LMFDB - Embedded newform 6400.2.a.cj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6400,2,Mod(1,6400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6400, 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("6400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6400 = 2^{8} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6400.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:	$51.1042572936$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	6400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} +2.73205 q^{7} -2.46410 q^{9} +O(q^{10})$	$q-0.732051 q^{3} +2.73205 q^{7} -2.46410 q^{9} -2.00000 q^{11} -3.46410 q^{13} +3.46410 q^{17} -7.46410 q^{19} -2.00000 q^{21} -4.19615 q^{23} +4.00000 q^{27} -6.92820 q^{29} -1.46410 q^{31} +1.46410 q^{33} -2.00000 q^{37} +2.53590 q^{39} +5.46410 q^{41} +8.73205 q^{43} +6.73205 q^{47} +0.464102 q^{49} -2.53590 q^{51} +4.53590 q^{53} +5.46410 q^{57} -0.535898 q^{59} +4.92820 q^{61} -6.73205 q^{63} +7.26795 q^{67} +3.07180 q^{69} -1.46410 q^{71} +0.535898 q^{73} -5.46410 q^{77} +14.9282 q^{79} +4.46410 q^{81} +4.73205 q^{83} +5.07180 q^{87} +4.92820 q^{89} -9.46410 q^{91} +1.07180 q^{93} -6.39230 q^{97} +4.92820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{11} - 8 q^{19} - 4 q^{21} + 2 q^{23} + 8 q^{27} + 4 q^{31} - 4 q^{33} - 4 q^{37} + 12 q^{39} + 4 q^{41} + 14 q^{43} + 10 q^{47} - 6 q^{49} - 12 q^{51} + 16 q^{53} + 4 q^{57} - 8 q^{59} - 4 q^{61} - 10 q^{63} + 18 q^{67} + 20 q^{69} + 4 q^{71} + 8 q^{73} - 4 q^{77} + 16 q^{79} + 2 q^{81} + 6 q^{83} + 24 q^{87} - 4 q^{89} - 12 q^{91} + 16 q^{93} + 8 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 - 4 * q^11 - 8 * q^19 - 4 * q^21 + 2 * q^23 + 8 * q^27 + 4 * q^31 - 4 * q^33 - 4 * q^37 + 12 * q^39 + 4 * q^41 + 14 * q^43 + 10 * q^47 - 6 * q^49 - 12 * q^51 + 16 * q^53 + 4 * q^57 - 8 * q^59 - 4 * q^61 - 10 * q^63 + 18 * q^67 + 20 * q^69 + 4 * q^71 + 8 * q^73 - 4 * q^77 + 16 * q^79 + 2 * q^81 + 6 * q^83 + 24 * q^87 - 4 * q^89 - 12 * q^91 + 16 * q^93 + 8 * q^97 - 4 * 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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.73205	1.03262	0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205	1.03262	0.516309	+	0.856402i	$0.327306\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	−7.46410	−1.71238	−0.856191	−	0.516659i	$-0.827175\pi$
$19$	−7.46410	−1.71238	−0.856191	+	0.516659i	$0.827175\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−4.19615	−0.874958	−0.437479	−	0.899229i	$-0.644129\pi$
$23$	−4.19615	−0.874958	−0.437479	+	0.899229i	$0.644129\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	0	0
$33$	1.46410	0.254867
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	2.53590	0.406069
$40$	0	0
$41$	5.46410	0.853349	0.426675	−	0.904405i	$-0.359685\pi$
$41$	5.46410	0.853349	0.426675	+	0.904405i	$0.359685\pi$
$42$	0	0
$43$	8.73205	1.33163	0.665813	−	0.746119i	$-0.268085\pi$
$43$	8.73205	1.33163	0.665813	+	0.746119i	$0.268085\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.73205	0.981971	0.490985	−	0.871168i	$-0.336637\pi$
$47$	6.73205	0.981971	0.490985	+	0.871168i	$0.336637\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	−2.53590	−0.355097
$52$	0	0
$53$	4.53590	0.623054	0.311527	−	0.950237i	$-0.399160\pi$
$53$	4.53590	0.623054	0.311527	+	0.950237i	$0.399160\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.46410	0.723738
$58$	0	0
$59$	−0.535898	−0.0697680	−0.0348840	−	0.999391i	$-0.511106\pi$
$59$	−0.535898	−0.0697680	−0.0348840	+	0.999391i	$0.511106\pi$
$60$	0	0
$61$	4.92820	0.630992	0.315496	−	0.948927i	$-0.397829\pi$
$61$	4.92820	0.630992	0.315496	+	0.948927i	$0.397829\pi$
$62$	0	0
$63$	−6.73205	−0.848159
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.26795	0.887921	0.443961	−	0.896046i	$-0.353573\pi$
$67$	7.26795	0.887921	0.443961	+	0.896046i	$0.353573\pi$
$68$	0	0
$69$	3.07180	0.369801
$70$	0	0
$71$	−1.46410	−0.173757	−0.0868784	−	0.996219i	$-0.527689\pi$
$71$	−1.46410	−0.173757	−0.0868784	+	0.996219i	$0.527689\pi$
$72$	0	0
$73$	0.535898	0.0627222	0.0313611	−	0.999508i	$-0.490016\pi$
$73$	0.535898	0.0627222	0.0313611	+	0.999508i	$0.490016\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.46410	−0.622692
$78$	0	0
$79$	14.9282	1.67955	0.839777	−	0.542931i	$-0.182686\pi$
$79$	14.9282	1.67955	0.839777	+	0.542931i	$0.182686\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	4.73205	0.519410	0.259705	−	0.965688i	$-0.416375\pi$
$83$	4.73205	0.519410	0.259705	+	0.965688i	$0.416375\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.07180	0.543754
$88$	0	0
$89$	4.92820	0.522388	0.261194	−	0.965286i	$-0.415884\pi$
$89$	4.92820	0.522388	0.261194	+	0.965286i	$0.415884\pi$
$90$	0	0
$91$	−9.46410	−0.992107
$92$	0	0
$93$	1.07180	0.111140
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.39230	−0.649040	−0.324520	−	0.945879i	$-0.605203\pi$
$97$	−6.39230	−0.649040	−0.324520	+	0.945879i	$0.605203\pi$
$98$	0	0
$99$	4.92820	0.495303
$100$	0	0
$101$	10.9282	1.08740	0.543698	−	0.839281i	$-0.317024\pi$
$101$	10.9282	1.08740	0.543698	+	0.839281i	$0.317024\pi$
$102$	0	0
$103$	1.66025	0.163590	0.0817948	−	0.996649i	$-0.473935\pi$
$103$	1.66025	0.163590	0.0817948	+	0.996649i	$0.473935\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.732051	0.0707700	0.0353850	−	0.999374i	$-0.488734\pi$
$107$	0.732051	0.0707700	0.0353850	+	0.999374i	$0.488734\pi$
$108$	0	0
$109$	3.07180	0.294225	0.147112	−	0.989120i	$-0.453002\pi$
$109$	3.07180	0.294225	0.147112	+	0.989120i	$0.453002\pi$
$110$	0	0
$111$	1.46410	0.138966
$112$	0	0
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.53590	0.789144
$118$	0	0
$119$	9.46410	0.867573
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.2679	1.17734	0.588670	−	0.808373i	$-0.299652\pi$
$127$	13.2679	1.17734	0.588670	+	0.808373i	$0.299652\pi$
$128$	0	0
$129$	−6.39230	−0.562811
$130$	0	0
$131$	−7.85641	−0.686417	−0.343209	−	0.939259i	$-0.611514\pi$
$131$	−7.85641	−0.686417	−0.343209	+	0.939259i	$0.611514\pi$
$132$	0	0
$133$	−20.3923	−1.76824
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.92820	−0.762788	−0.381394	−	0.924413i	$-0.624556\pi$
$137$	−8.92820	−0.762788	−0.381394	+	0.924413i	$0.624556\pi$
$138$	0	0
$139$	−7.46410	−0.633097	−0.316548	−	0.948576i	$-0.602524\pi$
$139$	−7.46410	−0.633097	−0.316548	+	0.948576i	$0.602524\pi$
$140$	0	0
$141$	−4.92820	−0.415030
$142$	0	0
$143$	6.92820	0.579365
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.339746	−0.0280218
$148$	0	0
$149$	−19.8564	−1.62670	−0.813350	−	0.581775i	$-0.802359\pi$
$149$	−19.8564	−1.62670	−0.813350	+	0.581775i	$0.802359\pi$
$150$	0	0
$151$	8.39230	0.682956	0.341478	−	0.939890i	$-0.389073\pi$
$151$	8.39230	0.682956	0.341478	+	0.939890i	$0.389073\pi$
$152$	0	0
$153$	−8.53590	−0.690086
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.9282	1.35102	0.675509	−	0.737352i	$-0.263924\pi$
$157$	16.9282	1.35102	0.675509	+	0.737352i	$0.263924\pi$
$158$	0	0
$159$	−3.32051	−0.263333
$160$	0	0
$161$	−11.4641	−0.903498
$162$	0	0
$163$	10.1962	0.798624	0.399312	−	0.916815i	$-0.369249\pi$
$163$	10.1962	0.798624	0.399312	+	0.916815i	$0.369249\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.1962	1.56283	0.781413	−	0.624015i	$-0.214499\pi$
$167$	20.1962	1.56283	0.781413	+	0.624015i	$0.214499\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	18.3923	1.40649
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.392305	0.0294874
$178$	0	0
$179$	15.4641	1.15584	0.577921	−	0.816093i	$-0.303864\pi$
$179$	15.4641	1.15584	0.577921	+	0.816093i	$0.303864\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	−3.60770	−0.266688
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.92820	−0.506640
$188$	0	0
$189$	10.9282	0.794910
$190$	0	0
$191$	19.3205	1.39798	0.698991	−	0.715130i	$-0.253633\pi$
$191$	19.3205	1.39798	0.698991	+	0.715130i	$0.253633\pi$
$192$	0	0
$193$	−7.46410	−0.537278	−0.268639	−	0.963241i	$-0.586574\pi$
$193$	−7.46410	−0.537278	−0.268639	+	0.963241i	$0.586574\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.5359	−0.893146	−0.446573	−	0.894747i	$-0.647356\pi$
$197$	−12.5359	−0.893146	−0.446573	+	0.894747i	$0.647356\pi$
$198$	0	0
$199$	25.8564	1.83291	0.916456	−	0.400135i	$-0.131037\pi$
$199$	25.8564	1.83291	0.916456	+	0.400135i	$0.131037\pi$
$200$	0	0
$201$	−5.32051	−0.375280
$202$	0	0
$203$	−18.9282	−1.32850
$204$	0	0
$205$	0	0
$206$	0	0
$207$	10.3397	0.718662
$208$	0	0
$209$	14.9282	1.03261
$210$	0	0
$211$	14.7846	1.01781	0.508907	−	0.860821i	$-0.330050\pi$
$211$	14.7846	1.01781	0.508907	+	0.860821i	$0.330050\pi$
$212$	0	0
$213$	1.07180	0.0734383
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−0.392305	−0.0265095
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−16.1962	−1.08457	−0.542287	−	0.840193i	$-0.682442\pi$
$223$	−16.1962	−1.08457	−0.542287	+	0.840193i	$0.682442\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−28.0526	−1.86191	−0.930957	−	0.365129i	$-0.881025\pi$
$227$	−28.0526	−1.86191	−0.930957	+	0.365129i	$0.881025\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	29.3205	1.92085	0.960425	−	0.278538i	$-0.0898499\pi$
$233$	29.3205	1.92085	0.960425	+	0.278538i	$0.0898499\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.9282	−0.709863
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−4.39230	−0.282933	−0.141467	−	0.989943i	$-0.545182\pi$
$241$	−4.39230	−0.282933	−0.141467	+	0.989943i	$0.545182\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	0	0
$246$	0	0
$247$	25.8564	1.64520
$248$	0	0
$249$	−3.46410	−0.219529
$250$	0	0
$251$	11.0718	0.698846	0.349423	−	0.936965i	$-0.386378\pi$
$251$	11.0718	0.698846	0.349423	+	0.936965i	$0.386378\pi$
$252$	0	0
$253$	8.39230	0.527620
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−5.46410	−0.339523
$260$	0	0
$261$	17.0718	1.05672
$262$	0	0
$263$	−5.66025	−0.349026	−0.174513	−	0.984655i	$-0.555835\pi$
$263$	−5.66025	−0.349026	−0.174513	+	0.984655i	$0.555835\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.60770	−0.220787
$268$	0	0
$269$	4.92820	0.300478	0.150239	−	0.988650i	$-0.451996\pi$
$269$	4.92820	0.300478	0.150239	+	0.988650i	$0.451996\pi$
$270$	0	0
$271$	−15.3205	−0.930655	−0.465327	−	0.885139i	$-0.654063\pi$
$271$	−15.3205	−0.930655	−0.465327	+	0.885139i	$0.654063\pi$
$272$	0	0
$273$	6.92820	0.419314
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	3.60770	0.215987
$280$	0	0
$281$	−17.4641	−1.04182	−0.520910	−	0.853611i	$-0.674407\pi$
$281$	−17.4641	−1.04182	−0.520910	+	0.853611i	$0.674407\pi$
$282$	0	0
$283$	−7.66025	−0.455355	−0.227677	−	0.973737i	$-0.573113\pi$
$283$	−7.66025	−0.455355	−0.227677	+	0.973737i	$0.573113\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.9282	0.881184
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	4.67949	0.274317
$292$	0	0
$293$	11.8564	0.692659	0.346329	−	0.938113i	$-0.387428\pi$
$293$	11.8564	0.692659	0.346329	+	0.938113i	$0.387428\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	14.5359	0.840633
$300$	0	0
$301$	23.8564	1.37506
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.9808	1.53987	0.769937	−	0.638120i	$-0.220288\pi$
$307$	26.9808	1.53987	0.769937	+	0.638120i	$0.220288\pi$
$308$	0	0
$309$	−1.21539	−0.0691411
$310$	0	0
$311$	−3.32051	−0.188289	−0.0941444	−	0.995559i	$-0.530012\pi$
$311$	−3.32051	−0.188289	−0.0941444	+	0.995559i	$0.530012\pi$
$312$	0	0
$313$	−31.8564	−1.80063	−0.900315	−	0.435238i	$-0.856664\pi$
$313$	−31.8564	−1.80063	−0.900315	+	0.435238i	$0.856664\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.4641	−0.868550	−0.434275	−	0.900780i	$-0.642995\pi$
$317$	−15.4641	−0.868550	−0.434275	+	0.900780i	$0.642995\pi$
$318$	0	0
$319$	13.8564	0.775810
$320$	0	0
$321$	−0.535898	−0.0299109
$322$	0	0
$323$	−25.8564	−1.43869
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.24871	−0.124354
$328$	0	0
$329$	18.3923	1.01400
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	0	0
$333$	4.92820	0.270064
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.85641	−0.427966	−0.213983	−	0.976837i	$-0.568644\pi$
$337$	−7.85641	−0.427966	−0.213983	+	0.976837i	$0.568644\pi$
$338$	0	0
$339$	0.679492	0.0369049
$340$	0	0
$341$	2.92820	0.158571
$342$	0	0
$343$	−17.8564	−0.964155
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.6603	−0.840686	−0.420343	−	0.907365i	$-0.638090\pi$
$347$	−15.6603	−0.840686	−0.420343	+	0.907365i	$0.638090\pi$
$348$	0	0
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	−13.8564	−0.739600
$352$	0	0
$353$	0.928203	0.0494033	0.0247016	−	0.999695i	$-0.492136\pi$
$353$	0.928203	0.0494033	0.0247016	+	0.999695i	$0.492136\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.92820	−0.366679
$358$	0	0
$359$	5.07180	0.267679	0.133840	−	0.991003i	$-0.457269\pi$
$359$	5.07180	0.267679	0.133840	+	0.991003i	$0.457269\pi$
$360$	0	0
$361$	36.7128	1.93225
$362$	0	0
$363$	5.12436	0.268959
$364$	0	0
$365$	0	0
$366$	0	0
$367$	27.1244	1.41588	0.707940	−	0.706273i	$-0.249625\pi$
$367$	27.1244	1.41588	0.707940	+	0.706273i	$0.249625\pi$
$368$	0	0
$369$	−13.4641	−0.700913
$370$	0	0
$371$	12.3923	0.643376
$372$	0	0
$373$	29.7128	1.53847	0.769236	−	0.638965i	$-0.220637\pi$
$373$	29.7128	1.53847	0.769236	+	0.638965i	$0.220637\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	12.2487	0.629174	0.314587	−	0.949229i	$-0.398134\pi$
$379$	12.2487	0.629174	0.314587	+	0.949229i	$0.398134\pi$
$380$	0	0
$381$	−9.71281	−0.497602
$382$	0	0
$383$	−3.12436	−0.159647	−0.0798236	−	0.996809i	$-0.525436\pi$
$383$	−3.12436	−0.159647	−0.0798236	+	0.996809i	$0.525436\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−21.5167	−1.09375
$388$	0	0
$389$	34.7846	1.76365	0.881825	−	0.471577i	$-0.156315\pi$
$389$	34.7846	1.76365	0.881825	+	0.471577i	$0.156315\pi$
$390$	0	0
$391$	−14.5359	−0.735112
$392$	0	0
$393$	5.75129	0.290114
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.2487	0.815499	0.407750	−	0.913094i	$-0.366314\pi$
$397$	16.2487	0.815499	0.407750	+	0.913094i	$0.366314\pi$
$398$	0	0
$399$	14.9282	0.747345
$400$	0	0
$401$	19.8564	0.991582	0.495791	−	0.868442i	$-0.334878\pi$
$401$	19.8564	0.991582	0.495791	+	0.868442i	$0.334878\pi$
$402$	0	0
$403$	5.07180	0.252644
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−23.3205	−1.15312	−0.576562	−	0.817053i	$-0.695606\pi$
$409$	−23.3205	−1.15312	−0.576562	+	0.817053i	$0.695606\pi$
$410$	0	0
$411$	6.53590	0.322392
$412$	0	0
$413$	−1.46410	−0.0720437
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.46410	0.267578
$418$	0	0
$419$	2.39230	0.116872	0.0584359	−	0.998291i	$-0.481389\pi$
$419$	2.39230	0.116872	0.0584359	+	0.998291i	$0.481389\pi$
$420$	0	0
$421$	−27.8564	−1.35764	−0.678819	−	0.734306i	$-0.737508\pi$
$421$	−27.8564	−1.35764	−0.678819	+	0.734306i	$0.737508\pi$
$422$	0	0
$423$	−16.5885	−0.806558
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.4641	0.651574
$428$	0	0
$429$	−5.07180	−0.244869
$430$	0	0
$431$	14.5359	0.700170	0.350085	−	0.936718i	$-0.386153\pi$
$431$	14.5359	0.700170	0.350085	+	0.936718i	$0.386153\pi$
$432$	0	0
$433$	12.5359	0.602437	0.301218	−	0.953555i	$-0.402607\pi$
$433$	12.5359	0.602437	0.301218	+	0.953555i	$0.402607\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	31.3205	1.49826
$438$	0	0
$439$	−0.784610	−0.0374474	−0.0187237	−	0.999825i	$-0.505960\pi$
$439$	−0.784610	−0.0374474	−0.0187237	+	0.999825i	$0.505960\pi$
$440$	0	0
$441$	−1.14359	−0.0544568
$442$	0	0
$443$	30.9808	1.47194	0.735970	−	0.677014i	$-0.236726\pi$
$443$	30.9808	1.47194	0.735970	+	0.677014i	$0.236726\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.5359	0.687524
$448$	0	0
$449$	11.3205	0.534248	0.267124	−	0.963662i	$-0.413927\pi$
$449$	11.3205	0.534248	0.267124	+	0.963662i	$0.413927\pi$
$450$	0	0
$451$	−10.9282	−0.514589
$452$	0	0
$453$	−6.14359	−0.288651
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.7846	0.691595	0.345797	−	0.938309i	$-0.387608\pi$
$457$	14.7846	0.691595	0.345797	+	0.938309i	$0.387608\pi$
$458$	0	0
$459$	13.8564	0.646762
$460$	0	0
$461$	2.92820	0.136380	0.0681900	−	0.997672i	$-0.478278\pi$
$461$	2.92820	0.136380	0.0681900	+	0.997672i	$0.478278\pi$
$462$	0	0
$463$	−14.7321	−0.684656	−0.342328	−	0.939580i	$-0.611215\pi$
$463$	−14.7321	−0.684656	−0.342328	+	0.939580i	$0.611215\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.33975	−0.385917	−0.192959	−	0.981207i	$-0.561808\pi$
$467$	−8.33975	−0.385917	−0.192959	+	0.981207i	$0.561808\pi$
$468$	0	0
$469$	19.8564	0.916884
$470$	0	0
$471$	−12.3923	−0.571007
$472$	0	0
$473$	−17.4641	−0.803000
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−11.1769	−0.511756
$478$	0	0
$479$	21.8564	0.998645	0.499322	−	0.866416i	$-0.333582\pi$
$479$	21.8564	0.998645	0.499322	+	0.866416i	$0.333582\pi$
$480$	0	0
$481$	6.92820	0.315899
$482$	0	0
$483$	8.39230	0.381863
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.5885	1.11421	0.557105	−	0.830442i	$-0.311912\pi$
$487$	24.5885	1.11421	0.557105	+	0.830442i	$0.311912\pi$
$488$	0	0
$489$	−7.46410	−0.337538
$490$	0	0
$491$	−3.07180	−0.138628	−0.0693141	−	0.997595i	$-0.522081\pi$
$491$	−3.07180	−0.138628	−0.0693141	+	0.997595i	$0.522081\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	−24.5359	−1.09838	−0.549189	−	0.835698i	$-0.685063\pi$
$499$	−24.5359	−1.09838	−0.549189	+	0.835698i	$0.685063\pi$
$500$	0	0
$501$	−14.7846	−0.660528
$502$	0	0
$503$	17.6603	0.787432	0.393716	−	0.919232i	$-0.371189\pi$
$503$	17.6603	0.787432	0.393716	+	0.919232i	$0.371189\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.732051	0.0325115
$508$	0	0
$509$	25.8564	1.14607	0.573033	−	0.819533i	$-0.305767\pi$
$509$	25.8564	1.14607	0.573033	+	0.819533i	$0.305767\pi$
$510$	0	0
$511$	1.46410	0.0647680
$512$	0	0
$513$	−29.8564	−1.31819
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−13.4641	−0.592151
$518$	0	0
$519$	1.46410	0.0642669
$520$	0	0
$521$	16.1436	0.707264	0.353632	−	0.935385i	$-0.384947\pi$
$521$	16.1436	0.707264	0.353632	+	0.935385i	$0.384947\pi$
$522$	0	0
$523$	22.1962	0.970570	0.485285	−	0.874356i	$-0.338716\pi$
$523$	22.1962	0.970570	0.485285	+	0.874356i	$0.338716\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.07180	−0.220931
$528$	0	0
$529$	−5.39230	−0.234448
$530$	0	0
$531$	1.32051	0.0573052
$532$	0	0
$533$	−18.9282	−0.819871
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11.3205	−0.488516
$538$	0	0
$539$	−0.928203	−0.0399805
$540$	0	0
$541$	−13.0718	−0.562000	−0.281000	−	0.959708i	$-0.590666\pi$
$541$	−13.0718	−0.562000	−0.281000	+	0.959708i	$0.590666\pi$
$542$	0	0
$543$	11.7128	0.502645
$544$	0	0
$545$	0	0
$546$	0	0
$547$	36.7321	1.57055	0.785275	−	0.619148i	$-0.212522\pi$
$547$	36.7321	1.57055	0.785275	+	0.619148i	$0.212522\pi$
$548$	0	0
$549$	−12.1436	−0.518276
$550$	0	0
$551$	51.7128	2.20304
$552$	0	0
$553$	40.7846	1.73434
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.7846	1.13490	0.567450	−	0.823408i	$-0.307930\pi$
$557$	26.7846	1.13490	0.567450	+	0.823408i	$0.307930\pi$
$558$	0	0
$559$	−30.2487	−1.27938
$560$	0	0
$561$	5.07180	0.214131
$562$	0	0
$563$	16.0526	0.676535	0.338267	−	0.941050i	$-0.390159\pi$
$563$	16.0526	0.676535	0.338267	+	0.941050i	$0.390159\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.1962	0.512190
$568$	0	0
$569$	6.53590	0.273999	0.137000	−	0.990571i	$-0.456254\pi$
$569$	6.53590	0.273999	0.137000	+	0.990571i	$0.456254\pi$
$570$	0	0
$571$	−34.7846	−1.45569	−0.727845	−	0.685741i	$-0.759478\pi$
$571$	−34.7846	−1.45569	−0.727845	+	0.685741i	$0.759478\pi$
$572$	0	0
$573$	−14.1436	−0.590857
$574$	0	0
$575$	0	0
$576$	0	0
$577$	43.5692	1.81381	0.906905	−	0.421335i	$-0.138438\pi$
$577$	43.5692	1.81381	0.906905	+	0.421335i	$0.138438\pi$
$578$	0	0
$579$	5.46410	0.227080
$580$	0	0
$581$	12.9282	0.536352
$582$	0	0
$583$	−9.07180	−0.375715
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.1962	0.585938	0.292969	−	0.956122i	$-0.405357\pi$
$587$	14.1962	0.585938	0.292969	+	0.956122i	$0.405357\pi$
$588$	0	0
$589$	10.9282	0.450289
$590$	0	0
$591$	9.17691	0.377488
$592$	0	0
$593$	36.6410	1.50467	0.752333	−	0.658783i	$-0.228928\pi$
$593$	36.6410	1.50467	0.752333	+	0.658783i	$0.228928\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−18.9282	−0.774680
$598$	0	0
$599$	34.6410	1.41539	0.707697	−	0.706516i	$-0.249734\pi$
$599$	34.6410	1.41539	0.707697	+	0.706516i	$0.249734\pi$
$600$	0	0
$601$	−25.4641	−1.03870	−0.519351	−	0.854561i	$-0.673826\pi$
$601$	−25.4641	−1.03870	−0.519351	+	0.854561i	$0.673826\pi$
$602$	0	0
$603$	−17.9090	−0.729309
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.9808	−0.851583	−0.425791	−	0.904821i	$-0.640004\pi$
$607$	−20.9808	−0.851583	−0.425791	+	0.904821i	$0.640004\pi$
$608$	0	0
$609$	13.8564	0.561490
$610$	0	0
$611$	−23.3205	−0.943447
$612$	0	0
$613$	5.60770	0.226493	0.113246	−	0.993567i	$-0.463875\pi$
$613$	5.60770	0.226493	0.113246	+	0.993567i	$0.463875\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27.4641	−1.10566	−0.552832	−	0.833293i	$-0.686453\pi$
$617$	−27.4641	−1.10566	−0.552832	+	0.833293i	$0.686453\pi$
$618$	0	0
$619$	−33.3205	−1.33926	−0.669632	−	0.742693i	$-0.733548\pi$
$619$	−33.3205	−1.33926	−0.669632	+	0.742693i	$0.733548\pi$
$620$	0	0
$621$	−16.7846	−0.673543
$622$	0	0
$623$	13.4641	0.539428
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−10.9282	−0.436430
$628$	0	0
$629$	−6.92820	−0.276246
$630$	0	0
$631$	11.3205	0.450662	0.225331	−	0.974282i	$-0.427654\pi$
$631$	11.3205	0.450662	0.225331	+	0.974282i	$0.427654\pi$
$632$	0	0
$633$	−10.8231	−0.430179
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.60770	−0.0636992
$638$	0	0
$639$	3.60770	0.142718
$640$	0	0
$641$	−20.3923	−0.805448	−0.402724	−	0.915322i	$-0.631936\pi$
$641$	−20.3923	−0.805448	−0.402724	+	0.915322i	$0.631936\pi$
$642$	0	0
$643$	14.8756	0.586638	0.293319	−	0.956015i	$-0.405240\pi$
$643$	14.8756	0.586638	0.293319	+	0.956015i	$0.405240\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.2679	−0.521617	−0.260808	−	0.965391i	$-0.583989\pi$
$647$	−13.2679	−0.521617	−0.260808	+	0.965391i	$0.583989\pi$
$648$	0	0
$649$	1.07180	0.0420717
$650$	0	0
$651$	2.92820	0.114765
$652$	0	0
$653$	36.2487	1.41852	0.709261	−	0.704946i	$-0.249029\pi$
$653$	36.2487	1.41852	0.709261	+	0.704946i	$0.249029\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.32051	−0.0515179
$658$	0	0
$659$	−17.3205	−0.674711	−0.337356	−	0.941377i	$-0.609532\pi$
$659$	−17.3205	−0.674711	−0.337356	+	0.941377i	$0.609532\pi$
$660$	0	0
$661$	35.8564	1.39465	0.697326	−	0.716754i	$-0.254373\pi$
$661$	35.8564	1.39465	0.697326	+	0.716754i	$0.254373\pi$
$662$	0	0
$663$	8.78461	0.341166
$664$	0	0
$665$	0	0
$666$	0	0
$667$	29.0718	1.12566
$668$	0	0
$669$	11.8564	0.458395
$670$	0	0
$671$	−9.85641	−0.380502
$672$	0	0
$673$	−19.4641	−0.750286	−0.375143	−	0.926967i	$-0.622406\pi$
$673$	−19.4641	−0.750286	−0.375143	+	0.926967i	$0.622406\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.3923	1.47554	0.737768	−	0.675054i	$-0.235880\pi$
$677$	38.3923	1.47554	0.737768	+	0.675054i	$0.235880\pi$
$678$	0	0
$679$	−17.4641	−0.670211
$680$	0	0
$681$	20.5359	0.786937
$682$	0	0
$683$	−34.9808	−1.33850	−0.669251	−	0.743037i	$-0.733385\pi$
$683$	−34.9808	−1.33850	−0.669251	+	0.743037i	$0.733385\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.92820	−0.111718
$688$	0	0
$689$	−15.7128	−0.598610
$690$	0	0
$691$	−18.0000	−0.684752	−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000	−0.684752	−0.342376	+	0.939563i	$0.611232\pi$
$692$	0	0
$693$	13.4641	0.511459
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.9282	0.716957
$698$	0	0
$699$	−21.4641	−0.811847
$700$	0	0
$701$	32.9282	1.24368	0.621841	−	0.783144i	$-0.286385\pi$
$701$	32.9282	1.24368	0.621841	+	0.783144i	$0.286385\pi$
$702$	0	0
$703$	14.9282	0.563028
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.8564	1.12287
$708$	0	0
$709$	−28.7846	−1.08103	−0.540514	−	0.841335i	$-0.681770\pi$
$709$	−28.7846	−1.08103	−0.540514	+	0.841335i	$0.681770\pi$
$710$	0	0
$711$	−36.7846	−1.37953
$712$	0	0
$713$	6.14359	0.230079
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14.6410	−0.546779
$718$	0	0
$719$	−25.8564	−0.964281	−0.482141	−	0.876094i	$-0.660141\pi$
$719$	−25.8564	−0.964281	−0.482141	+	0.876094i	$0.660141\pi$
$720$	0	0
$721$	4.53590	0.168926
$722$	0	0
$723$	3.21539	0.119582
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−14.0526	−0.521181	−0.260590	−	0.965449i	$-0.583917\pi$
$727$	−14.0526	−0.521181	−0.260590	+	0.965449i	$0.583917\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	30.2487	1.11879
$732$	0	0
$733$	48.9282	1.80720	0.903602	−	0.428373i	$-0.140913\pi$
$733$	48.9282	1.80720	0.903602	+	0.428373i	$0.140913\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.5359	−0.535437
$738$	0	0
$739$	5.32051	0.195718	0.0978590	−	0.995200i	$-0.468801\pi$
$739$	5.32051	0.195718	0.0978590	+	0.995200i	$0.468801\pi$
$740$	0	0
$741$	−18.9282	−0.695345
$742$	0	0
$743$	40.9808	1.50344	0.751719	−	0.659483i	$-0.229225\pi$
$743$	40.9808	1.50344	0.751719	+	0.659483i	$0.229225\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−11.6603	−0.426626
$748$	0	0
$749$	2.00000	0.0730784
$750$	0	0
$751$	22.2487	0.811867	0.405934	−	0.913903i	$-0.366946\pi$
$751$	22.2487	0.811867	0.405934	+	0.913903i	$0.366946\pi$
$752$	0	0
$753$	−8.10512	−0.295367
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−32.9282	−1.19680	−0.598398	−	0.801199i	$-0.704196\pi$
$757$	−32.9282	−1.19680	−0.598398	+	0.801199i	$0.704196\pi$
$758$	0	0
$759$	−6.14359	−0.222998
$760$	0	0
$761$	−49.7128	−1.80209	−0.901044	−	0.433728i	$-0.857198\pi$
$761$	−49.7128	−1.80209	−0.901044	+	0.433728i	$0.857198\pi$
$762$	0	0
$763$	8.39230	0.303822
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.85641	0.0670310
$768$	0	0
$769$	−0.928203	−0.0334719	−0.0167359	−	0.999860i	$-0.505327\pi$
$769$	−0.928203	−0.0334719	−0.0167359	+	0.999860i	$0.505327\pi$
$770$	0	0
$771$	−1.46410	−0.0527283
$772$	0	0
$773$	−1.60770	−0.0578248	−0.0289124	−	0.999582i	$-0.509204\pi$
$773$	−1.60770	−0.0578248	−0.0289124	+	0.999582i	$0.509204\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	−40.7846	−1.46126
$780$	0	0
$781$	2.92820	0.104779
$782$	0	0
$783$	−27.7128	−0.990375
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−14.5885	−0.520022	−0.260011	−	0.965606i	$-0.583726\pi$
$787$	−14.5885	−0.520022	−0.260011	+	0.965606i	$0.583726\pi$
$788$	0	0
$789$	4.14359	0.147516
$790$	0	0
$791$	−2.53590	−0.0901662
$792$	0	0
$793$	−17.0718	−0.606237
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.1051	0.924691	0.462345	−	0.886700i	$-0.347008\pi$
$797$	26.1051	0.924691	0.462345	+	0.886700i	$0.347008\pi$
$798$	0	0
$799$	23.3205	0.825020
$800$	0	0
$801$	−12.1436	−0.429073
$802$	0	0
$803$	−1.07180	−0.0378229
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.60770	−0.126997
$808$	0	0
$809$	3.85641	0.135584	0.0677920	−	0.997699i	$-0.478405\pi$
$809$	3.85641	0.135584	0.0677920	+	0.997699i	$0.478405\pi$
$810$	0	0
$811$	−15.0718	−0.529242	−0.264621	−	0.964352i	$-0.585247\pi$
$811$	−15.0718	−0.529242	−0.264621	+	0.964352i	$0.585247\pi$
$812$	0	0
$813$	11.2154	0.393341
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−65.1769	−2.28025
$818$	0	0
$819$	23.3205	0.814885
$820$	0	0
$821$	6.78461	0.236785	0.118392	−	0.992967i	$-0.462226\pi$
$821$	6.78461	0.236785	0.118392	+	0.992967i	$0.462226\pi$
$822$	0	0
$823$	−15.1244	−0.527202	−0.263601	−	0.964632i	$-0.584910\pi$
$823$	−15.1244	−0.527202	−0.263601	+	0.964632i	$0.584910\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.12436	0.0390977	0.0195488	−	0.999809i	$-0.493777\pi$
$827$	1.12436	0.0390977	0.0195488	+	0.999809i	$0.493777\pi$
$828$	0	0
$829$	15.0718	0.523465	0.261733	−	0.965140i	$-0.415706\pi$
$829$	15.0718	0.523465	0.261733	+	0.965140i	$0.415706\pi$
$830$	0	0
$831$	−1.46410	−0.0507891
$832$	0	0
$833$	1.60770	0.0557033
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−5.85641	−0.202427
$838$	0	0
$839$	−16.7846	−0.579469	−0.289735	−	0.957107i	$-0.593567\pi$
$839$	−16.7846	−0.579469	−0.289735	+	0.957107i	$0.593567\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	0	0
$843$	12.7846	0.440325
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−19.1244	−0.657121
$848$	0	0
$849$	5.60770	0.192456
$850$	0	0
$851$	8.39230	0.287685
$852$	0	0
$853$	−42.3923	−1.45148	−0.725742	−	0.687967i	$-0.758504\pi$
$853$	−42.3923	−1.45148	−0.725742	+	0.687967i	$0.758504\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.85641	0.268370	0.134185	−	0.990956i	$-0.457158\pi$
$857$	7.85641	0.268370	0.134185	+	0.990956i	$0.457158\pi$
$858$	0	0
$859$	−20.2487	−0.690877	−0.345439	−	0.938441i	$-0.612270\pi$
$859$	−20.2487	−0.690877	−0.345439	+	0.938441i	$0.612270\pi$
$860$	0	0
$861$	−10.9282	−0.372432
$862$	0	0
$863$	30.3397	1.03278	0.516388	−	0.856354i	$-0.327276\pi$
$863$	30.3397	1.03278	0.516388	+	0.856354i	$0.327276\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.66025	0.124309
$868$	0	0
$869$	−29.8564	−1.01281
$870$	0	0
$871$	−25.1769	−0.853087
$872$	0	0
$873$	15.7513	0.533100
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−53.7128	−1.81375	−0.906876	−	0.421397i	$-0.861540\pi$
$877$	−53.7128	−1.81375	−0.906876	+	0.421397i	$0.861540\pi$
$878$	0	0
$879$	−8.67949	−0.292752
$880$	0	0
$881$	2.53590	0.0854366	0.0427183	−	0.999087i	$-0.486398\pi$
$881$	2.53590	0.0854366	0.0427183	+	0.999087i	$0.486398\pi$
$882$	0	0
$883$	37.9090	1.27574	0.637869	−	0.770145i	$-0.279816\pi$
$883$	37.9090	1.27574	0.637869	+	0.770145i	$0.279816\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−51.9090	−1.74293	−0.871466	−	0.490455i	$-0.836830\pi$
$887$	−51.9090	−1.74293	−0.871466	+	0.490455i	$0.836830\pi$
$888$	0	0
$889$	36.2487	1.21574
$890$	0	0
$891$	−8.92820	−0.299106
$892$	0	0
$893$	−50.2487	−1.68151
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−10.6410	−0.355293
$898$	0	0
$899$	10.1436	0.338308
$900$	0	0
$901$	15.7128	0.523470
$902$	0	0
$903$	−17.4641	−0.581169
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−29.1244	−0.967058	−0.483529	−	0.875328i	$-0.660645\pi$
$907$	−29.1244	−0.967058	−0.483529	+	0.875328i	$0.660645\pi$
$908$	0	0
$909$	−26.9282	−0.893152
$910$	0	0
$911$	−13.1769	−0.436571	−0.218285	−	0.975885i	$-0.570046\pi$
$911$	−13.1769	−0.436571	−0.218285	+	0.975885i	$0.570046\pi$
$912$	0	0
$913$	−9.46410	−0.313216
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21.4641	−0.708807
$918$	0	0
$919$	−25.0718	−0.827042	−0.413521	−	0.910495i	$-0.635701\pi$
$919$	−25.0718	−0.827042	−0.413521	+	0.910495i	$0.635701\pi$
$920$	0	0
$921$	−19.7513	−0.650827
$922$	0	0
$923$	5.07180	0.166940
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.09103	−0.134367
$928$	0	0
$929$	−10.5359	−0.345672	−0.172836	−	0.984951i	$-0.555293\pi$
$929$	−10.5359	−0.345672	−0.172836	+	0.984951i	$0.555293\pi$
$930$	0	0
$931$	−3.46410	−0.113531
$932$	0	0
$933$	2.43078	0.0795802
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44.2487	−1.44554	−0.722771	−	0.691087i	$-0.757132\pi$
$937$	−44.2487	−1.44554	−0.722771	+	0.691087i	$0.757132\pi$
$938$	0	0
$939$	23.3205	0.761036
$940$	0	0
$941$	−32.0000	−1.04317	−0.521585	−	0.853199i	$-0.674659\pi$
$941$	−32.0000	−1.04317	−0.521585	+	0.853199i	$0.674659\pi$
$942$	0	0
$943$	−22.9282	−0.746645
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.1244	0.686449	0.343225	−	0.939253i	$-0.388481\pi$
$947$	21.1244	0.686449	0.343225	+	0.939253i	$0.388481\pi$
$948$	0	0
$949$	−1.85641	−0.0602615
$950$	0	0
$951$	11.3205	0.367093
$952$	0	0
$953$	−58.7846	−1.90422	−0.952110	−	0.305755i	$-0.901091\pi$
$953$	−58.7846	−1.90422	−0.952110	+	0.305755i	$0.901091\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−10.1436	−0.327896
$958$	0	0
$959$	−24.3923	−0.787669
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	−1.80385	−0.0581282
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−33.6603	−1.08244	−0.541220	−	0.840881i	$-0.682038\pi$
$967$	−33.6603	−1.08244	−0.541220	+	0.840881i	$0.682038\pi$
$968$	0	0
$969$	18.9282	0.608061
$970$	0	0
$971$	23.0718	0.740409	0.370205	−	0.928950i	$-0.379288\pi$
$971$	23.0718	0.740409	0.370205	+	0.928950i	$0.379288\pi$
$972$	0	0
$973$	−20.3923	−0.653747
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.4641	1.00663	0.503313	−	0.864104i	$-0.332114\pi$
$977$	31.4641	1.00663	0.503313	+	0.864104i	$0.332114\pi$
$978$	0	0
$979$	−9.85641	−0.315012
$980$	0	0
$981$	−7.56922	−0.241667
$982$	0	0
$983$	45.2679	1.44382	0.721912	−	0.691985i	$-0.243264\pi$
$983$	45.2679	1.44382	0.721912	+	0.691985i	$0.243264\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−13.4641	−0.428567
$988$	0	0
$989$	−36.6410	−1.16512
$990$	0	0
$991$	−34.5359	−1.09707	−0.548534	−	0.836128i	$-0.684814\pi$
$991$	−34.5359	−1.09707	−0.548534	+	0.836128i	$0.684814\pi$
$992$	0	0
$993$	10.2487	0.325233
$994$	0	0
$995$	0	0
$996$	0	0
$997$	51.1769	1.62079	0.810395	−	0.585884i	$-0.199253\pi$
$997$	51.1769	1.62079	0.810395	+	0.585884i	$0.199253\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6400.2.a.cj.1.1		2
4.3	odd	2		6400.2.a.z.1.2		2
5.4	even	2		1280.2.a.d.1.2		2
8.3	odd	2		6400.2.a.ce.1.1		2
8.5	even	2		6400.2.a.be.1.2		2
16.3	odd	4		200.2.d.f.101.1		4
16.5	even	4		800.2.d.e.401.3		4
16.11	odd	4		200.2.d.f.101.2		4
16.13	even	4		800.2.d.e.401.2		4
20.19	odd	2		1280.2.a.o.1.1		2
40.19	odd	2		1280.2.a.a.1.2		2
40.29	even	2		1280.2.a.n.1.1		2
48.5	odd	4		7200.2.k.j.3601.2		4
48.11	even	4		1800.2.k.j.901.3		4
48.29	odd	4		7200.2.k.j.3601.1		4
48.35	even	4		1800.2.k.j.901.4		4
80.3	even	4		200.2.f.c.149.2		4
80.13	odd	4		800.2.f.c.49.4		4
80.19	odd	4		40.2.d.a.21.4	yes	4
80.27	even	4		200.2.f.c.149.1		4
80.29	even	4		160.2.d.a.81.3		4
80.37	odd	4		800.2.f.c.49.3		4
80.43	even	4		200.2.f.e.149.4		4
80.53	odd	4		800.2.f.e.49.2		4
80.59	odd	4		40.2.d.a.21.3	✓	4
80.67	even	4		200.2.f.e.149.3		4
80.69	even	4		160.2.d.a.81.2		4
80.77	odd	4		800.2.f.e.49.1		4
240.29	odd	4		1440.2.k.e.721.4		4
240.53	even	4		7200.2.d.n.2449.4		4
240.59	even	4		360.2.k.e.181.2		4
240.77	even	4		7200.2.d.n.2449.1		4
240.83	odd	4		1800.2.d.p.1549.3		4
240.107	odd	4		1800.2.d.p.1549.4		4
240.149	odd	4		1440.2.k.e.721.2		4
240.173	even	4		7200.2.d.o.2449.4		4
240.179	even	4		360.2.k.e.181.1		4
240.197	even	4		7200.2.d.o.2449.1		4
240.203	odd	4		1800.2.d.l.1549.1		4
240.227	odd	4		1800.2.d.l.1549.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.d.a.21.3	✓	4	80.59	odd	4
40.2.d.a.21.4	yes	4	80.19	odd	4
160.2.d.a.81.2		4	80.69	even	4
160.2.d.a.81.3		4	80.29	even	4
200.2.d.f.101.1		4	16.3	odd	4
200.2.d.f.101.2		4	16.11	odd	4
200.2.f.c.149.1		4	80.27	even	4
200.2.f.c.149.2		4	80.3	even	4
200.2.f.e.149.3		4	80.67	even	4
200.2.f.e.149.4		4	80.43	even	4
360.2.k.e.181.1		4	240.179	even	4
360.2.k.e.181.2		4	240.59	even	4
800.2.d.e.401.2		4	16.13	even	4
800.2.d.e.401.3		4	16.5	even	4
800.2.f.c.49.3		4	80.37	odd	4
800.2.f.c.49.4		4	80.13	odd	4
800.2.f.e.49.1		4	80.77	odd	4
800.2.f.e.49.2		4	80.53	odd	4
1280.2.a.a.1.2		2	40.19	odd	2
1280.2.a.d.1.2		2	5.4	even	2
1280.2.a.n.1.1		2	40.29	even	2
1280.2.a.o.1.1		2	20.19	odd	2
1440.2.k.e.721.2		4	240.149	odd	4
1440.2.k.e.721.4		4	240.29	odd	4
1800.2.d.l.1549.1		4	240.203	odd	4
1800.2.d.l.1549.2		4	240.227	odd	4
1800.2.d.p.1549.3		4	240.83	odd	4
1800.2.d.p.1549.4		4	240.107	odd	4
1800.2.k.j.901.3		4	48.11	even	4
1800.2.k.j.901.4		4	48.35	even	4
6400.2.a.z.1.2		2	4.3	odd	2
6400.2.a.be.1.2		2	8.5	even	2
6400.2.a.ce.1.1		2	8.3	odd	2
6400.2.a.cj.1.1		2	1.1	even	1	trivial
7200.2.d.n.2449.1		4	240.77	even	4
7200.2.d.n.2449.4		4	240.53	even	4
7200.2.d.o.2449.1		4	240.197	even	4
7200.2.d.o.2449.4		4	240.173	even	4
7200.2.k.j.3601.1		4	48.29	odd	4
7200.2.k.j.3601.2		4	48.5	odd	4

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:	\( N \)	\(=\)	\( 6400 = 2^{8} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6400.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} +2.73205 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q-0.732051 q^{3} +2.73205 q^{7} -2.46410 q^{9} -2.00000 q^{11} -3.46410 q^{13} +3.46410 q^{17} -7.46410 q^{19} -2.00000 q^{21} -4.19615 q^{23} +4.00000 q^{27} -6.92820 q^{29} -1.46410 q^{31} +1.46410 q^{33} -2.00000 q^{37} +2.53590 q^{39} +5.46410 q^{41} +8.73205 q^{43} +6.73205 q^{47} +0.464102 q^{49} -2.53590 q^{51} +4.53590 q^{53} +5.46410 q^{57} -0.535898 q^{59} +4.92820 q^{61} -6.73205 q^{63} +7.26795 q^{67} +3.07180 q^{69} -1.46410 q^{71} +0.535898 q^{73} -5.46410 q^{77} +14.9282 q^{79} +4.46410 q^{81} +4.73205 q^{83} +5.07180 q^{87} +4.92820 q^{89} -9.46410 q^{91} +1.07180 q^{93} -6.39230 q^{97} +4.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{11} - 8 q^{19} - 4 q^{21} + 2 q^{23} + 8 q^{27} + 4 q^{31} - 4 q^{33} - 4 q^{37} + 12 q^{39} + 4 q^{41} + 14 q^{43} + 10 q^{47} - 6 q^{49} - 12 q^{51} + 16 q^{53} + 4 q^{57} - 8 q^{59} - 4 q^{61} - 10 q^{63} + 18 q^{67} + 20 q^{69} + 4 q^{71} + 8 q^{73} - 4 q^{77} + 16 q^{79} + 2 q^{81} + 6 q^{83} + 24 q^{87} - 4 q^{89} - 12 q^{91} + 16 q^{93} + 8 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 - 4 * q^11 - 8 * q^19 - 4 * q^21 + 2 * q^23 + 8 * q^27 + 4 * q^31 - 4 * q^33 - 4 * q^37 + 12 * q^39 + 4 * q^41 + 14 * q^43 + 10 * q^47 - 6 * q^49 - 12 * q^51 + 16 * q^53 + 4 * q^57 - 8 * q^59 - 4 * q^61 - 10 * q^63 + 18 * q^67 + 20 * q^69 + 4 * q^71 + 8 * q^73 - 4 * q^77 + 16 * q^79 + 2 * q^81 + 6 * q^83 + 24 * q^87 - 4 * q^89 - 12 * q^91 + 16 * q^93 + 8 * q^97 - 4 * q^99

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