LMFDB - Embedded newform 4026.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4026,2,Mod(1,4026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4026, 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("4026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4026 = 2 \cdot 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4026.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:	$32.1477718538$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.41421 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.41421 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.41421 q^{10} -1.00000 q^{11} +1.00000 q^{12} +3.00000 q^{13} -2.00000 q^{14} -4.41421 q^{15} +1.00000 q^{16} +2.82843 q^{17} -1.00000 q^{18} +1.17157 q^{19} -4.41421 q^{20} +2.00000 q^{21} +1.00000 q^{22} -2.82843 q^{23} -1.00000 q^{24} +14.4853 q^{25} -3.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +2.65685 q^{29} +4.41421 q^{30} +3.58579 q^{31} -1.00000 q^{32} -1.00000 q^{33} -2.82843 q^{34} -8.82843 q^{35} +1.00000 q^{36} -8.48528 q^{37} -1.17157 q^{38} +3.00000 q^{39} +4.41421 q^{40} -10.8995 q^{41} -2.00000 q^{42} -6.00000 q^{43} -1.00000 q^{44} -4.41421 q^{45} +2.82843 q^{46} -6.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -14.4853 q^{50} +2.82843 q^{51} +3.00000 q^{52} +5.17157 q^{53} -1.00000 q^{54} +4.41421 q^{55} -2.00000 q^{56} +1.17157 q^{57} -2.65685 q^{58} +14.3137 q^{59} -4.41421 q^{60} -1.00000 q^{61} -3.58579 q^{62} +2.00000 q^{63} +1.00000 q^{64} -13.2426 q^{65} +1.00000 q^{66} +8.48528 q^{67} +2.82843 q^{68} -2.82843 q^{69} +8.82843 q^{70} +6.00000 q^{71} -1.00000 q^{72} -1.65685 q^{73} +8.48528 q^{74} +14.4853 q^{75} +1.17157 q^{76} -2.00000 q^{77} -3.00000 q^{78} -4.41421 q^{80} +1.00000 q^{81} +10.8995 q^{82} -5.31371 q^{83} +2.00000 q^{84} -12.4853 q^{85} +6.00000 q^{86} +2.65685 q^{87} +1.00000 q^{88} -11.4853 q^{89} +4.41421 q^{90} +6.00000 q^{91} -2.82843 q^{92} +3.58579 q^{93} +6.00000 q^{94} -5.17157 q^{95} -1.00000 q^{96} +10.6569 q^{97} +3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 6 q^{5} - 2 q^{6} + 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 6 * q^5 - 2 * q^6 + 4 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 6 q^{5} - 2 q^{6} + 4 q^{7} - 2 q^{8} + 2 q^{9} + 6 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} - 4 q^{14} - 6 q^{15} + 2 q^{16} - 2 q^{18} + 8 q^{19} - 6 q^{20} + 4 q^{21} + 2 q^{22} - 2 q^{24} + 12 q^{25} - 6 q^{26} + 2 q^{27} + 4 q^{28} - 6 q^{29} + 6 q^{30} + 10 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{35} + 2 q^{36} - 8 q^{38} + 6 q^{39} + 6 q^{40} - 2 q^{41} - 4 q^{42} - 12 q^{43} - 2 q^{44} - 6 q^{45} - 12 q^{47} + 2 q^{48} - 6 q^{49} - 12 q^{50} + 6 q^{52} + 16 q^{53} - 2 q^{54} + 6 q^{55} - 4 q^{56} + 8 q^{57} + 6 q^{58} + 6 q^{59} - 6 q^{60} - 2 q^{61} - 10 q^{62} + 4 q^{63} + 2 q^{64} - 18 q^{65} + 2 q^{66} + 12 q^{70} + 12 q^{71} - 2 q^{72} + 8 q^{73} + 12 q^{75} + 8 q^{76} - 4 q^{77} - 6 q^{78} - 6 q^{80} + 2 q^{81} + 2 q^{82} + 12 q^{83} + 4 q^{84} - 8 q^{85} + 12 q^{86} - 6 q^{87} + 2 q^{88} - 6 q^{89} + 6 q^{90} + 12 q^{91} + 10 q^{93} + 12 q^{94} - 16 q^{95} - 2 q^{96} + 10 q^{97} + 6 q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 6 * q^5 - 2 * q^6 + 4 * q^7 - 2 * q^8 + 2 * q^9 + 6 * q^10 - 2 * q^11 + 2 * q^12 + 6 * q^13 - 4 * q^14 - 6 * q^15 + 2 * q^16 - 2 * q^18 + 8 * q^19 - 6 * q^20 + 4 * q^21 + 2 * q^22 - 2 * q^24 + 12 * q^25 - 6 * q^26 + 2 * q^27 + 4 * q^28 - 6 * q^29 + 6 * q^30 + 10 * q^31 - 2 * q^32 - 2 * q^33 - 12 * q^35 + 2 * q^36 - 8 * q^38 + 6 * q^39 + 6 * q^40 - 2 * q^41 - 4 * q^42 - 12 * q^43 - 2 * q^44 - 6 * q^45 - 12 * q^47 + 2 * q^48 - 6 * q^49 - 12 * q^50 + 6 * q^52 + 16 * q^53 - 2 * q^54 + 6 * q^55 - 4 * q^56 + 8 * q^57 + 6 * q^58 + 6 * q^59 - 6 * q^60 - 2 * q^61 - 10 * q^62 + 4 * q^63 + 2 * q^64 - 18 * q^65 + 2 * q^66 + 12 * q^70 + 12 * q^71 - 2 * q^72 + 8 * q^73 + 12 * q^75 + 8 * q^76 - 4 * q^77 - 6 * q^78 - 6 * q^80 + 2 * q^81 + 2 * q^82 + 12 * q^83 + 4 * q^84 - 8 * q^85 + 12 * q^86 - 6 * q^87 + 2 * q^88 - 6 * q^89 + 6 * q^90 + 12 * q^91 + 10 * q^93 + 12 * q^94 - 16 * q^95 - 2 * q^96 + 10 * q^97 + 6 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−4.41421	−1.97410	−0.987048	−	0.160424i	$-0.948714\pi$
$5$	−4.41421	−1.97410	−0.987048	+	0.160424i	$0.948714\pi$
$6$	−1.00000	−0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	4.41421	1.39590
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	−2.00000	−0.534522
$15$	−4.41421	−1.13975
$16$	1.00000	0.250000
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	−1.00000	−0.235702
$19$	1.17157	0.268777	0.134389	−	0.990929i	$-0.457093\pi$
$19$	1.17157	0.268777	0.134389	+	0.990929i	$0.457093\pi$
$20$	−4.41421	−0.987048
$21$	2.00000	0.436436
$22$	1.00000	0.213201
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	−1.00000	−0.204124
$25$	14.4853	2.89706
$26$	−3.00000	−0.588348
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	2.65685	0.493365	0.246683	−	0.969096i	$-0.420659\pi$
$29$	2.65685	0.493365	0.246683	+	0.969096i	$0.420659\pi$
$30$	4.41421	0.805921
$31$	3.58579	0.644026	0.322013	−	0.946735i	$-0.395640\pi$
$31$	3.58579	0.644026	0.322013	+	0.946735i	$0.395640\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−2.82843	−0.485071
$35$	−8.82843	−1.49228
$36$	1.00000	0.166667
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	−1.17157	−0.190054
$39$	3.00000	0.480384
$40$	4.41421	0.697948
$41$	−10.8995	−1.70222	−0.851108	−	0.524991i	$-0.824069\pi$
$41$	−10.8995	−1.70222	−0.851108	+	0.524991i	$0.824069\pi$
$42$	−2.00000	−0.308607
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−1.00000	−0.150756
$45$	−4.41421	−0.658032
$46$	2.82843	0.417029
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	−14.4853	−2.04853
$51$	2.82843	0.396059
$52$	3.00000	0.416025
$53$	5.17157	0.710370	0.355185	−	0.934796i	$-0.384418\pi$
$53$	5.17157	0.710370	0.355185	+	0.934796i	$0.384418\pi$
$54$	−1.00000	−0.136083
$55$	4.41421	0.595212
$56$	−2.00000	−0.267261
$57$	1.17157	0.155179
$58$	−2.65685	−0.348862
$59$	14.3137	1.86349	0.931743	−	0.363118i	$-0.118288\pi$
$59$	14.3137	1.86349	0.931743	+	0.363118i	$0.118288\pi$
$60$	−4.41421	−0.569873
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−3.58579	−0.455395
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−13.2426	−1.64255
$66$	1.00000	0.123091
$67$	8.48528	1.03664	0.518321	−	0.855186i	$-0.326557\pi$
$67$	8.48528	1.03664	0.518321	+	0.855186i	$0.326557\pi$
$68$	2.82843	0.342997
$69$	−2.82843	−0.340503
$70$	8.82843	1.05520
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−1.00000	−0.117851
$73$	−1.65685	−0.193920	−0.0969601	−	0.995288i	$-0.530912\pi$
$73$	−1.65685	−0.193920	−0.0969601	+	0.995288i	$0.530912\pi$
$74$	8.48528	0.986394
$75$	14.4853	1.67262
$76$	1.17157	0.134389
$77$	−2.00000	−0.227921
$78$	−3.00000	−0.339683
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−4.41421	−0.493524
$81$	1.00000	0.111111
$82$	10.8995	1.20365
$83$	−5.31371	−0.583255	−0.291628	−	0.956532i	$-0.594197\pi$
$83$	−5.31371	−0.583255	−0.291628	+	0.956532i	$0.594197\pi$
$84$	2.00000	0.218218
$85$	−12.4853	−1.35422
$86$	6.00000	0.646997
$87$	2.65685	0.284845
$88$	1.00000	0.106600
$89$	−11.4853	−1.21744	−0.608719	−	0.793386i	$-0.708316\pi$
$89$	−11.4853	−1.21744	−0.608719	+	0.793386i	$0.708316\pi$
$90$	4.41421	0.465299
$91$	6.00000	0.628971
$92$	−2.82843	−0.294884
$93$	3.58579	0.371829
$94$	6.00000	0.618853
$95$	−5.17157	−0.530592
$96$	−1.00000	−0.102062
$97$	10.6569	1.08204	0.541020	−	0.841010i	$-0.318038\pi$
$97$	10.6569	1.08204	0.541020	+	0.841010i	$0.318038\pi$
$98$	3.00000	0.303046
$99$	−1.00000	−0.100504
$100$	14.4853	1.44853
$101$	11.4853	1.14283	0.571414	−	0.820662i	$-0.306395\pi$
$101$	11.4853	1.14283	0.571414	+	0.820662i	$0.306395\pi$
$102$	−2.82843	−0.280056
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−3.00000	−0.294174
$105$	−8.82843	−0.861566
$106$	−5.17157	−0.502308
$107$	6.41421	0.620085	0.310043	−	0.950723i	$-0.399657\pi$
$107$	6.41421	0.620085	0.310043	+	0.950723i	$0.399657\pi$
$108$	1.00000	0.0962250
$109$	−2.17157	−0.207999	−0.103999	−	0.994577i	$-0.533164\pi$
$109$	−2.17157	−0.207999	−0.103999	+	0.994577i	$0.533164\pi$
$110$	−4.41421	−0.420879
$111$	−8.48528	−0.805387
$112$	2.00000	0.188982
$113$	20.1421	1.89481	0.947406	−	0.320033i	$-0.103694\pi$
$113$	20.1421	1.89481	0.947406	+	0.320033i	$0.103694\pi$
$114$	−1.17157	−0.109728
$115$	12.4853	1.16426
$116$	2.65685	0.246683
$117$	3.00000	0.277350
$118$	−14.3137	−1.31768
$119$	5.65685	0.518563
$120$	4.41421	0.402961
$121$	1.00000	0.0909091
$122$	1.00000	0.0905357
$123$	−10.8995	−0.982774
$124$	3.58579	0.322013
$125$	−41.8701	−3.74497
$126$	−2.00000	−0.178174
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	−6.00000	−0.528271
$130$	13.2426	1.16146
$131$	10.0711	0.879913	0.439957	−	0.898019i	$-0.354994\pi$
$131$	10.0711	0.879913	0.439957	+	0.898019i	$0.354994\pi$
$132$	−1.00000	−0.0870388
$133$	2.34315	0.203177
$134$	−8.48528	−0.733017
$135$	−4.41421	−0.379915
$136$	−2.82843	−0.242536
$137$	15.1716	1.29619	0.648097	−	0.761557i	$-0.275565\pi$
$137$	15.1716	1.29619	0.648097	+	0.761557i	$0.275565\pi$
$138$	2.82843	0.240772
$139$	18.0711	1.53277	0.766384	−	0.642383i	$-0.222054\pi$
$139$	18.0711	1.53277	0.766384	+	0.642383i	$0.222054\pi$
$140$	−8.82843	−0.746138
$141$	−6.00000	−0.505291
$142$	−6.00000	−0.503509
$143$	−3.00000	−0.250873
$144$	1.00000	0.0833333
$145$	−11.7279	−0.973951
$146$	1.65685	0.137122
$147$	−3.00000	−0.247436
$148$	−8.48528	−0.697486
$149$	13.1716	1.07906	0.539529	−	0.841967i	$-0.318602\pi$
$149$	13.1716	1.07906	0.539529	+	0.841967i	$0.318602\pi$
$150$	−14.4853	−1.18272
$151$	−8.48528	−0.690522	−0.345261	−	0.938507i	$-0.612210\pi$
$151$	−8.48528	−0.690522	−0.345261	+	0.938507i	$0.612210\pi$
$152$	−1.17157	−0.0950271
$153$	2.82843	0.228665
$154$	2.00000	0.161165
$155$	−15.8284	−1.27137
$156$	3.00000	0.240192
$157$	6.41421	0.511910	0.255955	−	0.966689i	$-0.417610\pi$
$157$	6.41421	0.511910	0.255955	+	0.966689i	$0.417610\pi$
$158$	0	0
$159$	5.17157	0.410132
$160$	4.41421	0.348974
$161$	−5.65685	−0.445823
$162$	−1.00000	−0.0785674
$163$	6.65685	0.521405	0.260703	−	0.965419i	$-0.416046\pi$
$163$	6.65685	0.521405	0.260703	+	0.965419i	$0.416046\pi$
$164$	−10.8995	−0.851108
$165$	4.41421	0.343646
$166$	5.31371	0.412424
$167$	1.65685	0.128211	0.0641056	−	0.997943i	$-0.479581\pi$
$167$	1.65685	0.128211	0.0641056	+	0.997943i	$0.479581\pi$
$168$	−2.00000	−0.154303
$169$	−4.00000	−0.307692
$170$	12.4853	0.957577
$171$	1.17157	0.0895924
$172$	−6.00000	−0.457496
$173$	18.7990	1.42926	0.714630	−	0.699502i	$-0.246595\pi$
$173$	18.7990	1.42926	0.714630	+	0.699502i	$0.246595\pi$
$174$	−2.65685	−0.201416
$175$	28.9706	2.18997
$176$	−1.00000	−0.0753778
$177$	14.3137	1.07588
$178$	11.4853	0.860858
$179$	15.1716	1.13398	0.566988	−	0.823726i	$-0.308108\pi$
$179$	15.1716	1.13398	0.566988	+	0.823726i	$0.308108\pi$
$180$	−4.41421	−0.329016
$181$	24.4142	1.81469	0.907347	−	0.420382i	$-0.138104\pi$
$181$	24.4142	1.81469	0.907347	+	0.420382i	$0.138104\pi$
$182$	−6.00000	−0.444750
$183$	−1.00000	−0.0739221
$184$	2.82843	0.208514
$185$	37.4558	2.75381
$186$	−3.58579	−0.262923
$187$	−2.82843	−0.206835
$188$	−6.00000	−0.437595
$189$	2.00000	0.145479
$190$	5.17157	0.375185
$191$	−12.1421	−0.878574	−0.439287	−	0.898347i	$-0.644769\pi$
$191$	−12.1421	−0.878574	−0.439287	+	0.898347i	$0.644769\pi$
$192$	1.00000	0.0721688
$193$	0.757359	0.0545159	0.0272580	−	0.999628i	$-0.491322\pi$
$193$	0.757359	0.0545159	0.0272580	+	0.999628i	$0.491322\pi$
$194$	−10.6569	−0.765118
$195$	−13.2426	−0.948325
$196$	−3.00000	−0.214286
$197$	−13.7990	−0.983137	−0.491569	−	0.870839i	$-0.663576\pi$
$197$	−13.7990	−0.983137	−0.491569	+	0.870839i	$0.663576\pi$
$198$	1.00000	0.0710669
$199$	6.97056	0.494130	0.247065	−	0.968999i	$-0.420534\pi$
$199$	6.97056	0.494130	0.247065	+	0.968999i	$0.420534\pi$
$200$	−14.4853	−1.02426
$201$	8.48528	0.598506
$202$	−11.4853	−0.808102
$203$	5.31371	0.372949
$204$	2.82843	0.198030
$205$	48.1127	3.36034
$206$	4.00000	0.278693
$207$	−2.82843	−0.196589
$208$	3.00000	0.208013
$209$	−1.17157	−0.0810394
$210$	8.82843	0.609219
$211$	−3.72792	−0.256641	−0.128320	−	0.991733i	$-0.540959\pi$
$211$	−3.72792	−0.256641	−0.128320	+	0.991733i	$0.540959\pi$
$212$	5.17157	0.355185
$213$	6.00000	0.411113
$214$	−6.41421	−0.438467
$215$	26.4853	1.80628
$216$	−1.00000	−0.0680414
$217$	7.17157	0.486838
$218$	2.17157	0.147077
$219$	−1.65685	−0.111960
$220$	4.41421	0.297606
$221$	8.48528	0.570782
$222$	8.48528	0.569495
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−2.00000	−0.133631
$225$	14.4853	0.965685
$226$	−20.1421	−1.33983
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	1.17157	0.0775893
$229$	2.34315	0.154839	0.0774197	−	0.996999i	$-0.475332\pi$
$229$	2.34315	0.154839	0.0774197	+	0.996999i	$0.475332\pi$
$230$	−12.4853	−0.823255
$231$	−2.00000	−0.131590
$232$	−2.65685	−0.174431
$233$	8.48528	0.555889	0.277945	−	0.960597i	$-0.410347\pi$
$233$	8.48528	0.555889	0.277945	+	0.960597i	$0.410347\pi$
$234$	−3.00000	−0.196116
$235$	26.4853	1.72771
$236$	14.3137	0.931743
$237$	0	0
$238$	−5.65685	−0.366679
$239$	10.8284	0.700433	0.350216	−	0.936669i	$-0.386108\pi$
$239$	10.8284	0.700433	0.350216	+	0.936669i	$0.386108\pi$
$240$	−4.41421	−0.284936
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	−1.00000	−0.0640184
$245$	13.2426	0.846041
$246$	10.8995	0.694926
$247$	3.51472	0.223636
$248$	−3.58579	−0.227698
$249$	−5.31371	−0.336743
$250$	41.8701	2.64809
$251$	5.48528	0.346228	0.173114	−	0.984902i	$-0.444617\pi$
$251$	5.48528	0.346228	0.173114	+	0.984902i	$0.444617\pi$
$252$	2.00000	0.125988
$253$	2.82843	0.177822
$254$	0	0
$255$	−12.4853	−0.781859
$256$	1.00000	0.0625000
$257$	5.31371	0.331460	0.165730	−	0.986171i	$-0.447002\pi$
$257$	5.31371	0.331460	0.165730	+	0.986171i	$0.447002\pi$
$258$	6.00000	0.373544
$259$	−16.9706	−1.05450
$260$	−13.2426	−0.821274
$261$	2.65685	0.164455
$262$	−10.0711	−0.622193
$263$	−26.4853	−1.63315	−0.816576	−	0.577238i	$-0.804131\pi$
$263$	−26.4853	−1.63315	−0.816576	+	0.577238i	$0.804131\pi$
$264$	1.00000	0.0615457
$265$	−22.8284	−1.40234
$266$	−2.34315	−0.143667
$267$	−11.4853	−0.702888
$268$	8.48528	0.518321
$269$	−21.7279	−1.32477	−0.662387	−	0.749161i	$-0.730457\pi$
$269$	−21.7279	−1.32477	−0.662387	+	0.749161i	$0.730457\pi$
$270$	4.41421	0.268640
$271$	−1.65685	−0.100647	−0.0503234	−	0.998733i	$-0.516025\pi$
$271$	−1.65685	−0.100647	−0.0503234	+	0.998733i	$0.516025\pi$
$272$	2.82843	0.171499
$273$	6.00000	0.363137
$274$	−15.1716	−0.916548
$275$	−14.4853	−0.873495
$276$	−2.82843	−0.170251
$277$	22.9706	1.38017	0.690084	−	0.723730i	$-0.257574\pi$
$277$	22.9706	1.38017	0.690084	+	0.723730i	$0.257574\pi$
$278$	−18.0711	−1.08383
$279$	3.58579	0.214675
$280$	8.82843	0.527599
$281$	22.6274	1.34984	0.674919	−	0.737892i	$-0.264178\pi$
$281$	22.6274	1.34984	0.674919	+	0.737892i	$0.264178\pi$
$282$	6.00000	0.357295
$283$	20.4853	1.21772	0.608862	−	0.793276i	$-0.291626\pi$
$283$	20.4853	1.21772	0.608862	+	0.793276i	$0.291626\pi$
$284$	6.00000	0.356034
$285$	−5.17157	−0.306338
$286$	3.00000	0.177394
$287$	−21.7990	−1.28675
$288$	−1.00000	−0.0589256
$289$	−9.00000	−0.529412
$290$	11.7279	0.688687
$291$	10.6569	0.624716
$292$	−1.65685	−0.0969601
$293$	18.3431	1.07162	0.535809	−	0.844339i	$-0.320007\pi$
$293$	18.3431	1.07162	0.535809	+	0.844339i	$0.320007\pi$
$294$	3.00000	0.174964
$295$	−63.1838	−3.67870
$296$	8.48528	0.493197
$297$	−1.00000	−0.0580259
$298$	−13.1716	−0.763009
$299$	−8.48528	−0.490716
$300$	14.4853	0.836308
$301$	−12.0000	−0.691669
$302$	8.48528	0.488273
$303$	11.4853	0.659812
$304$	1.17157	0.0671943
$305$	4.41421	0.252757
$306$	−2.82843	−0.161690
$307$	−18.2132	−1.03948	−0.519741	−	0.854324i	$-0.673972\pi$
$307$	−18.2132	−1.03948	−0.519741	+	0.854324i	$0.673972\pi$
$308$	−2.00000	−0.113961
$309$	−4.00000	−0.227552
$310$	15.8284	0.898994
$311$	−18.6274	−1.05626	−0.528132	−	0.849162i	$-0.677107\pi$
$311$	−18.6274	−1.05626	−0.528132	+	0.849162i	$0.677107\pi$
$312$	−3.00000	−0.169842
$313$	−18.1421	−1.02545	−0.512727	−	0.858552i	$-0.671365\pi$
$313$	−18.1421	−1.02545	−0.512727	+	0.858552i	$0.671365\pi$
$314$	−6.41421	−0.361975
$315$	−8.82843	−0.497426
$316$	0	0
$317$	3.92893	0.220671	0.110335	−	0.993894i	$-0.464807\pi$
$317$	3.92893	0.220671	0.110335	+	0.993894i	$0.464807\pi$
$318$	−5.17157	−0.290007
$319$	−2.65685	−0.148755
$320$	−4.41421	−0.246762
$321$	6.41421	0.358006
$322$	5.65685	0.315244
$323$	3.31371	0.184380
$324$	1.00000	0.0555556
$325$	43.4558	2.41050
$326$	−6.65685	−0.368689
$327$	−2.17157	−0.120088
$328$	10.8995	0.601824
$329$	−12.0000	−0.661581
$330$	−4.41421	−0.242994
$331$	−16.4853	−0.906113	−0.453057	−	0.891482i	$-0.649666\pi$
$331$	−16.4853	−0.906113	−0.453057	+	0.891482i	$0.649666\pi$
$332$	−5.31371	−0.291628
$333$	−8.48528	−0.464991
$334$	−1.65685	−0.0906590
$335$	−37.4558	−2.04643
$336$	2.00000	0.109109
$337$	16.7574	0.912832	0.456416	−	0.889767i	$-0.349133\pi$
$337$	16.7574	0.912832	0.456416	+	0.889767i	$0.349133\pi$
$338$	4.00000	0.217571
$339$	20.1421	1.09397
$340$	−12.4853	−0.677109
$341$	−3.58579	−0.194181
$342$	−1.17157	−0.0633514
$343$	−20.0000	−1.07990
$344$	6.00000	0.323498
$345$	12.4853	0.672185
$346$	−18.7990	−1.01064
$347$	24.8995	1.33667	0.668337	−	0.743858i	$-0.267006\pi$
$347$	24.8995	1.33667	0.668337	+	0.743858i	$0.267006\pi$
$348$	2.65685	0.142422
$349$	15.7990	0.845701	0.422850	−	0.906200i	$-0.361030\pi$
$349$	15.7990	0.845701	0.422850	+	0.906200i	$0.361030\pi$
$350$	−28.9706	−1.54854
$351$	3.00000	0.160128
$352$	1.00000	0.0533002
$353$	−13.3137	−0.708617	−0.354309	−	0.935129i	$-0.615284\pi$
$353$	−13.3137	−0.708617	−0.354309	+	0.935129i	$0.615284\pi$
$354$	−14.3137	−0.760765
$355$	−26.4853	−1.40569
$356$	−11.4853	−0.608719
$357$	5.65685	0.299392
$358$	−15.1716	−0.801843
$359$	−1.97056	−0.104002	−0.0520012	−	0.998647i	$-0.516560\pi$
$359$	−1.97056	−0.104002	−0.0520012	+	0.998647i	$0.516560\pi$
$360$	4.41421	0.232649
$361$	−17.6274	−0.927759
$362$	−24.4142	−1.28318
$363$	1.00000	0.0524864
$364$	6.00000	0.314485
$365$	7.31371	0.382817
$366$	1.00000	0.0522708
$367$	16.8284	0.878437	0.439218	−	0.898380i	$-0.355255\pi$
$367$	16.8284	0.878437	0.439218	+	0.898380i	$0.355255\pi$
$368$	−2.82843	−0.147442
$369$	−10.8995	−0.567405
$370$	−37.4558	−1.94724
$371$	10.3431	0.536989
$372$	3.58579	0.185914
$373$	−32.9706	−1.70715	−0.853576	−	0.520969i	$-0.825571\pi$
$373$	−32.9706	−1.70715	−0.853576	+	0.520969i	$0.825571\pi$
$374$	2.82843	0.146254
$375$	−41.8701	−2.16216
$376$	6.00000	0.309426
$377$	7.97056	0.410505
$378$	−2.00000	−0.102869
$379$	18.7990	0.965639	0.482820	−	0.875720i	$-0.339613\pi$
$379$	18.7990	0.965639	0.482820	+	0.875720i	$0.339613\pi$
$380$	−5.17157	−0.265296
$381$	0	0
$382$	12.1421	0.621246
$383$	33.1127	1.69198	0.845990	−	0.533199i	$-0.179010\pi$
$383$	33.1127	1.69198	0.845990	+	0.533199i	$0.179010\pi$
$384$	−1.00000	−0.0510310
$385$	8.82843	0.449938
$386$	−0.757359	−0.0385486
$387$	−6.00000	−0.304997
$388$	10.6569	0.541020
$389$	17.6569	0.895238	0.447619	−	0.894224i	$-0.352272\pi$
$389$	17.6569	0.895238	0.447619	+	0.894224i	$0.352272\pi$
$390$	13.2426	0.670567
$391$	−8.00000	−0.404577
$392$	3.00000	0.151523
$393$	10.0711	0.508018
$394$	13.7990	0.695183
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	25.8701	1.29838	0.649190	−	0.760626i	$-0.275108\pi$
$397$	25.8701	1.29838	0.649190	+	0.760626i	$0.275108\pi$
$398$	−6.97056	−0.349403
$399$	2.34315	0.117304
$400$	14.4853	0.724264
$401$	16.3137	0.814668	0.407334	−	0.913279i	$-0.366459\pi$
$401$	16.3137	0.814668	0.407334	+	0.913279i	$0.366459\pi$
$402$	−8.48528	−0.423207
$403$	10.7574	0.535862
$404$	11.4853	0.571414
$405$	−4.41421	−0.219344
$406$	−5.31371	−0.263715
$407$	8.48528	0.420600
$408$	−2.82843	−0.140028
$409$	−15.5858	−0.770668	−0.385334	−	0.922777i	$-0.625914\pi$
$409$	−15.5858	−0.770668	−0.385334	+	0.922777i	$0.625914\pi$
$410$	−48.1127	−2.37612
$411$	15.1716	0.748359
$412$	−4.00000	−0.197066
$413$	28.6274	1.40866
$414$	2.82843	0.139010
$415$	23.4558	1.15140
$416$	−3.00000	−0.147087
$417$	18.0711	0.884944
$418$	1.17157	0.0573035
$419$	−14.3431	−0.700709	−0.350354	−	0.936617i	$-0.613939\pi$
$419$	−14.3431	−0.700709	−0.350354	+	0.936617i	$0.613939\pi$
$420$	−8.82843	−0.430783
$421$	−23.2426	−1.13278	−0.566388	−	0.824138i	$-0.691660\pi$
$421$	−23.2426	−1.13278	−0.566388	+	0.824138i	$0.691660\pi$
$422$	3.72792	0.181472
$423$	−6.00000	−0.291730
$424$	−5.17157	−0.251154
$425$	40.9706	1.98736
$426$	−6.00000	−0.290701
$427$	−2.00000	−0.0967868
$428$	6.41421	0.310043
$429$	−3.00000	−0.144841
$430$	−26.4853	−1.27723
$431$	−19.1716	−0.923462	−0.461731	−	0.887020i	$-0.652771\pi$
$431$	−19.1716	−0.923462	−0.461731	+	0.887020i	$0.652771\pi$
$432$	1.00000	0.0481125
$433$	−10.8284	−0.520381	−0.260190	−	0.965557i	$-0.583785\pi$
$433$	−10.8284	−0.520381	−0.260190	+	0.965557i	$0.583785\pi$
$434$	−7.17157	−0.344247
$435$	−11.7279	−0.562311
$436$	−2.17157	−0.103999
$437$	−3.31371	−0.158516
$438$	1.65685	0.0791676
$439$	−1.48528	−0.0708886	−0.0354443	−	0.999372i	$-0.511285\pi$
$439$	−1.48528	−0.0708886	−0.0354443	+	0.999372i	$0.511285\pi$
$440$	−4.41421	−0.210439
$441$	−3.00000	−0.142857
$442$	−8.48528	−0.403604
$443$	7.65685	0.363788	0.181894	−	0.983318i	$-0.441777\pi$
$443$	7.65685	0.363788	0.181894	+	0.983318i	$0.441777\pi$
$444$	−8.48528	−0.402694
$445$	50.6985	2.40334
$446$	2.00000	0.0947027
$447$	13.1716	0.622994
$448$	2.00000	0.0944911
$449$	0.201010	0.00948625	0.00474313	−	0.999989i	$-0.498490\pi$
$449$	0.201010	0.00948625	0.00474313	+	0.999989i	$0.498490\pi$
$450$	−14.4853	−0.682843
$451$	10.8995	0.513237
$452$	20.1421	0.947406
$453$	−8.48528	−0.398673
$454$	12.0000	0.563188
$455$	−26.4853	−1.24165
$456$	−1.17157	−0.0548639
$457$	−8.48528	−0.396925	−0.198462	−	0.980109i	$-0.563595\pi$
$457$	−8.48528	−0.396925	−0.198462	+	0.980109i	$0.563595\pi$
$458$	−2.34315	−0.109488
$459$	2.82843	0.132020
$460$	12.4853	0.582129
$461$	8.82843	0.411181	0.205590	−	0.978638i	$-0.434089\pi$
$461$	8.82843	0.411181	0.205590	+	0.978638i	$0.434089\pi$
$462$	2.00000	0.0930484
$463$	−30.4853	−1.41677	−0.708386	−	0.705826i	$-0.750576\pi$
$463$	−30.4853	−1.41677	−0.708386	+	0.705826i	$0.750576\pi$
$464$	2.65685	0.123341
$465$	−15.8284	−0.734026
$466$	−8.48528	−0.393073
$467$	2.31371	0.107066	0.0535328	−	0.998566i	$-0.482952\pi$
$467$	2.31371	0.107066	0.0535328	+	0.998566i	$0.482952\pi$
$468$	3.00000	0.138675
$469$	16.9706	0.783628
$470$	−26.4853	−1.22167
$471$	6.41421	0.295551
$472$	−14.3137	−0.658842
$473$	6.00000	0.275880
$474$	0	0
$475$	16.9706	0.778663
$476$	5.65685	0.259281
$477$	5.17157	0.236790
$478$	−10.8284	−0.495281
$479$	21.6569	0.989527	0.494763	−	0.869028i	$-0.335255\pi$
$479$	21.6569	0.989527	0.494763	+	0.869028i	$0.335255\pi$
$480$	4.41421	0.201480
$481$	−25.4558	−1.16069
$482$	−4.00000	−0.182195
$483$	−5.65685	−0.257396
$484$	1.00000	0.0454545
$485$	−47.0416	−2.13605
$486$	−1.00000	−0.0453609
$487$	2.34315	0.106178	0.0530890	−	0.998590i	$-0.483093\pi$
$487$	2.34315	0.106178	0.0530890	+	0.998590i	$0.483093\pi$
$488$	1.00000	0.0452679
$489$	6.65685	0.301033
$490$	−13.2426	−0.598242
$491$	−9.31371	−0.420322	−0.210161	−	0.977667i	$-0.567399\pi$
$491$	−9.31371	−0.420322	−0.210161	+	0.977667i	$0.567399\pi$
$492$	−10.8995	−0.491387
$493$	7.51472	0.338446
$494$	−3.51472	−0.158135
$495$	4.41421	0.198404
$496$	3.58579	0.161007
$497$	12.0000	0.538274
$498$	5.31371	0.238113
$499$	29.7990	1.33399	0.666993	−	0.745064i	$-0.267581\pi$
$499$	29.7990	1.33399	0.666993	+	0.745064i	$0.267581\pi$
$500$	−41.8701	−1.87249
$501$	1.65685	0.0740228
$502$	−5.48528	−0.244820
$503$	22.9706	1.02421	0.512103	−	0.858924i	$-0.328866\pi$
$503$	22.9706	1.02421	0.512103	+	0.858924i	$0.328866\pi$
$504$	−2.00000	−0.0890871
$505$	−50.6985	−2.25605
$506$	−2.82843	−0.125739
$507$	−4.00000	−0.177646
$508$	0	0
$509$	−9.17157	−0.406523	−0.203261	−	0.979125i	$-0.565154\pi$
$509$	−9.17157	−0.406523	−0.203261	+	0.979125i	$0.565154\pi$
$510$	12.4853	0.552858
$511$	−3.31371	−0.146590
$512$	−1.00000	−0.0441942
$513$	1.17157	0.0517262
$514$	−5.31371	−0.234378
$515$	17.6569	0.778054
$516$	−6.00000	−0.264135
$517$	6.00000	0.263880
$518$	16.9706	0.745644
$519$	18.7990	0.825184
$520$	13.2426	0.580728
$521$	24.6274	1.07895	0.539473	−	0.842003i	$-0.318623\pi$
$521$	24.6274	1.07895	0.539473	+	0.842003i	$0.318623\pi$
$522$	−2.65685	−0.116287
$523$	35.5858	1.55606	0.778029	−	0.628228i	$-0.216220\pi$
$523$	35.5858	1.55606	0.778029	+	0.628228i	$0.216220\pi$
$524$	10.0711	0.439957
$525$	28.9706	1.26438
$526$	26.4853	1.15481
$527$	10.1421	0.441798
$528$	−1.00000	−0.0435194
$529$	−15.0000	−0.652174
$530$	22.8284	0.991604
$531$	14.3137	0.621162
$532$	2.34315	0.101588
$533$	−32.6985	−1.41633
$534$	11.4853	0.497017
$535$	−28.3137	−1.22411
$536$	−8.48528	−0.366508
$537$	15.1716	0.654702
$538$	21.7279	0.936757
$539$	3.00000	0.129219
$540$	−4.41421	−0.189958
$541$	−42.4264	−1.82405	−0.912027	−	0.410130i	$-0.865483\pi$
$541$	−42.4264	−1.82405	−0.912027	+	0.410130i	$0.865483\pi$
$542$	1.65685	0.0711680
$543$	24.4142	1.04771
$544$	−2.82843	−0.121268
$545$	9.58579	0.410610
$546$	−6.00000	−0.256776
$547$	−22.6985	−0.970517	−0.485259	−	0.874371i	$-0.661275\pi$
$547$	−22.6985	−0.970517	−0.485259	+	0.874371i	$0.661275\pi$
$548$	15.1716	0.648097
$549$	−1.00000	−0.0426790
$550$	14.4853	0.617654
$551$	3.11270	0.132605
$552$	2.82843	0.120386
$553$	0	0
$554$	−22.9706	−0.975926
$555$	37.4558	1.58991
$556$	18.0711	0.766384
$557$	29.2843	1.24081	0.620407	−	0.784280i	$-0.286967\pi$
$557$	29.2843	1.24081	0.620407	+	0.784280i	$0.286967\pi$
$558$	−3.58579	−0.151798
$559$	−18.0000	−0.761319
$560$	−8.82843	−0.373069
$561$	−2.82843	−0.119416
$562$	−22.6274	−0.954480
$563$	19.6569	0.828438	0.414219	−	0.910177i	$-0.364055\pi$
$563$	19.6569	0.828438	0.414219	+	0.910177i	$0.364055\pi$
$564$	−6.00000	−0.252646
$565$	−88.9117	−3.74054
$566$	−20.4853	−0.861061
$567$	2.00000	0.0839921
$568$	−6.00000	−0.251754
$569$	9.92893	0.416242	0.208121	−	0.978103i	$-0.433265\pi$
$569$	9.92893	0.416242	0.208121	+	0.978103i	$0.433265\pi$
$570$	5.17157	0.216613
$571$	30.1421	1.26141	0.630705	−	0.776023i	$-0.282766\pi$
$571$	30.1421	1.26141	0.630705	+	0.776023i	$0.282766\pi$
$572$	−3.00000	−0.125436
$573$	−12.1421	−0.507245
$574$	21.7990	0.909872
$575$	−40.9706	−1.70859
$576$	1.00000	0.0416667
$577$	25.6569	1.06811	0.534054	−	0.845450i	$-0.320668\pi$
$577$	25.6569	1.06811	0.534054	+	0.845450i	$0.320668\pi$
$578$	9.00000	0.374351
$579$	0.757359	0.0314748
$580$	−11.7279	−0.486975
$581$	−10.6274	−0.440900
$582$	−10.6569	−0.441741
$583$	−5.17157	−0.214185
$584$	1.65685	0.0685611
$585$	−13.2426	−0.547516
$586$	−18.3431	−0.757748
$587$	−24.1127	−0.995238	−0.497619	−	0.867396i	$-0.665792\pi$
$587$	−24.1127	−0.995238	−0.497619	+	0.867396i	$0.665792\pi$
$588$	−3.00000	−0.123718
$589$	4.20101	0.173100
$590$	63.1838	2.60123
$591$	−13.7990	−0.567615
$592$	−8.48528	−0.348743
$593$	8.34315	0.342612	0.171306	−	0.985218i	$-0.445201\pi$
$593$	8.34315	0.342612	0.171306	+	0.985218i	$0.445201\pi$
$594$	1.00000	0.0410305
$595$	−24.9706	−1.02369
$596$	13.1716	0.539529
$597$	6.97056	0.285286
$598$	8.48528	0.346989
$599$	−6.82843	−0.279002	−0.139501	−	0.990222i	$-0.544550\pi$
$599$	−6.82843	−0.279002	−0.139501	+	0.990222i	$0.544550\pi$
$600$	−14.4853	−0.591359
$601$	9.02944	0.368318	0.184159	−	0.982896i	$-0.441044\pi$
$601$	9.02944	0.368318	0.184159	+	0.982896i	$0.441044\pi$
$602$	12.0000	0.489083
$603$	8.48528	0.345547
$604$	−8.48528	−0.345261
$605$	−4.41421	−0.179463
$606$	−11.4853	−0.466558
$607$	−39.2843	−1.59450	−0.797250	−	0.603650i	$-0.793713\pi$
$607$	−39.2843	−1.59450	−0.797250	+	0.603650i	$0.793713\pi$
$608$	−1.17157	−0.0475136
$609$	5.31371	0.215322
$610$	−4.41421	−0.178726
$611$	−18.0000	−0.728202
$612$	2.82843	0.114332
$613$	31.4853	1.27168	0.635839	−	0.771821i	$-0.280654\pi$
$613$	31.4853	1.27168	0.635839	+	0.771821i	$0.280654\pi$
$614$	18.2132	0.735025
$615$	48.1127	1.94009
$616$	2.00000	0.0805823
$617$	−3.82843	−0.154127	−0.0770633	−	0.997026i	$-0.524554\pi$
$617$	−3.82843	−0.154127	−0.0770633	+	0.997026i	$0.524554\pi$
$618$	4.00000	0.160904
$619$	44.9411	1.80634	0.903168	−	0.429287i	$-0.141235\pi$
$619$	44.9411	1.80634	0.903168	+	0.429287i	$0.141235\pi$
$620$	−15.8284	−0.635685
$621$	−2.82843	−0.113501
$622$	18.6274	0.746891
$623$	−22.9706	−0.920296
$624$	3.00000	0.120096
$625$	112.397	4.49588
$626$	18.1421	0.725106
$627$	−1.17157	−0.0467881
$628$	6.41421	0.255955
$629$	−24.0000	−0.956943
$630$	8.82843	0.351733
$631$	−5.02944	−0.200219	−0.100109	−	0.994976i	$-0.531919\pi$
$631$	−5.02944	−0.200219	−0.100109	+	0.994976i	$0.531919\pi$
$632$	0	0
$633$	−3.72792	−0.148172
$634$	−3.92893	−0.156038
$635$	0	0
$636$	5.17157	0.205066
$637$	−9.00000	−0.356593
$638$	2.65685	0.105186
$639$	6.00000	0.237356
$640$	4.41421	0.174487
$641$	23.6569	0.934390	0.467195	−	0.884154i	$-0.345265\pi$
$641$	23.6569	0.934390	0.467195	+	0.884154i	$0.345265\pi$
$642$	−6.41421	−0.253149
$643$	24.6274	0.971211	0.485605	−	0.874178i	$-0.338599\pi$
$643$	24.6274	0.971211	0.485605	+	0.874178i	$0.338599\pi$
$644$	−5.65685	−0.222911
$645$	26.4853	1.04286
$646$	−3.31371	−0.130376
$647$	−31.4558	−1.23666	−0.618328	−	0.785920i	$-0.712190\pi$
$647$	−31.4558	−1.23666	−0.618328	+	0.785920i	$0.712190\pi$
$648$	−1.00000	−0.0392837
$649$	−14.3137	−0.561862
$650$	−43.4558	−1.70448
$651$	7.17157	0.281076
$652$	6.65685	0.260703
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	2.17157	0.0849152
$655$	−44.4558	−1.73703
$656$	−10.8995	−0.425554
$657$	−1.65685	−0.0646400
$658$	12.0000	0.467809
$659$	−8.27208	−0.322234	−0.161117	−	0.986935i	$-0.551510\pi$
$659$	−8.27208	−0.322234	−0.161117	+	0.986935i	$0.551510\pi$
$660$	4.41421	0.171823
$661$	2.21320	0.0860836	0.0430418	−	0.999073i	$-0.486295\pi$
$661$	2.21320	0.0860836	0.0430418	+	0.999073i	$0.486295\pi$
$662$	16.4853	0.640719
$663$	8.48528	0.329541
$664$	5.31371	0.206212
$665$	−10.3431	−0.401090
$666$	8.48528	0.328798
$667$	−7.51472	−0.290971
$668$	1.65685	0.0641056
$669$	−2.00000	−0.0773245
$670$	37.4558	1.44705
$671$	1.00000	0.0386046
$672$	−2.00000	−0.0771517
$673$	29.3848	1.13270	0.566350	−	0.824165i	$-0.308355\pi$
$673$	29.3848	1.13270	0.566350	+	0.824165i	$0.308355\pi$
$674$	−16.7574	−0.645469
$675$	14.4853	0.557539
$676$	−4.00000	−0.153846
$677$	−15.6569	−0.601742	−0.300871	−	0.953665i	$-0.597277\pi$
$677$	−15.6569	−0.601742	−0.300871	+	0.953665i	$0.597277\pi$
$678$	−20.1421	−0.773554
$679$	21.3137	0.817945
$680$	12.4853	0.478789
$681$	−12.0000	−0.459841
$682$	3.58579	0.137307
$683$	−34.9706	−1.33811	−0.669056	−	0.743212i	$-0.733301\pi$
$683$	−34.9706	−1.33811	−0.669056	+	0.743212i	$0.733301\pi$
$684$	1.17157	0.0447962
$685$	−66.9706	−2.55881
$686$	20.0000	0.763604
$687$	2.34315	0.0893966
$688$	−6.00000	−0.228748
$689$	15.5147	0.591064
$690$	−12.4853	−0.475307
$691$	−6.79899	−0.258646	−0.129323	−	0.991603i	$-0.541280\pi$
$691$	−6.79899	−0.258646	−0.129323	+	0.991603i	$0.541280\pi$
$692$	18.7990	0.714630
$693$	−2.00000	−0.0759737
$694$	−24.8995	−0.945172
$695$	−79.7696	−3.02583
$696$	−2.65685	−0.100708
$697$	−30.8284	−1.16771
$698$	−15.7990	−0.598001
$699$	8.48528	0.320943
$700$	28.9706	1.09498
$701$	−6.31371	−0.238465	−0.119233	−	0.992866i	$-0.538043\pi$
$701$	−6.31371	−0.238465	−0.119233	+	0.992866i	$0.538043\pi$
$702$	−3.00000	−0.113228
$703$	−9.94113	−0.374937
$704$	−1.00000	−0.0376889
$705$	26.4853	0.997493
$706$	13.3137	0.501068
$707$	22.9706	0.863897
$708$	14.3137	0.537942
$709$	13.3848	0.502676	0.251338	−	0.967899i	$-0.419129\pi$
$709$	13.3848	0.502676	0.251338	+	0.967899i	$0.419129\pi$
$710$	26.4853	0.993975
$711$	0	0
$712$	11.4853	0.430429
$713$	−10.1421	−0.379826
$714$	−5.65685	−0.211702
$715$	13.2426	0.495247
$716$	15.1716	0.566988
$717$	10.8284	0.404395
$718$	1.97056	0.0735407
$719$	−36.3431	−1.35537	−0.677685	−	0.735352i	$-0.737017\pi$
$719$	−36.3431	−1.35537	−0.677685	+	0.735352i	$0.737017\pi$
$720$	−4.41421	−0.164508
$721$	−8.00000	−0.297936
$722$	17.6274	0.656025
$723$	4.00000	0.148762
$724$	24.4142	0.907347
$725$	38.4853	1.42931
$726$	−1.00000	−0.0371135
$727$	44.7696	1.66041	0.830205	−	0.557458i	$-0.188223\pi$
$727$	44.7696	1.66041	0.830205	+	0.557458i	$0.188223\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	−7.31371	−0.270692
$731$	−16.9706	−0.627679
$732$	−1.00000	−0.0369611
$733$	34.9706	1.29167	0.645834	−	0.763478i	$-0.276510\pi$
$733$	34.9706	1.29167	0.645834	+	0.763478i	$0.276510\pi$
$734$	−16.8284	−0.621149
$735$	13.2426	0.488462
$736$	2.82843	0.104257
$737$	−8.48528	−0.312559
$738$	10.8995	0.401216
$739$	33.7279	1.24070	0.620351	−	0.784324i	$-0.286990\pi$
$739$	33.7279	1.24070	0.620351	+	0.784324i	$0.286990\pi$
$740$	37.4558	1.37690
$741$	3.51472	0.129116
$742$	−10.3431	−0.379709
$743$	32.6569	1.19806	0.599032	−	0.800725i	$-0.295552\pi$
$743$	32.6569	1.19806	0.599032	+	0.800725i	$0.295552\pi$
$744$	−3.58579	−0.131461
$745$	−58.1421	−2.13016
$746$	32.9706	1.20714
$747$	−5.31371	−0.194418
$748$	−2.82843	−0.103418
$749$	12.8284	0.468741
$750$	41.8701	1.52888
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	−6.00000	−0.218797
$753$	5.48528	0.199895
$754$	−7.97056	−0.290271
$755$	37.4558	1.36316
$756$	2.00000	0.0727393
$757$	12.2843	0.446479	0.223240	−	0.974764i	$-0.428337\pi$
$757$	12.2843	0.446479	0.223240	+	0.974764i	$0.428337\pi$
$758$	−18.7990	−0.682810
$759$	2.82843	0.102665
$760$	5.17157	0.187593
$761$	28.9706	1.05018	0.525091	−	0.851046i	$-0.324031\pi$
$761$	28.9706	1.05018	0.525091	+	0.851046i	$0.324031\pi$
$762$	0	0
$763$	−4.34315	−0.157232
$764$	−12.1421	−0.439287
$765$	−12.4853	−0.451406
$766$	−33.1127	−1.19641
$767$	42.9411	1.55051
$768$	1.00000	0.0360844
$769$	39.5269	1.42538	0.712688	−	0.701481i	$-0.247477\pi$
$769$	39.5269	1.42538	0.712688	+	0.701481i	$0.247477\pi$
$770$	−8.82843	−0.318154
$771$	5.31371	0.191369
$772$	0.757359	0.0272580
$773$	−34.5563	−1.24291	−0.621453	−	0.783452i	$-0.713457\pi$
$773$	−34.5563	−1.24291	−0.621453	+	0.783452i	$0.713457\pi$
$774$	6.00000	0.215666
$775$	51.9411	1.86578
$776$	−10.6569	−0.382559
$777$	−16.9706	−0.608816
$778$	−17.6569	−0.633029
$779$	−12.7696	−0.457517
$780$	−13.2426	−0.474163
$781$	−6.00000	−0.214697
$782$	8.00000	0.286079
$783$	2.65685	0.0949482
$784$	−3.00000	−0.107143
$785$	−28.3137	−1.01056
$786$	−10.0711	−0.359223
$787$	−5.58579	−0.199112	−0.0995559	−	0.995032i	$-0.531742\pi$
$787$	−5.58579	−0.199112	−0.0995559	+	0.995032i	$0.531742\pi$
$788$	−13.7990	−0.491569
$789$	−26.4853	−0.942901
$790$	0	0
$791$	40.2843	1.43234
$792$	1.00000	0.0355335
$793$	−3.00000	−0.106533
$794$	−25.8701	−0.918094
$795$	−22.8284	−0.809641
$796$	6.97056	0.247065
$797$	−22.8284	−0.808624	−0.404312	−	0.914621i	$-0.632489\pi$
$797$	−22.8284	−0.808624	−0.404312	+	0.914621i	$0.632489\pi$
$798$	−2.34315	−0.0829465
$799$	−16.9706	−0.600375
$800$	−14.4853	−0.512132
$801$	−11.4853	−0.405812
$802$	−16.3137	−0.576057
$803$	1.65685	0.0584691
$804$	8.48528	0.299253
$805$	24.9706	0.880097
$806$	−10.7574	−0.378912
$807$	−21.7279	−0.764859
$808$	−11.4853	−0.404051
$809$	2.27208	0.0798820	0.0399410	−	0.999202i	$-0.487283\pi$
$809$	2.27208	0.0798820	0.0399410	+	0.999202i	$0.487283\pi$
$810$	4.41421	0.155100
$811$	43.9411	1.54298	0.771491	−	0.636240i	$-0.219511\pi$
$811$	43.9411	1.54298	0.771491	+	0.636240i	$0.219511\pi$
$812$	5.31371	0.186475
$813$	−1.65685	−0.0581084
$814$	−8.48528	−0.297409
$815$	−29.3848	−1.02930
$816$	2.82843	0.0990148
$817$	−7.02944	−0.245929
$818$	15.5858	0.544944
$819$	6.00000	0.209657
$820$	48.1127	1.68017
$821$	−25.3137	−0.883455	−0.441727	−	0.897149i	$-0.645634\pi$
$821$	−25.3137	−0.883455	−0.441727	+	0.897149i	$0.645634\pi$
$822$	−15.1716	−0.529169
$823$	2.07107	0.0721929	0.0360964	−	0.999348i	$-0.488508\pi$
$823$	2.07107	0.0721929	0.0360964	+	0.999348i	$0.488508\pi$
$824$	4.00000	0.139347
$825$	−14.4853	−0.504313
$826$	−28.6274	−0.996075
$827$	−43.6569	−1.51810	−0.759049	−	0.651034i	$-0.774336\pi$
$827$	−43.6569	−1.51810	−0.759049	+	0.651034i	$0.774336\pi$
$828$	−2.82843	−0.0982946
$829$	−38.3431	−1.33171	−0.665856	−	0.746080i	$-0.731934\pi$
$829$	−38.3431	−1.33171	−0.665856	+	0.746080i	$0.731934\pi$
$830$	−23.4558	−0.814164
$831$	22.9706	0.796840
$832$	3.00000	0.104006
$833$	−8.48528	−0.293998
$834$	−18.0711	−0.625750
$835$	−7.31371	−0.253101
$836$	−1.17157	−0.0405197
$837$	3.58579	0.123943
$838$	14.3431	0.495476
$839$	−42.4142	−1.46430	−0.732151	−	0.681143i	$-0.761483\pi$
$839$	−42.4142	−1.46430	−0.732151	+	0.681143i	$0.761483\pi$
$840$	8.82843	0.304610
$841$	−21.9411	−0.756591
$842$	23.2426	0.800994
$843$	22.6274	0.779330
$844$	−3.72792	−0.128320
$845$	17.6569	0.607414
$846$	6.00000	0.206284
$847$	2.00000	0.0687208
$848$	5.17157	0.177593
$849$	20.4853	0.703053
$850$	−40.9706	−1.40528
$851$	24.0000	0.822709
$852$	6.00000	0.205557
$853$	−43.1421	−1.47716	−0.738579	−	0.674167i	$-0.764503\pi$
$853$	−43.1421	−1.47716	−0.738579	+	0.674167i	$0.764503\pi$
$854$	2.00000	0.0684386
$855$	−5.17157	−0.176864
$856$	−6.41421	−0.219233
$857$	−49.0416	−1.67523	−0.837615	−	0.546261i	$-0.816051\pi$
$857$	−49.0416	−1.67523	−0.837615	+	0.546261i	$0.816051\pi$
$858$	3.00000	0.102418
$859$	21.8284	0.744776	0.372388	−	0.928077i	$-0.378539\pi$
$859$	21.8284	0.744776	0.372388	+	0.928077i	$0.378539\pi$
$860$	26.4853	0.903141
$861$	−21.7990	−0.742908
$862$	19.1716	0.652986
$863$	−28.7574	−0.978912	−0.489456	−	0.872028i	$-0.662805\pi$
$863$	−28.7574	−0.978912	−0.489456	+	0.872028i	$0.662805\pi$
$864$	−1.00000	−0.0340207
$865$	−82.9828	−2.82150
$866$	10.8284	0.367965
$867$	−9.00000	−0.305656
$868$	7.17157	0.243419
$869$	0	0
$870$	11.7279	0.397614
$871$	25.4558	0.862538
$872$	2.17157	0.0735387
$873$	10.6569	0.360680
$874$	3.31371	0.112088
$875$	−83.7401	−2.83093
$876$	−1.65685	−0.0559799
$877$	43.5980	1.47220	0.736100	−	0.676873i	$-0.236665\pi$
$877$	43.5980	1.47220	0.736100	+	0.676873i	$0.236665\pi$
$878$	1.48528	0.0501258
$879$	18.3431	0.618699
$880$	4.41421	0.148803
$881$	41.2548	1.38991	0.694955	−	0.719053i	$-0.255424\pi$
$881$	41.2548	1.38991	0.694955	+	0.719053i	$0.255424\pi$
$882$	3.00000	0.101015
$883$	−40.4853	−1.36244	−0.681219	−	0.732080i	$-0.738550\pi$
$883$	−40.4853	−1.36244	−0.681219	+	0.732080i	$0.738550\pi$
$884$	8.48528	0.285391
$885$	−63.1838	−2.12390
$886$	−7.65685	−0.257237
$887$	56.5685	1.89939	0.949693	−	0.313183i	$-0.101395\pi$
$887$	56.5685	1.89939	0.949693	+	0.313183i	$0.101395\pi$
$888$	8.48528	0.284747
$889$	0	0
$890$	−50.6985	−1.69942
$891$	−1.00000	−0.0335013
$892$	−2.00000	−0.0669650
$893$	−7.02944	−0.235231
$894$	−13.1716	−0.440523
$895$	−66.9706	−2.23858
$896$	−2.00000	−0.0668153
$897$	−8.48528	−0.283315
$898$	−0.201010	−0.00670779
$899$	9.52691	0.317740
$900$	14.4853	0.482843
$901$	14.6274	0.487310
$902$	−10.8995	−0.362913
$903$	−12.0000	−0.399335
$904$	−20.1421	−0.669917
$905$	−107.770	−3.58238
$906$	8.48528	0.281905
$907$	−13.1716	−0.437355	−0.218677	−	0.975797i	$-0.570174\pi$
$907$	−13.1716	−0.437355	−0.218677	+	0.975797i	$0.570174\pi$
$908$	−12.0000	−0.398234
$909$	11.4853	0.380943
$910$	26.4853	0.877979
$911$	−2.27208	−0.0752773	−0.0376387	−	0.999291i	$-0.511984\pi$
$911$	−2.27208	−0.0752773	−0.0376387	+	0.999291i	$0.511984\pi$
$912$	1.17157	0.0387947
$913$	5.31371	0.175858
$914$	8.48528	0.280668
$915$	4.41421	0.145929
$916$	2.34315	0.0774197
$917$	20.1421	0.665152
$918$	−2.82843	−0.0933520
$919$	−53.4264	−1.76237	−0.881187	−	0.472767i	$-0.843255\pi$
$919$	−53.4264	−1.76237	−0.881187	+	0.472767i	$0.843255\pi$
$920$	−12.4853	−0.411628
$921$	−18.2132	−0.600145
$922$	−8.82843	−0.290749
$923$	18.0000	0.592477
$924$	−2.00000	−0.0657952
$925$	−122.912	−4.04131
$926$	30.4853	1.00181
$927$	−4.00000	−0.131377
$928$	−2.65685	−0.0872155
$929$	−45.1716	−1.48203	−0.741016	−	0.671488i	$-0.765656\pi$
$929$	−45.1716	−1.48203	−0.741016	+	0.671488i	$0.765656\pi$
$930$	15.8284	0.519035
$931$	−3.51472	−0.115190
$932$	8.48528	0.277945
$933$	−18.6274	−0.609834
$934$	−2.31371	−0.0757069
$935$	12.4853	0.408312
$936$	−3.00000	−0.0980581
$937$	−6.48528	−0.211865	−0.105932	−	0.994373i	$-0.533783\pi$
$937$	−6.48528	−0.211865	−0.105932	+	0.994373i	$0.533783\pi$
$938$	−16.9706	−0.554109
$939$	−18.1421	−0.592046
$940$	26.4853	0.863855
$941$	50.7990	1.65600	0.828000	−	0.560728i	$-0.189479\pi$
$941$	50.7990	1.65600	0.828000	+	0.560728i	$0.189479\pi$
$942$	−6.41421	−0.208986
$943$	30.8284	1.00391
$944$	14.3137	0.465872
$945$	−8.82843	−0.287189
$946$	−6.00000	−0.195077
$947$	−9.94113	−0.323043	−0.161522	−	0.986869i	$-0.551640\pi$
$947$	−9.94113	−0.323043	−0.161522	+	0.986869i	$0.551640\pi$
$948$	0	0
$949$	−4.97056	−0.161351
$950$	−16.9706	−0.550598
$951$	3.92893	0.127404
$952$	−5.65685	−0.183340
$953$	0.627417	0.0203240	0.0101620	−	0.999948i	$-0.496765\pi$
$953$	0.627417	0.0203240	0.0101620	+	0.999948i	$0.496765\pi$
$954$	−5.17157	−0.167436
$955$	53.5980	1.73439
$956$	10.8284	0.350216
$957$	−2.65685	−0.0858839
$958$	−21.6569	−0.699701
$959$	30.3431	0.979831
$960$	−4.41421	−0.142468
$961$	−18.1421	−0.585230
$962$	25.4558	0.820729
$963$	6.41421	0.206695
$964$	4.00000	0.128831
$965$	−3.34315	−0.107620
$966$	5.65685	0.182006
$967$	−3.97056	−0.127685	−0.0638423	−	0.997960i	$-0.520335\pi$
$967$	−3.97056	−0.127685	−0.0638423	+	0.997960i	$0.520335\pi$
$968$	−1.00000	−0.0321412
$969$	3.31371	0.106452
$970$	47.0416	1.51042
$971$	10.3431	0.331927	0.165964	−	0.986132i	$-0.446927\pi$
$971$	10.3431	0.331927	0.165964	+	0.986132i	$0.446927\pi$
$972$	1.00000	0.0320750
$973$	36.1421	1.15866
$974$	−2.34315	−0.0750792
$975$	43.4558	1.39170
$976$	−1.00000	−0.0320092
$977$	19.3137	0.617900	0.308950	−	0.951078i	$-0.400022\pi$
$977$	19.3137	0.617900	0.308950	+	0.951078i	$0.400022\pi$
$978$	−6.65685	−0.212863
$979$	11.4853	0.367071
$980$	13.2426	0.423021
$981$	−2.17157	−0.0693330
$982$	9.31371	0.297212
$983$	−18.6863	−0.596000	−0.298000	−	0.954566i	$-0.596320\pi$
$983$	−18.6863	−0.596000	−0.298000	+	0.954566i	$0.596320\pi$
$984$	10.8995	0.347463
$985$	60.9117	1.94081
$986$	−7.51472	−0.239317
$987$	−12.0000	−0.381964
$988$	3.51472	0.111818
$989$	16.9706	0.539633
$990$	−4.41421	−0.140293
$991$	58.7696	1.86688	0.933438	−	0.358738i	$-0.116793\pi$
$991$	58.7696	1.86688	0.933438	+	0.358738i	$0.116793\pi$
$992$	−3.58579	−0.113849
$993$	−16.4853	−0.523145
$994$	−12.0000	−0.380617
$995$	−30.7696	−0.975460
$996$	−5.31371	−0.168371
$997$	−61.9411	−1.96170	−0.980848	−	0.194777i	$-0.937602\pi$
$997$	−61.9411	−1.96170	−0.980848	+	0.194777i	$0.937602\pi$
$998$	−29.7990	−0.943271
$999$	−8.48528	−0.268462

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.k.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.k.1.1	✓	2	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:	\( N \)	\(=\)	\( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4026.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.41421 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.41421 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.41421 q^{10} -1.00000 q^{11} +1.00000 q^{12} +3.00000 q^{13} -2.00000 q^{14} -4.41421 q^{15} +1.00000 q^{16} +2.82843 q^{17} -1.00000 q^{18} +1.17157 q^{19} -4.41421 q^{20} +2.00000 q^{21} +1.00000 q^{22} -2.82843 q^{23} -1.00000 q^{24} +14.4853 q^{25} -3.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +2.65685 q^{29} +4.41421 q^{30} +3.58579 q^{31} -1.00000 q^{32} -1.00000 q^{33} -2.82843 q^{34} -8.82843 q^{35} +1.00000 q^{36} -8.48528 q^{37} -1.17157 q^{38} +3.00000 q^{39} +4.41421 q^{40} -10.8995 q^{41} -2.00000 q^{42} -6.00000 q^{43} -1.00000 q^{44} -4.41421 q^{45} +2.82843 q^{46} -6.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -14.4853 q^{50} +2.82843 q^{51} +3.00000 q^{52} +5.17157 q^{53} -1.00000 q^{54} +4.41421 q^{55} -2.00000 q^{56} +1.17157 q^{57} -2.65685 q^{58} +14.3137 q^{59} -4.41421 q^{60} -1.00000 q^{61} -3.58579 q^{62} +2.00000 q^{63} +1.00000 q^{64} -13.2426 q^{65} +1.00000 q^{66} +8.48528 q^{67} +2.82843 q^{68} -2.82843 q^{69} +8.82843 q^{70} +6.00000 q^{71} -1.00000 q^{72} -1.65685 q^{73} +8.48528 q^{74} +14.4853 q^{75} +1.17157 q^{76} -2.00000 q^{77} -3.00000 q^{78} -4.41421 q^{80} +1.00000 q^{81} +10.8995 q^{82} -5.31371 q^{83} +2.00000 q^{84} -12.4853 q^{85} +6.00000 q^{86} +2.65685 q^{87} +1.00000 q^{88} -11.4853 q^{89} +4.41421 q^{90} +6.00000 q^{91} -2.82843 q^{92} +3.58579 q^{93} +6.00000 q^{94} -5.17157 q^{95} -1.00000 q^{96} +10.6569 q^{97} +3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 6 q^{5} - 2 q^{6} + 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 6 * q^5 - 2 * q^6 + 4 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 6 q^{5} - 2 q^{6} + 4 q^{7} - 2 q^{8} + 2 q^{9} + 6 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} - 4 q^{14} - 6 q^{15} + 2 q^{16} - 2 q^{18} + 8 q^{19} - 6 q^{20} + 4 q^{21} + 2 q^{22} - 2 q^{24} + 12 q^{25} - 6 q^{26} + 2 q^{27} + 4 q^{28} - 6 q^{29} + 6 q^{30} + 10 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{35} + 2 q^{36} - 8 q^{38} + 6 q^{39} + 6 q^{40} - 2 q^{41} - 4 q^{42} - 12 q^{43} - 2 q^{44} - 6 q^{45} - 12 q^{47} + 2 q^{48} - 6 q^{49} - 12 q^{50} + 6 q^{52} + 16 q^{53} - 2 q^{54} + 6 q^{55} - 4 q^{56} + 8 q^{57} + 6 q^{58} + 6 q^{59} - 6 q^{60} - 2 q^{61} - 10 q^{62} + 4 q^{63} + 2 q^{64} - 18 q^{65} + 2 q^{66} + 12 q^{70} + 12 q^{71} - 2 q^{72} + 8 q^{73} + 12 q^{75} + 8 q^{76} - 4 q^{77} - 6 q^{78} - 6 q^{80} + 2 q^{81} + 2 q^{82} + 12 q^{83} + 4 q^{84} - 8 q^{85} + 12 q^{86} - 6 q^{87} + 2 q^{88} - 6 q^{89} + 6 q^{90} + 12 q^{91} + 10 q^{93} + 12 q^{94} - 16 q^{95} - 2 q^{96} + 10 q^{97} + 6 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 6 * q^5 - 2 * q^6 + 4 * q^7 - 2 * q^8 + 2 * q^9 + 6 * q^10 - 2 * q^11 + 2 * q^12 + 6 * q^13 - 4 * q^14 - 6 * q^15 + 2 * q^16 - 2 * q^18 + 8 * q^19 - 6 * q^20 + 4 * q^21 + 2 * q^22 - 2 * q^24 + 12 * q^25 - 6 * q^26 + 2 * q^27 + 4 * q^28 - 6 * q^29 + 6 * q^30 + 10 * q^31 - 2 * q^32 - 2 * q^33 - 12 * q^35 + 2 * q^36 - 8 * q^38 + 6 * q^39 + 6 * q^40 - 2 * q^41 - 4 * q^42 - 12 * q^43 - 2 * q^44 - 6 * q^45 - 12 * q^47 + 2 * q^48 - 6 * q^49 - 12 * q^50 + 6 * q^52 + 16 * q^53 - 2 * q^54 + 6 * q^55 - 4 * q^56 + 8 * q^57 + 6 * q^58 + 6 * q^59 - 6 * q^60 - 2 * q^61 - 10 * q^62 + 4 * q^63 + 2 * q^64 - 18 * q^65 + 2 * q^66 + 12 * q^70 + 12 * q^71 - 2 * q^72 + 8 * q^73 + 12 * q^75 + 8 * q^76 - 4 * q^77 - 6 * q^78 - 6 * q^80 + 2 * q^81 + 2 * q^82 + 12 * q^83 + 4 * q^84 - 8 * q^85 + 12 * q^86 - 6 * q^87 + 2 * q^88 - 6 * q^89 + 6 * q^90 + 12 * q^91 + 10 * q^93 + 12 * q^94 - 16 * q^95 - 2 * q^96 + 10 * q^97 + 6 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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