LMFDB - Embedded newform 4014.2.a.v.1.5

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 = [7,-7,0,7,-6,0,3,-7,0,6,1] 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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 18x^{5} - 8x^{4} + 51x^{3} + 47x^{2} - 2x - 8 $ x^7 - 18x^5 - 8x^4 + 51x^3 + 47x^2 - 2*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1.5
Root			$4.01465$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.603045 q^{5} +4.55050 q^{7} -1.00000 q^{8} -0.603045 q^{10} -1.88517 q^{11} +0.332889 q^{13} -4.55050 q^{14} +1.00000 q^{16} +1.68398 q^{17} +2.93007 q^{19} +0.603045 q^{20} +1.88517 q^{22} +3.43212 q^{23} -4.63634 q^{25} -0.332889 q^{26} +4.55050 q^{28} +1.00355 q^{29} +4.33956 q^{31} -1.00000 q^{32} -1.68398 q^{34} +2.74416 q^{35} -4.29279 q^{37} -2.93007 q^{38} -0.603045 q^{40} -3.48120 q^{41} +12.6586 q^{43} -1.88517 q^{44} -3.43212 q^{46} -10.1829 q^{47} +13.7071 q^{49} +4.63634 q^{50} +0.332889 q^{52} -3.19702 q^{53} -1.13684 q^{55} -4.55050 q^{56} -1.00355 q^{58} +11.9123 q^{59} +5.00355 q^{61} -4.33956 q^{62} +1.00000 q^{64} +0.200747 q^{65} -0.731176 q^{67} +1.68398 q^{68} -2.74416 q^{70} -2.34043 q^{71} +11.5553 q^{73} +4.29279 q^{74} +2.93007 q^{76} -8.57845 q^{77} -6.48581 q^{79} +0.603045 q^{80} +3.48120 q^{82} +11.6246 q^{83} +1.01551 q^{85} -12.6586 q^{86} +1.88517 q^{88} +8.85405 q^{89} +1.51481 q^{91} +3.43212 q^{92} +10.1829 q^{94} +1.76696 q^{95} +7.27938 q^{97} -13.7071 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8} + 6 q^{10} + q^{11} + 8 q^{13} - 3 q^{14} + 7 q^{16} - 16 q^{17} + 2 q^{19} - 6 q^{20} - q^{22} - 8 q^{23} + 19 q^{25} - 8 q^{26} + 3 q^{28} - 4 q^{29}+ \cdots - 24 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 6 * q^5 + 3 * q^7 - 7 * q^8 + 6 * q^10 + q^11 + 8 * q^13 - 3 * q^14 + 7 * q^16 - 16 * q^17 + 2 * q^19 - 6 * q^20 - q^22 - 8 * q^23 + 19 * q^25 - 8 * q^26 + 3 * q^28 - 4 * q^29 + 11 * q^31 - 7 * q^32 + 16 * q^34 + 17 * q^37 - 2 * q^38 + 6 * q^40 - 18 * q^41 - q^43 + q^44 + 8 * q^46 - 5 * q^47 + 24 * q^49 - 19 * q^50 + 8 * q^52 - 6 * q^53 + 21 * q^55 - 3 * q^56 + 4 * q^58 + 7 * q^59 + 24 * q^61 - 11 * q^62 + 7 * q^64 - 11 * q^65 - 4 * q^67 - 16 * q^68 + 7 * q^71 + 28 * q^73 - 17 * q^74 + 2 * q^76 + 2 * q^77 - 6 * q^80 + 18 * q^82 + 2 * q^83 + 4 * q^85 + q^86 - q^88 - 5 * q^89 + 2 * q^91 - 8 * q^92 + 5 * q^94 + 14 * q^95 + 23 * q^97 - 24 * 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
$5$	0.603045	0.269690	0.134845	−	0.990867i	$-0.456946\pi$
$5$	0.603045	0.269690	0.134845	+	0.990867i	$0.456946\pi$
$6$	0	0
$7$	4.55050	1.71993	0.859964	−	0.510354i	$-0.170486\pi$
$7$	4.55050	1.71993	0.859964	+	0.510354i	$0.170486\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.603045	−0.190700
$11$	−1.88517	−0.568399	−0.284199	−	0.958765i	$-0.591728\pi$
$11$	−1.88517	−0.568399	−0.284199	+	0.958765i	$0.591728\pi$
$12$	0	0
$13$	0.332889	0.0923269	0.0461635	−	0.998934i	$-0.485300\pi$
$13$	0.332889	0.0923269	0.0461635	+	0.998934i	$0.485300\pi$
$14$	−4.55050	−1.21617
$15$	0	0
$16$	1.00000	0.250000
$17$	1.68398	0.408424	0.204212	−	0.978927i	$-0.434537\pi$
$17$	1.68398	0.408424	0.204212	+	0.978927i	$0.434537\pi$
$18$	0	0
$19$	2.93007	0.672204	0.336102	−	0.941826i	$-0.390891\pi$
$19$	2.93007	0.672204	0.336102	+	0.941826i	$0.390891\pi$
$20$	0.603045	0.134845
$21$	0	0
$22$	1.88517	0.401919
$23$	3.43212	0.715647	0.357824	−	0.933789i	$-0.383519\pi$
$23$	3.43212	0.715647	0.357824	+	0.933789i	$0.383519\pi$
$24$	0	0
$25$	−4.63634	−0.927267
$26$	−0.332889	−0.0652850
$27$	0	0
$28$	4.55050	0.859964
$29$	1.00355	0.186354	0.0931769	−	0.995650i	$-0.470298\pi$
$29$	1.00355	0.186354	0.0931769	+	0.995650i	$0.470298\pi$
$30$	0	0
$31$	4.33956	0.779409	0.389704	−	0.920940i	$-0.372577\pi$
$31$	4.33956	0.779409	0.389704	+	0.920940i	$0.372577\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.68398	−0.288800
$35$	2.74416	0.463847
$36$	0	0
$37$	−4.29279	−0.705730	−0.352865	−	0.935674i	$-0.614792\pi$
$37$	−4.29279	−0.705730	−0.352865	+	0.935674i	$0.614792\pi$
$38$	−2.93007	−0.475320
$39$	0	0
$40$	−0.603045	−0.0953498
$41$	−3.48120	−0.543672	−0.271836	−	0.962344i	$-0.587631\pi$
$41$	−3.48120	−0.543672	−0.271836	+	0.962344i	$0.587631\pi$
$42$	0	0
$43$	12.6586	1.93042	0.965211	−	0.261471i	$-0.0842075\pi$
$43$	12.6586	1.93042	0.965211	+	0.261471i	$0.0842075\pi$
$44$	−1.88517	−0.284199
$45$	0	0
$46$	−3.43212	−0.506039
$47$	−10.1829	−1.48532	−0.742661	−	0.669668i	$-0.766437\pi$
$47$	−10.1829	−1.48532	−0.742661	+	0.669668i	$0.766437\pi$
$48$	0	0
$49$	13.7071	1.95815
$50$	4.63634	0.655677
$51$	0	0
$52$	0.332889	0.0461635
$53$	−3.19702	−0.439144	−0.219572	−	0.975596i	$-0.570466\pi$
$53$	−3.19702	−0.439144	−0.219572	+	0.975596i	$0.570466\pi$
$54$	0	0
$55$	−1.13684	−0.153291
$56$	−4.55050	−0.608087
$57$	0	0
$58$	−1.00355	−0.131772
$59$	11.9123	1.55084	0.775422	−	0.631444i	$-0.217537\pi$
$59$	11.9123	1.55084	0.775422	+	0.631444i	$0.217537\pi$
$60$	0	0
$61$	5.00355	0.640638	0.320319	−	0.947310i	$-0.396210\pi$
$61$	5.00355	0.640638	0.320319	+	0.947310i	$0.396210\pi$
$62$	−4.33956	−0.551125
$63$	0	0
$64$	1.00000	0.125000
$65$	0.200747	0.0248996
$66$	0	0
$67$	−0.731176	−0.0893274	−0.0446637	−	0.999002i	$-0.514222\pi$
$67$	−0.731176	−0.0893274	−0.0446637	+	0.999002i	$0.514222\pi$
$68$	1.68398	0.204212
$69$	0	0
$70$	−2.74416	−0.327990
$71$	−2.34043	−0.277757	−0.138879	−	0.990309i	$-0.544350\pi$
$71$	−2.34043	−0.277757	−0.138879	+	0.990309i	$0.544350\pi$
$72$	0	0
$73$	11.5553	1.35245	0.676223	−	0.736697i	$-0.263616\pi$
$73$	11.5553	1.35245	0.676223	+	0.736697i	$0.263616\pi$
$74$	4.29279	0.499026
$75$	0	0
$76$	2.93007	0.336102
$77$	−8.57845	−0.977605
$78$	0	0
$79$	−6.48581	−0.729711	−0.364856	−	0.931064i	$-0.618882\pi$
$79$	−6.48581	−0.729711	−0.364856	+	0.931064i	$0.618882\pi$
$80$	0.603045	0.0674225
$81$	0	0
$82$	3.48120	0.384434
$83$	11.6246	1.27597	0.637983	−	0.770050i	$-0.279769\pi$
$83$	11.6246	1.27597	0.637983	+	0.770050i	$0.279769\pi$
$84$	0	0
$85$	1.01551	0.110148
$86$	−12.6586	−1.36501
$87$	0	0
$88$	1.88517	0.200959
$89$	8.85405	0.938528	0.469264	−	0.883058i	$-0.344519\pi$
$89$	8.85405	0.938528	0.469264	+	0.883058i	$0.344519\pi$
$90$	0	0
$91$	1.51481	0.158796
$92$	3.43212	0.357824
$93$	0	0
$94$	10.1829	1.05028
$95$	1.76696	0.181287
$96$	0	0
$97$	7.27938	0.739109	0.369555	−	0.929209i	$-0.379510\pi$
$97$	7.27938	0.739109	0.369555	+	0.929209i	$0.379510\pi$
$98$	−13.7071	−1.38462
$99$	0	0
$100$	−4.63634	−0.463634
$101$	−3.91896	−0.389952	−0.194976	−	0.980808i	$-0.562463\pi$
$101$	−3.91896	−0.389952	−0.194976	+	0.980808i	$0.562463\pi$
$102$	0	0
$103$	−3.75915	−0.370400	−0.185200	−	0.982701i	$-0.559293\pi$
$103$	−3.75915	−0.370400	−0.185200	+	0.982701i	$0.559293\pi$
$104$	−0.332889	−0.0326425
$105$	0	0
$106$	3.19702	0.310522
$107$	13.6322	1.31788	0.658939	−	0.752196i	$-0.271005\pi$
$107$	13.6322	1.31788	0.658939	+	0.752196i	$0.271005\pi$
$108$	0	0
$109$	−12.8655	−1.23229	−0.616144	−	0.787633i	$-0.711306\pi$
$109$	−12.8655	−1.23229	−0.616144	+	0.787633i	$0.711306\pi$
$110$	1.13684	0.108393
$111$	0	0
$112$	4.55050	0.429982
$113$	−2.44489	−0.229995	−0.114998	−	0.993366i	$-0.536686\pi$
$113$	−2.44489	−0.229995	−0.114998	+	0.993366i	$0.536686\pi$
$114$	0	0
$115$	2.06972	0.193003
$116$	1.00355	0.0931769
$117$	0	0
$118$	−11.9123	−1.09661
$119$	7.66294	0.702461
$120$	0	0
$121$	−7.44615	−0.676923
$122$	−5.00355	−0.453000
$123$	0	0
$124$	4.33956	0.389704
$125$	−5.81115	−0.519765
$126$	0	0
$127$	10.4761	0.929602	0.464801	−	0.885415i	$-0.346126\pi$
$127$	10.4761	0.929602	0.464801	+	0.885415i	$0.346126\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.200747	−0.0176067
$131$	−13.8235	−1.20776	−0.603881	−	0.797075i	$-0.706380\pi$
$131$	−13.8235	−1.20776	−0.603881	+	0.797075i	$0.706380\pi$
$132$	0	0
$133$	13.3333	1.15614
$134$	0.731176	0.0631640
$135$	0	0
$136$	−1.68398	−0.144400
$137$	−9.36895	−0.800443	−0.400222	−	0.916418i	$-0.631067\pi$
$137$	−9.36895	−0.800443	−0.400222	+	0.916418i	$0.631067\pi$
$138$	0	0
$139$	−0.00638381	−0.000541467 0	−0.000270734	−	1.00000i	$-0.500086\pi$
$139$	−0.00638381	−0.000541467 0	−0.000270734		1.00000i	$0.500086\pi$
$140$	2.74416	0.231924
$141$	0	0
$142$	2.34043	0.196404
$143$	−0.627552	−0.0524785
$144$	0	0
$145$	0.605183	0.0502577
$146$	−11.5553	−0.956324
$147$	0	0
$148$	−4.29279	−0.352865
$149$	−5.33903	−0.437391	−0.218695	−	0.975793i	$-0.570180\pi$
$149$	−5.33903	−0.437391	−0.218695	+	0.975793i	$0.570180\pi$
$150$	0	0
$151$	−6.50484	−0.529357	−0.264678	−	0.964337i	$-0.585266\pi$
$151$	−6.50484	−0.529357	−0.264678	+	0.964337i	$0.585266\pi$
$152$	−2.93007	−0.237660
$153$	0	0
$154$	8.57845	0.691271
$155$	2.61695	0.210199
$156$	0	0
$157$	−10.2900	−0.821235	−0.410618	−	0.911808i	$-0.634687\pi$
$157$	−10.2900	−0.821235	−0.410618	+	0.911808i	$0.634687\pi$
$158$	6.48581	0.515984
$159$	0	0
$160$	−0.603045	−0.0476749
$161$	15.6179	1.23086
$162$	0	0
$163$	−22.8100	−1.78661	−0.893307	−	0.449447i	$-0.851621\pi$
$163$	−22.8100	−1.78661	−0.893307	+	0.449447i	$0.851621\pi$
$164$	−3.48120	−0.271836
$165$	0	0
$166$	−11.6246	−0.902245
$167$	7.09832	0.549285	0.274642	−	0.961546i	$-0.411441\pi$
$167$	7.09832	0.549285	0.274642	+	0.961546i	$0.411441\pi$
$168$	0	0
$169$	−12.8892	−0.991476
$170$	−1.01551	−0.0778864
$171$	0	0
$172$	12.6586	0.965211
$173$	17.2397	1.31071	0.655356	−	0.755320i	$-0.272519\pi$
$173$	17.2397	1.31071	0.655356	+	0.755320i	$0.272519\pi$
$174$	0	0
$175$	−21.0977	−1.59483
$176$	−1.88517	−0.142100
$177$	0	0
$178$	−8.85405	−0.663639
$179$	8.87461	0.663319	0.331660	−	0.943399i	$-0.392391\pi$
$179$	8.87461	0.663319	0.331660	+	0.943399i	$0.392391\pi$
$180$	0	0
$181$	−2.91525	−0.216689	−0.108344	−	0.994113i	$-0.534555\pi$
$181$	−2.91525	−0.216689	−0.108344	+	0.994113i	$0.534555\pi$
$182$	−1.51481	−0.112286
$183$	0	0
$184$	−3.43212	−0.253019
$185$	−2.58874	−0.190328
$186$	0	0
$187$	−3.17457	−0.232148
$188$	−10.1829	−0.742661
$189$	0	0
$190$	−1.76696	−0.128189
$191$	3.79232	0.274403	0.137201	−	0.990543i	$-0.456189\pi$
$191$	3.79232	0.274403	0.137201	+	0.990543i	$0.456189\pi$
$192$	0	0
$193$	−0.652041	−0.0469349	−0.0234675	−	0.999725i	$-0.507471\pi$
$193$	−0.652041	−0.0469349	−0.0234675	+	0.999725i	$0.507471\pi$
$194$	−7.27938	−0.522629
$195$	0	0
$196$	13.7071	0.979077
$197$	13.0157	0.927327	0.463664	−	0.886011i	$-0.346535\pi$
$197$	13.0157	0.927327	0.463664	+	0.886011i	$0.346535\pi$
$198$	0	0
$199$	−2.95663	−0.209590	−0.104795	−	0.994494i	$-0.533419\pi$
$199$	−2.95663	−0.209590	−0.104795	+	0.994494i	$0.533419\pi$
$200$	4.63634	0.327839
$201$	0	0
$202$	3.91896	0.275737
$203$	4.56664	0.320515
$204$	0	0
$205$	−2.09932	−0.146623
$206$	3.75915	0.261912
$207$	0	0
$208$	0.332889	0.0230817
$209$	−5.52367	−0.382080
$210$	0	0
$211$	8.43942	0.580994	0.290497	−	0.956876i	$-0.406179\pi$
$211$	8.43942	0.580994	0.290497	+	0.956876i	$0.406179\pi$
$212$	−3.19702	−0.219572
$213$	0	0
$214$	−13.6322	−0.931881
$215$	7.63372	0.520616
$216$	0	0
$217$	19.7472	1.34053
$218$	12.8655	0.871360
$219$	0	0
$220$	−1.13684	−0.0766457
$221$	0.560578	0.0377086
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−4.55050	−0.304043
$225$	0	0
$226$	2.44489	0.162631
$227$	−16.7885	−1.11429	−0.557147	−	0.830414i	$-0.688104\pi$
$227$	−16.7885	−1.11429	−0.557147	+	0.830414i	$0.688104\pi$
$228$	0	0
$229$	19.3526	1.27886	0.639428	−	0.768851i	$-0.279171\pi$
$229$	19.3526	1.27886	0.639428	+	0.768851i	$0.279171\pi$
$230$	−2.06972	−0.136474
$231$	0	0
$232$	−1.00355	−0.0658860
$233$	−16.7988	−1.10052	−0.550262	−	0.834992i	$-0.685472\pi$
$233$	−16.7988	−1.10052	−0.550262	+	0.834992i	$0.685472\pi$
$234$	0	0
$235$	−6.14072	−0.400576
$236$	11.9123	0.775422
$237$	0	0
$238$	−7.66294	−0.496715
$239$	9.59882	0.620896	0.310448	−	0.950590i	$-0.399521\pi$
$239$	9.59882	0.620896	0.310448	+	0.950590i	$0.399521\pi$
$240$	0	0
$241$	26.8255	1.72798	0.863990	−	0.503508i	$-0.167958\pi$
$241$	26.8255	1.72798	0.863990	+	0.503508i	$0.167958\pi$
$242$	7.44615	0.478657
$243$	0	0
$244$	5.00355	0.320319
$245$	8.26599	0.528094
$246$	0	0
$247$	0.975390	0.0620626
$248$	−4.33956	−0.275563
$249$	0	0
$250$	5.81115	0.367529
$251$	8.58922	0.542147	0.271073	−	0.962559i	$-0.412621\pi$
$251$	8.58922	0.542147	0.271073	+	0.962559i	$0.412621\pi$
$252$	0	0
$253$	−6.47012	−0.406773
$254$	−10.4761	−0.657328
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.4827	0.903408	0.451704	−	0.892168i	$-0.350816\pi$
$257$	14.4827	0.903408	0.451704	+	0.892168i	$0.350816\pi$
$258$	0	0
$259$	−19.5343	−1.21380
$260$	0.200747	0.0124498
$261$	0	0
$262$	13.8235	0.854016
$263$	15.0239	0.926413	0.463206	−	0.886250i	$-0.346699\pi$
$263$	15.0239	0.926413	0.463206	+	0.886250i	$0.346699\pi$
$264$	0	0
$265$	−1.92795	−0.118433
$266$	−13.3333	−0.817517
$267$	0	0
$268$	−0.731176	−0.0446637
$269$	15.5993	0.951103	0.475552	−	0.879688i	$-0.342248\pi$
$269$	15.5993	0.951103	0.475552	+	0.879688i	$0.342248\pi$
$270$	0	0
$271$	10.4110	0.632423	0.316211	−	0.948689i	$-0.397589\pi$
$271$	10.4110	0.632423	0.316211	+	0.948689i	$0.397589\pi$
$272$	1.68398	0.102106
$273$	0	0
$274$	9.36895	0.565999
$275$	8.74026	0.527058
$276$	0	0
$277$	−5.58823	−0.335764	−0.167882	−	0.985807i	$-0.553693\pi$
$277$	−5.58823	−0.335764	−0.167882	+	0.985807i	$0.553693\pi$
$278$	0.00638381	0.000382875 0
$279$	0	0
$280$	−2.74416	−0.163995
$281$	11.0448	0.658879	0.329439	−	0.944177i	$-0.393140\pi$
$281$	11.0448	0.658879	0.329439	+	0.944177i	$0.393140\pi$
$282$	0	0
$283$	−10.9022	−0.648071	−0.324035	−	0.946045i	$-0.605040\pi$
$283$	−10.9022	−0.648071	−0.324035	+	0.946045i	$0.605040\pi$
$284$	−2.34043	−0.138879
$285$	0	0
$286$	0.627552	0.0371079
$287$	−15.8412	−0.935077
$288$	0	0
$289$	−14.1642	−0.833190
$290$	−0.605183	−0.0355376
$291$	0	0
$292$	11.5553	0.676223
$293$	6.25282	0.365294	0.182647	−	0.983179i	$-0.441533\pi$
$293$	6.25282	0.365294	0.182647	+	0.983179i	$0.441533\pi$
$294$	0	0
$295$	7.18363	0.418247
$296$	4.29279	0.249513
$297$	0	0
$298$	5.33903	0.309282
$299$	1.14252	0.0660735
$300$	0	0
$301$	57.6031	3.32019
$302$	6.50484	0.374312
$303$	0	0
$304$	2.93007	0.168051
$305$	3.01736	0.172774
$306$	0	0
$307$	−9.92547	−0.566476	−0.283238	−	0.959050i	$-0.591409\pi$
$307$	−9.92547	−0.566476	−0.283238	+	0.959050i	$0.591409\pi$
$308$	−8.57845	−0.488803
$309$	0	0
$310$	−2.61695	−0.148633
$311$	3.49848	0.198381	0.0991903	−	0.995068i	$-0.468375\pi$
$311$	3.49848	0.198381	0.0991903	+	0.995068i	$0.468375\pi$
$312$	0	0
$313$	11.6644	0.659310	0.329655	−	0.944101i	$-0.393068\pi$
$313$	11.6644	0.659310	0.329655	+	0.944101i	$0.393068\pi$
$314$	10.2900	0.580701
$315$	0	0
$316$	−6.48581	−0.364856
$317$	17.6419	0.990870	0.495435	−	0.868645i	$-0.335009\pi$
$317$	17.6419	0.990870	0.495435	+	0.868645i	$0.335009\pi$
$318$	0	0
$319$	−1.89185	−0.105923
$320$	0.603045	0.0337112
$321$	0	0
$322$	−15.6179	−0.870351
$323$	4.93417	0.274545
$324$	0	0
$325$	−1.54339	−0.0856117
$326$	22.8100	1.26333
$327$	0	0
$328$	3.48120	0.192217
$329$	−46.3371	−2.55465
$330$	0	0
$331$	−9.07997	−0.499080	−0.249540	−	0.968364i	$-0.580279\pi$
$331$	−9.07997	−0.499080	−0.249540	+	0.968364i	$0.580279\pi$
$332$	11.6246	0.637983
$333$	0	0
$334$	−7.09832	−0.388403
$335$	−0.440932	−0.0240907
$336$	0	0
$337$	27.6737	1.50748	0.753741	−	0.657172i	$-0.228247\pi$
$337$	27.6737	1.50748	0.753741	+	0.657172i	$0.228247\pi$
$338$	12.8892	0.701079
$339$	0	0
$340$	1.01551	0.0550740
$341$	−8.18080	−0.443015
$342$	0	0
$343$	30.5206	1.64796
$344$	−12.6586	−0.682507
$345$	0	0
$346$	−17.2397	−0.926813
$347$	9.40234	0.504744	0.252372	−	0.967630i	$-0.418789\pi$
$347$	9.40234	0.504744	0.252372	+	0.967630i	$0.418789\pi$
$348$	0	0
$349$	−9.45111	−0.505906	−0.252953	−	0.967479i	$-0.581402\pi$
$349$	−9.45111	−0.505906	−0.252953	+	0.967479i	$0.581402\pi$
$350$	21.0977	1.12772
$351$	0	0
$352$	1.88517	0.100480
$353$	16.6377	0.885537	0.442769	−	0.896636i	$-0.353996\pi$
$353$	16.6377	0.885537	0.442769	+	0.896636i	$0.353996\pi$
$354$	0	0
$355$	−1.41138	−0.0749084
$356$	8.85405	0.469264
$357$	0	0
$358$	−8.87461	−0.469038
$359$	30.6733	1.61888	0.809438	−	0.587206i	$-0.199772\pi$
$359$	30.6733	1.61888	0.809438	+	0.587206i	$0.199772\pi$
$360$	0	0
$361$	−10.4147	−0.548141
$362$	2.91525	0.153222
$363$	0	0
$364$	1.51481	0.0793978
$365$	6.96837	0.364741
$366$	0	0
$367$	19.8301	1.03512	0.517561	−	0.855647i	$-0.326840\pi$
$367$	19.8301	1.03512	0.517561	+	0.855647i	$0.326840\pi$
$368$	3.43212	0.178912
$369$	0	0
$370$	2.58874	0.134582
$371$	−14.5480	−0.755297
$372$	0	0
$373$	27.1464	1.40558	0.702792	−	0.711395i	$-0.251936\pi$
$373$	27.1464	1.40558	0.702792	+	0.711395i	$0.251936\pi$
$374$	3.17457	0.164153
$375$	0	0
$376$	10.1829	0.525141
$377$	0.334070	0.0172055
$378$	0	0
$379$	−33.9956	−1.74623	−0.873117	−	0.487510i	$-0.837905\pi$
$379$	−33.9956	−1.74623	−0.873117	+	0.487510i	$0.837905\pi$
$380$	1.76696	0.0906434
$381$	0	0
$382$	−3.79232	−0.194032
$383$	12.7407	0.651019	0.325509	−	0.945539i	$-0.394464\pi$
$383$	12.7407	0.651019	0.325509	+	0.945539i	$0.394464\pi$
$384$	0	0
$385$	−5.17319	−0.263650
$386$	0.652041	0.0331880
$387$	0	0
$388$	7.27938	0.369555
$389$	−16.0790	−0.815237	−0.407619	−	0.913152i	$-0.633641\pi$
$389$	−16.0790	−0.815237	−0.407619	+	0.913152i	$0.633641\pi$
$390$	0	0
$391$	5.77961	0.292288
$392$	−13.7071	−0.692312
$393$	0	0
$394$	−13.0157	−0.655719
$395$	−3.91124	−0.196796
$396$	0	0
$397$	−30.5096	−1.53123	−0.765616	−	0.643298i	$-0.777566\pi$
$397$	−30.5096	−1.53123	−0.765616	+	0.643298i	$0.777566\pi$
$398$	2.95663	0.148203
$399$	0	0
$400$	−4.63634	−0.231817
$401$	−26.5433	−1.32551	−0.662755	−	0.748836i	$-0.730613\pi$
$401$	−26.5433	−1.32551	−0.662755	+	0.748836i	$0.730613\pi$
$402$	0	0
$403$	1.44459	0.0719604
$404$	−3.91896	−0.194976
$405$	0	0
$406$	−4.56664	−0.226638
$407$	8.09261	0.401136
$408$	0	0
$409$	40.2002	1.98777	0.993887	−	0.110406i	$-0.0352150\pi$
$409$	40.2002	1.98777	0.993887	+	0.110406i	$0.0352150\pi$
$410$	2.09932	0.103678
$411$	0	0
$412$	−3.75915	−0.185200
$413$	54.2068	2.66734
$414$	0	0
$415$	7.01016	0.344115
$416$	−0.332889	−0.0163212
$417$	0	0
$418$	5.52367	0.270171
$419$	−0.648474	−0.0316800	−0.0158400	−	0.999875i	$-0.505042\pi$
$419$	−0.648474	−0.0316800	−0.0158400	+	0.999875i	$0.505042\pi$
$420$	0	0
$421$	−21.3460	−1.04034	−0.520171	−	0.854062i	$-0.674132\pi$
$421$	−21.3460	−1.04034	−0.520171	+	0.854062i	$0.674132\pi$
$422$	−8.43942	−0.410825
$423$	0	0
$424$	3.19702	0.155261
$425$	−7.80748	−0.378719
$426$	0	0
$427$	22.7686	1.10185
$428$	13.6322	0.658939
$429$	0	0
$430$	−7.63372	−0.368131
$431$	17.4639	0.841207	0.420603	−	0.907245i	$-0.361818\pi$
$431$	17.4639	0.841207	0.420603	+	0.907245i	$0.361818\pi$
$432$	0	0
$433$	−8.87301	−0.426410	−0.213205	−	0.977008i	$-0.568390\pi$
$433$	−8.87301	−0.426410	−0.213205	+	0.977008i	$0.568390\pi$
$434$	−19.7472	−0.947896
$435$	0	0
$436$	−12.8655	−0.616144
$437$	10.0564	0.481061
$438$	0	0
$439$	−8.89218	−0.424400	−0.212200	−	0.977226i	$-0.568063\pi$
$439$	−8.89218	−0.424400	−0.212200	+	0.977226i	$0.568063\pi$
$440$	1.13684	0.0541967
$441$	0	0
$442$	−0.560578	−0.0266640
$443$	7.45500	0.354198	0.177099	−	0.984193i	$-0.443329\pi$
$443$	7.45500	0.354198	0.177099	+	0.984193i	$0.443329\pi$
$444$	0	0
$445$	5.33939	0.253112
$446$	1.00000	0.0473514
$447$	0	0
$448$	4.55050	0.214991
$449$	−19.7555	−0.932318	−0.466159	−	0.884701i	$-0.654363\pi$
$449$	−19.7555	−0.932318	−0.466159	+	0.884701i	$0.654363\pi$
$450$	0	0
$451$	6.56264	0.309022
$452$	−2.44489	−0.114998
$453$	0	0
$454$	16.7885	0.787925
$455$	0.913501	0.0428256
$456$	0	0
$457$	25.8231	1.20795	0.603976	−	0.797003i	$-0.293582\pi$
$457$	25.8231	1.20795	0.603976	+	0.797003i	$0.293582\pi$
$458$	−19.3526	−0.904287
$459$	0	0
$460$	2.06972	0.0965014
$461$	−42.3213	−1.97110	−0.985549	−	0.169388i	$-0.945821\pi$
$461$	−42.3213	−1.97110	−0.985549	+	0.169388i	$0.945821\pi$
$462$	0	0
$463$	4.10285	0.190675	0.0953377	−	0.995445i	$-0.469607\pi$
$463$	4.10285	0.190675	0.0953377	+	0.995445i	$0.469607\pi$
$464$	1.00355	0.0465884
$465$	0	0
$466$	16.7988	0.778188
$467$	−1.40172	−0.0648639	−0.0324320	−	0.999474i	$-0.510325\pi$
$467$	−1.40172	−0.0648639	−0.0324320	+	0.999474i	$0.510325\pi$
$468$	0	0
$469$	−3.32722	−0.153637
$470$	6.14072	0.283250
$471$	0	0
$472$	−11.9123	−0.548306
$473$	−23.8636	−1.09725
$474$	0	0
$475$	−13.5848	−0.623313
$476$	7.66294	0.351230
$477$	0	0
$478$	−9.59882	−0.439040
$479$	−1.60498	−0.0733332	−0.0366666	−	0.999328i	$-0.511674\pi$
$479$	−1.60498	−0.0733332	−0.0366666	+	0.999328i	$0.511674\pi$
$480$	0	0
$481$	−1.42902	−0.0651578
$482$	−26.8255	−1.22187
$483$	0	0
$484$	−7.44615	−0.338461
$485$	4.38980	0.199330
$486$	0	0
$487$	−1.70191	−0.0771210	−0.0385605	−	0.999256i	$-0.512277\pi$
$487$	−1.70191	−0.0771210	−0.0385605	+	0.999256i	$0.512277\pi$
$488$	−5.00355	−0.226500
$489$	0	0
$490$	−8.26599	−0.373419
$491$	27.4254	1.23769	0.618846	−	0.785512i	$-0.287600\pi$
$491$	27.4254	1.23769	0.618846	+	0.785512i	$0.287600\pi$
$492$	0	0
$493$	1.68995	0.0761114
$494$	−0.975390	−0.0438849
$495$	0	0
$496$	4.33956	0.194852
$497$	−10.6501	−0.477723
$498$	0	0
$499$	−34.5307	−1.54581	−0.772903	−	0.634524i	$-0.781196\pi$
$499$	−34.5307	−1.54581	−0.772903	+	0.634524i	$0.781196\pi$
$500$	−5.81115	−0.259882
$501$	0	0
$502$	−8.58922	−0.383356
$503$	−22.3773	−0.997754	−0.498877	−	0.866673i	$-0.666254\pi$
$503$	−22.3773	−0.997754	−0.498877	+	0.866673i	$0.666254\pi$
$504$	0	0
$505$	−2.36331	−0.105166
$506$	6.47012	0.287632
$507$	0	0
$508$	10.4761	0.464801
$509$	−24.9622	−1.10643	−0.553214	−	0.833039i	$-0.686599\pi$
$509$	−24.9622	−1.10643	−0.553214	+	0.833039i	$0.686599\pi$
$510$	0	0
$511$	52.5824	2.32611
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−14.4827	−0.638806
$515$	−2.26694	−0.0998931
$516$	0	0
$517$	19.1964	0.844255
$518$	19.5343	0.858289
$519$	0	0
$520$	−0.200747	−0.00880335
$521$	29.1552	1.27731	0.638656	−	0.769492i	$-0.279491\pi$
$521$	29.1552	1.27731	0.638656	+	0.769492i	$0.279491\pi$
$522$	0	0
$523$	−28.0352	−1.22589	−0.612946	−	0.790125i	$-0.710016\pi$
$523$	−28.0352	−1.22589	−0.612946	+	0.790125i	$0.710016\pi$
$524$	−13.8235	−0.603881
$525$	0	0
$526$	−15.0239	−0.655073
$527$	7.30772	0.318329
$528$	0	0
$529$	−11.2205	−0.487849
$530$	1.92795	0.0837446
$531$	0	0
$532$	13.3333	0.578072
$533$	−1.15885	−0.0501955
$534$	0	0
$535$	8.22086	0.355419
$536$	0.731176	0.0315820
$537$	0	0
$538$	−15.5993	−0.672532
$539$	−25.8401	−1.11301
$540$	0	0
$541$	5.54743	0.238503	0.119251	−	0.992864i	$-0.461951\pi$
$541$	5.54743	0.238503	0.119251	+	0.992864i	$0.461951\pi$
$542$	−10.4110	−0.447191
$543$	0	0
$544$	−1.68398	−0.0721999
$545$	−7.75846	−0.332336
$546$	0	0
$547$	−16.3410	−0.698689	−0.349344	−	0.936994i	$-0.613596\pi$
$547$	−16.3410	−0.698689	−0.349344	+	0.936994i	$0.613596\pi$
$548$	−9.36895	−0.400222
$549$	0	0
$550$	−8.74026	−0.372686
$551$	2.94046	0.125268
$552$	0	0
$553$	−29.5137	−1.25505
$554$	5.58823	0.237421
$555$	0	0
$556$	−0.00638381	−0.000270734 0
$557$	−7.64612	−0.323977	−0.161988	−	0.986793i	$-0.551791\pi$
$557$	−7.64612	−0.323977	−0.161988	+	0.986793i	$0.551791\pi$
$558$	0	0
$559$	4.21392	0.178230
$560$	2.74416	0.115962
$561$	0	0
$562$	−11.0448	−0.465898
$563$	−23.1760	−0.976752	−0.488376	−	0.872633i	$-0.662411\pi$
$563$	−23.1760	−0.976752	−0.488376	+	0.872633i	$0.662411\pi$
$564$	0	0
$565$	−1.47438	−0.0620275
$566$	10.9022	0.458255
$567$	0	0
$568$	2.34043	0.0982021
$569$	9.86719	0.413654	0.206827	−	0.978378i	$-0.433686\pi$
$569$	9.86719	0.413654	0.206827	+	0.978378i	$0.433686\pi$
$570$	0	0
$571$	−15.7512	−0.659168	−0.329584	−	0.944126i	$-0.606909\pi$
$571$	−15.7512	−0.659168	−0.329584	+	0.944126i	$0.606909\pi$
$572$	−0.627552	−0.0262393
$573$	0	0
$574$	15.8412	0.661199
$575$	−15.9125	−0.663596
$576$	0	0
$577$	31.8526	1.32604	0.663021	−	0.748601i	$-0.269274\pi$
$577$	31.8526	1.32604	0.663021	+	0.748601i	$0.269274\pi$
$578$	14.1642	0.589154
$579$	0	0
$580$	0.605183	0.0251289
$581$	52.8978	2.19457
$582$	0	0
$583$	6.02691	0.249609
$584$	−11.5553	−0.478162
$585$	0	0
$586$	−6.25282	−0.258302
$587$	−6.90003	−0.284795	−0.142397	−	0.989810i	$-0.545481\pi$
$587$	−6.90003	−0.284795	−0.142397	+	0.989810i	$0.545481\pi$
$588$	0	0
$589$	12.7152	0.523922
$590$	−7.18363	−0.295745
$591$	0	0
$592$	−4.29279	−0.176432
$593$	−13.2970	−0.546044	−0.273022	−	0.962008i	$-0.588023\pi$
$593$	−13.2970	−0.546044	−0.273022	+	0.962008i	$0.588023\pi$
$594$	0	0
$595$	4.62110	0.189447
$596$	−5.33903	−0.218695
$597$	0	0
$598$	−1.14252	−0.0467210
$599$	34.6141	1.41429	0.707146	−	0.707067i	$-0.249982\pi$
$599$	34.6141	1.41429	0.707146	+	0.707067i	$0.249982\pi$
$600$	0	0
$601$	28.6231	1.16756	0.583781	−	0.811911i	$-0.301573\pi$
$601$	28.6231	1.16756	0.583781	+	0.811911i	$0.301573\pi$
$602$	−57.6031	−2.34773
$603$	0	0
$604$	−6.50484	−0.264678
$605$	−4.49037	−0.182559
$606$	0	0
$607$	−4.34821	−0.176488	−0.0882442	−	0.996099i	$-0.528126\pi$
$607$	−4.34821	−0.176488	−0.0882442	+	0.996099i	$0.528126\pi$
$608$	−2.93007	−0.118830
$609$	0	0
$610$	−3.01736	−0.122169
$611$	−3.38976	−0.137135
$612$	0	0
$613$	−21.0395	−0.849777	−0.424889	−	0.905246i	$-0.639687\pi$
$613$	−21.0395	−0.849777	−0.424889	+	0.905246i	$0.639687\pi$
$614$	9.92547	0.400559
$615$	0	0
$616$	8.57845	0.345636
$617$	−23.2087	−0.934348	−0.467174	−	0.884166i	$-0.654728\pi$
$617$	−23.2087	−0.934348	−0.467174	+	0.884166i	$0.654728\pi$
$618$	0	0
$619$	−17.9895	−0.723060	−0.361530	−	0.932360i	$-0.617746\pi$
$619$	−17.9895	−0.723060	−0.361530	+	0.932360i	$0.617746\pi$
$620$	2.61695	0.105099
$621$	0	0
$622$	−3.49848	−0.140276
$623$	40.2904	1.61420
$624$	0	0
$625$	19.6773	0.787092
$626$	−11.6644	−0.466203
$627$	0	0
$628$	−10.2900	−0.410618
$629$	−7.22895	−0.288237
$630$	0	0
$631$	29.6983	1.18227	0.591134	−	0.806573i	$-0.298680\pi$
$631$	29.6983	1.18227	0.591134	+	0.806573i	$0.298680\pi$
$632$	6.48581	0.257992
$633$	0	0
$634$	−17.6419	−0.700651
$635$	6.31755	0.250704
$636$	0	0
$637$	4.56294	0.180790
$638$	1.89185	0.0748991
$639$	0	0
$640$	−0.603045	−0.0238375
$641$	44.9700	1.77621	0.888103	−	0.459644i	$-0.152023\pi$
$641$	44.9700	1.77621	0.888103	+	0.459644i	$0.152023\pi$
$642$	0	0
$643$	−18.2522	−0.719798	−0.359899	−	0.932991i	$-0.617189\pi$
$643$	−18.2522	−0.719798	−0.359899	+	0.932991i	$0.617189\pi$
$644$	15.6179	0.615431
$645$	0	0
$646$	−4.93417	−0.194132
$647$	−25.6912	−1.01002	−0.505012	−	0.863112i	$-0.668512\pi$
$647$	−25.6912	−1.01002	−0.505012	+	0.863112i	$0.668512\pi$
$648$	0	0
$649$	−22.4566	−0.881498
$650$	1.54339	0.0605366
$651$	0	0
$652$	−22.8100	−0.893307
$653$	39.6426	1.55134	0.775668	−	0.631141i	$-0.217413\pi$
$653$	39.6426	1.55134	0.775668	+	0.631141i	$0.217413\pi$
$654$	0	0
$655$	−8.33617	−0.325721
$656$	−3.48120	−0.135918
$657$	0	0
$658$	46.3371	1.80641
$659$	11.9261	0.464576	0.232288	−	0.972647i	$-0.425379\pi$
$659$	11.9261	0.464576	0.232288	+	0.972647i	$0.425379\pi$
$660$	0	0
$661$	−10.6868	−0.415667	−0.207833	−	0.978164i	$-0.566641\pi$
$661$	−10.6868	−0.415667	−0.207833	+	0.978164i	$0.566641\pi$
$662$	9.07997	0.352903
$663$	0	0
$664$	−11.6246	−0.451122
$665$	8.04058	0.311800
$666$	0	0
$667$	3.44429	0.133364
$668$	7.09832	0.274642
$669$	0	0
$670$	0.440932	0.0170347
$671$	−9.43251	−0.364138
$672$	0	0
$673$	13.8040	0.532106	0.266053	−	0.963958i	$-0.414280\pi$
$673$	13.8040	0.532106	0.266053	+	0.963958i	$0.414280\pi$
$674$	−27.6737	−1.06595
$675$	0	0
$676$	−12.8892	−0.495738
$677$	24.6499	0.947373	0.473687	−	0.880693i	$-0.342923\pi$
$677$	24.6499	0.947373	0.473687	+	0.880693i	$0.342923\pi$
$678$	0	0
$679$	33.1248	1.27122
$680$	−1.01551	−0.0389432
$681$	0	0
$682$	8.18080	0.313259
$683$	−25.3859	−0.971366	−0.485683	−	0.874135i	$-0.661429\pi$
$683$	−25.3859	−0.971366	−0.485683	+	0.874135i	$0.661429\pi$
$684$	0	0
$685$	−5.64990	−0.215872
$686$	−30.5206	−1.16528
$687$	0	0
$688$	12.6586	0.482606
$689$	−1.06425	−0.0405448
$690$	0	0
$691$	19.8537	0.755271	0.377635	−	0.925954i	$-0.376737\pi$
$691$	19.8537	0.755271	0.377635	+	0.925954i	$0.376737\pi$
$692$	17.2397	0.655356
$693$	0	0
$694$	−9.40234	−0.356908
$695$	−0.00384972	−0.000146028 0
$696$	0	0
$697$	−5.86226	−0.222049
$698$	9.45111	0.357730
$699$	0	0
$700$	−21.0977	−0.797417
$701$	−45.1071	−1.70367	−0.851836	−	0.523809i	$-0.824511\pi$
$701$	−45.1071	−1.70367	−0.851836	+	0.523809i	$0.824511\pi$
$702$	0	0
$703$	−12.5782	−0.474394
$704$	−1.88517	−0.0710499
$705$	0	0
$706$	−16.6377	−0.626169
$707$	−17.8333	−0.670689
$708$	0	0
$709$	−1.83520	−0.0689225	−0.0344612	−	0.999406i	$-0.510972\pi$
$709$	−1.83520	−0.0689225	−0.0344612	+	0.999406i	$0.510972\pi$
$710$	1.41138	0.0529682
$711$	0	0
$712$	−8.85405	−0.331820
$713$	14.8939	0.557782
$714$	0	0
$715$	−0.378442	−0.0141529
$716$	8.87461	0.331660
$717$	0	0
$718$	−30.6733	−1.14472
$719$	−42.7799	−1.59542	−0.797711	−	0.603040i	$-0.793956\pi$
$719$	−42.7799	−1.59542	−0.797711	+	0.603040i	$0.793956\pi$
$720$	0	0
$721$	−17.1060	−0.637061
$722$	10.4147	0.387594
$723$	0	0
$724$	−2.91525	−0.108344
$725$	−4.65278	−0.172800
$726$	0	0
$727$	38.7699	1.43790	0.718948	−	0.695064i	$-0.244624\pi$
$727$	38.7699	1.43790	0.718948	+	0.695064i	$0.244624\pi$
$728$	−1.51481	−0.0561428
$729$	0	0
$730$	−6.96837	−0.257911
$731$	21.3168	0.788432
$732$	0	0
$733$	9.59371	0.354352	0.177176	−	0.984179i	$-0.443304\pi$
$733$	9.59371	0.354352	0.177176	+	0.984179i	$0.443304\pi$
$734$	−19.8301	−0.731941
$735$	0	0
$736$	−3.43212	−0.126510
$737$	1.37839	0.0507736
$738$	0	0
$739$	38.5762	1.41905	0.709525	−	0.704680i	$-0.248910\pi$
$739$	38.5762	1.41905	0.709525	+	0.704680i	$0.248910\pi$
$740$	−2.58874	−0.0951641
$741$	0	0
$742$	14.5480	0.534075
$743$	−7.58479	−0.278259	−0.139129	−	0.990274i	$-0.544430\pi$
$743$	−7.58479	−0.278259	−0.139129	+	0.990274i	$0.544430\pi$
$744$	0	0
$745$	−3.21968	−0.117960
$746$	−27.1464	−0.993898
$747$	0	0
$748$	−3.17457	−0.116074
$749$	62.0336	2.26666
$750$	0	0
$751$	−32.2006	−1.17502	−0.587509	−	0.809217i	$-0.699891\pi$
$751$	−32.2006	−1.17502	−0.587509	+	0.809217i	$0.699891\pi$
$752$	−10.1829	−0.371330
$753$	0	0
$754$	−0.334070	−0.0121661
$755$	−3.92271	−0.142762
$756$	0	0
$757$	−50.1275	−1.82192	−0.910958	−	0.412499i	$-0.864656\pi$
$757$	−50.1275	−1.82192	−0.910958	+	0.412499i	$0.864656\pi$
$758$	33.9956	1.23477
$759$	0	0
$760$	−1.76696	−0.0640946
$761$	11.9580	0.433477	0.216739	−	0.976230i	$-0.430458\pi$
$761$	11.9580	0.433477	0.216739	+	0.976230i	$0.430458\pi$
$762$	0	0
$763$	−58.5444	−2.11945
$764$	3.79232	0.137201
$765$	0	0
$766$	−12.7407	−0.460340
$767$	3.96546	0.143185
$768$	0	0
$769$	−2.70203	−0.0974376	−0.0487188	−	0.998813i	$-0.515514\pi$
$769$	−2.70203	−0.0974376	−0.0487188	+	0.998813i	$0.515514\pi$
$770$	5.17319	0.186429
$771$	0	0
$772$	−0.652041	−0.0234675
$773$	−46.4393	−1.67031	−0.835153	−	0.550018i	$-0.814621\pi$
$773$	−46.4393	−1.67031	−0.835153	+	0.550018i	$0.814621\pi$
$774$	0	0
$775$	−20.1197	−0.722720
$776$	−7.27938	−0.261315
$777$	0	0
$778$	16.0790	0.576460
$779$	−10.2002	−0.365459
$780$	0	0
$781$	4.41209	0.157877
$782$	−5.77961	−0.206679
$783$	0	0
$784$	13.7071	0.489538
$785$	−6.20536	−0.221479
$786$	0	0
$787$	−16.2157	−0.578028	−0.289014	−	0.957325i	$-0.593327\pi$
$787$	−16.2157	−0.578028	−0.289014	+	0.957325i	$0.593327\pi$
$788$	13.0157	0.463664
$789$	0	0
$790$	3.91124	0.139156
$791$	−11.1255	−0.395576
$792$	0	0
$793$	1.66563	0.0591482
$794$	30.5096	1.08274
$795$	0	0
$796$	−2.95663	−0.104795
$797$	−16.9400	−0.600044	−0.300022	−	0.953932i	$-0.596994\pi$
$797$	−16.9400	−0.600044	−0.300022	+	0.953932i	$0.596994\pi$
$798$	0	0
$799$	−17.1477	−0.606642
$800$	4.63634	0.163919
$801$	0	0
$802$	26.5433	0.937277
$803$	−21.7837	−0.768729
$804$	0	0
$805$	9.41829	0.331951
$806$	−1.44459	−0.0508837
$807$	0	0
$808$	3.91896	0.137869
$809$	11.7475	0.413019	0.206510	−	0.978445i	$-0.433790\pi$
$809$	11.7475	0.413019	0.206510	+	0.978445i	$0.433790\pi$
$810$	0	0
$811$	41.3075	1.45050	0.725251	−	0.688485i	$-0.241724\pi$
$811$	41.3075	1.45050	0.725251	+	0.688485i	$0.241724\pi$
$812$	4.56664	0.160258
$813$	0	0
$814$	−8.09261	−0.283646
$815$	−13.7554	−0.481832
$816$	0	0
$817$	37.0907	1.29764
$818$	−40.2002	−1.40557
$819$	0	0
$820$	−2.09932	−0.0733114
$821$	−14.0695	−0.491029	−0.245515	−	0.969393i	$-0.578957\pi$
$821$	−14.0695	−0.491029	−0.245515	+	0.969393i	$0.578957\pi$
$822$	0	0
$823$	25.0661	0.873751	0.436876	−	0.899522i	$-0.356085\pi$
$823$	25.0661	0.873751	0.436876	+	0.899522i	$0.356085\pi$
$824$	3.75915	0.130956
$825$	0	0
$826$	−54.2068	−1.88609
$827$	−52.7969	−1.83593	−0.917965	−	0.396662i	$-0.870169\pi$
$827$	−52.7969	−1.83593	−0.917965	+	0.396662i	$0.870169\pi$
$828$	0	0
$829$	−17.3437	−0.602371	−0.301186	−	0.953566i	$-0.597382\pi$
$829$	−17.3437	−0.602371	−0.301186	+	0.953566i	$0.597382\pi$
$830$	−7.01016	−0.243326
$831$	0	0
$832$	0.332889	0.0115409
$833$	23.0824	0.799758
$834$	0	0
$835$	4.28061	0.148137
$836$	−5.52367	−0.191040
$837$	0	0
$838$	0.648474	0.0224012
$839$	15.0814	0.520669	0.260335	−	0.965518i	$-0.416167\pi$
$839$	15.0814	0.520669	0.260335	+	0.965518i	$0.416167\pi$
$840$	0	0
$841$	−27.9929	−0.965272
$842$	21.3460	0.735632
$843$	0	0
$844$	8.43942	0.290497
$845$	−7.77276	−0.267391
$846$	0	0
$847$	−33.8837	−1.16426
$848$	−3.19702	−0.109786
$849$	0	0
$850$	7.80748	0.267794
$851$	−14.7334	−0.505053
$852$	0	0
$853$	−45.9297	−1.57260	−0.786302	−	0.617843i	$-0.788007\pi$
$853$	−45.9297	−1.57260	−0.786302	+	0.617843i	$0.788007\pi$
$854$	−22.7686	−0.779127
$855$	0	0
$856$	−13.6322	−0.465941
$857$	11.7131	0.400112	0.200056	−	0.979784i	$-0.435888\pi$
$857$	11.7131	0.400112	0.200056	+	0.979784i	$0.435888\pi$
$858$	0	0
$859$	1.69590	0.0578633	0.0289317	−	0.999581i	$-0.490789\pi$
$859$	1.69590	0.0578633	0.0289317	+	0.999581i	$0.490789\pi$
$860$	7.63372	0.260308
$861$	0	0
$862$	−17.4639	−0.594823
$863$	−33.6266	−1.14466	−0.572331	−	0.820023i	$-0.693961\pi$
$863$	−33.6266	−1.14466	−0.572331	+	0.820023i	$0.693961\pi$
$864$	0	0
$865$	10.3963	0.353486
$866$	8.87301	0.301517
$867$	0	0
$868$	19.7472	0.670264
$869$	12.2268	0.414767
$870$	0	0
$871$	−0.243401	−0.00824732
$872$	12.8655	0.435680
$873$	0	0
$874$	−10.0564	−0.340162
$875$	−26.4436	−0.893958
$876$	0	0
$877$	13.1145	0.442846	0.221423	−	0.975178i	$-0.428930\pi$
$877$	13.1145	0.442846	0.221423	+	0.975178i	$0.428930\pi$
$878$	8.89218	0.300096
$879$	0	0
$880$	−1.13684	−0.0383229
$881$	30.7658	1.03653	0.518263	−	0.855221i	$-0.326579\pi$
$881$	30.7658	1.03653	0.518263	+	0.855221i	$0.326579\pi$
$882$	0	0
$883$	15.9997	0.538433	0.269216	−	0.963080i	$-0.413235\pi$
$883$	15.9997	0.538433	0.269216	+	0.963080i	$0.413235\pi$
$884$	0.560578	0.0188543
$885$	0	0
$886$	−7.45500	−0.250456
$887$	−23.3155	−0.782859	−0.391430	−	0.920208i	$-0.628019\pi$
$887$	−23.3155	−0.782859	−0.391430	+	0.920208i	$0.628019\pi$
$888$	0	0
$889$	47.6714	1.59885
$890$	−5.33939	−0.178977
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−29.8365	−0.998440
$894$	0	0
$895$	5.35179	0.178891
$896$	−4.55050	−0.152022
$897$	0	0
$898$	19.7555	0.659248
$899$	4.35495	0.145246
$900$	0	0
$901$	−5.38370	−0.179357
$902$	−6.56264	−0.218512
$903$	0	0
$904$	2.44489	0.0813157
$905$	−1.75803	−0.0584388
$906$	0	0
$907$	−23.5650	−0.782463	−0.391232	−	0.920292i	$-0.627951\pi$
$907$	−23.5650	−0.782463	−0.391232	+	0.920292i	$0.627951\pi$
$908$	−16.7885	−0.557147
$909$	0	0
$910$	−0.913501	−0.0302823
$911$	−12.2454	−0.405709	−0.202854	−	0.979209i	$-0.565022\pi$
$911$	−12.2454	−0.405709	−0.202854	+	0.979209i	$0.565022\pi$
$912$	0	0
$913$	−21.9143	−0.725258
$914$	−25.8231	−0.854151
$915$	0	0
$916$	19.3526	0.639428
$917$	−62.9037	−2.07726
$918$	0	0
$919$	−3.89961	−0.128636	−0.0643181	−	0.997929i	$-0.520487\pi$
$919$	−3.89961	−0.128636	−0.0643181	+	0.997929i	$0.520487\pi$
$920$	−2.06972	−0.0682368
$921$	0	0
$922$	42.3213	1.39378
$923$	−0.779103	−0.0256445
$924$	0	0
$925$	19.9028	0.654400
$926$	−4.10285	−0.134828
$927$	0	0
$928$	−1.00355	−0.0329430
$929$	0.784472	0.0257377	0.0128688	−	0.999917i	$-0.495904\pi$
$929$	0.784472	0.0257377	0.0128688	+	0.999917i	$0.495904\pi$
$930$	0	0
$931$	40.1627	1.31628
$932$	−16.7988	−0.550262
$933$	0	0
$934$	1.40172	0.0458657
$935$	−1.91441	−0.0626080
$936$	0	0
$937$	−0.197469	−0.00645102	−0.00322551	−	0.999995i	$-0.501027\pi$
$937$	−0.197469	−0.00645102	−0.00322551	+	0.999995i	$0.501027\pi$
$938$	3.32722	0.108638
$939$	0	0
$940$	−6.14072	−0.200288
$941$	40.6552	1.32532	0.662660	−	0.748920i	$-0.269427\pi$
$941$	40.6552	1.32532	0.662660	+	0.748920i	$0.269427\pi$
$942$	0	0
$943$	−11.9479	−0.389077
$944$	11.9123	0.387711
$945$	0	0
$946$	23.8636	0.775873
$947$	−34.3853	−1.11737	−0.558686	−	0.829379i	$-0.688694\pi$
$947$	−34.3853	−1.11737	−0.558686	+	0.829379i	$0.688694\pi$
$948$	0	0
$949$	3.84664	0.124867
$950$	13.5848	0.440749
$951$	0	0
$952$	−7.66294	−0.248357
$953$	−51.8834	−1.68067	−0.840334	−	0.542069i	$-0.817641\pi$
$953$	−51.8834	−1.68067	−0.840334	+	0.542069i	$0.817641\pi$
$954$	0	0
$955$	2.28694	0.0740036
$956$	9.59882	0.310448
$957$	0	0
$958$	1.60498	0.0518544
$959$	−42.6334	−1.37671
$960$	0	0
$961$	−12.1682	−0.392522
$962$	1.42902	0.0460735
$963$	0	0
$964$	26.8255	0.863990
$965$	−0.393210	−0.0126579
$966$	0	0
$967$	30.3219	0.975087	0.487544	−	0.873099i	$-0.337893\pi$
$967$	30.3219	0.975087	0.487544	+	0.873099i	$0.337893\pi$
$968$	7.44615	0.239328
$969$	0	0
$970$	−4.38980	−0.140948
$971$	−4.91934	−0.157869	−0.0789345	−	0.996880i	$-0.525152\pi$
$971$	−4.91934	−0.157869	−0.0789345	+	0.996880i	$0.525152\pi$
$972$	0	0
$973$	−0.0290495	−0.000931285 0
$974$	1.70191	0.0545328
$975$	0	0
$976$	5.00355	0.160160
$977$	19.5573	0.625694	0.312847	−	0.949804i	$-0.398717\pi$
$977$	19.5573	0.625694	0.312847	+	0.949804i	$0.398717\pi$
$978$	0	0
$979$	−16.6914	−0.533458
$980$	8.26599	0.264047
$981$	0	0
$982$	−27.4254	−0.875181
$983$	−37.0151	−1.18060	−0.590299	−	0.807185i	$-0.700990\pi$
$983$	−37.0151	−1.18060	−0.590299	+	0.807185i	$0.700990\pi$
$984$	0	0
$985$	7.84903	0.250091
$986$	−1.68995	−0.0538189
$987$	0	0
$988$	0.975390	0.0310313
$989$	43.4460	1.38150
$990$	0	0
$991$	−27.3983	−0.870335	−0.435168	−	0.900349i	$-0.643311\pi$
$991$	−27.3983	−0.870335	−0.435168	+	0.900349i	$0.643311\pi$
$992$	−4.33956	−0.137781
$993$	0	0
$994$	10.6501	0.337801
$995$	−1.78298	−0.0565244
$996$	0	0
$997$	50.7751	1.60806	0.804032	−	0.594586i	$-0.202684\pi$
$997$	50.7751	1.60806	0.804032	+	0.594586i	$0.202684\pi$
$998$	34.5307	1.09305
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.v.1.5		7
3.2	odd	2		1338.2.a.j.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.j.1.3	✓	7	3.2	odd	2
4014.2.a.v.1.5		7	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.603045 q^{5} +4.55050 q^{7} -1.00000 q^{8} -0.603045 q^{10} -1.88517 q^{11} +0.332889 q^{13} -4.55050 q^{14} +1.00000 q^{16} +1.68398 q^{17} +2.93007 q^{19} +0.603045 q^{20} +1.88517 q^{22} +3.43212 q^{23} -4.63634 q^{25} -0.332889 q^{26} +4.55050 q^{28} +1.00355 q^{29} +4.33956 q^{31} -1.00000 q^{32} -1.68398 q^{34} +2.74416 q^{35} -4.29279 q^{37} -2.93007 q^{38} -0.603045 q^{40} -3.48120 q^{41} +12.6586 q^{43} -1.88517 q^{44} -3.43212 q^{46} -10.1829 q^{47} +13.7071 q^{49} +4.63634 q^{50} +0.332889 q^{52} -3.19702 q^{53} -1.13684 q^{55} -4.55050 q^{56} -1.00355 q^{58} +11.9123 q^{59} +5.00355 q^{61} -4.33956 q^{62} +1.00000 q^{64} +0.200747 q^{65} -0.731176 q^{67} +1.68398 q^{68} -2.74416 q^{70} -2.34043 q^{71} +11.5553 q^{73} +4.29279 q^{74} +2.93007 q^{76} -8.57845 q^{77} -6.48581 q^{79} +0.603045 q^{80} +3.48120 q^{82} +11.6246 q^{83} +1.01551 q^{85} -12.6586 q^{86} +1.88517 q^{88} +8.85405 q^{89} +1.51481 q^{91} +3.43212 q^{92} +10.1829 q^{94} +1.76696 q^{95} +7.27938 q^{97} -13.7071 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8} + 6 q^{10} + q^{11} + 8 q^{13} - 3 q^{14} + 7 q^{16} - 16 q^{17} + 2 q^{19} - 6 q^{20} - q^{22} - 8 q^{23} + 19 q^{25} - 8 q^{26} + 3 q^{28} - 4 q^{29}+ \cdots - 24 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 6 * q^5 + 3 * q^7 - 7 * q^8 + 6 * q^10 + q^11 + 8 * q^13 - 3 * q^14 + 7 * q^16 - 16 * q^17 + 2 * q^19 - 6 * q^20 - q^22 - 8 * q^23 + 19 * q^25 - 8 * q^26 + 3 * q^28 - 4 * q^29 + 11 * q^31 - 7 * q^32 + 16 * q^34 + 17 * q^37 - 2 * q^38 + 6 * q^40 - 18 * q^41 - q^43 + q^44 + 8 * q^46 - 5 * q^47 + 24 * q^49 - 19 * q^50 + 8 * q^52 - 6 * q^53 + 21 * q^55 - 3 * q^56 + 4 * q^58 + 7 * q^59 + 24 * q^61 - 11 * q^62 + 7 * q^64 - 11 * q^65 - 4 * q^67 - 16 * q^68 + 7 * q^71 + 28 * q^73 - 17 * q^74 + 2 * q^76 + 2 * q^77 - 6 * q^80 + 18 * q^82 + 2 * q^83 + 4 * q^85 + q^86 - q^88 - 5 * q^89 + 2 * q^91 - 8 * q^92 + 5 * q^94 + 14 * q^95 + 23 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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