LMFDB - Embedded newform 1296.4.a.n.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:	$2$
Coefficient field:	$\Q(\sqrt{129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 32 $ x^2 - x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 648)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.17891$ 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)

$f(q)$	$=$	$q+9.35782 q^{5} -8.35782 q^{7} +O(q^{10})$	$q+9.35782 q^{5} -8.35782 q^{7} -29.0735 q^{11} +36.7891 q^{13} -28.4313 q^{17} -73.7891 q^{19} +30.5047 q^{23} -37.4313 q^{25} +184.936 q^{29} +190.863 q^{31} -78.2109 q^{35} +160.495 q^{37} +209.578 q^{41} -116.945 q^{43} +281.597 q^{47} -273.147 q^{49} +397.450 q^{53} -272.064 q^{55} -378.147 q^{59} -297.524 q^{61} +344.265 q^{65} +827.936 q^{67} -729.073 q^{71} +1105.44 q^{73} +242.991 q^{77} +1064.27 q^{79} -590.478 q^{83} -266.055 q^{85} -227.588 q^{89} -307.476 q^{91} -690.505 q^{95} +1285.65 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 6 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 + 6 * q^7	$ 2 q - 4 q^{5} + 6 q^{7} + 10 q^{11} - 40 q^{13} + 34 q^{17} - 34 q^{19} - 98 q^{23} + 16 q^{25} + 120 q^{29} + 200 q^{31} - 270 q^{35} + 480 q^{37} + 192 q^{41} + 334 q^{43} - 300 q^{47} - 410 q^{49} + 68 q^{53} - 794 q^{55} - 620 q^{59} + 200 q^{61} + 1370 q^{65} + 1406 q^{67} - 1390 q^{71} + 1802 q^{73} + 804 q^{77} + 334 q^{79} + 500 q^{83} - 1100 q^{85} + 90 q^{89} - 1410 q^{91} - 1222 q^{95} - 200 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 6 * q^7 + 10 * q^11 - 40 * q^13 + 34 * q^17 - 34 * q^19 - 98 * q^23 + 16 * q^25 + 120 * q^29 + 200 * q^31 - 270 * q^35 + 480 * q^37 + 192 * q^41 + 334 * q^43 - 300 * q^47 - 410 * q^49 + 68 * q^53 - 794 * q^55 - 620 * q^59 + 200 * q^61 + 1370 * q^65 + 1406 * q^67 - 1390 * q^71 + 1802 * q^73 + 804 * q^77 + 334 * q^79 + 500 * q^83 - 1100 * q^85 + 90 * q^89 - 1410 * q^91 - 1222 * q^95 - 200 * 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
$5$	9.35782	0.836989	0.418494	−	0.908219i	$-0.362558\pi$
$5$	9.35782	0.836989	0.418494	+	0.908219i	$0.362558\pi$
$6$	0	0
$7$	−8.35782	−0.451280	−0.225640	−	0.974211i	$-0.572447\pi$
$7$	−8.35782	−0.451280	−0.225640	+	0.974211i	$0.572447\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−29.0735	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$11$	−29.0735	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$12$	0	0
$13$	36.7891	0.784881	0.392441	−	0.919777i	$-0.371631\pi$
$13$	36.7891	0.784881	0.392441	+	0.919777i	$0.371631\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−28.4313	−0.405623	−0.202812	−	0.979218i	$-0.565008\pi$
$17$	−28.4313	−0.405623	−0.202812	+	0.979218i	$0.565008\pi$
$18$	0	0
$19$	−73.7891	−0.890967	−0.445484	−	0.895290i	$-0.646968\pi$
$19$	−73.7891	−0.890967	−0.445484	+	0.895290i	$0.646968\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	30.5047	0.276551	0.138276	−	0.990394i	$-0.455844\pi$
$23$	30.5047	0.276551	0.138276	+	0.990394i	$0.455844\pi$
$24$	0	0
$25$	−37.4313	−0.299450
$26$	0	0
$27$	0	0
$28$	0	0
$29$	184.936	1.18420	0.592099	−	0.805865i	$-0.298299\pi$
$29$	184.936	1.18420	0.592099	+	0.805865i	$0.298299\pi$
$30$	0	0
$31$	190.863	1.10580	0.552902	−	0.833246i	$-0.313520\pi$
$31$	190.863	1.10580	0.552902	+	0.833246i	$0.313520\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−78.2109	−0.377716
$36$	0	0
$37$	160.495	0.713115	0.356558	−	0.934273i	$-0.383950\pi$
$37$	160.495	0.713115	0.356558	+	0.934273i	$0.383950\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	209.578	0.798308	0.399154	−	0.916884i	$-0.369304\pi$
$41$	209.578	0.798308	0.399154	+	0.916884i	$0.369304\pi$
$42$	0	0
$43$	−116.945	−0.414744	−0.207372	−	0.978262i	$-0.566491\pi$
$43$	−116.945	−0.414744	−0.207372	+	0.978262i	$0.566491\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	281.597	0.873939	0.436970	−	0.899476i	$-0.356052\pi$
$47$	281.597	0.873939	0.436970	+	0.899476i	$0.356052\pi$
$48$	0	0
$49$	−273.147	−0.796347
$50$	0	0
$51$	0	0
$52$	0	0
$53$	397.450	1.03007	0.515037	−	0.857168i	$-0.327778\pi$
$53$	397.450	1.03007	0.515037	+	0.857168i	$0.327778\pi$
$54$	0	0
$55$	−272.064	−0.667002
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−378.147	−0.834416	−0.417208	−	0.908811i	$-0.636991\pi$
$59$	−378.147	−0.834416	−0.417208	+	0.908811i	$0.636991\pi$
$60$	0	0
$61$	−297.524	−0.624492	−0.312246	−	0.950001i	$-0.601081\pi$
$61$	−297.524	−0.624492	−0.312246	+	0.950001i	$0.601081\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	344.265	0.656937
$66$	0	0
$67$	827.936	1.50968	0.754839	−	0.655910i	$-0.227715\pi$
$67$	827.936	1.50968	0.754839	+	0.655910i	$0.227715\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−729.073	−1.21866	−0.609332	−	0.792915i	$-0.708562\pi$
$71$	−729.073	−1.21866	−0.609332	+	0.792915i	$0.708562\pi$
$72$	0	0
$73$	1105.44	1.77236	0.886178	−	0.463344i	$-0.153351\pi$
$73$	1105.44	1.77236	0.886178	+	0.463344i	$0.153351\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	242.991	0.359628
$78$	0	0
$79$	1064.27	1.51569	0.757845	−	0.652435i	$-0.226252\pi$
$79$	1064.27	1.51569	0.757845	+	0.652435i	$0.226252\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−590.478	−0.780885	−0.390442	−	0.920627i	$-0.627678\pi$
$83$	−590.478	−0.780885	−0.390442	+	0.920627i	$0.627678\pi$
$84$	0	0
$85$	−266.055	−0.339502
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−227.588	−0.271059	−0.135529	−	0.990773i	$-0.543274\pi$
$89$	−227.588	−0.271059	−0.135529	+	0.990773i	$0.543274\pi$
$90$	0	0
$91$	−307.476	−0.354201
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−690.505	−0.745729
$96$	0	0
$97$	1285.65	1.34576	0.672878	−	0.739753i	$-0.265058\pi$
$97$	1285.65	1.34576	0.672878	+	0.739753i	$0.265058\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1684.73	1.65978	0.829888	−	0.557930i	$-0.188404\pi$
$101$	1684.73	1.65978	0.829888	+	0.557930i	$0.188404\pi$
$102$	0	0
$103$	854.403	0.817348	0.408674	−	0.912680i	$-0.365991\pi$
$103$	854.403	0.817348	0.408674	+	0.912680i	$0.365991\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1369.73	−1.23754	−0.618768	−	0.785574i	$-0.712368\pi$
$107$	−1369.73	−1.23754	−0.618768	+	0.785574i	$0.712368\pi$
$108$	0	0
$109$	1475.06	1.29620	0.648099	−	0.761556i	$-0.275564\pi$
$109$	1475.06	1.29620	0.648099	+	0.761556i	$0.275564\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.01887	−0.00750817	−0.00375409	−	0.999993i	$-0.501195\pi$
$113$	−9.01887	−0.00750817	−0.00375409	+	0.999993i	$0.501195\pi$
$114$	0	0
$115$	285.458	0.231470
$116$	0	0
$117$	0	0
$118$	0	0
$119$	237.623	0.183050
$120$	0	0
$121$	−485.735	−0.364940
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1520.00	−1.08762
$126$	0	0
$127$	−2076.05	−1.45055	−0.725274	−	0.688460i	$-0.758287\pi$
$127$	−2076.05	−1.45055	−0.725274	+	0.688460i	$0.758287\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1774.41	−1.18345	−0.591723	−	0.806142i	$-0.701552\pi$
$131$	−1774.41	−1.18345	−0.591723	+	0.806142i	$0.701552\pi$
$132$	0	0
$133$	616.716	0.402075
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1210.69	0.755007	0.377504	−	0.926008i	$-0.376783\pi$
$137$	1210.69	0.755007	0.377504	+	0.926008i	$0.376783\pi$
$138$	0	0
$139$	733.228	0.447421	0.223711	−	0.974656i	$-0.428183\pi$
$139$	733.228	0.447421	0.223711	+	0.974656i	$0.428183\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1069.59	−0.625477
$144$	0	0
$145$	1730.60	0.991160
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−515.561	−0.283466	−0.141733	−	0.989905i	$-0.545267\pi$
$149$	−515.561	−0.283466	−0.141733	+	0.989905i	$0.545267\pi$
$150$	0	0
$151$	1153.89	0.621870	0.310935	−	0.950431i	$-0.399358\pi$
$151$	1153.89	0.621870	0.310935	+	0.950431i	$0.399358\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1786.06	0.925545
$156$	0	0
$157$	−1369.51	−0.696170	−0.348085	−	0.937463i	$-0.613168\pi$
$157$	−1369.51	−0.696170	−0.348085	+	0.937463i	$0.613168\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−254.953	−0.124802
$162$	0	0
$163$	3796.90	1.82452	0.912259	−	0.409613i	$-0.134336\pi$
$163$	3796.90	1.82452	0.912259	+	0.409613i	$0.134336\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2048.89	0.949390	0.474695	−	0.880150i	$-0.342558\pi$
$167$	2048.89	0.949390	0.474695	+	0.880150i	$0.342558\pi$
$168$	0	0
$169$	−843.563	−0.383961
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2025.23	0.890031	0.445016	−	0.895523i	$-0.353198\pi$
$173$	2025.23	0.890031	0.445016	+	0.895523i	$0.353198\pi$
$174$	0	0
$175$	312.844	0.135136
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3142.50	−1.31219	−0.656094	−	0.754679i	$-0.727792\pi$
$179$	−3142.50	−1.31219	−0.656094	+	0.754679i	$0.727792\pi$
$180$	0	0
$181$	37.6685	0.0154689	0.00773446	−	0.999970i	$-0.497538\pi$
$181$	37.6685	0.0154689	0.00773446	+	0.999970i	$0.497538\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1501.89	0.596869
$186$	0	0
$187$	826.595	0.323244
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4234.12	1.60403	0.802016	−	0.597302i	$-0.203761\pi$
$191$	4234.12	1.60403	0.802016	+	0.597302i	$0.203761\pi$
$192$	0	0
$193$	3829.30	1.42818	0.714090	−	0.700054i	$-0.246841\pi$
$193$	3829.30	1.42818	0.714090	+	0.700054i	$0.246841\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1072.38	−0.387838	−0.193919	−	0.981018i	$-0.562120\pi$
$197$	−1072.38	−0.387838	−0.193919	+	0.981018i	$0.562120\pi$
$198$	0	0
$199$	1078.53	0.384196	0.192098	−	0.981376i	$-0.438471\pi$
$199$	1078.53	0.384196	0.192098	+	0.981376i	$0.438471\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1545.66	−0.534405
$204$	0	0
$205$	1961.19	0.668174
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2145.30	0.710018
$210$	0	0
$211$	3616.16	1.17984	0.589922	−	0.807461i	$-0.299159\pi$
$211$	3616.16	1.17984	0.589922	+	0.807461i	$0.299159\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1094.35	−0.347136
$216$	0	0
$217$	−1595.19	−0.499027
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1045.96	−0.318366
$222$	0	0
$223$	861.149	0.258596	0.129298	−	0.991606i	$-0.458728\pi$
$223$	861.149	0.258596	0.129298	+	0.991606i	$0.458728\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3085.95	0.902300	0.451150	−	0.892448i	$-0.351014\pi$
$227$	3085.95	0.902300	0.451150	+	0.892448i	$0.351014\pi$
$228$	0	0
$229$	6191.29	1.78660	0.893301	−	0.449458i	$-0.148383\pi$
$229$	6191.29	1.78660	0.893301	+	0.449458i	$0.148383\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4325.48	−1.21619	−0.608093	−	0.793866i	$-0.708065\pi$
$233$	−4325.48	−1.21619	−0.608093	+	0.793866i	$0.708065\pi$
$234$	0	0
$235$	2635.13	0.731477
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3118.48	0.844007	0.422003	−	0.906594i	$-0.361327\pi$
$239$	3118.48	0.844007	0.422003	+	0.906594i	$0.361327\pi$
$240$	0	0
$241$	722.925	0.193227	0.0966134	−	0.995322i	$-0.469199\pi$
$241$	722.925	0.193227	0.0966134	+	0.995322i	$0.469199\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2556.06	−0.666533
$246$	0	0
$247$	−2714.63	−0.699303
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6373.89	1.60285	0.801427	−	0.598093i	$-0.204075\pi$
$251$	6373.89	1.60285	0.801427	+	0.598093i	$0.204075\pi$
$252$	0	0
$253$	−886.877	−0.220385
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3804.82	−0.923496	−0.461748	−	0.887011i	$-0.652777\pi$
$257$	−3804.82	−0.923496	−0.461748	+	0.887011i	$0.652777\pi$
$258$	0	0
$259$	−1341.39	−0.321814
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1458.62	0.341986	0.170993	−	0.985272i	$-0.445302\pi$
$263$	1458.62	0.341986	0.170993	+	0.985272i	$0.445302\pi$
$264$	0	0
$265$	3719.27	0.862161
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2240.99	−0.507940	−0.253970	−	0.967212i	$-0.581736\pi$
$269$	−2240.99	−0.507940	−0.253970	+	0.967212i	$0.581736\pi$
$270$	0	0
$271$	3077.92	0.689927	0.344963	−	0.938616i	$-0.387891\pi$
$271$	3077.92	0.689927	0.344963	+	0.938616i	$0.387891\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1088.26	0.238634
$276$	0	0
$277$	3090.70	0.670405	0.335203	−	0.942146i	$-0.391195\pi$
$277$	3090.70	0.670405	0.335203	+	0.942146i	$0.391195\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4841.80	1.02789	0.513946	−	0.857823i	$-0.328183\pi$
$281$	4841.80	1.02789	0.513946	+	0.857823i	$0.328183\pi$
$282$	0	0
$283$	−719.450	−0.151120	−0.0755598	−	0.997141i	$-0.524074\pi$
$283$	−719.450	−0.151120	−0.0755598	+	0.997141i	$0.524074\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1751.62	−0.360260
$288$	0	0
$289$	−4104.66	−0.835470
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5250.02	−1.04679	−0.523395	−	0.852090i	$-0.675335\pi$
$293$	−5250.02	−1.04679	−0.523395	+	0.852090i	$0.675335\pi$
$294$	0	0
$295$	−3538.63	−0.698397
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1122.24	0.217060
$300$	0	0
$301$	977.408	0.187166
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2784.17	−0.522692
$306$	0	0
$307$	5823.45	1.08261	0.541306	−	0.840826i	$-0.317930\pi$
$307$	5823.45	1.08261	0.541306	+	0.840826i	$0.317930\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−392.814	−0.0716220	−0.0358110	−	0.999359i	$-0.511401\pi$
$311$	−392.814	−0.0716220	−0.0358110	+	0.999359i	$0.511401\pi$
$312$	0	0
$313$	1116.47	0.201618	0.100809	−	0.994906i	$-0.467857\pi$
$313$	1116.47	0.201618	0.100809	+	0.994906i	$0.467857\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1308.68	0.231870	0.115935	−	0.993257i	$-0.463014\pi$
$317$	1308.68	0.231870	0.115935	+	0.993257i	$0.463014\pi$
$318$	0	0
$319$	−5376.73	−0.943696
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2097.92	0.361397
$324$	0	0
$325$	−1377.06	−0.235033
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2353.54	−0.394391
$330$	0	0
$331$	6199.10	1.02941	0.514703	−	0.857369i	$-0.327902\pi$
$331$	6199.10	1.02941	0.514703	+	0.857369i	$0.327902\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7747.67	1.26358
$336$	0	0
$337$	807.702	0.130559	0.0652794	−	0.997867i	$-0.479206\pi$
$337$	807.702	0.130559	0.0652794	+	0.997867i	$0.479206\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5549.03	−0.881223
$342$	0	0
$343$	5149.64	0.810655
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1493.22	−0.231009	−0.115504	−	0.993307i	$-0.536848\pi$
$347$	−1493.22	−0.231009	−0.115504	+	0.993307i	$0.536848\pi$
$348$	0	0
$349$	−498.611	−0.0764757	−0.0382378	−	0.999269i	$-0.512174\pi$
$349$	−498.611	−0.0764757	−0.0382378	+	0.999269i	$0.512174\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12005.3	−1.81014	−0.905069	−	0.425266i	$-0.860181\pi$
$353$	−12005.3	−1.81014	−0.905069	+	0.425266i	$0.860181\pi$
$354$	0	0
$355$	−6822.54	−1.02001
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6842.79	−1.00599	−0.502993	−	0.864291i	$-0.667768\pi$
$359$	−6842.79	−1.00599	−0.502993	+	0.864291i	$0.667768\pi$
$360$	0	0
$361$	−1414.17	−0.206177
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10344.5	1.48344
$366$	0	0
$367$	112.121	0.0159474	0.00797368	−	0.999968i	$-0.497462\pi$
$367$	112.121	0.0159474	0.00797368	+	0.999968i	$0.497462\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3321.82	−0.464852
$372$	0	0
$373$	1976.17	0.274322	0.137161	−	0.990549i	$-0.456202\pi$
$373$	1976.17	0.274322	0.137161	+	0.990549i	$0.456202\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6803.63	0.929455
$378$	0	0
$379$	669.717	0.0907679	0.0453840	−	0.998970i	$-0.485549\pi$
$379$	669.717	0.0907679	0.0453840	+	0.998970i	$0.485549\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11240.5	1.49964	0.749821	−	0.661640i	$-0.230139\pi$
$383$	11240.5	1.49964	0.749821	+	0.661640i	$0.230139\pi$
$384$	0	0
$385$	2273.86	0.301004
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3765.97	−0.490854	−0.245427	−	0.969415i	$-0.578928\pi$
$389$	−3765.97	−0.490854	−0.245427	+	0.969415i	$0.578928\pi$
$390$	0	0
$391$	−867.288	−0.112176
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9959.22	1.26861
$396$	0	0
$397$	−2005.69	−0.253558	−0.126779	−	0.991931i	$-0.540464\pi$
$397$	−2005.69	−0.253558	−0.126779	+	0.991931i	$0.540464\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5555.66	−0.691861	−0.345930	−	0.938260i	$-0.612437\pi$
$401$	−5555.66	−0.691861	−0.345930	+	0.938260i	$0.612437\pi$
$402$	0	0
$403$	7021.66	0.867925
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4666.15	−0.568286
$408$	0	0
$409$	−14876.5	−1.79852	−0.899260	−	0.437414i	$-0.855895\pi$
$409$	−14876.5	−1.79852	−0.899260	+	0.437414i	$0.855895\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3160.48	0.376555
$414$	0	0
$415$	−5525.59	−0.653592
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9499.63	−1.10761	−0.553803	−	0.832647i	$-0.686824\pi$
$419$	−9499.63	−1.10761	−0.553803	+	0.832647i	$0.686824\pi$
$420$	0	0
$421$	−5997.40	−0.694288	−0.347144	−	0.937812i	$-0.612848\pi$
$421$	−5997.40	−0.694288	−0.347144	+	0.937812i	$0.612848\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1064.22	0.121464
$426$	0	0
$427$	2486.65	0.281820
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15607.1	−1.74424	−0.872121	−	0.489291i	$-0.837256\pi$
$431$	−15607.1	−1.74424	−0.872121	+	0.489291i	$0.837256\pi$
$432$	0	0
$433$	−1204.46	−0.133678	−0.0668390	−	0.997764i	$-0.521291\pi$
$433$	−1204.46	−0.133678	−0.0668390	+	0.997764i	$0.521291\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2250.92	−0.246398
$438$	0	0
$439$	−16984.6	−1.84654	−0.923269	−	0.384154i	$-0.874493\pi$
$439$	−16984.6	−1.84654	−0.923269	+	0.384154i	$0.874493\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7169.70	−0.768945	−0.384473	−	0.923136i	$-0.625617\pi$
$443$	−7169.70	−0.768945	−0.384473	+	0.923136i	$0.625617\pi$
$444$	0	0
$445$	−2129.72	−0.226873
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4648.71	−0.488611	−0.244305	−	0.969698i	$-0.578560\pi$
$449$	−4648.71	−0.488611	−0.244305	+	0.969698i	$0.578560\pi$
$450$	0	0
$451$	−6093.16	−0.636177
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2877.31	−0.296462
$456$	0	0
$457$	944.047	0.0966316	0.0483158	−	0.998832i	$-0.484615\pi$
$457$	944.047	0.0966316	0.0483158	+	0.998832i	$0.484615\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5099.02	−0.515152	−0.257576	−	0.966258i	$-0.582924\pi$
$461$	−5099.02	−0.515152	−0.257576	+	0.966258i	$0.582924\pi$
$462$	0	0
$463$	917.718	0.0921166	0.0460583	−	0.998939i	$-0.485334\pi$
$463$	917.718	0.0921166	0.0460583	+	0.998939i	$0.485334\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19073.2	1.88994	0.944969	−	0.327160i	$-0.106092\pi$
$467$	19073.2	1.88994	0.944969	+	0.327160i	$0.106092\pi$
$468$	0	0
$469$	−6919.74	−0.681287
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3400.01	0.330513
$474$	0	0
$475$	2762.02	0.266800
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18326.1	−1.74810	−0.874052	−	0.485832i	$-0.838517\pi$
$479$	−18326.1	−1.74810	−0.874052	+	0.485832i	$0.838517\pi$
$480$	0	0
$481$	5904.47	0.559711
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12030.9	1.12638
$486$	0	0
$487$	−18087.7	−1.68302	−0.841512	−	0.540238i	$-0.818334\pi$
$487$	−18087.7	−1.68302	−0.841512	+	0.540238i	$0.818334\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3860.28	0.354810	0.177405	−	0.984138i	$-0.443230\pi$
$491$	3860.28	0.354810	0.177405	+	0.984138i	$0.443230\pi$
$492$	0	0
$493$	−5257.96	−0.480338
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6093.46	0.549958
$498$	0	0
$499$	14444.7	1.29586	0.647928	−	0.761702i	$-0.275636\pi$
$499$	14444.7	1.29586	0.647928	+	0.761702i	$0.275636\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2074.88	0.183925	0.0919625	−	0.995762i	$-0.470686\pi$
$503$	2074.88	0.183925	0.0919625	+	0.995762i	$0.470686\pi$
$504$	0	0
$505$	15765.4	1.38921
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4979.72	0.433639	0.216819	−	0.976212i	$-0.430432\pi$
$509$	4979.72	0.433639	0.216819	+	0.976212i	$0.430432\pi$
$510$	0	0
$511$	−9239.07	−0.799829
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7995.35	0.684111
$516$	0	0
$517$	−8187.00	−0.696448
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8212.49	0.690587	0.345293	−	0.938495i	$-0.387779\pi$
$521$	8212.49	0.690587	0.345293	+	0.938495i	$0.387779\pi$
$522$	0	0
$523$	−12630.6	−1.05602	−0.528009	−	0.849239i	$-0.677061\pi$
$523$	−12630.6	−1.05602	−0.528009	+	0.849239i	$0.677061\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5426.46	−0.448540
$528$	0	0
$529$	−11236.5	−0.923520
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7710.19	0.626577
$534$	0	0
$535$	−12817.6	−1.03580
$536$	0	0
$537$	0	0
$538$	0	0
$539$	7941.32	0.634614
$540$	0	0
$541$	22132.0	1.75883	0.879416	−	0.476054i	$-0.157933\pi$
$541$	22132.0	1.75883	0.879416	+	0.476054i	$0.157933\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13803.4	1.08490
$546$	0	0
$547$	4542.68	0.355084	0.177542	−	0.984113i	$-0.443185\pi$
$547$	4542.68	0.355084	0.177542	+	0.984113i	$0.443185\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−13646.3	−1.05508
$552$	0	0
$553$	−8894.95	−0.684000
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22330.0	1.69866	0.849330	−	0.527862i	$-0.177006\pi$
$557$	22330.0	1.69866	0.849330	+	0.527862i	$0.177006\pi$
$558$	0	0
$559$	−4302.31	−0.325525
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14186.8	−1.06199	−0.530996	−	0.847375i	$-0.678182\pi$
$563$	−14186.8	−1.06199	−0.530996	+	0.847375i	$0.678182\pi$
$564$	0	0
$565$	−84.3969	−0.00628426
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10938.3	−0.805902	−0.402951	−	0.915222i	$-0.632016\pi$
$569$	−10938.3	−0.805902	−0.402951	+	0.915222i	$0.632016\pi$
$570$	0	0
$571$	12141.6	0.889861	0.444931	−	0.895565i	$-0.353228\pi$
$571$	12141.6	0.889861	0.444931	+	0.895565i	$0.353228\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1141.83	−0.0828132
$576$	0	0
$577$	−14080.0	−1.01587	−0.507936	−	0.861395i	$-0.669592\pi$
$577$	−14080.0	−1.01587	−0.507936	+	0.861395i	$0.669592\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4935.11	0.352397
$582$	0	0
$583$	−11555.2	−0.820874
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25194.9	−1.77156	−0.885781	−	0.464104i	$-0.846377\pi$
$587$	−25194.9	−1.77156	−0.885781	+	0.464104i	$0.846377\pi$
$588$	0	0
$589$	−14083.6	−0.985235
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24086.1	1.66796	0.833978	−	0.551798i	$-0.186058\pi$
$593$	24086.1	1.66796	0.833978	+	0.551798i	$0.186058\pi$
$594$	0	0
$595$	2223.64	0.153210
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27834.8	1.89866	0.949332	−	0.314275i	$-0.101762\pi$
$599$	27834.8	1.89866	0.949332	+	0.314275i	$0.101762\pi$
$600$	0	0
$601$	−16113.8	−1.09367	−0.546833	−	0.837241i	$-0.684167\pi$
$601$	−16113.8	−1.09367	−0.546833	+	0.837241i	$0.684167\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4545.41	−0.305450
$606$	0	0
$607$	2925.98	0.195654	0.0978268	−	0.995203i	$-0.468811\pi$
$607$	2925.98	0.195654	0.0978268	+	0.995203i	$0.468811\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10359.7	0.685939
$612$	0	0
$613$	−11350.9	−0.747896	−0.373948	−	0.927450i	$-0.621996\pi$
$613$	−11350.9	−0.747896	−0.373948	+	0.927450i	$0.621996\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15815.2	1.03192	0.515961	−	0.856612i	$-0.327435\pi$
$617$	15815.2	1.03192	0.515961	+	0.856612i	$0.327435\pi$
$618$	0	0
$619$	12328.8	0.800544	0.400272	−	0.916396i	$-0.368916\pi$
$619$	12328.8	0.800544	0.400272	+	0.916396i	$0.368916\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1902.14	0.122323
$624$	0	0
$625$	−9544.99	−0.610879
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4563.08	−0.289256
$630$	0	0
$631$	−25545.6	−1.61165	−0.805826	−	0.592152i	$-0.798278\pi$
$631$	−25545.6	−1.61165	−0.805826	+	0.592152i	$0.798278\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19427.3	−1.21409
$636$	0	0
$637$	−10048.8	−0.625038
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12723.3	−0.783995	−0.391998	−	0.919966i	$-0.628216\pi$
$641$	−12723.3	−0.783995	−0.391998	+	0.919966i	$0.628216\pi$
$642$	0	0
$643$	14176.4	0.869457	0.434729	−	0.900561i	$-0.356844\pi$
$643$	14176.4	0.869457	0.434729	+	0.900561i	$0.356844\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25409.1	−1.54395	−0.771975	−	0.635653i	$-0.780731\pi$
$647$	−25409.1	−1.54395	−0.771975	+	0.635653i	$0.780731\pi$
$648$	0	0
$649$	10994.0	0.664952
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13496.4	−0.808816	−0.404408	−	0.914579i	$-0.632522\pi$
$653$	−13496.4	−0.808816	−0.404408	+	0.914579i	$0.632522\pi$
$654$	0	0
$655$	−16604.6	−0.990530
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12095.1	−0.714958	−0.357479	−	0.933921i	$-0.616364\pi$
$659$	−12095.1	−0.714958	−0.357479	+	0.933921i	$0.616364\pi$
$660$	0	0
$661$	−10004.7	−0.588713	−0.294356	−	0.955696i	$-0.595105\pi$
$661$	−10004.7	−0.588713	−0.294356	+	0.955696i	$0.595105\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5771.11	0.336533
$666$	0	0
$667$	5641.42	0.327491
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8650.04	0.497662
$672$	0	0
$673$	9911.06	0.567672	0.283836	−	0.958873i	$-0.408393\pi$
$673$	9911.06	0.567672	0.283836	+	0.958873i	$0.408393\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18849.5	−1.07008	−0.535040	−	0.844827i	$-0.679703\pi$
$677$	−18849.5	−1.07008	−0.535040	+	0.844827i	$0.679703\pi$
$678$	0	0
$679$	−10745.3	−0.607312
$680$	0	0
$681$	0	0
$682$	0	0
$683$	17224.5	0.964973	0.482486	−	0.875903i	$-0.339734\pi$
$683$	17224.5	0.964973	0.482486	+	0.875903i	$0.339734\pi$
$684$	0	0
$685$	11329.4	0.631932
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14621.8	0.808487
$690$	0	0
$691$	7954.63	0.437928	0.218964	−	0.975733i	$-0.429732\pi$
$691$	7954.63	0.437928	0.218964	+	0.975733i	$0.429732\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6861.41	0.374487
$696$	0	0
$697$	−5958.57	−0.323812
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3771.49	0.203206	0.101603	−	0.994825i	$-0.467603\pi$
$701$	3771.49	0.203206	0.101603	+	0.994825i	$0.467603\pi$
$702$	0	0
$703$	−11842.8	−0.635362
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14080.7	−0.749023
$708$	0	0
$709$	282.665	0.0149728	0.00748640	−	0.999972i	$-0.497617\pi$
$709$	282.665	0.0149728	0.00748640	+	0.999972i	$0.497617\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5822.21	0.305811
$714$	0	0
$715$	−10009.0	−0.523517
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8779.02	−0.455358	−0.227679	−	0.973736i	$-0.573114\pi$
$719$	−8779.02	−0.455358	−0.227679	+	0.973736i	$0.573114\pi$
$720$	0	0
$721$	−7140.94	−0.368852
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6922.39	−0.354608
$726$	0	0
$727$	2485.48	0.126797	0.0633985	−	0.997988i	$-0.479806\pi$
$727$	2485.48	0.126797	0.0633985	+	0.997988i	$0.479806\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3324.91	0.168230
$732$	0	0
$733$	−10179.6	−0.512947	−0.256474	−	0.966551i	$-0.582561\pi$
$733$	−10179.6	−0.512947	−0.256474	+	0.966551i	$0.582561\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24071.0	−1.20307
$738$	0	0
$739$	−3318.26	−0.165175	−0.0825874	−	0.996584i	$-0.526318\pi$
$739$	−3318.26	−0.165175	−0.0825874	+	0.996584i	$0.526318\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2666.67	0.131670	0.0658350	−	0.997831i	$-0.479029\pi$
$743$	2666.67	0.131670	0.0658350	+	0.997831i	$0.479029\pi$
$744$	0	0
$745$	−4824.53	−0.237258
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11447.9	0.558475
$750$	0	0
$751$	−19537.6	−0.949317	−0.474659	−	0.880170i	$-0.657428\pi$
$751$	−19537.6	−0.949317	−0.474659	+	0.880170i	$0.657428\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10797.9	0.520498
$756$	0	0
$757$	−30037.2	−1.44217	−0.721084	−	0.692848i	$-0.756356\pi$
$757$	−30037.2	−1.44217	−0.721084	+	0.692848i	$0.756356\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−41154.3	−1.96037	−0.980184	−	0.198087i	$-0.936527\pi$
$761$	−41154.3	−1.96037	−0.980184	+	0.198087i	$0.936527\pi$
$762$	0	0
$763$	−12328.3	−0.584948
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13911.7	−0.654917
$768$	0	0
$769$	21878.1	1.02594	0.512969	−	0.858407i	$-0.328546\pi$
$769$	21878.1	1.02594	0.512969	+	0.858407i	$0.328546\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−23321.1	−1.08512	−0.542561	−	0.840016i	$-0.682545\pi$
$773$	−23321.1	−1.08512	−0.542561	+	0.840016i	$0.682545\pi$
$774$	0	0
$775$	−7144.23	−0.331133
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−15464.6	−0.711266
$780$	0	0
$781$	21196.7	0.971161
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12815.6	−0.582686
$786$	0	0
$787$	−16311.6	−0.738811	−0.369406	−	0.929268i	$-0.620439\pi$
$787$	−16311.6	−0.738811	−0.369406	+	0.929268i	$0.620439\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	75.3780	0.00338829
$792$	0	0
$793$	−10945.6	−0.490152
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22961.5	1.02050	0.510249	−	0.860027i	$-0.329553\pi$
$797$	22961.5	1.02050	0.510249	+	0.860027i	$0.329553\pi$
$798$	0	0
$799$	−8006.16	−0.354490
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−32139.0	−1.41240
$804$	0	0
$805$	−2385.80	−0.104458
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30987.4	1.34667	0.673337	−	0.739336i	$-0.264860\pi$
$809$	30987.4	1.34667	0.673337	+	0.739336i	$0.264860\pi$
$810$	0	0
$811$	40596.9	1.75777	0.878884	−	0.477036i	$-0.158289\pi$
$811$	40596.9	1.75777	0.878884	+	0.477036i	$0.158289\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	35530.7	1.52710
$816$	0	0
$817$	8629.30	0.369524
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7873.46	−0.334696	−0.167348	−	0.985898i	$-0.553520\pi$
$821$	−7873.46	−0.334696	−0.167348	+	0.985898i	$0.553520\pi$
$822$	0	0
$823$	−19772.7	−0.837463	−0.418732	−	0.908110i	$-0.637525\pi$
$823$	−19772.7	−0.837463	−0.418732	+	0.908110i	$0.637525\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33412.4	1.40492	0.702458	−	0.711726i	$-0.252086\pi$
$827$	33412.4	1.40492	0.702458	+	0.711726i	$0.252086\pi$
$828$	0	0
$829$	681.873	0.0285675	0.0142837	−	0.999898i	$-0.495453\pi$
$829$	681.873	0.0285675	0.0142837	+	0.999898i	$0.495453\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7765.91	0.323017
$834$	0	0
$835$	19173.2	0.794628
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25234.6	1.03837	0.519187	−	0.854661i	$-0.326235\pi$
$839$	25234.6	1.03837	0.519187	+	0.854661i	$0.326235\pi$
$840$	0	0
$841$	9812.32	0.402326
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7893.91	−0.321371
$846$	0	0
$847$	4059.68	0.164690
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4895.86	0.197213
$852$	0	0
$853$	−15976.7	−0.641305	−0.320652	−	0.947197i	$-0.603902\pi$
$853$	−15976.7	−0.641305	−0.320652	+	0.947197i	$0.603902\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19615.2	0.781845	0.390923	−	0.920424i	$-0.372156\pi$
$857$	19615.2	0.781845	0.390923	+	0.920424i	$0.372156\pi$
$858$	0	0
$859$	15753.6	0.625735	0.312867	−	0.949797i	$-0.398710\pi$
$859$	15753.6	0.625735	0.312867	+	0.949797i	$0.398710\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−34943.9	−1.37834	−0.689169	−	0.724601i	$-0.742024\pi$
$863$	−34943.9	−1.37834	−0.689169	+	0.724601i	$0.742024\pi$
$864$	0	0
$865$	18951.7	0.744946
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−30941.9	−1.20786
$870$	0	0
$871$	30459.0	1.18492
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12703.9	0.490823
$876$	0	0
$877$	25979.3	1.00029	0.500147	−	0.865941i	$-0.333279\pi$
$877$	25979.3	1.00029	0.500147	+	0.865941i	$0.333279\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−36174.8	−1.38338	−0.691692	−	0.722193i	$-0.743134\pi$
$881$	−36174.8	−1.38338	−0.691692	+	0.722193i	$0.743134\pi$
$882$	0	0
$883$	33756.1	1.28651	0.643253	−	0.765654i	$-0.277584\pi$
$883$	33756.1	1.28651	0.643253	+	0.765654i	$0.277584\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31394.4	1.18841	0.594206	−	0.804313i	$-0.297466\pi$
$887$	31394.4	1.18841	0.594206	+	0.804313i	$0.297466\pi$
$888$	0	0
$889$	17351.2	0.654603
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−20778.8	−0.778651
$894$	0	0
$895$	−29407.0	−1.09829
$896$	0	0
$897$	0	0
$898$	0	0
$899$	35297.4	1.30949
$900$	0	0
$901$	−11300.0	−0.417822
$902$	0	0
$903$	0	0
$904$	0	0
$905$	352.495	0.0129473
$906$	0	0
$907$	−46300.1	−1.69500	−0.847502	−	0.530792i	$-0.821894\pi$
$907$	−46300.1	−1.69500	−0.847502	+	0.530792i	$0.821894\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	53195.3	1.93462	0.967310	−	0.253597i	$-0.0816137\pi$
$911$	53195.3	1.93462	0.967310	+	0.253597i	$0.0816137\pi$
$912$	0	0
$913$	17167.2	0.622292
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14830.2	0.534065
$918$	0	0
$919$	32346.8	1.16107	0.580534	−	0.814236i	$-0.302844\pi$
$919$	32346.8	1.16107	0.580534	+	0.814236i	$0.302844\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−26821.9	−0.956506
$924$	0	0
$925$	−6007.54	−0.213542
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28956.5	−1.02264	−0.511319	−	0.859391i	$-0.670843\pi$
$929$	−28956.5	−1.02264	−0.511319	+	0.859391i	$0.670843\pi$
$930$	0	0
$931$	20155.3	0.709519
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7735.12	0.270551
$936$	0	0
$937$	−20401.2	−0.711290	−0.355645	−	0.934621i	$-0.615739\pi$
$937$	−20401.2	−0.711290	−0.355645	+	0.934621i	$0.615739\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12805.8	0.443632	0.221816	−	0.975089i	$-0.428802\pi$
$941$	12805.8	0.443632	0.221816	+	0.975089i	$0.428802\pi$
$942$	0	0
$943$	6393.12	0.220773
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47164.7	1.61842	0.809211	−	0.587518i	$-0.199895\pi$
$947$	47164.7	1.61842	0.809211	+	0.587518i	$0.199895\pi$
$948$	0	0
$949$	40668.2	1.39109
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−22392.5	−0.761138	−0.380569	−	0.924752i	$-0.624272\pi$
$953$	−22392.5	−0.761138	−0.380569	+	0.924752i	$0.624272\pi$
$954$	0	0
$955$	39622.1	1.34256
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10118.7	−0.340719
$960$	0	0
$961$	6637.51	0.222802
$962$	0	0
$963$	0	0
$964$	0	0
$965$	35833.9	1.19537
$966$	0	0
$967$	25368.9	0.843648	0.421824	−	0.906678i	$-0.361390\pi$
$967$	25368.9	0.843648	0.421824	+	0.906678i	$0.361390\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19705.7	−0.651271	−0.325636	−	0.945495i	$-0.605578\pi$
$971$	−19705.7	−0.651271	−0.325636	+	0.945495i	$0.605578\pi$
$972$	0	0
$973$	−6128.18	−0.201912
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41854.6	−1.37057	−0.685285	−	0.728275i	$-0.740322\pi$
$977$	−41854.6	−1.37057	−0.685285	+	0.728275i	$0.740322\pi$
$978$	0	0
$979$	6616.76	0.216009
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17509.1	0.568112	0.284056	−	0.958808i	$-0.408320\pi$
$983$	17509.1	0.568112	0.284056	+	0.958808i	$0.408320\pi$
$984$	0	0
$985$	−10035.2	−0.324616
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3567.39	−0.114698
$990$	0	0
$991$	−32528.3	−1.04268	−0.521340	−	0.853349i	$-0.674568\pi$
$991$	−32528.3	−1.04268	−0.521340	+	0.853349i	$0.674568\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10092.7	0.321568
$996$	0	0
$997$	17951.6	0.570243	0.285121	−	0.958491i	$-0.407966\pi$
$997$	17951.6	0.570243	0.285121	+	0.958491i	$0.407966\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.n.1.2		2
3.2	odd	2		1296.4.a.p.1.1		2
4.3	odd	2		648.4.a.d.1.2	✓	2
12.11	even	2		648.4.a.e.1.1	yes	2
36.7	odd	6		648.4.i.q.433.1		4
36.11	even	6		648.4.i.p.433.2		4
36.23	even	6		648.4.i.p.217.2		4
36.31	odd	6		648.4.i.q.217.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.d.1.2	✓	2	4.3	odd	2
648.4.a.e.1.1	yes	2	12.11	even	2
648.4.i.p.217.2		4	36.23	even	6
648.4.i.p.433.2		4	36.11	even	6
648.4.i.q.217.1		4	36.31	odd	6
648.4.i.q.433.1		4	36.7	odd	6
1296.4.a.n.1.2		2	1.1	even	1	trivial
1296.4.a.p.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

\(f(q)\)	\(=\)	\(q+9.35782 q^{5} -8.35782 q^{7} +O(q^{10})\)	\(q+9.35782 q^{5} -8.35782 q^{7} -29.0735 q^{11} +36.7891 q^{13} -28.4313 q^{17} -73.7891 q^{19} +30.5047 q^{23} -37.4313 q^{25} +184.936 q^{29} +190.863 q^{31} -78.2109 q^{35} +160.495 q^{37} +209.578 q^{41} -116.945 q^{43} +281.597 q^{47} -273.147 q^{49} +397.450 q^{53} -272.064 q^{55} -378.147 q^{59} -297.524 q^{61} +344.265 q^{65} +827.936 q^{67} -729.073 q^{71} +1105.44 q^{73} +242.991 q^{77} +1064.27 q^{79} -590.478 q^{83} -266.055 q^{85} -227.588 q^{89} -307.476 q^{91} -690.505 q^{95} +1285.65 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 6 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 + 6 * q^7	\( 2 q - 4 q^{5} + 6 q^{7} + 10 q^{11} - 40 q^{13} + 34 q^{17} - 34 q^{19} - 98 q^{23} + 16 q^{25} + 120 q^{29} + 200 q^{31} - 270 q^{35} + 480 q^{37} + 192 q^{41} + 334 q^{43} - 300 q^{47} - 410 q^{49} + 68 q^{53} - 794 q^{55} - 620 q^{59} + 200 q^{61} + 1370 q^{65} + 1406 q^{67} - 1390 q^{71} + 1802 q^{73} + 804 q^{77} + 334 q^{79} + 500 q^{83} - 1100 q^{85} + 90 q^{89} - 1410 q^{91} - 1222 q^{95} - 200 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 6 * q^7 + 10 * q^11 - 40 * q^13 + 34 * q^17 - 34 * q^19 - 98 * q^23 + 16 * q^25 + 120 * q^29 + 200 * q^31 - 270 * q^35 + 480 * q^37 + 192 * q^41 + 334 * q^43 - 300 * q^47 - 410 * q^49 + 68 * q^53 - 794 * q^55 - 620 * q^59 + 200 * q^61 + 1370 * q^65 + 1406 * q^67 - 1390 * q^71 + 1802 * q^73 + 804 * q^77 + 334 * q^79 + 500 * q^83 - 1100 * q^85 + 90 * q^89 - 1410 * q^91 - 1222 * q^95 - 200 * q^97

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