LMFDB - Embedded newform 2400.4.a.bi.1.1

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.115636.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 69x - 81 $ x^3 - x^2 - 69*x - 81
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			1.1
Root			$-7.10600$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -21.6128 q^{7} +9.00000 q^{9} -21.8112 q^{11} +70.6495 q^{13} -54.6592 q^{17} -76.0368 q^{19} +64.8383 q^{21} -89.0560 q^{23} -27.0000 q^{27} -51.8305 q^{29} -110.045 q^{31} +65.4336 q^{33} +9.56818 q^{37} -211.949 q^{39} +317.184 q^{41} +374.312 q^{43} -347.142 q^{47} +124.112 q^{49} +163.978 q^{51} +159.395 q^{53} +228.110 q^{57} -693.462 q^{59} -684.924 q^{61} -194.515 q^{63} +559.521 q^{67} +267.168 q^{69} -517.600 q^{71} -745.471 q^{73} +471.401 q^{77} +120.093 q^{79} +81.0000 q^{81} -341.353 q^{83} +155.491 q^{87} -706.931 q^{89} -1526.93 q^{91} +330.134 q^{93} -171.217 q^{97} -196.301 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 3 q^{7} + 27 q^{9} - 44 q^{11} - 13 q^{13} + 36 q^{17} - 71 q^{19} - 9 q^{21} - 160 q^{23} - 81 q^{27} - 184 q^{29} + 59 q^{31} + 132 q^{33} + 350 q^{37} + 39 q^{39} + 166 q^{41} - 341 q^{43}+ \cdots - 396 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 + 3 * q^7 + 27 * q^9 - 44 * q^11 - 13 * q^13 + 36 * q^17 - 71 * q^19 - 9 * q^21 - 160 * q^23 - 81 * q^27 - 184 * q^29 + 59 * q^31 + 132 * q^33 + 350 * q^37 + 39 * q^39 + 166 * q^41 - 341 * q^43 + 394 * q^47 + 158 * q^49 - 108 * q^51 + 314 * q^53 + 213 * q^57 - 538 * q^59 - 523 * q^61 + 27 * q^63 + 543 * q^67 + 480 * q^69 - 910 * q^71 + 870 * q^73 - 828 * q^77 + 96 * q^79 + 243 * q^81 - 210 * q^83 + 552 * q^87 - 64 * q^89 - 2781 * q^91 - 177 * q^93 + 943 * q^97 - 396 * q^99

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
$7$	−21.6128	−1.16698	−0.583490	−	0.812120i	$-0.698313\pi$
$7$	−21.6128	−1.16698	−0.583490	+	0.812120i	$0.698313\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−21.8112	−0.597848	−0.298924	−	0.954277i	$-0.596628\pi$
$11$	−21.8112	−0.597848	−0.298924	+	0.954277i	$0.596628\pi$
$12$	0	0
$13$	70.6495	1.50728	0.753641	−	0.657287i	$-0.228296\pi$
$13$	70.6495	1.50728	0.753641	+	0.657287i	$0.228296\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−54.6592	−0.779812	−0.389906	−	0.920855i	$-0.627492\pi$
$17$	−54.6592	−0.779812	−0.389906	+	0.920855i	$0.627492\pi$
$18$	0	0
$19$	−76.0368	−0.918107	−0.459053	−	0.888409i	$-0.651811\pi$
$19$	−76.0368	−0.918107	−0.459053	+	0.888409i	$0.651811\pi$
$20$	0	0
$21$	64.8383	0.673756
$22$	0	0
$23$	−89.0560	−0.807368	−0.403684	−	0.914898i	$-0.632271\pi$
$23$	−89.0560	−0.807368	−0.403684	+	0.914898i	$0.632271\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−51.8305	−0.331886	−0.165943	−	0.986135i	$-0.553067\pi$
$29$	−51.8305	−0.331886	−0.165943	+	0.986135i	$0.553067\pi$
$30$	0	0
$31$	−110.045	−0.637568	−0.318784	−	0.947827i	$-0.603274\pi$
$31$	−110.045	−0.637568	−0.318784	+	0.947827i	$0.603274\pi$
$32$	0	0
$33$	65.4336	0.345168
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.56818	0.0425135	0.0212567	−	0.999774i	$-0.493233\pi$
$37$	9.56818	0.0425135	0.0212567	+	0.999774i	$0.493233\pi$
$38$	0	0
$39$	−211.949	−0.870229
$40$	0	0
$41$	317.184	1.20819	0.604095	−	0.796912i	$-0.293535\pi$
$41$	317.184	1.20819	0.604095	+	0.796912i	$0.293535\pi$
$42$	0	0
$43$	374.312	1.32749	0.663744	−	0.747960i	$-0.268966\pi$
$43$	374.312	1.32749	0.663744	+	0.747960i	$0.268966\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−347.142	−1.07736	−0.538679	−	0.842511i	$-0.681077\pi$
$47$	−347.142	−1.07736	−0.538679	+	0.842511i	$0.681077\pi$
$48$	0	0
$49$	124.112	0.361843
$50$	0	0
$51$	163.978	0.450224
$52$	0	0
$53$	159.395	0.413106	0.206553	−	0.978435i	$-0.433775\pi$
$53$	159.395	0.413106	0.206553	+	0.978435i	$0.433775\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	228.110	0.530069
$58$	0	0
$59$	−693.462	−1.53019	−0.765094	−	0.643919i	$-0.777307\pi$
$59$	−693.462	−1.53019	−0.765094	+	0.643919i	$0.777307\pi$
$60$	0	0
$61$	−684.924	−1.43763	−0.718816	−	0.695200i	$-0.755316\pi$
$61$	−684.924	−1.43763	−0.718816	+	0.695200i	$0.755316\pi$
$62$	0	0
$63$	−194.515	−0.388993
$64$	0	0
$65$	0	0
$66$	0	0
$67$	559.521	1.02025	0.510123	−	0.860102i	$-0.329600\pi$
$67$	559.521	1.02025	0.510123	+	0.860102i	$0.329600\pi$
$68$	0	0
$69$	267.168	0.466134
$70$	0	0
$71$	−517.600	−0.865180	−0.432590	−	0.901591i	$-0.642400\pi$
$71$	−517.600	−0.865180	−0.432590	+	0.901591i	$0.642400\pi$
$72$	0	0
$73$	−745.471	−1.19522	−0.597608	−	0.801788i	$-0.703882\pi$
$73$	−745.471	−1.19522	−0.597608	+	0.801788i	$0.703882\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	471.401	0.697677
$78$	0	0
$79$	120.093	0.171032	0.0855158	−	0.996337i	$-0.472746\pi$
$79$	120.093	0.171032	0.0855158	+	0.996337i	$0.472746\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−341.353	−0.451426	−0.225713	−	0.974194i	$-0.572471\pi$
$83$	−341.353	−0.451426	−0.225713	+	0.974194i	$0.572471\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	155.491	0.191614
$88$	0	0
$89$	−706.931	−0.841961	−0.420980	−	0.907070i	$-0.638314\pi$
$89$	−706.931	−0.841961	−0.420980	+	0.907070i	$0.638314\pi$
$90$	0	0
$91$	−1526.93	−1.75897
$92$	0	0
$93$	330.134	0.368100
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−171.217	−0.179221	−0.0896106	−	0.995977i	$-0.528562\pi$
$97$	−171.217	−0.179221	−0.0896106	+	0.995977i	$0.528562\pi$
$98$	0	0
$99$	−196.301	−0.199283
$100$	0	0
$101$	−1658.89	−1.63431	−0.817157	−	0.576416i	$-0.804451\pi$
$101$	−1658.89	−1.63431	−0.817157	+	0.576416i	$0.804451\pi$
$102$	0	0
$103$	684.412	0.654729	0.327365	−	0.944898i	$-0.393839\pi$
$103$	684.412	0.654729	0.327365	+	0.944898i	$0.393839\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	115.479	0.104334	0.0521670	−	0.998638i	$-0.483387\pi$
$107$	115.479	0.104334	0.0521670	+	0.998638i	$0.483387\pi$
$108$	0	0
$109$	1740.97	1.52986	0.764930	−	0.644113i	$-0.222773\pi$
$109$	1740.97	1.52986	0.764930	+	0.644113i	$0.222773\pi$
$110$	0	0
$111$	−28.7045	−0.0245452
$112$	0	0
$113$	1715.05	1.42777	0.713885	−	0.700263i	$-0.246934\pi$
$113$	1715.05	1.42777	0.713885	+	0.700263i	$0.246934\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	635.846	0.502427
$118$	0	0
$119$	1181.34	0.910025
$120$	0	0
$121$	−855.271	−0.642578
$122$	0	0
$123$	−951.551	−0.697549
$124$	0	0
$125$	0	0
$126$	0	0
$127$	892.595	0.623661	0.311831	−	0.950138i	$-0.399058\pi$
$127$	892.595	0.623661	0.311831	+	0.950138i	$0.399058\pi$
$128$	0	0
$129$	−1122.93	−0.766425
$130$	0	0
$131$	−48.6563	−0.0324513	−0.0162256	−	0.999868i	$-0.505165\pi$
$131$	−48.6563	−0.0324513	−0.0162256	+	0.999868i	$0.505165\pi$
$132$	0	0
$133$	1643.37	1.07141
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.2998	0.00954121	0.00477061	−	0.999989i	$-0.498481\pi$
$137$	15.2998	0.00954121	0.00477061	+	0.999989i	$0.498481\pi$
$138$	0	0
$139$	2069.83	1.26302	0.631512	−	0.775366i	$-0.282435\pi$
$139$	2069.83	1.26302	0.631512	+	0.775366i	$0.282435\pi$
$140$	0	0
$141$	1041.43	0.622013
$142$	0	0
$143$	−1540.95	−0.901125
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−372.336	−0.208910
$148$	0	0
$149$	1176.74	0.646997	0.323498	−	0.946229i	$-0.395141\pi$
$149$	1176.74	0.646997	0.323498	+	0.946229i	$0.395141\pi$
$150$	0	0
$151$	982.925	0.529730	0.264865	−	0.964285i	$-0.414673\pi$
$151$	982.925	0.529730	0.264865	+	0.964285i	$0.414673\pi$
$152$	0	0
$153$	−491.933	−0.259937
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3886.91	−1.97585	−0.987926	−	0.154925i	$-0.950487\pi$
$157$	−3886.91	−1.97585	−0.987926	+	0.154925i	$0.950487\pi$
$158$	0	0
$159$	−478.185	−0.238507
$160$	0	0
$161$	1924.75	0.942183
$162$	0	0
$163$	−3080.24	−1.48014	−0.740070	−	0.672530i	$-0.765207\pi$
$163$	−3080.24	−1.48014	−0.740070	+	0.672530i	$0.765207\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3183.64	1.47520	0.737598	−	0.675240i	$-0.235960\pi$
$167$	3183.64	1.47520	0.737598	+	0.675240i	$0.235960\pi$
$168$	0	0
$169$	2794.36	1.27190
$170$	0	0
$171$	−684.331	−0.306036
$172$	0	0
$173$	2375.87	1.04413	0.522064	−	0.852907i	$-0.325162\pi$
$173$	2375.87	1.04413	0.522064	+	0.852907i	$0.325162\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2080.39	0.883454
$178$	0	0
$179$	−1555.05	−0.649329	−0.324665	−	0.945829i	$-0.605251\pi$
$179$	−1555.05	−0.649329	−0.324665	+	0.945829i	$0.605251\pi$
$180$	0	0
$181$	−468.515	−0.192400	−0.0962002	−	0.995362i	$-0.530669\pi$
$181$	−468.515	−0.192400	−0.0962002	+	0.995362i	$0.530669\pi$
$182$	0	0
$183$	2054.77	0.830017
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1192.18	0.466209
$188$	0	0
$189$	583.545	0.224585
$190$	0	0
$191$	1816.53	0.688165	0.344082	−	0.938939i	$-0.388190\pi$
$191$	1816.53	0.688165	0.344082	+	0.938939i	$0.388190\pi$
$192$	0	0
$193$	2556.92	0.953634	0.476817	−	0.879003i	$-0.341790\pi$
$193$	2556.92	0.953634	0.476817	+	0.879003i	$0.341790\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3625.58	1.31123	0.655614	−	0.755097i	$-0.272410\pi$
$197$	3625.58	1.31123	0.655614	+	0.755097i	$0.272410\pi$
$198$	0	0
$199$	−1681.18	−0.598874	−0.299437	−	0.954116i	$-0.596799\pi$
$199$	−1681.18	−0.598874	−0.299437	+	0.954116i	$0.596799\pi$
$200$	0	0
$201$	−1678.56	−0.589039
$202$	0	0
$203$	1120.20	0.387304
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−801.504	−0.269123
$208$	0	0
$209$	1658.45	0.548888
$210$	0	0
$211$	1087.29	0.354749	0.177375	−	0.984143i	$-0.443240\pi$
$211$	1087.29	0.354749	0.177375	+	0.984143i	$0.443240\pi$
$212$	0	0
$213$	1552.80	0.499512
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2378.37	0.744029
$218$	0	0
$219$	2236.41	0.690058
$220$	0	0
$221$	−3861.65	−1.17540
$222$	0	0
$223$	−4220.26	−1.26731	−0.633654	−	0.773617i	$-0.718446\pi$
$223$	−4220.26	−1.26731	−0.633654	+	0.773617i	$0.718446\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−494.926	−0.144711	−0.0723555	−	0.997379i	$-0.523052\pi$
$227$	−494.926	−0.144711	−0.0723555	+	0.997379i	$0.523052\pi$
$228$	0	0
$229$	2070.82	0.597570	0.298785	−	0.954320i	$-0.403419\pi$
$229$	2070.82	0.597570	0.298785	+	0.954320i	$0.403419\pi$
$230$	0	0
$231$	−1414.20	−0.402804
$232$	0	0
$233$	3690.67	1.03770	0.518850	−	0.854865i	$-0.326360\pi$
$233$	3690.67	1.03770	0.518850	+	0.854865i	$0.326360\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−360.278	−0.0987451
$238$	0	0
$239$	655.698	0.177463	0.0887313	−	0.996056i	$-0.471719\pi$
$239$	655.698	0.177463	0.0887313	+	0.996056i	$0.471719\pi$
$240$	0	0
$241$	−4806.12	−1.28460	−0.642302	−	0.766452i	$-0.722021\pi$
$241$	−4806.12	−1.28460	−0.642302	+	0.766452i	$0.722021\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5371.96	−1.38385
$248$	0	0
$249$	1024.06	0.260631
$250$	0	0
$251$	−6493.30	−1.63288	−0.816441	−	0.577428i	$-0.804056\pi$
$251$	−6493.30	−1.63288	−0.816441	+	0.577428i	$0.804056\pi$
$252$	0	0
$253$	1942.42	0.482683
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6662.85	1.61719	0.808594	−	0.588367i	$-0.200229\pi$
$257$	6662.85	1.61719	0.808594	+	0.588367i	$0.200229\pi$
$258$	0	0
$259$	−206.795	−0.0496124
$260$	0	0
$261$	−466.474	−0.110629
$262$	0	0
$263$	6203.15	1.45438	0.727192	−	0.686434i	$-0.240825\pi$
$263$	6203.15	1.45438	0.727192	+	0.686434i	$0.240825\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2120.79	0.486106
$268$	0	0
$269$	1569.29	0.355692	0.177846	−	0.984058i	$-0.443087\pi$
$269$	1569.29	0.355692	0.177846	+	0.984058i	$0.443087\pi$
$270$	0	0
$271$	3259.38	0.730603	0.365301	−	0.930889i	$-0.380966\pi$
$271$	3259.38	0.730603	0.365301	+	0.930889i	$0.380966\pi$
$272$	0	0
$273$	4580.80	1.01554
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1475.24	0.319995	0.159997	−	0.987117i	$-0.448851\pi$
$277$	1475.24	0.319995	0.159997	+	0.987117i	$0.448851\pi$
$278$	0	0
$279$	−990.401	−0.212523
$280$	0	0
$281$	−3948.50	−0.838249	−0.419124	−	0.907929i	$-0.637663\pi$
$281$	−3948.50	−0.838249	−0.419124	+	0.907929i	$0.637663\pi$
$282$	0	0
$283$	808.764	0.169880	0.0849400	−	0.996386i	$-0.472930\pi$
$283$	808.764	0.169880	0.0849400	+	0.996386i	$0.472930\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6855.22	−1.40993
$288$	0	0
$289$	−1925.37	−0.391894
$290$	0	0
$291$	513.651	0.103473
$292$	0	0
$293$	−1665.17	−0.332014	−0.166007	−	0.986125i	$-0.553087\pi$
$293$	−1665.17	−0.332014	−0.166007	+	0.986125i	$0.553087\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	588.903	0.115056
$298$	0	0
$299$	−6291.77	−1.21693
$300$	0	0
$301$	−8089.91	−1.54915
$302$	0	0
$303$	4976.67	0.943571
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2642.13	0.491187	0.245594	−	0.969373i	$-0.421017\pi$
$307$	2642.13	0.491187	0.245594	+	0.969373i	$0.421017\pi$
$308$	0	0
$309$	−2053.24	−0.378008
$310$	0	0
$311$	5165.24	0.941781	0.470891	−	0.882192i	$-0.343933\pi$
$311$	5165.24	0.941781	0.470891	+	0.882192i	$0.343933\pi$
$312$	0	0
$313$	10137.3	1.83065	0.915327	−	0.402712i	$-0.131932\pi$
$313$	10137.3	1.83065	0.915327	+	0.402712i	$0.131932\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6427.21	−1.13876	−0.569381	−	0.822073i	$-0.692817\pi$
$317$	−6427.21	−1.13876	−0.569381	+	0.822073i	$0.692817\pi$
$318$	0	0
$319$	1130.49	0.198417
$320$	0	0
$321$	−346.436	−0.0602372
$322$	0	0
$323$	4156.11	0.715950
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5222.91	−0.883265
$328$	0	0
$329$	7502.70	1.25726
$330$	0	0
$331$	3345.03	0.555466	0.277733	−	0.960658i	$-0.410417\pi$
$331$	3345.03	0.555466	0.277733	+	0.960658i	$0.410417\pi$
$332$	0	0
$333$	86.1136	0.0141712
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7127.89	1.15217	0.576085	−	0.817390i	$-0.304580\pi$
$337$	7127.89	1.15217	0.576085	+	0.817390i	$0.304580\pi$
$338$	0	0
$339$	−5145.14	−0.824323
$340$	0	0
$341$	2400.21	0.381168
$342$	0	0
$343$	4730.78	0.744717
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2033.40	−0.314579	−0.157289	−	0.987553i	$-0.550276\pi$
$347$	−2033.40	−0.314579	−0.157289	+	0.987553i	$0.550276\pi$
$348$	0	0
$349$	6306.79	0.967321	0.483660	−	0.875256i	$-0.339307\pi$
$349$	6306.79	0.967321	0.483660	+	0.875256i	$0.339307\pi$
$350$	0	0
$351$	−1907.54	−0.290076
$352$	0	0
$353$	10350.3	1.56059	0.780296	−	0.625411i	$-0.215069\pi$
$353$	10350.3	1.56059	0.780296	+	0.625411i	$0.215069\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3544.01	−0.525403
$358$	0	0
$359$	9255.41	1.36067	0.680336	−	0.732900i	$-0.261834\pi$
$359$	9255.41	1.36067	0.680336	+	0.732900i	$0.261834\pi$
$360$	0	0
$361$	−1077.41	−0.157080
$362$	0	0
$363$	2565.81	0.370993
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−208.453	−0.0296490	−0.0148245	−	0.999890i	$-0.504719\pi$
$367$	−208.453	−0.0296490	−0.0148245	+	0.999890i	$0.504719\pi$
$368$	0	0
$369$	2854.65	0.402730
$370$	0	0
$371$	−3444.97	−0.482086
$372$	0	0
$373$	3208.18	0.445344	0.222672	−	0.974893i	$-0.428522\pi$
$373$	3208.18	0.445344	0.222672	+	0.974893i	$0.428522\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3661.80	−0.500245
$378$	0	0
$379$	3209.82	0.435033	0.217516	−	0.976057i	$-0.430204\pi$
$379$	3209.82	0.435033	0.217516	+	0.976057i	$0.430204\pi$
$380$	0	0
$381$	−2677.78	−0.360071
$382$	0	0
$383$	972.105	0.129693	0.0648463	−	0.997895i	$-0.479344\pi$
$383$	972.105	0.129693	0.0648463	+	0.997895i	$0.479344\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3368.80	0.442496
$388$	0	0
$389$	10508.9	1.36973	0.684863	−	0.728671i	$-0.259862\pi$
$389$	10508.9	1.36973	0.684863	+	0.728671i	$0.259862\pi$
$390$	0	0
$391$	4867.73	0.629595
$392$	0	0
$393$	145.969	0.0187358
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11952.2	1.51099	0.755496	−	0.655153i	$-0.227396\pi$
$397$	11952.2	1.51099	0.755496	+	0.655153i	$0.227396\pi$
$398$	0	0
$399$	−4930.10	−0.618580
$400$	0	0
$401$	−3349.39	−0.417108	−0.208554	−	0.978011i	$-0.566876\pi$
$401$	−3349.39	−0.417108	−0.208554	+	0.978011i	$0.566876\pi$
$402$	0	0
$403$	−7774.60	−0.960994
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−208.694	−0.0254166
$408$	0	0
$409$	6695.81	0.809502	0.404751	−	0.914427i	$-0.367358\pi$
$409$	6695.81	0.809502	0.404751	+	0.914427i	$0.367358\pi$
$410$	0	0
$411$	−45.8993	−0.00550862
$412$	0	0
$413$	14987.6	1.78570
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6209.48	−0.729207
$418$	0	0
$419$	−8413.46	−0.980965	−0.490483	−	0.871451i	$-0.663179\pi$
$419$	−8413.46	−0.980965	−0.490483	+	0.871451i	$0.663179\pi$
$420$	0	0
$421$	8129.10	0.941064	0.470532	−	0.882383i	$-0.344062\pi$
$421$	8129.10	0.941064	0.470532	+	0.882383i	$0.344062\pi$
$422$	0	0
$423$	−3124.28	−0.359120
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14803.1	1.67769
$428$	0	0
$429$	4622.86	0.520265
$430$	0	0
$431$	10644.8	1.18966	0.594828	−	0.803853i	$-0.297220\pi$
$431$	10644.8	1.18966	0.594828	+	0.803853i	$0.297220\pi$
$432$	0	0
$433$	15087.7	1.67452	0.837260	−	0.546805i	$-0.184156\pi$
$433$	15087.7	1.67452	0.837260	+	0.546805i	$0.184156\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6771.53	0.741250
$438$	0	0
$439$	15203.8	1.65293	0.826467	−	0.562985i	$-0.190347\pi$
$439$	15203.8	1.65293	0.826467	+	0.562985i	$0.190347\pi$
$440$	0	0
$441$	1117.01	0.120614
$442$	0	0
$443$	−11860.1	−1.27198	−0.635991	−	0.771696i	$-0.719409\pi$
$443$	−11860.1	−1.27198	−0.635991	+	0.771696i	$0.719409\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3530.23	−0.373544
$448$	0	0
$449$	4128.01	0.433881	0.216941	−	0.976185i	$-0.430392\pi$
$449$	4128.01	0.433881	0.216941	+	0.976185i	$0.430392\pi$
$450$	0	0
$451$	−6918.16	−0.722314
$452$	0	0
$453$	−2948.77	−0.305840
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4368.36	−0.447141	−0.223570	−	0.974688i	$-0.571771\pi$
$457$	−4368.36	−0.447141	−0.223570	+	0.974688i	$0.571771\pi$
$458$	0	0
$459$	1475.80	0.150075
$460$	0	0
$461$	−16287.4	−1.64551	−0.822755	−	0.568395i	$-0.807564\pi$
$461$	−16287.4	−1.64551	−0.822755	+	0.568395i	$0.807564\pi$
$462$	0	0
$463$	11145.3	1.11871	0.559357	−	0.828927i	$-0.311048\pi$
$463$	11145.3	1.11871	0.559357	+	0.828927i	$0.311048\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14600.2	−1.44671	−0.723356	−	0.690475i	$-0.757401\pi$
$467$	−14600.2	−1.44671	−0.723356	+	0.690475i	$0.757401\pi$
$468$	0	0
$469$	−12092.8	−1.19061
$470$	0	0
$471$	11660.7	1.14076
$472$	0	0
$473$	−8164.19	−0.793636
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1434.56	0.137702
$478$	0	0
$479$	−68.5375	−0.00653770	−0.00326885	−	0.999995i	$-0.501041\pi$
$479$	−68.5375	−0.00653770	−0.00326885	+	0.999995i	$0.501041\pi$
$480$	0	0
$481$	675.987	0.0640798
$482$	0	0
$483$	−5774.24	−0.543969
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−296.399	−0.0275793	−0.0137897	−	0.999905i	$-0.504390\pi$
$487$	−296.399	−0.0275793	−0.0137897	+	0.999905i	$0.504390\pi$
$488$	0	0
$489$	9240.71	0.854559
$490$	0	0
$491$	−3332.21	−0.306274	−0.153137	−	0.988205i	$-0.548938\pi$
$491$	−3332.21	−0.306274	−0.153137	+	0.988205i	$0.548938\pi$
$492$	0	0
$493$	2833.01	0.258808
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11186.8	1.00965
$498$	0	0
$499$	21280.8	1.90914	0.954568	−	0.297995i	$-0.0963178\pi$
$499$	21280.8	1.90914	0.954568	+	0.297995i	$0.0963178\pi$
$500$	0	0
$501$	−9550.93	−0.851705
$502$	0	0
$503$	−4862.81	−0.431058	−0.215529	−	0.976497i	$-0.569148\pi$
$503$	−4862.81	−0.431058	−0.215529	+	0.976497i	$0.569148\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−8383.07	−0.734330
$508$	0	0
$509$	7058.61	0.614670	0.307335	−	0.951601i	$-0.400563\pi$
$509$	7058.61	0.614670	0.307335	+	0.951601i	$0.400563\pi$
$510$	0	0
$511$	16111.7	1.39479
$512$	0	0
$513$	2052.99	0.176690
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7571.59	0.644097
$518$	0	0
$519$	−7127.61	−0.602827
$520$	0	0
$521$	−18033.6	−1.51644	−0.758222	−	0.651996i	$-0.773932\pi$
$521$	−18033.6	−1.51644	−0.758222	+	0.651996i	$0.773932\pi$
$522$	0	0
$523$	−15339.4	−1.28249	−0.641247	−	0.767334i	$-0.721583\pi$
$523$	−15339.4	−1.28249	−0.641247	+	0.767334i	$0.721583\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6014.95	0.497183
$528$	0	0
$529$	−4236.02	−0.348157
$530$	0	0
$531$	−6241.16	−0.510062
$532$	0	0
$533$	22408.9	1.82108
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4665.15	0.374890
$538$	0	0
$539$	−2707.03	−0.216327
$540$	0	0
$541$	−7640.12	−0.607162	−0.303581	−	0.952806i	$-0.598182\pi$
$541$	−7640.12	−0.607162	−0.303581	+	0.952806i	$0.598182\pi$
$542$	0	0
$543$	1405.55	0.111082
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10201.5	0.797415	0.398708	−	0.917078i	$-0.369459\pi$
$547$	10201.5	0.797415	0.398708	+	0.917078i	$0.369459\pi$
$548$	0	0
$549$	−6164.32	−0.479211
$550$	0	0
$551$	3941.02	0.304706
$552$	0	0
$553$	−2595.54	−0.199590
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11581.5	−0.881013	−0.440506	−	0.897749i	$-0.645201\pi$
$557$	−11581.5	−0.881013	−0.440506	+	0.897749i	$0.645201\pi$
$558$	0	0
$559$	26444.9	2.00090
$560$	0	0
$561$	−3576.55	−0.269166
$562$	0	0
$563$	13471.7	1.00846	0.504230	−	0.863569i	$-0.331776\pi$
$563$	13471.7	1.00846	0.504230	+	0.863569i	$0.331776\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1750.63	−0.129664
$568$	0	0
$569$	−22707.8	−1.67304	−0.836522	−	0.547934i	$-0.815415\pi$
$569$	−22707.8	−1.67304	−0.836522	+	0.547934i	$0.815415\pi$
$570$	0	0
$571$	−16646.0	−1.21999	−0.609993	−	0.792407i	$-0.708828\pi$
$571$	−16646.0	−1.21999	−0.609993	+	0.792407i	$0.708828\pi$
$572$	0	0
$573$	−5449.59	−0.397312
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16852.5	1.21591	0.607954	−	0.793972i	$-0.291990\pi$
$577$	16852.5	1.21591	0.607954	+	0.793972i	$0.291990\pi$
$578$	0	0
$579$	−7670.77	−0.550581
$580$	0	0
$581$	7377.59	0.526806
$582$	0	0
$583$	−3476.60	−0.246974
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24449.0	1.71911	0.859555	−	0.511044i	$-0.170741\pi$
$587$	24449.0	1.71911	0.859555	+	0.511044i	$0.170741\pi$
$588$	0	0
$589$	8367.43	0.585355
$590$	0	0
$591$	−10876.7	−0.757037
$592$	0	0
$593$	−25721.2	−1.78118	−0.890592	−	0.454802i	$-0.849710\pi$
$593$	−25721.2	−1.78118	−0.890592	+	0.454802i	$0.849710\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5043.55	0.345760
$598$	0	0
$599$	−26522.0	−1.80911	−0.904556	−	0.426355i	$-0.859797\pi$
$599$	−26522.0	−1.80911	−0.904556	+	0.426355i	$0.859797\pi$
$600$	0	0
$601$	13663.6	0.927373	0.463687	−	0.885999i	$-0.346526\pi$
$601$	13663.6	0.927373	0.463687	+	0.885999i	$0.346526\pi$
$602$	0	0
$603$	5035.69	0.340082
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20022.6	1.33887	0.669434	−	0.742871i	$-0.266537\pi$
$607$	20022.6	1.33887	0.669434	+	0.742871i	$0.266537\pi$
$608$	0	0
$609$	−3360.60	−0.223610
$610$	0	0
$611$	−24525.4	−1.62388
$612$	0	0
$613$	16164.9	1.06508	0.532539	−	0.846406i	$-0.321238\pi$
$613$	16164.9	1.06508	0.532539	+	0.846406i	$0.321238\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20653.5	−1.34762	−0.673808	−	0.738907i	$-0.735342\pi$
$617$	−20653.5	−1.34762	−0.673808	+	0.738907i	$0.735342\pi$
$618$	0	0
$619$	−7727.54	−0.501771	−0.250885	−	0.968017i	$-0.580722\pi$
$619$	−7727.54	−0.501771	−0.250885	+	0.968017i	$0.580722\pi$
$620$	0	0
$621$	2404.51	0.155378
$622$	0	0
$623$	15278.7	0.982552
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4975.36	−0.316901
$628$	0	0
$629$	−522.989	−0.0331525
$630$	0	0
$631$	14814.4	0.934631	0.467315	−	0.884091i	$-0.345221\pi$
$631$	14814.4	0.934631	0.467315	+	0.884091i	$0.345221\pi$
$632$	0	0
$633$	−3261.87	−0.204815
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8768.46	0.545399
$638$	0	0
$639$	−4658.40	−0.288393
$640$	0	0
$641$	2953.56	0.181995	0.0909973	−	0.995851i	$-0.470995\pi$
$641$	2953.56	0.181995	0.0909973	+	0.995851i	$0.470995\pi$
$642$	0	0
$643$	−15566.3	−0.954703	−0.477351	−	0.878712i	$-0.658403\pi$
$643$	−15566.3	−0.954703	−0.477351	+	0.878712i	$0.658403\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−31671.2	−1.92445	−0.962227	−	0.272248i	$-0.912233\pi$
$647$	−31671.2	−1.92445	−0.962227	+	0.272248i	$0.912233\pi$
$648$	0	0
$649$	15125.2	0.914819
$650$	0	0
$651$	−7135.11	−0.429565
$652$	0	0
$653$	10925.8	0.654765	0.327382	−	0.944892i	$-0.393833\pi$
$653$	10925.8	0.654765	0.327382	+	0.944892i	$0.393833\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6709.24	−0.398405
$658$	0	0
$659$	−12037.7	−0.711566	−0.355783	−	0.934569i	$-0.615786\pi$
$659$	−12037.7	−0.711566	−0.355783	+	0.934569i	$0.615786\pi$
$660$	0	0
$661$	2116.75	0.124557	0.0622785	−	0.998059i	$-0.480163\pi$
$661$	2116.75	0.124557	0.0622785	+	0.998059i	$0.480163\pi$
$662$	0	0
$663$	11584.9	0.678615
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4615.82	0.267954
$668$	0	0
$669$	12660.8	0.731680
$670$	0	0
$671$	14939.0	0.859485
$672$	0	0
$673$	482.210	0.0276193	0.0138097	−	0.999905i	$-0.495604\pi$
$673$	482.210	0.0276193	0.0138097	+	0.999905i	$0.495604\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16056.6	0.911527	0.455763	−	0.890101i	$-0.349366\pi$
$677$	16056.6	0.911527	0.455763	+	0.890101i	$0.349366\pi$
$678$	0	0
$679$	3700.47	0.209147
$680$	0	0
$681$	1484.78	0.0835489
$682$	0	0
$683$	−18345.4	−1.02777	−0.513884	−	0.857860i	$-0.671794\pi$
$683$	−18345.4	−1.02777	−0.513884	+	0.857860i	$0.671794\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6212.45	−0.345007
$688$	0	0
$689$	11261.2	0.622666
$690$	0	0
$691$	−20257.2	−1.11522	−0.557611	−	0.830102i	$-0.688282\pi$
$691$	−20257.2	−1.11522	−0.557611	+	0.830102i	$0.688282\pi$
$692$	0	0
$693$	4242.61	0.232559
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−17337.0	−0.942161
$698$	0	0
$699$	−11072.0	−0.599116
$700$	0	0
$701$	16341.0	0.880446	0.440223	−	0.897888i	$-0.354899\pi$
$701$	16341.0	0.880446	0.440223	+	0.897888i	$0.354899\pi$
$702$	0	0
$703$	−727.533	−0.0390319
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35853.2	1.90721
$708$	0	0
$709$	4826.19	0.255644	0.127822	−	0.991797i	$-0.459201\pi$
$709$	4826.19	0.255644	0.127822	+	0.991797i	$0.459201\pi$
$710$	0	0
$711$	1080.84	0.0570105
$712$	0	0
$713$	9800.14	0.514752
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1967.09	−0.102458
$718$	0	0
$719$	15545.4	0.806322	0.403161	−	0.915129i	$-0.367911\pi$
$719$	15545.4	0.806322	0.403161	+	0.915129i	$0.367911\pi$
$720$	0	0
$721$	−14792.0	−0.764056
$722$	0	0
$723$	14418.4	0.741667
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−17551.5	−0.895389	−0.447694	−	0.894187i	$-0.647755\pi$
$727$	−17551.5	−0.895389	−0.447694	+	0.894187i	$0.647755\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−20459.6	−1.03519
$732$	0	0
$733$	−5663.10	−0.285363	−0.142682	−	0.989769i	$-0.545573\pi$
$733$	−5663.10	−0.285363	−0.142682	+	0.989769i	$0.545573\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12203.8	−0.609951
$738$	0	0
$739$	−14966.6	−0.745000	−0.372500	−	0.928032i	$-0.621499\pi$
$739$	−14966.6	−0.745000	−0.372500	+	0.928032i	$0.621499\pi$
$740$	0	0
$741$	16115.9	0.798963
$742$	0	0
$743$	28113.0	1.38811	0.694055	−	0.719922i	$-0.255822\pi$
$743$	28113.0	1.38811	0.694055	+	0.719922i	$0.255822\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3072.18	−0.150475
$748$	0	0
$749$	−2495.81	−0.121756
$750$	0	0
$751$	1320.93	0.0641831	0.0320915	−	0.999485i	$-0.489783\pi$
$751$	1320.93	0.0641831	0.0320915	+	0.999485i	$0.489783\pi$
$752$	0	0
$753$	19479.9	0.942745
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25204.8	−1.21015	−0.605075	−	0.796169i	$-0.706857\pi$
$757$	−25204.8	−1.21015	−0.605075	+	0.796169i	$0.706857\pi$
$758$	0	0
$759$	−5827.26	−0.278677
$760$	0	0
$761$	24919.5	1.18703	0.593516	−	0.804822i	$-0.297739\pi$
$761$	24919.5	1.18703	0.593516	+	0.804822i	$0.297739\pi$
$762$	0	0
$763$	−37627.2	−1.78532
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−48992.8	−2.30642
$768$	0	0
$769$	3859.10	0.180966	0.0904830	−	0.995898i	$-0.471159\pi$
$769$	3859.10	0.180966	0.0904830	+	0.995898i	$0.471159\pi$
$770$	0	0
$771$	−19988.6	−0.933684
$772$	0	0
$773$	−32502.5	−1.51233	−0.756167	−	0.654379i	$-0.772930\pi$
$773$	−32502.5	−1.51233	−0.756167	+	0.654379i	$0.772930\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	620.385	0.0286437
$778$	0	0
$779$	−24117.6	−1.10925
$780$	0	0
$781$	11289.5	0.517246
$782$	0	0
$783$	1399.42	0.0638714
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14105.9	0.638907	0.319453	−	0.947602i	$-0.396501\pi$
$787$	14105.9	0.638907	0.319453	+	0.947602i	$0.396501\pi$
$788$	0	0
$789$	−18609.5	−0.839689
$790$	0	0
$791$	−37066.9	−1.66618
$792$	0	0
$793$	−48389.6	−2.16692
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23808.9	1.05816	0.529080	−	0.848572i	$-0.322537\pi$
$797$	23808.9	1.05816	0.529080	+	0.848572i	$0.322537\pi$
$798$	0	0
$799$	18974.5	0.840137
$800$	0	0
$801$	−6362.38	−0.280654
$802$	0	0
$803$	16259.6	0.714558
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4707.86	−0.205359
$808$	0	0
$809$	−33146.0	−1.44049	−0.720243	−	0.693722i	$-0.755970\pi$
$809$	−33146.0	−1.44049	−0.720243	+	0.693722i	$0.755970\pi$
$810$	0	0
$811$	40932.2	1.77229	0.886143	−	0.463413i	$-0.153375\pi$
$811$	40932.2	1.77229	0.886143	+	0.463413i	$0.153375\pi$
$812$	0	0
$813$	−9778.15	−0.421814
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−28461.4	−1.21878
$818$	0	0
$819$	−13742.4	−0.586322
$820$	0	0
$821$	−15389.1	−0.654181	−0.327090	−	0.944993i	$-0.606068\pi$
$821$	−15389.1	−0.654181	−0.327090	+	0.944993i	$0.606068\pi$
$822$	0	0
$823$	−34659.0	−1.46797	−0.733984	−	0.679167i	$-0.762341\pi$
$823$	−34659.0	−1.46797	−0.733984	+	0.679167i	$0.762341\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17437.2	−0.733194	−0.366597	−	0.930380i	$-0.619477\pi$
$827$	−17437.2	−0.733194	−0.366597	+	0.930380i	$0.619477\pi$
$828$	0	0
$829$	−28007.6	−1.17339	−0.586696	−	0.809807i	$-0.699572\pi$
$829$	−28007.6	−1.17339	−0.586696	+	0.809807i	$0.699572\pi$
$830$	0	0
$831$	−4425.72	−0.184749
$832$	0	0
$833$	−6783.86	−0.282169
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2971.20	0.122700
$838$	0	0
$839$	−33889.0	−1.39449	−0.697245	−	0.716833i	$-0.745591\pi$
$839$	−33889.0	−1.39449	−0.697245	+	0.716833i	$0.745591\pi$
$840$	0	0
$841$	−21702.6	−0.889852
$842$	0	0
$843$	11845.5	0.483963
$844$	0	0
$845$	0	0
$846$	0	0
$847$	18484.8	0.749876
$848$	0	0
$849$	−2426.29	−0.0980803
$850$	0	0
$851$	−852.104	−0.0343240
$852$	0	0
$853$	−43681.8	−1.75338	−0.876691	−	0.481054i	$-0.840254\pi$
$853$	−43681.8	−1.75338	−0.876691	+	0.481054i	$0.840254\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9095.47	−0.362538	−0.181269	−	0.983434i	$-0.558021\pi$
$857$	−9095.47	−0.362538	−0.181269	+	0.983434i	$0.558021\pi$
$858$	0	0
$859$	31215.0	1.23986	0.619932	−	0.784655i	$-0.287160\pi$
$859$	31215.0	1.23986	0.619932	+	0.784655i	$0.287160\pi$
$860$	0	0
$861$	20565.7	0.814026
$862$	0	0
$863$	22777.1	0.898427	0.449213	−	0.893424i	$-0.351704\pi$
$863$	22777.1	0.898427	0.449213	+	0.893424i	$0.351704\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5776.12	0.226260
$868$	0	0
$869$	−2619.37	−0.102251
$870$	0	0
$871$	39529.9	1.53780
$872$	0	0
$873$	−1540.95	−0.0597404
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12582.9	−0.484485	−0.242243	−	0.970216i	$-0.577883\pi$
$877$	−12582.9	−0.484485	−0.242243	+	0.970216i	$0.577883\pi$
$878$	0	0
$879$	4995.50	0.191688
$880$	0	0
$881$	30067.9	1.14984	0.574922	−	0.818208i	$-0.305032\pi$
$881$	30067.9	1.14984	0.574922	+	0.818208i	$0.305032\pi$
$882$	0	0
$883$	−37957.3	−1.44662	−0.723309	−	0.690524i	$-0.757380\pi$
$883$	−37957.3	−1.44662	−0.723309	+	0.690524i	$0.757380\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14575.0	0.551725	0.275862	−	0.961197i	$-0.411037\pi$
$887$	14575.0	0.551725	0.275862	+	0.961197i	$0.411037\pi$
$888$	0	0
$889$	−19291.4	−0.727800
$890$	0	0
$891$	−1766.71	−0.0664275
$892$	0	0
$893$	26395.6	0.989131
$894$	0	0
$895$	0	0
$896$	0	0
$897$	18875.3	0.702595
$898$	0	0
$899$	5703.67	0.211599
$900$	0	0
$901$	−8712.40	−0.322145
$902$	0	0
$903$	24269.7	0.894403
$904$	0	0
$905$	0	0
$906$	0	0
$907$	50932.7	1.86460	0.932301	−	0.361684i	$-0.117798\pi$
$907$	50932.7	1.86460	0.932301	+	0.361684i	$0.117798\pi$
$908$	0	0
$909$	−14930.0	−0.544771
$910$	0	0
$911$	20903.2	0.760212	0.380106	−	0.924943i	$-0.375887\pi$
$911$	20903.2	0.760212	0.380106	+	0.924943i	$0.375887\pi$
$912$	0	0
$913$	7445.33	0.269884
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1051.60	0.0378700
$918$	0	0
$919$	41545.1	1.49124	0.745619	−	0.666373i	$-0.232154\pi$
$919$	41545.1	1.49124	0.745619	+	0.666373i	$0.232154\pi$
$920$	0	0
$921$	−7926.39	−0.283587
$922$	0	0
$923$	−36568.2	−1.30407
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6159.71	0.218243
$928$	0	0
$929$	37671.2	1.33041	0.665205	−	0.746661i	$-0.268344\pi$
$929$	37671.2	1.33041	0.665205	+	0.746661i	$0.268344\pi$
$930$	0	0
$931$	−9437.08	−0.332210
$932$	0	0
$933$	−15495.7	−0.543738
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14680.4	0.511834	0.255917	−	0.966699i	$-0.417623\pi$
$937$	14680.4	0.511834	0.255917	+	0.966699i	$0.417623\pi$
$938$	0	0
$939$	−30411.9	−1.05693
$940$	0	0
$941$	−15985.8	−0.553797	−0.276898	−	0.960899i	$-0.589307\pi$
$941$	−15985.8	−0.553797	−0.276898	+	0.960899i	$0.589307\pi$
$942$	0	0
$943$	−28247.1	−0.975454
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6083.09	−0.208737	−0.104368	−	0.994539i	$-0.533282\pi$
$947$	−6083.09	−0.208737	−0.104368	+	0.994539i	$0.533282\pi$
$948$	0	0
$949$	−52667.2	−1.80153
$950$	0	0
$951$	19281.6	0.657465
$952$	0	0
$953$	−57758.8	−1.96326	−0.981632	−	0.190786i	$-0.938896\pi$
$953$	−57758.8	−1.96326	−0.981632	+	0.190786i	$0.938896\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3391.46	−0.114556
$958$	0	0
$959$	−330.670	−0.0111344
$960$	0	0
$961$	−17681.2	−0.593508
$962$	0	0
$963$	1039.31	0.0347780
$964$	0	0
$965$	0	0
$966$	0	0
$967$	10663.7	0.354625	0.177313	−	0.984155i	$-0.443260\pi$
$967$	10663.7	0.354625	0.177313	+	0.984155i	$0.443260\pi$
$968$	0	0
$969$	−12468.3	−0.413354
$970$	0	0
$971$	−40563.2	−1.34061	−0.670306	−	0.742085i	$-0.733837\pi$
$971$	−40563.2	−1.34061	−0.670306	+	0.742085i	$0.733837\pi$
$972$	0	0
$973$	−44734.7	−1.47392
$974$	0	0
$975$	0	0
$976$	0	0
$977$	573.645	0.0187846	0.00939228	−	0.999956i	$-0.497010\pi$
$977$	573.645	0.0187846	0.00939228	+	0.999956i	$0.497010\pi$
$978$	0	0
$979$	15419.0	0.503364
$980$	0	0
$981$	15668.7	0.509954
$982$	0	0
$983$	11416.8	0.370438	0.185219	−	0.982697i	$-0.440701\pi$
$983$	11416.8	0.370438	0.185219	+	0.982697i	$0.440701\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−22508.1	−0.725877
$988$	0	0
$989$	−33334.7	−1.07177
$990$	0	0
$991$	31656.9	1.01475	0.507373	−	0.861726i	$-0.330617\pi$
$991$	31656.9	1.01475	0.507373	+	0.861726i	$0.330617\pi$
$992$	0	0
$993$	−10035.1	−0.320699
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−42673.1	−1.35554	−0.677768	−	0.735276i	$-0.737053\pi$
$997$	−42673.1	−1.35554	−0.677768	+	0.735276i	$0.737053\pi$
$998$	0	0
$999$	−258.341	−0.00818172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bi.1.1	✓	3
4.3	odd	2		2400.4.a.bt.1.3	yes	3
5.4	even	2		2400.4.a.bs.1.3	yes	3
20.19	odd	2		2400.4.a.bj.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.4.a.bi.1.1	✓	3	1.1	even	1	trivial
2400.4.a.bj.1.1	yes	3	20.19	odd	2
2400.4.a.bs.1.3	yes	3	5.4	even	2
2400.4.a.bt.1.3	yes	3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -21.6128 q^{7} +9.00000 q^{9} -21.8112 q^{11} +70.6495 q^{13} -54.6592 q^{17} -76.0368 q^{19} +64.8383 q^{21} -89.0560 q^{23} -27.0000 q^{27} -51.8305 q^{29} -110.045 q^{31} +65.4336 q^{33} +9.56818 q^{37} -211.949 q^{39} +317.184 q^{41} +374.312 q^{43} -347.142 q^{47} +124.112 q^{49} +163.978 q^{51} +159.395 q^{53} +228.110 q^{57} -693.462 q^{59} -684.924 q^{61} -194.515 q^{63} +559.521 q^{67} +267.168 q^{69} -517.600 q^{71} -745.471 q^{73} +471.401 q^{77} +120.093 q^{79} +81.0000 q^{81} -341.353 q^{83} +155.491 q^{87} -706.931 q^{89} -1526.93 q^{91} +330.134 q^{93} -171.217 q^{97} -196.301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 3 q^{7} + 27 q^{9} - 44 q^{11} - 13 q^{13} + 36 q^{17} - 71 q^{19} - 9 q^{21} - 160 q^{23} - 81 q^{27} - 184 q^{29} + 59 q^{31} + 132 q^{33} + 350 q^{37} + 39 q^{39} + 166 q^{41} - 341 q^{43}+ \cdots - 396 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 + 3 * q^7 + 27 * q^9 - 44 * q^11 - 13 * q^13 + 36 * q^17 - 71 * q^19 - 9 * q^21 - 160 * q^23 - 81 * q^27 - 184 * q^29 + 59 * q^31 + 132 * q^33 + 350 * q^37 + 39 * q^39 + 166 * q^41 - 341 * q^43 + 394 * q^47 + 158 * q^49 - 108 * q^51 + 314 * q^53 + 213 * q^57 - 538 * q^59 - 523 * q^61 + 27 * q^63 + 543 * q^67 + 480 * q^69 - 910 * q^71 + 870 * q^73 - 828 * q^77 + 96 * q^79 + 243 * q^81 - 210 * q^83 + 552 * q^87 - 64 * q^89 - 2781 * q^91 - 177 * q^93 + 943 * q^97 - 396 * q^99

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