LMFDB - Embedded newform 4830.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4830,2,Mod(1,4830)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4830, 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("4830.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4830.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:	$38.5677441763$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	4830.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.12311 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +3.12311 q^{17} -1.00000 q^{18} +1.12311 q^{19} +1.00000 q^{20} -1.00000 q^{21} +3.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} -8.24621 q^{31} -1.00000 q^{32} -3.12311 q^{33} -3.12311 q^{34} -1.00000 q^{35} +1.00000 q^{36} +5.12311 q^{37} -1.12311 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} +10.2462 q^{43} -3.12311 q^{44} +1.00000 q^{45} +1.00000 q^{46} -2.87689 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +3.12311 q^{51} -4.00000 q^{52} -5.12311 q^{53} -1.00000 q^{54} -3.12311 q^{55} +1.00000 q^{56} +1.12311 q^{57} -2.00000 q^{58} +7.12311 q^{59} +1.00000 q^{60} -6.00000 q^{61} +8.24621 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +3.12311 q^{66} -4.00000 q^{67} +3.12311 q^{68} -1.00000 q^{69} +1.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} -13.1231 q^{73} -5.12311 q^{74} +1.00000 q^{75} +1.12311 q^{76} +3.12311 q^{77} +4.00000 q^{78} +17.3693 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -9.12311 q^{83} -1.00000 q^{84} +3.12311 q^{85} -10.2462 q^{86} +2.00000 q^{87} +3.12311 q^{88} -7.12311 q^{89} -1.00000 q^{90} +4.00000 q^{91} -1.00000 q^{92} -8.24621 q^{93} +2.87689 q^{94} +1.12311 q^{95} -1.00000 q^{96} -15.1231 q^{97} -1.00000 q^{98} -3.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 6 q^{19} + 2 q^{20} - 2 q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} + 2 q^{25} + 8 q^{26} + 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} - 2 q^{32} + 2 q^{33} + 2 q^{34} - 2 q^{35} + 2 q^{36} + 2 q^{37} + 6 q^{38} - 8 q^{39} - 2 q^{40} - 4 q^{41} + 2 q^{42} + 4 q^{43} + 2 q^{44} + 2 q^{45} + 2 q^{46} - 14 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 2 q^{51} - 8 q^{52} - 2 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{56} - 6 q^{57} - 4 q^{58} + 6 q^{59} + 2 q^{60} - 12 q^{61} - 2 q^{63} + 2 q^{64} - 8 q^{65} - 2 q^{66} - 8 q^{67} - 2 q^{68} - 2 q^{69} + 2 q^{70} - 8 q^{71} - 2 q^{72} - 18 q^{73} - 2 q^{74} + 2 q^{75} - 6 q^{76} - 2 q^{77} + 8 q^{78} + 10 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 10 q^{83} - 2 q^{84} - 2 q^{85} - 4 q^{86} + 4 q^{87} - 2 q^{88} - 6 q^{89} - 2 q^{90} + 8 q^{91} - 2 q^{92} + 14 q^{94} - 6 q^{95} - 2 q^{96} - 22 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 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 - 8 * q^13 + 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 6 * q^19 + 2 * q^20 - 2 * q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 + 2 * q^25 + 8 * q^26 + 2 * q^27 - 2 * q^28 + 4 * q^29 - 2 * q^30 - 2 * q^32 + 2 * q^33 + 2 * q^34 - 2 * q^35 + 2 * q^36 + 2 * q^37 + 6 * q^38 - 8 * q^39 - 2 * q^40 - 4 * q^41 + 2 * q^42 + 4 * q^43 + 2 * q^44 + 2 * q^45 + 2 * q^46 - 14 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 - 2 * q^51 - 8 * q^52 - 2 * q^53 - 2 * q^54 + 2 * q^55 + 2 * q^56 - 6 * q^57 - 4 * q^58 + 6 * q^59 + 2 * q^60 - 12 * q^61 - 2 * q^63 + 2 * q^64 - 8 * q^65 - 2 * q^66 - 8 * q^67 - 2 * q^68 - 2 * q^69 + 2 * q^70 - 8 * q^71 - 2 * q^72 - 18 * q^73 - 2 * q^74 + 2 * q^75 - 6 * q^76 - 2 * q^77 + 8 * q^78 + 10 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 - 10 * q^83 - 2 * q^84 - 2 * q^85 - 4 * q^86 + 4 * q^87 - 2 * q^88 - 6 * q^89 - 2 * q^90 + 8 * q^91 - 2 * q^92 + 14 * q^94 - 6 * q^95 - 2 * q^96 - 22 * q^97 - 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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	1.00000	0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	3.12311	0.757464	0.378732	−	0.925506i	$-0.376360\pi$
$17$	3.12311	0.757464	0.378732	+	0.925506i	$0.376360\pi$
$18$	−1.00000	−0.235702
$19$	1.12311	0.257658	0.128829	−	0.991667i	$-0.458878\pi$
$19$	1.12311	0.257658	0.128829	+	0.991667i	$0.458878\pi$
$20$	1.00000	0.223607
$21$	−1.00000	−0.218218
$22$	3.12311	0.665848
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	−8.24621	−1.48106	−0.740532	−	0.672022i	$-0.765426\pi$
$31$	−8.24621	−1.48106	−0.740532	+	0.672022i	$0.765426\pi$
$32$	−1.00000	−0.176777
$33$	−3.12311	−0.543663
$34$	−3.12311	−0.535608
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	5.12311	0.842233	0.421117	−	0.907006i	$-0.361638\pi$
$37$	5.12311	0.842233	0.421117	+	0.907006i	$0.361638\pi$
$38$	−1.12311	−0.182192
$39$	−4.00000	−0.640513
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	1.00000	0.154303
$43$	10.2462	1.56253	0.781266	−	0.624198i	$-0.214574\pi$
$43$	10.2462	1.56253	0.781266	+	0.624198i	$0.214574\pi$
$44$	−3.12311	−0.470826
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	−2.87689	−0.419638	−0.209819	−	0.977740i	$-0.567288\pi$
$47$	−2.87689	−0.419638	−0.209819	+	0.977740i	$0.567288\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	3.12311	0.437322
$52$	−4.00000	−0.554700
$53$	−5.12311	−0.703713	−0.351856	−	0.936054i	$-0.614449\pi$
$53$	−5.12311	−0.703713	−0.351856	+	0.936054i	$0.614449\pi$
$54$	−1.00000	−0.136083
$55$	−3.12311	−0.421119
$56$	1.00000	0.133631
$57$	1.12311	0.148759
$58$	−2.00000	−0.262613
$59$	7.12311	0.927349	0.463675	−	0.886006i	$-0.346531\pi$
$59$	7.12311	0.927349	0.463675	+	0.886006i	$0.346531\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	8.24621	1.04727
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	3.12311	0.384428
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.12311	0.378732
$69$	−1.00000	−0.120386
$70$	1.00000	0.119523
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−1.00000	−0.117851
$73$	−13.1231	−1.53594	−0.767972	−	0.640484i	$-0.778734\pi$
$73$	−13.1231	−1.53594	−0.767972	+	0.640484i	$0.778734\pi$
$74$	−5.12311	−0.595549
$75$	1.00000	0.115470
$76$	1.12311	0.128829
$77$	3.12311	0.355911
$78$	4.00000	0.452911
$79$	17.3693	1.95420	0.977100	−	0.212779i	$-0.0682513\pi$
$79$	17.3693	1.95420	0.977100	+	0.212779i	$0.0682513\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−9.12311	−1.00139	−0.500695	−	0.865624i	$-0.666922\pi$
$83$	−9.12311	−1.00139	−0.500695	+	0.865624i	$0.666922\pi$
$84$	−1.00000	−0.109109
$85$	3.12311	0.338748
$86$	−10.2462	−1.10488
$87$	2.00000	0.214423
$88$	3.12311	0.332924
$89$	−7.12311	−0.755048	−0.377524	−	0.926000i	$-0.623224\pi$
$89$	−7.12311	−0.755048	−0.377524	+	0.926000i	$0.623224\pi$
$90$	−1.00000	−0.105409
$91$	4.00000	0.419314
$92$	−1.00000	−0.104257
$93$	−8.24621	−0.855092
$94$	2.87689	0.296729
$95$	1.12311	0.115228
$96$	−1.00000	−0.102062
$97$	−15.1231	−1.53552	−0.767759	−	0.640738i	$-0.778628\pi$
$97$	−15.1231	−1.53552	−0.767759	+	0.640738i	$0.778628\pi$
$98$	−1.00000	−0.101015
$99$	−3.12311	−0.313884
$100$	1.00000	0.100000
$101$	−3.12311	−0.310761	−0.155380	−	0.987855i	$-0.549660\pi$
$101$	−3.12311	−0.310761	−0.155380	+	0.987855i	$0.549660\pi$
$102$	−3.12311	−0.309234
$103$	2.24621	0.221326	0.110663	−	0.993858i	$-0.464703\pi$
$103$	2.24621	0.221326	0.110663	+	0.993858i	$0.464703\pi$
$104$	4.00000	0.392232
$105$	−1.00000	−0.0975900
$106$	5.12311	0.497600
$107$	−14.2462	−1.37723	−0.688617	−	0.725126i	$-0.741782\pi$
$107$	−14.2462	−1.37723	−0.688617	+	0.725126i	$0.741782\pi$
$108$	1.00000	0.0962250
$109$	−16.2462	−1.55610	−0.778052	−	0.628199i	$-0.783792\pi$
$109$	−16.2462	−1.55610	−0.778052	+	0.628199i	$0.783792\pi$
$110$	3.12311	0.297776
$111$	5.12311	0.486264
$112$	−1.00000	−0.0944911
$113$	−15.3693	−1.44582	−0.722912	−	0.690940i	$-0.757197\pi$
$113$	−15.3693	−1.44582	−0.722912	+	0.690940i	$0.757197\pi$
$114$	−1.12311	−0.105188
$115$	−1.00000	−0.0932505
$116$	2.00000	0.185695
$117$	−4.00000	−0.369800
$118$	−7.12311	−0.655735
$119$	−3.12311	−0.286295
$120$	−1.00000	−0.0912871
$121$	−1.24621	−0.113292
$122$	6.00000	0.543214
$123$	−2.00000	−0.180334
$124$	−8.24621	−0.740532
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	5.75379	0.510566	0.255283	−	0.966866i	$-0.417831\pi$
$127$	5.75379	0.510566	0.255283	+	0.966866i	$0.417831\pi$
$128$	−1.00000	−0.0883883
$129$	10.2462	0.902129
$130$	4.00000	0.350823
$131$	−8.87689	−0.775578	−0.387789	−	0.921748i	$-0.626761\pi$
$131$	−8.87689	−0.775578	−0.387789	+	0.921748i	$0.626761\pi$
$132$	−3.12311	−0.271831
$133$	−1.12311	−0.0973856
$134$	4.00000	0.345547
$135$	1.00000	0.0860663
$136$	−3.12311	−0.267804
$137$	−6.87689	−0.587533	−0.293766	−	0.955877i	$-0.594909\pi$
$137$	−6.87689	−0.587533	−0.293766	+	0.955877i	$0.594909\pi$
$138$	1.00000	0.0851257
$139$	−3.12311	−0.264898	−0.132449	−	0.991190i	$-0.542284\pi$
$139$	−3.12311	−0.264898	−0.132449	+	0.991190i	$0.542284\pi$
$140$	−1.00000	−0.0845154
$141$	−2.87689	−0.242278
$142$	4.00000	0.335673
$143$	12.4924	1.04467
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	13.1231	1.08608
$147$	1.00000	0.0824786
$148$	5.12311	0.421117
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	−1.00000	−0.0816497
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	−1.12311	−0.0910959
$153$	3.12311	0.252488
$154$	−3.12311	−0.251667
$155$	−8.24621	−0.662352
$156$	−4.00000	−0.320256
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	−17.3693	−1.38183
$159$	−5.12311	−0.406289
$160$	−1.00000	−0.0790569
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	−24.4924	−1.91839	−0.959197	−	0.282738i	$-0.908757\pi$
$163$	−24.4924	−1.91839	−0.959197	+	0.282738i	$0.908757\pi$
$164$	−2.00000	−0.156174
$165$	−3.12311	−0.243133
$166$	9.12311	0.708090
$167$	−21.1231	−1.63455	−0.817277	−	0.576244i	$-0.804518\pi$
$167$	−21.1231	−1.63455	−0.817277	+	0.576244i	$0.804518\pi$
$168$	1.00000	0.0771517
$169$	3.00000	0.230769
$170$	−3.12311	−0.239531
$171$	1.12311	0.0858860
$172$	10.2462	0.781266
$173$	−8.00000	−0.608229	−0.304114	−	0.952636i	$-0.598361\pi$
$173$	−8.00000	−0.608229	−0.304114	+	0.952636i	$0.598361\pi$
$174$	−2.00000	−0.151620
$175$	−1.00000	−0.0755929
$176$	−3.12311	−0.235413
$177$	7.12311	0.535405
$178$	7.12311	0.533899
$179$	2.24621	0.167890	0.0839449	−	0.996470i	$-0.473248\pi$
$179$	2.24621	0.167890	0.0839449	+	0.996470i	$0.473248\pi$
$180$	1.00000	0.0745356
$181$	3.75379	0.279017	0.139508	−	0.990221i	$-0.455448\pi$
$181$	3.75379	0.279017	0.139508	+	0.990221i	$0.455448\pi$
$182$	−4.00000	−0.296500
$183$	−6.00000	−0.443533
$184$	1.00000	0.0737210
$185$	5.12311	0.376658
$186$	8.24621	0.604642
$187$	−9.75379	−0.713268
$188$	−2.87689	−0.209819
$189$	−1.00000	−0.0727393
$190$	−1.12311	−0.0814786
$191$	17.3693	1.25680	0.628400	−	0.777891i	$-0.283710\pi$
$191$	17.3693	1.25680	0.628400	+	0.777891i	$0.283710\pi$
$192$	1.00000	0.0721688
$193$	−6.49242	−0.467335	−0.233667	−	0.972317i	$-0.575073\pi$
$193$	−6.49242	−0.467335	−0.233667	+	0.972317i	$0.575073\pi$
$194$	15.1231	1.08578
$195$	−4.00000	−0.286446
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	3.12311	0.221949
$199$	10.2462	0.726335	0.363167	−	0.931724i	$-0.381695\pi$
$199$	10.2462	0.726335	0.363167	+	0.931724i	$0.381695\pi$
$200$	−1.00000	−0.0707107
$201$	−4.00000	−0.282138
$202$	3.12311	0.219741
$203$	−2.00000	−0.140372
$204$	3.12311	0.218661
$205$	−2.00000	−0.139686
$206$	−2.24621	−0.156501
$207$	−1.00000	−0.0695048
$208$	−4.00000	−0.277350
$209$	−3.50758	−0.242624
$210$	1.00000	0.0690066
$211$	10.2462	0.705378	0.352689	−	0.935741i	$-0.385267\pi$
$211$	10.2462	0.705378	0.352689	+	0.935741i	$0.385267\pi$
$212$	−5.12311	−0.351856
$213$	−4.00000	−0.274075
$214$	14.2462	0.973851
$215$	10.2462	0.698786
$216$	−1.00000	−0.0680414
$217$	8.24621	0.559789
$218$	16.2462	1.10033
$219$	−13.1231	−0.886777
$220$	−3.12311	−0.210560
$221$	−12.4924	−0.840331
$222$	−5.12311	−0.343840
$223$	25.6155	1.71534	0.857671	−	0.514198i	$-0.171910\pi$
$223$	25.6155	1.71534	0.857671	+	0.514198i	$0.171910\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	15.3693	1.02235
$227$	−19.3693	−1.28559	−0.642793	−	0.766040i	$-0.722225\pi$
$227$	−19.3693	−1.28559	−0.642793	+	0.766040i	$0.722225\pi$
$228$	1.12311	0.0743795
$229$	14.4924	0.957686	0.478843	−	0.877900i	$-0.341056\pi$
$229$	14.4924	0.957686	0.478843	+	0.877900i	$0.341056\pi$
$230$	1.00000	0.0659380
$231$	3.12311	0.205485
$232$	−2.00000	−0.131306
$233$	12.2462	0.802276	0.401138	−	0.916018i	$-0.368615\pi$
$233$	12.2462	0.802276	0.401138	+	0.916018i	$0.368615\pi$
$234$	4.00000	0.261488
$235$	−2.87689	−0.187668
$236$	7.12311	0.463675
$237$	17.3693	1.12826
$238$	3.12311	0.202441
$239$	−16.4924	−1.06681	−0.533403	−	0.845861i	$-0.679087\pi$
$239$	−16.4924	−1.06681	−0.533403	+	0.845861i	$0.679087\pi$
$240$	1.00000	0.0645497
$241$	20.8769	1.34480	0.672399	−	0.740188i	$-0.265264\pi$
$241$	20.8769	1.34480	0.672399	+	0.740188i	$0.265264\pi$
$242$	1.24621	0.0801095
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	1.00000	0.0638877
$246$	2.00000	0.127515
$247$	−4.49242	−0.285846
$248$	8.24621	0.523635
$249$	−9.12311	−0.578153
$250$	−1.00000	−0.0632456
$251$	5.12311	0.323368	0.161684	−	0.986843i	$-0.448308\pi$
$251$	5.12311	0.323368	0.161684	+	0.986843i	$0.448308\pi$
$252$	−1.00000	−0.0629941
$253$	3.12311	0.196348
$254$	−5.75379	−0.361025
$255$	3.12311	0.195576
$256$	1.00000	0.0625000
$257$	−17.1231	−1.06811	−0.534055	−	0.845450i	$-0.679332\pi$
$257$	−17.1231	−1.06811	−0.534055	+	0.845450i	$0.679332\pi$
$258$	−10.2462	−0.637901
$259$	−5.12311	−0.318334
$260$	−4.00000	−0.248069
$261$	2.00000	0.123797
$262$	8.87689	0.548416
$263$	6.24621	0.385158	0.192579	−	0.981281i	$-0.438315\pi$
$263$	6.24621	0.385158	0.192579	+	0.981281i	$0.438315\pi$
$264$	3.12311	0.192214
$265$	−5.12311	−0.314710
$266$	1.12311	0.0688620
$267$	−7.12311	−0.435927
$268$	−4.00000	−0.244339
$269$	−27.1231	−1.65372	−0.826862	−	0.562404i	$-0.809877\pi$
$269$	−27.1231	−1.65372	−0.826862	+	0.562404i	$0.809877\pi$
$270$	−1.00000	−0.0608581
$271$	−3.75379	−0.228026	−0.114013	−	0.993479i	$-0.536371\pi$
$271$	−3.75379	−0.228026	−0.114013	+	0.993479i	$0.536371\pi$
$272$	3.12311	0.189366
$273$	4.00000	0.242091
$274$	6.87689	0.415448
$275$	−3.12311	−0.188330
$276$	−1.00000	−0.0601929
$277$	20.2462	1.21648	0.608238	−	0.793754i	$-0.291876\pi$
$277$	20.2462	1.21648	0.608238	+	0.793754i	$0.291876\pi$
$278$	3.12311	0.187311
$279$	−8.24621	−0.493688
$280$	1.00000	0.0597614
$281$	2.49242	0.148685	0.0743427	−	0.997233i	$-0.476314\pi$
$281$	2.49242	0.148685	0.0743427	+	0.997233i	$0.476314\pi$
$282$	2.87689	0.171317
$283$	17.1231	1.01786	0.508931	−	0.860807i	$-0.330041\pi$
$283$	17.1231	1.01786	0.508931	+	0.860807i	$0.330041\pi$
$284$	−4.00000	−0.237356
$285$	1.12311	0.0665270
$286$	−12.4924	−0.738692
$287$	2.00000	0.118056
$288$	−1.00000	−0.0589256
$289$	−7.24621	−0.426248
$290$	−2.00000	−0.117444
$291$	−15.1231	−0.886532
$292$	−13.1231	−0.767972
$293$	18.4924	1.08034	0.540169	−	0.841556i	$-0.318360\pi$
$293$	18.4924	1.08034	0.540169	+	0.841556i	$0.318360\pi$
$294$	−1.00000	−0.0583212
$295$	7.12311	0.414723
$296$	−5.12311	−0.297774
$297$	−3.12311	−0.181221
$298$	14.0000	0.810998
$299$	4.00000	0.231326
$300$	1.00000	0.0577350
$301$	−10.2462	−0.590582
$302$	−12.0000	−0.690522
$303$	−3.12311	−0.179418
$304$	1.12311	0.0644145
$305$	−6.00000	−0.343559
$306$	−3.12311	−0.178536
$307$	−14.2462	−0.813074	−0.406537	−	0.913634i	$-0.633264\pi$
$307$	−14.2462	−0.813074	−0.406537	+	0.913634i	$0.633264\pi$
$308$	3.12311	0.177955
$309$	2.24621	0.127782
$310$	8.24621	0.468353
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	4.00000	0.226455
$313$	−3.12311	−0.176528	−0.0882642	−	0.996097i	$-0.528132\pi$
$313$	−3.12311	−0.176528	−0.0882642	+	0.996097i	$0.528132\pi$
$314$	−6.00000	−0.338600
$315$	−1.00000	−0.0563436
$316$	17.3693	0.977100
$317$	−14.4924	−0.813976	−0.406988	−	0.913434i	$-0.633421\pi$
$317$	−14.4924	−0.813976	−0.406988	+	0.913434i	$0.633421\pi$
$318$	5.12311	0.287289
$319$	−6.24621	−0.349721
$320$	1.00000	0.0559017
$321$	−14.2462	−0.795146
$322$	−1.00000	−0.0557278
$323$	3.50758	0.195167
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	24.4924	1.35651
$327$	−16.2462	−0.898418
$328$	2.00000	0.110432
$329$	2.87689	0.158608
$330$	3.12311	0.171921
$331$	−0.492423	−0.0270660	−0.0135330	−	0.999908i	$-0.504308\pi$
$331$	−0.492423	−0.0270660	−0.0135330	+	0.999908i	$0.504308\pi$
$332$	−9.12311	−0.500695
$333$	5.12311	0.280744
$334$	21.1231	1.15580
$335$	−4.00000	−0.218543
$336$	−1.00000	−0.0545545
$337$	−27.3693	−1.49090	−0.745451	−	0.666561i	$-0.767766\pi$
$337$	−27.3693	−1.49090	−0.745451	+	0.666561i	$0.767766\pi$
$338$	−3.00000	−0.163178
$339$	−15.3693	−0.834747
$340$	3.12311	0.169374
$341$	25.7538	1.39465
$342$	−1.12311	−0.0607306
$343$	−1.00000	−0.0539949
$344$	−10.2462	−0.552439
$345$	−1.00000	−0.0538382
$346$	8.00000	0.430083
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	2.00000	0.107211
$349$	−33.3693	−1.78622	−0.893109	−	0.449840i	$-0.851481\pi$
$349$	−33.3693	−1.78622	−0.893109	+	0.449840i	$0.851481\pi$
$350$	1.00000	0.0534522
$351$	−4.00000	−0.213504
$352$	3.12311	0.166462
$353$	−5.12311	−0.272675	−0.136338	−	0.990662i	$-0.543533\pi$
$353$	−5.12311	−0.272675	−0.136338	+	0.990662i	$0.543533\pi$
$354$	−7.12311	−0.378589
$355$	−4.00000	−0.212298
$356$	−7.12311	−0.377524
$357$	−3.12311	−0.165292
$358$	−2.24621	−0.118716
$359$	23.6155	1.24638	0.623190	−	0.782071i	$-0.285836\pi$
$359$	23.6155	1.24638	0.623190	+	0.782071i	$0.285836\pi$
$360$	−1.00000	−0.0527046
$361$	−17.7386	−0.933612
$362$	−3.75379	−0.197295
$363$	−1.24621	−0.0654091
$364$	4.00000	0.209657
$365$	−13.1231	−0.686895
$366$	6.00000	0.313625
$367$	2.24621	0.117251	0.0586256	−	0.998280i	$-0.481328\pi$
$367$	2.24621	0.117251	0.0586256	+	0.998280i	$0.481328\pi$
$368$	−1.00000	−0.0521286
$369$	−2.00000	−0.104116
$370$	−5.12311	−0.266338
$371$	5.12311	0.265978
$372$	−8.24621	−0.427546
$373$	0.630683	0.0326555	0.0163278	−	0.999867i	$-0.494802\pi$
$373$	0.630683	0.0326555	0.0163278	+	0.999867i	$0.494802\pi$
$374$	9.75379	0.504356
$375$	1.00000	0.0516398
$376$	2.87689	0.148364
$377$	−8.00000	−0.412021
$378$	1.00000	0.0514344
$379$	−15.6155	−0.802116	−0.401058	−	0.916053i	$-0.631357\pi$
$379$	−15.6155	−0.802116	−0.401058	+	0.916053i	$0.631357\pi$
$380$	1.12311	0.0576141
$381$	5.75379	0.294776
$382$	−17.3693	−0.888692
$383$	−12.4924	−0.638333	−0.319166	−	0.947699i	$-0.603403\pi$
$383$	−12.4924	−0.638333	−0.319166	+	0.947699i	$0.603403\pi$
$384$	−1.00000	−0.0510310
$385$	3.12311	0.159168
$386$	6.49242	0.330456
$387$	10.2462	0.520844
$388$	−15.1231	−0.767759
$389$	4.24621	0.215291	0.107646	−	0.994189i	$-0.465669\pi$
$389$	4.24621	0.215291	0.107646	+	0.994189i	$0.465669\pi$
$390$	4.00000	0.202548
$391$	−3.12311	−0.157942
$392$	−1.00000	−0.0505076
$393$	−8.87689	−0.447780
$394$	−6.00000	−0.302276
$395$	17.3693	0.873945
$396$	−3.12311	−0.156942
$397$	34.2462	1.71877	0.859384	−	0.511331i	$-0.170847\pi$
$397$	34.2462	1.71877	0.859384	+	0.511331i	$0.170847\pi$
$398$	−10.2462	−0.513596
$399$	−1.12311	−0.0562256
$400$	1.00000	0.0500000
$401$	16.7386	0.835887	0.417944	−	0.908473i	$-0.362751\pi$
$401$	16.7386	0.835887	0.417944	+	0.908473i	$0.362751\pi$
$402$	4.00000	0.199502
$403$	32.9848	1.64309
$404$	−3.12311	−0.155380
$405$	1.00000	0.0496904
$406$	2.00000	0.0992583
$407$	−16.0000	−0.793091
$408$	−3.12311	−0.154617
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	2.00000	0.0987730
$411$	−6.87689	−0.339212
$412$	2.24621	0.110663
$413$	−7.12311	−0.350505
$414$	1.00000	0.0491473
$415$	−9.12311	−0.447836
$416$	4.00000	0.196116
$417$	−3.12311	−0.152939
$418$	3.50758	0.171561
$419$	11.3693	0.555427	0.277714	−	0.960664i	$-0.410423\pi$
$419$	11.3693	0.555427	0.277714	+	0.960664i	$0.410423\pi$
$420$	−1.00000	−0.0487950
$421$	18.4924	0.901266	0.450633	−	0.892709i	$-0.351198\pi$
$421$	18.4924	0.901266	0.450633	+	0.892709i	$0.351198\pi$
$422$	−10.2462	−0.498778
$423$	−2.87689	−0.139879
$424$	5.12311	0.248800
$425$	3.12311	0.151493
$426$	4.00000	0.193801
$427$	6.00000	0.290360
$428$	−14.2462	−0.688617
$429$	12.4924	0.603140
$430$	−10.2462	−0.494116
$431$	−25.3693	−1.22200	−0.610998	−	0.791632i	$-0.709232\pi$
$431$	−25.3693	−1.22200	−0.610998	+	0.791632i	$0.709232\pi$
$432$	1.00000	0.0481125
$433$	21.3693	1.02694	0.513472	−	0.858106i	$-0.328359\pi$
$433$	21.3693	1.02694	0.513472	+	0.858106i	$0.328359\pi$
$434$	−8.24621	−0.395831
$435$	2.00000	0.0958927
$436$	−16.2462	−0.778052
$437$	−1.12311	−0.0537254
$438$	13.1231	0.627046
$439$	20.7386	0.989801	0.494900	−	0.868950i	$-0.335205\pi$
$439$	20.7386	0.989801	0.494900	+	0.868950i	$0.335205\pi$
$440$	3.12311	0.148888
$441$	1.00000	0.0476190
$442$	12.4924	0.594204
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	5.12311	0.243132
$445$	−7.12311	−0.337668
$446$	−25.6155	−1.21293
$447$	−14.0000	−0.662177
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−1.00000	−0.0471405
$451$	6.24621	0.294123
$452$	−15.3693	−0.722912
$453$	12.0000	0.563809
$454$	19.3693	0.909047
$455$	4.00000	0.187523
$456$	−1.12311	−0.0525942
$457$	9.61553	0.449795	0.224898	−	0.974382i	$-0.427795\pi$
$457$	9.61553	0.449795	0.224898	+	0.974382i	$0.427795\pi$
$458$	−14.4924	−0.677186
$459$	3.12311	0.145774
$460$	−1.00000	−0.0466252
$461$	−6.63068	−0.308822	−0.154411	−	0.988007i	$-0.549348\pi$
$461$	−6.63068	−0.308822	−0.154411	+	0.988007i	$0.549348\pi$
$462$	−3.12311	−0.145300
$463$	36.4924	1.69595	0.847973	−	0.530039i	$-0.177823\pi$
$463$	36.4924	1.69595	0.847973	+	0.530039i	$0.177823\pi$
$464$	2.00000	0.0928477
$465$	−8.24621	−0.382409
$466$	−12.2462	−0.567295
$467$	−23.8617	−1.10419	−0.552095	−	0.833781i	$-0.686171\pi$
$467$	−23.8617	−1.10419	−0.552095	+	0.833781i	$0.686171\pi$
$468$	−4.00000	−0.184900
$469$	4.00000	0.184703
$470$	2.87689	0.132701
$471$	6.00000	0.276465
$472$	−7.12311	−0.327868
$473$	−32.0000	−1.47136
$474$	−17.3693	−0.797799
$475$	1.12311	0.0515316
$476$	−3.12311	−0.143147
$477$	−5.12311	−0.234571
$478$	16.4924	0.754346
$479$	7.50758	0.343030	0.171515	−	0.985182i	$-0.445134\pi$
$479$	7.50758	0.343030	0.171515	+	0.985182i	$0.445134\pi$
$480$	−1.00000	−0.0456435
$481$	−20.4924	−0.934374
$482$	−20.8769	−0.950916
$483$	1.00000	0.0455016
$484$	−1.24621	−0.0566460
$485$	−15.1231	−0.686705
$486$	−1.00000	−0.0453609
$487$	36.4924	1.65363	0.826815	−	0.562474i	$-0.190150\pi$
$487$	36.4924	1.65363	0.826815	+	0.562474i	$0.190150\pi$
$488$	6.00000	0.271607
$489$	−24.4924	−1.10759
$490$	−1.00000	−0.0451754
$491$	18.2462	0.823440	0.411720	−	0.911310i	$-0.364928\pi$
$491$	18.2462	0.823440	0.411720	+	0.911310i	$0.364928\pi$
$492$	−2.00000	−0.0901670
$493$	6.24621	0.281315
$494$	4.49242	0.202124
$495$	−3.12311	−0.140373
$496$	−8.24621	−0.370266
$497$	4.00000	0.179425
$498$	9.12311	0.408816
$499$	38.7386	1.73418	0.867090	−	0.498152i	$-0.165988\pi$
$499$	38.7386	1.73418	0.867090	+	0.498152i	$0.165988\pi$
$500$	1.00000	0.0447214
$501$	−21.1231	−0.943711
$502$	−5.12311	−0.228655
$503$	−36.4924	−1.62712	−0.813558	−	0.581483i	$-0.802473\pi$
$503$	−36.4924	−1.62712	−0.813558	+	0.581483i	$0.802473\pi$
$504$	1.00000	0.0445435
$505$	−3.12311	−0.138976
$506$	−3.12311	−0.138839
$507$	3.00000	0.133235
$508$	5.75379	0.255283
$509$	19.1231	0.847617	0.423808	−	0.905752i	$-0.360693\pi$
$509$	19.1231	0.847617	0.423808	+	0.905752i	$0.360693\pi$
$510$	−3.12311	−0.138293
$511$	13.1231	0.580532
$512$	−1.00000	−0.0441942
$513$	1.12311	0.0495863
$514$	17.1231	0.755268
$515$	2.24621	0.0989799
$516$	10.2462	0.451064
$517$	8.98485	0.395153
$518$	5.12311	0.225096
$519$	−8.00000	−0.351161
$520$	4.00000	0.175412
$521$	41.8617	1.83400	0.916998	−	0.398892i	$-0.130605\pi$
$521$	41.8617	1.83400	0.916998	+	0.398892i	$0.130605\pi$
$522$	−2.00000	−0.0875376
$523$	−35.3693	−1.54659	−0.773296	−	0.634045i	$-0.781393\pi$
$523$	−35.3693	−1.54659	−0.773296	+	0.634045i	$0.781393\pi$
$524$	−8.87689	−0.387789
$525$	−1.00000	−0.0436436
$526$	−6.24621	−0.272348
$527$	−25.7538	−1.12185
$528$	−3.12311	−0.135916
$529$	1.00000	0.0434783
$530$	5.12311	0.222533
$531$	7.12311	0.309116
$532$	−1.12311	−0.0486928
$533$	8.00000	0.346518
$534$	7.12311	0.308247
$535$	−14.2462	−0.615917
$536$	4.00000	0.172774
$537$	2.24621	0.0969312
$538$	27.1231	1.16936
$539$	−3.12311	−0.134522
$540$	1.00000	0.0430331
$541$	37.2311	1.60069	0.800344	−	0.599541i	$-0.204650\pi$
$541$	37.2311	1.60069	0.800344	+	0.599541i	$0.204650\pi$
$542$	3.75379	0.161239
$543$	3.75379	0.161090
$544$	−3.12311	−0.133902
$545$	−16.2462	−0.695911
$546$	−4.00000	−0.171184
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−6.87689	−0.293766
$549$	−6.00000	−0.256074
$550$	3.12311	0.133170
$551$	2.24621	0.0956918
$552$	1.00000	0.0425628
$553$	−17.3693	−0.738618
$554$	−20.2462	−0.860179
$555$	5.12311	0.217464
$556$	−3.12311	−0.132449
$557$	−34.8769	−1.47778	−0.738891	−	0.673825i	$-0.764650\pi$
$557$	−34.8769	−1.47778	−0.738891	+	0.673825i	$0.764650\pi$
$558$	8.24621	0.349090
$559$	−40.9848	−1.73347
$560$	−1.00000	−0.0422577
$561$	−9.75379	−0.411805
$562$	−2.49242	−0.105136
$563$	−17.1231	−0.721653	−0.360826	−	0.932633i	$-0.617505\pi$
$563$	−17.1231	−0.721653	−0.360826	+	0.932633i	$0.617505\pi$
$564$	−2.87689	−0.121139
$565$	−15.3693	−0.646592
$566$	−17.1231	−0.719738
$567$	−1.00000	−0.0419961
$568$	4.00000	0.167836
$569$	8.73863	0.366343	0.183171	−	0.983081i	$-0.441364\pi$
$569$	8.73863	0.366343	0.183171	+	0.983081i	$0.441364\pi$
$570$	−1.12311	−0.0470417
$571$	−0.876894	−0.0366969	−0.0183484	−	0.999832i	$-0.505841\pi$
$571$	−0.876894	−0.0366969	−0.0183484	+	0.999832i	$0.505841\pi$
$572$	12.4924	0.522334
$573$	17.3693	0.725614
$574$	−2.00000	−0.0834784
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	−21.1231	−0.879366	−0.439683	−	0.898153i	$-0.644909\pi$
$577$	−21.1231	−0.879366	−0.439683	+	0.898153i	$0.644909\pi$
$578$	7.24621	0.301403
$579$	−6.49242	−0.269816
$580$	2.00000	0.0830455
$581$	9.12311	0.378490
$582$	15.1231	0.626873
$583$	16.0000	0.662652
$584$	13.1231	0.543038
$585$	−4.00000	−0.165380
$586$	−18.4924	−0.763915
$587$	−26.7386	−1.10362	−0.551811	−	0.833969i	$-0.686063\pi$
$587$	−26.7386	−1.10362	−0.551811	+	0.833969i	$0.686063\pi$
$588$	1.00000	0.0412393
$589$	−9.26137	−0.381608
$590$	−7.12311	−0.293254
$591$	6.00000	0.246807
$592$	5.12311	0.210558
$593$	−35.8617	−1.47267	−0.736333	−	0.676620i	$-0.763444\pi$
$593$	−35.8617	−1.47267	−0.736333	+	0.676620i	$0.763444\pi$
$594$	3.12311	0.128143
$595$	−3.12311	−0.128035
$596$	−14.0000	−0.573462
$597$	10.2462	0.419350
$598$	−4.00000	−0.163572
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	−1.00000	−0.0408248
$601$	−6.49242	−0.264831	−0.132416	−	0.991194i	$-0.542273\pi$
$601$	−6.49242	−0.264831	−0.132416	+	0.991194i	$0.542273\pi$
$602$	10.2462	0.417604
$603$	−4.00000	−0.162893
$604$	12.0000	0.488273
$605$	−1.24621	−0.0506657
$606$	3.12311	0.126867
$607$	−3.36932	−0.136756	−0.0683782	−	0.997659i	$-0.521782\pi$
$607$	−3.36932	−0.136756	−0.0683782	+	0.997659i	$0.521782\pi$
$608$	−1.12311	−0.0455479
$609$	−2.00000	−0.0810441
$610$	6.00000	0.242933
$611$	11.5076	0.465547
$612$	3.12311	0.126244
$613$	−47.8617	−1.93312	−0.966559	−	0.256445i	$-0.917449\pi$
$613$	−47.8617	−1.93312	−0.966559	+	0.256445i	$0.917449\pi$
$614$	14.2462	0.574930
$615$	−2.00000	−0.0806478
$616$	−3.12311	−0.125834
$617$	10.8769	0.437887	0.218944	−	0.975738i	$-0.429739\pi$
$617$	10.8769	0.437887	0.218944	+	0.975738i	$0.429739\pi$
$618$	−2.24621	−0.0903559
$619$	39.8617	1.60218	0.801089	−	0.598545i	$-0.204254\pi$
$619$	39.8617	1.60218	0.801089	+	0.598545i	$0.204254\pi$
$620$	−8.24621	−0.331176
$621$	−1.00000	−0.0401286
$622$	−26.0000	−1.04251
$623$	7.12311	0.285381
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	3.12311	0.124824
$627$	−3.50758	−0.140079
$628$	6.00000	0.239426
$629$	16.0000	0.637962
$630$	1.00000	0.0398410
$631$	−23.6155	−0.940119	−0.470060	−	0.882635i	$-0.655768\pi$
$631$	−23.6155	−0.940119	−0.470060	+	0.882635i	$0.655768\pi$
$632$	−17.3693	−0.690914
$633$	10.2462	0.407250
$634$	14.4924	0.575568
$635$	5.75379	0.228332
$636$	−5.12311	−0.203144
$637$	−4.00000	−0.158486
$638$	6.24621	0.247290
$639$	−4.00000	−0.158238
$640$	−1.00000	−0.0395285
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	14.2462	0.562253
$643$	11.3693	0.448362	0.224181	−	0.974548i	$-0.428029\pi$
$643$	11.3693	0.448362	0.224181	+	0.974548i	$0.428029\pi$
$644$	1.00000	0.0394055
$645$	10.2462	0.403444
$646$	−3.50758	−0.138004
$647$	−15.8617	−0.623589	−0.311795	−	0.950150i	$-0.600930\pi$
$647$	−15.8617	−0.623589	−0.311795	+	0.950150i	$0.600930\pi$
$648$	−1.00000	−0.0392837
$649$	−22.2462	−0.873240
$650$	4.00000	0.156893
$651$	8.24621	0.323195
$652$	−24.4924	−0.959197
$653$	36.7386	1.43769	0.718847	−	0.695168i	$-0.244670\pi$
$653$	36.7386	1.43769	0.718847	+	0.695168i	$0.244670\pi$
$654$	16.2462	0.635277
$655$	−8.87689	−0.346849
$656$	−2.00000	−0.0780869
$657$	−13.1231	−0.511981
$658$	−2.87689	−0.112153
$659$	−7.12311	−0.277477	−0.138738	−	0.990329i	$-0.544305\pi$
$659$	−7.12311	−0.277477	−0.138738	+	0.990329i	$0.544305\pi$
$660$	−3.12311	−0.121567
$661$	27.7538	1.07950	0.539749	−	0.841826i	$-0.318519\pi$
$661$	27.7538	1.07950	0.539749	+	0.841826i	$0.318519\pi$
$662$	0.492423	0.0191385
$663$	−12.4924	−0.485165
$664$	9.12311	0.354045
$665$	−1.12311	−0.0435522
$666$	−5.12311	−0.198516
$667$	−2.00000	−0.0774403
$668$	−21.1231	−0.817277
$669$	25.6155	0.990354
$670$	4.00000	0.154533
$671$	18.7386	0.723397
$672$	1.00000	0.0385758
$673$	−5.50758	−0.212302	−0.106151	−	0.994350i	$-0.533853\pi$
$673$	−5.50758	−0.212302	−0.106151	+	0.994350i	$0.533853\pi$
$674$	27.3693	1.05423
$675$	1.00000	0.0384900
$676$	3.00000	0.115385
$677$	−19.7538	−0.759200	−0.379600	−	0.925151i	$-0.623938\pi$
$677$	−19.7538	−0.759200	−0.379600	+	0.925151i	$0.623938\pi$
$678$	15.3693	0.590255
$679$	15.1231	0.580372
$680$	−3.12311	−0.119766
$681$	−19.3693	−0.742234
$682$	−25.7538	−0.986164
$683$	0.492423	0.0188420	0.00942101	−	0.999956i	$-0.497001\pi$
$683$	0.492423	0.0188420	0.00942101	+	0.999956i	$0.497001\pi$
$684$	1.12311	0.0429430
$685$	−6.87689	−0.262753
$686$	1.00000	0.0381802
$687$	14.4924	0.552920
$688$	10.2462	0.390633
$689$	20.4924	0.780699
$690$	1.00000	0.0380693
$691$	33.3693	1.26943	0.634714	−	0.772747i	$-0.281118\pi$
$691$	33.3693	1.26943	0.634714	+	0.772747i	$0.281118\pi$
$692$	−8.00000	−0.304114
$693$	3.12311	0.118637
$694$	12.0000	0.455514
$695$	−3.12311	−0.118466
$696$	−2.00000	−0.0758098
$697$	−6.24621	−0.236592
$698$	33.3693	1.26305
$699$	12.2462	0.463194
$700$	−1.00000	−0.0377964
$701$	−45.2311	−1.70835	−0.854177	−	0.519983i	$-0.825938\pi$
$701$	−45.2311	−1.70835	−0.854177	+	0.519983i	$0.825938\pi$
$702$	4.00000	0.150970
$703$	5.75379	0.217008
$704$	−3.12311	−0.117706
$705$	−2.87689	−0.108350
$706$	5.12311	0.192811
$707$	3.12311	0.117456
$708$	7.12311	0.267703
$709$	−37.2311	−1.39824	−0.699121	−	0.715004i	$-0.746425\pi$
$709$	−37.2311	−1.39824	−0.699121	+	0.715004i	$0.746425\pi$
$710$	4.00000	0.150117
$711$	17.3693	0.651400
$712$	7.12311	0.266950
$713$	8.24621	0.308823
$714$	3.12311	0.116879
$715$	12.4924	0.467190
$716$	2.24621	0.0839449
$717$	−16.4924	−0.615921
$718$	−23.6155	−0.881324
$719$	1.50758	0.0562232	0.0281116	−	0.999605i	$-0.491051\pi$
$719$	1.50758	0.0562232	0.0281116	+	0.999605i	$0.491051\pi$
$720$	1.00000	0.0372678
$721$	−2.24621	−0.0836533
$722$	17.7386	0.660164
$723$	20.8769	0.776420
$724$	3.75379	0.139508
$725$	2.00000	0.0742781
$726$	1.24621	0.0462512
$727$	−21.7538	−0.806803	−0.403402	−	0.915023i	$-0.632172\pi$
$727$	−21.7538	−0.806803	−0.403402	+	0.915023i	$0.632172\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	13.1231	0.485708
$731$	32.0000	1.18356
$732$	−6.00000	−0.221766
$733$	23.7538	0.877366	0.438683	−	0.898642i	$-0.355445\pi$
$733$	23.7538	0.877366	0.438683	+	0.898642i	$0.355445\pi$
$734$	−2.24621	−0.0829092
$735$	1.00000	0.0368856
$736$	1.00000	0.0368605
$737$	12.4924	0.460164
$738$	2.00000	0.0736210
$739$	−1.26137	−0.0464001	−0.0232001	−	0.999731i	$-0.507385\pi$
$739$	−1.26137	−0.0464001	−0.0232001	+	0.999731i	$0.507385\pi$
$740$	5.12311	0.188329
$741$	−4.49242	−0.165033
$742$	−5.12311	−0.188075
$743$	11.5076	0.422172	0.211086	−	0.977467i	$-0.432300\pi$
$743$	11.5076	0.422172	0.211086	+	0.977467i	$0.432300\pi$
$744$	8.24621	0.302321
$745$	−14.0000	−0.512920
$746$	−0.630683	−0.0230909
$747$	−9.12311	−0.333797
$748$	−9.75379	−0.356634
$749$	14.2462	0.520545
$750$	−1.00000	−0.0365148
$751$	0.384472	0.0140296	0.00701479	−	0.999975i	$-0.497767\pi$
$751$	0.384472	0.0140296	0.00701479	+	0.999975i	$0.497767\pi$
$752$	−2.87689	−0.104910
$753$	5.12311	0.186696
$754$	8.00000	0.291343
$755$	12.0000	0.436725
$756$	−1.00000	−0.0363696
$757$	−31.8617	−1.15803	−0.579017	−	0.815315i	$-0.696564\pi$
$757$	−31.8617	−1.15803	−0.579017	+	0.815315i	$0.696564\pi$
$758$	15.6155	0.567182
$759$	3.12311	0.113362
$760$	−1.12311	−0.0407393
$761$	−1.50758	−0.0546496	−0.0273248	−	0.999627i	$-0.508699\pi$
$761$	−1.50758	−0.0546496	−0.0273248	+	0.999627i	$0.508699\pi$
$762$	−5.75379	−0.208438
$763$	16.2462	0.588152
$764$	17.3693	0.628400
$765$	3.12311	0.112916
$766$	12.4924	0.451370
$767$	−28.4924	−1.02880
$768$	1.00000	0.0360844
$769$	−15.6155	−0.563110	−0.281555	−	0.959545i	$-0.590850\pi$
$769$	−15.6155	−0.563110	−0.281555	+	0.959545i	$0.590850\pi$
$770$	−3.12311	−0.112549
$771$	−17.1231	−0.616674
$772$	−6.49242	−0.233667
$773$	−10.4924	−0.377386	−0.188693	−	0.982036i	$-0.560425\pi$
$773$	−10.4924	−0.377386	−0.188693	+	0.982036i	$0.560425\pi$
$774$	−10.2462	−0.368292
$775$	−8.24621	−0.296213
$776$	15.1231	0.542888
$777$	−5.12311	−0.183790
$778$	−4.24621	−0.152234
$779$	−2.24621	−0.0804789
$780$	−4.00000	−0.143223
$781$	12.4924	0.447014
$782$	3.12311	0.111682
$783$	2.00000	0.0714742
$784$	1.00000	0.0357143
$785$	6.00000	0.214149
$786$	8.87689	0.316628
$787$	11.3693	0.405272	0.202636	−	0.979254i	$-0.435049\pi$
$787$	11.3693	0.405272	0.202636	+	0.979254i	$0.435049\pi$
$788$	6.00000	0.213741
$789$	6.24621	0.222371
$790$	−17.3693	−0.617973
$791$	15.3693	0.546470
$792$	3.12311	0.110975
$793$	24.0000	0.852265
$794$	−34.2462	−1.21535
$795$	−5.12311	−0.181698
$796$	10.2462	0.363167
$797$	−15.7538	−0.558028	−0.279014	−	0.960287i	$-0.590008\pi$
$797$	−15.7538	−0.558028	−0.279014	+	0.960287i	$0.590008\pi$
$798$	1.12311	0.0397575
$799$	−8.98485	−0.317861
$800$	−1.00000	−0.0353553
$801$	−7.12311	−0.251683
$802$	−16.7386	−0.591062
$803$	40.9848	1.44632
$804$	−4.00000	−0.141069
$805$	1.00000	0.0352454
$806$	−32.9848	−1.16184
$807$	−27.1231	−0.954779
$808$	3.12311	0.109870
$809$	−20.7386	−0.729132	−0.364566	−	0.931178i	$-0.618783\pi$
$809$	−20.7386	−0.729132	−0.364566	+	0.931178i	$0.618783\pi$
$810$	−1.00000	−0.0351364
$811$	−16.8769	−0.592628	−0.296314	−	0.955091i	$-0.595757\pi$
$811$	−16.8769	−0.592628	−0.296314	+	0.955091i	$0.595757\pi$
$812$	−2.00000	−0.0701862
$813$	−3.75379	−0.131651
$814$	16.0000	0.560800
$815$	−24.4924	−0.857932
$816$	3.12311	0.109331
$817$	11.5076	0.402599
$818$	22.0000	0.769212
$819$	4.00000	0.139771
$820$	−2.00000	−0.0698430
$821$	17.5076	0.611019	0.305509	−	0.952189i	$-0.401173\pi$
$821$	17.5076	0.611019	0.305509	+	0.952189i	$0.401173\pi$
$822$	6.87689	0.239859
$823$	−22.7386	−0.792619	−0.396309	−	0.918117i	$-0.629709\pi$
$823$	−22.7386	−0.792619	−0.396309	+	0.918117i	$0.629709\pi$
$824$	−2.24621	−0.0782505
$825$	−3.12311	−0.108733
$826$	7.12311	0.247845
$827$	2.73863	0.0952316	0.0476158	−	0.998866i	$-0.484838\pi$
$827$	2.73863	0.0952316	0.0476158	+	0.998866i	$0.484838\pi$
$828$	−1.00000	−0.0347524
$829$	15.1231	0.525247	0.262624	−	0.964898i	$-0.415412\pi$
$829$	15.1231	0.525247	0.262624	+	0.964898i	$0.415412\pi$
$830$	9.12311	0.316668
$831$	20.2462	0.702333
$832$	−4.00000	−0.138675
$833$	3.12311	0.108209
$834$	3.12311	0.108144
$835$	−21.1231	−0.730995
$836$	−3.50758	−0.121312
$837$	−8.24621	−0.285031
$838$	−11.3693	−0.392747
$839$	−47.2311	−1.63060	−0.815299	−	0.579041i	$-0.803427\pi$
$839$	−47.2311	−1.63060	−0.815299	+	0.579041i	$0.803427\pi$
$840$	1.00000	0.0345033
$841$	−25.0000	−0.862069
$842$	−18.4924	−0.637291
$843$	2.49242	0.0858436
$844$	10.2462	0.352689
$845$	3.00000	0.103203
$846$	2.87689	0.0989097
$847$	1.24621	0.0428203
$848$	−5.12311	−0.175928
$849$	17.1231	0.587663
$850$	−3.12311	−0.107122
$851$	−5.12311	−0.175618
$852$	−4.00000	−0.137038
$853$	2.24621	0.0769088	0.0384544	−	0.999260i	$-0.487757\pi$
$853$	2.24621	0.0769088	0.0384544	+	0.999260i	$0.487757\pi$
$854$	−6.00000	−0.205316
$855$	1.12311	0.0384094
$856$	14.2462	0.486925
$857$	35.8617	1.22501	0.612507	−	0.790465i	$-0.290161\pi$
$857$	35.8617	1.22501	0.612507	+	0.790465i	$0.290161\pi$
$858$	−12.4924	−0.426484
$859$	33.3693	1.13855	0.569273	−	0.822148i	$-0.307225\pi$
$859$	33.3693	1.13855	0.569273	+	0.822148i	$0.307225\pi$
$860$	10.2462	0.349393
$861$	2.00000	0.0681598
$862$	25.3693	0.864082
$863$	−26.2462	−0.893431	−0.446716	−	0.894676i	$-0.647406\pi$
$863$	−26.2462	−0.893431	−0.446716	+	0.894676i	$0.647406\pi$
$864$	−1.00000	−0.0340207
$865$	−8.00000	−0.272008
$866$	−21.3693	−0.726159
$867$	−7.24621	−0.246094
$868$	8.24621	0.279895
$869$	−54.2462	−1.84018
$870$	−2.00000	−0.0678064
$871$	16.0000	0.542139
$872$	16.2462	0.550166
$873$	−15.1231	−0.511840
$874$	1.12311	0.0379896
$875$	−1.00000	−0.0338062
$876$	−13.1231	−0.443389
$877$	27.7538	0.937179	0.468589	−	0.883416i	$-0.344762\pi$
$877$	27.7538	0.937179	0.468589	+	0.883416i	$0.344762\pi$
$878$	−20.7386	−0.699895
$879$	18.4924	0.623734
$880$	−3.12311	−0.105280
$881$	−43.1231	−1.45285	−0.726427	−	0.687243i	$-0.758821\pi$
$881$	−43.1231	−1.45285	−0.726427	+	0.687243i	$0.758821\pi$
$882$	−1.00000	−0.0336718
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	−12.4924	−0.420166
$885$	7.12311	0.239441
$886$	−4.00000	−0.134383
$887$	−6.87689	−0.230904	−0.115452	−	0.993313i	$-0.536832\pi$
$887$	−6.87689	−0.230904	−0.115452	+	0.993313i	$0.536832\pi$
$888$	−5.12311	−0.171920
$889$	−5.75379	−0.192976
$890$	7.12311	0.238767
$891$	−3.12311	−0.104628
$892$	25.6155	0.857671
$893$	−3.23106	−0.108123
$894$	14.0000	0.468230
$895$	2.24621	0.0750826
$896$	1.00000	0.0334077
$897$	4.00000	0.133556
$898$	18.0000	0.600668
$899$	−16.4924	−0.550053
$900$	1.00000	0.0333333
$901$	−16.0000	−0.533037
$902$	−6.24621	−0.207976
$903$	−10.2462	−0.340973
$904$	15.3693	0.511176
$905$	3.75379	0.124780
$906$	−12.0000	−0.398673
$907$	52.4924	1.74298	0.871491	−	0.490411i	$-0.163153\pi$
$907$	52.4924	1.74298	0.871491	+	0.490411i	$0.163153\pi$
$908$	−19.3693	−0.642793
$909$	−3.12311	−0.103587
$910$	−4.00000	−0.132599
$911$	−5.86174	−0.194208	−0.0971040	−	0.995274i	$-0.530958\pi$
$911$	−5.86174	−0.194208	−0.0971040	+	0.995274i	$0.530958\pi$
$912$	1.12311	0.0371897
$913$	28.4924	0.942962
$914$	−9.61553	−0.318053
$915$	−6.00000	−0.198354
$916$	14.4924	0.478843
$917$	8.87689	0.293141
$918$	−3.12311	−0.103078
$919$	−37.8617	−1.24894	−0.624472	−	0.781047i	$-0.714686\pi$
$919$	−37.8617	−1.24894	−0.624472	+	0.781047i	$0.714686\pi$
$920$	1.00000	0.0329690
$921$	−14.2462	−0.469429
$922$	6.63068	0.218370
$923$	16.0000	0.526646
$924$	3.12311	0.102743
$925$	5.12311	0.168447
$926$	−36.4924	−1.19922
$927$	2.24621	0.0737753
$928$	−2.00000	−0.0656532
$929$	52.2462	1.71414	0.857071	−	0.515198i	$-0.172282\pi$
$929$	52.2462	1.71414	0.857071	+	0.515198i	$0.172282\pi$
$930$	8.24621	0.270404
$931$	1.12311	0.0368083
$932$	12.2462	0.401138
$933$	26.0000	0.851202
$934$	23.8617	0.780780
$935$	−9.75379	−0.318983
$936$	4.00000	0.130744
$937$	−44.8769	−1.46606	−0.733032	−	0.680194i	$-0.761896\pi$
$937$	−44.8769	−1.46606	−0.733032	+	0.680194i	$0.761896\pi$
$938$	−4.00000	−0.130605
$939$	−3.12311	−0.101919
$940$	−2.87689	−0.0938339
$941$	16.7386	0.545664	0.272832	−	0.962062i	$-0.412040\pi$
$941$	16.7386	0.545664	0.272832	+	0.962062i	$0.412040\pi$
$942$	−6.00000	−0.195491
$943$	2.00000	0.0651290
$944$	7.12311	0.231837
$945$	−1.00000	−0.0325300
$946$	32.0000	1.04041
$947$	−36.9848	−1.20185	−0.600923	−	0.799307i	$-0.705200\pi$
$947$	−36.9848	−1.20185	−0.600923	+	0.799307i	$0.705200\pi$
$948$	17.3693	0.564129
$949$	52.4924	1.70398
$950$	−1.12311	−0.0364384
$951$	−14.4924	−0.469949
$952$	3.12311	0.101220
$953$	19.3693	0.627434	0.313717	−	0.949517i	$-0.398426\pi$
$953$	19.3693	0.627434	0.313717	+	0.949517i	$0.398426\pi$
$954$	5.12311	0.165867
$955$	17.3693	0.562058
$956$	−16.4924	−0.533403
$957$	−6.24621	−0.201911
$958$	−7.50758	−0.242559
$959$	6.87689	0.222067
$960$	1.00000	0.0322749
$961$	37.0000	1.19355
$962$	20.4924	0.660702
$963$	−14.2462	−0.459078
$964$	20.8769	0.672399
$965$	−6.49242	−0.208998
$966$	−1.00000	−0.0321745
$967$	52.4924	1.68804	0.844021	−	0.536310i	$-0.180182\pi$
$967$	52.4924	1.68804	0.844021	+	0.536310i	$0.180182\pi$
$968$	1.24621	0.0400547
$969$	3.50758	0.112680
$970$	15.1231	0.485574
$971$	45.1231	1.44807	0.724035	−	0.689764i	$-0.242286\pi$
$971$	45.1231	1.44807	0.724035	+	0.689764i	$0.242286\pi$
$972$	1.00000	0.0320750
$973$	3.12311	0.100122
$974$	−36.4924	−1.16929
$975$	−4.00000	−0.128103
$976$	−6.00000	−0.192055
$977$	27.8617	0.891376	0.445688	−	0.895188i	$-0.352959\pi$
$977$	27.8617	0.891376	0.445688	+	0.895188i	$0.352959\pi$
$978$	24.4924	0.783181
$979$	22.2462	0.710992
$980$	1.00000	0.0319438
$981$	−16.2462	−0.518702
$982$	−18.2462	−0.582260
$983$	28.4924	0.908767	0.454384	−	0.890806i	$-0.349860\pi$
$983$	28.4924	0.908767	0.454384	+	0.890806i	$0.349860\pi$
$984$	2.00000	0.0637577
$985$	6.00000	0.191176
$986$	−6.24621	−0.198920
$987$	2.87689	0.0915726
$988$	−4.49242	−0.142923
$989$	−10.2462	−0.325811
$990$	3.12311	0.0992588
$991$	0.492423	0.0156423	0.00782116	−	0.999969i	$-0.497510\pi$
$991$	0.492423	0.0156423	0.00782116	+	0.999969i	$0.497510\pi$
$992$	8.24621	0.261817
$993$	−0.492423	−0.0156266
$994$	−4.00000	−0.126872
$995$	10.2462	0.324827
$996$	−9.12311	−0.289077
$997$	16.4924	0.522320	0.261160	−	0.965295i	$-0.415895\pi$
$997$	16.4924	0.522320	0.261160	+	0.965295i	$0.415895\pi$
$998$	−38.7386	−1.22625
$999$	5.12311	0.162088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4830.2.a.br.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4830.2.a.br.1.1	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4830.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.12311 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +3.12311 q^{17} -1.00000 q^{18} +1.12311 q^{19} +1.00000 q^{20} -1.00000 q^{21} +3.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} -8.24621 q^{31} -1.00000 q^{32} -3.12311 q^{33} -3.12311 q^{34} -1.00000 q^{35} +1.00000 q^{36} +5.12311 q^{37} -1.12311 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} +10.2462 q^{43} -3.12311 q^{44} +1.00000 q^{45} +1.00000 q^{46} -2.87689 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +3.12311 q^{51} -4.00000 q^{52} -5.12311 q^{53} -1.00000 q^{54} -3.12311 q^{55} +1.00000 q^{56} +1.12311 q^{57} -2.00000 q^{58} +7.12311 q^{59} +1.00000 q^{60} -6.00000 q^{61} +8.24621 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +3.12311 q^{66} -4.00000 q^{67} +3.12311 q^{68} -1.00000 q^{69} +1.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} -13.1231 q^{73} -5.12311 q^{74} +1.00000 q^{75} +1.12311 q^{76} +3.12311 q^{77} +4.00000 q^{78} +17.3693 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -9.12311 q^{83} -1.00000 q^{84} +3.12311 q^{85} -10.2462 q^{86} +2.00000 q^{87} +3.12311 q^{88} -7.12311 q^{89} -1.00000 q^{90} +4.00000 q^{91} -1.00000 q^{92} -8.24621 q^{93} +2.87689 q^{94} +1.12311 q^{95} -1.00000 q^{96} -15.1231 q^{97} -1.00000 q^{98} -3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 6 q^{19} + 2 q^{20} - 2 q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} + 2 q^{25} + 8 q^{26} + 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} - 2 q^{32} + 2 q^{33} + 2 q^{34} - 2 q^{35} + 2 q^{36} + 2 q^{37} + 6 q^{38} - 8 q^{39} - 2 q^{40} - 4 q^{41} + 2 q^{42} + 4 q^{43} + 2 q^{44} + 2 q^{45} + 2 q^{46} - 14 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 2 q^{51} - 8 q^{52} - 2 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{56} - 6 q^{57} - 4 q^{58} + 6 q^{59} + 2 q^{60} - 12 q^{61} - 2 q^{63} + 2 q^{64} - 8 q^{65} - 2 q^{66} - 8 q^{67} - 2 q^{68} - 2 q^{69} + 2 q^{70} - 8 q^{71} - 2 q^{72} - 18 q^{73} - 2 q^{74} + 2 q^{75} - 6 q^{76} - 2 q^{77} + 8 q^{78} + 10 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 10 q^{83} - 2 q^{84} - 2 q^{85} - 4 q^{86} + 4 q^{87} - 2 q^{88} - 6 q^{89} - 2 q^{90} + 8 q^{91} - 2 q^{92} + 14 q^{94} - 6 q^{95} - 2 q^{96} - 22 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 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 - 8 * q^13 + 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 6 * q^19 + 2 * q^20 - 2 * q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 + 2 * q^25 + 8 * q^26 + 2 * q^27 - 2 * q^28 + 4 * q^29 - 2 * q^30 - 2 * q^32 + 2 * q^33 + 2 * q^34 - 2 * q^35 + 2 * q^36 + 2 * q^37 + 6 * q^38 - 8 * q^39 - 2 * q^40 - 4 * q^41 + 2 * q^42 + 4 * q^43 + 2 * q^44 + 2 * q^45 + 2 * q^46 - 14 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 - 2 * q^51 - 8 * q^52 - 2 * q^53 - 2 * q^54 + 2 * q^55 + 2 * q^56 - 6 * q^57 - 4 * q^58 + 6 * q^59 + 2 * q^60 - 12 * q^61 - 2 * q^63 + 2 * q^64 - 8 * q^65 - 2 * q^66 - 8 * q^67 - 2 * q^68 - 2 * q^69 + 2 * q^70 - 8 * q^71 - 2 * q^72 - 18 * q^73 - 2 * q^74 + 2 * q^75 - 6 * q^76 - 2 * q^77 + 8 * q^78 + 10 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 - 10 * q^83 - 2 * q^84 - 2 * q^85 - 4 * q^86 + 4 * q^87 - 2 * q^88 - 6 * q^89 - 2 * q^90 + 8 * q^91 - 2 * q^92 + 14 * q^94 - 6 * q^95 - 2 * q^96 - 22 * q^97 - 2 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more