LMFDB - Embedded newform 6422.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.18479 q^{3} +1.00000 q^{4} +4.29086 q^{5} -2.18479 q^{6} +2.69459 q^{7} +1.00000 q^{8} +1.77332 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.18479 q^{3} +1.00000 q^{4} +4.29086 q^{5} -2.18479 q^{6} +2.69459 q^{7} +1.00000 q^{8} +1.77332 q^{9} +4.29086 q^{10} +3.71688 q^{11} -2.18479 q^{12} +2.69459 q^{14} -9.37464 q^{15} +1.00000 q^{16} -0.184793 q^{17} +1.77332 q^{18} -1.00000 q^{19} +4.29086 q^{20} -5.88713 q^{21} +3.71688 q^{22} -6.17024 q^{23} -2.18479 q^{24} +13.4115 q^{25} +2.68004 q^{27} +2.69459 q^{28} -1.34730 q^{29} -9.37464 q^{30} +1.92127 q^{31} +1.00000 q^{32} -8.12061 q^{33} -0.184793 q^{34} +11.5621 q^{35} +1.77332 q^{36} +7.75877 q^{37} -1.00000 q^{38} +4.29086 q^{40} -11.2490 q^{41} -5.88713 q^{42} +6.82295 q^{43} +3.71688 q^{44} +7.60906 q^{45} -6.17024 q^{46} +1.17705 q^{47} -2.18479 q^{48} +0.260830 q^{49} +13.4115 q^{50} +0.403733 q^{51} -5.59627 q^{53} +2.68004 q^{54} +15.9486 q^{55} +2.69459 q^{56} +2.18479 q^{57} -1.34730 q^{58} +2.45336 q^{59} -9.37464 q^{60} +7.30541 q^{61} +1.92127 q^{62} +4.77837 q^{63} +1.00000 q^{64} -8.12061 q^{66} +15.0351 q^{67} -0.184793 q^{68} +13.4807 q^{69} +11.5621 q^{70} +7.31046 q^{71} +1.77332 q^{72} -4.04458 q^{73} +7.75877 q^{74} -29.3013 q^{75} -1.00000 q^{76} +10.0155 q^{77} +6.12836 q^{79} +4.29086 q^{80} -11.1753 q^{81} -11.2490 q^{82} +7.49020 q^{83} -5.88713 q^{84} -0.792919 q^{85} +6.82295 q^{86} +2.94356 q^{87} +3.71688 q^{88} -13.1898 q^{89} +7.60906 q^{90} -6.17024 q^{92} -4.19759 q^{93} +1.17705 q^{94} -4.29086 q^{95} -2.18479 q^{96} -7.23442 q^{97} +0.260830 q^{98} +6.59121 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 12 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 12 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 12 q^{9} - 3 q^{10} + 3 q^{11} - 3 q^{12} + 6 q^{14} - 6 q^{15} + 3 q^{16} + 3 q^{17} + 12 q^{18} - 3 q^{19} - 3 q^{20} + 12 q^{21} + 3 q^{22} + 3 q^{23} - 3 q^{24} + 30 q^{25} - 12 q^{27} + 6 q^{28} - 3 q^{29} - 6 q^{30} - 3 q^{31} + 3 q^{32} - 30 q^{33} + 3 q^{34} + 12 q^{36} + 12 q^{37} - 3 q^{38} - 3 q^{40} - 21 q^{41} + 12 q^{42} + 3 q^{44} - 30 q^{45} + 3 q^{46} + 24 q^{47} - 3 q^{48} + 15 q^{49} + 30 q^{50} + 15 q^{51} - 3 q^{53} - 12 q^{54} + 18 q^{55} + 6 q^{56} + 3 q^{57} - 3 q^{58} - 6 q^{59} - 6 q^{60} + 24 q^{61} - 3 q^{62} + 6 q^{63} + 3 q^{64} - 30 q^{66} + 3 q^{68} + 6 q^{69} + 9 q^{71} + 12 q^{72} + 12 q^{74} - 39 q^{75} - 3 q^{76} - 18 q^{77} - 3 q^{80} + 3 q^{81} - 21 q^{82} + 21 q^{83} + 12 q^{84} - 12 q^{85} - 6 q^{87} + 3 q^{88} - 21 q^{89} - 30 q^{90} + 3 q^{92} + 30 q^{93} + 24 q^{94} + 3 q^{95} - 3 q^{96} + 9 q^{97} + 15 q^{98} + 21 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 12 * q^9 - 3 * q^10 + 3 * q^11 - 3 * q^12 + 6 * q^14 - 6 * q^15 + 3 * q^16 + 3 * q^17 + 12 * q^18 - 3 * q^19 - 3 * q^20 + 12 * q^21 + 3 * q^22 + 3 * q^23 - 3 * q^24 + 30 * q^25 - 12 * q^27 + 6 * q^28 - 3 * q^29 - 6 * q^30 - 3 * q^31 + 3 * q^32 - 30 * q^33 + 3 * q^34 + 12 * q^36 + 12 * q^37 - 3 * q^38 - 3 * q^40 - 21 * q^41 + 12 * q^42 + 3 * q^44 - 30 * q^45 + 3 * q^46 + 24 * q^47 - 3 * q^48 + 15 * q^49 + 30 * q^50 + 15 * q^51 - 3 * q^53 - 12 * q^54 + 18 * q^55 + 6 * q^56 + 3 * q^57 - 3 * q^58 - 6 * q^59 - 6 * q^60 + 24 * q^61 - 3 * q^62 + 6 * q^63 + 3 * q^64 - 30 * q^66 + 3 * q^68 + 6 * q^69 + 9 * q^71 + 12 * q^72 + 12 * q^74 - 39 * q^75 - 3 * q^76 - 18 * q^77 - 3 * q^80 + 3 * q^81 - 21 * q^82 + 21 * q^83 + 12 * q^84 - 12 * q^85 - 6 * q^87 + 3 * q^88 - 21 * q^89 - 30 * q^90 + 3 * q^92 + 30 * q^93 + 24 * q^94 + 3 * q^95 - 3 * q^96 + 9 * q^97 + 15 * q^98 + 21 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.18479	−1.26139	−0.630695	−	0.776031i	$-0.717230\pi$
$3$	−2.18479	−1.26139	−0.630695	+	0.776031i	$0.717230\pi$
$4$	1.00000	0.500000
$5$	4.29086	1.91893	0.959465	−	0.281827i	$-0.0909403\pi$
$5$	4.29086	1.91893	0.959465	+	0.281827i	$0.0909403\pi$
$6$	−2.18479	−0.891938
$7$	2.69459	1.01846	0.509230	−	0.860630i	$-0.329930\pi$
$7$	2.69459	1.01846	0.509230	+	0.860630i	$0.329930\pi$
$8$	1.00000	0.353553
$9$	1.77332	0.591106
$10$	4.29086	1.35689
$11$	3.71688	1.12068	0.560341	−	0.828262i	$-0.310670\pi$
$11$	3.71688	1.12068	0.560341	+	0.828262i	$0.310670\pi$
$12$	−2.18479	−0.630695
$13$	0	0
$13$	0	0
$14$	2.69459	0.720160
$15$	−9.37464	−2.42052
$16$	1.00000	0.250000
$17$	−0.184793	−0.0448188	−0.0224094	−	0.999749i	$-0.507134\pi$
$17$	−0.184793	−0.0448188	−0.0224094	+	0.999749i	$0.507134\pi$
$18$	1.77332	0.417975
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.29086	0.959465
$21$	−5.88713	−1.28468
$22$	3.71688	0.792442
$23$	−6.17024	−1.28658	−0.643292	−	0.765621i	$-0.722432\pi$
$23$	−6.17024	−1.28658	−0.643292	+	0.765621i	$0.722432\pi$
$24$	−2.18479	−0.445969
$25$	13.4115	2.68229
$26$	0	0
$27$	2.68004	0.515775
$28$	2.69459	0.509230
$29$	−1.34730	−0.250187	−0.125093	−	0.992145i	$-0.539923\pi$
$29$	−1.34730	−0.250187	−0.125093	+	0.992145i	$0.539923\pi$
$30$	−9.37464	−1.71157
$31$	1.92127	0.345071	0.172536	−	0.985003i	$-0.444804\pi$
$31$	1.92127	0.345071	0.172536	+	0.985003i	$0.444804\pi$
$32$	1.00000	0.176777
$33$	−8.12061	−1.41362
$34$	−0.184793	−0.0316917
$35$	11.5621	1.95435
$36$	1.77332	0.295553
$37$	7.75877	1.27553	0.637767	−	0.770229i	$-0.279858\pi$
$37$	7.75877	1.27553	0.637767	+	0.770229i	$0.279858\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	4.29086	0.678444
$41$	−11.2490	−1.75679	−0.878397	−	0.477932i	$-0.841387\pi$
$41$	−11.2490	−1.75679	−0.878397	+	0.477932i	$0.841387\pi$
$42$	−5.88713	−0.908403
$43$	6.82295	1.04049	0.520245	−	0.854017i	$-0.325841\pi$
$43$	6.82295	1.04049	0.520245	+	0.854017i	$0.325841\pi$
$44$	3.71688	0.560341
$45$	7.60906	1.13429
$46$	−6.17024	−0.909753
$47$	1.17705	0.171691	0.0858453	−	0.996308i	$-0.472641\pi$
$47$	1.17705	0.171691	0.0858453	+	0.996308i	$0.472641\pi$
$48$	−2.18479	−0.315348
$49$	0.260830	0.0372614
$50$	13.4115	1.89667
$51$	0.403733	0.0565340
$52$	0	0
$53$	−5.59627	−0.768706	−0.384353	−	0.923186i	$-0.625576\pi$
$53$	−5.59627	−0.768706	−0.384353	+	0.923186i	$0.625576\pi$
$54$	2.68004	0.364708
$55$	15.9486	2.15051
$56$	2.69459	0.360080
$57$	2.18479	0.289383
$58$	−1.34730	−0.176909
$59$	2.45336	0.319401	0.159700	−	0.987166i	$-0.448947\pi$
$59$	2.45336	0.319401	0.159700	+	0.987166i	$0.448947\pi$
$60$	−9.37464	−1.21026
$61$	7.30541	0.935362	0.467681	−	0.883897i	$-0.345090\pi$
$61$	7.30541	0.935362	0.467681	+	0.883897i	$0.345090\pi$
$62$	1.92127	0.244002
$63$	4.77837	0.602018
$64$	1.00000	0.125000
$65$	0	0
$66$	−8.12061	−0.999579
$67$	15.0351	1.83683	0.918414	−	0.395621i	$-0.129471\pi$
$67$	15.0351	1.83683	0.918414	+	0.395621i	$0.129471\pi$
$68$	−0.184793	−0.0224094
$69$	13.4807	1.62289
$70$	11.5621	1.38194
$71$	7.31046	0.867592	0.433796	−	0.901011i	$-0.357174\pi$
$71$	7.31046	0.867592	0.433796	+	0.901011i	$0.357174\pi$
$72$	1.77332	0.208988
$73$	−4.04458	−0.473382	−0.236691	−	0.971585i	$-0.576063\pi$
$73$	−4.04458	−0.473382	−0.236691	+	0.971585i	$0.576063\pi$
$74$	7.75877	0.901939
$75$	−29.3013	−3.38342
$76$	−1.00000	−0.114708
$77$	10.0155	1.14137
$78$	0	0
$79$	6.12836	0.689494	0.344747	−	0.938696i	$-0.387965\pi$
$79$	6.12836	0.689494	0.344747	+	0.938696i	$0.387965\pi$
$80$	4.29086	0.479733
$81$	−11.1753	−1.24170
$82$	−11.2490	−1.24224
$83$	7.49020	0.822156	0.411078	−	0.911600i	$-0.365152\pi$
$83$	7.49020	0.822156	0.411078	+	0.911600i	$0.365152\pi$
$84$	−5.88713	−0.642338
$85$	−0.792919	−0.0860041
$86$	6.82295	0.735737
$87$	2.94356	0.315583
$88$	3.71688	0.396221
$89$	−13.1898	−1.39812	−0.699060	−	0.715063i	$-0.746398\pi$
$89$	−13.1898	−1.39812	−0.699060	+	0.715063i	$0.746398\pi$
$90$	7.60906	0.802065
$91$	0	0
$92$	−6.17024	−0.643292
$93$	−4.19759	−0.435269
$94$	1.17705	0.121404
$95$	−4.29086	−0.440233
$96$	−2.18479	−0.222984
$97$	−7.23442	−0.734544	−0.367272	−	0.930114i	$-0.619708\pi$
$97$	−7.23442	−0.734544	−0.367272	+	0.930114i	$0.619708\pi$
$98$	0.260830	0.0263478
$99$	6.59121	0.662442
$100$	13.4115	1.34115
$101$	9.27631	0.923027	0.461514	−	0.887133i	$-0.347307\pi$
$101$	9.27631	0.923027	0.461514	+	0.887133i	$0.347307\pi$
$102$	0.403733	0.0399756
$103$	−4.34049	−0.427681	−0.213841	−	0.976869i	$-0.568597\pi$
$103$	−4.34049	−0.427681	−0.213841	+	0.976869i	$0.568597\pi$
$104$	0	0
$105$	−25.2608	−2.46520
$106$	−5.59627	−0.543557
$107$	−20.1780	−1.95068	−0.975340	−	0.220709i	$-0.929163\pi$
$107$	−20.1780	−1.95068	−0.975340	+	0.220709i	$0.929163\pi$
$108$	2.68004	0.257887
$109$	18.7939	1.80012	0.900062	−	0.435761i	$-0.143521\pi$
$109$	18.7939	1.80012	0.900062	+	0.435761i	$0.143521\pi$
$110$	15.9486	1.52064
$111$	−16.9513	−1.60895
$112$	2.69459	0.254615
$113$	−8.32501	−0.783151	−0.391575	−	0.920146i	$-0.628070\pi$
$113$	−8.32501	−0.783151	−0.391575	+	0.920146i	$0.628070\pi$
$114$	2.18479	0.204625
$115$	−26.4757	−2.46887
$116$	−1.34730	−0.125093
$117$	0	0
$118$	2.45336	0.225850
$119$	−0.497941	−0.0456461
$120$	−9.37464	−0.855783
$121$	2.81521	0.255928
$122$	7.30541	0.661400
$123$	24.5767	2.21600
$124$	1.92127	0.172536
$125$	36.0925	3.22821
$126$	4.77837	0.425691
$127$	−2.56624	−0.227717	−0.113858	−	0.993497i	$-0.536321\pi$
$127$	−2.56624	−0.227717	−0.113858	+	0.993497i	$0.536321\pi$
$128$	1.00000	0.0883883
$129$	−14.9067	−1.31246
$130$	0	0
$131$	−21.7297	−1.89853	−0.949265	−	0.314477i	$-0.898171\pi$
$131$	−21.7297	−1.89853	−0.949265	+	0.314477i	$0.898171\pi$
$132$	−8.12061	−0.706809
$133$	−2.69459	−0.233651
$134$	15.0351	1.29883
$135$	11.4997	0.989736
$136$	−0.184793	−0.0158458
$137$	−15.6013	−1.33291	−0.666455	−	0.745545i	$-0.732189\pi$
$137$	−15.6013	−1.33291	−0.666455	+	0.745545i	$0.732189\pi$
$138$	13.4807	1.14755
$139$	9.14796	0.775919	0.387960	−	0.921676i	$-0.373180\pi$
$139$	9.14796	0.775919	0.387960	+	0.921676i	$0.373180\pi$
$140$	11.5621	0.977177
$141$	−2.57161	−0.216569
$142$	7.31046	0.613480
$143$	0	0
$144$	1.77332	0.147777
$145$	−5.78106	−0.480091
$146$	−4.04458	−0.334732
$147$	−0.569859	−0.0470012
$148$	7.75877	0.637767
$149$	−11.6578	−0.955041	−0.477520	−	0.878621i	$-0.658464\pi$
$149$	−11.6578	−0.955041	−0.477520	+	0.878621i	$0.658464\pi$
$150$	−29.3013	−2.39244
$151$	−2.04189	−0.166167	−0.0830833	−	0.996543i	$-0.526477\pi$
$151$	−2.04189	−0.166167	−0.0830833	+	0.996543i	$0.526477\pi$
$152$	−1.00000	−0.0811107
$153$	−0.327696	−0.0264927
$154$	10.0155	0.807071
$155$	8.24392	0.662167
$156$	0	0
$157$	16.0000	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$158$	6.12836	0.487546
$159$	12.2267	0.969639
$160$	4.29086	0.339222
$161$	−16.6263	−1.31034
$162$	−11.1753	−0.878014
$163$	6.75103	0.528781	0.264391	−	0.964416i	$-0.414829\pi$
$163$	6.75103	0.528781	0.264391	+	0.964416i	$0.414829\pi$
$164$	−11.2490	−0.878397
$165$	−34.8444	−2.71263
$166$	7.49020	0.581352
$167$	−20.8949	−1.61689	−0.808447	−	0.588569i	$-0.799691\pi$
$167$	−20.8949	−1.61689	−0.808447	+	0.588569i	$0.799691\pi$
$168$	−5.88713	−0.454202
$169$	0	0
$170$	−0.792919	−0.0608141
$171$	−1.77332	−0.135609
$172$	6.82295	0.520245
$173$	−3.03684	−0.230886	−0.115443	−	0.993314i	$-0.536829\pi$
$173$	−3.03684	−0.230886	−0.115443	+	0.993314i	$0.536829\pi$
$174$	2.94356	0.223151
$175$	36.1385	2.73181
$176$	3.71688	0.280170
$177$	−5.36009	−0.402889
$178$	−13.1898	−0.988621
$179$	9.89218	0.739376	0.369688	−	0.929156i	$-0.379465\pi$
$179$	9.89218	0.739376	0.369688	+	0.929156i	$0.379465\pi$
$180$	7.60906	0.567146
$181$	5.70233	0.423851	0.211926	−	0.977286i	$-0.432027\pi$
$181$	5.70233	0.423851	0.211926	+	0.977286i	$0.432027\pi$
$182$	0	0
$183$	−15.9608	−1.17986
$184$	−6.17024	−0.454876
$185$	33.2918	2.44766
$186$	−4.19759	−0.307782
$187$	−0.686852	−0.0502276
$188$	1.17705	0.0858453
$189$	7.22163	0.525296
$190$	−4.29086	−0.311292
$191$	11.9659	0.865819	0.432909	−	0.901437i	$-0.357487\pi$
$191$	11.9659	0.865819	0.432909	+	0.901437i	$0.357487\pi$
$192$	−2.18479	−0.157674
$193$	−3.78611	−0.272530	−0.136265	−	0.990672i	$-0.543510\pi$
$193$	−3.78611	−0.272530	−0.136265	+	0.990672i	$0.543510\pi$
$194$	−7.23442	−0.519401
$195$	0	0
$196$	0.260830	0.0186307
$197$	−16.1284	−1.14910	−0.574549	−	0.818470i	$-0.694822\pi$
$197$	−16.1284	−1.14910	−0.574549	+	0.818470i	$0.694822\pi$
$198$	6.59121	0.468417
$199$	1.82976	0.129708	0.0648540	−	0.997895i	$-0.479342\pi$
$199$	1.82976	0.129708	0.0648540	+	0.997895i	$0.479342\pi$
$200$	13.4115	0.948334
$201$	−32.8485	−2.31696
$202$	9.27631	0.652679
$203$	−3.63041	−0.254805
$204$	0.403733	0.0282670
$205$	−48.2677	−3.37117
$206$	−4.34049	−0.302416
$207$	−10.9418	−0.760508
$208$	0	0
$209$	−3.71688	−0.257102
$210$	−25.2608	−1.74316
$211$	6.62361	0.455988	0.227994	−	0.973663i	$-0.426783\pi$
$211$	6.62361	0.455988	0.227994	+	0.973663i	$0.426783\pi$
$212$	−5.59627	−0.384353
$213$	−15.9718	−1.09437
$214$	−20.1780	−1.37934
$215$	29.2763	1.99663
$216$	2.68004	0.182354
$217$	5.17705	0.351441
$218$	18.7939	1.27288
$219$	8.83656	0.597120
$220$	15.9486	1.07526
$221$	0	0
$222$	−16.9513	−1.13770
$223$	11.1361	0.745728	0.372864	−	0.927886i	$-0.378376\pi$
$223$	11.1361	0.745728	0.372864	+	0.927886i	$0.378376\pi$
$224$	2.69459	0.180040
$225$	23.7828	1.58552
$226$	−8.32501	−0.553771
$227$	−19.3601	−1.28497	−0.642487	−	0.766296i	$-0.722097\pi$
$227$	−19.3601	−1.28497	−0.642487	+	0.766296i	$0.722097\pi$
$228$	2.18479	0.144691
$229$	−5.13516	−0.339341	−0.169671	−	0.985501i	$-0.554270\pi$
$229$	−5.13516	−0.339341	−0.169671	+	0.985501i	$0.554270\pi$
$230$	−26.4757	−1.74575
$231$	−21.8817	−1.43971
$232$	−1.34730	−0.0884543
$233$	16.0051	1.04853	0.524263	−	0.851556i	$-0.324341\pi$
$233$	16.0051	1.04853	0.524263	+	0.851556i	$0.324341\pi$
$234$	0	0
$235$	5.05056	0.329462
$236$	2.45336	0.159700
$237$	−13.3892	−0.869721
$238$	−0.497941	−0.0322767
$239$	18.1284	1.17263	0.586313	−	0.810085i	$-0.300579\pi$
$239$	18.1284	1.17263	0.586313	+	0.810085i	$0.300579\pi$
$240$	−9.37464	−0.605130
$241$	−0.177985	−0.0114650	−0.00573252	−	0.999984i	$-0.501825\pi$
$241$	−0.177985	−0.0114650	−0.00573252	+	0.999984i	$0.501825\pi$
$242$	2.81521	0.180968
$243$	16.3756	1.05049
$244$	7.30541	0.467681
$245$	1.11918	0.0715021
$246$	24.5767	1.56695
$247$	0	0
$248$	1.92127	0.122001
$249$	−16.3645	−1.03706
$250$	36.0925	2.28269
$251$	11.1925	0.706466	0.353233	−	0.935535i	$-0.385082\pi$
$251$	11.1925	0.706466	0.353233	+	0.935535i	$0.385082\pi$
$252$	4.77837	0.301009
$253$	−22.9341	−1.44185
$254$	−2.56624	−0.161020
$255$	1.73236	0.108485
$256$	1.00000	0.0625000
$257$	−7.51754	−0.468931	−0.234466	−	0.972124i	$-0.575334\pi$
$257$	−7.51754	−0.468931	−0.234466	+	0.972124i	$0.575334\pi$
$258$	−14.9067	−0.928052
$259$	20.9067	1.29908
$260$	0	0
$261$	−2.38919	−0.147887
$262$	−21.7297	−1.34246
$263$	20.9240	1.29023	0.645113	−	0.764087i	$-0.276810\pi$
$263$	20.9240	1.29023	0.645113	+	0.764087i	$0.276810\pi$
$264$	−8.12061	−0.499789
$265$	−24.0128	−1.47509
$266$	−2.69459	−0.165216
$267$	28.8171	1.76358
$268$	15.0351	0.918414
$269$	−25.5175	−1.55583	−0.777916	−	0.628368i	$-0.783723\pi$
$269$	−25.5175	−1.55583	−0.777916	+	0.628368i	$0.783723\pi$
$270$	11.4997	0.699849
$271$	−12.6946	−0.771142	−0.385571	−	0.922678i	$-0.625995\pi$
$271$	−12.6946	−0.771142	−0.385571	+	0.922678i	$0.625995\pi$
$272$	−0.184793	−0.0112047
$273$	0	0
$274$	−15.6013	−0.942510
$275$	49.8489	3.00600
$276$	13.4807	0.811443
$277$	22.9513	1.37901	0.689505	−	0.724281i	$-0.257828\pi$
$277$	22.9513	1.37901	0.689505	+	0.724281i	$0.257828\pi$
$278$	9.14796	0.548658
$279$	3.40703	0.203974
$280$	11.5621	0.690969
$281$	−29.6141	−1.76663	−0.883315	−	0.468780i	$-0.844694\pi$
$281$	−29.6141	−1.76663	−0.883315	+	0.468780i	$0.844694\pi$
$282$	−2.57161	−0.153137
$283$	−2.34049	−0.139128	−0.0695638	−	0.997578i	$-0.522161\pi$
$283$	−2.34049	−0.139128	−0.0695638	+	0.997578i	$0.522161\pi$
$284$	7.31046	0.433796
$285$	9.37464	0.555306
$286$	0	0
$287$	−30.3114	−1.78922
$288$	1.77332	0.104494
$289$	−16.9659	−0.997991
$290$	−5.78106	−0.339475
$291$	15.8057	0.926547
$292$	−4.04458	−0.236691
$293$	13.9709	0.816189	0.408094	−	0.912940i	$-0.366193\pi$
$293$	13.9709	0.816189	0.408094	+	0.912940i	$0.366193\pi$
$294$	−0.569859	−0.0332349
$295$	10.5270	0.612908
$296$	7.75877	0.450969
$297$	9.96141	0.578020
$298$	−11.6578	−0.675316
$299$	0	0
$300$	−29.3013	−1.69171
$301$	18.3851	1.05970
$302$	−2.04189	−0.117498
$303$	−20.2668	−1.16430
$304$	−1.00000	−0.0573539
$305$	31.3465	1.79489
$306$	−0.327696	−0.0187331
$307$	11.2216	0.640452	0.320226	−	0.947341i	$-0.396241\pi$
$307$	11.2216	0.640452	0.320226	+	0.947341i	$0.396241\pi$
$308$	10.0155	0.570685
$309$	9.48307	0.539473
$310$	8.24392	0.468223
$311$	−28.6091	−1.62227	−0.811135	−	0.584858i	$-0.801150\pi$
$311$	−28.6091	−1.62227	−0.811135	+	0.584858i	$0.801150\pi$
$312$	0	0
$313$	−29.0009	−1.63923	−0.819615	−	0.572915i	$-0.805813\pi$
$313$	−29.0009	−1.63923	−0.819615	+	0.572915i	$0.805813\pi$
$314$	16.0000	0.902932
$315$	20.5033	1.15523
$316$	6.12836	0.344747
$317$	−27.1634	−1.52565	−0.762825	−	0.646605i	$-0.776188\pi$
$317$	−27.1634	−1.52565	−0.762825	+	0.646605i	$0.776188\pi$
$318$	12.2267	0.685638
$319$	−5.00774	−0.280380
$320$	4.29086	0.239866
$321$	44.0847	2.46057
$322$	−16.6263	−0.926547
$323$	0.184793	0.0102821
$324$	−11.1753	−0.620850
$325$	0	0
$326$	6.75103	0.373905
$327$	−41.0607	−2.27066
$328$	−11.2490	−0.621120
$329$	3.17168	0.174860
$330$	−34.8444	−1.91812
$331$	−25.8135	−1.41884	−0.709418	−	0.704788i	$-0.751042\pi$
$331$	−25.8135	−1.41884	−0.709418	+	0.704788i	$0.751042\pi$
$332$	7.49020	0.411078
$333$	13.7588	0.753976
$334$	−20.8949	−1.14332
$335$	64.5134	3.52474
$336$	−5.88713	−0.321169
$337$	−4.40879	−0.240162	−0.120081	−	0.992764i	$-0.538315\pi$
$337$	−4.40879	−0.240162	−0.120081	+	0.992764i	$0.538315\pi$
$338$	0	0
$339$	18.1884	0.987859
$340$	−0.792919	−0.0430021
$341$	7.14115	0.386715
$342$	−1.77332	−0.0958901
$343$	−18.1593	−0.980511
$344$	6.82295	0.367869
$345$	57.8438	3.11421
$346$	−3.03684	−0.163261
$347$	32.1830	1.72768	0.863838	−	0.503770i	$-0.168054\pi$
$347$	32.1830	1.72768	0.863838	+	0.503770i	$0.168054\pi$
$348$	2.94356	0.157792
$349$	28.6536	1.53379	0.766897	−	0.641770i	$-0.221800\pi$
$349$	28.6536	1.53379	0.766897	+	0.641770i	$0.221800\pi$
$350$	36.1385	1.93168
$351$	0	0
$352$	3.71688	0.198110
$353$	32.6810	1.73943	0.869716	−	0.493552i	$-0.164302\pi$
$353$	32.6810	1.73943	0.869716	+	0.493552i	$0.164302\pi$
$354$	−5.36009	−0.284886
$355$	31.3682	1.66485
$356$	−13.1898	−0.699060
$357$	1.08790	0.0575776
$358$	9.89218	0.522818
$359$	11.1480	0.588367	0.294183	−	0.955749i	$-0.404952\pi$
$359$	11.1480	0.588367	0.294183	+	0.955749i	$0.404952\pi$
$360$	7.60906	0.401033
$361$	1.00000	0.0526316
$362$	5.70233	0.299708
$363$	−6.15064	−0.322825
$364$	0	0
$365$	−17.3547	−0.908387
$366$	−15.9608	−0.834284
$367$	8.54757	0.446180	0.223090	−	0.974798i	$-0.428386\pi$
$367$	8.54757	0.446180	0.223090	+	0.974798i	$0.428386\pi$
$368$	−6.17024	−0.321646
$369$	−19.9480	−1.03845
$370$	33.2918	1.73076
$371$	−15.0797	−0.782897
$372$	−4.19759	−0.217635
$373$	−21.7915	−1.12832	−0.564160	−	0.825665i	$-0.690800\pi$
$373$	−21.7915	−1.12832	−0.564160	+	0.825665i	$0.690800\pi$
$374$	−0.686852	−0.0355163
$375$	−78.8545	−4.07203
$376$	1.17705	0.0607018
$377$	0	0
$378$	7.22163	0.371441
$379$	1.91622	0.0984297	0.0492149	−	0.998788i	$-0.484328\pi$
$379$	1.91622	0.0984297	0.0492149	+	0.998788i	$0.484328\pi$
$380$	−4.29086	−0.220116
$381$	5.60670	0.287240
$382$	11.9659	0.612226
$383$	30.9273	1.58031	0.790155	−	0.612908i	$-0.210000\pi$
$383$	30.9273	1.58031	0.790155	+	0.612908i	$0.210000\pi$
$384$	−2.18479	−0.111492
$385$	42.9750	2.19021
$386$	−3.78611	−0.192708
$387$	12.0993	0.615040
$388$	−7.23442	−0.367272
$389$	7.47296	0.378894	0.189447	−	0.981891i	$-0.439330\pi$
$389$	7.47296	0.378894	0.189447	+	0.981891i	$0.439330\pi$
$390$	0	0
$391$	1.14022	0.0576632
$392$	0.260830	0.0131739
$393$	47.4748	2.39479
$394$	−16.1284	−0.812535
$395$	26.2959	1.32309
$396$	6.59121	0.331221
$397$	−15.1584	−0.760778	−0.380389	−	0.924827i	$-0.624210\pi$
$397$	−15.1584	−0.760778	−0.380389	+	0.924827i	$0.624210\pi$
$398$	1.82976	0.0917174
$399$	5.88713	0.294725
$400$	13.4115	0.670574
$401$	20.8280	1.04010	0.520050	−	0.854136i	$-0.325913\pi$
$401$	20.8280	1.04010	0.520050	+	0.854136i	$0.325913\pi$
$402$	−32.8485	−1.63834
$403$	0	0
$404$	9.27631	0.461514
$405$	−47.9516	−2.38274
$406$	−3.63041	−0.180174
$407$	28.8384	1.42947
$408$	0.403733	0.0199878
$409$	−0.778371	−0.0384880	−0.0192440	−	0.999815i	$-0.506126\pi$
$409$	−0.778371	−0.0384880	−0.0192440	+	0.999815i	$0.506126\pi$
$410$	−48.2677	−2.38377
$411$	34.0856	1.68132
$412$	−4.34049	−0.213841
$413$	6.61081	0.325297
$414$	−10.9418	−0.537761
$415$	32.1394	1.57766
$416$	0	0
$417$	−19.9864	−0.978738
$418$	−3.71688	−0.181799
$419$	2.73379	0.133555	0.0667773	−	0.997768i	$-0.478728\pi$
$419$	2.73379	0.133555	0.0667773	+	0.997768i	$0.478728\pi$
$420$	−25.2608	−1.23260
$421$	23.1789	1.12967	0.564836	−	0.825203i	$-0.308940\pi$
$421$	23.1789	1.12967	0.564836	+	0.825203i	$0.308940\pi$
$422$	6.62361	0.322432
$423$	2.08729	0.101487
$424$	−5.59627	−0.271779
$425$	−2.47834	−0.120217
$426$	−15.9718	−0.773838
$427$	19.6851	0.952629
$428$	−20.1780	−0.975340
$429$	0	0
$430$	29.2763	1.41183
$431$	−17.6578	−0.850544	−0.425272	−	0.905066i	$-0.639822\pi$
$431$	−17.6578	−0.850544	−0.425272	+	0.905066i	$0.639822\pi$
$432$	2.68004	0.128944
$433$	9.54664	0.458782	0.229391	−	0.973334i	$-0.426327\pi$
$433$	9.54664	0.458782	0.229391	+	0.973334i	$0.426327\pi$
$434$	5.17705	0.248506
$435$	12.6304	0.605582
$436$	18.7939	0.900062
$437$	6.17024	0.295163
$438$	8.83656	0.422227
$439$	−1.34461	−0.0641746	−0.0320873	−	0.999485i	$-0.510215\pi$
$439$	−1.34461	−0.0641746	−0.0320873	+	0.999485i	$0.510215\pi$
$440$	15.9486	0.760320
$441$	0.462534	0.0220254
$442$	0	0
$443$	−14.4688	−0.687436	−0.343718	−	0.939073i	$-0.611686\pi$
$443$	−14.4688	−0.687436	−0.343718	+	0.939073i	$0.611686\pi$
$444$	−16.9513	−0.804473
$445$	−56.5958	−2.68290
$446$	11.1361	0.527309
$447$	25.4698	1.20468
$448$	2.69459	0.127308
$449$	3.36184	0.158655	0.0793276	−	0.996849i	$-0.474723\pi$
$449$	3.36184	0.158655	0.0793276	+	0.996849i	$0.474723\pi$
$450$	23.7828	1.12113
$451$	−41.8111	−1.96881
$452$	−8.32501	−0.391575
$453$	4.46110	0.209601
$454$	−19.3601	−0.908614
$455$	0	0
$456$	2.18479	0.102312
$457$	30.0993	1.40798	0.703992	−	0.710208i	$-0.251399\pi$
$457$	30.0993	1.40798	0.703992	+	0.710208i	$0.251399\pi$
$458$	−5.13516	−0.239950
$459$	−0.495252	−0.0231164
$460$	−26.4757	−1.23443
$461$	−25.1607	−1.17185	−0.585926	−	0.810364i	$-0.699269\pi$
$461$	−25.1607	−1.17185	−0.585926	+	0.810364i	$0.699269\pi$
$462$	−21.8817	−1.01803
$463$	−11.3601	−0.527948	−0.263974	−	0.964530i	$-0.585033\pi$
$463$	−11.3601	−0.527948	−0.263974	+	0.964530i	$0.585033\pi$
$464$	−1.34730	−0.0625467
$465$	−18.0113	−0.835252
$466$	16.0051	0.741420
$467$	30.1884	1.39695	0.698477	−	0.715633i	$-0.253862\pi$
$467$	30.1884	1.39695	0.698477	+	0.715633i	$0.253862\pi$
$468$	0	0
$469$	40.5134	1.87074
$470$	5.05056	0.232965
$471$	−34.9567	−1.61072
$472$	2.45336	0.112925
$473$	25.3601	1.16606
$474$	−13.3892	−0.614986
$475$	−13.4115	−0.615361
$476$	−0.497941	−0.0228231
$477$	−9.92396	−0.454387
$478$	18.1284	0.829172
$479$	41.2526	1.88488	0.942440	−	0.334377i	$-0.108526\pi$
$479$	41.2526	1.88488	0.942440	+	0.334377i	$0.108526\pi$
$480$	−9.37464	−0.427892
$481$	0	0
$482$	−0.177985	−0.00810701
$483$	36.3250	1.65285
$484$	2.81521	0.127964
$485$	−31.0419	−1.40954
$486$	16.3756	0.742811
$487$	9.22432	0.417994	0.208997	−	0.977916i	$-0.432980\pi$
$487$	9.22432	0.417994	0.208997	+	0.977916i	$0.432980\pi$
$488$	7.30541	0.330700
$489$	−14.7496	−0.667000
$490$	1.11918	0.0505596
$491$	25.3155	1.14247	0.571237	−	0.820785i	$-0.306464\pi$
$491$	25.3155	1.14247	0.571237	+	0.820785i	$0.306464\pi$
$492$	24.5767	1.10800
$493$	0.248970	0.0112131
$494$	0	0
$495$	28.2820	1.27118
$496$	1.92127	0.0862678
$497$	19.6987	0.883608
$498$	−16.3645	−0.733312
$499$	21.6186	0.967779	0.483890	−	0.875129i	$-0.339224\pi$
$499$	21.6186	0.967779	0.483890	+	0.875129i	$0.339224\pi$
$500$	36.0925	1.61410
$501$	45.6509	2.03953
$502$	11.1925	0.499547
$503$	11.7743	0.524988	0.262494	−	0.964934i	$-0.415455\pi$
$503$	11.7743	0.524988	0.262494	+	0.964934i	$0.415455\pi$
$504$	4.77837	0.212846
$505$	39.8033	1.77123
$506$	−22.9341	−1.01954
$507$	0	0
$508$	−2.56624	−0.113858
$509$	7.59121	0.336475	0.168237	−	0.985747i	$-0.446193\pi$
$509$	7.59121	0.336475	0.168237	+	0.985747i	$0.446193\pi$
$510$	1.73236	0.0767103
$511$	−10.8985	−0.482121
$512$	1.00000	0.0441942
$513$	−2.68004	−0.118327
$514$	−7.51754	−0.331585
$515$	−18.6244	−0.820690
$516$	−14.9067	−0.656232
$517$	4.37496	0.192411
$518$	20.9067	0.918589
$519$	6.63486	0.291238
$520$	0	0
$521$	−32.8776	−1.44040	−0.720198	−	0.693769i	$-0.755949\pi$
$521$	−32.8776	−1.44040	−0.720198	+	0.693769i	$0.755949\pi$
$522$	−2.38919	−0.104572
$523$	−20.2463	−0.885308	−0.442654	−	0.896692i	$-0.645963\pi$
$523$	−20.2463	−0.885308	−0.442654	+	0.896692i	$0.645963\pi$
$524$	−21.7297	−0.949265
$525$	−78.9550	−3.44588
$526$	20.9240	0.912328
$527$	−0.355037	−0.0154657
$528$	−8.12061	−0.353404
$529$	15.0719	0.655301
$530$	−24.0128	−1.04305
$531$	4.35059	0.188800
$532$	−2.69459	−0.116825
$533$	0	0
$534$	28.8171	1.24704
$535$	−86.5809	−3.74322
$536$	15.0351	0.649417
$537$	−21.6124	−0.932642
$538$	−25.5175	−1.10014
$539$	0.969474	0.0417582
$540$	11.4997	0.494868
$541$	3.05913	0.131522	0.0657610	−	0.997835i	$-0.479052\pi$
$541$	3.05913	0.131522	0.0657610	+	0.997835i	$0.479052\pi$
$542$	−12.6946	−0.545279
$543$	−12.4584	−0.534642
$544$	−0.184793	−0.00792291
$545$	80.6418	3.45431
$546$	0	0
$547$	10.6996	0.457484	0.228742	−	0.973487i	$-0.426539\pi$
$547$	10.6996	0.457484	0.228742	+	0.973487i	$0.426539\pi$
$548$	−15.6013	−0.666455
$549$	12.9548	0.552898
$550$	49.8489	2.12556
$551$	1.34730	0.0573968
$552$	13.4807	0.573777
$553$	16.5134	0.702222
$554$	22.9513	0.975107
$555$	−72.7357	−3.08746
$556$	9.14796	0.387960
$557$	−13.1352	−0.556555	−0.278277	−	0.960501i	$-0.589763\pi$
$557$	−13.1352	−0.556555	−0.278277	+	0.960501i	$0.589763\pi$
$558$	3.40703	0.144231
$559$	0	0
$560$	11.5621	0.488589
$561$	1.50063	0.0633566
$562$	−29.6141	−1.24920
$563$	17.9736	0.757497	0.378748	−	0.925500i	$-0.376355\pi$
$563$	17.9736	0.757497	0.378748	+	0.925500i	$0.376355\pi$
$564$	−2.57161	−0.108284
$565$	−35.7214	−1.50281
$566$	−2.34049	−0.0983781
$567$	−30.1129	−1.26462
$568$	7.31046	0.306740
$569$	9.43376	0.395484	0.197742	−	0.980254i	$-0.436639\pi$
$569$	9.43376	0.395484	0.197742	+	0.980254i	$0.436639\pi$
$570$	9.37464	0.392660
$571$	−13.2080	−0.552738	−0.276369	−	0.961052i	$-0.589131\pi$
$571$	−13.2080	−0.552738	−0.276369	+	0.961052i	$0.589131\pi$
$572$	0	0
$573$	−26.1429	−1.09214
$574$	−30.3114	−1.26517
$575$	−82.7521	−3.45100
$576$	1.77332	0.0738883
$577$	−28.6810	−1.19400	−0.597002	−	0.802239i	$-0.703642\pi$
$577$	−28.6810	−1.19400	−0.597002	+	0.802239i	$0.703642\pi$
$578$	−16.9659	−0.705686
$579$	8.27187	0.343767
$580$	−5.78106	−0.240045
$581$	20.1830	0.837334
$582$	15.8057	0.655168
$583$	−20.8007	−0.861475
$584$	−4.04458	−0.167366
$585$	0	0
$586$	13.9709	0.577133
$587$	−16.6500	−0.687220	−0.343610	−	0.939112i	$-0.611650\pi$
$587$	−16.6500	−0.687220	−0.343610	+	0.939112i	$0.611650\pi$
$588$	−0.569859	−0.0235006
$589$	−1.92127	−0.0791647
$590$	10.5270	0.433391
$591$	35.2371	1.44946
$592$	7.75877	0.318884
$593$	−43.6459	−1.79232	−0.896161	−	0.443729i	$-0.853655\pi$
$593$	−43.6459	−1.79232	−0.896161	+	0.443729i	$0.853655\pi$
$594$	9.96141	0.408722
$595$	−2.13659	−0.0875918
$596$	−11.6578	−0.477520
$597$	−3.99764	−0.163612
$598$	0	0
$599$	7.22163	0.295068	0.147534	−	0.989057i	$-0.452866\pi$
$599$	7.22163	0.295068	0.147534	+	0.989057i	$0.452866\pi$
$600$	−29.3013	−1.19622
$601$	−29.2526	−1.19324	−0.596619	−	0.802525i	$-0.703490\pi$
$601$	−29.2526	−1.19324	−0.596619	+	0.802525i	$0.703490\pi$
$602$	18.3851	0.749319
$603$	26.6620	1.08576
$604$	−2.04189	−0.0830833
$605$	12.0797	0.491108
$606$	−20.2668	−0.823283
$607$	11.8425	0.480674	0.240337	−	0.970689i	$-0.422742\pi$
$607$	11.8425	0.480674	0.240337	+	0.970689i	$0.422742\pi$
$608$	−1.00000	−0.0405554
$609$	7.93170	0.321409
$610$	31.3465	1.26918
$611$	0	0
$612$	−0.327696	−0.0132463
$613$	25.1976	1.01772	0.508860	−	0.860849i	$-0.330067\pi$
$613$	25.1976	1.01772	0.508860	+	0.860849i	$0.330067\pi$
$614$	11.2216	0.452868
$615$	105.455	4.25236
$616$	10.0155	0.403535
$617$	−32.6263	−1.31349	−0.656743	−	0.754115i	$-0.728066\pi$
$617$	−32.6263	−1.31349	−0.656743	+	0.754115i	$0.728066\pi$
$618$	9.48307	0.381465
$619$	−1.06923	−0.0429760	−0.0214880	−	0.999769i	$-0.506840\pi$
$619$	−1.06923	−0.0429760	−0.0214880	+	0.999769i	$0.506840\pi$
$620$	8.24392	0.331084
$621$	−16.5365	−0.663588
$622$	−28.6091	−1.14712
$623$	−35.5413	−1.42393
$624$	0	0
$625$	87.8103	3.51241
$626$	−29.0009	−1.15911
$627$	8.12061	0.324306
$628$	16.0000	0.638470
$629$	−1.43376	−0.0571679
$630$	20.5033	0.816872
$631$	−20.4142	−0.812675	−0.406337	−	0.913723i	$-0.633194\pi$
$631$	−20.4142	−0.812675	−0.406337	+	0.913723i	$0.633194\pi$
$632$	6.12836	0.243773
$633$	−14.4712	−0.575179
$634$	−27.1634	−1.07880
$635$	−11.0114	−0.436973
$636$	12.2267	0.484819
$637$	0	0
$638$	−5.00774	−0.198258
$639$	12.9638	0.512839
$640$	4.29086	0.169611
$641$	−33.4439	−1.32095	−0.660477	−	0.750847i	$-0.729646\pi$
$641$	−33.4439	−1.32095	−0.660477	+	0.750847i	$0.729646\pi$
$642$	44.0847	1.73988
$643$	4.59896	0.181365	0.0906825	−	0.995880i	$-0.471095\pi$
$643$	4.59896	0.181365	0.0906825	+	0.995880i	$0.471095\pi$
$644$	−16.6263	−0.655168
$645$	−63.9627	−2.51853
$646$	0.184793	0.00727056
$647$	22.6759	0.891483	0.445741	−	0.895162i	$-0.352940\pi$
$647$	22.6759	0.891483	0.445741	+	0.895162i	$0.352940\pi$
$648$	−11.1753	−0.439007
$649$	9.11886	0.357947
$650$	0	0
$651$	−11.3108	−0.443305
$652$	6.75103	0.264391
$653$	15.2763	0.597808	0.298904	−	0.954283i	$-0.403379\pi$
$653$	15.2763	0.597808	0.298904	+	0.954283i	$0.403379\pi$
$654$	−41.0607	−1.60560
$655$	−93.2390	−3.64315
$656$	−11.2490	−0.439199
$657$	−7.17232	−0.279819
$658$	3.17168	0.123645
$659$	−31.1462	−1.21328	−0.606642	−	0.794975i	$-0.707484\pi$
$659$	−31.1462	−1.21328	−0.606642	+	0.794975i	$0.707484\pi$
$660$	−34.8444	−1.35632
$661$	−11.3345	−0.440861	−0.220431	−	0.975403i	$-0.570746\pi$
$661$	−11.3345	−0.440861	−0.220431	+	0.975403i	$0.570746\pi$
$662$	−25.8135	−1.00327
$663$	0	0
$664$	7.49020	0.290676
$665$	−11.5621	−0.448360
$666$	13.7588	0.533142
$667$	8.31315	0.321886
$668$	−20.8949	−0.808447
$669$	−24.3301	−0.940654
$670$	64.5134	2.49237
$671$	27.1533	1.04824
$672$	−5.88713	−0.227101
$673$	−3.34998	−0.129132	−0.0645662	−	0.997913i	$-0.520566\pi$
$673$	−3.34998	−0.129132	−0.0645662	+	0.997913i	$0.520566\pi$
$674$	−4.40879	−0.169820
$675$	35.9434	1.38346
$676$	0	0
$677$	14.1702	0.544607	0.272303	−	0.962211i	$-0.412215\pi$
$677$	14.1702	0.544607	0.272303	+	0.962211i	$0.412215\pi$
$678$	18.1884	0.698522
$679$	−19.4938	−0.748104
$680$	−0.792919	−0.0304070
$681$	42.2978	1.62085
$682$	7.14115	0.273449
$683$	−44.2978	−1.69501	−0.847504	−	0.530789i	$-0.821895\pi$
$683$	−44.2978	−1.69501	−0.847504	+	0.530789i	$0.821895\pi$
$684$	−1.77332	−0.0678045
$685$	−66.9431	−2.55776
$686$	−18.1593	−0.693326
$687$	11.2193	0.428042
$688$	6.82295	0.260122
$689$	0	0
$690$	57.8438	2.20208
$691$	−7.37639	−0.280611	−0.140306	−	0.990108i	$-0.544808\pi$
$691$	−7.37639	−0.280611	−0.140306	+	0.990108i	$0.544808\pi$
$692$	−3.03684	−0.115443
$693$	17.7606	0.674671
$694$	32.1830	1.22165
$695$	39.2526	1.48894
$696$	2.94356	0.111575
$697$	2.07873	0.0787374
$698$	28.6536	1.08456
$699$	−34.9677	−1.32260
$700$	36.1385	1.36591
$701$	−20.2567	−0.765085	−0.382543	−	0.923938i	$-0.624952\pi$
$701$	−20.2567	−0.765085	−0.382543	+	0.923938i	$0.624952\pi$
$702$	0	0
$703$	−7.75877	−0.292628
$704$	3.71688	0.140085
$705$	−11.0344	−0.415581
$706$	32.6810	1.22996
$707$	24.9959	0.940067
$708$	−5.36009	−0.201445
$709$	16.7784	0.630125	0.315062	−	0.949071i	$-0.397975\pi$
$709$	16.7784	0.630125	0.315062	+	0.949071i	$0.397975\pi$
$710$	31.3682	1.17723
$711$	10.8675	0.407564
$712$	−13.1898	−0.494310
$713$	−11.8547	−0.443963
$714$	1.08790	0.0407135
$715$	0	0
$716$	9.89218	0.369688
$717$	−39.6067	−1.47914
$718$	11.1480	0.416038
$719$	4.90673	0.182990	0.0914950	−	0.995806i	$-0.470835\pi$
$719$	4.90673	0.182990	0.0914950	+	0.995806i	$0.470835\pi$
$720$	7.60906	0.283573
$721$	−11.6959	−0.435576
$722$	1.00000	0.0372161
$723$	0.388861	0.0144619
$724$	5.70233	0.211926
$725$	−18.0692	−0.671074
$726$	−6.15064	−0.228272
$727$	−26.5354	−0.984143	−0.492072	−	0.870555i	$-0.663760\pi$
$727$	−26.5354	−0.984143	−0.492072	+	0.870555i	$0.663760\pi$
$728$	0	0
$729$	−2.25133	−0.0833828
$730$	−17.3547	−0.642327
$731$	−1.26083	−0.0466335
$732$	−15.9608	−0.589928
$733$	24.0027	0.886560	0.443280	−	0.896383i	$-0.353815\pi$
$733$	24.0027	0.886560	0.443280	+	0.896383i	$0.353815\pi$
$734$	8.54757	0.315497
$735$	−2.44519	−0.0901920
$736$	−6.17024	−0.227438
$737$	55.8836	2.05850
$738$	−19.9480	−0.734296
$739$	−41.4466	−1.52464	−0.762318	−	0.647203i	$-0.775939\pi$
$739$	−41.4466	−1.52464	−0.762318	+	0.647203i	$0.775939\pi$
$740$	33.2918	1.22383
$741$	0	0
$742$	−15.0797	−0.553592
$743$	7.62536	0.279747	0.139874	−	0.990169i	$-0.455330\pi$
$743$	7.62536	0.279747	0.139874	+	0.990169i	$0.455330\pi$
$744$	−4.19759	−0.153891
$745$	−50.0218	−1.83266
$746$	−21.7915	−0.797843
$747$	13.2825	0.485982
$748$	−0.686852	−0.0251138
$749$	−54.3715	−1.98669
$750$	−78.8545	−2.87936
$751$	6.17293	0.225254	0.112627	−	0.993637i	$-0.464074\pi$
$751$	6.17293	0.225254	0.112627	+	0.993637i	$0.464074\pi$
$752$	1.17705	0.0429227
$753$	−24.4534	−0.891130
$754$	0	0
$755$	−8.76146	−0.318862
$756$	7.22163	0.262648
$757$	−20.3560	−0.739850	−0.369925	−	0.929062i	$-0.620617\pi$
$757$	−20.3560	−0.739850	−0.369925	+	0.929062i	$0.620617\pi$
$758$	1.91622	0.0696003
$759$	50.1062	1.81874
$760$	−4.29086	−0.155646
$761$	−27.9026	−1.01147	−0.505734	−	0.862689i	$-0.668778\pi$
$761$	−27.9026	−1.01147	−0.505734	+	0.862689i	$0.668778\pi$
$762$	5.60670	0.203109
$763$	50.6418	1.83336
$764$	11.9659	0.432909
$765$	−1.40610	−0.0508376
$766$	30.9273	1.11745
$767$	0	0
$768$	−2.18479	−0.0788369
$769$	−2.53714	−0.0914917	−0.0457458	−	0.998953i	$-0.514566\pi$
$769$	−2.53714	−0.0914917	−0.0457458	+	0.998953i	$0.514566\pi$
$770$	42.9750	1.54871
$771$	16.4243	0.591506
$772$	−3.78611	−0.136265
$773$	4.05468	0.145837	0.0729184	−	0.997338i	$-0.476769\pi$
$773$	4.05468	0.145837	0.0729184	+	0.997338i	$0.476769\pi$
$774$	12.0993	0.434899
$775$	25.7671	0.925582
$776$	−7.23442	−0.259701
$777$	−45.6769	−1.63865
$778$	7.47296	0.267919
$779$	11.2490	0.403036
$780$	0	0
$781$	27.1721	0.972295
$782$	1.14022	0.0407740
$783$	−3.61081	−0.129040
$784$	0.260830	0.00931535
$785$	68.6537	2.45036
$786$	47.4748	1.69337
$787$	−15.0642	−0.536980	−0.268490	−	0.963282i	$-0.586525\pi$
$787$	−15.0642	−0.536980	−0.268490	+	0.963282i	$0.586525\pi$
$788$	−16.1284	−0.574549
$789$	−45.7145	−1.62748
$790$	26.2959	0.935567
$791$	−22.4325	−0.797608
$792$	6.59121	0.234209
$793$	0	0
$794$	−15.1584	−0.537951
$795$	52.4630	1.86067
$796$	1.82976	0.0648540
$797$	32.4715	1.15020	0.575100	−	0.818083i	$-0.304963\pi$
$797$	32.4715	1.15020	0.575100	+	0.818083i	$0.304963\pi$
$798$	5.88713	0.208402
$799$	−0.217510	−0.00769496
$800$	13.4115	0.474167
$801$	−23.3898	−0.826438
$802$	20.8280	0.735462
$803$	−15.0332	−0.530511
$804$	−32.8485	−1.15848
$805$	−71.3411	−2.51444
$806$	0	0
$807$	55.7505	1.96251
$808$	9.27631	0.326339
$809$	−19.4210	−0.682805	−0.341402	−	0.939917i	$-0.610902\pi$
$809$	−19.4210	−0.682805	−0.341402	+	0.939917i	$0.610902\pi$
$810$	−47.9516	−1.68485
$811$	−22.8621	−0.802799	−0.401399	−	0.915903i	$-0.631476\pi$
$811$	−22.8621	−0.802799	−0.401399	+	0.915903i	$0.631476\pi$
$812$	−3.63041	−0.127403
$813$	27.7351	0.972711
$814$	28.8384	1.01079
$815$	28.9677	1.01469
$816$	0.403733	0.0141335
$817$	−6.82295	−0.238705
$818$	−0.778371	−0.0272151
$819$	0	0
$820$	−48.2677	−1.68558
$821$	4.20676	0.146817	0.0734084	−	0.997302i	$-0.476612\pi$
$821$	4.20676	0.146817	0.0734084	+	0.997302i	$0.476612\pi$
$822$	34.0856	1.18887
$823$	−29.2945	−1.02114	−0.510571	−	0.859836i	$-0.670566\pi$
$823$	−29.2945	−1.02114	−0.510571	+	0.859836i	$0.670566\pi$
$824$	−4.34049	−0.151208
$825$	−108.909	−3.79174
$826$	6.61081	0.230020
$827$	−8.39868	−0.292051	−0.146025	−	0.989281i	$-0.546648\pi$
$827$	−8.39868	−0.292051	−0.146025	+	0.989281i	$0.546648\pi$
$828$	−10.9418	−0.380254
$829$	34.9172	1.21272	0.606361	−	0.795189i	$-0.292628\pi$
$829$	34.9172	1.21272	0.606361	+	0.795189i	$0.292628\pi$
$830$	32.1394	1.11557
$831$	−50.1438	−1.73947
$832$	0	0
$833$	−0.0481994	−0.00167001
$834$	−19.9864	−0.692072
$835$	−89.6569	−3.10271
$836$	−3.71688	−0.128551
$837$	5.14910	0.177979
$838$	2.73379	0.0944373
$839$	−42.3296	−1.46138	−0.730689	−	0.682710i	$-0.760801\pi$
$839$	−42.3296	−1.46138	−0.730689	+	0.682710i	$0.760801\pi$
$840$	−25.2608	−0.871581
$841$	−27.1848	−0.937407
$842$	23.1789	0.798798
$843$	64.7007	2.22841
$844$	6.62361	0.227994
$845$	0	0
$846$	2.08729	0.0717624
$847$	7.58584	0.260652
$848$	−5.59627	−0.192177
$849$	5.11348	0.175494
$850$	−2.47834	−0.0850064
$851$	−47.8735	−1.64108
$852$	−15.9718	−0.547186
$853$	−17.0232	−0.582864	−0.291432	−	0.956592i	$-0.594132\pi$
$853$	−17.0232	−0.582864	−0.291432	+	0.956592i	$0.594132\pi$
$854$	19.6851	0.673610
$855$	−7.60906	−0.260224
$856$	−20.1780	−0.689669
$857$	56.5289	1.93099	0.965495	−	0.260421i	$-0.0838612\pi$
$857$	56.5289	1.93099	0.965495	+	0.260421i	$0.0838612\pi$
$858$	0	0
$859$	46.2567	1.57826	0.789129	−	0.614227i	$-0.210532\pi$
$859$	46.2567	1.57826	0.789129	+	0.614227i	$0.210532\pi$
$860$	29.2763	0.998314
$861$	66.2241	2.25691
$862$	−17.6578	−0.601426
$863$	20.0918	0.683934	0.341967	−	0.939712i	$-0.388907\pi$
$863$	20.0918	0.683934	0.341967	+	0.939712i	$0.388907\pi$
$864$	2.68004	0.0911770
$865$	−13.0306	−0.443055
$866$	9.54664	0.324408
$867$	37.0669	1.25886
$868$	5.17705	0.175721
$869$	22.7784	0.772703
$870$	12.6304	0.428211
$871$	0	0
$872$	18.7939	0.636440
$873$	−12.8289	−0.434194
$874$	6.17024	0.208712
$875$	97.2545	3.28780
$876$	8.83656	0.298560
$877$	34.2668	1.15711	0.578554	−	0.815644i	$-0.303617\pi$
$877$	34.2668	1.15711	0.578554	+	0.815644i	$0.303617\pi$
$878$	−1.34461	−0.0453783
$879$	−30.5235	−1.02953
$880$	15.9486	0.537628
$881$	7.34493	0.247457	0.123729	−	0.992316i	$-0.460515\pi$
$881$	7.34493	0.247457	0.123729	+	0.992316i	$0.460515\pi$
$882$	0.462534	0.0155743
$883$	−55.0369	−1.85214	−0.926070	−	0.377351i	$-0.876835\pi$
$883$	−55.0369	−1.85214	−0.926070	+	0.377351i	$0.876835\pi$
$884$	0	0
$885$	−22.9994	−0.773116
$886$	−14.4688	−0.486090
$887$	−30.7047	−1.03096	−0.515481	−	0.856901i	$-0.672387\pi$
$887$	−30.7047	−1.03096	−0.515481	+	0.856901i	$0.672387\pi$
$888$	−16.9513	−0.568849
$889$	−6.91496	−0.231920
$890$	−56.5958	−1.89709
$891$	−41.5373	−1.39155
$892$	11.1361	0.372864
$893$	−1.17705	−0.0393885
$894$	25.4698	0.851837
$895$	42.4459	1.41881
$896$	2.69459	0.0900200
$897$	0	0
$898$	3.36184	0.112186
$899$	−2.58853	−0.0863322
$900$	23.7828	0.792760
$901$	1.03415	0.0344525
$902$	−41.8111	−1.39216
$903$	−40.1676	−1.33669
$904$	−8.32501	−0.276886
$905$	24.4679	0.813341
$906$	4.46110	0.148210
$907$	27.5381	0.914387	0.457193	−	0.889367i	$-0.348855\pi$
$907$	27.5381	0.914387	0.457193	+	0.889367i	$0.348855\pi$
$908$	−19.3601	−0.642487
$909$	16.4499	0.545607
$910$	0	0
$911$	−25.6459	−0.849686	−0.424843	−	0.905267i	$-0.639671\pi$
$911$	−25.6459	−0.849686	−0.424843	+	0.905267i	$0.639671\pi$
$912$	2.18479	0.0723457
$913$	27.8402	0.921376
$914$	30.0993	0.995595
$915$	−68.4855	−2.26406
$916$	−5.13516	−0.169671
$917$	−58.5526	−1.93358
$918$	−0.495252	−0.0163458
$919$	−33.7178	−1.11225	−0.556124	−	0.831099i	$-0.687712\pi$
$919$	−33.7178	−1.11225	−0.556124	+	0.831099i	$0.687712\pi$
$920$	−26.4757	−0.872876
$921$	−24.5169	−0.807860
$922$	−25.1607	−0.828625
$923$	0	0
$924$	−21.8817	−0.719857
$925$	104.057	3.42136
$926$	−11.3601	−0.373316
$927$	−7.69707	−0.252805
$928$	−1.34730	−0.0442272
$929$	−19.4100	−0.636823	−0.318411	−	0.947953i	$-0.603149\pi$
$929$	−19.4100	−0.636823	−0.318411	+	0.947953i	$0.603149\pi$
$930$	−18.0113	−0.590612
$931$	−0.260830	−0.00854835
$932$	16.0051	0.524263
$933$	62.5049	2.04632
$934$	30.1884	0.987795
$935$	−2.94719	−0.0963833
$936$	0	0
$937$	−2.45842	−0.0803129	−0.0401565	−	0.999193i	$-0.512786\pi$
$937$	−2.45842	−0.0803129	−0.0401565	+	0.999193i	$0.512786\pi$
$938$	40.5134	1.32281
$939$	63.3610	2.06771
$940$	5.05056	0.164731
$941$	13.7196	0.447245	0.223623	−	0.974676i	$-0.428212\pi$
$941$	13.7196	0.447245	0.223623	+	0.974676i	$0.428212\pi$
$942$	−34.9567	−1.13895
$943$	69.4089	2.26026
$944$	2.45336	0.0798502
$945$	30.9870	1.00801
$946$	25.3601	0.824528
$947$	−1.06923	−0.0347453	−0.0173727	−	0.999849i	$-0.505530\pi$
$947$	−1.06923	−0.0347453	−0.0173727	+	0.999849i	$0.505530\pi$
$948$	−13.3892	−0.434861
$949$	0	0
$950$	−13.4115	−0.435126
$951$	59.3465	1.92444
$952$	−0.497941	−0.0161383
$953$	−22.9513	−0.743466	−0.371733	−	0.928340i	$-0.621236\pi$
$953$	−22.9513	−0.743466	−0.371733	+	0.928340i	$0.621236\pi$
$954$	−9.92396	−0.321300
$955$	51.3438	1.66145
$956$	18.1284	0.586313
$957$	10.9409	0.353668
$958$	41.2526	1.33281
$959$	−42.0392	−1.35752
$960$	−9.37464	−0.302565
$961$	−27.3087	−0.880926
$962$	0	0
$963$	−35.7820	−1.15306
$964$	−0.177985	−0.00573252
$965$	−16.2457	−0.522967
$966$	36.3250	1.16874
$967$	6.81757	0.219238	0.109619	−	0.993974i	$-0.465037\pi$
$967$	6.81757	0.219238	0.109619	+	0.993974i	$0.465037\pi$
$968$	2.81521	0.0904842
$969$	−0.403733	−0.0129698
$970$	−31.0419	−0.996695
$971$	−27.7948	−0.891977	−0.445989	−	0.895039i	$-0.647148\pi$
$971$	−27.7948	−0.891977	−0.445989	+	0.895039i	$0.647148\pi$
$972$	16.3756	0.525247
$973$	24.6500	0.790243
$974$	9.22432	0.295566
$975$	0	0
$976$	7.30541	0.233840
$977$	46.7573	1.49590	0.747950	−	0.663755i	$-0.231038\pi$
$977$	46.7573	1.49590	0.747950	+	0.663755i	$0.231038\pi$
$978$	−14.7496	−0.471640
$979$	−49.0251	−1.56685
$980$	1.11918	0.0357510
$981$	33.3275	1.06406
$982$	25.3155	0.807850
$983$	−19.2490	−0.613947	−0.306973	−	0.951718i	$-0.599316\pi$
$983$	−19.2490	−0.613947	−0.306973	+	0.951718i	$0.599316\pi$
$984$	24.5767	0.783476
$985$	−69.2045	−2.20504
$986$	0.248970	0.00792883
$987$	−6.92945	−0.220567
$988$	0	0
$989$	−42.0993	−1.33868
$990$	28.2820	0.898860
$991$	15.6513	0.497179	0.248590	−	0.968609i	$-0.420033\pi$
$991$	15.6513	0.497179	0.248590	+	0.968609i	$0.420033\pi$
$992$	1.92127	0.0610005
$993$	56.3970	1.78971
$994$	19.6987	0.624805
$995$	7.85122	0.248901
$996$	−16.3645	−0.518530
$997$	−49.2336	−1.55924	−0.779622	−	0.626251i	$-0.784589\pi$
$997$	−49.2336	−1.55924	−0.779622	+	0.626251i	$0.784589\pi$
$998$	21.6186	0.684323
$999$	20.7939	0.657888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.s.1.2		3
13.12	even	2		494.2.a.e.1.2	✓	3
39.38	odd	2		4446.2.a.bh.1.3		3
52.51	odd	2		3952.2.a.s.1.2		3
247.246	odd	2		9386.2.a.be.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.a.e.1.2	✓	3	13.12	even	2
3952.2.a.s.1.2		3	52.51	odd	2
4446.2.a.bh.1.3		3	39.38	odd	2
6422.2.a.s.1.2		3	1.1	even	1	trivial
9386.2.a.be.1.2		3	247.246	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.18479 q^{3} +1.00000 q^{4} +4.29086 q^{5} -2.18479 q^{6} +2.69459 q^{7} +1.00000 q^{8} +1.77332 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.18479 q^{3} +1.00000 q^{4} +4.29086 q^{5} -2.18479 q^{6} +2.69459 q^{7} +1.00000 q^{8} +1.77332 q^{9} +4.29086 q^{10} +3.71688 q^{11} -2.18479 q^{12} +2.69459 q^{14} -9.37464 q^{15} +1.00000 q^{16} -0.184793 q^{17} +1.77332 q^{18} -1.00000 q^{19} +4.29086 q^{20} -5.88713 q^{21} +3.71688 q^{22} -6.17024 q^{23} -2.18479 q^{24} +13.4115 q^{25} +2.68004 q^{27} +2.69459 q^{28} -1.34730 q^{29} -9.37464 q^{30} +1.92127 q^{31} +1.00000 q^{32} -8.12061 q^{33} -0.184793 q^{34} +11.5621 q^{35} +1.77332 q^{36} +7.75877 q^{37} -1.00000 q^{38} +4.29086 q^{40} -11.2490 q^{41} -5.88713 q^{42} +6.82295 q^{43} +3.71688 q^{44} +7.60906 q^{45} -6.17024 q^{46} +1.17705 q^{47} -2.18479 q^{48} +0.260830 q^{49} +13.4115 q^{50} +0.403733 q^{51} -5.59627 q^{53} +2.68004 q^{54} +15.9486 q^{55} +2.69459 q^{56} +2.18479 q^{57} -1.34730 q^{58} +2.45336 q^{59} -9.37464 q^{60} +7.30541 q^{61} +1.92127 q^{62} +4.77837 q^{63} +1.00000 q^{64} -8.12061 q^{66} +15.0351 q^{67} -0.184793 q^{68} +13.4807 q^{69} +11.5621 q^{70} +7.31046 q^{71} +1.77332 q^{72} -4.04458 q^{73} +7.75877 q^{74} -29.3013 q^{75} -1.00000 q^{76} +10.0155 q^{77} +6.12836 q^{79} +4.29086 q^{80} -11.1753 q^{81} -11.2490 q^{82} +7.49020 q^{83} -5.88713 q^{84} -0.792919 q^{85} +6.82295 q^{86} +2.94356 q^{87} +3.71688 q^{88} -13.1898 q^{89} +7.60906 q^{90} -6.17024 q^{92} -4.19759 q^{93} +1.17705 q^{94} -4.29086 q^{95} -2.18479 q^{96} -7.23442 q^{97} +0.260830 q^{98} +6.59121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 12 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 12 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 12 q^{9} - 3 q^{10} + 3 q^{11} - 3 q^{12} + 6 q^{14} - 6 q^{15} + 3 q^{16} + 3 q^{17} + 12 q^{18} - 3 q^{19} - 3 q^{20} + 12 q^{21} + 3 q^{22} + 3 q^{23} - 3 q^{24} + 30 q^{25} - 12 q^{27} + 6 q^{28} - 3 q^{29} - 6 q^{30} - 3 q^{31} + 3 q^{32} - 30 q^{33} + 3 q^{34} + 12 q^{36} + 12 q^{37} - 3 q^{38} - 3 q^{40} - 21 q^{41} + 12 q^{42} + 3 q^{44} - 30 q^{45} + 3 q^{46} + 24 q^{47} - 3 q^{48} + 15 q^{49} + 30 q^{50} + 15 q^{51} - 3 q^{53} - 12 q^{54} + 18 q^{55} + 6 q^{56} + 3 q^{57} - 3 q^{58} - 6 q^{59} - 6 q^{60} + 24 q^{61} - 3 q^{62} + 6 q^{63} + 3 q^{64} - 30 q^{66} + 3 q^{68} + 6 q^{69} + 9 q^{71} + 12 q^{72} + 12 q^{74} - 39 q^{75} - 3 q^{76} - 18 q^{77} - 3 q^{80} + 3 q^{81} - 21 q^{82} + 21 q^{83} + 12 q^{84} - 12 q^{85} - 6 q^{87} + 3 q^{88} - 21 q^{89} - 30 q^{90} + 3 q^{92} + 30 q^{93} + 24 q^{94} + 3 q^{95} - 3 q^{96} + 9 q^{97} + 15 q^{98} + 21 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 12 * q^9 - 3 * q^10 + 3 * q^11 - 3 * q^12 + 6 * q^14 - 6 * q^15 + 3 * q^16 + 3 * q^17 + 12 * q^18 - 3 * q^19 - 3 * q^20 + 12 * q^21 + 3 * q^22 + 3 * q^23 - 3 * q^24 + 30 * q^25 - 12 * q^27 + 6 * q^28 - 3 * q^29 - 6 * q^30 - 3 * q^31 + 3 * q^32 - 30 * q^33 + 3 * q^34 + 12 * q^36 + 12 * q^37 - 3 * q^38 - 3 * q^40 - 21 * q^41 + 12 * q^42 + 3 * q^44 - 30 * q^45 + 3 * q^46 + 24 * q^47 - 3 * q^48 + 15 * q^49 + 30 * q^50 + 15 * q^51 - 3 * q^53 - 12 * q^54 + 18 * q^55 + 6 * q^56 + 3 * q^57 - 3 * q^58 - 6 * q^59 - 6 * q^60 + 24 * q^61 - 3 * q^62 + 6 * q^63 + 3 * q^64 - 30 * q^66 + 3 * q^68 + 6 * q^69 + 9 * q^71 + 12 * q^72 + 12 * q^74 - 39 * q^75 - 3 * q^76 - 18 * q^77 - 3 * q^80 + 3 * q^81 - 21 * q^82 + 21 * q^83 + 12 * q^84 - 12 * q^85 - 6 * q^87 + 3 * q^88 - 21 * q^89 - 30 * q^90 + 3 * q^92 + 30 * q^93 + 24 * q^94 + 3 * q^95 - 3 * q^96 + 9 * q^97 + 15 * q^98 + 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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