LMFDB - Embedded newform 1296.4.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,4,Mod(1,1296)]

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, 0]))

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

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

chi := DirichletCharacter("1296.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1296.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:	$76.4664753674$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 19x^{3} + 4x^{2} + 81x + 12 $ x^5 - x^4 - 19x^3 + 4x^2 + 81*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.50994$ of defining polynomial
Character	$\chi$	$=$	1296.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.31871 q^{5} +29.5796 q^{7} +O(q^{10})$	$q-6.31871 q^{5} +29.5796 q^{7} +49.1523 q^{11} +18.2656 q^{13} +63.7873 q^{17} +52.0413 q^{19} +149.388 q^{23} -85.0739 q^{25} +119.415 q^{29} -303.693 q^{31} -186.905 q^{35} +170.764 q^{37} +43.6906 q^{41} -446.868 q^{43} +221.999 q^{47} +531.953 q^{49} -67.3089 q^{53} -310.579 q^{55} +467.796 q^{59} -475.008 q^{61} -115.415 q^{65} +240.980 q^{67} -550.486 q^{71} +552.924 q^{73} +1453.91 q^{77} -913.314 q^{79} -667.905 q^{83} -403.054 q^{85} -1430.73 q^{89} +540.288 q^{91} -328.834 q^{95} +1665.72 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} - 3 q^{7}+O(q^{10}) $ 5 * q + 5 * q^5 - 3 * q^7	$ 5 q + 5 q^{5} - 3 q^{7} + 25 q^{11} + 29 q^{13} + 28 q^{17} - 64 q^{19} - 89 q^{23} + 322 q^{25} + 129 q^{29} - 241 q^{31} + 243 q^{35} + 366 q^{37} + 171 q^{41} - 803 q^{43} + 477 q^{47} + 1072 q^{49} + 374 q^{53} - 1469 q^{55} + 607 q^{59} + 1349 q^{61} + 527 q^{65} - 1549 q^{67} + 812 q^{71} + 1928 q^{73} + 903 q^{77} - 1727 q^{79} + 1025 q^{83} + 2902 q^{85} + 2310 q^{89} - 2985 q^{91} + 2012 q^{95} + 2875 q^{97}+O(q^{100}) $ 5 * q + 5 * q^5 - 3 * q^7 + 25 * q^11 + 29 * q^13 + 28 * q^17 - 64 * q^19 - 89 * q^23 + 322 * q^25 + 129 * q^29 - 241 * q^31 + 243 * q^35 + 366 * q^37 + 171 * q^41 - 803 * q^43 + 477 * q^47 + 1072 * q^49 + 374 * q^53 - 1469 * q^55 + 607 * q^59 + 1349 * q^61 + 527 * q^65 - 1549 * q^67 + 812 * q^71 + 1928 * q^73 + 903 * q^77 - 1727 * q^79 + 1025 * q^83 + 2902 * q^85 + 2310 * q^89 - 2985 * q^91 + 2012 * q^95 + 2875 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−6.31871	−0.565163	−0.282581	−	0.959243i	$-0.591191\pi$
$5$	−6.31871	−0.565163	−0.282581	+	0.959243i	$0.591191\pi$
$6$	0	0
$7$	29.5796	1.59715	0.798574	−	0.601896i	$-0.205588\pi$
$7$	29.5796	1.59715	0.798574	+	0.601896i	$0.205588\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	49.1523	1.34727	0.673636	−	0.739064i	$-0.264732\pi$
$11$	49.1523	1.34727	0.673636	+	0.739064i	$0.264732\pi$
$12$	0	0
$13$	18.2656	0.389689	0.194844	−	0.980834i	$-0.437580\pi$
$13$	18.2656	0.389689	0.194844	+	0.980834i	$0.437580\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	63.7873	0.910041	0.455021	−	0.890481i	$-0.349632\pi$
$17$	63.7873	0.910041	0.455021	+	0.890481i	$0.349632\pi$
$18$	0	0
$19$	52.0413	0.628374	0.314187	−	0.949361i	$-0.398268\pi$
$19$	52.0413	0.628374	0.314187	+	0.949361i	$0.398268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	149.388	1.35433	0.677164	−	0.735832i	$-0.263209\pi$
$23$	149.388	1.35433	0.677164	+	0.735832i	$0.263209\pi$
$24$	0	0
$25$	−85.0739	−0.680591
$26$	0	0
$27$	0	0
$28$	0	0
$29$	119.415	0.764651	0.382326	−	0.924028i	$-0.375123\pi$
$29$	119.415	0.764651	0.382326	+	0.924028i	$0.375123\pi$
$30$	0	0
$31$	−303.693	−1.75951	−0.879755	−	0.475428i	$-0.842293\pi$
$31$	−303.693	−1.75951	−0.879755	+	0.475428i	$0.842293\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−186.905	−0.902649
$36$	0	0
$37$	170.764	0.758743	0.379371	−	0.925244i	$-0.376140\pi$
$37$	170.764	0.758743	0.379371	+	0.925244i	$0.376140\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	43.6906	0.166422	0.0832112	−	0.996532i	$-0.473482\pi$
$41$	43.6906	0.166422	0.0832112	+	0.996532i	$0.473482\pi$
$42$	0	0
$43$	−446.868	−1.58481	−0.792403	−	0.609998i	$-0.791170\pi$
$43$	−446.868	−1.58481	−0.792403	+	0.609998i	$0.791170\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	221.999	0.688977	0.344488	−	0.938791i	$-0.388052\pi$
$47$	221.999	0.688977	0.344488	+	0.938791i	$0.388052\pi$
$48$	0	0
$49$	531.953	1.55088
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−67.3089	−0.174445	−0.0872226	−	0.996189i	$-0.527799\pi$
$53$	−67.3089	−0.174445	−0.0872226	+	0.996189i	$0.527799\pi$
$54$	0	0
$55$	−310.579	−0.761428
$56$	0	0
$57$	0	0
$58$	0	0
$59$	467.796	1.03223	0.516117	−	0.856518i	$-0.327377\pi$
$59$	467.796	1.03223	0.516117	+	0.856518i	$0.327377\pi$
$60$	0	0
$61$	−475.008	−0.997025	−0.498512	−	0.866883i	$-0.666120\pi$
$61$	−475.008	−0.997025	−0.498512	+	0.866883i	$0.666120\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−115.415	−0.220238
$66$	0	0
$67$	240.980	0.439409	0.219704	−	0.975567i	$-0.429491\pi$
$67$	240.980	0.439409	0.219704	+	0.975567i	$0.429491\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−550.486	−0.920150	−0.460075	−	0.887880i	$-0.652177\pi$
$71$	−550.486	−0.920150	−0.460075	+	0.887880i	$0.652177\pi$
$72$	0	0
$73$	552.924	0.886505	0.443252	−	0.896397i	$-0.353825\pi$
$73$	552.924	0.886505	0.443252	+	0.896397i	$0.353825\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1453.91	2.15179
$78$	0	0
$79$	−913.314	−1.30071	−0.650354	−	0.759632i	$-0.725379\pi$
$79$	−913.314	−1.30071	−0.650354	+	0.759632i	$0.725379\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−667.905	−0.883278	−0.441639	−	0.897193i	$-0.645603\pi$
$83$	−667.905	−0.883278	−0.441639	+	0.897193i	$0.645603\pi$
$84$	0	0
$85$	−403.054	−0.514322
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1430.73	−1.70402	−0.852008	−	0.523529i	$-0.824615\pi$
$89$	−1430.73	−1.70402	−0.852008	+	0.523529i	$0.824615\pi$
$90$	0	0
$91$	540.288	0.622391
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−328.834	−0.355134
$96$	0	0
$97$	1665.72	1.74359	0.871796	−	0.489870i	$-0.162956\pi$
$97$	1665.72	1.74359	0.871796	+	0.489870i	$0.162956\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	684.686	0.674542	0.337271	−	0.941408i	$-0.390496\pi$
$101$	684.686	0.674542	0.337271	+	0.941408i	$0.390496\pi$
$102$	0	0
$103$	300.692	0.287651	0.143825	−	0.989603i	$-0.454060\pi$
$103$	300.692	0.287651	0.143825	+	0.989603i	$0.454060\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1950.79	1.76252	0.881261	−	0.472631i	$-0.156696\pi$
$107$	1950.79	1.76252	0.881261	+	0.472631i	$0.156696\pi$
$108$	0	0
$109$	1637.17	1.43865	0.719323	−	0.694676i	$-0.244452\pi$
$109$	1637.17	1.43865	0.719323	+	0.694676i	$0.244452\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1381.14	−1.14979	−0.574897	−	0.818226i	$-0.694958\pi$
$113$	−1381.14	−1.14979	−0.574897	+	0.818226i	$0.694958\pi$
$114$	0	0
$115$	−943.939	−0.765416
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1886.80	1.45347
$120$	0	0
$121$	1084.95	0.815139
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1327.40	0.949808
$126$	0	0
$127$	−1571.28	−1.09786	−0.548931	−	0.835868i	$-0.684965\pi$
$127$	−1571.28	−1.09786	−0.548931	+	0.835868i	$0.684965\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	144.380	0.0962944	0.0481472	−	0.998840i	$-0.484668\pi$
$131$	144.380	0.0962944	0.0481472	+	0.998840i	$0.484668\pi$
$132$	0	0
$133$	1539.36	1.00361
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1522.38	0.949387	0.474693	−	0.880151i	$-0.342559\pi$
$137$	1522.38	0.949387	0.474693	+	0.880151i	$0.342559\pi$
$138$	0	0
$139$	−1577.76	−0.962764	−0.481382	−	0.876511i	$-0.659865\pi$
$139$	−1577.76	−0.962764	−0.481382	+	0.876511i	$0.659865\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	897.795	0.525017
$144$	0	0
$145$	−754.552	−0.432152
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2335.25	1.28397	0.641984	−	0.766718i	$-0.278111\pi$
$149$	2335.25	1.28397	0.641984	+	0.766718i	$0.278111\pi$
$150$	0	0
$151$	1725.84	0.930109	0.465055	−	0.885282i	$-0.346035\pi$
$151$	1725.84	0.930109	0.465055	+	0.885282i	$0.346035\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1918.95	0.994409
$156$	0	0
$157$	−36.9669	−0.0187916	−0.00939579	−	0.999956i	$-0.502991\pi$
$157$	−36.9669	−0.0187916	−0.00939579	+	0.999956i	$0.502991\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4418.83	2.16306
$162$	0	0
$163$	−1112.17	−0.534430	−0.267215	−	0.963637i	$-0.586103\pi$
$163$	−1112.17	−0.534430	−0.267215	+	0.963637i	$0.586103\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1656.49	0.767561	0.383781	−	0.923424i	$-0.374622\pi$
$167$	1656.49	0.767561	0.383781	+	0.923424i	$0.374622\pi$
$168$	0	0
$169$	−1863.37	−0.848143
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−634.217	−0.278720	−0.139360	−	0.990242i	$-0.544505\pi$
$173$	−634.217	−0.278720	−0.139360	+	0.990242i	$0.544505\pi$
$174$	0	0
$175$	−2516.45	−1.08700
$176$	0	0
$177$	0	0
$178$	0	0
$179$	374.805	0.156504	0.0782521	−	0.996934i	$-0.475066\pi$
$179$	374.805	0.156504	0.0782521	+	0.996934i	$0.475066\pi$
$180$	0	0
$181$	−859.130	−0.352810	−0.176405	−	0.984318i	$-0.556447\pi$
$181$	−859.130	−0.352810	−0.176405	+	0.984318i	$0.556447\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1079.01	−0.428813
$186$	0	0
$187$	3135.30	1.22607
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3076.74	−1.16558	−0.582789	−	0.812624i	$-0.698038\pi$
$191$	−3076.74	−1.16558	−0.582789	+	0.812624i	$0.698038\pi$
$192$	0	0
$193$	136.616	0.0509526	0.0254763	−	0.999675i	$-0.491890\pi$
$193$	136.616	0.0509526	0.0254763	+	0.999675i	$0.491890\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3635.05	1.31465	0.657326	−	0.753606i	$-0.271687\pi$
$197$	3635.05	1.31465	0.657326	+	0.753606i	$0.271687\pi$
$198$	0	0
$199$	1611.10	0.573909	0.286955	−	0.957944i	$-0.407357\pi$
$199$	1611.10	0.573909	0.286955	+	0.957944i	$0.407357\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3532.26	1.22126
$204$	0	0
$205$	−276.068	−0.0940558
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2557.95	0.846590
$210$	0	0
$211$	3293.93	1.07471	0.537355	−	0.843356i	$-0.319424\pi$
$211$	3293.93	1.07471	0.537355	+	0.843356i	$0.319424\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2823.63	0.895674
$216$	0	0
$217$	−8983.10	−2.81020
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1165.11	0.354633
$222$	0	0
$223$	−1038.45	−0.311837	−0.155919	−	0.987770i	$-0.549834\pi$
$223$	−1038.45	−0.311837	−0.155919	+	0.987770i	$0.549834\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4628.67	1.35337	0.676686	−	0.736272i	$-0.263416\pi$
$227$	4628.67	1.35337	0.676686	+	0.736272i	$0.263416\pi$
$228$	0	0
$229$	−344.313	−0.0993574	−0.0496787	−	0.998765i	$-0.515820\pi$
$229$	−344.313	−0.0993574	−0.0496787	+	0.998765i	$0.515820\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	401.177	0.112798	0.0563991	−	0.998408i	$-0.482038\pi$
$233$	401.177	0.112798	0.0563991	+	0.998408i	$0.482038\pi$
$234$	0	0
$235$	−1402.75	−0.389384
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2276.32	−0.616078	−0.308039	−	0.951374i	$-0.599673\pi$
$239$	−2276.32	−0.616078	−0.308039	+	0.951374i	$0.599673\pi$
$240$	0	0
$241$	215.656	0.0576416	0.0288208	−	0.999585i	$-0.490825\pi$
$241$	215.656	0.0576416	0.0288208	+	0.999585i	$0.490825\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3361.26	−0.876501
$246$	0	0
$247$	950.564	0.244870
$248$	0	0
$249$	0	0
$250$	0	0
$251$	335.402	0.0843443	0.0421721	−	0.999110i	$-0.486572\pi$
$251$	335.402	0.0843443	0.0421721	+	0.999110i	$0.486572\pi$
$252$	0	0
$253$	7342.76	1.82465
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4507.60	−1.09407	−0.547035	−	0.837110i	$-0.684244\pi$
$257$	−4507.60	−1.09407	−0.547035	+	0.837110i	$0.684244\pi$
$258$	0	0
$259$	5051.14	1.21182
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2155.07	−0.505276	−0.252638	−	0.967561i	$-0.581298\pi$
$263$	−2155.07	−0.505276	−0.252638	+	0.967561i	$0.581298\pi$
$264$	0	0
$265$	425.306	0.0985899
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4048.11	−0.917537	−0.458768	−	0.888556i	$-0.651709\pi$
$269$	−4048.11	−0.917537	−0.458768	+	0.888556i	$0.651709\pi$
$270$	0	0
$271$	290.780	0.0651794	0.0325897	−	0.999469i	$-0.489625\pi$
$271$	290.780	0.0651794	0.0325897	+	0.999469i	$0.489625\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4181.58	−0.916940
$276$	0	0
$277$	−2151.90	−0.466770	−0.233385	−	0.972384i	$-0.574980\pi$
$277$	−2151.90	−0.466770	−0.233385	+	0.972384i	$0.574980\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4271.13	0.906742	0.453371	−	0.891322i	$-0.350221\pi$
$281$	4271.13	0.906742	0.453371	+	0.891322i	$0.350221\pi$
$282$	0	0
$283$	164.805	0.0346171	0.0173086	−	0.999850i	$-0.494490\pi$
$283$	164.805	0.0346171	0.0173086	+	0.999850i	$0.494490\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1292.35	0.265801
$288$	0	0
$289$	−844.176	−0.171825
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5489.72	1.09458	0.547292	−	0.836942i	$-0.315659\pi$
$293$	5489.72	1.09458	0.547292	+	0.836942i	$0.315659\pi$
$294$	0	0
$295$	−2955.87	−0.583381
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2728.65	0.527766
$300$	0	0
$301$	−13218.2	−2.53117
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3001.44	0.563481
$306$	0	0
$307$	6126.58	1.13897	0.569483	−	0.822003i	$-0.307143\pi$
$307$	6126.58	1.13897	0.569483	+	0.822003i	$0.307143\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	146.154	0.0266484	0.0133242	−	0.999911i	$-0.495759\pi$
$311$	146.154	0.0266484	0.0133242	+	0.999911i	$0.495759\pi$
$312$	0	0
$313$	4247.80	0.767093	0.383546	−	0.923522i	$-0.374703\pi$
$313$	4247.80	0.767093	0.383546	+	0.923522i	$0.374703\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3587.37	−0.635605	−0.317802	−	0.948157i	$-0.602945\pi$
$317$	−3587.37	−0.635605	−0.317802	+	0.948157i	$0.602945\pi$
$318$	0	0
$319$	5869.54	1.03019
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3319.58	0.571846
$324$	0	0
$325$	−1553.92	−0.265219
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6566.65	1.10040
$330$	0	0
$331$	−2492.92	−0.413968	−0.206984	−	0.978344i	$-0.566365\pi$
$331$	−2492.92	−0.413968	−0.206984	+	0.978344i	$0.566365\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1522.68	−0.248337
$336$	0	0
$337$	−1068.15	−0.172658	−0.0863288	−	0.996267i	$-0.527514\pi$
$337$	−1068.15	−0.172658	−0.0863288	+	0.996267i	$0.527514\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−14927.2	−2.37054
$342$	0	0
$343$	5589.14	0.879841
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6085.23	0.941418	0.470709	−	0.882288i	$-0.343998\pi$
$347$	6085.23	0.941418	0.470709	+	0.882288i	$0.343998\pi$
$348$	0	0
$349$	−7067.26	−1.08396	−0.541979	−	0.840392i	$-0.682325\pi$
$349$	−7067.26	−1.08396	−0.541979	+	0.840392i	$0.682325\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10032.2	1.51263	0.756316	−	0.654207i	$-0.226997\pi$
$353$	10032.2	1.51263	0.756316	+	0.654207i	$0.226997\pi$
$354$	0	0
$355$	3478.36	0.520035
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7628.71	−1.12153	−0.560763	−	0.827976i	$-0.689492\pi$
$359$	−7628.71	−1.12153	−0.560763	+	0.827976i	$0.689492\pi$
$360$	0	0
$361$	−4150.70	−0.605146
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3493.77	−0.501020
$366$	0	0
$367$	1346.98	0.191585	0.0957924	−	0.995401i	$-0.469462\pi$
$367$	1346.98	0.191585	0.0957924	+	0.995401i	$0.469462\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1990.97	−0.278615
$372$	0	0
$373$	−133.012	−0.0184641	−0.00923206	−	0.999957i	$-0.502939\pi$
$373$	−133.012	−0.0184641	−0.00923206	+	0.999957i	$0.502939\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2181.19	0.297976
$378$	0	0
$379$	−2109.40	−0.285891	−0.142945	−	0.989731i	$-0.545657\pi$
$379$	−2109.40	−0.285891	−0.142945	+	0.989731i	$0.545657\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	238.108	0.0317669	0.0158835	−	0.999874i	$-0.494944\pi$
$383$	238.108	0.0317669	0.0158835	+	0.999874i	$0.494944\pi$
$384$	0	0
$385$	−9186.81	−1.21611
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.15727	0.00106321	0.000531607	−	1.00000i	$-0.499831\pi$
$389$	8.15727	0.00106321	0.000531607		1.00000i	$0.499831\pi$
$390$	0	0
$391$	9529.06	1.23249
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5770.97	0.735112
$396$	0	0
$397$	347.061	0.0438753	0.0219376	−	0.999759i	$-0.493016\pi$
$397$	347.061	0.0438753	0.0219376	+	0.999759i	$0.493016\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7030.71	0.875553	0.437777	−	0.899084i	$-0.355766\pi$
$401$	7030.71	0.875553	0.437777	+	0.899084i	$0.355766\pi$
$402$	0	0
$403$	−5547.12	−0.685661
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8393.46	1.02223
$408$	0	0
$409$	14999.5	1.81339	0.906697	−	0.421783i	$-0.138596\pi$
$409$	14999.5	1.81339	0.906697	+	0.421783i	$0.138596\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13837.2	1.64863
$414$	0	0
$415$	4220.30	0.499196
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10468.3	1.22055	0.610275	−	0.792189i	$-0.291059\pi$
$419$	10468.3	1.22055	0.610275	+	0.792189i	$0.291059\pi$
$420$	0	0
$421$	−3797.08	−0.439568	−0.219784	−	0.975549i	$-0.570535\pi$
$421$	−3797.08	−0.439568	−0.219784	+	0.975549i	$0.570535\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5426.63	−0.619366
$426$	0	0
$427$	−14050.5	−1.59240
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16434.0	−1.83665	−0.918326	−	0.395825i	$-0.870459\pi$
$431$	−16434.0	−1.83665	−0.918326	+	0.395825i	$0.870459\pi$
$432$	0	0
$433$	15688.1	1.74116	0.870581	−	0.492024i	$-0.163743\pi$
$433$	15688.1	1.74116	0.870581	+	0.492024i	$0.163743\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7774.35	0.851024
$438$	0	0
$439$	799.797	0.0869527	0.0434763	−	0.999054i	$-0.486157\pi$
$439$	799.797	0.0869527	0.0434763	+	0.999054i	$0.486157\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1962.96	0.210526	0.105263	−	0.994444i	$-0.466432\pi$
$443$	1962.96	0.210526	0.105263	+	0.994444i	$0.466432\pi$
$444$	0	0
$445$	9040.39	0.963047
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4590.96	−0.482541	−0.241270	−	0.970458i	$-0.577564\pi$
$449$	−4590.96	−0.482541	−0.241270	+	0.970458i	$0.577564\pi$
$450$	0	0
$451$	2147.49	0.224216
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3413.93	−0.351752
$456$	0	0
$457$	1594.70	0.163232	0.0816161	−	0.996664i	$-0.473992\pi$
$457$	1594.70	0.163232	0.0816161	+	0.996664i	$0.473992\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5574.24	−0.563163	−0.281582	−	0.959537i	$-0.590859\pi$
$461$	−5574.24	−0.563163	−0.281582	+	0.959537i	$0.590859\pi$
$462$	0	0
$463$	−13289.6	−1.33395	−0.666977	−	0.745078i	$-0.732412\pi$
$463$	−13289.6	−1.33395	−0.666977	+	0.745078i	$0.732412\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18331.8	−1.81648	−0.908238	−	0.418455i	$-0.862572\pi$
$467$	−18331.8	−1.81648	−0.908238	+	0.418455i	$0.862572\pi$
$468$	0	0
$469$	7128.09	0.701801
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−21964.6	−2.13516
$474$	0	0
$475$	−4427.36	−0.427665
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−19923.5	−1.90048	−0.950240	−	0.311519i	$-0.899162\pi$
$479$	−19923.5	−1.90048	−0.950240	+	0.311519i	$0.899162\pi$
$480$	0	0
$481$	3119.11	0.295674
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10525.2	−0.985413
$486$	0	0
$487$	−6788.63	−0.631668	−0.315834	−	0.948814i	$-0.602284\pi$
$487$	−6788.63	−0.631668	−0.315834	+	0.948814i	$0.602284\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6071.56	0.558056	0.279028	−	0.960283i	$-0.409988\pi$
$491$	6071.56	0.558056	0.279028	+	0.960283i	$0.409988\pi$
$492$	0	0
$493$	7617.19	0.695864
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16283.1	−1.46962
$498$	0	0
$499$	7327.27	0.657342	0.328671	−	0.944444i	$-0.393399\pi$
$499$	7327.27	0.657342	0.328671	+	0.944444i	$0.393399\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2121.18	0.188029	0.0940145	−	0.995571i	$-0.470030\pi$
$503$	2121.18	0.188029	0.0940145	+	0.995571i	$0.470030\pi$
$504$	0	0
$505$	−4326.33	−0.381226
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11360.8	−0.989309	−0.494655	−	0.869090i	$-0.664705\pi$
$509$	−11360.8	−0.989309	−0.494655	+	0.869090i	$0.664705\pi$
$510$	0	0
$511$	16355.3	1.41588
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1899.98	−0.162569
$516$	0	0
$517$	10911.8	0.928238
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9012.94	−0.757897	−0.378948	−	0.925418i	$-0.623714\pi$
$521$	−9012.94	−0.757897	−0.378948	+	0.925418i	$0.623714\pi$
$522$	0	0
$523$	−7991.81	−0.668178	−0.334089	−	0.942541i	$-0.608429\pi$
$523$	−7991.81	−0.668178	−0.334089	+	0.942541i	$0.608429\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19371.7	−1.60123
$528$	0	0
$529$	10149.7	0.834203
$530$	0	0
$531$	0	0
$532$	0	0
$533$	798.033	0.0648530
$534$	0	0
$535$	−12326.5	−0.996112
$536$	0	0
$537$	0	0
$538$	0	0
$539$	26146.7	2.08946
$540$	0	0
$541$	−19447.1	−1.54546	−0.772732	−	0.634732i	$-0.781111\pi$
$541$	−19447.1	−1.54546	−0.772732	+	0.634732i	$0.781111\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10344.8	−0.813069
$546$	0	0
$547$	−16772.9	−1.31107	−0.655537	−	0.755163i	$-0.727558\pi$
$547$	−16772.9	−1.31107	−0.655537	+	0.755163i	$0.727558\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6214.54	0.480487
$552$	0	0
$553$	−27015.5	−2.07742
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7836.27	0.596110	0.298055	−	0.954549i	$-0.403662\pi$
$557$	7836.27	0.596110	0.298055	+	0.954549i	$0.403662\pi$
$558$	0	0
$559$	−8162.29	−0.617581
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24777.9	−1.85482	−0.927409	−	0.374050i	$-0.877969\pi$
$563$	−24777.9	−1.85482	−0.927409	+	0.374050i	$0.877969\pi$
$564$	0	0
$565$	8727.03	0.649821
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11571.3	0.852540	0.426270	−	0.904596i	$-0.359827\pi$
$569$	11571.3	0.852540	0.426270	+	0.904596i	$0.359827\pi$
$570$	0	0
$571$	195.293	0.0143131	0.00715653	−	0.999974i	$-0.497722\pi$
$571$	195.293	0.0143131	0.00715653	+	0.999974i	$0.497722\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12709.0	−0.921743
$576$	0	0
$577$	−5912.60	−0.426594	−0.213297	−	0.976987i	$-0.568420\pi$
$577$	−5912.60	−0.426594	−0.213297	+	0.976987i	$0.568420\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−19756.4	−1.41073
$582$	0	0
$583$	−3308.39	−0.235025
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1149.51	−0.0808270	−0.0404135	−	0.999183i	$-0.512868\pi$
$587$	−1149.51	−0.0808270	−0.0404135	+	0.999183i	$0.512868\pi$
$588$	0	0
$589$	−15804.6	−1.10563
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6387.66	−0.442344	−0.221172	−	0.975235i	$-0.570988\pi$
$593$	−6387.66	−0.442344	−0.221172	+	0.975235i	$0.570988\pi$
$594$	0	0
$595$	−11922.2	−0.821448
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17252.8	−1.17684	−0.588421	−	0.808555i	$-0.700250\pi$
$599$	−17252.8	−1.17684	−0.588421	+	0.808555i	$0.700250\pi$
$600$	0	0
$601$	7581.03	0.514537	0.257268	−	0.966340i	$-0.417178\pi$
$601$	7581.03	0.514537	0.257268	+	0.966340i	$0.417178\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6855.49	−0.460686
$606$	0	0
$607$	−15791.2	−1.05593	−0.527963	−	0.849268i	$-0.677044\pi$
$607$	−15791.2	−1.05593	−0.527963	+	0.849268i	$0.677044\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4054.94	0.268487
$612$	0	0
$613$	16043.4	1.05708	0.528538	−	0.848910i	$-0.322740\pi$
$613$	16043.4	1.05708	0.528538	+	0.848910i	$0.322740\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25164.5	1.64195	0.820975	−	0.570964i	$-0.193431\pi$
$617$	25164.5	1.64195	0.820975	+	0.570964i	$0.193431\pi$
$618$	0	0
$619$	−316.919	−0.0205785	−0.0102892	−	0.999947i	$-0.503275\pi$
$619$	−316.919	−0.0205785	−0.0102892	+	0.999947i	$0.503275\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−42320.5	−2.72157
$624$	0	0
$625$	2246.79	0.143795
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10892.6	0.690487
$630$	0	0
$631$	−24816.8	−1.56567	−0.782837	−	0.622227i	$-0.786228\pi$
$631$	−24816.8	−1.56567	−0.782837	+	0.622227i	$0.786228\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9928.46	0.620471
$636$	0	0
$637$	9716.42	0.604362
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19823.2	−1.22148	−0.610742	−	0.791830i	$-0.709129\pi$
$641$	−19823.2	−1.22148	−0.610742	+	0.791830i	$0.709129\pi$
$642$	0	0
$643$	−6095.53	−0.373848	−0.186924	−	0.982374i	$-0.559852\pi$
$643$	−6095.53	−0.373848	−0.186924	+	0.982374i	$0.559852\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7321.69	−0.444892	−0.222446	−	0.974945i	$-0.571404\pi$
$647$	−7321.69	−0.444892	−0.222446	+	0.974945i	$0.571404\pi$
$648$	0	0
$649$	22993.3	1.39070
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6735.73	0.403659	0.201830	−	0.979421i	$-0.435311\pi$
$653$	6735.73	0.403659	0.201830	+	0.979421i	$0.435311\pi$
$654$	0	0
$655$	−912.297	−0.0544220
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23001.2	1.35964	0.679818	−	0.733381i	$-0.262059\pi$
$659$	23001.2	1.35964	0.679818	+	0.733381i	$0.262059\pi$
$660$	0	0
$661$	25712.9	1.51303	0.756517	−	0.653974i	$-0.226899\pi$
$661$	25712.9	1.51303	0.756517	+	0.653974i	$0.226899\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9726.79	−0.567201
$666$	0	0
$667$	17839.2	1.03559
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−23347.7	−1.34326
$672$	0	0
$673$	4621.78	0.264720	0.132360	−	0.991202i	$-0.457745\pi$
$673$	4621.78	0.264720	0.132360	+	0.991202i	$0.457745\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18272.0	1.03730	0.518650	−	0.854987i	$-0.326435\pi$
$677$	18272.0	1.03730	0.518650	+	0.854987i	$0.326435\pi$
$678$	0	0
$679$	49271.4	2.78477
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30350.5	1.70034	0.850169	−	0.526510i	$-0.176500\pi$
$683$	30350.5	1.70034	0.850169	+	0.526510i	$0.176500\pi$
$684$	0	0
$685$	−9619.50	−0.536558
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1229.44	−0.0679793
$690$	0	0
$691$	10834.6	0.596482	0.298241	−	0.954491i	$-0.403600\pi$
$691$	10834.6	0.596482	0.298241	+	0.954491i	$0.403600\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9969.44	0.544118
$696$	0	0
$697$	2786.91	0.151451
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12323.5	−0.663982	−0.331991	−	0.943283i	$-0.607720\pi$
$701$	−12323.5	−0.663982	−0.331991	+	0.943283i	$0.607720\pi$
$702$	0	0
$703$	8886.81	0.476774
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20252.7	1.07734
$708$	0	0
$709$	32902.5	1.74285	0.871424	−	0.490531i	$-0.163197\pi$
$709$	32902.5	1.74285	0.871424	+	0.490531i	$0.163197\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−45368.0	−2.38295
$714$	0	0
$715$	−5672.91	−0.296720
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8518.05	0.441821	0.220911	−	0.975294i	$-0.429097\pi$
$719$	8518.05	0.441821	0.220911	+	0.975294i	$0.429097\pi$
$720$	0	0
$721$	8894.34	0.459421
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10159.1	−0.520415
$726$	0	0
$727$	−37212.3	−1.89839	−0.949193	−	0.314693i	$-0.898098\pi$
$727$	−37212.3	−1.89839	−0.949193	+	0.314693i	$0.898098\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−28504.5	−1.44224
$732$	0	0
$733$	20954.1	1.05588	0.527938	−	0.849283i	$-0.322965\pi$
$733$	20954.1	1.05588	0.527938	+	0.849283i	$0.322965\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11844.7	0.592002
$738$	0	0
$739$	−28544.4	−1.42087	−0.710435	−	0.703763i	$-0.751502\pi$
$739$	−28544.4	−1.42087	−0.710435	+	0.703763i	$0.751502\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15010.8	0.741176	0.370588	−	0.928797i	$-0.379156\pi$
$743$	15010.8	0.741176	0.370588	+	0.928797i	$0.379156\pi$
$744$	0	0
$745$	−14755.8	−0.725652
$746$	0	0
$747$	0	0
$748$	0	0
$749$	57703.5	2.81501
$750$	0	0
$751$	−14662.5	−0.712442	−0.356221	−	0.934402i	$-0.615935\pi$
$751$	−14662.5	−0.712442	−0.356221	+	0.934402i	$0.615935\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10905.1	−0.525663
$756$	0	0
$757$	−36100.7	−1.73329	−0.866646	−	0.498924i	$-0.833729\pi$
$757$	−36100.7	−1.73329	−0.866646	+	0.498924i	$0.833729\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−16830.6	−0.801718	−0.400859	−	0.916140i	$-0.631288\pi$
$761$	−16830.6	−0.801718	−0.400859	+	0.916140i	$0.631288\pi$
$762$	0	0
$763$	48426.8	2.29773
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8544.56	0.402250
$768$	0	0
$769$	−7423.79	−0.348126	−0.174063	−	0.984735i	$-0.555690\pi$
$769$	−7423.79	−0.348126	−0.174063	+	0.984735i	$0.555690\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13516.7	0.628929	0.314464	−	0.949269i	$-0.398175\pi$
$773$	13516.7	0.628929	0.314464	+	0.949269i	$0.398175\pi$
$774$	0	0
$775$	25836.3	1.19751
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2273.72	0.104576
$780$	0	0
$781$	−27057.6	−1.23969
$782$	0	0
$783$	0	0
$784$	0	0
$785$	233.583	0.0106203
$786$	0	0
$787$	4382.18	0.198485	0.0992426	−	0.995063i	$-0.468358\pi$
$787$	4382.18	0.198485	0.0992426	+	0.995063i	$0.468358\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−40853.6	−1.83639
$792$	0	0
$793$	−8676.28	−0.388529
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4343.69	0.193051	0.0965254	−	0.995331i	$-0.469227\pi$
$797$	4343.69	0.193051	0.0965254	+	0.995331i	$0.469227\pi$
$798$	0	0
$799$	14160.7	0.626997
$800$	0	0
$801$	0	0
$802$	0	0
$803$	27177.5	1.19436
$804$	0	0
$805$	−27921.3	−1.22248
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−14546.1	−0.632155	−0.316077	−	0.948733i	$-0.602366\pi$
$809$	−14546.1	−0.632155	−0.316077	+	0.948733i	$0.602366\pi$
$810$	0	0
$811$	22663.2	0.981272	0.490636	−	0.871365i	$-0.336764\pi$
$811$	22663.2	0.981272	0.490636	+	0.871365i	$0.336764\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7027.51	0.302040
$816$	0	0
$817$	−23255.6	−0.995851
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36422.0	−1.54828	−0.774139	−	0.633015i	$-0.781817\pi$
$821$	−36422.0	−1.54828	−0.774139	+	0.633015i	$0.781817\pi$
$822$	0	0
$823$	−7869.56	−0.333312	−0.166656	−	0.986015i	$-0.553297\pi$
$823$	−7869.56	−0.333312	−0.166656	+	0.986015i	$0.553297\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22101.6	−0.929322	−0.464661	−	0.885489i	$-0.653824\pi$
$827$	−22101.6	−0.929322	−0.464661	+	0.885489i	$0.653824\pi$
$828$	0	0
$829$	17056.2	0.714580	0.357290	−	0.933994i	$-0.383701\pi$
$829$	17056.2	0.714580	0.357290	+	0.933994i	$0.383701\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	33931.8	1.41137
$834$	0	0
$835$	−10466.9	−0.433797
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22944.6	−0.944144	−0.472072	−	0.881560i	$-0.656494\pi$
$839$	−22944.6	−0.944144	−0.472072	+	0.881560i	$0.656494\pi$
$840$	0	0
$841$	−10129.0	−0.415309
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11774.1	0.479339
$846$	0	0
$847$	32092.4	1.30190
$848$	0	0
$849$	0	0
$850$	0	0
$851$	25510.1	1.02759
$852$	0	0
$853$	−30871.2	−1.23917	−0.619584	−	0.784930i	$-0.712699\pi$
$853$	−30871.2	−1.23917	−0.619584	+	0.784930i	$0.712699\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15089.9	−0.601472	−0.300736	−	0.953707i	$-0.597232\pi$
$857$	−15089.9	−0.601472	−0.300736	+	0.953707i	$0.597232\pi$
$858$	0	0
$859$	−7752.69	−0.307938	−0.153969	−	0.988076i	$-0.549206\pi$
$859$	−7752.69	−0.307938	−0.153969	+	0.988076i	$0.549206\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22050.5	0.869765	0.434883	−	0.900487i	$-0.356790\pi$
$863$	22050.5	0.869765	0.434883	+	0.900487i	$0.356790\pi$
$864$	0	0
$865$	4007.44	0.157522
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−44891.5	−1.75241
$870$	0	0
$871$	4401.63	0.171233
$872$	0	0
$873$	0	0
$874$	0	0
$875$	39263.9	1.51698
$876$	0	0
$877$	34373.4	1.32350	0.661749	−	0.749725i	$-0.269814\pi$
$877$	34373.4	1.32350	0.661749	+	0.749725i	$0.269814\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14030.0	−0.536531	−0.268266	−	0.963345i	$-0.586450\pi$
$881$	−14030.0	−0.536531	−0.268266	+	0.963345i	$0.586450\pi$
$882$	0	0
$883$	14608.8	0.556767	0.278383	−	0.960470i	$-0.410201\pi$
$883$	14608.8	0.556767	0.278383	+	0.960470i	$0.410201\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24653.5	−0.933240	−0.466620	−	0.884458i	$-0.654528\pi$
$887$	−24653.5	−0.933240	−0.466620	+	0.884458i	$0.654528\pi$
$888$	0	0
$889$	−46477.8	−1.75345
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11553.1	0.432935
$894$	0	0
$895$	−2368.29	−0.0884504
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−36265.6	−1.34541
$900$	0	0
$901$	−4293.46	−0.158752
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5428.59	0.199395
$906$	0	0
$907$	16302.0	0.596802	0.298401	−	0.954441i	$-0.403547\pi$
$907$	16302.0	0.596802	0.298401	+	0.954441i	$0.403547\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31028.4	−1.12845	−0.564225	−	0.825621i	$-0.690825\pi$
$911$	−31028.4	−1.12845	−0.564225	+	0.825621i	$0.690825\pi$
$912$	0	0
$913$	−32829.1	−1.19002
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4270.71	0.153796
$918$	0	0
$919$	−21538.9	−0.773126	−0.386563	−	0.922263i	$-0.626338\pi$
$919$	−21538.9	−0.773126	−0.386563	+	0.922263i	$0.626338\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10054.9	−0.358572
$924$	0	0
$925$	−14527.6	−0.516393
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46927.2	1.65730	0.828650	−	0.559767i	$-0.189109\pi$
$929$	46927.2	1.65730	0.828650	+	0.559767i	$0.189109\pi$
$930$	0	0
$931$	27683.5	0.974534
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19811.0	−0.692931
$936$	0	0
$937$	35694.0	1.24447	0.622237	−	0.782829i	$-0.286224\pi$
$937$	35694.0	1.24447	0.622237	+	0.782829i	$0.286224\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6478.29	0.224427	0.112214	−	0.993684i	$-0.464206\pi$
$941$	6478.29	0.224427	0.112214	+	0.993684i	$0.464206\pi$
$942$	0	0
$943$	6526.84	0.225391
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51553.4	1.76902	0.884509	−	0.466523i	$-0.154494\pi$
$947$	51553.4	1.76902	0.884509	+	0.466523i	$0.154494\pi$
$948$	0	0
$949$	10099.5	0.345461
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14395.3	0.489306	0.244653	−	0.969611i	$-0.421326\pi$
$953$	14395.3	0.489306	0.244653	+	0.969611i	$0.421326\pi$
$954$	0	0
$955$	19441.1	0.658741
$956$	0	0
$957$	0	0
$958$	0	0
$959$	45031.5	1.51631
$960$	0	0
$961$	62438.2	2.09587
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−863.238	−0.0287965
$966$	0	0
$967$	−14676.5	−0.488070	−0.244035	−	0.969766i	$-0.578471\pi$
$967$	−14676.5	−0.488070	−0.244035	+	0.969766i	$0.578471\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36879.2	1.21886	0.609429	−	0.792841i	$-0.291399\pi$
$971$	36879.2	1.21886	0.609429	+	0.792841i	$0.291399\pi$
$972$	0	0
$973$	−46669.6	−1.53768
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32512.9	1.06467	0.532333	−	0.846535i	$-0.321315\pi$
$977$	32512.9	1.06467	0.532333	+	0.846535i	$0.321315\pi$
$978$	0	0
$979$	−70323.8	−2.29577
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19545.1	−0.634174	−0.317087	−	0.948396i	$-0.602705\pi$
$983$	−19545.1	−0.634174	−0.317087	+	0.948396i	$0.602705\pi$
$984$	0	0
$985$	−22968.8	−0.742993
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−66756.6	−2.14635
$990$	0	0
$991$	9818.92	0.314741	0.157370	−	0.987540i	$-0.449698\pi$
$991$	9818.92	0.314741	0.157370	+	0.987540i	$0.449698\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10180.1	−0.324352
$996$	0	0
$997$	11425.9	0.362949	0.181475	−	0.983396i	$-0.441913\pi$
$997$	11425.9	0.362949	0.181475	+	0.983396i	$0.441913\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.bd.1.2		5
3.2	odd	2		1296.4.a.bc.1.4		5
4.3	odd	2		648.4.a.l.1.2		5
9.2	odd	6		432.4.i.f.145.2		10
9.4	even	3		144.4.i.f.97.5		10
9.5	odd	6		432.4.i.f.289.2		10
9.7	even	3		144.4.i.f.49.5		10
12.11	even	2		648.4.a.k.1.4		5
36.7	odd	6		72.4.i.b.49.1	yes	10
36.11	even	6		216.4.i.b.145.2		10
36.23	even	6		216.4.i.b.73.2		10
36.31	odd	6		72.4.i.b.25.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.i.b.25.1	✓	10	36.31	odd	6
72.4.i.b.49.1	yes	10	36.7	odd	6
144.4.i.f.49.5		10	9.7	even	3
144.4.i.f.97.5		10	9.4	even	3
216.4.i.b.73.2		10	36.23	even	6
216.4.i.b.145.2		10	36.11	even	6
432.4.i.f.145.2		10	9.2	odd	6
432.4.i.f.289.2		10	9.5	odd	6
648.4.a.k.1.4		5	12.11	even	2
648.4.a.l.1.2		5	4.3	odd	2
1296.4.a.bc.1.4		5	3.2	odd	2
1296.4.a.bd.1.2		5	1.1	even	1	trivial

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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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

\(f(q)\)	\(=\)	\(q-6.31871 q^{5} +29.5796 q^{7} +O(q^{10})\)	\(q-6.31871 q^{5} +29.5796 q^{7} +49.1523 q^{11} +18.2656 q^{13} +63.7873 q^{17} +52.0413 q^{19} +149.388 q^{23} -85.0739 q^{25} +119.415 q^{29} -303.693 q^{31} -186.905 q^{35} +170.764 q^{37} +43.6906 q^{41} -446.868 q^{43} +221.999 q^{47} +531.953 q^{49} -67.3089 q^{53} -310.579 q^{55} +467.796 q^{59} -475.008 q^{61} -115.415 q^{65} +240.980 q^{67} -550.486 q^{71} +552.924 q^{73} +1453.91 q^{77} -913.314 q^{79} -667.905 q^{83} -403.054 q^{85} -1430.73 q^{89} +540.288 q^{91} -328.834 q^{95} +1665.72 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} - 3 q^{7}+O(q^{10}) \) 5 * q + 5 * q^5 - 3 * q^7	\( 5 q + 5 q^{5} - 3 q^{7} + 25 q^{11} + 29 q^{13} + 28 q^{17} - 64 q^{19} - 89 q^{23} + 322 q^{25} + 129 q^{29} - 241 q^{31} + 243 q^{35} + 366 q^{37} + 171 q^{41} - 803 q^{43} + 477 q^{47} + 1072 q^{49} + 374 q^{53} - 1469 q^{55} + 607 q^{59} + 1349 q^{61} + 527 q^{65} - 1549 q^{67} + 812 q^{71} + 1928 q^{73} + 903 q^{77} - 1727 q^{79} + 1025 q^{83} + 2902 q^{85} + 2310 q^{89} - 2985 q^{91} + 2012 q^{95} + 2875 q^{97}+O(q^{100}) \) 5 * q + 5 * q^5 - 3 * q^7 + 25 * q^11 + 29 * q^13 + 28 * q^17 - 64 * q^19 - 89 * q^23 + 322 * q^25 + 129 * q^29 - 241 * q^31 + 243 * q^35 + 366 * q^37 + 171 * q^41 - 803 * q^43 + 477 * q^47 + 1072 * q^49 + 374 * q^53 - 1469 * q^55 + 607 * q^59 + 1349 * q^61 + 527 * q^65 - 1549 * q^67 + 812 * q^71 + 1928 * q^73 + 903 * q^77 - 1727 * q^79 + 1025 * q^83 + 2902 * q^85 + 2310 * q^89 - 2985 * q^91 + 2012 * q^95 + 2875 * q^97

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