LMFDB - Embedded newform 67.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [67,2,Mod(1,67)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(67, 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("67.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	67.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:	$0.534997693543$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	67.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{2} -2.61803 q^{3} -1.85410 q^{4} -3.00000 q^{5} +1.00000 q^{6} +2.85410 q^{7} +1.47214 q^{8} +3.85410 q^{9} +O(q^{10})$	\(q-0.381966 q^{2} -2.61803 q^{3} -1.85410 q^{4} -3.00000 q^{5} +1.00000 q^{6} +2.85410 q^{7} +1.47214 q^{8} +3.85410 q^{9} +1.14590 q^{10} -2.23607 q^{11} +4.85410 q^{12} -6.85410 q^{13} -1.09017 q^{14} +7.85410 q^{15} +3.14590 q^{16} -5.23607 q^{17} -1.47214 q^{18} +3.85410 q^{19} +5.56231 q^{20} -7.47214 q^{21} +0.854102 q^{22} -1.47214 q^{23} -3.85410 q^{24} +4.00000 q^{25} +2.61803 q^{26} -2.23607 q^{27} -5.29180 q^{28} +1.47214 q^{29} -3.00000 q^{30} -1.00000 q^{31} -4.14590 q^{32} +5.85410 q^{33} +2.00000 q^{34} -8.56231 q^{35} -7.14590 q^{36} +2.85410 q^{37} -1.47214 q^{38} +17.9443 q^{39} -4.41641 q^{40} -2.61803 q^{41} +2.85410 q^{42} -1.85410 q^{43} +4.14590 q^{44} -11.5623 q^{45} +0.562306 q^{46} -6.38197 q^{47} -8.23607 q^{48} +1.14590 q^{49} -1.52786 q^{50} +13.7082 q^{51} +12.7082 q^{52} -9.00000 q^{53} +0.854102 q^{54} +6.70820 q^{55} +4.20163 q^{56} -10.0902 q^{57} -0.562306 q^{58} +6.00000 q^{59} -14.5623 q^{60} +6.56231 q^{61} +0.381966 q^{62} +11.0000 q^{63} -4.70820 q^{64} +20.5623 q^{65} -2.23607 q^{66} -1.00000 q^{67} +9.70820 q^{68} +3.85410 q^{69} +3.27051 q^{70} +8.23607 q^{71} +5.67376 q^{72} -4.00000 q^{73} -1.09017 q^{74} -10.4721 q^{75} -7.14590 q^{76} -6.38197 q^{77} -6.85410 q^{78} -13.5623 q^{79} -9.43769 q^{80} -5.70820 q^{81} +1.00000 q^{82} +0.326238 q^{83} +13.8541 q^{84} +15.7082 q^{85} +0.708204 q^{86} -3.85410 q^{87} -3.29180 q^{88} +2.23607 q^{89} +4.41641 q^{90} -19.5623 q^{91} +2.72949 q^{92} +2.61803 q^{93} +2.43769 q^{94} -11.5623 q^{95} +10.8541 q^{96} -12.4164 q^{97} -0.437694 q^{98} -8.61803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} - q^{7} - 6 q^{8} + q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 6 * q^5 + 2 * q^6 - q^7 - 6 * q^8 + q^9	\( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} - q^{7} - 6 q^{8} + q^{9} + 9 q^{10} + 3 q^{12} - 7 q^{13} + 9 q^{14} + 9 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} + q^{19} - 9 q^{20} - 6 q^{21} - 5 q^{22} + 6 q^{23} - q^{24} + 8 q^{25} + 3 q^{26} - 24 q^{28} - 6 q^{29} - 6 q^{30} - 2 q^{31} - 15 q^{32} + 5 q^{33} + 4 q^{34} + 3 q^{35} - 21 q^{36} - q^{37} + 6 q^{38} + 18 q^{39} + 18 q^{40} - 3 q^{41} - q^{42} + 3 q^{43} + 15 q^{44} - 3 q^{45} - 19 q^{46} - 15 q^{47} - 12 q^{48} + 9 q^{49} - 12 q^{50} + 14 q^{51} + 12 q^{52} - 18 q^{53} - 5 q^{54} + 33 q^{56} - 9 q^{57} + 19 q^{58} + 12 q^{59} - 9 q^{60} - 7 q^{61} + 3 q^{62} + 22 q^{63} + 4 q^{64} + 21 q^{65} - 2 q^{67} + 6 q^{68} + q^{69} - 27 q^{70} + 12 q^{71} + 27 q^{72} - 8 q^{73} + 9 q^{74} - 12 q^{75} - 21 q^{76} - 15 q^{77} - 7 q^{78} - 7 q^{79} - 39 q^{80} + 2 q^{81} + 2 q^{82} - 15 q^{83} + 21 q^{84} + 18 q^{85} - 12 q^{86} - q^{87} - 20 q^{88} - 18 q^{90} - 19 q^{91} + 39 q^{92} + 3 q^{93} + 25 q^{94} - 3 q^{95} + 15 q^{96} + 2 q^{97} - 21 q^{98} - 15 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 6 * q^5 + 2 * q^6 - q^7 - 6 * q^8 + q^9 + 9 * q^10 + 3 * q^12 - 7 * q^13 + 9 * q^14 + 9 * q^15 + 13 * q^16 - 6 * q^17 + 6 * q^18 + q^19 - 9 * q^20 - 6 * q^21 - 5 * q^22 + 6 * q^23 - q^24 + 8 * q^25 + 3 * q^26 - 24 * q^28 - 6 * q^29 - 6 * q^30 - 2 * q^31 - 15 * q^32 + 5 * q^33 + 4 * q^34 + 3 * q^35 - 21 * q^36 - q^37 + 6 * q^38 + 18 * q^39 + 18 * q^40 - 3 * q^41 - q^42 + 3 * q^43 + 15 * q^44 - 3 * q^45 - 19 * q^46 - 15 * q^47 - 12 * q^48 + 9 * q^49 - 12 * q^50 + 14 * q^51 + 12 * q^52 - 18 * q^53 - 5 * q^54 + 33 * q^56 - 9 * q^57 + 19 * q^58 + 12 * q^59 - 9 * q^60 - 7 * q^61 + 3 * q^62 + 22 * q^63 + 4 * q^64 + 21 * q^65 - 2 * q^67 + 6 * q^68 + q^69 - 27 * q^70 + 12 * q^71 + 27 * q^72 - 8 * q^73 + 9 * q^74 - 12 * q^75 - 21 * q^76 - 15 * q^77 - 7 * q^78 - 7 * q^79 - 39 * q^80 + 2 * q^81 + 2 * q^82 - 15 * q^83 + 21 * q^84 + 18 * q^85 - 12 * q^86 - q^87 - 20 * q^88 - 18 * q^90 - 19 * q^91 + 39 * q^92 + 3 * q^93 + 25 * q^94 - 3 * q^95 + 15 * q^96 + 2 * q^97 - 21 * q^98 - 15 * 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 $

$2$	−0.381966	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$2$	−0.381966	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$3$	−2.61803	−1.51152	−0.755761	−	0.654847i	$-0.772733\pi$
$3$	−2.61803	−1.51152	−0.755761	+	0.654847i	$0.772733\pi$
$4$	−1.85410	−0.927051
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	1.00000	0.408248
$7$	2.85410	1.07875	0.539375	−	0.842066i	$-0.318661\pi$
$7$	2.85410	1.07875	0.539375	+	0.842066i	$0.318661\pi$
$8$	1.47214	0.520479
$9$	3.85410	1.28470
$10$	1.14590	0.362365
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	4.85410	1.40126
$13$	−6.85410	−1.90099	−0.950493	−	0.310746i	$-0.899421\pi$
$13$	−6.85410	−1.90099	−0.950493	+	0.310746i	$0.899421\pi$
$14$	−1.09017	−0.291360
$15$	7.85410	2.02792
$16$	3.14590	0.786475
$17$	−5.23607	−1.26993	−0.634967	−	0.772540i	$-0.718986\pi$
$17$	−5.23607	−1.26993	−0.634967	+	0.772540i	$0.718986\pi$
$18$	−1.47214	−0.346986
$19$	3.85410	0.884192	0.442096	−	0.896968i	$-0.354235\pi$
$19$	3.85410	0.884192	0.442096	+	0.896968i	$0.354235\pi$
$20$	5.56231	1.24377
$21$	−7.47214	−1.63055
$22$	0.854102	0.182095
$23$	−1.47214	−0.306962	−0.153481	−	0.988152i	$-0.549048\pi$
$23$	−1.47214	−0.306962	−0.153481	+	0.988152i	$0.549048\pi$
$24$	−3.85410	−0.786715
$25$	4.00000	0.800000
$26$	2.61803	0.513439
$27$	−2.23607	−0.430331
$28$	−5.29180	−1.00006
$29$	1.47214	0.273369	0.136684	−	0.990615i	$-0.456355\pi$
$29$	1.47214	0.273369	0.136684	+	0.990615i	$0.456355\pi$
$30$	−3.00000	−0.547723
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	−4.14590	−0.732898
$33$	5.85410	1.01907
$34$	2.00000	0.342997
$35$	−8.56231	−1.44729
$36$	−7.14590	−1.19098
$37$	2.85410	0.469211	0.234606	−	0.972091i	$-0.424620\pi$
$37$	2.85410	0.469211	0.234606	+	0.972091i	$0.424620\pi$
$38$	−1.47214	−0.238812
$39$	17.9443	2.87338
$40$	−4.41641	−0.698295
$41$	−2.61803	−0.408868	−0.204434	−	0.978880i	$-0.565535\pi$
$41$	−2.61803	−0.408868	−0.204434	+	0.978880i	$0.565535\pi$
$42$	2.85410	0.440397
$43$	−1.85410	−0.282748	−0.141374	−	0.989956i	$-0.545152\pi$
$43$	−1.85410	−0.282748	−0.141374	+	0.989956i	$0.545152\pi$
$44$	4.14590	0.625018
$45$	−11.5623	−1.72361
$46$	0.562306	0.0829075
$47$	−6.38197	−0.930905	−0.465453	−	0.885073i	$-0.654108\pi$
$47$	−6.38197	−0.930905	−0.465453	+	0.885073i	$0.654108\pi$
$48$	−8.23607	−1.18877
$49$	1.14590	0.163700
$50$	−1.52786	−0.216073
$51$	13.7082	1.91953
$52$	12.7082	1.76231
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0.854102	0.116229
$55$	6.70820	0.904534
$56$	4.20163	0.561466
$57$	−10.0902	−1.33648
$58$	−0.562306	−0.0738344
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	−14.5623	−1.87999
$61$	6.56231	0.840217	0.420109	−	0.907474i	$-0.361992\pi$
$61$	6.56231	0.840217	0.420109	+	0.907474i	$0.361992\pi$
$62$	0.381966	0.0485097
$63$	11.0000	1.38587
$64$	−4.70820	−0.588525
$65$	20.5623	2.55044
$66$	−2.23607	−0.275241
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	9.70820	1.17729
$69$	3.85410	0.463979
$70$	3.27051	0.390901
$71$	8.23607	0.977441	0.488721	−	0.872440i	$-0.337464\pi$
$71$	8.23607	0.977441	0.488721	+	0.872440i	$0.337464\pi$
$72$	5.67376	0.668659
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−1.09017	−0.126730
$75$	−10.4721	−1.20922
$76$	−7.14590	−0.819691
$77$	−6.38197	−0.727293
$78$	−6.85410	−0.776074
$79$	−13.5623	−1.52588	−0.762939	−	0.646470i	$-0.776245\pi$
$79$	−13.5623	−1.52588	−0.762939	+	0.646470i	$0.776245\pi$
$80$	−9.43769	−1.05517
$81$	−5.70820	−0.634245
$82$	1.00000	0.110432
$83$	0.326238	0.0358093	0.0179046	−	0.999840i	$-0.494300\pi$
$83$	0.326238	0.0358093	0.0179046	+	0.999840i	$0.494300\pi$
$84$	13.8541	1.51161
$85$	15.7082	1.70379
$86$	0.708204	0.0763676
$87$	−3.85410	−0.413203
$88$	−3.29180	−0.350907
$89$	2.23607	0.237023	0.118511	−	0.992953i	$-0.462188\pi$
$89$	2.23607	0.237023	0.118511	+	0.992953i	$0.462188\pi$
$90$	4.41641	0.465530
$91$	−19.5623	−2.05069
$92$	2.72949	0.284569
$93$	2.61803	0.271477
$94$	2.43769	0.251429
$95$	−11.5623	−1.18627
$96$	10.8541	1.10779
$97$	−12.4164	−1.26070	−0.630348	−	0.776313i	$-0.717088\pi$
$97$	−12.4164	−1.26070	−0.630348	+	0.776313i	$0.717088\pi$
$98$	−0.437694	−0.0442138
$99$	−8.61803	−0.866145
$100$	−7.41641	−0.741641
$101$	1.09017	0.108476	0.0542380	−	0.998528i	$-0.482727\pi$
$101$	1.09017	0.108476	0.0542380	+	0.998528i	$0.482727\pi$
$102$	−5.23607	−0.518448
$103$	−12.5623	−1.23780	−0.618900	−	0.785469i	$-0.712422\pi$
$103$	−12.5623	−1.23780	−0.618900	+	0.785469i	$0.712422\pi$
$104$	−10.0902	−0.989423
$105$	22.4164	2.18762
$106$	3.43769	0.333898
$107$	18.7082	1.80859	0.904295	−	0.426908i	$-0.140397\pi$
$107$	18.7082	1.80859	0.904295	+	0.426908i	$0.140397\pi$
$108$	4.14590	0.398939
$109$	1.85410	0.177591	0.0887954	−	0.996050i	$-0.471698\pi$
$109$	1.85410	0.177591	0.0887954	+	0.996050i	$0.471698\pi$
$110$	−2.56231	−0.244306
$111$	−7.47214	−0.709224
$112$	8.97871	0.848409
$113$	17.6180	1.65737	0.828683	−	0.559719i	$-0.189091\pi$
$113$	17.6180	1.65737	0.828683	+	0.559719i	$0.189091\pi$
$114$	3.85410	0.360970
$115$	4.41641	0.411832
$116$	−2.72949	−0.253427
$117$	−26.4164	−2.44220
$118$	−2.29180	−0.210977
$119$	−14.9443	−1.36994
$120$	11.5623	1.05549
$121$	−6.00000	−0.545455
$122$	−2.50658	−0.226935
$123$	6.85410	0.618014
$124$	1.85410	0.166503
$125$	3.00000	0.268328
$126$	−4.20163	−0.374311
$127$	0.854102	0.0757893	0.0378946	−	0.999282i	$-0.487935\pi$
$127$	0.854102	0.0757893	0.0378946	+	0.999282i	$0.487935\pi$
$128$	10.0902	0.891853
$129$	4.85410	0.427380
$130$	−7.85410	−0.688850
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	−10.8541	−0.944728
$133$	11.0000	0.953821
$134$	0.381966	0.0329968
$135$	6.70820	0.577350
$136$	−7.70820	−0.660973
$137$	−12.7082	−1.08574	−0.542868	−	0.839818i	$-0.682661\pi$
$137$	−12.7082	−1.08574	−0.542868	+	0.839818i	$0.682661\pi$
$138$	−1.47214	−0.125317
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	15.8754	1.34172
$141$	16.7082	1.40708
$142$	−3.14590	−0.263998
$143$	15.3262	1.28164
$144$	12.1246	1.01038
$145$	−4.41641	−0.366763
$146$	1.52786	0.126447
$147$	−3.00000	−0.247436
$148$	−5.29180	−0.434983
$149$	−10.0344	−0.822054	−0.411027	−	0.911623i	$-0.634830\pi$
$149$	−10.0344	−0.822054	−0.411027	+	0.911623i	$0.634830\pi$
$150$	4.00000	0.326599
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	5.67376	0.460203
$153$	−20.1803	−1.63148
$154$	2.43769	0.196435
$155$	3.00000	0.240966
$156$	−33.2705	−2.66377
$157$	16.5623	1.32182	0.660908	−	0.750467i	$-0.270171\pi$
$157$	16.5623	1.32182	0.660908	+	0.750467i	$0.270171\pi$
$158$	5.18034	0.412126
$159$	23.5623	1.86861
$160$	12.4377	0.983286
$161$	−4.20163	−0.331135
$162$	2.18034	0.171304
$163$	−0.145898	−0.0114276	−0.00571381	−	0.999984i	$-0.501819\pi$
$163$	−0.145898	−0.0114276	−0.00571381	+	0.999984i	$0.501819\pi$
$164$	4.85410	0.379042
$165$	−17.5623	−1.36722
$166$	−0.124612	−0.00967175
$167$	23.8885	1.84855	0.924276	−	0.381726i	$-0.124670\pi$
$167$	23.8885	1.84855	0.924276	+	0.381726i	$0.124670\pi$
$168$	−11.0000	−0.848668
$169$	33.9787	2.61375
$170$	−6.00000	−0.460179
$171$	14.8541	1.13592
$172$	3.43769	0.262122
$173$	−25.0902	−1.90757	−0.953785	−	0.300489i	$-0.902850\pi$
$173$	−25.0902	−1.90757	−0.953785	+	0.300489i	$0.902850\pi$
$174$	1.47214	0.111602
$175$	11.4164	0.862999
$176$	−7.03444	−0.530241
$177$	−15.7082	−1.18070
$178$	−0.854102	−0.0640176
$179$	−11.2361	−0.839823	−0.419912	−	0.907565i	$-0.637939\pi$
$179$	−11.2361	−0.839823	−0.419912	+	0.907565i	$0.637939\pi$
$180$	21.4377	1.59787
$181$	−13.5623	−1.00808	−0.504039	−	0.863681i	$-0.668153\pi$
$181$	−13.5623	−1.00808	−0.504039	+	0.863681i	$0.668153\pi$
$182$	7.47214	0.553872
$183$	−17.1803	−1.27001
$184$	−2.16718	−0.159767
$185$	−8.56231	−0.629513
$186$	−1.00000	−0.0733236
$187$	11.7082	0.856189
$188$	11.8328	0.862997
$189$	−6.38197	−0.464220
$190$	4.41641	0.320400
$191$	−7.47214	−0.540665	−0.270332	−	0.962767i	$-0.587134\pi$
$191$	−7.47214	−0.540665	−0.270332	+	0.962767i	$0.587134\pi$
$192$	12.3262	0.889570
$193$	10.1459	0.730318	0.365159	−	0.930945i	$-0.381015\pi$
$193$	10.1459	0.730318	0.365159	+	0.930945i	$0.381015\pi$
$194$	4.74265	0.340502
$195$	−53.8328	−3.85505
$196$	−2.12461	−0.151758
$197$	−13.4721	−0.959850	−0.479925	−	0.877310i	$-0.659336\pi$
$197$	−13.4721	−0.959850	−0.479925	+	0.877310i	$0.659336\pi$
$198$	3.29180	0.233938
$199$	−23.2705	−1.64960	−0.824801	−	0.565423i	$-0.808713\pi$
$199$	−23.2705	−1.64960	−0.824801	+	0.565423i	$0.808713\pi$
$200$	5.88854	0.416383
$201$	2.61803	0.184662
$202$	−0.416408	−0.0292984
$203$	4.20163	0.294896
$204$	−25.4164	−1.77950
$205$	7.85410	0.548554
$206$	4.79837	0.334319
$207$	−5.67376	−0.394354
$208$	−21.5623	−1.49508
$209$	−8.61803	−0.596122
$210$	−8.56231	−0.590855
$211$	−8.85410	−0.609542	−0.304771	−	0.952426i	$-0.598580\pi$
$211$	−8.85410	−0.609542	−0.304771	+	0.952426i	$0.598580\pi$
$212$	16.6869	1.14606
$213$	−21.5623	−1.47742
$214$	−7.14590	−0.488484
$215$	5.56231	0.379346
$216$	−3.29180	−0.223978
$217$	−2.85410	−0.193749
$218$	−0.708204	−0.0479656
$219$	10.4721	0.707641
$220$	−12.4377	−0.838549
$221$	35.8885	2.41412
$222$	2.85410	0.191555
$223$	2.29180	0.153470	0.0767350	−	0.997052i	$-0.475550\pi$
$223$	2.29180	0.153470	0.0767350	+	0.997052i	$0.475550\pi$
$224$	−11.8328	−0.790613
$225$	15.4164	1.02776
$226$	−6.72949	−0.447639
$227$	17.5066	1.16195	0.580976	−	0.813921i	$-0.302671\pi$
$227$	17.5066	1.16195	0.580976	+	0.813921i	$0.302671\pi$
$228$	18.7082	1.23898
$229$	−12.1459	−0.802624	−0.401312	−	0.915942i	$-0.631446\pi$
$229$	−12.1459	−0.802624	−0.401312	+	0.915942i	$0.631446\pi$
$230$	−1.68692	−0.111232
$231$	16.7082	1.09932
$232$	2.16718	0.142283
$233$	10.4721	0.686052	0.343026	−	0.939326i	$-0.388548\pi$
$233$	10.4721	0.686052	0.343026	+	0.939326i	$0.388548\pi$
$234$	10.0902	0.659615
$235$	19.1459	1.24894
$236$	−11.1246	−0.724151
$237$	35.5066	2.30640
$238$	5.70820	0.370008
$239$	−7.90983	−0.511644	−0.255822	−	0.966724i	$-0.582346\pi$
$239$	−7.90983	−0.511644	−0.255822	+	0.966724i	$0.582346\pi$
$240$	24.7082	1.59491
$241$	19.1459	1.23330	0.616648	−	0.787239i	$-0.288490\pi$
$241$	19.1459	1.23330	0.616648	+	0.787239i	$0.288490\pi$
$242$	2.29180	0.147322
$243$	21.6525	1.38901
$244$	−12.1672	−0.778924
$245$	−3.43769	−0.219626
$246$	−2.61803	−0.166920
$247$	−26.4164	−1.68084
$248$	−1.47214	−0.0934807
$249$	−0.854102	−0.0541265
$250$	−1.14590	−0.0724730
$251$	6.76393	0.426936	0.213468	−	0.976950i	$-0.431524\pi$
$251$	6.76393	0.426936	0.213468	+	0.976950i	$0.431524\pi$
$252$	−20.3951	−1.28477
$253$	3.29180	0.206953
$254$	−0.326238	−0.0204700
$255$	−41.1246	−2.57532
$256$	5.56231	0.347644
$257$	8.88854	0.554452	0.277226	−	0.960805i	$-0.410585\pi$
$257$	8.88854	0.554452	0.277226	+	0.960805i	$0.410585\pi$
$258$	−1.85410	−0.115431
$259$	8.14590	0.506161
$260$	−38.1246	−2.36439
$261$	5.67376	0.351197
$262$	1.14590	0.0707938
$263$	−11.5623	−0.712962	−0.356481	−	0.934303i	$-0.616024\pi$
$263$	−11.5623	−0.712962	−0.356481	+	0.934303i	$0.616024\pi$
$264$	8.61803	0.530403
$265$	27.0000	1.65860
$266$	−4.20163	−0.257618
$267$	−5.85410	−0.358265
$268$	1.85410	0.113257
$269$	−7.79837	−0.475475	−0.237738	−	0.971329i	$-0.576406\pi$
$269$	−7.79837	−0.475475	−0.237738	+	0.971329i	$0.576406\pi$
$270$	−2.56231	−0.155937
$271$	11.2918	0.685928	0.342964	−	0.939349i	$-0.388569\pi$
$271$	11.2918	0.685928	0.342964	+	0.939349i	$0.388569\pi$
$272$	−16.4721	−0.998770
$273$	51.2148	3.09966
$274$	4.85410	0.293247
$275$	−8.94427	−0.539360
$276$	−7.14590	−0.430133
$277$	−20.8328	−1.25172	−0.625861	−	0.779934i	$-0.715252\pi$
$277$	−20.8328	−1.25172	−0.625861	+	0.779934i	$0.715252\pi$
$278$	1.14590	0.0687264
$279$	−3.85410	−0.230739
$280$	−12.6049	−0.753286
$281$	6.38197	0.380716	0.190358	−	0.981715i	$-0.439035\pi$
$281$	6.38197	0.380716	0.190358	+	0.981715i	$0.439035\pi$
$282$	−6.38197	−0.380041
$283$	4.41641	0.262528	0.131264	−	0.991347i	$-0.458096\pi$
$283$	4.41641	0.262528	0.131264	+	0.991347i	$0.458096\pi$
$284$	−15.2705	−0.906138
$285$	30.2705	1.79307
$286$	−5.85410	−0.346160
$287$	−7.47214	−0.441066
$288$	−15.9787	−0.941555
$289$	10.4164	0.612730
$290$	1.68692	0.0990592
$291$	32.5066	1.90557
$292$	7.41641	0.434012
$293$	−22.4164	−1.30958	−0.654790	−	0.755811i	$-0.727243\pi$
$293$	−22.4164	−1.30958	−0.654790	+	0.755811i	$0.727243\pi$
$294$	1.14590	0.0668301
$295$	−18.0000	−1.04800
$296$	4.20163	0.244215
$297$	5.00000	0.290129
$298$	3.83282	0.222029
$299$	10.0902	0.583530
$300$	19.4164	1.12101
$301$	−5.29180	−0.305014
$302$	0.381966	0.0219797
$303$	−2.85410	−0.163964
$304$	12.1246	0.695394
$305$	−19.6869	−1.12727
$306$	7.70820	0.440649
$307$	−9.41641	−0.537423	−0.268711	−	0.963221i	$-0.586598\pi$
$307$	−9.41641	−0.537423	−0.268711	+	0.963221i	$0.586598\pi$
$308$	11.8328	0.674237
$309$	32.8885	1.87096
$310$	−1.14590	−0.0650826
$311$	26.8328	1.52155	0.760775	−	0.649016i	$-0.224819\pi$
$311$	26.8328	1.52155	0.760775	+	0.649016i	$0.224819\pi$
$312$	26.4164	1.49553
$313$	−7.14590	−0.403910	−0.201955	−	0.979395i	$-0.564729\pi$
$313$	−7.14590	−0.403910	−0.201955	+	0.979395i	$0.564729\pi$
$314$	−6.32624	−0.357010
$315$	−33.0000	−1.85934
$316$	25.1459	1.41457
$317$	−30.5967	−1.71848	−0.859242	−	0.511569i	$-0.829065\pi$
$317$	−30.5967	−1.71848	−0.859242	+	0.511569i	$0.829065\pi$
$318$	−9.00000	−0.504695
$319$	−3.29180	−0.184305
$320$	14.1246	0.789590
$321$	−48.9787	−2.73373
$322$	1.60488	0.0894364
$323$	−20.1803	−1.12286
$324$	10.5836	0.587977
$325$	−27.4164	−1.52079
$326$	0.0557281	0.00308649
$327$	−4.85410	−0.268432
$328$	−3.85410	−0.212807
$329$	−18.2148	−1.00421
$330$	6.70820	0.369274
$331$	23.4164	1.28708	0.643541	−	0.765412i	$-0.277465\pi$
$331$	23.4164	1.28708	0.643541	+	0.765412i	$0.277465\pi$
$332$	−0.604878	−0.0331970
$333$	11.0000	0.602796
$334$	−9.12461	−0.499277
$335$	3.00000	0.163908
$336$	−23.5066	−1.28239
$337$	25.2705	1.37657	0.688286	−	0.725439i	$-0.258363\pi$
$337$	25.2705	1.37657	0.688286	+	0.725439i	$0.258363\pi$
$338$	−12.9787	−0.705949
$339$	−46.1246	−2.50515
$340$	−29.1246	−1.57950
$341$	2.23607	0.121090
$342$	−5.67376	−0.306802
$343$	−16.7082	−0.902158
$344$	−2.72949	−0.147164
$345$	−11.5623	−0.622494
$346$	9.58359	0.515217
$347$	11.1803	0.600192	0.300096	−	0.953909i	$-0.402981\pi$
$347$	11.1803	0.600192	0.300096	+	0.953909i	$0.402981\pi$
$348$	7.14590	0.383060
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	−4.36068	−0.233088
$351$	15.3262	0.818054
$352$	9.27051	0.494120
$353$	20.5066	1.09146	0.545728	−	0.837963i	$-0.316253\pi$
$353$	20.5066	1.09146	0.545728	+	0.837963i	$0.316253\pi$
$354$	6.00000	0.318896
$355$	−24.7082	−1.31138
$356$	−4.14590	−0.219732
$357$	39.1246	2.07069
$358$	4.29180	0.226828
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	−17.0213	−0.897101
$361$	−4.14590	−0.218205
$362$	5.18034	0.272273
$363$	15.7082	0.824467
$364$	36.2705	1.90109
$365$	12.0000	0.628109
$366$	6.56231	0.343017
$367$	21.1246	1.10270	0.551348	−	0.834275i	$-0.314114\pi$
$367$	21.1246	1.10270	0.551348	+	0.834275i	$0.314114\pi$
$368$	−4.63119	−0.241417
$369$	−10.0902	−0.525273
$370$	3.27051	0.170026
$371$	−25.6869	−1.33360
$372$	−4.85410	−0.251673
$373$	1.56231	0.0808931	0.0404466	−	0.999182i	$-0.487122\pi$
$373$	1.56231	0.0808931	0.0404466	+	0.999182i	$0.487122\pi$
$374$	−4.47214	−0.231249
$375$	−7.85410	−0.405584
$376$	−9.39512	−0.484516
$377$	−10.0902	−0.519670
$378$	2.43769	0.125381
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	21.4377	1.09973
$381$	−2.23607	−0.114557
$382$	2.85410	0.146029
$383$	−31.7984	−1.62482	−0.812411	−	0.583086i	$-0.801845\pi$
$383$	−31.7984	−1.62482	−0.812411	+	0.583086i	$0.801845\pi$
$384$	−26.4164	−1.34806
$385$	19.1459	0.975765
$386$	−3.87539	−0.197252
$387$	−7.14590	−0.363246
$388$	23.0213	1.16873
$389$	12.6525	0.641506	0.320753	−	0.947163i	$-0.396064\pi$
$389$	12.6525	0.641506	0.320753	+	0.947163i	$0.396064\pi$
$390$	20.5623	1.04121
$391$	7.70820	0.389821
$392$	1.68692	0.0852022
$393$	7.85410	0.396187
$394$	5.14590	0.259247
$395$	40.6869	2.04718
$396$	15.9787	0.802961
$397$	−15.4377	−0.774796	−0.387398	−	0.921913i	$-0.626626\pi$
$397$	−15.4377	−0.774796	−0.387398	+	0.921913i	$0.626626\pi$
$398$	8.88854	0.445542
$399$	−28.7984	−1.44172
$400$	12.5836	0.629180
$401$	1.14590	0.0572234	0.0286117	−	0.999591i	$-0.490891\pi$
$401$	1.14590	0.0572234	0.0286117	+	0.999591i	$0.490891\pi$
$402$	−1.00000	−0.0498755
$403$	6.85410	0.341427
$404$	−2.02129	−0.100563
$405$	17.1246	0.850929
$406$	−1.60488	−0.0796488
$407$	−6.38197	−0.316342
$408$	20.1803	0.999076
$409$	−27.1246	−1.34123	−0.670613	−	0.741808i	$-0.733969\pi$
$409$	−27.1246	−1.34123	−0.670613	+	0.741808i	$0.733969\pi$
$410$	−3.00000	−0.148159
$411$	33.2705	1.64111
$412$	23.2918	1.14750
$413$	17.1246	0.842647
$414$	2.16718	0.106511
$415$	−0.978714	−0.0480432
$416$	28.4164	1.39323
$417$	7.85410	0.384617
$418$	3.29180	0.161007
$419$	−22.3607	−1.09239	−0.546195	−	0.837658i	$-0.683924\pi$
$419$	−22.3607	−1.09239	−0.546195	+	0.837658i	$0.683924\pi$
$420$	−41.5623	−2.02803
$421$	−24.1459	−1.17680	−0.588400	−	0.808570i	$-0.700242\pi$
$421$	−24.1459	−1.17680	−0.588400	+	0.808570i	$0.700242\pi$
$422$	3.38197	0.164632
$423$	−24.5967	−1.19593
$424$	−13.2492	−0.643439
$425$	−20.9443	−1.01595
$426$	8.23607	0.399039
$427$	18.7295	0.906384
$428$	−34.6869	−1.67666
$429$	−40.1246	−1.93723
$430$	−2.12461	−0.102458
$431$	32.2361	1.55276	0.776378	−	0.630267i	$-0.217055\pi$
$431$	32.2361	1.55276	0.776378	+	0.630267i	$0.217055\pi$
$432$	−7.03444	−0.338445
$433$	33.3951	1.60487	0.802434	−	0.596741i	$-0.203538\pi$
$433$	33.3951	1.60487	0.802434	+	0.596741i	$0.203538\pi$
$434$	1.09017	0.0523298
$435$	11.5623	0.554370
$436$	−3.43769	−0.164636
$437$	−5.67376	−0.271413
$438$	−4.00000	−0.191127
$439$	21.6869	1.03506	0.517530	−	0.855665i	$-0.326852\pi$
$439$	21.6869	1.03506	0.517530	+	0.855665i	$0.326852\pi$
$440$	9.87539	0.470791
$441$	4.41641	0.210305
$442$	−13.7082	−0.652033
$443$	−19.4721	−0.925149	−0.462575	−	0.886580i	$-0.653074\pi$
$443$	−19.4721	−0.925149	−0.462575	+	0.886580i	$0.653074\pi$
$444$	13.8541	0.657487
$445$	−6.70820	−0.317999
$446$	−0.875388	−0.0414508
$447$	26.2705	1.24255
$448$	−13.4377	−0.634871
$449$	24.3262	1.14803	0.574013	−	0.818846i	$-0.305386\pi$
$449$	24.3262	1.14803	0.574013	+	0.818846i	$0.305386\pi$
$450$	−5.88854	−0.277589
$451$	5.85410	0.275659
$452$	−32.6656	−1.53646
$453$	2.61803	0.123006
$454$	−6.68692	−0.313833
$455$	58.6869	2.75129
$456$	−14.8541	−0.695607
$457$	−36.4164	−1.70349	−0.851744	−	0.523958i	$-0.824455\pi$
$457$	−36.4164	−1.70349	−0.851744	+	0.523958i	$0.824455\pi$
$458$	4.63932	0.216781
$459$	11.7082	0.546492
$460$	−8.18847	−0.381789
$461$	−7.14590	−0.332818	−0.166409	−	0.986057i	$-0.553217\pi$
$461$	−7.14590	−0.332818	−0.166409	+	0.986057i	$0.553217\pi$
$462$	−6.38197	−0.296916
$463$	−7.43769	−0.345659	−0.172829	−	0.984952i	$-0.555291\pi$
$463$	−7.43769	−0.345659	−0.172829	+	0.984952i	$0.555291\pi$
$464$	4.63119	0.214998
$465$	−7.85410	−0.364225
$466$	−4.00000	−0.185296
$467$	0.819660	0.0379293	0.0189647	−	0.999820i	$-0.493963\pi$
$467$	0.819660	0.0379293	0.0189647	+	0.999820i	$0.493963\pi$
$468$	48.9787	2.26404
$469$	−2.85410	−0.131790
$470$	−7.31308	−0.337327
$471$	−43.3607	−1.99795
$472$	8.83282	0.406563
$473$	4.14590	0.190629
$474$	−13.5623	−0.622937
$475$	15.4164	0.707353
$476$	27.7082	1.27000
$477$	−34.6869	−1.58820
$478$	3.02129	0.138190
$479$	−0.652476	−0.0298124	−0.0149062	−	0.999889i	$-0.504745\pi$
$479$	−0.652476	−0.0298124	−0.0149062	+	0.999889i	$0.504745\pi$
$480$	−32.5623	−1.48626
$481$	−19.5623	−0.891964
$482$	−7.31308	−0.333102
$483$	11.0000	0.500517
$484$	11.1246	0.505664
$485$	37.2492	1.69140
$486$	−8.27051	−0.375158
$487$	−13.8328	−0.626825	−0.313412	−	0.949617i	$-0.601472\pi$
$487$	−13.8328	−0.626825	−0.313412	+	0.949617i	$0.601472\pi$
$488$	9.66061	0.437315
$489$	0.381966	0.0172731
$490$	1.31308	0.0593190
$491$	1.58359	0.0714665	0.0357333	−	0.999361i	$-0.488623\pi$
$491$	1.58359	0.0714665	0.0357333	+	0.999361i	$0.488623\pi$
$492$	−12.7082	−0.572930
$493$	−7.70820	−0.347160
$494$	10.0902	0.453978
$495$	25.8541	1.16206
$496$	−3.14590	−0.141255
$497$	23.5066	1.05441
$498$	0.326238	0.0146191
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	−5.56231	−0.248754
$501$	−62.5410	−2.79413
$502$	−2.58359	−0.115311
$503$	24.0557	1.07259	0.536296	−	0.844030i	$-0.319823\pi$
$503$	24.0557	1.07259	0.536296	+	0.844030i	$0.319823\pi$
$504$	16.1935	0.721316
$505$	−3.27051	−0.145536
$506$	−1.25735	−0.0558962
$507$	−88.9574	−3.95074
$508$	−1.58359	−0.0702605
$509$	−19.4164	−0.860617	−0.430309	−	0.902682i	$-0.641595\pi$
$509$	−19.4164	−0.860617	−0.430309	+	0.902682i	$0.641595\pi$
$510$	15.7082	0.695571
$511$	−11.4164	−0.505032
$512$	−22.3050	−0.985749
$513$	−8.61803	−0.380495
$514$	−3.39512	−0.149752
$515$	37.6869	1.66068
$516$	−9.00000	−0.396203
$517$	14.2705	0.627616
$518$	−3.11146	−0.136710
$519$	65.6869	2.88334
$520$	30.2705	1.32745
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−2.16718	−0.0948551
$523$	18.5623	0.811673	0.405836	−	0.913946i	$-0.366980\pi$
$523$	18.5623	0.811673	0.405836	+	0.913946i	$0.366980\pi$
$524$	5.56231	0.242990
$525$	−29.8885	−1.30444
$526$	4.41641	0.192565
$527$	5.23607	0.228087
$528$	18.4164	0.801471
$529$	−20.8328	−0.905775
$530$	−10.3131	−0.447972
$531$	23.1246	1.00352
$532$	−20.3951	−0.884241
$533$	17.9443	0.777253
$534$	2.23607	0.0967641
$535$	−56.1246	−2.42648
$536$	−1.47214	−0.0635866
$537$	29.4164	1.26941
$538$	2.97871	0.128421
$539$	−2.56231	−0.110366
$540$	−12.4377	−0.535233
$541$	4.58359	0.197064	0.0985320	−	0.995134i	$-0.468585\pi$
$541$	4.58359	0.197064	0.0985320	+	0.995134i	$0.468585\pi$
$542$	−4.31308	−0.185263
$543$	35.5066	1.52373
$544$	21.7082	0.930732
$545$	−5.56231	−0.238263
$546$	−19.5623	−0.837189
$547$	25.5623	1.09297	0.546483	−	0.837470i	$-0.315966\pi$
$547$	25.5623	1.09297	0.546483	+	0.837470i	$0.315966\pi$
$548$	23.5623	1.00653
$549$	25.2918	1.07943
$550$	3.41641	0.145676
$551$	5.67376	0.241710
$552$	5.67376	0.241491
$553$	−38.7082	−1.64604
$554$	7.95743	0.338079
$555$	22.4164	0.951524
$556$	5.56231	0.235894
$557$	−32.6180	−1.38207	−0.691035	−	0.722821i	$-0.742845\pi$
$557$	−32.6180	−1.38207	−0.691035	+	0.722821i	$0.742845\pi$
$558$	1.47214	0.0623205
$559$	12.7082	0.537500
$560$	−26.9361	−1.13826
$561$	−30.6525	−1.29415
$562$	−2.43769	−0.102828
$563$	3.70820	0.156282	0.0781411	−	0.996942i	$-0.475102\pi$
$563$	3.70820	0.156282	0.0781411	+	0.996942i	$0.475102\pi$
$564$	−30.9787	−1.30444
$565$	−52.8541	−2.22359
$566$	−1.68692	−0.0709064
$567$	−16.2918	−0.684191
$568$	12.1246	0.508737
$569$	−2.18034	−0.0914046	−0.0457023	−	0.998955i	$-0.514553\pi$
$569$	−2.18034	−0.0914046	−0.0457023	+	0.998955i	$0.514553\pi$
$570$	−11.5623	−0.484292
$571$	−20.9787	−0.877932	−0.438966	−	0.898504i	$-0.644655\pi$
$571$	−20.9787	−0.877932	−0.438966	+	0.898504i	$0.644655\pi$
$572$	−28.4164	−1.18815
$573$	19.5623	0.817227
$574$	2.85410	0.119128
$575$	−5.88854	−0.245569
$576$	−18.1459	−0.756079
$577$	22.7082	0.945355	0.472677	−	0.881236i	$-0.343288\pi$
$577$	22.7082	0.945355	0.472677	+	0.881236i	$0.343288\pi$
$578$	−3.97871	−0.165493
$579$	−26.5623	−1.10389
$580$	8.18847	0.340008
$581$	0.931116	0.0386292
$582$	−12.4164	−0.514677
$583$	20.1246	0.833476
$584$	−5.88854	−0.243670
$585$	79.2492	3.27655
$586$	8.56231	0.353706
$587$	34.0902	1.40705	0.703526	−	0.710670i	$-0.251608\pi$
$587$	34.0902	1.40705	0.703526	+	0.710670i	$0.251608\pi$
$588$	5.56231	0.229386
$589$	−3.85410	−0.158806
$590$	6.87539	0.283055
$591$	35.2705	1.45083
$592$	8.97871	0.369023
$593$	−25.5279	−1.04830	−0.524152	−	0.851625i	$-0.675618\pi$
$593$	−25.5279	−1.04830	−0.524152	+	0.851625i	$0.675618\pi$
$594$	−1.90983	−0.0783613
$595$	44.8328	1.83797
$596$	18.6049	0.762086
$597$	60.9230	2.49341
$598$	−3.85410	−0.157606
$599$	1.52786	0.0624268	0.0312134	−	0.999513i	$-0.490063\pi$
$599$	1.52786	0.0624268	0.0312134	+	0.999513i	$0.490063\pi$
$600$	−15.4164	−0.629372
$601$	33.1246	1.35118	0.675591	−	0.737277i	$-0.263889\pi$
$601$	33.1246	1.35118	0.675591	+	0.737277i	$0.263889\pi$
$602$	2.02129	0.0823815
$603$	−3.85410	−0.156951
$604$	1.85410	0.0754423
$605$	18.0000	0.731804
$606$	1.09017	0.0442851
$607$	43.5623	1.76814	0.884070	−	0.467355i	$-0.154793\pi$
$607$	43.5623	1.76814	0.884070	+	0.467355i	$0.154793\pi$
$608$	−15.9787	−0.648022
$609$	−11.0000	−0.445742
$610$	7.51973	0.304465
$611$	43.7426	1.76964
$612$	37.4164	1.51247
$613$	12.4164	0.501494	0.250747	−	0.968053i	$-0.419324\pi$
$613$	12.4164	0.501494	0.250747	+	0.968053i	$0.419324\pi$
$614$	3.59675	0.145153
$615$	−20.5623	−0.829152
$616$	−9.39512	−0.378540
$617$	−45.9787	−1.85103	−0.925517	−	0.378707i	$-0.876369\pi$
$617$	−45.9787	−1.85103	−0.925517	+	0.378707i	$0.876369\pi$
$618$	−12.5623	−0.505330
$619$	−47.7082	−1.91755	−0.958777	−	0.284159i	$-0.908286\pi$
$619$	−47.7082	−1.91755	−0.958777	+	0.284159i	$0.908286\pi$
$620$	−5.56231	−0.223388
$621$	3.29180	0.132095
$622$	−10.2492	−0.410956
$623$	6.38197	0.255688
$624$	56.4508	2.25984
$625$	−29.0000	−1.16000
$626$	2.72949	0.109092
$627$	22.5623	0.901052
$628$	−30.7082	−1.22539
$629$	−14.9443	−0.595867
$630$	12.6049	0.502190
$631$	−24.5623	−0.977810	−0.488905	−	0.872337i	$-0.662604\pi$
$631$	−24.5623	−0.977810	−0.488905	+	0.872337i	$0.662604\pi$
$632$	−19.9656	−0.794187
$633$	23.1803	0.921336
$634$	11.6869	0.464147
$635$	−2.56231	−0.101682
$636$	−43.6869	−1.73230
$637$	−7.85410	−0.311191
$638$	1.25735	0.0497791
$639$	31.7426	1.25572
$640$	−30.2705	−1.19655
$641$	41.0689	1.62212	0.811062	−	0.584961i	$-0.198890\pi$
$641$	41.0689	1.62212	0.811062	+	0.584961i	$0.198890\pi$
$642$	18.7082	0.738354
$643$	−0.437694	−0.0172610	−0.00863049	−	0.999963i	$-0.502747\pi$
$643$	−0.437694	−0.0172610	−0.00863049	+	0.999963i	$0.502747\pi$
$644$	7.79024	0.306979
$645$	−14.5623	−0.573390
$646$	7.70820	0.303275
$647$	−29.2361	−1.14939	−0.574694	−	0.818368i	$-0.694879\pi$
$647$	−29.2361	−1.14939	−0.574694	+	0.818368i	$0.694879\pi$
$648$	−8.40325	−0.330111
$649$	−13.4164	−0.526640
$650$	10.4721	0.410751
$651$	7.47214	0.292856
$652$	0.270510	0.0105940
$653$	−4.47214	−0.175008	−0.0875041	−	0.996164i	$-0.527889\pi$
$653$	−4.47214	−0.175008	−0.0875041	+	0.996164i	$0.527889\pi$
$654$	1.85410	0.0725011
$655$	9.00000	0.351659
$656$	−8.23607	−0.321564
$657$	−15.4164	−0.601451
$658$	6.95743	0.271229
$659$	−12.0557	−0.469624	−0.234812	−	0.972041i	$-0.575448\pi$
$659$	−12.0557	−0.469624	−0.234812	+	0.972041i	$0.575448\pi$
$660$	32.5623	1.26749
$661$	9.87539	0.384108	0.192054	−	0.981384i	$-0.438485\pi$
$661$	9.87539	0.384108	0.192054	+	0.981384i	$0.438485\pi$
$662$	−8.94427	−0.347629
$663$	−93.9574	−3.64900
$664$	0.480267	0.0186380
$665$	−33.0000	−1.27969
$666$	−4.20163	−0.162810
$667$	−2.16718	−0.0839137
$668$	−44.2918	−1.71370
$669$	−6.00000	−0.231973
$670$	−1.14590	−0.0442699
$671$	−14.6738	−0.566474
$672$	30.9787	1.19503
$673$	24.6869	0.951611	0.475805	−	0.879551i	$-0.342157\pi$
$673$	24.6869	0.951611	0.475805	+	0.879551i	$0.342157\pi$
$674$	−9.65248	−0.371799
$675$	−8.94427	−0.344265
$676$	−63.0000	−2.42308
$677$	−11.1803	−0.429695	−0.214848	−	0.976648i	$-0.568926\pi$
$677$	−11.1803	−0.429695	−0.214848	+	0.976648i	$0.568926\pi$
$678$	17.6180	0.676617
$679$	−35.4377	−1.35997
$680$	23.1246	0.886788
$681$	−45.8328	−1.75632
$682$	−0.854102	−0.0327053
$683$	11.6180	0.444552	0.222276	−	0.974984i	$-0.428651\pi$
$683$	11.6180	0.444552	0.222276	+	0.974984i	$0.428651\pi$
$684$	−27.5410	−1.05306
$685$	38.1246	1.45667
$686$	6.38197	0.243665
$687$	31.7984	1.21318
$688$	−5.83282	−0.222374
$689$	61.6869	2.35008
$690$	4.41641	0.168130
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	46.5197	1.76841
$693$	−24.5967	−0.934353
$694$	−4.27051	−0.162106
$695$	9.00000	0.341389
$696$	−5.67376	−0.215063
$697$	13.7082	0.519235
$698$	2.29180	0.0867458
$699$	−27.4164	−1.03698
$700$	−21.1672	−0.800044
$701$	−4.36068	−0.164701	−0.0823503	−	0.996603i	$-0.526243\pi$
$701$	−4.36068	−0.164701	−0.0823503	+	0.996603i	$0.526243\pi$
$702$	−5.85410	−0.220949
$703$	11.0000	0.414873
$704$	10.5279	0.396784
$705$	−50.1246	−1.88780
$706$	−7.83282	−0.294792
$707$	3.11146	0.117018
$708$	29.1246	1.09457
$709$	21.4164	0.804310	0.402155	−	0.915572i	$-0.368261\pi$
$709$	21.4164	0.804310	0.402155	+	0.915572i	$0.368261\pi$
$710$	9.43769	0.354190
$711$	−52.2705	−1.96030
$712$	3.29180	0.123365
$713$	1.47214	0.0551319
$714$	−14.9443	−0.559275
$715$	−45.9787	−1.71951
$716$	20.8328	0.778559
$717$	20.7082	0.773362
$718$	6.87539	0.256587
$719$	49.4721	1.84500	0.922500	−	0.385998i	$-0.126143\pi$
$719$	49.4721	1.84500	0.922500	+	0.385998i	$0.126143\pi$
$720$	−36.3738	−1.35557
$721$	−35.8541	−1.33528
$722$	1.58359	0.0589352
$723$	−50.1246	−1.86415
$724$	25.1459	0.934540
$725$	5.88854	0.218695
$726$	−6.00000	−0.222681
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	−28.7984	−1.06734
$729$	−39.5623	−1.46527
$730$	−4.58359	−0.169646
$731$	9.70820	0.359071
$732$	31.8541	1.17736
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	−8.06888	−0.297828
$735$	9.00000	0.331970
$736$	6.10333	0.224972
$737$	2.23607	0.0823666
$738$	3.85410	0.141871
$739$	22.2705	0.819234	0.409617	−	0.912258i	$-0.365662\pi$
$739$	22.2705	0.819234	0.409617	+	0.912258i	$0.365662\pi$
$740$	15.8754	0.583591
$741$	69.1591	2.54062
$742$	9.81153	0.360193
$743$	−24.5967	−0.902367	−0.451184	−	0.892431i	$-0.648998\pi$
$743$	−24.5967	−0.902367	−0.451184	+	0.892431i	$0.648998\pi$
$744$	3.85410	0.141298
$745$	30.1033	1.10290
$746$	−0.596748	−0.0218485
$747$	1.25735	0.0460042
$748$	−21.7082	−0.793731
$749$	53.3951	1.95102
$750$	3.00000	0.109545
$751$	3.87539	0.141415	0.0707075	−	0.997497i	$-0.477474\pi$
$751$	3.87539	0.141415	0.0707075	+	0.997497i	$0.477474\pi$
$752$	−20.0770	−0.732133
$753$	−17.7082	−0.645323
$754$	3.85410	0.140358
$755$	3.00000	0.109181
$756$	11.8328	0.430355
$757$	11.5836	0.421013	0.210506	−	0.977592i	$-0.432489\pi$
$757$	11.5836	0.421013	0.210506	+	0.977592i	$0.432489\pi$
$758$	−1.52786	−0.0554945
$759$	−8.61803	−0.312815
$760$	−17.0213	−0.617427
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0.854102	0.0309408
$763$	5.29180	0.191576
$764$	13.8541	0.501224
$765$	60.5410	2.18887
$766$	12.1459	0.438849
$767$	−41.1246	−1.48492
$768$	−14.5623	−0.525472
$769$	2.70820	0.0976603	0.0488302	−	0.998807i	$-0.484451\pi$
$769$	2.70820	0.0976603	0.0488302	+	0.998807i	$0.484451\pi$
$770$	−7.31308	−0.263545
$771$	−23.2705	−0.838067
$772$	−18.8115	−0.677042
$773$	19.3607	0.696355	0.348178	−	0.937429i	$-0.386801\pi$
$773$	19.3607	0.696355	0.348178	+	0.937429i	$0.386801\pi$
$774$	2.72949	0.0981095
$775$	−4.00000	−0.143684
$776$	−18.2786	−0.656165
$777$	−21.3262	−0.765075
$778$	−4.83282	−0.173265
$779$	−10.0902	−0.361518
$780$	99.8115	3.57383
$781$	−18.4164	−0.658991
$782$	−2.94427	−0.105287
$783$	−3.29180	−0.117639
$784$	3.60488	0.128746
$785$	−49.6869	−1.77340
$786$	−3.00000	−0.107006
$787$	−20.4164	−0.727766	−0.363883	−	0.931445i	$-0.618549\pi$
$787$	−20.4164	−0.727766	−0.363883	+	0.931445i	$0.618549\pi$
$788$	24.9787	0.889830
$789$	30.2705	1.07766
$790$	−15.5410	−0.552925
$791$	50.2837	1.78788
$792$	−12.6869	−0.450810
$793$	−44.9787	−1.59724
$794$	5.89667	0.209265
$795$	−70.6869	−2.50701
$796$	43.1459	1.52927
$797$	43.6312	1.54550	0.772748	−	0.634713i	$-0.218882\pi$
$797$	43.6312	1.54550	0.772748	+	0.634713i	$0.218882\pi$
$798$	11.0000	0.389396
$799$	33.4164	1.18219
$800$	−16.5836	−0.586319
$801$	8.61803	0.304503
$802$	−0.437694	−0.0154555
$803$	8.94427	0.315637
$804$	−4.85410	−0.171191
$805$	12.6049	0.444264
$806$	−2.61803	−0.0922163
$807$	20.4164	0.718692
$808$	1.60488	0.0564594
$809$	−17.5623	−0.617458	−0.308729	−	0.951150i	$-0.599904\pi$
$809$	−17.5623	−0.617458	−0.308729	+	0.951150i	$0.599904\pi$
$810$	−6.54102	−0.229828
$811$	−7.87539	−0.276542	−0.138271	−	0.990394i	$-0.544155\pi$
$811$	−7.87539	−0.276542	−0.138271	+	0.990394i	$0.544155\pi$
$812$	−7.79024	−0.273384
$813$	−29.5623	−1.03680
$814$	2.43769	0.0854411
$815$	0.437694	0.0153318
$816$	43.1246	1.50966
$817$	−7.14590	−0.250003
$818$	10.3607	0.362253
$819$	−75.3951	−2.63452
$820$	−14.5623	−0.508538
$821$	−20.2361	−0.706244	−0.353122	−	0.935577i	$-0.614880\pi$
$821$	−20.2361	−0.706244	−0.353122	+	0.935577i	$0.614880\pi$
$822$	−12.7082	−0.443250
$823$	26.6869	0.930247	0.465124	−	0.885246i	$-0.346010\pi$
$823$	26.6869	0.930247	0.465124	+	0.885246i	$0.346010\pi$
$824$	−18.4934	−0.644249
$825$	23.4164	0.815255
$826$	−6.54102	−0.227591
$827$	17.2361	0.599357	0.299678	−	0.954040i	$-0.403121\pi$
$827$	17.2361	0.599357	0.299678	+	0.954040i	$0.403121\pi$
$828$	10.5197	0.365586
$829$	36.1246	1.25466	0.627330	−	0.778754i	$-0.284148\pi$
$829$	36.1246	1.25466	0.627330	+	0.778754i	$0.284148\pi$
$830$	0.373835	0.0129760
$831$	54.5410	1.89201
$832$	32.2705	1.11878
$833$	−6.00000	−0.207888
$834$	−3.00000	−0.103882
$835$	−71.6656	−2.48009
$836$	15.9787	0.552635
$837$	2.23607	0.0772898
$838$	8.54102	0.295045
$839$	43.3050	1.49505	0.747526	−	0.664232i	$-0.231241\pi$
$839$	43.3050	1.49505	0.747526	+	0.664232i	$0.231241\pi$
$840$	33.0000	1.13861
$841$	−26.8328	−0.925270
$842$	9.22291	0.317843
$843$	−16.7082	−0.575461
$844$	16.4164	0.565076
$845$	−101.936	−3.50671
$846$	9.39512	0.323011
$847$	−17.1246	−0.588409
$848$	−28.3131	−0.972275
$849$	−11.5623	−0.396817
$850$	8.00000	0.274398
$851$	−4.20163	−0.144030
$852$	39.9787	1.36965
$853$	−20.6869	−0.708307	−0.354153	−	0.935187i	$-0.615231\pi$
$853$	−20.6869	−0.708307	−0.354153	+	0.935187i	$0.615231\pi$
$854$	−7.15403	−0.244806
$855$	−44.5623	−1.52400
$856$	27.5410	0.941333
$857$	4.41641	0.150862	0.0754308	−	0.997151i	$-0.475967\pi$
$857$	4.41641	0.150862	0.0754308	+	0.997151i	$0.475967\pi$
$858$	15.3262	0.523229
$859$	44.9787	1.53465	0.767327	−	0.641256i	$-0.221586\pi$
$859$	44.9787	1.53465	0.767327	+	0.641256i	$0.221586\pi$
$860$	−10.3131	−0.351673
$861$	19.5623	0.666682
$862$	−12.3131	−0.419385
$863$	15.4377	0.525505	0.262753	−	0.964863i	$-0.415370\pi$
$863$	15.4377	0.525505	0.262753	+	0.964863i	$0.415370\pi$
$864$	9.27051	0.315389
$865$	75.2705	2.55927
$866$	−12.7558	−0.433460
$867$	−27.2705	−0.926155
$868$	5.29180	0.179615
$869$	30.3262	1.02875
$870$	−4.41641	−0.149730
$871$	6.85410	0.232242
$872$	2.72949	0.0924322
$873$	−47.8541	−1.61962
$874$	2.16718	0.0733061
$875$	8.56231	0.289459
$876$	−19.4164	−0.656020
$877$	28.1246	0.949701	0.474850	−	0.880067i	$-0.342502\pi$
$877$	28.1246	0.949701	0.474850	+	0.880067i	$0.342502\pi$
$878$	−8.28367	−0.279560
$879$	58.6869	1.97946
$880$	21.1033	0.711393
$881$	−21.7639	−0.733246	−0.366623	−	0.930370i	$-0.619486\pi$
$881$	−21.7639	−0.733246	−0.366623	+	0.930370i	$0.619486\pi$
$882$	−1.68692	−0.0568015
$883$	7.29180	0.245388	0.122694	−	0.992445i	$-0.460847\pi$
$883$	7.29180	0.245388	0.122694	+	0.992445i	$0.460847\pi$
$884$	−66.5410	−2.23802
$885$	47.1246	1.58408
$886$	7.43769	0.249874
$887$	−35.0689	−1.17750	−0.588749	−	0.808316i	$-0.700379\pi$
$887$	−35.0689	−1.17750	−0.588749	+	0.808316i	$0.700379\pi$
$888$	−11.0000	−0.369136
$889$	2.43769	0.0817576
$890$	2.56231	0.0858887
$891$	12.7639	0.427608
$892$	−4.24922	−0.142275
$893$	−24.5967	−0.823099
$894$	−10.0344	−0.335602
$895$	33.7082	1.12674
$896$	28.7984	0.962086
$897$	−26.4164	−0.882018
$898$	−9.29180	−0.310071
$899$	−1.47214	−0.0490985
$900$	−28.5836	−0.952786
$901$	47.1246	1.56995
$902$	−2.23607	−0.0744529
$903$	13.8541	0.461036
$904$	25.9361	0.862623
$905$	40.6869	1.35248
$906$	−1.00000	−0.0332228
$907$	−27.8541	−0.924880	−0.462440	−	0.886651i	$-0.653026\pi$
$907$	−27.8541	−0.924880	−0.462440	+	0.886651i	$0.653026\pi$
$908$	−32.4590	−1.07719
$909$	4.20163	0.139359
$910$	−22.4164	−0.743097
$911$	−49.8541	−1.65174	−0.825870	−	0.563861i	$-0.809316\pi$
$911$	−49.8541	−1.65174	−0.825870	+	0.563861i	$0.809316\pi$
$912$	−31.7426	−1.05110
$913$	−0.729490	−0.0241426
$914$	13.9098	0.460096
$915$	51.5410	1.70389
$916$	22.5197	0.744073
$917$	−8.56231	−0.282752
$918$	−4.47214	−0.147602
$919$	−23.5836	−0.777951	−0.388975	−	0.921248i	$-0.627171\pi$
$919$	−23.5836	−0.777951	−0.388975	+	0.921248i	$0.627171\pi$
$920$	6.50155	0.214350
$921$	24.6525	0.812327
$922$	2.72949	0.0898910
$923$	−56.4508	−1.85810
$924$	−30.9787	−1.01912
$925$	11.4164	0.375369
$926$	2.84095	0.0933593
$927$	−48.4164	−1.59020
$928$	−6.10333	−0.200351
$929$	0.0557281	0.00182838	0.000914190	−	1.00000i	$-0.499709\pi$
$929$	0.0557281	0.00182838	0.000914190		1.00000i	$0.499709\pi$
$930$	3.00000	0.0983739
$931$	4.41641	0.144742
$932$	−19.4164	−0.636006
$933$	−70.2492	−2.29986
$934$	−0.313082	−0.0102444
$935$	−35.1246	−1.14870
$936$	−38.8885	−1.27111
$937$	24.7082	0.807182	0.403591	−	0.914940i	$-0.367762\pi$
$937$	24.7082	0.807182	0.403591	+	0.914940i	$0.367762\pi$
$938$	1.09017	0.0355953
$939$	18.7082	0.610519
$940$	−35.4984	−1.15783
$941$	−53.8328	−1.75490	−0.877450	−	0.479668i	$-0.840757\pi$
$941$	−53.8328	−1.75490	−0.877450	+	0.479668i	$0.840757\pi$
$942$	16.5623	0.539629
$943$	3.85410	0.125507
$944$	18.8754	0.614342
$945$	19.1459	0.622816
$946$	−1.58359	−0.0514870
$947$	−36.2705	−1.17863	−0.589317	−	0.807902i	$-0.700603\pi$
$947$	−36.2705	−1.17863	−0.589317	+	0.807902i	$0.700603\pi$
$948$	−65.8328	−2.13815
$949$	27.4164	0.889974
$950$	−5.88854	−0.191050
$951$	80.1033	2.59753
$952$	−22.0000	−0.713024
$953$	−13.3607	−0.432795	−0.216397	−	0.976305i	$-0.569431\pi$
$953$	−13.3607	−0.432795	−0.216397	+	0.976305i	$0.569431\pi$
$954$	13.2492	0.428959
$955$	22.4164	0.725378
$956$	14.6656	0.474320
$957$	8.61803	0.278581
$958$	0.249224	0.00805205
$959$	−36.2705	−1.17124
$960$	−36.9787	−1.19348
$961$	−30.0000	−0.967742
$962$	7.47214	0.240911
$963$	72.1033	2.32350
$964$	−35.4984	−1.14333
$965$	−30.4377	−0.979824
$966$	−4.20163	−0.135185
$967$	−57.8328	−1.85978	−0.929889	−	0.367840i	$-0.880097\pi$
$967$	−57.8328	−1.85978	−0.929889	+	0.367840i	$0.880097\pi$
$968$	−8.83282	−0.283897
$969$	52.8328	1.69723
$970$	−14.2279	−0.456832
$971$	−20.7771	−0.666768	−0.333384	−	0.942791i	$-0.608191\pi$
$971$	−20.7771	−0.666768	−0.333384	+	0.942791i	$0.608191\pi$
$972$	−40.1459	−1.28768
$973$	−8.56231	−0.274495
$974$	5.28367	0.169300
$975$	71.7771	2.29871
$976$	20.6443	0.660809
$977$	−27.6525	−0.884681	−0.442341	−	0.896847i	$-0.645852\pi$
$977$	−27.6525	−0.884681	−0.442341	+	0.896847i	$0.645852\pi$
$978$	−0.145898	−0.00466530
$979$	−5.00000	−0.159801
$980$	6.37384	0.203605
$981$	7.14590	0.228151
$982$	−0.604878	−0.0193024
$983$	23.8328	0.760149	0.380074	−	0.924956i	$-0.375898\pi$
$983$	23.8328	0.760149	0.380074	+	0.924956i	$0.375898\pi$
$984$	10.0902	0.321663
$985$	40.4164	1.28777
$986$	2.94427	0.0937647
$987$	47.6869	1.51789
$988$	48.9787	1.55822
$989$	2.72949	0.0867927
$990$	−9.87539	−0.313860
$991$	30.4377	0.966885	0.483443	−	0.875376i	$-0.339386\pi$
$991$	30.4377	0.966885	0.483443	+	0.875376i	$0.339386\pi$
$992$	4.14590	0.131632
$993$	−61.3050	−1.94545
$994$	−8.97871	−0.284788
$995$	69.8115	2.21317
$996$	1.58359	0.0501780
$997$	−39.5410	−1.25228	−0.626138	−	0.779712i	$-0.715365\pi$
$997$	−39.5410	−1.25228	−0.626138	+	0.779712i	$0.715365\pi$
$998$	9.54915	0.302273
$999$	−6.38197	−0.201916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.2.a.b.1.2	✓	2
3.2	odd	2		603.2.a.h.1.1		2
4.3	odd	2		1072.2.a.f.1.2		2
5.2	odd	4		1675.2.c.d.1274.2		4
5.3	odd	4		1675.2.c.d.1274.3		4
5.4	even	2		1675.2.a.h.1.1		2
7.6	odd	2		3283.2.a.f.1.2		2
8.3	odd	2		4288.2.a.h.1.1		2
8.5	even	2		4288.2.a.p.1.2		2
11.10	odd	2		8107.2.a.i.1.1		2
12.11	even	2		9648.2.a.bi.1.1		2
67.66	odd	2		4489.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.a.b.1.2	✓	2	1.1	even	1	trivial
603.2.a.h.1.1		2	3.2	odd	2
1072.2.a.f.1.2		2	4.3	odd	2
1675.2.a.h.1.1		2	5.4	even	2
1675.2.c.d.1274.2		4	5.2	odd	4
1675.2.c.d.1274.3		4	5.3	odd	4
3283.2.a.f.1.2		2	7.6	odd	2
4288.2.a.h.1.1		2	8.3	odd	2
4288.2.a.p.1.2		2	8.5	even	2
4489.2.a.d.1.1		2	67.66	odd	2
8107.2.a.i.1.1		2	11.10	odd	2
9648.2.a.bi.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	67.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} -2.61803 q^{3} -1.85410 q^{4} -3.00000 q^{5} +1.00000 q^{6} +2.85410 q^{7} +1.47214 q^{8} +3.85410 q^{9} +O(q^{10})\)	\(q-0.381966 q^{2} -2.61803 q^{3} -1.85410 q^{4} -3.00000 q^{5} +1.00000 q^{6} +2.85410 q^{7} +1.47214 q^{8} +3.85410 q^{9} +1.14590 q^{10} -2.23607 q^{11} +4.85410 q^{12} -6.85410 q^{13} -1.09017 q^{14} +7.85410 q^{15} +3.14590 q^{16} -5.23607 q^{17} -1.47214 q^{18} +3.85410 q^{19} +5.56231 q^{20} -7.47214 q^{21} +0.854102 q^{22} -1.47214 q^{23} -3.85410 q^{24} +4.00000 q^{25} +2.61803 q^{26} -2.23607 q^{27} -5.29180 q^{28} +1.47214 q^{29} -3.00000 q^{30} -1.00000 q^{31} -4.14590 q^{32} +5.85410 q^{33} +2.00000 q^{34} -8.56231 q^{35} -7.14590 q^{36} +2.85410 q^{37} -1.47214 q^{38} +17.9443 q^{39} -4.41641 q^{40} -2.61803 q^{41} +2.85410 q^{42} -1.85410 q^{43} +4.14590 q^{44} -11.5623 q^{45} +0.562306 q^{46} -6.38197 q^{47} -8.23607 q^{48} +1.14590 q^{49} -1.52786 q^{50} +13.7082 q^{51} +12.7082 q^{52} -9.00000 q^{53} +0.854102 q^{54} +6.70820 q^{55} +4.20163 q^{56} -10.0902 q^{57} -0.562306 q^{58} +6.00000 q^{59} -14.5623 q^{60} +6.56231 q^{61} +0.381966 q^{62} +11.0000 q^{63} -4.70820 q^{64} +20.5623 q^{65} -2.23607 q^{66} -1.00000 q^{67} +9.70820 q^{68} +3.85410 q^{69} +3.27051 q^{70} +8.23607 q^{71} +5.67376 q^{72} -4.00000 q^{73} -1.09017 q^{74} -10.4721 q^{75} -7.14590 q^{76} -6.38197 q^{77} -6.85410 q^{78} -13.5623 q^{79} -9.43769 q^{80} -5.70820 q^{81} +1.00000 q^{82} +0.326238 q^{83} +13.8541 q^{84} +15.7082 q^{85} +0.708204 q^{86} -3.85410 q^{87} -3.29180 q^{88} +2.23607 q^{89} +4.41641 q^{90} -19.5623 q^{91} +2.72949 q^{92} +2.61803 q^{93} +2.43769 q^{94} -11.5623 q^{95} +10.8541 q^{96} -12.4164 q^{97} -0.437694 q^{98} -8.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} - q^{7} - 6 q^{8} + q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 6 * q^5 + 2 * q^6 - q^7 - 6 * q^8 + q^9	\( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} - q^{7} - 6 q^{8} + q^{9} + 9 q^{10} + 3 q^{12} - 7 q^{13} + 9 q^{14} + 9 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} + q^{19} - 9 q^{20} - 6 q^{21} - 5 q^{22} + 6 q^{23} - q^{24} + 8 q^{25} + 3 q^{26} - 24 q^{28} - 6 q^{29} - 6 q^{30} - 2 q^{31} - 15 q^{32} + 5 q^{33} + 4 q^{34} + 3 q^{35} - 21 q^{36} - q^{37} + 6 q^{38} + 18 q^{39} + 18 q^{40} - 3 q^{41} - q^{42} + 3 q^{43} + 15 q^{44} - 3 q^{45} - 19 q^{46} - 15 q^{47} - 12 q^{48} + 9 q^{49} - 12 q^{50} + 14 q^{51} + 12 q^{52} - 18 q^{53} - 5 q^{54} + 33 q^{56} - 9 q^{57} + 19 q^{58} + 12 q^{59} - 9 q^{60} - 7 q^{61} + 3 q^{62} + 22 q^{63} + 4 q^{64} + 21 q^{65} - 2 q^{67} + 6 q^{68} + q^{69} - 27 q^{70} + 12 q^{71} + 27 q^{72} - 8 q^{73} + 9 q^{74} - 12 q^{75} - 21 q^{76} - 15 q^{77} - 7 q^{78} - 7 q^{79} - 39 q^{80} + 2 q^{81} + 2 q^{82} - 15 q^{83} + 21 q^{84} + 18 q^{85} - 12 q^{86} - q^{87} - 20 q^{88} - 18 q^{90} - 19 q^{91} + 39 q^{92} + 3 q^{93} + 25 q^{94} - 3 q^{95} + 15 q^{96} + 2 q^{97} - 21 q^{98} - 15 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 6 * q^5 + 2 * q^6 - q^7 - 6 * q^8 + q^9 + 9 * q^10 + 3 * q^12 - 7 * q^13 + 9 * q^14 + 9 * q^15 + 13 * q^16 - 6 * q^17 + 6 * q^18 + q^19 - 9 * q^20 - 6 * q^21 - 5 * q^22 + 6 * q^23 - q^24 + 8 * q^25 + 3 * q^26 - 24 * q^28 - 6 * q^29 - 6 * q^30 - 2 * q^31 - 15 * q^32 + 5 * q^33 + 4 * q^34 + 3 * q^35 - 21 * q^36 - q^37 + 6 * q^38 + 18 * q^39 + 18 * q^40 - 3 * q^41 - q^42 + 3 * q^43 + 15 * q^44 - 3 * q^45 - 19 * q^46 - 15 * q^47 - 12 * q^48 + 9 * q^49 - 12 * q^50 + 14 * q^51 + 12 * q^52 - 18 * q^53 - 5 * q^54 + 33 * q^56 - 9 * q^57 + 19 * q^58 + 12 * q^59 - 9 * q^60 - 7 * q^61 + 3 * q^62 + 22 * q^63 + 4 * q^64 + 21 * q^65 - 2 * q^67 + 6 * q^68 + q^69 - 27 * q^70 + 12 * q^71 + 27 * q^72 - 8 * q^73 + 9 * q^74 - 12 * q^75 - 21 * q^76 - 15 * q^77 - 7 * q^78 - 7 * q^79 - 39 * q^80 + 2 * q^81 + 2 * q^82 - 15 * q^83 + 21 * q^84 + 18 * q^85 - 12 * q^86 - q^87 - 20 * q^88 - 18 * q^90 - 19 * q^91 + 39 * q^92 + 3 * q^93 + 25 * q^94 - 3 * q^95 + 15 * q^96 + 2 * q^97 - 21 * q^98 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more