LMFDB - Embedded newform 1520.2.a.p.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:	$3$
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:	$ 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			$-1.48119$ 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)

$f(q)$	$=$	$q-0.806063 q^{3} +1.00000 q^{5} -3.35026 q^{7} -2.35026 q^{9} +O(q^{10})$	$q-0.806063 q^{3} +1.00000 q^{5} -3.35026 q^{7} -2.35026 q^{9} -0.962389 q^{11} +6.15633 q^{13} -0.806063 q^{15} -6.31265 q^{17} +1.00000 q^{19} +2.70052 q^{21} +4.96239 q^{23} +1.00000 q^{25} +4.31265 q^{27} -3.61213 q^{29} +5.92478 q^{31} +0.775746 q^{33} -3.35026 q^{35} +10.1563 q^{37} -4.96239 q^{39} +6.31265 q^{41} +4.12601 q^{43} -2.35026 q^{45} -3.35026 q^{47} +4.22425 q^{49} +5.08840 q^{51} +1.84367 q^{53} -0.962389 q^{55} -0.806063 q^{57} +6.38787 q^{59} -11.2750 q^{61} +7.87399 q^{63} +6.15633 q^{65} +6.73084 q^{67} -4.00000 q^{69} +0.775746 q^{71} +0.387873 q^{73} -0.806063 q^{75} +3.22425 q^{77} +0.836381 q^{79} +3.57452 q^{81} +7.03761 q^{83} -6.31265 q^{85} +2.91160 q^{87} +7.08840 q^{89} -20.6253 q^{91} -4.77575 q^{93} +1.00000 q^{95} +10.9927 q^{97} +2.26187 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9	$ 3 q - 2 q^{3} + 3 q^{5} + 3 q^{9} + 8 q^{11} + 8 q^{13} - 2 q^{15} + 2 q^{17} + 3 q^{19} - 12 q^{21} + 4 q^{23} + 3 q^{25} - 8 q^{27} - 10 q^{29} - 4 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{39} - 2 q^{41} + 4 q^{43} + 3 q^{45} + 11 q^{49} - 4 q^{51} + 16 q^{53} + 8 q^{55} - 2 q^{57} + 20 q^{59} - 2 q^{61} + 32 q^{63} + 8 q^{65} - 2 q^{67} - 12 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 8 q^{77} - q^{81} + 32 q^{83} + 2 q^{85} + 28 q^{87} + 2 q^{89} - 20 q^{91} - 16 q^{93} + 3 q^{95} + 20 q^{97} + 16 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9 + 8 * q^11 + 8 * q^13 - 2 * q^15 + 2 * q^17 + 3 * q^19 - 12 * q^21 + 4 * q^23 + 3 * q^25 - 8 * q^27 - 10 * q^29 - 4 * q^31 + 4 * q^33 + 20 * q^37 - 4 * q^39 - 2 * q^41 + 4 * q^43 + 3 * q^45 + 11 * q^49 - 4 * q^51 + 16 * q^53 + 8 * q^55 - 2 * q^57 + 20 * q^59 - 2 * q^61 + 32 * q^63 + 8 * q^65 - 2 * q^67 - 12 * q^69 + 4 * q^71 + 2 * q^73 - 2 * q^75 + 8 * q^77 - q^81 + 32 * q^83 + 2 * q^85 + 28 * q^87 + 2 * q^89 - 20 * q^91 - 16 * q^93 + 3 * q^95 + 20 * q^97 + 16 * 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
$3$	−0.806063	−0.465381	−0.232690	−	0.972551i	$-0.574753\pi$
$3$	−0.806063	−0.465381	−0.232690	+	0.972551i	$0.574753\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.35026	−1.26628	−0.633140	−	0.774037i	$-0.718234\pi$
$7$	−3.35026	−1.26628	−0.633140	+	0.774037i	$0.718234\pi$
$8$	0	0
$9$	−2.35026	−0.783421
$10$	0	0
$11$	−0.962389	−0.290171	−0.145086	−	0.989419i	$-0.546346\pi$
$11$	−0.962389	−0.290171	−0.145086	+	0.989419i	$0.546346\pi$
$12$	0	0
$13$	6.15633	1.70746	0.853729	−	0.520718i	$-0.174336\pi$
$13$	6.15633	1.70746	0.853729	+	0.520718i	$0.174336\pi$
$14$	0	0
$15$	−0.806063	−0.208125
$16$	0	0
$17$	−6.31265	−1.53104	−0.765521	−	0.643411i	$-0.777519\pi$
$17$	−6.31265	−1.53104	−0.765521	+	0.643411i	$0.777519\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.70052	0.589303
$22$	0	0
$23$	4.96239	1.03473	0.517365	−	0.855765i	$-0.326913\pi$
$23$	4.96239	1.03473	0.517365	+	0.855765i	$0.326913\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.31265	0.829970
$28$	0	0
$29$	−3.61213	−0.670755	−0.335378	−	0.942084i	$-0.608864\pi$
$29$	−3.61213	−0.670755	−0.335378	+	0.942084i	$0.608864\pi$
$30$	0	0
$31$	5.92478	1.06412	0.532061	−	0.846706i	$-0.321418\pi$
$31$	5.92478	1.06412	0.532061	+	0.846706i	$0.321418\pi$
$32$	0	0
$33$	0.775746	0.135040
$34$	0	0
$35$	−3.35026	−0.566298
$36$	0	0
$37$	10.1563	1.66969	0.834845	−	0.550485i	$-0.185557\pi$
$37$	10.1563	1.66969	0.834845	+	0.550485i	$0.185557\pi$
$38$	0	0
$39$	−4.96239	−0.794618
$40$	0	0
$41$	6.31265	0.985870	0.492935	−	0.870066i	$-0.335924\pi$
$41$	6.31265	0.985870	0.492935	+	0.870066i	$0.335924\pi$
$42$	0	0
$43$	4.12601	0.629210	0.314605	−	0.949223i	$-0.398128\pi$
$43$	4.12601	0.629210	0.314605	+	0.949223i	$0.398128\pi$
$44$	0	0
$45$	−2.35026	−0.350356
$46$	0	0
$47$	−3.35026	−0.488686	−0.244343	−	0.969689i	$-0.578572\pi$
$47$	−3.35026	−0.488686	−0.244343	+	0.969689i	$0.578572\pi$
$48$	0	0
$49$	4.22425	0.603465
$50$	0	0
$51$	5.08840	0.712518
$52$	0	0
$53$	1.84367	0.253248	0.126624	−	0.991951i	$-0.459586\pi$
$53$	1.84367	0.253248	0.126624	+	0.991951i	$0.459586\pi$
$54$	0	0
$55$	−0.962389	−0.129768
$56$	0	0
$57$	−0.806063	−0.106766
$58$	0	0
$59$	6.38787	0.831630	0.415815	−	0.909449i	$-0.363496\pi$
$59$	6.38787	0.831630	0.415815	+	0.909449i	$0.363496\pi$
$60$	0	0
$61$	−11.2750	−1.44362	−0.721810	−	0.692091i	$-0.756690\pi$
$61$	−11.2750	−1.44362	−0.721810	+	0.692091i	$0.756690\pi$
$62$	0	0
$63$	7.87399	0.992030
$64$	0	0
$65$	6.15633	0.763598
$66$	0	0
$67$	6.73084	0.822303	0.411152	−	0.911567i	$-0.365127\pi$
$67$	6.73084	0.822303	0.411152	+	0.911567i	$0.365127\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0.775746	0.0920641	0.0460321	−	0.998940i	$-0.485342\pi$
$71$	0.775746	0.0920641	0.0460321	+	0.998940i	$0.485342\pi$
$72$	0	0
$73$	0.387873	0.0453971	0.0226986	−	0.999742i	$-0.492774\pi$
$73$	0.387873	0.0453971	0.0226986	+	0.999742i	$0.492774\pi$
$74$	0	0
$75$	−0.806063	−0.0930762
$76$	0	0
$77$	3.22425	0.367438
$78$	0	0
$79$	0.836381	0.0941002	0.0470501	−	0.998893i	$-0.485018\pi$
$79$	0.836381	0.0941002	0.0470501	+	0.998893i	$0.485018\pi$
$80$	0	0
$81$	3.57452	0.397168
$82$	0	0
$83$	7.03761	0.772478	0.386239	−	0.922399i	$-0.373774\pi$
$83$	7.03761	0.772478	0.386239	+	0.922399i	$0.373774\pi$
$84$	0	0
$85$	−6.31265	−0.684703
$86$	0	0
$87$	2.91160	0.312157
$88$	0	0
$89$	7.08840	0.751369	0.375684	−	0.926748i	$-0.377408\pi$
$89$	7.08840	0.751369	0.375684	+	0.926748i	$0.377408\pi$
$90$	0	0
$91$	−20.6253	−2.16212
$92$	0	0
$93$	−4.77575	−0.495222
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	10.9927	1.11614	0.558070	−	0.829794i	$-0.311542\pi$
$97$	10.9927	1.11614	0.558070	+	0.829794i	$0.311542\pi$
$98$	0	0
$99$	2.26187	0.227326
$100$	0	0
$101$	−2.64974	−0.263659	−0.131829	−	0.991272i	$-0.542085\pi$
$101$	−2.64974	−0.263659	−0.131829	+	0.991272i	$0.542085\pi$
$102$	0	0
$103$	10.7308	1.05734	0.528671	−	0.848827i	$-0.322691\pi$
$103$	10.7308	1.05734	0.528671	+	0.848827i	$0.322691\pi$
$104$	0	0
$105$	2.70052	0.263544
$106$	0	0
$107$	−4.80606	−0.464620	−0.232310	−	0.972642i	$-0.574628\pi$
$107$	−4.80606	−0.464620	−0.232310	+	0.972642i	$0.574628\pi$
$108$	0	0
$109$	−2.77575	−0.265868	−0.132934	−	0.991125i	$-0.542440\pi$
$109$	−2.77575	−0.265868	−0.132934	+	0.991125i	$0.542440\pi$
$110$	0	0
$111$	−8.18664	−0.777042
$112$	0	0
$113$	6.99271	0.657818	0.328909	−	0.944362i	$-0.393319\pi$
$113$	6.99271	0.657818	0.328909	+	0.944362i	$0.393319\pi$
$114$	0	0
$115$	4.96239	0.462745
$116$	0	0
$117$	−14.4690	−1.33766
$118$	0	0
$119$	21.1490	1.93873
$120$	0	0
$121$	−10.0738	−0.915801
$122$	0	0
$123$	−5.08840	−0.458805
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.4314	−1.19184	−0.595920	−	0.803043i	$-0.703213\pi$
$127$	−13.4314	−1.19184	−0.595920	+	0.803043i	$0.703213\pi$
$128$	0	0
$129$	−3.32582	−0.292822
$130$	0	0
$131$	−20.6253	−1.80204	−0.901020	−	0.433777i	$-0.857181\pi$
$131$	−20.6253	−1.80204	−0.901020	+	0.433777i	$0.857181\pi$
$132$	0	0
$133$	−3.35026	−0.290505
$134$	0	0
$135$	4.31265	0.371174
$136$	0	0
$137$	20.2374	1.72900	0.864500	−	0.502633i	$-0.167635\pi$
$137$	20.2374	1.72900	0.864500	+	0.502633i	$0.167635\pi$
$138$	0	0
$139$	17.5877	1.49177	0.745884	−	0.666076i	$-0.232027\pi$
$139$	17.5877	1.49177	0.745884	+	0.666076i	$0.232027\pi$
$140$	0	0
$141$	2.70052	0.227425
$142$	0	0
$143$	−5.92478	−0.495455
$144$	0	0
$145$	−3.61213	−0.299971
$146$	0	0
$147$	−3.40502	−0.280841
$148$	0	0
$149$	−7.42548	−0.608319	−0.304160	−	0.952621i	$-0.598376\pi$
$149$	−7.42548	−0.608319	−0.304160	+	0.952621i	$0.598376\pi$
$150$	0	0
$151$	1.61213	0.131193	0.0655965	−	0.997846i	$-0.479105\pi$
$151$	1.61213	0.131193	0.0655965	+	0.997846i	$0.479105\pi$
$152$	0	0
$153$	14.8364	1.19945
$154$	0	0
$155$	5.92478	0.475890
$156$	0	0
$157$	4.38787	0.350190	0.175095	−	0.984552i	$-0.443977\pi$
$157$	4.38787	0.350190	0.175095	+	0.984552i	$0.443977\pi$
$158$	0	0
$159$	−1.48612	−0.117857
$160$	0	0
$161$	−16.6253	−1.31026
$162$	0	0
$163$	0.649738	0.0508914	0.0254457	−	0.999676i	$-0.491900\pi$
$163$	0.649738	0.0508914	0.0254457	+	0.999676i	$0.491900\pi$
$164$	0	0
$165$	0.775746	0.0603918
$166$	0	0
$167$	15.3561	1.18829	0.594147	−	0.804357i	$-0.297490\pi$
$167$	15.3561	1.18829	0.594147	+	0.804357i	$0.297490\pi$
$168$	0	0
$169$	24.9003	1.91541
$170$	0	0
$171$	−2.35026	−0.179729
$172$	0	0
$173$	3.24472	0.246692	0.123346	−	0.992364i	$-0.460638\pi$
$173$	3.24472	0.246692	0.123346	+	0.992364i	$0.460638\pi$
$174$	0	0
$175$	−3.35026	−0.253256
$176$	0	0
$177$	−5.14903	−0.387025
$178$	0	0
$179$	−15.0132	−1.12214	−0.561069	−	0.827769i	$-0.689610\pi$
$179$	−15.0132	−1.12214	−0.561069	+	0.827769i	$0.689610\pi$
$180$	0	0
$181$	9.22425	0.685633	0.342817	−	0.939402i	$-0.388619\pi$
$181$	9.22425	0.685633	0.342817	+	0.939402i	$0.388619\pi$
$182$	0	0
$183$	9.08840	0.671834
$184$	0	0
$185$	10.1563	0.746708
$186$	0	0
$187$	6.07522	0.444264
$188$	0	0
$189$	−14.4485	−1.05097
$190$	0	0
$191$	21.7743	1.57554	0.787768	−	0.615972i	$-0.211237\pi$
$191$	21.7743	1.57554	0.787768	+	0.615972i	$0.211237\pi$
$192$	0	0
$193$	12.5442	0.902951	0.451476	−	0.892283i	$-0.350898\pi$
$193$	12.5442	0.902951	0.451476	+	0.892283i	$0.350898\pi$
$194$	0	0
$195$	−4.96239	−0.355364
$196$	0	0
$197$	24.5501	1.74912	0.874560	−	0.484917i	$-0.161150\pi$
$197$	24.5501	1.74912	0.874560	+	0.484917i	$0.161150\pi$
$198$	0	0
$199$	−23.0738	−1.63566	−0.817829	−	0.575461i	$-0.804823\pi$
$199$	−23.0738	−1.63566	−0.817829	+	0.575461i	$0.804823\pi$
$200$	0	0
$201$	−5.42548	−0.382684
$202$	0	0
$203$	12.1016	0.849364
$204$	0	0
$205$	6.31265	0.440895
$206$	0	0
$207$	−11.6629	−0.810628
$208$	0	0
$209$	−0.962389	−0.0665698
$210$	0	0
$211$	20.9380	1.44143	0.720714	−	0.693233i	$-0.243814\pi$
$211$	20.9380	1.44143	0.720714	+	0.693233i	$0.243814\pi$
$212$	0	0
$213$	−0.625301	−0.0428449
$214$	0	0
$215$	4.12601	0.281391
$216$	0	0
$217$	−19.8496	−1.34748
$218$	0	0
$219$	−0.312650	−0.0211270
$220$	0	0
$221$	−38.8627	−2.61419
$222$	0	0
$223$	0.0303172	0.00203019	0.00101509	−	0.999999i	$-0.499677\pi$
$223$	0.0303172	0.00203019	0.00101509	+	0.999999i	$0.499677\pi$
$224$	0	0
$225$	−2.35026	−0.156684
$226$	0	0
$227$	−4.80606	−0.318990	−0.159495	−	0.987199i	$-0.550987\pi$
$227$	−4.80606	−0.318990	−0.159495	+	0.987199i	$0.550987\pi$
$228$	0	0
$229$	1.87399	0.123837	0.0619184	−	0.998081i	$-0.480278\pi$
$229$	1.87399	0.123837	0.0619184	+	0.998081i	$0.480278\pi$
$230$	0	0
$231$	−2.59895	−0.170999
$232$	0	0
$233$	−11.1490	−0.730397	−0.365199	−	0.930930i	$-0.618999\pi$
$233$	−11.1490	−0.730397	−0.365199	+	0.930930i	$0.618999\pi$
$234$	0	0
$235$	−3.35026	−0.218547
$236$	0	0
$237$	−0.674176	−0.0437924
$238$	0	0
$239$	−9.29948	−0.601533	−0.300767	−	0.953698i	$-0.597243\pi$
$239$	−9.29948	−0.601533	−0.300767	+	0.953698i	$0.597243\pi$
$240$	0	0
$241$	−2.31265	−0.148971	−0.0744855	−	0.997222i	$-0.523731\pi$
$241$	−2.31265	−0.148971	−0.0744855	+	0.997222i	$0.523731\pi$
$242$	0	0
$243$	−15.8192	−1.01480
$244$	0	0
$245$	4.22425	0.269878
$246$	0	0
$247$	6.15633	0.391718
$248$	0	0
$249$	−5.67276	−0.359497
$250$	0	0
$251$	−24.1016	−1.52128	−0.760639	−	0.649175i	$-0.775114\pi$
$251$	−24.1016	−1.52128	−0.760639	+	0.649175i	$0.775114\pi$
$252$	0	0
$253$	−4.77575	−0.300249
$254$	0	0
$255$	5.08840	0.318648
$256$	0	0
$257$	−13.3199	−0.830875	−0.415438	−	0.909622i	$-0.636372\pi$
$257$	−13.3199	−0.830875	−0.415438	+	0.909622i	$0.636372\pi$
$258$	0	0
$259$	−34.0263	−2.11429
$260$	0	0
$261$	8.48944	0.525483
$262$	0	0
$263$	−12.9624	−0.799295	−0.399648	−	0.916669i	$-0.630867\pi$
$263$	−12.9624	−0.799295	−0.399648	+	0.916669i	$0.630867\pi$
$264$	0	0
$265$	1.84367	0.113256
$266$	0	0
$267$	−5.71370	−0.349673
$268$	0	0
$269$	−11.4010	−0.695134	−0.347567	−	0.937655i	$-0.612992\pi$
$269$	−11.4010	−0.695134	−0.347567	+	0.937655i	$0.612992\pi$
$270$	0	0
$271$	−16.8119	−1.02125	−0.510626	−	0.859803i	$-0.670586\pi$
$271$	−16.8119	−1.02125	−0.510626	+	0.859803i	$0.670586\pi$
$272$	0	0
$273$	16.6253	1.00621
$274$	0	0
$275$	−0.962389	−0.0580342
$276$	0	0
$277$	−29.7889	−1.78984	−0.894921	−	0.446224i	$-0.852769\pi$
$277$	−29.7889	−1.78984	−0.894921	+	0.446224i	$0.852769\pi$
$278$	0	0
$279$	−13.9248	−0.833655
$280$	0	0
$281$	−11.6121	−0.692721	−0.346361	−	0.938101i	$-0.612583\pi$
$281$	−11.6121	−0.692721	−0.346361	+	0.938101i	$0.612583\pi$
$282$	0	0
$283$	−2.26187	−0.134454	−0.0672270	−	0.997738i	$-0.521415\pi$
$283$	−2.26187	−0.134454	−0.0672270	+	0.997738i	$0.521415\pi$
$284$	0	0
$285$	−0.806063	−0.0477471
$286$	0	0
$287$	−21.1490	−1.24839
$288$	0	0
$289$	22.8496	1.34409
$290$	0	0
$291$	−8.86082	−0.519430
$292$	0	0
$293$	1.84367	0.107709	0.0538543	−	0.998549i	$-0.482849\pi$
$293$	1.84367	0.107709	0.0538543	+	0.998549i	$0.482849\pi$
$294$	0	0
$295$	6.38787	0.371916
$296$	0	0
$297$	−4.15045	−0.240833
$298$	0	0
$299$	30.5501	1.76676
$300$	0	0
$301$	−13.8232	−0.796756
$302$	0	0
$303$	2.13586	0.122702
$304$	0	0
$305$	−11.2750	−0.645607
$306$	0	0
$307$	−26.2071	−1.49572	−0.747859	−	0.663857i	$-0.768918\pi$
$307$	−26.2071	−1.49572	−0.747859	+	0.663857i	$0.768918\pi$
$308$	0	0
$309$	−8.64974	−0.492066
$310$	0	0
$311$	−6.51388	−0.369368	−0.184684	−	0.982798i	$-0.559126\pi$
$311$	−6.51388	−0.369368	−0.184684	+	0.982798i	$0.559126\pi$
$312$	0	0
$313$	16.0752	0.908625	0.454313	−	0.890842i	$-0.349885\pi$
$313$	16.0752	0.908625	0.454313	+	0.890842i	$0.349885\pi$
$314$	0	0
$315$	7.87399	0.443649
$316$	0	0
$317$	−5.69323	−0.319764	−0.159882	−	0.987136i	$-0.551111\pi$
$317$	−5.69323	−0.319764	−0.159882	+	0.987136i	$0.551111\pi$
$318$	0	0
$319$	3.47627	0.194634
$320$	0	0
$321$	3.87399	0.216225
$322$	0	0
$323$	−6.31265	−0.351245
$324$	0	0
$325$	6.15633	0.341491
$326$	0	0
$327$	2.23743	0.123730
$328$	0	0
$329$	11.2243	0.618813
$330$	0	0
$331$	12.3127	0.676764	0.338382	−	0.941009i	$-0.390120\pi$
$331$	12.3127	0.676764	0.338382	+	0.941009i	$0.390120\pi$
$332$	0	0
$333$	−23.8700	−1.30807
$334$	0	0
$335$	6.73084	0.367745
$336$	0	0
$337$	3.76845	0.205281	0.102640	−	0.994719i	$-0.467271\pi$
$337$	3.76845	0.205281	0.102640	+	0.994719i	$0.467271\pi$
$338$	0	0
$339$	−5.63656	−0.306136
$340$	0	0
$341$	−5.70194	−0.308777
$342$	0	0
$343$	9.29948	0.502125
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	22.3634	1.20053	0.600266	−	0.799800i	$-0.295061\pi$
$347$	22.3634	1.20053	0.600266	+	0.799800i	$0.295061\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	26.5501	1.41714
$352$	0	0
$353$	5.53690	0.294700	0.147350	−	0.989084i	$-0.452926\pi$
$353$	5.53690	0.294700	0.147350	+	0.989084i	$0.452926\pi$
$354$	0	0
$355$	0.775746	0.0411723
$356$	0	0
$357$	−17.0475	−0.902247
$358$	0	0
$359$	10.3634	0.546961	0.273481	−	0.961878i	$-0.411825\pi$
$359$	10.3634	0.546961	0.273481	+	0.961878i	$0.411825\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.12013	0.426196
$364$	0	0
$365$	0.387873	0.0203022
$366$	0	0
$367$	−3.35026	−0.174882	−0.0874411	−	0.996170i	$-0.527869\pi$
$367$	−3.35026	−0.174882	−0.0874411	+	0.996170i	$0.527869\pi$
$368$	0	0
$369$	−14.8364	−0.772351
$370$	0	0
$371$	−6.17679	−0.320683
$372$	0	0
$373$	−12.6048	−0.652653	−0.326327	−	0.945257i	$-0.605811\pi$
$373$	−12.6048	−0.652653	−0.326327	+	0.945257i	$0.605811\pi$
$374$	0	0
$375$	−0.806063	−0.0416249
$376$	0	0
$377$	−22.2374	−1.14529
$378$	0	0
$379$	37.2506	1.91343	0.956717	−	0.291018i	$-0.0939941\pi$
$379$	37.2506	1.91343	0.956717	+	0.291018i	$0.0939941\pi$
$380$	0	0
$381$	10.8265	0.554660
$382$	0	0
$383$	−30.8324	−1.57546	−0.787731	−	0.616019i	$-0.788744\pi$
$383$	−30.8324	−1.57546	−0.787731	+	0.616019i	$0.788744\pi$
$384$	0	0
$385$	3.22425	0.164323
$386$	0	0
$387$	−9.69720	−0.492936
$388$	0	0
$389$	−1.37470	−0.0697000	−0.0348500	−	0.999393i	$-0.511095\pi$
$389$	−1.37470	−0.0697000	−0.0348500	+	0.999393i	$0.511095\pi$
$390$	0	0
$391$	−31.3258	−1.58422
$392$	0	0
$393$	16.6253	0.838635
$394$	0	0
$395$	0.836381	0.0420829
$396$	0	0
$397$	−9.38646	−0.471093	−0.235546	−	0.971863i	$-0.575688\pi$
$397$	−9.38646	−0.471093	−0.235546	+	0.971863i	$0.575688\pi$
$398$	0	0
$399$	2.70052	0.135195
$400$	0	0
$401$	14.1016	0.704199	0.352099	−	0.935963i	$-0.385468\pi$
$401$	14.1016	0.704199	0.352099	+	0.935963i	$0.385468\pi$
$402$	0	0
$403$	36.4749	1.81694
$404$	0	0
$405$	3.57452	0.177619
$406$	0	0
$407$	−9.77433	−0.484496
$408$	0	0
$409$	35.1490	1.73801	0.869004	−	0.494805i	$-0.164761\pi$
$409$	35.1490	1.73801	0.869004	+	0.494805i	$0.164761\pi$
$410$	0	0
$411$	−16.3127	−0.804644
$412$	0	0
$413$	−21.4010	−1.05308
$414$	0	0
$415$	7.03761	0.345463
$416$	0	0
$417$	−14.1768	−0.694241
$418$	0	0
$419$	−9.02776	−0.441035	−0.220518	−	0.975383i	$-0.570775\pi$
$419$	−9.02776	−0.441035	−0.220518	+	0.975383i	$0.570775\pi$
$420$	0	0
$421$	15.2097	0.741274	0.370637	−	0.928778i	$-0.379139\pi$
$421$	15.2097	0.741274	0.370637	+	0.928778i	$0.379139\pi$
$422$	0	0
$423$	7.87399	0.382847
$424$	0	0
$425$	−6.31265	−0.306209
$426$	0	0
$427$	37.7743	1.82803
$428$	0	0
$429$	4.77575	0.230575
$430$	0	0
$431$	−16.3127	−0.785753	−0.392876	−	0.919591i	$-0.628520\pi$
$431$	−16.3127	−0.785753	−0.392876	+	0.919591i	$0.628520\pi$
$432$	0	0
$433$	11.1432	0.535506	0.267753	−	0.963488i	$-0.413719\pi$
$433$	11.1432	0.535506	0.267753	+	0.963488i	$0.413719\pi$
$434$	0	0
$435$	2.91160	0.139601
$436$	0	0
$437$	4.96239	0.237383
$438$	0	0
$439$	−27.3865	−1.30708	−0.653542	−	0.756890i	$-0.726718\pi$
$439$	−27.3865	−1.30708	−0.653542	+	0.756890i	$0.726718\pi$
$440$	0	0
$441$	−9.92810	−0.472767
$442$	0	0
$443$	−19.5125	−0.927065	−0.463533	−	0.886080i	$-0.653418\pi$
$443$	−19.5125	−0.927065	−0.463533	+	0.886080i	$0.653418\pi$
$444$	0	0
$445$	7.08840	0.336022
$446$	0	0
$447$	5.98541	0.283100
$448$	0	0
$449$	−22.1016	−1.04304	−0.521519	−	0.853240i	$-0.674634\pi$
$449$	−22.1016	−1.04304	−0.521519	+	0.853240i	$0.674634\pi$
$450$	0	0
$451$	−6.07522	−0.286071
$452$	0	0
$453$	−1.29948	−0.0610547
$454$	0	0
$455$	−20.6253	−0.966929
$456$	0	0
$457$	−17.8496	−0.834967	−0.417483	−	0.908685i	$-0.637088\pi$
$457$	−17.8496	−0.834967	−0.417483	+	0.908685i	$0.637088\pi$
$458$	0	0
$459$	−27.2243	−1.27072
$460$	0	0
$461$	−35.2506	−1.64178	−0.820892	−	0.571083i	$-0.806523\pi$
$461$	−35.2506	−1.64178	−0.820892	+	0.571083i	$0.806523\pi$
$462$	0	0
$463$	26.3634	1.22521	0.612606	−	0.790388i	$-0.290121\pi$
$463$	26.3634	1.22521	0.612606	+	0.790388i	$0.290121\pi$
$464$	0	0
$465$	−4.77575	−0.221470
$466$	0	0
$467$	6.78560	0.314000	0.157000	−	0.987599i	$-0.449818\pi$
$467$	6.78560	0.314000	0.157000	+	0.987599i	$0.449818\pi$
$468$	0	0
$469$	−22.5501	−1.04127
$470$	0	0
$471$	−3.53690	−0.162972
$472$	0	0
$473$	−3.97082	−0.182579
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−4.33312	−0.198400
$478$	0	0
$479$	−12.7104	−0.580752	−0.290376	−	0.956913i	$-0.593780\pi$
$479$	−12.7104	−0.580752	−0.290376	+	0.956913i	$0.593780\pi$
$480$	0	0
$481$	62.5256	2.85092
$482$	0	0
$483$	13.4010	0.609769
$484$	0	0
$485$	10.9927	0.499153
$486$	0	0
$487$	15.7586	0.714090	0.357045	−	0.934087i	$-0.383784\pi$
$487$	15.7586	0.714090	0.357045	+	0.934087i	$0.383784\pi$
$488$	0	0
$489$	−0.523730	−0.0236839
$490$	0	0
$491$	14.5501	0.656636	0.328318	−	0.944567i	$-0.393518\pi$
$491$	14.5501	0.656636	0.328318	+	0.944567i	$0.393518\pi$
$492$	0	0
$493$	22.8021	1.02695
$494$	0	0
$495$	2.26187	0.101663
$496$	0	0
$497$	−2.59895	−0.116579
$498$	0	0
$499$	5.48612	0.245592	0.122796	−	0.992432i	$-0.460814\pi$
$499$	5.48612	0.245592	0.122796	+	0.992432i	$0.460814\pi$
$500$	0	0
$501$	−12.3780	−0.553009
$502$	0	0
$503$	36.6615	1.63466	0.817328	−	0.576173i	$-0.195455\pi$
$503$	36.6615	1.63466	0.817328	+	0.576173i	$0.195455\pi$
$504$	0	0
$505$	−2.64974	−0.117912
$506$	0	0
$507$	−20.0713	−0.891396
$508$	0	0
$509$	39.1900	1.73706	0.868532	−	0.495632i	$-0.165064\pi$
$509$	39.1900	1.73706	0.868532	+	0.495632i	$0.165064\pi$
$510$	0	0
$511$	−1.29948	−0.0574855
$512$	0	0
$513$	4.31265	0.190408
$514$	0	0
$515$	10.7308	0.472857
$516$	0	0
$517$	3.22425	0.141803
$518$	0	0
$519$	−2.61545	−0.114806
$520$	0	0
$521$	−17.7283	−0.776690	−0.388345	−	0.921514i	$-0.626953\pi$
$521$	−17.7283	−0.776690	−0.388345	+	0.921514i	$0.626953\pi$
$522$	0	0
$523$	40.7572	1.78219	0.891094	−	0.453819i	$-0.149939\pi$
$523$	40.7572	1.78219	0.891094	+	0.453819i	$0.149939\pi$
$524$	0	0
$525$	2.70052	0.117861
$526$	0	0
$527$	−37.4010	−1.62922
$528$	0	0
$529$	1.62530	0.0706652
$530$	0	0
$531$	−15.0132	−0.651516
$532$	0	0
$533$	38.8627	1.68333
$534$	0	0
$535$	−4.80606	−0.207784
$536$	0	0
$537$	12.1016	0.522221
$538$	0	0
$539$	−4.06537	−0.175108
$540$	0	0
$541$	23.9003	1.02756	0.513778	−	0.857923i	$-0.328246\pi$
$541$	23.9003	1.02756	0.513778	+	0.857923i	$0.328246\pi$
$542$	0	0
$543$	−7.43533	−0.319081
$544$	0	0
$545$	−2.77575	−0.118900
$546$	0	0
$547$	−8.55405	−0.365745	−0.182872	−	0.983137i	$-0.558539\pi$
$547$	−8.55405	−0.365745	−0.182872	+	0.983137i	$0.558539\pi$
$548$	0	0
$549$	26.4993	1.13096
$550$	0	0
$551$	−3.61213	−0.153882
$552$	0	0
$553$	−2.80209	−0.119157
$554$	0	0
$555$	−8.18664	−0.347504
$556$	0	0
$557$	−4.23743	−0.179546	−0.0897728	−	0.995962i	$-0.528614\pi$
$557$	−4.23743	−0.179546	−0.0897728	+	0.995962i	$0.528614\pi$
$558$	0	0
$559$	25.4010	1.07435
$560$	0	0
$561$	−4.89701	−0.206752
$562$	0	0
$563$	−16.4934	−0.695114	−0.347557	−	0.937659i	$-0.612989\pi$
$563$	−16.4934	−0.695114	−0.347557	+	0.937659i	$0.612989\pi$
$564$	0	0
$565$	6.99271	0.294185
$566$	0	0
$567$	−11.9756	−0.502926
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	26.2619	1.09902	0.549512	−	0.835486i	$-0.314814\pi$
$571$	26.2619	1.09902	0.549512	+	0.835486i	$0.314814\pi$
$572$	0	0
$573$	−17.5515	−0.733224
$574$	0	0
$575$	4.96239	0.206946
$576$	0	0
$577$	30.1016	1.25314	0.626572	−	0.779363i	$-0.284457\pi$
$577$	30.1016	1.25314	0.626572	+	0.779363i	$0.284457\pi$
$578$	0	0
$579$	−10.1114	−0.420216
$580$	0	0
$581$	−23.5778	−0.978174
$582$	0	0
$583$	−1.77433	−0.0734853
$584$	0	0
$585$	−14.4690	−0.598219
$586$	0	0
$587$	35.1392	1.45035	0.725175	−	0.688565i	$-0.241759\pi$
$587$	35.1392	1.45035	0.725175	+	0.688565i	$0.241759\pi$
$588$	0	0
$589$	5.92478	0.244126
$590$	0	0
$591$	−19.7889	−0.814007
$592$	0	0
$593$	34.3244	1.40953	0.704767	−	0.709439i	$-0.251051\pi$
$593$	34.3244	1.40953	0.704767	+	0.709439i	$0.251051\pi$
$594$	0	0
$595$	21.1490	0.867026
$596$	0	0
$597$	18.5990	0.761204
$598$	0	0
$599$	−14.6107	−0.596978	−0.298489	−	0.954413i	$-0.596483\pi$
$599$	−14.6107	−0.596978	−0.298489	+	0.954413i	$0.596483\pi$
$600$	0	0
$601$	23.5633	0.961165	0.480583	−	0.876949i	$-0.340425\pi$
$601$	23.5633	0.961165	0.480583	+	0.876949i	$0.340425\pi$
$602$	0	0
$603$	−15.8192	−0.644209
$604$	0	0
$605$	−10.0738	−0.409559
$606$	0	0
$607$	8.80606	0.357427	0.178714	−	0.983901i	$-0.442806\pi$
$607$	8.80606	0.357427	0.178714	+	0.983901i	$0.442806\pi$
$608$	0	0
$609$	−9.75463	−0.395278
$610$	0	0
$611$	−20.6253	−0.834410
$612$	0	0
$613$	10.4142	0.420626	0.210313	−	0.977634i	$-0.432552\pi$
$613$	10.4142	0.420626	0.210313	+	0.977634i	$0.432552\pi$
$614$	0	0
$615$	−5.08840	−0.205184
$616$	0	0
$617$	−17.2849	−0.695863	−0.347932	−	0.937520i	$-0.613116\pi$
$617$	−17.2849	−0.695863	−0.347932	+	0.937520i	$0.613116\pi$
$618$	0	0
$619$	10.6351	0.427463	0.213731	−	0.976892i	$-0.431438\pi$
$619$	10.6351	0.427463	0.213731	+	0.976892i	$0.431438\pi$
$620$	0	0
$621$	21.4010	0.858794
$622$	0	0
$623$	−23.7480	−0.951443
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.775746	0.0309803
$628$	0	0
$629$	−64.1133	−2.55637
$630$	0	0
$631$	16.5599	0.659240	0.329620	−	0.944114i	$-0.393079\pi$
$631$	16.5599	0.659240	0.329620	+	0.944114i	$0.393079\pi$
$632$	0	0
$633$	−16.8773	−0.670813
$634$	0	0
$635$	−13.4314	−0.533007
$636$	0	0
$637$	26.0059	1.03039
$638$	0	0
$639$	−1.82321	−0.0721249
$640$	0	0
$641$	−16.7612	−0.662026	−0.331013	−	0.943626i	$-0.607390\pi$
$641$	−16.7612	−0.662026	−0.331013	+	0.943626i	$0.607390\pi$
$642$	0	0
$643$	−5.73813	−0.226290	−0.113145	−	0.993578i	$-0.536092\pi$
$643$	−5.73813	−0.226290	−0.113145	+	0.993578i	$0.536092\pi$
$644$	0	0
$645$	−3.32582	−0.130954
$646$	0	0
$647$	37.2144	1.46305	0.731525	−	0.681815i	$-0.238809\pi$
$647$	37.2144	1.46305	0.731525	+	0.681815i	$0.238809\pi$
$648$	0	0
$649$	−6.14762	−0.241315
$650$	0	0
$651$	16.0000	0.627089
$652$	0	0
$653$	11.7626	0.460305	0.230153	−	0.973155i	$-0.426077\pi$
$653$	11.7626	0.460305	0.230153	+	0.973155i	$0.426077\pi$
$654$	0	0
$655$	−20.6253	−0.805897
$656$	0	0
$657$	−0.911603	−0.0355650
$658$	0	0
$659$	−1.23884	−0.0482584	−0.0241292	−	0.999709i	$-0.507681\pi$
$659$	−1.23884	−0.0482584	−0.0241292	+	0.999709i	$0.507681\pi$
$660$	0	0
$661$	−9.53690	−0.370943	−0.185471	−	0.982650i	$-0.559381\pi$
$661$	−9.53690	−0.370943	−0.185471	+	0.982650i	$0.559381\pi$
$662$	0	0
$663$	31.3258	1.21659
$664$	0	0
$665$	−3.35026	−0.129918
$666$	0	0
$667$	−17.9248	−0.694050
$668$	0	0
$669$	−0.0244376	−0.000944811 0
$670$	0	0
$671$	10.8510	0.418897
$672$	0	0
$673$	39.9307	1.53921	0.769607	−	0.638518i	$-0.220452\pi$
$673$	39.9307	1.53921	0.769607	+	0.638518i	$0.220452\pi$
$674$	0	0
$675$	4.31265	0.165994
$676$	0	0
$677$	−3.05334	−0.117349	−0.0586747	−	0.998277i	$-0.518687\pi$
$677$	−3.05334	−0.117349	−0.0586747	+	0.998277i	$0.518687\pi$
$678$	0	0
$679$	−36.8284	−1.41335
$680$	0	0
$681$	3.87399	0.148452
$682$	0	0
$683$	−29.2692	−1.11995	−0.559977	−	0.828508i	$-0.689190\pi$
$683$	−29.2692	−1.11995	−0.559977	+	0.828508i	$0.689190\pi$
$684$	0	0
$685$	20.2374	0.773232
$686$	0	0
$687$	−1.51056	−0.0576313
$688$	0	0
$689$	11.3503	0.432411
$690$	0	0
$691$	−2.63515	−0.100246	−0.0501229	−	0.998743i	$-0.515961\pi$
$691$	−2.63515	−0.100246	−0.0501229	+	0.998743i	$0.515961\pi$
$692$	0	0
$693$	−7.57784	−0.287858
$694$	0	0
$695$	17.5877	0.667139
$696$	0	0
$697$	−39.8496	−1.50941
$698$	0	0
$699$	8.98683	0.339913
$700$	0	0
$701$	−25.0494	−0.946102	−0.473051	−	0.881035i	$-0.656847\pi$
$701$	−25.0494	−0.946102	−0.473051	+	0.881035i	$0.656847\pi$
$702$	0	0
$703$	10.1563	0.383053
$704$	0	0
$705$	2.70052	0.101708
$706$	0	0
$707$	8.87732	0.333866
$708$	0	0
$709$	−41.6991	−1.56604	−0.783021	−	0.621995i	$-0.786323\pi$
$709$	−41.6991	−1.56604	−0.783021	+	0.621995i	$0.786323\pi$
$710$	0	0
$711$	−1.96571	−0.0737200
$712$	0	0
$713$	29.4010	1.10108
$714$	0	0
$715$	−5.92478	−0.221574
$716$	0	0
$717$	7.49597	0.279942
$718$	0	0
$719$	−30.6351	−1.14250	−0.571249	−	0.820777i	$-0.693541\pi$
$719$	−30.6351	−1.14250	−0.571249	+	0.820777i	$0.693541\pi$
$720$	0	0
$721$	−35.9511	−1.33889
$722$	0	0
$723$	1.86414	0.0693282
$724$	0	0
$725$	−3.61213	−0.134151
$726$	0	0
$727$	−7.50071	−0.278186	−0.139093	−	0.990279i	$-0.544419\pi$
$727$	−7.50071	−0.278186	−0.139093	+	0.990279i	$0.544419\pi$
$728$	0	0
$729$	2.02776	0.0751023
$730$	0	0
$731$	−26.0460	−0.963348
$732$	0	0
$733$	9.84955	0.363802	0.181901	−	0.983317i	$-0.441775\pi$
$733$	9.84955	0.363802	0.181901	+	0.983317i	$0.441775\pi$
$734$	0	0
$735$	−3.40502	−0.125596
$736$	0	0
$737$	−6.47768	−0.238609
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−4.96239	−0.182298
$742$	0	0
$743$	15.7177	0.576625	0.288313	−	0.957536i	$-0.406906\pi$
$743$	15.7177	0.576625	0.288313	+	0.957536i	$0.406906\pi$
$744$	0	0
$745$	−7.42548	−0.272049
$746$	0	0
$747$	−16.5402	−0.605175
$748$	0	0
$749$	16.1016	0.588339
$750$	0	0
$751$	43.7400	1.59610	0.798048	−	0.602593i	$-0.205866\pi$
$751$	43.7400	1.59610	0.798048	+	0.602593i	$0.205866\pi$
$752$	0	0
$753$	19.4274	0.707974
$754$	0	0
$755$	1.61213	0.0586713
$756$	0	0
$757$	−15.7743	−0.573328	−0.286664	−	0.958031i	$-0.592546\pi$
$757$	−15.7743	−0.573328	−0.286664	+	0.958031i	$0.592546\pi$
$758$	0	0
$759$	3.84955	0.139730
$760$	0	0
$761$	43.2262	1.56695	0.783474	−	0.621425i	$-0.213446\pi$
$761$	43.2262	1.56695	0.783474	+	0.621425i	$0.213446\pi$
$762$	0	0
$763$	9.29948	0.336664
$764$	0	0
$765$	14.8364	0.536410
$766$	0	0
$767$	39.3258	1.41997
$768$	0	0
$769$	−19.1246	−0.689650	−0.344825	−	0.938667i	$-0.612062\pi$
$769$	−19.1246	−0.689650	−0.344825	+	0.938667i	$0.612062\pi$
$770$	0	0
$771$	10.7367	0.386674
$772$	0	0
$773$	25.8846	0.931005	0.465502	−	0.885047i	$-0.345874\pi$
$773$	25.8846	0.931005	0.465502	+	0.885047i	$0.345874\pi$
$774$	0	0
$775$	5.92478	0.212824
$776$	0	0
$777$	27.4274	0.983952
$778$	0	0
$779$	6.31265	0.226174
$780$	0	0
$781$	−0.746569	−0.0267144
$782$	0	0
$783$	−15.5778	−0.556707
$784$	0	0
$785$	4.38787	0.156610
$786$	0	0
$787$	−19.9814	−0.712261	−0.356131	−	0.934436i	$-0.615904\pi$
$787$	−19.9814	−0.712261	−0.356131	+	0.934436i	$0.615904\pi$
$788$	0	0
$789$	10.4485	0.371977
$790$	0	0
$791$	−23.4274	−0.832982
$792$	0	0
$793$	−69.4128	−2.46492
$794$	0	0
$795$	−1.48612	−0.0527072
$796$	0	0
$797$	28.6458	1.01469	0.507343	−	0.861744i	$-0.330628\pi$
$797$	28.6458	1.01469	0.507343	+	0.861744i	$0.330628\pi$
$798$	0	0
$799$	21.1490	0.748199
$800$	0	0
$801$	−16.6596	−0.588638
$802$	0	0
$803$	−0.373285	−0.0131729
$804$	0	0
$805$	−16.6253	−0.585965
$806$	0	0
$807$	9.18997	0.323502
$808$	0	0
$809$	−17.2243	−0.605573	−0.302786	−	0.953058i	$-0.597917\pi$
$809$	−17.2243	−0.605573	−0.302786	+	0.953058i	$0.597917\pi$
$810$	0	0
$811$	15.6267	0.548728	0.274364	−	0.961626i	$-0.411533\pi$
$811$	15.6267	0.548728	0.274364	+	0.961626i	$0.411533\pi$
$812$	0	0
$813$	13.5515	0.475272
$814$	0	0
$815$	0.649738	0.0227593
$816$	0	0
$817$	4.12601	0.144351
$818$	0	0
$819$	48.4749	1.69385
$820$	0	0
$821$	39.2506	1.36986	0.684928	−	0.728611i	$-0.259834\pi$
$821$	39.2506	1.36986	0.684928	+	0.728611i	$0.259834\pi$
$822$	0	0
$823$	45.5271	1.58697	0.793487	−	0.608588i	$-0.208264\pi$
$823$	45.5271	1.58697	0.793487	+	0.608588i	$0.208264\pi$
$824$	0	0
$825$	0.775746	0.0270080
$826$	0	0
$827$	−17.0698	−0.593576	−0.296788	−	0.954943i	$-0.595916\pi$
$827$	−17.0698	−0.593576	−0.296788	+	0.954943i	$0.595916\pi$
$828$	0	0
$829$	1.69911	0.0590125	0.0295062	−	0.999565i	$-0.490607\pi$
$829$	1.69911	0.0590125	0.0295062	+	0.999565i	$0.490607\pi$
$830$	0	0
$831$	24.0118	0.832959
$832$	0	0
$833$	−26.6662	−0.923930
$834$	0	0
$835$	15.3561	0.531421
$836$	0	0
$837$	25.5515	0.883189
$838$	0	0
$839$	50.5910	1.74660	0.873298	−	0.487187i	$-0.161977\pi$
$839$	50.5910	1.74660	0.873298	+	0.487187i	$0.161977\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	0	0
$843$	9.36011	0.322379
$844$	0	0
$845$	24.9003	0.856598
$846$	0	0
$847$	33.7499	1.15966
$848$	0	0
$849$	1.82321	0.0625723
$850$	0	0
$851$	50.3996	1.72768
$852$	0	0
$853$	−22.5237	−0.771198	−0.385599	−	0.922667i	$-0.626005\pi$
$853$	−22.5237	−0.771198	−0.385599	+	0.922667i	$0.626005\pi$
$854$	0	0
$855$	−2.35026	−0.0803773
$856$	0	0
$857$	−23.6180	−0.806776	−0.403388	−	0.915029i	$-0.632167\pi$
$857$	−23.6180	−0.806776	−0.403388	+	0.915029i	$0.632167\pi$
$858$	0	0
$859$	−15.1754	−0.517777	−0.258889	−	0.965907i	$-0.583356\pi$
$859$	−15.1754	−0.517777	−0.258889	+	0.965907i	$0.583356\pi$
$860$	0	0
$861$	17.0475	0.580976
$862$	0	0
$863$	−30.1055	−1.02480	−0.512402	−	0.858746i	$-0.671244\pi$
$863$	−30.1055	−1.02480	−0.512402	+	0.858746i	$0.671244\pi$
$864$	0	0
$865$	3.24472	0.110324
$866$	0	0
$867$	−18.4182	−0.625515
$868$	0	0
$869$	−0.804923	−0.0273051
$870$	0	0
$871$	41.4372	1.40405
$872$	0	0
$873$	−25.8357	−0.874407
$874$	0	0
$875$	−3.35026	−0.113260
$876$	0	0
$877$	−5.53102	−0.186769	−0.0933847	−	0.995630i	$-0.529769\pi$
$877$	−5.53102	−0.186769	−0.0933847	+	0.995630i	$0.529769\pi$
$878$	0	0
$879$	−1.48612	−0.0501255
$880$	0	0
$881$	20.8265	0.701664	0.350832	−	0.936438i	$-0.385899\pi$
$881$	20.8265	0.701664	0.350832	+	0.936438i	$0.385899\pi$
$882$	0	0
$883$	−43.1509	−1.45214	−0.726072	−	0.687618i	$-0.758656\pi$
$883$	−43.1509	−1.45214	−0.726072	+	0.687618i	$0.758656\pi$
$884$	0	0
$885$	−5.14903	−0.173083
$886$	0	0
$887$	44.5461	1.49571	0.747856	−	0.663861i	$-0.231083\pi$
$887$	44.5461	1.49571	0.747856	+	0.663861i	$0.231083\pi$
$888$	0	0
$889$	44.9986	1.50920
$890$	0	0
$891$	−3.44007	−0.115247
$892$	0	0
$893$	−3.35026	−0.112112
$894$	0	0
$895$	−15.0132	−0.501835
$896$	0	0
$897$	−24.6253	−0.822215
$898$	0	0
$899$	−21.4010	−0.713765
$900$	0	0
$901$	−11.6385	−0.387734
$902$	0	0
$903$	11.1424	0.370795
$904$	0	0
$905$	9.22425	0.306625
$906$	0	0
$907$	−53.7558	−1.78493	−0.892466	−	0.451115i	$-0.851026\pi$
$907$	−53.7558	−1.78493	−0.892466	+	0.451115i	$0.851026\pi$
$908$	0	0
$909$	6.22758	0.206556
$910$	0	0
$911$	2.28630	0.0757486	0.0378743	−	0.999283i	$-0.487941\pi$
$911$	2.28630	0.0757486	0.0378743	+	0.999283i	$0.487941\pi$
$912$	0	0
$913$	−6.77292	−0.224151
$914$	0	0
$915$	9.08840	0.300453
$916$	0	0
$917$	69.1002	2.28189
$918$	0	0
$919$	34.8510	1.14963	0.574814	−	0.818284i	$-0.305075\pi$
$919$	34.8510	1.14963	0.574814	+	0.818284i	$0.305075\pi$
$920$	0	0
$921$	21.1246	0.696079
$922$	0	0
$923$	4.77575	0.157196
$924$	0	0
$925$	10.1563	0.333938
$926$	0	0
$927$	−25.2203	−0.828343
$928$	0	0
$929$	41.6991	1.36810	0.684052	−	0.729434i	$-0.260216\pi$
$929$	41.6991	1.36810	0.684052	+	0.729434i	$0.260216\pi$
$930$	0	0
$931$	4.22425	0.138444
$932$	0	0
$933$	5.25060	0.171897
$934$	0	0
$935$	6.07522	0.198681
$936$	0	0
$937$	21.9102	0.715775	0.357887	−	0.933765i	$-0.383497\pi$
$937$	21.9102	0.715775	0.357887	+	0.933765i	$0.383497\pi$
$938$	0	0
$939$	−12.9576	−0.422857
$940$	0	0
$941$	−3.55149	−0.115775	−0.0578877	−	0.998323i	$-0.518437\pi$
$941$	−3.55149	−0.115775	−0.0578877	+	0.998323i	$0.518437\pi$
$942$	0	0
$943$	31.3258	1.02011
$944$	0	0
$945$	−14.4485	−0.470010
$946$	0	0
$947$	−38.3634	−1.24664	−0.623322	−	0.781965i	$-0.714217\pi$
$947$	−38.3634	−1.24664	−0.623322	+	0.781965i	$0.714217\pi$
$948$	0	0
$949$	2.38787	0.0775136
$950$	0	0
$951$	4.58910	0.148812
$952$	0	0
$953$	−22.8714	−0.740879	−0.370439	−	0.928857i	$-0.620793\pi$
$953$	−22.8714	−0.740879	−0.370439	+	0.928857i	$0.620793\pi$
$954$	0	0
$955$	21.7743	0.704601
$956$	0	0
$957$	−2.80209	−0.0905788
$958$	0	0
$959$	−67.8007	−2.18940
$960$	0	0
$961$	4.10299	0.132354
$962$	0	0
$963$	11.2955	0.363993
$964$	0	0
$965$	12.5442	0.403812
$966$	0	0
$967$	−27.6629	−0.889579	−0.444790	−	0.895635i	$-0.646722\pi$
$967$	−27.6629	−0.889579	−0.444790	+	0.895635i	$0.646722\pi$
$968$	0	0
$969$	5.08840	0.163463
$970$	0	0
$971$	42.1768	1.35352	0.676759	−	0.736205i	$-0.263384\pi$
$971$	42.1768	1.35352	0.676759	+	0.736205i	$0.263384\pi$
$972$	0	0
$973$	−58.9234	−1.88900
$974$	0	0
$975$	−4.96239	−0.158924
$976$	0	0
$977$	0.856849	0.0274130	0.0137065	−	0.999906i	$-0.495637\pi$
$977$	0.856849	0.0274130	0.0137065	+	0.999906i	$0.495637\pi$
$978$	0	0
$979$	−6.82179	−0.218025
$980$	0	0
$981$	6.52373	0.208287
$982$	0	0
$983$	−45.2809	−1.44424	−0.722119	−	0.691769i	$-0.756831\pi$
$983$	−45.2809	−1.44424	−0.722119	+	0.691769i	$0.756831\pi$
$984$	0	0
$985$	24.5501	0.782231
$986$	0	0
$987$	−9.04746	−0.287984
$988$	0	0
$989$	20.4749	0.651063
$990$	0	0
$991$	−47.0132	−1.49342	−0.746711	−	0.665148i	$-0.768368\pi$
$991$	−47.0132	−1.49342	−0.746711	+	0.665148i	$0.768368\pi$
$992$	0	0
$993$	−9.92478	−0.314953
$994$	0	0
$995$	−23.0738	−0.731489
$996$	0	0
$997$	13.6873	0.433483	0.216741	−	0.976229i	$-0.430457\pi$
$997$	13.6873	0.433483	0.216741	+	0.976229i	$0.430457\pi$
$998$	0	0
$999$	43.8007	1.38579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.a.p.1.2		3
4.3	odd	2		95.2.a.a.1.1	✓	3
5.4	even	2		7600.2.a.bx.1.2		3
8.3	odd	2		6080.2.a.bo.1.2		3
8.5	even	2		6080.2.a.by.1.2		3
12.11	even	2		855.2.a.i.1.3		3
20.3	even	4		475.2.b.d.324.5		6
20.7	even	4		475.2.b.d.324.2		6
20.19	odd	2		475.2.a.f.1.3		3
28.27	even	2		4655.2.a.u.1.1		3
60.59	even	2		4275.2.a.bk.1.1		3
76.75	even	2		1805.2.a.f.1.3		3
380.379	even	2		9025.2.a.bb.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.1	✓	3	4.3	odd	2
475.2.a.f.1.3		3	20.19	odd	2
475.2.b.d.324.2		6	20.7	even	4
475.2.b.d.324.5		6	20.3	even	4
855.2.a.i.1.3		3	12.11	even	2
1520.2.a.p.1.2		3	1.1	even	1	trivial
1805.2.a.f.1.3		3	76.75	even	2
4275.2.a.bk.1.1		3	60.59	even	2
4655.2.a.u.1.1		3	28.27	even	2
6080.2.a.bo.1.2		3	8.3	odd	2
6080.2.a.by.1.2		3	8.5	even	2
7600.2.a.bx.1.2		3	5.4	even	2
9025.2.a.bb.1.1		3	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)

\(f(q)\)	\(=\)	\(q-0.806063 q^{3} +1.00000 q^{5} -3.35026 q^{7} -2.35026 q^{9} +O(q^{10})\)	\(q-0.806063 q^{3} +1.00000 q^{5} -3.35026 q^{7} -2.35026 q^{9} -0.962389 q^{11} +6.15633 q^{13} -0.806063 q^{15} -6.31265 q^{17} +1.00000 q^{19} +2.70052 q^{21} +4.96239 q^{23} +1.00000 q^{25} +4.31265 q^{27} -3.61213 q^{29} +5.92478 q^{31} +0.775746 q^{33} -3.35026 q^{35} +10.1563 q^{37} -4.96239 q^{39} +6.31265 q^{41} +4.12601 q^{43} -2.35026 q^{45} -3.35026 q^{47} +4.22425 q^{49} +5.08840 q^{51} +1.84367 q^{53} -0.962389 q^{55} -0.806063 q^{57} +6.38787 q^{59} -11.2750 q^{61} +7.87399 q^{63} +6.15633 q^{65} +6.73084 q^{67} -4.00000 q^{69} +0.775746 q^{71} +0.387873 q^{73} -0.806063 q^{75} +3.22425 q^{77} +0.836381 q^{79} +3.57452 q^{81} +7.03761 q^{83} -6.31265 q^{85} +2.91160 q^{87} +7.08840 q^{89} -20.6253 q^{91} -4.77575 q^{93} +1.00000 q^{95} +10.9927 q^{97} +2.26187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9	\( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{9} + 8 q^{11} + 8 q^{13} - 2 q^{15} + 2 q^{17} + 3 q^{19} - 12 q^{21} + 4 q^{23} + 3 q^{25} - 8 q^{27} - 10 q^{29} - 4 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{39} - 2 q^{41} + 4 q^{43} + 3 q^{45} + 11 q^{49} - 4 q^{51} + 16 q^{53} + 8 q^{55} - 2 q^{57} + 20 q^{59} - 2 q^{61} + 32 q^{63} + 8 q^{65} - 2 q^{67} - 12 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 8 q^{77} - q^{81} + 32 q^{83} + 2 q^{85} + 28 q^{87} + 2 q^{89} - 20 q^{91} - 16 q^{93} + 3 q^{95} + 20 q^{97} + 16 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^9 + 8 * q^11 + 8 * q^13 - 2 * q^15 + 2 * q^17 + 3 * q^19 - 12 * q^21 + 4 * q^23 + 3 * q^25 - 8 * q^27 - 10 * q^29 - 4 * q^31 + 4 * q^33 + 20 * q^37 - 4 * q^39 - 2 * q^41 + 4 * q^43 + 3 * q^45 + 11 * q^49 - 4 * q^51 + 16 * q^53 + 8 * q^55 - 2 * q^57 + 20 * q^59 - 2 * q^61 + 32 * q^63 + 8 * q^65 - 2 * q^67 - 12 * q^69 + 4 * q^71 + 2 * q^73 - 2 * q^75 + 8 * q^77 - q^81 + 32 * q^83 + 2 * q^85 + 28 * q^87 + 2 * q^89 - 20 * q^91 - 16 * q^93 + 3 * q^95 + 20 * q^97 + 16 * q^99

Introduction

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