LMFDB - Embedded newform 3040.2.a.p.1.1

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:	$0$
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:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ 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)

$f(q)$	$=$	$q-0.481194 q^{3} +1.00000 q^{5} -4.15633 q^{7} -2.76845 q^{9} +3.19394 q^{11} -6.63752 q^{13} -0.481194 q^{15} +2.00000 q^{17} -1.00000 q^{19} +2.00000 q^{21} +4.15633 q^{23} +1.00000 q^{25} +2.77575 q^{27} -7.73813 q^{29} +4.57452 q^{31} -1.53690 q^{33} -4.15633 q^{35} -4.63752 q^{37} +3.19394 q^{39} -1.73813 q^{41} +12.3430 q^{43} -2.76845 q^{45} -4.15633 q^{47} +10.2750 q^{49} -0.962389 q^{51} +0.637519 q^{53} +3.19394 q^{55} +0.481194 q^{57} +3.61213 q^{59} +13.2447 q^{61} +11.5066 q^{63} -6.63752 q^{65} -3.25694 q^{67} -2.00000 q^{69} +4.57452 q^{71} +0.261865 q^{73} -0.481194 q^{75} -13.2750 q^{77} +11.0884 q^{79} +6.96968 q^{81} +13.4314 q^{83} +2.00000 q^{85} +3.72355 q^{87} +3.73813 q^{89} +27.5877 q^{91} -2.20123 q^{93} -1.00000 q^{95} +1.10062 q^{97} -8.84226 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} + 10 q^{11} - 4 q^{13} + 4 q^{15} + 6 q^{17} - 3 q^{19} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 10 q^{27} - 14 q^{29} + 2 q^{31} + 18 q^{33} - 2 q^{35} + 2 q^{37}+ \cdots + 30 q^{99}+O(q^{100}) $ 3 * q + 4 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 + 10 * q^11 - 4 * q^13 + 4 * q^15 + 6 * q^17 - 3 * q^19 + 6 * q^21 + 2 * q^23 + 3 * q^25 + 10 * q^27 - 14 * q^29 + 2 * q^31 + 18 * q^33 - 2 * q^35 + 2 * q^37 + 10 * q^39 + 4 * q^41 + 14 * q^43 + 3 * q^45 - 2 * q^47 - q^49 + 8 * q^51 - 14 * q^53 + 10 * q^55 - 4 * q^57 + 10 * q^59 + 10 * q^61 + 14 * q^63 - 4 * q^65 - 6 * q^67 - 6 * q^69 + 2 * q^71 + 10 * q^73 + 4 * q^75 - 8 * q^77 + 14 * q^79 + 23 * q^81 - 2 * q^83 + 6 * q^85 - 24 * q^87 + 2 * q^89 + 30 * q^91 - 20 * q^93 - 3 * q^95 + 10 * q^97 + 30 * 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
$3$	−0.481194	−0.277818	−0.138909	−	0.990305i	$-0.544359\pi$
$3$	−0.481194	−0.277818	−0.138909	+	0.990305i	$0.544359\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.15633	−1.57094	−0.785472	−	0.618898i	$-0.787580\pi$
$7$	−4.15633	−1.57094	−0.785472	+	0.618898i	$0.787580\pi$
$8$	0	0
$9$	−2.76845	−0.922817
$10$	0	0
$11$	3.19394	0.963008	0.481504	−	0.876444i	$-0.340091\pi$
$11$	3.19394	0.963008	0.481504	+	0.876444i	$0.340091\pi$
$12$	0	0
$13$	−6.63752	−1.84092	−0.920458	−	0.390841i	$-0.872184\pi$
$13$	−6.63752	−1.84092	−0.920458	+	0.390841i	$0.872184\pi$
$14$	0	0
$15$	−0.481194	−0.124244
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	4.15633	0.866654	0.433327	−	0.901237i	$-0.357340\pi$
$23$	4.15633	0.866654	0.433327	+	0.901237i	$0.357340\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.77575	0.534193
$28$	0	0
$29$	−7.73813	−1.43694	−0.718468	−	0.695560i	$-0.755156\pi$
$29$	−7.73813	−1.43694	−0.718468	+	0.695560i	$0.755156\pi$
$30$	0	0
$31$	4.57452	0.821607	0.410804	−	0.911724i	$-0.365248\pi$
$31$	4.57452	0.821607	0.410804	+	0.911724i	$0.365248\pi$
$32$	0	0
$33$	−1.53690	−0.267541
$34$	0	0
$35$	−4.15633	−0.702547
$36$	0	0
$37$	−4.63752	−0.762404	−0.381202	−	0.924492i	$-0.624490\pi$
$37$	−4.63752	−0.762404	−0.381202	+	0.924492i	$0.624490\pi$
$38$	0	0
$39$	3.19394	0.511439
$40$	0	0
$41$	−1.73813	−0.271451	−0.135726	−	0.990746i	$-0.543337\pi$
$41$	−1.73813	−0.271451	−0.135726	+	0.990746i	$0.543337\pi$
$42$	0	0
$43$	12.3430	1.88228	0.941142	−	0.338010i	$-0.109754\pi$
$43$	12.3430	1.88228	0.941142	+	0.338010i	$0.109754\pi$
$44$	0	0
$45$	−2.76845	−0.412696
$46$	0	0
$47$	−4.15633	−0.606262	−0.303131	−	0.952949i	$-0.598032\pi$
$47$	−4.15633	−0.606262	−0.303131	+	0.952949i	$0.598032\pi$
$48$	0	0
$49$	10.2750	1.46786
$50$	0	0
$51$	−0.962389	−0.134761
$52$	0	0
$53$	0.637519	0.0875700	0.0437850	−	0.999041i	$-0.486058\pi$
$53$	0.637519	0.0875700	0.0437850	+	0.999041i	$0.486058\pi$
$54$	0	0
$55$	3.19394	0.430670
$56$	0	0
$57$	0.481194	0.0637357
$58$	0	0
$59$	3.61213	0.470259	0.235129	−	0.971964i	$-0.424449\pi$
$59$	3.61213	0.470259	0.235129	+	0.971964i	$0.424449\pi$
$60$	0	0
$61$	13.2447	1.69581	0.847906	−	0.530146i	$-0.177863\pi$
$61$	13.2447	1.69581	0.847906	+	0.530146i	$0.177863\pi$
$62$	0	0
$63$	11.5066	1.44969
$64$	0	0
$65$	−6.63752	−0.823283
$66$	0	0
$67$	−3.25694	−0.397899	−0.198949	−	0.980010i	$-0.563753\pi$
$67$	−3.25694	−0.397899	−0.198949	+	0.980010i	$0.563753\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	4.57452	0.542895	0.271448	−	0.962453i	$-0.412498\pi$
$71$	4.57452	0.542895	0.271448	+	0.962453i	$0.412498\pi$
$72$	0	0
$73$	0.261865	0.0306490	0.0153245	−	0.999883i	$-0.495122\pi$
$73$	0.261865	0.0306490	0.0153245	+	0.999883i	$0.495122\pi$
$74$	0	0
$75$	−0.481194	−0.0555635
$76$	0	0
$77$	−13.2750	−1.51283
$78$	0	0
$79$	11.0884	1.24754	0.623771	−	0.781607i	$-0.285600\pi$
$79$	11.0884	1.24754	0.623771	+	0.781607i	$0.285600\pi$
$80$	0	0
$81$	6.96968	0.774409
$82$	0	0
$83$	13.4314	1.47428	0.737142	−	0.675738i	$-0.236175\pi$
$83$	13.4314	1.47428	0.737142	+	0.675738i	$0.236175\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	3.72355	0.399206
$88$	0	0
$89$	3.73813	0.396242	0.198121	−	0.980178i	$-0.436516\pi$
$89$	3.73813	0.396242	0.198121	+	0.980178i	$0.436516\pi$
$90$	0	0
$91$	27.5877	2.89198
$92$	0	0
$93$	−2.20123	−0.228257
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	1.10062	0.111751	0.0558753	−	0.998438i	$-0.482205\pi$
$97$	1.10062	0.111751	0.0558753	+	0.998438i	$0.482205\pi$
$98$	0	0
$99$	−8.84226	−0.888681
$100$	0	0
$101$	15.0435	1.49688	0.748442	−	0.663201i	$-0.230802\pi$
$101$	15.0435	1.49688	0.748442	+	0.663201i	$0.230802\pi$
$102$	0	0
$103$	−19.8822	−1.95906	−0.979528	−	0.201309i	$-0.935480\pi$
$103$	−19.8822	−1.95906	−0.979528	+	0.201309i	$0.935480\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−6.14411	−0.593973	−0.296987	−	0.954882i	$-0.595982\pi$
$107$	−6.14411	−0.593973	−0.296987	+	0.954882i	$0.595982\pi$
$108$	0	0
$109$	7.53690	0.721904	0.360952	−	0.932584i	$-0.382452\pi$
$109$	7.53690	0.721904	0.360952	+	0.932584i	$0.382452\pi$
$110$	0	0
$111$	2.23155	0.211809
$112$	0	0
$113$	−16.3757	−1.54049	−0.770246	−	0.637747i	$-0.779867\pi$
$113$	−16.3757	−1.54049	−0.770246	+	0.637747i	$0.779867\pi$
$114$	0	0
$115$	4.15633	0.387579
$116$	0	0
$117$	18.3757	1.69883
$118$	0	0
$119$	−8.31265	−0.762019
$120$	0	0
$121$	−0.798769	−0.0726154
$122$	0	0
$123$	0.836381	0.0754139
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.31757	−0.116916	−0.0584579	−	0.998290i	$-0.518618\pi$
$127$	−1.31757	−0.116916	−0.0584579	+	0.998290i	$0.518618\pi$
$128$	0	0
$129$	−5.93937	−0.522932
$130$	0	0
$131$	−19.5125	−1.70481	−0.852406	−	0.522880i	$-0.824857\pi$
$131$	−19.5125	−1.70481	−0.852406	+	0.522880i	$0.824857\pi$
$132$	0	0
$133$	4.15633	0.360399
$134$	0	0
$135$	2.77575	0.238898
$136$	0	0
$137$	−1.53690	−0.131307	−0.0656533	−	0.997842i	$-0.520913\pi$
$137$	−1.53690	−0.131307	−0.0656533	+	0.997842i	$0.520913\pi$
$138$	0	0
$139$	9.58181	0.812718	0.406359	−	0.913713i	$-0.366798\pi$
$139$	9.58181	0.812718	0.406359	+	0.913713i	$0.366798\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	−21.1998	−1.77282
$144$	0	0
$145$	−7.73813	−0.642617
$146$	0	0
$147$	−4.94429	−0.407798
$148$	0	0
$149$	5.24472	0.429664	0.214832	−	0.976651i	$-0.431080\pi$
$149$	5.24472	0.429664	0.214832	+	0.976651i	$0.431080\pi$
$150$	0	0
$151$	7.35026	0.598156	0.299078	−	0.954229i	$-0.403321\pi$
$151$	7.35026	0.598156	0.299078	+	0.954229i	$0.403321\pi$
$152$	0	0
$153$	−5.53690	−0.447632
$154$	0	0
$155$	4.57452	0.367434
$156$	0	0
$157$	17.5369	1.39960	0.699799	−	0.714340i	$-0.253273\pi$
$157$	17.5369	1.39960	0.699799	+	0.714340i	$0.253273\pi$
$158$	0	0
$159$	−0.306771	−0.0243285
$160$	0	0
$161$	−17.2750	−1.36146
$162$	0	0
$163$	11.5066	0.901265	0.450633	−	0.892709i	$-0.351198\pi$
$163$	11.5066	0.901265	0.450633	+	0.892709i	$0.351198\pi$
$164$	0	0
$165$	−1.53690	−0.119648
$166$	0	0
$167$	14.3453	1.11008	0.555038	−	0.831825i	$-0.312704\pi$
$167$	14.3453	1.11008	0.555038	+	0.831825i	$0.312704\pi$
$168$	0	0
$169$	31.0567	2.38897
$170$	0	0
$171$	2.76845	0.211709
$172$	0	0
$173$	−12.8994	−0.980722	−0.490361	−	0.871519i	$-0.663135\pi$
$173$	−12.8994	−0.980722	−0.490361	+	0.871519i	$0.663135\pi$
$174$	0	0
$175$	−4.15633	−0.314189
$176$	0	0
$177$	−1.73813	−0.130646
$178$	0	0
$179$	3.61213	0.269983	0.134992	−	0.990847i	$-0.456899\pi$
$179$	3.61213	0.269983	0.134992	+	0.990847i	$0.456899\pi$
$180$	0	0
$181$	12.5501	0.932840	0.466420	−	0.884563i	$-0.345544\pi$
$181$	12.5501	0.932840	0.466420	+	0.884563i	$0.345544\pi$
$182$	0	0
$183$	−6.37328	−0.471127
$184$	0	0
$185$	−4.63752	−0.340957
$186$	0	0
$187$	6.38787	0.467128
$188$	0	0
$189$	−11.5369	−0.839186
$190$	0	0
$191$	−1.08840	−0.0787536	−0.0393768	−	0.999224i	$-0.512537\pi$
$191$	−1.08840	−0.0787536	−0.0393768	+	0.999224i	$0.512537\pi$
$192$	0	0
$193$	15.6507	1.12656	0.563281	−	0.826266i	$-0.309539\pi$
$193$	15.6507	1.12656	0.563281	+	0.826266i	$0.309539\pi$
$194$	0	0
$195$	3.19394	0.228723
$196$	0	0
$197$	−7.73813	−0.551319	−0.275660	−	0.961255i	$-0.588896\pi$
$197$	−7.73813	−0.551319	−0.275660	+	0.961255i	$0.588896\pi$
$198$	0	0
$199$	−14.7005	−1.04209	−0.521046	−	0.853528i	$-0.674458\pi$
$199$	−14.7005	−1.04209	−0.521046	+	0.853528i	$0.674458\pi$
$200$	0	0
$201$	1.56722	0.110543
$202$	0	0
$203$	32.1622	2.25734
$204$	0	0
$205$	−1.73813	−0.121397
$206$	0	0
$207$	−11.5066	−0.799763
$208$	0	0
$209$	−3.19394	−0.220929
$210$	0	0
$211$	27.3503	1.88287	0.941435	−	0.337195i	$-0.109478\pi$
$211$	27.3503	1.88287	0.941435	+	0.337195i	$0.109478\pi$
$212$	0	0
$213$	−2.20123	−0.150826
$214$	0	0
$215$	12.3430	0.841783
$216$	0	0
$217$	−19.0132	−1.29070
$218$	0	0
$219$	−0.126008	−0.00851483
$220$	0	0
$221$	−13.2750	−0.892976
$222$	0	0
$223$	−23.3684	−1.56486	−0.782431	−	0.622738i	$-0.786020\pi$
$223$	−23.3684	−1.56486	−0.782431	+	0.622738i	$0.786020\pi$
$224$	0	0
$225$	−2.76845	−0.184563
$226$	0	0
$227$	28.1949	1.87136	0.935680	−	0.352849i	$-0.114787\pi$
$227$	28.1949	1.87136	0.935680	+	0.352849i	$0.114787\pi$
$228$	0	0
$229$	−23.0435	−1.52276	−0.761378	−	0.648308i	$-0.775477\pi$
$229$	−23.0435	−1.52276	−0.761378	+	0.648308i	$0.775477\pi$
$230$	0	0
$231$	6.38787	0.420291
$232$	0	0
$233$	−3.73813	−0.244893	−0.122447	−	0.992475i	$-0.539074\pi$
$233$	−3.73813	−0.244893	−0.122447	+	0.992475i	$0.539074\pi$
$234$	0	0
$235$	−4.15633	−0.271129
$236$	0	0
$237$	−5.33567	−0.346589
$238$	0	0
$239$	0.126008	0.00815078	0.00407539	−	0.999992i	$-0.498703\pi$
$239$	0.126008	0.00815078	0.00407539	+	0.999992i	$0.498703\pi$
$240$	0	0
$241$	−2.20123	−0.141794	−0.0708969	−	0.997484i	$-0.522586\pi$
$241$	−2.20123	−0.141794	−0.0708969	+	0.997484i	$0.522586\pi$
$242$	0	0
$243$	−11.6810	−0.749337
$244$	0	0
$245$	10.2750	0.656448
$246$	0	0
$247$	6.63752	0.422335
$248$	0	0
$249$	−6.46310	−0.409582
$250$	0	0
$251$	22.7612	1.43667	0.718336	−	0.695697i	$-0.244904\pi$
$251$	22.7612	1.43667	0.718336	+	0.695697i	$0.244904\pi$
$252$	0	0
$253$	13.2750	0.834595
$254$	0	0
$255$	−0.962389	−0.0602671
$256$	0	0
$257$	2.43629	0.151972	0.0759858	−	0.997109i	$-0.475790\pi$
$257$	2.43629	0.151972	0.0759858	+	0.997109i	$0.475790\pi$
$258$	0	0
$259$	19.2750	1.19769
$260$	0	0
$261$	21.4227	1.32603
$262$	0	0
$263$	12.4690	0.768870	0.384435	−	0.923152i	$-0.374396\pi$
$263$	12.4690	0.768870	0.384435	+	0.923152i	$0.374396\pi$
$264$	0	0
$265$	0.637519	0.0391625
$266$	0	0
$267$	−1.79877	−0.110083
$268$	0	0
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	−2.10554	−0.127902	−0.0639512	−	0.997953i	$-0.520370\pi$
$271$	−2.10554	−0.127902	−0.0639512	+	0.997953i	$0.520370\pi$
$272$	0	0
$273$	−13.2750	−0.803442
$274$	0	0
$275$	3.19394	0.192602
$276$	0	0
$277$	−24.0870	−1.44725	−0.723623	−	0.690195i	$-0.757525\pi$
$277$	−24.0870	−1.44725	−0.723623	+	0.690195i	$0.757525\pi$
$278$	0	0
$279$	−12.6643	−0.758193
$280$	0	0
$281$	−24.8119	−1.48016	−0.740078	−	0.672521i	$-0.765212\pi$
$281$	−24.8119	−1.48016	−0.740078	+	0.672521i	$0.765212\pi$
$282$	0	0
$283$	1.26916	0.0754437	0.0377218	−	0.999288i	$-0.487990\pi$
$283$	1.26916	0.0754437	0.0377218	+	0.999288i	$0.487990\pi$
$284$	0	0
$285$	0.481194	0.0285035
$286$	0	0
$287$	7.22425	0.426434
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−0.529610	−0.0310463
$292$	0	0
$293$	0.174424	0.0101899	0.00509497	−	0.999987i	$-0.498378\pi$
$293$	0.174424	0.0101899	0.00509497	+	0.999987i	$0.498378\pi$
$294$	0	0
$295$	3.61213	0.210306
$296$	0	0
$297$	8.86556	0.514432
$298$	0	0
$299$	−27.5877	−1.59544
$300$	0	0
$301$	−51.3014	−2.95696
$302$	0	0
$303$	−7.23884	−0.415861
$304$	0	0
$305$	13.2447	0.758391
$306$	0	0
$307$	−16.9951	−0.969960	−0.484980	−	0.874525i	$-0.661173\pi$
$307$	−16.9951	−0.969960	−0.484980	+	0.874525i	$0.661173\pi$
$308$	0	0
$309$	9.56722	0.544260
$310$	0	0
$311$	27.0435	1.53350	0.766748	−	0.641948i	$-0.221874\pi$
$311$	27.0435	1.53350	0.766748	+	0.641948i	$0.221874\pi$
$312$	0	0
$313$	7.73813	0.437385	0.218693	−	0.975794i	$-0.429821\pi$
$313$	7.73813	0.437385	0.218693	+	0.975794i	$0.429821\pi$
$314$	0	0
$315$	11.5066	0.648323
$316$	0	0
$317$	−10.1744	−0.571453	−0.285726	−	0.958311i	$-0.592235\pi$
$317$	−10.1744	−0.571453	−0.285726	+	0.958311i	$0.592235\pi$
$318$	0	0
$319$	−24.7151	−1.38378
$320$	0	0
$321$	2.95651	0.165016
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	−6.63752	−0.368183
$326$	0	0
$327$	−3.62672	−0.200558
$328$	0	0
$329$	17.2750	0.952404
$330$	0	0
$331$	19.0376	1.04640	0.523201	−	0.852209i	$-0.324738\pi$
$331$	19.0376	1.04640	0.523201	+	0.852209i	$0.324738\pi$
$332$	0	0
$333$	12.8388	0.703559
$334$	0	0
$335$	−3.25694	−0.177946
$336$	0	0
$337$	21.1876	1.15416	0.577081	−	0.816687i	$-0.304192\pi$
$337$	21.1876	1.15416	0.577081	+	0.816687i	$0.304192\pi$
$338$	0	0
$339$	7.87987	0.427976
$340$	0	0
$341$	14.6107	0.791214
$342$	0	0
$343$	−13.6121	−0.734986
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	−3.06793	−0.164695	−0.0823475	−	0.996604i	$-0.526242\pi$
$347$	−3.06793	−0.164695	−0.0823475	+	0.996604i	$0.526242\pi$
$348$	0	0
$349$	15.2750	0.817654	0.408827	−	0.912612i	$-0.365938\pi$
$349$	15.2750	0.817654	0.408827	+	0.912612i	$0.365938\pi$
$350$	0	0
$351$	−18.4241	−0.983404
$352$	0	0
$353$	28.5501	1.51957	0.759784	−	0.650176i	$-0.225305\pi$
$353$	28.5501	1.51957	0.759784	+	0.650176i	$0.225305\pi$
$354$	0	0
$355$	4.57452	0.242790
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	−30.8930	−1.63047	−0.815236	−	0.579128i	$-0.803393\pi$
$359$	−30.8930	−1.63047	−0.815236	+	0.579128i	$0.803393\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.384363	0.0201738
$364$	0	0
$365$	0.261865	0.0137066
$366$	0	0
$367$	−1.26916	−0.0662496	−0.0331248	−	0.999451i	$-0.510546\pi$
$367$	−1.26916	−0.0662496	−0.0331248	+	0.999451i	$0.510546\pi$
$368$	0	0
$369$	4.81194	0.250500
$370$	0	0
$371$	−2.64974	−0.137568
$372$	0	0
$373$	27.1270	1.40458	0.702290	−	0.711891i	$-0.252161\pi$
$373$	27.1270	1.40458	0.702290	+	0.711891i	$0.252161\pi$
$374$	0	0
$375$	−0.481194	−0.0248488
$376$	0	0
$377$	51.3620	2.64528
$378$	0	0
$379$	16.3733	0.841039	0.420520	−	0.907283i	$-0.361848\pi$
$379$	16.3733	0.841039	0.420520	+	0.907283i	$0.361848\pi$
$380$	0	0
$381$	0.634010	0.0324813
$382$	0	0
$383$	27.1065	1.38508	0.692539	−	0.721381i	$-0.256492\pi$
$383$	27.1065	1.38508	0.692539	+	0.721381i	$0.256492\pi$
$384$	0	0
$385$	−13.2750	−0.676559
$386$	0	0
$387$	−34.1709	−1.73701
$388$	0	0
$389$	−21.8251	−1.10658	−0.553289	−	0.832990i	$-0.686627\pi$
$389$	−21.8251	−1.10658	−0.553289	+	0.832990i	$0.686627\pi$
$390$	0	0
$391$	8.31265	0.420389
$392$	0	0
$393$	9.38929	0.473627
$394$	0	0
$395$	11.0884	0.557918
$396$	0	0
$397$	−3.27504	−0.164369	−0.0821847	−	0.996617i	$-0.526190\pi$
$397$	−3.27504	−0.164369	−0.0821847	+	0.996617i	$0.526190\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	5.01317	0.250346	0.125173	−	0.992135i	$-0.460051\pi$
$401$	5.01317	0.250346	0.125173	+	0.992135i	$0.460051\pi$
$402$	0	0
$403$	−30.3634	−1.51251
$404$	0	0
$405$	6.96968	0.346326
$406$	0	0
$407$	−14.8119	−0.734201
$408$	0	0
$409$	3.07381	0.151990	0.0759950	−	0.997108i	$-0.475787\pi$
$409$	3.07381	0.151990	0.0759950	+	0.997108i	$0.475787\pi$
$410$	0	0
$411$	0.739549	0.0364793
$412$	0	0
$413$	−15.0132	−0.738750
$414$	0	0
$415$	13.4314	0.659320
$416$	0	0
$417$	−4.61071	−0.225788
$418$	0	0
$419$	31.1998	1.52421	0.762105	−	0.647453i	$-0.224166\pi$
$419$	31.1998	1.52421	0.762105	+	0.647453i	$0.224166\pi$
$420$	0	0
$421$	−32.3488	−1.57659	−0.788293	−	0.615300i	$-0.789035\pi$
$421$	−32.3488	−1.57659	−0.788293	+	0.615300i	$0.789035\pi$
$422$	0	0
$423$	11.5066	0.559469
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−55.0494	−2.66403
$428$	0	0
$429$	10.2012	0.492520
$430$	0	0
$431$	0.962389	0.0463566	0.0231783	−	0.999731i	$-0.492621\pi$
$431$	0.962389	0.0463566	0.0231783	+	0.999731i	$0.492621\pi$
$432$	0	0
$433$	24.7245	1.18818	0.594092	−	0.804397i	$-0.297512\pi$
$433$	24.7245	1.18818	0.594092	+	0.804397i	$0.297512\pi$
$434$	0	0
$435$	3.72355	0.178530
$436$	0	0
$437$	−4.15633	−0.198824
$438$	0	0
$439$	1.10299	0.0526426	0.0263213	−	0.999654i	$-0.491621\pi$
$439$	1.10299	0.0526426	0.0263213	+	0.999654i	$0.491621\pi$
$440$	0	0
$441$	−28.4460	−1.35457
$442$	0	0
$443$	−21.8700	−1.03908	−0.519538	−	0.854447i	$-0.673896\pi$
$443$	−21.8700	−1.03908	−0.519538	+	0.854447i	$0.673896\pi$
$444$	0	0
$445$	3.73813	0.177205
$446$	0	0
$447$	−2.52373	−0.119368
$448$	0	0
$449$	−19.5369	−0.922004	−0.461002	−	0.887399i	$-0.652510\pi$
$449$	−19.5369	−0.922004	−0.461002	+	0.887399i	$0.652510\pi$
$450$	0	0
$451$	−5.55149	−0.261410
$452$	0	0
$453$	−3.53690	−0.166178
$454$	0	0
$455$	27.5877	1.29333
$456$	0	0
$457$	−28.0263	−1.31102	−0.655509	−	0.755188i	$-0.727546\pi$
$457$	−28.0263	−1.31102	−0.655509	+	0.755188i	$0.727546\pi$
$458$	0	0
$459$	5.55149	0.259121
$460$	0	0
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	0	0
$463$	−24.9927	−1.16151	−0.580755	−	0.814079i	$-0.697242\pi$
$463$	−24.9927	−1.16151	−0.580755	+	0.814079i	$0.697242\pi$
$464$	0	0
$465$	−2.20123	−0.102080
$466$	0	0
$467$	−10.2922	−0.476265	−0.238133	−	0.971233i	$-0.576535\pi$
$467$	−10.2922	−0.476265	−0.238133	+	0.971233i	$0.576535\pi$
$468$	0	0
$469$	13.5369	0.625076
$470$	0	0
$471$	−8.43866	−0.388833
$472$	0	0
$473$	39.4227	1.81266
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−1.76494	−0.0808111
$478$	0	0
$479$	24.7572	1.13118	0.565592	−	0.824685i	$-0.308648\pi$
$479$	24.7572	1.13118	0.565592	+	0.824685i	$0.308648\pi$
$480$	0	0
$481$	30.7816	1.40352
$482$	0	0
$483$	8.31265	0.378239
$484$	0	0
$485$	1.10062	0.0499764
$486$	0	0
$487$	−14.9443	−0.677190	−0.338595	−	0.940932i	$-0.609952\pi$
$487$	−14.9443	−0.677190	−0.338595	+	0.940932i	$0.609952\pi$
$488$	0	0
$489$	−5.53690	−0.250387
$490$	0	0
$491$	−4.93795	−0.222847	−0.111423	−	0.993773i	$-0.535541\pi$
$491$	−4.93795	−0.222847	−0.111423	+	0.993773i	$0.535541\pi$
$492$	0	0
$493$	−15.4763	−0.697016
$494$	0	0
$495$	−8.84226	−0.397430
$496$	0	0
$497$	−19.0132	−0.852857
$498$	0	0
$499$	3.44595	0.154262	0.0771310	−	0.997021i	$-0.475424\pi$
$499$	3.44595	0.154262	0.0771310	+	0.997021i	$0.475424\pi$
$500$	0	0
$501$	−6.90289	−0.308399
$502$	0	0
$503$	7.53102	0.335792	0.167896	−	0.985805i	$-0.446303\pi$
$503$	7.53102	0.335792	0.167896	+	0.985805i	$0.446303\pi$
$504$	0	0
$505$	15.0435	0.669427
$506$	0	0
$507$	−14.9443	−0.663699
$508$	0	0
$509$	12.2012	0.540810	0.270405	−	0.962747i	$-0.412842\pi$
$509$	12.2012	0.540810	0.270405	+	0.962747i	$0.412842\pi$
$510$	0	0
$511$	−1.08840	−0.0481478
$512$	0	0
$513$	−2.77575	−0.122552
$514$	0	0
$515$	−19.8822	−0.876116
$516$	0	0
$517$	−13.2750	−0.583836
$518$	0	0
$519$	6.20711	0.272462
$520$	0	0
$521$	−16.6107	−0.727729	−0.363864	−	0.931452i	$-0.618543\pi$
$521$	−16.6107	−0.727729	−0.363864	+	0.931452i	$0.618543\pi$
$522$	0	0
$523$	−34.4422	−1.50605	−0.753025	−	0.657991i	$-0.771406\pi$
$523$	−34.4422	−1.50605	−0.753025	+	0.657991i	$0.771406\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	9.14903	0.398538
$528$	0	0
$529$	−5.72496	−0.248911
$530$	0	0
$531$	−10.0000	−0.433963
$532$	0	0
$533$	11.5369	0.499719
$534$	0	0
$535$	−6.14411	−0.265633
$536$	0	0
$537$	−1.73813	−0.0750061
$538$	0	0
$539$	32.8178	1.41356
$540$	0	0
$541$	12.9565	0.557044	0.278522	−	0.960430i	$-0.410156\pi$
$541$	12.9565	0.557044	0.278522	+	0.960430i	$0.410156\pi$
$542$	0	0
$543$	−6.03903	−0.259159
$544$	0	0
$545$	7.53690	0.322845
$546$	0	0
$547$	14.6942	0.628278	0.314139	−	0.949377i	$-0.398284\pi$
$547$	14.6942	0.628278	0.314139	+	0.949377i	$0.398284\pi$
$548$	0	0
$549$	−36.6674	−1.56493
$550$	0	0
$551$	7.73813	0.329656
$552$	0	0
$553$	−46.0870	−1.95982
$554$	0	0
$555$	2.23155	0.0947239
$556$	0	0
$557$	37.0132	1.56830	0.784149	−	0.620572i	$-0.213100\pi$
$557$	37.0132	1.56830	0.784149	+	0.620572i	$0.213100\pi$
$558$	0	0
$559$	−81.9267	−3.46513
$560$	0	0
$561$	−3.07381	−0.129776
$562$	0	0
$563$	−19.9937	−0.842632	−0.421316	−	0.906914i	$-0.638432\pi$
$563$	−19.9937	−0.842632	−0.421316	+	0.906914i	$0.638432\pi$
$564$	0	0
$565$	−16.3757	−0.688929
$566$	0	0
$567$	−28.9683	−1.21655
$568$	0	0
$569$	11.4763	0.481110	0.240555	−	0.970635i	$-0.422671\pi$
$569$	11.4763	0.481110	0.240555	+	0.970635i	$0.422671\pi$
$570$	0	0
$571$	27.0435	1.13173	0.565867	−	0.824496i	$-0.308541\pi$
$571$	27.0435	1.13173	0.565867	+	0.824496i	$0.308541\pi$
$572$	0	0
$573$	0.523730	0.0218791
$574$	0	0
$575$	4.15633	0.173331
$576$	0	0
$577$	7.21440	0.300340	0.150170	−	0.988660i	$-0.452018\pi$
$577$	7.21440	0.300340	0.150170	+	0.988660i	$0.452018\pi$
$578$	0	0
$579$	−7.53102	−0.312979
$580$	0	0
$581$	−55.8251	−2.31602
$582$	0	0
$583$	2.03620	0.0843307
$584$	0	0
$585$	18.3757	0.759740
$586$	0	0
$587$	−28.1319	−1.16113	−0.580564	−	0.814215i	$-0.697168\pi$
$587$	−28.1319	−1.16113	−0.580564	+	0.814215i	$0.697168\pi$
$588$	0	0
$589$	−4.57452	−0.188490
$590$	0	0
$591$	3.72355	0.153166
$592$	0	0
$593$	−25.5975	−1.05116	−0.525582	−	0.850743i	$-0.676153\pi$
$593$	−25.5975	−1.05116	−0.525582	+	0.850743i	$0.676153\pi$
$594$	0	0
$595$	−8.31265	−0.340785
$596$	0	0
$597$	7.07381	0.289512
$598$	0	0
$599$	−11.9248	−0.487233	−0.243617	−	0.969872i	$-0.578334\pi$
$599$	−11.9248	−0.487233	−0.243617	+	0.969872i	$0.578334\pi$
$600$	0	0
$601$	−13.6775	−0.557917	−0.278958	−	0.960303i	$-0.589989\pi$
$601$	−13.6775	−0.557917	−0.278958	+	0.960303i	$0.589989\pi$
$602$	0	0
$603$	9.01668	0.367188
$604$	0	0
$605$	−0.798769	−0.0324746
$606$	0	0
$607$	−37.9283	−1.53946	−0.769731	−	0.638369i	$-0.779610\pi$
$607$	−37.9283	−1.53946	−0.769731	+	0.638369i	$0.779610\pi$
$608$	0	0
$609$	−15.4763	−0.627130
$610$	0	0
$611$	27.5877	1.11608
$612$	0	0
$613$	38.8119	1.56760	0.783800	−	0.621014i	$-0.213279\pi$
$613$	38.8119	1.56760	0.783800	+	0.621014i	$0.213279\pi$
$614$	0	0
$615$	0.836381	0.0337261
$616$	0	0
$617$	−14.8119	−0.596306	−0.298153	−	0.954518i	$-0.596371\pi$
$617$	−14.8119	−0.596306	−0.298153	+	0.954518i	$0.596371\pi$
$618$	0	0
$619$	20.6556	0.830219	0.415110	−	0.909771i	$-0.363743\pi$
$619$	20.6556	0.830219	0.415110	+	0.909771i	$0.363743\pi$
$620$	0	0
$621$	11.5369	0.462960
$622$	0	0
$623$	−15.5369	−0.622473
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.53690	0.0613780
$628$	0	0
$629$	−9.27504	−0.369820
$630$	0	0
$631$	−25.9551	−1.03326	−0.516628	−	0.856210i	$-0.672813\pi$
$631$	−25.9551	−1.03326	−0.516628	+	0.856210i	$0.672813\pi$
$632$	0	0
$633$	−13.1608	−0.523094
$634$	0	0
$635$	−1.31757	−0.0522864
$636$	0	0
$637$	−68.2008	−2.70221
$638$	0	0
$639$	−12.6643	−0.500993
$640$	0	0
$641$	−16.7513	−0.661637	−0.330818	−	0.943694i	$-0.607325\pi$
$641$	−16.7513	−0.661637	−0.330818	+	0.943694i	$0.607325\pi$
$642$	0	0
$643$	10.7962	0.425761	0.212881	−	0.977078i	$-0.431715\pi$
$643$	10.7962	0.425761	0.212881	+	0.977078i	$0.431715\pi$
$644$	0	0
$645$	−5.93937	−0.233862
$646$	0	0
$647$	−25.5936	−1.00619	−0.503094	−	0.864232i	$-0.667805\pi$
$647$	−25.5936	−1.00619	−0.503094	+	0.864232i	$0.667805\pi$
$648$	0	0
$649$	11.5369	0.452863
$650$	0	0
$651$	9.14903	0.358579
$652$	0	0
$653$	−39.7381	−1.55507	−0.777537	−	0.628838i	$-0.783531\pi$
$653$	−39.7381	−1.55507	−0.777537	+	0.628838i	$0.783531\pi$
$654$	0	0
$655$	−19.5125	−0.762415
$656$	0	0
$657$	−0.724961	−0.0282834
$658$	0	0
$659$	7.71370	0.300483	0.150241	−	0.988649i	$-0.451995\pi$
$659$	7.71370	0.300483	0.150241	+	0.988649i	$0.451995\pi$
$660$	0	0
$661$	−6.20123	−0.241200	−0.120600	−	0.992701i	$-0.538482\pi$
$661$	−6.20123	−0.241200	−0.120600	+	0.992701i	$0.538482\pi$
$662$	0	0
$663$	6.38787	0.248084
$664$	0	0
$665$	4.15633	0.161175
$666$	0	0
$667$	−32.1622	−1.24533
$668$	0	0
$669$	11.2447	0.434746
$670$	0	0
$671$	42.3028	1.63308
$672$	0	0
$673$	−10.1138	−0.389858	−0.194929	−	0.980817i	$-0.562448\pi$
$673$	−10.1138	−0.389858	−0.194929	+	0.980817i	$0.562448\pi$
$674$	0	0
$675$	2.77575	0.106839
$676$	0	0
$677$	−30.3757	−1.16743	−0.583716	−	0.811958i	$-0.698402\pi$
$677$	−30.3757	−1.16743	−0.583716	+	0.811958i	$0.698402\pi$
$678$	0	0
$679$	−4.57452	−0.175554
$680$	0	0
$681$	−13.5672	−0.519897
$682$	0	0
$683$	28.4323	1.08793	0.543966	−	0.839107i	$-0.316922\pi$
$683$	28.4323	1.08793	0.543966	+	0.839107i	$0.316922\pi$
$684$	0	0
$685$	−1.53690	−0.0587221
$686$	0	0
$687$	11.0884	0.423049
$688$	0	0
$689$	−4.23155	−0.161209
$690$	0	0
$691$	−12.9565	−0.492889	−0.246444	−	0.969157i	$-0.579262\pi$
$691$	−12.9565	−0.492889	−0.246444	+	0.969157i	$0.579262\pi$
$692$	0	0
$693$	36.7513	1.39607
$694$	0	0
$695$	9.58181	0.363459
$696$	0	0
$697$	−3.47627	−0.131673
$698$	0	0
$699$	1.79877	0.0680357
$700$	0	0
$701$	28.7816	1.08707	0.543534	−	0.839387i	$-0.317086\pi$
$701$	28.7816	1.08707	0.543534	+	0.839387i	$0.317086\pi$
$702$	0	0
$703$	4.63752	0.174907
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	−62.5256	−2.35152
$708$	0	0
$709$	−30.2276	−1.13522	−0.567610	−	0.823297i	$-0.692132\pi$
$709$	−30.2276	−1.13522	−0.567610	+	0.823297i	$0.692132\pi$
$710$	0	0
$711$	−30.6977	−1.15125
$712$	0	0
$713$	19.0132	0.712049
$714$	0	0
$715$	−21.1998	−0.792828
$716$	0	0
$717$	−0.0606343	−0.00226443
$718$	0	0
$719$	−40.6556	−1.51620	−0.758099	−	0.652139i	$-0.773872\pi$
$719$	−40.6556	−1.51620	−0.758099	+	0.652139i	$0.773872\pi$
$720$	0	0
$721$	82.6371	3.07756
$722$	0	0
$723$	1.05922	0.0393928
$724$	0	0
$725$	−7.73813	−0.287387
$726$	0	0
$727$	24.5052	0.908847	0.454423	−	0.890786i	$-0.349845\pi$
$727$	24.5052	0.908847	0.454423	+	0.890786i	$0.349845\pi$
$728$	0	0
$729$	−15.2882	−0.566230
$730$	0	0
$731$	24.6859	0.913042
$732$	0	0
$733$	33.8251	1.24936	0.624680	−	0.780881i	$-0.285230\pi$
$733$	33.8251	1.24936	0.624680	+	0.780881i	$0.285230\pi$
$734$	0	0
$735$	−4.94429	−0.182373
$736$	0	0
$737$	−10.4025	−0.383180
$738$	0	0
$739$	−42.6516	−1.56897	−0.784483	−	0.620150i	$-0.787072\pi$
$739$	−42.6516	−1.56897	−0.784483	+	0.620150i	$0.787072\pi$
$740$	0	0
$741$	−3.19394	−0.117332
$742$	0	0
$743$	33.1309	1.21546	0.607728	−	0.794145i	$-0.292081\pi$
$743$	33.1309	1.21546	0.607728	+	0.794145i	$0.292081\pi$
$744$	0	0
$745$	5.24472	0.192152
$746$	0	0
$747$	−37.1841	−1.36049
$748$	0	0
$749$	25.5369	0.933098
$750$	0	0
$751$	−12.0654	−0.440272	−0.220136	−	0.975469i	$-0.570650\pi$
$751$	−12.0654	−0.440272	−0.220136	+	0.975469i	$0.570650\pi$
$752$	0	0
$753$	−10.9525	−0.399133
$754$	0	0
$755$	7.35026	0.267503
$756$	0	0
$757$	−23.7381	−0.862777	−0.431389	−	0.902166i	$-0.641976\pi$
$757$	−23.7381	−0.862777	−0.431389	+	0.902166i	$0.641976\pi$
$758$	0	0
$759$	−6.38787	−0.231865
$760$	0	0
$761$	3.04349	0.110326	0.0551632	−	0.998477i	$-0.482432\pi$
$761$	3.04349	0.110326	0.0551632	+	0.998477i	$0.482432\pi$
$762$	0	0
$763$	−31.3258	−1.13407
$764$	0	0
$765$	−5.53690	−0.200187
$766$	0	0
$767$	−23.9756	−0.865707
$768$	0	0
$769$	53.5936	1.93263	0.966317	−	0.257356i	$-0.0828512\pi$
$769$	53.5936	1.93263	0.966317	+	0.257356i	$0.0828512\pi$
$770$	0	0
$771$	−1.17233	−0.0422204
$772$	0	0
$773$	42.9257	1.54393	0.771966	−	0.635664i	$-0.219274\pi$
$773$	42.9257	1.54393	0.771966	+	0.635664i	$0.219274\pi$
$774$	0	0
$775$	4.57452	0.164321
$776$	0	0
$777$	−9.27504	−0.332740
$778$	0	0
$779$	1.73813	0.0622751
$780$	0	0
$781$	14.6107	0.522812
$782$	0	0
$783$	−21.4791	−0.767600
$784$	0	0
$785$	17.5369	0.625919
$786$	0	0
$787$	10.8446	0.386569	0.193285	−	0.981143i	$-0.438086\pi$
$787$	10.8446	0.386569	0.193285	+	0.981143i	$0.438086\pi$
$788$	0	0
$789$	−6.00000	−0.213606
$790$	0	0
$791$	68.0625	2.42003
$792$	0	0
$793$	−87.9121	−3.12185
$794$	0	0
$795$	−0.306771	−0.0108800
$796$	0	0
$797$	38.1401	1.35099	0.675496	−	0.737363i	$-0.263929\pi$
$797$	38.1401	1.35099	0.675496	+	0.737363i	$0.263929\pi$
$798$	0	0
$799$	−8.31265	−0.294080
$800$	0	0
$801$	−10.3488	−0.365659
$802$	0	0
$803$	0.836381	0.0295152
$804$	0	0
$805$	−17.2750	−0.608865
$806$	0	0
$807$	−5.77433	−0.203266
$808$	0	0
$809$	−42.2276	−1.48464	−0.742321	−	0.670044i	$-0.766275\pi$
$809$	−42.2276	−1.48464	−0.742321	+	0.670044i	$0.766275\pi$
$810$	0	0
$811$	52.5256	1.84442	0.922212	−	0.386684i	$-0.126380\pi$
$811$	52.5256	1.84442	0.922212	+	0.386684i	$0.126380\pi$
$812$	0	0
$813$	1.01317	0.0355336
$814$	0	0
$815$	11.5066	0.403058
$816$	0	0
$817$	−12.3430	−0.431826
$818$	0	0
$819$	−76.3752	−2.66877
$820$	0	0
$821$	19.2750	0.672703	0.336352	−	0.941736i	$-0.390807\pi$
$821$	19.2750	0.672703	0.336352	+	0.941736i	$0.390807\pi$
$822$	0	0
$823$	−7.40502	−0.258123	−0.129061	−	0.991637i	$-0.541196\pi$
$823$	−7.40502	−0.258123	−0.129061	+	0.991637i	$0.541196\pi$
$824$	0	0
$825$	−1.53690	−0.0535081
$826$	0	0
$827$	22.8808	0.795644	0.397822	−	0.917463i	$-0.369766\pi$
$827$	22.8808	0.795644	0.397822	+	0.917463i	$0.369766\pi$
$828$	0	0
$829$	7.41564	0.257556	0.128778	−	0.991673i	$-0.458895\pi$
$829$	7.41564	0.257556	0.128778	+	0.991673i	$0.458895\pi$
$830$	0	0
$831$	11.5905	0.402071
$832$	0	0
$833$	20.5501	0.712018
$834$	0	0
$835$	14.3453	0.496441
$836$	0	0
$837$	12.6977	0.438897
$838$	0	0
$839$	−10.4749	−0.361632	−0.180816	−	0.983517i	$-0.557874\pi$
$839$	−10.4749	−0.361632	−0.180816	+	0.983517i	$0.557874\pi$
$840$	0	0
$841$	30.8787	1.06478
$842$	0	0
$843$	11.9394	0.411214
$844$	0	0
$845$	31.0567	1.06838
$846$	0	0
$847$	3.31994	0.114075
$848$	0	0
$849$	−0.610712	−0.0209596
$850$	0	0
$851$	−19.2750	−0.660740
$852$	0	0
$853$	−17.1344	−0.586672	−0.293336	−	0.956009i	$-0.594765\pi$
$853$	−17.1344	−0.586672	−0.293336	+	0.956009i	$0.594765\pi$
$854$	0	0
$855$	2.76845	0.0946791
$856$	0	0
$857$	−43.3089	−1.47940	−0.739701	−	0.672935i	$-0.765033\pi$
$857$	−43.3089	−1.47940	−0.739701	+	0.672935i	$0.765033\pi$
$858$	0	0
$859$	22.0508	0.752363	0.376182	−	0.926546i	$-0.377237\pi$
$859$	22.0508	0.752363	0.376182	+	0.926546i	$0.377237\pi$
$860$	0	0
$861$	−3.47627	−0.118471
$862$	0	0
$863$	43.6204	1.48485	0.742427	−	0.669926i	$-0.233674\pi$
$863$	43.6204	1.48485	0.742427	+	0.669926i	$0.233674\pi$
$864$	0	0
$865$	−12.8994	−0.438592
$866$	0	0
$867$	6.25553	0.212449
$868$	0	0
$869$	35.4156	1.20139
$870$	0	0
$871$	21.6180	0.732498
$872$	0	0
$873$	−3.04700	−0.103125
$874$	0	0
$875$	−4.15633	−0.140509
$876$	0	0
$877$	45.2746	1.52881	0.764407	−	0.644734i	$-0.223032\pi$
$877$	45.2746	1.52881	0.764407	+	0.644734i	$0.223032\pi$
$878$	0	0
$879$	−0.0839316	−0.00283094
$880$	0	0
$881$	−5.76845	−0.194344	−0.0971720	−	0.995268i	$-0.530980\pi$
$881$	−5.76845	−0.194344	−0.0971720	+	0.995268i	$0.530980\pi$
$882$	0	0
$883$	−6.91748	−0.232792	−0.116396	−	0.993203i	$-0.537134\pi$
$883$	−6.91748	−0.232792	−0.116396	+	0.993203i	$0.537134\pi$
$884$	0	0
$885$	−1.73813	−0.0584268
$886$	0	0
$887$	−8.93444	−0.299989	−0.149995	−	0.988687i	$-0.547926\pi$
$887$	−8.93444	−0.299989	−0.149995	+	0.988687i	$0.547926\pi$
$888$	0	0
$889$	5.47627	0.183668
$890$	0	0
$891$	22.2607	0.745762
$892$	0	0
$893$	4.15633	0.139086
$894$	0	0
$895$	3.61213	0.120740
$896$	0	0
$897$	13.2750	0.443241
$898$	0	0
$899$	−35.3982	−1.18060
$900$	0	0
$901$	1.27504	0.0424777
$902$	0	0
$903$	24.6859	0.821496
$904$	0	0
$905$	12.5501	0.417179
$906$	0	0
$907$	9.28329	0.308247	0.154123	−	0.988052i	$-0.450745\pi$
$907$	9.28329	0.308247	0.154123	+	0.988052i	$0.450745\pi$
$908$	0	0
$909$	−41.6472	−1.38135
$910$	0	0
$911$	−34.4650	−1.14188	−0.570938	−	0.820993i	$-0.693420\pi$
$911$	−34.4650	−1.14188	−0.570938	+	0.820993i	$0.693420\pi$
$912$	0	0
$913$	42.8989	1.41975
$914$	0	0
$915$	−6.37328	−0.210694
$916$	0	0
$917$	81.1002	2.67816
$918$	0	0
$919$	55.5534	1.83254	0.916269	−	0.400564i	$-0.131186\pi$
$919$	55.5534	1.83254	0.916269	+	0.400564i	$0.131186\pi$
$920$	0	0
$921$	8.17793	0.269472
$922$	0	0
$923$	−30.3634	−0.999425
$924$	0	0
$925$	−4.63752	−0.152481
$926$	0	0
$927$	55.0430	1.80785
$928$	0	0
$929$	2.12127	0.0695966	0.0347983	−	0.999394i	$-0.488921\pi$
$929$	2.12127	0.0695966	0.0347983	+	0.999394i	$0.488921\pi$
$930$	0	0
$931$	−10.2750	−0.336751
$932$	0	0
$933$	−13.0132	−0.426032
$934$	0	0
$935$	6.38787	0.208906
$936$	0	0
$937$	−18.6907	−0.610598	−0.305299	−	0.952257i	$-0.598756\pi$
$937$	−18.6907	−0.610598	−0.305299	+	0.952257i	$0.598756\pi$
$938$	0	0
$939$	−3.72355	−0.121513
$940$	0	0
$941$	−36.0870	−1.17640	−0.588201	−	0.808715i	$-0.700164\pi$
$941$	−36.0870	−1.17640	−0.588201	+	0.808715i	$0.700164\pi$
$942$	0	0
$943$	−7.22425	−0.235254
$944$	0	0
$945$	−11.5369	−0.375296
$946$	0	0
$947$	−22.2315	−0.722428	−0.361214	−	0.932483i	$-0.617638\pi$
$947$	−22.2315	−0.722428	−0.361214	+	0.932483i	$0.617638\pi$
$948$	0	0
$949$	−1.73813	−0.0564222
$950$	0	0
$951$	4.89587	0.158760
$952$	0	0
$953$	14.8994	0.482638	0.241319	−	0.970446i	$-0.422420\pi$
$953$	14.8994	0.482638	0.241319	+	0.970446i	$0.422420\pi$
$954$	0	0
$955$	−1.08840	−0.0352197
$956$	0	0
$957$	11.8928	0.384439
$958$	0	0
$959$	6.38787	0.206275
$960$	0	0
$961$	−10.0738	−0.324962
$962$	0	0
$963$	17.0097	0.548129
$964$	0	0
$965$	15.6507	0.503814
$966$	0	0
$967$	54.5198	1.75324	0.876619	−	0.481186i	$-0.159794\pi$
$967$	54.5198	1.75324	0.876619	+	0.481186i	$0.159794\pi$
$968$	0	0
$969$	0.962389	0.0309164
$970$	0	0
$971$	5.18806	0.166493	0.0832463	−	0.996529i	$-0.473471\pi$
$971$	5.18806	0.166493	0.0832463	+	0.996529i	$0.473471\pi$
$972$	0	0
$973$	−39.8251	−1.27673
$974$	0	0
$975$	3.19394	0.102288
$976$	0	0
$977$	49.0127	1.56806	0.784028	−	0.620726i	$-0.213162\pi$
$977$	49.0127	1.56806	0.784028	+	0.620726i	$0.213162\pi$
$978$	0	0
$979$	11.9394	0.381584
$980$	0	0
$981$	−20.8656	−0.666186
$982$	0	0
$983$	−36.8564	−1.17554	−0.587768	−	0.809029i	$-0.699993\pi$
$983$	−36.8564	−1.17554	−0.587768	+	0.809029i	$0.699993\pi$
$984$	0	0
$985$	−7.73813	−0.246557
$986$	0	0
$987$	−8.31265	−0.264595
$988$	0	0
$989$	51.3014	1.63129
$990$	0	0
$991$	−22.2736	−0.707545	−0.353772	−	0.935332i	$-0.615101\pi$
$991$	−22.2736	−0.707545	−0.353772	+	0.935332i	$0.615101\pi$
$992$	0	0
$993$	−9.16079	−0.290709
$994$	0	0
$995$	−14.7005	−0.466038
$996$	0	0
$997$	41.0738	1.30082	0.650410	−	0.759583i	$-0.274597\pi$
$997$	41.0738	1.30082	0.650410	+	0.759583i	$0.274597\pi$
$998$	0	0
$999$	−12.8726	−0.407270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.a.p.1.1	yes	3
4.3	odd	2		3040.2.a.j.1.3	✓	3
8.3	odd	2		6080.2.a.ca.1.1		3
8.5	even	2		6080.2.a.bm.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.j.1.3	✓	3	4.3	odd	2
3040.2.a.p.1.1	yes	3	1.1	even	1	trivial
6080.2.a.bm.1.3		3	8.5	even	2
6080.2.a.ca.1.1		3	8.3	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:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.a (trivial)

\(f(q)\)	\(=\)	\(q-0.481194 q^{3} +1.00000 q^{5} -4.15633 q^{7} -2.76845 q^{9} +3.19394 q^{11} -6.63752 q^{13} -0.481194 q^{15} +2.00000 q^{17} -1.00000 q^{19} +2.00000 q^{21} +4.15633 q^{23} +1.00000 q^{25} +2.77575 q^{27} -7.73813 q^{29} +4.57452 q^{31} -1.53690 q^{33} -4.15633 q^{35} -4.63752 q^{37} +3.19394 q^{39} -1.73813 q^{41} +12.3430 q^{43} -2.76845 q^{45} -4.15633 q^{47} +10.2750 q^{49} -0.962389 q^{51} +0.637519 q^{53} +3.19394 q^{55} +0.481194 q^{57} +3.61213 q^{59} +13.2447 q^{61} +11.5066 q^{63} -6.63752 q^{65} -3.25694 q^{67} -2.00000 q^{69} +4.57452 q^{71} +0.261865 q^{73} -0.481194 q^{75} -13.2750 q^{77} +11.0884 q^{79} +6.96968 q^{81} +13.4314 q^{83} +2.00000 q^{85} +3.72355 q^{87} +3.73813 q^{89} +27.5877 q^{91} -2.20123 q^{93} -1.00000 q^{95} +1.10062 q^{97} -8.84226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} + 10 q^{11} - 4 q^{13} + 4 q^{15} + 6 q^{17} - 3 q^{19} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 10 q^{27} - 14 q^{29} + 2 q^{31} + 18 q^{33} - 2 q^{35} + 2 q^{37}+ \cdots + 30 q^{99}+O(q^{100}) \) 3 * q + 4 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 + 10 * q^11 - 4 * q^13 + 4 * q^15 + 6 * q^17 - 3 * q^19 + 6 * q^21 + 2 * q^23 + 3 * q^25 + 10 * q^27 - 14 * q^29 + 2 * q^31 + 18 * q^33 - 2 * q^35 + 2 * q^37 + 10 * q^39 + 4 * q^41 + 14 * q^43 + 3 * q^45 - 2 * q^47 - q^49 + 8 * q^51 - 14 * q^53 + 10 * q^55 - 4 * q^57 + 10 * q^59 + 10 * q^61 + 14 * q^63 - 4 * q^65 - 6 * q^67 - 6 * q^69 + 2 * q^71 + 10 * q^73 + 4 * q^75 - 8 * q^77 + 14 * q^79 + 23 * q^81 - 2 * q^83 + 6 * q^85 - 24 * q^87 + 2 * q^89 + 30 * q^91 - 20 * q^93 - 3 * q^95 + 10 * q^97 + 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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