LMFDB - Embedded newform 8040.2.a.bb.1.3

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 34x^{7} + 123x^{6} + 375x^{5} - 1146x^{4} - 1662x^{3} + 3086x^{2} + 3372x - 64 $ x^9 - 4x^8 - 34x^7 + 123x^6 + 375x^5 - 1146x^4 - 1662x^3 + 3086x^2 + 3372x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.03344$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -1.95262 q^{7} +1.00000 q^{9} -0.265154 q^{11} -3.83909 q^{13} +1.00000 q^{15} -0.435176 q^{17} +4.50625 q^{19} -1.95262 q^{21} +8.20289 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.01331 q^{29} -6.70915 q^{31} -0.265154 q^{33} -1.95262 q^{35} -1.27093 q^{37} -3.83909 q^{39} +6.36264 q^{41} -3.44095 q^{43} +1.00000 q^{45} +8.40121 q^{47} -3.18728 q^{49} -0.435176 q^{51} +1.15437 q^{53} -0.265154 q^{55} +4.50625 q^{57} -9.49481 q^{59} +9.06300 q^{61} -1.95262 q^{63} -3.83909 q^{65} -1.00000 q^{67} +8.20289 q^{69} -0.652581 q^{71} -2.37379 q^{73} +1.00000 q^{75} +0.517744 q^{77} -8.13215 q^{79} +1.00000 q^{81} -0.870056 q^{83} -0.435176 q^{85} +4.01331 q^{87} +16.3725 q^{89} +7.49628 q^{91} -6.70915 q^{93} +4.50625 q^{95} -6.86586 q^{97} -0.265154 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} + 9 q^{5} + 5 q^{7} + 9 q^{9} - 7 q^{11} - 3 q^{13} + 9 q^{15} + 7 q^{17} + 8 q^{19} + 5 q^{21} + 19 q^{23} + 9 q^{25} + 9 q^{27} + 14 q^{29} + 27 q^{31} - 7 q^{33} + 5 q^{35} + 15 q^{37}+ \cdots - 7 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 + 9 * q^5 + 5 * q^7 + 9 * q^9 - 7 * q^11 - 3 * q^13 + 9 * q^15 + 7 * q^17 + 8 * q^19 + 5 * q^21 + 19 * q^23 + 9 * q^25 + 9 * q^27 + 14 * q^29 + 27 * q^31 - 7 * q^33 + 5 * q^35 + 15 * q^37 - 3 * q^39 - 5 * q^41 + 11 * q^43 + 9 * q^45 + 6 * q^47 + 30 * q^49 + 7 * q^51 - 11 * q^53 - 7 * q^55 + 8 * q^57 + 14 * q^59 - 7 * q^61 + 5 * q^63 - 3 * q^65 - 9 * q^67 + 19 * q^69 - 2 * q^71 - 23 * q^73 + 9 * q^75 - 9 * q^77 + 13 * q^79 + 9 * q^81 + 48 * q^83 + 7 * q^85 + 14 * q^87 + 5 * q^89 + 41 * q^91 + 27 * q^93 + 8 * q^95 + 19 * q^97 - 7 * 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
$3$	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
$7$	−1.95262	−0.738020	−0.369010	−	0.929425i	$-0.620303\pi$
$7$	−1.95262	−0.738020	−0.369010	+	0.929425i	$0.620303\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.265154	−0.0799469	−0.0399734	−	0.999201i	$-0.512727\pi$
$11$	−0.265154	−0.0799469	−0.0399734	+	0.999201i	$0.512727\pi$
$12$	0	0
$13$	−3.83909	−1.06477	−0.532386	−	0.846502i	$-0.678704\pi$
$13$	−3.83909	−1.06477	−0.532386	+	0.846502i	$0.678704\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−0.435176	−0.105546	−0.0527729	−	0.998607i	$-0.516806\pi$
$17$	−0.435176	−0.105546	−0.0527729	+	0.998607i	$0.516806\pi$
$18$	0	0
$19$	4.50625	1.03381	0.516903	−	0.856044i	$-0.327085\pi$
$19$	4.50625	1.03381	0.516903	+	0.856044i	$0.327085\pi$
$20$	0	0
$21$	−1.95262	−0.426096
$22$	0	0
$23$	8.20289	1.71042	0.855211	−	0.518280i	$-0.173428\pi$
$23$	8.20289	1.71042	0.855211	+	0.518280i	$0.173428\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.01331	0.745253	0.372627	−	0.927981i	$-0.378457\pi$
$29$	4.01331	0.745253	0.372627	+	0.927981i	$0.378457\pi$
$30$	0	0
$31$	−6.70915	−1.20500	−0.602499	−	0.798120i	$-0.705828\pi$
$31$	−6.70915	−1.20500	−0.602499	+	0.798120i	$0.705828\pi$
$32$	0	0
$33$	−0.265154	−0.0461573
$34$	0	0
$35$	−1.95262	−0.330053
$36$	0	0
$37$	−1.27093	−0.208940	−0.104470	−	0.994528i	$-0.533315\pi$
$37$	−1.27093	−0.208940	−0.104470	+	0.994528i	$0.533315\pi$
$38$	0	0
$39$	−3.83909	−0.614746
$40$	0	0
$41$	6.36264	0.993677	0.496838	−	0.867843i	$-0.334494\pi$
$41$	6.36264	0.993677	0.496838	+	0.867843i	$0.334494\pi$
$42$	0	0
$43$	−3.44095	−0.524741	−0.262370	−	0.964967i	$-0.584504\pi$
$43$	−3.44095	−0.524741	−0.262370	+	0.964967i	$0.584504\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	8.40121	1.22544	0.612721	−	0.790299i	$-0.290075\pi$
$47$	8.40121	1.22544	0.612721	+	0.790299i	$0.290075\pi$
$48$	0	0
$49$	−3.18728	−0.455326
$50$	0	0
$51$	−0.435176	−0.0609369
$52$	0	0
$53$	1.15437	0.158565	0.0792825	−	0.996852i	$-0.474737\pi$
$53$	1.15437	0.158565	0.0792825	+	0.996852i	$0.474737\pi$
$54$	0	0
$55$	−0.265154	−0.0357533
$56$	0	0
$57$	4.50625	0.596868
$58$	0	0
$59$	−9.49481	−1.23612	−0.618060	−	0.786131i	$-0.712081\pi$
$59$	−9.49481	−1.23612	−0.618060	+	0.786131i	$0.712081\pi$
$60$	0	0
$61$	9.06300	1.16040	0.580199	−	0.814474i	$-0.302975\pi$
$61$	9.06300	1.16040	0.580199	+	0.814474i	$0.302975\pi$
$62$	0	0
$63$	−1.95262	−0.246007
$64$	0	0
$65$	−3.83909	−0.476181
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	8.20289	0.987512
$70$	0	0
$71$	−0.652581	−0.0774471	−0.0387236	−	0.999250i	$-0.512329\pi$
$71$	−0.652581	−0.0774471	−0.0387236	+	0.999250i	$0.512329\pi$
$72$	0	0
$73$	−2.37379	−0.277831	−0.138915	−	0.990304i	$-0.544362\pi$
$73$	−2.37379	−0.277831	−0.138915	+	0.990304i	$0.544362\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0.517744	0.0590024
$78$	0	0
$79$	−8.13215	−0.914938	−0.457469	−	0.889226i	$-0.651244\pi$
$79$	−8.13215	−0.914938	−0.457469	+	0.889226i	$0.651244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.870056	−0.0955010	−0.0477505	−	0.998859i	$-0.515205\pi$
$83$	−0.870056	−0.0955010	−0.0477505	+	0.998859i	$0.515205\pi$
$84$	0	0
$85$	−0.435176	−0.0472015
$86$	0	0
$87$	4.01331	0.430272
$88$	0	0
$89$	16.3725	1.73548	0.867740	−	0.497019i	$-0.165572\pi$
$89$	16.3725	1.73548	0.867740	+	0.497019i	$0.165572\pi$
$90$	0	0
$91$	7.49628	0.785823
$92$	0	0
$93$	−6.70915	−0.695706
$94$	0	0
$95$	4.50625	0.462332
$96$	0	0
$97$	−6.86586	−0.697122	−0.348561	−	0.937286i	$-0.613330\pi$
$97$	−6.86586	−0.697122	−0.348561	+	0.937286i	$0.613330\pi$
$98$	0	0
$99$	−0.265154	−0.0266490
$100$	0	0
$101$	−8.03360	−0.799373	−0.399687	−	0.916652i	$-0.630881\pi$
$101$	−8.03360	−0.799373	−0.399687	+	0.916652i	$0.630881\pi$
$102$	0	0
$103$	6.74596	0.664699	0.332349	−	0.943156i	$-0.392159\pi$
$103$	6.74596	0.664699	0.332349	+	0.943156i	$0.392159\pi$
$104$	0	0
$105$	−1.95262	−0.190556
$106$	0	0
$107$	2.98468	0.288539	0.144270	−	0.989538i	$-0.453917\pi$
$107$	2.98468	0.288539	0.144270	+	0.989538i	$0.453917\pi$
$108$	0	0
$109$	11.6639	1.11720	0.558602	−	0.829436i	$-0.311338\pi$
$109$	11.6639	1.11720	0.558602	+	0.829436i	$0.311338\pi$
$110$	0	0
$111$	−1.27093	−0.120631
$112$	0	0
$113$	−10.8134	−1.01724	−0.508619	−	0.860992i	$-0.669844\pi$
$113$	−10.8134	−1.01724	−0.508619	+	0.860992i	$0.669844\pi$
$114$	0	0
$115$	8.20289	0.764924
$116$	0	0
$117$	−3.83909	−0.354924
$118$	0	0
$119$	0.849733	0.0778949
$120$	0	0
$121$	−10.9297	−0.993608
$122$	0	0
$123$	6.36264	0.573700
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.7626	1.57618	0.788090	−	0.615560i	$-0.211070\pi$
$127$	17.7626	1.57618	0.788090	+	0.615560i	$0.211070\pi$
$128$	0	0
$129$	−3.44095	−0.302959
$130$	0	0
$131$	1.53643	0.134238	0.0671192	−	0.997745i	$-0.478619\pi$
$131$	1.53643	0.134238	0.0671192	+	0.997745i	$0.478619\pi$
$132$	0	0
$133$	−8.79899	−0.762969
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	17.1571	1.46583	0.732915	−	0.680321i	$-0.238159\pi$
$137$	17.1571	1.46583	0.732915	+	0.680321i	$0.238159\pi$
$138$	0	0
$139$	14.3721	1.21902	0.609512	−	0.792777i	$-0.291366\pi$
$139$	14.3721	1.21902	0.609512	+	0.792777i	$0.291366\pi$
$140$	0	0
$141$	8.40121	0.707510
$142$	0	0
$143$	1.01795	0.0851252
$144$	0	0
$145$	4.01331	0.333287
$146$	0	0
$147$	−3.18728	−0.262883
$148$	0	0
$149$	4.10765	0.336512	0.168256	−	0.985743i	$-0.446187\pi$
$149$	4.10765	0.336512	0.168256	+	0.985743i	$0.446187\pi$
$150$	0	0
$151$	15.9041	1.29425	0.647127	−	0.762382i	$-0.275970\pi$
$151$	15.9041	1.29425	0.647127	+	0.762382i	$0.275970\pi$
$152$	0	0
$153$	−0.435176	−0.0351819
$154$	0	0
$155$	−6.70915	−0.538892
$156$	0	0
$157$	−4.44619	−0.354844	−0.177422	−	0.984135i	$-0.556776\pi$
$157$	−4.44619	−0.354844	−0.177422	+	0.984135i	$0.556776\pi$
$158$	0	0
$159$	1.15437	0.0915475
$160$	0	0
$161$	−16.0171	−1.26233
$162$	0	0
$163$	−7.18537	−0.562802	−0.281401	−	0.959590i	$-0.590799\pi$
$163$	−7.18537	−0.562802	−0.281401	+	0.959590i	$0.590799\pi$
$164$	0	0
$165$	−0.265154	−0.0206422
$166$	0	0
$167$	20.4382	1.58156	0.790778	−	0.612103i	$-0.209676\pi$
$167$	20.4382	1.58156	0.790778	+	0.612103i	$0.209676\pi$
$168$	0	0
$169$	1.73861	0.133739
$170$	0	0
$171$	4.50625	0.344602
$172$	0	0
$173$	1.45657	0.110741	0.0553703	−	0.998466i	$-0.482366\pi$
$173$	1.45657	0.110741	0.0553703	+	0.998466i	$0.482366\pi$
$174$	0	0
$175$	−1.95262	−0.147604
$176$	0	0
$177$	−9.49481	−0.713674
$178$	0	0
$179$	11.2181	0.838483	0.419242	−	0.907875i	$-0.362296\pi$
$179$	11.2181	0.838483	0.419242	+	0.907875i	$0.362296\pi$
$180$	0	0
$181$	9.52874	0.708266	0.354133	−	0.935195i	$-0.384776\pi$
$181$	9.52874	0.708266	0.354133	+	0.935195i	$0.384776\pi$
$182$	0	0
$183$	9.06300	0.669957
$184$	0	0
$185$	−1.27093	−0.0934407
$186$	0	0
$187$	0.115389	0.00843805
$188$	0	0
$189$	−1.95262	−0.142032
$190$	0	0
$191$	26.0126	1.88221	0.941103	−	0.338121i	$-0.109791\pi$
$191$	26.0126	1.88221	0.941103	+	0.338121i	$0.109791\pi$
$192$	0	0
$193$	23.6361	1.70136	0.850680	−	0.525683i	$-0.176190\pi$
$193$	23.6361	1.70136	0.850680	+	0.525683i	$0.176190\pi$
$194$	0	0
$195$	−3.83909	−0.274923
$196$	0	0
$197$	−6.23755	−0.444407	−0.222204	−	0.975000i	$-0.571325\pi$
$197$	−6.23755	−0.444407	−0.222204	+	0.975000i	$0.571325\pi$
$198$	0	0
$199$	15.4983	1.09865	0.549323	−	0.835610i	$-0.314886\pi$
$199$	15.4983	1.09865	0.549323	+	0.835610i	$0.314886\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	−7.83646	−0.550012
$204$	0	0
$205$	6.36264	0.444386
$206$	0	0
$207$	8.20289	0.570140
$208$	0	0
$209$	−1.19485	−0.0826495
$210$	0	0
$211$	10.0400	0.691185	0.345592	−	0.938385i	$-0.387678\pi$
$211$	10.0400	0.691185	0.345592	+	0.938385i	$0.387678\pi$
$212$	0	0
$213$	−0.652581	−0.0447141
$214$	0	0
$215$	−3.44095	−0.234671
$216$	0	0
$217$	13.1004	0.889313
$218$	0	0
$219$	−2.37379	−0.160406
$220$	0	0
$221$	1.67068	0.112382
$222$	0	0
$223$	−26.5184	−1.77580	−0.887902	−	0.460033i	$-0.847838\pi$
$223$	−26.5184	−1.77580	−0.887902	+	0.460033i	$0.847838\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	9.91737	0.658239	0.329119	−	0.944288i	$-0.393248\pi$
$227$	9.91737	0.658239	0.329119	+	0.944288i	$0.393248\pi$
$228$	0	0
$229$	13.2499	0.875579	0.437790	−	0.899077i	$-0.355761\pi$
$229$	13.2499	0.875579	0.437790	+	0.899077i	$0.355761\pi$
$230$	0	0
$231$	0.517744	0.0340651
$232$	0	0
$233$	−10.7279	−0.702809	−0.351405	−	0.936224i	$-0.614296\pi$
$233$	−10.7279	−0.702809	−0.351405	+	0.936224i	$0.614296\pi$
$234$	0	0
$235$	8.40121	0.548035
$236$	0	0
$237$	−8.13215	−0.528240
$238$	0	0
$239$	16.2197	1.04916	0.524582	−	0.851360i	$-0.324222\pi$
$239$	16.2197	1.04916	0.524582	+	0.851360i	$0.324222\pi$
$240$	0	0
$241$	−4.04466	−0.260539	−0.130270	−	0.991479i	$-0.541584\pi$
$241$	−4.04466	−0.260539	−0.130270	+	0.991479i	$0.541584\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.18728	−0.203628
$246$	0	0
$247$	−17.2999	−1.10077
$248$	0	0
$249$	−0.870056	−0.0551376
$250$	0	0
$251$	0.0940397	0.00593573	0.00296787	−	0.999996i	$-0.499055\pi$
$251$	0.0940397	0.00593573	0.00296787	+	0.999996i	$0.499055\pi$
$252$	0	0
$253$	−2.17503	−0.136743
$254$	0	0
$255$	−0.435176	−0.0272518
$256$	0	0
$257$	−2.48830	−0.155216	−0.0776079	−	0.996984i	$-0.524728\pi$
$257$	−2.48830	−0.155216	−0.0776079	+	0.996984i	$0.524728\pi$
$258$	0	0
$259$	2.48164	0.154202
$260$	0	0
$261$	4.01331	0.248418
$262$	0	0
$263$	11.8054	0.727953	0.363977	−	0.931408i	$-0.381419\pi$
$263$	11.8054	0.727953	0.363977	+	0.931408i	$0.381419\pi$
$264$	0	0
$265$	1.15437	0.0709124
$266$	0	0
$267$	16.3725	1.00198
$268$	0	0
$269$	18.0241	1.09895	0.549474	−	0.835511i	$-0.314828\pi$
$269$	18.0241	1.09895	0.549474	+	0.835511i	$0.314828\pi$
$270$	0	0
$271$	24.8036	1.50671	0.753357	−	0.657612i	$-0.228433\pi$
$271$	24.8036	1.50671	0.753357	+	0.657612i	$0.228433\pi$
$272$	0	0
$273$	7.49628	0.453695
$274$	0	0
$275$	−0.265154	−0.0159894
$276$	0	0
$277$	−13.6414	−0.819632	−0.409816	−	0.912168i	$-0.634407\pi$
$277$	−13.6414	−0.819632	−0.409816	+	0.912168i	$0.634407\pi$
$278$	0	0
$279$	−6.70915	−0.401666
$280$	0	0
$281$	−10.3878	−0.619686	−0.309843	−	0.950788i	$-0.600277\pi$
$281$	−10.3878	−0.619686	−0.309843	+	0.950788i	$0.600277\pi$
$282$	0	0
$283$	21.1531	1.25742	0.628711	−	0.777639i	$-0.283583\pi$
$283$	21.1531	1.25742	0.628711	+	0.777639i	$0.283583\pi$
$284$	0	0
$285$	4.50625	0.266927
$286$	0	0
$287$	−12.4238	−0.733354
$288$	0	0
$289$	−16.8106	−0.988860
$290$	0	0
$291$	−6.86586	−0.402484
$292$	0	0
$293$	−8.21204	−0.479753	−0.239876	−	0.970803i	$-0.577107\pi$
$293$	−8.21204	−0.479753	−0.239876	+	0.970803i	$0.577107\pi$
$294$	0	0
$295$	−9.49481	−0.552809
$296$	0	0
$297$	−0.265154	−0.0153858
$298$	0	0
$299$	−31.4916	−1.82121
$300$	0	0
$301$	6.71887	0.387269
$302$	0	0
$303$	−8.03360	−0.461518
$304$	0	0
$305$	9.06300	0.518946
$306$	0	0
$307$	−12.3029	−0.702164	−0.351082	−	0.936345i	$-0.614186\pi$
$307$	−12.3029	−0.702164	−0.351082	+	0.936345i	$0.614186\pi$
$308$	0	0
$309$	6.74596	0.383764
$310$	0	0
$311$	32.7081	1.85471	0.927353	−	0.374187i	$-0.122078\pi$
$311$	32.7081	1.85471	0.927353	+	0.374187i	$0.122078\pi$
$312$	0	0
$313$	−9.45635	−0.534505	−0.267252	−	0.963627i	$-0.586116\pi$
$313$	−9.45635	−0.534505	−0.267252	+	0.963627i	$0.586116\pi$
$314$	0	0
$315$	−1.95262	−0.110018
$316$	0	0
$317$	22.9380	1.28833	0.644164	−	0.764888i	$-0.277206\pi$
$317$	22.9380	1.28833	0.644164	+	0.764888i	$0.277206\pi$
$318$	0	0
$319$	−1.06414	−0.0595807
$320$	0	0
$321$	2.98468	0.166588
$322$	0	0
$323$	−1.96102	−0.109114
$324$	0	0
$325$	−3.83909	−0.212954
$326$	0	0
$327$	11.6639	0.645018
$328$	0	0
$329$	−16.4044	−0.904401
$330$	0	0
$331$	31.5109	1.73200	0.865998	−	0.500048i	$-0.166684\pi$
$331$	31.5109	1.73200	0.865998	+	0.500048i	$0.166684\pi$
$332$	0	0
$333$	−1.27093	−0.0696466
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−21.1089	−1.14987	−0.574936	−	0.818198i	$-0.694973\pi$
$337$	−21.1089	−1.14987	−0.574936	+	0.818198i	$0.694973\pi$
$338$	0	0
$339$	−10.8134	−0.587303
$340$	0	0
$341$	1.77896	0.0963358
$342$	0	0
$343$	19.8919	1.07406
$344$	0	0
$345$	8.20289	0.441629
$346$	0	0
$347$	−27.9072	−1.49813	−0.749067	−	0.662494i	$-0.769498\pi$
$347$	−27.9072	−1.49813	−0.749067	+	0.662494i	$0.769498\pi$
$348$	0	0
$349$	−20.5585	−1.10047	−0.550235	−	0.835010i	$-0.685462\pi$
$349$	−20.5585	−1.10047	−0.550235	+	0.835010i	$0.685462\pi$
$350$	0	0
$351$	−3.83909	−0.204915
$352$	0	0
$353$	−24.7433	−1.31695	−0.658477	−	0.752601i	$-0.728799\pi$
$353$	−24.7433	−1.31695	−0.658477	+	0.752601i	$0.728799\pi$
$354$	0	0
$355$	−0.652581	−0.0346354
$356$	0	0
$357$	0.849733	0.0449727
$358$	0	0
$359$	−27.2071	−1.43593	−0.717967	−	0.696078i	$-0.754927\pi$
$359$	−27.2071	−1.43593	−0.717967	+	0.696078i	$0.754927\pi$
$360$	0	0
$361$	1.30632	0.0687535
$362$	0	0
$363$	−10.9297	−0.573660
$364$	0	0
$365$	−2.37379	−0.124250
$366$	0	0
$367$	−30.0167	−1.56686	−0.783428	−	0.621482i	$-0.786531\pi$
$367$	−30.0167	−1.56686	−0.783428	+	0.621482i	$0.786531\pi$
$368$	0	0
$369$	6.36264	0.331226
$370$	0	0
$371$	−2.25404	−0.117024
$372$	0	0
$373$	−16.5804	−0.858500	−0.429250	−	0.903186i	$-0.641222\pi$
$373$	−16.5804	−0.858500	−0.429250	+	0.903186i	$0.641222\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−15.4075	−0.793525
$378$	0	0
$379$	−15.6662	−0.804717	−0.402358	−	0.915482i	$-0.631809\pi$
$379$	−15.6662	−0.804717	−0.402358	+	0.915482i	$0.631809\pi$
$380$	0	0
$381$	17.7626	0.910008
$382$	0	0
$383$	30.2799	1.54723	0.773616	−	0.633655i	$-0.218446\pi$
$383$	30.2799	1.54723	0.773616	+	0.633655i	$0.218446\pi$
$384$	0	0
$385$	0.517744	0.0263867
$386$	0	0
$387$	−3.44095	−0.174914
$388$	0	0
$389$	24.1719	1.22557	0.612783	−	0.790251i	$-0.290050\pi$
$389$	24.1719	1.22557	0.612783	+	0.790251i	$0.290050\pi$
$390$	0	0
$391$	−3.56971	−0.180528
$392$	0	0
$393$	1.53643	0.0775026
$394$	0	0
$395$	−8.13215	−0.409173
$396$	0	0
$397$	−10.1981	−0.511830	−0.255915	−	0.966699i	$-0.582377\pi$
$397$	−10.1981	−0.511830	−0.255915	+	0.966699i	$0.582377\pi$
$398$	0	0
$399$	−8.79899	−0.440501
$400$	0	0
$401$	−11.8028	−0.589403	−0.294702	−	0.955589i	$-0.595220\pi$
$401$	−11.8028	−0.589403	−0.294702	+	0.955589i	$0.595220\pi$
$402$	0	0
$403$	25.7570	1.28305
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0.336992	0.0167041
$408$	0	0
$409$	−0.329324	−0.0162840	−0.00814201	−	0.999967i	$-0.502592\pi$
$409$	−0.329324	−0.0162840	−0.00814201	+	0.999967i	$0.502592\pi$
$410$	0	0
$411$	17.1571	0.846297
$412$	0	0
$413$	18.5397	0.912281
$414$	0	0
$415$	−0.870056	−0.0427094
$416$	0	0
$417$	14.3721	0.703803
$418$	0	0
$419$	−27.6549	−1.35103	−0.675516	−	0.737346i	$-0.736079\pi$
$419$	−27.6549	−1.35103	−0.675516	+	0.737346i	$0.736079\pi$
$420$	0	0
$421$	8.35013	0.406961	0.203480	−	0.979079i	$-0.434775\pi$
$421$	8.35013	0.406961	0.203480	+	0.979079i	$0.434775\pi$
$422$	0	0
$423$	8.40121	0.408481
$424$	0	0
$425$	−0.435176	−0.0211092
$426$	0	0
$427$	−17.6966	−0.856398
$428$	0	0
$429$	1.01795	0.0491470
$430$	0	0
$431$	−14.0248	−0.675553	−0.337776	−	0.941226i	$-0.609675\pi$
$431$	−14.0248	−0.675553	−0.337776	+	0.941226i	$0.609675\pi$
$432$	0	0
$433$	24.0503	1.15579	0.577893	−	0.816113i	$-0.303875\pi$
$433$	24.0503	1.15579	0.577893	+	0.816113i	$0.303875\pi$
$434$	0	0
$435$	4.01331	0.192424
$436$	0	0
$437$	36.9643	1.76824
$438$	0	0
$439$	−5.30282	−0.253090	−0.126545	−	0.991961i	$-0.540389\pi$
$439$	−5.30282	−0.253090	−0.126545	+	0.991961i	$0.540389\pi$
$440$	0	0
$441$	−3.18728	−0.151775
$442$	0	0
$443$	40.1451	1.90735	0.953675	−	0.300839i	$-0.0972668\pi$
$443$	40.1451	1.90735	0.953675	+	0.300839i	$0.0972668\pi$
$444$	0	0
$445$	16.3725	0.776130
$446$	0	0
$447$	4.10765	0.194285
$448$	0	0
$449$	9.47336	0.447075	0.223538	−	0.974695i	$-0.428239\pi$
$449$	9.47336	0.447075	0.223538	+	0.974695i	$0.428239\pi$
$450$	0	0
$451$	−1.68708	−0.0794414
$452$	0	0
$453$	15.9041	0.747238
$454$	0	0
$455$	7.49628	0.351431
$456$	0	0
$457$	14.4001	0.673609	0.336805	−	0.941575i	$-0.390654\pi$
$457$	14.4001	0.673609	0.336805	+	0.941575i	$0.390654\pi$
$458$	0	0
$459$	−0.435176	−0.0203123
$460$	0	0
$461$	17.2780	0.804717	0.402358	−	0.915482i	$-0.368191\pi$
$461$	17.2780	0.804717	0.402358	+	0.915482i	$0.368191\pi$
$462$	0	0
$463$	−39.0020	−1.81258	−0.906289	−	0.422658i	$-0.861097\pi$
$463$	−39.0020	−1.81258	−0.906289	+	0.422658i	$0.861097\pi$
$464$	0	0
$465$	−6.70915	−0.311129
$466$	0	0
$467$	−20.0667	−0.928575	−0.464287	−	0.885685i	$-0.653689\pi$
$467$	−20.0667	−0.928575	−0.464287	+	0.885685i	$0.653689\pi$
$468$	0	0
$469$	1.95262	0.0901635
$470$	0	0
$471$	−4.44619	−0.204869
$472$	0	0
$473$	0.912382	0.0419514
$474$	0	0
$475$	4.50625	0.206761
$476$	0	0
$477$	1.15437	0.0528550
$478$	0	0
$479$	2.61228	0.119358	0.0596791	−	0.998218i	$-0.480992\pi$
$479$	2.61228	0.119358	0.0596791	+	0.998218i	$0.480992\pi$
$480$	0	0
$481$	4.87922	0.222473
$482$	0	0
$483$	−16.0171	−0.728804
$484$	0	0
$485$	−6.86586	−0.311763
$486$	0	0
$487$	37.1890	1.68519	0.842597	−	0.538545i	$-0.181026\pi$
$487$	37.1890	1.68519	0.842597	+	0.538545i	$0.181026\pi$
$488$	0	0
$489$	−7.18537	−0.324934
$490$	0	0
$491$	−17.3174	−0.781522	−0.390761	−	0.920492i	$-0.627788\pi$
$491$	−17.3174	−0.781522	−0.390761	+	0.920492i	$0.627788\pi$
$492$	0	0
$493$	−1.74650	−0.0786583
$494$	0	0
$495$	−0.265154	−0.0119178
$496$	0	0
$497$	1.27424	0.0571575
$498$	0	0
$499$	7.62022	0.341128	0.170564	−	0.985347i	$-0.445441\pi$
$499$	7.62022	0.341128	0.170564	+	0.985347i	$0.445441\pi$
$500$	0	0
$501$	20.4382	0.913112
$502$	0	0
$503$	−26.6211	−1.18697	−0.593487	−	0.804844i	$-0.702249\pi$
$503$	−26.6211	−1.18697	−0.593487	+	0.804844i	$0.702249\pi$
$504$	0	0
$505$	−8.03360	−0.357491
$506$	0	0
$507$	1.73861	0.0772144
$508$	0	0
$509$	−38.0975	−1.68864	−0.844320	−	0.535839i	$-0.819995\pi$
$509$	−38.0975	−1.68864	−0.844320	+	0.535839i	$0.819995\pi$
$510$	0	0
$511$	4.63510	0.205045
$512$	0	0
$513$	4.50625	0.198956
$514$	0	0
$515$	6.74596	0.297262
$516$	0	0
$517$	−2.22761	−0.0979703
$518$	0	0
$519$	1.45657	0.0639361
$520$	0	0
$521$	−12.1998	−0.534482	−0.267241	−	0.963630i	$-0.586112\pi$
$521$	−12.1998	−0.534482	−0.267241	+	0.963630i	$0.586112\pi$
$522$	0	0
$523$	3.78464	0.165491	0.0827454	−	0.996571i	$-0.473631\pi$
$523$	3.78464	0.165491	0.0827454	+	0.996571i	$0.473631\pi$
$524$	0	0
$525$	−1.95262	−0.0852192
$526$	0	0
$527$	2.91966	0.127182
$528$	0	0
$529$	44.2875	1.92554
$530$	0	0
$531$	−9.49481	−0.412040
$532$	0	0
$533$	−24.4267	−1.05804
$534$	0	0
$535$	2.98468	0.129039
$536$	0	0
$537$	11.2181	0.484098
$538$	0	0
$539$	0.845120	0.0364019
$540$	0	0
$541$	11.7660	0.505859	0.252929	−	0.967485i	$-0.418606\pi$
$541$	11.7660	0.505859	0.252929	+	0.967485i	$0.418606\pi$
$542$	0	0
$543$	9.52874	0.408918
$544$	0	0
$545$	11.6639	0.499629
$546$	0	0
$547$	−8.76173	−0.374624	−0.187312	−	0.982300i	$-0.559978\pi$
$547$	−8.76173	−0.374624	−0.187312	+	0.982300i	$0.559978\pi$
$548$	0	0
$549$	9.06300	0.386800
$550$	0	0
$551$	18.0850	0.770447
$552$	0	0
$553$	15.8790	0.675243
$554$	0	0
$555$	−1.27093	−0.0539480
$556$	0	0
$557$	−39.9639	−1.69332	−0.846661	−	0.532132i	$-0.821391\pi$
$557$	−39.9639	−1.69332	−0.846661	+	0.532132i	$0.821391\pi$
$558$	0	0
$559$	13.2101	0.558729
$560$	0	0
$561$	0.115389	0.00487171
$562$	0	0
$563$	32.0908	1.35247	0.676233	−	0.736688i	$-0.263611\pi$
$563$	32.0908	1.35247	0.676233	+	0.736688i	$0.263611\pi$
$564$	0	0
$565$	−10.8134	−0.454923
$566$	0	0
$567$	−1.95262	−0.0820023
$568$	0	0
$569$	9.40326	0.394205	0.197102	−	0.980383i	$-0.436847\pi$
$569$	9.40326	0.394205	0.197102	+	0.980383i	$0.436847\pi$
$570$	0	0
$571$	29.9363	1.25280	0.626398	−	0.779503i	$-0.284529\pi$
$571$	29.9363	1.25280	0.626398	+	0.779503i	$0.284529\pi$
$572$	0	0
$573$	26.0126	1.08669
$574$	0	0
$575$	8.20289	0.342084
$576$	0	0
$577$	25.4189	1.05820	0.529102	−	0.848558i	$-0.322529\pi$
$577$	25.4189	1.05820	0.529102	+	0.848558i	$0.322529\pi$
$578$	0	0
$579$	23.6361	0.982281
$580$	0	0
$581$	1.69889	0.0704817
$582$	0	0
$583$	−0.306086	−0.0126768
$584$	0	0
$585$	−3.83909	−0.158727
$586$	0	0
$587$	−4.87534	−0.201227	−0.100613	−	0.994926i	$-0.532081\pi$
$587$	−4.87534	−0.201227	−0.100613	+	0.994926i	$0.532081\pi$
$588$	0	0
$589$	−30.2331	−1.24573
$590$	0	0
$591$	−6.23755	−0.256579
$592$	0	0
$593$	−33.1154	−1.35989	−0.679943	−	0.733265i	$-0.737995\pi$
$593$	−33.1154	−1.35989	−0.679943	+	0.733265i	$0.737995\pi$
$594$	0	0
$595$	0.849733	0.0348357
$596$	0	0
$597$	15.4983	0.634303
$598$	0	0
$599$	−31.5281	−1.28820	−0.644101	−	0.764940i	$-0.722769\pi$
$599$	−31.5281	−1.28820	−0.644101	+	0.764940i	$0.722769\pi$
$600$	0	0
$601$	4.79001	0.195389	0.0976943	−	0.995216i	$-0.468853\pi$
$601$	4.79001	0.195389	0.0976943	+	0.995216i	$0.468853\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−10.9297	−0.444355
$606$	0	0
$607$	26.5302	1.07683	0.538413	−	0.842681i	$-0.319024\pi$
$607$	26.5302	1.07683	0.538413	+	0.842681i	$0.319024\pi$
$608$	0	0
$609$	−7.83646	−0.317550
$610$	0	0
$611$	−32.2530	−1.30482
$612$	0	0
$613$	−22.9635	−0.927485	−0.463743	−	0.885970i	$-0.653494\pi$
$613$	−22.9635	−0.927485	−0.463743	+	0.885970i	$0.653494\pi$
$614$	0	0
$615$	6.36264	0.256566
$616$	0	0
$617$	−11.2855	−0.454338	−0.227169	−	0.973855i	$-0.572947\pi$
$617$	−11.2855	−0.454338	−0.227169	+	0.973855i	$0.572947\pi$
$618$	0	0
$619$	7.06201	0.283846	0.141923	−	0.989878i	$-0.454671\pi$
$619$	7.06201	0.283846	0.141923	+	0.989878i	$0.454671\pi$
$620$	0	0
$621$	8.20289	0.329171
$622$	0	0
$623$	−31.9692	−1.28082
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.19485	−0.0477177
$628$	0	0
$629$	0.553079	0.0220527
$630$	0	0
$631$	33.5015	1.33367	0.666836	−	0.745204i	$-0.267648\pi$
$631$	33.5015	1.33367	0.666836	+	0.745204i	$0.267648\pi$
$632$	0	0
$633$	10.0400	0.399056
$634$	0	0
$635$	17.7626	0.704889
$636$	0	0
$637$	12.2363	0.484818
$638$	0	0
$639$	−0.652581	−0.0258157
$640$	0	0
$641$	−4.26082	−0.168292	−0.0841462	−	0.996453i	$-0.526816\pi$
$641$	−4.26082	−0.168292	−0.0841462	+	0.996453i	$0.526816\pi$
$642$	0	0
$643$	3.23148	0.127437	0.0637185	−	0.997968i	$-0.479704\pi$
$643$	3.23148	0.127437	0.0637185	+	0.997968i	$0.479704\pi$
$644$	0	0
$645$	−3.44095	−0.135487
$646$	0	0
$647$	−26.1199	−1.02688	−0.513439	−	0.858126i	$-0.671629\pi$
$647$	−26.1199	−1.02688	−0.513439	+	0.858126i	$0.671629\pi$
$648$	0	0
$649$	2.51759	0.0988239
$650$	0	0
$651$	13.1004	0.513445
$652$	0	0
$653$	37.3911	1.46323	0.731613	−	0.681720i	$-0.238768\pi$
$653$	37.3911	1.46323	0.731613	+	0.681720i	$0.238768\pi$
$654$	0	0
$655$	1.53643	0.0600333
$656$	0	0
$657$	−2.37379	−0.0926103
$658$	0	0
$659$	−32.7456	−1.27559	−0.637794	−	0.770207i	$-0.720153\pi$
$659$	−32.7456	−1.27559	−0.637794	+	0.770207i	$0.720153\pi$
$660$	0	0
$661$	−4.79183	−0.186380	−0.0931902	−	0.995648i	$-0.529706\pi$
$661$	−4.79183	−0.186380	−0.0931902	+	0.995648i	$0.529706\pi$
$662$	0	0
$663$	1.67068	0.0648839
$664$	0	0
$665$	−8.79899	−0.341210
$666$	0	0
$667$	32.9208	1.27470
$668$	0	0
$669$	−26.5184	−1.02526
$670$	0	0
$671$	−2.40309	−0.0927702
$672$	0	0
$673$	−22.0693	−0.850709	−0.425354	−	0.905027i	$-0.639851\pi$
$673$	−22.0693	−0.850709	−0.425354	+	0.905027i	$0.639851\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−5.15103	−0.197970	−0.0989850	−	0.995089i	$-0.531560\pi$
$677$	−5.15103	−0.197970	−0.0989850	+	0.995089i	$0.531560\pi$
$678$	0	0
$679$	13.4064	0.514490
$680$	0	0
$681$	9.91737	0.380034
$682$	0	0
$683$	−19.5676	−0.748733	−0.374367	−	0.927281i	$-0.622140\pi$
$683$	−19.5676	−0.748733	−0.374367	+	0.927281i	$0.622140\pi$
$684$	0	0
$685$	17.1571	0.655539
$686$	0	0
$687$	13.2499	0.505516
$688$	0	0
$689$	−4.43173	−0.168836
$690$	0	0
$691$	7.57437	0.288142	0.144071	−	0.989567i	$-0.453981\pi$
$691$	7.57437	0.288142	0.144071	+	0.989567i	$0.453981\pi$
$692$	0	0
$693$	0.517744	0.0196675
$694$	0	0
$695$	14.3721	0.545164
$696$	0	0
$697$	−2.76887	−0.104878
$698$	0	0
$699$	−10.7279	−0.405767
$700$	0	0
$701$	−5.23090	−0.197568	−0.0987842	−	0.995109i	$-0.531495\pi$
$701$	−5.23090	−0.197568	−0.0987842	+	0.995109i	$0.531495\pi$
$702$	0	0
$703$	−5.72714	−0.216003
$704$	0	0
$705$	8.40121	0.316408
$706$	0	0
$707$	15.6866	0.589954
$708$	0	0
$709$	−43.1273	−1.61968	−0.809840	−	0.586651i	$-0.800446\pi$
$709$	−43.1273	−1.61968	−0.809840	+	0.586651i	$0.800446\pi$
$710$	0	0
$711$	−8.13215	−0.304979
$712$	0	0
$713$	−55.0344	−2.06105
$714$	0	0
$715$	1.01795	0.0380691
$716$	0	0
$717$	16.2197	0.605735
$718$	0	0
$719$	−9.06250	−0.337974	−0.168987	−	0.985618i	$-0.554050\pi$
$719$	−9.06250	−0.337974	−0.168987	+	0.985618i	$0.554050\pi$
$720$	0	0
$721$	−13.1723	−0.490561
$722$	0	0
$723$	−4.04466	−0.150423
$724$	0	0
$725$	4.01331	0.149051
$726$	0	0
$727$	9.01479	0.334340	0.167170	−	0.985928i	$-0.446537\pi$
$727$	9.01479	0.334340	0.167170	+	0.985928i	$0.446537\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.49742	0.0553842
$732$	0	0
$733$	−36.8909	−1.36260	−0.681298	−	0.732007i	$-0.738584\pi$
$733$	−36.8909	−1.36260	−0.681298	+	0.732007i	$0.738584\pi$
$734$	0	0
$735$	−3.18728	−0.117565
$736$	0	0
$737$	0.265154	0.00976706
$738$	0	0
$739$	20.3654	0.749154	0.374577	−	0.927196i	$-0.377788\pi$
$739$	20.3654	0.749154	0.374577	+	0.927196i	$0.377788\pi$
$740$	0	0
$741$	−17.2999	−0.635528
$742$	0	0
$743$	−27.9539	−1.02553	−0.512765	−	0.858529i	$-0.671379\pi$
$743$	−27.9539	−1.02553	−0.512765	+	0.858529i	$0.671379\pi$
$744$	0	0
$745$	4.10765	0.150493
$746$	0	0
$747$	−0.870056	−0.0318337
$748$	0	0
$749$	−5.82793	−0.212948
$750$	0	0
$751$	−12.4048	−0.452658	−0.226329	−	0.974051i	$-0.572672\pi$
$751$	−12.4048	−0.452658	−0.226329	+	0.974051i	$0.572672\pi$
$752$	0	0
$753$	0.0940397	0.00342700
$754$	0	0
$755$	15.9041	0.578808
$756$	0	0
$757$	−22.2592	−0.809023	−0.404512	−	0.914533i	$-0.632559\pi$
$757$	−22.2592	−0.809023	−0.404512	+	0.914533i	$0.632559\pi$
$758$	0	0
$759$	−2.17503	−0.0789485
$760$	0	0
$761$	−42.7567	−1.54993	−0.774964	−	0.632005i	$-0.782232\pi$
$761$	−42.7567	−1.54993	−0.774964	+	0.632005i	$0.782232\pi$
$762$	0	0
$763$	−22.7752	−0.824519
$764$	0	0
$765$	−0.435176	−0.0157338
$766$	0	0
$767$	36.4514	1.31619
$768$	0	0
$769$	−21.9473	−0.791439	−0.395719	−	0.918372i	$-0.629505\pi$
$769$	−21.9473	−0.791439	−0.395719	+	0.918372i	$0.629505\pi$
$770$	0	0
$771$	−2.48830	−0.0896139
$772$	0	0
$773$	31.2283	1.12320	0.561602	−	0.827408i	$-0.310185\pi$
$773$	31.2283	1.12320	0.561602	+	0.827408i	$0.310185\pi$
$774$	0	0
$775$	−6.70915	−0.241000
$776$	0	0
$777$	2.48164	0.0890285
$778$	0	0
$779$	28.6717	1.02727
$780$	0	0
$781$	0.173034	0.00619165
$782$	0	0
$783$	4.01331	0.143424
$784$	0	0
$785$	−4.44619	−0.158691
$786$	0	0
$787$	34.9543	1.24599	0.622994	−	0.782227i	$-0.285916\pi$
$787$	34.9543	1.24599	0.622994	+	0.782227i	$0.285916\pi$
$788$	0	0
$789$	11.8054	0.420284
$790$	0	0
$791$	21.1144	0.750742
$792$	0	0
$793$	−34.7937	−1.23556
$794$	0	0
$795$	1.15437	0.0409413
$796$	0	0
$797$	−16.1127	−0.570739	−0.285370	−	0.958418i	$-0.592116\pi$
$797$	−16.1127	−0.570739	−0.285370	+	0.958418i	$0.592116\pi$
$798$	0	0
$799$	−3.65601	−0.129340
$800$	0	0
$801$	16.3725	0.578493
$802$	0	0
$803$	0.629419	0.0222117
$804$	0	0
$805$	−16.0171	−0.564529
$806$	0	0
$807$	18.0241	0.634478
$808$	0	0
$809$	−40.2891	−1.41649	−0.708245	−	0.705967i	$-0.750513\pi$
$809$	−40.2891	−1.41649	−0.708245	+	0.705967i	$0.750513\pi$
$810$	0	0
$811$	52.6072	1.84729	0.923645	−	0.383249i	$-0.125195\pi$
$811$	52.6072	1.84729	0.923645	+	0.383249i	$0.125195\pi$
$812$	0	0
$813$	24.8036	0.869901
$814$	0	0
$815$	−7.18537	−0.251693
$816$	0	0
$817$	−15.5058	−0.542480
$818$	0	0
$819$	7.49628	0.261941
$820$	0	0
$821$	4.85404	0.169407	0.0847035	−	0.996406i	$-0.473006\pi$
$821$	4.85404	0.169407	0.0847035	+	0.996406i	$0.473006\pi$
$822$	0	0
$823$	24.6992	0.860959	0.430479	−	0.902600i	$-0.358345\pi$
$823$	24.6992	0.860959	0.430479	+	0.902600i	$0.358345\pi$
$824$	0	0
$825$	−0.265154	−0.00923147
$826$	0	0
$827$	−41.8198	−1.45422	−0.727109	−	0.686522i	$-0.759136\pi$
$827$	−41.8198	−1.45422	−0.727109	+	0.686522i	$0.759136\pi$
$828$	0	0
$829$	49.4070	1.71598	0.857988	−	0.513671i	$-0.171715\pi$
$829$	49.4070	1.71598	0.857988	+	0.513671i	$0.171715\pi$
$830$	0	0
$831$	−13.6414	−0.473215
$832$	0	0
$833$	1.38703	0.0480577
$834$	0	0
$835$	20.4382	0.707294
$836$	0	0
$837$	−6.70915	−0.231902
$838$	0	0
$839$	25.8109	0.891091	0.445546	−	0.895259i	$-0.353010\pi$
$839$	25.8109	0.891091	0.445546	+	0.895259i	$0.353010\pi$
$840$	0	0
$841$	−12.8933	−0.444598
$842$	0	0
$843$	−10.3878	−0.357776
$844$	0	0
$845$	1.73861	0.0598101
$846$	0	0
$847$	21.3415	0.733303
$848$	0	0
$849$	21.1531	0.725973
$850$	0	0
$851$	−10.4253	−0.357375
$852$	0	0
$853$	−3.86607	−0.132372	−0.0661859	−	0.997807i	$-0.521083\pi$
$853$	−3.86607	−0.132372	−0.0661859	+	0.997807i	$0.521083\pi$
$854$	0	0
$855$	4.50625	0.154111
$856$	0	0
$857$	−29.5851	−1.01061	−0.505304	−	0.862941i	$-0.668620\pi$
$857$	−29.5851	−1.01061	−0.505304	+	0.862941i	$0.668620\pi$
$858$	0	0
$859$	−31.1890	−1.06415	−0.532077	−	0.846696i	$-0.678588\pi$
$859$	−31.1890	−1.06415	−0.532077	+	0.846696i	$0.678588\pi$
$860$	0	0
$861$	−12.4238	−0.423402
$862$	0	0
$863$	14.2464	0.484953	0.242477	−	0.970157i	$-0.422040\pi$
$863$	14.2464	0.484953	0.242477	+	0.970157i	$0.422040\pi$
$864$	0	0
$865$	1.45657	0.0495247
$866$	0	0
$867$	−16.8106	−0.570919
$868$	0	0
$869$	2.15627	0.0731464
$870$	0	0
$871$	3.83909	0.130083
$872$	0	0
$873$	−6.86586	−0.232374
$874$	0	0
$875$	−1.95262	−0.0660105
$876$	0	0
$877$	−16.0045	−0.540433	−0.270216	−	0.962800i	$-0.587095\pi$
$877$	−16.0045	−0.540433	−0.270216	+	0.962800i	$0.587095\pi$
$878$	0	0
$879$	−8.21204	−0.276985
$880$	0	0
$881$	37.7708	1.27253	0.636265	−	0.771471i	$-0.280479\pi$
$881$	37.7708	1.27253	0.636265	+	0.771471i	$0.280479\pi$
$882$	0	0
$883$	−42.9590	−1.44568	−0.722842	−	0.691013i	$-0.757165\pi$
$883$	−42.9590	−1.44568	−0.722842	+	0.691013i	$0.757165\pi$
$884$	0	0
$885$	−9.49481	−0.319165
$886$	0	0
$887$	−23.0401	−0.773612	−0.386806	−	0.922161i	$-0.626422\pi$
$887$	−23.0401	−0.773612	−0.386806	+	0.922161i	$0.626422\pi$
$888$	0	0
$889$	−34.6837	−1.16325
$890$	0	0
$891$	−0.265154	−0.00888298
$892$	0	0
$893$	37.8580	1.26687
$894$	0	0
$895$	11.2181	0.374981
$896$	0	0
$897$	−31.4916	−1.05148
$898$	0	0
$899$	−26.9259	−0.898029
$900$	0	0
$901$	−0.502355	−0.0167359
$902$	0	0
$903$	6.71887	0.223590
$904$	0	0
$905$	9.52874	0.316746
$906$	0	0
$907$	−1.13418	−0.0376600	−0.0188300	−	0.999823i	$-0.505994\pi$
$907$	−1.13418	−0.0376600	−0.0188300	+	0.999823i	$0.505994\pi$
$908$	0	0
$909$	−8.03360	−0.266458
$910$	0	0
$911$	−25.3360	−0.839418	−0.419709	−	0.907659i	$-0.637868\pi$
$911$	−25.3360	−0.839418	−0.419709	+	0.907659i	$0.637868\pi$
$912$	0	0
$913$	0.230699	0.00763501
$914$	0	0
$915$	9.06300	0.299614
$916$	0	0
$917$	−3.00006	−0.0990707
$918$	0	0
$919$	20.6001	0.679534	0.339767	−	0.940510i	$-0.389652\pi$
$919$	20.6001	0.679534	0.339767	+	0.940510i	$0.389652\pi$
$920$	0	0
$921$	−12.3029	−0.405394
$922$	0	0
$923$	2.50532	0.0824635
$924$	0	0
$925$	−1.27093	−0.0417880
$926$	0	0
$927$	6.74596	0.221566
$928$	0	0
$929$	8.58650	0.281714	0.140857	−	0.990030i	$-0.455014\pi$
$929$	8.58650	0.281714	0.140857	+	0.990030i	$0.455014\pi$
$930$	0	0
$931$	−14.3627	−0.470718
$932$	0	0
$933$	32.7081	1.07082
$934$	0	0
$935$	0.115389	0.00377361
$936$	0	0
$937$	38.6534	1.26275	0.631375	−	0.775477i	$-0.282491\pi$
$937$	38.6534	1.26275	0.631375	+	0.775477i	$0.282491\pi$
$938$	0	0
$939$	−9.45635	−0.308596
$940$	0	0
$941$	−41.0594	−1.33850	−0.669249	−	0.743038i	$-0.733384\pi$
$941$	−41.0594	−1.33850	−0.669249	+	0.743038i	$0.733384\pi$
$942$	0	0
$943$	52.1920	1.69961
$944$	0	0
$945$	−1.95262	−0.0635187
$946$	0	0
$947$	41.3801	1.34467	0.672336	−	0.740246i	$-0.265291\pi$
$947$	41.3801	1.34467	0.672336	+	0.740246i	$0.265291\pi$
$948$	0	0
$949$	9.11319	0.295827
$950$	0	0
$951$	22.9380	0.743816
$952$	0	0
$953$	13.8455	0.448501	0.224251	−	0.974532i	$-0.428007\pi$
$953$	13.8455	0.448501	0.224251	+	0.974532i	$0.428007\pi$
$954$	0	0
$955$	26.0126	0.841748
$956$	0	0
$957$	−1.06414	−0.0343989
$958$	0	0
$959$	−33.5012	−1.08181
$960$	0	0
$961$	14.0126	0.452021
$962$	0	0
$963$	2.98468	0.0961798
$964$	0	0
$965$	23.6361	0.760872
$966$	0	0
$967$	23.2819	0.748697	0.374348	−	0.927288i	$-0.377866\pi$
$967$	23.2819	0.748697	0.374348	+	0.927288i	$0.377866\pi$
$968$	0	0
$969$	−1.96102	−0.0629969
$970$	0	0
$971$	37.2974	1.19693	0.598465	−	0.801149i	$-0.295778\pi$
$971$	37.2974	1.19693	0.598465	+	0.801149i	$0.295778\pi$
$972$	0	0
$973$	−28.0632	−0.899664
$974$	0	0
$975$	−3.83909	−0.122949
$976$	0	0
$977$	12.0411	0.385228	0.192614	−	0.981275i	$-0.438303\pi$
$977$	12.0411	0.385228	0.192614	+	0.981275i	$0.438303\pi$
$978$	0	0
$979$	−4.34122	−0.138746
$980$	0	0
$981$	11.6639	0.372401
$982$	0	0
$983$	31.8678	1.01642	0.508212	−	0.861232i	$-0.330307\pi$
$983$	31.8678	1.01642	0.508212	+	0.861232i	$0.330307\pi$
$984$	0	0
$985$	−6.23755	−0.198745
$986$	0	0
$987$	−16.4044	−0.522156
$988$	0	0
$989$	−28.2258	−0.897528
$990$	0	0
$991$	−3.85333	−0.122405	−0.0612026	−	0.998125i	$-0.519494\pi$
$991$	−3.85333	−0.122405	−0.0612026	+	0.998125i	$0.519494\pi$
$992$	0	0
$993$	31.5109	0.999968
$994$	0	0
$995$	15.4983	0.491329
$996$	0	0
$997$	5.42135	0.171696	0.0858480	−	0.996308i	$-0.472640\pi$
$997$	5.42135	0.171696	0.0858480	+	0.996308i	$0.472640\pi$
$998$	0	0
$999$	−1.27093	−0.0402105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.bb.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.bb.1.3	✓	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.95262 q^{7} +1.00000 q^{9} -0.265154 q^{11} -3.83909 q^{13} +1.00000 q^{15} -0.435176 q^{17} +4.50625 q^{19} -1.95262 q^{21} +8.20289 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.01331 q^{29} -6.70915 q^{31} -0.265154 q^{33} -1.95262 q^{35} -1.27093 q^{37} -3.83909 q^{39} +6.36264 q^{41} -3.44095 q^{43} +1.00000 q^{45} +8.40121 q^{47} -3.18728 q^{49} -0.435176 q^{51} +1.15437 q^{53} -0.265154 q^{55} +4.50625 q^{57} -9.49481 q^{59} +9.06300 q^{61} -1.95262 q^{63} -3.83909 q^{65} -1.00000 q^{67} +8.20289 q^{69} -0.652581 q^{71} -2.37379 q^{73} +1.00000 q^{75} +0.517744 q^{77} -8.13215 q^{79} +1.00000 q^{81} -0.870056 q^{83} -0.435176 q^{85} +4.01331 q^{87} +16.3725 q^{89} +7.49628 q^{91} -6.70915 q^{93} +4.50625 q^{95} -6.86586 q^{97} -0.265154 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + 9 q^{5} + 5 q^{7} + 9 q^{9} - 7 q^{11} - 3 q^{13} + 9 q^{15} + 7 q^{17} + 8 q^{19} + 5 q^{21} + 19 q^{23} + 9 q^{25} + 9 q^{27} + 14 q^{29} + 27 q^{31} - 7 q^{33} + 5 q^{35} + 15 q^{37}+ \cdots - 7 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 + 9 * q^5 + 5 * q^7 + 9 * q^9 - 7 * q^11 - 3 * q^13 + 9 * q^15 + 7 * q^17 + 8 * q^19 + 5 * q^21 + 19 * q^23 + 9 * q^25 + 9 * q^27 + 14 * q^29 + 27 * q^31 - 7 * q^33 + 5 * q^35 + 15 * q^37 - 3 * q^39 - 5 * q^41 + 11 * q^43 + 9 * q^45 + 6 * q^47 + 30 * q^49 + 7 * q^51 - 11 * q^53 - 7 * q^55 + 8 * q^57 + 14 * q^59 - 7 * q^61 + 5 * q^63 - 3 * q^65 - 9 * q^67 + 19 * q^69 - 2 * q^71 - 23 * q^73 + 9 * q^75 - 9 * q^77 + 13 * q^79 + 9 * q^81 + 48 * q^83 + 7 * q^85 + 14 * q^87 + 5 * q^89 + 41 * q^91 + 27 * q^93 + 8 * q^95 + 19 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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