LMFDB - Embedded newform 2280.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2280,2,Mod(1,2280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2280.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:	$18.2058916609$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.85577$ of defining polynomial
Character	$\chi$	$=$	2280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.29966 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +1.29966 q^{7} +1.00000 q^{9} -5.01121 q^{11} +3.29966 q^{13} -1.00000 q^{15} +5.71155 q^{17} -1.00000 q^{19} +1.29966 q^{21} -3.71155 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.41188 q^{29} +7.71155 q^{31} -5.01121 q^{33} -1.29966 q^{35} +3.29966 q^{37} +3.29966 q^{39} -7.01121 q^{41} +5.29966 q^{43} -1.00000 q^{45} +3.71155 q^{47} -5.31087 q^{49} +5.71155 q^{51} +5.71155 q^{53} +5.01121 q^{55} -1.00000 q^{57} +6.59933 q^{59} +7.11222 q^{61} +1.29966 q^{63} -3.29966 q^{65} +11.4231 q^{67} -3.71155 q^{69} -10.0224 q^{71} +10.0000 q^{73} +1.00000 q^{75} -6.51289 q^{77} -11.4231 q^{79} +1.00000 q^{81} +16.5353 q^{83} -5.71155 q^{85} +4.41188 q^{87} -9.61054 q^{89} +4.28845 q^{91} +7.71155 q^{93} +1.00000 q^{95} +11.2997 q^{97} -5.01121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{9} + 4 q^{11} + 6 q^{13} - 3 q^{15} + 2 q^{17} - 3 q^{19} + 4 q^{23} + 3 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 4 q^{33} + 6 q^{37} + 6 q^{39} - 2 q^{41} + 12 q^{43} - 3 q^{45} - 4 q^{47} + 7 q^{49} + 2 q^{51} + 2 q^{53} - 4 q^{55} - 3 q^{57} + 12 q^{59} + 14 q^{61} - 6 q^{65} + 4 q^{67} + 4 q^{69} + 8 q^{71} + 30 q^{73} + 3 q^{75} - 20 q^{77} - 4 q^{79} + 3 q^{81} + 12 q^{83} - 2 q^{85} + 2 q^{87} - 2 q^{89} + 28 q^{91} + 8 q^{93} + 3 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^9 + 4 * q^11 + 6 * q^13 - 3 * q^15 + 2 * q^17 - 3 * q^19 + 4 * q^23 + 3 * q^25 + 3 * q^27 + 2 * q^29 + 8 * q^31 + 4 * q^33 + 6 * q^37 + 6 * q^39 - 2 * q^41 + 12 * q^43 - 3 * q^45 - 4 * q^47 + 7 * q^49 + 2 * q^51 + 2 * q^53 - 4 * q^55 - 3 * q^57 + 12 * q^59 + 14 * q^61 - 6 * q^65 + 4 * q^67 + 4 * q^69 + 8 * q^71 + 30 * q^73 + 3 * q^75 - 20 * q^77 - 4 * q^79 + 3 * q^81 + 12 * q^83 - 2 * q^85 + 2 * q^87 - 2 * q^89 + 28 * q^91 + 8 * q^93 + 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.29966	0.491227	0.245613	−	0.969368i	$-0.421011\pi$
$7$	1.29966	0.491227	0.245613	+	0.969368i	$0.421011\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.01121	−1.51094	−0.755468	−	0.655185i	$-0.772591\pi$
$11$	−5.01121	−1.51094	−0.755468	+	0.655185i	$0.772591\pi$
$12$	0	0
$13$	3.29966	0.915162	0.457581	−	0.889168i	$-0.348716\pi$
$13$	3.29966	0.915162	0.457581	+	0.889168i	$0.348716\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.71155	1.38525	0.692627	−	0.721296i	$-0.256453\pi$
$17$	5.71155	1.38525	0.692627	+	0.721296i	$0.256453\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.29966	0.283610
$22$	0	0
$23$	−3.71155	−0.773911	−0.386955	−	0.922098i	$-0.626473\pi$
$23$	−3.71155	−0.773911	−0.386955	+	0.922098i	$0.626473\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.41188	0.819266	0.409633	−	0.912250i	$-0.365657\pi$
$29$	4.41188	0.819266	0.409633	+	0.912250i	$0.365657\pi$
$30$	0	0
$31$	7.71155	1.38503	0.692517	−	0.721401i	$-0.256502\pi$
$31$	7.71155	1.38503	0.692517	+	0.721401i	$0.256502\pi$
$32$	0	0
$33$	−5.01121	−0.872340
$34$	0	0
$35$	−1.29966	−0.219683
$36$	0	0
$37$	3.29966	0.542461	0.271231	−	0.962514i	$-0.412569\pi$
$37$	3.29966	0.542461	0.271231	+	0.962514i	$0.412569\pi$
$38$	0	0
$39$	3.29966	0.528369
$40$	0	0
$41$	−7.01121	−1.09497	−0.547483	−	0.836817i	$-0.684414\pi$
$41$	−7.01121	−1.09497	−0.547483	+	0.836817i	$0.684414\pi$
$42$	0	0
$43$	5.29966	0.808191	0.404096	−	0.914717i	$-0.367586\pi$
$43$	5.29966	0.808191	0.404096	+	0.914717i	$0.367586\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	3.71155	0.541384	0.270692	−	0.962666i	$-0.412747\pi$
$47$	3.71155	0.541384	0.270692	+	0.962666i	$0.412747\pi$
$48$	0	0
$49$	−5.31087	−0.758696
$50$	0	0
$51$	5.71155	0.799776
$52$	0	0
$53$	5.71155	0.784541	0.392271	−	0.919850i	$-0.371690\pi$
$53$	5.71155	0.784541	0.392271	+	0.919850i	$0.371690\pi$
$54$	0	0
$55$	5.01121	0.675711
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	6.59933	0.859159	0.429580	−	0.903029i	$-0.358662\pi$
$59$	6.59933	0.859159	0.429580	+	0.903029i	$0.358662\pi$
$60$	0	0
$61$	7.11222	0.910626	0.455313	−	0.890331i	$-0.349527\pi$
$61$	7.11222	0.910626	0.455313	+	0.890331i	$0.349527\pi$
$62$	0	0
$63$	1.29966	0.163742
$64$	0	0
$65$	−3.29966	−0.409273
$66$	0	0
$67$	11.4231	1.39555	0.697776	−	0.716316i	$-0.254173\pi$
$67$	11.4231	1.39555	0.697776	+	0.716316i	$0.254173\pi$
$68$	0	0
$69$	−3.71155	−0.446818
$70$	0	0
$71$	−10.0224	−1.18944	−0.594721	−	0.803932i	$-0.702738\pi$
$71$	−10.0224	−1.18944	−0.594721	+	0.803932i	$0.702738\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−6.51289	−0.742213
$78$	0	0
$79$	−11.4231	−1.28520	−0.642599	−	0.766203i	$-0.722144\pi$
$79$	−11.4231	−1.28520	−0.642599	+	0.766203i	$0.722144\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.5353	1.81499	0.907493	−	0.420068i	$-0.137994\pi$
$83$	16.5353	1.81499	0.907493	+	0.420068i	$0.137994\pi$
$84$	0	0
$85$	−5.71155	−0.619504
$86$	0	0
$87$	4.41188	0.473003
$88$	0	0
$89$	−9.61054	−1.01871	−0.509357	−	0.860555i	$-0.670117\pi$
$89$	−9.61054	−1.01871	−0.509357	+	0.860555i	$0.670117\pi$
$90$	0	0
$91$	4.28845	0.449552
$92$	0	0
$93$	7.71155	0.799650
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	11.2997	1.14731	0.573654	−	0.819098i	$-0.305526\pi$
$97$	11.2997	1.14731	0.573654	+	0.819098i	$0.305526\pi$
$98$	0	0
$99$	−5.01121	−0.503645
$100$	0	0
$101$	4.59933	0.457650	0.228825	−	0.973468i	$-0.426512\pi$
$101$	4.59933	0.457650	0.228825	+	0.973468i	$0.426512\pi$
$102$	0	0
$103$	−11.4231	−1.12555	−0.562775	−	0.826610i	$-0.690266\pi$
$103$	−11.4231	−1.12555	−0.562775	+	0.826610i	$0.690266\pi$
$104$	0	0
$105$	−1.29966	−0.126834
$106$	0	0
$107$	−11.4231	−1.10431	−0.552156	−	0.833741i	$-0.686195\pi$
$107$	−11.4231	−1.10431	−0.552156	+	0.833741i	$0.686195\pi$
$108$	0	0
$109$	7.40067	0.708856	0.354428	−	0.935083i	$-0.384676\pi$
$109$	7.40067	0.708856	0.354428	+	0.935083i	$0.384676\pi$
$110$	0	0
$111$	3.29966	0.313190
$112$	0	0
$113$	7.11222	0.669061	0.334531	−	0.942385i	$-0.391422\pi$
$113$	7.11222	0.669061	0.334531	+	0.942385i	$0.391422\pi$
$114$	0	0
$115$	3.71155	0.346103
$116$	0	0
$117$	3.29966	0.305054
$118$	0	0
$119$	7.42309	0.680474
$120$	0	0
$121$	14.1122	1.28293
$122$	0	0
$123$	−7.01121	−0.632179
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.4231	−1.72352	−0.861760	−	0.507316i	$-0.830638\pi$
$127$	−19.4231	−1.72352	−0.861760	+	0.507316i	$0.830638\pi$
$128$	0	0
$129$	5.29966	0.466609
$130$	0	0
$131$	2.41188	0.210727	0.105364	−	0.994434i	$-0.466399\pi$
$131$	2.41188	0.210727	0.105364	+	0.994434i	$0.466399\pi$
$132$	0	0
$133$	−1.29966	−0.112695
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	7.19866	0.615023	0.307511	−	0.951544i	$-0.400504\pi$
$137$	7.19866	0.615023	0.307511	+	0.951544i	$0.400504\pi$
$138$	0	0
$139$	22.0224	1.86792	0.933959	−	0.357381i	$-0.116330\pi$
$139$	22.0224	1.86792	0.933959	+	0.357381i	$0.116330\pi$
$140$	0	0
$141$	3.71155	0.312568
$142$	0	0
$143$	−16.5353	−1.38275
$144$	0	0
$145$	−4.41188	−0.366387
$146$	0	0
$147$	−5.31087	−0.438033
$148$	0	0
$149$	−8.22443	−0.673772	−0.336886	−	0.941545i	$-0.609374\pi$
$149$	−8.22443	−0.673772	−0.336886	+	0.941545i	$0.609374\pi$
$150$	0	0
$151$	−5.48711	−0.446535	−0.223267	−	0.974757i	$-0.571672\pi$
$151$	−5.48711	−0.446535	−0.223267	+	0.974757i	$0.571672\pi$
$152$	0	0
$153$	5.71155	0.461751
$154$	0	0
$155$	−7.71155	−0.619406
$156$	0	0
$157$	−12.0224	−0.959493	−0.479747	−	0.877407i	$-0.659271\pi$
$157$	−12.0224	−0.959493	−0.479747	+	0.877407i	$0.659271\pi$
$158$	0	0
$159$	5.71155	0.452955
$160$	0	0
$161$	−4.82376	−0.380166
$162$	0	0
$163$	−10.1234	−0.792928	−0.396464	−	0.918050i	$-0.629763\pi$
$163$	−10.1234	−0.792928	−0.396464	+	0.918050i	$0.629763\pi$
$164$	0	0
$165$	5.01121	0.390122
$166$	0	0
$167$	−6.88778	−0.532993	−0.266496	−	0.963836i	$-0.585866\pi$
$167$	−6.88778	−0.532993	−0.266496	+	0.963836i	$0.585866\pi$
$168$	0	0
$169$	−2.11222	−0.162478
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	21.5095	1.63534	0.817670	−	0.575688i	$-0.195266\pi$
$173$	21.5095	1.63534	0.817670	+	0.575688i	$0.195266\pi$
$174$	0	0
$175$	1.29966	0.0982454
$176$	0	0
$177$	6.59933	0.496036
$178$	0	0
$179$	19.2211	1.43665	0.718325	−	0.695707i	$-0.244909\pi$
$179$	19.2211	1.43665	0.718325	+	0.695707i	$0.244909\pi$
$180$	0	0
$181$	12.0224	0.893619	0.446810	−	0.894629i	$-0.352560\pi$
$181$	12.0224	0.893619	0.446810	+	0.894629i	$0.352560\pi$
$182$	0	0
$183$	7.11222	0.525750
$184$	0	0
$185$	−3.29966	−0.242596
$186$	0	0
$187$	−28.6217	−2.09303
$188$	0	0
$189$	1.29966	0.0945367
$190$	0	0
$191$	4.43430	0.320855	0.160427	−	0.987048i	$-0.448713\pi$
$191$	4.43430	0.320855	0.160427	+	0.987048i	$0.448713\pi$
$192$	0	0
$193$	24.1234	1.73644	0.868221	−	0.496178i	$-0.165263\pi$
$193$	24.1234	1.73644	0.868221	+	0.496178i	$0.165263\pi$
$194$	0	0
$195$	−3.29966	−0.236294
$196$	0	0
$197$	−7.48711	−0.533435	−0.266717	−	0.963775i	$-0.585939\pi$
$197$	−7.48711	−0.533435	−0.266717	+	0.963775i	$0.585939\pi$
$198$	0	0
$199$	−3.17624	−0.225158	−0.112579	−	0.993643i	$-0.535911\pi$
$199$	−3.17624	−0.225158	−0.112579	+	0.993643i	$0.535911\pi$
$200$	0	0
$201$	11.4231	0.805723
$202$	0	0
$203$	5.73396	0.402445
$204$	0	0
$205$	7.01121	0.489684
$206$	0	0
$207$	−3.71155	−0.257970
$208$	0	0
$209$	5.01121	0.346633
$210$	0	0
$211$	−18.8462	−1.29742	−0.648712	−	0.761034i	$-0.724692\pi$
$211$	−18.8462	−1.29742	−0.648712	+	0.761034i	$0.724692\pi$
$212$	0	0
$213$	−10.0224	−0.686725
$214$	0	0
$215$	−5.29966	−0.361434
$216$	0	0
$217$	10.0224	0.680366
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	18.8462	1.26773
$222$	0	0
$223$	3.42309	0.229227	0.114614	−	0.993410i	$-0.463437\pi$
$223$	3.42309	0.229227	0.114614	+	0.993410i	$0.463437\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	2.31087	0.153378	0.0766890	−	0.997055i	$-0.475565\pi$
$227$	2.31087	0.153378	0.0766890	+	0.997055i	$0.475565\pi$
$228$	0	0
$229$	11.7340	0.775402	0.387701	−	0.921785i	$-0.373269\pi$
$229$	11.7340	0.775402	0.387701	+	0.921785i	$0.373269\pi$
$230$	0	0
$231$	−6.51289	−0.428517
$232$	0	0
$233$	−18.0448	−1.18216	−0.591078	−	0.806614i	$-0.701298\pi$
$233$	−18.0448	−1.18216	−0.591078	+	0.806614i	$0.701298\pi$
$234$	0	0
$235$	−3.71155	−0.242115
$236$	0	0
$237$	−11.4231	−0.742009
$238$	0	0
$239$	5.01121	0.324148	0.162074	−	0.986779i	$-0.448182\pi$
$239$	5.01121	0.324148	0.162074	+	0.986779i	$0.448182\pi$
$240$	0	0
$241$	−18.0448	−1.16237	−0.581185	−	0.813771i	$-0.697411\pi$
$241$	−18.0448	−1.16237	−0.581185	+	0.813771i	$0.697411\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.31087	0.339299
$246$	0	0
$247$	−3.29966	−0.209953
$248$	0	0
$249$	16.5353	1.04788
$250$	0	0
$251$	−7.61054	−0.480373	−0.240186	−	0.970727i	$-0.577209\pi$
$251$	−7.61054	−0.480373	−0.240186	+	0.970727i	$0.577209\pi$
$252$	0	0
$253$	18.5993	1.16933
$254$	0	0
$255$	−5.71155	−0.357671
$256$	0	0
$257$	−16.8878	−1.05343	−0.526715	−	0.850042i	$-0.676577\pi$
$257$	−16.8878	−1.05343	−0.526715	+	0.850042i	$0.676577\pi$
$258$	0	0
$259$	4.28845	0.266472
$260$	0	0
$261$	4.41188	0.273089
$262$	0	0
$263$	−27.5095	−1.69631	−0.848155	−	0.529748i	$-0.822287\pi$
$263$	−27.5095	−1.69631	−0.848155	+	0.529748i	$0.822287\pi$
$264$	0	0
$265$	−5.71155	−0.350857
$266$	0	0
$267$	−9.61054	−0.588155
$268$	0	0
$269$	−5.61054	−0.342080	−0.171040	−	0.985264i	$-0.554713\pi$
$269$	−5.61054	−0.342080	−0.171040	+	0.985264i	$0.554713\pi$
$270$	0	0
$271$	22.6442	1.37554	0.687768	−	0.725931i	$-0.258591\pi$
$271$	22.6442	1.37554	0.687768	+	0.725931i	$0.258591\pi$
$272$	0	0
$273$	4.28845	0.259549
$274$	0	0
$275$	−5.01121	−0.302187
$276$	0	0
$277$	−22.0448	−1.32455	−0.662273	−	0.749263i	$-0.730408\pi$
$277$	−22.0448	−1.32455	−0.662273	+	0.749263i	$0.730408\pi$
$278$	0	0
$279$	7.71155	0.461678
$280$	0	0
$281$	−17.2356	−1.02819	−0.514096	−	0.857733i	$-0.671873\pi$
$281$	−17.2356	−1.02819	−0.514096	+	0.857733i	$0.671873\pi$
$282$	0	0
$283$	2.49832	0.148510	0.0742549	−	0.997239i	$-0.476342\pi$
$283$	2.49832	0.148510	0.0742549	+	0.997239i	$0.476342\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	−9.11222	−0.537877
$288$	0	0
$289$	15.6217	0.918926
$290$	0	0
$291$	11.2997	0.662398
$292$	0	0
$293$	−4.31087	−0.251844	−0.125922	−	0.992040i	$-0.540189\pi$
$293$	−4.31087	−0.251844	−0.125922	+	0.992040i	$0.540189\pi$
$294$	0	0
$295$	−6.59933	−0.384228
$296$	0	0
$297$	−5.01121	−0.290780
$298$	0	0
$299$	−12.2469	−0.708254
$300$	0	0
$301$	6.88778	0.397005
$302$	0	0
$303$	4.59933	0.264225
$304$	0	0
$305$	−7.11222	−0.407244
$306$	0	0
$307$	−30.0224	−1.71347	−0.856735	−	0.515757i	$-0.827511\pi$
$307$	−30.0224	−1.71347	−0.856735	+	0.515757i	$0.827511\pi$
$308$	0	0
$309$	−11.4231	−0.649837
$310$	0	0
$311$	−30.6587	−1.73850	−0.869249	−	0.494375i	$-0.835397\pi$
$311$	−30.6587	−1.73850	−0.869249	+	0.494375i	$0.835397\pi$
$312$	0	0
$313$	20.0224	1.13173	0.565867	−	0.824497i	$-0.308542\pi$
$313$	20.0224	1.13173	0.565867	+	0.824497i	$0.308542\pi$
$314$	0	0
$315$	−1.29966	−0.0732278
$316$	0	0
$317$	−20.1089	−1.12943	−0.564713	−	0.825287i	$-0.691013\pi$
$317$	−20.1089	−1.12943	−0.564713	+	0.825287i	$0.691013\pi$
$318$	0	0
$319$	−22.1089	−1.23786
$320$	0	0
$321$	−11.4231	−0.637575
$322$	0	0
$323$	−5.71155	−0.317799
$324$	0	0
$325$	3.29966	0.183032
$326$	0	0
$327$	7.40067	0.409258
$328$	0	0
$329$	4.82376	0.265943
$330$	0	0
$331$	28.3333	1.55734	0.778669	−	0.627435i	$-0.215895\pi$
$331$	28.3333	1.55734	0.778669	+	0.627435i	$0.215895\pi$
$332$	0	0
$333$	3.29966	0.180820
$334$	0	0
$335$	−11.4231	−0.624110
$336$	0	0
$337$	2.92477	0.159322	0.0796612	−	0.996822i	$-0.474616\pi$
$337$	2.92477	0.159322	0.0796612	+	0.996822i	$0.474616\pi$
$338$	0	0
$339$	7.11222	0.386283
$340$	0	0
$341$	−38.6442	−2.09270
$342$	0	0
$343$	−16.0000	−0.863919
$344$	0	0
$345$	3.71155	0.199823
$346$	0	0
$347$	1.11222	0.0597069	0.0298535	−	0.999554i	$-0.490496\pi$
$347$	1.11222	0.0597069	0.0298535	+	0.999554i	$0.490496\pi$
$348$	0	0
$349$	24.2244	1.29670	0.648352	−	0.761341i	$-0.275458\pi$
$349$	24.2244	1.29670	0.648352	+	0.761341i	$0.275458\pi$
$350$	0	0
$351$	3.29966	0.176123
$352$	0	0
$353$	20.2244	1.07644	0.538219	−	0.842805i	$-0.319097\pi$
$353$	20.2244	1.07644	0.538219	+	0.842805i	$0.319097\pi$
$354$	0	0
$355$	10.0224	0.531935
$356$	0	0
$357$	7.42309	0.392872
$358$	0	0
$359$	−2.41188	−0.127294	−0.0636471	−	0.997972i	$-0.520273\pi$
$359$	−2.41188	−0.127294	−0.0636471	+	0.997972i	$0.520273\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	14.1122	0.740699
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	23.5689	1.23029	0.615144	−	0.788415i	$-0.289098\pi$
$367$	23.5689	1.23029	0.615144	+	0.788415i	$0.289098\pi$
$368$	0	0
$369$	−7.01121	−0.364989
$370$	0	0
$371$	7.42309	0.385388
$372$	0	0
$373$	−4.12343	−0.213503	−0.106751	−	0.994286i	$-0.534045\pi$
$373$	−4.12343	−0.213503	−0.106751	+	0.994286i	$0.534045\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	14.5577	0.749761
$378$	0	0
$379$	−22.5577	−1.15871	−0.579356	−	0.815074i	$-0.696696\pi$
$379$	−22.5577	−1.15871	−0.579356	+	0.815074i	$0.696696\pi$
$380$	0	0
$381$	−19.4231	−0.995075
$382$	0	0
$383$	19.4679	0.994765	0.497382	−	0.867531i	$-0.334295\pi$
$383$	19.4679	0.994765	0.497382	+	0.867531i	$0.334295\pi$
$384$	0	0
$385$	6.51289	0.331928
$386$	0	0
$387$	5.29966	0.269397
$388$	0	0
$389$	−23.8204	−1.20774	−0.603871	−	0.797082i	$-0.706376\pi$
$389$	−23.8204	−1.20774	−0.603871	+	0.797082i	$0.706376\pi$
$390$	0	0
$391$	−21.1987	−1.07206
$392$	0	0
$393$	2.41188	0.121663
$394$	0	0
$395$	11.4231	0.574758
$396$	0	0
$397$	−25.4231	−1.27595	−0.637974	−	0.770058i	$-0.720227\pi$
$397$	−25.4231	−1.27595	−0.637974	+	0.770058i	$0.720227\pi$
$398$	0	0
$399$	−1.29966	−0.0650646
$400$	0	0
$401$	28.8316	1.43978	0.719891	−	0.694087i	$-0.244192\pi$
$401$	28.8316	1.43978	0.719891	+	0.694087i	$0.244192\pi$
$402$	0	0
$403$	25.4455	1.26753
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−16.5353	−0.819625
$408$	0	0
$409$	9.22107	0.455953	0.227976	−	0.973667i	$-0.426789\pi$
$409$	9.22107	0.455953	0.227976	+	0.973667i	$0.426789\pi$
$410$	0	0
$411$	7.19866	0.355084
$412$	0	0
$413$	8.57691	0.422042
$414$	0	0
$415$	−16.5353	−0.811686
$416$	0	0
$417$	22.0224	1.07844
$418$	0	0
$419$	23.0336	1.12527	0.562633	−	0.826707i	$-0.309788\pi$
$419$	23.0336	1.12527	0.562633	+	0.826707i	$0.309788\pi$
$420$	0	0
$421$	−17.4679	−0.851335	−0.425667	−	0.904880i	$-0.639961\pi$
$421$	−17.4679	−0.851335	−0.425667	+	0.904880i	$0.639961\pi$
$422$	0	0
$423$	3.71155	0.180461
$424$	0	0
$425$	5.71155	0.277051
$426$	0	0
$427$	9.24349	0.447324
$428$	0	0
$429$	−16.5353	−0.798332
$430$	0	0
$431$	17.4455	0.840321	0.420160	−	0.907450i	$-0.361974\pi$
$431$	17.4455	0.840321	0.420160	+	0.907450i	$0.361974\pi$
$432$	0	0
$433$	18.5207	0.890050	0.445025	−	0.895518i	$-0.353195\pi$
$433$	18.5207	0.890050	0.445025	+	0.895518i	$0.353195\pi$
$434$	0	0
$435$	−4.41188	−0.211533
$436$	0	0
$437$	3.71155	0.177547
$438$	0	0
$439$	−0.621746	−0.0296743	−0.0148372	−	0.999890i	$-0.504723\pi$
$439$	−0.621746	−0.0296743	−0.0148372	+	0.999890i	$0.504723\pi$
$440$	0	0
$441$	−5.31087	−0.252899
$442$	0	0
$443$	−8.91020	−0.423336	−0.211668	−	0.977342i	$-0.567890\pi$
$443$	−8.91020	−0.423336	−0.211668	+	0.977342i	$0.567890\pi$
$444$	0	0
$445$	9.61054	0.455583
$446$	0	0
$447$	−8.22443	−0.389002
$448$	0	0
$449$	−30.4343	−1.43628	−0.718142	−	0.695897i	$-0.755007\pi$
$449$	−30.4343	−1.43628	−0.718142	+	0.695897i	$0.755007\pi$
$450$	0	0
$451$	35.1346	1.65443
$452$	0	0
$453$	−5.48711	−0.257807
$454$	0	0
$455$	−4.28845	−0.201046
$456$	0	0
$457$	11.4455	0.535398	0.267699	−	0.963503i	$-0.413737\pi$
$457$	11.4455	0.535398	0.267699	+	0.963503i	$0.413737\pi$
$458$	0	0
$459$	5.71155	0.266592
$460$	0	0
$461$	22.0448	1.02673	0.513365	−	0.858170i	$-0.328399\pi$
$461$	22.0448	1.02673	0.513365	+	0.858170i	$0.328399\pi$
$462$	0	0
$463$	−16.5207	−0.767784	−0.383892	−	0.923378i	$-0.625416\pi$
$463$	−16.5207	−0.767784	−0.383892	+	0.923378i	$0.625416\pi$
$464$	0	0
$465$	−7.71155	−0.357614
$466$	0	0
$467$	0.910201	0.0421191	0.0210595	−	0.999778i	$-0.493296\pi$
$467$	0.910201	0.0421191	0.0210595	+	0.999778i	$0.493296\pi$
$468$	0	0
$469$	14.8462	0.685533
$470$	0	0
$471$	−12.0224	−0.553964
$472$	0	0
$473$	−26.5577	−1.22113
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	5.71155	0.261514
$478$	0	0
$479$	−9.63296	−0.440141	−0.220070	−	0.975484i	$-0.570629\pi$
$479$	−9.63296	−0.440141	−0.220070	+	0.975484i	$0.570629\pi$
$480$	0	0
$481$	10.8878	0.496440
$482$	0	0
$483$	−4.82376	−0.219489
$484$	0	0
$485$	−11.2997	−0.513091
$486$	0	0
$487$	−30.5993	−1.38659	−0.693294	−	0.720655i	$-0.743841\pi$
$487$	−30.5993	−1.38659	−0.693294	+	0.720655i	$0.743841\pi$
$488$	0	0
$489$	−10.1234	−0.457797
$490$	0	0
$491$	−26.4119	−1.19195	−0.595976	−	0.803002i	$-0.703235\pi$
$491$	−26.4119	−1.19195	−0.595976	+	0.803002i	$0.703235\pi$
$492$	0	0
$493$	25.1987	1.13489
$494$	0	0
$495$	5.01121	0.225237
$496$	0	0
$497$	−13.0258	−0.584286
$498$	0	0
$499$	−3.22107	−0.144195	−0.0720976	−	0.997398i	$-0.522969\pi$
$499$	−3.22107	−0.144195	−0.0720976	+	0.997398i	$0.522969\pi$
$500$	0	0
$501$	−6.88778	−0.307723
$502$	0	0
$503$	−31.9584	−1.42495	−0.712477	−	0.701695i	$-0.752427\pi$
$503$	−31.9584	−1.42495	−0.712477	+	0.701695i	$0.752427\pi$
$504$	0	0
$505$	−4.59933	−0.204667
$506$	0	0
$507$	−2.11222	−0.0938068
$508$	0	0
$509$	−16.4119	−0.727444	−0.363722	−	0.931508i	$-0.618494\pi$
$509$	−16.4119	−0.727444	−0.363722	+	0.931508i	$0.618494\pi$
$510$	0	0
$511$	12.9966	0.574938
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	11.4231	0.503361
$516$	0	0
$517$	−18.5993	−0.817998
$518$	0	0
$519$	21.5095	0.944164
$520$	0	0
$521$	−7.01121	−0.307167	−0.153583	−	0.988136i	$-0.549081\pi$
$521$	−7.01121	−0.307167	−0.153583	+	0.988136i	$0.549081\pi$
$522$	0	0
$523$	−17.4007	−0.760878	−0.380439	−	0.924806i	$-0.624227\pi$
$523$	−17.4007	−0.760878	−0.380439	+	0.924806i	$0.624227\pi$
$524$	0	0
$525$	1.29966	0.0567220
$526$	0	0
$527$	44.0448	1.91862
$528$	0	0
$529$	−9.22443	−0.401062
$530$	0	0
$531$	6.59933	0.286386
$532$	0	0
$533$	−23.1346	−1.00207
$534$	0	0
$535$	11.4231	0.493863
$536$	0	0
$537$	19.2211	0.829451
$538$	0	0
$539$	26.6139	1.14634
$540$	0	0
$541$	20.8462	0.896247	0.448124	−	0.893972i	$-0.352092\pi$
$541$	20.8462	0.896247	0.448124	+	0.893972i	$0.352092\pi$
$542$	0	0
$543$	12.0224	0.515931
$544$	0	0
$545$	−7.40067	−0.317010
$546$	0	0
$547$	−30.0224	−1.28367	−0.641833	−	0.766844i	$-0.721826\pi$
$547$	−30.0224	−1.28367	−0.641833	+	0.766844i	$0.721826\pi$
$548$	0	0
$549$	7.11222	0.303542
$550$	0	0
$551$	−4.41188	−0.187952
$552$	0	0
$553$	−14.8462	−0.631324
$554$	0	0
$555$	−3.29966	−0.140063
$556$	0	0
$557$	−34.0448	−1.44253	−0.721263	−	0.692661i	$-0.756438\pi$
$557$	−34.0448	−1.44253	−0.721263	+	0.692661i	$0.756438\pi$
$558$	0	0
$559$	17.4871	0.739626
$560$	0	0
$561$	−28.6217	−1.20841
$562$	0	0
$563$	−30.9326	−1.30365	−0.651827	−	0.758367i	$-0.725997\pi$
$563$	−30.9326	−1.30365	−0.651827	+	0.758367i	$0.725997\pi$
$564$	0	0
$565$	−7.11222	−0.299213
$566$	0	0
$567$	1.29966	0.0545808
$568$	0	0
$569$	39.2581	1.64578	0.822892	−	0.568198i	$-0.192359\pi$
$569$	39.2581	1.64578	0.822892	+	0.568198i	$0.192359\pi$
$570$	0	0
$571$	−41.4903	−1.73632	−0.868158	−	0.496287i	$-0.834696\pi$
$571$	−41.4903	−1.73632	−0.868158	+	0.496287i	$0.834696\pi$
$572$	0	0
$573$	4.43430	0.185246
$574$	0	0
$575$	−3.71155	−0.154782
$576$	0	0
$577$	−28.4713	−1.18528	−0.592638	−	0.805469i	$-0.701913\pi$
$577$	−28.4713	−1.18528	−0.592638	+	0.805469i	$0.701913\pi$
$578$	0	0
$579$	24.1234	1.00254
$580$	0	0
$581$	21.4903	0.891570
$582$	0	0
$583$	−28.6217	−1.18539
$584$	0	0
$585$	−3.29966	−0.136424
$586$	0	0
$587$	6.51289	0.268816	0.134408	−	0.990926i	$-0.457087\pi$
$587$	6.51289	0.268816	0.134408	+	0.990926i	$0.457087\pi$
$588$	0	0
$589$	−7.71155	−0.317749
$590$	0	0
$591$	−7.48711	−0.307979
$592$	0	0
$593$	21.4679	0.881582	0.440791	−	0.897610i	$-0.354698\pi$
$593$	21.4679	0.881582	0.440791	+	0.897610i	$0.354698\pi$
$594$	0	0
$595$	−7.42309	−0.304317
$596$	0	0
$597$	−3.17624	−0.129995
$598$	0	0
$599$	−25.2435	−1.03142	−0.515711	−	0.856763i	$-0.672472\pi$
$599$	−25.2435	−1.03142	−0.515711	+	0.856763i	$0.672472\pi$
$600$	0	0
$601$	27.4455	1.11953	0.559763	−	0.828653i	$-0.310892\pi$
$601$	27.4455	1.11953	0.559763	+	0.828653i	$0.310892\pi$
$602$	0	0
$603$	11.4231	0.465184
$604$	0	0
$605$	−14.1122	−0.573743
$606$	0	0
$607$	−35.2211	−1.42958	−0.714790	−	0.699340i	$-0.753478\pi$
$607$	−35.2211	−1.42958	−0.714790	+	0.699340i	$0.753478\pi$
$608$	0	0
$609$	5.73396	0.232352
$610$	0	0
$611$	12.2469	0.495455
$612$	0	0
$613$	−11.4455	−0.462280	−0.231140	−	0.972921i	$-0.574245\pi$
$613$	−11.4455	−0.462280	−0.231140	+	0.972921i	$0.574245\pi$
$614$	0	0
$615$	7.01121	0.282719
$616$	0	0
$617$	−20.6858	−0.832778	−0.416389	−	0.909187i	$-0.636704\pi$
$617$	−20.6858	−0.832778	−0.416389	+	0.909187i	$0.636704\pi$
$618$	0	0
$619$	1.40067	0.0562978	0.0281489	−	0.999604i	$-0.491039\pi$
$619$	1.40067	0.0562978	0.0281489	+	0.999604i	$0.491039\pi$
$620$	0	0
$621$	−3.71155	−0.148939
$622$	0	0
$623$	−12.4905	−0.500420
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.01121	0.200128
$628$	0	0
$629$	18.8462	0.751446
$630$	0	0
$631$	−23.5960	−0.939341	−0.469670	−	0.882842i	$-0.655627\pi$
$631$	−23.5960	−0.939341	−0.469670	+	0.882842i	$0.655627\pi$
$632$	0	0
$633$	−18.8462	−0.749068
$634$	0	0
$635$	19.4231	0.770782
$636$	0	0
$637$	−17.5241	−0.694330
$638$	0	0
$639$	−10.0224	−0.396481
$640$	0	0
$641$	39.6330	1.56541	0.782704	−	0.622394i	$-0.213840\pi$
$641$	39.6330	1.56541	0.782704	+	0.622394i	$0.213840\pi$
$642$	0	0
$643$	7.14920	0.281937	0.140969	−	0.990014i	$-0.454978\pi$
$643$	7.14920	0.281937	0.140969	+	0.990014i	$0.454978\pi$
$644$	0	0
$645$	−5.29966	−0.208674
$646$	0	0
$647$	−31.9584	−1.25641	−0.628207	−	0.778046i	$-0.716211\pi$
$647$	−31.9584	−1.25641	−0.628207	+	0.778046i	$0.716211\pi$
$648$	0	0
$649$	−33.0706	−1.29814
$650$	0	0
$651$	10.0224	0.392810
$652$	0	0
$653$	11.4871	0.449525	0.224763	−	0.974414i	$-0.427839\pi$
$653$	11.4871	0.449525	0.224763	+	0.974414i	$0.427839\pi$
$654$	0	0
$655$	−2.41188	−0.0942400
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	−24.0448	−0.936654	−0.468327	−	0.883555i	$-0.655143\pi$
$659$	−24.0448	−0.936654	−0.468327	+	0.883555i	$0.655143\pi$
$660$	0	0
$661$	6.82376	0.265414	0.132707	−	0.991155i	$-0.457633\pi$
$661$	6.82376	0.265414	0.132707	+	0.991155i	$0.457633\pi$
$662$	0	0
$663$	18.8462	0.731925
$664$	0	0
$665$	1.29966	0.0503988
$666$	0	0
$667$	−16.3749	−0.634038
$668$	0	0
$669$	3.42309	0.132344
$670$	0	0
$671$	−35.6408	−1.37590
$672$	0	0
$673$	−9.69698	−0.373791	−0.186895	−	0.982380i	$-0.559843\pi$
$673$	−9.69698	−0.373791	−0.186895	+	0.982380i	$0.559843\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−21.0898	−0.810547	−0.405273	−	0.914196i	$-0.632824\pi$
$677$	−21.0898	−0.810547	−0.405273	+	0.914196i	$0.632824\pi$
$678$	0	0
$679$	14.6858	0.563588
$680$	0	0
$681$	2.31087	0.0885529
$682$	0	0
$683$	16.6217	0.636013	0.318007	−	0.948088i	$-0.396987\pi$
$683$	16.6217	0.636013	0.318007	+	0.948088i	$0.396987\pi$
$684$	0	0
$685$	−7.19866	−0.275047
$686$	0	0
$687$	11.7340	0.447679
$688$	0	0
$689$	18.8462	0.717982
$690$	0	0
$691$	2.55449	0.0971774	0.0485887	−	0.998819i	$-0.484528\pi$
$691$	2.55449	0.0971774	0.0485887	+	0.998819i	$0.484528\pi$
$692$	0	0
$693$	−6.51289	−0.247404
$694$	0	0
$695$	−22.0224	−0.835358
$696$	0	0
$697$	−40.0448	−1.51681
$698$	0	0
$699$	−18.0448	−0.682518
$700$	0	0
$701$	−45.0191	−1.70035	−0.850173	−	0.526503i	$-0.823503\pi$
$701$	−45.0191	−1.70035	−0.850173	+	0.526503i	$0.823503\pi$
$702$	0	0
$703$	−3.29966	−0.124449
$704$	0	0
$705$	−3.71155	−0.139785
$706$	0	0
$707$	5.97758	0.224810
$708$	0	0
$709$	−25.3815	−0.953222	−0.476611	−	0.879114i	$-0.658135\pi$
$709$	−25.3815	−0.953222	−0.476611	+	0.879114i	$0.658135\pi$
$710$	0	0
$711$	−11.4231	−0.428399
$712$	0	0
$713$	−28.6217	−1.07189
$714$	0	0
$715$	16.5353	0.618385
$716$	0	0
$717$	5.01121	0.187147
$718$	0	0
$719$	9.63296	0.359249	0.179624	−	0.983735i	$-0.442512\pi$
$719$	9.63296	0.359249	0.179624	+	0.983735i	$0.442512\pi$
$720$	0	0
$721$	−14.8462	−0.552901
$722$	0	0
$723$	−18.0448	−0.671095
$724$	0	0
$725$	4.41188	0.163853
$726$	0	0
$727$	16.5207	0.612720	0.306360	−	0.951916i	$-0.400889\pi$
$727$	16.5207	0.612720	0.306360	+	0.951916i	$0.400889\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.2693	1.11955
$732$	0	0
$733$	−14.6217	−0.540067	−0.270033	−	0.962851i	$-0.587035\pi$
$733$	−14.6217	−0.540067	−0.270033	+	0.962851i	$0.587035\pi$
$734$	0	0
$735$	5.31087	0.195895
$736$	0	0
$737$	−57.2435	−2.10859
$738$	0	0
$739$	34.8462	1.28184	0.640919	−	0.767609i	$-0.278554\pi$
$739$	34.8462	1.28184	0.640919	+	0.767609i	$0.278554\pi$
$740$	0	0
$741$	−3.29966	−0.121216
$742$	0	0
$743$	4.62175	0.169555	0.0847777	−	0.996400i	$-0.472982\pi$
$743$	4.62175	0.169555	0.0847777	+	0.996400i	$0.472982\pi$
$744$	0	0
$745$	8.22443	0.301320
$746$	0	0
$747$	16.5353	0.604995
$748$	0	0
$749$	−14.8462	−0.542468
$750$	0	0
$751$	5.48711	0.200228	0.100114	−	0.994976i	$-0.468079\pi$
$751$	5.48711	0.200228	0.100114	+	0.994976i	$0.468079\pi$
$752$	0	0
$753$	−7.61054	−0.277343
$754$	0	0
$755$	5.48711	0.199696
$756$	0	0
$757$	38.2917	1.39174	0.695868	−	0.718170i	$-0.255020\pi$
$757$	38.2917	1.39174	0.695868	+	0.718170i	$0.255020\pi$
$758$	0	0
$759$	18.5993	0.675113
$760$	0	0
$761$	−19.7756	−0.716864	−0.358432	−	0.933556i	$-0.616688\pi$
$761$	−19.7756	−0.716864	−0.358432	+	0.933556i	$0.616688\pi$
$762$	0	0
$763$	9.61839	0.348209
$764$	0	0
$765$	−5.71155	−0.206501
$766$	0	0
$767$	21.7756	0.786270
$768$	0	0
$769$	−9.95516	−0.358992	−0.179496	−	0.983759i	$-0.557447\pi$
$769$	−9.95516	−0.358992	−0.179496	+	0.983759i	$0.557447\pi$
$770$	0	0
$771$	−16.8878	−0.608199
$772$	0	0
$773$	54.3781	1.95585	0.977923	−	0.208967i	$-0.0670102\pi$
$773$	54.3781	1.95585	0.977923	+	0.208967i	$0.0670102\pi$
$774$	0	0
$775$	7.71155	0.277007
$776$	0	0
$777$	4.28845	0.153847
$778$	0	0
$779$	7.01121	0.251203
$780$	0	0
$781$	50.2244	1.79717
$782$	0	0
$783$	4.41188	0.157668
$784$	0	0
$785$	12.0224	0.429099
$786$	0	0
$787$	13.4455	0.479281	0.239640	−	0.970862i	$-0.422970\pi$
$787$	13.4455	0.479281	0.239640	+	0.970862i	$0.422970\pi$
$788$	0	0
$789$	−27.5095	−0.979365
$790$	0	0
$791$	9.24349	0.328661
$792$	0	0
$793$	23.4679	0.833371
$794$	0	0
$795$	−5.71155	−0.202568
$796$	0	0
$797$	−19.7340	−0.699013	−0.349506	−	0.936934i	$-0.613651\pi$
$797$	−19.7340	−0.699013	−0.349506	+	0.936934i	$0.613651\pi$
$798$	0	0
$799$	21.1987	0.749955
$800$	0	0
$801$	−9.61054	−0.339572
$802$	0	0
$803$	−50.1121	−1.76842
$804$	0	0
$805$	4.82376	0.170015
$806$	0	0
$807$	−5.61054	−0.197500
$808$	0	0
$809$	15.7756	0.554639	0.277320	−	0.960778i	$-0.410554\pi$
$809$	15.7756	0.554639	0.277320	+	0.960778i	$0.410554\pi$
$810$	0	0
$811$	−5.64752	−0.198311	−0.0991557	−	0.995072i	$-0.531614\pi$
$811$	−5.64752	−0.198311	−0.0991557	+	0.995072i	$0.531614\pi$
$812$	0	0
$813$	22.6442	0.794166
$814$	0	0
$815$	10.1234	0.354608
$816$	0	0
$817$	−5.29966	−0.185412
$818$	0	0
$819$	4.28845	0.149851
$820$	0	0
$821$	−49.6699	−1.73349	−0.866746	−	0.498749i	$-0.833793\pi$
$821$	−49.6699	−1.73349	−0.866746	+	0.498749i	$0.833793\pi$
$822$	0	0
$823$	−46.7900	−1.63100	−0.815499	−	0.578759i	$-0.803537\pi$
$823$	−46.7900	−1.63100	−0.815499	+	0.578759i	$0.803537\pi$
$824$	0	0
$825$	−5.01121	−0.174468
$826$	0	0
$827$	16.0864	0.559380	0.279690	−	0.960090i	$-0.409768\pi$
$827$	16.0864	0.559380	0.279690	+	0.960090i	$0.409768\pi$
$828$	0	0
$829$	18.8686	0.655334	0.327667	−	0.944793i	$-0.393738\pi$
$829$	18.8686	0.655334	0.327667	+	0.944793i	$0.393738\pi$
$830$	0	0
$831$	−22.0448	−0.764727
$832$	0	0
$833$	−30.3333	−1.05099
$834$	0	0
$835$	6.88778	0.238362
$836$	0	0
$837$	7.71155	0.266550
$838$	0	0
$839$	18.0224	0.622203	0.311101	−	0.950377i	$-0.399302\pi$
$839$	18.0224	0.622203	0.311101	+	0.950377i	$0.399302\pi$
$840$	0	0
$841$	−9.53531	−0.328804
$842$	0	0
$843$	−17.2356	−0.593627
$844$	0	0
$845$	2.11222	0.0726625
$846$	0	0
$847$	18.3411	0.630209
$848$	0	0
$849$	2.49832	0.0857421
$850$	0	0
$851$	−12.2469	−0.419817
$852$	0	0
$853$	41.0930	1.40700	0.703499	−	0.710696i	$-0.251620\pi$
$853$	41.0930	1.40700	0.703499	+	0.710696i	$0.251620\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	0.640931	0.0218938	0.0109469	−	0.999940i	$-0.496515\pi$
$857$	0.640931	0.0218938	0.0109469	+	0.999940i	$0.496515\pi$
$858$	0	0
$859$	47.4679	1.61958	0.809792	−	0.586717i	$-0.199580\pi$
$859$	47.4679	1.61958	0.809792	+	0.586717i	$0.199580\pi$
$860$	0	0
$861$	−9.11222	−0.310544
$862$	0	0
$863$	49.0706	1.67038	0.835192	−	0.549959i	$-0.185357\pi$
$863$	49.0706	1.67038	0.835192	+	0.549959i	$0.185357\pi$
$864$	0	0
$865$	−21.5095	−0.731346
$866$	0	0
$867$	15.6217	0.530542
$868$	0	0
$869$	57.2435	1.94185
$870$	0	0
$871$	37.6924	1.27716
$872$	0	0
$873$	11.2997	0.382436
$874$	0	0
$875$	−1.29966	−0.0439367
$876$	0	0
$877$	49.1940	1.66116	0.830582	−	0.556896i	$-0.188008\pi$
$877$	49.1940	1.66116	0.830582	+	0.556896i	$0.188008\pi$
$878$	0	0
$879$	−4.31087	−0.145402
$880$	0	0
$881$	25.2211	0.849720	0.424860	−	0.905259i	$-0.360323\pi$
$881$	25.2211	0.849720	0.424860	+	0.905259i	$0.360323\pi$
$882$	0	0
$883$	7.12007	0.239609	0.119805	−	0.992797i	$-0.461773\pi$
$883$	7.12007	0.239609	0.119805	+	0.992797i	$0.461773\pi$
$884$	0	0
$885$	−6.59933	−0.221834
$886$	0	0
$887$	10.8013	0.362674	0.181337	−	0.983421i	$-0.441958\pi$
$887$	10.8013	0.362674	0.181337	+	0.983421i	$0.441958\pi$
$888$	0	0
$889$	−25.2435	−0.846640
$890$	0	0
$891$	−5.01121	−0.167882
$892$	0	0
$893$	−3.71155	−0.124202
$894$	0	0
$895$	−19.2211	−0.642490
$896$	0	0
$897$	−12.2469	−0.408910
$898$	0	0
$899$	34.0224	1.13471
$900$	0	0
$901$	32.6217	1.08679
$902$	0	0
$903$	6.88778	0.229211
$904$	0	0
$905$	−12.0224	−0.399639
$906$	0	0
$907$	−3.62511	−0.120370	−0.0601848	−	0.998187i	$-0.519169\pi$
$907$	−3.62511	−0.120370	−0.0601848	+	0.998187i	$0.519169\pi$
$908$	0	0
$909$	4.59933	0.152550
$910$	0	0
$911$	−5.77557	−0.191353	−0.0956765	−	0.995412i	$-0.530501\pi$
$911$	−5.77557	−0.191353	−0.0956765	+	0.995412i	$0.530501\pi$
$912$	0	0
$913$	−82.8619	−2.74233
$914$	0	0
$915$	−7.11222	−0.235123
$916$	0	0
$917$	3.13464	0.103515
$918$	0	0
$919$	40.6666	1.34147	0.670733	−	0.741699i	$-0.265979\pi$
$919$	40.6666	1.34147	0.670733	+	0.741699i	$0.265979\pi$
$920$	0	0
$921$	−30.0224	−0.989272
$922$	0	0
$923$	−33.0706	−1.08853
$924$	0	0
$925$	3.29966	0.108492
$926$	0	0
$927$	−11.4231	−0.375184
$928$	0	0
$929$	30.4197	0.998039	0.499020	−	0.866591i	$-0.333694\pi$
$929$	30.4197	0.998039	0.499020	+	0.866591i	$0.333694\pi$
$930$	0	0
$931$	5.31087	0.174057
$932$	0	0
$933$	−30.6587	−1.00372
$934$	0	0
$935$	28.6217	0.936031
$936$	0	0
$937$	−25.6699	−0.838600	−0.419300	−	0.907848i	$-0.637725\pi$
$937$	−25.6699	−0.838600	−0.419300	+	0.907848i	$0.637725\pi$
$938$	0	0
$939$	20.0224	0.653407
$940$	0	0
$941$	−51.3029	−1.67243	−0.836213	−	0.548404i	$-0.815236\pi$
$941$	−51.3029	−1.67243	−0.836213	+	0.548404i	$0.815236\pi$
$942$	0	0
$943$	26.0224	0.847407
$944$	0	0
$945$	−1.29966	−0.0422781
$946$	0	0
$947$	6.31087	0.205076	0.102538	−	0.994729i	$-0.467304\pi$
$947$	6.31087	0.205076	0.102538	+	0.994729i	$0.467304\pi$
$948$	0	0
$949$	32.9966	1.07112
$950$	0	0
$951$	−20.1089	−0.652074
$952$	0	0
$953$	58.0032	1.87891	0.939455	−	0.342674i	$-0.111332\pi$
$953$	58.0032	1.87891	0.939455	+	0.342674i	$0.111332\pi$
$954$	0	0
$955$	−4.43430	−0.143491
$956$	0	0
$957$	−22.1089	−0.714678
$958$	0	0
$959$	9.35584	0.302116
$960$	0	0
$961$	28.4679	0.918320
$962$	0	0
$963$	−11.4231	−0.368104
$964$	0	0
$965$	−24.1234	−0.776561
$966$	0	0
$967$	−27.6970	−0.890675	−0.445337	−	0.895363i	$-0.646916\pi$
$967$	−27.6970	−0.890675	−0.445337	+	0.895363i	$0.646916\pi$
$968$	0	0
$969$	−5.71155	−0.183481
$970$	0	0
$971$	30.3973	0.975496	0.487748	−	0.872984i	$-0.337818\pi$
$971$	30.3973	0.975496	0.487748	+	0.872984i	$0.337818\pi$
$972$	0	0
$973$	28.6217	0.917571
$974$	0	0
$975$	3.29966	0.105674
$976$	0	0
$977$	−9.17947	−0.293677	−0.146839	−	0.989160i	$-0.546910\pi$
$977$	−9.17947	−0.293677	−0.146839	+	0.989160i	$0.546910\pi$
$978$	0	0
$979$	48.1604	1.53921
$980$	0	0
$981$	7.40067	0.236285
$982$	0	0
$983$	60.5801	1.93221	0.966103	−	0.258156i	$-0.0831149\pi$
$983$	60.5801	1.93221	0.966103	+	0.258156i	$0.0831149\pi$
$984$	0	0
$985$	7.48711	0.238559
$986$	0	0
$987$	4.82376	0.153542
$988$	0	0
$989$	−19.6699	−0.625468
$990$	0	0
$991$	−43.4231	−1.37938	−0.689690	−	0.724105i	$-0.742253\pi$
$991$	−43.4231	−1.37938	−0.689690	+	0.724105i	$0.742253\pi$
$992$	0	0
$993$	28.3333	0.899130
$994$	0	0
$995$	3.17624	0.100694
$996$	0	0
$997$	−25.6251	−0.811555	−0.405778	−	0.913972i	$-0.632999\pi$
$997$	−25.6251	−0.811555	−0.405778	+	0.913972i	$0.632999\pi$
$998$	0	0
$999$	3.29966	0.104397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2280.2.a.t.1.2	✓	3
3.2	odd	2		6840.2.a.bn.1.2		3
4.3	odd	2		4560.2.a.bq.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.t.1.2	✓	3	1.1	even	1	trivial
4560.2.a.bq.1.2		3	4.3	odd	2
6840.2.a.bn.1.2		3	3.2	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:	\( N \)	\(=\)	\( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.29966 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.29966 q^{7} +1.00000 q^{9} -5.01121 q^{11} +3.29966 q^{13} -1.00000 q^{15} +5.71155 q^{17} -1.00000 q^{19} +1.29966 q^{21} -3.71155 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.41188 q^{29} +7.71155 q^{31} -5.01121 q^{33} -1.29966 q^{35} +3.29966 q^{37} +3.29966 q^{39} -7.01121 q^{41} +5.29966 q^{43} -1.00000 q^{45} +3.71155 q^{47} -5.31087 q^{49} +5.71155 q^{51} +5.71155 q^{53} +5.01121 q^{55} -1.00000 q^{57} +6.59933 q^{59} +7.11222 q^{61} +1.29966 q^{63} -3.29966 q^{65} +11.4231 q^{67} -3.71155 q^{69} -10.0224 q^{71} +10.0000 q^{73} +1.00000 q^{75} -6.51289 q^{77} -11.4231 q^{79} +1.00000 q^{81} +16.5353 q^{83} -5.71155 q^{85} +4.41188 q^{87} -9.61054 q^{89} +4.28845 q^{91} +7.71155 q^{93} +1.00000 q^{95} +11.2997 q^{97} -5.01121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{9} + 4 q^{11} + 6 q^{13} - 3 q^{15} + 2 q^{17} - 3 q^{19} + 4 q^{23} + 3 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 4 q^{33} + 6 q^{37} + 6 q^{39} - 2 q^{41} + 12 q^{43} - 3 q^{45} - 4 q^{47} + 7 q^{49} + 2 q^{51} + 2 q^{53} - 4 q^{55} - 3 q^{57} + 12 q^{59} + 14 q^{61} - 6 q^{65} + 4 q^{67} + 4 q^{69} + 8 q^{71} + 30 q^{73} + 3 q^{75} - 20 q^{77} - 4 q^{79} + 3 q^{81} + 12 q^{83} - 2 q^{85} + 2 q^{87} - 2 q^{89} + 28 q^{91} + 8 q^{93} + 3 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^9 + 4 * q^11 + 6 * q^13 - 3 * q^15 + 2 * q^17 - 3 * q^19 + 4 * q^23 + 3 * q^25 + 3 * q^27 + 2 * q^29 + 8 * q^31 + 4 * q^33 + 6 * q^37 + 6 * q^39 - 2 * q^41 + 12 * q^43 - 3 * q^45 - 4 * q^47 + 7 * q^49 + 2 * q^51 + 2 * q^53 - 4 * q^55 - 3 * q^57 + 12 * q^59 + 14 * q^61 - 6 * q^65 + 4 * q^67 + 4 * q^69 + 8 * q^71 + 30 * q^73 + 3 * q^75 - 20 * q^77 - 4 * q^79 + 3 * q^81 + 12 * q^83 - 2 * q^85 + 2 * q^87 - 2 * q^89 + 28 * q^91 + 8 * q^93 + 3 * q^95 + 30 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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