LMFDB - Embedded newform 2400.4.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.32340.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 42x - 14 $ x^3 - 42*x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.64138$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -4.63836 q^{7} +9.00000 q^{9} -58.7815 q^{11} -33.5655 q^{13} -44.2039 q^{17} +133.466 q^{19} +13.9151 q^{21} +90.0824 q^{23} -27.0000 q^{27} +154.466 q^{29} -21.0703 q^{31} +176.345 q^{33} +38.0923 q^{37} +100.697 q^{39} +335.184 q^{41} +388.354 q^{43} +267.316 q^{47} -321.486 q^{49} +132.612 q^{51} -445.403 q^{53} -400.398 q^{57} -36.6018 q^{59} -813.303 q^{61} -41.7453 q^{63} -41.5968 q^{67} -270.247 q^{69} +999.660 q^{71} -56.3644 q^{73} +272.650 q^{77} +80.5101 q^{79} +81.0000 q^{81} -577.640 q^{83} -463.398 q^{87} +679.631 q^{89} +155.689 q^{91} +63.2110 q^{93} +193.914 q^{97} -529.034 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 7 q^{7} + 27 q^{9} - 21 q^{13} - 32 q^{17} - 19 q^{19} - 21 q^{21} + 60 q^{23} - 81 q^{27} + 44 q^{29} + 151 q^{31} - 330 q^{37} + 63 q^{39} + 82 q^{41} + 367 q^{43} - 246 q^{47} - 42 q^{49}+ \cdots + 351 q^{97}+O(q^{100}) $ 3 * q - 9 * q^3 + 7 * q^7 + 27 * q^9 - 21 * q^13 - 32 * q^17 - 19 * q^19 - 21 * q^21 + 60 * q^23 - 81 * q^27 + 44 * q^29 + 151 * q^31 - 330 * q^37 + 63 * q^39 + 82 * q^41 + 367 * q^43 - 246 * q^47 - 42 * q^49 + 96 * q^51 - 554 * q^53 + 57 * q^57 - 42 * q^59 - 27 * q^61 + 63 * q^63 - 837 * q^67 - 180 * q^69 + 1422 * q^71 - 498 * q^73 - 560 * q^77 + 688 * q^79 + 243 * q^81 + 702 * q^83 - 132 * q^87 - 1488 * q^89 + 735 * q^91 - 453 * q^93 + 351 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.63836	−0.250448	−0.125224	−	0.992128i	$-0.539965\pi$
$7$	−4.63836	−0.250448	−0.125224	+	0.992128i	$0.539965\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−58.7815	−1.61121	−0.805604	−	0.592454i	$-0.798159\pi$
$11$	−58.7815	−1.61121	−0.805604	+	0.592454i	$0.798159\pi$
$12$	0	0
$13$	−33.5655	−0.716108	−0.358054	−	0.933701i	$-0.616560\pi$
$13$	−33.5655	−0.716108	−0.358054	+	0.933701i	$0.616560\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−44.2039	−0.630648	−0.315324	−	0.948984i	$-0.602113\pi$
$17$	−44.2039	−0.630648	−0.315324	+	0.948984i	$0.602113\pi$
$18$	0	0
$19$	133.466	1.61154	0.805769	−	0.592230i	$-0.201752\pi$
$19$	133.466	1.61154	0.805769	+	0.592230i	$0.201752\pi$
$20$	0	0
$21$	13.9151	0.144596
$22$	0	0
$23$	90.0824	0.816673	0.408337	−	0.912831i	$-0.366109\pi$
$23$	90.0824	0.816673	0.408337	+	0.912831i	$0.366109\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	154.466	0.989091	0.494545	−	0.869152i	$-0.335335\pi$
$29$	154.466	0.989091	0.494545	+	0.869152i	$0.335335\pi$
$30$	0	0
$31$	−21.0703	−0.122076	−0.0610378	−	0.998135i	$-0.519441\pi$
$31$	−21.0703	−0.122076	−0.0610378	+	0.998135i	$0.519441\pi$
$32$	0	0
$33$	176.345	0.930232
$34$	0	0
$35$	0	0
$36$	0	0
$37$	38.0923	0.169252	0.0846262	−	0.996413i	$-0.473030\pi$
$37$	38.0923	0.169252	0.0846262	+	0.996413i	$0.473030\pi$
$38$	0	0
$39$	100.697	0.413445
$40$	0	0
$41$	335.184	1.27676	0.638378	−	0.769723i	$-0.279606\pi$
$41$	335.184	1.27676	0.638378	+	0.769723i	$0.279606\pi$
$42$	0	0
$43$	388.354	1.37729	0.688645	−	0.725099i	$-0.258206\pi$
$43$	388.354	1.37729	0.688645	+	0.725099i	$0.258206\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	267.316	0.829617	0.414809	−	0.909909i	$-0.363849\pi$
$47$	267.316	0.829617	0.414809	+	0.909909i	$0.363849\pi$
$48$	0	0
$49$	−321.486	−0.937276
$50$	0	0
$51$	132.612	0.364105
$52$	0	0
$53$	−445.403	−1.15435	−0.577177	−	0.816619i	$-0.695846\pi$
$53$	−445.403	−1.15435	−0.577177	+	0.816619i	$0.695846\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−400.398	−0.930422
$58$	0	0
$59$	−36.6018	−0.0807652	−0.0403826	−	0.999184i	$-0.512858\pi$
$59$	−36.6018	−0.0807652	−0.0403826	+	0.999184i	$0.512858\pi$
$60$	0	0
$61$	−813.303	−1.70710	−0.853548	−	0.521015i	$-0.825554\pi$
$61$	−813.303	−1.70710	−0.853548	+	0.521015i	$0.825554\pi$
$62$	0	0
$63$	−41.7453	−0.0834827
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−41.5968	−0.0758487	−0.0379244	−	0.999281i	$-0.512075\pi$
$67$	−41.5968	−0.0758487	−0.0379244	+	0.999281i	$0.512075\pi$
$68$	0	0
$69$	−270.247	−0.471507
$70$	0	0
$71$	999.660	1.67096	0.835478	−	0.549524i	$-0.185191\pi$
$71$	999.660	1.67096	0.835478	+	0.549524i	$0.185191\pi$
$72$	0	0
$73$	−56.3644	−0.0903693	−0.0451846	−	0.998979i	$-0.514388\pi$
$73$	−56.3644	−0.0903693	−0.0451846	+	0.998979i	$0.514388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	272.650	0.403524
$78$	0	0
$79$	80.5101	0.114659	0.0573297	−	0.998355i	$-0.481741\pi$
$79$	80.5101	0.114659	0.0573297	+	0.998355i	$0.481741\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−577.640	−0.763907	−0.381954	−	0.924182i	$-0.624749\pi$
$83$	−577.640	−0.763907	−0.381954	+	0.924182i	$0.624749\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−463.398	−0.571052
$88$	0	0
$89$	679.631	0.809446	0.404723	−	0.914439i	$-0.367368\pi$
$89$	679.631	0.809446	0.404723	+	0.914439i	$0.367368\pi$
$90$	0	0
$91$	155.689	0.179348
$92$	0	0
$93$	63.2110	0.0704804
$94$	0	0
$95$	0	0
$96$	0	0
$97$	193.914	0.202979	0.101489	−	0.994837i	$-0.467639\pi$
$97$	193.914	0.202979	0.101489	+	0.994837i	$0.467639\pi$
$98$	0	0
$99$	−529.034	−0.537070
$100$	0	0
$101$	757.519	0.746297	0.373148	−	0.927772i	$-0.378278\pi$
$101$	757.519	0.746297	0.373148	+	0.927772i	$0.378278\pi$
$102$	0	0
$103$	−499.587	−0.477920	−0.238960	−	0.971029i	$-0.576806\pi$
$103$	−499.587	−0.477920	−0.238960	+	0.971029i	$0.576806\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1466.02	−1.32454	−0.662271	−	0.749265i	$-0.730407\pi$
$107$	−1466.02	−1.32454	−0.662271	+	0.749265i	$0.730407\pi$
$108$	0	0
$109$	−1546.95	−1.35937	−0.679685	−	0.733504i	$-0.737883\pi$
$109$	−1546.95	−1.35937	−0.679685	+	0.733504i	$0.737883\pi$
$110$	0	0
$111$	−114.277	−0.0977180
$112$	0	0
$113$	−1583.68	−1.31841	−0.659204	−	0.751964i	$-0.729107\pi$
$113$	−1583.68	−1.31841	−0.659204	+	0.751964i	$0.729107\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−302.090	−0.238703
$118$	0	0
$119$	205.034	0.157945
$120$	0	0
$121$	2124.27	1.59599
$122$	0	0
$123$	−1005.55	−0.737136
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1522.46	−1.06375	−0.531875	−	0.846823i	$-0.678512\pi$
$127$	−1522.46	−1.06375	−0.531875	+	0.846823i	$0.678512\pi$
$128$	0	0
$129$	−1165.06	−0.795179
$130$	0	0
$131$	637.626	0.425264	0.212632	−	0.977132i	$-0.431796\pi$
$131$	637.626	0.425264	0.212632	+	0.977132i	$0.431796\pi$
$132$	0	0
$133$	−619.064	−0.403607
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1342.34	0.837107	0.418554	−	0.908192i	$-0.362537\pi$
$137$	1342.34	0.837107	0.418554	+	0.908192i	$0.362537\pi$
$138$	0	0
$139$	203.460	0.124153	0.0620765	−	0.998071i	$-0.480228\pi$
$139$	203.460	0.124153	0.0620765	+	0.998071i	$0.480228\pi$
$140$	0	0
$141$	−801.947	−0.478980
$142$	0	0
$143$	1973.03	1.15380
$144$	0	0
$145$	0	0
$146$	0	0
$147$	964.457	0.541136
$148$	0	0
$149$	1804.64	0.992225	0.496112	−	0.868258i	$-0.334760\pi$
$149$	1804.64	0.992225	0.496112	+	0.868258i	$0.334760\pi$
$150$	0	0
$151$	1527.91	0.823440	0.411720	−	0.911310i	$-0.364928\pi$
$151$	1527.91	0.823440	0.411720	+	0.911310i	$0.364928\pi$
$152$	0	0
$153$	−397.835	−0.210216
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2867.85	−1.45783	−0.728914	−	0.684605i	$-0.759975\pi$
$157$	−2867.85	−1.45783	−0.728914	+	0.684605i	$0.759975\pi$
$158$	0	0
$159$	1336.21	0.666466
$160$	0	0
$161$	−417.835	−0.204534
$162$	0	0
$163$	3499.31	1.68152	0.840759	−	0.541409i	$-0.182109\pi$
$163$	3499.31	1.68152	0.840759	+	0.541409i	$0.182109\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	288.699	0.133774	0.0668868	−	0.997761i	$-0.478693\pi$
$167$	288.699	0.133774	0.0668868	+	0.997761i	$0.478693\pi$
$168$	0	0
$169$	−1070.35	−0.487189
$170$	0	0
$171$	1201.19	0.537179
$172$	0	0
$173$	−989.001	−0.434638	−0.217319	−	0.976101i	$-0.569731\pi$
$173$	−989.001	−0.434638	−0.217319	+	0.976101i	$0.569731\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	109.805	0.0466298
$178$	0	0
$179$	172.882	0.0721888	0.0360944	−	0.999348i	$-0.488508\pi$
$179$	172.882	0.0721888	0.0360944	+	0.999348i	$0.488508\pi$
$180$	0	0
$181$	4329.68	1.77803	0.889013	−	0.457882i	$-0.151392\pi$
$181$	4329.68	1.77803	0.889013	+	0.457882i	$0.151392\pi$
$182$	0	0
$183$	2439.91	0.985592
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2598.37	1.01611
$188$	0	0
$189$	125.236	0.0481988
$190$	0	0
$191$	−1399.28	−0.530095	−0.265047	−	0.964235i	$-0.585388\pi$
$191$	−1399.28	−0.530095	−0.265047	+	0.964235i	$0.585388\pi$
$192$	0	0
$193$	−1222.42	−0.455916	−0.227958	−	0.973671i	$-0.573205\pi$
$193$	−1222.42	−0.455916	−0.227958	+	0.973671i	$0.573205\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3867.75	−1.39881	−0.699406	−	0.714725i	$-0.746552\pi$
$197$	−3867.75	−1.39881	−0.699406	+	0.714725i	$0.746552\pi$
$198$	0	0
$199$	−2184.64	−0.778218	−0.389109	−	0.921192i	$-0.627217\pi$
$199$	−2184.64	−0.778218	−0.389109	+	0.921192i	$0.627217\pi$
$200$	0	0
$201$	124.791	0.0437913
$202$	0	0
$203$	−716.470	−0.247716
$204$	0	0
$205$	0	0
$206$	0	0
$207$	810.742	0.272224
$208$	0	0
$209$	−7845.34	−2.59652
$210$	0	0
$211$	−3404.41	−1.11075	−0.555377	−	0.831599i	$-0.687426\pi$
$211$	−3404.41	−1.11075	−0.555377	+	0.831599i	$0.687426\pi$
$212$	0	0
$213$	−2998.98	−0.964727
$214$	0	0
$215$	0	0
$216$	0	0
$217$	97.7319	0.0305736
$218$	0	0
$219$	169.093	0.0521747
$220$	0	0
$221$	1483.73	0.451612
$222$	0	0
$223$	−945.784	−0.284011	−0.142005	−	0.989866i	$-0.545355\pi$
$223$	−945.784	−0.284011	−0.142005	+	0.989866i	$0.545355\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1913.77	0.559566	0.279783	−	0.960063i	$-0.409738\pi$
$227$	1913.77	0.559566	0.279783	+	0.960063i	$0.409738\pi$
$228$	0	0
$229$	1269.96	0.366467	0.183234	−	0.983069i	$-0.441343\pi$
$229$	1269.96	0.366467	0.183234	+	0.983069i	$0.441343\pi$
$230$	0	0
$231$	−817.950	−0.232975
$232$	0	0
$233$	−4255.71	−1.19657	−0.598285	−	0.801284i	$-0.704151\pi$
$233$	−4255.71	−1.19657	−0.598285	+	0.801284i	$0.704151\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−241.530	−0.0661986
$238$	0	0
$239$	4803.26	1.29999	0.649995	−	0.759939i	$-0.274771\pi$
$239$	4803.26	1.29999	0.649995	+	0.759939i	$0.274771\pi$
$240$	0	0
$241$	−675.567	−0.180569	−0.0902844	−	0.995916i	$-0.528778\pi$
$241$	−675.567	−0.180569	−0.0902844	+	0.995916i	$0.528778\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4479.86	−1.15404
$248$	0	0
$249$	1732.92	0.441042
$250$	0	0
$251$	−3599.90	−0.905275	−0.452637	−	0.891695i	$-0.649517\pi$
$251$	−3599.90	−0.905275	−0.452637	+	0.891695i	$0.649517\pi$
$252$	0	0
$253$	−5295.18	−1.31583
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2090.93	−0.507504	−0.253752	−	0.967269i	$-0.581665\pi$
$257$	−2090.93	−0.507504	−0.253752	+	0.967269i	$0.581665\pi$
$258$	0	0
$259$	−176.686	−0.0423890
$260$	0	0
$261$	1390.19	0.329697
$262$	0	0
$263$	−1506.25	−0.353152	−0.176576	−	0.984287i	$-0.556502\pi$
$263$	−1506.25	−0.353152	−0.176576	+	0.984287i	$0.556502\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2038.89	−0.467334
$268$	0	0
$269$	−6100.29	−1.38268	−0.691340	−	0.722529i	$-0.742979\pi$
$269$	−6100.29	−1.38268	−0.691340	+	0.722529i	$0.742979\pi$
$270$	0	0
$271$	5675.56	1.27220	0.636099	−	0.771608i	$-0.280547\pi$
$271$	5675.56	1.27220	0.636099	+	0.771608i	$0.280547\pi$
$272$	0	0
$273$	−467.068	−0.103547
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2581.47	0.559947	0.279974	−	0.960008i	$-0.409674\pi$
$277$	2581.47	0.559947	0.279974	+	0.960008i	$0.409674\pi$
$278$	0	0
$279$	−189.633	−0.0406919
$280$	0	0
$281$	3975.57	0.843995	0.421998	−	0.906597i	$-0.361329\pi$
$281$	3975.57	0.843995	0.421998	+	0.906597i	$0.361329\pi$
$282$	0	0
$283$	−6441.12	−1.35295	−0.676475	−	0.736466i	$-0.736493\pi$
$283$	−6441.12	−1.35295	−0.676475	+	0.736466i	$0.736493\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1554.71	−0.319761
$288$	0	0
$289$	−2959.01	−0.602283
$290$	0	0
$291$	−581.741	−0.117190
$292$	0	0
$293$	−6129.25	−1.22210	−0.611049	−	0.791593i	$-0.709252\pi$
$293$	−6129.25	−1.22210	−0.611049	+	0.791593i	$0.709252\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1587.10	0.310077
$298$	0	0
$299$	−3023.67	−0.584826
$300$	0	0
$301$	−1801.33	−0.344940
$302$	0	0
$303$	−2272.56	−0.430875
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2666.78	−0.495770	−0.247885	−	0.968789i	$-0.579736\pi$
$307$	−2666.78	−0.495770	−0.247885	+	0.968789i	$0.579736\pi$
$308$	0	0
$309$	1498.76	0.275927
$310$	0	0
$311$	5196.76	0.947529	0.473764	−	0.880652i	$-0.342895\pi$
$311$	5196.76	0.947529	0.473764	+	0.880652i	$0.342895\pi$
$312$	0	0
$313$	−10688.1	−1.93011	−0.965056	−	0.262043i	$-0.915604\pi$
$313$	−10688.1	−1.93011	−0.965056	+	0.262043i	$0.915604\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8826.29	−1.56383	−0.781915	−	0.623385i	$-0.785757\pi$
$317$	−8826.29	−1.56383	−0.781915	+	0.623385i	$0.785757\pi$
$318$	0	0
$319$	−9079.75	−1.59363
$320$	0	0
$321$	4398.07	0.764725
$322$	0	0
$323$	−5899.72	−1.01631
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4640.86	0.784832
$328$	0	0
$329$	−1239.91	−0.207776
$330$	0	0
$331$	−7108.78	−1.18046	−0.590232	−	0.807234i	$-0.700964\pi$
$331$	−7108.78	−1.18046	−0.590232	+	0.807234i	$0.700964\pi$
$332$	0	0
$333$	342.831	0.0564175
$334$	0	0
$335$	0	0
$336$	0	0
$337$	85.5709	0.0138319	0.00691594	−	0.999976i	$-0.497799\pi$
$337$	85.5709	0.0138319	0.00691594	+	0.999976i	$0.497799\pi$
$338$	0	0
$339$	4751.04	0.761183
$340$	0	0
$341$	1238.55	0.196689
$342$	0	0
$343$	3082.13	0.485187
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9256.79	1.43208	0.716038	−	0.698061i	$-0.245954\pi$
$347$	9256.79	1.43208	0.716038	+	0.698061i	$0.245954\pi$
$348$	0	0
$349$	−5479.63	−0.840452	−0.420226	−	0.907419i	$-0.638049\pi$
$349$	−5479.63	−0.840452	−0.420226	+	0.907419i	$0.638049\pi$
$350$	0	0
$351$	906.270	0.137815
$352$	0	0
$353$	3747.74	0.565076	0.282538	−	0.959256i	$-0.408824\pi$
$353$	3747.74	0.565076	0.282538	+	0.959256i	$0.408824\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−615.101	−0.0911894
$358$	0	0
$359$	291.519	0.0428574	0.0214287	−	0.999770i	$-0.493179\pi$
$359$	291.519	0.0428574	0.0214287	+	0.999770i	$0.493179\pi$
$360$	0	0
$361$	10954.2	1.59705
$362$	0	0
$363$	−6372.80	−0.921448
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3190.36	0.453775	0.226888	−	0.973921i	$-0.427145\pi$
$367$	3190.36	0.453775	0.226888	+	0.973921i	$0.427145\pi$
$368$	0	0
$369$	3016.66	0.425585
$370$	0	0
$371$	2065.94	0.289106
$372$	0	0
$373$	−14191.4	−1.96998	−0.984991	−	0.172605i	$-0.944782\pi$
$373$	−14191.4	−1.96998	−0.984991	+	0.172605i	$0.944782\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5184.74	−0.708296
$378$	0	0
$379$	−7215.48	−0.977927	−0.488963	−	0.872304i	$-0.662625\pi$
$379$	−7215.48	−0.977927	−0.488963	+	0.872304i	$0.662625\pi$
$380$	0	0
$381$	4567.37	0.614156
$382$	0	0
$383$	2688.97	0.358747	0.179373	−	0.983781i	$-0.442593\pi$
$383$	2688.97	0.358747	0.179373	+	0.983781i	$0.442593\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3495.19	0.459097
$388$	0	0
$389$	−13512.8	−1.76125	−0.880623	−	0.473818i	$-0.842876\pi$
$389$	−13512.8	−1.76125	−0.880623	+	0.473818i	$0.842876\pi$
$390$	0	0
$391$	−3981.99	−0.515034
$392$	0	0
$393$	−1912.88	−0.245526
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5487.92	−0.693781	−0.346890	−	0.937906i	$-0.612762\pi$
$397$	−5487.92	−0.693781	−0.346890	+	0.937906i	$0.612762\pi$
$398$	0	0
$399$	1857.19	0.233022
$400$	0	0
$401$	13000.5	1.61899	0.809496	−	0.587125i	$-0.199740\pi$
$401$	13000.5	1.61899	0.809496	+	0.587125i	$0.199740\pi$
$402$	0	0
$403$	707.237	0.0874194
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2239.13	−0.272701
$408$	0	0
$409$	−5388.88	−0.651499	−0.325750	−	0.945456i	$-0.605617\pi$
$409$	−5388.88	−0.651499	−0.325750	+	0.945456i	$0.605617\pi$
$410$	0	0
$411$	−4027.01	−0.483304
$412$	0	0
$413$	169.772	0.0202275
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−610.380	−0.0716798
$418$	0	0
$419$	−7276.23	−0.848371	−0.424185	−	0.905575i	$-0.639440\pi$
$419$	−7276.23	−0.848371	−0.424185	+	0.905575i	$0.639440\pi$
$420$	0	0
$421$	−432.934	−0.0501186	−0.0250593	−	0.999686i	$-0.507977\pi$
$421$	−432.934	−0.0501186	−0.0250593	+	0.999686i	$0.507977\pi$
$422$	0	0
$423$	2405.84	0.276539
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3772.40	0.427539
$428$	0	0
$429$	−5919.10	−0.666147
$430$	0	0
$431$	−1014.21	−0.113347	−0.0566736	−	0.998393i	$-0.518049\pi$
$431$	−1014.21	−0.113347	−0.0566736	+	0.998393i	$0.518049\pi$
$432$	0	0
$433$	5855.29	0.649855	0.324928	−	0.945739i	$-0.394660\pi$
$433$	5855.29	0.649855	0.324928	+	0.945739i	$0.394660\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12022.9	1.31610
$438$	0	0
$439$	12647.5	1.37502	0.687510	−	0.726174i	$-0.258704\pi$
$439$	12647.5	1.37502	0.687510	+	0.726174i	$0.258704\pi$
$440$	0	0
$441$	−2893.37	−0.312425
$442$	0	0
$443$	2030.46	0.217765	0.108882	−	0.994055i	$-0.465273\pi$
$443$	2030.46	0.217765	0.108882	+	0.994055i	$0.465273\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5413.91	−0.572861
$448$	0	0
$449$	−1213.89	−0.127588	−0.0637940	−	0.997963i	$-0.520320\pi$
$449$	−1213.89	−0.127588	−0.0637940	+	0.997963i	$0.520320\pi$
$450$	0	0
$451$	−19702.6	−2.05712
$452$	0	0
$453$	−4583.73	−0.475414
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4975.74	−0.509312	−0.254656	−	0.967032i	$-0.581962\pi$
$457$	−4975.74	−0.509312	−0.254656	+	0.967032i	$0.581962\pi$
$458$	0	0
$459$	1193.51	0.121368
$460$	0	0
$461$	−3778.94	−0.381785	−0.190893	−	0.981611i	$-0.561138\pi$
$461$	−3778.94	−0.381785	−0.190893	+	0.981611i	$0.561138\pi$
$462$	0	0
$463$	19151.9	1.92239	0.961194	−	0.275875i	$-0.0889675\pi$
$463$	19151.9	1.92239	0.961194	+	0.275875i	$0.0889675\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6047.25	0.599215	0.299607	−	0.954063i	$-0.403144\pi$
$467$	6047.25	0.599215	0.299607	+	0.954063i	$0.403144\pi$
$468$	0	0
$469$	192.941	0.0189962
$470$	0	0
$471$	8603.54	0.841677
$472$	0	0
$473$	−22828.1	−2.21910
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4008.62	−0.384785
$478$	0	0
$479$	16423.8	1.56664	0.783320	−	0.621619i	$-0.213525\pi$
$479$	16423.8	1.56664	0.783320	+	0.621619i	$0.213525\pi$
$480$	0	0
$481$	−1278.59	−0.121203
$482$	0	0
$483$	1253.51	0.118088
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−14392.1	−1.33916	−0.669578	−	0.742742i	$-0.733525\pi$
$487$	−14392.1	−1.33916	−0.669578	+	0.742742i	$0.733525\pi$
$488$	0	0
$489$	−10497.9	−0.970825
$490$	0	0
$491$	−17510.6	−1.60945	−0.804727	−	0.593645i	$-0.797688\pi$
$491$	−17510.6	−1.60945	−0.804727	+	0.593645i	$0.797688\pi$
$492$	0	0
$493$	−6828.00	−0.623768
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4636.79	−0.418488
$498$	0	0
$499$	−16297.1	−1.46204	−0.731019	−	0.682358i	$-0.760955\pi$
$499$	−16297.1	−1.46204	−0.731019	+	0.682358i	$0.760955\pi$
$500$	0	0
$501$	−866.097	−0.0772343
$502$	0	0
$503$	−7602.90	−0.673949	−0.336975	−	0.941514i	$-0.609404\pi$
$503$	−7602.90	−0.673949	−0.336975	+	0.941514i	$0.609404\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3211.06	0.281279
$508$	0	0
$509$	22010.6	1.91670	0.958351	−	0.285593i	$-0.0921905\pi$
$509$	22010.6	1.91670	0.958351	+	0.285593i	$0.0921905\pi$
$510$	0	0
$511$	261.439	0.0226328
$512$	0	0
$513$	−3603.58	−0.310141
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−15713.2	−1.33669
$518$	0	0
$519$	2967.00	0.250938
$520$	0	0
$521$	2249.98	0.189200	0.0946001	−	0.995515i	$-0.469843\pi$
$521$	2249.98	0.189200	0.0946001	+	0.995515i	$0.469843\pi$
$522$	0	0
$523$	5448.89	0.455570	0.227785	−	0.973711i	$-0.426852\pi$
$523$	5448.89	0.455570	0.227785	+	0.973711i	$0.426852\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	931.391	0.0769868
$528$	0	0
$529$	−4052.16	−0.333045
$530$	0	0
$531$	−329.416	−0.0269217
$532$	0	0
$533$	−11250.6	−0.914295
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−518.646	−0.0416782
$538$	0	0
$539$	18897.4	1.51015
$540$	0	0
$541$	23423.6	1.86148	0.930738	−	0.365687i	$-0.119166\pi$
$541$	23423.6	1.86148	0.930738	+	0.365687i	$0.119166\pi$
$542$	0	0
$543$	−12989.0	−1.02654
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−11465.7	−0.896232	−0.448116	−	0.893975i	$-0.647905\pi$
$547$	−11465.7	−0.896232	−0.448116	+	0.893975i	$0.647905\pi$
$548$	0	0
$549$	−7319.73	−0.569032
$550$	0	0
$551$	20616.0	1.59396
$552$	0	0
$553$	−373.435	−0.0287162
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14682.2	1.11688	0.558441	−	0.829544i	$-0.311400\pi$
$557$	14682.2	1.11688	0.558441	+	0.829544i	$0.311400\pi$
$558$	0	0
$559$	−13035.3	−0.986288
$560$	0	0
$561$	−7795.12	−0.586649
$562$	0	0
$563$	−11593.8	−0.867889	−0.433944	−	0.900940i	$-0.642879\pi$
$563$	−11593.8	−0.867889	−0.433944	+	0.900940i	$0.642879\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−375.707	−0.0278276
$568$	0	0
$569$	−4743.55	−0.349491	−0.174745	−	0.984614i	$-0.555910\pi$
$569$	−4743.55	−0.349491	−0.174745	+	0.984614i	$0.555910\pi$
$570$	0	0
$571$	20469.9	1.50024	0.750122	−	0.661300i	$-0.229995\pi$
$571$	20469.9	1.50024	0.750122	+	0.661300i	$0.229995\pi$
$572$	0	0
$573$	4197.83	0.306050
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5814.12	−0.419489	−0.209744	−	0.977756i	$-0.567263\pi$
$577$	−5814.12	−0.419489	−0.209744	+	0.977756i	$0.567263\pi$
$578$	0	0
$579$	3667.26	0.263223
$580$	0	0
$581$	2679.31	0.191319
$582$	0	0
$583$	26181.4	1.85990
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5876.70	0.413215	0.206608	−	0.978424i	$-0.433758\pi$
$587$	5876.70	0.413215	0.206608	+	0.978424i	$0.433758\pi$
$588$	0	0
$589$	−2812.18	−0.196729
$590$	0	0
$591$	11603.3	0.807604
$592$	0	0
$593$	−11344.3	−0.785590	−0.392795	−	0.919626i	$-0.628492\pi$
$593$	−11344.3	−0.785590	−0.392795	+	0.919626i	$0.628492\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6553.93	0.449304
$598$	0	0
$599$	−14991.8	−1.02262	−0.511308	−	0.859398i	$-0.670839\pi$
$599$	−14991.8	−1.02262	−0.511308	+	0.859398i	$0.670839\pi$
$600$	0	0
$601$	−20684.8	−1.40391	−0.701955	−	0.712222i	$-0.747689\pi$
$601$	−20684.8	−1.40391	−0.701955	+	0.712222i	$0.747689\pi$
$602$	0	0
$603$	−374.372	−0.0252829
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7476.54	−0.499940	−0.249970	−	0.968254i	$-0.580421\pi$
$607$	−7476.54	−0.499940	−0.249970	+	0.968254i	$0.580421\pi$
$608$	0	0
$609$	2149.41	0.143019
$610$	0	0
$611$	−8972.60	−0.594096
$612$	0	0
$613$	19585.0	1.29043	0.645213	−	0.764003i	$-0.276769\pi$
$613$	19585.0	1.29043	0.645213	+	0.764003i	$0.276769\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4788.74	0.312460	0.156230	−	0.987721i	$-0.450066\pi$
$617$	4788.74	0.312460	0.156230	+	0.987721i	$0.450066\pi$
$618$	0	0
$619$	−29582.1	−1.92085	−0.960423	−	0.278544i	$-0.910148\pi$
$619$	−29582.1	−1.92085	−0.960423	+	0.278544i	$0.910148\pi$
$620$	0	0
$621$	−2432.23	−0.157169
$622$	0	0
$623$	−3152.37	−0.202724
$624$	0	0
$625$	0	0
$626$	0	0
$627$	23536.0	1.49910
$628$	0	0
$629$	−1683.83	−0.106739
$630$	0	0
$631$	−2289.96	−0.144472	−0.0722361	−	0.997388i	$-0.523013\pi$
$631$	−2289.96	−0.144472	−0.0722361	+	0.997388i	$0.523013\pi$
$632$	0	0
$633$	10213.2	0.641294
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10790.8	0.671191
$638$	0	0
$639$	8996.94	0.556985
$640$	0	0
$641$	−4134.97	−0.254792	−0.127396	−	0.991852i	$-0.540662\pi$
$641$	−4134.97	−0.254792	−0.127396	+	0.991852i	$0.540662\pi$
$642$	0	0
$643$	−21728.1	−1.33262	−0.666308	−	0.745677i	$-0.732126\pi$
$643$	−21728.1	−1.33262	−0.666308	+	0.745677i	$0.732126\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16550.0	−1.00564	−0.502820	−	0.864391i	$-0.667704\pi$
$647$	−16550.0	−1.00564	−0.502820	+	0.864391i	$0.667704\pi$
$648$	0	0
$649$	2151.51	0.130130
$650$	0	0
$651$	−293.196	−0.0176517
$652$	0	0
$653$	1997.70	0.119718	0.0598592	−	0.998207i	$-0.480935\pi$
$653$	1997.70	0.119718	0.0598592	+	0.998207i	$0.480935\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−507.280	−0.0301231
$658$	0	0
$659$	−6464.65	−0.382135	−0.191068	−	0.981577i	$-0.561195\pi$
$659$	−6464.65	−0.382135	−0.191068	+	0.981577i	$0.561195\pi$
$660$	0	0
$661$	13815.9	0.812972	0.406486	−	0.913657i	$-0.366754\pi$
$661$	13815.9	0.812972	0.406486	+	0.913657i	$0.366754\pi$
$662$	0	0
$663$	−4451.18	−0.260739
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13914.7	0.807764
$668$	0	0
$669$	2837.35	0.163974
$670$	0	0
$671$	47807.2	2.75049
$672$	0	0
$673$	−24162.3	−1.38393	−0.691966	−	0.721930i	$-0.743255\pi$
$673$	−24162.3	−1.38393	−0.691966	+	0.721930i	$0.743255\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17605.0	−0.999432	−0.499716	−	0.866189i	$-0.666562\pi$
$677$	−17605.0	−0.999432	−0.499716	+	0.866189i	$0.666562\pi$
$678$	0	0
$679$	−899.442	−0.0508356
$680$	0	0
$681$	−5741.31	−0.323065
$682$	0	0
$683$	−16879.1	−0.945626	−0.472813	−	0.881163i	$-0.656761\pi$
$683$	−16879.1	−0.945626	−0.472813	+	0.881163i	$0.656761\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3809.87	−0.211580
$688$	0	0
$689$	14950.2	0.826642
$690$	0	0
$691$	−15617.5	−0.859795	−0.429897	−	0.902878i	$-0.641450\pi$
$691$	−15617.5	−0.859795	−0.429897	+	0.902878i	$0.641450\pi$
$692$	0	0
$693$	2453.85	0.134508
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−14816.5	−0.805184
$698$	0	0
$699$	12767.1	0.690840
$700$	0	0
$701$	15499.8	0.835123	0.417561	−	0.908649i	$-0.362885\pi$
$701$	15499.8	0.835123	0.417561	+	0.908649i	$0.362885\pi$
$702$	0	0
$703$	5084.03	0.272757
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3513.65	−0.186909
$708$	0	0
$709$	−28548.1	−1.51219	−0.756097	−	0.654460i	$-0.772896\pi$
$709$	−28548.1	−1.51219	−0.756097	+	0.654460i	$0.772896\pi$
$710$	0	0
$711$	724.591	0.0382198
$712$	0	0
$713$	−1898.07	−0.0996959
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14409.8	−0.750549
$718$	0	0
$719$	27794.4	1.44167	0.720833	−	0.693109i	$-0.243760\pi$
$719$	27794.4	1.44167	0.720833	+	0.693109i	$0.243760\pi$
$720$	0	0
$721$	2317.26	0.119694
$722$	0	0
$723$	2026.70	0.104251
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−35634.9	−1.81792	−0.908958	−	0.416888i	$-0.863121\pi$
$727$	−35634.9	−1.81792	−0.908958	+	0.416888i	$0.863121\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−17166.8	−0.868585
$732$	0	0
$733$	22550.8	1.13633	0.568166	−	0.822914i	$-0.307653\pi$
$733$	22550.8	1.13633	0.568166	+	0.822914i	$0.307653\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2445.13	0.122208
$738$	0	0
$739$	−9321.14	−0.463983	−0.231992	−	0.972718i	$-0.574524\pi$
$739$	−9321.14	−0.463983	−0.231992	+	0.972718i	$0.574524\pi$
$740$	0	0
$741$	13439.6	0.666283
$742$	0	0
$743$	34593.1	1.70807	0.854037	−	0.520212i	$-0.174147\pi$
$743$	34593.1	1.70807	0.854037	+	0.520212i	$0.174147\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5198.76	−0.254636
$748$	0	0
$749$	6799.96	0.331729
$750$	0	0
$751$	19968.6	0.970260	0.485130	−	0.874442i	$-0.338772\pi$
$751$	19968.6	0.970260	0.485130	+	0.874442i	$0.338772\pi$
$752$	0	0
$753$	10799.7	0.522661
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6549.67	0.314468	0.157234	−	0.987561i	$-0.449742\pi$
$757$	6549.67	0.314468	0.157234	+	0.987561i	$0.449742\pi$
$758$	0	0
$759$	15885.5	0.759695
$760$	0	0
$761$	13341.1	0.635497	0.317749	−	0.948175i	$-0.397073\pi$
$761$	13341.1	0.635497	0.317749	+	0.948175i	$0.397073\pi$
$762$	0	0
$763$	7175.33	0.340452
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1228.56	0.0578366
$768$	0	0
$769$	30907.6	1.44936	0.724679	−	0.689087i	$-0.241988\pi$
$769$	30907.6	1.44936	0.724679	+	0.689087i	$0.241988\pi$
$770$	0	0
$771$	6272.79	0.293008
$772$	0	0
$773$	40909.5	1.90351	0.951754	−	0.306861i	$-0.0992786\pi$
$773$	40909.5	1.90351	0.951754	+	0.306861i	$0.0992786\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	530.058	0.0244733
$778$	0	0
$779$	44735.7	2.05754
$780$	0	0
$781$	−58761.6	−2.69226
$782$	0	0
$783$	−4170.58	−0.190351
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−33628.8	−1.52317	−0.761586	−	0.648064i	$-0.775579\pi$
$787$	−33628.8	−1.52317	−0.761586	+	0.648064i	$0.775579\pi$
$788$	0	0
$789$	4518.74	0.203893
$790$	0	0
$791$	7345.68	0.330193
$792$	0	0
$793$	27299.0	1.22246
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25488.0	1.13279	0.566395	−	0.824134i	$-0.308338\pi$
$797$	25488.0	1.13279	0.566395	+	0.824134i	$0.308338\pi$
$798$	0	0
$799$	−11816.4	−0.523197
$800$	0	0
$801$	6116.68	0.269815
$802$	0	0
$803$	3313.19	0.145604
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18300.9	0.798291
$808$	0	0
$809$	−13832.0	−0.601122	−0.300561	−	0.953763i	$-0.597174\pi$
$809$	−13832.0	−0.601122	−0.300561	+	0.953763i	$0.597174\pi$
$810$	0	0
$811$	−29293.2	−1.26834	−0.634170	−	0.773194i	$-0.718658\pi$
$811$	−29293.2	−1.26834	−0.634170	+	0.773194i	$0.718658\pi$
$812$	0	0
$813$	−17026.7	−0.734504
$814$	0	0
$815$	0	0
$816$	0	0
$817$	51832.1	2.21955
$818$	0	0
$819$	1401.20	0.0597826
$820$	0	0
$821$	−18890.8	−0.803036	−0.401518	−	0.915851i	$-0.631517\pi$
$821$	−18890.8	−0.803036	−0.401518	+	0.915851i	$0.631517\pi$
$822$	0	0
$823$	37538.3	1.58992	0.794959	−	0.606663i	$-0.207492\pi$
$823$	37538.3	1.58992	0.794959	+	0.606663i	$0.207492\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19341.1	0.813247	0.406623	−	0.913596i	$-0.366706\pi$
$827$	19341.1	0.813247	0.406623	+	0.913596i	$0.366706\pi$
$828$	0	0
$829$	19921.4	0.834618	0.417309	−	0.908765i	$-0.362973\pi$
$829$	19921.4	0.834618	0.417309	+	0.908765i	$0.362973\pi$
$830$	0	0
$831$	−7744.40	−0.323286
$832$	0	0
$833$	14210.9	0.591091
$834$	0	0
$835$	0	0
$836$	0	0
$837$	568.899	0.0234935
$838$	0	0
$839$	18561.9	0.763798	0.381899	−	0.924204i	$-0.375270\pi$
$839$	18561.9	0.763798	0.381899	+	0.924204i	$0.375270\pi$
$840$	0	0
$841$	−529.235	−0.0216998
$842$	0	0
$843$	−11926.7	−0.487281
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9853.13	−0.399714
$848$	0	0
$849$	19323.4	0.781126
$850$	0	0
$851$	3431.45	0.138224
$852$	0	0
$853$	−41454.4	−1.66398	−0.831989	−	0.554793i	$-0.812798\pi$
$853$	−41454.4	−1.66398	−0.831989	+	0.554793i	$0.812798\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31013.0	−1.23615	−0.618077	−	0.786117i	$-0.712088\pi$
$857$	−31013.0	−1.23615	−0.618077	+	0.786117i	$0.712088\pi$
$858$	0	0
$859$	−42007.7	−1.66855	−0.834274	−	0.551350i	$-0.814113\pi$
$859$	−42007.7	−1.66855	−0.834274	+	0.551350i	$0.814113\pi$
$860$	0	0
$861$	4664.12	0.184614
$862$	0	0
$863$	6303.93	0.248654	0.124327	−	0.992241i	$-0.460323\pi$
$863$	6303.93	0.248654	0.124327	+	0.992241i	$0.460323\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8877.04	0.347728
$868$	0	0
$869$	−4732.51	−0.184740
$870$	0	0
$871$	1396.22	0.0543159
$872$	0	0
$873$	1745.22	0.0676596
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44647.5	1.71909	0.859544	−	0.511062i	$-0.170748\pi$
$877$	44647.5	1.71909	0.859544	+	0.511062i	$0.170748\pi$
$878$	0	0
$879$	18387.7	0.705578
$880$	0	0
$881$	−31955.0	−1.22201	−0.611004	−	0.791627i	$-0.709234\pi$
$881$	−31955.0	−1.22201	−0.611004	+	0.791627i	$0.709234\pi$
$882$	0	0
$883$	21030.0	0.801491	0.400746	−	0.916189i	$-0.368751\pi$
$883$	21030.0	0.801491	0.400746	+	0.916189i	$0.368751\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48751.9	1.84547	0.922734	−	0.385438i	$-0.125950\pi$
$887$	48751.9	1.84547	0.922734	+	0.385438i	$0.125950\pi$
$888$	0	0
$889$	7061.71	0.266414
$890$	0	0
$891$	−4761.30	−0.179023
$892$	0	0
$893$	35677.6	1.33696
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9071.00	0.337650
$898$	0	0
$899$	−3254.65	−0.120744
$900$	0	0
$901$	19688.5	0.727991
$902$	0	0
$903$	5403.98	0.199151
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7941.89	0.290746	0.145373	−	0.989377i	$-0.453562\pi$
$907$	7941.89	0.290746	0.145373	+	0.989377i	$0.453562\pi$
$908$	0	0
$909$	6817.67	0.248766
$910$	0	0
$911$	−16520.2	−0.600810	−0.300405	−	0.953812i	$-0.597122\pi$
$911$	−16520.2	−0.600810	−0.300405	+	0.953812i	$0.597122\pi$
$912$	0	0
$913$	33954.6	1.23081
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2957.54	−0.106507
$918$	0	0
$919$	−35905.8	−1.28882	−0.644408	−	0.764682i	$-0.722896\pi$
$919$	−35905.8	−1.28882	−0.644408	+	0.764682i	$0.722896\pi$
$920$	0	0
$921$	8000.35	0.286233
$922$	0	0
$923$	−33554.1	−1.19658
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4496.28	−0.159307
$928$	0	0
$929$	−10294.0	−0.363547	−0.181774	−	0.983340i	$-0.558184\pi$
$929$	−10294.0	−0.363547	−0.181774	+	0.983340i	$0.558184\pi$
$930$	0	0
$931$	−42907.4	−1.51046
$932$	0	0
$933$	−15590.3	−0.547056
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22550.3	−0.786219	−0.393110	−	0.919492i	$-0.628601\pi$
$937$	−22550.3	−0.786219	−0.393110	+	0.919492i	$0.628601\pi$
$938$	0	0
$939$	32064.2	1.11435
$940$	0	0
$941$	−55637.1	−1.92744	−0.963718	−	0.266923i	$-0.913993\pi$
$941$	−55637.1	−1.92744	−0.963718	+	0.266923i	$0.913993\pi$
$942$	0	0
$943$	30194.2	1.04269
$944$	0	0
$945$	0	0
$946$	0	0
$947$	57473.9	1.97218	0.986088	−	0.166224i	$-0.0531575\pi$
$947$	57473.9	1.97218	0.986088	+	0.166224i	$0.0531575\pi$
$948$	0	0
$949$	1891.90	0.0647142
$950$	0	0
$951$	26478.9	0.902877
$952$	0	0
$953$	19618.2	0.666837	0.333418	−	0.942779i	$-0.391798\pi$
$953$	19618.2	0.666837	0.333418	+	0.942779i	$0.391798\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	27239.3	0.920084
$958$	0	0
$959$	−6226.25	−0.209652
$960$	0	0
$961$	−29347.0	−0.985098
$962$	0	0
$963$	−13194.2	−0.441514
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−45483.5	−1.51256	−0.756282	−	0.654245i	$-0.772986\pi$
$967$	−45483.5	−1.51256	−0.756282	+	0.654245i	$0.772986\pi$
$968$	0	0
$969$	17699.2	0.586769
$970$	0	0
$971$	15711.6	0.519269	0.259635	−	0.965707i	$-0.416398\pi$
$971$	15711.6	0.519269	0.259635	+	0.965707i	$0.416398\pi$
$972$	0	0
$973$	−943.722	−0.0310939
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23201.6	−0.759761	−0.379880	−	0.925036i	$-0.624035\pi$
$977$	−23201.6	−0.759761	−0.379880	+	0.925036i	$0.624035\pi$
$978$	0	0
$979$	−39949.7	−1.30419
$980$	0	0
$981$	−13922.6	−0.453123
$982$	0	0
$983$	−29047.0	−0.942476	−0.471238	−	0.882006i	$-0.656193\pi$
$983$	−29047.0	−0.942476	−0.471238	+	0.882006i	$0.656193\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3719.72	0.119960
$988$	0	0
$989$	34983.9	1.12480
$990$	0	0
$991$	47593.1	1.52558	0.762788	−	0.646649i	$-0.223830\pi$
$991$	47593.1	1.52558	0.762788	+	0.646649i	$0.223830\pi$
$992$	0	0
$993$	21326.3	0.681541
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3012.91	−0.0957071	−0.0478535	−	0.998854i	$-0.515238\pi$
$997$	−3012.91	−0.0957071	−0.0478535	+	0.998854i	$0.515238\pi$
$998$	0	0
$999$	−1028.49	−0.0325727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bk.1.2	✓	3
4.3	odd	2		2400.4.a.bq.1.2	yes	3
5.4	even	2		2400.4.a.br.1.2	yes	3
20.19	odd	2		2400.4.a.bl.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.4.a.bk.1.2	✓	3	1.1	even	1	trivial
2400.4.a.bl.1.2	yes	3	20.19	odd	2
2400.4.a.bq.1.2	yes	3	4.3	odd	2
2400.4.a.br.1.2	yes	3	5.4	even	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -4.63836 q^{7} +9.00000 q^{9} -58.7815 q^{11} -33.5655 q^{13} -44.2039 q^{17} +133.466 q^{19} +13.9151 q^{21} +90.0824 q^{23} -27.0000 q^{27} +154.466 q^{29} -21.0703 q^{31} +176.345 q^{33} +38.0923 q^{37} +100.697 q^{39} +335.184 q^{41} +388.354 q^{43} +267.316 q^{47} -321.486 q^{49} +132.612 q^{51} -445.403 q^{53} -400.398 q^{57} -36.6018 q^{59} -813.303 q^{61} -41.7453 q^{63} -41.5968 q^{67} -270.247 q^{69} +999.660 q^{71} -56.3644 q^{73} +272.650 q^{77} +80.5101 q^{79} +81.0000 q^{81} -577.640 q^{83} -463.398 q^{87} +679.631 q^{89} +155.689 q^{91} +63.2110 q^{93} +193.914 q^{97} -529.034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 7 q^{7} + 27 q^{9} - 21 q^{13} - 32 q^{17} - 19 q^{19} - 21 q^{21} + 60 q^{23} - 81 q^{27} + 44 q^{29} + 151 q^{31} - 330 q^{37} + 63 q^{39} + 82 q^{41} + 367 q^{43} - 246 q^{47} - 42 q^{49}+ \cdots + 351 q^{97}+O(q^{100}) \) 3 * q - 9 * q^3 + 7 * q^7 + 27 * q^9 - 21 * q^13 - 32 * q^17 - 19 * q^19 - 21 * q^21 + 60 * q^23 - 81 * q^27 + 44 * q^29 + 151 * q^31 - 330 * q^37 + 63 * q^39 + 82 * q^41 + 367 * q^43 - 246 * q^47 - 42 * q^49 + 96 * q^51 - 554 * q^53 + 57 * q^57 - 42 * q^59 - 27 * q^61 + 63 * q^63 - 837 * q^67 - 180 * q^69 + 1422 * q^71 - 498 * q^73 - 560 * q^77 + 688 * q^79 + 243 * q^81 + 702 * q^83 - 132 * q^87 - 1488 * q^89 + 735 * q^91 - 453 * q^93 + 351 * q^97

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