LMFDB - Embedded newform 1440.4.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,4,Mod(1,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, 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("1440.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1440.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:	$84.9627504083$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{89}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 22 $ x^2 - x - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.21699$ of defining polynomial
Character	$\chi$	$=$	1440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} +12.8680 q^{7} +49.7359 q^{11} -52.6039 q^{13} -84.6039 q^{17} +26.6039 q^{19} +136.340 q^{23} +25.0000 q^{25} -6.00000 q^{29} -47.1320 q^{31} -64.3398 q^{35} +344.227 q^{37} +43.2078 q^{41} -252.000 q^{43} +306.076 q^{47} -177.416 q^{49} +455.208 q^{53} -248.680 q^{55} +708.151 q^{59} -652.039 q^{61} +263.019 q^{65} -704.528 q^{67} -531.775 q^{71} +57.6233 q^{73} +640.000 q^{77} -429.699 q^{79} +227.697 q^{83} +423.019 q^{85} -1032.87 q^{89} -676.905 q^{91} -133.019 q^{95} -152.416 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} - 12 q^{7} + 24 q^{11} + 8 q^{13} - 56 q^{17} - 60 q^{19} + 84 q^{23} + 50 q^{25} - 12 q^{29} - 132 q^{31} + 60 q^{35} - 104 q^{37} - 140 q^{41} - 504 q^{43} + 348 q^{47} + 98 q^{49} + 684 q^{53}+ \cdots + 148 q^{97}+O(q^{100}) $ 2 * q - 10 * q^5 - 12 * q^7 + 24 * q^11 + 8 * q^13 - 56 * q^17 - 60 * q^19 + 84 * q^23 + 50 * q^25 - 12 * q^29 - 132 * q^31 + 60 * q^35 - 104 * q^37 - 140 * q^41 - 504 * q^43 + 348 * q^47 + 98 * q^49 + 684 * q^53 - 120 * q^55 + 888 * q^59 - 172 * q^61 - 40 * q^65 - 1560 * q^67 + 144 * q^71 - 564 * q^73 + 1280 * q^77 + 84 * q^79 + 1512 * q^83 + 280 * q^85 - 28 * q^89 - 2184 * q^91 + 300 * q^95 + 148 * q^97

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
$7$	12.8680	0.694805	0.347402	−	0.937716i	$-0.387064\pi$
$7$	12.8680	0.694805	0.347402	+	0.937716i	$0.387064\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	49.7359	1.36327	0.681634	−	0.731693i	$-0.261270\pi$
$11$	49.7359	1.36327	0.681634	+	0.731693i	$0.261270\pi$
$12$	0	0
$13$	−52.6039	−1.12228	−0.561142	−	0.827720i	$-0.689638\pi$
$13$	−52.6039	−1.12228	−0.561142	+	0.827720i	$0.689638\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−84.6039	−1.20703	−0.603513	−	0.797353i	$-0.706233\pi$
$17$	−84.6039	−1.20703	−0.603513	+	0.797353i	$0.706233\pi$
$18$	0	0
$19$	26.6039	0.321229	0.160614	−	0.987017i	$-0.448652\pi$
$19$	26.6039	0.321229	0.160614	+	0.987017i	$0.448652\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	136.340	1.23604	0.618018	−	0.786164i	$-0.287936\pi$
$23$	136.340	1.23604	0.618018	+	0.786164i	$0.287936\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.00000	−0.0384197	−0.0192099	−	0.999815i	$-0.506115\pi$
$29$	−6.00000	−0.0384197	−0.0192099	+	0.999815i	$0.506115\pi$
$30$	0	0
$31$	−47.1320	−0.273070	−0.136535	−	0.990635i	$-0.543597\pi$
$31$	−47.1320	−0.273070	−0.136535	+	0.990635i	$0.543597\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−64.3398	−0.310726
$36$	0	0
$37$	344.227	1.52948	0.764738	−	0.644341i	$-0.222868\pi$
$37$	344.227	1.52948	0.764738	+	0.644341i	$0.222868\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	43.2078	0.164583	0.0822917	−	0.996608i	$-0.473776\pi$
$41$	43.2078	0.164583	0.0822917	+	0.996608i	$0.473776\pi$
$42$	0	0
$43$	−252.000	−0.893713	−0.446856	−	0.894606i	$-0.647456\pi$
$43$	−252.000	−0.893713	−0.446856	+	0.894606i	$0.647456\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	306.076	0.949909	0.474955	−	0.880010i	$-0.342464\pi$
$47$	306.076	0.949909	0.474955	+	0.880010i	$0.342464\pi$
$48$	0	0
$49$	−177.416	−0.517246
$50$	0	0
$51$	0	0
$52$	0	0
$53$	455.208	1.17977	0.589883	−	0.807489i	$-0.299174\pi$
$53$	455.208	1.17977	0.589883	+	0.807489i	$0.299174\pi$
$54$	0	0
$55$	−248.680	−0.609672
$56$	0	0
$57$	0	0
$58$	0	0
$59$	708.151	1.56260	0.781301	−	0.624155i	$-0.214557\pi$
$59$	708.151	1.56260	0.781301	+	0.624155i	$0.214557\pi$
$60$	0	0
$61$	−652.039	−1.36861	−0.684303	−	0.729197i	$-0.739894\pi$
$61$	−652.039	−1.36861	−0.684303	+	0.729197i	$0.739894\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	263.019	0.501901
$66$	0	0
$67$	−704.528	−1.28465	−0.642327	−	0.766431i	$-0.722031\pi$
$67$	−704.528	−1.28465	−0.642327	+	0.766431i	$0.722031\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−531.775	−0.888874	−0.444437	−	0.895810i	$-0.646596\pi$
$71$	−531.775	−0.888874	−0.444437	+	0.895810i	$0.646596\pi$
$72$	0	0
$73$	57.6233	0.0923877	0.0461938	−	0.998932i	$-0.485291\pi$
$73$	57.6233	0.0923877	0.0461938	+	0.998932i	$0.485291\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	640.000	0.947205
$78$	0	0
$79$	−429.699	−0.611961	−0.305981	−	0.952038i	$-0.598984\pi$
$79$	−429.699	−0.611961	−0.305981	+	0.952038i	$0.598984\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	227.697	0.301120	0.150560	−	0.988601i	$-0.451892\pi$
$83$	227.697	0.301120	0.150560	+	0.988601i	$0.451892\pi$
$84$	0	0
$85$	423.019	0.539799
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1032.87	−1.23016	−0.615079	−	0.788466i	$-0.710876\pi$
$89$	−1032.87	−1.23016	−0.615079	+	0.788466i	$0.710876\pi$
$90$	0	0
$91$	−676.905	−0.779768
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−133.019	−0.143658
$96$	0	0
$97$	−152.416	−0.159541	−0.0797704	−	0.996813i	$-0.525419\pi$
$97$	−152.416	−0.159541	−0.0797704	+	0.996813i	$0.525419\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	290.753	0.286446	0.143223	−	0.989690i	$-0.454253\pi$
$101$	290.753	0.286446	0.143223	+	0.989690i	$0.454253\pi$
$102$	0	0
$103$	5.62132	0.00537753	0.00268876	−	0.999996i	$-0.499144\pi$
$103$	5.62132	0.00537753	0.00268876	+	0.999996i	$0.499144\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2170.57	1.96109	0.980545	−	0.196294i	$-0.0628908\pi$
$107$	2170.57	1.96109	0.980545	+	0.196294i	$0.0628908\pi$
$108$	0	0
$109$	1411.58	1.24042	0.620208	−	0.784438i	$-0.287048\pi$
$109$	1411.58	1.24042	0.620208	+	0.784438i	$0.287048\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1557.81	1.29687	0.648436	−	0.761269i	$-0.275423\pi$
$113$	1557.81	1.29687	0.648436	+	0.761269i	$0.275423\pi$
$114$	0	0
$115$	−681.699	−0.552772
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1088.68	−0.838648
$120$	0	0
$121$	1142.66	0.858499
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1920.19	1.34165	0.670825	−	0.741616i	$-0.265940\pi$
$127$	1920.19	1.34165	0.670825	+	0.741616i	$0.265940\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	600.303	0.400372	0.200186	−	0.979758i	$-0.435845\pi$
$131$	600.303	0.400372	0.200186	+	0.979758i	$0.435845\pi$
$132$	0	0
$133$	342.338	0.223191
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1953.51	1.21825	0.609124	−	0.793075i	$-0.291521\pi$
$137$	1953.51	1.21825	0.609124	+	0.793075i	$0.291521\pi$
$138$	0	0
$139$	2261.25	1.37983	0.689916	−	0.723890i	$-0.257648\pi$
$139$	2261.25	1.37983	0.689916	+	0.723890i	$0.257648\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2616.30	−1.52997
$144$	0	0
$145$	30.0000	0.0171818
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2938.16	1.61546	0.807728	−	0.589555i	$-0.200697\pi$
$149$	2938.16	1.61546	0.807728	+	0.589555i	$0.200697\pi$
$150$	0	0
$151$	1053.40	0.567710	0.283855	−	0.958867i	$-0.408387\pi$
$151$	1053.40	0.567710	0.283855	+	0.958867i	$0.408387\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	235.660	0.122121
$156$	0	0
$157$	2789.43	1.41797	0.708985	−	0.705224i	$-0.249154\pi$
$157$	2789.43	1.41797	0.708985	+	0.705224i	$0.249154\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1754.42	0.858803
$162$	0	0
$163$	−2170.35	−1.04291	−0.521456	−	0.853278i	$-0.674611\pi$
$163$	−2170.35	−1.04291	−0.521456	+	0.853278i	$0.674611\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3175.74	1.47153	0.735767	−	0.677234i	$-0.236822\pi$
$167$	3175.74	1.47153	0.735767	+	0.677234i	$0.236822\pi$
$168$	0	0
$169$	570.169	0.259522
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−947.740	−0.416505	−0.208252	−	0.978075i	$-0.566778\pi$
$173$	−947.740	−0.416505	−0.208252	+	0.978075i	$0.566778\pi$
$174$	0	0
$175$	321.699	0.138961
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3003.47	−1.25413	−0.627067	−	0.778965i	$-0.715745\pi$
$179$	−3003.47	−1.25413	−0.627067	+	0.778965i	$0.715745\pi$
$180$	0	0
$181$	1502.91	0.617184	0.308592	−	0.951194i	$-0.400142\pi$
$181$	1502.91	0.617184	0.308592	+	0.951194i	$0.400142\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1721.14	−0.684002
$186$	0	0
$187$	−4207.85	−1.64550
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3144.00	−1.19106	−0.595528	−	0.803334i	$-0.703057\pi$
$191$	−3144.00	−1.19106	−0.595528	+	0.803334i	$0.703057\pi$
$192$	0	0
$193$	−3100.87	−1.15651	−0.578253	−	0.815858i	$-0.696265\pi$
$193$	−3100.87	−1.15651	−0.578253	+	0.815858i	$0.696265\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2563.13	0.926982	0.463491	−	0.886102i	$-0.346597\pi$
$197$	2563.13	0.926982	0.463491	+	0.886102i	$0.346597\pi$
$198$	0	0
$199$	2929.17	1.04343	0.521717	−	0.853119i	$-0.325292\pi$
$199$	2929.17	1.04343	0.521717	+	0.853119i	$0.325292\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−77.2078	−0.0266942
$204$	0	0
$205$	−216.039	−0.0736039
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1323.17	0.437921
$210$	0	0
$211$	3876.80	1.26488	0.632440	−	0.774609i	$-0.282053\pi$
$211$	3876.80	1.26488	0.632440	+	0.774609i	$0.282053\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1260.00	0.399680
$216$	0	0
$217$	−606.493	−0.189730
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4450.49	1.35463
$222$	0	0
$223$	4268.94	1.28193	0.640963	−	0.767572i	$-0.278535\pi$
$223$	4268.94	1.28193	0.640963	+	0.767572i	$0.278535\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4191.03	−1.22541	−0.612706	−	0.790311i	$-0.709919\pi$
$227$	−4191.03	−1.22541	−0.612706	+	0.790311i	$0.709919\pi$
$228$	0	0
$229$	2036.12	0.587556	0.293778	−	0.955874i	$-0.405087\pi$
$229$	2036.12	0.587556	0.293778	+	0.955874i	$0.405087\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4544.76	1.27784	0.638921	−	0.769273i	$-0.279381\pi$
$233$	4544.76	1.27784	0.638921	+	0.769273i	$0.279381\pi$
$234$	0	0
$235$	−1530.38	−0.424812
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2068.00	−0.559697	−0.279848	−	0.960044i	$-0.590284\pi$
$239$	−2068.00	−0.559697	−0.279848	+	0.960044i	$0.590284\pi$
$240$	0	0
$241$	−1901.25	−0.508175	−0.254087	−	0.967181i	$-0.581775\pi$
$241$	−1901.25	−0.508175	−0.254087	+	0.967181i	$0.581775\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	887.078	0.231320
$246$	0	0
$247$	−1399.47	−0.360510
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3794.42	0.954191	0.477095	−	0.878852i	$-0.341690\pi$
$251$	3794.42	0.954191	0.477095	+	0.878852i	$0.341690\pi$
$252$	0	0
$253$	6780.99	1.68505
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3103.33	−0.753231	−0.376616	−	0.926370i	$-0.622912\pi$
$257$	−3103.33	−0.753231	−0.376616	+	0.926370i	$0.622912\pi$
$258$	0	0
$259$	4429.50	1.06269
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4110.83	−0.963821	−0.481910	−	0.876221i	$-0.660057\pi$
$263$	−4110.83	−0.963821	−0.481910	+	0.876221i	$0.660057\pi$
$264$	0	0
$265$	−2276.04	−0.527607
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4272.34	0.968361	0.484180	−	0.874968i	$-0.339118\pi$
$269$	4272.34	0.968361	0.484180	+	0.874968i	$0.339118\pi$
$270$	0	0
$271$	4396.64	0.985524	0.492762	−	0.870164i	$-0.335987\pi$
$271$	4396.64	0.985524	0.492762	+	0.870164i	$0.335987\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1243.40	0.272654
$276$	0	0
$277$	−3068.54	−0.665598	−0.332799	−	0.942998i	$-0.607993\pi$
$277$	−3068.54	−0.665598	−0.332799	+	0.942998i	$0.607993\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2672.71	−0.567405	−0.283702	−	0.958912i	$-0.591563\pi$
$281$	−2672.71	−0.567405	−0.283702	+	0.958912i	$0.591563\pi$
$282$	0	0
$283$	6135.64	1.28878	0.644392	−	0.764696i	$-0.277111\pi$
$283$	6135.64	1.28878	0.644392	+	0.764696i	$0.277111\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	555.996	0.114353
$288$	0	0
$289$	2244.82	0.456914
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−40.7922	−0.00813347	−0.00406674	−	0.999992i	$-0.501294\pi$
$293$	−40.7922	−0.00813347	−0.00406674	+	0.999992i	$0.501294\pi$
$294$	0	0
$295$	−3540.76	−0.698816
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7172.00	−1.38718
$300$	0	0
$301$	−3242.73	−0.620956
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3260.19	0.612060
$306$	0	0
$307$	−7126.80	−1.32491	−0.662456	−	0.749101i	$-0.730486\pi$
$307$	−7126.80	−1.32491	−0.662456	+	0.749101i	$0.730486\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1704.52	0.310787	0.155393	−	0.987853i	$-0.450335\pi$
$311$	1704.52	0.310787	0.155393	+	0.987853i	$0.450335\pi$
$312$	0	0
$313$	5343.52	0.964963	0.482482	−	0.875906i	$-0.339735\pi$
$313$	5343.52	0.964963	0.482482	+	0.875906i	$0.339735\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5644.19	1.00003	0.500015	−	0.866017i	$-0.333328\pi$
$317$	5644.19	1.00003	0.500015	+	0.866017i	$0.333328\pi$
$318$	0	0
$319$	−298.416	−0.0523764
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2250.79	−0.387732
$324$	0	0
$325$	−1315.10	−0.224457
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3938.57	0.660001
$330$	0	0
$331$	−2361.55	−0.392152	−0.196076	−	0.980589i	$-0.562820\pi$
$331$	−2361.55	−0.392152	−0.196076	+	0.980589i	$0.562820\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3522.64	0.574515
$336$	0	0
$337$	5117.10	0.827141	0.413570	−	0.910472i	$-0.364282\pi$
$337$	5117.10	0.827141	0.413570	+	0.910472i	$0.364282\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2344.16	−0.372267
$342$	0	0
$343$	−6696.69	−1.05419
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6087.39	0.941753	0.470876	−	0.882199i	$-0.343938\pi$
$347$	6087.39	0.941753	0.470876	+	0.882199i	$0.343938\pi$
$348$	0	0
$349$	7827.52	1.20057	0.600283	−	0.799788i	$-0.295055\pi$
$349$	7827.52	1.20057	0.600283	+	0.799788i	$0.295055\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2515.47	−0.379278	−0.189639	−	0.981854i	$-0.560732\pi$
$353$	−2515.47	−0.379278	−0.189639	+	0.981854i	$0.560732\pi$
$354$	0	0
$355$	2658.87	0.397517
$356$	0	0
$357$	0	0
$358$	0	0
$359$	915.914	0.134652	0.0673261	−	0.997731i	$-0.478553\pi$
$359$	915.914	0.134652	0.0673261	+	0.997731i	$0.478553\pi$
$360$	0	0
$361$	−6151.23	−0.896812
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−288.117	−0.0413170
$366$	0	0
$367$	−12237.6	−1.74060	−0.870298	−	0.492525i	$-0.836074\pi$
$367$	−12237.6	−1.74060	−0.870298	+	0.492525i	$0.836074\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5857.60	0.819707
$372$	0	0
$373$	3825.20	0.530996	0.265498	−	0.964111i	$-0.414464\pi$
$373$	3825.20	0.530996	0.265498	+	0.964111i	$0.414464\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	315.623	0.0431178
$378$	0	0
$379$	8197.63	1.11104	0.555519	−	0.831504i	$-0.312519\pi$
$379$	8197.63	1.11104	0.555519	+	0.831504i	$0.312519\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	452.196	0.0603294	0.0301647	−	0.999545i	$-0.490397\pi$
$383$	452.196	0.0603294	0.0301647	+	0.999545i	$0.490397\pi$
$384$	0	0
$385$	−3200.00	−0.423603
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6816.13	−0.888410	−0.444205	−	0.895925i	$-0.646514\pi$
$389$	−6816.13	−0.888410	−0.444205	+	0.895925i	$0.646514\pi$
$390$	0	0
$391$	−11534.9	−1.49193
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2148.50	0.273677
$396$	0	0
$397$	13380.3	1.69153	0.845766	−	0.533554i	$-0.179144\pi$
$397$	13380.3	1.69153	0.845766	+	0.533554i	$0.179144\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6130.00	−0.763386	−0.381693	−	0.924289i	$-0.624659\pi$
$401$	−6130.00	−0.763386	−0.381693	+	0.924289i	$0.624659\pi$
$402$	0	0
$403$	2479.33	0.306462
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17120.5	2.08509
$408$	0	0
$409$	−15003.5	−1.81387	−0.906935	−	0.421270i	$-0.861584\pi$
$409$	−15003.5	−1.81387	−0.906935	+	0.421270i	$0.861584\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9112.47	1.08570
$414$	0	0
$415$	−1138.49	−0.134665
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−16300.7	−1.90057	−0.950287	−	0.311375i	$-0.899210\pi$
$419$	−16300.7	−1.90057	−0.950287	+	0.311375i	$0.899210\pi$
$420$	0	0
$421$	7663.91	0.887211	0.443606	−	0.896222i	$-0.353699\pi$
$421$	7663.91	0.887211	0.443606	+	0.896222i	$0.353699\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2115.10	−0.241405
$426$	0	0
$427$	−8390.41	−0.950914
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3702.20	−0.413755	−0.206878	−	0.978367i	$-0.566330\pi$
$431$	−3702.20	−0.413755	−0.206878	+	0.978367i	$0.566330\pi$
$432$	0	0
$433$	−10275.8	−1.14047	−0.570237	−	0.821481i	$-0.693148\pi$
$433$	−10275.8	−1.14047	−0.570237	+	0.821481i	$0.693148\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3627.17	0.397050
$438$	0	0
$439$	−14178.6	−1.54148	−0.770738	−	0.637152i	$-0.780112\pi$
$439$	−14178.6	−1.54148	−0.770738	+	0.637152i	$0.780112\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13395.6	1.43666	0.718332	−	0.695700i	$-0.244906\pi$
$443$	13395.6	1.43666	0.718332	+	0.695700i	$0.244906\pi$
$444$	0	0
$445$	5164.35	0.550143
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9490.39	0.997504	0.498752	−	0.866745i	$-0.333792\pi$
$449$	9490.39	0.997504	0.498752	+	0.866745i	$0.333792\pi$
$450$	0	0
$451$	2148.98	0.224371
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3384.52	0.348723
$456$	0	0
$457$	587.064	0.0600913	0.0300456	−	0.999549i	$-0.490435\pi$
$457$	587.064	0.0600913	0.0300456	+	0.999549i	$0.490435\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2800.42	−0.282925	−0.141462	−	0.989944i	$-0.545180\pi$
$461$	−2800.42	−0.282925	−0.141462	+	0.989944i	$0.545180\pi$
$462$	0	0
$463$	19745.6	1.98198	0.990988	−	0.133952i	$-0.0427668\pi$
$463$	19745.6	1.98198	0.990988	+	0.133952i	$0.0427668\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13552.0	1.34285	0.671427	−	0.741071i	$-0.265682\pi$
$467$	13552.0	1.34285	0.671427	+	0.741071i	$0.265682\pi$
$468$	0	0
$469$	−9065.84	−0.892584
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−12533.5	−1.21837
$474$	0	0
$475$	665.097	0.0642458
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14131.9	1.34803	0.674014	−	0.738719i	$-0.264569\pi$
$479$	14131.9	1.34803	0.674014	+	0.738719i	$0.264569\pi$
$480$	0	0
$481$	−18107.7	−1.71651
$482$	0	0
$483$	0	0
$484$	0	0
$485$	762.078	0.0713488
$486$	0	0
$487$	−7958.52	−0.740524	−0.370262	−	0.928927i	$-0.620732\pi$
$487$	−7958.52	−0.740524	−0.370262	+	0.928927i	$0.620732\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10984.3	−1.00960	−0.504799	−	0.863237i	$-0.668433\pi$
$491$	−10984.3	−1.00960	−0.504799	+	0.863237i	$0.668433\pi$
$492$	0	0
$493$	507.623	0.0463736
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6842.86	−0.617594
$498$	0	0
$499$	17710.3	1.58882	0.794411	−	0.607381i	$-0.207780\pi$
$499$	17710.3	1.58882	0.794411	+	0.607381i	$0.207780\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2279.89	−0.202098	−0.101049	−	0.994881i	$-0.532220\pi$
$503$	−2279.89	−0.202098	−0.101049	+	0.994881i	$0.532220\pi$
$504$	0	0
$505$	−1453.77	−0.128103
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3254.99	0.283447	0.141724	−	0.989906i	$-0.454736\pi$
$509$	3254.99	0.283447	0.141724	+	0.989906i	$0.454736\pi$
$510$	0	0
$511$	741.495	0.0641914
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−28.1066	−0.00240490
$516$	0	0
$517$	15223.0	1.29498
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21591.2	−1.81560	−0.907799	−	0.419406i	$-0.862238\pi$
$521$	−21591.2	−1.81560	−0.907799	+	0.419406i	$0.862238\pi$
$522$	0	0
$523$	−16868.8	−1.41037	−0.705185	−	0.709024i	$-0.749136\pi$
$523$	−16868.8	−1.41037	−0.705185	+	0.709024i	$0.749136\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3987.55	0.329603
$528$	0	0
$529$	6421.54	0.527784
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2272.90	−0.184709
$534$	0	0
$535$	−10852.8	−0.877026
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−8823.93	−0.705145
$540$	0	0
$541$	−4366.23	−0.346985	−0.173493	−	0.984835i	$-0.555505\pi$
$541$	−4366.23	−0.346985	−0.173493	+	0.984835i	$0.555505\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7057.92	−0.554731
$546$	0	0
$547$	16579.7	1.29597	0.647987	−	0.761652i	$-0.275611\pi$
$547$	16579.7	1.29597	0.647987	+	0.761652i	$0.275611\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−159.623	−0.0123415
$552$	0	0
$553$	−5529.35	−0.425193
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5971.66	0.454268	0.227134	−	0.973863i	$-0.427064\pi$
$557$	5971.66	0.454268	0.227134	+	0.973863i	$0.427064\pi$
$558$	0	0
$559$	13256.2	1.00300
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15561.2	−1.16488	−0.582441	−	0.812873i	$-0.697902\pi$
$563$	−15561.2	−1.16488	−0.582441	+	0.812873i	$0.697902\pi$
$564$	0	0
$565$	−7789.06	−0.579979
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11871.6	−0.874662	−0.437331	−	0.899301i	$-0.644076\pi$
$569$	−11871.6	−0.874662	−0.437331	+	0.899301i	$0.644076\pi$
$570$	0	0
$571$	−18660.0	−1.36759	−0.683797	−	0.729672i	$-0.739673\pi$
$571$	−18660.0	−1.36759	−0.683797	+	0.729672i	$0.739673\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3408.50	0.247207
$576$	0	0
$577$	−7549.12	−0.544669	−0.272334	−	0.962203i	$-0.587796\pi$
$577$	−7549.12	−0.544669	−0.272334	+	0.962203i	$0.587796\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2930.00	0.209220
$582$	0	0
$583$	22640.2	1.60834
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4513.39	−0.317355	−0.158678	−	0.987330i	$-0.550723\pi$
$587$	−4513.39	−0.317355	−0.158678	+	0.987330i	$0.550723\pi$
$588$	0	0
$589$	−1253.90	−0.0877179
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8335.46	−0.577228	−0.288614	−	0.957445i	$-0.593194\pi$
$593$	−8335.46	−0.577228	−0.288614	+	0.957445i	$0.593194\pi$
$594$	0	0
$595$	5443.40	0.375055
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10806.9	0.737162	0.368581	−	0.929596i	$-0.379844\pi$
$599$	10806.9	0.737162	0.368581	+	0.929596i	$0.379844\pi$
$600$	0	0
$601$	−25653.1	−1.74112	−0.870559	−	0.492063i	$-0.836243\pi$
$601$	−25653.1	−1.74112	−0.870559	+	0.492063i	$0.836243\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5713.31	−0.383932
$606$	0	0
$607$	−8879.17	−0.593730	−0.296865	−	0.954919i	$-0.595941\pi$
$607$	−8879.17	−0.593730	−0.296865	+	0.954919i	$0.595941\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16100.8	−1.06607
$612$	0	0
$613$	3846.28	0.253425	0.126713	−	0.991939i	$-0.459557\pi$
$613$	3846.28	0.253425	0.126713	+	0.991939i	$0.459557\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15821.2	1.03231	0.516156	−	0.856495i	$-0.327363\pi$
$617$	15821.2	1.03231	0.516156	+	0.856495i	$0.327363\pi$
$618$	0	0
$619$	−11518.7	−0.747942	−0.373971	−	0.927440i	$-0.622004\pi$
$619$	−11518.7	−0.747942	−0.373971	+	0.927440i	$0.622004\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13290.9	−0.854719
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29123.0	−1.84612
$630$	0	0
$631$	9053.33	0.571169	0.285584	−	0.958354i	$-0.407812\pi$
$631$	9053.33	0.571169	0.285584	+	0.958354i	$0.407812\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9600.96	−0.600004
$636$	0	0
$637$	9332.75	0.580498
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1700.77	−0.104799	−0.0523996	−	0.998626i	$-0.516687\pi$
$641$	−1700.77	−0.104799	−0.0523996	+	0.998626i	$0.516687\pi$
$642$	0	0
$643$	−17275.1	−1.05951	−0.529755	−	0.848151i	$-0.677716\pi$
$643$	−17275.1	−1.05951	−0.529755	+	0.848151i	$0.677716\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11182.2	−0.679468	−0.339734	−	0.940522i	$-0.610337\pi$
$647$	−11182.2	−0.679468	−0.339734	+	0.940522i	$0.610337\pi$
$648$	0	0
$649$	35220.6	2.13024
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20720.5	−1.24174	−0.620869	−	0.783915i	$-0.713220\pi$
$653$	−20720.5	−1.24174	−0.620869	+	0.783915i	$0.713220\pi$
$654$	0	0
$655$	−3001.51	−0.179052
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8745.26	−0.516945	−0.258473	−	0.966019i	$-0.583219\pi$
$659$	−8745.26	−0.516945	−0.258473	+	0.966019i	$0.583219\pi$
$660$	0	0
$661$	−25921.8	−1.52533	−0.762663	−	0.646796i	$-0.776108\pi$
$661$	−25921.8	−1.52533	−0.762663	+	0.646796i	$0.776108\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1711.69	−0.0998142
$666$	0	0
$667$	−818.039	−0.0474881
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−32429.8	−1.86578
$672$	0	0
$673$	−12111.4	−0.693702	−0.346851	−	0.937920i	$-0.612749\pi$
$673$	−12111.4	−0.693702	−0.346851	+	0.937920i	$0.612749\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4961.50	−0.281663	−0.140832	−	0.990034i	$-0.544978\pi$
$677$	−4961.50	−0.281663	−0.140832	+	0.990034i	$0.544978\pi$
$678$	0	0
$679$	−1961.28	−0.110850
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24838.1	1.39151	0.695755	−	0.718279i	$-0.255070\pi$
$683$	24838.1	1.39151	0.695755	+	0.718279i	$0.255070\pi$
$684$	0	0
$685$	−9767.56	−0.544817
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−23945.7	−1.32403
$690$	0	0
$691$	6857.65	0.377536	0.188768	−	0.982022i	$-0.439551\pi$
$691$	6857.65	0.377536	0.188768	+	0.982022i	$0.439551\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11306.2	−0.617079
$696$	0	0
$697$	−3655.55	−0.198657
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−32760.0	−1.76509	−0.882545	−	0.470228i	$-0.844172\pi$
$701$	−32760.0	−1.76509	−0.882545	+	0.470228i	$0.844172\pi$
$702$	0	0
$703$	9157.78	0.491312
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3741.40	0.199024
$708$	0	0
$709$	8678.69	0.459711	0.229855	−	0.973225i	$-0.426175\pi$
$709$	8678.69	0.459711	0.229855	+	0.973225i	$0.426175\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6425.97	−0.337524
$714$	0	0
$715$	13081.5	0.684225
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8160.85	−0.423294	−0.211647	−	0.977346i	$-0.567883\pi$
$719$	−8160.85	−0.423294	−0.211647	+	0.977346i	$0.567883\pi$
$720$	0	0
$721$	72.3349	0.00373633
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−150.000	−0.00768395
$726$	0	0
$727$	23907.3	1.21963	0.609816	−	0.792543i	$-0.291243\pi$
$727$	23907.3	1.21963	0.609816	+	0.792543i	$0.291243\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	21320.2	1.07874
$732$	0	0
$733$	−28946.6	−1.45862	−0.729309	−	0.684185i	$-0.760158\pi$
$733$	−28946.6	−1.45862	−0.729309	+	0.684185i	$0.760158\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35040.4	−1.75133
$738$	0	0
$739$	39038.5	1.94324	0.971620	−	0.236547i	$-0.0760157\pi$
$739$	39038.5	1.94324	0.971620	+	0.236547i	$0.0760157\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4175.77	0.206183	0.103092	−	0.994672i	$-0.467126\pi$
$743$	4175.77	0.206183	0.103092	+	0.994672i	$0.467126\pi$
$744$	0	0
$745$	−14690.8	−0.722454
$746$	0	0
$747$	0	0
$748$	0	0
$749$	27930.8	1.36257
$750$	0	0
$751$	22219.1	1.07961	0.539805	−	0.841790i	$-0.318498\pi$
$751$	22219.1	1.07961	0.539805	+	0.841790i	$0.318498\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5266.98	−0.253887
$756$	0	0
$757$	22514.1	1.08096	0.540482	−	0.841355i	$-0.318242\pi$
$757$	22514.1	1.08096	0.540482	+	0.841355i	$0.318242\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26354.3	1.25538	0.627689	−	0.778465i	$-0.284001\pi$
$761$	26354.3	1.25538	0.627689	+	0.778465i	$0.284001\pi$
$762$	0	0
$763$	18164.2	0.861846
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−37251.5	−1.75368
$768$	0	0
$769$	−26226.3	−1.22983	−0.614917	−	0.788591i	$-0.710811\pi$
$769$	−26226.3	−1.22983	−0.614917	+	0.788591i	$0.710811\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28023.6	1.30393	0.651965	−	0.758249i	$-0.273945\pi$
$773$	28023.6	1.30393	0.651965	+	0.758249i	$0.273945\pi$
$774$	0	0
$775$	−1178.30	−0.0546140
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1149.49	0.0528690
$780$	0	0
$781$	−26448.3	−1.21177
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13947.2	−0.634135
$786$	0	0
$787$	16361.2	0.741061	0.370530	−	0.928820i	$-0.379176\pi$
$787$	16361.2	0.741061	0.370530	+	0.928820i	$0.379176\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	20045.9	0.901073
$792$	0	0
$793$	34299.8	1.53597
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20999.9	0.933319	0.466660	−	0.884437i	$-0.345457\pi$
$797$	20999.9	0.933319	0.466660	+	0.884437i	$0.345457\pi$
$798$	0	0
$799$	−25895.2	−1.14657
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2865.95	0.125949
$804$	0	0
$805$	−8772.08	−0.384068
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29390.4	1.27727	0.638635	−	0.769510i	$-0.279499\pi$
$809$	29390.4	1.27727	0.638635	+	0.769510i	$0.279499\pi$
$810$	0	0
$811$	−17296.9	−0.748924	−0.374462	−	0.927242i	$-0.622173\pi$
$811$	−17296.9	−0.748924	−0.374462	+	0.927242i	$0.622173\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	10851.7	0.466404
$816$	0	0
$817$	−6704.18	−0.287086
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19989.1	−0.849725	−0.424862	−	0.905258i	$-0.639678\pi$
$821$	−19989.1	−0.849725	−0.424862	+	0.905258i	$0.639678\pi$
$822$	0	0
$823$	−20226.7	−0.856694	−0.428347	−	0.903614i	$-0.640904\pi$
$823$	−20226.7	−0.856694	−0.428347	+	0.903614i	$0.640904\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24124.7	1.01439	0.507193	−	0.861833i	$-0.330683\pi$
$827$	24124.7	1.01439	0.507193	+	0.861833i	$0.330683\pi$
$828$	0	0
$829$	43594.3	1.82641	0.913204	−	0.407503i	$-0.133600\pi$
$829$	43594.3	1.82641	0.913204	+	0.407503i	$0.133600\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	15010.0	0.624330
$834$	0	0
$835$	−15878.7	−0.658090
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29937.7	1.23190	0.615949	−	0.787786i	$-0.288773\pi$
$839$	29937.7	1.23190	0.615949	+	0.787786i	$0.288773\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2850.84	−0.116062
$846$	0	0
$847$	14703.7	0.596489
$848$	0	0
$849$	0	0
$850$	0	0
$851$	46931.9	1.89049
$852$	0	0
$853$	−2147.83	−0.0862136	−0.0431068	−	0.999070i	$-0.513726\pi$
$853$	−2147.83	−0.0862136	−0.0431068	+	0.999070i	$0.513726\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15771.5	−0.628638	−0.314319	−	0.949317i	$-0.601776\pi$
$857$	−15771.5	−0.628638	−0.314319	+	0.949317i	$0.601776\pi$
$858$	0	0
$859$	−29364.8	−1.16637	−0.583187	−	0.812338i	$-0.698194\pi$
$859$	−29364.8	−1.16637	−0.583187	+	0.812338i	$0.698194\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44989.3	−1.77457	−0.887285	−	0.461221i	$-0.847412\pi$
$863$	−44989.3	−1.77457	−0.887285	+	0.461221i	$0.847412\pi$
$864$	0	0
$865$	4738.70	0.186267
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−21371.5	−0.834267
$870$	0	0
$871$	37060.9	1.44175
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1608.50	−0.0621452
$876$	0	0
$877$	36547.6	1.40721	0.703606	−	0.710590i	$-0.251572\pi$
$877$	36547.6	1.40721	0.703606	+	0.710590i	$0.251572\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−21146.4	−0.808672	−0.404336	−	0.914610i	$-0.632497\pi$
$881$	−21146.4	−0.808672	−0.404336	+	0.914610i	$0.632497\pi$
$882$	0	0
$883$	−39103.3	−1.49030	−0.745148	−	0.666899i	$-0.767621\pi$
$883$	−39103.3	−1.49030	−0.745148	+	0.666899i	$0.767621\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32169.3	−1.21775	−0.608873	−	0.793268i	$-0.708378\pi$
$887$	−32169.3	−1.21775	−0.608873	+	0.793268i	$0.708378\pi$
$888$	0	0
$889$	24709.0	0.932184
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8142.80	0.305138
$894$	0	0
$895$	15017.4	0.560866
$896$	0	0
$897$	0	0
$898$	0	0
$899$	282.792	0.0104913
$900$	0	0
$901$	−38512.3	−1.42401
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7514.54	−0.276013
$906$	0	0
$907$	−10810.4	−0.395759	−0.197880	−	0.980226i	$-0.563405\pi$
$907$	−10810.4	−0.395759	−0.197880	+	0.980226i	$0.563405\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	244.183	0.00888053	0.00444026	−	0.999990i	$-0.498587\pi$
$911$	244.183	0.00888053	0.00444026	+	0.999990i	$0.498587\pi$
$912$	0	0
$913$	11324.7	0.410508
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7724.68	0.278180
$918$	0	0
$919$	−34164.1	−1.22630	−0.613151	−	0.789966i	$-0.710098\pi$
$919$	−34164.1	−1.22630	−0.613151	+	0.789966i	$0.710098\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	27973.4	0.997569
$924$	0	0
$925$	8605.68	0.305895
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35758.6	1.26287	0.631433	−	0.775431i	$-0.282467\pi$
$929$	35758.6	1.26287	0.631433	+	0.775431i	$0.282467\pi$
$930$	0	0
$931$	−4719.94	−0.166155
$932$	0	0
$933$	0	0
$934$	0	0
$935$	21039.3	0.735890
$936$	0	0
$937$	40301.0	1.40510	0.702549	−	0.711635i	$-0.252045\pi$
$937$	40301.0	1.40510	0.702549	+	0.711635i	$0.252045\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−48866.6	−1.69289	−0.846443	−	0.532480i	$-0.821260\pi$
$941$	−48866.6	−1.69289	−0.846443	+	0.532480i	$0.821260\pi$
$942$	0	0
$943$	5890.94	0.203431
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42669.4	−1.46417	−0.732084	−	0.681214i	$-0.761452\pi$
$947$	−42669.4	−1.46417	−0.732084	+	0.681214i	$0.761452\pi$
$948$	0	0
$949$	−3031.21	−0.103685
$950$	0	0
$951$	0	0
$952$	0	0
$953$	42523.5	1.44541	0.722703	−	0.691158i	$-0.242899\pi$
$953$	42523.5	1.44541	0.722703	+	0.691158i	$0.242899\pi$
$954$	0	0
$955$	15720.0	0.532657
$956$	0	0
$957$	0	0
$958$	0	0
$959$	25137.7	0.846444
$960$	0	0
$961$	−27569.6	−0.925433
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15504.3	0.517205
$966$	0	0
$967$	13205.1	0.439138	0.219569	−	0.975597i	$-0.429535\pi$
$967$	13205.1	0.439138	0.219569	+	0.975597i	$0.429535\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3545.55	−0.117180	−0.0585901	−	0.998282i	$-0.518661\pi$
$971$	−3545.55	−0.117180	−0.0585901	+	0.998282i	$0.518661\pi$
$972$	0	0
$973$	29097.7	0.958713
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13070.8	0.428015	0.214008	−	0.976832i	$-0.431348\pi$
$977$	13070.8	0.428015	0.214008	+	0.976832i	$0.431348\pi$
$978$	0	0
$979$	−51370.7	−1.67703
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12342.3	−0.400467	−0.200234	−	0.979748i	$-0.564170\pi$
$983$	−12342.3	−0.400467	−0.200234	+	0.979748i	$0.564170\pi$
$984$	0	0
$985$	−12815.7	−0.414559
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−34357.6	−1.10466
$990$	0	0
$991$	−35095.4	−1.12497	−0.562484	−	0.826808i	$-0.690154\pi$
$991$	−35095.4	−1.12497	−0.562484	+	0.826808i	$0.690154\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14645.9	−0.466638
$996$	0	0
$997$	−46486.0	−1.47666	−0.738328	−	0.674442i	$-0.764384\pi$
$997$	−46486.0	−1.47666	−0.738328	+	0.674442i	$0.764384\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.4.a.s.1.2		2
3.2	odd	2		480.4.a.o.1.2	✓	2
4.3	odd	2		1440.4.a.y.1.1		2
12.11	even	2		480.4.a.r.1.1	yes	2
15.14	odd	2		2400.4.a.bb.1.1		2
24.5	odd	2		960.4.a.bn.1.2		2
24.11	even	2		960.4.a.bk.1.1		2
60.59	even	2		2400.4.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.o.1.2	✓	2	3.2	odd	2
480.4.a.r.1.1	yes	2	12.11	even	2
960.4.a.bk.1.1		2	24.11	even	2
960.4.a.bn.1.2		2	24.5	odd	2
1440.4.a.s.1.2		2	1.1	even	1	trivial
1440.4.a.y.1.1		2	4.3	odd	2
2400.4.a.y.1.2		2	60.59	even	2
2400.4.a.bb.1.1		2	15.14	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +12.8680 q^{7} +49.7359 q^{11} -52.6039 q^{13} -84.6039 q^{17} +26.6039 q^{19} +136.340 q^{23} +25.0000 q^{25} -6.00000 q^{29} -47.1320 q^{31} -64.3398 q^{35} +344.227 q^{37} +43.2078 q^{41} -252.000 q^{43} +306.076 q^{47} -177.416 q^{49} +455.208 q^{53} -248.680 q^{55} +708.151 q^{59} -652.039 q^{61} +263.019 q^{65} -704.528 q^{67} -531.775 q^{71} +57.6233 q^{73} +640.000 q^{77} -429.699 q^{79} +227.697 q^{83} +423.019 q^{85} -1032.87 q^{89} -676.905 q^{91} -133.019 q^{95} -152.416 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} - 12 q^{7} + 24 q^{11} + 8 q^{13} - 56 q^{17} - 60 q^{19} + 84 q^{23} + 50 q^{25} - 12 q^{29} - 132 q^{31} + 60 q^{35} - 104 q^{37} - 140 q^{41} - 504 q^{43} + 348 q^{47} + 98 q^{49} + 684 q^{53}+ \cdots + 148 q^{97}+O(q^{100}) \) 2 * q - 10 * q^5 - 12 * q^7 + 24 * q^11 + 8 * q^13 - 56 * q^17 - 60 * q^19 + 84 * q^23 + 50 * q^25 - 12 * q^29 - 132 * q^31 + 60 * q^35 - 104 * q^37 - 140 * q^41 - 504 * q^43 + 348 * q^47 + 98 * q^49 + 684 * q^53 - 120 * q^55 + 888 * q^59 - 172 * q^61 - 40 * q^65 - 1560 * q^67 + 144 * q^71 - 564 * q^73 + 1280 * q^77 + 84 * q^79 + 1512 * q^83 + 280 * q^85 - 28 * q^89 - 2184 * q^91 + 300 * q^95 + 148 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more