 q^{98} - 54 q^{99}+O(q^{100}) \) 8 * q + 16 * q^2 - 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 + 64 * q^8 + 72 * q^9 - 60 * q^10 - 6 * q^11 - 96 * q^12 + 48 * q^13 - 68 * q^14 + 90 * q^15 + 128 * q^16 - 36 * q^17 + 144 * q^18 - 120 * q^20 + 102 * q^21 - 12 * q^22 - 282 * q^23 - 192 * q^24 + 396 * q^25 + 96 * q^26 - 216 * q^27 - 136 * q^28 - 380 * q^29 + 180 * q^30 - 48 * q^31 + 256 * q^32 + 18 * q^33 - 72 * q^34 + 762 * q^35 + 288 * q^36 - 168 * q^37 - 144 * q^39 - 240 * q^40 + 342 * q^41 + 204 * q^42 - 788 * q^43 - 24 * q^44 - 270 * q^45 - 564 * q^46 - 468 * q^47 - 384 * q^48 + 222 * q^49 + 792 * q^50 + 108 * q^51 + 192 * q^52 + 1682 * q^53 - 432 * q^54 - 46 * q^55 - 272 * q^56 - 760 * q^58 + 292 * q^59 + 360 * q^60 + 522 * q^61 - 96 * q^62 - 306 * q^63 + 512 * q^64 - 1120 * q^65 + 36 * q^66 - 2484 * q^67 - 144 * q^68 + 846 * q^69 + 1524 * q^70 + 1182 * q^71 + 576 * q^72 + 182 * q^73 - 336 * q^74 - 1188 * q^75 - 504 * q^77 - 288 * q^78 - 2232 * q^79 - 480 * q^80 + 648 * q^81 + 684 * q^82 - 750 * q^83 + 408 * q^84 - 2238 * q^85 - 1576 * q^86 + 1140 * q^87 - 48 * q^88 + 1304 * q^89 - 540 * q^90 - 624 * q^91 - 1128 * q^92 + 144 * q^93 - 936 * q^94 - 768 * q^96 - 1248 * q^97 + 444 * q^98 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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