LMFDB - Embedded newform 1368.4.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,4,Mod(1,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, 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("1368.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1368.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:	$80.7146128879$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 429x^{3} + 1657x^{2} + 46980x - 289104 $ x^5 - x^4 - 429x^3 + 1657x^2 + 46980*x - 289104
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$7.82361$ of defining polynomial
Character	$\chi$	$=$	1368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.82361 q^{5} -24.1803 q^{7} +O(q^{10})$	$q-8.82361 q^{5} -24.1803 q^{7} -24.0756 q^{11} -50.0259 q^{13} -101.956 q^{17} -19.0000 q^{19} +29.7923 q^{23} -47.1440 q^{25} -255.503 q^{29} -46.1772 q^{31} +213.357 q^{35} +396.193 q^{37} +125.698 q^{41} -252.083 q^{43} -489.537 q^{47} +241.687 q^{49} -217.053 q^{53} +212.434 q^{55} -344.556 q^{59} +112.344 q^{61} +441.409 q^{65} +427.069 q^{67} -161.232 q^{71} +437.176 q^{73} +582.155 q^{77} +777.704 q^{79} -527.588 q^{83} +899.623 q^{85} -1301.52 q^{89} +1209.64 q^{91} +167.649 q^{95} +711.890 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 6 q^{5} + 10 q^{7}+O(q^{10}) $ 5 * q - 6 * q^5 + 10 * q^7	$ 5 q - 6 q^{5} + 10 q^{7} + 32 q^{11} + 36 q^{13} + 8 q^{17} - 95 q^{19} + 152 q^{23} + 241 q^{25} - 248 q^{29} + 118 q^{31} - 162 q^{35} + 472 q^{37} - 944 q^{41} + 150 q^{43} - 38 q^{47} + 1719 q^{49} - 788 q^{53} + 1270 q^{55} - 396 q^{59} + 1724 q^{61} - 2200 q^{65} + 204 q^{67} - 480 q^{71} + 2608 q^{73} - 2458 q^{77} + 786 q^{79} + 1118 q^{83} + 1798 q^{85} - 792 q^{89} + 1040 q^{91} + 114 q^{95} + 4638 q^{97}+O(q^{100}) $ 5 * q - 6 * q^5 + 10 * q^7 + 32 * q^11 + 36 * q^13 + 8 * q^17 - 95 * q^19 + 152 * q^23 + 241 * q^25 - 248 * q^29 + 118 * q^31 - 162 * q^35 + 472 * q^37 - 944 * q^41 + 150 * q^43 - 38 * q^47 + 1719 * q^49 - 788 * q^53 + 1270 * q^55 - 396 * q^59 + 1724 * q^61 - 2200 * q^65 + 204 * q^67 - 480 * q^71 + 2608 * q^73 - 2458 * q^77 + 786 * q^79 + 1118 * q^83 + 1798 * q^85 - 792 * q^89 + 1040 * q^91 + 114 * q^95 + 4638 * 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
$5$	−8.82361	−0.789207	−0.394604	−	0.918851i	$-0.629118\pi$
$5$	−8.82361	−0.789207	−0.394604	+	0.918851i	$0.629118\pi$
$6$	0	0
$7$	−24.1803	−1.30561	−0.652807	−	0.757525i	$-0.726409\pi$
$7$	−24.1803	−1.30561	−0.652807	+	0.757525i	$0.726409\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−24.0756	−0.659915	−0.329958	−	0.943996i	$-0.607034\pi$
$11$	−24.0756	−0.659915	−0.329958	+	0.943996i	$0.607034\pi$
$12$	0	0
$13$	−50.0259	−1.06728	−0.533642	−	0.845710i	$-0.679177\pi$
$13$	−50.0259	−1.06728	−0.533642	+	0.845710i	$0.679177\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−101.956	−1.45459	−0.727296	−	0.686324i	$-0.759223\pi$
$17$	−101.956	−1.45459	−0.727296	+	0.686324i	$0.759223\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	29.7923	0.270093	0.135046	−	0.990839i	$-0.456882\pi$
$23$	29.7923	0.270093	0.135046	+	0.990839i	$0.456882\pi$
$24$	0	0
$25$	−47.1440	−0.377152
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−255.503	−1.63606	−0.818031	−	0.575174i	$-0.804934\pi$
$29$	−255.503	−1.63606	−0.818031	+	0.575174i	$0.804934\pi$
$30$	0	0
$31$	−46.1772	−0.267538	−0.133769	−	0.991013i	$-0.542708\pi$
$31$	−46.1772	−0.267538	−0.133769	+	0.991013i	$0.542708\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	213.357	1.03040
$36$	0	0
$37$	396.193	1.76037	0.880185	−	0.474631i	$-0.157419\pi$
$37$	396.193	1.76037	0.880185	+	0.474631i	$0.157419\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	125.698	0.478799	0.239399	−	0.970921i	$-0.423049\pi$
$41$	125.698	0.478799	0.239399	+	0.970921i	$0.423049\pi$
$42$	0	0
$43$	−252.083	−0.894007	−0.447003	−	0.894532i	$-0.647509\pi$
$43$	−252.083	−0.894007	−0.447003	+	0.894532i	$0.647509\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−489.537	−1.51928	−0.759642	−	0.650342i	$-0.774626\pi$
$47$	−489.537	−1.51928	−0.759642	+	0.650342i	$0.774626\pi$
$48$	0	0
$49$	241.687	0.704626
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−217.053	−0.562538	−0.281269	−	0.959629i	$-0.590755\pi$
$53$	−217.053	−0.562538	−0.281269	+	0.959629i	$0.590755\pi$
$54$	0	0
$55$	212.434	0.520810
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−344.556	−0.760295	−0.380148	−	0.924926i	$-0.624127\pi$
$59$	−344.556	−0.760295	−0.380148	+	0.924926i	$0.624127\pi$
$60$	0	0
$61$	112.344	0.235806	0.117903	−	0.993025i	$-0.462383\pi$
$61$	112.344	0.235806	0.117903	+	0.993025i	$0.462383\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	441.409	0.842309
$66$	0	0
$67$	427.069	0.778729	0.389364	−	0.921084i	$-0.372695\pi$
$67$	427.069	0.778729	0.389364	+	0.921084i	$0.372695\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−161.232	−0.269503	−0.134752	−	0.990879i	$-0.543024\pi$
$71$	−161.232	−0.269503	−0.134752	+	0.990879i	$0.543024\pi$
$72$	0	0
$73$	437.176	0.700926	0.350463	−	0.936577i	$-0.386024\pi$
$73$	437.176	0.700926	0.350463	+	0.936577i	$0.386024\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	582.155	0.861594
$78$	0	0
$79$	777.704	1.10758	0.553788	−	0.832658i	$-0.313182\pi$
$79$	777.704	1.10758	0.553788	+	0.832658i	$0.313182\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−527.588	−0.697715	−0.348858	−	0.937176i	$-0.613430\pi$
$83$	−527.588	−0.697715	−0.348858	+	0.937176i	$0.613430\pi$
$84$	0	0
$85$	899.623	1.14797
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1301.52	−1.55012	−0.775062	−	0.631885i	$-0.782282\pi$
$89$	−1301.52	−1.55012	−0.775062	+	0.631885i	$0.782282\pi$
$90$	0	0
$91$	1209.64	1.39346
$92$	0	0
$93$	0	0
$94$	0	0
$95$	167.649	0.181057
$96$	0	0
$97$	711.890	0.745170	0.372585	−	0.927998i	$-0.378471\pi$
$97$	711.890	0.745170	0.372585	+	0.927998i	$0.378471\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1775.92	−1.74961	−0.874807	−	0.484472i	$-0.839012\pi$
$101$	−1775.92	−1.74961	−0.874807	+	0.484472i	$0.839012\pi$
$102$	0	0
$103$	−541.739	−0.518244	−0.259122	−	0.965845i	$-0.583433\pi$
$103$	−541.739	−0.518244	−0.259122	+	0.965845i	$0.583433\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	743.032	0.671323	0.335662	−	0.941983i	$-0.391040\pi$
$107$	743.032	0.671323	0.335662	+	0.941983i	$0.391040\pi$
$108$	0	0
$109$	1282.23	1.12675	0.563375	−	0.826202i	$-0.309503\pi$
$109$	1282.23	1.12675	0.563375	+	0.826202i	$0.309503\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	734.934	0.611830	0.305915	−	0.952059i	$-0.401038\pi$
$113$	734.934	0.611830	0.305915	+	0.952059i	$0.401038\pi$
$114$	0	0
$115$	−262.876	−0.213159
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2465.34	1.89913
$120$	0	0
$121$	−751.365	−0.564512
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1518.93	1.08686
$126$	0	0
$127$	−1700.77	−1.18834	−0.594171	−	0.804339i	$-0.702520\pi$
$127$	−1700.77	−1.18834	−0.594171	+	0.804339i	$0.702520\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1159.99	0.773654	0.386827	−	0.922152i	$-0.373571\pi$
$131$	1159.99	0.773654	0.386827	+	0.922152i	$0.373571\pi$
$132$	0	0
$133$	459.426	0.299528
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2853.28	1.77936	0.889680	−	0.456584i	$-0.150927\pi$
$137$	2853.28	1.77936	0.889680	+	0.456584i	$0.150927\pi$
$138$	0	0
$139$	273.612	0.166960	0.0834800	−	0.996509i	$-0.473397\pi$
$139$	273.612	0.166960	0.0834800	+	0.996509i	$0.473397\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1204.40	0.704317
$144$	0	0
$145$	2254.46	1.29119
$146$	0	0
$147$	0	0
$148$	0	0
$149$	598.545	0.329092	0.164546	−	0.986369i	$-0.447384\pi$
$149$	598.545	0.329092	0.164546	+	0.986369i	$0.447384\pi$
$150$	0	0
$151$	3201.96	1.72564	0.862820	−	0.505512i	$-0.168696\pi$
$151$	3201.96	1.72564	0.862820	+	0.505512i	$0.168696\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	407.450	0.211143
$156$	0	0
$157$	−2317.11	−1.17787	−0.588936	−	0.808180i	$-0.700453\pi$
$157$	−2317.11	−1.17787	−0.588936	+	0.808180i	$0.700453\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−720.387	−0.352637
$162$	0	0
$163$	−2648.65	−1.27275	−0.636375	−	0.771380i	$-0.719567\pi$
$163$	−2648.65	−1.27275	−0.636375	+	0.771380i	$0.719567\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−975.898	−0.452199	−0.226099	−	0.974104i	$-0.572597\pi$
$167$	−975.898	−0.452199	−0.226099	+	0.974104i	$0.572597\pi$
$168$	0	0
$169$	305.594	0.139096
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1461.16	−0.642139	−0.321069	−	0.947056i	$-0.604042\pi$
$173$	−1461.16	−0.642139	−0.321069	+	0.947056i	$0.604042\pi$
$174$	0	0
$175$	1139.95	0.492414
$176$	0	0
$177$	0	0
$178$	0	0
$179$	782.639	0.326800	0.163400	−	0.986560i	$-0.447754\pi$
$179$	782.639	0.326800	0.163400	+	0.986560i	$0.447754\pi$
$180$	0	0
$181$	−2664.69	−1.09428	−0.547139	−	0.837041i	$-0.684283\pi$
$181$	−2664.69	−1.09428	−0.547139	+	0.837041i	$0.684283\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3495.85	−1.38930
$186$	0	0
$187$	2454.66	0.959908
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3545.07	−1.34299	−0.671497	−	0.741007i	$-0.734348\pi$
$191$	−3545.07	−1.34299	−0.671497	+	0.741007i	$0.734348\pi$
$192$	0	0
$193$	4312.86	1.60853	0.804265	−	0.594271i	$-0.202559\pi$
$193$	4312.86	1.60853	0.804265	+	0.594271i	$0.202559\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−313.121	−0.113243	−0.0566217	−	0.998396i	$-0.518033\pi$
$197$	−313.121	−0.113243	−0.0566217	+	0.998396i	$0.518033\pi$
$198$	0	0
$199$	−2834.47	−1.00970	−0.504850	−	0.863207i	$-0.668452\pi$
$199$	−2834.47	−1.00970	−0.504850	+	0.863207i	$0.668452\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6178.15	2.13606
$204$	0	0
$205$	−1109.11	−0.377872
$206$	0	0
$207$	0	0
$208$	0	0
$209$	457.437	0.151395
$210$	0	0
$211$	−4126.31	−1.34629	−0.673145	−	0.739511i	$-0.735057\pi$
$211$	−4126.31	−1.34629	−0.673145	+	0.739511i	$0.735057\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2224.28	0.705557
$216$	0	0
$217$	1116.58	0.349301
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5100.46	1.55246
$222$	0	0
$223$	−2178.02	−0.654040	−0.327020	−	0.945017i	$-0.606045\pi$
$223$	−2178.02	−0.654040	−0.327020	+	0.945017i	$0.606045\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1023.66	0.299307	0.149653	−	0.988739i	$-0.452184\pi$
$227$	1023.66	0.299307	0.149653	+	0.988739i	$0.452184\pi$
$228$	0	0
$229$	6468.43	1.86657	0.933287	−	0.359130i	$-0.116927\pi$
$229$	6468.43	1.86657	0.933287	+	0.359130i	$0.116927\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4926.43	1.38516	0.692578	−	0.721343i	$-0.256475\pi$
$233$	4926.43	1.38516	0.692578	+	0.721343i	$0.256475\pi$
$234$	0	0
$235$	4319.48	1.19903
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4198.26	−1.13625	−0.568123	−	0.822944i	$-0.692330\pi$
$239$	−4198.26	−1.13625	−0.568123	+	0.822944i	$0.692330\pi$
$240$	0	0
$241$	4833.87	1.29202	0.646010	−	0.763329i	$-0.276436\pi$
$241$	4833.87	1.29202	0.646010	+	0.763329i	$0.276436\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2132.55	−0.556096
$246$	0	0
$247$	950.493	0.244852
$248$	0	0
$249$	0	0
$250$	0	0
$251$	498.862	0.125450	0.0627249	−	0.998031i	$-0.480021\pi$
$251$	498.862	0.125450	0.0627249	+	0.998031i	$0.480021\pi$
$252$	0	0
$253$	−717.269	−0.178238
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4884.53	1.18556	0.592779	−	0.805365i	$-0.298031\pi$
$257$	4884.53	1.18556	0.592779	+	0.805365i	$0.298031\pi$
$258$	0	0
$259$	−9580.06	−2.29836
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4660.53	−1.09270	−0.546351	−	0.837556i	$-0.683984\pi$
$263$	−4660.53	−1.09270	−0.546351	+	0.837556i	$0.683984\pi$
$264$	0	0
$265$	1915.19	0.443959
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4555.63	1.03257	0.516286	−	0.856416i	$-0.327314\pi$
$269$	4555.63	1.03257	0.516286	+	0.856416i	$0.327314\pi$
$270$	0	0
$271$	−1896.85	−0.425186	−0.212593	−	0.977141i	$-0.568191\pi$
$271$	−1896.85	−0.425186	−0.212593	+	0.977141i	$0.568191\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1135.02	0.248888
$276$	0	0
$277$	−7651.70	−1.65973	−0.829866	−	0.557962i	$-0.811583\pi$
$277$	−7651.70	−1.65973	−0.829866	+	0.557962i	$0.811583\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3242.89	−0.688450	−0.344225	−	0.938887i	$-0.611858\pi$
$281$	−3242.89	−0.688450	−0.344225	+	0.938887i	$0.611858\pi$
$282$	0	0
$283$	1610.97	0.338383	0.169191	−	0.985583i	$-0.445884\pi$
$283$	1610.97	0.338383	0.169191	+	0.985583i	$0.445884\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3039.42	−0.625126
$288$	0	0
$289$	5482.11	1.11584
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8133.22	1.62166	0.810832	−	0.585279i	$-0.199015\pi$
$293$	8133.22	1.62166	0.810832	+	0.585279i	$0.199015\pi$
$294$	0	0
$295$	3040.23	0.600031
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1490.39	−0.288266
$300$	0	0
$301$	6095.44	1.16723
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−991.279	−0.186100
$306$	0	0
$307$	−1325.24	−0.246370	−0.123185	−	0.992384i	$-0.539311\pi$
$307$	−1325.24	−0.246370	−0.123185	+	0.992384i	$0.539311\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9980.42	−1.81974	−0.909868	−	0.414898i	$-0.863817\pi$
$311$	−9980.42	−1.81974	−0.909868	+	0.414898i	$0.863817\pi$
$312$	0	0
$313$	1038.44	0.187527	0.0937635	−	0.995594i	$-0.470110\pi$
$313$	1038.44	0.187527	0.0937635	+	0.995594i	$0.470110\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−939.566	−0.166471	−0.0832355	−	0.996530i	$-0.526525\pi$
$317$	−939.566	−0.166471	−0.0832355	+	0.996530i	$0.526525\pi$
$318$	0	0
$319$	6151.40	1.07966
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1937.17	0.333706
$324$	0	0
$325$	2358.42	0.402528
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11837.2	1.98360
$330$	0	0
$331$	11237.1	1.86600	0.933000	−	0.359877i	$-0.117181\pi$
$331$	11237.1	1.86600	0.933000	+	0.359877i	$0.117181\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3768.29	−0.614578
$336$	0	0
$337$	5175.27	0.836542	0.418271	−	0.908322i	$-0.362636\pi$
$337$	5175.27	0.836542	0.418271	+	0.908322i	$0.362636\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1111.74	0.176552
$342$	0	0
$343$	2449.79	0.385645
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1592.71	−0.246400	−0.123200	−	0.992382i	$-0.539316\pi$
$347$	−1592.71	−0.246400	−0.123200	+	0.992382i	$0.539316\pi$
$348$	0	0
$349$	−4765.31	−0.730892	−0.365446	−	0.930833i	$-0.619083\pi$
$349$	−4765.31	−0.730892	−0.365446	+	0.930833i	$0.619083\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2992.35	−0.451181	−0.225590	−	0.974222i	$-0.572431\pi$
$353$	−2992.35	−0.451181	−0.225590	+	0.974222i	$0.572431\pi$
$354$	0	0
$355$	1422.65	0.212694
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1245.64	−0.183126	−0.0915631	−	0.995799i	$-0.529186\pi$
$359$	−1245.64	−0.183126	−0.0915631	+	0.995799i	$0.529186\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3857.47	−0.553176
$366$	0	0
$367$	3704.86	0.526953	0.263477	−	0.964666i	$-0.415131\pi$
$367$	3704.86	0.526953	0.263477	+	0.964666i	$0.415131\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5248.40	0.734457
$372$	0	0
$373$	2262.19	0.314026	0.157013	−	0.987597i	$-0.449814\pi$
$373$	2262.19	0.314026	0.157013	+	0.987597i	$0.449814\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12781.8	1.74614
$378$	0	0
$379$	−11283.2	−1.52923	−0.764617	−	0.644484i	$-0.777072\pi$
$379$	−11283.2	−1.52923	−0.764617	+	0.644484i	$0.777072\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5942.77	−0.792849	−0.396425	−	0.918067i	$-0.629749\pi$
$383$	−5942.77	−0.792849	−0.396425	+	0.918067i	$0.629749\pi$
$384$	0	0
$385$	−5136.71	−0.679976
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4611.96	−0.601120	−0.300560	−	0.953763i	$-0.597174\pi$
$389$	−4611.96	−0.601120	−0.300560	+	0.953763i	$0.597174\pi$
$390$	0	0
$391$	−3037.52	−0.392875
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6862.15	−0.874107
$396$	0	0
$397$	−12705.9	−1.60627	−0.803135	−	0.595797i	$-0.796836\pi$
$397$	−12705.9	−1.60627	−0.803135	+	0.595797i	$0.796836\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	602.894	0.0750801	0.0375400	−	0.999295i	$-0.488048\pi$
$401$	602.894	0.0750801	0.0375400	+	0.999295i	$0.488048\pi$
$402$	0	0
$403$	2310.06	0.285539
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9538.58	−1.16170
$408$	0	0
$409$	9184.61	1.11039	0.555195	−	0.831720i	$-0.312644\pi$
$409$	9184.61	1.11039	0.555195	+	0.831720i	$0.312644\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8331.47	0.992651
$414$	0	0
$415$	4655.23	0.550642
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7833.90	−0.913392	−0.456696	−	0.889623i	$-0.650967\pi$
$419$	−7833.90	−0.913392	−0.456696	+	0.889623i	$0.650967\pi$
$420$	0	0
$421$	−8654.64	−1.00190	−0.500952	−	0.865475i	$-0.667017\pi$
$421$	−8654.64	−1.00190	−0.500952	+	0.865475i	$0.667017\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4806.63	0.548602
$426$	0	0
$427$	−2716.51	−0.307872
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2995.47	−0.334772	−0.167386	−	0.985891i	$-0.553533\pi$
$431$	−2995.47	−0.334772	−0.167386	+	0.985891i	$0.553533\pi$
$432$	0	0
$433$	−3788.26	−0.420444	−0.210222	−	0.977654i	$-0.567419\pi$
$433$	−3788.26	−0.420444	−0.210222	+	0.977654i	$0.567419\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−566.054	−0.0619635
$438$	0	0
$439$	11685.1	1.27039	0.635193	−	0.772353i	$-0.280921\pi$
$439$	11685.1	1.27039	0.635193	+	0.772353i	$0.280921\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4982.47	0.534366	0.267183	−	0.963646i	$-0.413907\pi$
$443$	4982.47	0.534366	0.267183	+	0.963646i	$0.413907\pi$
$444$	0	0
$445$	11484.1	1.22337
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6375.86	−0.670145	−0.335073	−	0.942192i	$-0.608761\pi$
$449$	−6375.86	−0.670145	−0.335073	+	0.942192i	$0.608761\pi$
$450$	0	0
$451$	−3026.26	−0.315967
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10673.4	−1.09973
$456$	0	0
$457$	−9674.17	−0.990238	−0.495119	−	0.868825i	$-0.664875\pi$
$457$	−9674.17	−0.990238	−0.495119	+	0.868825i	$0.664875\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13016.4	−1.31504	−0.657520	−	0.753437i	$-0.728394\pi$
$461$	−13016.4	−1.31504	−0.657520	+	0.753437i	$0.728394\pi$
$462$	0	0
$463$	7318.81	0.734631	0.367315	−	0.930096i	$-0.380277\pi$
$463$	7318.81	0.734631	0.367315	+	0.930096i	$0.380277\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5152.83	−0.510588	−0.255294	−	0.966864i	$-0.582172\pi$
$467$	−5152.83	−0.510588	−0.255294	+	0.966864i	$0.582172\pi$
$468$	0	0
$469$	−10326.7	−1.01672
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6069.05	0.589969
$474$	0	0
$475$	895.735	0.0865245
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11557.6	−1.10246	−0.551230	−	0.834353i	$-0.685841\pi$
$479$	−11557.6	−1.10246	−0.551230	+	0.834353i	$0.685841\pi$
$480$	0	0
$481$	−19819.9	−1.87882
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6281.44	−0.588094
$486$	0	0
$487$	−2038.74	−0.189700	−0.0948502	−	0.995492i	$-0.530237\pi$
$487$	−2038.74	−0.189700	−0.0948502	+	0.995492i	$0.530237\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2753.89	−0.253119	−0.126559	−	0.991959i	$-0.540393\pi$
$491$	−2753.89	−0.253119	−0.126559	+	0.991959i	$0.540393\pi$
$492$	0	0
$493$	26050.2	2.37980
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3898.64	0.351867
$498$	0	0
$499$	−18455.6	−1.65568	−0.827842	−	0.560962i	$-0.810431\pi$
$499$	−18455.6	−1.65568	−0.827842	+	0.560962i	$0.810431\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−979.494	−0.0868260	−0.0434130	−	0.999057i	$-0.513823\pi$
$503$	−979.494	−0.0868260	−0.0434130	+	0.999057i	$0.513823\pi$
$504$	0	0
$505$	15670.1	1.38081
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2222.56	−0.193543	−0.0967714	−	0.995307i	$-0.530852\pi$
$509$	−2222.56	−0.193543	−0.0967714	+	0.995307i	$0.530852\pi$
$510$	0	0
$511$	−10571.0	−0.915138
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4780.09	0.409002
$516$	0	0
$517$	11785.9	1.00260
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6535.90	0.549602	0.274801	−	0.961501i	$-0.411388\pi$
$521$	6535.90	0.549602	0.274801	+	0.961501i	$0.411388\pi$
$522$	0	0
$523$	4577.47	0.382713	0.191356	−	0.981521i	$-0.438711\pi$
$523$	4577.47	0.382713	0.191356	+	0.981521i	$0.438711\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4708.06	0.389158
$528$	0	0
$529$	−11279.4	−0.927050
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6288.17	−0.511015
$534$	0	0
$535$	−6556.22	−0.529813
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5818.75	−0.464993
$540$	0	0
$541$	−10063.8	−0.799770	−0.399885	−	0.916565i	$-0.630950\pi$
$541$	−10063.8	−0.799770	−0.399885	+	0.916565i	$0.630950\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11313.9	−0.889239
$546$	0	0
$547$	18255.8	1.42699	0.713494	−	0.700661i	$-0.247112\pi$
$547$	18255.8	1.42699	0.713494	+	0.700661i	$0.247112\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4854.56	0.375338
$552$	0	0
$553$	−18805.1	−1.44607
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19426.0	1.47775	0.738873	−	0.673844i	$-0.235358\pi$
$557$	19426.0	1.47775	0.738873	+	0.673844i	$0.235358\pi$
$558$	0	0
$559$	12610.7	0.954160
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6064.76	−0.453995	−0.226997	−	0.973895i	$-0.572891\pi$
$563$	−6064.76	−0.453995	−0.226997	+	0.973895i	$0.572891\pi$
$564$	0	0
$565$	−6484.77	−0.482861
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3009.73	0.221748	0.110874	−	0.993834i	$-0.464635\pi$
$569$	3009.73	0.221748	0.110874	+	0.993834i	$0.464635\pi$
$570$	0	0
$571$	−21598.8	−1.58298	−0.791489	−	0.611183i	$-0.790694\pi$
$571$	−21598.8	−1.58298	−0.791489	+	0.611183i	$0.790694\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1404.53	−0.101866
$576$	0	0
$577$	−17221.5	−1.24253	−0.621264	−	0.783601i	$-0.713380\pi$
$577$	−17221.5	−1.24253	−0.621264	+	0.783601i	$0.713380\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12757.2	0.910946
$582$	0	0
$583$	5225.68	0.371228
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8929.29	−0.627856	−0.313928	−	0.949447i	$-0.601645\pi$
$587$	−8929.29	−0.627856	−0.313928	+	0.949447i	$0.601645\pi$
$588$	0	0
$589$	877.367	0.0613774
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14535.3	1.00657	0.503284	−	0.864121i	$-0.332125\pi$
$593$	14535.3	1.00657	0.503284	+	0.864121i	$0.332125\pi$
$594$	0	0
$595$	−21753.2	−1.49881
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4287.81	0.292479	0.146240	−	0.989249i	$-0.453283\pi$
$599$	4287.81	0.292479	0.146240	+	0.989249i	$0.453283\pi$
$600$	0	0
$601$	−56.4493	−0.00383131	−0.00191565	−	0.999998i	$-0.500610\pi$
$601$	−56.4493	−0.00383131	−0.00191565	+	0.999998i	$0.500610\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6629.75	0.445517
$606$	0	0
$607$	2577.16	0.172329	0.0861644	−	0.996281i	$-0.472539\pi$
$607$	2577.16	0.172329	0.0861644	+	0.996281i	$0.472539\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24489.6	1.62151
$612$	0	0
$613$	15.8421	0.00104381	0.000521906	−	1.00000i	$-0.499834\pi$
$613$	15.8421	0.00104381	0.000521906		1.00000i	$0.499834\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12748.5	−0.831822	−0.415911	−	0.909405i	$-0.636537\pi$
$617$	−12748.5	−0.831822	−0.415911	+	0.909405i	$0.636537\pi$
$618$	0	0
$619$	1706.51	0.110808	0.0554041	−	0.998464i	$-0.482355\pi$
$619$	1706.51	0.110808	0.0554041	+	0.998464i	$0.482355\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	31471.2	2.02386
$624$	0	0
$625$	−7509.45	−0.480605
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−40394.4	−2.56062
$630$	0	0
$631$	−22509.0	−1.42008	−0.710038	−	0.704164i	$-0.751322\pi$
$631$	−22509.0	−1.42008	−0.710038	+	0.704164i	$0.751322\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15007.0	0.937848
$636$	0	0
$637$	−12090.6	−0.752036
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18892.6	1.16414	0.582070	−	0.813139i	$-0.302243\pi$
$641$	18892.6	1.16414	0.582070	+	0.813139i	$0.302243\pi$
$642$	0	0
$643$	−388.125	−0.0238043	−0.0119022	−	0.999929i	$-0.503789\pi$
$643$	−388.125	−0.0238043	−0.0119022	+	0.999929i	$0.503789\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16580.3	−1.00748	−0.503739	−	0.863856i	$-0.668043\pi$
$647$	−16580.3	−1.00748	−0.503739	+	0.863856i	$0.668043\pi$
$648$	0	0
$649$	8295.40	0.501731
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11319.0	0.678328	0.339164	−	0.940727i	$-0.389856\pi$
$653$	11319.0	0.678328	0.339164	+	0.940727i	$0.389856\pi$
$654$	0	0
$655$	−10235.3	−0.610573
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−31740.2	−1.87621	−0.938104	−	0.346354i	$-0.887420\pi$
$659$	−31740.2	−1.87621	−0.938104	+	0.346354i	$0.887420\pi$
$660$	0	0
$661$	−10761.2	−0.633228	−0.316614	−	0.948554i	$-0.602546\pi$
$661$	−10761.2	−0.633228	−0.316614	+	0.948554i	$0.602546\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4053.79	−0.236390
$666$	0	0
$667$	−7612.04	−0.441888
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2704.75	−0.155612
$672$	0	0
$673$	25585.2	1.46544	0.732718	−	0.680533i	$-0.238252\pi$
$673$	25585.2	1.46544	0.732718	+	0.680533i	$0.238252\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34707.6	−1.97034	−0.985170	−	0.171582i	$-0.945112\pi$
$677$	−34707.6	−1.97034	−0.985170	+	0.171582i	$0.945112\pi$
$678$	0	0
$679$	−17213.7	−0.972904
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−22863.7	−1.28090	−0.640450	−	0.768000i	$-0.721252\pi$
$683$	−22863.7	−1.28090	−0.640450	+	0.768000i	$0.721252\pi$
$684$	0	0
$685$	−25176.2	−1.40428
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10858.3	0.600388
$690$	0	0
$691$	−18617.7	−1.02496	−0.512482	−	0.858698i	$-0.671274\pi$
$691$	−18617.7	−1.02496	−0.512482	+	0.858698i	$0.671274\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2414.24	−0.131766
$696$	0	0
$697$	−12815.7	−0.696457
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13292.8	0.716209	0.358104	−	0.933682i	$-0.383423\pi$
$701$	13292.8	0.716209	0.358104	+	0.933682i	$0.383423\pi$
$702$	0	0
$703$	−7527.66	−0.403856
$704$	0	0
$705$	0	0
$706$	0	0
$707$	42942.4	2.28432
$708$	0	0
$709$	4278.41	0.226628	0.113314	−	0.993559i	$-0.463853\pi$
$709$	4278.41	0.226628	0.113314	+	0.993559i	$0.463853\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1375.73	−0.0722600
$714$	0	0
$715$	−10627.2	−0.555853
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−29106.8	−1.50974	−0.754869	−	0.655876i	$-0.772299\pi$
$719$	−29106.8	−1.50974	−0.754869	+	0.655876i	$0.772299\pi$
$720$	0	0
$721$	13099.4	0.676626
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12045.4	0.617043
$726$	0	0
$727$	−20330.8	−1.03718	−0.518589	−	0.855024i	$-0.673542\pi$
$727$	−20330.8	−1.03718	−0.518589	+	0.855024i	$0.673542\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	25701.5	1.30042
$732$	0	0
$733$	−11390.3	−0.573959	−0.286980	−	0.957937i	$-0.592651\pi$
$733$	−11390.3	−0.573959	−0.286980	+	0.957937i	$0.592651\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10282.0	−0.513895
$738$	0	0
$739$	−20573.7	−1.02411	−0.512053	−	0.858954i	$-0.671115\pi$
$739$	−20573.7	−1.02411	−0.512053	+	0.858954i	$0.671115\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26553.7	1.31112	0.655560	−	0.755143i	$-0.272433\pi$
$743$	26553.7	1.31112	0.655560	+	0.755143i	$0.272433\pi$
$744$	0	0
$745$	−5281.32	−0.259722
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−17966.7	−0.876488
$750$	0	0
$751$	9028.29	0.438678	0.219339	−	0.975649i	$-0.429610\pi$
$751$	9028.29	0.438678	0.219339	+	0.975649i	$0.429610\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−28252.8	−1.36189
$756$	0	0
$757$	34686.7	1.66540	0.832701	−	0.553722i	$-0.186793\pi$
$757$	34686.7	1.66540	0.832701	+	0.553722i	$0.186793\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35291.5	−1.68110	−0.840550	−	0.541734i	$-0.817768\pi$
$761$	−35291.5	−1.68110	−0.840550	+	0.541734i	$0.817768\pi$
$762$	0	0
$763$	−31004.8	−1.47110
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17236.8	0.811451
$768$	0	0
$769$	20828.5	0.976717	0.488359	−	0.872643i	$-0.337596\pi$
$769$	20828.5	0.976717	0.488359	+	0.872643i	$0.337596\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−17535.7	−0.815931	−0.407966	−	0.912997i	$-0.633762\pi$
$773$	−17535.7	−0.815931	−0.407966	+	0.912997i	$0.633762\pi$
$774$	0	0
$775$	2176.98	0.100902
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2388.27	−0.109844
$780$	0	0
$781$	3881.76	0.177849
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20445.3	0.929585
$786$	0	0
$787$	20214.9	0.915607	0.457803	−	0.889053i	$-0.348636\pi$
$787$	20214.9	0.915607	0.457803	+	0.889053i	$0.348636\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17770.9	−0.798813
$792$	0	0
$793$	−5620.11	−0.251672
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14467.5	0.642993	0.321497	−	0.946911i	$-0.395814\pi$
$797$	14467.5	0.642993	0.321497	+	0.946911i	$0.395814\pi$
$798$	0	0
$799$	49911.5	2.20994
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10525.3	−0.462552
$804$	0	0
$805$	6356.42	0.278303
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32041.4	1.39248	0.696240	−	0.717809i	$-0.254855\pi$
$809$	32041.4	1.39248	0.696240	+	0.717809i	$0.254855\pi$
$810$	0	0
$811$	−36356.3	−1.57416	−0.787078	−	0.616853i	$-0.788407\pi$
$811$	−36356.3	−1.57416	−0.787078	+	0.616853i	$0.788407\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	23370.6	1.00446
$816$	0	0
$817$	4789.58	0.205099
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15245.3	−0.648069	−0.324035	−	0.946045i	$-0.605039\pi$
$821$	−15245.3	−0.648069	−0.324035	+	0.946045i	$0.605039\pi$
$822$	0	0
$823$	−10663.4	−0.451646	−0.225823	−	0.974168i	$-0.572507\pi$
$823$	−10663.4	−0.451646	−0.225823	+	0.974168i	$0.572507\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	416.078	0.0174951	0.00874756	−	0.999962i	$-0.497216\pi$
$827$	416.078	0.0174951	0.00874756	+	0.999962i	$0.497216\pi$
$828$	0	0
$829$	−10759.9	−0.450793	−0.225397	−	0.974267i	$-0.572368\pi$
$829$	−10759.9	−0.450793	−0.225397	+	0.974267i	$0.572368\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−24641.5	−1.02494
$834$	0	0
$835$	8610.94	0.356879
$836$	0	0
$837$	0	0
$838$	0	0
$839$	28212.4	1.16090	0.580452	−	0.814294i	$-0.302876\pi$
$839$	28212.4	1.16090	0.580452	+	0.814294i	$0.302876\pi$
$840$	0	0
$841$	40893.0	1.67670
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2696.44	−0.109776
$846$	0	0
$847$	18168.2	0.737034
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11803.5	0.475463
$852$	0	0
$853$	−29230.7	−1.17332	−0.586659	−	0.809834i	$-0.699557\pi$
$853$	−29230.7	−1.17332	−0.586659	+	0.809834i	$0.699557\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32364.3	1.29002	0.645008	−	0.764176i	$-0.276854\pi$
$857$	32364.3	1.29002	0.645008	+	0.764176i	$0.276854\pi$
$858$	0	0
$859$	−17886.4	−0.710449	−0.355225	−	0.934781i	$-0.615596\pi$
$859$	−17886.4	−0.710449	−0.355225	+	0.934781i	$0.615596\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29845.7	−1.17724	−0.588620	−	0.808410i	$-0.700329\pi$
$863$	−29845.7	−1.17724	−0.588620	+	0.808410i	$0.700329\pi$
$864$	0	0
$865$	12892.7	0.506781
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−18723.7	−0.730906
$870$	0	0
$871$	−21364.5	−0.831125
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−36728.2	−1.41902
$876$	0	0
$877$	−7487.19	−0.288283	−0.144142	−	0.989557i	$-0.546042\pi$
$877$	−7487.19	−0.288283	−0.144142	+	0.989557i	$0.546042\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5472.71	−0.209285	−0.104643	−	0.994510i	$-0.533370\pi$
$881$	−5472.71	−0.209285	−0.104643	+	0.994510i	$0.533370\pi$
$882$	0	0
$883$	46818.2	1.78432	0.892161	−	0.451717i	$-0.149188\pi$
$883$	46818.2	1.78432	0.892161	+	0.451717i	$0.149188\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42502.5	1.60890	0.804449	−	0.594021i	$-0.202460\pi$
$887$	42502.5	1.60890	0.804449	+	0.594021i	$0.202460\pi$
$888$	0	0
$889$	41125.2	1.55151
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9301.21	0.348548
$894$	0	0
$895$	−6905.70	−0.257913
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11798.4	0.437708
$900$	0	0
$901$	22129.9	0.818263
$902$	0	0
$903$	0	0
$904$	0	0
$905$	23512.1	0.863613
$906$	0	0
$907$	−40983.4	−1.50037	−0.750184	−	0.661230i	$-0.770035\pi$
$907$	−40983.4	−1.50037	−0.750184	+	0.661230i	$0.770035\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−33343.3	−1.21264	−0.606320	−	0.795221i	$-0.707355\pi$
$911$	−33343.3	−1.21264	−0.606320	+	0.795221i	$0.707355\pi$
$912$	0	0
$913$	12702.0	0.460433
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−28048.9	−1.01009
$918$	0	0
$919$	44388.2	1.59329	0.796645	−	0.604448i	$-0.206606\pi$
$919$	44388.2	1.59329	0.796645	+	0.604448i	$0.206606\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8065.78	0.287637
$924$	0	0
$925$	−18678.1	−0.663926
$926$	0	0
$927$	0	0
$928$	0	0
$929$	41919.2	1.48043	0.740217	−	0.672368i	$-0.234723\pi$
$929$	41919.2	1.48043	0.740217	+	0.672368i	$0.234723\pi$
$930$	0	0
$931$	−4592.04	−0.161652
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−21659.0	−0.757566
$936$	0	0
$937$	4569.21	0.159306	0.0796530	−	0.996823i	$-0.474619\pi$
$937$	4569.21	0.159306	0.0796530	+	0.996823i	$0.474619\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	170.608	0.00591037	0.00295518	−	0.999996i	$-0.499059\pi$
$941$	170.608	0.00591037	0.00295518	+	0.999996i	$0.499059\pi$
$942$	0	0
$943$	3744.84	0.129320
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10906.7	0.374255	0.187128	−	0.982336i	$-0.440082\pi$
$947$	10906.7	0.374255	0.187128	+	0.982336i	$0.440082\pi$
$948$	0	0
$949$	−21870.1	−0.748087
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21798.9	−0.740960	−0.370480	−	0.928841i	$-0.620807\pi$
$953$	−21798.9	−0.740960	−0.370480	+	0.928841i	$0.620807\pi$
$954$	0	0
$955$	31280.3	1.05990
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−68993.2	−2.32316
$960$	0	0
$961$	−27658.7	−0.928424
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−38055.0	−1.26946
$966$	0	0
$967$	14160.7	0.470919	0.235459	−	0.971884i	$-0.424341\pi$
$967$	14160.7	0.470919	0.235459	+	0.971884i	$0.424341\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26224.3	0.866711	0.433356	−	0.901223i	$-0.357329\pi$
$971$	26224.3	0.866711	0.433356	+	0.901223i	$0.357329\pi$
$972$	0	0
$973$	−6616.01	−0.217985
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4063.56	0.133065	0.0665327	−	0.997784i	$-0.478806\pi$
$977$	4063.56	0.133065	0.0665327	+	0.997784i	$0.478806\pi$
$978$	0	0
$979$	31334.9	1.02295
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18314.4	0.594241	0.297120	−	0.954840i	$-0.403974\pi$
$983$	18314.4	0.594241	0.297120	+	0.954840i	$0.403974\pi$
$984$	0	0
$985$	2762.86	0.0893725
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7510.14	−0.241465
$990$	0	0
$991$	60771.2	1.94799	0.973996	−	0.226564i	$-0.0727492\pi$
$991$	60771.2	1.94799	0.973996	+	0.226564i	$0.0727492\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25010.2	0.796862
$996$	0	0
$997$	−11518.4	−0.365889	−0.182945	−	0.983123i	$-0.558563\pi$
$997$	−11518.4	−0.365889	−0.182945	+	0.983123i	$0.558563\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.4.a.l.1.3		5
3.2	odd	2		456.4.a.h.1.3	✓	5
12.11	even	2		912.4.a.w.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.4.a.h.1.3	✓	5	3.2	odd	2
912.4.a.w.1.3		5	12.11	even	2
1368.4.a.l.1.3		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)

\(f(q)\)	\(=\)	\(q-8.82361 q^{5} -24.1803 q^{7} +O(q^{10})\)	\(q-8.82361 q^{5} -24.1803 q^{7} -24.0756 q^{11} -50.0259 q^{13} -101.956 q^{17} -19.0000 q^{19} +29.7923 q^{23} -47.1440 q^{25} -255.503 q^{29} -46.1772 q^{31} +213.357 q^{35} +396.193 q^{37} +125.698 q^{41} -252.083 q^{43} -489.537 q^{47} +241.687 q^{49} -217.053 q^{53} +212.434 q^{55} -344.556 q^{59} +112.344 q^{61} +441.409 q^{65} +427.069 q^{67} -161.232 q^{71} +437.176 q^{73} +582.155 q^{77} +777.704 q^{79} -527.588 q^{83} +899.623 q^{85} -1301.52 q^{89} +1209.64 q^{91} +167.649 q^{95} +711.890 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 6 q^{5} + 10 q^{7}+O(q^{10}) \) 5 * q - 6 * q^5 + 10 * q^7	\( 5 q - 6 q^{5} + 10 q^{7} + 32 q^{11} + 36 q^{13} + 8 q^{17} - 95 q^{19} + 152 q^{23} + 241 q^{25} - 248 q^{29} + 118 q^{31} - 162 q^{35} + 472 q^{37} - 944 q^{41} + 150 q^{43} - 38 q^{47} + 1719 q^{49} - 788 q^{53} + 1270 q^{55} - 396 q^{59} + 1724 q^{61} - 2200 q^{65} + 204 q^{67} - 480 q^{71} + 2608 q^{73} - 2458 q^{77} + 786 q^{79} + 1118 q^{83} + 1798 q^{85} - 792 q^{89} + 1040 q^{91} + 114 q^{95} + 4638 q^{97}+O(q^{100}) \) 5 * q - 6 * q^5 + 10 * q^7 + 32 * q^11 + 36 * q^13 + 8 * q^17 - 95 * q^19 + 152 * q^23 + 241 * q^25 - 248 * q^29 + 118 * q^31 - 162 * q^35 + 472 * q^37 - 944 * q^41 + 150 * q^43 - 38 * q^47 + 1719 * q^49 - 788 * q^53 + 1270 * q^55 - 396 * q^59 + 1724 * q^61 - 2200 * q^65 + 204 * q^67 - 480 * q^71 + 2608 * q^73 - 2458 * q^77 + 786 * q^79 + 1118 * q^83 + 1798 * q^85 - 792 * q^89 + 1040 * q^91 + 114 * q^95 + 4638 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more