LMFDB - Embedded newform 2960.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.935504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 4x^{2} + 8x + 2 $ x^5 - x^4 - 8x^3 + 4x^2 + 8*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44705$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44705 q^{3} -1.00000 q^{5} +3.62974 q^{7} +2.98807 q^{9} +O(q^{10})$	$q-2.44705 q^{3} -1.00000 q^{5} +3.62974 q^{7} +2.98807 q^{9} +6.10600 q^{11} +4.48331 q^{13} +2.44705 q^{15} -2.11793 q^{17} +4.61005 q^{19} -8.88217 q^{21} +6.89411 q^{23} +1.00000 q^{25} +0.0292026 q^{27} +5.25949 q^{29} -5.03811 q^{31} -14.9417 q^{33} -3.62974 q^{35} +1.00000 q^{37} -10.9709 q^{39} -10.0476 q^{41} +5.28335 q^{43} -2.98807 q^{45} +1.98101 q^{47} +6.17503 q^{49} +5.18269 q^{51} -4.20238 q^{53} -6.10600 q^{55} -11.2810 q^{57} -5.93222 q^{59} -8.17261 q^{61} +10.8459 q^{63} -4.48331 q^{65} +1.71594 q^{67} -16.8702 q^{69} +4.16300 q^{71} +11.0571 q^{73} -2.44705 q^{75} +22.1632 q^{77} -5.37036 q^{79} -9.03566 q^{81} +8.73991 q^{83} +2.11793 q^{85} -12.8702 q^{87} -6.62026 q^{89} +16.2733 q^{91} +12.3285 q^{93} -4.61005 q^{95} -6.44645 q^{97} +18.2451 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9	$ 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 4 q^{13} - q^{15} + 10 q^{19} - 5 q^{21} + 8 q^{23} + 5 q^{25} + 7 q^{27} - 8 q^{29} + 2 q^{31} - 11 q^{33} - q^{35} + 5 q^{37} + 2 q^{39} - 13 q^{41} + 18 q^{43} - 2 q^{45} + 9 q^{47} - 6 q^{49} + 22 q^{51} - 13 q^{53} - 7 q^{55} + 4 q^{57} + 24 q^{59} - 2 q^{61} + 20 q^{63} - 4 q^{65} + 22 q^{67} - 32 q^{69} + 21 q^{71} + 29 q^{73} + q^{75} + 11 q^{77} + 6 q^{79} + 5 q^{81} + 33 q^{83} - 12 q^{87} - 26 q^{89} + 38 q^{91} + 14 q^{93} - 10 q^{95} + 26 q^{97} + 42 q^{99}+O(q^{100}) $ 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 4 * q^13 - q^15 + 10 * q^19 - 5 * q^21 + 8 * q^23 + 5 * q^25 + 7 * q^27 - 8 * q^29 + 2 * q^31 - 11 * q^33 - q^35 + 5 * q^37 + 2 * q^39 - 13 * q^41 + 18 * q^43 - 2 * q^45 + 9 * q^47 - 6 * q^49 + 22 * q^51 - 13 * q^53 - 7 * q^55 + 4 * q^57 + 24 * q^59 - 2 * q^61 + 20 * q^63 - 4 * q^65 + 22 * q^67 - 32 * q^69 + 21 * q^71 + 29 * q^73 + q^75 + 11 * q^77 + 6 * q^79 + 5 * q^81 + 33 * q^83 - 12 * q^87 - 26 * q^89 + 38 * q^91 + 14 * q^93 - 10 * q^95 + 26 * q^97 + 42 * 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 $

$2$	0	0
$2$	0	0
$3$	−2.44705	−1.41281	−0.706403	−	0.707810i	$-0.749683\pi$
$3$	−2.44705	−1.41281	−0.706403	+	0.707810i	$0.749683\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.62974	1.37191	0.685957	−	0.727642i	$-0.259384\pi$
$7$	3.62974	1.37191	0.685957	+	0.727642i	$0.259384\pi$
$8$	0	0
$9$	2.98807	0.996022
$10$	0	0
$11$	6.10600	1.84103	0.920514	−	0.390710i	$-0.127770\pi$
$11$	6.10600	1.84103	0.920514	+	0.390710i	$0.127770\pi$
$12$	0	0
$13$	4.48331	1.24345	0.621724	−	0.783237i	$-0.286433\pi$
$13$	4.48331	1.24345	0.621724	+	0.783237i	$0.286433\pi$
$14$	0	0
$15$	2.44705	0.631826
$16$	0	0
$17$	−2.11793	−0.513674	−0.256837	−	0.966455i	$-0.582680\pi$
$17$	−2.11793	−0.513674	−0.256837	+	0.966455i	$0.582680\pi$
$18$	0	0
$19$	4.61005	1.05762	0.528809	−	0.848741i	$-0.322639\pi$
$19$	4.61005	1.05762	0.528809	+	0.848741i	$0.322639\pi$
$20$	0	0
$21$	−8.88217	−1.93825
$22$	0	0
$23$	6.89411	1.43752	0.718760	−	0.695258i	$-0.244710\pi$
$23$	6.89411	1.43752	0.718760	+	0.695258i	$0.244710\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.0292026	0.00562004
$28$	0	0
$29$	5.25949	0.976662	0.488331	−	0.872659i	$-0.337606\pi$
$29$	5.25949	0.976662	0.488331	+	0.872659i	$0.337606\pi$
$30$	0	0
$31$	−5.03811	−0.904871	−0.452436	−	0.891797i	$-0.649445\pi$
$31$	−5.03811	−0.904871	−0.452436	+	0.891797i	$0.649445\pi$
$32$	0	0
$33$	−14.9417	−2.60102
$34$	0	0
$35$	−3.62974	−0.613539
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−10.9709	−1.75675
$40$	0	0
$41$	−10.0476	−1.56917	−0.784585	−	0.620021i	$-0.787124\pi$
$41$	−10.0476	−1.56917	−0.784585	+	0.620021i	$0.787124\pi$
$42$	0	0
$43$	5.28335	0.805704	0.402852	−	0.915265i	$-0.368019\pi$
$43$	5.28335	0.805704	0.402852	+	0.915265i	$0.368019\pi$
$44$	0	0
$45$	−2.98807	−0.445435
$46$	0	0
$47$	1.98101	0.288960	0.144480	−	0.989508i	$-0.453849\pi$
$47$	1.98101	0.288960	0.144480	+	0.989508i	$0.453849\pi$
$48$	0	0
$49$	6.17503	0.882148
$50$	0	0
$51$	5.18269	0.725722
$52$	0	0
$53$	−4.20238	−0.577242	−0.288621	−	0.957443i	$-0.593197\pi$
$53$	−4.20238	−0.577242	−0.288621	+	0.957443i	$0.593197\pi$
$54$	0	0
$55$	−6.10600	−0.823333
$56$	0	0
$57$	−11.2810	−1.49421
$58$	0	0
$59$	−5.93222	−0.772309	−0.386154	−	0.922434i	$-0.626197\pi$
$59$	−5.93222	−0.772309	−0.386154	+	0.922434i	$0.626197\pi$
$60$	0	0
$61$	−8.17261	−1.04640	−0.523198	−	0.852211i	$-0.675261\pi$
$61$	−8.17261	−1.04640	−0.523198	+	0.852211i	$0.675261\pi$
$62$	0	0
$63$	10.8459	1.36646
$64$	0	0
$65$	−4.48331	−0.556086
$66$	0	0
$67$	1.71594	0.209636	0.104818	−	0.994491i	$-0.466574\pi$
$67$	1.71594	0.209636	0.104818	+	0.994491i	$0.466574\pi$
$68$	0	0
$69$	−16.8702	−2.03094
$70$	0	0
$71$	4.16300	0.494057	0.247028	−	0.969008i	$-0.420546\pi$
$71$	4.16300	0.494057	0.247028	+	0.969008i	$0.420546\pi$
$72$	0	0
$73$	11.0571	1.29414	0.647068	−	0.762432i	$-0.275995\pi$
$73$	11.0571	1.29414	0.647068	+	0.762432i	$0.275995\pi$
$74$	0	0
$75$	−2.44705	−0.282561
$76$	0	0
$77$	22.1632	2.52573
$78$	0	0
$79$	−5.37036	−0.604213	−0.302106	−	0.953274i	$-0.597690\pi$
$79$	−5.37036	−0.604213	−0.302106	+	0.953274i	$0.597690\pi$
$80$	0	0
$81$	−9.03566	−1.00396
$82$	0	0
$83$	8.73991	0.959330	0.479665	−	0.877452i	$-0.340758\pi$
$83$	8.73991	0.959330	0.479665	+	0.877452i	$0.340758\pi$
$84$	0	0
$85$	2.11793	0.229722
$86$	0	0
$87$	−12.8702	−1.37983
$88$	0	0
$89$	−6.62026	−0.701746	−0.350873	−	0.936423i	$-0.614115\pi$
$89$	−6.62026	−0.701746	−0.350873	+	0.936423i	$0.614115\pi$
$90$	0	0
$91$	16.2733	1.70590
$92$	0	0
$93$	12.3285	1.27841
$94$	0	0
$95$	−4.61005	−0.472981
$96$	0	0
$97$	−6.44645	−0.654538	−0.327269	−	0.944931i	$-0.606128\pi$
$97$	−6.44645	−0.654538	−0.327269	+	0.944931i	$0.606128\pi$
$98$	0	0
$99$	18.2451	1.83370
$100$	0	0
$101$	10.1261	1.00759	0.503794	−	0.863824i	$-0.331937\pi$
$101$	10.1261	1.00759	0.503794	+	0.863824i	$0.331937\pi$
$102$	0	0
$103$	−15.4701	−1.52431	−0.762156	−	0.647394i	$-0.775859\pi$
$103$	−15.4701	−1.52431	−0.762156	+	0.647394i	$0.775859\pi$
$104$	0	0
$105$	8.88217	0.866811
$106$	0	0
$107$	−17.5571	−1.69731	−0.848654	−	0.528949i	$-0.822586\pi$
$107$	−17.5571	−1.69731	−0.848654	+	0.528949i	$0.822586\pi$
$108$	0	0
$109$	10.7739	1.03195	0.515974	−	0.856604i	$-0.327430\pi$
$109$	10.7739	1.03195	0.515974	+	0.856604i	$0.327430\pi$
$110$	0	0
$111$	−2.44705	−0.232264
$112$	0	0
$113$	0.157321	0.0147995	0.00739974	−	0.999973i	$-0.497645\pi$
$113$	0.157321	0.0147995	0.00739974	+	0.999973i	$0.497645\pi$
$114$	0	0
$115$	−6.89411	−0.642879
$116$	0	0
$117$	13.3964	1.23850
$118$	0	0
$119$	−7.68755	−0.704716
$120$	0	0
$121$	26.2832	2.38938
$122$	0	0
$123$	24.5870	2.21693
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.7411	1.21932	0.609662	−	0.792662i	$-0.291305\pi$
$127$	13.7411	1.21932	0.609662	+	0.792662i	$0.291305\pi$
$128$	0	0
$129$	−12.9286	−1.13830
$130$	0	0
$131$	−12.0090	−1.04923	−0.524616	−	0.851339i	$-0.675791\pi$
$131$	−12.0090	−1.04923	−0.524616	+	0.851339i	$0.675791\pi$
$132$	0	0
$133$	16.7333	1.45096
$134$	0	0
$135$	−0.0292026	−0.00251336
$136$	0	0
$137$	−9.82134	−0.839094	−0.419547	−	0.907734i	$-0.637811\pi$
$137$	−9.82134	−0.839094	−0.419547	+	0.907734i	$0.637811\pi$
$138$	0	0
$139$	7.65014	0.648876	0.324438	−	0.945907i	$-0.394825\pi$
$139$	7.65014	0.648876	0.324438	+	0.945907i	$0.394825\pi$
$140$	0	0
$141$	−4.84763	−0.408244
$142$	0	0
$143$	27.3751	2.28922
$144$	0	0
$145$	−5.25949	−0.436777
$146$	0	0
$147$	−15.1106	−1.24630
$148$	0	0
$149$	−11.5010	−0.942201	−0.471100	−	0.882080i	$-0.656143\pi$
$149$	−11.5010	−0.942201	−0.471100	+	0.882080i	$0.656143\pi$
$150$	0	0
$151$	−0.332250	−0.0270381	−0.0135191	−	0.999909i	$-0.504303\pi$
$151$	−0.332250	−0.0270381	−0.0135191	+	0.999909i	$0.504303\pi$
$152$	0	0
$153$	−6.32852	−0.511631
$154$	0	0
$155$	5.03811	0.404671
$156$	0	0
$157$	13.1549	1.04987	0.524937	−	0.851141i	$-0.324089\pi$
$157$	13.1549	1.04987	0.524937	+	0.851141i	$0.324089\pi$
$158$	0	0
$159$	10.2835	0.815531
$160$	0	0
$161$	25.0238	1.97215
$162$	0	0
$163$	18.5238	1.45090	0.725449	−	0.688276i	$-0.241632\pi$
$163$	18.5238	1.45090	0.725449	+	0.688276i	$0.241632\pi$
$164$	0	0
$165$	14.9417	1.16321
$166$	0	0
$167$	−5.30378	−0.410419	−0.205209	−	0.978718i	$-0.565788\pi$
$167$	−5.30378	−0.410419	−0.205209	+	0.978718i	$0.565788\pi$
$168$	0	0
$169$	7.10009	0.546161
$170$	0	0
$171$	13.7751	1.05341
$172$	0	0
$173$	22.4561	1.70730	0.853651	−	0.520845i	$-0.174383\pi$
$173$	22.4561	1.70730	0.853651	+	0.520845i	$0.174383\pi$
$174$	0	0
$175$	3.62974	0.274383
$176$	0	0
$177$	14.5164	1.09112
$178$	0	0
$179$	15.9279	1.19051	0.595255	−	0.803537i	$-0.297051\pi$
$179$	15.9279	1.19051	0.595255	+	0.803537i	$0.297051\pi$
$180$	0	0
$181$	−10.8252	−0.804629	−0.402314	−	0.915502i	$-0.631794\pi$
$181$	−10.8252	−0.804629	−0.402314	+	0.915502i	$0.631794\pi$
$182$	0	0
$183$	19.9988	1.47835
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−12.9321	−0.945688
$188$	0	0
$189$	0.105998	0.00771021
$190$	0	0
$191$	−3.03441	−0.219562	−0.109781	−	0.993956i	$-0.535015\pi$
$191$	−3.03441	−0.219562	−0.109781	+	0.993956i	$0.535015\pi$
$192$	0	0
$193$	6.06696	0.436709	0.218355	−	0.975869i	$-0.429931\pi$
$193$	6.06696	0.436709	0.218355	+	0.975869i	$0.429931\pi$
$194$	0	0
$195$	10.9709	0.785642
$196$	0	0
$197$	2.75498	0.196284	0.0981420	−	0.995172i	$-0.468710\pi$
$197$	2.75498	0.196284	0.0981420	+	0.995172i	$0.468710\pi$
$198$	0	0
$199$	−0.0481934	−0.00341634	−0.00170817	−	0.999999i	$-0.500544\pi$
$199$	−0.0481934	−0.00341634	−0.00170817	+	0.999999i	$0.500544\pi$
$200$	0	0
$201$	−4.19900	−0.296175
$202$	0	0
$203$	19.0906	1.33990
$204$	0	0
$205$	10.0476	0.701754
$206$	0	0
$207$	20.6000	1.43180
$208$	0	0
$209$	28.1489	1.94710
$210$	0	0
$211$	−10.7165	−0.737755	−0.368877	−	0.929478i	$-0.620258\pi$
$211$	−10.7165	−0.737755	−0.368877	+	0.929478i	$0.620258\pi$
$212$	0	0
$213$	−10.1871	−0.698006
$214$	0	0
$215$	−5.28335	−0.360322
$216$	0	0
$217$	−18.2870	−1.24141
$218$	0	0
$219$	−27.0573	−1.82836
$220$	0	0
$221$	−9.49535	−0.638726
$222$	0	0
$223$	18.5184	1.24008	0.620041	−	0.784569i	$-0.287116\pi$
$223$	18.5184	1.24008	0.620041	+	0.784569i	$0.287116\pi$
$224$	0	0
$225$	2.98807	0.199204
$226$	0	0
$227$	−29.9812	−1.98992	−0.994961	−	0.100265i	$-0.968031\pi$
$227$	−29.9812	−1.98992	−0.994961	+	0.100265i	$0.968031\pi$
$228$	0	0
$229$	0.519113	0.0343040	0.0171520	−	0.999853i	$-0.494540\pi$
$229$	0.519113	0.0343040	0.0171520	+	0.999853i	$0.494540\pi$
$230$	0	0
$231$	−54.2345	−3.56837
$232$	0	0
$233$	−4.30353	−0.281934	−0.140967	−	0.990014i	$-0.545021\pi$
$233$	−4.30353	−0.281934	−0.140967	+	0.990014i	$0.545021\pi$
$234$	0	0
$235$	−1.98101	−0.129227
$236$	0	0
$237$	13.1416	0.853636
$238$	0	0
$239$	0.519920	0.0336308	0.0168154	−	0.999859i	$-0.494647\pi$
$239$	0.519920	0.0336308	0.0168154	+	0.999859i	$0.494647\pi$
$240$	0	0
$241$	−30.5051	−1.96500	−0.982502	−	0.186252i	$-0.940366\pi$
$241$	−30.5051	−1.96500	−0.982502	+	0.186252i	$0.940366\pi$
$242$	0	0
$243$	22.0231	1.41278
$244$	0	0
$245$	−6.17503	−0.394508
$246$	0	0
$247$	20.6683	1.31509
$248$	0	0
$249$	−21.3870	−1.35535
$250$	0	0
$251$	22.2673	1.40550	0.702751	−	0.711436i	$-0.251955\pi$
$251$	22.2673	1.40550	0.702751	+	0.711436i	$0.251955\pi$
$252$	0	0
$253$	42.0954	2.64651
$254$	0	0
$255$	−5.18269	−0.324553
$256$	0	0
$257$	−31.3051	−1.95276	−0.976379	−	0.216063i	$-0.930678\pi$
$257$	−31.3051	−1.95276	−0.976379	+	0.216063i	$0.930678\pi$
$258$	0	0
$259$	3.62974	0.225541
$260$	0	0
$261$	15.7157	0.972777
$262$	0	0
$263$	−20.0805	−1.23821	−0.619107	−	0.785306i	$-0.712505\pi$
$263$	−20.0805	−1.23821	−0.619107	+	0.785306i	$0.712505\pi$
$264$	0	0
$265$	4.20238	0.258150
$266$	0	0
$267$	16.2001	0.991432
$268$	0	0
$269$	−9.80955	−0.598099	−0.299049	−	0.954238i	$-0.596670\pi$
$269$	−9.80955	−0.598099	−0.299049	+	0.954238i	$0.596670\pi$
$270$	0	0
$271$	−3.37383	−0.204946	−0.102473	−	0.994736i	$-0.532675\pi$
$271$	−3.37383	−0.204946	−0.102473	+	0.994736i	$0.532675\pi$
$272$	0	0
$273$	−39.8215	−2.41011
$274$	0	0
$275$	6.10600	0.368206
$276$	0	0
$277$	−6.58896	−0.395892	−0.197946	−	0.980213i	$-0.563427\pi$
$277$	−6.58896	−0.395892	−0.197946	+	0.980213i	$0.563427\pi$
$278$	0	0
$279$	−15.0542	−0.901272
$280$	0	0
$281$	−7.85006	−0.468295	−0.234148	−	0.972201i	$-0.575230\pi$
$281$	−7.85006	−0.468295	−0.234148	+	0.972201i	$0.575230\pi$
$282$	0	0
$283$	15.2694	0.907674	0.453837	−	0.891085i	$-0.350055\pi$
$283$	15.2694	0.907674	0.453837	+	0.891085i	$0.350055\pi$
$284$	0	0
$285$	11.2810	0.668231
$286$	0	0
$287$	−36.4702	−2.15277
$288$	0	0
$289$	−12.5144	−0.736139
$290$	0	0
$291$	15.7748	0.924736
$292$	0	0
$293$	−25.5679	−1.49369	−0.746845	−	0.664998i	$-0.768433\pi$
$293$	−25.5679	−1.49369	−0.746845	+	0.664998i	$0.768433\pi$
$294$	0	0
$295$	5.93222	0.345387
$296$	0	0
$297$	0.178311	0.0103467
$298$	0	0
$299$	30.9084	1.78748
$300$	0	0
$301$	19.1772	1.10536
$302$	0	0
$303$	−24.7792	−1.42353
$304$	0	0
$305$	8.17261	0.467962
$306$	0	0
$307$	16.1156	0.919765	0.459882	−	0.887980i	$-0.347892\pi$
$307$	16.1156	0.919765	0.459882	+	0.887980i	$0.347892\pi$
$308$	0	0
$309$	37.8561	2.15356
$310$	0	0
$311$	13.9463	0.790824	0.395412	−	0.918504i	$-0.370602\pi$
$311$	13.9463	0.790824	0.395412	+	0.918504i	$0.370602\pi$
$312$	0	0
$313$	28.4274	1.60681	0.803407	−	0.595430i	$-0.203018\pi$
$313$	28.4274	1.60681	0.803407	+	0.595430i	$0.203018\pi$
$314$	0	0
$315$	−10.8459	−0.611098
$316$	0	0
$317$	−5.03429	−0.282754	−0.141377	−	0.989956i	$-0.545153\pi$
$317$	−5.03429	−0.282754	−0.141377	+	0.989956i	$0.545153\pi$
$318$	0	0
$319$	32.1144	1.79806
$320$	0	0
$321$	42.9631	2.39797
$322$	0	0
$323$	−9.76377	−0.543271
$324$	0	0
$325$	4.48331	0.248689
$326$	0	0
$327$	−26.3642	−1.45794
$328$	0	0
$329$	7.19055	0.396428
$330$	0	0
$331$	−16.5102	−0.907481	−0.453740	−	0.891134i	$-0.649911\pi$
$331$	−16.5102	−0.907481	−0.453740	+	0.891134i	$0.649911\pi$
$332$	0	0
$333$	2.98807	0.163745
$334$	0	0
$335$	−1.71594	−0.0937520
$336$	0	0
$337$	35.7106	1.94528	0.972641	−	0.232312i	$-0.0746289\pi$
$337$	35.7106	1.94528	0.972641	+	0.232312i	$0.0746289\pi$
$338$	0	0
$339$	−0.384972	−0.0209088
$340$	0	0
$341$	−30.7627	−1.66589
$342$	0	0
$343$	−2.99442	−0.161683
$344$	0	0
$345$	16.8702	0.908263
$346$	0	0
$347$	9.73121	0.522399	0.261199	−	0.965285i	$-0.415882\pi$
$347$	9.73121	0.522399	0.261199	+	0.965285i	$0.415882\pi$
$348$	0	0
$349$	−31.5783	−1.69035	−0.845173	−	0.534494i	$-0.820502\pi$
$349$	−31.5783	−1.69035	−0.845173	+	0.534494i	$0.820502\pi$
$350$	0	0
$351$	0.130924	0.00698823
$352$	0	0
$353$	−13.8860	−0.739077	−0.369539	−	0.929215i	$-0.620484\pi$
$353$	−13.8860	−0.739077	−0.369539	+	0.929215i	$0.620484\pi$
$354$	0	0
$355$	−4.16300	−0.220949
$356$	0	0
$357$	18.8118	0.995628
$358$	0	0
$359$	−30.6722	−1.61882	−0.809409	−	0.587246i	$-0.800212\pi$
$359$	−30.6722	−1.61882	−0.809409	+	0.587246i	$0.800212\pi$
$360$	0	0
$361$	2.25255	0.118555
$362$	0	0
$363$	−64.3164	−3.37574
$364$	0	0
$365$	−11.0571	−0.578755
$366$	0	0
$367$	−29.5282	−1.54136	−0.770680	−	0.637222i	$-0.780084\pi$
$367$	−29.5282	−1.54136	−0.770680	+	0.637222i	$0.780084\pi$
$368$	0	0
$369$	−30.0229	−1.56293
$370$	0	0
$371$	−15.2536	−0.791926
$372$	0	0
$373$	22.4158	1.16065	0.580323	−	0.814387i	$-0.302926\pi$
$373$	22.4158	1.16065	0.580323	+	0.814387i	$0.302926\pi$
$374$	0	0
$375$	2.44705	0.126365
$376$	0	0
$377$	23.5799	1.21443
$378$	0	0
$379$	12.7072	0.652727	0.326364	−	0.945244i	$-0.394177\pi$
$379$	12.7072	0.652727	0.326364	+	0.945244i	$0.394177\pi$
$380$	0	0
$381$	−33.6251	−1.72267
$382$	0	0
$383$	27.9559	1.42848	0.714240	−	0.699901i	$-0.246772\pi$
$383$	27.9559	1.42848	0.714240	+	0.699901i	$0.246772\pi$
$384$	0	0
$385$	−22.1632	−1.12954
$386$	0	0
$387$	15.7870	0.802499
$388$	0	0
$389$	−0.121662	−0.00616849	−0.00308425	−	0.999995i	$-0.500982\pi$
$389$	−0.121662	−0.00616849	−0.00308425	+	0.999995i	$0.500982\pi$
$390$	0	0
$391$	−14.6012	−0.738417
$392$	0	0
$393$	29.3867	1.48236
$394$	0	0
$395$	5.37036	0.270212
$396$	0	0
$397$	16.4728	0.826745	0.413373	−	0.910562i	$-0.364351\pi$
$397$	16.4728	0.826745	0.413373	+	0.910562i	$0.364351\pi$
$398$	0	0
$399$	−40.9472	−2.04993
$400$	0	0
$401$	17.6552	0.881659	0.440829	−	0.897591i	$-0.354684\pi$
$401$	17.6552	0.881659	0.440829	+	0.897591i	$0.354684\pi$
$402$	0	0
$403$	−22.5874	−1.12516
$404$	0	0
$405$	9.03566	0.448986
$406$	0	0
$407$	6.10600	0.302663
$408$	0	0
$409$	11.7502	0.581011	0.290506	−	0.956873i	$-0.406176\pi$
$409$	11.7502	0.581011	0.290506	+	0.956873i	$0.406176\pi$
$410$	0	0
$411$	24.0333	1.18548
$412$	0	0
$413$	−21.5324	−1.05954
$414$	0	0
$415$	−8.73991	−0.429025
$416$	0	0
$417$	−18.7203	−0.916737
$418$	0	0
$419$	−37.9421	−1.85360	−0.926798	−	0.375561i	$-0.877450\pi$
$419$	−37.9421	−1.85360	−0.926798	+	0.375561i	$0.877450\pi$
$420$	0	0
$421$	−29.1596	−1.42115	−0.710576	−	0.703620i	$-0.751566\pi$
$421$	−29.1596	−1.42115	−0.710576	+	0.703620i	$0.751566\pi$
$422$	0	0
$423$	5.91939	0.287810
$424$	0	0
$425$	−2.11793	−0.102735
$426$	0	0
$427$	−29.6645	−1.43556
$428$	0	0
$429$	−66.9883	−3.23423
$430$	0	0
$431$	19.5473	0.941562	0.470781	−	0.882250i	$-0.343972\pi$
$431$	19.5473	0.941562	0.470781	+	0.882250i	$0.343972\pi$
$432$	0	0
$433$	23.6966	1.13879	0.569393	−	0.822066i	$-0.307178\pi$
$433$	23.6966	1.13879	0.569393	+	0.822066i	$0.307178\pi$
$434$	0	0
$435$	12.8702	0.617081
$436$	0	0
$437$	31.7822	1.52035
$438$	0	0
$439$	27.3303	1.30441	0.652203	−	0.758044i	$-0.273845\pi$
$439$	27.3303	1.30441	0.652203	+	0.758044i	$0.273845\pi$
$440$	0	0
$441$	18.4514	0.878638
$442$	0	0
$443$	−19.5299	−0.927892	−0.463946	−	0.885864i	$-0.653567\pi$
$443$	−19.5299	−0.927892	−0.463946	+	0.885864i	$0.653567\pi$
$444$	0	0
$445$	6.62026	0.313830
$446$	0	0
$447$	28.1436	1.33115
$448$	0	0
$449$	−12.1154	−0.571761	−0.285881	−	0.958265i	$-0.592286\pi$
$449$	−12.1154	−0.571761	−0.285881	+	0.958265i	$0.592286\pi$
$450$	0	0
$451$	−61.3506	−2.88889
$452$	0	0
$453$	0.813034	0.0381997
$454$	0	0
$455$	−16.2733	−0.762903
$456$	0	0
$457$	14.4942	0.678009	0.339005	−	0.940785i	$-0.389910\pi$
$457$	14.4942	0.678009	0.339005	+	0.940785i	$0.389910\pi$
$458$	0	0
$459$	−0.0618491	−0.00288687
$460$	0	0
$461$	−9.55375	−0.444963	−0.222481	−	0.974937i	$-0.571416\pi$
$461$	−9.55375	−0.444963	−0.222481	+	0.974937i	$0.571416\pi$
$462$	0	0
$463$	−26.8322	−1.24700	−0.623499	−	0.781824i	$-0.714290\pi$
$463$	−26.8322	−1.24700	−0.623499	+	0.781824i	$0.714290\pi$
$464$	0	0
$465$	−12.3285	−0.571721
$466$	0	0
$467$	39.3746	1.82204	0.911020	−	0.412363i	$-0.135296\pi$
$467$	39.3746	1.82204	0.911020	+	0.412363i	$0.135296\pi$
$468$	0	0
$469$	6.22843	0.287602
$470$	0	0
$471$	−32.1907	−1.48327
$472$	0	0
$473$	32.2601	1.48332
$474$	0	0
$475$	4.61005	0.211524
$476$	0	0
$477$	−12.5570	−0.574946
$478$	0	0
$479$	4.33615	0.198124	0.0990619	−	0.995081i	$-0.468416\pi$
$479$	4.33615	0.198124	0.0990619	+	0.995081i	$0.468416\pi$
$480$	0	0
$481$	4.48331	0.204421
$482$	0	0
$483$	−61.2346	−2.78627
$484$	0	0
$485$	6.44645	0.292718
$486$	0	0
$487$	−21.0463	−0.953698	−0.476849	−	0.878985i	$-0.658221\pi$
$487$	−21.0463	−0.953698	−0.476849	+	0.878985i	$0.658221\pi$
$488$	0	0
$489$	−45.3288	−2.04984
$490$	0	0
$491$	28.3895	1.28120	0.640599	−	0.767876i	$-0.278686\pi$
$491$	28.3895	1.28120	0.640599	+	0.767876i	$0.278686\pi$
$492$	0	0
$493$	−11.1392	−0.501686
$494$	0	0
$495$	−18.2451	−0.820057
$496$	0	0
$497$	15.1106	0.677803
$498$	0	0
$499$	−20.7426	−0.928568	−0.464284	−	0.885686i	$-0.653688\pi$
$499$	−20.7426	−0.928568	−0.464284	+	0.885686i	$0.653688\pi$
$500$	0	0
$501$	12.9786	0.579842
$502$	0	0
$503$	25.1012	1.11921	0.559605	−	0.828760i	$-0.310953\pi$
$503$	25.1012	1.11921	0.559605	+	0.828760i	$0.310953\pi$
$504$	0	0
$505$	−10.1261	−0.450607
$506$	0	0
$507$	−17.3743	−0.771619
$508$	0	0
$509$	−42.2457	−1.87251	−0.936253	−	0.351325i	$-0.885731\pi$
$509$	−42.2457	−1.87251	−0.936253	+	0.351325i	$0.885731\pi$
$510$	0	0
$511$	40.1344	1.77544
$512$	0	0
$513$	0.134625	0.00594386
$514$	0	0
$515$	15.4701	0.681693
$516$	0	0
$517$	12.0960	0.531983
$518$	0	0
$519$	−54.9512	−2.41209
$520$	0	0
$521$	−22.9681	−1.00625	−0.503125	−	0.864214i	$-0.667817\pi$
$521$	−22.9681	−1.00625	−0.503125	+	0.864214i	$0.667817\pi$
$522$	0	0
$523$	27.8714	1.21873	0.609366	−	0.792889i	$-0.291424\pi$
$523$	27.8714	1.21873	0.609366	+	0.792889i	$0.291424\pi$
$524$	0	0
$525$	−8.88217	−0.387650
$526$	0	0
$527$	10.6704	0.464809
$528$	0	0
$529$	24.5287	1.06646
$530$	0	0
$531$	−17.7259	−0.769236
$532$	0	0
$533$	−45.0465	−1.95118
$534$	0	0
$535$	17.5571	0.759059
$536$	0	0
$537$	−38.9765	−1.68196
$538$	0	0
$539$	37.7047	1.62406
$540$	0	0
$541$	17.5095	0.752793	0.376396	−	0.926459i	$-0.377163\pi$
$541$	17.5095	0.752793	0.376396	+	0.926459i	$0.377163\pi$
$542$	0	0
$543$	26.4898	1.13678
$544$	0	0
$545$	−10.7739	−0.461501
$546$	0	0
$547$	−9.07526	−0.388030	−0.194015	−	0.980999i	$-0.562151\pi$
$547$	−9.07526	−0.388030	−0.194015	+	0.980999i	$0.562151\pi$
$548$	0	0
$549$	−24.4203	−1.04223
$550$	0	0
$551$	24.2465	1.03293
$552$	0	0
$553$	−19.4930	−0.828928
$554$	0	0
$555$	2.44705	0.103872
$556$	0	0
$557$	−27.3204	−1.15760	−0.578802	−	0.815468i	$-0.696479\pi$
$557$	−27.3204	−1.15760	−0.578802	+	0.815468i	$0.696479\pi$
$558$	0	0
$559$	23.6869	1.00185
$560$	0	0
$561$	31.6455	1.33607
$562$	0	0
$563$	1.77389	0.0747606	0.0373803	−	0.999301i	$-0.488099\pi$
$563$	1.77389	0.0747606	0.0373803	+	0.999301i	$0.488099\pi$
$564$	0	0
$565$	−0.157321	−0.00661853
$566$	0	0
$567$	−32.7971	−1.37735
$568$	0	0
$569$	16.4657	0.690277	0.345138	−	0.938552i	$-0.387832\pi$
$569$	16.4657	0.690277	0.345138	+	0.938552i	$0.387832\pi$
$570$	0	0
$571$	31.8456	1.33270	0.666348	−	0.745641i	$-0.267857\pi$
$571$	31.8456	1.33270	0.666348	+	0.745641i	$0.267857\pi$
$572$	0	0
$573$	7.42536	0.310199
$574$	0	0
$575$	6.89411	0.287504
$576$	0	0
$577$	5.02130	0.209039	0.104520	−	0.994523i	$-0.466669\pi$
$577$	5.02130	0.209039	0.104520	+	0.994523i	$0.466669\pi$
$578$	0	0
$579$	−14.8462	−0.616986
$580$	0	0
$581$	31.7236	1.31612
$582$	0	0
$583$	−25.6598	−1.06272
$584$	0	0
$585$	−13.3964	−0.553874
$586$	0	0
$587$	−24.3748	−1.00606	−0.503029	−	0.864270i	$-0.667781\pi$
$587$	−24.3748	−1.00606	−0.503029	+	0.864270i	$0.667781\pi$
$588$	0	0
$589$	−23.2259	−0.957008
$590$	0	0
$591$	−6.74157	−0.277311
$592$	0	0
$593$	25.4882	1.04668	0.523338	−	0.852125i	$-0.324686\pi$
$593$	25.4882	1.04668	0.523338	+	0.852125i	$0.324686\pi$
$594$	0	0
$595$	7.68755	0.315159
$596$	0	0
$597$	0.117932	0.00482662
$598$	0	0
$599$	9.38591	0.383498	0.191749	−	0.981444i	$-0.438584\pi$
$599$	9.38591	0.383498	0.191749	+	0.981444i	$0.438584\pi$
$600$	0	0
$601$	−29.3825	−1.19854	−0.599268	−	0.800548i	$-0.704542\pi$
$601$	−29.3825	−1.19854	−0.599268	+	0.800548i	$0.704542\pi$
$602$	0	0
$603$	5.12735	0.208802
$604$	0	0
$605$	−26.2832	−1.06856
$606$	0	0
$607$	−46.0138	−1.86764	−0.933822	−	0.357738i	$-0.883548\pi$
$607$	−46.0138	−1.86764	−0.933822	+	0.357738i	$0.883548\pi$
$608$	0	0
$609$	−46.7157	−1.89301
$610$	0	0
$611$	8.88148	0.359306
$612$	0	0
$613$	41.7943	1.68806	0.844028	−	0.536298i	$-0.180178\pi$
$613$	41.7943	1.68806	0.844028	+	0.536298i	$0.180178\pi$
$614$	0	0
$615$	−24.5870	−0.991443
$616$	0	0
$617$	22.0541	0.887863	0.443931	−	0.896061i	$-0.353583\pi$
$617$	22.0541	0.887863	0.443931	+	0.896061i	$0.353583\pi$
$618$	0	0
$619$	−14.7290	−0.592009	−0.296005	−	0.955186i	$-0.595654\pi$
$619$	−14.7290	−0.592009	−0.296005	+	0.955186i	$0.595654\pi$
$620$	0	0
$621$	0.201326	0.00807893
$622$	0	0
$623$	−24.0298	−0.962735
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−68.8819	−2.75088
$628$	0	0
$629$	−2.11793	−0.0844475
$630$	0	0
$631$	37.2342	1.48227	0.741134	−	0.671357i	$-0.234288\pi$
$631$	37.2342	1.48227	0.741134	+	0.671357i	$0.234288\pi$
$632$	0	0
$633$	26.2238	1.04230
$634$	0	0
$635$	−13.7411	−0.545298
$636$	0	0
$637$	27.6846	1.09690
$638$	0	0
$639$	12.4393	0.492091
$640$	0	0
$641$	11.5214	0.455068	0.227534	−	0.973770i	$-0.426934\pi$
$641$	11.5214	0.455068	0.227534	+	0.973770i	$0.426934\pi$
$642$	0	0
$643$	25.4075	1.00197	0.500987	−	0.865455i	$-0.332970\pi$
$643$	25.4075	1.00197	0.500987	+	0.865455i	$0.332970\pi$
$644$	0	0
$645$	12.9286	0.509065
$646$	0	0
$647$	26.3248	1.03493	0.517467	−	0.855703i	$-0.326875\pi$
$647$	26.3248	1.03493	0.517467	+	0.855703i	$0.326875\pi$
$648$	0	0
$649$	−36.2221	−1.42184
$650$	0	0
$651$	44.7494	1.75387
$652$	0	0
$653$	−42.6339	−1.66839	−0.834196	−	0.551468i	$-0.814068\pi$
$653$	−42.6339	−1.66839	−0.834196	+	0.551468i	$0.814068\pi$
$654$	0	0
$655$	12.0090	0.469231
$656$	0	0
$657$	33.0393	1.28899
$658$	0	0
$659$	14.1158	0.549872	0.274936	−	0.961463i	$-0.411343\pi$
$659$	14.1158	0.549872	0.274936	+	0.961463i	$0.411343\pi$
$660$	0	0
$661$	−32.9119	−1.28013	−0.640063	−	0.768323i	$-0.721092\pi$
$661$	−32.9119	−1.28013	−0.640063	+	0.768323i	$0.721092\pi$
$662$	0	0
$663$	23.2356	0.902397
$664$	0	0
$665$	−16.7333	−0.648889
$666$	0	0
$667$	36.2594	1.40397
$668$	0	0
$669$	−45.3154	−1.75200
$670$	0	0
$671$	−49.9019	−1.92644
$672$	0	0
$673$	14.6500	0.564717	0.282359	−	0.959309i	$-0.408883\pi$
$673$	14.6500	0.564717	0.282359	+	0.959309i	$0.408883\pi$
$674$	0	0
$675$	0.0292026	0.00112401
$676$	0	0
$677$	−35.0265	−1.34618	−0.673088	−	0.739562i	$-0.735033\pi$
$677$	−35.0265	−1.34618	−0.673088	+	0.739562i	$0.735033\pi$
$678$	0	0
$679$	−23.3990	−0.897970
$680$	0	0
$681$	73.3655	2.81137
$682$	0	0
$683$	−0.519022	−0.0198598	−0.00992991	−	0.999951i	$-0.503161\pi$
$683$	−0.519022	−0.0198598	−0.00992991	+	0.999951i	$0.503161\pi$
$684$	0	0
$685$	9.82134	0.375254
$686$	0	0
$687$	−1.27030	−0.0484649
$688$	0	0
$689$	−18.8406	−0.717770
$690$	0	0
$691$	−48.0331	−1.82726	−0.913632	−	0.406543i	$-0.866734\pi$
$691$	−48.0331	−1.82726	−0.913632	+	0.406543i	$0.866734\pi$
$692$	0	0
$693$	66.2251	2.51568
$694$	0	0
$695$	−7.65014	−0.290186
$696$	0	0
$697$	21.2801	0.806042
$698$	0	0
$699$	10.5310	0.398318
$700$	0	0
$701$	−34.2465	−1.29347	−0.646737	−	0.762713i	$-0.723867\pi$
$701$	−34.2465	−1.29347	−0.646737	+	0.762713i	$0.723867\pi$
$702$	0	0
$703$	4.61005	0.173871
$704$	0	0
$705$	4.84763	0.182572
$706$	0	0
$707$	36.7553	1.38232
$708$	0	0
$709$	−20.8014	−0.781212	−0.390606	−	0.920558i	$-0.627734\pi$
$709$	−20.8014	−0.781212	−0.390606	+	0.920558i	$0.627734\pi$
$710$	0	0
$711$	−16.0470	−0.601809
$712$	0	0
$713$	−34.7333	−1.30077
$714$	0	0
$715$	−27.3751	−1.02377
$716$	0	0
$717$	−1.27227	−0.0475139
$718$	0	0
$719$	16.2049	0.604340	0.302170	−	0.953254i	$-0.402289\pi$
$719$	16.2049	0.604340	0.302170	+	0.953254i	$0.402289\pi$
$720$	0	0
$721$	−56.1524	−2.09122
$722$	0	0
$723$	74.6475	2.77617
$724$	0	0
$725$	5.25949	0.195332
$726$	0	0
$727$	−29.1999	−1.08297	−0.541483	−	0.840712i	$-0.682137\pi$
$727$	−29.1999	−1.08297	−0.541483	+	0.840712i	$0.682137\pi$
$728$	0	0
$729$	−26.7848	−0.992028
$730$	0	0
$731$	−11.1898	−0.413869
$732$	0	0
$733$	6.53323	0.241310	0.120655	−	0.992694i	$-0.461500\pi$
$733$	6.53323	0.241310	0.120655	+	0.992694i	$0.461500\pi$
$734$	0	0
$735$	15.1106	0.557364
$736$	0	0
$737$	10.4775	0.385945
$738$	0	0
$739$	29.7656	1.09495	0.547473	−	0.836823i	$-0.315590\pi$
$739$	29.7656	1.09495	0.547473	+	0.836823i	$0.315590\pi$
$740$	0	0
$741$	−50.5764	−1.85797
$742$	0	0
$743$	18.5786	0.681581	0.340790	−	0.940139i	$-0.389305\pi$
$743$	18.5786	0.681581	0.340790	+	0.940139i	$0.389305\pi$
$744$	0	0
$745$	11.5010	0.421365
$746$	0	0
$747$	26.1154	0.955514
$748$	0	0
$749$	−63.7277	−2.32856
$750$	0	0
$751$	−7.25684	−0.264806	−0.132403	−	0.991196i	$-0.542269\pi$
$751$	−7.25684	−0.264806	−0.132403	+	0.991196i	$0.542269\pi$
$752$	0	0
$753$	−54.4893	−1.98570
$754$	0	0
$755$	0.332250	0.0120918
$756$	0	0
$757$	12.2973	0.446952	0.223476	−	0.974709i	$-0.428260\pi$
$757$	12.2973	0.446952	0.223476	+	0.974709i	$0.428260\pi$
$758$	0	0
$759$	−103.010	−3.73901
$760$	0	0
$761$	24.0499	0.871808	0.435904	−	0.899993i	$-0.356429\pi$
$761$	24.0499	0.871808	0.435904	+	0.899993i	$0.356429\pi$
$762$	0	0
$763$	39.1063	1.41574
$764$	0	0
$765$	6.32852	0.228808
$766$	0	0
$767$	−26.5960	−0.960325
$768$	0	0
$769$	2.34706	0.0846372	0.0423186	−	0.999104i	$-0.486526\pi$
$769$	2.34706	0.0846372	0.0423186	+	0.999104i	$0.486526\pi$
$770$	0	0
$771$	76.6052	2.75887
$772$	0	0
$773$	−33.4859	−1.20441	−0.602203	−	0.798343i	$-0.705710\pi$
$773$	−33.4859	−1.20441	−0.602203	+	0.798343i	$0.705710\pi$
$774$	0	0
$775$	−5.03811	−0.180974
$776$	0	0
$777$	−8.88217	−0.318646
$778$	0	0
$779$	−46.3199	−1.65958
$780$	0	0
$781$	25.4192	0.909572
$782$	0	0
$783$	0.153591	0.00548888
$784$	0	0
$785$	−13.1549	−0.469518
$786$	0	0
$787$	18.4889	0.659057	0.329529	−	0.944146i	$-0.393110\pi$
$787$	18.4889	0.659057	0.329529	+	0.944146i	$0.393110\pi$
$788$	0	0
$789$	49.1380	1.74936
$790$	0	0
$791$	0.571033	0.0203036
$792$	0	0
$793$	−36.6403	−1.30114
$794$	0	0
$795$	−10.2835	−0.364717
$796$	0	0
$797$	41.6392	1.47493	0.737467	−	0.675383i	$-0.236022\pi$
$797$	41.6392	1.47493	0.737467	+	0.675383i	$0.236022\pi$
$798$	0	0
$799$	−4.19564	−0.148431
$800$	0	0
$801$	−19.7818	−0.698955
$802$	0	0
$803$	67.5146	2.38254
$804$	0	0
$805$	−25.0238	−0.881974
$806$	0	0
$807$	24.0045	0.844998
$808$	0	0
$809$	43.9415	1.54490	0.772451	−	0.635075i	$-0.219031\pi$
$809$	43.9415	1.54490	0.772451	+	0.635075i	$0.219031\pi$
$810$	0	0
$811$	44.4181	1.55973	0.779865	−	0.625948i	$-0.215288\pi$
$811$	44.4181	1.55973	0.779865	+	0.625948i	$0.215288\pi$
$812$	0	0
$813$	8.25594	0.289548
$814$	0	0
$815$	−18.5238	−0.648861
$816$	0	0
$817$	24.3565	0.852127
$818$	0	0
$819$	48.6256	1.69912
$820$	0	0
$821$	−31.5116	−1.09976	−0.549881	−	0.835243i	$-0.685327\pi$
$821$	−31.5116	−1.09976	−0.549881	+	0.835243i	$0.685327\pi$
$822$	0	0
$823$	−29.8191	−1.03943	−0.519714	−	0.854340i	$-0.673962\pi$
$823$	−29.8191	−1.03943	−0.519714	+	0.854340i	$0.673962\pi$
$824$	0	0
$825$	−14.9417	−0.520203
$826$	0	0
$827$	22.5669	0.784728	0.392364	−	0.919810i	$-0.371657\pi$
$827$	22.5669	0.784728	0.392364	+	0.919810i	$0.371657\pi$
$828$	0	0
$829$	−20.0790	−0.697372	−0.348686	−	0.937240i	$-0.613372\pi$
$829$	−20.0790	−0.697372	−0.348686	+	0.937240i	$0.613372\pi$
$830$	0	0
$831$	16.1235	0.559319
$832$	0	0
$833$	−13.0783	−0.453136
$834$	0	0
$835$	5.30378	0.183545
$836$	0	0
$837$	−0.147126	−0.00508542
$838$	0	0
$839$	37.1070	1.28108	0.640538	−	0.767926i	$-0.278711\pi$
$839$	37.1070	1.28108	0.640538	+	0.767926i	$0.278711\pi$
$840$	0	0
$841$	−1.33781	−0.0461314
$842$	0	0
$843$	19.2095	0.661611
$844$	0	0
$845$	−7.10009	−0.244250
$846$	0	0
$847$	95.4013	3.27803
$848$	0	0
$849$	−37.3651	−1.28237
$850$	0	0
$851$	6.89411	0.236327
$852$	0	0
$853$	−10.5445	−0.361035	−0.180518	−	0.983572i	$-0.557777\pi$
$853$	−10.5445	−0.361035	−0.180518	+	0.983572i	$0.557777\pi$
$854$	0	0
$855$	−13.7751	−0.471099
$856$	0	0
$857$	−52.3394	−1.78788	−0.893939	−	0.448188i	$-0.852070\pi$
$857$	−52.3394	−1.78788	−0.893939	+	0.448188i	$0.852070\pi$
$858$	0	0
$859$	26.9834	0.920660	0.460330	−	0.887748i	$-0.347731\pi$
$859$	26.9834	0.920660	0.460330	+	0.887748i	$0.347731\pi$
$860$	0	0
$861$	89.2444	3.04144
$862$	0	0
$863$	−23.5268	−0.800862	−0.400431	−	0.916327i	$-0.631140\pi$
$863$	−23.5268	−0.800862	−0.400431	+	0.916327i	$0.631140\pi$
$864$	0	0
$865$	−22.4561	−0.763529
$866$	0	0
$867$	30.6233	1.04002
$868$	0	0
$869$	−32.7914	−1.11237
$870$	0	0
$871$	7.69311	0.260671
$872$	0	0
$873$	−19.2624	−0.651934
$874$	0	0
$875$	−3.62974	−0.122708
$876$	0	0
$877$	−29.0625	−0.981372	−0.490686	−	0.871337i	$-0.663254\pi$
$877$	−29.0625	−0.981372	−0.490686	+	0.871337i	$0.663254\pi$
$878$	0	0
$879$	62.5659	2.11030
$880$	0	0
$881$	−3.44698	−0.116132	−0.0580659	−	0.998313i	$-0.518493\pi$
$881$	−3.44698	−0.116132	−0.0580659	+	0.998313i	$0.518493\pi$
$882$	0	0
$883$	1.20228	0.0404600	0.0202300	−	0.999795i	$-0.493560\pi$
$883$	1.20228	0.0404600	0.0202300	+	0.999795i	$0.493560\pi$
$884$	0	0
$885$	−14.5164	−0.487965
$886$	0	0
$887$	3.99441	0.134119	0.0670596	−	0.997749i	$-0.478638\pi$
$887$	3.99441	0.134119	0.0670596	+	0.997749i	$0.478638\pi$
$888$	0	0
$889$	49.8766	1.67281
$890$	0	0
$891$	−55.1717	−1.84832
$892$	0	0
$893$	9.13255	0.305609
$894$	0	0
$895$	−15.9279	−0.532412
$896$	0	0
$897$	−75.6345	−2.52536
$898$	0	0
$899$	−26.4979	−0.883753
$900$	0	0
$901$	8.90036	0.296514
$902$	0	0
$903$	−46.9276	−1.56165
$904$	0	0
$905$	10.8252	0.359841
$906$	0	0
$907$	34.1540	1.13407	0.567033	−	0.823695i	$-0.308091\pi$
$907$	34.1540	1.13407	0.567033	+	0.823695i	$0.308091\pi$
$908$	0	0
$909$	30.2576	1.00358
$910$	0	0
$911$	−0.654218	−0.0216752	−0.0108376	−	0.999941i	$-0.503450\pi$
$911$	−0.654218	−0.0216752	−0.0108376	+	0.999941i	$0.503450\pi$
$912$	0	0
$913$	53.3659	1.76615
$914$	0	0
$915$	−19.9988	−0.661140
$916$	0	0
$917$	−43.5896	−1.43946
$918$	0	0
$919$	−24.0668	−0.793890	−0.396945	−	0.917842i	$-0.629930\pi$
$919$	−24.0668	−0.793890	−0.396945	+	0.917842i	$0.629930\pi$
$920$	0	0
$921$	−39.4357	−1.29945
$922$	0	0
$923$	18.6640	0.614333
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−46.2256	−1.51825
$928$	0	0
$929$	48.4744	1.59039	0.795197	−	0.606351i	$-0.207367\pi$
$929$	48.4744	1.59039	0.795197	+	0.606351i	$0.207367\pi$
$930$	0	0
$931$	28.4672	0.932975
$932$	0	0
$933$	−34.1274	−1.11728
$934$	0	0
$935$	12.9321	0.422924
$936$	0	0
$937$	22.2644	0.727348	0.363674	−	0.931526i	$-0.381522\pi$
$937$	22.2644	0.727348	0.363674	+	0.931526i	$0.381522\pi$
$938$	0	0
$939$	−69.5634	−2.27012
$940$	0	0
$941$	31.0299	1.01155	0.505773	−	0.862667i	$-0.331207\pi$
$941$	31.0299	1.01155	0.505773	+	0.862667i	$0.331207\pi$
$942$	0	0
$943$	−69.2692	−2.25571
$944$	0	0
$945$	−0.105998	−0.00344811
$946$	0	0
$947$	−2.02892	−0.0659312	−0.0329656	−	0.999456i	$-0.510495\pi$
$947$	−2.02892	−0.0659312	−0.0329656	+	0.999456i	$0.510495\pi$
$948$	0	0
$949$	49.5724	1.60919
$950$	0	0
$951$	12.3192	0.399477
$952$	0	0
$953$	36.1774	1.17190	0.585950	−	0.810347i	$-0.300722\pi$
$953$	36.1774	1.17190	0.585950	+	0.810347i	$0.300722\pi$
$954$	0	0
$955$	3.03441	0.0981912
$956$	0	0
$957$	−78.5856	−2.54031
$958$	0	0
$959$	−35.6489	−1.15116
$960$	0	0
$961$	−5.61745	−0.181208
$962$	0	0
$963$	−52.4617	−1.69056
$964$	0	0
$965$	−6.06696	−0.195302
$966$	0	0
$967$	17.6536	0.567703	0.283851	−	0.958868i	$-0.408388\pi$
$967$	17.6536	0.567703	0.283851	+	0.958868i	$0.408388\pi$
$968$	0	0
$969$	23.8925	0.767536
$970$	0	0
$971$	8.85612	0.284207	0.142103	−	0.989852i	$-0.454613\pi$
$971$	8.85612	0.284207	0.142103	+	0.989852i	$0.454613\pi$
$972$	0	0
$973$	27.7680	0.890203
$974$	0	0
$975$	−10.9709	−0.351350
$976$	0	0
$977$	−22.8512	−0.731075	−0.365538	−	0.930797i	$-0.619115\pi$
$977$	−22.8512	−0.731075	−0.365538	+	0.930797i	$0.619115\pi$
$978$	0	0
$979$	−40.4233	−1.29193
$980$	0	0
$981$	32.1930	1.02784
$982$	0	0
$983$	−22.1776	−0.707355	−0.353677	−	0.935368i	$-0.615069\pi$
$983$	−22.1776	−0.707355	−0.353677	+	0.935368i	$0.615069\pi$
$984$	0	0
$985$	−2.75498	−0.0877809
$986$	0	0
$987$	−17.5957	−0.560076
$988$	0	0
$989$	36.4240	1.15822
$990$	0	0
$991$	−9.46148	−0.300554	−0.150277	−	0.988644i	$-0.548017\pi$
$991$	−9.46148	−0.300554	−0.150277	+	0.988644i	$0.548017\pi$
$992$	0	0
$993$	40.4012	1.28209
$994$	0	0
$995$	0.0481934	0.00152783
$996$	0	0
$997$	25.5308	0.808569	0.404284	−	0.914633i	$-0.367521\pi$
$997$	25.5308	0.808569	0.404284	+	0.914633i	$0.367521\pi$
$998$	0	0
$999$	0.0292026	0.000923929 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.z.1.1		5
4.3	odd	2		1480.2.a.h.1.5	✓	5
20.19	odd	2		7400.2.a.q.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.h.1.5	✓	5	4.3	odd	2
2960.2.a.z.1.1		5	1.1	even	1	trivial
7400.2.a.q.1.1		5	20.19	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:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44705 q^{3} -1.00000 q^{5} +3.62974 q^{7} +2.98807 q^{9} +O(q^{10})\)	\(q-2.44705 q^{3} -1.00000 q^{5} +3.62974 q^{7} +2.98807 q^{9} +6.10600 q^{11} +4.48331 q^{13} +2.44705 q^{15} -2.11793 q^{17} +4.61005 q^{19} -8.88217 q^{21} +6.89411 q^{23} +1.00000 q^{25} +0.0292026 q^{27} +5.25949 q^{29} -5.03811 q^{31} -14.9417 q^{33} -3.62974 q^{35} +1.00000 q^{37} -10.9709 q^{39} -10.0476 q^{41} +5.28335 q^{43} -2.98807 q^{45} +1.98101 q^{47} +6.17503 q^{49} +5.18269 q^{51} -4.20238 q^{53} -6.10600 q^{55} -11.2810 q^{57} -5.93222 q^{59} -8.17261 q^{61} +10.8459 q^{63} -4.48331 q^{65} +1.71594 q^{67} -16.8702 q^{69} +4.16300 q^{71} +11.0571 q^{73} -2.44705 q^{75} +22.1632 q^{77} -5.37036 q^{79} -9.03566 q^{81} +8.73991 q^{83} +2.11793 q^{85} -12.8702 q^{87} -6.62026 q^{89} +16.2733 q^{91} +12.3285 q^{93} -4.61005 q^{95} -6.44645 q^{97} +18.2451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9	\( 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 4 q^{13} - q^{15} + 10 q^{19} - 5 q^{21} + 8 q^{23} + 5 q^{25} + 7 q^{27} - 8 q^{29} + 2 q^{31} - 11 q^{33} - q^{35} + 5 q^{37} + 2 q^{39} - 13 q^{41} + 18 q^{43} - 2 q^{45} + 9 q^{47} - 6 q^{49} + 22 q^{51} - 13 q^{53} - 7 q^{55} + 4 q^{57} + 24 q^{59} - 2 q^{61} + 20 q^{63} - 4 q^{65} + 22 q^{67} - 32 q^{69} + 21 q^{71} + 29 q^{73} + q^{75} + 11 q^{77} + 6 q^{79} + 5 q^{81} + 33 q^{83} - 12 q^{87} - 26 q^{89} + 38 q^{91} + 14 q^{93} - 10 q^{95} + 26 q^{97} + 42 q^{99}+O(q^{100}) \) 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 4 * q^13 - q^15 + 10 * q^19 - 5 * q^21 + 8 * q^23 + 5 * q^25 + 7 * q^27 - 8 * q^29 + 2 * q^31 - 11 * q^33 - q^35 + 5 * q^37 + 2 * q^39 - 13 * q^41 + 18 * q^43 - 2 * q^45 + 9 * q^47 - 6 * q^49 + 22 * q^51 - 13 * q^53 - 7 * q^55 + 4 * q^57 + 24 * q^59 - 2 * q^61 + 20 * q^63 - 4 * q^65 + 22 * q^67 - 32 * q^69 + 21 * q^71 + 29 * q^73 + q^75 + 11 * q^77 + 6 * q^79 + 5 * q^81 + 33 * q^83 - 12 * q^87 - 26 * q^89 + 38 * q^91 + 14 * q^93 - 10 * q^95 + 26 * q^97 + 42 * q^99

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