LMFDB - Embedded newform 4029.2.a.f.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4029,2,Mod(1,4029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4029, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.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:	$32.1717269744$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.843162 q^{2} -1.00000 q^{3} -1.28908 q^{4} -2.14706 q^{5} -0.843162 q^{6} +4.16350 q^{7} -2.77323 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.843162 q^{2} -1.00000 q^{3} -1.28908 q^{4} -2.14706 q^{5} -0.843162 q^{6} +4.16350 q^{7} -2.77323 q^{8} +1.00000 q^{9} -1.81032 q^{10} +1.87230 q^{11} +1.28908 q^{12} -2.91586 q^{13} +3.51051 q^{14} +2.14706 q^{15} +0.239876 q^{16} +1.00000 q^{17} +0.843162 q^{18} -2.76731 q^{19} +2.76773 q^{20} -4.16350 q^{21} +1.57865 q^{22} -7.37894 q^{23} +2.77323 q^{24} -0.390122 q^{25} -2.45854 q^{26} -1.00000 q^{27} -5.36708 q^{28} +6.85810 q^{29} +1.81032 q^{30} +5.01726 q^{31} +5.74871 q^{32} -1.87230 q^{33} +0.843162 q^{34} -8.93930 q^{35} -1.28908 q^{36} +3.03422 q^{37} -2.33329 q^{38} +2.91586 q^{39} +5.95429 q^{40} +11.8115 q^{41} -3.51051 q^{42} -8.69283 q^{43} -2.41354 q^{44} -2.14706 q^{45} -6.22164 q^{46} +7.35486 q^{47} -0.239876 q^{48} +10.3348 q^{49} -0.328936 q^{50} -1.00000 q^{51} +3.75877 q^{52} -7.09810 q^{53} -0.843162 q^{54} -4.01994 q^{55} -11.5463 q^{56} +2.76731 q^{57} +5.78249 q^{58} -12.1533 q^{59} -2.76773 q^{60} +2.95471 q^{61} +4.23036 q^{62} +4.16350 q^{63} +4.36734 q^{64} +6.26053 q^{65} -1.57865 q^{66} -11.9789 q^{67} -1.28908 q^{68} +7.37894 q^{69} -7.53728 q^{70} -1.56592 q^{71} -2.77323 q^{72} -4.76855 q^{73} +2.55834 q^{74} +0.390122 q^{75} +3.56728 q^{76} +7.79532 q^{77} +2.45854 q^{78} +1.00000 q^{79} -0.515029 q^{80} +1.00000 q^{81} +9.95903 q^{82} +9.50659 q^{83} +5.36708 q^{84} -2.14706 q^{85} -7.32946 q^{86} -6.85810 q^{87} -5.19230 q^{88} -10.9267 q^{89} -1.81032 q^{90} -12.1402 q^{91} +9.51203 q^{92} -5.01726 q^{93} +6.20134 q^{94} +5.94159 q^{95} -5.74871 q^{96} -10.7747 q^{97} +8.71387 q^{98} +1.87230 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + q^{2} - 22 q^{3} + 19 q^{4} + q^{5} - q^{6} - 15 q^{7} + 15 q^{8} + 22 q^{9}+O(q^{10}) $ 22 * q + q^2 - 22 * q^3 + 19 * q^4 + q^5 - q^6 - 15 * q^7 + 15 * q^8 + 22 * q^9	\( 22 q + q^{2} - 22 q^{3} + 19 q^{4} + q^{5} - q^{6} - 15 q^{7} + 15 q^{8} + 22 q^{9} - 13 q^{10} - 23 q^{11} - 19 q^{12} - 18 q^{13} - 9 q^{14} - q^{15} + 21 q^{16} + 22 q^{17} + q^{18} - 30 q^{19} - 7 q^{20} + 15 q^{21} + 4 q^{22} - 3 q^{23} - 15 q^{24} + 19 q^{25} - 7 q^{26} - 22 q^{27} - 25 q^{28} - 7 q^{29} + 13 q^{30} - 10 q^{31} + 31 q^{32} + 23 q^{33} + q^{34} - 11 q^{35} + 19 q^{36} - q^{37} - 29 q^{38} + 18 q^{39} - 59 q^{40} + 9 q^{42} - 43 q^{43} - 80 q^{44} + q^{45} - 43 q^{46} + 2 q^{47} - 21 q^{48} + 43 q^{49} + 25 q^{50} - 22 q^{51} - 5 q^{52} - q^{53} - q^{54} - 19 q^{55} - 8 q^{56} + 30 q^{57} - 43 q^{58} - 28 q^{59} + 7 q^{60} - 29 q^{61} - 3 q^{62} - 15 q^{63} + 23 q^{64} + 19 q^{65} - 4 q^{66} - 16 q^{67} + 19 q^{68} + 3 q^{69} - 5 q^{70} - q^{71} + 15 q^{72} - 19 q^{73} - 24 q^{74} - 19 q^{75} - 72 q^{76} + 24 q^{77} + 7 q^{78} + 22 q^{79} - 82 q^{80} + 22 q^{81} - 81 q^{82} - 29 q^{83} + 25 q^{84} + q^{85} - 42 q^{86} + 7 q^{87} - 43 q^{88} - 28 q^{89} - 13 q^{90} - 96 q^{91} - 11 q^{92} + 10 q^{93} - 63 q^{94} - 23 q^{95} - 31 q^{96} - 51 q^{97} + 12 q^{98} - 23 q^{99}+O(q^{100}) \) 22 * q + q^2 - 22 * q^3 + 19 * q^4 + q^5 - q^6 - 15 * q^7 + 15 * q^8 + 22 * q^9 - 13 * q^10 - 23 * q^11 - 19 * q^12 - 18 * q^13 - 9 * q^14 - q^15 + 21 * q^16 + 22 * q^17 + q^18 - 30 * q^19 - 7 * q^20 + 15 * q^21 + 4 * q^22 - 3 * q^23 - 15 * q^24 + 19 * q^25 - 7 * q^26 - 22 * q^27 - 25 * q^28 - 7 * q^29 + 13 * q^30 - 10 * q^31 + 31 * q^32 + 23 * q^33 + q^34 - 11 * q^35 + 19 * q^36 - q^37 - 29 * q^38 + 18 * q^39 - 59 * q^40 + 9 * q^42 - 43 * q^43 - 80 * q^44 + q^45 - 43 * q^46 + 2 * q^47 - 21 * q^48 + 43 * q^49 + 25 * q^50 - 22 * q^51 - 5 * q^52 - q^53 - q^54 - 19 * q^55 - 8 * q^56 + 30 * q^57 - 43 * q^58 - 28 * q^59 + 7 * q^60 - 29 * q^61 - 3 * q^62 - 15 * q^63 + 23 * q^64 + 19 * q^65 - 4 * q^66 - 16 * q^67 + 19 * q^68 + 3 * q^69 - 5 * q^70 - q^71 + 15 * q^72 - 19 * q^73 - 24 * q^74 - 19 * q^75 - 72 * q^76 + 24 * q^77 + 7 * q^78 + 22 * q^79 - 82 * q^80 + 22 * q^81 - 81 * q^82 - 29 * q^83 + 25 * q^84 + q^85 - 42 * q^86 + 7 * q^87 - 43 * q^88 - 28 * q^89 - 13 * q^90 - 96 * q^91 - 11 * q^92 + 10 * q^93 - 63 * q^94 - 23 * q^95 - 31 * q^96 - 51 * q^97 + 12 * 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 $

$2$	0.843162	0.596206	0.298103	−	0.954534i	$-0.403646\pi$
$2$	0.843162	0.596206	0.298103	+	0.954534i	$0.403646\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.28908	−0.644539
$5$	−2.14706	−0.960196	−0.480098	−	0.877215i	$-0.659399\pi$
$5$	−2.14706	−0.960196	−0.480098	+	0.877215i	$0.659399\pi$
$6$	−0.843162	−0.344220
$7$	4.16350	1.57366	0.786828	−	0.617172i	$-0.211722\pi$
$7$	4.16350	1.57366	0.786828	+	0.617172i	$0.211722\pi$
$8$	−2.77323	−0.980483
$9$	1.00000	0.333333
$10$	−1.81032	−0.572474
$11$	1.87230	0.564519	0.282260	−	0.959338i	$-0.408916\pi$
$11$	1.87230	0.564519	0.282260	+	0.959338i	$0.408916\pi$
$12$	1.28908	0.372125
$13$	−2.91586	−0.808713	−0.404357	−	0.914601i	$-0.632505\pi$
$13$	−2.91586	−0.808713	−0.404357	+	0.914601i	$0.632505\pi$
$14$	3.51051	0.938223
$15$	2.14706	0.554369
$16$	0.239876	0.0599690
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0.843162	0.198735
$19$	−2.76731	−0.634864	−0.317432	−	0.948281i	$-0.602821\pi$
$19$	−2.76731	−0.634864	−0.317432	+	0.948281i	$0.602821\pi$
$20$	2.76773	0.618883
$21$	−4.16350	−0.908551
$22$	1.57865	0.336569
$23$	−7.37894	−1.53862	−0.769308	−	0.638878i	$-0.779399\pi$
$23$	−7.37894	−1.53862	−0.769308	+	0.638878i	$0.779399\pi$
$24$	2.77323	0.566082
$25$	−0.390122	−0.0780244
$26$	−2.45854	−0.482159
$27$	−1.00000	−0.192450
$28$	−5.36708	−1.01428
$29$	6.85810	1.27352	0.636758	−	0.771063i	$-0.280275\pi$
$29$	6.85810	1.27352	0.636758	+	0.771063i	$0.280275\pi$
$30$	1.81032	0.330518
$31$	5.01726	0.901127	0.450563	−	0.892744i	$-0.351223\pi$
$31$	5.01726	0.901127	0.450563	+	0.892744i	$0.351223\pi$
$32$	5.74871	1.01624
$33$	−1.87230	−0.325925
$34$	0.843162	0.144601
$35$	−8.93930	−1.51102
$36$	−1.28908	−0.214846
$37$	3.03422	0.498822	0.249411	−	0.968398i	$-0.419763\pi$
$37$	3.03422	0.498822	0.249411	+	0.968398i	$0.419763\pi$
$38$	−2.33329	−0.378510
$39$	2.91586	0.466911
$40$	5.95429	0.941456
$41$	11.8115	1.84465	0.922325	−	0.386415i	$-0.126287\pi$
$41$	11.8115	1.84465	0.922325	+	0.386415i	$0.126287\pi$
$42$	−3.51051	−0.541683
$43$	−8.69283	−1.32564	−0.662822	−	0.748777i	$-0.730641\pi$
$43$	−8.69283	−1.32564	−0.662822	+	0.748777i	$0.730641\pi$
$44$	−2.41354	−0.363854
$45$	−2.14706	−0.320065
$46$	−6.22164	−0.917331
$47$	7.35486	1.07282	0.536408	−	0.843959i	$-0.319781\pi$
$47$	7.35486	1.07282	0.536408	+	0.843959i	$0.319781\pi$
$48$	−0.239876	−0.0346231
$49$	10.3348	1.47639
$50$	−0.328936	−0.0465186
$51$	−1.00000	−0.140028
$52$	3.75877	0.521247
$53$	−7.09810	−0.975000	−0.487500	−	0.873123i	$-0.662091\pi$
$53$	−7.09810	−0.975000	−0.487500	+	0.873123i	$0.662091\pi$
$54$	−0.843162	−0.114740
$55$	−4.01994	−0.542049
$56$	−11.5463	−1.54294
$57$	2.76731	0.366539
$58$	5.78249	0.759278
$59$	−12.1533	−1.58223	−0.791115	−	0.611667i	$-0.790499\pi$
$59$	−12.1533	−1.58223	−0.791115	+	0.611667i	$0.790499\pi$
$60$	−2.76773	−0.357312
$61$	2.95471	0.378312	0.189156	−	0.981947i	$-0.439425\pi$
$61$	2.95471	0.378312	0.189156	+	0.981947i	$0.439425\pi$
$62$	4.23036	0.537257
$63$	4.16350	0.524552
$64$	4.36734	0.545917
$65$	6.26053	0.776523
$66$	−1.57865	−0.194318
$67$	−11.9789	−1.46346	−0.731729	−	0.681595i	$-0.761287\pi$
$67$	−11.9789	−1.46346	−0.731729	+	0.681595i	$0.761287\pi$
$68$	−1.28908	−0.156324
$69$	7.37894	0.888320
$70$	−7.53728	−0.900877
$71$	−1.56592	−0.185841	−0.0929203	−	0.995674i	$-0.529620\pi$
$71$	−1.56592	−0.185841	−0.0929203	+	0.995674i	$0.529620\pi$
$72$	−2.77323	−0.326828
$73$	−4.76855	−0.558117	−0.279059	−	0.960274i	$-0.590022\pi$
$73$	−4.76855	−0.558117	−0.279059	+	0.960274i	$0.590022\pi$
$74$	2.55834	0.297401
$75$	0.390122	0.0450474
$76$	3.56728	0.409195
$77$	7.79532	0.888359
$78$	2.45854	0.278375
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	−0.515029	−0.0575820
$81$	1.00000	0.111111
$82$	9.95903	1.09979
$83$	9.50659	1.04348	0.521742	−	0.853103i	$-0.325282\pi$
$83$	9.50659	1.04348	0.521742	+	0.853103i	$0.325282\pi$
$84$	5.36708	0.585596
$85$	−2.14706	−0.232882
$86$	−7.32946	−0.790356
$87$	−6.85810	−0.735265
$88$	−5.19230	−0.553502
$89$	−10.9267	−1.15823	−0.579117	−	0.815245i	$-0.696602\pi$
$89$	−10.9267	−1.15823	−0.579117	+	0.815245i	$0.696602\pi$
$90$	−1.81032	−0.190825
$91$	−12.1402	−1.27264
$92$	9.51203	0.991698
$93$	−5.01726	−0.520266
$94$	6.20134	0.639619
$95$	5.94159	0.609594
$96$	−5.74871	−0.586725
$97$	−10.7747	−1.09401	−0.547004	−	0.837130i	$-0.684232\pi$
$97$	−10.7747	−1.09401	−0.547004	+	0.837130i	$0.684232\pi$
$98$	8.71387	0.880234
$99$	1.87230	0.188173
$100$	0.502897	0.0502897
$101$	−15.7899	−1.57115	−0.785577	−	0.618764i	$-0.787634\pi$
$101$	−15.7899	−1.57115	−0.785577	+	0.618764i	$0.787634\pi$
$102$	−0.843162	−0.0834855
$103$	2.32112	0.228707	0.114353	−	0.993440i	$-0.463520\pi$
$103$	2.32112	0.228707	0.114353	+	0.993440i	$0.463520\pi$
$104$	8.08633	0.792930
$105$	8.93930	0.872386
$106$	−5.98485	−0.581300
$107$	17.1250	1.65553	0.827766	−	0.561073i	$-0.189611\pi$
$107$	17.1250	1.65553	0.827766	+	0.561073i	$0.189611\pi$
$108$	1.28908	0.124042
$109$	−9.41208	−0.901514	−0.450757	−	0.892647i	$-0.648846\pi$
$109$	−9.41208	−0.901514	−0.450757	+	0.892647i	$0.648846\pi$
$110$	−3.38946	−0.323173
$111$	−3.03422	−0.287995
$112$	0.998725	0.0943706
$113$	4.11457	0.387066	0.193533	−	0.981094i	$-0.438005\pi$
$113$	4.11457	0.387066	0.193533	+	0.981094i	$0.438005\pi$
$114$	2.33329	0.218533
$115$	15.8431	1.47737
$116$	−8.84062	−0.820831
$117$	−2.91586	−0.269571
$118$	−10.2472	−0.943334
$119$	4.16350	0.381668
$120$	−5.95429	−0.543550
$121$	−7.49450	−0.681318
$122$	2.49130	0.225552
$123$	−11.8115	−1.06501
$124$	−6.46764	−0.580811
$125$	11.5729	1.03511
$126$	3.51051	0.312741
$127$	−11.6609	−1.03474	−0.517371	−	0.855761i	$-0.673089\pi$
$127$	−11.6609	−1.03474	−0.517371	+	0.855761i	$0.673089\pi$
$128$	−7.81504	−0.690758
$129$	8.69283	0.765361
$130$	5.27864	0.462967
$131$	−9.26392	−0.809393	−0.404696	−	0.914451i	$-0.632623\pi$
$131$	−9.26392	−0.809393	−0.404696	+	0.914451i	$0.632623\pi$
$132$	2.41354	0.210071
$133$	−11.5217	−0.999058
$134$	−10.1002	−0.872523
$135$	2.14706	0.184790
$136$	−2.77323	−0.237802
$137$	17.9746	1.53567	0.767836	−	0.640646i	$-0.221334\pi$
$137$	17.9746	1.53567	0.767836	+	0.640646i	$0.221334\pi$
$138$	6.22164	0.529622
$139$	−9.57294	−0.811966	−0.405983	−	0.913881i	$-0.633071\pi$
$139$	−9.57294	−0.811966	−0.405983	+	0.913881i	$0.633071\pi$
$140$	11.5235	0.973909
$141$	−7.35486	−0.619391
$142$	−1.32033	−0.110799
$143$	−5.45935	−0.456534
$144$	0.239876	0.0199897
$145$	−14.7248	−1.22283
$146$	−4.02066	−0.332753
$147$	−10.3348	−0.852396
$148$	−3.91134	−0.321510
$149$	−15.1933	−1.24468	−0.622342	−	0.782746i	$-0.713819\pi$
$149$	−15.1933	−1.24468	−0.622342	+	0.782746i	$0.713819\pi$
$150$	0.328936	0.0268575
$151$	−6.87636	−0.559590	−0.279795	−	0.960060i	$-0.590267\pi$
$151$	−6.87636	−0.559590	−0.279795	+	0.960060i	$0.590267\pi$
$152$	7.67437	0.622474
$153$	1.00000	0.0808452
$154$	6.57272	0.529645
$155$	−10.7724	−0.865258
$156$	−3.75877	−0.300942
$157$	1.44538	0.115354	0.0576769	−	0.998335i	$-0.481631\pi$
$157$	1.44538	0.115354	0.0576769	+	0.998335i	$0.481631\pi$
$158$	0.843162	0.0670784
$159$	7.09810	0.562916
$160$	−12.3428	−0.975787
$161$	−30.7222	−2.42125
$162$	0.843162	0.0662451
$163$	−14.2228	−1.11402	−0.557008	−	0.830507i	$-0.688051\pi$
$163$	−14.2228	−1.11402	−0.557008	+	0.830507i	$0.688051\pi$
$164$	−15.2260	−1.18895
$165$	4.01994	0.312952
$166$	8.01560	0.622131
$167$	2.16612	0.167620	0.0838098	−	0.996482i	$-0.473291\pi$
$167$	2.16612	0.167620	0.0838098	+	0.996482i	$0.473291\pi$
$168$	11.5463	0.890819
$169$	−4.49778	−0.345983
$170$	−1.81032	−0.138845
$171$	−2.76731	−0.211621
$172$	11.2057	0.854429
$173$	−0.160343	−0.0121906	−0.00609532	−	0.999981i	$-0.501940\pi$
$173$	−0.160343	−0.0121906	−0.00609532	+	0.999981i	$0.501940\pi$
$174$	−5.78249	−0.438369
$175$	−1.62427	−0.122784
$176$	0.449119	0.0338536
$177$	12.1533	0.913501
$178$	−9.21302	−0.690545
$179$	−0.312158	−0.0233318	−0.0116659	−	0.999932i	$-0.503713\pi$
$179$	−0.312158	−0.0233318	−0.0116659	+	0.999932i	$0.503713\pi$
$180$	2.76773	0.206294
$181$	−15.3850	−1.14356	−0.571779	−	0.820408i	$-0.693746\pi$
$181$	−15.3850	−1.14356	−0.571779	+	0.820408i	$0.693746\pi$
$182$	−10.2361	−0.758753
$183$	−2.95471	−0.218419
$184$	20.4635	1.50859
$185$	−6.51466	−0.478967
$186$	−4.23036	−0.310185
$187$	1.87230	0.136916
$188$	−9.48099	−0.691472
$189$	−4.16350	−0.302850
$190$	5.00972	0.363443
$191$	−0.652531	−0.0472155	−0.0236077	−	0.999721i	$-0.507515\pi$
$191$	−0.652531	−0.0472155	−0.0236077	+	0.999721i	$0.507515\pi$
$192$	−4.36734	−0.315186
$193$	0.00722550	0.000520103 0	0.000260051	−	1.00000i	$-0.499917\pi$
$193$	0.00722550	0.000520103 0	0.000260051		1.00000i	$0.499917\pi$
$194$	−9.08485	−0.652254
$195$	−6.26053	−0.448326
$196$	−13.3223	−0.951593
$197$	−23.2500	−1.65649	−0.828246	−	0.560364i	$-0.810661\pi$
$197$	−23.2500	−1.65649	−0.828246	+	0.560364i	$0.810661\pi$
$198$	1.57865	0.112190
$199$	−18.5927	−1.31800	−0.658999	−	0.752143i	$-0.729020\pi$
$199$	−18.5927	−1.31800	−0.658999	+	0.752143i	$0.729020\pi$
$200$	1.08190	0.0765016
$201$	11.9789	0.844928
$202$	−13.3134	−0.936731
$203$	28.5537	2.00408
$204$	1.28908	0.0902535
$205$	−25.3601	−1.77122
$206$	1.95708	0.136356
$207$	−7.37894	−0.512872
$208$	−0.699444	−0.0484977
$209$	−5.18123	−0.358393
$210$	7.53728	0.520122
$211$	−3.58122	−0.246541	−0.123271	−	0.992373i	$-0.539338\pi$
$211$	−3.58122	−0.246541	−0.123271	+	0.992373i	$0.539338\pi$
$212$	9.15001	0.628425
$213$	1.56592	0.107295
$214$	14.4391	0.987038
$215$	18.6640	1.27288
$216$	2.77323	0.188694
$217$	20.8894	1.41806
$218$	−7.93591	−0.537488
$219$	4.76855	0.322229
$220$	5.18202	0.349371
$221$	−2.91586	−0.196142
$222$	−2.55834	−0.171704
$223$	−0.681804	−0.0456570	−0.0228285	−	0.999739i	$-0.507267\pi$
$223$	−0.681804	−0.0456570	−0.0228285	+	0.999739i	$0.507267\pi$
$224$	23.9348	1.59921
$225$	−0.390122	−0.0260081
$226$	3.46925	0.230771
$227$	12.8655	0.853913	0.426957	−	0.904272i	$-0.359586\pi$
$227$	12.8655	0.853913	0.426957	+	0.904272i	$0.359586\pi$
$228$	−3.56728	−0.236249
$229$	−0.970739	−0.0641482	−0.0320741	−	0.999485i	$-0.510211\pi$
$229$	−0.970739	−0.0641482	−0.0320741	+	0.999485i	$0.510211\pi$
$230$	13.3583	0.880818
$231$	−7.79532	−0.512894
$232$	−19.0191	−1.24866
$233$	5.70269	0.373596	0.186798	−	0.982398i	$-0.440189\pi$
$233$	5.70269	0.373596	0.186798	+	0.982398i	$0.440189\pi$
$234$	−2.45854	−0.160720
$235$	−15.7913	−1.03011
$236$	15.6666	1.01981
$237$	−1.00000	−0.0649570
$238$	3.51051	0.227552
$239$	8.17111	0.528545	0.264273	−	0.964448i	$-0.414868\pi$
$239$	8.17111	0.528545	0.264273	+	0.964448i	$0.414868\pi$
$240$	0.515029	0.0332450
$241$	−5.03943	−0.324618	−0.162309	−	0.986740i	$-0.551894\pi$
$241$	−5.03943	−0.324618	−0.162309	+	0.986740i	$0.551894\pi$
$242$	−6.31908	−0.406206
$243$	−1.00000	−0.0641500
$244$	−3.80885	−0.243837
$245$	−22.1894	−1.41763
$246$	−9.95903	−0.634964
$247$	8.06908	0.513423
$248$	−13.9140	−0.883540
$249$	−9.50659	−0.602456
$250$	9.75786	0.617141
$251$	−14.5242	−0.916759	−0.458380	−	0.888756i	$-0.651570\pi$
$251$	−14.5242	−0.916759	−0.458380	+	0.888756i	$0.651570\pi$
$252$	−5.36708	−0.338094
$253$	−13.8156	−0.868578
$254$	−9.83207	−0.616919
$255$	2.14706	0.134454
$256$	−15.3240	−0.957751
$257$	−8.98592	−0.560526	−0.280263	−	0.959923i	$-0.590422\pi$
$257$	−8.98592	−0.560526	−0.280263	+	0.959923i	$0.590422\pi$
$258$	7.32946	0.456312
$259$	12.6330	0.784975
$260$	−8.07030	−0.500499
$261$	6.85810	0.424506
$262$	−7.81099	−0.482565
$263$	−21.7306	−1.33996	−0.669982	−	0.742378i	$-0.733698\pi$
$263$	−21.7306	−1.33996	−0.669982	+	0.742378i	$0.733698\pi$
$264$	5.19230	0.319564
$265$	15.2401	0.936190
$266$	−9.71466	−0.595644
$267$	10.9267	0.668706
$268$	15.4418	0.943256
$269$	−17.3754	−1.05940	−0.529698	−	0.848186i	$-0.677695\pi$
$269$	−17.3754	−1.05940	−0.529698	+	0.848186i	$0.677695\pi$
$270$	1.81032	0.110173
$271$	−21.9913	−1.33588	−0.667939	−	0.744216i	$-0.732823\pi$
$271$	−21.9913	−1.33588	−0.667939	+	0.744216i	$0.732823\pi$
$272$	0.239876	0.0145446
$273$	12.1402	0.734757
$274$	15.1555	0.915576
$275$	−0.730425	−0.0440463
$276$	−9.51203	−0.572557
$277$	15.4228	0.926667	0.463334	−	0.886184i	$-0.346653\pi$
$277$	15.4228	0.926667	0.463334	+	0.886184i	$0.346653\pi$
$278$	−8.07154	−0.484099
$279$	5.01726	0.300376
$280$	24.7907	1.48153
$281$	3.27667	0.195470	0.0977350	−	0.995212i	$-0.468840\pi$
$281$	3.27667	0.195470	0.0977350	+	0.995212i	$0.468840\pi$
$282$	−6.20134	−0.369284
$283$	14.2273	0.845725	0.422862	−	0.906194i	$-0.361025\pi$
$283$	14.2273	0.845725	0.422862	+	0.906194i	$0.361025\pi$
$284$	2.01859	0.119782
$285$	−5.94159	−0.351949
$286$	−4.60312	−0.272188
$287$	49.1773	2.90284
$288$	5.74871	0.338746
$289$	1.00000	0.0588235
$290$	−12.4154	−0.729055
$291$	10.7747	0.631626
$292$	6.14704	0.359728
$293$	10.7205	0.626301	0.313150	−	0.949704i	$-0.398616\pi$
$293$	10.7205	0.626301	0.313150	+	0.949704i	$0.398616\pi$
$294$	−8.71387	−0.508203
$295$	26.0940	1.51925
$296$	−8.41457	−0.489087
$297$	−1.87230	−0.108642
$298$	−12.8104	−0.742087
$299$	21.5159	1.24430
$300$	−0.502897	−0.0290348
$301$	−36.1926	−2.08611
$302$	−5.79789	−0.333631
$303$	15.7899	0.907106
$304$	−0.663811	−0.0380722
$305$	−6.34395	−0.363254
$306$	0.843162	0.0482004
$307$	−12.2762	−0.700638	−0.350319	−	0.936630i	$-0.613927\pi$
$307$	−12.2762	−0.700638	−0.350319	+	0.936630i	$0.613927\pi$
$308$	−10.0488	−0.572582
$309$	−2.32112	−0.132044
$310$	−9.08286	−0.515872
$311$	−0.884493	−0.0501550	−0.0250775	−	0.999686i	$-0.507983\pi$
$311$	−0.884493	−0.0501550	−0.0250775	+	0.999686i	$0.507983\pi$
$312$	−8.08633	−0.457798
$313$	−10.9444	−0.618615	−0.309307	−	0.950962i	$-0.600097\pi$
$313$	−10.9444	−0.618615	−0.309307	+	0.950962i	$0.600097\pi$
$314$	1.21869	0.0687746
$315$	−8.93930	−0.503673
$316$	−1.28908	−0.0725163
$317$	−8.05618	−0.452480	−0.226240	−	0.974072i	$-0.572643\pi$
$317$	−8.05618	−0.452480	−0.226240	+	0.974072i	$0.572643\pi$
$318$	5.98485	0.335614
$319$	12.8404	0.718925
$320$	−9.37695	−0.524188
$321$	−17.1250	−0.955822
$322$	−25.9038	−1.44356
$323$	−2.76731	−0.153977
$324$	−1.28908	−0.0716154
$325$	1.13754	0.0630994
$326$	−11.9921	−0.664183
$327$	9.41208	0.520489
$328$	−32.7560	−1.80865
$329$	30.6220	1.68824
$330$	3.38946	0.186584
$331$	31.0903	1.70888	0.854440	−	0.519550i	$-0.173900\pi$
$331$	31.0903	1.70888	0.854440	+	0.519550i	$0.173900\pi$
$332$	−12.2547	−0.672566
$333$	3.03422	0.166274
$334$	1.82639	0.0999357
$335$	25.7195	1.40521
$336$	−0.998725	−0.0544849
$337$	13.2475	0.721637	0.360818	−	0.932636i	$-0.382497\pi$
$337$	13.2475	0.721637	0.360818	+	0.932636i	$0.382497\pi$
$338$	−3.79236	−0.206277
$339$	−4.11457	−0.223473
$340$	2.76773	0.150101
$341$	9.39381	0.508703
$342$	−2.33329	−0.126170
$343$	13.8842	0.749679
$344$	24.1072	1.29977
$345$	−15.8431	−0.852961
$346$	−0.135195	−0.00726813
$347$	15.9484	0.856158	0.428079	−	0.903741i	$-0.359191\pi$
$347$	15.9484	0.856158	0.428079	+	0.903741i	$0.359191\pi$
$348$	8.84062	0.473907
$349$	−35.6193	−1.90666	−0.953330	−	0.301931i	$-0.902369\pi$
$349$	−35.6193	−1.90666	−0.953330	+	0.301931i	$0.902369\pi$
$350$	−1.36953	−0.0732043
$351$	2.91586	0.155637
$352$	10.7633	0.573685
$353$	−10.1588	−0.540699	−0.270350	−	0.962762i	$-0.587139\pi$
$353$	−10.1588	−0.540699	−0.270350	+	0.962762i	$0.587139\pi$
$354$	10.2472	0.544634
$355$	3.36213	0.178443
$356$	14.0854	0.746526
$357$	−4.16350	−0.220356
$358$	−0.263200	−0.0139106
$359$	−3.67208	−0.193805	−0.0969025	−	0.995294i	$-0.530894\pi$
$359$	−3.67208	−0.193805	−0.0969025	+	0.995294i	$0.530894\pi$
$360$	5.95429	0.313819
$361$	−11.3420	−0.596947
$362$	−12.9720	−0.681795
$363$	7.49450	0.393359
$364$	15.6496	0.820263
$365$	10.2384	0.535902
$366$	−2.49130	−0.130222
$367$	6.77614	0.353712	0.176856	−	0.984237i	$-0.443407\pi$
$367$	6.77614	0.353712	0.176856	+	0.984237i	$0.443407\pi$
$368$	−1.77003	−0.0922693
$369$	11.8115	0.614883
$370$	−5.49291	−0.285563
$371$	−29.5530	−1.53431
$372$	6.46764	0.335331
$373$	18.8542	0.976232	0.488116	−	0.872779i	$-0.337684\pi$
$373$	18.8542	0.976232	0.488116	+	0.872779i	$0.337684\pi$
$374$	1.57865	0.0816301
$375$	−11.5729	−0.597624
$376$	−20.3967	−1.05188
$377$	−19.9972	−1.02991
$378$	−3.51051	−0.180561
$379$	17.0654	0.876591	0.438295	−	0.898831i	$-0.355582\pi$
$379$	17.0654	0.876591	0.438295	+	0.898831i	$0.355582\pi$
$380$	−7.65917	−0.392907
$381$	11.6609	0.597408
$382$	−0.550189	−0.0281501
$383$	4.31584	0.220529	0.110265	−	0.993902i	$-0.464830\pi$
$383$	4.31584	0.220529	0.110265	+	0.993902i	$0.464830\pi$
$384$	7.81504	0.398809
$385$	−16.7370	−0.852998
$386$	0.00609227	0.000310088 0
$387$	−8.69283	−0.441881
$388$	13.8895	0.705131
$389$	−22.3100	−1.13116	−0.565581	−	0.824693i	$-0.691348\pi$
$389$	−22.3100	−1.13116	−0.565581	+	0.824693i	$0.691348\pi$
$390$	−5.27864	−0.267294
$391$	−7.37894	−0.373169
$392$	−28.6606	−1.44758
$393$	9.26392	0.467303
$394$	−19.6035	−0.987610
$395$	−2.14706	−0.108030
$396$	−2.41354	−0.121285
$397$	5.51364	0.276722	0.138361	−	0.990382i	$-0.455817\pi$
$397$	5.51364	0.276722	0.138361	+	0.990382i	$0.455817\pi$
$398$	−15.6766	−0.785798
$399$	11.5217	0.576806
$400$	−0.0935809	−0.00467905
$401$	−7.78284	−0.388657	−0.194328	−	0.980937i	$-0.562253\pi$
$401$	−7.78284	−0.388657	−0.194328	+	0.980937i	$0.562253\pi$
$402$	10.1002	0.503751
$403$	−14.6296	−0.728753
$404$	20.3544	1.01267
$405$	−2.14706	−0.106688
$406$	24.0754	1.19484
$407$	5.68096	0.281595
$408$	2.77323	0.137295
$409$	19.8646	0.982241	0.491120	−	0.871092i	$-0.336588\pi$
$409$	19.8646	0.982241	0.491120	+	0.871092i	$0.336588\pi$
$410$	−21.3827	−1.05601
$411$	−17.9746	−0.886621
$412$	−2.99211	−0.147410
$413$	−50.6005	−2.48989
$414$	−6.22164	−0.305777
$415$	−20.4113	−1.00195
$416$	−16.7624	−0.821844
$417$	9.57294	0.468789
$418$	−4.36861	−0.213676
$419$	−19.9777	−0.975973	−0.487986	−	0.872851i	$-0.662268\pi$
$419$	−19.9777	−0.975973	−0.487986	+	0.872851i	$0.662268\pi$
$420$	−11.5235	−0.562287
$421$	−0.00271669	−0.000132403 0	−6.62017e−5	−	1.00000i	$-0.500021\pi$
$421$	−0.00271669	−0.000132403 0	−6.62017e−5		1.00000i	$0.500021\pi$
$422$	−3.01955	−0.146989
$423$	7.35486	0.357606
$424$	19.6846	0.955971
$425$	−0.390122	−0.0189237
$426$	1.32033	0.0639700
$427$	12.3020	0.595333
$428$	−22.0754	−1.06705
$429$	5.45935	0.263580
$430$	15.7368	0.758897
$431$	39.8717	1.92055	0.960277	−	0.279048i	$-0.0900187\pi$
$431$	39.8717	1.92055	0.960277	+	0.279048i	$0.0900187\pi$
$432$	−0.239876	−0.0115410
$433$	17.7803	0.854469	0.427234	−	0.904141i	$-0.359488\pi$
$433$	17.7803	0.854469	0.427234	+	0.904141i	$0.359488\pi$
$434$	17.6131	0.845457
$435$	14.7248	0.705999
$436$	12.1329	0.581061
$437$	20.4198	0.976812
$438$	4.02066	0.192115
$439$	3.02968	0.144599	0.0722994	−	0.997383i	$-0.476966\pi$
$439$	3.02968	0.144599	0.0722994	+	0.997383i	$0.476966\pi$
$440$	11.1482	0.531470
$441$	10.3348	0.492131
$442$	−2.45854	−0.116941
$443$	−21.2920	−1.01162	−0.505808	−	0.862646i	$-0.668805\pi$
$443$	−21.2920	−1.01162	−0.505808	+	0.862646i	$0.668805\pi$
$444$	3.91134	0.185624
$445$	23.4604	1.11213
$446$	−0.574871	−0.0272209
$447$	15.1933	0.718618
$448$	18.1834	0.859086
$449$	8.08909	0.381748	0.190874	−	0.981615i	$-0.438868\pi$
$449$	8.08909	0.381748	0.190874	+	0.981615i	$0.438868\pi$
$450$	−0.328936	−0.0155062
$451$	22.1147	1.04134
$452$	−5.30399	−0.249479
$453$	6.87636	0.323080
$454$	10.8477	0.509108
$455$	26.0657	1.22198
$456$	−7.67437	−0.359385
$457$	23.3479	1.09217	0.546085	−	0.837730i	$-0.316118\pi$
$457$	23.3479	1.09217	0.546085	+	0.837730i	$0.316118\pi$
$458$	−0.818490	−0.0382455
$459$	−1.00000	−0.0466760
$460$	−20.4229	−0.952224
$461$	37.1311	1.72937	0.864685	−	0.502315i	$-0.167518\pi$
$461$	37.1311	1.72937	0.864685	+	0.502315i	$0.167518\pi$
$462$	−6.57272	−0.305790
$463$	−29.8387	−1.38672	−0.693361	−	0.720590i	$-0.743871\pi$
$463$	−29.8387	−1.38672	−0.693361	+	0.720590i	$0.743871\pi$
$464$	1.64509	0.0763716
$465$	10.7724	0.499557
$466$	4.80829	0.222740
$467$	−16.6691	−0.771356	−0.385678	−	0.922634i	$-0.626032\pi$
$467$	−16.6691	−0.771356	−0.385678	+	0.922634i	$0.626032\pi$
$468$	3.75877	0.173749
$469$	−49.8743	−2.30298
$470$	−13.3147	−0.614160
$471$	−1.44538	−0.0665996
$472$	33.7040	1.55135
$473$	−16.2756	−0.748351
$474$	−0.843162	−0.0387277
$475$	1.07959	0.0495349
$476$	−5.36708	−0.246000
$477$	−7.09810	−0.325000
$478$	6.88957	0.315122
$479$	−26.2023	−1.19721	−0.598607	−	0.801043i	$-0.704279\pi$
$479$	−26.2023	−1.19721	−0.598607	+	0.801043i	$0.704279\pi$
$480$	12.3428	0.563371
$481$	−8.84734	−0.403404
$482$	−4.24906	−0.193539
$483$	30.7222	1.39791
$484$	9.66099	0.439136
$485$	23.1340	1.05046
$486$	−0.843162	−0.0382466
$487$	22.5981	1.02402	0.512008	−	0.858981i	$-0.328902\pi$
$487$	22.5981	1.02402	0.512008	+	0.858981i	$0.328902\pi$
$488$	−8.19409	−0.370929
$489$	14.2228	0.643178
$490$	−18.7092	−0.845197
$491$	38.3488	1.73066	0.865329	−	0.501204i	$-0.167109\pi$
$491$	38.3488	1.73066	0.865329	+	0.501204i	$0.167109\pi$
$492$	15.2260	0.686440
$493$	6.85810	0.308873
$494$	6.80354	0.306106
$495$	−4.01994	−0.180683
$496$	1.20352	0.0540397
$497$	−6.51972	−0.292449
$498$	−8.01560	−0.359188
$499$	25.7083	1.15086	0.575431	−	0.817850i	$-0.304834\pi$
$499$	25.7083	1.15086	0.575431	+	0.817850i	$0.304834\pi$
$500$	−14.9184	−0.667171
$501$	−2.16612	−0.0967752
$502$	−12.2463	−0.546577
$503$	36.9791	1.64882	0.824408	−	0.565996i	$-0.191508\pi$
$503$	36.9791	1.64882	0.824408	+	0.565996i	$0.191508\pi$
$504$	−11.5463	−0.514315
$505$	33.9019	1.50862
$506$	−11.6488	−0.517851
$507$	4.49778	0.199753
$508$	15.0319	0.666931
$509$	−23.0926	−1.02356	−0.511780	−	0.859116i	$-0.671014\pi$
$509$	−23.0926	−1.02356	−0.511780	+	0.859116i	$0.671014\pi$
$510$	1.81032	0.0801624
$511$	−19.8539	−0.878284
$512$	2.70944	0.119741
$513$	2.76731	0.122180
$514$	−7.57659	−0.334189
$515$	−4.98359	−0.219603
$516$	−11.2057	−0.493305
$517$	13.7705	0.605626
$518$	10.6516	0.468006
$519$	0.160343	0.00703827
$520$	−17.3619	−0.761368
$521$	1.79648	0.0787051	0.0393526	−	0.999225i	$-0.487470\pi$
$521$	1.79648	0.0787051	0.0393526	+	0.999225i	$0.487470\pi$
$522$	5.78249	0.253093
$523$	−8.18551	−0.357927	−0.178964	−	0.983856i	$-0.557274\pi$
$523$	−8.18551	−0.357927	−0.178964	+	0.983856i	$0.557274\pi$
$524$	11.9419	0.521685
$525$	1.62427	0.0708891
$526$	−18.3224	−0.798894
$527$	5.01726	0.218555
$528$	−0.449119	−0.0195454
$529$	31.4488	1.36734
$530$	12.8499	0.558162
$531$	−12.1533	−0.527410
$532$	14.8524	0.643932
$533$	−34.4407	−1.49179
$534$	9.21302	0.398686
$535$	−36.7684	−1.58963
$536$	33.2203	1.43490
$537$	0.312158	0.0134706
$538$	−14.6503	−0.631618
$539$	19.3497	0.833452
$540$	−2.76773	−0.119104
$541$	18.2277	0.783670	0.391835	−	0.920035i	$-0.371840\pi$
$541$	18.2277	0.783670	0.391835	+	0.920035i	$0.371840\pi$
$542$	−18.5422	−0.796458
$543$	15.3850	0.660233
$544$	5.74871	0.246474
$545$	20.2083	0.865630
$546$	10.2361	0.438066
$547$	19.5268	0.834905	0.417453	−	0.908699i	$-0.362923\pi$
$547$	19.5268	0.834905	0.417453	+	0.908699i	$0.362923\pi$
$548$	−23.1706	−0.989800
$549$	2.95471	0.126104
$550$	−0.615866	−0.0262606
$551$	−18.9785	−0.808510
$552$	−20.4635	−0.870983
$553$	4.16350	0.177050
$554$	13.0039	0.552484
$555$	6.51466	0.276532
$556$	12.3403	0.523344
$557$	−11.5373	−0.488852	−0.244426	−	0.969668i	$-0.578599\pi$
$557$	−11.5373	−0.488852	−0.244426	+	0.969668i	$0.578599\pi$
$558$	4.23036	0.179086
$559$	25.3470	1.07207
$560$	−2.14432	−0.0906142
$561$	−1.87230	−0.0790485
$562$	2.76277	0.116540
$563$	6.12945	0.258325	0.129163	−	0.991623i	$-0.458771\pi$
$563$	6.12945	0.258325	0.129163	+	0.991623i	$0.458771\pi$
$564$	9.48099	0.399222
$565$	−8.83423	−0.371659
$566$	11.9959	0.504226
$567$	4.16350	0.174851
$568$	4.34265	0.182214
$569$	32.3134	1.35465	0.677324	−	0.735685i	$-0.263140\pi$
$569$	32.3134	1.35465	0.677324	+	0.735685i	$0.263140\pi$
$570$	−5.00972	−0.209834
$571$	−9.67794	−0.405009	−0.202505	−	0.979281i	$-0.564908\pi$
$571$	−9.67794	−0.405009	−0.202505	+	0.979281i	$0.564908\pi$
$572$	7.03753	0.294254
$573$	0.652531	0.0272599
$574$	41.4644	1.73069
$575$	2.87869	0.120050
$576$	4.36734	0.181972
$577$	−13.9634	−0.581305	−0.290653	−	0.956829i	$-0.593872\pi$
$577$	−13.9634	−0.581305	−0.290653	+	0.956829i	$0.593872\pi$
$578$	0.843162	0.0350709
$579$	−0.00722550	−0.000300281 0
$580$	18.9814	0.788158
$581$	39.5807	1.64209
$582$	9.08485	0.376579
$583$	−13.2898	−0.550406
$584$	13.2243	0.547224
$585$	6.26053	0.258841
$586$	9.03916	0.373404
$587$	−22.1703	−0.915065	−0.457533	−	0.889193i	$-0.651267\pi$
$587$	−22.1703	−0.915065	−0.457533	+	0.889193i	$0.651267\pi$
$588$	13.3223	0.549402
$589$	−13.8843	−0.572093
$590$	22.0015	0.905786
$591$	23.2500	0.956377
$592$	0.727836	0.0299139
$593$	31.0958	1.27695	0.638475	−	0.769643i	$-0.279566\pi$
$593$	31.0958	1.27695	0.638475	+	0.769643i	$0.279566\pi$
$594$	−1.57865	−0.0647728
$595$	−8.93930	−0.366476
$596$	19.5853	0.802247
$597$	18.5927	0.760947
$598$	18.1414	0.741858
$599$	10.2257	0.417810	0.208905	−	0.977936i	$-0.433010\pi$
$599$	10.2257	0.417810	0.208905	+	0.977936i	$0.433010\pi$
$600$	−1.08190	−0.0441682
$601$	−0.886250	−0.0361509	−0.0180754	−	0.999837i	$-0.505754\pi$
$601$	−0.886250	−0.0361509	−0.0180754	+	0.999837i	$0.505754\pi$
$602$	−30.5162	−1.24375
$603$	−11.9789	−0.487820
$604$	8.86416	0.360678
$605$	16.0912	0.654199
$606$	13.3134	0.540822
$607$	2.18449	0.0886655	0.0443328	−	0.999017i	$-0.485884\pi$
$607$	2.18449	0.0886655	0.0443328	+	0.999017i	$0.485884\pi$
$608$	−15.9084	−0.645173
$609$	−28.5537	−1.15705
$610$	−5.34898	−0.216574
$611$	−21.4457	−0.867601
$612$	−1.28908	−0.0521079
$613$	5.74362	0.231983	0.115991	−	0.993250i	$-0.462996\pi$
$613$	5.74362	0.231983	0.115991	+	0.993250i	$0.462996\pi$
$614$	−10.3508	−0.417725
$615$	25.3601	1.02262
$616$	−21.6182	−0.871021
$617$	19.8661	0.799780	0.399890	−	0.916563i	$-0.369048\pi$
$617$	19.8661	0.799780	0.399890	+	0.916563i	$0.369048\pi$
$618$	−1.95708	−0.0787254
$619$	−43.9713	−1.76736	−0.883678	−	0.468095i	$-0.844941\pi$
$619$	−43.9713	−1.76736	−0.883678	+	0.468095i	$0.844941\pi$
$620$	13.8864	0.557692
$621$	7.37894	0.296107
$622$	−0.745771	−0.0299027
$623$	−45.4935	−1.82266
$624$	0.699444	0.0280002
$625$	−22.8972	−0.915888
$626$	−9.22792	−0.368822
$627$	5.18123	0.206918
$628$	−1.86321	−0.0743500
$629$	3.03422	0.120982
$630$	−7.53728	−0.300292
$631$	24.8952	0.991063	0.495532	−	0.868590i	$-0.334973\pi$
$631$	24.8952	0.991063	0.495532	+	0.868590i	$0.334973\pi$
$632$	−2.77323	−0.110313
$633$	3.58122	0.142341
$634$	−6.79267	−0.269771
$635$	25.0368	0.993555
$636$	−9.15001	−0.362821
$637$	−30.1347	−1.19398
$638$	10.8265	0.428627
$639$	−1.56592	−0.0619469
$640$	16.7794	0.663263
$641$	−11.0436	−0.436198	−0.218099	−	0.975927i	$-0.569986\pi$
$641$	−11.0436	−0.436198	−0.218099	+	0.975927i	$0.569986\pi$
$642$	−14.4391	−0.569867
$643$	−18.3554	−0.723868	−0.361934	−	0.932204i	$-0.617883\pi$
$643$	−18.3554	−0.723868	−0.361934	+	0.932204i	$0.617883\pi$
$644$	39.6034	1.56059
$645$	−18.6640	−0.734896
$646$	−2.33329	−0.0918021
$647$	35.0275	1.37707	0.688537	−	0.725201i	$-0.258253\pi$
$647$	35.0275	1.37707	0.688537	+	0.725201i	$0.258253\pi$
$648$	−2.77323	−0.108943
$649$	−22.7547	−0.893199
$650$	0.959130	0.0376202
$651$	−20.8894	−0.818719
$652$	18.3343	0.718027
$653$	37.3889	1.46314	0.731570	−	0.681766i	$-0.238788\pi$
$653$	37.3889	1.46314	0.731570	+	0.681766i	$0.238788\pi$
$654$	7.93591	0.310319
$655$	19.8902	0.777175
$656$	2.83330	0.110622
$657$	−4.76855	−0.186039
$658$	25.8193	1.00654
$659$	−42.8908	−1.67079	−0.835394	−	0.549651i	$-0.814761\pi$
$659$	−42.8908	−1.67079	−0.835394	+	0.549651i	$0.814761\pi$
$660$	−5.18202	−0.201710
$661$	10.2551	0.398876	0.199438	−	0.979910i	$-0.436088\pi$
$661$	10.2551	0.398876	0.199438	+	0.979910i	$0.436088\pi$
$662$	26.2142	1.01884
$663$	2.91586	0.113242
$664$	−26.3639	−1.02312
$665$	24.7378	0.959291
$666$	2.55834	0.0991336
$667$	−50.6055	−1.95945
$668$	−2.79230	−0.108037
$669$	0.681804	0.0263601
$670$	21.6857	0.837792
$671$	5.53210	0.213564
$672$	−23.9348	−0.923303
$673$	−24.3334	−0.937983	−0.468991	−	0.883203i	$-0.655382\pi$
$673$	−24.3334	−0.937983	−0.468991	+	0.883203i	$0.655382\pi$
$674$	11.1698	0.430244
$675$	0.390122	0.0150158
$676$	5.79799	0.223000
$677$	−16.4734	−0.633126	−0.316563	−	0.948572i	$-0.602529\pi$
$677$	−16.4734	−0.633126	−0.316563	+	0.948572i	$0.602529\pi$
$678$	−3.46925	−0.133236
$679$	−44.8607	−1.72159
$680$	5.95429	0.228337
$681$	−12.8655	−0.493007
$682$	7.92050	0.303292
$683$	45.4072	1.73746	0.868729	−	0.495288i	$-0.164938\pi$
$683$	45.4072	1.73746	0.868729	+	0.495288i	$0.164938\pi$
$684$	3.56728	0.136398
$685$	−38.5926	−1.47455
$686$	11.7067	0.446963
$687$	0.970739	0.0370360
$688$	−2.08520	−0.0794975
$689$	20.6971	0.788495
$690$	−13.3583	−0.508540
$691$	−34.6060	−1.31647	−0.658237	−	0.752811i	$-0.728697\pi$
$691$	−34.6060	−1.31647	−0.658237	+	0.752811i	$0.728697\pi$
$692$	0.206694	0.00785734
$693$	7.79532	0.296120
$694$	13.4471	0.510446
$695$	20.5537	0.779647
$696$	19.0191	0.720915
$697$	11.8115	0.447393
$698$	−30.0329	−1.13676
$699$	−5.70269	−0.215696
$700$	2.09381	0.0791388
$701$	−16.3614	−0.617960	−0.308980	−	0.951068i	$-0.599988\pi$
$701$	−16.3614	−0.617960	−0.308980	+	0.951068i	$0.599988\pi$
$702$	2.45854	0.0927916
$703$	−8.39662	−0.316684
$704$	8.17696	0.308181
$705$	15.7913	0.594737
$706$	−8.56553	−0.322368
$707$	−65.7413	−2.47246
$708$	−15.6666	−0.588787
$709$	−10.8331	−0.406845	−0.203423	−	0.979091i	$-0.565207\pi$
$709$	−10.8331	−0.406845	−0.203423	+	0.979091i	$0.565207\pi$
$710$	2.83482	0.106389
$711$	1.00000	0.0375029
$712$	30.3023	1.13563
$713$	−37.0221	−1.38649
$714$	−3.51051	−0.131377
$715$	11.7216	0.438362
$716$	0.402396	0.0150383
$717$	−8.17111	−0.305156
$718$	−3.09616	−0.115548
$719$	−14.4208	−0.537804	−0.268902	−	0.963168i	$-0.586661\pi$
$719$	−14.4208	−0.537804	−0.268902	+	0.963168i	$0.586661\pi$
$720$	−0.515029	−0.0191940
$721$	9.66399	0.359906
$722$	−9.56315	−0.355903
$723$	5.03943	0.187418
$724$	19.8324	0.737067
$725$	−2.67549	−0.0993654
$726$	6.31908	0.234523
$727$	−48.3154	−1.79192	−0.895960	−	0.444135i	$-0.853511\pi$
$727$	−48.3154	−1.79192	−0.895960	+	0.444135i	$0.853511\pi$
$728$	33.6674	1.24780
$729$	1.00000	0.0370370
$730$	8.63262	0.319508
$731$	−8.69283	−0.321516
$732$	3.80885	0.140779
$733$	18.5511	0.685200	0.342600	−	0.939481i	$-0.388692\pi$
$733$	18.5511	0.685200	0.342600	+	0.939481i	$0.388692\pi$
$734$	5.71339	0.210885
$735$	22.1894	0.818467
$736$	−42.4194	−1.56360
$737$	−22.4281	−0.826150
$738$	9.95903	0.366597
$739$	24.9544	0.917963	0.458982	−	0.888446i	$-0.348214\pi$
$739$	24.9544	0.917963	0.458982	+	0.888446i	$0.348214\pi$
$740$	8.39790	0.308713
$741$	−8.06908	−0.296425
$742$	−24.9180	−0.914767
$743$	−10.5540	−0.387191	−0.193595	−	0.981081i	$-0.562015\pi$
$743$	−10.5540	−0.387191	−0.193595	+	0.981081i	$0.562015\pi$
$744$	13.9140	0.510112
$745$	32.6210	1.19514
$746$	15.8971	0.582035
$747$	9.50659	0.347828
$748$	−2.41354	−0.0882477
$749$	71.2998	2.60524
$750$	−9.75786	−0.356307
$751$	−12.2246	−0.446080	−0.223040	−	0.974809i	$-0.571598\pi$
$751$	−12.2246	−0.446080	−0.223040	+	0.974809i	$0.571598\pi$
$752$	1.76426	0.0643358
$753$	14.5242	0.529291
$754$	−16.8609	−0.614038
$755$	14.7640	0.537316
$756$	5.36708	0.195199
$757$	23.7078	0.861675	0.430838	−	0.902429i	$-0.358218\pi$
$757$	23.7078	0.861675	0.430838	+	0.902429i	$0.358218\pi$
$758$	14.3889	0.522628
$759$	13.8156	0.501474
$760$	−16.4774	−0.597697
$761$	−9.22881	−0.334544	−0.167272	−	0.985911i	$-0.553496\pi$
$761$	−9.22881	−0.334544	−0.167272	+	0.985911i	$0.553496\pi$
$762$	9.83207	0.356178
$763$	−39.1872	−1.41867
$764$	0.841162	0.0304322
$765$	−2.14706	−0.0776272
$766$	3.63895	0.131481
$767$	35.4374	1.27957
$768$	15.3240	0.552958
$769$	−32.8449	−1.18442	−0.592209	−	0.805785i	$-0.701744\pi$
$769$	−32.8449	−1.18442	−0.592209	+	0.805785i	$0.701744\pi$
$770$	−14.1120	−0.508562
$771$	8.98592	0.323620
$772$	−0.00931423	−0.000335226 0
$773$	20.0747	0.722038	0.361019	−	0.932558i	$-0.382429\pi$
$773$	20.0747	0.722038	0.361019	+	0.932558i	$0.382429\pi$
$774$	−7.32946	−0.263452
$775$	−1.95734	−0.0703099
$776$	29.8808	1.07266
$777$	−12.6330	−0.453205
$778$	−18.8109	−0.674405
$779$	−32.6861	−1.17110
$780$	8.07030	0.288963
$781$	−2.93187	−0.104911
$782$	−6.22164	−0.222486
$783$	−6.85810	−0.245088
$784$	2.47906	0.0885378
$785$	−3.10332	−0.110762
$786$	7.81099	0.278609
$787$	−26.9405	−0.960327	−0.480163	−	0.877179i	$-0.659423\pi$
$787$	−26.9405	−0.960327	−0.480163	+	0.877179i	$0.659423\pi$
$788$	29.9710	1.06767
$789$	21.7306	0.773628
$790$	−1.81032	−0.0644084
$791$	17.1310	0.609108
$792$	−5.19230	−0.184501
$793$	−8.61552	−0.305946
$794$	4.64889	0.164983
$795$	−15.2401	−0.540510
$796$	23.9674	0.849501
$797$	−23.9010	−0.846618	−0.423309	−	0.905985i	$-0.639132\pi$
$797$	−23.9010	−0.846618	−0.423309	+	0.905985i	$0.639132\pi$
$798$	9.71466	0.343895
$799$	7.35486	0.260196
$800$	−2.24270	−0.0792913
$801$	−10.9267	−0.386078
$802$	−6.56220	−0.231719
$803$	−8.92815	−0.315068
$804$	−15.4418	−0.544589
$805$	65.9626	2.32488
$806$	−12.3351	−0.434487
$807$	17.3754	0.611642
$808$	43.7890	1.54049
$809$	−10.2039	−0.358752	−0.179376	−	0.983781i	$-0.557408\pi$
$809$	−10.2039	−0.358752	−0.179376	+	0.983781i	$0.557408\pi$
$810$	−1.81032	−0.0636082
$811$	−52.7315	−1.85165	−0.925827	−	0.377949i	$-0.876630\pi$
$811$	−52.7315	−1.85165	−0.925827	+	0.377949i	$0.876630\pi$
$812$	−36.8079	−1.29171
$813$	21.9913	0.771269
$814$	4.78997	0.167888
$815$	30.5373	1.06967
$816$	−0.239876	−0.00839734
$817$	24.0557	0.841604
$818$	16.7491	0.585617
$819$	−12.1402	−0.424212
$820$	32.6911	1.14162
$821$	39.4134	1.37554	0.687768	−	0.725930i	$-0.258591\pi$
$821$	39.4134	1.37554	0.687768	+	0.725930i	$0.258591\pi$
$822$	−15.1555	−0.528608
$823$	−18.4120	−0.641803	−0.320901	−	0.947113i	$-0.603986\pi$
$823$	−18.4120	−0.641803	−0.320901	+	0.947113i	$0.603986\pi$
$824$	−6.43699	−0.224243
$825$	0.730425	0.0254301
$826$	−42.6644	−1.48448
$827$	51.8488	1.80296	0.901480	−	0.432821i	$-0.142482\pi$
$827$	51.8488	1.80296	0.901480	+	0.432821i	$0.142482\pi$
$828$	9.51203	0.330566
$829$	−11.5296	−0.400438	−0.200219	−	0.979751i	$-0.564165\pi$
$829$	−11.5296	−0.400438	−0.200219	+	0.979751i	$0.564165\pi$
$830$	−17.2100	−0.597368
$831$	−15.4228	−0.535012
$832$	−12.7345	−0.441491
$833$	10.3348	0.358078
$834$	8.07154	0.279495
$835$	−4.65080	−0.160948
$836$	6.67900	0.230998
$837$	−5.01726	−0.173422
$838$	−16.8444	−0.581881
$839$	−27.0698	−0.934552	−0.467276	−	0.884111i	$-0.654765\pi$
$839$	−27.0698	−0.934552	−0.467276	+	0.884111i	$0.654765\pi$
$840$	−24.7907	−0.855360
$841$	18.0335	0.621846
$842$	−0.00229061	−7.89396e−5 0
$843$	−3.27667	−0.112855
$844$	4.61647	0.158905
$845$	9.65702	0.332211
$846$	6.20134	0.213206
$847$	−31.2034	−1.07216
$848$	−1.70267	−0.0584698
$849$	−14.2273	−0.488279
$850$	−0.328936	−0.0112824
$851$	−22.3893	−0.767496
$852$	−2.01859	−0.0691559
$853$	28.3842	0.971858	0.485929	−	0.873998i	$-0.338481\pi$
$853$	28.3842	0.971858	0.485929	+	0.873998i	$0.338481\pi$
$854$	10.3725	0.354941
$855$	5.94159	0.203198
$856$	−47.4914	−1.62322
$857$	−35.4163	−1.20980	−0.604899	−	0.796302i	$-0.706787\pi$
$857$	−35.4163	−1.20980	−0.604899	+	0.796302i	$0.706787\pi$
$858$	4.60312	0.157148
$859$	−2.08367	−0.0710940	−0.0355470	−	0.999368i	$-0.511317\pi$
$859$	−2.08367	−0.0710940	−0.0355470	+	0.999368i	$0.511317\pi$
$860$	−24.0594	−0.820419
$861$	−49.1773	−1.67596
$862$	33.6183	1.14505
$863$	−38.7689	−1.31971	−0.659855	−	0.751393i	$-0.729382\pi$
$863$	−38.7689	−1.31971	−0.659855	+	0.751393i	$0.729382\pi$
$864$	−5.74871	−0.195575
$865$	0.344266	0.0117054
$866$	14.9917	0.509439
$867$	−1.00000	−0.0339618
$868$	−26.9280	−0.913997
$869$	1.87230	0.0635134
$870$	12.4154	0.420920
$871$	34.9288	1.18352
$872$	26.1018	0.883919
$873$	−10.7747	−0.364670
$874$	17.2172	0.582381
$875$	48.1839	1.62891
$876$	−6.14704	−0.207689
$877$	−26.0227	−0.878723	−0.439361	−	0.898310i	$-0.644795\pi$
$877$	−26.0227	−0.878723	−0.439361	+	0.898310i	$0.644795\pi$
$878$	2.55451	0.0862106
$879$	−10.7205	−0.361595
$880$	−0.964288	−0.0325061
$881$	44.2866	1.49205	0.746026	−	0.665916i	$-0.231959\pi$
$881$	44.2866	1.49205	0.746026	+	0.665916i	$0.231959\pi$
$882$	8.71387	0.293411
$883$	−34.6603	−1.16641	−0.583206	−	0.812324i	$-0.698202\pi$
$883$	−34.6603	−1.16641	−0.583206	+	0.812324i	$0.698202\pi$
$884$	3.75877	0.126421
$885$	−26.0940	−0.877140
$886$	−17.9526	−0.603131
$887$	−52.6316	−1.76720	−0.883599	−	0.468245i	$-0.844887\pi$
$887$	−52.6316	−1.76720	−0.883599	+	0.468245i	$0.844887\pi$
$888$	8.41457	0.282374
$889$	−48.5504	−1.62833
$890$	19.7809	0.663058
$891$	1.87230	0.0627243
$892$	0.878898	0.0294277
$893$	−20.3532	−0.681093
$894$	12.8104	0.428444
$895$	0.670224	0.0224031
$896$	−32.5379	−1.08702
$897$	−21.5159	−0.718396
$898$	6.82041	0.227600
$899$	34.4089	1.14760
$900$	0.502897	0.0167632
$901$	−7.09810	−0.236472
$902$	18.6463	0.620853
$903$	36.1926	1.20441
$904$	−11.4106	−0.379512
$905$	33.0325	1.09804
$906$	5.79789	0.192622
$907$	44.7629	1.48633	0.743164	−	0.669109i	$-0.233324\pi$
$907$	44.7629	1.48633	0.743164	+	0.669109i	$0.233324\pi$
$908$	−16.5846	−0.550380
$909$	−15.7899	−0.523718
$910$	21.9776	0.728551
$911$	54.6402	1.81031	0.905155	−	0.425082i	$-0.139755\pi$
$911$	54.6402	1.81031	0.905155	+	0.425082i	$0.139755\pi$
$912$	0.663811	0.0219810
$913$	17.7992	0.589067
$914$	19.6861	0.651158
$915$	6.34395	0.209725
$916$	1.25136	0.0413460
$917$	−38.5704	−1.27371
$918$	−0.843162	−0.0278285
$919$	−36.2707	−1.19646	−0.598230	−	0.801325i	$-0.704129\pi$
$919$	−36.2707	−1.19646	−0.598230	+	0.801325i	$0.704129\pi$
$920$	−43.9364	−1.44854
$921$	12.2762	0.404514
$922$	31.3076	1.03106
$923$	4.56600	0.150292
$924$	10.0488	0.330580
$925$	−1.18371	−0.0389203
$926$	−25.1589	−0.826772
$927$	2.32112	0.0762356
$928$	39.4252	1.29420
$929$	−31.2646	−1.02576	−0.512880	−	0.858460i	$-0.671421\pi$
$929$	−31.2646	−1.02576	−0.512880	+	0.858460i	$0.671421\pi$
$930$	9.08286	0.297839
$931$	−28.5995	−0.937309
$932$	−7.35121	−0.240797
$933$	0.884493	0.0289570
$934$	−14.0548	−0.459887
$935$	−4.01994	−0.131466
$936$	8.08633	0.264310
$937$	−34.6479	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$937$	−34.6479	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$938$	−42.0521	−1.37305
$939$	10.9444	0.357157
$940$	20.3563	0.663948
$941$	−52.3598	−1.70688	−0.853440	−	0.521190i	$-0.825488\pi$
$941$	−52.3598	−1.70688	−0.853440	+	0.521190i	$0.825488\pi$
$942$	−1.21869	−0.0397070
$943$	−87.1565	−2.83821
$944$	−2.91529	−0.0948848
$945$	8.93930	0.290795
$946$	−13.7229	−0.446171
$947$	−5.76547	−0.187353	−0.0936763	−	0.995603i	$-0.529862\pi$
$947$	−5.76547	−0.187353	−0.0936763	+	0.995603i	$0.529862\pi$
$948$	1.28908	0.0418673
$949$	13.9044	0.451357
$950$	0.910268	0.0295330
$951$	8.05618	0.261240
$952$	−11.5463	−0.374219
$953$	34.1809	1.10723	0.553614	−	0.832774i	$-0.313248\pi$
$953$	34.1809	1.10723	0.553614	+	0.832774i	$0.313248\pi$
$954$	−5.98485	−0.193767
$955$	1.40102	0.0453361
$956$	−10.5332	−0.340668
$957$	−12.8404	−0.415071
$958$	−22.0928	−0.713786
$959$	74.8372	2.41662
$960$	9.37695	0.302640
$961$	−5.82710	−0.187971
$962$	−7.45975	−0.240512
$963$	17.1250	0.551844
$964$	6.49622	0.209229
$965$	−0.0155136	−0.000499400 0
$966$	25.9038	0.833442
$967$	4.98512	0.160311	0.0801553	−	0.996782i	$-0.474458\pi$
$967$	4.98512	0.160311	0.0801553	+	0.996782i	$0.474458\pi$
$968$	20.7839	0.668021
$969$	2.76731	0.0888988
$970$	19.5057	0.626292
$971$	6.50782	0.208846	0.104423	−	0.994533i	$-0.466700\pi$
$971$	6.50782	0.208846	0.104423	+	0.994533i	$0.466700\pi$
$972$	1.28908	0.0413472
$973$	−39.8570	−1.27776
$974$	19.0538	0.610525
$975$	−1.13754	−0.0364304
$976$	0.708765	0.0226870
$977$	−33.7790	−1.08069	−0.540343	−	0.841445i	$-0.681705\pi$
$977$	−33.7790	−1.08069	−0.540343	+	0.841445i	$0.681705\pi$
$978$	11.9921	0.383466
$979$	−20.4581	−0.653845
$980$	28.6038	0.913715
$981$	−9.41208	−0.300505
$982$	32.3343	1.03183
$983$	−8.85241	−0.282348	−0.141174	−	0.989985i	$-0.545088\pi$
$983$	−8.85241	−0.282348	−0.141174	+	0.989985i	$0.545088\pi$
$984$	32.7560	1.04422
$985$	49.9192	1.59056
$986$	5.78249	0.184152
$987$	−30.6220	−0.974708
$988$	−10.4017	−0.330921
$989$	64.1439	2.03966
$990$	−3.38946	−0.107724
$991$	−31.5519	−1.00228	−0.501139	−	0.865367i	$-0.667086\pi$
$991$	−31.5519	−1.00228	−0.501139	+	0.865367i	$0.667086\pi$
$992$	28.8428	0.915758
$993$	−31.0903	−0.986622
$994$	−5.49718	−0.174360
$995$	39.9196	1.26554
$996$	12.2547	0.388306
$997$	−45.3867	−1.43741	−0.718706	−	0.695314i	$-0.755265\pi$
$997$	−45.3867	−1.43741	−0.718706	+	0.695314i	$0.755265\pi$
$998$	21.6763	0.686151
$999$	−3.03422	−0.0959984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.f.1.16	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.f.1.16	✓	22	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:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q+0.843162 q^{2} -1.00000 q^{3} -1.28908 q^{4} -2.14706 q^{5} -0.843162 q^{6} +4.16350 q^{7} -2.77323 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.843162 q^{2} -1.00000 q^{3} -1.28908 q^{4} -2.14706 q^{5} -0.843162 q^{6} +4.16350 q^{7} -2.77323 q^{8} +1.00000 q^{9} -1.81032 q^{10} +1.87230 q^{11} +1.28908 q^{12} -2.91586 q^{13} +3.51051 q^{14} +2.14706 q^{15} +0.239876 q^{16} +1.00000 q^{17} +0.843162 q^{18} -2.76731 q^{19} +2.76773 q^{20} -4.16350 q^{21} +1.57865 q^{22} -7.37894 q^{23} +2.77323 q^{24} -0.390122 q^{25} -2.45854 q^{26} -1.00000 q^{27} -5.36708 q^{28} +6.85810 q^{29} +1.81032 q^{30} +5.01726 q^{31} +5.74871 q^{32} -1.87230 q^{33} +0.843162 q^{34} -8.93930 q^{35} -1.28908 q^{36} +3.03422 q^{37} -2.33329 q^{38} +2.91586 q^{39} +5.95429 q^{40} +11.8115 q^{41} -3.51051 q^{42} -8.69283 q^{43} -2.41354 q^{44} -2.14706 q^{45} -6.22164 q^{46} +7.35486 q^{47} -0.239876 q^{48} +10.3348 q^{49} -0.328936 q^{50} -1.00000 q^{51} +3.75877 q^{52} -7.09810 q^{53} -0.843162 q^{54} -4.01994 q^{55} -11.5463 q^{56} +2.76731 q^{57} +5.78249 q^{58} -12.1533 q^{59} -2.76773 q^{60} +2.95471 q^{61} +4.23036 q^{62} +4.16350 q^{63} +4.36734 q^{64} +6.26053 q^{65} -1.57865 q^{66} -11.9789 q^{67} -1.28908 q^{68} +7.37894 q^{69} -7.53728 q^{70} -1.56592 q^{71} -2.77323 q^{72} -4.76855 q^{73} +2.55834 q^{74} +0.390122 q^{75} +3.56728 q^{76} +7.79532 q^{77} +2.45854 q^{78} +1.00000 q^{79} -0.515029 q^{80} +1.00000 q^{81} +9.95903 q^{82} +9.50659 q^{83} +5.36708 q^{84} -2.14706 q^{85} -7.32946 q^{86} -6.85810 q^{87} -5.19230 q^{88} -10.9267 q^{89} -1.81032 q^{90} -12.1402 q^{91} +9.51203 q^{92} -5.01726 q^{93} +6.20134 q^{94} +5.94159 q^{95} -5.74871 q^{96} -10.7747 q^{97} +8.71387 q^{98} +1.87230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + q^{2} - 22 q^{3} + 19 q^{4} + q^{5} - q^{6} - 15 q^{7} + 15 q^{8} + 22 q^{9}+O(q^{10}) \) 22 * q + q^2 - 22 * q^3 + 19 * q^4 + q^5 - q^6 - 15 * q^7 + 15 * q^8 + 22 * q^9	\( 22 q + q^{2} - 22 q^{3} + 19 q^{4} + q^{5} - q^{6} - 15 q^{7} + 15 q^{8} + 22 q^{9} - 13 q^{10} - 23 q^{11} - 19 q^{12} - 18 q^{13} - 9 q^{14} - q^{15} + 21 q^{16} + 22 q^{17} + q^{18} - 30 q^{19} - 7 q^{20} + 15 q^{21} + 4 q^{22} - 3 q^{23} - 15 q^{24} + 19 q^{25} - 7 q^{26} - 22 q^{27} - 25 q^{28} - 7 q^{29} + 13 q^{30} - 10 q^{31} + 31 q^{32} + 23 q^{33} + q^{34} - 11 q^{35} + 19 q^{36} - q^{37} - 29 q^{38} + 18 q^{39} - 59 q^{40} + 9 q^{42} - 43 q^{43} - 80 q^{44} + q^{45} - 43 q^{46} + 2 q^{47} - 21 q^{48} + 43 q^{49} + 25 q^{50} - 22 q^{51} - 5 q^{52} - q^{53} - q^{54} - 19 q^{55} - 8 q^{56} + 30 q^{57} - 43 q^{58} - 28 q^{59} + 7 q^{60} - 29 q^{61} - 3 q^{62} - 15 q^{63} + 23 q^{64} + 19 q^{65} - 4 q^{66} - 16 q^{67} + 19 q^{68} + 3 q^{69} - 5 q^{70} - q^{71} + 15 q^{72} - 19 q^{73} - 24 q^{74} - 19 q^{75} - 72 q^{76} + 24 q^{77} + 7 q^{78} + 22 q^{79} - 82 q^{80} + 22 q^{81} - 81 q^{82} - 29 q^{83} + 25 q^{84} + q^{85} - 42 q^{86} + 7 q^{87} - 43 q^{88} - 28 q^{89} - 13 q^{90} - 96 q^{91} - 11 q^{92} + 10 q^{93} - 63 q^{94} - 23 q^{95} - 31 q^{96} - 51 q^{97} + 12 q^{98} - 23 q^{99}+O(q^{100}) \) 22 * q + q^2 - 22 * q^3 + 19 * q^4 + q^5 - q^6 - 15 * q^7 + 15 * q^8 + 22 * q^9 - 13 * q^10 - 23 * q^11 - 19 * q^12 - 18 * q^13 - 9 * q^14 - q^15 + 21 * q^16 + 22 * q^17 + q^18 - 30 * q^19 - 7 * q^20 + 15 * q^21 + 4 * q^22 - 3 * q^23 - 15 * q^24 + 19 * q^25 - 7 * q^26 - 22 * q^27 - 25 * q^28 - 7 * q^29 + 13 * q^30 - 10 * q^31 + 31 * q^32 + 23 * q^33 + q^34 - 11 * q^35 + 19 * q^36 - q^37 - 29 * q^38 + 18 * q^39 - 59 * q^40 + 9 * q^42 - 43 * q^43 - 80 * q^44 + q^45 - 43 * q^46 + 2 * q^47 - 21 * q^48 + 43 * q^49 + 25 * q^50 - 22 * q^51 - 5 * q^52 - q^53 - q^54 - 19 * q^55 - 8 * q^56 + 30 * q^57 - 43 * q^58 - 28 * q^59 + 7 * q^60 - 29 * q^61 - 3 * q^62 - 15 * q^63 + 23 * q^64 + 19 * q^65 - 4 * q^66 - 16 * q^67 + 19 * q^68 + 3 * q^69 - 5 * q^70 - q^71 + 15 * q^72 - 19 * q^73 - 24 * q^74 - 19 * q^75 - 72 * q^76 + 24 * q^77 + 7 * q^78 + 22 * q^79 - 82 * q^80 + 22 * q^81 - 81 * q^82 - 29 * q^83 + 25 * q^84 + q^85 - 42 * q^86 + 7 * q^87 - 43 * q^88 - 28 * q^89 - 13 * q^90 - 96 * q^91 - 11 * q^92 + 10 * q^93 - 63 * q^94 - 23 * q^95 - 31 * q^96 - 51 * q^97 + 12 * q^98 - 23 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more