LMFDB - Embedded newform 4012.2.a.j.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.03698 q^{3} +0.719057 q^{5} -1.71880 q^{7} -1.92468 q^{9} +O(q^{10})$	$q-1.03698 q^{3} +0.719057 q^{5} -1.71880 q^{7} -1.92468 q^{9} -3.90029 q^{11} +1.22436 q^{13} -0.745644 q^{15} +1.00000 q^{17} +5.77793 q^{19} +1.78235 q^{21} +0.202808 q^{23} -4.48296 q^{25} +5.10678 q^{27} -7.59135 q^{29} -0.922961 q^{31} +4.04451 q^{33} -1.23591 q^{35} -8.43911 q^{37} -1.26963 q^{39} +8.86332 q^{41} +2.34033 q^{43} -1.38395 q^{45} -5.88414 q^{47} -4.04573 q^{49} -1.03698 q^{51} +3.80241 q^{53} -2.80453 q^{55} -5.99157 q^{57} +1.00000 q^{59} +0.251704 q^{61} +3.30814 q^{63} +0.880381 q^{65} +2.99471 q^{67} -0.210307 q^{69} +12.7551 q^{71} +0.790326 q^{73} +4.64872 q^{75} +6.70381 q^{77} +12.6793 q^{79} +0.478436 q^{81} +0.815103 q^{83} +0.719057 q^{85} +7.87205 q^{87} -11.7513 q^{89} -2.10442 q^{91} +0.957088 q^{93} +4.15466 q^{95} +8.60682 q^{97} +7.50682 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.03698	−0.598698	−0.299349	−	0.954144i	$-0.596770\pi$
$3$	−1.03698	−0.598698	−0.299349	+	0.954144i	$0.596770\pi$
$4$	0	0
$5$	0.719057	0.321572	0.160786	−	0.986989i	$-0.448597\pi$
$5$	0.719057	0.321572	0.160786	+	0.986989i	$0.448597\pi$
$6$	0	0
$7$	−1.71880	−0.649644	−0.324822	−	0.945775i	$-0.605304\pi$
$7$	−1.71880	−0.649644	−0.324822	+	0.945775i	$0.605304\pi$
$8$	0	0
$9$	−1.92468	−0.641560
$10$	0	0
$11$	−3.90029	−1.17598	−0.587991	−	0.808867i	$-0.700081\pi$
$11$	−3.90029	−1.17598	−0.587991	+	0.808867i	$0.700081\pi$
$12$	0	0
$13$	1.22436	0.339575	0.169788	−	0.985481i	$-0.445692\pi$
$13$	1.22436	0.339575	0.169788	+	0.985481i	$0.445692\pi$
$14$	0	0
$15$	−0.745644	−0.192525
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.77793	1.32555	0.662774	−	0.748820i	$-0.269379\pi$
$19$	5.77793	1.32555	0.662774	+	0.748820i	$0.269379\pi$
$20$	0	0
$21$	1.78235	0.388941
$22$	0	0
$23$	0.202808	0.0422884	0.0211442	−	0.999776i	$-0.493269\pi$
$23$	0.202808	0.0422884	0.0211442	+	0.999776i	$0.493269\pi$
$24$	0	0
$25$	−4.48296	−0.896592
$26$	0	0
$27$	5.10678	0.982800
$28$	0	0
$29$	−7.59135	−1.40968	−0.704839	−	0.709367i	$-0.748981\pi$
$29$	−7.59135	−1.40968	−0.704839	+	0.709367i	$0.748981\pi$
$30$	0	0
$31$	−0.922961	−0.165769	−0.0828843	−	0.996559i	$-0.526413\pi$
$31$	−0.922961	−0.165769	−0.0828843	+	0.996559i	$0.526413\pi$
$32$	0	0
$33$	4.04451	0.704059
$34$	0	0
$35$	−1.23591	−0.208907
$36$	0	0
$37$	−8.43911	−1.38738	−0.693690	−	0.720273i	$-0.744016\pi$
$37$	−8.43911	−1.38738	−0.693690	+	0.720273i	$0.744016\pi$
$38$	0	0
$39$	−1.26963	−0.203303
$40$	0	0
$41$	8.86332	1.38422	0.692109	−	0.721793i	$-0.256682\pi$
$41$	8.86332	1.38422	0.692109	+	0.721793i	$0.256682\pi$
$42$	0	0
$43$	2.34033	0.356897	0.178449	−	0.983949i	$-0.442892\pi$
$43$	2.34033	0.356897	0.178449	+	0.983949i	$0.442892\pi$
$44$	0	0
$45$	−1.38395	−0.206308
$46$	0	0
$47$	−5.88414	−0.858290	−0.429145	−	0.903236i	$-0.641185\pi$
$47$	−5.88414	−0.858290	−0.429145	+	0.903236i	$0.641185\pi$
$48$	0	0
$49$	−4.04573	−0.577962
$50$	0	0
$51$	−1.03698	−0.145206
$52$	0	0
$53$	3.80241	0.522302	0.261151	−	0.965298i	$-0.415898\pi$
$53$	3.80241	0.522302	0.261151	+	0.965298i	$0.415898\pi$
$54$	0	0
$55$	−2.80453	−0.378163
$56$	0	0
$57$	−5.99157	−0.793603
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	0.251704	0.0322274	0.0161137	−	0.999870i	$-0.494871\pi$
$61$	0.251704	0.0322274	0.0161137	+	0.999870i	$0.494871\pi$
$62$	0	0
$63$	3.30814	0.416786
$64$	0	0
$65$	0.880381	0.109198
$66$	0	0
$67$	2.99471	0.365862	0.182931	−	0.983126i	$-0.441442\pi$
$67$	2.99471	0.365862	0.182931	+	0.983126i	$0.441442\pi$
$68$	0	0
$69$	−0.210307	−0.0253180
$70$	0	0
$71$	12.7551	1.51375	0.756875	−	0.653559i	$-0.226725\pi$
$71$	12.7551	1.51375	0.756875	+	0.653559i	$0.226725\pi$
$72$	0	0
$73$	0.790326	0.0925007	0.0462503	−	0.998930i	$-0.485273\pi$
$73$	0.790326	0.0925007	0.0462503	+	0.998930i	$0.485273\pi$
$74$	0	0
$75$	4.64872	0.536788
$76$	0	0
$77$	6.70381	0.763970
$78$	0	0
$79$	12.6793	1.42653	0.713265	−	0.700895i	$-0.247216\pi$
$79$	12.6793	1.42653	0.713265	+	0.700895i	$0.247216\pi$
$80$	0	0
$81$	0.478436	0.0531596
$82$	0	0
$83$	0.815103	0.0894692	0.0447346	−	0.998999i	$-0.485756\pi$
$83$	0.815103	0.0894692	0.0447346	+	0.998999i	$0.485756\pi$
$84$	0	0
$85$	0.719057	0.0779926
$86$	0	0
$87$	7.87205	0.843972
$88$	0	0
$89$	−11.7513	−1.24564	−0.622820	−	0.782365i	$-0.714013\pi$
$89$	−11.7513	−1.24564	−0.622820	+	0.782365i	$0.714013\pi$
$90$	0	0
$91$	−2.10442	−0.220603
$92$	0	0
$93$	0.957088	0.0992454
$94$	0	0
$95$	4.15466	0.426259
$96$	0	0
$97$	8.60682	0.873890	0.436945	−	0.899488i	$-0.356060\pi$
$97$	8.60682	0.873890	0.436945	+	0.899488i	$0.356060\pi$
$98$	0	0
$99$	7.50682	0.754463
$100$	0	0
$101$	−2.21815	−0.220715	−0.110357	−	0.993892i	$-0.535200\pi$
$101$	−2.21815	−0.220715	−0.110357	+	0.993892i	$0.535200\pi$
$102$	0	0
$103$	−10.4844	−1.03306	−0.516528	−	0.856270i	$-0.672776\pi$
$103$	−10.4844	−1.03306	−0.516528	+	0.856270i	$0.672776\pi$
$104$	0	0
$105$	1.28161	0.125073
$106$	0	0
$107$	16.6357	1.60823	0.804116	−	0.594472i	$-0.202639\pi$
$107$	16.6357	1.60823	0.804116	+	0.594472i	$0.202639\pi$
$108$	0	0
$109$	3.62189	0.346914	0.173457	−	0.984841i	$-0.444506\pi$
$109$	3.62189	0.346914	0.173457	+	0.984841i	$0.444506\pi$
$110$	0	0
$111$	8.75115	0.830623
$112$	0	0
$113$	9.32263	0.877000	0.438500	−	0.898731i	$-0.355510\pi$
$113$	9.32263	0.877000	0.438500	+	0.898731i	$0.355510\pi$
$114$	0	0
$115$	0.145831	0.0135988
$116$	0	0
$117$	−2.35649	−0.217858
$118$	0	0
$119$	−1.71880	−0.157562
$120$	0	0
$121$	4.21228	0.382934
$122$	0	0
$123$	−9.19105	−0.828729
$124$	0	0
$125$	−6.81878	−0.609890
$126$	0	0
$127$	−3.30690	−0.293440	−0.146720	−	0.989178i	$-0.546872\pi$
$127$	−3.30690	−0.293440	−0.146720	+	0.989178i	$0.546872\pi$
$128$	0	0
$129$	−2.42687	−0.213674
$130$	0	0
$131$	20.1314	1.75889	0.879443	−	0.476005i	$-0.157916\pi$
$131$	20.1314	1.75889	0.879443	+	0.476005i	$0.157916\pi$
$132$	0	0
$133$	−9.93109	−0.861134
$134$	0	0
$135$	3.67206	0.316041
$136$	0	0
$137$	10.5220	0.898955	0.449477	−	0.893292i	$-0.351610\pi$
$137$	10.5220	0.898955	0.449477	+	0.893292i	$0.351610\pi$
$138$	0	0
$139$	7.91096	0.670999	0.335499	−	0.942040i	$-0.391095\pi$
$139$	7.91096	0.670999	0.335499	+	0.942040i	$0.391095\pi$
$140$	0	0
$141$	6.10171	0.513857
$142$	0	0
$143$	−4.77535	−0.399335
$144$	0	0
$145$	−5.45861	−0.453313
$146$	0	0
$147$	4.19533	0.346025
$148$	0	0
$149$	1.32886	0.108864	0.0544322	−	0.998517i	$-0.482665\pi$
$149$	1.32886	0.108864	0.0544322	+	0.998517i	$0.482665\pi$
$150$	0	0
$151$	8.09812	0.659016	0.329508	−	0.944153i	$-0.393117\pi$
$151$	8.09812	0.659016	0.329508	+	0.944153i	$0.393117\pi$
$152$	0	0
$153$	−1.92468	−0.155601
$154$	0	0
$155$	−0.663661	−0.0533065
$156$	0	0
$157$	0.0212338	0.00169464	0.000847322	−	1.00000i	$-0.499730\pi$
$157$	0.0212338	0.00169464	0.000847322		1.00000i	$0.499730\pi$
$158$	0	0
$159$	−3.94301	−0.312701
$160$	0	0
$161$	−0.348586	−0.0274725
$162$	0	0
$163$	18.6687	1.46225	0.731123	−	0.682246i	$-0.238997\pi$
$163$	18.6687	1.46225	0.731123	+	0.682246i	$0.238997\pi$
$164$	0	0
$165$	2.90823	0.226405
$166$	0	0
$167$	13.6632	1.05729	0.528644	−	0.848843i	$-0.322701\pi$
$167$	13.6632	1.05729	0.528644	+	0.848843i	$0.322701\pi$
$168$	0	0
$169$	−11.5010	−0.884689
$170$	0	0
$171$	−11.1207	−0.850418
$172$	0	0
$173$	−2.89118	−0.219812	−0.109906	−	0.993942i	$-0.535055\pi$
$173$	−2.89118	−0.219812	−0.109906	+	0.993942i	$0.535055\pi$
$174$	0	0
$175$	7.70530	0.582466
$176$	0	0
$177$	−1.03698	−0.0779439
$178$	0	0
$179$	9.50199	0.710212	0.355106	−	0.934826i	$-0.384445\pi$
$179$	9.50199	0.710212	0.355106	+	0.934826i	$0.384445\pi$
$180$	0	0
$181$	−11.6226	−0.863900	−0.431950	−	0.901898i	$-0.642174\pi$
$181$	−11.6226	−0.863900	−0.431950	+	0.901898i	$0.642174\pi$
$182$	0	0
$183$	−0.261011	−0.0192945
$184$	0	0
$185$	−6.06819	−0.446142
$186$	0	0
$187$	−3.90029	−0.285218
$188$	0	0
$189$	−8.77752	−0.638470
$190$	0	0
$191$	−16.4746	−1.19206	−0.596030	−	0.802962i	$-0.703256\pi$
$191$	−16.4746	−1.19206	−0.596030	+	0.802962i	$0.703256\pi$
$192$	0	0
$193$	−19.3759	−1.39471	−0.697354	−	0.716726i	$-0.745640\pi$
$193$	−19.3759	−1.39471	−0.697354	+	0.716726i	$0.745640\pi$
$194$	0	0
$195$	−0.912934	−0.0653766
$196$	0	0
$197$	18.8448	1.34263	0.671317	−	0.741171i	$-0.265729\pi$
$197$	18.8448	1.34263	0.671317	+	0.741171i	$0.265729\pi$
$198$	0	0
$199$	16.9689	1.20289	0.601445	−	0.798914i	$-0.294592\pi$
$199$	16.9689	1.20289	0.601445	+	0.798914i	$0.294592\pi$
$200$	0	0
$201$	−3.10544	−0.219041
$202$	0	0
$203$	13.0480	0.915790
$204$	0	0
$205$	6.37323	0.445126
$206$	0	0
$207$	−0.390341	−0.0271306
$208$	0	0
$209$	−22.5356	−1.55882
$210$	0	0
$211$	12.7192	0.875626	0.437813	−	0.899066i	$-0.355753\pi$
$211$	12.7192	0.875626	0.437813	+	0.899066i	$0.355753\pi$
$212$	0	0
$213$	−13.2267	−0.906280
$214$	0	0
$215$	1.68283	0.114768
$216$	0	0
$217$	1.58638	0.107691
$218$	0	0
$219$	−0.819549	−0.0553800
$220$	0	0
$221$	1.22436	0.0823591
$222$	0	0
$223$	−12.2016	−0.817082	−0.408541	−	0.912740i	$-0.633962\pi$
$223$	−12.2016	−0.817082	−0.408541	+	0.912740i	$0.633962\pi$
$224$	0	0
$225$	8.62826	0.575217
$226$	0	0
$227$	−24.5907	−1.63214	−0.816070	−	0.577953i	$-0.803852\pi$
$227$	−24.5907	−1.63214	−0.816070	+	0.577953i	$0.803852\pi$
$228$	0	0
$229$	−17.9280	−1.18471	−0.592357	−	0.805676i	$-0.701802\pi$
$229$	−17.9280	−1.18471	−0.592357	+	0.805676i	$0.701802\pi$
$230$	0	0
$231$	−6.95169	−0.457388
$232$	0	0
$233$	8.02286	0.525595	0.262798	−	0.964851i	$-0.415355\pi$
$233$	8.02286	0.525595	0.262798	+	0.964851i	$0.415355\pi$
$234$	0	0
$235$	−4.23103	−0.276002
$236$	0	0
$237$	−13.1481	−0.854061
$238$	0	0
$239$	6.15117	0.397886	0.198943	−	0.980011i	$-0.436249\pi$
$239$	6.15117	0.397886	0.198943	+	0.980011i	$0.436249\pi$
$240$	0	0
$241$	9.88317	0.636631	0.318315	−	0.947985i	$-0.396883\pi$
$241$	9.88317	0.636631	0.318315	+	0.947985i	$0.396883\pi$
$242$	0	0
$243$	−15.8165	−1.01463
$244$	0	0
$245$	−2.90911	−0.185856
$246$	0	0
$247$	7.07424	0.450123
$248$	0	0
$249$	−0.845243	−0.0535651
$250$	0	0
$251$	1.36437	0.0861180	0.0430590	−	0.999073i	$-0.486290\pi$
$251$	1.36437	0.0861180	0.0430590	+	0.999073i	$0.486290\pi$
$252$	0	0
$253$	−0.791011	−0.0497305
$254$	0	0
$255$	−0.745644	−0.0466941
$256$	0	0
$257$	13.0187	0.812083	0.406041	−	0.913855i	$-0.366909\pi$
$257$	13.0187	0.812083	0.406041	+	0.913855i	$0.366909\pi$
$258$	0	0
$259$	14.5051	0.901304
$260$	0	0
$261$	14.6109	0.904393
$262$	0	0
$263$	12.6010	0.777009	0.388504	−	0.921447i	$-0.372992\pi$
$263$	12.6010	0.777009	0.388504	+	0.921447i	$0.372992\pi$
$264$	0	0
$265$	2.73415	0.167957
$266$	0	0
$267$	12.1859	0.745763
$268$	0	0
$269$	23.3585	1.42419	0.712096	−	0.702082i	$-0.247746\pi$
$269$	23.3585	1.42419	0.712096	+	0.702082i	$0.247746\pi$
$270$	0	0
$271$	0.463409	0.0281501	0.0140750	−	0.999901i	$-0.495520\pi$
$271$	0.463409	0.0281501	0.0140750	+	0.999901i	$0.495520\pi$
$272$	0	0
$273$	2.18223	0.132075
$274$	0	0
$275$	17.4848	1.05438
$276$	0	0
$277$	15.2741	0.917734	0.458867	−	0.888505i	$-0.348255\pi$
$277$	15.2741	0.917734	0.458867	+	0.888505i	$0.348255\pi$
$278$	0	0
$279$	1.77640	0.106351
$280$	0	0
$281$	30.1581	1.79908	0.899542	−	0.436834i	$-0.143900\pi$
$281$	30.1581	1.79908	0.899542	+	0.436834i	$0.143900\pi$
$282$	0	0
$283$	−2.97000	−0.176548	−0.0882740	−	0.996096i	$-0.528135\pi$
$283$	−2.97000	−0.176548	−0.0882740	+	0.996096i	$0.528135\pi$
$284$	0	0
$285$	−4.30828	−0.255200
$286$	0	0
$287$	−15.2343	−0.899250
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−8.92506	−0.523197
$292$	0	0
$293$	18.1364	1.05954	0.529770	−	0.848141i	$-0.322278\pi$
$293$	18.1364	1.05954	0.529770	+	0.848141i	$0.322278\pi$
$294$	0	0
$295$	0.719057	0.0418651
$296$	0	0
$297$	−19.9179	−1.15575
$298$	0	0
$299$	0.248310	0.0143601
$300$	0	0
$301$	−4.02256	−0.231856
$302$	0	0
$303$	2.30017	0.132141
$304$	0	0
$305$	0.180989	0.0103634
$306$	0	0
$307$	1.04618	0.0597089	0.0298544	−	0.999554i	$-0.490496\pi$
$307$	1.04618	0.0597089	0.0298544	+	0.999554i	$0.490496\pi$
$308$	0	0
$309$	10.8721	0.618490
$310$	0	0
$311$	−12.6754	−0.718759	−0.359379	−	0.933192i	$-0.617012\pi$
$311$	−12.6754	−0.718759	−0.359379	+	0.933192i	$0.617012\pi$
$312$	0	0
$313$	−2.27153	−0.128395	−0.0641973	−	0.997937i	$-0.520449\pi$
$313$	−2.27153	−0.128395	−0.0641973	+	0.997937i	$0.520449\pi$
$314$	0	0
$315$	2.37874	0.134027
$316$	0	0
$317$	6.80793	0.382371	0.191186	−	0.981554i	$-0.438767\pi$
$317$	6.80793	0.382371	0.191186	+	0.981554i	$0.438767\pi$
$318$	0	0
$319$	29.6085	1.65776
$320$	0	0
$321$	−17.2508	−0.962847
$322$	0	0
$323$	5.77793	0.321492
$324$	0	0
$325$	−5.48874	−0.304460
$326$	0	0
$327$	−3.75581	−0.207697
$328$	0	0
$329$	10.1136	0.557583
$330$	0	0
$331$	−1.61406	−0.0887167	−0.0443583	−	0.999016i	$-0.514124\pi$
$331$	−1.61406	−0.0887167	−0.0443583	+	0.999016i	$0.514124\pi$
$332$	0	0
$333$	16.2426	0.890088
$334$	0	0
$335$	2.15336	0.117651
$336$	0	0
$337$	0.323486	0.0176214	0.00881071	−	0.999961i	$-0.497195\pi$
$337$	0.323486	0.0176214	0.00881071	+	0.999961i	$0.497195\pi$
$338$	0	0
$339$	−9.66735	−0.525058
$340$	0	0
$341$	3.59982	0.194941
$342$	0	0
$343$	18.9854	1.02511
$344$	0	0
$345$	−0.151223	−0.00814156
$346$	0	0
$347$	−24.8078	−1.33175	−0.665876	−	0.746062i	$-0.731942\pi$
$347$	−24.8078	−1.33175	−0.665876	+	0.746062i	$0.731942\pi$
$348$	0	0
$349$	9.54711	0.511045	0.255522	−	0.966803i	$-0.417753\pi$
$349$	9.54711	0.511045	0.255522	+	0.966803i	$0.417753\pi$
$350$	0	0
$351$	6.25251	0.333734
$352$	0	0
$353$	13.2751	0.706560	0.353280	−	0.935518i	$-0.385066\pi$
$353$	13.2751	0.706560	0.353280	+	0.935518i	$0.385066\pi$
$354$	0	0
$355$	9.17163	0.486780
$356$	0	0
$357$	1.78235	0.0943321
$358$	0	0
$359$	−21.6457	−1.14242	−0.571208	−	0.820805i	$-0.693525\pi$
$359$	−21.6457	−1.14242	−0.571208	+	0.820805i	$0.693525\pi$
$360$	0	0
$361$	14.3844	0.757076
$362$	0	0
$363$	−4.36803	−0.229262
$364$	0	0
$365$	0.568289	0.0297456
$366$	0	0
$367$	−1.58591	−0.0827836	−0.0413918	−	0.999143i	$-0.513179\pi$
$367$	−1.58591	−0.0827836	−0.0413918	+	0.999143i	$0.513179\pi$
$368$	0	0
$369$	−17.0591	−0.888059
$370$	0	0
$371$	−6.53558	−0.339310
$372$	0	0
$373$	10.6022	0.548963	0.274482	−	0.961592i	$-0.411494\pi$
$373$	10.6022	0.548963	0.274482	+	0.961592i	$0.411494\pi$
$374$	0	0
$375$	7.07092	0.365140
$376$	0	0
$377$	−9.29451	−0.478692
$378$	0	0
$379$	14.6400	0.752006	0.376003	−	0.926618i	$-0.377298\pi$
$379$	14.6400	0.752006	0.376003	+	0.926618i	$0.377298\pi$
$380$	0	0
$381$	3.42917	0.175682
$382$	0	0
$383$	4.50405	0.230146	0.115073	−	0.993357i	$-0.463290\pi$
$383$	4.50405	0.230146	0.115073	+	0.993357i	$0.463290\pi$
$384$	0	0
$385$	4.82042	0.245671
$386$	0	0
$387$	−4.50439	−0.228971
$388$	0	0
$389$	−8.15256	−0.413351	−0.206676	−	0.978410i	$-0.566264\pi$
$389$	−8.15256	−0.413351	−0.206676	+	0.978410i	$0.566264\pi$
$390$	0	0
$391$	0.202808	0.0102565
$392$	0	0
$393$	−20.8757	−1.05304
$394$	0	0
$395$	9.11711	0.458732
$396$	0	0
$397$	8.87286	0.445316	0.222658	−	0.974897i	$-0.428527\pi$
$397$	8.87286	0.445316	0.222658	+	0.974897i	$0.428527\pi$
$398$	0	0
$399$	10.2983	0.515560
$400$	0	0
$401$	−7.74156	−0.386595	−0.193297	−	0.981140i	$-0.561918\pi$
$401$	−7.74156	−0.386595	−0.193297	+	0.981140i	$0.561918\pi$
$402$	0	0
$403$	−1.13003	−0.0562909
$404$	0	0
$405$	0.344023	0.0170946
$406$	0	0
$407$	32.9150	1.63153
$408$	0	0
$409$	34.0532	1.68382	0.841912	−	0.539616i	$-0.181430\pi$
$409$	34.0532	1.68382	0.841912	+	0.539616i	$0.181430\pi$
$410$	0	0
$411$	−10.9111	−0.538203
$412$	0	0
$413$	−1.71880	−0.0845765
$414$	0	0
$415$	0.586105	0.0287708
$416$	0	0
$417$	−8.20347	−0.401726
$418$	0	0
$419$	−6.98132	−0.341060	−0.170530	−	0.985353i	$-0.554548\pi$
$419$	−6.98132	−0.341060	−0.170530	+	0.985353i	$0.554548\pi$
$420$	0	0
$421$	−3.96586	−0.193284	−0.0966421	−	0.995319i	$-0.530810\pi$
$421$	−3.96586	−0.193284	−0.0966421	+	0.995319i	$0.530810\pi$
$422$	0	0
$423$	11.3251	0.550645
$424$	0	0
$425$	−4.48296	−0.217455
$426$	0	0
$427$	−0.432628	−0.0209363
$428$	0	0
$429$	4.95192	0.239081
$430$	0	0
$431$	−11.6854	−0.562865	−0.281432	−	0.959581i	$-0.590809\pi$
$431$	−11.6854	−0.562865	−0.281432	+	0.959581i	$0.590809\pi$
$432$	0	0
$433$	−9.75663	−0.468873	−0.234437	−	0.972131i	$-0.575325\pi$
$433$	−9.75663	−0.468873	−0.234437	+	0.972131i	$0.575325\pi$
$434$	0	0
$435$	5.66045	0.271398
$436$	0	0
$437$	1.17181	0.0560553
$438$	0	0
$439$	28.6436	1.36709	0.683543	−	0.729910i	$-0.260438\pi$
$439$	28.6436	1.36709	0.683543	+	0.729910i	$0.260438\pi$
$440$	0	0
$441$	7.78675	0.370797
$442$	0	0
$443$	−12.0294	−0.571533	−0.285766	−	0.958299i	$-0.592248\pi$
$443$	−12.0294	−0.571533	−0.285766	+	0.958299i	$0.592248\pi$
$444$	0	0
$445$	−8.44988	−0.400563
$446$	0	0
$447$	−1.37800	−0.0651770
$448$	0	0
$449$	−13.1502	−0.620598	−0.310299	−	0.950639i	$-0.600429\pi$
$449$	−13.1502	−0.620598	−0.310299	+	0.950639i	$0.600429\pi$
$450$	0	0
$451$	−34.5695	−1.62782
$452$	0	0
$453$	−8.39756	−0.394552
$454$	0	0
$455$	−1.51320	−0.0709398
$456$	0	0
$457$	−11.1908	−0.523484	−0.261742	−	0.965138i	$-0.584297\pi$
$457$	−11.1908	−0.523484	−0.261742	+	0.965138i	$0.584297\pi$
$458$	0	0
$459$	5.10678	0.238364
$460$	0	0
$461$	−0.786864	−0.0366479	−0.0183239	−	0.999832i	$-0.505833\pi$
$461$	−0.786864	−0.0366479	−0.0183239	+	0.999832i	$0.505833\pi$
$462$	0	0
$463$	16.4889	0.766304	0.383152	−	0.923685i	$-0.374839\pi$
$463$	16.4889	0.766304	0.383152	+	0.923685i	$0.374839\pi$
$464$	0	0
$465$	0.688201	0.0319145
$466$	0	0
$467$	−11.4660	−0.530583	−0.265292	−	0.964168i	$-0.585468\pi$
$467$	−11.4660	−0.530583	−0.265292	+	0.964168i	$0.585468\pi$
$468$	0	0
$469$	−5.14729	−0.237680
$470$	0	0
$471$	−0.0220190	−0.00101458
$472$	0	0
$473$	−9.12797	−0.419705
$474$	0	0
$475$	−25.9022	−1.18847
$476$	0	0
$477$	−7.31843	−0.335088
$478$	0	0
$479$	−0.173148	−0.00791132	−0.00395566	−	0.999992i	$-0.501259\pi$
$479$	−0.173148	−0.00791132	−0.00395566	+	0.999992i	$0.501259\pi$
$480$	0	0
$481$	−10.3325	−0.471120
$482$	0	0
$483$	0.361476	0.0164477
$484$	0	0
$485$	6.18879	0.281018
$486$	0	0
$487$	13.2466	0.600260	0.300130	−	0.953898i	$-0.402970\pi$
$487$	13.2466	0.600260	0.300130	+	0.953898i	$0.402970\pi$
$488$	0	0
$489$	−19.3590	−0.875444
$490$	0	0
$491$	43.3524	1.95647	0.978234	−	0.207505i	$-0.0665342\pi$
$491$	43.3524	1.95647	0.978234	+	0.207505i	$0.0665342\pi$
$492$	0	0
$493$	−7.59135	−0.341897
$494$	0	0
$495$	5.39782	0.242614
$496$	0	0
$497$	−21.9234	−0.983400
$498$	0	0
$499$	−11.2348	−0.502938	−0.251469	−	0.967865i	$-0.580914\pi$
$499$	−11.2348	−0.502938	−0.251469	+	0.967865i	$0.580914\pi$
$500$	0	0
$501$	−14.1684	−0.632997
$502$	0	0
$503$	31.4953	1.40431	0.702154	−	0.712025i	$-0.252222\pi$
$503$	31.4953	1.40431	0.702154	+	0.712025i	$0.252222\pi$
$504$	0	0
$505$	−1.59498	−0.0709756
$506$	0	0
$507$	11.9262	0.529662
$508$	0	0
$509$	−14.2272	−0.630609	−0.315305	−	0.948990i	$-0.602107\pi$
$509$	−14.2272	−0.630609	−0.315305	+	0.948990i	$0.602107\pi$
$510$	0	0
$511$	−1.35841	−0.0600925
$512$	0	0
$513$	29.5066	1.30275
$514$	0	0
$515$	−7.53886	−0.332202
$516$	0	0
$517$	22.9499	1.00933
$518$	0	0
$519$	2.99808	0.131601
$520$	0	0
$521$	−21.2228	−0.929787	−0.464893	−	0.885367i	$-0.653907\pi$
$521$	−21.2228	−0.929787	−0.464893	+	0.885367i	$0.653907\pi$
$522$	0	0
$523$	−9.43216	−0.412439	−0.206220	−	0.978506i	$-0.566116\pi$
$523$	−9.43216	−0.412439	−0.206220	+	0.978506i	$0.566116\pi$
$524$	0	0
$525$	−7.99021	−0.348721
$526$	0	0
$527$	−0.922961	−0.0402048
$528$	0	0
$529$	−22.9589	−0.998212
$530$	0	0
$531$	−1.92468	−0.0835240
$532$	0	0
$533$	10.8519	0.470046
$534$	0	0
$535$	11.9620	0.517162
$536$	0	0
$537$	−9.85334	−0.425203
$538$	0	0
$539$	15.7795	0.679673
$540$	0	0
$541$	20.7300	0.891252	0.445626	−	0.895219i	$-0.352981\pi$
$541$	20.7300	0.891252	0.445626	+	0.895219i	$0.352981\pi$
$542$	0	0
$543$	12.0523	0.517215
$544$	0	0
$545$	2.60434	0.111558
$546$	0	0
$547$	7.33188	0.313488	0.156744	−	0.987639i	$-0.449900\pi$
$547$	7.33188	0.313488	0.156744	+	0.987639i	$0.449900\pi$
$548$	0	0
$549$	−0.484449	−0.0206758
$550$	0	0
$551$	−43.8623	−1.86859
$552$	0	0
$553$	−21.7931	−0.926737
$554$	0	0
$555$	6.29257	0.267105
$556$	0	0
$557$	−0.447993	−0.0189821	−0.00949103	−	0.999955i	$-0.503021\pi$
$557$	−0.447993	−0.0189821	−0.00949103	+	0.999955i	$0.503021\pi$
$558$	0	0
$559$	2.86540	0.121193
$560$	0	0
$561$	4.04451	0.170759
$562$	0	0
$563$	−0.739280	−0.0311569	−0.0155785	−	0.999879i	$-0.504959\pi$
$563$	−0.739280	−0.0311569	−0.0155785	+	0.999879i	$0.504959\pi$
$564$	0	0
$565$	6.70350	0.282018
$566$	0	0
$567$	−0.822335	−0.0345348
$568$	0	0
$569$	−23.8636	−1.00041	−0.500207	−	0.865906i	$-0.666743\pi$
$569$	−23.8636	−1.00041	−0.500207	+	0.865906i	$0.666743\pi$
$570$	0	0
$571$	−27.3542	−1.14474	−0.572370	−	0.819996i	$-0.693976\pi$
$571$	−27.3542	−1.14474	−0.572370	+	0.819996i	$0.693976\pi$
$572$	0	0
$573$	17.0838	0.713685
$574$	0	0
$575$	−0.909181	−0.0379155
$576$	0	0
$577$	−18.8574	−0.785045	−0.392522	−	0.919742i	$-0.628397\pi$
$577$	−18.8574	−0.785045	−0.392522	+	0.919742i	$0.628397\pi$
$578$	0	0
$579$	20.0924	0.835010
$580$	0	0
$581$	−1.40100	−0.0581232
$582$	0	0
$583$	−14.8305	−0.614217
$584$	0	0
$585$	−1.69445	−0.0700570
$586$	0	0
$587$	−32.1011	−1.32495	−0.662477	−	0.749082i	$-0.730495\pi$
$587$	−32.1011	−1.32495	−0.662477	+	0.749082i	$0.730495\pi$
$588$	0	0
$589$	−5.33280	−0.219734
$590$	0	0
$591$	−19.5416	−0.803833
$592$	0	0
$593$	−1.31539	−0.0540166	−0.0270083	−	0.999635i	$-0.508598\pi$
$593$	−1.31539	−0.0540166	−0.0270083	+	0.999635i	$0.508598\pi$
$594$	0	0
$595$	−1.23591	−0.0506675
$596$	0	0
$597$	−17.5963	−0.720169
$598$	0	0
$599$	41.3892	1.69112	0.845558	−	0.533884i	$-0.179268\pi$
$599$	41.3892	1.69112	0.845558	+	0.533884i	$0.179268\pi$
$600$	0	0
$601$	−18.2643	−0.745016	−0.372508	−	0.928029i	$-0.621502\pi$
$601$	−18.2643	−0.745016	−0.372508	+	0.928029i	$0.621502\pi$
$602$	0	0
$603$	−5.76385	−0.234722
$604$	0	0
$605$	3.02887	0.123141
$606$	0	0
$607$	−21.4968	−0.872528	−0.436264	−	0.899819i	$-0.643699\pi$
$607$	−21.4968	−0.872528	−0.436264	+	0.899819i	$0.643699\pi$
$608$	0	0
$609$	−13.5305	−0.548282
$610$	0	0
$611$	−7.20428	−0.291454
$612$	0	0
$613$	−22.3217	−0.901564	−0.450782	−	0.892634i	$-0.648855\pi$
$613$	−22.3217	−0.901564	−0.450782	+	0.892634i	$0.648855\pi$
$614$	0	0
$615$	−6.60889	−0.266496
$616$	0	0
$617$	−13.8264	−0.556629	−0.278315	−	0.960490i	$-0.589776\pi$
$617$	−13.8264	−0.556629	−0.278315	+	0.960490i	$0.589776\pi$
$618$	0	0
$619$	−0.374291	−0.0150440	−0.00752201	−	0.999972i	$-0.502394\pi$
$619$	−0.374291	−0.0150440	−0.00752201	+	0.999972i	$0.502394\pi$
$620$	0	0
$621$	1.03570	0.0415611
$622$	0	0
$623$	20.1982	0.809223
$624$	0	0
$625$	17.5117	0.700468
$626$	0	0
$627$	23.3689	0.933263
$628$	0	0
$629$	−8.43911	−0.336489
$630$	0	0
$631$	19.9257	0.793229	0.396614	−	0.917985i	$-0.370185\pi$
$631$	19.9257	0.793229	0.396614	+	0.917985i	$0.370185\pi$
$632$	0	0
$633$	−13.1895	−0.524236
$634$	0	0
$635$	−2.37785	−0.0943619
$636$	0	0
$637$	−4.95342	−0.196262
$638$	0	0
$639$	−24.5495	−0.971162
$640$	0	0
$641$	16.2686	0.642571	0.321285	−	0.946982i	$-0.395885\pi$
$641$	16.2686	0.642571	0.321285	+	0.946982i	$0.395885\pi$
$642$	0	0
$643$	14.7165	0.580363	0.290181	−	0.956972i	$-0.406284\pi$
$643$	14.7165	0.580363	0.290181	+	0.956972i	$0.406284\pi$
$644$	0	0
$645$	−1.74505	−0.0687115
$646$	0	0
$647$	5.45410	0.214423	0.107211	−	0.994236i	$-0.465808\pi$
$647$	5.45410	0.214423	0.107211	+	0.994236i	$0.465808\pi$
$648$	0	0
$649$	−3.90029	−0.153100
$650$	0	0
$651$	−1.64504	−0.0644743
$652$	0	0
$653$	−43.9287	−1.71906	−0.859531	−	0.511083i	$-0.829244\pi$
$653$	−43.9287	−1.71906	−0.859531	+	0.511083i	$0.829244\pi$
$654$	0	0
$655$	14.4756	0.565608
$656$	0	0
$657$	−1.52113	−0.0593447
$658$	0	0
$659$	24.8191	0.966813	0.483407	−	0.875396i	$-0.339399\pi$
$659$	24.8191	0.966813	0.483407	+	0.875396i	$0.339399\pi$
$660$	0	0
$661$	−16.7496	−0.651483	−0.325741	−	0.945459i	$-0.605614\pi$
$661$	−16.7496	−0.651483	−0.325741	+	0.945459i	$0.605614\pi$
$662$	0	0
$663$	−1.26963	−0.0493083
$664$	0	0
$665$	−7.14101	−0.276917
$666$	0	0
$667$	−1.53959	−0.0596131
$668$	0	0
$669$	12.6528	0.489186
$670$	0	0
$671$	−0.981718	−0.0378988
$672$	0	0
$673$	34.0081	1.31092	0.655459	−	0.755231i	$-0.272475\pi$
$673$	34.0081	1.31092	0.655459	+	0.755231i	$0.272475\pi$
$674$	0	0
$675$	−22.8935	−0.881170
$676$	0	0
$677$	−6.53430	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$677$	−6.53430	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$678$	0	0
$679$	−14.7934	−0.567718
$680$	0	0
$681$	25.4999	0.977160
$682$	0	0
$683$	17.7857	0.680550	0.340275	−	0.940326i	$-0.389480\pi$
$683$	17.7857	0.680550	0.340275	+	0.940326i	$0.389480\pi$
$684$	0	0
$685$	7.56591	0.289078
$686$	0	0
$687$	18.5909	0.709286
$688$	0	0
$689$	4.65551	0.177361
$690$	0	0
$691$	−21.9647	−0.835577	−0.417789	−	0.908544i	$-0.637195\pi$
$691$	−21.9647	−0.835577	−0.417789	+	0.908544i	$0.637195\pi$
$692$	0	0
$693$	−12.9027	−0.490133
$694$	0	0
$695$	5.68842	0.215774
$696$	0	0
$697$	8.86332	0.335722
$698$	0	0
$699$	−8.31952	−0.314673
$700$	0	0
$701$	−35.2009	−1.32952	−0.664759	−	0.747058i	$-0.731466\pi$
$701$	−35.2009	−1.32952	−0.664759	+	0.747058i	$0.731466\pi$
$702$	0	0
$703$	−48.7605	−1.83904
$704$	0	0
$705$	4.38748	0.165242
$706$	0	0
$707$	3.81256	0.143386
$708$	0	0
$709$	−51.5579	−1.93630	−0.968148	−	0.250377i	$-0.919445\pi$
$709$	−51.5579	−1.93630	−0.968148	+	0.250377i	$0.919445\pi$
$710$	0	0
$711$	−24.4035	−0.915205
$712$	0	0
$713$	−0.187184	−0.00701010
$714$	0	0
$715$	−3.43374	−0.128415
$716$	0	0
$717$	−6.37862	−0.238214
$718$	0	0
$719$	−30.5180	−1.13813	−0.569066	−	0.822292i	$-0.692695\pi$
$719$	−30.5180	−1.13813	−0.569066	+	0.822292i	$0.692695\pi$
$720$	0	0
$721$	18.0205	0.671120
$722$	0	0
$723$	−10.2486	−0.381150
$724$	0	0
$725$	34.0317	1.26391
$726$	0	0
$727$	−18.4999	−0.686124	−0.343062	−	0.939313i	$-0.611464\pi$
$727$	−18.4999	−0.686124	−0.343062	+	0.939313i	$0.611464\pi$
$728$	0	0
$729$	14.9660	0.554295
$730$	0	0
$731$	2.34033	0.0865603
$732$	0	0
$733$	−15.4154	−0.569380	−0.284690	−	0.958620i	$-0.591891\pi$
$733$	−15.4154	−0.569380	−0.284690	+	0.958620i	$0.591891\pi$
$734$	0	0
$735$	3.01668	0.111272
$736$	0	0
$737$	−11.6802	−0.430247
$738$	0	0
$739$	−18.6664	−0.686654	−0.343327	−	0.939216i	$-0.611554\pi$
$739$	−18.6664	−0.686654	−0.343327	+	0.939216i	$0.611554\pi$
$740$	0	0
$741$	−7.33582	−0.269488
$742$	0	0
$743$	47.8158	1.75419	0.877095	−	0.480316i	$-0.159478\pi$
$743$	47.8158	1.75419	0.877095	+	0.480316i	$0.159478\pi$
$744$	0	0
$745$	0.955525	0.0350077
$746$	0	0
$747$	−1.56881	−0.0573999
$748$	0	0
$749$	−28.5934	−1.04478
$750$	0	0
$751$	39.2227	1.43126	0.715629	−	0.698481i	$-0.246140\pi$
$751$	39.2227	1.43126	0.715629	+	0.698481i	$0.246140\pi$
$752$	0	0
$753$	−1.41482	−0.0515587
$754$	0	0
$755$	5.82301	0.211921
$756$	0	0
$757$	−53.6293	−1.94919	−0.974594	−	0.223978i	$-0.928096\pi$
$757$	−53.6293	−1.94919	−0.974594	+	0.223978i	$0.928096\pi$
$758$	0	0
$759$	0.820260	0.0297735
$760$	0	0
$761$	−12.5163	−0.453714	−0.226857	−	0.973928i	$-0.572845\pi$
$761$	−12.5163	−0.453714	−0.226857	+	0.973928i	$0.572845\pi$
$762$	0	0
$763$	−6.22529	−0.225371
$764$	0	0
$765$	−1.38395	−0.0500370
$766$	0	0
$767$	1.22436	0.0442089
$768$	0	0
$769$	26.1264	0.942142	0.471071	−	0.882095i	$-0.343868\pi$
$769$	26.1264	0.942142	0.471071	+	0.882095i	$0.343868\pi$
$770$	0	0
$771$	−13.5001	−0.486193
$772$	0	0
$773$	21.2774	0.765296	0.382648	−	0.923894i	$-0.375012\pi$
$773$	21.2774	0.765296	0.382648	+	0.923894i	$0.375012\pi$
$774$	0	0
$775$	4.13759	0.148627
$776$	0	0
$777$	−15.0415	−0.539609
$778$	0	0
$779$	51.2116	1.83485
$780$	0	0
$781$	−49.7486	−1.78014
$782$	0	0
$783$	−38.7673	−1.38543
$784$	0	0
$785$	0.0152683	0.000544950 0
$786$	0	0
$787$	21.6402	0.771389	0.385694	−	0.922627i	$-0.373962\pi$
$787$	21.6402	0.771389	0.385694	+	0.922627i	$0.373962\pi$
$788$	0	0
$789$	−13.0669	−0.465194
$790$	0	0
$791$	−16.0237	−0.569738
$792$	0	0
$793$	0.308175	0.0109436
$794$	0	0
$795$	−2.83525	−0.100556
$796$	0	0
$797$	5.82460	0.206318	0.103159	−	0.994665i	$-0.467105\pi$
$797$	5.82460	0.206318	0.103159	+	0.994665i	$0.467105\pi$
$798$	0	0
$799$	−5.88414	−0.208166
$800$	0	0
$801$	22.6176	0.799153
$802$	0	0
$803$	−3.08250	−0.108779
$804$	0	0
$805$	−0.250653	−0.00883437
$806$	0	0
$807$	−24.2222	−0.852662
$808$	0	0
$809$	−1.68422	−0.0592140	−0.0296070	−	0.999562i	$-0.509426\pi$
$809$	−1.68422	−0.0592140	−0.0296070	+	0.999562i	$0.509426\pi$
$810$	0	0
$811$	−39.4173	−1.38413	−0.692065	−	0.721836i	$-0.743299\pi$
$811$	−39.4173	−1.38413	−0.692065	+	0.721836i	$0.743299\pi$
$812$	0	0
$813$	−0.480544	−0.0168534
$814$	0	0
$815$	13.4238	0.470217
$816$	0	0
$817$	13.5223	0.473084
$818$	0	0
$819$	4.05034	0.141530
$820$	0	0
$821$	−29.4778	−1.02878	−0.514391	−	0.857556i	$-0.671982\pi$
$821$	−29.4778	−1.02878	−0.514391	+	0.857556i	$0.671982\pi$
$822$	0	0
$823$	−4.95491	−0.172717	−0.0863587	−	0.996264i	$-0.527523\pi$
$823$	−4.95491	−0.172717	−0.0863587	+	0.996264i	$0.527523\pi$
$824$	0	0
$825$	−18.1314	−0.631253
$826$	0	0
$827$	4.28994	0.149176	0.0745879	−	0.997214i	$-0.476236\pi$
$827$	4.28994	0.149176	0.0745879	+	0.997214i	$0.476236\pi$
$828$	0	0
$829$	−21.8277	−0.758106	−0.379053	−	0.925375i	$-0.623750\pi$
$829$	−21.8277	−0.758106	−0.379053	+	0.925375i	$0.623750\pi$
$830$	0	0
$831$	−15.8389	−0.549446
$832$	0	0
$833$	−4.04573	−0.140176
$834$	0	0
$835$	9.82460	0.339994
$836$	0	0
$837$	−4.71335	−0.162917
$838$	0	0
$839$	−44.6572	−1.54174	−0.770869	−	0.636994i	$-0.780178\pi$
$839$	−44.6572	−1.54174	−0.770869	+	0.636994i	$0.780178\pi$
$840$	0	0
$841$	28.6286	0.987192
$842$	0	0
$843$	−31.2733	−1.07711
$844$	0	0
$845$	−8.26983	−0.284491
$846$	0	0
$847$	−7.24005	−0.248771
$848$	0	0
$849$	3.07982	0.105699
$850$	0	0
$851$	−1.71152	−0.0586702
$852$	0	0
$853$	−0.296088	−0.0101379	−0.00506893	−	0.999987i	$-0.501613\pi$
$853$	−0.296088	−0.0101379	−0.00506893	+	0.999987i	$0.501613\pi$
$854$	0	0
$855$	−7.99639	−0.273471
$856$	0	0
$857$	−4.28483	−0.146367	−0.0731835	−	0.997318i	$-0.523316\pi$
$857$	−4.28483	−0.146367	−0.0731835	+	0.997318i	$0.523316\pi$
$858$	0	0
$859$	14.0579	0.479648	0.239824	−	0.970816i	$-0.422910\pi$
$859$	14.0579	0.479648	0.239824	+	0.970816i	$0.422910\pi$
$860$	0	0
$861$	15.7976	0.538380
$862$	0	0
$863$	−18.0299	−0.613746	−0.306873	−	0.951750i	$-0.599283\pi$
$863$	−18.0299	−0.613746	−0.306873	+	0.951750i	$0.599283\pi$
$864$	0	0
$865$	−2.07892	−0.0706854
$866$	0	0
$867$	−1.03698	−0.0352176
$868$	0	0
$869$	−49.4529	−1.67757
$870$	0	0
$871$	3.66659	0.124238
$872$	0	0
$873$	−16.5654	−0.560653
$874$	0	0
$875$	11.7201	0.396212
$876$	0	0
$877$	5.17546	0.174763	0.0873815	−	0.996175i	$-0.472150\pi$
$877$	5.17546	0.174763	0.0873815	+	0.996175i	$0.472150\pi$
$878$	0	0
$879$	−18.8070	−0.634345
$880$	0	0
$881$	58.5456	1.97245	0.986226	−	0.165403i	$-0.0528925\pi$
$881$	58.5456	1.97245	0.986226	+	0.165403i	$0.0528925\pi$
$882$	0	0
$883$	−36.0853	−1.21437	−0.607184	−	0.794561i	$-0.707701\pi$
$883$	−36.0853	−1.21437	−0.607184	+	0.794561i	$0.707701\pi$
$884$	0	0
$885$	−0.745644	−0.0250646
$886$	0	0
$887$	56.6358	1.90164	0.950821	−	0.309740i	$-0.100242\pi$
$887$	56.6358	1.90164	0.950821	+	0.309740i	$0.100242\pi$
$888$	0	0
$889$	5.68389	0.190631
$890$	0	0
$891$	−1.86604	−0.0625147
$892$	0	0
$893$	−33.9981	−1.13770
$894$	0	0
$895$	6.83247	0.228384
$896$	0	0
$897$	−0.257491	−0.00859738
$898$	0	0
$899$	7.00652	0.233680
$900$	0	0
$901$	3.80241	0.126677
$902$	0	0
$903$	4.17129	0.138812
$904$	0	0
$905$	−8.35729	−0.277806
$906$	0	0
$907$	−27.1879	−0.902760	−0.451380	−	0.892332i	$-0.649068\pi$
$907$	−27.1879	−0.902760	−0.451380	+	0.892332i	$0.649068\pi$
$908$	0	0
$909$	4.26924	0.141602
$910$	0	0
$911$	0.331822	0.0109938	0.00549688	−	0.999985i	$-0.498250\pi$
$911$	0.331822	0.0109938	0.00549688	+	0.999985i	$0.498250\pi$
$912$	0	0
$913$	−3.17914	−0.105214
$914$	0	0
$915$	−0.187682	−0.00620456
$916$	0	0
$917$	−34.6017	−1.14265
$918$	0	0
$919$	−53.5715	−1.76716	−0.883580	−	0.468281i	$-0.844874\pi$
$919$	−53.5715	−1.76716	−0.883580	+	0.468281i	$0.844874\pi$
$920$	0	0
$921$	−1.08487	−0.0357476
$922$	0	0
$923$	15.6168	0.514032
$924$	0	0
$925$	37.8322	1.24391
$926$	0	0
$927$	20.1791	0.662768
$928$	0	0
$929$	−14.7924	−0.485323	−0.242661	−	0.970111i	$-0.578020\pi$
$929$	−14.7924	−0.485323	−0.242661	+	0.970111i	$0.578020\pi$
$930$	0	0
$931$	−23.3760	−0.766116
$932$	0	0
$933$	13.1441	0.430320
$934$	0	0
$935$	−2.80453	−0.0917180
$936$	0	0
$937$	60.5047	1.97660	0.988301	−	0.152518i	$-0.0487382\pi$
$937$	60.5047	1.97660	0.988301	+	0.152518i	$0.0487382\pi$
$938$	0	0
$939$	2.35553	0.0768697
$940$	0	0
$941$	47.6771	1.55423	0.777114	−	0.629360i	$-0.216683\pi$
$941$	47.6771	1.55423	0.777114	+	0.629360i	$0.216683\pi$
$942$	0	0
$943$	1.79756	0.0585364
$944$	0	0
$945$	−6.31153	−0.205314
$946$	0	0
$947$	30.7128	0.998033	0.499017	−	0.866592i	$-0.333695\pi$
$947$	30.7128	0.998033	0.499017	+	0.866592i	$0.333695\pi$
$948$	0	0
$949$	0.967641	0.0314109
$950$	0	0
$951$	−7.05966	−0.228925
$952$	0	0
$953$	20.6616	0.669294	0.334647	−	0.942343i	$-0.391383\pi$
$953$	20.6616	0.669294	0.334647	+	0.942343i	$0.391383\pi$
$954$	0	0
$955$	−11.8462	−0.383333
$956$	0	0
$957$	−30.7033	−0.992496
$958$	0	0
$959$	−18.0852	−0.584001
$960$	0	0
$961$	−30.1481	−0.972521
$962$	0	0
$963$	−32.0184	−1.03178
$964$	0	0
$965$	−13.9324	−0.448499
$966$	0	0
$967$	51.2704	1.64875	0.824373	−	0.566047i	$-0.191528\pi$
$967$	51.2704	1.64875	0.824373	+	0.566047i	$0.191528\pi$
$968$	0	0
$969$	−5.99157	−0.192477
$970$	0	0
$971$	39.3634	1.26323	0.631616	−	0.775282i	$-0.282392\pi$
$971$	39.3634	1.26323	0.631616	+	0.775282i	$0.282392\pi$
$972$	0	0
$973$	−13.5973	−0.435911
$974$	0	0
$975$	5.69169	0.182280
$976$	0	0
$977$	40.5285	1.29662	0.648310	−	0.761376i	$-0.275476\pi$
$977$	40.5285	1.29662	0.648310	+	0.761376i	$0.275476\pi$
$978$	0	0
$979$	45.8337	1.46485
$980$	0	0
$981$	−6.97097	−0.222566
$982$	0	0
$983$	−11.6222	−0.370692	−0.185346	−	0.982673i	$-0.559341\pi$
$983$	−11.6222	−0.370692	−0.185346	+	0.982673i	$0.559341\pi$
$984$	0	0
$985$	13.5504	0.431753
$986$	0	0
$987$	−10.4876	−0.333824
$988$	0	0
$989$	0.474638	0.0150926
$990$	0	0
$991$	2.27502	0.0722685	0.0361343	−	0.999347i	$-0.488496\pi$
$991$	2.27502	0.0722685	0.0361343	+	0.999347i	$0.488496\pi$
$992$	0	0
$993$	1.67374	0.0531146
$994$	0	0
$995$	12.2016	0.386816
$996$	0	0
$997$	−14.5470	−0.460709	−0.230354	−	0.973107i	$-0.573989\pi$
$997$	−14.5470	−0.460709	−0.230354	+	0.973107i	$0.573989\pi$
$998$	0	0
$999$	−43.0966	−1.36352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.7	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.7	✓	21	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-1.03698 q^{3} +0.719057 q^{5} -1.71880 q^{7} -1.92468 q^{9} +O(q^{10})\)	\(q-1.03698 q^{3} +0.719057 q^{5} -1.71880 q^{7} -1.92468 q^{9} -3.90029 q^{11} +1.22436 q^{13} -0.745644 q^{15} +1.00000 q^{17} +5.77793 q^{19} +1.78235 q^{21} +0.202808 q^{23} -4.48296 q^{25} +5.10678 q^{27} -7.59135 q^{29} -0.922961 q^{31} +4.04451 q^{33} -1.23591 q^{35} -8.43911 q^{37} -1.26963 q^{39} +8.86332 q^{41} +2.34033 q^{43} -1.38395 q^{45} -5.88414 q^{47} -4.04573 q^{49} -1.03698 q^{51} +3.80241 q^{53} -2.80453 q^{55} -5.99157 q^{57} +1.00000 q^{59} +0.251704 q^{61} +3.30814 q^{63} +0.880381 q^{65} +2.99471 q^{67} -0.210307 q^{69} +12.7551 q^{71} +0.790326 q^{73} +4.64872 q^{75} +6.70381 q^{77} +12.6793 q^{79} +0.478436 q^{81} +0.815103 q^{83} +0.719057 q^{85} +7.87205 q^{87} -11.7513 q^{89} -2.10442 q^{91} +0.957088 q^{93} +4.15466 q^{95} +8.60682 q^{97} +7.50682 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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