LMFDB - Embedded newform 1080.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,4,Mod(1,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1080.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.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:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.985.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 1 $ x^3 - x^2 - 6*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.09376$ of defining polynomial
Character	$\chi$	$=$	1080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} -12.8657 q^{7} +O(q^{10})$	$q+5.00000 q^{5} -12.8657 q^{7} +18.2595 q^{11} +20.2595 q^{13} +6.65336 q^{17} +150.972 q^{19} -88.1068 q^{23} +25.0000 q^{25} -201.867 q^{29} -268.330 q^{31} -64.3283 q^{35} -123.539 q^{37} +275.174 q^{41} +488.679 q^{43} +436.725 q^{47} -177.475 q^{49} +340.573 q^{53} +91.2975 q^{55} +548.360 q^{59} -206.793 q^{61} +101.298 q^{65} +499.621 q^{67} -460.900 q^{71} -416.818 q^{73} -234.921 q^{77} -289.912 q^{79} +909.471 q^{83} +33.2668 q^{85} +186.814 q^{89} -260.652 q^{91} +754.862 q^{95} +648.440 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} + 6 q^{7}+O(q^{10}) $ 3 * q + 15 * q^5 + 6 * q^7	$ 3 q + 15 q^{5} + 6 q^{7} + 12 q^{11} + 18 q^{13} - 21 q^{17} + 57 q^{19} + 87 q^{23} + 75 q^{25} + 138 q^{29} + 117 q^{31} + 30 q^{35} + 150 q^{37} + 180 q^{43} + 684 q^{47} - 81 q^{49} + 87 q^{53} + 60 q^{55} + 714 q^{59} - 513 q^{61} + 90 q^{65} - 174 q^{67} + 768 q^{71} - 252 q^{73} + 888 q^{77} + 207 q^{79} + 1689 q^{83} - 105 q^{85} + 312 q^{89} + 900 q^{91} + 285 q^{95} - 1080 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 + 6 * q^7 + 12 * q^11 + 18 * q^13 - 21 * q^17 + 57 * q^19 + 87 * q^23 + 75 * q^25 + 138 * q^29 + 117 * q^31 + 30 * q^35 + 150 * q^37 + 180 * q^43 + 684 * q^47 - 81 * q^49 + 87 * q^53 + 60 * q^55 + 714 * q^59 - 513 * q^61 + 90 * q^65 - 174 * q^67 + 768 * q^71 - 252 * q^73 + 888 * q^77 + 207 * q^79 + 1689 * q^83 - 105 * q^85 + 312 * q^89 + 900 * q^91 + 285 * q^95 - 1080 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−12.8657	−0.694680	−0.347340	−	0.937739i	$-0.612915\pi$
$7$	−12.8657	−0.694680	−0.347340	+	0.937739i	$0.612915\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	18.2595	0.500495	0.250248	−	0.968182i	$-0.419488\pi$
$11$	18.2595	0.500495	0.250248	+	0.968182i	$0.419488\pi$
$12$	0	0
$13$	20.2595	0.432229	0.216114	−	0.976368i	$-0.430662\pi$
$13$	20.2595	0.432229	0.216114	+	0.976368i	$0.430662\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.65336	0.0949222	0.0474611	−	0.998873i	$-0.484887\pi$
$17$	6.65336	0.0949222	0.0474611	+	0.998873i	$0.484887\pi$
$18$	0	0
$19$	150.972	1.82292	0.911459	−	0.411390i	$-0.134957\pi$
$19$	150.972	1.82292	0.911459	+	0.411390i	$0.134957\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−88.1068	−0.798762	−0.399381	−	0.916785i	$-0.630775\pi$
$23$	−88.1068	−0.798762	−0.399381	+	0.916785i	$0.630775\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−201.867	−1.29261	−0.646306	−	0.763078i	$-0.723687\pi$
$29$	−201.867	−1.29261	−0.646306	+	0.763078i	$0.723687\pi$
$30$	0	0
$31$	−268.330	−1.55463	−0.777313	−	0.629114i	$-0.783418\pi$
$31$	−268.330	−1.55463	−0.777313	+	0.629114i	$0.783418\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−64.3283	−0.310670
$36$	0	0
$37$	−123.539	−0.548909	−0.274455	−	0.961600i	$-0.588497\pi$
$37$	−123.539	−0.548909	−0.274455	+	0.961600i	$0.588497\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	275.174	1.04817	0.524084	−	0.851666i	$-0.324408\pi$
$41$	275.174	1.04817	0.524084	+	0.851666i	$0.324408\pi$
$42$	0	0
$43$	488.679	1.73309	0.866545	−	0.499098i	$-0.166335\pi$
$43$	488.679	1.73309	0.866545	+	0.499098i	$0.166335\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	436.725	1.35538	0.677691	−	0.735347i	$-0.262981\pi$
$47$	436.725	1.35538	0.677691	+	0.735347i	$0.262981\pi$
$48$	0	0
$49$	−177.475	−0.517420
$50$	0	0
$51$	0	0
$52$	0	0
$53$	340.573	0.882665	0.441332	−	0.897344i	$-0.354506\pi$
$53$	340.573	0.882665	0.441332	+	0.897344i	$0.354506\pi$
$54$	0	0
$55$	91.2975	0.223828
$56$	0	0
$57$	0	0
$58$	0	0
$59$	548.360	1.21001	0.605004	−	0.796223i	$-0.293172\pi$
$59$	548.360	1.21001	0.605004	+	0.796223i	$0.293172\pi$
$60$	0	0
$61$	−206.793	−0.434051	−0.217025	−	0.976166i	$-0.569635\pi$
$61$	−206.793	−0.434051	−0.217025	+	0.976166i	$0.569635\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	101.298	0.193299
$66$	0	0
$67$	499.621	0.911021	0.455511	−	0.890230i	$-0.349457\pi$
$67$	499.621	0.911021	0.455511	+	0.890230i	$0.349457\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−460.900	−0.770406	−0.385203	−	0.922832i	$-0.625869\pi$
$71$	−460.900	−0.770406	−0.385203	+	0.922832i	$0.625869\pi$
$72$	0	0
$73$	−416.818	−0.668285	−0.334143	−	0.942522i	$-0.608447\pi$
$73$	−416.818	−0.668285	−0.334143	+	0.942522i	$0.608447\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−234.921	−0.347684
$78$	0	0
$79$	−289.912	−0.412882	−0.206441	−	0.978459i	$-0.566188\pi$
$79$	−289.912	−0.412882	−0.206441	+	0.978459i	$0.566188\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	909.471	1.20274	0.601370	−	0.798971i	$-0.294622\pi$
$83$	909.471	1.20274	0.601370	+	0.798971i	$0.294622\pi$
$84$	0	0
$85$	33.2668	0.0424505
$86$	0	0
$87$	0	0
$88$	0	0
$89$	186.814	0.222497	0.111249	−	0.993793i	$-0.464515\pi$
$89$	186.814	0.222497	0.111249	+	0.993793i	$0.464515\pi$
$90$	0	0
$91$	−260.652	−0.300261
$92$	0	0
$93$	0	0
$94$	0	0
$95$	754.862	0.815234
$96$	0	0
$97$	648.440	0.678754	0.339377	−	0.940651i	$-0.389784\pi$
$97$	648.440	0.678754	0.339377	+	0.940651i	$0.389784\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1415.88	1.39490	0.697450	−	0.716634i	$-0.254318\pi$
$101$	1415.88	1.39490	0.697450	+	0.716634i	$0.254318\pi$
$102$	0	0
$103$	−1199.94	−1.14790	−0.573949	−	0.818891i	$-0.694589\pi$
$103$	−1199.94	−1.14790	−0.573949	+	0.818891i	$0.694589\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1597.84	1.44363	0.721817	−	0.692084i	$-0.243307\pi$
$107$	1597.84	1.44363	0.721817	+	0.692084i	$0.243307\pi$
$108$	0	0
$109$	1316.75	1.15708	0.578541	−	0.815653i	$-0.303622\pi$
$109$	1316.75	1.15708	0.578541	+	0.815653i	$0.303622\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−806.519	−0.671425	−0.335712	−	0.941965i	$-0.608977\pi$
$113$	−806.519	−0.671425	−0.335712	+	0.941965i	$0.608977\pi$
$114$	0	0
$115$	−440.534	−0.357217
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−85.5999	−0.0659406
$120$	0	0
$121$	−997.590	−0.749504
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1480.02	1.03410	0.517050	−	0.855955i	$-0.327030\pi$
$127$	1480.02	1.03410	0.517050	+	0.855955i	$0.327030\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−95.0432	−0.0633890	−0.0316945	−	0.999498i	$-0.510090\pi$
$131$	−95.0432	−0.0633890	−0.0316945	+	0.999498i	$0.510090\pi$
$132$	0	0
$133$	−1942.36	−1.26635
$134$	0	0
$135$	0	0
$136$	0	0
$137$	793.417	0.494790	0.247395	−	0.968915i	$-0.420425\pi$
$137$	793.417	0.494790	0.247395	+	0.968915i	$0.420425\pi$
$138$	0	0
$139$	−24.6768	−0.0150580	−0.00752900	−	0.999972i	$-0.502397\pi$
$139$	−24.6768	−0.0150580	−0.00752900	+	0.999972i	$0.502397\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	369.929	0.216329
$144$	0	0
$145$	−1009.33	−0.578074
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3092.28	1.70020	0.850098	−	0.526624i	$-0.176543\pi$
$149$	3092.28	1.70020	0.850098	+	0.526624i	$0.176543\pi$
$150$	0	0
$151$	2056.52	1.10832	0.554162	−	0.832409i	$-0.313039\pi$
$151$	2056.52	1.10832	0.554162	+	0.832409i	$0.313039\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1341.65	−0.695250
$156$	0	0
$157$	−649.944	−0.330390	−0.165195	−	0.986261i	$-0.552825\pi$
$157$	−649.944	−0.330390	−0.165195	+	0.986261i	$0.552825\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1133.55	0.554884
$162$	0	0
$163$	3183.46	1.52974	0.764870	−	0.644185i	$-0.222803\pi$
$163$	3183.46	1.52974	0.764870	+	0.644185i	$0.222803\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	743.307	0.344424	0.172212	−	0.985060i	$-0.444909\pi$
$167$	743.307	0.344424	0.172212	+	0.985060i	$0.444909\pi$
$168$	0	0
$169$	−1786.55	−0.813178
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2152.84	0.946113	0.473057	−	0.881032i	$-0.343151\pi$
$173$	2152.84	0.946113	0.473057	+	0.881032i	$0.343151\pi$
$174$	0	0
$175$	−321.641	−0.138936
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−597.624	−0.249545	−0.124772	−	0.992185i	$-0.539820\pi$
$179$	−597.624	−0.249545	−0.124772	+	0.992185i	$0.539820\pi$
$180$	0	0
$181$	−736.338	−0.302384	−0.151192	−	0.988504i	$-0.548311\pi$
$181$	−736.338	−0.302384	−0.151192	+	0.988504i	$0.548311\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−617.693	−0.245480
$186$	0	0
$187$	121.487	0.0475081
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3355.19	1.27106	0.635531	−	0.772075i	$-0.280781\pi$
$191$	3355.19	1.27106	0.635531	+	0.772075i	$0.280781\pi$
$192$	0	0
$193$	1521.54	0.567475	0.283737	−	0.958902i	$-0.408426\pi$
$193$	1521.54	0.567475	0.283737	+	0.958902i	$0.408426\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	364.341	0.131768	0.0658838	−	0.997827i	$-0.479013\pi$
$197$	364.341	0.131768	0.0658838	+	0.997827i	$0.479013\pi$
$198$	0	0
$199$	−171.730	−0.0611738	−0.0305869	−	0.999532i	$-0.509738\pi$
$199$	−171.730	−0.0611738	−0.0305869	+	0.999532i	$0.509738\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2597.15	0.897952
$204$	0	0
$205$	1375.87	0.468755
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2756.68	0.912362
$210$	0	0
$211$	904.632	0.295154	0.147577	−	0.989051i	$-0.452853\pi$
$211$	904.632	0.295154	0.147577	+	0.989051i	$0.452853\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2443.40	0.775062
$216$	0	0
$217$	3452.24	1.07997
$218$	0	0
$219$	0	0
$220$	0	0
$221$	134.794	0.0410281
$222$	0	0
$223$	405.423	0.121745	0.0608724	−	0.998146i	$-0.480612\pi$
$223$	405.423	0.121745	0.0608724	+	0.998146i	$0.480612\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	556.762	0.162791	0.0813956	−	0.996682i	$-0.474062\pi$
$227$	556.762	0.162791	0.0813956	+	0.996682i	$0.474062\pi$
$228$	0	0
$229$	−4584.45	−1.32292	−0.661461	−	0.749979i	$-0.730063\pi$
$229$	−4584.45	−1.32292	−0.661461	+	0.749979i	$0.730063\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4593.97	1.29168	0.645839	−	0.763473i	$-0.276508\pi$
$233$	4593.97	1.29168	0.645839	+	0.763473i	$0.276508\pi$
$234$	0	0
$235$	2183.63	0.606145
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1754.22	0.474775	0.237387	−	0.971415i	$-0.423709\pi$
$239$	1754.22	0.474775	0.237387	+	0.971415i	$0.423709\pi$
$240$	0	0
$241$	−4394.90	−1.17469	−0.587346	−	0.809336i	$-0.699827\pi$
$241$	−4394.90	−1.17469	−0.587346	+	0.809336i	$0.699827\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−887.375	−0.231397
$246$	0	0
$247$	3058.63	0.787918
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2260.13	0.568360	0.284180	−	0.958771i	$-0.408279\pi$
$251$	2260.13	0.568360	0.284180	+	0.958771i	$0.408279\pi$
$252$	0	0
$253$	−1608.79	−0.399777
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−570.878	−0.138562	−0.0692809	−	0.997597i	$-0.522070\pi$
$257$	−570.878	−0.138562	−0.0692809	+	0.997597i	$0.522070\pi$
$258$	0	0
$259$	1589.41	0.381316
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6248.85	−1.46510	−0.732549	−	0.680715i	$-0.761669\pi$
$263$	−6248.85	−1.46510	−0.732549	+	0.680715i	$0.761669\pi$
$264$	0	0
$265$	1702.86	0.394740
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1718.42	−0.389494	−0.194747	−	0.980853i	$-0.562389\pi$
$269$	−1718.42	−0.389494	−0.194747	+	0.980853i	$0.562389\pi$
$270$	0	0
$271$	883.253	0.197985	0.0989923	−	0.995088i	$-0.468438\pi$
$271$	883.253	0.197985	0.0989923	+	0.995088i	$0.468438\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	456.488	0.100099
$276$	0	0
$277$	−4889.06	−1.06049	−0.530244	−	0.847845i	$-0.677900\pi$
$277$	−4889.06	−1.06049	−0.530244	+	0.847845i	$0.677900\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3590.34	−0.762212	−0.381106	−	0.924531i	$-0.624457\pi$
$281$	−3590.34	−0.762212	−0.381106	+	0.924531i	$0.624457\pi$
$282$	0	0
$283$	−2003.82	−0.420900	−0.210450	−	0.977605i	$-0.567493\pi$
$283$	−2003.82	−0.420900	−0.210450	+	0.977605i	$0.567493\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3540.29	−0.728142
$288$	0	0
$289$	−4868.73	−0.990990
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4348.62	0.867062	0.433531	−	0.901139i	$-0.357267\pi$
$293$	4348.62	0.867062	0.433531	+	0.901139i	$0.357267\pi$
$294$	0	0
$295$	2741.80	0.541132
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1785.00	−0.345248
$300$	0	0
$301$	−6287.18	−1.20394
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1033.96	−0.194113
$306$	0	0
$307$	7513.36	1.39678	0.698388	−	0.715719i	$-0.253901\pi$
$307$	7513.36	1.39678	0.698388	+	0.715719i	$0.253901\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−624.265	−0.113823	−0.0569113	−	0.998379i	$-0.518125\pi$
$311$	−624.265	−0.113823	−0.0569113	+	0.998379i	$0.518125\pi$
$312$	0	0
$313$	−1963.18	−0.354521	−0.177261	−	0.984164i	$-0.556724\pi$
$313$	−1963.18	−0.354521	−0.177261	+	0.984164i	$0.556724\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2928.29	0.518829	0.259415	−	0.965766i	$-0.416470\pi$
$317$	2928.29	0.518829	0.259415	+	0.965766i	$0.416470\pi$
$318$	0	0
$319$	−3685.99	−0.646946
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1004.47	0.173035
$324$	0	0
$325$	506.488	0.0864458
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5618.76	−0.941556
$330$	0	0
$331$	3619.06	0.600972	0.300486	−	0.953786i	$-0.402851\pi$
$331$	3619.06	0.600972	0.300486	+	0.953786i	$0.402851\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2498.11	0.407421
$336$	0	0
$337$	−9440.02	−1.52591	−0.762954	−	0.646453i	$-0.776251\pi$
$337$	−9440.02	−1.52591	−0.762954	+	0.646453i	$0.776251\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4899.57	−0.778083
$342$	0	0
$343$	6696.25	1.05412
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1657.99	0.256500	0.128250	−	0.991742i	$-0.459064\pi$
$347$	1657.99	0.256500	0.128250	+	0.991742i	$0.459064\pi$
$348$	0	0
$349$	−4537.03	−0.695879	−0.347939	−	0.937517i	$-0.613119\pi$
$349$	−4537.03	−0.695879	−0.347939	+	0.937517i	$0.613119\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8627.43	−1.30083	−0.650413	−	0.759580i	$-0.725404\pi$
$353$	−8627.43	−1.30083	−0.650413	+	0.759580i	$0.725404\pi$
$354$	0	0
$355$	−2304.50	−0.344536
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10116.5	1.48726	0.743631	−	0.668590i	$-0.233102\pi$
$359$	10116.5	1.48726	0.743631	+	0.668590i	$0.233102\pi$
$360$	0	0
$361$	15933.7	2.32303
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2084.09	−0.298866
$366$	0	0
$367$	4707.74	0.669597	0.334798	−	0.942290i	$-0.391332\pi$
$367$	4707.74	0.669597	0.334798	+	0.942290i	$0.391332\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4381.69	−0.613170
$372$	0	0
$373$	−14191.2	−1.96995	−0.984973	−	0.172706i	$-0.944749\pi$
$373$	−14191.2	−1.96995	−0.984973	+	0.172706i	$0.944749\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4089.73	−0.558704
$378$	0	0
$379$	8841.08	1.19825	0.599124	−	0.800656i	$-0.295516\pi$
$379$	8841.08	1.19825	0.599124	+	0.800656i	$0.295516\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6225.37	−0.830553	−0.415276	−	0.909695i	$-0.636315\pi$
$383$	−6225.37	−0.830553	−0.415276	+	0.909695i	$0.636315\pi$
$384$	0	0
$385$	−1174.60	−0.155489
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3967.13	0.517074	0.258537	−	0.966001i	$-0.416760\pi$
$389$	3967.13	0.517074	0.258537	+	0.966001i	$0.416760\pi$
$390$	0	0
$391$	−586.206	−0.0758203
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1449.56	−0.184646
$396$	0	0
$397$	10510.6	1.32875	0.664375	−	0.747399i	$-0.268698\pi$
$397$	10510.6	1.32875	0.664375	+	0.747399i	$0.268698\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6440.92	0.802105	0.401052	−	0.916055i	$-0.368645\pi$
$401$	6440.92	0.802105	0.401052	+	0.916055i	$0.368645\pi$
$402$	0	0
$403$	−5436.22	−0.671954
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2255.76	−0.274726
$408$	0	0
$409$	−4820.75	−0.582813	−0.291407	−	0.956599i	$-0.594123\pi$
$409$	−4820.75	−0.582813	−0.291407	+	0.956599i	$0.594123\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7055.01	−0.840568
$414$	0	0
$415$	4547.36	0.537882
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17003.8	−1.98255	−0.991276	−	0.131803i	$-0.957923\pi$
$419$	−17003.8	−1.98255	−0.991276	+	0.131803i	$0.957923\pi$
$420$	0	0
$421$	1582.86	0.183239	0.0916197	−	0.995794i	$-0.470796\pi$
$421$	1582.86	0.183239	0.0916197	+	0.995794i	$0.470796\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	166.334	0.0189844
$426$	0	0
$427$	2660.52	0.301526
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10518.7	−1.17557	−0.587784	−	0.809018i	$-0.699999\pi$
$431$	−10518.7	−1.17557	−0.587784	+	0.809018i	$0.699999\pi$
$432$	0	0
$433$	−5469.93	−0.607086	−0.303543	−	0.952818i	$-0.598170\pi$
$433$	−5469.93	−0.607086	−0.303543	+	0.952818i	$0.598170\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13301.7	−1.45608
$438$	0	0
$439$	−12068.3	−1.31205	−0.656024	−	0.754740i	$-0.727763\pi$
$439$	−12068.3	−1.31205	−0.656024	+	0.754740i	$0.727763\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9948.23	−1.06694	−0.533470	−	0.845819i	$-0.679112\pi$
$443$	−9948.23	−1.06694	−0.533470	+	0.845819i	$0.679112\pi$
$444$	0	0
$445$	934.071	0.0995039
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12449.4	−1.30852	−0.654258	−	0.756271i	$-0.727019\pi$
$449$	−12449.4	−1.30852	−0.654258	+	0.756271i	$0.727019\pi$
$450$	0	0
$451$	5024.54	0.524603
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1303.26	−0.134281
$456$	0	0
$457$	15270.3	1.56305	0.781524	−	0.623875i	$-0.214442\pi$
$457$	15270.3	1.56305	0.781524	+	0.623875i	$0.214442\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11670.3	1.17905	0.589524	−	0.807751i	$-0.299315\pi$
$461$	11670.3	1.17905	0.589524	+	0.807751i	$0.299315\pi$
$462$	0	0
$463$	−9205.75	−0.924033	−0.462017	−	0.886871i	$-0.652874\pi$
$463$	−9205.75	−0.924033	−0.462017	+	0.886871i	$0.652874\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12090.0	1.19799	0.598994	−	0.800754i	$-0.295567\pi$
$467$	12090.0	1.19799	0.598994	+	0.800754i	$0.295567\pi$
$468$	0	0
$469$	−6427.95	−0.632868
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8923.04	0.867404
$474$	0	0
$475$	3774.31	0.364584
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3804.88	0.362942	0.181471	−	0.983396i	$-0.441914\pi$
$479$	3804.88	0.362942	0.181471	+	0.983396i	$0.441914\pi$
$480$	0	0
$481$	−2502.83	−0.237254
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3242.20	0.303548
$486$	0	0
$487$	5679.19	0.528437	0.264218	−	0.964463i	$-0.414886\pi$
$487$	5679.19	0.528437	0.264218	+	0.964463i	$0.414886\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7710.36	0.708684	0.354342	−	0.935116i	$-0.384705\pi$
$491$	7710.36	0.708684	0.354342	+	0.935116i	$0.384705\pi$
$492$	0	0
$493$	−1343.09	−0.122698
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5929.78	0.535185
$498$	0	0
$499$	−7766.60	−0.696755	−0.348378	−	0.937354i	$-0.613267\pi$
$499$	−7766.60	−0.696755	−0.348378	+	0.937354i	$0.613267\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8393.87	0.744064	0.372032	−	0.928220i	$-0.378661\pi$
$503$	8393.87	0.744064	0.372032	+	0.928220i	$0.378661\pi$
$504$	0	0
$505$	7079.38	0.623818
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14775.6	−1.28667	−0.643337	−	0.765583i	$-0.722451\pi$
$509$	−14775.6	−1.28667	−0.643337	+	0.765583i	$0.722451\pi$
$510$	0	0
$511$	5362.63	0.464245
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5999.70	−0.513356
$516$	0	0
$517$	7974.39	0.678362
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21089.0	−1.77337	−0.886684	−	0.462376i	$-0.846997\pi$
$521$	−21089.0	−1.77337	−0.886684	+	0.462376i	$0.846997\pi$
$522$	0	0
$523$	860.536	0.0719476	0.0359738	−	0.999353i	$-0.488547\pi$
$523$	860.536	0.0719476	0.0359738	+	0.999353i	$0.488547\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1785.29	−0.147569
$528$	0	0
$529$	−4404.19	−0.361979
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5574.88	0.453049
$534$	0	0
$535$	7989.19	0.645613
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3240.61	−0.258966
$540$	0	0
$541$	−23226.2	−1.84579	−0.922895	−	0.385051i	$-0.874184\pi$
$541$	−23226.2	−1.84579	−0.922895	+	0.385051i	$0.874184\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6583.76	0.517463
$546$	0	0
$547$	11800.1	0.922371	0.461185	−	0.887304i	$-0.347424\pi$
$547$	11800.1	0.922371	0.461185	+	0.887304i	$0.347424\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−30476.3	−2.35633
$552$	0	0
$553$	3729.91	0.286821
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13654.7	−1.03872	−0.519361	−	0.854555i	$-0.673830\pi$
$557$	−13654.7	−1.03872	−0.519361	+	0.854555i	$0.673830\pi$
$558$	0	0
$559$	9900.40	0.749092
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12390.2	−0.927502	−0.463751	−	0.885966i	$-0.653497\pi$
$563$	−12390.2	−0.927502	−0.463751	+	0.885966i	$0.653497\pi$
$564$	0	0
$565$	−4032.60	−0.300270
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12361.4	−0.910753	−0.455376	−	0.890299i	$-0.650495\pi$
$569$	−12361.4	−0.910753	−0.455376	+	0.890299i	$0.650495\pi$
$570$	0	0
$571$	−20577.3	−1.50812	−0.754058	−	0.656807i	$-0.771906\pi$
$571$	−20577.3	−1.50812	−0.754058	+	0.656807i	$0.771906\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2202.67	−0.159752
$576$	0	0
$577$	18345.2	1.32361	0.661803	−	0.749678i	$-0.269791\pi$
$577$	18345.2	1.32361	0.661803	+	0.749678i	$0.269791\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−11700.9	−0.835520
$582$	0	0
$583$	6218.69	0.441770
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19713.9	−1.38617	−0.693085	−	0.720856i	$-0.743749\pi$
$587$	−19713.9	−1.38617	−0.693085	+	0.720856i	$0.743749\pi$
$588$	0	0
$589$	−40510.4	−2.83396
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6553.49	−0.453827	−0.226914	−	0.973915i	$-0.572864\pi$
$593$	−6553.49	−0.453827	−0.226914	+	0.973915i	$0.572864\pi$
$594$	0	0
$595$	−427.999	−0.0294895
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15005.2	1.02354	0.511768	−	0.859124i	$-0.328991\pi$
$599$	15005.2	1.02354	0.511768	+	0.859124i	$0.328991\pi$
$600$	0	0
$601$	−20290.4	−1.37714	−0.688572	−	0.725168i	$-0.741762\pi$
$601$	−20290.4	−1.37714	−0.688572	+	0.725168i	$0.741762\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4987.95	−0.335189
$606$	0	0
$607$	11867.6	0.793563	0.396781	−	0.917913i	$-0.370127\pi$
$607$	11867.6	0.793563	0.396781	+	0.917913i	$0.370127\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8847.84	0.585835
$612$	0	0
$613$	27850.2	1.83501	0.917503	−	0.397730i	$-0.130202\pi$
$613$	27850.2	1.83501	0.917503	+	0.397730i	$0.130202\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4116.33	0.268586	0.134293	−	0.990942i	$-0.457124\pi$
$617$	4116.33	0.268586	0.134293	+	0.990942i	$0.457124\pi$
$618$	0	0
$619$	−25022.4	−1.62478	−0.812388	−	0.583117i	$-0.801833\pi$
$619$	−25022.4	−1.62478	−0.812388	+	0.583117i	$0.801833\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2403.49	−0.154565
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−821.948	−0.0521037
$630$	0	0
$631$	17801.6	1.12309	0.561545	−	0.827446i	$-0.310207\pi$
$631$	17801.6	1.12309	0.561545	+	0.827446i	$0.310207\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7400.10	0.462463
$636$	0	0
$637$	−3595.56	−0.223644
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19325.4	1.19081	0.595405	−	0.803426i	$-0.296992\pi$
$641$	19325.4	1.19081	0.595405	+	0.803426i	$0.296992\pi$
$642$	0	0
$643$	267.444	0.0164028	0.00820138	−	0.999966i	$-0.497389\pi$
$643$	267.444	0.0164028	0.00820138	+	0.999966i	$0.497389\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18749.2	−1.13927	−0.569634	−	0.821898i	$-0.692915\pi$
$647$	−18749.2	−1.13927	−0.569634	+	0.821898i	$0.692915\pi$
$648$	0	0
$649$	10012.8	0.605603
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14996.0	−0.898684	−0.449342	−	0.893360i	$-0.648341\pi$
$653$	−14996.0	−0.898684	−0.449342	+	0.893360i	$0.648341\pi$
$654$	0	0
$655$	−475.216	−0.0283484
$656$	0	0
$657$	0	0
$658$	0	0
$659$	19249.4	1.13786	0.568929	−	0.822386i	$-0.307358\pi$
$659$	19249.4	1.13786	0.568929	+	0.822386i	$0.307358\pi$
$660$	0	0
$661$	12989.7	0.764357	0.382178	−	0.924089i	$-0.375174\pi$
$661$	12989.7	0.764357	0.382178	+	0.924089i	$0.375174\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9711.80	−0.566327
$666$	0	0
$667$	17785.8	1.03249
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3775.93	−0.217240
$672$	0	0
$673$	−5909.54	−0.338478	−0.169239	−	0.985575i	$-0.554131\pi$
$673$	−5909.54	−0.338478	−0.169239	+	0.985575i	$0.554131\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23548.5	1.33684	0.668420	−	0.743784i	$-0.266971\pi$
$677$	23548.5	1.33684	0.668420	+	0.743784i	$0.266971\pi$
$678$	0	0
$679$	−8342.60	−0.471516
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10065.3	−0.563890	−0.281945	−	0.959431i	$-0.590980\pi$
$683$	−10065.3	−0.563890	−0.281945	+	0.959431i	$0.590980\pi$
$684$	0	0
$685$	3967.09	0.221277
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6899.83	0.381513
$690$	0	0
$691$	−28648.5	−1.57720	−0.788598	−	0.614910i	$-0.789193\pi$
$691$	−28648.5	−1.57720	−0.788598	+	0.614910i	$0.789193\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−123.384	−0.00673414
$696$	0	0
$697$	1830.83	0.0994945
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30910.9	1.66546	0.832730	−	0.553679i	$-0.186777\pi$
$701$	30910.9	1.66546	0.832730	+	0.553679i	$0.186777\pi$
$702$	0	0
$703$	−18650.9	−1.00062
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18216.2	−0.969009
$708$	0	0
$709$	11680.0	0.618691	0.309345	−	0.950950i	$-0.399890\pi$
$709$	11680.0	0.618691	0.309345	+	0.950950i	$0.399890\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	23641.7	1.24178
$714$	0	0
$715$	1849.64	0.0967451
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9205.71	0.477489	0.238745	−	0.971082i	$-0.423264\pi$
$719$	9205.71	0.477489	0.238745	+	0.971082i	$0.423264\pi$
$720$	0	0
$721$	15438.0	0.797422
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5046.67	−0.258522
$726$	0	0
$727$	8102.51	0.413350	0.206675	−	0.978410i	$-0.433736\pi$
$727$	8102.51	0.413350	0.206675	+	0.978410i	$0.433736\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3251.36	0.164509
$732$	0	0
$733$	−33738.0	−1.70006	−0.850028	−	0.526737i	$-0.823415\pi$
$733$	−33738.0	−1.70006	−0.850028	+	0.526737i	$0.823415\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9122.84	0.455962
$738$	0	0
$739$	−28667.6	−1.42700	−0.713502	−	0.700653i	$-0.752892\pi$
$739$	−28667.6	−1.42700	−0.713502	+	0.700653i	$0.752892\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2344.60	0.115767	0.0578837	−	0.998323i	$-0.481565\pi$
$743$	2344.60	0.115767	0.0578837	+	0.998323i	$0.481565\pi$
$744$	0	0
$745$	15461.4	0.760351
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20557.2	−1.00286
$750$	0	0
$751$	−23466.1	−1.14020	−0.570101	−	0.821575i	$-0.693096\pi$
$751$	−23466.1	−1.14020	−0.570101	+	0.821575i	$0.693096\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10282.6	0.495658
$756$	0	0
$757$	5711.42	0.274221	0.137110	−	0.990556i	$-0.456219\pi$
$757$	5711.42	0.274221	0.137110	+	0.990556i	$0.456219\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8101.10	0.385893	0.192947	−	0.981209i	$-0.438196\pi$
$761$	8101.10	0.385893	0.192947	+	0.981209i	$0.438196\pi$
$762$	0	0
$763$	−16940.9	−0.803802
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11109.5	0.523000
$768$	0	0
$769$	19468.4	0.912937	0.456468	−	0.889740i	$-0.349114\pi$
$769$	19468.4	0.912937	0.456468	+	0.889740i	$0.349114\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9632.96	0.448219	0.224110	−	0.974564i	$-0.428053\pi$
$773$	9632.96	0.448219	0.224110	+	0.974564i	$0.428053\pi$
$774$	0	0
$775$	−6708.24	−0.310925
$776$	0	0
$777$	0	0
$778$	0	0
$779$	41543.6	1.91073
$780$	0	0
$781$	−8415.81	−0.385584
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3249.72	−0.147755
$786$	0	0
$787$	38781.5	1.75656	0.878278	−	0.478150i	$-0.158692\pi$
$787$	38781.5	1.75656	0.878278	+	0.478150i	$0.158692\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10376.4	0.466425
$792$	0	0
$793$	−4189.52	−0.187609
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7148.58	0.317711	0.158856	−	0.987302i	$-0.449220\pi$
$797$	7148.58	0.317711	0.158856	+	0.987302i	$0.449220\pi$
$798$	0	0
$799$	2905.69	0.128656
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7610.89	−0.334474
$804$	0	0
$805$	5667.76	0.248152
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9156.29	−0.397921	−0.198960	−	0.980008i	$-0.563757\pi$
$809$	−9156.29	−0.397921	−0.198960	+	0.980008i	$0.563757\pi$
$810$	0	0
$811$	21877.2	0.947241	0.473621	−	0.880729i	$-0.342947\pi$
$811$	21877.2	0.947241	0.473621	+	0.880729i	$0.342947\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	15917.3	0.684120
$816$	0	0
$817$	73777.1	3.15928
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33923.3	−1.44206	−0.721030	−	0.692903i	$-0.756331\pi$
$821$	−33923.3	−1.44206	−0.721030	+	0.692903i	$0.756331\pi$
$822$	0	0
$823$	−6357.76	−0.269280	−0.134640	−	0.990895i	$-0.542988\pi$
$823$	−6357.76	−0.269280	−0.134640	+	0.990895i	$0.542988\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8059.61	0.338887	0.169444	−	0.985540i	$-0.445803\pi$
$827$	8059.61	0.338887	0.169444	+	0.985540i	$0.445803\pi$
$828$	0	0
$829$	−25496.5	−1.06819	−0.534095	−	0.845425i	$-0.679347\pi$
$829$	−25496.5	−1.06819	−0.534095	+	0.845425i	$0.679347\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1180.81	−0.0491146
$834$	0	0
$835$	3716.54	0.154031
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31917.6	−1.31337	−0.656686	−	0.754164i	$-0.728042\pi$
$839$	−31917.6	−1.31337	−0.656686	+	0.754164i	$0.728042\pi$
$840$	0	0
$841$	16361.3	0.670846
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8932.76	−0.363664
$846$	0	0
$847$	12834.7	0.520666
$848$	0	0
$849$	0	0
$850$	0	0
$851$	10884.6	0.438448
$852$	0	0
$853$	−28295.7	−1.13579	−0.567893	−	0.823102i	$-0.692241\pi$
$853$	−28295.7	−1.13579	−0.567893	+	0.823102i	$0.692241\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41896.2	1.66995	0.834975	−	0.550288i	$-0.185482\pi$
$857$	41896.2	1.66995	0.834975	+	0.550288i	$0.185482\pi$
$858$	0	0
$859$	−18124.5	−0.719907	−0.359953	−	0.932970i	$-0.617207\pi$
$859$	−18124.5	−0.719907	−0.359953	+	0.932970i	$0.617207\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25361.4	1.00036	0.500182	−	0.865921i	$-0.333267\pi$
$863$	25361.4	1.00036	0.500182	+	0.865921i	$0.333267\pi$
$864$	0	0
$865$	10764.2	0.423115
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5293.66	−0.206646
$870$	0	0
$871$	10122.1	0.393770
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1608.21	−0.0621341
$876$	0	0
$877$	28128.7	1.08306	0.541528	−	0.840683i	$-0.317846\pi$
$877$	28128.7	1.08306	0.541528	+	0.840683i	$0.317846\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12198.2	−0.466481	−0.233240	−	0.972419i	$-0.574933\pi$
$881$	−12198.2	−0.466481	−0.233240	+	0.972419i	$0.574933\pi$
$882$	0	0
$883$	−15725.2	−0.599317	−0.299658	−	0.954047i	$-0.596873\pi$
$883$	−15725.2	−0.599317	−0.299658	+	0.954047i	$0.596873\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3176.83	0.120257	0.0601283	−	0.998191i	$-0.480849\pi$
$887$	3176.83	0.120257	0.0601283	+	0.998191i	$0.480849\pi$
$888$	0	0
$889$	−19041.4	−0.718368
$890$	0	0
$891$	0	0
$892$	0	0
$893$	65933.5	2.47075
$894$	0	0
$895$	−2988.12	−0.111600
$896$	0	0
$897$	0	0
$898$	0	0
$899$	54166.9	2.00953
$900$	0	0
$901$	2265.95	0.0837845
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3681.69	−0.135230
$906$	0	0
$907$	−11400.8	−0.417374	−0.208687	−	0.977982i	$-0.566919\pi$
$907$	−11400.8	−0.417374	−0.208687	+	0.977982i	$0.566919\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9547.78	−0.347236	−0.173618	−	0.984813i	$-0.555546\pi$
$911$	−9547.78	−0.347236	−0.173618	+	0.984813i	$0.555546\pi$
$912$	0	0
$913$	16606.5	0.601966
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1222.79	0.0440351
$918$	0	0
$919$	−47127.5	−1.69161	−0.845807	−	0.533488i	$-0.820881\pi$
$919$	−47127.5	−1.69161	−0.845807	+	0.533488i	$0.820881\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9337.61	−0.332992
$924$	0	0
$925$	−3088.47	−0.109782
$926$	0	0
$927$	0	0
$928$	0	0
$929$	25591.7	0.903807	0.451904	−	0.892067i	$-0.350745\pi$
$929$	25591.7	0.903807	0.451904	+	0.892067i	$0.350745\pi$
$930$	0	0
$931$	−26793.8	−0.943214
$932$	0	0
$933$	0	0
$934$	0	0
$935$	607.436	0.0212463
$936$	0	0
$937$	47575.6	1.65873	0.829364	−	0.558709i	$-0.188703\pi$
$937$	47575.6	1.65873	0.829364	+	0.558709i	$0.188703\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42722.0	1.48002	0.740009	−	0.672597i	$-0.234821\pi$
$941$	42722.0	1.48002	0.740009	+	0.672597i	$0.234821\pi$
$942$	0	0
$943$	−24244.7	−0.837238
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19523.9	−0.669948	−0.334974	−	0.942227i	$-0.608728\pi$
$947$	−19523.9	−0.669948	−0.334974	+	0.942227i	$0.608728\pi$
$948$	0	0
$949$	−8444.52	−0.288852
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6129.73	0.208354	0.104177	−	0.994559i	$-0.466779\pi$
$953$	6129.73	0.208354	0.104177	+	0.994559i	$0.466779\pi$
$954$	0	0
$955$	16775.9	0.568436
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10207.8	−0.343721
$960$	0	0
$961$	42209.8	1.41686
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7607.68	0.253782
$966$	0	0
$967$	−33974.3	−1.12982	−0.564912	−	0.825151i	$-0.691090\pi$
$967$	−33974.3	−1.12982	−0.564912	+	0.825151i	$0.691090\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28328.7	0.936262	0.468131	−	0.883659i	$-0.344928\pi$
$971$	28328.7	0.936262	0.468131	+	0.883659i	$0.344928\pi$
$972$	0	0
$973$	317.484	0.0104605
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37587.6	−1.23084	−0.615421	−	0.788198i	$-0.711014\pi$
$977$	−37587.6	−1.23084	−0.615421	+	0.788198i	$0.711014\pi$
$978$	0	0
$979$	3411.14	0.111359
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5777.92	−0.187474	−0.0937371	−	0.995597i	$-0.529881\pi$
$983$	−5777.92	−0.187474	−0.0937371	+	0.995597i	$0.529881\pi$
$984$	0	0
$985$	1821.71	0.0589283
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−43056.0	−1.38433
$990$	0	0
$991$	−45377.5	−1.45455	−0.727277	−	0.686344i	$-0.759215\pi$
$991$	−45377.5	−1.45455	−0.727277	+	0.686344i	$0.759215\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−858.648	−0.0273578
$996$	0	0
$997$	−6994.90	−0.222197	−0.111099	−	0.993809i	$-0.535437\pi$
$997$	−6994.90	−0.222197	−0.111099	+	0.993809i	$0.535437\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.a.m.1.1	yes	3
3.2	odd	2		1080.4.a.g.1.1	✓	3
4.3	odd	2		2160.4.a.bo.1.3		3
12.11	even	2		2160.4.a.bg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.g.1.1	✓	3	3.2	odd	2
1080.4.a.m.1.1	yes	3	1.1	even	1	trivial
2160.4.a.bg.1.3		3	12.11	even	2
2160.4.a.bo.1.3		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -12.8657 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} -12.8657 q^{7} +18.2595 q^{11} +20.2595 q^{13} +6.65336 q^{17} +150.972 q^{19} -88.1068 q^{23} +25.0000 q^{25} -201.867 q^{29} -268.330 q^{31} -64.3283 q^{35} -123.539 q^{37} +275.174 q^{41} +488.679 q^{43} +436.725 q^{47} -177.475 q^{49} +340.573 q^{53} +91.2975 q^{55} +548.360 q^{59} -206.793 q^{61} +101.298 q^{65} +499.621 q^{67} -460.900 q^{71} -416.818 q^{73} -234.921 q^{77} -289.912 q^{79} +909.471 q^{83} +33.2668 q^{85} +186.814 q^{89} -260.652 q^{91} +754.862 q^{95} +648.440 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} + 6 q^{7}+O(q^{10}) \) 3 * q + 15 * q^5 + 6 * q^7	\( 3 q + 15 q^{5} + 6 q^{7} + 12 q^{11} + 18 q^{13} - 21 q^{17} + 57 q^{19} + 87 q^{23} + 75 q^{25} + 138 q^{29} + 117 q^{31} + 30 q^{35} + 150 q^{37} + 180 q^{43} + 684 q^{47} - 81 q^{49} + 87 q^{53} + 60 q^{55} + 714 q^{59} - 513 q^{61} + 90 q^{65} - 174 q^{67} + 768 q^{71} - 252 q^{73} + 888 q^{77} + 207 q^{79} + 1689 q^{83} - 105 q^{85} + 312 q^{89} + 900 q^{91} + 285 q^{95} - 1080 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 + 6 * q^7 + 12 * q^11 + 18 * q^13 - 21 * q^17 + 57 * q^19 + 87 * q^23 + 75 * q^25 + 138 * q^29 + 117 * q^31 + 30 * q^35 + 150 * q^37 + 180 * q^43 + 684 * q^47 - 81 * q^49 + 87 * q^53 + 60 * q^55 + 714 * q^59 - 513 * q^61 + 90 * q^65 - 174 * q^67 + 768 * q^71 - 252 * q^73 + 888 * q^77 + 207 * q^79 + 1689 * q^83 - 105 * q^85 + 312 * q^89 + 900 * q^91 + 285 * q^95 - 1080 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more