LMFDB - Embedded newform 1296.4.a.v.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.1509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 4 $ x^3 - x^2 - 7*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 36)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.92542$ 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)

$f(q)$	$=$	$q-4.89803 q^{5} +10.6545 q^{7} +O(q^{10})$	$q-4.89803 q^{5} +10.6545 q^{7} -35.4537 q^{11} +72.7003 q^{13} -127.417 q^{17} +46.3913 q^{19} +131.148 q^{23} -101.009 q^{25} -137.510 q^{29} +106.128 q^{31} -52.1861 q^{35} +137.401 q^{37} -71.7971 q^{41} -376.919 q^{43} +613.626 q^{47} -229.481 q^{49} -431.757 q^{53} +173.653 q^{55} +285.755 q^{59} -43.9364 q^{61} -356.088 q^{65} +45.2103 q^{67} -357.328 q^{71} +530.718 q^{73} -377.741 q^{77} +195.108 q^{79} -760.704 q^{83} +624.093 q^{85} -1214.67 q^{89} +774.586 q^{91} -227.226 q^{95} -1104.80 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{5} - 6 q^{7}+O(q^{10}) $ 3 * q - 6 * q^5 - 6 * q^7	$ 3 q - 6 q^{5} - 6 q^{7} + 51 q^{11} - 12 q^{13} - 111 q^{17} - 15 q^{19} + 210 q^{23} + 3 q^{25} - 456 q^{29} + 48 q^{31} + 552 q^{35} - 48 q^{37} - 897 q^{41} + 129 q^{43} + 522 q^{47} + 225 q^{49} - 1104 q^{53} + 108 q^{55} + 453 q^{59} + 402 q^{61} - 1110 q^{65} - 213 q^{67} - 60 q^{71} + 375 q^{73} - 1128 q^{77} + 552 q^{79} - 612 q^{83} - 1188 q^{85} - 462 q^{89} + 132 q^{91} - 2184 q^{95} - 93 q^{97}+O(q^{100}) $ 3 * q - 6 * q^5 - 6 * q^7 + 51 * q^11 - 12 * q^13 - 111 * q^17 - 15 * q^19 + 210 * q^23 + 3 * q^25 - 456 * q^29 + 48 * q^31 + 552 * q^35 - 48 * q^37 - 897 * q^41 + 129 * q^43 + 522 * q^47 + 225 * q^49 - 1104 * q^53 + 108 * q^55 + 453 * q^59 + 402 * q^61 - 1110 * q^65 - 213 * q^67 - 60 * q^71 + 375 * q^73 - 1128 * q^77 + 552 * q^79 - 612 * q^83 - 1188 * q^85 - 462 * q^89 + 132 * q^91 - 2184 * q^95 - 93 * 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
$5$	−4.89803	−0.438093	−0.219047	−	0.975714i	$-0.570295\pi$
$5$	−4.89803	−0.438093	−0.219047	+	0.975714i	$0.570295\pi$
$6$	0	0
$7$	10.6545	0.575289	0.287645	−	0.957737i	$-0.407128\pi$
$7$	10.6545	0.575289	0.287645	+	0.957737i	$0.407128\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−35.4537	−0.971789	−0.485895	−	0.874017i	$-0.661506\pi$
$11$	−35.4537	−0.971789	−0.485895	+	0.874017i	$0.661506\pi$
$12$	0	0
$13$	72.7003	1.55103	0.775517	−	0.631327i	$-0.217489\pi$
$13$	72.7003	1.55103	0.775517	+	0.631327i	$0.217489\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−127.417	−1.81784	−0.908918	−	0.416975i	$-0.863090\pi$
$17$	−127.417	−1.81784	−0.908918	+	0.416975i	$0.863090\pi$
$18$	0	0
$19$	46.3913	0.560152	0.280076	−	0.959978i	$-0.409640\pi$
$19$	46.3913	0.560152	0.280076	+	0.959978i	$0.409640\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	131.148	1.18897	0.594483	−	0.804109i	$-0.297357\pi$
$23$	131.148	1.18897	0.594483	+	0.804109i	$0.297357\pi$
$24$	0	0
$25$	−101.009	−0.808074
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−137.510	−0.880515	−0.440258	−	0.897872i	$-0.645113\pi$
$29$	−137.510	−0.880515	−0.440258	+	0.897872i	$0.645113\pi$
$30$	0	0
$31$	106.128	0.614876	0.307438	−	0.951568i	$-0.400528\pi$
$31$	106.128	0.614876	0.307438	+	0.951568i	$0.400528\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−52.1861	−0.252030
$36$	0	0
$37$	137.401	0.610500	0.305250	−	0.952272i	$-0.401260\pi$
$37$	137.401	0.610500	0.305250	+	0.952272i	$0.401260\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−71.7971	−0.273484	−0.136742	−	0.990607i	$-0.543663\pi$
$41$	−71.7971	−0.273484	−0.136742	+	0.990607i	$0.543663\pi$
$42$	0	0
$43$	−376.919	−1.33673	−0.668367	−	0.743832i	$-0.733006\pi$
$43$	−376.919	−1.33673	−0.668367	+	0.743832i	$0.733006\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	613.626	1.90440	0.952198	−	0.305482i	$-0.0988174\pi$
$47$	613.626	1.90440	0.952198	+	0.305482i	$0.0988174\pi$
$48$	0	0
$49$	−229.481	−0.669042
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−431.757	−1.11899	−0.559494	−	0.828835i	$-0.689004\pi$
$53$	−431.757	−1.11899	−0.559494	+	0.828835i	$0.689004\pi$
$54$	0	0
$55$	173.653	0.425734
$56$	0	0
$57$	0	0
$58$	0	0
$59$	285.755	0.630545	0.315272	−	0.949001i	$-0.397904\pi$
$59$	285.755	0.630545	0.315272	+	0.949001i	$0.397904\pi$
$60$	0	0
$61$	−43.9364	−0.0922209	−0.0461104	−	0.998936i	$-0.514683\pi$
$61$	−43.9364	−0.0922209	−0.0461104	+	0.998936i	$0.514683\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−356.088	−0.679497
$66$	0	0
$67$	45.2103	0.0824376	0.0412188	−	0.999150i	$-0.486876\pi$
$67$	45.2103	0.0824376	0.0412188	+	0.999150i	$0.486876\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−357.328	−0.597282	−0.298641	−	0.954366i	$-0.596533\pi$
$71$	−357.328	−0.597282	−0.298641	+	0.954366i	$0.596533\pi$
$72$	0	0
$73$	530.718	0.850901	0.425451	−	0.904982i	$-0.360116\pi$
$73$	530.718	0.850901	0.425451	+	0.904982i	$0.360116\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−377.741	−0.559060
$78$	0	0
$79$	195.108	0.277865	0.138933	−	0.990302i	$-0.455633\pi$
$79$	195.108	0.277865	0.138933	+	0.990302i	$0.455633\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−760.704	−1.00600	−0.503001	−	0.864286i	$-0.667771\pi$
$83$	−760.704	−1.00600	−0.503001	+	0.864286i	$0.667771\pi$
$84$	0	0
$85$	624.093	0.796381
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1214.67	−1.44668	−0.723339	−	0.690493i	$-0.757393\pi$
$89$	−1214.67	−1.44668	−0.723339	+	0.690493i	$0.757393\pi$
$90$	0	0
$91$	774.586	0.892293
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−227.226	−0.245399
$96$	0	0
$97$	−1104.80	−1.15645	−0.578226	−	0.815877i	$-0.696255\pi$
$97$	−1104.80	−1.15645	−0.578226	+	0.815877i	$0.696255\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1017.92	1.00284	0.501421	−	0.865203i	$-0.332811\pi$
$101$	1017.92	1.00284	0.501421	+	0.865203i	$0.332811\pi$
$102$	0	0
$103$	−832.257	−0.796162	−0.398081	−	0.917350i	$-0.630324\pi$
$103$	−832.257	−0.796162	−0.398081	+	0.917350i	$0.630324\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−481.992	−0.435476	−0.217738	−	0.976007i	$-0.569868\pi$
$107$	−481.992	−0.435476	−0.217738	+	0.976007i	$0.569868\pi$
$108$	0	0
$109$	−904.531	−0.794847	−0.397424	−	0.917635i	$-0.630096\pi$
$109$	−904.531	−0.794847	−0.397424	+	0.917635i	$0.630096\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	800.987	0.666819	0.333410	−	0.942782i	$-0.391801\pi$
$113$	800.987	0.666819	0.333410	+	0.942782i	$0.391801\pi$
$114$	0	0
$115$	−642.366	−0.520877
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1357.57	−1.04578
$120$	0	0
$121$	−74.0372	−0.0556253
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1107.00	0.792105
$126$	0	0
$127$	−1755.04	−1.22626	−0.613129	−	0.789982i	$-0.710090\pi$
$127$	−1755.04	−1.22626	−0.613129	+	0.789982i	$0.710090\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1103.74	−0.736140	−0.368070	−	0.929798i	$-0.619981\pi$
$131$	−1103.74	−0.736140	−0.368070	+	0.929798i	$0.619981\pi$
$132$	0	0
$133$	494.276	0.322250
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2126.51	−1.32613	−0.663064	−	0.748563i	$-0.730744\pi$
$137$	−2126.51	−1.32613	−0.663064	+	0.748563i	$0.730744\pi$
$138$	0	0
$139$	2255.47	1.37630	0.688151	−	0.725567i	$-0.258423\pi$
$139$	2255.47	1.37630	0.688151	+	0.725567i	$0.258423\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2577.49	−1.50728
$144$	0	0
$145$	673.527	0.385748
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−685.450	−0.376874	−0.188437	−	0.982085i	$-0.560342\pi$
$149$	−685.450	−0.376874	−0.188437	+	0.982085i	$0.560342\pi$
$150$	0	0
$151$	−540.330	−0.291201	−0.145601	−	0.989343i	$-0.546511\pi$
$151$	−540.330	−0.291201	−0.145601	+	0.989343i	$0.546511\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−519.818	−0.269373
$156$	0	0
$157$	−473.327	−0.240609	−0.120304	−	0.992737i	$-0.538387\pi$
$157$	−473.327	−0.240609	−0.120304	+	0.992737i	$0.538387\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1397.31	0.683999
$162$	0	0
$163$	198.981	0.0956160	0.0478080	−	0.998857i	$-0.484776\pi$
$163$	198.981	0.0956160	0.0478080	+	0.998857i	$0.484776\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4275.65	−1.98120	−0.990598	−	0.136808i	$-0.956316\pi$
$167$	−4275.65	−1.98120	−0.990598	+	0.136808i	$0.956316\pi$
$168$	0	0
$169$	3088.33	1.40570
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.1047	0.00663811	0.00331905	−	0.999994i	$-0.498944\pi$
$173$	15.1047	0.00663811	0.00331905	+	0.999994i	$0.498944\pi$
$174$	0	0
$175$	−1076.20	−0.464877
$176$	0	0
$177$	0	0
$178$	0	0
$179$	309.915	0.129408	0.0647042	−	0.997904i	$-0.479390\pi$
$179$	309.915	0.129408	0.0647042	+	0.997904i	$0.479390\pi$
$180$	0	0
$181$	−2253.32	−0.925348	−0.462674	−	0.886529i	$-0.653110\pi$
$181$	−2253.32	−0.925348	−0.462674	+	0.886529i	$0.653110\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−672.992	−0.267456
$186$	0	0
$187$	4517.41	1.76655
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3698.56	−1.40114	−0.700571	−	0.713583i	$-0.747071\pi$
$191$	−3698.56	−1.40114	−0.700571	+	0.713583i	$0.747071\pi$
$192$	0	0
$193$	−3563.28	−1.32896	−0.664482	−	0.747304i	$-0.731348\pi$
$193$	−3563.28	−1.32896	−0.664482	+	0.747304i	$0.731348\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	89.0014	0.0321883	0.0160941	−	0.999870i	$-0.494877\pi$
$197$	89.0014	0.0321883	0.0160941	+	0.999870i	$0.494877\pi$
$198$	0	0
$199$	−287.103	−0.102272	−0.0511362	−	0.998692i	$-0.516284\pi$
$199$	−287.103	−0.102272	−0.0511362	+	0.998692i	$0.516284\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1465.10	−0.506551
$204$	0	0
$205$	351.664	0.119811
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1644.74	−0.544350
$210$	0	0
$211$	−5059.86	−1.65088	−0.825438	−	0.564492i	$-0.809072\pi$
$211$	−5059.86	−1.65088	−0.825438	+	0.564492i	$0.809072\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1846.16	0.585614
$216$	0	0
$217$	1130.74	0.353732
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9263.27	−2.81952
$222$	0	0
$223$	399.714	0.120031	0.0600153	−	0.998197i	$-0.480885\pi$
$223$	399.714	0.120031	0.0600153	+	0.998197i	$0.480885\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	375.418	0.109768	0.0548840	−	0.998493i	$-0.482521\pi$
$227$	375.418	0.109768	0.0548840	+	0.998493i	$0.482521\pi$
$228$	0	0
$229$	4826.56	1.39279	0.696394	−	0.717660i	$-0.254787\pi$
$229$	4826.56	1.39279	0.696394	+	0.717660i	$0.254787\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3858.19	−1.08480	−0.542401	−	0.840120i	$-0.682484\pi$
$233$	−3858.19	−1.08480	−0.542401	+	0.840120i	$0.682484\pi$
$234$	0	0
$235$	−3005.56	−0.834303
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1305.21	−0.353250	−0.176625	−	0.984278i	$-0.556518\pi$
$239$	−1305.21	−0.353250	−0.176625	+	0.984278i	$0.556518\pi$
$240$	0	0
$241$	−2216.14	−0.592340	−0.296170	−	0.955135i	$-0.595710\pi$
$241$	−2216.14	−0.592340	−0.296170	+	0.955135i	$0.595710\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1124.01	0.293103
$246$	0	0
$247$	3372.66	0.868815
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2993.80	0.752856	0.376428	−	0.926446i	$-0.377152\pi$
$251$	2993.80	0.752856	0.376428	+	0.926446i	$0.377152\pi$
$252$	0	0
$253$	−4649.67	−1.15542
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2935.03	0.712382	0.356191	−	0.934413i	$-0.384075\pi$
$257$	2935.03	0.712382	0.356191	+	0.934413i	$0.384075\pi$
$258$	0	0
$259$	1463.94	0.351214
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1553.77	−0.364294	−0.182147	−	0.983271i	$-0.558305\pi$
$263$	−1553.77	−0.364294	−0.182147	+	0.983271i	$0.558305\pi$
$264$	0	0
$265$	2114.76	0.490221
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1762.62	−0.399513	−0.199756	−	0.979846i	$-0.564015\pi$
$269$	−1762.62	−0.399513	−0.199756	+	0.979846i	$0.564015\pi$
$270$	0	0
$271$	6924.63	1.55218	0.776091	−	0.630621i	$-0.217200\pi$
$271$	6924.63	1.55218	0.776091	+	0.630621i	$0.217200\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3581.15	0.785278
$276$	0	0
$277$	6024.95	1.30687	0.653437	−	0.756981i	$-0.273326\pi$
$277$	6024.95	1.30687	0.653437	+	0.756981i	$0.273326\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6337.48	−1.34542	−0.672709	−	0.739907i	$-0.734869\pi$
$281$	−6337.48	−1.34542	−0.672709	+	0.739907i	$0.734869\pi$
$282$	0	0
$283$	1968.68	0.413518	0.206759	−	0.978392i	$-0.433708\pi$
$283$	1968.68	0.413518	0.206759	+	0.978392i	$0.433708\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−764.963	−0.157332
$288$	0	0
$289$	11322.1	2.30453
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1174.42	−0.234164	−0.117082	−	0.993122i	$-0.537354\pi$
$293$	−1174.42	−0.234164	−0.117082	+	0.993122i	$0.537354\pi$
$294$	0	0
$295$	−1399.64	−0.276237
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9534.48	1.84412
$300$	0	0
$301$	−4015.88	−0.769009
$302$	0	0
$303$	0	0
$304$	0	0
$305$	215.202	0.0404013
$306$	0	0
$307$	−3258.92	−0.605851	−0.302926	−	0.953014i	$-0.597963\pi$
$307$	−3258.92	−0.605851	−0.302926	+	0.953014i	$0.597963\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7958.02	1.45099	0.725495	−	0.688228i	$-0.241611\pi$
$311$	7958.02	1.45099	0.725495	+	0.688228i	$0.241611\pi$
$312$	0	0
$313$	−4196.25	−0.757784	−0.378892	−	0.925441i	$-0.623695\pi$
$313$	−4196.25	−0.757784	−0.378892	+	0.925441i	$0.623695\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4833.56	−0.856404	−0.428202	−	0.903683i	$-0.640853\pi$
$317$	−4833.56	−0.856404	−0.428202	+	0.903683i	$0.640853\pi$
$318$	0	0
$319$	4875.23	0.855675
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5911.05	−1.01826
$324$	0	0
$325$	−7343.41	−1.25335
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6537.89	1.09558
$330$	0	0
$331$	−10540.7	−1.75036	−0.875179	−	0.483800i	$-0.839256\pi$
$331$	−10540.7	−1.75036	−0.875179	+	0.483800i	$0.839256\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−221.441	−0.0361153
$336$	0	0
$337$	−4874.90	−0.787990	−0.393995	−	0.919112i	$-0.628907\pi$
$337$	−4874.90	−0.787990	−0.393995	+	0.919112i	$0.628907\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3762.63	−0.597530
$342$	0	0
$343$	−6099.51	−0.960182
$344$	0	0
$345$	0	0
$346$	0	0
$347$	270.756	0.0418875	0.0209437	−	0.999781i	$-0.493333\pi$
$347$	270.756	0.0418875	0.0209437	+	0.999781i	$0.493333\pi$
$348$	0	0
$349$	−10440.0	−1.60126	−0.800630	−	0.599159i	$-0.795502\pi$
$349$	−10440.0	−1.60126	−0.800630	+	0.599159i	$0.795502\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	65.9254	0.00994010	0.00497005	−	0.999988i	$-0.498418\pi$
$353$	65.9254	0.00994010	0.00497005	+	0.999988i	$0.498418\pi$
$354$	0	0
$355$	1750.20	0.261665
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1973.98	−0.290203	−0.145101	−	0.989417i	$-0.546351\pi$
$359$	−1973.98	−0.290203	−0.145101	+	0.989417i	$0.546351\pi$
$360$	0	0
$361$	−4706.85	−0.686230
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2599.47	−0.372774
$366$	0	0
$367$	−4375.94	−0.622404	−0.311202	−	0.950344i	$-0.600732\pi$
$367$	−4375.94	−0.622404	−0.311202	+	0.950344i	$0.600732\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4600.15	−0.643742
$372$	0	0
$373$	10834.2	1.50395	0.751975	−	0.659192i	$-0.229102\pi$
$373$	10834.2	1.50395	0.751975	+	0.659192i	$0.229102\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9997.01	−1.36571
$378$	0	0
$379$	9390.04	1.27265	0.636325	−	0.771421i	$-0.280454\pi$
$379$	9390.04	1.27265	0.636325	+	0.771421i	$0.280454\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7133.22	−0.951673	−0.475836	−	0.879534i	$-0.657855\pi$
$383$	−7133.22	−0.951673	−0.475836	+	0.879534i	$0.657855\pi$
$384$	0	0
$385$	1850.19	0.244920
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8486.58	1.10614	0.553068	−	0.833136i	$-0.313457\pi$
$389$	8486.58	1.10614	0.553068	+	0.833136i	$0.313457\pi$
$390$	0	0
$391$	−16710.5	−2.16134
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−955.645	−0.121731
$396$	0	0
$397$	13046.4	1.64932	0.824660	−	0.565628i	$-0.191366\pi$
$397$	13046.4	1.64932	0.824660	+	0.565628i	$0.191366\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2604.06	−0.324291	−0.162146	−	0.986767i	$-0.551841\pi$
$401$	−2604.06	−0.324291	−0.162146	+	0.986767i	$0.551841\pi$
$402$	0	0
$403$	7715.54	0.953693
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4871.36	−0.593278
$408$	0	0
$409$	−7993.05	−0.966334	−0.483167	−	0.875528i	$-0.660514\pi$
$409$	−7993.05	−0.966334	−0.483167	+	0.875528i	$0.660514\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3044.58	0.362746
$414$	0	0
$415$	3725.95	0.440722
$416$	0	0
$417$	0	0
$418$	0	0
$419$	278.400	0.0324600	0.0162300	−	0.999868i	$-0.494834\pi$
$419$	278.400	0.0324600	0.0162300	+	0.999868i	$0.494834\pi$
$420$	0	0
$421$	−859.241	−0.0994699	−0.0497350	−	0.998762i	$-0.515838\pi$
$421$	−859.241	−0.0994699	−0.0497350	+	0.998762i	$0.515838\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12870.3	1.46895
$426$	0	0
$427$	−468.120	−0.0530537
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9455.34	−1.05672	−0.528362	−	0.849019i	$-0.677193\pi$
$431$	−9455.34	−1.05672	−0.528362	+	0.849019i	$0.677193\pi$
$432$	0	0
$433$	1527.61	0.169544	0.0847718	−	0.996400i	$-0.472984\pi$
$433$	1527.61	0.169544	0.0847718	+	0.996400i	$0.472984\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6084.11	0.666001
$438$	0	0
$439$	2606.05	0.283326	0.141663	−	0.989915i	$-0.454755\pi$
$439$	2606.05	0.283326	0.141663	+	0.989915i	$0.454755\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1452.44	−0.155774	−0.0778868	−	0.996962i	$-0.524817\pi$
$443$	−1452.44	−0.155774	−0.0778868	+	0.996962i	$0.524817\pi$
$444$	0	0
$445$	5949.47	0.633779
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12241.5	1.28666	0.643331	−	0.765589i	$-0.277552\pi$
$449$	12241.5	1.28666	0.643331	+	0.765589i	$0.277552\pi$
$450$	0	0
$451$	2545.47	0.265769
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3793.94	−0.390907
$456$	0	0
$457$	−9756.25	−0.998639	−0.499320	−	0.866418i	$-0.666417\pi$
$457$	−9756.25	−0.998639	−0.499320	+	0.866418i	$0.666417\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10244.0	1.03495	0.517474	−	0.855699i	$-0.326873\pi$
$461$	10244.0	1.03495	0.517474	+	0.855699i	$0.326873\pi$
$462$	0	0
$463$	13887.3	1.39395	0.696976	−	0.717095i	$-0.254529\pi$
$463$	13887.3	1.39395	0.696976	+	0.717095i	$0.254529\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14367.2	1.42363	0.711815	−	0.702367i	$-0.247874\pi$
$467$	14367.2	1.42363	0.711815	+	0.702367i	$0.247874\pi$
$468$	0	0
$469$	481.694	0.0474255
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13363.1	1.29902
$474$	0	0
$475$	−4685.95	−0.452645
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10371.2	0.989293	0.494647	−	0.869094i	$-0.335297\pi$
$479$	10371.2	0.989293	0.494647	+	0.869094i	$0.335297\pi$
$480$	0	0
$481$	9989.06	0.946907
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5411.36	0.506634
$486$	0	0
$487$	4805.47	0.447139	0.223569	−	0.974688i	$-0.428229\pi$
$487$	4805.47	0.447139	0.223569	+	0.974688i	$0.428229\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17549.3	1.61302	0.806508	−	0.591223i	$-0.201355\pi$
$491$	17549.3	1.61302	0.806508	+	0.591223i	$0.201355\pi$
$492$	0	0
$493$	17521.1	1.60063
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3807.15	−0.343610
$498$	0	0
$499$	794.160	0.0712454	0.0356227	−	0.999365i	$-0.488659\pi$
$499$	794.160	0.0712454	0.0356227	+	0.999365i	$0.488659\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6389.32	−0.566374	−0.283187	−	0.959065i	$-0.591392\pi$
$503$	−6389.32	−0.566374	−0.283187	+	0.959065i	$0.591392\pi$
$504$	0	0
$505$	−4985.82	−0.439338
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13871.1	1.20791	0.603953	−	0.797020i	$-0.293591\pi$
$509$	13871.1	1.20791	0.603953	+	0.797020i	$0.293591\pi$
$510$	0	0
$511$	5654.53	0.489514
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4076.42	0.348793
$516$	0	0
$517$	−21755.3	−1.85067
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13682.5	1.15056	0.575280	−	0.817956i	$-0.304893\pi$
$521$	13682.5	1.15056	0.575280	+	0.817956i	$0.304893\pi$
$522$	0	0
$523$	−8390.18	−0.701485	−0.350743	−	0.936472i	$-0.614071\pi$
$523$	−8390.18	−0.701485	−0.350743	+	0.936472i	$0.614071\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13522.5	−1.11774
$528$	0	0
$529$	5032.73	0.413638
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5219.67	−0.424182
$534$	0	0
$535$	2360.81	0.190779
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8135.96	0.650168
$540$	0	0
$541$	−3447.33	−0.273960	−0.136980	−	0.990574i	$-0.543740\pi$
$541$	−3447.33	−0.273960	−0.136980	+	0.990574i	$0.543740\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4430.42	0.348217
$546$	0	0
$547$	−14538.5	−1.13642	−0.568208	−	0.822885i	$-0.692363\pi$
$547$	−14538.5	−1.13642	−0.568208	+	0.822885i	$0.692363\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6379.26	−0.493222
$552$	0	0
$553$	2078.78	0.159853
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14894.7	1.13305	0.566525	−	0.824044i	$-0.308287\pi$
$557$	14894.7	1.13305	0.566525	+	0.824044i	$0.308287\pi$
$558$	0	0
$559$	−27402.1	−2.07332
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12137.3	0.908574	0.454287	−	0.890855i	$-0.349894\pi$
$563$	12137.3	0.908574	0.454287	+	0.890855i	$0.349894\pi$
$564$	0	0
$565$	−3923.26	−0.292129
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−520.484	−0.0383477	−0.0191738	−	0.999816i	$-0.506104\pi$
$569$	−520.484	−0.0383477	−0.0191738	+	0.999816i	$0.506104\pi$
$570$	0	0
$571$	4591.47	0.336510	0.168255	−	0.985744i	$-0.446187\pi$
$571$	4591.47	0.336510	0.168255	+	0.985744i	$0.446187\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13247.1	−0.960772
$576$	0	0
$577$	−5429.84	−0.391763	−0.195881	−	0.980628i	$-0.562757\pi$
$577$	−5429.84	−0.391763	−0.195881	+	0.980628i	$0.562757\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8104.93	−0.578742
$582$	0	0
$583$	15307.4	1.08742
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19786.7	1.39128	0.695641	−	0.718390i	$-0.255120\pi$
$587$	19786.7	1.39128	0.695641	+	0.718390i	$0.255120\pi$
$588$	0	0
$589$	4923.42	0.344424
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2181.13	0.151042	0.0755212	−	0.997144i	$-0.475938\pi$
$593$	2181.13	0.151042	0.0755212	+	0.997144i	$0.475938\pi$
$594$	0	0
$595$	6649.41	0.458150
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4087.24	−0.278798	−0.139399	−	0.990236i	$-0.544517\pi$
$599$	−4087.24	−0.278798	−0.139399	+	0.990236i	$0.544517\pi$
$600$	0	0
$601$	11684.9	0.793074	0.396537	−	0.918019i	$-0.370212\pi$
$601$	11684.9	0.793074	0.396537	+	0.918019i	$0.370212\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	362.637	0.0243690
$606$	0	0
$607$	−3730.75	−0.249467	−0.124733	−	0.992190i	$-0.539808\pi$
$607$	−3730.75	−0.249467	−0.124733	+	0.992190i	$0.539808\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	44610.8	2.95378
$612$	0	0
$613$	−2259.79	−0.148894	−0.0744470	−	0.997225i	$-0.523719\pi$
$613$	−2259.79	−0.148894	−0.0744470	+	0.997225i	$0.523719\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23227.6	−1.51557	−0.757787	−	0.652502i	$-0.773719\pi$
$617$	−23227.6	−1.51557	−0.757787	+	0.652502i	$0.773719\pi$
$618$	0	0
$619$	8399.62	0.545411	0.272705	−	0.962098i	$-0.412082\pi$
$619$	8399.62	0.545411	0.272705	+	0.962098i	$0.412082\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12941.7	−0.832258
$624$	0	0
$625$	7204.05	0.461059
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−17507.2	−1.10979
$630$	0	0
$631$	22475.2	1.41795	0.708973	−	0.705236i	$-0.249159\pi$
$631$	22475.2	1.41795	0.708973	+	0.705236i	$0.249159\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8596.25	0.537215
$636$	0	0
$637$	−16683.4	−1.03771
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8874.07	0.546809	0.273405	−	0.961899i	$-0.411850\pi$
$641$	8874.07	0.546809	0.273405	+	0.961899i	$0.411850\pi$
$642$	0	0
$643$	−21353.1	−1.30962	−0.654810	−	0.755794i	$-0.727251\pi$
$643$	−21353.1	−1.30962	−0.654810	+	0.755794i	$0.727251\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24145.6	1.46718	0.733588	−	0.679595i	$-0.237844\pi$
$647$	24145.6	1.46718	0.733588	+	0.679595i	$0.237844\pi$
$648$	0	0
$649$	−10131.1	−0.612757
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11695.3	0.700875	0.350438	−	0.936586i	$-0.386033\pi$
$653$	11695.3	0.700875	0.350438	+	0.936586i	$0.386033\pi$
$654$	0	0
$655$	5406.16	0.322498
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5828.89	0.344554	0.172277	−	0.985049i	$-0.444888\pi$
$659$	5828.89	0.344554	0.172277	+	0.985049i	$0.444888\pi$
$660$	0	0
$661$	−1621.31	−0.0954034	−0.0477017	−	0.998862i	$-0.515190\pi$
$661$	−1621.31	−0.0954034	−0.0477017	+	0.998862i	$0.515190\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2420.98	−0.141175
$666$	0	0
$667$	−18034.1	−1.04690
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1557.71	0.0896193
$672$	0	0
$673$	20376.4	1.16709	0.583545	−	0.812081i	$-0.301665\pi$
$673$	20376.4	1.16709	0.583545	+	0.812081i	$0.301665\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11350.2	−0.644348	−0.322174	−	0.946680i	$-0.604414\pi$
$677$	−11350.2	−0.644348	−0.322174	+	0.946680i	$0.604414\pi$
$678$	0	0
$679$	−11771.1	−0.665295
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3508.08	0.196534	0.0982670	−	0.995160i	$-0.468670\pi$
$683$	3508.08	0.196534	0.0982670	+	0.995160i	$0.468670\pi$
$684$	0	0
$685$	10415.7	0.580968
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−31388.8	−1.73559
$690$	0	0
$691$	17538.8	0.965565	0.482783	−	0.875740i	$-0.339626\pi$
$691$	17538.8	0.965565	0.482783	+	0.875740i	$0.339626\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11047.3	−0.602949
$696$	0	0
$697$	9148.19	0.497148
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15208.5	0.819428	0.409714	−	0.912214i	$-0.365629\pi$
$701$	15208.5	0.819428	0.409714	+	0.912214i	$0.365629\pi$
$702$	0	0
$703$	6374.19	0.341973
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10845.5	0.576925
$708$	0	0
$709$	194.480	0.0103016	0.00515081	−	0.999987i	$-0.498360\pi$
$709$	194.480	0.0103016	0.00515081	+	0.999987i	$0.498360\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13918.5	0.731066
$714$	0	0
$715$	12624.6	0.660328
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1001.11	−0.0519266	−0.0259633	−	0.999663i	$-0.508265\pi$
$719$	−1001.11	−0.0519266	−0.0259633	+	0.999663i	$0.508265\pi$
$720$	0	0
$721$	−8867.29	−0.458024
$722$	0	0
$723$	0	0
$724$	0	0
$725$	13889.8	0.711522
$726$	0	0
$727$	14985.0	0.764461	0.382231	−	0.924067i	$-0.375156\pi$
$727$	14985.0	0.764461	0.382231	+	0.924067i	$0.375156\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	48025.9	2.42996
$732$	0	0
$733$	−31698.0	−1.59726	−0.798632	−	0.601820i	$-0.794442\pi$
$733$	−31698.0	−1.59726	−0.798632	+	0.601820i	$0.794442\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1602.87	−0.0801120
$738$	0	0
$739$	1333.07	0.0663570	0.0331785	−	0.999449i	$-0.489437\pi$
$739$	1333.07	0.0663570	0.0331785	+	0.999449i	$0.489437\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2206.65	−0.108956	−0.0544780	−	0.998515i	$-0.517349\pi$
$743$	−2206.65	−0.108956	−0.0544780	+	0.998515i	$0.517349\pi$
$744$	0	0
$745$	3357.36	0.165106
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5135.39	−0.250525
$750$	0	0
$751$	16688.0	0.810859	0.405430	−	0.914126i	$-0.367122\pi$
$751$	16688.0	0.810859	0.405430	+	0.914126i	$0.367122\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2646.55	0.127573
$756$	0	0
$757$	17183.8	0.825039	0.412519	−	0.910949i	$-0.364649\pi$
$757$	17183.8	0.825039	0.412519	+	0.910949i	$0.364649\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18953.9	−0.902862	−0.451431	−	0.892306i	$-0.649086\pi$
$761$	−18953.9	−0.902862	−0.451431	+	0.892306i	$0.649086\pi$
$762$	0	0
$763$	−9637.33	−0.457267
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20774.5	0.977996
$768$	0	0
$769$	6248.76	0.293025	0.146512	−	0.989209i	$-0.453195\pi$
$769$	6248.76	0.293025	0.146512	+	0.989209i	$0.453195\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4856.58	0.225975	0.112988	−	0.993596i	$-0.463958\pi$
$773$	4856.58	0.225975	0.112988	+	0.993596i	$0.463958\pi$
$774$	0	0
$775$	−10719.9	−0.496866
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3330.76	−0.153192
$780$	0	0
$781$	12668.6	0.580432
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2318.37	0.105409
$786$	0	0
$787$	−16478.0	−0.746351	−0.373175	−	0.927761i	$-0.621731\pi$
$787$	−16478.0	−0.746351	−0.373175	+	0.927761i	$0.621731\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8534.13	0.383614
$792$	0	0
$793$	−3194.19	−0.143038
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32746.3	1.45537	0.727687	−	0.685910i	$-0.240596\pi$
$797$	32746.3	1.45537	0.727687	+	0.685910i	$0.240596\pi$
$798$	0	0
$799$	−78186.6	−3.46188
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−18815.9	−0.826897
$804$	0	0
$805$	−6844.09	−0.299655
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3011.82	0.130890	0.0654451	−	0.997856i	$-0.479153\pi$
$809$	3011.82	0.130890	0.0654451	+	0.997856i	$0.479153\pi$
$810$	0	0
$811$	−3560.24	−0.154151	−0.0770757	−	0.997025i	$-0.524558\pi$
$811$	−3560.24	−0.154151	−0.0770757	+	0.997025i	$0.524558\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−974.616	−0.0418887
$816$	0	0
$817$	−17485.7	−0.748774
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38081.7	−1.61883	−0.809415	−	0.587237i	$-0.800216\pi$
$821$	−38081.7	−1.61883	−0.809415	+	0.587237i	$0.800216\pi$
$822$	0	0
$823$	−5121.76	−0.216930	−0.108465	−	0.994100i	$-0.534594\pi$
$823$	−5121.76	−0.216930	−0.108465	+	0.994100i	$0.534594\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−43712.2	−1.83800	−0.918998	−	0.394261i	$-0.871000\pi$
$827$	−43712.2	−1.83800	−0.918998	+	0.394261i	$0.871000\pi$
$828$	0	0
$829$	−14707.1	−0.616161	−0.308080	−	0.951360i	$-0.599687\pi$
$829$	−14707.1	−0.616161	−0.308080	+	0.951360i	$0.599687\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	29239.9	1.21621
$834$	0	0
$835$	20942.2	0.867948
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34649.5	−1.42579	−0.712893	−	0.701273i	$-0.752616\pi$
$839$	−34649.5	−1.42579	−0.712893	+	0.701273i	$0.752616\pi$
$840$	0	0
$841$	−5480.04	−0.224693
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−15126.7	−0.615829
$846$	0	0
$847$	−788.830	−0.0320006
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18019.8	0.725864
$852$	0	0
$853$	−29740.1	−1.19377	−0.596883	−	0.802328i	$-0.703594\pi$
$853$	−29740.1	−1.19377	−0.596883	+	0.802328i	$0.703594\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1854.26	−0.0739093	−0.0369547	−	0.999317i	$-0.511766\pi$
$857$	−1854.26	−0.0739093	−0.0369547	+	0.999317i	$0.511766\pi$
$858$	0	0
$859$	46002.5	1.82722	0.913611	−	0.406589i	$-0.133282\pi$
$859$	46002.5	1.82722	0.913611	+	0.406589i	$0.133282\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38032.8	−1.50018	−0.750088	−	0.661338i	$-0.769989\pi$
$863$	−38032.8	−1.50018	−0.750088	+	0.661338i	$0.769989\pi$
$864$	0	0
$865$	−73.9835	−0.00290811
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6917.30	−0.270027
$870$	0	0
$871$	3286.80	0.127863
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11794.5	0.455690
$876$	0	0
$877$	11507.2	0.443069	0.221534	−	0.975153i	$-0.428893\pi$
$877$	11507.2	0.443069	0.221534	+	0.975153i	$0.428893\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30567.4	−1.16895	−0.584473	−	0.811413i	$-0.698699\pi$
$881$	−30567.4	−1.16895	−0.584473	+	0.811413i	$0.698699\pi$
$882$	0	0
$883$	17998.2	0.685944	0.342972	−	0.939346i	$-0.388566\pi$
$883$	17998.2	0.685944	0.342972	+	0.939346i	$0.388566\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11792.5	0.446396	0.223198	−	0.974773i	$-0.428350\pi$
$887$	11792.5	0.446396	0.223198	+	0.974773i	$0.428350\pi$
$888$	0	0
$889$	−18699.1	−0.705454
$890$	0	0
$891$	0	0
$892$	0	0
$893$	28466.9	1.06675
$894$	0	0
$895$	−1517.97	−0.0566929
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14593.7	−0.541408
$900$	0	0
$901$	55013.2	2.03414
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11036.8	0.405388
$906$	0	0
$907$	−18620.0	−0.681661	−0.340830	−	0.940125i	$-0.610708\pi$
$907$	−18620.0	−0.681661	−0.340830	+	0.940125i	$0.610708\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13597.7	−0.494526	−0.247263	−	0.968948i	$-0.579531\pi$
$911$	−13597.7	−0.494526	−0.247263	+	0.968948i	$0.579531\pi$
$912$	0	0
$913$	26969.8	0.977622
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11759.8	−0.423493
$918$	0	0
$919$	18959.3	0.680532	0.340266	−	0.940329i	$-0.389483\pi$
$919$	18959.3	0.680532	0.340266	+	0.940329i	$0.389483\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−25977.8	−0.926404
$924$	0	0
$925$	−13878.7	−0.493330
$926$	0	0
$927$	0	0
$928$	0	0
$929$	51163.3	1.80690	0.903452	−	0.428689i	$-0.141024\pi$
$929$	51163.3	1.80690	0.903452	+	0.428689i	$0.141024\pi$
$930$	0	0
$931$	−10645.9	−0.374765
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−22126.4	−0.773915
$936$	0	0
$937$	−33124.9	−1.15490	−0.577452	−	0.816425i	$-0.695953\pi$
$937$	−33124.9	−1.15490	−0.577452	+	0.816425i	$0.695953\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	16745.9	0.580128	0.290064	−	0.957007i	$-0.406323\pi$
$941$	16745.9	0.580128	0.290064	+	0.957007i	$0.406323\pi$
$942$	0	0
$943$	−9416.03	−0.325163
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31645.0	−1.08587	−0.542937	−	0.839773i	$-0.682688\pi$
$947$	−31645.0	−1.08587	−0.542937	+	0.839773i	$0.682688\pi$
$948$	0	0
$949$	38583.3	1.31978
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44345.0	1.50732	0.753660	−	0.657264i	$-0.228286\pi$
$953$	44345.0	1.50732	0.753660	+	0.657264i	$0.228286\pi$
$954$	0	0
$955$	18115.6	0.613830
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22656.9	−0.762908
$960$	0	0
$961$	−18527.8	−0.621927
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17453.0	0.582210
$966$	0	0
$967$	−11352.3	−0.377522	−0.188761	−	0.982023i	$-0.560447\pi$
$967$	−11352.3	−0.377522	−0.188761	+	0.982023i	$0.560447\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38458.3	1.27105	0.635523	−	0.772082i	$-0.280785\pi$
$971$	38458.3	1.27105	0.635523	+	0.772082i	$0.280785\pi$
$972$	0	0
$973$	24030.9	0.791772
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7365.24	−0.241182	−0.120591	−	0.992702i	$-0.538479\pi$
$977$	−7365.24	−0.241182	−0.120591	+	0.992702i	$0.538479\pi$
$978$	0	0
$979$	43064.4	1.40587
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−51741.4	−1.67884	−0.839418	−	0.543487i	$-0.817104\pi$
$983$	−51741.4	−1.67884	−0.839418	+	0.543487i	$0.817104\pi$
$984$	0	0
$985$	−435.931	−0.0141015
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−49432.0	−1.58933
$990$	0	0
$991$	−7427.40	−0.238082	−0.119041	−	0.992889i	$-0.537982\pi$
$991$	−7427.40	−0.238082	−0.119041	+	0.992889i	$0.537982\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1406.24	0.0448048
$996$	0	0
$997$	42586.2	1.35278	0.676389	−	0.736545i	$-0.263544\pi$
$997$	42586.2	1.35278	0.676389	+	0.736545i	$0.263544\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.v.1.2		3
3.2	odd	2		1296.4.a.w.1.2		3
4.3	odd	2		324.4.a.c.1.2		3
9.2	odd	6		432.4.i.d.145.2		6
9.4	even	3		144.4.i.d.97.1		6
9.5	odd	6		432.4.i.d.289.2		6
9.7	even	3		144.4.i.d.49.1		6
12.11	even	2		324.4.a.d.1.2		3
36.7	odd	6		36.4.e.a.13.3	✓	6
36.11	even	6		108.4.e.a.37.2		6
36.23	even	6		108.4.e.a.73.2		6
36.31	odd	6		36.4.e.a.25.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.4.e.a.13.3	✓	6	36.7	odd	6
36.4.e.a.25.3	yes	6	36.31	odd	6
108.4.e.a.37.2		6	36.11	even	6
108.4.e.a.73.2		6	36.23	even	6
144.4.i.d.49.1		6	9.7	even	3
144.4.i.d.97.1		6	9.4	even	3
324.4.a.c.1.2		3	4.3	odd	2
324.4.a.d.1.2		3	12.11	even	2
432.4.i.d.145.2		6	9.2	odd	6
432.4.i.d.289.2		6	9.5	odd	6
1296.4.a.v.1.2		3	1.1	even	1	trivial
1296.4.a.w.1.2		3	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)

\(f(q)\)	\(=\)	\(q-4.89803 q^{5} +10.6545 q^{7} +O(q^{10})\)	\(q-4.89803 q^{5} +10.6545 q^{7} -35.4537 q^{11} +72.7003 q^{13} -127.417 q^{17} +46.3913 q^{19} +131.148 q^{23} -101.009 q^{25} -137.510 q^{29} +106.128 q^{31} -52.1861 q^{35} +137.401 q^{37} -71.7971 q^{41} -376.919 q^{43} +613.626 q^{47} -229.481 q^{49} -431.757 q^{53} +173.653 q^{55} +285.755 q^{59} -43.9364 q^{61} -356.088 q^{65} +45.2103 q^{67} -357.328 q^{71} +530.718 q^{73} -377.741 q^{77} +195.108 q^{79} -760.704 q^{83} +624.093 q^{85} -1214.67 q^{89} +774.586 q^{91} -227.226 q^{95} -1104.80 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{5} - 6 q^{7}+O(q^{10}) \) 3 * q - 6 * q^5 - 6 * q^7	\( 3 q - 6 q^{5} - 6 q^{7} + 51 q^{11} - 12 q^{13} - 111 q^{17} - 15 q^{19} + 210 q^{23} + 3 q^{25} - 456 q^{29} + 48 q^{31} + 552 q^{35} - 48 q^{37} - 897 q^{41} + 129 q^{43} + 522 q^{47} + 225 q^{49} - 1104 q^{53} + 108 q^{55} + 453 q^{59} + 402 q^{61} - 1110 q^{65} - 213 q^{67} - 60 q^{71} + 375 q^{73} - 1128 q^{77} + 552 q^{79} - 612 q^{83} - 1188 q^{85} - 462 q^{89} + 132 q^{91} - 2184 q^{95} - 93 q^{97}+O(q^{100}) \) 3 * q - 6 * q^5 - 6 * q^7 + 51 * q^11 - 12 * q^13 - 111 * q^17 - 15 * q^19 + 210 * q^23 + 3 * q^25 - 456 * q^29 + 48 * q^31 + 552 * q^35 - 48 * q^37 - 897 * q^41 + 129 * q^43 + 522 * q^47 + 225 * q^49 - 1104 * q^53 + 108 * q^55 + 453 * q^59 + 402 * q^61 - 1110 * q^65 - 213 * q^67 - 60 * q^71 + 375 * q^73 - 1128 * q^77 + 552 * q^79 - 612 * q^83 - 1188 * q^85 - 462 * q^89 + 132 * q^91 - 2184 * q^95 - 93 * q^97

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