LMFDB - Embedded newform 2034.2.a.r.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2034,2,Mod(1,2034)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2034.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2034, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2034 = 2 \cdot 3^{2} \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2034.a (trivial)

Newform invariants

comment:select newform

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

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$16.2415717711$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 226)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.90211$ of defining polynomial
Character	$\chi$	$=$	2034.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -0.273457 q^{5} +1.23607 q^{7} -1.00000 q^{8} +0.273457 q^{10} +2.35114 q^{11} -5.04029 q^{13} -1.23607 q^{14} +1.00000 q^{16} -4.47214 q^{17} +3.75621 q^{19} -0.273457 q^{20} -2.35114 q^{22} +1.32739 q^{23} -4.92522 q^{25} +5.04029 q^{26} +1.23607 q^{28} -5.31375 q^{29} -6.82328 q^{31} -1.00000 q^{32} +4.47214 q^{34} -0.338012 q^{35} +1.86067 q^{37} -3.75621 q^{38} +0.273457 q^{40} -1.10194 q^{41} +5.43945 q^{43} +2.35114 q^{44} -1.32739 q^{46} +6.02967 q^{47} -5.47214 q^{49} +4.92522 q^{50} -5.04029 q^{52} -8.49338 q^{53} -0.642937 q^{55} -1.23607 q^{56} +5.31375 q^{58} -0.243785 q^{59} -6.85765 q^{61} +6.82328 q^{62} +1.00000 q^{64} +1.37831 q^{65} -5.28408 q^{67} -4.47214 q^{68} +0.338012 q^{70} -1.08540 q^{71} +13.7638 q^{73} -1.86067 q^{74} +3.75621 q^{76} +2.90617 q^{77} -5.03186 q^{79} -0.273457 q^{80} +1.10194 q^{82} +0.919299 q^{83} +1.22294 q^{85} -5.43945 q^{86} -2.35114 q^{88} -14.0000 q^{89} -6.23015 q^{91} +1.32739 q^{92} -6.02967 q^{94} -1.02717 q^{95} +13.2917 q^{97} +5.47214 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{7} - 4 q^{8} + 4 q^{10} + 4 q^{13} + 4 q^{14} + 4 q^{16} - 6 q^{19} - 4 q^{20} + 6 q^{23} + 4 q^{25} - 4 q^{26} - 4 q^{28} - 4 q^{32} + 4 q^{35} - 8 q^{37} + 6 q^{38}+ \cdots + 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^7 - 4 * q^8 + 4 * q^10 + 4 * q^13 + 4 * q^14 + 4 * q^16 - 6 * q^19 - 4 * q^20 + 6 * q^23 + 4 * q^25 - 4 * q^26 - 4 * q^28 - 4 * q^32 + 4 * q^35 - 8 * q^37 + 6 * q^38 + 4 * q^40 - 8 * q^41 - 6 * q^43 - 6 * q^46 + 6 * q^47 - 4 * q^49 - 4 * q^50 + 4 * q^52 - 4 * q^53 - 20 * q^55 + 4 * q^56 - 22 * q^59 + 16 * q^61 + 4 * q^64 - 24 * q^65 - 18 * q^67 - 4 * q^70 - 22 * q^71 + 8 * q^74 - 6 * q^76 - 2 * q^79 - 4 * q^80 + 8 * q^82 - 12 * q^83 + 6 * q^86 - 56 * q^89 - 24 * q^91 + 6 * q^92 - 6 * q^94 + 16 * q^95 + 16 * q^97 + 4 * 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
$5$	−0.273457	−0.122294	−0.0611469	−	0.998129i	$-0.519476\pi$
$5$	−0.273457	−0.122294	−0.0611469	+	0.998129i	$0.519476\pi$
$6$	0	0
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.273457	0.0864748
$11$	2.35114	0.708896	0.354448	−	0.935076i	$-0.384669\pi$
$11$	2.35114	0.708896	0.354448	+	0.935076i	$0.384669\pi$
$12$	0	0
$13$	−5.04029	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$13$	−5.04029	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$14$	−1.23607	−0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	3.75621	0.861735	0.430867	−	0.902415i	$-0.358208\pi$
$19$	3.75621	0.861735	0.430867	+	0.902415i	$0.358208\pi$
$20$	−0.273457	−0.0611469
$21$	0	0
$22$	−2.35114	−0.501265
$23$	1.32739	0.276780	0.138390	−	0.990378i	$-0.455807\pi$
$23$	1.32739	0.276780	0.138390	+	0.990378i	$0.455807\pi$
$24$	0	0
$25$	−4.92522	−0.985044
$26$	5.04029	0.988483
$27$	0	0
$28$	1.23607	0.233595
$29$	−5.31375	−0.986739	−0.493369	−	0.869820i	$-0.664235\pi$
$29$	−5.31375	−0.986739	−0.493369	+	0.869820i	$0.664235\pi$
$30$	0	0
$31$	−6.82328	−1.22550	−0.612748	−	0.790278i	$-0.709936\pi$
$31$	−6.82328	−1.22550	−0.612748	+	0.790278i	$0.709936\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.47214	0.766965
$35$	−0.338012	−0.0571345
$36$	0	0
$37$	1.86067	0.305892	0.152946	−	0.988235i	$-0.451124\pi$
$37$	1.86067	0.305892	0.152946	+	0.988235i	$0.451124\pi$
$38$	−3.75621	−0.609339
$39$	0	0
$40$	0.273457	0.0432374
$41$	−1.10194	−0.172095	−0.0860474	−	0.996291i	$-0.527424\pi$
$41$	−1.10194	−0.172095	−0.0860474	+	0.996291i	$0.527424\pi$
$42$	0	0
$43$	5.43945	0.829508	0.414754	−	0.909934i	$-0.363868\pi$
$43$	5.43945	0.829508	0.414754	+	0.909934i	$0.363868\pi$
$44$	2.35114	0.354448
$45$	0	0
$46$	−1.32739	−0.195713
$47$	6.02967	0.879518	0.439759	−	0.898116i	$-0.355064\pi$
$47$	6.02967	0.879518	0.439759	+	0.898116i	$0.355064\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	4.92522	0.696531
$51$	0	0
$52$	−5.04029	−0.698963
$53$	−8.49338	−1.16666	−0.583328	−	0.812237i	$-0.698250\pi$
$53$	−8.49338	−1.16666	−0.583328	+	0.812237i	$0.698250\pi$
$54$	0	0
$55$	−0.642937	−0.0866936
$56$	−1.23607	−0.165177
$57$	0	0
$58$	5.31375	0.697730
$59$	−0.243785	−0.0317381	−0.0158691	−	0.999874i	$-0.505051\pi$
$59$	−0.243785	−0.0317381	−0.0158691	+	0.999874i	$0.505051\pi$
$60$	0	0
$61$	−6.85765	−0.878032	−0.439016	−	0.898479i	$-0.644673\pi$
$61$	−6.85765	−0.878032	−0.439016	+	0.898479i	$0.644673\pi$
$62$	6.82328	0.866557
$63$	0	0
$64$	1.00000	0.125000
$65$	1.37831	0.170958
$66$	0	0
$67$	−5.28408	−0.645553	−0.322777	−	0.946475i	$-0.604616\pi$
$67$	−5.28408	−0.645553	−0.322777	+	0.946475i	$0.604616\pi$
$68$	−4.47214	−0.542326
$69$	0	0
$70$	0.338012	0.0404002
$71$	−1.08540	−0.128813	−0.0644067	−	0.997924i	$-0.520515\pi$
$71$	−1.08540	−0.128813	−0.0644067	+	0.997924i	$0.520515\pi$
$72$	0	0
$73$	13.7638	1.61093	0.805467	−	0.592641i	$-0.201915\pi$
$73$	13.7638	1.61093	0.805467	+	0.592641i	$0.201915\pi$
$74$	−1.86067	−0.216298
$75$	0	0
$76$	3.75621	0.430867
$77$	2.90617	0.331189
$78$	0	0
$79$	−5.03186	−0.566129	−0.283065	−	0.959101i	$-0.591351\pi$
$79$	−5.03186	−0.566129	−0.283065	+	0.959101i	$0.591351\pi$
$80$	−0.273457	−0.0305735
$81$	0	0
$82$	1.10194	0.121689
$83$	0.919299	0.100906	0.0504531	−	0.998726i	$-0.483933\pi$
$83$	0.919299	0.100906	0.0504531	+	0.998726i	$0.483933\pi$
$84$	0	0
$85$	1.22294	0.132646
$86$	−5.43945	−0.586551
$87$	0	0
$88$	−2.35114	−0.250632
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−6.23015	−0.653097
$92$	1.32739	0.138390
$93$	0	0
$94$	−6.02967	−0.621913
$95$	−1.02717	−0.105385
$96$	0	0
$97$	13.2917	1.34957	0.674783	−	0.738016i	$-0.264237\pi$
$97$	13.2917	1.34957	0.674783	+	0.738016i	$0.264237\pi$
$98$	5.47214	0.552769
$99$	0	0
$100$	−4.92522	−0.492522
$101$	−13.1311	−1.30659	−0.653297	−	0.757102i	$-0.726615\pi$
$101$	−13.1311	−1.30659	−0.653297	+	0.757102i	$0.726615\pi$
$102$	0	0
$103$	10.3404	1.01887	0.509435	−	0.860509i	$-0.329854\pi$
$103$	10.3404	1.01887	0.509435	+	0.860509i	$0.329854\pi$
$104$	5.04029	0.494241
$105$	0	0
$106$	8.49338	0.824950
$107$	5.43945	0.525851	0.262926	−	0.964816i	$-0.415313\pi$
$107$	5.43945	0.525851	0.262926	+	0.964816i	$0.415313\pi$
$108$	0	0
$109$	2.02124	0.193600	0.0968000	−	0.995304i	$-0.469139\pi$
$109$	2.02124	0.193600	0.0968000	+	0.995304i	$0.469139\pi$
$110$	0.642937	0.0613016
$111$	0	0
$112$	1.23607	0.116797
$113$	1.00000	0.0940721
$113$	1.00000	0.0940721
$114$	0	0
$115$	−0.362985	−0.0338485
$116$	−5.31375	−0.493369
$117$	0	0
$118$	0.243785	0.0224422
$119$	−5.52786	−0.506738
$120$	0	0
$121$	−5.47214	−0.497467
$122$	6.85765	0.620862
$123$	0	0
$124$	−6.82328	−0.612748
$125$	2.71413	0.242759
$126$	0	0
$127$	−21.5468	−1.91197	−0.955985	−	0.293416i	$-0.905208\pi$
$127$	−21.5468	−1.91197	−0.955985	+	0.293416i	$0.905208\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.37831	−0.120885
$131$	−18.4449	−1.61153	−0.805767	−	0.592232i	$-0.798247\pi$
$131$	−18.4449	−1.61153	−0.805767	+	0.592232i	$0.798247\pi$
$132$	0	0
$133$	4.64294	0.402594
$134$	5.28408	0.456475
$135$	0	0
$136$	4.47214	0.383482
$137$	−13.2169	−1.12920	−0.564598	−	0.825366i	$-0.690969\pi$
$137$	−13.2169	−1.12920	−0.564598	+	0.825366i	$0.690969\pi$
$138$	0	0
$139$	−0.313039	−0.0265516	−0.0132758	−	0.999912i	$-0.504226\pi$
$139$	−0.313039	−0.0265516	−0.0132758	+	0.999912i	$0.504226\pi$
$140$	−0.338012	−0.0285672
$141$	0	0
$142$	1.08540	0.0910848
$143$	−11.8504	−0.990984
$144$	0	0
$145$	1.45309	0.120672
$146$	−13.7638	−1.13910
$147$	0	0
$148$	1.86067	0.152946
$149$	−12.5850	−1.03100	−0.515502	−	0.856888i	$-0.672395\pi$
$149$	−12.5850	−1.03100	−0.515502	+	0.856888i	$0.672395\pi$
$150$	0	0
$151$	−8.44246	−0.687038	−0.343519	−	0.939146i	$-0.611619\pi$
$151$	−8.44246	−0.687038	−0.343519	+	0.939146i	$0.611619\pi$
$152$	−3.75621	−0.304669
$153$	0	0
$154$	−2.90617	−0.234186
$155$	1.86588	0.149871
$156$	0	0
$157$	−0.997808	−0.0796337	−0.0398169	−	0.999207i	$-0.512677\pi$
$157$	−0.997808	−0.0796337	−0.0398169	+	0.999207i	$0.512677\pi$
$158$	5.03186	0.400314
$159$	0	0
$160$	0.273457	0.0216187
$161$	1.64074	0.129309
$162$	0	0
$163$	3.90398	0.305783	0.152892	−	0.988243i	$-0.451141\pi$
$163$	3.90398	0.305783	0.152892	+	0.988243i	$0.451141\pi$
$164$	−1.10194	−0.0860474
$165$	0	0
$166$	−0.919299	−0.0713514
$167$	14.2275	1.10096	0.550480	−	0.834849i	$-0.314445\pi$
$167$	14.2275	1.10096	0.550480	+	0.834849i	$0.314445\pi$
$168$	0	0
$169$	12.4046	0.954197
$170$	−1.22294	−0.0937951
$171$	0	0
$172$	5.43945	0.414754
$173$	−0.884927	−0.0672798	−0.0336399	−	0.999434i	$-0.510710\pi$
$173$	−0.884927	−0.0672798	−0.0336399	+	0.999434i	$0.510710\pi$
$174$	0	0
$175$	−6.08791	−0.460203
$176$	2.35114	0.177224
$177$	0	0
$178$	14.0000	1.04934
$179$	−15.7027	−1.17367	−0.586837	−	0.809705i	$-0.699627\pi$
$179$	−15.7027	−1.17367	−0.586837	+	0.809705i	$0.699627\pi$
$180$	0	0
$181$	−10.4501	−0.776747	−0.388374	−	0.921502i	$-0.626963\pi$
$181$	−10.4501	−0.776747	−0.388374	+	0.921502i	$0.626963\pi$
$182$	6.23015	0.461809
$183$	0	0
$184$	−1.32739	−0.0978565
$185$	−0.508813	−0.0374087
$186$	0	0
$187$	−10.5146	−0.768905
$188$	6.02967	0.439759
$189$	0	0
$190$	1.02717	0.0745184
$191$	14.9833	1.08416	0.542078	−	0.840328i	$-0.317638\pi$
$191$	14.9833	1.08416	0.542078	+	0.840328i	$0.317638\pi$
$192$	0	0
$193$	3.49119	0.251301	0.125651	−	0.992075i	$-0.459898\pi$
$193$	3.49119	0.251301	0.125651	+	0.992075i	$0.459898\pi$
$194$	−13.2917	−0.954287
$195$	0	0
$196$	−5.47214	−0.390867
$197$	12.0279	0.856951	0.428475	−	0.903553i	$-0.359051\pi$
$197$	12.0279	0.856951	0.428475	+	0.903553i	$0.359051\pi$
$198$	0	0
$199$	−5.44027	−0.385651	−0.192825	−	0.981233i	$-0.561765\pi$
$199$	−5.44027	−0.385651	−0.192825	+	0.981233i	$0.561765\pi$
$200$	4.92522	0.348266
$201$	0	0
$202$	13.1311	0.923901
$203$	−6.56816	−0.460994
$204$	0	0
$205$	0.301335	0.0210461
$206$	−10.3404	−0.720450
$207$	0	0
$208$	−5.04029	−0.349482
$209$	8.83139	0.610880
$210$	0	0
$211$	1.83860	0.126574	0.0632872	−	0.997995i	$-0.479842\pi$
$211$	1.83860	0.126574	0.0632872	+	0.997995i	$0.479842\pi$
$212$	−8.49338	−0.583328
$213$	0	0
$214$	−5.43945	−0.371833
$215$	−1.48746	−0.101444
$216$	0	0
$217$	−8.43403	−0.572540
$218$	−2.02124	−0.136896
$219$	0	0
$220$	−0.642937	−0.0433468
$221$	22.5409	1.51626
$222$	0	0
$223$	−20.1103	−1.34668	−0.673341	−	0.739332i	$-0.735142\pi$
$223$	−20.1103	−1.34668	−0.673341	+	0.739332i	$0.735142\pi$
$224$	−1.23607	−0.0825883
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	15.1744	1.00716	0.503581	−	0.863948i	$-0.332016\pi$
$227$	15.1744	1.00716	0.503581	+	0.863948i	$0.332016\pi$
$228$	0	0
$229$	−7.08361	−0.468098	−0.234049	−	0.972225i	$-0.575198\pi$
$229$	−7.08361	−0.468098	−0.234049	+	0.972225i	$0.575198\pi$
$230$	0.362985	0.0239345
$231$	0	0
$232$	5.31375	0.348865
$233$	19.8848	1.30270	0.651349	−	0.758779i	$-0.274204\pi$
$233$	19.8848	1.30270	0.651349	+	0.758779i	$0.274204\pi$
$234$	0	0
$235$	−1.64886	−0.107560
$236$	−0.243785	−0.0158691
$237$	0	0
$238$	5.52786	0.358318
$239$	5.09591	0.329627	0.164813	−	0.986325i	$-0.447298\pi$
$239$	5.09591	0.329627	0.164813	+	0.986325i	$0.447298\pi$
$240$	0	0
$241$	−3.01532	−0.194234	−0.0971170	−	0.995273i	$-0.530962\pi$
$241$	−3.01532	−0.194234	−0.0971170	+	0.995273i	$0.530962\pi$
$242$	5.47214	0.351762
$243$	0	0
$244$	−6.85765	−0.439016
$245$	1.49640	0.0956013
$246$	0	0
$247$	−18.9324	−1.20464
$248$	6.82328	0.433279
$249$	0	0
$250$	−2.71413	−0.171656
$251$	−26.9288	−1.69973	−0.849867	−	0.526998i	$-0.823318\pi$
$251$	−26.9288	−1.69973	−0.849867	+	0.526998i	$0.823318\pi$
$252$	0	0
$253$	3.12088	0.196208
$254$	21.5468	1.35197
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.2741	−1.26466	−0.632330	−	0.774699i	$-0.717901\pi$
$257$	−20.2741	−1.26466	−0.632330	+	0.774699i	$0.717901\pi$
$258$	0	0
$259$	2.29991	0.142909
$260$	1.37831	0.0854789
$261$	0	0
$262$	18.4449	1.13953
$263$	9.93365	0.612535	0.306268	−	0.951945i	$-0.400920\pi$
$263$	9.93365	0.612535	0.306268	+	0.951945i	$0.400920\pi$
$264$	0	0
$265$	2.32258	0.142675
$266$	−4.64294	−0.284677
$267$	0	0
$268$	−5.28408	−0.322777
$269$	−13.5924	−0.828744	−0.414372	−	0.910108i	$-0.635999\pi$
$269$	−13.5924	−0.828744	−0.414372	+	0.910108i	$0.635999\pi$
$270$	0	0
$271$	−29.1506	−1.77077	−0.885385	−	0.464858i	$-0.846105\pi$
$271$	−29.1506	−1.77077	−0.885385	+	0.464858i	$0.846105\pi$
$272$	−4.47214	−0.271163
$273$	0	0
$274$	13.2169	0.798462
$275$	−11.5799	−0.698294
$276$	0	0
$277$	−29.8825	−1.79547	−0.897733	−	0.440540i	$-0.854787\pi$
$277$	−29.8825	−1.79547	−0.897733	+	0.440540i	$0.854787\pi$
$278$	0.313039	0.0187748
$279$	0	0
$280$	0.338012	0.0202001
$281$	21.2432	1.26726	0.633630	−	0.773636i	$-0.281564\pi$
$281$	21.2432	1.26726	0.633630	+	0.773636i	$0.281564\pi$
$282$	0	0
$283$	−13.9061	−0.826629	−0.413315	−	0.910588i	$-0.635629\pi$
$283$	−13.9061	−0.826629	−0.413315	+	0.910588i	$0.635629\pi$
$284$	−1.08540	−0.0644067
$285$	0	0
$286$	11.8504	0.700731
$287$	−1.36208	−0.0804009
$288$	0	0
$289$	3.00000	0.176471
$290$	−1.45309	−0.0853281
$291$	0	0
$292$	13.7638	0.805467
$293$	4.27346	0.249658	0.124829	−	0.992178i	$-0.460162\pi$
$293$	4.27346	0.249658	0.124829	+	0.992178i	$0.460162\pi$
$294$	0	0
$295$	0.0666648	0.00388138
$296$	−1.86067	−0.108149
$297$	0	0
$298$	12.5850	0.729030
$299$	−6.69044	−0.386918
$300$	0	0
$301$	6.72353	0.387538
$302$	8.44246	0.485809
$303$	0	0
$304$	3.75621	0.215434
$305$	1.87528	0.107378
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	2.90617	0.165594
$309$	0	0
$310$	−1.86588	−0.105975
$311$	−9.69636	−0.549830	−0.274915	−	0.961469i	$-0.588650\pi$
$311$	−9.69636	−0.549830	−0.274915	+	0.961469i	$0.588650\pi$
$312$	0	0
$313$	18.5550	1.04879	0.524396	−	0.851474i	$-0.324291\pi$
$313$	18.5550	1.04879	0.524396	+	0.851474i	$0.324291\pi$
$314$	0.997808	0.0563095
$315$	0	0
$316$	−5.03186	−0.283065
$317$	10.8364	0.608633	0.304317	−	0.952571i	$-0.401572\pi$
$317$	10.8364	0.608633	0.304317	+	0.952571i	$0.401572\pi$
$318$	0	0
$319$	−12.4934	−0.699495
$320$	−0.273457	−0.0152867
$321$	0	0
$322$	−1.64074	−0.0914351
$323$	−16.7983	−0.934683
$324$	0	0
$325$	24.8246	1.37702
$326$	−3.90398	−0.216221
$327$	0	0
$328$	1.10194	0.0608447
$329$	7.45309	0.410902
$330$	0	0
$331$	23.3914	1.28571	0.642855	−	0.765988i	$-0.277750\pi$
$331$	23.3914	1.28571	0.642855	+	0.765988i	$0.277750\pi$
$332$	0.919299	0.0504531
$333$	0	0
$334$	−14.2275	−0.778496
$335$	1.44497	0.0789472
$336$	0	0
$337$	−30.6939	−1.67201	−0.836003	−	0.548725i	$-0.815113\pi$
$337$	−30.6939	−1.67201	−0.836003	+	0.548725i	$0.815113\pi$
$338$	−12.4046	−0.674719
$339$	0	0
$340$	1.22294	0.0663232
$341$	−16.0425	−0.868749
$342$	0	0
$343$	−15.4164	−0.832408
$344$	−5.43945	−0.293275
$345$	0	0
$346$	0.884927	0.0475740
$347$	−3.41641	−0.183402	−0.0917012	−	0.995787i	$-0.529230\pi$
$347$	−3.41641	−0.183402	−0.0917012	+	0.995787i	$0.529230\pi$
$348$	0	0
$349$	36.2851	1.94230	0.971148	−	0.238478i	$-0.0766486\pi$
$349$	36.2851	1.94230	0.971148	+	0.238478i	$0.0766486\pi$
$350$	6.08791	0.325412
$351$	0	0
$352$	−2.35114	−0.125316
$353$	16.4684	0.876525	0.438262	−	0.898847i	$-0.355594\pi$
$353$	16.4684	0.876525	0.438262	+	0.898847i	$0.355594\pi$
$354$	0	0
$355$	0.296811	0.0157531
$356$	−14.0000	−0.741999
$357$	0	0
$358$	15.7027	0.829912
$359$	−24.1197	−1.27299	−0.636493	−	0.771282i	$-0.719616\pi$
$359$	−24.1197	−1.27299	−0.636493	+	0.771282i	$0.719616\pi$
$360$	0	0
$361$	−4.89085	−0.257413
$362$	10.4501	0.549243
$363$	0	0
$364$	−6.23015	−0.326548
$365$	−3.76382	−0.197007
$366$	0	0
$367$	34.4413	1.79782	0.898910	−	0.438134i	$-0.144360\pi$
$367$	34.4413	1.79782	0.898910	+	0.438134i	$0.144360\pi$
$368$	1.32739	0.0691950
$369$	0	0
$370$	0.508813	0.0264519
$371$	−10.4984	−0.545049
$372$	0	0
$373$	26.3540	1.36456	0.682280	−	0.731091i	$-0.260988\pi$
$373$	26.3540	1.36456	0.682280	+	0.731091i	$0.260988\pi$
$374$	10.5146	0.543698
$375$	0	0
$376$	−6.02967	−0.310957
$377$	26.7829	1.37939
$378$	0	0
$379$	28.5509	1.46656	0.733281	−	0.679925i	$-0.237988\pi$
$379$	28.5509	1.46656	0.733281	+	0.679925i	$0.237988\pi$
$380$	−1.02717	−0.0526925
$381$	0	0
$382$	−14.9833	−0.766615
$383$	−22.7425	−1.16209	−0.581043	−	0.813873i	$-0.697355\pi$
$383$	−22.7425	−1.16209	−0.581043	+	0.813873i	$0.697355\pi$
$384$	0	0
$385$	−0.794714	−0.0405024
$386$	−3.49119	−0.177697
$387$	0	0
$388$	13.2917	0.674783
$389$	36.7249	1.86203	0.931014	−	0.364982i	$-0.118925\pi$
$389$	36.7249	1.86203	0.931014	+	0.364982i	$0.118925\pi$
$390$	0	0
$391$	−5.93627	−0.300210
$392$	5.47214	0.276385
$393$	0	0
$394$	−12.0279	−0.605956
$395$	1.37600	0.0692341
$396$	0	0
$397$	−14.7183	−0.738691	−0.369346	−	0.929292i	$-0.620418\pi$
$397$	−14.7183	−0.738691	−0.369346	+	0.929292i	$0.620418\pi$
$398$	5.44027	0.272696
$399$	0	0
$400$	−4.92522	−0.246261
$401$	6.79702	0.339427	0.169713	−	0.985493i	$-0.445716\pi$
$401$	6.79702	0.339427	0.169713	+	0.985493i	$0.445716\pi$
$402$	0	0
$403$	34.3913	1.71315
$404$	−13.1311	−0.653297
$405$	0	0
$406$	6.56816	0.325972
$407$	4.37469	0.216845
$408$	0	0
$409$	9.91919	0.490472	0.245236	−	0.969463i	$-0.421135\pi$
$409$	9.91919	0.490472	0.245236	+	0.969463i	$0.421135\pi$
$410$	−0.301335	−0.0148819
$411$	0	0
$412$	10.3404	0.509435
$413$	−0.301335	−0.0148277
$414$	0	0
$415$	−0.251389	−0.0123402
$416$	5.04029	0.247121
$417$	0	0
$418$	−8.83139	−0.431957
$419$	14.8498	0.725461	0.362731	−	0.931894i	$-0.381845\pi$
$419$	14.8498	0.725461	0.362731	+	0.931894i	$0.381845\pi$
$420$	0	0
$421$	3.49119	0.170150	0.0850750	−	0.996375i	$-0.472887\pi$
$421$	3.49119	0.170150	0.0850750	+	0.996375i	$0.472887\pi$
$422$	−1.83860	−0.0895016
$423$	0	0
$424$	8.49338	0.412475
$425$	22.0263	1.06843
$426$	0	0
$427$	−8.47652	−0.410208
$428$	5.43945	0.262926
$429$	0	0
$430$	1.48746	0.0717316
$431$	38.2963	1.84467	0.922333	−	0.386395i	$-0.126280\pi$
$431$	38.2963	1.84467	0.922333	+	0.386395i	$0.126280\pi$
$432$	0	0
$433$	−39.4104	−1.89394	−0.946971	−	0.321320i	$-0.895874\pi$
$433$	−39.4104	−1.89394	−0.946971	+	0.321320i	$0.895874\pi$
$434$	8.43403	0.404847
$435$	0	0
$436$	2.02124	0.0968000
$437$	4.98596	0.238511
$438$	0	0
$439$	32.1518	1.53452	0.767260	−	0.641336i	$-0.221619\pi$
$439$	32.1518	1.53452	0.767260	+	0.641336i	$0.221619\pi$
$440$	0.642937	0.0306508
$441$	0	0
$442$	−22.5409	−1.07216
$443$	40.0402	1.90237	0.951183	−	0.308627i	$-0.0998694\pi$
$443$	40.0402	1.90237	0.951183	+	0.308627i	$0.0998694\pi$
$444$	0	0
$445$	3.82840	0.181484
$446$	20.1103	0.952248
$447$	0	0
$448$	1.23607	0.0583987
$449$	−4.86368	−0.229531	−0.114766	−	0.993393i	$-0.536612\pi$
$449$	−4.86368	−0.229531	−0.114766	+	0.993393i	$0.536612\pi$
$450$	0	0
$451$	−2.59083	−0.121997
$452$	1.00000	0.0470360
$453$	0	0
$454$	−15.1744	−0.712171
$455$	1.70368	0.0798697
$456$	0	0
$457$	8.12472	0.380059	0.190029	−	0.981778i	$-0.439142\pi$
$457$	8.12472	0.380059	0.190029	+	0.981778i	$0.439142\pi$
$458$	7.08361	0.330995
$459$	0	0
$460$	−0.362985	−0.0169243
$461$	−5.57177	−0.259503	−0.129752	−	0.991547i	$-0.541418\pi$
$461$	−5.57177	−0.259503	−0.129752	+	0.991547i	$0.541418\pi$
$462$	0	0
$463$	−26.7235	−1.24195	−0.620974	−	0.783831i	$-0.713263\pi$
$463$	−26.7235	−1.24195	−0.620974	+	0.783831i	$0.713263\pi$
$464$	−5.31375	−0.246685
$465$	0	0
$466$	−19.8848	−0.921146
$467$	28.4469	1.31637	0.658184	−	0.752857i	$-0.271325\pi$
$467$	28.4469	1.31637	0.658184	+	0.752857i	$0.271325\pi$
$468$	0	0
$469$	−6.53148	−0.301596
$470$	1.64886	0.0760562
$471$	0	0
$472$	0.243785	0.0112211
$473$	12.7889	0.588034
$474$	0	0
$475$	−18.5002	−0.848847
$476$	−5.52786	−0.253369
$477$	0	0
$478$	−5.09591	−0.233081
$479$	−5.05812	−0.231112	−0.115556	−	0.993301i	$-0.536865\pi$
$479$	−5.05812	−0.231112	−0.115556	+	0.993301i	$0.536865\pi$
$480$	0	0
$481$	−9.37831	−0.427614
$482$	3.01532	0.137344
$483$	0	0
$484$	−5.47214	−0.248733
$485$	−3.63471	−0.165044
$486$	0	0
$487$	−11.1660	−0.505979	−0.252990	−	0.967469i	$-0.581414\pi$
$487$	−11.1660	−0.505979	−0.252990	+	0.967469i	$0.581414\pi$
$488$	6.85765	0.310431
$489$	0	0
$490$	−1.49640	−0.0676003
$491$	32.1169	1.44942	0.724708	−	0.689057i	$-0.241975\pi$
$491$	32.1169	1.44942	0.724708	+	0.689057i	$0.241975\pi$
$492$	0	0
$493$	23.7638	1.07027
$494$	18.9324	0.851810
$495$	0	0
$496$	−6.82328	−0.306374
$497$	−1.34163	−0.0601803
$498$	0	0
$499$	36.9671	1.65487	0.827437	−	0.561559i	$-0.189798\pi$
$499$	36.9671	1.65487	0.827437	+	0.561559i	$0.189798\pi$
$500$	2.71413	0.121379
$501$	0	0
$502$	26.9288	1.20189
$503$	13.7793	0.614387	0.307193	−	0.951647i	$-0.400610\pi$
$503$	13.7793	0.614387	0.307193	+	0.951647i	$0.400610\pi$
$504$	0	0
$505$	3.59080	0.159788
$506$	−3.12088	−0.138740
$507$	0	0
$508$	−21.5468	−0.955985
$509$	−13.6772	−0.606231	−0.303116	−	0.952954i	$-0.598027\pi$
$509$	−13.6772	−0.606231	−0.303116	+	0.952954i	$0.598027\pi$
$510$	0	0
$511$	17.0130	0.752612
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	20.2741	0.894250
$515$	−2.82766	−0.124602
$516$	0	0
$517$	14.1766	0.623487
$518$	−2.29991	−0.101052
$519$	0	0
$520$	−1.37831	−0.0604427
$521$	−30.2360	−1.32466	−0.662331	−	0.749212i	$-0.730433\pi$
$521$	−30.2360	−1.32466	−0.662331	+	0.749212i	$0.730433\pi$
$522$	0	0
$523$	25.1287	1.09880	0.549401	−	0.835559i	$-0.314856\pi$
$523$	25.1287	1.09880	0.549401	+	0.835559i	$0.314856\pi$
$524$	−18.4449	−0.805767
$525$	0	0
$526$	−9.93365	−0.433128
$527$	30.5146	1.32924
$528$	0	0
$529$	−21.2380	−0.923393
$530$	−2.32258	−0.100886
$531$	0	0
$532$	4.64294	0.201297
$533$	5.55412	0.240576
$534$	0	0
$535$	−1.48746	−0.0643084
$536$	5.28408	0.228237
$537$	0	0
$538$	13.5924	0.586011
$539$	−12.8658	−0.554168
$540$	0	0
$541$	43.1046	1.85321	0.926606	−	0.376033i	$-0.122712\pi$
$541$	43.1046	1.85321	0.926606	+	0.376033i	$0.122712\pi$
$542$	29.1506	1.25212
$543$	0	0
$544$	4.47214	0.191741
$545$	−0.552724	−0.0236761
$546$	0	0
$547$	−19.3217	−0.826135	−0.413067	−	0.910700i	$-0.635543\pi$
$547$	−19.3217	−0.826135	−0.413067	+	0.910700i	$0.635543\pi$
$548$	−13.2169	−0.564598
$549$	0	0
$550$	11.5799	0.493768
$551$	−19.9596	−0.850307
$552$	0	0
$553$	−6.21973	−0.264490
$554$	29.8825	1.26959
$555$	0	0
$556$	−0.313039	−0.0132758
$557$	6.89677	0.292226	0.146113	−	0.989268i	$-0.453324\pi$
$557$	6.89677	0.292226	0.146113	+	0.989268i	$0.453324\pi$
$558$	0	0
$559$	−27.4164	−1.15959
$560$	−0.338012	−0.0142836
$561$	0	0
$562$	−21.2432	−0.896089
$563$	−1.36208	−0.0574047	−0.0287024	−	0.999588i	$-0.509138\pi$
$563$	−1.36208	−0.0574047	−0.0287024	+	0.999588i	$0.509138\pi$
$564$	0	0
$565$	−0.273457	−0.0115044
$566$	13.9061	0.584515
$567$	0	0
$568$	1.08540	0.0455424
$569$	−23.9633	−1.00459	−0.502297	−	0.864695i	$-0.667512\pi$
$569$	−23.9633	−1.00459	−0.502297	+	0.864695i	$0.667512\pi$
$570$	0	0
$571$	25.4028	1.06307	0.531536	−	0.847035i	$-0.321615\pi$
$571$	25.4028	1.06307	0.531536	+	0.847035i	$0.321615\pi$
$572$	−11.8504	−0.495492
$573$	0	0
$574$	1.36208	0.0568520
$575$	−6.53769	−0.272641
$576$	0	0
$577$	32.6333	1.35854	0.679271	−	0.733887i	$-0.262296\pi$
$577$	32.6333	1.35854	0.679271	+	0.733887i	$0.262296\pi$
$578$	−3.00000	−0.124784
$579$	0	0
$580$	1.45309	0.0603361
$581$	1.13632	0.0471423
$582$	0	0
$583$	−19.9691	−0.827037
$584$	−13.7638	−0.569551
$585$	0	0
$586$	−4.27346	−0.176535
$587$	−14.2953	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$587$	−14.2953	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$588$	0	0
$589$	−25.6297	−1.05605
$590$	−0.0666648	−0.00274455
$591$	0	0
$592$	1.86067	0.0764729
$593$	−4.34741	−0.178527	−0.0892634	−	0.996008i	$-0.528451\pi$
$593$	−4.34741	−0.178527	−0.0892634	+	0.996008i	$0.528451\pi$
$594$	0	0
$595$	1.51164	0.0619710
$596$	−12.5850	−0.515502
$597$	0	0
$598$	6.69044	0.273592
$599$	−18.1776	−0.742716	−0.371358	−	0.928490i	$-0.621108\pi$
$599$	−18.1776	−0.742716	−0.371358	+	0.928490i	$0.621108\pi$
$600$	0	0
$601$	25.6145	1.04484	0.522418	−	0.852689i	$-0.325030\pi$
$601$	25.6145	1.04484	0.522418	+	0.852689i	$0.325030\pi$
$602$	−6.72353	−0.274030
$603$	0	0
$604$	−8.44246	−0.343519
$605$	1.49640	0.0608372
$606$	0	0
$607$	−13.2849	−0.539218	−0.269609	−	0.962970i	$-0.586894\pi$
$607$	−13.2849	−0.539218	−0.269609	+	0.962970i	$0.586894\pi$
$608$	−3.75621	−0.152335
$609$	0	0
$610$	−1.87528	−0.0759277
$611$	−30.3913	−1.22950
$612$	0	0
$613$	22.3871	0.904208	0.452104	−	0.891965i	$-0.350674\pi$
$613$	22.3871	0.904208	0.452104	+	0.891965i	$0.350674\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−2.90617	−0.117093
$617$	21.4201	0.862342	0.431171	−	0.902270i	$-0.358101\pi$
$617$	21.4201	0.862342	0.431171	+	0.902270i	$0.358101\pi$
$618$	0	0
$619$	6.03050	0.242386	0.121193	−	0.992629i	$-0.461328\pi$
$619$	6.03050	0.242386	0.121193	+	0.992629i	$0.461328\pi$
$620$	1.86588	0.0749354
$621$	0	0
$622$	9.69636	0.388789
$623$	−17.3050	−0.693308
$624$	0	0
$625$	23.8839	0.955356
$626$	−18.5550	−0.741608
$627$	0	0
$628$	−0.997808	−0.0398169
$629$	−8.32115	−0.331786
$630$	0	0
$631$	6.17923	0.245991	0.122996	−	0.992407i	$-0.460750\pi$
$631$	6.17923	0.245991	0.122996	+	0.992407i	$0.460750\pi$
$632$	5.03186	0.200157
$633$	0	0
$634$	−10.8364	−0.430369
$635$	5.89213	0.233822
$636$	0	0
$637$	27.5812	1.09281
$638$	12.4934	0.494618
$639$	0	0
$640$	0.273457	0.0108094
$641$	14.3549	0.566983	0.283492	−	0.958975i	$-0.408507\pi$
$641$	14.3549	0.566983	0.283492	+	0.958975i	$0.408507\pi$
$642$	0	0
$643$	−38.5024	−1.51839	−0.759193	−	0.650865i	$-0.774406\pi$
$643$	−38.5024	−1.51839	−0.759193	+	0.650865i	$0.774406\pi$
$644$	1.64074	0.0646544
$645$	0	0
$646$	16.7983	0.660920
$647$	6.76601	0.265999	0.133000	−	0.991116i	$-0.457539\pi$
$647$	6.76601	0.265999	0.133000	+	0.991116i	$0.457539\pi$
$648$	0	0
$649$	−0.573173	−0.0224990
$650$	−24.8246	−0.973699
$651$	0	0
$652$	3.90398	0.152892
$653$	29.5718	1.15723	0.578616	−	0.815600i	$-0.303593\pi$
$653$	29.5718	1.15723	0.578616	+	0.815600i	$0.303593\pi$
$654$	0	0
$655$	5.04388	0.197081
$656$	−1.10194	−0.0430237
$657$	0	0
$658$	−7.45309	−0.290552
$659$	10.1953	0.397151	0.198576	−	0.980086i	$-0.436368\pi$
$659$	10.1953	0.397151	0.198576	+	0.980086i	$0.436368\pi$
$660$	0	0
$661$	−9.35624	−0.363915	−0.181958	−	0.983306i	$-0.558243\pi$
$661$	−9.35624	−0.363915	−0.181958	+	0.983306i	$0.558243\pi$
$662$	−23.3914	−0.909134
$663$	0	0
$664$	−0.919299	−0.0356757
$665$	−1.26965	−0.0492348
$666$	0	0
$667$	−7.05342	−0.273110
$668$	14.2275	0.550480
$669$	0	0
$670$	−1.44497	−0.0558241
$671$	−16.1233	−0.622433
$672$	0	0
$673$	−28.4280	−1.09582	−0.547909	−	0.836538i	$-0.684576\pi$
$673$	−28.4280	−1.09582	−0.547909	+	0.836538i	$0.684576\pi$
$674$	30.6939	1.18229
$675$	0	0
$676$	12.4046	0.477099
$677$	−26.5453	−1.02022	−0.510109	−	0.860110i	$-0.670395\pi$
$677$	−26.5453	−1.02022	−0.510109	+	0.860110i	$0.670395\pi$
$678$	0	0
$679$	16.4294	0.630503
$680$	−1.22294	−0.0468976
$681$	0	0
$682$	16.0425	0.614299
$683$	45.0433	1.72353	0.861767	−	0.507305i	$-0.169358\pi$
$683$	45.0433	1.72353	0.861767	+	0.507305i	$0.169358\pi$
$684$	0	0
$685$	3.61426	0.138094
$686$	15.4164	0.588601
$687$	0	0
$688$	5.43945	0.207377
$689$	42.8091	1.63090
$690$	0	0
$691$	18.9214	0.719803	0.359902	−	0.932990i	$-0.382810\pi$
$691$	18.9214	0.719803	0.359902	+	0.932990i	$0.382810\pi$
$692$	−0.884927	−0.0336399
$693$	0	0
$694$	3.41641	0.129685
$695$	0.0856029	0.00324710
$696$	0	0
$697$	4.92804	0.186663
$698$	−36.2851	−1.37341
$699$	0	0
$700$	−6.08791	−0.230101
$701$	29.5285	1.11527	0.557637	−	0.830085i	$-0.311708\pi$
$701$	29.5285	1.11527	0.557637	+	0.830085i	$0.311708\pi$
$702$	0	0
$703$	6.98906	0.263598
$704$	2.35114	0.0886120
$705$	0	0
$706$	−16.4684	−0.619797
$707$	−16.2309	−0.610427
$708$	0	0
$709$	37.3085	1.40115	0.700576	−	0.713578i	$-0.252927\pi$
$709$	37.3085	1.40115	0.700576	+	0.713578i	$0.252927\pi$
$710$	−0.296811	−0.0111391
$711$	0	0
$712$	14.0000	0.524672
$713$	−9.05715	−0.339193
$714$	0	0
$715$	3.24059	0.121191
$716$	−15.7027	−0.586837
$717$	0	0
$718$	24.1197	0.900138
$719$	−3.72967	−0.139093	−0.0695467	−	0.997579i	$-0.522155\pi$
$719$	−3.72967	−0.139093	−0.0695467	+	0.997579i	$0.522155\pi$
$720$	0	0
$721$	12.7814	0.476006
$722$	4.89085	0.182019
$723$	0	0
$724$	−10.4501	−0.388374
$725$	26.1714	0.971981
$726$	0	0
$727$	−50.7068	−1.88061	−0.940306	−	0.340331i	$-0.889461\pi$
$727$	−50.7068	−1.88061	−0.940306	+	0.340331i	$0.889461\pi$
$728$	6.23015	0.230905
$729$	0	0
$730$	3.76382	0.139305
$731$	−24.3259	−0.899727
$732$	0	0
$733$	10.5036	0.387959	0.193980	−	0.981006i	$-0.437860\pi$
$733$	10.5036	0.387959	0.193980	+	0.981006i	$0.437860\pi$
$734$	−34.4413	−1.27125
$735$	0	0
$736$	−1.32739	−0.0489283
$737$	−12.4236	−0.457630
$738$	0	0
$739$	36.9036	1.35752	0.678761	−	0.734359i	$-0.262517\pi$
$739$	36.9036	1.35752	0.678761	+	0.734359i	$0.262517\pi$
$740$	−0.508813	−0.0187043
$741$	0	0
$742$	10.4984	0.385408
$743$	−27.4306	−1.00633	−0.503166	−	0.864190i	$-0.667832\pi$
$743$	−27.4306	−1.00633	−0.503166	+	0.864190i	$0.667832\pi$
$744$	0	0
$745$	3.44147	0.126086
$746$	−26.3540	−0.964890
$747$	0	0
$748$	−10.5146	−0.384453
$749$	6.72353	0.245672
$750$	0	0
$751$	−15.0962	−0.550869	−0.275435	−	0.961320i	$-0.588822\pi$
$751$	−15.0962	−0.550869	−0.275435	+	0.961320i	$0.588822\pi$
$752$	6.02967	0.219880
$753$	0	0
$754$	−26.7829	−0.975375
$755$	2.30865	0.0840205
$756$	0	0
$757$	0.208303	0.00757092	0.00378546	−	0.999993i	$-0.498795\pi$
$757$	0.208303	0.00757092	0.00378546	+	0.999993i	$0.498795\pi$
$758$	−28.5509	−1.03702
$759$	0	0
$760$	1.02717	0.0372592
$761$	41.3758	1.49987	0.749935	−	0.661511i	$-0.230085\pi$
$761$	41.3758	1.49987	0.749935	+	0.661511i	$0.230085\pi$
$762$	0	0
$763$	2.49839	0.0904479
$764$	14.9833	0.542078
$765$	0	0
$766$	22.7425	0.821719
$767$	1.22875	0.0443675
$768$	0	0
$769$	−45.5051	−1.64096	−0.820478	−	0.571678i	$-0.806293\pi$
$769$	−45.5051	−1.64096	−0.820478	+	0.571678i	$0.806293\pi$
$770$	0.794714	0.0286395
$771$	0	0
$772$	3.49119	0.125651
$773$	−46.8456	−1.68492	−0.842459	−	0.538760i	$-0.818893\pi$
$773$	−46.8456	−1.68492	−0.842459	+	0.538760i	$0.818893\pi$
$774$	0	0
$775$	33.6061	1.20717
$776$	−13.2917	−0.477144
$777$	0	0
$778$	−36.7249	−1.31665
$779$	−4.13914	−0.148300
$780$	0	0
$781$	−2.55193	−0.0913152
$782$	5.93627	0.212281
$783$	0	0
$784$	−5.47214	−0.195433
$785$	0.272858	0.00973872
$786$	0	0
$787$	−48.3311	−1.72282	−0.861409	−	0.507912i	$-0.830417\pi$
$787$	−48.3311	−1.72282	−0.861409	+	0.507912i	$0.830417\pi$
$788$	12.0279	0.428475
$789$	0	0
$790$	−1.37600	−0.0489559
$791$	1.23607	0.0439495
$792$	0	0
$793$	34.5646	1.22742
$794$	14.7183	0.522333
$795$	0	0
$796$	−5.44027	−0.192825
$797$	−34.9772	−1.23895	−0.619477	−	0.785015i	$-0.712655\pi$
$797$	−34.9772	−1.23895	−0.619477	+	0.785015i	$0.712655\pi$
$798$	0	0
$799$	−26.9655	−0.953971
$800$	4.92522	0.174133
$801$	0	0
$802$	−6.79702	−0.240011
$803$	32.3607	1.14198
$804$	0	0
$805$	−0.448674	−0.0158137
$806$	−34.3913	−1.21138
$807$	0	0
$808$	13.1311	0.461951
$809$	−47.8115	−1.68096	−0.840481	−	0.541841i	$-0.817727\pi$
$809$	−47.8115	−1.68096	−0.840481	+	0.541841i	$0.817727\pi$
$810$	0	0
$811$	−12.7598	−0.448058	−0.224029	−	0.974582i	$-0.571921\pi$
$811$	−12.7598	−0.448058	−0.224029	+	0.974582i	$0.571921\pi$
$812$	−6.56816	−0.230497
$813$	0	0
$814$	−4.37469	−0.153333
$815$	−1.06757	−0.0373954
$816$	0	0
$817$	20.4317	0.714816
$818$	−9.91919	−0.346816
$819$	0	0
$820$	0.301335	0.0105231
$821$	−43.5657	−1.52045	−0.760227	−	0.649657i	$-0.774912\pi$
$821$	−43.5657	−1.52045	−0.760227	+	0.649657i	$0.774912\pi$
$822$	0	0
$823$	−1.44289	−0.0502960	−0.0251480	−	0.999684i	$-0.508006\pi$
$823$	−1.44289	−0.0502960	−0.0251480	+	0.999684i	$0.508006\pi$
$824$	−10.3404	−0.360225
$825$	0	0
$826$	0.301335	0.0104848
$827$	−28.2397	−0.981990	−0.490995	−	0.871162i	$-0.663367\pi$
$827$	−28.2397	−0.981990	−0.490995	+	0.871162i	$0.663367\pi$
$828$	0	0
$829$	−6.33036	−0.219862	−0.109931	−	0.993939i	$-0.535063\pi$
$829$	−6.33036	−0.219862	−0.109931	+	0.993939i	$0.535063\pi$
$830$	0.251389	0.00872585
$831$	0	0
$832$	−5.04029	−0.174741
$833$	24.4721	0.847909
$834$	0	0
$835$	−3.89062	−0.134641
$836$	8.83139	0.305440
$837$	0	0
$838$	−14.8498	−0.512978
$839$	52.6519	1.81775	0.908873	−	0.417072i	$-0.136944\pi$
$839$	52.6519	1.81775	0.908873	+	0.417072i	$0.136944\pi$
$840$	0	0
$841$	−0.764045	−0.0263464
$842$	−3.49119	−0.120314
$843$	0	0
$844$	1.83860	0.0632872
$845$	−3.39212	−0.116693
$846$	0	0
$847$	−6.76393	−0.232411
$848$	−8.49338	−0.291664
$849$	0	0
$850$	−22.0263	−0.755494
$851$	2.46983	0.0846647
$852$	0	0
$853$	21.1090	0.722760	0.361380	−	0.932419i	$-0.382306\pi$
$853$	21.1090	0.722760	0.361380	+	0.932419i	$0.382306\pi$
$854$	8.47652	0.290061
$855$	0	0
$856$	−5.43945	−0.185916
$857$	−7.29333	−0.249136	−0.124568	−	0.992211i	$-0.539754\pi$
$857$	−7.29333	−0.249136	−0.124568	+	0.992211i	$0.539754\pi$
$858$	0	0
$859$	−1.13617	−0.0387657	−0.0193828	−	0.999812i	$-0.506170\pi$
$859$	−1.13617	−0.0387657	−0.0193828	+	0.999812i	$0.506170\pi$
$860$	−1.48746	−0.0507219
$861$	0	0
$862$	−38.2963	−1.30438
$863$	−1.73677	−0.0591202	−0.0295601	−	0.999563i	$-0.509411\pi$
$863$	−1.73677	−0.0591202	−0.0295601	+	0.999563i	$0.509411\pi$
$864$	0	0
$865$	0.241990	0.00822790
$866$	39.4104	1.33922
$867$	0	0
$868$	−8.43403	−0.286270
$869$	−11.8306	−0.401326
$870$	0	0
$871$	26.6333	0.902435
$872$	−2.02124	−0.0684479
$873$	0	0
$874$	−4.98596	−0.168653
$875$	3.35484	0.113414
$876$	0	0
$877$	2.27847	0.0769385	0.0384693	−	0.999260i	$-0.487752\pi$
$877$	2.27847	0.0769385	0.0384693	+	0.999260i	$0.487752\pi$
$878$	−32.1518	−1.08507
$879$	0	0
$880$	−0.642937	−0.0216734
$881$	−28.9120	−0.974069	−0.487035	−	0.873383i	$-0.661921\pi$
$881$	−28.9120	−0.974069	−0.487035	+	0.873383i	$0.661921\pi$
$882$	0	0
$883$	−41.0479	−1.38137	−0.690686	−	0.723155i	$-0.742691\pi$
$883$	−41.0479	−1.38137	−0.690686	+	0.723155i	$0.742691\pi$
$884$	22.5409	0.758132
$885$	0	0
$886$	−40.0402	−1.34518
$887$	−23.4411	−0.787074	−0.393537	−	0.919309i	$-0.628749\pi$
$887$	−23.4411	−0.787074	−0.393537	+	0.919309i	$0.628749\pi$
$888$	0	0
$889$	−26.6333	−0.893253
$890$	−3.82840	−0.128328
$891$	0	0
$892$	−20.1103	−0.673341
$893$	22.6487	0.757911
$894$	0	0
$895$	4.29401	0.143533
$896$	−1.23607	−0.0412941
$897$	0	0
$898$	4.86368	0.162303
$899$	36.2572	1.20925
$900$	0	0
$901$	37.9835	1.26542
$902$	2.59083	0.0862651
$903$	0	0
$904$	−1.00000	−0.0332595
$905$	2.85765	0.0949915
$906$	0	0
$907$	−5.39798	−0.179237	−0.0896185	−	0.995976i	$-0.528565\pi$
$907$	−5.39798	−0.179237	−0.0896185	+	0.995976i	$0.528565\pi$
$908$	15.1744	0.503581
$909$	0	0
$910$	−1.70368	−0.0564764
$911$	−5.06757	−0.167896	−0.0839481	−	0.996470i	$-0.526753\pi$
$911$	−5.06757	−0.167896	−0.0839481	+	0.996470i	$0.526753\pi$
$912$	0	0
$913$	2.16140	0.0715320
$914$	−8.12472	−0.268742
$915$	0	0
$916$	−7.08361	−0.234049
$917$	−22.7991	−0.752893
$918$	0	0
$919$	−45.2550	−1.49282	−0.746412	−	0.665484i	$-0.768225\pi$
$919$	−45.2550	−1.49282	−0.746412	+	0.665484i	$0.768225\pi$
$920$	0.362985	0.0119673
$921$	0	0
$922$	5.57177	0.183497
$923$	5.47074	0.180072
$924$	0	0
$925$	−9.16419	−0.301317
$926$	26.7235	0.878190
$927$	0	0
$928$	5.31375	0.174432
$929$	−8.62169	−0.282869	−0.141434	−	0.989948i	$-0.545171\pi$
$929$	−8.62169	−0.282869	−0.141434	+	0.989948i	$0.545171\pi$
$930$	0	0
$931$	−20.5545	−0.673647
$932$	19.8848	0.651349
$933$	0	0
$934$	−28.4469	−0.930812
$935$	2.87530	0.0940324
$936$	0	0
$937$	−17.5481	−0.573271	−0.286636	−	0.958040i	$-0.592537\pi$
$937$	−17.5481	−0.573271	−0.286636	+	0.958040i	$0.592537\pi$
$938$	6.53148	0.213260
$939$	0	0
$940$	−1.64886	−0.0537799
$941$	32.7029	1.06608	0.533042	−	0.846089i	$-0.321049\pi$
$941$	32.7029	1.06608	0.533042	+	0.846089i	$0.321049\pi$
$942$	0	0
$943$	−1.46271	−0.0476324
$944$	−0.243785	−0.00793453
$945$	0	0
$946$	−12.7889	−0.415803
$947$	−59.7085	−1.94027	−0.970133	−	0.242575i	$-0.922008\pi$
$947$	−59.7085	−1.94027	−0.970133	+	0.242575i	$0.922008\pi$
$948$	0	0
$949$	−69.3737	−2.25197
$950$	18.5002	0.600225
$951$	0	0
$952$	5.52786	0.179159
$953$	58.8800	1.90731	0.953654	−	0.300904i	$-0.0972885\pi$
$953$	58.8800	1.90731	0.953654	+	0.300904i	$0.0972885\pi$
$954$	0	0
$955$	−4.09731	−0.132586
$956$	5.09591	0.164813
$957$	0	0
$958$	5.05812	0.163421
$959$	−16.3370	−0.527549
$960$	0	0
$961$	15.5571	0.501842
$962$	9.37831	0.302369
$963$	0	0
$964$	−3.01532	−0.0971170
$965$	−0.954691	−0.0307326
$966$	0	0
$967$	51.4799	1.65548	0.827741	−	0.561110i	$-0.189626\pi$
$967$	51.4799	1.65548	0.827741	+	0.561110i	$0.189626\pi$
$968$	5.47214	0.175881
$969$	0	0
$970$	3.63471	0.116704
$971$	−27.1359	−0.870834	−0.435417	−	0.900229i	$-0.643399\pi$
$971$	−27.1359	−0.870834	−0.435417	+	0.900229i	$0.643399\pi$
$972$	0	0
$973$	−0.386938	−0.0124047
$974$	11.1660	0.357781
$975$	0	0
$976$	−6.85765	−0.219508
$977$	−25.8667	−0.827548	−0.413774	−	0.910380i	$-0.635790\pi$
$977$	−25.8667	−0.827548	−0.413774	+	0.910380i	$0.635790\pi$
$978$	0	0
$979$	−32.9160	−1.05200
$980$	1.49640	0.0478006
$981$	0	0
$982$	−32.1169	−1.02489
$983$	−49.7983	−1.58832	−0.794159	−	0.607710i	$-0.792088\pi$
$983$	−49.7983	−1.58832	−0.794159	+	0.607710i	$0.792088\pi$
$984$	0	0
$985$	−3.28911	−0.104800
$986$	−23.7638	−0.756794
$987$	0	0
$988$	−18.9324	−0.602321
$989$	7.22027	0.229591
$990$	0	0
$991$	36.7879	1.16861	0.584303	−	0.811536i	$-0.301368\pi$
$991$	36.7879	1.16861	0.584303	+	0.811536i	$0.301368\pi$
$992$	6.82328	0.216639
$993$	0	0
$994$	1.34163	0.0425539
$995$	1.48768	0.0471627
$996$	0	0
$997$	6.45172	0.204328	0.102164	−	0.994768i	$-0.467423\pi$
$997$	6.45172	0.204328	0.102164	+	0.994768i	$0.467423\pi$
$998$	−36.9671	−1.17017
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2034.2.a.r.1.3		4
3.2	odd	2		226.2.a.d.1.3	✓	4
12.11	even	2		1808.2.a.j.1.2		4
15.14	odd	2		5650.2.a.o.1.2		4
24.5	odd	2		7232.2.a.u.1.2		4
24.11	even	2		7232.2.a.v.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
226.2.a.d.1.3	✓	4	3.2	odd	2
1808.2.a.j.1.2		4	12.11	even	2
2034.2.a.r.1.3		4	1.1	even	1	trivial
5650.2.a.o.1.2		4	15.14	odd	2
7232.2.a.u.1.2		4	24.5	odd	2
7232.2.a.v.1.3		4	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2034 = 2 \cdot 3^{2} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.273457 q^{5} +1.23607 q^{7} -1.00000 q^{8} +0.273457 q^{10} +2.35114 q^{11} -5.04029 q^{13} -1.23607 q^{14} +1.00000 q^{16} -4.47214 q^{17} +3.75621 q^{19} -0.273457 q^{20} -2.35114 q^{22} +1.32739 q^{23} -4.92522 q^{25} +5.04029 q^{26} +1.23607 q^{28} -5.31375 q^{29} -6.82328 q^{31} -1.00000 q^{32} +4.47214 q^{34} -0.338012 q^{35} +1.86067 q^{37} -3.75621 q^{38} +0.273457 q^{40} -1.10194 q^{41} +5.43945 q^{43} +2.35114 q^{44} -1.32739 q^{46} +6.02967 q^{47} -5.47214 q^{49} +4.92522 q^{50} -5.04029 q^{52} -8.49338 q^{53} -0.642937 q^{55} -1.23607 q^{56} +5.31375 q^{58} -0.243785 q^{59} -6.85765 q^{61} +6.82328 q^{62} +1.00000 q^{64} +1.37831 q^{65} -5.28408 q^{67} -4.47214 q^{68} +0.338012 q^{70} -1.08540 q^{71} +13.7638 q^{73} -1.86067 q^{74} +3.75621 q^{76} +2.90617 q^{77} -5.03186 q^{79} -0.273457 q^{80} +1.10194 q^{82} +0.919299 q^{83} +1.22294 q^{85} -5.43945 q^{86} -2.35114 q^{88} -14.0000 q^{89} -6.23015 q^{91} +1.32739 q^{92} -6.02967 q^{94} -1.02717 q^{95} +13.2917 q^{97} +5.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{7} - 4 q^{8} + 4 q^{10} + 4 q^{13} + 4 q^{14} + 4 q^{16} - 6 q^{19} - 4 q^{20} + 6 q^{23} + 4 q^{25} - 4 q^{26} - 4 q^{28} - 4 q^{32} + 4 q^{35} - 8 q^{37} + 6 q^{38}+ \cdots + 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 - 4 * q^7 - 4 * q^8 + 4 * q^10 + 4 * q^13 + 4 * q^14 + 4 * q^16 - 6 * q^19 - 4 * q^20 + 6 * q^23 + 4 * q^25 - 4 * q^26 - 4 * q^28 - 4 * q^32 + 4 * q^35 - 8 * q^37 + 6 * q^38 + 4 * q^40 - 8 * q^41 - 6 * q^43 - 6 * q^46 + 6 * q^47 - 4 * q^49 - 4 * q^50 + 4 * q^52 - 4 * q^53 - 20 * q^55 + 4 * q^56 - 22 * q^59 + 16 * q^61 + 4 * q^64 - 24 * q^65 - 18 * q^67 - 4 * q^70 - 22 * q^71 + 8 * q^74 - 6 * q^76 - 2 * q^79 - 4 * q^80 + 8 * q^82 - 12 * q^83 + 6 * q^86 - 56 * q^89 - 24 * q^91 + 6 * q^92 - 6 * q^94 + 16 * q^95 + 16 * q^97 + 4 * q^98

Introduction

L-functions

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