LMFDB - Embedded newform 1520.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.51658$ of defining polynomial
Character	$\chi$	$=$	1520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53844 q^{3} -1.00000 q^{5} -5.03316 q^{7} -0.633188 q^{9} +O(q^{10})$	$q-1.53844 q^{3} -1.00000 q^{5} -5.03316 q^{7} -0.633188 q^{9} +3.03316 q^{11} -4.57160 q^{13} +1.53844 q^{15} -1.07689 q^{17} -1.00000 q^{19} +7.74324 q^{21} -4.11005 q^{23} +1.00000 q^{25} +5.58946 q^{27} -1.07689 q^{29} -5.58946 q^{31} -4.66635 q^{33} +5.03316 q^{35} +0.0947438 q^{37} +7.03316 q^{39} +10.6663 q^{41} -5.03316 q^{43} +0.633188 q^{45} +12.2995 q^{47} +18.3327 q^{49} +1.65673 q^{51} -4.09474 q^{53} -3.03316 q^{55} +1.53844 q^{57} +1.39997 q^{59} -5.69951 q^{61} +3.18694 q^{63} +4.57160 q^{65} -5.28168 q^{67} +6.32308 q^{69} +5.67692 q^{71} +9.07689 q^{73} -1.53844 q^{75} -15.2664 q^{77} +5.39997 q^{79} -6.69951 q^{81} -1.95627 q^{83} +1.07689 q^{85} +1.65673 q^{87} -2.18949 q^{89} +23.0096 q^{91} +8.59907 q^{93} +1.00000 q^{95} -2.16106 q^{97} -1.92056 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9	$ 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{35} - 6 q^{37} + 12 q^{39} + 16 q^{41} - 4 q^{43} - 8 q^{45} + 12 q^{47} + 20 q^{49} + 36 q^{51} - 10 q^{53} + 4 q^{55} + 2 q^{57} + 20 q^{61} - 20 q^{63} - 2 q^{65} + 18 q^{67} + 28 q^{69} + 20 q^{71} + 28 q^{73} - 2 q^{75} - 40 q^{77} + 16 q^{79} + 16 q^{81} - 4 q^{85} + 36 q^{87} + 4 q^{89} + 36 q^{91} - 40 q^{93} + 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9 - 4 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^21 + 8 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 - 4 * q^31 + 8 * q^33 + 4 * q^35 - 6 * q^37 + 12 * q^39 + 16 * q^41 - 4 * q^43 - 8 * q^45 + 12 * q^47 + 20 * q^49 + 36 * q^51 - 10 * q^53 + 4 * q^55 + 2 * q^57 + 20 * q^61 - 20 * q^63 - 2 * q^65 + 18 * q^67 + 28 * q^69 + 20 * q^71 + 28 * q^73 - 2 * q^75 - 40 * q^77 + 16 * q^79 + 16 * q^81 - 4 * q^85 + 36 * q^87 + 4 * q^89 + 36 * q^91 - 40 * q^93 + 4 * q^95 + 30 * q^97 + 4 * 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
$3$	−1.53844	−0.888221	−0.444111	−	0.895972i	$-0.646480\pi$
$3$	−1.53844	−0.888221	−0.444111	+	0.895972i	$0.646480\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−5.03316	−1.90236	−0.951178	−	0.308644i	$-0.900125\pi$
$7$	−5.03316	−1.90236	−0.951178	+	0.308644i	$0.900125\pi$
$8$	0	0
$9$	−0.633188	−0.211063
$10$	0	0
$11$	3.03316	0.914532	0.457266	−	0.889330i	$-0.348829\pi$
$11$	3.03316	0.914532	0.457266	+	0.889330i	$0.348829\pi$
$12$	0	0
$13$	−4.57160	−1.26793	−0.633967	−	0.773360i	$-0.718575\pi$
$13$	−4.57160	−1.26793	−0.633967	+	0.773360i	$0.718575\pi$
$14$	0	0
$15$	1.53844	0.397225
$16$	0	0
$17$	−1.07689	−0.261184	−0.130592	−	0.991436i	$-0.541688\pi$
$17$	−1.07689	−0.261184	−0.130592	+	0.991436i	$0.541688\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	7.74324	1.68971
$22$	0	0
$23$	−4.11005	−0.857004	−0.428502	−	0.903541i	$-0.640959\pi$
$23$	−4.11005	−0.857004	−0.428502	+	0.903541i	$0.640959\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.58946	1.07569
$28$	0	0
$29$	−1.07689	−0.199973	−0.0999866	−	0.994989i	$-0.531880\pi$
$29$	−1.07689	−0.199973	−0.0999866	+	0.994989i	$0.531880\pi$
$30$	0	0
$31$	−5.58946	−1.00390	−0.501948	−	0.864898i	$-0.667383\pi$
$31$	−5.58946	−1.00390	−0.501948	+	0.864898i	$0.667383\pi$
$32$	0	0
$33$	−4.66635	−0.812307
$34$	0	0
$35$	5.03316	0.850759
$36$	0	0
$37$	0.0947438	0.0155758	0.00778789	−	0.999970i	$-0.497521\pi$
$37$	0.0947438	0.0155758	0.00778789	+	0.999970i	$0.497521\pi$
$38$	0	0
$39$	7.03316	1.12621
$40$	0	0
$41$	10.6663	1.66580	0.832902	−	0.553421i	$-0.186678\pi$
$41$	10.6663	1.66580	0.832902	+	0.553421i	$0.186678\pi$
$42$	0	0
$43$	−5.03316	−0.767550	−0.383775	−	0.923427i	$-0.625376\pi$
$43$	−5.03316	−0.767550	−0.383775	+	0.923427i	$0.625376\pi$
$44$	0	0
$45$	0.633188	0.0943901
$46$	0	0
$47$	12.2995	1.79407	0.897036	−	0.441958i	$-0.145716\pi$
$47$	12.2995	1.79407	0.897036	+	0.441958i	$0.145716\pi$
$48$	0	0
$49$	18.3327	2.61896
$50$	0	0
$51$	1.65673	0.231989
$52$	0	0
$53$	−4.09474	−0.562456	−0.281228	−	0.959641i	$-0.590742\pi$
$53$	−4.09474	−0.562456	−0.281228	+	0.959641i	$0.590742\pi$
$54$	0	0
$55$	−3.03316	−0.408991
$56$	0	0
$57$	1.53844	0.203772
$58$	0	0
$59$	1.39997	0.182261	0.0911304	−	0.995839i	$-0.470952\pi$
$59$	1.39997	0.182261	0.0911304	+	0.995839i	$0.470952\pi$
$60$	0	0
$61$	−5.69951	−0.729747	−0.364874	−	0.931057i	$-0.618888\pi$
$61$	−5.69951	−0.729747	−0.364874	+	0.931057i	$0.618888\pi$
$62$	0	0
$63$	3.18694	0.401516
$64$	0	0
$65$	4.57160	0.567038
$66$	0	0
$67$	−5.28168	−0.645260	−0.322630	−	0.946525i	$-0.604567\pi$
$67$	−5.28168	−0.645260	−0.322630	+	0.946525i	$0.604567\pi$
$68$	0	0
$69$	6.32308	0.761210
$70$	0	0
$71$	5.67692	0.673726	0.336863	−	0.941554i	$-0.390634\pi$
$71$	5.67692	0.673726	0.336863	+	0.941554i	$0.390634\pi$
$72$	0	0
$73$	9.07689	1.06237	0.531185	−	0.847256i	$-0.321747\pi$
$73$	9.07689	1.06237	0.531185	+	0.847256i	$0.321747\pi$
$74$	0	0
$75$	−1.53844	−0.177644
$76$	0	0
$77$	−15.2664	−1.73977
$78$	0	0
$79$	5.39997	0.607544	0.303772	−	0.952745i	$-0.401754\pi$
$79$	5.39997	0.607544	0.303772	+	0.952745i	$0.401754\pi$
$80$	0	0
$81$	−6.69951	−0.744390
$82$	0	0
$83$	−1.95627	−0.214729	−0.107364	−	0.994220i	$-0.534241\pi$
$83$	−1.95627	−0.214729	−0.107364	+	0.994220i	$0.534241\pi$
$84$	0	0
$85$	1.07689	0.116805
$86$	0	0
$87$	1.65673	0.177621
$88$	0	0
$89$	−2.18949	−0.232085	−0.116043	−	0.993244i	$-0.537021\pi$
$89$	−2.18949	−0.232085	−0.116043	+	0.993244i	$0.537021\pi$
$90$	0	0
$91$	23.0096	2.41206
$92$	0	0
$93$	8.59907	0.891682
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−2.16106	−0.219423	−0.109711	−	0.993963i	$-0.534993\pi$
$97$	−2.16106	−0.219423	−0.109711	+	0.993963i	$0.534993\pi$
$98$	0	0
$99$	−1.92056	−0.193024
$100$	0	0
$101$	12.5869	1.25244	0.626222	−	0.779645i	$-0.284600\pi$
$101$	12.5869	1.25244	0.626222	+	0.779645i	$0.284600\pi$
$102$	0	0
$103$	6.20479	0.611376	0.305688	−	0.952132i	$-0.401113\pi$
$103$	6.20479	0.611376	0.305688	+	0.952132i	$0.401113\pi$
$104$	0	0
$105$	−7.74324	−0.755663
$106$	0	0
$107$	12.5481	1.21307	0.606533	−	0.795058i	$-0.292560\pi$
$107$	12.5481	1.21307	0.606533	+	0.795058i	$0.292560\pi$
$108$	0	0
$109$	−15.8096	−1.51428	−0.757140	−	0.653252i	$-0.773404\pi$
$109$	−15.8096	−1.51428	−0.757140	+	0.653252i	$0.773404\pi$
$110$	0	0
$111$	−0.145758	−0.0138347
$112$	0	0
$113$	−5.49472	−0.516899	−0.258450	−	0.966025i	$-0.583212\pi$
$113$	−5.49472	−0.516899	−0.258450	+	0.966025i	$0.583212\pi$
$114$	0	0
$115$	4.11005	0.383264
$116$	0	0
$117$	2.89469	0.267614
$118$	0	0
$119$	5.42015	0.496865
$120$	0	0
$121$	−1.79994	−0.163631
$122$	0	0
$123$	−16.4096	−1.47960
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.61533	0.764487	0.382244	−	0.924062i	$-0.375152\pi$
$127$	8.61533	0.764487	0.382244	+	0.924062i	$0.375152\pi$
$128$	0	0
$129$	7.74324	0.681754
$130$	0	0
$131$	2.15378	0.188176	0.0940882	−	0.995564i	$-0.470006\pi$
$131$	2.15378	0.188176	0.0940882	+	0.995564i	$0.470006\pi$
$132$	0	0
$133$	5.03316	0.436430
$134$	0	0
$135$	−5.58946	−0.481064
$136$	0	0
$137$	2.18949	0.187061	0.0935303	−	0.995616i	$-0.470185\pi$
$137$	2.18949	0.187061	0.0935303	+	0.995616i	$0.470185\pi$
$138$	0	0
$139$	−22.5196	−1.91009	−0.955045	−	0.296460i	$-0.904194\pi$
$139$	−22.5196	−1.91009	−0.955045	+	0.296460i	$0.904194\pi$
$140$	0	0
$141$	−18.9222	−1.59353
$142$	0	0
$143$	−13.8664	−1.15957
$144$	0	0
$145$	1.07689	0.0894308
$146$	0	0
$147$	−28.2038	−2.32621
$148$	0	0
$149$	9.78697	0.801780	0.400890	−	0.916126i	$-0.368701\pi$
$149$	9.78697	0.801780	0.400890	+	0.916126i	$0.368701\pi$
$150$	0	0
$151$	−5.87683	−0.478250	−0.239125	−	0.970989i	$-0.576861\pi$
$151$	−5.87683	−0.478250	−0.239125	+	0.970989i	$0.576861\pi$
$152$	0	0
$153$	0.681874	0.0551262
$154$	0	0
$155$	5.58946	0.448956
$156$	0	0
$157$	10.1895	0.813210	0.406605	−	0.913604i	$-0.366713\pi$
$157$	10.1895	0.813210	0.406605	+	0.913604i	$0.366713\pi$
$158$	0	0
$159$	6.29954	0.499586
$160$	0	0
$161$	20.6865	1.63033
$162$	0	0
$163$	10.3659	0.811916	0.405958	−	0.913892i	$-0.366938\pi$
$163$	10.3659	0.811916	0.405958	+	0.913892i	$0.366938\pi$
$164$	0	0
$165$	4.66635	0.363275
$166$	0	0
$167$	15.6048	1.20753	0.603766	−	0.797161i	$-0.293666\pi$
$167$	15.6048	1.20753	0.603766	+	0.797161i	$0.293666\pi$
$168$	0	0
$169$	7.89956	0.607659
$170$	0	0
$171$	0.633188	0.0484211
$172$	0	0
$173$	0.571604	0.0434583	0.0217291	−	0.999764i	$-0.493083\pi$
$173$	0.571604	0.0434583	0.0217291	+	0.999764i	$0.493083\pi$
$174$	0	0
$175$	−5.03316	−0.380471
$176$	0	0
$177$	−2.15378	−0.161888
$178$	0	0
$179$	−24.8864	−1.86010	−0.930050	−	0.367433i	$-0.880237\pi$
$179$	−24.8864	−1.86010	−0.930050	+	0.367433i	$0.880237\pi$
$180$	0	0
$181$	6.95372	0.516866	0.258433	−	0.966029i	$-0.416794\pi$
$181$	6.95372	0.516866	0.258433	+	0.966029i	$0.416794\pi$
$182$	0	0
$183$	8.76838	0.648177
$184$	0	0
$185$	−0.0947438	−0.00696570
$186$	0	0
$187$	−3.26638	−0.238861
$188$	0	0
$189$	−28.1326	−2.04635
$190$	0	0
$191$	3.91254	0.283102	0.141551	−	0.989931i	$-0.454791\pi$
$191$	3.91254	0.283102	0.141551	+	0.989931i	$0.454791\pi$
$192$	0	0
$193$	−9.20734	−0.662759	−0.331379	−	0.943498i	$-0.607514\pi$
$193$	−9.20734	−0.662759	−0.331379	+	0.943498i	$0.607514\pi$
$194$	0	0
$195$	−7.03316	−0.503655
$196$	0	0
$197$	−3.84622	−0.274032	−0.137016	−	0.990569i	$-0.543751\pi$
$197$	−3.84622	−0.274032	−0.137016	+	0.990569i	$0.543751\pi$
$198$	0	0
$199$	18.2864	1.29629	0.648145	−	0.761517i	$-0.275545\pi$
$199$	18.2864	1.29629	0.648145	+	0.761517i	$0.275545\pi$
$200$	0	0
$201$	8.12557	0.573134
$202$	0	0
$203$	5.42015	0.380420
$204$	0	0
$205$	−10.6663	−0.744970
$206$	0	0
$207$	2.60243	0.180882
$208$	0	0
$209$	−3.03316	−0.209808
$210$	0	0
$211$	19.2970	1.32846	0.664230	−	0.747529i	$-0.268760\pi$
$211$	19.2970	1.32846	0.664230	+	0.747529i	$0.268760\pi$
$212$	0	0
$213$	−8.73362	−0.598418
$214$	0	0
$215$	5.03316	0.343259
$216$	0	0
$217$	28.1326	1.90977
$218$	0	0
$219$	−13.9643	−0.943619
$220$	0	0
$221$	4.92311	0.331164
$222$	0	0
$223$	−0.615334	−0.0412058	−0.0206029	−	0.999788i	$-0.506559\pi$
$223$	−0.615334	−0.0412058	−0.0206029	+	0.999788i	$0.506559\pi$
$224$	0	0
$225$	−0.633188	−0.0422126
$226$	0	0
$227$	−17.4712	−1.15960	−0.579801	−	0.814758i	$-0.696870\pi$
$227$	−17.4712	−1.15960	−0.579801	+	0.814758i	$0.696870\pi$
$228$	0	0
$229$	9.69951	0.640962	0.320481	−	0.947255i	$-0.396156\pi$
$229$	9.69951	0.640962	0.320481	+	0.947255i	$0.396156\pi$
$230$	0	0
$231$	23.4865	1.54530
$232$	0	0
$233$	−8.15378	−0.534172	−0.267086	−	0.963673i	$-0.586061\pi$
$233$	−8.15378	−0.534172	−0.267086	+	0.963673i	$0.586061\pi$
$234$	0	0
$235$	−12.2995	−0.802333
$236$	0	0
$237$	−8.30756	−0.539634
$238$	0	0
$239$	−24.5991	−1.59118	−0.795591	−	0.605834i	$-0.792839\pi$
$239$	−24.5991	−1.59118	−0.795591	+	0.605834i	$0.792839\pi$
$240$	0	0
$241$	25.7738	1.66024	0.830120	−	0.557585i	$-0.188272\pi$
$241$	25.7738	1.66024	0.830120	+	0.557585i	$0.188272\pi$
$242$	0	0
$243$	−6.46156	−0.414509
$244$	0	0
$245$	−18.3327	−1.17123
$246$	0	0
$247$	4.57160	0.290884
$248$	0	0
$249$	3.00961	0.190727
$250$	0	0
$251$	−12.5991	−0.795246	−0.397623	−	0.917549i	$-0.630165\pi$
$251$	−12.5991	−0.795246	−0.397623	+	0.917549i	$0.630165\pi$
$252$	0	0
$253$	−12.4664	−0.783758
$254$	0	0
$255$	−1.65673	−0.103749
$256$	0	0
$257$	0.182203	0.0113655	0.00568275	−	0.999984i	$-0.498191\pi$
$257$	0.182203	0.0113655	0.00568275	+	0.999984i	$0.498191\pi$
$258$	0	0
$259$	−0.476860	−0.0296307
$260$	0	0
$261$	0.681874	0.0422069
$262$	0	0
$263$	−7.37643	−0.454850	−0.227425	−	0.973796i	$-0.573031\pi$
$263$	−7.37643	−0.454850	−0.227425	+	0.973796i	$0.573031\pi$
$264$	0	0
$265$	4.09474	0.251538
$266$	0	0
$267$	3.36841	0.206143
$268$	0	0
$269$	4.70206	0.286689	0.143345	−	0.989673i	$-0.454214\pi$
$269$	4.70206	0.286689	0.143345	+	0.989673i	$0.454214\pi$
$270$	0	0
$271$	−9.92056	−0.602631	−0.301316	−	0.953524i	$-0.597426\pi$
$271$	−9.92056	−0.602631	−0.301316	+	0.953524i	$0.597426\pi$
$272$	0	0
$273$	−35.3990	−2.14245
$274$	0	0
$275$	3.03316	0.182906
$276$	0	0
$277$	−22.6297	−1.35969	−0.679843	−	0.733358i	$-0.737952\pi$
$277$	−22.6297	−1.35969	−0.679843	+	0.733358i	$0.737952\pi$
$278$	0	0
$279$	3.53918	0.211885
$280$	0	0
$281$	−3.95386	−0.235868	−0.117934	−	0.993021i	$-0.537627\pi$
$281$	−3.95386	−0.235868	−0.117934	+	0.993021i	$0.537627\pi$
$282$	0	0
$283$	13.0638	0.776561	0.388280	−	0.921541i	$-0.373069\pi$
$283$	13.0638	0.776561	0.388280	+	0.921541i	$0.373069\pi$
$284$	0	0
$285$	−1.53844	−0.0911296
$286$	0	0
$287$	−53.6854	−3.16895
$288$	0	0
$289$	−15.8403	−0.931783
$290$	0	0
$291$	3.32468	0.194896
$292$	0	0
$293$	−9.69463	−0.566366	−0.283183	−	0.959066i	$-0.591390\pi$
$293$	−9.69463	−0.566366	−0.283183	+	0.959066i	$0.591390\pi$
$294$	0	0
$295$	−1.39997	−0.0815095
$296$	0	0
$297$	16.9537	0.983755
$298$	0	0
$299$	18.7895	1.08663
$300$	0	0
$301$	25.3327	1.46015
$302$	0	0
$303$	−19.3643	−1.11245
$304$	0	0
$305$	5.69951	0.326353
$306$	0	0
$307$	28.5481	1.62932	0.814662	−	0.579936i	$-0.196923\pi$
$307$	28.5481	1.62932	0.814662	+	0.579936i	$0.196923\pi$
$308$	0	0
$309$	−9.54573	−0.543038
$310$	0	0
$311$	−25.4070	−1.44070	−0.720350	−	0.693610i	$-0.756019\pi$
$311$	−25.4070	−1.44070	−0.720350	+	0.693610i	$0.756019\pi$
$312$	0	0
$313$	16.7999	0.949589	0.474794	−	0.880097i	$-0.342522\pi$
$313$	16.7999	0.949589	0.474794	+	0.880097i	$0.342522\pi$
$314$	0	0
$315$	−3.18694	−0.179564
$316$	0	0
$317$	−31.7505	−1.78329	−0.891643	−	0.452738i	$-0.850447\pi$
$317$	−31.7505	−1.78329	−0.891643	+	0.452738i	$0.850447\pi$
$318$	0	0
$319$	−3.26638	−0.182882
$320$	0	0
$321$	−19.3045	−1.07747
$322$	0	0
$323$	1.07689	0.0599197
$324$	0	0
$325$	−4.57160	−0.253587
$326$	0	0
$327$	24.3221	1.34502
$328$	0	0
$329$	−61.9055	−3.41296
$330$	0	0
$331$	9.96429	0.547687	0.273843	−	0.961774i	$-0.411705\pi$
$331$	9.96429	0.547687	0.273843	+	0.961774i	$0.411705\pi$
$332$	0	0
$333$	−0.0599906	−0.00328747
$334$	0	0
$335$	5.28168	0.288569
$336$	0	0
$337$	23.1918	1.26334	0.631669	−	0.775238i	$-0.282370\pi$
$337$	23.1918	1.26334	0.631669	+	0.775238i	$0.282370\pi$
$338$	0	0
$339$	8.45331	0.459121
$340$	0	0
$341$	−16.9537	−0.918095
$342$	0	0
$343$	−57.0393	−3.07983
$344$	0	0
$345$	−6.32308	−0.340423
$346$	0	0
$347$	−21.1352	−1.13460	−0.567298	−	0.823512i	$-0.692011\pi$
$347$	−21.1352	−1.13460	−0.567298	+	0.823512i	$0.692011\pi$
$348$	0	0
$349$	−5.04628	−0.270121	−0.135061	−	0.990837i	$-0.543123\pi$
$349$	−5.04628	−0.270121	−0.135061	+	0.990837i	$0.543123\pi$
$350$	0	0
$351$	−25.5528	−1.36391
$352$	0	0
$353$	12.5634	0.668680	0.334340	−	0.942452i	$-0.391487\pi$
$353$	12.5634	0.668680	0.334340	+	0.942452i	$0.391487\pi$
$354$	0	0
$355$	−5.67692	−0.301300
$356$	0	0
$357$	−8.33861	−0.441326
$358$	0	0
$359$	17.6534	0.931709	0.465855	−	0.884861i	$-0.345747\pi$
$359$	17.6534	0.931709	0.465855	+	0.884861i	$0.345747\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.76911	0.145341
$364$	0	0
$365$	−9.07689	−0.475106
$366$	0	0
$367$	−4.43409	−0.231457	−0.115729	−	0.993281i	$-0.536920\pi$
$367$	−4.43409	−0.231457	−0.115729	+	0.993281i	$0.536920\pi$
$368$	0	0
$369$	−6.75381	−0.351589
$370$	0	0
$371$	20.6095	1.06999
$372$	0	0
$373$	−20.8735	−1.08079	−0.540396	−	0.841411i	$-0.681725\pi$
$373$	−20.8735	−1.08079	−0.540396	+	0.841411i	$0.681725\pi$
$374$	0	0
$375$	1.53844	0.0794449
$376$	0	0
$377$	4.92311	0.253553
$378$	0	0
$379$	6.79994	0.349290	0.174645	−	0.984632i	$-0.444122\pi$
$379$	6.79994	0.349290	0.174645	+	0.984632i	$0.444122\pi$
$380$	0	0
$381$	−13.2542	−0.679034
$382$	0	0
$383$	10.9078	0.557363	0.278681	−	0.960384i	$-0.410103\pi$
$383$	10.9078	0.557363	0.278681	+	0.960384i	$0.410103\pi$
$384$	0	0
$385$	15.2664	0.778047
$386$	0	0
$387$	3.18694	0.162001
$388$	0	0
$389$	20.9326	1.06132	0.530662	−	0.847584i	$-0.321943\pi$
$389$	20.9326	1.06132	0.530662	+	0.847584i	$0.321943\pi$
$390$	0	0
$391$	4.42607	0.223836
$392$	0	0
$393$	−3.31347	−0.167142
$394$	0	0
$395$	−5.39997	−0.271702
$396$	0	0
$397$	13.1221	0.658578	0.329289	−	0.944229i	$-0.393191\pi$
$397$	13.1221	0.658578	0.329289	+	0.944229i	$0.393191\pi$
$398$	0	0
$399$	−7.74324	−0.387647
$400$	0	0
$401$	−11.5528	−0.576919	−0.288459	−	0.957492i	$-0.593143\pi$
$401$	−11.5528	−0.576919	−0.288459	+	0.957492i	$0.593143\pi$
$402$	0	0
$403$	25.5528	1.27288
$404$	0	0
$405$	6.69951	0.332901
$406$	0	0
$407$	0.287373	0.0142445
$408$	0	0
$409$	−18.9433	−0.936686	−0.468343	−	0.883547i	$-0.655149\pi$
$409$	−18.9433	−0.936686	−0.468343	+	0.883547i	$0.655149\pi$
$410$	0	0
$411$	−3.36841	−0.166151
$412$	0	0
$413$	−7.04628	−0.346725
$414$	0	0
$415$	1.95627	0.0960295
$416$	0	0
$417$	34.6452	1.69658
$418$	0	0
$419$	−27.2664	−1.33205	−0.666025	−	0.745930i	$-0.732006\pi$
$419$	−27.2664	−1.33205	−0.666025	+	0.745930i	$0.732006\pi$
$420$	0	0
$421$	6.66635	0.324898	0.162449	−	0.986717i	$-0.448061\pi$
$421$	6.66635	0.324898	0.162449	+	0.986717i	$0.448061\pi$
$422$	0	0
$423$	−7.78792	−0.378662
$424$	0	0
$425$	−1.07689	−0.0522368
$426$	0	0
$427$	28.6865	1.38824
$428$	0	0
$429$	21.3327	1.02995
$430$	0	0
$431$	32.4548	1.56329	0.781646	−	0.623723i	$-0.214381\pi$
$431$	32.4548	1.56329	0.781646	+	0.623723i	$0.214381\pi$
$432$	0	0
$433$	33.8168	1.62513	0.812567	−	0.582868i	$-0.198070\pi$
$433$	33.8168	1.62513	0.812567	+	0.582868i	$0.198070\pi$
$434$	0	0
$435$	−1.65673	−0.0794343
$436$	0	0
$437$	4.11005	0.196610
$438$	0	0
$439$	27.8453	1.32898	0.664491	−	0.747296i	$-0.268648\pi$
$439$	27.8453	1.32898	0.664491	+	0.747296i	$0.268648\pi$
$440$	0	0
$441$	−11.6080	−0.552764
$442$	0	0
$443$	−22.7754	−1.08209	−0.541047	−	0.840992i	$-0.681972\pi$
$443$	−22.7754	−1.08209	−0.541047	+	0.840992i	$0.681972\pi$
$444$	0	0
$445$	2.18949	0.103792
$446$	0	0
$447$	−15.0567	−0.712158
$448$	0	0
$449$	24.8507	1.17278	0.586389	−	0.810029i	$-0.300549\pi$
$449$	24.8507	1.17278	0.586389	+	0.810029i	$0.300549\pi$
$450$	0	0
$451$	32.3527	1.52343
$452$	0	0
$453$	9.04118	0.424792
$454$	0	0
$455$	−23.0096	−1.07871
$456$	0	0
$457$	−7.33270	−0.343009	−0.171505	−	0.985183i	$-0.554863\pi$
$457$	−7.33270	−0.343009	−0.171505	+	0.985183i	$0.554863\pi$
$458$	0	0
$459$	−6.01923	−0.280953
$460$	0	0
$461$	10.3076	0.480071	0.240035	−	0.970764i	$-0.422841\pi$
$461$	10.3076	0.480071	0.240035	+	0.970764i	$0.422841\pi$
$462$	0	0
$463$	−7.80249	−0.362613	−0.181306	−	0.983427i	$-0.558033\pi$
$463$	−7.80249	−0.362613	−0.181306	+	0.983427i	$0.558033\pi$
$464$	0	0
$465$	−8.59907	−0.398772
$466$	0	0
$467$	7.37643	0.341340	0.170670	−	0.985328i	$-0.445407\pi$
$467$	7.37643	0.341340	0.170670	+	0.985328i	$0.445407\pi$
$468$	0	0
$469$	26.5835	1.22751
$470$	0	0
$471$	−15.6760	−0.722310
$472$	0	0
$473$	−15.2664	−0.701949
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	2.59274	0.118714
$478$	0	0
$479$	16.1458	0.737719	0.368859	−	0.929485i	$-0.379748\pi$
$479$	16.1458	0.737719	0.368859	+	0.929485i	$0.379748\pi$
$480$	0	0
$481$	−0.433131	−0.0197491
$482$	0	0
$483$	−31.8251	−1.44809
$484$	0	0
$485$	2.16106	0.0981288
$486$	0	0
$487$	8.40566	0.380897	0.190448	−	0.981697i	$-0.439006\pi$
$487$	8.40566	0.380897	0.190448	+	0.981697i	$0.439006\pi$
$488$	0	0
$489$	−15.9473	−0.721162
$490$	0	0
$491$	36.0855	1.62852	0.814259	−	0.580502i	$-0.197144\pi$
$491$	36.0855	1.62852	0.814259	+	0.580502i	$0.197144\pi$
$492$	0	0
$493$	1.15969	0.0522298
$494$	0	0
$495$	1.92056	0.0863228
$496$	0	0
$497$	−28.5728	−1.28167
$498$	0	0
$499$	3.30035	0.147744	0.0738720	−	0.997268i	$-0.476464\pi$
$499$	3.30035	0.147744	0.0738720	+	0.997268i	$0.476464\pi$
$500$	0	0
$501$	−24.0071	−1.07256
$502$	0	0
$503$	−3.53530	−0.157631	−0.0788157	−	0.996889i	$-0.525114\pi$
$503$	−3.53530	−0.157631	−0.0788157	+	0.996889i	$0.525114\pi$
$504$	0	0
$505$	−12.5869	−0.560110
$506$	0	0
$507$	−12.1530	−0.539736
$508$	0	0
$509$	22.4096	0.993287	0.496644	−	0.867955i	$-0.334566\pi$
$509$	22.4096	0.993287	0.496644	+	0.867955i	$0.334566\pi$
$510$	0	0
$511$	−45.6854	−2.02100
$512$	0	0
$513$	−5.58946	−0.246781
$514$	0	0
$515$	−6.20479	−0.273416
$516$	0	0
$517$	37.3065	1.64074
$518$	0	0
$519$	−0.879381	−0.0386006
$520$	0	0
$521$	34.4402	1.50885	0.754426	−	0.656385i	$-0.227915\pi$
$521$	34.4402	1.50885	0.754426	+	0.656385i	$0.227915\pi$
$522$	0	0
$523$	30.4451	1.33127	0.665635	−	0.746277i	$-0.268161\pi$
$523$	30.4451	1.33127	0.665635	+	0.746277i	$0.268161\pi$
$524$	0	0
$525$	7.74324	0.337943
$526$	0	0
$527$	6.01923	0.262202
$528$	0	0
$529$	−6.10750	−0.265543
$530$	0	0
$531$	−0.886445	−0.0384685
$532$	0	0
$533$	−48.7623	−2.11213
$534$	0	0
$535$	−12.5481	−0.542500
$536$	0	0
$537$	38.2864	1.65218
$538$	0	0
$539$	55.6060	2.39512
$540$	0	0
$541$	−12.2794	−0.527931	−0.263965	−	0.964532i	$-0.585030\pi$
$541$	−12.2794	−0.527931	−0.263965	+	0.964532i	$0.585030\pi$
$542$	0	0
$543$	−10.6979	−0.459091
$544$	0	0
$545$	15.8096	0.677207
$546$	0	0
$547$	−0.250928	−0.0107289	−0.00536446	−	0.999986i	$-0.501708\pi$
$547$	−0.250928	−0.0107289	−0.00536446	+	0.999986i	$0.501708\pi$
$548$	0	0
$549$	3.60886	0.154022
$550$	0	0
$551$	1.07689	0.0458770
$552$	0	0
$553$	−27.1789	−1.15577
$554$	0	0
$555$	0.145758	0.00618708
$556$	0	0
$557$	1.67596	0.0710128	0.0355064	−	0.999369i	$-0.488696\pi$
$557$	1.67596	0.0710128	0.0355064	+	0.999369i	$0.488696\pi$
$558$	0	0
$559$	23.0096	0.973203
$560$	0	0
$561$	5.02514	0.212162
$562$	0	0
$563$	−20.6605	−0.870737	−0.435368	−	0.900252i	$-0.643382\pi$
$563$	−20.6605	−0.870737	−0.435368	+	0.900252i	$0.643382\pi$
$564$	0	0
$565$	5.49472	0.231164
$566$	0	0
$567$	33.7197	1.41609
$568$	0	0
$569$	40.9673	1.71744	0.858720	−	0.512445i	$-0.171260\pi$
$569$	40.9673	1.71744	0.858720	+	0.512445i	$0.171260\pi$
$570$	0	0
$571$	−24.2070	−1.01303	−0.506515	−	0.862231i	$-0.669067\pi$
$571$	−24.2070	−1.01303	−0.506515	+	0.862231i	$0.669067\pi$
$572$	0	0
$573$	−6.01923	−0.251457
$574$	0	0
$575$	−4.11005	−0.171401
$576$	0	0
$577$	40.2864	1.67715	0.838573	−	0.544790i	$-0.183391\pi$
$577$	40.2864	1.67715	0.838573	+	0.544790i	$0.183391\pi$
$578$	0	0
$579$	14.1650	0.588677
$580$	0	0
$581$	9.84622	0.408490
$582$	0	0
$583$	−12.4200	−0.514384
$584$	0	0
$585$	−2.89469	−0.119681
$586$	0	0
$587$	34.7754	1.43534	0.717668	−	0.696385i	$-0.245210\pi$
$587$	34.7754	1.43534	0.717668	+	0.696385i	$0.245210\pi$
$588$	0	0
$589$	5.58946	0.230310
$590$	0	0
$591$	5.91720	0.243401
$592$	0	0
$593$	−35.0864	−1.44082	−0.720412	−	0.693546i	$-0.756047\pi$
$593$	−35.0864	−1.44082	−0.720412	+	0.693546i	$0.756047\pi$
$594$	0	0
$595$	−5.42015	−0.222205
$596$	0	0
$597$	−28.1326	−1.15139
$598$	0	0
$599$	−30.8201	−1.25928	−0.629638	−	0.776889i	$-0.716797\pi$
$599$	−30.8201	−1.25928	−0.629638	+	0.776889i	$0.716797\pi$
$600$	0	0
$601$	−7.10654	−0.289882	−0.144941	−	0.989440i	$-0.546299\pi$
$601$	−7.10654	−0.289882	−0.144941	+	0.989440i	$0.546299\pi$
$602$	0	0
$603$	3.34430	0.136190
$604$	0	0
$605$	1.79994	0.0731781
$606$	0	0
$607$	−2.01531	−0.0817987	−0.0408994	−	0.999163i	$-0.513022\pi$
$607$	−2.01531	−0.0817987	−0.0408994	+	0.999163i	$0.513022\pi$
$608$	0	0
$609$	−8.33861	−0.337897
$610$	0	0
$611$	−56.2286	−2.27477
$612$	0	0
$613$	−11.6096	−0.468909	−0.234455	−	0.972127i	$-0.575330\pi$
$613$	−11.6096	−0.468909	−0.234455	+	0.972127i	$0.575330\pi$
$614$	0	0
$615$	16.4096	0.661698
$616$	0	0
$617$	12.4307	0.500442	0.250221	−	0.968189i	$-0.419497\pi$
$617$	12.4307	0.500442	0.250221	+	0.968189i	$0.419497\pi$
$618$	0	0
$619$	−3.85424	−0.154915	−0.0774575	−	0.996996i	$-0.524680\pi$
$619$	−3.85424	−0.154915	−0.0774575	+	0.996996i	$0.524680\pi$
$620$	0	0
$621$	−22.9729	−0.921873
$622$	0	0
$623$	11.0200	0.441509
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.66635	0.186356
$628$	0	0
$629$	−0.102029	−0.00406814
$630$	0	0
$631$	16.8794	0.671958	0.335979	−	0.941870i	$-0.390933\pi$
$631$	16.8794	0.671958	0.335979	+	0.941870i	$0.390933\pi$
$632$	0	0
$633$	−29.6873	−1.17997
$634$	0	0
$635$	−8.61533	−0.341889
$636$	0	0
$637$	−83.8098	−3.32067
$638$	0	0
$639$	−3.59456	−0.142199
$640$	0	0
$641$	43.4855	1.71757	0.858787	−	0.512332i	$-0.171218\pi$
$641$	43.4855	1.71757	0.858787	+	0.512332i	$0.171218\pi$
$642$	0	0
$643$	−11.0689	−0.436514	−0.218257	−	0.975891i	$-0.570037\pi$
$643$	−11.0689	−0.436514	−0.218257	+	0.975891i	$0.570037\pi$
$644$	0	0
$645$	−7.74324	−0.304890
$646$	0	0
$647$	−5.61766	−0.220853	−0.110427	−	0.993884i	$-0.535222\pi$
$647$	−5.61766	−0.220853	−0.110427	+	0.993884i	$0.535222\pi$
$648$	0	0
$649$	4.24634	0.166683
$650$	0	0
$651$	−43.2805	−1.69630
$652$	0	0
$653$	44.5211	1.74224	0.871122	−	0.491066i	$-0.163393\pi$
$653$	44.5211	1.74224	0.871122	+	0.491066i	$0.163393\pi$
$654$	0	0
$655$	−2.15378	−0.0841551
$656$	0	0
$657$	−5.74738	−0.224227
$658$	0	0
$659$	6.89154	0.268456	0.134228	−	0.990950i	$-0.457144\pi$
$659$	6.89154	0.268456	0.134228	+	0.990950i	$0.457144\pi$
$660$	0	0
$661$	−44.3989	−1.72692	−0.863458	−	0.504421i	$-0.831706\pi$
$661$	−44.3989	−1.72692	−0.863458	+	0.504421i	$0.831706\pi$
$662$	0	0
$663$	−7.57393	−0.294147
$664$	0	0
$665$	−5.03316	−0.195178
$666$	0	0
$667$	4.42607	0.171378
$668$	0	0
$669$	0.946657	0.0365999
$670$	0	0
$671$	−17.2875	−0.667377
$672$	0	0
$673$	38.6214	1.48875	0.744374	−	0.667763i	$-0.232748\pi$
$673$	38.6214	1.48875	0.744374	+	0.667763i	$0.232748\pi$
$674$	0	0
$675$	5.58946	0.215138
$676$	0	0
$677$	43.3917	1.66768	0.833840	−	0.552006i	$-0.186138\pi$
$677$	43.3917	1.66768	0.833840	+	0.552006i	$0.186138\pi$
$678$	0	0
$679$	10.8770	0.417420
$680$	0	0
$681$	26.8784	1.02998
$682$	0	0
$683$	34.7123	1.32823	0.664114	−	0.747631i	$-0.268809\pi$
$683$	34.7123	1.32823	0.664114	+	0.747631i	$0.268809\pi$
$684$	0	0
$685$	−2.18949	−0.0836560
$686$	0	0
$687$	−14.9222	−0.569316
$688$	0	0
$689$	18.7195	0.713158
$690$	0	0
$691$	−2.94570	−0.112060	−0.0560299	−	0.998429i	$-0.517844\pi$
$691$	−2.94570	−0.112060	−0.0560299	+	0.998429i	$0.517844\pi$
$692$	0	0
$693$	9.66649	0.367200
$694$	0	0
$695$	22.5196	0.854218
$696$	0	0
$697$	−11.4865	−0.435081
$698$	0	0
$699$	12.5441	0.474463
$700$	0	0
$701$	18.5920	0.702210	0.351105	−	0.936336i	$-0.385806\pi$
$701$	18.5920	0.702210	0.351105	+	0.936336i	$0.385806\pi$
$702$	0	0
$703$	−0.0947438	−0.00357333
$704$	0	0
$705$	18.9222	0.712650
$706$	0	0
$707$	−63.3519	−2.38259
$708$	0	0
$709$	39.8443	1.49638	0.748192	−	0.663482i	$-0.230922\pi$
$709$	39.8443	1.49638	0.748192	+	0.663482i	$0.230922\pi$
$710$	0	0
$711$	−3.41920	−0.128230
$712$	0	0
$713$	22.9729	0.860344
$714$	0	0
$715$	13.8664	0.518574
$716$	0	0
$717$	37.8443	1.41332
$718$	0	0
$719$	−21.2321	−0.791824	−0.395912	−	0.918288i	$-0.629572\pi$
$719$	−21.2321	−0.791824	−0.395912	+	0.918288i	$0.629572\pi$
$720$	0	0
$721$	−31.2297	−1.16306
$722$	0	0
$723$	−39.6516	−1.47466
$724$	0	0
$725$	−1.07689	−0.0399947
$726$	0	0
$727$	49.1658	1.82346	0.911729	−	0.410792i	$-0.134748\pi$
$727$	49.1658	1.82346	0.911729	+	0.410792i	$0.134748\pi$
$728$	0	0
$729$	30.0393	1.11257
$730$	0	0
$731$	5.42015	0.200472
$732$	0	0
$733$	1.60498	0.0592815	0.0296407	−	0.999561i	$-0.490564\pi$
$733$	1.60498	0.0592815	0.0296407	+	0.999561i	$0.490564\pi$
$734$	0	0
$735$	28.2038	1.04031
$736$	0	0
$737$	−16.0202	−0.590111
$738$	0	0
$739$	−8.13264	−0.299164	−0.149582	−	0.988749i	$-0.547793\pi$
$739$	−8.13264	−0.299164	−0.149582	+	0.988749i	$0.547793\pi$
$740$	0	0
$741$	−7.03316	−0.258370
$742$	0	0
$743$	42.9261	1.57481	0.787403	−	0.616439i	$-0.211425\pi$
$743$	42.9261	1.57481	0.787403	+	0.616439i	$0.211425\pi$
$744$	0	0
$745$	−9.78697	−0.358567
$746$	0	0
$747$	1.23869	0.0453212
$748$	0	0
$749$	−63.1564	−2.30768
$750$	0	0
$751$	−4.05685	−0.148037	−0.0740183	−	0.997257i	$-0.523582\pi$
$751$	−4.05685	−0.148037	−0.0740183	+	0.997257i	$0.523582\pi$
$752$	0	0
$753$	19.3830	0.706355
$754$	0	0
$755$	5.87683	0.213880
$756$	0	0
$757$	22.6151	0.821960	0.410980	−	0.911644i	$-0.365187\pi$
$757$	22.6151	0.821960	0.410980	+	0.911644i	$0.365187\pi$
$758$	0	0
$759$	19.1789	0.696151
$760$	0	0
$761$	−41.3609	−1.49933	−0.749666	−	0.661817i	$-0.769786\pi$
$761$	−41.3609	−1.49933	−0.749666	+	0.661817i	$0.769786\pi$
$762$	0	0
$763$	79.5720	2.88070
$764$	0	0
$765$	−0.681874	−0.0246532
$766$	0	0
$767$	−6.40011	−0.231095
$768$	0	0
$769$	17.9196	0.646197	0.323099	−	0.946365i	$-0.395275\pi$
$769$	17.9196	0.646197	0.323099	+	0.946365i	$0.395275\pi$
$770$	0	0
$771$	−0.280309	−0.0100951
$772$	0	0
$773$	−21.9043	−0.787843	−0.393921	−	0.919144i	$-0.628882\pi$
$773$	−21.9043	−0.787843	−0.393921	+	0.919144i	$0.628882\pi$
$774$	0	0
$775$	−5.58946	−0.200779
$776$	0	0
$777$	0.733623	0.0263186
$778$	0	0
$779$	−10.6663	−0.382162
$780$	0	0
$781$	17.2190	0.616144
$782$	0	0
$783$	−6.01923	−0.215110
$784$	0	0
$785$	−10.1895	−0.363678
$786$	0	0
$787$	31.6354	1.12768	0.563840	−	0.825884i	$-0.309324\pi$
$787$	31.6354	1.12768	0.563840	+	0.825884i	$0.309324\pi$
$788$	0	0
$789$	11.3482	0.404007
$790$	0	0
$791$	27.6558	0.983326
$792$	0	0
$793$	26.0559	0.925272
$794$	0	0
$795$	−6.29954	−0.223422
$796$	0	0
$797$	−15.4486	−0.547217	−0.273608	−	0.961841i	$-0.588217\pi$
$797$	−15.4486	−0.547217	−0.273608	+	0.961841i	$0.588217\pi$
$798$	0	0
$799$	−13.2452	−0.468583
$800$	0	0
$801$	1.38636	0.0489845
$802$	0	0
$803$	27.5317	0.971571
$804$	0	0
$805$	−20.6865	−0.729104
$806$	0	0
$807$	−7.23385	−0.254644
$808$	0	0
$809$	−32.9326	−1.15785	−0.578924	−	0.815382i	$-0.696527\pi$
$809$	−32.9326	−1.15785	−0.578924	+	0.815382i	$0.696527\pi$
$810$	0	0
$811$	25.5895	0.898567	0.449284	−	0.893389i	$-0.351679\pi$
$811$	25.5895	0.898567	0.449284	+	0.893389i	$0.351679\pi$
$812$	0	0
$813$	15.2622	0.535270
$814$	0	0
$815$	−10.3659	−0.363100
$816$	0	0
$817$	5.03316	0.176088
$818$	0	0
$819$	−14.5694	−0.509097
$820$	0	0
$821$	17.8674	0.623575	0.311788	−	0.950152i	$-0.399072\pi$
$821$	17.8674	0.623575	0.311788	+	0.950152i	$0.399072\pi$
$822$	0	0
$823$	18.4533	0.643242	0.321621	−	0.946868i	$-0.395772\pi$
$823$	18.4533	0.643242	0.321621	+	0.946868i	$0.395772\pi$
$824$	0	0
$825$	−4.66635	−0.162461
$826$	0	0
$827$	−47.5460	−1.65334	−0.826668	−	0.562690i	$-0.809767\pi$
$827$	−47.5460	−1.65334	−0.826668	+	0.562690i	$0.809767\pi$
$828$	0	0
$829$	19.0308	0.660965	0.330483	−	0.943812i	$-0.392788\pi$
$829$	19.0308	0.660965	0.330483	+	0.943812i	$0.392788\pi$
$830$	0	0
$831$	34.8145	1.20770
$832$	0	0
$833$	−19.7423	−0.684029
$834$	0	0
$835$	−15.6048	−0.540025
$836$	0	0
$837$	−31.2420	−1.07988
$838$	0	0
$839$	−3.04022	−0.104960	−0.0524801	−	0.998622i	$-0.516713\pi$
$839$	−3.04022	−0.104960	−0.0524801	+	0.998622i	$0.516713\pi$
$840$	0	0
$841$	−27.8403	−0.960011
$842$	0	0
$843$	6.08280	0.209503
$844$	0	0
$845$	−7.89956	−0.271753
$846$	0	0
$847$	9.05940	0.311285
$848$	0	0
$849$	−20.0979	−0.689758
$850$	0	0
$851$	−0.389401	−0.0133485
$852$	0	0
$853$	−18.2201	−0.623844	−0.311922	−	0.950108i	$-0.600973\pi$
$853$	−18.2201	−0.623844	−0.311922	+	0.950108i	$0.600973\pi$
$854$	0	0
$855$	−0.633188	−0.0216546
$856$	0	0
$857$	19.3611	0.661363	0.330682	−	0.943742i	$-0.392721\pi$
$857$	19.3611	0.661363	0.330682	+	0.943742i	$0.392721\pi$
$858$	0	0
$859$	−3.44129	−0.117415	−0.0587077	−	0.998275i	$-0.518698\pi$
$859$	−3.44129	−0.117415	−0.0587077	+	0.998275i	$0.518698\pi$
$860$	0	0
$861$	82.5921	2.81473
$862$	0	0
$863$	26.5471	0.903674	0.451837	−	0.892101i	$-0.350769\pi$
$863$	26.5471	0.903674	0.451837	+	0.892101i	$0.350769\pi$
$864$	0	0
$865$	−0.571604	−0.0194351
$866$	0	0
$867$	24.3694	0.827630
$868$	0	0
$869$	16.3790	0.555619
$870$	0	0
$871$	24.1458	0.818148
$872$	0	0
$873$	1.36836	0.0463120
$874$	0	0
$875$	5.03316	0.170152
$876$	0	0
$877$	20.5495	0.693908	0.346954	−	0.937882i	$-0.387216\pi$
$877$	20.5495	0.693908	0.346954	+	0.937882i	$0.387216\pi$
$878$	0	0
$879$	14.9146	0.503059
$880$	0	0
$881$	−31.1911	−1.05085	−0.525427	−	0.850839i	$-0.676094\pi$
$881$	−31.1911	−1.05085	−0.525427	+	0.850839i	$0.676094\pi$
$882$	0	0
$883$	−14.1861	−0.477401	−0.238701	−	0.971093i	$-0.576721\pi$
$883$	−14.1861	−0.477401	−0.238701	+	0.971093i	$0.576721\pi$
$884$	0	0
$885$	2.15378	0.0723985
$886$	0	0
$887$	−46.6046	−1.56483	−0.782415	−	0.622757i	$-0.786012\pi$
$887$	−46.6046	−1.56483	−0.782415	+	0.622757i	$0.786012\pi$
$888$	0	0
$889$	−43.3623	−1.45433
$890$	0	0
$891$	−20.3207	−0.680768
$892$	0	0
$893$	−12.2995	−0.411588
$894$	0	0
$895$	24.8864	0.831862
$896$	0	0
$897$	−28.9066	−0.965164
$898$	0	0
$899$	6.01923	0.200752
$900$	0	0
$901$	4.40959	0.146905
$902$	0	0
$903$	−38.9729	−1.29694
$904$	0	0
$905$	−6.95372	−0.231150
$906$	0	0
$907$	0.754028	0.0250371	0.0125185	−	0.999922i	$-0.496015\pi$
$907$	0.754028	0.0250371	0.0125185	+	0.999922i	$0.496015\pi$
$908$	0	0
$909$	−7.96988	−0.264344
$910$	0	0
$911$	53.1413	1.76065	0.880325	−	0.474371i	$-0.157325\pi$
$911$	53.1413	1.76065	0.880325	+	0.474371i	$0.157325\pi$
$912$	0	0
$913$	−5.93368	−0.196376
$914$	0	0
$915$	−8.76838	−0.289874
$916$	0	0
$917$	−10.8403	−0.357979
$918$	0	0
$919$	−37.4202	−1.23438	−0.617189	−	0.786815i	$-0.711728\pi$
$919$	−37.4202	−1.23438	−0.617189	+	0.786815i	$0.711728\pi$
$920$	0	0
$921$	−43.9196	−1.44720
$922$	0	0
$923$	−25.9526	−0.854241
$924$	0	0
$925$	0.0947438	0.00311516
$926$	0	0
$927$	−3.92880	−0.129039
$928$	0	0
$929$	−19.1981	−0.629871	−0.314935	−	0.949113i	$-0.601983\pi$
$929$	−19.1981	−0.629871	−0.314935	+	0.949113i	$0.601983\pi$
$930$	0	0
$931$	−18.3327	−0.600830
$932$	0	0
$933$	39.0873	1.27966
$934$	0	0
$935$	3.26638	0.106822
$936$	0	0
$937$	−20.0095	−0.653681	−0.326840	−	0.945080i	$-0.605984\pi$
$937$	−20.0095	−0.653681	−0.326840	+	0.945080i	$0.605984\pi$
$938$	0	0
$939$	−25.8458	−0.843445
$940$	0	0
$941$	15.3327	0.499832	0.249916	−	0.968268i	$-0.419597\pi$
$941$	15.3327	0.499832	0.249916	+	0.968268i	$0.419597\pi$
$942$	0	0
$943$	−43.8392	−1.42760
$944$	0	0
$945$	28.1326	0.915155
$946$	0	0
$947$	29.8217	0.969076	0.484538	−	0.874770i	$-0.338988\pi$
$947$	29.8217	0.969076	0.484538	+	0.874770i	$0.338988\pi$
$948$	0	0
$949$	−41.4959	−1.34702
$950$	0	0
$951$	48.8464	1.58395
$952$	0	0
$953$	31.4938	1.02018	0.510091	−	0.860120i	$-0.329612\pi$
$953$	31.4938	1.02018	0.510091	+	0.860120i	$0.329612\pi$
$954$	0	0
$955$	−3.91254	−0.126607
$956$	0	0
$957$	5.02514	0.162440
$958$	0	0
$959$	−11.0200	−0.355856
$960$	0	0
$961$	0.242050	0.00780806
$962$	0	0
$963$	−7.94528	−0.256033
$964$	0	0
$965$	9.20734	0.296395
$966$	0	0
$967$	−1.16143	−0.0373490	−0.0186745	−	0.999826i	$-0.505945\pi$
$967$	−1.16143	−0.0373490	−0.0186745	+	0.999826i	$0.505945\pi$
$968$	0	0
$969$	−1.65673	−0.0532220
$970$	0	0
$971$	−18.4557	−0.592272	−0.296136	−	0.955146i	$-0.595698\pi$
$971$	−18.4557	−0.592272	−0.296136	+	0.955146i	$0.595698\pi$
$972$	0	0
$973$	113.345	3.63367
$974$	0	0
$975$	7.03316	0.225241
$976$	0	0
$977$	5.62735	0.180035	0.0900175	−	0.995940i	$-0.471308\pi$
$977$	5.62735	0.180035	0.0900175	+	0.995940i	$0.471308\pi$
$978$	0	0
$979$	−6.64107	−0.212249
$980$	0	0
$981$	10.0104	0.319608
$982$	0	0
$983$	25.1228	0.801293	0.400647	−	0.916233i	$-0.368786\pi$
$983$	25.1228	0.801293	0.400647	+	0.916233i	$0.368786\pi$
$984$	0	0
$985$	3.84622	0.122551
$986$	0	0
$987$	95.2382	3.03147
$988$	0	0
$989$	20.6865	0.657793
$990$	0	0
$991$	41.8749	1.33020	0.665100	−	0.746754i	$-0.268389\pi$
$991$	41.8749	1.33020	0.665100	+	0.746754i	$0.268389\pi$
$992$	0	0
$993$	−15.3295	−0.486467
$994$	0	0
$995$	−18.2864	−0.579718
$996$	0	0
$997$	−37.0346	−1.17290	−0.586449	−	0.809986i	$-0.699475\pi$
$997$	−37.0346	−1.17290	−0.586449	+	0.809986i	$0.699475\pi$
$998$	0	0
$999$	0.529566	0.0167547

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.a.t.1.2		4
4.3	odd	2		95.2.a.b.1.3	✓	4
5.4	even	2		7600.2.a.cf.1.3		4
8.3	odd	2		6080.2.a.cc.1.2		4
8.5	even	2		6080.2.a.ch.1.3		4
12.11	even	2		855.2.a.m.1.2		4
20.3	even	4		475.2.b.e.324.4		8
20.7	even	4		475.2.b.e.324.5		8
20.19	odd	2		475.2.a.i.1.2		4
28.27	even	2		4655.2.a.y.1.3		4
60.59	even	2		4275.2.a.bo.1.3		4
76.75	even	2		1805.2.a.p.1.2		4
380.379	even	2		9025.2.a.bf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.3	✓	4	4.3	odd	2
475.2.a.i.1.2		4	20.19	odd	2
475.2.b.e.324.4		8	20.3	even	4
475.2.b.e.324.5		8	20.7	even	4
855.2.a.m.1.2		4	12.11	even	2
1520.2.a.t.1.2		4	1.1	even	1	trivial
1805.2.a.p.1.2		4	76.75	even	2
4275.2.a.bo.1.3		4	60.59	even	2
4655.2.a.y.1.3		4	28.27	even	2
6080.2.a.cc.1.2		4	8.3	odd	2
6080.2.a.ch.1.3		4	8.5	even	2
7600.2.a.cf.1.3		4	5.4	even	2
9025.2.a.bf.1.3		4	380.379	even	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53844 q^{3} -1.00000 q^{5} -5.03316 q^{7} -0.633188 q^{9} +O(q^{10})\)	\(q-1.53844 q^{3} -1.00000 q^{5} -5.03316 q^{7} -0.633188 q^{9} +3.03316 q^{11} -4.57160 q^{13} +1.53844 q^{15} -1.07689 q^{17} -1.00000 q^{19} +7.74324 q^{21} -4.11005 q^{23} +1.00000 q^{25} +5.58946 q^{27} -1.07689 q^{29} -5.58946 q^{31} -4.66635 q^{33} +5.03316 q^{35} +0.0947438 q^{37} +7.03316 q^{39} +10.6663 q^{41} -5.03316 q^{43} +0.633188 q^{45} +12.2995 q^{47} +18.3327 q^{49} +1.65673 q^{51} -4.09474 q^{53} -3.03316 q^{55} +1.53844 q^{57} +1.39997 q^{59} -5.69951 q^{61} +3.18694 q^{63} +4.57160 q^{65} -5.28168 q^{67} +6.32308 q^{69} +5.67692 q^{71} +9.07689 q^{73} -1.53844 q^{75} -15.2664 q^{77} +5.39997 q^{79} -6.69951 q^{81} -1.95627 q^{83} +1.07689 q^{85} +1.65673 q^{87} -2.18949 q^{89} +23.0096 q^{91} +8.59907 q^{93} +1.00000 q^{95} -2.16106 q^{97} -1.92056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9	\( 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{35} - 6 q^{37} + 12 q^{39} + 16 q^{41} - 4 q^{43} - 8 q^{45} + 12 q^{47} + 20 q^{49} + 36 q^{51} - 10 q^{53} + 4 q^{55} + 2 q^{57} + 20 q^{61} - 20 q^{63} - 2 q^{65} + 18 q^{67} + 28 q^{69} + 20 q^{71} + 28 q^{73} - 2 q^{75} - 40 q^{77} + 16 q^{79} + 16 q^{81} - 4 q^{85} + 36 q^{87} + 4 q^{89} + 36 q^{91} - 40 q^{93} + 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9 - 4 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^21 + 8 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 - 4 * q^31 + 8 * q^33 + 4 * q^35 - 6 * q^37 + 12 * q^39 + 16 * q^41 - 4 * q^43 - 8 * q^45 + 12 * q^47 + 20 * q^49 + 36 * q^51 - 10 * q^53 + 4 * q^55 + 2 * q^57 + 20 * q^61 - 20 * q^63 - 2 * q^65 + 18 * q^67 + 28 * q^69 + 20 * q^71 + 28 * q^73 - 2 * q^75 - 40 * q^77 + 16 * q^79 + 16 * q^81 - 4 * q^85 + 36 * q^87 + 4 * q^89 + 36 * q^91 - 40 * q^93 + 4 * q^95 + 30 * q^97 + 4 * q^99

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