LMFDB - Embedded newform 170.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [170,2,Mod(1,170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(170, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("170.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 170 = 2 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	170.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:	$1.35745683436$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.56155 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.56155 q^{6} +5.12311 q^{7} +1.00000 q^{8} +3.56155 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.56155 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.56155 q^{6} +5.12311 q^{7} +1.00000 q^{8} +3.56155 q^{9} +1.00000 q^{10} -4.00000 q^{11} -2.56155 q^{12} +4.56155 q^{13} +5.12311 q^{14} -2.56155 q^{15} +1.00000 q^{16} +1.00000 q^{17} +3.56155 q^{18} -2.56155 q^{19} +1.00000 q^{20} -13.1231 q^{21} -4.00000 q^{22} -5.12311 q^{23} -2.56155 q^{24} +1.00000 q^{25} +4.56155 q^{26} -1.43845 q^{27} +5.12311 q^{28} -5.68466 q^{29} -2.56155 q^{30} -6.56155 q^{31} +1.00000 q^{32} +10.2462 q^{33} +1.00000 q^{34} +5.12311 q^{35} +3.56155 q^{36} -7.12311 q^{37} -2.56155 q^{38} -11.6847 q^{39} +1.00000 q^{40} +4.24621 q^{41} -13.1231 q^{42} -1.12311 q^{43} -4.00000 q^{44} +3.56155 q^{45} -5.12311 q^{46} +6.56155 q^{47} -2.56155 q^{48} +19.2462 q^{49} +1.00000 q^{50} -2.56155 q^{51} +4.56155 q^{52} -0.561553 q^{53} -1.43845 q^{54} -4.00000 q^{55} +5.12311 q^{56} +6.56155 q^{57} -5.68466 q^{58} -0.315342 q^{59} -2.56155 q^{60} +7.43845 q^{61} -6.56155 q^{62} +18.2462 q^{63} +1.00000 q^{64} +4.56155 q^{65} +10.2462 q^{66} -9.12311 q^{67} +1.00000 q^{68} +13.1231 q^{69} +5.12311 q^{70} -4.31534 q^{71} +3.56155 q^{72} -6.80776 q^{73} -7.12311 q^{74} -2.56155 q^{75} -2.56155 q^{76} -20.4924 q^{77} -11.6847 q^{78} +1.00000 q^{80} -7.00000 q^{81} +4.24621 q^{82} -6.24621 q^{83} -13.1231 q^{84} +1.00000 q^{85} -1.12311 q^{86} +14.5616 q^{87} -4.00000 q^{88} -9.68466 q^{89} +3.56155 q^{90} +23.3693 q^{91} -5.12311 q^{92} +16.8078 q^{93} +6.56155 q^{94} -2.56155 q^{95} -2.56155 q^{96} -1.68466 q^{97} +19.2462 q^{98} -14.2462 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 2 * q^5 - q^6 + 2 * q^7 + 2 * q^8 + 3 * q^9	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} + 3 q^{9} + 2 q^{10} - 8 q^{11} - q^{12} + 5 q^{13} + 2 q^{14} - q^{15} + 2 q^{16} + 2 q^{17} + 3 q^{18} - q^{19} + 2 q^{20} - 18 q^{21} - 8 q^{22} - 2 q^{23} - q^{24} + 2 q^{25} + 5 q^{26} - 7 q^{27} + 2 q^{28} + q^{29} - q^{30} - 9 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{34} + 2 q^{35} + 3 q^{36} - 6 q^{37} - q^{38} - 11 q^{39} + 2 q^{40} - 8 q^{41} - 18 q^{42} + 6 q^{43} - 8 q^{44} + 3 q^{45} - 2 q^{46} + 9 q^{47} - q^{48} + 22 q^{49} + 2 q^{50} - q^{51} + 5 q^{52} + 3 q^{53} - 7 q^{54} - 8 q^{55} + 2 q^{56} + 9 q^{57} + q^{58} - 13 q^{59} - q^{60} + 19 q^{61} - 9 q^{62} + 20 q^{63} + 2 q^{64} + 5 q^{65} + 4 q^{66} - 10 q^{67} + 2 q^{68} + 18 q^{69} + 2 q^{70} - 21 q^{71} + 3 q^{72} + 7 q^{73} - 6 q^{74} - q^{75} - q^{76} - 8 q^{77} - 11 q^{78} + 2 q^{80} - 14 q^{81} - 8 q^{82} + 4 q^{83} - 18 q^{84} + 2 q^{85} + 6 q^{86} + 25 q^{87} - 8 q^{88} - 7 q^{89} + 3 q^{90} + 22 q^{91} - 2 q^{92} + 13 q^{93} + 9 q^{94} - q^{95} - q^{96} + 9 q^{97} + 22 q^{98} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 2 * q^5 - q^6 + 2 * q^7 + 2 * q^8 + 3 * q^9 + 2 * q^10 - 8 * q^11 - q^12 + 5 * q^13 + 2 * q^14 - q^15 + 2 * q^16 + 2 * q^17 + 3 * q^18 - q^19 + 2 * q^20 - 18 * q^21 - 8 * q^22 - 2 * q^23 - q^24 + 2 * q^25 + 5 * q^26 - 7 * q^27 + 2 * q^28 + q^29 - q^30 - 9 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^34 + 2 * q^35 + 3 * q^36 - 6 * q^37 - q^38 - 11 * q^39 + 2 * q^40 - 8 * q^41 - 18 * q^42 + 6 * q^43 - 8 * q^44 + 3 * q^45 - 2 * q^46 + 9 * q^47 - q^48 + 22 * q^49 + 2 * q^50 - q^51 + 5 * q^52 + 3 * q^53 - 7 * q^54 - 8 * q^55 + 2 * q^56 + 9 * q^57 + q^58 - 13 * q^59 - q^60 + 19 * q^61 - 9 * q^62 + 20 * q^63 + 2 * q^64 + 5 * q^65 + 4 * q^66 - 10 * q^67 + 2 * q^68 + 18 * q^69 + 2 * q^70 - 21 * q^71 + 3 * q^72 + 7 * q^73 - 6 * q^74 - q^75 - q^76 - 8 * q^77 - 11 * q^78 + 2 * q^80 - 14 * q^81 - 8 * q^82 + 4 * q^83 - 18 * q^84 + 2 * q^85 + 6 * q^86 + 25 * q^87 - 8 * q^88 - 7 * q^89 + 3 * q^90 + 22 * q^91 - 2 * q^92 + 13 * q^93 + 9 * q^94 - q^95 - q^96 + 9 * q^97 + 22 * q^98 - 12 * 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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.56155	−1.04575
$7$	5.12311	1.93635	0.968176	−	0.250270i	$-0.0805195\pi$
$7$	5.12311	1.93635	0.968176	+	0.250270i	$0.0805195\pi$
$8$	1.00000	0.353553
$9$	3.56155	1.18718
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−2.56155	−0.739457
$13$	4.56155	1.26515	0.632574	−	0.774500i	$-0.281999\pi$
$13$	4.56155	1.26515	0.632574	+	0.774500i	$0.281999\pi$
$14$	5.12311	1.36921
$15$	−2.56155	−0.661390
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	3.56155	0.839466
$19$	−2.56155	−0.587661	−0.293830	−	0.955858i	$-0.594930\pi$
$19$	−2.56155	−0.587661	−0.293830	+	0.955858i	$0.594930\pi$
$20$	1.00000	0.223607
$21$	−13.1231	−2.86370
$22$	−4.00000	−0.852803
$23$	−5.12311	−1.06824	−0.534121	−	0.845408i	$-0.679357\pi$
$23$	−5.12311	−1.06824	−0.534121	+	0.845408i	$0.679357\pi$
$24$	−2.56155	−0.522875
$25$	1.00000	0.200000
$26$	4.56155	0.894594
$27$	−1.43845	−0.276829
$28$	5.12311	0.968176
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	−2.56155	−0.467673
$31$	−6.56155	−1.17849	−0.589245	−	0.807955i	$-0.700575\pi$
$31$	−6.56155	−1.17849	−0.589245	+	0.807955i	$0.700575\pi$
$32$	1.00000	0.176777
$33$	10.2462	1.78364
$34$	1.00000	0.171499
$35$	5.12311	0.865963
$36$	3.56155	0.593592
$37$	−7.12311	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$37$	−7.12311	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$38$	−2.56155	−0.415539
$39$	−11.6847	−1.87104
$40$	1.00000	0.158114
$41$	4.24621	0.663147	0.331573	−	0.943429i	$-0.392421\pi$
$41$	4.24621	0.663147	0.331573	+	0.943429i	$0.392421\pi$
$42$	−13.1231	−2.02494
$43$	−1.12311	−0.171272	−0.0856360	−	0.996326i	$-0.527292\pi$
$43$	−1.12311	−0.171272	−0.0856360	+	0.996326i	$0.527292\pi$
$44$	−4.00000	−0.603023
$45$	3.56155	0.530925
$46$	−5.12311	−0.755361
$47$	6.56155	0.957101	0.478550	−	0.878060i	$-0.341162\pi$
$47$	6.56155	0.957101	0.478550	+	0.878060i	$0.341162\pi$
$48$	−2.56155	−0.369728
$49$	19.2462	2.74946
$50$	1.00000	0.141421
$51$	−2.56155	−0.358689
$52$	4.56155	0.632574
$53$	−0.561553	−0.0771352	−0.0385676	−	0.999256i	$-0.512279\pi$
$53$	−0.561553	−0.0771352	−0.0385676	+	0.999256i	$0.512279\pi$
$54$	−1.43845	−0.195748
$55$	−4.00000	−0.539360
$56$	5.12311	0.684604
$57$	6.56155	0.869099
$58$	−5.68466	−0.746432
$59$	−0.315342	−0.0410540	−0.0205270	−	0.999789i	$-0.506534\pi$
$59$	−0.315342	−0.0410540	−0.0205270	+	0.999789i	$0.506534\pi$
$60$	−2.56155	−0.330695
$61$	7.43845	0.952396	0.476198	−	0.879338i	$-0.342015\pi$
$61$	7.43845	0.952396	0.476198	+	0.879338i	$0.342015\pi$
$62$	−6.56155	−0.833318
$63$	18.2462	2.29881
$64$	1.00000	0.125000
$65$	4.56155	0.565791
$66$	10.2462	1.26122
$67$	−9.12311	−1.11456	−0.557282	−	0.830323i	$-0.688156\pi$
$67$	−9.12311	−1.11456	−0.557282	+	0.830323i	$0.688156\pi$
$68$	1.00000	0.121268
$69$	13.1231	1.57984
$70$	5.12311	0.612328
$71$	−4.31534	−0.512137	−0.256068	−	0.966659i	$-0.582427\pi$
$71$	−4.31534	−0.512137	−0.256068	+	0.966659i	$0.582427\pi$
$72$	3.56155	0.419733
$73$	−6.80776	−0.796789	−0.398394	−	0.917214i	$-0.630432\pi$
$73$	−6.80776	−0.796789	−0.398394	+	0.917214i	$0.630432\pi$
$74$	−7.12311	−0.828044
$75$	−2.56155	−0.295783
$76$	−2.56155	−0.293830
$77$	−20.4924	−2.33533
$78$	−11.6847	−1.32303
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	1.00000	0.111803
$81$	−7.00000	−0.777778
$82$	4.24621	0.468916
$83$	−6.24621	−0.685611	−0.342805	−	0.939406i	$-0.611377\pi$
$83$	−6.24621	−0.685611	−0.342805	+	0.939406i	$0.611377\pi$
$84$	−13.1231	−1.43185
$85$	1.00000	0.108465
$86$	−1.12311	−0.121108
$87$	14.5616	1.56116
$88$	−4.00000	−0.426401
$89$	−9.68466	−1.02657	−0.513286	−	0.858218i	$-0.671572\pi$
$89$	−9.68466	−1.02657	−0.513286	+	0.858218i	$0.671572\pi$
$90$	3.56155	0.375421
$91$	23.3693	2.44977
$92$	−5.12311	−0.534121
$93$	16.8078	1.74288
$94$	6.56155	0.676772
$95$	−2.56155	−0.262810
$96$	−2.56155	−0.261437
$97$	−1.68466	−0.171051	−0.0855256	−	0.996336i	$-0.527257\pi$
$97$	−1.68466	−0.171051	−0.0855256	+	0.996336i	$0.527257\pi$
$98$	19.2462	1.94416
$99$	−14.2462	−1.43180
$100$	1.00000	0.100000
$101$	−14.4924	−1.44205	−0.721025	−	0.692909i	$-0.756329\pi$
$101$	−14.4924	−1.44205	−0.721025	+	0.692909i	$0.756329\pi$
$102$	−2.56155	−0.253632
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	4.56155	0.447297
$105$	−13.1231	−1.28068
$106$	−0.561553	−0.0545428
$107$	16.4924	1.59438	0.797191	−	0.603727i	$-0.206318\pi$
$107$	16.4924	1.59438	0.797191	+	0.603727i	$0.206318\pi$
$108$	−1.43845	−0.138415
$109$	−8.56155	−0.820048	−0.410024	−	0.912075i	$-0.634480\pi$
$109$	−8.56155	−0.820048	−0.410024	+	0.912075i	$0.634480\pi$
$110$	−4.00000	−0.381385
$111$	18.2462	1.73185
$112$	5.12311	0.484088
$113$	2.80776	0.264132	0.132066	−	0.991241i	$-0.457839\pi$
$113$	2.80776	0.264132	0.132066	+	0.991241i	$0.457839\pi$
$114$	6.56155	0.614546
$115$	−5.12311	−0.477732
$116$	−5.68466	−0.527807
$117$	16.2462	1.50196
$118$	−0.315342	−0.0290295
$119$	5.12311	0.469634
$120$	−2.56155	−0.233837
$121$	5.00000	0.454545
$122$	7.43845	0.673445
$123$	−10.8769	−0.980737
$124$	−6.56155	−0.589245
$125$	1.00000	0.0894427
$126$	18.2462	1.62550
$127$	9.43845	0.837527	0.418763	−	0.908095i	$-0.362464\pi$
$127$	9.43845	0.837527	0.418763	+	0.908095i	$0.362464\pi$
$128$	1.00000	0.0883883
$129$	2.87689	0.253296
$130$	4.56155	0.400075
$131$	1.12311	0.0981262	0.0490631	−	0.998796i	$-0.484376\pi$
$131$	1.12311	0.0981262	0.0490631	+	0.998796i	$0.484376\pi$
$132$	10.2462	0.891818
$133$	−13.1231	−1.13792
$134$	−9.12311	−0.788116
$135$	−1.43845	−0.123802
$136$	1.00000	0.0857493
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	13.1231	1.11711
$139$	16.4924	1.39887	0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	5.12311	0.432981
$141$	−16.8078	−1.41547
$142$	−4.31534	−0.362135
$143$	−18.2462	−1.52582
$144$	3.56155	0.296796
$145$	−5.68466	−0.472085
$146$	−6.80776	−0.563415
$147$	−49.3002	−4.06621
$148$	−7.12311	−0.585516
$149$	3.12311	0.255855	0.127927	−	0.991784i	$-0.459168\pi$
$149$	3.12311	0.255855	0.127927	+	0.991784i	$0.459168\pi$
$150$	−2.56155	−0.209150
$151$	−15.3693	−1.25074	−0.625369	−	0.780329i	$-0.715051\pi$
$151$	−15.3693	−1.25074	−0.625369	+	0.780329i	$0.715051\pi$
$152$	−2.56155	−0.207769
$153$	3.56155	0.287934
$154$	−20.4924	−1.65133
$155$	−6.56155	−0.527037
$156$	−11.6847	−0.935521
$157$	18.4924	1.47586	0.737928	−	0.674879i	$-0.235804\pi$
$157$	18.4924	1.47586	0.737928	+	0.674879i	$0.235804\pi$
$158$	0	0
$159$	1.43845	0.114076
$160$	1.00000	0.0790569
$161$	−26.2462	−2.06849
$162$	−7.00000	−0.549972
$163$	14.2462	1.11585	0.557925	−	0.829892i	$-0.311598\pi$
$163$	14.2462	1.11585	0.557925	+	0.829892i	$0.311598\pi$
$164$	4.24621	0.331573
$165$	10.2462	0.797666
$166$	−6.24621	−0.484800
$167$	5.12311	0.396438	0.198219	−	0.980158i	$-0.436484\pi$
$167$	5.12311	0.396438	0.198219	+	0.980158i	$0.436484\pi$
$168$	−13.1231	−1.01247
$169$	7.80776	0.600597
$170$	1.00000	0.0766965
$171$	−9.12311	−0.697661
$172$	−1.12311	−0.0856360
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	14.5616	1.10391
$175$	5.12311	0.387270
$176$	−4.00000	−0.301511
$177$	0.807764	0.0607153
$178$	−9.68466	−0.725896
$179$	8.49242	0.634753	0.317377	−	0.948300i	$-0.397198\pi$
$179$	8.49242	0.634753	0.317377	+	0.948300i	$0.397198\pi$
$180$	3.56155	0.265462
$181$	16.2462	1.20757	0.603786	−	0.797147i	$-0.293658\pi$
$181$	16.2462	1.20757	0.603786	+	0.797147i	$0.293658\pi$
$182$	23.3693	1.73225
$183$	−19.0540	−1.40851
$184$	−5.12311	−0.377680
$185$	−7.12311	−0.523701
$186$	16.8078	1.23241
$187$	−4.00000	−0.292509
$188$	6.56155	0.478550
$189$	−7.36932	−0.536039
$190$	−2.56155	−0.185835
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−2.56155	−0.184864
$193$	−24.2462	−1.74528	−0.872640	−	0.488364i	$-0.837594\pi$
$193$	−24.2462	−1.74528	−0.872640	+	0.488364i	$0.837594\pi$
$194$	−1.68466	−0.120951
$195$	−11.6847	−0.836756
$196$	19.2462	1.37473
$197$	13.3693	0.952524	0.476262	−	0.879303i	$-0.341991\pi$
$197$	13.3693	0.952524	0.476262	+	0.879303i	$0.341991\pi$
$198$	−14.2462	−1.01243
$199$	14.5616	1.03224	0.516121	−	0.856516i	$-0.327376\pi$
$199$	14.5616	1.03224	0.516121	+	0.856516i	$0.327376\pi$
$200$	1.00000	0.0707107
$201$	23.3693	1.64834
$202$	−14.4924	−1.01968
$203$	−29.1231	−2.04404
$204$	−2.56155	−0.179345
$205$	4.24621	0.296568
$206$	8.00000	0.557386
$207$	−18.2462	−1.26820
$208$	4.56155	0.316287
$209$	10.2462	0.708745
$210$	−13.1231	−0.905580
$211$	−3.36932	−0.231953	−0.115977	−	0.993252i	$-0.537000\pi$
$211$	−3.36932	−0.231953	−0.115977	+	0.993252i	$0.537000\pi$
$212$	−0.561553	−0.0385676
$213$	11.0540	0.757406
$214$	16.4924	1.12740
$215$	−1.12311	−0.0765952
$216$	−1.43845	−0.0978739
$217$	−33.6155	−2.28197
$218$	−8.56155	−0.579862
$219$	17.4384	1.17838
$220$	−4.00000	−0.269680
$221$	4.56155	0.306843
$222$	18.2462	1.22461
$223$	3.68466	0.246743	0.123371	−	0.992361i	$-0.460629\pi$
$223$	3.68466	0.246743	0.123371	+	0.992361i	$0.460629\pi$
$224$	5.12311	0.342302
$225$	3.56155	0.237437
$226$	2.80776	0.186770
$227$	16.3153	1.08289	0.541444	−	0.840737i	$-0.317878\pi$
$227$	16.3153	1.08289	0.541444	+	0.840737i	$0.317878\pi$
$228$	6.56155	0.434549
$229$	−14.4924	−0.957686	−0.478843	−	0.877900i	$-0.658944\pi$
$229$	−14.4924	−0.957686	−0.478843	+	0.877900i	$0.658944\pi$
$230$	−5.12311	−0.337808
$231$	52.4924	3.45375
$232$	−5.68466	−0.373216
$233$	6.31534	0.413732	0.206866	−	0.978369i	$-0.433674\pi$
$233$	6.31534	0.413732	0.206866	+	0.978369i	$0.433674\pi$
$234$	16.2462	1.06205
$235$	6.56155	0.428029
$236$	−0.315342	−0.0205270
$237$	0	0
$238$	5.12311	0.332082
$239$	−23.3693	−1.51164	−0.755818	−	0.654782i	$-0.772760\pi$
$239$	−23.3693	−1.51164	−0.755818	+	0.654782i	$0.772760\pi$
$240$	−2.56155	−0.165348
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	5.00000	0.321412
$243$	22.2462	1.42710
$244$	7.43845	0.476198
$245$	19.2462	1.22960
$246$	−10.8769	−0.693485
$247$	−11.6847	−0.743477
$248$	−6.56155	−0.416659
$249$	16.0000	1.01396
$250$	1.00000	0.0632456
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	18.2462	1.14940
$253$	20.4924	1.28835
$254$	9.43845	0.592221
$255$	−2.56155	−0.160411
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	2.87689	0.179108
$259$	−36.4924	−2.26753
$260$	4.56155	0.282895
$261$	−20.2462	−1.25321
$262$	1.12311	0.0693857
$263$	24.8078	1.52971	0.764856	−	0.644201i	$-0.222810\pi$
$263$	24.8078	1.52971	0.764856	+	0.644201i	$0.222810\pi$
$264$	10.2462	0.630611
$265$	−0.561553	−0.0344959
$266$	−13.1231	−0.804629
$267$	24.8078	1.51821
$268$	−9.12311	−0.557282
$269$	−2.80776	−0.171192	−0.0855962	−	0.996330i	$-0.527279\pi$
$269$	−2.80776	−0.171192	−0.0855962	+	0.996330i	$0.527279\pi$
$270$	−1.43845	−0.0875411
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	1.00000	0.0606339
$273$	−59.8617	−3.62300
$274$	10.0000	0.604122
$275$	−4.00000	−0.241209
$276$	13.1231	0.789918
$277$	−15.7538	−0.946553	−0.473277	−	0.880914i	$-0.656929\pi$
$277$	−15.7538	−0.946553	−0.473277	+	0.880914i	$0.656929\pi$
$278$	16.4924	0.989150
$279$	−23.3693	−1.39908
$280$	5.12311	0.306164
$281$	16.5616	0.987979	0.493990	−	0.869468i	$-0.335538\pi$
$281$	16.5616	0.987979	0.493990	+	0.869468i	$0.335538\pi$
$282$	−16.8078	−1.00089
$283$	8.31534	0.494296	0.247148	−	0.968978i	$-0.420507\pi$
$283$	8.31534	0.494296	0.247148	+	0.968978i	$0.420507\pi$
$284$	−4.31534	−0.256068
$285$	6.56155	0.388673
$286$	−18.2462	−1.07892
$287$	21.7538	1.28409
$288$	3.56155	0.209867
$289$	1.00000	0.0588235
$290$	−5.68466	−0.333815
$291$	4.31534	0.252970
$292$	−6.80776	−0.398394
$293$	−16.5616	−0.967536	−0.483768	−	0.875196i	$-0.660732\pi$
$293$	−16.5616	−0.967536	−0.483768	+	0.875196i	$0.660732\pi$
$294$	−49.3002	−2.87525
$295$	−0.315342	−0.0183599
$296$	−7.12311	−0.414022
$297$	5.75379	0.333869
$298$	3.12311	0.180917
$299$	−23.3693	−1.35148
$300$	−2.56155	−0.147891
$301$	−5.75379	−0.331643
$302$	−15.3693	−0.884405
$303$	37.1231	2.13267
$304$	−2.56155	−0.146915
$305$	7.43845	0.425924
$306$	3.56155	0.203600
$307$	−17.7538	−1.01326	−0.506631	−	0.862163i	$-0.669109\pi$
$307$	−17.7538	−1.01326	−0.506631	+	0.862163i	$0.669109\pi$
$308$	−20.4924	−1.16766
$309$	−20.4924	−1.16577
$310$	−6.56155	−0.372671
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−11.6847	−0.661514
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	18.4924	1.04359
$315$	18.2462	1.02806
$316$	0	0
$317$	31.6155	1.77570	0.887852	−	0.460128i	$-0.152197\pi$
$317$	31.6155	1.77570	0.887852	+	0.460128i	$0.152197\pi$
$318$	1.43845	0.0806641
$319$	22.7386	1.27312
$320$	1.00000	0.0559017
$321$	−42.2462	−2.35795
$322$	−26.2462	−1.46264
$323$	−2.56155	−0.142529
$324$	−7.00000	−0.388889
$325$	4.56155	0.253029
$326$	14.2462	0.789025
$327$	21.9309	1.21278
$328$	4.24621	0.234458
$329$	33.6155	1.85328
$330$	10.2462	0.564035
$331$	23.0540	1.26716	0.633581	−	0.773677i	$-0.281585\pi$
$331$	23.0540	1.26716	0.633581	+	0.773677i	$0.281585\pi$
$332$	−6.24621	−0.342805
$333$	−25.3693	−1.39023
$334$	5.12311	0.280324
$335$	−9.12311	−0.498449
$336$	−13.1231	−0.715924
$337$	−10.3153	−0.561912	−0.280956	−	0.959721i	$-0.590652\pi$
$337$	−10.3153	−0.561912	−0.280956	+	0.959721i	$0.590652\pi$
$338$	7.80776	0.424686
$339$	−7.19224	−0.390629
$340$	1.00000	0.0542326
$341$	26.2462	1.42131
$342$	−9.12311	−0.493321
$343$	62.7386	3.38757
$344$	−1.12311	−0.0605538
$345$	13.1231	0.706524
$346$	−18.0000	−0.967686
$347$	−10.5616	−0.566974	−0.283487	−	0.958976i	$-0.591491\pi$
$347$	−10.5616	−0.566974	−0.283487	+	0.958976i	$0.591491\pi$
$348$	14.5616	0.780581
$349$	−15.1231	−0.809521	−0.404761	−	0.914423i	$-0.632645\pi$
$349$	−15.1231	−0.809521	−0.404761	+	0.914423i	$0.632645\pi$
$350$	5.12311	0.273842
$351$	−6.56155	−0.350230
$352$	−4.00000	−0.213201
$353$	22.4924	1.19715	0.598575	−	0.801066i	$-0.295734\pi$
$353$	22.4924	1.19715	0.598575	+	0.801066i	$0.295734\pi$
$354$	0.807764	0.0429322
$355$	−4.31534	−0.229035
$356$	−9.68466	−0.513286
$357$	−13.1231	−0.694548
$358$	8.49242	0.448838
$359$	−10.8769	−0.574061	−0.287030	−	0.957922i	$-0.592668\pi$
$359$	−10.8769	−0.574061	−0.287030	+	0.957922i	$0.592668\pi$
$360$	3.56155	0.187710
$361$	−12.4384	−0.654655
$362$	16.2462	0.853882
$363$	−12.8078	−0.672233
$364$	23.3693	1.22489
$365$	−6.80776	−0.356335
$366$	−19.0540	−0.995967
$367$	−33.6155	−1.75472	−0.877358	−	0.479836i	$-0.840696\pi$
$367$	−33.6155	−1.75472	−0.877358	+	0.479836i	$0.840696\pi$
$368$	−5.12311	−0.267060
$369$	15.1231	0.787277
$370$	−7.12311	−0.370313
$371$	−2.87689	−0.149361
$372$	16.8078	0.871442
$373$	−24.7386	−1.28092	−0.640459	−	0.767992i	$-0.721256\pi$
$373$	−24.7386	−1.28092	−0.640459	+	0.767992i	$0.721256\pi$
$374$	−4.00000	−0.206835
$375$	−2.56155	−0.132278
$376$	6.56155	0.338386
$377$	−25.9309	−1.33551
$378$	−7.36932	−0.379037
$379$	−14.2462	−0.731779	−0.365889	−	0.930658i	$-0.619235\pi$
$379$	−14.2462	−0.731779	−0.365889	+	0.930658i	$0.619235\pi$
$380$	−2.56155	−0.131405
$381$	−24.1771	−1.23863
$382$	0	0
$383$	−16.8078	−0.858837	−0.429418	−	0.903106i	$-0.641281\pi$
$383$	−16.8078	−0.858837	−0.429418	+	0.903106i	$0.641281\pi$
$384$	−2.56155	−0.130719
$385$	−20.4924	−1.04439
$386$	−24.2462	−1.23410
$387$	−4.00000	−0.203331
$388$	−1.68466	−0.0855256
$389$	13.3693	0.677851	0.338926	−	0.940813i	$-0.389936\pi$
$389$	13.3693	0.677851	0.338926	+	0.940813i	$0.389936\pi$
$390$	−11.6847	−0.591676
$391$	−5.12311	−0.259087
$392$	19.2462	0.972080
$393$	−2.87689	−0.145120
$394$	13.3693	0.673536
$395$	0	0
$396$	−14.2462	−0.715899
$397$	21.3693	1.07250	0.536248	−	0.844061i	$-0.319841\pi$
$397$	21.3693	1.07250	0.536248	+	0.844061i	$0.319841\pi$
$398$	14.5616	0.729905
$399$	33.6155	1.68288
$400$	1.00000	0.0500000
$401$	10.6307	0.530871	0.265435	−	0.964129i	$-0.414484\pi$
$401$	10.6307	0.530871	0.265435	+	0.964129i	$0.414484\pi$
$402$	23.3693	1.16556
$403$	−29.9309	−1.49096
$404$	−14.4924	−0.721025
$405$	−7.00000	−0.347833
$406$	−29.1231	−1.44536
$407$	28.4924	1.41232
$408$	−2.56155	−0.126816
$409$	−2.31534	−0.114486	−0.0572431	−	0.998360i	$-0.518231\pi$
$409$	−2.31534	−0.114486	−0.0572431	+	0.998360i	$0.518231\pi$
$410$	4.24621	0.209705
$411$	−25.6155	−1.26352
$412$	8.00000	0.394132
$413$	−1.61553	−0.0794949
$414$	−18.2462	−0.896752
$415$	−6.24621	−0.306614
$416$	4.56155	0.223649
$417$	−42.2462	−2.06881
$418$	10.2462	0.501159
$419$	−25.1231	−1.22734	−0.613672	−	0.789561i	$-0.710308\pi$
$419$	−25.1231	−1.22734	−0.613672	+	0.789561i	$0.710308\pi$
$420$	−13.1231	−0.640342
$421$	7.61553	0.371158	0.185579	−	0.982629i	$-0.440584\pi$
$421$	7.61553	0.371158	0.185579	+	0.982629i	$0.440584\pi$
$422$	−3.36932	−0.164016
$423$	23.3693	1.13626
$424$	−0.561553	−0.0272714
$425$	1.00000	0.0485071
$426$	11.0540	0.535567
$427$	38.1080	1.84417
$428$	16.4924	0.797191
$429$	46.7386	2.25656
$430$	−1.12311	−0.0541610
$431$	−20.4924	−0.987085	−0.493543	−	0.869722i	$-0.664298\pi$
$431$	−20.4924	−0.987085	−0.493543	+	0.869722i	$0.664298\pi$
$432$	−1.43845	−0.0692073
$433$	−2.49242	−0.119778	−0.0598891	−	0.998205i	$-0.519075\pi$
$433$	−2.49242	−0.119778	−0.0598891	+	0.998205i	$0.519075\pi$
$434$	−33.6155	−1.61360
$435$	14.5616	0.698173
$436$	−8.56155	−0.410024
$437$	13.1231	0.627763
$438$	17.4384	0.833241
$439$	−32.9848	−1.57428	−0.787140	−	0.616774i	$-0.788439\pi$
$439$	−32.9848	−1.57428	−0.787140	+	0.616774i	$0.788439\pi$
$440$	−4.00000	−0.190693
$441$	68.5464	3.26411
$442$	4.56155	0.216971
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	18.2462	0.865927
$445$	−9.68466	−0.459097
$446$	3.68466	0.174474
$447$	−8.00000	−0.378387
$448$	5.12311	0.242044
$449$	−16.8769	−0.796470	−0.398235	−	0.917283i	$-0.630377\pi$
$449$	−16.8769	−0.796470	−0.398235	+	0.917283i	$0.630377\pi$
$450$	3.56155	0.167893
$451$	−16.9848	−0.799785
$452$	2.80776	0.132066
$453$	39.3693	1.84973
$454$	16.3153	0.765717
$455$	23.3693	1.09557
$456$	6.56155	0.307273
$457$	−16.2462	−0.759966	−0.379983	−	0.924994i	$-0.624070\pi$
$457$	−16.2462	−0.759966	−0.379983	+	0.924994i	$0.624070\pi$
$458$	−14.4924	−0.677186
$459$	−1.43845	−0.0671410
$460$	−5.12311	−0.238866
$461$	2.49242	0.116084	0.0580418	−	0.998314i	$-0.481514\pi$
$461$	2.49242	0.116084	0.0580418	+	0.998314i	$0.481514\pi$
$462$	52.4924	2.44217
$463$	−13.9309	−0.647422	−0.323711	−	0.946156i	$-0.604931\pi$
$463$	−13.9309	−0.647422	−0.323711	+	0.946156i	$0.604931\pi$
$464$	−5.68466	−0.263904
$465$	16.8078	0.779441
$466$	6.31534	0.292553
$467$	−32.4924	−1.50357	−0.751785	−	0.659408i	$-0.770807\pi$
$467$	−32.4924	−1.50357	−0.751785	+	0.659408i	$0.770807\pi$
$468$	16.2462	0.750981
$469$	−46.7386	−2.15819
$470$	6.56155	0.302662
$471$	−47.3693	−2.18266
$472$	−0.315342	−0.0145148
$473$	4.49242	0.206562
$474$	0	0
$475$	−2.56155	−0.117532
$476$	5.12311	0.234817
$477$	−2.00000	−0.0915737
$478$	−23.3693	−1.06889
$479$	38.5616	1.76192	0.880961	−	0.473189i	$-0.156897\pi$
$479$	38.5616	1.76192	0.880961	+	0.473189i	$0.156897\pi$
$480$	−2.56155	−0.116918
$481$	−32.4924	−1.48153
$482$	12.2462	0.557800
$483$	67.2311	3.05912
$484$	5.00000	0.227273
$485$	−1.68466	−0.0764964
$486$	22.2462	1.00911
$487$	25.6155	1.16075	0.580375	−	0.814349i	$-0.302906\pi$
$487$	25.6155	1.16075	0.580375	+	0.814349i	$0.302906\pi$
$488$	7.43845	0.336723
$489$	−36.4924	−1.65024
$490$	19.2462	0.869455
$491$	2.56155	0.115601	0.0578006	−	0.998328i	$-0.481591\pi$
$491$	2.56155	0.115601	0.0578006	+	0.998328i	$0.481591\pi$
$492$	−10.8769	−0.490368
$493$	−5.68466	−0.256024
$494$	−11.6847	−0.525718
$495$	−14.2462	−0.640320
$496$	−6.56155	−0.294622
$497$	−22.1080	−0.991677
$498$	16.0000	0.716977
$499$	−9.12311	−0.408406	−0.204203	−	0.978929i	$-0.565460\pi$
$499$	−9.12311	−0.408406	−0.204203	+	0.978929i	$0.565460\pi$
$500$	1.00000	0.0447214
$501$	−13.1231	−0.586297
$502$	−4.00000	−0.178529
$503$	22.7386	1.01387	0.506933	−	0.861986i	$-0.330779\pi$
$503$	22.7386	1.01387	0.506933	+	0.861986i	$0.330779\pi$
$504$	18.2462	0.812751
$505$	−14.4924	−0.644904
$506$	20.4924	0.910999
$507$	−20.0000	−0.888231
$508$	9.43845	0.418763
$509$	16.8769	0.748055	0.374028	−	0.927418i	$-0.377977\pi$
$509$	16.8769	0.748055	0.374028	+	0.927418i	$0.377977\pi$
$510$	−2.56155	−0.113427
$511$	−34.8769	−1.54286
$512$	1.00000	0.0441942
$513$	3.68466	0.162682
$514$	2.00000	0.0882162
$515$	8.00000	0.352522
$516$	2.87689	0.126648
$517$	−26.2462	−1.15431
$518$	−36.4924	−1.60338
$519$	46.1080	2.02391
$520$	4.56155	0.200037
$521$	43.6155	1.91083	0.955415	−	0.295265i	$-0.0954078\pi$
$521$	43.6155	1.91083	0.955415	+	0.295265i	$0.0954078\pi$
$522$	−20.2462	−0.886153
$523$	39.8617	1.74303	0.871516	−	0.490367i	$-0.163137\pi$
$523$	39.8617	1.74303	0.871516	+	0.490367i	$0.163137\pi$
$524$	1.12311	0.0490631
$525$	−13.1231	−0.572739
$526$	24.8078	1.08167
$527$	−6.56155	−0.285826
$528$	10.2462	0.445909
$529$	3.24621	0.141140
$530$	−0.561553	−0.0243923
$531$	−1.12311	−0.0487386
$532$	−13.1231	−0.568959
$533$	19.3693	0.838978
$534$	24.8078	1.07354
$535$	16.4924	0.713030
$536$	−9.12311	−0.394058
$537$	−21.7538	−0.938745
$538$	−2.80776	−0.121051
$539$	−76.9848	−3.31597
$540$	−1.43845	−0.0619009
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	16.0000	0.687259
$543$	−41.6155	−1.78589
$544$	1.00000	0.0428746
$545$	−8.56155	−0.366737
$546$	−59.8617	−2.56185
$547$	29.4384	1.25870	0.629349	−	0.777123i	$-0.283322\pi$
$547$	29.4384	1.25870	0.629349	+	0.777123i	$0.283322\pi$
$548$	10.0000	0.427179
$549$	26.4924	1.13067
$550$	−4.00000	−0.170561
$551$	14.5616	0.620343
$552$	13.1231	0.558556
$553$	0	0
$554$	−15.7538	−0.669314
$555$	18.2462	0.774509
$556$	16.4924	0.699435
$557$	36.5616	1.54916	0.774581	−	0.632474i	$-0.217961\pi$
$557$	36.5616	1.54916	0.774581	+	0.632474i	$0.217961\pi$
$558$	−23.3693	−0.989302
$559$	−5.12311	−0.216684
$560$	5.12311	0.216491
$561$	10.2462	0.432595
$562$	16.5616	0.698607
$563$	−4.63068	−0.195160	−0.0975800	−	0.995228i	$-0.531110\pi$
$563$	−4.63068	−0.195160	−0.0975800	+	0.995228i	$0.531110\pi$
$564$	−16.8078	−0.707735
$565$	2.80776	0.118124
$566$	8.31534	0.349520
$567$	−35.8617	−1.50605
$568$	−4.31534	−0.181068
$569$	−3.93087	−0.164791	−0.0823953	−	0.996600i	$-0.526257\pi$
$569$	−3.93087	−0.164791	−0.0823953	+	0.996600i	$0.526257\pi$
$570$	6.56155	0.274833
$571$	25.1231	1.05137	0.525685	−	0.850680i	$-0.323809\pi$
$571$	25.1231	1.05137	0.525685	+	0.850680i	$0.323809\pi$
$572$	−18.2462	−0.762912
$573$	0	0
$574$	21.7538	0.907986
$575$	−5.12311	−0.213648
$576$	3.56155	0.148398
$577$	−5.36932	−0.223528	−0.111764	−	0.993735i	$-0.535650\pi$
$577$	−5.36932	−0.223528	−0.111764	+	0.993735i	$0.535650\pi$
$578$	1.00000	0.0415945
$579$	62.1080	2.58112
$580$	−5.68466	−0.236043
$581$	−32.0000	−1.32758
$582$	4.31534	0.178877
$583$	2.24621	0.0930286
$584$	−6.80776	−0.281707
$585$	16.2462	0.671698
$586$	−16.5616	−0.684151
$587$	−6.87689	−0.283840	−0.141920	−	0.989878i	$-0.545328\pi$
$587$	−6.87689	−0.283840	−0.141920	+	0.989878i	$0.545328\pi$
$588$	−49.3002	−2.03311
$589$	16.8078	0.692552
$590$	−0.315342	−0.0129824
$591$	−34.2462	−1.40870
$592$	−7.12311	−0.292758
$593$	−8.24621	−0.338631	−0.169316	−	0.985562i	$-0.554156\pi$
$593$	−8.24621	−0.338631	−0.169316	+	0.985562i	$0.554156\pi$
$594$	5.75379	0.236081
$595$	5.12311	0.210027
$596$	3.12311	0.127927
$597$	−37.3002	−1.52660
$598$	−23.3693	−0.955642
$599$	−9.61553	−0.392880	−0.196440	−	0.980516i	$-0.562938\pi$
$599$	−9.61553	−0.392880	−0.196440	+	0.980516i	$0.562938\pi$
$600$	−2.56155	−0.104575
$601$	−7.61553	−0.310644	−0.155322	−	0.987864i	$-0.549641\pi$
$601$	−7.61553	−0.310644	−0.155322	+	0.987864i	$0.549641\pi$
$602$	−5.75379	−0.234507
$603$	−32.4924	−1.32319
$604$	−15.3693	−0.625369
$605$	5.00000	0.203279
$606$	37.1231	1.50802
$607$	−43.8617	−1.78029	−0.890147	−	0.455674i	$-0.849398\pi$
$607$	−43.8617	−1.78029	−0.890147	+	0.455674i	$0.849398\pi$
$608$	−2.56155	−0.103885
$609$	74.6004	3.02296
$610$	7.43845	0.301174
$611$	29.9309	1.21087
$612$	3.56155	0.143967
$613$	−23.9309	−0.966559	−0.483279	−	0.875466i	$-0.660554\pi$
$613$	−23.9309	−0.966559	−0.483279	+	0.875466i	$0.660554\pi$
$614$	−17.7538	−0.716485
$615$	−10.8769	−0.438599
$616$	−20.4924	−0.825663
$617$	6.31534	0.254246	0.127123	−	0.991887i	$-0.459426\pi$
$617$	6.31534	0.254246	0.127123	+	0.991887i	$0.459426\pi$
$618$	−20.4924	−0.824326
$619$	9.12311	0.366689	0.183344	−	0.983049i	$-0.441308\pi$
$619$	9.12311	0.366689	0.183344	+	0.983049i	$0.441308\pi$
$620$	−6.56155	−0.263518
$621$	7.36932	0.295720
$622$	8.00000	0.320771
$623$	−49.6155	−1.98780
$624$	−11.6847	−0.467761
$625$	1.00000	0.0400000
$626$	−6.00000	−0.239808
$627$	−26.2462	−1.04817
$628$	18.4924	0.737928
$629$	−7.12311	−0.284017
$630$	18.2462	0.726946
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	8.63068	0.343039
$634$	31.6155	1.25561
$635$	9.43845	0.374553
$636$	1.43845	0.0570381
$637$	87.7926	3.47847
$638$	22.7386	0.900231
$639$	−15.3693	−0.608001
$640$	1.00000	0.0395285
$641$	−24.2462	−0.957668	−0.478834	−	0.877906i	$-0.658940\pi$
$641$	−24.2462	−0.957668	−0.478834	+	0.877906i	$0.658940\pi$
$642$	−42.2462	−1.66732
$643$	40.4924	1.59687	0.798433	−	0.602084i	$-0.205663\pi$
$643$	40.4924	1.59687	0.798433	+	0.602084i	$0.205663\pi$
$644$	−26.2462	−1.03425
$645$	2.87689	0.113278
$646$	−2.56155	−0.100783
$647$	11.6847	0.459371	0.229686	−	0.973265i	$-0.426230\pi$
$647$	11.6847	0.459371	0.229686	+	0.973265i	$0.426230\pi$
$648$	−7.00000	−0.274986
$649$	1.26137	0.0495130
$650$	4.56155	0.178919
$651$	86.1080	3.37484
$652$	14.2462	0.557925
$653$	−38.4924	−1.50632	−0.753162	−	0.657835i	$-0.771473\pi$
$653$	−38.4924	−1.50632	−0.753162	+	0.657835i	$0.771473\pi$
$654$	21.9309	0.857565
$655$	1.12311	0.0438834
$656$	4.24621	0.165787
$657$	−24.2462	−0.945935
$658$	33.6155	1.31047
$659$	−41.9309	−1.63339	−0.816697	−	0.577066i	$-0.804197\pi$
$659$	−41.9309	−1.63339	−0.816697	+	0.577066i	$0.804197\pi$
$660$	10.2462	0.398833
$661$	17.8617	0.694741	0.347371	−	0.937728i	$-0.387075\pi$
$661$	17.8617	0.694741	0.347371	+	0.937728i	$0.387075\pi$
$662$	23.0540	0.896018
$663$	−11.6847	−0.453795
$664$	−6.24621	−0.242400
$665$	−13.1231	−0.508892
$666$	−25.3693	−0.983041
$667$	29.1231	1.12765
$668$	5.12311	0.198219
$669$	−9.43845	−0.364911
$670$	−9.12311	−0.352456
$671$	−29.7538	−1.14863
$672$	−13.1231	−0.506235
$673$	20.4233	0.787260	0.393630	−	0.919269i	$-0.371219\pi$
$673$	20.4233	0.787260	0.393630	+	0.919269i	$0.371219\pi$
$674$	−10.3153	−0.397332
$675$	−1.43845	−0.0553659
$676$	7.80776	0.300299
$677$	−40.7386	−1.56571	−0.782856	−	0.622202i	$-0.786238\pi$
$677$	−40.7386	−1.56571	−0.782856	+	0.622202i	$0.786238\pi$
$678$	−7.19224	−0.276216
$679$	−8.63068	−0.331215
$680$	1.00000	0.0383482
$681$	−41.7926	−1.60150
$682$	26.2462	1.00502
$683$	−41.3002	−1.58031	−0.790154	−	0.612909i	$-0.789999\pi$
$683$	−41.3002	−1.58031	−0.790154	+	0.612909i	$0.789999\pi$
$684$	−9.12311	−0.348831
$685$	10.0000	0.382080
$686$	62.7386	2.39537
$687$	37.1231	1.41633
$688$	−1.12311	−0.0428180
$689$	−2.56155	−0.0975874
$690$	13.1231	0.499588
$691$	9.75379	0.371052	0.185526	−	0.982639i	$-0.440601\pi$
$691$	9.75379	0.371052	0.185526	+	0.982639i	$0.440601\pi$
$692$	−18.0000	−0.684257
$693$	−72.9848	−2.77247
$694$	−10.5616	−0.400911
$695$	16.4924	0.625593
$696$	14.5616	0.551954
$697$	4.24621	0.160837
$698$	−15.1231	−0.572418
$699$	−16.1771	−0.611873
$700$	5.12311	0.193635
$701$	46.3542	1.75077	0.875386	−	0.483424i	$-0.160607\pi$
$701$	46.3542	1.75077	0.875386	+	0.483424i	$0.160607\pi$
$702$	−6.56155	−0.247650
$703$	18.2462	0.688169
$704$	−4.00000	−0.150756
$705$	−16.8078	−0.633017
$706$	22.4924	0.846513
$707$	−74.2462	−2.79232
$708$	0.807764	0.0303576
$709$	37.5464	1.41008	0.705042	−	0.709165i	$-0.250928\pi$
$709$	37.5464	1.41008	0.705042	+	0.709165i	$0.250928\pi$
$710$	−4.31534	−0.161952
$711$	0	0
$712$	−9.68466	−0.362948
$713$	33.6155	1.25891
$714$	−13.1231	−0.491120
$715$	−18.2462	−0.682370
$716$	8.49242	0.317377
$717$	59.8617	2.23558
$718$	−10.8769	−0.405922
$719$	15.1922	0.566575	0.283287	−	0.959035i	$-0.408575\pi$
$719$	15.1922	0.566575	0.283287	+	0.959035i	$0.408575\pi$
$720$	3.56155	0.132731
$721$	40.9848	1.52636
$722$	−12.4384	−0.462911
$723$	−31.3693	−1.16664
$724$	16.2462	0.603786
$725$	−5.68466	−0.211123
$726$	−12.8078	−0.475341
$727$	46.5616	1.72687	0.863436	−	0.504458i	$-0.168308\pi$
$727$	46.5616	1.72687	0.863436	+	0.504458i	$0.168308\pi$
$728$	23.3693	0.866125
$729$	−35.9848	−1.33277
$730$	−6.80776	−0.251967
$731$	−1.12311	−0.0415396
$732$	−19.0540	−0.704255
$733$	−22.4924	−0.830777	−0.415388	−	0.909644i	$-0.636354\pi$
$733$	−22.4924	−0.830777	−0.415388	+	0.909644i	$0.636354\pi$
$734$	−33.6155	−1.24077
$735$	−49.3002	−1.81846
$736$	−5.12311	−0.188840
$737$	36.4924	1.34422
$738$	15.1231	0.556689
$739$	−20.1771	−0.742226	−0.371113	−	0.928588i	$-0.621024\pi$
$739$	−20.1771	−0.742226	−0.371113	+	0.928588i	$0.621024\pi$
$740$	−7.12311	−0.261851
$741$	29.9309	1.09954
$742$	−2.87689	−0.105614
$743$	−21.1231	−0.774932	−0.387466	−	0.921884i	$-0.626649\pi$
$743$	−21.1231	−0.774932	−0.387466	+	0.921884i	$0.626649\pi$
$744$	16.8078	0.616203
$745$	3.12311	0.114422
$746$	−24.7386	−0.905746
$747$	−22.2462	−0.813946
$748$	−4.00000	−0.146254
$749$	84.4924	3.08729
$750$	−2.56155	−0.0935347
$751$	−24.1771	−0.882234	−0.441117	−	0.897450i	$-0.645418\pi$
$751$	−24.1771	−0.882234	−0.441117	+	0.897450i	$0.645418\pi$
$752$	6.56155	0.239275
$753$	10.2462	0.373393
$754$	−25.9309	−0.944347
$755$	−15.3693	−0.559347
$756$	−7.36932	−0.268019
$757$	47.7926	1.73705	0.868526	−	0.495644i	$-0.165068\pi$
$757$	47.7926	1.73705	0.868526	+	0.495644i	$0.165068\pi$
$758$	−14.2462	−0.517446
$759$	−52.4924	−1.90535
$760$	−2.56155	−0.0929173
$761$	−20.7386	−0.751775	−0.375887	−	0.926665i	$-0.622662\pi$
$761$	−20.7386	−0.751775	−0.375887	+	0.926665i	$0.622662\pi$
$762$	−24.1771	−0.875843
$763$	−43.8617	−1.58790
$764$	0	0
$765$	3.56155	0.128768
$766$	−16.8078	−0.607289
$767$	−1.43845	−0.0519393
$768$	−2.56155	−0.0924321
$769$	11.4384	0.412481	0.206240	−	0.978501i	$-0.433877\pi$
$769$	11.4384	0.412481	0.206240	+	0.978501i	$0.433877\pi$
$770$	−20.4924	−0.738496
$771$	−5.12311	−0.184504
$772$	−24.2462	−0.872640
$773$	20.7386	0.745917	0.372958	−	0.927848i	$-0.378343\pi$
$773$	20.7386	0.745917	0.372958	+	0.927848i	$0.378343\pi$
$774$	−4.00000	−0.143777
$775$	−6.56155	−0.235698
$776$	−1.68466	−0.0604757
$777$	93.4773	3.35348
$778$	13.3693	0.479313
$779$	−10.8769	−0.389705
$780$	−11.6847	−0.418378
$781$	17.2614	0.617660
$782$	−5.12311	−0.183202
$783$	8.17708	0.292225
$784$	19.2462	0.687365
$785$	18.4924	0.660023
$786$	−2.87689	−0.102615
$787$	−27.5464	−0.981923	−0.490962	−	0.871181i	$-0.663354\pi$
$787$	−27.5464	−0.981923	−0.490962	+	0.871181i	$0.663354\pi$
$788$	13.3693	0.476262
$789$	−63.5464	−2.26231
$790$	0	0
$791$	14.3845	0.511453
$792$	−14.2462	−0.506217
$793$	33.9309	1.20492
$794$	21.3693	0.758369
$795$	1.43845	0.0510165
$796$	14.5616	0.516121
$797$	18.4924	0.655035	0.327518	−	0.944845i	$-0.393788\pi$
$797$	18.4924	0.655035	0.327518	+	0.944845i	$0.393788\pi$
$798$	33.6155	1.18998
$799$	6.56155	0.232131
$800$	1.00000	0.0353553
$801$	−34.4924	−1.21873
$802$	10.6307	0.375382
$803$	27.2311	0.960963
$804$	23.3693	0.824172
$805$	−26.2462	−0.925057
$806$	−29.9309	−1.05427
$807$	7.19224	0.253179
$808$	−14.4924	−0.509842
$809$	18.9848	0.667472	0.333736	−	0.942667i	$-0.391691\pi$
$809$	18.9848	0.667472	0.333736	+	0.942667i	$0.391691\pi$
$810$	−7.00000	−0.245955
$811$	3.36932	0.118313	0.0591564	−	0.998249i	$-0.481159\pi$
$811$	3.36932	0.118313	0.0591564	+	0.998249i	$0.481159\pi$
$812$	−29.1231	−1.02202
$813$	−40.9848	−1.43740
$814$	28.4924	0.998659
$815$	14.2462	0.499023
$816$	−2.56155	−0.0896723
$817$	2.87689	0.100650
$818$	−2.31534	−0.0809540
$819$	83.2311	2.90833
$820$	4.24621	0.148284
$821$	−15.3002	−0.533980	−0.266990	−	0.963699i	$-0.586029\pi$
$821$	−15.3002	−0.533980	−0.266990	+	0.963699i	$0.586029\pi$
$822$	−25.6155	−0.893444
$823$	0.630683	0.0219842	0.0109921	−	0.999940i	$-0.496501\pi$
$823$	0.630683	0.0219842	0.0109921	+	0.999940i	$0.496501\pi$
$824$	8.00000	0.278693
$825$	10.2462	0.356727
$826$	−1.61553	−0.0562114
$827$	0.492423	0.0171232	0.00856160	−	0.999963i	$-0.497275\pi$
$827$	0.492423	0.0171232	0.00856160	+	0.999963i	$0.497275\pi$
$828$	−18.2462	−0.634100
$829$	−9.36932	−0.325410	−0.162705	−	0.986675i	$-0.552022\pi$
$829$	−9.36932	−0.325410	−0.162705	+	0.986675i	$0.552022\pi$
$830$	−6.24621	−0.216809
$831$	40.3542	1.39987
$832$	4.56155	0.158143
$833$	19.2462	0.666842
$834$	−42.2462	−1.46287
$835$	5.12311	0.177292
$836$	10.2462	0.354373
$837$	9.43845	0.326240
$838$	−25.1231	−0.867863
$839$	35.0540	1.21020	0.605099	−	0.796150i	$-0.293134\pi$
$839$	35.0540	1.21020	0.605099	+	0.796150i	$0.293134\pi$
$840$	−13.1231	−0.452790
$841$	3.31534	0.114322
$842$	7.61553	0.262448
$843$	−42.4233	−1.46114
$844$	−3.36932	−0.115977
$845$	7.80776	0.268595
$846$	23.3693	0.803454
$847$	25.6155	0.880160
$848$	−0.561553	−0.0192838
$849$	−21.3002	−0.731021
$850$	1.00000	0.0342997
$851$	36.4924	1.25094
$852$	11.0540	0.378703
$853$	48.2462	1.65192	0.825959	−	0.563730i	$-0.190634\pi$
$853$	48.2462	1.65192	0.825959	+	0.563730i	$0.190634\pi$
$854$	38.1080	1.30403
$855$	−9.12311	−0.312004
$856$	16.4924	0.563699
$857$	1.82292	0.0622697	0.0311349	−	0.999515i	$-0.490088\pi$
$857$	1.82292	0.0622697	0.0311349	+	0.999515i	$0.490088\pi$
$858$	46.7386	1.59563
$859$	−35.5464	−1.21283	−0.606414	−	0.795149i	$-0.707392\pi$
$859$	−35.5464	−1.21283	−0.606414	+	0.795149i	$0.707392\pi$
$860$	−1.12311	−0.0382976
$861$	−55.7235	−1.89905
$862$	−20.4924	−0.697975
$863$	11.5076	0.391722	0.195861	−	0.980632i	$-0.437250\pi$
$863$	11.5076	0.391722	0.195861	+	0.980632i	$0.437250\pi$
$864$	−1.43845	−0.0489370
$865$	−18.0000	−0.612018
$866$	−2.49242	−0.0846960
$867$	−2.56155	−0.0869949
$868$	−33.6155	−1.14099
$869$	0	0
$870$	14.5616	0.493683
$871$	−41.6155	−1.41009
$872$	−8.56155	−0.289931
$873$	−6.00000	−0.203069
$874$	13.1231	0.443896
$875$	5.12311	0.173193
$876$	17.4384	0.589191
$877$	−51.6155	−1.74293	−0.871466	−	0.490455i	$-0.836830\pi$
$877$	−51.6155	−1.74293	−0.871466	+	0.490455i	$0.836830\pi$
$878$	−32.9848	−1.11318
$879$	42.4233	1.43090
$880$	−4.00000	−0.134840
$881$	−28.7386	−0.968229	−0.484115	−	0.875005i	$-0.660858\pi$
$881$	−28.7386	−0.968229	−0.484115	+	0.875005i	$0.660858\pi$
$882$	68.5464	2.30808
$883$	27.3693	0.921051	0.460525	−	0.887647i	$-0.347661\pi$
$883$	27.3693	0.921051	0.460525	+	0.887647i	$0.347661\pi$
$884$	4.56155	0.153422
$885$	0.807764	0.0271527
$886$	28.0000	0.940678
$887$	50.6004	1.69899	0.849497	−	0.527593i	$-0.176905\pi$
$887$	50.6004	1.69899	0.849497	+	0.527593i	$0.176905\pi$
$888$	18.2462	0.612303
$889$	48.3542	1.62175
$890$	−9.68466	−0.324630
$891$	28.0000	0.938035
$892$	3.68466	0.123371
$893$	−16.8078	−0.562450
$894$	−8.00000	−0.267560
$895$	8.49242	0.283870
$896$	5.12311	0.171151
$897$	59.8617	1.99873
$898$	−16.8769	−0.563189
$899$	37.3002	1.24403
$900$	3.56155	0.118718
$901$	−0.561553	−0.0187080
$902$	−16.9848	−0.565533
$903$	14.7386	0.490471
$904$	2.80776	0.0933848
$905$	16.2462	0.540042
$906$	39.3693	1.30796
$907$	18.5616	0.616326	0.308163	−	0.951334i	$-0.400286\pi$
$907$	18.5616	0.616326	0.308163	+	0.951334i	$0.400286\pi$
$908$	16.3153	0.541444
$909$	−51.6155	−1.71198
$910$	23.3693	0.774685
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	6.56155	0.217275
$913$	24.9848	0.826878
$914$	−16.2462	−0.537377
$915$	−19.0540	−0.629905
$916$	−14.4924	−0.478843
$917$	5.75379	0.190007
$918$	−1.43845	−0.0474758
$919$	−21.1231	−0.696787	−0.348393	−	0.937348i	$-0.613273\pi$
$919$	−21.1231	−0.696787	−0.348393	+	0.937348i	$0.613273\pi$
$920$	−5.12311	−0.168904
$921$	45.4773	1.49853
$922$	2.49242	0.0820836
$923$	−19.6847	−0.647928
$924$	52.4924	1.72687
$925$	−7.12311	−0.234206
$926$	−13.9309	−0.457797
$927$	28.4924	0.935814
$928$	−5.68466	−0.186608
$929$	−59.4773	−1.95139	−0.975693	−	0.219142i	$-0.929674\pi$
$929$	−59.4773	−1.95139	−0.975693	+	0.219142i	$0.929674\pi$
$930$	16.8078	0.551148
$931$	−49.3002	−1.61575
$932$	6.31534	0.206866
$933$	−20.4924	−0.670892
$934$	−32.4924	−1.06318
$935$	−4.00000	−0.130814
$936$	16.2462	0.531024
$937$	−44.1080	−1.44094	−0.720472	−	0.693484i	$-0.756075\pi$
$937$	−44.1080	−1.44094	−0.720472	+	0.693484i	$0.756075\pi$
$938$	−46.7386	−1.52607
$939$	15.3693	0.501559
$940$	6.56155	0.214014
$941$	−27.4384	−0.894468	−0.447234	−	0.894417i	$-0.647591\pi$
$941$	−27.4384	−0.894468	−0.447234	+	0.894417i	$0.647591\pi$
$942$	−47.3693	−1.54338
$943$	−21.7538	−0.708401
$944$	−0.315342	−0.0102635
$945$	−7.36932	−0.239724
$946$	4.49242	0.146061
$947$	52.8078	1.71602	0.858011	−	0.513632i	$-0.171700\pi$
$947$	52.8078	1.71602	0.858011	+	0.513632i	$0.171700\pi$
$948$	0	0
$949$	−31.0540	−1.00805
$950$	−2.56155	−0.0831077
$951$	−80.9848	−2.62611
$952$	5.12311	0.166041
$953$	39.1231	1.26732	0.633661	−	0.773611i	$-0.281551\pi$
$953$	39.1231	1.26732	0.633661	+	0.773611i	$0.281551\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−23.3693	−0.755818
$957$	−58.2462	−1.88283
$958$	38.5616	1.24587
$959$	51.2311	1.65434
$960$	−2.56155	−0.0826738
$961$	12.0540	0.388838
$962$	−32.4924	−1.04760
$963$	58.7386	1.89283
$964$	12.2462	0.394424
$965$	−24.2462	−0.780513
$966$	67.2311	2.16312
$967$	19.5076	0.627321	0.313661	−	0.949535i	$-0.398445\pi$
$967$	19.5076	0.627321	0.313661	+	0.949535i	$0.398445\pi$
$968$	5.00000	0.160706
$969$	6.56155	0.210787
$970$	−1.68466	−0.0540911
$971$	15.6847	0.503345	0.251672	−	0.967813i	$-0.419019\pi$
$971$	15.6847	0.503345	0.251672	+	0.967813i	$0.419019\pi$
$972$	22.2462	0.713548
$973$	84.4924	2.70870
$974$	25.6155	0.820774
$975$	−11.6847	−0.374209
$976$	7.43845	0.238099
$977$	−27.1231	−0.867745	−0.433873	−	0.900974i	$-0.642853\pi$
$977$	−27.1231	−0.867745	−0.433873	+	0.900974i	$0.642853\pi$
$978$	−36.4924	−1.16690
$979$	38.7386	1.23809
$980$	19.2462	0.614798
$981$	−30.4924	−0.973548
$982$	2.56155	0.0817424
$983$	−37.1231	−1.18404	−0.592022	−	0.805922i	$-0.701670\pi$
$983$	−37.1231	−1.18404	−0.592022	+	0.805922i	$0.701670\pi$
$984$	−10.8769	−0.346743
$985$	13.3693	0.425982
$986$	−5.68466	−0.181036
$987$	−86.1080	−2.74085
$988$	−11.6847	−0.371739
$989$	5.75379	0.182960
$990$	−14.2462	−0.452774
$991$	34.4233	1.09349	0.546746	−	0.837299i	$-0.315866\pi$
$991$	34.4233	1.09349	0.546746	+	0.837299i	$0.315866\pi$
$992$	−6.56155	−0.208330
$993$	−59.0540	−1.87402
$994$	−22.1080	−0.701222
$995$	14.5616	0.461632
$996$	16.0000	0.506979
$997$	11.7538	0.372246	0.186123	−	0.982526i	$-0.440408\pi$
$997$	11.7538	0.372246	0.186123	+	0.982526i	$0.440408\pi$
$998$	−9.12311	−0.288787
$999$	10.2462	0.324176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.2.a.f.1.1	✓	2
3.2	odd	2		1530.2.a.r.1.2		2
4.3	odd	2		1360.2.a.m.1.2		2
5.2	odd	4		850.2.c.i.749.4		4
5.3	odd	4		850.2.c.i.749.1		4
5.4	even	2		850.2.a.n.1.2		2
7.6	odd	2		8330.2.a.bq.1.2		2
8.3	odd	2		5440.2.a.bd.1.1		2
8.5	even	2		5440.2.a.bj.1.2		2
15.14	odd	2		7650.2.a.de.1.1		2
17.4	even	4		2890.2.b.i.2311.4		4
17.13	even	4		2890.2.b.i.2311.1		4
17.16	even	2		2890.2.a.u.1.2		2
20.19	odd	2		6800.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.f.1.1	✓	2	1.1	even	1	trivial
850.2.a.n.1.2		2	5.4	even	2
850.2.c.i.749.1		4	5.3	odd	4
850.2.c.i.749.4		4	5.2	odd	4
1360.2.a.m.1.2		2	4.3	odd	2
1530.2.a.r.1.2		2	3.2	odd	2
2890.2.a.u.1.2		2	17.16	even	2
2890.2.b.i.2311.1		4	17.13	even	4
2890.2.b.i.2311.4		4	17.4	even	4
5440.2.a.bd.1.1		2	8.3	odd	2
5440.2.a.bj.1.2		2	8.5	even	2
6800.2.a.be.1.1		2	20.19	odd	2
7650.2.a.de.1.1		2	15.14	odd	2
8330.2.a.bq.1.2		2	7.6	odd	2

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:	\( N \)	\(=\)	\( 170 = 2 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	170.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.56155 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.56155 q^{6} +5.12311 q^{7} +1.00000 q^{8} +3.56155 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.56155 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.56155 q^{6} +5.12311 q^{7} +1.00000 q^{8} +3.56155 q^{9} +1.00000 q^{10} -4.00000 q^{11} -2.56155 q^{12} +4.56155 q^{13} +5.12311 q^{14} -2.56155 q^{15} +1.00000 q^{16} +1.00000 q^{17} +3.56155 q^{18} -2.56155 q^{19} +1.00000 q^{20} -13.1231 q^{21} -4.00000 q^{22} -5.12311 q^{23} -2.56155 q^{24} +1.00000 q^{25} +4.56155 q^{26} -1.43845 q^{27} +5.12311 q^{28} -5.68466 q^{29} -2.56155 q^{30} -6.56155 q^{31} +1.00000 q^{32} +10.2462 q^{33} +1.00000 q^{34} +5.12311 q^{35} +3.56155 q^{36} -7.12311 q^{37} -2.56155 q^{38} -11.6847 q^{39} +1.00000 q^{40} +4.24621 q^{41} -13.1231 q^{42} -1.12311 q^{43} -4.00000 q^{44} +3.56155 q^{45} -5.12311 q^{46} +6.56155 q^{47} -2.56155 q^{48} +19.2462 q^{49} +1.00000 q^{50} -2.56155 q^{51} +4.56155 q^{52} -0.561553 q^{53} -1.43845 q^{54} -4.00000 q^{55} +5.12311 q^{56} +6.56155 q^{57} -5.68466 q^{58} -0.315342 q^{59} -2.56155 q^{60} +7.43845 q^{61} -6.56155 q^{62} +18.2462 q^{63} +1.00000 q^{64} +4.56155 q^{65} +10.2462 q^{66} -9.12311 q^{67} +1.00000 q^{68} +13.1231 q^{69} +5.12311 q^{70} -4.31534 q^{71} +3.56155 q^{72} -6.80776 q^{73} -7.12311 q^{74} -2.56155 q^{75} -2.56155 q^{76} -20.4924 q^{77} -11.6847 q^{78} +1.00000 q^{80} -7.00000 q^{81} +4.24621 q^{82} -6.24621 q^{83} -13.1231 q^{84} +1.00000 q^{85} -1.12311 q^{86} +14.5616 q^{87} -4.00000 q^{88} -9.68466 q^{89} +3.56155 q^{90} +23.3693 q^{91} -5.12311 q^{92} +16.8078 q^{93} +6.56155 q^{94} -2.56155 q^{95} -2.56155 q^{96} -1.68466 q^{97} +19.2462 q^{98} -14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 2 * q^5 - q^6 + 2 * q^7 + 2 * q^8 + 3 * q^9	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} + 3 q^{9} + 2 q^{10} - 8 q^{11} - q^{12} + 5 q^{13} + 2 q^{14} - q^{15} + 2 q^{16} + 2 q^{17} + 3 q^{18} - q^{19} + 2 q^{20} - 18 q^{21} - 8 q^{22} - 2 q^{23} - q^{24} + 2 q^{25} + 5 q^{26} - 7 q^{27} + 2 q^{28} + q^{29} - q^{30} - 9 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{34} + 2 q^{35} + 3 q^{36} - 6 q^{37} - q^{38} - 11 q^{39} + 2 q^{40} - 8 q^{41} - 18 q^{42} + 6 q^{43} - 8 q^{44} + 3 q^{45} - 2 q^{46} + 9 q^{47} - q^{48} + 22 q^{49} + 2 q^{50} - q^{51} + 5 q^{52} + 3 q^{53} - 7 q^{54} - 8 q^{55} + 2 q^{56} + 9 q^{57} + q^{58} - 13 q^{59} - q^{60} + 19 q^{61} - 9 q^{62} + 20 q^{63} + 2 q^{64} + 5 q^{65} + 4 q^{66} - 10 q^{67} + 2 q^{68} + 18 q^{69} + 2 q^{70} - 21 q^{71} + 3 q^{72} + 7 q^{73} - 6 q^{74} - q^{75} - q^{76} - 8 q^{77} - 11 q^{78} + 2 q^{80} - 14 q^{81} - 8 q^{82} + 4 q^{83} - 18 q^{84} + 2 q^{85} + 6 q^{86} + 25 q^{87} - 8 q^{88} - 7 q^{89} + 3 q^{90} + 22 q^{91} - 2 q^{92} + 13 q^{93} + 9 q^{94} - q^{95} - q^{96} + 9 q^{97} + 22 q^{98} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 2 * q^5 - q^6 + 2 * q^7 + 2 * q^8 + 3 * q^9 + 2 * q^10 - 8 * q^11 - q^12 + 5 * q^13 + 2 * q^14 - q^15 + 2 * q^16 + 2 * q^17 + 3 * q^18 - q^19 + 2 * q^20 - 18 * q^21 - 8 * q^22 - 2 * q^23 - q^24 + 2 * q^25 + 5 * q^26 - 7 * q^27 + 2 * q^28 + q^29 - q^30 - 9 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^34 + 2 * q^35 + 3 * q^36 - 6 * q^37 - q^38 - 11 * q^39 + 2 * q^40 - 8 * q^41 - 18 * q^42 + 6 * q^43 - 8 * q^44 + 3 * q^45 - 2 * q^46 + 9 * q^47 - q^48 + 22 * q^49 + 2 * q^50 - q^51 + 5 * q^52 + 3 * q^53 - 7 * q^54 - 8 * q^55 + 2 * q^56 + 9 * q^57 + q^58 - 13 * q^59 - q^60 + 19 * q^61 - 9 * q^62 + 20 * q^63 + 2 * q^64 + 5 * q^65 + 4 * q^66 - 10 * q^67 + 2 * q^68 + 18 * q^69 + 2 * q^70 - 21 * q^71 + 3 * q^72 + 7 * q^73 - 6 * q^74 - q^75 - q^76 - 8 * q^77 - 11 * q^78 + 2 * q^80 - 14 * q^81 - 8 * q^82 + 4 * q^83 - 18 * q^84 + 2 * q^85 + 6 * q^86 + 25 * q^87 - 8 * q^88 - 7 * q^89 + 3 * q^90 + 22 * q^91 - 2 * q^92 + 13 * q^93 + 9 * q^94 - q^95 - q^96 + 9 * q^97 + 22 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more