LMFDB - Embedded newform 648.4.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(1,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, 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("648.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.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:	$38.2332376837$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.29952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 13 $ x^4 - 8*x^2 + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.50597$ of defining polynomial
Character	$\chi$	$=$	648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.5206 q^{5} -0.354888 q^{7} +O(q^{10})$	$q+10.5206 q^{5} -0.354888 q^{7} -8.61468 q^{11} -34.7884 q^{13} -72.5080 q^{17} -78.3905 q^{19} +49.2244 q^{23} -14.3167 q^{25} -138.807 q^{29} -62.4287 q^{31} -3.73364 q^{35} +36.0226 q^{37} -190.987 q^{41} +532.379 q^{43} +70.6836 q^{47} -342.874 q^{49} -245.898 q^{53} -90.6318 q^{55} -835.717 q^{59} +480.757 q^{61} -365.996 q^{65} +110.541 q^{67} +58.4968 q^{71} -313.784 q^{73} +3.05725 q^{77} +995.222 q^{79} -733.292 q^{83} -762.828 q^{85} -994.244 q^{89} +12.3460 q^{91} -824.716 q^{95} +1058.18 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5}+O(q^{10}) $ 4 * q - 8 * q^5	$ 4 q - 8 q^{5} - 8 q^{11} + 4 q^{13} - 16 q^{17} + 80 q^{19} - 200 q^{23} + 8 q^{25} - 216 q^{29} + 80 q^{31} - 408 q^{35} - 276 q^{37} - 384 q^{41} + 160 q^{43} - 768 q^{47} - 268 q^{49} - 944 q^{53} + 304 q^{55} - 992 q^{59} - 548 q^{61} - 1328 q^{65} + 464 q^{67} - 1720 q^{71} - 764 q^{73} - 1728 q^{77} + 688 q^{79} - 2128 q^{83} - 1324 q^{85} - 2112 q^{89} + 1776 q^{91} - 2056 q^{95} - 1816 q^{97}+O(q^{100}) $ 4 * q - 8 * q^5 - 8 * q^11 + 4 * q^13 - 16 * q^17 + 80 * q^19 - 200 * q^23 + 8 * q^25 - 216 * q^29 + 80 * q^31 - 408 * q^35 - 276 * q^37 - 384 * q^41 + 160 * q^43 - 768 * q^47 - 268 * q^49 - 944 * q^53 + 304 * q^55 - 992 * q^59 - 548 * q^61 - 1328 * q^65 + 464 * q^67 - 1720 * q^71 - 764 * q^73 - 1728 * q^77 + 688 * q^79 - 2128 * q^83 - 1324 * q^85 - 2112 * q^89 + 1776 * q^91 - 2056 * q^95 - 1816 * 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
$5$	10.5206	0.940992	0.470496	−	0.882402i	$-0.344075\pi$
$5$	10.5206	0.940992	0.470496	+	0.882402i	$0.344075\pi$
$6$	0	0
$7$	−0.354888	−0.0191622	−0.00958108	−	0.999954i	$-0.503050\pi$
$7$	−0.354888	−0.0191622	−0.00958108	+	0.999954i	$0.503050\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−8.61468	−0.236130	−0.118065	−	0.993006i	$-0.537669\pi$
$11$	−8.61468	−0.236130	−0.118065	+	0.993006i	$0.537669\pi$
$12$	0	0
$13$	−34.7884	−0.742198	−0.371099	−	0.928593i	$-0.621019\pi$
$13$	−34.7884	−0.742198	−0.371099	+	0.928593i	$0.621019\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−72.5080	−1.03446	−0.517228	−	0.855847i	$-0.673036\pi$
$17$	−72.5080	−1.03446	−0.517228	+	0.855847i	$0.673036\pi$
$18$	0	0
$19$	−78.3905	−0.946527	−0.473263	−	0.880921i	$-0.656924\pi$
$19$	−78.3905	−0.946527	−0.473263	+	0.880921i	$0.656924\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	49.2244	0.446260	0.223130	−	0.974789i	$-0.428373\pi$
$23$	49.2244	0.446260	0.223130	+	0.974789i	$0.428373\pi$
$24$	0	0
$25$	−14.3167	−0.114534
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−138.807	−0.888823	−0.444411	−	0.895823i	$-0.646587\pi$
$29$	−138.807	−0.888823	−0.444411	+	0.895823i	$0.646587\pi$
$30$	0	0
$31$	−62.4287	−0.361694	−0.180847	−	0.983511i	$-0.557884\pi$
$31$	−62.4287	−0.361694	−0.180847	+	0.983511i	$0.557884\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.73364	−0.0180314
$36$	0	0
$37$	36.0226	0.160056	0.0800281	−	0.996793i	$-0.474499\pi$
$37$	36.0226	0.160056	0.0800281	+	0.996793i	$0.474499\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−190.987	−0.727492	−0.363746	−	0.931498i	$-0.618502\pi$
$41$	−190.987	−0.727492	−0.363746	+	0.931498i	$0.618502\pi$
$42$	0	0
$43$	532.379	1.88807	0.944035	−	0.329844i	$-0.106996\pi$
$43$	532.379	1.88807	0.944035	+	0.329844i	$0.106996\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	70.6836	0.219367	0.109684	−	0.993967i	$-0.465016\pi$
$47$	70.6836	0.219367	0.109684	+	0.993967i	$0.465016\pi$
$48$	0	0
$49$	−342.874	−0.999633
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−245.898	−0.637296	−0.318648	−	0.947873i	$-0.603229\pi$
$53$	−245.898	−0.637296	−0.318648	+	0.947873i	$0.603229\pi$
$54$	0	0
$55$	−90.6318	−0.222196
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−835.717	−1.84409	−0.922043	−	0.387087i	$-0.873481\pi$
$59$	−835.717	−1.84409	−0.922043	+	0.387087i	$0.873481\pi$
$60$	0	0
$61$	480.757	1.00909	0.504547	−	0.863384i	$-0.331660\pi$
$61$	480.757	1.00909	0.504547	+	0.863384i	$0.331660\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−365.996	−0.698402
$66$	0	0
$67$	110.541	0.201564	0.100782	−	0.994909i	$-0.467866\pi$
$67$	110.541	0.201564	0.100782	+	0.994909i	$0.467866\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	58.4968	0.0977787	0.0488894	−	0.998804i	$-0.484432\pi$
$71$	58.4968	0.0977787	0.0488894	+	0.998804i	$0.484432\pi$
$72$	0	0
$73$	−313.784	−0.503090	−0.251545	−	0.967846i	$-0.580939\pi$
$73$	−313.784	−0.503090	−0.251545	+	0.967846i	$0.580939\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.05725	0.00452475
$78$	0	0
$79$	995.222	1.41736	0.708679	−	0.705531i	$-0.249292\pi$
$79$	995.222	1.41736	0.708679	+	0.705531i	$0.249292\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−733.292	−0.969750	−0.484875	−	0.874583i	$-0.661135\pi$
$83$	−733.292	−0.969750	−0.484875	+	0.874583i	$0.661135\pi$
$84$	0	0
$85$	−762.828	−0.973416
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−994.244	−1.18415	−0.592077	−	0.805881i	$-0.701692\pi$
$89$	−994.244	−1.18415	−0.592077	+	0.805881i	$0.701692\pi$
$90$	0	0
$91$	12.3460	0.0142221
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−824.716	−0.890674
$96$	0	0
$97$	1058.18	1.10765	0.553823	−	0.832635i	$-0.313169\pi$
$97$	1058.18	1.10765	0.553823	+	0.832635i	$0.313169\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−608.930	−0.599909	−0.299955	−	0.953953i	$-0.596972\pi$
$101$	−608.930	−0.599909	−0.299955	+	0.953953i	$0.596972\pi$
$102$	0	0
$103$	−1812.61	−1.73400	−0.867000	−	0.498308i	$-0.833955\pi$
$103$	−1812.61	−1.73400	−0.867000	+	0.498308i	$0.833955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−707.990	−0.639664	−0.319832	−	0.947474i	$-0.603626\pi$
$107$	−707.990	−0.639664	−0.319832	+	0.947474i	$0.603626\pi$
$108$	0	0
$109$	−2101.35	−1.84654	−0.923270	−	0.384153i	$-0.874494\pi$
$109$	−2101.35	−1.84654	−0.923270	+	0.384153i	$0.874494\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1364.66	1.13607	0.568035	−	0.823004i	$-0.307704\pi$
$113$	1364.66	1.13607	0.568035	+	0.823004i	$0.307704\pi$
$114$	0	0
$115$	517.870	0.419927
$116$	0	0
$117$	0	0
$118$	0	0
$119$	25.7322	0.0198224
$120$	0	0
$121$	−1256.79	−0.944243
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1465.70	−1.04877
$126$	0	0
$127$	1473.64	1.02964	0.514819	−	0.857299i	$-0.327859\pi$
$127$	1473.64	1.02964	0.514819	+	0.857299i	$0.327859\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1723.71	−1.14963	−0.574813	−	0.818285i	$-0.694925\pi$
$131$	−1723.71	−1.14963	−0.574813	+	0.818285i	$0.694925\pi$
$132$	0	0
$133$	27.8199	0.0181375
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1332.04	−0.830686	−0.415343	−	0.909665i	$-0.636338\pi$
$137$	−1332.04	−0.830686	−0.415343	+	0.909665i	$0.636338\pi$
$138$	0	0
$139$	2092.69	1.27698	0.638488	−	0.769632i	$-0.279560\pi$
$139$	2092.69	1.27698	0.638488	+	0.769632i	$0.279560\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	299.691	0.175255
$144$	0	0
$145$	−1460.34	−0.836375
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1417.23	0.779224	0.389612	−	0.920979i	$-0.372609\pi$
$149$	1417.23	0.779224	0.389612	+	0.920979i	$0.372609\pi$
$150$	0	0
$151$	2530.84	1.36395	0.681976	−	0.731375i	$-0.261121\pi$
$151$	2530.84	1.36395	0.681976	+	0.731375i	$0.261121\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−656.788	−0.340351
$156$	0	0
$157$	−1609.99	−0.818416	−0.409208	−	0.912441i	$-0.634195\pi$
$157$	−1609.99	−0.818416	−0.409208	+	0.912441i	$0.634195\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−17.4691	−0.00855131
$162$	0	0
$163$	−162.622	−0.0781445	−0.0390722	−	0.999236i	$-0.512440\pi$
$163$	−162.622	−0.0781445	−0.0390722	+	0.999236i	$0.512440\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3759.66	−1.74211	−0.871053	−	0.491190i	$-0.836562\pi$
$167$	−3759.66	−1.74211	−0.871053	+	0.491190i	$0.836562\pi$
$168$	0	0
$169$	−986.766	−0.449142
$170$	0	0
$171$	0	0
$172$	0	0
$173$	239.409	0.105213	0.0526067	−	0.998615i	$-0.483247\pi$
$173$	239.409	0.105213	0.0526067	+	0.998615i	$0.483247\pi$
$174$	0	0
$175$	5.08083	0.00219471
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2212.57	0.923885	0.461943	−	0.886910i	$-0.347153\pi$
$179$	2212.57	0.923885	0.461943	+	0.886910i	$0.347153\pi$
$180$	0	0
$181$	1754.76	0.720608	0.360304	−	0.932835i	$-0.382673\pi$
$181$	1754.76	0.720608	0.360304	+	0.932835i	$0.382673\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	378.980	0.150612
$186$	0	0
$187$	624.633	0.244266
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4001.96	1.51608	0.758040	−	0.652208i	$-0.226157\pi$
$191$	4001.96	1.51608	0.758040	+	0.652208i	$0.226157\pi$
$192$	0	0
$193$	634.410	0.236610	0.118305	−	0.992977i	$-0.462254\pi$
$193$	634.410	0.236610	0.118305	+	0.992977i	$0.462254\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−889.895	−0.321839	−0.160920	−	0.986967i	$-0.551446\pi$
$197$	−889.895	−0.321839	−0.160920	+	0.986967i	$0.551446\pi$
$198$	0	0
$199$	1653.18	0.588900	0.294450	−	0.955667i	$-0.404864\pi$
$199$	1653.18	0.588900	0.294450	+	0.955667i	$0.404864\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	49.2611	0.0170318
$204$	0	0
$205$	−2009.30	−0.684565
$206$	0	0
$207$	0	0
$208$	0	0
$209$	675.309	0.223503
$210$	0	0
$211$	3370.63	1.09973	0.549866	−	0.835253i	$-0.314679\pi$
$211$	3370.63	1.09973	0.549866	+	0.835253i	$0.314679\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5600.95	1.77666
$216$	0	0
$217$	22.1552	0.00693084
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2522.44	0.767772
$222$	0	0
$223$	−1310.16	−0.393429	−0.196715	−	0.980461i	$-0.563027\pi$
$223$	−1310.16	−0.393429	−0.196715	+	0.980461i	$0.563027\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−713.869	−0.208727	−0.104364	−	0.994539i	$-0.533281\pi$
$227$	−713.869	−0.208727	−0.104364	+	0.994539i	$0.533281\pi$
$228$	0	0
$229$	1343.62	0.387724	0.193862	−	0.981029i	$-0.437899\pi$
$229$	1343.62	0.387724	0.193862	+	0.981029i	$0.437899\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3886.80	−1.09284	−0.546422	−	0.837510i	$-0.684011\pi$
$233$	−3886.80	−1.09284	−0.546422	+	0.837510i	$0.684011\pi$
$234$	0	0
$235$	743.635	0.206423
$236$	0	0
$237$	0	0
$238$	0	0
$239$	969.166	0.262302	0.131151	−	0.991362i	$-0.458133\pi$
$239$	969.166	0.262302	0.131151	+	0.991362i	$0.458133\pi$
$240$	0	0
$241$	−5971.78	−1.59617	−0.798084	−	0.602546i	$-0.794153\pi$
$241$	−5971.78	−1.59617	−0.798084	+	0.602546i	$0.794153\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3607.25	−0.940647
$246$	0	0
$247$	2727.08	0.702510
$248$	0	0
$249$	0	0
$250$	0	0
$251$	605.752	0.152330	0.0761648	−	0.997095i	$-0.475732\pi$
$251$	605.752	0.152330	0.0761648	+	0.997095i	$0.475732\pi$
$252$	0	0
$253$	−424.052	−0.105375
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5977.42	1.45082	0.725411	−	0.688316i	$-0.241650\pi$
$257$	5977.42	1.45082	0.725411	+	0.688316i	$0.241650\pi$
$258$	0	0
$259$	−12.7840	−0.00306702
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6859.40	1.60825	0.804124	−	0.594462i	$-0.202635\pi$
$263$	6859.40	1.60825	0.804124	+	0.594462i	$0.202635\pi$
$264$	0	0
$265$	−2587.00	−0.599690
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1166.26	0.264342	0.132171	−	0.991227i	$-0.457805\pi$
$269$	1166.26	0.264342	0.132171	+	0.991227i	$0.457805\pi$
$270$	0	0
$271$	−1286.99	−0.288483	−0.144241	−	0.989543i	$-0.546074\pi$
$271$	−1286.99	−0.288483	−0.144241	+	0.989543i	$0.546074\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	123.334	0.0270448
$276$	0	0
$277$	−7632.64	−1.65560	−0.827800	−	0.561024i	$-0.810408\pi$
$277$	−7632.64	−1.65560	−0.827800	+	0.561024i	$0.810408\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3064.38	0.650554	0.325277	−	0.945619i	$-0.394542\pi$
$281$	3064.38	0.650554	0.325277	+	0.945619i	$0.394542\pi$
$282$	0	0
$283$	4238.63	0.890319	0.445159	−	0.895451i	$-0.353147\pi$
$283$	4238.63	0.890319	0.445159	+	0.895451i	$0.353147\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	67.7791	0.0139403
$288$	0	0
$289$	344.406	0.0701011
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5560.35	1.10867	0.554333	−	0.832295i	$-0.312973\pi$
$293$	5560.35	1.10867	0.554333	+	0.832295i	$0.312973\pi$
$294$	0	0
$295$	−8792.26	−1.73527
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1712.44	−0.331213
$300$	0	0
$301$	−188.935	−0.0361795
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5057.86	0.949549
$306$	0	0
$307$	6287.70	1.16892	0.584459	−	0.811423i	$-0.301307\pi$
$307$	6287.70	1.16892	0.584459	+	0.811423i	$0.301307\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2457.49	0.448075	0.224038	−	0.974580i	$-0.428076\pi$
$311$	2457.49	0.448075	0.224038	+	0.974580i	$0.428076\pi$
$312$	0	0
$313$	−1606.68	−0.290143	−0.145072	−	0.989421i	$-0.546341\pi$
$313$	−1606.68	−0.290143	−0.145072	+	0.989421i	$0.546341\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4597.78	0.814629	0.407314	−	0.913288i	$-0.366465\pi$
$317$	4597.78	0.814629	0.407314	+	0.913288i	$0.366465\pi$
$318$	0	0
$319$	1195.78	0.209877
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5683.93	0.979141
$324$	0	0
$325$	498.055	0.0850065
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−25.0848	−0.00420355
$330$	0	0
$331$	949.340	0.157645	0.0788224	−	0.996889i	$-0.474884\pi$
$331$	949.340	0.157645	0.0788224	+	0.996889i	$0.474884\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1162.96	0.189670
$336$	0	0
$337$	7500.54	1.21240	0.606202	−	0.795311i	$-0.292692\pi$
$337$	7500.54	1.21240	0.606202	+	0.795311i	$0.292692\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	537.803	0.0854067
$342$	0	0
$343$	243.409	0.0383173
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2268.73	0.350985	0.175493	−	0.984481i	$-0.443848\pi$
$347$	2268.73	0.350985	0.175493	+	0.984481i	$0.443848\pi$
$348$	0	0
$349$	3743.08	0.574104	0.287052	−	0.957915i	$-0.407325\pi$
$349$	3743.08	0.574104	0.287052	+	0.957915i	$0.407325\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2331.04	−0.351469	−0.175735	−	0.984438i	$-0.556230\pi$
$353$	−2331.04	−0.351469	−0.175735	+	0.984438i	$0.556230\pi$
$354$	0	0
$355$	615.422	0.0920090
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7364.86	−1.08274	−0.541368	−	0.840785i	$-0.682094\pi$
$359$	−7364.86	−1.08274	−0.541368	+	0.840785i	$0.682094\pi$
$360$	0	0
$361$	−713.935	−0.104087
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3301.19	−0.473404
$366$	0	0
$367$	−6382.15	−0.907754	−0.453877	−	0.891064i	$-0.649959\pi$
$367$	−6382.15	−0.907754	−0.453877	+	0.891064i	$0.649959\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	87.2663	0.0122120
$372$	0	0
$373$	8799.53	1.22151	0.610754	−	0.791820i	$-0.290866\pi$
$373$	8799.53	1.22151	0.610754	+	0.791820i	$0.290866\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4828.88	0.659682
$378$	0	0
$379$	8742.11	1.18483	0.592417	−	0.805631i	$-0.298174\pi$
$379$	8742.11	1.18483	0.592417	+	0.805631i	$0.298174\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14398.7	−1.92099	−0.960495	−	0.278298i	$-0.910230\pi$
$383$	−14398.7	−1.92099	−0.960495	+	0.278298i	$0.910230\pi$
$384$	0	0
$385$	32.1642	0.00425776
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10244.0	1.33520	0.667600	−	0.744520i	$-0.267321\pi$
$389$	10244.0	1.33520	0.667600	+	0.744520i	$0.267321\pi$
$390$	0	0
$391$	−3569.16	−0.461637
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10470.3	1.33372
$396$	0	0
$397$	−3791.18	−0.479279	−0.239639	−	0.970862i	$-0.577029\pi$
$397$	−3791.18	−0.479279	−0.239639	+	0.970862i	$0.577029\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1368.79	−0.170459	−0.0852295	−	0.996361i	$-0.527162\pi$
$401$	−1368.79	−0.170459	−0.0852295	+	0.996361i	$0.527162\pi$
$402$	0	0
$403$	2171.79	0.268449
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−310.323	−0.0377940
$408$	0	0
$409$	−1355.18	−0.163837	−0.0819184	−	0.996639i	$-0.526105\pi$
$409$	−1355.18	−0.163837	−0.0819184	+	0.996639i	$0.526105\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	296.586	0.0353367
$414$	0	0
$415$	−7714.68	−0.912527
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14112.0	1.64539	0.822694	−	0.568485i	$-0.192470\pi$
$419$	14112.0	1.64539	0.822694	+	0.568485i	$0.192470\pi$
$420$	0	0
$421$	−243.132	−0.0281462	−0.0140731	−	0.999901i	$-0.504480\pi$
$421$	−243.132	−0.0281462	−0.0140731	+	0.999901i	$0.504480\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1038.07	0.118480
$426$	0	0
$427$	−170.615	−0.0193364
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6214.92	−0.694575	−0.347288	−	0.937759i	$-0.612897\pi$
$431$	−6214.92	−0.694575	−0.347288	+	0.937759i	$0.612897\pi$
$432$	0	0
$433$	7922.06	0.879238	0.439619	−	0.898184i	$-0.355114\pi$
$433$	7922.06	0.879238	0.439619	+	0.898184i	$0.355114\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3858.72	−0.422397
$438$	0	0
$439$	10.1112	0.00109927	0.000549636	−	1.00000i	$-0.499825\pi$
$439$	10.1112	0.00109927	0.000549636		1.00000i	$0.499825\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8628.26	−0.925375	−0.462687	−	0.886521i	$-0.653115\pi$
$443$	−8628.26	−0.925375	−0.462687	+	0.886521i	$0.653115\pi$
$444$	0	0
$445$	−10460.1	−1.11428
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3543.11	−0.372405	−0.186202	−	0.982511i	$-0.559618\pi$
$449$	−3543.11	−0.372405	−0.186202	+	0.982511i	$0.559618\pi$
$450$	0	0
$451$	1645.29	0.171782
$452$	0	0
$453$	0	0
$454$	0	0
$455$	129.888	0.0133829
$456$	0	0
$457$	14667.7	1.50137	0.750685	−	0.660660i	$-0.229724\pi$
$457$	14667.7	1.50137	0.750685	+	0.660660i	$0.229724\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14576.1	−1.47262	−0.736311	−	0.676643i	$-0.763434\pi$
$461$	−14576.1	−1.47262	−0.736311	+	0.676643i	$0.763434\pi$
$462$	0	0
$463$	−5077.19	−0.509626	−0.254813	−	0.966990i	$-0.582014\pi$
$463$	−5077.19	−0.509626	−0.254813	+	0.966990i	$0.582014\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12469.1	−1.23555	−0.617774	−	0.786356i	$-0.711965\pi$
$467$	−12469.1	−1.23555	−0.617774	+	0.786356i	$0.711965\pi$
$468$	0	0
$469$	−39.2298	−0.00386240
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4586.28	−0.445829
$474$	0	0
$475$	1122.29	0.108409
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3931.05	0.374978	0.187489	−	0.982267i	$-0.439965\pi$
$479$	3931.05	0.374978	0.187489	+	0.982267i	$0.439965\pi$
$480$	0	0
$481$	−1253.17	−0.118793
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11132.7	1.04229
$486$	0	0
$487$	−20710.1	−1.92703	−0.963516	−	0.267652i	$-0.913752\pi$
$487$	−20710.1	−1.92703	−0.963516	+	0.267652i	$0.913752\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12496.8	1.14862	0.574308	−	0.818639i	$-0.305271\pi$
$491$	12496.8	1.14862	0.574308	+	0.818639i	$0.305271\pi$
$492$	0	0
$493$	10064.6	0.919449
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.7598	−0.00187365
$498$	0	0
$499$	−10154.9	−0.911018	−0.455509	−	0.890231i	$-0.650543\pi$
$499$	−10154.9	−0.911018	−0.455509	+	0.890231i	$0.650543\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20309.2	−1.80029	−0.900143	−	0.435594i	$-0.856538\pi$
$503$	−20309.2	−1.80029	−0.900143	+	0.435594i	$0.856538\pi$
$504$	0	0
$505$	−6406.32	−0.564510
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14194.6	−1.23608	−0.618039	−	0.786148i	$-0.712072\pi$
$509$	−14194.6	−1.23608	−0.618039	+	0.786148i	$0.712072\pi$
$510$	0	0
$511$	111.358	0.00964030
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−19069.8	−1.63168
$516$	0	0
$517$	−608.917	−0.0517991
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10644.1	−0.895057	−0.447528	−	0.894270i	$-0.647696\pi$
$521$	−10644.1	−0.895057	−0.447528	+	0.894270i	$0.647696\pi$
$522$	0	0
$523$	−2761.82	−0.230910	−0.115455	−	0.993313i	$-0.536833\pi$
$523$	−2761.82	−0.230910	−0.115455	+	0.993313i	$0.536833\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4526.58	0.374157
$528$	0	0
$529$	−9743.96	−0.800852
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6644.14	0.539943
$534$	0	0
$535$	−7448.49	−0.601919
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2953.75	0.236043
$540$	0	0
$541$	13261.6	1.05390	0.526952	−	0.849895i	$-0.323335\pi$
$541$	13261.6	1.05390	0.526952	+	0.849895i	$0.323335\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−22107.5	−1.73758
$546$	0	0
$547$	20447.6	1.59831	0.799155	−	0.601125i	$-0.205281\pi$
$547$	20447.6	1.59831	0.799155	+	0.601125i	$0.205281\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10881.2	0.841294
$552$	0	0
$553$	−353.193	−0.0271596
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13888.2	−1.05649	−0.528243	−	0.849093i	$-0.677149\pi$
$557$	−13888.2	−1.05649	−0.528243	+	0.849093i	$0.677149\pi$
$558$	0	0
$559$	−18520.6	−1.40132
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16878.2	1.26347	0.631734	−	0.775186i	$-0.282344\pi$
$563$	16878.2	1.26347	0.631734	+	0.775186i	$0.282344\pi$
$564$	0	0
$565$	14357.0	1.06903
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11853.7	−0.873348	−0.436674	−	0.899620i	$-0.643844\pi$
$569$	−11853.7	−0.873348	−0.436674	+	0.899620i	$0.643844\pi$
$570$	0	0
$571$	−14012.8	−1.02700	−0.513499	−	0.858090i	$-0.671651\pi$
$571$	−14012.8	−1.02700	−0.513499	+	0.858090i	$0.671651\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−704.730	−0.0511118
$576$	0	0
$577$	2886.05	0.208228	0.104114	−	0.994565i	$-0.466799\pi$
$577$	2886.05	0.208228	0.104114	+	0.994565i	$0.466799\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	260.237	0.0185825
$582$	0	0
$583$	2118.33	0.150484
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14040.0	−0.987213	−0.493607	−	0.869685i	$-0.664322\pi$
$587$	−14040.0	−0.987213	−0.493607	+	0.869685i	$0.664322\pi$
$588$	0	0
$589$	4893.81	0.342353
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18394.8	1.27383	0.636917	−	0.770932i	$-0.280209\pi$
$593$	18394.8	1.27383	0.636917	+	0.770932i	$0.280209\pi$
$594$	0	0
$595$	270.719	0.0186528
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−28197.6	−1.92341	−0.961704	−	0.274089i	$-0.911624\pi$
$599$	−28197.6	−1.92341	−0.961704	+	0.274089i	$0.911624\pi$
$600$	0	0
$601$	26542.2	1.80146	0.900731	−	0.434376i	$-0.143031\pi$
$601$	26542.2	1.80146	0.900731	+	0.434376i	$0.143031\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−13222.2	−0.888525
$606$	0	0
$607$	−10892.1	−0.728329	−0.364165	−	0.931335i	$-0.618645\pi$
$607$	−10892.1	−0.728329	−0.364165	+	0.931335i	$0.618645\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2458.97	−0.162814
$612$	0	0
$613$	6843.92	0.450935	0.225467	−	0.974251i	$-0.427609\pi$
$613$	6843.92	0.450935	0.225467	+	0.974251i	$0.427609\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16419.4	−1.07135	−0.535673	−	0.844425i	$-0.679942\pi$
$617$	−16419.4	−1.07135	−0.535673	+	0.844425i	$0.679942\pi$
$618$	0	0
$619$	14067.4	0.913438	0.456719	−	0.889611i	$-0.349024\pi$
$619$	14067.4	0.913438	0.456719	+	0.889611i	$0.349024\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	352.846	0.0226910
$624$	0	0
$625$	−13630.4	−0.872349
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2611.92	−0.165571
$630$	0	0
$631$	91.1166	0.00574848	0.00287424	−	0.999996i	$-0.499085\pi$
$631$	91.1166	0.00574848	0.00287424	+	0.999996i	$0.499085\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15503.6	0.968882
$636$	0	0
$637$	11928.0	0.741925
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24629.9	1.51766	0.758831	−	0.651287i	$-0.225771\pi$
$641$	24629.9	1.51766	0.758831	+	0.651287i	$0.225771\pi$
$642$	0	0
$643$	6067.58	0.372133	0.186067	−	0.982537i	$-0.440426\pi$
$643$	6067.58	0.372133	0.186067	+	0.982537i	$0.440426\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	890.884	0.0541334	0.0270667	−	0.999634i	$-0.491383\pi$
$647$	890.884	0.0541334	0.0270667	+	0.999634i	$0.491383\pi$
$648$	0	0
$649$	7199.44	0.435443
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27701.1	−1.66007	−0.830036	−	0.557709i	$-0.811680\pi$
$653$	−27701.1	−1.66007	−0.830036	+	0.557709i	$0.811680\pi$
$654$	0	0
$655$	−18134.4	−1.08179
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−19578.2	−1.15730	−0.578648	−	0.815577i	$-0.696420\pi$
$659$	−19578.2	−1.15730	−0.578648	+	0.815577i	$0.696420\pi$
$660$	0	0
$661$	11008.3	0.647764	0.323882	−	0.946097i	$-0.395012\pi$
$661$	11008.3	0.647764	0.323882	+	0.946097i	$0.395012\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	292.682	0.0170672
$666$	0	0
$667$	−6832.70	−0.396646
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4141.57	−0.238277
$672$	0	0
$673$	4492.07	0.257290	0.128645	−	0.991691i	$-0.458937\pi$
$673$	4492.07	0.257290	0.128645	+	0.991691i	$0.458937\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6072.04	0.344708	0.172354	−	0.985035i	$-0.444863\pi$
$677$	6072.04	0.344708	0.172354	+	0.985035i	$0.444863\pi$
$678$	0	0
$679$	−375.535	−0.0212249
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20709.0	−1.16018	−0.580092	−	0.814551i	$-0.696984\pi$
$683$	−20709.0	−1.16018	−0.580092	+	0.814551i	$0.696984\pi$
$684$	0	0
$685$	−14013.9	−0.781669
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8554.40	0.473000
$690$	0	0
$691$	7117.61	0.391847	0.195924	−	0.980619i	$-0.437230\pi$
$691$	7117.61	0.391847	0.195924	+	0.980619i	$0.437230\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22016.4	1.20162
$696$	0	0
$697$	13848.1	0.752560
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26295.0	1.41676	0.708379	−	0.705833i	$-0.249427\pi$
$701$	26295.0	1.41676	0.708379	+	0.705833i	$0.249427\pi$
$702$	0	0
$703$	−2823.83	−0.151497
$704$	0	0
$705$	0	0
$706$	0	0
$707$	216.102	0.0114956
$708$	0	0
$709$	−8729.67	−0.462411	−0.231206	−	0.972905i	$-0.574267\pi$
$709$	−8729.67	−0.462411	−0.231206	+	0.972905i	$0.574267\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3073.01	−0.161410
$714$	0	0
$715$	3152.94	0.164913
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32340.7	−1.67747	−0.838736	−	0.544538i	$-0.816705\pi$
$719$	−32340.7	−1.67747	−0.838736	+	0.544538i	$0.816705\pi$
$720$	0	0
$721$	643.275	0.0332272
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1987.26	0.101800
$726$	0	0
$727$	−26305.9	−1.34199	−0.670997	−	0.741460i	$-0.734134\pi$
$727$	−26305.9	−1.34199	−0.670997	+	0.741460i	$0.734134\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−38601.7	−1.95313
$732$	0	0
$733$	−16373.5	−0.825058	−0.412529	−	0.910944i	$-0.635354\pi$
$733$	−16373.5	−0.825058	−0.412529	+	0.910944i	$0.635354\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−952.278	−0.0475951
$738$	0	0
$739$	17567.9	0.874486	0.437243	−	0.899343i	$-0.355955\pi$
$739$	17567.9	0.874486	0.437243	+	0.899343i	$0.355955\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22821.4	1.12683	0.563416	−	0.826174i	$-0.309487\pi$
$743$	22821.4	1.12683	0.563416	+	0.826174i	$0.309487\pi$
$744$	0	0
$745$	14910.2	0.733244
$746$	0	0
$747$	0	0
$748$	0	0
$749$	251.257	0.0122573
$750$	0	0
$751$	−35497.1	−1.72478	−0.862388	−	0.506248i	$-0.831032\pi$
$751$	−35497.1	−1.72478	−0.862388	+	0.506248i	$0.831032\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	26626.0	1.28347
$756$	0	0
$757$	−2676.83	−0.128522	−0.0642609	−	0.997933i	$-0.520469\pi$
$757$	−2676.83	−0.128522	−0.0642609	+	0.997933i	$0.520469\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3047.53	0.145168	0.0725841	−	0.997362i	$-0.476875\pi$
$761$	3047.53	0.145168	0.0725841	+	0.997362i	$0.476875\pi$
$762$	0	0
$763$	745.744	0.0353837
$764$	0	0
$765$	0	0
$766$	0	0
$767$	29073.3	1.36868
$768$	0	0
$769$	20592.4	0.965645	0.482823	−	0.875718i	$-0.339612\pi$
$769$	20592.4	0.965645	0.482823	+	0.875718i	$0.339612\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3252.88	−0.151356	−0.0756779	−	0.997132i	$-0.524112\pi$
$773$	−3252.88	−0.151356	−0.0756779	+	0.997132i	$0.524112\pi$
$774$	0	0
$775$	893.772	0.0414261
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14971.6	0.688591
$780$	0	0
$781$	−503.931	−0.0230884
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16938.1	−0.770123
$786$	0	0
$787$	38336.3	1.73639	0.868196	−	0.496222i	$-0.165280\pi$
$787$	38336.3	1.73639	0.868196	+	0.496222i	$0.165280\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−484.300	−0.0217696
$792$	0	0
$793$	−16724.8	−0.748947
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25157.8	1.11811	0.559056	−	0.829130i	$-0.311164\pi$
$797$	25157.8	1.11811	0.559056	+	0.829130i	$0.311164\pi$
$798$	0	0
$799$	−5125.13	−0.226926
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2703.15	0.118794
$804$	0	0
$805$	−183.786	−0.00804672
$806$	0	0
$807$	0	0
$808$	0	0
$809$	159.016	0.00691064	0.00345532	−	0.999994i	$-0.498900\pi$
$809$	159.016	0.00691064	0.00345532	+	0.999994i	$0.498900\pi$
$810$	0	0
$811$	16146.0	0.699093	0.349547	−	0.936919i	$-0.386336\pi$
$811$	16146.0	0.699093	0.349547	+	0.936919i	$0.386336\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1710.89	−0.0735334
$816$	0	0
$817$	−41733.4	−1.78711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32727.6	1.39123	0.695616	−	0.718413i	$-0.255131\pi$
$821$	32727.6	1.39123	0.695616	+	0.718413i	$0.255131\pi$
$822$	0	0
$823$	−32838.4	−1.39086	−0.695428	−	0.718596i	$-0.744785\pi$
$823$	−32838.4	−1.39086	−0.695428	+	0.718596i	$0.744785\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4725.60	0.198700	0.0993502	−	0.995053i	$-0.468324\pi$
$827$	4725.60	0.198700	0.0993502	+	0.995053i	$0.468324\pi$
$828$	0	0
$829$	2080.19	0.0871510	0.0435755	−	0.999050i	$-0.486125\pi$
$829$	2080.19	0.0871510	0.0435755	+	0.999050i	$0.486125\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	24861.1	1.03408
$834$	0	0
$835$	−39554.0	−1.63931
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−328.744	−0.0135274	−0.00676371	−	0.999977i	$-0.502153\pi$
$839$	−328.744	−0.0135274	−0.00676371	+	0.999977i	$0.502153\pi$
$840$	0	0
$841$	−5121.55	−0.209994
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10381.4	−0.422640
$846$	0	0
$847$	446.019	0.0180937
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1773.19	0.0714267
$852$	0	0
$853$	−30712.6	−1.23280	−0.616400	−	0.787433i	$-0.711409\pi$
$853$	−30712.6	−1.23280	−0.616400	+	0.787433i	$0.711409\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10969.1	0.437219	0.218610	−	0.975812i	$-0.429848\pi$
$857$	10969.1	0.437219	0.218610	+	0.975812i	$0.429848\pi$
$858$	0	0
$859$	39265.8	1.55964	0.779821	−	0.626003i	$-0.215310\pi$
$859$	39265.8	1.55964	0.779821	+	0.626003i	$0.215310\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23715.3	0.935433	0.467716	−	0.883879i	$-0.345077\pi$
$863$	23715.3	0.935433	0.467716	+	0.883879i	$0.345077\pi$
$864$	0	0
$865$	2518.73	0.0990050
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8573.52	−0.334680
$870$	0	0
$871$	−3845.56	−0.149600
$872$	0	0
$873$	0	0
$874$	0	0
$875$	520.159	0.0200967
$876$	0	0
$877$	28687.0	1.10455	0.552275	−	0.833662i	$-0.313760\pi$
$877$	28687.0	1.10455	0.552275	+	0.833662i	$0.313760\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	27975.8	1.06984	0.534920	−	0.844903i	$-0.320342\pi$
$881$	27975.8	1.06984	0.534920	+	0.844903i	$0.320342\pi$
$882$	0	0
$883$	4610.87	0.175728	0.0878642	−	0.996132i	$-0.471996\pi$
$883$	4610.87	0.175728	0.0878642	+	0.996132i	$0.471996\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25663.8	−0.971483	−0.485741	−	0.874103i	$-0.661450\pi$
$887$	−25663.8	−0.971483	−0.485741	+	0.874103i	$0.661450\pi$
$888$	0	0
$889$	−522.977	−0.0197301
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5540.92	−0.207637
$894$	0	0
$895$	23277.6	0.869369
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8665.55	0.321482
$900$	0	0
$901$	17829.6	0.659255
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18461.1	0.678086
$906$	0	0
$907$	9333.72	0.341699	0.170850	−	0.985297i	$-0.445349\pi$
$907$	9333.72	0.341699	0.170850	+	0.985297i	$0.445349\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	48138.0	1.75070	0.875348	−	0.483493i	$-0.160632\pi$
$911$	48138.0	1.75070	0.875348	+	0.483493i	$0.160632\pi$
$912$	0	0
$913$	6317.08	0.228987
$914$	0	0
$915$	0	0
$916$	0	0
$917$	611.723	0.0220293
$918$	0	0
$919$	−32300.1	−1.15939	−0.579696	−	0.814833i	$-0.696829\pi$
$919$	−32300.1	−1.15939	−0.579696	+	0.814833i	$0.696829\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2035.01	−0.0725712
$924$	0	0
$925$	−515.724	−0.0183318
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9013.54	0.318326	0.159163	−	0.987252i	$-0.449120\pi$
$929$	9013.54	0.318326	0.159163	+	0.987252i	$0.449120\pi$
$930$	0	0
$931$	26878.1	0.946179
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6571.53	0.229852
$936$	0	0
$937$	47418.3	1.65324	0.826622	−	0.562758i	$-0.190260\pi$
$937$	47418.3	1.65324	0.826622	+	0.562758i	$0.190260\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21893.5	0.758458	0.379229	−	0.925303i	$-0.376189\pi$
$941$	21893.5	0.758458	0.379229	+	0.925303i	$0.376189\pi$
$942$	0	0
$943$	−9401.22	−0.324651
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44069.5	−1.51221	−0.756107	−	0.654448i	$-0.772901\pi$
$947$	−44069.5	−1.51221	−0.756107	+	0.654448i	$0.772901\pi$
$948$	0	0
$949$	10916.0	0.373392
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−22429.1	−0.762382	−0.381191	−	0.924496i	$-0.624486\pi$
$953$	−22429.1	−0.762382	−0.381191	+	0.924496i	$0.624486\pi$
$954$	0	0
$955$	42103.0	1.42662
$956$	0	0
$957$	0	0
$958$	0	0
$959$	472.726	0.0159177
$960$	0	0
$961$	−25893.7	−0.869177
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6674.38	0.222649
$966$	0	0
$967$	−5277.40	−0.175501	−0.0877507	−	0.996142i	$-0.527968\pi$
$967$	−5277.40	−0.175501	−0.0877507	+	0.996142i	$0.527968\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	53211.8	1.75865	0.879325	−	0.476223i	$-0.157995\pi$
$971$	53211.8	1.75865	0.879325	+	0.476223i	$0.157995\pi$
$972$	0	0
$973$	−742.671	−0.0244696
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40093.7	−1.31291	−0.656453	−	0.754367i	$-0.727944\pi$
$977$	−40093.7	−1.31291	−0.656453	+	0.754367i	$0.727944\pi$
$978$	0	0
$979$	8565.10	0.279614
$980$	0	0
$981$	0	0
$982$	0	0
$983$	50247.4	1.63036	0.815180	−	0.579207i	$-0.196638\pi$
$983$	50247.4	1.63036	0.815180	+	0.579207i	$0.196638\pi$
$984$	0	0
$985$	−9362.24	−0.302848
$986$	0	0
$987$	0	0
$988$	0	0
$989$	26206.0	0.842571
$990$	0	0
$991$	32333.6	1.03644	0.518219	−	0.855248i	$-0.326595\pi$
$991$	32333.6	1.03644	0.518219	+	0.855248i	$0.326595\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17392.5	0.554151
$996$	0	0
$997$	−21239.1	−0.674672	−0.337336	−	0.941384i	$-0.609526\pi$
$997$	−21239.1	−0.674672	−0.337336	+	0.941384i	$0.609526\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.a.g.1.4	✓	4
3.2	odd	2		648.4.a.j.1.1	yes	4
4.3	odd	2		1296.4.a.x.1.4		4
9.2	odd	6		648.4.i.u.433.4		8
9.4	even	3		648.4.i.v.217.1		8
9.5	odd	6		648.4.i.u.217.4		8
9.7	even	3		648.4.i.v.433.1		8
12.11	even	2		1296.4.a.bb.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.g.1.4	✓	4	1.1	even	1	trivial
648.4.a.j.1.1	yes	4	3.2	odd	2
648.4.i.u.217.4		8	9.5	odd	6
648.4.i.u.433.4		8	9.2	odd	6
648.4.i.v.217.1		8	9.4	even	3
648.4.i.v.433.1		8	9.7	even	3
1296.4.a.x.1.4		4	4.3	odd	2
1296.4.a.bb.1.1		4	12.11	even	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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q+10.5206 q^{5} -0.354888 q^{7} +O(q^{10})\)	\(q+10.5206 q^{5} -0.354888 q^{7} -8.61468 q^{11} -34.7884 q^{13} -72.5080 q^{17} -78.3905 q^{19} +49.2244 q^{23} -14.3167 q^{25} -138.807 q^{29} -62.4287 q^{31} -3.73364 q^{35} +36.0226 q^{37} -190.987 q^{41} +532.379 q^{43} +70.6836 q^{47} -342.874 q^{49} -245.898 q^{53} -90.6318 q^{55} -835.717 q^{59} +480.757 q^{61} -365.996 q^{65} +110.541 q^{67} +58.4968 q^{71} -313.784 q^{73} +3.05725 q^{77} +995.222 q^{79} -733.292 q^{83} -762.828 q^{85} -994.244 q^{89} +12.3460 q^{91} -824.716 q^{95} +1058.18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5}+O(q^{10}) \) 4 * q - 8 * q^5	\( 4 q - 8 q^{5} - 8 q^{11} + 4 q^{13} - 16 q^{17} + 80 q^{19} - 200 q^{23} + 8 q^{25} - 216 q^{29} + 80 q^{31} - 408 q^{35} - 276 q^{37} - 384 q^{41} + 160 q^{43} - 768 q^{47} - 268 q^{49} - 944 q^{53} + 304 q^{55} - 992 q^{59} - 548 q^{61} - 1328 q^{65} + 464 q^{67} - 1720 q^{71} - 764 q^{73} - 1728 q^{77} + 688 q^{79} - 2128 q^{83} - 1324 q^{85} - 2112 q^{89} + 1776 q^{91} - 2056 q^{95} - 1816 q^{97}+O(q^{100}) \) 4 * q - 8 * q^5 - 8 * q^11 + 4 * q^13 - 16 * q^17 + 80 * q^19 - 200 * q^23 + 8 * q^25 - 216 * q^29 + 80 * q^31 - 408 * q^35 - 276 * q^37 - 384 * q^41 + 160 * q^43 - 768 * q^47 - 268 * q^49 - 944 * q^53 + 304 * q^55 - 992 * q^59 - 548 * q^61 - 1328 * q^65 + 464 * q^67 - 1720 * q^71 - 764 * q^73 - 1728 * q^77 + 688 * q^79 - 2128 * q^83 - 1324 * q^85 - 2112 * q^89 + 1776 * q^91 - 2056 * q^95 - 1816 * q^97

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