LMFDB - Embedded newform 1380.4.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,4,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, 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("1380.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1380.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:	$81.4226358079$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 1420x^{5} - 7866x^{4} + 519199x^{3} + 5329890x^{2} - 8528484x - 84125016 $ x^7 - 1420x^5 - 7866x^4 + 519199x^3 + 5329890x^2 - 8528484*x - 84125016
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.05705$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} +0.942946 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.00000 q^{5} +0.942946 q^{7} +9.00000 q^{9} +64.4328 q^{11} -77.8548 q^{13} +15.0000 q^{15} +73.5860 q^{17} +72.8975 q^{19} +2.82884 q^{21} +23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +18.1971 q^{29} -95.4235 q^{31} +193.298 q^{33} +4.71473 q^{35} -244.386 q^{37} -233.564 q^{39} +102.506 q^{41} +213.253 q^{43} +45.0000 q^{45} +465.449 q^{47} -342.111 q^{49} +220.758 q^{51} +227.980 q^{53} +322.164 q^{55} +218.692 q^{57} +345.582 q^{59} -152.916 q^{61} +8.48652 q^{63} -389.274 q^{65} -200.796 q^{67} +69.0000 q^{69} +787.807 q^{71} +167.692 q^{73} +75.0000 q^{75} +60.7566 q^{77} -1370.69 q^{79} +81.0000 q^{81} +824.879 q^{83} +367.930 q^{85} +54.5913 q^{87} -1170.58 q^{89} -73.4129 q^{91} -286.270 q^{93} +364.487 q^{95} +1366.38 q^{97} +579.895 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9}+O(q^{10}) $ 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9	$ 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9} + 48 q^{11} + 90 q^{13} + 105 q^{15} + 163 q^{17} + 200 q^{19} + 105 q^{21} + 161 q^{23} + 175 q^{25} + 189 q^{27} + 81 q^{29} + 125 q^{31} + 144 q^{33} + 175 q^{35} + 5 q^{37} + 270 q^{39} + 369 q^{41} + 462 q^{43} + 315 q^{45} + 134 q^{47} + 614 q^{49} + 489 q^{51} + 561 q^{53} + 240 q^{55} + 600 q^{57} + 951 q^{59} + 860 q^{61} + 315 q^{63} + 450 q^{65} + 447 q^{67} + 483 q^{69} + 735 q^{71} + 1460 q^{73} + 525 q^{75} + 496 q^{77} + 18 q^{79} + 567 q^{81} + 261 q^{83} + 815 q^{85} + 243 q^{87} + 2024 q^{89} + 692 q^{91} + 375 q^{93} + 1000 q^{95} + 668 q^{97} + 432 q^{99}+O(q^{100}) $ 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9 + 48 * q^11 + 90 * q^13 + 105 * q^15 + 163 * q^17 + 200 * q^19 + 105 * q^21 + 161 * q^23 + 175 * q^25 + 189 * q^27 + 81 * q^29 + 125 * q^31 + 144 * q^33 + 175 * q^35 + 5 * q^37 + 270 * q^39 + 369 * q^41 + 462 * q^43 + 315 * q^45 + 134 * q^47 + 614 * q^49 + 489 * q^51 + 561 * q^53 + 240 * q^55 + 600 * q^57 + 951 * q^59 + 860 * q^61 + 315 * q^63 + 450 * q^65 + 447 * q^67 + 483 * q^69 + 735 * q^71 + 1460 * q^73 + 525 * q^75 + 496 * q^77 + 18 * q^79 + 567 * q^81 + 261 * q^83 + 815 * q^85 + 243 * q^87 + 2024 * q^89 + 692 * q^91 + 375 * q^93 + 1000 * q^95 + 668 * q^97 + 432 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0.942946	0.0509143	0.0254572	−	0.999676i	$-0.491896\pi$
$7$	0.942946	0.0509143	0.0254572	+	0.999676i	$0.491896\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	64.4328	1.76611	0.883055	−	0.469270i	$-0.155483\pi$
$11$	64.4328	1.76611	0.883055	+	0.469270i	$0.155483\pi$
$12$	0	0
$13$	−77.8548	−1.66100	−0.830501	−	0.557017i	$-0.811946\pi$
$13$	−77.8548	−1.66100	−0.830501	+	0.557017i	$0.811946\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	73.5860	1.04984	0.524919	−	0.851152i	$-0.324096\pi$
$17$	73.5860	1.04984	0.524919	+	0.851152i	$0.324096\pi$
$18$	0	0
$19$	72.8975	0.880201	0.440101	−	0.897948i	$-0.354943\pi$
$19$	72.8975	0.880201	0.440101	+	0.897948i	$0.354943\pi$
$20$	0	0
$21$	2.82884	0.0293954
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	18.1971	0.116521	0.0582606	−	0.998301i	$-0.481445\pi$
$29$	18.1971	0.116521	0.0582606	+	0.998301i	$0.481445\pi$
$30$	0	0
$31$	−95.4235	−0.552857	−0.276428	−	0.961035i	$-0.589151\pi$
$31$	−95.4235	−0.552857	−0.276428	+	0.961035i	$0.589151\pi$
$32$	0	0
$33$	193.298	1.01966
$34$	0	0
$35$	4.71473	0.0227696
$36$	0	0
$37$	−244.386	−1.08586	−0.542930	−	0.839778i	$-0.682685\pi$
$37$	−244.386	−1.08586	−0.542930	+	0.839778i	$0.682685\pi$
$38$	0	0
$39$	−233.564	−0.958980
$40$	0	0
$41$	102.506	0.390457	0.195228	−	0.980758i	$-0.437455\pi$
$41$	102.506	0.390457	0.195228	+	0.980758i	$0.437455\pi$
$42$	0	0
$43$	213.253	0.756296	0.378148	−	0.925745i	$-0.376561\pi$
$43$	213.253	0.756296	0.378148	+	0.925745i	$0.376561\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	465.449	1.44453	0.722263	−	0.691619i	$-0.243102\pi$
$47$	465.449	1.44453	0.722263	+	0.691619i	$0.243102\pi$
$48$	0	0
$49$	−342.111	−0.997408
$50$	0	0
$51$	220.758	0.606124
$52$	0	0
$53$	227.980	0.590859	0.295429	−	0.955365i	$-0.404537\pi$
$53$	227.980	0.590859	0.295429	+	0.955365i	$0.404537\pi$
$54$	0	0
$55$	322.164	0.789828
$56$	0	0
$57$	218.692	0.508184
$58$	0	0
$59$	345.582	0.762557	0.381279	−	0.924460i	$-0.375484\pi$
$59$	345.582	0.762557	0.381279	+	0.924460i	$0.375484\pi$
$60$	0	0
$61$	−152.916	−0.320964	−0.160482	−	0.987039i	$-0.551305\pi$
$61$	−152.916	−0.320964	−0.160482	+	0.987039i	$0.551305\pi$
$62$	0	0
$63$	8.48652	0.0169714
$64$	0	0
$65$	−389.274	−0.742823
$66$	0	0
$67$	−200.796	−0.366136	−0.183068	−	0.983100i	$-0.558603\pi$
$67$	−200.796	−0.366136	−0.183068	+	0.983100i	$0.558603\pi$
$68$	0	0
$69$	69.0000	0.120386
$70$	0	0
$71$	787.807	1.31684	0.658419	−	0.752652i	$-0.271226\pi$
$71$	787.807	1.31684	0.658419	+	0.752652i	$0.271226\pi$
$72$	0	0
$73$	167.692	0.268861	0.134431	−	0.990923i	$-0.457079\pi$
$73$	167.692	0.268861	0.134431	+	0.990923i	$0.457079\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	60.7566	0.0899203
$78$	0	0
$79$	−1370.69	−1.95209	−0.976046	−	0.217566i	$-0.930188\pi$
$79$	−1370.69	−1.95209	−0.976046	+	0.217566i	$0.930188\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	824.879	1.09087	0.545435	−	0.838153i	$-0.316364\pi$
$83$	824.879	1.09087	0.545435	+	0.838153i	$0.316364\pi$
$84$	0	0
$85$	367.930	0.469502
$86$	0	0
$87$	54.5913	0.0672736
$88$	0	0
$89$	−1170.58	−1.39417	−0.697083	−	0.716990i	$-0.745519\pi$
$89$	−1170.58	−1.39417	−0.697083	+	0.716990i	$0.745519\pi$
$90$	0	0
$91$	−73.4129	−0.0845688
$92$	0	0
$93$	−286.270	−0.319192
$94$	0	0
$95$	364.487	0.393638
$96$	0	0
$97$	1366.38	1.43026	0.715128	−	0.698994i	$-0.246369\pi$
$97$	1366.38	1.43026	0.715128	+	0.698994i	$0.246369\pi$
$98$	0	0
$99$	579.895	0.588703
$100$	0	0
$101$	−807.846	−0.795878	−0.397939	−	0.917412i	$-0.630274\pi$
$101$	−807.846	−0.795878	−0.397939	+	0.917412i	$0.630274\pi$
$102$	0	0
$103$	1525.79	1.45962	0.729810	−	0.683650i	$-0.239609\pi$
$103$	1525.79	1.45962	0.729810	+	0.683650i	$0.239609\pi$
$104$	0	0
$105$	14.1442	0.0131460
$106$	0	0
$107$	1871.13	1.69055	0.845277	−	0.534328i	$-0.179435\pi$
$107$	1871.13	1.69055	0.845277	+	0.534328i	$0.179435\pi$
$108$	0	0
$109$	−1908.47	−1.67705	−0.838523	−	0.544867i	$-0.816580\pi$
$109$	−1908.47	−1.67705	−0.838523	+	0.544867i	$0.816580\pi$
$110$	0	0
$111$	−733.158	−0.626922
$112$	0	0
$113$	2.00132	0.00166609	0.000833045	−	1.00000i	$-0.499735\pi$
$113$	2.00132	0.00166609	0.000833045		1.00000i	$0.499735\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−700.693	−0.553667
$118$	0	0
$119$	69.3877	0.0534518
$120$	0	0
$121$	2820.58	2.11914
$122$	0	0
$123$	307.518	0.225430
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	148.820	0.103981	0.0519905	−	0.998648i	$-0.483443\pi$
$127$	148.820	0.103981	0.0519905	+	0.998648i	$0.483443\pi$
$128$	0	0
$129$	639.758	0.436647
$130$	0	0
$131$	−15.2357	−0.0101614	−0.00508072	−	0.999987i	$-0.501617\pi$
$131$	−15.2357	−0.0101614	−0.00508072	+	0.999987i	$0.501617\pi$
$132$	0	0
$133$	68.7384	0.0448148
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	1131.43	0.705584	0.352792	−	0.935702i	$-0.385232\pi$
$137$	1131.43	0.705584	0.352792	+	0.935702i	$0.385232\pi$
$138$	0	0
$139$	1018.18	0.621302	0.310651	−	0.950524i	$-0.399453\pi$
$139$	1018.18	0.621302	0.310651	+	0.950524i	$0.399453\pi$
$140$	0	0
$141$	1396.35	0.833997
$142$	0	0
$143$	−5016.40	−2.93351
$144$	0	0
$145$	90.9855	0.0521099
$146$	0	0
$147$	−1026.33	−0.575854
$148$	0	0
$149$	1902.09	1.04581	0.522905	−	0.852391i	$-0.324848\pi$
$149$	1902.09	1.04581	0.522905	+	0.852391i	$0.324848\pi$
$150$	0	0
$151$	190.612	0.102727	0.0513635	−	0.998680i	$-0.483643\pi$
$151$	190.612	0.102727	0.0513635	+	0.998680i	$0.483643\pi$
$152$	0	0
$153$	662.274	0.349946
$154$	0	0
$155$	−477.117	−0.247245
$156$	0	0
$157$	3248.52	1.65134	0.825670	−	0.564154i	$-0.190797\pi$
$157$	3248.52	1.65134	0.825670	+	0.564154i	$0.190797\pi$
$158$	0	0
$159$	683.941	0.341132
$160$	0	0
$161$	21.6878	0.0106164
$162$	0	0
$163$	1547.34	0.743540	0.371770	−	0.928325i	$-0.378751\pi$
$163$	1547.34	0.743540	0.371770	+	0.928325i	$0.378751\pi$
$164$	0	0
$165$	966.491	0.456008
$166$	0	0
$167$	−1698.55	−0.787052	−0.393526	−	0.919313i	$-0.628745\pi$
$167$	−1698.55	−0.787052	−0.393526	+	0.919313i	$0.628745\pi$
$168$	0	0
$169$	3864.36	1.75893
$170$	0	0
$171$	656.077	0.293400
$172$	0	0
$173$	12.5208	0.00550252	0.00275126	−	0.999996i	$-0.499124\pi$
$173$	12.5208	0.00550252	0.00275126	+	0.999996i	$0.499124\pi$
$174$	0	0
$175$	23.5737	0.0101829
$176$	0	0
$177$	1036.74	0.440263
$178$	0	0
$179$	728.836	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$179$	728.836	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$180$	0	0
$181$	−4258.47	−1.74878	−0.874392	−	0.485220i	$-0.838739\pi$
$181$	−4258.47	−1.74878	−0.874392	+	0.485220i	$0.838739\pi$
$182$	0	0
$183$	−458.747	−0.185309
$184$	0	0
$185$	−1221.93	−0.485611
$186$	0	0
$187$	4741.35	1.85413
$188$	0	0
$189$	25.4595	0.00979846
$190$	0	0
$191$	−2095.29	−0.793771	−0.396885	−	0.917868i	$-0.629909\pi$
$191$	−2095.29	−0.793771	−0.396885	+	0.917868i	$0.629909\pi$
$192$	0	0
$193$	−2684.69	−1.00128	−0.500642	−	0.865654i	$-0.666903\pi$
$193$	−2684.69	−1.00128	−0.500642	+	0.865654i	$0.666903\pi$
$194$	0	0
$195$	−1167.82	−0.428869
$196$	0	0
$197$	−702.701	−0.254139	−0.127069	−	0.991894i	$-0.540557\pi$
$197$	−702.701	−0.254139	−0.127069	+	0.991894i	$0.540557\pi$
$198$	0	0
$199$	−4579.76	−1.63141	−0.815705	−	0.578468i	$-0.803651\pi$
$199$	−4579.76	−1.63141	−0.815705	+	0.578468i	$0.803651\pi$
$200$	0	0
$201$	−602.387	−0.211389
$202$	0	0
$203$	17.1589	0.00593260
$204$	0	0
$205$	512.529	0.174618
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	4696.99	1.55453
$210$	0	0
$211$	2218.99	0.723988	0.361994	−	0.932181i	$-0.382096\pi$
$211$	2218.99	0.723988	0.361994	+	0.932181i	$0.382096\pi$
$212$	0	0
$213$	2363.42	0.760277
$214$	0	0
$215$	1066.26	0.338226
$216$	0	0
$217$	−89.9792	−0.0281483
$218$	0	0
$219$	503.076	0.155227
$220$	0	0
$221$	−5729.02	−1.74378
$222$	0	0
$223$	−5099.34	−1.53129	−0.765643	−	0.643266i	$-0.777579\pi$
$223$	−5099.34	−1.53129	−0.765643	+	0.643266i	$0.777579\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−1510.64	−0.441695	−0.220847	−	0.975308i	$-0.570882\pi$
$227$	−1510.64	−0.441695	−0.220847	+	0.975308i	$0.570882\pi$
$228$	0	0
$229$	2446.95	0.706109	0.353055	−	0.935603i	$-0.385143\pi$
$229$	2446.95	0.706109	0.353055	+	0.935603i	$0.385143\pi$
$230$	0	0
$231$	182.270	0.0519155
$232$	0	0
$233$	849.689	0.238906	0.119453	−	0.992840i	$-0.461886\pi$
$233$	849.689	0.238906	0.119453	+	0.992840i	$0.461886\pi$
$234$	0	0
$235$	2327.25	0.646012
$236$	0	0
$237$	−4112.08	−1.12704
$238$	0	0
$239$	6348.24	1.71813	0.859066	−	0.511865i	$-0.171045\pi$
$239$	6348.24	1.71813	0.859066	+	0.511865i	$0.171045\pi$
$240$	0	0
$241$	3708.14	0.991131	0.495565	−	0.868571i	$-0.334961\pi$
$241$	3708.14	0.991131	0.495565	+	0.868571i	$0.334961\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1710.55	−0.446054
$246$	0	0
$247$	−5675.42	−1.46202
$248$	0	0
$249$	2474.64	0.629814
$250$	0	0
$251$	−5700.84	−1.43360	−0.716801	−	0.697278i	$-0.754394\pi$
$251$	−5700.84	−1.43360	−0.716801	+	0.697278i	$0.754394\pi$
$252$	0	0
$253$	1481.95	0.368259
$254$	0	0
$255$	1103.79	0.271067
$256$	0	0
$257$	1873.58	0.454751	0.227375	−	0.973807i	$-0.426986\pi$
$257$	1873.58	0.454751	0.227375	+	0.973807i	$0.426986\pi$
$258$	0	0
$259$	−230.443	−0.0552858
$260$	0	0
$261$	163.774	0.0388404
$262$	0	0
$263$	7536.93	1.76710	0.883549	−	0.468338i	$-0.155147\pi$
$263$	7536.93	1.76710	0.883549	+	0.468338i	$0.155147\pi$
$264$	0	0
$265$	1139.90	0.264240
$266$	0	0
$267$	−3511.73	−0.804922
$268$	0	0
$269$	6155.01	1.39508	0.697542	−	0.716544i	$-0.254277\pi$
$269$	6155.01	1.39508	0.697542	+	0.716544i	$0.254277\pi$
$270$	0	0
$271$	7399.73	1.65868	0.829339	−	0.558746i	$-0.188717\pi$
$271$	7399.73	1.65868	0.829339	+	0.558746i	$0.188717\pi$
$272$	0	0
$273$	−220.239	−0.0488258
$274$	0	0
$275$	1610.82	0.353222
$276$	0	0
$277$	3930.22	0.852504	0.426252	−	0.904604i	$-0.359834\pi$
$277$	3930.22	0.852504	0.426252	+	0.904604i	$0.359834\pi$
$278$	0	0
$279$	−858.811	−0.184286
$280$	0	0
$281$	4381.55	0.930182	0.465091	−	0.885263i	$-0.346022\pi$
$281$	4381.55	0.930182	0.465091	+	0.885263i	$0.346022\pi$
$282$	0	0
$283$	6092.22	1.27966	0.639832	−	0.768515i	$-0.279004\pi$
$283$	6092.22	1.27966	0.639832	+	0.768515i	$0.279004\pi$
$284$	0	0
$285$	1093.46	0.227267
$286$	0	0
$287$	96.6575	0.0198798
$288$	0	0
$289$	501.907	0.102159
$290$	0	0
$291$	4099.14	0.825758
$292$	0	0
$293$	−5656.94	−1.12792	−0.563962	−	0.825801i	$-0.690724\pi$
$293$	−5656.94	−1.12792	−0.563962	+	0.825801i	$0.690724\pi$
$294$	0	0
$295$	1727.91	0.341026
$296$	0	0
$297$	1739.68	0.339888
$298$	0	0
$299$	−1790.66	−0.346343
$300$	0	0
$301$	201.086	0.0385063
$302$	0	0
$303$	−2423.54	−0.459500
$304$	0	0
$305$	−764.578	−0.143540
$306$	0	0
$307$	−3384.86	−0.629264	−0.314632	−	0.949214i	$-0.601881\pi$
$307$	−3384.86	−0.629264	−0.314632	+	0.949214i	$0.601881\pi$
$308$	0	0
$309$	4577.38	0.842712
$310$	0	0
$311$	9891.69	1.80356	0.901779	−	0.432198i	$-0.142262\pi$
$311$	9891.69	1.80356	0.901779	+	0.432198i	$0.142262\pi$
$312$	0	0
$313$	−7806.19	−1.40969	−0.704843	−	0.709363i	$-0.748983\pi$
$313$	−7806.19	−1.40969	−0.704843	+	0.709363i	$0.748983\pi$
$314$	0	0
$315$	42.4326	0.00758986
$316$	0	0
$317$	7866.20	1.39372	0.696861	−	0.717206i	$-0.254579\pi$
$317$	7866.20	1.39372	0.696861	+	0.717206i	$0.254579\pi$
$318$	0	0
$319$	1172.49	0.205789
$320$	0	0
$321$	5613.40	0.976042
$322$	0	0
$323$	5364.24	0.924068
$324$	0	0
$325$	−1946.37	−0.332200
$326$	0	0
$327$	−5725.40	−0.968242
$328$	0	0
$329$	438.893	0.0735470
$330$	0	0
$331$	−9895.27	−1.64318	−0.821591	−	0.570077i	$-0.806913\pi$
$331$	−9895.27	−1.64318	−0.821591	+	0.570077i	$0.806913\pi$
$332$	0	0
$333$	−2199.47	−0.361953
$334$	0	0
$335$	−1003.98	−0.163741
$336$	0	0
$337$	4608.21	0.744882	0.372441	−	0.928056i	$-0.378521\pi$
$337$	4608.21	0.744882	0.372441	+	0.928056i	$0.378521\pi$
$338$	0	0
$339$	6.00396	0.000961918 0
$340$	0	0
$341$	−6148.40	−0.976406
$342$	0	0
$343$	−646.023	−0.101697
$344$	0	0
$345$	345.000	0.0538382
$346$	0	0
$347$	614.450	0.0950588	0.0475294	−	0.998870i	$-0.484865\pi$
$347$	614.450	0.0950588	0.0475294	+	0.998870i	$0.484865\pi$
$348$	0	0
$349$	2596.93	0.398311	0.199155	−	0.979968i	$-0.436180\pi$
$349$	2596.93	0.398311	0.199155	+	0.979968i	$0.436180\pi$
$350$	0	0
$351$	−2102.08	−0.319660
$352$	0	0
$353$	−12663.7	−1.90941	−0.954706	−	0.297551i	$-0.903830\pi$
$353$	−12663.7	−1.90941	−0.954706	+	0.297551i	$0.903830\pi$
$354$	0	0
$355$	3939.03	0.588908
$356$	0	0
$357$	208.163	0.0308604
$358$	0	0
$359$	−8199.72	−1.20547	−0.602736	−	0.797940i	$-0.705923\pi$
$359$	−8199.72	−1.20547	−0.602736	+	0.797940i	$0.705923\pi$
$360$	0	0
$361$	−1544.96	−0.225246
$362$	0	0
$363$	8461.74	1.22349
$364$	0	0
$365$	838.461	0.120238
$366$	0	0
$367$	11029.4	1.56875	0.784376	−	0.620286i	$-0.212983\pi$
$367$	11029.4	1.56875	0.784376	+	0.620286i	$0.212983\pi$
$368$	0	0
$369$	922.553	0.130152
$370$	0	0
$371$	214.973	0.0300832
$372$	0	0
$373$	−4798.37	−0.666086	−0.333043	−	0.942912i	$-0.608075\pi$
$373$	−4798.37	−0.666086	−0.333043	+	0.942912i	$0.608075\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−1416.73	−0.193542
$378$	0	0
$379$	−3895.63	−0.527982	−0.263991	−	0.964525i	$-0.585039\pi$
$379$	−3895.63	−0.527982	−0.263991	+	0.964525i	$0.585039\pi$
$380$	0	0
$381$	446.459	0.0600335
$382$	0	0
$383$	−6836.90	−0.912139	−0.456069	−	0.889944i	$-0.650743\pi$
$383$	−6836.90	−0.912139	−0.456069	+	0.889944i	$0.650743\pi$
$384$	0	0
$385$	303.783	0.0402136
$386$	0	0
$387$	1919.27	0.252099
$388$	0	0
$389$	−3657.21	−0.476678	−0.238339	−	0.971182i	$-0.576603\pi$
$389$	−3657.21	−0.476678	−0.238339	+	0.971182i	$0.576603\pi$
$390$	0	0
$391$	1692.48	0.218906
$392$	0	0
$393$	−45.7071	−0.00586671
$394$	0	0
$395$	−6853.47	−0.873002
$396$	0	0
$397$	−4008.23	−0.506718	−0.253359	−	0.967372i	$-0.581535\pi$
$397$	−4008.23	−0.506718	−0.253359	+	0.967372i	$0.581535\pi$
$398$	0	0
$399$	206.215	0.0258739
$400$	0	0
$401$	−4299.95	−0.535484	−0.267742	−	0.963491i	$-0.586277\pi$
$401$	−4299.95	−0.535484	−0.267742	+	0.963491i	$0.586277\pi$
$402$	0	0
$403$	7429.17	0.918296
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−15746.5	−1.91775
$408$	0	0
$409$	11022.1	1.33253	0.666267	−	0.745713i	$-0.267891\pi$
$409$	11022.1	1.33253	0.666267	+	0.745713i	$0.267891\pi$
$410$	0	0
$411$	3394.30	0.407369
$412$	0	0
$413$	325.865	0.0388251
$414$	0	0
$415$	4124.39	0.487852
$416$	0	0
$417$	3054.54	0.358709
$418$	0	0
$419$	−6437.61	−0.750591	−0.375296	−	0.926905i	$-0.622459\pi$
$419$	−6437.61	−0.750591	−0.375296	+	0.926905i	$0.622459\pi$
$420$	0	0
$421$	−3875.63	−0.448662	−0.224331	−	0.974513i	$-0.572020\pi$
$421$	−3875.63	−0.448662	−0.224331	+	0.974513i	$0.572020\pi$
$422$	0	0
$423$	4189.04	0.481509
$424$	0	0
$425$	1839.65	0.209968
$426$	0	0
$427$	−144.191	−0.0163417
$428$	0	0
$429$	−15049.2	−1.69366
$430$	0	0
$431$	−17007.7	−1.90077	−0.950386	−	0.311074i	$-0.899311\pi$
$431$	−17007.7	−1.90077	−0.950386	+	0.311074i	$0.899311\pi$
$432$	0	0
$433$	−3726.94	−0.413639	−0.206819	−	0.978379i	$-0.566311\pi$
$433$	−3726.94	−0.413639	−0.206819	+	0.978379i	$0.566311\pi$
$434$	0	0
$435$	272.956	0.0300857
$436$	0	0
$437$	1676.64	0.183535
$438$	0	0
$439$	−7087.29	−0.770519	−0.385260	−	0.922808i	$-0.625888\pi$
$439$	−7087.29	−0.770519	−0.385260	+	0.922808i	$0.625888\pi$
$440$	0	0
$441$	−3079.00	−0.332469
$442$	0	0
$443$	−3114.03	−0.333977	−0.166989	−	0.985959i	$-0.553404\pi$
$443$	−3114.03	−0.333977	−0.166989	+	0.985959i	$0.553404\pi$
$444$	0	0
$445$	−5852.88	−0.623490
$446$	0	0
$447$	5706.28	0.603798
$448$	0	0
$449$	−13555.4	−1.42476	−0.712381	−	0.701793i	$-0.752383\pi$
$449$	−13555.4	−1.42476	−0.712381	+	0.701793i	$0.752383\pi$
$450$	0	0
$451$	6604.73	0.689589
$452$	0	0
$453$	571.836	0.0593095
$454$	0	0
$455$	−367.064	−0.0378203
$456$	0	0
$457$	−8181.63	−0.837463	−0.418732	−	0.908110i	$-0.637525\pi$
$457$	−8181.63	−0.837463	−0.418732	+	0.908110i	$0.637525\pi$
$458$	0	0
$459$	1986.82	0.202041
$460$	0	0
$461$	697.891	0.0705076	0.0352538	−	0.999378i	$-0.488776\pi$
$461$	697.891	0.0705076	0.0352538	+	0.999378i	$0.488776\pi$
$462$	0	0
$463$	9558.85	0.959476	0.479738	−	0.877412i	$-0.340732\pi$
$463$	9558.85	0.959476	0.479738	+	0.877412i	$0.340732\pi$
$464$	0	0
$465$	−1431.35	−0.142747
$466$	0	0
$467$	16344.8	1.61959	0.809795	−	0.586712i	$-0.199578\pi$
$467$	16344.8	1.61959	0.809795	+	0.586712i	$0.199578\pi$
$468$	0	0
$469$	−189.340	−0.0186416
$470$	0	0
$471$	9745.57	0.953401
$472$	0	0
$473$	13740.5	1.33570
$474$	0	0
$475$	1822.44	0.176040
$476$	0	0
$477$	2051.82	0.196953
$478$	0	0
$479$	13536.5	1.29123	0.645616	−	0.763662i	$-0.276601\pi$
$479$	13536.5	1.29123	0.645616	+	0.763662i	$0.276601\pi$
$480$	0	0
$481$	19026.6	1.80362
$482$	0	0
$483$	65.0633	0.00612936
$484$	0	0
$485$	6831.89	0.639630
$486$	0	0
$487$	15422.7	1.43505	0.717527	−	0.696531i	$-0.245274\pi$
$487$	15422.7	1.43505	0.717527	+	0.696531i	$0.245274\pi$
$488$	0	0
$489$	4642.02	0.429283
$490$	0	0
$491$	−11092.0	−1.01950	−0.509750	−	0.860322i	$-0.670262\pi$
$491$	−11092.0	−1.01950	−0.509750	+	0.860322i	$0.670262\pi$
$492$	0	0
$493$	1339.05	0.122328
$494$	0	0
$495$	2899.47	0.263276
$496$	0	0
$497$	742.860	0.0670459
$498$	0	0
$499$	6769.41	0.607296	0.303648	−	0.952784i	$-0.401795\pi$
$499$	6769.41	0.607296	0.303648	+	0.952784i	$0.401795\pi$
$500$	0	0
$501$	−5095.65	−0.454405
$502$	0	0
$503$	−21613.2	−1.91587	−0.957937	−	0.286977i	$-0.907350\pi$
$503$	−21613.2	−1.91587	−0.957937	+	0.286977i	$0.907350\pi$
$504$	0	0
$505$	−4039.23	−0.355928
$506$	0	0
$507$	11593.1	1.01552
$508$	0	0
$509$	−17753.1	−1.54595	−0.772977	−	0.634434i	$-0.781233\pi$
$509$	−17753.1	−1.54595	−0.772977	+	0.634434i	$0.781233\pi$
$510$	0	0
$511$	158.125	0.0136889
$512$	0	0
$513$	1968.23	0.169395
$514$	0	0
$515$	7628.96	0.652762
$516$	0	0
$517$	29990.2	2.55119
$518$	0	0
$519$	37.5623	0.00317688
$520$	0	0
$521$	−271.844	−0.0228593	−0.0114296	−	0.999935i	$-0.503638\pi$
$521$	−271.844	−0.0228593	−0.0114296	+	0.999935i	$0.503638\pi$
$522$	0	0
$523$	−16046.1	−1.34158	−0.670792	−	0.741645i	$-0.734046\pi$
$523$	−16046.1	−1.34158	−0.670792	+	0.741645i	$0.734046\pi$
$524$	0	0
$525$	70.7210	0.00587908
$526$	0	0
$527$	−7021.84	−0.580410
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	3110.23	0.254186
$532$	0	0
$533$	−7980.57	−0.648549
$534$	0	0
$535$	9355.67	0.756039
$536$	0	0
$537$	2186.51	0.175707
$538$	0	0
$539$	−22043.1	−1.76153
$540$	0	0
$541$	−9777.45	−0.777015	−0.388508	−	0.921445i	$-0.627009\pi$
$541$	−9777.45	−0.777015	−0.388508	+	0.921445i	$0.627009\pi$
$542$	0	0
$543$	−12775.4	−1.00966
$544$	0	0
$545$	−9542.33	−0.749997
$546$	0	0
$547$	−23418.2	−1.83051	−0.915255	−	0.402876i	$-0.868011\pi$
$547$	−23418.2	−1.83051	−0.915255	+	0.402876i	$0.868011\pi$
$548$	0	0
$549$	−1376.24	−0.106988
$550$	0	0
$551$	1326.52	0.102562
$552$	0	0
$553$	−1292.49	−0.0993894
$554$	0	0
$555$	−3665.79	−0.280368
$556$	0	0
$557$	17021.0	1.29480	0.647399	−	0.762152i	$-0.275857\pi$
$557$	17021.0	1.29480	0.647399	+	0.762152i	$0.275857\pi$
$558$	0	0
$559$	−16602.7	−1.25621
$560$	0	0
$561$	14224.1	1.07048
$562$	0	0
$563$	−19913.5	−1.49068	−0.745341	−	0.666684i	$-0.767713\pi$
$563$	−19913.5	−1.49068	−0.745341	+	0.666684i	$0.767713\pi$
$564$	0	0
$565$	10.0066	0.000745098 0
$566$	0	0
$567$	76.3786	0.00565715
$568$	0	0
$569$	−18054.8	−1.33022	−0.665109	−	0.746746i	$-0.731615\pi$
$569$	−18054.8	−1.33022	−0.665109	+	0.746746i	$0.731615\pi$
$570$	0	0
$571$	12319.8	0.902924	0.451462	−	0.892290i	$-0.350903\pi$
$571$	12319.8	0.902924	0.451462	+	0.892290i	$0.350903\pi$
$572$	0	0
$573$	−6285.88	−0.458284
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	21311.6	1.53763	0.768814	−	0.639472i	$-0.220847\pi$
$577$	21311.6	1.53763	0.768814	+	0.639472i	$0.220847\pi$
$578$	0	0
$579$	−8054.06	−0.578092
$580$	0	0
$581$	777.816	0.0555409
$582$	0	0
$583$	14689.4	1.04352
$584$	0	0
$585$	−3503.46	−0.247608
$586$	0	0
$587$	−7504.89	−0.527700	−0.263850	−	0.964564i	$-0.584992\pi$
$587$	−7504.89	−0.527700	−0.263850	+	0.964564i	$0.584992\pi$
$588$	0	0
$589$	−6956.13	−0.486625
$590$	0	0
$591$	−2108.10	−0.146727
$592$	0	0
$593$	13414.9	0.928978	0.464489	−	0.885579i	$-0.346238\pi$
$593$	13414.9	0.928978	0.464489	+	0.885579i	$0.346238\pi$
$594$	0	0
$595$	346.938	0.0239044
$596$	0	0
$597$	−13739.3	−0.941895
$598$	0	0
$599$	−8136.63	−0.555015	−0.277507	−	0.960724i	$-0.589508\pi$
$599$	−8136.63	−0.555015	−0.277507	+	0.960724i	$0.589508\pi$
$600$	0	0
$601$	−18669.6	−1.26714	−0.633568	−	0.773687i	$-0.718410\pi$
$601$	−18669.6	−1.26714	−0.633568	+	0.773687i	$0.718410\pi$
$602$	0	0
$603$	−1807.16	−0.122045
$604$	0	0
$605$	14102.9	0.947710
$606$	0	0
$607$	−23620.8	−1.57947	−0.789734	−	0.613449i	$-0.789782\pi$
$607$	−23620.8	−1.57947	−0.789734	+	0.613449i	$0.789782\pi$
$608$	0	0
$609$	51.4767	0.00342519
$610$	0	0
$611$	−36237.4	−2.39936
$612$	0	0
$613$	−19823.8	−1.30616	−0.653079	−	0.757290i	$-0.726523\pi$
$613$	−19823.8	−1.30616	−0.653079	+	0.757290i	$0.726523\pi$
$614$	0	0
$615$	1537.59	0.100815
$616$	0	0
$617$	5538.37	0.361372	0.180686	−	0.983541i	$-0.442168\pi$
$617$	5538.37	0.361372	0.180686	+	0.983541i	$0.442168\pi$
$618$	0	0
$619$	−19409.4	−1.26031	−0.630153	−	0.776471i	$-0.717008\pi$
$619$	−19409.4	−1.26031	−0.630153	+	0.776471i	$0.717008\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	0	0
$623$	−1103.79	−0.0709830
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	14091.0	0.897510
$628$	0	0
$629$	−17983.4	−1.13998
$630$	0	0
$631$	−16713.0	−1.05441	−0.527207	−	0.849737i	$-0.676761\pi$
$631$	−16713.0	−1.05441	−0.527207	+	0.849737i	$0.676761\pi$
$632$	0	0
$633$	6656.96	0.417994
$634$	0	0
$635$	744.098	0.0465017
$636$	0	0
$637$	26635.0	1.65670
$638$	0	0
$639$	7090.26	0.438946
$640$	0	0
$641$	−6463.75	−0.398288	−0.199144	−	0.979970i	$-0.563816\pi$
$641$	−6463.75	−0.398288	−0.199144	+	0.979970i	$0.563816\pi$
$642$	0	0
$643$	−16858.5	−1.03396	−0.516978	−	0.855999i	$-0.672943\pi$
$643$	−16858.5	−1.03396	−0.516978	+	0.855999i	$0.672943\pi$
$644$	0	0
$645$	3198.79	0.195275
$646$	0	0
$647$	−8859.27	−0.538321	−0.269161	−	0.963095i	$-0.586746\pi$
$647$	−8859.27	−0.538321	−0.269161	+	0.963095i	$0.586746\pi$
$648$	0	0
$649$	22266.8	1.34676
$650$	0	0
$651$	−269.938	−0.0162514
$652$	0	0
$653$	7265.32	0.435397	0.217698	−	0.976016i	$-0.430145\pi$
$653$	7265.32	0.435397	0.217698	+	0.976016i	$0.430145\pi$
$654$	0	0
$655$	−76.1785	−0.00454433
$656$	0	0
$657$	1509.23	0.0896204
$658$	0	0
$659$	−10204.4	−0.603197	−0.301599	−	0.953435i	$-0.597520\pi$
$659$	−10204.4	−0.603197	−0.301599	+	0.953435i	$0.597520\pi$
$660$	0	0
$661$	−12515.5	−0.736454	−0.368227	−	0.929736i	$-0.620035\pi$
$661$	−12515.5	−0.736454	−0.368227	+	0.929736i	$0.620035\pi$
$662$	0	0
$663$	−17187.1	−1.00677
$664$	0	0
$665$	343.692	0.0200418
$666$	0	0
$667$	418.533	0.0242964
$668$	0	0
$669$	−15298.0	−0.884088
$670$	0	0
$671$	−9852.77	−0.566858
$672$	0	0
$673$	6147.87	0.352129	0.176065	−	0.984379i	$-0.443663\pi$
$673$	6147.87	0.352129	0.176065	+	0.984379i	$0.443663\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	10493.5	0.595716	0.297858	−	0.954610i	$-0.403728\pi$
$677$	10493.5	0.595716	0.297858	+	0.954610i	$0.403728\pi$
$678$	0	0
$679$	1288.42	0.0728205
$680$	0	0
$681$	−4531.92	−0.255012
$682$	0	0
$683$	−9582.26	−0.536830	−0.268415	−	0.963303i	$-0.586500\pi$
$683$	−9582.26	−0.536830	−0.268415	+	0.963303i	$0.586500\pi$
$684$	0	0
$685$	5657.17	0.315547
$686$	0	0
$687$	7340.85	0.407672
$688$	0	0
$689$	−17749.4	−0.981418
$690$	0	0
$691$	−24153.4	−1.32972	−0.664861	−	0.746967i	$-0.731509\pi$
$691$	−24153.4	−1.32972	−0.664861	+	0.746967i	$0.731509\pi$
$692$	0	0
$693$	546.810	0.0299734
$694$	0	0
$695$	5090.90	0.277855
$696$	0	0
$697$	7543.00	0.409916
$698$	0	0
$699$	2549.07	0.137932
$700$	0	0
$701$	2841.56	0.153102	0.0765508	−	0.997066i	$-0.475609\pi$
$701$	2841.56	0.153102	0.0765508	+	0.997066i	$0.475609\pi$
$702$	0	0
$703$	−17815.1	−0.955775
$704$	0	0
$705$	6981.74	0.372975
$706$	0	0
$707$	−761.755	−0.0405216
$708$	0	0
$709$	17105.7	0.906091	0.453046	−	0.891487i	$-0.350337\pi$
$709$	17105.7	0.906091	0.453046	+	0.891487i	$0.350337\pi$
$710$	0	0
$711$	−12336.3	−0.650697
$712$	0	0
$713$	−2194.74	−0.115279
$714$	0	0
$715$	−25082.0	−1.31191
$716$	0	0
$717$	19044.7	0.991964
$718$	0	0
$719$	14061.6	0.729360	0.364680	−	0.931133i	$-0.381178\pi$
$719$	14061.6	0.729360	0.364680	+	0.931133i	$0.381178\pi$
$720$	0	0
$721$	1438.74	0.0743155
$722$	0	0
$723$	11124.4	0.572230
$724$	0	0
$725$	454.927	0.0233042
$726$	0	0
$727$	5014.56	0.255818	0.127909	−	0.991786i	$-0.459173\pi$
$727$	5014.56	0.255818	0.127909	+	0.991786i	$0.459173\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	15692.4	0.793988
$732$	0	0
$733$	6849.91	0.345167	0.172583	−	0.984995i	$-0.444789\pi$
$733$	6849.91	0.345167	0.172583	+	0.984995i	$0.444789\pi$
$734$	0	0
$735$	−5131.66	−0.257530
$736$	0	0
$737$	−12937.8	−0.646636
$738$	0	0
$739$	−14467.4	−0.720149	−0.360075	−	0.932924i	$-0.617249\pi$
$739$	−14467.4	−0.720149	−0.360075	+	0.932924i	$0.617249\pi$
$740$	0	0
$741$	−17026.2	−0.844095
$742$	0	0
$743$	−15367.2	−0.758771	−0.379385	−	0.925239i	$-0.623865\pi$
$743$	−15367.2	−0.758771	−0.379385	+	0.925239i	$0.623865\pi$
$744$	0	0
$745$	9510.47	0.467700
$746$	0	0
$747$	7423.91	0.363623
$748$	0	0
$749$	1764.38	0.0860734
$750$	0	0
$751$	−19683.9	−0.956428	−0.478214	−	0.878243i	$-0.658716\pi$
$751$	−19683.9	−0.956428	−0.478214	+	0.878243i	$0.658716\pi$
$752$	0	0
$753$	−17102.5	−0.827690
$754$	0	0
$755$	953.060	0.0459410
$756$	0	0
$757$	−24442.5	−1.17355	−0.586776	−	0.809749i	$-0.699603\pi$
$757$	−24442.5	−1.17355	−0.586776	+	0.809749i	$0.699603\pi$
$758$	0	0
$759$	4445.86	0.212615
$760$	0	0
$761$	21937.1	1.04497	0.522483	−	0.852650i	$-0.325006\pi$
$761$	21937.1	1.04497	0.522483	+	0.852650i	$0.325006\pi$
$762$	0	0
$763$	−1799.58	−0.0853856
$764$	0	0
$765$	3311.37	0.156501
$766$	0	0
$767$	−26905.2	−1.26661
$768$	0	0
$769$	−26912.7	−1.26203	−0.631013	−	0.775772i	$-0.717361\pi$
$769$	−26912.7	−1.26203	−0.631013	+	0.775772i	$0.717361\pi$
$770$	0	0
$771$	5620.75	0.262551
$772$	0	0
$773$	3142.56	0.146223	0.0731113	−	0.997324i	$-0.476707\pi$
$773$	3142.56	0.146223	0.0731113	+	0.997324i	$0.476707\pi$
$774$	0	0
$775$	−2385.59	−0.110571
$776$	0	0
$777$	−691.329	−0.0319193
$778$	0	0
$779$	7472.42	0.343680
$780$	0	0
$781$	50760.6	2.32568
$782$	0	0
$783$	491.322	0.0224245
$784$	0	0
$785$	16242.6	0.738502
$786$	0	0
$787$	−5454.44	−0.247052	−0.123526	−	0.992341i	$-0.539420\pi$
$787$	−5454.44	−0.247052	−0.123526	+	0.992341i	$0.539420\pi$
$788$	0	0
$789$	22610.8	1.02023
$790$	0	0
$791$	1.88714	8.48279e−5 0
$792$	0	0
$793$	11905.2	0.533122
$794$	0	0
$795$	3419.71	0.152559
$796$	0	0
$797$	28621.4	1.27205	0.636023	−	0.771670i	$-0.280578\pi$
$797$	28621.4	1.27205	0.636023	+	0.771670i	$0.280578\pi$
$798$	0	0
$799$	34250.6	1.51652
$800$	0	0
$801$	−10535.2	−0.464722
$802$	0	0
$803$	10804.9	0.474839
$804$	0	0
$805$	108.439	0.00474778
$806$	0	0
$807$	18465.0	0.805452
$808$	0	0
$809$	−4844.33	−0.210528	−0.105264	−	0.994444i	$-0.533569\pi$
$809$	−4844.33	−0.210528	−0.105264	+	0.994444i	$0.533569\pi$
$810$	0	0
$811$	−27660.8	−1.19766	−0.598829	−	0.800877i	$-0.704367\pi$
$811$	−27660.8	−1.19766	−0.598829	+	0.800877i	$0.704367\pi$
$812$	0	0
$813$	22199.2	0.957638
$814$	0	0
$815$	7736.70	0.332521
$816$	0	0
$817$	15545.6	0.665692
$818$	0	0
$819$	−660.716	−0.0281896
$820$	0	0
$821$	5732.06	0.243667	0.121833	−	0.992551i	$-0.461123\pi$
$821$	5732.06	0.243667	0.121833	+	0.992551i	$0.461123\pi$
$822$	0	0
$823$	2283.67	0.0967240	0.0483620	−	0.998830i	$-0.484600\pi$
$823$	2283.67	0.0967240	0.0483620	+	0.998830i	$0.484600\pi$
$824$	0	0
$825$	4832.46	0.203933
$826$	0	0
$827$	−40141.9	−1.68787	−0.843937	−	0.536443i	$-0.819768\pi$
$827$	−40141.9	−1.68787	−0.843937	+	0.536443i	$0.819768\pi$
$828$	0	0
$829$	6609.31	0.276901	0.138450	−	0.990369i	$-0.455788\pi$
$829$	6609.31	0.276901	0.138450	+	0.990369i	$0.455788\pi$
$830$	0	0
$831$	11790.6	0.492194
$832$	0	0
$833$	−25174.6	−1.04712
$834$	0	0
$835$	−8492.75	−0.351980
$836$	0	0
$837$	−2576.43	−0.106397
$838$	0	0
$839$	−5075.16	−0.208837	−0.104418	−	0.994533i	$-0.533298\pi$
$839$	−5075.16	−0.208837	−0.104418	+	0.994533i	$0.533298\pi$
$840$	0	0
$841$	−24057.9	−0.986423
$842$	0	0
$843$	13144.6	0.537041
$844$	0	0
$845$	19321.8	0.786616
$846$	0	0
$847$	2659.66	0.107895
$848$	0	0
$849$	18276.7	0.738814
$850$	0	0
$851$	−5620.88	−0.226417
$852$	0	0
$853$	−20985.6	−0.842360	−0.421180	−	0.906977i	$-0.638384\pi$
$853$	−20985.6	−0.842360	−0.421180	+	0.906977i	$0.638384\pi$
$854$	0	0
$855$	3280.39	0.131213
$856$	0	0
$857$	−21990.4	−0.876521	−0.438260	−	0.898848i	$-0.644405\pi$
$857$	−21990.4	−0.876521	−0.438260	+	0.898848i	$0.644405\pi$
$858$	0	0
$859$	−3490.96	−0.138661	−0.0693307	−	0.997594i	$-0.522086\pi$
$859$	−3490.96	−0.138661	−0.0693307	+	0.997594i	$0.522086\pi$
$860$	0	0
$861$	289.973	0.0114776
$862$	0	0
$863$	−43776.0	−1.72671	−0.863357	−	0.504594i	$-0.831642\pi$
$863$	−43776.0	−1.72671	−0.863357	+	0.504594i	$0.831642\pi$
$864$	0	0
$865$	62.6038	0.00246080
$866$	0	0
$867$	1505.72	0.0589815
$868$	0	0
$869$	−88317.6	−3.44761
$870$	0	0
$871$	15632.9	0.608152
$872$	0	0
$873$	12297.4	0.476752
$874$	0	0
$875$	117.868	0.00455391
$876$	0	0
$877$	−6001.84	−0.231092	−0.115546	−	0.993302i	$-0.536862\pi$
$877$	−6001.84	−0.231092	−0.115546	+	0.993302i	$0.536862\pi$
$878$	0	0
$879$	−16970.8	−0.651207
$880$	0	0
$881$	16032.8	0.613121	0.306560	−	0.951851i	$-0.400822\pi$
$881$	16032.8	0.613121	0.306560	+	0.951851i	$0.400822\pi$
$882$	0	0
$883$	−13915.6	−0.530347	−0.265173	−	0.964201i	$-0.585429\pi$
$883$	−13915.6	−0.530347	−0.265173	+	0.964201i	$0.585429\pi$
$884$	0	0
$885$	5183.72	0.196891
$886$	0	0
$887$	−6908.27	−0.261507	−0.130754	−	0.991415i	$-0.541740\pi$
$887$	−6908.27	−0.261507	−0.130754	+	0.991415i	$0.541740\pi$
$888$	0	0
$889$	140.329	0.00529412
$890$	0	0
$891$	5219.05	0.196234
$892$	0	0
$893$	33930.1	1.27147
$894$	0	0
$895$	3644.18	0.136102
$896$	0	0
$897$	−5371.98	−0.199961
$898$	0	0
$899$	−1736.43	−0.0644196
$900$	0	0
$901$	16776.2	0.620306
$902$	0	0
$903$	603.257	0.0222316
$904$	0	0
$905$	−21292.4	−0.782080
$906$	0	0
$907$	10153.3	0.371704	0.185852	−	0.982578i	$-0.440496\pi$
$907$	10153.3	0.371704	0.185852	+	0.982578i	$0.440496\pi$
$908$	0	0
$909$	−7270.61	−0.265293
$910$	0	0
$911$	21003.6	0.763865	0.381932	−	0.924190i	$-0.375259\pi$
$911$	21003.6	0.763865	0.381932	+	0.924190i	$0.375259\pi$
$912$	0	0
$913$	53149.2	1.92660
$914$	0	0
$915$	−2293.73	−0.0828726
$916$	0	0
$917$	−14.3664	−0.000517363 0
$918$	0	0
$919$	8960.82	0.321643	0.160822	−	0.986983i	$-0.448586\pi$
$919$	8960.82	0.321643	0.160822	+	0.986983i	$0.448586\pi$
$920$	0	0
$921$	−10154.6	−0.363306
$922$	0	0
$923$	−61334.5	−2.18727
$924$	0	0
$925$	−6109.65	−0.217172
$926$	0	0
$927$	13732.1	0.486540
$928$	0	0
$929$	−2361.14	−0.0833869	−0.0416934	−	0.999130i	$-0.513275\pi$
$929$	−2361.14	−0.0833869	−0.0416934	+	0.999130i	$0.513275\pi$
$930$	0	0
$931$	−24939.0	−0.877920
$932$	0	0
$933$	29675.1	1.04128
$934$	0	0
$935$	23706.8	0.829191
$936$	0	0
$937$	−3888.65	−0.135578	−0.0677890	−	0.997700i	$-0.521594\pi$
$937$	−3888.65	−0.135578	−0.0677890	+	0.997700i	$0.521594\pi$
$938$	0	0
$939$	−23418.6	−0.813883
$940$	0	0
$941$	15170.1	0.525539	0.262769	−	0.964859i	$-0.415364\pi$
$941$	15170.1	0.525539	0.262769	+	0.964859i	$0.415364\pi$
$942$	0	0
$943$	2357.63	0.0814158
$944$	0	0
$945$	127.298	0.00438201
$946$	0	0
$947$	−11026.2	−0.378357	−0.189179	−	0.981943i	$-0.560583\pi$
$947$	−11026.2	−0.378357	−0.189179	+	0.981943i	$0.560583\pi$
$948$	0	0
$949$	−13055.6	−0.446579
$950$	0	0
$951$	23598.6	0.804666
$952$	0	0
$953$	4468.68	0.151894	0.0759469	−	0.997112i	$-0.475802\pi$
$953$	4468.68	0.151894	0.0759469	+	0.997112i	$0.475802\pi$
$954$	0	0
$955$	−10476.5	−0.354985
$956$	0	0
$957$	3517.47	0.118813
$958$	0	0
$959$	1066.88	0.0359243
$960$	0	0
$961$	−20685.4	−0.694349
$962$	0	0
$963$	16840.2	0.563518
$964$	0	0
$965$	−13423.4	−0.447788
$966$	0	0
$967$	23859.5	0.793454	0.396727	−	0.917937i	$-0.370146\pi$
$967$	23859.5	0.793454	0.396727	+	0.917937i	$0.370146\pi$
$968$	0	0
$969$	16092.7	0.533511
$970$	0	0
$971$	51435.8	1.69995	0.849977	−	0.526820i	$-0.176616\pi$
$971$	51435.8	1.69995	0.849977	+	0.526820i	$0.176616\pi$
$972$	0	0
$973$	960.089	0.0316331
$974$	0	0
$975$	−5839.11	−0.191796
$976$	0	0
$977$	−24105.1	−0.789345	−0.394673	−	0.918822i	$-0.629142\pi$
$977$	−24105.1	−0.789345	−0.394673	+	0.918822i	$0.629142\pi$
$978$	0	0
$979$	−75423.4	−2.46225
$980$	0	0
$981$	−17176.2	−0.559015
$982$	0	0
$983$	39751.1	1.28979	0.644895	−	0.764271i	$-0.276901\pi$
$983$	39751.1	1.28979	0.644895	+	0.764271i	$0.276901\pi$
$984$	0	0
$985$	−3513.50	−0.113654
$986$	0	0
$987$	1316.68	0.0424624
$988$	0	0
$989$	4904.81	0.157699
$990$	0	0
$991$	22409.1	0.718312	0.359156	−	0.933278i	$-0.383065\pi$
$991$	22409.1	0.718312	0.359156	+	0.933278i	$0.383065\pi$
$992$	0	0
$993$	−29685.8	−0.948692
$994$	0	0
$995$	−22898.8	−0.729589
$996$	0	0
$997$	4790.72	0.152180	0.0760900	−	0.997101i	$-0.475756\pi$
$997$	4790.72	0.152180	0.0760900	+	0.997101i	$0.475756\pi$
$998$	0	0
$999$	−6598.42	−0.208974

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.j.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.j.1.3	✓	7	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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} +0.942946 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.00000 q^{5} +0.942946 q^{7} +9.00000 q^{9} +64.4328 q^{11} -77.8548 q^{13} +15.0000 q^{15} +73.5860 q^{17} +72.8975 q^{19} +2.82884 q^{21} +23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +18.1971 q^{29} -95.4235 q^{31} +193.298 q^{33} +4.71473 q^{35} -244.386 q^{37} -233.564 q^{39} +102.506 q^{41} +213.253 q^{43} +45.0000 q^{45} +465.449 q^{47} -342.111 q^{49} +220.758 q^{51} +227.980 q^{53} +322.164 q^{55} +218.692 q^{57} +345.582 q^{59} -152.916 q^{61} +8.48652 q^{63} -389.274 q^{65} -200.796 q^{67} +69.0000 q^{69} +787.807 q^{71} +167.692 q^{73} +75.0000 q^{75} +60.7566 q^{77} -1370.69 q^{79} +81.0000 q^{81} +824.879 q^{83} +367.930 q^{85} +54.5913 q^{87} -1170.58 q^{89} -73.4129 q^{91} -286.270 q^{93} +364.487 q^{95} +1366.38 q^{97} +579.895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9}+O(q^{10}) \) 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9	\( 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9} + 48 q^{11} + 90 q^{13} + 105 q^{15} + 163 q^{17} + 200 q^{19} + 105 q^{21} + 161 q^{23} + 175 q^{25} + 189 q^{27} + 81 q^{29} + 125 q^{31} + 144 q^{33} + 175 q^{35} + 5 q^{37} + 270 q^{39} + 369 q^{41} + 462 q^{43} + 315 q^{45} + 134 q^{47} + 614 q^{49} + 489 q^{51} + 561 q^{53} + 240 q^{55} + 600 q^{57} + 951 q^{59} + 860 q^{61} + 315 q^{63} + 450 q^{65} + 447 q^{67} + 483 q^{69} + 735 q^{71} + 1460 q^{73} + 525 q^{75} + 496 q^{77} + 18 q^{79} + 567 q^{81} + 261 q^{83} + 815 q^{85} + 243 q^{87} + 2024 q^{89} + 692 q^{91} + 375 q^{93} + 1000 q^{95} + 668 q^{97} + 432 q^{99}+O(q^{100}) \) 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9 + 48 * q^11 + 90 * q^13 + 105 * q^15 + 163 * q^17 + 200 * q^19 + 105 * q^21 + 161 * q^23 + 175 * q^25 + 189 * q^27 + 81 * q^29 + 125 * q^31 + 144 * q^33 + 175 * q^35 + 5 * q^37 + 270 * q^39 + 369 * q^41 + 462 * q^43 + 315 * q^45 + 134 * q^47 + 614 * q^49 + 489 * q^51 + 561 * q^53 + 240 * q^55 + 600 * q^57 + 951 * q^59 + 860 * q^61 + 315 * q^63 + 450 * q^65 + 447 * q^67 + 483 * q^69 + 735 * q^71 + 1460 * q^73 + 525 * q^75 + 496 * q^77 + 18 * q^79 + 567 * q^81 + 261 * q^83 + 815 * q^85 + 243 * q^87 + 2024 * q^89 + 692 * q^91 + 375 * q^93 + 1000 * q^95 + 668 * q^97 + 432 * q^99

Introduction

L-functions

Modular forms

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