LMFDB - Embedded newform 1296.3.e.g.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(161,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1296.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.3.e.g.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615i q^{5} +8.34847 q^{7} +O(q^{10})$	$q+5.19615i q^{5} +8.34847 q^{7} -0.953512i q^{11} -9.69694 q^{13} +18.8776i q^{17} +24.6969 q^{19} +0.953512i q^{23} -2.00000 q^{25} +13.6814i q^{29} -3.04541 q^{31} +43.3799i q^{35} +46.6969 q^{37} +10.9172i q^{41} -45.0454 q^{43} -45.2869i q^{47} +20.6969 q^{49} +94.3879i q^{53} +4.95459 q^{55} -18.7813i q^{59} +13.0908 q^{61} -50.3868i q^{65} -75.0454 q^{67} +18.0204i q^{71} -7.90918 q^{73} -7.96036i q^{77} +43.7423 q^{79} +130.332i q^{83} -98.0908 q^{85} -145.300i q^{89} -80.9546 q^{91} +128.329i q^{95} -109.879 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 20 q^{13} + 40 q^{19} - 8 q^{25} + 76 q^{31} + 128 q^{37} - 92 q^{43} + 24 q^{49} + 108 q^{55} - 124 q^{61} - 212 q^{67} - 208 q^{73} + 28 q^{79} - 216 q^{85} - 412 q^{91} - 28 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 20 * q^13 + 40 * q^19 - 8 * q^25 + 76 * q^31 + 128 * q^37 - 92 * q^43 + 24 * q^49 + 108 * q^55 - 124 * q^61 - 212 * q^67 - 208 * q^73 + 28 * q^79 - 216 * q^85 - 412 * q^91 - 28 * q^97

Display 100 coefficients 10 coefficients

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
$5$	5.19615i	1.03923i	0.854400	+	0.519615i	$0.173925\pi$
$5$	5.19615i	1.03923i	−0.854400	+	0.519615i	$0.826075\pi$
$6$	0	0
$7$	8.34847	1.19264	0.596319	−	0.802747i	$-0.296629\pi$
$7$	8.34847	1.19264	0.596319	+	0.802747i	$0.296629\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 0.953512i	− 0.0866829i	−0.999060	−	0.0433414i	$-0.986200\pi$
$11$	− 0.953512i	− 0.0866829i	0.999060	−	0.0433414i	$-0.0138003\pi$
$12$	0	0
$13$	−9.69694	−0.745918	−0.372959	−	0.927848i	$-0.621657\pi$
$13$	−9.69694	−0.745918	−0.372959	+	0.927848i	$0.621657\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.8776i	1.11045i	0.831701	+	0.555223i	$0.187367\pi$
$17$	18.8776i	1.11045i	−0.831701	+	0.555223i	$0.812633\pi$
$18$	0	0
$19$	24.6969	1.29984	0.649919	−	0.760003i	$-0.274803\pi$
$19$	24.6969	1.29984	0.649919	+	0.760003i	$0.274803\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.953512i	0.0414570i	0.999785	+	0.0207285i	$0.00659856\pi$
$23$	0.953512i	0.0414570i	−0.999785	+	0.0207285i	$0.993401\pi$
$24$	0	0
$25$	−2.00000	−0.0800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	13.6814i	0.471774i	0.971781	+	0.235887i	$0.0757995\pi$
$29$	13.6814i	0.471774i	−0.971781	+	0.235887i	$0.924201\pi$
$30$	0	0
$31$	−3.04541	−0.0982390	−0.0491195	−	0.998793i	$-0.515642\pi$
$31$	−3.04541	−0.0982390	−0.0491195	+	0.998793i	$0.515642\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	43.3799i	1.23943i
$36$	0	0
$37$	46.6969	1.26208	0.631040	−	0.775751i	$-0.282628\pi$
$37$	46.6969	1.26208	0.631040	+	0.775751i	$0.282628\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.9172i	0.266274i	0.991098	+	0.133137i	$0.0425050\pi$
$41$	10.9172i	0.266274i	−0.991098	+	0.133137i	$0.957495\pi$
$42$	0	0
$43$	−45.0454	−1.04757	−0.523784	−	0.851851i	$-0.675480\pi$
$43$	−45.0454	−1.04757	−0.523784	+	0.851851i	$0.675480\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 45.2869i	− 0.963552i	−0.876294	−	0.481776i	$-0.839992\pi$
$47$	− 45.2869i	− 0.963552i	0.876294	−	0.481776i	$-0.160008\pi$
$48$	0	0
$49$	20.6969	0.422386
$50$	0	0
$51$	0	0
$52$	0	0
$53$	94.3879i	1.78090i	0.455077	+	0.890452i	$0.349612\pi$
$53$	94.3879i	1.78090i	−0.455077	+	0.890452i	$0.650388\pi$
$54$	0	0
$55$	4.95459	0.0900835
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 18.7813i	− 0.318326i	−0.987252	−	0.159163i	$-0.949120\pi$
$59$	− 18.7813i	− 0.318326i	0.987252	−	0.159163i	$-0.0508796\pi$
$60$	0	0
$61$	13.0908	0.214604	0.107302	−	0.994226i	$-0.465779\pi$
$61$	13.0908	0.214604	0.107302	+	0.994226i	$0.465779\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 50.3868i	− 0.775181i
$66$	0	0
$67$	−75.0454	−1.12008	−0.560040	−	0.828465i	$-0.689214\pi$
$67$	−75.0454	−1.12008	−0.560040	+	0.828465i	$0.689214\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	18.0204i	0.253808i	0.991915	+	0.126904i	$0.0405041\pi$
$71$	18.0204i	0.253808i	−0.991915	+	0.126904i	$0.959496\pi$
$72$	0	0
$73$	−7.90918	−0.108345	−0.0541725	−	0.998532i	$-0.517252\pi$
$73$	−7.90918	−0.108345	−0.0541725	+	0.998532i	$0.517252\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 7.96036i	− 0.103381i
$78$	0	0
$79$	43.7423	0.553701	0.276850	−	0.960913i	$-0.410709\pi$
$79$	43.7423	0.553701	0.276850	+	0.960913i	$0.410709\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	130.332i	1.57027i	0.619325	+	0.785135i	$0.287406\pi$
$83$	130.332i	1.57027i	−0.619325	+	0.785135i	$0.712594\pi$
$84$	0	0
$85$	−98.0908	−1.15401
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 145.300i	− 1.63258i	−0.577642	−	0.816290i	$-0.696027\pi$
$89$	− 145.300i	− 1.63258i	0.577642	−	0.816290i	$-0.303973\pi$
$90$	0	0
$91$	−80.9546	−0.889611
$92$	0	0
$93$	0	0
$94$	0	0
$95$	128.329i	1.35083i
$96$	0	0
$97$	−109.879	−1.13277	−0.566384	−	0.824141i	$-0.691658\pi$
$97$	−109.879	−1.13277	−0.566384	+	0.824141i	$0.691658\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	147.539i	1.46078i	0.683030	+	0.730391i	$0.260662\pi$
$101$	147.539i	1.46078i	−0.683030	+	0.730391i	$0.739338\pi$
$102$	0	0
$103$	103.136	1.00132	0.500661	−	0.865643i	$-0.333090\pi$
$103$	103.136	1.00132	0.500661	+	0.865643i	$0.333090\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	36.0408i	0.336830i	0.985716	+	0.168415i	$0.0538649\pi$
$107$	36.0408i	0.336830i	−0.985716	+	0.168415i	$0.946135\pi$
$108$	0	0
$109$	−148.272	−1.36030	−0.680149	−	0.733074i	$-0.738085\pi$
$109$	−148.272	−1.36030	−0.680149	+	0.733074i	$0.738085\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	171.088i	1.51405i	0.653385	+	0.757025i	$0.273348\pi$
$113$	171.088i	1.51405i	−0.653385	+	0.757025i	$0.726652\pi$
$114$	0	0
$115$	−4.95459	−0.0430834
$116$	0	0
$117$	0	0
$118$	0	0
$119$	157.599i	1.32436i
$120$	0	0
$121$	120.091	0.992486
$122$	0	0
$123$	0	0
$124$	0	0
$125$	119.512i	0.956092i
$126$	0	0
$127$	78.0908	0.614888	0.307444	−	0.951566i	$-0.400526\pi$
$127$	78.0908	0.614888	0.307444	+	0.951566i	$0.400526\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	234.063i	1.78674i	0.449323	+	0.893369i	$0.351665\pi$
$131$	234.063i	1.78674i	−0.449323	+	0.893369i	$0.648335\pi$
$132$	0	0
$133$	206.182	1.55024
$134$	0	0
$135$	0	0
$136$	0	0
$137$	149.831i	1.09366i	0.837244	+	0.546829i	$0.184165\pi$
$137$	149.831i	1.09366i	−0.837244	+	0.546829i	$0.815835\pi$
$138$	0	0
$139$	84.5301	0.608130	0.304065	−	0.952651i	$-0.401656\pi$
$139$	84.5301	0.608130	0.304065	+	0.952651i	$0.401656\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.24614i	0.0646584i
$144$	0	0
$145$	−71.0908	−0.490281
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 115.505i	− 0.775200i	−0.921828	−	0.387600i	$-0.873304\pi$
$149$	− 115.505i	− 0.775200i	0.921828	−	0.387600i	$-0.126696\pi$
$150$	0	0
$151$	64.7730	0.428960	0.214480	−	0.976728i	$-0.431194\pi$
$151$	64.7730	0.428960	0.214480	+	0.976728i	$0.431194\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 15.8244i	− 0.102093i
$156$	0	0
$157$	−20.8184	−0.132601	−0.0663005	−	0.997800i	$-0.521120\pi$
$157$	−20.8184	−0.132601	−0.0663005	+	0.997800i	$0.521120\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.96036i	0.0494433i
$162$	0	0
$163$	−133.060	−0.816320	−0.408160	−	0.912910i	$-0.633829\pi$
$163$	−133.060	−0.816320	−0.408160	+	0.912910i	$0.633829\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 294.510i	− 1.76353i	−0.471688	−	0.881765i	$-0.656355\pi$
$167$	− 294.510i	− 1.76353i	0.471688	−	0.881765i	$-0.343645\pi$
$168$	0	0
$169$	−74.9694	−0.443606
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 69.2644i	− 0.400372i	−0.979758	−	0.200186i	$-0.935845\pi$
$173$	− 69.2644i	− 0.400372i	0.979758	−	0.200186i	$-0.0641546\pi$
$174$	0	0
$175$	−16.6969	−0.0954111
$176$	0	0
$177$	0	0
$178$	0	0
$179$	47.4829i	0.265268i	0.991165	+	0.132634i	$0.0423435\pi$
$179$	47.4829i	0.265268i	−0.991165	+	0.132634i	$0.957657\pi$
$180$	0	0
$181$	242.879	1.34187	0.670935	−	0.741516i	$-0.265893\pi$
$181$	242.879	1.34187	0.670935	+	0.741516i	$0.265893\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	242.644i	1.31159i
$186$	0	0
$187$	18.0000	0.0962567
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 7.53177i	− 0.0394333i	−0.999806	−	0.0197167i	$-0.993724\pi$
$191$	− 7.53177i	− 0.0394333i	0.999806	−	0.0197167i	$-0.00627642\pi$
$192$	0	0
$193$	345.454	1.78992	0.894959	−	0.446149i	$-0.147205\pi$
$193$	345.454	1.78992	0.894959	+	0.446149i	$0.147205\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 77.2247i	− 0.392004i	−0.980604	−	0.196002i	$-0.937204\pi$
$197$	− 77.2247i	− 0.392004i	0.980604	−	0.196002i	$-0.0627959\pi$
$198$	0	0
$199$	−153.485	−0.771280	−0.385640	−	0.922649i	$-0.626019\pi$
$199$	−153.485	−0.771280	−0.385640	+	0.922649i	$0.626019\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	114.219i	0.562655i
$204$	0	0
$205$	−56.7276	−0.276720
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 23.5488i	− 0.112674i
$210$	0	0
$211$	51.5607	0.244364	0.122182	−	0.992508i	$-0.461011\pi$
$211$	51.5607	0.244364	0.122182	+	0.992508i	$0.461011\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 234.063i	− 1.08866i
$216$	0	0
$217$	−25.4245	−0.117164
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 183.055i	− 0.828302i
$222$	0	0
$223$	−313.227	−1.40461	−0.702303	−	0.711878i	$-0.747845\pi$
$223$	−313.227	−1.40461	−0.702303	+	0.711878i	$0.747845\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	76.2712i	0.335997i	0.985787	+	0.167998i	$0.0537303\pi$
$227$	76.2712i	0.335997i	−0.985787	+	0.167998i	$0.946270\pi$
$228$	0	0
$229$	−121.545	−0.530764	−0.265382	−	0.964143i	$-0.585498\pi$
$229$	−121.545	−0.530764	−0.265382	+	0.964143i	$0.585498\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	151.021i	0.648157i	0.946030	+	0.324079i	$0.105054\pi$
$233$	151.021i	0.648157i	−0.946030	+	0.324079i	$0.894946\pi$
$234$	0	0
$235$	235.318	1.00135
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 87.7133i	− 0.367001i	−0.983020	−	0.183501i	$-0.941257\pi$
$239$	− 87.7133i	− 0.367001i	0.983020	−	0.183501i	$-0.0587430\pi$
$240$	0	0
$241$	201.788	0.837294	0.418647	−	0.908149i	$-0.362505\pi$
$241$	201.788	0.837294	0.418647	+	0.908149i	$0.362505\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	107.544i	0.438957i
$246$	0	0
$247$	−239.485	−0.969574
$248$	0	0
$249$	0	0
$250$	0	0
$251$	52.6261i	0.209666i	0.994490	+	0.104833i	$0.0334307\pi$
$251$	52.6261i	0.209666i	−0.994490	+	0.104833i	$0.966569\pi$
$252$	0	0
$253$	0.909185	0.00359362
$254$	0	0
$255$	0	0
$256$	0	0
$257$	80.7065i	0.314033i	0.987596	+	0.157017i	$0.0501876\pi$
$257$	80.7065i	0.314033i	−0.987596	+	0.157017i	$0.949812\pi$
$258$	0	0
$259$	389.848	1.50520
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 463.743i	− 1.76328i	−0.471920	−	0.881641i	$-0.656439\pi$
$263$	− 463.743i	− 1.76328i	0.471920	−	0.881641i	$-0.343561\pi$
$264$	0	0
$265$	−490.454	−1.85077
$266$	0	0
$267$	0	0
$268$	0	0
$269$	43.4762i	0.161622i	0.996729	+	0.0808109i	$0.0257510\pi$
$269$	43.4762i	0.161622i	−0.996729	+	0.0808109i	$0.974249\pi$
$270$	0	0
$271$	342.636	1.26434	0.632169	−	0.774830i	$-0.282165\pi$
$271$	342.636	1.26434	0.632169	+	0.774830i	$0.282165\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.90702i	0.00693463i
$276$	0	0
$277$	−49.0000	−0.176895	−0.0884477	−	0.996081i	$-0.528191\pi$
$277$	−49.0000	−0.176895	−0.0884477	+	0.996081i	$0.528191\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 20.6450i	− 0.0734697i	−0.999325	−	0.0367349i	$-0.988304\pi$
$281$	− 20.6450i	− 0.0734697i	0.999325	−	0.0367349i	$-0.0116957\pi$
$282$	0	0
$283$	−53.4087	−0.188723	−0.0943616	−	0.995538i	$-0.530081\pi$
$283$	−53.4087	−0.188723	−0.0943616	+	0.995538i	$0.530081\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	91.1421i	0.317568i
$288$	0	0
$289$	−67.3633	−0.233091
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.9239i	0.0509349i	0.999676	+	0.0254674i	$0.00810741\pi$
$293$	14.9239i	0.0509349i	−0.999676	+	0.0254674i	$0.991893\pi$
$294$	0	0
$295$	97.5903	0.330815
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 9.24614i	− 0.0309236i
$300$	0	0
$301$	−376.060	−1.24937
$302$	0	0
$303$	0	0
$304$	0	0
$305$	68.0219i	0.223023i
$306$	0	0
$307$	−65.9092	−0.214688	−0.107344	−	0.994222i	$-0.534235\pi$
$307$	−65.9092	−0.214688	−0.107344	+	0.994222i	$0.534235\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 250.176i	− 0.804425i	−0.915546	−	0.402213i	$-0.868241\pi$
$311$	− 250.176i	− 0.804425i	0.915546	−	0.402213i	$-0.131759\pi$
$312$	0	0
$313$	−426.394	−1.36228	−0.681140	−	0.732153i	$-0.738516\pi$
$313$	−426.394	−1.36228	−0.681140	+	0.732153i	$0.738516\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 463.979i	− 1.46366i	−0.681489	−	0.731829i	$-0.738667\pi$
$317$	− 463.979i	− 1.46366i	0.681489	−	0.731829i	$-0.261333\pi$
$318$	0	0
$319$	13.0454	0.0408947
$320$	0	0
$321$	0	0
$322$	0	0
$323$	466.219i	1.44340i
$324$	0	0
$325$	19.3939	0.0596735
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 378.077i	− 1.14917i
$330$	0	0
$331$	−472.803	−1.42841	−0.714203	−	0.699938i	$-0.753211\pi$
$331$	−472.803	−1.42841	−0.714203	+	0.699938i	$0.753211\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 389.947i	− 1.16402i
$336$	0	0
$337$	305.606	0.906843	0.453422	−	0.891296i	$-0.350203\pi$
$337$	305.606	0.906843	0.453422	+	0.891296i	$0.350203\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.90383i	0.00851564i
$342$	0	0
$343$	−236.287	−0.688884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	133.675i	0.385229i	0.981274	+	0.192615i	$0.0616967\pi$
$347$	133.675i	0.385229i	−0.981274	+	0.192615i	$0.938303\pi$
$348$	0	0
$349$	−98.7571	−0.282972	−0.141486	−	0.989940i	$-0.545188\pi$
$349$	−98.7571	−0.282972	−0.141486	+	0.989940i	$0.545188\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 326.115i	− 0.923839i	−0.886922	−	0.461919i	$-0.847161\pi$
$353$	− 326.115i	− 0.923839i	0.886922	−	0.461919i	$-0.152839\pi$
$354$	0	0
$355$	−93.6367	−0.263765
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 418.736i	− 1.16639i	−0.812331	−	0.583197i	$-0.801801\pi$
$359$	− 418.736i	− 1.16639i	0.812331	−	0.583197i	$-0.198199\pi$
$360$	0	0
$361$	248.939	0.689581
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 41.0973i	− 0.112595i
$366$	0	0
$367$	−187.227	−0.510155	−0.255078	−	0.966921i	$-0.582101\pi$
$367$	−187.227	−0.510155	−0.255078	+	0.966921i	$0.582101\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	787.995i	2.12398i
$372$	0	0
$373$	451.030	1.20919	0.604597	−	0.796531i	$-0.293334\pi$
$373$	451.030	1.20919	0.604597	+	0.796531i	$0.293334\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 132.668i	− 0.351905i
$378$	0	0
$379$	489.666	1.29200	0.645998	−	0.763339i	$-0.276442\pi$
$379$	489.666	1.29200	0.645998	+	0.763339i	$0.276442\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	103.056i	0.269076i	0.990908	+	0.134538i	$0.0429550\pi$
$383$	103.056i	0.269076i	−0.990908	+	0.134538i	$0.957045\pi$
$384$	0	0
$385$	41.3633	0.107437
$386$	0	0
$387$	0	0
$388$	0	0
$389$	34.2734i	0.0881064i	0.999029	+	0.0440532i	$0.0140271\pi$
$389$	34.2734i	0.0881064i	−0.999029	+	0.0440532i	$0.985973\pi$
$390$	0	0
$391$	−18.0000	−0.0460358
$392$	0	0
$393$	0	0
$394$	0	0
$395$	227.292i	0.575423i
$396$	0	0
$397$	8.27245	0.0208374	0.0104187	−	0.999946i	$-0.496684\pi$
$397$	8.27245	0.0208374	0.0104187	+	0.999946i	$0.496684\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 414.117i	− 1.03271i	−0.856374	−	0.516356i	$-0.827288\pi$
$401$	− 414.117i	− 1.03271i	0.856374	−	0.516356i	$-0.172712\pi$
$402$	0	0
$403$	29.5311	0.0732782
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 44.5261i	− 0.109401i
$408$	0	0
$409$	−326.212	−0.797585	−0.398792	−	0.917041i	$-0.630571\pi$
$409$	−326.212	−0.797585	−0.398792	+	0.917041i	$0.630571\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 156.795i	− 0.379648i
$414$	0	0
$415$	−677.227	−1.63187
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 540.775i	− 1.29063i	−0.763915	−	0.645317i	$-0.776725\pi$
$419$	− 540.775i	− 1.29063i	0.763915	−	0.645317i	$-0.223275\pi$
$420$	0	0
$421$	283.697	0.673864	0.336932	−	0.941529i	$-0.390611\pi$
$421$	283.697	0.673864	0.336932	+	0.941529i	$0.390611\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 37.7552i	− 0.0888357i
$426$	0	0
$427$	109.288	0.255944
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 257.429i	− 0.597282i	−0.954365	−	0.298641i	$-0.903467\pi$
$431$	− 257.429i	− 0.597282i	0.954365	−	0.298641i	$-0.0965334\pi$
$432$	0	0
$433$	476.272	1.09994	0.549968	−	0.835186i	$-0.314640\pi$
$433$	476.272	1.09994	0.549968	+	0.835186i	$0.314640\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.5488i	0.0538875i
$438$	0	0
$439$	557.863	1.27076	0.635379	−	0.772200i	$-0.280844\pi$
$439$	557.863	1.27076	0.635379	+	0.772200i	$0.280844\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 831.847i	− 1.87776i	−0.344248	−	0.938879i	$-0.611866\pi$
$443$	− 831.847i	− 1.87776i	0.344248	−	0.938879i	$-0.388134\pi$
$444$	0	0
$445$	754.999	1.69663
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 729.927i	− 1.62567i	−0.582492	−	0.812836i	$-0.697922\pi$
$449$	− 729.927i	− 1.62567i	0.582492	−	0.812836i	$-0.302078\pi$
$450$	0	0
$451$	10.4097	0.0230814
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 420.652i	− 0.924511i
$456$	0	0
$457$	709.636	1.55281	0.776407	−	0.630232i	$-0.217040\pi$
$457$	709.636	1.55281	0.776407	+	0.630232i	$0.217040\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 9.20285i	− 0.0199628i	−0.999950	−	0.00998140i	$-0.996823\pi$
$461$	− 9.20285i	− 0.0199628i	0.999950	−	0.00998140i	$-0.00317723\pi$
$462$	0	0
$463$	55.1975	0.119217	0.0596085	−	0.998222i	$-0.481015\pi$
$463$	55.1975	0.119217	0.0596085	+	0.998222i	$0.481015\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 625.811i	− 1.34007i	−0.742331	−	0.670033i	$-0.766280\pi$
$467$	− 625.811i	− 1.34007i	0.742331	−	0.670033i	$-0.233720\pi$
$468$	0	0
$469$	−626.514	−1.33585
$470$	0	0
$471$	0	0
$472$	0	0
$473$	42.9513i	0.0908062i
$474$	0	0
$475$	−49.3939	−0.103987
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 309.294i	− 0.645708i	−0.946449	−	0.322854i	$-0.895358\pi$
$479$	− 309.294i	− 0.645708i	0.946449	−	0.322854i	$-0.104642\pi$
$480$	0	0
$481$	−452.817	−0.941408
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 570.946i	− 1.17721i
$486$	0	0
$487$	28.3337	0.0581800	0.0290900	−	0.999577i	$-0.490739\pi$
$487$	28.3337	0.0581800	0.0290900	+	0.999577i	$0.490739\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	949.697i	1.93421i	0.254379	+	0.967105i	$0.418129\pi$
$491$	949.697i	1.93421i	−0.254379	+	0.967105i	$0.581871\pi$
$492$	0	0
$493$	−258.272	−0.523879
$494$	0	0
$495$	0	0
$496$	0	0
$497$	150.443i	0.302702i
$498$	0	0
$499$	560.226	1.12270	0.561349	−	0.827579i	$-0.310283\pi$
$499$	560.226	1.12270	0.561349	+	0.827579i	$0.310283\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 897.832i	− 1.78495i	−0.451094	−	0.892477i	$-0.648966\pi$
$503$	− 897.832i	− 1.78495i	0.451094	−	0.892477i	$-0.351034\pi$
$504$	0	0
$505$	−766.635	−1.51809
$506$	0	0
$507$	0	0
$508$	0	0
$509$	196.823i	0.386685i	0.981131	+	0.193343i	$0.0619329\pi$
$509$	196.823i	0.386685i	−0.981131	+	0.193343i	$0.938067\pi$
$510$	0	0
$511$	−66.0296	−0.129216
$512$	0	0
$513$	0	0
$514$	0	0
$515$	535.912i	1.04060i
$516$	0	0
$517$	−43.1816	−0.0835235
$518$	0	0
$519$	0	0
$520$	0	0
$521$	375.837i	0.721377i	0.932686	+	0.360688i	$0.117458\pi$
$521$	375.837i	0.721377i	−0.932686	+	0.360688i	$0.882542\pi$
$522$	0	0
$523$	−91.1827	−0.174345	−0.0871727	−	0.996193i	$-0.527783\pi$
$523$	−91.1827	−0.174345	−0.0871727	+	0.996193i	$0.527783\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 57.4899i	− 0.109089i
$528$	0	0
$529$	528.091	0.998281
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 105.864i	− 0.198618i
$534$	0	0
$535$	−187.273	−0.350044
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 19.7348i	− 0.0366137i
$540$	0	0
$541$	−38.8490	−0.0718096	−0.0359048	−	0.999355i	$-0.511431\pi$
$541$	−38.8490	−0.0718096	−0.0359048	+	0.999355i	$0.511431\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 770.446i	− 1.41366i
$546$	0	0
$547$	466.044	0.852001	0.426000	−	0.904723i	$-0.359922\pi$
$547$	466.044	0.852001	0.426000	+	0.904723i	$0.359922\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	337.890i	0.613230i
$552$	0	0
$553$	365.182	0.660365
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 695.042i	− 1.24783i	−0.781492	−	0.623916i	$-0.785541\pi$
$557$	− 695.042i	− 1.24783i	0.781492	−	0.623916i	$-0.214459\pi$
$558$	0	0
$559$	436.803	0.781400
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 547.074i	− 0.971713i	−0.874039	−	0.485857i	$-0.838508\pi$
$563$	− 547.074i	− 0.971713i	0.874039	−	0.485857i	$-0.161492\pi$
$564$	0	0
$565$	−888.998	−1.57345
$566$	0	0
$567$	0	0
$568$	0	0
$569$	249.362i	0.438247i	0.975697	+	0.219123i	$0.0703197\pi$
$569$	249.362i	0.438247i	−0.975697	+	0.219123i	$0.929680\pi$
$570$	0	0
$571$	−73.8332	−0.129305	−0.0646525	−	0.997908i	$-0.520594\pi$
$571$	−73.8332	−0.129305	−0.0646525	+	0.997908i	$0.520594\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1.90702i	− 0.00331656i
$576$	0	0
$577$	−43.9092	−0.0760991	−0.0380496	−	0.999276i	$-0.512114\pi$
$577$	−43.9092	−0.0760991	−0.0380496	+	0.999276i	$0.512114\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1088.08i	1.87276i
$582$	0	0
$583$	90.0000	0.154374
$584$	0	0
$585$	0	0
$586$	0	0
$587$	440.387i	0.750234i	0.926978	+	0.375117i	$0.122397\pi$
$587$	440.387i	0.750234i	−0.926978	+	0.375117i	$0.877603\pi$
$588$	0	0
$589$	−75.2122	−0.127695
$590$	0	0
$591$	0	0
$592$	0	0
$593$	347.232i	0.585551i	0.956181	+	0.292776i	$0.0945789\pi$
$593$	347.232i	0.585551i	−0.956181	+	0.292776i	$0.905421\pi$
$594$	0	0
$595$	−818.908	−1.37632
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 789.911i	− 1.31872i	−0.751829	−	0.659359i	$-0.770828\pi$
$599$	− 789.911i	− 1.31872i	0.751829	−	0.659359i	$-0.229172\pi$
$600$	0	0
$601$	−706.909	−1.17622	−0.588111	−	0.808780i	$-0.700128\pi$
$601$	−706.909	−1.17622	−0.588111	+	0.808780i	$0.700128\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	624.010i	1.03142i
$606$	0	0
$607$	1193.26	1.96583	0.982913	−	0.184070i	$-0.0589272\pi$
$607$	1193.26	1.96583	0.982913	+	0.184070i	$0.0589272\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	439.145i	0.718731i
$612$	0	0
$613$	629.181	1.02640	0.513198	−	0.858270i	$-0.328461\pi$
$613$	629.181	1.02640	0.513198	+	0.858270i	$0.328461\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 192.730i	− 0.312366i	−0.987728	−	0.156183i	$-0.950081\pi$
$617$	− 192.730i	− 0.312366i	0.987728	−	0.156183i	$-0.0499189\pi$
$618$	0	0
$619$	152.955	0.247100	0.123550	−	0.992338i	$-0.460572\pi$
$619$	152.955	0.247100	0.123550	+	0.992338i	$0.460572\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1213.03i	− 1.94708i
$624$	0	0
$625$	−671.000	−1.07360
$626$	0	0
$627$	0	0
$628$	0	0
$629$	881.525i	1.40147i
$630$	0	0
$631$	−44.8786	−0.0711229	−0.0355615	−	0.999367i	$-0.511322\pi$
$631$	−44.8786	−0.0711229	−0.0355615	+	0.999367i	$0.511322\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	405.772i	0.639011i
$636$	0	0
$637$	−200.697	−0.315066
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 241.455i	− 0.376685i	−0.982103	−	0.188342i	$-0.939689\pi$
$641$	− 241.455i	− 0.376685i	0.982103	−	0.188342i	$-0.0603115\pi$
$642$	0	0
$643$	−791.409	−1.23081	−0.615403	−	0.788212i	$-0.711007\pi$
$643$	−791.409	−1.23081	−0.615403	+	0.788212i	$0.711007\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	294.028i	0.454448i	0.973842	+	0.227224i	$0.0729650\pi$
$647$	294.028i	0.454448i	−0.973842	+	0.227224i	$0.927035\pi$
$648$	0	0
$649$	−17.9082	−0.0275935
$650$	0	0
$651$	0	0
$652$	0	0
$653$	768.313i	1.17659i	0.808647	+	0.588295i	$0.200201\pi$
$653$	768.313i	1.17659i	−0.808647	+	0.588295i	$0.799799\pi$
$654$	0	0
$655$	−1216.23	−1.85683
$656$	0	0
$657$	0	0
$658$	0	0
$659$	430.939i	0.653928i	0.945037	+	0.326964i	$0.106026\pi$
$659$	430.939i	0.653928i	−0.945037	+	0.326964i	$0.893974\pi$
$660$	0	0
$661$	1012.27	1.53142	0.765712	−	0.643183i	$-0.222387\pi$
$661$	1012.27	1.53142	0.765712	+	0.643183i	$0.222387\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1071.35i	1.61105i
$666$	0	0
$667$	−13.0454	−0.0195583
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 12.4822i	− 0.0186025i
$672$	0	0
$673$	563.211	0.836867	0.418433	−	0.908248i	$-0.362579\pi$
$673$	563.211	0.836867	0.418433	+	0.908248i	$0.362579\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 350.136i	− 0.517187i	−0.965986	−	0.258594i	$-0.916741\pi$
$677$	− 350.136i	− 0.517187i	0.965986	−	0.258594i	$-0.0832591\pi$
$678$	0	0
$679$	−917.318	−1.35098
$680$	0	0
$681$	0	0
$682$	0	0
$683$	502.818i	0.736190i	0.929788	+	0.368095i	$0.119990\pi$
$683$	502.818i	0.736190i	−0.929788	+	0.368095i	$0.880010\pi$
$684$	0	0
$685$	−778.546	−1.13656
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 915.274i	− 1.32841i
$690$	0	0
$691$	−376.319	−0.544600	−0.272300	−	0.962212i	$-0.587784\pi$
$691$	−376.319	−0.544600	−0.272300	+	0.962212i	$0.587784\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	439.231i	0.631987i
$696$	0	0
$697$	−206.091	−0.295683
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 489.681i	− 0.698546i	−0.937021	−	0.349273i	$-0.886429\pi$
$701$	− 489.681i	− 0.698546i	0.937021	−	0.349273i	$-0.113571\pi$
$702$	0	0
$703$	1153.27	1.64050
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1231.72i	1.74218i
$708$	0	0
$709$	474.029	0.668588	0.334294	−	0.942469i	$-0.391502\pi$
$709$	474.029	0.668588	0.334294	+	0.942469i	$0.391502\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.90383i	− 0.00407270i
$714$	0	0
$715$	−48.0444	−0.0671949
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 108.122i	− 0.150379i	−0.997169	−	0.0751894i	$-0.976044\pi$
$719$	− 108.122i	− 0.150379i	0.997169	−	0.0751894i	$-0.0239561\pi$
$720$	0	0
$721$	861.030	1.19422
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 27.3629i	− 0.0377419i
$726$	0	0
$727$	444.591	0.611542	0.305771	−	0.952105i	$-0.401086\pi$
$727$	444.591	0.611542	0.305771	+	0.952105i	$0.401086\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 850.349i	− 1.16327i
$732$	0	0
$733$	−716.362	−0.977302	−0.488651	−	0.872479i	$-0.662511\pi$
$733$	−716.362	−0.977302	−0.488651	+	0.872479i	$0.662511\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	71.5567i	0.0970918i
$738$	0	0
$739$	−933.362	−1.26301	−0.631504	−	0.775373i	$-0.717562\pi$
$739$	−933.362	−1.26301	−0.631504	+	0.775373i	$0.717562\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 15.9110i	− 0.0214145i	−0.999943	−	0.0107073i	$-0.996592\pi$
$743$	− 15.9110i	− 0.0214145i	0.999943	−	0.0107073i	$-0.00340829\pi$
$744$	0	0
$745$	600.181	0.805612
$746$	0	0
$747$	0	0
$748$	0	0
$749$	300.885i	0.401716i
$750$	0	0
$751$	−809.831	−1.07834	−0.539169	−	0.842198i	$-0.681261\pi$
$751$	−809.831	−1.07834	−0.539169	+	0.842198i	$0.681261\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	336.570i	0.445788i
$756$	0	0
$757$	689.637	0.911013	0.455506	−	0.890232i	$-0.349458\pi$
$757$	689.637	0.911013	0.455506	+	0.890232i	$0.349458\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	953.082i	1.25241i	0.779659	+	0.626204i	$0.215392\pi$
$761$	953.082i	1.25241i	−0.779659	+	0.626204i	$0.784608\pi$
$762$	0	0
$763$	−1237.85	−1.62234
$764$	0	0
$765$	0	0
$766$	0	0
$767$	182.121i	0.237446i
$768$	0	0
$769$	−656.696	−0.853961	−0.426980	−	0.904261i	$-0.640423\pi$
$769$	−656.696	−0.853961	−0.426980	+	0.904261i	$0.640423\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	278.021i	0.359665i	0.983697	+	0.179832i	$0.0575555\pi$
$773$	278.021i	0.359665i	−0.983697	+	0.179832i	$0.942445\pi$
$774$	0	0
$775$	6.09082	0.00785912
$776$	0	0
$777$	0	0
$778$	0	0
$779$	269.622i	0.346113i
$780$	0	0
$781$	17.1827	0.0220008
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 108.175i	− 0.137803i
$786$	0	0
$787$	−821.954	−1.04441	−0.522207	−	0.852819i	$-0.674891\pi$
$787$	−821.954	−1.04441	−0.522207	+	0.852819i	$0.674891\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1428.32i	1.80572i
$792$	0	0
$793$	−126.941	−0.160077
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1322.51i	− 1.65937i	−0.558235	−	0.829683i	$-0.688521\pi$
$797$	− 1322.51i	− 1.65937i	0.558235	−	0.829683i	$-0.311479\pi$
$798$	0	0
$799$	854.908	1.06997
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.54150i	0.00939166i
$804$	0	0
$805$	−41.3633	−0.0513829
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 235.681i	− 0.291324i	−0.989334	−	0.145662i	$-0.953469\pi$
$809$	− 235.681i	− 0.291324i	0.989334	−	0.145662i	$-0.0465311\pi$
$810$	0	0
$811$	587.362	0.724244	0.362122	−	0.932131i	$-0.382052\pi$
$811$	587.362	0.724244	0.362122	+	0.932131i	$0.382052\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 691.401i	− 0.848345i
$816$	0	0
$817$	−1112.48	−1.36167
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 943.913i	− 1.14971i	−0.818255	−	0.574856i	$-0.805058\pi$
$821$	− 943.913i	− 1.14971i	0.818255	−	0.574856i	$-0.194942\pi$
$822$	0	0
$823$	1615.74	1.96323	0.981617	−	0.190859i	$-0.0611273\pi$
$823$	1615.74	1.96323	0.981617	+	0.190859i	$0.0611273\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	582.354i	0.704177i	0.935967	+	0.352088i	$0.114528\pi$
$827$	582.354i	0.704177i	−0.935967	+	0.352088i	$0.885472\pi$
$828$	0	0
$829$	877.121	1.05805	0.529024	−	0.848607i	$-0.322558\pi$
$829$	877.121	1.05805	0.529024	+	0.848607i	$0.322558\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	390.708i	0.469038i
$834$	0	0
$835$	1530.32	1.83271
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1137.12i	1.35533i	0.735370	+	0.677666i	$0.237009\pi$
$839$	1137.12i	1.35533i	−0.735370	+	0.677666i	$0.762991\pi$
$840$	0	0
$841$	653.818	0.777430
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 389.552i	− 0.461009i
$846$	0	0
$847$	1002.57	1.18368
$848$	0	0
$849$	0	0
$850$	0	0
$851$	44.5261i	0.0523221i
$852$	0	0
$853$	−319.817	−0.374932	−0.187466	−	0.982271i	$-0.560028\pi$
$853$	−319.817	−0.374932	−0.187466	+	0.982271i	$0.560028\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	797.968i	0.931118i	0.885017	+	0.465559i	$0.154147\pi$
$857$	797.968i	0.931118i	−0.885017	+	0.465559i	$0.845853\pi$
$858$	0	0
$859$	467.802	0.544588	0.272294	−	0.962214i	$-0.412218\pi$
$859$	467.802	0.544588	0.272294	+	0.962214i	$0.412218\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1304.85i	− 1.51199i	−0.654578	−	0.755994i	$-0.727154\pi$
$863$	− 1304.85i	− 1.51199i	0.654578	−	0.755994i	$-0.272846\pi$
$864$	0	0
$865$	359.908	0.416079
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 41.7088i	− 0.0479964i
$870$	0	0
$871$	727.711	0.835489
$872$	0	0
$873$	0	0
$874$	0	0
$875$	997.738i	1.14027i
$876$	0	0
$877$	−373.756	−0.426176	−0.213088	−	0.977033i	$-0.568352\pi$
$877$	−373.756	−0.426176	−0.213088	+	0.977033i	$0.568352\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 229.979i	− 0.261043i	−0.991445	−	0.130522i	$-0.958335\pi$
$881$	− 229.979i	− 0.261043i	0.991445	−	0.130522i	$-0.0416652\pi$
$882$	0	0
$883$	1381.79	1.56488	0.782439	−	0.622728i	$-0.213976\pi$
$883$	1381.79	1.56488	0.782439	+	0.622728i	$0.213976\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	876.180i	0.987802i	0.869518	+	0.493901i	$0.164429\pi$
$887$	876.180i	0.987802i	−0.869518	+	0.493901i	$0.835571\pi$
$888$	0	0
$889$	651.939	0.733339
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1118.45i	− 1.25246i
$894$	0	0
$895$	−246.729	−0.275674
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 41.6655i	− 0.0463465i
$900$	0	0
$901$	−1781.82	−1.97760
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1262.03i	1.39451i
$906$	0	0
$907$	−1180.07	−1.30107	−0.650537	−	0.759475i	$-0.725456\pi$
$907$	−1180.07	−1.30107	−0.650537	+	0.759475i	$0.725456\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1270.32i	− 1.39442i	−0.716867	−	0.697210i	$-0.754424\pi$
$911$	− 1270.32i	− 1.39442i	0.716867	−	0.697210i	$-0.245576\pi$
$912$	0	0
$913$	124.273	0.136116
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1954.07i	2.13093i
$918$	0	0
$919$	−1316.63	−1.43268	−0.716340	−	0.697751i	$-0.754184\pi$
$919$	−1316.63	−1.43268	−0.716340	+	0.697751i	$0.754184\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 174.743i	− 0.189320i
$924$	0	0
$925$	−93.3939	−0.100966
$926$	0	0
$927$	0	0
$928$	0	0
$929$	627.492i	0.675449i	0.941245	+	0.337724i	$0.109657\pi$
$929$	627.492i	0.675449i	−0.941245	+	0.337724i	$0.890343\pi$
$930$	0	0
$931$	511.151	0.549034
$932$	0	0
$933$	0	0
$934$	0	0
$935$	93.5307i	0.100033i
$936$	0	0
$937$	469.789	0.501375	0.250688	−	0.968068i	$-0.419343\pi$
$937$	469.789	0.501375	0.250688	+	0.968068i	$0.419343\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 930.670i	− 0.989022i	−0.869171	−	0.494511i	$-0.835347\pi$
$941$	− 930.670i	− 0.989022i	0.869171	−	0.494511i	$-0.164653\pi$
$942$	0	0
$943$	−10.4097	−0.0110389
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.63113i	0.00383435i	0.999998	+	0.00191717i	$0.000610256\pi$
$947$	3.63113i	0.00383435i	−0.999998	+	0.00191717i	$0.999390\pi$
$948$	0	0
$949$	76.6949	0.0808165
$950$	0	0
$951$	0	0
$952$	0	0
$953$	719.641i	0.755132i	0.925983	+	0.377566i	$0.123239\pi$
$953$	719.641i	0.755132i	−0.925983	+	0.377566i	$0.876761\pi$
$954$	0	0
$955$	39.1362	0.0409803
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1250.86i	1.30434i
$960$	0	0
$961$	−951.725	−0.990349
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1795.03i	1.86014i
$966$	0	0
$967$	−33.7740	−0.0349266	−0.0174633	−	0.999848i	$-0.505559\pi$
$967$	−33.7740	−0.0349266	−0.0174633	+	0.999848i	$0.505559\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	970.472i	0.999456i	0.866182	+	0.499728i	$0.166567\pi$
$971$	970.472i	0.999456i	−0.866182	+	0.499728i	$0.833433\pi$
$972$	0	0
$973$	705.697	0.725279
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1570.30i	1.60727i	0.595123	+	0.803635i	$0.297103\pi$
$977$	1570.30i	1.60727i	−0.595123	+	0.803635i	$0.702897\pi$
$978$	0	0
$979$	−138.545	−0.141517
$980$	0	0
$981$	0	0
$982$	0	0
$983$	775.878i	0.789296i	0.918832	+	0.394648i	$0.129134\pi$
$983$	775.878i	0.789296i	−0.918832	+	0.394648i	$0.870866\pi$
$984$	0	0
$985$	401.271	0.407382
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 42.9513i	− 0.0434290i
$990$	0	0
$991$	−870.454	−0.878359	−0.439180	−	0.898399i	$-0.644731\pi$
$991$	−870.454	−0.878359	−0.439180	+	0.898399i	$0.644731\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 797.530i	− 0.801538i
$996$	0	0
$997$	1245.00	1.24874	0.624372	−	0.781127i	$-0.285355\pi$
$997$	1245.00	1.24874	0.624372	+	0.781127i	$0.285355\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.e.g.161.4		4
3.2	odd	2	inner	1296.3.e.g.161.2		4
4.3	odd	2		162.3.b.a.161.4		4
9.2	odd	6		432.3.q.d.17.1		4
9.4	even	3		432.3.q.d.305.1		4
9.5	odd	6		144.3.q.c.65.2		4
9.7	even	3		144.3.q.c.113.2		4
12.11	even	2		162.3.b.a.161.1		4
36.7	odd	6		18.3.d.a.5.2	✓	4
36.11	even	6		54.3.d.a.17.1		4
36.23	even	6		18.3.d.a.11.2	yes	4
36.31	odd	6		54.3.d.a.35.1		4
72.5	odd	6		576.3.q.e.65.1		4
72.11	even	6		1728.3.q.d.449.2		4
72.13	even	6		1728.3.q.c.1601.1		4
72.29	odd	6		1728.3.q.c.449.1		4
72.43	odd	6		576.3.q.f.257.2		4
72.59	even	6		576.3.q.f.65.2		4
72.61	even	6		576.3.q.e.257.1		4
72.67	odd	6		1728.3.q.d.1601.2		4
180.7	even	12		450.3.k.a.149.4		8
180.23	odd	12		450.3.k.a.299.4		8
180.43	even	12		450.3.k.a.149.1		8
180.47	odd	12		1350.3.k.a.449.1		8
180.59	even	6		450.3.i.b.101.1		4
180.67	even	12		1350.3.k.a.899.4		8
180.79	odd	6		450.3.i.b.401.1		4
180.83	odd	12		1350.3.k.a.449.4		8
180.103	even	12		1350.3.k.a.899.1		8
180.119	even	6		1350.3.i.b.1151.2		4
180.139	odd	6		1350.3.i.b.251.2		4
180.167	odd	12		450.3.k.a.299.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.d.a.5.2	✓	4	36.7	odd	6
18.3.d.a.11.2	yes	4	36.23	even	6
54.3.d.a.17.1		4	36.11	even	6
54.3.d.a.35.1		4	36.31	odd	6
144.3.q.c.65.2		4	9.5	odd	6
144.3.q.c.113.2		4	9.7	even	3
162.3.b.a.161.1		4	12.11	even	2
162.3.b.a.161.4		4	4.3	odd	2
432.3.q.d.17.1		4	9.2	odd	6
432.3.q.d.305.1		4	9.4	even	3
450.3.i.b.101.1		4	180.59	even	6
450.3.i.b.401.1		4	180.79	odd	6
450.3.k.a.149.1		8	180.43	even	12
450.3.k.a.149.4		8	180.7	even	12
450.3.k.a.299.1		8	180.167	odd	12
450.3.k.a.299.4		8	180.23	odd	12
576.3.q.e.65.1		4	72.5	odd	6
576.3.q.e.257.1		4	72.61	even	6
576.3.q.f.65.2		4	72.59	even	6
576.3.q.f.257.2		4	72.43	odd	6
1296.3.e.g.161.2		4	3.2	odd	2	inner
1296.3.e.g.161.4		4	1.1	even	1	trivial
1350.3.i.b.251.2		4	180.139	odd	6
1350.3.i.b.1151.2		4	180.119	even	6
1350.3.k.a.449.1		8	180.47	odd	12
1350.3.k.a.449.4		8	180.83	odd	12
1350.3.k.a.899.1		8	180.103	even	12
1350.3.k.a.899.4		8	180.67	even	12
1728.3.q.c.449.1		4	72.29	odd	6
1728.3.q.c.1601.1		4	72.13	even	6
1728.3.q.d.449.2		4	72.11	even	6
1728.3.q.d.1601.2		4	72.67	odd	6

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615i q^{5} +8.34847 q^{7} +O(q^{10})\)	\(q+5.19615i q^{5} +8.34847 q^{7} -0.953512i q^{11} -9.69694 q^{13} +18.8776i q^{17} +24.6969 q^{19} +0.953512i q^{23} -2.00000 q^{25} +13.6814i q^{29} -3.04541 q^{31} +43.3799i q^{35} +46.6969 q^{37} +10.9172i q^{41} -45.0454 q^{43} -45.2869i q^{47} +20.6969 q^{49} +94.3879i q^{53} +4.95459 q^{55} -18.7813i q^{59} +13.0908 q^{61} -50.3868i q^{65} -75.0454 q^{67} +18.0204i q^{71} -7.90918 q^{73} -7.96036i q^{77} +43.7423 q^{79} +130.332i q^{83} -98.0908 q^{85} -145.300i q^{89} -80.9546 q^{91} +128.329i q^{95} -109.879 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 20 q^{13} + 40 q^{19} - 8 q^{25} + 76 q^{31} + 128 q^{37} - 92 q^{43} + 24 q^{49} + 108 q^{55} - 124 q^{61} - 212 q^{67} - 208 q^{73} + 28 q^{79} - 216 q^{85} - 412 q^{91} - 28 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 20 * q^13 + 40 * q^19 - 8 * q^25 + 76 * q^31 + 128 * q^37 - 92 * q^43 + 24 * q^49 + 108 * q^55 - 124 * q^61 - 212 * q^67 - 208 * q^73 + 28 * q^79 - 216 * q^85 - 412 * q^91 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more