LMFDB - Embedded newform 4014.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1338)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.44504 q^{5} -4.35690 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.44504 q^{5} -4.35690 q^{7} -1.00000 q^{8} -2.44504 q^{10} -6.34481 q^{11} -5.15883 q^{13} +4.35690 q^{14} +1.00000 q^{16} +6.82908 q^{17} -5.18598 q^{19} +2.44504 q^{20} +6.34481 q^{22} +5.35690 q^{23} +0.978230 q^{25} +5.15883 q^{26} -4.35690 q^{28} -1.96077 q^{29} -0.198062 q^{31} -1.00000 q^{32} -6.82908 q^{34} -10.6528 q^{35} +3.63102 q^{37} +5.18598 q^{38} -2.44504 q^{40} -10.2838 q^{41} -3.69202 q^{43} -6.34481 q^{44} -5.35690 q^{46} +7.25667 q^{47} +11.9825 q^{49} -0.978230 q^{50} -5.15883 q^{52} +5.62565 q^{53} -15.5133 q^{55} +4.35690 q^{56} +1.96077 q^{58} +6.92692 q^{59} +3.12498 q^{61} +0.198062 q^{62} +1.00000 q^{64} -12.6136 q^{65} -5.03684 q^{67} +6.82908 q^{68} +10.6528 q^{70} +6.16421 q^{71} +8.65279 q^{73} -3.63102 q^{74} -5.18598 q^{76} +27.6437 q^{77} +4.78448 q^{79} +2.44504 q^{80} +10.2838 q^{82} -0.911854 q^{83} +16.6974 q^{85} +3.69202 q^{86} +6.34481 q^{88} +13.7409 q^{89} +22.4765 q^{91} +5.35690 q^{92} -7.25667 q^{94} -12.6799 q^{95} -9.50365 q^{97} -11.9825 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 7 q^{5} - 9 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 7 * q^5 - 9 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 7 q^{5} - 9 q^{7} - 3 q^{8} - 7 q^{10} + 4 q^{11} - 7 q^{13} + 9 q^{14} + 3 q^{16} + 10 q^{17} - q^{19} + 7 q^{20} - 4 q^{22} + 12 q^{23} + 6 q^{25} + 7 q^{26} - 9 q^{28} + 7 q^{29} - 5 q^{31} - 3 q^{32} - 10 q^{34} - 14 q^{35} - 4 q^{37} + q^{38} - 7 q^{40} + 2 q^{41} - 6 q^{43} + 4 q^{44} - 12 q^{46} - 5 q^{47} + 20 q^{49} - 6 q^{50} - 7 q^{52} + 5 q^{53} + 7 q^{55} + 9 q^{56} - 7 q^{58} - 8 q^{59} - 15 q^{61} + 5 q^{62} + 3 q^{64} - 7 q^{65} + 13 q^{67} + 10 q^{68} + 14 q^{70} + 7 q^{71} + 8 q^{73} + 4 q^{74} - q^{76} + 2 q^{77} - 6 q^{79} + 7 q^{80} - 2 q^{82} + q^{83} + 42 q^{85} + 6 q^{86} - 4 q^{88} + 27 q^{89} + 42 q^{91} + 12 q^{92} + 5 q^{94} - 14 q^{95} + 3 q^{97} - 20 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 7 * q^5 - 9 * q^7 - 3 * q^8 - 7 * q^10 + 4 * q^11 - 7 * q^13 + 9 * q^14 + 3 * q^16 + 10 * q^17 - q^19 + 7 * q^20 - 4 * q^22 + 12 * q^23 + 6 * q^25 + 7 * q^26 - 9 * q^28 + 7 * q^29 - 5 * q^31 - 3 * q^32 - 10 * q^34 - 14 * q^35 - 4 * q^37 + q^38 - 7 * q^40 + 2 * q^41 - 6 * q^43 + 4 * q^44 - 12 * q^46 - 5 * q^47 + 20 * q^49 - 6 * q^50 - 7 * q^52 + 5 * q^53 + 7 * q^55 + 9 * q^56 - 7 * q^58 - 8 * q^59 - 15 * q^61 + 5 * q^62 + 3 * q^64 - 7 * q^65 + 13 * q^67 + 10 * q^68 + 14 * q^70 + 7 * q^71 + 8 * q^73 + 4 * q^74 - q^76 + 2 * q^77 - 6 * q^79 + 7 * q^80 - 2 * q^82 + q^83 + 42 * q^85 + 6 * q^86 - 4 * q^88 + 27 * q^89 + 42 * q^91 + 12 * q^92 + 5 * q^94 - 14 * q^95 + 3 * q^97 - 20 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.44504	1.09346	0.546728	−	0.837310i	$-0.315873\pi$
$5$	2.44504	1.09346	0.546728	+	0.837310i	$0.315873\pi$
$6$	0	0
$7$	−4.35690	−1.64675	−0.823376	−	0.567496i	$-0.807912\pi$
$7$	−4.35690	−1.64675	−0.823376	+	0.567496i	$0.807912\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.44504	−0.773190
$11$	−6.34481	−1.91303	−0.956517	−	0.291677i	$-0.905787\pi$
$11$	−6.34481	−1.91303	−0.956517	+	0.291677i	$0.905787\pi$
$12$	0	0
$13$	−5.15883	−1.43080	−0.715402	−	0.698714i	$-0.753756\pi$
$13$	−5.15883	−1.43080	−0.715402	+	0.698714i	$0.753756\pi$
$14$	4.35690	1.16443
$15$	0	0
$16$	1.00000	0.250000
$17$	6.82908	1.65630	0.828148	−	0.560509i	$-0.189395\pi$
$17$	6.82908	1.65630	0.828148	+	0.560509i	$0.189395\pi$
$18$	0	0
$19$	−5.18598	−1.18975	−0.594873	−	0.803820i	$-0.702798\pi$
$19$	−5.18598	−1.18975	−0.594873	+	0.803820i	$0.702798\pi$
$20$	2.44504	0.546728
$21$	0	0
$22$	6.34481	1.35272
$23$	5.35690	1.11699	0.558495	−	0.829508i	$-0.311379\pi$
$23$	5.35690	1.11699	0.558495	+	0.829508i	$0.311379\pi$
$24$	0	0
$25$	0.978230	0.195646
$26$	5.15883	1.01173
$27$	0	0
$28$	−4.35690	−0.823376
$29$	−1.96077	−0.364106	−0.182053	−	0.983289i	$-0.558274\pi$
$29$	−1.96077	−0.364106	−0.182053	+	0.983289i	$0.558274\pi$
$30$	0	0
$31$	−0.198062	−0.0355730	−0.0177865	−	0.999842i	$-0.505662\pi$
$31$	−0.198062	−0.0355730	−0.0177865	+	0.999842i	$0.505662\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.82908	−1.17118
$35$	−10.6528	−1.80065
$36$	0	0
$37$	3.63102	0.596936	0.298468	−	0.954420i	$-0.403524\pi$
$37$	3.63102	0.596936	0.298468	+	0.954420i	$0.403524\pi$
$38$	5.18598	0.841277
$39$	0	0
$40$	−2.44504	−0.386595
$41$	−10.2838	−1.60606	−0.803031	−	0.595937i	$-0.796781\pi$
$41$	−10.2838	−1.60606	−0.803031	+	0.595937i	$0.796781\pi$
$42$	0	0
$43$	−3.69202	−0.563028	−0.281514	−	0.959557i	$-0.590837\pi$
$43$	−3.69202	−0.563028	−0.281514	+	0.959557i	$0.590837\pi$
$44$	−6.34481	−0.956517
$45$	0	0
$46$	−5.35690	−0.789831
$47$	7.25667	1.05849	0.529247	−	0.848468i	$-0.322475\pi$
$47$	7.25667	1.05849	0.529247	+	0.848468i	$0.322475\pi$
$48$	0	0
$49$	11.9825	1.71179
$50$	−0.978230	−0.138343
$51$	0	0
$52$	−5.15883	−0.715402
$53$	5.62565	0.772742	0.386371	−	0.922343i	$-0.373728\pi$
$53$	5.62565	0.772742	0.386371	+	0.922343i	$0.373728\pi$
$54$	0	0
$55$	−15.5133	−2.09182
$56$	4.35690	0.582215
$57$	0	0
$58$	1.96077	0.257462
$59$	6.92692	0.901808	0.450904	−	0.892572i	$-0.351102\pi$
$59$	6.92692	0.901808	0.450904	+	0.892572i	$0.351102\pi$
$60$	0	0
$61$	3.12498	0.400113	0.200056	−	0.979784i	$-0.435887\pi$
$61$	3.12498	0.400113	0.200056	+	0.979784i	$0.435887\pi$
$62$	0.198062	0.0251539
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.6136	−1.56452
$66$	0	0
$67$	−5.03684	−0.615347	−0.307674	−	0.951492i	$-0.599551\pi$
$67$	−5.03684	−0.615347	−0.307674	+	0.951492i	$0.599551\pi$
$68$	6.82908	0.828148
$69$	0	0
$70$	10.6528	1.27325
$71$	6.16421	0.731557	0.365779	−	0.930702i	$-0.380803\pi$
$71$	6.16421	0.731557	0.365779	+	0.930702i	$0.380803\pi$
$72$	0	0
$73$	8.65279	1.01273	0.506366	−	0.862318i	$-0.330988\pi$
$73$	8.65279	1.01273	0.506366	+	0.862318i	$0.330988\pi$
$74$	−3.63102	−0.422098
$75$	0	0
$76$	−5.18598	−0.594873
$77$	27.6437	3.15029
$78$	0	0
$79$	4.78448	0.538296	0.269148	−	0.963099i	$-0.413258\pi$
$79$	4.78448	0.538296	0.269148	+	0.963099i	$0.413258\pi$
$80$	2.44504	0.273364
$81$	0	0
$82$	10.2838	1.13566
$83$	−0.911854	−0.100089	−0.0500445	−	0.998747i	$-0.515936\pi$
$83$	−0.911854	−0.100089	−0.0500445	+	0.998747i	$0.515936\pi$
$84$	0	0
$85$	16.6974	1.81109
$86$	3.69202	0.398121
$87$	0	0
$88$	6.34481	0.676359
$89$	13.7409	1.45654	0.728268	−	0.685292i	$-0.240325\pi$
$89$	13.7409	1.45654	0.728268	+	0.685292i	$0.240325\pi$
$90$	0	0
$91$	22.4765	2.35618
$92$	5.35690	0.558495
$93$	0	0
$94$	−7.25667	−0.748468
$95$	−12.6799	−1.30093
$96$	0	0
$97$	−9.50365	−0.964949	−0.482475	−	0.875910i	$-0.660262\pi$
$97$	−9.50365	−0.964949	−0.482475	+	0.875910i	$0.660262\pi$
$98$	−11.9825	−1.21042
$99$	0	0
$100$	0.978230	0.0978230
$101$	10.2784	1.02274	0.511371	−	0.859360i	$-0.329138\pi$
$101$	10.2784	1.02274	0.511371	+	0.859360i	$0.329138\pi$
$102$	0	0
$103$	19.6233	1.93354	0.966768	−	0.255654i	$-0.0822909\pi$
$103$	19.6233	1.93354	0.966768	+	0.255654i	$0.0822909\pi$
$104$	5.15883	0.505865
$105$	0	0
$106$	−5.62565	−0.546411
$107$	−2.25236	−0.217744	−0.108872	−	0.994056i	$-0.534724\pi$
$107$	−2.25236	−0.217744	−0.108872	+	0.994056i	$0.534724\pi$
$108$	0	0
$109$	−9.77777	−0.936541	−0.468270	−	0.883585i	$-0.655123\pi$
$109$	−9.77777	−0.936541	−0.468270	+	0.883585i	$0.655123\pi$
$110$	15.5133	1.47914
$111$	0	0
$112$	−4.35690	−0.411688
$113$	−0.807315	−0.0759458	−0.0379729	−	0.999279i	$-0.512090\pi$
$113$	−0.807315	−0.0759458	−0.0379729	+	0.999279i	$0.512090\pi$
$114$	0	0
$115$	13.0978	1.22138
$116$	−1.96077	−0.182053
$117$	0	0
$118$	−6.92692	−0.637675
$119$	−29.7536	−2.72751
$120$	0	0
$121$	29.2567	2.65970
$122$	−3.12498	−0.282923
$123$	0	0
$124$	−0.198062	−0.0177865
$125$	−9.83340	−0.879526
$126$	0	0
$127$	2.40581	0.213481	0.106741	−	0.994287i	$-0.465959\pi$
$127$	2.40581	0.213481	0.106741	+	0.994287i	$0.465959\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	12.6136	1.10628
$131$	−3.79225	−0.331330	−0.165665	−	0.986182i	$-0.552977\pi$
$131$	−3.79225	−0.331330	−0.165665	+	0.986182i	$0.552977\pi$
$132$	0	0
$133$	22.5948	1.95922
$134$	5.03684	0.435116
$135$	0	0
$136$	−6.82908	−0.585589
$137$	−5.74632	−0.490941	−0.245470	−	0.969404i	$-0.578942\pi$
$137$	−5.74632	−0.490941	−0.245470	+	0.969404i	$0.578942\pi$
$138$	0	0
$139$	1.34721	0.114269	0.0571343	−	0.998367i	$-0.481804\pi$
$139$	1.34721	0.114269	0.0571343	+	0.998367i	$0.481804\pi$
$140$	−10.6528	−0.900325
$141$	0	0
$142$	−6.16421	−0.517289
$143$	32.7318	2.73717
$144$	0	0
$145$	−4.79417	−0.398134
$146$	−8.65279	−0.716110
$147$	0	0
$148$	3.63102	0.298468
$149$	13.8998	1.13871	0.569357	−	0.822090i	$-0.307192\pi$
$149$	13.8998	1.13871	0.569357	+	0.822090i	$0.307192\pi$
$150$	0	0
$151$	14.0761	1.14549	0.572747	−	0.819732i	$-0.305878\pi$
$151$	14.0761	1.14549	0.572747	+	0.819732i	$0.305878\pi$
$152$	5.18598	0.420639
$153$	0	0
$154$	−27.6437	−2.22759
$155$	−0.484271	−0.0388975
$156$	0	0
$157$	13.2784	1.05973	0.529867	−	0.848081i	$-0.322242\pi$
$157$	13.2784	1.05973	0.529867	+	0.848081i	$0.322242\pi$
$158$	−4.78448	−0.380633
$159$	0	0
$160$	−2.44504	−0.193298
$161$	−23.3394	−1.83941
$162$	0	0
$163$	−24.7439	−1.93809	−0.969047	−	0.246877i	$-0.920596\pi$
$163$	−24.7439	−1.93809	−0.969047	+	0.246877i	$0.920596\pi$
$164$	−10.2838	−0.803031
$165$	0	0
$166$	0.911854	0.0707736
$167$	−21.4373	−1.65887	−0.829433	−	0.558606i	$-0.811336\pi$
$167$	−21.4373	−1.65887	−0.829433	+	0.558606i	$0.811336\pi$
$168$	0	0
$169$	13.6136	1.04720
$170$	−16.6974	−1.28063
$171$	0	0
$172$	−3.69202	−0.281514
$173$	9.93900	0.755648	0.377824	−	0.925877i	$-0.376672\pi$
$173$	9.93900	0.755648	0.377824	+	0.925877i	$0.376672\pi$
$174$	0	0
$175$	−4.26205	−0.322180
$176$	−6.34481	−0.478258
$177$	0	0
$178$	−13.7409	−1.02993
$179$	−22.3870	−1.67328	−0.836642	−	0.547749i	$-0.815485\pi$
$179$	−22.3870	−1.67328	−0.836642	+	0.547749i	$0.815485\pi$
$180$	0	0
$181$	−10.2252	−0.760034	−0.380017	−	0.924980i	$-0.624082\pi$
$181$	−10.2252	−0.760034	−0.380017	+	0.924980i	$0.624082\pi$
$182$	−22.4765	−1.66607
$183$	0	0
$184$	−5.35690	−0.394916
$185$	8.87800	0.652724
$186$	0	0
$187$	−43.3293	−3.16855
$188$	7.25667	0.529247
$189$	0	0
$190$	12.6799	0.919900
$191$	24.4373	1.76822	0.884109	−	0.467280i	$-0.154766\pi$
$191$	24.4373	1.76822	0.884109	+	0.467280i	$0.154766\pi$
$192$	0	0
$193$	7.97152	0.573803	0.286901	−	0.957960i	$-0.407375\pi$
$193$	7.97152	0.573803	0.286901	+	0.957960i	$0.407375\pi$
$194$	9.50365	0.682322
$195$	0	0
$196$	11.9825	0.855896
$197$	−13.1468	−0.936667	−0.468334	−	0.883552i	$-0.655145\pi$
$197$	−13.1468	−0.936667	−0.468334	+	0.883552i	$0.655145\pi$
$198$	0	0
$199$	−18.1371	−1.28570	−0.642851	−	0.765991i	$-0.722249\pi$
$199$	−18.1371	−1.28570	−0.642851	+	0.765991i	$0.722249\pi$
$200$	−0.978230	−0.0691713
$201$	0	0
$202$	−10.2784	−0.723188
$203$	8.54288	0.599592
$204$	0	0
$205$	−25.1444	−1.75616
$206$	−19.6233	−1.36722
$207$	0	0
$208$	−5.15883	−0.357701
$209$	32.9041	2.27602
$210$	0	0
$211$	19.1521	1.31849	0.659243	−	0.751930i	$-0.270877\pi$
$211$	19.1521	1.31849	0.659243	+	0.751930i	$0.270877\pi$
$212$	5.62565	0.386371
$213$	0	0
$214$	2.25236	0.153968
$215$	−9.02715	−0.615646
$216$	0	0
$217$	0.862937	0.0585800
$218$	9.77777	0.662234
$219$	0	0
$220$	−15.5133	−1.04591
$221$	−35.2301	−2.36983
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	4.35690	0.291107
$225$	0	0
$226$	0.807315	0.0537018
$227$	−1.31336	−0.0871705	−0.0435852	−	0.999050i	$-0.513878\pi$
$227$	−1.31336	−0.0871705	−0.0435852	+	0.999050i	$0.513878\pi$
$228$	0	0
$229$	6.63773	0.438633	0.219317	−	0.975654i	$-0.429617\pi$
$229$	6.63773	0.438633	0.219317	+	0.975654i	$0.429617\pi$
$230$	−13.0978	−0.863646
$231$	0	0
$232$	1.96077	0.128731
$233$	17.9463	1.17570	0.587850	−	0.808970i	$-0.299974\pi$
$233$	17.9463	1.17570	0.587850	+	0.808970i	$0.299974\pi$
$234$	0	0
$235$	17.7429	1.15742
$236$	6.92692	0.450904
$237$	0	0
$238$	29.7536	1.92864
$239$	−1.57002	−0.101556	−0.0507782	−	0.998710i	$-0.516170\pi$
$239$	−1.57002	−0.101556	−0.0507782	+	0.998710i	$0.516170\pi$
$240$	0	0
$241$	22.9409	1.47775	0.738877	−	0.673840i	$-0.235356\pi$
$241$	22.9409	1.47775	0.738877	+	0.673840i	$0.235356\pi$
$242$	−29.2567	−1.88069
$243$	0	0
$244$	3.12498	0.200056
$245$	29.2978	1.87177
$246$	0	0
$247$	26.7536	1.70229
$248$	0.198062	0.0125770
$249$	0	0
$250$	9.83340	0.621919
$251$	−22.1540	−1.39835	−0.699176	−	0.714950i	$-0.746449\pi$
$251$	−22.1540	−1.39835	−0.699176	+	0.714950i	$0.746449\pi$
$252$	0	0
$253$	−33.9885	−2.13684
$254$	−2.40581	−0.150954
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.3230	1.76674	0.883371	−	0.468673i	$-0.155268\pi$
$257$	28.3230	1.76674	0.883371	+	0.468673i	$0.155268\pi$
$258$	0	0
$259$	−15.8200	−0.983006
$260$	−12.6136	−0.782260
$261$	0	0
$262$	3.79225	0.234286
$263$	−25.5840	−1.57758	−0.788789	−	0.614664i	$-0.789292\pi$
$263$	−25.5840	−1.57758	−0.788789	+	0.614664i	$0.789292\pi$
$264$	0	0
$265$	13.7549	0.844959
$266$	−22.5948	−1.38537
$267$	0	0
$268$	−5.03684	−0.307674
$269$	11.0218	0.672009	0.336005	−	0.941860i	$-0.390924\pi$
$269$	11.0218	0.672009	0.336005	+	0.941860i	$0.390924\pi$
$270$	0	0
$271$	7.03146	0.427131	0.213565	−	0.976929i	$-0.431492\pi$
$271$	7.03146	0.427131	0.213565	+	0.976929i	$0.431492\pi$
$272$	6.82908	0.414074
$273$	0	0
$274$	5.74632	0.347148
$275$	−6.20669	−0.374277
$276$	0	0
$277$	−3.57242	−0.214646	−0.107323	−	0.994224i	$-0.534228\pi$
$277$	−3.57242	−0.214646	−0.107323	+	0.994224i	$0.534228\pi$
$278$	−1.34721	−0.0808001
$279$	0	0
$280$	10.6528	0.636626
$281$	9.50902	0.567261	0.283630	−	0.958934i	$-0.408461\pi$
$281$	9.50902	0.567261	0.283630	+	0.958934i	$0.408461\pi$
$282$	0	0
$283$	−11.2078	−0.666232	−0.333116	−	0.942886i	$-0.608100\pi$
$283$	−11.2078	−0.666232	−0.333116	+	0.942886i	$0.608100\pi$
$284$	6.16421	0.365779
$285$	0	0
$286$	−32.7318	−1.93547
$287$	44.8055	2.64479
$288$	0	0
$289$	29.6364	1.74332
$290$	4.79417	0.281523
$291$	0	0
$292$	8.65279	0.506366
$293$	25.1782	1.47093	0.735463	−	0.677564i	$-0.236965\pi$
$293$	25.1782	1.47093	0.735463	+	0.677564i	$0.236965\pi$
$294$	0	0
$295$	16.9366	0.986087
$296$	−3.63102	−0.211049
$297$	0	0
$298$	−13.8998	−0.805192
$299$	−27.6353	−1.59819
$300$	0	0
$301$	16.0858	0.927167
$302$	−14.0761	−0.809986
$303$	0	0
$304$	−5.18598	−0.297436
$305$	7.64071	0.437506
$306$	0	0
$307$	−6.37329	−0.363743	−0.181871	−	0.983322i	$-0.558215\pi$
$307$	−6.37329	−0.363743	−0.181871	+	0.983322i	$0.558215\pi$
$308$	27.6437	1.57515
$309$	0	0
$310$	0.484271	0.0275047
$311$	2.26875	0.128649	0.0643245	−	0.997929i	$-0.479511\pi$
$311$	2.26875	0.128649	0.0643245	+	0.997929i	$0.479511\pi$
$312$	0	0
$313$	−20.4843	−1.15784	−0.578920	−	0.815385i	$-0.696526\pi$
$313$	−20.4843	−1.15784	−0.578920	+	0.815385i	$0.696526\pi$
$314$	−13.2784	−0.749346
$315$	0	0
$316$	4.78448	0.269148
$317$	−23.0780	−1.29619	−0.648094	−	0.761560i	$-0.724434\pi$
$317$	−23.0780	−1.29619	−0.648094	+	0.761560i	$0.724434\pi$
$318$	0	0
$319$	12.4407	0.696547
$320$	2.44504	0.136682
$321$	0	0
$322$	23.3394	1.30066
$323$	−35.4155	−1.97057
$324$	0	0
$325$	−5.04652	−0.279931
$326$	24.7439	1.37044
$327$	0	0
$328$	10.2838	0.567829
$329$	−31.6165	−1.74308
$330$	0	0
$331$	0.953476	0.0524078	0.0262039	−	0.999657i	$-0.491658\pi$
$331$	0.953476	0.0524078	0.0262039	+	0.999657i	$0.491658\pi$
$332$	−0.911854	−0.0500445
$333$	0	0
$334$	21.4373	1.17300
$335$	−12.3153	−0.672855
$336$	0	0
$337$	−5.23251	−0.285033	−0.142516	−	0.989792i	$-0.545519\pi$
$337$	−5.23251	−0.285033	−0.142516	+	0.989792i	$0.545519\pi$
$338$	−13.6136	−0.740480
$339$	0	0
$340$	16.6974	0.905544
$341$	1.25667	0.0680524
$342$	0	0
$343$	−21.7084	−1.17214
$344$	3.69202	0.199060
$345$	0	0
$346$	−9.93900	−0.534324
$347$	26.4983	1.42250	0.711251	−	0.702938i	$-0.248129\pi$
$347$	26.4983	1.42250	0.711251	+	0.702938i	$0.248129\pi$
$348$	0	0
$349$	−12.6853	−0.679029	−0.339515	−	0.940601i	$-0.610263\pi$
$349$	−12.6853	−0.679029	−0.339515	+	0.940601i	$0.610263\pi$
$350$	4.26205	0.227816
$351$	0	0
$352$	6.34481	0.338180
$353$	11.4125	0.607427	0.303714	−	0.952763i	$-0.401773\pi$
$353$	11.4125	0.607427	0.303714	+	0.952763i	$0.401773\pi$
$354$	0	0
$355$	15.0718	0.799926
$356$	13.7409	0.728268
$357$	0	0
$358$	22.3870	1.18319
$359$	31.7754	1.67704	0.838520	−	0.544870i	$-0.183421\pi$
$359$	31.7754	1.67704	0.838520	+	0.544870i	$0.183421\pi$
$360$	0	0
$361$	7.89440	0.415495
$362$	10.2252	0.537425
$363$	0	0
$364$	22.4765	1.17809
$365$	21.1564	1.10738
$366$	0	0
$367$	−36.0901	−1.88389	−0.941943	−	0.335773i	$-0.891003\pi$
$367$	−36.0901	−1.88389	−0.941943	+	0.335773i	$0.891003\pi$
$368$	5.35690	0.279248
$369$	0	0
$370$	−8.87800	−0.461545
$371$	−24.5104	−1.27251
$372$	0	0
$373$	−26.3575	−1.36474	−0.682369	−	0.731007i	$-0.739050\pi$
$373$	−26.3575	−1.36474	−0.682369	+	0.731007i	$0.739050\pi$
$374$	43.3293	2.24050
$375$	0	0
$376$	−7.25667	−0.374234
$377$	10.1153	0.520964
$378$	0	0
$379$	6.29888	0.323552	0.161776	−	0.986828i	$-0.448278\pi$
$379$	6.29888	0.323552	0.161776	+	0.986828i	$0.448278\pi$
$380$	−12.6799	−0.650467
$381$	0	0
$382$	−24.4373	−1.25032
$383$	26.0411	1.33064	0.665320	−	0.746558i	$-0.268295\pi$
$383$	26.0411	1.33064	0.665320	+	0.746558i	$0.268295\pi$
$384$	0	0
$385$	67.5900	3.44470
$386$	−7.97152	−0.405740
$387$	0	0
$388$	−9.50365	−0.482475
$389$	28.5211	1.44608	0.723039	−	0.690807i	$-0.242745\pi$
$389$	28.5211	1.44608	0.723039	+	0.690807i	$0.242745\pi$
$390$	0	0
$391$	36.5827	1.85007
$392$	−11.9825	−0.605210
$393$	0	0
$394$	13.1468	0.662324
$395$	11.6983	0.588603
$396$	0	0
$397$	−6.67994	−0.335257	−0.167628	−	0.985850i	$-0.553611\pi$
$397$	−6.67994	−0.335257	−0.167628	+	0.985850i	$0.553611\pi$
$398$	18.1371	0.909129
$399$	0	0
$400$	0.978230	0.0489115
$401$	17.7313	0.885456	0.442728	−	0.896656i	$-0.354011\pi$
$401$	17.7313	0.885456	0.442728	+	0.896656i	$0.354011\pi$
$402$	0	0
$403$	1.02177	0.0508980
$404$	10.2784	0.511371
$405$	0	0
$406$	−8.54288	−0.423976
$407$	−23.0382	−1.14196
$408$	0	0
$409$	−21.8907	−1.08242	−0.541212	−	0.840886i	$-0.682034\pi$
$409$	−21.8907	−1.08242	−0.541212	+	0.840886i	$0.682034\pi$
$410$	25.1444	1.24179
$411$	0	0
$412$	19.6233	0.966768
$413$	−30.1799	−1.48505
$414$	0	0
$415$	−2.22952	−0.109443
$416$	5.15883	0.252933
$417$	0	0
$418$	−32.9041	−1.60939
$419$	−9.37867	−0.458178	−0.229089	−	0.973406i	$-0.573575\pi$
$419$	−9.37867	−0.458178	−0.229089	+	0.973406i	$0.573575\pi$
$420$	0	0
$421$	−26.3672	−1.28506	−0.642529	−	0.766262i	$-0.722115\pi$
$421$	−26.3672	−1.28506	−0.642529	+	0.766262i	$0.722115\pi$
$422$	−19.1521	−0.932311
$423$	0	0
$424$	−5.62565	−0.273206
$425$	6.68041	0.324048
$426$	0	0
$427$	−13.6152	−0.658887
$428$	−2.25236	−0.108872
$429$	0	0
$430$	9.02715	0.435328
$431$	6.70709	0.323069	0.161535	−	0.986867i	$-0.448356\pi$
$431$	6.70709	0.323069	0.161535	+	0.986867i	$0.448356\pi$
$432$	0	0
$433$	9.57540	0.460164	0.230082	−	0.973171i	$-0.426100\pi$
$433$	9.57540	0.460164	0.230082	+	0.973171i	$0.426100\pi$
$434$	−0.862937	−0.0414223
$435$	0	0
$436$	−9.77777	−0.468270
$437$	−27.7808	−1.32893
$438$	0	0
$439$	−35.4426	−1.69159	−0.845793	−	0.533512i	$-0.820872\pi$
$439$	−35.4426	−1.69159	−0.845793	+	0.533512i	$0.820872\pi$
$440$	15.5133	0.739569
$441$	0	0
$442$	35.2301	1.67573
$443$	−22.8485	−1.08556	−0.542782	−	0.839874i	$-0.682629\pi$
$443$	−22.8485	−1.08556	−0.542782	+	0.839874i	$0.682629\pi$
$444$	0	0
$445$	33.5972	1.59266
$446$	1.00000	0.0473514
$447$	0	0
$448$	−4.35690	−0.205844
$449$	−7.67887	−0.362389	−0.181194	−	0.983447i	$-0.557996\pi$
$449$	−7.67887	−0.362389	−0.181194	+	0.983447i	$0.557996\pi$
$450$	0	0
$451$	65.2489	3.07245
$452$	−0.807315	−0.0379729
$453$	0	0
$454$	1.31336	0.0616388
$455$	54.9560	2.57638
$456$	0	0
$457$	8.03146	0.375696	0.187848	−	0.982198i	$-0.439849\pi$
$457$	8.03146	0.375696	0.187848	+	0.982198i	$0.439849\pi$
$458$	−6.63773	−0.310161
$459$	0	0
$460$	13.0978	0.610690
$461$	15.3666	0.715693	0.357847	−	0.933780i	$-0.383511\pi$
$461$	15.3666	0.715693	0.357847	+	0.933780i	$0.383511\pi$
$462$	0	0
$463$	14.7627	0.686081	0.343041	−	0.939321i	$-0.388543\pi$
$463$	14.7627	0.686081	0.343041	+	0.939321i	$0.388543\pi$
$464$	−1.96077	−0.0910265
$465$	0	0
$466$	−17.9463	−0.831346
$467$	36.8471	1.70508	0.852541	−	0.522660i	$-0.175060\pi$
$467$	36.8471	1.70508	0.852541	+	0.522660i	$0.175060\pi$
$468$	0	0
$469$	21.9450	1.01332
$470$	−17.7429	−0.818417
$471$	0	0
$472$	−6.92692	−0.318837
$473$	23.4252	1.07709
$474$	0	0
$475$	−5.07308	−0.232769
$476$	−29.7536	−1.36375
$477$	0	0
$478$	1.57002	0.0718112
$479$	−12.8334	−0.586373	−0.293186	−	0.956055i	$-0.594716\pi$
$479$	−12.8334	−0.586373	−0.293186	+	0.956055i	$0.594716\pi$
$480$	0	0
$481$	−18.7318	−0.854098
$482$	−22.9409	−1.04493
$483$	0	0
$484$	29.2567	1.32985
$485$	−23.2368	−1.05513
$486$	0	0
$487$	22.7265	1.02983	0.514917	−	0.857240i	$-0.327823\pi$
$487$	22.7265	1.02983	0.514917	+	0.857240i	$0.327823\pi$
$488$	−3.12498	−0.141461
$489$	0	0
$490$	−29.2978	−1.32354
$491$	9.43967	0.426006	0.213003	−	0.977052i	$-0.431676\pi$
$491$	9.43967	0.426006	0.213003	+	0.977052i	$0.431676\pi$
$492$	0	0
$493$	−13.3903	−0.603068
$494$	−26.7536	−1.20370
$495$	0	0
$496$	−0.198062	−0.00889326
$497$	−26.8568	−1.20469
$498$	0	0
$499$	−9.79715	−0.438581	−0.219290	−	0.975660i	$-0.570374\pi$
$499$	−9.79715	−0.438581	−0.219290	+	0.975660i	$0.570374\pi$
$500$	−9.83340	−0.439763
$501$	0	0
$502$	22.1540	0.988784
$503$	−20.9028	−0.932008	−0.466004	−	0.884783i	$-0.654307\pi$
$503$	−20.9028	−0.932008	−0.466004	+	0.884783i	$0.654307\pi$
$504$	0	0
$505$	25.1312	1.11832
$506$	33.9885	1.51097
$507$	0	0
$508$	2.40581	0.106741
$509$	25.0121	1.10864	0.554321	−	0.832303i	$-0.312978\pi$
$509$	25.0121	1.10864	0.554321	+	0.832303i	$0.312978\pi$
$510$	0	0
$511$	−37.6993	−1.66772
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−28.3230	−1.24928
$515$	47.9797	2.11424
$516$	0	0
$517$	−46.0422	−2.02493
$518$	15.8200	0.695090
$519$	0	0
$520$	12.6136	0.553141
$521$	36.5144	1.59972	0.799862	−	0.600183i	$-0.204906\pi$
$521$	36.5144	1.59972	0.799862	+	0.600183i	$0.204906\pi$
$522$	0	0
$523$	34.0006	1.48674	0.743371	−	0.668879i	$-0.233226\pi$
$523$	34.0006	1.48674	0.743371	+	0.668879i	$0.233226\pi$
$524$	−3.79225	−0.165665
$525$	0	0
$526$	25.5840	1.11552
$527$	−1.35258	−0.0589195
$528$	0	0
$529$	5.69633	0.247667
$530$	−13.7549	−0.597476
$531$	0	0
$532$	22.5948	0.979608
$533$	53.0525	2.29796
$534$	0	0
$535$	−5.50711	−0.238093
$536$	5.03684	0.217558
$537$	0	0
$538$	−11.0218	−0.475182
$539$	−76.0270	−3.27471
$540$	0	0
$541$	25.4577	1.09451	0.547256	−	0.836965i	$-0.315672\pi$
$541$	25.4577	1.09451	0.547256	+	0.836965i	$0.315672\pi$
$542$	−7.03146	−0.302027
$543$	0	0
$544$	−6.82908	−0.292795
$545$	−23.9071	−1.02407
$546$	0	0
$547$	3.17821	0.135890	0.0679452	−	0.997689i	$-0.478356\pi$
$547$	3.17821	0.135890	0.0679452	+	0.997689i	$0.478356\pi$
$548$	−5.74632	−0.245470
$549$	0	0
$550$	6.20669	0.264654
$551$	10.1685	0.433194
$552$	0	0
$553$	−20.8455	−0.886440
$554$	3.57242	0.151777
$555$	0	0
$556$	1.34721	0.0571343
$557$	−26.5429	−1.12466	−0.562329	−	0.826914i	$-0.690094\pi$
$557$	−26.5429	−1.12466	−0.562329	+	0.826914i	$0.690094\pi$
$558$	0	0
$559$	19.0465	0.805582
$560$	−10.6528	−0.450163
$561$	0	0
$562$	−9.50902	−0.401114
$563$	−12.3647	−0.521109	−0.260554	−	0.965459i	$-0.583905\pi$
$563$	−12.3647	−0.521109	−0.260554	+	0.965459i	$0.583905\pi$
$564$	0	0
$565$	−1.97392	−0.0830433
$566$	11.2078	0.471097
$567$	0	0
$568$	−6.16421	−0.258645
$569$	36.2282	1.51876	0.759382	−	0.650645i	$-0.225501\pi$
$569$	36.2282	1.51876	0.759382	+	0.650645i	$0.225501\pi$
$570$	0	0
$571$	−10.8280	−0.453139	−0.226569	−	0.973995i	$-0.572751\pi$
$571$	−10.8280	−0.453139	−0.226569	+	0.973995i	$0.572751\pi$
$572$	32.7318	1.36859
$573$	0	0
$574$	−44.8055	−1.87015
$575$	5.24027	0.218535
$576$	0	0
$577$	6.48427	0.269944	0.134972	−	0.990849i	$-0.456906\pi$
$577$	6.48427	0.269944	0.134972	+	0.990849i	$0.456906\pi$
$578$	−29.6364	−1.23271
$579$	0	0
$580$	−4.79417	−0.199067
$581$	3.97285	0.164822
$582$	0	0
$583$	−35.6937	−1.47828
$584$	−8.65279	−0.358055
$585$	0	0
$586$	−25.1782	−1.04010
$587$	27.7651	1.14599	0.572994	−	0.819559i	$-0.305782\pi$
$587$	27.7651	1.14599	0.572994	+	0.819559i	$0.305782\pi$
$588$	0	0
$589$	1.02715	0.0423229
$590$	−16.9366	−0.697269
$591$	0	0
$592$	3.63102	0.149234
$593$	−36.2446	−1.48839	−0.744193	−	0.667964i	$-0.767166\pi$
$593$	−36.2446	−1.48839	−0.744193	+	0.667964i	$0.767166\pi$
$594$	0	0
$595$	−72.7488	−2.98241
$596$	13.8998	0.569357
$597$	0	0
$598$	27.6353	1.13009
$599$	30.8950	1.26233	0.631167	−	0.775647i	$-0.282576\pi$
$599$	30.8950	1.26233	0.631167	+	0.775647i	$0.282576\pi$
$600$	0	0
$601$	−17.5646	−0.716477	−0.358238	−	0.933630i	$-0.616623\pi$
$601$	−17.5646	−0.716477	−0.358238	+	0.933630i	$0.616623\pi$
$602$	−16.0858	−0.655606
$603$	0	0
$604$	14.0761	0.572747
$605$	71.5338	2.90826
$606$	0	0
$607$	16.0164	0.650085	0.325043	−	0.945699i	$-0.394621\pi$
$607$	16.0164	0.650085	0.325043	+	0.945699i	$0.394621\pi$
$608$	5.18598	0.210319
$609$	0	0
$610$	−7.64071	−0.309363
$611$	−37.4359	−1.51450
$612$	0	0
$613$	−7.43834	−0.300432	−0.150216	−	0.988653i	$-0.547997\pi$
$613$	−7.43834	−0.300432	−0.150216	+	0.988653i	$0.547997\pi$
$614$	6.37329	0.257205
$615$	0	0
$616$	−27.6437	−1.11380
$617$	−22.1987	−0.893684	−0.446842	−	0.894613i	$-0.647451\pi$
$617$	−22.1987	−0.893684	−0.446842	+	0.894613i	$0.647451\pi$
$618$	0	0
$619$	27.5157	1.10595	0.552975	−	0.833198i	$-0.313492\pi$
$619$	27.5157	1.10595	0.552975	+	0.833198i	$0.313492\pi$
$620$	−0.484271	−0.0194488
$621$	0	0
$622$	−2.26875	−0.0909686
$623$	−59.8678	−2.39855
$624$	0	0
$625$	−28.9342	−1.15737
$626$	20.4843	0.818716
$627$	0	0
$628$	13.2784	0.529867
$629$	24.7966	0.988704
$630$	0	0
$631$	−0.361208	−0.0143795	−0.00718973	−	0.999974i	$-0.502289\pi$
$631$	−0.361208	−0.0143795	−0.00718973	+	0.999974i	$0.502289\pi$
$632$	−4.78448	−0.190316
$633$	0	0
$634$	23.0780	0.916544
$635$	5.88231	0.233433
$636$	0	0
$637$	−61.8159	−2.44924
$638$	−12.4407	−0.492533
$639$	0	0
$640$	−2.44504	−0.0966488
$641$	30.0877	1.18839	0.594196	−	0.804320i	$-0.297470\pi$
$641$	30.0877	1.18839	0.594196	+	0.804320i	$0.297470\pi$
$642$	0	0
$643$	22.9420	0.904744	0.452372	−	0.891829i	$-0.350578\pi$
$643$	22.9420	0.904744	0.452372	+	0.891829i	$0.350578\pi$
$644$	−23.3394	−0.919703
$645$	0	0
$646$	35.4155	1.39340
$647$	−34.7899	−1.36773	−0.683865	−	0.729608i	$-0.739702\pi$
$647$	−34.7899	−1.36773	−0.683865	+	0.729608i	$0.739702\pi$
$648$	0	0
$649$	−43.9500	−1.72519
$650$	5.04652	0.197941
$651$	0	0
$652$	−24.7439	−0.969047
$653$	−5.04115	−0.197275	−0.0986377	−	0.995123i	$-0.531448\pi$
$653$	−5.04115	−0.197275	−0.0986377	+	0.995123i	$0.531448\pi$
$654$	0	0
$655$	−9.27221	−0.362295
$656$	−10.2838	−0.401516
$657$	0	0
$658$	31.6165	1.23254
$659$	9.41789	0.366869	0.183435	−	0.983032i	$-0.441278\pi$
$659$	9.41789	0.366869	0.183435	+	0.983032i	$0.441278\pi$
$660$	0	0
$661$	3.58509	0.139444	0.0697219	−	0.997566i	$-0.477789\pi$
$661$	3.58509	0.139444	0.0697219	+	0.997566i	$0.477789\pi$
$662$	−0.953476	−0.0370579
$663$	0	0
$664$	0.911854	0.0353868
$665$	55.2452	2.14232
$666$	0	0
$667$	−10.5036	−0.406703
$668$	−21.4373	−0.829433
$669$	0	0
$670$	12.3153	0.475781
$671$	−19.8274	−0.765429
$672$	0	0
$673$	16.4566	0.634357	0.317179	−	0.948366i	$-0.397265\pi$
$673$	16.4566	0.634357	0.317179	+	0.948366i	$0.397265\pi$
$674$	5.23251	0.201549
$675$	0	0
$676$	13.6136	0.523599
$677$	−29.2626	−1.12465	−0.562327	−	0.826915i	$-0.690094\pi$
$677$	−29.2626	−1.12465	−0.562327	+	0.826915i	$0.690094\pi$
$678$	0	0
$679$	41.4064	1.58903
$680$	−16.6974	−0.640316
$681$	0	0
$682$	−1.25667	−0.0481203
$683$	35.2083	1.34721	0.673605	−	0.739092i	$-0.264745\pi$
$683$	35.2083	1.34721	0.673605	+	0.739092i	$0.264745\pi$
$684$	0	0
$685$	−14.0500	−0.536822
$686$	21.7084	0.828831
$687$	0	0
$688$	−3.69202	−0.140757
$689$	−29.0218	−1.10564
$690$	0	0
$691$	−25.6273	−0.974909	−0.487454	−	0.873149i	$-0.662074\pi$
$691$	−25.6273	−0.974909	−0.487454	+	0.873149i	$0.662074\pi$
$692$	9.93900	0.377824
$693$	0	0
$694$	−26.4983	−1.00586
$695$	3.29398	0.124948
$696$	0	0
$697$	−70.2290	−2.66011
$698$	12.6853	0.480146
$699$	0	0
$700$	−4.26205	−0.161090
$701$	−10.4668	−0.395326	−0.197663	−	0.980270i	$-0.563335\pi$
$701$	−10.4668	−0.395326	−0.197663	+	0.980270i	$0.563335\pi$
$702$	0	0
$703$	−18.8304	−0.710202
$704$	−6.34481	−0.239129
$705$	0	0
$706$	−11.4125	−0.429516
$707$	−44.7821	−1.68420
$708$	0	0
$709$	27.3532	1.02727	0.513635	−	0.858009i	$-0.328299\pi$
$709$	27.3532	1.02727	0.513635	+	0.858009i	$0.328299\pi$
$710$	−15.0718	−0.565633
$711$	0	0
$712$	−13.7409	−0.514963
$713$	−1.06100	−0.0397347
$714$	0	0
$715$	80.0307	2.99298
$716$	−22.3870	−0.836642
$717$	0	0
$718$	−31.7754	−1.18585
$719$	29.1250	1.08618	0.543089	−	0.839675i	$-0.317255\pi$
$719$	29.1250	1.08618	0.543089	+	0.839675i	$0.317255\pi$
$720$	0	0
$721$	−85.4965	−3.18405
$722$	−7.89440	−0.293799
$723$	0	0
$724$	−10.2252	−0.380017
$725$	−1.91808	−0.0712359
$726$	0	0
$727$	2.82669	0.104836	0.0524181	−	0.998625i	$-0.483307\pi$
$727$	2.82669	0.104836	0.0524181	+	0.998625i	$0.483307\pi$
$728$	−22.4765	−0.833035
$729$	0	0
$730$	−21.1564	−0.783035
$731$	−25.2131	−0.932541
$732$	0	0
$733$	17.5163	0.646980	0.323490	−	0.946232i	$-0.395144\pi$
$733$	17.5163	0.646980	0.323490	+	0.946232i	$0.395144\pi$
$734$	36.0901	1.33211
$735$	0	0
$736$	−5.35690	−0.197458
$737$	31.9578	1.17718
$738$	0	0
$739$	29.0672	1.06926	0.534628	−	0.845088i	$-0.320452\pi$
$739$	29.0672	1.06926	0.534628	+	0.845088i	$0.320452\pi$
$740$	8.87800	0.326362
$741$	0	0
$742$	24.5104	0.899803
$743$	−22.3351	−0.819396	−0.409698	−	0.912221i	$-0.634366\pi$
$743$	−22.3351	−0.819396	−0.409698	+	0.912221i	$0.634366\pi$
$744$	0	0
$745$	33.9855	1.24513
$746$	26.3575	0.965016
$747$	0	0
$748$	−43.3293	−1.58428
$749$	9.81328	0.358570
$750$	0	0
$751$	−29.1715	−1.06448	−0.532242	−	0.846592i	$-0.678650\pi$
$751$	−29.1715	−1.06448	−0.532242	+	0.846592i	$0.678650\pi$
$752$	7.25667	0.264623
$753$	0	0
$754$	−10.1153	−0.368377
$755$	34.4166	1.25255
$756$	0	0
$757$	2.74525	0.0997778	0.0498889	−	0.998755i	$-0.484113\pi$
$757$	2.74525	0.0997778	0.0498889	+	0.998755i	$0.484113\pi$
$758$	−6.29888	−0.228786
$759$	0	0
$760$	12.6799	0.459950
$761$	−8.47086	−0.307068	−0.153534	−	0.988143i	$-0.549066\pi$
$761$	−8.47086	−0.307068	−0.153534	+	0.988143i	$0.549066\pi$
$762$	0	0
$763$	42.6007	1.54225
$764$	24.4373	0.884109
$765$	0	0
$766$	−26.0411	−0.940905
$767$	−35.7348	−1.29031
$768$	0	0
$769$	42.4922	1.53231	0.766153	−	0.642658i	$-0.222168\pi$
$769$	42.4922	1.53231	0.766153	+	0.642658i	$0.222168\pi$
$770$	−67.5900	−2.43577
$771$	0	0
$772$	7.97152	0.286901
$773$	−32.4547	−1.16732	−0.583658	−	0.812000i	$-0.698379\pi$
$773$	−32.4547	−1.16732	−0.583658	+	0.812000i	$0.698379\pi$
$774$	0	0
$775$	−0.193750	−0.00695972
$776$	9.50365	0.341161
$777$	0	0
$778$	−28.5211	−1.02253
$779$	53.3317	1.91081
$780$	0	0
$781$	−39.1108	−1.39949
$782$	−36.5827	−1.30819
$783$	0	0
$784$	11.9825	0.427948
$785$	32.4663	1.15877
$786$	0	0
$787$	32.4956	1.15834	0.579172	−	0.815206i	$-0.303376\pi$
$787$	32.4956	1.15834	0.579172	+	0.815206i	$0.303376\pi$
$788$	−13.1468	−0.468334
$789$	0	0
$790$	−11.6983	−0.416205
$791$	3.51739	0.125064
$792$	0	0
$793$	−16.1213	−0.572483
$794$	6.67994	0.237062
$795$	0	0
$796$	−18.1371	−0.642851
$797$	3.42566	0.121343	0.0606716	−	0.998158i	$-0.480676\pi$
$797$	3.42566	0.121343	0.0606716	+	0.998158i	$0.480676\pi$
$798$	0	0
$799$	49.5564	1.75318
$800$	−0.978230	−0.0345856
$801$	0	0
$802$	−17.7313	−0.626112
$803$	−54.9004	−1.93739
$804$	0	0
$805$	−57.0659	−2.01131
$806$	−1.02177	−0.0359903
$807$	0	0
$808$	−10.2784	−0.361594
$809$	38.7066	1.36085	0.680426	−	0.732817i	$-0.261795\pi$
$809$	38.7066	1.36085	0.680426	+	0.732817i	$0.261795\pi$
$810$	0	0
$811$	37.8678	1.32972	0.664860	−	0.746968i	$-0.268491\pi$
$811$	37.8678	1.32972	0.664860	+	0.746968i	$0.268491\pi$
$812$	8.54288	0.299796
$813$	0	0
$814$	23.0382	0.807487
$815$	−60.4999	−2.11922
$816$	0	0
$817$	19.1468	0.669860
$818$	21.8907	0.765389
$819$	0	0
$820$	−25.1444	−0.878079
$821$	1.79331	0.0625871	0.0312935	−	0.999510i	$-0.490037\pi$
$821$	1.79331	0.0625871	0.0312935	+	0.999510i	$0.490037\pi$
$822$	0	0
$823$	18.7834	0.654749	0.327374	−	0.944895i	$-0.393836\pi$
$823$	18.7834	0.654749	0.327374	+	0.944895i	$0.393836\pi$
$824$	−19.6233	−0.683608
$825$	0	0
$826$	30.1799	1.05009
$827$	−20.1782	−0.701665	−0.350833	−	0.936438i	$-0.614101\pi$
$827$	−20.1782	−0.701665	−0.350833	+	0.936438i	$0.614101\pi$
$828$	0	0
$829$	−1.82238	−0.0632939	−0.0316469	−	0.999499i	$-0.510075\pi$
$829$	−1.82238	−0.0632939	−0.0316469	+	0.999499i	$0.510075\pi$
$830$	2.22952	0.0773878
$831$	0	0
$832$	−5.15883	−0.178850
$833$	81.8298	2.83523
$834$	0	0
$835$	−52.4150	−1.81390
$836$	32.9041	1.13801
$837$	0	0
$838$	9.37867	0.323981
$839$	14.3002	0.493698	0.246849	−	0.969054i	$-0.420605\pi$
$839$	14.3002	0.493698	0.246849	+	0.969054i	$0.420605\pi$
$840$	0	0
$841$	−25.1554	−0.867427
$842$	26.3672	0.908673
$843$	0	0
$844$	19.1521	0.659243
$845$	33.2857	1.14506
$846$	0	0
$847$	−127.468	−4.37986
$848$	5.62565	0.193185
$849$	0	0
$850$	−6.68041	−0.229136
$851$	19.4510	0.666772
$852$	0	0
$853$	−32.1280	−1.10004	−0.550020	−	0.835151i	$-0.685380\pi$
$853$	−32.1280	−1.10004	−0.550020	+	0.835151i	$0.685380\pi$
$854$	13.6152	0.465903
$855$	0	0
$856$	2.25236	0.0769840
$857$	20.6926	0.706846	0.353423	−	0.935464i	$-0.385018\pi$
$857$	20.6926	0.706846	0.353423	+	0.935464i	$0.385018\pi$
$858$	0	0
$859$	46.7676	1.59569	0.797845	−	0.602862i	$-0.205973\pi$
$859$	46.7676	1.59569	0.797845	+	0.602862i	$0.205973\pi$
$860$	−9.02715	−0.307823
$861$	0	0
$862$	−6.70709	−0.228444
$863$	−23.8683	−0.812487	−0.406243	−	0.913765i	$-0.633161\pi$
$863$	−23.8683	−0.812487	−0.406243	+	0.913765i	$0.633161\pi$
$864$	0	0
$865$	24.3013	0.826268
$866$	−9.57540	−0.325385
$867$	0	0
$868$	0.862937	0.0292900
$869$	−30.3566	−1.02978
$870$	0	0
$871$	25.9842	0.880441
$872$	9.77777	0.331117
$873$	0	0
$874$	27.7808	0.939698
$875$	42.8431	1.44836
$876$	0	0
$877$	−8.63235	−0.291494	−0.145747	−	0.989322i	$-0.546559\pi$
$877$	−8.63235	−0.291494	−0.145747	+	0.989322i	$0.546559\pi$
$878$	35.4426	1.19613
$879$	0	0
$880$	−15.5133	−0.522954
$881$	−15.0785	−0.508006	−0.254003	−	0.967203i	$-0.581747\pi$
$881$	−15.0785	−0.508006	−0.254003	+	0.967203i	$0.581747\pi$
$882$	0	0
$883$	−56.4174	−1.89860	−0.949299	−	0.314376i	$-0.898205\pi$
$883$	−56.4174	−1.89860	−0.949299	+	0.314376i	$0.898205\pi$
$884$	−35.2301	−1.18492
$885$	0	0
$886$	22.8485	0.767609
$887$	0.0422126	0.00141736	0.000708680	−	1.00000i	$-0.499774\pi$
$887$	0.0422126	0.00141736	0.000708680		1.00000i	$0.499774\pi$
$888$	0	0
$889$	−10.4819	−0.351551
$890$	−33.5972	−1.12618
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−37.6329	−1.25934
$894$	0	0
$895$	−54.7372	−1.82966
$896$	4.35690	0.145554
$897$	0	0
$898$	7.67887	0.256247
$899$	0.388355	0.0129524
$900$	0	0
$901$	38.4180	1.27989
$902$	−65.2489	−2.17255
$903$	0	0
$904$	0.807315	0.0268509
$905$	−25.0011	−0.831064
$906$	0	0
$907$	22.0194	0.731141	0.365571	−	0.930784i	$-0.380874\pi$
$907$	22.0194	0.731141	0.365571	+	0.930784i	$0.380874\pi$
$908$	−1.31336	−0.0435852
$909$	0	0
$910$	−54.9560	−1.82177
$911$	18.5114	0.613311	0.306655	−	0.951821i	$-0.400790\pi$
$911$	18.5114	0.613311	0.306655	+	0.951821i	$0.400790\pi$
$912$	0	0
$913$	5.78554	0.191474
$914$	−8.03146	−0.265657
$915$	0	0
$916$	6.63773	0.219317
$917$	16.5224	0.545619
$918$	0	0
$919$	20.4728	0.675335	0.337667	−	0.941266i	$-0.390362\pi$
$919$	20.4728	0.675335	0.337667	+	0.941266i	$0.390362\pi$
$920$	−13.0978	−0.431823
$921$	0	0
$922$	−15.3666	−0.506072
$923$	−31.8001	−1.04671
$924$	0	0
$925$	3.55197	0.116788
$926$	−14.7627	−0.485133
$927$	0	0
$928$	1.96077	0.0643655
$929$	−3.02608	−0.0992825	−0.0496413	−	0.998767i	$-0.515808\pi$
$929$	−3.02608	−0.0992825	−0.0496413	+	0.998767i	$0.515808\pi$
$930$	0	0
$931$	−62.1412	−2.03660
$932$	17.9463	0.587850
$933$	0	0
$934$	−36.8471	−1.20568
$935$	−105.942	−3.46467
$936$	0	0
$937$	−41.3217	−1.34992	−0.674961	−	0.737854i	$-0.735839\pi$
$937$	−41.3217	−1.34992	−0.674961	+	0.737854i	$0.735839\pi$
$938$	−21.9450	−0.716529
$939$	0	0
$940$	17.7429	0.578708
$941$	−44.1564	−1.43946	−0.719729	−	0.694255i	$-0.755734\pi$
$941$	−44.1564	−1.43946	−0.719729	+	0.694255i	$0.755734\pi$
$942$	0	0
$943$	−55.0893	−1.79396
$944$	6.92692	0.225452
$945$	0	0
$946$	−23.4252	−0.761619
$947$	13.6504	0.443578	0.221789	−	0.975095i	$-0.428810\pi$
$947$	13.6504	0.443578	0.221789	+	0.975095i	$0.428810\pi$
$948$	0	0
$949$	−44.6383	−1.44902
$950$	5.07308	0.164592
$951$	0	0
$952$	29.7536	0.964320
$953$	12.3653	0.400550	0.200275	−	0.979740i	$-0.435816\pi$
$953$	12.3653	0.400550	0.200275	+	0.979740i	$0.435816\pi$
$954$	0	0
$955$	59.7502	1.93347
$956$	−1.57002	−0.0507782
$957$	0	0
$958$	12.8334	0.414628
$959$	25.0361	0.808458
$960$	0	0
$961$	−30.9608	−0.998735
$962$	18.7318	0.603939
$963$	0	0
$964$	22.9409	0.738877
$965$	19.4907	0.627428
$966$	0	0
$967$	13.5308	0.435121	0.217561	−	0.976047i	$-0.430190\pi$
$967$	13.5308	0.435121	0.217561	+	0.976047i	$0.430190\pi$
$968$	−29.2567	−0.940345
$969$	0	0
$970$	23.2368	0.746089
$971$	35.8297	1.14983	0.574915	−	0.818213i	$-0.305035\pi$
$971$	35.8297	1.14983	0.574915	+	0.818213i	$0.305035\pi$
$972$	0	0
$973$	−5.86964	−0.188172
$974$	−22.7265	−0.728203
$975$	0	0
$976$	3.12498	0.100028
$977$	−40.9661	−1.31062	−0.655312	−	0.755359i	$-0.727463\pi$
$977$	−40.9661	−1.31062	−0.655312	+	0.755359i	$0.727463\pi$
$978$	0	0
$979$	−87.1837	−2.78640
$980$	29.2978	0.935884
$981$	0	0
$982$	−9.43967	−0.301232
$983$	−10.2150	−0.325809	−0.162905	−	0.986642i	$-0.552086\pi$
$983$	−10.2150	−0.325809	−0.162905	+	0.986642i	$0.552086\pi$
$984$	0	0
$985$	−32.1444	−1.02420
$986$	13.3903	0.426433
$987$	0	0
$988$	26.7536	0.851146
$989$	−19.7778	−0.628897
$990$	0	0
$991$	−51.3051	−1.62976	−0.814880	−	0.579629i	$-0.803197\pi$
$991$	−51.3051	−1.62976	−0.814880	+	0.579629i	$0.803197\pi$
$992$	0.198062	0.00628848
$993$	0	0
$994$	26.8568	0.851847
$995$	−44.3459	−1.40586
$996$	0	0
$997$	42.5743	1.34834	0.674171	−	0.738575i	$-0.264501\pi$
$997$	42.5743	1.34834	0.674171	+	0.738575i	$0.264501\pi$
$998$	9.79715	0.310123
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.n.1.2		3
3.2	odd	2		1338.2.a.f.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.f.1.2	✓	3	3.2	odd	2
4014.2.a.n.1.2		3	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.44504 q^{5} -4.35690 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.44504 q^{5} -4.35690 q^{7} -1.00000 q^{8} -2.44504 q^{10} -6.34481 q^{11} -5.15883 q^{13} +4.35690 q^{14} +1.00000 q^{16} +6.82908 q^{17} -5.18598 q^{19} +2.44504 q^{20} +6.34481 q^{22} +5.35690 q^{23} +0.978230 q^{25} +5.15883 q^{26} -4.35690 q^{28} -1.96077 q^{29} -0.198062 q^{31} -1.00000 q^{32} -6.82908 q^{34} -10.6528 q^{35} +3.63102 q^{37} +5.18598 q^{38} -2.44504 q^{40} -10.2838 q^{41} -3.69202 q^{43} -6.34481 q^{44} -5.35690 q^{46} +7.25667 q^{47} +11.9825 q^{49} -0.978230 q^{50} -5.15883 q^{52} +5.62565 q^{53} -15.5133 q^{55} +4.35690 q^{56} +1.96077 q^{58} +6.92692 q^{59} +3.12498 q^{61} +0.198062 q^{62} +1.00000 q^{64} -12.6136 q^{65} -5.03684 q^{67} +6.82908 q^{68} +10.6528 q^{70} +6.16421 q^{71} +8.65279 q^{73} -3.63102 q^{74} -5.18598 q^{76} +27.6437 q^{77} +4.78448 q^{79} +2.44504 q^{80} +10.2838 q^{82} -0.911854 q^{83} +16.6974 q^{85} +3.69202 q^{86} +6.34481 q^{88} +13.7409 q^{89} +22.4765 q^{91} +5.35690 q^{92} -7.25667 q^{94} -12.6799 q^{95} -9.50365 q^{97} -11.9825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 7 q^{5} - 9 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 7 * q^5 - 9 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 7 q^{5} - 9 q^{7} - 3 q^{8} - 7 q^{10} + 4 q^{11} - 7 q^{13} + 9 q^{14} + 3 q^{16} + 10 q^{17} - q^{19} + 7 q^{20} - 4 q^{22} + 12 q^{23} + 6 q^{25} + 7 q^{26} - 9 q^{28} + 7 q^{29} - 5 q^{31} - 3 q^{32} - 10 q^{34} - 14 q^{35} - 4 q^{37} + q^{38} - 7 q^{40} + 2 q^{41} - 6 q^{43} + 4 q^{44} - 12 q^{46} - 5 q^{47} + 20 q^{49} - 6 q^{50} - 7 q^{52} + 5 q^{53} + 7 q^{55} + 9 q^{56} - 7 q^{58} - 8 q^{59} - 15 q^{61} + 5 q^{62} + 3 q^{64} - 7 q^{65} + 13 q^{67} + 10 q^{68} + 14 q^{70} + 7 q^{71} + 8 q^{73} + 4 q^{74} - q^{76} + 2 q^{77} - 6 q^{79} + 7 q^{80} - 2 q^{82} + q^{83} + 42 q^{85} + 6 q^{86} - 4 q^{88} + 27 q^{89} + 42 q^{91} + 12 q^{92} + 5 q^{94} - 14 q^{95} + 3 q^{97} - 20 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 7 * q^5 - 9 * q^7 - 3 * q^8 - 7 * q^10 + 4 * q^11 - 7 * q^13 + 9 * q^14 + 3 * q^16 + 10 * q^17 - q^19 + 7 * q^20 - 4 * q^22 + 12 * q^23 + 6 * q^25 + 7 * q^26 - 9 * q^28 + 7 * q^29 - 5 * q^31 - 3 * q^32 - 10 * q^34 - 14 * q^35 - 4 * q^37 + q^38 - 7 * q^40 + 2 * q^41 - 6 * q^43 + 4 * q^44 - 12 * q^46 - 5 * q^47 + 20 * q^49 - 6 * q^50 - 7 * q^52 + 5 * q^53 + 7 * q^55 + 9 * q^56 - 7 * q^58 - 8 * q^59 - 15 * q^61 + 5 * q^62 + 3 * q^64 - 7 * q^65 + 13 * q^67 + 10 * q^68 + 14 * q^70 + 7 * q^71 + 8 * q^73 + 4 * q^74 - q^76 + 2 * q^77 - 6 * q^79 + 7 * q^80 - 2 * q^82 + q^83 + 42 * q^85 + 6 * q^86 - 4 * q^88 + 27 * q^89 + 42 * q^91 + 12 * q^92 + 5 * q^94 - 14 * q^95 + 3 * q^97 - 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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