LMFDB - Embedded newform 2400.4.a.bm.1.3

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:	3.3.28724.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 19x + 9 $ x^3 - x^2 - 19*x + 9
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			1.3
Root			$0.467558$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} +32.4330 q^{7} +9.00000 q^{9} -41.5628 q^{11} -19.3893 q^{13} -132.073 q^{17} -80.4669 q^{19} -97.2990 q^{21} -134.420 q^{23} -27.0000 q^{27} +217.123 q^{29} -129.375 q^{31} +124.688 q^{33} -239.828 q^{37} +58.1679 q^{39} +371.270 q^{41} +476.442 q^{43} +44.5275 q^{47} +708.900 q^{49} +396.220 q^{51} +14.7052 q^{53} +241.401 q^{57} -186.711 q^{59} -609.872 q^{61} +291.897 q^{63} +332.142 q^{67} +403.261 q^{69} +386.062 q^{71} +1136.65 q^{73} -1348.01 q^{77} +1313.21 q^{79} +81.0000 q^{81} -430.813 q^{83} -651.368 q^{87} -1525.80 q^{89} -628.853 q^{91} +388.126 q^{93} +462.295 q^{97} -374.065 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 19 q^{7} + 27 q^{9} - 48 q^{11} - 63 q^{13} - 16 q^{17} - 21 q^{19} - 57 q^{21} + 140 q^{23} - 81 q^{27} + 92 q^{29} - 179 q^{31} + 144 q^{33} - 550 q^{37} + 189 q^{39} - 206 q^{41} + 279 q^{43}+ \cdots - 432 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 + 19 * q^7 + 27 * q^9 - 48 * q^11 - 63 * q^13 - 16 * q^17 - 21 * q^19 - 57 * q^21 + 140 * q^23 - 81 * q^27 + 92 * q^29 - 179 * q^31 + 144 * q^33 - 550 * q^37 + 189 * q^39 - 206 * q^41 + 279 * q^43 + 450 * q^47 + 446 * q^49 + 48 * q^51 + 434 * q^53 + 63 * q^57 - 854 * q^59 - 1103 * q^61 + 171 * q^63 + 555 * q^67 - 420 * q^69 + 970 * q^71 + 722 * q^73 - 1184 * q^77 + 856 * q^79 + 243 * q^81 + 222 * q^83 - 276 * q^87 - 1216 * q^89 + 1025 * q^91 + 537 * q^93 + 89 * q^97 - 432 * 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
$7$	32.4330	1.75122	0.875609	−	0.483021i	$-0.160460\pi$
$7$	32.4330	1.75122	0.875609	+	0.483021i	$0.160460\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−41.5628	−1.13924	−0.569620	−	0.821908i	$-0.692910\pi$
$11$	−41.5628	−1.13924	−0.569620	+	0.821908i	$0.692910\pi$
$12$	0	0
$13$	−19.3893	−0.413663	−0.206832	−	0.978377i	$-0.566315\pi$
$13$	−19.3893	−0.413663	−0.206832	+	0.978377i	$0.566315\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−132.073	−1.88427	−0.942133	−	0.335240i	$-0.891183\pi$
$17$	−132.073	−1.88427	−0.942133	+	0.335240i	$0.891183\pi$
$18$	0	0
$19$	−80.4669	−0.971599	−0.485800	−	0.874070i	$-0.661472\pi$
$19$	−80.4669	−0.971599	−0.485800	+	0.874070i	$0.661472\pi$
$20$	0	0
$21$	−97.2990	−1.01107
$22$	0	0
$23$	−134.420	−1.21863	−0.609317	−	0.792927i	$-0.708556\pi$
$23$	−134.420	−1.21863	−0.609317	+	0.792927i	$0.708556\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	217.123	1.39030	0.695150	−	0.718865i	$-0.255338\pi$
$29$	217.123	1.39030	0.695150	+	0.718865i	$0.255338\pi$
$30$	0	0
$31$	−129.375	−0.749564	−0.374782	−	0.927113i	$-0.622282\pi$
$31$	−129.375	−0.749564	−0.374782	+	0.927113i	$0.622282\pi$
$32$	0	0
$33$	124.688	0.657741
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−239.828	−1.06561	−0.532804	−	0.846239i	$-0.678862\pi$
$37$	−239.828	−1.06561	−0.532804	+	0.846239i	$0.678862\pi$
$38$	0	0
$39$	58.1679	0.238829
$40$	0	0
$41$	371.270	1.41421	0.707104	−	0.707109i	$-0.250001\pi$
$41$	371.270	1.41421	0.707104	+	0.707109i	$0.250001\pi$
$42$	0	0
$43$	476.442	1.68969	0.844845	−	0.535011i	$-0.179693\pi$
$43$	476.442	1.68969	0.844845	+	0.535011i	$0.179693\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	44.5275	0.138192	0.0690958	−	0.997610i	$-0.477989\pi$
$47$	44.5275	0.138192	0.0690958	+	0.997610i	$0.477989\pi$
$48$	0	0
$49$	708.900	2.06676
$50$	0	0
$51$	396.220	1.08788
$52$	0	0
$53$	14.7052	0.0381115	0.0190558	−	0.999818i	$-0.493934\pi$
$53$	14.7052	0.0381115	0.0190558	+	0.999818i	$0.493934\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	241.401	0.560953
$58$	0	0
$59$	−186.711	−0.411994	−0.205997	−	0.978553i	$-0.566044\pi$
$59$	−186.711	−0.411994	−0.205997	+	0.978553i	$0.566044\pi$
$60$	0	0
$61$	−609.872	−1.28010	−0.640050	−	0.768333i	$-0.721086\pi$
$61$	−609.872	−1.28010	−0.640050	+	0.768333i	$0.721086\pi$
$62$	0	0
$63$	291.897	0.583739
$64$	0	0
$65$	0	0
$66$	0	0
$67$	332.142	0.605636	0.302818	−	0.953048i	$-0.402073\pi$
$67$	332.142	0.605636	0.302818	+	0.953048i	$0.402073\pi$
$68$	0	0
$69$	403.261	0.703579
$70$	0	0
$71$	386.062	0.645312	0.322656	−	0.946516i	$-0.395424\pi$
$71$	386.062	0.645312	0.322656	+	0.946516i	$0.395424\pi$
$72$	0	0
$73$	1136.65	1.82239	0.911194	−	0.411977i	$-0.135162\pi$
$73$	1136.65	1.82239	0.911194	+	0.411977i	$0.135162\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1348.01	−1.99506
$78$	0	0
$79$	1313.21	1.87023	0.935114	−	0.354346i	$-0.115297\pi$
$79$	1313.21	1.87023	0.935114	+	0.354346i	$0.115297\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−430.813	−0.569733	−0.284867	−	0.958567i	$-0.591949\pi$
$83$	−430.813	−0.569733	−0.284867	+	0.958567i	$0.591949\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−651.368	−0.802690
$88$	0	0
$89$	−1525.80	−1.81724	−0.908621	−	0.417622i	$-0.862864\pi$
$89$	−1525.80	−1.81724	−0.908621	+	0.417622i	$0.862864\pi$
$90$	0	0
$91$	−628.853	−0.724415
$92$	0	0
$93$	388.126	0.432761
$94$	0	0
$95$	0	0
$96$	0	0
$97$	462.295	0.483906	0.241953	−	0.970288i	$-0.422212\pi$
$97$	462.295	0.483906	0.241953	+	0.970288i	$0.422212\pi$
$98$	0	0
$99$	−374.065	−0.379747
$100$	0	0
$101$	−555.771	−0.547537	−0.273769	−	0.961796i	$-0.588270\pi$
$101$	−555.771	−0.547537	−0.273769	+	0.961796i	$0.588270\pi$
$102$	0	0
$103$	861.117	0.823771	0.411885	−	0.911236i	$-0.364870\pi$
$103$	861.117	0.823771	0.411885	+	0.911236i	$0.364870\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	55.5126	0.0501551	0.0250776	−	0.999686i	$-0.492017\pi$
$107$	55.5126	0.0501551	0.0250776	+	0.999686i	$0.492017\pi$
$108$	0	0
$109$	−955.494	−0.839631	−0.419815	−	0.907610i	$-0.637905\pi$
$109$	−955.494	−0.839631	−0.419815	+	0.907610i	$0.637905\pi$
$110$	0	0
$111$	719.484	0.615229
$112$	0	0
$113$	1389.52	1.15677	0.578387	−	0.815763i	$-0.303683\pi$
$113$	1389.52	1.15677	0.578387	+	0.815763i	$0.303683\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−174.504	−0.137888
$118$	0	0
$119$	−4283.54	−3.29976
$120$	0	0
$121$	396.465	0.297870
$122$	0	0
$123$	−1113.81	−0.816494
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2465.40	1.72259	0.861295	−	0.508106i	$-0.169654\pi$
$127$	2465.40	1.72259	0.861295	+	0.508106i	$0.169654\pi$
$128$	0	0
$129$	−1429.33	−0.975543
$130$	0	0
$131$	858.874	0.572826	0.286413	−	0.958106i	$-0.407537\pi$
$131$	858.874	0.572826	0.286413	+	0.958106i	$0.407537\pi$
$132$	0	0
$133$	−2609.79	−1.70148
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1812.82	1.13051	0.565253	−	0.824918i	$-0.308779\pi$
$137$	1812.82	1.13051	0.565253	+	0.824918i	$0.308779\pi$
$138$	0	0
$139$	−409.704	−0.250004	−0.125002	−	0.992156i	$-0.539894\pi$
$139$	−409.704	−0.250004	−0.125002	+	0.992156i	$0.539894\pi$
$140$	0	0
$141$	−133.582	−0.0797849
$142$	0	0
$143$	805.873	0.471262
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2126.70	−1.19325
$148$	0	0
$149$	−326.225	−0.179365	−0.0896824	−	0.995970i	$-0.528585\pi$
$149$	−326.225	−0.179365	−0.0896824	+	0.995970i	$0.528585\pi$
$150$	0	0
$151$	744.061	0.400999	0.200499	−	0.979694i	$-0.435744\pi$
$151$	744.061	0.400999	0.200499	+	0.979694i	$0.435744\pi$
$152$	0	0
$153$	−1188.66	−0.628088
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−738.867	−0.375592	−0.187796	−	0.982208i	$-0.560134\pi$
$157$	−738.867	−0.375592	−0.187796	+	0.982208i	$0.560134\pi$
$158$	0	0
$159$	−44.1155	−0.0220037
$160$	0	0
$161$	−4359.66	−2.13409
$162$	0	0
$163$	3805.76	1.82878	0.914388	−	0.404839i	$-0.132672\pi$
$163$	3805.76	1.82878	0.914388	+	0.404839i	$0.132672\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−85.0480	−0.0394085	−0.0197042	−	0.999806i	$-0.506272\pi$
$167$	−85.0480	−0.0394085	−0.0197042	+	0.999806i	$0.506272\pi$
$168$	0	0
$169$	−1821.05	−0.828883
$170$	0	0
$171$	−724.202	−0.323866
$172$	0	0
$173$	−2036.93	−0.895171	−0.447585	−	0.894241i	$-0.647716\pi$
$173$	−2036.93	−0.895171	−0.447585	+	0.894241i	$0.647716\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	560.132	0.237865
$178$	0	0
$179$	770.892	0.321895	0.160947	−	0.986963i	$-0.448545\pi$
$179$	770.892	0.321895	0.160947	+	0.986963i	$0.448545\pi$
$180$	0	0
$181$	−1771.66	−0.727551	−0.363775	−	0.931487i	$-0.618512\pi$
$181$	−1771.66	−0.727551	−0.363775	+	0.931487i	$0.618512\pi$
$182$	0	0
$183$	1829.62	0.739066
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5489.34	2.14663
$188$	0	0
$189$	−875.691	−0.337022
$190$	0	0
$191$	2885.05	1.09296	0.546479	−	0.837473i	$-0.315968\pi$
$191$	2885.05	1.09296	0.546479	+	0.837473i	$0.315968\pi$
$192$	0	0
$193$	1467.60	0.547357	0.273678	−	0.961821i	$-0.411760\pi$
$193$	1467.60	0.547357	0.273678	+	0.961821i	$0.411760\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4211.06	1.52297	0.761487	−	0.648180i	$-0.224470\pi$
$197$	4211.06	1.52297	0.761487	+	0.648180i	$0.224470\pi$
$198$	0	0
$199$	3373.36	1.20166	0.600832	−	0.799375i	$-0.294836\pi$
$199$	3373.36	1.20166	0.600832	+	0.799375i	$0.294836\pi$
$200$	0	0
$201$	−996.427	−0.349664
$202$	0	0
$203$	7041.94	2.43472
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1209.78	−0.406211
$208$	0	0
$209$	3344.43	1.10689
$210$	0	0
$211$	31.9571	0.0104266	0.00521331	−	0.999986i	$-0.498341\pi$
$211$	31.9571	0.0104266	0.00521331	+	0.999986i	$0.498341\pi$
$212$	0	0
$213$	−1158.19	−0.372571
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4196.03	−1.31265
$218$	0	0
$219$	−3409.94	−1.05216
$220$	0	0
$221$	2560.81	0.779452
$222$	0	0
$223$	608.934	0.182857	0.0914287	−	0.995812i	$-0.470857\pi$
$223$	608.934	0.182857	0.0914287	+	0.995812i	$0.470857\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5737.79	−1.67767	−0.838833	−	0.544388i	$-0.816762\pi$
$227$	−5737.79	−1.67767	−0.838833	+	0.544388i	$0.816762\pi$
$228$	0	0
$229$	−523.868	−0.151171	−0.0755856	−	0.997139i	$-0.524083\pi$
$229$	−523.868	−0.151171	−0.0755856	+	0.997139i	$0.524083\pi$
$230$	0	0
$231$	4044.02	1.15185
$232$	0	0
$233$	−2583.73	−0.726462	−0.363231	−	0.931699i	$-0.618326\pi$
$233$	−2583.73	−0.726462	−0.363231	+	0.931699i	$0.618326\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3939.64	−1.07978
$238$	0	0
$239$	−1364.75	−0.369364	−0.184682	−	0.982798i	$-0.559126\pi$
$239$	−1364.75	−0.369364	−0.184682	+	0.982798i	$0.559126\pi$
$240$	0	0
$241$	−609.549	−0.162923	−0.0814616	−	0.996676i	$-0.525959\pi$
$241$	−609.549	−0.162923	−0.0814616	+	0.996676i	$0.525959\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1560.20	0.401915
$248$	0	0
$249$	1292.44	0.328936
$250$	0	0
$251$	2981.89	0.749862	0.374931	−	0.927053i	$-0.377666\pi$
$251$	2981.89	0.749862	0.374931	+	0.927053i	$0.377666\pi$
$252$	0	0
$253$	5586.88	1.38832
$254$	0	0
$255$	0	0
$256$	0	0
$257$	81.7584	0.0198442	0.00992208	−	0.999951i	$-0.496842\pi$
$257$	81.7584	0.0198442	0.00992208	+	0.999951i	$0.496842\pi$
$258$	0	0
$259$	−7778.34	−1.86611
$260$	0	0
$261$	1954.10	0.463433
$262$	0	0
$263$	−2777.11	−0.651118	−0.325559	−	0.945522i	$-0.605552\pi$
$263$	−2777.11	−0.651118	−0.325559	+	0.945522i	$0.605552\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4577.40	1.04919
$268$	0	0
$269$	−1858.78	−0.421307	−0.210654	−	0.977561i	$-0.567559\pi$
$269$	−1858.78	−0.421307	−0.210654	+	0.977561i	$0.567559\pi$
$270$	0	0
$271$	124.578	0.0279247	0.0139623	−	0.999903i	$-0.495556\pi$
$271$	124.578	0.0279247	0.0139623	+	0.999903i	$0.495556\pi$
$272$	0	0
$273$	1886.56	0.418241
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−570.149	−0.123671	−0.0618356	−	0.998086i	$-0.519695\pi$
$277$	−570.149	−0.123671	−0.0618356	+	0.998086i	$0.519695\pi$
$278$	0	0
$279$	−1164.38	−0.249855
$280$	0	0
$281$	2281.46	0.484344	0.242172	−	0.970233i	$-0.422140\pi$
$281$	2281.46	0.484344	0.242172	+	0.970233i	$0.422140\pi$
$282$	0	0
$283$	5726.88	1.20292	0.601462	−	0.798901i	$-0.294585\pi$
$283$	5726.88	1.20292	0.601462	+	0.798901i	$0.294585\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12041.4	2.47659
$288$	0	0
$289$	12530.4	2.55046
$290$	0	0
$291$	−1386.88	−0.279383
$292$	0	0
$293$	6061.02	1.20849	0.604247	−	0.796797i	$-0.293474\pi$
$293$	6061.02	1.20849	0.604247	+	0.796797i	$0.293474\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1122.20	0.219247
$298$	0	0
$299$	2606.32	0.504104
$300$	0	0
$301$	15452.4	2.95902
$302$	0	0
$303$	1667.31	0.316121
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3224.65	0.599480	0.299740	−	0.954021i	$-0.403100\pi$
$307$	3224.65	0.599480	0.299740	+	0.954021i	$0.403100\pi$
$308$	0	0
$309$	−2583.35	−0.475604
$310$	0	0
$311$	3698.99	0.674438	0.337219	−	0.941426i	$-0.390514\pi$
$311$	3698.99	0.674438	0.337219	+	0.941426i	$0.390514\pi$
$312$	0	0
$313$	−2624.15	−0.473885	−0.236942	−	0.971524i	$-0.576145\pi$
$313$	−2624.15	−0.473885	−0.236942	+	0.971524i	$0.576145\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8976.36	−1.59042	−0.795209	−	0.606335i	$-0.792639\pi$
$317$	−8976.36	−1.59042	−0.795209	+	0.606335i	$0.792639\pi$
$318$	0	0
$319$	−9024.22	−1.58389
$320$	0	0
$321$	−166.538	−0.0289571
$322$	0	0
$323$	10627.5	1.83075
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2866.48	0.484761
$328$	0	0
$329$	1444.16	0.242004
$330$	0	0
$331$	6008.69	0.997787	0.498893	−	0.866663i	$-0.333740\pi$
$331$	6008.69	0.997787	0.498893	+	0.866663i	$0.333740\pi$
$332$	0	0
$333$	−2158.45	−0.355202
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10418.4	1.68406	0.842029	−	0.539432i	$-0.181361\pi$
$337$	10418.4	1.68406	0.842029	+	0.539432i	$0.181361\pi$
$338$	0	0
$339$	−4168.57	−0.667863
$340$	0	0
$341$	5377.20	0.853934
$342$	0	0
$343$	11867.2	1.86814
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7282.16	−1.12659	−0.563295	−	0.826256i	$-0.690467\pi$
$347$	−7282.16	−1.12659	−0.563295	+	0.826256i	$0.690467\pi$
$348$	0	0
$349$	9693.32	1.48674	0.743369	−	0.668881i	$-0.233227\pi$
$349$	9693.32	1.48674	0.743369	+	0.668881i	$0.233227\pi$
$350$	0	0
$351$	523.511	0.0796096
$352$	0	0
$353$	−2310.21	−0.348329	−0.174164	−	0.984717i	$-0.555722\pi$
$353$	−2310.21	−0.348329	−0.174164	+	0.984717i	$0.555722\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	12850.6	1.90512
$358$	0	0
$359$	−5461.69	−0.802944	−0.401472	−	0.915871i	$-0.631501\pi$
$359$	−5461.69	−0.802944	−0.401472	+	0.915871i	$0.631501\pi$
$360$	0	0
$361$	−384.071	−0.0559952
$362$	0	0
$363$	−1189.39	−0.171975
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7719.14	−1.09792	−0.548959	−	0.835849i	$-0.684976\pi$
$367$	−7719.14	−1.09792	−0.548959	+	0.835849i	$0.684976\pi$
$368$	0	0
$369$	3341.43	0.471403
$370$	0	0
$371$	476.933	0.0667416
$372$	0	0
$373$	13893.4	1.92862	0.964310	−	0.264775i	$-0.0852978\pi$
$373$	13893.4	1.92862	0.964310	+	0.264775i	$0.0852978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4209.86	−0.575116
$378$	0	0
$379$	−11232.3	−1.52233	−0.761165	−	0.648558i	$-0.775372\pi$
$379$	−11232.3	−1.52233	−0.761165	+	0.648558i	$0.775372\pi$
$380$	0	0
$381$	−7396.20	−0.994538
$382$	0	0
$383$	9634.60	1.28539	0.642696	−	0.766121i	$-0.277816\pi$
$383$	9634.60	1.28539	0.642696	+	0.766121i	$0.277816\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4287.98	0.563230
$388$	0	0
$389$	−8090.48	−1.05451	−0.527254	−	0.849708i	$-0.676778\pi$
$389$	−8090.48	−1.05451	−0.527254	+	0.849708i	$0.676778\pi$
$390$	0	0
$391$	17753.4	2.29623
$392$	0	0
$393$	−2576.62	−0.330721
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9653.73	−1.22042	−0.610210	−	0.792240i	$-0.708915\pi$
$397$	−9653.73	−1.22042	−0.610210	+	0.792240i	$0.708915\pi$
$398$	0	0
$399$	7829.36	0.982351
$400$	0	0
$401$	1285.98	0.160147	0.0800734	−	0.996789i	$-0.474485\pi$
$401$	1285.98	0.160147	0.0800734	+	0.996789i	$0.474485\pi$
$402$	0	0
$403$	2508.50	0.310067
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9967.91	1.21398
$408$	0	0
$409$	−8627.56	−1.04304	−0.521522	−	0.853238i	$-0.674636\pi$
$409$	−8627.56	−1.04304	−0.521522	+	0.853238i	$0.674636\pi$
$410$	0	0
$411$	−5438.45	−0.652698
$412$	0	0
$413$	−6055.59	−0.721492
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1229.11	0.144340
$418$	0	0
$419$	5097.58	0.594352	0.297176	−	0.954823i	$-0.403955\pi$
$419$	5097.58	0.594352	0.297176	+	0.954823i	$0.403955\pi$
$420$	0	0
$421$	−12069.2	−1.39718	−0.698592	−	0.715520i	$-0.746190\pi$
$421$	−12069.2	−1.39718	−0.698592	+	0.715520i	$0.746190\pi$
$422$	0	0
$423$	400.747	0.0460638
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−19780.0	−2.24173
$428$	0	0
$429$	−2417.62	−0.272083
$430$	0	0
$431$	13667.3	1.52745	0.763723	−	0.645545i	$-0.223370\pi$
$431$	13667.3	1.52745	0.763723	+	0.645545i	$0.223370\pi$
$432$	0	0
$433$	−10004.1	−1.11032	−0.555160	−	0.831744i	$-0.687343\pi$
$433$	−10004.1	−1.11032	−0.555160	+	0.831744i	$0.687343\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10816.4	1.18402
$438$	0	0
$439$	4222.00	0.459009	0.229505	−	0.973308i	$-0.426289\pi$
$439$	4222.00	0.459009	0.229505	+	0.973308i	$0.426289\pi$
$440$	0	0
$441$	6380.10	0.688921
$442$	0	0
$443$	13188.3	1.41444	0.707220	−	0.706994i	$-0.249949\pi$
$443$	13188.3	1.41444	0.707220	+	0.706994i	$0.249949\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	978.674	0.103556
$448$	0	0
$449$	−147.427	−0.0154956	−0.00774780	−	0.999970i	$-0.502466\pi$
$449$	−147.427	−0.0154956	−0.00774780	+	0.999970i	$0.502466\pi$
$450$	0	0
$451$	−15431.0	−1.61112
$452$	0	0
$453$	−2232.18	−0.231517
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5076.79	−0.519654	−0.259827	−	0.965655i	$-0.583666\pi$
$457$	−5076.79	−0.519654	−0.259827	+	0.965655i	$0.583666\pi$
$458$	0	0
$459$	3565.98	0.362627
$460$	0	0
$461$	−5915.32	−0.597622	−0.298811	−	0.954312i	$-0.596590\pi$
$461$	−5915.32	−0.597622	−0.298811	+	0.954312i	$0.596590\pi$
$462$	0	0
$463$	−11149.7	−1.11916	−0.559581	−	0.828775i	$-0.689038\pi$
$463$	−11149.7	−1.11916	−0.559581	+	0.828775i	$0.689038\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9833.80	0.974420	0.487210	−	0.873285i	$-0.338015\pi$
$467$	9833.80	0.974420	0.487210	+	0.873285i	$0.338015\pi$
$468$	0	0
$469$	10772.4	1.06060
$470$	0	0
$471$	2216.60	0.216848
$472$	0	0
$473$	−19802.2	−1.92496
$474$	0	0
$475$	0	0
$476$	0	0
$477$	132.347	0.0127038
$478$	0	0
$479$	−8465.11	−0.807476	−0.403738	−	0.914875i	$-0.632289\pi$
$479$	−8465.11	−0.807476	−0.403738	+	0.914875i	$0.632289\pi$
$480$	0	0
$481$	4650.10	0.440803
$482$	0	0
$483$	13079.0	1.23212
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7987.08	0.743181	0.371590	−	0.928397i	$-0.378813\pi$
$487$	7987.08	0.743181	0.371590	+	0.928397i	$0.378813\pi$
$488$	0	0
$489$	−11417.3	−1.05584
$490$	0	0
$491$	−7316.92	−0.672521	−0.336261	−	0.941769i	$-0.609162\pi$
$491$	−7316.92	−0.672521	−0.336261	+	0.941769i	$0.609162\pi$
$492$	0	0
$493$	−28676.1	−2.61969
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12521.2	1.13008
$498$	0	0
$499$	−12522.5	−1.12342	−0.561709	−	0.827335i	$-0.689856\pi$
$499$	−12522.5	−1.12342	−0.561709	+	0.827335i	$0.689856\pi$
$500$	0	0
$501$	255.144	0.0227525
$502$	0	0
$503$	6666.15	0.590912	0.295456	−	0.955356i	$-0.404528\pi$
$503$	6666.15	0.590912	0.295456	+	0.955356i	$0.404528\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5463.16	0.478556
$508$	0	0
$509$	−20896.1	−1.81965	−0.909826	−	0.414991i	$-0.863785\pi$
$509$	−20896.1	−1.81965	−0.909826	+	0.414991i	$0.863785\pi$
$510$	0	0
$511$	36864.9	3.19140
$512$	0	0
$513$	2172.61	0.186984
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1850.69	−0.157433
$518$	0	0
$519$	6110.78	0.516827
$520$	0	0
$521$	2513.34	0.211346	0.105673	−	0.994401i	$-0.466300\pi$
$521$	2513.34	0.211346	0.105673	+	0.994401i	$0.466300\pi$
$522$	0	0
$523$	−1587.26	−0.132708	−0.0663540	−	0.997796i	$-0.521137\pi$
$523$	−1587.26	−0.132708	−0.0663540	+	0.997796i	$0.521137\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17087.0	1.41238
$528$	0	0
$529$	5901.84	0.485069
$530$	0	0
$531$	−1680.40	−0.137331
$532$	0	0
$533$	−7198.66	−0.585007
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2312.68	−0.185846
$538$	0	0
$539$	−29463.9	−2.35454
$540$	0	0
$541$	6560.86	0.521393	0.260696	−	0.965421i	$-0.416048\pi$
$541$	6560.86	0.521393	0.260696	+	0.965421i	$0.416048\pi$
$542$	0	0
$543$	5314.99	0.420052
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−34.5479	−0.00270048	−0.00135024	−	0.999999i	$-0.500430\pi$
$547$	−34.5479	−0.00270048	−0.00135024	+	0.999999i	$0.500430\pi$
$548$	0	0
$549$	−5488.85	−0.426700
$550$	0	0
$551$	−17471.2	−1.35081
$552$	0	0
$553$	42591.5	3.27518
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19210.6	1.46136	0.730682	−	0.682718i	$-0.239202\pi$
$557$	19210.6	1.46136	0.730682	+	0.682718i	$0.239202\pi$
$558$	0	0
$559$	−9237.87	−0.698963
$560$	0	0
$561$	−16468.0	−1.23936
$562$	0	0
$563$	9681.71	0.724752	0.362376	−	0.932032i	$-0.381966\pi$
$563$	9681.71	0.724752	0.362376	+	0.932032i	$0.381966\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2627.07	0.194580
$568$	0	0
$569$	−15583.4	−1.14813	−0.574067	−	0.818808i	$-0.694635\pi$
$569$	−15583.4	−1.14813	−0.574067	+	0.818808i	$0.694635\pi$
$570$	0	0
$571$	3486.63	0.255536	0.127768	−	0.991804i	$-0.459219\pi$
$571$	3486.63	0.255536	0.127768	+	0.991804i	$0.459219\pi$
$572$	0	0
$573$	−8655.15	−0.631019
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4525.47	−0.326512	−0.163256	−	0.986584i	$-0.552200\pi$
$577$	−4525.47	−0.326512	−0.163256	+	0.986584i	$0.552200\pi$
$578$	0	0
$579$	−4402.79	−0.316017
$580$	0	0
$581$	−13972.6	−0.997727
$582$	0	0
$583$	−611.188	−0.0434182
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13553.4	0.952995	0.476498	−	0.879176i	$-0.341906\pi$
$587$	13553.4	0.952995	0.476498	+	0.879176i	$0.341906\pi$
$588$	0	0
$589$	10410.4	0.728276
$590$	0	0
$591$	−12633.2	−0.879289
$592$	0	0
$593$	19931.6	1.38026	0.690130	−	0.723686i	$-0.257554\pi$
$593$	19931.6	1.38026	0.690130	+	0.723686i	$0.257554\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10120.1	−0.693781
$598$	0	0
$599$	13496.2	0.920601	0.460300	−	0.887763i	$-0.347742\pi$
$599$	13496.2	0.920601	0.460300	+	0.887763i	$0.347742\pi$
$600$	0	0
$601$	5435.31	0.368903	0.184452	−	0.982842i	$-0.440949\pi$
$601$	5435.31	0.368903	0.184452	+	0.982842i	$0.440949\pi$
$602$	0	0
$603$	2989.28	0.201879
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3262.62	0.218164	0.109082	−	0.994033i	$-0.465209\pi$
$607$	3262.62	0.218164	0.109082	+	0.994033i	$0.465209\pi$
$608$	0	0
$609$	−21125.8	−1.40568
$610$	0	0
$611$	−863.357	−0.0571648
$612$	0	0
$613$	14063.3	0.926611	0.463305	−	0.886199i	$-0.346663\pi$
$613$	14063.3	0.926611	0.463305	+	0.886199i	$0.346663\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−282.414	−0.0184272	−0.00921358	−	0.999958i	$-0.502933\pi$
$617$	−282.414	−0.0184272	−0.00921358	+	0.999958i	$0.502933\pi$
$618$	0	0
$619$	5009.10	0.325255	0.162627	−	0.986688i	$-0.448003\pi$
$619$	5009.10	0.325255	0.162627	+	0.986688i	$0.448003\pi$
$620$	0	0
$621$	3629.35	0.234526
$622$	0	0
$623$	−49486.3	−3.18239
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−10033.3	−0.639061
$628$	0	0
$629$	31674.9	2.00789
$630$	0	0
$631$	7151.52	0.451184	0.225592	−	0.974222i	$-0.427568\pi$
$631$	7151.52	0.451184	0.225592	+	0.974222i	$0.427568\pi$
$632$	0	0
$633$	−95.8713	−0.00601981
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−13745.1	−0.854945
$638$	0	0
$639$	3474.56	0.215104
$640$	0	0
$641$	−16142.0	−0.994653	−0.497326	−	0.867564i	$-0.665685\pi$
$641$	−16142.0	−0.994653	−0.497326	+	0.867564i	$0.665685\pi$
$642$	0	0
$643$	30122.8	1.84748	0.923739	−	0.383022i	$-0.125117\pi$
$643$	30122.8	1.84748	0.923739	+	0.383022i	$0.125117\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21556.4	−1.30985	−0.654923	−	0.755696i	$-0.727299\pi$
$647$	−21556.4	−1.30985	−0.654923	+	0.755696i	$0.727299\pi$
$648$	0	0
$649$	7760.22	0.469361
$650$	0	0
$651$	12588.1	0.757859
$652$	0	0
$653$	16249.1	0.973779	0.486890	−	0.873464i	$-0.338131\pi$
$653$	16249.1	0.973779	0.486890	+	0.873464i	$0.338131\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10229.8	0.607463
$658$	0	0
$659$	11872.5	0.701802	0.350901	−	0.936413i	$-0.385875\pi$
$659$	11872.5	0.701802	0.350901	+	0.936413i	$0.385875\pi$
$660$	0	0
$661$	−666.152	−0.0391986	−0.0195993	−	0.999808i	$-0.506239\pi$
$661$	−666.152	−0.0391986	−0.0195993	+	0.999808i	$0.506239\pi$
$662$	0	0
$663$	−7682.44	−0.450017
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−29185.7	−1.69427
$668$	0	0
$669$	−1826.80	−0.105573
$670$	0	0
$671$	25348.0	1.45834
$672$	0	0
$673$	13054.4	0.747711	0.373855	−	0.927487i	$-0.378036\pi$
$673$	13054.4	0.747711	0.373855	+	0.927487i	$0.378036\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	637.818	0.0362088	0.0181044	−	0.999836i	$-0.494237\pi$
$677$	637.818	0.0362088	0.0181044	+	0.999836i	$0.494237\pi$
$678$	0	0
$679$	14993.6	0.847425
$680$	0	0
$681$	17213.4	0.968601
$682$	0	0
$683$	16617.1	0.930947	0.465474	−	0.885062i	$-0.345884\pi$
$683$	16617.1	0.930947	0.465474	+	0.885062i	$0.345884\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1571.60	0.0872787
$688$	0	0
$689$	−285.123	−0.0157654
$690$	0	0
$691$	−10972.7	−0.604083	−0.302041	−	0.953295i	$-0.597668\pi$
$691$	−10972.7	−0.604083	−0.302041	+	0.953295i	$0.597668\pi$
$692$	0	0
$693$	−12132.1	−0.665020
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−49034.8	−2.66474
$698$	0	0
$699$	7751.18	0.419423
$700$	0	0
$701$	−9279.63	−0.499981	−0.249990	−	0.968248i	$-0.580427\pi$
$701$	−9279.63	−0.499981	−0.249990	+	0.968248i	$0.580427\pi$
$702$	0	0
$703$	19298.2	1.03534
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18025.3	−0.958857
$708$	0	0
$709$	24908.0	1.31938	0.659691	−	0.751537i	$-0.270687\pi$
$709$	24908.0	1.31938	0.659691	+	0.751537i	$0.270687\pi$
$710$	0	0
$711$	11818.9	0.623410
$712$	0	0
$713$	17390.7	0.913445
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4094.24	0.213253
$718$	0	0
$719$	10625.1	0.551110	0.275555	−	0.961285i	$-0.411138\pi$
$719$	10625.1	0.551110	0.275555	+	0.961285i	$0.411138\pi$
$720$	0	0
$721$	27928.6	1.44260
$722$	0	0
$723$	1828.65	0.0940638
$724$	0	0
$725$	0	0
$726$	0	0
$727$	33844.7	1.72659	0.863295	−	0.504700i	$-0.168397\pi$
$727$	33844.7	1.72659	0.863295	+	0.504700i	$0.168397\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−62925.3	−3.18382
$732$	0	0
$733$	16748.1	0.843937	0.421969	−	0.906610i	$-0.361339\pi$
$733$	16748.1	0.843937	0.421969	+	0.906610i	$0.361339\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13804.8	−0.689966
$738$	0	0
$739$	26797.0	1.33389	0.666943	−	0.745109i	$-0.267602\pi$
$739$	26797.0	1.33389	0.666943	+	0.745109i	$0.267602\pi$
$740$	0	0
$741$	−4680.59	−0.232046
$742$	0	0
$743$	1006.34	0.0496889	0.0248445	−	0.999691i	$-0.492091\pi$
$743$	1006.34	0.0496889	0.0248445	+	0.999691i	$0.492091\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3877.32	−0.189911
$748$	0	0
$749$	1800.44	0.0878326
$750$	0	0
$751$	−24409.4	−1.18603	−0.593016	−	0.805190i	$-0.702063\pi$
$751$	−24409.4	−1.18603	−0.593016	+	0.805190i	$0.702063\pi$
$752$	0	0
$753$	−8945.68	−0.432933
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−548.936	−0.0263559	−0.0131780	−	0.999913i	$-0.504195\pi$
$757$	−548.936	−0.0263559	−0.0131780	+	0.999913i	$0.504195\pi$
$758$	0	0
$759$	−16760.7	−0.801546
$760$	0	0
$761$	−5262.50	−0.250677	−0.125339	−	0.992114i	$-0.540002\pi$
$761$	−5262.50	−0.250677	−0.125339	+	0.992114i	$0.540002\pi$
$762$	0	0
$763$	−30989.5	−1.47038
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3620.19	0.170427
$768$	0	0
$769$	21244.0	0.996202	0.498101	−	0.867119i	$-0.334031\pi$
$769$	21244.0	0.996202	0.498101	+	0.867119i	$0.334031\pi$
$770$	0	0
$771$	−245.275	−0.0114570
$772$	0	0
$773$	8223.96	0.382659	0.191329	−	0.981526i	$-0.438720\pi$
$773$	8223.96	0.382659	0.191329	+	0.981526i	$0.438720\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	23335.0	1.07740
$778$	0	0
$779$	−29874.9	−1.37404
$780$	0	0
$781$	−16045.8	−0.735166
$782$	0	0
$783$	−5862.31	−0.267563
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9000.26	−0.407655	−0.203827	−	0.979007i	$-0.565338\pi$
$787$	−9000.26	−0.407655	−0.203827	+	0.979007i	$0.565338\pi$
$788$	0	0
$789$	8331.34	0.375923
$790$	0	0
$791$	45066.4	2.02576
$792$	0	0
$793$	11825.0	0.529531
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31608.0	−1.40478	−0.702391	−	0.711791i	$-0.747884\pi$
$797$	−31608.0	−1.40478	−0.702391	+	0.711791i	$0.747884\pi$
$798$	0	0
$799$	−5880.90	−0.260390
$800$	0	0
$801$	−13732.2	−0.605747
$802$	0	0
$803$	−47242.2	−2.07614
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5576.33	0.243242
$808$	0	0
$809$	−25647.4	−1.11460	−0.557301	−	0.830310i	$-0.688163\pi$
$809$	−25647.4	−1.11460	−0.557301	+	0.830310i	$0.688163\pi$
$810$	0	0
$811$	−44184.0	−1.91308	−0.956541	−	0.291597i	$-0.905813\pi$
$811$	−44184.0	−1.91308	−0.956541	+	0.291597i	$0.905813\pi$
$812$	0	0
$813$	−373.735	−0.0161223
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−38337.8	−1.64170
$818$	0	0
$819$	−5659.68	−0.241472
$820$	0	0
$821$	30634.2	1.30224	0.651122	−	0.758973i	$-0.274299\pi$
$821$	30634.2	1.30224	0.651122	+	0.758973i	$0.274299\pi$
$822$	0	0
$823$	−38535.0	−1.63213	−0.816067	−	0.577957i	$-0.803850\pi$
$823$	−38535.0	−1.63213	−0.816067	+	0.577957i	$0.803850\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13462.0	−0.566045	−0.283022	−	0.959113i	$-0.591337\pi$
$827$	−13462.0	−0.566045	−0.283022	+	0.959113i	$0.591337\pi$
$828$	0	0
$829$	28650.0	1.20031	0.600155	−	0.799884i	$-0.295106\pi$
$829$	28650.0	1.20031	0.600155	+	0.799884i	$0.295106\pi$
$830$	0	0
$831$	1710.45	0.0714016
$832$	0	0
$833$	−93626.9	−3.89433
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3493.13	0.144254
$838$	0	0
$839$	31639.5	1.30193	0.650964	−	0.759108i	$-0.274365\pi$
$839$	31639.5	1.30193	0.650964	+	0.759108i	$0.274365\pi$
$840$	0	0
$841$	22753.3	0.932932
$842$	0	0
$843$	−6844.39	−0.279636
$844$	0	0
$845$	0	0
$846$	0	0
$847$	12858.5	0.521635
$848$	0	0
$849$	−17180.6	−0.694508
$850$	0	0
$851$	32237.8	1.29859
$852$	0	0
$853$	31676.5	1.27149	0.635745	−	0.771899i	$-0.280693\pi$
$853$	31676.5	1.27149	0.635745	+	0.771899i	$0.280693\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7503.48	0.299083	0.149542	−	0.988755i	$-0.452220\pi$
$857$	7503.48	0.299083	0.149542	+	0.988755i	$0.452220\pi$
$858$	0	0
$859$	−4817.58	−0.191355	−0.0956773	−	0.995412i	$-0.530502\pi$
$859$	−4817.58	−0.191355	−0.0956773	+	0.995412i	$0.530502\pi$
$860$	0	0
$861$	−36124.2	−1.42986
$862$	0	0
$863$	29704.2	1.17166	0.585829	−	0.810434i	$-0.300769\pi$
$863$	29704.2	1.17166	0.585829	+	0.810434i	$0.300769\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−37591.2	−1.47251
$868$	0	0
$869$	−54580.8	−2.13064
$870$	0	0
$871$	−6440.01	−0.250530
$872$	0	0
$873$	4160.65	0.161302
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27429.4	1.05613	0.528065	−	0.849204i	$-0.322918\pi$
$877$	27429.4	1.05613	0.528065	+	0.849204i	$0.322918\pi$
$878$	0	0
$879$	−18183.1	−0.697724
$880$	0	0
$881$	−42751.1	−1.63487	−0.817434	−	0.576022i	$-0.804604\pi$
$881$	−42751.1	−1.63487	−0.817434	+	0.576022i	$0.804604\pi$
$882$	0	0
$883$	9697.41	0.369585	0.184793	−	0.982778i	$-0.440839\pi$
$883$	9697.41	0.369585	0.184793	+	0.982778i	$0.440839\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19990.8	−0.756736	−0.378368	−	0.925655i	$-0.623515\pi$
$887$	−19990.8	−0.756736	−0.378368	+	0.925655i	$0.623515\pi$
$888$	0	0
$889$	79960.3	3.01663
$890$	0	0
$891$	−3366.59	−0.126582
$892$	0	0
$893$	−3582.99	−0.134267
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7818.95	−0.291045
$898$	0	0
$899$	−28090.3	−1.04212
$900$	0	0
$901$	−1942.16	−0.0718122
$902$	0	0
$903$	−46357.3	−1.70839
$904$	0	0
$905$	0	0
$906$	0	0
$907$	47525.0	1.73985	0.869923	−	0.493187i	$-0.164168\pi$
$907$	47525.0	1.73985	0.869923	+	0.493187i	$0.164168\pi$
$908$	0	0
$909$	−5001.94	−0.182512
$910$	0	0
$911$	20992.8	0.763470	0.381735	−	0.924272i	$-0.375327\pi$
$911$	20992.8	0.763470	0.381735	+	0.924272i	$0.375327\pi$
$912$	0	0
$913$	17905.8	0.649063
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27855.9	1.00314
$918$	0	0
$919$	8141.30	0.292227	0.146113	−	0.989268i	$-0.453324\pi$
$919$	8141.30	0.292227	0.146113	+	0.989268i	$0.453324\pi$
$920$	0	0
$921$	−9673.95	−0.346110
$922$	0	0
$923$	−7485.48	−0.266942
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7750.05	0.274590
$928$	0	0
$929$	−5796.81	−0.204723	−0.102361	−	0.994747i	$-0.532640\pi$
$929$	−5796.81	−0.204723	−0.102361	+	0.994747i	$0.532640\pi$
$930$	0	0
$931$	−57043.0	−2.00807
$932$	0	0
$933$	−11097.0	−0.389387
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42608.3	1.48554	0.742771	−	0.669545i	$-0.233511\pi$
$937$	42608.3	1.48554	0.742771	+	0.669545i	$0.233511\pi$
$938$	0	0
$939$	7872.46	0.273597
$940$	0	0
$941$	−4263.36	−0.147696	−0.0738478	−	0.997270i	$-0.523528\pi$
$941$	−4263.36	−0.147696	−0.0738478	+	0.997270i	$0.523528\pi$
$942$	0	0
$943$	−49906.2	−1.72340
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39212.3	−1.34554	−0.672771	−	0.739851i	$-0.734896\pi$
$947$	−39212.3	−1.34554	−0.672771	+	0.739851i	$0.734896\pi$
$948$	0	0
$949$	−22038.8	−0.753856
$950$	0	0
$951$	26929.1	0.918228
$952$	0	0
$953$	−34874.5	−1.18541	−0.592705	−	0.805419i	$-0.701940\pi$
$953$	−34874.5	−1.18541	−0.592705	+	0.805419i	$0.701940\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	27072.7	0.914457
$958$	0	0
$959$	58795.1	1.97976
$960$	0	0
$961$	−13053.0	−0.438153
$962$	0	0
$963$	499.613	0.0167184
$964$	0	0
$965$	0	0
$966$	0	0
$967$	35343.7	1.17536	0.587682	−	0.809092i	$-0.300040\pi$
$967$	35343.7	1.17536	0.587682	+	0.809092i	$0.300040\pi$
$968$	0	0
$969$	−31882.6	−1.05698
$970$	0	0
$971$	−47231.9	−1.56101	−0.780507	−	0.625147i	$-0.785039\pi$
$971$	−47231.9	−1.56101	−0.780507	+	0.625147i	$0.785039\pi$
$972$	0	0
$973$	−13287.9	−0.437812
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18439.1	−0.603807	−0.301903	−	0.953339i	$-0.597622\pi$
$977$	−18439.1	−0.603807	−0.301903	+	0.953339i	$0.597622\pi$
$978$	0	0
$979$	63416.5	2.07028
$980$	0	0
$981$	−8599.45	−0.279877
$982$	0	0
$983$	−30278.7	−0.982443	−0.491222	−	0.871035i	$-0.663449\pi$
$983$	−30278.7	−0.982443	−0.491222	+	0.871035i	$0.663449\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4332.48	−0.139721
$988$	0	0
$989$	−64043.5	−2.05911
$990$	0	0
$991$	−46613.9	−1.49419	−0.747094	−	0.664718i	$-0.768552\pi$
$991$	−46613.9	−1.49419	−0.747094	+	0.664718i	$0.768552\pi$
$992$	0	0
$993$	−18026.1	−0.576072
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−44805.8	−1.42328	−0.711642	−	0.702543i	$-0.752048\pi$
$997$	−44805.8	−1.42328	−0.711642	+	0.702543i	$0.752048\pi$
$998$	0	0
$999$	6475.35	0.205076

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bm.1.3	✓	3
4.3	odd	2		2400.4.a.bp.1.1	yes	3
5.4	even	2		2400.4.a.bo.1.1	yes	3
20.19	odd	2		2400.4.a.bn.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.4.a.bm.1.3	✓	3	1.1	even	1	trivial
2400.4.a.bn.1.3	yes	3	20.19	odd	2
2400.4.a.bo.1.1	yes	3	5.4	even	2
2400.4.a.bp.1.1	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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +32.4330 q^{7} +9.00000 q^{9} -41.5628 q^{11} -19.3893 q^{13} -132.073 q^{17} -80.4669 q^{19} -97.2990 q^{21} -134.420 q^{23} -27.0000 q^{27} +217.123 q^{29} -129.375 q^{31} +124.688 q^{33} -239.828 q^{37} +58.1679 q^{39} +371.270 q^{41} +476.442 q^{43} +44.5275 q^{47} +708.900 q^{49} +396.220 q^{51} +14.7052 q^{53} +241.401 q^{57} -186.711 q^{59} -609.872 q^{61} +291.897 q^{63} +332.142 q^{67} +403.261 q^{69} +386.062 q^{71} +1136.65 q^{73} -1348.01 q^{77} +1313.21 q^{79} +81.0000 q^{81} -430.813 q^{83} -651.368 q^{87} -1525.80 q^{89} -628.853 q^{91} +388.126 q^{93} +462.295 q^{97} -374.065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 19 q^{7} + 27 q^{9} - 48 q^{11} - 63 q^{13} - 16 q^{17} - 21 q^{19} - 57 q^{21} + 140 q^{23} - 81 q^{27} + 92 q^{29} - 179 q^{31} + 144 q^{33} - 550 q^{37} + 189 q^{39} - 206 q^{41} + 279 q^{43}+ \cdots - 432 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 + 19 * q^7 + 27 * q^9 - 48 * q^11 - 63 * q^13 - 16 * q^17 - 21 * q^19 - 57 * q^21 + 140 * q^23 - 81 * q^27 + 92 * q^29 - 179 * q^31 + 144 * q^33 - 550 * q^37 + 189 * q^39 - 206 * q^41 + 279 * q^43 + 450 * q^47 + 446 * q^49 + 48 * q^51 + 434 * q^53 + 63 * q^57 - 854 * q^59 - 1103 * q^61 + 171 * q^63 + 555 * q^67 - 420 * q^69 + 970 * q^71 + 722 * q^73 - 1184 * q^77 + 856 * q^79 + 243 * q^81 + 222 * q^83 - 276 * q^87 - 1216 * q^89 + 1025 * q^91 + 537 * q^93 + 89 * q^97 - 432 * q^99

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