LMFDB - Embedded newform 6027.2.a.bm.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 64 x^{12} - 456 x^{11} - 54 x^{10} + 1532 x^{9} - 400 x^{8} + \cdots - 49 $ x^16 - 4x^15 - 14x^14 + 68x^13 + 64x^12 - 456x^11 - 54x^10 + 1532x^9 - 400x^8 - 2708x^7 + 1218x^6 + 2424x^5 - 1276x^4 - 960x^3 + 500x^2 + 112*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.631061$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.631061 q^{2} +1.00000 q^{3} -1.60176 q^{4} -1.65938 q^{5} -0.631061 q^{6} +2.27293 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.631061 q^{2} +1.00000 q^{3} -1.60176 q^{4} -1.65938 q^{5} -0.631061 q^{6} +2.27293 q^{8} +1.00000 q^{9} +1.04717 q^{10} +2.82239 q^{11} -1.60176 q^{12} -3.15999 q^{13} -1.65938 q^{15} +1.76917 q^{16} +1.11961 q^{17} -0.631061 q^{18} +3.11082 q^{19} +2.65794 q^{20} -1.78110 q^{22} -6.51222 q^{23} +2.27293 q^{24} -2.24644 q^{25} +1.99415 q^{26} +1.00000 q^{27} +9.16578 q^{29} +1.04717 q^{30} -4.35118 q^{31} -5.66231 q^{32} +2.82239 q^{33} -0.706545 q^{34} -1.60176 q^{36} -11.1252 q^{37} -1.96312 q^{38} -3.15999 q^{39} -3.77167 q^{40} +1.00000 q^{41} -0.684567 q^{43} -4.52079 q^{44} -1.65938 q^{45} +4.10961 q^{46} +6.17414 q^{47} +1.76917 q^{48} +1.41764 q^{50} +1.11961 q^{51} +5.06155 q^{52} -13.0735 q^{53} -0.631061 q^{54} -4.68342 q^{55} +3.11082 q^{57} -5.78417 q^{58} +7.71075 q^{59} +2.65794 q^{60} +0.419560 q^{61} +2.74586 q^{62} +0.0349308 q^{64} +5.24364 q^{65} -1.78110 q^{66} +9.95583 q^{67} -1.79336 q^{68} -6.51222 q^{69} +7.39023 q^{71} +2.27293 q^{72} +13.9295 q^{73} +7.02071 q^{74} -2.24644 q^{75} -4.98280 q^{76} +1.99415 q^{78} -0.195831 q^{79} -2.93573 q^{80} +1.00000 q^{81} -0.631061 q^{82} -2.50464 q^{83} -1.85787 q^{85} +0.432003 q^{86} +9.16578 q^{87} +6.41509 q^{88} -12.4999 q^{89} +1.04717 q^{90} +10.4310 q^{92} -4.35118 q^{93} -3.89626 q^{94} -5.16206 q^{95} -5.66231 q^{96} +11.3773 q^{97} +2.82239 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9	$ 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) $ 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9 - 4 * q^10 - 4 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^17 - 4 * q^18 + 4 * q^19 - 20 * q^20 - 16 * q^22 - 12 * q^23 - 12 * q^24 - 8 * q^25 - 8 * q^26 + 16 * q^27 - 16 * q^29 - 4 * q^30 - 4 * q^31 - 48 * q^32 - 4 * q^33 + 16 * q^34 + 12 * q^36 - 48 * q^37 - 4 * q^38 + 56 * q^40 + 16 * q^41 - 16 * q^43 - 12 * q^45 - 4 * q^46 - 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 - 4 * q^54 + 8 * q^55 + 4 * q^57 - 36 * q^58 - 36 * q^59 - 20 * q^60 - 4 * q^61 - 12 * q^62 + 52 * q^64 - 36 * q^65 - 16 * q^66 - 52 * q^67 - 8 * q^68 - 12 * q^69 - 12 * q^71 - 12 * q^72 - 16 * q^73 + 4 * q^74 - 8 * q^75 + 16 * q^76 - 8 * q^78 - 36 * q^79 - 68 * q^80 + 16 * q^81 - 4 * q^82 - 32 * q^83 - 28 * q^85 - 8 * q^86 - 16 * q^87 - 36 * q^88 - 12 * q^89 - 4 * q^90 - 36 * q^92 - 4 * q^93 + 24 * q^94 - 20 * q^95 - 48 * q^96 + 48 * q^97 - 4 * 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 $

$2$	−0.631061	−0.446227	−0.223114	−	0.974792i	$-0.571622\pi$
$2$	−0.631061	−0.446227	−0.223114	+	0.974792i	$0.571622\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.60176	−0.800881
$5$	−1.65938	−0.742100	−0.371050	−	0.928613i	$-0.621002\pi$
$5$	−1.65938	−0.742100	−0.371050	+	0.928613i	$0.621002\pi$
$6$	−0.631061	−0.257629
$7$	0	0
$7$	0	0
$8$	2.27293	0.803602
$9$	1.00000	0.333333
$10$	1.04717	0.331145
$11$	2.82239	0.850981	0.425491	−	0.904963i	$-0.360102\pi$
$11$	2.82239	0.850981	0.425491	+	0.904963i	$0.360102\pi$
$12$	−1.60176	−0.462389
$13$	−3.15999	−0.876423	−0.438212	−	0.898872i	$-0.644388\pi$
$13$	−3.15999	−0.876423	−0.438212	+	0.898872i	$0.644388\pi$
$14$	0	0
$15$	−1.65938	−0.428451
$16$	1.76917	0.442292
$17$	1.11961	0.271547	0.135773	−	0.990740i	$-0.456648\pi$
$17$	1.11961	0.271547	0.135773	+	0.990740i	$0.456648\pi$
$18$	−0.631061	−0.148742
$19$	3.11082	0.713672	0.356836	−	0.934167i	$-0.383855\pi$
$19$	3.11082	0.713672	0.356836	+	0.934167i	$0.383855\pi$
$20$	2.65794	0.594333
$21$	0	0
$22$	−1.78110	−0.379731
$23$	−6.51222	−1.35789	−0.678946	−	0.734188i	$-0.737563\pi$
$23$	−6.51222	−1.35789	−0.678946	+	0.734188i	$0.737563\pi$
$24$	2.27293	0.463960
$25$	−2.24644	−0.449288
$26$	1.99415	0.391084
$27$	1.00000	0.192450
$28$	0	0
$29$	9.16578	1.70204	0.851021	−	0.525131i	$-0.175984\pi$
$29$	9.16578	1.70204	0.851021	+	0.525131i	$0.175984\pi$
$30$	1.04717	0.191187
$31$	−4.35118	−0.781495	−0.390748	−	0.920498i	$-0.627783\pi$
$31$	−4.35118	−0.781495	−0.390748	+	0.920498i	$0.627783\pi$
$32$	−5.66231	−1.00097
$33$	2.82239	0.491314
$34$	−0.706545	−0.121171
$35$	0	0
$36$	−1.60176	−0.266960
$37$	−11.1252	−1.82898	−0.914490	−	0.404609i	$-0.867407\pi$
$37$	−11.1252	−1.82898	−0.914490	+	0.404609i	$0.867407\pi$
$38$	−1.96312	−0.318460
$39$	−3.15999	−0.506003
$40$	−3.77167	−0.596353
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−0.684567	−0.104395	−0.0521977	−	0.998637i	$-0.516623\pi$
$43$	−0.684567	−0.104395	−0.0521977	+	0.998637i	$0.516623\pi$
$44$	−4.52079	−0.681535
$45$	−1.65938	−0.247367
$46$	4.10961	0.605929
$47$	6.17414	0.900591	0.450296	−	0.892880i	$-0.351319\pi$
$47$	6.17414	0.900591	0.450296	+	0.892880i	$0.351319\pi$
$48$	1.76917	0.255357
$49$	0	0
$50$	1.41764	0.200485
$51$	1.11961	0.156777
$52$	5.06155	0.701911
$53$	−13.0735	−1.79579	−0.897894	−	0.440212i	$-0.854903\pi$
$53$	−13.0735	−1.79579	−0.897894	+	0.440212i	$0.854903\pi$
$54$	−0.631061	−0.0858765
$55$	−4.68342	−0.631513
$56$	0	0
$57$	3.11082	0.412039
$58$	−5.78417	−0.759498
$59$	7.71075	1.00385	0.501927	−	0.864910i	$-0.332625\pi$
$59$	7.71075	1.00385	0.501927	+	0.864910i	$0.332625\pi$
$60$	2.65794	0.343139
$61$	0.419560	0.0537192	0.0268596	−	0.999639i	$-0.491449\pi$
$61$	0.419560	0.0537192	0.0268596	+	0.999639i	$0.491449\pi$
$62$	2.74586	0.348725
$63$	0	0
$64$	0.0349308	0.00436635
$65$	5.24364	0.650393
$66$	−1.78110	−0.219238
$67$	9.95583	1.21630	0.608149	−	0.793823i	$-0.291912\pi$
$67$	9.95583	1.21630	0.608149	+	0.793823i	$0.291912\pi$
$68$	−1.79336	−0.217476
$69$	−6.51222	−0.783979
$70$	0	0
$71$	7.39023	0.877059	0.438529	−	0.898717i	$-0.355499\pi$
$71$	7.39023	0.877059	0.438529	+	0.898717i	$0.355499\pi$
$72$	2.27293	0.267867
$73$	13.9295	1.63033	0.815165	−	0.579228i	$-0.196646\pi$
$73$	13.9295	1.63033	0.815165	+	0.579228i	$0.196646\pi$
$74$	7.02071	0.816141
$75$	−2.24644	−0.259397
$76$	−4.98280	−0.571567
$77$	0	0
$78$	1.99415	0.225792
$79$	−0.195831	−0.0220327	−0.0110164	−	0.999939i	$-0.503507\pi$
$79$	−0.195831	−0.0220327	−0.0110164	+	0.999939i	$0.503507\pi$
$80$	−2.93573	−0.328224
$81$	1.00000	0.111111
$82$	−0.631061	−0.0696890
$83$	−2.50464	−0.274919	−0.137460	−	0.990507i	$-0.543894\pi$
$83$	−2.50464	−0.274919	−0.137460	+	0.990507i	$0.543894\pi$
$84$	0	0
$85$	−1.85787	−0.201515
$86$	0.432003	0.0465841
$87$	9.16578	0.982675
$88$	6.41509	0.683851
$89$	−12.4999	−1.32499	−0.662493	−	0.749068i	$-0.730502\pi$
$89$	−12.4999	−1.32499	−0.662493	+	0.749068i	$0.730502\pi$
$90$	1.04717	0.110382
$91$	0	0
$92$	10.4310	1.08751
$93$	−4.35118	−0.451196
$94$	−3.89626	−0.401868
$95$	−5.16206	−0.529616
$96$	−5.66231	−0.577907
$97$	11.3773	1.15519	0.577593	−	0.816325i	$-0.303992\pi$
$97$	11.3773	1.15519	0.577593	+	0.816325i	$0.303992\pi$
$98$	0	0
$99$	2.82239	0.283660
$100$	3.59827	0.359827
$101$	14.9268	1.48527	0.742634	−	0.669697i	$-0.233576\pi$
$101$	14.9268	1.48527	0.742634	+	0.669697i	$0.233576\pi$
$102$	−0.706545	−0.0699584
$103$	−9.93429	−0.978854	−0.489427	−	0.872044i	$-0.662794\pi$
$103$	−9.93429	−0.978854	−0.489427	+	0.872044i	$0.662794\pi$
$104$	−7.18244	−0.704296
$105$	0	0
$106$	8.25019	0.801330
$107$	−16.6479	−1.60941	−0.804707	−	0.593673i	$-0.797677\pi$
$107$	−16.6479	−1.60941	−0.804707	+	0.593673i	$0.797677\pi$
$108$	−1.60176	−0.154130
$109$	17.3263	1.65956	0.829779	−	0.558092i	$-0.188466\pi$
$109$	17.3263	1.65956	0.829779	+	0.558092i	$0.188466\pi$
$110$	2.95553	0.281798
$111$	−11.1252	−1.05596
$112$	0	0
$113$	−10.5963	−0.996820	−0.498410	−	0.866941i	$-0.666083\pi$
$113$	−10.5963	−0.996820	−0.498410	+	0.866941i	$0.666083\pi$
$114$	−1.96312	−0.183863
$115$	10.8063	1.00769
$116$	−14.6814	−1.36313
$117$	−3.15999	−0.292141
$118$	−4.86595	−0.447947
$119$	0	0
$120$	−3.77167	−0.344305
$121$	−3.03414	−0.275831
$122$	−0.264768	−0.0239710
$123$	1.00000	0.0901670
$124$	6.96956	0.625885
$125$	12.0246	1.07552
$126$	0	0
$127$	−6.33759	−0.562371	−0.281185	−	0.959653i	$-0.590728\pi$
$127$	−6.33759	−0.562371	−0.281185	+	0.959653i	$0.590728\pi$
$128$	11.3026	0.999017
$129$	−0.684567	−0.0602727
$130$	−3.30905	−0.290223
$131$	−5.07981	−0.443825	−0.221913	−	0.975067i	$-0.571230\pi$
$131$	−5.07981	−0.443825	−0.221913	+	0.975067i	$0.571230\pi$
$132$	−4.52079	−0.393484
$133$	0	0
$134$	−6.28273	−0.542745
$135$	−1.65938	−0.142817
$136$	2.54481	0.218215
$137$	−8.11444	−0.693263	−0.346632	−	0.938001i	$-0.612675\pi$
$137$	−8.11444	−0.693263	−0.346632	+	0.938001i	$0.612675\pi$
$138$	4.10961	0.349833
$139$	−21.7599	−1.84565	−0.922826	−	0.385216i	$-0.874127\pi$
$139$	−21.7599	−1.84565	−0.922826	+	0.385216i	$0.874127\pi$
$140$	0	0
$141$	6.17414	0.519957
$142$	−4.66368	−0.391368
$143$	−8.91871	−0.745820
$144$	1.76917	0.147431
$145$	−15.2096	−1.26309
$146$	−8.79039	−0.727498
$147$	0	0
$148$	17.8200	1.46479
$149$	14.2646	1.16860	0.584301	−	0.811537i	$-0.301369\pi$
$149$	14.2646	1.16860	0.584301	+	0.811537i	$0.301369\pi$
$150$	1.41764	0.115750
$151$	2.29265	0.186573	0.0932867	−	0.995639i	$-0.470263\pi$
$151$	2.29265	0.186573	0.0932867	+	0.995639i	$0.470263\pi$
$152$	7.07069	0.573509
$153$	1.11961	0.0905155
$154$	0	0
$155$	7.22028	0.579947
$156$	5.06155	0.405248
$157$	−24.1517	−1.92751	−0.963756	−	0.266786i	$-0.914038\pi$
$157$	−24.1517	−1.92751	−0.963756	+	0.266786i	$0.914038\pi$
$158$	0.123581	0.00983161
$159$	−13.0735	−1.03680
$160$	9.39596	0.742816
$161$	0	0
$162$	−0.631061	−0.0495808
$163$	−0.309243	−0.0242218	−0.0121109	−	0.999927i	$-0.503855\pi$
$163$	−0.309243	−0.0242218	−0.0121109	+	0.999927i	$0.503855\pi$
$164$	−1.60176	−0.125077
$165$	−4.68342	−0.364604
$166$	1.58058	0.122677
$167$	−3.45676	−0.267492	−0.133746	−	0.991016i	$-0.542701\pi$
$167$	−3.45676	−0.267492	−0.133746	+	0.991016i	$0.542701\pi$
$168$	0	0
$169$	−3.01447	−0.231882
$170$	1.17243	0.0899213
$171$	3.11082	0.237891
$172$	1.09651	0.0836083
$173$	−15.7701	−1.19898	−0.599489	−	0.800383i	$-0.704630\pi$
$173$	−15.7701	−1.19898	−0.599489	+	0.800383i	$0.704630\pi$
$174$	−5.78417	−0.438496
$175$	0	0
$176$	4.99327	0.376382
$177$	7.71075	0.579575
$178$	7.88820	0.591245
$179$	16.6359	1.24343	0.621714	−	0.783244i	$-0.286437\pi$
$179$	16.6359	1.24343	0.621714	+	0.783244i	$0.286437\pi$
$180$	2.65794	0.198111
$181$	15.8774	1.18016	0.590078	−	0.807346i	$-0.299097\pi$
$181$	15.8774	1.18016	0.590078	+	0.807346i	$0.299097\pi$
$182$	0	0
$183$	0.419560	0.0310148
$184$	−14.8018	−1.09121
$185$	18.4611	1.35728
$186$	2.74586	0.201336
$187$	3.15999	0.231081
$188$	−9.88951	−0.721267
$189$	0	0
$190$	3.25757	0.236329
$191$	−21.0219	−1.52109	−0.760545	−	0.649285i	$-0.775068\pi$
$191$	−21.0219	−1.52109	−0.760545	+	0.649285i	$0.775068\pi$
$192$	0.0349308	0.00252091
$193$	−19.9012	−1.43252	−0.716261	−	0.697833i	$-0.754148\pi$
$193$	−19.9012	−1.43252	−0.716261	+	0.697833i	$0.754148\pi$
$194$	−7.17975	−0.515476
$195$	5.24364	0.375505
$196$	0	0
$197$	−20.1886	−1.43838	−0.719189	−	0.694814i	$-0.755487\pi$
$197$	−20.1886	−1.43838	−0.719189	+	0.694814i	$0.755487\pi$
$198$	−1.78110	−0.126577
$199$	−22.6327	−1.60439	−0.802195	−	0.597062i	$-0.796335\pi$
$199$	−22.6327	−1.60439	−0.802195	+	0.597062i	$0.796335\pi$
$200$	−5.10601	−0.361049
$201$	9.95583	0.702230
$202$	−9.41969	−0.662767
$203$	0	0
$204$	−1.79336	−0.125560
$205$	−1.65938	−0.115896
$206$	6.26914	0.436792
$207$	−6.51222	−0.452631
$208$	−5.59055	−0.387635
$209$	8.77995	0.607322
$210$	0	0
$211$	−4.48974	−0.309086	−0.154543	−	0.987986i	$-0.549391\pi$
$211$	−4.48974	−0.309086	−0.154543	+	0.987986i	$0.549391\pi$
$212$	20.9407	1.43821
$213$	7.39023	0.506370
$214$	10.5058	0.718164
$215$	1.13596	0.0774718
$216$	2.27293	0.154653
$217$	0	0
$218$	−10.9340	−0.740541
$219$	13.9295	0.941272
$220$	7.50173	0.505767
$221$	−3.53797	−0.237990
$222$	7.02071	0.471199
$223$	22.8364	1.52924	0.764620	−	0.644481i	$-0.222926\pi$
$223$	22.8364	1.52924	0.764620	+	0.644481i	$0.222926\pi$
$224$	0	0
$225$	−2.24644	−0.149763
$226$	6.68694	0.444808
$227$	−14.6770	−0.974148	−0.487074	−	0.873361i	$-0.661936\pi$
$227$	−14.6770	−0.974148	−0.487074	+	0.873361i	$0.661936\pi$
$228$	−4.98280	−0.329994
$229$	10.8858	0.719352	0.359676	−	0.933077i	$-0.382887\pi$
$229$	10.8858	0.719352	0.359676	+	0.933077i	$0.382887\pi$
$230$	−6.81942	−0.449659
$231$	0	0
$232$	20.8332	1.36777
$233$	−10.6391	−0.696994	−0.348497	−	0.937310i	$-0.613308\pi$
$233$	−10.6391	−0.696994	−0.348497	+	0.937310i	$0.613308\pi$
$234$	1.99415	0.130361
$235$	−10.2453	−0.668328
$236$	−12.3508	−0.803968
$237$	−0.195831	−0.0127206
$238$	0	0
$239$	3.02612	0.195743	0.0978716	−	0.995199i	$-0.468797\pi$
$239$	3.02612	0.195743	0.0978716	+	0.995199i	$0.468797\pi$
$240$	−2.93573	−0.189500
$241$	18.4381	1.18770	0.593851	−	0.804575i	$-0.297607\pi$
$241$	18.4381	1.18770	0.593851	+	0.804575i	$0.297607\pi$
$242$	1.91473	0.123083
$243$	1.00000	0.0641500
$244$	−0.672036	−0.0430227
$245$	0	0
$246$	−0.631061	−0.0402350
$247$	−9.83017	−0.625479
$248$	−9.88993	−0.628011
$249$	−2.50464	−0.158725
$250$	−7.58828	−0.479925
$251$	1.85759	0.117250	0.0586252	−	0.998280i	$-0.481328\pi$
$251$	1.85759	0.117250	0.0586252	+	0.998280i	$0.481328\pi$
$252$	0	0
$253$	−18.3800	−1.15554
$254$	3.99941	0.250945
$255$	−1.85787	−0.116344
$256$	−7.20248	−0.450155
$257$	−10.6305	−0.663113	−0.331557	−	0.943435i	$-0.607574\pi$
$257$	−10.6305	−0.663113	−0.331557	+	0.943435i	$0.607574\pi$
$258$	0.432003	0.0268953
$259$	0	0
$260$	−8.39906	−0.520888
$261$	9.16578	0.567348
$262$	3.20567	0.198047
$263$	−29.4520	−1.81609	−0.908043	−	0.418878i	$-0.862424\pi$
$263$	−29.4520	−1.81609	−0.908043	+	0.418878i	$0.862424\pi$
$264$	6.41509	0.394821
$265$	21.6940	1.33265
$266$	0	0
$267$	−12.4999	−0.764981
$268$	−15.9469	−0.974110
$269$	−9.23627	−0.563145	−0.281573	−	0.959540i	$-0.590856\pi$
$269$	−9.23627	−0.563145	−0.281573	+	0.959540i	$0.590856\pi$
$270$	1.04717	0.0637289
$271$	−7.69672	−0.467542	−0.233771	−	0.972292i	$-0.575107\pi$
$271$	−7.69672	−0.467542	−0.233771	+	0.972292i	$0.575107\pi$
$272$	1.98079	0.120103
$273$	0	0
$274$	5.12070	0.309353
$275$	−6.34032	−0.382336
$276$	10.4310	0.627874
$277$	−19.4269	−1.16725	−0.583624	−	0.812024i	$-0.698366\pi$
$277$	−19.4269	−1.16725	−0.583624	+	0.812024i	$0.698366\pi$
$278$	13.7318	0.823581
$279$	−4.35118	−0.260498
$280$	0	0
$281$	0.197677	0.0117924	0.00589622	−	0.999983i	$-0.498123\pi$
$281$	0.197677	0.0117924	0.00589622	+	0.999983i	$0.498123\pi$
$282$	−3.89626	−0.232019
$283$	31.9961	1.90197	0.950985	−	0.309238i	$-0.100074\pi$
$283$	31.9961	1.90197	0.950985	+	0.309238i	$0.100074\pi$
$284$	−11.8374	−0.702420
$285$	−5.16206	−0.305774
$286$	5.62825	0.332805
$287$	0	0
$288$	−5.66231	−0.333655
$289$	−15.7465	−0.926262
$290$	9.59816	0.563623
$291$	11.3773	0.666947
$292$	−22.3118	−1.30570
$293$	−17.2175	−1.00586	−0.502930	−	0.864327i	$-0.667745\pi$
$293$	−17.2175	−1.00586	−0.502930	+	0.864327i	$0.667745\pi$
$294$	0	0
$295$	−12.7951	−0.744960
$296$	−25.2869	−1.46977
$297$	2.82239	0.163771
$298$	−9.00183	−0.521462
$299$	20.5786	1.19009
$300$	3.59827	0.207746
$301$	0	0
$302$	−1.44680	−0.0832542
$303$	14.9268	0.857520
$304$	5.50357	0.315651
$305$	−0.696212	−0.0398650
$306$	−0.706545	−0.0403905
$307$	−24.2479	−1.38390	−0.691950	−	0.721946i	$-0.743248\pi$
$307$	−24.2479	−1.38390	−0.691950	+	0.721946i	$0.743248\pi$
$308$	0	0
$309$	−9.93429	−0.565142
$310$	−4.55644	−0.258788
$311$	10.2592	0.581744	0.290872	−	0.956762i	$-0.406055\pi$
$311$	10.2592	0.581744	0.290872	+	0.956762i	$0.406055\pi$
$312$	−7.18244	−0.406625
$313$	−25.6802	−1.45153	−0.725765	−	0.687943i	$-0.758514\pi$
$313$	−25.6802	−1.45153	−0.725765	+	0.687943i	$0.758514\pi$
$314$	15.2412	0.860109
$315$	0	0
$316$	0.313675	0.0176456
$317$	−19.2173	−1.07935	−0.539675	−	0.841874i	$-0.681453\pi$
$317$	−19.2173	−1.07935	−0.539675	+	0.841874i	$0.681453\pi$
$318$	8.25019	0.462648
$319$	25.8694	1.44841
$320$	−0.0579636	−0.00324026
$321$	−16.6479	−0.929195
$322$	0	0
$323$	3.48293	0.193795
$324$	−1.60176	−0.0889868
$325$	7.09873	0.393767
$326$	0.195151	0.0108084
$327$	17.3263	0.958147
$328$	2.27293	0.125502
$329$	0	0
$330$	2.95553	0.162696
$331$	0.536324	0.0294790	0.0147395	−	0.999891i	$-0.495308\pi$
$331$	0.536324	0.0294790	0.0147395	+	0.999891i	$0.495308\pi$
$332$	4.01183	0.220178
$333$	−11.1252	−0.609660
$334$	2.18143	0.119362
$335$	−16.5205	−0.902614
$336$	0	0
$337$	11.5251	0.627810	0.313905	−	0.949454i	$-0.398363\pi$
$337$	11.5251	0.627810	0.313905	+	0.949454i	$0.398363\pi$
$338$	1.90231	0.103472
$339$	−10.5963	−0.575514
$340$	2.97587	0.161389
$341$	−12.2807	−0.665038
$342$	−1.96312	−0.106153
$343$	0	0
$344$	−1.55597	−0.0838924
$345$	10.8063	0.581791
$346$	9.95189	0.535017
$347$	−21.9533	−1.17852	−0.589258	−	0.807945i	$-0.700580\pi$
$347$	−21.9533	−1.17852	−0.589258	+	0.807945i	$0.700580\pi$
$348$	−14.6814	−0.787006
$349$	5.35886	0.286853	0.143427	−	0.989661i	$-0.454188\pi$
$349$	5.35886	0.286853	0.143427	+	0.989661i	$0.454188\pi$
$350$	0	0
$351$	−3.15999	−0.168668
$352$	−15.9812	−0.851803
$353$	9.44187	0.502540	0.251270	−	0.967917i	$-0.419152\pi$
$353$	9.44187	0.502540	0.251270	+	0.967917i	$0.419152\pi$
$354$	−4.86595	−0.258622
$355$	−12.2632	−0.650865
$356$	20.0219	1.06116
$357$	0	0
$358$	−10.4983	−0.554852
$359$	−4.43594	−0.234120	−0.117060	−	0.993125i	$-0.537347\pi$
$359$	−4.43594	−0.234120	−0.117060	+	0.993125i	$0.537347\pi$
$360$	−3.77167	−0.198784
$361$	−9.32277	−0.490672
$362$	−10.0196	−0.526618
$363$	−3.03414	−0.159251
$364$	0	0
$365$	−23.1145	−1.20987
$366$	−0.264768	−0.0138397
$367$	−7.96777	−0.415914	−0.207957	−	0.978138i	$-0.566681\pi$
$367$	−7.96777	−0.415914	−0.207957	+	0.978138i	$0.566681\pi$
$368$	−11.5212	−0.600584
$369$	1.00000	0.0520579
$370$	−11.6501	−0.605658
$371$	0	0
$372$	6.96956	0.361355
$373$	−16.7465	−0.867103	−0.433552	−	0.901129i	$-0.642740\pi$
$373$	−16.7465	−0.867103	−0.433552	+	0.901129i	$0.642740\pi$
$374$	−1.99414	−0.103115
$375$	12.0246	0.620950
$376$	14.0334	0.723717
$377$	−28.9638	−1.49171
$378$	0	0
$379$	−7.02596	−0.360899	−0.180450	−	0.983584i	$-0.557755\pi$
$379$	−7.02596	−0.360899	−0.180450	+	0.983584i	$0.557755\pi$
$380$	8.26839	0.424159
$381$	−6.33759	−0.324685
$382$	13.2661	0.678752
$383$	−23.3436	−1.19280	−0.596401	−	0.802687i	$-0.703403\pi$
$383$	−23.3436	−1.19280	−0.596401	+	0.802687i	$0.703403\pi$
$384$	11.3026	0.576783
$385$	0	0
$386$	12.5589	0.639230
$387$	−0.684567	−0.0347985
$388$	−18.2237	−0.925167
$389$	1.37396	0.0696626	0.0348313	−	0.999393i	$-0.488911\pi$
$389$	1.37396	0.0696626	0.0348313	+	0.999393i	$0.488911\pi$
$390$	−3.30905	−0.167560
$391$	−7.29118	−0.368731
$392$	0	0
$393$	−5.07981	−0.256243
$394$	12.7402	0.641844
$395$	0.324960	0.0163505
$396$	−4.52079	−0.227178
$397$	27.4009	1.37521	0.687605	−	0.726085i	$-0.258662\pi$
$397$	27.4009	1.37521	0.687605	+	0.726085i	$0.258662\pi$
$398$	14.2826	0.715923
$399$	0	0
$400$	−3.97433	−0.198716
$401$	−2.84574	−0.142109	−0.0710547	−	0.997472i	$-0.522636\pi$
$401$	−2.84574	−0.142109	−0.0710547	+	0.997472i	$0.522636\pi$
$402$	−6.28273	−0.313354
$403$	13.7497	0.684921
$404$	−23.9091	−1.18952
$405$	−1.65938	−0.0824555
$406$	0	0
$407$	−31.3997	−1.55643
$408$	2.54481	0.125987
$409$	28.4383	1.40619	0.703093	−	0.711098i	$-0.251802\pi$
$409$	28.4383	1.40619	0.703093	+	0.711098i	$0.251802\pi$
$410$	1.04717	0.0517162
$411$	−8.11444	−0.400256
$412$	15.9124	0.783946
$413$	0	0
$414$	4.10961	0.201976
$415$	4.15615	0.204018
$416$	17.8928	0.877269
$417$	−21.7599	−1.06559
$418$	−5.54068	−0.271004
$419$	10.4579	0.510904	0.255452	−	0.966822i	$-0.417776\pi$
$419$	10.4579	0.510904	0.255452	+	0.966822i	$0.417776\pi$
$420$	0	0
$421$	−18.4016	−0.896839	−0.448419	−	0.893823i	$-0.648013\pi$
$421$	−18.4016	−0.896839	−0.448419	+	0.893823i	$0.648013\pi$
$422$	2.83330	0.137923
$423$	6.17414	0.300197
$424$	−29.7152	−1.44310
$425$	−2.51515	−0.122003
$426$	−4.66368	−0.225956
$427$	0	0
$428$	26.6660	1.28895
$429$	−8.91871	−0.430599
$430$	−0.716860	−0.0345700
$431$	−7.16006	−0.344888	−0.172444	−	0.985019i	$-0.555166\pi$
$431$	−7.16006	−0.344888	−0.172444	+	0.985019i	$0.555166\pi$
$432$	1.76917	0.0851191
$433$	32.8811	1.58016	0.790082	−	0.613001i	$-0.210038\pi$
$433$	32.8811	1.58016	0.790082	+	0.613001i	$0.210038\pi$
$434$	0	0
$435$	−15.2096	−0.729243
$436$	−27.7526	−1.32911
$437$	−20.2584	−0.969090
$438$	−8.79039	−0.420021
$439$	3.76719	0.179798	0.0898991	−	0.995951i	$-0.471346\pi$
$439$	3.76719	0.179798	0.0898991	+	0.995951i	$0.471346\pi$
$440$	−10.6451	−0.507485
$441$	0	0
$442$	2.23267	0.106198
$443$	30.3572	1.44232	0.721158	−	0.692770i	$-0.243610\pi$
$443$	30.3572	1.44232	0.721158	+	0.692770i	$0.243610\pi$
$444$	17.8200	0.845700
$445$	20.7421	0.983272
$446$	−14.4112	−0.682389
$447$	14.2646	0.674692
$448$	0	0
$449$	22.5523	1.06431	0.532154	−	0.846648i	$-0.321383\pi$
$449$	22.5523	1.06431	0.532154	+	0.846648i	$0.321383\pi$
$450$	1.41764	0.0668282
$451$	2.82239	0.132901
$452$	16.9728	0.798334
$453$	2.29265	0.107718
$454$	9.26209	0.434691
$455$	0	0
$456$	7.07069	0.331115
$457$	−14.6973	−0.687509	−0.343755	−	0.939060i	$-0.611699\pi$
$457$	−14.6973	−0.687509	−0.343755	+	0.939060i	$0.611699\pi$
$458$	−6.86959	−0.320995
$459$	1.11961	0.0522591
$460$	−17.3091	−0.807041
$461$	−19.3426	−0.900874	−0.450437	−	0.892808i	$-0.648732\pi$
$461$	−19.3426	−0.900874	−0.450437	+	0.892808i	$0.648732\pi$
$462$	0	0
$463$	3.19576	0.148520	0.0742598	−	0.997239i	$-0.476341\pi$
$463$	3.19576	0.148520	0.0742598	+	0.997239i	$0.476341\pi$
$464$	16.2158	0.752799
$465$	7.22028	0.334833
$466$	6.71395	0.311018
$467$	−17.7807	−0.822791	−0.411396	−	0.911457i	$-0.634958\pi$
$467$	−17.7807	−0.822791	−0.411396	+	0.911457i	$0.634958\pi$
$468$	5.06155	0.233970
$469$	0	0
$470$	6.46539	0.298226
$471$	−24.1517	−1.11285
$472$	17.5260	0.806700
$473$	−1.93211	−0.0888386
$474$	0.123581	0.00567628
$475$	−6.98829	−0.320645
$476$	0	0
$477$	−13.0735	−0.598596
$478$	−1.90966	−0.0873460
$479$	−34.2608	−1.56542	−0.782709	−	0.622388i	$-0.786163\pi$
$479$	−34.2608	−1.56542	−0.782709	+	0.622388i	$0.786163\pi$
$480$	9.39596	0.428865
$481$	35.1557	1.60296
$482$	−11.6356	−0.529985
$483$	0	0
$484$	4.85997	0.220908
$485$	−18.8793	−0.857263
$486$	−0.631061	−0.0286255
$487$	29.6230	1.34234	0.671172	−	0.741301i	$-0.265791\pi$
$487$	29.6230	1.34234	0.671172	+	0.741301i	$0.265791\pi$
$488$	0.953632	0.0431689
$489$	−0.309243	−0.0139845
$490$	0	0
$491$	−15.7558	−0.711049	−0.355525	−	0.934667i	$-0.615698\pi$
$491$	−15.7558	−0.711049	−0.355525	+	0.934667i	$0.615698\pi$
$492$	−1.60176	−0.0722130
$493$	10.2621	0.462184
$494$	6.20344	0.279106
$495$	−4.68342	−0.210504
$496$	−7.69796	−0.345649
$497$	0	0
$498$	1.58058	0.0708274
$499$	3.95585	0.177088	0.0885441	−	0.996072i	$-0.471779\pi$
$499$	3.95585	0.177088	0.0885441	+	0.996072i	$0.471779\pi$
$500$	−19.2606	−0.861361
$501$	−3.45676	−0.154437
$502$	−1.17226	−0.0523203
$503$	−33.0299	−1.47273	−0.736366	−	0.676583i	$-0.763460\pi$
$503$	−33.0299	−1.47273	−0.736366	+	0.676583i	$0.763460\pi$
$504$	0	0
$505$	−24.7692	−1.10222
$506$	11.5989	0.515634
$507$	−3.01447	−0.133877
$508$	10.1513	0.450392
$509$	25.8609	1.14626	0.573131	−	0.819463i	$-0.305728\pi$
$509$	25.8609	1.14626	0.573131	+	0.819463i	$0.305728\pi$
$510$	1.17243	0.0519161
$511$	0	0
$512$	−18.0600	−0.798145
$513$	3.11082	0.137346
$514$	6.70850	0.295899
$515$	16.4848	0.726407
$516$	1.09651	0.0482713
$517$	17.4258	0.766386
$518$	0	0
$519$	−15.7701	−0.692230
$520$	11.9184	0.522658
$521$	1.48131	0.0648972	0.0324486	−	0.999473i	$-0.489669\pi$
$521$	1.48131	0.0648972	0.0324486	+	0.999473i	$0.489669\pi$
$522$	−5.78417	−0.253166
$523$	30.6601	1.34067	0.670336	−	0.742057i	$-0.266150\pi$
$523$	30.6601	1.34067	0.670336	+	0.742057i	$0.266150\pi$
$524$	8.13665	0.355451
$525$	0	0
$526$	18.5860	0.810387
$527$	−4.87165	−0.212212
$528$	4.99327	0.217304
$529$	19.4090	0.843871
$530$	−13.6902	−0.594666
$531$	7.71075	0.334618
$532$	0	0
$533$	−3.15999	−0.136874
$534$	7.88820	0.341356
$535$	27.6253	1.19434
$536$	22.6289	0.977420
$537$	16.6359	0.717894
$538$	5.82865	0.251291
$539$	0	0
$540$	2.65794	0.114380
$541$	−6.28709	−0.270303	−0.135151	−	0.990825i	$-0.543152\pi$
$541$	−6.28709	−0.270303	−0.135151	+	0.990825i	$0.543152\pi$
$542$	4.85710	0.208630
$543$	15.8774	0.681363
$544$	−6.33961	−0.271809
$545$	−28.7510	−1.23156
$546$	0	0
$547$	13.2731	0.567515	0.283758	−	0.958896i	$-0.408419\pi$
$547$	13.2731	0.567515	0.283758	+	0.958896i	$0.408419\pi$
$548$	12.9974	0.555221
$549$	0.419560	0.0179064
$550$	4.00113	0.170609
$551$	28.5131	1.21470
$552$	−14.8018	−0.630008
$553$	0	0
$554$	12.2595	0.520858
$555$	18.4611	0.783629
$556$	34.8542	1.47815
$557$	−31.4107	−1.33091	−0.665457	−	0.746436i	$-0.731763\pi$
$557$	−31.4107	−1.33091	−0.665457	+	0.746436i	$0.731763\pi$
$558$	2.74586	0.116242
$559$	2.16322	0.0914946
$560$	0	0
$561$	3.15999	0.133415
$562$	−0.124746	−0.00526211
$563$	0.492351	0.0207501	0.0103751	−	0.999946i	$-0.496697\pi$
$563$	0.492351	0.0207501	0.0103751	+	0.999946i	$0.496697\pi$
$564$	−9.88951	−0.416423
$565$	17.5834	0.739740
$566$	−20.1915	−0.848711
$567$	0	0
$568$	16.7975	0.704807
$569$	8.70460	0.364916	0.182458	−	0.983214i	$-0.441595\pi$
$569$	8.70460	0.364916	0.182458	+	0.983214i	$0.441595\pi$
$570$	3.25757	0.136445
$571$	−36.8663	−1.54281	−0.771403	−	0.636347i	$-0.780445\pi$
$571$	−36.8663	−1.54281	−0.771403	+	0.636347i	$0.780445\pi$
$572$	14.2856	0.597313
$573$	−21.0219	−0.878202
$574$	0	0
$575$	14.6293	0.610085
$576$	0.0349308	0.00145545
$577$	0.227862	0.00948604	0.00474302	−	0.999989i	$-0.498490\pi$
$577$	0.227862	0.00948604	0.00474302	+	0.999989i	$0.498490\pi$
$578$	9.93698	0.413324
$579$	−19.9012	−0.827067
$580$	24.3621	1.01158
$581$	0	0
$582$	−7.17975	−0.297610
$583$	−36.8985	−1.52818
$584$	31.6609	1.31014
$585$	5.24364	0.216798
$586$	10.8653	0.448842
$587$	−4.58547	−0.189262	−0.0946312	−	0.995512i	$-0.530167\pi$
$587$	−4.58547	−0.189262	−0.0946312	+	0.995512i	$0.530167\pi$
$588$	0	0
$589$	−13.5358	−0.557731
$590$	8.07449	0.332421
$591$	−20.1886	−0.830448
$592$	−19.6824	−0.808942
$593$	−9.17184	−0.376642	−0.188321	−	0.982108i	$-0.560305\pi$
$593$	−9.17184	−0.376642	−0.188321	+	0.982108i	$0.560305\pi$
$594$	−1.78110	−0.0730793
$595$	0	0
$596$	−22.8485	−0.935911
$597$	−22.6327	−0.926295
$598$	−12.9863	−0.531050
$599$	−36.1676	−1.47777	−0.738884	−	0.673832i	$-0.764647\pi$
$599$	−36.1676	−1.47777	−0.738884	+	0.673832i	$0.764647\pi$
$600$	−5.10601	−0.208452
$601$	1.30057	0.0530515	0.0265258	−	0.999648i	$-0.491556\pi$
$601$	1.30057	0.0530515	0.0265258	+	0.999648i	$0.491556\pi$
$602$	0	0
$603$	9.95583	0.405433
$604$	−3.67229	−0.149423
$605$	5.03480	0.204694
$606$	−9.41969	−0.382649
$607$	19.2109	0.779745	0.389872	−	0.920869i	$-0.372519\pi$
$607$	19.2109	0.779745	0.389872	+	0.920869i	$0.372519\pi$
$608$	−17.6145	−0.714361
$609$	0	0
$610$	0.439352	0.0177889
$611$	−19.5102	−0.789299
$612$	−1.79336	−0.0724922
$613$	8.59390	0.347104	0.173552	−	0.984825i	$-0.444475\pi$
$613$	8.59390	0.347104	0.173552	+	0.984825i	$0.444475\pi$
$614$	15.3019	0.617534
$615$	−1.65938	−0.0669129
$616$	0	0
$617$	1.93896	0.0780595	0.0390298	−	0.999238i	$-0.487573\pi$
$617$	1.93896	0.0780595	0.0390298	+	0.999238i	$0.487573\pi$
$618$	6.26914	0.252182
$619$	4.54259	0.182582	0.0912910	−	0.995824i	$-0.470901\pi$
$619$	4.54259	0.182582	0.0912910	+	0.995824i	$0.470901\pi$
$620$	−11.5652	−0.464469
$621$	−6.51222	−0.261326
$622$	−6.47416	−0.259590
$623$	0	0
$624$	−5.59055	−0.223801
$625$	−8.72129	−0.348852
$626$	16.2058	0.647712
$627$	8.77995	0.350637
$628$	38.6852	1.54371
$629$	−12.4560	−0.496653
$630$	0	0
$631$	−42.7975	−1.70374	−0.851870	−	0.523754i	$-0.824531\pi$
$631$	−42.7975	−1.70374	−0.851870	+	0.523754i	$0.824531\pi$
$632$	−0.445111	−0.0177056
$633$	−4.48974	−0.178451
$634$	12.1273	0.481635
$635$	10.5165	0.417335
$636$	20.9407	0.830352
$637$	0	0
$638$	−16.3251	−0.646319
$639$	7.39023	0.292353
$640$	−18.7553	−0.741370
$641$	30.8886	1.22003	0.610013	−	0.792391i	$-0.291164\pi$
$641$	30.8886	1.22003	0.610013	+	0.792391i	$0.291164\pi$
$642$	10.5058	0.414632
$643$	16.5004	0.650712	0.325356	−	0.945592i	$-0.394516\pi$
$643$	16.5004	0.650712	0.325356	+	0.945592i	$0.394516\pi$
$644$	0	0
$645$	1.13596	0.0447284
$646$	−2.19794	−0.0864767
$647$	−22.9982	−0.904152	−0.452076	−	0.891979i	$-0.649316\pi$
$647$	−22.9982	−0.904152	−0.452076	+	0.891979i	$0.649316\pi$
$648$	2.27293	0.0892892
$649$	21.7627	0.854261
$650$	−4.47973	−0.175709
$651$	0	0
$652$	0.495334	0.0193988
$653$	26.0313	1.01868	0.509342	−	0.860564i	$-0.329889\pi$
$653$	26.0313	1.01868	0.509342	+	0.860564i	$0.329889\pi$
$654$	−10.9340	−0.427551
$655$	8.42936	0.329362
$656$	1.76917	0.0690744
$657$	13.9295	0.543443
$658$	0	0
$659$	13.0247	0.507369	0.253684	−	0.967287i	$-0.418358\pi$
$659$	13.0247	0.507369	0.253684	+	0.967287i	$0.418358\pi$
$660$	7.50173	0.292005
$661$	−26.1475	−1.01702	−0.508510	−	0.861056i	$-0.669803\pi$
$661$	−26.1475	−1.01702	−0.508510	+	0.861056i	$0.669803\pi$
$662$	−0.338453	−0.0131544
$663$	−3.53797	−0.137403
$664$	−5.69286	−0.220926
$665$	0	0
$666$	7.02071	0.272047
$667$	−59.6896	−2.31119
$668$	5.53691	0.214229
$669$	22.8364	0.882908
$670$	10.4255	0.402771
$671$	1.18416	0.0457140
$672$	0	0
$673$	10.6856	0.411901	0.205951	−	0.978562i	$-0.433971\pi$
$673$	10.6856	0.411901	0.205951	+	0.978562i	$0.433971\pi$
$674$	−7.27302	−0.280146
$675$	−2.24644	−0.0864656
$676$	4.82846	0.185710
$677$	−16.2993	−0.626431	−0.313216	−	0.949682i	$-0.601406\pi$
$677$	−16.2993	−0.626431	−0.313216	+	0.949682i	$0.601406\pi$
$678$	6.68694	0.256810
$679$	0	0
$680$	−4.22282	−0.161938
$681$	−14.6770	−0.562424
$682$	7.74987	0.296758
$683$	6.89890	0.263979	0.131990	−	0.991251i	$-0.457863\pi$
$683$	6.89890	0.263979	0.131990	+	0.991251i	$0.457863\pi$
$684$	−4.98280	−0.190522
$685$	13.4650	0.514470
$686$	0	0
$687$	10.8858	0.415318
$688$	−1.21111	−0.0461732
$689$	41.3122	1.57387
$690$	−6.81942	−0.259611
$691$	10.9192	0.415385	0.207692	−	0.978194i	$-0.433405\pi$
$691$	10.9192	0.415385	0.207692	+	0.978194i	$0.433405\pi$
$692$	25.2599	0.960239
$693$	0	0
$694$	13.8539	0.525886
$695$	36.1081	1.36966
$696$	20.8332	0.789680
$697$	1.11961	0.0424084
$698$	−3.38177	−0.128002
$699$	−10.6391	−0.402409
$700$	0	0
$701$	−34.1965	−1.29158	−0.645791	−	0.763514i	$-0.723472\pi$
$701$	−34.1965	−1.29158	−0.645791	+	0.763514i	$0.723472\pi$
$702$	1.99415	0.0752642
$703$	−34.6087	−1.30529
$704$	0.0985881	0.00371568
$705$	−10.2453	−0.385860
$706$	−5.95840	−0.224247
$707$	0	0
$708$	−12.3508	−0.464171
$709$	16.4538	0.617934	0.308967	−	0.951073i	$-0.400017\pi$
$709$	16.4538	0.617934	0.308967	+	0.951073i	$0.400017\pi$
$710$	7.73885	0.290434
$711$	−0.195831	−0.00734425
$712$	−28.4114	−1.06476
$713$	28.3359	1.06119
$714$	0	0
$715$	14.7996	0.553473
$716$	−26.6468	−0.995838
$717$	3.02612	0.113012
$718$	2.79935	0.104471
$719$	17.6106	0.656764	0.328382	−	0.944545i	$-0.393497\pi$
$719$	17.6106	0.656764	0.328382	+	0.944545i	$0.393497\pi$
$720$	−2.93573	−0.109408
$721$	0	0
$722$	5.88323	0.218951
$723$	18.4381	0.685720
$724$	−25.4318	−0.945164
$725$	−20.5904	−0.764708
$726$	1.91473	0.0710622
$727$	34.6841	1.28636	0.643181	−	0.765714i	$-0.277614\pi$
$727$	34.6841	1.28636	0.643181	+	0.765714i	$0.277614\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	14.5866	0.539876
$731$	−0.766451	−0.0283482
$732$	−0.672036	−0.0248392
$733$	14.9917	0.553731	0.276865	−	0.960909i	$-0.410704\pi$
$733$	14.9917	0.553731	0.276865	+	0.960909i	$0.410704\pi$
$734$	5.02814	0.185592
$735$	0	0
$736$	36.8742	1.35920
$737$	28.0992	1.03505
$738$	−0.631061	−0.0232297
$739$	11.5757	0.425820	0.212910	−	0.977072i	$-0.431706\pi$
$739$	11.5757	0.425820	0.212910	+	0.977072i	$0.431706\pi$
$740$	−29.5702	−1.08702
$741$	−9.83017	−0.361120
$742$	0	0
$743$	25.7503	0.944687	0.472343	−	0.881415i	$-0.343408\pi$
$743$	25.7503	0.944687	0.472343	+	0.881415i	$0.343408\pi$
$744$	−9.88993	−0.362583
$745$	−23.6705	−0.867218
$746$	10.5681	0.386925
$747$	−2.50464	−0.0916398
$748$	−5.06154	−0.185068
$749$	0	0
$750$	−7.58828	−0.277085
$751$	−14.3476	−0.523550	−0.261775	−	0.965129i	$-0.584308\pi$
$751$	−14.3476	−0.523550	−0.261775	+	0.965129i	$0.584308\pi$
$752$	10.9231	0.398324
$753$	1.85759	0.0676945
$754$	18.2779	0.665642
$755$	−3.80439	−0.138456
$756$	0	0
$757$	52.9415	1.92419	0.962095	−	0.272714i	$-0.0879213\pi$
$757$	52.9415	1.92419	0.962095	+	0.272714i	$0.0879213\pi$
$758$	4.43381	0.161043
$759$	−18.3800	−0.667152
$760$	−11.7330	−0.425601
$761$	23.8612	0.864969	0.432485	−	0.901641i	$-0.357637\pi$
$761$	23.8612	0.864969	0.432485	+	0.901641i	$0.357637\pi$
$762$	3.99941	0.144883
$763$	0	0
$764$	33.6721	1.21821
$765$	−1.85787	−0.0671715
$766$	14.7312	0.532261
$767$	−24.3659	−0.879801
$768$	−7.20248	−0.259897
$769$	40.3431	1.45481	0.727405	−	0.686208i	$-0.240726\pi$
$769$	40.3431	1.45481	0.727405	+	0.686208i	$0.240726\pi$
$770$	0	0
$771$	−10.6305	−0.382849
$772$	31.8770	1.14728
$773$	−23.3193	−0.838736	−0.419368	−	0.907816i	$-0.637748\pi$
$773$	−23.3193	−0.838736	−0.419368	+	0.907816i	$0.637748\pi$
$774$	0.432003	0.0155280
$775$	9.77467	0.351117
$776$	25.8597	0.928311
$777$	0	0
$778$	−0.867053	−0.0310854
$779$	3.11082	0.111457
$780$	−8.39906	−0.300735
$781$	20.8581	0.746361
$782$	4.60118	0.164538
$783$	9.16578	0.327558
$784$	0	0
$785$	40.0769	1.43041
$786$	3.20567	0.114342
$787$	−38.4755	−1.37150	−0.685752	−	0.727835i	$-0.740527\pi$
$787$	−38.4755	−1.37150	−0.685752	+	0.727835i	$0.740527\pi$
$788$	32.3373	1.15197
$789$	−29.4520	−1.04852
$790$	−0.205069	−0.00729604
$791$	0	0
$792$	6.41509	0.227950
$793$	−1.32581	−0.0470808
$794$	−17.2916	−0.613657
$795$	21.6940	0.769408
$796$	36.2522	1.28493
$797$	8.86744	0.314101	0.157050	−	0.987591i	$-0.449802\pi$
$797$	8.86744	0.314101	0.157050	+	0.987591i	$0.449802\pi$
$798$	0	0
$799$	6.91266	0.244552
$800$	12.7201	0.449722
$801$	−12.4999	−0.441662
$802$	1.79583	0.0634131
$803$	39.3146	1.38738
$804$	−15.9469	−0.562403
$805$	0	0
$806$	−8.67689	−0.305630
$807$	−9.23627	−0.325132
$808$	33.9275	1.19357
$809$	−8.17161	−0.287299	−0.143649	−	0.989629i	$-0.545884\pi$
$809$	−8.17161	−0.287299	−0.143649	+	0.989629i	$0.545884\pi$
$810$	1.04717	0.0367939
$811$	5.53278	0.194282	0.0971411	−	0.995271i	$-0.469030\pi$
$811$	5.53278	0.194282	0.0971411	+	0.995271i	$0.469030\pi$
$812$	0	0
$813$	−7.69672	−0.269936
$814$	19.8151	0.694520
$815$	0.513153	0.0179750
$816$	1.98079	0.0693414
$817$	−2.12957	−0.0745041
$818$	−17.9463	−0.627478
$819$	0	0
$820$	2.65794	0.0928193
$821$	6.89627	0.240681	0.120341	−	0.992733i	$-0.461601\pi$
$821$	6.89627	0.240681	0.120341	+	0.992733i	$0.461601\pi$
$822$	5.12070	0.178605
$823$	−37.9259	−1.32201	−0.661006	−	0.750380i	$-0.729870\pi$
$823$	−37.9259	−1.32201	−0.661006	+	0.750380i	$0.729870\pi$
$824$	−22.5800	−0.786610
$825$	−6.34032	−0.220742
$826$	0	0
$827$	22.6357	0.787121	0.393561	−	0.919299i	$-0.371243\pi$
$827$	22.6357	0.787121	0.393561	+	0.919299i	$0.371243\pi$
$828$	10.4310	0.362503
$829$	34.6081	1.20199	0.600995	−	0.799253i	$-0.294771\pi$
$829$	34.6081	1.20199	0.600995	+	0.799253i	$0.294771\pi$
$830$	−2.62279	−0.0910382
$831$	−19.4269	−0.673910
$832$	−0.110381	−0.00382677
$833$	0	0
$834$	13.7318	0.475495
$835$	5.73610	0.198506
$836$	−14.0634	−0.486392
$837$	−4.35118	−0.150399
$838$	−6.59960	−0.227979
$839$	−53.3934	−1.84334	−0.921672	−	0.387969i	$-0.873177\pi$
$839$	−53.3934	−1.84334	−0.921672	+	0.387969i	$0.873177\pi$
$840$	0	0
$841$	55.0116	1.89695
$842$	11.6125	0.400194
$843$	0.197677	0.00680837
$844$	7.19149	0.247541
$845$	5.00216	0.172080
$846$	−3.89626	−0.133956
$847$	0	0
$848$	−23.1293	−0.794262
$849$	31.9961	1.09810
$850$	1.58721	0.0544409
$851$	72.4501	2.48356
$852$	−11.8374	−0.405542
$853$	9.56536	0.327512	0.163756	−	0.986501i	$-0.447639\pi$
$853$	9.56536	0.327512	0.163756	+	0.986501i	$0.447639\pi$
$854$	0	0
$855$	−5.16206	−0.176539
$856$	−37.8395	−1.29333
$857$	2.78473	0.0951246	0.0475623	−	0.998868i	$-0.484855\pi$
$857$	2.78473	0.0951246	0.0475623	+	0.998868i	$0.484855\pi$
$858$	5.62825	0.192145
$859$	−11.2578	−0.384110	−0.192055	−	0.981384i	$-0.561515\pi$
$859$	−11.2578	−0.384110	−0.192055	+	0.981384i	$0.561515\pi$
$860$	−1.81954	−0.0620457
$861$	0	0
$862$	4.51843	0.153899
$863$	−5.08316	−0.173033	−0.0865163	−	0.996250i	$-0.527573\pi$
$863$	−5.08316	−0.173033	−0.0865163	+	0.996250i	$0.527573\pi$
$864$	−5.66231	−0.192636
$865$	26.1687	0.889761
$866$	−20.7500	−0.705112
$867$	−15.7465	−0.534778
$868$	0	0
$869$	−0.552711	−0.0187495
$870$	9.59816	0.325408
$871$	−31.4603	−1.06599
$872$	39.3815	1.33363
$873$	11.3773	0.385062
$874$	12.7843	0.432434
$875$	0	0
$876$	−22.3118	−0.753847
$877$	−24.7658	−0.836282	−0.418141	−	0.908382i	$-0.637318\pi$
$877$	−24.7658	−0.836282	−0.418141	+	0.908382i	$0.637318\pi$
$878$	−2.37733	−0.0802308
$879$	−17.2175	−0.580733
$880$	−8.28576	−0.279313
$881$	−13.0909	−0.441044	−0.220522	−	0.975382i	$-0.570776\pi$
$881$	−13.0909	−0.441044	−0.220522	+	0.975382i	$0.570776\pi$
$882$	0	0
$883$	−44.5002	−1.49755	−0.748775	−	0.662824i	$-0.769358\pi$
$883$	−44.5002	−1.49755	−0.748775	+	0.662824i	$0.769358\pi$
$884$	5.66699	0.190601
$885$	−12.7951	−0.430103
$886$	−19.1573	−0.643601
$887$	22.0697	0.741028	0.370514	−	0.928827i	$-0.379181\pi$
$887$	22.0697	0.741028	0.370514	+	0.928827i	$0.379181\pi$
$888$	−25.2869	−0.848573
$889$	0	0
$890$	−13.0896	−0.438763
$891$	2.82239	0.0945535
$892$	−36.5785	−1.22474
$893$	19.2067	0.642727
$894$	−9.00183	−0.301066
$895$	−27.6054	−0.922747
$896$	0	0
$897$	20.5786	0.687098
$898$	−14.2319	−0.474923
$899$	−39.8820	−1.33014
$900$	3.59827	0.119942
$901$	−14.6373	−0.487640
$902$	−1.78110	−0.0593040
$903$	0	0
$904$	−24.0848	−0.801047
$905$	−26.3467	−0.875793
$906$	−1.44680	−0.0480668
$907$	−9.62739	−0.319672	−0.159836	−	0.987144i	$-0.551097\pi$
$907$	−9.62739	−0.319672	−0.159836	+	0.987144i	$0.551097\pi$
$908$	23.5091	0.780176
$909$	14.9268	0.495089
$910$	0	0
$911$	−55.0618	−1.82428	−0.912140	−	0.409880i	$-0.865571\pi$
$911$	−55.0618	−1.82428	−0.912140	+	0.409880i	$0.865571\pi$
$912$	5.50357	0.182241
$913$	−7.06905	−0.233951
$914$	9.27487	0.306785
$915$	−0.696212	−0.0230161
$916$	−17.4364	−0.576116
$917$	0	0
$918$	−0.706545	−0.0233195
$919$	−18.9673	−0.625673	−0.312837	−	0.949807i	$-0.601279\pi$
$919$	−18.9673	−0.625673	−0.312837	+	0.949807i	$0.601279\pi$
$920$	24.5619	0.809783
$921$	−24.2479	−0.798995
$922$	12.2064	0.401995
$923$	−23.3530	−0.768675
$924$	0	0
$925$	24.9922	0.821739
$926$	−2.01672	−0.0662735
$927$	−9.93429	−0.326285
$928$	−51.8995	−1.70369
$929$	25.2606	0.828773	0.414386	−	0.910101i	$-0.363996\pi$
$929$	25.2606	0.828773	0.414386	+	0.910101i	$0.363996\pi$
$930$	−4.55644	−0.149411
$931$	0	0
$932$	17.0414	0.558209
$933$	10.2592	0.335870
$934$	11.2207	0.367152
$935$	−5.24363	−0.171485
$936$	−7.18244	−0.234765
$937$	13.5468	0.442554	0.221277	−	0.975211i	$-0.428978\pi$
$937$	13.5468	0.442554	0.221277	+	0.975211i	$0.428978\pi$
$938$	0	0
$939$	−25.6802	−0.838041
$940$	16.4105	0.535252
$941$	−33.7758	−1.10106	−0.550529	−	0.834816i	$-0.685574\pi$
$941$	−33.7758	−1.10106	−0.550529	+	0.834816i	$0.685574\pi$
$942$	15.2412	0.496584
$943$	−6.51222	−0.212067
$944$	13.6416	0.443996
$945$	0	0
$946$	1.21928	0.0396422
$947$	23.8236	0.774162	0.387081	−	0.922046i	$-0.373483\pi$
$947$	23.8236	0.774162	0.387081	+	0.922046i	$0.373483\pi$
$948$	0.313675	0.0101877
$949$	−44.0172	−1.42886
$950$	4.41003	0.143080
$951$	−19.2173	−0.623163
$952$	0	0
$953$	−18.8482	−0.610553	−0.305277	−	0.952264i	$-0.598749\pi$
$953$	−18.8482	−0.610553	−0.305277	+	0.952264i	$0.598749\pi$
$954$	8.25019	0.267110
$955$	34.8834	1.12880
$956$	−4.84712	−0.156767
$957$	25.8694	0.836238
$958$	21.6207	0.698532
$959$	0	0
$960$	−0.0579636	−0.00187077
$961$	−12.0672	−0.389265
$962$	−22.1854	−0.715285
$963$	−16.6479	−0.536471
$964$	−29.5334	−0.951208
$965$	33.0238	1.06307
$966$	0	0
$967$	−45.4505	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$967$	−45.4505	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$968$	−6.89639	−0.221658
$969$	3.48293	0.111888
$970$	11.9140	0.382534
$971$	−57.0677	−1.83139	−0.915695	−	0.401874i	$-0.868359\pi$
$971$	−57.0677	−1.83139	−0.915695	+	0.401874i	$0.868359\pi$
$972$	−1.60176	−0.0513765
$973$	0	0
$974$	−18.6939	−0.598991
$975$	7.09873	0.227341
$976$	0.742272	0.0237596
$977$	−15.5246	−0.496675	−0.248337	−	0.968674i	$-0.579884\pi$
$977$	−15.5246	−0.496675	−0.248337	+	0.968674i	$0.579884\pi$
$978$	0.195151	0.00624025
$979$	−35.2795	−1.12754
$980$	0	0
$981$	17.3263	0.553186
$982$	9.94287	0.317290
$983$	−10.3050	−0.328678	−0.164339	−	0.986404i	$-0.552549\pi$
$983$	−10.3050	−0.328678	−0.164339	+	0.986404i	$0.552549\pi$
$984$	2.27293	0.0724584
$985$	33.5007	1.06742
$986$	−6.47604	−0.206239
$987$	0	0
$988$	15.7456	0.500934
$989$	4.45805	0.141758
$990$	2.95553	0.0939328
$991$	44.2241	1.40483	0.702413	−	0.711770i	$-0.252106\pi$
$991$	44.2241	1.40483	0.702413	+	0.711770i	$0.252106\pi$
$992$	24.6378	0.782249
$993$	0.536324	0.0170197
$994$	0	0
$995$	37.5564	1.19062
$996$	4.01183	0.127120
$997$	−33.4105	−1.05812	−0.529061	−	0.848584i	$-0.677456\pi$
$997$	−33.4105	−1.05812	−0.529061	+	0.848584i	$0.677456\pi$
$998$	−2.49638	−0.0790216
$999$	−11.1252	−0.351987

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.bm.1.8	yes	16
7.6	odd	2		6027.2.a.bl.1.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.bl.1.8	✓	16	7.6	odd	2
6027.2.a.bm.1.8	yes	16	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q-0.631061 q^{2} +1.00000 q^{3} -1.60176 q^{4} -1.65938 q^{5} -0.631061 q^{6} +2.27293 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.631061 q^{2} +1.00000 q^{3} -1.60176 q^{4} -1.65938 q^{5} -0.631061 q^{6} +2.27293 q^{8} +1.00000 q^{9} +1.04717 q^{10} +2.82239 q^{11} -1.60176 q^{12} -3.15999 q^{13} -1.65938 q^{15} +1.76917 q^{16} +1.11961 q^{17} -0.631061 q^{18} +3.11082 q^{19} +2.65794 q^{20} -1.78110 q^{22} -6.51222 q^{23} +2.27293 q^{24} -2.24644 q^{25} +1.99415 q^{26} +1.00000 q^{27} +9.16578 q^{29} +1.04717 q^{30} -4.35118 q^{31} -5.66231 q^{32} +2.82239 q^{33} -0.706545 q^{34} -1.60176 q^{36} -11.1252 q^{37} -1.96312 q^{38} -3.15999 q^{39} -3.77167 q^{40} +1.00000 q^{41} -0.684567 q^{43} -4.52079 q^{44} -1.65938 q^{45} +4.10961 q^{46} +6.17414 q^{47} +1.76917 q^{48} +1.41764 q^{50} +1.11961 q^{51} +5.06155 q^{52} -13.0735 q^{53} -0.631061 q^{54} -4.68342 q^{55} +3.11082 q^{57} -5.78417 q^{58} +7.71075 q^{59} +2.65794 q^{60} +0.419560 q^{61} +2.74586 q^{62} +0.0349308 q^{64} +5.24364 q^{65} -1.78110 q^{66} +9.95583 q^{67} -1.79336 q^{68} -6.51222 q^{69} +7.39023 q^{71} +2.27293 q^{72} +13.9295 q^{73} +7.02071 q^{74} -2.24644 q^{75} -4.98280 q^{76} +1.99415 q^{78} -0.195831 q^{79} -2.93573 q^{80} +1.00000 q^{81} -0.631061 q^{82} -2.50464 q^{83} -1.85787 q^{85} +0.432003 q^{86} +9.16578 q^{87} +6.41509 q^{88} -12.4999 q^{89} +1.04717 q^{90} +10.4310 q^{92} -4.35118 q^{93} -3.89626 q^{94} -5.16206 q^{95} -5.66231 q^{96} +11.3773 q^{97} +2.82239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) \) 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9 - 4 * q^10 - 4 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^17 - 4 * q^18 + 4 * q^19 - 20 * q^20 - 16 * q^22 - 12 * q^23 - 12 * q^24 - 8 * q^25 - 8 * q^26 + 16 * q^27 - 16 * q^29 - 4 * q^30 - 4 * q^31 - 48 * q^32 - 4 * q^33 + 16 * q^34 + 12 * q^36 - 48 * q^37 - 4 * q^38 + 56 * q^40 + 16 * q^41 - 16 * q^43 - 12 * q^45 - 4 * q^46 - 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 - 4 * q^54 + 8 * q^55 + 4 * q^57 - 36 * q^58 - 36 * q^59 - 20 * q^60 - 4 * q^61 - 12 * q^62 + 52 * q^64 - 36 * q^65 - 16 * q^66 - 52 * q^67 - 8 * q^68 - 12 * q^69 - 12 * q^71 - 12 * q^72 - 16 * q^73 + 4 * q^74 - 8 * q^75 + 16 * q^76 - 8 * q^78 - 36 * q^79 - 68 * q^80 + 16 * q^81 - 4 * q^82 - 32 * q^83 - 28 * q^85 - 8 * q^86 - 16 * q^87 - 36 * q^88 - 12 * q^89 - 4 * q^90 - 36 * q^92 - 4 * q^93 + 24 * q^94 - 20 * q^95 - 48 * q^96 + 48 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more