LMFDB - Embedded newform 1296.5.g.e.1135.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,5,Mod(1135,1296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1296.1135"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1296.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$133.967472157$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-19}, \sqrt{-39})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 25 $ x^4 + 29*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1135.1
Root			$-0.943050i$ of defining polynomial
Character	$\chi$	$=$	1296.1135
Dual form			1296.5.g.e.1135.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-15.2213 q^{5} -16.9749i q^{7} +173.895i q^{11} +43.6639 q^{13} +69.8853 q^{17} +453.786i q^{19} +500.565i q^{23} -393.312 q^{25} +290.533 q^{29} -1281.02i q^{31} +258.380i q^{35} -437.648 q^{37} +893.951 q^{41} -895.133i q^{43} -390.033i q^{47} +2112.85 q^{49} +2381.90 q^{53} -2646.91i q^{55} -474.518i q^{59} -2705.55 q^{61} -664.623 q^{65} +6781.91i q^{67} -5454.12i q^{71} -1054.36 q^{73} +2951.85 q^{77} +4360.21i q^{79} -874.402i q^{83} -1063.75 q^{85} +2992.18 q^{89} -741.191i q^{91} -6907.22i q^{95} -6541.85 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 48 q^{5} - 152 q^{13} - 156 q^{17} + 1040 q^{25} - 1560 q^{29} + 536 q^{37} - 2304 q^{41} - 9188 q^{49} - 2232 q^{53} + 3224 q^{61} - 10716 q^{65} - 7484 q^{73} - 5832 q^{77} - 13728 q^{85} + 20244 q^{89}+ \cdots - 8528 q^{97}+O(q^{100}) $ 4 * q + 48 * q^5 - 152 * q^13 - 156 * q^17 + 1040 * q^25 - 1560 * q^29 + 536 * q^37 - 2304 * q^41 - 9188 * q^49 - 2232 * q^53 + 3224 * q^61 - 10716 * q^65 - 7484 * q^73 - 5832 * q^77 - 13728 * q^85 + 20244 * q^89 - 8528 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−15.2213	−0.608853	−0.304426	−	0.952536i	$-0.598465\pi$
$5$	−15.2213	−0.608853	−0.304426	+	0.952536i	$0.598465\pi$
$6$	0	0
$7$	− 16.9749i	− 0.346426i	−0.984884	−	0.173213i	$-0.944585\pi$
$7$	− 16.9749i	− 0.346426i	0.984884	−	0.173213i	$-0.0554150\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	173.895i	1.43715i	0.695449	+	0.718575i	$0.255206\pi$
$11$	173.895i	1.43715i	−0.695449	+	0.718575i	$0.744794\pi$
$12$	0	0
$13$	43.6639	0.258367	0.129183	−	0.991621i	$-0.458764\pi$
$13$	43.6639	0.258367	0.129183	+	0.991621i	$0.458764\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	69.8853	0.241818	0.120909	−	0.992664i	$-0.461419\pi$
$17$	69.8853	0.241818	0.120909	+	0.992664i	$0.461419\pi$
$18$	0	0
$19$	453.786i	1.25703i	0.777799	+	0.628513i	$0.216336\pi$
$19$	453.786i	1.25703i	−0.777799	+	0.628513i	$0.783664\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	500.565i	0.946247i	0.880996	+	0.473123i	$0.156874\pi$
$23$	500.565i	0.946247i	−0.880996	+	0.473123i	$0.843126\pi$
$24$	0	0
$25$	−393.312	−0.629299
$26$	0	0
$27$	0	0
$28$	0	0
$29$	290.533	0.345461	0.172731	−	0.984969i	$-0.444741\pi$
$29$	290.533	0.345461	0.172731	+	0.984969i	$0.444741\pi$
$30$	0	0
$31$	− 1281.02i	− 1.33301i	−0.745502	−	0.666504i	$-0.767790\pi$
$31$	− 1281.02i	− 1.33301i	0.745502	−	0.666504i	$-0.232210\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	258.380i	0.210923i
$36$	0	0
$37$	−437.648	−0.319684	−0.159842	−	0.987143i	$-0.551099\pi$
$37$	−437.648	−0.319684	−0.159842	+	0.987143i	$0.551099\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	893.951	0.531797	0.265899	−	0.964001i	$-0.414331\pi$
$41$	893.951	0.531797	0.265899	+	0.964001i	$0.414331\pi$
$42$	0	0
$43$	− 895.133i	− 0.484118i	−0.970262	−	0.242059i	$-0.922177\pi$
$43$	− 895.133i	− 0.484118i	0.970262	−	0.242059i	$-0.0778227\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 390.033i	− 0.176565i	−0.996095	−	0.0882827i	$-0.971862\pi$
$47$	− 390.033i	− 0.176565i	0.996095	−	0.0882827i	$-0.0281379\pi$
$48$	0	0
$49$	2112.85	0.879989
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2381.90	0.847954	0.423977	−	0.905673i	$-0.360634\pi$
$53$	2381.90	0.847954	0.423977	+	0.905673i	$0.360634\pi$
$54$	0	0
$55$	− 2646.91i	− 0.875013i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 474.518i	− 0.136317i	−0.997675	−	0.0681583i	$-0.978288\pi$
$59$	− 474.518i	− 0.136317i	0.997675	−	0.0681583i	$-0.0217123\pi$
$60$	0	0
$61$	−2705.55	−0.727103	−0.363551	−	0.931574i	$-0.618436\pi$
$61$	−2705.55	−0.727103	−0.363551	+	0.931574i	$0.618436\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−664.623	−0.157307
$66$	0	0
$67$	6781.91i	1.51079i	0.655272	+	0.755393i	$0.272554\pi$
$67$	6781.91i	1.51079i	−0.655272	+	0.755393i	$0.727446\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 5454.12i	− 1.08195i	−0.841038	−	0.540976i	$-0.818055\pi$
$71$	− 5454.12i	− 1.08195i	0.841038	−	0.540976i	$-0.181945\pi$
$72$	0	0
$73$	−1054.36	−0.197853	−0.0989267	−	0.995095i	$-0.531541\pi$
$73$	−1054.36	−0.197853	−0.0989267	+	0.995095i	$0.531541\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2951.85	0.497867
$78$	0	0
$79$	4360.21i	0.698640i	0.937004	+	0.349320i	$0.113587\pi$
$79$	4360.21i	0.698640i	−0.937004	+	0.349320i	$0.886413\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 874.402i	− 0.126927i	−0.997984	−	0.0634636i	$-0.979785\pi$
$83$	− 874.402i	− 0.126927i	0.997984	−	0.0634636i	$-0.0202147\pi$
$84$	0	0
$85$	−1063.75	−0.147231
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2992.18	0.377753	0.188876	−	0.982001i	$-0.439515\pi$
$89$	2992.18	0.377753	0.188876	+	0.982001i	$0.439515\pi$
$90$	0	0
$91$	− 741.191i	− 0.0895050i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 6907.22i	− 0.765343i
$96$	0	0
$97$	−6541.85	−0.695276	−0.347638	−	0.937629i	$-0.613016\pi$
$97$	−6541.85	−0.695276	−0.347638	+	0.937629i	$0.613016\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7530.05	−0.738168	−0.369084	−	0.929396i	$-0.620328\pi$
$101$	−7530.05	−0.738168	−0.369084	+	0.929396i	$0.620328\pi$
$102$	0	0
$103$	− 11400.1i	− 1.07457i	−0.843401	−	0.537285i	$-0.819450\pi$
$103$	− 11400.1i	− 1.07457i	0.843401	−	0.537285i	$-0.180550\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14838.1i	1.29602i	0.761632	+	0.648009i	$0.224398\pi$
$107$	14838.1i	1.29602i	−0.761632	+	0.648009i	$0.775602\pi$
$108$	0	0
$109$	−8488.14	−0.714430	−0.357215	−	0.934022i	$-0.616274\pi$
$109$	−8488.14	−0.714430	−0.357215	+	0.934022i	$0.616274\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5048.31	−0.395357	−0.197678	−	0.980267i	$-0.563340\pi$
$113$	−5048.31	−0.395357	−0.197678	+	0.980267i	$0.563340\pi$
$114$	0	0
$115$	− 7619.25i	− 0.576125i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1186.29i	− 0.0837720i
$120$	0	0
$121$	−15598.6	−1.06540
$122$	0	0
$123$	0	0
$124$	0	0
$125$	15500.0	0.992003
$126$	0	0
$127$	19816.2i	1.22860i	0.789071	+	0.614302i	$0.210562\pi$
$127$	19816.2i	1.22860i	−0.789071	+	0.614302i	$0.789438\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	22569.1i	1.31514i	0.753395	+	0.657568i	$0.228415\pi$
$131$	22569.1i	1.31514i	−0.753395	+	0.657568i	$0.771585\pi$
$132$	0	0
$133$	7702.97	0.435467
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−28814.1	−1.53520	−0.767599	−	0.640931i	$-0.778549\pi$
$137$	−28814.1	−1.53520	−0.767599	+	0.640931i	$0.778549\pi$
$138$	0	0
$139$	1894.45i	0.0980515i	0.998798	+	0.0490258i	$0.0156116\pi$
$139$	1894.45i	0.0980515i	−0.998798	+	0.0490258i	$0.984388\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7592.95i	0.371312i
$144$	0	0
$145$	−4422.29	−0.210335
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5057.14	0.227789	0.113894	−	0.993493i	$-0.463667\pi$
$149$	5057.14	0.227789	0.113894	+	0.993493i	$0.463667\pi$
$150$	0	0
$151$	4755.03i	0.208545i	0.994549	+	0.104272i	$0.0332514\pi$
$151$	4755.03i	0.208545i	−0.994549	+	0.104272i	$0.966749\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	19498.8i	0.811605i
$156$	0	0
$157$	−31828.7	−1.29128	−0.645639	−	0.763643i	$-0.723409\pi$
$157$	−31828.7	−1.29128	−0.645639	+	0.763643i	$0.723409\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8497.03	0.327805
$162$	0	0
$163$	− 5158.03i	− 0.194137i	−0.995278	−	0.0970686i	$-0.969053\pi$
$163$	− 5158.03i	− 0.194137i	0.995278	−	0.0970686i	$-0.0309466\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 27153.7i	− 0.973635i	−0.873504	−	0.486817i	$-0.838158\pi$
$167$	− 27153.7i	− 0.973635i	0.873504	−	0.486817i	$-0.161842\pi$
$168$	0	0
$169$	−26654.5	−0.933247
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−49969.0	−1.66959	−0.834793	−	0.550565i	$-0.814412\pi$
$173$	−49969.0	−1.66959	−0.834793	+	0.550565i	$0.814412\pi$
$174$	0	0
$175$	6676.42i	0.218006i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 51470.2i	− 1.60638i	−0.595720	−	0.803192i	$-0.703133\pi$
$179$	− 51470.2i	− 1.60638i	0.595720	−	0.803192i	$-0.296867\pi$
$180$	0	0
$181$	−32941.4	−1.00551	−0.502753	−	0.864430i	$-0.667680\pi$
$181$	−32941.4	−1.00551	−0.502753	+	0.864430i	$0.667680\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6661.57	0.194641
$186$	0	0
$187$	12152.7i	0.347528i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 19641.7i	− 0.538410i	−0.963083	−	0.269205i	$-0.913239\pi$
$191$	− 19641.7i	− 0.538410i	0.963083	−	0.269205i	$-0.0867609\pi$
$192$	0	0
$193$	−31311.8	−0.840608	−0.420304	−	0.907383i	$-0.638077\pi$
$193$	−31311.8	−0.840608	−0.420304	+	0.907383i	$0.638077\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	53935.8	1.38978	0.694888	−	0.719118i	$-0.255454\pi$
$197$	53935.8	1.38978	0.694888	+	0.719118i	$0.255454\pi$
$198$	0	0
$199$	− 60022.4i	− 1.51568i	−0.652441	−	0.757839i	$-0.726255\pi$
$199$	− 60022.4i	− 1.51568i	0.652441	−	0.757839i	$-0.273745\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4931.76i	− 0.119677i
$204$	0	0
$205$	−13607.1	−0.323786
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−78911.3	−1.80654
$210$	0	0
$211$	10362.4i	0.232754i	0.993205	+	0.116377i	$0.0371281\pi$
$211$	10362.4i	0.232754i	−0.993205	+	0.116377i	$0.962872\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	13625.1i	0.294756i
$216$	0	0
$217$	−21745.2	−0.461789
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3051.47	0.0624776
$222$	0	0
$223$	− 32916.1i	− 0.661910i	−0.943647	−	0.330955i	$-0.892629\pi$
$223$	− 32916.1i	− 0.661910i	0.943647	−	0.330955i	$-0.107371\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 69605.3i	− 1.35080i	−0.737453	−	0.675399i	$-0.763971\pi$
$227$	− 69605.3i	− 1.35080i	0.737453	−	0.675399i	$-0.236029\pi$
$228$	0	0
$229$	−24729.1	−0.471560	−0.235780	−	0.971807i	$-0.575764\pi$
$229$	−24729.1	−0.471560	−0.235780	+	0.971807i	$0.575764\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	85422.6	1.57348	0.786739	−	0.617285i	$-0.211768\pi$
$233$	85422.6	1.57348	0.786739	+	0.617285i	$0.211768\pi$
$234$	0	0
$235$	5936.81i	0.107502i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	59241.3i	1.03712i	0.855042	+	0.518559i	$0.173531\pi$
$239$	59241.3i	1.03712i	−0.855042	+	0.518559i	$0.826469\pi$
$240$	0	0
$241$	−98255.5	−1.69170	−0.845849	−	0.533423i	$-0.820906\pi$
$241$	−98255.5	−1.69170	−0.845849	+	0.533423i	$0.820906\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−32160.4	−0.535783
$246$	0	0
$247$	19814.1i	0.324773i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	39009.6i	0.619189i	0.950869	+	0.309595i	$0.100193\pi$
$251$	39009.6i	0.619189i	−0.950869	+	0.309595i	$0.899807\pi$
$252$	0	0
$253$	−87045.8	−1.35990
$254$	0	0
$255$	0	0
$256$	0	0
$257$	47777.8	0.723369	0.361685	−	0.932301i	$-0.382202\pi$
$257$	47777.8	0.723369	0.361685	+	0.932301i	$0.382202\pi$
$258$	0	0
$259$	7429.02i	0.110747i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	129957.i	1.87883i	0.342782	+	0.939415i	$0.388631\pi$
$263$	129957.i	1.87883i	−0.342782	+	0.939415i	$0.611369\pi$
$264$	0	0
$265$	−36255.7	−0.516279
$266$	0	0
$267$	0	0
$268$	0	0
$269$	125441.	1.73354	0.866772	−	0.498705i	$-0.166191\pi$
$269$	125441.	1.73354	0.866772	+	0.498705i	$0.166191\pi$
$270$	0	0
$271$	69954.5i	0.952526i	0.879303	+	0.476263i	$0.158009\pi$
$271$	69954.5i	0.952526i	−0.879303	+	0.476263i	$0.841991\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 68395.0i	− 0.904397i
$276$	0	0
$277$	−62557.4	−0.815303	−0.407652	−	0.913138i	$-0.633652\pi$
$277$	−62557.4	−0.815303	−0.407652	+	0.913138i	$0.633652\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−134250.	−1.70021	−0.850104	−	0.526615i	$-0.823461\pi$
$281$	−134250.	−1.70021	−0.850104	+	0.526615i	$0.823461\pi$
$282$	0	0
$283$	− 117926.i	− 1.47244i	−0.676743	−	0.736219i	$-0.736609\pi$
$283$	− 117926.i	− 1.47244i	0.676743	−	0.736219i	$-0.263391\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 15174.7i	− 0.184229i
$288$	0	0
$289$	−78637.1	−0.941524
$290$	0	0
$291$	0	0
$292$	0	0
$293$	65257.2	0.760140	0.380070	−	0.924958i	$-0.375900\pi$
$293$	65257.2	0.760140	0.380070	+	0.924958i	$0.375900\pi$
$294$	0	0
$295$	7222.79i	0.0829967i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21856.6i	0.244479i
$300$	0	0
$301$	−15194.8	−0.167711
$302$	0	0
$303$	0	0
$304$	0	0
$305$	41182.0	0.442698
$306$	0	0
$307$	122733.i	1.30222i	0.758983	+	0.651110i	$0.225696\pi$
$307$	122733.i	1.30222i	−0.758983	+	0.651110i	$0.774304\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	51056.2i	0.527871i	0.964540	+	0.263936i	$0.0850207\pi$
$311$	51056.2i	0.527871i	−0.964540	+	0.263936i	$0.914979\pi$
$312$	0	0
$313$	−132325.	−1.35068	−0.675342	−	0.737505i	$-0.736004\pi$
$313$	−132325.	−1.35068	−0.675342	+	0.737505i	$0.736004\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−133273.	−1.32624	−0.663120	−	0.748513i	$-0.730768\pi$
$317$	−133273.	−1.32624	−0.663120	+	0.748513i	$0.730768\pi$
$318$	0	0
$319$	50522.3i	0.496480i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	31713.0i	0.303971i
$324$	0	0
$325$	−17173.5	−0.162590
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6620.77	−0.0611669
$330$	0	0
$331$	106484.i	0.971915i	0.873982	+	0.485958i	$0.161529\pi$
$331$	106484.i	0.971915i	−0.873982	+	0.485958i	$0.838471\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 103230.i	− 0.919846i
$336$	0	0
$337$	−51135.6	−0.450260	−0.225130	−	0.974329i	$-0.572281\pi$
$337$	−51135.6	−0.450260	−0.225130	+	0.974329i	$0.572281\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	222763.	1.91573
$342$	0	0
$343$	− 76622.2i	− 0.651278i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 14131.9i	− 0.117366i	−0.998277	−	0.0586830i	$-0.981310\pi$
$347$	− 14131.9i	− 0.117366i	0.998277	−	0.0586830i	$-0.0186901\pi$
$348$	0	0
$349$	−4439.91	−0.0364522	−0.0182261	−	0.999834i	$-0.505802\pi$
$349$	−4439.91	−0.0364522	−0.0182261	+	0.999834i	$0.505802\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−89684.0	−0.719723	−0.359861	−	0.933006i	$-0.617176\pi$
$353$	−89684.0	−0.719723	−0.359861	+	0.933006i	$0.617176\pi$
$354$	0	0
$355$	83018.8i	0.658749i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	48287.3i	0.374666i	0.982297	+	0.187333i	$0.0599843\pi$
$359$	48287.3i	0.374666i	−0.982297	+	0.187333i	$0.940016\pi$
$360$	0	0
$361$	−75600.9	−0.580113
$362$	0	0
$363$	0	0
$364$	0	0
$365$	16048.8	0.120464
$366$	0	0
$367$	− 159123.i	− 1.18141i	−0.806888	−	0.590704i	$-0.798850\pi$
$367$	− 159123.i	− 1.18141i	0.806888	−	0.590704i	$-0.201150\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 40432.5i	− 0.293754i
$372$	0	0
$373$	−96570.2	−0.694105	−0.347053	−	0.937846i	$-0.612817\pi$
$373$	−96570.2	−0.694105	−0.347053	+	0.937846i	$0.612817\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12685.8	0.0892556
$378$	0	0
$379$	− 273247.i	− 1.90229i	−0.308748	−	0.951144i	$-0.599910\pi$
$379$	− 273247.i	− 1.90229i	0.308748	−	0.951144i	$-0.400090\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 86237.4i	− 0.587892i	−0.955822	−	0.293946i	$-0.905031\pi$
$383$	− 86237.4i	− 0.587892i	0.955822	−	0.293946i	$-0.0949687\pi$
$384$	0	0
$385$	−44931.1	−0.303128
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−222348.	−1.46938	−0.734690	−	0.678403i	$-0.762672\pi$
$389$	−222348.	−1.46938	−0.734690	+	0.678403i	$0.762672\pi$
$390$	0	0
$391$	34982.1i	0.228819i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 66368.1i	− 0.425369i
$396$	0	0
$397$	−53463.7	−0.339218	−0.169609	−	0.985511i	$-0.554250\pi$
$397$	−53463.7	−0.339218	−0.169609	+	0.985511i	$0.554250\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	62063.6	0.385965	0.192983	−	0.981202i	$-0.438184\pi$
$401$	62063.6	0.385965	0.192983	+	0.981202i	$0.438184\pi$
$402$	0	0
$403$	− 55934.4i	− 0.344404i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 76104.8i	− 0.459434i
$408$	0	0
$409$	−223564.	−1.33646	−0.668228	−	0.743956i	$-0.732947\pi$
$409$	−223564.	−1.33646	−0.668228	+	0.743956i	$0.732947\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8054.89	−0.0472236
$414$	0	0
$415$	13309.5i	0.0772800i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 12540.2i	− 0.0714291i	−0.999362	−	0.0357145i	$-0.988629\pi$
$419$	− 12540.2i	− 0.0714291i	0.999362	−	0.0357145i	$-0.0113707\pi$
$420$	0	0
$421$	13898.0	0.0784131	0.0392065	−	0.999231i	$-0.487517\pi$
$421$	13898.0	0.0784131	0.0392065	+	0.999231i	$0.487517\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−27486.7	−0.152175
$426$	0	0
$427$	45926.4i	0.251888i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 161097.i	− 0.867229i	−0.901099	−	0.433614i	$-0.857238\pi$
$431$	− 161097.i	− 0.867229i	0.901099	−	0.433614i	$-0.142762\pi$
$432$	0	0
$433$	−133403.	−0.711524	−0.355762	−	0.934577i	$-0.615779\pi$
$433$	−133403.	−0.711524	−0.355762	+	0.934577i	$0.615779\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−227149.	−1.18946
$438$	0	0
$439$	302327.i	1.56873i	0.620302	+	0.784363i	$0.287010\pi$
$439$	302327.i	1.56873i	−0.620302	+	0.784363i	$0.712990\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	226223.i	1.15273i	0.817191	+	0.576367i	$0.195530\pi$
$443$	226223.i	1.15273i	−0.817191	+	0.576367i	$0.804470\pi$
$444$	0	0
$445$	−45544.9	−0.229996
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2138.33	0.0106067	0.00530336	−	0.999986i	$-0.498312\pi$
$449$	2138.33	0.0106067	0.00530336	+	0.999986i	$0.498312\pi$
$450$	0	0
$451$	155454.i	0.764273i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	11281.9i	0.0544953i
$456$	0	0
$457$	77351.2	0.370369	0.185185	−	0.982704i	$-0.440712\pi$
$457$	77351.2	0.370369	0.185185	+	0.982704i	$0.440712\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11587.3	−0.0545230	−0.0272615	−	0.999628i	$-0.508679\pi$
$461$	−11587.3	−0.0545230	−0.0272615	+	0.999628i	$0.508679\pi$
$462$	0	0
$463$	− 151981.i	− 0.708971i	−0.935062	−	0.354485i	$-0.884656\pi$
$463$	− 151981.i	− 0.708971i	0.935062	−	0.354485i	$-0.115344\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 123392.i	− 0.565789i	−0.959151	−	0.282894i	$-0.908705\pi$
$467$	− 123392.i	− 0.565789i	0.959151	−	0.282894i	$-0.0912946\pi$
$468$	0	0
$469$	115122.	0.523376
$470$	0	0
$471$	0	0
$472$	0	0
$473$	155659.	0.695750
$474$	0	0
$475$	− 178479.i	− 0.791044i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 91907.4i	− 0.400571i	−0.979738	−	0.200285i	$-0.935813\pi$
$479$	− 91907.4i	− 0.400571i	0.979738	−	0.200285i	$-0.0641869\pi$
$480$	0	0
$481$	−19109.4	−0.0825957
$482$	0	0
$483$	0	0
$484$	0	0
$485$	99575.6	0.423321
$486$	0	0
$487$	257669.i	1.08644i	0.839591	+	0.543219i	$0.182795\pi$
$487$	257669.i	1.08644i	−0.839591	+	0.543219i	$0.817205\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	366045.i	1.51835i	0.650887	+	0.759175i	$0.274397\pi$
$491$	366045.i	1.51835i	−0.650887	+	0.759175i	$0.725603\pi$
$492$	0	0
$493$	20304.0	0.0835386
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−92583.0	−0.374816
$498$	0	0
$499$	− 116180.i	− 0.466586i	−0.972406	−	0.233293i	$-0.925050\pi$
$499$	− 116180.i	− 0.466586i	0.972406	−	0.233293i	$-0.0749502\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	272217.i	1.07592i	0.842971	+	0.537960i	$0.180805\pi$
$503$	272217.i	1.07592i	−0.842971	+	0.537960i	$0.819195\pi$
$504$	0	0
$505$	114617.	0.449435
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−172673.	−0.666481	−0.333240	−	0.942842i	$-0.608142\pi$
$509$	−172673.	−0.666481	−0.333240	+	0.942842i	$0.608142\pi$
$510$	0	0
$511$	17897.7i	0.0685416i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	173525.i	0.654255i
$516$	0	0
$517$	67824.9	0.253751
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−277263.	−1.02145	−0.510725	−	0.859744i	$-0.670623\pi$
$521$	−277263.	−1.02145	−0.510725	+	0.859744i	$0.670623\pi$
$522$	0	0
$523$	− 105568.i	− 0.385948i	−0.981204	−	0.192974i	$-0.938187\pi$
$523$	− 105568.i	− 0.385948i	0.981204	−	0.192974i	$-0.0618133\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 89524.4i	− 0.322345i
$528$	0	0
$529$	29276.2	0.104617
$530$	0	0
$531$	0	0
$532$	0	0
$533$	39033.4	0.137399
$534$	0	0
$535$	− 225856.i	− 0.789084i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	367415.i	1.26468i
$540$	0	0
$541$	144736.	0.494518	0.247259	−	0.968949i	$-0.420470\pi$
$541$	144736.	0.494518	0.247259	+	0.968949i	$0.420470\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	129201.	0.434982
$546$	0	0
$547$	− 521543.i	− 1.74307i	−0.490332	−	0.871536i	$-0.663124\pi$
$547$	− 521543.i	− 1.74307i	0.490332	−	0.871536i	$-0.336876\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	131840.i	0.434254i
$552$	0	0
$553$	74014.1	0.242027
$554$	0	0
$555$	0	0
$556$	0	0
$557$	65953.4	0.212582	0.106291	−	0.994335i	$-0.466102\pi$
$557$	65953.4	0.212582	0.106291	+	0.994335i	$0.466102\pi$
$558$	0	0
$559$	− 39085.1i	− 0.125080i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 263260.i	− 0.830556i	−0.909695	−	0.415278i	$-0.863684\pi$
$563$	− 263260.i	− 0.830556i	0.909695	−	0.415278i	$-0.136316\pi$
$564$	0	0
$565$	76842.0	0.240714
$566$	0	0
$567$	0	0
$568$	0	0
$569$	574551.	1.77462	0.887308	−	0.461178i	$-0.152573\pi$
$569$	574551.	1.77462	0.887308	+	0.461178i	$0.152573\pi$
$570$	0	0
$571$	257488.i	0.789740i	0.918737	+	0.394870i	$0.129210\pi$
$571$	257488.i	0.789740i	−0.918737	+	0.394870i	$0.870790\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 196878.i	− 0.595472i
$576$	0	0
$577$	−300676.	−0.903124	−0.451562	−	0.892240i	$-0.649133\pi$
$577$	−300676.	−0.903124	−0.451562	+	0.892240i	$0.649133\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14842.9	−0.0439709
$582$	0	0
$583$	414201.i	1.21864i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 587555.i	− 1.70519i	−0.522573	−	0.852594i	$-0.675028\pi$
$587$	− 587555.i	− 1.70519i	0.522573	−	0.852594i	$-0.324972\pi$
$588$	0	0
$589$	581309.	1.67562
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−433537.	−1.23287	−0.616434	−	0.787406i	$-0.711423\pi$
$593$	−433537.	−1.23287	−0.616434	+	0.787406i	$0.711423\pi$
$594$	0	0
$595$	18057.0i	0.0510048i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	180877.i	0.504114i	0.967712	+	0.252057i	$0.0811071\pi$
$599$	180877.i	0.504114i	−0.967712	+	0.252057i	$0.918893\pi$
$600$	0	0
$601$	−9091.93	−0.0251714	−0.0125857	−	0.999921i	$-0.504006\pi$
$601$	−9091.93	−0.0251714	−0.0125857	+	0.999921i	$0.504006\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	237431.	0.648673
$606$	0	0
$607$	219816.i	0.596599i	0.954472	+	0.298300i	$0.0964194\pi$
$607$	219816.i	0.596599i	−0.954472	+	0.298300i	$0.903581\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 17030.4i	− 0.0456186i
$612$	0	0
$613$	486280.	1.29409	0.647047	−	0.762450i	$-0.276004\pi$
$613$	486280.	1.29409	0.647047	+	0.762450i	$0.276004\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	372548.	0.978616	0.489308	−	0.872111i	$-0.337249\pi$
$617$	372548.	0.978616	0.489308	+	0.872111i	$0.337249\pi$
$618$	0	0
$619$	593912.i	1.55003i	0.631942	+	0.775016i	$0.282258\pi$
$619$	593912.i	1.55003i	−0.631942	+	0.775016i	$0.717742\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 50791.9i	− 0.130864i
$624$	0	0
$625$	9888.71	0.0253151
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−30585.1	−0.0773052
$630$	0	0
$631$	− 356435.i	− 0.895204i	−0.894233	−	0.447602i	$-0.852278\pi$
$631$	− 356435.i	− 0.895204i	0.894233	−	0.447602i	$-0.147722\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 301628.i	− 0.748039i
$636$	0	0
$637$	92255.5	0.227360
$638$	0	0
$639$	0	0
$640$	0	0
$641$	509005.	1.23881	0.619407	−	0.785070i	$-0.287373\pi$
$641$	509005.	1.23881	0.619407	+	0.785070i	$0.287373\pi$
$642$	0	0
$643$	280745.i	0.679031i	0.940600	+	0.339516i	$0.110263\pi$
$643$	280745.i	0.679031i	−0.940600	+	0.339516i	$0.889737\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 636655.i	− 1.52088i	−0.649407	−	0.760441i	$-0.724983\pi$
$647$	− 636655.i	− 1.52088i	0.649407	−	0.760441i	$-0.275017\pi$
$648$	0	0
$649$	82516.4	0.195907
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−649824.	−1.52395	−0.761973	−	0.647609i	$-0.775769\pi$
$653$	−649824.	−1.52395	−0.761973	+	0.647609i	$0.775769\pi$
$654$	0	0
$655$	− 343531.i	− 0.800725i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	412172.i	0.949090i	0.880231	+	0.474545i	$0.157387\pi$
$659$	412172.i	0.949090i	−0.880231	+	0.474545i	$0.842613\pi$
$660$	0	0
$661$	600902.	1.37531	0.687655	−	0.726038i	$-0.258640\pi$
$661$	600902.	1.37531	0.687655	+	0.726038i	$0.258640\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−117249.	−0.265135
$666$	0	0
$667$	145430.i	0.326892i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 470482.i	− 1.04496i
$672$	0	0
$673$	210251.	0.464203	0.232102	−	0.972692i	$-0.425440\pi$
$673$	210251.	0.464203	0.232102	+	0.972692i	$0.425440\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	629967.	1.37449	0.687243	−	0.726428i	$-0.258821\pi$
$677$	629967.	1.37449	0.687243	+	0.726428i	$0.258821\pi$
$678$	0	0
$679$	111047.i	0.240862i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 32843.1i	− 0.0704048i	−0.999380	−	0.0352024i	$-0.988792\pi$
$683$	− 32843.1i	− 0.0704048i	0.999380	−	0.0352024i	$-0.0112076\pi$
$684$	0	0
$685$	438589.	0.934709
$686$	0	0
$687$	0	0
$688$	0	0
$689$	104003.	0.219083
$690$	0	0
$691$	301198.i	0.630807i	0.948958	+	0.315403i	$0.102140\pi$
$691$	301198.i	0.630807i	−0.948958	+	0.315403i	$0.897860\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 28836.1i	− 0.0596989i
$696$	0	0
$697$	62474.0	0.128598
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−336422.	−0.684618	−0.342309	−	0.939587i	$-0.611209\pi$
$701$	−336422.	−0.684618	−0.342309	+	0.939587i	$0.611209\pi$
$702$	0	0
$703$	− 198598.i	− 0.401851i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	127822.i	0.255721i
$708$	0	0
$709$	338659.	0.673706	0.336853	−	0.941557i	$-0.390637\pi$
$709$	338659.	0.673706	0.336853	+	0.941557i	$0.390637\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	641233.	1.26135
$714$	0	0
$715$	− 115575.i	− 0.226074i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	908694.i	1.75776i	0.477042	+	0.878881i	$0.341709\pi$
$719$	908694.i	1.75776i	−0.477042	+	0.878881i	$0.658291\pi$
$720$	0	0
$721$	−193516.	−0.372259
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−114270.	−0.217398
$726$	0	0
$727$	− 929452.i	− 1.75856i	−0.476302	−	0.879282i	$-0.658023\pi$
$727$	− 929452.i	− 1.75856i	0.476302	−	0.879282i	$-0.341977\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 62556.6i	− 0.117068i
$732$	0	0
$733$	442345.	0.823290	0.411645	−	0.911344i	$-0.364954\pi$
$733$	442345.	0.823290	0.411645	+	0.911344i	$0.364954\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.17934e6	−2.17123
$738$	0	0
$739$	45905.6i	0.0840576i	0.999116	+	0.0420288i	$0.0133821\pi$
$739$	45905.6i	0.0840576i	−0.999116	+	0.0420288i	$0.986618\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	804334.i	1.45700i	0.685047	+	0.728499i	$0.259782\pi$
$743$	804334.i	1.45700i	−0.685047	+	0.728499i	$0.740218\pi$
$744$	0	0
$745$	−76976.3	−0.138690
$746$	0	0
$747$	0	0
$748$	0	0
$749$	251875.	0.448975
$750$	0	0
$751$	− 379318.i	− 0.672548i	−0.941764	−	0.336274i	$-0.890833\pi$
$751$	− 379318.i	− 0.672548i	0.941764	−	0.336274i	$-0.109167\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 72377.8i	− 0.126973i
$756$	0	0
$757$	1.06087e6	1.85126	0.925632	−	0.378424i	$-0.123534\pi$
$757$	1.06087e6	1.85126	0.925632	+	0.378424i	$0.123534\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	212245.	0.366494	0.183247	−	0.983067i	$-0.441339\pi$
$761$	212245.	0.366494	0.183247	+	0.983067i	$0.441339\pi$
$762$	0	0
$763$	144085.i	0.247497i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 20719.3i	− 0.0352196i
$768$	0	0
$769$	375985.	0.635796	0.317898	−	0.948125i	$-0.397023\pi$
$769$	375985.	0.635796	0.317898	+	0.948125i	$0.397023\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−177912.	−0.297747	−0.148873	−	0.988856i	$-0.547565\pi$
$773$	−177912.	−0.297747	−0.148873	+	0.988856i	$0.547565\pi$
$774$	0	0
$775$	503840.i	0.838859i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	405663.i	0.668483i
$780$	0	0
$781$	948445.	1.55493
$782$	0	0
$783$	0	0
$784$	0	0
$785$	484475.	0.786198
$786$	0	0
$787$	− 1.02528e6i	− 1.65535i	−0.561205	−	0.827677i	$-0.689662\pi$
$787$	− 1.02528e6i	− 1.65535i	0.561205	−	0.827677i	$-0.310338\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	85694.6i	0.136962i
$792$	0	0
$793$	−118135.	−0.187859
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−609217.	−0.959082	−0.479541	−	0.877520i	$-0.659197\pi$
$797$	−609217.	−0.959082	−0.479541	+	0.877520i	$0.659197\pi$
$798$	0	0
$799$	− 27257.6i	− 0.0426966i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 183348.i	− 0.284345i
$804$	0	0
$805$	−129336.	−0.199585
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.02958e6	−1.57313	−0.786566	−	0.617507i	$-0.788143\pi$
$809$	−1.02958e6	−1.57313	−0.786566	+	0.617507i	$0.788143\pi$
$810$	0	0
$811$	397632.i	0.604561i	0.953219	+	0.302280i	$0.0977479\pi$
$811$	397632.i	0.604561i	−0.953219	+	0.302280i	$0.902252\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	78512.0i	0.118201i
$816$	0	0
$817$	406199.	0.608548
$818$	0	0
$819$	0	0
$820$	0	0
$821$	444223.	0.659045	0.329523	−	0.944148i	$-0.393112\pi$
$821$	444223.	0.659045	0.329523	+	0.944148i	$0.393112\pi$
$822$	0	0
$823$	448550.i	0.662234i	0.943590	+	0.331117i	$0.107425\pi$
$823$	448550.i	0.662234i	−0.943590	+	0.331117i	$0.892575\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 472019.i	− 0.690158i	−0.938574	−	0.345079i	$-0.887852\pi$
$827$	− 472019.i	− 0.690158i	0.938574	−	0.345079i	$-0.112148\pi$
$828$	0	0
$829$	262562.	0.382053	0.191026	−	0.981585i	$-0.438818\pi$
$829$	262562.	0.382053	0.191026	+	0.981585i	$0.438818\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	147657.	0.212797
$834$	0	0
$835$	413315.i	0.592800i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	680311.i	0.966459i	0.875494	+	0.483230i	$0.160536\pi$
$839$	680311.i	0.966459i	−0.875494	+	0.483230i	$0.839464\pi$
$840$	0	0
$841$	−622872.	−0.880657
$842$	0	0
$843$	0	0
$844$	0	0
$845$	405716.	0.568210
$846$	0	0
$847$	264784.i	0.369084i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 219071.i	− 0.302500i
$852$	0	0
$853$	827500.	1.13729	0.568643	−	0.822584i	$-0.307468\pi$
$853$	827500.	1.13729	0.568643	+	0.822584i	$0.307468\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	609260.	0.829547	0.414773	−	0.909925i	$-0.363861\pi$
$857$	609260.	0.829547	0.414773	+	0.909925i	$0.363861\pi$
$858$	0	0
$859$	− 378716.i	− 0.513247i	−0.966511	−	0.256624i	$-0.917390\pi$
$859$	− 378716.i	− 0.513247i	0.966511	−	0.256624i	$-0.0826101\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 109339.i	− 0.146810i	−0.997302	−	0.0734050i	$-0.976613\pi$
$863$	− 109339.i	− 0.146810i	0.997302	−	0.0734050i	$-0.0233866\pi$
$864$	0	0
$865$	760594.	1.01653
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−758220.	−1.00405
$870$	0	0
$871$	296125.i	0.390336i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 263112.i	− 0.343656i
$876$	0	0
$877$	902691.	1.17365	0.586827	−	0.809712i	$-0.300377\pi$
$877$	902691.	1.17365	0.586827	+	0.809712i	$0.300377\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.33896e6	−1.72510	−0.862552	−	0.505969i	$-0.831135\pi$
$881$	−1.33896e6	−1.72510	−0.862552	+	0.505969i	$0.831135\pi$
$882$	0	0
$883$	− 914655.i	− 1.17310i	−0.809912	−	0.586551i	$-0.800485\pi$
$883$	− 914655.i	− 1.17310i	0.809912	−	0.586551i	$-0.199515\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 977441.i	− 1.24235i	−0.783673	−	0.621174i	$-0.786656\pi$
$887$	− 977441.i	− 1.24235i	0.783673	−	0.621174i	$-0.213344\pi$
$888$	0	0
$889$	336377.	0.425621
$890$	0	0
$891$	0	0
$892$	0	0
$893$	176992.	0.221947
$894$	0	0
$895$	783443.i	0.978051i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 372178.i	− 0.460502i
$900$	0	0
$901$	166460.	0.205050
$902$	0	0
$903$	0	0
$904$	0	0
$905$	501412.	0.612206
$906$	0	0
$907$	− 848344.i	− 1.03124i	−0.856819	−	0.515618i	$-0.827563\pi$
$907$	− 848344.i	− 1.03124i	0.856819	−	0.515618i	$-0.172437\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	931493.i	1.12239i	0.827684	+	0.561194i	$0.189658\pi$
$911$	931493.i	1.12239i	−0.827684	+	0.561194i	$0.810342\pi$
$912$	0	0
$913$	152054.	0.182414
$914$	0	0
$915$	0	0
$916$	0	0
$917$	383107.	0.455598
$918$	0	0
$919$	− 599481.i	− 0.709814i	−0.934902	−	0.354907i	$-0.884513\pi$
$919$	− 599481.i	− 0.709814i	0.934902	−	0.354907i	$-0.115487\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 238148.i	− 0.279540i
$924$	0	0
$925$	172132.	0.201177
$926$	0	0
$927$	0	0
$928$	0	0
$929$	278737.	0.322971	0.161486	−	0.986875i	$-0.448371\pi$
$929$	278737.	0.322971	0.161486	+	0.986875i	$0.448371\pi$
$930$	0	0
$931$	958784.i	1.10617i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 184980.i	− 0.211593i
$936$	0	0
$937$	−20618.2	−0.0234840	−0.0117420	−	0.999931i	$-0.503738\pi$
$937$	−20618.2	−0.0234840	−0.0117420	+	0.999931i	$0.503738\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.14309e6	−1.29093	−0.645465	−	0.763790i	$-0.723337\pi$
$941$	−1.14309e6	−1.29093	−0.645465	+	0.763790i	$0.723337\pi$
$942$	0	0
$943$	447480.i	0.503211i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 936609.i	− 1.04438i	−0.852829	−	0.522190i	$-0.825115\pi$
$947$	− 936609.i	− 1.04438i	0.852829	−	0.522190i	$-0.174885\pi$
$948$	0	0
$949$	−46037.5	−0.0511187
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.17574e6	−1.29457	−0.647284	−	0.762249i	$-0.724095\pi$
$953$	−1.17574e6	−1.29457	−0.647284	+	0.762249i	$0.724095\pi$
$954$	0	0
$955$	298973.i	0.327812i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	489117.i	0.531833i
$960$	0	0
$961$	−717491.	−0.776908
$962$	0	0
$963$	0	0
$964$	0	0
$965$	476607.	0.511806
$966$	0	0
$967$	780606.i	0.834793i	0.908724	+	0.417396i	$0.137057\pi$
$967$	780606.i	0.834793i	−0.908724	+	0.417396i	$0.862943\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	299591.i	0.317753i	0.987298	+	0.158877i	$0.0507872\pi$
$971$	299591.i	0.317753i	−0.987298	+	0.158877i	$0.949213\pi$
$972$	0	0
$973$	32158.1	0.0339676
$974$	0	0
$975$	0	0
$976$	0	0
$977$	730501.	0.765300	0.382650	−	0.923893i	$-0.375012\pi$
$977$	730501.	0.765300	0.382650	+	0.923893i	$0.375012\pi$
$978$	0	0
$979$	520326.i	0.542888i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.78186e6i	− 1.84403i	−0.387159	−	0.922013i	$-0.626544\pi$
$983$	− 1.78186e6i	− 1.84403i	0.387159	−	0.922013i	$-0.373456\pi$
$984$	0	0
$985$	−820974.	−0.846169
$986$	0	0
$987$	0	0
$988$	0	0
$989$	448072.	0.458095
$990$	0	0
$991$	− 1.06455e6i	− 1.08397i	−0.840388	−	0.541986i	$-0.817673\pi$
$991$	− 1.06455e6i	− 1.08397i	0.840388	−	0.541986i	$-0.182327\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	913620.i	0.922825i
$996$	0	0
$997$	996516.	1.00252	0.501261	−	0.865296i	$-0.332870\pi$
$997$	996516.	1.00252	0.501261	+	0.865296i	$0.332870\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.g.e.1135.1	yes	4
3.2	odd	2		1296.5.g.c.1135.3	✓	4
4.3	odd	2	inner	1296.5.g.e.1135.2	yes	4
12.11	even	2		1296.5.g.c.1135.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1296.5.g.c.1135.3	✓	4	3.2	odd	2
1296.5.g.c.1135.4	yes	4	12.11	even	2
1296.5.g.e.1135.1	yes	4	1.1	even	1	trivial
1296.5.g.e.1135.2	yes	4	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1296.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15.2213 q^{5} -16.9749i q^{7} +173.895i q^{11} +43.6639 q^{13} +69.8853 q^{17} +453.786i q^{19} +500.565i q^{23} -393.312 q^{25} +290.533 q^{29} -1281.02i q^{31} +258.380i q^{35} -437.648 q^{37} +893.951 q^{41} -895.133i q^{43} -390.033i q^{47} +2112.85 q^{49} +2381.90 q^{53} -2646.91i q^{55} -474.518i q^{59} -2705.55 q^{61} -664.623 q^{65} +6781.91i q^{67} -5454.12i q^{71} -1054.36 q^{73} +2951.85 q^{77} +4360.21i q^{79} -874.402i q^{83} -1063.75 q^{85} +2992.18 q^{89} -741.191i q^{91} -6907.22i q^{95} -6541.85 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 48 q^{5} - 152 q^{13} - 156 q^{17} + 1040 q^{25} - 1560 q^{29} + 536 q^{37} - 2304 q^{41} - 9188 q^{49} - 2232 q^{53} + 3224 q^{61} - 10716 q^{65} - 7484 q^{73} - 5832 q^{77} - 13728 q^{85} + 20244 q^{89}+ \cdots - 8528 q^{97}+O(q^{100}) \) 4 * q + 48 * q^5 - 152 * q^13 - 156 * q^17 + 1040 * q^25 - 1560 * q^29 + 536 * q^37 - 2304 * q^41 - 9188 * q^49 - 2232 * q^53 + 3224 * q^61 - 10716 * q^65 - 7484 * q^73 - 5832 * q^77 - 13728 * q^85 + 20244 * q^89 - 8528 * q^97

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