LMFDB - Embedded newform 8038.2.a.c.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$84$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.89385 q^{3} +1.00000 q^{4} +0.388076 q^{5} +2.89385 q^{6} -0.0680600 q^{7} -1.00000 q^{8} +5.37435 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.89385 q^{3} +1.00000 q^{4} +0.388076 q^{5} +2.89385 q^{6} -0.0680600 q^{7} -1.00000 q^{8} +5.37435 q^{9} -0.388076 q^{10} -1.33476 q^{11} -2.89385 q^{12} +4.96616 q^{13} +0.0680600 q^{14} -1.12303 q^{15} +1.00000 q^{16} +1.01322 q^{17} -5.37435 q^{18} +2.47528 q^{19} +0.388076 q^{20} +0.196955 q^{21} +1.33476 q^{22} +2.97391 q^{23} +2.89385 q^{24} -4.84940 q^{25} -4.96616 q^{26} -6.87101 q^{27} -0.0680600 q^{28} -1.88752 q^{29} +1.12303 q^{30} +4.91110 q^{31} -1.00000 q^{32} +3.86260 q^{33} -1.01322 q^{34} -0.0264124 q^{35} +5.37435 q^{36} -5.29437 q^{37} -2.47528 q^{38} -14.3713 q^{39} -0.388076 q^{40} -9.86257 q^{41} -0.196955 q^{42} +5.73499 q^{43} -1.33476 q^{44} +2.08566 q^{45} -2.97391 q^{46} -3.29931 q^{47} -2.89385 q^{48} -6.99537 q^{49} +4.84940 q^{50} -2.93209 q^{51} +4.96616 q^{52} -9.21569 q^{53} +6.87101 q^{54} -0.517989 q^{55} +0.0680600 q^{56} -7.16308 q^{57} +1.88752 q^{58} +0.297629 q^{59} -1.12303 q^{60} +8.19406 q^{61} -4.91110 q^{62} -0.365778 q^{63} +1.00000 q^{64} +1.92725 q^{65} -3.86260 q^{66} +6.98306 q^{67} +1.01322 q^{68} -8.60605 q^{69} +0.0264124 q^{70} -3.66674 q^{71} -5.37435 q^{72} -0.321746 q^{73} +5.29437 q^{74} +14.0334 q^{75} +2.47528 q^{76} +0.0908439 q^{77} +14.3713 q^{78} -0.108097 q^{79} +0.388076 q^{80} +3.76059 q^{81} +9.86257 q^{82} -6.16548 q^{83} +0.196955 q^{84} +0.393205 q^{85} -5.73499 q^{86} +5.46218 q^{87} +1.33476 q^{88} +1.40125 q^{89} -2.08566 q^{90} -0.337997 q^{91} +2.97391 q^{92} -14.2120 q^{93} +3.29931 q^{94} +0.960596 q^{95} +2.89385 q^{96} -7.73175 q^{97} +6.99537 q^{98} -7.17348 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) $ 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.89385	−1.67076	−0.835382	−	0.549670i	$-0.814753\pi$
$3$	−2.89385	−1.67076	−0.835382	+	0.549670i	$0.814753\pi$
$4$	1.00000	0.500000
$5$	0.388076	0.173553	0.0867764	−	0.996228i	$-0.472343\pi$
$5$	0.388076	0.173553	0.0867764	+	0.996228i	$0.472343\pi$
$6$	2.89385	1.18141
$7$	−0.0680600	−0.0257242	−0.0128621	−	0.999917i	$-0.504094\pi$
$7$	−0.0680600	−0.0257242	−0.0128621	+	0.999917i	$0.504094\pi$
$8$	−1.00000	−0.353553
$9$	5.37435	1.79145
$10$	−0.388076	−0.122720
$11$	−1.33476	−0.402446	−0.201223	−	0.979545i	$-0.564492\pi$
$11$	−1.33476	−0.402446	−0.201223	+	0.979545i	$0.564492\pi$
$12$	−2.89385	−0.835382
$13$	4.96616	1.37737	0.688683	−	0.725063i	$-0.258189\pi$
$13$	4.96616	1.37737	0.688683	+	0.725063i	$0.258189\pi$
$14$	0.0680600	0.0181898
$15$	−1.12303	−0.289966
$16$	1.00000	0.250000
$17$	1.01322	0.245741	0.122871	−	0.992423i	$-0.460790\pi$
$17$	1.01322	0.245741	0.122871	+	0.992423i	$0.460790\pi$
$18$	−5.37435	−1.26675
$19$	2.47528	0.567868	0.283934	−	0.958844i	$-0.408360\pi$
$19$	2.47528	0.567868	0.283934	+	0.958844i	$0.408360\pi$
$20$	0.388076	0.0867764
$21$	0.196955	0.0429791
$22$	1.33476	0.284572
$23$	2.97391	0.620104	0.310052	−	0.950720i	$-0.399654\pi$
$23$	2.97391	0.620104	0.310052	+	0.950720i	$0.399654\pi$
$24$	2.89385	0.590704
$25$	−4.84940	−0.969879
$26$	−4.96616	−0.973945
$27$	−6.87101	−1.32233
$28$	−0.0680600	−0.0128621
$29$	−1.88752	−0.350503	−0.175252	−	0.984524i	$-0.556074\pi$
$29$	−1.88752	−0.350503	−0.175252	+	0.984524i	$0.556074\pi$
$30$	1.12303	0.205037
$31$	4.91110	0.882060	0.441030	−	0.897492i	$-0.354613\pi$
$31$	4.91110	0.882060	0.441030	+	0.897492i	$0.354613\pi$
$32$	−1.00000	−0.176777
$33$	3.86260	0.672392
$34$	−1.01322	−0.173765
$35$	−0.0264124	−0.00446452
$36$	5.37435	0.895725
$37$	−5.29437	−0.870389	−0.435195	−	0.900336i	$-0.643320\pi$
$37$	−5.29437	−0.870389	−0.435195	+	0.900336i	$0.643320\pi$
$38$	−2.47528	−0.401543
$39$	−14.3713	−2.30125
$40$	−0.388076	−0.0613602
$41$	−9.86257	−1.54028	−0.770138	−	0.637878i	$-0.779812\pi$
$41$	−9.86257	−1.54028	−0.770138	+	0.637878i	$0.779812\pi$
$42$	−0.196955	−0.0303908
$43$	5.73499	0.874578	0.437289	−	0.899321i	$-0.355939\pi$
$43$	5.73499	0.874578	0.437289	+	0.899321i	$0.355939\pi$
$44$	−1.33476	−0.201223
$45$	2.08566	0.310911
$46$	−2.97391	−0.438479
$47$	−3.29931	−0.481253	−0.240627	−	0.970618i	$-0.577353\pi$
$47$	−3.29931	−0.481253	−0.240627	+	0.970618i	$0.577353\pi$
$48$	−2.89385	−0.417691
$49$	−6.99537	−0.999338
$50$	4.84940	0.685808
$51$	−2.93209	−0.410575
$52$	4.96616	0.688683
$53$	−9.21569	−1.26587	−0.632936	−	0.774204i	$-0.718150\pi$
$53$	−9.21569	−1.26587	−0.632936	+	0.774204i	$0.718150\pi$
$54$	6.87101	0.935026
$55$	−0.517989	−0.0698456
$56$	0.0680600	0.00909490
$57$	−7.16308	−0.948773
$58$	1.88752	0.247843
$59$	0.297629	0.0387480	0.0193740	−	0.999812i	$-0.493833\pi$
$59$	0.297629	0.0387480	0.0193740	+	0.999812i	$0.493833\pi$
$60$	−1.12303	−0.144983
$61$	8.19406	1.04914	0.524571	−	0.851367i	$-0.324226\pi$
$61$	8.19406	1.04914	0.524571	+	0.851367i	$0.324226\pi$
$62$	−4.91110	−0.623710
$63$	−0.365778	−0.0460837
$64$	1.00000	0.125000
$65$	1.92725	0.239046
$66$	−3.86260	−0.475453
$67$	6.98306	0.853116	0.426558	−	0.904460i	$-0.359726\pi$
$67$	6.98306	0.853116	0.426558	+	0.904460i	$0.359726\pi$
$68$	1.01322	0.122871
$69$	−8.60605	−1.03605
$70$	0.0264124	0.00315689
$71$	−3.66674	−0.435162	−0.217581	−	0.976042i	$-0.569817\pi$
$71$	−3.66674	−0.435162	−0.217581	+	0.976042i	$0.569817\pi$
$72$	−5.37435	−0.633373
$73$	−0.321746	−0.0376576	−0.0188288	−	0.999823i	$-0.505994\pi$
$73$	−0.321746	−0.0376576	−0.0188288	+	0.999823i	$0.505994\pi$
$74$	5.29437	0.615458
$75$	14.0334	1.62044
$76$	2.47528	0.283934
$77$	0.0908439	0.0103526
$78$	14.3713	1.62723
$79$	−0.108097	−0.0121619	−0.00608095	−	0.999982i	$-0.501936\pi$
$79$	−0.108097	−0.0121619	−0.00608095	+	0.999982i	$0.501936\pi$
$80$	0.388076	0.0433882
$81$	3.76059	0.417844
$82$	9.86257	1.08914
$83$	−6.16548	−0.676749	−0.338375	−	0.941012i	$-0.609877\pi$
$83$	−6.16548	−0.676749	−0.338375	+	0.941012i	$0.609877\pi$
$84$	0.196955	0.0214896
$85$	0.393205	0.0426491
$86$	−5.73499	−0.618420
$87$	5.46218	0.585608
$88$	1.33476	0.142286
$89$	1.40125	0.148532	0.0742659	−	0.997238i	$-0.476339\pi$
$89$	1.40125	0.148532	0.0742659	+	0.997238i	$0.476339\pi$
$90$	−2.08566	−0.219847
$91$	−0.337997	−0.0354317
$92$	2.97391	0.310052
$93$	−14.2120	−1.47371
$94$	3.29931	0.340297
$95$	0.960596	0.0985551
$96$	2.89385	0.295352
$97$	−7.73175	−0.785040	−0.392520	−	0.919743i	$-0.628397\pi$
$97$	−7.73175	−0.785040	−0.392520	+	0.919743i	$0.628397\pi$
$98$	6.99537	0.706639
$99$	−7.17348	−0.720962
$100$	−4.84940	−0.484940
$101$	8.28509	0.824397	0.412199	−	0.911094i	$-0.364761\pi$
$101$	8.28509	0.824397	0.412199	+	0.911094i	$0.364761\pi$
$102$	2.93209	0.290320
$103$	−4.16518	−0.410407	−0.205204	−	0.978719i	$-0.565786\pi$
$103$	−4.16518	−0.410407	−0.205204	+	0.978719i	$0.565786\pi$
$104$	−4.96616	−0.486972
$105$	0.0764336	0.00745915
$106$	9.21569	0.895107
$107$	−9.61822	−0.929829	−0.464914	−	0.885356i	$-0.653915\pi$
$107$	−9.61822	−0.929829	−0.464914	+	0.885356i	$0.653915\pi$
$108$	−6.87101	−0.661163
$109$	0.712207	0.0682170	0.0341085	−	0.999418i	$-0.489141\pi$
$109$	0.712207	0.0682170	0.0341085	+	0.999418i	$0.489141\pi$
$110$	0.517989	0.0493883
$111$	15.3211	1.45421
$112$	−0.0680600	−0.00643106
$113$	7.30367	0.687072	0.343536	−	0.939140i	$-0.388375\pi$
$113$	7.30367	0.687072	0.343536	+	0.939140i	$0.388375\pi$
$114$	7.16308	0.670884
$115$	1.15410	0.107621
$116$	−1.88752	−0.175252
$117$	26.6899	2.46748
$118$	−0.297629	−0.0273990
$119$	−0.0689595	−0.00632150
$120$	1.12303	0.102518
$121$	−9.21841	−0.838037
$122$	−8.19406	−0.741855
$123$	28.5408	2.57344
$124$	4.91110	0.441030
$125$	−3.82231	−0.341878
$126$	0.365778	0.0325861
$127$	6.75714	0.599599	0.299799	−	0.954002i	$-0.403080\pi$
$127$	6.75714	0.599599	0.299799	+	0.954002i	$0.403080\pi$
$128$	−1.00000	−0.0883883
$129$	−16.5962	−1.46121
$130$	−1.92725	−0.169031
$131$	8.90379	0.777928	0.388964	−	0.921253i	$-0.372833\pi$
$131$	8.90379	0.777928	0.388964	+	0.921253i	$0.372833\pi$
$132$	3.86260	0.336196
$133$	−0.168467	−0.0146080
$134$	−6.98306	−0.603244
$135$	−2.66647	−0.229493
$136$	−1.01322	−0.0868826
$137$	−8.37034	−0.715126	−0.357563	−	0.933889i	$-0.616392\pi$
$137$	−8.37034	−0.715126	−0.357563	+	0.933889i	$0.616392\pi$
$138$	8.60605	0.732595
$139$	−20.2832	−1.72040	−0.860198	−	0.509959i	$-0.829660\pi$
$139$	−20.2832	−1.72040	−0.860198	+	0.509959i	$0.829660\pi$
$140$	−0.0264124	−0.00223226
$141$	9.54769	0.804060
$142$	3.66674	0.307706
$143$	−6.62865	−0.554315
$144$	5.37435	0.447863
$145$	−0.732500	−0.0608308
$146$	0.321746	0.0266279
$147$	20.2435	1.66966
$148$	−5.29437	−0.435195
$149$	−19.5752	−1.60366	−0.801831	−	0.597551i	$-0.796141\pi$
$149$	−19.5752	−1.60366	−0.801831	+	0.597551i	$0.796141\pi$
$150$	−14.0334	−1.14582
$151$	23.6680	1.92607	0.963035	−	0.269375i	$-0.0868173\pi$
$151$	23.6680	1.92607	0.963035	+	0.269375i	$0.0868173\pi$
$152$	−2.47528	−0.200772
$153$	5.44538	0.440233
$154$	−0.0908439	−0.00732041
$155$	1.90588	0.153084
$156$	−14.3713	−1.15063
$157$	−1.90077	−0.151698	−0.0758492	−	0.997119i	$-0.524167\pi$
$157$	−1.90077	−0.151698	−0.0758492	+	0.997119i	$0.524167\pi$
$158$	0.108097	0.00859977
$159$	26.6688	2.11497
$160$	−0.388076	−0.0306801
$161$	−0.202404	−0.0159517
$162$	−3.76059	−0.295460
$163$	−24.9279	−1.95251	−0.976253	−	0.216635i	$-0.930492\pi$
$163$	−24.9279	−1.95251	−0.976253	+	0.216635i	$0.930492\pi$
$164$	−9.86257	−0.770138
$165$	1.49898	0.116696
$166$	6.16548	0.478534
$167$	0.548094	0.0424128	0.0212064	−	0.999775i	$-0.493249\pi$
$167$	0.548094	0.0424128	0.0212064	+	0.999775i	$0.493249\pi$
$168$	−0.196955	−0.0151954
$169$	11.6628	0.897136
$170$	−0.393205	−0.0301574
$171$	13.3030	1.01731
$172$	5.73499	0.437289
$173$	−4.07323	−0.309682	−0.154841	−	0.987939i	$-0.549487\pi$
$173$	−4.07323	−0.309682	−0.154841	+	0.987939i	$0.549487\pi$
$174$	−5.46218	−0.414087
$175$	0.330050	0.0249494
$176$	−1.33476	−0.100611
$177$	−0.861293	−0.0647388
$178$	−1.40125	−0.105028
$179$	16.1988	1.21076	0.605379	−	0.795938i	$-0.293022\pi$
$179$	16.1988	1.21076	0.605379	+	0.795938i	$0.293022\pi$
$180$	2.08566	0.155456
$181$	6.42636	0.477668	0.238834	−	0.971060i	$-0.423235\pi$
$181$	6.42636	0.477668	0.238834	+	0.971060i	$0.423235\pi$
$182$	0.337997	0.0250540
$183$	−23.7123	−1.75287
$184$	−2.97391	−0.219240
$185$	−2.05462	−0.151059
$186$	14.2120	1.04207
$187$	−1.35240	−0.0988975
$188$	−3.29931	−0.240627
$189$	0.467641	0.0340158
$190$	−0.960596	−0.0696890
$191$	−6.61620	−0.478731	−0.239366	−	0.970930i	$-0.576939\pi$
$191$	−6.61620	−0.478731	−0.239366	+	0.970930i	$0.576939\pi$
$192$	−2.89385	−0.208845
$193$	−10.6879	−0.769333	−0.384666	−	0.923056i	$-0.625684\pi$
$193$	−10.6879	−0.769333	−0.384666	+	0.923056i	$0.625684\pi$
$194$	7.73175	0.555107
$195$	−5.57716	−0.399389
$196$	−6.99537	−0.499669
$197$	−1.50034	−0.106895	−0.0534474	−	0.998571i	$-0.517021\pi$
$197$	−1.50034	−0.106895	−0.0534474	+	0.998571i	$0.517021\pi$
$198$	7.17348	0.509797
$199$	23.3134	1.65264	0.826322	−	0.563198i	$-0.190429\pi$
$199$	23.3134	1.65264	0.826322	+	0.563198i	$0.190429\pi$
$200$	4.84940	0.342904
$201$	−20.2079	−1.42536
$202$	−8.28509	−0.582937
$203$	0.128464	0.00901643
$204$	−2.93209	−0.205288
$205$	−3.82743	−0.267319
$206$	4.16518	0.290202
$207$	15.9828	1.11088
$208$	4.96616	0.344341
$209$	−3.30391	−0.228536
$210$	−0.0764336	−0.00527442
$211$	−5.00071	−0.344263	−0.172131	−	0.985074i	$-0.555065\pi$
$211$	−5.00071	−0.344263	−0.172131	+	0.985074i	$0.555065\pi$
$212$	−9.21569	−0.632936
$213$	10.6110	0.727052
$214$	9.61822	0.657488
$215$	2.22561	0.151785
$216$	6.87101	0.467513
$217$	−0.334249	−0.0226903
$218$	−0.712207	−0.0482367
$219$	0.931085	0.0629169
$220$	−0.517989	−0.0349228
$221$	5.03180	0.338475
$222$	−15.3211	−1.02829
$223$	6.53290	0.437475	0.218738	−	0.975784i	$-0.429806\pi$
$223$	6.53290	0.437475	0.218738	+	0.975784i	$0.429806\pi$
$224$	0.0680600	0.00454745
$225$	−26.0624	−1.73749
$226$	−7.30367	−0.485833
$227$	21.8118	1.44770	0.723851	−	0.689957i	$-0.242370\pi$
$227$	21.8118	1.44770	0.723851	+	0.689957i	$0.242370\pi$
$228$	−7.16308	−0.474386
$229$	23.4525	1.54978	0.774892	−	0.632093i	$-0.217804\pi$
$229$	23.4525	1.54978	0.774892	+	0.632093i	$0.217804\pi$
$230$	−1.15410	−0.0760994
$231$	−0.262888	−0.0172968
$232$	1.88752	0.123922
$233$	11.6249	0.761573	0.380786	−	0.924663i	$-0.375653\pi$
$233$	11.6249	0.761573	0.380786	+	0.924663i	$0.375653\pi$
$234$	−26.6899	−1.74477
$235$	−1.28038	−0.0835229
$236$	0.297629	0.0193740
$237$	0.312817	0.0203197
$238$	0.0689595	0.00446998
$239$	22.7372	1.47075	0.735373	−	0.677662i	$-0.237007\pi$
$239$	22.7372	1.47075	0.735373	+	0.677662i	$0.237007\pi$
$240$	−1.12303	−0.0724914
$241$	14.0966	0.908039	0.454020	−	0.890992i	$-0.349990\pi$
$241$	14.0966	0.908039	0.454020	+	0.890992i	$0.349990\pi$
$242$	9.21841	0.592582
$243$	9.73044	0.624208
$244$	8.19406	0.524571
$245$	−2.71473	−0.173438
$246$	−28.5408	−1.81969
$247$	12.2926	0.782162
$248$	−4.91110	−0.311855
$249$	17.8419	1.13069
$250$	3.82231	0.241744
$251$	17.2916	1.09144	0.545718	−	0.837969i	$-0.316257\pi$
$251$	17.2916	1.09144	0.545718	+	0.837969i	$0.316257\pi$
$252$	−0.365778	−0.0230419
$253$	−3.96946	−0.249558
$254$	−6.75714	−0.423980
$255$	−1.13787	−0.0712565
$256$	1.00000	0.0625000
$257$	16.9294	1.05603	0.528015	−	0.849235i	$-0.322937\pi$
$257$	16.9294	1.05603	0.528015	+	0.849235i	$0.322937\pi$
$258$	16.5962	1.03323
$259$	0.360335	0.0223901
$260$	1.92725	0.119523
$261$	−10.1442	−0.627909
$262$	−8.90379	−0.550078
$263$	−18.2746	−1.12686	−0.563430	−	0.826164i	$-0.690518\pi$
$263$	−18.2746	−1.12686	−0.563430	+	0.826164i	$0.690518\pi$
$264$	−3.86260	−0.237726
$265$	−3.57639	−0.219696
$266$	0.168467	0.0103294
$267$	−4.05499	−0.248162
$268$	6.98306	0.426558
$269$	−24.9803	−1.52308	−0.761539	−	0.648119i	$-0.775556\pi$
$269$	−24.9803	−1.52308	−0.761539	+	0.648119i	$0.775556\pi$
$270$	2.66647	0.162276
$271$	−13.0423	−0.792261	−0.396131	−	0.918194i	$-0.629647\pi$
$271$	−13.0423	−0.792261	−0.396131	+	0.918194i	$0.629647\pi$
$272$	1.01322	0.0614353
$273$	0.978111	0.0591980
$274$	8.37034	0.505671
$275$	6.47279	0.390324
$276$	−8.60605	−0.518023
$277$	2.48420	0.149261	0.0746307	−	0.997211i	$-0.476222\pi$
$277$	2.48420	0.149261	0.0746307	+	0.997211i	$0.476222\pi$
$278$	20.2832	1.21650
$279$	26.3940	1.58017
$280$	0.0264124	0.00157845
$281$	−4.19181	−0.250062	−0.125031	−	0.992153i	$-0.539903\pi$
$281$	−4.19181	−0.250062	−0.125031	+	0.992153i	$0.539903\pi$
$282$	−9.54769	−0.568556
$283$	26.8758	1.59760	0.798799	−	0.601598i	$-0.205469\pi$
$283$	26.8758	1.59760	0.798799	+	0.601598i	$0.205469\pi$
$284$	−3.66674	−0.217581
$285$	−2.77982	−0.164662
$286$	6.62865	0.391960
$287$	0.671246	0.0396224
$288$	−5.37435	−0.316687
$289$	−15.9734	−0.939611
$290$	0.732500	0.0430139
$291$	22.3745	1.31162
$292$	−0.321746	−0.0188288
$293$	−29.1856	−1.70504	−0.852521	−	0.522694i	$-0.824927\pi$
$293$	−29.1856	−1.70504	−0.852521	+	0.522694i	$0.824927\pi$
$294$	−20.2435	−1.18063
$295$	0.115503	0.00672483
$296$	5.29437	0.307729
$297$	9.17116	0.532165
$298$	19.5752	1.13396
$299$	14.7689	0.854109
$300$	14.0334	0.810219
$301$	−0.390323	−0.0224979
$302$	−23.6680	−1.36194
$303$	−23.9758	−1.37737
$304$	2.47528	0.141967
$305$	3.17992	0.182081
$306$	−5.44538	−0.311292
$307$	−5.73196	−0.327140	−0.163570	−	0.986532i	$-0.552301\pi$
$307$	−5.73196	−0.327140	−0.163570	+	0.986532i	$0.552301\pi$
$308$	0.0908439	0.00517631
$309$	12.0534	0.685693
$310$	−1.90588	−0.108247
$311$	4.72317	0.267827	0.133913	−	0.990993i	$-0.457246\pi$
$311$	4.72317	0.267827	0.133913	+	0.990993i	$0.457246\pi$
$312$	14.3713	0.813615
$313$	−19.3658	−1.09462	−0.547309	−	0.836931i	$-0.684348\pi$
$313$	−19.3658	−1.09462	−0.547309	+	0.836931i	$0.684348\pi$
$314$	1.90077	0.107267
$315$	−0.141950	−0.00799796
$316$	−0.108097	−0.00608095
$317$	−18.3211	−1.02902	−0.514509	−	0.857485i	$-0.672026\pi$
$317$	−18.3211	−1.02902	−0.514509	+	0.857485i	$0.672026\pi$
$318$	−26.6688	−1.49551
$319$	2.51939	0.141059
$320$	0.388076	0.0216941
$321$	27.8337	1.55352
$322$	0.202404	0.0112796
$323$	2.50799	0.139548
$324$	3.76059	0.208922
$325$	−24.0829	−1.33588
$326$	24.9279	1.38063
$327$	−2.06102	−0.113974
$328$	9.86257	0.544570
$329$	0.224551	0.0123799
$330$	−1.49898	−0.0825162
$331$	−11.9774	−0.658337	−0.329168	−	0.944271i	$-0.606768\pi$
$331$	−11.9774	−0.658337	−0.329168	+	0.944271i	$0.606768\pi$
$332$	−6.16548	−0.338375
$333$	−28.4538	−1.55926
$334$	−0.548094	−0.0299904
$335$	2.70996	0.148061
$336$	0.196955	0.0107448
$337$	18.7454	1.02113	0.510564	−	0.859840i	$-0.329437\pi$
$337$	18.7454	1.02113	0.510564	+	0.859840i	$0.329437\pi$
$338$	−11.6628	−0.634371
$339$	−21.1357	−1.14793
$340$	0.393205	0.0213245
$341$	−6.55515	−0.354981
$342$	−13.3030	−0.719345
$343$	0.952524	0.0514315
$344$	−5.73499	−0.309210
$345$	−3.33980	−0.179809
$346$	4.07323	0.218978
$347$	−31.8677	−1.71075	−0.855373	−	0.518012i	$-0.826672\pi$
$347$	−31.8677	−1.71075	−0.855373	+	0.518012i	$0.826672\pi$
$348$	5.46218	0.292804
$349$	23.0028	1.23131	0.615655	−	0.788016i	$-0.288891\pi$
$349$	23.0028	1.23131	0.615655	+	0.788016i	$0.288891\pi$
$350$	−0.330050	−0.0176419
$351$	−34.1225	−1.82133
$352$	1.33476	0.0711431
$353$	1.93095	0.102774	0.0513872	−	0.998679i	$-0.483636\pi$
$353$	1.93095	0.102774	0.0513872	+	0.998679i	$0.483636\pi$
$354$	0.861293	0.0457772
$355$	−1.42297	−0.0755235
$356$	1.40125	0.0742659
$357$	0.199558	0.0105617
$358$	−16.1988	−0.856135
$359$	23.6807	1.24982	0.624909	−	0.780698i	$-0.285136\pi$
$359$	23.6807	1.24982	0.624909	+	0.780698i	$0.285136\pi$
$360$	−2.08566	−0.109924
$361$	−12.8730	−0.677526
$362$	−6.42636	−0.337762
$363$	26.6767	1.40016
$364$	−0.337997	−0.0177158
$365$	−0.124862	−0.00653558
$366$	23.7123	1.23946
$367$	12.2231	0.638039	0.319020	−	0.947748i	$-0.396646\pi$
$367$	12.2231	0.638039	0.319020	+	0.947748i	$0.396646\pi$
$368$	2.97391	0.155026
$369$	−53.0049	−2.75933
$370$	2.05462	0.106815
$371$	0.627219	0.0325636
$372$	−14.2120	−0.736856
$373$	−26.2708	−1.36025	−0.680125	−	0.733096i	$-0.738075\pi$
$373$	−26.2708	−1.36025	−0.680125	+	0.733096i	$0.738075\pi$
$374$	1.35240	0.0699311
$375$	11.0612	0.571198
$376$	3.29931	0.170149
$377$	−9.37371	−0.482771
$378$	−0.467641	−0.0240528
$379$	14.5058	0.745111	0.372555	−	0.928010i	$-0.378482\pi$
$379$	14.5058	0.745111	0.372555	+	0.928010i	$0.378482\pi$
$380$	0.960596	0.0492776
$381$	−19.5541	−1.00179
$382$	6.61620	0.338514
$383$	−4.00387	−0.204588	−0.102294	−	0.994754i	$-0.532618\pi$
$383$	−4.00387	−0.204588	−0.102294	+	0.994754i	$0.532618\pi$
$384$	2.89385	0.147676
$385$	0.0352543	0.00179673
$386$	10.6879	0.544000
$387$	30.8218	1.56676
$388$	−7.73175	−0.392520
$389$	11.8450	0.600563	0.300282	−	0.953851i	$-0.402919\pi$
$389$	11.8450	0.600563	0.300282	+	0.953851i	$0.402919\pi$
$390$	5.57716	0.282411
$391$	3.01322	0.152385
$392$	6.99537	0.353319
$393$	−25.7662	−1.29973
$394$	1.50034	0.0755860
$395$	−0.0419500	−0.00211073
$396$	−7.17348	−0.360481
$397$	−25.5019	−1.27990	−0.639951	−	0.768416i	$-0.721045\pi$
$397$	−25.5019	−1.27990	−0.639951	+	0.768416i	$0.721045\pi$
$398$	−23.3134	−1.16860
$399$	0.487519	0.0244065
$400$	−4.84940	−0.242470
$401$	−8.96805	−0.447843	−0.223921	−	0.974607i	$-0.571886\pi$
$401$	−8.96805	−0.447843	−0.223921	+	0.974607i	$0.571886\pi$
$402$	20.2079	1.00788
$403$	24.3893	1.21492
$404$	8.28509	0.412199
$405$	1.45940	0.0725179
$406$	−0.128464	−0.00637558
$407$	7.06673	0.350285
$408$	2.93209	0.145160
$409$	−24.5973	−1.21626	−0.608130	−	0.793837i	$-0.708080\pi$
$409$	−24.5973	−1.21626	−0.608130	+	0.793837i	$0.708080\pi$
$410$	3.82743	0.189023
$411$	24.2225	1.19481
$412$	−4.16518	−0.205204
$413$	−0.0202566	−0.000996763 0
$414$	−15.9828	−0.785514
$415$	−2.39267	−0.117452
$416$	−4.96616	−0.243486
$417$	58.6964	2.87438
$418$	3.30391	0.161599
$419$	−30.5651	−1.49320	−0.746602	−	0.665271i	$-0.768316\pi$
$419$	−30.5651	−1.49320	−0.746602	+	0.665271i	$0.768316\pi$
$420$	0.0764336	0.00372958
$421$	32.6070	1.58917	0.794585	−	0.607153i	$-0.207689\pi$
$421$	32.6070	1.58917	0.794585	+	0.607153i	$0.207689\pi$
$422$	5.00071	0.243431
$423$	−17.7316	−0.862141
$424$	9.21569	0.447553
$425$	−4.91349	−0.238339
$426$	−10.6110	−0.514103
$427$	−0.557687	−0.0269884
$428$	−9.61822	−0.464914
$429$	19.1823	0.926129
$430$	−2.22561	−0.107329
$431$	−27.9692	−1.34723	−0.673614	−	0.739083i	$-0.735259\pi$
$431$	−27.9692	−1.34723	−0.673614	+	0.739083i	$0.735259\pi$
$432$	−6.87101	−0.330581
$433$	20.0078	0.961515	0.480758	−	0.876854i	$-0.340362\pi$
$433$	20.0078	0.961515	0.480758	+	0.876854i	$0.340362\pi$
$434$	0.334249	0.0160445
$435$	2.11974	0.101634
$436$	0.712207	0.0341085
$437$	7.36126	0.352137
$438$	−0.931085	−0.0444890
$439$	−24.3952	−1.16432	−0.582159	−	0.813075i	$-0.697792\pi$
$439$	−24.3952	−1.16432	−0.582159	+	0.813075i	$0.697792\pi$
$440$	0.517989	0.0246942
$441$	−37.5956	−1.79026
$442$	−5.03180	−0.239338
$443$	2.72675	0.129552	0.0647759	−	0.997900i	$-0.479367\pi$
$443$	2.72675	0.129552	0.0647759	+	0.997900i	$0.479367\pi$
$444$	15.3211	0.727107
$445$	0.543790	0.0257781
$446$	−6.53290	−0.309342
$447$	56.6476	2.67934
$448$	−0.0680600	−0.00321553
$449$	10.2651	0.484440	0.242220	−	0.970221i	$-0.422124\pi$
$449$	10.2651	0.484440	0.242220	+	0.970221i	$0.422124\pi$
$450$	26.0624	1.22859
$451$	13.1642	0.619877
$452$	7.30367	0.343536
$453$	−68.4914	−3.21801
$454$	−21.8118	−1.02368
$455$	−0.131168	−0.00614927
$456$	7.16308	0.335442
$457$	−27.1882	−1.27181	−0.635905	−	0.771767i	$-0.719373\pi$
$457$	−27.1882	−1.27181	−0.635905	+	0.771767i	$0.719373\pi$
$458$	−23.4525	−1.09586
$459$	−6.96182	−0.324950
$460$	1.15410	0.0538104
$461$	−6.77414	−0.315503	−0.157752	−	0.987479i	$-0.550424\pi$
$461$	−6.77414	−0.315503	−0.157752	+	0.987479i	$0.550424\pi$
$462$	0.262888	0.0122307
$463$	−3.89722	−0.181119	−0.0905595	−	0.995891i	$-0.528866\pi$
$463$	−3.89722	−0.181119	−0.0905595	+	0.995891i	$0.528866\pi$
$464$	−1.88752	−0.0876258
$465$	−5.51533	−0.255767
$466$	−11.6249	−0.538513
$467$	−24.3340	−1.12604	−0.563021	−	0.826442i	$-0.690361\pi$
$467$	−24.3340	−1.12604	−0.563021	+	0.826442i	$0.690361\pi$
$468$	26.6899	1.23374
$469$	−0.475267	−0.0219458
$470$	1.28038	0.0590596
$471$	5.50055	0.253452
$472$	−0.297629	−0.0136995
$473$	−7.65485	−0.351970
$474$	−0.312817	−0.0143682
$475$	−12.0036	−0.550763
$476$	−0.0689595	−0.00316075
$477$	−49.5283	−2.26775
$478$	−22.7372	−1.03997
$479$	23.5302	1.07512	0.537562	−	0.843224i	$-0.319345\pi$
$479$	23.5302	1.07512	0.537562	+	0.843224i	$0.319345\pi$
$480$	1.12303	0.0512592
$481$	−26.2927	−1.19884
$482$	−14.0966	−0.642081
$483$	0.585727	0.0266515
$484$	−9.21841	−0.419019
$485$	−3.00051	−0.136246
$486$	−9.73044	−0.441382
$487$	−0.527114	−0.0238858	−0.0119429	−	0.999929i	$-0.503802\pi$
$487$	−0.527114	−0.0238858	−0.0119429	+	0.999929i	$0.503802\pi$
$488$	−8.19406	−0.370927
$489$	72.1376	3.26217
$490$	2.71473	0.122639
$491$	−0.932548	−0.0420853	−0.0210427	−	0.999779i	$-0.506699\pi$
$491$	−0.932548	−0.0420853	−0.0210427	+	0.999779i	$0.506699\pi$
$492$	28.5408	1.28672
$493$	−1.91246	−0.0861330
$494$	−12.2926	−0.553072
$495$	−2.78385	−0.125125
$496$	4.91110	0.220515
$497$	0.249558	0.0111942
$498$	−17.8419	−0.799517
$499$	6.47097	0.289680	0.144840	−	0.989455i	$-0.453733\pi$
$499$	6.47097	0.289680	0.144840	+	0.989455i	$0.453733\pi$
$500$	−3.82231	−0.170939
$501$	−1.58610	−0.0708618
$502$	−17.2916	−0.771762
$503$	−17.4125	−0.776386	−0.388193	−	0.921578i	$-0.626901\pi$
$503$	−17.4125	−0.776386	−0.388193	+	0.921578i	$0.626901\pi$
$504$	0.365778	0.0162931
$505$	3.21525	0.143077
$506$	3.96946	0.176464
$507$	−33.7503	−1.49890
$508$	6.75714	0.299799
$509$	−10.4993	−0.465374	−0.232687	−	0.972552i	$-0.574752\pi$
$509$	−10.4993	−0.465374	−0.232687	+	0.972552i	$0.574752\pi$
$510$	1.13787	0.0503859
$511$	0.0218981	0.000968713 0
$512$	−1.00000	−0.0441942
$513$	−17.0077	−0.750907
$514$	−16.9294	−0.746726
$515$	−1.61641	−0.0712273
$516$	−16.5962	−0.730606
$517$	4.40379	0.193678
$518$	−0.360335	−0.0158322
$519$	11.7873	0.517405
$520$	−1.92725	−0.0845154
$521$	−7.01253	−0.307225	−0.153612	−	0.988131i	$-0.549091\pi$
$521$	−7.01253	−0.307225	−0.153612	+	0.988131i	$0.549091\pi$
$522$	10.1442	0.443998
$523$	−37.9479	−1.65935	−0.829673	−	0.558249i	$-0.811473\pi$
$523$	−37.9479	−1.65935	−0.829673	+	0.558249i	$0.811473\pi$
$524$	8.90379	0.388964
$525$	−0.955114	−0.0416846
$526$	18.2746	0.796810
$527$	4.97601	0.216758
$528$	3.86260	0.168098
$529$	−14.1558	−0.615472
$530$	3.57639	0.155348
$531$	1.59956	0.0694151
$532$	−0.168467	−0.00730399
$533$	−48.9791	−2.12152
$534$	4.05499	0.175477
$535$	−3.73260	−0.161374
$536$	−6.98306	−0.301622
$537$	−46.8769	−2.02289
$538$	24.9803	1.07698
$539$	9.33715	0.402180
$540$	−2.66647	−0.114747
$541$	31.2301	1.34269	0.671343	−	0.741147i	$-0.265718\pi$
$541$	31.2301	1.34269	0.671343	+	0.741147i	$0.265718\pi$
$542$	13.0423	0.560213
$543$	−18.5969	−0.798070
$544$	−1.01322	−0.0434413
$545$	0.276390	0.0118393
$546$	−0.978111	−0.0418593
$547$	−39.5893	−1.69271	−0.846357	−	0.532616i	$-0.821209\pi$
$547$	−39.5893	−1.69271	−0.846357	+	0.532616i	$0.821209\pi$
$548$	−8.37034	−0.357563
$549$	44.0377	1.87948
$550$	−6.47279	−0.276001
$551$	−4.67213	−0.199039
$552$	8.60605	0.366298
$553$	0.00735711	0.000312856 0
$554$	−2.48420	−0.105544
$555$	5.94575	0.252383
$556$	−20.2832	−0.860198
$557$	−9.00964	−0.381751	−0.190875	−	0.981614i	$-0.561133\pi$
$557$	−9.00964	−0.381751	−0.190875	+	0.981614i	$0.561133\pi$
$558$	−26.3940	−1.11735
$559$	28.4809	1.20461
$560$	−0.0264124	−0.00111613
$561$	3.91365	0.165234
$562$	4.19181	0.176821
$563$	−10.0454	−0.423363	−0.211682	−	0.977339i	$-0.567894\pi$
$563$	−10.0454	−0.423363	−0.211682	+	0.977339i	$0.567894\pi$
$564$	9.54769	0.402030
$565$	2.83438	0.119243
$566$	−26.8758	−1.12967
$567$	−0.255946	−0.0107487
$568$	3.66674	0.153853
$569$	15.5247	0.650827	0.325414	−	0.945572i	$-0.394496\pi$
$569$	15.5247	0.650827	0.325414	+	0.945572i	$0.394496\pi$
$570$	2.77982	0.116434
$571$	−3.62037	−0.151508	−0.0757540	−	0.997127i	$-0.524136\pi$
$571$	−3.62037	−0.151508	−0.0757540	+	0.997127i	$0.524136\pi$
$572$	−6.62865	−0.277158
$573$	19.1463	0.799847
$574$	−0.671246	−0.0280173
$575$	−14.4217	−0.601426
$576$	5.37435	0.223931
$577$	10.3785	0.432064	0.216032	−	0.976386i	$-0.430688\pi$
$577$	10.3785	0.432064	0.216032	+	0.976386i	$0.430688\pi$
$578$	15.9734	0.664406
$579$	30.9292	1.28537
$580$	−0.732500	−0.0304154
$581$	0.419622	0.0174089
$582$	−22.3745	−0.927453
$583$	12.3007	0.509445
$584$	0.321746	0.0133140
$585$	10.3577	0.428239
$586$	29.1856	1.20565
$587$	13.5239	0.558193	0.279096	−	0.960263i	$-0.409965\pi$
$587$	13.5239	0.558193	0.279096	+	0.960263i	$0.409965\pi$
$588$	20.2435	0.834829
$589$	12.1563	0.500893
$590$	−0.115503	−0.00475517
$591$	4.34175	0.178596
$592$	−5.29437	−0.217597
$593$	−19.0962	−0.784187	−0.392093	−	0.919925i	$-0.628249\pi$
$593$	−19.0962	−0.784187	−0.392093	+	0.919925i	$0.628249\pi$
$594$	−9.17116	−0.376297
$595$	−0.0267615	−0.00109711
$596$	−19.5752	−0.801831
$597$	−67.4655	−2.76118
$598$	−14.7689	−0.603946
$599$	−27.8313	−1.13715	−0.568577	−	0.822630i	$-0.692506\pi$
$599$	−27.8313	−1.13715	−0.568577	+	0.822630i	$0.692506\pi$
$600$	−14.0334	−0.572912
$601$	−29.0890	−1.18657	−0.593283	−	0.804994i	$-0.702168\pi$
$601$	−29.0890	−1.18657	−0.593283	+	0.804994i	$0.702168\pi$
$602$	0.390323	0.0159084
$603$	37.5294	1.52831
$604$	23.6680	0.963035
$605$	−3.57744	−0.145444
$606$	23.9758	0.973950
$607$	−3.04614	−0.123639	−0.0618196	−	0.998087i	$-0.519690\pi$
$607$	−3.04614	−0.123639	−0.0618196	+	0.998087i	$0.519690\pi$
$608$	−2.47528	−0.100386
$609$	−0.371756	−0.0150643
$610$	−3.17992	−0.128751
$611$	−16.3849	−0.662862
$612$	5.44538	0.220116
$613$	2.60015	0.105019	0.0525096	−	0.998620i	$-0.483278\pi$
$613$	2.60015	0.105019	0.0525096	+	0.998620i	$0.483278\pi$
$614$	5.73196	0.231323
$615$	11.0760	0.446627
$616$	−0.0908439	−0.00366020
$617$	−16.5382	−0.665802	−0.332901	−	0.942962i	$-0.608028\pi$
$617$	−16.5382	−0.665802	−0.332901	+	0.942962i	$0.608028\pi$
$618$	−12.0534	−0.484858
$619$	2.81881	0.113298	0.0566488	−	0.998394i	$-0.481958\pi$
$619$	2.81881	0.113298	0.0566488	+	0.998394i	$0.481958\pi$
$620$	1.90588	0.0765420
$621$	−20.4338	−0.819979
$622$	−4.72317	−0.189382
$623$	−0.0953688	−0.00382087
$624$	−14.3713	−0.575313
$625$	22.7636	0.910545
$626$	19.3658	0.774011
$627$	9.56101	0.381830
$628$	−1.90077	−0.0758492
$629$	−5.36434	−0.213890
$630$	0.141950	0.00565541
$631$	23.7440	0.945235	0.472617	−	0.881268i	$-0.343309\pi$
$631$	23.7440	0.945235	0.472617	+	0.881268i	$0.343309\pi$
$632$	0.108097	0.00429988
$633$	14.4713	0.575182
$634$	18.3211	0.727625
$635$	2.62228	0.104062
$636$	26.6688	1.05749
$637$	−34.7401	−1.37645
$638$	−2.51939	−0.0997434
$639$	−19.7063	−0.779570
$640$	−0.388076	−0.0153401
$641$	−11.2004	−0.442388	−0.221194	−	0.975230i	$-0.570995\pi$
$641$	−11.2004	−0.442388	−0.221194	+	0.975230i	$0.570995\pi$
$642$	−27.8337	−1.09851
$643$	−40.6994	−1.60503	−0.802514	−	0.596633i	$-0.796505\pi$
$643$	−40.6994	−1.60503	−0.802514	+	0.596633i	$0.796505\pi$
$644$	−0.202404	−0.00797585
$645$	−6.44058	−0.253598
$646$	−2.50799	−0.0986757
$647$	35.4713	1.39452	0.697259	−	0.716819i	$-0.254403\pi$
$647$	35.4713	1.39452	0.697259	+	0.716819i	$0.254403\pi$
$648$	−3.76059	−0.147730
$649$	−0.397264	−0.0155940
$650$	24.0829	0.944609
$651$	0.967266	0.0379102
$652$	−24.9279	−0.976253
$653$	−24.5536	−0.960855	−0.480428	−	0.877034i	$-0.659519\pi$
$653$	−24.5536	−0.960855	−0.480428	+	0.877034i	$0.659519\pi$
$654$	2.06102	0.0805921
$655$	3.45535	0.135012
$656$	−9.86257	−0.385069
$657$	−1.72918	−0.0674617
$658$	−0.224551	−0.00875390
$659$	−31.5548	−1.22920	−0.614601	−	0.788838i	$-0.710683\pi$
$659$	−31.5548	−1.22920	−0.614601	+	0.788838i	$0.710683\pi$
$660$	1.49898	0.0583478
$661$	28.5975	1.11231	0.556157	−	0.831078i	$-0.312275\pi$
$661$	28.5975	1.11231	0.556157	+	0.831078i	$0.312275\pi$
$662$	11.9774	0.465514
$663$	−14.5612	−0.565512
$664$	6.16548	0.239267
$665$	−0.0653782	−0.00253526
$666$	28.4538	1.10256
$667$	−5.61331	−0.217348
$668$	0.548094	0.0212064
$669$	−18.9052	−0.730917
$670$	−2.70996	−0.104695
$671$	−10.9371	−0.422223
$672$	−0.196955	−0.00759771
$673$	39.7713	1.53307	0.766536	−	0.642202i	$-0.221979\pi$
$673$	39.7713	1.53307	0.766536	+	0.642202i	$0.221979\pi$
$674$	−18.7454	−0.722046
$675$	33.3202	1.28250
$676$	11.6628	0.448568
$677$	−34.6862	−1.33310	−0.666550	−	0.745460i	$-0.732230\pi$
$677$	−34.6862	−1.33310	−0.666550	+	0.745460i	$0.732230\pi$
$678$	21.1357	0.811712
$679$	0.526223	0.0201946
$680$	−0.393205	−0.0150787
$681$	−63.1201	−2.41877
$682$	6.55515	0.251010
$683$	−3.64563	−0.139496	−0.0697481	−	0.997565i	$-0.522220\pi$
$683$	−3.64563	−0.139496	−0.0697481	+	0.997565i	$0.522220\pi$
$684$	13.3030	0.508654
$685$	−3.24833	−0.124112
$686$	−0.952524	−0.0363675
$687$	−67.8679	−2.58932
$688$	5.73499	0.218644
$689$	−45.7666	−1.74357
$690$	3.33980	0.127144
$691$	−15.3850	−0.585273	−0.292636	−	0.956224i	$-0.594533\pi$
$691$	−15.3850	−0.585273	−0.292636	+	0.956224i	$0.594533\pi$
$692$	−4.07323	−0.154841
$693$	0.488227	0.0185462
$694$	31.8677	1.20968
$695$	−7.87142	−0.298580
$696$	−5.46218	−0.207044
$697$	−9.99292	−0.378509
$698$	−23.0028	−0.870668
$699$	−33.6407	−1.27241
$700$	0.330050	0.0124747
$701$	−3.47814	−0.131368	−0.0656838	−	0.997840i	$-0.520923\pi$
$701$	−3.47814	−0.131368	−0.0656838	+	0.997840i	$0.520923\pi$
$702$	34.1225	1.28787
$703$	−13.1050	−0.494266
$704$	−1.33476	−0.0503057
$705$	3.70523	0.139547
$706$	−1.93095	−0.0726724
$707$	−0.563883	−0.0212070
$708$	−0.861293	−0.0323694
$709$	10.2150	0.383632	0.191816	−	0.981431i	$-0.438562\pi$
$709$	10.2150	0.383632	0.191816	+	0.981431i	$0.438562\pi$
$710$	1.42297	0.0534032
$711$	−0.580953	−0.0217875
$712$	−1.40125	−0.0525140
$713$	14.6052	0.546968
$714$	−0.199558	−0.00746828
$715$	−2.57242	−0.0962030
$716$	16.1988	0.605379
$717$	−65.7979	−2.45727
$718$	−23.6807	−0.883755
$719$	8.70551	0.324661	0.162330	−	0.986736i	$-0.448099\pi$
$719$	8.70551	0.324661	0.162330	+	0.986736i	$0.448099\pi$
$720$	2.08566	0.0777278
$721$	0.283482	0.0105574
$722$	12.8730	0.479083
$723$	−40.7933	−1.51712
$724$	6.42636	0.238834
$725$	9.15332	0.339946
$726$	−26.6767	−0.990064
$727$	15.8507	0.587870	0.293935	−	0.955825i	$-0.405035\pi$
$727$	15.8507	0.587870	0.293935	+	0.955825i	$0.405035\pi$
$728$	0.337997	0.0125270
$729$	−39.4402	−1.46075
$730$	0.124862	0.00462135
$731$	5.81079	0.214920
$732$	−23.7123	−0.876433
$733$	−10.7723	−0.397886	−0.198943	−	0.980011i	$-0.563751\pi$
$733$	−10.7723	−0.397886	−0.198943	+	0.980011i	$0.563751\pi$
$734$	−12.2231	−0.451162
$735$	7.85603	0.289774
$736$	−2.97391	−0.109620
$737$	−9.32072	−0.343333
$738$	53.0049	1.95114
$739$	−34.9195	−1.28453	−0.642267	−	0.766481i	$-0.722006\pi$
$739$	−34.9195	−1.28453	−0.642267	+	0.766481i	$0.722006\pi$
$740$	−2.05462	−0.0755293
$741$	−35.5730	−1.30681
$742$	−0.627219	−0.0230259
$743$	−10.5125	−0.385667	−0.192833	−	0.981232i	$-0.561768\pi$
$743$	−10.5125	−0.385667	−0.192833	+	0.981232i	$0.561768\pi$
$744$	14.2120	0.521036
$745$	−7.59666	−0.278320
$746$	26.2708	0.961842
$747$	−33.1354	−1.21236
$748$	−1.35240	−0.0494487
$749$	0.654616	0.0239191
$750$	−11.0612	−0.403898
$751$	−39.6998	−1.44866	−0.724332	−	0.689451i	$-0.757852\pi$
$751$	−39.6998	−1.44866	−0.724332	+	0.689451i	$0.757852\pi$
$752$	−3.29931	−0.120313
$753$	−50.0393	−1.82353
$754$	9.37371	0.341371
$755$	9.18496	0.334275
$756$	0.467641	0.0170079
$757$	32.4306	1.17871	0.589356	−	0.807874i	$-0.299382\pi$
$757$	32.4306	1.17871	0.589356	+	0.807874i	$0.299382\pi$
$758$	−14.5058	−0.526873
$759$	11.4870	0.416953
$760$	−0.960596	−0.0348445
$761$	18.0835	0.655525	0.327763	−	0.944760i	$-0.393705\pi$
$761$	18.0835	0.655525	0.327763	+	0.944760i	$0.393705\pi$
$762$	19.5541	0.708371
$763$	−0.0484728	−0.00175483
$764$	−6.61620	−0.239366
$765$	2.11322	0.0764037
$766$	4.00387	0.144666
$767$	1.47807	0.0533702
$768$	−2.89385	−0.104423
$769$	26.8761	0.969177	0.484589	−	0.874742i	$-0.338969\pi$
$769$	26.8761	0.969177	0.484589	+	0.874742i	$0.338969\pi$
$770$	−0.0352543	−0.00127048
$771$	−48.9912	−1.76438
$772$	−10.6879	−0.384666
$773$	−37.3992	−1.34515	−0.672577	−	0.740027i	$-0.734813\pi$
$773$	−37.3992	−1.34515	−0.672577	+	0.740027i	$0.734813\pi$
$774$	−30.8218	−1.10787
$775$	−23.8159	−0.855492
$776$	7.73175	0.277554
$777$	−1.04275	−0.0374086
$778$	−11.8450	−0.424662
$779$	−24.4126	−0.874673
$780$	−5.57716	−0.199694
$781$	4.89422	0.175129
$782$	−3.01322	−0.107752
$783$	12.9691	0.463479
$784$	−6.99537	−0.249835
$785$	−0.737645	−0.0263277
$786$	25.7662	0.919051
$787$	−39.4703	−1.40696	−0.703482	−	0.710713i	$-0.748373\pi$
$787$	−39.4703	−1.40696	−0.703482	+	0.710713i	$0.748373\pi$
$788$	−1.50034	−0.0534474
$789$	52.8839	1.88271
$790$	0.0419500	0.00149251
$791$	−0.497088	−0.0176744
$792$	7.17348	0.254898
$793$	40.6930	1.44505
$794$	25.5019	0.905028
$795$	10.3495	0.367060
$796$	23.3134	0.826322
$797$	−5.29383	−0.187517	−0.0937586	−	0.995595i	$-0.529888\pi$
$797$	−5.29383	−0.187517	−0.0937586	+	0.995595i	$0.529888\pi$
$798$	−0.487519	−0.0172580
$799$	−3.34291	−0.118264
$800$	4.84940	0.171452
$801$	7.53079	0.266087
$802$	8.96805	0.316673
$803$	0.429455	0.0151551
$804$	−20.2079	−0.712678
$805$	−0.0785483	−0.00276846
$806$	−24.3893	−0.859077
$807$	72.2893	2.54470
$808$	−8.28509	−0.291469
$809$	16.5101	0.580464	0.290232	−	0.956956i	$-0.406268\pi$
$809$	16.5101	0.580464	0.290232	+	0.956956i	$0.406268\pi$
$810$	−1.45940	−0.0512779
$811$	10.9391	0.384124	0.192062	−	0.981383i	$-0.438483\pi$
$811$	10.9391	0.384124	0.192062	+	0.981383i	$0.438483\pi$
$812$	0.128464	0.00450821
$813$	37.7423	1.32368
$814$	−7.06673	−0.247689
$815$	−9.67393	−0.338863
$816$	−2.93209	−0.102644
$817$	14.1957	0.496645
$818$	24.5973	0.860026
$819$	−1.81651	−0.0634741
$820$	−3.82743	−0.133660
$821$	31.8604	1.11194	0.555968	−	0.831203i	$-0.312348\pi$
$821$	31.8604	1.11194	0.555968	+	0.831203i	$0.312348\pi$
$822$	−24.2225	−0.844856
$823$	31.3043	1.09120	0.545600	−	0.838045i	$-0.316302\pi$
$823$	31.3043	1.09120	0.545600	+	0.838045i	$0.316302\pi$
$824$	4.16518	0.145101
$825$	−18.7313	−0.652139
$826$	0.0202566	0.000704818 0
$827$	−27.4813	−0.955617	−0.477808	−	0.878464i	$-0.658569\pi$
$827$	−27.4813	−0.955617	−0.477808	+	0.878464i	$0.658569\pi$
$828$	15.9828	0.555442
$829$	5.63613	0.195751	0.0978754	−	0.995199i	$-0.468795\pi$
$829$	5.63613	0.195751	0.0978754	+	0.995199i	$0.468795\pi$
$830$	2.39267	0.0830509
$831$	−7.18891	−0.249380
$832$	4.96616	0.172171
$833$	−7.08782	−0.245578
$834$	−58.6964	−2.03249
$835$	0.212702	0.00736087
$836$	−3.30391	−0.114268
$837$	−33.7442	−1.16637
$838$	30.5651	1.05585
$839$	−35.4002	−1.22215	−0.611075	−	0.791572i	$-0.709263\pi$
$839$	−35.4002	−1.22215	−0.611075	+	0.791572i	$0.709263\pi$
$840$	−0.0764336	−0.00263721
$841$	−25.4373	−0.877148
$842$	−32.6070	−1.12371
$843$	12.1304	0.417795
$844$	−5.00071	−0.172131
$845$	4.52604	0.155701
$846$	17.7316	0.609626
$847$	0.627405	0.0215579
$848$	−9.21569	−0.316468
$849$	−77.7744	−2.66921
$850$	4.91349	0.168531
$851$	−15.7450	−0.539732
$852$	10.6110	0.363526
$853$	34.4081	1.17811	0.589055	−	0.808093i	$-0.299500\pi$
$853$	34.4081	1.17811	0.589055	+	0.808093i	$0.299500\pi$
$854$	0.557687	0.0190837
$855$	5.16258	0.176557
$856$	9.61822	0.328744
$857$	2.00790	0.0685884	0.0342942	−	0.999412i	$-0.489082\pi$
$857$	2.00790	0.0685884	0.0342942	+	0.999412i	$0.489082\pi$
$858$	−19.1823	−0.654872
$859$	25.4969	0.869941	0.434971	−	0.900445i	$-0.356759\pi$
$859$	25.4969	0.869941	0.434971	+	0.900445i	$0.356759\pi$
$860$	2.22561	0.0758927
$861$	−1.94248	−0.0661997
$862$	27.9692	0.952634
$863$	57.9588	1.97294	0.986470	−	0.163945i	$-0.0524219\pi$
$863$	57.9588	1.97294	0.986470	+	0.163945i	$0.0524219\pi$
$864$	6.87101	0.233756
$865$	−1.58072	−0.0537462
$866$	−20.0078	−0.679894
$867$	46.2246	1.56987
$868$	−0.334249	−0.0113452
$869$	0.144284	0.00489451
$870$	−2.11974	−0.0718660
$871$	34.6790	1.17505
$872$	−0.712207	−0.0241184
$873$	−41.5531	−1.40636
$874$	−7.36126	−0.248998
$875$	0.260147	0.00879456
$876$	0.931085	0.0314584
$877$	−46.6893	−1.57659	−0.788293	−	0.615300i	$-0.789035\pi$
$877$	−46.6893	−1.57659	−0.788293	+	0.615300i	$0.789035\pi$
$878$	24.3952	0.823297
$879$	84.4587	2.84872
$880$	−0.517989	−0.0174614
$881$	10.8430	0.365311	0.182655	−	0.983177i	$-0.441531\pi$
$881$	10.8430	0.365311	0.182655	+	0.983177i	$0.441531\pi$
$882$	37.5956	1.26591
$883$	48.6717	1.63793	0.818966	−	0.573842i	$-0.194547\pi$
$883$	48.6717	1.63793	0.818966	+	0.573842i	$0.194547\pi$
$884$	5.03180	0.169238
$885$	−0.334247	−0.0112356
$886$	−2.72675	−0.0916069
$887$	29.6219	0.994605	0.497302	−	0.867577i	$-0.334324\pi$
$887$	29.6219	0.994605	0.497302	+	0.867577i	$0.334324\pi$
$888$	−15.3211	−0.514143
$889$	−0.459890	−0.0154242
$890$	−0.543790	−0.0182279
$891$	−5.01950	−0.168159
$892$	6.53290	0.218738
$893$	−8.16670	−0.273288
$894$	−56.6476	−1.89458
$895$	6.28638	0.210130
$896$	0.0680600	0.00227372
$897$	−42.7390	−1.42701
$898$	−10.2651	−0.342551
$899$	−9.26978	−0.309165
$900$	−26.0624	−0.868745
$901$	−9.33748	−0.311077
$902$	−13.1642	−0.438320
$903$	1.12954	0.0375886
$904$	−7.30367	−0.242916
$905$	2.49392	0.0829006
$906$	68.4914	2.27548
$907$	−54.7542	−1.81808	−0.909041	−	0.416706i	$-0.863185\pi$
$907$	−54.7542	−1.81808	−0.909041	+	0.416706i	$0.863185\pi$
$908$	21.8118	0.723851
$909$	44.5270	1.47687
$910$	0.131168	0.00434819
$911$	31.2240	1.03450	0.517248	−	0.855835i	$-0.326956\pi$
$911$	31.2240	1.03450	0.517248	+	0.855835i	$0.326956\pi$
$912$	−7.16308	−0.237193
$913$	8.22945	0.272355
$914$	27.1882	0.899306
$915$	−9.20219	−0.304215
$916$	23.4525	0.774892
$917$	−0.605992	−0.0200116
$918$	6.96182	0.229774
$919$	6.52704	0.215307	0.107654	−	0.994188i	$-0.465666\pi$
$919$	6.52704	0.215307	0.107654	+	0.994188i	$0.465666\pi$
$920$	−1.15410	−0.0380497
$921$	16.5874	0.546574
$922$	6.77414	0.223094
$923$	−18.2096	−0.599377
$924$	−0.262888	−0.00864839
$925$	25.6745	0.844173
$926$	3.89722	0.128070
$927$	−22.3851	−0.735224
$928$	1.88752	0.0619608
$929$	−23.4736	−0.770144	−0.385072	−	0.922887i	$-0.625823\pi$
$929$	−23.4736	−0.770144	−0.385072	+	0.922887i	$0.625823\pi$
$930$	5.51533	0.180855
$931$	−17.3155	−0.567492
$932$	11.6249	0.380786
$933$	−13.6681	−0.447475
$934$	24.3340	0.796233
$935$	−0.524835	−0.0171639
$936$	−26.6899	−0.872387
$937$	−11.6505	−0.380606	−0.190303	−	0.981725i	$-0.560947\pi$
$937$	−11.6505	−0.380606	−0.190303	+	0.981725i	$0.560947\pi$
$938$	0.475267	0.0155180
$939$	56.0415	1.82885
$940$	−1.28038	−0.0417614
$941$	−1.89081	−0.0616386	−0.0308193	−	0.999525i	$-0.509812\pi$
$941$	−1.89081	−0.0616386	−0.0308193	+	0.999525i	$0.509812\pi$
$942$	−5.50055	−0.179218
$943$	−29.3304	−0.955130
$944$	0.297629	0.00968700
$945$	0.181480	0.00590355
$946$	7.65485	0.248881
$947$	−23.6901	−0.769825	−0.384913	−	0.922953i	$-0.625768\pi$
$947$	−23.6901	−0.769825	−0.384913	+	0.922953i	$0.625768\pi$
$948$	0.312817	0.0101598
$949$	−1.59785	−0.0518683
$950$	12.0036	0.389449
$951$	53.0186	1.71924
$952$	0.0689595	0.00223499
$953$	−49.5909	−1.60641	−0.803204	−	0.595704i	$-0.796873\pi$
$953$	−49.5909	−1.60641	−0.803204	+	0.595704i	$0.796873\pi$
$954$	49.5283	1.60354
$955$	−2.56759	−0.0830852
$956$	22.7372	0.735373
$957$	−7.29072	−0.235675
$958$	−23.5302	−0.760227
$959$	0.569685	0.0183961
$960$	−1.12303	−0.0362457
$961$	−6.88109	−0.221971
$962$	26.2927	0.847711
$963$	−51.6917	−1.66574
$964$	14.0966	0.454020
$965$	−4.14772	−0.133520
$966$	−0.585727	−0.0188455
$967$	−34.1232	−1.09733	−0.548665	−	0.836043i	$-0.684864\pi$
$967$	−34.1232	−1.09733	−0.548665	+	0.836043i	$0.684864\pi$
$968$	9.21841	0.296291
$969$	−7.25775	−0.233152
$970$	3.00051	0.0963405
$971$	3.71501	0.119220	0.0596102	−	0.998222i	$-0.481014\pi$
$971$	3.71501	0.119220	0.0596102	+	0.998222i	$0.481014\pi$
$972$	9.73044	0.312104
$973$	1.38047	0.0442559
$974$	0.527114	0.0168898
$975$	69.6922	2.23194
$976$	8.19406	0.262285
$977$	6.99521	0.223796	0.111898	−	0.993720i	$-0.464307\pi$
$977$	6.99521	0.223796	0.111898	+	0.993720i	$0.464307\pi$
$978$	−72.1376	−2.30671
$979$	−1.87033	−0.0597760
$980$	−2.71473	−0.0867190
$981$	3.82765	0.122207
$982$	0.932548	0.0297588
$983$	−55.5810	−1.77276	−0.886379	−	0.462960i	$-0.846787\pi$
$983$	−55.5810	−1.77276	−0.886379	+	0.462960i	$0.846787\pi$
$984$	−28.5408	−0.909847
$985$	−0.582245	−0.0185519
$986$	1.91246	0.0609052
$987$	−0.649815	−0.0206838
$988$	12.2926	0.391081
$989$	17.0554	0.542329
$990$	2.78385	0.0884767
$991$	29.5627	0.939090	0.469545	−	0.882909i	$-0.344418\pi$
$991$	29.5627	0.939090	0.469545	+	0.882909i	$0.344418\pi$
$992$	−4.91110	−0.155928
$993$	34.6607	1.09992
$994$	−0.249558	−0.00791550
$995$	9.04738	0.286821
$996$	17.8419	0.565344
$997$	−5.25980	−0.166580	−0.0832898	−	0.996525i	$-0.526543\pi$
$997$	−5.25980	−0.166580	−0.0832898	+	0.996525i	$0.526543\pi$
$998$	−6.47097	−0.204835
$999$	36.3777	1.15094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.c.1.9	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.c.1.9	✓	84	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.89385 q^{3} +1.00000 q^{4} +0.388076 q^{5} +2.89385 q^{6} -0.0680600 q^{7} -1.00000 q^{8} +5.37435 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.89385 q^{3} +1.00000 q^{4} +0.388076 q^{5} +2.89385 q^{6} -0.0680600 q^{7} -1.00000 q^{8} +5.37435 q^{9} -0.388076 q^{10} -1.33476 q^{11} -2.89385 q^{12} +4.96616 q^{13} +0.0680600 q^{14} -1.12303 q^{15} +1.00000 q^{16} +1.01322 q^{17} -5.37435 q^{18} +2.47528 q^{19} +0.388076 q^{20} +0.196955 q^{21} +1.33476 q^{22} +2.97391 q^{23} +2.89385 q^{24} -4.84940 q^{25} -4.96616 q^{26} -6.87101 q^{27} -0.0680600 q^{28} -1.88752 q^{29} +1.12303 q^{30} +4.91110 q^{31} -1.00000 q^{32} +3.86260 q^{33} -1.01322 q^{34} -0.0264124 q^{35} +5.37435 q^{36} -5.29437 q^{37} -2.47528 q^{38} -14.3713 q^{39} -0.388076 q^{40} -9.86257 q^{41} -0.196955 q^{42} +5.73499 q^{43} -1.33476 q^{44} +2.08566 q^{45} -2.97391 q^{46} -3.29931 q^{47} -2.89385 q^{48} -6.99537 q^{49} +4.84940 q^{50} -2.93209 q^{51} +4.96616 q^{52} -9.21569 q^{53} +6.87101 q^{54} -0.517989 q^{55} +0.0680600 q^{56} -7.16308 q^{57} +1.88752 q^{58} +0.297629 q^{59} -1.12303 q^{60} +8.19406 q^{61} -4.91110 q^{62} -0.365778 q^{63} +1.00000 q^{64} +1.92725 q^{65} -3.86260 q^{66} +6.98306 q^{67} +1.01322 q^{68} -8.60605 q^{69} +0.0264124 q^{70} -3.66674 q^{71} -5.37435 q^{72} -0.321746 q^{73} +5.29437 q^{74} +14.0334 q^{75} +2.47528 q^{76} +0.0908439 q^{77} +14.3713 q^{78} -0.108097 q^{79} +0.388076 q^{80} +3.76059 q^{81} +9.86257 q^{82} -6.16548 q^{83} +0.196955 q^{84} +0.393205 q^{85} -5.73499 q^{86} +5.46218 q^{87} +1.33476 q^{88} +1.40125 q^{89} -2.08566 q^{90} -0.337997 q^{91} +2.97391 q^{92} -14.2120 q^{93} +3.29931 q^{94} +0.960596 q^{95} +2.89385 q^{96} -7.73175 q^{97} +6.99537 q^{98} -7.17348 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more