LMFDB - Embedded newform 8040.2.a.bc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 37x^{8} + 132x^{7} + 358x^{6} - 1708x^{5} - 92x^{4} + 5969x^{3} - 3864x^{2} - 4752x + 3524 $ x^10 - 3x^9 - 37x^8 + 132x^7 + 358x^6 - 1708x^5 - 92x^4 + 5969x^3 - 3864x^2 - 4752*x + 3524
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.04435$ of defining polynomial
Character	$\chi$	$=$	8040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -1.71989 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -1.71989 q^{7} +1.00000 q^{9} -1.36824 q^{11} +0.754504 q^{13} -1.00000 q^{15} -5.06685 q^{17} -5.43216 q^{19} +1.71989 q^{21} -4.81345 q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.51684 q^{29} +2.98501 q^{31} +1.36824 q^{33} -1.71989 q^{35} -3.95474 q^{37} -0.754504 q^{39} -4.98254 q^{41} -4.84665 q^{43} +1.00000 q^{45} +7.20514 q^{47} -4.04198 q^{49} +5.06685 q^{51} +3.05763 q^{53} -1.36824 q^{55} +5.43216 q^{57} -12.7030 q^{59} +5.80714 q^{61} -1.71989 q^{63} +0.754504 q^{65} -1.00000 q^{67} +4.81345 q^{69} +2.01465 q^{71} +7.75941 q^{73} -1.00000 q^{75} +2.35322 q^{77} +14.6271 q^{79} +1.00000 q^{81} -1.49185 q^{83} -5.06685 q^{85} -9.51684 q^{87} +1.85489 q^{89} -1.29766 q^{91} -2.98501 q^{93} -5.43216 q^{95} +11.5865 q^{97} -1.36824 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9	$ 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9} + 7 q^{11} - q^{13} - 10 q^{15} + 3 q^{17} + 4 q^{19} - q^{21} - 13 q^{23} + 10 q^{25} - 10 q^{27} + 18 q^{29} - 9 q^{31} - 7 q^{33} + q^{35} - 3 q^{37} + q^{39} + 19 q^{41} + 5 q^{43} + 10 q^{45} - 8 q^{47} + 43 q^{49} - 3 q^{51} + 17 q^{53} + 7 q^{55} - 4 q^{57} + 24 q^{59} + 21 q^{61} + q^{63} - q^{65} - 10 q^{67} + 13 q^{69} + 2 q^{71} + 25 q^{73} - 10 q^{75} + 15 q^{77} - q^{79} + 10 q^{81} - 6 q^{83} + 3 q^{85} - 18 q^{87} + 23 q^{89} + 29 q^{91} + 9 q^{93} + 4 q^{95} + 21 q^{97} + 7 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9 + 7 * q^11 - q^13 - 10 * q^15 + 3 * q^17 + 4 * q^19 - q^21 - 13 * q^23 + 10 * q^25 - 10 * q^27 + 18 * q^29 - 9 * q^31 - 7 * q^33 + q^35 - 3 * q^37 + q^39 + 19 * q^41 + 5 * q^43 + 10 * q^45 - 8 * q^47 + 43 * q^49 - 3 * q^51 + 17 * q^53 + 7 * q^55 - 4 * q^57 + 24 * q^59 + 21 * q^61 + q^63 - q^65 - 10 * q^67 + 13 * q^69 + 2 * q^71 + 25 * q^73 - 10 * q^75 + 15 * q^77 - q^79 + 10 * q^81 - 6 * q^83 + 3 * q^85 - 18 * q^87 + 23 * q^89 + 29 * q^91 + 9 * q^93 + 4 * q^95 + 21 * q^97 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.71989	−0.650057	−0.325028	−	0.945704i	$-0.605374\pi$
$7$	−1.71989	−0.650057	−0.325028	+	0.945704i	$0.605374\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.36824	−0.412539	−0.206269	−	0.978495i	$-0.566132\pi$
$11$	−1.36824	−0.412539	−0.206269	+	0.978495i	$0.566132\pi$
$12$	0	0
$13$	0.754504	0.209262	0.104631	−	0.994511i	$-0.466634\pi$
$13$	0.754504	0.209262	0.104631	+	0.994511i	$0.466634\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−5.06685	−1.22889	−0.614446	−	0.788959i	$-0.710620\pi$
$17$	−5.06685	−1.22889	−0.614446	+	0.788959i	$0.710620\pi$
$18$	0	0
$19$	−5.43216	−1.24622	−0.623111	−	0.782133i	$-0.714132\pi$
$19$	−5.43216	−1.24622	−0.623111	+	0.782133i	$0.714132\pi$
$20$	0	0
$21$	1.71989	0.375310
$22$	0	0
$23$	−4.81345	−1.00367	−0.501837	−	0.864962i	$-0.667342\pi$
$23$	−4.81345	−1.00367	−0.501837	+	0.864962i	$0.667342\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.51684	1.76723	0.883616	−	0.468212i	$-0.155102\pi$
$29$	9.51684	1.76723	0.883616	+	0.468212i	$0.155102\pi$
$30$	0	0
$31$	2.98501	0.536124	0.268062	−	0.963402i	$-0.413617\pi$
$31$	2.98501	0.536124	0.268062	+	0.963402i	$0.413617\pi$
$32$	0	0
$33$	1.36824	0.238179
$34$	0	0
$35$	−1.71989	−0.290714
$36$	0	0
$37$	−3.95474	−0.650156	−0.325078	−	0.945687i	$-0.605391\pi$
$37$	−3.95474	−0.650156	−0.325078	+	0.945687i	$0.605391\pi$
$38$	0	0
$39$	−0.754504	−0.120817
$40$	0	0
$41$	−4.98254	−0.778141	−0.389071	−	0.921208i	$-0.627204\pi$
$41$	−4.98254	−0.778141	−0.389071	+	0.921208i	$0.627204\pi$
$42$	0	0
$43$	−4.84665	−0.739106	−0.369553	−	0.929210i	$-0.620489\pi$
$43$	−4.84665	−0.739106	−0.369553	+	0.929210i	$0.620489\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	7.20514	1.05098	0.525489	−	0.850800i	$-0.323882\pi$
$47$	7.20514	1.05098	0.525489	+	0.850800i	$0.323882\pi$
$48$	0	0
$49$	−4.04198	−0.577426
$50$	0	0
$51$	5.06685	0.709501
$52$	0	0
$53$	3.05763	0.419998	0.209999	−	0.977702i	$-0.432654\pi$
$53$	3.05763	0.419998	0.209999	+	0.977702i	$0.432654\pi$
$54$	0	0
$55$	−1.36824	−0.184493
$56$	0	0
$57$	5.43216	0.719507
$58$	0	0
$59$	−12.7030	−1.65379	−0.826894	−	0.562358i	$-0.809894\pi$
$59$	−12.7030	−1.65379	−0.826894	+	0.562358i	$0.809894\pi$
$60$	0	0
$61$	5.80714	0.743528	0.371764	−	0.928327i	$-0.378753\pi$
$61$	5.80714	0.743528	0.371764	+	0.928327i	$0.378753\pi$
$62$	0	0
$63$	−1.71989	−0.216686
$64$	0	0
$65$	0.754504	0.0935847
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	4.81345	0.579471
$70$	0	0
$71$	2.01465	0.239095	0.119548	−	0.992828i	$-0.461856\pi$
$71$	2.01465	0.239095	0.119548	+	0.992828i	$0.461856\pi$
$72$	0	0
$73$	7.75941	0.908170	0.454085	−	0.890958i	$-0.349966\pi$
$73$	7.75941	0.908170	0.454085	+	0.890958i	$0.349966\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	2.35322	0.268174
$78$	0	0
$79$	14.6271	1.64568	0.822838	−	0.568275i	$-0.192389\pi$
$79$	14.6271	1.64568	0.822838	+	0.568275i	$0.192389\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.49185	−0.163752	−0.0818762	−	0.996643i	$-0.526091\pi$
$83$	−1.49185	−0.163752	−0.0818762	+	0.996643i	$0.526091\pi$
$84$	0	0
$85$	−5.06685	−0.549577
$86$	0	0
$87$	−9.51684	−1.02031
$88$	0	0
$89$	1.85489	0.196618	0.0983091	−	0.995156i	$-0.468657\pi$
$89$	1.85489	0.196618	0.0983091	+	0.995156i	$0.468657\pi$
$90$	0	0
$91$	−1.29766	−0.136032
$92$	0	0
$93$	−2.98501	−0.309531
$94$	0	0
$95$	−5.43216	−0.557328
$96$	0	0
$97$	11.5865	1.17643	0.588215	−	0.808704i	$-0.299831\pi$
$97$	11.5865	1.17643	0.588215	+	0.808704i	$0.299831\pi$
$98$	0	0
$99$	−1.36824	−0.137513
$100$	0	0
$101$	10.9950	1.09404	0.547021	−	0.837119i	$-0.315762\pi$
$101$	10.9950	1.09404	0.547021	+	0.837119i	$0.315762\pi$
$102$	0	0
$103$	−0.717921	−0.0707389	−0.0353695	−	0.999374i	$-0.511261\pi$
$103$	−0.717921	−0.0707389	−0.0353695	+	0.999374i	$0.511261\pi$
$104$	0	0
$105$	1.71989	0.167844
$106$	0	0
$107$	−1.94199	−0.187739	−0.0938695	−	0.995585i	$-0.529924\pi$
$107$	−1.94199	−0.187739	−0.0938695	+	0.995585i	$0.529924\pi$
$108$	0	0
$109$	14.3548	1.37494	0.687472	−	0.726211i	$-0.258720\pi$
$109$	14.3548	1.37494	0.687472	+	0.726211i	$0.258720\pi$
$110$	0	0
$111$	3.95474	0.375368
$112$	0	0
$113$	−12.0303	−1.13172	−0.565858	−	0.824502i	$-0.691455\pi$
$113$	−12.0303	−1.13172	−0.565858	+	0.824502i	$0.691455\pi$
$114$	0	0
$115$	−4.81345	−0.448857
$116$	0	0
$117$	0.754504	0.0697539
$118$	0	0
$119$	8.71442	0.798849
$120$	0	0
$121$	−9.12793	−0.829812
$122$	0	0
$123$	4.98254	0.449260
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.9050	−1.05640	−0.528201	−	0.849120i	$-0.677133\pi$
$127$	−11.9050	−1.05640	−0.528201	+	0.849120i	$0.677133\pi$
$128$	0	0
$129$	4.84665	0.426723
$130$	0	0
$131$	21.4533	1.87439	0.937193	−	0.348813i	$-0.113415\pi$
$131$	21.4533	1.87439	0.937193	+	0.348813i	$0.113415\pi$
$132$	0	0
$133$	9.34271	0.810116
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	13.6702	1.16793	0.583964	−	0.811780i	$-0.301501\pi$
$137$	13.6702	1.16793	0.583964	+	0.811780i	$0.301501\pi$
$138$	0	0
$139$	−16.1327	−1.36835	−0.684177	−	0.729316i	$-0.739838\pi$
$139$	−16.1327	−1.36835	−0.684177	+	0.729316i	$0.739838\pi$
$140$	0	0
$141$	−7.20514	−0.606783
$142$	0	0
$143$	−1.03234	−0.0863286
$144$	0	0
$145$	9.51684	0.790330
$146$	0	0
$147$	4.04198	0.333377
$148$	0	0
$149$	5.37315	0.440185	0.220093	−	0.975479i	$-0.429364\pi$
$149$	5.37315	0.440185	0.220093	+	0.975479i	$0.429364\pi$
$150$	0	0
$151$	19.3795	1.57708	0.788541	−	0.614982i	$-0.210837\pi$
$151$	19.3795	1.57708	0.788541	+	0.614982i	$0.210837\pi$
$152$	0	0
$153$	−5.06685	−0.409630
$154$	0	0
$155$	2.98501	0.239762
$156$	0	0
$157$	−18.2618	−1.45745	−0.728725	−	0.684807i	$-0.759887\pi$
$157$	−18.2618	−1.45745	−0.728725	+	0.684807i	$0.759887\pi$
$158$	0	0
$159$	−3.05763	−0.242486
$160$	0	0
$161$	8.27860	0.652445
$162$	0	0
$163$	−7.05074	−0.552257	−0.276128	−	0.961121i	$-0.589052\pi$
$163$	−7.05074	−0.552257	−0.276128	+	0.961121i	$0.589052\pi$
$164$	0	0
$165$	1.36824	0.106517
$166$	0	0
$167$	9.01474	0.697581	0.348791	−	0.937201i	$-0.386592\pi$
$167$	9.01474	0.697581	0.348791	+	0.937201i	$0.386592\pi$
$168$	0	0
$169$	−12.4307	−0.956210
$170$	0	0
$171$	−5.43216	−0.415408
$172$	0	0
$173$	15.2730	1.16118	0.580592	−	0.814195i	$-0.302821\pi$
$173$	15.2730	1.16118	0.580592	+	0.814195i	$0.302821\pi$
$174$	0	0
$175$	−1.71989	−0.130011
$176$	0	0
$177$	12.7030	0.954815
$178$	0	0
$179$	−1.58759	−0.118662	−0.0593309	−	0.998238i	$-0.518897\pi$
$179$	−1.58759	−0.118662	−0.0593309	+	0.998238i	$0.518897\pi$
$180$	0	0
$181$	−16.6047	−1.23422	−0.617110	−	0.786877i	$-0.711697\pi$
$181$	−16.6047	−1.23422	−0.617110	+	0.786877i	$0.711697\pi$
$182$	0	0
$183$	−5.80714	−0.429276
$184$	0	0
$185$	−3.95474	−0.290759
$186$	0	0
$187$	6.93265	0.506966
$188$	0	0
$189$	1.71989	0.125103
$190$	0	0
$191$	2.61235	0.189023	0.0945114	−	0.995524i	$-0.469871\pi$
$191$	2.61235	0.189023	0.0945114	+	0.995524i	$0.469871\pi$
$192$	0	0
$193$	4.63709	0.333785	0.166893	−	0.985975i	$-0.446627\pi$
$193$	4.63709	0.333785	0.166893	+	0.985975i	$0.446627\pi$
$194$	0	0
$195$	−0.754504	−0.0540311
$196$	0	0
$197$	19.5837	1.39528	0.697641	−	0.716448i	$-0.254233\pi$
$197$	19.5837	1.39528	0.697641	+	0.716448i	$0.254233\pi$
$198$	0	0
$199$	26.3243	1.86608	0.933039	−	0.359774i	$-0.117146\pi$
$199$	26.3243	1.86608	0.933039	+	0.359774i	$0.117146\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−16.3679	−1.14880
$204$	0	0
$205$	−4.98254	−0.347995
$206$	0	0
$207$	−4.81345	−0.334558
$208$	0	0
$209$	7.43248	0.514116
$210$	0	0
$211$	21.0604	1.44986	0.724930	−	0.688823i	$-0.241872\pi$
$211$	21.0604	1.44986	0.724930	+	0.688823i	$0.241872\pi$
$212$	0	0
$213$	−2.01465	−0.138042
$214$	0	0
$215$	−4.84665	−0.330538
$216$	0	0
$217$	−5.13389	−0.348511
$218$	0	0
$219$	−7.75941	−0.524332
$220$	0	0
$221$	−3.82296	−0.257160
$222$	0	0
$223$	−5.36967	−0.359580	−0.179790	−	0.983705i	$-0.557542\pi$
$223$	−5.36967	−0.359580	−0.179790	+	0.983705i	$0.557542\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−20.5942	−1.36689	−0.683444	−	0.730003i	$-0.739519\pi$
$227$	−20.5942	−1.36689	−0.683444	+	0.730003i	$0.739519\pi$
$228$	0	0
$229$	18.7086	1.23630	0.618149	−	0.786061i	$-0.287883\pi$
$229$	18.7086	1.23630	0.618149	+	0.786061i	$0.287883\pi$
$230$	0	0
$231$	−2.35322	−0.154830
$232$	0	0
$233$	−26.4981	−1.73595	−0.867974	−	0.496610i	$-0.834578\pi$
$233$	−26.4981	−1.73595	−0.867974	+	0.496610i	$0.834578\pi$
$234$	0	0
$235$	7.20514	0.470012
$236$	0	0
$237$	−14.6271	−0.950132
$238$	0	0
$239$	−22.9183	−1.48246	−0.741231	−	0.671250i	$-0.765758\pi$
$239$	−22.9183	−1.48246	−0.741231	+	0.671250i	$0.765758\pi$
$240$	0	0
$241$	3.11815	0.200858	0.100429	−	0.994944i	$-0.467978\pi$
$241$	3.11815	0.200858	0.100429	+	0.994944i	$0.467978\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.04198	−0.258233
$246$	0	0
$247$	−4.09858	−0.260787
$248$	0	0
$249$	1.49185	0.0945425
$250$	0	0
$251$	−9.02391	−0.569584	−0.284792	−	0.958589i	$-0.591925\pi$
$251$	−9.02391	−0.569584	−0.284792	+	0.958589i	$0.591925\pi$
$252$	0	0
$253$	6.58594	0.414055
$254$	0	0
$255$	5.06685	0.317298
$256$	0	0
$257$	2.34236	0.146112	0.0730561	−	0.997328i	$-0.476725\pi$
$257$	2.34236	0.146112	0.0730561	+	0.997328i	$0.476725\pi$
$258$	0	0
$259$	6.80172	0.422638
$260$	0	0
$261$	9.51684	0.589077
$262$	0	0
$263$	−4.11019	−0.253445	−0.126722	−	0.991938i	$-0.540446\pi$
$263$	−4.11019	−0.253445	−0.126722	+	0.991938i	$0.540446\pi$
$264$	0	0
$265$	3.05763	0.187829
$266$	0	0
$267$	−1.85489	−0.113518
$268$	0	0
$269$	9.65412	0.588622	0.294311	−	0.955710i	$-0.404910\pi$
$269$	9.65412	0.588622	0.294311	+	0.955710i	$0.404910\pi$
$270$	0	0
$271$	−3.71013	−0.225375	−0.112687	−	0.993631i	$-0.535946\pi$
$271$	−3.71013	−0.225375	−0.112687	+	0.993631i	$0.535946\pi$
$272$	0	0
$273$	1.29766	0.0785381
$274$	0	0
$275$	−1.36824	−0.0825078
$276$	0	0
$277$	−29.6352	−1.78061	−0.890304	−	0.455366i	$-0.849508\pi$
$277$	−29.6352	−1.78061	−0.890304	+	0.455366i	$0.849508\pi$
$278$	0	0
$279$	2.98501	0.178708
$280$	0	0
$281$	14.8004	0.882916	0.441458	−	0.897282i	$-0.354461\pi$
$281$	14.8004	0.882916	0.441458	+	0.897282i	$0.354461\pi$
$282$	0	0
$283$	4.04399	0.240390	0.120195	−	0.992750i	$-0.461648\pi$
$283$	4.04399	0.240390	0.120195	+	0.992750i	$0.461648\pi$
$284$	0	0
$285$	5.43216	0.321773
$286$	0	0
$287$	8.56941	0.505836
$288$	0	0
$289$	8.67296	0.510174
$290$	0	0
$291$	−11.5865	−0.679212
$292$	0	0
$293$	−13.9583	−0.815453	−0.407727	−	0.913104i	$-0.633678\pi$
$293$	−13.9583	−0.815453	−0.407727	+	0.913104i	$0.633678\pi$
$294$	0	0
$295$	−12.7030	−0.739596
$296$	0	0
$297$	1.36824	0.0793932
$298$	0	0
$299$	−3.63177	−0.210030
$300$	0	0
$301$	8.33569	0.480461
$302$	0	0
$303$	−10.9950	−0.631645
$304$	0	0
$305$	5.80714	0.332516
$306$	0	0
$307$	14.8483	0.847436	0.423718	−	0.905794i	$-0.360725\pi$
$307$	14.8483	0.847436	0.423718	+	0.905794i	$0.360725\pi$
$308$	0	0
$309$	0.717921	0.0408411
$310$	0	0
$311$	15.1526	0.859228	0.429614	−	0.903013i	$-0.358650\pi$
$311$	15.1526	0.859228	0.429614	+	0.903013i	$0.358650\pi$
$312$	0	0
$313$	31.6317	1.78793	0.893964	−	0.448139i	$-0.147913\pi$
$313$	31.6317	1.78793	0.893964	+	0.448139i	$0.147913\pi$
$314$	0	0
$315$	−1.71989	−0.0969047
$316$	0	0
$317$	23.4467	1.31690	0.658448	−	0.752626i	$-0.271213\pi$
$317$	23.4467	1.31690	0.658448	+	0.752626i	$0.271213\pi$
$318$	0	0
$319$	−13.0213	−0.729052
$320$	0	0
$321$	1.94199	0.108391
$322$	0	0
$323$	27.5239	1.53147
$324$	0	0
$325$	0.754504	0.0418523
$326$	0	0
$327$	−14.3548	−0.793825
$328$	0	0
$329$	−12.3920	−0.683196
$330$	0	0
$331$	−33.3397	−1.83252	−0.916258	−	0.400589i	$-0.868805\pi$
$331$	−33.3397	−1.83252	−0.916258	+	0.400589i	$0.868805\pi$
$332$	0	0
$333$	−3.95474	−0.216719
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	25.9088	1.41134	0.705672	−	0.708539i	$-0.250645\pi$
$337$	25.9088	1.41134	0.705672	+	0.708539i	$0.250645\pi$
$338$	0	0
$339$	12.0303	0.653397
$340$	0	0
$341$	−4.08420	−0.221172
$342$	0	0
$343$	18.9910	1.02542
$344$	0	0
$345$	4.81345	0.259147
$346$	0	0
$347$	13.2461	0.711090	0.355545	−	0.934659i	$-0.384295\pi$
$347$	13.2461	0.711090	0.355545	+	0.934659i	$0.384295\pi$
$348$	0	0
$349$	−34.0854	−1.82455	−0.912275	−	0.409578i	$-0.865676\pi$
$349$	−34.0854	−1.82455	−0.912275	+	0.409578i	$0.865676\pi$
$350$	0	0
$351$	−0.754504	−0.0402724
$352$	0	0
$353$	33.2009	1.76711	0.883553	−	0.468331i	$-0.155144\pi$
$353$	33.2009	1.76711	0.883553	+	0.468331i	$0.155144\pi$
$354$	0	0
$355$	2.01465	0.106927
$356$	0	0
$357$	−8.71442	−0.461216
$358$	0	0
$359$	−24.7500	−1.30625	−0.653127	−	0.757249i	$-0.726543\pi$
$359$	−24.7500	−1.30625	−0.653127	+	0.757249i	$0.726543\pi$
$360$	0	0
$361$	10.5084	0.553072
$362$	0	0
$363$	9.12793	0.479092
$364$	0	0
$365$	7.75941	0.406146
$366$	0	0
$367$	−13.8649	−0.723744	−0.361872	−	0.932228i	$-0.617862\pi$
$367$	−13.8649	−0.723744	−0.361872	+	0.932228i	$0.617862\pi$
$368$	0	0
$369$	−4.98254	−0.259380
$370$	0	0
$371$	−5.25879	−0.273023
$372$	0	0
$373$	38.2670	1.98139	0.990696	−	0.136092i	$-0.0434543\pi$
$373$	38.2670	1.98139	0.990696	+	0.136092i	$0.0434543\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	7.18049	0.369814
$378$	0	0
$379$	11.3110	0.581006	0.290503	−	0.956874i	$-0.406177\pi$
$379$	11.3110	0.581006	0.290503	+	0.956874i	$0.406177\pi$
$380$	0	0
$381$	11.9050	0.609914
$382$	0	0
$383$	−6.43861	−0.328998	−0.164499	−	0.986377i	$-0.552601\pi$
$383$	−6.43861	−0.328998	−0.164499	+	0.986377i	$0.552601\pi$
$384$	0	0
$385$	2.35322	0.119931
$386$	0	0
$387$	−4.84665	−0.246369
$388$	0	0
$389$	0.522124	0.0264728	0.0132364	−	0.999912i	$-0.495787\pi$
$389$	0.522124	0.0264728	0.0132364	+	0.999912i	$0.495787\pi$
$390$	0	0
$391$	24.3890	1.23341
$392$	0	0
$393$	−21.4533	−1.08218
$394$	0	0
$395$	14.6271	0.735969
$396$	0	0
$397$	13.1055	0.657745	0.328873	−	0.944374i	$-0.393331\pi$
$397$	13.1055	0.657745	0.328873	+	0.944374i	$0.393331\pi$
$398$	0	0
$399$	−9.34271	−0.467721
$400$	0	0
$401$	−5.01035	−0.250205	−0.125102	−	0.992144i	$-0.539926\pi$
$401$	−5.01035	−0.250205	−0.125102	+	0.992144i	$0.539926\pi$
$402$	0	0
$403$	2.25220	0.112190
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	5.41103	0.268215
$408$	0	0
$409$	3.64571	0.180269	0.0901345	−	0.995930i	$-0.471270\pi$
$409$	3.64571	0.180269	0.0901345	+	0.995930i	$0.471270\pi$
$410$	0	0
$411$	−13.6702	−0.674303
$412$	0	0
$413$	21.8477	1.07506
$414$	0	0
$415$	−1.49185	−0.0732323
$416$	0	0
$417$	16.1327	0.790019
$418$	0	0
$419$	27.0408	1.32103	0.660514	−	0.750813i	$-0.270338\pi$
$419$	27.0408	1.32103	0.660514	+	0.750813i	$0.270338\pi$
$420$	0	0
$421$	37.5688	1.83099	0.915495	−	0.402330i	$-0.131800\pi$
$421$	37.5688	1.83099	0.915495	+	0.402330i	$0.131800\pi$
$422$	0	0
$423$	7.20514	0.350326
$424$	0	0
$425$	−5.06685	−0.245778
$426$	0	0
$427$	−9.98763	−0.483335
$428$	0	0
$429$	1.03234	0.0498418
$430$	0	0
$431$	18.6543	0.898546	0.449273	−	0.893395i	$-0.351683\pi$
$431$	18.6543	0.898546	0.449273	+	0.893395i	$0.351683\pi$
$432$	0	0
$433$	28.7430	1.38130	0.690650	−	0.723189i	$-0.257325\pi$
$433$	28.7430	1.38130	0.690650	+	0.723189i	$0.257325\pi$
$434$	0	0
$435$	−9.51684	−0.456297
$436$	0	0
$437$	26.1474	1.25080
$438$	0	0
$439$	33.0191	1.57592	0.787958	−	0.615729i	$-0.211138\pi$
$439$	33.0191	1.57592	0.787958	+	0.615729i	$0.211138\pi$
$440$	0	0
$441$	−4.04198	−0.192475
$442$	0	0
$443$	−2.04372	−0.0971002	−0.0485501	−	0.998821i	$-0.515460\pi$
$443$	−2.04372	−0.0971002	−0.0485501	+	0.998821i	$0.515460\pi$
$444$	0	0
$445$	1.85489	0.0879304
$446$	0	0
$447$	−5.37315	−0.254141
$448$	0	0
$449$	−14.8844	−0.702439	−0.351219	−	0.936293i	$-0.614233\pi$
$449$	−14.8844	−0.702439	−0.351219	+	0.936293i	$0.614233\pi$
$450$	0	0
$451$	6.81729	0.321014
$452$	0	0
$453$	−19.3795	−0.910529
$454$	0	0
$455$	−1.29766	−0.0608353
$456$	0	0
$457$	−40.2813	−1.88428	−0.942140	−	0.335220i	$-0.891190\pi$
$457$	−40.2813	−1.88428	−0.942140	+	0.335220i	$0.891190\pi$
$458$	0	0
$459$	5.06685	0.236500
$460$	0	0
$461$	37.6402	1.75308	0.876539	−	0.481331i	$-0.159847\pi$
$461$	37.6402	1.75308	0.876539	+	0.481331i	$0.159847\pi$
$462$	0	0
$463$	−17.1907	−0.798918	−0.399459	−	0.916751i	$-0.630802\pi$
$463$	−17.1907	−0.798918	−0.399459	+	0.916751i	$0.630802\pi$
$464$	0	0
$465$	−2.98501	−0.138427
$466$	0	0
$467$	−22.4785	−1.04018	−0.520091	−	0.854111i	$-0.674102\pi$
$467$	−22.4785	−1.04018	−0.520091	+	0.854111i	$0.674102\pi$
$468$	0	0
$469$	1.71989	0.0794171
$470$	0	0
$471$	18.2618	0.841459
$472$	0	0
$473$	6.63136	0.304910
$474$	0	0
$475$	−5.43216	−0.249245
$476$	0	0
$477$	3.05763	0.139999
$478$	0	0
$479$	28.2970	1.29292	0.646462	−	0.762946i	$-0.276248\pi$
$479$	28.2970	1.29292	0.646462	+	0.762946i	$0.276248\pi$
$480$	0	0
$481$	−2.98387	−0.136053
$482$	0	0
$483$	−8.27860	−0.376689
$484$	0	0
$485$	11.5865	0.526116
$486$	0	0
$487$	−28.3675	−1.28545	−0.642727	−	0.766095i	$-0.722197\pi$
$487$	−28.3675	−1.28545	−0.642727	+	0.766095i	$0.722197\pi$
$488$	0	0
$489$	7.05074	0.318846
$490$	0	0
$491$	0.757327	0.0341777	0.0170888	−	0.999854i	$-0.494560\pi$
$491$	0.757327	0.0341777	0.0170888	+	0.999854i	$0.494560\pi$
$492$	0	0
$493$	−48.2204	−2.17174
$494$	0	0
$495$	−1.36824	−0.0614977
$496$	0	0
$497$	−3.46497	−0.155425
$498$	0	0
$499$	32.1005	1.43702	0.718508	−	0.695519i	$-0.244825\pi$
$499$	32.1005	1.43702	0.718508	+	0.695519i	$0.244825\pi$
$500$	0	0
$501$	−9.01474	−0.402749
$502$	0	0
$503$	31.0881	1.38615	0.693076	−	0.720865i	$-0.256255\pi$
$503$	31.0881	1.38615	0.693076	+	0.720865i	$0.256255\pi$
$504$	0	0
$505$	10.9950	0.489270
$506$	0	0
$507$	12.4307	0.552068
$508$	0	0
$509$	−10.4264	−0.462144	−0.231072	−	0.972937i	$-0.574223\pi$
$509$	−10.4264	−0.462144	−0.231072	+	0.972937i	$0.574223\pi$
$510$	0	0
$511$	−13.3453	−0.590362
$512$	0	0
$513$	5.43216	0.239836
$514$	0	0
$515$	−0.717921	−0.0316354
$516$	0	0
$517$	−9.85835	−0.433570
$518$	0	0
$519$	−15.2730	−0.670409
$520$	0	0
$521$	20.0524	0.878513	0.439256	−	0.898362i	$-0.355242\pi$
$521$	20.0524	0.878513	0.439256	+	0.898362i	$0.355242\pi$
$522$	0	0
$523$	23.5070	1.02789	0.513944	−	0.857824i	$-0.328184\pi$
$523$	23.5070	1.02789	0.513944	+	0.857824i	$0.328184\pi$
$524$	0	0
$525$	1.71989	0.0750621
$526$	0	0
$527$	−15.1246	−0.658838
$528$	0	0
$529$	0.169308	0.00736120
$530$	0	0
$531$	−12.7030	−0.551262
$532$	0	0
$533$	−3.75934	−0.162835
$534$	0	0
$535$	−1.94199	−0.0839595
$536$	0	0
$537$	1.58759	0.0685095
$538$	0	0
$539$	5.53039	0.238211
$540$	0	0
$541$	11.8202	0.508190	0.254095	−	0.967179i	$-0.418222\pi$
$541$	11.8202	0.508190	0.254095	+	0.967179i	$0.418222\pi$
$542$	0	0
$543$	16.6047	0.712577
$544$	0	0
$545$	14.3548	0.614894
$546$	0	0
$547$	23.0282	0.984616	0.492308	−	0.870421i	$-0.336153\pi$
$547$	23.0282	0.984616	0.492308	+	0.870421i	$0.336153\pi$
$548$	0	0
$549$	5.80714	0.247843
$550$	0	0
$551$	−51.6970	−2.20237
$552$	0	0
$553$	−25.1570	−1.06978
$554$	0	0
$555$	3.95474	0.167870
$556$	0	0
$557$	28.1962	1.19471	0.597356	−	0.801976i	$-0.296218\pi$
$557$	28.1962	1.19471	0.597356	+	0.801976i	$0.296218\pi$
$558$	0	0
$559$	−3.65681	−0.154667
$560$	0	0
$561$	−6.93265	−0.292697
$562$	0	0
$563$	28.8141	1.21437	0.607186	−	0.794560i	$-0.292298\pi$
$563$	28.8141	1.21437	0.607186	+	0.794560i	$0.292298\pi$
$564$	0	0
$565$	−12.0303	−0.506119
$566$	0	0
$567$	−1.71989	−0.0722285
$568$	0	0
$569$	9.60550	0.402683	0.201342	−	0.979521i	$-0.435470\pi$
$569$	9.60550	0.402683	0.201342	+	0.979521i	$0.435470\pi$
$570$	0	0
$571$	40.3784	1.68978	0.844892	−	0.534936i	$-0.179664\pi$
$571$	40.3784	1.68978	0.844892	+	0.534936i	$0.179664\pi$
$572$	0	0
$573$	−2.61235	−0.109132
$574$	0	0
$575$	−4.81345	−0.200735
$576$	0	0
$577$	−27.0331	−1.12540	−0.562701	−	0.826660i	$-0.690238\pi$
$577$	−27.0331	−1.12540	−0.562701	+	0.826660i	$0.690238\pi$
$578$	0	0
$579$	−4.63709	−0.192711
$580$	0	0
$581$	2.56582	0.106448
$582$	0	0
$583$	−4.18357	−0.173266
$584$	0	0
$585$	0.754504	0.0311949
$586$	0	0
$587$	33.3210	1.37531	0.687653	−	0.726040i	$-0.258641\pi$
$587$	33.3210	1.37531	0.687653	+	0.726040i	$0.258641\pi$
$588$	0	0
$589$	−16.2151	−0.668130
$590$	0	0
$591$	−19.5837	−0.805566
$592$	0	0
$593$	−8.78429	−0.360728	−0.180364	−	0.983600i	$-0.557728\pi$
$593$	−8.78429	−0.360728	−0.180364	+	0.983600i	$0.557728\pi$
$594$	0	0
$595$	8.71442	0.357256
$596$	0	0
$597$	−26.3243	−1.07738
$598$	0	0
$599$	20.8749	0.852923	0.426462	−	0.904506i	$-0.359760\pi$
$599$	20.8749	0.852923	0.426462	+	0.904506i	$0.359760\pi$
$600$	0	0
$601$	−24.3387	−0.992795	−0.496397	−	0.868095i	$-0.665344\pi$
$601$	−24.3387	−0.992795	−0.496397	+	0.868095i	$0.665344\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−9.12793	−0.371103
$606$	0	0
$607$	−25.4838	−1.03436	−0.517178	−	0.855878i	$-0.673017\pi$
$607$	−25.4838	−1.03436	−0.517178	+	0.855878i	$0.673017\pi$
$608$	0	0
$609$	16.3679	0.663261
$610$	0	0
$611$	5.43631	0.219929
$612$	0	0
$613$	−18.3392	−0.740713	−0.370357	−	0.928890i	$-0.620765\pi$
$613$	−18.3392	−0.740713	−0.370357	+	0.928890i	$0.620765\pi$
$614$	0	0
$615$	4.98254	0.200915
$616$	0	0
$617$	−40.7369	−1.64000	−0.820002	−	0.572360i	$-0.806028\pi$
$617$	−40.7369	−1.64000	−0.820002	+	0.572360i	$0.806028\pi$
$618$	0	0
$619$	−6.46679	−0.259922	−0.129961	−	0.991519i	$-0.541485\pi$
$619$	−6.46679	−0.259922	−0.129961	+	0.991519i	$0.541485\pi$
$620$	0	0
$621$	4.81345	0.193157
$622$	0	0
$623$	−3.19021	−0.127813
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.43248	−0.296825
$628$	0	0
$629$	20.0381	0.798971
$630$	0	0
$631$	−17.7277	−0.705727	−0.352864	−	0.935675i	$-0.614792\pi$
$631$	−17.7277	−0.705727	−0.352864	+	0.935675i	$0.614792\pi$
$632$	0	0
$633$	−21.0604	−0.837077
$634$	0	0
$635$	−11.9050	−0.472437
$636$	0	0
$637$	−3.04969	−0.120833
$638$	0	0
$639$	2.01465	0.0796984
$640$	0	0
$641$	−17.9193	−0.707768	−0.353884	−	0.935289i	$-0.615139\pi$
$641$	−17.9193	−0.707768	−0.353884	+	0.935289i	$0.615139\pi$
$642$	0	0
$643$	−24.8301	−0.979202	−0.489601	−	0.871946i	$-0.662858\pi$
$643$	−24.8301	−0.979202	−0.489601	+	0.871946i	$0.662858\pi$
$644$	0	0
$645$	4.84665	0.190836
$646$	0	0
$647$	−18.9630	−0.745511	−0.372756	−	0.927930i	$-0.621587\pi$
$647$	−18.9630	−0.745511	−0.372756	+	0.927930i	$0.621587\pi$
$648$	0	0
$649$	17.3807	0.682252
$650$	0	0
$651$	5.13389	0.201213
$652$	0	0
$653$	−8.53575	−0.334030	−0.167015	−	0.985954i	$-0.553413\pi$
$653$	−8.53575	−0.334030	−0.167015	+	0.985954i	$0.553413\pi$
$654$	0	0
$655$	21.4533	0.838250
$656$	0	0
$657$	7.75941	0.302723
$658$	0	0
$659$	26.0784	1.01587	0.507934	−	0.861396i	$-0.330409\pi$
$659$	26.0784	1.01587	0.507934	+	0.861396i	$0.330409\pi$
$660$	0	0
$661$	−11.5678	−0.449933	−0.224967	−	0.974366i	$-0.572227\pi$
$661$	−11.5678	−0.449933	−0.224967	+	0.974366i	$0.572227\pi$
$662$	0	0
$663$	3.82296	0.148471
$664$	0	0
$665$	9.34271	0.362295
$666$	0	0
$667$	−45.8088	−1.77372
$668$	0	0
$669$	5.36967	0.207604
$670$	0	0
$671$	−7.94554	−0.306734
$672$	0	0
$673$	−41.7060	−1.60765	−0.803824	−	0.594867i	$-0.797205\pi$
$673$	−41.7060	−1.60765	−0.803824	+	0.594867i	$0.797205\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−37.2132	−1.43022	−0.715110	−	0.699012i	$-0.753623\pi$
$677$	−37.2132	−1.43022	−0.715110	+	0.699012i	$0.753623\pi$
$678$	0	0
$679$	−19.9275	−0.764747
$680$	0	0
$681$	20.5942	0.789173
$682$	0	0
$683$	12.6013	0.482176	0.241088	−	0.970503i	$-0.422496\pi$
$683$	12.6013	0.482176	0.241088	+	0.970503i	$0.422496\pi$
$684$	0	0
$685$	13.6702	0.522313
$686$	0	0
$687$	−18.7086	−0.713776
$688$	0	0
$689$	2.30700	0.0878896
$690$	0	0
$691$	−11.5941	−0.441061	−0.220530	−	0.975380i	$-0.570779\pi$
$691$	−11.5941	−0.441061	−0.220530	+	0.975380i	$0.570779\pi$
$692$	0	0
$693$	2.35322	0.0893913
$694$	0	0
$695$	−16.1327	−0.611946
$696$	0	0
$697$	25.2458	0.956251
$698$	0	0
$699$	26.4981	1.00225
$700$	0	0
$701$	14.4870	0.547166	0.273583	−	0.961848i	$-0.411791\pi$
$701$	14.4870	0.547166	0.273583	+	0.961848i	$0.411791\pi$
$702$	0	0
$703$	21.4828	0.810239
$704$	0	0
$705$	−7.20514	−0.271361
$706$	0	0
$707$	−18.9101	−0.711189
$708$	0	0
$709$	16.8859	0.634162	0.317081	−	0.948398i	$-0.397297\pi$
$709$	16.8859	0.634162	0.317081	+	0.948398i	$0.397297\pi$
$710$	0	0
$711$	14.6271	0.548559
$712$	0	0
$713$	−14.3682	−0.538094
$714$	0	0
$715$	−1.03234	−0.0386073
$716$	0	0
$717$	22.9183	0.855900
$718$	0	0
$719$	−12.5734	−0.468908	−0.234454	−	0.972127i	$-0.575330\pi$
$719$	−12.5734	−0.468908	−0.234454	+	0.972127i	$0.575330\pi$
$720$	0	0
$721$	1.23475	0.0459843
$722$	0	0
$723$	−3.11815	−0.115965
$724$	0	0
$725$	9.51684	0.353446
$726$	0	0
$727$	44.5862	1.65361	0.826806	−	0.562488i	$-0.190156\pi$
$727$	44.5862	1.65361	0.826806	+	0.562488i	$0.190156\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.5572	0.908282
$732$	0	0
$733$	−2.34599	−0.0866511	−0.0433256	−	0.999061i	$-0.513795\pi$
$733$	−2.34599	−0.0866511	−0.0433256	+	0.999061i	$0.513795\pi$
$734$	0	0
$735$	4.04198	0.149091
$736$	0	0
$737$	1.36824	0.0503997
$738$	0	0
$739$	−42.7112	−1.57116	−0.785578	−	0.618763i	$-0.787634\pi$
$739$	−42.7112	−1.57116	−0.785578	+	0.618763i	$0.787634\pi$
$740$	0	0
$741$	4.09858	0.150565
$742$	0	0
$743$	18.1785	0.666905	0.333452	−	0.942767i	$-0.391786\pi$
$743$	18.1785	0.666905	0.333452	+	0.942767i	$0.391786\pi$
$744$	0	0
$745$	5.37315	0.196857
$746$	0	0
$747$	−1.49185	−0.0545841
$748$	0	0
$749$	3.34000	0.122041
$750$	0	0
$751$	−9.46805	−0.345494	−0.172747	−	0.984966i	$-0.555264\pi$
$751$	−9.46805	−0.345494	−0.172747	+	0.984966i	$0.555264\pi$
$752$	0	0
$753$	9.02391	0.328850
$754$	0	0
$755$	19.3795	0.705293
$756$	0	0
$757$	15.3848	0.559170	0.279585	−	0.960121i	$-0.409803\pi$
$757$	15.3848	0.559170	0.279585	+	0.960121i	$0.409803\pi$
$758$	0	0
$759$	−6.58594	−0.239055
$760$	0	0
$761$	46.5394	1.68705	0.843527	−	0.537087i	$-0.180476\pi$
$761$	46.5394	1.68705	0.843527	+	0.537087i	$0.180476\pi$
$762$	0	0
$763$	−24.6887	−0.893792
$764$	0	0
$765$	−5.06685	−0.183192
$766$	0	0
$767$	−9.58445	−0.346074
$768$	0	0
$769$	−0.914621	−0.0329821	−0.0164910	−	0.999864i	$-0.505249\pi$
$769$	−0.914621	−0.0329821	−0.0164910	+	0.999864i	$0.505249\pi$
$770$	0	0
$771$	−2.34236	−0.0843579
$772$	0	0
$773$	48.9216	1.75959	0.879793	−	0.475356i	$-0.157681\pi$
$773$	48.9216	1.75959	0.879793	+	0.475356i	$0.157681\pi$
$774$	0	0
$775$	2.98501	0.107225
$776$	0	0
$777$	−6.80172	−0.244010
$778$	0	0
$779$	27.0659	0.969738
$780$	0	0
$781$	−2.75652	−0.0986360
$782$	0	0
$783$	−9.51684	−0.340104
$784$	0	0
$785$	−18.2618	−0.651791
$786$	0	0
$787$	−31.5292	−1.12389	−0.561947	−	0.827173i	$-0.689947\pi$
$787$	−31.5292	−1.12389	−0.561947	+	0.827173i	$0.689947\pi$
$788$	0	0
$789$	4.11019	0.146327
$790$	0	0
$791$	20.6908	0.735680
$792$	0	0
$793$	4.38151	0.155592
$794$	0	0
$795$	−3.05763	−0.108443
$796$	0	0
$797$	12.0467	0.426716	0.213358	−	0.976974i	$-0.431560\pi$
$797$	12.0467	0.426716	0.213358	+	0.976974i	$0.431560\pi$
$798$	0	0
$799$	−36.5074	−1.29154
$800$	0	0
$801$	1.85489	0.0655394
$802$	0	0
$803$	−10.6167	−0.374655
$804$	0	0
$805$	8.27860	0.291782
$806$	0	0
$807$	−9.65412	−0.339841
$808$	0	0
$809$	35.6649	1.25391	0.626955	−	0.779056i	$-0.284301\pi$
$809$	35.6649	1.25391	0.626955	+	0.779056i	$0.284301\pi$
$810$	0	0
$811$	20.7024	0.726958	0.363479	−	0.931602i	$-0.381589\pi$
$811$	20.7024	0.726958	0.363479	+	0.931602i	$0.381589\pi$
$812$	0	0
$813$	3.71013	0.130120
$814$	0	0
$815$	−7.05074	−0.246977
$816$	0	0
$817$	26.3278	0.921091
$818$	0	0
$819$	−1.29766	−0.0453440
$820$	0	0
$821$	39.4137	1.37555	0.687773	−	0.725926i	$-0.258588\pi$
$821$	39.4137	1.37555	0.687773	+	0.725926i	$0.258588\pi$
$822$	0	0
$823$	22.3950	0.780639	0.390319	−	0.920679i	$-0.372365\pi$
$823$	22.3950	0.780639	0.390319	+	0.920679i	$0.372365\pi$
$824$	0	0
$825$	1.36824	0.0476359
$826$	0	0
$827$	7.52347	0.261617	0.130808	−	0.991408i	$-0.458243\pi$
$827$	7.52347	0.261617	0.130808	+	0.991408i	$0.458243\pi$
$828$	0	0
$829$	13.4237	0.466225	0.233112	−	0.972450i	$-0.425109\pi$
$829$	13.4237	0.466225	0.233112	+	0.972450i	$0.425109\pi$
$830$	0	0
$831$	29.6352	1.02803
$832$	0	0
$833$	20.4801	0.709594
$834$	0	0
$835$	9.01474	0.311968
$836$	0	0
$837$	−2.98501	−0.103177
$838$	0	0
$839$	36.9948	1.27720	0.638601	−	0.769538i	$-0.279513\pi$
$839$	36.9948	1.27720	0.638601	+	0.769538i	$0.279513\pi$
$840$	0	0
$841$	61.5702	2.12311
$842$	0	0
$843$	−14.8004	−0.509752
$844$	0	0
$845$	−12.4307	−0.427630
$846$	0	0
$847$	15.6990	0.539425
$848$	0	0
$849$	−4.04399	−0.138789
$850$	0	0
$851$	19.0360	0.652544
$852$	0	0
$853$	17.8030	0.609563	0.304781	−	0.952422i	$-0.401417\pi$
$853$	17.8030	0.609563	0.304781	+	0.952422i	$0.401417\pi$
$854$	0	0
$855$	−5.43216	−0.185776
$856$	0	0
$857$	22.0508	0.753240	0.376620	−	0.926368i	$-0.377086\pi$
$857$	22.0508	0.753240	0.376620	+	0.926368i	$0.377086\pi$
$858$	0	0
$859$	−36.6739	−1.25130	−0.625649	−	0.780104i	$-0.715166\pi$
$859$	−36.6739	−1.25130	−0.625649	+	0.780104i	$0.715166\pi$
$860$	0	0
$861$	−8.56941	−0.292045
$862$	0	0
$863$	6.08355	0.207086	0.103543	−	0.994625i	$-0.466982\pi$
$863$	6.08355	0.207086	0.103543	+	0.994625i	$0.466982\pi$
$864$	0	0
$865$	15.2730	0.519297
$866$	0	0
$867$	−8.67296	−0.294549
$868$	0	0
$869$	−20.0133	−0.678906
$870$	0	0
$871$	−0.754504	−0.0255654
$872$	0	0
$873$	11.5865	0.392144
$874$	0	0
$875$	−1.71989	−0.0581428
$876$	0	0
$877$	−13.9243	−0.470191	−0.235095	−	0.971972i	$-0.575540\pi$
$877$	−13.9243	−0.470191	−0.235095	+	0.971972i	$0.575540\pi$
$878$	0	0
$879$	13.9583	0.470802
$880$	0	0
$881$	34.9017	1.17587	0.587935	−	0.808908i	$-0.299941\pi$
$881$	34.9017	1.17587	0.587935	+	0.808908i	$0.299941\pi$
$882$	0	0
$883$	−54.6242	−1.83825	−0.919126	−	0.393964i	$-0.871104\pi$
$883$	−54.6242	−1.83825	−0.919126	+	0.393964i	$0.871104\pi$
$884$	0	0
$885$	12.7030	0.427006
$886$	0	0
$887$	−32.3645	−1.08669	−0.543347	−	0.839508i	$-0.682843\pi$
$887$	−32.3645	−1.08669	−0.543347	+	0.839508i	$0.682843\pi$
$888$	0	0
$889$	20.4753	0.686721
$890$	0	0
$891$	−1.36824	−0.0458377
$892$	0	0
$893$	−39.1395	−1.30975
$894$	0	0
$895$	−1.58759	−0.0530672
$896$	0	0
$897$	3.63177	0.121261
$898$	0	0
$899$	28.4079	0.947455
$900$	0	0
$901$	−15.4926	−0.516132
$902$	0	0
$903$	−8.33569	−0.277394
$904$	0	0
$905$	−16.6047	−0.551960
$906$	0	0
$907$	25.8402	0.858011	0.429005	−	0.903302i	$-0.358864\pi$
$907$	25.8402	0.858011	0.429005	+	0.903302i	$0.358864\pi$
$908$	0	0
$909$	10.9950	0.364680
$910$	0	0
$911$	5.29053	0.175283	0.0876415	−	0.996152i	$-0.472067\pi$
$911$	5.29053	0.175283	0.0876415	+	0.996152i	$0.472067\pi$
$912$	0	0
$913$	2.04121	0.0675542
$914$	0	0
$915$	−5.80714	−0.191978
$916$	0	0
$917$	−36.8973	−1.21846
$918$	0	0
$919$	−57.8088	−1.90694	−0.953468	−	0.301495i	$-0.902514\pi$
$919$	−57.8088	−1.90694	−0.953468	+	0.301495i	$0.902514\pi$
$920$	0	0
$921$	−14.8483	−0.489268
$922$	0	0
$923$	1.52006	0.0500334
$924$	0	0
$925$	−3.95474	−0.130031
$926$	0	0
$927$	−0.717921	−0.0235796
$928$	0	0
$929$	−36.6506	−1.20247	−0.601234	−	0.799073i	$-0.705324\pi$
$929$	−36.6506	−1.20247	−0.601234	+	0.799073i	$0.705324\pi$
$930$	0	0
$931$	21.9567	0.719602
$932$	0	0
$933$	−15.1526	−0.496075
$934$	0	0
$935$	6.93265	0.226722
$936$	0	0
$937$	−28.5202	−0.931714	−0.465857	−	0.884860i	$-0.654254\pi$
$937$	−28.5202	−0.931714	−0.465857	+	0.884860i	$0.654254\pi$
$938$	0	0
$939$	−31.6317	−1.03226
$940$	0	0
$941$	18.6007	0.606367	0.303183	−	0.952932i	$-0.401951\pi$
$941$	18.6007	0.606367	0.303183	+	0.952932i	$0.401951\pi$
$942$	0	0
$943$	23.9832	0.781000
$944$	0	0
$945$	1.71989	0.0559480
$946$	0	0
$947$	−32.3409	−1.05094	−0.525469	−	0.850813i	$-0.676110\pi$
$947$	−32.3409	−1.05094	−0.525469	+	0.850813i	$0.676110\pi$
$948$	0	0
$949$	5.85450	0.190045
$950$	0	0
$951$	−23.4467	−0.760310
$952$	0	0
$953$	18.8516	0.610664	0.305332	−	0.952246i	$-0.401233\pi$
$953$	18.8516	0.610664	0.305332	+	0.952246i	$0.401233\pi$
$954$	0	0
$955$	2.61235	0.0845336
$956$	0	0
$957$	13.0213	0.420918
$958$	0	0
$959$	−23.5113	−0.759219
$960$	0	0
$961$	−22.0897	−0.712571
$962$	0	0
$963$	−1.94199	−0.0625797
$964$	0	0
$965$	4.63709	0.149273
$966$	0	0
$967$	−42.5615	−1.36869	−0.684343	−	0.729160i	$-0.739911\pi$
$967$	−42.5615	−1.36869	−0.684343	+	0.729160i	$0.739911\pi$
$968$	0	0
$969$	−27.5239	−0.884196
$970$	0	0
$971$	29.6151	0.950393	0.475197	−	0.879880i	$-0.342377\pi$
$971$	29.6151	0.950393	0.475197	+	0.879880i	$0.342377\pi$
$972$	0	0
$973$	27.7464	0.889508
$974$	0	0
$975$	−0.754504	−0.0241635
$976$	0	0
$977$	−35.8943	−1.14836	−0.574181	−	0.818729i	$-0.694679\pi$
$977$	−35.8943	−1.14836	−0.574181	+	0.818729i	$0.694679\pi$
$978$	0	0
$979$	−2.53793	−0.0811127
$980$	0	0
$981$	14.3548	0.458315
$982$	0	0
$983$	−29.7359	−0.948427	−0.474214	−	0.880410i	$-0.657268\pi$
$983$	−29.7359	−0.948427	−0.474214	+	0.880410i	$0.657268\pi$
$984$	0	0
$985$	19.5837	0.623989
$986$	0	0
$987$	12.3920	0.394443
$988$	0	0
$989$	23.3291	0.741822
$990$	0	0
$991$	−37.9422	−1.20527	−0.602637	−	0.798016i	$-0.705883\pi$
$991$	−37.9422	−1.20527	−0.602637	+	0.798016i	$0.705883\pi$
$992$	0	0
$993$	33.3397	1.05800
$994$	0	0
$995$	26.3243	0.834536
$996$	0	0
$997$	−42.3747	−1.34202	−0.671010	−	0.741448i	$-0.734139\pi$
$997$	−42.3747	−1.34202	−0.671010	+	0.741448i	$0.734139\pi$
$998$	0	0
$999$	3.95474	0.125123

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.bc.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.bc.1.4	✓	10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.71989 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.71989 q^{7} +1.00000 q^{9} -1.36824 q^{11} +0.754504 q^{13} -1.00000 q^{15} -5.06685 q^{17} -5.43216 q^{19} +1.71989 q^{21} -4.81345 q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.51684 q^{29} +2.98501 q^{31} +1.36824 q^{33} -1.71989 q^{35} -3.95474 q^{37} -0.754504 q^{39} -4.98254 q^{41} -4.84665 q^{43} +1.00000 q^{45} +7.20514 q^{47} -4.04198 q^{49} +5.06685 q^{51} +3.05763 q^{53} -1.36824 q^{55} +5.43216 q^{57} -12.7030 q^{59} +5.80714 q^{61} -1.71989 q^{63} +0.754504 q^{65} -1.00000 q^{67} +4.81345 q^{69} +2.01465 q^{71} +7.75941 q^{73} -1.00000 q^{75} +2.35322 q^{77} +14.6271 q^{79} +1.00000 q^{81} -1.49185 q^{83} -5.06685 q^{85} -9.51684 q^{87} +1.85489 q^{89} -1.29766 q^{91} -2.98501 q^{93} -5.43216 q^{95} +11.5865 q^{97} -1.36824 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9	\( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9} + 7 q^{11} - q^{13} - 10 q^{15} + 3 q^{17} + 4 q^{19} - q^{21} - 13 q^{23} + 10 q^{25} - 10 q^{27} + 18 q^{29} - 9 q^{31} - 7 q^{33} + q^{35} - 3 q^{37} + q^{39} + 19 q^{41} + 5 q^{43} + 10 q^{45} - 8 q^{47} + 43 q^{49} - 3 q^{51} + 17 q^{53} + 7 q^{55} - 4 q^{57} + 24 q^{59} + 21 q^{61} + q^{63} - q^{65} - 10 q^{67} + 13 q^{69} + 2 q^{71} + 25 q^{73} - 10 q^{75} + 15 q^{77} - q^{79} + 10 q^{81} - 6 q^{83} + 3 q^{85} - 18 q^{87} + 23 q^{89} + 29 q^{91} + 9 q^{93} + 4 q^{95} + 21 q^{97} + 7 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9 + 7 * q^11 - q^13 - 10 * q^15 + 3 * q^17 + 4 * q^19 - q^21 - 13 * q^23 + 10 * q^25 - 10 * q^27 + 18 * q^29 - 9 * q^31 - 7 * q^33 + q^35 - 3 * q^37 + q^39 + 19 * q^41 + 5 * q^43 + 10 * q^45 - 8 * q^47 + 43 * q^49 - 3 * q^51 + 17 * q^53 + 7 * q^55 - 4 * q^57 + 24 * q^59 + 21 * q^61 + q^63 - q^65 - 10 * q^67 + 13 * q^69 + 2 * q^71 + 25 * q^73 - 10 * q^75 + 15 * q^77 - q^79 + 10 * q^81 - 6 * q^83 + 3 * q^85 - 18 * q^87 + 23 * q^89 + 29 * q^91 + 9 * q^93 + 4 * q^95 + 21 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

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