LMFDB - Embedded newform 4029.2.a.j.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.1717269744$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.782307 q^{2} +1.00000 q^{3} -1.38800 q^{4} -1.22158 q^{5} +0.782307 q^{6} -4.37722 q^{7} -2.65045 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.782307 q^{2} +1.00000 q^{3} -1.38800 q^{4} -1.22158 q^{5} +0.782307 q^{6} -4.37722 q^{7} -2.65045 q^{8} +1.00000 q^{9} -0.955651 q^{10} +4.99582 q^{11} -1.38800 q^{12} -6.13291 q^{13} -3.42433 q^{14} -1.22158 q^{15} +0.702526 q^{16} -1.00000 q^{17} +0.782307 q^{18} +2.65366 q^{19} +1.69555 q^{20} -4.37722 q^{21} +3.90827 q^{22} -5.45308 q^{23} -2.65045 q^{24} -3.50774 q^{25} -4.79781 q^{26} +1.00000 q^{27} +6.07557 q^{28} +0.0104062 q^{29} -0.955651 q^{30} +8.85303 q^{31} +5.85050 q^{32} +4.99582 q^{33} -0.782307 q^{34} +5.34713 q^{35} -1.38800 q^{36} -0.0975274 q^{37} +2.07598 q^{38} -6.13291 q^{39} +3.23774 q^{40} +3.72791 q^{41} -3.42433 q^{42} -9.44869 q^{43} -6.93418 q^{44} -1.22158 q^{45} -4.26598 q^{46} -7.93648 q^{47} +0.702526 q^{48} +12.1601 q^{49} -2.74413 q^{50} -1.00000 q^{51} +8.51245 q^{52} -1.17673 q^{53} +0.782307 q^{54} -6.10280 q^{55} +11.6016 q^{56} +2.65366 q^{57} +0.00814086 q^{58} -0.500591 q^{59} +1.69555 q^{60} +11.9871 q^{61} +6.92579 q^{62} -4.37722 q^{63} +3.17183 q^{64} +7.49184 q^{65} +3.90827 q^{66} +14.1874 q^{67} +1.38800 q^{68} -5.45308 q^{69} +4.18309 q^{70} +11.2130 q^{71} -2.65045 q^{72} +8.66706 q^{73} -0.0762963 q^{74} -3.50774 q^{75} -3.68327 q^{76} -21.8678 q^{77} -4.79781 q^{78} -1.00000 q^{79} -0.858192 q^{80} +1.00000 q^{81} +2.91637 q^{82} +8.03271 q^{83} +6.07557 q^{84} +1.22158 q^{85} -7.39177 q^{86} +0.0104062 q^{87} -13.2412 q^{88} +0.310737 q^{89} -0.955651 q^{90} +26.8451 q^{91} +7.56886 q^{92} +8.85303 q^{93} -6.20876 q^{94} -3.24166 q^{95} +5.85050 q^{96} +11.4371 q^{97} +9.51290 q^{98} +4.99582 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 6 q^{2} + 25 q^{3} + 26 q^{4} + 6 q^{5} + 6 q^{6} + 4 q^{7} + 18 q^{8} + 25 q^{9}+O(q^{10}) $ 25 * q + 6 * q^2 + 25 * q^3 + 26 * q^4 + 6 * q^5 + 6 * q^6 + 4 * q^7 + 18 * q^8 + 25 * q^9	\( 25 q + 6 q^{2} + 25 q^{3} + 26 q^{4} + 6 q^{5} + 6 q^{6} + 4 q^{7} + 18 q^{8} + 25 q^{9} + 13 q^{10} + 19 q^{11} + 26 q^{12} + 17 q^{14} + 6 q^{15} + 16 q^{16} - 25 q^{17} + 6 q^{18} + 25 q^{19} + 32 q^{20} + 4 q^{21} - 7 q^{22} + 8 q^{23} + 18 q^{24} + 15 q^{25} + 20 q^{26} + 25 q^{27} + 9 q^{28} + 21 q^{29} + 13 q^{30} + 4 q^{31} + 27 q^{32} + 19 q^{33} - 6 q^{34} + 50 q^{35} + 26 q^{36} - 8 q^{37} + 31 q^{38} + 52 q^{40} + 40 q^{41} + 17 q^{42} + 21 q^{43} + 34 q^{44} + 6 q^{45} + 29 q^{46} + 43 q^{47} + 16 q^{48} + 21 q^{49} + 13 q^{50} - 25 q^{51} + 3 q^{52} + 44 q^{53} + 6 q^{54} + 13 q^{55} + 38 q^{56} + 25 q^{57} - 5 q^{58} + 45 q^{59} + 32 q^{60} + 22 q^{61} + 4 q^{62} + 4 q^{63} + 26 q^{64} + 43 q^{65} - 7 q^{66} + 8 q^{67} - 26 q^{68} + 8 q^{69} + 29 q^{70} + 9 q^{71} + 18 q^{72} - 7 q^{73} + 18 q^{74} + 15 q^{75} + 33 q^{76} + 20 q^{77} + 20 q^{78} - 25 q^{79} + 42 q^{80} + 25 q^{81} - 43 q^{82} + 41 q^{83} + 9 q^{84} - 6 q^{85} - 12 q^{86} + 21 q^{87} - 43 q^{88} + 68 q^{89} + 13 q^{90} + 10 q^{91} + 2 q^{92} + 4 q^{93} - 17 q^{94} + 8 q^{95} + 27 q^{96} + 15 q^{97} + 11 q^{98} + 19 q^{99}+O(q^{100}) \) 25 * q + 6 * q^2 + 25 * q^3 + 26 * q^4 + 6 * q^5 + 6 * q^6 + 4 * q^7 + 18 * q^8 + 25 * q^9 + 13 * q^10 + 19 * q^11 + 26 * q^12 + 17 * q^14 + 6 * q^15 + 16 * q^16 - 25 * q^17 + 6 * q^18 + 25 * q^19 + 32 * q^20 + 4 * q^21 - 7 * q^22 + 8 * q^23 + 18 * q^24 + 15 * q^25 + 20 * q^26 + 25 * q^27 + 9 * q^28 + 21 * q^29 + 13 * q^30 + 4 * q^31 + 27 * q^32 + 19 * q^33 - 6 * q^34 + 50 * q^35 + 26 * q^36 - 8 * q^37 + 31 * q^38 + 52 * q^40 + 40 * q^41 + 17 * q^42 + 21 * q^43 + 34 * q^44 + 6 * q^45 + 29 * q^46 + 43 * q^47 + 16 * q^48 + 21 * q^49 + 13 * q^50 - 25 * q^51 + 3 * q^52 + 44 * q^53 + 6 * q^54 + 13 * q^55 + 38 * q^56 + 25 * q^57 - 5 * q^58 + 45 * q^59 + 32 * q^60 + 22 * q^61 + 4 * q^62 + 4 * q^63 + 26 * q^64 + 43 * q^65 - 7 * q^66 + 8 * q^67 - 26 * q^68 + 8 * q^69 + 29 * q^70 + 9 * q^71 + 18 * q^72 - 7 * q^73 + 18 * q^74 + 15 * q^75 + 33 * q^76 + 20 * q^77 + 20 * q^78 - 25 * q^79 + 42 * q^80 + 25 * q^81 - 43 * q^82 + 41 * q^83 + 9 * q^84 - 6 * q^85 - 12 * q^86 + 21 * q^87 - 43 * q^88 + 68 * q^89 + 13 * q^90 + 10 * q^91 + 2 * q^92 + 4 * q^93 - 17 * q^94 + 8 * q^95 + 27 * q^96 + 15 * q^97 + 11 * q^98 + 19 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.782307	0.553174	0.276587	−	0.960989i	$-0.410797\pi$
$2$	0.782307	0.553174	0.276587	+	0.960989i	$0.410797\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.38800	−0.693998
$5$	−1.22158	−0.546307	−0.273154	−	0.961970i	$-0.588067\pi$
$5$	−1.22158	−0.546307	−0.273154	+	0.961970i	$0.588067\pi$
$6$	0.782307	0.319375
$7$	−4.37722	−1.65443	−0.827217	−	0.561883i	$-0.810077\pi$
$7$	−4.37722	−1.65443	−0.827217	+	0.561883i	$0.810077\pi$
$8$	−2.65045	−0.937076
$9$	1.00000	0.333333
$10$	−0.955651	−0.302203
$11$	4.99582	1.50630	0.753149	−	0.657851i	$-0.228534\pi$
$11$	4.99582	1.50630	0.753149	+	0.657851i	$0.228534\pi$
$12$	−1.38800	−0.400680
$13$	−6.13291	−1.70096	−0.850481	−	0.526006i	$-0.823689\pi$
$13$	−6.13291	−1.70096	−0.850481	+	0.526006i	$0.823689\pi$
$14$	−3.42433	−0.915190
$15$	−1.22158	−0.315411
$16$	0.702526	0.175631
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.782307	0.184391
$19$	2.65366	0.608792	0.304396	−	0.952546i	$-0.401545\pi$
$19$	2.65366	0.608792	0.304396	+	0.952546i	$0.401545\pi$
$20$	1.69555	0.379136
$21$	−4.37722	−0.955188
$22$	3.90827	0.833245
$23$	−5.45308	−1.13705	−0.568523	−	0.822667i	$-0.692485\pi$
$23$	−5.45308	−1.13705	−0.568523	+	0.822667i	$0.692485\pi$
$24$	−2.65045	−0.541021
$25$	−3.50774	−0.701548
$26$	−4.79781	−0.940929
$27$	1.00000	0.192450
$28$	6.07557	1.14817
$29$	0.0104062	0.00193239	0.000966194	−	1.00000i	$-0.499692\pi$
$29$	0.0104062	0.00193239	0.000966194		1.00000i	$0.499692\pi$
$30$	−0.955651	−0.174477
$31$	8.85303	1.59005	0.795026	−	0.606576i	$-0.207457\pi$
$31$	8.85303	1.59005	0.795026	+	0.606576i	$0.207457\pi$
$32$	5.85050	1.03423
$33$	4.99582	0.869661
$34$	−0.782307	−0.134164
$35$	5.34713	0.903829
$36$	−1.38800	−0.231333
$37$	−0.0975274	−0.0160334	−0.00801670	−	0.999968i	$-0.502552\pi$
$37$	−0.0975274	−0.0160334	−0.00801670	+	0.999968i	$0.502552\pi$
$38$	2.07598	0.336768
$39$	−6.13291	−0.982051
$40$	3.23774	0.511932
$41$	3.72791	0.582202	0.291101	−	0.956692i	$-0.405978\pi$
$41$	3.72791	0.582202	0.291101	+	0.956692i	$0.405978\pi$
$42$	−3.42433	−0.528385
$43$	−9.44869	−1.44091	−0.720456	−	0.693501i	$-0.756067\pi$
$43$	−9.44869	−1.44091	−0.720456	+	0.693501i	$0.756067\pi$
$44$	−6.93418	−1.04537
$45$	−1.22158	−0.182102
$46$	−4.26598	−0.628985
$47$	−7.93648	−1.15766	−0.578828	−	0.815450i	$-0.696490\pi$
$47$	−7.93648	−1.15766	−0.578828	+	0.815450i	$0.696490\pi$
$48$	0.702526	0.101401
$49$	12.1601	1.73715
$50$	−2.74413	−0.388079
$51$	−1.00000	−0.140028
$52$	8.51245	1.18046
$53$	−1.17673	−0.161637	−0.0808184	−	0.996729i	$-0.525753\pi$
$53$	−1.17673	−0.161637	−0.0808184	+	0.996729i	$0.525753\pi$
$54$	0.782307	0.106458
$55$	−6.10280	−0.822901
$56$	11.6016	1.55033
$57$	2.65366	0.351486
$58$	0.00814086	0.00106895
$59$	−0.500591	−0.0651714	−0.0325857	−	0.999469i	$-0.510374\pi$
$59$	−0.500591	−0.0651714	−0.0325857	+	0.999469i	$0.510374\pi$
$60$	1.69555	0.218894
$61$	11.9871	1.53479	0.767394	−	0.641176i	$-0.221553\pi$
$61$	11.9871	1.53479	0.767394	+	0.641176i	$0.221553\pi$
$62$	6.92579	0.879576
$63$	−4.37722	−0.551478
$64$	3.17183	0.396479
$65$	7.49184	0.929248
$66$	3.90827	0.481074
$67$	14.1874	1.73327	0.866633	−	0.498945i	$-0.166279\pi$
$67$	14.1874	1.73327	0.866633	+	0.498945i	$0.166279\pi$
$68$	1.38800	0.168319
$69$	−5.45308	−0.656474
$70$	4.18309	0.499975
$71$	11.2130	1.33074	0.665369	−	0.746515i	$-0.268274\pi$
$71$	11.2130	1.33074	0.665369	+	0.746515i	$0.268274\pi$
$72$	−2.65045	−0.312359
$73$	8.66706	1.01440	0.507202	−	0.861827i	$-0.330680\pi$
$73$	8.66706	1.01440	0.507202	+	0.861827i	$0.330680\pi$
$74$	−0.0762963	−0.00886927
$75$	−3.50774	−0.405039
$76$	−3.68327	−0.422500
$77$	−21.8678	−2.49207
$78$	−4.79781	−0.543245
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	−0.858192	−0.0959487
$81$	1.00000	0.111111
$82$	2.91637	0.322059
$83$	8.03271	0.881705	0.440852	−	0.897580i	$-0.354676\pi$
$83$	8.03271	0.881705	0.440852	+	0.897580i	$0.354676\pi$
$84$	6.07557	0.662899
$85$	1.22158	0.132499
$86$	−7.39177	−0.797076
$87$	0.0104062	0.00111566
$88$	−13.2412	−1.41152
$89$	0.310737	0.0329381	0.0164690	−	0.999864i	$-0.494758\pi$
$89$	0.310737	0.0329381	0.0164690	+	0.999864i	$0.494758\pi$
$90$	−0.955651	−0.100734
$91$	26.8451	2.81413
$92$	7.56886	0.789108
$93$	8.85303	0.918016
$94$	−6.20876	−0.640385
$95$	−3.24166	−0.332587
$96$	5.85050	0.597114
$97$	11.4371	1.16127	0.580633	−	0.814166i	$-0.302805\pi$
$97$	11.4371	1.16127	0.580633	+	0.814166i	$0.302805\pi$
$98$	9.51290	0.960948
$99$	4.99582	0.502099
$100$	4.86873	0.486873
$101$	−5.84010	−0.581111	−0.290556	−	0.956858i	$-0.593840\pi$
$101$	−5.84010	−0.581111	−0.290556	+	0.956858i	$0.593840\pi$
$102$	−0.782307	−0.0774599
$103$	−2.62656	−0.258803	−0.129401	−	0.991592i	$-0.541306\pi$
$103$	−2.62656	−0.258803	−0.129401	+	0.991592i	$0.541306\pi$
$104$	16.2550	1.59393
$105$	5.34713	0.521826
$106$	−0.920567	−0.0894134
$107$	0.0459284	0.00444007	0.00222003	−	0.999998i	$-0.499293\pi$
$107$	0.0459284	0.00444007	0.00222003	+	0.999998i	$0.499293\pi$
$108$	−1.38800	−0.133560
$109$	−3.37105	−0.322888	−0.161444	−	0.986882i	$-0.551615\pi$
$109$	−3.37105	−0.322888	−0.161444	+	0.986882i	$0.551615\pi$
$110$	−4.77426	−0.455208
$111$	−0.0975274	−0.00925689
$112$	−3.07511	−0.290571
$113$	−4.68436	−0.440668	−0.220334	−	0.975425i	$-0.570715\pi$
$113$	−4.68436	−0.440668	−0.220334	+	0.975425i	$0.570715\pi$
$114$	2.07598	0.194433
$115$	6.66138	0.621177
$116$	−0.0144438	−0.00134107
$117$	−6.13291	−0.566987
$118$	−0.391616	−0.0360511
$119$	4.37722	0.401259
$120$	3.23774	0.295564
$121$	13.9582	1.26893
$122$	9.37757	0.849005
$123$	3.72791	0.336135
$124$	−12.2880	−1.10349
$125$	10.3929	0.929568
$126$	−3.42433	−0.305063
$127$	1.85778	0.164851	0.0824257	−	0.996597i	$-0.473733\pi$
$127$	1.85778	0.164851	0.0824257	+	0.996597i	$0.473733\pi$
$128$	−9.21965	−0.814909
$129$	−9.44869	−0.831911
$130$	5.86091	0.514036
$131$	−4.44758	−0.388586	−0.194293	−	0.980943i	$-0.562241\pi$
$131$	−4.44758	−0.388586	−0.194293	+	0.980943i	$0.562241\pi$
$132$	−6.93418	−0.603543
$133$	−11.6157	−1.00721
$134$	11.0989	0.958799
$135$	−1.22158	−0.105137
$136$	2.65045	0.227274
$137$	0.133961	0.0114451	0.00572254	−	0.999984i	$-0.498178\pi$
$137$	0.133961	0.0114451	0.00572254	+	0.999984i	$0.498178\pi$
$138$	−4.26598	−0.363145
$139$	4.71679	0.400073	0.200036	−	0.979788i	$-0.435894\pi$
$139$	4.71679	0.400073	0.200036	+	0.979788i	$0.435894\pi$
$140$	−7.42179	−0.627256
$141$	−7.93648	−0.668372
$142$	8.77200	0.736130
$143$	−30.6389	−2.56215
$144$	0.702526	0.0585438
$145$	−0.0127120	−0.00105568
$146$	6.78030	0.561142
$147$	12.1601	1.00294
$148$	0.135368	0.0111272
$149$	16.1434	1.32252	0.661260	−	0.750157i	$-0.270022\pi$
$149$	16.1434	1.32252	0.661260	+	0.750157i	$0.270022\pi$
$150$	−2.74413	−0.224057
$151$	−13.2517	−1.07841	−0.539204	−	0.842175i	$-0.681275\pi$
$151$	−13.2517	−1.07841	−0.539204	+	0.842175i	$0.681275\pi$
$152$	−7.03341	−0.570485
$153$	−1.00000	−0.0808452
$154$	−17.1073	−1.37855
$155$	−10.8147	−0.868657
$156$	8.51245	0.681541
$157$	15.6578	1.24963	0.624814	−	0.780774i	$-0.285175\pi$
$157$	15.6578	1.24963	0.624814	+	0.780774i	$0.285175\pi$
$158$	−0.782307	−0.0622370
$159$	−1.17673	−0.0933211
$160$	−7.14685	−0.565008
$161$	23.8694	1.88117
$162$	0.782307	0.0614638
$163$	8.43386	0.660591	0.330296	−	0.943878i	$-0.392852\pi$
$163$	8.43386	0.660591	0.330296	+	0.943878i	$0.392852\pi$
$164$	−5.17433	−0.404047
$165$	−6.10280	−0.475102
$166$	6.28404	0.487736
$167$	−4.23429	−0.327659	−0.163830	−	0.986489i	$-0.552385\pi$
$167$	−4.23429	−0.327659	−0.163830	+	0.986489i	$0.552385\pi$
$168$	11.6016	0.895084
$169$	24.6125	1.89327
$170$	0.955651	0.0732950
$171$	2.65366	0.202931
$172$	13.1147	0.999990
$173$	−7.29649	−0.554742	−0.277371	−	0.960763i	$-0.589463\pi$
$173$	−7.29649	−0.554742	−0.277371	+	0.960763i	$0.589463\pi$
$174$	0.00814086	0.000617157 0
$175$	15.3542	1.16067
$176$	3.50969	0.264553
$177$	−0.500591	−0.0376267
$178$	0.243092	0.0182205
$179$	−8.67736	−0.648577	−0.324288	−	0.945958i	$-0.605125\pi$
$179$	−8.67736	−0.648577	−0.324288	+	0.945958i	$0.605125\pi$
$180$	1.69555	0.126379
$181$	−10.6265	−0.789865	−0.394932	−	0.918710i	$-0.629232\pi$
$181$	−10.6265	−0.789865	−0.394932	+	0.918710i	$0.629232\pi$
$182$	21.0011	1.55670
$183$	11.9871	0.886110
$184$	14.4531	1.06550
$185$	0.119138	0.00875917
$186$	6.92579	0.507823
$187$	−4.99582	−0.365331
$188$	11.0158	0.803410
$189$	−4.37722	−0.318396
$190$	−2.53597	−0.183979
$191$	2.27510	0.164620	0.0823101	−	0.996607i	$-0.473770\pi$
$191$	2.27510	0.164620	0.0823101	+	0.996607i	$0.473770\pi$
$192$	3.17183	0.228907
$193$	16.3549	1.17725	0.588626	−	0.808405i	$-0.299669\pi$
$193$	16.3549	1.17725	0.588626	+	0.808405i	$0.299669\pi$
$194$	8.94735	0.642382
$195$	7.49184	0.536502
$196$	−16.8781	−1.20558
$197$	27.6431	1.96949	0.984746	−	0.173996i	$-0.0556679\pi$
$197$	27.6431	1.96949	0.984746	+	0.173996i	$0.0556679\pi$
$198$	3.90827	0.277748
$199$	−6.70016	−0.474962	−0.237481	−	0.971392i	$-0.576322\pi$
$199$	−6.70016	−0.474962	−0.237481	+	0.971392i	$0.576322\pi$
$200$	9.29710	0.657404
$201$	14.1874	1.00070
$202$	−4.56875	−0.321456
$203$	−0.0455503	−0.00319701
$204$	1.38800	0.0971792
$205$	−4.55394	−0.318061
$206$	−2.05478	−0.143163
$207$	−5.45308	−0.379016
$208$	−4.30852	−0.298742
$209$	13.2572	0.917021
$210$	4.18309	0.288661
$211$	8.18209	0.563278	0.281639	−	0.959520i	$-0.409122\pi$
$211$	8.18209	0.563278	0.281639	+	0.959520i	$0.409122\pi$
$212$	1.63330	0.112176
$213$	11.2130	0.768302
$214$	0.0359301	0.00245613
$215$	11.5423	0.787181
$216$	−2.65045	−0.180340
$217$	−38.7517	−2.63063
$218$	−2.63719	−0.178613
$219$	8.66706	0.585666
$220$	8.47066	0.571092
$221$	6.13291	0.412544
$222$	−0.0762963	−0.00512067
$223$	2.33087	0.156087	0.0780433	−	0.996950i	$-0.475133\pi$
$223$	2.33087	0.156087	0.0780433	+	0.996950i	$0.475133\pi$
$224$	−25.6089	−1.71107
$225$	−3.50774	−0.233849
$226$	−3.66461	−0.243766
$227$	11.0937	0.736312	0.368156	−	0.929764i	$-0.379989\pi$
$227$	11.0937	0.736312	0.368156	+	0.929764i	$0.379989\pi$
$228$	−3.68327	−0.243931
$229$	−6.79443	−0.448989	−0.224494	−	0.974475i	$-0.572073\pi$
$229$	−6.79443	−0.448989	−0.224494	+	0.974475i	$0.572073\pi$
$230$	5.21124	0.343619
$231$	−21.8678	−1.43880
$232$	−0.0275812	−0.00181079
$233$	25.9571	1.70051	0.850254	−	0.526372i	$-0.176448\pi$
$233$	25.9571	1.70051	0.850254	+	0.526372i	$0.176448\pi$
$234$	−4.79781	−0.313643
$235$	9.69505	0.632435
$236$	0.694818	0.0452288
$237$	−1.00000	−0.0649570
$238$	3.42433	0.221966
$239$	−2.22663	−0.144029	−0.0720143	−	0.997404i	$-0.522943\pi$
$239$	−2.22663	−0.144029	−0.0720143	+	0.997404i	$0.522943\pi$
$240$	−0.858192	−0.0553960
$241$	−21.7463	−1.40080	−0.700401	−	0.713750i	$-0.746995\pi$
$241$	−21.7463	−1.40080	−0.700401	+	0.713750i	$0.746995\pi$
$242$	10.9196	0.701940
$243$	1.00000	0.0641500
$244$	−16.6380	−1.06514
$245$	−14.8545	−0.949018
$246$	2.91637	0.185941
$247$	−16.2747	−1.03553
$248$	−23.4645	−1.49000
$249$	8.03271	0.509052
$250$	8.13043	0.514213
$251$	−13.3387	−0.841933	−0.420966	−	0.907076i	$-0.638309\pi$
$251$	−13.3387	−0.841933	−0.420966	+	0.907076i	$0.638309\pi$
$252$	6.07557	0.382725
$253$	−27.2426	−1.71273
$254$	1.45335	0.0911916
$255$	1.22158	0.0764983
$256$	−13.5563	−0.847266
$257$	2.18520	0.136309	0.0681547	−	0.997675i	$-0.478289\pi$
$257$	2.18520	0.136309	0.0681547	+	0.997675i	$0.478289\pi$
$258$	−7.39177	−0.460192
$259$	0.426899	0.0265262
$260$	−10.3986	−0.644896
$261$	0.0104062	0.000644129 0
$262$	−3.47937	−0.214956
$263$	12.0026	0.740109	0.370054	−	0.929010i	$-0.379339\pi$
$263$	12.0026	0.740109	0.370054	+	0.929010i	$0.379339\pi$
$264$	−13.2412	−0.814939
$265$	1.43748	0.0883034
$266$	−9.08701	−0.557161
$267$	0.310737	0.0190168
$268$	−19.6921	−1.20288
$269$	−12.5232	−0.763554	−0.381777	−	0.924255i	$-0.624688\pi$
$269$	−12.5232	−0.763554	−0.381777	+	0.924255i	$0.624688\pi$
$270$	−0.955651	−0.0581590
$271$	7.38672	0.448711	0.224356	−	0.974507i	$-0.427972\pi$
$271$	7.38672	0.448711	0.224356	+	0.974507i	$0.427972\pi$
$272$	−0.702526	−0.0425969
$273$	26.8451	1.62474
$274$	0.104799	0.00633112
$275$	−17.5241	−1.05674
$276$	7.56886	0.455592
$277$	−14.1184	−0.848293	−0.424147	−	0.905594i	$-0.639426\pi$
$277$	−14.1184	−0.848293	−0.424147	+	0.905594i	$0.639426\pi$
$278$	3.68998	0.221310
$279$	8.85303	0.530017
$280$	−14.1723	−0.846957
$281$	7.36176	0.439166	0.219583	−	0.975594i	$-0.429530\pi$
$281$	7.36176	0.439166	0.219583	+	0.975594i	$0.429530\pi$
$282$	−6.20876	−0.369727
$283$	−23.7506	−1.41182	−0.705912	−	0.708300i	$-0.749463\pi$
$283$	−23.7506	−1.41182	−0.705912	+	0.708300i	$0.749463\pi$
$284$	−15.5636	−0.923530
$285$	−3.24166	−0.192019
$286$	−23.9690	−1.41732
$287$	−16.3179	−0.963215
$288$	5.85050	0.344744
$289$	1.00000	0.0588235
$290$	−0.00994472	−0.000583974 0
$291$	11.4371	0.670457
$292$	−12.0299	−0.703994
$293$	10.4712	0.611734	0.305867	−	0.952074i	$-0.401054\pi$
$293$	10.4712	0.611734	0.305867	+	0.952074i	$0.401054\pi$
$294$	9.51290	0.554803
$295$	0.611512	0.0356036
$296$	0.258492	0.0150245
$297$	4.99582	0.289887
$298$	12.6291	0.731585
$299$	33.4432	1.93407
$300$	4.86873	0.281096
$301$	41.3590	2.38389
$302$	−10.3669	−0.596548
$303$	−5.84010	−0.335505
$304$	1.86427	0.106923
$305$	−14.6432	−0.838466
$306$	−0.782307	−0.0447215
$307$	−3.71235	−0.211875	−0.105938	−	0.994373i	$-0.533784\pi$
$307$	−3.71235	−0.211875	−0.105938	+	0.994373i	$0.533784\pi$
$308$	30.3524	1.72949
$309$	−2.62656	−0.149420
$310$	−8.46040	−0.480519
$311$	−26.6037	−1.50856	−0.754278	−	0.656556i	$-0.772013\pi$
$311$	−26.6037	−1.50856	−0.754278	+	0.656556i	$0.772013\pi$
$312$	16.2550	0.920257
$313$	−28.2075	−1.59438	−0.797190	−	0.603729i	$-0.793681\pi$
$313$	−28.2075	−1.59438	−0.797190	+	0.603729i	$0.793681\pi$
$314$	12.2492	0.691262
$315$	5.34713	0.301276
$316$	1.38800	0.0780809
$317$	−11.2060	−0.629391	−0.314695	−	0.949193i	$-0.601902\pi$
$317$	−11.2060	−0.629391	−0.314695	+	0.949193i	$0.601902\pi$
$318$	−0.920567	−0.0516228
$319$	0.0519877	0.00291075
$320$	−3.87465	−0.216599
$321$	0.0459284	0.00256347
$322$	18.6732	1.04061
$323$	−2.65366	−0.147654
$324$	−1.38800	−0.0771109
$325$	21.5126	1.19331
$326$	6.59787	0.365422
$327$	−3.37105	−0.186419
$328$	−9.88065	−0.545568
$329$	34.7397	1.91526
$330$	−4.77426	−0.262814
$331$	14.0636	0.773003	0.386502	−	0.922289i	$-0.373683\pi$
$331$	14.0636	0.773003	0.386502	+	0.922289i	$0.373683\pi$
$332$	−11.1494	−0.611901
$333$	−0.0975274	−0.00534447
$334$	−3.31251	−0.181253
$335$	−17.3310	−0.946896
$336$	−3.07511	−0.167761
$337$	16.3194	0.888974	0.444487	−	0.895785i	$-0.353386\pi$
$337$	16.3194	0.888974	0.444487	+	0.895785i	$0.353386\pi$
$338$	19.2545	1.04731
$339$	−4.68436	−0.254420
$340$	−1.69555	−0.0919540
$341$	44.2282	2.39509
$342$	2.07598	0.112256
$343$	−22.5867	−1.21957
$344$	25.0433	1.35024
$345$	6.66138	0.358637
$346$	−5.70810	−0.306869
$347$	20.0069	1.07403	0.537014	−	0.843573i	$-0.319552\pi$
$347$	20.0069	1.07403	0.537014	+	0.843573i	$0.319552\pi$
$348$	−0.0144438	−0.000774269 0
$349$	−2.40976	−0.128991	−0.0644957	−	0.997918i	$-0.520544\pi$
$349$	−2.40976	−0.128991	−0.0644957	+	0.997918i	$0.520544\pi$
$350$	12.0117	0.642050
$351$	−6.13291	−0.327350
$352$	29.2280	1.55786
$353$	−21.8484	−1.16288	−0.581438	−	0.813591i	$-0.697510\pi$
$353$	−21.8484	−1.16288	−0.581438	+	0.813591i	$0.697510\pi$
$354$	−0.391616	−0.0208141
$355$	−13.6976	−0.726992
$356$	−0.431302	−0.0228590
$357$	4.37722	0.231667
$358$	−6.78836	−0.358776
$359$	−32.5925	−1.72016	−0.860082	−	0.510156i	$-0.829588\pi$
$359$	−32.5925	−1.72016	−0.860082	+	0.510156i	$0.829588\pi$
$360$	3.23774	0.170644
$361$	−11.9581	−0.629372
$362$	−8.31321	−0.436933
$363$	13.9582	0.732618
$364$	−37.2609	−1.95300
$365$	−10.5875	−0.554176
$366$	9.37757	0.490173
$367$	−29.8211	−1.55665	−0.778323	−	0.627864i	$-0.783930\pi$
$367$	−29.8211	−1.55665	−0.778323	+	0.627864i	$0.783930\pi$
$368$	−3.83093	−0.199701
$369$	3.72791	0.194067
$370$	0.0932021	0.00484535
$371$	5.15083	0.267418
$372$	−12.2880	−0.637102
$373$	9.94660	0.515015	0.257508	−	0.966276i	$-0.417099\pi$
$373$	9.94660	0.515015	0.257508	+	0.966276i	$0.417099\pi$
$374$	−3.90827	−0.202092
$375$	10.3929	0.536687
$376$	21.0353	1.08481
$377$	−0.0638204	−0.00328692
$378$	−3.42433	−0.176128
$379$	−28.8886	−1.48391	−0.741954	−	0.670451i	$-0.766101\pi$
$379$	−28.8886	−1.48391	−0.741954	+	0.670451i	$0.766101\pi$
$380$	4.49941	0.230815
$381$	1.85778	0.0951770
$382$	1.77982	0.0910637
$383$	18.7653	0.958864	0.479432	−	0.877579i	$-0.340843\pi$
$383$	18.7653	0.958864	0.479432	+	0.877579i	$0.340843\pi$
$384$	−9.21965	−0.470488
$385$	26.7133	1.36144
$386$	12.7946	0.651226
$387$	−9.44869	−0.480304
$388$	−15.8747	−0.805916
$389$	−28.5324	−1.44665	−0.723326	−	0.690507i	$-0.757387\pi$
$389$	−28.5324	−1.44665	−0.723326	+	0.690507i	$0.757387\pi$
$390$	5.86091	0.296779
$391$	5.45308	0.275774
$392$	−32.2297	−1.62784
$393$	−4.44758	−0.224350
$394$	21.6254	1.08947
$395$	1.22158	0.0614644
$396$	−6.93418	−0.348456
$397$	13.3623	0.670633	0.335316	−	0.942106i	$-0.391157\pi$
$397$	13.3623	0.670633	0.335316	+	0.942106i	$0.391157\pi$
$398$	−5.24158	−0.262737
$399$	−11.6157	−0.581511
$400$	−2.46428	−0.123214
$401$	4.59531	0.229479	0.114739	−	0.993396i	$-0.463397\pi$
$401$	4.59531	0.229479	0.114739	+	0.993396i	$0.463397\pi$
$402$	11.0989	0.553563
$403$	−54.2948	−2.70462
$404$	8.10603	0.403290
$405$	−1.22158	−0.0607008
$406$	−0.0356343	−0.00176850
$407$	−0.487230	−0.0241511
$408$	2.65045	0.131217
$409$	14.6912	0.726435	0.363217	−	0.931704i	$-0.381678\pi$
$409$	14.6912	0.726435	0.363217	+	0.931704i	$0.381678\pi$
$410$	−3.56258	−0.175943
$411$	0.133961	0.00660781
$412$	3.64566	0.179609
$413$	2.19120	0.107822
$414$	−4.26598	−0.209662
$415$	−9.81260	−0.481682
$416$	−35.8805	−1.75919
$417$	4.71679	0.230982
$418$	10.3712	0.507273
$419$	15.6903	0.766520	0.383260	−	0.923641i	$-0.374801\pi$
$419$	15.6903	0.766520	0.383260	+	0.923641i	$0.374801\pi$
$420$	−7.42179	−0.362146
$421$	5.84185	0.284714	0.142357	−	0.989815i	$-0.454532\pi$
$421$	5.84185	0.284714	0.142357	+	0.989815i	$0.454532\pi$
$422$	6.40091	0.311591
$423$	−7.93648	−0.385885
$424$	3.11888	0.151466
$425$	3.50774	0.170150
$426$	8.77200	0.425005
$427$	−52.4701	−2.53920
$428$	−0.0637484	−0.00308140
$429$	−30.6389	−1.47926
$430$	9.02965	0.435448
$431$	9.00499	0.433755	0.216877	−	0.976199i	$-0.430413\pi$
$431$	9.00499	0.433755	0.216877	+	0.976199i	$0.430413\pi$
$432$	0.702526	0.0338003
$433$	−18.6699	−0.897220	−0.448610	−	0.893728i	$-0.648081\pi$
$433$	−18.6699	−0.897220	−0.448610	+	0.893728i	$0.648081\pi$
$434$	−30.3157	−1.45520
$435$	−0.0127120	−0.000609496 0
$436$	4.67900	0.224083
$437$	−14.4706	−0.692225
$438$	6.78030	0.323975
$439$	27.9072	1.33194	0.665969	−	0.745979i	$-0.268018\pi$
$439$	27.9072	1.33194	0.665969	+	0.745979i	$0.268018\pi$
$440$	16.1752	0.771121
$441$	12.1601	0.579050
$442$	4.79781	0.228209
$443$	−38.9206	−1.84917	−0.924586	−	0.380972i	$-0.875589\pi$
$443$	−38.9206	−1.84917	−0.924586	+	0.380972i	$0.875589\pi$
$444$	0.135368	0.00642426
$445$	−0.379590	−0.0179943
$446$	1.82346	0.0863431
$447$	16.1434	0.763558
$448$	−13.8838	−0.655948
$449$	30.4843	1.43864	0.719321	−	0.694678i	$-0.244453\pi$
$449$	30.4843	1.43864	0.719321	+	0.694678i	$0.244453\pi$
$450$	−2.74413	−0.129360
$451$	18.6240	0.876969
$452$	6.50187	0.305822
$453$	−13.2517	−0.622620
$454$	8.67865	0.407309
$455$	−32.7934	−1.53738
$456$	−7.03341	−0.329369
$457$	12.5016	0.584802	0.292401	−	0.956296i	$-0.405546\pi$
$457$	12.5016	0.584802	0.292401	+	0.956296i	$0.405546\pi$
$458$	−5.31533	−0.248369
$459$	−1.00000	−0.0466760
$460$	−9.24597	−0.431096
$461$	23.8290	1.10983	0.554914	−	0.831907i	$-0.312751\pi$
$461$	23.8290	1.10983	0.554914	+	0.831907i	$0.312751\pi$
$462$	−17.1073	−0.795905
$463$	36.5022	1.69640	0.848200	−	0.529676i	$-0.177687\pi$
$463$	36.5022	1.69640	0.848200	+	0.529676i	$0.177687\pi$
$464$	0.00731064	0.000339388 0
$465$	−10.8147	−0.501519
$466$	20.3064	0.940678
$467$	27.6858	1.28114	0.640572	−	0.767898i	$-0.278697\pi$
$467$	27.6858	1.28114	0.640572	+	0.767898i	$0.278697\pi$
$468$	8.51245	0.393488
$469$	−62.1014	−2.86758
$470$	7.58450	0.349847
$471$	15.6578	0.721473
$472$	1.32679	0.0610706
$473$	−47.2040	−2.17044
$474$	−0.782307	−0.0359325
$475$	−9.30836	−0.427097
$476$	−6.07557	−0.278473
$477$	−1.17673	−0.0538790
$478$	−1.74191	−0.0796730
$479$	12.0551	0.550813	0.275407	−	0.961328i	$-0.411188\pi$
$479$	12.0551	0.550813	0.275407	+	0.961328i	$0.411188\pi$
$480$	−7.14685	−0.326208
$481$	0.598126	0.0272722
$482$	−17.0123	−0.774887
$483$	23.8694	1.08609
$484$	−19.3740	−0.880636
$485$	−13.9714	−0.634408
$486$	0.782307	0.0354862
$487$	−22.1408	−1.00330	−0.501649	−	0.865071i	$-0.667273\pi$
$487$	−22.1408	−1.00330	−0.501649	+	0.865071i	$0.667273\pi$
$488$	−31.7712	−1.43821
$489$	8.43386	0.381392
$490$	−11.6208	−0.524973
$491$	−34.2705	−1.54661	−0.773304	−	0.634035i	$-0.781397\pi$
$491$	−34.2705	−1.54661	−0.773304	+	0.634035i	$0.781397\pi$
$492$	−5.17433	−0.233277
$493$	−0.0104062	−0.000468673 0
$494$	−12.7318	−0.572830
$495$	−6.10280	−0.274300
$496$	6.21948	0.279263
$497$	−49.0818	−2.20162
$498$	6.28404	0.281595
$499$	38.5284	1.72477	0.862385	−	0.506254i	$-0.168970\pi$
$499$	38.5284	1.72477	0.862385	+	0.506254i	$0.168970\pi$
$500$	−14.4253	−0.645119
$501$	−4.23429	−0.189174
$502$	−10.4350	−0.465736
$503$	27.5842	1.22992	0.614959	−	0.788559i	$-0.289172\pi$
$503$	27.5842	1.22992	0.614959	+	0.788559i	$0.289172\pi$
$504$	11.6016	0.516777
$505$	7.13415	0.317465
$506$	−21.3121	−0.947438
$507$	24.6125	1.09308
$508$	−2.57859	−0.114407
$509$	−12.9094	−0.572200	−0.286100	−	0.958200i	$-0.592359\pi$
$509$	−12.9094	−0.572200	−0.286100	+	0.958200i	$0.592359\pi$
$510$	0.955651	0.0423169
$511$	−37.9377	−1.67826
$512$	7.83414	0.346224
$513$	2.65366	0.117162
$514$	1.70950	0.0754028
$515$	3.20856	0.141386
$516$	13.1147	0.577345
$517$	−39.6493	−1.74377
$518$	0.333966	0.0146736
$519$	−7.29649	−0.320281
$520$	−19.8568	−0.870776
$521$	14.7776	0.647419	0.323709	−	0.946157i	$-0.395070\pi$
$521$	14.7776	0.647419	0.323709	+	0.946157i	$0.395070\pi$
$522$	0.00814086	0.000356316 0
$523$	−2.47651	−0.108290	−0.0541451	−	0.998533i	$-0.517243\pi$
$523$	−2.47651	−0.108290	−0.0541451	+	0.998533i	$0.517243\pi$
$524$	6.17322	0.269678
$525$	15.3542	0.670110
$526$	9.38968	0.409409
$527$	−8.85303	−0.385644
$528$	3.50969	0.152740
$529$	6.73613	0.292875
$530$	1.12455	0.0488472
$531$	−0.500591	−0.0217238
$532$	16.1225	0.698999
$533$	−22.8629	−0.990304
$534$	0.243092	0.0105196
$535$	−0.0561052	−0.00242564
$536$	−37.6030	−1.62420
$537$	−8.67736	−0.374456
$538$	−9.79699	−0.422378
$539$	60.7495	2.61667
$540$	1.69555	0.0729648
$541$	9.78524	0.420700	0.210350	−	0.977626i	$-0.432540\pi$
$541$	9.78524	0.420700	0.210350	+	0.977626i	$0.432540\pi$
$542$	5.77868	0.248216
$543$	−10.6265	−0.456029
$544$	−5.85050	−0.250838
$545$	4.11800	0.176396
$546$	21.0011	0.898763
$547$	−25.5682	−1.09322	−0.546609	−	0.837388i	$-0.684081\pi$
$547$	−25.5682	−1.09322	−0.546609	+	0.837388i	$0.684081\pi$
$548$	−0.185938	−0.00794286
$549$	11.9871	0.511596
$550$	−13.7092	−0.584562
$551$	0.0276146	0.00117642
$552$	14.4531	0.615166
$553$	4.37722	0.186138
$554$	−11.0449	−0.469254
$555$	0.119138	0.00505711
$556$	−6.54688	−0.277650
$557$	0.650061	0.0275440	0.0137720	−	0.999905i	$-0.495616\pi$
$557$	0.650061	0.0275440	0.0137720	+	0.999905i	$0.495616\pi$
$558$	6.92579	0.293192
$559$	57.9479	2.45094
$560$	3.75649	0.158741
$561$	−4.99582	−0.210924
$562$	5.75916	0.242935
$563$	10.7297	0.452201	0.226100	−	0.974104i	$-0.427402\pi$
$563$	10.7297	0.452201	0.226100	+	0.974104i	$0.427402\pi$
$564$	11.0158	0.463849
$565$	5.72232	0.240740
$566$	−18.5802	−0.780985
$567$	−4.37722	−0.183826
$568$	−29.7195	−1.24700
$569$	27.9819	1.17306	0.586531	−	0.809927i	$-0.300493\pi$
$569$	27.9819	1.17306	0.586531	+	0.809927i	$0.300493\pi$
$570$	−2.53597	−0.106220
$571$	30.3737	1.27110	0.635550	−	0.772060i	$-0.280774\pi$
$571$	30.3737	1.27110	0.635550	+	0.772060i	$0.280774\pi$
$572$	42.5267	1.77813
$573$	2.27510	0.0950435
$574$	−12.7656	−0.532826
$575$	19.1280	0.797693
$576$	3.17183	0.132160
$577$	10.8744	0.452705	0.226353	−	0.974045i	$-0.427320\pi$
$577$	10.8744	0.452705	0.226353	+	0.974045i	$0.427320\pi$
$578$	0.782307	0.0325397
$579$	16.3549	0.679687
$580$	0.0176443	0.000732638 0
$581$	−35.1609	−1.45872
$582$	8.94735	0.370880
$583$	−5.87876	−0.243473
$584$	−22.9716	−0.950573
$585$	7.49184	0.309749
$586$	8.19169	0.338395
$587$	−3.08837	−0.127471	−0.0637353	−	0.997967i	$-0.520301\pi$
$587$	−3.08837	−0.127471	−0.0637353	+	0.997967i	$0.520301\pi$
$588$	−16.8781	−0.696042
$589$	23.4930	0.968010
$590$	0.478390	0.0196950
$591$	27.6431	1.13709
$592$	−0.0685155	−0.00281597
$593$	−20.2125	−0.830027	−0.415014	−	0.909815i	$-0.636223\pi$
$593$	−20.2125	−0.830027	−0.415014	+	0.909815i	$0.636223\pi$
$594$	3.90827	0.160358
$595$	−5.34713	−0.219211
$596$	−22.4070	−0.917827
$597$	−6.70016	−0.274219
$598$	26.1629	1.06988
$599$	25.5708	1.04479	0.522397	−	0.852702i	$-0.325038\pi$
$599$	25.5708	1.04479	0.522397	+	0.852702i	$0.325038\pi$
$600$	9.29710	0.379553
$601$	−17.4710	−0.712657	−0.356328	−	0.934361i	$-0.615972\pi$
$601$	−17.4710	−0.712657	−0.356328	+	0.934361i	$0.615972\pi$
$602$	32.3554	1.31871
$603$	14.1874	0.577756
$604$	18.3933	0.748414
$605$	−17.0511	−0.693226
$606$	−4.56875	−0.185593
$607$	38.3048	1.55474	0.777371	−	0.629042i	$-0.216553\pi$
$607$	38.3048	1.55474	0.777371	+	0.629042i	$0.216553\pi$
$608$	15.5252	0.629632
$609$	−0.0455503	−0.00184579
$610$	−11.4555	−0.463818
$611$	48.6737	1.96913
$612$	1.38800	0.0561064
$613$	42.0277	1.69748	0.848741	−	0.528809i	$-0.177361\pi$
$613$	42.0277	1.69748	0.848741	+	0.528809i	$0.177361\pi$
$614$	−2.90420	−0.117204
$615$	−4.55394	−0.183633
$616$	57.9596	2.33526
$617$	20.6741	0.832309	0.416154	−	0.909294i	$-0.363378\pi$
$617$	20.6741	0.832309	0.416154	+	0.909294i	$0.363378\pi$
$618$	−2.05478	−0.0826552
$619$	46.9816	1.88835	0.944176	−	0.329442i	$-0.106861\pi$
$619$	46.9816	1.88835	0.944176	+	0.329442i	$0.106861\pi$
$620$	15.0107	0.602846
$621$	−5.45308	−0.218825
$622$	−20.8122	−0.834494
$623$	−1.36016	−0.0544939
$624$	−4.30852	−0.172479
$625$	4.84296	0.193718
$626$	−22.0669	−0.881970
$627$	13.2572	0.529443
$628$	−21.7329	−0.867239
$629$	0.0975274	0.00388867
$630$	4.18309	0.166658
$631$	24.9937	0.994983	0.497491	−	0.867469i	$-0.334255\pi$
$631$	24.9937	0.994983	0.497491	+	0.867469i	$0.334255\pi$
$632$	2.65045	0.105429
$633$	8.18209	0.325209
$634$	−8.76652	−0.348163
$635$	−2.26943	−0.0900595
$636$	1.63330	0.0647647
$637$	−74.5765	−2.95483
$638$	0.0406703	0.00161015
$639$	11.2130	0.443579
$640$	11.2625	0.445191
$641$	−42.6559	−1.68481	−0.842404	−	0.538847i	$-0.818860\pi$
$641$	−42.6559	−1.68481	−0.842404	+	0.538847i	$0.818860\pi$
$642$	0.0359301	0.00141805
$643$	−8.97105	−0.353784	−0.176892	−	0.984230i	$-0.556604\pi$
$643$	−8.97105	−0.353784	−0.176892	+	0.984230i	$0.556604\pi$
$644$	−33.1306	−1.30553
$645$	11.5423	0.454479
$646$	−2.07598	−0.0816783
$647$	48.1117	1.89146	0.945732	−	0.324947i	$-0.105347\pi$
$647$	48.1117	1.89146	0.945732	+	0.324947i	$0.105347\pi$
$648$	−2.65045	−0.104120
$649$	−2.50086	−0.0981675
$650$	16.8295	0.660107
$651$	−38.7517	−1.51880
$652$	−11.7062	−0.458449
$653$	−19.7113	−0.771362	−0.385681	−	0.922632i	$-0.626033\pi$
$653$	−19.7113	−0.771362	−0.385681	+	0.922632i	$0.626033\pi$
$654$	−2.63719	−0.103122
$655$	5.43307	0.212288
$656$	2.61895	0.102253
$657$	8.66706	0.338134
$658$	27.1771	1.05947
$659$	2.30492	0.0897869	0.0448934	−	0.998992i	$-0.485705\pi$
$659$	2.30492	0.0897869	0.0448934	+	0.998992i	$0.485705\pi$
$660$	8.47066	0.329720
$661$	19.0893	0.742489	0.371244	−	0.928535i	$-0.378931\pi$
$661$	19.0893	0.742489	0.371244	+	0.928535i	$0.378931\pi$
$662$	11.0020	0.427606
$663$	6.13291	0.238182
$664$	−21.2903	−0.826224
$665$	14.1895	0.550244
$666$	−0.0762963	−0.00295642
$667$	−0.0567460	−0.00219722
$668$	5.87717	0.227395
$669$	2.33087	0.0901167
$670$	−13.5582	−0.523799
$671$	59.8853	2.31185
$672$	−25.6089	−0.987885
$673$	−18.8737	−0.727527	−0.363763	−	0.931491i	$-0.618508\pi$
$673$	−18.8737	−0.727527	−0.363763	+	0.931491i	$0.618508\pi$
$674$	12.7668	0.491757
$675$	−3.50774	−0.135013
$676$	−34.1621	−1.31393
$677$	17.7551	0.682384	0.341192	−	0.939994i	$-0.389169\pi$
$677$	17.7551	0.682384	0.341192	+	0.939994i	$0.389169\pi$
$678$	−3.66461	−0.140738
$679$	−50.0629	−1.92124
$680$	−3.23774	−0.124162
$681$	11.0937	0.425110
$682$	34.6000	1.32490
$683$	−6.41330	−0.245398	−0.122699	−	0.992444i	$-0.539155\pi$
$683$	−6.41330	−0.245398	−0.122699	+	0.992444i	$0.539155\pi$
$684$	−3.68327	−0.140833
$685$	−0.163644	−0.00625253
$686$	−17.6697	−0.674634
$687$	−6.79443	−0.259224
$688$	−6.63795	−0.253069
$689$	7.21680	0.274938
$690$	5.21124	0.198389
$691$	−29.8006	−1.13367	−0.566834	−	0.823832i	$-0.691832\pi$
$691$	−29.8006	−1.13367	−0.566834	+	0.823832i	$0.691832\pi$
$692$	10.1275	0.384990
$693$	−21.8678	−0.830690
$694$	15.6515	0.594125
$695$	−5.76194	−0.218563
$696$	−0.0275812	−0.00104546
$697$	−3.72791	−0.141205
$698$	−1.88517	−0.0713548
$699$	25.9571	0.981789
$700$	−21.3115	−0.805499
$701$	33.0534	1.24841	0.624205	−	0.781261i	$-0.285423\pi$
$701$	33.0534	1.24841	0.624205	+	0.781261i	$0.285423\pi$
$702$	−4.79781	−0.181082
$703$	−0.258805	−0.00976101
$704$	15.8459	0.597215
$705$	9.69505	0.365137
$706$	−17.0922	−0.643273
$707$	25.5634	0.961410
$708$	0.694818	0.0261129
$709$	−37.6086	−1.41242	−0.706211	−	0.708002i	$-0.749597\pi$
$709$	−37.6086	−1.41242	−0.706211	+	0.708002i	$0.749597\pi$
$710$	−10.7157	−0.402153
$711$	−1.00000	−0.0375029
$712$	−0.823594	−0.0308655
$713$	−48.2763	−1.80796
$714$	3.42433	0.128152
$715$	37.4279	1.39972
$716$	12.0441	0.450111
$717$	−2.22663	−0.0831550
$718$	−25.4973	−0.951551
$719$	−39.0489	−1.45628	−0.728138	−	0.685430i	$-0.759614\pi$
$719$	−39.0489	−1.45628	−0.728138	+	0.685430i	$0.759614\pi$
$720$	−0.858192	−0.0319829
$721$	11.4970	0.428172
$722$	−9.35488	−0.348153
$723$	−21.7463	−0.808753
$724$	14.7496	0.548164
$725$	−0.0365024	−0.00135566
$726$	10.9196	0.405265
$727$	−2.98972	−0.110883	−0.0554413	−	0.998462i	$-0.517657\pi$
$727$	−2.98972	−0.110883	−0.0554413	+	0.998462i	$0.517657\pi$
$728$	−71.1516	−2.63705
$729$	1.00000	0.0370370
$730$	−8.28268	−0.306556
$731$	9.44869	0.349472
$732$	−16.6380	−0.614959
$733$	−38.9641	−1.43917	−0.719585	−	0.694404i	$-0.755668\pi$
$733$	−38.9641	−1.43917	−0.719585	+	0.694404i	$0.755668\pi$
$734$	−23.3292	−0.861097
$735$	−14.8545	−0.547916
$736$	−31.9032	−1.17597
$737$	70.8777	2.61081
$738$	2.91637	0.107353
$739$	19.4505	0.715498	0.357749	−	0.933818i	$-0.383544\pi$
$739$	19.4505	0.715498	0.357749	+	0.933818i	$0.383544\pi$
$740$	−0.165362	−0.00607884
$741$	−16.2747	−0.597865
$742$	4.02953	0.147929
$743$	23.6876	0.869014	0.434507	−	0.900669i	$-0.356923\pi$
$743$	23.6876	0.869014	0.434507	+	0.900669i	$0.356923\pi$
$744$	−23.4645	−0.860252
$745$	−19.7205	−0.722503
$746$	7.78129	0.284893
$747$	8.03271	0.293902
$748$	6.93418	0.253539
$749$	−0.201039	−0.00734580
$750$	8.13043	0.296881
$751$	−15.6483	−0.571014	−0.285507	−	0.958377i	$-0.592162\pi$
$751$	−15.6483	−0.571014	−0.285507	+	0.958377i	$0.592162\pi$
$752$	−5.57558	−0.203321
$753$	−13.3387	−0.486090
$754$	−0.0499271	−0.00181824
$755$	16.1880	0.589143
$756$	6.07557	0.220966
$757$	−41.7547	−1.51760	−0.758800	−	0.651324i	$-0.774214\pi$
$757$	−41.7547	−1.51760	−0.758800	+	0.651324i	$0.774214\pi$
$758$	−22.5997	−0.820860
$759$	−27.2426	−0.988845
$760$	8.59187	0.311660
$761$	54.2052	1.96494	0.982468	−	0.186430i	$-0.0596918\pi$
$761$	54.2052	1.96494	0.982468	+	0.186430i	$0.0596918\pi$
$762$	1.45335	0.0526495
$763$	14.7558	0.534196
$764$	−3.15783	−0.114246
$765$	1.22158	0.0441663
$766$	14.6803	0.530419
$767$	3.07008	0.110854
$768$	−13.5563	−0.489169
$769$	−49.8946	−1.79924	−0.899622	−	0.436669i	$-0.856158\pi$
$769$	−49.8946	−1.79924	−0.899622	+	0.436669i	$0.856158\pi$
$770$	20.8980	0.753111
$771$	2.18520	0.0786982
$772$	−22.7006	−0.817011
$773$	14.0559	0.505555	0.252777	−	0.967524i	$-0.418656\pi$
$773$	14.0559	0.505555	0.252777	+	0.967524i	$0.418656\pi$
$774$	−7.39177	−0.265692
$775$	−31.0541	−1.11550
$776$	−30.3136	−1.08819
$777$	0.426899	0.0153149
$778$	−22.3211	−0.800250
$779$	9.89262	0.354440
$780$	−10.3986	−0.372331
$781$	56.0181	2.00449
$782$	4.26598	0.152551
$783$	0.0104062	0.000371888 0
$784$	8.54275	0.305098
$785$	−19.1272	−0.682681
$786$	−3.47937	−0.124105
$787$	47.1422	1.68044	0.840220	−	0.542246i	$-0.182426\pi$
$787$	47.1422	1.68044	0.840220	+	0.542246i	$0.182426\pi$
$788$	−38.3686	−1.36682
$789$	12.0026	0.427302
$790$	0.955651	0.0340005
$791$	20.5045	0.729055
$792$	−13.2412	−0.470505
$793$	−73.5156	−2.61061
$794$	10.4534	0.370977
$795$	1.43748	0.0509820
$796$	9.29980	0.329623
$797$	55.5959	1.96931	0.984654	−	0.174518i	$-0.0558366\pi$
$797$	55.5959	1.96931	0.984654	+	0.174518i	$0.0558366\pi$
$798$	−9.08701	−0.321677
$799$	7.93648	0.280773
$800$	−20.5220	−0.725563
$801$	0.310737	0.0109794
$802$	3.59494	0.126942
$803$	43.2991	1.52799
$804$	−19.6921	−0.694485
$805$	−29.1583	−1.02770
$806$	−42.4752	−1.49612
$807$	−12.5232	−0.440838
$808$	15.4789	0.544546
$809$	−37.4399	−1.31632	−0.658159	−	0.752879i	$-0.728665\pi$
$809$	−37.4399	−1.31632	−0.658159	+	0.752879i	$0.728665\pi$
$810$	−0.955651	−0.0335781
$811$	29.7547	1.04483	0.522414	−	0.852692i	$-0.325031\pi$
$811$	29.7547	1.04483	0.522414	+	0.852692i	$0.325031\pi$
$812$	0.0632237	0.00221872
$813$	7.38672	0.259063
$814$	−0.381163	−0.0133598
$815$	−10.3026	−0.360886
$816$	−0.702526	−0.0245933
$817$	−25.0736	−0.877216
$818$	11.4930	0.401845
$819$	26.8451	0.938043
$820$	6.32086	0.220734
$821$	−10.3943	−0.362762	−0.181381	−	0.983413i	$-0.558057\pi$
$821$	−10.3943	−0.362762	−0.181381	+	0.983413i	$0.558057\pi$
$822$	0.104799	0.00365527
$823$	−4.75968	−0.165912	−0.0829561	−	0.996553i	$-0.526436\pi$
$823$	−4.75968	−0.165912	−0.0829561	+	0.996553i	$0.526436\pi$
$824$	6.96158	0.242518
$825$	−17.5241	−0.610109
$826$	1.71419	0.0596442
$827$	−31.4720	−1.09439	−0.547194	−	0.837006i	$-0.684304\pi$
$827$	−31.4720	−1.09439	−0.547194	+	0.837006i	$0.684304\pi$
$828$	7.56886	0.263036
$829$	40.9206	1.42123	0.710616	−	0.703580i	$-0.248416\pi$
$829$	40.9206	1.42123	0.710616	+	0.703580i	$0.248416\pi$
$830$	−7.67646	−0.266454
$831$	−14.1184	−0.489762
$832$	−19.4525	−0.674395
$833$	−12.1601	−0.421321
$834$	3.68998	0.127773
$835$	5.17252	0.179003
$836$	−18.4010	−0.636411
$837$	8.85303	0.306005
$838$	12.2746	0.424019
$839$	52.0774	1.79791	0.898956	−	0.438038i	$-0.144327\pi$
$839$	52.0774	1.79791	0.898956	+	0.438038i	$0.144327\pi$
$840$	−14.1723	−0.488991
$841$	−28.9999	−0.999996
$842$	4.57012	0.157497
$843$	7.36176	0.253553
$844$	−11.3567	−0.390914
$845$	−30.0662	−1.03431
$846$	−6.20876	−0.213462
$847$	−61.0983	−2.09936
$848$	−0.826686	−0.0283885
$849$	−23.7506	−0.815117
$850$	2.74413	0.0941229
$851$	0.531825	0.0182307
$852$	−15.5636	−0.533200
$853$	−3.96655	−0.135812	−0.0679060	−	0.997692i	$-0.521632\pi$
$853$	−3.96655	−0.135812	−0.0679060	+	0.997692i	$0.521632\pi$
$854$	−41.0477	−1.40462
$855$	−3.24166	−0.110862
$856$	−0.121731	−0.00416068
$857$	27.0403	0.923679	0.461839	−	0.886964i	$-0.347190\pi$
$857$	27.0403	0.923679	0.461839	+	0.886964i	$0.347190\pi$
$858$	−23.9690	−0.818289
$859$	−16.5534	−0.564795	−0.282397	−	0.959298i	$-0.591130\pi$
$859$	−16.5534	−0.564795	−0.282397	+	0.959298i	$0.591130\pi$
$860$	−16.0207	−0.546302
$861$	−16.3179	−0.556112
$862$	7.04466	0.239942
$863$	−23.2481	−0.791376	−0.395688	−	0.918385i	$-0.629494\pi$
$863$	−23.2481	−0.791376	−0.395688	+	0.918385i	$0.629494\pi$
$864$	5.85050	0.199038
$865$	8.91325	0.303060
$866$	−14.6056	−0.496319
$867$	1.00000	0.0339618
$868$	53.7872	1.82566
$869$	−4.99582	−0.169472
$870$	−0.00994472	−0.000337157 0
$871$	−87.0100	−2.94822
$872$	8.93479	0.302570
$873$	11.4371	0.387088
$874$	−11.3205	−0.382921
$875$	−45.4920	−1.53791
$876$	−12.0299	−0.406451
$877$	−47.4161	−1.60113	−0.800565	−	0.599246i	$-0.795467\pi$
$877$	−47.4161	−1.60113	−0.800565	+	0.599246i	$0.795467\pi$
$878$	21.8320	0.736794
$879$	10.4712	0.353185
$880$	−4.28737	−0.144527
$881$	29.3512	0.988868	0.494434	−	0.869215i	$-0.335375\pi$
$881$	29.3512	0.988868	0.494434	+	0.869215i	$0.335375\pi$
$882$	9.51290	0.320316
$883$	35.1207	1.18191	0.590953	−	0.806706i	$-0.298752\pi$
$883$	35.1207	1.18191	0.590953	+	0.806706i	$0.298752\pi$
$884$	−8.51245	−0.286305
$885$	0.611512	0.0205557
$886$	−30.4478	−1.02292
$887$	37.0827	1.24512	0.622558	−	0.782574i	$-0.286094\pi$
$887$	37.0827	1.24512	0.622558	+	0.782574i	$0.286094\pi$
$888$	0.258492	0.00867441
$889$	−8.13192	−0.272736
$890$	−0.296956	−0.00995399
$891$	4.99582	0.167366
$892$	−3.23524	−0.108324
$893$	−21.0607	−0.704771
$894$	12.6291	0.422381
$895$	10.6001	0.354322
$896$	40.3564	1.34821
$897$	33.4432	1.11664
$898$	23.8481	0.795820
$899$	0.0921266	0.00307260
$900$	4.86873	0.162291
$901$	1.17673	0.0392027
$902$	14.5697	0.485117
$903$	41.3590	1.37634
$904$	12.4157	0.412939
$905$	12.9812	0.431509
$906$	−10.3669	−0.344417
$907$	−56.3502	−1.87108	−0.935539	−	0.353224i	$-0.885085\pi$
$907$	−56.3502	−1.87108	−0.935539	+	0.353224i	$0.885085\pi$
$908$	−15.3980	−0.510999
$909$	−5.84010	−0.193704
$910$	−25.6545	−0.850439
$911$	32.5116	1.07716	0.538578	−	0.842576i	$-0.318962\pi$
$911$	32.5116	1.07716	0.538578	+	0.842576i	$0.318962\pi$
$912$	1.86427	0.0617320
$913$	40.1300	1.32811
$914$	9.78012	0.323498
$915$	−14.6432	−0.484088
$916$	9.43064	0.311597
$917$	19.4680	0.642891
$918$	−0.782307	−0.0258200
$919$	52.1894	1.72157	0.860785	−	0.508968i	$-0.169973\pi$
$919$	52.1894	1.72157	0.860785	+	0.508968i	$0.169973\pi$
$920$	−17.6557	−0.582090
$921$	−3.71235	−0.122326
$922$	18.6416	0.613929
$923$	−68.7682	−2.26353
$924$	30.3524	0.998522
$925$	0.342101	0.0112482
$926$	28.5559	0.938405
$927$	−2.62656	−0.0862676
$928$	0.0608816	0.00199854
$929$	35.3722	1.16052	0.580261	−	0.814430i	$-0.302950\pi$
$929$	35.3722	1.16052	0.580261	+	0.814430i	$0.302950\pi$
$930$	−8.46040	−0.277428
$931$	32.2687	1.05756
$932$	−36.0284	−1.18015
$933$	−26.6037	−0.870965
$934$	21.6588	0.708697
$935$	6.10280	0.199583
$936$	16.2550	0.531310
$937$	−17.1784	−0.561194	−0.280597	−	0.959826i	$-0.590532\pi$
$937$	−17.1784	−0.561194	−0.280597	+	0.959826i	$0.590532\pi$
$938$	−48.5823	−1.58627
$939$	−28.2075	−0.920515
$940$	−13.4567	−0.438909
$941$	−23.7882	−0.775473	−0.387737	−	0.921770i	$-0.626743\pi$
$941$	−23.7882	−0.775473	−0.387737	+	0.921770i	$0.626743\pi$
$942$	12.2492	0.399100
$943$	−20.3286	−0.661991
$944$	−0.351678	−0.0114461
$945$	5.34713	0.173942
$946$	−36.9280	−1.20063
$947$	−32.5193	−1.05674	−0.528368	−	0.849015i	$-0.677196\pi$
$947$	−32.5193	−1.05674	−0.528368	+	0.849015i	$0.677196\pi$
$948$	1.38800	0.0450800
$949$	−53.1543	−1.72546
$950$	−7.28199	−0.236259
$951$	−11.2060	−0.363379
$952$	−11.6016	−0.376010
$953$	−19.0376	−0.616688	−0.308344	−	0.951275i	$-0.599775\pi$
$953$	−19.0376	−0.616688	−0.308344	+	0.951275i	$0.599775\pi$
$954$	−0.920567	−0.0298045
$955$	−2.77921	−0.0899332
$956$	3.09055	0.0999556
$957$	0.0519877	0.00168052
$958$	9.43081	0.304696
$959$	−0.586377	−0.0189351
$960$	−3.87465	−0.125054
$961$	47.3761	1.52826
$962$	0.467918	0.0150863
$963$	0.0459284	0.00148002
$964$	30.1838	0.972153
$965$	−19.9788	−0.643141
$966$	18.6732	0.600799
$967$	15.8995	0.511294	0.255647	−	0.966770i	$-0.417712\pi$
$967$	15.8995	0.511294	0.255647	+	0.966770i	$0.417712\pi$
$968$	−36.9956	−1.18909
$969$	−2.65366	−0.0852479
$970$	−10.9299	−0.350938
$971$	−4.50944	−0.144715	−0.0723574	−	0.997379i	$-0.523052\pi$
$971$	−4.50944	−0.144715	−0.0723574	+	0.997379i	$0.523052\pi$
$972$	−1.38800	−0.0445200
$973$	−20.6464	−0.661894
$974$	−17.3209	−0.554998
$975$	21.5126	0.688956
$976$	8.42123	0.269557
$977$	−24.3965	−0.780512	−0.390256	−	0.920706i	$-0.627614\pi$
$977$	−24.3965	−0.780512	−0.390256	+	0.920706i	$0.627614\pi$
$978$	6.59787	0.210977
$979$	1.55239	0.0496145
$980$	20.6180	0.658617
$981$	−3.37105	−0.107629
$982$	−26.8101	−0.855544
$983$	−0.323198	−0.0103084	−0.00515421	−	0.999987i	$-0.501641\pi$
$983$	−0.323198	−0.0103084	−0.00515421	+	0.999987i	$0.501641\pi$
$984$	−9.88065	−0.314984
$985$	−33.7683	−1.07595
$986$	−0.00814086	−0.000259258 0
$987$	34.7397	1.10578
$988$	22.5892	0.718657
$989$	51.5245	1.63838
$990$	−4.77426	−0.151736
$991$	−55.9539	−1.77743	−0.888717	−	0.458457i	$-0.848402\pi$
$991$	−55.9539	−1.77743	−0.888717	+	0.458457i	$0.848402\pi$
$992$	51.7946	1.64448
$993$	14.0636	0.446294
$994$	−38.3970	−1.21788
$995$	8.18479	0.259475
$996$	−11.1494	−0.353281
$997$	−8.49957	−0.269184	−0.134592	−	0.990901i	$-0.542972\pi$
$997$	−8.49957	−0.269184	−0.134592	+	0.990901i	$0.542972\pi$
$998$	30.1410	0.954098
$999$	−0.0975274	−0.00308563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.j.1.15	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.j.1.15	✓	25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q+0.782307 q^{2} +1.00000 q^{3} -1.38800 q^{4} -1.22158 q^{5} +0.782307 q^{6} -4.37722 q^{7} -2.65045 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.782307 q^{2} +1.00000 q^{3} -1.38800 q^{4} -1.22158 q^{5} +0.782307 q^{6} -4.37722 q^{7} -2.65045 q^{8} +1.00000 q^{9} -0.955651 q^{10} +4.99582 q^{11} -1.38800 q^{12} -6.13291 q^{13} -3.42433 q^{14} -1.22158 q^{15} +0.702526 q^{16} -1.00000 q^{17} +0.782307 q^{18} +2.65366 q^{19} +1.69555 q^{20} -4.37722 q^{21} +3.90827 q^{22} -5.45308 q^{23} -2.65045 q^{24} -3.50774 q^{25} -4.79781 q^{26} +1.00000 q^{27} +6.07557 q^{28} +0.0104062 q^{29} -0.955651 q^{30} +8.85303 q^{31} +5.85050 q^{32} +4.99582 q^{33} -0.782307 q^{34} +5.34713 q^{35} -1.38800 q^{36} -0.0975274 q^{37} +2.07598 q^{38} -6.13291 q^{39} +3.23774 q^{40} +3.72791 q^{41} -3.42433 q^{42} -9.44869 q^{43} -6.93418 q^{44} -1.22158 q^{45} -4.26598 q^{46} -7.93648 q^{47} +0.702526 q^{48} +12.1601 q^{49} -2.74413 q^{50} -1.00000 q^{51} +8.51245 q^{52} -1.17673 q^{53} +0.782307 q^{54} -6.10280 q^{55} +11.6016 q^{56} +2.65366 q^{57} +0.00814086 q^{58} -0.500591 q^{59} +1.69555 q^{60} +11.9871 q^{61} +6.92579 q^{62} -4.37722 q^{63} +3.17183 q^{64} +7.49184 q^{65} +3.90827 q^{66} +14.1874 q^{67} +1.38800 q^{68} -5.45308 q^{69} +4.18309 q^{70} +11.2130 q^{71} -2.65045 q^{72} +8.66706 q^{73} -0.0762963 q^{74} -3.50774 q^{75} -3.68327 q^{76} -21.8678 q^{77} -4.79781 q^{78} -1.00000 q^{79} -0.858192 q^{80} +1.00000 q^{81} +2.91637 q^{82} +8.03271 q^{83} +6.07557 q^{84} +1.22158 q^{85} -7.39177 q^{86} +0.0104062 q^{87} -13.2412 q^{88} +0.310737 q^{89} -0.955651 q^{90} +26.8451 q^{91} +7.56886 q^{92} +8.85303 q^{93} -6.20876 q^{94} -3.24166 q^{95} +5.85050 q^{96} +11.4371 q^{97} +9.51290 q^{98} +4.99582 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 6 q^{2} + 25 q^{3} + 26 q^{4} + 6 q^{5} + 6 q^{6} + 4 q^{7} + 18 q^{8} + 25 q^{9}+O(q^{10}) \) 25 * q + 6 * q^2 + 25 * q^3 + 26 * q^4 + 6 * q^5 + 6 * q^6 + 4 * q^7 + 18 * q^8 + 25 * q^9	\( 25 q + 6 q^{2} + 25 q^{3} + 26 q^{4} + 6 q^{5} + 6 q^{6} + 4 q^{7} + 18 q^{8} + 25 q^{9} + 13 q^{10} + 19 q^{11} + 26 q^{12} + 17 q^{14} + 6 q^{15} + 16 q^{16} - 25 q^{17} + 6 q^{18} + 25 q^{19} + 32 q^{20} + 4 q^{21} - 7 q^{22} + 8 q^{23} + 18 q^{24} + 15 q^{25} + 20 q^{26} + 25 q^{27} + 9 q^{28} + 21 q^{29} + 13 q^{30} + 4 q^{31} + 27 q^{32} + 19 q^{33} - 6 q^{34} + 50 q^{35} + 26 q^{36} - 8 q^{37} + 31 q^{38} + 52 q^{40} + 40 q^{41} + 17 q^{42} + 21 q^{43} + 34 q^{44} + 6 q^{45} + 29 q^{46} + 43 q^{47} + 16 q^{48} + 21 q^{49} + 13 q^{50} - 25 q^{51} + 3 q^{52} + 44 q^{53} + 6 q^{54} + 13 q^{55} + 38 q^{56} + 25 q^{57} - 5 q^{58} + 45 q^{59} + 32 q^{60} + 22 q^{61} + 4 q^{62} + 4 q^{63} + 26 q^{64} + 43 q^{65} - 7 q^{66} + 8 q^{67} - 26 q^{68} + 8 q^{69} + 29 q^{70} + 9 q^{71} + 18 q^{72} - 7 q^{73} + 18 q^{74} + 15 q^{75} + 33 q^{76} + 20 q^{77} + 20 q^{78} - 25 q^{79} + 42 q^{80} + 25 q^{81} - 43 q^{82} + 41 q^{83} + 9 q^{84} - 6 q^{85} - 12 q^{86} + 21 q^{87} - 43 q^{88} + 68 q^{89} + 13 q^{90} + 10 q^{91} + 2 q^{92} + 4 q^{93} - 17 q^{94} + 8 q^{95} + 27 q^{96} + 15 q^{97} + 11 q^{98} + 19 q^{99}+O(q^{100}) \) 25 * q + 6 * q^2 + 25 * q^3 + 26 * q^4 + 6 * q^5 + 6 * q^6 + 4 * q^7 + 18 * q^8 + 25 * q^9 + 13 * q^10 + 19 * q^11 + 26 * q^12 + 17 * q^14 + 6 * q^15 + 16 * q^16 - 25 * q^17 + 6 * q^18 + 25 * q^19 + 32 * q^20 + 4 * q^21 - 7 * q^22 + 8 * q^23 + 18 * q^24 + 15 * q^25 + 20 * q^26 + 25 * q^27 + 9 * q^28 + 21 * q^29 + 13 * q^30 + 4 * q^31 + 27 * q^32 + 19 * q^33 - 6 * q^34 + 50 * q^35 + 26 * q^36 - 8 * q^37 + 31 * q^38 + 52 * q^40 + 40 * q^41 + 17 * q^42 + 21 * q^43 + 34 * q^44 + 6 * q^45 + 29 * q^46 + 43 * q^47 + 16 * q^48 + 21 * q^49 + 13 * q^50 - 25 * q^51 + 3 * q^52 + 44 * q^53 + 6 * q^54 + 13 * q^55 + 38 * q^56 + 25 * q^57 - 5 * q^58 + 45 * q^59 + 32 * q^60 + 22 * q^61 + 4 * q^62 + 4 * q^63 + 26 * q^64 + 43 * q^65 - 7 * q^66 + 8 * q^67 - 26 * q^68 + 8 * q^69 + 29 * q^70 + 9 * q^71 + 18 * q^72 - 7 * q^73 + 18 * q^74 + 15 * q^75 + 33 * q^76 + 20 * q^77 + 20 * q^78 - 25 * q^79 + 42 * q^80 + 25 * q^81 - 43 * q^82 + 41 * q^83 + 9 * q^84 - 6 * q^85 - 12 * q^86 + 21 * q^87 - 43 * q^88 + 68 * q^89 + 13 * q^90 + 10 * q^91 + 2 * q^92 + 4 * q^93 - 17 * q^94 + 8 * q^95 + 27 * q^96 + 15 * q^97 + 11 * q^98 + 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more