LMFDB - Embedded newform 6005.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6005,2,Mod(1,6005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6005, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6005 = 5 \cdot 1201 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6005.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:	$47.9501664138$
Analytic rank:	$0$
Dimension:	$111$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71589 q^{2} +0.174668 q^{3} +5.37604 q^{4} +1.00000 q^{5} -0.474377 q^{6} +2.58754 q^{7} -9.16893 q^{8} -2.96949 q^{9} +O(q^{10})$	\(q-2.71589 q^{2} +0.174668 q^{3} +5.37604 q^{4} +1.00000 q^{5} -0.474377 q^{6} +2.58754 q^{7} -9.16893 q^{8} -2.96949 q^{9} -2.71589 q^{10} -4.89698 q^{11} +0.939019 q^{12} -6.47521 q^{13} -7.02747 q^{14} +0.174668 q^{15} +14.1497 q^{16} -2.72774 q^{17} +8.06480 q^{18} -2.42289 q^{19} +5.37604 q^{20} +0.451960 q^{21} +13.2996 q^{22} -5.41691 q^{23} -1.60151 q^{24} +1.00000 q^{25} +17.5859 q^{26} -1.04268 q^{27} +13.9107 q^{28} +0.00262486 q^{29} -0.474377 q^{30} -8.15861 q^{31} -20.0911 q^{32} -0.855344 q^{33} +7.40822 q^{34} +2.58754 q^{35} -15.9641 q^{36} +9.14813 q^{37} +6.58029 q^{38} -1.13101 q^{39} -9.16893 q^{40} -4.73959 q^{41} -1.22747 q^{42} -1.41359 q^{43} -26.3263 q^{44} -2.96949 q^{45} +14.7117 q^{46} +11.3149 q^{47} +2.47149 q^{48} -0.304616 q^{49} -2.71589 q^{50} -0.476447 q^{51} -34.8109 q^{52} -12.5462 q^{53} +2.83179 q^{54} -4.89698 q^{55} -23.7250 q^{56} -0.423200 q^{57} -0.00712881 q^{58} +1.08610 q^{59} +0.939019 q^{60} -3.10876 q^{61} +22.1579 q^{62} -7.68369 q^{63} +26.2657 q^{64} -6.47521 q^{65} +2.32302 q^{66} +11.7848 q^{67} -14.6644 q^{68} -0.946159 q^{69} -7.02747 q^{70} +5.00848 q^{71} +27.2270 q^{72} +3.22703 q^{73} -24.8453 q^{74} +0.174668 q^{75} -13.0255 q^{76} -12.6711 q^{77} +3.07169 q^{78} -15.2470 q^{79} +14.1497 q^{80} +8.72635 q^{81} +12.8722 q^{82} -17.7693 q^{83} +2.42975 q^{84} -2.72774 q^{85} +3.83916 q^{86} +0.000458477 q^{87} +44.9000 q^{88} +0.0257273 q^{89} +8.06480 q^{90} -16.7549 q^{91} -29.1215 q^{92} -1.42505 q^{93} -30.7300 q^{94} -2.42289 q^{95} -3.50926 q^{96} +11.8140 q^{97} +0.827301 q^{98} +14.5415 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 111 q + 20 q^{2} + 40 q^{3} + 136 q^{4} + 111 q^{5} + 3 q^{6} + 39 q^{7} + 45 q^{8} + 139 q^{9}+O(q^{10}) $ 111 * q + 20 * q^2 + 40 * q^3 + 136 * q^4 + 111 * q^5 + 3 * q^6 + 39 * q^7 + 45 * q^8 + 139 * q^9	\( 111 q + 20 q^{2} + 40 q^{3} + 136 q^{4} + 111 q^{5} + 3 q^{6} + 39 q^{7} + 45 q^{8} + 139 q^{9} + 20 q^{10} + 36 q^{11} + 80 q^{12} + 36 q^{13} + 7 q^{14} + 40 q^{15} + 190 q^{16} + 38 q^{17} + 48 q^{18} + 77 q^{19} + 136 q^{20} + 11 q^{21} + 39 q^{22} + 82 q^{23} - 3 q^{24} + 111 q^{25} - 3 q^{26} + 130 q^{27} + 87 q^{28} + 20 q^{29} + 3 q^{30} + 41 q^{31} + 85 q^{32} + 33 q^{33} + 7 q^{34} + 39 q^{35} + 191 q^{36} + 80 q^{37} + 42 q^{38} + 21 q^{39} + 45 q^{40} + 16 q^{41} + 33 q^{42} + 164 q^{43} + 37 q^{44} + 139 q^{45} + 32 q^{46} + 148 q^{47} + 149 q^{48} + 160 q^{49} + 20 q^{50} + 51 q^{51} + 87 q^{52} + 83 q^{53} - 6 q^{54} + 36 q^{55} - 10 q^{56} + 28 q^{57} + 47 q^{58} + 14 q^{59} + 80 q^{60} + 20 q^{61} + 14 q^{62} + 120 q^{63} + 231 q^{64} + 36 q^{65} - 4 q^{66} + 253 q^{67} + 80 q^{68} + 6 q^{69} + 7 q^{70} + 5 q^{71} + 124 q^{72} + 64 q^{73} - 37 q^{74} + 40 q^{75} + 92 q^{76} + 63 q^{77} + 29 q^{78} + 91 q^{79} + 190 q^{80} + 187 q^{81} - 7 q^{82} + 63 q^{83} - 69 q^{84} + 38 q^{85} - 22 q^{86} + 57 q^{87} + 121 q^{88} - 6 q^{89} + 48 q^{90} + 119 q^{91} + 104 q^{92} + 14 q^{93} - q^{94} + 77 q^{95} - 38 q^{96} + 96 q^{97} + 81 q^{98} + 106 q^{99}+O(q^{100}) \) 111 * q + 20 * q^2 + 40 * q^3 + 136 * q^4 + 111 * q^5 + 3 * q^6 + 39 * q^7 + 45 * q^8 + 139 * q^9 + 20 * q^10 + 36 * q^11 + 80 * q^12 + 36 * q^13 + 7 * q^14 + 40 * q^15 + 190 * q^16 + 38 * q^17 + 48 * q^18 + 77 * q^19 + 136 * q^20 + 11 * q^21 + 39 * q^22 + 82 * q^23 - 3 * q^24 + 111 * q^25 - 3 * q^26 + 130 * q^27 + 87 * q^28 + 20 * q^29 + 3 * q^30 + 41 * q^31 + 85 * q^32 + 33 * q^33 + 7 * q^34 + 39 * q^35 + 191 * q^36 + 80 * q^37 + 42 * q^38 + 21 * q^39 + 45 * q^40 + 16 * q^41 + 33 * q^42 + 164 * q^43 + 37 * q^44 + 139 * q^45 + 32 * q^46 + 148 * q^47 + 149 * q^48 + 160 * q^49 + 20 * q^50 + 51 * q^51 + 87 * q^52 + 83 * q^53 - 6 * q^54 + 36 * q^55 - 10 * q^56 + 28 * q^57 + 47 * q^58 + 14 * q^59 + 80 * q^60 + 20 * q^61 + 14 * q^62 + 120 * q^63 + 231 * q^64 + 36 * q^65 - 4 * q^66 + 253 * q^67 + 80 * q^68 + 6 * q^69 + 7 * q^70 + 5 * q^71 + 124 * q^72 + 64 * q^73 - 37 * q^74 + 40 * q^75 + 92 * q^76 + 63 * q^77 + 29 * q^78 + 91 * q^79 + 190 * q^80 + 187 * q^81 - 7 * q^82 + 63 * q^83 - 69 * q^84 + 38 * q^85 - 22 * q^86 + 57 * q^87 + 121 * q^88 - 6 * q^89 + 48 * q^90 + 119 * q^91 + 104 * q^92 + 14 * q^93 - q^94 + 77 * q^95 - 38 * q^96 + 96 * q^97 + 81 * q^98 + 106 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.71589	−1.92042	−0.960211	−	0.279277i	$-0.909905\pi$
$2$	−2.71589	−1.92042	−0.960211	+	0.279277i	$0.909905\pi$
$3$	0.174668	0.100844	0.0504222	−	0.998728i	$-0.483943\pi$
$3$	0.174668	0.100844	0.0504222	+	0.998728i	$0.483943\pi$
$4$	5.37604	2.68802
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.474377	−0.193664
$7$	2.58754	0.978000	0.489000	−	0.872284i	$-0.337362\pi$
$7$	2.58754	0.978000	0.489000	+	0.872284i	$0.337362\pi$
$8$	−9.16893	−3.24171
$9$	−2.96949	−0.989830
$10$	−2.71589	−0.858839
$11$	−4.89698	−1.47649	−0.738247	−	0.674530i	$-0.764346\pi$
$11$	−4.89698	−1.47649	−0.738247	+	0.674530i	$0.764346\pi$
$12$	0.939019	0.271072
$13$	−6.47521	−1.79590	−0.897950	−	0.440098i	$-0.854944\pi$
$13$	−6.47521	−1.79590	−0.897950	+	0.440098i	$0.854944\pi$
$14$	−7.02747	−1.87817
$15$	0.174668	0.0450990
$16$	14.1497	3.53742
$17$	−2.72774	−0.661573	−0.330787	−	0.943706i	$-0.607314\pi$
$17$	−2.72774	−0.661573	−0.330787	+	0.943706i	$0.607314\pi$
$18$	8.06480	1.90089
$19$	−2.42289	−0.555849	−0.277924	−	0.960603i	$-0.589647\pi$
$19$	−2.42289	−0.555849	−0.277924	+	0.960603i	$0.589647\pi$
$20$	5.37604	1.20212
$21$	0.451960	0.0986258
$22$	13.2996	2.83549
$23$	−5.41691	−1.12950	−0.564752	−	0.825261i	$-0.691028\pi$
$23$	−5.41691	−1.12950	−0.564752	+	0.825261i	$0.691028\pi$
$24$	−1.60151	−0.326908
$25$	1.00000	0.200000
$26$	17.5859	3.44888
$27$	−1.04268	−0.200663
$28$	13.9107	2.62888
$29$	0.00262486	0.000487424 0	0.000243712	−	1.00000i	$-0.499922\pi$
$29$	0.00262486	0.000487424 0	0.000243712		1.00000i	$0.499922\pi$
$30$	−0.474377	−0.0866090
$31$	−8.15861	−1.46533	−0.732665	−	0.680589i	$-0.761724\pi$
$31$	−8.15861	−1.46533	−0.732665	+	0.680589i	$0.761724\pi$
$32$	−20.0911	−3.55164
$33$	−0.855344	−0.148896
$34$	7.40822	1.27050
$35$	2.58754	0.437375
$36$	−15.9641	−2.66068
$37$	9.14813	1.50394	0.751971	−	0.659196i	$-0.229103\pi$
$37$	9.14813	1.50394	0.751971	+	0.659196i	$0.229103\pi$
$38$	6.58029	1.06746
$39$	−1.13101	−0.181106
$40$	−9.16893	−1.44973
$41$	−4.73959	−0.740199	−0.370099	−	0.928992i	$-0.620676\pi$
$41$	−4.73959	−0.740199	−0.370099	+	0.928992i	$0.620676\pi$
$42$	−1.22747	−0.189403
$43$	−1.41359	−0.215571	−0.107785	−	0.994174i	$-0.534376\pi$
$43$	−1.41359	−0.215571	−0.107785	+	0.994174i	$0.534376\pi$
$44$	−26.3263	−3.96884
$45$	−2.96949	−0.442666
$46$	14.7117	2.16912
$47$	11.3149	1.65045	0.825224	−	0.564805i	$-0.191049\pi$
$47$	11.3149	1.65045	0.825224	+	0.564805i	$0.191049\pi$
$48$	2.47149	0.356729
$49$	−0.304616	−0.0435165
$50$	−2.71589	−0.384084
$51$	−0.476447	−0.0667160
$52$	−34.8109	−4.82741
$53$	−12.5462	−1.72335	−0.861676	−	0.507458i	$-0.830585\pi$
$53$	−12.5462	−1.72335	−0.861676	+	0.507458i	$0.830585\pi$
$54$	2.83179	0.385358
$55$	−4.89698	−0.660309
$56$	−23.7250	−3.17039
$57$	−0.423200	−0.0560542
$58$	−0.00712881	−0.000936059 0
$59$	1.08610	0.141399	0.0706993	−	0.997498i	$-0.477477\pi$
$59$	1.08610	0.141399	0.0706993	+	0.997498i	$0.477477\pi$
$60$	0.939019	0.121227
$61$	−3.10876	−0.398036	−0.199018	−	0.979996i	$-0.563775\pi$
$61$	−3.10876	−0.398036	−0.199018	+	0.979996i	$0.563775\pi$
$62$	22.1579	2.81405
$63$	−7.68369	−0.968054
$64$	26.2657	3.28321
$65$	−6.47521	−0.803151
$66$	2.32302	0.285943
$67$	11.7848	1.43975	0.719873	−	0.694106i	$-0.244200\pi$
$67$	11.7848	1.43975	0.719873	+	0.694106i	$0.244200\pi$
$68$	−14.6644	−1.77832
$69$	−0.946159	−0.113904
$70$	−7.02747	−0.839944
$71$	5.00848	0.594397	0.297199	−	0.954816i	$-0.403948\pi$
$71$	5.00848	0.594397	0.297199	+	0.954816i	$0.403948\pi$
$72$	27.2270	3.20874
$73$	3.22703	0.377696	0.188848	−	0.982006i	$-0.439525\pi$
$73$	3.22703	0.377696	0.188848	+	0.982006i	$0.439525\pi$
$74$	−24.8453	−2.88820
$75$	0.174668	0.0201689
$76$	−13.0255	−1.49413
$77$	−12.6711	−1.44401
$78$	3.07169	0.347801
$79$	−15.2470	−1.71542	−0.857710	−	0.514134i	$-0.828113\pi$
$79$	−15.2470	−1.71542	−0.857710	+	0.514134i	$0.828113\pi$
$80$	14.1497	1.58198
$81$	8.72635	0.969595
$82$	12.8722	1.42149
$83$	−17.7693	−1.95043	−0.975216	−	0.221253i	$-0.928985\pi$
$83$	−17.7693	−1.95043	−0.975216	+	0.221253i	$0.928985\pi$
$84$	2.42975	0.265108
$85$	−2.72774	−0.295865
$86$	3.83916	0.413987
$87$	0.000458477 0	4.91539e−5 0
$88$	44.9000	4.78636
$89$	0.0257273	0.00272709	0.00136355	−	0.999999i	$-0.499566\pi$
$89$	0.0257273	0.00272709	0.00136355	+	0.999999i	$0.499566\pi$
$90$	8.06480	0.850104
$91$	−16.7549	−1.75639
$92$	−29.1215	−3.03613
$93$	−1.42505	−0.147770
$94$	−30.7300	−3.16956
$95$	−2.42289	−0.248583
$96$	−3.50926	−0.358162
$97$	11.8140	1.19953	0.599764	−	0.800177i	$-0.295261\pi$
$97$	11.8140	1.19953	0.599764	+	0.800177i	$0.295261\pi$
$98$	0.827301	0.0835700
$99$	14.5415	1.46148
$100$	5.37604	0.537604
$101$	6.28142	0.625025	0.312512	−	0.949914i	$-0.398829\pi$
$101$	6.28142	0.625025	0.312512	+	0.949914i	$0.398829\pi$
$102$	1.29398	0.128123
$103$	10.6371	1.04811	0.524055	−	0.851685i	$-0.324419\pi$
$103$	10.6371	1.04811	0.524055	+	0.851685i	$0.324419\pi$
$104$	59.3707	5.82178
$105$	0.451960	0.0441068
$106$	34.0740	3.30956
$107$	8.99179	0.869269	0.434635	−	0.900607i	$-0.356878\pi$
$107$	8.99179	0.869269	0.434635	+	0.900607i	$0.356878\pi$
$108$	−5.60547	−0.539386
$109$	5.83676	0.559060	0.279530	−	0.960137i	$-0.409821\pi$
$109$	5.83676	0.559060	0.279530	+	0.960137i	$0.409821\pi$
$110$	13.2996	1.26807
$111$	1.59788	0.151664
$112$	36.6129	3.45960
$113$	2.85508	0.268583	0.134291	−	0.990942i	$-0.457124\pi$
$113$	2.85508	0.268583	0.134291	+	0.990942i	$0.457124\pi$
$114$	1.14936	0.107648
$115$	−5.41691	−0.505130
$116$	0.0141113	0.00131020
$117$	19.2281	1.77764
$118$	−2.94973	−0.271545
$119$	−7.05814	−0.647019
$120$	−1.60151	−0.146198
$121$	12.9804	1.18004
$122$	8.44304	0.764397
$123$	−0.827852	−0.0746449
$124$	−43.8610	−3.93883
$125$	1.00000	0.0894427
$126$	20.8680	1.85907
$127$	7.47096	0.662941	0.331470	−	0.943466i	$-0.392455\pi$
$127$	7.47096	0.662941	0.331470	+	0.943466i	$0.392455\pi$
$128$	−31.1525	−2.75352
$129$	−0.246909	−0.0217391
$130$	17.5859	1.54239
$131$	14.9836	1.30913	0.654564	−	0.756007i	$-0.272852\pi$
$131$	14.9836	1.30913	0.654564	+	0.756007i	$0.272852\pi$
$132$	−4.59836	−0.400236
$133$	−6.26933	−0.543620
$134$	−32.0063	−2.76492
$135$	−1.04268	−0.0897393
$136$	25.0104	2.14463
$137$	−19.7650	−1.68864	−0.844319	−	0.535840i	$-0.819995\pi$
$137$	−19.7650	−1.68864	−0.844319	+	0.535840i	$0.819995\pi$
$138$	2.56966	0.218744
$139$	20.4697	1.73622	0.868109	−	0.496374i	$-0.165336\pi$
$139$	20.4697	1.73622	0.868109	+	0.496374i	$0.165336\pi$
$140$	13.9107	1.17567
$141$	1.97635	0.166439
$142$	−13.6025	−1.14149
$143$	31.7090	2.65164
$144$	−42.0174	−3.50145
$145$	0.00262486	0.000217982 0
$146$	−8.76425	−0.725335
$147$	−0.0532065	−0.00438840
$148$	49.1807	4.04263
$149$	−7.06798	−0.579031	−0.289516	−	0.957173i	$-0.593494\pi$
$149$	−7.06798	−0.579031	−0.289516	+	0.957173i	$0.593494\pi$
$150$	−0.474377	−0.0387327
$151$	5.03282	0.409565	0.204782	−	0.978808i	$-0.434351\pi$
$151$	5.03282	0.409565	0.204782	+	0.978808i	$0.434351\pi$
$152$	22.2153	1.80190
$153$	8.09999	0.654845
$154$	34.4134	2.77311
$155$	−8.15861	−0.655316
$156$	−6.08035	−0.486817
$157$	20.8907	1.66726	0.833631	−	0.552322i	$-0.186258\pi$
$157$	20.8907	1.66726	0.833631	+	0.552322i	$0.186258\pi$
$158$	41.4091	3.29433
$159$	−2.19141	−0.173790
$160$	−20.0911	−1.58834
$161$	−14.0165	−1.10465
$162$	−23.6998	−1.86203
$163$	9.21687	0.721921	0.360960	−	0.932581i	$-0.382449\pi$
$163$	9.21687	0.721921	0.360960	+	0.932581i	$0.382449\pi$
$164$	−25.4802	−1.98967
$165$	−0.855344	−0.0665884
$166$	48.2594	3.74565
$167$	7.08288	0.548090	0.274045	−	0.961717i	$-0.411638\pi$
$167$	7.08288	0.548090	0.274045	+	0.961717i	$0.411638\pi$
$168$	−4.14399	−0.319716
$169$	28.9283	2.22526
$170$	7.40822	0.568185
$171$	7.19475	0.550196
$172$	−7.59953	−0.579458
$173$	18.4928	1.40598	0.702991	−	0.711199i	$-0.251847\pi$
$173$	18.4928	1.40598	0.702991	+	0.711199i	$0.251847\pi$
$174$	−0.00124517	−9.43963e−5 0
$175$	2.58754	0.195600
$176$	−69.2907	−5.22299
$177$	0.189707	0.0142593
$178$	−0.0698725	−0.00523717
$179$	−14.0869	−1.05290	−0.526452	−	0.850205i	$-0.676478\pi$
$179$	−14.0869	−1.05290	−0.526452	+	0.850205i	$0.676478\pi$
$180$	−15.9641	−1.18989
$181$	−21.8257	−1.62229	−0.811147	−	0.584842i	$-0.801156\pi$
$181$	−21.8257	−1.62229	−0.811147	+	0.584842i	$0.801156\pi$
$182$	45.5044	3.37301
$183$	−0.543000	−0.0401397
$184$	49.6673	3.66152
$185$	9.14813	0.672584
$186$	3.87026	0.283781
$187$	13.3577	0.976810
$188$	60.8293	4.43644
$189$	−2.69797	−0.196249
$190$	6.58029	0.477384
$191$	−20.6935	−1.49733	−0.748666	−	0.662947i	$-0.769305\pi$
$191$	−20.6935	−1.49733	−0.748666	+	0.662947i	$0.769305\pi$
$192$	4.58777	0.331094
$193$	1.05805	0.0761598	0.0380799	−	0.999275i	$-0.487876\pi$
$193$	1.05805	0.0761598	0.0380799	+	0.999275i	$0.487876\pi$
$194$	−32.0854	−2.30360
$195$	−1.13101	−0.0809932
$196$	−1.63762	−0.116973
$197$	9.16932	0.653287	0.326643	−	0.945148i	$-0.394082\pi$
$197$	9.16932	0.653287	0.326643	+	0.945148i	$0.394082\pi$
$198$	−39.4932	−2.80666
$199$	−15.1743	−1.07568	−0.537839	−	0.843048i	$-0.680759\pi$
$199$	−15.1743	−1.07568	−0.537839	+	0.843048i	$0.680759\pi$
$200$	−9.16893	−0.648341
$201$	2.05843	0.145190
$202$	−17.0596	−1.20031
$203$	0.00679193	0.000476700 0
$204$	−2.56140	−0.179334
$205$	−4.73959	−0.331027
$206$	−28.8893	−2.01281
$207$	16.0855	1.11802
$208$	−91.6222	−6.35285
$209$	11.8648	0.820708
$210$	−1.22747	−0.0847036
$211$	−10.7680	−0.741302	−0.370651	−	0.928772i	$-0.620865\pi$
$211$	−10.7680	−0.741302	−0.370651	+	0.928772i	$0.620865\pi$
$212$	−67.4488	−4.63240
$213$	0.874819	0.0599416
$214$	−24.4207	−1.66936
$215$	−1.41359	−0.0964062
$216$	9.56023	0.650491
$217$	−21.1108	−1.43309
$218$	−15.8520	−1.07363
$219$	0.563658	0.0380885
$220$	−26.3263	−1.77492
$221$	17.6627	1.18812
$222$	−4.33966	−0.291259
$223$	26.7509	1.79137	0.895686	−	0.444688i	$-0.146685\pi$
$223$	26.7509	1.79137	0.895686	+	0.444688i	$0.146685\pi$
$224$	−51.9866	−3.47350
$225$	−2.96949	−0.197966
$226$	−7.75406	−0.515792
$227$	−16.5535	−1.09869	−0.549346	−	0.835595i	$-0.685123\pi$
$227$	−16.5535	−1.09869	−0.549346	+	0.835595i	$0.685123\pi$
$228$	−2.27514	−0.150675
$229$	14.4288	0.953483	0.476742	−	0.879044i	$-0.341818\pi$
$229$	14.4288	0.953483	0.476742	+	0.879044i	$0.341818\pi$
$230$	14.7117	0.970061
$231$	−2.21324	−0.145620
$232$	−0.0240671	−0.00158008
$233$	2.87023	0.188035	0.0940176	−	0.995571i	$-0.470029\pi$
$233$	2.87023	0.188035	0.0940176	+	0.995571i	$0.470029\pi$
$234$	−52.2213	−3.41381
$235$	11.3149	0.738103
$236$	5.83893	0.380082
$237$	−2.66315	−0.172990
$238$	19.1691	1.24255
$239$	−18.4467	−1.19322	−0.596608	−	0.802533i	$-0.703485\pi$
$239$	−18.4467	−1.19322	−0.596608	+	0.802533i	$0.703485\pi$
$240$	2.47149	0.159534
$241$	20.9417	1.34897	0.674486	−	0.738287i	$-0.264365\pi$
$241$	20.9417	1.34897	0.674486	+	0.738287i	$0.264365\pi$
$242$	−35.2533	−2.26617
$243$	4.65224	0.298441
$244$	−16.7128	−1.06993
$245$	−0.304616	−0.0194612
$246$	2.24835	0.143350
$247$	15.6887	0.998249
$248$	74.8057	4.75017
$249$	−3.10372	−0.196690
$250$	−2.71589	−0.171768
$251$	25.4351	1.60545	0.802723	−	0.596352i	$-0.203384\pi$
$251$	25.4351	1.60545	0.802723	+	0.596352i	$0.203384\pi$
$252$	−41.3078	−2.60215
$253$	26.5265	1.66771
$254$	−20.2903	−1.27313
$255$	−0.476447	−0.0298363
$256$	32.0752	2.00470
$257$	−4.00613	−0.249896	−0.124948	−	0.992163i	$-0.539876\pi$
$257$	−4.00613	−0.249896	−0.124948	+	0.992163i	$0.539876\pi$
$258$	0.670576	0.0417483
$259$	23.6712	1.47086
$260$	−34.8109	−2.15888
$261$	−0.00779449	−0.000482467 0
$262$	−40.6939	−2.51408
$263$	−11.0780	−0.683099	−0.341550	−	0.939864i	$-0.610952\pi$
$263$	−11.0780	−0.683099	−0.341550	+	0.939864i	$0.610952\pi$
$264$	7.84258	0.482678
$265$	−12.5462	−0.770707
$266$	17.0268	1.04398
$267$	0.00449373	0.000275012 0
$268$	63.3557	3.87006
$269$	−2.36181	−0.144002	−0.0720010	−	0.997405i	$-0.522938\pi$
$269$	−2.36181	−0.144002	−0.0720010	+	0.997405i	$0.522938\pi$
$270$	2.83179	0.172337
$271$	−18.2530	−1.10879	−0.554395	−	0.832254i	$-0.687050\pi$
$271$	−18.2530	−1.10879	−0.554395	+	0.832254i	$0.687050\pi$
$272$	−38.5966	−2.34026
$273$	−2.92654	−0.177122
$274$	53.6795	3.24290
$275$	−4.89698	−0.295299
$276$	−5.08658	−0.306176
$277$	−20.3634	−1.22352	−0.611760	−	0.791044i	$-0.709538\pi$
$277$	−20.3634	−1.22352	−0.611760	+	0.791044i	$0.709538\pi$
$278$	−55.5934	−3.33427
$279$	24.2269	1.45043
$280$	−23.7250	−1.41784
$281$	14.1460	0.843880	0.421940	−	0.906624i	$-0.361349\pi$
$281$	14.1460	0.843880	0.421940	+	0.906624i	$0.361349\pi$
$282$	−5.36754	−0.319632
$283$	−19.3248	−1.14874	−0.574371	−	0.818595i	$-0.694753\pi$
$283$	−19.3248	−1.14874	−0.574371	+	0.818595i	$0.694753\pi$
$284$	26.9258	1.59775
$285$	−0.423200	−0.0250682
$286$	−86.1179	−5.09226
$287$	−12.2639	−0.723914
$288$	59.6603	3.51552
$289$	−9.55945	−0.562321
$290$	−0.00712881	−0.000418618 0
$291$	2.06352	0.120966
$292$	17.3486	1.01525
$293$	0.325614	0.0190226	0.00951129	−	0.999955i	$-0.496972\pi$
$293$	0.325614	0.0190226	0.00951129	+	0.999955i	$0.496972\pi$
$294$	0.144503	0.00842757
$295$	1.08610	0.0632354
$296$	−83.8785	−4.87534
$297$	5.10597	0.296278
$298$	19.1958	1.11198
$299$	35.0756	2.02848
$300$	0.939019	0.0542143
$301$	−3.65773	−0.210828
$302$	−13.6686	−0.786537
$303$	1.09716	0.0630303
$304$	−34.2831	−1.96627
$305$	−3.10876	−0.178007
$306$	−21.9986	−1.25758
$307$	6.63838	0.378872	0.189436	−	0.981893i	$-0.439334\pi$
$307$	6.63838	0.378872	0.189436	+	0.981893i	$0.439334\pi$
$308$	−68.1206	−3.88153
$309$	1.85796	0.105696
$310$	22.1579	1.25848
$311$	7.63138	0.432736	0.216368	−	0.976312i	$-0.430579\pi$
$311$	7.63138	0.432736	0.216368	+	0.976312i	$0.430579\pi$
$312$	10.3701	0.587094
$313$	21.0993	1.19260	0.596302	−	0.802760i	$-0.296636\pi$
$313$	21.0993	1.19260	0.596302	+	0.802760i	$0.296636\pi$
$314$	−56.7369	−3.20185
$315$	−7.68369	−0.432927
$316$	−81.9683	−4.61108
$317$	24.2542	1.36225	0.681125	−	0.732167i	$-0.261491\pi$
$317$	24.2542	1.36225	0.681125	+	0.732167i	$0.261491\pi$
$318$	5.95163	0.333751
$319$	−0.0128539	−0.000719678 0
$320$	26.2657	1.46830
$321$	1.57057	0.0876609
$322$	38.0672	2.12140
$323$	6.60900	0.367735
$324$	46.9132	2.60629
$325$	−6.47521	−0.359180
$326$	−25.0320	−1.38639
$327$	1.01949	0.0563781
$328$	43.4569	2.39951
$329$	29.2778	1.61414
$330$	2.32302	0.127878
$331$	−20.3300	−1.11744	−0.558718	−	0.829358i	$-0.688706\pi$
$331$	−20.3300	−1.11744	−0.558718	+	0.829358i	$0.688706\pi$
$332$	−95.5283	−5.24280
$333$	−27.1653	−1.48865
$334$	−19.2363	−1.05256
$335$	11.7848	0.643874
$336$	6.39509	0.348881
$337$	−20.1816	−1.09936	−0.549681	−	0.835375i	$-0.685251\pi$
$337$	−20.1816	−1.09936	−0.549681	+	0.835375i	$0.685251\pi$
$338$	−78.5660	−4.27343
$339$	0.498689	0.0270851
$340$	−14.6644	−0.795289
$341$	39.9526	2.16355
$342$	−19.5401	−1.05661
$343$	−18.9010	−1.02056
$344$	12.9611	0.698817
$345$	−0.946159	−0.0509395
$346$	−50.2244	−2.70008
$347$	17.2750	0.927369	0.463684	−	0.886000i	$-0.346527\pi$
$347$	17.2750	0.927369	0.463684	+	0.886000i	$0.346527\pi$
$348$	0.00246479	0.000132127 0
$349$	−29.0310	−1.55399	−0.776997	−	0.629504i	$-0.783258\pi$
$349$	−29.0310	−1.55399	−0.776997	+	0.629504i	$0.783258\pi$
$350$	−7.02747	−0.375634
$351$	6.75155	0.360371
$352$	98.3856	5.24397
$353$	−7.27365	−0.387137	−0.193569	−	0.981087i	$-0.562006\pi$
$353$	−7.27365	−0.387137	−0.193569	+	0.981087i	$0.562006\pi$
$354$	−0.515223	−0.0273838
$355$	5.00848	0.265823
$356$	0.138311	0.00733047
$357$	−1.23283	−0.0652482
$358$	38.2584	2.02202
$359$	15.5123	0.818710	0.409355	−	0.912375i	$-0.365754\pi$
$359$	15.5123	0.818710	0.409355	+	0.912375i	$0.365754\pi$
$360$	27.2270	1.43499
$361$	−13.1296	−0.691032
$362$	59.2762	3.11549
$363$	2.26726	0.119000
$364$	−90.0749	−4.72121
$365$	3.22703	0.168911
$366$	1.47472	0.0770851
$367$	25.4483	1.32839	0.664195	−	0.747560i	$-0.268775\pi$
$367$	25.4483	1.32839	0.664195	+	0.747560i	$0.268775\pi$
$368$	−76.6476	−3.99553
$369$	14.0742	0.732671
$370$	−24.8453	−1.29164
$371$	−32.4638	−1.68544
$372$	−7.66109	−0.397209
$373$	−8.21754	−0.425488	−0.212744	−	0.977108i	$-0.568240\pi$
$373$	−8.21754	−0.425488	−0.212744	+	0.977108i	$0.568240\pi$
$374$	−36.2779	−1.87589
$375$	0.174668	0.00901980
$376$	−103.746	−5.35027
$377$	−0.0169965	−0.000875364 0
$378$	7.32738	0.376880
$379$	8.99337	0.461958	0.230979	−	0.972959i	$-0.425807\pi$
$379$	8.99337	0.461958	0.230979	+	0.972959i	$0.425807\pi$
$380$	−13.0255	−0.668196
$381$	1.30493	0.0668538
$382$	56.2013	2.87551
$383$	−2.51265	−0.128390	−0.0641951	−	0.997937i	$-0.520448\pi$
$383$	−2.51265	−0.128390	−0.0641951	+	0.997937i	$0.520448\pi$
$384$	−5.44133	−0.277677
$385$	−12.6711	−0.645782
$386$	−2.87353	−0.146259
$387$	4.19765	0.213379
$388$	63.5124	3.22435
$389$	2.07809	0.105364	0.0526818	−	0.998611i	$-0.483223\pi$
$389$	2.07809	0.105364	0.0526818	+	0.998611i	$0.483223\pi$
$390$	3.07169	0.155541
$391$	14.7759	0.747250
$392$	2.79300	0.141068
$393$	2.61716	0.132018
$394$	−24.9028	−1.25459
$395$	−15.2470	−0.767159
$396$	78.1758	3.92848
$397$	2.69996	0.135507	0.0677535	−	0.997702i	$-0.478417\pi$
$397$	2.69996	0.135507	0.0677535	+	0.997702i	$0.478417\pi$
$398$	41.2117	2.06575
$399$	−1.09505	−0.0548210
$400$	14.1497	0.707484
$401$	−9.36531	−0.467681	−0.233841	−	0.972275i	$-0.575129\pi$
$401$	−9.36531	−0.467681	−0.233841	+	0.972275i	$0.575129\pi$
$402$	−5.59046	−0.278827
$403$	52.8287	2.63159
$404$	33.7692	1.68008
$405$	8.72635	0.433616
$406$	−0.0184461	−0.000915465 0
$407$	−44.7982	−2.22056
$408$	4.36851	0.216273
$409$	−6.68063	−0.330336	−0.165168	−	0.986265i	$-0.552817\pi$
$409$	−6.68063	−0.330336	−0.165168	+	0.986265i	$0.552817\pi$
$410$	12.8722	0.635711
$411$	−3.45231	−0.170290
$412$	57.1857	2.81734
$413$	2.81034	0.138288
$414$	−43.6863	−2.14706
$415$	−17.7693	−0.872260
$416$	130.094	6.37838
$417$	3.57539	0.175088
$418$	−32.2235	−1.57611
$419$	−8.09617	−0.395524	−0.197762	−	0.980250i	$-0.563367\pi$
$419$	−8.09617	−0.395524	−0.197762	+	0.980250i	$0.563367\pi$
$420$	2.42975	0.118560
$421$	−12.9728	−0.632258	−0.316129	−	0.948716i	$-0.602383\pi$
$421$	−12.9728	−0.632258	−0.316129	+	0.948716i	$0.602383\pi$
$422$	29.2448	1.42361
$423$	−33.5995	−1.63366
$424$	115.035	5.58660
$425$	−2.72774	−0.132315
$426$	−2.37591	−0.115113
$427$	−8.04405	−0.389279
$428$	48.3402	2.33661
$429$	5.53853	0.267403
$430$	3.83916	0.185141
$431$	−6.43078	−0.309760	−0.154880	−	0.987933i	$-0.549499\pi$
$431$	−6.43078	−0.309760	−0.154880	+	0.987933i	$0.549499\pi$
$432$	−14.7536	−0.709831
$433$	−6.32634	−0.304025	−0.152012	−	0.988379i	$-0.548575\pi$
$433$	−6.32634	−0.304025	−0.152012	+	0.988379i	$0.548575\pi$
$434$	57.3344	2.75214
$435$	0.000458477 0	2.19823e−5 0
$436$	31.3786	1.50276
$437$	13.1246	0.627834
$438$	−1.53083	−0.0731459
$439$	−17.4875	−0.834634	−0.417317	−	0.908761i	$-0.637030\pi$
$439$	−17.4875	−0.834634	−0.417317	+	0.908761i	$0.637030\pi$
$440$	44.9000	2.14053
$441$	0.904553	0.0430740
$442$	−47.9698	−2.28169
$443$	−15.7926	−0.750329	−0.375164	−	0.926958i	$-0.622414\pi$
$443$	−15.7926	−0.750329	−0.375164	+	0.926958i	$0.622414\pi$
$444$	8.59027	0.407676
$445$	0.0257273	0.00121959
$446$	−72.6523	−3.44019
$447$	−1.23455	−0.0583921
$448$	67.9637	3.21098
$449$	36.4201	1.71877	0.859385	−	0.511330i	$-0.170847\pi$
$449$	36.4201	1.71877	0.859385	+	0.511330i	$0.170847\pi$
$450$	8.06480	0.380178
$451$	23.2097	1.09290
$452$	15.3490	0.721956
$453$	0.879070	0.0413023
$454$	44.9573	2.10995
$455$	−16.7549	−0.785481
$456$	3.88029	0.181711
$457$	−13.0075	−0.608467	−0.304233	−	0.952598i	$-0.598400\pi$
$457$	−13.0075	−0.608467	−0.304233	+	0.952598i	$0.598400\pi$
$458$	−39.1870	−1.83109
$459$	2.84415	0.132753
$460$	−29.1215	−1.35780
$461$	−19.2126	−0.894818	−0.447409	−	0.894329i	$-0.647653\pi$
$461$	−19.2126	−0.894818	−0.447409	+	0.894329i	$0.647653\pi$
$462$	6.01091	0.279653
$463$	28.6460	1.33129	0.665646	−	0.746268i	$-0.268156\pi$
$463$	28.6460	1.33129	0.665646	+	0.746268i	$0.268156\pi$
$464$	0.0371409	0.00172422
$465$	−1.42505	−0.0660849
$466$	−7.79523	−0.361107
$467$	−14.7511	−0.682598	−0.341299	−	0.939955i	$-0.610867\pi$
$467$	−14.7511	−0.682598	−0.341299	+	0.939955i	$0.610867\pi$
$468$	103.371	4.77832
$469$	30.4938	1.40807
$470$	−30.7300	−1.41747
$471$	3.64893	0.168134
$472$	−9.95841	−0.458373
$473$	6.92233	0.318289
$474$	7.23282	0.332214
$475$	−2.42289	−0.111170
$476$	−37.9448	−1.73920
$477$	37.2558	1.70583
$478$	50.0991	2.29148
$479$	−16.7800	−0.766697	−0.383349	−	0.923604i	$-0.625229\pi$
$479$	−16.7800	−0.766697	−0.383349	+	0.923604i	$0.625229\pi$
$480$	−3.50926	−0.160175
$481$	−59.2360	−2.70093
$482$	−56.8752	−2.59060
$483$	−2.44823	−0.111398
$484$	69.7831	3.17196
$485$	11.8140	0.536445
$486$	−12.6350	−0.573133
$487$	17.3481	0.786119	0.393060	−	0.919513i	$-0.371417\pi$
$487$	17.3481	0.786119	0.393060	+	0.919513i	$0.371417\pi$
$488$	28.5040	1.29032
$489$	1.60989	0.0728017
$490$	0.827301	0.0373737
$491$	−27.9889	−1.26312	−0.631561	−	0.775326i	$-0.717585\pi$
$491$	−27.9889	−1.26312	−0.631561	+	0.775326i	$0.717585\pi$
$492$	−4.45056	−0.200647
$493$	−0.00715992	−0.000322466 0
$494$	−42.6088	−1.91706
$495$	14.5415	0.653593
$496$	−115.442	−5.18349
$497$	12.9597	0.581320
$498$	8.42935	0.377728
$499$	−21.6292	−0.968257	−0.484128	−	0.874997i	$-0.660863\pi$
$499$	−21.6292	−0.968257	−0.484128	+	0.874997i	$0.660863\pi$
$500$	5.37604	0.240424
$501$	1.23715	0.0552718
$502$	−69.0787	−3.08313
$503$	−31.1649	−1.38957	−0.694787	−	0.719215i	$-0.744502\pi$
$503$	−31.1649	−1.38957	−0.694787	+	0.719215i	$0.744502\pi$
$504$	70.4512	3.13815
$505$	6.28142	0.279520
$506$	−72.0429	−3.20270
$507$	5.05284	0.224405
$508$	40.1641	1.78200
$509$	−25.5017	−1.13034	−0.565172	−	0.824973i	$-0.691190\pi$
$509$	−25.5017	−1.13034	−0.565172	+	0.824973i	$0.691190\pi$
$510$	1.29398	0.0572982
$511$	8.35009	0.369386
$512$	−24.8076	−1.09635
$513$	2.52629	0.111538
$514$	10.8802	0.479905
$515$	10.6371	0.468729
$516$	−1.32739	−0.0584351
$517$	−55.4089	−2.43688
$518$	−64.2882	−2.82466
$519$	3.23009	0.141785
$520$	59.3707	2.60358
$521$	35.1980	1.54205	0.771026	−	0.636804i	$-0.219744\pi$
$521$	35.1980	1.54205	0.771026	+	0.636804i	$0.219744\pi$
$522$	0.0211689	0.000926539 0
$523$	15.3784	0.672451	0.336225	−	0.941782i	$-0.390850\pi$
$523$	15.3784	0.672451	0.336225	+	0.941782i	$0.390850\pi$
$524$	80.5526	3.51896
$525$	0.451960	0.0197252
$526$	30.0866	1.31184
$527$	22.2545	0.969423
$528$	−12.1028	−0.526709
$529$	6.34292	0.275779
$530$	34.0740	1.48008
$531$	−3.22518	−0.139961
$532$	−33.7042	−1.46126
$533$	30.6898	1.32932
$534$	−0.0122045	−0.000528139 0
$535$	8.99179	0.388749
$536$	−108.054	−4.66723
$537$	−2.46052	−0.106179
$538$	6.41441	0.276545
$539$	1.49170	0.0642519
$540$	−5.60547	−0.241221
$541$	3.77569	0.162330	0.0811648	−	0.996701i	$-0.474136\pi$
$541$	3.77569	0.162330	0.0811648	+	0.996701i	$0.474136\pi$
$542$	49.5730	2.12934
$543$	−3.81225	−0.163599
$544$	54.8032	2.34967
$545$	5.83676	0.250019
$546$	7.94814	0.340149
$547$	29.3461	1.25475	0.627374	−	0.778718i	$-0.284130\pi$
$547$	29.3461	1.25475	0.627374	+	0.778718i	$0.284130\pi$
$548$	−106.257	−4.53909
$549$	9.23143	0.393988
$550$	13.2996	0.567098
$551$	−0.00635974	−0.000270934 0
$552$	8.67526	0.369244
$553$	−39.4522	−1.67768
$554$	55.3047	2.34967
$555$	1.59788	0.0678263
$556$	110.046	4.66698
$557$	−7.81349	−0.331068	−0.165534	−	0.986204i	$-0.552935\pi$
$557$	−7.81349	−0.331068	−0.165534	+	0.986204i	$0.552935\pi$
$558$	−65.7976	−2.78543
$559$	9.15331	0.387144
$560$	36.6129	1.54718
$561$	2.33315	0.0985058
$562$	−38.4189	−1.62061
$563$	32.5142	1.37031	0.685156	−	0.728397i	$-0.259734\pi$
$563$	32.5142	1.37031	0.685156	+	0.728397i	$0.259734\pi$
$564$	10.6249	0.447390
$565$	2.85508	0.120114
$566$	52.4840	2.20607
$567$	22.5798	0.948263
$568$	−45.9224	−1.92686
$569$	−7.82177	−0.327906	−0.163953	−	0.986468i	$-0.552424\pi$
$569$	−7.82177	−0.327906	−0.163953	+	0.986468i	$0.552424\pi$
$570$	1.14936	0.0481415
$571$	24.3260	1.01801	0.509006	−	0.860763i	$-0.330013\pi$
$571$	24.3260	1.01801	0.509006	+	0.860763i	$0.330013\pi$
$572$	170.468	7.12765
$573$	−3.61449	−0.150998
$574$	33.3073	1.39022
$575$	−5.41691	−0.225901
$576$	−77.9958	−3.24982
$577$	20.2548	0.843216	0.421608	−	0.906778i	$-0.361466\pi$
$577$	20.2548	0.843216	0.421608	+	0.906778i	$0.361466\pi$
$578$	25.9624	1.07989
$579$	0.184806	0.00768028
$580$	0.0141113	0.000585941 0
$581$	−45.9788	−1.90752
$582$	−5.60428	−0.232305
$583$	61.4385	2.54452
$584$	−29.5884	−1.22438
$585$	19.2281	0.794983
$586$	−0.884331	−0.0365314
$587$	−2.24610	−0.0927067	−0.0463533	−	0.998925i	$-0.514760\pi$
$587$	−2.24610	−0.0927067	−0.0463533	+	0.998925i	$0.514760\pi$
$588$	−0.286040	−0.0117961
$589$	19.7674	0.814502
$590$	−2.94973	−0.121439
$591$	1.60158	0.0658803
$592$	129.443	5.32008
$593$	−18.3956	−0.755418	−0.377709	−	0.925924i	$-0.623288\pi$
$593$	−18.3956	−0.755418	−0.377709	+	0.925924i	$0.623288\pi$
$594$	−13.8672	−0.568979
$595$	−7.05814	−0.289356
$596$	−37.9977	−1.55645
$597$	−2.65046	−0.108476
$598$	−95.2614	−3.89553
$599$	40.3387	1.64819	0.824097	−	0.566449i	$-0.191683\pi$
$599$	40.3387	1.64819	0.824097	+	0.566449i	$0.191683\pi$
$600$	−1.60151	−0.0653816
$601$	23.5105	0.959015	0.479508	−	0.877538i	$-0.340815\pi$
$601$	23.5105	0.959015	0.479508	+	0.877538i	$0.340815\pi$
$602$	9.93399	0.404879
$603$	−34.9950	−1.42510
$604$	27.0566	1.10092
$605$	12.9804	0.527729
$606$	−2.97976	−0.121045
$607$	−23.8414	−0.967693	−0.483847	−	0.875153i	$-0.660761\pi$
$607$	−23.8414	−0.967693	−0.483847	+	0.875153i	$0.660761\pi$
$608$	48.6785	1.97417
$609$	0.00118633	4.80725e−5 0
$610$	8.44304	0.341849
$611$	−73.2664	−2.96404
$612$	43.5458	1.76024
$613$	−0.482585	−0.0194914	−0.00974572	−	0.999953i	$-0.503102\pi$
$613$	−0.482585	−0.0194914	−0.00974572	+	0.999953i	$0.503102\pi$
$614$	−18.0291	−0.727594
$615$	−0.827852	−0.0333822
$616$	116.181	4.68106
$617$	−21.7068	−0.873884	−0.436942	−	0.899490i	$-0.643938\pi$
$617$	−21.7068	−0.873884	−0.436942	+	0.899490i	$0.643938\pi$
$618$	−5.04602	−0.202981
$619$	−18.6560	−0.749849	−0.374925	−	0.927055i	$-0.622331\pi$
$619$	−18.6560	−0.749849	−0.374925	+	0.927055i	$0.622331\pi$
$620$	−43.8610	−1.76150
$621$	5.64809	0.226650
$622$	−20.7260	−0.831035
$623$	0.0665706	0.00266710
$624$	−16.0034	−0.640650
$625$	1.00000	0.0400000
$626$	−57.3033	−2.29030
$627$	2.07240	0.0827638
$628$	112.309	4.48163
$629$	−24.9537	−0.994969
$630$	20.8680	0.831402
$631$	−4.50327	−0.179272	−0.0896361	−	0.995975i	$-0.528570\pi$
$631$	−4.50327	−0.179272	−0.0896361	+	0.995975i	$0.528570\pi$
$632$	139.798	5.56088
$633$	−1.88083	−0.0747561
$634$	−65.8715	−2.61609
$635$	7.47096	0.296476
$636$	−11.7811	−0.467152
$637$	1.97245	0.0781513
$638$	0.0349096	0.00138209
$639$	−14.8726	−0.588353
$640$	−31.1525	−1.23141
$641$	9.27400	0.366301	0.183150	−	0.983085i	$-0.441370\pi$
$641$	9.27400	0.366301	0.183150	+	0.983085i	$0.441370\pi$
$642$	−4.26550	−0.168346
$643$	−31.9911	−1.26161	−0.630803	−	0.775943i	$-0.717274\pi$
$643$	−31.9911	−1.26161	−0.630803	+	0.775943i	$0.717274\pi$
$644$	−75.3532	−2.96933
$645$	−0.246909	−0.00972203
$646$	−17.9493	−0.706206
$647$	−15.6125	−0.613791	−0.306896	−	0.951743i	$-0.599290\pi$
$647$	−15.6125	−0.613791	−0.306896	+	0.951743i	$0.599290\pi$
$648$	−80.0113	−3.14314
$649$	−5.31863	−0.208774
$650$	17.5859	0.689777
$651$	−3.68737	−0.144519
$652$	49.5502	1.94054
$653$	8.09704	0.316862	0.158431	−	0.987370i	$-0.449356\pi$
$653$	8.09704	0.316862	0.158431	+	0.987370i	$0.449356\pi$
$654$	−2.76883	−0.108270
$655$	14.9836	0.585460
$656$	−67.0637	−2.61840
$657$	−9.58264	−0.373855
$658$	−79.5152	−3.09983
$659$	13.7030	0.533794	0.266897	−	0.963725i	$-0.414002\pi$
$659$	13.7030	0.533794	0.266897	+	0.963725i	$0.414002\pi$
$660$	−4.59836	−0.178991
$661$	−37.7852	−1.46967	−0.734837	−	0.678244i	$-0.762741\pi$
$661$	−37.7852	−1.46967	−0.734837	+	0.678244i	$0.762741\pi$
$662$	55.2138	2.14595
$663$	3.08510	0.119815
$664$	162.925	6.32273
$665$	−6.26933	−0.243114
$666$	73.7778	2.85883
$667$	−0.0142186	−0.000550547 0
$668$	38.0778	1.47328
$669$	4.67251	0.180650
$670$	−32.0063	−1.23651
$671$	15.2235	0.587698
$672$	−9.08037	−0.350283
$673$	7.65209	0.294966	0.147483	−	0.989065i	$-0.452883\pi$
$673$	7.65209	0.294966	0.147483	+	0.989065i	$0.452883\pi$
$674$	54.8109	2.11124
$675$	−1.04268	−0.0401326
$676$	155.520	5.98153
$677$	−10.2035	−0.392154	−0.196077	−	0.980589i	$-0.562820\pi$
$677$	−10.2035	−0.392154	−0.196077	+	0.980589i	$0.562820\pi$
$678$	−1.35438	−0.0520148
$679$	30.5692	1.17314
$680$	25.0104	0.959106
$681$	−2.89135	−0.110797
$682$	−108.507	−4.15493
$683$	−16.7070	−0.639276	−0.319638	−	0.947540i	$-0.603561\pi$
$683$	−16.7070	−0.639276	−0.319638	+	0.947540i	$0.603561\pi$
$684$	38.6792	1.47894
$685$	−19.7650	−0.755182
$686$	51.3330	1.95990
$687$	2.52025	0.0961534
$688$	−20.0019	−0.762565
$689$	81.2392	3.09497
$690$	2.56966	0.0978253
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	99.4180	3.77931
$693$	37.6269	1.42933
$694$	−46.9168	−1.78094
$695$	20.4697	0.776460
$696$	−0.00420375	−0.000159343 0
$697$	12.9283	0.489696
$698$	78.8449	2.98432
$699$	0.501337	0.0189623
$700$	13.9107	0.525776
$701$	3.46496	0.130870	0.0654350	−	0.997857i	$-0.479157\pi$
$701$	3.46496	0.130870	0.0654350	+	0.997857i	$0.479157\pi$
$702$	−18.3364	−0.692064
$703$	−22.1649	−0.835965
$704$	−128.623	−4.84765
$705$	1.97635	0.0744336
$706$	19.7544	0.743466
$707$	16.2535	0.611274
$708$	1.01987	0.0383292
$709$	−36.9938	−1.38933	−0.694666	−	0.719332i	$-0.744448\pi$
$709$	−36.9938	−1.38933	−0.694666	+	0.719332i	$0.744448\pi$
$710$	−13.6025	−0.510491
$711$	45.2758	1.69797
$712$	−0.235892	−0.00884043
$713$	44.1945	1.65510
$714$	3.34822	0.125304
$715$	31.7090	1.18585
$716$	−75.7316	−2.83022
$717$	−3.22204	−0.120329
$718$	−42.1298	−1.57227
$719$	−52.0350	−1.94058	−0.970288	−	0.241951i	$-0.922213\pi$
$719$	−52.0350	−1.94058	−0.970288	+	0.241951i	$0.922213\pi$
$720$	−42.0174	−1.56590
$721$	27.5241	1.02505
$722$	35.6585	1.32707
$723$	3.65783	0.136036
$724$	−117.336	−4.36076
$725$	0.00262486	9.74847e−5 0
$726$	−6.15761	−0.228530
$727$	−20.1475	−0.747230	−0.373615	−	0.927584i	$-0.621882\pi$
$727$	−20.1475	−0.747230	−0.373615	+	0.927584i	$0.621882\pi$
$728$	153.624	5.69370
$729$	−25.3665	−0.939499
$730$	−8.76425	−0.324379
$731$	3.85591	0.142616
$732$	−2.91919	−0.107896
$733$	22.0107	0.812984	0.406492	−	0.913654i	$-0.366752\pi$
$733$	22.0107	0.812984	0.406492	+	0.913654i	$0.366752\pi$
$734$	−69.1146	−2.55107
$735$	−0.0532065	−0.00196255
$736$	108.832	4.01159
$737$	−57.7101	−2.12578
$738$	−38.2238	−1.40704
$739$	−0.153455	−0.00564492	−0.00282246	−	0.999996i	$-0.500898\pi$
$739$	−0.153455	−0.00564492	−0.00282246	+	0.999996i	$0.500898\pi$
$740$	49.1807	1.80792
$741$	2.74031	0.100668
$742$	88.1681	3.23675
$743$	36.1059	1.32460	0.662298	−	0.749240i	$-0.269581\pi$
$743$	36.1059	1.32460	0.662298	+	0.749240i	$0.269581\pi$
$744$	13.0661	0.479028
$745$	−7.06798	−0.258951
$746$	22.3179	0.817116
$747$	52.7657	1.93060
$748$	71.8113	2.62568
$749$	23.2667	0.850145
$750$	−0.474377	−0.0173218
$751$	5.73747	0.209363	0.104682	−	0.994506i	$-0.466618\pi$
$751$	5.73747	0.209363	0.104682	+	0.994506i	$0.466618\pi$
$752$	160.102	5.83833
$753$	4.44268	0.161900
$754$	0.0461605	0.00168107
$755$	5.03282	0.183163
$756$	−14.5044	−0.527520
$757$	35.2394	1.28080	0.640399	−	0.768042i	$-0.278769\pi$
$757$	35.2394	1.28080	0.640399	+	0.768042i	$0.278769\pi$
$758$	−24.4250	−0.887154
$759$	4.63332	0.168179
$760$	22.2153	0.805834
$761$	−0.112812	−0.00408943	−0.00204472	−	0.999998i	$-0.500651\pi$
$761$	−0.112812	−0.00408943	−0.00204472	+	0.999998i	$0.500651\pi$
$762$	−3.54405	−0.128388
$763$	15.1029	0.546761
$764$	−111.249	−4.02486
$765$	8.09999	0.292856
$766$	6.82406	0.246563
$767$	−7.03275	−0.253938
$768$	5.60250	0.202163
$769$	−7.99710	−0.288383	−0.144191	−	0.989550i	$-0.546058\pi$
$769$	−7.99710	−0.288383	−0.144191	+	0.989550i	$0.546058\pi$
$770$	34.4134	1.24017
$771$	−0.699742	−0.0252006
$772$	5.68809	0.204719
$773$	11.5222	0.414423	0.207211	−	0.978296i	$-0.433561\pi$
$773$	11.5222	0.414423	0.207211	+	0.978296i	$0.433561\pi$
$774$	−11.4003	−0.409777
$775$	−8.15861	−0.293066
$776$	−108.322	−3.88852
$777$	4.13459	0.148328
$778$	−5.64387	−0.202342
$779$	11.4835	0.411439
$780$	−6.08035	−0.217711
$781$	−24.5264	−0.877625
$782$	−40.1297	−1.43503
$783$	−0.00273688	−9.78080e−5 0
$784$	−4.31022	−0.153936
$785$	20.8907	0.745622
$786$	−7.10790	−0.253530
$787$	−30.2569	−1.07854	−0.539271	−	0.842132i	$-0.681300\pi$
$787$	−30.2569	−1.07854	−0.539271	+	0.842132i	$0.681300\pi$
$788$	49.2946	1.75605
$789$	−1.93497	−0.0688867
$790$	41.4091	1.47327
$791$	7.38763	0.262674
$792$	−133.330	−4.73769
$793$	20.1299	0.714832
$794$	−7.33277	−0.260230
$795$	−2.19141	−0.0777214
$796$	−81.5776	−2.89144
$797$	−11.8443	−0.419547	−0.209773	−	0.977750i	$-0.567273\pi$
$797$	−11.8443	−0.419547	−0.209773	+	0.977750i	$0.567273\pi$
$798$	2.97403	0.105279
$799$	−30.8641	−1.09189
$800$	−20.0911	−0.710327
$801$	−0.0763971	−0.00269936
$802$	25.4351	0.898145
$803$	−15.8027	−0.557665
$804$	11.0662	0.390274
$805$	−14.0165	−0.494017
$806$	−143.477	−5.05375
$807$	−0.412532	−0.0145218
$808$	−57.5939	−2.02615
$809$	19.6495	0.690839	0.345420	−	0.938448i	$-0.387737\pi$
$809$	19.6495	0.690839	0.345420	+	0.938448i	$0.387737\pi$
$810$	−23.6998	−0.832725
$811$	−0.283304	−0.00994814	−0.00497407	−	0.999988i	$-0.501583\pi$
$811$	−0.283304	−0.00994814	−0.00497407	+	0.999988i	$0.501583\pi$
$812$	0.0365137	0.00128138
$813$	−3.18820	−0.111815
$814$	121.667	4.26442
$815$	9.21687	0.322853
$816$	−6.74158	−0.236003
$817$	3.42498	0.119825
$818$	18.1438	0.634384
$819$	49.7535	1.73853
$820$	−25.4802	−0.889806
$821$	12.2821	0.428649	0.214325	−	0.976762i	$-0.431245\pi$
$821$	12.2821	0.428649	0.214325	+	0.976762i	$0.431245\pi$
$822$	9.37607	0.327028
$823$	−39.0050	−1.35963	−0.679815	−	0.733384i	$-0.737940\pi$
$823$	−39.0050	−1.35963	−0.679815	+	0.733384i	$0.737940\pi$
$824$	−97.5312	−3.39766
$825$	−0.855344	−0.0297792
$826$	−7.63257	−0.265571
$827$	4.38474	0.152472	0.0762361	−	0.997090i	$-0.475710\pi$
$827$	4.38474	0.152472	0.0762361	+	0.997090i	$0.475710\pi$
$828$	86.4761	3.00525
$829$	21.9278	0.761584	0.380792	−	0.924661i	$-0.375651\pi$
$829$	21.9278	0.761584	0.380792	+	0.924661i	$0.375651\pi$
$830$	48.2594	1.67511
$831$	−3.55683	−0.123385
$832$	−170.076	−5.89632
$833$	0.830911	0.0287894
$834$	−9.71036	−0.336242
$835$	7.08288	0.245113
$836$	63.7858	2.20608
$837$	8.50680	0.294038
$838$	21.9883	0.759572
$839$	−35.9919	−1.24258	−0.621290	−	0.783581i	$-0.713391\pi$
$839$	−35.9919	−1.24258	−0.621290	+	0.783581i	$0.713391\pi$
$840$	−4.14399	−0.142981
$841$	−29.0000	−1.00000
$842$	35.2328	1.21420
$843$	2.47085	0.0851006
$844$	−57.8893	−1.99263
$845$	28.9283	0.995164
$846$	91.2525	3.13732
$847$	33.5874	1.15408
$848$	−177.525	−6.09623
$849$	−3.37542	−0.115844
$850$	7.40822	0.254100
$851$	−49.5546	−1.69871
$852$	4.70306	0.161124
$853$	35.8042	1.22591	0.612957	−	0.790116i	$-0.289980\pi$
$853$	35.8042	1.22591	0.612957	+	0.790116i	$0.289980\pi$
$854$	21.8467	0.747580
$855$	7.19475	0.246055
$856$	−82.4451	−2.81791
$857$	−34.0725	−1.16389	−0.581946	−	0.813227i	$-0.697709\pi$
$857$	−34.0725	−1.16389	−0.581946	+	0.813227i	$0.697709\pi$
$858$	−15.0420	−0.513526
$859$	44.7457	1.52670	0.763351	−	0.645983i	$-0.223553\pi$
$859$	44.7457	1.52670	0.763351	+	0.645983i	$0.223553\pi$
$860$	−7.59953	−0.259142
$861$	−2.14210	−0.0730027
$862$	17.4653	0.594870
$863$	13.4861	0.459072	0.229536	−	0.973300i	$-0.426279\pi$
$863$	13.4861	0.459072	0.229536	+	0.973300i	$0.426279\pi$
$864$	20.9485	0.712683
$865$	18.4928	0.628774
$866$	17.1816	0.583855
$867$	−1.66973	−0.0567069
$868$	−113.492	−3.85218
$869$	74.6641	2.53281
$870$	−0.00124517	−4.22153e−5 0
$871$	−76.3092	−2.58564
$872$	−53.5168	−1.81231
$873$	−35.0815	−1.18733
$874$	−35.6448	−1.20570
$875$	2.58754	0.0874750
$876$	3.03025	0.102383
$877$	36.7954	1.24249	0.621246	−	0.783616i	$-0.286627\pi$
$877$	36.7954	1.24249	0.621246	+	0.783616i	$0.286627\pi$
$878$	47.4941	1.60285
$879$	0.0568742	0.00191832
$880$	−69.2907	−2.33579
$881$	28.8631	0.972423	0.486211	−	0.873841i	$-0.338378\pi$
$881$	28.8631	0.972423	0.486211	+	0.873841i	$0.338378\pi$
$882$	−2.45666	−0.0827202
$883$	−54.5127	−1.83450	−0.917249	−	0.398313i	$-0.869596\pi$
$883$	−54.5127	−1.83450	−0.917249	+	0.398313i	$0.869596\pi$
$884$	94.9551	3.19369
$885$	0.189707	0.00637694
$886$	42.8909	1.44095
$887$	43.4254	1.45808	0.729041	−	0.684470i	$-0.239966\pi$
$887$	43.4254	1.45808	0.729041	+	0.684470i	$0.239966\pi$
$888$	−14.6509	−0.491651
$889$	19.3314	0.648356
$890$	−0.0698725	−0.00234213
$891$	−42.7328	−1.43160
$892$	143.814	4.81524
$893$	−27.4148	−0.917400
$894$	3.35289	0.112137
$895$	−14.0869	−0.470873
$896$	−80.6085	−2.69294
$897$	6.12657	0.204560
$898$	−98.9127	−3.30076
$899$	−0.0214152	−0.000714236 0
$900$	−15.9641	−0.532136
$901$	34.2227	1.14012
$902$	−63.0348	−2.09883
$903$	−0.638888	−0.0212608
$904$	−26.1780	−0.870667
$905$	−21.8257	−0.725512
$906$	−2.38745	−0.0793178
$907$	−0.0699457	−0.00232251	−0.00116125	−	0.999999i	$-0.500370\pi$
$907$	−0.0699457	−0.00232251	−0.00116125	+	0.999999i	$0.500370\pi$
$908$	−88.9920	−2.95330
$909$	−18.6526	−0.618669
$910$	45.5044	1.50845
$911$	39.5296	1.30967	0.654837	−	0.755770i	$-0.272737\pi$
$911$	39.5296	1.30967	0.654837	+	0.755770i	$0.272737\pi$
$912$	−5.98815	−0.198288
$913$	87.0158	2.87980
$914$	35.3270	1.16851
$915$	−0.543000	−0.0179510
$916$	77.5698	2.56298
$917$	38.7708	1.28033
$918$	−7.72438	−0.254943
$919$	−20.9183	−0.690032	−0.345016	−	0.938597i	$-0.612127\pi$
$919$	−20.9183	−0.690032	−0.345016	+	0.938597i	$0.612127\pi$
$920$	49.6673	1.63748
$921$	1.15951	0.0382071
$922$	52.1791	1.71843
$923$	−32.4310	−1.06748
$924$	−11.8985	−0.391430
$925$	9.14813	0.300789
$926$	−77.7993	−2.55664
$927$	−31.5869	−1.03745
$928$	−0.0527362	−0.00173115
$929$	35.4784	1.16401	0.582005	−	0.813185i	$-0.302269\pi$
$929$	35.4784	1.16401	0.582005	+	0.813185i	$0.302269\pi$
$930$	3.87026	0.126911
$931$	0.738050	0.0241886
$932$	15.4305	0.505442
$933$	1.33296	0.0436390
$934$	40.0622	1.31088
$935$	13.3577	0.436843
$936$	−176.301	−5.76257
$937$	−39.4933	−1.29019	−0.645095	−	0.764103i	$-0.723182\pi$
$937$	−39.4933	−1.29019	−0.645095	+	0.764103i	$0.723182\pi$
$938$	−82.8176	−2.70409
$939$	3.68537	0.120267
$940$	60.8293	1.98403
$941$	40.1633	1.30929	0.654644	−	0.755938i	$-0.272819\pi$
$941$	40.1633	1.30929	0.654644	+	0.755938i	$0.272819\pi$
$942$	−9.91009	−0.322888
$943$	25.6739	0.836058
$944$	15.3680	0.500187
$945$	−2.69797	−0.0877650
$946$	−18.8003	−0.611250
$947$	50.0731	1.62716	0.813578	−	0.581455i	$-0.197516\pi$
$947$	50.0731	1.62716	0.813578	+	0.581455i	$0.197516\pi$
$948$	−14.3172	−0.465001
$949$	−20.8957	−0.678303
$950$	6.58029	0.213493
$951$	4.23642	0.137375
$952$	64.7156	2.09744
$953$	−52.5508	−1.70229	−0.851143	−	0.524934i	$-0.824090\pi$
$953$	−52.5508	−1.70229	−0.851143	+	0.524934i	$0.824090\pi$
$954$	−101.183	−3.27591
$955$	−20.6935	−0.669627
$956$	−99.1700	−3.20739
$957$	−0.00224515	−7.25755e−5 0
$958$	45.5725	1.47238
$959$	−51.1428	−1.65149
$960$	4.58777	0.148070
$961$	35.5630	1.14719
$962$	160.878	5.18692
$963$	−26.7010	−0.860429
$964$	112.583	3.62606
$965$	1.05805	0.0340597
$966$	6.64911	0.213931
$967$	−11.8722	−0.381784	−0.190892	−	0.981611i	$-0.561138\pi$
$967$	−11.8722	−0.381784	−0.190892	+	0.981611i	$0.561138\pi$
$968$	−119.016	−3.82533
$969$	1.15438	0.0370840
$970$	−32.0854	−1.03020
$971$	27.8580	0.894006	0.447003	−	0.894533i	$-0.352491\pi$
$971$	27.8580	0.894006	0.447003	+	0.894533i	$0.352491\pi$
$972$	25.0106	0.802216
$973$	52.9663	1.69802
$974$	−47.1156	−1.50968
$975$	−1.13101	−0.0362213
$976$	−43.9880	−1.40802
$977$	−50.9198	−1.62907	−0.814535	−	0.580114i	$-0.803008\pi$
$977$	−50.9198	−1.62907	−0.814535	+	0.580114i	$0.803008\pi$
$978$	−4.37227	−0.139810
$979$	−0.125986	−0.00402654
$980$	−1.63762	−0.0523120
$981$	−17.3322	−0.553375
$982$	76.0147	2.42573
$983$	−8.76186	−0.279460	−0.139730	−	0.990190i	$-0.544623\pi$
$983$	−8.76186	−0.279460	−0.139730	+	0.990190i	$0.544623\pi$
$984$	7.59052	0.241977
$985$	9.16932	0.292159
$986$	0.0194455	0.000619271 0
$987$	5.11389	0.162777
$988$	84.3431	2.68331
$989$	7.65731	0.243488
$990$	−39.4932	−1.25517
$991$	−2.37246	−0.0753636	−0.0376818	−	0.999290i	$-0.511997\pi$
$991$	−2.37246	−0.0753636	−0.0376818	+	0.999290i	$0.511997\pi$
$992$	163.915	5.20432
$993$	−3.55098	−0.112687
$994$	−35.1970	−1.11638
$995$	−15.1743	−0.481058
$996$	−16.6857	−0.528707
$997$	15.9261	0.504386	0.252193	−	0.967677i	$-0.418848\pi$
$997$	15.9261	0.504386	0.252193	+	0.967677i	$0.418848\pi$
$998$	58.7425	1.85946
$999$	−9.53854	−0.301786

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6005.2.a.f.1.3	✓	111

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6005.2.a.f.1.3	✓	111	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:	\( N \)	\(=\)	\( 6005 = 5 \cdot 1201 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71589 q^{2} +0.174668 q^{3} +5.37604 q^{4} +1.00000 q^{5} -0.474377 q^{6} +2.58754 q^{7} -9.16893 q^{8} -2.96949 q^{9} +O(q^{10})\)	\(q-2.71589 q^{2} +0.174668 q^{3} +5.37604 q^{4} +1.00000 q^{5} -0.474377 q^{6} +2.58754 q^{7} -9.16893 q^{8} -2.96949 q^{9} -2.71589 q^{10} -4.89698 q^{11} +0.939019 q^{12} -6.47521 q^{13} -7.02747 q^{14} +0.174668 q^{15} +14.1497 q^{16} -2.72774 q^{17} +8.06480 q^{18} -2.42289 q^{19} +5.37604 q^{20} +0.451960 q^{21} +13.2996 q^{22} -5.41691 q^{23} -1.60151 q^{24} +1.00000 q^{25} +17.5859 q^{26} -1.04268 q^{27} +13.9107 q^{28} +0.00262486 q^{29} -0.474377 q^{30} -8.15861 q^{31} -20.0911 q^{32} -0.855344 q^{33} +7.40822 q^{34} +2.58754 q^{35} -15.9641 q^{36} +9.14813 q^{37} +6.58029 q^{38} -1.13101 q^{39} -9.16893 q^{40} -4.73959 q^{41} -1.22747 q^{42} -1.41359 q^{43} -26.3263 q^{44} -2.96949 q^{45} +14.7117 q^{46} +11.3149 q^{47} +2.47149 q^{48} -0.304616 q^{49} -2.71589 q^{50} -0.476447 q^{51} -34.8109 q^{52} -12.5462 q^{53} +2.83179 q^{54} -4.89698 q^{55} -23.7250 q^{56} -0.423200 q^{57} -0.00712881 q^{58} +1.08610 q^{59} +0.939019 q^{60} -3.10876 q^{61} +22.1579 q^{62} -7.68369 q^{63} +26.2657 q^{64} -6.47521 q^{65} +2.32302 q^{66} +11.7848 q^{67} -14.6644 q^{68} -0.946159 q^{69} -7.02747 q^{70} +5.00848 q^{71} +27.2270 q^{72} +3.22703 q^{73} -24.8453 q^{74} +0.174668 q^{75} -13.0255 q^{76} -12.6711 q^{77} +3.07169 q^{78} -15.2470 q^{79} +14.1497 q^{80} +8.72635 q^{81} +12.8722 q^{82} -17.7693 q^{83} +2.42975 q^{84} -2.72774 q^{85} +3.83916 q^{86} +0.000458477 q^{87} +44.9000 q^{88} +0.0257273 q^{89} +8.06480 q^{90} -16.7549 q^{91} -29.1215 q^{92} -1.42505 q^{93} -30.7300 q^{94} -2.42289 q^{95} -3.50926 q^{96} +11.8140 q^{97} +0.827301 q^{98} +14.5415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 111 q + 20 q^{2} + 40 q^{3} + 136 q^{4} + 111 q^{5} + 3 q^{6} + 39 q^{7} + 45 q^{8} + 139 q^{9}+O(q^{10}) \) 111 * q + 20 * q^2 + 40 * q^3 + 136 * q^4 + 111 * q^5 + 3 * q^6 + 39 * q^7 + 45 * q^8 + 139 * q^9	\( 111 q + 20 q^{2} + 40 q^{3} + 136 q^{4} + 111 q^{5} + 3 q^{6} + 39 q^{7} + 45 q^{8} + 139 q^{9} + 20 q^{10} + 36 q^{11} + 80 q^{12} + 36 q^{13} + 7 q^{14} + 40 q^{15} + 190 q^{16} + 38 q^{17} + 48 q^{18} + 77 q^{19} + 136 q^{20} + 11 q^{21} + 39 q^{22} + 82 q^{23} - 3 q^{24} + 111 q^{25} - 3 q^{26} + 130 q^{27} + 87 q^{28} + 20 q^{29} + 3 q^{30} + 41 q^{31} + 85 q^{32} + 33 q^{33} + 7 q^{34} + 39 q^{35} + 191 q^{36} + 80 q^{37} + 42 q^{38} + 21 q^{39} + 45 q^{40} + 16 q^{41} + 33 q^{42} + 164 q^{43} + 37 q^{44} + 139 q^{45} + 32 q^{46} + 148 q^{47} + 149 q^{48} + 160 q^{49} + 20 q^{50} + 51 q^{51} + 87 q^{52} + 83 q^{53} - 6 q^{54} + 36 q^{55} - 10 q^{56} + 28 q^{57} + 47 q^{58} + 14 q^{59} + 80 q^{60} + 20 q^{61} + 14 q^{62} + 120 q^{63} + 231 q^{64} + 36 q^{65} - 4 q^{66} + 253 q^{67} + 80 q^{68} + 6 q^{69} + 7 q^{70} + 5 q^{71} + 124 q^{72} + 64 q^{73} - 37 q^{74} + 40 q^{75} + 92 q^{76} + 63 q^{77} + 29 q^{78} + 91 q^{79} + 190 q^{80} + 187 q^{81} - 7 q^{82} + 63 q^{83} - 69 q^{84} + 38 q^{85} - 22 q^{86} + 57 q^{87} + 121 q^{88} - 6 q^{89} + 48 q^{90} + 119 q^{91} + 104 q^{92} + 14 q^{93} - q^{94} + 77 q^{95} - 38 q^{96} + 96 q^{97} + 81 q^{98} + 106 q^{99}+O(q^{100}) \) 111 * q + 20 * q^2 + 40 * q^3 + 136 * q^4 + 111 * q^5 + 3 * q^6 + 39 * q^7 + 45 * q^8 + 139 * q^9 + 20 * q^10 + 36 * q^11 + 80 * q^12 + 36 * q^13 + 7 * q^14 + 40 * q^15 + 190 * q^16 + 38 * q^17 + 48 * q^18 + 77 * q^19 + 136 * q^20 + 11 * q^21 + 39 * q^22 + 82 * q^23 - 3 * q^24 + 111 * q^25 - 3 * q^26 + 130 * q^27 + 87 * q^28 + 20 * q^29 + 3 * q^30 + 41 * q^31 + 85 * q^32 + 33 * q^33 + 7 * q^34 + 39 * q^35 + 191 * q^36 + 80 * q^37 + 42 * q^38 + 21 * q^39 + 45 * q^40 + 16 * q^41 + 33 * q^42 + 164 * q^43 + 37 * q^44 + 139 * q^45 + 32 * q^46 + 148 * q^47 + 149 * q^48 + 160 * q^49 + 20 * q^50 + 51 * q^51 + 87 * q^52 + 83 * q^53 - 6 * q^54 + 36 * q^55 - 10 * q^56 + 28 * q^57 + 47 * q^58 + 14 * q^59 + 80 * q^60 + 20 * q^61 + 14 * q^62 + 120 * q^63 + 231 * q^64 + 36 * q^65 - 4 * q^66 + 253 * q^67 + 80 * q^68 + 6 * q^69 + 7 * q^70 + 5 * q^71 + 124 * q^72 + 64 * q^73 - 37 * q^74 + 40 * q^75 + 92 * q^76 + 63 * q^77 + 29 * q^78 + 91 * q^79 + 190 * q^80 + 187 * q^81 - 7 * q^82 + 63 * q^83 - 69 * q^84 + 38 * q^85 - 22 * q^86 + 57 * q^87 + 121 * q^88 - 6 * q^89 + 48 * q^90 + 119 * q^91 + 104 * q^92 + 14 * q^93 - q^94 + 77 * q^95 - 38 * q^96 + 96 * q^97 + 81 * q^98 + 106 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more