LMFDB - Embedded newform 3520.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3520,2,Mod(1,3520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3520, 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("3520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3520 = 2^{6} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3520.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:	$28.1073415115$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	3520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.37228 q^{3} -1.00000 q^{5} -3.37228 q^{7} +8.37228 q^{9} -1.00000 q^{11} -2.00000 q^{13} +3.37228 q^{15} +1.37228 q^{17} +0.627719 q^{19} +11.3723 q^{21} -2.74456 q^{23} +1.00000 q^{25} -18.1168 q^{27} -1.37228 q^{29} -3.37228 q^{31} +3.37228 q^{33} +3.37228 q^{35} -9.37228 q^{37} +6.74456 q^{39} -11.4891 q^{41} -4.00000 q^{43} -8.37228 q^{45} -2.74456 q^{47} +4.37228 q^{49} -4.62772 q^{51} +4.11684 q^{53} +1.00000 q^{55} -2.11684 q^{57} -2.74456 q^{59} +5.37228 q^{61} -28.2337 q^{63} +2.00000 q^{65} +8.00000 q^{67} +9.25544 q^{69} -10.1168 q^{71} -15.4891 q^{73} -3.37228 q^{75} +3.37228 q^{77} +1.25544 q^{79} +35.9783 q^{81} -2.74456 q^{83} -1.37228 q^{85} +4.62772 q^{87} -1.37228 q^{89} +6.74456 q^{91} +11.3723 q^{93} -0.627719 q^{95} -12.7446 q^{97} -8.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} - q^{7} + 11 q^{9} - 2 q^{11} - 4 q^{13} + q^{15} - 3 q^{17} + 7 q^{19} + 17 q^{21} + 6 q^{23} + 2 q^{25} - 19 q^{27} + 3 q^{29} - q^{31} + q^{33} + q^{35} - 13 q^{37} + 2 q^{39}+ \cdots - 11 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 - q^7 + 11 * q^9 - 2 * q^11 - 4 * q^13 + q^15 - 3 * q^17 + 7 * q^19 + 17 * q^21 + 6 * q^23 + 2 * q^25 - 19 * q^27 + 3 * q^29 - q^31 + q^33 + q^35 - 13 * q^37 + 2 * q^39 - 8 * q^43 - 11 * q^45 + 6 * q^47 + 3 * q^49 - 15 * q^51 - 9 * q^53 + 2 * q^55 + 13 * q^57 + 6 * q^59 + 5 * q^61 - 22 * q^63 + 4 * q^65 + 16 * q^67 + 30 * q^69 - 3 * q^71 - 8 * q^73 - q^75 + q^77 + 14 * q^79 + 26 * q^81 + 6 * q^83 + 3 * q^85 + 15 * q^87 + 3 * q^89 + 2 * q^91 + 17 * q^93 - 7 * q^95 - 14 * q^97 - 11 * 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
$3$	−3.37228	−1.94699	−0.973494	−	0.228714i	$-0.926548\pi$
$3$	−3.37228	−1.94699	−0.973494	+	0.228714i	$0.926548\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	0	0
$9$	8.37228	2.79076
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	3.37228	0.870719
$16$	0	0
$17$	1.37228	0.332827	0.166414	−	0.986056i	$-0.446781\pi$
$17$	1.37228	0.332827	0.166414	+	0.986056i	$0.446781\pi$
$18$	0	0
$19$	0.627719	0.144009	0.0720043	−	0.997404i	$-0.477060\pi$
$19$	0.627719	0.144009	0.0720043	+	0.997404i	$0.477060\pi$
$20$	0	0
$21$	11.3723	2.48164
$22$	0	0
$23$	−2.74456	−0.572281	−0.286140	−	0.958188i	$-0.592372\pi$
$23$	−2.74456	−0.572281	−0.286140	+	0.958188i	$0.592372\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−18.1168	−3.48659
$28$	0	0
$29$	−1.37228	−0.254826	−0.127413	−	0.991850i	$-0.540667\pi$
$29$	−1.37228	−0.254826	−0.127413	+	0.991850i	$0.540667\pi$
$30$	0	0
$31$	−3.37228	−0.605680	−0.302840	−	0.953041i	$-0.597935\pi$
$31$	−3.37228	−0.605680	−0.302840	+	0.953041i	$0.597935\pi$
$32$	0	0
$33$	3.37228	0.587039
$34$	0	0
$35$	3.37228	0.570020
$36$	0	0
$37$	−9.37228	−1.54079	−0.770397	−	0.637565i	$-0.779942\pi$
$37$	−9.37228	−1.54079	−0.770397	+	0.637565i	$0.779942\pi$
$38$	0	0
$39$	6.74456	1.07999
$40$	0	0
$41$	−11.4891	−1.79430	−0.897150	−	0.441726i	$-0.854366\pi$
$41$	−11.4891	−1.79430	−0.897150	+	0.441726i	$0.854366\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−8.37228	−1.24807
$46$	0	0
$47$	−2.74456	−0.400336	−0.200168	−	0.979762i	$-0.564149\pi$
$47$	−2.74456	−0.400336	−0.200168	+	0.979762i	$0.564149\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	−4.62772	−0.648010
$52$	0	0
$53$	4.11684	0.565492	0.282746	−	0.959195i	$-0.408755\pi$
$53$	4.11684	0.565492	0.282746	+	0.959195i	$0.408755\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−2.11684	−0.280383
$58$	0	0
$59$	−2.74456	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$59$	−2.74456	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$60$	0	0
$61$	5.37228	0.687850	0.343925	−	0.938997i	$-0.388243\pi$
$61$	5.37228	0.687850	0.343925	+	0.938997i	$0.388243\pi$
$62$	0	0
$63$	−28.2337	−3.55711
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	9.25544	1.11422
$70$	0	0
$71$	−10.1168	−1.20065	−0.600324	−	0.799757i	$-0.704962\pi$
$71$	−10.1168	−1.20065	−0.600324	+	0.799757i	$0.704962\pi$
$72$	0	0
$73$	−15.4891	−1.81286	−0.906432	−	0.422351i	$-0.861205\pi$
$73$	−15.4891	−1.81286	−0.906432	+	0.422351i	$0.861205\pi$
$74$	0	0
$75$	−3.37228	−0.389398
$76$	0	0
$77$	3.37228	0.384307
$78$	0	0
$79$	1.25544	0.141248	0.0706239	−	0.997503i	$-0.477501\pi$
$79$	1.25544	0.141248	0.0706239	+	0.997503i	$0.477501\pi$
$80$	0	0
$81$	35.9783	3.99758
$82$	0	0
$83$	−2.74456	−0.301255	−0.150627	−	0.988591i	$-0.548129\pi$
$83$	−2.74456	−0.301255	−0.150627	+	0.988591i	$0.548129\pi$
$84$	0	0
$85$	−1.37228	−0.148845
$86$	0	0
$87$	4.62772	0.496144
$88$	0	0
$89$	−1.37228	−0.145462	−0.0727308	−	0.997352i	$-0.523171\pi$
$89$	−1.37228	−0.145462	−0.0727308	+	0.997352i	$0.523171\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	0	0
$93$	11.3723	1.17925
$94$	0	0
$95$	−0.627719	−0.0644026
$96$	0	0
$97$	−12.7446	−1.29401	−0.647007	−	0.762484i	$-0.723980\pi$
$97$	−12.7446	−1.29401	−0.647007	+	0.762484i	$0.723980\pi$
$98$	0	0
$99$	−8.37228	−0.841446
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	9.48913	0.934991	0.467496	−	0.883995i	$-0.345156\pi$
$103$	9.48913	0.934991	0.467496	+	0.883995i	$0.345156\pi$
$104$	0	0
$105$	−11.3723	−1.10982
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	15.4891	1.48359	0.741795	−	0.670627i	$-0.233975\pi$
$109$	15.4891	1.48359	0.741795	+	0.670627i	$0.233975\pi$
$110$	0	0
$111$	31.6060	2.99991
$112$	0	0
$113$	3.25544	0.306246	0.153123	−	0.988207i	$-0.451067\pi$
$113$	3.25544	0.306246	0.153123	+	0.988207i	$0.451067\pi$
$114$	0	0
$115$	2.74456	0.255932
$116$	0	0
$117$	−16.7446	−1.54804
$118$	0	0
$119$	−4.62772	−0.424222
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	38.7446	3.49348
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	13.4891	1.18765
$130$	0	0
$131$	22.1168	1.93236	0.966179	−	0.257873i	$-0.0830216\pi$
$131$	22.1168	1.93236	0.966179	+	0.257873i	$0.0830216\pi$
$132$	0	0
$133$	−2.11684	−0.183554
$134$	0	0
$135$	18.1168	1.55925
$136$	0	0
$137$	8.74456	0.747098	0.373549	−	0.927610i	$-0.378141\pi$
$137$	8.74456	0.747098	0.373549	+	0.927610i	$0.378141\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	9.25544	0.779448
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	1.37228	0.113962
$146$	0	0
$147$	−14.7446	−1.21611
$148$	0	0
$149$	−21.6060	−1.77003	−0.885015	−	0.465563i	$-0.845852\pi$
$149$	−21.6060	−1.77003	−0.885015	+	0.465563i	$0.845852\pi$
$150$	0	0
$151$	12.2337	0.995563	0.497782	−	0.867302i	$-0.334148\pi$
$151$	12.2337	0.995563	0.497782	+	0.867302i	$0.334148\pi$
$152$	0	0
$153$	11.4891	0.928841
$154$	0	0
$155$	3.37228	0.270868
$156$	0	0
$157$	−9.37228	−0.747989	−0.373995	−	0.927431i	$-0.622012\pi$
$157$	−9.37228	−0.747989	−0.373995	+	0.927431i	$0.622012\pi$
$158$	0	0
$159$	−13.8832	−1.10101
$160$	0	0
$161$	9.25544	0.729431
$162$	0	0
$163$	−5.88316	−0.460804	−0.230402	−	0.973095i	$-0.574004\pi$
$163$	−5.88316	−0.460804	−0.230402	+	0.973095i	$0.574004\pi$
$164$	0	0
$165$	−3.37228	−0.262532
$166$	0	0
$167$	4.62772	0.358104	0.179052	−	0.983840i	$-0.442697\pi$
$167$	4.62772	0.358104	0.179052	+	0.983840i	$0.442697\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	5.25544	0.401893
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−3.37228	−0.254921
$176$	0	0
$177$	9.25544	0.695681
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−18.1168	−1.33924
$184$	0	0
$185$	9.37228	0.689064
$186$	0	0
$187$	−1.37228	−0.100351
$188$	0	0
$189$	61.0951	4.44401
$190$	0	0
$191$	5.48913	0.397179	0.198590	−	0.980083i	$-0.436364\pi$
$191$	5.48913	0.397179	0.198590	+	0.980083i	$0.436364\pi$
$192$	0	0
$193$	14.8614	1.06975	0.534874	−	0.844932i	$-0.320359\pi$
$193$	14.8614	1.06975	0.534874	+	0.844932i	$0.320359\pi$
$194$	0	0
$195$	−6.74456	−0.482988
$196$	0	0
$197$	20.7446	1.47799	0.738994	−	0.673712i	$-0.235301\pi$
$197$	20.7446	1.47799	0.738994	+	0.673712i	$0.235301\pi$
$198$	0	0
$199$	−18.1168	−1.28427	−0.642135	−	0.766592i	$-0.721951\pi$
$199$	−18.1168	−1.28427	−0.642135	+	0.766592i	$0.721951\pi$
$200$	0	0
$201$	−26.9783	−1.90290
$202$	0	0
$203$	4.62772	0.324802
$204$	0	0
$205$	11.4891	0.802435
$206$	0	0
$207$	−22.9783	−1.59710
$208$	0	0
$209$	−0.627719	−0.0434202
$210$	0	0
$211$	6.11684	0.421101	0.210550	−	0.977583i	$-0.432474\pi$
$211$	6.11684	0.421101	0.210550	+	0.977583i	$0.432474\pi$
$212$	0	0
$213$	34.1168	2.33765
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	11.3723	0.772001
$218$	0	0
$219$	52.2337	3.52963
$220$	0	0
$221$	−2.74456	−0.184619
$222$	0	0
$223$	18.7446	1.25523	0.627614	−	0.778524i	$-0.284031\pi$
$223$	18.7446	1.25523	0.627614	+	0.778524i	$0.284031\pi$
$224$	0	0
$225$	8.37228	0.558152
$226$	0	0
$227$	−2.74456	−0.182163	−0.0910815	−	0.995843i	$-0.529032\pi$
$227$	−2.74456	−0.182163	−0.0910815	+	0.995843i	$0.529032\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−11.3723	−0.748241
$232$	0	0
$233$	1.37228	0.0899011	0.0449506	−	0.998989i	$-0.485687\pi$
$233$	1.37228	0.0899011	0.0449506	+	0.998989i	$0.485687\pi$
$234$	0	0
$235$	2.74456	0.179036
$236$	0	0
$237$	−4.23369	−0.275008
$238$	0	0
$239$	14.7446	0.953746	0.476873	−	0.878972i	$-0.341770\pi$
$239$	14.7446	0.953746	0.476873	+	0.878972i	$0.341770\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	−66.9783	−4.29666
$244$	0	0
$245$	−4.37228	−0.279335
$246$	0	0
$247$	−1.25544	−0.0798816
$248$	0	0
$249$	9.25544	0.586540
$250$	0	0
$251$	2.74456	0.173235	0.0866176	−	0.996242i	$-0.472394\pi$
$251$	2.74456	0.173235	0.0866176	+	0.996242i	$0.472394\pi$
$252$	0	0
$253$	2.74456	0.172549
$254$	0	0
$255$	4.62772	0.289799
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	31.6060	1.96390
$260$	0	0
$261$	−11.4891	−0.711159
$262$	0	0
$263$	24.8614	1.53302	0.766510	−	0.642232i	$-0.221992\pi$
$263$	24.8614	1.53302	0.766510	+	0.642232i	$0.221992\pi$
$264$	0	0
$265$	−4.11684	−0.252896
$266$	0	0
$267$	4.62772	0.283212
$268$	0	0
$269$	8.74456	0.533165	0.266583	−	0.963812i	$-0.414105\pi$
$269$	8.74456	0.533165	0.266583	+	0.963812i	$0.414105\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	−22.7446	−1.37656
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	12.7446	0.765747	0.382873	−	0.923801i	$-0.374935\pi$
$277$	12.7446	0.765747	0.382873	+	0.923801i	$0.374935\pi$
$278$	0	0
$279$	−28.2337	−1.69031
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	5.25544	0.312403	0.156202	−	0.987725i	$-0.450075\pi$
$283$	5.25544	0.312403	0.156202	+	0.987725i	$0.450075\pi$
$284$	0	0
$285$	2.11684	0.125391
$286$	0	0
$287$	38.7446	2.28702
$288$	0	0
$289$	−15.1168	−0.889226
$290$	0	0
$291$	42.9783	2.51943
$292$	0	0
$293$	−23.4891	−1.37225	−0.686125	−	0.727484i	$-0.740690\pi$
$293$	−23.4891	−1.37225	−0.686125	+	0.727484i	$0.740690\pi$
$294$	0	0
$295$	2.74456	0.159795
$296$	0	0
$297$	18.1168	1.05125
$298$	0	0
$299$	5.48913	0.317444
$300$	0	0
$301$	13.4891	0.777500
$302$	0	0
$303$	20.2337	1.16240
$304$	0	0
$305$	−5.37228	−0.307616
$306$	0	0
$307$	5.25544	0.299944	0.149972	−	0.988690i	$-0.452082\pi$
$307$	5.25544	0.299944	0.149972	+	0.988690i	$0.452082\pi$
$308$	0	0
$309$	−32.0000	−1.82042
$310$	0	0
$311$	−19.3723	−1.09850	−0.549251	−	0.835658i	$-0.685087\pi$
$311$	−19.3723	−1.09850	−0.549251	+	0.835658i	$0.685087\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	28.2337	1.59079
$316$	0	0
$317$	24.3505	1.36766	0.683831	−	0.729640i	$-0.260313\pi$
$317$	24.3505	1.36766	0.683831	+	0.729640i	$0.260313\pi$
$318$	0	0
$319$	1.37228	0.0768330
$320$	0	0
$321$	40.4674	2.25867
$322$	0	0
$323$	0.861407	0.0479299
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	−52.2337	−2.88853
$328$	0	0
$329$	9.25544	0.510269
$330$	0	0
$331$	30.9783	1.70272	0.851359	−	0.524583i	$-0.175779\pi$
$331$	30.9783	1.70272	0.851359	+	0.524583i	$0.175779\pi$
$332$	0	0
$333$	−78.4674	−4.29999
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	24.1168	1.31373	0.656864	−	0.754009i	$-0.271883\pi$
$337$	24.1168	1.31373	0.656864	+	0.754009i	$0.271883\pi$
$338$	0	0
$339$	−10.9783	−0.596257
$340$	0	0
$341$	3.37228	0.182619
$342$	0	0
$343$	8.86141	0.478471
$344$	0	0
$345$	−9.25544	−0.498296
$346$	0	0
$347$	−32.2337	−1.73040	−0.865198	−	0.501431i	$-0.832807\pi$
$347$	−32.2337	−1.73040	−0.865198	+	0.501431i	$0.832807\pi$
$348$	0	0
$349$	−19.4891	−1.04323	−0.521614	−	0.853181i	$-0.674670\pi$
$349$	−19.4891	−1.04323	−0.521614	+	0.853181i	$0.674670\pi$
$350$	0	0
$351$	36.2337	1.93401
$352$	0	0
$353$	−0.510875	−0.0271911	−0.0135956	−	0.999908i	$-0.504328\pi$
$353$	−0.510875	−0.0271911	−0.0135956	+	0.999908i	$0.504328\pi$
$354$	0	0
$355$	10.1168	0.536946
$356$	0	0
$357$	15.6060	0.825955
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−18.6060	−0.979262
$362$	0	0
$363$	−3.37228	−0.176999
$364$	0	0
$365$	15.4891	0.810738
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	−96.1902	−5.00746
$370$	0	0
$371$	−13.8832	−0.720778
$372$	0	0
$373$	−31.4891	−1.63045	−0.815223	−	0.579148i	$-0.803385\pi$
$373$	−31.4891	−1.63045	−0.815223	+	0.579148i	$0.803385\pi$
$374$	0	0
$375$	3.37228	0.174144
$376$	0	0
$377$	2.74456	0.141352
$378$	0	0
$379$	−0.233688	−0.0120037	−0.00600187	−	0.999982i	$-0.501910\pi$
$379$	−0.233688	−0.0120037	−0.00600187	+	0.999982i	$0.501910\pi$
$380$	0	0
$381$	26.9783	1.38214
$382$	0	0
$383$	−32.2337	−1.64706	−0.823532	−	0.567269i	$-0.808000\pi$
$383$	−32.2337	−1.64706	−0.823532	+	0.567269i	$0.808000\pi$
$384$	0	0
$385$	−3.37228	−0.171867
$386$	0	0
$387$	−33.4891	−1.70235
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−3.76631	−0.190471
$392$	0	0
$393$	−74.5842	−3.76228
$394$	0	0
$395$	−1.25544	−0.0631679
$396$	0	0
$397$	−24.9783	−1.25362	−0.626811	−	0.779171i	$-0.715640\pi$
$397$	−24.9783	−1.25362	−0.626811	+	0.779171i	$0.715640\pi$
$398$	0	0
$399$	7.13859	0.357377
$400$	0	0
$401$	13.3723	0.667780	0.333890	−	0.942612i	$-0.391639\pi$
$401$	13.3723	0.667780	0.333890	+	0.942612i	$0.391639\pi$
$402$	0	0
$403$	6.74456	0.335971
$404$	0	0
$405$	−35.9783	−1.78777
$406$	0	0
$407$	9.37228	0.464567
$408$	0	0
$409$	−1.76631	−0.0873385	−0.0436693	−	0.999046i	$-0.513905\pi$
$409$	−1.76631	−0.0873385	−0.0436693	+	0.999046i	$0.513905\pi$
$410$	0	0
$411$	−29.4891	−1.45459
$412$	0	0
$413$	9.25544	0.455430
$414$	0	0
$415$	2.74456	0.134725
$416$	0	0
$417$	13.4891	0.660565
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−10.2337	−0.498759	−0.249380	−	0.968406i	$-0.580227\pi$
$421$	−10.2337	−0.498759	−0.249380	+	0.968406i	$0.580227\pi$
$422$	0	0
$423$	−22.9783	−1.11724
$424$	0	0
$425$	1.37228	0.0665654
$426$	0	0
$427$	−18.1168	−0.876736
$428$	0	0
$429$	−6.74456	−0.325631
$430$	0	0
$431$	34.9783	1.68484	0.842422	−	0.538819i	$-0.181129\pi$
$431$	34.9783	1.68484	0.842422	+	0.538819i	$0.181129\pi$
$432$	0	0
$433$	27.7228	1.33227	0.666137	−	0.745830i	$-0.267947\pi$
$433$	27.7228	1.33227	0.666137	+	0.745830i	$0.267947\pi$
$434$	0	0
$435$	−4.62772	−0.221882
$436$	0	0
$437$	−1.72281	−0.0824133
$438$	0	0
$439$	−18.9783	−0.905782	−0.452891	−	0.891566i	$-0.649607\pi$
$439$	−18.9783	−0.905782	−0.452891	+	0.891566i	$0.649607\pi$
$440$	0	0
$441$	36.6060	1.74314
$442$	0	0
$443$	29.4891	1.40107	0.700535	−	0.713618i	$-0.252945\pi$
$443$	29.4891	1.40107	0.700535	+	0.713618i	$0.252945\pi$
$444$	0	0
$445$	1.37228	0.0650524
$446$	0	0
$447$	72.8614	3.44623
$448$	0	0
$449$	28.9783	1.36757	0.683784	−	0.729684i	$-0.260333\pi$
$449$	28.9783	1.36757	0.683784	+	0.729684i	$0.260333\pi$
$450$	0	0
$451$	11.4891	0.541002
$452$	0	0
$453$	−41.2554	−1.93835
$454$	0	0
$455$	−6.74456	−0.316190
$456$	0	0
$457$	−16.3505	−0.764846	−0.382423	−	0.923987i	$-0.624910\pi$
$457$	−16.3505	−0.764846	−0.382423	+	0.923987i	$0.624910\pi$
$458$	0	0
$459$	−24.8614	−1.16043
$460$	0	0
$461$	−16.1168	−0.750636	−0.375318	−	0.926896i	$-0.622467\pi$
$461$	−16.1168	−0.750636	−0.375318	+	0.926896i	$0.622467\pi$
$462$	0	0
$463$	0.233688	0.0108604	0.00543020	−	0.999985i	$-0.498272\pi$
$463$	0.233688	0.0108604	0.00543020	+	0.999985i	$0.498272\pi$
$464$	0	0
$465$	−11.3723	−0.527377
$466$	0	0
$467$	−19.3723	−0.896442	−0.448221	−	0.893923i	$-0.647942\pi$
$467$	−19.3723	−0.896442	−0.448221	+	0.893923i	$0.647942\pi$
$468$	0	0
$469$	−26.9783	−1.24574
$470$	0	0
$471$	31.6060	1.45633
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	0.627719	0.0288017
$476$	0	0
$477$	34.4674	1.57815
$478$	0	0
$479$	−5.48913	−0.250805	−0.125402	−	0.992106i	$-0.540022\pi$
$479$	−5.48913	−0.250805	−0.125402	+	0.992106i	$0.540022\pi$
$480$	0	0
$481$	18.7446	0.854678
$482$	0	0
$483$	−31.2119	−1.42019
$484$	0	0
$485$	12.7446	0.578701
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	19.8397	0.897180
$490$	0	0
$491$	−7.37228	−0.332706	−0.166353	−	0.986066i	$-0.553199\pi$
$491$	−7.37228	−0.332706	−0.166353	+	0.986066i	$0.553199\pi$
$492$	0	0
$493$	−1.88316	−0.0848131
$494$	0	0
$495$	8.37228	0.376306
$496$	0	0
$497$	34.1168	1.53035
$498$	0	0
$499$	−33.4891	−1.49918	−0.749590	−	0.661903i	$-0.769749\pi$
$499$	−33.4891	−1.49918	−0.749590	+	0.661903i	$0.769749\pi$
$500$	0	0
$501$	−15.6060	−0.697223
$502$	0	0
$503$	34.9783	1.55960	0.779802	−	0.626027i	$-0.215320\pi$
$503$	34.9783	1.55960	0.779802	+	0.626027i	$0.215320\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	30.3505	1.34791
$508$	0	0
$509$	−9.76631	−0.432884	−0.216442	−	0.976295i	$-0.569445\pi$
$509$	−9.76631	−0.432884	−0.216442	+	0.976295i	$0.569445\pi$
$510$	0	0
$511$	52.2337	2.31068
$512$	0	0
$513$	−11.3723	−0.502098
$514$	0	0
$515$	−9.48913	−0.418141
$516$	0	0
$517$	2.74456	0.120706
$518$	0	0
$519$	−20.2337	−0.888160
$520$	0	0
$521$	12.5109	0.548111	0.274056	−	0.961714i	$-0.411635\pi$
$521$	12.5109	0.548111	0.274056	+	0.961714i	$0.411635\pi$
$522$	0	0
$523$	30.9783	1.35458	0.677292	−	0.735714i	$-0.263153\pi$
$523$	30.9783	1.35458	0.677292	+	0.735714i	$0.263153\pi$
$524$	0	0
$525$	11.3723	0.496327
$526$	0	0
$527$	−4.62772	−0.201587
$528$	0	0
$529$	−15.4674	−0.672495
$530$	0	0
$531$	−22.9783	−0.997171
$532$	0	0
$533$	22.9783	0.995299
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	40.4674	1.74630
$538$	0	0
$539$	−4.37228	−0.188327
$540$	0	0
$541$	20.1168	0.864891	0.432445	−	0.901660i	$-0.357651\pi$
$541$	20.1168	0.864891	0.432445	+	0.901660i	$0.357651\pi$
$542$	0	0
$543$	−33.7228	−1.44718
$544$	0	0
$545$	−15.4891	−0.663481
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	44.9783	1.91962
$550$	0	0
$551$	−0.861407	−0.0366972
$552$	0	0
$553$	−4.23369	−0.180035
$554$	0	0
$555$	−31.6060	−1.34160
$556$	0	0
$557$	−4.97825	−0.210935	−0.105468	−	0.994423i	$-0.533634\pi$
$557$	−4.97825	−0.210935	−0.105468	+	0.994423i	$0.533634\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	4.62772	0.195382
$562$	0	0
$563$	−8.23369	−0.347009	−0.173504	−	0.984833i	$-0.555509\pi$
$563$	−8.23369	−0.347009	−0.173504	+	0.984833i	$0.555509\pi$
$564$	0	0
$565$	−3.25544	−0.136957
$566$	0	0
$567$	−121.329	−5.09533
$568$	0	0
$569$	−15.2554	−0.639541	−0.319771	−	0.947495i	$-0.603606\pi$
$569$	−15.2554	−0.639541	−0.319771	+	0.947495i	$0.603606\pi$
$570$	0	0
$571$	15.3723	0.643310	0.321655	−	0.946857i	$-0.395761\pi$
$571$	15.3723	0.643310	0.321655	+	0.946857i	$0.395761\pi$
$572$	0	0
$573$	−18.5109	−0.773303
$574$	0	0
$575$	−2.74456	−0.114456
$576$	0	0
$577$	36.9783	1.53942	0.769712	−	0.638391i	$-0.220400\pi$
$577$	36.9783	1.53942	0.769712	+	0.638391i	$0.220400\pi$
$578$	0	0
$579$	−50.1168	−2.08278
$580$	0	0
$581$	9.25544	0.383980
$582$	0	0
$583$	−4.11684	−0.170502
$584$	0	0
$585$	16.7446	0.692302
$586$	0	0
$587$	24.8614	1.02614	0.513070	−	0.858347i	$-0.328508\pi$
$587$	24.8614	1.02614	0.513070	+	0.858347i	$0.328508\pi$
$588$	0	0
$589$	−2.11684	−0.0872230
$590$	0	0
$591$	−69.9565	−2.87763
$592$	0	0
$593$	−12.5109	−0.513760	−0.256880	−	0.966443i	$-0.582695\pi$
$593$	−12.5109	−0.513760	−0.256880	+	0.966443i	$0.582695\pi$
$594$	0	0
$595$	4.62772	0.189718
$596$	0	0
$597$	61.0951	2.50046
$598$	0	0
$599$	39.6060	1.61826	0.809128	−	0.587632i	$-0.199940\pi$
$599$	39.6060	1.61826	0.809128	+	0.587632i	$0.199940\pi$
$600$	0	0
$601$	−16.5109	−0.673493	−0.336746	−	0.941595i	$-0.609326\pi$
$601$	−16.5109	−0.673493	−0.336746	+	0.941595i	$0.609326\pi$
$602$	0	0
$603$	66.9783	2.72757
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	5.88316	0.238790	0.119395	−	0.992847i	$-0.461905\pi$
$607$	5.88316	0.238790	0.119395	+	0.992847i	$0.461905\pi$
$608$	0	0
$609$	−15.6060	−0.632386
$610$	0	0
$611$	5.48913	0.222066
$612$	0	0
$613$	−20.5109	−0.828426	−0.414213	−	0.910180i	$-0.635943\pi$
$613$	−20.5109	−0.828426	−0.414213	+	0.910180i	$0.635943\pi$
$614$	0	0
$615$	−38.7446	−1.56233
$616$	0	0
$617$	−2.23369	−0.0899249	−0.0449624	−	0.998989i	$-0.514317\pi$
$617$	−2.23369	−0.0899249	−0.0449624	+	0.998989i	$0.514317\pi$
$618$	0	0
$619$	−44.4674	−1.78729	−0.893647	−	0.448770i	$-0.851862\pi$
$619$	−44.4674	−1.78729	−0.893647	+	0.448770i	$0.851862\pi$
$620$	0	0
$621$	49.7228	1.99531
$622$	0	0
$623$	4.62772	0.185406
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.11684	0.0845386
$628$	0	0
$629$	−12.8614	−0.512818
$630$	0	0
$631$	−42.1168	−1.67665	−0.838323	−	0.545175i	$-0.816463\pi$
$631$	−42.1168	−1.67665	−0.838323	+	0.545175i	$0.816463\pi$
$632$	0	0
$633$	−20.6277	−0.819878
$634$	0	0
$635$	8.00000	0.317470
$636$	0	0
$637$	−8.74456	−0.346472
$638$	0	0
$639$	−84.7011	−3.35072
$640$	0	0
$641$	−27.0951	−1.07019	−0.535096	−	0.844791i	$-0.679725\pi$
$641$	−27.0951	−1.07019	−0.535096	+	0.844791i	$0.679725\pi$
$642$	0	0
$643$	−5.88316	−0.232009	−0.116005	−	0.993249i	$-0.537009\pi$
$643$	−5.88316	−0.232009	−0.116005	+	0.993249i	$0.537009\pi$
$644$	0	0
$645$	−13.4891	−0.531134
$646$	0	0
$647$	37.7228	1.48304	0.741518	−	0.670933i	$-0.234106\pi$
$647$	37.7228	1.48304	0.741518	+	0.670933i	$0.234106\pi$
$648$	0	0
$649$	2.74456	0.107734
$650$	0	0
$651$	−38.3505	−1.50308
$652$	0	0
$653$	−10.6277	−0.415895	−0.207947	−	0.978140i	$-0.566678\pi$
$653$	−10.6277	−0.415895	−0.207947	+	0.978140i	$0.566678\pi$
$654$	0	0
$655$	−22.1168	−0.864177
$656$	0	0
$657$	−129.679	−5.05927
$658$	0	0
$659$	12.8614	0.501009	0.250505	−	0.968115i	$-0.419403\pi$
$659$	12.8614	0.501009	0.250505	+	0.968115i	$0.419403\pi$
$660$	0	0
$661$	−8.51087	−0.331035	−0.165517	−	0.986207i	$-0.552929\pi$
$661$	−8.51087	−0.331035	−0.165517	+	0.986207i	$0.552929\pi$
$662$	0	0
$663$	9.25544	0.359451
$664$	0	0
$665$	2.11684	0.0820877
$666$	0	0
$667$	3.76631	0.145832
$668$	0	0
$669$	−63.2119	−2.44391
$670$	0	0
$671$	−5.37228	−0.207395
$672$	0	0
$673$	14.8614	0.572865	0.286433	−	0.958100i	$-0.407531\pi$
$673$	14.8614	0.572865	0.286433	+	0.958100i	$0.407531\pi$
$674$	0	0
$675$	−18.1168	−0.697318
$676$	0	0
$677$	−3.25544	−0.125117	−0.0625583	−	0.998041i	$-0.519926\pi$
$677$	−3.25544	−0.125117	−0.0625583	+	0.998041i	$0.519926\pi$
$678$	0	0
$679$	42.9783	1.64935
$680$	0	0
$681$	9.25544	0.354669
$682$	0	0
$683$	−28.6277	−1.09541	−0.547705	−	0.836672i	$-0.684498\pi$
$683$	−28.6277	−1.09541	−0.547705	+	0.836672i	$0.684498\pi$
$684$	0	0
$685$	−8.74456	−0.334113
$686$	0	0
$687$	−33.7228	−1.28661
$688$	0	0
$689$	−8.23369	−0.313679
$690$	0	0
$691$	40.2337	1.53056	0.765281	−	0.643697i	$-0.222600\pi$
$691$	40.2337	1.53056	0.765281	+	0.643697i	$0.222600\pi$
$692$	0	0
$693$	28.2337	1.07251
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−15.7663	−0.597192
$698$	0	0
$699$	−4.62772	−0.175036
$700$	0	0
$701$	37.3723	1.41153	0.705766	−	0.708445i	$-0.250603\pi$
$701$	37.3723	1.41153	0.705766	+	0.708445i	$0.250603\pi$
$702$	0	0
$703$	−5.88316	−0.221887
$704$	0	0
$705$	−9.25544	−0.348580
$706$	0	0
$707$	20.2337	0.760966
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	10.5109	0.394189
$712$	0	0
$713$	9.25544	0.346619
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	0	0
$717$	−49.7228	−1.85693
$718$	0	0
$719$	−13.8832	−0.517754	−0.258877	−	0.965910i	$-0.583352\pi$
$719$	−13.8832	−0.517754	−0.258877	+	0.965910i	$0.583352\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	0	0
$723$	74.1902	2.75916
$724$	0	0
$725$	−1.37228	−0.0509652
$726$	0	0
$727$	24.2337	0.898778	0.449389	−	0.893336i	$-0.351642\pi$
$727$	24.2337	0.898778	0.449389	+	0.893336i	$0.351642\pi$
$728$	0	0
$729$	117.935	4.36795
$730$	0	0
$731$	−5.48913	−0.203023
$732$	0	0
$733$	−46.2337	−1.70768	−0.853840	−	0.520535i	$-0.825732\pi$
$733$	−46.2337	−1.70768	−0.853840	+	0.520535i	$0.825732\pi$
$734$	0	0
$735$	14.7446	0.543861
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−20.4674	−0.752905	−0.376452	−	0.926436i	$-0.622856\pi$
$739$	−20.4674	−0.752905	−0.376452	+	0.926436i	$0.622856\pi$
$740$	0	0
$741$	4.23369	0.155528
$742$	0	0
$743$	4.62772	0.169775	0.0848873	−	0.996391i	$-0.472947\pi$
$743$	4.62772	0.169775	0.0848873	+	0.996391i	$0.472947\pi$
$744$	0	0
$745$	21.6060	0.791581
$746$	0	0
$747$	−22.9783	−0.840730
$748$	0	0
$749$	40.4674	1.47865
$750$	0	0
$751$	−8.86141	−0.323357	−0.161679	−	0.986843i	$-0.551691\pi$
$751$	−8.86141	−0.323357	−0.161679	+	0.986843i	$0.551691\pi$
$752$	0	0
$753$	−9.25544	−0.337287
$754$	0	0
$755$	−12.2337	−0.445229
$756$	0	0
$757$	20.9783	0.762467	0.381234	−	0.924479i	$-0.375499\pi$
$757$	20.9783	0.762467	0.381234	+	0.924479i	$0.375499\pi$
$758$	0	0
$759$	−9.25544	−0.335951
$760$	0	0
$761$	4.97825	0.180461	0.0902307	−	0.995921i	$-0.471240\pi$
$761$	4.97825	0.180461	0.0902307	+	0.995921i	$0.471240\pi$
$762$	0	0
$763$	−52.2337	−1.89099
$764$	0	0
$765$	−11.4891	−0.415390
$766$	0	0
$767$	5.48913	0.198201
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	−60.7011	−2.18610
$772$	0	0
$773$	33.6060	1.20872	0.604361	−	0.796710i	$-0.293428\pi$
$773$	33.6060	1.20872	0.604361	+	0.796710i	$0.293428\pi$
$774$	0	0
$775$	−3.37228	−0.121136
$776$	0	0
$777$	−106.584	−3.82369
$778$	0	0
$779$	−7.21194	−0.258395
$780$	0	0
$781$	10.1168	0.362009
$782$	0	0
$783$	24.8614	0.888474
$784$	0	0
$785$	9.37228	0.334511
$786$	0	0
$787$	−44.4674	−1.58509	−0.792545	−	0.609813i	$-0.791245\pi$
$787$	−44.4674	−1.58509	−0.792545	+	0.609813i	$0.791245\pi$
$788$	0	0
$789$	−83.8397	−2.98477
$790$	0	0
$791$	−10.9783	−0.390342
$792$	0	0
$793$	−10.7446	−0.381551
$794$	0	0
$795$	13.8832	0.492385
$796$	0	0
$797$	−11.4891	−0.406966	−0.203483	−	0.979079i	$-0.565226\pi$
$797$	−11.4891	−0.406966	−0.203483	+	0.979079i	$0.565226\pi$
$798$	0	0
$799$	−3.76631	−0.133243
$800$	0	0
$801$	−11.4891	−0.405948
$802$	0	0
$803$	15.4891	0.546599
$804$	0	0
$805$	−9.25544	−0.326211
$806$	0	0
$807$	−29.4891	−1.03807
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	44.8614	1.57530	0.787649	−	0.616125i	$-0.211298\pi$
$811$	44.8614	1.57530	0.787649	+	0.616125i	$0.211298\pi$
$812$	0	0
$813$	−53.9565	−1.89234
$814$	0	0
$815$	5.88316	0.206078
$816$	0	0
$817$	−2.51087	−0.0878444
$818$	0	0
$819$	56.4674	1.97313
$820$	0	0
$821$	−11.4891	−0.400973	−0.200487	−	0.979696i	$-0.564252\pi$
$821$	−11.4891	−0.400973	−0.200487	+	0.979696i	$0.564252\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	0	0
$825$	3.37228	0.117408
$826$	0	0
$827$	46.9783	1.63359	0.816797	−	0.576925i	$-0.195748\pi$
$827$	46.9783	1.63359	0.816797	+	0.576925i	$0.195748\pi$
$828$	0	0
$829$	24.7446	0.859414	0.429707	−	0.902968i	$-0.358617\pi$
$829$	24.7446	0.859414	0.429707	+	0.902968i	$0.358617\pi$
$830$	0	0
$831$	−42.9783	−1.49090
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	−4.62772	−0.160149
$836$	0	0
$837$	61.0951	2.11176
$838$	0	0
$839$	10.9783	0.379011	0.189506	−	0.981880i	$-0.439311\pi$
$839$	10.9783	0.379011	0.189506	+	0.981880i	$0.439311\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	0	0
$843$	−60.7011	−2.09066
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	−3.37228	−0.115873
$848$	0	0
$849$	−17.7228	−0.608245
$850$	0	0
$851$	25.7228	0.881767
$852$	0	0
$853$	38.4674	1.31710	0.658549	−	0.752538i	$-0.271171\pi$
$853$	38.4674	1.31710	0.658549	+	0.752538i	$0.271171\pi$
$854$	0	0
$855$	−5.25544	−0.179732
$856$	0	0
$857$	36.3505	1.24171	0.620855	−	0.783925i	$-0.286785\pi$
$857$	36.3505	1.24171	0.620855	+	0.783925i	$0.286785\pi$
$858$	0	0
$859$	−42.7446	−1.45843	−0.729213	−	0.684287i	$-0.760114\pi$
$859$	−42.7446	−1.45843	−0.729213	+	0.684287i	$0.760114\pi$
$860$	0	0
$861$	−130.658	−4.45280
$862$	0	0
$863$	−21.2554	−0.723544	−0.361772	−	0.932267i	$-0.617828\pi$
$863$	−21.2554	−0.723544	−0.361772	+	0.932267i	$0.617828\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	50.9783	1.73131
$868$	0	0
$869$	−1.25544	−0.0425878
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	−106.701	−3.61128
$874$	0	0
$875$	3.37228	0.114004
$876$	0	0
$877$	−36.9783	−1.24867	−0.624333	−	0.781158i	$-0.714629\pi$
$877$	−36.9783	−1.24867	−0.624333	+	0.781158i	$0.714629\pi$
$878$	0	0
$879$	79.2119	2.67175
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	3.37228	0.113486	0.0567432	−	0.998389i	$-0.481928\pi$
$883$	3.37228	0.113486	0.0567432	+	0.998389i	$0.481928\pi$
$884$	0	0
$885$	−9.25544	−0.311118
$886$	0	0
$887$	10.9783	0.368614	0.184307	−	0.982869i	$-0.440996\pi$
$887$	10.9783	0.368614	0.184307	+	0.982869i	$0.440996\pi$
$888$	0	0
$889$	26.9783	0.904821
$890$	0	0
$891$	−35.9783	−1.20532
$892$	0	0
$893$	−1.72281	−0.0576517
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−18.5109	−0.618060
$898$	0	0
$899$	4.62772	0.154343
$900$	0	0
$901$	5.64947	0.188211
$902$	0	0
$903$	−45.4891	−1.51378
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−0.394031	−0.0130836	−0.00654179	−	0.999979i	$-0.502082\pi$
$907$	−0.394031	−0.0130836	−0.00654179	+	0.999979i	$0.502082\pi$
$908$	0	0
$909$	−50.2337	−1.66615
$910$	0	0
$911$	8.39403	0.278107	0.139053	−	0.990285i	$-0.455594\pi$
$911$	8.39403	0.278107	0.139053	+	0.990285i	$0.455594\pi$
$912$	0	0
$913$	2.74456	0.0908318
$914$	0	0
$915$	18.1168	0.598924
$916$	0	0
$917$	−74.5842	−2.46299
$918$	0	0
$919$	−18.9783	−0.626035	−0.313017	−	0.949747i	$-0.601340\pi$
$919$	−18.9783	−0.626035	−0.313017	+	0.949747i	$0.601340\pi$
$920$	0	0
$921$	−17.7228	−0.583987
$922$	0	0
$923$	20.2337	0.666000
$924$	0	0
$925$	−9.37228	−0.308159
$926$	0	0
$927$	79.4456	2.60934
$928$	0	0
$929$	24.3505	0.798915	0.399458	−	0.916752i	$-0.369199\pi$
$929$	24.3505	0.798915	0.399458	+	0.916752i	$0.369199\pi$
$930$	0	0
$931$	2.74456	0.0899494
$932$	0	0
$933$	65.3288	2.13877
$934$	0	0
$935$	1.37228	0.0448784
$936$	0	0
$937$	−28.5109	−0.931410	−0.465705	−	0.884940i	$-0.654199\pi$
$937$	−28.5109	−0.931410	−0.465705	+	0.884940i	$0.654199\pi$
$938$	0	0
$939$	74.1902	2.42111
$940$	0	0
$941$	15.0951	0.492086	0.246043	−	0.969259i	$-0.420870\pi$
$941$	15.0951	0.492086	0.246043	+	0.969259i	$0.420870\pi$
$942$	0	0
$943$	31.5326	1.02684
$944$	0	0
$945$	−61.0951	−1.98742
$946$	0	0
$947$	48.8614	1.58778	0.793891	−	0.608060i	$-0.208052\pi$
$947$	48.8614	1.58778	0.793891	+	0.608060i	$0.208052\pi$
$948$	0	0
$949$	30.9783	1.00560
$950$	0	0
$951$	−82.1168	−2.66282
$952$	0	0
$953$	40.1168	1.29951	0.649756	−	0.760143i	$-0.274871\pi$
$953$	40.1168	1.29951	0.649756	+	0.760143i	$0.274871\pi$
$954$	0	0
$955$	−5.48913	−0.177624
$956$	0	0
$957$	−4.62772	−0.149593
$958$	0	0
$959$	−29.4891	−0.952254
$960$	0	0
$961$	−19.6277	−0.633152
$962$	0	0
$963$	−100.467	−3.23752
$964$	0	0
$965$	−14.8614	−0.478406
$966$	0	0
$967$	−47.6060	−1.53090	−0.765452	−	0.643493i	$-0.777485\pi$
$967$	−47.6060	−1.53090	−0.765452	+	0.643493i	$0.777485\pi$
$968$	0	0
$969$	−2.90491	−0.0933190
$970$	0	0
$971$	−1.02175	−0.0327895	−0.0163947	−	0.999866i	$-0.505219\pi$
$971$	−1.02175	−0.0327895	−0.0163947	+	0.999866i	$0.505219\pi$
$972$	0	0
$973$	13.4891	0.432442
$974$	0	0
$975$	6.74456	0.215999
$976$	0	0
$977$	14.2337	0.455376	0.227688	−	0.973734i	$-0.426883\pi$
$977$	14.2337	0.455376	0.227688	+	0.973734i	$0.426883\pi$
$978$	0	0
$979$	1.37228	0.0438583
$980$	0	0
$981$	129.679	4.14034
$982$	0	0
$983$	13.7228	0.437690	0.218845	−	0.975760i	$-0.429771\pi$
$983$	13.7228	0.437690	0.218845	+	0.975760i	$0.429771\pi$
$984$	0	0
$985$	−20.7446	−0.660977
$986$	0	0
$987$	−31.2119	−0.993487
$988$	0	0
$989$	10.9783	0.349088
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	−104.467	−3.31517
$994$	0	0
$995$	18.1168	0.574343
$996$	0	0
$997$	−22.2337	−0.704148	−0.352074	−	0.935972i	$-0.614523\pi$
$997$	−22.2337	−0.704148	−0.352074	+	0.935972i	$0.614523\pi$
$998$	0	0
$999$	169.796	5.37211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3520.2.a.bj.1.1		2
4.3	odd	2		3520.2.a.bq.1.2		2
8.3	odd	2		110.2.a.d.1.1	✓	2
8.5	even	2		880.2.a.n.1.2		2
24.5	odd	2		7920.2.a.bq.1.1		2
24.11	even	2		990.2.a.m.1.2		2
40.3	even	4		550.2.b.f.199.3		4
40.13	odd	4		4400.2.b.p.4049.4		4
40.19	odd	2		550.2.a.n.1.2		2
40.27	even	4		550.2.b.f.199.2		4
40.29	even	2		4400.2.a.bl.1.1		2
40.37	odd	4		4400.2.b.p.4049.1		4
56.27	even	2		5390.2.a.bp.1.2		2
88.21	odd	2		9680.2.a.bt.1.2		2
88.43	even	2		1210.2.a.r.1.1		2
120.59	even	2		4950.2.a.bw.1.1		2
120.83	odd	4		4950.2.c.bc.199.1		4
120.107	odd	4		4950.2.c.bc.199.4		4
440.219	even	2		6050.2.a.cb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.d.1.1	✓	2	8.3	odd	2
550.2.a.n.1.2		2	40.19	odd	2
550.2.b.f.199.2		4	40.27	even	4
550.2.b.f.199.3		4	40.3	even	4
880.2.a.n.1.2		2	8.5	even	2
990.2.a.m.1.2		2	24.11	even	2
1210.2.a.r.1.1		2	88.43	even	2
3520.2.a.bj.1.1		2	1.1	even	1	trivial
3520.2.a.bq.1.2		2	4.3	odd	2
4400.2.a.bl.1.1		2	40.29	even	2
4400.2.b.p.4049.1		4	40.37	odd	4
4400.2.b.p.4049.4		4	40.13	odd	4
4950.2.a.bw.1.1		2	120.59	even	2
4950.2.c.bc.199.1		4	120.83	odd	4
4950.2.c.bc.199.4		4	120.107	odd	4
5390.2.a.bp.1.2		2	56.27	even	2
6050.2.a.cb.1.2		2	440.219	even	2
7920.2.a.bq.1.1		2	24.5	odd	2
9680.2.a.bt.1.2		2	88.21	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3520.a (trivial)

\(f(q)\)	\(=\)	\(q-3.37228 q^{3} -1.00000 q^{5} -3.37228 q^{7} +8.37228 q^{9} -1.00000 q^{11} -2.00000 q^{13} +3.37228 q^{15} +1.37228 q^{17} +0.627719 q^{19} +11.3723 q^{21} -2.74456 q^{23} +1.00000 q^{25} -18.1168 q^{27} -1.37228 q^{29} -3.37228 q^{31} +3.37228 q^{33} +3.37228 q^{35} -9.37228 q^{37} +6.74456 q^{39} -11.4891 q^{41} -4.00000 q^{43} -8.37228 q^{45} -2.74456 q^{47} +4.37228 q^{49} -4.62772 q^{51} +4.11684 q^{53} +1.00000 q^{55} -2.11684 q^{57} -2.74456 q^{59} +5.37228 q^{61} -28.2337 q^{63} +2.00000 q^{65} +8.00000 q^{67} +9.25544 q^{69} -10.1168 q^{71} -15.4891 q^{73} -3.37228 q^{75} +3.37228 q^{77} +1.25544 q^{79} +35.9783 q^{81} -2.74456 q^{83} -1.37228 q^{85} +4.62772 q^{87} -1.37228 q^{89} +6.74456 q^{91} +11.3723 q^{93} -0.627719 q^{95} -12.7446 q^{97} -8.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} - q^{7} + 11 q^{9} - 2 q^{11} - 4 q^{13} + q^{15} - 3 q^{17} + 7 q^{19} + 17 q^{21} + 6 q^{23} + 2 q^{25} - 19 q^{27} + 3 q^{29} - q^{31} + q^{33} + q^{35} - 13 q^{37} + 2 q^{39}+ \cdots - 11 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 - q^7 + 11 * q^9 - 2 * q^11 - 4 * q^13 + q^15 - 3 * q^17 + 7 * q^19 + 17 * q^21 + 6 * q^23 + 2 * q^25 - 19 * q^27 + 3 * q^29 - q^31 + q^33 + q^35 - 13 * q^37 + 2 * q^39 - 8 * q^43 - 11 * q^45 + 6 * q^47 + 3 * q^49 - 15 * q^51 - 9 * q^53 + 2 * q^55 + 13 * q^57 + 6 * q^59 + 5 * q^61 - 22 * q^63 + 4 * q^65 + 16 * q^67 + 30 * q^69 - 3 * q^71 - 8 * q^73 - q^75 + q^77 + 14 * q^79 + 26 * q^81 + 6 * q^83 + 3 * q^85 + 15 * q^87 + 3 * q^89 + 2 * q^91 + 17 * q^93 - 7 * q^95 - 14 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more