LMFDB - Embedded newform 1672.2.a.j.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1672,2,Mod(1,1672)] 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("1672.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1672 = 2^{3} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1672.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,4,0,1] 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:	$13.3509872180$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.576096652.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 11x^{3} + 16x^{2} - 6x - 4 $ x^6 - 2x^5 - 8x^4 + 11x^3 + 16x^2 - 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.634105$ of defining polynomial
Character	$\chi$	$=$	1672.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.365895 q^{3} +3.51994 q^{5} +1.20230 q^{7} -2.86612 q^{9} +1.00000 q^{11} +1.20230 q^{13} +1.28793 q^{15} -3.15405 q^{17} +1.00000 q^{19} +0.439914 q^{21} +6.54966 q^{23} +7.39001 q^{25} -2.14638 q^{27} +4.07236 q^{29} +5.34737 q^{31} +0.365895 q^{33} +4.23202 q^{35} +1.39561 q^{37} +0.439914 q^{39} -0.683544 q^{41} +3.62600 q^{43} -10.0886 q^{45} +5.67280 q^{47} -5.55448 q^{49} -1.15405 q^{51} -0.523891 q^{53} +3.51994 q^{55} +0.365895 q^{57} -8.37564 q^{59} -6.55777 q^{61} -3.44593 q^{63} +4.23202 q^{65} +1.71120 q^{67} +2.39649 q^{69} +14.1389 q^{71} -14.5048 q^{73} +2.70397 q^{75} +1.20230 q^{77} +10.6190 q^{79} +7.81301 q^{81} -7.55086 q^{83} -11.1021 q^{85} +1.49006 q^{87} -10.6447 q^{89} +1.44552 q^{91} +1.95657 q^{93} +3.51994 q^{95} +12.8239 q^{97} -2.86612 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + q^{5} + 4 q^{9} + 6 q^{11} + 7 q^{15} + 3 q^{17} + 6 q^{19} + 7 q^{21} + 7 q^{23} + 3 q^{25} + 13 q^{27} - 4 q^{29} + 7 q^{31} + 4 q^{33} + 6 q^{35} - 2 q^{37} + 7 q^{39} + 7 q^{41} + 21 q^{43}+ \cdots + 4 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + q^5 + 4 * q^9 + 6 * q^11 + 7 * q^15 + 3 * q^17 + 6 * q^19 + 7 * q^21 + 7 * q^23 + 3 * q^25 + 13 * q^27 - 4 * q^29 + 7 * q^31 + 4 * q^33 + 6 * q^35 - 2 * q^37 + 7 * q^39 + 7 * q^41 + 21 * q^43 + 11 * q^45 + 16 * q^47 + 2 * q^49 + 15 * q^51 + 17 * q^53 + q^55 + 4 * q^57 + 19 * q^59 - 6 * q^61 + 2 * q^63 + 6 * q^65 + 14 * q^67 + q^69 + q^71 - 5 * q^73 + 18 * q^75 + 2 * q^81 + 11 * q^83 - 26 * q^85 + 14 * q^87 + 12 * q^89 + 44 * q^91 - 6 * q^93 + q^95 + 14 * q^97 + 4 * q^99

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
$3$	0.365895	0.211249	0.105625	−	0.994406i	$-0.466316\pi$
$3$	0.365895	0.211249	0.105625	+	0.994406i	$0.466316\pi$
$4$	0	0
$5$	3.51994	1.57417	0.787084	−	0.616846i	$-0.211590\pi$
$5$	3.51994	1.57417	0.787084	+	0.616846i	$0.211590\pi$
$6$	0	0
$7$	1.20230	0.454425	0.227213	−	0.973845i	$-0.427039\pi$
$7$	1.20230	0.454425	0.227213	+	0.973845i	$0.427039\pi$
$8$	0	0
$9$	−2.86612	−0.955374
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.20230	0.333457	0.166728	−	0.986003i	$-0.446680\pi$
$13$	1.20230	0.333457	0.166728	+	0.986003i	$0.446680\pi$
$14$	0	0
$15$	1.28793	0.332542
$16$	0	0
$17$	−3.15405	−0.764970	−0.382485	−	0.923962i	$-0.624932\pi$
$17$	−3.15405	−0.764970	−0.382485	+	0.923962i	$0.624932\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.439914	0.0959970
$22$	0	0
$23$	6.54966	1.36570	0.682850	−	0.730559i	$-0.260740\pi$
$23$	6.54966	1.36570	0.682850	+	0.730559i	$0.260740\pi$
$24$	0	0
$25$	7.39001	1.47800
$26$	0	0
$27$	−2.14638	−0.413072
$28$	0	0
$29$	4.07236	0.756219	0.378109	−	0.925761i	$-0.376574\pi$
$29$	4.07236	0.756219	0.378109	+	0.925761i	$0.376574\pi$
$30$	0	0
$31$	5.34737	0.960416	0.480208	−	0.877155i	$-0.340561\pi$
$31$	5.34737	0.960416	0.480208	+	0.877155i	$0.340561\pi$
$32$	0	0
$33$	0.365895	0.0636941
$34$	0	0
$35$	4.23202	0.715341
$36$	0	0
$37$	1.39561	0.229438	0.114719	−	0.993398i	$-0.463403\pi$
$37$	1.39561	0.229438	0.114719	+	0.993398i	$0.463403\pi$
$38$	0	0
$39$	0.439914	0.0704426
$40$	0	0
$41$	−0.683544	−0.106752	−0.0533758	−	0.998574i	$-0.516998\pi$
$41$	−0.683544	−0.106752	−0.0533758	+	0.998574i	$0.516998\pi$
$42$	0	0
$43$	3.62600	0.552960	0.276480	−	0.961020i	$-0.410832\pi$
$43$	3.62600	0.552960	0.276480	+	0.961020i	$0.410832\pi$
$44$	0	0
$45$	−10.0886	−1.50392
$46$	0	0
$47$	5.67280	0.827463	0.413732	−	0.910399i	$-0.364225\pi$
$47$	5.67280	0.827463	0.413732	+	0.910399i	$0.364225\pi$
$48$	0	0
$49$	−5.55448	−0.793498
$50$	0	0
$51$	−1.15405	−0.161599
$52$	0	0
$53$	−0.523891	−0.0719620	−0.0359810	−	0.999352i	$-0.511456\pi$
$53$	−0.523891	−0.0719620	−0.0359810	+	0.999352i	$0.511456\pi$
$54$	0	0
$55$	3.51994	0.474629
$56$	0	0
$57$	0.365895	0.0484639
$58$	0	0
$59$	−8.37564	−1.09042	−0.545208	−	0.838301i	$-0.683549\pi$
$59$	−8.37564	−1.09042	−0.545208	+	0.838301i	$0.683549\pi$
$60$	0	0
$61$	−6.55777	−0.839636	−0.419818	−	0.907608i	$-0.637906\pi$
$61$	−6.55777	−0.839636	−0.419818	+	0.907608i	$0.637906\pi$
$62$	0	0
$63$	−3.44593	−0.434146
$64$	0	0
$65$	4.23202	0.524917
$66$	0	0
$67$	1.71120	0.209056	0.104528	−	0.994522i	$-0.466667\pi$
$67$	1.71120	0.209056	0.104528	+	0.994522i	$0.466667\pi$
$68$	0	0
$69$	2.39649	0.288503
$70$	0	0
$71$	14.1389	1.67798	0.838989	−	0.544148i	$-0.183147\pi$
$71$	14.1389	1.67798	0.838989	+	0.544148i	$0.183147\pi$
$72$	0	0
$73$	−14.5048	−1.69766	−0.848829	−	0.528668i	$-0.822692\pi$
$73$	−14.5048	−1.69766	−0.848829	+	0.528668i	$0.822692\pi$
$74$	0	0
$75$	2.70397	0.312227
$76$	0	0
$77$	1.20230	0.137014
$78$	0	0
$79$	10.6190	1.19473	0.597363	−	0.801971i	$-0.296215\pi$
$79$	10.6190	1.19473	0.597363	+	0.801971i	$0.296215\pi$
$80$	0	0
$81$	7.81301	0.868113
$82$	0	0
$83$	−7.55086	−0.828814	−0.414407	−	0.910092i	$-0.636011\pi$
$83$	−7.55086	−0.828814	−0.414407	+	0.910092i	$0.636011\pi$
$84$	0	0
$85$	−11.1021	−1.20419
$86$	0	0
$87$	1.49006	0.159751
$88$	0	0
$89$	−10.6447	−1.12834	−0.564169	−	0.825659i	$-0.690804\pi$
$89$	−10.6447	−1.12834	−0.564169	+	0.825659i	$0.690804\pi$
$90$	0	0
$91$	1.44552	0.151531
$92$	0	0
$93$	1.95657	0.202887
$94$	0	0
$95$	3.51994	0.361139
$96$	0	0
$97$	12.8239	1.30207	0.651034	−	0.759048i	$-0.274335\pi$
$97$	12.8239	1.30207	0.651034	+	0.759048i	$0.274335\pi$
$98$	0	0
$99$	−2.86612	−0.288056
$100$	0	0
$101$	5.82598	0.579707	0.289853	−	0.957071i	$-0.406393\pi$
$101$	5.82598	0.579707	0.289853	+	0.957071i	$0.406393\pi$
$102$	0	0
$103$	−8.49462	−0.837000	−0.418500	−	0.908217i	$-0.637444\pi$
$103$	−8.49462	−0.837000	−0.418500	+	0.908217i	$0.637444\pi$
$104$	0	0
$105$	1.54847	0.151115
$106$	0	0
$107$	−7.56378	−0.731218	−0.365609	−	0.930768i	$-0.619139\pi$
$107$	−7.56378	−0.731218	−0.365609	+	0.930768i	$0.619139\pi$
$108$	0	0
$109$	9.77301	0.936084	0.468042	−	0.883706i	$-0.344960\pi$
$109$	9.77301	0.936084	0.468042	+	0.883706i	$0.344960\pi$
$110$	0	0
$111$	0.510648	0.0484686
$112$	0	0
$113$	−6.28747	−0.591475	−0.295738	−	0.955269i	$-0.595565\pi$
$113$	−6.28747	−0.591475	−0.295738	+	0.955269i	$0.595565\pi$
$114$	0	0
$115$	23.0545	2.14984
$116$	0	0
$117$	−3.44593	−0.318576
$118$	0	0
$119$	−3.79210	−0.347621
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.250105	−0.0225512
$124$	0	0
$125$	8.41271	0.752456
$126$	0	0
$127$	0.482121	0.0427813	0.0213906	−	0.999771i	$-0.493191\pi$
$127$	0.482121	0.0427813	0.0213906	+	0.999771i	$0.493191\pi$
$128$	0	0
$129$	1.32673	0.116812
$130$	0	0
$131$	−10.5164	−0.918823	−0.459411	−	0.888224i	$-0.651940\pi$
$131$	−10.5164	−0.918823	−0.459411	+	0.888224i	$0.651940\pi$
$132$	0	0
$133$	1.20230	0.104252
$134$	0	0
$135$	−7.55515	−0.650244
$136$	0	0
$137$	−0.538471	−0.0460047	−0.0230023	−	0.999735i	$-0.507323\pi$
$137$	−0.538471	−0.0460047	−0.0230023	+	0.999735i	$0.507323\pi$
$138$	0	0
$139$	−4.33970	−0.368089	−0.184044	−	0.982918i	$-0.558919\pi$
$139$	−4.33970	−0.368089	−0.184044	+	0.982918i	$0.558919\pi$
$140$	0	0
$141$	2.07565	0.174801
$142$	0	0
$143$	1.20230	0.100541
$144$	0	0
$145$	14.3345	1.19042
$146$	0	0
$147$	−2.03236	−0.167626
$148$	0	0
$149$	−15.3721	−1.25933	−0.629665	−	0.776866i	$-0.716808\pi$
$149$	−15.3721	−1.25933	−0.629665	+	0.776866i	$0.716808\pi$
$150$	0	0
$151$	5.48533	0.446390	0.223195	−	0.974774i	$-0.428351\pi$
$151$	5.48533	0.446390	0.223195	+	0.974774i	$0.428351\pi$
$152$	0	0
$153$	9.03989	0.730832
$154$	0	0
$155$	18.8224	1.51186
$156$	0	0
$157$	−7.33247	−0.585195	−0.292597	−	0.956236i	$-0.594520\pi$
$157$	−7.33247	−0.585195	−0.292597	+	0.956236i	$0.594520\pi$
$158$	0	0
$159$	−0.191689	−0.0152019
$160$	0	0
$161$	7.87464	0.620608
$162$	0	0
$163$	7.08074	0.554606	0.277303	−	0.960783i	$-0.410559\pi$
$163$	7.08074	0.554606	0.277303	+	0.960783i	$0.410559\pi$
$164$	0	0
$165$	1.28793	0.100265
$166$	0	0
$167$	18.0849	1.39945	0.699727	−	0.714411i	$-0.253305\pi$
$167$	18.0849	1.39945	0.699727	+	0.714411i	$0.253305\pi$
$168$	0	0
$169$	−11.5545	−0.888806
$170$	0	0
$171$	−2.86612	−0.219178
$172$	0	0
$173$	−20.2877	−1.54244	−0.771221	−	0.636568i	$-0.780354\pi$
$173$	−20.2877	−1.54244	−0.771221	+	0.636568i	$0.780354\pi$
$174$	0	0
$175$	8.88498	0.671642
$176$	0	0
$177$	−3.06460	−0.230350
$178$	0	0
$179$	17.2173	1.28688	0.643440	−	0.765496i	$-0.277507\pi$
$179$	17.2173	1.28688	0.643440	+	0.765496i	$0.277507\pi$
$180$	0	0
$181$	10.5723	0.785836	0.392918	−	0.919574i	$-0.371466\pi$
$181$	10.5723	0.785836	0.392918	+	0.919574i	$0.371466\pi$
$182$	0	0
$183$	−2.39945	−0.177373
$184$	0	0
$185$	4.91249	0.361173
$186$	0	0
$187$	−3.15405	−0.230647
$188$	0	0
$189$	−2.58059	−0.187710
$190$	0	0
$191$	8.57377	0.620376	0.310188	−	0.950675i	$-0.399608\pi$
$191$	8.57377	0.620376	0.310188	+	0.950675i	$0.399608\pi$
$192$	0	0
$193$	9.75494	0.702176	0.351088	−	0.936342i	$-0.385812\pi$
$193$	9.75494	0.702176	0.351088	+	0.936342i	$0.385812\pi$
$194$	0	0
$195$	1.54847	0.110888
$196$	0	0
$197$	18.1258	1.29141	0.645704	−	0.763588i	$-0.276564\pi$
$197$	18.1258	1.29141	0.645704	+	0.763588i	$0.276564\pi$
$198$	0	0
$199$	−18.3156	−1.29836	−0.649178	−	0.760636i	$-0.724887\pi$
$199$	−18.3156	−1.29836	−0.649178	+	0.760636i	$0.724887\pi$
$200$	0	0
$201$	0.626118	0.0441630
$202$	0	0
$203$	4.89619	0.343645
$204$	0	0
$205$	−2.40604	−0.168045
$206$	0	0
$207$	−18.7721	−1.30475
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−8.94748	−0.615970	−0.307985	−	0.951391i	$-0.599655\pi$
$211$	−8.94748	−0.615970	−0.307985	+	0.951391i	$0.599655\pi$
$212$	0	0
$213$	5.17335	0.354472
$214$	0	0
$215$	12.7633	0.870451
$216$	0	0
$217$	6.42912	0.436437
$218$	0	0
$219$	−5.30723	−0.358629
$220$	0	0
$221$	−3.79210	−0.255084
$222$	0	0
$223$	−5.86139	−0.392508	−0.196254	−	0.980553i	$-0.562878\pi$
$223$	−5.86139	−0.392508	−0.196254	+	0.980553i	$0.562878\pi$
$224$	0	0
$225$	−21.1807	−1.41204
$226$	0	0
$227$	−7.61983	−0.505746	−0.252873	−	0.967500i	$-0.581375\pi$
$227$	−7.61983	−0.505746	−0.252873	+	0.967500i	$0.581375\pi$
$228$	0	0
$229$	18.5863	1.22822	0.614108	−	0.789222i	$-0.289516\pi$
$229$	18.5863	1.22822	0.614108	+	0.789222i	$0.289516\pi$
$230$	0	0
$231$	0.439914	0.0289442
$232$	0	0
$233$	19.6588	1.28789	0.643947	−	0.765070i	$-0.277296\pi$
$233$	19.6588	1.28789	0.643947	+	0.765070i	$0.277296\pi$
$234$	0	0
$235$	19.9680	1.30257
$236$	0	0
$237$	3.88542	0.252385
$238$	0	0
$239$	−12.2588	−0.792958	−0.396479	−	0.918044i	$-0.629768\pi$
$239$	−12.2588	−0.792958	−0.396479	+	0.918044i	$0.629768\pi$
$240$	0	0
$241$	−19.1499	−1.23356	−0.616778	−	0.787138i	$-0.711562\pi$
$241$	−19.1499	−1.23356	−0.616778	+	0.787138i	$0.711562\pi$
$242$	0	0
$243$	9.29789	0.596460
$244$	0	0
$245$	−19.5515	−1.24910
$246$	0	0
$247$	1.20230	0.0765003
$248$	0	0
$249$	−2.76282	−0.175087
$250$	0	0
$251$	7.54090	0.475977	0.237989	−	0.971268i	$-0.423512\pi$
$251$	7.54090	0.475977	0.237989	+	0.971268i	$0.423512\pi$
$252$	0	0
$253$	6.54966	0.411774
$254$	0	0
$255$	−4.06219	−0.254384
$256$	0	0
$257$	−27.4533	−1.71249	−0.856244	−	0.516571i	$-0.827208\pi$
$257$	−27.4533	−1.71249	−0.856244	+	0.516571i	$0.827208\pi$
$258$	0	0
$259$	1.67794	0.104262
$260$	0	0
$261$	−11.6719	−0.722472
$262$	0	0
$263$	−22.1136	−1.36358	−0.681792	−	0.731546i	$-0.738799\pi$
$263$	−22.1136	−1.36358	−0.681792	+	0.731546i	$0.738799\pi$
$264$	0	0
$265$	−1.84407	−0.113280
$266$	0	0
$267$	−3.89485	−0.238361
$268$	0	0
$269$	−13.6931	−0.834886	−0.417443	−	0.908703i	$-0.637074\pi$
$269$	−13.6931	−0.834886	−0.417443	+	0.908703i	$0.637074\pi$
$270$	0	0
$271$	−19.8960	−1.20860	−0.604299	−	0.796758i	$-0.706547\pi$
$271$	−19.8960	−1.20860	−0.604299	+	0.796758i	$0.706547\pi$
$272$	0	0
$273$	0.528907	0.0320109
$274$	0	0
$275$	7.39001	0.445635
$276$	0	0
$277$	−26.9072	−1.61670	−0.808349	−	0.588703i	$-0.799639\pi$
$277$	−26.9072	−1.61670	−0.808349	+	0.588703i	$0.799639\pi$
$278$	0	0
$279$	−15.3262	−0.917556
$280$	0	0
$281$	−21.5531	−1.28575	−0.642875	−	0.765971i	$-0.722259\pi$
$281$	−21.5531	−1.28575	−0.642875	+	0.765971i	$0.722259\pi$
$282$	0	0
$283$	−6.76939	−0.402398	−0.201199	−	0.979550i	$-0.564484\pi$
$283$	−6.76939	−0.402398	−0.201199	+	0.979550i	$0.564484\pi$
$284$	0	0
$285$	1.28793	0.0762903
$286$	0	0
$287$	−0.821822	−0.0485106
$288$	0	0
$289$	−7.05197	−0.414822
$290$	0	0
$291$	4.69219	0.275061
$292$	0	0
$293$	14.9628	0.874134	0.437067	−	0.899429i	$-0.356017\pi$
$293$	14.9628	0.874134	0.437067	+	0.899429i	$0.356017\pi$
$294$	0	0
$295$	−29.4818	−1.71650
$296$	0	0
$297$	−2.14638	−0.124546
$298$	0	0
$299$	7.87464	0.455402
$300$	0	0
$301$	4.35953	0.251279
$302$	0	0
$303$	2.13170	0.122463
$304$	0	0
$305$	−23.0830	−1.32173
$306$	0	0
$307$	−12.6139	−0.719913	−0.359957	−	0.932969i	$-0.617208\pi$
$307$	−12.6139	−0.719913	−0.359957	+	0.932969i	$0.617208\pi$
$308$	0	0
$309$	−3.10814	−0.176816
$310$	0	0
$311$	29.0144	1.64526	0.822628	−	0.568580i	$-0.192507\pi$
$311$	29.0144	1.64526	0.822628	+	0.568580i	$0.192507\pi$
$312$	0	0
$313$	−6.20308	−0.350619	−0.175309	−	0.984513i	$-0.556093\pi$
$313$	−6.20308	−0.350619	−0.175309	+	0.984513i	$0.556093\pi$
$314$	0	0
$315$	−12.1295	−0.683418
$316$	0	0
$317$	−13.7236	−0.770796	−0.385398	−	0.922750i	$-0.625936\pi$
$317$	−13.7236	−0.770796	−0.385398	+	0.922750i	$0.625936\pi$
$318$	0	0
$319$	4.07236	0.228009
$320$	0	0
$321$	−2.76755	−0.154469
$322$	0	0
$323$	−3.15405	−0.175496
$324$	0	0
$325$	8.88498	0.492850
$326$	0	0
$327$	3.57589	0.197747
$328$	0	0
$329$	6.82039	0.376020
$330$	0	0
$331$	9.96456	0.547702	0.273851	−	0.961772i	$-0.411703\pi$
$331$	9.96456	0.547702	0.273851	+	0.961772i	$0.411703\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	6.02332	0.329089
$336$	0	0
$337$	−28.8617	−1.57220	−0.786099	−	0.618100i	$-0.787903\pi$
$337$	−28.8617	−1.57220	−0.786099	+	0.618100i	$0.787903\pi$
$338$	0	0
$339$	−2.30055	−0.124949
$340$	0	0
$341$	5.34737	0.289576
$342$	0	0
$343$	−15.0942	−0.815011
$344$	0	0
$345$	8.43550	0.454152
$346$	0	0
$347$	3.06265	0.164411	0.0822057	−	0.996615i	$-0.473804\pi$
$347$	3.06265	0.164411	0.0822057	+	0.996615i	$0.473804\pi$
$348$	0	0
$349$	−22.8672	−1.22405	−0.612026	−	0.790837i	$-0.709645\pi$
$349$	−22.8672	−1.22405	−0.612026	+	0.790837i	$0.709645\pi$
$350$	0	0
$351$	−2.58059	−0.137742
$352$	0	0
$353$	22.7933	1.21316	0.606582	−	0.795021i	$-0.292540\pi$
$353$	22.7933	1.21316	0.606582	+	0.795021i	$0.292540\pi$
$354$	0	0
$355$	49.7681	2.64142
$356$	0	0
$357$	−1.38751	−0.0734348
$358$	0	0
$359$	−26.7211	−1.41028	−0.705142	−	0.709066i	$-0.749117\pi$
$359$	−26.7211	−1.41028	−0.705142	+	0.709066i	$0.749117\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.365895	0.0192045
$364$	0	0
$365$	−51.0561	−2.67240
$366$	0	0
$367$	0.803413	0.0419378	0.0209689	−	0.999780i	$-0.493325\pi$
$367$	0.803413	0.0419378	0.0209689	+	0.999780i	$0.493325\pi$
$368$	0	0
$369$	1.95912	0.101988
$370$	0	0
$371$	−0.629873	−0.0327014
$372$	0	0
$373$	26.1021	1.35151	0.675757	−	0.737125i	$-0.263817\pi$
$373$	26.1021	1.35151	0.675757	+	0.737125i	$0.263817\pi$
$374$	0	0
$375$	3.07817	0.158956
$376$	0	0
$377$	4.89619	0.252166
$378$	0	0
$379$	5.29257	0.271861	0.135930	−	0.990718i	$-0.456598\pi$
$379$	5.29257	0.271861	0.135930	+	0.990718i	$0.456598\pi$
$380$	0	0
$381$	0.176405	0.00903752
$382$	0	0
$383$	20.7583	1.06070	0.530350	−	0.847779i	$-0.322061\pi$
$383$	20.7583	1.06070	0.530350	+	0.847779i	$0.322061\pi$
$384$	0	0
$385$	4.23202	0.215683
$386$	0	0
$387$	−10.3926	−0.528283
$388$	0	0
$389$	−0.361176	−0.0183124	−0.00915618	−	0.999958i	$-0.502915\pi$
$389$	−0.361176	−0.0183124	−0.00915618	+	0.999958i	$0.502915\pi$
$390$	0	0
$391$	−20.6580	−1.04472
$392$	0	0
$393$	−3.84790	−0.194101
$394$	0	0
$395$	37.3781	1.88070
$396$	0	0
$397$	−34.7796	−1.74554	−0.872768	−	0.488135i	$-0.837677\pi$
$397$	−34.7796	−1.74554	−0.872768	+	0.488135i	$0.837677\pi$
$398$	0	0
$399$	0.439914	0.0220232
$400$	0	0
$401$	9.71865	0.485326	0.242663	−	0.970111i	$-0.421979\pi$
$401$	9.71865	0.485326	0.242663	+	0.970111i	$0.421979\pi$
$402$	0	0
$403$	6.42912	0.320257
$404$	0	0
$405$	27.5014	1.36655
$406$	0	0
$407$	1.39561	0.0691780
$408$	0	0
$409$	−17.2083	−0.850897	−0.425448	−	0.904983i	$-0.639884\pi$
$409$	−17.2083	−0.850897	−0.425448	+	0.904983i	$0.639884\pi$
$410$	0	0
$411$	−0.197024	−0.00971846
$412$	0	0
$413$	−10.0700	−0.495512
$414$	0	0
$415$	−26.5786	−1.30469
$416$	0	0
$417$	−1.58787	−0.0777585
$418$	0	0
$419$	26.5600	1.29754	0.648771	−	0.760984i	$-0.275283\pi$
$419$	26.5600	1.29754	0.648771	+	0.760984i	$0.275283\pi$
$420$	0	0
$421$	17.2380	0.840128	0.420064	−	0.907494i	$-0.362008\pi$
$421$	17.2380	0.840128	0.420064	+	0.907494i	$0.362008\pi$
$422$	0	0
$423$	−16.2589	−0.790537
$424$	0	0
$425$	−23.3085	−1.13063
$426$	0	0
$427$	−7.88438	−0.381552
$428$	0	0
$429$	0.439914	0.0212392
$430$	0	0
$431$	7.01404	0.337855	0.168927	−	0.985629i	$-0.445970\pi$
$431$	7.01404	0.337855	0.168927	+	0.985629i	$0.445970\pi$
$432$	0	0
$433$	−17.6289	−0.847193	−0.423596	−	0.905851i	$-0.639233\pi$
$433$	−17.6289	−0.847193	−0.423596	+	0.905851i	$0.639233\pi$
$434$	0	0
$435$	5.24492	0.251474
$436$	0	0
$437$	6.54966	0.313313
$438$	0	0
$439$	3.49465	0.166791	0.0833953	−	0.996517i	$-0.473424\pi$
$439$	3.49465	0.166791	0.0833953	+	0.996517i	$0.473424\pi$
$440$	0	0
$441$	15.9198	0.758087
$442$	0	0
$443$	3.47907	0.165296	0.0826479	−	0.996579i	$-0.473662\pi$
$443$	3.47907	0.165296	0.0826479	+	0.996579i	$0.473662\pi$
$444$	0	0
$445$	−37.4689	−1.77619
$446$	0	0
$447$	−5.62457	−0.266033
$448$	0	0
$449$	24.8822	1.17426	0.587131	−	0.809492i	$-0.300257\pi$
$449$	24.8822	1.17426	0.587131	+	0.809492i	$0.300257\pi$
$450$	0	0
$451$	−0.683544	−0.0321868
$452$	0	0
$453$	2.00705	0.0942995
$454$	0	0
$455$	5.08814	0.238535
$456$	0	0
$457$	−9.90793	−0.463473	−0.231737	−	0.972779i	$-0.574441\pi$
$457$	−9.90793	−0.463473	−0.231737	+	0.972779i	$0.574441\pi$
$458$	0	0
$459$	6.76980	0.315987
$460$	0	0
$461$	2.11797	0.0986439	0.0493219	−	0.998783i	$-0.484294\pi$
$461$	2.11797	0.0986439	0.0493219	+	0.998783i	$0.484294\pi$
$462$	0	0
$463$	−14.1093	−0.655715	−0.327858	−	0.944727i	$-0.606327\pi$
$463$	−14.1093	−0.655715	−0.327858	+	0.944727i	$0.606327\pi$
$464$	0	0
$465$	6.88703	0.319378
$466$	0	0
$467$	3.62118	0.167568	0.0837841	−	0.996484i	$-0.473299\pi$
$467$	3.62118	0.167568	0.0837841	+	0.996484i	$0.473299\pi$
$468$	0	0
$469$	2.05737	0.0950003
$470$	0	0
$471$	−2.68291	−0.123622
$472$	0	0
$473$	3.62600	0.166724
$474$	0	0
$475$	7.39001	0.339077
$476$	0	0
$477$	1.50154	0.0687506
$478$	0	0
$479$	−17.0418	−0.778660	−0.389330	−	0.921098i	$-0.627293\pi$
$479$	−17.0418	−0.778660	−0.389330	+	0.921098i	$0.627293\pi$
$480$	0	0
$481$	1.67794	0.0765076
$482$	0	0
$483$	2.88129	0.131103
$484$	0	0
$485$	45.1394	2.04967
$486$	0	0
$487$	−0.323990	−0.0146814	−0.00734069	−	0.999973i	$-0.502337\pi$
$487$	−0.323990	−0.0146814	−0.00734069	+	0.999973i	$0.502337\pi$
$488$	0	0
$489$	2.59080	0.117160
$490$	0	0
$491$	−32.1744	−1.45201	−0.726005	−	0.687690i	$-0.758625\pi$
$491$	−32.1744	−1.45201	−0.726005	+	0.687690i	$0.758625\pi$
$492$	0	0
$493$	−12.8444	−0.578484
$494$	0	0
$495$	−10.0886	−0.453448
$496$	0	0
$497$	16.9991	0.762516
$498$	0	0
$499$	−26.7659	−1.19821	−0.599104	−	0.800671i	$-0.704476\pi$
$499$	−26.7659	−1.19821	−0.599104	+	0.800671i	$0.704476\pi$
$500$	0	0
$501$	6.61718	0.295634
$502$	0	0
$503$	−24.2384	−1.08074	−0.540369	−	0.841428i	$-0.681715\pi$
$503$	−24.2384	−1.08074	−0.540369	+	0.841428i	$0.681715\pi$
$504$	0	0
$505$	20.5071	0.912555
$506$	0	0
$507$	−4.22772	−0.187760
$508$	0	0
$509$	−2.51666	−0.111549	−0.0557745	−	0.998443i	$-0.517763\pi$
$509$	−2.51666	−0.111549	−0.0557745	+	0.998443i	$0.517763\pi$
$510$	0	0
$511$	−17.4391	−0.771458
$512$	0	0
$513$	−2.14638	−0.0947651
$514$	0	0
$515$	−29.9006	−1.31758
$516$	0	0
$517$	5.67280	0.249490
$518$	0	0
$519$	−7.42314	−0.325840
$520$	0	0
$521$	23.2024	1.01651	0.508257	−	0.861205i	$-0.330290\pi$
$521$	23.2024	1.01651	0.508257	+	0.861205i	$0.330290\pi$
$522$	0	0
$523$	7.29602	0.319033	0.159516	−	0.987195i	$-0.449007\pi$
$523$	7.29602	0.319033	0.159516	+	0.987195i	$0.449007\pi$
$524$	0	0
$525$	3.25097	0.141884
$526$	0	0
$527$	−16.8659	−0.734689
$528$	0	0
$529$	19.8981	0.865135
$530$	0	0
$531$	24.0056	1.04175
$532$	0	0
$533$	−0.821822	−0.0355971
$534$	0	0
$535$	−26.6241	−1.15106
$536$	0	0
$537$	6.29971	0.271853
$538$	0	0
$539$	−5.55448	−0.239249
$540$	0	0
$541$	−24.8917	−1.07018	−0.535089	−	0.844796i	$-0.679722\pi$
$541$	−24.8917	−1.07018	−0.535089	+	0.844796i	$0.679722\pi$
$542$	0	0
$543$	3.86836	0.166007
$544$	0	0
$545$	34.4004	1.47355
$546$	0	0
$547$	25.6886	1.09837	0.549184	−	0.835702i	$-0.314939\pi$
$547$	25.6886	1.09837	0.549184	+	0.835702i	$0.314939\pi$
$548$	0	0
$549$	18.7954	0.802166
$550$	0	0
$551$	4.07236	0.173489
$552$	0	0
$553$	12.7671	0.542913
$554$	0	0
$555$	1.79745	0.0762976
$556$	0	0
$557$	31.5541	1.33699	0.668494	−	0.743717i	$-0.266939\pi$
$557$	31.5541	1.33699	0.668494	+	0.743717i	$0.266939\pi$
$558$	0	0
$559$	4.35953	0.184388
$560$	0	0
$561$	−1.15405	−0.0487240
$562$	0	0
$563$	−34.3994	−1.44976	−0.724882	−	0.688873i	$-0.758106\pi$
$563$	−34.3994	−1.44976	−0.724882	+	0.688873i	$0.758106\pi$
$564$	0	0
$565$	−22.1315	−0.931081
$566$	0	0
$567$	9.39355	0.394492
$568$	0	0
$569$	6.34244	0.265889	0.132944	−	0.991123i	$-0.457557\pi$
$569$	6.34244	0.265889	0.132944	+	0.991123i	$0.457557\pi$
$570$	0	0
$571$	4.49002	0.187902	0.0939508	−	0.995577i	$-0.470050\pi$
$571$	4.49002	0.187902	0.0939508	+	0.995577i	$0.470050\pi$
$572$	0	0
$573$	3.13710	0.131054
$574$	0	0
$575$	48.4021	2.01851
$576$	0	0
$577$	−28.6227	−1.19158	−0.595789	−	0.803141i	$-0.703161\pi$
$577$	−28.6227	−1.19158	−0.595789	+	0.803141i	$0.703161\pi$
$578$	0	0
$579$	3.56928	0.148334
$580$	0	0
$581$	−9.07837	−0.376634
$582$	0	0
$583$	−0.523891	−0.0216974
$584$	0	0
$585$	−12.1295	−0.501492
$586$	0	0
$587$	−8.94511	−0.369204	−0.184602	−	0.982813i	$-0.559100\pi$
$587$	−8.94511	−0.369204	−0.184602	+	0.982813i	$0.559100\pi$
$588$	0	0
$589$	5.34737	0.220334
$590$	0	0
$591$	6.63212	0.272809
$592$	0	0
$593$	4.48487	0.184172	0.0920859	−	0.995751i	$-0.470647\pi$
$593$	4.48487	0.184172	0.0920859	+	0.995751i	$0.470647\pi$
$594$	0	0
$595$	−13.3480	−0.547214
$596$	0	0
$597$	−6.70157	−0.274277
$598$	0	0
$599$	2.13414	0.0871985	0.0435992	−	0.999049i	$-0.486118\pi$
$599$	2.13414	0.0871985	0.0435992	+	0.999049i	$0.486118\pi$
$600$	0	0
$601$	33.1661	1.35287	0.676436	−	0.736501i	$-0.263524\pi$
$601$	33.1661	1.35287	0.676436	+	0.736501i	$0.263524\pi$
$602$	0	0
$603$	−4.90450	−0.199727
$604$	0	0
$605$	3.51994	0.143106
$606$	0	0
$607$	−49.0969	−1.99278	−0.996391	−	0.0848855i	$-0.972948\pi$
$607$	−49.0969	−1.99278	−0.996391	+	0.0848855i	$0.972948\pi$
$608$	0	0
$609$	1.79149	0.0725948
$610$	0	0
$611$	6.82039	0.275923
$612$	0	0
$613$	34.6846	1.40090	0.700450	−	0.713702i	$-0.252983\pi$
$613$	34.6846	1.40090	0.700450	+	0.713702i	$0.252983\pi$
$614$	0	0
$615$	−0.880356	−0.0354994
$616$	0	0
$617$	−20.9248	−0.842400	−0.421200	−	0.906968i	$-0.638391\pi$
$617$	−20.9248	−0.842400	−0.421200	+	0.906968i	$0.638391\pi$
$618$	0	0
$619$	−23.8215	−0.957468	−0.478734	−	0.877960i	$-0.658904\pi$
$619$	−23.8215	−0.957468	−0.478734	+	0.877960i	$0.658904\pi$
$620$	0	0
$621$	−14.0581	−0.564132
$622$	0	0
$623$	−12.7981	−0.512746
$624$	0	0
$625$	−7.33778	−0.293511
$626$	0	0
$627$	0.365895	0.0146124
$628$	0	0
$629$	−4.40184	−0.175513
$630$	0	0
$631$	−44.1992	−1.75954	−0.879771	−	0.475398i	$-0.842304\pi$
$631$	−44.1992	−1.75954	−0.879771	+	0.475398i	$0.842304\pi$
$632$	0	0
$633$	−3.27383	−0.130123
$634$	0	0
$635$	1.69704	0.0673449
$636$	0	0
$637$	−6.67813	−0.264597
$638$	0	0
$639$	−40.5238	−1.60310
$640$	0	0
$641$	4.85043	0.191580	0.0957902	−	0.995402i	$-0.469462\pi$
$641$	4.85043	0.191580	0.0957902	+	0.995402i	$0.469462\pi$
$642$	0	0
$643$	25.8729	1.02033	0.510165	−	0.860077i	$-0.329584\pi$
$643$	25.8729	1.02033	0.510165	+	0.860077i	$0.329584\pi$
$644$	0	0
$645$	4.67003	0.183882
$646$	0	0
$647$	23.7816	0.934952	0.467476	−	0.884006i	$-0.345163\pi$
$647$	23.7816	0.934952	0.467476	+	0.884006i	$0.345163\pi$
$648$	0	0
$649$	−8.37564	−0.328773
$650$	0	0
$651$	2.35238	0.0921971
$652$	0	0
$653$	40.7334	1.59402	0.797011	−	0.603965i	$-0.206414\pi$
$653$	40.7334	1.59402	0.797011	+	0.603965i	$0.206414\pi$
$654$	0	0
$655$	−37.0172	−1.44638
$656$	0	0
$657$	41.5725	1.62190
$658$	0	0
$659$	−6.03351	−0.235032	−0.117516	−	0.993071i	$-0.537493\pi$
$659$	−6.03351	−0.235032	−0.117516	+	0.993071i	$0.537493\pi$
$660$	0	0
$661$	−6.42894	−0.250057	−0.125028	−	0.992153i	$-0.539902\pi$
$661$	−6.42894	−0.250057	−0.125028	+	0.992153i	$0.539902\pi$
$662$	0	0
$663$	−1.38751	−0.0538864
$664$	0	0
$665$	4.23202	0.164111
$666$	0	0
$667$	26.6726	1.03277
$668$	0	0
$669$	−2.14465	−0.0829170
$670$	0	0
$671$	−6.55777	−0.253160
$672$	0	0
$673$	23.1621	0.892832	0.446416	−	0.894826i	$-0.352700\pi$
$673$	23.1621	0.892832	0.446416	+	0.894826i	$0.352700\pi$
$674$	0	0
$675$	−15.8618	−0.610521
$676$	0	0
$677$	10.0519	0.386325	0.193163	−	0.981167i	$-0.438125\pi$
$677$	10.0519	0.386325	0.193163	+	0.981167i	$0.438125\pi$
$678$	0	0
$679$	15.4181	0.591693
$680$	0	0
$681$	−2.78805	−0.106838
$682$	0	0
$683$	30.1560	1.15389	0.576943	−	0.816784i	$-0.304245\pi$
$683$	30.1560	1.15389	0.576943	+	0.816784i	$0.304245\pi$
$684$	0	0
$685$	−1.89539	−0.0724191
$686$	0	0
$687$	6.80062	0.259460
$688$	0	0
$689$	−0.629873	−0.0239962
$690$	0	0
$691$	36.7673	1.39870	0.699348	−	0.714782i	$-0.253474\pi$
$691$	36.7673	1.39870	0.699348	+	0.714782i	$0.253474\pi$
$692$	0	0
$693$	−3.44593	−0.130900
$694$	0	0
$695$	−15.2755	−0.579433
$696$	0	0
$697$	2.15593	0.0816617
$698$	0	0
$699$	7.19307	0.272067
$700$	0	0
$701$	19.7296	0.745176	0.372588	−	0.927997i	$-0.378471\pi$
$701$	19.7296	0.745176	0.372588	+	0.927997i	$0.378471\pi$
$702$	0	0
$703$	1.39561	0.0526366
$704$	0	0
$705$	7.30617	0.275166
$706$	0	0
$707$	7.00455	0.263433
$708$	0	0
$709$	3.17620	0.119285	0.0596424	−	0.998220i	$-0.481004\pi$
$709$	3.17620	0.119285	0.0596424	+	0.998220i	$0.481004\pi$
$710$	0	0
$711$	−30.4352	−1.14141
$712$	0	0
$713$	35.0235	1.31164
$714$	0	0
$715$	4.23202	0.158268
$716$	0	0
$717$	−4.48544	−0.167512
$718$	0	0
$719$	−12.2220	−0.455805	−0.227903	−	0.973684i	$-0.573187\pi$
$719$	−12.2220	−0.455805	−0.227903	+	0.973684i	$0.573187\pi$
$720$	0	0
$721$	−10.2131	−0.380354
$722$	0	0
$723$	−7.00686	−0.260588
$724$	0	0
$725$	30.0948	1.11769
$726$	0	0
$727$	12.0446	0.446711	0.223355	−	0.974737i	$-0.428299\pi$
$727$	12.0446	0.446711	0.223355	+	0.974737i	$0.428299\pi$
$728$	0	0
$729$	−20.0370	−0.742111
$730$	0	0
$731$	−11.4366	−0.422998
$732$	0	0
$733$	0.0621934	0.00229717	0.00114858	−	0.999999i	$-0.499634\pi$
$733$	0.0621934	0.00229717	0.00114858	+	0.999999i	$0.499634\pi$
$734$	0	0
$735$	−7.15378	−0.263871
$736$	0	0
$737$	1.71120	0.0630328
$738$	0	0
$739$	−30.4907	−1.12162	−0.560810	−	0.827945i	$-0.689510\pi$
$739$	−30.4907	−1.12162	−0.560810	+	0.827945i	$0.689510\pi$
$740$	0	0
$741$	0.439914	0.0161606
$742$	0	0
$743$	1.38168	0.0506891	0.0253445	−	0.999679i	$-0.491932\pi$
$743$	1.38168	0.0506891	0.0253445	+	0.999679i	$0.491932\pi$
$744$	0	0
$745$	−54.1089	−1.98240
$746$	0	0
$747$	21.6417	0.791827
$748$	0	0
$749$	−9.09390	−0.332284
$750$	0	0
$751$	−12.6933	−0.463185	−0.231593	−	0.972813i	$-0.574394\pi$
$751$	−12.6933	−0.463185	−0.231593	+	0.972813i	$0.574394\pi$
$752$	0	0
$753$	2.75917	0.100550
$754$	0	0
$755$	19.3081	0.702692
$756$	0	0
$757$	13.3745	0.486103	0.243051	−	0.970013i	$-0.421852\pi$
$757$	13.3745	0.486103	0.243051	+	0.970013i	$0.421852\pi$
$758$	0	0
$759$	2.39649	0.0869870
$760$	0	0
$761$	−18.7046	−0.678041	−0.339021	−	0.940779i	$-0.610096\pi$
$761$	−18.7046	−0.678041	−0.339021	+	0.940779i	$0.610096\pi$
$762$	0	0
$763$	11.7500	0.425380
$764$	0	0
$765$	31.8199	1.15045
$766$	0	0
$767$	−10.0700	−0.363607
$768$	0	0
$769$	0.784395	0.0282860	0.0141430	−	0.999900i	$-0.495498\pi$
$769$	0.784395	0.0282860	0.0141430	+	0.999900i	$0.495498\pi$
$770$	0	0
$771$	−10.0450	−0.361762
$772$	0	0
$773$	49.7624	1.78983	0.894915	−	0.446236i	$-0.147236\pi$
$773$	49.7624	1.78983	0.894915	+	0.446236i	$0.147236\pi$
$774$	0	0
$775$	39.5171	1.41950
$776$	0	0
$777$	0.613950	0.0220253
$778$	0	0
$779$	−0.683544	−0.0244905
$780$	0	0
$781$	14.1389	0.505930
$782$	0	0
$783$	−8.74085	−0.312373
$784$	0	0
$785$	−25.8099	−0.921194
$786$	0	0
$787$	−10.4745	−0.373377	−0.186689	−	0.982419i	$-0.559776\pi$
$787$	−10.4745	−0.373377	−0.186689	+	0.982419i	$0.559776\pi$
$788$	0	0
$789$	−8.09126	−0.288056
$790$	0	0
$791$	−7.55940	−0.268781
$792$	0	0
$793$	−7.88438	−0.279983
$794$	0	0
$795$	−0.674735	−0.0239304
$796$	0	0
$797$	51.0562	1.80850	0.904252	−	0.427000i	$-0.140429\pi$
$797$	51.0562	1.80850	0.904252	+	0.427000i	$0.140429\pi$
$798$	0	0
$799$	−17.8923	−0.632984
$800$	0	0
$801$	30.5091	1.07799
$802$	0	0
$803$	−14.5048	−0.511863
$804$	0	0
$805$	27.7183	0.976941
$806$	0	0
$807$	−5.01025	−0.176369
$808$	0	0
$809$	53.6806	1.88731	0.943655	−	0.330929i	$-0.107362\pi$
$809$	53.6806	1.88731	0.943655	+	0.330929i	$0.107362\pi$
$810$	0	0
$811$	21.3619	0.750117	0.375058	−	0.927001i	$-0.377623\pi$
$811$	21.3619	0.750117	0.375058	+	0.927001i	$0.377623\pi$
$812$	0	0
$813$	−7.27985	−0.255315
$814$	0	0
$815$	24.9238	0.873043
$816$	0	0
$817$	3.62600	0.126858
$818$	0	0
$819$	−4.14302	−0.144769
$820$	0	0
$821$	7.58097	0.264578	0.132289	−	0.991211i	$-0.457767\pi$
$821$	7.58097	0.264578	0.132289	+	0.991211i	$0.457767\pi$
$822$	0	0
$823$	−31.9617	−1.11411	−0.557057	−	0.830474i	$-0.688070\pi$
$823$	−31.9617	−1.11411	−0.557057	+	0.830474i	$0.688070\pi$
$824$	0	0
$825$	2.70397	0.0941400
$826$	0	0
$827$	−5.13967	−0.178724	−0.0893620	−	0.995999i	$-0.528483\pi$
$827$	−5.13967	−0.178724	−0.0893620	+	0.995999i	$0.528483\pi$
$828$	0	0
$829$	−55.6227	−1.93186	−0.965928	−	0.258811i	$-0.916669\pi$
$829$	−55.6227	−1.93186	−0.965928	+	0.258811i	$0.916669\pi$
$830$	0	0
$831$	−9.84521	−0.341527
$832$	0	0
$833$	17.5191	0.607002
$834$	0	0
$835$	63.6579	2.20297
$836$	0	0
$837$	−11.4775	−0.396720
$838$	0	0
$839$	28.4332	0.981624	0.490812	−	0.871265i	$-0.336700\pi$
$839$	28.4332	0.981624	0.490812	+	0.871265i	$0.336700\pi$
$840$	0	0
$841$	−12.4159	−0.428133
$842$	0	0
$843$	−7.88617	−0.271614
$844$	0	0
$845$	−40.6711	−1.39913
$846$	0	0
$847$	1.20230	0.0413114
$848$	0	0
$849$	−2.47688	−0.0850064
$850$	0	0
$851$	9.14081	0.313343
$852$	0	0
$853$	27.7505	0.950160	0.475080	−	0.879943i	$-0.342419\pi$
$853$	27.7505	0.950160	0.475080	+	0.879943i	$0.342419\pi$
$854$	0	0
$855$	−10.0886	−0.345022
$856$	0	0
$857$	23.9861	0.819348	0.409674	−	0.912232i	$-0.365642\pi$
$857$	23.9861	0.819348	0.409674	+	0.912232i	$0.365642\pi$
$858$	0	0
$859$	−14.1447	−0.482609	−0.241305	−	0.970449i	$-0.577575\pi$
$859$	−14.1447	−0.482609	−0.241305	+	0.970449i	$0.577575\pi$
$860$	0	0
$861$	−0.300700	−0.0102478
$862$	0	0
$863$	−54.3214	−1.84912	−0.924561	−	0.381034i	$-0.875568\pi$
$863$	−54.3214	−1.84912	−0.924561	+	0.381034i	$0.875568\pi$
$864$	0	0
$865$	−71.4114	−2.42806
$866$	0	0
$867$	−2.58028	−0.0876308
$868$	0	0
$869$	10.6190	0.360223
$870$	0	0
$871$	2.05737	0.0697112
$872$	0	0
$873$	−36.7548	−1.24396
$874$	0	0
$875$	10.1146	0.341935
$876$	0	0
$877$	−11.4882	−0.387930	−0.193965	−	0.981008i	$-0.562135\pi$
$877$	−11.4882	−0.387930	−0.193965	+	0.981008i	$0.562135\pi$
$878$	0	0
$879$	5.47480	0.184660
$880$	0	0
$881$	5.50799	0.185569	0.0927844	−	0.995686i	$-0.470423\pi$
$881$	5.50799	0.185569	0.0927844	+	0.995686i	$0.470423\pi$
$882$	0	0
$883$	40.8837	1.37585	0.687923	−	0.725783i	$-0.258522\pi$
$883$	40.8837	1.37585	0.687923	+	0.725783i	$0.258522\pi$
$884$	0	0
$885$	−10.7872	−0.362609
$886$	0	0
$887$	47.7220	1.60235	0.801175	−	0.598430i	$-0.204209\pi$
$887$	47.7220	1.60235	0.801175	+	0.598430i	$0.204209\pi$
$888$	0	0
$889$	0.579652	0.0194409
$890$	0	0
$891$	7.81301	0.261746
$892$	0	0
$893$	5.67280	0.189833
$894$	0	0
$895$	60.6039	2.02576
$896$	0	0
$897$	2.88129	0.0962034
$898$	0	0
$899$	21.7764	0.726285
$900$	0	0
$901$	1.65238	0.0550488
$902$	0	0
$903$	1.59513	0.0530825
$904$	0	0
$905$	37.2140	1.23704
$906$	0	0
$907$	−26.2158	−0.870480	−0.435240	−	0.900314i	$-0.643337\pi$
$907$	−26.2158	−0.870480	−0.435240	+	0.900314i	$0.643337\pi$
$908$	0	0
$909$	−16.6980	−0.553836
$910$	0	0
$911$	−4.52294	−0.149852	−0.0749258	−	0.997189i	$-0.523872\pi$
$911$	−4.52294	−0.149852	−0.0749258	+	0.997189i	$0.523872\pi$
$912$	0	0
$913$	−7.55086	−0.249897
$914$	0	0
$915$	−8.44594	−0.279214
$916$	0	0
$917$	−12.6438	−0.417536
$918$	0	0
$919$	5.66305	0.186807	0.0934034	−	0.995628i	$-0.470225\pi$
$919$	5.66305	0.186807	0.0934034	+	0.995628i	$0.470225\pi$
$920$	0	0
$921$	−4.61536	−0.152081
$922$	0	0
$923$	16.9991	0.559534
$924$	0	0
$925$	10.3136	0.339109
$926$	0	0
$927$	24.3466	0.799648
$928$	0	0
$929$	−44.7181	−1.46715	−0.733577	−	0.679607i	$-0.762151\pi$
$929$	−44.7181	−1.46715	−0.733577	+	0.679607i	$0.762151\pi$
$930$	0	0
$931$	−5.55448	−0.182041
$932$	0	0
$933$	10.6162	0.347559
$934$	0	0
$935$	−11.1021	−0.363077
$936$	0	0
$937$	18.0094	0.588340	0.294170	−	0.955753i	$-0.404957\pi$
$937$	18.0094	0.588340	0.294170	+	0.955753i	$0.404957\pi$
$938$	0	0
$939$	−2.26967	−0.0740680
$940$	0	0
$941$	−2.92714	−0.0954220	−0.0477110	−	0.998861i	$-0.515193\pi$
$941$	−2.92714	−0.0954220	−0.0477110	+	0.998861i	$0.515193\pi$
$942$	0	0
$943$	−4.47698	−0.145791
$944$	0	0
$945$	−9.08352	−0.295487
$946$	0	0
$947$	25.0977	0.815564	0.407782	−	0.913079i	$-0.366302\pi$
$947$	25.0977	0.815564	0.407782	+	0.913079i	$0.366302\pi$
$948$	0	0
$949$	−17.4391	−0.566096
$950$	0	0
$951$	−5.02141	−0.162830
$952$	0	0
$953$	23.5782	0.763773	0.381886	−	0.924209i	$-0.375275\pi$
$953$	23.5782	0.763773	0.381886	+	0.924209i	$0.375275\pi$
$954$	0	0
$955$	30.1792	0.976576
$956$	0	0
$957$	1.49006	0.0481667
$958$	0	0
$959$	−0.647401	−0.0209057
$960$	0	0
$961$	−2.40565	−0.0776015
$962$	0	0
$963$	21.6787	0.698587
$964$	0	0
$965$	34.3369	1.10534
$966$	0	0
$967$	38.9377	1.25215	0.626076	−	0.779762i	$-0.284660\pi$
$967$	38.9377	1.25215	0.626076	+	0.779762i	$0.284660\pi$
$968$	0	0
$969$	−1.15405	−0.0370734
$970$	0	0
$971$	58.9978	1.89333	0.946665	−	0.322219i	$-0.104429\pi$
$971$	58.9978	1.89333	0.946665	+	0.322219i	$0.104429\pi$
$972$	0	0
$973$	−5.21761	−0.167269
$974$	0	0
$975$	3.25097	0.104114
$976$	0	0
$977$	23.7802	0.760795	0.380398	−	0.924823i	$-0.375787\pi$
$977$	23.7802	0.760795	0.380398	+	0.924823i	$0.375787\pi$
$978$	0	0
$979$	−10.6447	−0.340207
$980$	0	0
$981$	−28.0106	−0.894310
$982$	0	0
$983$	−39.0553	−1.24567	−0.622835	−	0.782353i	$-0.714019\pi$
$983$	−39.0553	−1.24567	−0.622835	+	0.782353i	$0.714019\pi$
$984$	0	0
$985$	63.8017	2.03289
$986$	0	0
$987$	2.49554	0.0794340
$988$	0	0
$989$	23.7491	0.755177
$990$	0	0
$991$	12.5250	0.397871	0.198935	−	0.980013i	$-0.436252\pi$
$991$	12.5250	0.397871	0.198935	+	0.980013i	$0.436252\pi$
$992$	0	0
$993$	3.64598	0.115702
$994$	0	0
$995$	−64.4698	−2.04383
$996$	0	0
$997$	23.8834	0.756393	0.378197	−	0.925725i	$-0.376544\pi$
$997$	23.8834	0.756393	0.378197	+	0.925725i	$0.376544\pi$
$998$	0	0
$999$	−2.99552	−0.0947741

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1672.2.a.j.1.3	✓	6
4.3	odd	2		3344.2.a.v.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.j.1.3	✓	6	1.1	even	1	trivial
3344.2.a.v.1.4		6	4.3	odd	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:	\( N \)	\(=\)	\( 1672 = 2^{3} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1672.a (trivial)

\(f(q)\)	\(=\)	\(q+0.365895 q^{3} +3.51994 q^{5} +1.20230 q^{7} -2.86612 q^{9} +1.00000 q^{11} +1.20230 q^{13} +1.28793 q^{15} -3.15405 q^{17} +1.00000 q^{19} +0.439914 q^{21} +6.54966 q^{23} +7.39001 q^{25} -2.14638 q^{27} +4.07236 q^{29} +5.34737 q^{31} +0.365895 q^{33} +4.23202 q^{35} +1.39561 q^{37} +0.439914 q^{39} -0.683544 q^{41} +3.62600 q^{43} -10.0886 q^{45} +5.67280 q^{47} -5.55448 q^{49} -1.15405 q^{51} -0.523891 q^{53} +3.51994 q^{55} +0.365895 q^{57} -8.37564 q^{59} -6.55777 q^{61} -3.44593 q^{63} +4.23202 q^{65} +1.71120 q^{67} +2.39649 q^{69} +14.1389 q^{71} -14.5048 q^{73} +2.70397 q^{75} +1.20230 q^{77} +10.6190 q^{79} +7.81301 q^{81} -7.55086 q^{83} -11.1021 q^{85} +1.49006 q^{87} -10.6447 q^{89} +1.44552 q^{91} +1.95657 q^{93} +3.51994 q^{95} +12.8239 q^{97} -2.86612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + q^{5} + 4 q^{9} + 6 q^{11} + 7 q^{15} + 3 q^{17} + 6 q^{19} + 7 q^{21} + 7 q^{23} + 3 q^{25} + 13 q^{27} - 4 q^{29} + 7 q^{31} + 4 q^{33} + 6 q^{35} - 2 q^{37} + 7 q^{39} + 7 q^{41} + 21 q^{43}+ \cdots + 4 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + q^5 + 4 * q^9 + 6 * q^11 + 7 * q^15 + 3 * q^17 + 6 * q^19 + 7 * q^21 + 7 * q^23 + 3 * q^25 + 13 * q^27 - 4 * q^29 + 7 * q^31 + 4 * q^33 + 6 * q^35 - 2 * q^37 + 7 * q^39 + 7 * q^41 + 21 * q^43 + 11 * q^45 + 16 * q^47 + 2 * q^49 + 15 * q^51 + 17 * q^53 + q^55 + 4 * q^57 + 19 * q^59 - 6 * q^61 + 2 * q^63 + 6 * q^65 + 14 * q^67 + q^69 + q^71 - 5 * q^73 + 18 * q^75 + 2 * q^81 + 11 * q^83 - 26 * q^85 + 14 * q^87 + 12 * q^89 + 44 * q^91 - 6 * q^93 + q^95 + 14 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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