LMFDB - Embedded newform 6003.2.a.r.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 $ x^16 - x^15 - 28x^14 + 27x^13 + 316x^12 - 295x^11 - 1835x^10 + 1665x^9 + 5778x^8 - 5124x^7 - 9405x^6 + 8288x^5 + 6405x^4 - 6032x^3 - 400x^2 + 1088x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.77164$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.77164 q^{2} +5.68196 q^{4} -2.29773 q^{5} -5.27354 q^{7} +10.2051 q^{8} +O(q^{10})$	\(q+2.77164 q^{2} +5.68196 q^{4} -2.29773 q^{5} -5.27354 q^{7} +10.2051 q^{8} -6.36848 q^{10} -1.67873 q^{11} +3.70083 q^{13} -14.6163 q^{14} +16.9208 q^{16} +2.24661 q^{17} +3.22268 q^{19} -13.0556 q^{20} -4.65284 q^{22} +1.00000 q^{23} +0.279573 q^{25} +10.2573 q^{26} -29.9640 q^{28} +1.00000 q^{29} +5.83534 q^{31} +26.4881 q^{32} +6.22679 q^{34} +12.1172 q^{35} -1.40968 q^{37} +8.93210 q^{38} -23.4485 q^{40} +7.14568 q^{41} +8.82562 q^{43} -9.53851 q^{44} +2.77164 q^{46} -10.7579 q^{47} +20.8102 q^{49} +0.774875 q^{50} +21.0280 q^{52} +10.6567 q^{53} +3.85728 q^{55} -53.8167 q^{56} +2.77164 q^{58} +5.86788 q^{59} +4.02077 q^{61} +16.1734 q^{62} +39.5738 q^{64} -8.50351 q^{65} -4.06540 q^{67} +12.7652 q^{68} +33.5844 q^{70} -8.18183 q^{71} -1.30400 q^{73} -3.90713 q^{74} +18.3112 q^{76} +8.85287 q^{77} +9.03636 q^{79} -38.8794 q^{80} +19.8052 q^{82} +1.52544 q^{83} -5.16211 q^{85} +24.4614 q^{86} -17.1316 q^{88} -14.5519 q^{89} -19.5164 q^{91} +5.68196 q^{92} -29.8170 q^{94} -7.40486 q^{95} +8.58188 q^{97} +57.6782 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

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 $

$2$	2.77164	1.95984	0.979921	−	0.199386i	$-0.0638947\pi$
$2$	2.77164	1.95984	0.979921	+	0.199386i	$0.0638947\pi$
$3$	0	0
$3$	0	0
$4$	5.68196	2.84098
$5$	−2.29773	−1.02758	−0.513789	−	0.857917i	$-0.671758\pi$
$5$	−2.29773	−1.02758	−0.513789	+	0.857917i	$0.671758\pi$
$6$	0	0
$7$	−5.27354	−1.99321	−0.996605	−	0.0823371i	$-0.973762\pi$
$7$	−5.27354	−1.99321	−0.996605	+	0.0823371i	$0.973762\pi$
$8$	10.2051	3.60803
$9$	0	0
$10$	−6.36848	−2.01389
$11$	−1.67873	−0.506157	−0.253079	−	0.967446i	$-0.581443\pi$
$11$	−1.67873	−0.506157	−0.253079	+	0.967446i	$0.581443\pi$
$12$	0	0
$13$	3.70083	1.02642	0.513212	−	0.858262i	$-0.328455\pi$
$13$	3.70083	1.02642	0.513212	+	0.858262i	$0.328455\pi$
$14$	−14.6163	−3.90638
$15$	0	0
$16$	16.9208	4.23019
$17$	2.24661	0.544883	0.272442	−	0.962172i	$-0.412169\pi$
$17$	2.24661	0.544883	0.272442	+	0.962172i	$0.412169\pi$
$18$	0	0
$19$	3.22268	0.739334	0.369667	−	0.929164i	$-0.379472\pi$
$19$	3.22268	0.739334	0.369667	+	0.929164i	$0.379472\pi$
$20$	−13.0556	−2.91933
$21$	0	0
$22$	−4.65284	−0.991989
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0.279573	0.0559147
$26$	10.2573	2.01163
$27$	0	0
$28$	−29.9640	−5.66267
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.83534	1.04806	0.524029	−	0.851700i	$-0.324428\pi$
$31$	5.83534	1.04806	0.524029	+	0.851700i	$0.324428\pi$
$32$	26.4881	4.68248
$33$	0	0
$34$	6.22679	1.06789
$35$	12.1172	2.04818
$36$	0	0
$37$	−1.40968	−0.231750	−0.115875	−	0.993264i	$-0.536967\pi$
$37$	−1.40968	−0.231750	−0.115875	+	0.993264i	$0.536967\pi$
$38$	8.93210	1.44898
$39$	0	0
$40$	−23.4485	−3.70753
$41$	7.14568	1.11597	0.557984	−	0.829852i	$-0.311575\pi$
$41$	7.14568	1.11597	0.557984	+	0.829852i	$0.311575\pi$
$42$	0	0
$43$	8.82562	1.34589	0.672947	−	0.739690i	$-0.265028\pi$
$43$	8.82562	1.34589	0.672947	+	0.739690i	$0.265028\pi$
$44$	−9.53851	−1.43798
$45$	0	0
$46$	2.77164	0.408655
$47$	−10.7579	−1.56920	−0.784600	−	0.620002i	$-0.787132\pi$
$47$	−10.7579	−1.56920	−0.784600	+	0.620002i	$0.787132\pi$
$48$	0	0
$49$	20.8102	2.97288
$50$	0.774875	0.109584
$51$	0	0
$52$	21.0280	2.91605
$53$	10.6567	1.46382	0.731908	−	0.681403i	$-0.238630\pi$
$53$	10.6567	1.46382	0.731908	+	0.681403i	$0.238630\pi$
$54$	0	0
$55$	3.85728	0.520116
$56$	−53.8167	−7.19156
$57$	0	0
$58$	2.77164	0.363934
$59$	5.86788	0.763933	0.381967	−	0.924176i	$-0.375247\pi$
$59$	5.86788	0.763933	0.381967	+	0.924176i	$0.375247\pi$
$60$	0	0
$61$	4.02077	0.514807	0.257404	−	0.966304i	$-0.417133\pi$
$61$	4.02077	0.514807	0.257404	+	0.966304i	$0.417133\pi$
$62$	16.1734	2.05403
$63$	0	0
$64$	39.5738	4.94672
$65$	−8.50351	−1.05473
$66$	0	0
$67$	−4.06540	−0.496667	−0.248334	−	0.968675i	$-0.579883\pi$
$67$	−4.06540	−0.496667	−0.248334	+	0.968675i	$0.579883\pi$
$68$	12.7652	1.54800
$69$	0	0
$70$	33.5844	4.01410
$71$	−8.18183	−0.971005	−0.485503	−	0.874235i	$-0.661363\pi$
$71$	−8.18183	−0.971005	−0.485503	+	0.874235i	$0.661363\pi$
$72$	0	0
$73$	−1.30400	−0.152622	−0.0763109	−	0.997084i	$-0.524314\pi$
$73$	−1.30400	−0.152622	−0.0763109	+	0.997084i	$0.524314\pi$
$74$	−3.90713	−0.454194
$75$	0	0
$76$	18.3112	2.10043
$77$	8.85287	1.00888
$78$	0	0
$79$	9.03636	1.01667	0.508335	−	0.861159i	$-0.330261\pi$
$79$	9.03636	1.01667	0.508335	+	0.861159i	$0.330261\pi$
$80$	−38.8794	−4.34685
$81$	0	0
$82$	19.8052	2.18712
$83$	1.52544	0.167438	0.0837191	−	0.996489i	$-0.473320\pi$
$83$	1.52544	0.167438	0.0837191	+	0.996489i	$0.473320\pi$
$84$	0	0
$85$	−5.16211	−0.559910
$86$	24.4614	2.63774
$87$	0	0
$88$	−17.1316	−1.82623
$89$	−14.5519	−1.54249	−0.771247	−	0.636535i	$-0.780367\pi$
$89$	−14.5519	−1.54249	−0.771247	+	0.636535i	$0.780367\pi$
$90$	0	0
$91$	−19.5164	−2.04588
$92$	5.68196	0.592385
$93$	0	0
$94$	−29.8170	−3.07538
$95$	−7.40486	−0.759723
$96$	0	0
$97$	8.58188	0.871358	0.435679	−	0.900102i	$-0.356508\pi$
$97$	8.58188	0.871358	0.435679	+	0.900102i	$0.356508\pi$
$98$	57.6782	5.82638
$99$	0	0
$100$	1.58853	0.158853
$101$	16.9500	1.68659	0.843295	−	0.537451i	$-0.180613\pi$
$101$	16.9500	1.68659	0.843295	+	0.537451i	$0.180613\pi$
$102$	0	0
$103$	11.1996	1.10353	0.551766	−	0.833999i	$-0.313954\pi$
$103$	11.1996	1.10353	0.551766	+	0.833999i	$0.313954\pi$
$104$	37.7671	3.70337
$105$	0	0
$106$	29.5366	2.86885
$107$	−10.1710	−0.983268	−0.491634	−	0.870802i	$-0.663600\pi$
$107$	−10.1710	−0.983268	−0.491634	+	0.870802i	$0.663600\pi$
$108$	0	0
$109$	2.78611	0.266861	0.133431	−	0.991058i	$-0.457401\pi$
$109$	2.78611	0.266861	0.133431	+	0.991058i	$0.457401\pi$
$110$	10.6910	1.01934
$111$	0	0
$112$	−89.2323	−8.43166
$113$	−14.9996	−1.41104	−0.705522	−	0.708688i	$-0.749287\pi$
$113$	−14.9996	−1.41104	−0.705522	+	0.708688i	$0.749287\pi$
$114$	0	0
$115$	−2.29773	−0.214265
$116$	5.68196	0.527557
$117$	0	0
$118$	16.2636	1.49719
$119$	−11.8476	−1.08607
$120$	0	0
$121$	−8.18185	−0.743805
$122$	11.1441	1.00894
$123$	0	0
$124$	33.1562	2.97751
$125$	10.8463	0.970120
$126$	0	0
$127$	9.47544	0.840809	0.420405	−	0.907337i	$-0.361888\pi$
$127$	9.47544	0.840809	0.420405	+	0.907337i	$0.361888\pi$
$128$	56.7079	5.01232
$129$	0	0
$130$	−23.5686	−2.06711
$131$	−10.2890	−0.898951	−0.449475	−	0.893293i	$-0.648389\pi$
$131$	−10.2890	−0.898951	−0.449475	+	0.893293i	$0.648389\pi$
$132$	0	0
$133$	−16.9949	−1.47365
$134$	−11.2678	−0.973389
$135$	0	0
$136$	22.9268	1.96596
$137$	9.72484	0.830849	0.415425	−	0.909628i	$-0.363633\pi$
$137$	9.72484	0.830849	0.415425	+	0.909628i	$0.363633\pi$
$138$	0	0
$139$	−9.72154	−0.824570	−0.412285	−	0.911055i	$-0.635269\pi$
$139$	−9.72154	−0.824570	−0.412285	+	0.911055i	$0.635269\pi$
$140$	68.8493	5.81883
$141$	0	0
$142$	−22.6771	−1.90302
$143$	−6.21271	−0.519533
$144$	0	0
$145$	−2.29773	−0.190816
$146$	−3.61421	−0.299115
$147$	0	0
$148$	−8.00977	−0.658399
$149$	−14.5112	−1.18880	−0.594400	−	0.804169i	$-0.702611\pi$
$149$	−14.5112	−1.18880	−0.594400	+	0.804169i	$0.702611\pi$
$150$	0	0
$151$	3.76576	0.306453	0.153226	−	0.988191i	$-0.451034\pi$
$151$	3.76576	0.306453	0.153226	+	0.988191i	$0.451034\pi$
$152$	32.8877	2.66754
$153$	0	0
$154$	24.5369	1.97724
$155$	−13.4081	−1.07696
$156$	0	0
$157$	17.5108	1.39752	0.698758	−	0.715358i	$-0.253737\pi$
$157$	17.5108	1.39752	0.698758	+	0.715358i	$0.253737\pi$
$158$	25.0455	1.99251
$159$	0	0
$160$	−60.8625	−4.81161
$161$	−5.27354	−0.415613
$162$	0	0
$163$	−17.2023	−1.34738	−0.673692	−	0.739012i	$-0.735293\pi$
$163$	−17.2023	−1.34738	−0.673692	+	0.739012i	$0.735293\pi$
$164$	40.6015	3.17044
$165$	0	0
$166$	4.22795	0.328153
$167$	9.74013	0.753714	0.376857	−	0.926271i	$-0.377005\pi$
$167$	9.74013	0.753714	0.376857	+	0.926271i	$0.377005\pi$
$168$	0	0
$169$	0.696119	0.0535476
$170$	−14.3075	−1.09733
$171$	0	0
$172$	50.1468	3.82366
$173$	−20.1382	−1.53108	−0.765541	−	0.643387i	$-0.777529\pi$
$173$	−20.1382	−1.53108	−0.765541	+	0.643387i	$0.777529\pi$
$174$	0	0
$175$	−1.47434	−0.111450
$176$	−28.4055	−2.14114
$177$	0	0
$178$	−40.3325	−3.02305
$179$	−10.6994	−0.799714	−0.399857	−	0.916578i	$-0.630940\pi$
$179$	−10.6994	−0.799714	−0.399857	+	0.916578i	$0.630940\pi$
$180$	0	0
$181$	−20.1216	−1.49563	−0.747814	−	0.663908i	$-0.768896\pi$
$181$	−20.1216	−1.49563	−0.747814	+	0.663908i	$0.768896\pi$
$182$	−54.0925	−4.00960
$183$	0	0
$184$	10.2051	0.752327
$185$	3.23907	0.238141
$186$	0	0
$187$	−3.77146	−0.275797
$188$	−61.1260	−4.45807
$189$	0	0
$190$	−20.5236	−1.48894
$191$	−16.9007	−1.22289	−0.611446	−	0.791286i	$-0.709412\pi$
$191$	−16.9007	−1.22289	−0.611446	+	0.791286i	$0.709412\pi$
$192$	0	0
$193$	−2.84220	−0.204586	−0.102293	−	0.994754i	$-0.532618\pi$
$193$	−2.84220	−0.204586	−0.102293	+	0.994754i	$0.532618\pi$
$194$	23.7859	1.70772
$195$	0	0
$196$	118.243	8.44590
$197$	1.41372	0.100723	0.0503616	−	0.998731i	$-0.483963\pi$
$197$	1.41372	0.100723	0.0503616	+	0.998731i	$0.483963\pi$
$198$	0	0
$199$	19.5480	1.38572	0.692860	−	0.721072i	$-0.256350\pi$
$199$	19.5480	1.38572	0.692860	+	0.721072i	$0.256350\pi$
$200$	2.85306	0.201742
$201$	0	0
$202$	46.9793	3.30545
$203$	−5.27354	−0.370130
$204$	0	0
$205$	−16.4189	−1.14674
$206$	31.0413	2.16275
$207$	0	0
$208$	62.6208	4.34197
$209$	−5.41003	−0.374220
$210$	0	0
$211$	15.2346	1.04879	0.524397	−	0.851474i	$-0.324291\pi$
$211$	15.2346	1.04879	0.524397	+	0.851474i	$0.324291\pi$
$212$	60.5512	4.15867
$213$	0	0
$214$	−28.1903	−1.92705
$215$	−20.2789	−1.38301
$216$	0	0
$217$	−30.7729	−2.08900
$218$	7.72208	0.523006
$219$	0	0
$220$	21.9169	1.47764
$221$	8.31432	0.559282
$222$	0	0
$223$	−3.31580	−0.222042	−0.111021	−	0.993818i	$-0.535412\pi$
$223$	−3.31580	−0.222042	−0.111021	+	0.993818i	$0.535412\pi$
$224$	−139.686	−9.33316
$225$	0	0
$226$	−41.5734	−2.76542
$227$	6.38701	0.423921	0.211960	−	0.977278i	$-0.432015\pi$
$227$	6.38701	0.423921	0.211960	+	0.977278i	$0.432015\pi$
$228$	0	0
$229$	−1.71280	−0.113185	−0.0565926	−	0.998397i	$-0.518024\pi$
$229$	−1.71280	−0.113185	−0.0565926	+	0.998397i	$0.518024\pi$
$230$	−6.36848	−0.419925
$231$	0	0
$232$	10.2051	0.669995
$233$	10.2185	0.669434	0.334717	−	0.942319i	$-0.391359\pi$
$233$	10.2185	0.669434	0.334717	+	0.942319i	$0.391359\pi$
$234$	0	0
$235$	24.7188	1.61247
$236$	33.3411	2.17032
$237$	0	0
$238$	−32.8372	−2.12852
$239$	18.5912	1.20256	0.601282	−	0.799037i	$-0.294657\pi$
$239$	18.5912	1.20256	0.601282	+	0.799037i	$0.294657\pi$
$240$	0	0
$241$	22.8983	1.47501	0.737504	−	0.675343i	$-0.236004\pi$
$241$	22.8983	1.47501	0.737504	+	0.675343i	$0.236004\pi$
$242$	−22.6771	−1.45774
$243$	0	0
$244$	22.8459	1.46256
$245$	−47.8162	−3.05487
$246$	0	0
$247$	11.9266	0.758871
$248$	59.5500	3.78143
$249$	0	0
$250$	30.0619	1.90128
$251$	−18.8395	−1.18914	−0.594568	−	0.804045i	$-0.702677\pi$
$251$	−18.8395	−1.18914	−0.594568	+	0.804045i	$0.702677\pi$
$252$	0	0
$253$	−1.67873	−0.105541
$254$	26.2625	1.64785
$255$	0	0
$256$	78.0261	4.87663
$257$	14.3969	0.898054	0.449027	−	0.893518i	$-0.351771\pi$
$257$	14.3969	0.898054	0.449027	+	0.893518i	$0.351771\pi$
$258$	0	0
$259$	7.43401	0.461927
$260$	−48.3166	−2.99647
$261$	0	0
$262$	−28.5173	−1.76180
$263$	−1.78382	−0.109995	−0.0549976	−	0.998486i	$-0.517515\pi$
$263$	−1.78382	−0.109995	−0.0549976	+	0.998486i	$0.517515\pi$
$264$	0	0
$265$	−24.4863	−1.50418
$266$	−47.1038	−2.88812
$267$	0	0
$268$	−23.0994	−1.41102
$269$	2.57069	0.156738	0.0783689	−	0.996924i	$-0.475029\pi$
$269$	2.57069	0.156738	0.0783689	+	0.996924i	$0.475029\pi$
$270$	0	0
$271$	−5.38793	−0.327293	−0.163647	−	0.986519i	$-0.552326\pi$
$271$	−5.38793	−0.327293	−0.163647	+	0.986519i	$0.552326\pi$
$272$	38.0144	2.30496
$273$	0	0
$274$	26.9537	1.62833
$275$	−0.469329	−0.0283016
$276$	0	0
$277$	25.2470	1.51694	0.758472	−	0.651705i	$-0.225946\pi$
$277$	25.2470	1.51694	0.758472	+	0.651705i	$0.225946\pi$
$278$	−26.9446	−1.61603
$279$	0	0
$280$	123.656	7.38988
$281$	12.6699	0.755823	0.377912	−	0.925842i	$-0.376642\pi$
$281$	12.6699	0.755823	0.377912	+	0.925842i	$0.376642\pi$
$282$	0	0
$283$	22.1074	1.31415	0.657073	−	0.753827i	$-0.271794\pi$
$283$	22.1074	1.31415	0.657073	+	0.753827i	$0.271794\pi$
$284$	−46.4889	−2.75861
$285$	0	0
$286$	−17.2194	−1.01820
$287$	−37.6830	−2.22436
$288$	0	0
$289$	−11.9527	−0.703102
$290$	−6.36848	−0.373970
$291$	0	0
$292$	−7.40928	−0.433595
$293$	22.1453	1.29374	0.646870	−	0.762600i	$-0.276078\pi$
$293$	22.1453	1.29374	0.646870	+	0.762600i	$0.276078\pi$
$294$	0	0
$295$	−13.4828	−0.785000
$296$	−14.3859	−0.836163
$297$	0	0
$298$	−40.2197	−2.32986
$299$	3.70083	0.214024
$300$	0	0
$301$	−46.5422	−2.68265
$302$	10.4373	0.600599
$303$	0	0
$304$	54.5303	3.12753
$305$	−9.23866	−0.529004
$306$	0	0
$307$	13.9191	0.794406	0.397203	−	0.917731i	$-0.369981\pi$
$307$	13.9191	0.794406	0.397203	+	0.917731i	$0.369981\pi$
$308$	50.3016	2.86620
$309$	0	0
$310$	−37.1622	−2.11067
$311$	5.38930	0.305599	0.152799	−	0.988257i	$-0.451171\pi$
$311$	5.38930	0.305599	0.152799	+	0.988257i	$0.451171\pi$
$312$	0	0
$313$	−22.3895	−1.26553	−0.632764	−	0.774345i	$-0.718080\pi$
$313$	−22.3895	−1.26553	−0.632764	+	0.774345i	$0.718080\pi$
$314$	48.5336	2.73891
$315$	0	0
$316$	51.3443	2.88834
$317$	−3.60388	−0.202414	−0.101207	−	0.994865i	$-0.532270\pi$
$317$	−3.60388	−0.202414	−0.101207	+	0.994865i	$0.532270\pi$
$318$	0	0
$319$	−1.67873	−0.0939911
$320$	−90.9300	−5.08314
$321$	0	0
$322$	−14.6163	−0.814536
$323$	7.24012	0.402851
$324$	0	0
$325$	1.03465	0.0573922
$326$	−47.6784	−2.64066
$327$	0	0
$328$	72.9221	4.02645
$329$	56.7321	3.12774
$330$	0	0
$331$	−28.4927	−1.56610	−0.783051	−	0.621958i	$-0.786338\pi$
$331$	−28.4927	−1.56610	−0.783051	+	0.621958i	$0.786338\pi$
$332$	8.66747	0.475689
$333$	0	0
$334$	26.9961	1.47716
$335$	9.34119	0.510364
$336$	0	0
$337$	24.0084	1.30782	0.653911	−	0.756572i	$-0.273127\pi$
$337$	24.0084	1.30782	0.653911	+	0.756572i	$0.273127\pi$
$338$	1.92939	0.104945
$339$	0	0
$340$	−29.3309	−1.59069
$341$	−9.79599	−0.530483
$342$	0	0
$343$	−72.8285	−3.93237
$344$	90.0660	4.85603
$345$	0	0
$346$	−55.8159	−3.00068
$347$	−30.5553	−1.64029	−0.820146	−	0.572154i	$-0.806108\pi$
$347$	−30.5553	−1.64029	−0.820146	+	0.572154i	$0.806108\pi$
$348$	0	0
$349$	−10.0999	−0.540635	−0.270317	−	0.962771i	$-0.587129\pi$
$349$	−10.0999	−0.540635	−0.270317	+	0.962771i	$0.587129\pi$
$350$	−4.08633	−0.218424
$351$	0	0
$352$	−44.4665	−2.37007
$353$	−19.2053	−1.02220	−0.511098	−	0.859522i	$-0.670761\pi$
$353$	−19.2053	−1.02220	−0.511098	+	0.859522i	$0.670761\pi$
$354$	0	0
$355$	18.7997	0.997783
$356$	−82.6832	−4.38220
$357$	0	0
$358$	−29.6550	−1.56731
$359$	−22.4237	−1.18348	−0.591740	−	0.806129i	$-0.701559\pi$
$359$	−22.4237	−1.18348	−0.591740	+	0.806129i	$0.701559\pi$
$360$	0	0
$361$	−8.61432	−0.453385
$362$	−55.7698	−2.93119
$363$	0	0
$364$	−110.892	−5.81230
$365$	2.99624	0.156831
$366$	0	0
$367$	22.1059	1.15392	0.576960	−	0.816772i	$-0.304239\pi$
$367$	22.1059	1.15392	0.576960	+	0.816772i	$0.304239\pi$
$368$	16.9208	0.882056
$369$	0	0
$370$	8.97753	0.466720
$371$	−56.1987	−2.91769
$372$	0	0
$373$	−14.0351	−0.726709	−0.363354	−	0.931651i	$-0.618369\pi$
$373$	−14.0351	−0.726709	−0.363354	+	0.931651i	$0.618369\pi$
$374$	−10.4531	−0.540518
$375$	0	0
$376$	−109.785	−5.66173
$377$	3.70083	0.190602
$378$	0	0
$379$	−21.1240	−1.08507	−0.542534	−	0.840034i	$-0.682535\pi$
$379$	−21.1240	−1.08507	−0.542534	+	0.840034i	$0.682535\pi$
$380$	−42.0741	−2.15836
$381$	0	0
$382$	−46.8426	−2.39668
$383$	−16.0170	−0.818431	−0.409216	−	0.912438i	$-0.634198\pi$
$383$	−16.0170	−0.818431	−0.409216	+	0.912438i	$0.634198\pi$
$384$	0	0
$385$	−20.3415	−1.03670
$386$	−7.87754	−0.400956
$387$	0	0
$388$	48.7619	2.47551
$389$	−9.48004	−0.480657	−0.240329	−	0.970692i	$-0.577255\pi$
$389$	−9.48004	−0.480657	−0.240329	+	0.970692i	$0.577255\pi$
$390$	0	0
$391$	2.24661	0.113616
$392$	212.369	10.7263
$393$	0	0
$394$	3.91831	0.197402
$395$	−20.7631	−1.04471
$396$	0	0
$397$	10.8768	0.545892	0.272946	−	0.962029i	$-0.412002\pi$
$397$	10.8768	0.545892	0.272946	+	0.962029i	$0.412002\pi$
$398$	54.1799	2.71579
$399$	0	0
$400$	4.73060	0.236530
$401$	17.0386	0.850869	0.425435	−	0.904989i	$-0.360121\pi$
$401$	17.0386	0.850869	0.425435	+	0.904989i	$0.360121\pi$
$402$	0	0
$403$	21.5956	1.07575
$404$	96.3094	4.79157
$405$	0	0
$406$	−14.6163	−0.725396
$407$	2.36648	0.117302
$408$	0	0
$409$	7.07988	0.350077	0.175039	−	0.984562i	$-0.443995\pi$
$409$	7.07988	0.350077	0.175039	+	0.984562i	$0.443995\pi$
$410$	−45.5071	−2.24743
$411$	0	0
$412$	63.6359	3.13512
$413$	−30.9445	−1.52268
$414$	0	0
$415$	−3.50504	−0.172056
$416$	98.0278	4.80621
$417$	0	0
$418$	−14.9946	−0.733411
$419$	−16.0016	−0.781730	−0.390865	−	0.920448i	$-0.627824\pi$
$419$	−16.0016	−0.781730	−0.390865	+	0.920448i	$0.627824\pi$
$420$	0	0
$421$	12.4817	0.608322	0.304161	−	0.952621i	$-0.401624\pi$
$421$	12.4817	0.608322	0.304161	+	0.952621i	$0.401624\pi$
$422$	42.2248	2.05547
$423$	0	0
$424$	108.753	5.28150
$425$	0.628093	0.0304670
$426$	0	0
$427$	−21.2037	−1.02612
$428$	−57.7912	−2.79344
$429$	0	0
$430$	−56.2058	−2.71048
$431$	−6.72467	−0.323916	−0.161958	−	0.986798i	$-0.551781\pi$
$431$	−6.72467	−0.323916	−0.161958	+	0.986798i	$0.551781\pi$
$432$	0	0
$433$	8.87260	0.426390	0.213195	−	0.977010i	$-0.431613\pi$
$433$	8.87260	0.426390	0.213195	+	0.977010i	$0.431613\pi$
$434$	−85.2912	−4.09411
$435$	0	0
$436$	15.8306	0.758147
$437$	3.22268	0.154162
$438$	0	0
$439$	28.8671	1.37775	0.688877	−	0.724878i	$-0.258104\pi$
$439$	28.8671	1.37775	0.688877	+	0.724878i	$0.258104\pi$
$440$	39.3638	1.87659
$441$	0	0
$442$	23.0443	1.09610
$443$	2.97745	0.141463	0.0707314	−	0.997495i	$-0.477467\pi$
$443$	2.97745	0.141463	0.0707314	+	0.997495i	$0.477467\pi$
$444$	0	0
$445$	33.4363	1.58503
$446$	−9.19018	−0.435168
$447$	0	0
$448$	−208.694	−9.85985
$449$	−6.31768	−0.298150	−0.149075	−	0.988826i	$-0.547630\pi$
$449$	−6.31768	−0.298150	−0.149075	+	0.988826i	$0.547630\pi$
$450$	0	0
$451$	−11.9957	−0.564855
$452$	−85.2271	−4.00875
$453$	0	0
$454$	17.7025	0.830818
$455$	44.8436	2.10230
$456$	0	0
$457$	7.06653	0.330558	0.165279	−	0.986247i	$-0.447148\pi$
$457$	7.06653	0.330558	0.165279	+	0.986247i	$0.447148\pi$
$458$	−4.74726	−0.221825
$459$	0	0
$460$	−13.0556	−0.608722
$461$	−12.9326	−0.602332	−0.301166	−	0.953572i	$-0.597376\pi$
$461$	−12.9326	−0.602332	−0.301166	+	0.953572i	$0.597376\pi$
$462$	0	0
$463$	−42.0848	−1.95585	−0.977924	−	0.208963i	$-0.932991\pi$
$463$	−42.0848	−1.95585	−0.977924	+	0.208963i	$0.932991\pi$
$464$	16.9208	0.785527
$465$	0	0
$466$	28.3219	1.31198
$467$	−5.03119	−0.232816	−0.116408	−	0.993202i	$-0.537138\pi$
$467$	−5.03119	−0.232816	−0.116408	+	0.993202i	$0.537138\pi$
$468$	0	0
$469$	21.4390	0.989961
$470$	68.5114	3.16019
$471$	0	0
$472$	59.8821	2.75630
$473$	−14.8159	−0.681235
$474$	0	0
$475$	0.900976	0.0413396
$476$	−67.3175	−3.08549
$477$	0	0
$478$	51.5280	2.35684
$479$	−11.6047	−0.530231	−0.265115	−	0.964217i	$-0.585410\pi$
$479$	−11.6047	−0.530231	−0.265115	+	0.964217i	$0.585410\pi$
$480$	0	0
$481$	−5.21699	−0.237874
$482$	63.4656	2.89078
$483$	0	0
$484$	−46.4890	−2.11313
$485$	−19.7189	−0.895388
$486$	0	0
$487$	8.92181	0.404286	0.202143	−	0.979356i	$-0.435209\pi$
$487$	8.92181	0.404286	0.202143	+	0.979356i	$0.435209\pi$
$488$	41.0322	1.85744
$489$	0	0
$490$	−132.529	−5.98705
$491$	−12.3990	−0.559558	−0.279779	−	0.960064i	$-0.590261\pi$
$491$	−12.3990	−0.559558	−0.279779	+	0.960064i	$0.590261\pi$
$492$	0	0
$493$	2.24661	0.101182
$494$	33.0562	1.48727
$495$	0	0
$496$	98.7385	4.43349
$497$	43.1472	1.93542
$498$	0	0
$499$	−35.9590	−1.60974	−0.804872	−	0.593448i	$-0.797766\pi$
$499$	−35.9590	−1.60974	−0.804872	+	0.593448i	$0.797766\pi$
$500$	61.6281	2.75609
$501$	0	0
$502$	−52.2161	−2.33052
$503$	24.7041	1.10150	0.550751	−	0.834669i	$-0.314341\pi$
$503$	24.7041	1.10150	0.550751	+	0.834669i	$0.314341\pi$
$504$	0	0
$505$	−38.9466	−1.73310
$506$	−4.65284	−0.206844
$507$	0	0
$508$	53.8391	2.38872
$509$	33.8638	1.50099	0.750494	−	0.660877i	$-0.229816\pi$
$509$	33.8638	1.50099	0.750494	+	0.660877i	$0.229816\pi$
$510$	0	0
$511$	6.87669	0.304207
$512$	102.844	4.54511
$513$	0	0
$514$	39.9030	1.76004
$515$	−25.7338	−1.13397
$516$	0	0
$517$	18.0596	0.794262
$518$	20.6044	0.905304
$519$	0	0
$520$	−86.7788	−3.80550
$521$	−13.8126	−0.605140	−0.302570	−	0.953127i	$-0.597845\pi$
$521$	−13.8126	−0.605140	−0.302570	+	0.953127i	$0.597845\pi$
$522$	0	0
$523$	4.38916	0.191925	0.0959624	−	0.995385i	$-0.469407\pi$
$523$	4.38916	0.191925	0.0959624	+	0.995385i	$0.469407\pi$
$524$	−58.4615	−2.55390
$525$	0	0
$526$	−4.94410	−0.215573
$527$	13.1097	0.571070
$528$	0	0
$529$	1.00000	0.0434783
$530$	−67.8672	−2.94796
$531$	0	0
$532$	−96.5646	−4.18660
$533$	26.4449	1.14546
$534$	0	0
$535$	23.3702	1.01038
$536$	−41.4876	−1.79199
$537$	0	0
$538$	7.12502	0.307181
$539$	−34.9348	−1.50475
$540$	0	0
$541$	−30.7399	−1.32161	−0.660806	−	0.750556i	$-0.729786\pi$
$541$	−30.7399	−1.32161	−0.660806	+	0.750556i	$0.729786\pi$
$542$	−14.9334	−0.641443
$543$	0	0
$544$	59.5084	2.55140
$545$	−6.40174	−0.274220
$546$	0	0
$547$	−7.41249	−0.316935	−0.158468	−	0.987364i	$-0.550655\pi$
$547$	−7.41249	−0.316935	−0.158468	+	0.987364i	$0.550655\pi$
$548$	55.2562	2.36043
$549$	0	0
$550$	−1.30081	−0.0554667
$551$	3.22268	0.137291
$552$	0	0
$553$	−47.6536	−2.02644
$554$	69.9755	2.97297
$555$	0	0
$556$	−55.2374	−2.34259
$557$	5.78672	0.245191	0.122596	−	0.992457i	$-0.460878\pi$
$557$	5.78672	0.245191	0.122596	+	0.992457i	$0.460878\pi$
$558$	0	0
$559$	32.6621	1.38146
$560$	205.032	8.66418
$561$	0	0
$562$	35.1164	1.48129
$563$	22.3792	0.943170	0.471585	−	0.881821i	$-0.343682\pi$
$563$	22.3792	0.943170	0.471585	+	0.881821i	$0.343682\pi$
$564$	0	0
$565$	34.4651	1.44996
$566$	61.2735	2.57552
$567$	0	0
$568$	−83.4961	−3.50342
$569$	−17.3697	−0.728175	−0.364088	−	0.931365i	$-0.618619\pi$
$569$	−17.3697	−0.728175	−0.364088	+	0.931365i	$0.618619\pi$
$570$	0	0
$571$	−30.1747	−1.26277	−0.631386	−	0.775469i	$-0.717514\pi$
$571$	−30.1747	−1.26277	−0.631386	+	0.775469i	$0.717514\pi$
$572$	−35.3004	−1.47598
$573$	0	0
$574$	−104.444	−4.35939
$575$	0.279573	0.0116590
$576$	0	0
$577$	7.97999	0.332212	0.166106	−	0.986108i	$-0.446881\pi$
$577$	7.97999	0.332212	0.166106	+	0.986108i	$0.446881\pi$
$578$	−33.1286	−1.37797
$579$	0	0
$580$	−13.0556	−0.542105
$581$	−8.04444	−0.333739
$582$	0	0
$583$	−17.8898	−0.740921
$584$	−13.3074	−0.550664
$585$	0	0
$586$	61.3786	2.53553
$587$	−19.7257	−0.814169	−0.407084	−	0.913391i	$-0.633454\pi$
$587$	−19.7257	−0.814169	−0.407084	+	0.913391i	$0.633454\pi$
$588$	0	0
$589$	18.8055	0.774866
$590$	−37.3695	−1.53848
$591$	0	0
$592$	−23.8529	−0.980349
$593$	37.3676	1.53450	0.767252	−	0.641345i	$-0.221623\pi$
$593$	37.3676	1.53450	0.767252	+	0.641345i	$0.221623\pi$
$594$	0	0
$595$	27.2226	1.11602
$596$	−82.4519	−3.37736
$597$	0	0
$598$	10.2573	0.419454
$599$	16.4020	0.670167	0.335083	−	0.942189i	$-0.391236\pi$
$599$	16.4020	0.670167	0.335083	+	0.942189i	$0.391236\pi$
$600$	0	0
$601$	−38.6689	−1.57734	−0.788668	−	0.614819i	$-0.789229\pi$
$601$	−38.6689	−1.57734	−0.788668	+	0.614819i	$0.789229\pi$
$602$	−128.998	−5.25757
$603$	0	0
$604$	21.3969	0.870627
$605$	18.7997	0.764317
$606$	0	0
$607$	25.8529	1.04934	0.524669	−	0.851307i	$-0.324189\pi$
$607$	25.8529	1.04934	0.524669	+	0.851307i	$0.324189\pi$
$608$	85.3627	3.46191
$609$	0	0
$610$	−25.6062	−1.03676
$611$	−39.8131	−1.61067
$612$	0	0
$613$	−12.9124	−0.521527	−0.260764	−	0.965403i	$-0.583974\pi$
$613$	−12.9124	−0.521527	−0.260764	+	0.965403i	$0.583974\pi$
$614$	38.5787	1.55691
$615$	0	0
$616$	90.3440	3.64006
$617$	26.7525	1.07701	0.538507	−	0.842621i	$-0.318988\pi$
$617$	26.7525	1.07701	0.538507	+	0.842621i	$0.318988\pi$
$618$	0	0
$619$	−30.7646	−1.23653	−0.618266	−	0.785969i	$-0.712165\pi$
$619$	−30.7646	−1.23653	−0.618266	+	0.785969i	$0.712165\pi$
$620$	−76.1841	−3.05963
$621$	0	0
$622$	14.9372	0.598926
$623$	76.7398	3.07451
$624$	0	0
$625$	−26.3197	−1.05279
$626$	−62.0554	−2.48023
$627$	0	0
$628$	99.4958	3.97031
$629$	−3.16701	−0.126277
$630$	0	0
$631$	11.5124	0.458302	0.229151	−	0.973391i	$-0.426405\pi$
$631$	11.5124	0.458302	0.229151	+	0.973391i	$0.426405\pi$
$632$	92.2166	3.66818
$633$	0	0
$634$	−9.98863	−0.396699
$635$	−21.7720	−0.863996
$636$	0	0
$637$	77.0149	3.05144
$638$	−4.65284	−0.184208
$639$	0	0
$640$	−130.300	−5.15054
$641$	−30.2176	−1.19353	−0.596763	−	0.802418i	$-0.703547\pi$
$641$	−30.2176	−1.19353	−0.596763	+	0.802418i	$0.703547\pi$
$642$	0	0
$643$	0.0750745	0.00296065	0.00148033	−	0.999999i	$-0.499529\pi$
$643$	0.0750745	0.00296065	0.00148033	+	0.999999i	$0.499529\pi$
$644$	−29.9640	−1.18075
$645$	0	0
$646$	20.0670	0.789524
$647$	−28.4909	−1.12009	−0.560045	−	0.828462i	$-0.689216\pi$
$647$	−28.4909	−1.12009	−0.560045	+	0.828462i	$0.689216\pi$
$648$	0	0
$649$	−9.85062	−0.386670
$650$	2.86768	0.112480
$651$	0	0
$652$	−97.7426	−3.82789
$653$	20.8705	0.816726	0.408363	−	0.912820i	$-0.366100\pi$
$653$	20.8705	0.816726	0.408363	+	0.912820i	$0.366100\pi$
$654$	0	0
$655$	23.6413	0.923741
$656$	120.910	4.72076
$657$	0	0
$658$	157.241	6.12988
$659$	50.7944	1.97867	0.989334	−	0.145664i	$-0.0465318\pi$
$659$	50.7944	1.97867	0.989334	+	0.145664i	$0.0465318\pi$
$660$	0	0
$661$	−3.65855	−0.142301	−0.0711505	−	0.997466i	$-0.522667\pi$
$661$	−3.65855	−0.142301	−0.0711505	+	0.997466i	$0.522667\pi$
$662$	−78.9714	−3.06931
$663$	0	0
$664$	15.5672	0.604123
$665$	39.0498	1.51429
$666$	0	0
$667$	1.00000	0.0387202
$668$	55.3431	2.14129
$669$	0	0
$670$	25.8904	1.00023
$671$	−6.74981	−0.260573
$672$	0	0
$673$	−14.6688	−0.565440	−0.282720	−	0.959202i	$-0.591237\pi$
$673$	−14.6688	−0.565440	−0.282720	+	0.959202i	$0.591237\pi$
$674$	66.5426	2.56312
$675$	0	0
$676$	3.95532	0.152128
$677$	35.1696	1.35168	0.675839	−	0.737050i	$-0.263782\pi$
$677$	35.1696	1.35168	0.675839	+	0.737050i	$0.263782\pi$
$678$	0	0
$679$	−45.2569	−1.73680
$680$	−52.6796	−2.02017
$681$	0	0
$682$	−27.1509	−1.03966
$683$	44.8115	1.71466	0.857332	−	0.514763i	$-0.172120\pi$
$683$	44.8115	1.71466	0.857332	+	0.514763i	$0.172120\pi$
$684$	0	0
$685$	−22.3451	−0.853761
$686$	−201.854	−7.70682
$687$	0	0
$688$	149.336	5.69339
$689$	39.4388	1.50250
$690$	0	0
$691$	−3.58113	−0.136233	−0.0681163	−	0.997677i	$-0.521699\pi$
$691$	−3.58113	−0.136233	−0.0681163	+	0.997677i	$0.521699\pi$
$692$	−114.425	−4.34978
$693$	0	0
$694$	−84.6881	−3.21471
$695$	22.3375	0.847310
$696$	0	0
$697$	16.0536	0.608072
$698$	−27.9932	−1.05956
$699$	0	0
$700$	−8.37714	−0.316626
$701$	−17.0168	−0.642714	−0.321357	−	0.946958i	$-0.604139\pi$
$701$	−17.0168	−0.642714	−0.321357	+	0.946958i	$0.604139\pi$
$702$	0	0
$703$	−4.54296	−0.171341
$704$	−66.4339	−2.50382
$705$	0	0
$706$	−53.2302	−2.00334
$707$	−89.3865	−3.36173
$708$	0	0
$709$	21.7959	0.818561	0.409281	−	0.912409i	$-0.365780\pi$
$709$	21.7959	0.818561	0.409281	+	0.912409i	$0.365780\pi$
$710$	52.1058	1.95550
$711$	0	0
$712$	−148.503	−5.56537
$713$	5.83534	0.218535
$714$	0	0
$715$	14.2751	0.533860
$716$	−60.7938	−2.27197
$717$	0	0
$718$	−62.1504	−2.31943
$719$	50.2327	1.87336	0.936682	−	0.350182i	$-0.113880\pi$
$719$	50.2327	1.87336	0.936682	+	0.350182i	$0.113880\pi$
$720$	0	0
$721$	−59.0617	−2.19957
$722$	−23.8757	−0.888563
$723$	0	0
$724$	−114.330	−4.24905
$725$	0.279573	0.0103831
$726$	0	0
$727$	−25.4213	−0.942823	−0.471411	−	0.881913i	$-0.656255\pi$
$727$	−25.4213	−0.942823	−0.471411	+	0.881913i	$0.656255\pi$
$728$	−199.166	−7.38160
$729$	0	0
$730$	8.30450	0.307363
$731$	19.8277	0.733355
$732$	0	0
$733$	27.3676	1.01085	0.505423	−	0.862872i	$-0.331336\pi$
$733$	27.3676	1.01085	0.505423	+	0.862872i	$0.331336\pi$
$734$	61.2696	2.26150
$735$	0	0
$736$	26.4881	0.976364
$737$	6.82472	0.251392
$738$	0	0
$739$	−10.5300	−0.387351	−0.193675	−	0.981066i	$-0.562041\pi$
$739$	−10.5300	−0.387351	−0.193675	+	0.981066i	$0.562041\pi$
$740$	18.4043	0.676555
$741$	0	0
$742$	−155.762	−5.71821
$743$	−37.2813	−1.36772	−0.683860	−	0.729613i	$-0.739700\pi$
$743$	−37.2813	−1.36772	−0.683860	+	0.729613i	$0.739700\pi$
$744$	0	0
$745$	33.3428	1.22158
$746$	−38.9001	−1.42423
$747$	0	0
$748$	−21.4293	−0.783533
$749$	53.6371	1.95986
$750$	0	0
$751$	24.2787	0.885943	0.442972	−	0.896536i	$-0.353924\pi$
$751$	24.2787	0.885943	0.442972	+	0.896536i	$0.353924\pi$
$752$	−182.032	−6.63802
$753$	0	0
$754$	10.2573	0.373550
$755$	−8.65270	−0.314904
$756$	0	0
$757$	−22.0047	−0.799773	−0.399887	−	0.916565i	$-0.630951\pi$
$757$	−22.0047	−0.799773	−0.399887	+	0.916565i	$0.630951\pi$
$758$	−58.5481	−2.12656
$759$	0	0
$760$	−75.5670	−2.74110
$761$	−15.8149	−0.573291	−0.286646	−	0.958037i	$-0.592540\pi$
$761$	−15.8149	−0.573291	−0.286646	+	0.958037i	$0.592540\pi$
$762$	0	0
$763$	−14.6927	−0.531910
$764$	−96.0292	−3.47422
$765$	0	0
$766$	−44.3933	−1.60400
$767$	21.7160	0.784120
$768$	0	0
$769$	−7.73217	−0.278829	−0.139415	−	0.990234i	$-0.544522\pi$
$769$	−7.73217	−0.278829	−0.139415	+	0.990234i	$0.544522\pi$
$770$	−56.3793	−2.03177
$771$	0	0
$772$	−16.1493	−0.581225
$773$	1.43601	0.0516496	0.0258248	−	0.999666i	$-0.491779\pi$
$773$	1.43601	0.0516496	0.0258248	+	0.999666i	$0.491779\pi$
$774$	0	0
$775$	1.63141	0.0586019
$776$	87.5786	3.14389
$777$	0	0
$778$	−26.2752	−0.942012
$779$	23.0283	0.825073
$780$	0	0
$781$	13.7351	0.491482
$782$	6.22679	0.222669
$783$	0	0
$784$	352.124	12.5759
$785$	−40.2352	−1.43605
$786$	0	0
$787$	−26.9478	−0.960586	−0.480293	−	0.877108i	$-0.659470\pi$
$787$	−26.9478	−0.960586	−0.480293	+	0.877108i	$0.659470\pi$
$788$	8.03269	0.286153
$789$	0	0
$790$	−57.5478	−2.04746
$791$	79.1009	2.81250
$792$	0	0
$793$	14.8802	0.528411
$794$	30.1466	1.06986
$795$	0	0
$796$	111.071	3.93681
$797$	−25.1365	−0.890380	−0.445190	−	0.895436i	$-0.646864\pi$
$797$	−25.1365	−0.890380	−0.445190	+	0.895436i	$0.646864\pi$
$798$	0	0
$799$	−24.1688	−0.855031
$800$	7.40536	0.261819
$801$	0	0
$802$	47.2249	1.66757
$803$	2.18907	0.0772506
$804$	0	0
$805$	12.1172	0.427074
$806$	59.8551	2.10831
$807$	0	0
$808$	172.976	6.08527
$809$	−6.20511	−0.218160	−0.109080	−	0.994033i	$-0.534790\pi$
$809$	−6.20511	−0.218160	−0.109080	+	0.994033i	$0.534790\pi$
$810$	0	0
$811$	−53.5107	−1.87901	−0.939507	−	0.342529i	$-0.888716\pi$
$811$	−53.5107	−1.87901	−0.939507	+	0.342529i	$0.888716\pi$
$812$	−29.9640	−1.05153
$813$	0	0
$814$	6.55903	0.229894
$815$	39.5262	1.38454
$816$	0	0
$817$	28.4422	0.995066
$818$	19.6228	0.686096
$819$	0	0
$820$	−93.2913	−3.25787
$821$	43.8028	1.52873	0.764364	−	0.644785i	$-0.223053\pi$
$821$	43.8028	1.52873	0.764364	+	0.644785i	$0.223053\pi$
$822$	0	0
$823$	9.16598	0.319506	0.159753	−	0.987157i	$-0.448930\pi$
$823$	9.16598	0.319506	0.159753	+	0.987157i	$0.448930\pi$
$824$	114.293	3.98158
$825$	0	0
$826$	−85.7668	−2.98421
$827$	7.52174	0.261557	0.130778	−	0.991412i	$-0.458252\pi$
$827$	7.52174	0.261557	0.130778	+	0.991412i	$0.458252\pi$
$828$	0	0
$829$	15.6528	0.543644	0.271822	−	0.962348i	$-0.412374\pi$
$829$	15.6528	0.543644	0.271822	+	0.962348i	$0.412374\pi$
$830$	−9.71470	−0.337202
$831$	0	0
$832$	146.456	5.07744
$833$	46.7524	1.61987
$834$	0	0
$835$	−22.3802	−0.774499
$836$	−30.7396	−1.06315
$837$	0	0
$838$	−44.3507	−1.53207
$839$	−35.7749	−1.23509	−0.617544	−	0.786536i	$-0.711872\pi$
$839$	−35.7749	−1.23509	−0.617544	+	0.786536i	$0.711872\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	34.5948	1.19221
$843$	0	0
$844$	86.5624	2.97960
$845$	−1.59949	−0.0550243
$846$	0	0
$847$	43.1473	1.48256
$848$	180.320	6.19222
$849$	0	0
$850$	1.74084	0.0597105
$851$	−1.40968	−0.0483233
$852$	0	0
$853$	40.3320	1.38094	0.690471	−	0.723360i	$-0.257403\pi$
$853$	40.3320	1.38094	0.690471	+	0.723360i	$0.257403\pi$
$854$	−58.7689	−2.01103
$855$	0	0
$856$	−103.796	−3.54766
$857$	−12.9572	−0.442609	−0.221305	−	0.975205i	$-0.571032\pi$
$857$	−12.9572	−0.442609	−0.221305	+	0.975205i	$0.571032\pi$
$858$	0	0
$859$	−34.2116	−1.16728	−0.583642	−	0.812011i	$-0.698373\pi$
$859$	−34.2116	−1.16728	−0.583642	+	0.812011i	$0.698373\pi$
$860$	−115.224	−3.92911
$861$	0	0
$862$	−18.6383	−0.634824
$863$	−46.0580	−1.56783	−0.783916	−	0.620867i	$-0.786781\pi$
$863$	−46.0580	−1.56783	−0.783916	+	0.620867i	$0.786781\pi$
$864$	0	0
$865$	46.2723	1.57331
$866$	24.5916	0.835657
$867$	0	0
$868$	−174.850	−5.93481
$869$	−15.1697	−0.514595
$870$	0	0
$871$	−15.0453	−0.509791
$872$	28.4324	0.962843
$873$	0	0
$874$	8.93210	0.302133
$875$	−57.1982	−1.93365
$876$	0	0
$877$	55.0308	1.85826	0.929129	−	0.369756i	$-0.120559\pi$
$877$	55.0308	1.85826	0.929129	+	0.369756i	$0.120559\pi$
$878$	80.0092	2.70018
$879$	0	0
$880$	65.2682	2.20019
$881$	34.6195	1.16636	0.583180	−	0.812343i	$-0.301808\pi$
$881$	34.6195	1.16636	0.583180	+	0.812343i	$0.301808\pi$
$882$	0	0
$883$	25.0742	0.843814	0.421907	−	0.906639i	$-0.361361\pi$
$883$	25.0742	0.843814	0.421907	+	0.906639i	$0.361361\pi$
$884$	47.2416	1.58891
$885$	0	0
$886$	8.25240	0.277245
$887$	30.2600	1.01603	0.508015	−	0.861348i	$-0.330379\pi$
$887$	30.2600	1.01603	0.508015	+	0.861348i	$0.330379\pi$
$888$	0	0
$889$	−49.9691	−1.67591
$890$	92.6732	3.10641
$891$	0	0
$892$	−18.8402	−0.630818
$893$	−34.6693	−1.16016
$894$	0	0
$895$	24.5845	0.821768
$896$	−299.051	−9.99060
$897$	0	0
$898$	−17.5103	−0.584327
$899$	5.83534	0.194620
$900$	0	0
$901$	23.9416	0.797609
$902$	−33.2477	−1.10703
$903$	0	0
$904$	−153.072	−5.09109
$905$	46.2341	1.53687
$906$	0	0
$907$	−36.3480	−1.20691	−0.603457	−	0.797395i	$-0.706211\pi$
$907$	−36.3480	−1.20691	−0.603457	+	0.797395i	$0.706211\pi$
$908$	36.2907	1.20435
$909$	0	0
$910$	124.290	4.12017
$911$	24.5626	0.813796	0.406898	−	0.913474i	$-0.366610\pi$
$911$	24.5626	0.813796	0.406898	+	0.913474i	$0.366610\pi$
$912$	0	0
$913$	−2.56080	−0.0847501
$914$	19.5858	0.647842
$915$	0	0
$916$	−9.73207	−0.321557
$917$	54.2592	1.79180
$918$	0	0
$919$	0.797896	0.0263202	0.0131601	−	0.999913i	$-0.495811\pi$
$919$	0.797896	0.0263202	0.0131601	+	0.999913i	$0.495811\pi$
$920$	−23.4485	−0.773074
$921$	0	0
$922$	−35.8445	−1.18048
$923$	−30.2796	−0.996664
$924$	0	0
$925$	−0.394110	−0.0129583
$926$	−116.644	−3.83315
$927$	0	0
$928$	26.4881	0.869514
$929$	−8.85813	−0.290626	−0.145313	−	0.989386i	$-0.546419\pi$
$929$	−8.85813	−0.290626	−0.145313	+	0.989386i	$0.546419\pi$
$930$	0	0
$931$	67.0646	2.19795
$932$	58.0609	1.90185
$933$	0	0
$934$	−13.9446	−0.456282
$935$	8.66581	0.283402
$936$	0	0
$937$	14.0445	0.458814	0.229407	−	0.973331i	$-0.426321\pi$
$937$	14.0445	0.458814	0.229407	+	0.973331i	$0.426321\pi$
$938$	59.4211	1.94017
$939$	0	0
$940$	140.451	4.58101
$941$	8.17124	0.266375	0.133187	−	0.991091i	$-0.457479\pi$
$941$	8.17124	0.266375	0.133187	+	0.991091i	$0.457479\pi$
$942$	0	0
$943$	7.14568	0.232695
$944$	99.2891	3.23158
$945$	0	0
$946$	−41.0642	−1.33511
$947$	42.8749	1.39325	0.696623	−	0.717437i	$-0.254685\pi$
$947$	42.8749	1.39325	0.696623	+	0.717437i	$0.254685\pi$
$948$	0	0
$949$	−4.82588	−0.156655
$950$	2.49718	0.0810191
$951$	0	0
$952$	−120.905	−3.91856
$953$	20.8388	0.675034	0.337517	−	0.941319i	$-0.390413\pi$
$953$	20.8388	0.675034	0.337517	+	0.941319i	$0.390413\pi$
$954$	0	0
$955$	38.8333	1.25662
$956$	105.634	3.41646
$957$	0	0
$958$	−32.1639	−1.03917
$959$	−51.2843	−1.65606
$960$	0	0
$961$	3.05124	0.0984270
$962$	−14.4596	−0.466196
$963$	0	0
$964$	130.107	4.19047
$965$	6.53061	0.210228
$966$	0	0
$967$	46.9763	1.51066	0.755328	−	0.655347i	$-0.227477\pi$
$967$	46.9763	1.51066	0.755328	+	0.655347i	$0.227477\pi$
$968$	−83.4962	−2.68367
$969$	0	0
$970$	−54.6535	−1.75482
$971$	−5.28608	−0.169638	−0.0848192	−	0.996396i	$-0.527031\pi$
$971$	−5.28608	−0.169638	−0.0848192	+	0.996396i	$0.527031\pi$
$972$	0	0
$973$	51.2669	1.64354
$974$	24.7280	0.792336
$975$	0	0
$976$	68.0345	2.17773
$977$	10.4569	0.334547	0.167274	−	0.985911i	$-0.446504\pi$
$977$	10.4569	0.334547	0.167274	+	0.985911i	$0.446504\pi$
$978$	0	0
$979$	24.4287	0.780745
$980$	−271.690	−8.67882
$981$	0	0
$982$	−34.3655	−1.09665
$983$	−27.7660	−0.885597	−0.442798	−	0.896621i	$-0.646014\pi$
$983$	−27.7660	−0.885597	−0.442798	+	0.896621i	$0.646014\pi$
$984$	0	0
$985$	−3.24834	−0.103501
$986$	6.22679	0.198301
$987$	0	0
$988$	67.7664	2.15594
$989$	8.82562	0.280638
$990$	0	0
$991$	−12.8880	−0.409400	−0.204700	−	0.978825i	$-0.565622\pi$
$991$	−12.8880	−0.409400	−0.204700	+	0.978825i	$0.565622\pi$
$992$	154.567	4.90751
$993$	0	0
$994$	119.588	3.79311
$995$	−44.9161	−1.42394
$996$	0	0
$997$	−12.5755	−0.398271	−0.199135	−	0.979972i	$-0.563813\pi$
$997$	−12.5755	−0.398271	−0.199135	+	0.979972i	$0.563813\pi$
$998$	−99.6651	−3.15484
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.r.1.16		16
3.2	odd	2		2001.2.a.n.1.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.n.1.1	✓	16	3.2	odd	2
6003.2.a.r.1.16		16	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+2.77164 q^{2} +5.68196 q^{4} -2.29773 q^{5} -5.27354 q^{7} +10.2051 q^{8} +O(q^{10})\)	\(q+2.77164 q^{2} +5.68196 q^{4} -2.29773 q^{5} -5.27354 q^{7} +10.2051 q^{8} -6.36848 q^{10} -1.67873 q^{11} +3.70083 q^{13} -14.6163 q^{14} +16.9208 q^{16} +2.24661 q^{17} +3.22268 q^{19} -13.0556 q^{20} -4.65284 q^{22} +1.00000 q^{23} +0.279573 q^{25} +10.2573 q^{26} -29.9640 q^{28} +1.00000 q^{29} +5.83534 q^{31} +26.4881 q^{32} +6.22679 q^{34} +12.1172 q^{35} -1.40968 q^{37} +8.93210 q^{38} -23.4485 q^{40} +7.14568 q^{41} +8.82562 q^{43} -9.53851 q^{44} +2.77164 q^{46} -10.7579 q^{47} +20.8102 q^{49} +0.774875 q^{50} +21.0280 q^{52} +10.6567 q^{53} +3.85728 q^{55} -53.8167 q^{56} +2.77164 q^{58} +5.86788 q^{59} +4.02077 q^{61} +16.1734 q^{62} +39.5738 q^{64} -8.50351 q^{65} -4.06540 q^{67} +12.7652 q^{68} +33.5844 q^{70} -8.18183 q^{71} -1.30400 q^{73} -3.90713 q^{74} +18.3112 q^{76} +8.85287 q^{77} +9.03636 q^{79} -38.8794 q^{80} +19.8052 q^{82} +1.52544 q^{83} -5.16211 q^{85} +24.4614 q^{86} -17.1316 q^{88} -14.5519 q^{89} -19.5164 q^{91} +5.68196 q^{92} -29.8170 q^{94} -7.40486 q^{95} +8.58188 q^{97} +57.6782 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more