LMFDB - Embedded newform 3080.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3080,2,Mod(1,3080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3080, 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("3080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3080.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:	$24.5939238226$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	3080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.481194 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.76845 q^{9} +O(q^{10})$	$q+0.481194 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.76845 q^{9} +1.00000 q^{11} -2.86907 q^{13} +0.481194 q^{15} -4.09332 q^{17} -5.35026 q^{19} +0.481194 q^{21} +8.46898 q^{23} +1.00000 q^{25} -2.77575 q^{27} -0.387873 q^{29} -7.28726 q^{31} +0.481194 q^{33} +1.00000 q^{35} -3.76845 q^{37} -1.38058 q^{39} -1.28726 q^{41} +7.50659 q^{43} -2.76845 q^{45} -13.1817 q^{47} +1.00000 q^{49} -1.96968 q^{51} -8.85685 q^{53} +1.00000 q^{55} -2.57452 q^{57} -8.24965 q^{59} -3.28726 q^{61} -2.76845 q^{63} -2.86907 q^{65} +3.50659 q^{67} +4.07522 q^{69} -14.2374 q^{71} +6.79384 q^{73} +0.481194 q^{75} +1.00000 q^{77} +10.2823 q^{79} +6.96968 q^{81} -6.77575 q^{83} -4.09332 q^{85} -0.186642 q^{87} +7.22425 q^{89} -2.86907 q^{91} -3.50659 q^{93} -5.35026 q^{95} -1.61213 q^{97} -2.76845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 4 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 4 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} - 6 q^{17} - 6 q^{19} - 4 q^{21} - 6 q^{23} + 3 q^{25} - 10 q^{27} - 2 q^{29} - 16 q^{31} - 4 q^{33} + 3 q^{35} + 8 q^{39} + 2 q^{41} + 2 q^{43} + 3 q^{45} - 14 q^{47} + 3 q^{49} - 8 q^{51} + 4 q^{53} + 3 q^{55} + 4 q^{57} - 8 q^{59} - 4 q^{61} + 3 q^{63} - 4 q^{65} - 10 q^{67} + 34 q^{69} - 6 q^{73} - 4 q^{75} + 3 q^{77} + 12 q^{79} + 23 q^{81} - 22 q^{83} - 6 q^{85} + 12 q^{87} + 20 q^{89} - 4 q^{91} + 10 q^{93} - 6 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 4 * q^15 - 6 * q^17 - 6 * q^19 - 4 * q^21 - 6 * q^23 + 3 * q^25 - 10 * q^27 - 2 * q^29 - 16 * q^31 - 4 * q^33 + 3 * q^35 + 8 * q^39 + 2 * q^41 + 2 * q^43 + 3 * q^45 - 14 * q^47 + 3 * q^49 - 8 * q^51 + 4 * q^53 + 3 * q^55 + 4 * q^57 - 8 * q^59 - 4 * q^61 + 3 * q^63 - 4 * q^65 - 10 * q^67 + 34 * q^69 - 6 * q^73 - 4 * q^75 + 3 * q^77 + 12 * q^79 + 23 * q^81 - 22 * q^83 - 6 * q^85 + 12 * q^87 + 20 * q^89 - 4 * q^91 + 10 * q^93 - 6 * q^95 - 4 * q^97 + 3 * 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
$3$	0.481194	0.277818	0.138909	−	0.990305i	$-0.455641\pi$
$3$	0.481194	0.277818	0.138909	+	0.990305i	$0.455641\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.76845	−0.922817
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.86907	−0.795736	−0.397868	−	0.917443i	$-0.630250\pi$
$13$	−2.86907	−0.795736	−0.397868	+	0.917443i	$0.630250\pi$
$14$	0	0
$15$	0.481194	0.124244
$16$	0	0
$17$	−4.09332	−0.992776	−0.496388	−	0.868101i	$-0.665341\pi$
$17$	−4.09332	−0.992776	−0.496388	+	0.868101i	$0.665341\pi$
$18$	0	0
$19$	−5.35026	−1.22743	−0.613717	−	0.789526i	$-0.710326\pi$
$19$	−5.35026	−1.22743	−0.613717	+	0.789526i	$0.710326\pi$
$20$	0	0
$21$	0.481194	0.105005
$22$	0	0
$23$	8.46898	1.76590	0.882952	−	0.469464i	$-0.155553\pi$
$23$	8.46898	1.76590	0.882952	+	0.469464i	$0.155553\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.77575	−0.534193
$28$	0	0
$29$	−0.387873	−0.0720262	−0.0360131	−	0.999351i	$-0.511466\pi$
$29$	−0.387873	−0.0720262	−0.0360131	+	0.999351i	$0.511466\pi$
$30$	0	0
$31$	−7.28726	−1.30883	−0.654415	−	0.756136i	$-0.727085\pi$
$31$	−7.28726	−1.30883	−0.654415	+	0.756136i	$0.727085\pi$
$32$	0	0
$33$	0.481194	0.0837652
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.76845	−0.619530	−0.309765	−	0.950813i	$-0.600250\pi$
$37$	−3.76845	−0.619530	−0.309765	+	0.950813i	$0.600250\pi$
$38$	0	0
$39$	−1.38058	−0.221070
$40$	0	0
$41$	−1.28726	−0.201036	−0.100518	−	0.994935i	$-0.532050\pi$
$41$	−1.28726	−0.201036	−0.100518	+	0.994935i	$0.532050\pi$
$42$	0	0
$43$	7.50659	1.14474	0.572372	−	0.819994i	$-0.306023\pi$
$43$	7.50659	1.14474	0.572372	+	0.819994i	$0.306023\pi$
$44$	0	0
$45$	−2.76845	−0.412696
$46$	0	0
$47$	−13.1817	−1.92275	−0.961376	−	0.275240i	$-0.911243\pi$
$47$	−13.1817	−1.92275	−0.961376	+	0.275240i	$0.911243\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.96968	−0.275811
$52$	0	0
$53$	−8.85685	−1.21658	−0.608291	−	0.793714i	$-0.708145\pi$
$53$	−8.85685	−1.21658	−0.608291	+	0.793714i	$0.708145\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−2.57452	−0.341003
$58$	0	0
$59$	−8.24965	−1.07401	−0.537006	−	0.843578i	$-0.680445\pi$
$59$	−8.24965	−1.07401	−0.537006	+	0.843578i	$0.680445\pi$
$60$	0	0
$61$	−3.28726	−0.420890	−0.210445	−	0.977606i	$-0.567491\pi$
$61$	−3.28726	−0.420890	−0.210445	+	0.977606i	$0.567491\pi$
$62$	0	0
$63$	−2.76845	−0.348792
$64$	0	0
$65$	−2.86907	−0.355864
$66$	0	0
$67$	3.50659	0.428398	0.214199	−	0.976790i	$-0.431286\pi$
$67$	3.50659	0.428398	0.214199	+	0.976790i	$0.431286\pi$
$68$	0	0
$69$	4.07522	0.490599
$70$	0	0
$71$	−14.2374	−1.68967	−0.844836	−	0.535026i	$-0.820302\pi$
$71$	−14.2374	−1.68967	−0.844836	+	0.535026i	$0.820302\pi$
$72$	0	0
$73$	6.79384	0.795159	0.397580	−	0.917568i	$-0.369850\pi$
$73$	6.79384	0.795159	0.397580	+	0.917568i	$0.369850\pi$
$74$	0	0
$75$	0.481194	0.0555635
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	10.2823	1.15685	0.578426	−	0.815735i	$-0.303667\pi$
$79$	10.2823	1.15685	0.578426	+	0.815735i	$0.303667\pi$
$80$	0	0
$81$	6.96968	0.774409
$82$	0	0
$83$	−6.77575	−0.743735	−0.371867	−	0.928286i	$-0.621282\pi$
$83$	−6.77575	−0.743735	−0.371867	+	0.928286i	$0.621282\pi$
$84$	0	0
$85$	−4.09332	−0.443983
$86$	0	0
$87$	−0.186642	−0.0200102
$88$	0	0
$89$	7.22425	0.765769	0.382885	−	0.923796i	$-0.374931\pi$
$89$	7.22425	0.765769	0.382885	+	0.923796i	$0.374931\pi$
$90$	0	0
$91$	−2.86907	−0.300760
$92$	0	0
$93$	−3.50659	−0.363616
$94$	0	0
$95$	−5.35026	−0.548925
$96$	0	0
$97$	−1.61213	−0.163687	−0.0818433	−	0.996645i	$-0.526081\pi$
$97$	−1.61213	−0.163687	−0.0818433	+	0.996645i	$0.526081\pi$
$98$	0	0
$99$	−2.76845	−0.278240
$100$	0	0
$101$	−5.10062	−0.507530	−0.253765	−	0.967266i	$-0.581669\pi$
$101$	−5.10062	−0.507530	−0.253765	+	0.967266i	$0.581669\pi$
$102$	0	0
$103$	−1.59403	−0.157064	−0.0785321	−	0.996912i	$-0.525023\pi$
$103$	−1.59403	−0.157064	−0.0785321	+	0.996912i	$0.525023\pi$
$104$	0	0
$105$	0.481194	0.0469598
$106$	0	0
$107$	0.312650	0.0302251	0.0151125	−	0.999886i	$-0.495189\pi$
$107$	0.312650	0.0302251	0.0151125	+	0.999886i	$0.495189\pi$
$108$	0	0
$109$	11.7381	1.12431	0.562155	−	0.827032i	$-0.309973\pi$
$109$	11.7381	1.12431	0.562155	+	0.827032i	$0.309973\pi$
$110$	0	0
$111$	−1.81336	−0.172116
$112$	0	0
$113$	8.57452	0.806623	0.403311	−	0.915063i	$-0.367859\pi$
$113$	8.57452	0.806623	0.403311	+	0.915063i	$0.367859\pi$
$114$	0	0
$115$	8.46898	0.789736
$116$	0	0
$117$	7.94288	0.734319
$118$	0	0
$119$	−4.09332	−0.375234
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.619421	−0.0558513
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.8119	−1.84676	−0.923381	−	0.383886i	$-0.874586\pi$
$127$	−20.8119	−1.84676	−0.923381	+	0.383886i	$0.874586\pi$
$128$	0	0
$129$	3.61213	0.318030
$130$	0	0
$131$	−13.9248	−1.21661	−0.608307	−	0.793702i	$-0.708151\pi$
$131$	−13.9248	−1.21661	−0.608307	+	0.793702i	$0.708151\pi$
$132$	0	0
$133$	−5.35026	−0.463927
$134$	0	0
$135$	−2.77575	−0.238898
$136$	0	0
$137$	−4.54420	−0.388237	−0.194119	−	0.980978i	$-0.562185\pi$
$137$	−4.54420	−0.388237	−0.194119	+	0.980978i	$0.562185\pi$
$138$	0	0
$139$	10.1114	0.857639	0.428820	−	0.903390i	$-0.358930\pi$
$139$	10.1114	0.857639	0.428820	+	0.903390i	$0.358930\pi$
$140$	0	0
$141$	−6.34297	−0.534174
$142$	0	0
$143$	−2.86907	−0.239923
$144$	0	0
$145$	−0.387873	−0.0322111
$146$	0	0
$147$	0.481194	0.0396882
$148$	0	0
$149$	13.6629	1.11931	0.559655	−	0.828726i	$-0.310934\pi$
$149$	13.6629	1.11931	0.559655	+	0.828726i	$0.310934\pi$
$150$	0	0
$151$	7.24472	0.589567	0.294784	−	0.955564i	$-0.404752\pi$
$151$	7.24472	0.589567	0.294784	+	0.955564i	$0.404752\pi$
$152$	0	0
$153$	11.3322	0.916151
$154$	0	0
$155$	−7.28726	−0.585327
$156$	0	0
$157$	−2.72496	−0.217476	−0.108738	−	0.994070i	$-0.534681\pi$
$157$	−2.72496	−0.217476	−0.108738	+	0.994070i	$0.534681\pi$
$158$	0	0
$159$	−4.26187	−0.337988
$160$	0	0
$161$	8.46898	0.667449
$162$	0	0
$163$	−13.5818	−1.06381	−0.531905	−	0.846804i	$-0.678524\pi$
$163$	−13.5818	−1.06381	−0.531905	+	0.846804i	$0.678524\pi$
$164$	0	0
$165$	0.481194	0.0374609
$166$	0	0
$167$	16.8265	1.30208	0.651038	−	0.759045i	$-0.274334\pi$
$167$	16.8265	1.30208	0.651038	+	0.759045i	$0.274334\pi$
$168$	0	0
$169$	−4.76845	−0.366804
$170$	0	0
$171$	14.8119	1.13270
$172$	0	0
$173$	11.8822	0.903390	0.451695	−	0.892172i	$-0.350820\pi$
$173$	11.8822	0.903390	0.451695	+	0.892172i	$0.350820\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−3.96968	−0.298380
$178$	0	0
$179$	−18.2374	−1.36313	−0.681565	−	0.731758i	$-0.738700\pi$
$179$	−18.2374	−1.36313	−0.681565	+	0.731758i	$0.738700\pi$
$180$	0	0
$181$	−15.7235	−1.16872	−0.584361	−	0.811494i	$-0.698655\pi$
$181$	−15.7235	−1.16872	−0.584361	+	0.811494i	$0.698655\pi$
$182$	0	0
$183$	−1.58181	−0.116931
$184$	0	0
$185$	−3.76845	−0.277062
$186$	0	0
$187$	−4.09332	−0.299333
$188$	0	0
$189$	−2.77575	−0.201906
$190$	0	0
$191$	7.19982	0.520960	0.260480	−	0.965479i	$-0.416119\pi$
$191$	7.19982	0.520960	0.260480	+	0.965479i	$0.416119\pi$
$192$	0	0
$193$	−22.5950	−1.62642	−0.813211	−	0.581969i	$-0.802283\pi$
$193$	−22.5950	−1.62642	−0.813211	+	0.581969i	$0.802283\pi$
$194$	0	0
$195$	−1.38058	−0.0988653
$196$	0	0
$197$	−3.11871	−0.222199	−0.111100	−	0.993809i	$-0.535437\pi$
$197$	−3.11871	−0.222199	−0.111100	+	0.993809i	$0.535437\pi$
$198$	0	0
$199$	−14.5017	−1.02800	−0.513998	−	0.857792i	$-0.671836\pi$
$199$	−14.5017	−1.02800	−0.513998	+	0.857792i	$0.671836\pi$
$200$	0	0
$201$	1.68735	0.119016
$202$	0	0
$203$	−0.387873	−0.0272234
$204$	0	0
$205$	−1.28726	−0.0899060
$206$	0	0
$207$	−23.4460	−1.62961
$208$	0	0
$209$	−5.35026	−0.370085
$210$	0	0
$211$	−7.66291	−0.527537	−0.263768	−	0.964586i	$-0.584965\pi$
$211$	−7.66291	−0.527537	−0.263768	+	0.964586i	$0.584965\pi$
$212$	0	0
$213$	−6.85097	−0.469421
$214$	0	0
$215$	7.50659	0.511945
$216$	0	0
$217$	−7.28726	−0.494691
$218$	0	0
$219$	3.26916	0.220909
$220$	0	0
$221$	11.7440	0.789988
$222$	0	0
$223$	3.95746	0.265011	0.132506	−	0.991182i	$-0.457698\pi$
$223$	3.95746	0.265011	0.132506	+	0.991182i	$0.457698\pi$
$224$	0	0
$225$	−2.76845	−0.184563
$226$	0	0
$227$	−6.07522	−0.403227	−0.201613	−	0.979465i	$-0.564618\pi$
$227$	−6.07522	−0.403227	−0.201613	+	0.979465i	$0.564618\pi$
$228$	0	0
$229$	10.4993	0.693813	0.346906	−	0.937900i	$-0.387232\pi$
$229$	10.4993	0.693813	0.346906	+	0.937900i	$0.387232\pi$
$230$	0	0
$231$	0.481194	0.0316603
$232$	0	0
$233$	13.6932	0.897073	0.448537	−	0.893764i	$-0.351945\pi$
$233$	13.6932	0.897073	0.448537	+	0.893764i	$0.351945\pi$
$234$	0	0
$235$	−13.1817	−0.859880
$236$	0	0
$237$	4.94780	0.321394
$238$	0	0
$239$	−3.28963	−0.212788	−0.106394	−	0.994324i	$-0.533931\pi$
$239$	−3.28963	−0.212788	−0.106394	+	0.994324i	$0.533931\pi$
$240$	0	0
$241$	−5.22662	−0.336676	−0.168338	−	0.985729i	$-0.553840\pi$
$241$	−5.22662	−0.336676	−0.168338	+	0.985729i	$0.553840\pi$
$242$	0	0
$243$	11.6810	0.749337
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	15.3503	0.976714
$248$	0	0
$249$	−3.26045	−0.206623
$250$	0	0
$251$	15.7259	0.992611	0.496306	−	0.868148i	$-0.334690\pi$
$251$	15.7259	0.992611	0.496306	+	0.868148i	$0.334690\pi$
$252$	0	0
$253$	8.46898	0.532440
$254$	0	0
$255$	−1.96968	−0.123346
$256$	0	0
$257$	−0.589104	−0.0367473	−0.0183736	−	0.999831i	$-0.505849\pi$
$257$	−0.589104	−0.0367473	−0.0183736	+	0.999831i	$0.505849\pi$
$258$	0	0
$259$	−3.76845	−0.234160
$260$	0	0
$261$	1.07381	0.0664671
$262$	0	0
$263$	−2.88717	−0.178030	−0.0890151	−	0.996030i	$-0.528372\pi$
$263$	−2.88717	−0.178030	−0.0890151	+	0.996030i	$0.528372\pi$
$264$	0	0
$265$	−8.85685	−0.544072
$266$	0	0
$267$	3.47627	0.212744
$268$	0	0
$269$	6.88717	0.419918	0.209959	−	0.977710i	$-0.432667\pi$
$269$	6.88717	0.419918	0.209959	+	0.977710i	$0.432667\pi$
$270$	0	0
$271$	−16.1866	−0.983268	−0.491634	−	0.870802i	$-0.663600\pi$
$271$	−16.1866	−0.983268	−0.491634	+	0.870802i	$0.663600\pi$
$272$	0	0
$273$	−1.38058	−0.0835564
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	10.4690	0.629020	0.314510	−	0.949254i	$-0.398160\pi$
$277$	10.4690	0.629020	0.314510	+	0.949254i	$0.398160\pi$
$278$	0	0
$279$	20.1744	1.20781
$280$	0	0
$281$	−3.92478	−0.234133	−0.117066	−	0.993124i	$-0.537349\pi$
$281$	−3.92478	−0.234133	−0.117066	+	0.993124i	$0.537349\pi$
$282$	0	0
$283$	−19.5369	−1.16135	−0.580674	−	0.814136i	$-0.697211\pi$
$283$	−19.5369	−1.16135	−0.580674	+	0.814136i	$0.697211\pi$
$284$	0	0
$285$	−2.57452	−0.152501
$286$	0	0
$287$	−1.28726	−0.0759844
$288$	0	0
$289$	−0.244722	−0.0143954
$290$	0	0
$291$	−0.775746	−0.0454751
$292$	0	0
$293$	27.5452	1.60921	0.804603	−	0.593814i	$-0.202378\pi$
$293$	27.5452	1.60921	0.804603	+	0.593814i	$0.202378\pi$
$294$	0	0
$295$	−8.24965	−0.480313
$296$	0	0
$297$	−2.77575	−0.161065
$298$	0	0
$299$	−24.2981	−1.40519
$300$	0	0
$301$	7.50659	0.432672
$302$	0	0
$303$	−2.45439	−0.141001
$304$	0	0
$305$	−3.28726	−0.188228
$306$	0	0
$307$	−1.66291	−0.0949074	−0.0474537	−	0.998873i	$-0.515111\pi$
$307$	−1.66291	−0.0949074	−0.0474537	+	0.998873i	$0.515111\pi$
$308$	0	0
$309$	−0.767037	−0.0436352
$310$	0	0
$311$	31.7767	1.80189	0.900946	−	0.433932i	$-0.142874\pi$
$311$	31.7767	1.80189	0.900946	+	0.433932i	$0.142874\pi$
$312$	0	0
$313$	−20.7612	−1.17349	−0.586745	−	0.809772i	$-0.699591\pi$
$313$	−20.7612	−1.17349	−0.586745	+	0.809772i	$0.699591\pi$
$314$	0	0
$315$	−2.76845	−0.155985
$316$	0	0
$317$	25.3258	1.42244	0.711220	−	0.702969i	$-0.248143\pi$
$317$	25.3258	1.42244	0.711220	+	0.702969i	$0.248143\pi$
$318$	0	0
$319$	−0.387873	−0.0217167
$320$	0	0
$321$	0.150446	0.00839705
$322$	0	0
$323$	21.9003	1.21857
$324$	0	0
$325$	−2.86907	−0.159147
$326$	0	0
$327$	5.64832	0.312353
$328$	0	0
$329$	−13.1817	−0.726732
$330$	0	0
$331$	6.11142	0.335914	0.167957	−	0.985794i	$-0.446283\pi$
$331$	6.11142	0.335914	0.167957	+	0.985794i	$0.446283\pi$
$332$	0	0
$333$	10.4328	0.571713
$334$	0	0
$335$	3.50659	0.191585
$336$	0	0
$337$	−19.2692	−1.04966	−0.524829	−	0.851208i	$-0.675871\pi$
$337$	−19.2692	−1.04966	−0.524829	+	0.851208i	$0.675871\pi$
$338$	0	0
$339$	4.12601	0.224094
$340$	0	0
$341$	−7.28726	−0.394627
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	4.07522	0.219403
$346$	0	0
$347$	−21.4920	−1.15375	−0.576875	−	0.816832i	$-0.695728\pi$
$347$	−21.4920	−1.15375	−0.576875	+	0.816832i	$0.695728\pi$
$348$	0	0
$349$	3.10062	0.165972	0.0829861	−	0.996551i	$-0.473554\pi$
$349$	3.10062	0.165972	0.0829861	+	0.996551i	$0.473554\pi$
$350$	0	0
$351$	7.96380	0.425076
$352$	0	0
$353$	−9.53690	−0.507598	−0.253799	−	0.967257i	$-0.581680\pi$
$353$	−9.53690	−0.507598	−0.253799	+	0.967257i	$0.581680\pi$
$354$	0	0
$355$	−14.2374	−0.755644
$356$	0	0
$357$	−1.96968	−0.104247
$358$	0	0
$359$	−3.05808	−0.161399	−0.0806996	−	0.996738i	$-0.525715\pi$
$359$	−3.05808	−0.161399	−0.0806996	+	0.996738i	$0.525715\pi$
$360$	0	0
$361$	9.62530	0.506595
$362$	0	0
$363$	0.481194	0.0252562
$364$	0	0
$365$	6.79384	0.355606
$366$	0	0
$367$	−28.8691	−1.50695	−0.753477	−	0.657475i	$-0.771625\pi$
$367$	−28.8691	−1.50695	−0.753477	+	0.657475i	$0.771625\pi$
$368$	0	0
$369$	3.56371	0.185519
$370$	0	0
$371$	−8.85685	−0.459825
$372$	0	0
$373$	−7.43136	−0.384781	−0.192391	−	0.981318i	$-0.561624\pi$
$373$	−7.43136	−0.384781	−0.192391	+	0.981318i	$0.561624\pi$
$374$	0	0
$375$	0.481194	0.0248488
$376$	0	0
$377$	1.11283	0.0573139
$378$	0	0
$379$	−7.16362	−0.367970	−0.183985	−	0.982929i	$-0.558900\pi$
$379$	−7.16362	−0.367970	−0.183985	+	0.982929i	$0.558900\pi$
$380$	0	0
$381$	−10.0146	−0.513063
$382$	0	0
$383$	−1.56959	−0.0802023	−0.0401012	−	0.999196i	$-0.512768\pi$
$383$	−1.56959	−0.0802023	−0.0401012	+	0.999196i	$0.512768\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−20.7816	−1.05639
$388$	0	0
$389$	−25.4518	−1.29046	−0.645230	−	0.763989i	$-0.723238\pi$
$389$	−25.4518	−1.29046	−0.645230	+	0.763989i	$0.723238\pi$
$390$	0	0
$391$	−34.6662	−1.75315
$392$	0	0
$393$	−6.70052	−0.337997
$394$	0	0
$395$	10.2823	0.517360
$396$	0	0
$397$	0.962389	0.0483009	0.0241504	−	0.999708i	$-0.492312\pi$
$397$	0.962389	0.0483009	0.0241504	+	0.999708i	$0.492312\pi$
$398$	0	0
$399$	−2.57452	−0.128887
$400$	0	0
$401$	1.47627	0.0737214	0.0368607	−	0.999320i	$-0.488264\pi$
$401$	1.47627	0.0737214	0.0368607	+	0.999320i	$0.488264\pi$
$402$	0	0
$403$	20.9076	1.04148
$404$	0	0
$405$	6.96968	0.346326
$406$	0	0
$407$	−3.76845	−0.186795
$408$	0	0
$409$	38.9864	1.92775	0.963876	−	0.266352i	$-0.0858183\pi$
$409$	38.9864	1.92775	0.963876	+	0.266352i	$0.0858183\pi$
$410$	0	0
$411$	−2.18664	−0.107859
$412$	0	0
$413$	−8.24965	−0.405939
$414$	0	0
$415$	−6.77575	−0.332608
$416$	0	0
$417$	4.86556	0.238267
$418$	0	0
$419$	−14.4119	−0.704065	−0.352033	−	0.935988i	$-0.614509\pi$
$419$	−14.4119	−0.704065	−0.352033	+	0.935988i	$0.614509\pi$
$420$	0	0
$421$	12.4036	0.604515	0.302257	−	0.953226i	$-0.402260\pi$
$421$	12.4036	0.604515	0.302257	+	0.953226i	$0.402260\pi$
$422$	0	0
$423$	36.4930	1.77435
$424$	0	0
$425$	−4.09332	−0.198555
$426$	0	0
$427$	−3.28726	−0.159082
$428$	0	0
$429$	−1.38058	−0.0666550
$430$	0	0
$431$	27.4314	1.32132	0.660661	−	0.750684i	$-0.270276\pi$
$431$	27.4314	1.32132	0.660661	+	0.750684i	$0.270276\pi$
$432$	0	0
$433$	9.81336	0.471600	0.235800	−	0.971802i	$-0.424229\pi$
$433$	9.81336	0.471600	0.235800	+	0.971802i	$0.424229\pi$
$434$	0	0
$435$	−0.186642	−0.00894882
$436$	0	0
$437$	−45.3112	−2.16753
$438$	0	0
$439$	22.4241	1.07024	0.535121	−	0.844775i	$-0.320266\pi$
$439$	22.4241	1.07024	0.535121	+	0.844775i	$0.320266\pi$
$440$	0	0
$441$	−2.76845	−0.131831
$442$	0	0
$443$	−17.8700	−0.849030	−0.424515	−	0.905421i	$-0.639555\pi$
$443$	−17.8700	−0.849030	−0.424515	+	0.905421i	$0.639555\pi$
$444$	0	0
$445$	7.22425	0.342462
$446$	0	0
$447$	6.57452	0.310964
$448$	0	0
$449$	31.3561	1.47979	0.739894	−	0.672724i	$-0.234876\pi$
$449$	31.3561	1.47979	0.739894	+	0.672724i	$0.234876\pi$
$450$	0	0
$451$	−1.28726	−0.0606146
$452$	0	0
$453$	3.48612	0.163792
$454$	0	0
$455$	−2.86907	−0.134504
$456$	0	0
$457$	16.8324	0.787387	0.393693	−	0.919242i	$-0.371197\pi$
$457$	16.8324	0.787387	0.393693	+	0.919242i	$0.371197\pi$
$458$	0	0
$459$	11.3620	0.530334
$460$	0	0
$461$	0.385503	0.0179547	0.00897734	−	0.999960i	$-0.497142\pi$
$461$	0.385503	0.0179547	0.00897734	+	0.999960i	$0.497142\pi$
$462$	0	0
$463$	32.8061	1.52463	0.762314	−	0.647208i	$-0.224063\pi$
$463$	32.8061	1.52463	0.762314	+	0.647208i	$0.224063\pi$
$464$	0	0
$465$	−3.50659	−0.162614
$466$	0	0
$467$	−26.4060	−1.22192	−0.610961	−	0.791660i	$-0.709217\pi$
$467$	−26.4060	−1.22192	−0.610961	+	0.791660i	$0.709217\pi$
$468$	0	0
$469$	3.50659	0.161919
$470$	0	0
$471$	−1.31124	−0.0604186
$472$	0	0
$473$	7.50659	0.345153
$474$	0	0
$475$	−5.35026	−0.245487
$476$	0	0
$477$	24.5198	1.12268
$478$	0	0
$479$	33.1392	1.51417	0.757084	−	0.653318i	$-0.226623\pi$
$479$	33.1392	1.51417	0.757084	+	0.653318i	$0.226623\pi$
$480$	0	0
$481$	10.8119	0.492982
$482$	0	0
$483$	4.07522	0.185429
$484$	0	0
$485$	−1.61213	−0.0732029
$486$	0	0
$487$	−32.3792	−1.46724	−0.733620	−	0.679560i	$-0.762171\pi$
$487$	−32.3792	−1.46724	−0.733620	+	0.679560i	$0.762171\pi$
$488$	0	0
$489$	−6.53549	−0.295545
$490$	0	0
$491$	39.7440	1.79362	0.896811	−	0.442414i	$-0.145878\pi$
$491$	39.7440	1.79362	0.896811	+	0.442414i	$0.145878\pi$
$492$	0	0
$493$	1.58769	0.0715059
$494$	0	0
$495$	−2.76845	−0.124433
$496$	0	0
$497$	−14.2374	−0.638636
$498$	0	0
$499$	−2.55008	−0.114157	−0.0570786	−	0.998370i	$-0.518179\pi$
$499$	−2.55008	−0.114157	−0.0570786	+	0.998370i	$0.518179\pi$
$500$	0	0
$501$	8.09683	0.361740
$502$	0	0
$503$	−8.55008	−0.381229	−0.190615	−	0.981665i	$-0.561048\pi$
$503$	−8.55008	−0.381229	−0.190615	+	0.981665i	$0.561048\pi$
$504$	0	0
$505$	−5.10062	−0.226974
$506$	0	0
$507$	−2.29455	−0.101905
$508$	0	0
$509$	24.1114	1.06872	0.534360	−	0.845257i	$-0.320553\pi$
$509$	24.1114	1.06872	0.534360	+	0.845257i	$0.320553\pi$
$510$	0	0
$511$	6.79384	0.300542
$512$	0	0
$513$	14.8510	0.655686
$514$	0	0
$515$	−1.59403	−0.0702413
$516$	0	0
$517$	−13.1817	−0.579731
$518$	0	0
$519$	5.71767	0.250978
$520$	0	0
$521$	23.1998	1.01640	0.508201	−	0.861238i	$-0.330311\pi$
$521$	23.1998	1.01640	0.508201	+	0.861238i	$0.330311\pi$
$522$	0	0
$523$	−3.58769	−0.156879	−0.0784393	−	0.996919i	$-0.524994\pi$
$523$	−3.58769	−0.156879	−0.0784393	+	0.996919i	$0.524994\pi$
$524$	0	0
$525$	0.481194	0.0210010
$526$	0	0
$527$	29.8291	1.29938
$528$	0	0
$529$	48.7235	2.11842
$530$	0	0
$531$	22.8388	0.991117
$532$	0	0
$533$	3.69323	0.159972
$534$	0	0
$535$	0.312650	0.0135171
$536$	0	0
$537$	−8.77575	−0.378701
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	26.5599	1.14190	0.570950	−	0.820985i	$-0.306575\pi$
$541$	26.5599	1.14190	0.570950	+	0.820985i	$0.306575\pi$
$542$	0	0
$543$	−7.56608	−0.324692
$544$	0	0
$545$	11.7381	0.502806
$546$	0	0
$547$	9.02302	0.385797	0.192898	−	0.981219i	$-0.438211\pi$
$547$	9.02302	0.385797	0.192898	+	0.981219i	$0.438211\pi$
$548$	0	0
$549$	9.10062	0.388405
$550$	0	0
$551$	2.07522	0.0884075
$552$	0	0
$553$	10.2823	0.437249
$554$	0	0
$555$	−1.81336	−0.0769727
$556$	0	0
$557$	−46.5705	−1.97326	−0.986629	−	0.162984i	$-0.947888\pi$
$557$	−46.5705	−1.97326	−0.986629	+	0.162984i	$0.947888\pi$
$558$	0	0
$559$	−21.5369	−0.910914
$560$	0	0
$561$	−1.96968	−0.0831601
$562$	0	0
$563$	−19.9756	−0.841870	−0.420935	−	0.907091i	$-0.638298\pi$
$563$	−19.9756	−0.841870	−0.420935	+	0.907091i	$0.638298\pi$
$564$	0	0
$565$	8.57452	0.360733
$566$	0	0
$567$	6.96968	0.292699
$568$	0	0
$569$	25.1392	1.05389	0.526945	−	0.849900i	$-0.323338\pi$
$569$	25.1392	1.05389	0.526945	+	0.849900i	$0.323338\pi$
$570$	0	0
$571$	−9.58769	−0.401232	−0.200616	−	0.979670i	$-0.564294\pi$
$571$	−9.58769	−0.401232	−0.200616	+	0.979670i	$0.564294\pi$
$572$	0	0
$573$	3.46451	0.144732
$574$	0	0
$575$	8.46898	0.353181
$576$	0	0
$577$	2.55008	0.106161	0.0530806	−	0.998590i	$-0.483096\pi$
$577$	2.55008	0.106161	0.0530806	+	0.998590i	$0.483096\pi$
$578$	0	0
$579$	−10.8726	−0.451849
$580$	0	0
$581$	−6.77575	−0.281105
$582$	0	0
$583$	−8.85685	−0.366813
$584$	0	0
$585$	7.94288	0.328397
$586$	0	0
$587$	−3.91653	−0.161652	−0.0808262	−	0.996728i	$-0.525756\pi$
$587$	−3.91653	−0.161652	−0.0808262	+	0.996728i	$0.525756\pi$
$588$	0	0
$589$	38.9887	1.60650
$590$	0	0
$591$	−1.50071	−0.0617309
$592$	0	0
$593$	−27.3439	−1.12288	−0.561440	−	0.827517i	$-0.689753\pi$
$593$	−27.3439	−1.12288	−0.561440	+	0.827517i	$0.689753\pi$
$594$	0	0
$595$	−4.09332	−0.167810
$596$	0	0
$597$	−6.97812	−0.285595
$598$	0	0
$599$	−11.0738	−0.452464	−0.226232	−	0.974074i	$-0.572641\pi$
$599$	−11.0738	−0.452464	−0.226232	+	0.974074i	$0.572641\pi$
$600$	0	0
$601$	1.47390	0.0601216	0.0300608	−	0.999548i	$-0.490430\pi$
$601$	1.47390	0.0601216	0.0300608	+	0.999548i	$0.490430\pi$
$602$	0	0
$603$	−9.70782	−0.395333
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	11.1246	0.451533	0.225767	−	0.974181i	$-0.427511\pi$
$607$	11.1246	0.451533	0.225767	+	0.974181i	$0.427511\pi$
$608$	0	0
$609$	−0.186642	−0.00756313
$610$	0	0
$611$	37.8192	1.53000
$612$	0	0
$613$	−27.9307	−1.12811	−0.564054	−	0.825738i	$-0.690759\pi$
$613$	−27.9307	−1.12811	−0.564054	+	0.825738i	$0.690759\pi$
$614$	0	0
$615$	−0.619421	−0.0249775
$616$	0	0
$617$	32.4544	1.30656	0.653282	−	0.757114i	$-0.273392\pi$
$617$	32.4544	1.30656	0.653282	+	0.757114i	$0.273392\pi$
$618$	0	0
$619$	−42.8505	−1.72231	−0.861154	−	0.508345i	$-0.830258\pi$
$619$	−42.8505	−1.72231	−0.861154	+	0.508345i	$0.830258\pi$
$620$	0	0
$621$	−23.5077	−0.943333
$622$	0	0
$623$	7.22425	0.289434
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.57452	−0.102816
$628$	0	0
$629$	15.4255	0.615054
$630$	0	0
$631$	−22.9234	−0.912565	−0.456282	−	0.889835i	$-0.650819\pi$
$631$	−22.9234	−0.912565	−0.456282	+	0.889835i	$0.650819\pi$
$632$	0	0
$633$	−3.68735	−0.146559
$634$	0	0
$635$	−20.8119	−0.825897
$636$	0	0
$637$	−2.86907	−0.113677
$638$	0	0
$639$	39.4156	1.55926
$640$	0	0
$641$	9.79289	0.386796	0.193398	−	0.981120i	$-0.438049\pi$
$641$	9.79289	0.386796	0.193398	+	0.981120i	$0.438049\pi$
$642$	0	0
$643$	45.1935	1.78226	0.891128	−	0.453751i	$-0.149915\pi$
$643$	45.1935	1.78226	0.891128	+	0.453751i	$0.149915\pi$
$644$	0	0
$645$	3.61213	0.142227
$646$	0	0
$647$	−28.6335	−1.12570	−0.562851	−	0.826559i	$-0.690295\pi$
$647$	−28.6335	−1.12570	−0.562851	+	0.826559i	$0.690295\pi$
$648$	0	0
$649$	−8.24965	−0.323827
$650$	0	0
$651$	−3.50659	−0.137434
$652$	0	0
$653$	31.6483	1.23849	0.619247	−	0.785196i	$-0.287438\pi$
$653$	31.6483	1.23849	0.619247	+	0.785196i	$0.287438\pi$
$654$	0	0
$655$	−13.9248	−0.544086
$656$	0	0
$657$	−18.8084	−0.733787
$658$	0	0
$659$	−3.39517	−0.132257	−0.0661285	−	0.997811i	$-0.521065\pi$
$659$	−3.39517	−0.132257	−0.0661285	+	0.997811i	$0.521065\pi$
$660$	0	0
$661$	−34.6107	−1.34620	−0.673100	−	0.739551i	$-0.735038\pi$
$661$	−34.6107	−1.34620	−0.673100	+	0.739551i	$0.735038\pi$
$662$	0	0
$663$	5.65115	0.219473
$664$	0	0
$665$	−5.35026	−0.207474
$666$	0	0
$667$	−3.28489	−0.127191
$668$	0	0
$669$	1.90431	0.0736248
$670$	0	0
$671$	−3.28726	−0.126903
$672$	0	0
$673$	31.9062	1.22989	0.614947	−	0.788568i	$-0.289177\pi$
$673$	31.9062	1.22989	0.614947	+	0.788568i	$0.289177\pi$
$674$	0	0
$675$	−2.77575	−0.106839
$676$	0	0
$677$	34.0689	1.30937	0.654687	−	0.755900i	$-0.272801\pi$
$677$	34.0689	1.30937	0.654687	+	0.755900i	$0.272801\pi$
$678$	0	0
$679$	−1.61213	−0.0618678
$680$	0	0
$681$	−2.92336	−0.112023
$682$	0	0
$683$	−8.43278	−0.322671	−0.161336	−	0.986900i	$-0.551580\pi$
$683$	−8.43278	−0.322671	−0.161336	+	0.986900i	$0.551580\pi$
$684$	0	0
$685$	−4.54420	−0.173625
$686$	0	0
$687$	5.05220	0.192753
$688$	0	0
$689$	25.4109	0.968078
$690$	0	0
$691$	−35.2482	−1.34091	−0.670453	−	0.741952i	$-0.733900\pi$
$691$	−35.2482	−1.34091	−0.670453	+	0.741952i	$0.733900\pi$
$692$	0	0
$693$	−2.76845	−0.105165
$694$	0	0
$695$	10.1114	0.383548
$696$	0	0
$697$	5.26916	0.199584
$698$	0	0
$699$	6.58910	0.249223
$700$	0	0
$701$	2.02444	0.0764620	0.0382310	−	0.999269i	$-0.487828\pi$
$701$	2.02444	0.0764620	0.0382310	+	0.999269i	$0.487828\pi$
$702$	0	0
$703$	20.1622	0.760432
$704$	0	0
$705$	−6.34297	−0.238890
$706$	0	0
$707$	−5.10062	−0.191828
$708$	0	0
$709$	14.2228	0.534150	0.267075	−	0.963676i	$-0.413943\pi$
$709$	14.2228	0.534150	0.267075	+	0.963676i	$0.413943\pi$
$710$	0	0
$711$	−28.4661	−1.06756
$712$	0	0
$713$	−61.7156	−2.31127
$714$	0	0
$715$	−2.86907	−0.107297
$716$	0	0
$717$	−1.58295	−0.0591163
$718$	0	0
$719$	15.9633	0.595332	0.297666	−	0.954670i	$-0.403792\pi$
$719$	15.9633	0.595332	0.297666	+	0.954670i	$0.403792\pi$
$720$	0	0
$721$	−1.59403	−0.0593647
$722$	0	0
$723$	−2.51502	−0.0935347
$724$	0	0
$725$	−0.387873	−0.0144052
$726$	0	0
$727$	9.64481	0.357706	0.178853	−	0.983876i	$-0.442761\pi$
$727$	9.64481	0.357706	0.178853	+	0.983876i	$0.442761\pi$
$728$	0	0
$729$	−15.2882	−0.566230
$730$	0	0
$731$	−30.7269	−1.13647
$732$	0	0
$733$	−48.2965	−1.78387	−0.891935	−	0.452163i	$-0.850652\pi$
$733$	−48.2965	−1.78387	−0.891935	+	0.452163i	$0.850652\pi$
$734$	0	0
$735$	0.481194	0.0177491
$736$	0	0
$737$	3.50659	0.129167
$738$	0	0
$739$	13.2506	0.487431	0.243716	−	0.969847i	$-0.421634\pi$
$739$	13.2506	0.487431	0.243716	+	0.969847i	$0.421634\pi$
$740$	0	0
$741$	7.38646	0.271348
$742$	0	0
$743$	51.6385	1.89443	0.947216	−	0.320596i	$-0.103883\pi$
$743$	51.6385	1.89443	0.947216	+	0.320596i	$0.103883\pi$
$744$	0	0
$745$	13.6629	0.500570
$746$	0	0
$747$	18.7583	0.686331
$748$	0	0
$749$	0.312650	0.0114240
$750$	0	0
$751$	27.3620	0.998454	0.499227	−	0.866471i	$-0.333617\pi$
$751$	27.3620	0.998454	0.499227	+	0.866471i	$0.333617\pi$
$752$	0	0
$753$	7.56722	0.275765
$754$	0	0
$755$	7.24472	0.263662
$756$	0	0
$757$	−51.4372	−1.86952	−0.934759	−	0.355282i	$-0.884385\pi$
$757$	−51.4372	−1.86952	−0.934759	+	0.355282i	$0.884385\pi$
$758$	0	0
$759$	4.07522	0.147921
$760$	0	0
$761$	54.4020	1.97207	0.986036	−	0.166535i	$-0.0532579\pi$
$761$	54.4020	1.97207	0.986036	+	0.166535i	$0.0532579\pi$
$762$	0	0
$763$	11.7381	0.424949
$764$	0	0
$765$	11.3322	0.409715
$766$	0	0
$767$	23.6688	0.854631
$768$	0	0
$769$	24.5524	0.885384	0.442692	−	0.896674i	$-0.354024\pi$
$769$	24.5524	0.885384	0.442692	+	0.896674i	$0.354024\pi$
$770$	0	0
$771$	−0.283473	−0.0102090
$772$	0	0
$773$	−33.0592	−1.18906	−0.594529	−	0.804074i	$-0.702661\pi$
$773$	−33.0592	−1.18906	−0.594529	+	0.804074i	$0.702661\pi$
$774$	0	0
$775$	−7.28726	−0.261766
$776$	0	0
$777$	−1.81336	−0.0650538
$778$	0	0
$779$	6.88717	0.246758
$780$	0	0
$781$	−14.2374	−0.509455
$782$	0	0
$783$	1.07664	0.0384759
$784$	0	0
$785$	−2.72496	−0.0972580
$786$	0	0
$787$	23.7235	0.845653	0.422827	−	0.906211i	$-0.361038\pi$
$787$	23.7235	0.845653	0.422827	+	0.906211i	$0.361038\pi$
$788$	0	0
$789$	−1.38929	−0.0494600
$790$	0	0
$791$	8.57452	0.304875
$792$	0	0
$793$	9.43136	0.334918
$794$	0	0
$795$	−4.26187	−0.151153
$796$	0	0
$797$	−1.67750	−0.0594201	−0.0297101	−	0.999559i	$-0.509458\pi$
$797$	−1.67750	−0.0594201	−0.0297101	+	0.999559i	$0.509458\pi$
$798$	0	0
$799$	53.9570	1.90886
$800$	0	0
$801$	−20.0000	−0.706665
$802$	0	0
$803$	6.79384	0.239750
$804$	0	0
$805$	8.46898	0.298492
$806$	0	0
$807$	3.31406	0.116661
$808$	0	0
$809$	−6.83638	−0.240354	−0.120177	−	0.992752i	$-0.538346\pi$
$809$	−6.83638	−0.240354	−0.120177	+	0.992752i	$0.538346\pi$
$810$	0	0
$811$	−1.00141	−0.0351644	−0.0175822	−	0.999845i	$-0.505597\pi$
$811$	−1.00141	−0.0351644	−0.0175822	+	0.999845i	$0.505597\pi$
$812$	0	0
$813$	−7.78892	−0.273169
$814$	0	0
$815$	−13.5818	−0.475750
$816$	0	0
$817$	−40.1622	−1.40510
$818$	0	0
$819$	7.94288	0.277547
$820$	0	0
$821$	31.9003	1.11333	0.556665	−	0.830737i	$-0.312081\pi$
$821$	31.9003	1.11333	0.556665	+	0.830737i	$0.312081\pi$
$822$	0	0
$823$	−9.64244	−0.336114	−0.168057	−	0.985777i	$-0.553749\pi$
$823$	−9.64244	−0.336114	−0.168057	+	0.985777i	$0.553749\pi$
$824$	0	0
$825$	0.481194	0.0167530
$826$	0	0
$827$	−16.9076	−0.587936	−0.293968	−	0.955815i	$-0.594976\pi$
$827$	−16.9076	−0.587936	−0.293968	+	0.955815i	$0.594976\pi$
$828$	0	0
$829$	21.6267	0.751127	0.375563	−	0.926797i	$-0.377449\pi$
$829$	21.6267	0.751127	0.375563	+	0.926797i	$0.377449\pi$
$830$	0	0
$831$	5.03761	0.174753
$832$	0	0
$833$	−4.09332	−0.141825
$834$	0	0
$835$	16.8265	0.582306
$836$	0	0
$837$	20.2276	0.699167
$838$	0	0
$839$	44.3366	1.53067	0.765335	−	0.643632i	$-0.222573\pi$
$839$	44.3366	1.53067	0.765335	+	0.643632i	$0.222573\pi$
$840$	0	0
$841$	−28.8496	−0.994812
$842$	0	0
$843$	−1.88858	−0.0650462
$844$	0	0
$845$	−4.76845	−0.164040
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−9.40105	−0.322643
$850$	0	0
$851$	−31.9149	−1.09403
$852$	0	0
$853$	−2.34534	−0.0803028	−0.0401514	−	0.999194i	$-0.512784\pi$
$853$	−2.34534	−0.0803028	−0.0401514	+	0.999194i	$0.512784\pi$
$854$	0	0
$855$	14.8119	0.506558
$856$	0	0
$857$	7.78067	0.265783	0.132891	−	0.991131i	$-0.457574\pi$
$857$	7.78067	0.265783	0.132891	+	0.991131i	$0.457574\pi$
$858$	0	0
$859$	−14.7635	−0.503725	−0.251863	−	0.967763i	$-0.581043\pi$
$859$	−14.7635	−0.503725	−0.251863	+	0.967763i	$0.581043\pi$
$860$	0	0
$861$	−0.619421	−0.0211098
$862$	0	0
$863$	58.5560	1.99327	0.996634	−	0.0819798i	$-0.0261243\pi$
$863$	58.5560	1.99327	0.996634	+	0.0819798i	$0.0261243\pi$
$864$	0	0
$865$	11.8822	0.404008
$866$	0	0
$867$	−0.117759	−0.00399930
$868$	0	0
$869$	10.2823	0.348804
$870$	0	0
$871$	−10.0606	−0.340892
$872$	0	0
$873$	4.46310	0.151053
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	0.815913	0.0275514	0.0137757	−	0.999905i	$-0.495615\pi$
$877$	0.815913	0.0275514	0.0137757	+	0.999905i	$0.495615\pi$
$878$	0	0
$879$	13.2546	0.447066
$880$	0	0
$881$	26.6761	0.898740	0.449370	−	0.893346i	$-0.351648\pi$
$881$	26.6761	0.898740	0.449370	+	0.893346i	$0.351648\pi$
$882$	0	0
$883$	−32.1925	−1.08336	−0.541682	−	0.840583i	$-0.682212\pi$
$883$	−32.1925	−1.08336	−0.541682	+	0.840583i	$0.682212\pi$
$884$	0	0
$885$	−3.96968	−0.133439
$886$	0	0
$887$	13.8886	0.466333	0.233166	−	0.972437i	$-0.425091\pi$
$887$	13.8886	0.466333	0.233166	+	0.972437i	$0.425091\pi$
$888$	0	0
$889$	−20.8119	−0.698010
$890$	0	0
$891$	6.96968	0.233493
$892$	0	0
$893$	70.5256	2.36005
$894$	0	0
$895$	−18.2374	−0.609610
$896$	0	0
$897$	−11.6921	−0.390387
$898$	0	0
$899$	2.82653	0.0942701
$900$	0	0
$901$	36.2539	1.20779
$902$	0	0
$903$	3.61213	0.120204
$904$	0	0
$905$	−15.7235	−0.522668
$906$	0	0
$907$	−22.2024	−0.737218	−0.368609	−	0.929585i	$-0.620166\pi$
$907$	−22.2024	−0.737218	−0.368609	+	0.929585i	$0.620166\pi$
$908$	0	0
$909$	14.1208	0.468358
$910$	0	0
$911$	−34.7123	−1.15007	−0.575035	−	0.818129i	$-0.695011\pi$
$911$	−34.7123	−1.15007	−0.575035	+	0.818129i	$0.695011\pi$
$912$	0	0
$913$	−6.77575	−0.224244
$914$	0	0
$915$	−1.58181	−0.0522930
$916$	0	0
$917$	−13.9248	−0.459837
$918$	0	0
$919$	−10.6155	−0.350171	−0.175086	−	0.984553i	$-0.556020\pi$
$919$	−10.6155	−0.350171	−0.175086	+	0.984553i	$0.556020\pi$
$920$	0	0
$921$	−0.800184	−0.0263670
$922$	0	0
$923$	40.8481	1.34453
$924$	0	0
$925$	−3.76845	−0.123906
$926$	0	0
$927$	4.41299	0.144942
$928$	0	0
$929$	−13.0033	−0.426625	−0.213313	−	0.976984i	$-0.568425\pi$
$929$	−13.0033	−0.426625	−0.213313	+	0.976984i	$0.568425\pi$
$930$	0	0
$931$	−5.35026	−0.175348
$932$	0	0
$933$	15.2908	0.500597
$934$	0	0
$935$	−4.09332	−0.133866
$936$	0	0
$937$	−33.7200	−1.10159	−0.550793	−	0.834642i	$-0.685675\pi$
$937$	−33.7200	−1.10159	−0.550793	+	0.834642i	$0.685675\pi$
$938$	0	0
$939$	−9.99015	−0.326016
$940$	0	0
$941$	10.1138	0.329700	0.164850	−	0.986319i	$-0.447286\pi$
$941$	10.1138	0.329700	0.164850	+	0.986319i	$0.447286\pi$
$942$	0	0
$943$	−10.9018	−0.355010
$944$	0	0
$945$	−2.77575	−0.0902950
$946$	0	0
$947$	−44.5099	−1.44638	−0.723189	−	0.690650i	$-0.757324\pi$
$947$	−44.5099	−1.44638	−0.723189	+	0.690650i	$0.757324\pi$
$948$	0	0
$949$	−19.4920	−0.632737
$950$	0	0
$951$	12.1866	0.395179
$952$	0	0
$953$	−42.4807	−1.37609	−0.688043	−	0.725670i	$-0.741530\pi$
$953$	−42.4807	−1.37609	−0.688043	+	0.725670i	$0.741530\pi$
$954$	0	0
$955$	7.19982	0.232981
$956$	0	0
$957$	−0.186642	−0.00603329
$958$	0	0
$959$	−4.54420	−0.146740
$960$	0	0
$961$	22.1041	0.713036
$962$	0	0
$963$	−0.865557	−0.0278922
$964$	0	0
$965$	−22.5950	−0.727358
$966$	0	0
$967$	23.4050	0.752655	0.376327	−	0.926487i	$-0.377187\pi$
$967$	23.4050	0.752655	0.376327	+	0.926487i	$0.377187\pi$
$968$	0	0
$969$	10.5383	0.338540
$970$	0	0
$971$	6.83401	0.219314	0.109657	−	0.993969i	$-0.465025\pi$
$971$	6.83401	0.219314	0.109657	+	0.993969i	$0.465025\pi$
$972$	0	0
$973$	10.1114	0.324157
$974$	0	0
$975$	−1.38058	−0.0442139
$976$	0	0
$977$	19.0033	0.607970	0.303985	−	0.952677i	$-0.401683\pi$
$977$	19.0033	0.607970	0.303985	+	0.952677i	$0.401683\pi$
$978$	0	0
$979$	7.22425	0.230888
$980$	0	0
$981$	−32.4965	−1.03753
$982$	0	0
$983$	−14.0063	−0.446733	−0.223366	−	0.974735i	$-0.571705\pi$
$983$	−14.0063	−0.446733	−0.223366	+	0.974735i	$0.571705\pi$
$984$	0	0
$985$	−3.11871	−0.0993705
$986$	0	0
$987$	−6.34297	−0.201899
$988$	0	0
$989$	63.5731	2.02151
$990$	0	0
$991$	22.6761	0.720330	0.360165	−	0.932889i	$-0.382720\pi$
$991$	22.6761	0.720330	0.360165	+	0.932889i	$0.382720\pi$
$992$	0	0
$993$	2.94078	0.0933228
$994$	0	0
$995$	−14.5017	−0.459734
$996$	0	0
$997$	−61.2785	−1.94071	−0.970356	−	0.241682i	$-0.922301\pi$
$997$	−61.2785	−1.94071	−0.970356	+	0.241682i	$0.922301\pi$
$998$	0	0
$999$	10.4603	0.330948

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3080.2.a.j.1.3	✓	3
4.3	odd	2		6160.2.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.j.1.3	✓	3	1.1	even	1	trivial
6160.2.a.bp.1.1		3	4.3	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:	\( N \)	\(=\)	\( 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3080.a (trivial)

\(f(q)\)	\(=\)	\(q+0.481194 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.76845 q^{9} +O(q^{10})\)	\(q+0.481194 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.76845 q^{9} +1.00000 q^{11} -2.86907 q^{13} +0.481194 q^{15} -4.09332 q^{17} -5.35026 q^{19} +0.481194 q^{21} +8.46898 q^{23} +1.00000 q^{25} -2.77575 q^{27} -0.387873 q^{29} -7.28726 q^{31} +0.481194 q^{33} +1.00000 q^{35} -3.76845 q^{37} -1.38058 q^{39} -1.28726 q^{41} +7.50659 q^{43} -2.76845 q^{45} -13.1817 q^{47} +1.00000 q^{49} -1.96968 q^{51} -8.85685 q^{53} +1.00000 q^{55} -2.57452 q^{57} -8.24965 q^{59} -3.28726 q^{61} -2.76845 q^{63} -2.86907 q^{65} +3.50659 q^{67} +4.07522 q^{69} -14.2374 q^{71} +6.79384 q^{73} +0.481194 q^{75} +1.00000 q^{77} +10.2823 q^{79} +6.96968 q^{81} -6.77575 q^{83} -4.09332 q^{85} -0.186642 q^{87} +7.22425 q^{89} -2.86907 q^{91} -3.50659 q^{93} -5.35026 q^{95} -1.61213 q^{97} -2.76845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 4 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 4 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} - 6 q^{17} - 6 q^{19} - 4 q^{21} - 6 q^{23} + 3 q^{25} - 10 q^{27} - 2 q^{29} - 16 q^{31} - 4 q^{33} + 3 q^{35} + 8 q^{39} + 2 q^{41} + 2 q^{43} + 3 q^{45} - 14 q^{47} + 3 q^{49} - 8 q^{51} + 4 q^{53} + 3 q^{55} + 4 q^{57} - 8 q^{59} - 4 q^{61} + 3 q^{63} - 4 q^{65} - 10 q^{67} + 34 q^{69} - 6 q^{73} - 4 q^{75} + 3 q^{77} + 12 q^{79} + 23 q^{81} - 22 q^{83} - 6 q^{85} + 12 q^{87} + 20 q^{89} - 4 q^{91} + 10 q^{93} - 6 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 4 * q^15 - 6 * q^17 - 6 * q^19 - 4 * q^21 - 6 * q^23 + 3 * q^25 - 10 * q^27 - 2 * q^29 - 16 * q^31 - 4 * q^33 + 3 * q^35 + 8 * q^39 + 2 * q^41 + 2 * q^43 + 3 * q^45 - 14 * q^47 + 3 * q^49 - 8 * q^51 + 4 * q^53 + 3 * q^55 + 4 * q^57 - 8 * q^59 - 4 * q^61 + 3 * q^63 - 4 * q^65 - 10 * q^67 + 34 * q^69 - 6 * q^73 - 4 * q^75 + 3 * q^77 + 12 * q^79 + 23 * q^81 - 22 * q^83 - 6 * q^85 + 12 * q^87 + 20 * q^89 - 4 * q^91 + 10 * q^93 - 6 * q^95 - 4 * q^97 + 3 * q^99

Introduction

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