LMFDB - Embedded newform 5610.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5610.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:	$44.7960755339$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +1.23607 q^{13} +3.23607 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -7.70820 q^{19} -1.00000 q^{20} -3.23607 q^{21} +1.00000 q^{22} -5.23607 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.23607 q^{26} -1.00000 q^{27} +3.23607 q^{28} -8.47214 q^{29} +1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} -3.23607 q^{35} +1.00000 q^{36} -6.00000 q^{37} -7.70820 q^{38} -1.23607 q^{39} -1.00000 q^{40} -8.94427 q^{41} -3.23607 q^{42} -3.23607 q^{43} +1.00000 q^{44} -1.00000 q^{45} -5.23607 q^{46} +2.47214 q^{47} -1.00000 q^{48} +3.47214 q^{49} +1.00000 q^{50} +1.00000 q^{51} +1.23607 q^{52} +3.23607 q^{53} -1.00000 q^{54} -1.00000 q^{55} +3.23607 q^{56} +7.70820 q^{57} -8.47214 q^{58} -3.23607 q^{59} +1.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} +3.23607 q^{63} +1.00000 q^{64} -1.23607 q^{65} -1.00000 q^{66} +6.00000 q^{67} -1.00000 q^{68} +5.23607 q^{69} -3.23607 q^{70} -0.763932 q^{71} +1.00000 q^{72} +1.23607 q^{73} -6.00000 q^{74} -1.00000 q^{75} -7.70820 q^{76} +3.23607 q^{77} -1.23607 q^{78} +3.70820 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.94427 q^{82} +4.94427 q^{83} -3.23607 q^{84} +1.00000 q^{85} -3.23607 q^{86} +8.47214 q^{87} +1.00000 q^{88} -6.94427 q^{89} -1.00000 q^{90} +4.00000 q^{91} -5.23607 q^{92} +4.00000 q^{93} +2.47214 q^{94} +7.70820 q^{95} -1.00000 q^{96} +3.47214 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{21} + 2 q^{22} - 6 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} + 2 q^{28} - 8 q^{29} + 2 q^{30} - 8 q^{31} + 2 q^{32} - 2 q^{33} - 2 q^{34} - 2 q^{35} + 2 q^{36} - 12 q^{37} - 2 q^{38} + 2 q^{39} - 2 q^{40} - 2 q^{42} - 2 q^{43} + 2 q^{44} - 2 q^{45} - 6 q^{46} - 4 q^{47} - 2 q^{48} - 2 q^{49} + 2 q^{50} + 2 q^{51} - 2 q^{52} + 2 q^{53} - 2 q^{54} - 2 q^{55} + 2 q^{56} + 2 q^{57} - 8 q^{58} - 2 q^{59} + 2 q^{60} + 12 q^{61} - 8 q^{62} + 2 q^{63} + 2 q^{64} + 2 q^{65} - 2 q^{66} + 12 q^{67} - 2 q^{68} + 6 q^{69} - 2 q^{70} - 6 q^{71} + 2 q^{72} - 2 q^{73} - 12 q^{74} - 2 q^{75} - 2 q^{76} + 2 q^{77} + 2 q^{78} - 6 q^{79} - 2 q^{80} + 2 q^{81} - 8 q^{83} - 2 q^{84} + 2 q^{85} - 2 q^{86} + 8 q^{87} + 2 q^{88} + 4 q^{89} - 2 q^{90} + 8 q^{91} - 6 q^{92} + 8 q^{93} - 4 q^{94} + 2 q^{95} - 2 q^{96} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 - 2 * q^12 - 2 * q^13 + 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 2 * q^21 + 2 * q^22 - 6 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^26 - 2 * q^27 + 2 * q^28 - 8 * q^29 + 2 * q^30 - 8 * q^31 + 2 * q^32 - 2 * q^33 - 2 * q^34 - 2 * q^35 + 2 * q^36 - 12 * q^37 - 2 * q^38 + 2 * q^39 - 2 * q^40 - 2 * q^42 - 2 * q^43 + 2 * q^44 - 2 * q^45 - 6 * q^46 - 4 * q^47 - 2 * q^48 - 2 * q^49 + 2 * q^50 + 2 * q^51 - 2 * q^52 + 2 * q^53 - 2 * q^54 - 2 * q^55 + 2 * q^56 + 2 * q^57 - 8 * q^58 - 2 * q^59 + 2 * q^60 + 12 * q^61 - 8 * q^62 + 2 * q^63 + 2 * q^64 + 2 * q^65 - 2 * q^66 + 12 * q^67 - 2 * q^68 + 6 * q^69 - 2 * q^70 - 6 * q^71 + 2 * q^72 - 2 * q^73 - 12 * q^74 - 2 * q^75 - 2 * q^76 + 2 * q^77 + 2 * q^78 - 6 * q^79 - 2 * q^80 + 2 * q^81 - 8 * q^83 - 2 * q^84 + 2 * q^85 - 2 * q^86 + 8 * q^87 + 2 * q^88 + 4 * q^89 - 2 * q^90 + 8 * q^91 - 6 * q^92 + 8 * q^93 - 4 * q^94 + 2 * q^95 - 2 * q^96 - 2 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	3.23607	0.864876
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	−7.70820	−1.76838	−0.884192	−	0.467124i	$-0.845290\pi$
$19$	−7.70820	−1.76838	−0.884192	+	0.467124i	$0.845290\pi$
$20$	−1.00000	−0.223607
$21$	−3.23607	−0.706168
$22$	1.00000	0.213201
$23$	−5.23607	−1.09180	−0.545898	−	0.837852i	$-0.683811\pi$
$23$	−5.23607	−1.09180	−0.545898	+	0.837852i	$0.683811\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	1.23607	0.242413
$27$	−1.00000	−0.192450
$28$	3.23607	0.611559
$29$	−8.47214	−1.57324	−0.786618	−	0.617440i	$-0.788170\pi$
$29$	−8.47214	−1.57324	−0.786618	+	0.617440i	$0.788170\pi$
$30$	1.00000	0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−1.00000	−0.171499
$35$	−3.23607	−0.546995
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−7.70820	−1.25044
$39$	−1.23607	−0.197929
$40$	−1.00000	−0.158114
$41$	−8.94427	−1.39686	−0.698430	−	0.715678i	$-0.746118\pi$
$41$	−8.94427	−1.39686	−0.698430	+	0.715678i	$0.746118\pi$
$42$	−3.23607	−0.499336
$43$	−3.23607	−0.493496	−0.246748	−	0.969080i	$-0.579362\pi$
$43$	−3.23607	−0.493496	−0.246748	+	0.969080i	$0.579362\pi$
$44$	1.00000	0.150756
$45$	−1.00000	−0.149071
$46$	−5.23607	−0.772016
$47$	2.47214	0.360598	0.180299	−	0.983612i	$-0.442293\pi$
$47$	2.47214	0.360598	0.180299	+	0.983612i	$0.442293\pi$
$48$	−1.00000	−0.144338
$49$	3.47214	0.496019
$50$	1.00000	0.141421
$51$	1.00000	0.140028
$52$	1.23607	0.171412
$53$	3.23607	0.444508	0.222254	−	0.974989i	$-0.428659\pi$
$53$	3.23607	0.444508	0.222254	+	0.974989i	$0.428659\pi$
$54$	−1.00000	−0.136083
$55$	−1.00000	−0.134840
$56$	3.23607	0.432438
$57$	7.70820	1.02098
$58$	−8.47214	−1.11245
$59$	−3.23607	−0.421300	−0.210650	−	0.977562i	$-0.567558\pi$
$59$	−3.23607	−0.421300	−0.210650	+	0.977562i	$0.567558\pi$
$60$	1.00000	0.129099
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	3.23607	0.407706
$64$	1.00000	0.125000
$65$	−1.23607	−0.153315
$66$	−1.00000	−0.123091
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	−1.00000	−0.121268
$69$	5.23607	0.630349
$70$	−3.23607	−0.386784
$71$	−0.763932	−0.0906621	−0.0453310	−	0.998972i	$-0.514434\pi$
$71$	−0.763932	−0.0906621	−0.0453310	+	0.998972i	$0.514434\pi$
$72$	1.00000	0.117851
$73$	1.23607	0.144671	0.0723354	−	0.997380i	$-0.476955\pi$
$73$	1.23607	0.144671	0.0723354	+	0.997380i	$0.476955\pi$
$74$	−6.00000	−0.697486
$75$	−1.00000	−0.115470
$76$	−7.70820	−0.884192
$77$	3.23607	0.368784
$78$	−1.23607	−0.139957
$79$	3.70820	0.417206	0.208603	−	0.978000i	$-0.433108\pi$
$79$	3.70820	0.417206	0.208603	+	0.978000i	$0.433108\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−8.94427	−0.987730
$83$	4.94427	0.542704	0.271352	−	0.962480i	$-0.412529\pi$
$83$	4.94427	0.542704	0.271352	+	0.962480i	$0.412529\pi$
$84$	−3.23607	−0.353084
$85$	1.00000	0.108465
$86$	−3.23607	−0.348954
$87$	8.47214	0.908308
$88$	1.00000	0.106600
$89$	−6.94427	−0.736091	−0.368046	−	0.929808i	$-0.619973\pi$
$89$	−6.94427	−0.736091	−0.368046	+	0.929808i	$0.619973\pi$
$90$	−1.00000	−0.105409
$91$	4.00000	0.419314
$92$	−5.23607	−0.545898
$93$	4.00000	0.414781
$94$	2.47214	0.254981
$95$	7.70820	0.790845
$96$	−1.00000	−0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	3.47214	0.350739
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	8.94427	0.889988	0.444994	−	0.895533i	$-0.353206\pi$
$101$	8.94427	0.889988	0.444994	+	0.895533i	$0.353206\pi$
$102$	1.00000	0.0990148
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	1.23607	0.121206
$105$	3.23607	0.315808
$106$	3.23607	0.314315
$107$	0.944272	0.0912862	0.0456431	−	0.998958i	$-0.485466\pi$
$107$	0.944272	0.0912862	0.0456431	+	0.998958i	$0.485466\pi$
$108$	−1.00000	−0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	−1.00000	−0.0953463
$111$	6.00000	0.569495
$112$	3.23607	0.305780
$113$	−15.2361	−1.43329	−0.716644	−	0.697439i	$-0.754323\pi$
$113$	−15.2361	−1.43329	−0.716644	+	0.697439i	$0.754323\pi$
$114$	7.70820	0.721939
$115$	5.23607	0.488266
$116$	−8.47214	−0.786618
$117$	1.23607	0.114275
$118$	−3.23607	−0.297904
$119$	−3.23607	−0.296650
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	6.00000	0.543214
$123$	8.94427	0.806478
$124$	−4.00000	−0.359211
$125$	−1.00000	−0.0894427
$126$	3.23607	0.288292
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	1.00000	0.0883883
$129$	3.23607	0.284920
$130$	−1.23607	−0.108410
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−1.00000	−0.0870388
$133$	−24.9443	−2.16294
$134$	6.00000	0.518321
$135$	1.00000	0.0860663
$136$	−1.00000	−0.0857493
$137$	−15.8885	−1.35745	−0.678725	−	0.734393i	$-0.737467\pi$
$137$	−15.8885	−1.35745	−0.678725	+	0.734393i	$0.737467\pi$
$138$	5.23607	0.445724
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−3.23607	−0.273498
$141$	−2.47214	−0.208191
$142$	−0.763932	−0.0641078
$143$	1.23607	0.103365
$144$	1.00000	0.0833333
$145$	8.47214	0.703573
$146$	1.23607	0.102298
$147$	−3.47214	−0.286377
$148$	−6.00000	−0.493197
$149$	−9.52786	−0.780553	−0.390277	−	0.920698i	$-0.627621\pi$
$149$	−9.52786	−0.780553	−0.390277	+	0.920698i	$0.627621\pi$
$150$	−1.00000	−0.0816497
$151$	−14.4721	−1.17773	−0.588863	−	0.808233i	$-0.700424\pi$
$151$	−14.4721	−1.17773	−0.588863	+	0.808233i	$0.700424\pi$
$152$	−7.70820	−0.625218
$153$	−1.00000	−0.0808452
$154$	3.23607	0.260770
$155$	4.00000	0.321288
$156$	−1.23607	−0.0989646
$157$	19.4164	1.54960	0.774799	−	0.632208i	$-0.217851\pi$
$157$	19.4164	1.54960	0.774799	+	0.632208i	$0.217851\pi$
$158$	3.70820	0.295009
$159$	−3.23607	−0.256637
$160$	−1.00000	−0.0790569
$161$	−16.9443	−1.33540
$162$	1.00000	0.0785674
$163$	−6.47214	−0.506937	−0.253468	−	0.967344i	$-0.581571\pi$
$163$	−6.47214	−0.506937	−0.253468	+	0.967344i	$0.581571\pi$
$164$	−8.94427	−0.698430
$165$	1.00000	0.0778499
$166$	4.94427	0.383750
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−3.23607	−0.249668
$169$	−11.4721	−0.882472
$170$	1.00000	0.0766965
$171$	−7.70820	−0.589461
$172$	−3.23607	−0.246748
$173$	13.4164	1.02003	0.510015	−	0.860165i	$-0.329640\pi$
$173$	13.4164	1.02003	0.510015	+	0.860165i	$0.329640\pi$
$174$	8.47214	0.642271
$175$	3.23607	0.244624
$176$	1.00000	0.0753778
$177$	3.23607	0.243238
$178$	−6.94427	−0.520495
$179$	−12.7639	−0.954021	−0.477011	−	0.878898i	$-0.658280\pi$
$179$	−12.7639	−0.954021	−0.477011	+	0.878898i	$0.658280\pi$
$180$	−1.00000	−0.0745356
$181$	8.47214	0.629729	0.314864	−	0.949137i	$-0.398041\pi$
$181$	8.47214	0.629729	0.314864	+	0.949137i	$0.398041\pi$
$182$	4.00000	0.296500
$183$	−6.00000	−0.443533
$184$	−5.23607	−0.386008
$185$	6.00000	0.441129
$186$	4.00000	0.293294
$187$	−1.00000	−0.0731272
$188$	2.47214	0.180299
$189$	−3.23607	−0.235389
$190$	7.70820	0.559212
$191$	1.52786	0.110552	0.0552762	−	0.998471i	$-0.482396\pi$
$191$	1.52786	0.110552	0.0552762	+	0.998471i	$0.482396\pi$
$192$	−1.00000	−0.0721688
$193$	7.70820	0.554849	0.277424	−	0.960747i	$-0.410519\pi$
$193$	7.70820	0.554849	0.277424	+	0.960747i	$0.410519\pi$
$194$	0	0
$195$	1.23607	0.0885167
$196$	3.47214	0.248010
$197$	14.9443	1.06474	0.532368	−	0.846513i	$-0.321302\pi$
$197$	14.9443	1.06474	0.532368	+	0.846513i	$0.321302\pi$
$198$	1.00000	0.0710669
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	1.00000	0.0707107
$201$	−6.00000	−0.423207
$202$	8.94427	0.629317
$203$	−27.4164	−1.92425
$204$	1.00000	0.0700140
$205$	8.94427	0.624695
$206$	4.00000	0.278693
$207$	−5.23607	−0.363932
$208$	1.23607	0.0857059
$209$	−7.70820	−0.533188
$210$	3.23607	0.223310
$211$	20.3607	1.40169	0.700844	−	0.713315i	$-0.252807\pi$
$211$	20.3607	1.40169	0.700844	+	0.713315i	$0.252807\pi$
$212$	3.23607	0.222254
$213$	0.763932	0.0523438
$214$	0.944272	0.0645491
$215$	3.23607	0.220698
$216$	−1.00000	−0.0680414
$217$	−12.9443	−0.878714
$218$	−6.00000	−0.406371
$219$	−1.23607	−0.0835257
$220$	−1.00000	−0.0674200
$221$	−1.23607	−0.0831469
$222$	6.00000	0.402694
$223$	−23.4164	−1.56808	−0.784039	−	0.620711i	$-0.786844\pi$
$223$	−23.4164	−1.56808	−0.784039	+	0.620711i	$0.786844\pi$
$224$	3.23607	0.216219
$225$	1.00000	0.0666667
$226$	−15.2361	−1.01349
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	7.70820	0.510488
$229$	29.4164	1.94389	0.971945	−	0.235206i	$-0.0755766\pi$
$229$	29.4164	1.94389	0.971945	+	0.235206i	$0.0755766\pi$
$230$	5.23607	0.345256
$231$	−3.23607	−0.212918
$232$	−8.47214	−0.556223
$233$	−7.52786	−0.493167	−0.246583	−	0.969122i	$-0.579308\pi$
$233$	−7.52786	−0.493167	−0.246583	+	0.969122i	$0.579308\pi$
$234$	1.23607	0.0808043
$235$	−2.47214	−0.161264
$236$	−3.23607	−0.210650
$237$	−3.70820	−0.240874
$238$	−3.23607	−0.209763
$239$	−16.3607	−1.05828	−0.529142	−	0.848533i	$-0.677486\pi$
$239$	−16.3607	−1.05828	−0.529142	+	0.848533i	$0.677486\pi$
$240$	1.00000	0.0645497
$241$	4.18034	0.269279	0.134640	−	0.990895i	$-0.457012\pi$
$241$	4.18034	0.269279	0.134640	+	0.990895i	$0.457012\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	−3.47214	−0.221827
$246$	8.94427	0.570266
$247$	−9.52786	−0.606243
$248$	−4.00000	−0.254000
$249$	−4.94427	−0.313331
$250$	−1.00000	−0.0632456
$251$	23.2361	1.46665	0.733324	−	0.679880i	$-0.237968\pi$
$251$	23.2361	1.46665	0.733324	+	0.679880i	$0.237968\pi$
$252$	3.23607	0.203853
$253$	−5.23607	−0.329189
$254$	−12.0000	−0.752947
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	−4.47214	−0.278964	−0.139482	−	0.990225i	$-0.544544\pi$
$257$	−4.47214	−0.278964	−0.139482	+	0.990225i	$0.544544\pi$
$258$	3.23607	0.201469
$259$	−19.4164	−1.20648
$260$	−1.23607	−0.0766577
$261$	−8.47214	−0.524412
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	−1.00000	−0.0615457
$265$	−3.23607	−0.198790
$266$	−24.9443	−1.52943
$267$	6.94427	0.424983
$268$	6.00000	0.366508
$269$	−24.4721	−1.49209	−0.746046	−	0.665894i	$-0.768050\pi$
$269$	−24.4721	−1.49209	−0.746046	+	0.665894i	$0.768050\pi$
$270$	1.00000	0.0608581
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−1.00000	−0.0606339
$273$	−4.00000	−0.242091
$274$	−15.8885	−0.959862
$275$	1.00000	0.0603023
$276$	5.23607	0.315174
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	8.00000	0.479808
$279$	−4.00000	−0.239474
$280$	−3.23607	−0.193392
$281$	−16.4721	−0.982645	−0.491323	−	0.870978i	$-0.663486\pi$
$281$	−16.4721	−0.982645	−0.491323	+	0.870978i	$0.663486\pi$
$282$	−2.47214	−0.147214
$283$	−3.41641	−0.203084	−0.101542	−	0.994831i	$-0.532378\pi$
$283$	−3.41641	−0.203084	−0.101542	+	0.994831i	$0.532378\pi$
$284$	−0.763932	−0.0453310
$285$	−7.70820	−0.456595
$286$	1.23607	0.0730902
$287$	−28.9443	−1.70853
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	8.47214	0.497501
$291$	0	0
$292$	1.23607	0.0723354
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	−3.47214	−0.202499
$295$	3.23607	0.188411
$296$	−6.00000	−0.348743
$297$	−1.00000	−0.0580259
$298$	−9.52786	−0.551934
$299$	−6.47214	−0.374293
$300$	−1.00000	−0.0577350
$301$	−10.4721	−0.603604
$302$	−14.4721	−0.832778
$303$	−8.94427	−0.513835
$304$	−7.70820	−0.442096
$305$	−6.00000	−0.343559
$306$	−1.00000	−0.0571662
$307$	10.2918	0.587384	0.293692	−	0.955900i	$-0.405116\pi$
$307$	10.2918	0.587384	0.293692	+	0.955900i	$0.405116\pi$
$308$	3.23607	0.184392
$309$	−4.00000	−0.227552
$310$	4.00000	0.227185
$311$	11.2361	0.637139	0.318569	−	0.947900i	$-0.396798\pi$
$311$	11.2361	0.637139	0.318569	+	0.947900i	$0.396798\pi$
$312$	−1.23607	−0.0699786
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	19.4164	1.09573
$315$	−3.23607	−0.182332
$316$	3.70820	0.208603
$317$	−13.0557	−0.733283	−0.366641	−	0.930362i	$-0.619492\pi$
$317$	−13.0557	−0.733283	−0.366641	+	0.930362i	$0.619492\pi$
$318$	−3.23607	−0.181470
$319$	−8.47214	−0.474349
$320$	−1.00000	−0.0559017
$321$	−0.944272	−0.0527041
$322$	−16.9443	−0.944267
$323$	7.70820	0.428896
$324$	1.00000	0.0555556
$325$	1.23607	0.0685647
$326$	−6.47214	−0.358458
$327$	6.00000	0.331801
$328$	−8.94427	−0.493865
$329$	8.00000	0.441054
$330$	1.00000	0.0550482
$331$	33.8885	1.86268	0.931341	−	0.364147i	$-0.118639\pi$
$331$	33.8885	1.86268	0.931341	+	0.364147i	$0.118639\pi$
$332$	4.94427	0.271352
$333$	−6.00000	−0.328798
$334$	−12.0000	−0.656611
$335$	−6.00000	−0.327815
$336$	−3.23607	−0.176542
$337$	−3.70820	−0.201999	−0.100999	−	0.994886i	$-0.532204\pi$
$337$	−3.70820	−0.201999	−0.100999	+	0.994886i	$0.532204\pi$
$338$	−11.4721	−0.624002
$339$	15.2361	0.827510
$340$	1.00000	0.0542326
$341$	−4.00000	−0.216612
$342$	−7.70820	−0.416812
$343$	−11.4164	−0.616428
$344$	−3.23607	−0.174477
$345$	−5.23607	−0.281900
$346$	13.4164	0.721271
$347$	12.9443	0.694885	0.347442	−	0.937701i	$-0.387050\pi$
$347$	12.9443	0.694885	0.347442	+	0.937701i	$0.387050\pi$
$348$	8.47214	0.454154
$349$	1.70820	0.0914381	0.0457190	−	0.998954i	$-0.485442\pi$
$349$	1.70820	0.0914381	0.0457190	+	0.998954i	$0.485442\pi$
$350$	3.23607	0.172975
$351$	−1.23607	−0.0659764
$352$	1.00000	0.0533002
$353$	−4.47214	−0.238028	−0.119014	−	0.992893i	$-0.537973\pi$
$353$	−4.47214	−0.238028	−0.119014	+	0.992893i	$0.537973\pi$
$354$	3.23607	0.171995
$355$	0.763932	0.0405453
$356$	−6.94427	−0.368046
$357$	3.23607	0.171271
$358$	−12.7639	−0.674595
$359$	−7.41641	−0.391423	−0.195712	−	0.980662i	$-0.562702\pi$
$359$	−7.41641	−0.391423	−0.195712	+	0.980662i	$0.562702\pi$
$360$	−1.00000	−0.0527046
$361$	40.4164	2.12718
$362$	8.47214	0.445286
$363$	−1.00000	−0.0524864
$364$	4.00000	0.209657
$365$	−1.23607	−0.0646988
$366$	−6.00000	−0.313625
$367$	28.8328	1.50506	0.752530	−	0.658558i	$-0.228833\pi$
$367$	28.8328	1.50506	0.752530	+	0.658558i	$0.228833\pi$
$368$	−5.23607	−0.272949
$369$	−8.94427	−0.465620
$370$	6.00000	0.311925
$371$	10.4721	0.543686
$372$	4.00000	0.207390
$373$	−15.7082	−0.813340	−0.406670	−	0.913575i	$-0.633310\pi$
$373$	−15.7082	−0.813340	−0.406670	+	0.913575i	$0.633310\pi$
$374$	−1.00000	−0.0517088
$375$	1.00000	0.0516398
$376$	2.47214	0.127491
$377$	−10.4721	−0.539342
$378$	−3.23607	−0.166445
$379$	−24.9443	−1.28130	−0.640651	−	0.767833i	$-0.721335\pi$
$379$	−24.9443	−1.28130	−0.640651	+	0.767833i	$0.721335\pi$
$380$	7.70820	0.395423
$381$	12.0000	0.614779
$382$	1.52786	0.0781723
$383$	10.4721	0.535101	0.267551	−	0.963544i	$-0.413786\pi$
$383$	10.4721	0.535101	0.267551	+	0.963544i	$0.413786\pi$
$384$	−1.00000	−0.0510310
$385$	−3.23607	−0.164925
$386$	7.70820	0.392337
$387$	−3.23607	−0.164499
$388$	0	0
$389$	25.5967	1.29781	0.648903	−	0.760871i	$-0.275228\pi$
$389$	25.5967	1.29781	0.648903	+	0.760871i	$0.275228\pi$
$390$	1.23607	0.0625907
$391$	5.23607	0.264799
$392$	3.47214	0.175369
$393$	0	0
$394$	14.9443	0.752882
$395$	−3.70820	−0.186580
$396$	1.00000	0.0502519
$397$	−19.8885	−0.998177	−0.499089	−	0.866551i	$-0.666332\pi$
$397$	−19.8885	−0.998177	−0.499089	+	0.866551i	$0.666332\pi$
$398$	−12.0000	−0.601506
$399$	24.9443	1.24878
$400$	1.00000	0.0500000
$401$	−20.2918	−1.01332	−0.506662	−	0.862145i	$-0.669121\pi$
$401$	−20.2918	−1.01332	−0.506662	+	0.862145i	$0.669121\pi$
$402$	−6.00000	−0.299253
$403$	−4.94427	−0.246292
$404$	8.94427	0.444994
$405$	−1.00000	−0.0496904
$406$	−27.4164	−1.36065
$407$	−6.00000	−0.297409
$408$	1.00000	0.0495074
$409$	10.9443	0.541159	0.270580	−	0.962698i	$-0.412785\pi$
$409$	10.9443	0.541159	0.270580	+	0.962698i	$0.412785\pi$
$410$	8.94427	0.441726
$411$	15.8885	0.783724
$412$	4.00000	0.197066
$413$	−10.4721	−0.515300
$414$	−5.23607	−0.257339
$415$	−4.94427	−0.242705
$416$	1.23607	0.0606032
$417$	−8.00000	−0.391762
$418$	−7.70820	−0.377021
$419$	10.4721	0.511597	0.255799	−	0.966730i	$-0.417662\pi$
$419$	10.4721	0.511597	0.255799	+	0.966730i	$0.417662\pi$
$420$	3.23607	0.157904
$421$	−6.94427	−0.338443	−0.169222	−	0.985578i	$-0.554125\pi$
$421$	−6.94427	−0.338443	−0.169222	+	0.985578i	$0.554125\pi$
$422$	20.3607	0.991142
$423$	2.47214	0.120199
$424$	3.23607	0.157157
$425$	−1.00000	−0.0485071
$426$	0.763932	0.0370126
$427$	19.4164	0.939626
$428$	0.944272	0.0456431
$429$	−1.23607	−0.0596779
$430$	3.23607	0.156057
$431$	−5.05573	−0.243526	−0.121763	−	0.992559i	$-0.538855\pi$
$431$	−5.05573	−0.243526	−0.121763	+	0.992559i	$0.538855\pi$
$432$	−1.00000	−0.0481125
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	−12.9443	−0.621345
$435$	−8.47214	−0.406208
$436$	−6.00000	−0.287348
$437$	40.3607	1.93071
$438$	−1.23607	−0.0590616
$439$	−4.29180	−0.204836	−0.102418	−	0.994741i	$-0.532658\pi$
$439$	−4.29180	−0.204836	−0.102418	+	0.994741i	$0.532658\pi$
$440$	−1.00000	−0.0476731
$441$	3.47214	0.165340
$442$	−1.23607	−0.0587938
$443$	−3.34752	−0.159046	−0.0795228	−	0.996833i	$-0.525340\pi$
$443$	−3.34752	−0.159046	−0.0795228	+	0.996833i	$0.525340\pi$
$444$	6.00000	0.284747
$445$	6.94427	0.329190
$446$	−23.4164	−1.10880
$447$	9.52786	0.450653
$448$	3.23607	0.152890
$449$	6.18034	0.291668	0.145834	−	0.989309i	$-0.453413\pi$
$449$	6.18034	0.291668	0.145834	+	0.989309i	$0.453413\pi$
$450$	1.00000	0.0471405
$451$	−8.94427	−0.421169
$452$	−15.2361	−0.716644
$453$	14.4721	0.679960
$454$	0	0
$455$	−4.00000	−0.187523
$456$	7.70820	0.360970
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	29.4164	1.37454
$459$	1.00000	0.0466760
$460$	5.23607	0.244133
$461$	7.41641	0.345417	0.172708	−	0.984973i	$-0.444748\pi$
$461$	7.41641	0.345417	0.172708	+	0.984973i	$0.444748\pi$
$462$	−3.23607	−0.150556
$463$	20.3607	0.946241	0.473121	−	0.880998i	$-0.343128\pi$
$463$	20.3607	0.946241	0.473121	+	0.880998i	$0.343128\pi$
$464$	−8.47214	−0.393309
$465$	−4.00000	−0.185496
$466$	−7.52786	−0.348722
$467$	14.7639	0.683193	0.341597	−	0.939847i	$-0.389032\pi$
$467$	14.7639	0.683193	0.341597	+	0.939847i	$0.389032\pi$
$468$	1.23607	0.0571373
$469$	19.4164	0.896566
$470$	−2.47214	−0.114031
$471$	−19.4164	−0.894661
$472$	−3.23607	−0.148952
$473$	−3.23607	−0.148795
$474$	−3.70820	−0.170323
$475$	−7.70820	−0.353677
$476$	−3.23607	−0.148325
$477$	3.23607	0.148169
$478$	−16.3607	−0.748320
$479$	−27.5279	−1.25778	−0.628890	−	0.777494i	$-0.716490\pi$
$479$	−27.5279	−1.25778	−0.628890	+	0.777494i	$0.716490\pi$
$480$	1.00000	0.0456435
$481$	−7.41641	−0.338159
$482$	4.18034	0.190409
$483$	16.9443	0.770991
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−20.8328	−0.944025	−0.472012	−	0.881592i	$-0.656472\pi$
$487$	−20.8328	−0.944025	−0.472012	+	0.881592i	$0.656472\pi$
$488$	6.00000	0.271607
$489$	6.47214	0.292680
$490$	−3.47214	−0.156855
$491$	1.05573	0.0476443	0.0238222	−	0.999716i	$-0.492416\pi$
$491$	1.05573	0.0476443	0.0238222	+	0.999716i	$0.492416\pi$
$492$	8.94427	0.403239
$493$	8.47214	0.381566
$494$	−9.52786	−0.428679
$495$	−1.00000	−0.0449467
$496$	−4.00000	−0.179605
$497$	−2.47214	−0.110890
$498$	−4.94427	−0.221558
$499$	−22.8328	−1.02214	−0.511069	−	0.859540i	$-0.670750\pi$
$499$	−22.8328	−1.02214	−0.511069	+	0.859540i	$0.670750\pi$
$500$	−1.00000	−0.0447214
$501$	12.0000	0.536120
$502$	23.2361	1.03708
$503$	18.4721	0.823632	0.411816	−	0.911267i	$-0.364895\pi$
$503$	18.4721	0.823632	0.411816	+	0.911267i	$0.364895\pi$
$504$	3.23607	0.144146
$505$	−8.94427	−0.398015
$506$	−5.23607	−0.232772
$507$	11.4721	0.509495
$508$	−12.0000	−0.532414
$509$	34.7639	1.54088	0.770442	−	0.637510i	$-0.220036\pi$
$509$	34.7639	1.54088	0.770442	+	0.637510i	$0.220036\pi$
$510$	−1.00000	−0.0442807
$511$	4.00000	0.176950
$512$	1.00000	0.0441942
$513$	7.70820	0.340326
$514$	−4.47214	−0.197257
$515$	−4.00000	−0.176261
$516$	3.23607	0.142460
$517$	2.47214	0.108724
$518$	−19.4164	−0.853108
$519$	−13.4164	−0.588915
$520$	−1.23607	−0.0542052
$521$	−35.1246	−1.53884	−0.769419	−	0.638745i	$-0.779454\pi$
$521$	−35.1246	−1.53884	−0.769419	+	0.638745i	$0.779454\pi$
$522$	−8.47214	−0.370815
$523$	21.1246	0.923715	0.461857	−	0.886954i	$-0.347183\pi$
$523$	21.1246	0.923715	0.461857	+	0.886954i	$0.347183\pi$
$524$	0	0
$525$	−3.23607	−0.141234
$526$	−12.0000	−0.523225
$527$	4.00000	0.174243
$528$	−1.00000	−0.0435194
$529$	4.41641	0.192018
$530$	−3.23607	−0.140566
$531$	−3.23607	−0.140433
$532$	−24.9443	−1.08147
$533$	−11.0557	−0.478877
$534$	6.94427	0.300508
$535$	−0.944272	−0.0408244
$536$	6.00000	0.259161
$537$	12.7639	0.550804
$538$	−24.4721	−1.05507
$539$	3.47214	0.149555
$540$	1.00000	0.0430331
$541$	−45.7771	−1.96811	−0.984055	−	0.177862i	$-0.943082\pi$
$541$	−45.7771	−1.96811	−0.984055	+	0.177862i	$0.943082\pi$
$542$	20.0000	0.859074
$543$	−8.47214	−0.363574
$544$	−1.00000	−0.0428746
$545$	6.00000	0.257012
$546$	−4.00000	−0.171184
$547$	4.58359	0.195980	0.0979901	−	0.995187i	$-0.468759\pi$
$547$	4.58359	0.195980	0.0979901	+	0.995187i	$0.468759\pi$
$548$	−15.8885	−0.678725
$549$	6.00000	0.256074
$550$	1.00000	0.0426401
$551$	65.3050	2.78208
$552$	5.23607	0.222862
$553$	12.0000	0.510292
$554$	14.0000	0.594803
$555$	−6.00000	−0.254686
$556$	8.00000	0.339276
$557$	−20.8328	−0.882715	−0.441357	−	0.897331i	$-0.645503\pi$
$557$	−20.8328	−0.882715	−0.441357	+	0.897331i	$0.645503\pi$
$558$	−4.00000	−0.169334
$559$	−4.00000	−0.169182
$560$	−3.23607	−0.136749
$561$	1.00000	0.0422200
$562$	−16.4721	−0.694835
$563$	−35.7771	−1.50782	−0.753912	−	0.656975i	$-0.771836\pi$
$563$	−35.7771	−1.50782	−0.753912	+	0.656975i	$0.771836\pi$
$564$	−2.47214	−0.104096
$565$	15.2361	0.640986
$566$	−3.41641	−0.143602
$567$	3.23607	0.135902
$568$	−0.763932	−0.0320539
$569$	19.3050	0.809306	0.404653	−	0.914470i	$-0.367392\pi$
$569$	19.3050	0.809306	0.404653	+	0.914470i	$0.367392\pi$
$570$	−7.70820	−0.322861
$571$	−25.8885	−1.08340	−0.541701	−	0.840571i	$-0.682219\pi$
$571$	−25.8885	−1.08340	−0.541701	+	0.840571i	$0.682219\pi$
$572$	1.23607	0.0516826
$573$	−1.52786	−0.0638274
$574$	−28.9443	−1.20811
$575$	−5.23607	−0.218359
$576$	1.00000	0.0416667
$577$	−15.8885	−0.661449	−0.330724	−	0.943727i	$-0.607293\pi$
$577$	−15.8885	−0.661449	−0.330724	+	0.943727i	$0.607293\pi$
$578$	1.00000	0.0415945
$579$	−7.70820	−0.320342
$580$	8.47214	0.351786
$581$	16.0000	0.663792
$582$	0	0
$583$	3.23607	0.134024
$584$	1.23607	0.0511489
$585$	−1.23607	−0.0511051
$586$	−18.0000	−0.743573
$587$	−2.18034	−0.0899923	−0.0449961	−	0.998987i	$-0.514328\pi$
$587$	−2.18034	−0.0899923	−0.0449961	+	0.998987i	$0.514328\pi$
$588$	−3.47214	−0.143188
$589$	30.8328	1.27044
$590$	3.23607	0.133227
$591$	−14.9443	−0.614725
$592$	−6.00000	−0.246598
$593$	13.4164	0.550946	0.275473	−	0.961309i	$-0.411166\pi$
$593$	13.4164	0.550946	0.275473	+	0.961309i	$0.411166\pi$
$594$	−1.00000	−0.0410305
$595$	3.23607	0.132666
$596$	−9.52786	−0.390277
$597$	12.0000	0.491127
$598$	−6.47214	−0.264665
$599$	4.58359	0.187280	0.0936402	−	0.995606i	$-0.470150\pi$
$599$	4.58359	0.187280	0.0936402	+	0.995606i	$0.470150\pi$
$600$	−1.00000	−0.0408248
$601$	−1.12461	−0.0458739	−0.0229369	−	0.999737i	$-0.507302\pi$
$601$	−1.12461	−0.0458739	−0.0229369	+	0.999737i	$0.507302\pi$
$602$	−10.4721	−0.426812
$603$	6.00000	0.244339
$604$	−14.4721	−0.588863
$605$	−1.00000	−0.0406558
$606$	−8.94427	−0.363336
$607$	29.1246	1.18213	0.591066	−	0.806623i	$-0.298707\pi$
$607$	29.1246	1.18213	0.591066	+	0.806623i	$0.298707\pi$
$608$	−7.70820	−0.312609
$609$	27.4164	1.11097
$610$	−6.00000	−0.242933
$611$	3.05573	0.123622
$612$	−1.00000	−0.0404226
$613$	7.12461	0.287760	0.143880	−	0.989595i	$-0.454042\pi$
$613$	7.12461	0.287760	0.143880	+	0.989595i	$0.454042\pi$
$614$	10.2918	0.415343
$615$	−8.94427	−0.360668
$616$	3.23607	0.130385
$617$	−0.763932	−0.0307547	−0.0153774	−	0.999882i	$-0.504895\pi$
$617$	−0.763932	−0.0307547	−0.0153774	+	0.999882i	$0.504895\pi$
$618$	−4.00000	−0.160904
$619$	−9.88854	−0.397454	−0.198727	−	0.980055i	$-0.563681\pi$
$619$	−9.88854	−0.397454	−0.198727	+	0.980055i	$0.563681\pi$
$620$	4.00000	0.160644
$621$	5.23607	0.210116
$622$	11.2361	0.450525
$623$	−22.4721	−0.900327
$624$	−1.23607	−0.0494823
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.70820	0.307836
$628$	19.4164	0.774799
$629$	6.00000	0.239236
$630$	−3.23607	−0.128928
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	3.70820	0.147504
$633$	−20.3607	−0.809264
$634$	−13.0557	−0.518509
$635$	12.0000	0.476205
$636$	−3.23607	−0.128318
$637$	4.29180	0.170047
$638$	−8.47214	−0.335415
$639$	−0.763932	−0.0302207
$640$	−1.00000	−0.0395285
$641$	8.65248	0.341752	0.170876	−	0.985293i	$-0.445340\pi$
$641$	8.65248	0.341752	0.170876	+	0.985293i	$0.445340\pi$
$642$	−0.944272	−0.0372674
$643$	−36.9443	−1.45694	−0.728470	−	0.685078i	$-0.759768\pi$
$643$	−36.9443	−1.45694	−0.728470	+	0.685078i	$0.759768\pi$
$644$	−16.9443	−0.667698
$645$	−3.23607	−0.127420
$646$	7.70820	0.303275
$647$	−14.8328	−0.583138	−0.291569	−	0.956550i	$-0.594177\pi$
$647$	−14.8328	−0.583138	−0.291569	+	0.956550i	$0.594177\pi$
$648$	1.00000	0.0392837
$649$	−3.23607	−0.127027
$650$	1.23607	0.0484826
$651$	12.9443	0.507326
$652$	−6.47214	−0.253468
$653$	11.8885	0.465235	0.232617	−	0.972568i	$-0.425271\pi$
$653$	11.8885	0.465235	0.232617	+	0.972568i	$0.425271\pi$
$654$	6.00000	0.234619
$655$	0	0
$656$	−8.94427	−0.349215
$657$	1.23607	0.0482236
$658$	8.00000	0.311872
$659$	44.8328	1.74644	0.873219	−	0.487328i	$-0.162028\pi$
$659$	44.8328	1.74644	0.873219	+	0.487328i	$0.162028\pi$
$660$	1.00000	0.0389249
$661$	1.41641	0.0550919	0.0275459	−	0.999621i	$-0.491231\pi$
$661$	1.41641	0.0550919	0.0275459	+	0.999621i	$0.491231\pi$
$662$	33.8885	1.31712
$663$	1.23607	0.0480049
$664$	4.94427	0.191875
$665$	24.9443	0.967297
$666$	−6.00000	−0.232495
$667$	44.3607	1.71765
$668$	−12.0000	−0.464294
$669$	23.4164	0.905331
$670$	−6.00000	−0.231800
$671$	6.00000	0.231627
$672$	−3.23607	−0.124834
$673$	−45.2361	−1.74372	−0.871861	−	0.489753i	$-0.837087\pi$
$673$	−45.2361	−1.74372	−0.871861	+	0.489753i	$0.837087\pi$
$674$	−3.70820	−0.142835
$675$	−1.00000	−0.0384900
$676$	−11.4721	−0.441236
$677$	−39.8885	−1.53304	−0.766521	−	0.642220i	$-0.778014\pi$
$677$	−39.8885	−1.53304	−0.766521	+	0.642220i	$0.778014\pi$
$678$	15.2361	0.585138
$679$	0	0
$680$	1.00000	0.0383482
$681$	0	0
$682$	−4.00000	−0.153168
$683$	−23.7771	−0.909805	−0.454902	−	0.890541i	$-0.650326\pi$
$683$	−23.7771	−0.909805	−0.454902	+	0.890541i	$0.650326\pi$
$684$	−7.70820	−0.294731
$685$	15.8885	0.607070
$686$	−11.4164	−0.435880
$687$	−29.4164	−1.12231
$688$	−3.23607	−0.123374
$689$	4.00000	0.152388
$690$	−5.23607	−0.199334
$691$	42.8328	1.62944	0.814719	−	0.579857i	$-0.196891\pi$
$691$	42.8328	1.62944	0.814719	+	0.579857i	$0.196891\pi$
$692$	13.4164	0.510015
$693$	3.23607	0.122928
$694$	12.9443	0.491358
$695$	−8.00000	−0.303457
$696$	8.47214	0.321135
$697$	8.94427	0.338788
$698$	1.70820	0.0646565
$699$	7.52786	0.284730
$700$	3.23607	0.122312
$701$	26.4721	0.999839	0.499919	−	0.866072i	$-0.333363\pi$
$701$	26.4721	0.999839	0.499919	+	0.866072i	$0.333363\pi$
$702$	−1.23607	−0.0466524
$703$	46.2492	1.74432
$704$	1.00000	0.0376889
$705$	2.47214	0.0931060
$706$	−4.47214	−0.168311
$707$	28.9443	1.08856
$708$	3.23607	0.121619
$709$	−24.8328	−0.932616	−0.466308	−	0.884622i	$-0.654416\pi$
$709$	−24.8328	−0.932616	−0.466308	+	0.884622i	$0.654416\pi$
$710$	0.763932	0.0286699
$711$	3.70820	0.139069
$712$	−6.94427	−0.260248
$713$	20.9443	0.784369
$714$	3.23607	0.121107
$715$	−1.23607	−0.0462263
$716$	−12.7639	−0.477011
$717$	16.3607	0.611001
$718$	−7.41641	−0.276778
$719$	48.5410	1.81027	0.905137	−	0.425119i	$-0.139768\pi$
$719$	48.5410	1.81027	0.905137	+	0.425119i	$0.139768\pi$
$720$	−1.00000	−0.0372678
$721$	12.9443	0.482070
$722$	40.4164	1.50414
$723$	−4.18034	−0.155469
$724$	8.47214	0.314864
$725$	−8.47214	−0.314647
$726$	−1.00000	−0.0371135
$727$	−18.8328	−0.698470	−0.349235	−	0.937035i	$-0.613559\pi$
$727$	−18.8328	−0.698470	−0.349235	+	0.937035i	$0.613559\pi$
$728$	4.00000	0.148250
$729$	1.00000	0.0370370
$730$	−1.23607	−0.0457489
$731$	3.23607	0.119690
$732$	−6.00000	−0.221766
$733$	3.34752	0.123644	0.0618218	−	0.998087i	$-0.480309\pi$
$733$	3.34752	0.123644	0.0618218	+	0.998087i	$0.480309\pi$
$734$	28.8328	1.06424
$735$	3.47214	0.128072
$736$	−5.23607	−0.193004
$737$	6.00000	0.221013
$738$	−8.94427	−0.329243
$739$	28.2918	1.04073	0.520365	−	0.853944i	$-0.325796\pi$
$739$	28.2918	1.04073	0.520365	+	0.853944i	$0.325796\pi$
$740$	6.00000	0.220564
$741$	9.52786	0.350015
$742$	10.4721	0.384444
$743$	−22.4721	−0.824423	−0.412211	−	0.911088i	$-0.635243\pi$
$743$	−22.4721	−0.824423	−0.412211	+	0.911088i	$0.635243\pi$
$744$	4.00000	0.146647
$745$	9.52786	0.349074
$746$	−15.7082	−0.575118
$747$	4.94427	0.180901
$748$	−1.00000	−0.0365636
$749$	3.05573	0.111654
$750$	1.00000	0.0365148
$751$	7.41641	0.270629	0.135314	−	0.990803i	$-0.456796\pi$
$751$	7.41641	0.270629	0.135314	+	0.990803i	$0.456796\pi$
$752$	2.47214	0.0901495
$753$	−23.2361	−0.846769
$754$	−10.4721	−0.381373
$755$	14.4721	0.526695
$756$	−3.23607	−0.117695
$757$	21.8885	0.795553	0.397776	−	0.917482i	$-0.369782\pi$
$757$	21.8885	0.795553	0.397776	+	0.917482i	$0.369782\pi$
$758$	−24.9443	−0.906017
$759$	5.23607	0.190057
$760$	7.70820	0.279606
$761$	−36.4721	−1.32211	−0.661057	−	0.750336i	$-0.729892\pi$
$761$	−36.4721	−1.32211	−0.661057	+	0.750336i	$0.729892\pi$
$762$	12.0000	0.434714
$763$	−19.4164	−0.702921
$764$	1.52786	0.0552762
$765$	1.00000	0.0361551
$766$	10.4721	0.378374
$767$	−4.00000	−0.144432
$768$	−1.00000	−0.0360844
$769$	24.4721	0.882488	0.441244	−	0.897387i	$-0.354537\pi$
$769$	24.4721	0.882488	0.441244	+	0.897387i	$0.354537\pi$
$770$	−3.23607	−0.116620
$771$	4.47214	0.161060
$772$	7.70820	0.277424
$773$	−31.2361	−1.12348	−0.561742	−	0.827313i	$-0.689868\pi$
$773$	−31.2361	−1.12348	−0.561742	+	0.827313i	$0.689868\pi$
$774$	−3.23607	−0.116318
$775$	−4.00000	−0.143684
$776$	0	0
$777$	19.4164	0.696560
$778$	25.5967	0.917688
$779$	68.9443	2.47018
$780$	1.23607	0.0442583
$781$	−0.763932	−0.0273356
$782$	5.23607	0.187241
$783$	8.47214	0.302769
$784$	3.47214	0.124005
$785$	−19.4164	−0.693001
$786$	0	0
$787$	19.0557	0.679263	0.339632	−	0.940559i	$-0.389698\pi$
$787$	19.0557	0.679263	0.339632	+	0.940559i	$0.389698\pi$
$788$	14.9443	0.532368
$789$	12.0000	0.427211
$790$	−3.70820	−0.131932
$791$	−49.3050	−1.75308
$792$	1.00000	0.0355335
$793$	7.41641	0.263364
$794$	−19.8885	−0.705818
$795$	3.23607	0.114772
$796$	−12.0000	−0.425329
$797$	35.0132	1.24023	0.620115	−	0.784511i	$-0.287086\pi$
$797$	35.0132	1.24023	0.620115	+	0.784511i	$0.287086\pi$
$798$	24.9443	0.883018
$799$	−2.47214	−0.0874579
$800$	1.00000	0.0353553
$801$	−6.94427	−0.245364
$802$	−20.2918	−0.716528
$803$	1.23607	0.0436199
$804$	−6.00000	−0.211604
$805$	16.9443	0.597207
$806$	−4.94427	−0.174155
$807$	24.4721	0.861460
$808$	8.94427	0.314658
$809$	−16.3607	−0.575211	−0.287605	−	0.957749i	$-0.592859\pi$
$809$	−16.3607	−0.575211	−0.287605	+	0.957749i	$0.592859\pi$
$810$	−1.00000	−0.0351364
$811$	−30.2492	−1.06219	−0.531097	−	0.847311i	$-0.678220\pi$
$811$	−30.2492	−1.06219	−0.531097	+	0.847311i	$0.678220\pi$
$812$	−27.4164	−0.962127
$813$	−20.0000	−0.701431
$814$	−6.00000	−0.210300
$815$	6.47214	0.226709
$816$	1.00000	0.0350070
$817$	24.9443	0.872690
$818$	10.9443	0.382657
$819$	4.00000	0.139771
$820$	8.94427	0.312348
$821$	−10.3607	−0.361590	−0.180795	−	0.983521i	$-0.557867\pi$
$821$	−10.3607	−0.361590	−0.180795	+	0.983521i	$0.557867\pi$
$822$	15.8885	0.554177
$823$	37.4164	1.30425	0.652127	−	0.758110i	$-0.273877\pi$
$823$	37.4164	1.30425	0.652127	+	0.758110i	$0.273877\pi$
$824$	4.00000	0.139347
$825$	−1.00000	−0.0348155
$826$	−10.4721	−0.364372
$827$	−32.9443	−1.14558	−0.572792	−	0.819701i	$-0.694140\pi$
$827$	−32.9443	−1.14558	−0.572792	+	0.819701i	$0.694140\pi$
$828$	−5.23607	−0.181966
$829$	−49.7771	−1.72883	−0.864415	−	0.502779i	$-0.832311\pi$
$829$	−49.7771	−1.72883	−0.864415	+	0.502779i	$0.832311\pi$
$830$	−4.94427	−0.171618
$831$	−14.0000	−0.485655
$832$	1.23607	0.0428529
$833$	−3.47214	−0.120302
$834$	−8.00000	−0.277017
$835$	12.0000	0.415277
$836$	−7.70820	−0.266594
$837$	4.00000	0.138260
$838$	10.4721	0.361754
$839$	14.2918	0.493408	0.246704	−	0.969091i	$-0.420653\pi$
$839$	14.2918	0.493408	0.246704	+	0.969091i	$0.420653\pi$
$840$	3.23607	0.111655
$841$	42.7771	1.47507
$842$	−6.94427	−0.239315
$843$	16.4721	0.567330
$844$	20.3607	0.700844
$845$	11.4721	0.394653
$846$	2.47214	0.0849938
$847$	3.23607	0.111193
$848$	3.23607	0.111127
$849$	3.41641	0.117251
$850$	−1.00000	−0.0342997
$851$	31.4164	1.07694
$852$	0.763932	0.0261719
$853$	−14.3607	−0.491700	−0.245850	−	0.969308i	$-0.579067\pi$
$853$	−14.3607	−0.491700	−0.245850	+	0.969308i	$0.579067\pi$
$854$	19.4164	0.664416
$855$	7.70820	0.263615
$856$	0.944272	0.0322745
$857$	−19.8885	−0.679380	−0.339690	−	0.940538i	$-0.610322\pi$
$857$	−19.8885	−0.679380	−0.339690	+	0.940538i	$0.610322\pi$
$858$	−1.23607	−0.0421987
$859$	12.9443	0.441653	0.220826	−	0.975313i	$-0.429125\pi$
$859$	12.9443	0.441653	0.220826	+	0.975313i	$0.429125\pi$
$860$	3.23607	0.110349
$861$	28.9443	0.986418
$862$	−5.05573	−0.172199
$863$	−52.9443	−1.80224	−0.901122	−	0.433566i	$-0.857255\pi$
$863$	−52.9443	−1.80224	−0.901122	+	0.433566i	$0.857255\pi$
$864$	−1.00000	−0.0340207
$865$	−13.4164	−0.456172
$866$	−6.00000	−0.203888
$867$	−1.00000	−0.0339618
$868$	−12.9443	−0.439357
$869$	3.70820	0.125792
$870$	−8.47214	−0.287232
$871$	7.41641	0.251295
$872$	−6.00000	−0.203186
$873$	0	0
$874$	40.3607	1.36522
$875$	−3.23607	−0.109399
$876$	−1.23607	−0.0417629
$877$	42.3607	1.43042	0.715209	−	0.698910i	$-0.246331\pi$
$877$	42.3607	1.43042	0.715209	+	0.698910i	$0.246331\pi$
$878$	−4.29180	−0.144841
$879$	18.0000	0.607125
$880$	−1.00000	−0.0337100
$881$	14.1803	0.477748	0.238874	−	0.971051i	$-0.423222\pi$
$881$	14.1803	0.477748	0.238874	+	0.971051i	$0.423222\pi$
$882$	3.47214	0.116913
$883$	33.4164	1.12455	0.562276	−	0.826950i	$-0.309926\pi$
$883$	33.4164	1.12455	0.562276	+	0.826950i	$0.309926\pi$
$884$	−1.23607	−0.0415735
$885$	−3.23607	−0.108779
$886$	−3.34752	−0.112462
$887$	−56.7214	−1.90452	−0.952258	−	0.305293i	$-0.901246\pi$
$887$	−56.7214	−1.90452	−0.952258	+	0.305293i	$0.901246\pi$
$888$	6.00000	0.201347
$889$	−38.8328	−1.30241
$890$	6.94427	0.232773
$891$	1.00000	0.0335013
$892$	−23.4164	−0.784039
$893$	−19.0557	−0.637676
$894$	9.52786	0.318659
$895$	12.7639	0.426651
$896$	3.23607	0.108109
$897$	6.47214	0.216098
$898$	6.18034	0.206241
$899$	33.8885	1.13025
$900$	1.00000	0.0333333
$901$	−3.23607	−0.107809
$902$	−8.94427	−0.297812
$903$	10.4721	0.348491
$904$	−15.2361	−0.506744
$905$	−8.47214	−0.281623
$906$	14.4721	0.480805
$907$	−38.8328	−1.28942	−0.644711	−	0.764426i	$-0.723022\pi$
$907$	−38.8328	−1.28942	−0.644711	+	0.764426i	$0.723022\pi$
$908$	0	0
$909$	8.94427	0.296663
$910$	−4.00000	−0.132599
$911$	31.2361	1.03490	0.517449	−	0.855714i	$-0.326882\pi$
$911$	31.2361	1.03490	0.517449	+	0.855714i	$0.326882\pi$
$912$	7.70820	0.255244
$913$	4.94427	0.163632
$914$	18.0000	0.595387
$915$	6.00000	0.198354
$916$	29.4164	0.971945
$917$	0	0
$918$	1.00000	0.0330049
$919$	−6.47214	−0.213496	−0.106748	−	0.994286i	$-0.534044\pi$
$919$	−6.47214	−0.213496	−0.106748	+	0.994286i	$0.534044\pi$
$920$	5.23607	0.172628
$921$	−10.2918	−0.339126
$922$	7.41641	0.244246
$923$	−0.944272	−0.0310811
$924$	−3.23607	−0.106459
$925$	−6.00000	−0.197279
$926$	20.3607	0.669093
$927$	4.00000	0.131377
$928$	−8.47214	−0.278111
$929$	33.0132	1.08313	0.541563	−	0.840660i	$-0.317833\pi$
$929$	33.0132	1.08313	0.541563	+	0.840660i	$0.317833\pi$
$930$	−4.00000	−0.131165
$931$	−26.7639	−0.877152
$932$	−7.52786	−0.246583
$933$	−11.2361	−0.367852
$934$	14.7639	0.483091
$935$	1.00000	0.0327035
$936$	1.23607	0.0404021
$937$	−60.2492	−1.96826	−0.984128	−	0.177459i	$-0.943212\pi$
$937$	−60.2492	−1.96826	−0.984128	+	0.177459i	$0.943212\pi$
$938$	19.4164	0.633968
$939$	0	0
$940$	−2.47214	−0.0806322
$941$	12.4721	0.406580	0.203290	−	0.979119i	$-0.434837\pi$
$941$	12.4721	0.406580	0.203290	+	0.979119i	$0.434837\pi$
$942$	−19.4164	−0.632621
$943$	46.8328	1.52509
$944$	−3.23607	−0.105325
$945$	3.23607	0.105269
$946$	−3.23607	−0.105214
$947$	44.7214	1.45325	0.726624	−	0.687035i	$-0.241088\pi$
$947$	44.7214	1.45325	0.726624	+	0.687035i	$0.241088\pi$
$948$	−3.70820	−0.120437
$949$	1.52786	0.0495966
$950$	−7.70820	−0.250087
$951$	13.0557	0.423361
$952$	−3.23607	−0.104882
$953$	43.3050	1.40278	0.701392	−	0.712775i	$-0.252562\pi$
$953$	43.3050	1.40278	0.701392	+	0.712775i	$0.252562\pi$
$954$	3.23607	0.104772
$955$	−1.52786	−0.0494405
$956$	−16.3607	−0.529142
$957$	8.47214	0.273865
$958$	−27.5279	−0.889385
$959$	−51.4164	−1.66032
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	−7.41641	−0.239115
$963$	0.944272	0.0304287
$964$	4.18034	0.134640
$965$	−7.70820	−0.248136
$966$	16.9443	0.545173
$967$	−7.41641	−0.238496	−0.119248	−	0.992865i	$-0.538048\pi$
$967$	−7.41641	−0.238496	−0.119248	+	0.992865i	$0.538048\pi$
$968$	1.00000	0.0321412
$969$	−7.70820	−0.247623
$970$	0	0
$971$	48.7639	1.56491	0.782455	−	0.622708i	$-0.213967\pi$
$971$	48.7639	1.56491	0.782455	+	0.622708i	$0.213967\pi$
$972$	−1.00000	−0.0320750
$973$	25.8885	0.829949
$974$	−20.8328	−0.667526
$975$	−1.23607	−0.0395859
$976$	6.00000	0.192055
$977$	−7.52786	−0.240838	−0.120419	−	0.992723i	$-0.538424\pi$
$977$	−7.52786	−0.240838	−0.120419	+	0.992723i	$0.538424\pi$
$978$	6.47214	0.206956
$979$	−6.94427	−0.221940
$980$	−3.47214	−0.110913
$981$	−6.00000	−0.191565
$982$	1.05573	0.0336896
$983$	0.652476	0.0208107	0.0104054	−	0.999946i	$-0.496688\pi$
$983$	0.652476	0.0208107	0.0104054	+	0.999946i	$0.496688\pi$
$984$	8.94427	0.285133
$985$	−14.9443	−0.476164
$986$	8.47214	0.269808
$987$	−8.00000	−0.254643
$988$	−9.52786	−0.303122
$989$	16.9443	0.538797
$990$	−1.00000	−0.0317821
$991$	34.2492	1.08796	0.543981	−	0.839097i	$-0.316916\pi$
$991$	34.2492	1.08796	0.543981	+	0.839097i	$0.316916\pi$
$992$	−4.00000	−0.127000
$993$	−33.8885	−1.07542
$994$	−2.47214	−0.0784114
$995$	12.0000	0.380426
$996$	−4.94427	−0.156665
$997$	5.63932	0.178599	0.0892995	−	0.996005i	$-0.471537\pi$
$997$	5.63932	0.178599	0.0892995	+	0.996005i	$0.471537\pi$
$998$	−22.8328	−0.722760
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5610.2.a.bu.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.bu.1.2	✓	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:	\( N \)	\(=\)	\( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5610.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +1.23607 q^{13} +3.23607 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -7.70820 q^{19} -1.00000 q^{20} -3.23607 q^{21} +1.00000 q^{22} -5.23607 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.23607 q^{26} -1.00000 q^{27} +3.23607 q^{28} -8.47214 q^{29} +1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} -3.23607 q^{35} +1.00000 q^{36} -6.00000 q^{37} -7.70820 q^{38} -1.23607 q^{39} -1.00000 q^{40} -8.94427 q^{41} -3.23607 q^{42} -3.23607 q^{43} +1.00000 q^{44} -1.00000 q^{45} -5.23607 q^{46} +2.47214 q^{47} -1.00000 q^{48} +3.47214 q^{49} +1.00000 q^{50} +1.00000 q^{51} +1.23607 q^{52} +3.23607 q^{53} -1.00000 q^{54} -1.00000 q^{55} +3.23607 q^{56} +7.70820 q^{57} -8.47214 q^{58} -3.23607 q^{59} +1.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} +3.23607 q^{63} +1.00000 q^{64} -1.23607 q^{65} -1.00000 q^{66} +6.00000 q^{67} -1.00000 q^{68} +5.23607 q^{69} -3.23607 q^{70} -0.763932 q^{71} +1.00000 q^{72} +1.23607 q^{73} -6.00000 q^{74} -1.00000 q^{75} -7.70820 q^{76} +3.23607 q^{77} -1.23607 q^{78} +3.70820 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.94427 q^{82} +4.94427 q^{83} -3.23607 q^{84} +1.00000 q^{85} -3.23607 q^{86} +8.47214 q^{87} +1.00000 q^{88} -6.94427 q^{89} -1.00000 q^{90} +4.00000 q^{91} -5.23607 q^{92} +4.00000 q^{93} +2.47214 q^{94} +7.70820 q^{95} -1.00000 q^{96} +3.47214 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{21} + 2 q^{22} - 6 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} + 2 q^{28} - 8 q^{29} + 2 q^{30} - 8 q^{31} + 2 q^{32} - 2 q^{33} - 2 q^{34} - 2 q^{35} + 2 q^{36} - 12 q^{37} - 2 q^{38} + 2 q^{39} - 2 q^{40} - 2 q^{42} - 2 q^{43} + 2 q^{44} - 2 q^{45} - 6 q^{46} - 4 q^{47} - 2 q^{48} - 2 q^{49} + 2 q^{50} + 2 q^{51} - 2 q^{52} + 2 q^{53} - 2 q^{54} - 2 q^{55} + 2 q^{56} + 2 q^{57} - 8 q^{58} - 2 q^{59} + 2 q^{60} + 12 q^{61} - 8 q^{62} + 2 q^{63} + 2 q^{64} + 2 q^{65} - 2 q^{66} + 12 q^{67} - 2 q^{68} + 6 q^{69} - 2 q^{70} - 6 q^{71} + 2 q^{72} - 2 q^{73} - 12 q^{74} - 2 q^{75} - 2 q^{76} + 2 q^{77} + 2 q^{78} - 6 q^{79} - 2 q^{80} + 2 q^{81} - 8 q^{83} - 2 q^{84} + 2 q^{85} - 2 q^{86} + 8 q^{87} + 2 q^{88} + 4 q^{89} - 2 q^{90} + 8 q^{91} - 6 q^{92} + 8 q^{93} - 4 q^{94} + 2 q^{95} - 2 q^{96} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 - 2 * q^12 - 2 * q^13 + 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 2 * q^21 + 2 * q^22 - 6 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^26 - 2 * q^27 + 2 * q^28 - 8 * q^29 + 2 * q^30 - 8 * q^31 + 2 * q^32 - 2 * q^33 - 2 * q^34 - 2 * q^35 + 2 * q^36 - 12 * q^37 - 2 * q^38 + 2 * q^39 - 2 * q^40 - 2 * q^42 - 2 * q^43 + 2 * q^44 - 2 * q^45 - 6 * q^46 - 4 * q^47 - 2 * q^48 - 2 * q^49 + 2 * q^50 + 2 * q^51 - 2 * q^52 + 2 * q^53 - 2 * q^54 - 2 * q^55 + 2 * q^56 + 2 * q^57 - 8 * q^58 - 2 * q^59 + 2 * q^60 + 12 * q^61 - 8 * q^62 + 2 * q^63 + 2 * q^64 + 2 * q^65 - 2 * q^66 + 12 * q^67 - 2 * q^68 + 6 * q^69 - 2 * q^70 - 6 * q^71 + 2 * q^72 - 2 * q^73 - 12 * q^74 - 2 * q^75 - 2 * q^76 + 2 * q^77 + 2 * q^78 - 6 * q^79 - 2 * q^80 + 2 * q^81 - 8 * q^83 - 2 * q^84 + 2 * q^85 - 2 * q^86 + 8 * q^87 + 2 * q^88 + 4 * q^89 - 2 * q^90 + 8 * q^91 - 6 * q^92 + 8 * q^93 - 4 * q^94 + 2 * q^95 - 2 * q^96 - 2 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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