LMFDB - Embedded newform 6036.2.a.f.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 12 x^{12} + 112 x^{11} + 7 x^{10} - 710 x^{9} + 281 x^{8} + 1850 x^{7} - 830 x^{6} + \cdots + 1 $ x^14 - 6x^13 - 12x^12 + 112x^11 + 7x^10 - 710x^9 + 281x^8 + 1850x^7 - 830x^6 - 1918x^5 + 623x^4 + 480x^3 - 29x^2 - 17*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.38751$ of defining polynomial
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.38751 q^{5} -0.715830 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.38751 q^{5} -0.715830 q^{7} +1.00000 q^{9} +3.44457 q^{11} -0.0881957 q^{13} -1.38751 q^{15} +3.27833 q^{17} +1.97176 q^{19} +0.715830 q^{21} -5.87454 q^{23} -3.07480 q^{25} -1.00000 q^{27} -8.26295 q^{29} -3.72516 q^{31} -3.44457 q^{33} -0.993224 q^{35} -1.33627 q^{37} +0.0881957 q^{39} -9.85619 q^{41} -9.08085 q^{43} +1.38751 q^{45} +2.51582 q^{47} -6.48759 q^{49} -3.27833 q^{51} -9.50918 q^{53} +4.77938 q^{55} -1.97176 q^{57} +6.47108 q^{59} +8.48036 q^{61} -0.715830 q^{63} -0.122373 q^{65} +1.30841 q^{67} +5.87454 q^{69} +0.137896 q^{71} +15.4177 q^{73} +3.07480 q^{75} -2.46572 q^{77} +3.27933 q^{79} +1.00000 q^{81} +8.19491 q^{83} +4.54873 q^{85} +8.26295 q^{87} -1.28588 q^{89} +0.0631331 q^{91} +3.72516 q^{93} +2.73585 q^{95} -0.299309 q^{97} +3.44457 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.38751	0.620515	0.310258	−	0.950653i	$-0.399585\pi$
$5$	1.38751	0.620515	0.310258	+	0.950653i	$0.399585\pi$
$6$	0	0
$7$	−0.715830	−0.270558	−0.135279	−	0.990808i	$-0.543193\pi$
$7$	−0.715830	−0.270558	−0.135279	+	0.990808i	$0.543193\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.44457	1.03858	0.519288	−	0.854599i	$-0.326197\pi$
$11$	3.44457	1.03858	0.519288	+	0.854599i	$0.326197\pi$
$12$	0	0
$13$	−0.0881957	−0.0244611	−0.0122305	−	0.999925i	$-0.503893\pi$
$13$	−0.0881957	−0.0244611	−0.0122305	+	0.999925i	$0.503893\pi$
$14$	0	0
$15$	−1.38751	−0.358255
$16$	0	0
$17$	3.27833	0.795111	0.397556	−	0.917578i	$-0.369859\pi$
$17$	3.27833	0.795111	0.397556	+	0.917578i	$0.369859\pi$
$18$	0	0
$19$	1.97176	0.452353	0.226176	−	0.974086i	$-0.427377\pi$
$19$	1.97176	0.452353	0.226176	+	0.974086i	$0.427377\pi$
$20$	0	0
$21$	0.715830	0.156207
$22$	0	0
$23$	−5.87454	−1.22493	−0.612463	−	0.790499i	$-0.709821\pi$
$23$	−5.87454	−1.22493	−0.612463	+	0.790499i	$0.709821\pi$
$24$	0	0
$25$	−3.07480	−0.614961
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.26295	−1.53439	−0.767196	−	0.641413i	$-0.778348\pi$
$29$	−8.26295	−1.53439	−0.767196	+	0.641413i	$0.778348\pi$
$30$	0	0
$31$	−3.72516	−0.669059	−0.334530	−	0.942385i	$-0.608577\pi$
$31$	−3.72516	−0.669059	−0.334530	+	0.942385i	$0.608577\pi$
$32$	0	0
$33$	−3.44457	−0.599622
$34$	0	0
$35$	−0.993224	−0.167885
$36$	0	0
$37$	−1.33627	−0.219681	−0.109841	−	0.993949i	$-0.535034\pi$
$37$	−1.33627	−0.219681	−0.109841	+	0.993949i	$0.535034\pi$
$38$	0	0
$39$	0.0881957	0.0141226
$40$	0	0
$41$	−9.85619	−1.53928	−0.769639	−	0.638479i	$-0.779564\pi$
$41$	−9.85619	−1.53928	−0.769639	+	0.638479i	$0.779564\pi$
$42$	0	0
$43$	−9.08085	−1.38482	−0.692408	−	0.721506i	$-0.743450\pi$
$43$	−9.08085	−1.38482	−0.692408	+	0.721506i	$0.743450\pi$
$44$	0	0
$45$	1.38751	0.206838
$46$	0	0
$47$	2.51582	0.366970	0.183485	−	0.983022i	$-0.441262\pi$
$47$	2.51582	0.366970	0.183485	+	0.983022i	$0.441262\pi$
$48$	0	0
$49$	−6.48759	−0.926798
$50$	0	0
$51$	−3.27833	−0.459058
$52$	0	0
$53$	−9.50918	−1.30619	−0.653093	−	0.757278i	$-0.726529\pi$
$53$	−9.50918	−1.30619	−0.653093	+	0.757278i	$0.726529\pi$
$54$	0	0
$55$	4.77938	0.644452
$56$	0	0
$57$	−1.97176	−0.261166
$58$	0	0
$59$	6.47108	0.842463	0.421231	−	0.906953i	$-0.361598\pi$
$59$	6.47108	0.842463	0.421231	+	0.906953i	$0.361598\pi$
$60$	0	0
$61$	8.48036	1.08580	0.542899	−	0.839798i	$-0.317327\pi$
$61$	8.48036	1.08580	0.542899	+	0.839798i	$0.317327\pi$
$62$	0	0
$63$	−0.715830	−0.0901861
$64$	0	0
$65$	−0.122373	−0.0151785
$66$	0	0
$67$	1.30841	0.159848	0.0799239	−	0.996801i	$-0.474532\pi$
$67$	1.30841	0.159848	0.0799239	+	0.996801i	$0.474532\pi$
$68$	0	0
$69$	5.87454	0.707211
$70$	0	0
$71$	0.137896	0.0163653	0.00818263	−	0.999967i	$-0.497395\pi$
$71$	0.137896	0.0163653	0.00818263	+	0.999967i	$0.497395\pi$
$72$	0	0
$73$	15.4177	1.80450	0.902252	−	0.431209i	$-0.141913\pi$
$73$	15.4177	1.80450	0.902252	+	0.431209i	$0.141913\pi$
$74$	0	0
$75$	3.07480	0.355048
$76$	0	0
$77$	−2.46572	−0.280995
$78$	0	0
$79$	3.27933	0.368953	0.184476	−	0.982837i	$-0.440941\pi$
$79$	3.27933	0.368953	0.184476	+	0.982837i	$0.440941\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.19491	0.899508	0.449754	−	0.893153i	$-0.351512\pi$
$83$	8.19491	0.899508	0.449754	+	0.893153i	$0.351512\pi$
$84$	0	0
$85$	4.54873	0.493379
$86$	0	0
$87$	8.26295	0.885881
$88$	0	0
$89$	−1.28588	−0.136303	−0.0681513	−	0.997675i	$-0.521710\pi$
$89$	−1.28588	−0.136303	−0.0681513	+	0.997675i	$0.521710\pi$
$90$	0	0
$91$	0.0631331	0.00661815
$92$	0	0
$93$	3.72516	0.386282
$94$	0	0
$95$	2.73585	0.280692
$96$	0	0
$97$	−0.299309	−0.0303902	−0.0151951	−	0.999885i	$-0.504837\pi$
$97$	−0.299309	−0.0303902	−0.0151951	+	0.999885i	$0.504837\pi$
$98$	0	0
$99$	3.44457	0.346192
$100$	0	0
$101$	−9.14982	−0.910441	−0.455220	−	0.890379i	$-0.650440\pi$
$101$	−9.14982	−0.910441	−0.455220	+	0.890379i	$0.650440\pi$
$102$	0	0
$103$	6.91023	0.680885	0.340442	−	0.940265i	$-0.389423\pi$
$103$	6.91023	0.680885	0.340442	+	0.940265i	$0.389423\pi$
$104$	0	0
$105$	0.993224	0.0969287
$106$	0	0
$107$	−4.08299	−0.394717	−0.197359	−	0.980331i	$-0.563236\pi$
$107$	−4.08299	−0.394717	−0.197359	+	0.980331i	$0.563236\pi$
$108$	0	0
$109$	−19.5423	−1.87181	−0.935904	−	0.352255i	$-0.885415\pi$
$109$	−19.5423	−1.87181	−0.935904	+	0.352255i	$0.885415\pi$
$110$	0	0
$111$	1.33627	0.126833
$112$	0	0
$113$	−1.74166	−0.163842	−0.0819209	−	0.996639i	$-0.526105\pi$
$113$	−1.74166	−0.163842	−0.0819209	+	0.996639i	$0.526105\pi$
$114$	0	0
$115$	−8.15101	−0.760085
$116$	0	0
$117$	−0.0881957	−0.00815370
$118$	0	0
$119$	−2.34673	−0.215124
$120$	0	0
$121$	0.865036	0.0786396
$122$	0	0
$123$	9.85619	0.888703
$124$	0	0
$125$	−11.2039	−1.00211
$126$	0	0
$127$	−9.99683	−0.887076	−0.443538	−	0.896256i	$-0.646277\pi$
$127$	−9.99683	−0.887076	−0.443538	+	0.896256i	$0.646277\pi$
$128$	0	0
$129$	9.08085	0.799524
$130$	0	0
$131$	2.77078	0.242085	0.121042	−	0.992647i	$-0.461376\pi$
$131$	2.77078	0.242085	0.121042	+	0.992647i	$0.461376\pi$
$132$	0	0
$133$	−1.41144	−0.122388
$134$	0	0
$135$	−1.38751	−0.119418
$136$	0	0
$137$	−0.970330	−0.0829009	−0.0414505	−	0.999141i	$-0.513198\pi$
$137$	−0.970330	−0.0829009	−0.0414505	+	0.999141i	$0.513198\pi$
$138$	0	0
$139$	−12.2109	−1.03571	−0.517856	−	0.855468i	$-0.673270\pi$
$139$	−12.2109	−1.03571	−0.517856	+	0.855468i	$0.673270\pi$
$140$	0	0
$141$	−2.51582	−0.211870
$142$	0	0
$143$	−0.303796	−0.0254047
$144$	0	0
$145$	−11.4650	−0.952113
$146$	0	0
$147$	6.48759	0.535087
$148$	0	0
$149$	−1.48212	−0.121420	−0.0607102	−	0.998155i	$-0.519337\pi$
$149$	−1.48212	−0.121420	−0.0607102	+	0.998155i	$0.519337\pi$
$150$	0	0
$151$	20.5369	1.67127	0.835633	−	0.549289i	$-0.185101\pi$
$151$	20.5369	1.67127	0.835633	+	0.549289i	$0.185101\pi$
$152$	0	0
$153$	3.27833	0.265037
$154$	0	0
$155$	−5.16872	−0.415161
$156$	0	0
$157$	−7.51339	−0.599634	−0.299817	−	0.953997i	$-0.596926\pi$
$157$	−7.51339	−0.599634	−0.299817	+	0.953997i	$0.596926\pi$
$158$	0	0
$159$	9.50918	0.754127
$160$	0	0
$161$	4.20517	0.331414
$162$	0	0
$163$	6.80379	0.532914	0.266457	−	0.963847i	$-0.414147\pi$
$163$	6.80379	0.532914	0.266457	+	0.963847i	$0.414147\pi$
$164$	0	0
$165$	−4.77938	−0.372075
$166$	0	0
$167$	−7.37877	−0.570987	−0.285493	−	0.958381i	$-0.592157\pi$
$167$	−7.37877	−0.570987	−0.285493	+	0.958381i	$0.592157\pi$
$168$	0	0
$169$	−12.9922	−0.999402
$170$	0	0
$171$	1.97176	0.150784
$172$	0	0
$173$	19.0277	1.44665	0.723324	−	0.690509i	$-0.242613\pi$
$173$	19.0277	1.44665	0.723324	+	0.690509i	$0.242613\pi$
$174$	0	0
$175$	2.20104	0.166383
$176$	0	0
$177$	−6.47108	−0.486396
$178$	0	0
$179$	−17.8938	−1.33744	−0.668721	−	0.743513i	$-0.733158\pi$
$179$	−17.8938	−1.33744	−0.668721	+	0.743513i	$0.733158\pi$
$180$	0	0
$181$	2.33363	0.173457	0.0867287	−	0.996232i	$-0.472359\pi$
$181$	2.33363	0.173457	0.0867287	+	0.996232i	$0.472359\pi$
$182$	0	0
$183$	−8.48036	−0.626886
$184$	0	0
$185$	−1.85409	−0.136315
$186$	0	0
$187$	11.2924	0.825783
$188$	0	0
$189$	0.715830	0.0520690
$190$	0	0
$191$	2.29851	0.166315	0.0831574	−	0.996536i	$-0.473500\pi$
$191$	2.29851	0.166315	0.0831574	+	0.996536i	$0.473500\pi$
$192$	0	0
$193$	−9.74298	−0.701315	−0.350658	−	0.936504i	$-0.614042\pi$
$193$	−9.74298	−0.701315	−0.350658	+	0.936504i	$0.614042\pi$
$194$	0	0
$195$	0.122373	0.00876330
$196$	0	0
$197$	−27.8344	−1.98312	−0.991558	−	0.129666i	$-0.958609\pi$
$197$	−27.8344	−1.98312	−0.991558	+	0.129666i	$0.958609\pi$
$198$	0	0
$199$	−2.38050	−0.168749	−0.0843744	−	0.996434i	$-0.526889\pi$
$199$	−2.38050	−0.168749	−0.0843744	+	0.996434i	$0.526889\pi$
$200$	0	0
$201$	−1.30841	−0.0922882
$202$	0	0
$203$	5.91486	0.415142
$204$	0	0
$205$	−13.6756	−0.955145
$206$	0	0
$207$	−5.87454	−0.408309
$208$	0	0
$209$	6.79186	0.469803
$210$	0	0
$211$	0.871429	0.0599917	0.0299958	−	0.999550i	$-0.490451\pi$
$211$	0.871429	0.0599917	0.0299958	+	0.999550i	$0.490451\pi$
$212$	0	0
$213$	−0.137896	−0.00944849
$214$	0	0
$215$	−12.5998	−0.859299
$216$	0	0
$217$	2.66658	0.181019
$218$	0	0
$219$	−15.4177	−1.04183
$220$	0	0
$221$	−0.289135	−0.0194493
$222$	0	0
$223$	4.35552	0.291667	0.145834	−	0.989309i	$-0.453414\pi$
$223$	4.35552	0.291667	0.145834	+	0.989309i	$0.453414\pi$
$224$	0	0
$225$	−3.07480	−0.204987
$226$	0	0
$227$	8.53086	0.566213	0.283106	−	0.959088i	$-0.408635\pi$
$227$	8.53086	0.566213	0.283106	+	0.959088i	$0.408635\pi$
$228$	0	0
$229$	−3.44604	−0.227721	−0.113860	−	0.993497i	$-0.536322\pi$
$229$	−3.44604	−0.227721	−0.113860	+	0.993497i	$0.536322\pi$
$230$	0	0
$231$	2.46572	0.162233
$232$	0	0
$233$	0.0961489	0.00629892	0.00314946	−	0.999995i	$-0.498997\pi$
$233$	0.0961489	0.00629892	0.00314946	+	0.999995i	$0.498997\pi$
$234$	0	0
$235$	3.49074	0.227711
$236$	0	0
$237$	−3.27933	−0.213015
$238$	0	0
$239$	−1.22608	−0.0793082	−0.0396541	−	0.999213i	$-0.512626\pi$
$239$	−1.22608	−0.0793082	−0.0396541	+	0.999213i	$0.512626\pi$
$240$	0	0
$241$	14.6416	0.943147	0.471573	−	0.881827i	$-0.343686\pi$
$241$	14.6416	0.943147	0.471573	+	0.881827i	$0.343686\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−9.00162	−0.575092
$246$	0	0
$247$	−0.173901	−0.0110650
$248$	0	0
$249$	−8.19491	−0.519331
$250$	0	0
$251$	7.80044	0.492359	0.246180	−	0.969224i	$-0.420825\pi$
$251$	7.80044	0.492359	0.246180	+	0.969224i	$0.420825\pi$
$252$	0	0
$253$	−20.2352	−1.27218
$254$	0	0
$255$	−4.54873	−0.284852
$256$	0	0
$257$	0.264056	0.0164714	0.00823568	−	0.999966i	$-0.497378\pi$
$257$	0.264056	0.0164714	0.00823568	+	0.999966i	$0.497378\pi$
$258$	0	0
$259$	0.956541	0.0594365
$260$	0	0
$261$	−8.26295	−0.511464
$262$	0	0
$263$	1.75704	0.108344	0.0541718	−	0.998532i	$-0.482748\pi$
$263$	1.75704	0.108344	0.0541718	+	0.998532i	$0.482748\pi$
$264$	0	0
$265$	−13.1941	−0.810508
$266$	0	0
$267$	1.28588	0.0786944
$268$	0	0
$269$	3.95509	0.241146	0.120573	−	0.992704i	$-0.461527\pi$
$269$	3.95509	0.241146	0.120573	+	0.992704i	$0.461527\pi$
$270$	0	0
$271$	−14.3749	−0.873210	−0.436605	−	0.899653i	$-0.643819\pi$
$271$	−14.3749	−0.873210	−0.436605	+	0.899653i	$0.643819\pi$
$272$	0	0
$273$	−0.0631331	−0.00382099
$274$	0	0
$275$	−10.5914	−0.638683
$276$	0	0
$277$	−7.53875	−0.452959	−0.226480	−	0.974016i	$-0.572722\pi$
$277$	−7.53875	−0.452959	−0.226480	+	0.974016i	$0.572722\pi$
$278$	0	0
$279$	−3.72516	−0.223020
$280$	0	0
$281$	21.7342	1.29655	0.648277	−	0.761404i	$-0.275490\pi$
$281$	21.7342	1.29655	0.648277	+	0.761404i	$0.275490\pi$
$282$	0	0
$283$	−10.7499	−0.639014	−0.319507	−	0.947584i	$-0.603517\pi$
$283$	−10.7499	−0.639014	−0.319507	+	0.947584i	$0.603517\pi$
$284$	0	0
$285$	−2.73585	−0.162057
$286$	0	0
$287$	7.05535	0.416464
$288$	0	0
$289$	−6.25256	−0.367798
$290$	0	0
$291$	0.299309	0.0175458
$292$	0	0
$293$	−4.26334	−0.249067	−0.124533	−	0.992215i	$-0.539743\pi$
$293$	−4.26334	−0.249067	−0.124533	+	0.992215i	$0.539743\pi$
$294$	0	0
$295$	8.97871	0.522761
$296$	0	0
$297$	−3.44457	−0.199874
$298$	0	0
$299$	0.518109	0.0299630
$300$	0	0
$301$	6.50034	0.374673
$302$	0	0
$303$	9.14982	0.525643
$304$	0	0
$305$	11.7666	0.673754
$306$	0	0
$307$	−0.156797	−0.00894890	−0.00447445	−	0.999990i	$-0.501424\pi$
$307$	−0.156797	−0.00894890	−0.00447445	+	0.999990i	$0.501424\pi$
$308$	0	0
$309$	−6.91023	−0.393109
$310$	0	0
$311$	−21.1425	−1.19888	−0.599440	−	0.800420i	$-0.704610\pi$
$311$	−21.1425	−1.19888	−0.599440	+	0.800420i	$0.704610\pi$
$312$	0	0
$313$	5.94596	0.336085	0.168043	−	0.985780i	$-0.446255\pi$
$313$	5.94596	0.336085	0.168043	+	0.985780i	$0.446255\pi$
$314$	0	0
$315$	−0.993224	−0.0559618
$316$	0	0
$317$	14.5465	0.817011	0.408506	−	0.912756i	$-0.366050\pi$
$317$	14.5465	0.817011	0.408506	+	0.912756i	$0.366050\pi$
$318$	0	0
$319$	−28.4623	−1.59358
$320$	0	0
$321$	4.08299	0.227890
$322$	0	0
$323$	6.46408	0.359671
$324$	0	0
$325$	0.271185	0.0150426
$326$	0	0
$327$	19.5423	1.08069
$328$	0	0
$329$	−1.80090	−0.0992869
$330$	0	0
$331$	−3.52374	−0.193682	−0.0968412	−	0.995300i	$-0.530874\pi$
$331$	−3.52374	−0.193682	−0.0968412	+	0.995300i	$0.530874\pi$
$332$	0	0
$333$	−1.33627	−0.0732271
$334$	0	0
$335$	1.81544	0.0991880
$336$	0	0
$337$	−16.2080	−0.882904	−0.441452	−	0.897285i	$-0.645537\pi$
$337$	−16.2080	−0.882904	−0.441452	+	0.897285i	$0.645537\pi$
$338$	0	0
$339$	1.74166	0.0945941
$340$	0	0
$341$	−12.8316	−0.694869
$342$	0	0
$343$	9.65482	0.521311
$344$	0	0
$345$	8.15101	0.438835
$346$	0	0
$347$	15.6193	0.838488	0.419244	−	0.907874i	$-0.362295\pi$
$347$	15.6193	0.838488	0.419244	+	0.907874i	$0.362295\pi$
$348$	0	0
$349$	−31.2751	−1.67412	−0.837060	−	0.547112i	$-0.815727\pi$
$349$	−31.2751	−1.67412	−0.837060	+	0.547112i	$0.815727\pi$
$350$	0	0
$351$	0.0881957	0.00470754
$352$	0	0
$353$	15.7067	0.835982	0.417991	−	0.908451i	$-0.362734\pi$
$353$	15.7067	0.835982	0.417991	+	0.908451i	$0.362734\pi$
$354$	0	0
$355$	0.191333	0.0101549
$356$	0	0
$357$	2.34673	0.124202
$358$	0	0
$359$	−19.4188	−1.02488	−0.512442	−	0.858722i	$-0.671259\pi$
$359$	−19.4188	−1.02488	−0.512442	+	0.858722i	$0.671259\pi$
$360$	0	0
$361$	−15.1122	−0.795377
$362$	0	0
$363$	−0.865036	−0.0454026
$364$	0	0
$365$	21.3923	1.11972
$366$	0	0
$367$	−0.133976	−0.00699348	−0.00349674	−	0.999994i	$-0.501113\pi$
$367$	−0.133976	−0.00699348	−0.00349674	+	0.999994i	$0.501113\pi$
$368$	0	0
$369$	−9.85619	−0.513093
$370$	0	0
$371$	6.80695	0.353399
$372$	0	0
$373$	−2.07758	−0.107573	−0.0537864	−	0.998552i	$-0.517129\pi$
$373$	−2.07758	−0.107573	−0.0537864	+	0.998552i	$0.517129\pi$
$374$	0	0
$375$	11.2039	0.578567
$376$	0	0
$377$	0.728757	0.0375329
$378$	0	0
$379$	−11.2114	−0.575893	−0.287946	−	0.957647i	$-0.592973\pi$
$379$	−11.2114	−0.575893	−0.287946	+	0.957647i	$0.592973\pi$
$380$	0	0
$381$	9.99683	0.512153
$382$	0	0
$383$	16.6938	0.853016	0.426508	−	0.904484i	$-0.359744\pi$
$383$	16.6938	0.853016	0.426508	+	0.904484i	$0.359744\pi$
$384$	0	0
$385$	−3.42123	−0.174362
$386$	0	0
$387$	−9.08085	−0.461605
$388$	0	0
$389$	−18.8536	−0.955915	−0.477958	−	0.878383i	$-0.658623\pi$
$389$	−18.8536	−0.955915	−0.477958	+	0.878383i	$0.658623\pi$
$390$	0	0
$391$	−19.2587	−0.973953
$392$	0	0
$393$	−2.77078	−0.139768
$394$	0	0
$395$	4.55011	0.228941
$396$	0	0
$397$	−3.10156	−0.155663	−0.0778313	−	0.996967i	$-0.524800\pi$
$397$	−3.10156	−0.155663	−0.0778313	+	0.996967i	$0.524800\pi$
$398$	0	0
$399$	1.41144	0.0706606
$400$	0	0
$401$	2.96630	0.148130	0.0740650	−	0.997253i	$-0.476403\pi$
$401$	2.96630	0.148130	0.0740650	+	0.997253i	$0.476403\pi$
$402$	0	0
$403$	0.328544	0.0163659
$404$	0	0
$405$	1.38751	0.0689461
$406$	0	0
$407$	−4.60286	−0.228156
$408$	0	0
$409$	−14.1887	−0.701586	−0.350793	−	0.936453i	$-0.614088\pi$
$409$	−14.1887	−0.701586	−0.350793	+	0.936453i	$0.614088\pi$
$410$	0	0
$411$	0.970330	0.0478629
$412$	0	0
$413$	−4.63219	−0.227935
$414$	0	0
$415$	11.3705	0.558158
$416$	0	0
$417$	12.2109	0.597969
$418$	0	0
$419$	17.5145	0.855638	0.427819	−	0.903864i	$-0.359282\pi$
$419$	17.5145	0.855638	0.427819	+	0.903864i	$0.359282\pi$
$420$	0	0
$421$	14.1393	0.689108	0.344554	−	0.938767i	$-0.388030\pi$
$421$	14.1393	0.689108	0.344554	+	0.938767i	$0.388030\pi$
$422$	0	0
$423$	2.51582	0.122323
$424$	0	0
$425$	−10.0802	−0.488962
$426$	0	0
$427$	−6.07049	−0.293772
$428$	0	0
$429$	0.303796	0.0146674
$430$	0	0
$431$	1.30842	0.0630245	0.0315122	−	0.999503i	$-0.489968\pi$
$431$	1.30842	0.0630245	0.0315122	+	0.999503i	$0.489968\pi$
$432$	0	0
$433$	4.62017	0.222031	0.111016	−	0.993819i	$-0.464590\pi$
$433$	4.62017	0.222031	0.111016	+	0.993819i	$0.464590\pi$
$434$	0	0
$435$	11.4650	0.549703
$436$	0	0
$437$	−11.5832	−0.554099
$438$	0	0
$439$	−14.4110	−0.687799	−0.343899	−	0.939007i	$-0.611748\pi$
$439$	−14.4110	−0.687799	−0.343899	+	0.939007i	$0.611748\pi$
$440$	0	0
$441$	−6.48759	−0.308933
$442$	0	0
$443$	30.2387	1.43669	0.718343	−	0.695690i	$-0.244901\pi$
$443$	30.2387	1.43669	0.718343	+	0.695690i	$0.244901\pi$
$444$	0	0
$445$	−1.78417	−0.0845778
$446$	0	0
$447$	1.48212	0.0701021
$448$	0	0
$449$	−3.73872	−0.176441	−0.0882205	−	0.996101i	$-0.528118\pi$
$449$	−3.73872	−0.176441	−0.0882205	+	0.996101i	$0.528118\pi$
$450$	0	0
$451$	−33.9503	−1.59866
$452$	0	0
$453$	−20.5369	−0.964906
$454$	0	0
$455$	0.0875981	0.00410666
$456$	0	0
$457$	−14.1656	−0.662639	−0.331319	−	0.943519i	$-0.607494\pi$
$457$	−14.1656	−0.662639	−0.331319	+	0.943519i	$0.607494\pi$
$458$	0	0
$459$	−3.27833	−0.153019
$460$	0	0
$461$	−16.7592	−0.780556	−0.390278	−	0.920697i	$-0.627621\pi$
$461$	−16.7592	−0.780556	−0.390278	+	0.920697i	$0.627621\pi$
$462$	0	0
$463$	−38.0954	−1.77044	−0.885222	−	0.465168i	$-0.845994\pi$
$463$	−38.0954	−1.77044	−0.885222	+	0.465168i	$0.845994\pi$
$464$	0	0
$465$	5.16872	0.239694
$466$	0	0
$467$	26.4456	1.22375	0.611877	−	0.790953i	$-0.290415\pi$
$467$	26.4456	1.22375	0.611877	+	0.790953i	$0.290415\pi$
$468$	0	0
$469$	−0.936599	−0.0432481
$470$	0	0
$471$	7.51339	0.346199
$472$	0	0
$473$	−31.2796	−1.43824
$474$	0	0
$475$	−6.06278	−0.278179
$476$	0	0
$477$	−9.50918	−0.435395
$478$	0	0
$479$	−20.3724	−0.930837	−0.465418	−	0.885091i	$-0.654096\pi$
$479$	−20.3724	−0.930837	−0.465418	+	0.885091i	$0.654096\pi$
$480$	0	0
$481$	0.117853	0.00537364
$482$	0	0
$483$	−4.20517	−0.191342
$484$	0	0
$485$	−0.415295	−0.0188576
$486$	0	0
$487$	22.3545	1.01298	0.506490	−	0.862246i	$-0.330943\pi$
$487$	22.3545	1.01298	0.506490	+	0.862246i	$0.330943\pi$
$488$	0	0
$489$	−6.80379	−0.307678
$490$	0	0
$491$	17.8324	0.804765	0.402383	−	0.915472i	$-0.368182\pi$
$491$	17.8324	0.804765	0.402383	+	0.915472i	$0.368182\pi$
$492$	0	0
$493$	−27.0887	−1.22001
$494$	0	0
$495$	4.77938	0.214817
$496$	0	0
$497$	−0.0987102	−0.00442776
$498$	0	0
$499$	8.18013	0.366193	0.183096	−	0.983095i	$-0.441388\pi$
$499$	8.18013	0.366193	0.183096	+	0.983095i	$0.441388\pi$
$500$	0	0
$501$	7.37877	0.329659
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−12.6955	−0.564942
$506$	0	0
$507$	12.9922	0.577005
$508$	0	0
$509$	2.95538	0.130995	0.0654974	−	0.997853i	$-0.479137\pi$
$509$	2.95538	0.130995	0.0654974	+	0.997853i	$0.479137\pi$
$510$	0	0
$511$	−11.0364	−0.488224
$512$	0	0
$513$	−1.97176	−0.0870553
$514$	0	0
$515$	9.58804	0.422499
$516$	0	0
$517$	8.66592	0.381127
$518$	0	0
$519$	−19.0277	−0.835223
$520$	0	0
$521$	−7.51971	−0.329445	−0.164722	−	0.986340i	$-0.552673\pi$
$521$	−7.51971	−0.329445	−0.164722	+	0.986340i	$0.552673\pi$
$522$	0	0
$523$	−1.73286	−0.0757725	−0.0378862	−	0.999282i	$-0.512062\pi$
$523$	−1.73286	−0.0757725	−0.0378862	+	0.999282i	$0.512062\pi$
$524$	0	0
$525$	−2.20104	−0.0960611
$526$	0	0
$527$	−12.2123	−0.531977
$528$	0	0
$529$	11.5102	0.500444
$530$	0	0
$531$	6.47108	0.280821
$532$	0	0
$533$	0.869274	0.0376524
$534$	0	0
$535$	−5.66521	−0.244928
$536$	0	0
$537$	17.8938	0.772173
$538$	0	0
$539$	−22.3469	−0.962550
$540$	0	0
$541$	−26.8655	−1.15504	−0.577519	−	0.816377i	$-0.695979\pi$
$541$	−26.8655	−1.15504	−0.577519	+	0.816377i	$0.695979\pi$
$542$	0	0
$543$	−2.33363	−0.100146
$544$	0	0
$545$	−27.1152	−1.16149
$546$	0	0
$547$	29.7434	1.27174	0.635869	−	0.771797i	$-0.280642\pi$
$547$	29.7434	1.27174	0.635869	+	0.771797i	$0.280642\pi$
$548$	0	0
$549$	8.48036	0.361933
$550$	0	0
$551$	−16.2926	−0.694086
$552$	0	0
$553$	−2.34744	−0.0998233
$554$	0	0
$555$	1.85409	0.0787018
$556$	0	0
$557$	−29.9108	−1.26736	−0.633681	−	0.773595i	$-0.718457\pi$
$557$	−29.9108	−1.26736	−0.633681	+	0.773595i	$0.718457\pi$
$558$	0	0
$559$	0.800892	0.0338741
$560$	0	0
$561$	−11.2924	−0.476766
$562$	0	0
$563$	13.0743	0.551017	0.275509	−	0.961299i	$-0.411154\pi$
$563$	13.0743	0.551017	0.275509	+	0.961299i	$0.411154\pi$
$564$	0	0
$565$	−2.41658	−0.101666
$566$	0	0
$567$	−0.715830	−0.0300620
$568$	0	0
$569$	19.4827	0.816759	0.408379	−	0.912812i	$-0.366094\pi$
$569$	19.4827	0.816759	0.408379	+	0.912812i	$0.366094\pi$
$570$	0	0
$571$	27.1383	1.13570	0.567852	−	0.823131i	$-0.307775\pi$
$571$	27.1383	1.13570	0.567852	+	0.823131i	$0.307775\pi$
$572$	0	0
$573$	−2.29851	−0.0960218
$574$	0	0
$575$	18.0631	0.753282
$576$	0	0
$577$	−0.387613	−0.0161365	−0.00806827	−	0.999967i	$-0.502568\pi$
$577$	−0.387613	−0.0161365	−0.00806827	+	0.999967i	$0.502568\pi$
$578$	0	0
$579$	9.74298	0.404905
$580$	0	0
$581$	−5.86616	−0.243369
$582$	0	0
$583$	−32.7550	−1.35657
$584$	0	0
$585$	−0.122373	−0.00505949
$586$	0	0
$587$	11.0240	0.455008	0.227504	−	0.973777i	$-0.426944\pi$
$587$	11.0240	0.455008	0.227504	+	0.973777i	$0.426944\pi$
$588$	0	0
$589$	−7.34513	−0.302651
$590$	0	0
$591$	27.8344	1.14495
$592$	0	0
$593$	34.5818	1.42010	0.710052	−	0.704149i	$-0.248671\pi$
$593$	34.5818	1.42010	0.710052	+	0.704149i	$0.248671\pi$
$594$	0	0
$595$	−3.25611	−0.133488
$596$	0	0
$597$	2.38050	0.0974272
$598$	0	0
$599$	42.0567	1.71839	0.859196	−	0.511647i	$-0.170965\pi$
$599$	42.0567	1.71839	0.859196	+	0.511647i	$0.170965\pi$
$600$	0	0
$601$	−6.96774	−0.284220	−0.142110	−	0.989851i	$-0.545389\pi$
$601$	−6.96774	−0.284220	−0.142110	+	0.989851i	$0.545389\pi$
$602$	0	0
$603$	1.30841	0.0532826
$604$	0	0
$605$	1.20025	0.0487971
$606$	0	0
$607$	13.0077	0.527965	0.263983	−	0.964527i	$-0.414964\pi$
$607$	13.0077	0.527965	0.263983	+	0.964527i	$0.414964\pi$
$608$	0	0
$609$	−5.91486	−0.239682
$610$	0	0
$611$	−0.221885	−0.00897650
$612$	0	0
$613$	−32.7834	−1.32411	−0.662055	−	0.749456i	$-0.730315\pi$
$613$	−32.7834	−1.32411	−0.662055	+	0.749456i	$0.730315\pi$
$614$	0	0
$615$	13.6756	0.551453
$616$	0	0
$617$	0.679271	0.0273464	0.0136732	−	0.999907i	$-0.495648\pi$
$617$	0.679271	0.0273464	0.0136732	+	0.999907i	$0.495648\pi$
$618$	0	0
$619$	15.8171	0.635744	0.317872	−	0.948134i	$-0.397032\pi$
$619$	15.8171	0.635744	0.317872	+	0.948134i	$0.397032\pi$
$620$	0	0
$621$	5.87454	0.235737
$622$	0	0
$623$	0.920469	0.0368778
$624$	0	0
$625$	−0.171555	−0.00686221
$626$	0	0
$627$	−6.79186	−0.271241
$628$	0	0
$629$	−4.38073	−0.174671
$630$	0	0
$631$	−23.0580	−0.917926	−0.458963	−	0.888455i	$-0.651779\pi$
$631$	−23.0580	−0.917926	−0.458963	+	0.888455i	$0.651779\pi$
$632$	0	0
$633$	−0.871429	−0.0346362
$634$	0	0
$635$	−13.8707	−0.550444
$636$	0	0
$637$	0.572178	0.0226705
$638$	0	0
$639$	0.137896	0.00545509
$640$	0	0
$641$	−7.51935	−0.296997	−0.148498	−	0.988913i	$-0.547444\pi$
$641$	−7.51935	−0.296997	−0.148498	+	0.988913i	$0.547444\pi$
$642$	0	0
$643$	−12.0036	−0.473377	−0.236688	−	0.971586i	$-0.576062\pi$
$643$	−12.0036	−0.473377	−0.236688	+	0.971586i	$0.576062\pi$
$644$	0	0
$645$	12.5998	0.496117
$646$	0	0
$647$	8.24193	0.324024	0.162012	−	0.986789i	$-0.448202\pi$
$647$	8.24193	0.324024	0.162012	+	0.986789i	$0.448202\pi$
$648$	0	0
$649$	22.2901	0.874961
$650$	0	0
$651$	−2.66658	−0.104512
$652$	0	0
$653$	−19.2674	−0.753991	−0.376995	−	0.926215i	$-0.623043\pi$
$653$	−19.2674	−0.753991	−0.376995	+	0.926215i	$0.623043\pi$
$654$	0	0
$655$	3.84450	0.150217
$656$	0	0
$657$	15.4177	0.601501
$658$	0	0
$659$	−3.91492	−0.152504	−0.0762518	−	0.997089i	$-0.524295\pi$
$659$	−3.91492	−0.152504	−0.0762518	+	0.997089i	$0.524295\pi$
$660$	0	0
$661$	40.6474	1.58100	0.790501	−	0.612461i	$-0.209820\pi$
$661$	40.6474	1.58100	0.790501	+	0.612461i	$0.209820\pi$
$662$	0	0
$663$	0.289135	0.0112291
$664$	0	0
$665$	−1.95840	−0.0759435
$666$	0	0
$667$	48.5410	1.87952
$668$	0	0
$669$	−4.35552	−0.168394
$670$	0	0
$671$	29.2111	1.12768
$672$	0	0
$673$	−9.77915	−0.376958	−0.188479	−	0.982077i	$-0.560356\pi$
$673$	−9.77915	−0.376958	−0.188479	+	0.982077i	$0.560356\pi$
$674$	0	0
$675$	3.07480	0.118349
$676$	0	0
$677$	−11.7108	−0.450083	−0.225041	−	0.974349i	$-0.572252\pi$
$677$	−11.7108	−0.450083	−0.225041	+	0.974349i	$0.572252\pi$
$678$	0	0
$679$	0.214254	0.00822232
$680$	0	0
$681$	−8.53086	−0.326903
$682$	0	0
$683$	−45.7366	−1.75006	−0.875031	−	0.484066i	$-0.839159\pi$
$683$	−45.7366	−1.75006	−0.875031	+	0.484066i	$0.839159\pi$
$684$	0	0
$685$	−1.34635	−0.0514413
$686$	0	0
$687$	3.44604	0.131475
$688$	0	0
$689$	0.838669	0.0319507
$690$	0	0
$691$	−24.2119	−0.921064	−0.460532	−	0.887643i	$-0.652341\pi$
$691$	−24.2119	−0.921064	−0.460532	+	0.887643i	$0.652341\pi$
$692$	0	0
$693$	−2.46572	−0.0936651
$694$	0	0
$695$	−16.9428	−0.642675
$696$	0	0
$697$	−32.3118	−1.22390
$698$	0	0
$699$	−0.0961489	−0.00363668
$700$	0	0
$701$	4.35228	0.164383	0.0821917	−	0.996617i	$-0.473808\pi$
$701$	4.35228	0.164383	0.0821917	+	0.996617i	$0.473808\pi$
$702$	0	0
$703$	−2.63480	−0.0993734
$704$	0	0
$705$	−3.49074	−0.131469
$706$	0	0
$707$	6.54971	0.246327
$708$	0	0
$709$	30.9952	1.16405	0.582024	−	0.813172i	$-0.302261\pi$
$709$	30.9952	1.16405	0.582024	+	0.813172i	$0.302261\pi$
$710$	0	0
$711$	3.27933	0.122984
$712$	0	0
$713$	21.8836	0.819548
$714$	0	0
$715$	−0.421521	−0.0157640
$716$	0	0
$717$	1.22608	0.0457886
$718$	0	0
$719$	4.01190	0.149619	0.0748094	−	0.997198i	$-0.476165\pi$
$719$	4.01190	0.149619	0.0748094	+	0.997198i	$0.476165\pi$
$720$	0	0
$721$	−4.94655	−0.184219
$722$	0	0
$723$	−14.6416	−0.544526
$724$	0	0
$725$	25.4070	0.943591
$726$	0	0
$727$	−7.86851	−0.291827	−0.145913	−	0.989297i	$-0.546612\pi$
$727$	−7.86851	−0.291827	−0.145913	+	0.989297i	$0.546612\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−29.7700	−1.10108
$732$	0	0
$733$	38.1604	1.40949	0.704743	−	0.709463i	$-0.251062\pi$
$733$	38.1604	1.40949	0.704743	+	0.709463i	$0.251062\pi$
$734$	0	0
$735$	9.00162	0.332030
$736$	0	0
$737$	4.50691	0.166014
$738$	0	0
$739$	−35.1194	−1.29189	−0.645944	−	0.763385i	$-0.723536\pi$
$739$	−35.1194	−1.29189	−0.645944	+	0.763385i	$0.723536\pi$
$740$	0	0
$741$	0.173901	0.00638841
$742$	0	0
$743$	24.6694	0.905034	0.452517	−	0.891756i	$-0.350526\pi$
$743$	24.6694	0.905034	0.452517	+	0.891756i	$0.350526\pi$
$744$	0	0
$745$	−2.05647	−0.0753432
$746$	0	0
$747$	8.19491	0.299836
$748$	0	0
$749$	2.92273	0.106794
$750$	0	0
$751$	−24.4670	−0.892813	−0.446407	−	0.894830i	$-0.647296\pi$
$751$	−24.4670	−0.892813	−0.446407	+	0.894830i	$0.647296\pi$
$752$	0	0
$753$	−7.80044	−0.284264
$754$	0	0
$755$	28.4952	1.03705
$756$	0	0
$757$	−9.74911	−0.354337	−0.177169	−	0.984180i	$-0.556694\pi$
$757$	−9.74911	−0.354337	−0.177169	+	0.984180i	$0.556694\pi$
$758$	0	0
$759$	20.2352	0.734493
$760$	0	0
$761$	2.04415	0.0741003	0.0370502	−	0.999313i	$-0.488204\pi$
$761$	2.04415	0.0741003	0.0370502	+	0.999313i	$0.488204\pi$
$762$	0	0
$763$	13.9889	0.506433
$764$	0	0
$765$	4.54873	0.164460
$766$	0	0
$767$	−0.570722	−0.0206076
$768$	0	0
$769$	−20.3889	−0.735243	−0.367622	−	0.929975i	$-0.619828\pi$
$769$	−20.3889	−0.735243	−0.367622	+	0.929975i	$0.619828\pi$
$770$	0	0
$771$	−0.264056	−0.00950974
$772$	0	0
$773$	33.9673	1.22172	0.610859	−	0.791739i	$-0.290824\pi$
$773$	33.9673	1.22172	0.610859	+	0.791739i	$0.290824\pi$
$774$	0	0
$775$	11.4542	0.411445
$776$	0	0
$777$	−0.956541	−0.0343157
$778$	0	0
$779$	−19.4340	−0.696297
$780$	0	0
$781$	0.474993	0.0169966
$782$	0	0
$783$	8.26295	0.295294
$784$	0	0
$785$	−10.4249	−0.372082
$786$	0	0
$787$	−17.9106	−0.638445	−0.319223	−	0.947680i	$-0.603422\pi$
$787$	−17.9106	−0.638445	−0.319223	+	0.947680i	$0.603422\pi$
$788$	0	0
$789$	−1.75704	−0.0625522
$790$	0	0
$791$	1.24673	0.0443287
$792$	0	0
$793$	−0.747931	−0.0265598
$794$	0	0
$795$	13.1941	0.467947
$796$	0	0
$797$	−22.7754	−0.806748	−0.403374	−	0.915035i	$-0.632163\pi$
$797$	−22.7754	−0.806748	−0.403374	+	0.915035i	$0.632163\pi$
$798$	0	0
$799$	8.24769	0.291782
$800$	0	0
$801$	−1.28588	−0.0454342
$802$	0	0
$803$	53.1073	1.87411
$804$	0	0
$805$	5.83473	0.205647
$806$	0	0
$807$	−3.95509	−0.139226
$808$	0	0
$809$	−15.1845	−0.533858	−0.266929	−	0.963716i	$-0.586009\pi$
$809$	−15.1845	−0.533858	−0.266929	+	0.963716i	$0.586009\pi$
$810$	0	0
$811$	−2.58068	−0.0906198	−0.0453099	−	0.998973i	$-0.514428\pi$
$811$	−2.58068	−0.0906198	−0.0453099	+	0.998973i	$0.514428\pi$
$812$	0	0
$813$	14.3749	0.504148
$814$	0	0
$815$	9.44036	0.330681
$816$	0	0
$817$	−17.9052	−0.626425
$818$	0	0
$819$	0.0631331	0.00220605
$820$	0	0
$821$	6.81864	0.237972	0.118986	−	0.992896i	$-0.462036\pi$
$821$	6.81864	0.237972	0.118986	+	0.992896i	$0.462036\pi$
$822$	0	0
$823$	−25.9331	−0.903971	−0.451986	−	0.892025i	$-0.649284\pi$
$823$	−25.9331	−0.903971	−0.451986	+	0.892025i	$0.649284\pi$
$824$	0	0
$825$	10.5914	0.368744
$826$	0	0
$827$	34.6733	1.20571	0.602854	−	0.797852i	$-0.294030\pi$
$827$	34.6733	1.20571	0.602854	+	0.797852i	$0.294030\pi$
$828$	0	0
$829$	1.23878	0.0430245	0.0215122	−	0.999769i	$-0.493152\pi$
$829$	1.23878	0.0430245	0.0215122	+	0.999769i	$0.493152\pi$
$830$	0	0
$831$	7.53875	0.261516
$832$	0	0
$833$	−21.2684	−0.736908
$834$	0	0
$835$	−10.2382	−0.354306
$836$	0	0
$837$	3.72516	0.128761
$838$	0	0
$839$	−49.1532	−1.69696	−0.848478	−	0.529230i	$-0.822481\pi$
$839$	−49.1532	−1.69696	−0.848478	+	0.529230i	$0.822481\pi$
$840$	0	0
$841$	39.2763	1.35436
$842$	0	0
$843$	−21.7342	−0.748566
$844$	0	0
$845$	−18.0269	−0.620144
$846$	0	0
$847$	−0.619219	−0.0212766
$848$	0	0
$849$	10.7499	0.368935
$850$	0	0
$851$	7.84996	0.269093
$852$	0	0
$853$	−56.1677	−1.92315	−0.961573	−	0.274551i	$-0.911471\pi$
$853$	−56.1677	−1.92315	−0.961573	+	0.274551i	$0.911471\pi$
$854$	0	0
$855$	2.73585	0.0935639
$856$	0	0
$857$	−37.3161	−1.27469	−0.637346	−	0.770578i	$-0.719968\pi$
$857$	−37.3161	−1.27469	−0.637346	+	0.770578i	$0.719968\pi$
$858$	0	0
$859$	−32.6533	−1.11412	−0.557058	−	0.830474i	$-0.688070\pi$
$859$	−32.6533	−1.11412	−0.557058	+	0.830474i	$0.688070\pi$
$860$	0	0
$861$	−7.05535	−0.240446
$862$	0	0
$863$	25.8672	0.880530	0.440265	−	0.897868i	$-0.354885\pi$
$863$	25.8672	0.880530	0.440265	+	0.897868i	$0.354885\pi$
$864$	0	0
$865$	26.4012	0.897667
$866$	0	0
$867$	6.25256	0.212348
$868$	0	0
$869$	11.2959	0.383186
$870$	0	0
$871$	−0.115396	−0.00391005
$872$	0	0
$873$	−0.299309	−0.0101301
$874$	0	0
$875$	8.02009	0.271129
$876$	0	0
$877$	39.0511	1.31866	0.659331	−	0.751853i	$-0.270840\pi$
$877$	39.0511	1.31866	0.659331	+	0.751853i	$0.270840\pi$
$878$	0	0
$879$	4.26334	0.143799
$880$	0	0
$881$	−41.7538	−1.40672	−0.703361	−	0.710833i	$-0.748318\pi$
$881$	−41.7538	−1.40672	−0.703361	+	0.710833i	$0.748318\pi$
$882$	0	0
$883$	−37.7537	−1.27051	−0.635256	−	0.772301i	$-0.719105\pi$
$883$	−37.7537	−1.27051	−0.635256	+	0.772301i	$0.719105\pi$
$884$	0	0
$885$	−8.97871	−0.301816
$886$	0	0
$887$	42.7224	1.43448	0.717240	−	0.696827i	$-0.245405\pi$
$887$	42.7224	1.43448	0.717240	+	0.696827i	$0.245405\pi$
$888$	0	0
$889$	7.15603	0.240006
$890$	0	0
$891$	3.44457	0.115397
$892$	0	0
$893$	4.96060	0.166000
$894$	0	0
$895$	−24.8279	−0.829904
$896$	0	0
$897$	−0.518109	−0.0172992
$898$	0	0
$899$	30.7808	1.02660
$900$	0	0
$901$	−31.1742	−1.03856
$902$	0	0
$903$	−6.50034	−0.216318
$904$	0	0
$905$	3.23795	0.107633
$906$	0	0
$907$	6.68346	0.221921	0.110960	−	0.993825i	$-0.464607\pi$
$907$	6.68346	0.221921	0.110960	+	0.993825i	$0.464607\pi$
$908$	0	0
$909$	−9.14982	−0.303480
$910$	0	0
$911$	33.3645	1.10542	0.552708	−	0.833375i	$-0.313595\pi$
$911$	33.3645	1.10542	0.552708	+	0.833375i	$0.313595\pi$
$912$	0	0
$913$	28.2279	0.934207
$914$	0	0
$915$	−11.7666	−0.388992
$916$	0	0
$917$	−1.98341	−0.0654980
$918$	0	0
$919$	7.40599	0.244301	0.122151	−	0.992512i	$-0.461021\pi$
$919$	7.40599	0.244301	0.122151	+	0.992512i	$0.461021\pi$
$920$	0	0
$921$	0.156797	0.00516665
$922$	0	0
$923$	−0.0121619	−0.000400312 0
$924$	0	0
$925$	4.10876	0.135095
$926$	0	0
$927$	6.91023	0.226962
$928$	0	0
$929$	−2.69524	−0.0884279	−0.0442139	−	0.999022i	$-0.514078\pi$
$929$	−2.69524	−0.0884279	−0.0442139	+	0.999022i	$0.514078\pi$
$930$	0	0
$931$	−12.7920	−0.419240
$932$	0	0
$933$	21.1425	0.692174
$934$	0	0
$935$	15.6684	0.512411
$936$	0	0
$937$	−4.20240	−0.137287	−0.0686433	−	0.997641i	$-0.521867\pi$
$937$	−4.20240	−0.137287	−0.0686433	+	0.997641i	$0.521867\pi$
$938$	0	0
$939$	−5.94596	−0.194039
$940$	0	0
$941$	5.98711	0.195174	0.0975871	−	0.995227i	$-0.468888\pi$
$941$	5.98711	0.195174	0.0975871	+	0.995227i	$0.468888\pi$
$942$	0	0
$943$	57.9006	1.88550
$944$	0	0
$945$	0.993224	0.0323096
$946$	0	0
$947$	−24.2097	−0.786709	−0.393355	−	0.919387i	$-0.628686\pi$
$947$	−24.2097	−0.786709	−0.393355	+	0.919387i	$0.628686\pi$
$948$	0	0
$949$	−1.35977	−0.0441402
$950$	0	0
$951$	−14.5465	−0.471702
$952$	0	0
$953$	−39.3840	−1.27577	−0.637887	−	0.770130i	$-0.720191\pi$
$953$	−39.3840	−1.27577	−0.637887	+	0.770130i	$0.720191\pi$
$954$	0	0
$955$	3.18922	0.103201
$956$	0	0
$957$	28.4623	0.920055
$958$	0	0
$959$	0.694591	0.0224295
$960$	0	0
$961$	−17.1232	−0.552360
$962$	0	0
$963$	−4.08299	−0.131572
$964$	0	0
$965$	−13.5185	−0.435177
$966$	0	0
$967$	−9.06896	−0.291638	−0.145819	−	0.989311i	$-0.546582\pi$
$967$	−9.06896	−0.291638	−0.145819	+	0.989311i	$0.546582\pi$
$968$	0	0
$969$	−6.46408	−0.207656
$970$	0	0
$971$	−13.5719	−0.435544	−0.217772	−	0.976000i	$-0.569879\pi$
$971$	−13.5719	−0.435544	−0.217772	+	0.976000i	$0.569879\pi$
$972$	0	0
$973$	8.74090	0.280221
$974$	0	0
$975$	−0.271185	−0.00868486
$976$	0	0
$977$	57.7012	1.84602	0.923012	−	0.384771i	$-0.125720\pi$
$977$	57.7012	1.84602	0.923012	+	0.384771i	$0.125720\pi$
$978$	0	0
$979$	−4.42929	−0.141561
$980$	0	0
$981$	−19.5423	−0.623936
$982$	0	0
$983$	18.5212	0.590734	0.295367	−	0.955384i	$-0.404558\pi$
$983$	18.5212	0.590734	0.295367	+	0.955384i	$0.404558\pi$
$984$	0	0
$985$	−38.6206	−1.23055
$986$	0	0
$987$	1.80090	0.0573233
$988$	0	0
$989$	53.3458	1.69630
$990$	0	0
$991$	37.0323	1.17637	0.588186	−	0.808726i	$-0.299842\pi$
$991$	37.0323	1.17637	0.588186	+	0.808726i	$0.299842\pi$
$992$	0	0
$993$	3.52374	0.111823
$994$	0	0
$995$	−3.30297	−0.104711
$996$	0	0
$997$	26.3675	0.835067	0.417533	−	0.908662i	$-0.362895\pi$
$997$	26.3675	0.835067	0.417533	+	0.908662i	$0.362895\pi$
$998$	0	0
$999$	1.33627	0.0422777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.f.1.11	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.f.1.11	✓	14	1.1	even	1	trivial

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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.38751 q^{5} -0.715830 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.38751 q^{5} -0.715830 q^{7} +1.00000 q^{9} +3.44457 q^{11} -0.0881957 q^{13} -1.38751 q^{15} +3.27833 q^{17} +1.97176 q^{19} +0.715830 q^{21} -5.87454 q^{23} -3.07480 q^{25} -1.00000 q^{27} -8.26295 q^{29} -3.72516 q^{31} -3.44457 q^{33} -0.993224 q^{35} -1.33627 q^{37} +0.0881957 q^{39} -9.85619 q^{41} -9.08085 q^{43} +1.38751 q^{45} +2.51582 q^{47} -6.48759 q^{49} -3.27833 q^{51} -9.50918 q^{53} +4.77938 q^{55} -1.97176 q^{57} +6.47108 q^{59} +8.48036 q^{61} -0.715830 q^{63} -0.122373 q^{65} +1.30841 q^{67} +5.87454 q^{69} +0.137896 q^{71} +15.4177 q^{73} +3.07480 q^{75} -2.46572 q^{77} +3.27933 q^{79} +1.00000 q^{81} +8.19491 q^{83} +4.54873 q^{85} +8.26295 q^{87} -1.28588 q^{89} +0.0631331 q^{91} +3.72516 q^{93} +2.73585 q^{95} -0.299309 q^{97} +3.44457 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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