LMFDB - Embedded newform 572.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(1,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, 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("572.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.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:	$4.56744299562$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	572.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.30278 q^{3} +0.302776 q^{7} +7.90833 q^{9} +O(q^{10})$	$q-3.30278 q^{3} +0.302776 q^{7} +7.90833 q^{9} -1.00000 q^{11} +1.00000 q^{13} +2.60555 q^{17} -5.30278 q^{19} -1.00000 q^{21} -1.69722 q^{23} -5.00000 q^{25} -16.2111 q^{27} -5.21110 q^{29} +4.60555 q^{31} +3.30278 q^{33} -9.21110 q^{37} -3.30278 q^{39} +1.69722 q^{41} -6.60555 q^{43} -6.00000 q^{47} -6.90833 q^{49} -8.60555 q^{51} -4.69722 q^{53} +17.5139 q^{57} +14.6056 q^{59} +9.81665 q^{61} +2.39445 q^{63} -1.39445 q^{67} +5.60555 q^{69} -3.39445 q^{71} -10.5139 q^{73} +16.5139 q^{75} -0.302776 q^{77} -15.2111 q^{79} +29.8167 q^{81} -9.51388 q^{83} +17.2111 q^{87} +0.302776 q^{91} -15.2111 q^{93} +2.78890 q^{97} -7.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 3 * q^7 + 5 * q^9	$ 2 q - 3 q^{3} - 3 q^{7} + 5 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{17} - 7 q^{19} - 2 q^{21} - 7 q^{23} - 10 q^{25} - 18 q^{27} + 4 q^{29} + 2 q^{31} + 3 q^{33} - 4 q^{37} - 3 q^{39} + 7 q^{41} - 6 q^{43} - 12 q^{47} - 3 q^{49} - 10 q^{51} - 13 q^{53} + 17 q^{57} + 22 q^{59} - 2 q^{61} + 12 q^{63} - 10 q^{67} + 4 q^{69} - 14 q^{71} - 3 q^{73} + 15 q^{75} + 3 q^{77} - 16 q^{79} + 38 q^{81} - q^{83} + 20 q^{87} - 3 q^{91} - 16 q^{93} + 20 q^{97} - 5 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 3 * q^7 + 5 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^17 - 7 * q^19 - 2 * q^21 - 7 * q^23 - 10 * q^25 - 18 * q^27 + 4 * q^29 + 2 * q^31 + 3 * q^33 - 4 * q^37 - 3 * q^39 + 7 * q^41 - 6 * q^43 - 12 * q^47 - 3 * q^49 - 10 * q^51 - 13 * q^53 + 17 * q^57 + 22 * q^59 - 2 * q^61 + 12 * q^63 - 10 * q^67 + 4 * q^69 - 14 * q^71 - 3 * q^73 + 15 * q^75 + 3 * q^77 - 16 * q^79 + 38 * q^81 - q^83 + 20 * q^87 - 3 * q^91 - 16 * q^93 + 20 * q^97 - 5 * 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
$3$	−3.30278	−1.90686	−0.953429	−	0.301617i	$-0.902474\pi$
$3$	−3.30278	−1.90686	−0.953429	+	0.301617i	$0.902474\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0.302776	0.114438	0.0572192	−	0.998362i	$-0.481777\pi$
$7$	0.302776	0.114438	0.0572192	+	0.998362i	$0.481777\pi$
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.60555	0.631939	0.315970	−	0.948769i	$-0.397670\pi$
$17$	2.60555	0.631939	0.315970	+	0.948769i	$0.397670\pi$
$18$	0	0
$19$	−5.30278	−1.21654	−0.608270	−	0.793730i	$-0.708136\pi$
$19$	−5.30278	−1.21654	−0.608270	+	0.793730i	$0.708136\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.69722	−0.353896	−0.176948	−	0.984220i	$-0.556622\pi$
$23$	−1.69722	−0.353896	−0.176948	+	0.984220i	$0.556622\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−16.2111	−3.11983
$28$	0	0
$29$	−5.21110	−0.967677	−0.483839	−	0.875157i	$-0.660758\pi$
$29$	−5.21110	−0.967677	−0.483839	+	0.875157i	$0.660758\pi$
$30$	0	0
$31$	4.60555	0.827181	0.413591	−	0.910463i	$-0.364274\pi$
$31$	4.60555	0.827181	0.413591	+	0.910463i	$0.364274\pi$
$32$	0	0
$33$	3.30278	0.574939
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.21110	−1.51430	−0.757148	−	0.653243i	$-0.773408\pi$
$37$	−9.21110	−1.51430	−0.757148	+	0.653243i	$0.773408\pi$
$38$	0	0
$39$	−3.30278	−0.528867
$40$	0	0
$41$	1.69722	0.265062	0.132531	−	0.991179i	$-0.457690\pi$
$41$	1.69722	0.265062	0.132531	+	0.991179i	$0.457690\pi$
$42$	0	0
$43$	−6.60555	−1.00734	−0.503669	−	0.863897i	$-0.668017\pi$
$43$	−6.60555	−1.00734	−0.503669	+	0.863897i	$0.668017\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−6.90833	−0.986904
$50$	0	0
$51$	−8.60555	−1.20502
$52$	0	0
$53$	−4.69722	−0.645213	−0.322607	−	0.946533i	$-0.604559\pi$
$53$	−4.69722	−0.645213	−0.322607	+	0.946533i	$0.604559\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	17.5139	2.31977
$58$	0	0
$59$	14.6056	1.90148	0.950740	−	0.309988i	$-0.100325\pi$
$59$	14.6056	1.90148	0.950740	+	0.309988i	$0.100325\pi$
$60$	0	0
$61$	9.81665	1.25689	0.628447	−	0.777853i	$-0.283691\pi$
$61$	9.81665	1.25689	0.628447	+	0.777853i	$0.283691\pi$
$62$	0	0
$63$	2.39445	0.301672
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.39445	−0.170359	−0.0851795	−	0.996366i	$-0.527146\pi$
$67$	−1.39445	−0.170359	−0.0851795	+	0.996366i	$0.527146\pi$
$68$	0	0
$69$	5.60555	0.674829
$70$	0	0
$71$	−3.39445	−0.402847	−0.201423	−	0.979504i	$-0.564557\pi$
$71$	−3.39445	−0.402847	−0.201423	+	0.979504i	$0.564557\pi$
$72$	0	0
$73$	−10.5139	−1.23056	−0.615278	−	0.788310i	$-0.710956\pi$
$73$	−10.5139	−1.23056	−0.615278	+	0.788310i	$0.710956\pi$
$74$	0	0
$75$	16.5139	1.90686
$76$	0	0
$77$	−0.302776	−0.0345045
$78$	0	0
$79$	−15.2111	−1.71138	−0.855691	−	0.517486i	$-0.826868\pi$
$79$	−15.2111	−1.71138	−0.855691	+	0.517486i	$0.826868\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	−9.51388	−1.04428	−0.522142	−	0.852859i	$-0.674867\pi$
$83$	−9.51388	−1.04428	−0.522142	+	0.852859i	$0.674867\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	17.2111	1.84522
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0.302776	0.0317395
$92$	0	0
$93$	−15.2111	−1.57732
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.78890	0.283170	0.141585	−	0.989926i	$-0.454780\pi$
$97$	2.78890	0.283170	0.141585	+	0.989926i	$0.454780\pi$
$98$	0	0
$99$	−7.90833	−0.794817
$100$	0	0
$101$	13.8167	1.37481	0.687404	−	0.726275i	$-0.258750\pi$
$101$	13.8167	1.37481	0.687404	+	0.726275i	$0.258750\pi$
$102$	0	0
$103$	2.90833	0.286566	0.143283	−	0.989682i	$-0.454234\pi$
$103$	2.90833	0.286566	0.143283	+	0.989682i	$0.454234\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	4.09167	0.391911	0.195956	−	0.980613i	$-0.437219\pi$
$109$	4.09167	0.391911	0.195956	+	0.980613i	$0.437219\pi$
$110$	0	0
$111$	30.4222	2.88755
$112$	0	0
$113$	2.09167	0.196768	0.0983840	−	0.995149i	$-0.468633\pi$
$113$	2.09167	0.196768	0.0983840	+	0.995149i	$0.468633\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.90833	0.731125
$118$	0	0
$119$	0.788897	0.0723181
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.60555	−0.505436
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	0	0
$129$	21.8167	1.92085
$130$	0	0
$131$	−5.21110	−0.455296	−0.227648	−	0.973743i	$-0.573104\pi$
$131$	−5.21110	−0.455296	−0.227648	+	0.973743i	$0.573104\pi$
$132$	0	0
$133$	−1.60555	−0.139219
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−18.6056	−1.57810	−0.789051	−	0.614328i	$-0.789427\pi$
$139$	−18.6056	−1.57810	−0.789051	+	0.614328i	$0.789427\pi$
$140$	0	0
$141$	19.8167	1.66886
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0	0
$146$	0	0
$147$	22.8167	1.88189
$148$	0	0
$149$	−17.3305	−1.41977	−0.709886	−	0.704316i	$-0.751254\pi$
$149$	−17.3305	−1.41977	−0.709886	+	0.704316i	$0.751254\pi$
$150$	0	0
$151$	23.6333	1.92325	0.961626	−	0.274365i	$-0.0884676\pi$
$151$	23.6333	1.92325	0.961626	+	0.274365i	$0.0884676\pi$
$152$	0	0
$153$	20.6056	1.66586
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.09167	−0.246742	−0.123371	−	0.992361i	$-0.539371\pi$
$157$	−3.09167	−0.246742	−0.123371	+	0.992361i	$0.539371\pi$
$158$	0	0
$159$	15.5139	1.23033
$160$	0	0
$161$	−0.513878	−0.0404993
$162$	0	0
$163$	−18.6056	−1.45730	−0.728650	−	0.684887i	$-0.759852\pi$
$163$	−18.6056	−1.45730	−0.728650	+	0.684887i	$0.759852\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.9083	1.23102	0.615512	−	0.788128i	$-0.288949\pi$
$167$	15.9083	1.23102	0.615512	+	0.788128i	$0.288949\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−41.9361	−3.20693
$172$	0	0
$173$	−11.2111	−0.852364	−0.426182	−	0.904637i	$-0.640142\pi$
$173$	−11.2111	−0.852364	−0.426182	+	0.904637i	$0.640142\pi$
$174$	0	0
$175$	−1.51388	−0.114438
$176$	0	0
$177$	−48.2389	−3.62585
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−8.69722	−0.646460	−0.323230	−	0.946321i	$-0.604769\pi$
$181$	−8.69722	−0.646460	−0.323230	+	0.946321i	$0.604769\pi$
$182$	0	0
$183$	−32.4222	−2.39672
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.60555	−0.190537
$188$	0	0
$189$	−4.90833	−0.357028
$190$	0	0
$191$	−0.513878	−0.0371829	−0.0185915	−	0.999827i	$-0.505918\pi$
$191$	−0.513878	−0.0371829	−0.0185915	+	0.999827i	$0.505918\pi$
$192$	0	0
$193$	−17.6972	−1.27387	−0.636937	−	0.770916i	$-0.719799\pi$
$193$	−17.6972	−1.27387	−0.636937	+	0.770916i	$0.719799\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.9083	−1.13342	−0.566711	−	0.823917i	$-0.691784\pi$
$197$	−15.9083	−1.13342	−0.566711	+	0.823917i	$0.691784\pi$
$198$	0	0
$199$	18.9361	1.34234	0.671172	−	0.741302i	$-0.265791\pi$
$199$	18.9361	1.34234	0.671172	+	0.741302i	$0.265791\pi$
$200$	0	0
$201$	4.60555	0.324851
$202$	0	0
$203$	−1.57779	−0.110739
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−13.4222	−0.932908
$208$	0	0
$209$	5.30278	0.366801
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	0	0
$213$	11.2111	0.768172
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.39445	0.0946613
$218$	0	0
$219$	34.7250	2.34650
$220$	0	0
$221$	2.60555	0.175268
$222$	0	0
$223$	−2.18335	−0.146208	−0.0731038	−	0.997324i	$-0.523290\pi$
$223$	−2.18335	−0.146208	−0.0731038	+	0.997324i	$0.523290\pi$
$224$	0	0
$225$	−39.5416	−2.63611
$226$	0	0
$227$	21.1194	1.40175	0.700873	−	0.713286i	$-0.252794\pi$
$227$	21.1194	1.40175	0.700873	+	0.713286i	$0.252794\pi$
$228$	0	0
$229$	−15.2111	−1.00518	−0.502589	−	0.864525i	$-0.667619\pi$
$229$	−15.2111	−1.00518	−0.502589	+	0.864525i	$0.667619\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−3.39445	−0.222378	−0.111189	−	0.993799i	$-0.535466\pi$
$233$	−3.39445	−0.222378	−0.111189	+	0.993799i	$0.535466\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	50.2389	3.26336
$238$	0	0
$239$	30.5139	1.97378	0.986889	−	0.161398i	$-0.0516004\pi$
$239$	30.5139	1.97378	0.986889	+	0.161398i	$0.0516004\pi$
$240$	0	0
$241$	−9.33053	−0.601032	−0.300516	−	0.953777i	$-0.597159\pi$
$241$	−9.33053	−0.601032	−0.300516	+	0.953777i	$0.597159\pi$
$242$	0	0
$243$	−49.8444	−3.19752
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.30278	−0.337408
$248$	0	0
$249$	31.4222	1.99130
$250$	0	0
$251$	8.72498	0.550716	0.275358	−	0.961342i	$-0.411204\pi$
$251$	8.72498	0.550716	0.275358	+	0.961342i	$0.411204\pi$
$252$	0	0
$253$	1.69722	0.106704
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.1194	−1.13026	−0.565129	−	0.825002i	$-0.691174\pi$
$257$	−18.1194	−1.13026	−0.565129	+	0.825002i	$0.691174\pi$
$258$	0	0
$259$	−2.78890	−0.173294
$260$	0	0
$261$	−41.2111	−2.55090
$262$	0	0
$263$	20.6056	1.27059	0.635296	−	0.772268i	$-0.280878\pi$
$263$	20.6056	1.27059	0.635296	+	0.772268i	$0.280878\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.7250	0.897798	0.448899	−	0.893583i	$-0.351816\pi$
$269$	14.7250	0.897798	0.448899	+	0.893583i	$0.351816\pi$
$270$	0	0
$271$	−5.30278	−0.322121	−0.161060	−	0.986945i	$-0.551491\pi$
$271$	−5.30278	−0.322121	−0.161060	+	0.986945i	$0.551491\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	−16.7889	−1.00875	−0.504374	−	0.863486i	$-0.668277\pi$
$277$	−16.7889	−1.00875	−0.504374	+	0.863486i	$0.668277\pi$
$278$	0	0
$279$	36.4222	2.18054
$280$	0	0
$281$	26.3305	1.57075	0.785374	−	0.619022i	$-0.212471\pi$
$281$	26.3305	1.57075	0.785374	+	0.619022i	$0.212471\pi$
$282$	0	0
$283$	−3.21110	−0.190880	−0.0954401	−	0.995435i	$-0.530426\pi$
$283$	−3.21110	−0.190880	−0.0954401	+	0.995435i	$0.530426\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.513878	0.0303333
$288$	0	0
$289$	−10.2111	−0.600653
$290$	0	0
$291$	−9.21110	−0.539964
$292$	0	0
$293$	21.6333	1.26383	0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333	1.26383	0.631916	+	0.775037i	$0.282269\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.2111	0.940664
$298$	0	0
$299$	−1.69722	−0.0981530
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	−45.6333	−2.62157
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.4222	1.05141	0.525705	−	0.850667i	$-0.323801\pi$
$307$	18.4222	1.05141	0.525705	+	0.850667i	$0.323801\pi$
$308$	0	0
$309$	−9.60555	−0.546441
$310$	0	0
$311$	−32.3305	−1.83330	−0.916648	−	0.399695i	$-0.869116\pi$
$311$	−32.3305	−1.83330	−0.916648	+	0.399695i	$0.869116\pi$
$312$	0	0
$313$	10.7250	0.606212	0.303106	−	0.952957i	$-0.401976\pi$
$313$	10.7250	0.606212	0.303106	+	0.952957i	$0.401976\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.60555	0.483336	0.241668	−	0.970359i	$-0.422306\pi$
$317$	8.60555	0.483336	0.241668	+	0.970359i	$0.422306\pi$
$318$	0	0
$319$	5.21110	0.291766
$320$	0	0
$321$	−19.8167	−1.10606
$322$	0	0
$323$	−13.8167	−0.768779
$324$	0	0
$325$	−5.00000	−0.277350
$326$	0	0
$327$	−13.5139	−0.747319
$328$	0	0
$329$	−1.81665	−0.100155
$330$	0	0
$331$	10.6056	0.582934	0.291467	−	0.956581i	$-0.405857\pi$
$331$	10.6056	0.582934	0.291467	+	0.956581i	$0.405857\pi$
$332$	0	0
$333$	−72.8444	−3.99185
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.8167	0.861588	0.430794	−	0.902450i	$-0.358234\pi$
$337$	15.8167	0.861588	0.430794	+	0.902450i	$0.358234\pi$
$338$	0	0
$339$	−6.90833	−0.375209
$340$	0	0
$341$	−4.60555	−0.249405
$342$	0	0
$343$	−4.21110	−0.227378
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.4500	1.58096	0.790478	−	0.612490i	$-0.209832\pi$
$347$	29.4500	1.58096	0.790478	+	0.612490i	$0.209832\pi$
$348$	0	0
$349$	−7.51388	−0.402209	−0.201104	−	0.979570i	$-0.564453\pi$
$349$	−7.51388	−0.402209	−0.201104	+	0.979570i	$0.564453\pi$
$350$	0	0
$351$	−16.2111	−0.865285
$352$	0	0
$353$	30.2389	1.60945	0.804726	−	0.593646i	$-0.202312\pi$
$353$	30.2389	1.60945	0.804726	+	0.593646i	$0.202312\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.60555	−0.137900
$358$	0	0
$359$	−4.30278	−0.227092	−0.113546	−	0.993533i	$-0.536221\pi$
$359$	−4.30278	−0.227092	−0.113546	+	0.993533i	$0.536221\pi$
$360$	0	0
$361$	9.11943	0.479970
$362$	0	0
$363$	−3.30278	−0.173351
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−29.5416	−1.54206	−0.771030	−	0.636798i	$-0.780258\pi$
$367$	−29.5416	−1.54206	−0.771030	+	0.636798i	$0.780258\pi$
$368$	0	0
$369$	13.4222	0.698732
$370$	0	0
$371$	−1.42221	−0.0738372
$372$	0	0
$373$	34.8444	1.80418	0.902088	−	0.431553i	$-0.142034\pi$
$373$	34.8444	1.80418	0.902088	+	0.431553i	$0.142034\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.21110	−0.268385
$378$	0	0
$379$	9.81665	0.504248	0.252124	−	0.967695i	$-0.418871\pi$
$379$	9.81665	0.504248	0.252124	+	0.967695i	$0.418871\pi$
$380$	0	0
$381$	33.0278	1.69206
$382$	0	0
$383$	−21.6333	−1.10541	−0.552705	−	0.833377i	$-0.686404\pi$
$383$	−21.6333	−1.10541	−0.552705	+	0.833377i	$0.686404\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−52.2389	−2.65545
$388$	0	0
$389$	−17.7250	−0.898692	−0.449346	−	0.893358i	$-0.648343\pi$
$389$	−17.7250	−0.898692	−0.449346	+	0.893358i	$0.648343\pi$
$390$	0	0
$391$	−4.42221	−0.223641
$392$	0	0
$393$	17.2111	0.868185
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.0278	0.754221	0.377111	−	0.926168i	$-0.376918\pi$
$397$	15.0278	0.754221	0.377111	+	0.926168i	$0.376918\pi$
$398$	0	0
$399$	5.30278	0.265471
$400$	0	0
$401$	−25.8167	−1.28922	−0.644611	−	0.764511i	$-0.722981\pi$
$401$	−25.8167	−1.28922	−0.644611	+	0.764511i	$0.722981\pi$
$402$	0	0
$403$	4.60555	0.229419
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.21110	0.456577
$408$	0	0
$409$	31.2111	1.54329	0.771645	−	0.636054i	$-0.219434\pi$
$409$	31.2111	1.54329	0.771645	+	0.636054i	$0.219434\pi$
$410$	0	0
$411$	19.8167	0.977483
$412$	0	0
$413$	4.42221	0.217602
$414$	0	0
$415$	0	0
$416$	0	0
$417$	61.4500	3.00922
$418$	0	0
$419$	−39.7527	−1.94205	−0.971024	−	0.238981i	$-0.923187\pi$
$419$	−39.7527	−1.94205	−0.971024	+	0.238981i	$0.923187\pi$
$420$	0	0
$421$	−29.0278	−1.41473	−0.707363	−	0.706850i	$-0.750115\pi$
$421$	−29.0278	−1.41473	−0.707363	+	0.706850i	$0.750115\pi$
$422$	0	0
$423$	−47.4500	−2.30710
$424$	0	0
$425$	−13.0278	−0.631939
$426$	0	0
$427$	2.97224	0.143837
$428$	0	0
$429$	3.30278	0.159460
$430$	0	0
$431$	26.7250	1.28730	0.643649	−	0.765321i	$-0.277420\pi$
$431$	26.7250	1.28730	0.643649	+	0.765321i	$0.277420\pi$
$432$	0	0
$433$	−19.1194	−0.918821	−0.459411	−	0.888224i	$-0.651939\pi$
$433$	−19.1194	−0.918821	−0.459411	+	0.888224i	$0.651939\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.00000	0.430528
$438$	0	0
$439$	−27.2111	−1.29872	−0.649358	−	0.760483i	$-0.724962\pi$
$439$	−27.2111	−1.29872	−0.649358	+	0.760483i	$0.724962\pi$
$440$	0	0
$441$	−54.6333	−2.60159
$442$	0	0
$443$	36.3583	1.72743	0.863717	−	0.503977i	$-0.168130\pi$
$443$	36.3583	1.72743	0.863717	+	0.503977i	$0.168130\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	57.2389	2.70731
$448$	0	0
$449$	−21.3944	−1.00967	−0.504833	−	0.863217i	$-0.668446\pi$
$449$	−21.3944	−1.00967	−0.504833	+	0.863217i	$0.668446\pi$
$450$	0	0
$451$	−1.69722	−0.0799192
$452$	0	0
$453$	−78.0555	−3.66737
$454$	0	0
$455$	0	0
$456$	0	0
$457$	41.1194	1.92349	0.961743	−	0.273954i	$-0.0883315\pi$
$457$	41.1194	1.92349	0.961743	+	0.273954i	$0.0883315\pi$
$458$	0	0
$459$	−42.2389	−1.97154
$460$	0	0
$461$	−1.30278	−0.0606763	−0.0303382	−	0.999540i	$-0.509658\pi$
$461$	−1.30278	−0.0606763	−0.0303382	+	0.999540i	$0.509658\pi$
$462$	0	0
$463$	−34.2389	−1.59121	−0.795607	−	0.605813i	$-0.792848\pi$
$463$	−34.2389	−1.59121	−0.795607	+	0.605813i	$0.792848\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	39.6333	1.83401	0.917005	−	0.398875i	$-0.130599\pi$
$467$	39.6333	1.83401	0.917005	+	0.398875i	$0.130599\pi$
$468$	0	0
$469$	−0.422205	−0.0194956
$470$	0	0
$471$	10.2111	0.470503
$472$	0	0
$473$	6.60555	0.303724
$474$	0	0
$475$	26.5139	1.21654
$476$	0	0
$477$	−37.1472	−1.70085
$478$	0	0
$479$	13.5778	0.620385	0.310193	−	0.950674i	$-0.399606\pi$
$479$	13.5778	0.620385	0.310193	+	0.950674i	$0.399606\pi$
$480$	0	0
$481$	−9.21110	−0.419990
$482$	0	0
$483$	1.69722	0.0772264
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.02776	0.137201	0.0686004	−	0.997644i	$-0.478147\pi$
$487$	3.02776	0.137201	0.0686004	+	0.997644i	$0.478147\pi$
$488$	0	0
$489$	61.4500	2.77886
$490$	0	0
$491$	−42.2389	−1.90621	−0.953107	−	0.302635i	$-0.902134\pi$
$491$	−42.2389	−1.90621	−0.953107	+	0.302635i	$0.902134\pi$
$492$	0	0
$493$	−13.5778	−0.611513
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.02776	−0.0461012
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	−52.5416	−2.34739
$502$	0	0
$503$	19.8167	0.883581	0.441790	−	0.897118i	$-0.354343\pi$
$503$	19.8167	0.883581	0.441790	+	0.897118i	$0.354343\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.30278	−0.146681
$508$	0	0
$509$	19.8167	0.878358	0.439179	−	0.898400i	$-0.355269\pi$
$509$	19.8167	0.878358	0.439179	+	0.898400i	$0.355269\pi$
$510$	0	0
$511$	−3.18335	−0.140823
$512$	0	0
$513$	85.9638	3.79540
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	37.0278	1.62534
$520$	0	0
$521$	−23.0917	−1.01166	−0.505832	−	0.862632i	$-0.668815\pi$
$521$	−23.0917	−1.01166	−0.505832	+	0.862632i	$0.668815\pi$
$522$	0	0
$523$	−12.6056	−0.551202	−0.275601	−	0.961272i	$-0.588877\pi$
$523$	−12.6056	−0.551202	−0.275601	+	0.961272i	$0.588877\pi$
$524$	0	0
$525$	5.00000	0.218218
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−20.1194	−0.874758
$530$	0	0
$531$	115.505	5.01251
$532$	0	0
$533$	1.69722	0.0735149
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−39.6333	−1.71030
$538$	0	0
$539$	6.90833	0.297563
$540$	0	0
$541$	8.11943	0.349082	0.174541	−	0.984650i	$-0.444156\pi$
$541$	8.11943	0.349082	0.174541	+	0.984650i	$0.444156\pi$
$542$	0	0
$543$	28.7250	1.23271
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.8167	0.676271	0.338136	−	0.941097i	$-0.390204\pi$
$547$	15.8167	0.676271	0.338136	+	0.941097i	$0.390204\pi$
$548$	0	0
$549$	77.6333	3.31331
$550$	0	0
$551$	27.6333	1.17722
$552$	0	0
$553$	−4.60555	−0.195848
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.3305	−1.24277	−0.621387	−	0.783504i	$-0.713431\pi$
$557$	−29.3305	−1.24277	−0.621387	+	0.783504i	$0.713431\pi$
$558$	0	0
$559$	−6.60555	−0.279385
$560$	0	0
$561$	8.60555	0.363327
$562$	0	0
$563$	22.1833	0.934917	0.467458	−	0.884015i	$-0.345170\pi$
$563$	22.1833	0.934917	0.467458	+	0.884015i	$0.345170\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.02776	0.379130
$568$	0	0
$569$	21.3944	0.896902	0.448451	−	0.893807i	$-0.351976\pi$
$569$	21.3944	0.896902	0.448451	+	0.893807i	$0.351976\pi$
$570$	0	0
$571$	−11.8167	−0.494512	−0.247256	−	0.968950i	$-0.579529\pi$
$571$	−11.8167	−0.494512	−0.247256	+	0.968950i	$0.579529\pi$
$572$	0	0
$573$	1.69722	0.0709026
$574$	0	0
$575$	8.48612	0.353896
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	58.4500	2.42910
$580$	0	0
$581$	−2.88057	−0.119506
$582$	0	0
$583$	4.69722	0.194539
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.8167	1.06557	0.532784	−	0.846251i	$-0.321146\pi$
$587$	25.8167	1.06557	0.532784	+	0.846251i	$0.321146\pi$
$588$	0	0
$589$	−24.4222	−1.00630
$590$	0	0
$591$	52.5416	2.16127
$592$	0	0
$593$	−24.1194	−0.990466	−0.495233	−	0.868760i	$-0.664917\pi$
$593$	−24.1194	−0.990466	−0.495233	+	0.868760i	$0.664917\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−62.5416	−2.55966
$598$	0	0
$599$	−20.8806	−0.853157	−0.426578	−	0.904451i	$-0.640281\pi$
$599$	−20.8806	−0.853157	−0.426578	+	0.904451i	$0.640281\pi$
$600$	0	0
$601$	3.57779	0.145941	0.0729706	−	0.997334i	$-0.476752\pi$
$601$	3.57779	0.145941	0.0729706	+	0.997334i	$0.476752\pi$
$602$	0	0
$603$	−11.0278	−0.449085
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−29.8167	−1.21022	−0.605110	−	0.796142i	$-0.706871\pi$
$607$	−29.8167	−1.21022	−0.605110	+	0.796142i	$0.706871\pi$
$608$	0	0
$609$	5.21110	0.211165
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	34.3305	1.38660	0.693299	−	0.720650i	$-0.256157\pi$
$613$	34.3305	1.38660	0.693299	+	0.720650i	$0.256157\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.8444	1.32227	0.661133	−	0.750269i	$-0.270076\pi$
$617$	32.8444	1.32227	0.661133	+	0.750269i	$0.270076\pi$
$618$	0	0
$619$	−18.8444	−0.757421	−0.378710	−	0.925515i	$-0.623632\pi$
$619$	−18.8444	−0.757421	−0.378710	+	0.925515i	$0.623632\pi$
$620$	0	0
$621$	27.5139	1.10409
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−17.5139	−0.699437
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−36.0555	−1.43535	−0.717674	−	0.696380i	$-0.754793\pi$
$631$	−36.0555	−1.43535	−0.717674	+	0.696380i	$0.754793\pi$
$632$	0	0
$633$	−6.60555	−0.262547
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.90833	−0.273718
$638$	0	0
$639$	−26.8444	−1.06195
$640$	0	0
$641$	0.908327	0.0358768	0.0179384	−	0.999839i	$-0.494290\pi$
$641$	0.908327	0.0358768	0.0179384	+	0.999839i	$0.494290\pi$
$642$	0	0
$643$	9.81665	0.387131	0.193566	−	0.981087i	$-0.437995\pi$
$643$	9.81665	0.387131	0.193566	+	0.981087i	$0.437995\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−47.9638	−1.88565	−0.942827	−	0.333284i	$-0.891843\pi$
$647$	−47.9638	−1.88565	−0.942827	+	0.333284i	$0.891843\pi$
$648$	0	0
$649$	−14.6056	−0.573318
$650$	0	0
$651$	−4.60555	−0.180506
$652$	0	0
$653$	−0.788897	−0.0308719	−0.0154360	−	0.999881i	$-0.504914\pi$
$653$	−0.788897	−0.0308719	−0.0154360	+	0.999881i	$0.504914\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−83.1472	−3.24388
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	34.8444	1.35529	0.677645	−	0.735389i	$-0.263001\pi$
$661$	34.8444	1.35529	0.677645	+	0.735389i	$0.263001\pi$
$662$	0	0
$663$	−8.60555	−0.334212
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.84441	0.342457
$668$	0	0
$669$	7.21110	0.278797
$670$	0	0
$671$	−9.81665	−0.378968
$672$	0	0
$673$	−7.39445	−0.285035	−0.142518	−	0.989792i	$-0.545520\pi$
$673$	−7.39445	−0.285035	−0.142518	+	0.989792i	$0.545520\pi$
$674$	0	0
$675$	81.0555	3.11983
$676$	0	0
$677$	−43.2666	−1.66287	−0.831436	−	0.555621i	$-0.812480\pi$
$677$	−43.2666	−1.66287	−0.831436	+	0.555621i	$0.812480\pi$
$678$	0	0
$679$	0.844410	0.0324055
$680$	0	0
$681$	−69.7527	−2.67293
$682$	0	0
$683$	−15.6333	−0.598192	−0.299096	−	0.954223i	$-0.596685\pi$
$683$	−15.6333	−0.598192	−0.299096	+	0.954223i	$0.596685\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	50.2389	1.91673
$688$	0	0
$689$	−4.69722	−0.178950
$690$	0	0
$691$	12.1833	0.463476	0.231738	−	0.972778i	$-0.425559\pi$
$691$	12.1833	0.463476	0.231738	+	0.972778i	$0.425559\pi$
$692$	0	0
$693$	−2.39445	−0.0909576
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.42221	0.167503
$698$	0	0
$699$	11.2111	0.424043
$700$	0	0
$701$	13.0278	0.492052	0.246026	−	0.969263i	$-0.420875\pi$
$701$	13.0278	0.492052	0.246026	+	0.969263i	$0.420875\pi$
$702$	0	0
$703$	48.8444	1.84220
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.18335	0.157331
$708$	0	0
$709$	4.60555	0.172965	0.0864826	−	0.996253i	$-0.472437\pi$
$709$	4.60555	0.172965	0.0864826	+	0.996253i	$0.472437\pi$
$710$	0	0
$711$	−120.294	−4.51139
$712$	0	0
$713$	−7.81665	−0.292736
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−100.780	−3.76372
$718$	0	0
$719$	−34.4222	−1.28373	−0.641866	−	0.766817i	$-0.721839\pi$
$719$	−34.4222	−1.28373	−0.641866	+	0.766817i	$0.721839\pi$
$720$	0	0
$721$	0.880571	0.0327942
$722$	0	0
$723$	30.8167	1.14608
$724$	0	0
$725$	26.0555	0.967677
$726$	0	0
$727$	16.4861	0.611436	0.305718	−	0.952122i	$-0.401103\pi$
$727$	16.4861	0.611436	0.305718	+	0.952122i	$0.401103\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	−17.2111	−0.636576
$732$	0	0
$733$	−15.8806	−0.586562	−0.293281	−	0.956026i	$-0.594747\pi$
$733$	−15.8806	−0.586562	−0.293281	+	0.956026i	$0.594747\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.39445	0.0513652
$738$	0	0
$739$	−25.5139	−0.938543	−0.469272	−	0.883054i	$-0.655483\pi$
$739$	−25.5139	−0.938543	−0.469272	+	0.883054i	$0.655483\pi$
$740$	0	0
$741$	17.5139	0.643388
$742$	0	0
$743$	−44.8444	−1.64518	−0.822591	−	0.568634i	$-0.807472\pi$
$743$	−44.8444	−1.64518	−0.822591	+	0.568634i	$0.807472\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−75.2389	−2.75285
$748$	0	0
$749$	1.81665	0.0663791
$750$	0	0
$751$	−18.0917	−0.660175	−0.330087	−	0.943950i	$-0.607078\pi$
$751$	−18.0917	−0.660175	−0.330087	+	0.943950i	$0.607078\pi$
$752$	0	0
$753$	−28.8167	−1.05014
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−28.9083	−1.05069	−0.525346	−	0.850889i	$-0.676064\pi$
$757$	−28.9083	−1.05069	−0.525346	+	0.850889i	$0.676064\pi$
$758$	0	0
$759$	−5.60555	−0.203469
$760$	0	0
$761$	32.7250	1.18628	0.593140	−	0.805099i	$-0.297888\pi$
$761$	32.7250	1.18628	0.593140	+	0.805099i	$0.297888\pi$
$762$	0	0
$763$	1.23886	0.0448497
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.6056	0.527376
$768$	0	0
$769$	1.09167	0.0393667	0.0196834	−	0.999806i	$-0.493734\pi$
$769$	1.09167	0.0393667	0.0196834	+	0.999806i	$0.493734\pi$
$770$	0	0
$771$	59.8444	2.15524
$772$	0	0
$773$	−42.2389	−1.51923	−0.759613	−	0.650375i	$-0.774612\pi$
$773$	−42.2389	−1.51923	−0.759613	+	0.650375i	$0.774612\pi$
$774$	0	0
$775$	−23.0278	−0.827181
$776$	0	0
$777$	9.21110	0.330446
$778$	0	0
$779$	−9.00000	−0.322458
$780$	0	0
$781$	3.39445	0.121463
$782$	0	0
$783$	84.4777	3.01899
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−27.0917	−0.965714	−0.482857	−	0.875699i	$-0.660401\pi$
$787$	−27.0917	−0.965714	−0.482857	+	0.875699i	$0.660401\pi$
$788$	0	0
$789$	−68.0555	−2.42284
$790$	0	0
$791$	0.633308	0.0225178
$792$	0	0
$793$	9.81665	0.348600
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	−15.6333	−0.553067
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.5139	0.371027
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−48.6333	−1.71197
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	15.6972	0.551204	0.275602	−	0.961272i	$-0.411123\pi$
$811$	15.6972	0.551204	0.275602	+	0.961272i	$0.411123\pi$
$812$	0	0
$813$	17.5139	0.614239
$814$	0	0
$815$	0	0
$816$	0	0
$817$	35.0278	1.22547
$818$	0	0
$819$	2.39445	0.0836688
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−12.4861	−0.435239	−0.217619	−	0.976034i	$-0.569829\pi$
$823$	−12.4861	−0.435239	−0.217619	+	0.976034i	$0.569829\pi$
$824$	0	0
$825$	−16.5139	−0.574939
$826$	0	0
$827$	28.9361	1.00621	0.503103	−	0.864226i	$-0.332192\pi$
$827$	28.9361	1.00621	0.503103	+	0.864226i	$0.332192\pi$
$828$	0	0
$829$	−6.72498	−0.233568	−0.116784	−	0.993157i	$-0.537259\pi$
$829$	−6.72498	−0.233568	−0.116784	+	0.993157i	$0.537259\pi$
$830$	0	0
$831$	55.4500	1.92354
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−74.6611	−2.58066
$838$	0	0
$839$	7.57779	0.261615	0.130807	−	0.991408i	$-0.458243\pi$
$839$	7.57779	0.261615	0.130807	+	0.991408i	$0.458243\pi$
$840$	0	0
$841$	−1.84441	−0.0636004
$842$	0	0
$843$	−86.9638	−2.99519
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.302776	0.0104035
$848$	0	0
$849$	10.6056	0.363982
$850$	0	0
$851$	15.6333	0.535903
$852$	0	0
$853$	23.9083	0.818606	0.409303	−	0.912399i	$-0.365772\pi$
$853$	23.9083	0.818606	0.409303	+	0.912399i	$0.365772\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−49.0278	−1.67476	−0.837378	−	0.546624i	$-0.815913\pi$
$857$	−49.0278	−1.67476	−0.837378	+	0.546624i	$0.815913\pi$
$858$	0	0
$859$	19.4861	0.664858	0.332429	−	0.943128i	$-0.392132\pi$
$859$	19.4861	0.664858	0.332429	+	0.943128i	$0.392132\pi$
$860$	0	0
$861$	−1.69722	−0.0578413
$862$	0	0
$863$	−33.3944	−1.13676	−0.568380	−	0.822766i	$-0.692430\pi$
$863$	−33.3944	−1.13676	−0.568380	+	0.822766i	$0.692430\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	33.7250	1.14536
$868$	0	0
$869$	15.2111	0.516001
$870$	0	0
$871$	−1.39445	−0.0472491
$872$	0	0
$873$	22.0555	0.746466
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.11943	−0.0378004	−0.0189002	−	0.999821i	$-0.506016\pi$
$877$	−1.11943	−0.0378004	−0.0189002	+	0.999821i	$0.506016\pi$
$878$	0	0
$879$	−71.4500	−2.40995
$880$	0	0
$881$	−51.1194	−1.72226	−0.861129	−	0.508387i	$-0.830242\pi$
$881$	−51.1194	−1.72226	−0.861129	+	0.508387i	$0.830242\pi$
$882$	0	0
$883$	−14.9361	−0.502639	−0.251320	−	0.967904i	$-0.580865\pi$
$883$	−14.9361	−0.502639	−0.251320	+	0.967904i	$0.580865\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.7889	−1.03379	−0.516895	−	0.856049i	$-0.672912\pi$
$887$	−30.7889	−1.03379	−0.516895	+	0.856049i	$0.672912\pi$
$888$	0	0
$889$	−3.02776	−0.101548
$890$	0	0
$891$	−29.8167	−0.998895
$892$	0	0
$893$	31.8167	1.06470
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.60555	0.187164
$898$	0	0
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−12.2389	−0.407736
$902$	0	0
$903$	6.60555	0.219819
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−16.1194	−0.535237	−0.267618	−	0.963525i	$-0.586237\pi$
$907$	−16.1194	−0.535237	−0.267618	+	0.963525i	$0.586237\pi$
$908$	0	0
$909$	109.267	3.62414
$910$	0	0
$911$	−1.69722	−0.0562316	−0.0281158	−	0.999605i	$-0.508951\pi$
$911$	−1.69722	−0.0562316	−0.0281158	+	0.999605i	$0.508951\pi$
$912$	0	0
$913$	9.51388	0.314863
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.57779	−0.0521034
$918$	0	0
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	0	0
$921$	−60.8444	−2.00489
$922$	0	0
$923$	−3.39445	−0.111730
$924$	0	0
$925$	46.0555	1.51430
$926$	0	0
$927$	23.0000	0.755419
$928$	0	0
$929$	−24.2389	−0.795251	−0.397626	−	0.917548i	$-0.630166\pi$
$929$	−24.2389	−0.795251	−0.397626	+	0.917548i	$0.630166\pi$
$930$	0	0
$931$	36.6333	1.20061
$932$	0	0
$933$	106.780	3.49584
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	−35.4222	−1.15596
$940$	0	0
$941$	31.5416	1.02823	0.514114	−	0.857722i	$-0.328121\pi$
$941$	31.5416	1.02823	0.514114	+	0.857722i	$0.328121\pi$
$942$	0	0
$943$	−2.88057	−0.0938043
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.7889	−0.610557	−0.305279	−	0.952263i	$-0.598750\pi$
$947$	−18.7889	−0.610557	−0.305279	+	0.952263i	$0.598750\pi$
$948$	0	0
$949$	−10.5139	−0.341295
$950$	0	0
$951$	−28.4222	−0.921653
$952$	0	0
$953$	−27.6333	−0.895131	−0.447565	−	0.894251i	$-0.647709\pi$
$953$	−27.6333	−0.895131	−0.447565	+	0.894251i	$0.647709\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−17.2111	−0.556356
$958$	0	0
$959$	−1.81665	−0.0586628
$960$	0	0
$961$	−9.78890	−0.315771
$962$	0	0
$963$	47.4500	1.52905
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−13.6695	−0.439580	−0.219790	−	0.975547i	$-0.570537\pi$
$967$	−13.6695	−0.439580	−0.219790	+	0.975547i	$0.570537\pi$
$968$	0	0
$969$	45.6333	1.46595
$970$	0	0
$971$	24.5139	0.786688	0.393344	−	0.919391i	$-0.371318\pi$
$971$	24.5139	0.786688	0.393344	+	0.919391i	$0.371318\pi$
$972$	0	0
$973$	−5.63331	−0.180596
$974$	0	0
$975$	16.5139	0.528867
$976$	0	0
$977$	−34.4222	−1.10126	−0.550632	−	0.834748i	$-0.685613\pi$
$977$	−34.4222	−1.10126	−0.550632	+	0.834748i	$0.685613\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	32.3583	1.03312
$982$	0	0
$983$	−16.4222	−0.523787	−0.261893	−	0.965097i	$-0.584347\pi$
$983$	−16.4222	−0.523787	−0.261893	+	0.965097i	$0.584347\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.00000	0.190982
$988$	0	0
$989$	11.2111	0.356492
$990$	0	0
$991$	41.9083	1.33126	0.665631	−	0.746281i	$-0.268163\pi$
$991$	41.9083	1.33126	0.665631	+	0.746281i	$0.268163\pi$
$992$	0	0
$993$	−35.0278	−1.11157
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16.6056	0.525903	0.262952	−	0.964809i	$-0.415304\pi$
$997$	16.6056	0.525903	0.262952	+	0.964809i	$0.415304\pi$
$998$	0	0
$999$	149.322	4.72434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.a.b.1.1	✓	2
3.2	odd	2		5148.2.a.h.1.2		2
4.3	odd	2		2288.2.a.r.1.2		2
8.3	odd	2		9152.2.a.bg.1.1		2
8.5	even	2		9152.2.a.bq.1.2		2
11.10	odd	2		6292.2.a.k.1.1		2
13.12	even	2		7436.2.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.a.b.1.1	✓	2	1.1	even	1	trivial
2288.2.a.r.1.2		2	4.3	odd	2
5148.2.a.h.1.2		2	3.2	odd	2
6292.2.a.k.1.1		2	11.10	odd	2
7436.2.a.e.1.1		2	13.12	even	2
9152.2.a.bg.1.1		2	8.3	odd	2
9152.2.a.bq.1.2		2	8.5	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.a (trivial)

\(f(q)\)	\(=\)	\(q-3.30278 q^{3} +0.302776 q^{7} +7.90833 q^{9} +O(q^{10})\)	\(q-3.30278 q^{3} +0.302776 q^{7} +7.90833 q^{9} -1.00000 q^{11} +1.00000 q^{13} +2.60555 q^{17} -5.30278 q^{19} -1.00000 q^{21} -1.69722 q^{23} -5.00000 q^{25} -16.2111 q^{27} -5.21110 q^{29} +4.60555 q^{31} +3.30278 q^{33} -9.21110 q^{37} -3.30278 q^{39} +1.69722 q^{41} -6.60555 q^{43} -6.00000 q^{47} -6.90833 q^{49} -8.60555 q^{51} -4.69722 q^{53} +17.5139 q^{57} +14.6056 q^{59} +9.81665 q^{61} +2.39445 q^{63} -1.39445 q^{67} +5.60555 q^{69} -3.39445 q^{71} -10.5139 q^{73} +16.5139 q^{75} -0.302776 q^{77} -15.2111 q^{79} +29.8167 q^{81} -9.51388 q^{83} +17.2111 q^{87} +0.302776 q^{91} -15.2111 q^{93} +2.78890 q^{97} -7.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 3 * q^7 + 5 * q^9	\( 2 q - 3 q^{3} - 3 q^{7} + 5 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{17} - 7 q^{19} - 2 q^{21} - 7 q^{23} - 10 q^{25} - 18 q^{27} + 4 q^{29} + 2 q^{31} + 3 q^{33} - 4 q^{37} - 3 q^{39} + 7 q^{41} - 6 q^{43} - 12 q^{47} - 3 q^{49} - 10 q^{51} - 13 q^{53} + 17 q^{57} + 22 q^{59} - 2 q^{61} + 12 q^{63} - 10 q^{67} + 4 q^{69} - 14 q^{71} - 3 q^{73} + 15 q^{75} + 3 q^{77} - 16 q^{79} + 38 q^{81} - q^{83} + 20 q^{87} - 3 q^{91} - 16 q^{93} + 20 q^{97} - 5 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 3 * q^7 + 5 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^17 - 7 * q^19 - 2 * q^21 - 7 * q^23 - 10 * q^25 - 18 * q^27 + 4 * q^29 + 2 * q^31 + 3 * q^33 - 4 * q^37 - 3 * q^39 + 7 * q^41 - 6 * q^43 - 12 * q^47 - 3 * q^49 - 10 * q^51 - 13 * q^53 + 17 * q^57 + 22 * q^59 - 2 * q^61 + 12 * q^63 - 10 * q^67 + 4 * q^69 - 14 * q^71 - 3 * q^73 + 15 * q^75 + 3 * q^77 - 16 * q^79 + 38 * q^81 - q^83 + 20 * q^87 - 3 * q^91 - 16 * q^93 + 20 * q^97 - 5 * q^99

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