LMFDB - Embedded newform 672.4.a.q.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,4,Mod(1,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,9,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.22700.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 28x + 12 $ x^3 - x^2 - 28*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.03475$ of defining polynomial
Character	$\chi$	$=$	672.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -4.31394 q^{5} -7.00000 q^{7} +9.00000 q^{9} +41.9640 q^{11} +26.2780 q^{13} -12.9418 q^{15} +57.8477 q^{17} -2.16162 q^{19} -21.0000 q^{21} -200.960 q^{23} -106.390 q^{25} +27.0000 q^{27} -121.274 q^{29} +279.830 q^{31} +125.892 q^{33} +30.1976 q^{35} +124.322 q^{37} +78.8340 q^{39} +363.355 q^{41} +253.488 q^{43} -38.8255 q^{45} +84.6279 q^{47} +49.0000 q^{49} +173.543 q^{51} +507.104 q^{53} -181.030 q^{55} -6.48487 q^{57} +338.027 q^{59} -529.831 q^{61} -63.0000 q^{63} -113.362 q^{65} +429.884 q^{67} -602.879 q^{69} +1147.54 q^{71} +490.466 q^{73} -319.170 q^{75} -293.748 q^{77} +645.131 q^{79} +81.0000 q^{81} +974.314 q^{83} -249.551 q^{85} -363.821 q^{87} -302.494 q^{89} -183.946 q^{91} +839.489 q^{93} +9.32512 q^{95} -1142.65 q^{97} +377.676 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 10 q^{5} - 21 q^{7} + 27 q^{9} - 50 q^{13} - 30 q^{15} + 30 q^{17} + 140 q^{19} - 63 q^{21} + 56 q^{23} + 325 q^{25} + 81 q^{27} + 298 q^{29} - 80 q^{31} + 70 q^{35} + 10 q^{37} - 150 q^{39}+ \cdots - 130 q^{97}+O(q^{100}) $ 3 * q + 9 * q^3 - 10 * q^5 - 21 * q^7 + 27 * q^9 - 50 * q^13 - 30 * q^15 + 30 * q^17 + 140 * q^19 - 63 * q^21 + 56 * q^23 + 325 * q^25 + 81 * q^27 + 298 * q^29 - 80 * q^31 + 70 * q^35 + 10 * q^37 - 150 * q^39 + 390 * q^41 + 784 * q^43 - 90 * q^45 + 248 * q^47 + 147 * q^49 + 90 * q^51 + 10 * q^53 + 1360 * q^55 + 420 * q^57 + 1500 * q^59 - 810 * q^61 - 189 * q^63 + 860 * q^65 + 1272 * q^67 + 168 * q^69 + 160 * q^71 - 1170 * q^73 + 975 * q^75 + 840 * q^79 + 243 * q^81 + 1564 * q^83 - 2740 * q^85 + 894 * q^87 - 178 * q^89 + 350 * q^91 - 240 * q^93 + 2840 * q^95 - 130 * q^97

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−4.31394	−0.385851	−0.192925	−	0.981213i	$-0.561797\pi$
$5$	−4.31394	−0.385851	−0.192925	+	0.981213i	$0.561797\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	41.9640	1.15024	0.575120	−	0.818069i	$-0.304955\pi$
$11$	41.9640	1.15024	0.575120	+	0.818069i	$0.304955\pi$
$12$	0	0
$13$	26.2780	0.560631	0.280315	−	0.959908i	$-0.409561\pi$
$13$	26.2780	0.560631	0.280315	+	0.959908i	$0.409561\pi$
$14$	0	0
$15$	−12.9418	−0.222771
$16$	0	0
$17$	57.8477	0.825301	0.412651	−	0.910889i	$-0.364603\pi$
$17$	57.8477	0.825301	0.412651	+	0.910889i	$0.364603\pi$
$18$	0	0
$19$	−2.16162	−0.0261006	−0.0130503	−	0.999915i	$-0.504154\pi$
$19$	−2.16162	−0.0261006	−0.0130503	+	0.999915i	$0.504154\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−200.960	−1.82187	−0.910934	−	0.412551i	$-0.864638\pi$
$23$	−200.960	−1.82187	−0.910934	+	0.412551i	$0.864638\pi$
$24$	0	0
$25$	−106.390	−0.851119
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−121.274	−0.776549	−0.388275	−	0.921544i	$-0.626929\pi$
$29$	−121.274	−0.776549	−0.388275	+	0.921544i	$0.626929\pi$
$30$	0	0
$31$	279.830	1.62125	0.810627	−	0.585563i	$-0.199127\pi$
$31$	279.830	1.62125	0.810627	+	0.585563i	$0.199127\pi$
$32$	0	0
$33$	125.892	0.664091
$34$	0	0
$35$	30.1976	0.145838
$36$	0	0
$37$	124.322	0.552391	0.276196	−	0.961101i	$-0.410926\pi$
$37$	124.322	0.552391	0.276196	+	0.961101i	$0.410926\pi$
$38$	0	0
$39$	78.8340	0.323680
$40$	0	0
$41$	363.355	1.38406	0.692030	−	0.721868i	$-0.256716\pi$
$41$	363.355	1.38406	0.692030	+	0.721868i	$0.256716\pi$
$42$	0	0
$43$	253.488	0.898991	0.449496	−	0.893282i	$-0.351604\pi$
$43$	253.488	0.898991	0.449496	+	0.893282i	$0.351604\pi$
$44$	0	0
$45$	−38.8255	−0.128617
$46$	0	0
$47$	84.6279	0.262644	0.131322	−	0.991340i	$-0.458078\pi$
$47$	84.6279	0.262644	0.131322	+	0.991340i	$0.458078\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	173.543	0.476488
$52$	0	0
$53$	507.104	1.31427	0.657133	−	0.753775i	$-0.271769\pi$
$53$	507.104	1.31427	0.657133	+	0.753775i	$0.271769\pi$
$54$	0	0
$55$	−181.030	−0.443821
$56$	0	0
$57$	−6.48487	−0.0150692
$58$	0	0
$59$	338.027	0.745889	0.372944	−	0.927854i	$-0.378348\pi$
$59$	338.027	0.745889	0.372944	+	0.927854i	$0.378348\pi$
$60$	0	0
$61$	−529.831	−1.11210	−0.556049	−	0.831150i	$-0.687683\pi$
$61$	−529.831	−1.11210	−0.556049	+	0.831150i	$0.687683\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−113.362	−0.216320
$66$	0	0
$67$	429.884	0.783860	0.391930	−	0.919995i	$-0.371807\pi$
$67$	429.884	0.783860	0.391930	+	0.919995i	$0.371807\pi$
$68$	0	0
$69$	−602.879	−1.05186
$70$	0	0
$71$	1147.54	1.91814	0.959072	−	0.283162i	$-0.0913835\pi$
$71$	1147.54	1.91814	0.959072	+	0.283162i	$0.0913835\pi$
$72$	0	0
$73$	490.466	0.786366	0.393183	−	0.919460i	$-0.371374\pi$
$73$	490.466	0.786366	0.393183	+	0.919460i	$0.371374\pi$
$74$	0	0
$75$	−319.170	−0.491394
$76$	0	0
$77$	−293.748	−0.434750
$78$	0	0
$79$	645.131	0.918770	0.459385	−	0.888237i	$-0.348070\pi$
$79$	645.131	0.918770	0.459385	+	0.888237i	$0.348070\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	974.314	1.28849	0.644246	−	0.764818i	$-0.277171\pi$
$83$	974.314	1.28849	0.644246	+	0.764818i	$0.277171\pi$
$84$	0	0
$85$	−249.551	−0.318443
$86$	0	0
$87$	−363.821	−0.448341
$88$	0	0
$89$	−302.494	−0.360273	−0.180137	−	0.983642i	$-0.557654\pi$
$89$	−302.494	−0.360273	−0.180137	+	0.983642i	$0.557654\pi$
$90$	0	0
$91$	−183.946	−0.211899
$92$	0	0
$93$	839.489	0.936031
$94$	0	0
$95$	9.32512	0.0100709
$96$	0	0
$97$	−1142.65	−1.19606	−0.598032	−	0.801472i	$-0.704051\pi$
$97$	−1142.65	−1.19606	−0.598032	+	0.801472i	$0.704051\pi$
$98$	0	0
$99$	377.676	0.383413
$100$	0	0
$101$	−1022.18	−1.00703	−0.503517	−	0.863985i	$-0.667961\pi$
$101$	−1022.18	−1.00703	−0.503517	+	0.863985i	$0.667961\pi$
$102$	0	0
$103$	422.343	0.404026	0.202013	−	0.979383i	$-0.435252\pi$
$103$	422.343	0.404026	0.202013	+	0.979383i	$0.435252\pi$
$104$	0	0
$105$	90.5927	0.0841995
$106$	0	0
$107$	573.263	0.517938	0.258969	−	0.965886i	$-0.416617\pi$
$107$	573.263	0.517938	0.258969	+	0.965886i	$0.416617\pi$
$108$	0	0
$109$	−1568.49	−1.37829	−0.689147	−	0.724621i	$-0.742015\pi$
$109$	−1568.49	−1.37829	−0.689147	+	0.724621i	$0.742015\pi$
$110$	0	0
$111$	372.967	0.318923
$112$	0	0
$113$	236.274	0.196697	0.0983486	−	0.995152i	$-0.468644\pi$
$113$	236.274	0.196697	0.0983486	+	0.995152i	$0.468644\pi$
$114$	0	0
$115$	866.928	0.702969
$116$	0	0
$117$	236.502	0.186877
$118$	0	0
$119$	−404.934	−0.311935
$120$	0	0
$121$	429.981	0.323051
$122$	0	0
$123$	1090.06	0.799088
$124$	0	0
$125$	998.202	0.714255
$126$	0	0
$127$	−1266.32	−0.884782	−0.442391	−	0.896822i	$-0.645870\pi$
$127$	−1266.32	−0.884782	−0.442391	+	0.896822i	$0.645870\pi$
$128$	0	0
$129$	760.465	0.519033
$130$	0	0
$131$	−591.805	−0.394704	−0.197352	−	0.980333i	$-0.563234\pi$
$131$	−591.805	−0.394704	−0.197352	+	0.980333i	$0.563234\pi$
$132$	0	0
$133$	15.1314	0.00986508
$134$	0	0
$135$	−116.476	−0.0742570
$136$	0	0
$137$	2621.03	1.63452	0.817262	−	0.576266i	$-0.195491\pi$
$137$	2621.03	1.63452	0.817262	+	0.576266i	$0.195491\pi$
$138$	0	0
$139$	1947.48	1.18837	0.594183	−	0.804330i	$-0.297475\pi$
$139$	1947.48	1.18837	0.594183	+	0.804330i	$0.297475\pi$
$140$	0	0
$141$	253.884	0.151637
$142$	0	0
$143$	1102.73	0.644860
$144$	0	0
$145$	523.167	0.299632
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−660.857	−0.363353	−0.181676	−	0.983358i	$-0.558152\pi$
$149$	−660.857	−0.363353	−0.181676	+	0.983358i	$0.558152\pi$
$150$	0	0
$151$	2051.46	1.10560	0.552800	−	0.833314i	$-0.313559\pi$
$151$	2051.46	1.10560	0.552800	+	0.833314i	$0.313559\pi$
$152$	0	0
$153$	520.629	0.275100
$154$	0	0
$155$	−1207.17	−0.625562
$156$	0	0
$157$	−2673.81	−1.35919	−0.679596	−	0.733587i	$-0.737845\pi$
$157$	−2673.81	−1.35919	−0.679596	+	0.733587i	$0.737845\pi$
$158$	0	0
$159$	1521.31	0.758792
$160$	0	0
$161$	1406.72	0.688602
$162$	0	0
$163$	−3318.17	−1.59447	−0.797236	−	0.603667i	$-0.793705\pi$
$163$	−3318.17	−1.59447	−0.797236	+	0.603667i	$0.793705\pi$
$164$	0	0
$165$	−543.091	−0.256240
$166$	0	0
$167$	2239.35	1.03764	0.518821	−	0.854883i	$-0.326371\pi$
$167$	2239.35	1.03764	0.518821	+	0.854883i	$0.326371\pi$
$168$	0	0
$169$	−1506.47	−0.685693
$170$	0	0
$171$	−19.4546	−0.00870019
$172$	0	0
$173$	−1656.63	−0.728040	−0.364020	−	0.931391i	$-0.618596\pi$
$173$	−1656.63	−0.728040	−0.364020	+	0.931391i	$0.618596\pi$
$174$	0	0
$175$	744.729	0.321693
$176$	0	0
$177$	1014.08	0.430639
$178$	0	0
$179$	−814.515	−0.340110	−0.170055	−	0.985435i	$-0.554395\pi$
$179$	−814.515	−0.340110	−0.170055	+	0.985435i	$0.554395\pi$
$180$	0	0
$181$	−2612.41	−1.07281	−0.536406	−	0.843960i	$-0.680219\pi$
$181$	−2612.41	−1.07281	−0.536406	+	0.843960i	$0.680219\pi$
$182$	0	0
$183$	−1589.49	−0.642070
$184$	0	0
$185$	−536.320	−0.213141
$186$	0	0
$187$	2427.52	0.949294
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	252.729	0.0957425	0.0478712	−	0.998854i	$-0.484756\pi$
$191$	252.729	0.0957425	0.0478712	+	0.998854i	$0.484756\pi$
$192$	0	0
$193$	4460.67	1.66366	0.831829	−	0.555032i	$-0.187294\pi$
$193$	4460.67	1.66366	0.831829	+	0.555032i	$0.187294\pi$
$194$	0	0
$195$	−340.085	−0.124892
$196$	0	0
$197$	1499.31	0.542242	0.271121	−	0.962545i	$-0.412606\pi$
$197$	1499.31	0.542242	0.271121	+	0.962545i	$0.412606\pi$
$198$	0	0
$199$	454.108	0.161763	0.0808815	−	0.996724i	$-0.474226\pi$
$199$	454.108	0.161763	0.0808815	+	0.996724i	$0.474226\pi$
$200$	0	0
$201$	1289.65	0.452562
$202$	0	0
$203$	848.915	0.293508
$204$	0	0
$205$	−1567.49	−0.534040
$206$	0	0
$207$	−1808.64	−0.607290
$208$	0	0
$209$	−90.7105	−0.0300219
$210$	0	0
$211$	−2278.79	−0.743501	−0.371750	−	0.928333i	$-0.621242\pi$
$211$	−2278.79	−0.743501	−0.371750	+	0.928333i	$0.621242\pi$
$212$	0	0
$213$	3442.63	1.10744
$214$	0	0
$215$	−1093.53	−0.346876
$216$	0	0
$217$	−1958.81	−0.612776
$218$	0	0
$219$	1471.40	0.454009
$220$	0	0
$221$	1520.12	0.462689
$222$	0	0
$223$	−1412.43	−0.424141	−0.212070	−	0.977254i	$-0.568021\pi$
$223$	−1412.43	−0.424141	−0.212070	+	0.977254i	$0.568021\pi$
$224$	0	0
$225$	−957.509	−0.283706
$226$	0	0
$227$	−3027.07	−0.885082	−0.442541	−	0.896748i	$-0.645923\pi$
$227$	−3027.07	−0.885082	−0.442541	+	0.896748i	$0.645923\pi$
$228$	0	0
$229$	−6.14697	−0.00177381	−0.000886906	−	1.00000i	$-0.500282\pi$
$229$	−6.14697	−0.00177381	−0.000886906		1.00000i	$0.500282\pi$
$230$	0	0
$231$	−881.245	−0.251003
$232$	0	0
$233$	2385.66	0.670771	0.335386	−	0.942081i	$-0.391133\pi$
$233$	2385.66	0.670771	0.335386	+	0.942081i	$0.391133\pi$
$234$	0	0
$235$	−365.080	−0.101341
$236$	0	0
$237$	1935.39	0.530452
$238$	0	0
$239$	−1606.40	−0.434768	−0.217384	−	0.976086i	$-0.569752\pi$
$239$	−1606.40	−0.434768	−0.217384	+	0.976086i	$0.569752\pi$
$240$	0	0
$241$	835.200	0.223236	0.111618	−	0.993751i	$-0.464397\pi$
$241$	835.200	0.223236	0.111618	+	0.993751i	$0.464397\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−211.383	−0.0551215
$246$	0	0
$247$	−56.8031	−0.0146328
$248$	0	0
$249$	2922.94	0.743912
$250$	0	0
$251$	−1326.03	−0.333459	−0.166730	−	0.986003i	$-0.553321\pi$
$251$	−1326.03	−0.333459	−0.166730	+	0.986003i	$0.553321\pi$
$252$	0	0
$253$	−8433.08	−2.09559
$254$	0	0
$255$	−748.654	−0.183853
$256$	0	0
$257$	5210.12	1.26458	0.632292	−	0.774730i	$-0.282114\pi$
$257$	5210.12	1.26458	0.632292	+	0.774730i	$0.282114\pi$
$258$	0	0
$259$	−870.257	−0.208784
$260$	0	0
$261$	−1091.46	−0.258850
$262$	0	0
$263$	2478.00	0.580988	0.290494	−	0.956877i	$-0.406180\pi$
$263$	2478.00	0.580988	0.290494	+	0.956877i	$0.406180\pi$
$264$	0	0
$265$	−2187.62	−0.507110
$266$	0	0
$267$	−907.483	−0.208004
$268$	0	0
$269$	−4990.25	−1.13108	−0.565540	−	0.824721i	$-0.691332\pi$
$269$	−4990.25	−1.13108	−0.565540	+	0.824721i	$0.691332\pi$
$270$	0	0
$271$	−6860.80	−1.53787	−0.768937	−	0.639325i	$-0.779214\pi$
$271$	−6860.80	−1.53787	−0.768937	+	0.639325i	$0.779214\pi$
$272$	0	0
$273$	−551.838	−0.122340
$274$	0	0
$275$	−4464.55	−0.978991
$276$	0	0
$277$	781.578	0.169532	0.0847662	−	0.996401i	$-0.472986\pi$
$277$	781.578	0.169532	0.0847662	+	0.996401i	$0.472986\pi$
$278$	0	0
$279$	2518.47	0.540418
$280$	0	0
$281$	5447.06	1.15638	0.578192	−	0.815900i	$-0.303758\pi$
$281$	5447.06	1.15638	0.578192	+	0.815900i	$0.303758\pi$
$282$	0	0
$283$	−6008.83	−1.26215	−0.631074	−	0.775723i	$-0.717385\pi$
$283$	−6008.83	−1.26215	−0.631074	+	0.775723i	$0.717385\pi$
$284$	0	0
$285$	27.9753	0.00581444
$286$	0	0
$287$	−2543.48	−0.523126
$288$	0	0
$289$	−1566.65	−0.318878
$290$	0	0
$291$	−3427.94	−0.690548
$292$	0	0
$293$	−5840.09	−1.16444	−0.582221	−	0.813030i	$-0.697816\pi$
$293$	−5840.09	−1.16444	−0.582221	+	0.813030i	$0.697816\pi$
$294$	0	0
$295$	−1458.23	−0.287802
$296$	0	0
$297$	1133.03	0.221364
$298$	0	0
$299$	−5280.81	−1.02140
$300$	0	0
$301$	−1774.42	−0.339787
$302$	0	0
$303$	−3066.53	−0.581411
$304$	0	0
$305$	2285.66	0.429103
$306$	0	0
$307$	8443.25	1.56965	0.784824	−	0.619719i	$-0.212753\pi$
$307$	8443.25	1.56965	0.784824	+	0.619719i	$0.212753\pi$
$308$	0	0
$309$	1267.03	0.233264
$310$	0	0
$311$	5326.08	0.971106	0.485553	−	0.874207i	$-0.338618\pi$
$311$	5326.08	0.971106	0.485553	+	0.874207i	$0.338618\pi$
$312$	0	0
$313$	−8820.06	−1.59278	−0.796388	−	0.604786i	$-0.793259\pi$
$313$	−8820.06	−1.59278	−0.796388	+	0.604786i	$0.793259\pi$
$314$	0	0
$315$	271.778	0.0486126
$316$	0	0
$317$	−31.9617	−0.00566293	−0.00283147	−	0.999996i	$-0.500901\pi$
$317$	−31.9617	−0.00566293	−0.00283147	+	0.999996i	$0.500901\pi$
$318$	0	0
$319$	−5089.13	−0.893218
$320$	0	0
$321$	1719.79	0.299032
$322$	0	0
$323$	−125.045	−0.0215408
$324$	0	0
$325$	−2795.71	−0.477164
$326$	0	0
$327$	−4705.47	−0.795759
$328$	0	0
$329$	−592.395	−0.0992699
$330$	0	0
$331$	943.305	0.156643	0.0783214	−	0.996928i	$-0.475044\pi$
$331$	943.305	0.156643	0.0783214	+	0.996928i	$0.475044\pi$
$332$	0	0
$333$	1118.90	0.184130
$334$	0	0
$335$	−1854.49	−0.302453
$336$	0	0
$337$	−2994.01	−0.483958	−0.241979	−	0.970281i	$-0.577797\pi$
$337$	−2994.01	−0.483958	−0.241979	+	0.970281i	$0.577797\pi$
$338$	0	0
$339$	708.822	0.113563
$340$	0	0
$341$	11742.8	1.86483
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	2600.78	0.405859
$346$	0	0
$347$	−756.781	−0.117078	−0.0585391	−	0.998285i	$-0.518644\pi$
$347$	−756.781	−0.117078	−0.0585391	+	0.998285i	$0.518644\pi$
$348$	0	0
$349$	−6388.02	−0.979778	−0.489889	−	0.871785i	$-0.662963\pi$
$349$	−6388.02	−0.979778	−0.489889	+	0.871785i	$0.662963\pi$
$350$	0	0
$351$	709.506	0.107893
$352$	0	0
$353$	315.429	0.0475597	0.0237799	−	0.999717i	$-0.492430\pi$
$353$	315.429	0.0475597	0.0237799	+	0.999717i	$0.492430\pi$
$354$	0	0
$355$	−4950.43	−0.740117
$356$	0	0
$357$	−1214.80	−0.180096
$358$	0	0
$359$	−6672.00	−0.980876	−0.490438	−	0.871476i	$-0.663163\pi$
$359$	−6672.00	−0.980876	−0.490438	+	0.871476i	$0.663163\pi$
$360$	0	0
$361$	−6854.33	−0.999319
$362$	0	0
$363$	1289.94	0.186514
$364$	0	0
$365$	−2115.84	−0.303420
$366$	0	0
$367$	23.3540	0.00332171	0.00166085	−	0.999999i	$-0.499471\pi$
$367$	23.3540	0.00332171	0.00166085	+	0.999999i	$0.499471\pi$
$368$	0	0
$369$	3270.19	0.461354
$370$	0	0
$371$	−3549.73	−0.496746
$372$	0	0
$373$	−6294.34	−0.873750	−0.436875	−	0.899522i	$-0.643915\pi$
$373$	−6294.34	−0.873750	−0.436875	+	0.899522i	$0.643915\pi$
$374$	0	0
$375$	2994.61	0.412376
$376$	0	0
$377$	−3186.82	−0.435358
$378$	0	0
$379$	−1311.59	−0.177762	−0.0888810	−	0.996042i	$-0.528329\pi$
$379$	−1311.59	−0.177762	−0.0888810	+	0.996042i	$0.528329\pi$
$380$	0	0
$381$	−3798.95	−0.510829
$382$	0	0
$383$	3519.66	0.469572	0.234786	−	0.972047i	$-0.424561\pi$
$383$	3519.66	0.469572	0.234786	+	0.972047i	$0.424561\pi$
$384$	0	0
$385$	1267.21	0.167748
$386$	0	0
$387$	2281.40	0.299664
$388$	0	0
$389$	12858.6	1.67598	0.837988	−	0.545689i	$-0.183732\pi$
$389$	12858.6	1.67598	0.837988	+	0.545689i	$0.183732\pi$
$390$	0	0
$391$	−11625.0	−1.50359
$392$	0	0
$393$	−1775.41	−0.227882
$394$	0	0
$395$	−2783.05	−0.354508
$396$	0	0
$397$	13317.7	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$397$	13317.7	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$398$	0	0
$399$	45.3941	0.00569561
$400$	0	0
$401$	12106.3	1.50763	0.753817	−	0.657085i	$-0.228211\pi$
$401$	12106.3	1.50763	0.753817	+	0.657085i	$0.228211\pi$
$402$	0	0
$403$	7353.36	0.908925
$404$	0	0
$405$	−349.429	−0.0428723
$406$	0	0
$407$	5217.07	0.635382
$408$	0	0
$409$	−12778.6	−1.54489	−0.772447	−	0.635079i	$-0.780968\pi$
$409$	−12778.6	−1.54489	−0.772447	+	0.635079i	$0.780968\pi$
$410$	0	0
$411$	7863.09	0.943693
$412$	0	0
$413$	−2366.19	−0.281919
$414$	0	0
$415$	−4203.13	−0.497166
$416$	0	0
$417$	5842.43	0.686104
$418$	0	0
$419$	−15045.6	−1.75423	−0.877117	−	0.480277i	$-0.840536\pi$
$419$	−15045.6	−1.75423	−0.877117	+	0.480277i	$0.840536\pi$
$420$	0	0
$421$	2782.26	0.322088	0.161044	−	0.986947i	$-0.448514\pi$
$421$	2782.26	0.322088	0.161044	+	0.986947i	$0.448514\pi$
$422$	0	0
$423$	761.651	0.0875478
$424$	0	0
$425$	−6154.41	−0.702430
$426$	0	0
$427$	3708.82	0.420333
$428$	0	0
$429$	3308.19	0.372310
$430$	0	0
$431$	5223.64	0.583791	0.291896	−	0.956450i	$-0.405714\pi$
$431$	5223.64	0.583791	0.291896	+	0.956450i	$0.405714\pi$
$432$	0	0
$433$	−4568.84	−0.507077	−0.253539	−	0.967325i	$-0.581594\pi$
$433$	−4568.84	−0.507077	−0.253539	+	0.967325i	$0.581594\pi$
$434$	0	0
$435$	1569.50	0.172993
$436$	0	0
$437$	434.399	0.0475518
$438$	0	0
$439$	380.473	0.0413644	0.0206822	−	0.999786i	$-0.493416\pi$
$439$	380.473	0.0413644	0.0206822	+	0.999786i	$0.493416\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−3397.79	−0.364411	−0.182205	−	0.983261i	$-0.558324\pi$
$443$	−3397.79	−0.364411	−0.182205	+	0.983261i	$0.558324\pi$
$444$	0	0
$445$	1304.94	0.139012
$446$	0	0
$447$	−1982.57	−0.209782
$448$	0	0
$449$	−16161.4	−1.69867	−0.849336	−	0.527853i	$-0.822997\pi$
$449$	−16161.4	−1.69867	−0.849336	+	0.527853i	$0.822997\pi$
$450$	0	0
$451$	15247.8	1.59200
$452$	0	0
$453$	6154.39	0.638319
$454$	0	0
$455$	793.532	0.0817612
$456$	0	0
$457$	8906.44	0.911653	0.455827	−	0.890069i	$-0.349344\pi$
$457$	8906.44	0.911653	0.455827	+	0.890069i	$0.349344\pi$
$458$	0	0
$459$	1561.89	0.158829
$460$	0	0
$461$	−3435.99	−0.347137	−0.173568	−	0.984822i	$-0.555530\pi$
$461$	−3435.99	−0.347137	−0.173568	+	0.984822i	$0.555530\pi$
$462$	0	0
$463$	7531.44	0.755974	0.377987	−	0.925811i	$-0.376616\pi$
$463$	7531.44	0.755974	0.377987	+	0.925811i	$0.376616\pi$
$464$	0	0
$465$	−3621.50	−0.361168
$466$	0	0
$467$	8808.26	0.872800	0.436400	−	0.899753i	$-0.356253\pi$
$467$	8808.26	0.872800	0.436400	+	0.899753i	$0.356253\pi$
$468$	0	0
$469$	−3009.19	−0.296271
$470$	0	0
$471$	−8021.43	−0.784730
$472$	0	0
$473$	10637.4	1.03406
$474$	0	0
$475$	229.975	0.0222147
$476$	0	0
$477$	4563.94	0.438089
$478$	0	0
$479$	978.352	0.0933237	0.0466618	−	0.998911i	$-0.485142\pi$
$479$	978.352	0.0933237	0.0466618	+	0.998911i	$0.485142\pi$
$480$	0	0
$481$	3266.94	0.309688
$482$	0	0
$483$	4220.15	0.397564
$484$	0	0
$485$	4929.31	0.461502
$486$	0	0
$487$	−7188.45	−0.668870	−0.334435	−	0.942419i	$-0.608546\pi$
$487$	−7188.45	−0.668870	−0.334435	+	0.942419i	$0.608546\pi$
$488$	0	0
$489$	−9954.51	−0.920569
$490$	0	0
$491$	6156.65	0.565877	0.282939	−	0.959138i	$-0.408691\pi$
$491$	6156.65	0.565877	0.282939	+	0.959138i	$0.408691\pi$
$492$	0	0
$493$	−7015.39	−0.640887
$494$	0	0
$495$	−1629.27	−0.147940
$496$	0	0
$497$	−8032.80	−0.724990
$498$	0	0
$499$	13832.0	1.24089	0.620445	−	0.784250i	$-0.286952\pi$
$499$	13832.0	1.24089	0.620445	+	0.784250i	$0.286952\pi$
$500$	0	0
$501$	6718.05	0.599083
$502$	0	0
$503$	16630.6	1.47420	0.737100	−	0.675783i	$-0.236195\pi$
$503$	16630.6	1.47420	0.737100	+	0.675783i	$0.236195\pi$
$504$	0	0
$505$	4409.61	0.388564
$506$	0	0
$507$	−4519.40	−0.395885
$508$	0	0
$509$	11795.0	1.02712	0.513562	−	0.858053i	$-0.328326\pi$
$509$	11795.0	1.02712	0.513562	+	0.858053i	$0.328326\pi$
$510$	0	0
$511$	−3433.26	−0.297219
$512$	0	0
$513$	−58.3638	−0.00502305
$514$	0	0
$515$	−1821.96	−0.155894
$516$	0	0
$517$	3551.33	0.302103
$518$	0	0
$519$	−4969.88	−0.420334
$520$	0	0
$521$	−597.161	−0.0502151	−0.0251076	−	0.999685i	$-0.507993\pi$
$521$	−597.161	−0.0502151	−0.0251076	+	0.999685i	$0.507993\pi$
$522$	0	0
$523$	−6287.96	−0.525724	−0.262862	−	0.964833i	$-0.584666\pi$
$523$	−6287.96	−0.525724	−0.262862	+	0.964833i	$0.584666\pi$
$524$	0	0
$525$	2234.19	0.185729
$526$	0	0
$527$	16187.5	1.33802
$528$	0	0
$529$	28217.8	2.31921
$530$	0	0
$531$	3042.25	0.248630
$532$	0	0
$533$	9548.23	0.775947
$534$	0	0
$535$	−2473.02	−0.199847
$536$	0	0
$537$	−2443.55	−0.196363
$538$	0	0
$539$	2056.24	0.164320
$540$	0	0
$541$	−6931.94	−0.550882	−0.275441	−	0.961318i	$-0.588824\pi$
$541$	−6931.94	−0.550882	−0.275441	+	0.961318i	$0.588824\pi$
$542$	0	0
$543$	−7837.24	−0.619389
$544$	0	0
$545$	6766.37	0.531816
$546$	0	0
$547$	−24713.6	−1.93177	−0.965884	−	0.258976i	$-0.916615\pi$
$547$	−24713.6	−1.93177	−0.965884	+	0.258976i	$0.916615\pi$
$548$	0	0
$549$	−4768.48	−0.370699
$550$	0	0
$551$	262.148	0.0202684
$552$	0	0
$553$	−4515.91	−0.347263
$554$	0	0
$555$	−1608.96	−0.123057
$556$	0	0
$557$	2598.91	0.197700	0.0988502	−	0.995102i	$-0.468484\pi$
$557$	2598.91	0.197700	0.0988502	+	0.995102i	$0.468484\pi$
$558$	0	0
$559$	6661.17	0.504002
$560$	0	0
$561$	7282.57	0.548075
$562$	0	0
$563$	−15673.3	−1.17327	−0.586636	−	0.809851i	$-0.699548\pi$
$563$	−15673.3	−1.17327	−0.586636	+	0.809851i	$0.699548\pi$
$564$	0	0
$565$	−1019.27	−0.0758957
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−23643.3	−1.74196	−0.870981	−	0.491316i	$-0.836516\pi$
$569$	−23643.3	−1.74196	−0.870981	+	0.491316i	$0.836516\pi$
$570$	0	0
$571$	11308.1	0.828772	0.414386	−	0.910101i	$-0.363996\pi$
$571$	11308.1	0.828772	0.414386	+	0.910101i	$0.363996\pi$
$572$	0	0
$573$	758.186	0.0552769
$574$	0	0
$575$	21380.1	1.55063
$576$	0	0
$577$	−26768.4	−1.93134	−0.965669	−	0.259776i	$-0.916351\pi$
$577$	−26768.4	−1.93134	−0.965669	+	0.259776i	$0.916351\pi$
$578$	0	0
$579$	13382.0	0.960513
$580$	0	0
$581$	−6820.20	−0.487004
$582$	0	0
$583$	21280.1	1.51172
$584$	0	0
$585$	−1020.25	−0.0721066
$586$	0	0
$587$	12495.4	0.878607	0.439304	−	0.898339i	$-0.355225\pi$
$587$	12495.4	0.878607	0.439304	+	0.898339i	$0.355225\pi$
$588$	0	0
$589$	−604.886	−0.0423156
$590$	0	0
$591$	4497.94	0.313063
$592$	0	0
$593$	1891.59	0.130992	0.0654959	−	0.997853i	$-0.479137\pi$
$593$	1891.59	0.130992	0.0654959	+	0.997853i	$0.479137\pi$
$594$	0	0
$595$	1746.86	0.120360
$596$	0	0
$597$	1362.32	0.0933939
$598$	0	0
$599$	6620.09	0.451569	0.225784	−	0.974177i	$-0.427506\pi$
$599$	6620.09	0.451569	0.225784	+	0.974177i	$0.427506\pi$
$600$	0	0
$601$	−28416.4	−1.92867	−0.964333	−	0.264690i	$-0.914730\pi$
$601$	−28416.4	−1.92867	−0.964333	+	0.264690i	$0.914730\pi$
$602$	0	0
$603$	3868.95	0.261287
$604$	0	0
$605$	−1854.91	−0.124649
$606$	0	0
$607$	8543.67	0.571297	0.285648	−	0.958335i	$-0.407791\pi$
$607$	8543.67	0.571297	0.285648	+	0.958335i	$0.407791\pi$
$608$	0	0
$609$	2546.74	0.169457
$610$	0	0
$611$	2223.85	0.147246
$612$	0	0
$613$	20334.9	1.33984	0.669918	−	0.742435i	$-0.266329\pi$
$613$	20334.9	1.33984	0.669918	+	0.742435i	$0.266329\pi$
$614$	0	0
$615$	−4702.47	−0.308328
$616$	0	0
$617$	−27044.6	−1.76463	−0.882315	−	0.470660i	$-0.844016\pi$
$617$	−27044.6	−1.76463	−0.882315	+	0.470660i	$0.844016\pi$
$618$	0	0
$619$	−5072.37	−0.329363	−0.164682	−	0.986347i	$-0.552660\pi$
$619$	−5072.37	−0.329363	−0.164682	+	0.986347i	$0.552660\pi$
$620$	0	0
$621$	−5425.91	−0.350619
$622$	0	0
$623$	2117.46	0.136170
$624$	0	0
$625$	8992.56	0.575524
$626$	0	0
$627$	−272.131	−0.0173331
$628$	0	0
$629$	7191.76	0.455889
$630$	0	0
$631$	15462.5	0.975517	0.487759	−	0.872979i	$-0.337815\pi$
$631$	15462.5	0.975517	0.487759	+	0.872979i	$0.337815\pi$
$632$	0	0
$633$	−6836.38	−0.429260
$634$	0	0
$635$	5462.81	0.341394
$636$	0	0
$637$	1287.62	0.0800901
$638$	0	0
$639$	10327.9	0.639381
$640$	0	0
$641$	7083.52	0.436478	0.218239	−	0.975895i	$-0.429969\pi$
$641$	7083.52	0.436478	0.218239	+	0.975895i	$0.429969\pi$
$642$	0	0
$643$	22814.3	1.39924	0.699619	−	0.714516i	$-0.253353\pi$
$643$	22814.3	1.39924	0.699619	+	0.714516i	$0.253353\pi$
$644$	0	0
$645$	−3280.60	−0.200269
$646$	0	0
$647$	18553.9	1.12740	0.563700	−	0.825980i	$-0.309377\pi$
$647$	18553.9	1.12740	0.563700	+	0.825980i	$0.309377\pi$
$648$	0	0
$649$	14185.0	0.857951
$650$	0	0
$651$	−5876.42	−0.353787
$652$	0	0
$653$	8293.14	0.496992	0.248496	−	0.968633i	$-0.420064\pi$
$653$	8293.14	0.496992	0.248496	+	0.968633i	$0.420064\pi$
$654$	0	0
$655$	2553.01	0.152297
$656$	0	0
$657$	4414.20	0.262122
$658$	0	0
$659$	−22379.6	−1.32289	−0.661447	−	0.749992i	$-0.730057\pi$
$659$	−22379.6	−1.32289	−0.661447	+	0.749992i	$0.730057\pi$
$660$	0	0
$661$	13096.5	0.770643	0.385322	−	0.922782i	$-0.374090\pi$
$661$	13096.5	0.770643	0.385322	+	0.922782i	$0.374090\pi$
$662$	0	0
$663$	4560.36	0.267134
$664$	0	0
$665$	−65.2758	−0.00380645
$666$	0	0
$667$	24371.1	1.41477
$668$	0	0
$669$	−4237.30	−0.244878
$670$	0	0
$671$	−22233.9	−1.27918
$672$	0	0
$673$	−27876.6	−1.59668	−0.798339	−	0.602209i	$-0.794288\pi$
$673$	−27876.6	−1.59668	−0.798339	+	0.602209i	$0.794288\pi$
$674$	0	0
$675$	−2872.53	−0.163798
$676$	0	0
$677$	2365.28	0.134276	0.0671381	−	0.997744i	$-0.478613\pi$
$677$	2365.28	0.134276	0.0671381	+	0.997744i	$0.478613\pi$
$678$	0	0
$679$	7998.53	0.452070
$680$	0	0
$681$	−9081.20	−0.511002
$682$	0	0
$683$	−31403.9	−1.75935	−0.879675	−	0.475576i	$-0.842240\pi$
$683$	−31403.9	−1.75935	−0.879675	+	0.475576i	$0.842240\pi$
$684$	0	0
$685$	−11307.0	−0.630682
$686$	0	0
$687$	−18.4409	−0.00102411
$688$	0	0
$689$	13325.7	0.736818
$690$	0	0
$691$	14302.5	0.787397	0.393698	−	0.919240i	$-0.371195\pi$
$691$	14302.5	0.787397	0.393698	+	0.919240i	$0.371195\pi$
$692$	0	0
$693$	−2643.73	−0.144917
$694$	0	0
$695$	−8401.30	−0.458532
$696$	0	0
$697$	21019.2	1.14227
$698$	0	0
$699$	7156.97	0.387270
$700$	0	0
$701$	−32812.2	−1.76790	−0.883950	−	0.467581i	$-0.845125\pi$
$701$	−32812.2	−1.76790	−0.883950	+	0.467581i	$0.845125\pi$
$702$	0	0
$703$	−268.738	−0.0144177
$704$	0	0
$705$	−1095.24	−0.0585093
$706$	0	0
$707$	7155.24	0.380623
$708$	0	0
$709$	34531.4	1.82913	0.914565	−	0.404439i	$-0.132533\pi$
$709$	34531.4	1.82913	0.914565	+	0.404439i	$0.132533\pi$
$710$	0	0
$711$	5806.17	0.306257
$712$	0	0
$713$	−56234.4	−2.95371
$714$	0	0
$715$	−4757.11	−0.248820
$716$	0	0
$717$	−4819.21	−0.251013
$718$	0	0
$719$	1577.27	0.0818113	0.0409056	−	0.999163i	$-0.486976\pi$
$719$	1577.27	0.0818113	0.0409056	+	0.999163i	$0.486976\pi$
$720$	0	0
$721$	−2956.40	−0.152707
$722$	0	0
$723$	2505.60	0.128886
$724$	0	0
$725$	12902.3	0.660936
$726$	0	0
$727$	−35968.9	−1.83496	−0.917479	−	0.397784i	$-0.869779\pi$
$727$	−35968.9	−1.83496	−0.917479	+	0.397784i	$0.869779\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	14663.7	0.741939
$732$	0	0
$733$	19579.7	0.986619	0.493310	−	0.869854i	$-0.335787\pi$
$733$	19579.7	0.986619	0.493310	+	0.869854i	$0.335787\pi$
$734$	0	0
$735$	−634.149	−0.0318244
$736$	0	0
$737$	18039.7	0.901627
$738$	0	0
$739$	963.684	0.0479698	0.0239849	−	0.999712i	$-0.492365\pi$
$739$	963.684	0.0479698	0.0239849	+	0.999712i	$0.492365\pi$
$740$	0	0
$741$	−170.409	−0.00844824
$742$	0	0
$743$	30102.0	1.48632	0.743158	−	0.669115i	$-0.233327\pi$
$743$	30102.0	1.48632	0.743158	+	0.669115i	$0.233327\pi$
$744$	0	0
$745$	2850.90	0.140200
$746$	0	0
$747$	8768.83	0.429498
$748$	0	0
$749$	−4012.84	−0.195762
$750$	0	0
$751$	−19002.8	−0.923331	−0.461666	−	0.887054i	$-0.652748\pi$
$751$	−19002.8	−0.923331	−0.461666	+	0.887054i	$0.652748\pi$
$752$	0	0
$753$	−3978.09	−0.192523
$754$	0	0
$755$	−8849.89	−0.426597
$756$	0	0
$757$	23766.6	1.14110	0.570550	−	0.821263i	$-0.306730\pi$
$757$	23766.6	1.14110	0.570550	+	0.821263i	$0.306730\pi$
$758$	0	0
$759$	−25299.2	−1.20989
$760$	0	0
$761$	7390.89	0.352063	0.176031	−	0.984385i	$-0.443674\pi$
$761$	7390.89	0.352063	0.176031	+	0.984385i	$0.443674\pi$
$762$	0	0
$763$	10979.4	0.520947
$764$	0	0
$765$	−2245.96	−0.106148
$766$	0	0
$767$	8882.68	0.418168
$768$	0	0
$769$	20847.7	0.977616	0.488808	−	0.872391i	$-0.337432\pi$
$769$	20847.7	0.977616	0.488808	+	0.872391i	$0.337432\pi$
$770$	0	0
$771$	15630.4	0.730108
$772$	0	0
$773$	−26362.7	−1.22665	−0.613325	−	0.789831i	$-0.710168\pi$
$773$	−26362.7	−1.22665	−0.613325	+	0.789831i	$0.710168\pi$
$774$	0	0
$775$	−29771.0	−1.37988
$776$	0	0
$777$	−2610.77	−0.120542
$778$	0	0
$779$	−785.436	−0.0361248
$780$	0	0
$781$	48155.5	2.20633
$782$	0	0
$783$	−3274.39	−0.149447
$784$	0	0
$785$	11534.6	0.524445
$786$	0	0
$787$	−9668.13	−0.437906	−0.218953	−	0.975735i	$-0.570264\pi$
$787$	−9668.13	−0.437906	−0.218953	+	0.975735i	$0.570264\pi$
$788$	0	0
$789$	7433.99	0.335434
$790$	0	0
$791$	−1653.92	−0.0743445
$792$	0	0
$793$	−13922.9	−0.623476
$794$	0	0
$795$	−6562.85	−0.292780
$796$	0	0
$797$	−19201.7	−0.853398	−0.426699	−	0.904394i	$-0.640324\pi$
$797$	−19201.7	−0.853398	−0.426699	+	0.904394i	$0.640324\pi$
$798$	0	0
$799$	4895.53	0.216760
$800$	0	0
$801$	−2722.45	−0.120091
$802$	0	0
$803$	20582.0	0.904510
$804$	0	0
$805$	−6068.49	−0.265697
$806$	0	0
$807$	−14970.7	−0.653030
$808$	0	0
$809$	15965.2	0.693829	0.346914	−	0.937897i	$-0.387229\pi$
$809$	15965.2	0.693829	0.346914	+	0.937897i	$0.387229\pi$
$810$	0	0
$811$	−40776.8	−1.76556	−0.882778	−	0.469791i	$-0.844329\pi$
$811$	−40776.8	−1.76556	−0.882778	+	0.469791i	$0.844329\pi$
$812$	0	0
$813$	−20582.4	−0.887892
$814$	0	0
$815$	14314.4	0.615228
$816$	0	0
$817$	−547.947	−0.0234642
$818$	0	0
$819$	−1655.51	−0.0706329
$820$	0	0
$821$	29419.1	1.25059	0.625296	−	0.780388i	$-0.284978\pi$
$821$	29419.1	1.25059	0.625296	+	0.780388i	$0.284978\pi$
$822$	0	0
$823$	21955.7	0.929925	0.464963	−	0.885330i	$-0.346068\pi$
$823$	21955.7	0.929925	0.464963	+	0.885330i	$0.346068\pi$
$824$	0	0
$825$	−13393.7	−0.565221
$826$	0	0
$827$	−31083.4	−1.30699	−0.653493	−	0.756933i	$-0.726697\pi$
$827$	−31083.4	−1.30699	−0.653493	+	0.756933i	$0.726697\pi$
$828$	0	0
$829$	−20566.2	−0.861635	−0.430817	−	0.902439i	$-0.641775\pi$
$829$	−20566.2	−0.861635	−0.430817	+	0.902439i	$0.641775\pi$
$830$	0	0
$831$	2344.73	0.0978796
$832$	0	0
$833$	2834.54	0.117900
$834$	0	0
$835$	−9660.43	−0.400375
$836$	0	0
$837$	7555.40	0.312010
$838$	0	0
$839$	−12908.4	−0.531165	−0.265582	−	0.964088i	$-0.585564\pi$
$839$	−12908.4	−0.531165	−0.265582	+	0.964088i	$0.585564\pi$
$840$	0	0
$841$	−9681.72	−0.396971
$842$	0	0
$843$	16341.2	0.667639
$844$	0	0
$845$	6498.81	0.264575
$846$	0	0
$847$	−3009.87	−0.122102
$848$	0	0
$849$	−18026.5	−0.728701
$850$	0	0
$851$	−24983.8	−1.00638
$852$	0	0
$853$	−19464.3	−0.781297	−0.390648	−	0.920540i	$-0.627749\pi$
$853$	−19464.3	−0.781297	−0.390648	+	0.920540i	$0.627749\pi$
$854$	0	0
$855$	83.9260	0.00335697
$856$	0	0
$857$	−17420.5	−0.694366	−0.347183	−	0.937797i	$-0.612862\pi$
$857$	−17420.5	−0.694366	−0.347183	+	0.937797i	$0.612862\pi$
$858$	0	0
$859$	−17576.1	−0.698126	−0.349063	−	0.937099i	$-0.613500\pi$
$859$	−17576.1	−0.698126	−0.349063	+	0.937099i	$0.613500\pi$
$860$	0	0
$861$	−7630.45	−0.302027
$862$	0	0
$863$	−16781.2	−0.661920	−0.330960	−	0.943645i	$-0.607373\pi$
$863$	−16781.2	−0.661920	−0.330960	+	0.943645i	$0.607373\pi$
$864$	0	0
$865$	7146.58	0.280915
$866$	0	0
$867$	−4699.94	−0.184104
$868$	0	0
$869$	27072.3	1.05681
$870$	0	0
$871$	11296.5	0.439456
$872$	0	0
$873$	−10283.8	−0.398688
$874$	0	0
$875$	−6987.42	−0.269963
$876$	0	0
$877$	−11052.6	−0.425564	−0.212782	−	0.977100i	$-0.568252\pi$
$877$	−11052.6	−0.425564	−0.212782	+	0.977100i	$0.568252\pi$
$878$	0	0
$879$	−17520.3	−0.672291
$880$	0	0
$881$	−27155.3	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$881$	−27155.3	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$882$	0	0
$883$	16920.5	0.644869	0.322435	−	0.946592i	$-0.395499\pi$
$883$	16920.5	0.644869	0.322435	+	0.946592i	$0.395499\pi$
$884$	0	0
$885$	−4374.69	−0.166162
$886$	0	0
$887$	−12728.0	−0.481809	−0.240904	−	0.970549i	$-0.577444\pi$
$887$	−12728.0	−0.481809	−0.240904	+	0.970549i	$0.577444\pi$
$888$	0	0
$889$	8864.21	0.334416
$890$	0	0
$891$	3399.09	0.127804
$892$	0	0
$893$	−182.934	−0.00685514
$894$	0	0
$895$	3513.77	0.131232
$896$	0	0
$897$	−15842.4	−0.589703
$898$	0	0
$899$	−33935.9	−1.25898
$900$	0	0
$901$	29334.8	1.08467
$902$	0	0
$903$	−5323.26	−0.196176
$904$	0	0
$905$	11269.8	0.413945
$906$	0	0
$907$	−7314.94	−0.267793	−0.133897	−	0.990995i	$-0.542749\pi$
$907$	−7314.94	−0.267793	−0.133897	+	0.990995i	$0.542749\pi$
$908$	0	0
$909$	−9199.59	−0.335678
$910$	0	0
$911$	33450.2	1.21652	0.608262	−	0.793736i	$-0.291867\pi$
$911$	33450.2	1.21652	0.608262	+	0.793736i	$0.291867\pi$
$912$	0	0
$913$	40886.2	1.48208
$914$	0	0
$915$	6856.98	0.247743
$916$	0	0
$917$	4142.63	0.149184
$918$	0	0
$919$	−10448.9	−0.375058	−0.187529	−	0.982259i	$-0.560048\pi$
$919$	−10448.9	−0.375058	−0.187529	+	0.982259i	$0.560048\pi$
$920$	0	0
$921$	25329.8	0.906237
$922$	0	0
$923$	30155.1	1.07537
$924$	0	0
$925$	−13226.7	−0.470151
$926$	0	0
$927$	3801.08	0.134675
$928$	0	0
$929$	11230.1	0.396606	0.198303	−	0.980141i	$-0.436457\pi$
$929$	11230.1	0.396606	0.198303	+	0.980141i	$0.436457\pi$
$930$	0	0
$931$	−105.920	−0.00372865
$932$	0	0
$933$	15978.2	0.560669
$934$	0	0
$935$	−10472.2	−0.366286
$936$	0	0
$937$	34110.1	1.18925	0.594625	−	0.804003i	$-0.297300\pi$
$937$	34110.1	1.18925	0.594625	+	0.804003i	$0.297300\pi$
$938$	0	0
$939$	−26460.2	−0.919590
$940$	0	0
$941$	−14232.0	−0.493039	−0.246520	−	0.969138i	$-0.579287\pi$
$941$	−14232.0	−0.493039	−0.246520	+	0.969138i	$0.579287\pi$
$942$	0	0
$943$	−73019.6	−2.52158
$944$	0	0
$945$	815.335	0.0280665
$946$	0	0
$947$	1587.95	0.0544893	0.0272446	−	0.999629i	$-0.491327\pi$
$947$	1587.95	0.0544893	0.0272446	+	0.999629i	$0.491327\pi$
$948$	0	0
$949$	12888.5	0.440861
$950$	0	0
$951$	−95.8852	−0.00326950
$952$	0	0
$953$	2167.64	0.0736796	0.0368398	−	0.999321i	$-0.488271\pi$
$953$	2167.64	0.0736796	0.0368398	+	0.999321i	$0.488271\pi$
$954$	0	0
$955$	−1090.26	−0.0369423
$956$	0	0
$957$	−15267.4	−0.515700
$958$	0	0
$959$	−18347.2	−0.617792
$960$	0	0
$961$	48513.6	1.62846
$962$	0	0
$963$	5159.36	0.172646
$964$	0	0
$965$	−19243.1	−0.641923
$966$	0	0
$967$	−36873.9	−1.22625	−0.613125	−	0.789986i	$-0.710088\pi$
$967$	−36873.9	−1.22625	−0.613125	+	0.789986i	$0.710088\pi$
$968$	0	0
$969$	−375.135	−0.0124366
$970$	0	0
$971$	−7783.35	−0.257240	−0.128620	−	0.991694i	$-0.541055\pi$
$971$	−7783.35	−0.257240	−0.128620	+	0.991694i	$0.541055\pi$
$972$	0	0
$973$	−13632.3	−0.449160
$974$	0	0
$975$	−8387.14	−0.275491
$976$	0	0
$977$	32743.7	1.07222	0.536112	−	0.844147i	$-0.319893\pi$
$977$	32743.7	1.07222	0.536112	+	0.844147i	$0.319893\pi$
$978$	0	0
$979$	−12693.9	−0.414401
$980$	0	0
$981$	−14116.4	−0.459432
$982$	0	0
$983$	−34153.1	−1.10815	−0.554076	−	0.832466i	$-0.686928\pi$
$983$	−34153.1	−1.10815	−0.554076	+	0.832466i	$0.686928\pi$
$984$	0	0
$985$	−6467.94	−0.209224
$986$	0	0
$987$	−1777.19	−0.0573135
$988$	0	0
$989$	−50940.9	−1.63784
$990$	0	0
$991$	52838.0	1.69370	0.846849	−	0.531833i	$-0.178497\pi$
$991$	52838.0	1.69370	0.846849	+	0.531833i	$0.178497\pi$
$992$	0	0
$993$	2829.92	0.0904377
$994$	0	0
$995$	−1958.99	−0.0624163
$996$	0	0
$997$	24190.0	0.768409	0.384204	−	0.923248i	$-0.374476\pi$
$997$	24190.0	0.768409	0.384204	+	0.923248i	$0.374476\pi$
$998$	0	0
$999$	3356.71	0.106308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.a.q.1.2	yes	3
3.2	odd	2		2016.4.a.y.1.2		3
4.3	odd	2		672.4.a.o.1.2	✓	3
8.3	odd	2		1344.4.a.bv.1.2		3
8.5	even	2		1344.4.a.bt.1.2		3
12.11	even	2		2016.4.a.z.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.o.1.2	✓	3	4.3	odd	2
672.4.a.q.1.2	yes	3	1.1	even	1	trivial
1344.4.a.bt.1.2		3	8.5	even	2
1344.4.a.bv.1.2		3	8.3	odd	2
2016.4.a.y.1.2		3	3.2	odd	2
2016.4.a.z.1.2		3	12.11	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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -4.31394 q^{5} -7.00000 q^{7} +9.00000 q^{9} +41.9640 q^{11} +26.2780 q^{13} -12.9418 q^{15} +57.8477 q^{17} -2.16162 q^{19} -21.0000 q^{21} -200.960 q^{23} -106.390 q^{25} +27.0000 q^{27} -121.274 q^{29} +279.830 q^{31} +125.892 q^{33} +30.1976 q^{35} +124.322 q^{37} +78.8340 q^{39} +363.355 q^{41} +253.488 q^{43} -38.8255 q^{45} +84.6279 q^{47} +49.0000 q^{49} +173.543 q^{51} +507.104 q^{53} -181.030 q^{55} -6.48487 q^{57} +338.027 q^{59} -529.831 q^{61} -63.0000 q^{63} -113.362 q^{65} +429.884 q^{67} -602.879 q^{69} +1147.54 q^{71} +490.466 q^{73} -319.170 q^{75} -293.748 q^{77} +645.131 q^{79} +81.0000 q^{81} +974.314 q^{83} -249.551 q^{85} -363.821 q^{87} -302.494 q^{89} -183.946 q^{91} +839.489 q^{93} +9.32512 q^{95} -1142.65 q^{97} +377.676 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 10 q^{5} - 21 q^{7} + 27 q^{9} - 50 q^{13} - 30 q^{15} + 30 q^{17} + 140 q^{19} - 63 q^{21} + 56 q^{23} + 325 q^{25} + 81 q^{27} + 298 q^{29} - 80 q^{31} + 70 q^{35} + 10 q^{37} - 150 q^{39}+ \cdots - 130 q^{97}+O(q^{100}) \) 3 * q + 9 * q^3 - 10 * q^5 - 21 * q^7 + 27 * q^9 - 50 * q^13 - 30 * q^15 + 30 * q^17 + 140 * q^19 - 63 * q^21 + 56 * q^23 + 325 * q^25 + 81 * q^27 + 298 * q^29 - 80 * q^31 + 70 * q^35 + 10 * q^37 - 150 * q^39 + 390 * q^41 + 784 * q^43 - 90 * q^45 + 248 * q^47 + 147 * q^49 + 90 * q^51 + 10 * q^53 + 1360 * q^55 + 420 * q^57 + 1500 * q^59 - 810 * q^61 - 189 * q^63 + 860 * q^65 + 1272 * q^67 + 168 * q^69 + 160 * q^71 - 1170 * q^73 + 975 * q^75 + 840 * q^79 + 243 * q^81 + 1564 * q^83 - 2740 * q^85 + 894 * q^87 - 178 * q^89 + 350 * q^91 - 240 * q^93 + 2840 * q^95 - 130 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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