LMFDB - Embedded newform 1521.4.a.bd.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 $ x^8 - 52x^6 + 805x^4 - 4210*x^2 + 4992
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$5.39589$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.39589 q^{2} +21.1156 q^{4} +13.0421 q^{5} +6.42494 q^{7} +70.7705 q^{8} +O(q^{10})$	$q+5.39589 q^{2} +21.1156 q^{4} +13.0421 q^{5} +6.42494 q^{7} +70.7705 q^{8} +70.3740 q^{10} +26.2056 q^{11} +34.6683 q^{14} +212.945 q^{16} +123.877 q^{17} -109.667 q^{19} +275.393 q^{20} +141.402 q^{22} +63.4094 q^{23} +45.0976 q^{25} +135.667 q^{28} -225.410 q^{29} -200.732 q^{31} +582.863 q^{32} +668.426 q^{34} +83.7950 q^{35} -252.509 q^{37} -591.749 q^{38} +922.999 q^{40} +227.423 q^{41} -384.032 q^{43} +553.348 q^{44} +342.150 q^{46} +34.6646 q^{47} -301.720 q^{49} +243.342 q^{50} +61.0601 q^{53} +341.777 q^{55} +454.696 q^{56} -1216.29 q^{58} -80.5562 q^{59} -26.1383 q^{61} -1083.13 q^{62} +1441.50 q^{64} +931.510 q^{67} +2615.74 q^{68} +452.149 q^{70} +427.608 q^{71} -108.518 q^{73} -1362.51 q^{74} -2315.68 q^{76} +168.369 q^{77} +384.590 q^{79} +2777.26 q^{80} +1227.15 q^{82} -85.9758 q^{83} +1615.62 q^{85} -2072.19 q^{86} +1854.58 q^{88} -495.903 q^{89} +1338.93 q^{92} +187.046 q^{94} -1430.29 q^{95} -190.857 q^{97} -1628.05 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 40 q^{4} + 22 q^{7}+O(q^{10}) $ 8 * q + 40 * q^4 + 22 * q^7	$ 8 q + 40 q^{4} + 22 q^{7} + 36 q^{10} + 204 q^{16} - 244 q^{19} + 136 q^{22} + 354 q^{25} + 452 q^{28} - 242 q^{31} + 1292 q^{34} - 1018 q^{37} + 1700 q^{40} + 74 q^{43} + 896 q^{46} + 298 q^{49} + 1300 q^{55} - 812 q^{58} + 1148 q^{61} + 3636 q^{64} + 2198 q^{67} - 2200 q^{70} + 2176 q^{73} - 6936 q^{76} + 1862 q^{79} + 5436 q^{82} + 890 q^{85} + 3528 q^{88} - 3104 q^{94} + 4370 q^{97}+O(q^{100}) $ 8 * q + 40 * q^4 + 22 * q^7 + 36 * q^10 + 204 * q^16 - 244 * q^19 + 136 * q^22 + 354 * q^25 + 452 * q^28 - 242 * q^31 + 1292 * q^34 - 1018 * q^37 + 1700 * q^40 + 74 * q^43 + 896 * q^46 + 298 * q^49 + 1300 * q^55 - 812 * q^58 + 1148 * q^61 + 3636 * q^64 + 2198 * q^67 - 2200 * q^70 + 2176 * q^73 - 6936 * q^76 + 1862 * q^79 + 5436 * q^82 + 890 * q^85 + 3528 * q^88 - 3104 * q^94 + 4370 * 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 $

$2$	5.39589	1.90774	0.953868	−	0.300228i	$-0.0970626\pi$
$2$	5.39589	1.90774	0.953868	+	0.300228i	$0.0970626\pi$
$3$	0	0
$3$	0	0
$4$	21.1156	2.63945
$5$	13.0421	1.16653	0.583263	−	0.812284i	$-0.301776\pi$
$5$	13.0421	1.16653	0.583263	+	0.812284i	$0.301776\pi$
$6$	0	0
$7$	6.42494	0.346914	0.173457	−	0.984841i	$-0.444506\pi$
$7$	6.42494	0.346914	0.173457	+	0.984841i	$0.444506\pi$
$8$	70.7705	3.12764
$9$	0	0
$10$	70.3740	2.22542
$11$	26.2056	0.718298	0.359149	−	0.933280i	$-0.383067\pi$
$11$	26.2056	0.718298	0.359149	+	0.933280i	$0.383067\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	34.6683	0.661820
$15$	0	0
$16$	212.945	3.32726
$17$	123.877	1.76733	0.883663	−	0.468124i	$-0.155070\pi$
$17$	123.877	1.76733	0.883663	+	0.468124i	$0.155070\pi$
$18$	0	0
$19$	−109.667	−1.32417	−0.662086	−	0.749428i	$-0.730329\pi$
$19$	−109.667	−1.32417	−0.662086	+	0.749428i	$0.730329\pi$
$20$	275.393	3.07899
$21$	0	0
$22$	141.402	1.37032
$23$	63.4094	0.574860	0.287430	−	0.957802i	$-0.407199\pi$
$23$	63.4094	0.574860	0.287430	+	0.957802i	$0.407199\pi$
$24$	0	0
$25$	45.0976	0.360781
$26$	0	0
$27$	0	0
$28$	135.667	0.915664
$29$	−225.410	−1.44337	−0.721683	−	0.692224i	$-0.756631\pi$
$29$	−225.410	−1.44337	−0.721683	+	0.692224i	$0.756631\pi$
$30$	0	0
$31$	−200.732	−1.16298	−0.581491	−	0.813553i	$-0.697530\pi$
$31$	−200.732	−1.16298	−0.581491	+	0.813553i	$0.697530\pi$
$32$	582.863	3.21989
$33$	0	0
$34$	668.426	3.37159
$35$	83.7950	0.404684
$36$	0	0
$37$	−252.509	−1.12195	−0.560976	−	0.827832i	$-0.689574\pi$
$37$	−252.509	−1.12195	−0.560976	+	0.827832i	$0.689574\pi$
$38$	−591.749	−2.52617
$39$	0	0
$40$	922.999	3.64847
$41$	227.423	0.866282	0.433141	−	0.901326i	$-0.357405\pi$
$41$	227.423	0.866282	0.433141	+	0.901326i	$0.357405\pi$
$42$	0	0
$43$	−384.032	−1.36196	−0.680980	−	0.732302i	$-0.738446\pi$
$43$	−384.032	−1.36196	−0.680980	+	0.732302i	$0.738446\pi$
$44$	553.348	1.89592
$45$	0	0
$46$	342.150	1.09668
$47$	34.6646	0.107582	0.0537910	−	0.998552i	$-0.482870\pi$
$47$	34.6646	0.107582	0.0537910	+	0.998552i	$0.482870\pi$
$48$	0	0
$49$	−301.720	−0.879651
$50$	243.342	0.688274
$51$	0	0
$52$	0	0
$53$	61.0601	0.158250	0.0791251	−	0.996865i	$-0.474787\pi$
$53$	61.0601	0.158250	0.0791251	+	0.996865i	$0.474787\pi$
$54$	0	0
$55$	341.777	0.837913
$56$	454.696	1.08502
$57$	0	0
$58$	−1216.29	−2.75356
$59$	−80.5562	−0.177755	−0.0888774	−	0.996043i	$-0.528328\pi$
$59$	−80.5562	−0.177755	−0.0888774	+	0.996043i	$0.528328\pi$
$60$	0	0
$61$	−26.1383	−0.0548634	−0.0274317	−	0.999624i	$-0.508733\pi$
$61$	−26.1383	−0.0548634	−0.0274317	+	0.999624i	$0.508733\pi$
$62$	−1083.13	−2.21866
$63$	0	0
$64$	1441.50	2.81544
$65$	0	0
$66$	0	0
$67$	931.510	1.69854	0.849269	−	0.527960i	$-0.177043\pi$
$67$	931.510	1.69854	0.849269	+	0.527960i	$0.177043\pi$
$68$	2615.74	4.66477
$69$	0	0
$70$	452.149	0.772030
$71$	427.608	0.714757	0.357379	−	0.933960i	$-0.383671\pi$
$71$	427.608	0.714757	0.357379	+	0.933960i	$0.383671\pi$
$72$	0	0
$73$	−108.518	−0.173987	−0.0869935	−	0.996209i	$-0.527726\pi$
$73$	−108.518	−0.173987	−0.0869935	+	0.996209i	$0.527726\pi$
$74$	−1362.51	−2.14039
$75$	0	0
$76$	−2315.68	−3.49509
$77$	168.369	0.249188
$78$	0	0
$79$	384.590	0.547718	0.273859	−	0.961770i	$-0.411700\pi$
$79$	384.590	0.547718	0.273859	+	0.961770i	$0.411700\pi$
$80$	2777.26	3.88133
$81$	0	0
$82$	1227.15	1.65264
$83$	−85.9758	−0.113700	−0.0568498	−	0.998383i	$-0.518106\pi$
$83$	−85.9758	−0.113700	−0.0568498	+	0.998383i	$0.518106\pi$
$84$	0	0
$85$	1615.62	2.06163
$86$	−2072.19	−2.59826
$87$	0	0
$88$	1854.58	2.24658
$89$	−495.903	−0.590625	−0.295313	−	0.955401i	$-0.595424\pi$
$89$	−495.903	−0.590625	−0.295313	+	0.955401i	$0.595424\pi$
$90$	0	0
$91$	0	0
$92$	1338.93	1.51732
$93$	0	0
$94$	187.046	0.205238
$95$	−1430.29	−1.54468
$96$	0	0
$97$	−190.857	−0.199780	−0.0998898	−	0.994999i	$-0.531849\pi$
$97$	−190.857	−0.199780	−0.0998898	+	0.994999i	$0.531849\pi$
$98$	−1628.05	−1.67814
$99$	0	0
$100$	952.264	0.952264
$101$	850.980	0.838373	0.419186	−	0.907900i	$-0.362315\pi$
$101$	850.980	0.838373	0.419186	+	0.907900i	$0.362315\pi$
$102$	0	0
$103$	−1309.50	−1.25270	−0.626352	−	0.779540i	$-0.715453\pi$
$103$	−1309.50	−1.25270	−0.626352	+	0.779540i	$0.715453\pi$
$104$	0	0
$105$	0	0
$106$	329.474	0.301899
$107$	962.302	0.869432	0.434716	−	0.900567i	$-0.356849\pi$
$107$	962.302	0.869432	0.434716	+	0.900567i	$0.356849\pi$
$108$	0	0
$109$	−891.973	−0.783812	−0.391906	−	0.920005i	$-0.628184\pi$
$109$	−891.973	−0.783812	−0.391906	+	0.920005i	$0.628184\pi$
$110$	1844.19	1.59852
$111$	0	0
$112$	1368.16	1.15427
$113$	263.839	0.219645	0.109823	−	0.993951i	$-0.464972\pi$
$113$	263.839	0.219645	0.109823	+	0.993951i	$0.464972\pi$
$114$	0	0
$115$	826.995	0.670588
$116$	−4759.68	−3.80970
$117$	0	0
$118$	−434.673	−0.339109
$119$	795.901	0.613110
$120$	0	0
$121$	−644.267	−0.484047
$122$	−141.039	−0.104665
$123$	0	0
$124$	−4238.57	−3.06964
$125$	−1042.10	−0.745665
$126$	0	0
$127$	−1543.97	−1.07878	−0.539391	−	0.842056i	$-0.681345\pi$
$127$	−1543.97	−1.07878	−0.539391	+	0.842056i	$0.681345\pi$
$128$	3115.30	2.15122
$129$	0	0
$130$	0	0
$131$	−776.790	−0.518080	−0.259040	−	0.965867i	$-0.583406\pi$
$131$	−776.790	−0.518080	−0.259040	+	0.965867i	$0.583406\pi$
$132$	0	0
$133$	−704.602	−0.459374
$134$	5026.33	3.24036
$135$	0	0
$136$	8766.82	5.52756
$137$	2434.71	1.51833	0.759166	−	0.650897i	$-0.225607\pi$
$137$	2434.71	1.51833	0.759166	+	0.650897i	$0.225607\pi$
$138$	0	0
$139$	1251.40	0.763615	0.381807	−	0.924242i	$-0.375302\pi$
$139$	1251.40	0.763615	0.381807	+	0.924242i	$0.375302\pi$
$140$	1769.38	1.06814
$141$	0	0
$142$	2307.33	1.36357
$143$	0	0
$144$	0	0
$145$	−2939.83	−1.68372
$146$	−585.550	−0.331921
$147$	0	0
$148$	−5331.88	−2.96134
$149$	783.619	0.430850	0.215425	−	0.976520i	$-0.430886\pi$
$149$	783.619	0.430850	0.215425	+	0.976520i	$0.430886\pi$
$150$	0	0
$151$	−162.220	−0.0874255	−0.0437128	−	0.999044i	$-0.513919\pi$
$151$	−162.220	−0.0874255	−0.0437128	+	0.999044i	$0.513919\pi$
$152$	−7761.16	−4.14154
$153$	0	0
$154$	908.503	0.475385
$155$	−2617.97	−1.35665
$156$	0	0
$157$	−2355.76	−1.19752	−0.598758	−	0.800930i	$-0.704339\pi$
$157$	−2355.76	−1.19752	−0.598758	+	0.800930i	$0.704339\pi$
$158$	2075.20	1.04490
$159$	0	0
$160$	7601.78	3.75608
$161$	407.402	0.199427
$162$	0	0
$163$	−1574.29	−0.756488	−0.378244	−	0.925706i	$-0.623472\pi$
$163$	−1574.29	−0.756488	−0.378244	+	0.925706i	$0.623472\pi$
$164$	4802.19	2.28651
$165$	0	0
$166$	−463.916	−0.216909
$167$	3574.56	1.65633	0.828166	−	0.560483i	$-0.189384\pi$
$167$	3574.56	1.65633	0.828166	+	0.560483i	$0.189384\pi$
$168$	0	0
$169$	0	0
$170$	8717.70	3.93304
$171$	0	0
$172$	−8109.07	−3.59483
$173$	−1.44108	−0.000633314 0	−0.000316657	−	1.00000i	$-0.500101\pi$
$173$	−1.44108	−0.000633314 0	−0.000316657		1.00000i	$0.500101\pi$
$174$	0	0
$175$	289.749	0.125160
$176$	5580.34	2.38997
$177$	0	0
$178$	−2675.84	−1.12676
$179$	2082.28	0.869478	0.434739	−	0.900556i	$-0.356841\pi$
$179$	2082.28	0.869478	0.434739	+	0.900556i	$0.356841\pi$
$180$	0	0
$181$	−464.500	−0.190751	−0.0953756	−	0.995441i	$-0.530405\pi$
$181$	−464.500	−0.190751	−0.0953756	+	0.995441i	$0.530405\pi$
$182$	0	0
$183$	0	0
$184$	4487.51	1.79796
$185$	−3293.26	−1.30878
$186$	0	0
$187$	3246.26	1.26947
$188$	731.965	0.283957
$189$	0	0
$190$	−7717.68	−2.94684
$191$	4866.28	1.84352	0.921758	−	0.387765i	$-0.126753\pi$
$191$	4866.28	1.84352	0.921758	+	0.387765i	$0.126753\pi$
$192$	0	0
$193$	856.339	0.319381	0.159691	−	0.987167i	$-0.448950\pi$
$193$	856.339	0.319381	0.159691	+	0.987167i	$0.448950\pi$
$194$	−1029.84	−0.381127
$195$	0	0
$196$	−6371.01	−2.32180
$197$	−1571.93	−0.568504	−0.284252	−	0.958750i	$-0.591745\pi$
$197$	−1571.93	−0.568504	−0.284252	+	0.958750i	$0.591745\pi$
$198$	0	0
$199$	3126.61	1.11377	0.556883	−	0.830591i	$-0.311997\pi$
$199$	3126.61	1.11377	0.556883	+	0.830591i	$0.311997\pi$
$200$	3191.58	1.12839
$201$	0	0
$202$	4591.79	1.59939
$203$	−1448.25	−0.500724
$204$	0	0
$205$	2966.09	1.01054
$206$	−7065.90	−2.38983
$207$	0	0
$208$	0	0
$209$	−2873.88	−0.951150
$210$	0	0
$211$	123.392	0.0402590	0.0201295	−	0.999797i	$-0.493592\pi$
$211$	123.392	0.0402590	0.0201295	+	0.999797i	$0.493592\pi$
$212$	1289.32	0.417694
$213$	0	0
$214$	5192.48	1.65865
$215$	−5008.60	−1.58876
$216$	0	0
$217$	−1289.69	−0.403455
$218$	−4812.99	−1.49531
$219$	0	0
$220$	7216.84	2.21163
$221$	0	0
$222$	0	0
$223$	2503.69	0.751836	0.375918	−	0.926653i	$-0.377327\pi$
$223$	2503.69	0.751836	0.375918	+	0.926653i	$0.377327\pi$
$224$	3744.86	1.11703
$225$	0	0
$226$	1423.65	0.419025
$227$	579.270	0.169372	0.0846861	−	0.996408i	$-0.473011\pi$
$227$	579.270	0.169372	0.0846861	+	0.996408i	$0.473011\pi$
$228$	0	0
$229$	768.922	0.221886	0.110943	−	0.993827i	$-0.464613\pi$
$229$	768.922	0.221886	0.110943	+	0.993827i	$0.464613\pi$
$230$	4462.37	1.27930
$231$	0	0
$232$	−15952.4	−4.51433
$233$	−845.695	−0.237783	−0.118891	−	0.992907i	$-0.537934\pi$
$233$	−845.695	−0.237783	−0.118891	+	0.992907i	$0.537934\pi$
$234$	0	0
$235$	452.101	0.125497
$236$	−1701.00	−0.469175
$237$	0	0
$238$	4294.59	1.16965
$239$	−6552.78	−1.77349	−0.886744	−	0.462260i	$-0.847039\pi$
$239$	−6552.78	−1.77349	−0.886744	+	0.462260i	$0.847039\pi$
$240$	0	0
$241$	−5176.75	−1.38367	−0.691834	−	0.722056i	$-0.743197\pi$
$241$	−5176.75	−1.38367	−0.691834	+	0.722056i	$0.743197\pi$
$242$	−3476.39	−0.923434
$243$	0	0
$244$	−551.927	−0.144809
$245$	−3935.08	−1.02613
$246$	0	0
$247$	0	0
$248$	−14205.9	−3.63739
$249$	0	0
$250$	−5623.05	−1.42253
$251$	−4296.82	−1.08053	−0.540264	−	0.841495i	$-0.681676\pi$
$251$	−4296.82	−1.08053	−0.540264	+	0.841495i	$0.681676\pi$
$252$	0	0
$253$	1661.68	0.412921
$254$	−8331.10	−2.05803
$255$	0	0
$256$	5277.77	1.28852
$257$	1383.94	0.335906	0.167953	−	0.985795i	$-0.446284\pi$
$257$	1383.94	0.335906	0.167953	+	0.985795i	$0.446284\pi$
$258$	0	0
$259$	−1622.35	−0.389221
$260$	0	0
$261$	0	0
$262$	−4191.47	−0.988359
$263$	2478.68	0.581147	0.290574	−	0.956853i	$-0.406154\pi$
$263$	2478.68	0.581147	0.290574	+	0.956853i	$0.406154\pi$
$264$	0	0
$265$	796.355	0.184603
$266$	−3801.95	−0.876364
$267$	0	0
$268$	19669.4	4.48321
$269$	2452.64	0.555911	0.277956	−	0.960594i	$-0.410343\pi$
$269$	2452.64	0.555911	0.277956	+	0.960594i	$0.410343\pi$
$270$	0	0
$271$	5980.74	1.34060	0.670302	−	0.742088i	$-0.266164\pi$
$271$	5980.74	1.34060	0.670302	+	0.742088i	$0.266164\pi$
$272$	26378.9	5.88036
$273$	0	0
$274$	13137.4	2.89658
$275$	1181.81	0.259148
$276$	0	0
$277$	−7377.99	−1.60036	−0.800182	−	0.599758i	$-0.795264\pi$
$277$	−7377.99	−1.60036	−0.800182	+	0.599758i	$0.795264\pi$
$278$	6752.42	1.45677
$279$	0	0
$280$	5930.22	1.26571
$281$	−5937.17	−1.26043	−0.630217	−	0.776419i	$-0.717034\pi$
$281$	−5937.17	−1.26043	−0.630217	+	0.776419i	$0.717034\pi$
$282$	0	0
$283$	1486.52	0.312241	0.156121	−	0.987738i	$-0.450101\pi$
$283$	1486.52	0.312241	0.156121	+	0.987738i	$0.450101\pi$
$284$	9029.22	1.88657
$285$	0	0
$286$	0	0
$287$	1461.18	0.300526
$288$	0	0
$289$	10432.5	2.12344
$290$	−15863.0	−3.21210
$291$	0	0
$292$	−2291.42	−0.459231
$293$	3580.05	0.713818	0.356909	−	0.934139i	$-0.383831\pi$
$293$	3580.05	0.713818	0.356909	+	0.934139i	$0.383831\pi$
$294$	0	0
$295$	−1050.63	−0.207355
$296$	−17870.2	−3.50906
$297$	0	0
$298$	4228.32	0.821947
$299$	0	0
$300$	0	0
$301$	−2467.38	−0.472483
$302$	−875.320	−0.166785
$303$	0	0
$304$	−23352.9	−4.40587
$305$	−340.900	−0.0639995
$306$	0	0
$307$	415.013	0.0771533	0.0385767	−	0.999256i	$-0.487718\pi$
$307$	415.013	0.0771533	0.0385767	+	0.999256i	$0.487718\pi$
$308$	3555.23	0.657720
$309$	0	0
$310$	−14126.3	−2.58812
$311$	3009.05	0.548642	0.274321	−	0.961638i	$-0.411547\pi$
$311$	3009.05	0.548642	0.274321	+	0.961638i	$0.411547\pi$
$312$	0	0
$313$	3760.84	0.679154	0.339577	−	0.940578i	$-0.389716\pi$
$313$	3760.84	0.679154	0.339577	+	0.940578i	$0.389716\pi$
$314$	−12711.4	−2.28454
$315$	0	0
$316$	8120.86	1.44568
$317$	−2772.04	−0.491146	−0.245573	−	0.969378i	$-0.578976\pi$
$317$	−2772.04	−0.491146	−0.245573	+	0.969378i	$0.578976\pi$
$318$	0	0
$319$	−5907.01	−1.03677
$320$	18800.3	3.28428
$321$	0	0
$322$	2198.29	0.380454
$323$	−13585.2	−2.34024
$324$	0	0
$325$	0	0
$326$	−8494.67	−1.44318
$327$	0	0
$328$	16094.9	2.70942
$329$	222.718	0.0373217
$330$	0	0
$331$	−6558.16	−1.08903	−0.544515	−	0.838751i	$-0.683286\pi$
$331$	−6558.16	−1.08903	−0.544515	+	0.838751i	$0.683286\pi$
$332$	−1815.43	−0.300105
$333$	0	0
$334$	19287.9	3.15984
$335$	12148.9	1.98139
$336$	0	0
$337$	9509.17	1.53708	0.768542	−	0.639799i	$-0.220983\pi$
$337$	9509.17	1.53708	0.768542	+	0.639799i	$0.220983\pi$
$338$	0	0
$339$	0	0
$340$	34114.8	5.44158
$341$	−5260.29	−0.835368
$342$	0	0
$343$	−4142.29	−0.652077
$344$	−27178.1	−4.25973
$345$	0	0
$346$	−7.77591	−0.00120820
$347$	9808.78	1.51747	0.758736	−	0.651398i	$-0.225817\pi$
$347$	9808.78	1.51747	0.758736	+	0.651398i	$0.225817\pi$
$348$	0	0
$349$	−794.952	−0.121928	−0.0609639	−	0.998140i	$-0.519417\pi$
$349$	−794.952	−0.121928	−0.0609639	+	0.998140i	$0.519417\pi$
$350$	1563.46	0.238772
$351$	0	0
$352$	15274.3	2.31284
$353$	−7115.60	−1.07288	−0.536438	−	0.843940i	$-0.680230\pi$
$353$	−7115.60	−1.07288	−0.536438	+	0.843940i	$0.680230\pi$
$354$	0	0
$355$	5576.93	0.833782
$356$	−10471.3	−1.55893
$357$	0	0
$358$	11235.7	1.65873
$359$	−6907.19	−1.01545	−0.507726	−	0.861518i	$-0.669514\pi$
$359$	−6907.19	−1.01545	−0.507726	+	0.861518i	$0.669514\pi$
$360$	0	0
$361$	5167.78	0.753430
$362$	−2506.39	−0.363903
$363$	0	0
$364$	0	0
$365$	−1415.31	−0.202960
$366$	0	0
$367$	−459.533	−0.0653609	−0.0326804	−	0.999466i	$-0.510404\pi$
$367$	−459.533	−0.0653609	−0.0326804	+	0.999466i	$0.510404\pi$
$368$	13502.7	1.91271
$369$	0	0
$370$	−17770.1	−2.49681
$371$	392.308	0.0548992
$372$	0	0
$373$	5359.92	0.744038	0.372019	−	0.928225i	$-0.378666\pi$
$373$	5359.92	0.744038	0.372019	+	0.928225i	$0.378666\pi$
$374$	17516.5	2.42181
$375$	0	0
$376$	2453.23	0.336478
$377$	0	0
$378$	0	0
$379$	−8594.06	−1.16477	−0.582384	−	0.812914i	$-0.697880\pi$
$379$	−8594.06	−1.16477	−0.582384	+	0.812914i	$0.697880\pi$
$380$	−30201.4	−4.07711
$381$	0	0
$382$	26257.9	3.51694
$383$	9336.49	1.24562	0.622810	−	0.782373i	$-0.285991\pi$
$383$	9336.49	1.24562	0.622810	+	0.782373i	$0.285991\pi$
$384$	0	0
$385$	2195.90	0.290684
$386$	4620.71	0.609295
$387$	0	0
$388$	−4030.07	−0.527309
$389$	−12792.1	−1.66732	−0.833659	−	0.552279i	$-0.813758\pi$
$389$	−12792.1	−1.66732	−0.833659	+	0.552279i	$0.813758\pi$
$390$	0	0
$391$	7854.95	1.01596
$392$	−21352.9	−2.75123
$393$	0	0
$394$	−8481.96	−1.08456
$395$	5015.88	0.638927
$396$	0	0
$397$	13384.5	1.69206	0.846030	−	0.533135i	$-0.178986\pi$
$397$	13384.5	1.69206	0.846030	+	0.533135i	$0.178986\pi$
$398$	16870.8	2.12477
$399$	0	0
$400$	9603.30	1.20041
$401$	−6357.41	−0.791706	−0.395853	−	0.918314i	$-0.629551\pi$
$401$	−6357.41	−0.791706	−0.395853	+	0.918314i	$0.629551\pi$
$402$	0	0
$403$	0	0
$404$	17969.0	2.21285
$405$	0	0
$406$	−7814.58	−0.955249
$407$	−6617.14	−0.805896
$408$	0	0
$409$	5301.63	0.640950	0.320475	−	0.947257i	$-0.396157\pi$
$409$	5301.63	0.640950	0.320475	+	0.947257i	$0.396157\pi$
$410$	16004.7	1.92784
$411$	0	0
$412$	−27650.8	−3.30645
$413$	−517.569	−0.0616656
$414$	0	0
$415$	−1121.31	−0.132633
$416$	0	0
$417$	0	0
$418$	−15507.1	−1.81454
$419$	−16395.8	−1.91167	−0.955835	−	0.293905i	$-0.905045\pi$
$419$	−16395.8	−1.91167	−0.955835	+	0.293905i	$0.905045\pi$
$420$	0	0
$421$	−8484.68	−0.982227	−0.491114	−	0.871095i	$-0.663410\pi$
$421$	−8484.68	−0.982227	−0.491114	+	0.871095i	$0.663410\pi$
$422$	665.808	0.0768034
$423$	0	0
$424$	4321.26	0.494950
$425$	5586.55	0.637617
$426$	0	0
$427$	−167.937	−0.0190329
$428$	20319.6	2.29483
$429$	0	0
$430$	−27025.8	−3.03094
$431$	1594.74	0.178227	0.0891136	−	0.996021i	$-0.471597\pi$
$431$	1594.74	0.178227	0.0891136	+	0.996021i	$0.471597\pi$
$432$	0	0
$433$	−3387.57	−0.375973	−0.187987	−	0.982172i	$-0.560196\pi$
$433$	−3387.57	−0.375973	−0.187987	+	0.982172i	$0.560196\pi$
$434$	−6959.02	−0.769685
$435$	0	0
$436$	−18834.6	−2.06884
$437$	−6953.90	−0.761213
$438$	0	0
$439$	−7211.54	−0.784027	−0.392014	−	0.919959i	$-0.628221\pi$
$439$	−7211.54	−0.784027	−0.392014	+	0.919959i	$0.628221\pi$
$440$	24187.7	2.62069
$441$	0	0
$442$	0	0
$443$	7500.66	0.804441	0.402220	−	0.915543i	$-0.368239\pi$
$443$	7500.66	0.804441	0.402220	+	0.915543i	$0.368239\pi$
$444$	0	0
$445$	−6467.65	−0.688979
$446$	13509.6	1.43430
$447$	0	0
$448$	9261.58	0.976715
$449$	−7649.53	−0.804017	−0.402009	−	0.915636i	$-0.631688\pi$
$449$	−7649.53	−0.804017	−0.402009	+	0.915636i	$0.631688\pi$
$450$	0	0
$451$	5959.77	0.622249
$452$	5571.13	0.579743
$453$	0	0
$454$	3125.67	0.323117
$455$	0	0
$456$	0	0
$457$	9001.79	0.921414	0.460707	−	0.887552i	$-0.347596\pi$
$457$	9001.79	0.921414	0.460707	+	0.887552i	$0.347596\pi$
$458$	4149.02	0.423299
$459$	0	0
$460$	17462.5	1.76999
$461$	−1347.83	−0.136171	−0.0680854	−	0.997679i	$-0.521689\pi$
$461$	−1347.83	−0.136171	−0.0680854	+	0.997679i	$0.521689\pi$
$462$	0	0
$463$	94.9035	0.00952600	0.00476300	−	0.999989i	$-0.498484\pi$
$463$	94.9035	0.00952600	0.00476300	+	0.999989i	$0.498484\pi$
$464$	−47999.9	−4.80246
$465$	0	0
$466$	−4563.28	−0.453626
$467$	−13290.9	−1.31698	−0.658489	−	0.752591i	$-0.728804\pi$
$467$	−13290.9	−1.31698	−0.658489	+	0.752591i	$0.728804\pi$
$468$	0	0
$469$	5984.90	0.589247
$470$	2439.49	0.239415
$471$	0	0
$472$	−5701.00	−0.555953
$473$	−10063.8	−0.978294
$474$	0	0
$475$	−4945.70	−0.477736
$476$	16806.0	1.61828
$477$	0	0
$478$	−35358.1	−3.38335
$479$	−3801.59	−0.362628	−0.181314	−	0.983425i	$-0.558035\pi$
$479$	−3801.59	−0.362628	−0.181314	+	0.983425i	$0.558035\pi$
$480$	0	0
$481$	0	0
$482$	−27933.2	−2.63967
$483$	0	0
$484$	−13604.1	−1.27762
$485$	−2489.19	−0.233048
$486$	0	0
$487$	8747.34	0.813922	0.406961	−	0.913446i	$-0.366588\pi$
$487$	8747.34	0.813922	0.406961	+	0.913446i	$0.366588\pi$
$488$	−1849.82	−0.171593
$489$	0	0
$490$	−21233.2	−1.95759
$491$	−15085.4	−1.38655	−0.693273	−	0.720675i	$-0.743832\pi$
$491$	−15085.4	−1.38655	−0.693273	+	0.720675i	$0.743832\pi$
$492$	0	0
$493$	−27923.1	−2.55090
$494$	0	0
$495$	0	0
$496$	−42744.7	−3.86955
$497$	2747.36	0.247959
$498$	0	0
$499$	−14593.9	−1.30925	−0.654623	−	0.755955i	$-0.727173\pi$
$499$	−14593.9	−1.30925	−0.654623	+	0.755955i	$0.727173\pi$
$500$	−22004.6	−1.96815
$501$	0	0
$502$	−23185.2	−2.06136
$503$	18204.9	1.61375	0.806876	−	0.590721i	$-0.201157\pi$
$503$	18204.9	1.61375	0.806876	+	0.590721i	$0.201157\pi$
$504$	0	0
$505$	11098.6	0.977983
$506$	8966.25	0.787744
$507$	0	0
$508$	−32601.9	−2.84739
$509$	−7891.92	−0.687236	−0.343618	−	0.939109i	$-0.611653\pi$
$509$	−7891.92	−0.687236	−0.343618	+	0.939109i	$0.611653\pi$
$510$	0	0
$511$	−697.221	−0.0603586
$512$	3555.88	0.306932
$513$	0	0
$514$	7467.59	0.640819
$515$	−17078.6	−1.46131
$516$	0	0
$517$	908.406	0.0772759
$518$	−8754.05	−0.742530
$519$	0	0
$520$	0	0
$521$	−647.902	−0.0544820	−0.0272410	−	0.999629i	$-0.508672\pi$
$521$	−647.902	−0.0544820	−0.0272410	+	0.999629i	$0.508672\pi$
$522$	0	0
$523$	9269.99	0.775044	0.387522	−	0.921860i	$-0.373331\pi$
$523$	9269.99	0.775044	0.387522	+	0.921860i	$0.373331\pi$
$524$	−16402.4	−1.36745
$525$	0	0
$526$	13374.7	1.10867
$527$	−24866.0	−2.05537
$528$	0	0
$529$	−8146.25	−0.669536
$530$	4297.05	0.352173
$531$	0	0
$532$	−14878.1	−1.21250
$533$	0	0
$534$	0	0
$535$	12550.5	1.01421
$536$	65923.4	5.31242
$537$	0	0
$538$	13234.2	1.06053
$539$	−7906.75	−0.631852
$540$	0	0
$541$	−3844.83	−0.305549	−0.152775	−	0.988261i	$-0.548821\pi$
$541$	−3844.83	−0.305549	−0.152775	+	0.988261i	$0.548821\pi$
$542$	32271.4	2.55752
$543$	0	0
$544$	72203.1	5.69060
$545$	−11633.2	−0.914337
$546$	0	0
$547$	−20245.4	−1.58251	−0.791253	−	0.611489i	$-0.790571\pi$
$547$	−20245.4	−1.58251	−0.791253	+	0.611489i	$0.790571\pi$
$548$	51410.5	4.00757
$549$	0	0
$550$	6376.91	0.494386
$551$	24720.0	1.91126
$552$	0	0
$553$	2470.97	0.190011
$554$	−39810.8	−3.05307
$555$	0	0
$556$	26424.1	2.01553
$557$	−7223.34	−0.549485	−0.274742	−	0.961518i	$-0.588593\pi$
$557$	−7223.34	−0.549485	−0.274742	+	0.961518i	$0.588593\pi$
$558$	0	0
$559$	0	0
$560$	17843.7	1.34649
$561$	0	0
$562$	−32036.3	−2.40458
$563$	−17290.3	−1.29431	−0.647156	−	0.762358i	$-0.724042\pi$
$563$	−17290.3	−1.29431	−0.647156	+	0.762358i	$0.724042\pi$
$564$	0	0
$565$	3441.03	0.256221
$566$	8021.09	0.595674
$567$	0	0
$568$	30262.0	2.23551
$569$	17364.0	1.27933	0.639663	−	0.768655i	$-0.279074\pi$
$569$	17364.0	1.27933	0.639663	+	0.768655i	$0.279074\pi$
$570$	0	0
$571$	158.149	0.0115908	0.00579540	−	0.999983i	$-0.498155\pi$
$571$	158.149	0.0115908	0.00579540	+	0.999983i	$0.498155\pi$
$572$	0	0
$573$	0	0
$574$	7884.38	0.573323
$575$	2859.61	0.207398
$576$	0	0
$577$	−9627.04	−0.694591	−0.347295	−	0.937756i	$-0.612900\pi$
$577$	−9627.04	−0.694591	−0.347295	+	0.937756i	$0.612900\pi$
$578$	56292.4	4.05096
$579$	0	0
$580$	−62076.4	−4.44411
$581$	−552.389	−0.0394440
$582$	0	0
$583$	1600.12	0.113671
$584$	−7679.86	−0.544169
$585$	0	0
$586$	19317.6	1.36178
$587$	13321.2	0.936668	0.468334	−	0.883551i	$-0.344854\pi$
$587$	13321.2	0.936668	0.468334	+	0.883551i	$0.344854\pi$
$588$	0	0
$589$	22013.6	1.53999
$590$	−5669.06	−0.395579
$591$	0	0
$592$	−53770.4	−3.73303
$593$	−16723.4	−1.15809	−0.579044	−	0.815296i	$-0.696574\pi$
$593$	−16723.4	−1.15809	−0.579044	+	0.815296i	$0.696574\pi$
$594$	0	0
$595$	10380.3	0.715209
$596$	16546.6	1.13721
$597$	0	0
$598$	0	0
$599$	17160.0	1.17051	0.585256	−	0.810849i	$-0.300994\pi$
$599$	17160.0	1.17051	0.585256	+	0.810849i	$0.300994\pi$
$600$	0	0
$601$	11048.6	0.749885	0.374942	−	0.927048i	$-0.377663\pi$
$601$	11048.6	0.749885	0.374942	+	0.927048i	$0.377663\pi$
$602$	−13313.7	−0.901373
$603$	0	0
$604$	−3425.37	−0.230756
$605$	−8402.62	−0.564653
$606$	0	0
$607$	21119.6	1.41222	0.706110	−	0.708102i	$-0.250448\pi$
$607$	21119.6	1.41222	0.706110	+	0.708102i	$0.250448\pi$
$608$	−63920.6	−4.26369
$609$	0	0
$610$	−1839.46	−0.122094
$611$	0	0
$612$	0	0
$613$	6233.83	0.410737	0.205369	−	0.978685i	$-0.434161\pi$
$613$	6233.83	0.410737	0.205369	+	0.978685i	$0.434161\pi$
$614$	2239.37	0.147188
$615$	0	0
$616$	11915.6	0.779371
$617$	5598.36	0.365286	0.182643	−	0.983179i	$-0.441535\pi$
$617$	5598.36	0.365286	0.182643	+	0.983179i	$0.441535\pi$
$618$	0	0
$619$	15874.3	1.03076	0.515382	−	0.856961i	$-0.327650\pi$
$619$	15874.3	1.03076	0.515382	+	0.856961i	$0.327650\pi$
$620$	−55280.1	−3.58081
$621$	0	0
$622$	16236.5	1.04666
$623$	−3186.15	−0.204896
$624$	0	0
$625$	−19228.4	−1.23062
$626$	20293.1	1.29565
$627$	0	0
$628$	−49743.3	−3.16079
$629$	−31280.0	−1.98285
$630$	0	0
$631$	−11249.0	−0.709693	−0.354846	−	0.934925i	$-0.615467\pi$
$631$	−11249.0	−0.709693	−0.354846	+	0.934925i	$0.615467\pi$
$632$	27217.6	1.71307
$633$	0	0
$634$	−14957.6	−0.936976
$635$	−20136.7	−1.25843
$636$	0	0
$637$	0	0
$638$	−31873.6	−1.97788
$639$	0	0
$640$	40630.2	2.50945
$641$	23768.3	1.46457	0.732287	−	0.680996i	$-0.238453\pi$
$641$	23768.3	1.46457	0.732287	+	0.680996i	$0.238453\pi$
$642$	0	0
$643$	4368.95	0.267954	0.133977	−	0.990984i	$-0.457225\pi$
$643$	4368.95	0.267954	0.133977	+	0.990984i	$0.457225\pi$
$644$	8602.54	0.526378
$645$	0	0
$646$	−73304.0	−4.46456
$647$	21305.2	1.29458	0.647290	−	0.762244i	$-0.275902\pi$
$647$	21305.2	1.29458	0.647290	+	0.762244i	$0.275902\pi$
$648$	0	0
$649$	−2111.02	−0.127681
$650$	0	0
$651$	0	0
$652$	−33242.0	−1.99672
$653$	27958.3	1.67549	0.837744	−	0.546064i	$-0.183874\pi$
$653$	27958.3	1.67549	0.837744	+	0.546064i	$0.183874\pi$
$654$	0	0
$655$	−10131.0	−0.604353
$656$	48428.6	2.88235
$657$	0	0
$658$	1201.76	0.0711999
$659$	−9619.46	−0.568621	−0.284310	−	0.958732i	$-0.591765\pi$
$659$	−9619.46	−0.568621	−0.284310	+	0.958732i	$0.591765\pi$
$660$	0	0
$661$	−4414.40	−0.259759	−0.129879	−	0.991530i	$-0.541459\pi$
$661$	−4414.40	−0.259759	−0.129879	+	0.991530i	$0.541459\pi$
$662$	−35387.1	−2.07758
$663$	0	0
$664$	−6084.55	−0.355612
$665$	−9189.52	−0.535871
$666$	0	0
$667$	−14293.1	−0.829733
$668$	75479.0	4.37181
$669$	0	0
$670$	65554.1	3.77996
$671$	−684.970	−0.0394083
$672$	0	0
$673$	−18001.8	−1.03108	−0.515540	−	0.856865i	$-0.672409\pi$
$673$	−18001.8	−1.03108	−0.515540	+	0.856865i	$0.672409\pi$
$674$	51310.4	2.93235
$675$	0	0
$676$	0	0
$677$	11192.0	0.635369	0.317684	−	0.948197i	$-0.397095\pi$
$677$	11192.0	0.635369	0.317684	+	0.948197i	$0.397095\pi$
$678$	0	0
$679$	−1226.25	−0.0693064
$680$	114338.	6.44804
$681$	0	0
$682$	−28383.9	−1.59366
$683$	35122.3	1.96767	0.983835	−	0.179076i	$-0.0573109\pi$
$683$	35122.3	1.96767	0.983835	+	0.179076i	$0.0573109\pi$
$684$	0	0
$685$	31753.9	1.77117
$686$	−22351.3	−1.24399
$687$	0	0
$688$	−81777.5	−4.53160
$689$	0	0
$690$	0	0
$691$	−29534.6	−1.62597	−0.812987	−	0.582282i	$-0.802160\pi$
$691$	−29534.6	−1.62597	−0.812987	+	0.582282i	$0.802160\pi$
$692$	−30.4293	−0.00167160
$693$	0	0
$694$	52927.1	2.89494
$695$	16321.0	0.890776
$696$	0	0
$697$	28172.5	1.53100
$698$	−4289.47	−0.232606
$699$	0	0
$700$	6118.24	0.330354
$701$	−8804.19	−0.474365	−0.237182	−	0.971465i	$-0.576224\pi$
$701$	−8804.19	−0.474365	−0.237182	+	0.971465i	$0.576224\pi$
$702$	0	0
$703$	27691.8	1.48566
$704$	37775.5	2.02232
$705$	0	0
$706$	−38395.0	−2.04676
$707$	5467.49	0.290843
$708$	0	0
$709$	−27138.2	−1.43752	−0.718758	−	0.695261i	$-0.755289\pi$
$709$	−27138.2	−1.43752	−0.718758	+	0.695261i	$0.755289\pi$
$710$	30092.5	1.59064
$711$	0	0
$712$	−35095.3	−1.84727
$713$	−12728.3	−0.668552
$714$	0	0
$715$	0	0
$716$	43968.6	2.29495
$717$	0	0
$718$	−37270.4	−1.93721
$719$	−17337.4	−0.899274	−0.449637	−	0.893211i	$-0.648447\pi$
$719$	−17337.4	−0.899274	−0.449637	+	0.893211i	$0.648447\pi$
$720$	0	0
$721$	−8413.44	−0.434581
$722$	27884.8	1.43735
$723$	0	0
$724$	−9808.20	−0.503479
$725$	−10165.5	−0.520739
$726$	0	0
$727$	25771.3	1.31473	0.657363	−	0.753574i	$-0.271672\pi$
$727$	25771.3	1.31473	0.657363	+	0.753574i	$0.271672\pi$
$728$	0	0
$729$	0	0
$730$	−7636.83	−0.387194
$731$	−47572.6	−2.40703
$732$	0	0
$733$	−21784.2	−1.09771	−0.548853	−	0.835919i	$-0.684935\pi$
$733$	−21784.2	−1.09771	−0.548853	+	0.835919i	$0.684935\pi$
$734$	−2479.59	−0.124691
$735$	0	0
$736$	36959.0	1.85099
$737$	24410.8	1.22006
$738$	0	0
$739$	−22353.8	−1.11272	−0.556360	−	0.830942i	$-0.687802\pi$
$739$	−22353.8	−1.11272	−0.556360	+	0.830942i	$0.687802\pi$
$740$	−69539.2	−3.45448
$741$	0	0
$742$	2116.85	0.104733
$743$	−18382.6	−0.907662	−0.453831	−	0.891088i	$-0.649943\pi$
$743$	−18382.6	−0.907662	−0.453831	+	0.891088i	$0.649943\pi$
$744$	0	0
$745$	10220.1	0.502597
$746$	28921.5	1.41943
$747$	0	0
$748$	68546.9	3.35070
$749$	6182.74	0.301618
$750$	0	0
$751$	13782.8	0.669694	0.334847	−	0.942272i	$-0.391315\pi$
$751$	13782.8	0.669694	0.334847	+	0.942272i	$0.391315\pi$
$752$	7381.64	0.357953
$753$	0	0
$754$	0	0
$755$	−2115.69	−0.101984
$756$	0	0
$757$	25639.5	1.23102	0.615510	−	0.788129i	$-0.288950\pi$
$757$	25639.5	1.23102	0.615510	+	0.788129i	$0.288950\pi$
$758$	−46372.6	−2.22207
$759$	0	0
$760$	−101222.	−4.83121
$761$	43.4757	0.00207095	0.00103548	−	0.999999i	$-0.499670\pi$
$761$	43.4757	0.00207095	0.00103548	+	0.999999i	$0.499670\pi$
$762$	0	0
$763$	−5730.88	−0.271916
$764$	102755.	4.86588
$765$	0	0
$766$	50378.7	2.37631
$767$	0	0
$768$	0	0
$769$	20485.7	0.960641	0.480321	−	0.877093i	$-0.340520\pi$
$769$	20485.7	0.960641	0.480321	+	0.877093i	$0.340520\pi$
$770$	11848.8	0.554548
$771$	0	0
$772$	18082.1	0.842992
$773$	29450.3	1.37031	0.685157	−	0.728395i	$-0.259734\pi$
$773$	29450.3	1.37031	0.685157	+	0.728395i	$0.259734\pi$
$774$	0	0
$775$	−9052.51	−0.419582
$776$	−13507.1	−0.624839
$777$	0	0
$778$	−69025.0	−3.18080
$779$	−24940.8	−1.14711
$780$	0	0
$781$	11205.7	0.513409
$782$	42384.5	1.93819
$783$	0	0
$784$	−64249.7	−2.92683
$785$	−30724.1	−1.39693
$786$	0	0
$787$	−9751.10	−0.441664	−0.220832	−	0.975312i	$-0.570877\pi$
$787$	−9751.10	−0.441664	−0.220832	+	0.975312i	$0.570877\pi$
$788$	−33192.3	−1.50054
$789$	0	0
$790$	27065.1	1.21890
$791$	1695.15	0.0761980
$792$	0	0
$793$	0	0
$794$	72221.2	3.22800
$795$	0	0
$796$	66020.3	2.93973
$797$	37750.8	1.67779	0.838896	−	0.544291i	$-0.183201\pi$
$797$	37750.8	1.67779	0.838896	+	0.544291i	$0.183201\pi$
$798$	0	0
$799$	4294.14	0.190132
$800$	26285.7	1.16167
$801$	0	0
$802$	−34303.9	−1.51037
$803$	−2843.77	−0.124975
$804$	0	0
$805$	5313.39	0.232637
$806$	0	0
$807$	0	0
$808$	60224.2	2.62213
$809$	6999.45	0.304187	0.152094	−	0.988366i	$-0.451398\pi$
$809$	6999.45	0.304187	0.152094	+	0.988366i	$0.451398\pi$
$810$	0	0
$811$	35792.1	1.54973	0.774864	−	0.632128i	$-0.217818\pi$
$811$	35792.1	1.54973	0.774864	+	0.632128i	$0.217818\pi$
$812$	−30580.6	−1.32164
$813$	0	0
$814$	−35705.4	−1.53744
$815$	−20532.1	−0.882463
$816$	0	0
$817$	42115.5	1.80347
$818$	28607.0	1.22276
$819$	0	0
$820$	62630.9	2.66727
$821$	9776.62	0.415599	0.207799	−	0.978171i	$-0.433370\pi$
$821$	9776.62	0.415599	0.207799	+	0.978171i	$0.433370\pi$
$822$	0	0
$823$	19662.0	0.832777	0.416389	−	0.909187i	$-0.363296\pi$
$823$	19662.0	0.832777	0.416389	+	0.909187i	$0.363296\pi$
$824$	−92673.7	−3.91801
$825$	0	0
$826$	−2792.75	−0.117642
$827$	5281.22	0.222063	0.111032	−	0.993817i	$-0.464585\pi$
$827$	5281.22	0.222063	0.111032	+	0.993817i	$0.464585\pi$
$828$	0	0
$829$	21735.5	0.910621	0.455310	−	0.890333i	$-0.349528\pi$
$829$	21735.5	0.910621	0.455310	+	0.890333i	$0.349528\pi$
$830$	−6050.46	−0.253030
$831$	0	0
$832$	0	0
$833$	−37376.1	−1.55463
$834$	0	0
$835$	46619.9	1.93215
$836$	−60683.8	−2.51052
$837$	0	0
$838$	−88470.2	−3.64696
$839$	−28574.7	−1.17582	−0.587908	−	0.808928i	$-0.700048\pi$
$839$	−28574.7	−1.17582	−0.587908	+	0.808928i	$0.700048\pi$
$840$	0	0
$841$	26420.7	1.08330
$842$	−45782.4	−1.87383
$843$	0	0
$844$	2605.49	0.106262
$845$	0	0
$846$	0	0
$847$	−4139.38	−0.167923
$848$	13002.4	0.526540
$849$	0	0
$850$	30144.4	1.21640
$851$	−16011.4	−0.644965
$852$	0	0
$853$	−5251.39	−0.210791	−0.105395	−	0.994430i	$-0.533611\pi$
$853$	−5251.39	−0.210791	−0.105395	+	0.994430i	$0.533611\pi$
$854$	−906.170	−0.0363097
$855$	0	0
$856$	68102.6	2.71927
$857$	34851.0	1.38913	0.694566	−	0.719429i	$-0.255596\pi$
$857$	34851.0	1.38913	0.694566	+	0.719429i	$0.255596\pi$
$858$	0	0
$859$	41697.5	1.65623	0.828114	−	0.560559i	$-0.189414\pi$
$859$	41697.5	1.65623	0.828114	+	0.560559i	$0.189414\pi$
$860$	−105760.	−4.19346
$861$	0	0
$862$	8605.05	0.340010
$863$	28648.9	1.13004	0.565018	−	0.825079i	$-0.308869\pi$
$863$	28648.9	1.13004	0.565018	+	0.825079i	$0.308869\pi$
$864$	0	0
$865$	−18.7948	−0.000738777 0
$866$	−18279.0	−0.717257
$867$	0	0
$868$	−27232.6	−1.06490
$869$	10078.4	0.393425
$870$	0	0
$871$	0	0
$872$	−63125.4	−2.45149
$873$	0	0
$874$	−37522.5	−1.45219
$875$	−6695.42	−0.258682
$876$	0	0
$877$	31516.8	1.21351	0.606754	−	0.794890i	$-0.292471\pi$
$877$	31516.8	1.21351	0.606754	+	0.794890i	$0.292471\pi$
$878$	−38912.7	−1.49572
$879$	0	0
$880$	72779.7	2.78796
$881$	−31246.1	−1.19490	−0.597451	−	0.801905i	$-0.703820\pi$
$881$	−31246.1	−1.19490	−0.597451	+	0.801905i	$0.703820\pi$
$882$	0	0
$883$	−16181.8	−0.616718	−0.308359	−	0.951270i	$-0.599780\pi$
$883$	−16181.8	−0.616718	−0.308359	+	0.951270i	$0.599780\pi$
$884$	0	0
$885$	0	0
$886$	40472.8	1.53466
$887$	13289.0	0.503046	0.251523	−	0.967851i	$-0.419069\pi$
$887$	13289.0	0.503046	0.251523	+	0.967851i	$0.419069\pi$
$888$	0	0
$889$	−9919.92	−0.374245
$890$	−34898.7	−1.31439
$891$	0	0
$892$	52867.0	1.98444
$893$	−3801.55	−0.142457
$894$	0	0
$895$	27157.4	1.01427
$896$	20015.6	0.746288
$897$	0	0
$898$	−41276.0	−1.53385
$899$	45246.9	1.67861
$900$	0	0
$901$	7563.94	0.279679
$902$	32158.2	1.18709
$903$	0	0
$904$	18672.0	0.686971
$905$	−6058.07	−0.222516
$906$	0	0
$907$	47128.4	1.72533	0.862664	−	0.505778i	$-0.168795\pi$
$907$	47128.4	1.72533	0.862664	+	0.505778i	$0.168795\pi$
$908$	12231.6	0.447050
$909$	0	0
$910$	0	0
$911$	2884.69	0.104911	0.0524556	−	0.998623i	$-0.483295\pi$
$911$	2884.69	0.104911	0.0524556	+	0.998623i	$0.483295\pi$
$912$	0	0
$913$	−2253.05	−0.0816703
$914$	48572.7	1.75781
$915$	0	0
$916$	16236.3	0.585657
$917$	−4990.83	−0.179729
$918$	0	0
$919$	−25752.2	−0.924359	−0.462179	−	0.886787i	$-0.652932\pi$
$919$	−25752.2	−0.924359	−0.462179	+	0.886787i	$0.652932\pi$
$920$	58526.8	2.09736
$921$	0	0
$922$	−7272.75	−0.259778
$923$	0	0
$924$	0	0
$925$	−11387.5	−0.404779
$926$	512.089	0.0181731
$927$	0	0
$928$	−131383.	−4.64748
$929$	−51394.7	−1.81508	−0.907539	−	0.419969i	$-0.862041\pi$
$929$	−51394.7	−1.81508	−0.907539	+	0.419969i	$0.862041\pi$
$930$	0	0
$931$	33088.6	1.16481
$932$	−17857.4	−0.627616
$933$	0	0
$934$	−71716.1	−2.51244
$935$	42338.3	1.48087
$936$	0	0
$937$	6781.86	0.236450	0.118225	−	0.992987i	$-0.462280\pi$
$937$	6781.86	0.236450	0.118225	+	0.992987i	$0.462280\pi$
$938$	32293.9	1.12413
$939$	0	0
$940$	9546.39	0.331244
$941$	22038.2	0.763470	0.381735	−	0.924272i	$-0.375327\pi$
$941$	22038.2	0.763470	0.381735	+	0.924272i	$0.375327\pi$
$942$	0	0
$943$	14420.8	0.497991
$944$	−17154.0	−0.591436
$945$	0	0
$946$	−54303.0	−1.86633
$947$	45531.0	1.56236	0.781182	−	0.624304i	$-0.214617\pi$
$947$	45531.0	1.56236	0.781182	+	0.624304i	$0.214617\pi$
$948$	0	0
$949$	0	0
$950$	−26686.5	−0.911393
$951$	0	0
$952$	56326.3	1.91759
$953$	−26541.2	−0.902154	−0.451077	−	0.892485i	$-0.648960\pi$
$953$	−26541.2	−0.902154	−0.451077	+	0.892485i	$0.648960\pi$
$954$	0	0
$955$	63466.7	2.15051
$956$	−138366.	−4.68104
$957$	0	0
$958$	−20513.0	−0.691799
$959$	15642.9	0.526731
$960$	0	0
$961$	10502.1	0.352527
$962$	0	0
$963$	0	0
$964$	−109310.	−3.65213
$965$	11168.5	0.372566
$966$	0	0
$967$	−48269.6	−1.60522	−0.802609	−	0.596506i	$-0.796555\pi$
$967$	−48269.6	−1.60522	−0.802609	+	0.596506i	$0.796555\pi$
$968$	−45595.1	−1.51393
$969$	0	0
$970$	−13431.4	−0.444594
$971$	34161.3	1.12903	0.564515	−	0.825423i	$-0.309063\pi$
$971$	34161.3	1.12903	0.564515	+	0.825423i	$0.309063\pi$
$972$	0	0
$973$	8040.18	0.264909
$974$	47199.7	1.55275
$975$	0	0
$976$	−5566.02	−0.182545
$977$	−24124.2	−0.789971	−0.394985	−	0.918687i	$-0.629250\pi$
$977$	−24124.2	−0.789971	−0.394985	+	0.918687i	$0.629250\pi$
$978$	0	0
$979$	−12995.4	−0.424245
$980$	−83091.7	−2.70843
$981$	0	0
$982$	−81399.1	−2.64516
$983$	19023.4	0.617246	0.308623	−	0.951185i	$-0.400132\pi$
$983$	19023.4	0.617246	0.308623	+	0.951185i	$0.400132\pi$
$984$	0	0
$985$	−20501.3	−0.663175
$986$	−150670.	−4.86644
$987$	0	0
$988$	0	0
$989$	−24351.2	−0.782936
$990$	0	0
$991$	−229.962	−0.00737131	−0.00368566	−	0.999993i	$-0.501173\pi$
$991$	−229.962	−0.00737131	−0.00368566	+	0.999993i	$0.501173\pi$
$992$	−116999.	−3.74468
$993$	0	0
$994$	14824.4	0.473041
$995$	40777.7	1.29924
$996$	0	0
$997$	26991.1	0.857389	0.428694	−	0.903450i	$-0.358974\pi$
$997$	26991.1	0.857389	0.428694	+	0.903450i	$0.358974\pi$
$998$	−78747.2	−2.49769
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bd.1.8		8
3.2	odd	2	inner	1521.4.a.bd.1.1		8
13.4	even	6		117.4.g.f.55.8	yes	16
13.10	even	6		117.4.g.f.100.8	yes	16
13.12	even	2		1521.4.a.bc.1.1		8
39.17	odd	6		117.4.g.f.55.1	✓	16
39.23	odd	6		117.4.g.f.100.1	yes	16
39.38	odd	2		1521.4.a.bc.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.g.f.55.1	✓	16	39.17	odd	6
117.4.g.f.55.8	yes	16	13.4	even	6
117.4.g.f.100.1	yes	16	39.23	odd	6
117.4.g.f.100.8	yes	16	13.10	even	6
1521.4.a.bc.1.1		8	13.12	even	2
1521.4.a.bc.1.8		8	39.38	odd	2
1521.4.a.bd.1.1		8	3.2	odd	2	inner
1521.4.a.bd.1.8		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+5.39589 q^{2} +21.1156 q^{4} +13.0421 q^{5} +6.42494 q^{7} +70.7705 q^{8} +O(q^{10})\)	\(q+5.39589 q^{2} +21.1156 q^{4} +13.0421 q^{5} +6.42494 q^{7} +70.7705 q^{8} +70.3740 q^{10} +26.2056 q^{11} +34.6683 q^{14} +212.945 q^{16} +123.877 q^{17} -109.667 q^{19} +275.393 q^{20} +141.402 q^{22} +63.4094 q^{23} +45.0976 q^{25} +135.667 q^{28} -225.410 q^{29} -200.732 q^{31} +582.863 q^{32} +668.426 q^{34} +83.7950 q^{35} -252.509 q^{37} -591.749 q^{38} +922.999 q^{40} +227.423 q^{41} -384.032 q^{43} +553.348 q^{44} +342.150 q^{46} +34.6646 q^{47} -301.720 q^{49} +243.342 q^{50} +61.0601 q^{53} +341.777 q^{55} +454.696 q^{56} -1216.29 q^{58} -80.5562 q^{59} -26.1383 q^{61} -1083.13 q^{62} +1441.50 q^{64} +931.510 q^{67} +2615.74 q^{68} +452.149 q^{70} +427.608 q^{71} -108.518 q^{73} -1362.51 q^{74} -2315.68 q^{76} +168.369 q^{77} +384.590 q^{79} +2777.26 q^{80} +1227.15 q^{82} -85.9758 q^{83} +1615.62 q^{85} -2072.19 q^{86} +1854.58 q^{88} -495.903 q^{89} +1338.93 q^{92} +187.046 q^{94} -1430.29 q^{95} -190.857 q^{97} -1628.05 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{4} + 22 q^{7}+O(q^{10}) \) 8 * q + 40 * q^4 + 22 * q^7	\( 8 q + 40 q^{4} + 22 q^{7} + 36 q^{10} + 204 q^{16} - 244 q^{19} + 136 q^{22} + 354 q^{25} + 452 q^{28} - 242 q^{31} + 1292 q^{34} - 1018 q^{37} + 1700 q^{40} + 74 q^{43} + 896 q^{46} + 298 q^{49} + 1300 q^{55} - 812 q^{58} + 1148 q^{61} + 3636 q^{64} + 2198 q^{67} - 2200 q^{70} + 2176 q^{73} - 6936 q^{76} + 1862 q^{79} + 5436 q^{82} + 890 q^{85} + 3528 q^{88} - 3104 q^{94} + 4370 q^{97}+O(q^{100}) \) 8 * q + 40 * q^4 + 22 * q^7 + 36 * q^10 + 204 * q^16 - 244 * q^19 + 136 * q^22 + 354 * q^25 + 452 * q^28 - 242 * q^31 + 1292 * q^34 - 1018 * q^37 + 1700 * q^40 + 74 * q^43 + 896 * q^46 + 298 * q^49 + 1300 * q^55 - 812 * q^58 + 1148 * q^61 + 3636 * q^64 + 2198 * q^67 - 2200 * q^70 + 2176 * q^73 - 6936 * q^76 + 1862 * q^79 + 5436 * q^82 + 890 * q^85 + 3528 * q^88 - 3104 * q^94 + 4370 * q^97

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