LMFDB - Embedded newform 4014.2.a.n.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4014,2,Mod(1,4014)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [3,-3,0,3,7,0,-9,-3,0,-7,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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			1.1
Root			$-1.24698$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.753020 q^{5} -4.69202 q^{7} -1.00000 q^{8} -0.753020 q^{10} +5.51573 q^{11} -4.13706 q^{13} +4.69202 q^{14} +1.00000 q^{16} -4.34481 q^{17} +5.65279 q^{19} +0.753020 q^{20} -5.51573 q^{22} +5.69202 q^{23} -4.43296 q^{25} +4.13706 q^{26} -4.69202 q^{28} +0.417895 q^{29} -1.55496 q^{31} -1.00000 q^{32} +4.34481 q^{34} -3.53319 q^{35} -8.89977 q^{37} -5.65279 q^{38} -0.753020 q^{40} +9.36658 q^{41} +1.04892 q^{43} +5.51573 q^{44} -5.69202 q^{46} -2.57673 q^{47} +15.0151 q^{49} +4.43296 q^{50} -4.13706 q^{52} +8.32304 q^{53} +4.15346 q^{55} +4.69202 q^{56} -0.417895 q^{58} -13.0586 q^{59} -15.5036 q^{61} +1.55496 q^{62} +1.00000 q^{64} -3.11529 q^{65} +11.5646 q^{67} -4.34481 q^{68} +3.53319 q^{70} -10.0858 q^{71} +1.53319 q^{73} +8.89977 q^{74} +5.65279 q^{76} -25.8799 q^{77} +6.46011 q^{79} +0.753020 q^{80} -9.36658 q^{82} -2.93900 q^{83} -3.27173 q^{85} -1.04892 q^{86} -5.51573 q^{88} +4.59419 q^{89} +19.4112 q^{91} +5.69202 q^{92} +2.57673 q^{94} +4.25667 q^{95} +3.37867 q^{97} -15.0151 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 7 q^{5} - 9 q^{7} - 3 q^{8} - 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}+ \cdots - 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

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
$5$	0.753020	0.336761	0.168380	−	0.985722i	$-0.446146\pi$
$5$	0.753020	0.336761	0.168380	+	0.985722i	$0.446146\pi$
$6$	0	0
$7$	−4.69202	−1.77342	−0.886709	−	0.462329i	$-0.847014\pi$
$7$	−4.69202	−1.77342	−0.886709	+	0.462329i	$0.847014\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.753020	−0.238126
$11$	5.51573	1.66306	0.831528	−	0.555484i	$-0.187467\pi$
$11$	5.51573	1.66306	0.831528	+	0.555484i	$0.187467\pi$
$12$	0	0
$13$	−4.13706	−1.14741	−0.573707	−	0.819060i	$-0.694495\pi$
$13$	−4.13706	−1.14741	−0.573707	+	0.819060i	$0.694495\pi$
$14$	4.69202	1.25400
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.34481	−1.05377	−0.526886	−	0.849936i	$-0.676641\pi$
$17$	−4.34481	−1.05377	−0.526886	+	0.849936i	$0.676641\pi$
$18$	0	0
$19$	5.65279	1.29684	0.648420	−	0.761283i	$-0.275430\pi$
$19$	5.65279	1.29684	0.648420	+	0.761283i	$0.275430\pi$
$20$	0.753020	0.168380
$21$	0	0
$22$	−5.51573	−1.17596
$23$	5.69202	1.18687	0.593434	−	0.804882i	$-0.297772\pi$
$23$	5.69202	1.18687	0.593434	+	0.804882i	$0.297772\pi$
$24$	0	0
$25$	−4.43296	−0.886592
$26$	4.13706	0.811345
$27$	0	0
$28$	−4.69202	−0.886709
$29$	0.417895	0.0776011	0.0388006	−	0.999247i	$-0.487646\pi$
$29$	0.417895	0.0776011	0.0388006	+	0.999247i	$0.487646\pi$
$30$	0	0
$31$	−1.55496	−0.279279	−0.139639	−	0.990202i	$-0.544594\pi$
$31$	−1.55496	−0.279279	−0.139639	+	0.990202i	$0.544594\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.34481	0.745130
$35$	−3.53319	−0.597218
$36$	0	0
$37$	−8.89977	−1.46311	−0.731557	−	0.681781i	$-0.761206\pi$
$37$	−8.89977	−1.46311	−0.731557	+	0.681781i	$0.761206\pi$
$38$	−5.65279	−0.917004
$39$	0	0
$40$	−0.753020	−0.119063
$41$	9.36658	1.46281	0.731407	−	0.681941i	$-0.238864\pi$
$41$	9.36658	1.46281	0.731407	+	0.681941i	$0.238864\pi$
$42$	0	0
$43$	1.04892	0.159958	0.0799792	−	0.996797i	$-0.474515\pi$
$43$	1.04892	0.159958	0.0799792	+	0.996797i	$0.474515\pi$
$44$	5.51573	0.831528
$45$	0	0
$46$	−5.69202	−0.839243
$47$	−2.57673	−0.375854	−0.187927	−	0.982183i	$-0.560177\pi$
$47$	−2.57673	−0.375854	−0.187927	+	0.982183i	$0.560177\pi$
$48$	0	0
$49$	15.0151	2.14501
$50$	4.43296	0.626915
$51$	0	0
$52$	−4.13706	−0.573707
$53$	8.32304	1.14326	0.571629	−	0.820512i	$-0.306312\pi$
$53$	8.32304	1.14326	0.571629	+	0.820512i	$0.306312\pi$
$54$	0	0
$55$	4.15346	0.560052
$56$	4.69202	0.626998
$57$	0	0
$58$	−0.417895	−0.0548723
$59$	−13.0586	−1.70009	−0.850043	−	0.526714i	$-0.823424\pi$
$59$	−13.0586	−1.70009	−0.850043	+	0.526714i	$0.823424\pi$
$60$	0	0
$61$	−15.5036	−1.98504	−0.992519	−	0.122087i	$-0.961041\pi$
$61$	−15.5036	−1.98504	−0.992519	+	0.122087i	$0.961041\pi$
$62$	1.55496	0.197480
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.11529	−0.386405
$66$	0	0
$67$	11.5646	1.41285	0.706423	−	0.707790i	$-0.250308\pi$
$67$	11.5646	1.41285	0.706423	+	0.707790i	$0.250308\pi$
$68$	−4.34481	−0.526886
$69$	0	0
$70$	3.53319	0.422297
$71$	−10.0858	−1.19696	−0.598479	−	0.801138i	$-0.704228\pi$
$71$	−10.0858	−1.19696	−0.598479	+	0.801138i	$0.704228\pi$
$72$	0	0
$73$	1.53319	0.179446	0.0897230	−	0.995967i	$-0.471402\pi$
$73$	1.53319	0.179446	0.0897230	+	0.995967i	$0.471402\pi$
$74$	8.89977	1.03458
$75$	0	0
$76$	5.65279	0.648420
$77$	−25.8799	−2.94929
$78$	0	0
$79$	6.46011	0.726819	0.363409	−	0.931630i	$-0.381613\pi$
$79$	6.46011	0.726819	0.363409	+	0.931630i	$0.381613\pi$
$80$	0.753020	0.0841902
$81$	0	0
$82$	−9.36658	−1.03437
$83$	−2.93900	−0.322597	−0.161299	−	0.986906i	$-0.551568\pi$
$83$	−2.93900	−0.322597	−0.161299	+	0.986906i	$0.551568\pi$
$84$	0	0
$85$	−3.27173	−0.354869
$86$	−1.04892	−0.113108
$87$	0	0
$88$	−5.51573	−0.587979
$89$	4.59419	0.486983	0.243491	−	0.969903i	$-0.421707\pi$
$89$	4.59419	0.486983	0.243491	+	0.969903i	$0.421707\pi$
$90$	0	0
$91$	19.4112	2.03485
$92$	5.69202	0.593434
$93$	0	0
$94$	2.57673	0.265769
$95$	4.25667	0.436725
$96$	0	0
$97$	3.37867	0.343052	0.171526	−	0.985180i	$-0.445130\pi$
$97$	3.37867	0.343052	0.171526	+	0.985180i	$0.445130\pi$
$98$	−15.0151	−1.51675
$99$	0	0
$100$	−4.43296	−0.443296
$101$	5.85623	0.582717	0.291358	−	0.956614i	$-0.405893\pi$
$101$	5.85623	0.582717	0.291358	+	0.956614i	$0.405893\pi$
$102$	0	0
$103$	3.34050	0.329149	0.164575	−	0.986365i	$-0.447375\pi$
$103$	3.34050	0.329149	0.164575	+	0.986365i	$0.447375\pi$
$104$	4.13706	0.405672
$105$	0	0
$106$	−8.32304	−0.808405
$107$	16.0248	1.54917	0.774586	−	0.632469i	$-0.217958\pi$
$107$	16.0248	1.54917	0.774586	+	0.632469i	$0.217958\pi$
$108$	0	0
$109$	15.9705	1.52969	0.764846	−	0.644213i	$-0.222815\pi$
$109$	15.9705	1.52969	0.764846	+	0.644213i	$0.222815\pi$
$110$	−4.15346	−0.396017
$111$	0	0
$112$	−4.69202	−0.443354
$113$	15.7778	1.48425	0.742124	−	0.670263i	$-0.233818\pi$
$113$	15.7778	1.48425	0.742124	+	0.670263i	$0.233818\pi$
$114$	0	0
$115$	4.28621	0.399691
$116$	0.417895	0.0388006
$117$	0	0
$118$	13.0586	1.20214
$119$	20.3860	1.86878
$120$	0	0
$121$	19.4233	1.76575
$122$	15.5036	1.40363
$123$	0	0
$124$	−1.55496	−0.139639
$125$	−7.10321	−0.635331
$126$	0	0
$127$	−1.66487	−0.147734	−0.0738669	−	0.997268i	$-0.523534\pi$
$127$	−1.66487	−0.147734	−0.0738669	+	0.997268i	$0.523534\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	3.11529	0.273229
$131$	−9.21983	−0.805541	−0.402770	−	0.915301i	$-0.631953\pi$
$131$	−9.21983	−0.805541	−0.402770	+	0.915301i	$0.631953\pi$
$132$	0	0
$133$	−26.5230	−2.29984
$134$	−11.5646	−0.999033
$135$	0	0
$136$	4.34481	0.372565
$137$	18.6286	1.59155	0.795776	−	0.605592i	$-0.207063\pi$
$137$	18.6286	1.59155	0.795776	+	0.605592i	$0.207063\pi$
$138$	0	0
$139$	8.46681	0.718146	0.359073	−	0.933310i	$-0.383093\pi$
$139$	8.46681	0.718146	0.359073	+	0.933310i	$0.383093\pi$
$140$	−3.53319	−0.298609
$141$	0	0
$142$	10.0858	0.846378
$143$	−22.8189	−1.90821
$144$	0	0
$145$	0.314683	0.0261330
$146$	−1.53319	−0.126888
$147$	0	0
$148$	−8.89977	−0.731557
$149$	3.73125	0.305676	0.152838	−	0.988251i	$-0.451159\pi$
$149$	3.73125	0.305676	0.152838	+	0.988251i	$0.451159\pi$
$150$	0	0
$151$	−0.146752	−0.0119425	−0.00597125	−	0.999982i	$-0.501901\pi$
$151$	−0.146752	−0.0119425	−0.00597125	+	0.999982i	$0.501901\pi$
$152$	−5.65279	−0.458502
$153$	0	0
$154$	25.8799	2.08546
$155$	−1.17092	−0.0940502
$156$	0	0
$157$	8.85623	0.706804	0.353402	−	0.935472i	$-0.385025\pi$
$157$	8.85623	0.706804	0.353402	+	0.935472i	$0.385025\pi$
$158$	−6.46011	−0.513939
$159$	0	0
$160$	−0.753020	−0.0595315
$161$	−26.7071	−2.10481
$162$	0	0
$163$	18.6112	1.45774	0.728870	−	0.684652i	$-0.240046\pi$
$163$	18.6112	1.45774	0.728870	+	0.684652i	$0.240046\pi$
$164$	9.36658	0.731407
$165$	0	0
$166$	2.93900	0.228111
$167$	−15.9933	−1.23760	−0.618799	−	0.785549i	$-0.712381\pi$
$167$	−15.9933	−1.23760	−0.618799	+	0.785549i	$0.712381\pi$
$168$	0	0
$169$	4.11529	0.316561
$170$	3.27173	0.250931
$171$	0	0
$172$	1.04892	0.0799792
$173$	2.14914	0.163396	0.0816982	−	0.996657i	$-0.473966\pi$
$173$	2.14914	0.163396	0.0816982	+	0.996657i	$0.473966\pi$
$174$	0	0
$175$	20.7995	1.57230
$176$	5.51573	0.415764
$177$	0	0
$178$	−4.59419	−0.344349
$179$	21.3032	1.59228	0.796138	−	0.605116i	$-0.206873\pi$
$179$	21.3032	1.59228	0.796138	+	0.605116i	$0.206873\pi$
$180$	0	0
$181$	−1.76510	−0.131199	−0.0655995	−	0.997846i	$-0.520896\pi$
$181$	−1.76510	−0.131199	−0.0655995	+	0.997846i	$0.520896\pi$
$182$	−19.4112	−1.43885
$183$	0	0
$184$	−5.69202	−0.419621
$185$	−6.70171	−0.492720
$186$	0	0
$187$	−23.9648	−1.75248
$188$	−2.57673	−0.187927
$189$	0	0
$190$	−4.25667	−0.308811
$191$	18.9933	1.37431	0.687153	−	0.726512i	$-0.258860\pi$
$191$	18.9933	1.37431	0.687153	+	0.726512i	$0.258860\pi$
$192$	0	0
$193$	−24.8635	−1.78972	−0.894858	−	0.446351i	$-0.852723\pi$
$193$	−24.8635	−1.78972	−0.894858	+	0.446351i	$0.852723\pi$
$194$	−3.37867	−0.242574
$195$	0	0
$196$	15.0151	1.07250
$197$	0.0706876	0.00503628	0.00251814	−	0.999997i	$-0.499198\pi$
$197$	0.0706876	0.00503628	0.00251814	+	0.999997i	$0.499198\pi$
$198$	0	0
$199$	−11.7041	−0.829682	−0.414841	−	0.909894i	$-0.636163\pi$
$199$	−11.7041	−0.829682	−0.414841	+	0.909894i	$0.636163\pi$
$200$	4.43296	0.313458
$201$	0	0
$202$	−5.85623	−0.412043
$203$	−1.96077	−0.137619
$204$	0	0
$205$	7.05323	0.492619
$206$	−3.34050	−0.232744
$207$	0	0
$208$	−4.13706	−0.286854
$209$	31.1793	2.15672
$210$	0	0
$211$	−9.29350	−0.639791	−0.319896	−	0.947453i	$-0.603648\pi$
$211$	−9.29350	−0.639791	−0.319896	+	0.947453i	$0.603648\pi$
$212$	8.32304	0.571629
$213$	0	0
$214$	−16.0248	−1.09543
$215$	0.789856	0.0538677
$216$	0	0
$217$	7.29590	0.495278
$218$	−15.9705	−1.08166
$219$	0	0
$220$	4.15346	0.280026
$221$	17.9748	1.20911
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	4.69202	0.313499
$225$	0	0
$226$	−15.7778	−1.04952
$227$	9.17390	0.608893	0.304447	−	0.952529i	$-0.401529\pi$
$227$	9.17390	0.608893	0.304447	+	0.952529i	$0.401529\pi$
$228$	0	0
$229$	21.5308	1.42279	0.711397	−	0.702790i	$-0.248063\pi$
$229$	21.5308	1.42279	0.711397	+	0.702790i	$0.248063\pi$
$230$	−4.28621	−0.282624
$231$	0	0
$232$	−0.417895	−0.0274361
$233$	−15.6082	−1.02253	−0.511263	−	0.859424i	$-0.670822\pi$
$233$	−15.6082	−1.02253	−0.511263	+	0.859424i	$0.670822\pi$
$234$	0	0
$235$	−1.94033	−0.126573
$236$	−13.0586	−0.850043
$237$	0	0
$238$	−20.3860	−1.32143
$239$	18.7506	1.21288	0.606439	−	0.795130i	$-0.292598\pi$
$239$	18.7506	1.21288	0.606439	+	0.795130i	$0.292598\pi$
$240$	0	0
$241$	4.61463	0.297254	0.148627	−	0.988893i	$-0.452515\pi$
$241$	4.61463	0.297254	0.148627	+	0.988893i	$0.452515\pi$
$242$	−19.4233	−1.24858
$243$	0	0
$244$	−15.5036	−0.992519
$245$	11.3067	0.722355
$246$	0	0
$247$	−23.3860	−1.48801
$248$	1.55496	0.0987399
$249$	0	0
$250$	7.10321	0.449247
$251$	16.8280	1.06218	0.531088	−	0.847317i	$-0.321784\pi$
$251$	16.8280	1.06218	0.531088	+	0.847317i	$0.321784\pi$
$252$	0	0
$253$	31.3957	1.97383
$254$	1.66487	0.104464
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.0513	0.689362	0.344681	−	0.938720i	$-0.387987\pi$
$257$	11.0513	0.689362	0.344681	+	0.938720i	$0.387987\pi$
$258$	0	0
$259$	41.7579	2.59471
$260$	−3.11529	−0.193202
$261$	0	0
$262$	9.21983	0.569603
$263$	−6.92261	−0.426866	−0.213433	−	0.976958i	$-0.568465\pi$
$263$	−6.92261	−0.426866	−0.213433	+	0.976958i	$0.568465\pi$
$264$	0	0
$265$	6.26742	0.385005
$266$	26.5230	1.62623
$267$	0	0
$268$	11.5646	0.706423
$269$	16.4330	1.00194	0.500968	−	0.865466i	$-0.332978\pi$
$269$	16.4330	1.00194	0.500968	+	0.865466i	$0.332978\pi$
$270$	0	0
$271$	5.65817	0.343709	0.171855	−	0.985122i	$-0.445024\pi$
$271$	5.65817	0.343709	0.171855	+	0.985122i	$0.445024\pi$
$272$	−4.34481	−0.263443
$273$	0	0
$274$	−18.6286	−1.12540
$275$	−24.4510	−1.47445
$276$	0	0
$277$	−2.23191	−0.134103	−0.0670514	−	0.997750i	$-0.521359\pi$
$277$	−2.23191	−0.134103	−0.0670514	+	0.997750i	$0.521359\pi$
$278$	−8.46681	−0.507806
$279$	0	0
$280$	3.53319	0.211148
$281$	−18.6015	−1.10967	−0.554836	−	0.831960i	$-0.687219\pi$
$281$	−18.6015	−1.10967	−0.554836	+	0.831960i	$0.687219\pi$
$282$	0	0
$283$	−5.78017	−0.343595	−0.171798	−	0.985132i	$-0.554958\pi$
$283$	−5.78017	−0.343595	−0.171798	+	0.985132i	$0.554958\pi$
$284$	−10.0858	−0.598479
$285$	0	0
$286$	22.8189	1.34931
$287$	−43.9482	−2.59418
$288$	0	0
$289$	1.87741	0.110436
$290$	−0.314683	−0.0184788
$291$	0	0
$292$	1.53319	0.0897230
$293$	10.5875	0.618527	0.309264	−	0.950976i	$-0.399917\pi$
$293$	10.5875	0.618527	0.309264	+	0.950976i	$0.399917\pi$
$294$	0	0
$295$	−9.83340	−0.572522
$296$	8.89977	0.517289
$297$	0	0
$298$	−3.73125	−0.216146
$299$	−23.5483	−1.36183
$300$	0	0
$301$	−4.92154	−0.283673
$302$	0.146752	0.00844463
$303$	0	0
$304$	5.65279	0.324210
$305$	−11.6746	−0.668484
$306$	0	0
$307$	−27.3478	−1.56082	−0.780411	−	0.625267i	$-0.784990\pi$
$307$	−27.3478	−1.56082	−0.780411	+	0.625267i	$0.784990\pi$
$308$	−25.8799	−1.47465
$309$	0	0
$310$	1.17092	0.0665035
$311$	4.63102	0.262601	0.131301	−	0.991343i	$-0.458085\pi$
$311$	4.63102	0.262601	0.131301	+	0.991343i	$0.458085\pi$
$312$	0	0
$313$	−21.1709	−1.19665	−0.598325	−	0.801253i	$-0.704167\pi$
$313$	−21.1709	−1.19665	−0.598325	+	0.801253i	$0.704167\pi$
$314$	−8.85623	−0.499786
$315$	0	0
$316$	6.46011	0.363409
$317$	1.68127	0.0944294	0.0472147	−	0.998885i	$-0.484966\pi$
$317$	1.68127	0.0944294	0.0472147	+	0.998885i	$0.484966\pi$
$318$	0	0
$319$	2.30499	0.129055
$320$	0.753020	0.0420951
$321$	0	0
$322$	26.7071	1.48833
$323$	−24.5603	−1.36657
$324$	0	0
$325$	18.3394	1.01729
$326$	−18.6112	−1.03078
$327$	0	0
$328$	−9.36658	−0.517183
$329$	12.0901	0.666547
$330$	0	0
$331$	24.3394	1.33782	0.668908	−	0.743345i	$-0.266762\pi$
$331$	24.3394	1.33782	0.668908	+	0.743345i	$0.266762\pi$
$332$	−2.93900	−0.161299
$333$	0	0
$334$	15.9933	0.875114
$335$	8.70841	0.475792
$336$	0	0
$337$	28.9922	1.57931	0.789654	−	0.613553i	$-0.210260\pi$
$337$	28.9922	1.57931	0.789654	+	0.613553i	$0.210260\pi$
$338$	−4.11529	−0.223842
$339$	0	0
$340$	−3.27173	−0.177435
$341$	−8.57673	−0.464456
$342$	0	0
$343$	−37.6069	−2.03058
$344$	−1.04892	−0.0565538
$345$	0	0
$346$	−2.14914	−0.115539
$347$	28.8442	1.54844	0.774218	−	0.632920i	$-0.218144\pi$
$347$	28.8442	1.54844	0.774218	+	0.632920i	$0.218144\pi$
$348$	0	0
$349$	19.4795	1.04271	0.521356	−	0.853339i	$-0.325426\pi$
$349$	19.4795	1.04271	0.521356	+	0.853339i	$0.325426\pi$
$350$	−20.7995	−1.11178
$351$	0	0
$352$	−5.51573	−0.293989
$353$	34.7657	1.85039	0.925196	−	0.379491i	$-0.123901\pi$
$353$	34.7657	1.85039	0.925196	+	0.379491i	$0.123901\pi$
$354$	0	0
$355$	−7.59478	−0.403089
$356$	4.59419	0.243491
$357$	0	0
$358$	−21.3032	−1.12591
$359$	−12.9530	−0.683633	−0.341817	−	0.939767i	$-0.611042\pi$
$359$	−12.9530	−0.683633	−0.341817	+	0.939767i	$0.611042\pi$
$360$	0	0
$361$	12.9541	0.681793
$362$	1.76510	0.0927717
$363$	0	0
$364$	19.4112	1.01742
$365$	1.15452	0.0604304
$366$	0	0
$367$	−23.5265	−1.22807	−0.614036	−	0.789278i	$-0.710455\pi$
$367$	−23.5265	−1.22807	−0.614036	+	0.789278i	$0.710455\pi$
$368$	5.69202	0.296717
$369$	0	0
$370$	6.70171	0.348405
$371$	−39.0519	−2.02747
$372$	0	0
$373$	26.4959	1.37190	0.685952	−	0.727647i	$-0.259386\pi$
$373$	26.4959	1.37190	0.685952	+	0.727647i	$0.259386\pi$
$374$	23.9648	1.23919
$375$	0	0
$376$	2.57673	0.132885
$377$	−1.72886	−0.0890407
$378$	0	0
$379$	−35.3642	−1.81654	−0.908268	−	0.418388i	$-0.862595\pi$
$379$	−35.3642	−1.81654	−0.908268	+	0.418388i	$0.862595\pi$
$380$	4.25667	0.218362
$381$	0	0
$382$	−18.9933	−0.971782
$383$	17.8834	0.913798	0.456899	−	0.889519i	$-0.348960\pi$
$383$	17.8834	0.913798	0.456899	+	0.889519i	$0.348960\pi$
$384$	0	0
$385$	−19.4881	−0.993206
$386$	24.8635	1.26552
$387$	0	0
$388$	3.37867	0.171526
$389$	12.6063	0.639163	0.319582	−	0.947559i	$-0.396458\pi$
$389$	12.6063	0.639163	0.319582	+	0.947559i	$0.396458\pi$
$390$	0	0
$391$	−24.7308	−1.25069
$392$	−15.0151	−0.758375
$393$	0	0
$394$	−0.0706876	−0.00356119
$395$	4.86459	0.244764
$396$	0	0
$397$	10.2567	0.514767	0.257384	−	0.966309i	$-0.417140\pi$
$397$	10.2567	0.514767	0.257384	+	0.966309i	$0.417140\pi$
$398$	11.7041	0.586674
$399$	0	0
$400$	−4.43296	−0.221648
$401$	15.3690	0.767490	0.383745	−	0.923439i	$-0.374634\pi$
$401$	15.3690	0.767490	0.383745	+	0.923439i	$0.374634\pi$
$402$	0	0
$403$	6.43296	0.320449
$404$	5.85623	0.291358
$405$	0	0
$406$	1.96077	0.0973114
$407$	−49.0887	−2.43324
$408$	0	0
$409$	34.6819	1.71491	0.857454	−	0.514561i	$-0.172045\pi$
$409$	34.6819	1.71491	0.857454	+	0.514561i	$0.172045\pi$
$410$	−7.05323	−0.348334
$411$	0	0
$412$	3.34050	0.164575
$413$	61.2713	3.01496
$414$	0	0
$415$	−2.21313	−0.108638
$416$	4.13706	0.202836
$417$	0	0
$418$	−31.1793	−1.52503
$419$	−15.1250	−0.738904	−0.369452	−	0.929250i	$-0.620455\pi$
$419$	−15.1250	−0.738904	−0.369452	+	0.929250i	$0.620455\pi$
$420$	0	0
$421$	33.2707	1.62151	0.810757	−	0.585383i	$-0.199056\pi$
$421$	33.2707	1.62151	0.810757	+	0.585383i	$0.199056\pi$
$422$	9.29350	0.452401
$423$	0	0
$424$	−8.32304	−0.404203
$425$	19.2604	0.934266
$426$	0	0
$427$	72.7434	3.52030
$428$	16.0248	0.774586
$429$	0	0
$430$	−0.789856	−0.0380902
$431$	−20.0465	−0.965607	−0.482804	−	0.875729i	$-0.660381\pi$
$431$	−20.0465	−0.965607	−0.482804	+	0.875729i	$0.660381\pi$
$432$	0	0
$433$	−25.9734	−1.24820	−0.624102	−	0.781343i	$-0.714535\pi$
$433$	−25.9734	−1.24820	−0.624102	+	0.781343i	$0.714535\pi$
$434$	−7.29590	−0.350214
$435$	0	0
$436$	15.9705	0.764846
$437$	32.1758	1.53918
$438$	0	0
$439$	−14.7705	−0.704956	−0.352478	−	0.935820i	$-0.614661\pi$
$439$	−14.7705	−0.704956	−0.352478	+	0.935820i	$0.614661\pi$
$440$	−4.15346	−0.198008
$441$	0	0
$442$	−17.9748	−0.854973
$443$	1.89440	0.0900055	0.0450027	−	0.998987i	$-0.485670\pi$
$443$	1.89440	0.0900055	0.0450027	+	0.998987i	$0.485670\pi$
$444$	0	0
$445$	3.45952	0.163997
$446$	1.00000	0.0473514
$447$	0	0
$448$	−4.69202	−0.221677
$449$	−14.4142	−0.680247	−0.340123	−	0.940381i	$-0.610469\pi$
$449$	−14.4142	−0.680247	−0.340123	+	0.940381i	$0.610469\pi$
$450$	0	0
$451$	51.6635	2.43274
$452$	15.7778	0.742124
$453$	0	0
$454$	−9.17390	−0.430552
$455$	14.6170	0.685257
$456$	0	0
$457$	6.65817	0.311456	0.155728	−	0.987800i	$-0.450228\pi$
$457$	6.65817	0.311456	0.155728	+	0.987800i	$0.450228\pi$
$458$	−21.5308	−1.00607
$459$	0	0
$460$	4.28621	0.199845
$461$	8.91723	0.415317	0.207658	−	0.978201i	$-0.433416\pi$
$461$	8.91723	0.415317	0.207658	+	0.978201i	$0.433416\pi$
$462$	0	0
$463$	11.0271	0.512475	0.256237	−	0.966614i	$-0.417517\pi$
$463$	11.0271	0.512475	0.256237	+	0.966614i	$0.417517\pi$
$464$	0.417895	0.0194003
$465$	0	0
$466$	15.6082	0.723035
$467$	−30.5478	−1.41358	−0.706791	−	0.707422i	$-0.749858\pi$
$467$	−30.5478	−1.41358	−0.706791	+	0.707422i	$0.749858\pi$
$468$	0	0
$469$	−54.2616	−2.50557
$470$	1.94033	0.0895007
$471$	0	0
$472$	13.0586	0.601071
$473$	5.78554	0.266020
$474$	0	0
$475$	−25.0586	−1.14977
$476$	20.3860	0.934389
$477$	0	0
$478$	−18.7506	−0.857634
$479$	−10.1032	−0.461628	−0.230814	−	0.972998i	$-0.574139\pi$
$479$	−10.1032	−0.461628	−0.230814	+	0.972998i	$0.574139\pi$
$480$	0	0
$481$	36.8189	1.67880
$482$	−4.61463	−0.210191
$483$	0	0
$484$	19.4233	0.882876
$485$	2.54420	0.115526
$486$	0	0
$487$	−17.5961	−0.797356	−0.398678	−	0.917091i	$-0.630531\pi$
$487$	−17.5961	−0.797356	−0.398678	+	0.917091i	$0.630531\pi$
$488$	15.5036	0.701817
$489$	0	0
$490$	−11.3067	−0.510782
$491$	22.9758	1.03689	0.518443	−	0.855112i	$-0.326512\pi$
$491$	22.9758	1.03689	0.518443	+	0.855112i	$0.326512\pi$
$492$	0	0
$493$	−1.81568	−0.0817739
$494$	23.3860	1.05218
$495$	0	0
$496$	−1.55496	−0.0698197
$497$	47.3226	2.12271
$498$	0	0
$499$	29.5200	1.32150	0.660749	−	0.750607i	$-0.270239\pi$
$499$	29.5200	1.32150	0.660749	+	0.750607i	$0.270239\pi$
$500$	−7.10321	−0.317665
$501$	0	0
$502$	−16.8280	−0.751071
$503$	23.4741	1.04666	0.523329	−	0.852130i	$-0.324690\pi$
$503$	23.4741	1.04666	0.523329	+	0.852130i	$0.324690\pi$
$504$	0	0
$505$	4.40986	0.196236
$506$	−31.3957	−1.39571
$507$	0	0
$508$	−1.66487	−0.0738669
$509$	37.2078	1.64920	0.824602	−	0.565713i	$-0.191399\pi$
$509$	37.2078	1.64920	0.824602	+	0.565713i	$0.191399\pi$
$510$	0	0
$511$	−7.19375	−0.318233
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−11.0513	−0.487452
$515$	2.51547	0.110845
$516$	0	0
$517$	−14.2125	−0.625067
$518$	−41.7579	−1.83474
$519$	0	0
$520$	3.11529	0.136615
$521$	−6.82430	−0.298978	−0.149489	−	0.988763i	$-0.547763\pi$
$521$	−6.82430	−0.298978	−0.149489	+	0.988763i	$0.547763\pi$
$522$	0	0
$523$	−19.1879	−0.839028	−0.419514	−	0.907749i	$-0.637800\pi$
$523$	−19.1879	−0.839028	−0.419514	+	0.907749i	$0.637800\pi$
$524$	−9.21983	−0.402770
$525$	0	0
$526$	6.92261	0.301840
$527$	6.75600	0.294296
$528$	0	0
$529$	9.39911	0.408657
$530$	−6.26742	−0.272239
$531$	0	0
$532$	−26.5230	−1.14992
$533$	−38.7502	−1.67846
$534$	0	0
$535$	12.0670	0.521700
$536$	−11.5646	−0.499517
$537$	0	0
$538$	−16.4330	−0.708475
$539$	82.8190	3.56727
$540$	0	0
$541$	−17.2271	−0.740652	−0.370326	−	0.928902i	$-0.620754\pi$
$541$	−17.2271	−0.740652	−0.370326	+	0.928902i	$0.620754\pi$
$542$	−5.65817	−0.243039
$543$	0	0
$544$	4.34481	0.186282
$545$	12.0261	0.515141
$546$	0	0
$547$	−11.4125	−0.487964	−0.243982	−	0.969780i	$-0.578454\pi$
$547$	−11.4125	−0.487964	−0.243982	+	0.969780i	$0.578454\pi$
$548$	18.6286	0.795776
$549$	0	0
$550$	24.4510	1.04259
$551$	2.36227	0.100636
$552$	0	0
$553$	−30.3110	−1.28895
$554$	2.23191	0.0948249
$555$	0	0
$556$	8.46681	0.359073
$557$	−16.0392	−0.679604	−0.339802	−	0.940497i	$-0.610360\pi$
$557$	−16.0392	−0.679604	−0.339802	+	0.940497i	$0.610360\pi$
$558$	0	0
$559$	−4.33944	−0.183539
$560$	−3.53319	−0.149304
$561$	0	0
$562$	18.6015	0.784656
$563$	−16.4517	−0.693358	−0.346679	−	0.937984i	$-0.612691\pi$
$563$	−16.4517	−0.693358	−0.346679	+	0.937984i	$0.612691\pi$
$564$	0	0
$565$	11.8810	0.499837
$566$	5.78017	0.242959
$567$	0	0
$568$	10.0858	0.423189
$569$	−6.44026	−0.269990	−0.134995	−	0.990846i	$-0.543102\pi$
$569$	−6.44026	−0.269990	−0.134995	+	0.990846i	$0.543102\pi$
$570$	0	0
$571$	−23.3260	−0.976164	−0.488082	−	0.872798i	$-0.662303\pi$
$571$	−23.3260	−0.976164	−0.488082	+	0.872798i	$0.662303\pi$
$572$	−22.8189	−0.954107
$573$	0	0
$574$	43.9482	1.83436
$575$	−25.2325	−1.05227
$576$	0	0
$577$	7.17092	0.298529	0.149265	−	0.988797i	$-0.452309\pi$
$577$	7.17092	0.298529	0.149265	+	0.988797i	$0.452309\pi$
$578$	−1.87741	−0.0780900
$579$	0	0
$580$	0.314683	0.0130665
$581$	13.7899	0.572100
$582$	0	0
$583$	45.9077	1.90130
$584$	−1.53319	−0.0634438
$585$	0	0
$586$	−10.5875	−0.437365
$587$	43.0097	1.77520	0.887600	−	0.460615i	$-0.152371\pi$
$587$	43.0097	1.77520	0.887600	+	0.460615i	$0.152371\pi$
$588$	0	0
$589$	−8.78986	−0.362180
$590$	9.83340	0.404835
$591$	0	0
$592$	−8.89977	−0.365778
$593$	−14.2155	−0.583761	−0.291881	−	0.956455i	$-0.594281\pi$
$593$	−14.2155	−0.583761	−0.291881	+	0.956455i	$0.594281\pi$
$594$	0	0
$595$	15.3510	0.629331
$596$	3.73125	0.152838
$597$	0	0
$598$	23.5483	0.962960
$599$	−17.2338	−0.704155	−0.352078	−	0.935971i	$-0.614525\pi$
$599$	−17.2338	−0.704155	−0.352078	+	0.935971i	$0.614525\pi$
$600$	0	0
$601$	−12.4722	−0.508751	−0.254376	−	0.967106i	$-0.581870\pi$
$601$	−12.4722	−0.508751	−0.254376	+	0.967106i	$0.581870\pi$
$602$	4.92154	0.200587
$603$	0	0
$604$	−0.146752	−0.00597125
$605$	14.6261	0.594636
$606$	0	0
$607$	36.6558	1.48781	0.743906	−	0.668284i	$-0.232971\pi$
$607$	36.6558	1.48781	0.743906	+	0.668284i	$0.232971\pi$
$608$	−5.65279	−0.229251
$609$	0	0
$610$	11.6746	0.472689
$611$	10.6601	0.431261
$612$	0	0
$613$	21.6775	0.875548	0.437774	−	0.899085i	$-0.355767\pi$
$613$	21.6775	0.875548	0.437774	+	0.899085i	$0.355767\pi$
$614$	27.3478	1.10367
$615$	0	0
$616$	25.8799	1.04273
$617$	29.6329	1.19298	0.596489	−	0.802622i	$-0.296562\pi$
$617$	29.6329	1.19298	0.596489	+	0.802622i	$0.296562\pi$
$618$	0	0
$619$	26.8291	1.07835	0.539176	−	0.842193i	$-0.318736\pi$
$619$	26.8291	1.07835	0.539176	+	0.842193i	$0.318736\pi$
$620$	−1.17092	−0.0470251
$621$	0	0
$622$	−4.63102	−0.185687
$623$	−21.5560	−0.863624
$624$	0	0
$625$	16.8159	0.672638
$626$	21.1709	0.846160
$627$	0	0
$628$	8.85623	0.353402
$629$	38.6679	1.54179
$630$	0	0
$631$	−9.14005	−0.363860	−0.181930	−	0.983312i	$-0.558234\pi$
$631$	−9.14005	−0.363860	−0.181930	+	0.983312i	$0.558234\pi$
$632$	−6.46011	−0.256969
$633$	0	0
$634$	−1.68127	−0.0667717
$635$	−1.25368	−0.0497509
$636$	0	0
$637$	−62.1183	−2.46122
$638$	−2.30499	−0.0912556
$639$	0	0
$640$	−0.753020	−0.0297657
$641$	−1.45606	−0.0575109	−0.0287554	−	0.999586i	$-0.509154\pi$
$641$	−1.45606	−0.0575109	−0.0287554	+	0.999586i	$0.509154\pi$
$642$	0	0
$643$	−19.0562	−0.751504	−0.375752	−	0.926720i	$-0.622615\pi$
$643$	−19.0562	−0.751504	−0.375752	+	0.926720i	$0.622615\pi$
$644$	−26.7071	−1.05241
$645$	0	0
$646$	24.5603	0.966313
$647$	−21.2373	−0.834924	−0.417462	−	0.908694i	$-0.637080\pi$
$647$	−21.2373	−0.834924	−0.417462	+	0.908694i	$0.637080\pi$
$648$	0	0
$649$	−72.0277	−2.82734
$650$	−18.3394	−0.719332
$651$	0	0
$652$	18.6112	0.728870
$653$	3.11662	0.121963	0.0609814	−	0.998139i	$-0.480577\pi$
$653$	3.11662	0.121963	0.0609814	+	0.998139i	$0.480577\pi$
$654$	0	0
$655$	−6.94272	−0.271275
$656$	9.36658	0.365704
$657$	0	0
$658$	−12.0901	−0.471320
$659$	17.5429	0.683373	0.341687	−	0.939814i	$-0.389002\pi$
$659$	17.5429	0.683373	0.341687	+	0.939814i	$0.389002\pi$
$660$	0	0
$661$	−38.7482	−1.50713	−0.753566	−	0.657372i	$-0.771668\pi$
$661$	−38.7482	−1.50713	−0.753566	+	0.657372i	$0.771668\pi$
$662$	−24.3394	−0.945979
$663$	0	0
$664$	2.93900	0.114055
$665$	−19.9724	−0.774496
$666$	0	0
$667$	2.37867	0.0921023
$668$	−15.9933	−0.618799
$669$	0	0
$670$	−8.70841	−0.336435
$671$	−85.5139	−3.30123
$672$	0	0
$673$	−2.55629	−0.0985376	−0.0492688	−	0.998786i	$-0.515689\pi$
$673$	−2.55629	−0.0985376	−0.0492688	+	0.998786i	$0.515689\pi$
$674$	−28.9922	−1.11674
$675$	0	0
$676$	4.11529	0.158281
$677$	48.9874	1.88274	0.941370	−	0.337375i	$-0.109539\pi$
$677$	48.9874	1.88274	0.941370	+	0.337375i	$0.109539\pi$
$678$	0	0
$679$	−15.8528	−0.608374
$680$	3.27173	0.125465
$681$	0	0
$682$	8.57673	0.328420
$683$	−23.4077	−0.895672	−0.447836	−	0.894116i	$-0.647805\pi$
$683$	−23.4077	−0.895672	−0.447836	+	0.894116i	$0.647805\pi$
$684$	0	0
$685$	14.0277	0.535972
$686$	37.6069	1.43584
$687$	0	0
$688$	1.04892	0.0399896
$689$	−34.4330	−1.31179
$690$	0	0
$691$	48.5357	1.84638	0.923192	−	0.384338i	$-0.125570\pi$
$691$	48.5357	1.84638	0.923192	+	0.384338i	$0.125570\pi$
$692$	2.14914	0.0816982
$693$	0	0
$694$	−28.8442	−1.09491
$695$	6.37568	0.241843
$696$	0	0
$697$	−40.6961	−1.54147
$698$	−19.4795	−0.737309
$699$	0	0
$700$	20.7995	0.786149
$701$	−14.1860	−0.535797	−0.267899	−	0.963447i	$-0.586329\pi$
$701$	−14.1860	−0.535797	−0.267899	+	0.963447i	$0.586329\pi$
$702$	0	0
$703$	−50.3086	−1.89742
$704$	5.51573	0.207882
$705$	0	0
$706$	−34.7657	−1.30842
$707$	−27.4776	−1.03340
$708$	0	0
$709$	−33.9439	−1.27479	−0.637395	−	0.770537i	$-0.719988\pi$
$709$	−33.9439	−1.27479	−0.637395	+	0.770537i	$0.719988\pi$
$710$	7.59478	0.285027
$711$	0	0
$712$	−4.59419	−0.172174
$713$	−8.85086	−0.331467
$714$	0	0
$715$	−17.1831	−0.642612
$716$	21.3032	0.796138
$717$	0	0
$718$	12.9530	0.483402
$719$	10.4964	0.391448	0.195724	−	0.980659i	$-0.437294\pi$
$719$	10.4964	0.391448	0.195724	+	0.980659i	$0.437294\pi$
$720$	0	0
$721$	−15.6737	−0.583719
$722$	−12.9541	−0.482100
$723$	0	0
$724$	−1.76510	−0.0655995
$725$	−1.85251	−0.0688005
$726$	0	0
$727$	−27.3274	−1.01352	−0.506758	−	0.862088i	$-0.669156\pi$
$727$	−27.3274	−1.01352	−0.506758	+	0.862088i	$0.669156\pi$
$728$	−19.4112	−0.719427
$729$	0	0
$730$	−1.15452	−0.0427308
$731$	−4.55735	−0.168560
$732$	0	0
$733$	−36.3588	−1.34294	−0.671472	−	0.741030i	$-0.734338\pi$
$733$	−36.3588	−1.34294	−0.671472	+	0.741030i	$0.734338\pi$
$734$	23.5265	0.868378
$735$	0	0
$736$	−5.69202	−0.209811
$737$	63.7875	2.34964
$738$	0	0
$739$	34.7644	1.27883	0.639414	−	0.768863i	$-0.279177\pi$
$739$	34.7644	1.27883	0.639414	+	0.768863i	$0.279177\pi$
$740$	−6.70171	−0.246360
$741$	0	0
$742$	39.0519	1.43364
$743$	−17.2591	−0.633174	−0.316587	−	0.948564i	$-0.602537\pi$
$743$	−17.2591	−0.633174	−0.316587	+	0.948564i	$0.602537\pi$
$744$	0	0
$745$	2.80971	0.102940
$746$	−26.4959	−0.970083
$747$	0	0
$748$	−23.9648	−0.876241
$749$	−75.1885	−2.74733
$750$	0	0
$751$	12.8431	0.468651	0.234325	−	0.972158i	$-0.424712\pi$
$751$	12.8431	0.468651	0.234325	+	0.972158i	$0.424712\pi$
$752$	−2.57673	−0.0939636
$753$	0	0
$754$	1.72886	0.0629613
$755$	−0.110507	−0.00402177
$756$	0	0
$757$	2.04221	0.0742255	0.0371127	−	0.999311i	$-0.488184\pi$
$757$	2.04221	0.0742255	0.0371127	+	0.999311i	$0.488184\pi$
$758$	35.3642	1.28449
$759$	0	0
$760$	−4.25667	−0.154406
$761$	45.6902	1.65627	0.828135	−	0.560529i	$-0.189402\pi$
$761$	45.6902	1.65627	0.828135	+	0.560529i	$0.189402\pi$
$762$	0	0
$763$	−74.9337	−2.71278
$764$	18.9933	0.687153
$765$	0	0
$766$	−17.8834	−0.646153
$767$	54.0243	1.95070
$768$	0	0
$769$	−35.7743	−1.29005	−0.645027	−	0.764159i	$-0.723154\pi$
$769$	−35.7743	−1.29005	−0.645027	+	0.764159i	$0.723154\pi$
$770$	19.4881	0.702303
$771$	0	0
$772$	−24.8635	−0.894858
$773$	−23.9782	−0.862437	−0.431218	−	0.902248i	$-0.641916\pi$
$773$	−23.9782	−0.862437	−0.431218	+	0.902248i	$0.641916\pi$
$774$	0	0
$775$	6.89307	0.247606
$776$	−3.37867	−0.121287
$777$	0	0
$778$	−12.6063	−0.451957
$779$	52.9474	1.89704
$780$	0	0
$781$	−55.6303	−1.99061
$782$	24.7308	0.884371
$783$	0	0
$784$	15.0151	0.536252
$785$	6.66892	0.238024
$786$	0	0
$787$	−50.4626	−1.79880	−0.899399	−	0.437129i	$-0.855995\pi$
$787$	−50.4626	−1.79880	−0.899399	+	0.437129i	$0.855995\pi$
$788$	0.0706876	0.00251814
$789$	0	0
$790$	−4.86459	−0.173074
$791$	−74.0297	−2.63219
$792$	0	0
$793$	64.1396	2.27766
$794$	−10.2567	−0.363996
$795$	0	0
$796$	−11.7041	−0.414841
$797$	15.3026	0.542046	0.271023	−	0.962573i	$-0.412638\pi$
$797$	15.3026	0.542046	0.271023	+	0.962573i	$0.412638\pi$
$798$	0	0
$799$	11.1954	0.396065
$800$	4.43296	0.156729
$801$	0	0
$802$	−15.3690	−0.542697
$803$	8.45665	0.298429
$804$	0	0
$805$	−20.1110	−0.708819
$806$	−6.43296	−0.226591
$807$	0	0
$808$	−5.85623	−0.206022
$809$	−17.5636	−0.617503	−0.308751	−	0.951143i	$-0.599911\pi$
$809$	−17.5636	−0.617503	−0.308751	+	0.951143i	$0.599911\pi$
$810$	0	0
$811$	−0.443977	−0.0155901	−0.00779507	−	0.999970i	$-0.502481\pi$
$811$	−0.443977	−0.0155901	−0.00779507	+	0.999970i	$0.502481\pi$
$812$	−1.96077	−0.0688096
$813$	0	0
$814$	49.0887	1.72056
$815$	14.0146	0.490910
$816$	0	0
$817$	5.92931	0.207440
$818$	−34.6819	−1.21262
$819$	0	0
$820$	7.05323	0.246309
$821$	−16.4510	−0.574144	−0.287072	−	0.957909i	$-0.592682\pi$
$821$	−16.4510	−0.574144	−0.287072	+	0.957909i	$0.592682\pi$
$822$	0	0
$823$	44.1309	1.53831	0.769154	−	0.639063i	$-0.220678\pi$
$823$	44.1309	1.53831	0.769154	+	0.639063i	$0.220678\pi$
$824$	−3.34050	−0.116372
$825$	0	0
$826$	−61.2713	−2.13190
$827$	−5.58748	−0.194296	−0.0971479	−	0.995270i	$-0.530972\pi$
$827$	−5.58748	−0.194296	−0.0971479	+	0.995270i	$0.530972\pi$
$828$	0	0
$829$	36.7754	1.27726	0.638631	−	0.769513i	$-0.279501\pi$
$829$	36.7754	1.27726	0.638631	+	0.769513i	$0.279501\pi$
$830$	2.21313	0.0768188
$831$	0	0
$832$	−4.13706	−0.143427
$833$	−65.2377	−2.26035
$834$	0	0
$835$	−12.0433	−0.416775
$836$	31.1793	1.07836
$837$	0	0
$838$	15.1250	0.522484
$839$	15.2892	0.527842	0.263921	−	0.964544i	$-0.414984\pi$
$839$	15.2892	0.527842	0.263921	+	0.964544i	$0.414984\pi$
$840$	0	0
$841$	−28.8254	−0.993978
$842$	−33.2707	−1.14658
$843$	0	0
$844$	−9.29350	−0.319896
$845$	3.09890	0.106605
$846$	0	0
$847$	−91.1344	−3.13142
$848$	8.32304	0.285815
$849$	0	0
$850$	−19.2604	−0.660626
$851$	−50.6577	−1.73652
$852$	0	0
$853$	20.7090	0.709063	0.354531	−	0.935044i	$-0.384640\pi$
$853$	20.7090	0.709063	0.354531	+	0.935044i	$0.384640\pi$
$854$	−72.7434	−2.48923
$855$	0	0
$856$	−16.0248	−0.547715
$857$	−37.2368	−1.27199	−0.635993	−	0.771695i	$-0.719409\pi$
$857$	−37.2368	−1.27199	−0.635993	+	0.771695i	$0.719409\pi$
$858$	0	0
$859$	−1.71273	−0.0584375	−0.0292187	−	0.999573i	$-0.509302\pi$
$859$	−1.71273	−0.0584375	−0.0292187	+	0.999573i	$0.509302\pi$
$860$	0.789856	0.0269339
$861$	0	0
$862$	20.0465	0.682787
$863$	−15.0731	−0.513094	−0.256547	−	0.966532i	$-0.582585\pi$
$863$	−15.0731	−0.513094	−0.256547	+	0.966532i	$0.582585\pi$
$864$	0	0
$865$	1.61835	0.0550255
$866$	25.9734	0.882614
$867$	0	0
$868$	7.29590	0.247639
$869$	35.6322	1.20874
$870$	0	0
$871$	−47.8437	−1.62112
$872$	−15.9705	−0.540828
$873$	0	0
$874$	−32.1758	−1.08836
$875$	33.3284	1.12671
$876$	0	0
$877$	−38.7536	−1.30862	−0.654308	−	0.756228i	$-0.727040\pi$
$877$	−38.7536	−1.30862	−0.654308	+	0.756228i	$0.727040\pi$
$878$	14.7705	0.498480
$879$	0	0
$880$	4.15346	0.140013
$881$	−19.8358	−0.668285	−0.334142	−	0.942523i	$-0.608447\pi$
$881$	−19.8358	−0.668285	−0.334142	+	0.942523i	$0.608447\pi$
$882$	0	0
$883$	−35.0258	−1.17871	−0.589356	−	0.807873i	$-0.700619\pi$
$883$	−35.0258	−1.17871	−0.589356	+	0.807873i	$0.700619\pi$
$884$	17.9748	0.604557
$885$	0	0
$886$	−1.89440	−0.0636435
$887$	−31.7875	−1.06732	−0.533659	−	0.845700i	$-0.679184\pi$
$887$	−31.7875	−1.06732	−0.533659	+	0.845700i	$0.679184\pi$
$888$	0	0
$889$	7.81163	0.261994
$890$	−3.45952	−0.115963
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−14.5657	−0.487423
$894$	0	0
$895$	16.0417	0.536216
$896$	4.69202	0.156749
$897$	0	0
$898$	14.4142	0.481007
$899$	−0.649809	−0.0216723
$900$	0	0
$901$	−36.1621	−1.20473
$902$	−51.6635	−1.72021
$903$	0	0
$904$	−15.7778	−0.524761
$905$	−1.32916	−0.0441827
$906$	0	0
$907$	8.45042	0.280592	0.140296	−	0.990110i	$-0.455195\pi$
$907$	8.45042	0.280592	0.140296	+	0.990110i	$0.455195\pi$
$908$	9.17390	0.304447
$909$	0	0
$910$	−14.6170	−0.484550
$911$	9.38106	0.310808	0.155404	−	0.987851i	$-0.450332\pi$
$911$	9.38106	0.310808	0.155404	+	0.987851i	$0.450332\pi$
$912$	0	0
$913$	−16.2107	−0.536497
$914$	−6.65817	−0.220233
$915$	0	0
$916$	21.5308	0.711397
$917$	43.2597	1.42856
$918$	0	0
$919$	−44.2247	−1.45884	−0.729420	−	0.684066i	$-0.760210\pi$
$919$	−44.2247	−1.45884	−0.729420	+	0.684066i	$0.760210\pi$
$920$	−4.28621	−0.141312
$921$	0	0
$922$	−8.91723	−0.293673
$923$	41.7254	1.37341
$924$	0	0
$925$	39.4523	1.29718
$926$	−11.0271	−0.362375
$927$	0	0
$928$	−0.417895	−0.0137181
$929$	−16.8810	−0.553847	−0.276924	−	0.960892i	$-0.589315\pi$
$929$	−16.8810	−0.553847	−0.276924	+	0.960892i	$0.589315\pi$
$930$	0	0
$931$	84.8771	2.78173
$932$	−15.6082	−0.511263
$933$	0	0
$934$	30.5478	0.999554
$935$	−18.0460	−0.590167
$936$	0	0
$937$	18.6021	0.607703	0.303852	−	0.952719i	$-0.401727\pi$
$937$	18.6021	0.607703	0.303852	+	0.952719i	$0.401727\pi$
$938$	54.2616	1.77170
$939$	0	0
$940$	−1.94033	−0.0632866
$941$	−24.1545	−0.787415	−0.393707	−	0.919236i	$-0.628808\pi$
$941$	−24.1545	−0.787415	−0.393707	+	0.919236i	$0.628808\pi$
$942$	0	0
$943$	53.3148	1.73617
$944$	−13.0586	−0.425021
$945$	0	0
$946$	−5.78554	−0.188104
$947$	−12.4494	−0.404550	−0.202275	−	0.979329i	$-0.564833\pi$
$947$	−12.4494	−0.404550	−0.202275	+	0.979329i	$0.564833\pi$
$948$	0	0
$949$	−6.34290	−0.205899
$950$	25.0586	0.813009
$951$	0	0
$952$	−20.3860	−0.660713
$953$	−36.7362	−1.19000	−0.595000	−	0.803725i	$-0.702848\pi$
$953$	−36.7362	−1.19000	−0.595000	+	0.803725i	$0.702848\pi$
$954$	0	0
$955$	14.3023	0.462813
$956$	18.7506	0.606439
$957$	0	0
$958$	10.1032	0.326420
$959$	−87.4059	−2.82248
$960$	0	0
$961$	−28.5821	−0.922003
$962$	−36.8189	−1.18709
$963$	0	0
$964$	4.61463	0.148627
$965$	−18.7227	−0.602706
$966$	0	0
$967$	−9.16852	−0.294840	−0.147420	−	0.989074i	$-0.547097\pi$
$967$	−9.16852	−0.294840	−0.147420	+	0.989074i	$0.547097\pi$
$968$	−19.4233	−0.624288
$969$	0	0
$970$	−2.54420	−0.0816895
$971$	−28.5327	−0.915658	−0.457829	−	0.889040i	$-0.651373\pi$
$971$	−28.5327	−0.915658	−0.457829	+	0.889040i	$0.651373\pi$
$972$	0	0
$973$	−39.7265	−1.27357
$974$	17.5961	0.563816
$975$	0	0
$976$	−15.5036	−0.496260
$977$	−23.3593	−0.747330	−0.373665	−	0.927564i	$-0.621899\pi$
$977$	−23.3593	−0.747330	−0.373665	+	0.927564i	$0.621899\pi$
$978$	0	0
$979$	25.3403	0.809879
$980$	11.3067	0.361178
$981$	0	0
$982$	−22.9758	−0.733188
$983$	20.9772	0.669068	0.334534	−	0.942384i	$-0.391421\pi$
$983$	20.9772	0.669068	0.334534	+	0.942384i	$0.391421\pi$
$984$	0	0
$985$	0.0532292	0.00169602
$986$	1.81568	0.0578229
$987$	0	0
$988$	−23.3860	−0.744007
$989$	5.97046	0.189850
$990$	0	0
$991$	−7.54932	−0.239812	−0.119906	−	0.992785i	$-0.538259\pi$
$991$	−7.54932	−0.239812	−0.119906	+	0.992785i	$0.538259\pi$
$992$	1.55496	0.0493700
$993$	0	0
$994$	−47.3226	−1.50098
$995$	−8.81343	−0.279404
$996$	0	0
$997$	30.6974	0.972196	0.486098	−	0.873904i	$-0.338420\pi$
$997$	30.6974	0.972196	0.486098	+	0.873904i	$0.338420\pi$
$998$	−29.5200	−0.934441
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.n.1.1		3
3.2	odd	2		1338.2.a.f.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.f.1.3	✓	3	3.2	odd	2
4014.2.a.n.1.1		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.753020 q^{5} -4.69202 q^{7} -1.00000 q^{8} -0.753020 q^{10} +5.51573 q^{11} -4.13706 q^{13} +4.69202 q^{14} +1.00000 q^{16} -4.34481 q^{17} +5.65279 q^{19} +0.753020 q^{20} -5.51573 q^{22} +5.69202 q^{23} -4.43296 q^{25} +4.13706 q^{26} -4.69202 q^{28} +0.417895 q^{29} -1.55496 q^{31} -1.00000 q^{32} +4.34481 q^{34} -3.53319 q^{35} -8.89977 q^{37} -5.65279 q^{38} -0.753020 q^{40} +9.36658 q^{41} +1.04892 q^{43} +5.51573 q^{44} -5.69202 q^{46} -2.57673 q^{47} +15.0151 q^{49} +4.43296 q^{50} -4.13706 q^{52} +8.32304 q^{53} +4.15346 q^{55} +4.69202 q^{56} -0.417895 q^{58} -13.0586 q^{59} -15.5036 q^{61} +1.55496 q^{62} +1.00000 q^{64} -3.11529 q^{65} +11.5646 q^{67} -4.34481 q^{68} +3.53319 q^{70} -10.0858 q^{71} +1.53319 q^{73} +8.89977 q^{74} +5.65279 q^{76} -25.8799 q^{77} +6.46011 q^{79} +0.753020 q^{80} -9.36658 q^{82} -2.93900 q^{83} -3.27173 q^{85} -1.04892 q^{86} -5.51573 q^{88} +4.59419 q^{89} +19.4112 q^{91} +5.69202 q^{92} +2.57673 q^{94} +4.25667 q^{95} +3.37867 q^{97} -15.0151 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 7 q^{5} - 9 q^{7} - 3 q^{8} - 7 q^{10} + 4 q^{11} - 7 q^{13} + 9 q^{14} + 3 q^{16} + 10 q^{17} - q^{19} + 7 q^{20} - 4 q^{22} + 12 q^{23} + 6 q^{25} + 7 q^{26} - 9 q^{28} + 7 q^{29}+ \cdots - 20 q^{98}+O(q^{100}) \) 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

Related objects

Downloads

Learn more