LMFDB - Embedded newform 4031.2.a.c.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.99079 q^{2} -3.03517 q^{3} +1.96324 q^{4} -1.96489 q^{5} +6.04239 q^{6} +0.618316 q^{7} +0.0731745 q^{8} +6.21228 q^{9} +O(q^{10})$	\(q-1.99079 q^{2} -3.03517 q^{3} +1.96324 q^{4} -1.96489 q^{5} +6.04239 q^{6} +0.618316 q^{7} +0.0731745 q^{8} +6.21228 q^{9} +3.91168 q^{10} -1.03647 q^{11} -5.95879 q^{12} -1.13046 q^{13} -1.23094 q^{14} +5.96377 q^{15} -4.07216 q^{16} +1.84344 q^{17} -12.3673 q^{18} +5.19120 q^{19} -3.85755 q^{20} -1.87670 q^{21} +2.06339 q^{22} -0.0424119 q^{23} -0.222097 q^{24} -1.13922 q^{25} +2.25052 q^{26} -9.74983 q^{27} +1.21390 q^{28} -1.00000 q^{29} -11.8726 q^{30} -5.48926 q^{31} +7.96047 q^{32} +3.14586 q^{33} -3.66990 q^{34} -1.21492 q^{35} +12.1962 q^{36} -2.74985 q^{37} -10.3346 q^{38} +3.43116 q^{39} -0.143780 q^{40} -4.79099 q^{41} +3.73611 q^{42} -0.286022 q^{43} -2.03484 q^{44} -12.2064 q^{45} +0.0844332 q^{46} -0.412310 q^{47} +12.3597 q^{48} -6.61769 q^{49} +2.26795 q^{50} -5.59516 q^{51} -2.21938 q^{52} +14.1406 q^{53} +19.4099 q^{54} +2.03654 q^{55} +0.0452449 q^{56} -15.7562 q^{57} +1.99079 q^{58} +5.80725 q^{59} +11.7083 q^{60} -11.5442 q^{61} +10.9280 q^{62} +3.84115 q^{63} -7.70330 q^{64} +2.22124 q^{65} -6.26274 q^{66} -9.91482 q^{67} +3.61912 q^{68} +0.128728 q^{69} +2.41865 q^{70} -2.35394 q^{71} +0.454580 q^{72} +14.8420 q^{73} +5.47437 q^{74} +3.45773 q^{75} +10.1916 q^{76} -0.640864 q^{77} -6.83071 q^{78} +3.71256 q^{79} +8.00134 q^{80} +10.9556 q^{81} +9.53785 q^{82} -1.88462 q^{83} -3.68441 q^{84} -3.62215 q^{85} +0.569409 q^{86} +3.03517 q^{87} -0.0758429 q^{88} -12.3638 q^{89} +24.3004 q^{90} -0.698984 q^{91} -0.0832649 q^{92} +16.6608 q^{93} +0.820822 q^{94} -10.2001 q^{95} -24.1614 q^{96} +0.989970 q^{97} +13.1744 q^{98} -6.43882 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.99079	−1.40770	−0.703850	−	0.710348i	$-0.748537\pi$
$2$	−1.99079	−1.40770	−0.703850	+	0.710348i	$0.748537\pi$
$3$	−3.03517	−1.75236	−0.876179	−	0.481985i	$-0.839916\pi$
$3$	−3.03517	−1.75236	−0.876179	+	0.481985i	$0.839916\pi$
$4$	1.96324	0.981622
$5$	−1.96489	−0.878724	−0.439362	−	0.898310i	$-0.644795\pi$
$5$	−1.96489	−0.878724	−0.439362	+	0.898310i	$0.644795\pi$
$6$	6.04239	2.46680
$7$	0.618316	0.233701	0.116851	−	0.993149i	$-0.462720\pi$
$7$	0.618316	0.233701	0.116851	+	0.993149i	$0.462720\pi$
$8$	0.0731745	0.0258711
$9$	6.21228	2.07076
$10$	3.91168	1.23698
$11$	−1.03647	−0.312507	−0.156253	−	0.987717i	$-0.549942\pi$
$11$	−1.03647	−0.312507	−0.156253	+	0.987717i	$0.549942\pi$
$12$	−5.95879	−1.72015
$13$	−1.13046	−0.313534	−0.156767	−	0.987636i	$-0.550107\pi$
$13$	−1.13046	−0.313534	−0.156767	+	0.987636i	$0.550107\pi$
$14$	−1.23094	−0.328982
$15$	5.96377	1.53984
$16$	−4.07216	−1.01804
$17$	1.84344	0.447100	0.223550	−	0.974693i	$-0.428235\pi$
$17$	1.84344	0.447100	0.223550	+	0.974693i	$0.428235\pi$
$18$	−12.3673	−2.91501
$19$	5.19120	1.19094	0.595472	−	0.803376i	$-0.296965\pi$
$19$	5.19120	1.19094	0.595472	+	0.803376i	$0.296965\pi$
$20$	−3.85755	−0.862575
$21$	−1.87670	−0.409529
$22$	2.06339	0.439916
$23$	−0.0424119	−0.00884350	−0.00442175	−	0.999990i	$-0.501407\pi$
$23$	−0.0424119	−0.00884350	−0.00442175	+	0.999990i	$0.501407\pi$
$24$	−0.222097	−0.0453354
$25$	−1.13922	−0.227844
$26$	2.25052	0.441363
$27$	−9.74983	−1.87636
$28$	1.21390	0.229406
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−11.8726	−2.16763
$31$	−5.48926	−0.985899	−0.492950	−	0.870058i	$-0.664081\pi$
$31$	−5.48926	−0.985899	−0.492950	+	0.870058i	$0.664081\pi$
$32$	7.96047	1.40723
$33$	3.14586	0.547623
$34$	−3.66990	−0.629383
$35$	−1.21492	−0.205359
$36$	12.1962	2.03270
$37$	−2.74985	−0.452072	−0.226036	−	0.974119i	$-0.572577\pi$
$37$	−2.74985	−0.452072	−0.226036	+	0.974119i	$0.572577\pi$
$38$	−10.3346	−1.67649
$39$	3.43116	0.549425
$40$	−0.143780	−0.0227335
$41$	−4.79099	−0.748226	−0.374113	−	0.927383i	$-0.622053\pi$
$41$	−4.79099	−0.748226	−0.374113	+	0.927383i	$0.622053\pi$
$42$	3.73611	0.576494
$43$	−0.286022	−0.0436179	−0.0218090	−	0.999762i	$-0.506943\pi$
$43$	−0.286022	−0.0436179	−0.0218090	+	0.999762i	$0.506943\pi$
$44$	−2.03484	−0.306763
$45$	−12.2064	−1.81963
$46$	0.0844332	0.0124490
$47$	−0.412310	−0.0601416	−0.0300708	−	0.999548i	$-0.509573\pi$
$47$	−0.412310	−0.0601416	−0.0300708	+	0.999548i	$0.509573\pi$
$48$	12.3597	1.78397
$49$	−6.61769	−0.945384
$50$	2.26795	0.320736
$51$	−5.59516	−0.783479
$52$	−2.21938	−0.307772
$53$	14.1406	1.94236	0.971180	−	0.238346i	$-0.0766052\pi$
$53$	14.1406	1.94236	0.971180	+	0.238346i	$0.0766052\pi$
$54$	19.4099	2.64135
$55$	2.03654	0.274607
$56$	0.0452449	0.00604611
$57$	−15.7562	−2.08696
$58$	1.99079	0.261403
$59$	5.80725	0.756039	0.378019	−	0.925798i	$-0.376605\pi$
$59$	5.80725	0.756039	0.378019	+	0.925798i	$0.376605\pi$
$60$	11.7083	1.51154
$61$	−11.5442	−1.47808	−0.739039	−	0.673662i	$-0.764720\pi$
$61$	−11.5442	−1.47808	−0.739039	+	0.673662i	$0.764720\pi$
$62$	10.9280	1.38785
$63$	3.84115	0.483939
$64$	−7.70330	−0.962912
$65$	2.22124	0.275510
$66$	−6.26274	−0.770890
$67$	−9.91482	−1.21129	−0.605644	−	0.795736i	$-0.707085\pi$
$67$	−9.91482	−1.21129	−0.605644	+	0.795736i	$0.707085\pi$
$68$	3.61912	0.438883
$69$	0.128728	0.0154970
$70$	2.41865	0.289084
$71$	−2.35394	−0.279361	−0.139681	−	0.990197i	$-0.544608\pi$
$71$	−2.35394	−0.279361	−0.139681	+	0.990197i	$0.544608\pi$
$72$	0.454580	0.0535728
$73$	14.8420	1.73712	0.868562	−	0.495580i	$-0.165044\pi$
$73$	14.8420	1.73712	0.868562	+	0.495580i	$0.165044\pi$
$74$	5.47437	0.636383
$75$	3.45773	0.399264
$76$	10.1916	1.16906
$77$	−0.640864	−0.0730332
$78$	−6.83071	−0.773426
$79$	3.71256	0.417695	0.208848	−	0.977948i	$-0.433029\pi$
$79$	3.71256	0.417695	0.208848	+	0.977948i	$0.433029\pi$
$80$	8.00134	0.894577
$81$	10.9556	1.21729
$82$	9.53785	1.05328
$83$	−1.88462	−0.206864	−0.103432	−	0.994637i	$-0.532982\pi$
$83$	−1.88462	−0.206864	−0.103432	+	0.994637i	$0.532982\pi$
$84$	−3.68441	−0.402002
$85$	−3.62215	−0.392877
$86$	0.569409	0.0614010
$87$	3.03517	0.325405
$88$	−0.0758429	−0.00808488
$89$	−12.3638	−1.31056	−0.655279	−	0.755387i	$-0.727449\pi$
$89$	−12.3638	−1.31056	−0.655279	+	0.755387i	$0.727449\pi$
$90$	24.3004	2.56149
$91$	−0.698984	−0.0732734
$92$	−0.0832649	−0.00868097
$93$	16.6608	1.72765
$94$	0.820822	0.0846613
$95$	−10.2001	−1.04651
$96$	−24.1614	−2.46596
$97$	0.989970	0.100516	0.0502581	−	0.998736i	$-0.483996\pi$
$97$	0.989970	0.100516	0.0502581	+	0.998736i	$0.483996\pi$
$98$	13.1744	1.33082
$99$	−6.43882	−0.647126
$100$	−2.23656	−0.223656
$101$	14.1373	1.40672	0.703359	−	0.710835i	$-0.251683\pi$
$101$	14.1373	1.40672	0.703359	+	0.710835i	$0.251683\pi$
$102$	11.1388	1.10290
$103$	12.2103	1.20311	0.601557	−	0.798830i	$-0.294547\pi$
$103$	12.2103	1.20311	0.601557	+	0.798830i	$0.294547\pi$
$104$	−0.0827212	−0.00811148
$105$	3.68750	0.359863
$106$	−28.1510	−2.73426
$107$	3.70346	0.358027	0.179014	−	0.983847i	$-0.442709\pi$
$107$	3.70346	0.358027	0.179014	+	0.983847i	$0.442709\pi$
$108$	−19.1413	−1.84187
$109$	10.0782	0.965313	0.482657	−	0.875810i	$-0.339672\pi$
$109$	10.0782	0.965313	0.482657	+	0.875810i	$0.339672\pi$
$110$	−4.05432	−0.386565
$111$	8.34627	0.792193
$112$	−2.51788	−0.237917
$113$	14.9360	1.40506	0.702531	−	0.711653i	$-0.252053\pi$
$113$	14.9360	1.40506	0.702531	+	0.711653i	$0.252053\pi$
$114$	31.3673	2.93781
$115$	0.0833346	0.00777099
$116$	−1.96324	−0.182283
$117$	−7.02276	−0.649255
$118$	−11.5610	−1.06428
$119$	1.13983	0.104488
$120$	0.436396	0.0398373
$121$	−9.92574	−0.902340
$122$	22.9820	2.08069
$123$	14.5415	1.31116
$124$	−10.7767	−0.967780
$125$	12.0629	1.07894
$126$	−7.64692	−0.681242
$127$	−3.94508	−0.350069	−0.175035	−	0.984562i	$-0.556004\pi$
$127$	−3.94508	−0.350069	−0.175035	+	0.984562i	$0.556004\pi$
$128$	−0.585297	−0.0517334
$129$	0.868126	0.0764342
$130$	−4.42201	−0.387836
$131$	17.2565	1.50771	0.753853	−	0.657043i	$-0.228193\pi$
$131$	17.2565	1.50771	0.753853	+	0.657043i	$0.228193\pi$
$132$	6.17608	0.537559
$133$	3.20980	0.278325
$134$	19.7383	1.70513
$135$	19.1573	1.64880
$136$	0.134893	0.0115670
$137$	−2.85946	−0.244300	−0.122150	−	0.992512i	$-0.538979\pi$
$137$	−2.85946	−0.244300	−0.122150	+	0.992512i	$0.538979\pi$
$138$	−0.256269	−0.0218151
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	−2.38519	−0.201585
$141$	1.25143	0.105390
$142$	4.68620	0.393257
$143$	1.17169	0.0979816
$144$	−25.2974	−2.10812
$145$	1.96489	0.163175
$146$	−29.5473	−2.44535
$147$	20.0858	1.65665
$148$	−5.39862	−0.443764
$149$	16.4280	1.34584	0.672919	−	0.739717i	$-0.265040\pi$
$149$	16.4280	1.34584	0.672919	+	0.739717i	$0.265040\pi$
$150$	−6.88361	−0.562044
$151$	−22.3254	−1.81681	−0.908407	−	0.418086i	$-0.862701\pi$
$151$	−22.3254	−1.81681	−0.908407	+	0.418086i	$0.862701\pi$
$152$	0.379863	0.0308110
$153$	11.4520	0.925836
$154$	1.27582	0.102809
$155$	10.7858	0.866334
$156$	6.73620	0.539327
$157$	7.22010	0.576227	0.288113	−	0.957596i	$-0.406972\pi$
$157$	7.22010	0.576227	0.288113	+	0.957596i	$0.406972\pi$
$158$	−7.39092	−0.587990
$159$	−42.9192	−3.40371
$160$	−15.6414	−1.23656
$161$	−0.0262240	−0.00206674
$162$	−21.8103	−1.71358
$163$	18.8468	1.47620	0.738098	−	0.674694i	$-0.235724\pi$
$163$	18.8468	1.47620	0.738098	+	0.674694i	$0.235724\pi$
$164$	−9.40587	−0.734475
$165$	−6.18125	−0.481210
$166$	3.75188	0.291202
$167$	16.9149	1.30891	0.654457	−	0.756100i	$-0.272897\pi$
$167$	16.9149	1.30891	0.654457	+	0.756100i	$0.272897\pi$
$168$	−0.137326	−0.0105949
$169$	−11.7220	−0.901696
$170$	7.21094	0.553054
$171$	32.2492	2.46616
$172$	−0.561531	−0.0428163
$173$	−12.6719	−0.963425	−0.481713	−	0.876329i	$-0.659985\pi$
$173$	−12.6719	−0.963425	−0.481713	+	0.876329i	$0.659985\pi$
$174$	−6.04239	−0.458073
$175$	−0.704397	−0.0532474
$176$	4.22066	0.318144
$177$	−17.6260	−1.32485
$178$	24.6137	1.84487
$179$	−3.59805	−0.268931	−0.134465	−	0.990918i	$-0.542932\pi$
$179$	−3.59805	−0.268931	−0.134465	+	0.990918i	$0.542932\pi$
$180$	−23.9642	−1.78619
$181$	13.5658	1.00834	0.504170	−	0.863604i	$-0.331798\pi$
$181$	13.5658	1.00834	0.504170	+	0.863604i	$0.331798\pi$
$182$	1.39153	0.103147
$183$	35.0385	2.59012
$184$	−0.00310347	−0.000228791 0
$185$	5.40314	0.397247
$186$	−33.1682	−2.43201
$187$	−1.91066	−0.139722
$188$	−0.809464	−0.0590363
$189$	−6.02847	−0.438507
$190$	20.3063	1.47317
$191$	−3.15942	−0.228607	−0.114304	−	0.993446i	$-0.536464\pi$
$191$	−3.15942	−0.228607	−0.114304	+	0.993446i	$0.536464\pi$
$192$	23.3808	1.68737
$193$	11.6283	0.837024	0.418512	−	0.908211i	$-0.362552\pi$
$193$	11.6283	0.837024	0.418512	+	0.908211i	$0.362552\pi$
$194$	−1.97082	−0.141497
$195$	−6.74183	−0.482793
$196$	−12.9921	−0.928009
$197$	−9.94991	−0.708902	−0.354451	−	0.935075i	$-0.615332\pi$
$197$	−9.94991	−0.708902	−0.354451	+	0.935075i	$0.615332\pi$
$198$	12.8183	0.910960
$199$	−26.7437	−1.89581	−0.947905	−	0.318552i	$-0.896803\pi$
$199$	−26.7437	−1.89581	−0.947905	+	0.318552i	$0.896803\pi$
$200$	−0.0833618	−0.00589457
$201$	30.0932	2.12261
$202$	−28.1445	−1.98024
$203$	−0.618316	−0.0433973
$204$	−10.9847	−0.769080
$205$	9.41375	0.657485
$206$	−24.3081	−1.69362
$207$	−0.263475	−0.0183128
$208$	4.60343	0.319191
$209$	−5.38051	−0.372177
$210$	−7.34103	−0.506579
$211$	7.89316	0.543388	0.271694	−	0.962384i	$-0.412416\pi$
$211$	7.89316	0.543388	0.271694	+	0.962384i	$0.412416\pi$
$212$	27.7614	1.90666
$213$	7.14461	0.489541
$214$	−7.37281	−0.503995
$215$	0.562001	0.0383281
$216$	−0.713439	−0.0485433
$217$	−3.39409	−0.230406
$218$	−20.0635	−1.35887
$219$	−45.0480	−3.04407
$220$	3.99822	0.269560
$221$	−2.08394	−0.140181
$222$	−16.6157	−1.11517
$223$	−14.2098	−0.951559	−0.475780	−	0.879565i	$-0.657834\pi$
$223$	−14.2098	−0.951559	−0.475780	+	0.879565i	$0.657834\pi$
$224$	4.92208	0.328871
$225$	−7.07715	−0.471810
$226$	−29.7345	−1.97791
$227$	24.4396	1.62211	0.811056	−	0.584968i	$-0.198893\pi$
$227$	24.4396	1.62211	0.811056	+	0.584968i	$0.198893\pi$
$228$	−30.9332	−2.04860
$229$	−7.78178	−0.514234	−0.257117	−	0.966380i	$-0.582773\pi$
$229$	−7.78178	−0.514234	−0.257117	+	0.966380i	$0.582773\pi$
$230$	−0.165902	−0.0109392
$231$	1.94513	0.127980
$232$	−0.0731745	−0.00480414
$233$	−28.1267	−1.84264	−0.921320	−	0.388805i	$-0.872888\pi$
$233$	−28.1267	−1.84264	−0.921320	+	0.388805i	$0.872888\pi$
$234$	13.9808	0.913956
$235$	0.810142	0.0528478
$236$	11.4010	0.742144
$237$	−11.2683	−0.731952
$238$	−2.26916	−0.147088
$239$	6.31416	0.408429	0.204215	−	0.978926i	$-0.434536\pi$
$239$	6.31416	0.408429	0.204215	+	0.978926i	$0.434536\pi$
$240$	−24.2855	−1.56762
$241$	−10.9833	−0.707498	−0.353749	−	0.935340i	$-0.615093\pi$
$241$	−10.9833	−0.707498	−0.353749	+	0.935340i	$0.615093\pi$
$242$	19.7601	1.27022
$243$	−4.00261	−0.256768
$244$	−22.6640	−1.45091
$245$	13.0030	0.830731
$246$	−28.9490	−1.84572
$247$	−5.86847	−0.373402
$248$	−0.401673	−0.0255063
$249$	5.72014	0.362499
$250$	−24.0146	−1.51882
$251$	5.12821	0.323690	0.161845	−	0.986816i	$-0.448256\pi$
$251$	5.12821	0.323690	0.161845	+	0.986816i	$0.448256\pi$
$252$	7.54111	0.475046
$253$	0.0439585	0.00276365
$254$	7.85383	0.492793
$255$	10.9939	0.688462
$256$	16.5718	1.03574
$257$	0.377940	0.0235753	0.0117876	−	0.999931i	$-0.496248\pi$
$257$	0.377940	0.0235753	0.0117876	+	0.999931i	$0.496248\pi$
$258$	−1.72826	−0.107597
$259$	−1.70028	−0.105650
$260$	4.36083	0.270447
$261$	−6.21228	−0.384530
$262$	−34.3540	−2.12240
$263$	−1.13347	−0.0698925	−0.0349463	−	0.999389i	$-0.511126\pi$
$263$	−1.13347	−0.0698925	−0.0349463	+	0.999389i	$0.511126\pi$
$264$	0.230196	0.0141676
$265$	−27.7847	−1.70680
$266$	−6.39004	−0.391798
$267$	37.5262	2.29657
$268$	−19.4652	−1.18903
$269$	−2.99493	−0.182604	−0.0913021	−	0.995823i	$-0.529103\pi$
$269$	−2.99493	−0.182604	−0.0913021	+	0.995823i	$0.529103\pi$
$270$	−38.1382	−2.32102
$271$	28.8420	1.75202	0.876012	−	0.482289i	$-0.160194\pi$
$271$	28.8420	1.75202	0.876012	+	0.482289i	$0.160194\pi$
$272$	−7.50678	−0.455166
$273$	2.12154	0.128401
$274$	5.69259	0.343902
$275$	1.18076	0.0712027
$276$	0.252724	0.0152122
$277$	−7.03322	−0.422585	−0.211293	−	0.977423i	$-0.567767\pi$
$277$	−7.03322	−0.422585	−0.211293	+	0.977423i	$0.567767\pi$
$278$	1.99079	0.119400
$279$	−34.1008	−2.04156
$280$	−0.0889012	−0.00531286
$281$	−12.3519	−0.736855	−0.368428	−	0.929656i	$-0.620104\pi$
$281$	−12.3519	−0.736855	−0.368428	+	0.929656i	$0.620104\pi$
$282$	−2.49134	−0.148357
$283$	7.82905	0.465389	0.232695	−	0.972550i	$-0.425246\pi$
$283$	7.82905	0.465389	0.232695	+	0.972550i	$0.425246\pi$
$284$	−4.62136	−0.274227
$285$	30.9591	1.83386
$286$	−2.33259	−0.137929
$287$	−2.96234	−0.174862
$288$	49.4527	2.91403
$289$	−13.6017	−0.800102
$290$	−3.91168	−0.229702
$291$	−3.00473	−0.176141
$292$	29.1385	1.70520
$293$	−9.40685	−0.549554	−0.274777	−	0.961508i	$-0.588604\pi$
$293$	−9.40685	−0.549554	−0.274777	+	0.961508i	$0.588604\pi$
$294$	−39.9867	−2.33207
$295$	−11.4106	−0.664350
$296$	−0.201219	−0.0116956
$297$	10.1054	0.586373
$298$	−32.7048	−1.89454
$299$	0.0479452	0.00277274
$300$	6.78836	0.391926
$301$	−0.176852	−0.0101936
$302$	44.4452	2.55753
$303$	−42.9093	−2.46507
$304$	−21.1394	−1.21243
$305$	22.6830	1.29882
$306$	−22.7984	−1.30330
$307$	−6.70926	−0.382918	−0.191459	−	0.981501i	$-0.561322\pi$
$307$	−6.70926	−0.382918	−0.191459	+	0.981501i	$0.561322\pi$
$308$	−1.25817	−0.0716910
$309$	−37.0603	−2.10829
$310$	−21.4722	−1.21954
$311$	−0.812658	−0.0460816	−0.0230408	−	0.999735i	$-0.507335\pi$
$311$	−0.812658	−0.0460816	−0.0230408	+	0.999735i	$0.507335\pi$
$312$	0.251073	0.0142142
$313$	−25.0974	−1.41859	−0.709295	−	0.704912i	$-0.750987\pi$
$313$	−25.0974	−1.41859	−0.709295	+	0.704912i	$0.750987\pi$
$314$	−14.3737	−0.811155
$315$	−7.54743	−0.425249
$316$	7.28865	0.410019
$317$	−15.3952	−0.864683	−0.432341	−	0.901710i	$-0.642312\pi$
$317$	−15.3952	−0.864683	−0.432341	+	0.901710i	$0.642312\pi$
$318$	85.4431	4.79141
$319$	1.03647	0.0580310
$320$	15.1361	0.846134
$321$	−11.2406	−0.627392
$322$	0.0522064	0.00290935
$323$	9.56966	0.532470
$324$	21.5085	1.19492
$325$	1.28785	0.0714369
$326$	−37.5200	−2.07804
$327$	−30.5890	−1.69157
$328$	−0.350578	−0.0193574
$329$	−0.254938	−0.0140552
$330$	12.3056	0.677400
$331$	20.5846	1.13143	0.565715	−	0.824601i	$-0.308600\pi$
$331$	20.5846	1.13143	0.565715	+	0.824601i	$0.308600\pi$
$332$	−3.69996	−0.203062
$333$	−17.0828	−0.936133
$334$	−33.6740	−1.84256
$335$	19.4815	1.06439
$336$	7.64221	0.416917
$337$	−21.7917	−1.18707	−0.593536	−	0.804808i	$-0.702268\pi$
$337$	−21.7917	−1.18707	−0.593536	+	0.804808i	$0.702268\pi$
$338$	23.3361	1.26932
$339$	−45.3334	−2.46217
$340$	−7.11116	−0.385657
$341$	5.68943	0.308100
$342$	−64.2014	−3.47161
$343$	−8.42003	−0.454639
$344$	−0.0209295	−0.00112844
$345$	−0.252935	−0.0136176
$346$	25.2270	1.35621
$347$	19.5433	1.04914	0.524568	−	0.851368i	$-0.324227\pi$
$347$	19.5433	1.04914	0.524568	+	0.851368i	$0.324227\pi$
$348$	5.95879	0.319424
$349$	−18.4308	−0.986576	−0.493288	−	0.869866i	$-0.664205\pi$
$349$	−18.4308	−0.986576	−0.493288	+	0.869866i	$0.664205\pi$
$350$	1.40231	0.0749564
$351$	11.0218	0.588302
$352$	−8.25076	−0.439767
$353$	12.1098	0.644538	0.322269	−	0.946648i	$-0.395554\pi$
$353$	12.1098	0.644538	0.322269	+	0.946648i	$0.395554\pi$
$354$	35.0897	1.86499
$355$	4.62522	0.245481
$356$	−24.2731	−1.28647
$357$	−3.45957	−0.183100
$358$	7.16296	0.378574
$359$	−8.32719	−0.439493	−0.219746	−	0.975557i	$-0.570523\pi$
$359$	−8.32719	−0.439493	−0.219746	+	0.975557i	$0.570523\pi$
$360$	−0.893199	−0.0470757
$361$	7.94856	0.418345
$362$	−27.0067	−1.41944
$363$	30.1263	1.58122
$364$	−1.37228	−0.0719268
$365$	−29.1629	−1.52645
$366$	−69.7544	−3.64612
$367$	23.5583	1.22973	0.614865	−	0.788632i	$-0.289210\pi$
$367$	23.5583	1.22973	0.614865	+	0.788632i	$0.289210\pi$
$368$	0.172708	0.00900304
$369$	−29.7629	−1.54940
$370$	−10.7565	−0.559205
$371$	8.74336	0.453932
$372$	32.7093	1.69590
$373$	−1.29756	−0.0671850	−0.0335925	−	0.999436i	$-0.510695\pi$
$373$	−1.29756	−0.0671850	−0.0335925	+	0.999436i	$0.510695\pi$
$374$	3.80373	0.196686
$375$	−36.6129	−1.89068
$376$	−0.0301705	−0.00155593
$377$	1.13046	0.0582219
$378$	12.0014	0.617286
$379$	31.4832	1.61718	0.808592	−	0.588370i	$-0.200230\pi$
$379$	31.4832	1.61718	0.808592	+	0.588370i	$0.200230\pi$
$380$	−20.0253	−1.02728
$381$	11.9740	0.613447
$382$	6.28973	0.321811
$383$	−8.37636	−0.428012	−0.214006	−	0.976832i	$-0.568651\pi$
$383$	−8.37636	−0.428012	−0.214006	+	0.976832i	$0.568651\pi$
$384$	1.77648	0.0906555
$385$	1.25922	0.0641760
$386$	−23.1495	−1.17828
$387$	−1.77685	−0.0903223
$388$	1.94355	0.0986690
$389$	10.2381	0.519091	0.259545	−	0.965731i	$-0.416427\pi$
$389$	10.2381	0.519091	0.259545	+	0.965731i	$0.416427\pi$
$390$	13.4216	0.679628
$391$	−0.0781838	−0.00395392
$392$	−0.484246	−0.0244581
$393$	−52.3764	−2.64204
$394$	19.8082	0.997922
$395$	−7.29476	−0.367039
$396$	−12.6410	−0.635233
$397$	21.3390	1.07098	0.535488	−	0.844543i	$-0.320128\pi$
$397$	21.3390	1.07098	0.535488	+	0.844543i	$0.320128\pi$
$398$	53.2411	2.66873
$399$	−9.74230	−0.487725
$400$	4.63908	0.231954
$401$	−28.7782	−1.43711	−0.718556	−	0.695469i	$-0.755197\pi$
$401$	−28.7782	−1.43711	−0.718556	+	0.695469i	$0.755197\pi$
$402$	−59.9093	−2.98800
$403$	6.20541	0.309113
$404$	27.7550	1.38086
$405$	−21.5265	−1.06966
$406$	1.23094	0.0610904
$407$	2.85013	0.141276
$408$	−0.409423	−0.0202694
$409$	32.3931	1.60174	0.800868	−	0.598841i	$-0.204372\pi$
$409$	32.3931	1.60174	0.800868	+	0.598841i	$0.204372\pi$
$410$	−18.7408	−0.925542
$411$	8.67897	0.428102
$412$	23.9717	1.18100
$413$	3.59071	0.176687
$414$	0.524523	0.0257789
$415$	3.70306	0.181776
$416$	−8.99903	−0.441214
$417$	3.03517	0.148633
$418$	10.7115	0.523915
$419$	13.0089	0.635526	0.317763	−	0.948170i	$-0.397068\pi$
$419$	13.0089	0.635526	0.317763	+	0.948170i	$0.397068\pi$
$420$	7.23945	0.353249
$421$	−32.0254	−1.56082	−0.780412	−	0.625265i	$-0.784991\pi$
$421$	−32.0254	−1.56082	−0.780412	+	0.625265i	$0.784991\pi$
$422$	−15.7136	−0.764927
$423$	−2.56138	−0.124539
$424$	1.03473	0.0502510
$425$	−2.10008	−0.101869
$426$	−14.2234	−0.689127
$427$	−7.13794	−0.345429
$428$	7.27080	0.351447
$429$	−3.55628	−0.171699
$430$	−1.11883	−0.0539545
$431$	10.4918	0.505371	0.252685	−	0.967548i	$-0.418686\pi$
$431$	10.4918	0.505371	0.252685	+	0.967548i	$0.418686\pi$
$432$	39.7029	1.91021
$433$	−5.51526	−0.265047	−0.132523	−	0.991180i	$-0.542308\pi$
$433$	−5.51526	−0.265047	−0.132523	+	0.991180i	$0.542308\pi$
$434$	6.75693	0.324343
$435$	−5.96377	−0.285941
$436$	19.7859	0.947572
$437$	−0.220169	−0.0105321
$438$	89.6812	4.28513
$439$	−4.89269	−0.233515	−0.116758	−	0.993160i	$-0.537250\pi$
$439$	−4.89269	−0.233515	−0.116758	+	0.993160i	$0.537250\pi$
$440$	0.149023	0.00710438
$441$	−41.1109	−1.95766
$442$	4.14869	0.197333
$443$	−10.1697	−0.483176	−0.241588	−	0.970379i	$-0.577668\pi$
$443$	−10.1697	−0.483176	−0.241588	+	0.970379i	$0.577668\pi$
$444$	16.3858	0.777634
$445$	24.2934	1.15162
$446$	28.2887	1.33951
$447$	−49.8619	−2.35839
$448$	−4.76307	−0.225034
$449$	−1.88070	−0.0887556	−0.0443778	−	0.999015i	$-0.514131\pi$
$449$	−1.88070	−0.0887556	−0.0443778	+	0.999015i	$0.514131\pi$
$450$	14.0891	0.664167
$451$	4.96570	0.233826
$452$	29.3230	1.37924
$453$	67.7614	3.18371
$454$	−48.6541	−2.28345
$455$	1.37342	0.0643871
$456$	−1.15295	−0.0539919
$457$	−1.08083	−0.0505593	−0.0252796	−	0.999680i	$-0.508048\pi$
$457$	−1.08083	−0.0505593	−0.0252796	+	0.999680i	$0.508048\pi$
$458$	15.4919	0.723888
$459$	−17.9732	−0.838918
$460$	0.163606	0.00762818
$461$	−20.7355	−0.965750	−0.482875	−	0.875689i	$-0.660408\pi$
$461$	−20.7355	−0.965750	−0.482875	+	0.875689i	$0.660408\pi$
$462$	−3.87235	−0.180158
$463$	−3.69998	−0.171953	−0.0859763	−	0.996297i	$-0.527401\pi$
$463$	−3.69998	−0.171953	−0.0859763	+	0.996297i	$0.527401\pi$
$464$	4.07216	0.189045
$465$	−32.7367	−1.51813
$466$	55.9943	2.59389
$467$	−23.3234	−1.07928	−0.539639	−	0.841896i	$-0.681439\pi$
$467$	−23.3234	−1.07928	−0.539639	+	0.841896i	$0.681439\pi$
$468$	−13.7874	−0.637322
$469$	−6.13049	−0.283080
$470$	−1.61282	−0.0743939
$471$	−21.9143	−1.00976
$472$	0.424942	0.0195595
$473$	0.296452	0.0136309
$474$	22.4327	1.03037
$475$	−5.91391	−0.271349
$476$	2.23776	0.102568
$477$	87.8454	4.02216
$478$	−12.5702	−0.574946
$479$	31.5230	1.44032	0.720161	−	0.693807i	$-0.244068\pi$
$479$	31.5230	1.44032	0.720161	+	0.693807i	$0.244068\pi$
$480$	47.4744	2.16690
$481$	3.10861	0.141740
$482$	21.8655	0.995945
$483$	0.0795943	0.00362166
$484$	−19.4866	−0.885756
$485$	−1.94518	−0.0883261
$486$	7.96836	0.361452
$487$	18.8807	0.855565	0.427783	−	0.903882i	$-0.359295\pi$
$487$	18.8807	0.855565	0.427783	+	0.903882i	$0.359295\pi$
$488$	−0.844738	−0.0382395
$489$	−57.2033	−2.58682
$490$	−25.8862	−1.16942
$491$	−0.141902	−0.00640394	−0.00320197	−	0.999995i	$-0.501019\pi$
$491$	−0.141902	−0.00640394	−0.00320197	+	0.999995i	$0.501019\pi$
$492$	28.5485	1.28706
$493$	−1.84344	−0.0830243
$494$	11.6829	0.525638
$495$	12.6516	0.568645
$496$	22.3531	1.00369
$497$	−1.45548	−0.0652871
$498$	−11.3876	−0.510291
$499$	−19.4327	−0.869927	−0.434963	−	0.900448i	$-0.643239\pi$
$499$	−19.4327	−0.869927	−0.434963	+	0.900448i	$0.643239\pi$
$500$	23.6824	1.05911
$501$	−51.3396	−2.29368
$502$	−10.2092	−0.455658
$503$	28.6588	1.27783	0.638916	−	0.769277i	$-0.279383\pi$
$503$	28.6588	1.27783	0.638916	+	0.769277i	$0.279383\pi$
$504$	0.281074	0.0125200
$505$	−27.7783	−1.23612
$506$	−0.0875122	−0.00389039
$507$	35.5785	1.58009
$508$	−7.74515	−0.343636
$509$	−7.94018	−0.351942	−0.175971	−	0.984395i	$-0.556307\pi$
$509$	−7.94018	−0.351942	−0.175971	+	0.984395i	$0.556307\pi$
$510$	−21.8864	−0.969148
$511$	9.17704	0.405968
$512$	−31.8204	−1.40627
$513$	−50.6133	−2.23463
$514$	−0.752399	−0.0331869
$515$	−23.9918	−1.05720
$516$	1.70434	0.0750295
$517$	0.427345	0.0187946
$518$	3.38489	0.148724
$519$	38.4614	1.68827
$520$	0.162538	0.00712775
$521$	−6.24814	−0.273736	−0.136868	−	0.990589i	$-0.543704\pi$
$521$	−6.24814	−0.273736	−0.136868	+	0.990589i	$0.543704\pi$
$522$	12.3673	0.541304
$523$	−35.1953	−1.53898	−0.769492	−	0.638657i	$-0.779490\pi$
$523$	−35.1953	−1.53898	−0.769492	+	0.638657i	$0.779490\pi$
$524$	33.8787	1.48000
$525$	2.13797	0.0933086
$526$	2.25649	0.0983877
$527$	−10.1191	−0.440795
$528$	−12.8104	−0.557503
$529$	−22.9982	−0.999922
$530$	55.3135	2.40266
$531$	36.0762	1.56558
$532$	6.30162	0.273210
$533$	5.41604	0.234595
$534$	−74.7068	−3.23288
$535$	−7.27688	−0.314607
$536$	−0.725512	−0.0313373
$537$	10.9207	0.471263
$538$	5.96228	0.257052
$539$	6.85901	0.295439
$540$	37.6105	1.61850
$541$	35.4575	1.52444	0.762219	−	0.647319i	$-0.224110\pi$
$541$	35.4575	1.52444	0.762219	+	0.647319i	$0.224110\pi$
$542$	−57.4183	−2.46633
$543$	−41.1747	−1.76697
$544$	14.6746	0.629170
$545$	−19.8025	−0.848244
$546$	−4.22354	−0.180751
$547$	−45.2653	−1.93540	−0.967702	−	0.252098i	$-0.918879\pi$
$547$	−45.2653	−1.93540	−0.967702	+	0.252098i	$0.918879\pi$
$548$	−5.61382	−0.239811
$549$	−71.7156	−3.06075
$550$	−2.35065	−0.100232
$551$	−5.19120	−0.221153
$552$	0.00941957	0.000400924 0
$553$	2.29553	0.0976160
$554$	14.0017	0.594874
$555$	−16.3995	−0.696119
$556$	−1.96324	−0.0832601
$557$	2.59079	0.109775	0.0548876	−	0.998493i	$-0.482520\pi$
$557$	2.59079	0.109775	0.0548876	+	0.998493i	$0.482520\pi$
$558$	67.8875	2.87391
$559$	0.323338	0.0136757
$560$	4.94735	0.209064
$561$	5.79920	0.244842
$562$	24.5901	1.03727
$563$	39.9166	1.68229	0.841143	−	0.540813i	$-0.181883\pi$
$563$	39.9166	1.68229	0.841143	+	0.540813i	$0.181883\pi$
$564$	2.45687	0.103453
$565$	−29.3476	−1.23466
$566$	−15.5860	−0.655129
$567$	6.77401	0.284482
$568$	−0.172248	−0.00722738
$569$	−10.1979	−0.427517	−0.213758	−	0.976887i	$-0.568570\pi$
$569$	−10.1979	−0.427517	−0.213758	+	0.976887i	$0.568570\pi$
$570$	−61.6331	−2.58153
$571$	26.3242	1.10163	0.550816	−	0.834627i	$-0.314317\pi$
$571$	26.3242	1.10163	0.550816	+	0.834627i	$0.314317\pi$
$572$	2.30031	0.0961808
$573$	9.58938	0.400602
$574$	5.89740	0.246153
$575$	0.0483165	0.00201494
$576$	−47.8550	−1.99396
$577$	−8.44929	−0.351749	−0.175874	−	0.984413i	$-0.556275\pi$
$577$	−8.44929	−0.351749	−0.175874	+	0.984413i	$0.556275\pi$
$578$	27.0782	1.12630
$579$	−35.2939	−1.46677
$580$	3.85755	0.160176
$581$	−1.16529	−0.0483443
$582$	5.98179	0.247953
$583$	−14.6563	−0.607000
$584$	1.08606	0.0449413
$585$	13.7989	0.570516
$586$	18.7271	0.773608
$587$	−5.18583	−0.214042	−0.107021	−	0.994257i	$-0.534131\pi$
$587$	−5.18583	−0.214042	−0.107021	+	0.994257i	$0.534131\pi$
$588$	39.4334	1.62620
$589$	−28.4958	−1.17415
$590$	22.7161	0.935206
$591$	30.1997	1.24225
$592$	11.1978	0.460228
$593$	37.8149	1.55287	0.776436	−	0.630196i	$-0.217025\pi$
$593$	37.8149	1.55287	0.776436	+	0.630196i	$0.217025\pi$
$594$	−20.1177	−0.825438
$595$	−2.23963	−0.0918160
$596$	32.2522	1.32110
$597$	81.1718	3.32214
$598$	−0.0954488	−0.00390319
$599$	−37.5096	−1.53260	−0.766300	−	0.642483i	$-0.777905\pi$
$599$	−37.5096	−1.53260	−0.766300	+	0.642483i	$0.777905\pi$
$600$	0.253017	0.0103294
$601$	−11.8715	−0.484249	−0.242125	−	0.970245i	$-0.577844\pi$
$601$	−11.8715	−0.484249	−0.242125	+	0.970245i	$0.577844\pi$
$602$	0.352075	0.0143495
$603$	−61.5937	−2.50829
$604$	−43.8302	−1.78342
$605$	19.5030	0.792908
$606$	85.4234	3.47009
$607$	−2.44019	−0.0990444	−0.0495222	−	0.998773i	$-0.515770\pi$
$607$	−2.44019	−0.0990444	−0.0495222	+	0.998773i	$0.515770\pi$
$608$	41.3244	1.67593
$609$	1.87670	0.0760476
$610$	−45.1570	−1.82835
$611$	0.466102	0.0188564
$612$	22.4830	0.908821
$613$	−9.08037	−0.366753	−0.183376	−	0.983043i	$-0.558703\pi$
$613$	−9.08037	−0.366753	−0.183376	+	0.983043i	$0.558703\pi$
$614$	13.3567	0.539034
$615$	−28.5724	−1.15215
$616$	−0.0468949	−0.00188945
$617$	−11.7093	−0.471400	−0.235700	−	0.971826i	$-0.575738\pi$
$617$	−11.7093	−0.471400	−0.235700	+	0.971826i	$0.575738\pi$
$618$	73.7792	2.96784
$619$	−1.12866	−0.0453647	−0.0226824	−	0.999743i	$-0.507221\pi$
$619$	−1.12866	−0.0453647	−0.0226824	+	0.999743i	$0.507221\pi$
$620$	21.1751	0.850412
$621$	0.413509	0.0165935
$622$	1.61783	0.0648691
$623$	−7.64472	−0.306279
$624$	−13.9722	−0.559337
$625$	−18.0061	−0.720243
$626$	49.9637	1.99695
$627$	16.3308	0.652188
$628$	14.1748	0.565637
$629$	−5.06918	−0.202121
$630$	15.0253	0.598624
$631$	−43.9511	−1.74967	−0.874833	−	0.484425i	$-0.839029\pi$
$631$	−43.9511	−1.74967	−0.874833	+	0.484425i	$0.839029\pi$
$632$	0.271664	0.0108062
$633$	−23.9571	−0.952210
$634$	30.6487	1.21721
$635$	7.75164	0.307614
$636$	−84.2608	−3.34116
$637$	7.48106	0.296410
$638$	−2.06339	−0.0816903
$639$	−14.6233	−0.578490
$640$	1.15004	0.0454594
$641$	−5.00176	−0.197558	−0.0987788	−	0.995109i	$-0.531494\pi$
$641$	−5.00176	−0.197558	−0.0987788	+	0.995109i	$0.531494\pi$
$642$	22.3778	0.883180
$643$	−21.2757	−0.839033	−0.419516	−	0.907748i	$-0.637800\pi$
$643$	−21.2757	−0.839033	−0.419516	+	0.907748i	$0.637800\pi$
$644$	−0.0514840	−0.00202875
$645$	−1.70577	−0.0671646
$646$	−19.0512	−0.749559
$647$	4.95949	0.194977	0.0974887	−	0.995237i	$-0.468919\pi$
$647$	4.95949	0.194977	0.0974887	+	0.995237i	$0.468919\pi$
$648$	0.801669	0.0314925
$649$	−6.01902	−0.236267
$650$	−2.56383	−0.100562
$651$	10.3017	0.403754
$652$	37.0009	1.44907
$653$	6.00556	0.235016	0.117508	−	0.993072i	$-0.462509\pi$
$653$	6.00556	0.235016	0.117508	+	0.993072i	$0.462509\pi$
$654$	60.8962	2.38123
$655$	−33.9070	−1.32486
$656$	19.5097	0.761725
$657$	92.2027	3.59717
$658$	0.507527	0.0197855
$659$	−2.75218	−0.107210	−0.0536048	−	0.998562i	$-0.517071\pi$
$659$	−2.75218	−0.107210	−0.0536048	+	0.998562i	$0.517071\pi$
$660$	−12.1353	−0.472366
$661$	−20.4997	−0.797348	−0.398674	−	0.917093i	$-0.630529\pi$
$661$	−20.4997	−0.797348	−0.398674	+	0.917093i	$0.630529\pi$
$662$	−40.9795	−1.59272
$663$	6.32513	0.245648
$664$	−0.137906	−0.00535179
$665$	−6.30690	−0.244571
$666$	34.0083	1.31780
$667$	0.0424119	0.00164220
$668$	33.2080	1.28486
$669$	43.1292	1.66747
$670$	−38.7836	−1.49834
$671$	11.9651	0.461909
$672$	−14.9394	−0.576299
$673$	45.5856	1.75719	0.878597	−	0.477563i	$-0.158480\pi$
$673$	45.5856	1.75719	0.878597	+	0.477563i	$0.158480\pi$
$674$	43.3827	1.67104
$675$	11.1072	0.427516
$676$	−23.0132	−0.885125
$677$	26.9528	1.03588	0.517940	−	0.855417i	$-0.326699\pi$
$677$	26.9528	1.03588	0.517940	+	0.855417i	$0.326699\pi$
$678$	90.2493	3.46600
$679$	0.612114	0.0234908
$680$	−0.265049	−0.0101642
$681$	−74.1784	−2.84252
$682$	−11.3265	−0.433713
$683$	12.1957	0.466655	0.233328	−	0.972398i	$-0.425039\pi$
$683$	12.1957	0.466655	0.233328	+	0.972398i	$0.425039\pi$
$684$	63.3130	2.42083
$685$	5.61852	0.214673
$686$	16.7625	0.639996
$687$	23.6191	0.901123
$688$	1.16473	0.0444048
$689$	−15.9854	−0.608997
$690$	0.503541	0.0191695
$691$	−47.8697	−1.82105	−0.910524	−	0.413455i	$-0.864322\pi$
$691$	−47.8697	−1.82105	−0.910524	+	0.413455i	$0.864322\pi$
$692$	−24.8780	−0.945719
$693$	−3.98123	−0.151234
$694$	−38.9065	−1.47687
$695$	1.96489	0.0745324
$696$	0.222097	0.00841857
$697$	−8.83189	−0.334532
$698$	36.6918	1.38880
$699$	85.3694	3.22897
$700$	−1.38290	−0.0522688
$701$	−45.4821	−1.71784	−0.858918	−	0.512114i	$-0.828863\pi$
$701$	−45.4821	−1.71784	−0.858918	+	0.512114i	$0.828863\pi$
$702$	−21.9422	−0.828153
$703$	−14.2750	−0.538393
$704$	7.98421	0.300916
$705$	−2.45892	−0.0926084
$706$	−24.1080	−0.907317
$707$	8.74134	0.328752
$708$	−34.6041	−1.30050
$709$	21.0823	0.791761	0.395881	−	0.918302i	$-0.370439\pi$
$709$	21.0823	0.791761	0.395881	+	0.918302i	$0.370439\pi$
$710$	−9.20785	−0.345564
$711$	23.0634	0.864947
$712$	−0.904713	−0.0339055
$713$	0.232810	0.00871880
$714$	6.88729	0.257750
$715$	−2.30224	−0.0860988
$716$	−7.06385	−0.263988
$717$	−19.1646	−0.715714
$718$	16.5777	0.618674
$719$	5.49866	0.205065	0.102533	−	0.994730i	$-0.467305\pi$
$719$	5.49866	0.205065	0.102533	+	0.994730i	$0.467305\pi$
$720$	49.7066	1.85245
$721$	7.54980	0.281169
$722$	−15.8239	−0.588905
$723$	33.3363	1.23979
$724$	26.6330	0.989809
$725$	1.13922	0.0423095
$726$	−59.9752	−2.22589
$727$	−6.24221	−0.231511	−0.115755	−	0.993278i	$-0.536929\pi$
$727$	−6.24221	−0.231511	−0.115755	+	0.993278i	$0.536929\pi$
$728$	−0.0511478	−0.00189566
$729$	−20.7181	−0.767338
$730$	58.0571	2.14879
$731$	−0.527264	−0.0195016
$732$	68.7892	2.54252
$733$	−23.9659	−0.885201	−0.442600	−	0.896719i	$-0.645944\pi$
$733$	−23.9659	−0.885201	−0.442600	+	0.896719i	$0.645944\pi$
$734$	−46.8995	−1.73109
$735$	−39.4664	−1.45574
$736$	−0.337619	−0.0124448
$737$	10.2764	0.378536
$738$	59.2518	2.18109
$739$	−35.7444	−1.31488	−0.657439	−	0.753508i	$-0.728360\pi$
$739$	−35.7444	−1.31488	−0.657439	+	0.753508i	$0.728360\pi$
$740$	10.6077	0.389946
$741$	17.8118	0.654334
$742$	−17.4062	−0.639001
$743$	15.5544	0.570636	0.285318	−	0.958433i	$-0.407901\pi$
$743$	15.5544	0.570636	0.285318	+	0.958433i	$0.407901\pi$
$744$	1.21915	0.0446962
$745$	−32.2792	−1.18262
$746$	2.58317	0.0945764
$747$	−11.7078	−0.428365
$748$	−3.75110	−0.137154
$749$	2.28991	0.0836714
$750$	72.8886	2.66152
$751$	−12.0401	−0.439350	−0.219675	−	0.975573i	$-0.570500\pi$
$751$	−12.0401	−0.439350	−0.219675	+	0.975573i	$0.570500\pi$
$752$	1.67899	0.0612265
$753$	−15.5650	−0.567221
$754$	−2.25052	−0.0819590
$755$	43.8669	1.59648
$756$	−11.8354	−0.430448
$757$	−32.8775	−1.19495	−0.597477	−	0.801886i	$-0.703830\pi$
$757$	−32.8775	−1.19495	−0.597477	+	0.801886i	$0.703830\pi$
$758$	−62.6764	−2.27651
$759$	−0.133422	−0.00484291
$760$	−0.746389	−0.0270744
$761$	−15.9564	−0.578419	−0.289210	−	0.957266i	$-0.593392\pi$
$761$	−15.9564	−0.578419	−0.289210	+	0.957266i	$0.593392\pi$
$762$	−23.8377	−0.863550
$763$	6.23149	0.225595
$764$	−6.20270	−0.224406
$765$	−22.5018	−0.813555
$766$	16.6756	0.602513
$767$	−6.56489	−0.237044
$768$	−50.2983	−1.81498
$769$	49.1859	1.77369	0.886844	−	0.462069i	$-0.152893\pi$
$769$	49.1859	1.77369	0.886844	+	0.462069i	$0.152893\pi$
$770$	−2.50685	−0.0903407
$771$	−1.14711	−0.0413123
$772$	22.8292	0.821641
$773$	−20.4919	−0.737042	−0.368521	−	0.929619i	$-0.620136\pi$
$773$	−20.4919	−0.737042	−0.368521	+	0.929619i	$0.620136\pi$
$774$	3.53733	0.127147
$775$	6.25346	0.224631
$776$	0.0724406	0.00260046
$777$	5.16063	0.185137
$778$	−20.3818	−0.730724
$779$	−24.8710	−0.891095
$780$	−13.2359	−0.473920
$781$	2.43978	0.0873022
$782$	0.155647	0.00556594
$783$	9.74983	0.348430
$784$	26.9483	0.962439
$785$	−14.1867	−0.506344
$786$	104.270	3.71920
$787$	−8.78693	−0.313220	−0.156610	−	0.987660i	$-0.550057\pi$
$787$	−8.78693	−0.313220	−0.156610	+	0.987660i	$0.550057\pi$
$788$	−19.5341	−0.695874
$789$	3.44027	0.122477
$790$	14.5223	0.516681
$791$	9.23518	0.328365
$792$	−0.471157	−0.0167418
$793$	13.0503	0.463429
$794$	−42.4815	−1.50761
$795$	84.3313	2.99092
$796$	−52.5044	−1.86097
$797$	23.2200	0.822496	0.411248	−	0.911523i	$-0.365093\pi$
$797$	23.2200	0.822496	0.411248	+	0.911523i	$0.365093\pi$
$798$	19.3949	0.686571
$799$	−0.760068	−0.0268893
$800$	−9.06872	−0.320628
$801$	−76.8072	−2.71385
$802$	57.2913	2.02303
$803$	−15.3832	−0.542863
$804$	59.0803	2.08360
$805$	0.0515271	0.00181609
$806$	−12.3537	−0.435139
$807$	9.09014	0.319988
$808$	1.03449	0.0363933
$809$	11.2625	0.395970	0.197985	−	0.980205i	$-0.436560\pi$
$809$	11.2625	0.395970	0.197985	+	0.980205i	$0.436560\pi$
$810$	42.8547	1.50576
$811$	30.3139	1.06446	0.532232	−	0.846599i	$-0.321353\pi$
$811$	30.3139	1.06446	0.532232	+	0.846599i	$0.321353\pi$
$812$	−1.21390	−0.0425997
$813$	−87.5404	−3.07017
$814$	−5.67400	−0.198874
$815$	−37.0318	−1.29717
$816$	22.7844	0.797613
$817$	−1.48480	−0.0519465
$818$	−64.4878	−2.25476
$819$	−4.34228	−0.151732
$820$	18.4815	0.645401
$821$	−30.3861	−1.06048	−0.530241	−	0.847847i	$-0.677899\pi$
$821$	−30.3861	−1.06048	−0.530241	+	0.847847i	$0.677899\pi$
$822$	−17.2780	−0.602639
$823$	9.63285	0.335780	0.167890	−	0.985806i	$-0.446305\pi$
$823$	9.63285	0.335780	0.167890	+	0.985806i	$0.446305\pi$
$824$	0.893480	0.0311258
$825$	−3.58382	−0.124773
$826$	−7.14835	−0.248723
$827$	3.32234	0.115529	0.0577646	−	0.998330i	$-0.481603\pi$
$827$	3.32234	0.115529	0.0577646	+	0.998330i	$0.481603\pi$
$828$	−0.517265	−0.0179762
$829$	41.1785	1.43019	0.715094	−	0.699028i	$-0.246384\pi$
$829$	41.1785	1.43019	0.715094	+	0.699028i	$0.246384\pi$
$830$	−7.37202	−0.255886
$831$	21.3470	0.740521
$832$	8.70830	0.301906
$833$	−12.1993	−0.422681
$834$	−6.04239	−0.209231
$835$	−33.2358	−1.15017
$836$	−10.5632	−0.365337
$837$	53.5193	1.84990
$838$	−25.8980	−0.894630
$839$	3.85828	0.133203	0.0666013	−	0.997780i	$-0.478784\pi$
$839$	3.85828	0.133203	0.0666013	+	0.997780i	$0.478784\pi$
$840$	0.269831	0.00931004
$841$	1.00000	0.0344828
$842$	63.7559	2.19717
$843$	37.4903	1.29123
$844$	15.4962	0.533401
$845$	23.0325	0.792342
$846$	5.09918	0.175313
$847$	−6.13724	−0.210878
$848$	−57.5828	−1.97740
$849$	−23.7625	−0.815528
$850$	4.18082	0.143401
$851$	0.116626	0.00399790
$852$	14.0266	0.480544
$853$	7.71719	0.264232	0.132116	−	0.991234i	$-0.457823\pi$
$853$	7.71719	0.264232	0.132116	+	0.991234i	$0.457823\pi$
$854$	14.2101	0.486261
$855$	−63.3660	−2.16707
$856$	0.270999	0.00926255
$857$	39.7219	1.35687	0.678437	−	0.734658i	$-0.262657\pi$
$857$	39.7219	1.35687	0.678437	+	0.734658i	$0.262657\pi$
$858$	7.07981	0.241701
$859$	−18.8539	−0.643286	−0.321643	−	0.946861i	$-0.604235\pi$
$859$	−18.8539	−0.643286	−0.321643	+	0.946861i	$0.604235\pi$
$860$	1.10334	0.0376237
$861$	8.99122	0.306420
$862$	−20.8869	−0.711411
$863$	−26.0840	−0.887911	−0.443955	−	0.896049i	$-0.646425\pi$
$863$	−26.0840	−0.887911	−0.443955	+	0.896049i	$0.646425\pi$
$864$	−77.6132	−2.64045
$865$	24.8988	0.846585
$866$	10.9797	0.373106
$867$	41.2836	1.40207
$868$	−6.66343	−0.226172
$869$	−3.84794	−0.130533
$870$	11.8726	0.402519
$871$	11.2084	0.379781
$872$	0.737464	0.0249737
$873$	6.14997	0.208145
$874$	0.438310	0.0148260
$875$	7.45866	0.252149
$876$	−88.4403	−2.98812
$877$	−13.4299	−0.453497	−0.226748	−	0.973953i	$-0.572809\pi$
$877$	−13.4299	−0.453497	−0.226748	+	0.973953i	$0.572809\pi$
$878$	9.74031	0.328720
$879$	28.5514	0.963016
$880$	−8.29312	−0.279561
$881$	20.9098	0.704470	0.352235	−	0.935911i	$-0.385422\pi$
$881$	20.9098	0.704470	0.352235	+	0.935911i	$0.385422\pi$
$882$	81.8432	2.75580
$883$	0.187748	0.00631821	0.00315910	−	0.999995i	$-0.498994\pi$
$883$	0.187748	0.00631821	0.00315910	+	0.999995i	$0.498994\pi$
$884$	−4.09129	−0.137605
$885$	34.6331	1.16418
$886$	20.2457	0.680167
$887$	30.2144	1.01450	0.507250	−	0.861799i	$-0.330662\pi$
$887$	30.2144	1.01450	0.507250	+	0.861799i	$0.330662\pi$
$888$	0.610734	0.0204949
$889$	−2.43931	−0.0818117
$890$	−48.3631	−1.62113
$891$	−11.3551	−0.380410
$892$	−27.8973	−0.934071
$893$	−2.14038	−0.0716252
$894$	99.2646	3.31991
$895$	7.06976	0.236316
$896$	−0.361898	−0.0120902
$897$	−0.145522	−0.00485884
$898$	3.74408	0.124941
$899$	5.48926	0.183077
$900$	−13.8942	−0.463139
$901$	26.0673	0.868429
$902$	−9.88566	−0.329157
$903$	0.536776	0.0178628
$904$	1.09294	0.0363505
$905$	−26.6553	−0.886053
$906$	−134.899	−4.48171
$907$	11.1751	0.371062	0.185531	−	0.982638i	$-0.440600\pi$
$907$	11.1751	0.371062	0.185531	+	0.982638i	$0.440600\pi$
$908$	47.9809	1.59230
$909$	87.8251	2.91298
$910$	−2.73420	−0.0906378
$911$	1.55333	0.0514641	0.0257320	−	0.999669i	$-0.491808\pi$
$911$	1.55333	0.0514641	0.0257320	+	0.999669i	$0.491808\pi$
$912$	64.1618	2.12461
$913$	1.95334	0.0646463
$914$	2.15171	0.0711723
$915$	−68.8468	−2.27600
$916$	−15.2775	−0.504784
$917$	10.6700	0.352353
$918$	35.7809	1.18095
$919$	31.0958	1.02575	0.512877	−	0.858462i	$-0.328580\pi$
$919$	31.0958	1.02575	0.512877	+	0.858462i	$0.328580\pi$
$920$	0.00609797	0.000201044 0
$921$	20.3638	0.671009
$922$	41.2801	1.35949
$923$	2.66104	0.0875894
$924$	3.81877	0.125628
$925$	3.13268	0.103002
$926$	7.36588	0.242058
$927$	75.8536	2.49136
$928$	−7.96047	−0.261315
$929$	−29.9104	−0.981327	−0.490664	−	0.871349i	$-0.663246\pi$
$929$	−29.9104	−0.981327	−0.490664	+	0.871349i	$0.663246\pi$
$930$	65.1718	2.13707
$931$	−34.3537	−1.12590
$932$	−55.2195	−1.80878
$933$	2.46656	0.0807515
$934$	46.4320	1.51930
$935$	3.75424	0.122777
$936$	−0.513887	−0.0167969
$937$	−34.1094	−1.11431	−0.557153	−	0.830410i	$-0.688106\pi$
$937$	−34.1094	−1.11431	−0.557153	+	0.830410i	$0.688106\pi$
$938$	12.2045	0.398492
$939$	76.1750	2.48588
$940$	1.59051	0.0518766
$941$	31.0678	1.01278	0.506391	−	0.862304i	$-0.330979\pi$
$941$	31.0678	1.01278	0.506391	+	0.862304i	$0.330979\pi$
$942$	43.6267	1.42143
$943$	0.203195	0.00661694
$944$	−23.6480	−0.769678
$945$	11.8453	0.385327
$946$	−0.590174	−0.0191882
$947$	2.27822	0.0740323	0.0370161	−	0.999315i	$-0.488215\pi$
$947$	2.27822	0.0740323	0.0370161	+	0.999315i	$0.488215\pi$
$948$	−22.1223	−0.718500
$949$	−16.7784	−0.544648
$950$	11.7734	0.381978
$951$	46.7272	1.51523
$952$	0.0834063	0.00270321
$953$	23.8262	0.771807	0.385904	−	0.922539i	$-0.373890\pi$
$953$	23.8262	0.771807	0.385904	+	0.922539i	$0.373890\pi$
$954$	−174.882	−5.66200
$955$	6.20790	0.200883
$956$	12.3962	0.400923
$957$	−3.14586	−0.101691
$958$	−62.7556	−2.02754
$959$	−1.76805	−0.0570933
$960$	−45.9407	−1.48273
$961$	−0.868077	−0.0280025
$962$	−6.18858	−0.199528
$963$	23.0069	0.741388
$964$	−21.5629	−0.694495
$965$	−22.8483	−0.735513
$966$	−0.158455	−0.00509822
$967$	−1.50665	−0.0484505	−0.0242253	−	0.999707i	$-0.507712\pi$
$967$	−1.50665	−0.0484505	−0.0242253	+	0.999707i	$0.507712\pi$
$968$	−0.726311	−0.0233445
$969$	−29.0456	−0.933079
$970$	3.87244	0.124337
$971$	−60.3746	−1.93751	−0.968756	−	0.248014i	$-0.920222\pi$
$971$	−60.3746	−1.93751	−0.968756	+	0.248014i	$0.920222\pi$
$972$	−7.85810	−0.252049
$973$	−0.618316	−0.0198223
$974$	−37.5875	−1.20438
$975$	−3.90884	−0.125183
$976$	47.0097	1.50474
$977$	9.69570	0.310193	0.155096	−	0.987899i	$-0.450431\pi$
$977$	9.69570	0.310193	0.155096	+	0.987899i	$0.450431\pi$
$978$	113.880	3.64147
$979$	12.8146	0.409558
$980$	25.5281	0.815464
$981$	62.6084	1.99893
$982$	0.282497	0.00901483
$983$	53.0689	1.69263	0.846317	−	0.532679i	$-0.178815\pi$
$983$	53.0689	1.69263	0.846317	+	0.532679i	$0.178815\pi$
$984$	1.06406	0.0339211
$985$	19.5505	0.622929
$986$	3.66990	0.116873
$987$	0.773780	0.0246297
$988$	−11.5212	−0.366539
$989$	0.0121307	0.000385735 0
$990$	−25.1866	−0.800482
$991$	32.4474	1.03073	0.515363	−	0.856972i	$-0.327657\pi$
$991$	32.4474	1.03073	0.515363	+	0.856972i	$0.327657\pi$
$992$	−43.6970	−1.38738
$993$	−62.4777	−1.98267
$994$	2.89755	0.0919047
$995$	52.5484	1.66589
$996$	11.2300	0.355837
$997$	−50.4827	−1.59880	−0.799402	−	0.600797i	$-0.794850\pi$
$997$	−50.4827	−1.59880	−0.799402	+	0.600797i	$0.794850\pi$
$998$	38.6864	1.22460
$999$	26.8106	0.848248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.10	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.10	✓	61	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-1.99079 q^{2} -3.03517 q^{3} +1.96324 q^{4} -1.96489 q^{5} +6.04239 q^{6} +0.618316 q^{7} +0.0731745 q^{8} +6.21228 q^{9} +O(q^{10})\)	\(q-1.99079 q^{2} -3.03517 q^{3} +1.96324 q^{4} -1.96489 q^{5} +6.04239 q^{6} +0.618316 q^{7} +0.0731745 q^{8} +6.21228 q^{9} +3.91168 q^{10} -1.03647 q^{11} -5.95879 q^{12} -1.13046 q^{13} -1.23094 q^{14} +5.96377 q^{15} -4.07216 q^{16} +1.84344 q^{17} -12.3673 q^{18} +5.19120 q^{19} -3.85755 q^{20} -1.87670 q^{21} +2.06339 q^{22} -0.0424119 q^{23} -0.222097 q^{24} -1.13922 q^{25} +2.25052 q^{26} -9.74983 q^{27} +1.21390 q^{28} -1.00000 q^{29} -11.8726 q^{30} -5.48926 q^{31} +7.96047 q^{32} +3.14586 q^{33} -3.66990 q^{34} -1.21492 q^{35} +12.1962 q^{36} -2.74985 q^{37} -10.3346 q^{38} +3.43116 q^{39} -0.143780 q^{40} -4.79099 q^{41} +3.73611 q^{42} -0.286022 q^{43} -2.03484 q^{44} -12.2064 q^{45} +0.0844332 q^{46} -0.412310 q^{47} +12.3597 q^{48} -6.61769 q^{49} +2.26795 q^{50} -5.59516 q^{51} -2.21938 q^{52} +14.1406 q^{53} +19.4099 q^{54} +2.03654 q^{55} +0.0452449 q^{56} -15.7562 q^{57} +1.99079 q^{58} +5.80725 q^{59} +11.7083 q^{60} -11.5442 q^{61} +10.9280 q^{62} +3.84115 q^{63} -7.70330 q^{64} +2.22124 q^{65} -6.26274 q^{66} -9.91482 q^{67} +3.61912 q^{68} +0.128728 q^{69} +2.41865 q^{70} -2.35394 q^{71} +0.454580 q^{72} +14.8420 q^{73} +5.47437 q^{74} +3.45773 q^{75} +10.1916 q^{76} -0.640864 q^{77} -6.83071 q^{78} +3.71256 q^{79} +8.00134 q^{80} +10.9556 q^{81} +9.53785 q^{82} -1.88462 q^{83} -3.68441 q^{84} -3.62215 q^{85} +0.569409 q^{86} +3.03517 q^{87} -0.0758429 q^{88} -12.3638 q^{89} +24.3004 q^{90} -0.698984 q^{91} -0.0832649 q^{92} +16.6608 q^{93} +0.820822 q^{94} -10.2001 q^{95} -24.1614 q^{96} +0.989970 q^{97} +13.1744 q^{98} -6.43882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more