LMFDB - Embedded newform 3040.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,2,Mod(1,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, 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("3040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.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:	$24.2745222145$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
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			1.4
Root			$-2.10710$ of defining polynomial
Character	$\chi$	$=$	3040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.10710 q^{3} +1.00000 q^{5} -2.43986 q^{7} +1.43986 q^{9} +1.36667 q^{11} -5.47377 q^{13} +2.10710 q^{15} -3.28738 q^{17} -1.00000 q^{19} -5.14101 q^{21} -3.41378 q^{23} +1.00000 q^{25} -3.28738 q^{27} -0.926817 q^{29} -11.0939 q^{31} +2.87971 q^{33} -2.43986 q^{35} -6.90751 q^{37} -11.5338 q^{39} +4.21419 q^{41} -0.486962 q^{43} +1.43986 q^{45} +10.4606 q^{47} -1.04711 q^{49} -6.92682 q^{51} +13.2332 q^{53} +1.36667 q^{55} -2.10710 q^{57} -10.0207 q^{59} -9.48665 q^{61} -3.51304 q^{63} -5.47377 q^{65} +4.01288 q^{67} -7.19316 q^{69} +4.94754 q^{71} +13.3552 q^{73} +2.10710 q^{75} -3.33448 q^{77} -6.97392 q^{79} -11.2464 q^{81} -3.51304 q^{83} -3.28738 q^{85} -1.95289 q^{87} +8.06783 q^{89} +13.3552 q^{91} -23.3759 q^{93} -1.00000 q^{95} -0.546952 q^{97} +1.96781 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9} - 6 q^{11} - q^{13} - q^{15} - q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 16 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} - 13 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 - 5 * q^7 + q^9 - 6 * q^11 - q^13 - q^15 - q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - q^27 + 3 * q^29 - 16 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 - 13 * q^39 - 2 * q^41 + q^45 + 2 * q^47 - 7 * q^49 - 21 * q^51 + 13 * q^53 - 6 * q^55 + q^57 - 5 * q^59 - 2 * q^61 - 16 * q^63 - q^65 + q^67 - 11 * q^69 - 22 * q^71 + 9 * q^73 - q^75 - 4 * q^77 - 24 * q^79 - 24 * q^81 - 16 * q^83 - q^85 - 5 * q^87 + 9 * q^91 - 14 * q^93 - 4 * q^95 + 12 * q^97 + 10 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.10710	1.21653	0.608266	−	0.793733i	$-0.291865\pi$
$3$	2.10710	1.21653	0.608266	+	0.793733i	$0.291865\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.43986	−0.922179	−0.461089	−	0.887354i	$-0.652541\pi$
$7$	−2.43986	−0.922179	−0.461089	+	0.887354i	$0.652541\pi$
$8$	0	0
$9$	1.43986	0.479952
$10$	0	0
$11$	1.36667	0.412067	0.206034	−	0.978545i	$-0.433944\pi$
$11$	1.36667	0.412067	0.206034	+	0.978545i	$0.433944\pi$
$12$	0	0
$13$	−5.47377	−1.51815	−0.759075	−	0.651003i	$-0.774349\pi$
$13$	−5.47377	−1.51815	−0.759075	+	0.651003i	$0.774349\pi$
$14$	0	0
$15$	2.10710	0.544050
$16$	0	0
$17$	−3.28738	−0.797306	−0.398653	−	0.917102i	$-0.630522\pi$
$17$	−3.28738	−0.797306	−0.398653	+	0.917102i	$0.630522\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−5.14101	−1.12186
$22$	0	0
$23$	−3.41378	−0.711822	−0.355911	−	0.934520i	$-0.615829\pi$
$23$	−3.41378	−0.711822	−0.355911	+	0.934520i	$0.615829\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.28738	−0.632656
$28$	0	0
$29$	−0.926817	−0.172106	−0.0860528	−	0.996291i	$-0.527425\pi$
$29$	−0.926817	−0.172106	−0.0860528	+	0.996291i	$0.527425\pi$
$30$	0	0
$31$	−11.0939	−1.99252	−0.996262	−	0.0863841i	$-0.972469\pi$
$31$	−11.0939	−1.99252	−0.996262	+	0.0863841i	$0.972469\pi$
$32$	0	0
$33$	2.87971	0.501293
$34$	0	0
$35$	−2.43986	−0.412411
$36$	0	0
$37$	−6.90751	−1.13559	−0.567794	−	0.823171i	$-0.692203\pi$
$37$	−6.90751	−1.13559	−0.567794	+	0.823171i	$0.692203\pi$
$38$	0	0
$39$	−11.5338	−1.84688
$40$	0	0
$41$	4.21419	0.658146	0.329073	−	0.944304i	$-0.393264\pi$
$41$	4.21419	0.658146	0.329073	+	0.944304i	$0.393264\pi$
$42$	0	0
$43$	−0.486962	−0.0742610	−0.0371305	−	0.999310i	$-0.511822\pi$
$43$	−0.486962	−0.0742610	−0.0371305	+	0.999310i	$0.511822\pi$
$44$	0	0
$45$	1.43986	0.214641
$46$	0	0
$47$	10.4606	1.52583	0.762916	−	0.646498i	$-0.223767\pi$
$47$	10.4606	1.52583	0.762916	+	0.646498i	$0.223767\pi$
$48$	0	0
$49$	−1.04711	−0.149587
$50$	0	0
$51$	−6.92682	−0.969948
$52$	0	0
$53$	13.2332	1.81772	0.908859	−	0.417103i	$-0.136955\pi$
$53$	13.2332	1.81772	0.908859	+	0.417103i	$0.136955\pi$
$54$	0	0
$55$	1.36667	0.184282
$56$	0	0
$57$	−2.10710	−0.279092
$58$	0	0
$59$	−10.0207	−1.30459	−0.652293	−	0.757967i	$-0.726193\pi$
$59$	−10.0207	−1.30459	−0.652293	+	0.757967i	$0.726193\pi$
$60$	0	0
$61$	−9.48665	−1.21464	−0.607321	−	0.794457i	$-0.707756\pi$
$61$	−9.48665	−1.21464	−0.607321	+	0.794457i	$0.707756\pi$
$62$	0	0
$63$	−3.51304	−0.442601
$64$	0	0
$65$	−5.47377	−0.678937
$66$	0	0
$67$	4.01288	0.490252	0.245126	−	0.969491i	$-0.421171\pi$
$67$	4.01288	0.490252	0.245126	+	0.969491i	$0.421171\pi$
$68$	0	0
$69$	−7.19316	−0.865955
$70$	0	0
$71$	4.94754	0.587165	0.293582	−	0.955934i	$-0.405153\pi$
$71$	4.94754	0.587165	0.293582	+	0.955934i	$0.405153\pi$
$72$	0	0
$73$	13.3552	1.56311	0.781554	−	0.623837i	$-0.214427\pi$
$73$	13.3552	1.56311	0.781554	+	0.623837i	$0.214427\pi$
$74$	0	0
$75$	2.10710	0.243307
$76$	0	0
$77$	−3.33448	−0.380000
$78$	0	0
$79$	−6.97392	−0.784628	−0.392314	−	0.919831i	$-0.628325\pi$
$79$	−6.97392	−0.784628	−0.392314	+	0.919831i	$0.628325\pi$
$80$	0	0
$81$	−11.2464	−1.24960
$82$	0	0
$83$	−3.51304	−0.385606	−0.192803	−	0.981237i	$-0.561758\pi$
$83$	−3.51304	−0.385606	−0.192803	+	0.981237i	$0.561758\pi$
$84$	0	0
$85$	−3.28738	−0.356566
$86$	0	0
$87$	−1.95289	−0.209372
$88$	0	0
$89$	8.06783	0.855188	0.427594	−	0.903971i	$-0.359361\pi$
$89$	8.06783	0.855188	0.427594	+	0.903971i	$0.359361\pi$
$90$	0	0
$91$	13.3552	1.40001
$92$	0	0
$93$	−23.3759	−2.42397
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−0.546952	−0.0555345	−0.0277673	−	0.999614i	$-0.508840\pi$
$97$	−0.546952	−0.0555345	−0.0277673	+	0.999614i	$0.508840\pi$
$98$	0	0
$99$	1.96781	0.197772
$100$	0	0
$101$	−14.7690	−1.46957	−0.734784	−	0.678301i	$-0.762717\pi$
$101$	−14.7690	−1.46957	−0.734784	+	0.678301i	$0.762717\pi$
$102$	0	0
$103$	−5.18028	−0.510428	−0.255214	−	0.966885i	$-0.582146\pi$
$103$	−5.18028	−0.510428	−0.255214	+	0.966885i	$0.582146\pi$
$104$	0	0
$105$	−5.14101	−0.501711
$106$	0	0
$107$	1.74654	0.168844	0.0844221	−	0.996430i	$-0.473096\pi$
$107$	1.74654	0.168844	0.0844221	+	0.996430i	$0.473096\pi$
$108$	0	0
$109$	8.83605	0.846340	0.423170	−	0.906050i	$-0.360917\pi$
$109$	8.83605	0.846340	0.423170	+	0.906050i	$0.360917\pi$
$110$	0	0
$111$	−14.5548	−1.38148
$112$	0	0
$113$	19.3481	1.82012	0.910059	−	0.414478i	$-0.136036\pi$
$113$	19.3481	1.82012	0.910059	+	0.414478i	$0.136036\pi$
$114$	0	0
$115$	−3.41378	−0.318337
$116$	0	0
$117$	−7.88143	−0.728639
$118$	0	0
$119$	8.02072	0.735258
$120$	0	0
$121$	−9.13221	−0.830201
$122$	0	0
$123$	8.87971	0.800657
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.27418	0.734215	0.367107	−	0.930179i	$-0.380348\pi$
$127$	8.27418	0.734215	0.367107	+	0.930179i	$0.380348\pi$
$128$	0	0
$129$	−1.02608	−0.0903409
$130$	0	0
$131$	−8.30841	−0.725909	−0.362954	−	0.931807i	$-0.618232\pi$
$131$	−8.30841	−0.725909	−0.362954	+	0.931807i	$0.618232\pi$
$132$	0	0
$133$	2.43986	0.211562
$134$	0	0
$135$	−3.28738	−0.282932
$136$	0	0
$137$	0.459817	0.0392848	0.0196424	−	0.999807i	$-0.493747\pi$
$137$	0.459817	0.0392848	0.0196424	+	0.999807i	$0.493747\pi$
$138$	0	0
$139$	−3.12609	−0.265152	−0.132576	−	0.991173i	$-0.542325\pi$
$139$	−3.12609	−0.265152	−0.132576	+	0.991173i	$0.542325\pi$
$140$	0	0
$141$	22.0414	1.85622
$142$	0	0
$143$	−7.48085	−0.625580
$144$	0	0
$145$	−0.926817	−0.0769680
$146$	0	0
$147$	−2.20636	−0.181977
$148$	0	0
$149$	15.1817	1.24373	0.621866	−	0.783123i	$-0.286375\pi$
$149$	15.1817	1.24373	0.621866	+	0.783123i	$0.286375\pi$
$150$	0	0
$151$	−22.9475	−1.86744	−0.933722	−	0.357999i	$-0.883459\pi$
$151$	−22.9475	−1.86744	−0.933722	+	0.357999i	$0.883459\pi$
$152$	0	0
$153$	−4.73334	−0.382668
$154$	0	0
$155$	−11.0939	−0.891084
$156$	0	0
$157$	−2.97392	−0.237345	−0.118672	−	0.992933i	$-0.537864\pi$
$157$	−2.97392	−0.237345	−0.118672	+	0.992933i	$0.537864\pi$
$158$	0	0
$159$	27.8836	2.21131
$160$	0	0
$161$	8.32913	0.656427
$162$	0	0
$163$	8.19423	0.641822	0.320911	−	0.947109i	$-0.396011\pi$
$163$	8.19423	0.641822	0.320911	+	0.947109i	$0.396011\pi$
$164$	0	0
$165$	2.87971	0.224185
$166$	0	0
$167$	−13.3125	−1.03015	−0.515076	−	0.857145i	$-0.672236\pi$
$167$	−13.3125	−1.03015	−0.515076	+	0.857145i	$0.672236\pi$
$168$	0	0
$169$	16.9621	1.30478
$170$	0	0
$171$	−1.43986	−0.110108
$172$	0	0
$173$	9.97565	0.758434	0.379217	−	0.925308i	$-0.376193\pi$
$173$	9.97565	0.758434	0.379217	+	0.925308i	$0.376193\pi$
$174$	0	0
$175$	−2.43986	−0.184436
$176$	0	0
$177$	−21.1146	−1.58707
$178$	0	0
$179$	−10.2698	−0.767600	−0.383800	−	0.923416i	$-0.625385\pi$
$179$	−10.2698	−0.767600	−0.383800	+	0.923416i	$0.625385\pi$
$180$	0	0
$181$	20.1878	1.50055	0.750274	−	0.661127i	$-0.229922\pi$
$181$	20.1878	1.50055	0.750274	+	0.661127i	$0.229922\pi$
$182$	0	0
$183$	−19.9893	−1.47765
$184$	0	0
$185$	−6.90751	−0.507850
$186$	0	0
$187$	−4.49277	−0.328544
$188$	0	0
$189$	8.02072	0.583422
$190$	0	0
$191$	−16.3135	−1.18040	−0.590200	−	0.807257i	$-0.700951\pi$
$191$	−16.3135	−1.18040	−0.590200	+	0.807257i	$0.700951\pi$
$192$	0	0
$193$	−9.58526	−0.689962	−0.344981	−	0.938610i	$-0.612115\pi$
$193$	−9.58526	−0.689962	−0.344981	+	0.938610i	$0.612115\pi$
$194$	0	0
$195$	−11.5338	−0.825950
$196$	0	0
$197$	−20.3078	−1.44687	−0.723435	−	0.690393i	$-0.757438\pi$
$197$	−20.3078	−1.44687	−0.723435	+	0.690393i	$0.757438\pi$
$198$	0	0
$199$	−13.2874	−0.941917	−0.470959	−	0.882155i	$-0.656092\pi$
$199$	−13.2874	−0.941917	−0.470959	+	0.882155i	$0.656092\pi$
$200$	0	0
$201$	8.45553	0.596407
$202$	0	0
$203$	2.26130	0.158712
$204$	0	0
$205$	4.21419	0.294332
$206$	0	0
$207$	−4.91535	−0.341640
$208$	0	0
$209$	−1.36667	−0.0945347
$210$	0	0
$211$	−7.80653	−0.537424	−0.268712	−	0.963221i	$-0.586598\pi$
$211$	−7.80653	−0.537424	−0.268712	+	0.963221i	$0.586598\pi$
$212$	0	0
$213$	10.4249	0.714305
$214$	0	0
$215$	−0.486962	−0.0332105
$216$	0	0
$217$	27.0675	1.83746
$218$	0	0
$219$	28.1407	1.90157
$220$	0	0
$221$	17.9943	1.21043
$222$	0	0
$223$	11.1275	0.745153	0.372576	−	0.928002i	$-0.378474\pi$
$223$	11.1275	0.745153	0.372576	+	0.928002i	$0.378474\pi$
$224$	0	0
$225$	1.43986	0.0959903
$226$	0	0
$227$	15.7758	1.04707	0.523537	−	0.852003i	$-0.324612\pi$
$227$	15.7758	1.04707	0.523537	+	0.852003i	$0.324612\pi$
$228$	0	0
$229$	17.3437	1.14611	0.573053	−	0.819518i	$-0.305759\pi$
$229$	17.3437	1.14611	0.573053	+	0.819518i	$0.305759\pi$
$230$	0	0
$231$	−7.02608	−0.462282
$232$	0	0
$233$	24.4434	1.60134	0.800671	−	0.599104i	$-0.204476\pi$
$233$	24.4434	1.60134	0.800671	+	0.599104i	$0.204476\pi$
$234$	0	0
$235$	10.4606	0.682373
$236$	0	0
$237$	−14.6947	−0.954525
$238$	0	0
$239$	−23.4196	−1.51489	−0.757443	−	0.652901i	$-0.773552\pi$
$239$	−23.4196	−1.51489	−0.757443	+	0.652901i	$0.773552\pi$
$240$	0	0
$241$	−19.7629	−1.27304	−0.636519	−	0.771261i	$-0.719626\pi$
$241$	−19.7629	−1.27304	−0.636519	+	0.771261i	$0.719626\pi$
$242$	0	0
$243$	−13.8351	−0.887521
$244$	0	0
$245$	−1.04711	−0.0668972
$246$	0	0
$247$	5.47377	0.348288
$248$	0	0
$249$	−7.40231	−0.469102
$250$	0	0
$251$	7.45446	0.470521	0.235261	−	0.971932i	$-0.424406\pi$
$251$	7.45446	0.470521	0.235261	+	0.971932i	$0.424406\pi$
$252$	0	0
$253$	−4.66552	−0.293319
$254$	0	0
$255$	−6.92682	−0.433774
$256$	0	0
$257$	2.85191	0.177897	0.0889486	−	0.996036i	$-0.471649\pi$
$257$	2.85191	0.177897	0.0889486	+	0.996036i	$0.471649\pi$
$258$	0	0
$259$	16.8533	1.04721
$260$	0	0
$261$	−1.33448	−0.0826024
$262$	0	0
$263$	17.1295	1.05625	0.528126	−	0.849166i	$-0.322895\pi$
$263$	17.1295	1.05625	0.528126	+	0.849166i	$0.322895\pi$
$264$	0	0
$265$	13.2332	0.812908
$266$	0	0
$267$	16.9997	1.04036
$268$	0	0
$269$	27.4315	1.67253	0.836265	−	0.548326i	$-0.184735\pi$
$269$	27.4315	1.67253	0.836265	+	0.548326i	$0.184735\pi$
$270$	0	0
$271$	−31.1468	−1.89203	−0.946017	−	0.324117i	$-0.894933\pi$
$271$	−31.1468	−1.89203	−0.946017	+	0.324117i	$0.894933\pi$
$272$	0	0
$273$	28.1407	1.70315
$274$	0	0
$275$	1.36667	0.0824134
$276$	0	0
$277$	−0.827558	−0.0497232	−0.0248616	−	0.999691i	$-0.507915\pi$
$277$	−0.827558	−0.0497232	−0.0248616	+	0.999691i	$0.507915\pi$
$278$	0	0
$279$	−15.9736	−0.956315
$280$	0	0
$281$	15.7330	0.938554	0.469277	−	0.883051i	$-0.344515\pi$
$281$	15.7330	0.938554	0.469277	+	0.883051i	$0.344515\pi$
$282$	0	0
$283$	5.03250	0.299151	0.149576	−	0.988750i	$-0.452209\pi$
$283$	5.03250	0.299151	0.149576	+	0.988750i	$0.452209\pi$
$284$	0	0
$285$	−2.10710	−0.124814
$286$	0	0
$287$	−10.2820	−0.606928
$288$	0	0
$289$	−6.19316	−0.364304
$290$	0	0
$291$	−1.15248	−0.0675595
$292$	0	0
$293$	2.59406	0.151547	0.0757733	−	0.997125i	$-0.475857\pi$
$293$	2.59406	0.151547	0.0757733	+	0.997125i	$0.475857\pi$
$294$	0	0
$295$	−10.0207	−0.583429
$296$	0	0
$297$	−4.49277	−0.260697
$298$	0	0
$299$	18.6862	1.08065
$300$	0	0
$301$	1.18812	0.0684819
$302$	0	0
$303$	−31.1197	−1.78778
$304$	0	0
$305$	−9.48665	−0.543204
$306$	0	0
$307$	−31.9164	−1.82157	−0.910784	−	0.412883i	$-0.864522\pi$
$307$	−31.9164	−1.82157	−0.910784	+	0.412883i	$0.864522\pi$
$308$	0	0
$309$	−10.9153	−0.620952
$310$	0	0
$311$	19.1239	1.08442	0.542208	−	0.840244i	$-0.317589\pi$
$311$	19.1239	1.08442	0.542208	+	0.840244i	$0.317589\pi$
$312$	0	0
$313$	−12.1115	−0.684582	−0.342291	−	0.939594i	$-0.611203\pi$
$313$	−12.1115	−0.684582	−0.342291	+	0.939594i	$0.611203\pi$
$314$	0	0
$315$	−3.51304	−0.197937
$316$	0	0
$317$	−10.3535	−0.581509	−0.290755	−	0.956798i	$-0.593906\pi$
$317$	−10.3535	−0.581509	−0.290755	+	0.956798i	$0.593906\pi$
$318$	0	0
$319$	−1.26666	−0.0709191
$320$	0	0
$321$	3.68012	0.205404
$322$	0	0
$323$	3.28738	0.182914
$324$	0	0
$325$	−5.47377	−0.303630
$326$	0	0
$327$	18.6184	1.02960
$328$	0	0
$329$	−25.5223	−1.40709
$330$	0	0
$331$	−4.55748	−0.250502	−0.125251	−	0.992125i	$-0.539974\pi$
$331$	−4.55748	−0.250502	−0.125251	+	0.992125i	$0.539974\pi$
$332$	0	0
$333$	−9.94581	−0.545027
$334$	0	0
$335$	4.01288	0.219247
$336$	0	0
$337$	−23.2451	−1.26624	−0.633121	−	0.774052i	$-0.718227\pi$
$337$	−23.2451	−1.26624	−0.633121	+	0.774052i	$0.718227\pi$
$338$	0	0
$339$	40.7684	2.21423
$340$	0	0
$341$	−15.1617	−0.821054
$342$	0	0
$343$	19.6338	1.06012
$344$	0	0
$345$	−7.19316	−0.387267
$346$	0	0
$347$	14.3020	0.767771	0.383885	−	0.923381i	$-0.374586\pi$
$347$	14.3020	0.767771	0.383885	+	0.923381i	$0.374586\pi$
$348$	0	0
$349$	10.4840	0.561195	0.280597	−	0.959826i	$-0.409467\pi$
$349$	10.4840	0.561195	0.280597	+	0.959826i	$0.409467\pi$
$350$	0	0
$351$	17.9943	0.960466
$352$	0	0
$353$	−10.9004	−0.580171	−0.290086	−	0.957001i	$-0.593684\pi$
$353$	−10.9004	−0.580171	−0.290086	+	0.957001i	$0.593684\pi$
$354$	0	0
$355$	4.94754	0.262588
$356$	0	0
$357$	16.9004	0.894466
$358$	0	0
$359$	16.4778	0.869668	0.434834	−	0.900511i	$-0.356807\pi$
$359$	16.4778	0.869668	0.434834	+	0.900511i	$0.356807\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−19.2424	−1.00997
$364$	0	0
$365$	13.3552	0.699043
$366$	0	0
$367$	−24.7040	−1.28954	−0.644769	−	0.764378i	$-0.723046\pi$
$367$	−24.7040	−1.28954	−0.644769	+	0.764378i	$0.723046\pi$
$368$	0	0
$369$	6.06783	0.315878
$370$	0	0
$371$	−32.2871	−1.67626
$372$	0	0
$373$	−1.40594	−0.0727969	−0.0363984	−	0.999337i	$-0.511589\pi$
$373$	−1.40594	−0.0727969	−0.0363984	+	0.999337i	$0.511589\pi$
$374$	0	0
$375$	2.10710	0.108810
$376$	0	0
$377$	5.07318	0.261282
$378$	0	0
$379$	19.5280	1.00308	0.501542	−	0.865133i	$-0.332766\pi$
$379$	19.5280	1.00308	0.501542	+	0.865133i	$0.332766\pi$
$380$	0	0
$381$	17.4345	0.893196
$382$	0	0
$383$	−19.9692	−1.02038	−0.510190	−	0.860062i	$-0.670425\pi$
$383$	−19.9692	−1.02038	−0.510190	+	0.860062i	$0.670425\pi$
$384$	0	0
$385$	−3.33448	−0.169941
$386$	0	0
$387$	−0.701155	−0.0356417
$388$	0	0
$389$	4.20196	0.213048	0.106524	−	0.994310i	$-0.466028\pi$
$389$	4.20196	0.213048	0.106524	+	0.994310i	$0.466028\pi$
$390$	0	0
$391$	11.2224	0.567540
$392$	0	0
$393$	−17.5066	−0.883092
$394$	0	0
$395$	−6.97392	−0.350896
$396$	0	0
$397$	−16.9475	−0.850573	−0.425286	−	0.905059i	$-0.639827\pi$
$397$	−16.9475	−0.850573	−0.425286	+	0.905059i	$0.639827\pi$
$398$	0	0
$399$	5.14101	0.257372
$400$	0	0
$401$	18.4928	0.923485	0.461742	−	0.887014i	$-0.347224\pi$
$401$	18.4928	0.923485	0.461742	+	0.887014i	$0.347224\pi$
$402$	0	0
$403$	60.7255	3.02495
$404$	0	0
$405$	−11.2464	−0.558837
$406$	0	0
$407$	−9.44030	−0.467938
$408$	0	0
$409$	5.42525	0.268261	0.134131	−	0.990964i	$-0.457176\pi$
$409$	5.42525	0.268261	0.134131	+	0.990964i	$0.457176\pi$
$410$	0	0
$411$	0.968879	0.0477913
$412$	0	0
$413$	24.4491	1.20306
$414$	0	0
$415$	−3.51304	−0.172448
$416$	0	0
$417$	−6.58698	−0.322566
$418$	0	0
$419$	−13.1353	−0.641704	−0.320852	−	0.947129i	$-0.603969\pi$
$419$	−13.1353	−0.641704	−0.320852	+	0.947129i	$0.603969\pi$
$420$	0	0
$421$	−20.9560	−1.02133	−0.510667	−	0.859778i	$-0.670602\pi$
$421$	−20.9560	−1.02133	−0.510667	+	0.859778i	$0.670602\pi$
$422$	0	0
$423$	15.0617	0.732326
$424$	0	0
$425$	−3.28738	−0.159461
$426$	0	0
$427$	23.1461	1.12012
$428$	0	0
$429$	−15.7629	−0.761038
$430$	0	0
$431$	−7.69159	−0.370491	−0.185246	−	0.982692i	$-0.559308\pi$
$431$	−7.69159	−0.370491	−0.185246	+	0.982692i	$0.559308\pi$
$432$	0	0
$433$	−24.1493	−1.16054	−0.580271	−	0.814424i	$-0.697053\pi$
$433$	−24.1493	−1.16054	−0.580271	+	0.814424i	$0.697053\pi$
$434$	0	0
$435$	−1.95289	−0.0936341
$436$	0	0
$437$	3.41378	0.163303
$438$	0	0
$439$	−29.3373	−1.40019	−0.700097	−	0.714048i	$-0.746860\pi$
$439$	−29.3373	−1.40019	−0.700097	+	0.714048i	$0.746860\pi$
$440$	0	0
$441$	−1.50768	−0.0717944
$442$	0	0
$443$	−23.2061	−1.10256	−0.551279	−	0.834321i	$-0.685860\pi$
$443$	−23.2061	−1.10256	−0.551279	+	0.834321i	$0.685860\pi$
$444$	0	0
$445$	8.06783	0.382452
$446$	0	0
$447$	31.9893	1.51304
$448$	0	0
$449$	−26.9104	−1.26998	−0.634991	−	0.772519i	$-0.718996\pi$
$449$	−26.9104	−1.26998	−0.634991	+	0.772519i	$0.718996\pi$
$450$	0	0
$451$	5.75942	0.271201
$452$	0	0
$453$	−48.3527	−2.27181
$454$	0	0
$455$	13.3552	0.626102
$456$	0	0
$457$	6.27353	0.293463	0.146732	−	0.989176i	$-0.453125\pi$
$457$	6.27353	0.293463	0.146732	+	0.989176i	$0.453125\pi$
$458$	0	0
$459$	10.8068	0.504420
$460$	0	0
$461$	−25.5188	−1.18853	−0.594265	−	0.804269i	$-0.702557\pi$
$461$	−25.5188	−1.18853	−0.594265	+	0.804269i	$0.702557\pi$
$462$	0	0
$463$	−7.07049	−0.328594	−0.164297	−	0.986411i	$-0.552535\pi$
$463$	−7.07049	−0.328594	−0.164297	+	0.986411i	$0.552535\pi$
$464$	0	0
$465$	−23.3759	−1.08403
$466$	0	0
$467$	−10.0322	−0.464234	−0.232117	−	0.972688i	$-0.574565\pi$
$467$	−10.0322	−0.464234	−0.232117	+	0.972688i	$0.574565\pi$
$468$	0	0
$469$	−9.79085	−0.452100
$470$	0	0
$471$	−6.26634	−0.288738
$472$	0	0
$473$	−0.665518	−0.0306005
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	19.0539	0.872417
$478$	0	0
$479$	−30.9690	−1.41501	−0.707505	−	0.706708i	$-0.750179\pi$
$479$	−30.9690	−1.41501	−0.707505	+	0.706708i	$0.750179\pi$
$480$	0	0
$481$	37.8101	1.72399
$482$	0	0
$483$	17.5503	0.798565
$484$	0	0
$485$	−0.546952	−0.0248358
$486$	0	0
$487$	41.1661	1.86541	0.932707	−	0.360634i	$-0.117440\pi$
$487$	41.1661	1.86541	0.932707	+	0.360634i	$0.117440\pi$
$488$	0	0
$489$	17.2660	0.780797
$490$	0	0
$491$	−14.3464	−0.647444	−0.323722	−	0.946152i	$-0.604934\pi$
$491$	−14.3464	−0.647444	−0.323722	+	0.946152i	$0.604934\pi$
$492$	0	0
$493$	3.04680	0.137221
$494$	0	0
$495$	1.96781	0.0884465
$496$	0	0
$497$	−12.0713	−0.541471
$498$	0	0
$499$	5.46089	0.244463	0.122231	−	0.992502i	$-0.460995\pi$
$499$	5.46089	0.244463	0.122231	+	0.992502i	$0.460995\pi$
$500$	0	0
$501$	−28.0507	−1.25321
$502$	0	0
$503$	−22.3227	−0.995320	−0.497660	−	0.867372i	$-0.665807\pi$
$503$	−22.3227	−0.995320	−0.497660	+	0.867372i	$0.665807\pi$
$504$	0	0
$505$	−14.7690	−0.657211
$506$	0	0
$507$	35.7409	1.58731
$508$	0	0
$509$	38.4557	1.70452	0.852259	−	0.523120i	$-0.175232\pi$
$509$	38.4557	1.70452	0.852259	+	0.523120i	$0.175232\pi$
$510$	0	0
$511$	−32.5848	−1.44146
$512$	0	0
$513$	3.28738	0.145141
$514$	0	0
$515$	−5.18028	−0.228270
$516$	0	0
$517$	14.2962	0.628745
$518$	0	0
$519$	21.0197	0.922660
$520$	0	0
$521$	−33.2032	−1.45466	−0.727329	−	0.686289i	$-0.759238\pi$
$521$	−33.2032	−1.45466	−0.727329	+	0.686289i	$0.759238\pi$
$522$	0	0
$523$	0.743401	0.0325067	0.0162533	−	0.999868i	$-0.494826\pi$
$523$	0.743401	0.0325067	0.0162533	+	0.999868i	$0.494826\pi$
$524$	0	0
$525$	−5.14101	−0.224372
$526$	0	0
$527$	36.4698	1.58865
$528$	0	0
$529$	−11.3461	−0.493309
$530$	0	0
$531$	−14.4284	−0.626139
$532$	0	0
$533$	−23.0675	−0.999165
$534$	0	0
$535$	1.74654	0.0755094
$536$	0	0
$537$	−21.6394	−0.933811
$538$	0	0
$539$	−1.43105	−0.0616398
$540$	0	0
$541$	33.8067	1.45346	0.726731	−	0.686922i	$-0.241039\pi$
$541$	33.8067	1.45346	0.726731	+	0.686922i	$0.241039\pi$
$542$	0	0
$543$	42.5377	1.82547
$544$	0	0
$545$	8.83605	0.378495
$546$	0	0
$547$	28.4397	1.21599	0.607996	−	0.793940i	$-0.291974\pi$
$547$	28.4397	1.21599	0.607996	+	0.793940i	$0.291974\pi$
$548$	0	0
$549$	−13.6594	−0.582969
$550$	0	0
$551$	0.926817	0.0394837
$552$	0	0
$553$	17.0154	0.723567
$554$	0	0
$555$	−14.5548	−0.617816
$556$	0	0
$557$	26.9626	1.14244	0.571221	−	0.820796i	$-0.306470\pi$
$557$	26.9626	1.14244	0.571221	+	0.820796i	$0.306470\pi$
$558$	0	0
$559$	2.66552	0.112739
$560$	0	0
$561$	−9.46669	−0.399684
$562$	0	0
$563$	−24.1400	−1.01738	−0.508691	−	0.860949i	$-0.669870\pi$
$563$	−24.1400	−1.01738	−0.508691	+	0.860949i	$0.669870\pi$
$564$	0	0
$565$	19.3481	0.813982
$566$	0	0
$567$	27.4395	1.15235
$568$	0	0
$569$	16.7211	0.700986	0.350493	−	0.936565i	$-0.386014\pi$
$569$	16.7211	0.700986	0.350493	+	0.936565i	$0.386014\pi$
$570$	0	0
$571$	−14.0648	−0.588596	−0.294298	−	0.955714i	$-0.595086\pi$
$571$	−14.0648	−0.588596	−0.294298	+	0.955714i	$0.595086\pi$
$572$	0	0
$573$	−34.3740	−1.43600
$574$	0	0
$575$	−3.41378	−0.142364
$576$	0	0
$577$	15.6303	0.650699	0.325350	−	0.945594i	$-0.394518\pi$
$577$	15.6303	0.650699	0.325350	+	0.945594i	$0.394518\pi$
$578$	0	0
$579$	−20.1971	−0.839361
$580$	0	0
$581$	8.57130	0.355598
$582$	0	0
$583$	18.0854	0.749022
$584$	0	0
$585$	−7.88143	−0.325857
$586$	0	0
$587$	−16.3054	−0.672997	−0.336499	−	0.941684i	$-0.609243\pi$
$587$	−16.3054	−0.672997	−0.336499	+	0.941684i	$0.609243\pi$
$588$	0	0
$589$	11.0939	0.457116
$590$	0	0
$591$	−42.7905	−1.76016
$592$	0	0
$593$	−29.3373	−1.20474	−0.602369	−	0.798217i	$-0.705777\pi$
$593$	−29.3373	−1.20474	−0.602369	+	0.798217i	$0.705777\pi$
$594$	0	0
$595$	8.02072	0.328817
$596$	0	0
$597$	−27.9978	−1.14587
$598$	0	0
$599$	32.9089	1.34462	0.672311	−	0.740269i	$-0.265302\pi$
$599$	32.9089	1.34462	0.672311	+	0.740269i	$0.265302\pi$
$600$	0	0
$601$	−9.42807	−0.384579	−0.192290	−	0.981338i	$-0.561591\pi$
$601$	−9.42807	−0.384579	−0.192290	+	0.981338i	$0.561591\pi$
$602$	0	0
$603$	5.77797	0.235297
$604$	0	0
$605$	−9.13221	−0.371277
$606$	0	0
$607$	29.8229	1.21047	0.605236	−	0.796046i	$-0.293079\pi$
$607$	29.8229	1.21047	0.605236	+	0.796046i	$0.293079\pi$
$608$	0	0
$609$	4.76478	0.193078
$610$	0	0
$611$	−57.2588	−2.31644
$612$	0	0
$613$	0.907613	0.0366582	0.0183291	−	0.999832i	$-0.494165\pi$
$613$	0.907613	0.0366582	0.0183291	+	0.999832i	$0.494165\pi$
$614$	0	0
$615$	8.87971	0.358064
$616$	0	0
$617$	−12.6533	−0.509402	−0.254701	−	0.967020i	$-0.581977\pi$
$617$	−12.6533	−0.509402	−0.254701	+	0.967020i	$0.581977\pi$
$618$	0	0
$619$	20.0756	0.806905	0.403452	−	0.915001i	$-0.367810\pi$
$619$	20.0756	0.806905	0.403452	+	0.915001i	$0.367810\pi$
$620$	0	0
$621$	11.2224	0.450338
$622$	0	0
$623$	−19.6843	−0.788636
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.87971	−0.115005
$628$	0	0
$629$	22.7076	0.905410
$630$	0	0
$631$	38.6763	1.53968	0.769839	−	0.638238i	$-0.220336\pi$
$631$	38.6763	1.53968	0.769839	+	0.638238i	$0.220336\pi$
$632$	0	0
$633$	−16.4491	−0.653793
$634$	0	0
$635$	8.27418	0.328351
$636$	0	0
$637$	5.73162	0.227095
$638$	0	0
$639$	7.12374	0.281811
$640$	0	0
$641$	−16.5084	−0.652044	−0.326022	−	0.945362i	$-0.605708\pi$
$641$	−16.5084	−0.652044	−0.326022	+	0.945362i	$0.605708\pi$
$642$	0	0
$643$	45.6521	1.80034	0.900172	−	0.435534i	$-0.143440\pi$
$643$	45.6521	1.80034	0.900172	+	0.435534i	$0.143440\pi$
$644$	0	0
$645$	−1.02608	−0.0404017
$646$	0	0
$647$	−11.4166	−0.448833	−0.224417	−	0.974493i	$-0.572048\pi$
$647$	−11.4166	−0.448833	−0.224417	+	0.974493i	$0.572048\pi$
$648$	0	0
$649$	−13.6950	−0.537577
$650$	0	0
$651$	57.0339	2.23533
$652$	0	0
$653$	19.3882	0.758717	0.379358	−	0.925250i	$-0.376145\pi$
$653$	19.3882	0.758717	0.379358	+	0.925250i	$0.376145\pi$
$654$	0	0
$655$	−8.30841	−0.324636
$656$	0	0
$657$	19.2296	0.750216
$658$	0	0
$659$	35.3303	1.37627	0.688137	−	0.725581i	$-0.258429\pi$
$659$	35.3303	1.37627	0.688137	+	0.725581i	$0.258429\pi$
$660$	0	0
$661$	−9.90012	−0.385070	−0.192535	−	0.981290i	$-0.561671\pi$
$661$	−9.90012	−0.385070	−0.192535	+	0.981290i	$0.561671\pi$
$662$	0	0
$663$	37.9158	1.47253
$664$	0	0
$665$	2.43986	0.0946135
$666$	0	0
$667$	3.16395	0.122509
$668$	0	0
$669$	23.4467	0.906503
$670$	0	0
$671$	−12.9651	−0.500514
$672$	0	0
$673$	21.2803	0.820295	0.410148	−	0.912019i	$-0.365477\pi$
$673$	21.2803	0.820295	0.410148	+	0.912019i	$0.365477\pi$
$674$	0	0
$675$	−3.28738	−0.126531
$676$	0	0
$677$	26.2241	1.00787	0.503937	−	0.863740i	$-0.331884\pi$
$677$	26.2241	1.00787	0.503937	+	0.863740i	$0.331884\pi$
$678$	0	0
$679$	1.33448	0.0512127
$680$	0	0
$681$	33.2410	1.27380
$682$	0	0
$683$	42.1551	1.61302	0.806510	−	0.591220i	$-0.201354\pi$
$683$	42.1551	1.61302	0.806510	+	0.591220i	$0.201354\pi$
$684$	0	0
$685$	0.459817	0.0175687
$686$	0	0
$687$	36.5449	1.39428
$688$	0	0
$689$	−72.4354	−2.75957
$690$	0	0
$691$	−18.9920	−0.722488	−0.361244	−	0.932471i	$-0.617648\pi$
$691$	−18.9920	−0.722488	−0.361244	+	0.932471i	$0.617648\pi$
$692$	0	0
$693$	−4.80117	−0.182381
$694$	0	0
$695$	−3.12609	−0.118579
$696$	0	0
$697$	−13.8536	−0.524744
$698$	0	0
$699$	51.5047	1.94809
$700$	0	0
$701$	−17.4223	−0.658030	−0.329015	−	0.944325i	$-0.606717\pi$
$701$	−17.4223	−0.658030	−0.329015	+	0.944325i	$0.606717\pi$
$702$	0	0
$703$	6.90751	0.260522
$704$	0	0
$705$	22.0414	0.830129
$706$	0	0
$707$	36.0342	1.35520
$708$	0	0
$709$	−12.4777	−0.468610	−0.234305	−	0.972163i	$-0.575281\pi$
$709$	−12.4777	−0.468610	−0.234305	+	0.972163i	$0.575281\pi$
$710$	0	0
$711$	−10.0414	−0.376583
$712$	0	0
$713$	37.8721	1.41832
$714$	0	0
$715$	−7.48085	−0.279768
$716$	0	0
$717$	−49.3473	−1.84291
$718$	0	0
$719$	−39.5507	−1.47499	−0.737497	−	0.675351i	$-0.763992\pi$
$719$	−39.5507	−1.47499	−0.737497	+	0.675351i	$0.763992\pi$
$720$	0	0
$721$	12.6391	0.470706
$722$	0	0
$723$	−41.6423	−1.54869
$724$	0	0
$725$	−0.926817	−0.0344211
$726$	0	0
$727$	28.2448	1.04754	0.523771	−	0.851859i	$-0.324525\pi$
$727$	28.2448	1.04754	0.523771	+	0.851859i	$0.324525\pi$
$728$	0	0
$729$	4.58729	0.169900
$730$	0	0
$731$	1.60083	0.0592087
$732$	0	0
$733$	22.5782	0.833945	0.416972	−	0.908919i	$-0.363091\pi$
$733$	22.5782	0.833945	0.416972	+	0.908919i	$0.363091\pi$
$734$	0	0
$735$	−2.20636	−0.0813826
$736$	0	0
$737$	5.48430	0.202017
$738$	0	0
$739$	2.01071	0.0739652	0.0369826	−	0.999316i	$-0.488225\pi$
$739$	2.01071	0.0739652	0.0369826	+	0.999316i	$0.488225\pi$
$740$	0	0
$741$	11.5338	0.423703
$742$	0	0
$743$	−36.3483	−1.33349	−0.666745	−	0.745286i	$-0.732313\pi$
$743$	−36.3483	−1.33349	−0.666745	+	0.745286i	$0.732313\pi$
$744$	0	0
$745$	15.1817	0.556214
$746$	0	0
$747$	−5.05827	−0.185072
$748$	0	0
$749$	−4.26130	−0.155705
$750$	0	0
$751$	47.4987	1.73325	0.866627	−	0.498957i	$-0.166284\pi$
$751$	47.4987	1.73325	0.866627	+	0.498957i	$0.166284\pi$
$752$	0	0
$753$	15.7073	0.572405
$754$	0	0
$755$	−22.9475	−0.835146
$756$	0	0
$757$	44.2550	1.60848	0.804238	−	0.594308i	$-0.202574\pi$
$757$	44.2550	1.60848	0.804238	+	0.594308i	$0.202574\pi$
$758$	0	0
$759$	−9.83070	−0.356832
$760$	0	0
$761$	7.46938	0.270765	0.135382	−	0.990793i	$-0.456774\pi$
$761$	7.46938	0.270765	0.135382	+	0.990793i	$0.456774\pi$
$762$	0	0
$763$	−21.5587	−0.780477
$764$	0	0
$765$	−4.73334	−0.171134
$766$	0	0
$767$	54.8511	1.98056
$768$	0	0
$769$	−37.2932	−1.34483	−0.672413	−	0.740176i	$-0.734742\pi$
$769$	−37.2932	−1.34483	−0.672413	+	0.740176i	$0.734742\pi$
$770$	0	0
$771$	6.00925	0.216418
$772$	0	0
$773$	−25.9564	−0.933588	−0.466794	−	0.884366i	$-0.654591\pi$
$773$	−25.9564	−0.933588	−0.466794	+	0.884366i	$0.654591\pi$
$774$	0	0
$775$	−11.0939	−0.398505
$776$	0	0
$777$	35.5116	1.27397
$778$	0	0
$779$	−4.21419	−0.150989
$780$	0	0
$781$	6.76166	0.241951
$782$	0	0
$783$	3.04680	0.108884
$784$	0	0
$785$	−2.97392	−0.106144
$786$	0	0
$787$	−25.4186	−0.906077	−0.453038	−	0.891491i	$-0.649660\pi$
$787$	−25.4186	−0.906077	−0.453038	+	0.891491i	$0.649660\pi$
$788$	0	0
$789$	36.0936	1.28497
$790$	0	0
$791$	−47.2066	−1.67847
$792$	0	0
$793$	51.9277	1.84401
$794$	0	0
$795$	27.8836	0.988930
$796$	0	0
$797$	−30.5501	−1.08214	−0.541069	−	0.840978i	$-0.681980\pi$
$797$	−30.5501	−1.08214	−0.541069	+	0.840978i	$0.681980\pi$
$798$	0	0
$799$	−34.3878	−1.21655
$800$	0	0
$801$	11.6165	0.410449
$802$	0	0
$803$	18.2522	0.644106
$804$	0	0
$805$	8.32913	0.293563
$806$	0	0
$807$	57.8009	2.03469
$808$	0	0
$809$	−23.1790	−0.814930	−0.407465	−	0.913221i	$-0.633587\pi$
$809$	−23.1790	−0.814930	−0.407465	+	0.913221i	$0.633587\pi$
$810$	0	0
$811$	7.92844	0.278405	0.139203	−	0.990264i	$-0.455546\pi$
$811$	7.92844	0.278405	0.139203	+	0.990264i	$0.455546\pi$
$812$	0	0
$813$	−65.6293	−2.30172
$814$	0	0
$815$	8.19423	0.287031
$816$	0	0
$817$	0.486962	0.0170366
$818$	0	0
$819$	19.2296	0.671935
$820$	0	0
$821$	14.5713	0.508542	0.254271	−	0.967133i	$-0.418164\pi$
$821$	14.5713	0.508542	0.254271	+	0.967133i	$0.418164\pi$
$822$	0	0
$823$	18.6886	0.651443	0.325722	−	0.945466i	$-0.394393\pi$
$823$	18.6886	0.651443	0.325722	+	0.945466i	$0.394393\pi$
$824$	0	0
$825$	2.87971	0.100259
$826$	0	0
$827$	−17.5317	−0.609638	−0.304819	−	0.952410i	$-0.598596\pi$
$827$	−17.5317	−0.609638	−0.304819	+	0.952410i	$0.598596\pi$
$828$	0	0
$829$	0.108039	0.00375236	0.00187618	−	0.999998i	$-0.499403\pi$
$829$	0.108039	0.00375236	0.00187618	+	0.999998i	$0.499403\pi$
$830$	0	0
$831$	−1.74375	−0.0604899
$832$	0	0
$833$	3.44223	0.119266
$834$	0	0
$835$	−13.3125	−0.460698
$836$	0	0
$837$	36.4698	1.26058
$838$	0	0
$839$	−13.1319	−0.453363	−0.226682	−	0.973969i	$-0.572788\pi$
$839$	−13.1319	−0.453363	−0.226682	+	0.973969i	$0.572788\pi$
$840$	0	0
$841$	−28.1410	−0.970380
$842$	0	0
$843$	33.1510	1.14178
$844$	0	0
$845$	16.9621	0.583515
$846$	0	0
$847$	22.2813	0.765593
$848$	0	0
$849$	10.6040	0.363927
$850$	0	0
$851$	23.5807	0.808336
$852$	0	0
$853$	27.1611	0.929979	0.464989	−	0.885316i	$-0.346058\pi$
$853$	27.1611	0.929979	0.464989	+	0.885316i	$0.346058\pi$
$854$	0	0
$855$	−1.43986	−0.0492420
$856$	0	0
$857$	−3.42321	−0.116935	−0.0584674	−	0.998289i	$-0.518621\pi$
$857$	−3.42321	−0.116935	−0.0584674	+	0.998289i	$0.518621\pi$
$858$	0	0
$859$	20.6724	0.705334	0.352667	−	0.935749i	$-0.385275\pi$
$859$	20.6724	0.705334	0.352667	+	0.935749i	$0.385275\pi$
$860$	0	0
$861$	−21.6652	−0.738348
$862$	0	0
$863$	−48.6884	−1.65737	−0.828686	−	0.559713i	$-0.810911\pi$
$863$	−48.6884	−1.65737	−0.828686	+	0.559713i	$0.810911\pi$
$864$	0	0
$865$	9.97565	0.339182
$866$	0	0
$867$	−13.0496	−0.443187
$868$	0	0
$869$	−9.53107	−0.323319
$870$	0	0
$871$	−21.9656	−0.744276
$872$	0	0
$873$	−0.787531	−0.0266539
$874$	0	0
$875$	−2.43986	−0.0824822
$876$	0	0
$877$	13.7159	0.463152	0.231576	−	0.972817i	$-0.425612\pi$
$877$	13.7159	0.463152	0.231576	+	0.972817i	$0.425612\pi$
$878$	0	0
$879$	5.46593	0.184361
$880$	0	0
$881$	11.7171	0.394761	0.197380	−	0.980327i	$-0.436757\pi$
$881$	11.7171	0.394761	0.197380	+	0.980327i	$0.436757\pi$
$882$	0	0
$883$	5.01683	0.168830	0.0844148	−	0.996431i	$-0.473098\pi$
$883$	5.01683	0.168830	0.0844148	+	0.996431i	$0.473098\pi$
$884$	0	0
$885$	−21.1146	−0.709760
$886$	0	0
$887$	34.0770	1.14419	0.572096	−	0.820186i	$-0.306130\pi$
$887$	34.0770	1.14419	0.572096	+	0.820186i	$0.306130\pi$
$888$	0	0
$889$	−20.1878	−0.677077
$890$	0	0
$891$	−15.3701	−0.514918
$892$	0	0
$893$	−10.4606	−0.350050
$894$	0	0
$895$	−10.2698	−0.343281
$896$	0	0
$897$	39.3737	1.31465
$898$	0	0
$899$	10.2820	0.342925
$900$	0	0
$901$	−43.5025	−1.44928
$902$	0	0
$903$	2.50348	0.0833105
$904$	0	0
$905$	20.1878	0.671065
$906$	0	0
$907$	−51.5103	−1.71037	−0.855186	−	0.518322i	$-0.826557\pi$
$907$	−51.5103	−1.71037	−0.855186	+	0.518322i	$0.826557\pi$
$908$	0	0
$909$	−21.2652	−0.705322
$910$	0	0
$911$	10.9246	0.361948	0.180974	−	0.983488i	$-0.442075\pi$
$911$	10.9246	0.361948	0.180974	+	0.983488i	$0.442075\pi$
$912$	0	0
$913$	−4.80117	−0.158896
$914$	0	0
$915$	−19.9893	−0.660825
$916$	0	0
$917$	20.2713	0.669418
$918$	0	0
$919$	21.2902	0.702299	0.351149	−	0.936319i	$-0.385791\pi$
$919$	21.2902	0.702299	0.351149	+	0.936319i	$0.385791\pi$
$920$	0	0
$921$	−67.2510	−2.21600
$922$	0	0
$923$	−27.0817	−0.891404
$924$	0	0
$925$	−6.90751	−0.227118
$926$	0	0
$927$	−7.45885	−0.244981
$928$	0	0
$929$	−6.37783	−0.209250	−0.104625	−	0.994512i	$-0.533364\pi$
$929$	−6.37783	−0.209250	−0.104625	+	0.994512i	$0.533364\pi$
$930$	0	0
$931$	1.04711	0.0343175
$932$	0	0
$933$	40.2958	1.31923
$934$	0	0
$935$	−4.49277	−0.146929
$936$	0	0
$937$	16.1354	0.527119	0.263560	−	0.964643i	$-0.415103\pi$
$937$	16.1354	0.527119	0.263560	+	0.964643i	$0.415103\pi$
$938$	0	0
$939$	−25.5201	−0.832816
$940$	0	0
$941$	5.52947	0.180256	0.0901278	−	0.995930i	$-0.471272\pi$
$941$	5.52947	0.180256	0.0901278	+	0.995930i	$0.471272\pi$
$942$	0	0
$943$	−14.3863	−0.468483
$944$	0	0
$945$	8.02072	0.260914
$946$	0	0
$947$	−46.2919	−1.50428	−0.752142	−	0.659001i	$-0.770979\pi$
$947$	−46.2919	−1.50428	−0.752142	+	0.659001i	$0.770979\pi$
$948$	0	0
$949$	−73.1033	−2.37303
$950$	0	0
$951$	−21.8158	−0.707425
$952$	0	0
$953$	−14.5090	−0.469991	−0.234996	−	0.971996i	$-0.575508\pi$
$953$	−14.5090	−0.469991	−0.234996	+	0.971996i	$0.575508\pi$
$954$	0	0
$955$	−16.3135	−0.527891
$956$	0	0
$957$	−2.66896	−0.0862754
$958$	0	0
$959$	−1.12189	−0.0362276
$960$	0	0
$961$	92.0747	2.97015
$962$	0	0
$963$	2.51476	0.0810371
$964$	0	0
$965$	−9.58526	−0.308560
$966$	0	0
$967$	−32.1829	−1.03493	−0.517466	−	0.855704i	$-0.673125\pi$
$967$	−32.1829	−1.03493	−0.517466	+	0.855704i	$0.673125\pi$
$968$	0	0
$969$	6.92682	0.222521
$970$	0	0
$971$	17.8687	0.573433	0.286717	−	0.958015i	$-0.407436\pi$
$971$	17.8687	0.573433	0.286717	+	0.958015i	$0.407436\pi$
$972$	0	0
$973$	7.62721	0.244517
$974$	0	0
$975$	−11.5338	−0.369376
$976$	0	0
$977$	−43.4295	−1.38943	−0.694716	−	0.719284i	$-0.744470\pi$
$977$	−43.4295	−1.38943	−0.694716	+	0.719284i	$0.744470\pi$
$978$	0	0
$979$	11.0261	0.352395
$980$	0	0
$981$	12.7226	0.406202
$982$	0	0
$983$	34.0669	1.08656	0.543282	−	0.839550i	$-0.317181\pi$
$983$	34.0669	1.08656	0.543282	+	0.839550i	$0.317181\pi$
$984$	0	0
$985$	−20.3078	−0.647060
$986$	0	0
$987$	−53.7779	−1.71177
$988$	0	0
$989$	1.66238	0.0528606
$990$	0	0
$991$	17.1881	0.545999	0.272999	−	0.962014i	$-0.411984\pi$
$991$	17.1881	0.545999	0.272999	+	0.962014i	$0.411984\pi$
$992$	0	0
$993$	−9.60305	−0.304743
$994$	0	0
$995$	−13.2874	−0.421238
$996$	0	0
$997$	−24.5050	−0.776081	−0.388040	−	0.921642i	$-0.626848\pi$
$997$	−24.5050	−0.776081	−0.388040	+	0.921642i	$0.626848\pi$
$998$	0	0
$999$	22.7076	0.718436

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.a.r.1.4	✓	4
4.3	odd	2		3040.2.a.t.1.1	yes	4
8.3	odd	2		6080.2.a.cd.1.4		4
8.5	even	2		6080.2.a.cf.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.r.1.4	✓	4	1.1	even	1	trivial
3040.2.a.t.1.1	yes	4	4.3	odd	2
6080.2.a.cd.1.4		4	8.3	odd	2
6080.2.a.cf.1.1		4	8.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.a (trivial)

\(f(q)\)	\(=\)	\(q+2.10710 q^{3} +1.00000 q^{5} -2.43986 q^{7} +1.43986 q^{9} +1.36667 q^{11} -5.47377 q^{13} +2.10710 q^{15} -3.28738 q^{17} -1.00000 q^{19} -5.14101 q^{21} -3.41378 q^{23} +1.00000 q^{25} -3.28738 q^{27} -0.926817 q^{29} -11.0939 q^{31} +2.87971 q^{33} -2.43986 q^{35} -6.90751 q^{37} -11.5338 q^{39} +4.21419 q^{41} -0.486962 q^{43} +1.43986 q^{45} +10.4606 q^{47} -1.04711 q^{49} -6.92682 q^{51} +13.2332 q^{53} +1.36667 q^{55} -2.10710 q^{57} -10.0207 q^{59} -9.48665 q^{61} -3.51304 q^{63} -5.47377 q^{65} +4.01288 q^{67} -7.19316 q^{69} +4.94754 q^{71} +13.3552 q^{73} +2.10710 q^{75} -3.33448 q^{77} -6.97392 q^{79} -11.2464 q^{81} -3.51304 q^{83} -3.28738 q^{85} -1.95289 q^{87} +8.06783 q^{89} +13.3552 q^{91} -23.3759 q^{93} -1.00000 q^{95} -0.546952 q^{97} +1.96781 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9} - 6 q^{11} - q^{13} - q^{15} - q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 16 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} - 13 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 - 5 * q^7 + q^9 - 6 * q^11 - q^13 - q^15 - q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - q^27 + 3 * q^29 - 16 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 - 13 * q^39 - 2 * q^41 + q^45 + 2 * q^47 - 7 * q^49 - 21 * q^51 + 13 * q^53 - 6 * q^55 + q^57 - 5 * q^59 - 2 * q^61 - 16 * q^63 - q^65 + q^67 - 11 * q^69 - 22 * q^71 + 9 * q^73 - q^75 - 4 * q^77 - 24 * q^79 - 24 * q^81 - 16 * q^83 - q^85 - 5 * q^87 + 9 * q^91 - 14 * q^93 - 4 * q^95 + 12 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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