LMFDB - Embedded newform 261.4.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,4,Mod(1,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("261.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	261.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:	$15.3994985115$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13458092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 $ x^5 - x^4 - 14x^3 + 18x^2 + 20*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.27399$ of defining polynomial
Character	$\chi$	$=$	261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.63099 q^{2} -5.33986 q^{4} +16.8209 q^{5} +5.21997 q^{7} +21.7572 q^{8} +O(q^{10})$	\(q-1.63099 q^{2} -5.33986 q^{4} +16.8209 q^{5} +5.21997 q^{7} +21.7572 q^{8} -27.4348 q^{10} +8.55158 q^{11} -11.3429 q^{13} -8.51374 q^{14} +7.23299 q^{16} -68.4740 q^{17} +6.93014 q^{19} -89.8214 q^{20} -13.9476 q^{22} +132.042 q^{23} +157.944 q^{25} +18.5001 q^{26} -27.8739 q^{28} +29.0000 q^{29} -0.419319 q^{31} -185.855 q^{32} +111.681 q^{34} +87.8048 q^{35} +395.483 q^{37} -11.3030 q^{38} +365.977 q^{40} +447.209 q^{41} +184.132 q^{43} -45.6642 q^{44} -215.360 q^{46} +97.2612 q^{47} -315.752 q^{49} -257.605 q^{50} +60.5693 q^{52} +209.547 q^{53} +143.845 q^{55} +113.572 q^{56} -47.2988 q^{58} -45.9651 q^{59} +427.655 q^{61} +0.683906 q^{62} +245.264 q^{64} -190.798 q^{65} -405.055 q^{67} +365.641 q^{68} -143.209 q^{70} +557.971 q^{71} -381.988 q^{73} -645.031 q^{74} -37.0060 q^{76} +44.6390 q^{77} +577.208 q^{79} +121.666 q^{80} -729.396 q^{82} +353.745 q^{83} -1151.80 q^{85} -300.318 q^{86} +186.059 q^{88} +277.871 q^{89} -59.2095 q^{91} -705.088 q^{92} -158.632 q^{94} +116.571 q^{95} +677.917 q^{97} +514.989 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8}+O(q^{10}) $ 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8	$ 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8} - 64 q^{10} - 12 q^{11} + 14 q^{13} + 192 q^{14} + 146 q^{16} - 66 q^{17} + 214 q^{19} - 6 q^{20} - 98 q^{22} - 164 q^{23} + 207 q^{25} - 56 q^{26} + 540 q^{28} + 145 q^{29} + 420 q^{31} + 652 q^{32} + 204 q^{34} + 52 q^{35} + 378 q^{37} + 496 q^{38} - 80 q^{40} + 1158 q^{41} - 204 q^{43} - 784 q^{44} + 580 q^{46} - 248 q^{47} - 283 q^{49} - 908 q^{50} + 1482 q^{52} + 554 q^{53} + 546 q^{55} + 608 q^{56} - 440 q^{59} + 618 q^{61} - 1250 q^{62} + 2594 q^{64} + 1656 q^{65} + 1164 q^{67} - 356 q^{68} - 2184 q^{70} + 692 q^{71} - 1950 q^{73} + 1832 q^{74} + 1376 q^{76} + 1616 q^{77} + 272 q^{79} + 890 q^{80} + 92 q^{82} - 512 q^{83} - 1628 q^{85} - 2446 q^{86} - 6954 q^{88} - 866 q^{89} + 2580 q^{91} - 3468 q^{92} - 5942 q^{94} - 2244 q^{95} + 1562 q^{97} + 3408 q^{98}+O(q^{100}) $ 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8 - 64 * q^10 - 12 * q^11 + 14 * q^13 + 192 * q^14 + 146 * q^16 - 66 * q^17 + 214 * q^19 - 6 * q^20 - 98 * q^22 - 164 * q^23 + 207 * q^25 - 56 * q^26 + 540 * q^28 + 145 * q^29 + 420 * q^31 + 652 * q^32 + 204 * q^34 + 52 * q^35 + 378 * q^37 + 496 * q^38 - 80 * q^40 + 1158 * q^41 - 204 * q^43 - 784 * q^44 + 580 * q^46 - 248 * q^47 - 283 * q^49 - 908 * q^50 + 1482 * q^52 + 554 * q^53 + 546 * q^55 + 608 * q^56 - 440 * q^59 + 618 * q^61 - 1250 * q^62 + 2594 * q^64 + 1656 * q^65 + 1164 * q^67 - 356 * q^68 - 2184 * q^70 + 692 * q^71 - 1950 * q^73 + 1832 * q^74 + 1376 * q^76 + 1616 * q^77 + 272 * q^79 + 890 * q^80 + 92 * q^82 - 512 * q^83 - 1628 * q^85 - 2446 * q^86 - 6954 * q^88 - 866 * q^89 + 2580 * q^91 - 3468 * q^92 - 5942 * q^94 - 2244 * q^95 + 1562 * q^97 + 3408 * q^98

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 $

$2$	−1.63099	−0.576643	−0.288322	−	0.957534i	$-0.593097\pi$
$2$	−1.63099	−0.576643	−0.288322	+	0.957534i	$0.593097\pi$
$3$	0	0
$3$	0	0
$4$	−5.33986	−0.667483
$5$	16.8209	1.50451	0.752255	−	0.658872i	$-0.228966\pi$
$5$	16.8209	1.50451	0.752255	+	0.658872i	$0.228966\pi$
$6$	0	0
$7$	5.21997	0.281852	0.140926	−	0.990020i	$-0.454992\pi$
$7$	5.21997	0.281852	0.140926	+	0.990020i	$0.454992\pi$
$8$	21.7572	0.961543
$9$	0	0
$10$	−27.4348	−0.867565
$11$	8.55158	0.234400	0.117200	−	0.993108i	$-0.462608\pi$
$11$	8.55158	0.234400	0.117200	+	0.993108i	$0.462608\pi$
$12$	0	0
$13$	−11.3429	−0.241996	−0.120998	−	0.992653i	$-0.538609\pi$
$13$	−11.3429	−0.241996	−0.120998	+	0.992653i	$0.538609\pi$
$14$	−8.51374	−0.162528
$15$	0	0
$16$	7.23299	0.113016
$17$	−68.4740	−0.976904	−0.488452	−	0.872591i	$-0.662438\pi$
$17$	−68.4740	−0.976904	−0.488452	+	0.872591i	$0.662438\pi$
$18$	0	0
$19$	6.93014	0.0836780	0.0418390	−	0.999124i	$-0.486678\pi$
$19$	6.93014	0.0836780	0.0418390	+	0.999124i	$0.486678\pi$
$20$	−89.8214	−1.00423
$21$	0	0
$22$	−13.9476	−0.135165
$23$	132.042	1.19708	0.598538	−	0.801095i	$-0.295749\pi$
$23$	132.042	1.19708	0.598538	+	0.801095i	$0.295749\pi$
$24$	0	0
$25$	157.944	1.26355
$26$	18.5001	0.139545
$27$	0	0
$28$	−27.8739	−0.188131
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−0.419319	−0.00242941	−0.00121471	−	0.999999i	$-0.500387\pi$
$31$	−0.419319	−0.00242941	−0.00121471	+	0.999999i	$0.500387\pi$
$32$	−185.855	−1.02671
$33$	0	0
$34$	111.681	0.563325
$35$	87.8048	0.424049
$36$	0	0
$37$	395.483	1.75722	0.878609	−	0.477542i	$-0.158472\pi$
$37$	395.483	1.75722	0.878609	+	0.477542i	$0.158472\pi$
$38$	−11.3030	−0.0482524
$39$	0	0
$40$	365.977	1.44665
$41$	447.209	1.70347	0.851736	−	0.523971i	$-0.175550\pi$
$41$	447.209	1.70347	0.851736	+	0.523971i	$0.175550\pi$
$42$	0	0
$43$	184.132	0.653020	0.326510	−	0.945194i	$-0.394127\pi$
$43$	184.132	0.653020	0.326510	+	0.945194i	$0.394127\pi$
$44$	−45.6642	−0.156458
$45$	0	0
$46$	−215.360	−0.690285
$47$	97.2612	0.301851	0.150926	−	0.988545i	$-0.451775\pi$
$47$	97.2612	0.301851	0.150926	+	0.988545i	$0.451775\pi$
$48$	0	0
$49$	−315.752	−0.920559
$50$	−257.605	−0.728617
$51$	0	0
$52$	60.5693	0.161528
$53$	209.547	0.543086	0.271543	−	0.962426i	$-0.412466\pi$
$53$	209.547	0.543086	0.271543	+	0.962426i	$0.412466\pi$
$54$	0	0
$55$	143.845	0.352657
$56$	113.572	0.271013
$57$	0	0
$58$	−47.2988	−0.107080
$59$	−45.9651	−0.101426	−0.0507131	−	0.998713i	$-0.516149\pi$
$59$	−45.9651	−0.101426	−0.0507131	+	0.998713i	$0.516149\pi$
$60$	0	0
$61$	427.655	0.897634	0.448817	−	0.893624i	$-0.351846\pi$
$61$	427.655	0.897634	0.448817	+	0.893624i	$0.351846\pi$
$62$	0.683906	0.00140091
$63$	0	0
$64$	245.264	0.479031
$65$	−190.798	−0.364085
$66$	0	0
$67$	−405.055	−0.738588	−0.369294	−	0.929313i	$-0.620400\pi$
$67$	−405.055	−0.738588	−0.369294	+	0.929313i	$0.620400\pi$
$68$	365.641	0.652067
$69$	0	0
$70$	−143.209	−0.244525
$71$	557.971	0.932662	0.466331	−	0.884610i	$-0.345576\pi$
$71$	557.971	0.932662	0.466331	+	0.884610i	$0.345576\pi$
$72$	0	0
$73$	−381.988	−0.612443	−0.306222	−	0.951960i	$-0.599065\pi$
$73$	−381.988	−0.612443	−0.306222	+	0.951960i	$0.599065\pi$
$74$	−645.031	−1.01329
$75$	0	0
$76$	−37.0060	−0.0558536
$77$	44.6390	0.0660660
$78$	0	0
$79$	577.208	0.822038	0.411019	−	0.911627i	$-0.365173\pi$
$79$	577.208	0.822038	0.411019	+	0.911627i	$0.365173\pi$
$80$	121.666	0.170033
$81$	0	0
$82$	−729.396	−0.982296
$83$	353.745	0.467813	0.233907	−	0.972259i	$-0.424849\pi$
$83$	353.745	0.467813	0.233907	+	0.972259i	$0.424849\pi$
$84$	0	0
$85$	−1151.80	−1.46976
$86$	−300.318	−0.376560
$87$	0	0
$88$	186.059	0.225385
$89$	277.871	0.330947	0.165473	−	0.986214i	$-0.447085\pi$
$89$	277.871	0.330947	0.165473	+	0.986214i	$0.447085\pi$
$90$	0	0
$91$	−59.2095	−0.0682070
$92$	−705.088	−0.799027
$93$	0	0
$94$	−158.632	−0.174060
$95$	116.571	0.125894
$96$	0	0
$97$	677.917	0.709609	0.354804	−	0.934941i	$-0.384547\pi$
$97$	677.917	0.709609	0.354804	+	0.934941i	$0.384547\pi$
$98$	514.989	0.530834
$99$	0	0
$100$	−843.397	−0.843397
$101$	−567.816	−0.559404	−0.279702	−	0.960087i	$-0.590236\pi$
$101$	−567.816	−0.559404	−0.279702	+	0.960087i	$0.590236\pi$
$102$	0	0
$103$	319.205	0.305362	0.152681	−	0.988276i	$-0.451209\pi$
$103$	319.205	0.305362	0.152681	+	0.988276i	$0.451209\pi$
$104$	−246.789	−0.232689
$105$	0	0
$106$	−341.771	−0.313167
$107$	79.6547	0.0719674	0.0359837	−	0.999352i	$-0.488544\pi$
$107$	79.6547	0.0719674	0.0359837	+	0.999352i	$0.488544\pi$
$108$	0	0
$109$	−1708.43	−1.50126	−0.750632	−	0.660721i	$-0.770251\pi$
$109$	−1708.43	−1.50126	−0.750632	+	0.660721i	$0.770251\pi$
$110$	−234.611	−0.203357
$111$	0	0
$112$	37.7560	0.0318536
$113$	−1050.31	−0.874383	−0.437191	−	0.899369i	$-0.644027\pi$
$113$	−1050.31	−0.874383	−0.437191	+	0.899369i	$0.644027\pi$
$114$	0	0
$115$	2221.08	1.80101
$116$	−154.856	−0.123948
$117$	0	0
$118$	74.9687	0.0584867
$119$	−357.432	−0.275342
$120$	0	0
$121$	−1257.87	−0.945057
$122$	−697.503	−0.517615
$123$	0	0
$124$	2.23910	0.00162159
$125$	554.143	0.396513
$126$	0	0
$127$	366.926	0.256373	0.128187	−	0.991750i	$-0.459084\pi$
$127$	366.926	0.256373	0.128187	+	0.991750i	$0.459084\pi$
$128$	1086.81	0.750482
$129$	0	0
$130$	311.190	0.209947
$131$	−2310.86	−1.54123	−0.770614	−	0.637302i	$-0.780050\pi$
$131$	−2310.86	−1.54123	−0.770614	+	0.637302i	$0.780050\pi$
$132$	0	0
$133$	36.1751	0.0235848
$134$	660.643	0.425902
$135$	0	0
$136$	−1489.80	−0.939335
$137$	−1899.65	−1.18466	−0.592329	−	0.805696i	$-0.701791\pi$
$137$	−1899.65	−1.18466	−0.592329	+	0.805696i	$0.701791\pi$
$138$	0	0
$139$	1309.49	0.799061	0.399531	−	0.916720i	$-0.369173\pi$
$139$	1309.49	0.799061	0.399531	+	0.916720i	$0.369173\pi$
$140$	−468.865	−0.283045
$141$	0	0
$142$	−910.047	−0.537813
$143$	−96.9994	−0.0567238
$144$	0	0
$145$	487.807	0.279380
$146$	623.020	0.353161
$147$	0	0
$148$	−2111.83	−1.17291
$149$	−1782.98	−0.980320	−0.490160	−	0.871632i	$-0.663062\pi$
$149$	−1782.98	−0.980320	−0.490160	+	0.871632i	$0.663062\pi$
$150$	0	0
$151$	−1631.96	−0.879515	−0.439758	−	0.898116i	$-0.644936\pi$
$151$	−1631.96	−0.879515	−0.439758	+	0.898116i	$0.644936\pi$
$152$	150.781	0.0804600
$153$	0	0
$154$	−72.8059	−0.0380965
$155$	−7.05333	−0.00365508
$156$	0	0
$157$	852.817	0.433517	0.216759	−	0.976225i	$-0.430452\pi$
$157$	852.817	0.433517	0.216759	+	0.976225i	$0.430452\pi$
$158$	−941.422	−0.474022
$159$	0	0
$160$	−3126.25	−1.54470
$161$	689.257	0.337398
$162$	0	0
$163$	3280.24	1.57625	0.788123	−	0.615518i	$-0.211053\pi$
$163$	3280.24	1.57625	0.788123	+	0.615518i	$0.211053\pi$
$164$	−2388.04	−1.13704
$165$	0	0
$166$	−576.955	−0.269761
$167$	1682.26	0.779504	0.389752	−	0.920920i	$-0.372561\pi$
$167$	1682.26	0.779504	0.389752	+	0.920920i	$0.372561\pi$
$168$	0	0
$169$	−2068.34	−0.941438
$170$	1878.57	0.847528
$171$	0	0
$172$	−983.239	−0.435879
$173$	1590.22	0.698854	0.349427	−	0.936964i	$-0.386376\pi$
$173$	1590.22	0.698854	0.349427	+	0.936964i	$0.386376\pi$
$174$	0	0
$175$	824.462	0.356134
$176$	61.8535	0.0264908
$177$	0	0
$178$	−453.206	−0.190838
$179$	1794.60	0.749356	0.374678	−	0.927155i	$-0.377753\pi$
$179$	1794.60	0.749356	0.374678	+	0.927155i	$0.377753\pi$
$180$	0	0
$181$	2353.41	0.966450	0.483225	−	0.875496i	$-0.339465\pi$
$181$	2353.41	0.966450	0.483225	+	0.875496i	$0.339465\pi$
$182$	96.5702	0.0393311
$183$	0	0
$184$	2872.88	1.15104
$185$	6652.40	2.64375
$186$	0	0
$187$	−585.560	−0.228986
$188$	−519.361	−0.201480
$189$	0	0
$190$	−190.127	−0.0725962
$191$	2184.35	0.827508	0.413754	−	0.910389i	$-0.364217\pi$
$191$	2184.35	0.827508	0.413754	+	0.910389i	$0.364217\pi$
$192$	0	0
$193$	−3109.71	−1.15980	−0.579901	−	0.814687i	$-0.696909\pi$
$193$	−3109.71	−1.15980	−0.579901	+	0.814687i	$0.696909\pi$
$194$	−1105.68	−0.409191
$195$	0	0
$196$	1686.07	0.614457
$197$	−923.756	−0.334086	−0.167043	−	0.985950i	$-0.553422\pi$
$197$	−923.756	−0.334086	−0.167043	+	0.985950i	$0.553422\pi$
$198$	0	0
$199$	4548.71	1.62035	0.810174	−	0.586189i	$-0.199372\pi$
$199$	4548.71	1.62035	0.810174	+	0.586189i	$0.199372\pi$
$200$	3436.42	1.21496
$201$	0	0
$202$	926.103	0.322576
$203$	151.379	0.0523386
$204$	0	0
$205$	7522.48	2.56289
$206$	−520.622	−0.176085
$207$	0	0
$208$	−82.0429	−0.0273493
$209$	59.2636	0.0196141
$210$	0	0
$211$	−318.664	−0.103970	−0.0519851	−	0.998648i	$-0.516555\pi$
$211$	−318.664	−0.103970	−0.0519851	+	0.998648i	$0.516555\pi$
$212$	−1118.95	−0.362500
$213$	0	0
$214$	−129.916	−0.0414995
$215$	3097.27	0.982475
$216$	0	0
$217$	−2.18883	−0.000684735 0
$218$	2786.44	0.865694
$219$	0	0
$220$	−768.115	−0.235392
$221$	776.691	0.236407
$222$	0	0
$223$	1706.70	0.512509	0.256254	−	0.966609i	$-0.417512\pi$
$223$	1706.70	0.512509	0.256254	+	0.966609i	$0.417512\pi$
$224$	−970.157	−0.289381
$225$	0	0
$226$	1713.06	0.504207
$227$	−3043.63	−0.889925	−0.444962	−	0.895549i	$-0.646783\pi$
$227$	−3043.63	−0.889925	−0.444962	+	0.895549i	$0.646783\pi$
$228$	0	0
$229$	−2621.57	−0.756500	−0.378250	−	0.925704i	$-0.623474\pi$
$229$	−2621.57	−0.756500	−0.378250	+	0.925704i	$0.623474\pi$
$230$	−3622.56	−1.03854
$231$	0	0
$232$	630.960	0.178554
$233$	2778.73	0.781291	0.390646	−	0.920541i	$-0.372252\pi$
$233$	2778.73	0.781291	0.390646	+	0.920541i	$0.372252\pi$
$234$	0	0
$235$	1636.02	0.454138
$236$	245.447	0.0677002
$237$	0	0
$238$	582.969	0.158774
$239$	−4722.67	−1.27818	−0.639088	−	0.769133i	$-0.720688\pi$
$239$	−4722.67	−1.27818	−0.639088	+	0.769133i	$0.720688\pi$
$240$	0	0
$241$	−4704.13	−1.25734	−0.628672	−	0.777671i	$-0.716401\pi$
$241$	−4704.13	−1.25734	−0.628672	+	0.777671i	$0.716401\pi$
$242$	2051.58	0.544961
$243$	0	0
$244$	−2283.62	−0.599155
$245$	−5311.24	−1.38499
$246$	0	0
$247$	−78.6076	−0.0202497
$248$	−9.12321	−0.00233599
$249$	0	0
$250$	−903.804	−0.228646
$251$	−4449.07	−1.11882	−0.559408	−	0.828893i	$-0.688971\pi$
$251$	−4449.07	−1.11882	−0.559408	+	0.828893i	$0.688971\pi$
$252$	0	0
$253$	1129.17	0.280594
$254$	−598.454	−0.147836
$255$	0	0
$256$	−3734.70	−0.911792
$257$	−7226.68	−1.75404	−0.877019	−	0.480456i	$-0.840471\pi$
$257$	−7226.68	−1.75404	−0.877019	+	0.480456i	$0.840471\pi$
$258$	0	0
$259$	2064.41	0.495275
$260$	1018.83	0.243020
$261$	0	0
$262$	3769.00	0.888739
$263$	4789.59	1.12296	0.561481	−	0.827490i	$-0.310232\pi$
$263$	4789.59	1.12296	0.561481	+	0.827490i	$0.310232\pi$
$264$	0	0
$265$	3524.78	0.817078
$266$	−59.0014	−0.0136000
$267$	0	0
$268$	2162.94	0.492995
$269$	47.0448	0.0106631	0.00533154	−	0.999986i	$-0.498303\pi$
$269$	47.0448	0.0106631	0.00533154	+	0.999986i	$0.498303\pi$
$270$	0	0
$271$	7721.78	1.73087	0.865433	−	0.501025i	$-0.167044\pi$
$271$	7721.78	1.73087	0.865433	+	0.501025i	$0.167044\pi$
$272$	−495.272	−0.110405
$273$	0	0
$274$	3098.32	0.683125
$275$	1350.67	0.296176
$276$	0	0
$277$	−7554.66	−1.63868	−0.819342	−	0.573305i	$-0.805661\pi$
$277$	−7554.66	−1.63868	−0.819342	+	0.573305i	$0.805661\pi$
$278$	−2135.77	−0.460773
$279$	0	0
$280$	1910.39	0.407741
$281$	2094.83	0.444723	0.222361	−	0.974964i	$-0.428623\pi$
$281$	2094.83	0.444723	0.222361	+	0.974964i	$0.428623\pi$
$282$	0	0
$283$	−8200.83	−1.72257	−0.861287	−	0.508119i	$-0.830341\pi$
$283$	−8200.83	−1.72257	−0.861287	+	0.508119i	$0.830341\pi$
$284$	−2979.49	−0.622536
$285$	0	0
$286$	158.205	0.0327094
$287$	2334.42	0.480127
$288$	0	0
$289$	−224.318	−0.0456580
$290$	−795.610	−0.161103
$291$	0	0
$292$	2039.76	0.408795
$293$	−3518.87	−0.701620	−0.350810	−	0.936447i	$-0.614094\pi$
$293$	−3518.87	−0.701620	−0.350810	+	0.936447i	$0.614094\pi$
$294$	0	0
$295$	−773.175	−0.152597
$296$	8604.62	1.68964
$297$	0	0
$298$	2908.03	0.565295
$299$	−1497.74	−0.289687
$300$	0	0
$301$	961.163	0.184055
$302$	2661.71	0.507167
$303$	0	0
$304$	50.1256	0.00945691
$305$	7193.56	1.35050
$306$	0	0
$307$	−8725.36	−1.62209	−0.811046	−	0.584982i	$-0.801102\pi$
$307$	−8725.36	−1.62209	−0.811046	+	0.584982i	$0.801102\pi$
$308$	−238.366	−0.0440979
$309$	0	0
$310$	11.5039	0.00210768
$311$	−2640.09	−0.481370	−0.240685	−	0.970603i	$-0.577372\pi$
$311$	−2640.09	−0.481370	−0.240685	+	0.970603i	$0.577372\pi$
$312$	0	0
$313$	−1938.00	−0.349976	−0.174988	−	0.984571i	$-0.555989\pi$
$313$	−1938.00	−0.349976	−0.174988	+	0.984571i	$0.555989\pi$
$314$	−1390.94	−0.249985
$315$	0	0
$316$	−3082.21	−0.548696
$317$	5546.11	0.982651	0.491326	−	0.870976i	$-0.336512\pi$
$317$	5546.11	0.982651	0.491326	+	0.870976i	$0.336512\pi$
$318$	0	0
$319$	247.996	0.0435270
$320$	4125.57	0.720707
$321$	0	0
$322$	−1124.17	−0.194558
$323$	−474.534	−0.0817454
$324$	0	0
$325$	−1791.53	−0.305774
$326$	−5350.04	−0.908931
$327$	0	0
$328$	9730.04	1.63796
$329$	507.701	0.0850773
$330$	0	0
$331$	185.492	0.0308023	0.0154012	−	0.999881i	$-0.495097\pi$
$331$	185.492	0.0308023	0.0154012	+	0.999881i	$0.495097\pi$
$332$	−1888.95	−0.312257
$333$	0	0
$334$	−2743.76	−0.449496
$335$	−6813.41	−1.11121
$336$	0	0
$337$	−8508.32	−1.37530	−0.687652	−	0.726040i	$-0.741359\pi$
$337$	−8508.32	−1.37530	−0.687652	+	0.726040i	$0.741359\pi$
$338$	3373.45	0.542874
$339$	0	0
$340$	6150.43	0.981041
$341$	−3.58584	−0.000569454 0
$342$	0	0
$343$	−3438.67	−0.541313
$344$	4006.20	0.627906
$345$	0	0
$346$	−2593.63	−0.402990
$347$	−7853.53	−1.21498	−0.607492	−	0.794326i	$-0.707824\pi$
$347$	−7853.53	−1.21498	−0.607492	+	0.794326i	$0.707824\pi$
$348$	0	0
$349$	6328.33	0.970624	0.485312	−	0.874341i	$-0.338706\pi$
$349$	6328.33	0.970624	0.485312	+	0.874341i	$0.338706\pi$
$350$	−1344.69	−0.205362
$351$	0	0
$352$	−1589.35	−0.240661
$353$	7973.45	1.20222	0.601111	−	0.799166i	$-0.294725\pi$
$353$	7973.45	1.20222	0.601111	+	0.799166i	$0.294725\pi$
$354$	0	0
$355$	9385.59	1.40320
$356$	−1483.79	−0.220901
$357$	0	0
$358$	−2926.98	−0.432111
$359$	10059.4	1.47887	0.739435	−	0.673228i	$-0.235093\pi$
$359$	10059.4	1.47887	0.739435	+	0.673228i	$0.235093\pi$
$360$	0	0
$361$	−6810.97	−0.992998
$362$	−3838.39	−0.557297
$363$	0	0
$364$	316.170	0.0455270
$365$	−6425.40	−0.921426
$366$	0	0
$367$	−3032.67	−0.431346	−0.215673	−	0.976466i	$-0.569195\pi$
$367$	−3032.67	−0.431346	−0.215673	+	0.976466i	$0.569195\pi$
$368$	955.061	0.135288
$369$	0	0
$370$	−10850.0	−1.52450
$371$	1093.83	0.153070
$372$	0	0
$373$	−11109.4	−1.54215	−0.771077	−	0.636742i	$-0.780282\pi$
$373$	−11109.4	−1.54215	−0.771077	+	0.636742i	$0.780282\pi$
$374$	955.045	0.132043
$375$	0	0
$376$	2116.13	0.290243
$377$	−328.943	−0.0449375
$378$	0	0
$379$	4510.20	0.611275	0.305638	−	0.952148i	$-0.401130\pi$
$379$	4510.20	0.611275	0.305638	+	0.952148i	$0.401130\pi$
$380$	−622.475	−0.0840323
$381$	0	0
$382$	−3562.66	−0.477177
$383$	−7810.96	−1.04209	−0.521047	−	0.853528i	$-0.674458\pi$
$383$	−7810.96	−1.04209	−0.521047	+	0.853528i	$0.674458\pi$
$384$	0	0
$385$	750.869	0.0993970
$386$	5071.92	0.668792
$387$	0	0
$388$	−3619.98	−0.473652
$389$	−9221.90	−1.20198	−0.600989	−	0.799258i	$-0.705226\pi$
$389$	−9221.90	−1.20198	−0.600989	+	0.799258i	$0.705226\pi$
$390$	0	0
$391$	−9041.46	−1.16943
$392$	−6869.88	−0.885157
$393$	0	0
$394$	1506.64	0.192648
$395$	9709.17	1.23676
$396$	0	0
$397$	−10034.3	−1.26854	−0.634268	−	0.773113i	$-0.718699\pi$
$397$	−10034.3	−1.26854	−0.634268	+	0.773113i	$0.718699\pi$
$398$	−7418.91	−0.934363
$399$	0	0
$400$	1142.41	0.142801
$401$	8193.64	1.02038	0.510188	−	0.860063i	$-0.329576\pi$
$401$	8193.64	1.02038	0.510188	+	0.860063i	$0.329576\pi$
$402$	0	0
$403$	4.75628	0.000587908 0
$404$	3032.06	0.373392
$405$	0	0
$406$	−246.898	−0.0301807
$407$	3382.01	0.411892
$408$	0	0
$409$	3921.69	0.474120	0.237060	−	0.971495i	$-0.423816\pi$
$409$	3921.69	0.474120	0.237060	+	0.971495i	$0.423816\pi$
$410$	−12269.1	−1.47787
$411$	0	0
$412$	−1704.51	−0.203824
$413$	−239.936	−0.0285872
$414$	0	0
$415$	5950.31	0.703830
$416$	2108.13	0.248460
$417$	0	0
$418$	−96.6586	−0.0113103
$419$	−3026.66	−0.352892	−0.176446	−	0.984310i	$-0.556460\pi$
$419$	−3026.66	−0.352892	−0.176446	+	0.984310i	$0.556460\pi$
$420$	0	0
$421$	4823.43	0.558384	0.279192	−	0.960235i	$-0.409933\pi$
$421$	4823.43	0.558384	0.279192	+	0.960235i	$0.409933\pi$
$422$	519.739	0.0599538
$423$	0	0
$424$	4559.17	0.522200
$425$	−10815.0	−1.23437
$426$	0	0
$427$	2232.35	0.253000
$428$	−425.345	−0.0480370
$429$	0	0
$430$	−5051.63	−0.566537
$431$	2029.51	0.226816	0.113408	−	0.993548i	$-0.463823\pi$
$431$	2029.51	0.226816	0.113408	+	0.993548i	$0.463823\pi$
$432$	0	0
$433$	−535.808	−0.0594672	−0.0297336	−	0.999558i	$-0.509466\pi$
$433$	−535.808	−0.0594672	−0.0297336	+	0.999558i	$0.509466\pi$
$434$	3.56997	0.000394848 0
$435$	0	0
$436$	9122.77	1.00207
$437$	915.072	0.100169
$438$	0	0
$439$	8791.74	0.955824	0.477912	−	0.878408i	$-0.341394\pi$
$439$	8791.74	0.955824	0.477912	+	0.878408i	$0.341394\pi$
$440$	3129.68	0.339095
$441$	0	0
$442$	−1266.78	−0.136322
$443$	−13916.9	−1.49258	−0.746291	−	0.665619i	$-0.768167\pi$
$443$	−13916.9	−1.49258	−0.746291	+	0.665619i	$0.768167\pi$
$444$	0	0
$445$	4674.05	0.497912
$446$	−2783.62	−0.295535
$447$	0	0
$448$	1280.27	0.135016
$449$	−2129.54	−0.223829	−0.111915	−	0.993718i	$-0.535698\pi$
$449$	−2129.54	−0.223829	−0.111915	+	0.993718i	$0.535698\pi$
$450$	0	0
$451$	3824.35	0.399294
$452$	5608.53	0.583635
$453$	0	0
$454$	4964.14	0.513169
$455$	−995.958	−0.102618
$456$	0	0
$457$	1932.25	0.197783	0.0988915	−	0.995098i	$-0.468470\pi$
$457$	1932.25	0.197783	0.0988915	+	0.995098i	$0.468470\pi$
$458$	4275.77	0.436231
$459$	0	0
$460$	−11860.2	−1.20214
$461$	16518.3	1.66884	0.834418	−	0.551132i	$-0.185804\pi$
$461$	16518.3	1.66884	0.834418	+	0.551132i	$0.185804\pi$
$462$	0	0
$463$	11535.2	1.15785	0.578926	−	0.815380i	$-0.303472\pi$
$463$	11535.2	1.15785	0.578926	+	0.815380i	$0.303472\pi$
$464$	209.757	0.0209865
$465$	0	0
$466$	−4532.10	−0.450526
$467$	−15667.8	−1.55250	−0.776250	−	0.630425i	$-0.782880\pi$
$467$	−15667.8	−1.55250	−0.776250	+	0.630425i	$0.782880\pi$
$468$	0	0
$469$	−2114.38	−0.208172
$470$	−2668.34	−0.261876
$471$	0	0
$472$	−1000.07	−0.0975255
$473$	1574.62	0.153068
$474$	0	0
$475$	1094.57	0.105731
$476$	1908.64	0.183786
$477$	0	0
$478$	7702.65	0.737052
$479$	−4672.28	−0.445682	−0.222841	−	0.974855i	$-0.571533\pi$
$479$	−4672.28	−0.445682	−0.222841	+	0.974855i	$0.571533\pi$
$480$	0	0
$481$	−4485.92	−0.425239
$482$	7672.41	0.725038
$483$	0	0
$484$	6716.85	0.630809
$485$	11403.2	1.06761
$486$	0	0
$487$	−8465.76	−0.787722	−0.393861	−	0.919170i	$-0.628861\pi$
$487$	−8465.76	−0.787722	−0.393861	+	0.919170i	$0.628861\pi$
$488$	9304.60	0.863113
$489$	0	0
$490$	8662.60	0.798646
$491$	12302.6	1.13077	0.565385	−	0.824827i	$-0.308728\pi$
$491$	12302.6	1.13077	0.565385	+	0.824827i	$0.308728\pi$
$492$	0	0
$493$	−1985.74	−0.181407
$494$	128.209	0.0116769
$495$	0	0
$496$	−3.03293	−0.000274562 0
$497$	2912.59	0.262873
$498$	0	0
$499$	−2781.31	−0.249516	−0.124758	−	0.992187i	$-0.539815\pi$
$499$	−2781.31	−0.249516	−0.124758	+	0.992187i	$0.539815\pi$
$500$	−2959.05	−0.264665
$501$	0	0
$502$	7256.40	0.645158
$503$	13673.9	1.21211	0.606054	−	0.795424i	$-0.292752\pi$
$503$	13673.9	1.21211	0.606054	+	0.795424i	$0.292752\pi$
$504$	0	0
$505$	−9551.18	−0.841628
$506$	−1841.67	−0.161803
$507$	0	0
$508$	−1959.33	−0.171125
$509$	−7431.68	−0.647158	−0.323579	−	0.946201i	$-0.604886\pi$
$509$	−7431.68	−0.647158	−0.323579	+	0.946201i	$0.604886\pi$
$510$	0	0
$511$	−1993.97	−0.172618
$512$	−2603.24	−0.224704
$513$	0	0
$514$	11786.7	1.01145
$515$	5369.33	0.459419
$516$	0	0
$517$	831.737	0.0707538
$518$	−3367.04	−0.285597
$519$	0	0
$520$	−4151.23	−0.350083
$521$	−7797.86	−0.655721	−0.327860	−	0.944726i	$-0.606328\pi$
$521$	−7797.86	−0.655721	−0.327860	+	0.944726i	$0.606328\pi$
$522$	0	0
$523$	−16670.9	−1.39382	−0.696908	−	0.717160i	$-0.745442\pi$
$523$	−16670.9	−1.39382	−0.696908	+	0.717160i	$0.745442\pi$
$524$	12339.7	1.02874
$525$	0	0
$526$	−7811.79	−0.647548
$527$	28.7124	0.00237331
$528$	0	0
$529$	5268.18	0.432990
$530$	−5748.90	−0.471163
$531$	0	0
$532$	−193.170	−0.0157425
$533$	−5072.64	−0.412233
$534$	0	0
$535$	1339.87	0.108276
$536$	−8812.88	−0.710184
$537$	0	0
$538$	−76.7297	−0.00614880
$539$	−2700.18	−0.215779
$540$	0	0
$541$	13400.1	1.06491	0.532455	−	0.846458i	$-0.321270\pi$
$541$	13400.1	1.06491	0.532455	+	0.846458i	$0.321270\pi$
$542$	−12594.2	−0.998092
$543$	0	0
$544$	12726.2	1.00300
$545$	−28737.4	−2.25867
$546$	0	0
$547$	−17339.4	−1.35535	−0.677677	−	0.735360i	$-0.737013\pi$
$547$	−17339.4	−1.35535	−0.677677	+	0.735360i	$0.737013\pi$
$548$	10143.9	0.790739
$549$	0	0
$550$	−2202.93	−0.170788
$551$	200.974	0.0155386
$552$	0	0
$553$	3013.01	0.231693
$554$	12321.6	0.944936
$555$	0	0
$556$	−6992.50	−0.533360
$557$	7790.90	0.592659	0.296330	−	0.955086i	$-0.404237\pi$
$557$	7790.90	0.592659	0.296330	+	0.955086i	$0.404237\pi$
$558$	0	0
$559$	−2088.58	−0.158028
$560$	635.091	0.0479241
$561$	0	0
$562$	−3416.66	−0.256447
$563$	21514.2	1.61051	0.805253	−	0.592932i	$-0.202030\pi$
$563$	21514.2	1.61051	0.805253	+	0.592932i	$0.202030\pi$
$564$	0	0
$565$	−17667.3	−1.31552
$566$	13375.5	0.993311
$567$	0	0
$568$	12139.9	0.896794
$569$	−16993.5	−1.25203	−0.626015	−	0.779811i	$-0.715315\pi$
$569$	−16993.5	−1.25203	−0.626015	+	0.779811i	$0.715315\pi$
$570$	0	0
$571$	16791.4	1.23064	0.615321	−	0.788277i	$-0.289027\pi$
$571$	16791.4	1.23064	0.615321	+	0.788277i	$0.289027\pi$
$572$	517.963	0.0378621
$573$	0	0
$574$	−3807.42	−0.276862
$575$	20855.3	1.51256
$576$	0	0
$577$	8108.87	0.585055	0.292527	−	0.956257i	$-0.405504\pi$
$577$	8108.87	0.585055	0.292527	+	0.956257i	$0.405504\pi$
$578$	365.860	0.0263284
$579$	0	0
$580$	−2604.82	−0.186482
$581$	1846.54	0.131854
$582$	0	0
$583$	1791.96	0.127299
$584$	−8311.00	−0.588890
$585$	0	0
$586$	5739.25	0.404584
$587$	−16076.3	−1.13039	−0.565197	−	0.824956i	$-0.691200\pi$
$587$	−16076.3	−1.13039	−0.565197	+	0.824956i	$0.691200\pi$
$588$	0	0
$589$	−2.90594	−0.000203289 0
$590$	1261.04	0.0879938
$591$	0	0
$592$	2860.53	0.198593
$593$	4341.83	0.300671	0.150335	−	0.988635i	$-0.451965\pi$
$593$	4341.83	0.300671	0.150335	+	0.988635i	$0.451965\pi$
$594$	0	0
$595$	−6012.34	−0.414255
$596$	9520.88	0.654346
$597$	0	0
$598$	2442.80	0.167046
$599$	10540.1	0.718960	0.359480	−	0.933153i	$-0.382954\pi$
$599$	10540.1	0.718960	0.359480	+	0.933153i	$0.382954\pi$
$600$	0	0
$601$	16485.6	1.11890	0.559451	−	0.828863i	$-0.311012\pi$
$601$	16485.6	1.11890	0.559451	+	0.828863i	$0.311012\pi$
$602$	−1567.65	−0.106134
$603$	0	0
$604$	8714.43	0.587061
$605$	−21158.6	−1.42185
$606$	0	0
$607$	13326.5	0.891112	0.445556	−	0.895254i	$-0.353006\pi$
$607$	13326.5	0.891112	0.445556	+	0.895254i	$0.353006\pi$
$608$	−1288.00	−0.0859132
$609$	0	0
$610$	−11732.7	−0.778756
$611$	−1103.22	−0.0730467
$612$	0	0
$613$	20459.4	1.34804	0.674019	−	0.738714i	$-0.264567\pi$
$613$	20459.4	1.34804	0.674019	+	0.738714i	$0.264567\pi$
$614$	14231.0	0.935369
$615$	0	0
$616$	971.221	0.0635253
$617$	3108.50	0.202826	0.101413	−	0.994844i	$-0.467664\pi$
$617$	3108.50	0.202826	0.101413	+	0.994844i	$0.467664\pi$
$618$	0	0
$619$	10426.3	0.677012	0.338506	−	0.940964i	$-0.390078\pi$
$619$	10426.3	0.677012	0.338506	+	0.940964i	$0.390078\pi$
$620$	37.6638	0.00243970
$621$	0	0
$622$	4305.97	0.277579
$623$	1450.48	0.0932780
$624$	0	0
$625$	−10421.8	−0.666992
$626$	3160.87	0.201811
$627$	0	0
$628$	−4553.92	−0.289365
$629$	−27080.3	−1.71663
$630$	0	0
$631$	18018.2	1.13675	0.568377	−	0.822768i	$-0.307572\pi$
$631$	18018.2	1.13675	0.568377	+	0.822768i	$0.307572\pi$
$632$	12558.4	0.790424
$633$	0	0
$634$	−9045.67	−0.566639
$635$	6172.04	0.385716
$636$	0	0
$637$	3581.53	0.222772
$638$	−404.479	−0.0250995
$639$	0	0
$640$	18281.2	1.12911
$641$	−15178.7	−0.935295	−0.467648	−	0.883915i	$-0.654898\pi$
$641$	−15178.7	−0.935295	−0.467648	+	0.883915i	$0.654898\pi$
$642$	0	0
$643$	23957.1	1.46932	0.734662	−	0.678433i	$-0.237341\pi$
$643$	23957.1	1.46932	0.734662	+	0.678433i	$0.237341\pi$
$644$	−3680.54	−0.225207
$645$	0	0
$646$	773.962	0.0471379
$647$	12835.2	0.779915	0.389958	−	0.920833i	$-0.372490\pi$
$647$	12835.2	0.779915	0.389958	+	0.920833i	$0.372490\pi$
$648$	0	0
$649$	−393.074	−0.0237743
$650$	2921.98	0.176322
$651$	0	0
$652$	−17516.0	−1.05212
$653$	4355.97	0.261045	0.130523	−	0.991445i	$-0.458335\pi$
$653$	4355.97	0.261045	0.130523	+	0.991445i	$0.458335\pi$
$654$	0	0
$655$	−38870.8	−2.31879
$656$	3234.66	0.192519
$657$	0	0
$658$	−828.056	−0.0490593
$659$	32704.1	1.93319	0.966594	−	0.256313i	$-0.0825079\pi$
$659$	32704.1	1.93319	0.966594	+	0.256313i	$0.0825079\pi$
$660$	0	0
$661$	−2839.66	−0.167095	−0.0835475	−	0.996504i	$-0.526625\pi$
$661$	−2839.66	−0.167095	−0.0835475	+	0.996504i	$0.526625\pi$
$662$	−302.536	−0.0177620
$663$	0	0
$664$	7696.50	0.449823
$665$	608.499	0.0354836
$666$	0	0
$667$	3829.23	0.222291
$668$	−8983.04	−0.520306
$669$	0	0
$670$	11112.6	0.640773
$671$	3657.13	0.210405
$672$	0	0
$673$	−18569.9	−1.06362	−0.531810	−	0.846864i	$-0.678488\pi$
$673$	−18569.9	−1.06362	−0.531810	+	0.846864i	$0.678488\pi$
$674$	13877.0	0.793060
$675$	0	0
$676$	11044.6	0.628393
$677$	−18106.5	−1.02790	−0.513952	−	0.857819i	$-0.671819\pi$
$677$	−18106.5	−1.02790	−0.513952	+	0.857819i	$0.671819\pi$
$678$	0	0
$679$	3538.71	0.200005
$680$	−25059.9	−1.41324
$681$	0	0
$682$	5.84848	0.000328372 0
$683$	5510.97	0.308743	0.154371	−	0.988013i	$-0.450665\pi$
$683$	5510.97	0.308743	0.154371	+	0.988013i	$0.450665\pi$
$684$	0	0
$685$	−31953.9	−1.78233
$686$	5608.44	0.312145
$687$	0	0
$688$	1331.82	0.0738014
$689$	−2376.87	−0.131425
$690$	0	0
$691$	24652.0	1.35717	0.678587	−	0.734520i	$-0.262592\pi$
$691$	24652.0	1.35717	0.678587	+	0.734520i	$0.262592\pi$
$692$	−8491.53	−0.466473
$693$	0	0
$694$	12809.1	0.700613
$695$	22026.9	1.20220
$696$	0	0
$697$	−30622.2	−1.66413
$698$	−10321.5	−0.559704
$699$	0	0
$700$	−4402.51	−0.237713
$701$	3399.10	0.183142	0.0915708	−	0.995799i	$-0.470811\pi$
$701$	3399.10	0.183142	0.0915708	+	0.995799i	$0.470811\pi$
$702$	0	0
$703$	2740.75	0.147041
$704$	2097.39	0.112285
$705$	0	0
$706$	−13004.7	−0.693253
$707$	−2963.98	−0.157669
$708$	0	0
$709$	26329.1	1.39465	0.697327	−	0.716753i	$-0.254373\pi$
$709$	26329.1	1.39465	0.697327	+	0.716753i	$0.254373\pi$
$710$	−15307.8	−0.809145
$711$	0	0
$712$	6045.70	0.318219
$713$	−55.3678	−0.00290819
$714$	0	0
$715$	−1631.62	−0.0853415
$716$	−9582.91	−0.500182
$717$	0	0
$718$	−16406.8	−0.852781
$719$	−22345.6	−1.15904	−0.579521	−	0.814957i	$-0.696760\pi$
$719$	−22345.6	−1.15904	−0.579521	+	0.814957i	$0.696760\pi$
$720$	0	0
$721$	1666.24	0.0860668
$722$	11108.7	0.572606
$723$	0	0
$724$	−12566.9	−0.645089
$725$	4580.37	0.234635
$726$	0	0
$727$	−29862.8	−1.52345	−0.761727	−	0.647898i	$-0.775648\pi$
$727$	−29862.8	−1.52345	−0.761727	+	0.647898i	$0.775648\pi$
$728$	−1288.23	−0.0655839
$729$	0	0
$730$	10479.8	0.531334
$731$	−12608.2	−0.637938
$732$	0	0
$733$	−987.919	−0.0497812	−0.0248906	−	0.999690i	$-0.507924\pi$
$733$	−987.919	−0.0497812	−0.0248906	+	0.999690i	$0.507924\pi$
$734$	4946.27	0.248733
$735$	0	0
$736$	−24540.7	−1.22905
$737$	−3463.86	−0.173125
$738$	0	0
$739$	−35870.9	−1.78556	−0.892782	−	0.450489i	$-0.851250\pi$
$739$	−35870.9	−1.78556	−0.892782	+	0.450489i	$0.851250\pi$
$740$	−35522.9	−1.76466
$741$	0	0
$742$	−1784.03	−0.0882667
$743$	−8322.20	−0.410918	−0.205459	−	0.978666i	$-0.565869\pi$
$743$	−8322.20	−0.410918	−0.205459	+	0.978666i	$0.565869\pi$
$744$	0	0
$745$	−29991.4	−1.47490
$746$	18119.4	0.889273
$747$	0	0
$748$	3126.81	0.152844
$749$	415.795	0.0202841
$750$	0	0
$751$	20781.2	1.00974	0.504872	−	0.863194i	$-0.331540\pi$
$751$	20781.2	1.00974	0.504872	+	0.863194i	$0.331540\pi$
$752$	703.489	0.0341139
$753$	0	0
$754$	536.504	0.0259129
$755$	−27451.0	−1.32324
$756$	0	0
$757$	−16424.7	−0.788596	−0.394298	−	0.918983i	$-0.629012\pi$
$757$	−16424.7	−0.788596	−0.394298	+	0.918983i	$0.629012\pi$
$758$	−7356.10	−0.352488
$759$	0	0
$760$	2536.27	0.121053
$761$	−416.312	−0.0198309	−0.00991543	−	0.999951i	$-0.503156\pi$
$761$	−416.312	−0.0198309	−0.00991543	+	0.999951i	$0.503156\pi$
$762$	0	0
$763$	−8917.95	−0.423134
$764$	−11664.1	−0.552347
$765$	0	0
$766$	12739.6	0.600916
$767$	521.376	0.0245447
$768$	0	0
$769$	−37129.2	−1.74111	−0.870554	−	0.492073i	$-0.836239\pi$
$769$	−37129.2	−1.74111	−0.870554	+	0.492073i	$0.836239\pi$
$770$	−1224.66	−0.0573166
$771$	0	0
$772$	16605.4	0.774148
$773$	2182.83	0.101567	0.0507833	−	0.998710i	$-0.483828\pi$
$773$	2182.83	0.101567	0.0507833	+	0.998710i	$0.483828\pi$
$774$	0	0
$775$	−66.2287	−0.00306969
$776$	14749.6	0.682319
$777$	0	0
$778$	15040.9	0.693112
$779$	3099.22	0.142543
$780$	0	0
$781$	4771.53	0.218616
$782$	14746.6	0.674343
$783$	0	0
$784$	−2283.83	−0.104037
$785$	14345.2	0.652231
$786$	0	0
$787$	7068.15	0.320143	0.160071	−	0.987105i	$-0.448828\pi$
$787$	7068.15	0.320143	0.160071	+	0.987105i	$0.448828\pi$
$788$	4932.73	0.222996
$789$	0	0
$790$	−15835.6	−0.713171
$791$	−5482.61	−0.246446
$792$	0	0
$793$	−4850.84	−0.217224
$794$	16365.9	0.731493
$795$	0	0
$796$	−24289.5	−1.08155
$797$	−15086.0	−0.670482	−0.335241	−	0.942132i	$-0.608818\pi$
$797$	−15086.0	−0.670482	−0.335241	+	0.942132i	$0.608818\pi$
$798$	0	0
$799$	−6659.86	−0.294880
$800$	−29354.6	−1.29730
$801$	0	0
$802$	−13363.8	−0.588393
$803$	−3266.60	−0.143557
$804$	0	0
$805$	11593.9	0.507619
$806$	−7.75746	−0.000339013 0
$807$	0	0
$808$	−12354.1	−0.537890
$809$	21798.3	0.947326	0.473663	−	0.880706i	$-0.342931\pi$
$809$	21798.3	0.947326	0.473663	+	0.880706i	$0.342931\pi$
$810$	0	0
$811$	5569.61	0.241153	0.120577	−	0.992704i	$-0.461526\pi$
$811$	5569.61	0.241153	0.120577	+	0.992704i	$0.461526\pi$
$812$	−808.344	−0.0349351
$813$	0	0
$814$	−5516.03	−0.237515
$815$	55176.6	2.37148
$816$	0	0
$817$	1276.06	0.0546434
$818$	−6396.25	−0.273398
$819$	0	0
$820$	−40169.0	−1.71069
$821$	−21281.8	−0.904676	−0.452338	−	0.891847i	$-0.649410\pi$
$821$	−21281.8	−0.904676	−0.452338	+	0.891847i	$0.649410\pi$
$822$	0	0
$823$	16799.0	0.711514	0.355757	−	0.934578i	$-0.384223\pi$
$823$	16799.0	0.711514	0.355757	+	0.934578i	$0.384223\pi$
$824$	6945.02	0.293618
$825$	0	0
$826$	391.335	0.0164846
$827$	−7360.62	−0.309497	−0.154748	−	0.987954i	$-0.549457\pi$
$827$	−7360.62	−0.309497	−0.154748	+	0.987954i	$0.549457\pi$
$828$	0	0
$829$	−11634.6	−0.487438	−0.243719	−	0.969846i	$-0.578367\pi$
$829$	−11634.6	−0.487438	−0.243719	+	0.969846i	$0.578367\pi$
$830$	−9704.92	−0.405859
$831$	0	0
$832$	−2782.00	−0.115924
$833$	21620.8	0.899299
$834$	0	0
$835$	28297.2	1.17277
$836$	−316.459	−0.0130921
$837$	0	0
$838$	4936.46	0.203493
$839$	32151.0	1.32297	0.661486	−	0.749957i	$-0.269926\pi$
$839$	32151.0	1.32297	0.661486	+	0.749957i	$0.269926\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−7866.99	−0.321988
$843$	0	0
$844$	1701.62	0.0693984
$845$	−34791.4	−1.41640
$846$	0	0
$847$	−6566.05	−0.266366
$848$	1515.66	0.0613771
$849$	0	0
$850$	17639.2	0.711789
$851$	52220.6	2.10352
$852$	0	0
$853$	35044.9	1.40670	0.703349	−	0.710844i	$-0.251687\pi$
$853$	35044.9	1.40670	0.703349	+	0.710844i	$0.251687\pi$
$854$	−3640.95	−0.145891
$855$	0	0
$856$	1733.07	0.0691997
$857$	−7143.05	−0.284716	−0.142358	−	0.989815i	$-0.545468\pi$
$857$	−7143.05	−0.284716	−0.142358	+	0.989815i	$0.545468\pi$
$858$	0	0
$859$	33960.3	1.34891	0.674454	−	0.738317i	$-0.264379\pi$
$859$	33960.3	1.34891	0.674454	+	0.738317i	$0.264379\pi$
$860$	−16539.0	−0.655785
$861$	0	0
$862$	−3310.11	−0.130792
$863$	−45823.4	−1.80747	−0.903736	−	0.428090i	$-0.859186\pi$
$863$	−45823.4	−1.80747	−0.903736	+	0.428090i	$0.859186\pi$
$864$	0	0
$865$	26748.9	1.05143
$866$	873.900	0.0342914
$867$	0	0
$868$	11.6881	0.000457049 0
$869$	4936.04	0.192685
$870$	0	0
$871$	4594.49	0.178735
$872$	−37170.7	−1.44353
$873$	0	0
$874$	−1492.48	−0.0577617
$875$	2892.61	0.111758
$876$	0	0
$877$	−21281.3	−0.819406	−0.409703	−	0.912219i	$-0.634368\pi$
$877$	−21281.3	−0.819406	−0.409703	+	0.912219i	$0.634368\pi$
$878$	−14339.3	−0.551169
$879$	0	0
$880$	1040.43	0.0398557
$881$	−1359.32	−0.0519826	−0.0259913	−	0.999662i	$-0.508274\pi$
$881$	−1359.32	−0.0519826	−0.0259913	+	0.999662i	$0.508274\pi$
$882$	0	0
$883$	−47928.2	−1.82663	−0.913313	−	0.407257i	$-0.866485\pi$
$883$	−47928.2	−1.82663	−0.913313	+	0.407257i	$0.866485\pi$
$884$	−4147.42	−0.157797
$885$	0	0
$886$	22698.5	0.860688
$887$	−3100.58	−0.117370	−0.0586851	−	0.998277i	$-0.518691\pi$
$887$	−3100.58	−0.117370	−0.0586851	+	0.998277i	$0.518691\pi$
$888$	0	0
$889$	1915.34	0.0722593
$890$	−7623.34	−0.287118
$891$	0	0
$892$	−9113.56	−0.342090
$893$	674.033	0.0252583
$894$	0	0
$895$	30186.8	1.12741
$896$	5673.14	0.211525
$897$	0	0
$898$	3473.27	0.129070
$899$	−12.1602	−0.000451131 0
$900$	0	0
$901$	−14348.5	−0.530543
$902$	−6237.48	−0.230250
$903$	0	0
$904$	−22851.9	−0.840756
$905$	39586.5	1.45403
$906$	0	0
$907$	23428.1	0.857680	0.428840	−	0.903380i	$-0.358922\pi$
$907$	23428.1	0.857680	0.428840	+	0.903380i	$0.358922\pi$
$908$	16252.6	0.594009
$909$	0	0
$910$	1624.40	0.0591740
$911$	12868.3	0.467999	0.234000	−	0.972237i	$-0.424819\pi$
$911$	12868.3	0.467999	0.234000	+	0.972237i	$0.424819\pi$
$912$	0	0
$913$	3025.07	0.109655
$914$	−3151.49	−0.114050
$915$	0	0
$916$	13998.8	0.504950
$917$	−12062.6	−0.434398
$918$	0	0
$919$	3938.41	0.141367	0.0706834	−	0.997499i	$-0.477482\pi$
$919$	3938.41	0.141367	0.0706834	+	0.997499i	$0.477482\pi$
$920$	48324.4	1.73175
$921$	0	0
$922$	−26941.2	−0.962323
$923$	−6328.99	−0.225700
$924$	0	0
$925$	62464.1	2.22033
$926$	−18813.8	−0.667667
$927$	0	0
$928$	−5389.79	−0.190656
$929$	15856.7	0.560001	0.280001	−	0.960000i	$-0.409665\pi$
$929$	15856.7	0.560001	0.280001	+	0.960000i	$0.409665\pi$
$930$	0	0
$931$	−2188.20	−0.0770306
$932$	−14838.0	−0.521498
$933$	0	0
$934$	25554.0	0.895238
$935$	−9849.67	−0.344512
$936$	0	0
$937$	−18559.7	−0.647085	−0.323542	−	0.946214i	$-0.604874\pi$
$937$	−18559.7	−0.647085	−0.323542	+	0.946214i	$0.604874\pi$
$938$	3448.54	0.120041
$939$	0	0
$940$	−8736.14	−0.303129
$941$	−24125.6	−0.835785	−0.417892	−	0.908497i	$-0.637231\pi$
$941$	−24125.6	−0.835785	−0.417892	+	0.908497i	$0.637231\pi$
$942$	0	0
$943$	59050.6	2.03919
$944$	−332.465	−0.0114627
$945$	0	0
$946$	−2568.19	−0.0882655
$947$	6883.31	0.236196	0.118098	−	0.993002i	$-0.462320\pi$
$947$	6883.31	0.236196	0.118098	+	0.993002i	$0.462320\pi$
$948$	0	0
$949$	4332.84	0.148209
$950$	−1785.24	−0.0609693
$951$	0	0
$952$	−7776.73	−0.264753
$953$	−37573.8	−1.27716	−0.638580	−	0.769555i	$-0.720478\pi$
$953$	−37573.8	−1.27716	−0.638580	+	0.769555i	$0.720478\pi$
$954$	0	0
$955$	36742.8	1.24499
$956$	25218.4	0.853161
$957$	0	0
$958$	7620.45	0.257000
$959$	−9916.13	−0.333898
$960$	0	0
$961$	−29790.8	−0.999994
$962$	7316.50	0.245211
$963$	0	0
$964$	25119.4	0.839255
$965$	−52308.2	−1.74493
$966$	0	0
$967$	−1584.28	−0.0526856	−0.0263428	−	0.999653i	$-0.508386\pi$
$967$	−1584.28	−0.0526856	−0.0263428	+	0.999653i	$0.508386\pi$
$968$	−27367.8	−0.908712
$969$	0	0
$970$	−18598.5	−0.615632
$971$	−36569.1	−1.20861	−0.604304	−	0.796754i	$-0.706549\pi$
$971$	−36569.1	−1.20861	−0.604304	+	0.796754i	$0.706549\pi$
$972$	0	0
$973$	6835.50	0.225217
$974$	13807.6	0.454234
$975$	0	0
$976$	3093.23	0.101447
$977$	33583.0	1.09971	0.549854	−	0.835261i	$-0.314683\pi$
$977$	33583.0	1.09971	0.549854	+	0.835261i	$0.314683\pi$
$978$	0	0
$979$	2376.23	0.0775738
$980$	28361.3	0.924457
$981$	0	0
$982$	−20065.4	−0.652051
$983$	25900.6	0.840387	0.420193	−	0.907435i	$-0.361962\pi$
$983$	25900.6	0.840387	0.420193	+	0.907435i	$0.361962\pi$
$984$	0	0
$985$	−15538.4	−0.502635
$986$	3238.74	0.104607
$987$	0	0
$988$	419.754	0.0135163
$989$	24313.2	0.781714
$990$	0	0
$991$	29604.3	0.948951	0.474475	−	0.880269i	$-0.342638\pi$
$991$	29604.3	0.948951	0.474475	+	0.880269i	$0.342638\pi$
$992$	77.9324	0.00249431
$993$	0	0
$994$	−4750.42	−0.151584
$995$	76513.5	2.43783
$996$	0	0
$997$	39267.2	1.24735	0.623673	−	0.781685i	$-0.285640\pi$
$997$	39267.2	1.24735	0.623673	+	0.781685i	$0.285640\pi$
$998$	4536.30	0.143882
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.4.a.f.1.3		5
3.2	odd	2		29.4.a.b.1.3	✓	5
12.11	even	2		464.4.a.l.1.1		5
15.14	odd	2		725.4.a.c.1.3		5
21.20	even	2		1421.4.a.e.1.3		5
24.5	odd	2		1856.4.a.y.1.1		5
24.11	even	2		1856.4.a.bb.1.5		5
87.86	odd	2		841.4.a.b.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.b.1.3	✓	5	3.2	odd	2
261.4.a.f.1.3		5	1.1	even	1	trivial
464.4.a.l.1.1		5	12.11	even	2
725.4.a.c.1.3		5	15.14	odd	2
841.4.a.b.1.3		5	87.86	odd	2
1421.4.a.e.1.3		5	21.20	even	2
1856.4.a.y.1.1		5	24.5	odd	2
1856.4.a.bb.1.5		5	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)

\(f(q)\)	\(=\)	\(q-1.63099 q^{2} -5.33986 q^{4} +16.8209 q^{5} +5.21997 q^{7} +21.7572 q^{8} +O(q^{10})\)	\(q-1.63099 q^{2} -5.33986 q^{4} +16.8209 q^{5} +5.21997 q^{7} +21.7572 q^{8} -27.4348 q^{10} +8.55158 q^{11} -11.3429 q^{13} -8.51374 q^{14} +7.23299 q^{16} -68.4740 q^{17} +6.93014 q^{19} -89.8214 q^{20} -13.9476 q^{22} +132.042 q^{23} +157.944 q^{25} +18.5001 q^{26} -27.8739 q^{28} +29.0000 q^{29} -0.419319 q^{31} -185.855 q^{32} +111.681 q^{34} +87.8048 q^{35} +395.483 q^{37} -11.3030 q^{38} +365.977 q^{40} +447.209 q^{41} +184.132 q^{43} -45.6642 q^{44} -215.360 q^{46} +97.2612 q^{47} -315.752 q^{49} -257.605 q^{50} +60.5693 q^{52} +209.547 q^{53} +143.845 q^{55} +113.572 q^{56} -47.2988 q^{58} -45.9651 q^{59} +427.655 q^{61} +0.683906 q^{62} +245.264 q^{64} -190.798 q^{65} -405.055 q^{67} +365.641 q^{68} -143.209 q^{70} +557.971 q^{71} -381.988 q^{73} -645.031 q^{74} -37.0060 q^{76} +44.6390 q^{77} +577.208 q^{79} +121.666 q^{80} -729.396 q^{82} +353.745 q^{83} -1151.80 q^{85} -300.318 q^{86} +186.059 q^{88} +277.871 q^{89} -59.2095 q^{91} -705.088 q^{92} -158.632 q^{94} +116.571 q^{95} +677.917 q^{97} +514.989 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8}+O(q^{10}) \) 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8	\( 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8} - 64 q^{10} - 12 q^{11} + 14 q^{13} + 192 q^{14} + 146 q^{16} - 66 q^{17} + 214 q^{19} - 6 q^{20} - 98 q^{22} - 164 q^{23} + 207 q^{25} - 56 q^{26} + 540 q^{28} + 145 q^{29} + 420 q^{31} + 652 q^{32} + 204 q^{34} + 52 q^{35} + 378 q^{37} + 496 q^{38} - 80 q^{40} + 1158 q^{41} - 204 q^{43} - 784 q^{44} + 580 q^{46} - 248 q^{47} - 283 q^{49} - 908 q^{50} + 1482 q^{52} + 554 q^{53} + 546 q^{55} + 608 q^{56} - 440 q^{59} + 618 q^{61} - 1250 q^{62} + 2594 q^{64} + 1656 q^{65} + 1164 q^{67} - 356 q^{68} - 2184 q^{70} + 692 q^{71} - 1950 q^{73} + 1832 q^{74} + 1376 q^{76} + 1616 q^{77} + 272 q^{79} + 890 q^{80} + 92 q^{82} - 512 q^{83} - 1628 q^{85} - 2446 q^{86} - 6954 q^{88} - 866 q^{89} + 2580 q^{91} - 3468 q^{92} - 5942 q^{94} - 2244 q^{95} + 1562 q^{97} + 3408 q^{98}+O(q^{100}) \) 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8 - 64 * q^10 - 12 * q^11 + 14 * q^13 + 192 * q^14 + 146 * q^16 - 66 * q^17 + 214 * q^19 - 6 * q^20 - 98 * q^22 - 164 * q^23 + 207 * q^25 - 56 * q^26 + 540 * q^28 + 145 * q^29 + 420 * q^31 + 652 * q^32 + 204 * q^34 + 52 * q^35 + 378 * q^37 + 496 * q^38 - 80 * q^40 + 1158 * q^41 - 204 * q^43 - 784 * q^44 + 580 * q^46 - 248 * q^47 - 283 * q^49 - 908 * q^50 + 1482 * q^52 + 554 * q^53 + 546 * q^55 + 608 * q^56 - 440 * q^59 + 618 * q^61 - 1250 * q^62 + 2594 * q^64 + 1656 * q^65 + 1164 * q^67 - 356 * q^68 - 2184 * q^70 + 692 * q^71 - 1950 * q^73 + 1832 * q^74 + 1376 * q^76 + 1616 * q^77 + 272 * q^79 + 890 * q^80 + 92 * q^82 - 512 * q^83 - 1628 * q^85 - 2446 * q^86 - 6954 * q^88 - 866 * q^89 + 2580 * q^91 - 3468 * q^92 - 5942 * q^94 - 2244 * q^95 + 1562 * q^97 + 3408 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more