LMFDB - Embedded newform 402.4.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [402,4,Mod(1,402)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(402, 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("402.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 402 = 2 \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	402.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:	$23.7187678223$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	402.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -7.58579 q^{5} +6.00000 q^{6} -7.34315 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -7.58579 q^{5} +6.00000 q^{6} -7.34315 q^{7} +8.00000 q^{8} +9.00000 q^{9} -15.1716 q^{10} -49.3137 q^{11} +12.0000 q^{12} -85.8823 q^{13} -14.6863 q^{14} -22.7574 q^{15} +16.0000 q^{16} +38.2254 q^{17} +18.0000 q^{18} -19.5147 q^{19} -30.3431 q^{20} -22.0294 q^{21} -98.6274 q^{22} -110.497 q^{23} +24.0000 q^{24} -67.4558 q^{25} -171.765 q^{26} +27.0000 q^{27} -29.3726 q^{28} +48.2498 q^{29} -45.5147 q^{30} +245.357 q^{31} +32.0000 q^{32} -147.941 q^{33} +76.4508 q^{34} +55.7035 q^{35} +36.0000 q^{36} +23.2843 q^{37} -39.0294 q^{38} -257.647 q^{39} -60.6863 q^{40} -129.762 q^{41} -44.0589 q^{42} -4.13708 q^{43} -197.255 q^{44} -68.2721 q^{45} -220.995 q^{46} -362.098 q^{47} +48.0000 q^{48} -289.078 q^{49} -134.912 q^{50} +114.676 q^{51} -343.529 q^{52} -87.2914 q^{53} +54.0000 q^{54} +374.083 q^{55} -58.7452 q^{56} -58.5442 q^{57} +96.4996 q^{58} -122.095 q^{59} -91.0294 q^{60} +432.804 q^{61} +490.715 q^{62} -66.0883 q^{63} +64.0000 q^{64} +651.484 q^{65} -295.882 q^{66} +67.0000 q^{67} +152.902 q^{68} -331.492 q^{69} +111.407 q^{70} +459.637 q^{71} +72.0000 q^{72} +245.128 q^{73} +46.5685 q^{74} -202.368 q^{75} -78.0589 q^{76} +362.118 q^{77} -515.294 q^{78} +509.019 q^{79} -121.373 q^{80} +81.0000 q^{81} -259.525 q^{82} -1435.72 q^{83} -88.1177 q^{84} -289.970 q^{85} -8.27417 q^{86} +144.749 q^{87} -394.510 q^{88} -130.686 q^{89} -136.544 q^{90} +630.646 q^{91} -441.990 q^{92} +736.072 q^{93} -724.195 q^{94} +148.034 q^{95} +96.0000 q^{96} -120.045 q^{97} -578.156 q^{98} -443.823 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 18 q^{5} + 12 q^{6} - 26 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 18 * q^5 + 12 * q^6 - 26 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 18 q^{5} + 12 q^{6} - 26 q^{7} + 16 q^{8} + 18 q^{9} - 36 q^{10} - 76 q^{11} + 24 q^{12} - 36 q^{13} - 52 q^{14} - 54 q^{15} + 32 q^{16} - 48 q^{17} + 36 q^{18} - 56 q^{19} - 72 q^{20} - 78 q^{21} - 152 q^{22} - 122 q^{23} + 48 q^{24} - 84 q^{25} - 72 q^{26} + 54 q^{27} - 104 q^{28} - 192 q^{29} - 108 q^{30} - 58 q^{31} + 64 q^{32} - 228 q^{33} - 96 q^{34} + 250 q^{35} + 72 q^{36} - 10 q^{37} - 112 q^{38} - 108 q^{39} - 144 q^{40} - 466 q^{41} - 156 q^{42} + 218 q^{43} - 304 q^{44} - 162 q^{45} - 244 q^{46} - 68 q^{47} + 96 q^{48} - 284 q^{49} - 168 q^{50} - 144 q^{51} - 144 q^{52} + 162 q^{53} + 108 q^{54} + 652 q^{55} - 208 q^{56} - 168 q^{57} - 384 q^{58} - 66 q^{59} - 216 q^{60} + 1024 q^{61} - 116 q^{62} - 234 q^{63} + 128 q^{64} + 132 q^{65} - 456 q^{66} + 134 q^{67} - 192 q^{68} - 366 q^{69} + 500 q^{70} + 116 q^{71} + 144 q^{72} + 1022 q^{73} - 20 q^{74} - 252 q^{75} - 224 q^{76} + 860 q^{77} - 216 q^{78} + 656 q^{79} - 288 q^{80} + 162 q^{81} - 932 q^{82} - 142 q^{83} - 312 q^{84} + 608 q^{85} + 436 q^{86} - 576 q^{87} - 608 q^{88} - 284 q^{89} - 324 q^{90} - 300 q^{91} - 488 q^{92} - 174 q^{93} - 136 q^{94} + 528 q^{95} + 192 q^{96} - 868 q^{97} - 568 q^{98} - 684 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 18 * q^5 + 12 * q^6 - 26 * q^7 + 16 * q^8 + 18 * q^9 - 36 * q^10 - 76 * q^11 + 24 * q^12 - 36 * q^13 - 52 * q^14 - 54 * q^15 + 32 * q^16 - 48 * q^17 + 36 * q^18 - 56 * q^19 - 72 * q^20 - 78 * q^21 - 152 * q^22 - 122 * q^23 + 48 * q^24 - 84 * q^25 - 72 * q^26 + 54 * q^27 - 104 * q^28 - 192 * q^29 - 108 * q^30 - 58 * q^31 + 64 * q^32 - 228 * q^33 - 96 * q^34 + 250 * q^35 + 72 * q^36 - 10 * q^37 - 112 * q^38 - 108 * q^39 - 144 * q^40 - 466 * q^41 - 156 * q^42 + 218 * q^43 - 304 * q^44 - 162 * q^45 - 244 * q^46 - 68 * q^47 + 96 * q^48 - 284 * q^49 - 168 * q^50 - 144 * q^51 - 144 * q^52 + 162 * q^53 + 108 * q^54 + 652 * q^55 - 208 * q^56 - 168 * q^57 - 384 * q^58 - 66 * q^59 - 216 * q^60 + 1024 * q^61 - 116 * q^62 - 234 * q^63 + 128 * q^64 + 132 * q^65 - 456 * q^66 + 134 * q^67 - 192 * q^68 - 366 * q^69 + 500 * q^70 + 116 * q^71 + 144 * q^72 + 1022 * q^73 - 20 * q^74 - 252 * q^75 - 224 * q^76 + 860 * q^77 - 216 * q^78 + 656 * q^79 - 288 * q^80 + 162 * q^81 - 932 * q^82 - 142 * q^83 - 312 * q^84 + 608 * q^85 + 436 * q^86 - 576 * q^87 - 608 * q^88 - 284 * q^89 - 324 * q^90 - 300 * q^91 - 488 * q^92 - 174 * q^93 - 136 * q^94 + 528 * q^95 + 192 * q^96 - 868 * q^97 - 568 * q^98 - 684 * 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
$5$	−7.58579	−0.678493	−0.339247	−	0.940697i	$-0.610172\pi$
$5$	−7.58579	−0.678493	−0.339247	+	0.940697i	$0.610172\pi$
$6$	6.00000	0.408248
$7$	−7.34315	−0.396493	−0.198246	−	0.980152i	$-0.563525\pi$
$7$	−7.34315	−0.396493	−0.198246	+	0.980152i	$0.563525\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−15.1716	−0.479767
$11$	−49.3137	−1.35169	−0.675847	−	0.737042i	$-0.736222\pi$
$11$	−49.3137	−1.35169	−0.675847	+	0.737042i	$0.736222\pi$
$12$	12.0000	0.288675
$13$	−85.8823	−1.83227	−0.916133	−	0.400875i	$-0.868706\pi$
$13$	−85.8823	−1.83227	−0.916133	+	0.400875i	$0.868706\pi$
$14$	−14.6863	−0.280363
$15$	−22.7574	−0.391728
$16$	16.0000	0.250000
$17$	38.2254	0.545354	0.272677	−	0.962106i	$-0.412091\pi$
$17$	38.2254	0.545354	0.272677	+	0.962106i	$0.412091\pi$
$18$	18.0000	0.235702
$19$	−19.5147	−0.235631	−0.117815	−	0.993036i	$-0.537589\pi$
$19$	−19.5147	−0.235631	−0.117815	+	0.993036i	$0.537589\pi$
$20$	−30.3431	−0.339247
$21$	−22.0294	−0.228915
$22$	−98.6274	−0.955793
$23$	−110.497	−1.00175	−0.500876	−	0.865519i	$-0.666989\pi$
$23$	−110.497	−1.00175	−0.500876	+	0.865519i	$0.666989\pi$
$24$	24.0000	0.204124
$25$	−67.4558	−0.539647
$26$	−171.765	−1.29561
$27$	27.0000	0.192450
$28$	−29.3726	−0.198246
$29$	48.2498	0.308957	0.154479	−	0.987996i	$-0.450630\pi$
$29$	48.2498	0.308957	0.154479	+	0.987996i	$0.450630\pi$
$30$	−45.5147	−0.276994
$31$	245.357	1.42153	0.710766	−	0.703428i	$-0.248348\pi$
$31$	245.357	1.42153	0.710766	+	0.703428i	$0.248348\pi$
$32$	32.0000	0.176777
$33$	−147.941	−0.780401
$34$	76.4508	0.385624
$35$	55.7035	0.269018
$36$	36.0000	0.166667
$37$	23.2843	0.103457	0.0517285	−	0.998661i	$-0.483527\pi$
$37$	23.2843	0.103457	0.0517285	+	0.998661i	$0.483527\pi$
$38$	−39.0294	−0.166616
$39$	−257.647	−1.05786
$40$	−60.6863	−0.239884
$41$	−129.762	−0.494280	−0.247140	−	0.968980i	$-0.579491\pi$
$41$	−129.762	−0.494280	−0.247140	+	0.968980i	$0.579491\pi$
$42$	−44.0589	−0.161867
$43$	−4.13708	−0.0146721	−0.00733604	−	0.999973i	$-0.502335\pi$
$43$	−4.13708	−0.0146721	−0.00733604	+	0.999973i	$0.502335\pi$
$44$	−197.255	−0.675847
$45$	−68.2721	−0.226164
$46$	−220.995	−0.708346
$47$	−362.098	−1.12377	−0.561887	−	0.827214i	$-0.689924\pi$
$47$	−362.098	−1.12377	−0.561887	+	0.827214i	$0.689924\pi$
$48$	48.0000	0.144338
$49$	−289.078	−0.842794
$50$	−134.912	−0.381588
$51$	114.676	0.314860
$52$	−343.529	−0.916133
$53$	−87.2914	−0.226234	−0.113117	−	0.993582i	$-0.536083\pi$
$53$	−87.2914	−0.226234	−0.113117	+	0.993582i	$0.536083\pi$
$54$	54.0000	0.136083
$55$	374.083	0.917116
$56$	−58.7452	−0.140181
$57$	−58.5442	−0.136041
$58$	96.4996	0.218466
$59$	−122.095	−0.269415	−0.134707	−	0.990885i	$-0.543009\pi$
$59$	−122.095	−0.269415	−0.134707	+	0.990885i	$0.543009\pi$
$60$	−91.0294	−0.195864
$61$	432.804	0.908441	0.454220	−	0.890889i	$-0.349918\pi$
$61$	432.804	0.908441	0.454220	+	0.890889i	$0.349918\pi$
$62$	490.715	1.00517
$63$	−66.0883	−0.132164
$64$	64.0000	0.125000
$65$	651.484	1.24318
$66$	−295.882	−0.551827
$67$	67.0000	0.122169
$67$	67.0000	0.122169
$68$	152.902	0.272677
$69$	−331.492	−0.578362
$70$	111.407	0.190224
$71$	459.637	0.768293	0.384147	−	0.923272i	$-0.374496\pi$
$71$	459.637	0.768293	0.384147	+	0.923272i	$0.374496\pi$
$72$	72.0000	0.117851
$73$	245.128	0.393014	0.196507	−	0.980502i	$-0.437040\pi$
$73$	245.128	0.393014	0.196507	+	0.980502i	$0.437040\pi$
$74$	46.5685	0.0731552
$75$	−202.368	−0.311565
$76$	−78.0589	−0.117815
$77$	362.118	0.535937
$78$	−515.294	−0.748019
$79$	509.019	0.724926	0.362463	−	0.931998i	$-0.381936\pi$
$79$	509.019	0.724926	0.362463	+	0.931998i	$0.381936\pi$
$80$	−121.373	−0.169623
$81$	81.0000	0.111111
$82$	−259.525	−0.349509
$83$	−1435.72	−1.89868	−0.949339	−	0.314253i	$-0.898246\pi$
$83$	−1435.72	−1.89868	−0.949339	+	0.314253i	$0.898246\pi$
$84$	−88.1177	−0.114458
$85$	−289.970	−0.370019
$86$	−8.27417	−0.0103747
$87$	144.749	0.178377
$88$	−394.510	−0.477896
$89$	−130.686	−0.155649	−0.0778243	−	0.996967i	$-0.524797\pi$
$89$	−130.686	−0.155649	−0.0778243	+	0.996967i	$0.524797\pi$
$90$	−136.544	−0.159922
$91$	630.646	0.726480
$92$	−441.990	−0.500876
$93$	736.072	0.820722
$94$	−724.195	−0.794628
$95$	148.034	0.159874
$96$	96.0000	0.102062
$97$	−120.045	−0.125657	−0.0628283	−	0.998024i	$-0.520012\pi$
$97$	−120.045	−0.125657	−0.0628283	+	0.998024i	$0.520012\pi$
$98$	−578.156	−0.595945
$99$	−443.823	−0.450565
$100$	−269.823	−0.269823
$101$	804.827	0.792903	0.396452	−	0.918056i	$-0.370241\pi$
$101$	804.827	0.792903	0.396452	+	0.918056i	$0.370241\pi$
$102$	229.352	0.222640
$103$	−290.014	−0.277436	−0.138718	−	0.990332i	$-0.544298\pi$
$103$	−290.014	−0.277436	−0.138718	+	0.990332i	$0.544298\pi$
$104$	−687.058	−0.647804
$105$	167.111	0.155317
$106$	−174.583	−0.159972
$107$	456.518	0.412460	0.206230	−	0.978504i	$-0.433880\pi$
$107$	456.518	0.412460	0.206230	+	0.978504i	$0.433880\pi$
$108$	108.000	0.0962250
$109$	−410.926	−0.361097	−0.180548	−	0.983566i	$-0.557787\pi$
$109$	−410.926	−0.361097	−0.180548	+	0.983566i	$0.557787\pi$
$110$	748.167	0.648499
$111$	69.8528	0.0597310
$112$	−117.490	−0.0991232
$113$	−819.856	−0.682527	−0.341264	−	0.939968i	$-0.610855\pi$
$113$	−819.856	−0.682527	−0.341264	+	0.939968i	$0.610855\pi$
$114$	−117.088	−0.0961958
$115$	838.210	0.679683
$116$	192.999	0.154479
$117$	−772.940	−0.610755
$118$	−244.191	−0.190505
$119$	−280.695	−0.216229
$120$	−182.059	−0.138497
$121$	1100.84	0.827079
$122$	865.608	0.642365
$123$	−389.287	−0.285373
$124$	981.430	0.710766
$125$	1459.93	1.04464
$126$	−132.177	−0.0934542
$127$	−657.765	−0.459584	−0.229792	−	0.973240i	$-0.573805\pi$
$127$	−657.765	−0.459584	−0.229792	+	0.973240i	$0.573805\pi$
$128$	128.000	0.0883883
$129$	−12.4113	−0.00847093
$130$	1302.97	0.879061
$131$	−2103.42	−1.40287	−0.701437	−	0.712732i	$-0.747458\pi$
$131$	−2103.42	−1.40287	−0.701437	+	0.712732i	$0.747458\pi$
$132$	−591.765	−0.390201
$133$	143.299	0.0934258
$134$	134.000	0.0863868
$135$	−204.816	−0.130576
$136$	305.803	0.192812
$137$	944.471	0.588990	0.294495	−	0.955653i	$-0.404849\pi$
$137$	944.471	0.588990	0.294495	+	0.955653i	$0.404849\pi$
$138$	−662.985	−0.408964
$139$	−1055.58	−0.644125	−0.322062	−	0.946718i	$-0.604376\pi$
$139$	−1055.58	−0.644125	−0.322062	+	0.946718i	$0.604376\pi$
$140$	222.814	0.134509
$141$	−1086.29	−0.648811
$142$	919.273	0.543266
$143$	4235.17	2.47666
$144$	144.000	0.0833333
$145$	−366.013	−0.209625
$146$	490.256	0.277903
$147$	−867.235	−0.486587
$148$	93.1371	0.0517285
$149$	−1715.84	−0.943401	−0.471701	−	0.881759i	$-0.656360\pi$
$149$	−1715.84	−0.943401	−0.471701	+	0.881759i	$0.656360\pi$
$150$	−404.735	−0.220310
$151$	2545.65	1.37193	0.685966	−	0.727634i	$-0.259380\pi$
$151$	2545.65	1.37193	0.685966	+	0.727634i	$0.259380\pi$
$152$	−156.118	−0.0833080
$153$	344.029	0.181785
$154$	724.235	0.378965
$155$	−1861.23	−0.964500
$156$	−1030.59	−0.528929
$157$	−1897.10	−0.964363	−0.482182	−	0.876071i	$-0.660156\pi$
$157$	−1897.10	−0.964363	−0.482182	+	0.876071i	$0.660156\pi$
$158$	1018.04	0.512600
$159$	−261.874	−0.130616
$160$	−242.745	−0.119942
$161$	811.399	0.397188
$162$	162.000	0.0785674
$163$	2201.01	1.05764	0.528822	−	0.848733i	$-0.322634\pi$
$163$	2201.01	1.05764	0.528822	+	0.848733i	$0.322634\pi$
$164$	−519.050	−0.247140
$165$	1122.25	0.529497
$166$	−2871.43	−1.34257
$167$	3423.45	1.58632	0.793158	−	0.609016i	$-0.208436\pi$
$167$	3423.45	1.58632	0.793158	+	0.609016i	$0.208436\pi$
$168$	−176.235	−0.0809337
$169$	5178.76	2.35720
$170$	−579.939	−0.261643
$171$	−175.632	−0.0785436
$172$	−16.5483	−0.00733604
$173$	−56.9075	−0.0250092	−0.0125046	−	0.999922i	$-0.503980\pi$
$173$	−56.9075	−0.0250092	−0.0125046	+	0.999922i	$0.503980\pi$
$174$	289.499	0.126131
$175$	495.338	0.213966
$176$	−789.019	−0.337924
$177$	−366.286	−0.155547
$178$	−261.373	−0.110060
$179$	−2929.89	−1.22341	−0.611705	−	0.791086i	$-0.709516\pi$
$179$	−2929.89	−1.22341	−0.611705	+	0.791086i	$0.709516\pi$
$180$	−273.088	−0.113082
$181$	3437.32	1.41157	0.705785	−	0.708426i	$-0.250595\pi$
$181$	3437.32	1.41157	0.705785	+	0.708426i	$0.250595\pi$
$182$	1261.29	0.513699
$183$	1298.41	0.524488
$184$	−883.980	−0.354173
$185$	−176.630	−0.0701949
$186$	1472.14	0.580338
$187$	−1885.04	−0.737152
$188$	−1448.39	−0.561887
$189$	−198.265	−0.0763050
$190$	296.069	0.113048
$191$	−169.996	−0.0644003	−0.0322002	−	0.999481i	$-0.510251\pi$
$191$	−169.996	−0.0644003	−0.0322002	+	0.999481i	$0.510251\pi$
$192$	192.000	0.0721688
$193$	1707.54	0.636846	0.318423	−	0.947949i	$-0.396847\pi$
$193$	1707.54	0.636846	0.318423	+	0.947949i	$0.396847\pi$
$194$	−240.089	−0.0888526
$195$	1954.45	0.717750
$196$	−1156.31	−0.421397
$197$	−1309.06	−0.473436	−0.236718	−	0.971578i	$-0.576072\pi$
$197$	−1309.06	−0.473436	−0.236718	+	0.971578i	$0.576072\pi$
$198$	−887.647	−0.318598
$199$	2704.01	0.963229	0.481614	−	0.876383i	$-0.340051\pi$
$199$	2704.01	0.963229	0.481614	+	0.876383i	$0.340051\pi$
$200$	−539.647	−0.190794
$201$	201.000	0.0705346
$202$	1609.65	0.560667
$203$	−354.305	−0.122499
$204$	458.705	0.157430
$205$	984.350	0.335366
$206$	−580.029	−0.196177
$207$	−994.477	−0.333918
$208$	−1374.12	−0.458066
$209$	962.343	0.318501
$210$	334.221	0.109826
$211$	−912.630	−0.297763	−0.148882	−	0.988855i	$-0.547567\pi$
$211$	−912.630	−0.297763	−0.148882	+	0.988855i	$0.547567\pi$
$212$	−349.166	−0.113117
$213$	1378.91	0.443574
$214$	913.036	0.291654
$215$	31.3830	0.00995491
$216$	216.000	0.0680414
$217$	−1801.70	−0.563627
$218$	−821.852	−0.255334
$219$	735.384	0.226907
$220$	1496.33	0.458558
$221$	−3282.88	−0.999234
$222$	139.706	0.0422362
$223$	979.086	0.294011	0.147005	−	0.989136i	$-0.453037\pi$
$223$	979.086	0.294011	0.147005	+	0.989136i	$0.453037\pi$
$224$	−234.981	−0.0700907
$225$	−607.103	−0.179882
$226$	−1639.71	−0.482620
$227$	−1021.83	−0.298771	−0.149386	−	0.988779i	$-0.547730\pi$
$227$	−1021.83	−0.298771	−0.149386	+	0.988779i	$0.547730\pi$
$228$	−234.177	−0.0680207
$229$	6305.79	1.81964	0.909822	−	0.414999i	$-0.136218\pi$
$229$	6305.79	1.81964	0.909822	+	0.414999i	$0.136218\pi$
$230$	1676.42	0.480608
$231$	1086.35	0.309423
$232$	385.998	0.109233
$233$	−4758.43	−1.33792	−0.668960	−	0.743299i	$-0.733260\pi$
$233$	−4758.43	−1.33792	−0.668960	+	0.743299i	$0.733260\pi$
$234$	−1545.88	−0.431869
$235$	2746.79	0.762473
$236$	−488.382	−0.134707
$237$	1527.06	0.418536
$238$	−561.389	−0.152897
$239$	−3542.42	−0.958744	−0.479372	−	0.877612i	$-0.659136\pi$
$239$	−3542.42	−0.958744	−0.479372	+	0.877612i	$0.659136\pi$
$240$	−364.118	−0.0979321
$241$	2926.20	0.782129	0.391065	−	0.920363i	$-0.372107\pi$
$241$	2926.20	0.782129	0.391065	+	0.920363i	$0.372107\pi$
$242$	2201.68	0.584833
$243$	243.000	0.0641500
$244$	1731.22	0.454220
$245$	2192.89	0.571830
$246$	−778.574	−0.201789
$247$	1675.97	0.431738
$248$	1962.86	0.502587
$249$	−4307.15	−1.09620
$250$	2919.86	0.738672
$251$	−3267.88	−0.821779	−0.410889	−	0.911685i	$-0.634782\pi$
$251$	−3267.88	−0.821779	−0.410889	+	0.911685i	$0.634782\pi$
$252$	−264.353	−0.0660821
$253$	5449.04	1.35406
$254$	−1315.53	−0.324975
$255$	−869.909	−0.213631
$256$	256.000	0.0625000
$257$	−3290.22	−0.798593	−0.399297	−	0.916822i	$-0.630746\pi$
$257$	−3290.22	−0.798593	−0.399297	+	0.916822i	$0.630746\pi$
$258$	−24.8225	−0.00598985
$259$	−170.980	−0.0410200
$260$	2605.94	0.621590
$261$	434.248	0.102986
$262$	−4206.84	−0.991981
$263$	−1573.04	−0.368813	−0.184407	−	0.982850i	$-0.559036\pi$
$263$	−1573.04	−0.368813	−0.184407	+	0.982850i	$0.559036\pi$
$264$	−1183.53	−0.275914
$265$	662.174	0.153498
$266$	286.599	0.0660620
$267$	−392.059	−0.0898637
$268$	268.000	0.0610847
$269$	−6568.74	−1.48886	−0.744430	−	0.667701i	$-0.767279\pi$
$269$	−6568.74	−1.48886	−0.744430	+	0.667701i	$0.767279\pi$
$270$	−409.632	−0.0923313
$271$	623.524	0.139765	0.0698826	−	0.997555i	$-0.477738\pi$
$271$	623.524	0.139765	0.0698826	+	0.997555i	$0.477738\pi$
$272$	611.606	0.136339
$273$	1891.94	0.419433
$274$	1888.94	0.416479
$275$	3326.50	0.729438
$276$	−1325.97	−0.289181
$277$	−7358.89	−1.59622	−0.798110	−	0.602512i	$-0.794167\pi$
$277$	−7358.89	−1.59622	−0.798110	+	0.602512i	$0.794167\pi$
$278$	−2111.17	−0.455465
$279$	2208.22	0.473844
$280$	445.628	0.0951121
$281$	−4795.86	−1.01814	−0.509070	−	0.860725i	$-0.670011\pi$
$281$	−4795.86	−1.01814	−0.509070	+	0.860725i	$0.670011\pi$
$282$	−2172.59	−0.458779
$283$	−1406.81	−0.295499	−0.147749	−	0.989025i	$-0.547203\pi$
$283$	−1406.81	−0.295499	−0.147749	+	0.989025i	$0.547203\pi$
$284$	1838.55	0.384147
$285$	444.103	0.0923032
$286$	8470.34	1.75127
$287$	952.864	0.195978
$288$	288.000	0.0589256
$289$	−3451.82	−0.702589
$290$	−732.025	−0.148228
$291$	−360.134	−0.0725478
$292$	980.511	0.196507
$293$	−5850.21	−1.16646	−0.583231	−	0.812307i	$-0.698212\pi$
$293$	−5850.21	−1.16646	−0.583231	+	0.812307i	$0.698212\pi$
$294$	−1734.47	−0.344069
$295$	926.190	0.182796
$296$	186.274	0.0365776
$297$	−1331.47	−0.260134
$298$	−3431.67	−0.667085
$299$	9489.77	1.83548
$300$	−809.470	−0.155783
$301$	30.3792	0.00581737
$302$	5091.29	0.970102
$303$	2414.48	0.457783
$304$	−312.235	−0.0589077
$305$	−3283.16	−0.616371
$306$	688.057	0.128541
$307$	7193.11	1.33724	0.668619	−	0.743605i	$-0.266886\pi$
$307$	7193.11	1.33724	0.668619	+	0.743605i	$0.266886\pi$
$308$	1448.47	0.267968
$309$	−870.043	−0.160178
$310$	−3722.46	−0.682005
$311$	6744.21	1.22968	0.614838	−	0.788653i	$-0.289221\pi$
$311$	6744.21	1.22968	0.614838	+	0.788653i	$0.289221\pi$
$312$	−2061.17	−0.374010
$313$	−7456.91	−1.34661	−0.673306	−	0.739364i	$-0.735126\pi$
$313$	−7456.91	−1.34661	−0.673306	+	0.739364i	$0.735126\pi$
$314$	−3794.20	−0.681908
$315$	501.332	0.0896725
$316$	2036.08	0.362463
$317$	−7782.36	−1.37887	−0.689433	−	0.724349i	$-0.742140\pi$
$317$	−7782.36	−1.37887	−0.689433	+	0.724349i	$0.742140\pi$
$318$	−523.748	−0.0923596
$319$	−2379.38	−0.417616
$320$	−485.490	−0.0848117
$321$	1369.55	0.238134
$322$	1622.80	0.280854
$323$	−745.958	−0.128502
$324$	324.000	0.0555556
$325$	5793.26	0.988776
$326$	4402.01	0.747867
$327$	−1232.78	−0.208479
$328$	−1038.10	−0.174754
$329$	2658.94	0.445568
$330$	2244.50	0.374411
$331$	−7079.55	−1.17561	−0.587805	−	0.809002i	$-0.700008\pi$
$331$	−7079.55	−1.17561	−0.587805	+	0.809002i	$0.700008\pi$
$332$	−5742.86	−0.949339
$333$	209.558	0.0344857
$334$	6846.91	1.12169
$335$	−508.248	−0.0828912
$336$	−352.471	−0.0572288
$337$	−1259.65	−0.203613	−0.101806	−	0.994804i	$-0.532462\pi$
$337$	−1259.65	−0.203613	−0.101806	+	0.994804i	$0.532462\pi$
$338$	10357.5	1.66679
$339$	−2459.57	−0.394057
$340$	−1159.88	−0.185010
$341$	−12099.5	−1.92148
$342$	−351.265	−0.0555387
$343$	4641.44	0.730654
$344$	−33.0967	−0.00518736
$345$	2514.63	0.392415
$346$	−113.815	−0.0176842
$347$	5571.15	0.861888	0.430944	−	0.902379i	$-0.358181\pi$
$347$	5571.15	0.861888	0.430944	+	0.902379i	$0.358181\pi$
$348$	578.997	0.0891883
$349$	−5949.15	−0.912467	−0.456233	−	0.889860i	$-0.650802\pi$
$349$	−5949.15	−0.912467	−0.456233	+	0.889860i	$0.650802\pi$
$350$	990.676	0.151297
$351$	−2318.82	−0.352620
$352$	−1578.04	−0.238948
$353$	6709.95	1.01171	0.505856	−	0.862618i	$-0.331177\pi$
$353$	6709.95	1.01171	0.505856	+	0.862618i	$0.331177\pi$
$354$	−732.573	−0.109988
$355$	−3486.71	−0.521282
$356$	−522.745	−0.0778243
$357$	−842.084	−0.124840
$358$	−5859.78	−0.865082
$359$	4616.63	0.678709	0.339355	−	0.940659i	$-0.389791\pi$
$359$	4616.63	0.678709	0.339355	+	0.940659i	$0.389791\pi$
$360$	−546.177	−0.0799612
$361$	−6478.18	−0.944478
$362$	6874.64	0.998130
$363$	3302.53	0.477514
$364$	2522.58	0.363240
$365$	−1859.49	−0.266658
$366$	2596.82	0.370869
$367$	−12424.2	−1.76713	−0.883565	−	0.468308i	$-0.844864\pi$
$367$	−12424.2	−1.76713	−0.883565	+	0.468308i	$0.844864\pi$
$368$	−1767.96	−0.250438
$369$	−1167.86	−0.164760
$370$	−353.259	−0.0496353
$371$	640.994	0.0897001
$372$	2944.29	0.410361
$373$	2319.28	0.321952	0.160976	−	0.986958i	$-0.448536\pi$
$373$	2319.28	0.321952	0.160976	+	0.986958i	$0.448536\pi$
$374$	−3770.07	−0.521245
$375$	4379.79	0.603123
$376$	−2896.78	−0.397314
$377$	−4143.80	−0.566092
$378$	−396.530	−0.0539558
$379$	5341.33	0.723920	0.361960	−	0.932194i	$-0.382108\pi$
$379$	5341.33	0.723920	0.361960	+	0.932194i	$0.382108\pi$
$380$	592.138	0.0799369
$381$	−1973.29	−0.265341
$382$	−339.992	−0.0455379
$383$	−8463.73	−1.12918	−0.564591	−	0.825371i	$-0.690966\pi$
$383$	−8463.73	−1.12918	−0.564591	+	0.825371i	$0.690966\pi$
$384$	384.000	0.0510310
$385$	−2746.95	−0.363630
$386$	3415.08	0.450318
$387$	−37.2338	−0.00489069
$388$	−480.178	−0.0628283
$389$	−4747.20	−0.618747	−0.309374	−	0.950941i	$-0.600119\pi$
$389$	−4747.20	−0.618747	−0.309374	+	0.950941i	$0.600119\pi$
$390$	3908.91	0.507526
$391$	−4223.81	−0.546310
$392$	−2312.63	−0.297973
$393$	−6310.25	−0.809949
$394$	−2618.12	−0.334769
$395$	−3861.31	−0.491857
$396$	−1775.29	−0.225282
$397$	−12501.0	−1.58037	−0.790187	−	0.612866i	$-0.790016\pi$
$397$	−12501.0	−1.58037	−0.790187	+	0.612866i	$0.790016\pi$
$398$	5408.03	0.681105
$399$	429.898	0.0539394
$400$	−1079.29	−0.134912
$401$	−12820.6	−1.59658	−0.798292	−	0.602270i	$-0.794263\pi$
$401$	−12820.6	−1.59658	−0.798292	+	0.602270i	$0.794263\pi$
$402$	402.000	0.0498755
$403$	−21071.8	−2.60462
$404$	3219.31	0.396452
$405$	−614.449	−0.0753882
$406$	−708.610	−0.0866201
$407$	−1148.23	−0.139842
$408$	917.410	0.111320
$409$	−5571.01	−0.673518	−0.336759	−	0.941591i	$-0.609331\pi$
$409$	−5571.01	−0.673518	−0.336759	+	0.941591i	$0.609331\pi$
$410$	1968.70	0.237139
$411$	2833.41	0.340054
$412$	−1160.06	−0.138718
$413$	896.565	0.106821
$414$	−1988.95	−0.236115
$415$	10891.0	1.28824
$416$	−2748.23	−0.323902
$417$	−3166.75	−0.371886
$418$	1924.69	0.225214
$419$	−3557.91	−0.414834	−0.207417	−	0.978253i	$-0.566506\pi$
$419$	−3557.91	−0.414834	−0.207417	+	0.978253i	$0.566506\pi$
$420$	668.442	0.0776587
$421$	−515.408	−0.0596662	−0.0298331	−	0.999555i	$-0.509498\pi$
$421$	−515.408	−0.0596662	−0.0298331	+	0.999555i	$0.509498\pi$
$422$	−1825.26	−0.210550
$423$	−3258.88	−0.374591
$424$	−698.331	−0.0799858
$425$	−2578.53	−0.294299
$426$	2757.82	0.313654
$427$	−3178.14	−0.360190
$428$	1826.07	0.206230
$429$	12705.5	1.42990
$430$	62.7661	0.00703918
$431$	−8536.99	−0.954089	−0.477044	−	0.878879i	$-0.658292\pi$
$431$	−8536.99	−0.954089	−0.477044	+	0.878879i	$0.658292\pi$
$432$	432.000	0.0481125
$433$	11219.4	1.24520	0.622598	−	0.782542i	$-0.286077\pi$
$433$	11219.4	1.24520	0.622598	+	0.782542i	$0.286077\pi$
$434$	−3603.39	−0.398544
$435$	−1098.04	−0.121027
$436$	−1643.70	−0.180548
$437$	2156.33	0.236044
$438$	1470.77	0.160447
$439$	12665.0	1.37692	0.688462	−	0.725273i	$-0.258286\pi$
$439$	12665.0	1.37692	0.688462	+	0.725273i	$0.258286\pi$
$440$	2992.67	0.324249
$441$	−2601.70	−0.280931
$442$	−6565.77	−0.706565
$443$	10382.6	1.11352	0.556762	−	0.830672i	$-0.312044\pi$
$443$	10382.6	1.11352	0.556762	+	0.830672i	$0.312044\pi$
$444$	279.411	0.0298655
$445$	991.358	0.105606
$446$	1958.17	0.207897
$447$	−5147.51	−0.544673
$448$	−469.961	−0.0495616
$449$	−15434.7	−1.62229	−0.811147	−	0.584843i	$-0.801156\pi$
$449$	−15434.7	−1.62229	−0.811147	+	0.584843i	$0.801156\pi$
$450$	−1214.21	−0.127196
$451$	6399.07	0.668116
$452$	−3279.42	−0.341264
$453$	7636.94	0.792085
$454$	−2043.66	−0.211263
$455$	−4783.95	−0.492912
$456$	−468.353	−0.0480979
$457$	−354.364	−0.0362723	−0.0181361	−	0.999836i	$-0.505773\pi$
$457$	−354.364	−0.0362723	−0.0181361	+	0.999836i	$0.505773\pi$
$458$	12611.6	1.28668
$459$	1032.09	0.104953
$460$	3352.84	0.339841
$461$	−367.463	−0.0371247	−0.0185623	−	0.999828i	$-0.505909\pi$
$461$	−367.463	−0.0371247	−0.0185623	+	0.999828i	$0.505909\pi$
$462$	2172.71	0.218795
$463$	−4909.26	−0.492770	−0.246385	−	0.969172i	$-0.579243\pi$
$463$	−4909.26	−0.492770	−0.246385	+	0.969172i	$0.579243\pi$
$464$	771.997	0.0772393
$465$	−5583.69	−0.556854
$466$	−9516.86	−0.946052
$467$	16158.5	1.60113	0.800565	−	0.599246i	$-0.204533\pi$
$467$	16158.5	1.60113	0.800565	+	0.599246i	$0.204533\pi$
$468$	−3091.76	−0.305378
$469$	−491.991	−0.0484393
$470$	5493.59	0.539150
$471$	−5691.30	−0.556776
$472$	−976.764	−0.0952525
$473$	204.015	0.0198322
$474$	3054.12	0.295950
$475$	1316.38	0.127157
$476$	−1122.78	−0.108114
$477$	−785.623	−0.0754113
$478$	−7084.83	−0.677934
$479$	18030.3	1.71988	0.859941	−	0.510393i	$-0.170500\pi$
$479$	18030.3	1.71988	0.859941	+	0.510393i	$0.170500\pi$
$480$	−728.235	−0.0692484
$481$	−1999.71	−0.189561
$482$	5852.40	0.553049
$483$	2434.20	0.229316
$484$	4403.37	0.413539
$485$	910.633	0.0852571
$486$	486.000	0.0453609
$487$	13975.5	1.30039	0.650196	−	0.759767i	$-0.274687\pi$
$487$	13975.5	1.30039	0.650196	+	0.759767i	$0.274687\pi$
$488$	3462.43	0.321182
$489$	6603.02	0.610631
$490$	4385.77	0.404345
$491$	1749.26	0.160780	0.0803898	−	0.996764i	$-0.474383\pi$
$491$	1749.26	0.160780	0.0803898	+	0.996764i	$0.474383\pi$
$492$	−1557.15	−0.142686
$493$	1844.37	0.168491
$494$	3351.94	0.305285
$495$	3366.75	0.305705
$496$	3925.72	0.355383
$497$	−3375.18	−0.304623
$498$	−8614.30	−0.775132
$499$	−1966.06	−0.176378	−0.0881891	−	0.996104i	$-0.528108\pi$
$499$	−1966.06	−0.176378	−0.0881891	+	0.996104i	$0.528108\pi$
$500$	5839.72	0.522320
$501$	10270.4	0.915860
$502$	−6535.75	−0.581085
$503$	−12092.8	−1.07195	−0.535975	−	0.844234i	$-0.680056\pi$
$503$	−12092.8	−1.07195	−0.535975	+	0.844234i	$0.680056\pi$
$504$	−528.706	−0.0467271
$505$	−6105.24	−0.537980
$506$	10898.1	0.957468
$507$	15536.3	1.36093
$508$	−2631.06	−0.229792
$509$	−13218.4	−1.15107	−0.575535	−	0.817777i	$-0.695206\pi$
$509$	−13218.4	−1.15107	−0.575535	+	0.817777i	$0.695206\pi$
$510$	−1739.82	−0.151060
$511$	−1800.01	−0.155827
$512$	512.000	0.0441942
$513$	−526.897	−0.0453472
$514$	−6580.45	−0.564691
$515$	2199.99	0.188239
$516$	−49.6450	−0.00423547
$517$	17856.4	1.51900
$518$	−341.960	−0.0290055
$519$	−170.723	−0.0144391
$520$	5211.88	0.439530
$521$	12583.5	1.05814	0.529071	−	0.848578i	$-0.322541\pi$
$521$	12583.5	1.05814	0.529071	+	0.848578i	$0.322541\pi$
$522$	868.496	0.0728219
$523$	−16948.7	−1.41704	−0.708521	−	0.705690i	$-0.750637\pi$
$523$	−16948.7	−1.41704	−0.708521	+	0.705690i	$0.750637\pi$
$524$	−8413.67	−0.701437
$525$	1486.01	0.123533
$526$	−3146.08	−0.260790
$527$	9378.89	0.775238
$528$	−2367.06	−0.195100
$529$	42.6919	0.00350883
$530$	1324.35	0.108540
$531$	−1098.86	−0.0898049
$532$	573.198	0.0467129
$533$	11144.3	0.905652
$534$	−784.118	−0.0635432
$535$	−3463.05	−0.279852
$536$	536.000	0.0431934
$537$	−8789.68	−0.706336
$538$	−13137.5	−1.05278
$539$	14255.5	1.13920
$540$	−819.265	−0.0652881
$541$	−2701.35	−0.214677	−0.107338	−	0.994223i	$-0.534233\pi$
$541$	−2701.35	−0.214677	−0.107338	+	0.994223i	$0.534233\pi$
$542$	1247.05	0.0988290
$543$	10312.0	0.814970
$544$	1223.21	0.0964059
$545$	3117.20	0.245002
$546$	3783.88	0.296584
$547$	−3787.38	−0.296045	−0.148023	−	0.988984i	$-0.547291\pi$
$547$	−3787.38	−0.296045	−0.148023	+	0.988984i	$0.547291\pi$
$548$	3777.89	0.294495
$549$	3895.24	0.302814
$550$	6653.00	0.515790
$551$	−941.581	−0.0727998
$552$	−2651.94	−0.204482
$553$	−3737.80	−0.287428
$554$	−14717.8	−1.12870
$555$	−529.889	−0.0405271
$556$	−4222.33	−0.322062
$557$	16331.9	1.24238	0.621189	−	0.783661i	$-0.286650\pi$
$557$	16331.9	1.24238	0.621189	+	0.783661i	$0.286650\pi$
$558$	4416.43	0.335058
$559$	355.302	0.0268831
$560$	891.257	0.0672544
$561$	−5655.11	−0.425595
$562$	−9591.73	−0.719934
$563$	358.864	0.0268638	0.0134319	−	0.999910i	$-0.495724\pi$
$563$	358.864	0.0268638	0.0134319	+	0.999910i	$0.495724\pi$
$564$	−4345.17	−0.324405
$565$	6219.25	0.463090
$566$	−2813.62	−0.208949
$567$	−594.795	−0.0440547
$568$	3677.09	0.271633
$569$	4110.83	0.302874	0.151437	−	0.988467i	$-0.451610\pi$
$569$	4110.83	0.302874	0.151437	+	0.988467i	$0.451610\pi$
$570$	888.207	0.0652682
$571$	−17546.5	−1.28598	−0.642992	−	0.765873i	$-0.722307\pi$
$571$	−17546.5	−1.28598	−0.642992	+	0.765873i	$0.722307\pi$
$572$	16940.7	1.23833
$573$	−509.987	−0.0371816
$574$	1905.73	0.138578
$575$	7453.70	0.540593
$576$	576.000	0.0416667
$577$	20266.8	1.46225	0.731124	−	0.682245i	$-0.238996\pi$
$577$	20266.8	1.46225	0.731124	+	0.682245i	$0.238996\pi$
$578$	−6903.64	−0.496805
$579$	5122.61	0.367683
$580$	−1464.05	−0.104813
$581$	10542.7	0.752812
$582$	−720.268	−0.0512991
$583$	4304.66	0.305799
$584$	1961.02	0.138952
$585$	5863.36	0.414393
$586$	−11700.4	−0.824813
$587$	−11811.6	−0.830522	−0.415261	−	0.909702i	$-0.636310\pi$
$587$	−11811.6	−0.830522	−0.415261	+	0.909702i	$0.636310\pi$
$588$	−3468.94	−0.243294
$589$	−4788.08	−0.334957
$590$	1852.38	0.129256
$591$	−3927.19	−0.273338
$592$	372.548	0.0258643
$593$	2288.09	0.158449	0.0792247	−	0.996857i	$-0.474756\pi$
$593$	2288.09	0.158449	0.0792247	+	0.996857i	$0.474756\pi$
$594$	−2662.94	−0.183942
$595$	2129.29	0.146710
$596$	−6863.34	−0.471701
$597$	8112.04	0.556120
$598$	18979.5	1.29788
$599$	7487.78	0.510755	0.255378	−	0.966841i	$-0.417800\pi$
$599$	7487.78	0.510755	0.255378	+	0.966841i	$0.417800\pi$
$600$	−1618.94	−0.110155
$601$	−25440.1	−1.72666	−0.863330	−	0.504639i	$-0.831626\pi$
$601$	−25440.1	−1.72666	−0.863330	+	0.504639i	$0.831626\pi$
$602$	60.7584	0.00411350
$603$	603.000	0.0407231
$604$	10182.6	0.685966
$605$	−8350.75	−0.561167
$606$	4828.96	0.323701
$607$	6300.94	0.421330	0.210665	−	0.977558i	$-0.432437\pi$
$607$	6300.94	0.421330	0.210665	+	0.977558i	$0.432437\pi$
$608$	−624.471	−0.0416540
$609$	−1062.92	−0.0707250
$610$	−6566.32	−0.435840
$611$	31097.8	2.05905
$612$	1376.11	0.0908924
$613$	−25317.6	−1.66814	−0.834070	−	0.551659i	$-0.813995\pi$
$613$	−25317.6	−1.66814	−0.834070	+	0.551659i	$0.813995\pi$
$614$	14386.2	0.945571
$615$	2953.05	0.193624
$616$	2896.94	0.189482
$617$	2752.37	0.179589	0.0897943	−	0.995960i	$-0.471379\pi$
$617$	2752.37	0.179589	0.0897943	+	0.995960i	$0.471379\pi$
$618$	−1740.09	−0.113263
$619$	14353.3	0.932002	0.466001	−	0.884784i	$-0.345694\pi$
$619$	14353.3	0.932002	0.466001	+	0.884784i	$0.345694\pi$
$620$	−7444.92	−0.482250
$621$	−2983.43	−0.192787
$622$	13488.4	0.869512
$623$	959.648	0.0617135
$624$	−4122.35	−0.264465
$625$	−2642.73	−0.169135
$626$	−14913.8	−0.952198
$627$	2887.03	0.183887
$628$	−7588.40	−0.482182
$629$	890.051	0.0564207
$630$	1002.66	0.0634081
$631$	20483.1	1.29226	0.646132	−	0.763226i	$-0.276386\pi$
$631$	20483.1	1.29226	0.646132	+	0.763226i	$0.276386\pi$
$632$	4072.15	0.256300
$633$	−2737.89	−0.171914
$634$	−15564.7	−0.975006
$635$	4989.66	0.311825
$636$	−1047.50	−0.0653081
$637$	24826.7	1.54422
$638$	−4758.75	−0.295299
$639$	4136.73	0.256098
$640$	−970.981	−0.0599709
$641$	−12312.3	−0.758670	−0.379335	−	0.925259i	$-0.623847\pi$
$641$	−12312.3	−0.758670	−0.379335	+	0.925259i	$0.623847\pi$
$642$	2739.11	0.168386
$643$	−151.612	−0.00929860	−0.00464930	−	0.999989i	$-0.501480\pi$
$643$	−151.612	−0.00929860	−0.00464930	+	0.999989i	$0.501480\pi$
$644$	3245.60	0.198594
$645$	94.1491	0.00574747
$646$	−1491.92	−0.0908648
$647$	19148.5	1.16353	0.581765	−	0.813357i	$-0.302362\pi$
$647$	19148.5	1.16353	0.581765	+	0.813357i	$0.302362\pi$
$648$	648.000	0.0392837
$649$	6020.98	0.364167
$650$	11586.5	0.699170
$651$	−5405.09	−0.325410
$652$	8804.02	0.528822
$653$	3586.14	0.214910	0.107455	−	0.994210i	$-0.465730\pi$
$653$	3586.14	0.214910	0.107455	+	0.994210i	$0.465730\pi$
$654$	−2465.56	−0.147417
$655$	15956.1	0.951840
$656$	−2076.20	−0.123570
$657$	2206.15	0.131005
$658$	5317.87	0.315064
$659$	2557.93	0.151203	0.0756016	−	0.997138i	$-0.475912\pi$
$659$	2557.93	0.151203	0.0756016	+	0.997138i	$0.475912\pi$
$660$	4489.00	0.264749
$661$	3819.81	0.224771	0.112385	−	0.993665i	$-0.464151\pi$
$661$	3819.81	0.224771	0.112385	+	0.993665i	$0.464151\pi$
$662$	−14159.1	−0.831282
$663$	−9848.65	−0.576908
$664$	−11485.7	−0.671284
$665$	−1087.04	−0.0633888
$666$	419.117	0.0243851
$667$	−5331.48	−0.309499
$668$	13693.8	0.793158
$669$	2937.26	0.169747
$670$	−1016.50	−0.0586129
$671$	−21343.2	−1.22793
$672$	−704.942	−0.0404669
$673$	−33525.5	−1.92023	−0.960114	−	0.279610i	$-0.909795\pi$
$673$	−33525.5	−1.92023	−0.960114	+	0.279610i	$0.909795\pi$
$674$	−2519.30	−0.143976
$675$	−1821.31	−0.103855
$676$	20715.0	1.17860
$677$	28910.7	1.64125	0.820626	−	0.571466i	$-0.193625\pi$
$677$	28910.7	1.64125	0.820626	+	0.571466i	$0.193625\pi$
$678$	−4919.14	−0.278641
$679$	881.505	0.0498219
$680$	−2319.76	−0.130822
$681$	−3065.48	−0.172496
$682$	−24199.0	−1.35869
$683$	−25271.7	−1.41580	−0.707902	−	0.706310i	$-0.750358\pi$
$683$	−25271.7	−1.41580	−0.707902	+	0.706310i	$0.750358\pi$
$684$	−702.530	−0.0392718
$685$	−7164.56	−0.399626
$686$	9282.88	0.516650
$687$	18917.4	1.05057
$688$	−66.1934	−0.00366802
$689$	7496.78	0.414521
$690$	5029.26	0.277479
$691$	29074.6	1.60065	0.800326	−	0.599564i	$-0.204660\pi$
$691$	29074.6	1.60065	0.800326	+	0.599564i	$0.204660\pi$
$692$	−227.630	−0.0125046
$693$	3259.06	0.178646
$694$	11142.3	0.609447
$695$	8007.43	0.437034
$696$	1157.99	0.0630656
$697$	−4960.22	−0.269558
$698$	−11898.3	−0.645212
$699$	−14275.3	−0.772448
$700$	1981.35	0.106983
$701$	33065.0	1.78152	0.890762	−	0.454469i	$-0.150171\pi$
$701$	33065.0	1.78152	0.890762	+	0.454469i	$0.150171\pi$
$702$	−4637.64	−0.249340
$703$	−454.386	−0.0243777
$704$	−3156.08	−0.168962
$705$	8240.38	0.440214
$706$	13419.9	0.715388
$707$	−5909.96	−0.314380
$708$	−1465.15	−0.0777734
$709$	−29313.8	−1.55276	−0.776378	−	0.630267i	$-0.782945\pi$
$709$	−29313.8	−1.55276	−0.776378	+	0.630267i	$0.782945\pi$
$710$	−6973.41	−0.368602
$711$	4581.17	0.241642
$712$	−1045.49	−0.0550301
$713$	−27111.4	−1.42402
$714$	−1684.17	−0.0882751
$715$	−32127.1	−1.68040
$716$	−11719.6	−0.611705
$717$	−10627.2	−0.553531
$718$	9233.27	0.479920
$719$	−9724.83	−0.504416	−0.252208	−	0.967673i	$-0.581157\pi$
$719$	−9724.83	−0.504416	−0.252208	+	0.967673i	$0.581157\pi$
$720$	−1092.35	−0.0565411
$721$	2129.62	0.110002
$722$	−12956.4	−0.667847
$723$	8778.60	0.451562
$724$	13749.3	0.705785
$725$	−3254.73	−0.166728
$726$	6605.05	0.337653
$727$	15422.0	0.786753	0.393376	−	0.919378i	$-0.371307\pi$
$727$	15422.0	0.786753	0.393376	+	0.919378i	$0.371307\pi$
$728$	5045.17	0.256849
$729$	729.000	0.0370370
$730$	−3718.98	−0.188555
$731$	−158.142	−0.00800148
$732$	5193.65	0.262244
$733$	3814.22	0.192198	0.0960991	−	0.995372i	$-0.469363\pi$
$733$	3814.22	0.192198	0.0960991	+	0.995372i	$0.469363\pi$
$734$	−24848.4	−1.24955
$735$	6578.66	0.330146
$736$	−3535.92	−0.177087
$737$	−3304.02	−0.165136
$738$	−2335.72	−0.116503
$739$	8246.03	0.410467	0.205234	−	0.978713i	$-0.434205\pi$
$739$	8246.03	0.410467	0.205234	+	0.978713i	$0.434205\pi$
$740$	−706.518	−0.0350975
$741$	5027.90	0.249264
$742$	1281.99	0.0634275
$743$	30927.7	1.52709	0.763545	−	0.645754i	$-0.223457\pi$
$743$	30927.7	1.52709	0.763545	+	0.645754i	$0.223457\pi$
$744$	5888.58	0.290169
$745$	13016.0	0.640091
$746$	4638.57	0.227654
$747$	−12921.4	−0.632893
$748$	−7540.14	−0.368576
$749$	−3352.28	−0.163538
$750$	8759.57	0.426473
$751$	39387.1	1.91379	0.956895	−	0.290434i	$-0.0937996\pi$
$751$	39387.1	1.91379	0.956895	+	0.290434i	$0.0937996\pi$
$752$	−5793.56	−0.280943
$753$	−9803.63	−0.474454
$754$	−8287.60	−0.400287
$755$	−19310.7	−0.930846
$756$	−793.060	−0.0381525
$757$	−13593.9	−0.652679	−0.326340	−	0.945253i	$-0.605815\pi$
$757$	−13593.9	−0.652679	−0.326340	+	0.945253i	$0.605815\pi$
$758$	10682.7	0.511889
$759$	16347.1	0.781769
$760$	1184.28	0.0565240
$761$	32192.5	1.53348	0.766740	−	0.641958i	$-0.221878\pi$
$761$	32192.5	1.53348	0.766740	+	0.641958i	$0.221878\pi$
$762$	−3946.59	−0.187624
$763$	3017.49	0.143172
$764$	−679.983	−0.0322002
$765$	−2609.73	−0.123340
$766$	−16927.5	−0.798452
$767$	10485.8	0.493639
$768$	768.000	0.0360844
$769$	−6182.46	−0.289916	−0.144958	−	0.989438i	$-0.546305\pi$
$769$	−6182.46	−0.289916	−0.144958	+	0.989438i	$0.546305\pi$
$770$	−5493.90	−0.257125
$771$	−9870.67	−0.461068
$772$	6830.15	0.318423
$773$	−6656.15	−0.309709	−0.154855	−	0.987937i	$-0.549491\pi$
$773$	−6656.15	−0.309709	−0.154855	+	0.987937i	$0.549491\pi$
$774$	−74.4675	−0.00345824
$775$	−16550.8	−0.767125
$776$	−960.357	−0.0444263
$777$	−512.939	−0.0236829
$778$	−9494.40	−0.437520
$779$	2532.28	0.116468
$780$	7817.81	0.358875
$781$	−22666.4	−1.03850
$782$	−8447.62	−0.386300
$783$	1302.74	0.0594588
$784$	−4625.25	−0.210698
$785$	14391.0	0.654314
$786$	−12620.5	−0.572721
$787$	28307.2	1.28214	0.641069	−	0.767483i	$-0.278491\pi$
$787$	28307.2	1.28214	0.641069	+	0.767483i	$0.278491\pi$
$788$	−5236.25	−0.236718
$789$	−4719.12	−0.212934
$790$	−7722.62	−0.347796
$791$	6020.32	0.270617
$792$	−3550.59	−0.159299
$793$	−37170.2	−1.66450
$794$	−25002.0	−1.11749
$795$	1986.52	0.0886222
$796$	10816.1	0.481614
$797$	−40495.2	−1.79977	−0.899883	−	0.436131i	$-0.856348\pi$
$797$	−40495.2	−1.79977	−0.899883	+	0.436131i	$0.856348\pi$
$798$	859.797	0.0381409
$799$	−13841.3	−0.612855
$800$	−2158.59	−0.0953970
$801$	−1176.18	−0.0518828
$802$	−25641.2	−1.12896
$803$	−12088.2	−0.531235
$804$	804.000	0.0352673
$805$	−6155.10	−0.269489
$806$	−42143.7	−1.84175
$807$	−19706.2	−0.859593
$808$	6438.61	0.280334
$809$	11327.0	0.492259	0.246129	−	0.969237i	$-0.420841\pi$
$809$	11327.0	0.492259	0.246129	+	0.969237i	$0.420841\pi$
$810$	−1228.90	−0.0533075
$811$	−35693.4	−1.54546	−0.772728	−	0.634738i	$-0.781108\pi$
$811$	−35693.4	−1.54546	−0.772728	+	0.634738i	$0.781108\pi$
$812$	−1417.22	−0.0612496
$813$	1870.57	0.0806935
$814$	−2296.47	−0.0988835
$815$	−16696.4	−0.717605
$816$	1834.82	0.0787151
$817$	80.7340	0.00345719
$818$	−11142.0	−0.476249
$819$	5675.81	0.242160
$820$	3937.40	0.167683
$821$	−29092.0	−1.23668	−0.618342	−	0.785909i	$-0.712195\pi$
$821$	−29092.0	−1.23668	−0.618342	+	0.785909i	$0.712195\pi$
$822$	5666.83	0.240454
$823$	−24127.3	−1.02190	−0.510951	−	0.859610i	$-0.670707\pi$
$823$	−24127.3	−1.02190	−0.510951	+	0.859610i	$0.670707\pi$
$824$	−2320.11	−0.0980886
$825$	9979.49	0.421141
$826$	1793.13	0.0755338
$827$	35157.3	1.47828	0.739141	−	0.673551i	$-0.235232\pi$
$827$	35157.3	1.47828	0.739141	+	0.673551i	$0.235232\pi$
$828$	−3977.91	−0.166959
$829$	−4301.14	−0.180199	−0.0900994	−	0.995933i	$-0.528718\pi$
$829$	−4301.14	−0.180199	−0.0900994	+	0.995933i	$0.528718\pi$
$830$	21782.1	0.910924
$831$	−22076.7	−0.921578
$832$	−5496.46	−0.229033
$833$	−11050.1	−0.459621
$834$	−6333.50	−0.262963
$835$	−25969.6	−1.07630
$836$	3849.37	0.159250
$837$	6624.65	0.273574
$838$	−7115.83	−0.293332
$839$	−23434.4	−0.964297	−0.482148	−	0.876090i	$-0.660143\pi$
$839$	−23434.4	−0.964297	−0.482148	+	0.876090i	$0.660143\pi$
$840$	1336.88	0.0549130
$841$	−22061.0	−0.904545
$842$	−1030.82	−0.0421904
$843$	−14387.6	−0.587823
$844$	−3650.52	−0.148882
$845$	−39285.0	−1.59934
$846$	−6517.76	−0.264876
$847$	−8083.64	−0.327931
$848$	−1396.66	−0.0565585
$849$	−4220.43	−0.170606
$850$	−5157.05	−0.208101
$851$	−2572.85	−0.103638
$852$	5515.64	0.221787
$853$	43169.4	1.73282	0.866409	−	0.499335i	$-0.166422\pi$
$853$	43169.4	1.73282	0.866409	+	0.499335i	$0.166422\pi$
$854$	−6356.29	−0.254693
$855$	1332.31	0.0532913
$856$	3652.14	0.145827
$857$	25259.7	1.00683	0.503417	−	0.864044i	$-0.332076\pi$
$857$	25259.7	1.00683	0.503417	+	0.864044i	$0.332076\pi$
$858$	25411.0	1.01109
$859$	−14689.7	−0.583477	−0.291738	−	0.956498i	$-0.594234\pi$
$859$	−14689.7	−0.583477	−0.291738	+	0.956498i	$0.594234\pi$
$860$	125.532	0.00497746
$861$	2858.59	0.113148
$862$	−17074.0	−0.674643
$863$	1606.89	0.0633825	0.0316912	−	0.999498i	$-0.489911\pi$
$863$	1606.89	0.0633825	0.0316912	+	0.999498i	$0.489911\pi$
$864$	864.000	0.0340207
$865$	431.688	0.0169686
$866$	22438.8	0.880486
$867$	−10355.5	−0.405640
$868$	−7206.78	−0.281813
$869$	−25101.6	−0.979879
$870$	−2196.08	−0.0855792
$871$	−5754.11	−0.223847
$872$	−3287.41	−0.127667
$873$	−1080.40	−0.0418855
$874$	4312.65	0.166908
$875$	−10720.5	−0.414192
$876$	2941.53	0.113453
$877$	25512.6	0.982327	0.491163	−	0.871067i	$-0.336572\pi$
$877$	25512.6	0.982327	0.491163	+	0.871067i	$0.336572\pi$
$878$	25330.1	0.973632
$879$	−17550.6	−0.673457
$880$	5985.33	0.229279
$881$	−10289.8	−0.393500	−0.196750	−	0.980454i	$-0.563039\pi$
$881$	−10289.8	−0.393500	−0.196750	+	0.980454i	$0.563039\pi$
$882$	−5203.41	−0.198648
$883$	46838.2	1.78509	0.892543	−	0.450962i	$-0.148919\pi$
$883$	46838.2	1.78509	0.892543	+	0.450962i	$0.148919\pi$
$884$	−13131.5	−0.499617
$885$	2778.57	0.105537
$886$	20765.1	0.787380
$887$	5464.28	0.206846	0.103423	−	0.994637i	$-0.467020\pi$
$887$	5464.28	0.206846	0.103423	+	0.994637i	$0.467020\pi$
$888$	558.823	0.0211181
$889$	4830.06	0.182222
$890$	1982.72	0.0746751
$891$	−3994.41	−0.150188
$892$	3916.34	0.147005
$893$	7066.23	0.264796
$894$	−10295.0	−0.385142
$895$	22225.5	0.830076
$896$	−939.923	−0.0350453
$897$	28469.3	1.05971
$898$	−30869.4	−1.14713
$899$	11838.4	0.439193
$900$	−2428.41	−0.0899411
$901$	−3336.75	−0.123378
$902$	12798.1	0.472429
$903$	91.1377	0.00335866
$904$	−6558.85	−0.241310
$905$	−26074.8	−0.957741
$906$	15273.9	0.560089
$907$	−53161.2	−1.94618	−0.973092	−	0.230418i	$-0.925991\pi$
$907$	−53161.2	−1.94618	−0.973092	+	0.230418i	$0.925991\pi$
$908$	−4087.31	−0.149386
$909$	7243.44	0.264301
$910$	−9567.89	−0.348541
$911$	−34785.9	−1.26510	−0.632551	−	0.774518i	$-0.717992\pi$
$911$	−34785.9	−1.26510	−0.632551	+	0.774518i	$0.717992\pi$
$912$	−936.706	−0.0340104
$913$	70800.5	2.56643
$914$	−708.727	−0.0256484
$915$	−9849.48	−0.355862
$916$	25223.2	0.909822
$917$	15445.7	0.556229
$918$	2064.17	0.0742133
$919$	41997.2	1.50747	0.753733	−	0.657181i	$-0.228251\pi$
$919$	41997.2	1.50747	0.753733	+	0.657181i	$0.228251\pi$
$920$	6705.68	0.240304
$921$	21579.3	0.772055
$922$	−734.927	−0.0262511
$923$	−39474.6	−1.40772
$924$	4345.41	0.154712
$925$	−1570.66	−0.0558303
$926$	−9818.51	−0.348441
$927$	−2610.13	−0.0924788
$928$	1543.99	0.0546164
$929$	9612.89	0.339493	0.169746	−	0.985488i	$-0.445705\pi$
$929$	9612.89	0.339493	0.169746	+	0.985488i	$0.445705\pi$
$930$	−11167.4	−0.393756
$931$	5641.28	0.198588
$932$	−19033.7	−0.668960
$933$	20232.6	0.709954
$934$	32317.1	1.13217
$935$	14299.5	0.500153
$936$	−6183.52	−0.215935
$937$	55264.0	1.92678	0.963391	−	0.268099i	$-0.0863955\pi$
$937$	55264.0	1.92678	0.963391	+	0.268099i	$0.0863955\pi$
$938$	−983.982	−0.0342517
$939$	−22370.7	−0.777467
$940$	10987.2	0.381236
$941$	−18748.5	−0.649503	−0.324752	−	0.945799i	$-0.605281\pi$
$941$	−18748.5	−0.649503	−0.324752	+	0.945799i	$0.605281\pi$
$942$	−11382.6	−0.393700
$943$	14338.4	0.495147
$944$	−1953.53	−0.0673537
$945$	1504.00	0.0517725
$946$	408.030	0.0140235
$947$	−2918.76	−0.100155	−0.0500776	−	0.998745i	$-0.515947\pi$
$947$	−2918.76	−0.100155	−0.0500776	+	0.998745i	$0.515947\pi$
$948$	6108.23	0.209268
$949$	−21052.1	−0.720107
$950$	2632.76	0.0899138
$951$	−23347.1	−0.796089
$952$	−2245.56	−0.0764485
$953$	−30385.2	−1.03282	−0.516408	−	0.856343i	$-0.672731\pi$
$953$	−30385.2	−1.03282	−0.516408	+	0.856343i	$0.672731\pi$
$954$	−1571.25	−0.0533238
$955$	1289.55	0.0436952
$956$	−14169.7	−0.479372
$957$	−7138.13	−0.241111
$958$	36060.5	1.21614
$959$	−6935.39	−0.233530
$960$	−1456.47	−0.0489660
$961$	30409.3	1.02075
$962$	−3999.41	−0.134040
$963$	4108.66	0.137487
$964$	11704.8	0.391065
$965$	−12953.0	−0.432096
$966$	4868.39	0.162151
$967$	−3072.95	−0.102192	−0.0510958	−	0.998694i	$-0.516271\pi$
$967$	−3072.95	−0.102192	−0.0510958	+	0.998694i	$0.516271\pi$
$968$	8806.73	0.292417
$969$	−2237.87	−0.0741908
$970$	1821.27	0.0602859
$971$	3272.57	0.108158	0.0540792	−	0.998537i	$-0.482778\pi$
$971$	3272.57	0.108158	0.0540792	+	0.998537i	$0.482778\pi$
$972$	972.000	0.0320750
$973$	7751.30	0.255391
$974$	27951.0	0.919516
$975$	17379.8	0.570870
$976$	6924.86	0.227110
$977$	17470.0	0.572073	0.286036	−	0.958219i	$-0.407662\pi$
$977$	17470.0	0.572073	0.286036	+	0.958219i	$0.407662\pi$
$978$	13206.0	0.431781
$979$	6444.63	0.210389
$980$	8771.54	0.285915
$981$	−3698.33	−0.120366
$982$	3498.51	0.113688
$983$	15870.9	0.514957	0.257478	−	0.966284i	$-0.417108\pi$
$983$	15870.9	0.514957	0.257478	+	0.966284i	$0.417108\pi$
$984$	−3114.30	−0.100895
$985$	9930.26	0.321223
$986$	3688.73	0.119141
$987$	7976.81	0.257249
$988$	6703.87	0.215869
$989$	457.137	0.0146978
$990$	6733.50	0.216166
$991$	56377.2	1.80715	0.903573	−	0.428434i	$-0.140934\pi$
$991$	56377.2	1.80715	0.903573	+	0.428434i	$0.140934\pi$
$992$	7851.44	0.251294
$993$	−21238.6	−0.678739
$994$	−6750.36	−0.215401
$995$	−20512.1	−0.653544
$996$	−17228.6	−0.548101
$997$	−3589.00	−0.114007	−0.0570033	−	0.998374i	$-0.518155\pi$
$997$	−3589.00	−0.114007	−0.0570033	+	0.998374i	$0.518155\pi$
$998$	−3932.11	−0.124718
$999$	628.675	0.0199103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.4.a.b.1.2	✓	2
3.2	odd	2		1206.4.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.4.a.b.1.2	✓	2	1.1	even	1	trivial
1206.4.a.d.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 402 = 2 \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	402.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -7.58579 q^{5} +6.00000 q^{6} -7.34315 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -7.58579 q^{5} +6.00000 q^{6} -7.34315 q^{7} +8.00000 q^{8} +9.00000 q^{9} -15.1716 q^{10} -49.3137 q^{11} +12.0000 q^{12} -85.8823 q^{13} -14.6863 q^{14} -22.7574 q^{15} +16.0000 q^{16} +38.2254 q^{17} +18.0000 q^{18} -19.5147 q^{19} -30.3431 q^{20} -22.0294 q^{21} -98.6274 q^{22} -110.497 q^{23} +24.0000 q^{24} -67.4558 q^{25} -171.765 q^{26} +27.0000 q^{27} -29.3726 q^{28} +48.2498 q^{29} -45.5147 q^{30} +245.357 q^{31} +32.0000 q^{32} -147.941 q^{33} +76.4508 q^{34} +55.7035 q^{35} +36.0000 q^{36} +23.2843 q^{37} -39.0294 q^{38} -257.647 q^{39} -60.6863 q^{40} -129.762 q^{41} -44.0589 q^{42} -4.13708 q^{43} -197.255 q^{44} -68.2721 q^{45} -220.995 q^{46} -362.098 q^{47} +48.0000 q^{48} -289.078 q^{49} -134.912 q^{50} +114.676 q^{51} -343.529 q^{52} -87.2914 q^{53} +54.0000 q^{54} +374.083 q^{55} -58.7452 q^{56} -58.5442 q^{57} +96.4996 q^{58} -122.095 q^{59} -91.0294 q^{60} +432.804 q^{61} +490.715 q^{62} -66.0883 q^{63} +64.0000 q^{64} +651.484 q^{65} -295.882 q^{66} +67.0000 q^{67} +152.902 q^{68} -331.492 q^{69} +111.407 q^{70} +459.637 q^{71} +72.0000 q^{72} +245.128 q^{73} +46.5685 q^{74} -202.368 q^{75} -78.0589 q^{76} +362.118 q^{77} -515.294 q^{78} +509.019 q^{79} -121.373 q^{80} +81.0000 q^{81} -259.525 q^{82} -1435.72 q^{83} -88.1177 q^{84} -289.970 q^{85} -8.27417 q^{86} +144.749 q^{87} -394.510 q^{88} -130.686 q^{89} -136.544 q^{90} +630.646 q^{91} -441.990 q^{92} +736.072 q^{93} -724.195 q^{94} +148.034 q^{95} +96.0000 q^{96} -120.045 q^{97} -578.156 q^{98} -443.823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 18 q^{5} + 12 q^{6} - 26 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 18 * q^5 + 12 * q^6 - 26 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 18 q^{5} + 12 q^{6} - 26 q^{7} + 16 q^{8} + 18 q^{9} - 36 q^{10} - 76 q^{11} + 24 q^{12} - 36 q^{13} - 52 q^{14} - 54 q^{15} + 32 q^{16} - 48 q^{17} + 36 q^{18} - 56 q^{19} - 72 q^{20} - 78 q^{21} - 152 q^{22} - 122 q^{23} + 48 q^{24} - 84 q^{25} - 72 q^{26} + 54 q^{27} - 104 q^{28} - 192 q^{29} - 108 q^{30} - 58 q^{31} + 64 q^{32} - 228 q^{33} - 96 q^{34} + 250 q^{35} + 72 q^{36} - 10 q^{37} - 112 q^{38} - 108 q^{39} - 144 q^{40} - 466 q^{41} - 156 q^{42} + 218 q^{43} - 304 q^{44} - 162 q^{45} - 244 q^{46} - 68 q^{47} + 96 q^{48} - 284 q^{49} - 168 q^{50} - 144 q^{51} - 144 q^{52} + 162 q^{53} + 108 q^{54} + 652 q^{55} - 208 q^{56} - 168 q^{57} - 384 q^{58} - 66 q^{59} - 216 q^{60} + 1024 q^{61} - 116 q^{62} - 234 q^{63} + 128 q^{64} + 132 q^{65} - 456 q^{66} + 134 q^{67} - 192 q^{68} - 366 q^{69} + 500 q^{70} + 116 q^{71} + 144 q^{72} + 1022 q^{73} - 20 q^{74} - 252 q^{75} - 224 q^{76} + 860 q^{77} - 216 q^{78} + 656 q^{79} - 288 q^{80} + 162 q^{81} - 932 q^{82} - 142 q^{83} - 312 q^{84} + 608 q^{85} + 436 q^{86} - 576 q^{87} - 608 q^{88} - 284 q^{89} - 324 q^{90} - 300 q^{91} - 488 q^{92} - 174 q^{93} - 136 q^{94} + 528 q^{95} + 192 q^{96} - 868 q^{97} - 568 q^{98} - 684 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 18 * q^5 + 12 * q^6 - 26 * q^7 + 16 * q^8 + 18 * q^9 - 36 * q^10 - 76 * q^11 + 24 * q^12 - 36 * q^13 - 52 * q^14 - 54 * q^15 + 32 * q^16 - 48 * q^17 + 36 * q^18 - 56 * q^19 - 72 * q^20 - 78 * q^21 - 152 * q^22 - 122 * q^23 + 48 * q^24 - 84 * q^25 - 72 * q^26 + 54 * q^27 - 104 * q^28 - 192 * q^29 - 108 * q^30 - 58 * q^31 + 64 * q^32 - 228 * q^33 - 96 * q^34 + 250 * q^35 + 72 * q^36 - 10 * q^37 - 112 * q^38 - 108 * q^39 - 144 * q^40 - 466 * q^41 - 156 * q^42 + 218 * q^43 - 304 * q^44 - 162 * q^45 - 244 * q^46 - 68 * q^47 + 96 * q^48 - 284 * q^49 - 168 * q^50 - 144 * q^51 - 144 * q^52 + 162 * q^53 + 108 * q^54 + 652 * q^55 - 208 * q^56 - 168 * q^57 - 384 * q^58 - 66 * q^59 - 216 * q^60 + 1024 * q^61 - 116 * q^62 - 234 * q^63 + 128 * q^64 + 132 * q^65 - 456 * q^66 + 134 * q^67 - 192 * q^68 - 366 * q^69 + 500 * q^70 + 116 * q^71 + 144 * q^72 + 1022 * q^73 - 20 * q^74 - 252 * q^75 - 224 * q^76 + 860 * q^77 - 216 * q^78 + 656 * q^79 - 288 * q^80 + 162 * q^81 - 932 * q^82 - 142 * q^83 - 312 * q^84 + 608 * q^85 + 436 * q^86 - 576 * q^87 - 608 * q^88 - 284 * q^89 - 324 * q^90 - 300 * q^91 - 488 * q^92 - 174 * q^93 - 136 * q^94 + 528 * q^95 + 192 * q^96 - 868 * q^97 - 568 * q^98 - 684 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more