LMFDB - Embedded newform 5712.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5712,2,Mod(1,5712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5712, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("5712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5712.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:	$45.6105496346$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 357)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	5712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.267949 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -7.19615 q^{13} +0.267949 q^{15} +1.00000 q^{17} -3.19615 q^{19} -1.00000 q^{21} +3.00000 q^{23} -4.92820 q^{25} -1.00000 q^{27} +4.00000 q^{29} +1.26795 q^{31} -5.00000 q^{33} -0.267949 q^{35} +1.26795 q^{37} +7.19615 q^{39} -8.26795 q^{41} +11.3923 q^{43} -0.267949 q^{45} -1.26795 q^{47} +1.00000 q^{49} -1.00000 q^{51} +11.6603 q^{53} -1.33975 q^{55} +3.19615 q^{57} +8.19615 q^{59} -3.26795 q^{61} +1.00000 q^{63} +1.92820 q^{65} +4.53590 q^{67} -3.00000 q^{69} -3.46410 q^{71} +3.46410 q^{73} +4.92820 q^{75} +5.00000 q^{77} -10.1962 q^{79} +1.00000 q^{81} -0.928203 q^{83} -0.267949 q^{85} -4.00000 q^{87} +8.39230 q^{89} -7.19615 q^{91} -1.26795 q^{93} +0.856406 q^{95} -13.8564 q^{97} +5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{7} + 2 q^{9} + 10 q^{11} - 4 q^{13} + 4 q^{15} + 2 q^{17} + 4 q^{19} - 2 q^{21} + 6 q^{23} + 4 q^{25} - 2 q^{27} + 8 q^{29} + 6 q^{31} - 10 q^{33} - 4 q^{35} + 6 q^{37} + 4 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^7 + 2 * q^9 + 10 * q^11 - 4 * q^13 + 4 * q^15 + 2 * q^17 + 4 * q^19 - 2 * q^21 + 6 * q^23 + 4 * q^25 - 2 * q^27 + 8 * q^29 + 6 * q^31 - 10 * q^33 - 4 * q^35 + 6 * q^37 + 4 * q^39 - 20 * q^41 + 2 * q^43 - 4 * q^45 - 6 * q^47 + 2 * q^49 - 2 * q^51 + 6 * q^53 - 20 * q^55 - 4 * q^57 + 6 * q^59 - 10 * q^61 + 2 * q^63 - 10 * q^65 + 16 * q^67 - 6 * q^69 - 4 * q^75 + 10 * q^77 - 10 * q^79 + 2 * q^81 + 12 * q^83 - 4 * q^85 - 8 * q^87 - 4 * q^89 - 4 * q^91 - 6 * q^93 - 26 * q^95 + 10 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.267949	−0.119831	−0.0599153	−	0.998203i	$-0.519083\pi$
$5$	−0.267949	−0.119831	−0.0599153	+	0.998203i	$0.519083\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	−7.19615	−1.99585	−0.997927	−	0.0643593i	$-0.979500\pi$
$13$	−7.19615	−1.99585	−0.997927	+	0.0643593i	$0.979500\pi$
$14$	0	0
$15$	0.267949	0.0691842
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−3.19615	−0.733248	−0.366624	−	0.930369i	$-0.619486\pi$
$19$	−3.19615	−0.733248	−0.366624	+	0.930369i	$0.619486\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.92820	−0.985641
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	1.26795	0.227730	0.113865	−	0.993496i	$-0.463677\pi$
$31$	1.26795	0.227730	0.113865	+	0.993496i	$0.463677\pi$
$32$	0	0
$33$	−5.00000	−0.870388
$34$	0	0
$35$	−0.267949	−0.0452917
$36$	0	0
$37$	1.26795	0.208450	0.104225	−	0.994554i	$-0.466764\pi$
$37$	1.26795	0.208450	0.104225	+	0.994554i	$0.466764\pi$
$38$	0	0
$39$	7.19615	1.15231
$40$	0	0
$41$	−8.26795	−1.29124	−0.645618	−	0.763660i	$-0.723400\pi$
$41$	−8.26795	−1.29124	−0.645618	+	0.763660i	$0.723400\pi$
$42$	0	0
$43$	11.3923	1.73731	0.868655	−	0.495417i	$-0.164985\pi$
$43$	11.3923	1.73731	0.868655	+	0.495417i	$0.164985\pi$
$44$	0	0
$45$	−0.267949	−0.0399435
$46$	0	0
$47$	−1.26795	−0.184949	−0.0924747	−	0.995715i	$-0.529478\pi$
$47$	−1.26795	−0.184949	−0.0924747	+	0.995715i	$0.529478\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	11.6603	1.60166	0.800830	−	0.598892i	$-0.204392\pi$
$53$	11.6603	1.60166	0.800830	+	0.598892i	$0.204392\pi$
$54$	0	0
$55$	−1.33975	−0.180651
$56$	0	0
$57$	3.19615	0.423341
$58$	0	0
$59$	8.19615	1.06705	0.533524	−	0.845785i	$-0.320867\pi$
$59$	8.19615	1.06705	0.533524	+	0.845785i	$0.320867\pi$
$60$	0	0
$61$	−3.26795	−0.418418	−0.209209	−	0.977871i	$-0.567089\pi$
$61$	−3.26795	−0.418418	−0.209209	+	0.977871i	$0.567089\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	1.92820	0.239164
$66$	0	0
$67$	4.53590	0.554148	0.277074	−	0.960849i	$-0.410635\pi$
$67$	4.53590	0.554148	0.277074	+	0.960849i	$0.410635\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	0	0
$73$	3.46410	0.405442	0.202721	−	0.979236i	$-0.435021\pi$
$73$	3.46410	0.405442	0.202721	+	0.979236i	$0.435021\pi$
$74$	0	0
$75$	4.92820	0.569060
$76$	0	0
$77$	5.00000	0.569803
$78$	0	0
$79$	−10.1962	−1.14716	−0.573578	−	0.819151i	$-0.694445\pi$
$79$	−10.1962	−1.14716	−0.573578	+	0.819151i	$0.694445\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.928203	−0.101884	−0.0509418	−	0.998702i	$-0.516222\pi$
$83$	−0.928203	−0.101884	−0.0509418	+	0.998702i	$0.516222\pi$
$84$	0	0
$85$	−0.267949	−0.0290632
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	8.39230	0.889583	0.444791	−	0.895634i	$-0.353278\pi$
$89$	8.39230	0.889583	0.444791	+	0.895634i	$0.353278\pi$
$90$	0	0
$91$	−7.19615	−0.754362
$92$	0	0
$93$	−1.26795	−0.131480
$94$	0	0
$95$	0.856406	0.0878654
$96$	0	0
$97$	−13.8564	−1.40690	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	−13.8564	−1.40690	−0.703452	+	0.710742i	$0.748359\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	0.732051	0.0728418	0.0364209	−	0.999337i	$-0.488404\pi$
$101$	0.732051	0.0728418	0.0364209	+	0.999337i	$0.488404\pi$
$102$	0	0
$103$	−7.19615	−0.709058	−0.354529	−	0.935045i	$-0.615359\pi$
$103$	−7.19615	−0.709058	−0.354529	+	0.935045i	$0.615359\pi$
$104$	0	0
$105$	0.267949	0.0261492
$106$	0	0
$107$	11.0000	1.06341	0.531705	−	0.846930i	$-0.321551\pi$
$107$	11.0000	1.06341	0.531705	+	0.846930i	$0.321551\pi$
$108$	0	0
$109$	−5.66025	−0.542154	−0.271077	−	0.962558i	$-0.587380\pi$
$109$	−5.66025	−0.542154	−0.271077	+	0.962558i	$0.587380\pi$
$110$	0	0
$111$	−1.26795	−0.120348
$112$	0	0
$113$	−3.39230	−0.319121	−0.159561	−	0.987188i	$-0.551008\pi$
$113$	−3.39230	−0.319121	−0.159561	+	0.987188i	$0.551008\pi$
$114$	0	0
$115$	−0.803848	−0.0749592
$116$	0	0
$117$	−7.19615	−0.665285
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	8.26795	0.745496
$124$	0	0
$125$	2.66025	0.237940
$126$	0	0
$127$	7.39230	0.655961	0.327980	−	0.944684i	$-0.393632\pi$
$127$	7.39230	0.655961	0.327980	+	0.944684i	$0.393632\pi$
$128$	0	0
$129$	−11.3923	−1.00304
$130$	0	0
$131$	2.80385	0.244973	0.122487	−	0.992470i	$-0.460913\pi$
$131$	2.80385	0.244973	0.122487	+	0.992470i	$0.460913\pi$
$132$	0	0
$133$	−3.19615	−0.277142
$134$	0	0
$135$	0.267949	0.0230614
$136$	0	0
$137$	−2.19615	−0.187630	−0.0938150	−	0.995590i	$-0.529906\pi$
$137$	−2.19615	−0.187630	−0.0938150	+	0.995590i	$0.529906\pi$
$138$	0	0
$139$	15.4641	1.31165	0.655824	−	0.754914i	$-0.272321\pi$
$139$	15.4641	1.31165	0.655824	+	0.754914i	$0.272321\pi$
$140$	0	0
$141$	1.26795	0.106781
$142$	0	0
$143$	−35.9808	−3.00886
$144$	0	0
$145$	−1.07180	−0.0890079
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	3.46410	0.283790	0.141895	−	0.989882i	$-0.454680\pi$
$149$	3.46410	0.283790	0.141895	+	0.989882i	$0.454680\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−0.339746	−0.0272891
$156$	0	0
$157$	19.5885	1.56333	0.781665	−	0.623699i	$-0.214371\pi$
$157$	19.5885	1.56333	0.781665	+	0.623699i	$0.214371\pi$
$158$	0	0
$159$	−11.6603	−0.924718
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	7.07180	0.553906	0.276953	−	0.960883i	$-0.410675\pi$
$163$	7.07180	0.553906	0.276953	+	0.960883i	$0.410675\pi$
$164$	0	0
$165$	1.33975	0.104299
$166$	0	0
$167$	−4.80385	−0.371733	−0.185866	−	0.982575i	$-0.559509\pi$
$167$	−4.80385	−0.371733	−0.185866	+	0.982575i	$0.559509\pi$
$168$	0	0
$169$	38.7846	2.98343
$170$	0	0
$171$	−3.19615	−0.244416
$172$	0	0
$173$	−3.19615	−0.242999	−0.121499	−	0.992591i	$-0.538770\pi$
$173$	−3.19615	−0.242999	−0.121499	+	0.992591i	$0.538770\pi$
$174$	0	0
$175$	−4.92820	−0.372537
$176$	0	0
$177$	−8.19615	−0.616061
$178$	0	0
$179$	−11.4641	−0.856867	−0.428434	−	0.903573i	$-0.640934\pi$
$179$	−11.4641	−0.856867	−0.428434	+	0.903573i	$0.640934\pi$
$180$	0	0
$181$	9.32051	0.692788	0.346394	−	0.938089i	$-0.387406\pi$
$181$	9.32051	0.692788	0.346394	+	0.938089i	$0.387406\pi$
$182$	0	0
$183$	3.26795	0.241574
$184$	0	0
$185$	−0.339746	−0.0249786
$186$	0	0
$187$	5.00000	0.365636
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−16.9282	−1.22488	−0.612441	−	0.790516i	$-0.709812\pi$
$191$	−16.9282	−1.22488	−0.612441	+	0.790516i	$0.709812\pi$
$192$	0	0
$193$	24.0526	1.73134	0.865671	−	0.500614i	$-0.166892\pi$
$193$	24.0526	1.73134	0.865671	+	0.500614i	$0.166892\pi$
$194$	0	0
$195$	−1.92820	−0.138082
$196$	0	0
$197$	−11.0000	−0.783718	−0.391859	−	0.920025i	$-0.628168\pi$
$197$	−11.0000	−0.783718	−0.391859	+	0.920025i	$0.628168\pi$
$198$	0	0
$199$	4.73205	0.335446	0.167723	−	0.985834i	$-0.446359\pi$
$199$	4.73205	0.335446	0.167723	+	0.985834i	$0.446359\pi$
$200$	0	0
$201$	−4.53590	−0.319938
$202$	0	0
$203$	4.00000	0.280745
$204$	0	0
$205$	2.21539	0.154730
$206$	0	0
$207$	3.00000	0.208514
$208$	0	0
$209$	−15.9808	−1.10541
$210$	0	0
$211$	−22.5885	−1.55505	−0.777527	−	0.628850i	$-0.783526\pi$
$211$	−22.5885	−1.55505	−0.777527	+	0.628850i	$0.783526\pi$
$212$	0	0
$213$	3.46410	0.237356
$214$	0	0
$215$	−3.05256	−0.208183
$216$	0	0
$217$	1.26795	0.0860740
$218$	0	0
$219$	−3.46410	−0.234082
$220$	0	0
$221$	−7.19615	−0.484066
$222$	0	0
$223$	14.1244	0.945837	0.472918	−	0.881106i	$-0.343201\pi$
$223$	14.1244	0.945837	0.472918	+	0.881106i	$0.343201\pi$
$224$	0	0
$225$	−4.92820	−0.328547
$226$	0	0
$227$	19.0526	1.26456	0.632281	−	0.774739i	$-0.282119\pi$
$227$	19.0526	1.26456	0.632281	+	0.774739i	$0.282119\pi$
$228$	0	0
$229$	−5.85641	−0.387002	−0.193501	−	0.981100i	$-0.561984\pi$
$229$	−5.85641	−0.387002	−0.193501	+	0.981100i	$0.561984\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	15.9282	1.04349	0.521746	−	0.853101i	$-0.325281\pi$
$233$	15.9282	1.04349	0.521746	+	0.853101i	$0.325281\pi$
$234$	0	0
$235$	0.339746	0.0221626
$236$	0	0
$237$	10.1962	0.662311
$238$	0	0
$239$	19.5167	1.26243	0.631214	−	0.775609i	$-0.282557\pi$
$239$	19.5167	1.26243	0.631214	+	0.775609i	$0.282557\pi$
$240$	0	0
$241$	14.9282	0.961610	0.480805	−	0.876828i	$-0.340344\pi$
$241$	14.9282	0.961610	0.480805	+	0.876828i	$0.340344\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.267949	−0.0171186
$246$	0	0
$247$	23.0000	1.46345
$248$	0	0
$249$	0.928203	0.0588225
$250$	0	0
$251$	29.1244	1.83831	0.919157	−	0.393893i	$-0.128872\pi$
$251$	29.1244	1.83831	0.919157	+	0.393893i	$0.128872\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	0	0
$255$	0.267949	0.0167796
$256$	0	0
$257$	−3.66025	−0.228320	−0.114160	−	0.993462i	$-0.536418\pi$
$257$	−3.66025	−0.228320	−0.114160	+	0.993462i	$0.536418\pi$
$258$	0	0
$259$	1.26795	0.0787865
$260$	0	0
$261$	4.00000	0.247594
$262$	0	0
$263$	−13.4641	−0.830232	−0.415116	−	0.909768i	$-0.636259\pi$
$263$	−13.4641	−0.830232	−0.415116	+	0.909768i	$0.636259\pi$
$264$	0	0
$265$	−3.12436	−0.191928
$266$	0	0
$267$	−8.39230	−0.513601
$268$	0	0
$269$	−23.1962	−1.41429	−0.707147	−	0.707066i	$-0.750018\pi$
$269$	−23.1962	−1.41429	−0.707147	+	0.707066i	$0.750018\pi$
$270$	0	0
$271$	15.7321	0.955654	0.477827	−	0.878454i	$-0.341425\pi$
$271$	15.7321	0.955654	0.477827	+	0.878454i	$0.341425\pi$
$272$	0	0
$273$	7.19615	0.435531
$274$	0	0
$275$	−24.6410	−1.48591
$276$	0	0
$277$	−6.39230	−0.384076	−0.192038	−	0.981387i	$-0.561510\pi$
$277$	−6.39230	−0.384076	−0.192038	+	0.981387i	$0.561510\pi$
$278$	0	0
$279$	1.26795	0.0759101
$280$	0	0
$281$	−11.6077	−0.692457	−0.346229	−	0.938150i	$-0.612538\pi$
$281$	−11.6077	−0.692457	−0.346229	+	0.938150i	$0.612538\pi$
$282$	0	0
$283$	30.7321	1.82683	0.913415	−	0.407029i	$-0.133435\pi$
$283$	30.7321	1.82683	0.913415	+	0.407029i	$0.133435\pi$
$284$	0	0
$285$	−0.856406	−0.0507291
$286$	0	0
$287$	−8.26795	−0.488042
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	13.8564	0.812277
$292$	0	0
$293$	14.9282	0.872115	0.436057	−	0.899919i	$-0.356374\pi$
$293$	14.9282	0.872115	0.436057	+	0.899919i	$0.356374\pi$
$294$	0	0
$295$	−2.19615	−0.127865
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	−21.5885	−1.24849
$300$	0	0
$301$	11.3923	0.656642
$302$	0	0
$303$	−0.732051	−0.0420552
$304$	0	0
$305$	0.875644	0.0501392
$306$	0	0
$307$	−22.9282	−1.30858	−0.654291	−	0.756243i	$-0.727033\pi$
$307$	−22.9282	−1.30858	−0.654291	+	0.756243i	$0.727033\pi$
$308$	0	0
$309$	7.19615	0.409375
$310$	0	0
$311$	27.7128	1.57145	0.785725	−	0.618576i	$-0.212290\pi$
$311$	27.7128	1.57145	0.785725	+	0.618576i	$0.212290\pi$
$312$	0	0
$313$	5.12436	0.289646	0.144823	−	0.989458i	$-0.453739\pi$
$313$	5.12436	0.289646	0.144823	+	0.989458i	$0.453739\pi$
$314$	0	0
$315$	−0.267949	−0.0150972
$316$	0	0
$317$	24.9282	1.40011	0.700054	−	0.714090i	$-0.253159\pi$
$317$	24.9282	1.40011	0.700054	+	0.714090i	$0.253159\pi$
$318$	0	0
$319$	20.0000	1.11979
$320$	0	0
$321$	−11.0000	−0.613960
$322$	0	0
$323$	−3.19615	−0.177839
$324$	0	0
$325$	35.4641	1.96719
$326$	0	0
$327$	5.66025	0.313013
$328$	0	0
$329$	−1.26795	−0.0699043
$330$	0	0
$331$	1.92820	0.105984	0.0529918	−	0.998595i	$-0.483124\pi$
$331$	1.92820	0.105984	0.0529918	+	0.998595i	$0.483124\pi$
$332$	0	0
$333$	1.26795	0.0694832
$334$	0	0
$335$	−1.21539	−0.0664039
$336$	0	0
$337$	13.8038	0.751943	0.375972	−	0.926631i	$-0.377309\pi$
$337$	13.8038	0.751943	0.375972	+	0.926631i	$0.377309\pi$
$338$	0	0
$339$	3.39230	0.184245
$340$	0	0
$341$	6.33975	0.343316
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.803848	0.0432777
$346$	0	0
$347$	22.0000	1.18102	0.590511	−	0.807030i	$-0.298926\pi$
$347$	22.0000	1.18102	0.590511	+	0.807030i	$0.298926\pi$
$348$	0	0
$349$	4.80385	0.257144	0.128572	−	0.991700i	$-0.458961\pi$
$349$	4.80385	0.257144	0.128572	+	0.991700i	$0.458961\pi$
$350$	0	0
$351$	7.19615	0.384102
$352$	0	0
$353$	−8.58846	−0.457117	−0.228559	−	0.973530i	$-0.573401\pi$
$353$	−8.58846	−0.457117	−0.228559	+	0.973530i	$0.573401\pi$
$354$	0	0
$355$	0.928203	0.0492639
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	16.5885	0.875505	0.437753	−	0.899095i	$-0.355775\pi$
$359$	16.5885	0.875505	0.437753	+	0.899095i	$0.355775\pi$
$360$	0	0
$361$	−8.78461	−0.462348
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−0.928203	−0.0485844
$366$	0	0
$367$	5.07180	0.264746	0.132373	−	0.991200i	$-0.457740\pi$
$367$	5.07180	0.264746	0.132373	+	0.991200i	$0.457740\pi$
$368$	0	0
$369$	−8.26795	−0.430412
$370$	0	0
$371$	11.6603	0.605370
$372$	0	0
$373$	29.4641	1.52559	0.762797	−	0.646638i	$-0.223826\pi$
$373$	29.4641	1.52559	0.762797	+	0.646638i	$0.223826\pi$
$374$	0	0
$375$	−2.66025	−0.137375
$376$	0	0
$377$	−28.7846	−1.48248
$378$	0	0
$379$	5.07180	0.260521	0.130260	−	0.991480i	$-0.458419\pi$
$379$	5.07180	0.260521	0.130260	+	0.991480i	$0.458419\pi$
$380$	0	0
$381$	−7.39230	−0.378719
$382$	0	0
$383$	14.1962	0.725390	0.362695	−	0.931908i	$-0.381857\pi$
$383$	14.1962	0.725390	0.362695	+	0.931908i	$0.381857\pi$
$384$	0	0
$385$	−1.33975	−0.0682798
$386$	0	0
$387$	11.3923	0.579103
$388$	0	0
$389$	−9.46410	−0.479849	−0.239924	−	0.970792i	$-0.577123\pi$
$389$	−9.46410	−0.479849	−0.239924	+	0.970792i	$0.577123\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	−2.80385	−0.141435
$394$	0	0
$395$	2.73205	0.137464
$396$	0	0
$397$	−21.1244	−1.06020	−0.530101	−	0.847935i	$-0.677846\pi$
$397$	−21.1244	−1.06020	−0.530101	+	0.847935i	$0.677846\pi$
$398$	0	0
$399$	3.19615	0.160008
$400$	0	0
$401$	33.9282	1.69429	0.847147	−	0.531359i	$-0.178318\pi$
$401$	33.9282	1.69429	0.847147	+	0.531359i	$0.178318\pi$
$402$	0	0
$403$	−9.12436	−0.454517
$404$	0	0
$405$	−0.267949	−0.0133145
$406$	0	0
$407$	6.33975	0.314250
$408$	0	0
$409$	−4.80385	−0.237535	−0.118767	−	0.992922i	$-0.537894\pi$
$409$	−4.80385	−0.237535	−0.118767	+	0.992922i	$0.537894\pi$
$410$	0	0
$411$	2.19615	0.108328
$412$	0	0
$413$	8.19615	0.403306
$414$	0	0
$415$	0.248711	0.0122088
$416$	0	0
$417$	−15.4641	−0.757280
$418$	0	0
$419$	29.3205	1.43240	0.716200	−	0.697895i	$-0.245880\pi$
$419$	29.3205	1.43240	0.716200	+	0.697895i	$0.245880\pi$
$420$	0	0
$421$	−2.60770	−0.127091	−0.0635456	−	0.997979i	$-0.520241\pi$
$421$	−2.60770	−0.127091	−0.0635456	+	0.997979i	$0.520241\pi$
$422$	0	0
$423$	−1.26795	−0.0616498
$424$	0	0
$425$	−4.92820	−0.239053
$426$	0	0
$427$	−3.26795	−0.158147
$428$	0	0
$429$	35.9808	1.73717
$430$	0	0
$431$	−18.9282	−0.911739	−0.455870	−	0.890047i	$-0.650672\pi$
$431$	−18.9282	−0.911739	−0.455870	+	0.890047i	$0.650672\pi$
$432$	0	0
$433$	5.05256	0.242810	0.121405	−	0.992603i	$-0.461260\pi$
$433$	5.05256	0.242810	0.121405	+	0.992603i	$0.461260\pi$
$434$	0	0
$435$	1.07180	0.0513887
$436$	0	0
$437$	−9.58846	−0.458678
$438$	0	0
$439$	22.9808	1.09681	0.548406	−	0.836212i	$-0.315235\pi$
$439$	22.9808	1.09681	0.548406	+	0.836212i	$0.315235\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−2.19615	−0.104342	−0.0521712	−	0.998638i	$-0.516614\pi$
$443$	−2.19615	−0.104342	−0.0521712	+	0.998638i	$0.516614\pi$
$444$	0	0
$445$	−2.24871	−0.106599
$446$	0	0
$447$	−3.46410	−0.163846
$448$	0	0
$449$	−40.3923	−1.90623	−0.953115	−	0.302607i	$-0.902143\pi$
$449$	−40.3923	−1.90623	−0.953115	+	0.302607i	$0.902143\pi$
$450$	0	0
$451$	−41.3397	−1.94661
$452$	0	0
$453$	2.00000	0.0939682
$454$	0	0
$455$	1.92820	0.0903956
$456$	0	0
$457$	23.3923	1.09425	0.547123	−	0.837052i	$-0.315723\pi$
$457$	23.3923	1.09425	0.547123	+	0.837052i	$0.315723\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−7.80385	−0.363461	−0.181731	−	0.983348i	$-0.558170\pi$
$461$	−7.80385	−0.363461	−0.181731	+	0.983348i	$0.558170\pi$
$462$	0	0
$463$	19.0718	0.886342	0.443171	−	0.896437i	$-0.353854\pi$
$463$	19.0718	0.886342	0.443171	+	0.896437i	$0.353854\pi$
$464$	0	0
$465$	0.339746	0.0157553
$466$	0	0
$467$	22.3923	1.03619	0.518096	−	0.855322i	$-0.326641\pi$
$467$	22.3923	1.03619	0.518096	+	0.855322i	$0.326641\pi$
$468$	0	0
$469$	4.53590	0.209448
$470$	0	0
$471$	−19.5885	−0.902588
$472$	0	0
$473$	56.9615	2.61909
$474$	0	0
$475$	15.7513	0.722719
$476$	0	0
$477$	11.6603	0.533886
$478$	0	0
$479$	−20.8038	−0.950552	−0.475276	−	0.879837i	$-0.657652\pi$
$479$	−20.8038	−0.950552	−0.475276	+	0.879837i	$0.657652\pi$
$480$	0	0
$481$	−9.12436	−0.416035
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	3.71281	0.168590
$486$	0	0
$487$	−16.9282	−0.767090	−0.383545	−	0.923522i	$-0.625297\pi$
$487$	−16.9282	−0.767090	−0.383545	+	0.923522i	$0.625297\pi$
$488$	0	0
$489$	−7.07180	−0.319798
$490$	0	0
$491$	−3.85641	−0.174037	−0.0870186	−	0.996207i	$-0.527734\pi$
$491$	−3.85641	−0.174037	−0.0870186	+	0.996207i	$0.527734\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	−1.33975	−0.0602171
$496$	0	0
$497$	−3.46410	−0.155386
$498$	0	0
$499$	0.732051	0.0327711	0.0163855	−	0.999866i	$-0.494784\pi$
$499$	0.732051	0.0327711	0.0163855	+	0.999866i	$0.494784\pi$
$500$	0	0
$501$	4.80385	0.214620
$502$	0	0
$503$	20.6603	0.921195	0.460598	−	0.887609i	$-0.347635\pi$
$503$	20.6603	0.921195	0.460598	+	0.887609i	$0.347635\pi$
$504$	0	0
$505$	−0.196152	−0.00872867
$506$	0	0
$507$	−38.7846	−1.72248
$508$	0	0
$509$	−12.7846	−0.566668	−0.283334	−	0.959021i	$-0.591440\pi$
$509$	−12.7846	−0.566668	−0.283334	+	0.959021i	$0.591440\pi$
$510$	0	0
$511$	3.46410	0.153243
$512$	0	0
$513$	3.19615	0.141114
$514$	0	0
$515$	1.92820	0.0849668
$516$	0	0
$517$	−6.33975	−0.278822
$518$	0	0
$519$	3.19615	0.140296
$520$	0	0
$521$	−32.6603	−1.43087	−0.715436	−	0.698678i	$-0.753772\pi$
$521$	−32.6603	−1.43087	−0.715436	+	0.698678i	$0.753772\pi$
$522$	0	0
$523$	38.2487	1.67250	0.836250	−	0.548349i	$-0.184743\pi$
$523$	38.2487	1.67250	0.836250	+	0.548349i	$0.184743\pi$
$524$	0	0
$525$	4.92820	0.215084
$526$	0	0
$527$	1.26795	0.0552327
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	8.19615	0.355683
$532$	0	0
$533$	59.4974	2.57712
$534$	0	0
$535$	−2.94744	−0.127429
$536$	0	0
$537$	11.4641	0.494713
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	−17.1769	−0.738493	−0.369247	−	0.929331i	$-0.620384\pi$
$541$	−17.1769	−0.738493	−0.369247	+	0.929331i	$0.620384\pi$
$542$	0	0
$543$	−9.32051	−0.399981
$544$	0	0
$545$	1.51666	0.0649666
$546$	0	0
$547$	7.41154	0.316895	0.158447	−	0.987367i	$-0.449351\pi$
$547$	7.41154	0.316895	0.158447	+	0.987367i	$0.449351\pi$
$548$	0	0
$549$	−3.26795	−0.139473
$550$	0	0
$551$	−12.7846	−0.544643
$552$	0	0
$553$	−10.1962	−0.433585
$554$	0	0
$555$	0.339746	0.0144214
$556$	0	0
$557$	3.66025	0.155090	0.0775450	−	0.996989i	$-0.475292\pi$
$557$	3.66025	0.155090	0.0775450	+	0.996989i	$0.475292\pi$
$558$	0	0
$559$	−81.9808	−3.46742
$560$	0	0
$561$	−5.00000	−0.211100
$562$	0	0
$563$	−2.39230	−0.100824	−0.0504118	−	0.998729i	$-0.516053\pi$
$563$	−2.39230	−0.100824	−0.0504118	+	0.998729i	$0.516053\pi$
$564$	0	0
$565$	0.908965	0.0382405
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−38.2487	−1.60347	−0.801735	−	0.597680i	$-0.796089\pi$
$569$	−38.2487	−1.60347	−0.801735	+	0.597680i	$0.796089\pi$
$570$	0	0
$571$	−23.7128	−0.992350	−0.496175	−	0.868222i	$-0.665263\pi$
$571$	−23.7128	−0.992350	−0.496175	+	0.868222i	$0.665263\pi$
$572$	0	0
$573$	16.9282	0.707186
$574$	0	0
$575$	−14.7846	−0.616561
$576$	0	0
$577$	22.1244	0.921049	0.460524	−	0.887647i	$-0.347661\pi$
$577$	22.1244	0.921049	0.460524	+	0.887647i	$0.347661\pi$
$578$	0	0
$579$	−24.0526	−0.999590
$580$	0	0
$581$	−0.928203	−0.0385084
$582$	0	0
$583$	58.3013	2.41459
$584$	0	0
$585$	1.92820	0.0797214
$586$	0	0
$587$	−17.5167	−0.722990	−0.361495	−	0.932374i	$-0.617734\pi$
$587$	−17.5167	−0.722990	−0.361495	+	0.932374i	$0.617734\pi$
$588$	0	0
$589$	−4.05256	−0.166983
$590$	0	0
$591$	11.0000	0.452480
$592$	0	0
$593$	20.8756	0.857260	0.428630	−	0.903480i	$-0.358996\pi$
$593$	20.8756	0.857260	0.428630	+	0.903480i	$0.358996\pi$
$594$	0	0
$595$	−0.267949	−0.0109848
$596$	0	0
$597$	−4.73205	−0.193670
$598$	0	0
$599$	−40.6410	−1.66055	−0.830273	−	0.557356i	$-0.811816\pi$
$599$	−40.6410	−1.66055	−0.830273	+	0.557356i	$0.811816\pi$
$600$	0	0
$601$	−26.4449	−1.07871	−0.539354	−	0.842079i	$-0.681332\pi$
$601$	−26.4449	−1.07871	−0.539354	+	0.842079i	$0.681332\pi$
$602$	0	0
$603$	4.53590	0.184716
$604$	0	0
$605$	−3.75129	−0.152512
$606$	0	0
$607$	−29.3731	−1.19222	−0.596108	−	0.802904i	$-0.703287\pi$
$607$	−29.3731	−1.19222	−0.596108	+	0.802904i	$0.703287\pi$
$608$	0	0
$609$	−4.00000	−0.162088
$610$	0	0
$611$	9.12436	0.369132
$612$	0	0
$613$	−10.0718	−0.406796	−0.203398	−	0.979096i	$-0.565199\pi$
$613$	−10.0718	−0.406796	−0.203398	+	0.979096i	$0.565199\pi$
$614$	0	0
$615$	−2.21539	−0.0893332
$616$	0	0
$617$	−18.9282	−0.762021	−0.381010	−	0.924571i	$-0.624424\pi$
$617$	−18.9282	−0.762021	−0.381010	+	0.924571i	$0.624424\pi$
$618$	0	0
$619$	31.9090	1.28253	0.641265	−	0.767320i	$-0.278410\pi$
$619$	31.9090	1.28253	0.641265	+	0.767320i	$0.278410\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	0	0
$623$	8.39230	0.336231
$624$	0	0
$625$	23.9282	0.957128
$626$	0	0
$627$	15.9808	0.638210
$628$	0	0
$629$	1.26795	0.0505564
$630$	0	0
$631$	−17.9282	−0.713711	−0.356855	−	0.934160i	$-0.616151\pi$
$631$	−17.9282	−0.713711	−0.356855	+	0.934160i	$0.616151\pi$
$632$	0	0
$633$	22.5885	0.897811
$634$	0	0
$635$	−1.98076	−0.0786041
$636$	0	0
$637$	−7.19615	−0.285122
$638$	0	0
$639$	−3.46410	−0.137038
$640$	0	0
$641$	−10.8564	−0.428802	−0.214401	−	0.976746i	$-0.568780\pi$
$641$	−10.8564	−0.428802	−0.214401	+	0.976746i	$0.568780\pi$
$642$	0	0
$643$	41.8564	1.65066	0.825328	−	0.564654i	$-0.190990\pi$
$643$	41.8564	1.65066	0.825328	+	0.564654i	$0.190990\pi$
$644$	0	0
$645$	3.05256	0.120194
$646$	0	0
$647$	−30.3923	−1.19484	−0.597422	−	0.801927i	$-0.703808\pi$
$647$	−30.3923	−1.19484	−0.597422	+	0.801927i	$0.703808\pi$
$648$	0	0
$649$	40.9808	1.60864
$650$	0	0
$651$	−1.26795	−0.0496948
$652$	0	0
$653$	−13.3923	−0.524081	−0.262041	−	0.965057i	$-0.584395\pi$
$653$	−13.3923	−0.524081	−0.262041	+	0.965057i	$0.584395\pi$
$654$	0	0
$655$	−0.751289	−0.0293553
$656$	0	0
$657$	3.46410	0.135147
$658$	0	0
$659$	−4.73205	−0.184335	−0.0921673	−	0.995744i	$-0.529379\pi$
$659$	−4.73205	−0.184335	−0.0921673	+	0.995744i	$0.529379\pi$
$660$	0	0
$661$	17.9808	0.699371	0.349685	−	0.936867i	$-0.386289\pi$
$661$	17.9808	0.699371	0.349685	+	0.936867i	$0.386289\pi$
$662$	0	0
$663$	7.19615	0.279475
$664$	0	0
$665$	0.856406	0.0332100
$666$	0	0
$667$	12.0000	0.464642
$668$	0	0
$669$	−14.1244	−0.546079
$670$	0	0
$671$	−16.3397	−0.630789
$672$	0	0
$673$	−8.92820	−0.344157	−0.172078	−	0.985083i	$-0.555048\pi$
$673$	−8.92820	−0.344157	−0.172078	+	0.985083i	$0.555048\pi$
$674$	0	0
$675$	4.92820	0.189687
$676$	0	0
$677$	−11.8756	−0.456418	−0.228209	−	0.973612i	$-0.573287\pi$
$677$	−11.8756	−0.456418	−0.228209	+	0.973612i	$0.573287\pi$
$678$	0	0
$679$	−13.8564	−0.531760
$680$	0	0
$681$	−19.0526	−0.730096
$682$	0	0
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	0	0
$685$	0.588457	0.0224838
$686$	0	0
$687$	5.85641	0.223436
$688$	0	0
$689$	−83.9090	−3.19668
$690$	0	0
$691$	39.6603	1.50875	0.754374	−	0.656445i	$-0.227941\pi$
$691$	39.6603	1.50875	0.754374	+	0.656445i	$0.227941\pi$
$692$	0	0
$693$	5.00000	0.189934
$694$	0	0
$695$	−4.14359	−0.157175
$696$	0	0
$697$	−8.26795	−0.313171
$698$	0	0
$699$	−15.9282	−0.602460
$700$	0	0
$701$	7.60770	0.287339	0.143669	−	0.989626i	$-0.454110\pi$
$701$	7.60770	0.287339	0.143669	+	0.989626i	$0.454110\pi$
$702$	0	0
$703$	−4.05256	−0.152845
$704$	0	0
$705$	−0.339746	−0.0127956
$706$	0	0
$707$	0.732051	0.0275316
$708$	0	0
$709$	−45.7128	−1.71678	−0.858390	−	0.512997i	$-0.828535\pi$
$709$	−45.7128	−1.71678	−0.858390	+	0.512997i	$0.828535\pi$
$710$	0	0
$711$	−10.1962	−0.382386
$712$	0	0
$713$	3.80385	0.142455
$714$	0	0
$715$	9.64102	0.360554
$716$	0	0
$717$	−19.5167	−0.728863
$718$	0	0
$719$	42.1244	1.57097	0.785487	−	0.618879i	$-0.212413\pi$
$719$	42.1244	1.57097	0.785487	+	0.618879i	$0.212413\pi$
$720$	0	0
$721$	−7.19615	−0.267999
$722$	0	0
$723$	−14.9282	−0.555186
$724$	0	0
$725$	−19.7128	−0.732115
$726$	0	0
$727$	−12.3923	−0.459605	−0.229803	−	0.973237i	$-0.573808\pi$
$727$	−12.3923	−0.459605	−0.229803	+	0.973237i	$0.573808\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.3923	0.421360
$732$	0	0
$733$	43.1769	1.59478	0.797388	−	0.603467i	$-0.206215\pi$
$733$	43.1769	1.59478	0.797388	+	0.603467i	$0.206215\pi$
$734$	0	0
$735$	0.267949	0.00988345
$736$	0	0
$737$	22.6795	0.835410
$738$	0	0
$739$	−31.3923	−1.15478	−0.577392	−	0.816467i	$-0.695930\pi$
$739$	−31.3923	−1.15478	−0.577392	+	0.816467i	$0.695930\pi$
$740$	0	0
$741$	−23.0000	−0.844926
$742$	0	0
$743$	43.8564	1.60894	0.804468	−	0.593996i	$-0.202451\pi$
$743$	43.8564	1.60894	0.804468	+	0.593996i	$0.202451\pi$
$744$	0	0
$745$	−0.928203	−0.0340067
$746$	0	0
$747$	−0.928203	−0.0339612
$748$	0	0
$749$	11.0000	0.401931
$750$	0	0
$751$	22.1962	0.809949	0.404975	−	0.914328i	$-0.367280\pi$
$751$	22.1962	0.809949	0.404975	+	0.914328i	$0.367280\pi$
$752$	0	0
$753$	−29.1244	−1.06135
$754$	0	0
$755$	0.535898	0.0195033
$756$	0	0
$757$	−22.4641	−0.816472	−0.408236	−	0.912877i	$-0.633856\pi$
$757$	−22.4641	−0.816472	−0.408236	+	0.912877i	$0.633856\pi$
$758$	0	0
$759$	−15.0000	−0.544466
$760$	0	0
$761$	48.9282	1.77365	0.886823	−	0.462109i	$-0.152907\pi$
$761$	48.9282	1.77365	0.886823	+	0.462109i	$0.152907\pi$
$762$	0	0
$763$	−5.66025	−0.204915
$764$	0	0
$765$	−0.267949	−0.00968772
$766$	0	0
$767$	−58.9808	−2.12967
$768$	0	0
$769$	26.1244	0.942068	0.471034	−	0.882115i	$-0.343881\pi$
$769$	26.1244	0.942068	0.471034	+	0.882115i	$0.343881\pi$
$770$	0	0
$771$	3.66025	0.131821
$772$	0	0
$773$	10.1962	0.366730	0.183365	−	0.983045i	$-0.441301\pi$
$773$	10.1962	0.366730	0.183365	+	0.983045i	$0.441301\pi$
$774$	0	0
$775$	−6.24871	−0.224460
$776$	0	0
$777$	−1.26795	−0.0454874
$778$	0	0
$779$	26.4256	0.946796
$780$	0	0
$781$	−17.3205	−0.619777
$782$	0	0
$783$	−4.00000	−0.142948
$784$	0	0
$785$	−5.24871	−0.187335
$786$	0	0
$787$	−33.9090	−1.20872	−0.604362	−	0.796710i	$-0.706572\pi$
$787$	−33.9090	−1.20872	−0.604362	+	0.796710i	$0.706572\pi$
$788$	0	0
$789$	13.4641	0.479335
$790$	0	0
$791$	−3.39230	−0.120616
$792$	0	0
$793$	23.5167	0.835101
$794$	0	0
$795$	3.12436	0.110809
$796$	0	0
$797$	5.85641	0.207445	0.103722	−	0.994606i	$-0.466925\pi$
$797$	5.85641	0.207445	0.103722	+	0.994606i	$0.466925\pi$
$798$	0	0
$799$	−1.26795	−0.0448568
$800$	0	0
$801$	8.39230	0.296528
$802$	0	0
$803$	17.3205	0.611227
$804$	0	0
$805$	−0.803848	−0.0283319
$806$	0	0
$807$	23.1962	0.816543
$808$	0	0
$809$	26.3205	0.925380	0.462690	−	0.886520i	$-0.346884\pi$
$809$	26.3205	0.925380	0.462690	+	0.886520i	$0.346884\pi$
$810$	0	0
$811$	44.6410	1.56756	0.783779	−	0.621040i	$-0.213289\pi$
$811$	44.6410	1.56756	0.783779	+	0.621040i	$0.213289\pi$
$812$	0	0
$813$	−15.7321	−0.551747
$814$	0	0
$815$	−1.89488	−0.0663748
$816$	0	0
$817$	−36.4115	−1.27388
$818$	0	0
$819$	−7.19615	−0.251454
$820$	0	0
$821$	34.3205	1.19779	0.598897	−	0.800826i	$-0.295606\pi$
$821$	34.3205	1.19779	0.598897	+	0.800826i	$0.295606\pi$
$822$	0	0
$823$	−28.9808	−1.01021	−0.505103	−	0.863059i	$-0.668545\pi$
$823$	−28.9808	−1.01021	−0.505103	+	0.863059i	$0.668545\pi$
$824$	0	0
$825$	24.6410	0.857890
$826$	0	0
$827$	21.2487	0.738890	0.369445	−	0.929253i	$-0.379548\pi$
$827$	21.2487	0.738890	0.369445	+	0.929253i	$0.379548\pi$
$828$	0	0
$829$	−51.5692	−1.79107	−0.895537	−	0.444988i	$-0.853208\pi$
$829$	−51.5692	−1.79107	−0.895537	+	0.444988i	$0.853208\pi$
$830$	0	0
$831$	6.39230	0.221747
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	1.28719	0.0445449
$836$	0	0
$837$	−1.26795	−0.0438267
$838$	0	0
$839$	−42.9090	−1.48138	−0.740691	−	0.671846i	$-0.765502\pi$
$839$	−42.9090	−1.48138	−0.740691	+	0.671846i	$0.765502\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	11.6077	0.399790
$844$	0	0
$845$	−10.3923	−0.357506
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	−30.7321	−1.05472
$850$	0	0
$851$	3.80385	0.130394
$852$	0	0
$853$	8.58846	0.294063	0.147032	−	0.989132i	$-0.453028\pi$
$853$	8.58846	0.294063	0.147032	+	0.989132i	$0.453028\pi$
$854$	0	0
$855$	0.856406	0.0292885
$856$	0	0
$857$	−37.3205	−1.27484	−0.637422	−	0.770515i	$-0.719999\pi$
$857$	−37.3205	−1.27484	−0.637422	+	0.770515i	$0.719999\pi$
$858$	0	0
$859$	22.2487	0.759116	0.379558	−	0.925168i	$-0.376076\pi$
$859$	22.2487	0.759116	0.379558	+	0.925168i	$0.376076\pi$
$860$	0	0
$861$	8.26795	0.281771
$862$	0	0
$863$	−21.3731	−0.727548	−0.363774	−	0.931487i	$-0.618512\pi$
$863$	−21.3731	−0.727548	−0.363774	+	0.931487i	$0.618512\pi$
$864$	0	0
$865$	0.856406	0.0291187
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−50.9808	−1.72940
$870$	0	0
$871$	−32.6410	−1.10600
$872$	0	0
$873$	−13.8564	−0.468968
$874$	0	0
$875$	2.66025	0.0899330
$876$	0	0
$877$	−36.7321	−1.24035	−0.620177	−	0.784462i	$-0.712939\pi$
$877$	−36.7321	−1.24035	−0.620177	+	0.784462i	$0.712939\pi$
$878$	0	0
$879$	−14.9282	−0.503516
$880$	0	0
$881$	10.3923	0.350126	0.175063	−	0.984557i	$-0.443987\pi$
$881$	10.3923	0.350126	0.175063	+	0.984557i	$0.443987\pi$
$882$	0	0
$883$	−26.4641	−0.890588	−0.445294	−	0.895384i	$-0.646901\pi$
$883$	−26.4641	−0.890588	−0.445294	+	0.895384i	$0.646901\pi$
$884$	0	0
$885$	2.19615	0.0738229
$886$	0	0
$887$	28.1244	0.944323	0.472162	−	0.881512i	$-0.343474\pi$
$887$	28.1244	0.944323	0.472162	+	0.881512i	$0.343474\pi$
$888$	0	0
$889$	7.39230	0.247930
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	4.05256	0.135614
$894$	0	0
$895$	3.07180	0.102679
$896$	0	0
$897$	21.5885	0.720818
$898$	0	0
$899$	5.07180	0.169154
$900$	0	0
$901$	11.6603	0.388459
$902$	0	0
$903$	−11.3923	−0.379112
$904$	0	0
$905$	−2.49742	−0.0830171
$906$	0	0
$907$	37.0333	1.22967	0.614836	−	0.788655i	$-0.289222\pi$
$907$	37.0333	1.22967	0.614836	+	0.788655i	$0.289222\pi$
$908$	0	0
$909$	0.732051	0.0242806
$910$	0	0
$911$	−9.24871	−0.306423	−0.153212	−	0.988193i	$-0.548962\pi$
$911$	−9.24871	−0.306423	−0.153212	+	0.988193i	$0.548962\pi$
$912$	0	0
$913$	−4.64102	−0.153595
$914$	0	0
$915$	−0.875644	−0.0289479
$916$	0	0
$917$	2.80385	0.0925912
$918$	0	0
$919$	13.6795	0.451245	0.225622	−	0.974215i	$-0.427558\pi$
$919$	13.6795	0.451245	0.225622	+	0.974215i	$0.427558\pi$
$920$	0	0
$921$	22.9282	0.755510
$922$	0	0
$923$	24.9282	0.820522
$924$	0	0
$925$	−6.24871	−0.205456
$926$	0	0
$927$	−7.19615	−0.236353
$928$	0	0
$929$	39.9808	1.31173	0.655863	−	0.754880i	$-0.272305\pi$
$929$	39.9808	1.31173	0.655863	+	0.754880i	$0.272305\pi$
$930$	0	0
$931$	−3.19615	−0.104750
$932$	0	0
$933$	−27.7128	−0.907277
$934$	0	0
$935$	−1.33975	−0.0438144
$936$	0	0
$937$	−3.21539	−0.105042	−0.0525211	−	0.998620i	$-0.516726\pi$
$937$	−3.21539	−0.105042	−0.0525211	+	0.998620i	$0.516726\pi$
$938$	0	0
$939$	−5.12436	−0.167227
$940$	0	0
$941$	39.8564	1.29928	0.649641	−	0.760241i	$-0.274919\pi$
$941$	39.8564	1.29928	0.649641	+	0.760241i	$0.274919\pi$
$942$	0	0
$943$	−24.8038	−0.807724
$944$	0	0
$945$	0.267949	0.00871639
$946$	0	0
$947$	−12.1436	−0.394614	−0.197307	−	0.980342i	$-0.563220\pi$
$947$	−12.1436	−0.394614	−0.197307	+	0.980342i	$0.563220\pi$
$948$	0	0
$949$	−24.9282	−0.809204
$950$	0	0
$951$	−24.9282	−0.808352
$952$	0	0
$953$	18.7321	0.606791	0.303395	−	0.952865i	$-0.401880\pi$
$953$	18.7321	0.606791	0.303395	+	0.952865i	$0.401880\pi$
$954$	0	0
$955$	4.53590	0.146778
$956$	0	0
$957$	−20.0000	−0.646508
$958$	0	0
$959$	−2.19615	−0.0709175
$960$	0	0
$961$	−29.3923	−0.948139
$962$	0	0
$963$	11.0000	0.354470
$964$	0	0
$965$	−6.44486	−0.207468
$966$	0	0
$967$	−3.78461	−0.121705	−0.0608524	−	0.998147i	$-0.519382\pi$
$967$	−3.78461	−0.121705	−0.0608524	+	0.998147i	$0.519382\pi$
$968$	0	0
$969$	3.19615	0.102675
$970$	0	0
$971$	−23.0718	−0.740409	−0.370205	−	0.928950i	$-0.620712\pi$
$971$	−23.0718	−0.740409	−0.370205	+	0.928950i	$0.620712\pi$
$972$	0	0
$973$	15.4641	0.495756
$974$	0	0
$975$	−35.4641	−1.13576
$976$	0	0
$977$	38.1051	1.21909	0.609545	−	0.792751i	$-0.291352\pi$
$977$	38.1051	1.21909	0.609545	+	0.792751i	$0.291352\pi$
$978$	0	0
$979$	41.9615	1.34110
$980$	0	0
$981$	−5.66025	−0.180718
$982$	0	0
$983$	−18.8038	−0.599750	−0.299875	−	0.953979i	$-0.596945\pi$
$983$	−18.8038	−0.599750	−0.299875	+	0.953979i	$0.596945\pi$
$984$	0	0
$985$	2.94744	0.0939133
$986$	0	0
$987$	1.26795	0.0403593
$988$	0	0
$989$	34.1769	1.08676
$990$	0	0
$991$	−11.2154	−0.356269	−0.178134	−	0.984006i	$-0.557006\pi$
$991$	−11.2154	−0.356269	−0.178134	+	0.984006i	$0.557006\pi$
$992$	0	0
$993$	−1.92820	−0.0611897
$994$	0	0
$995$	−1.26795	−0.0401967
$996$	0	0
$997$	−8.92820	−0.282759	−0.141380	−	0.989955i	$-0.545154\pi$
$997$	−8.92820	−0.282759	−0.141380	+	0.989955i	$0.545154\pi$
$998$	0	0
$999$	−1.26795	−0.0401161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5712.2.a.bc.1.2		2
4.3	odd	2		357.2.a.e.1.1	✓	2
12.11	even	2		1071.2.a.f.1.2		2
20.19	odd	2		8925.2.a.bl.1.2		2
28.27	even	2		2499.2.a.n.1.1		2
68.67	odd	2		6069.2.a.f.1.1		2
84.83	odd	2		7497.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.a.e.1.1	✓	2	4.3	odd	2
1071.2.a.f.1.2		2	12.11	even	2
2499.2.a.n.1.1		2	28.27	even	2
5712.2.a.bc.1.2		2	1.1	even	1	trivial
6069.2.a.f.1.1		2	68.67	odd	2
7497.2.a.y.1.2		2	84.83	odd	2
8925.2.a.bl.1.2		2	20.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5712.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.267949 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -7.19615 q^{13} +0.267949 q^{15} +1.00000 q^{17} -3.19615 q^{19} -1.00000 q^{21} +3.00000 q^{23} -4.92820 q^{25} -1.00000 q^{27} +4.00000 q^{29} +1.26795 q^{31} -5.00000 q^{33} -0.267949 q^{35} +1.26795 q^{37} +7.19615 q^{39} -8.26795 q^{41} +11.3923 q^{43} -0.267949 q^{45} -1.26795 q^{47} +1.00000 q^{49} -1.00000 q^{51} +11.6603 q^{53} -1.33975 q^{55} +3.19615 q^{57} +8.19615 q^{59} -3.26795 q^{61} +1.00000 q^{63} +1.92820 q^{65} +4.53590 q^{67} -3.00000 q^{69} -3.46410 q^{71} +3.46410 q^{73} +4.92820 q^{75} +5.00000 q^{77} -10.1962 q^{79} +1.00000 q^{81} -0.928203 q^{83} -0.267949 q^{85} -4.00000 q^{87} +8.39230 q^{89} -7.19615 q^{91} -1.26795 q^{93} +0.856406 q^{95} -13.8564 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{7} + 2 q^{9} + 10 q^{11} - 4 q^{13} + 4 q^{15} + 2 q^{17} + 4 q^{19} - 2 q^{21} + 6 q^{23} + 4 q^{25} - 2 q^{27} + 8 q^{29} + 6 q^{31} - 10 q^{33} - 4 q^{35} + 6 q^{37} + 4 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^7 + 2 * q^9 + 10 * q^11 - 4 * q^13 + 4 * q^15 + 2 * q^17 + 4 * q^19 - 2 * q^21 + 6 * q^23 + 4 * q^25 - 2 * q^27 + 8 * q^29 + 6 * q^31 - 10 * q^33 - 4 * q^35 + 6 * q^37 + 4 * q^39 - 20 * q^41 + 2 * q^43 - 4 * q^45 - 6 * q^47 + 2 * q^49 - 2 * q^51 + 6 * q^53 - 20 * q^55 - 4 * q^57 + 6 * q^59 - 10 * q^61 + 2 * q^63 - 10 * q^65 + 16 * q^67 - 6 * q^69 - 4 * q^75 + 10 * q^77 - 10 * q^79 + 2 * q^81 + 12 * q^83 - 4 * q^85 - 8 * q^87 - 4 * q^89 - 4 * q^91 - 6 * q^93 - 26 * q^95 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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