LMFDB - Embedded newform 2400.4.a.bn.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:	3.3.28724.1
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			1.1
Root			$-4.12921$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -19.6176 q^{7} +9.00000 q^{9} +7.89924 q^{11} +74.5505 q^{13} +0.529863 q^{17} +89.7988 q^{19} +58.8528 q^{21} +174.773 q^{23} -27.0000 q^{27} -275.941 q^{29} +267.417 q^{31} -23.6977 q^{33} -11.3700 q^{37} -223.652 q^{39} -384.881 q^{41} -28.5115 q^{43} +289.504 q^{47} +41.8506 q^{49} -1.58959 q^{51} -256.571 q^{53} -269.396 q^{57} +174.268 q^{59} -732.913 q^{61} -176.558 q^{63} -498.110 q^{67} -524.318 q^{69} -763.136 q^{71} +927.268 q^{73} -154.964 q^{77} +217.199 q^{79} +81.0000 q^{81} +21.8132 q^{83} +827.823 q^{87} -191.701 q^{89} -1462.50 q^{91} -802.250 q^{93} +885.353 q^{97} +71.0931 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−19.6176	−1.05925	−0.529626	−	0.848232i	$-0.677668\pi$
$7$	−19.6176	−1.05925	−0.529626	+	0.848232i	$0.677668\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	7.89924	0.216519	0.108260	−	0.994123i	$-0.465472\pi$
$11$	7.89924	0.216519	0.108260	+	0.994123i	$0.465472\pi$
$12$	0	0
$13$	74.5505	1.59051	0.795254	−	0.606277i	$-0.207338\pi$
$13$	74.5505	1.59051	0.795254	+	0.606277i	$0.207338\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.529863	0.00755946	0.00377973	−	0.999993i	$-0.498797\pi$
$17$	0.529863	0.00755946	0.00377973	+	0.999993i	$0.498797\pi$
$18$	0	0
$19$	89.7988	1.08428	0.542138	−	0.840289i	$-0.317615\pi$
$19$	89.7988	1.08428	0.542138	+	0.840289i	$0.317615\pi$
$20$	0	0
$21$	58.8528	0.611559
$22$	0	0
$23$	174.773	1.58446	0.792231	−	0.610221i	$-0.208919\pi$
$23$	174.773	1.58446	0.792231	+	0.610221i	$0.208919\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−275.941	−1.76693	−0.883465	−	0.468497i	$-0.844796\pi$
$29$	−275.941	−1.76693	−0.883465	+	0.468497i	$0.844796\pi$
$30$	0	0
$31$	267.417	1.54934	0.774669	−	0.632367i	$-0.217917\pi$
$31$	267.417	1.54934	0.774669	+	0.632367i	$0.217917\pi$
$32$	0	0
$33$	−23.6977	−0.125007
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.3700	−0.0505192	−0.0252596	−	0.999681i	$-0.508041\pi$
$37$	−11.3700	−0.0505192	−0.0252596	+	0.999681i	$0.508041\pi$
$38$	0	0
$39$	−223.652	−0.918280
$40$	0	0
$41$	−384.881	−1.46606	−0.733029	−	0.680197i	$-0.761894\pi$
$41$	−384.881	−1.46606	−0.733029	+	0.680197i	$0.761894\pi$
$42$	0	0
$43$	−28.5115	−0.101115	−0.0505577	−	0.998721i	$-0.516100\pi$
$43$	−28.5115	−0.101115	−0.0505577	+	0.998721i	$0.516100\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	289.504	0.898479	0.449240	−	0.893411i	$-0.351695\pi$
$47$	289.504	0.898479	0.449240	+	0.893411i	$0.351695\pi$
$48$	0	0
$49$	41.8506	0.122013
$50$	0	0
$51$	−1.58959	−0.00436445
$52$	0	0
$53$	−256.571	−0.664958	−0.332479	−	0.943111i	$-0.607885\pi$
$53$	−256.571	−0.664958	−0.332479	+	0.943111i	$0.607885\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−269.396	−0.626007
$58$	0	0
$59$	174.268	0.384539	0.192270	−	0.981342i	$-0.438415\pi$
$59$	174.268	0.384539	0.192270	+	0.981342i	$0.438415\pi$
$60$	0	0
$61$	−732.913	−1.53836	−0.769180	−	0.639032i	$-0.779335\pi$
$61$	−732.913	−1.53836	−0.769180	+	0.639032i	$0.779335\pi$
$62$	0	0
$63$	−176.558	−0.353084
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−498.110	−0.908266	−0.454133	−	0.890934i	$-0.650051\pi$
$67$	−498.110	−0.908266	−0.454133	+	0.890934i	$0.650051\pi$
$68$	0	0
$69$	−524.318	−0.914790
$70$	0	0
$71$	−763.136	−1.27560	−0.637800	−	0.770202i	$-0.720155\pi$
$71$	−763.136	−1.27560	−0.637800	+	0.770202i	$0.720155\pi$
$72$	0	0
$73$	927.268	1.48669	0.743346	−	0.668907i	$-0.233238\pi$
$73$	927.268	1.48669	0.743346	+	0.668907i	$0.233238\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−154.964	−0.229348
$78$	0	0
$79$	217.199	0.309327	0.154663	−	0.987967i	$-0.450571\pi$
$79$	217.199	0.309327	0.154663	+	0.987967i	$0.450571\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	21.8132	0.0288471	0.0144235	−	0.999896i	$-0.495409\pi$
$83$	21.8132	0.0288471	0.0144235	+	0.999896i	$0.495409\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	827.823	1.02014
$88$	0	0
$89$	−191.701	−0.228318	−0.114159	−	0.993463i	$-0.536417\pi$
$89$	−191.701	−0.228318	−0.114159	+	0.993463i	$0.536417\pi$
$90$	0	0
$91$	−1462.50	−1.68475
$92$	0	0
$93$	−802.250	−0.894510
$94$	0	0
$95$	0	0
$96$	0	0
$97$	885.353	0.926742	0.463371	−	0.886164i	$-0.346640\pi$
$97$	885.353	0.926742	0.463371	+	0.886164i	$0.346640\pi$
$98$	0	0
$99$	71.0931	0.0721730
$100$	0	0
$101$	1432.59	1.41137	0.705684	−	0.708527i	$-0.250640\pi$
$101$	1432.59	1.41137	0.705684	+	0.708527i	$0.250640\pi$
$102$	0	0
$103$	622.362	0.595370	0.297685	−	0.954664i	$-0.403785\pi$
$103$	622.362	0.595370	0.297685	+	0.954664i	$0.403785\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1204.84	1.08857	0.544283	−	0.838902i	$-0.316802\pi$
$107$	1204.84	1.08857	0.544283	+	0.838902i	$0.316802\pi$
$108$	0	0
$109$	109.573	0.0962866	0.0481433	−	0.998840i	$-0.484670\pi$
$109$	109.573	0.0962866	0.0481433	+	0.998840i	$0.484670\pi$
$110$	0	0
$111$	34.1099	0.0291673
$112$	0	0
$113$	−924.455	−0.769606	−0.384803	−	0.922999i	$-0.625731\pi$
$113$	−924.455	−0.769606	−0.384803	+	0.922999i	$0.625731\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	670.955	0.530169
$118$	0	0
$119$	−10.3947	−0.00800736
$120$	0	0
$121$	−1268.60	−0.953120
$122$	0	0
$123$	1154.64	0.846429
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1251.14	−0.874178	−0.437089	−	0.899418i	$-0.643991\pi$
$127$	−1251.14	−0.874178	−0.437089	+	0.899418i	$0.643991\pi$
$128$	0	0
$129$	85.5344	0.0583790
$130$	0	0
$131$	2391.14	1.59477	0.797387	−	0.603469i	$-0.206215\pi$
$131$	2391.14	1.59477	0.797387	+	0.603469i	$0.206215\pi$
$132$	0	0
$133$	−1761.64	−1.14852
$134$	0	0
$135$	0	0
$136$	0	0
$137$	568.517	0.354538	0.177269	−	0.984162i	$-0.443274\pi$
$137$	568.517	0.354538	0.177269	+	0.984162i	$0.443274\pi$
$138$	0	0
$139$	1615.92	0.986049	0.493024	−	0.870015i	$-0.335891\pi$
$139$	1615.92	0.986049	0.493024	+	0.870015i	$0.335891\pi$
$140$	0	0
$141$	−868.512	−0.518737
$142$	0	0
$143$	588.892	0.344375
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−125.552	−0.0704444
$148$	0	0
$149$	36.9552	0.0203187	0.0101593	−	0.999948i	$-0.496766\pi$
$149$	36.9552	0.0203187	0.0101593	+	0.999948i	$0.496766\pi$
$150$	0	0
$151$	−2566.94	−1.38341	−0.691704	−	0.722181i	$-0.743140\pi$
$151$	−2566.94	−1.38341	−0.691704	+	0.722181i	$0.743140\pi$
$152$	0	0
$153$	4.76877	0.00251982
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2649.19	1.34668	0.673338	−	0.739335i	$-0.264860\pi$
$157$	2649.19	1.34668	0.673338	+	0.739335i	$0.264860\pi$
$158$	0	0
$159$	769.714	0.383914
$160$	0	0
$161$	−3428.62	−1.67834
$162$	0	0
$163$	1787.40	0.858897	0.429448	−	0.903091i	$-0.358708\pi$
$163$	1787.40	0.858897	0.429448	+	0.903091i	$0.358708\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1223.60	−0.566974	−0.283487	−	0.958976i	$-0.591491\pi$
$167$	−1223.60	−0.566974	−0.283487	+	0.958976i	$0.591491\pi$
$168$	0	0
$169$	3360.78	1.52971
$170$	0	0
$171$	808.189	0.361425
$172$	0	0
$173$	778.379	0.342075	0.171038	−	0.985264i	$-0.445288\pi$
$173$	778.379	0.342075	0.171038	+	0.985264i	$0.445288\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−522.805	−0.222014
$178$	0	0
$179$	1316.25	0.549614	0.274807	−	0.961499i	$-0.411386\pi$
$179$	1316.25	0.549614	0.274807	+	0.961499i	$0.411386\pi$
$180$	0	0
$181$	1822.39	0.748381	0.374190	−	0.927352i	$-0.377921\pi$
$181$	1822.39	0.748381	0.374190	+	0.927352i	$0.377921\pi$
$182$	0	0
$183$	2198.74	0.888172
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.18552	0.00163677
$188$	0	0
$189$	529.675	0.203853
$190$	0	0
$191$	−1417.75	−0.537092	−0.268546	−	0.963267i	$-0.586543\pi$
$191$	−1417.75	−0.537092	−0.268546	+	0.963267i	$0.586543\pi$
$192$	0	0
$193$	−2798.24	−1.04364	−0.521818	−	0.853057i	$-0.674746\pi$
$193$	−2798.24	−1.04364	−0.521818	+	0.853057i	$0.674746\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	690.819	0.249842	0.124921	−	0.992167i	$-0.460132\pi$
$197$	690.819	0.249842	0.124921	+	0.992167i	$0.460132\pi$
$198$	0	0
$199$	5000.57	1.78131	0.890656	−	0.454679i	$-0.150246\pi$
$199$	5000.57	1.78131	0.890656	+	0.454679i	$0.150246\pi$
$200$	0	0
$201$	1494.33	0.524388
$202$	0	0
$203$	5413.31	1.87162
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1572.95	0.528154
$208$	0	0
$209$	709.342	0.234766
$210$	0	0
$211$	−1102.07	−0.359572	−0.179786	−	0.983706i	$-0.557541\pi$
$211$	−1102.07	−0.359572	−0.179786	+	0.983706i	$0.557541\pi$
$212$	0	0
$213$	2289.41	0.736467
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5246.08	−1.64114
$218$	0	0
$219$	−2781.80	−0.858342
$220$	0	0
$221$	39.5016	0.0120234
$222$	0	0
$223$	−1638.23	−0.491946	−0.245973	−	0.969277i	$-0.579107\pi$
$223$	−1638.23	−0.491946	−0.245973	+	0.969277i	$0.579107\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3682.55	1.07674	0.538368	−	0.842710i	$-0.319041\pi$
$227$	3682.55	1.07674	0.538368	+	0.842710i	$0.319041\pi$
$228$	0	0
$229$	−3932.80	−1.13488	−0.567439	−	0.823415i	$-0.692066\pi$
$229$	−3932.80	−1.13488	−0.567439	+	0.823415i	$0.692066\pi$
$230$	0	0
$231$	464.892	0.132414
$232$	0	0
$233$	−830.393	−0.233480	−0.116740	−	0.993163i	$-0.537244\pi$
$233$	−830.393	−0.233480	−0.116740	+	0.993163i	$0.537244\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−651.597	−0.178590
$238$	0	0
$239$	−1673.62	−0.452959	−0.226479	−	0.974016i	$-0.572722\pi$
$239$	−1673.62	−0.452959	−0.226479	+	0.974016i	$0.572722\pi$
$240$	0	0
$241$	4585.84	1.22572	0.612862	−	0.790190i	$-0.290018\pi$
$241$	4585.84	1.22572	0.612862	+	0.790190i	$0.290018\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6694.55	1.72455
$248$	0	0
$249$	−65.4395	−0.0166549
$250$	0	0
$251$	−6139.35	−1.54387	−0.771937	−	0.635699i	$-0.780712\pi$
$251$	−6139.35	−1.54387	−0.771937	+	0.635699i	$0.780712\pi$
$252$	0	0
$253$	1380.57	0.343066
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2218.01	0.538349	0.269174	−	0.963091i	$-0.413249\pi$
$257$	2218.01	0.538349	0.269174	+	0.963091i	$0.413249\pi$
$258$	0	0
$259$	223.051	0.0535125
$260$	0	0
$261$	−2483.47	−0.588977
$262$	0	0
$263$	7479.49	1.75363	0.876815	−	0.480827i	$-0.159664\pi$
$263$	7479.49	1.75363	0.876815	+	0.480827i	$0.159664\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	575.103	0.131819
$268$	0	0
$269$	−294.700	−0.0667962	−0.0333981	−	0.999442i	$-0.510633\pi$
$269$	−294.700	−0.0667962	−0.0333981	+	0.999442i	$0.510633\pi$
$270$	0	0
$271$	−6961.39	−1.56042	−0.780211	−	0.625517i	$-0.784888\pi$
$271$	−6961.39	−1.56042	−0.780211	+	0.625517i	$0.784888\pi$
$272$	0	0
$273$	4387.51	0.972689
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2035.72	0.441570	0.220785	−	0.975323i	$-0.429138\pi$
$277$	2035.72	0.441570	0.220785	+	0.975323i	$0.429138\pi$
$278$	0	0
$279$	2406.75	0.516446
$280$	0	0
$281$	−1058.00	−0.224609	−0.112305	−	0.993674i	$-0.535823\pi$
$281$	−1058.00	−0.224609	−0.112305	+	0.993674i	$0.535823\pi$
$282$	0	0
$283$	9252.98	1.94358	0.971789	−	0.235853i	$-0.0757883\pi$
$283$	9252.98	1.94358	0.971789	+	0.235853i	$0.0757883\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7550.45	1.55292
$288$	0	0
$289$	−4912.72	−0.999943
$290$	0	0
$291$	−2656.06	−0.535055
$292$	0	0
$293$	7009.84	1.39768	0.698838	−	0.715280i	$-0.253701\pi$
$293$	7009.84	1.39768	0.698838	+	0.715280i	$0.253701\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−213.279	−0.0416691
$298$	0	0
$299$	13029.4	2.52010
$300$	0	0
$301$	559.327	0.107107
$302$	0	0
$303$	−4297.77	−0.814853
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7972.73	−1.48218	−0.741088	−	0.671408i	$-0.765690\pi$
$307$	−7972.73	−1.48218	−0.741088	+	0.671408i	$0.765690\pi$
$308$	0	0
$309$	−1867.09	−0.343737
$310$	0	0
$311$	−8775.06	−1.59996	−0.799981	−	0.600025i	$-0.795157\pi$
$311$	−8775.06	−1.59996	−0.799981	+	0.600025i	$0.795157\pi$
$312$	0	0
$313$	−7595.33	−1.37161	−0.685804	−	0.727786i	$-0.740549\pi$
$313$	−7595.33	−1.37161	−0.685804	+	0.727786i	$0.740549\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5442.28	−0.964256	−0.482128	−	0.876101i	$-0.660136\pi$
$317$	−5442.28	−0.964256	−0.482128	+	0.876101i	$0.660136\pi$
$318$	0	0
$319$	−2179.72	−0.382574
$320$	0	0
$321$	−3614.53	−0.628484
$322$	0	0
$323$	47.5811	0.00819654
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−328.720	−0.0555911
$328$	0	0
$329$	−5679.38	−0.951715
$330$	0	0
$331$	6541.66	1.08629	0.543146	−	0.839639i	$-0.317233\pi$
$331$	6541.66	1.08629	0.543146	+	0.839639i	$0.317233\pi$
$332$	0	0
$333$	−102.330	−0.0168397
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−512.773	−0.0828859	−0.0414429	−	0.999141i	$-0.513195\pi$
$337$	−512.773	−0.0828859	−0.0414429	+	0.999141i	$0.513195\pi$
$338$	0	0
$339$	2773.37	0.444332
$340$	0	0
$341$	2112.39	0.335461
$342$	0	0
$343$	5907.83	0.930009
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5367.42	−0.830370	−0.415185	−	0.909737i	$-0.636283\pi$
$347$	−5367.42	−0.830370	−0.415185	+	0.909737i	$0.636283\pi$
$348$	0	0
$349$	12534.3	1.92249	0.961243	−	0.275702i	$-0.0889103\pi$
$349$	12534.3	1.92249	0.961243	+	0.275702i	$0.0889103\pi$
$350$	0	0
$351$	−2012.86	−0.306093
$352$	0	0
$353$	7701.70	1.16125	0.580623	−	0.814172i	$-0.302809\pi$
$353$	7701.70	1.16125	0.580623	+	0.814172i	$0.302809\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	31.1840	0.00462305
$358$	0	0
$359$	−3344.13	−0.491633	−0.245817	−	0.969316i	$-0.579056\pi$
$359$	−3344.13	−0.491633	−0.245817	+	0.969316i	$0.579056\pi$
$360$	0	0
$361$	1204.82	0.175655
$362$	0	0
$363$	3805.81	0.550284
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5289.16	0.752293	0.376147	−	0.926560i	$-0.377249\pi$
$367$	5289.16	0.752293	0.376147	+	0.926560i	$0.377249\pi$
$368$	0	0
$369$	−3463.93	−0.488686
$370$	0	0
$371$	5033.31	0.704357
$372$	0	0
$373$	2109.80	0.292872	0.146436	−	0.989220i	$-0.453220\pi$
$373$	2109.80	0.292872	0.146436	+	0.989220i	$0.453220\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−20571.6	−2.81032
$378$	0	0
$379$	10081.8	1.36641	0.683205	−	0.730227i	$-0.260586\pi$
$379$	10081.8	1.36641	0.683205	+	0.730227i	$0.260586\pi$
$380$	0	0
$381$	3753.42	0.504707
$382$	0	0
$383$	10499.4	1.40076	0.700381	−	0.713769i	$-0.253013\pi$
$383$	10499.4	1.40076	0.700381	+	0.713769i	$0.253013\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−256.603	−0.0337051
$388$	0	0
$389$	−3812.31	−0.496893	−0.248447	−	0.968646i	$-0.579920\pi$
$389$	−3812.31	−0.496893	−0.248447	+	0.968646i	$0.579920\pi$
$390$	0	0
$391$	92.6057	0.0119777
$392$	0	0
$393$	−7173.43	−0.920743
$394$	0	0
$395$	0	0
$396$	0	0
$397$	227.796	0.0287979	0.0143989	−	0.999896i	$-0.495417\pi$
$397$	227.796	0.0287979	0.0143989	+	0.999896i	$0.495417\pi$
$398$	0	0
$399$	5284.91	0.663099
$400$	0	0
$401$	12646.6	1.57492	0.787460	−	0.616366i	$-0.211396\pi$
$401$	12646.6	1.57492	0.787460	+	0.616366i	$0.211396\pi$
$402$	0	0
$403$	19936.1	2.46423
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−89.8140	−0.0109384
$408$	0	0
$409$	10306.0	1.24596	0.622982	−	0.782236i	$-0.285921\pi$
$409$	10306.0	1.24596	0.622982	+	0.782236i	$0.285921\pi$
$410$	0	0
$411$	−1705.55	−0.204692
$412$	0	0
$413$	−3418.73	−0.407323
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4847.77	−0.569296
$418$	0	0
$419$	5722.65	0.667231	0.333616	−	0.942709i	$-0.391731\pi$
$419$	5722.65	0.667231	0.333616	+	0.942709i	$0.391731\pi$
$420$	0	0
$421$	4459.44	0.516247	0.258123	−	0.966112i	$-0.416896\pi$
$421$	4459.44	0.516247	0.258123	+	0.966112i	$0.416896\pi$
$422$	0	0
$423$	2605.54	0.299493
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14378.0	1.62951
$428$	0	0
$429$	−1766.68	−0.198825
$430$	0	0
$431$	−9956.60	−1.11274	−0.556372	−	0.830933i	$-0.687807\pi$
$431$	−9956.60	−1.11274	−0.556372	+	0.830933i	$0.687807\pi$
$432$	0	0
$433$	7033.60	0.780631	0.390316	−	0.920681i	$-0.372366\pi$
$433$	7033.60	0.780631	0.390316	+	0.920681i	$0.372366\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15694.4	1.71799
$438$	0	0
$439$	4855.75	0.527910	0.263955	−	0.964535i	$-0.414973\pi$
$439$	4855.75	0.527910	0.263955	+	0.964535i	$0.414973\pi$
$440$	0	0
$441$	376.655	0.0406711
$442$	0	0
$443$	3421.49	0.366952	0.183476	−	0.983024i	$-0.441265\pi$
$443$	3421.49	0.366952	0.183476	+	0.983024i	$0.441265\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−110.865	−0.0117310
$448$	0	0
$449$	13938.3	1.46501	0.732505	−	0.680761i	$-0.238351\pi$
$449$	13938.3	1.46501	0.732505	+	0.680761i	$0.238351\pi$
$450$	0	0
$451$	−3040.27	−0.317429
$452$	0	0
$453$	7700.82	0.798711
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18167.8	1.85964	0.929820	−	0.368015i	$-0.119963\pi$
$457$	18167.8	1.85964	0.929820	+	0.368015i	$0.119963\pi$
$458$	0	0
$459$	−14.3063	−0.00145482
$460$	0	0
$461$	−5727.65	−0.578662	−0.289331	−	0.957229i	$-0.593433\pi$
$461$	−5727.65	−0.578662	−0.289331	+	0.957229i	$0.593433\pi$
$462$	0	0
$463$	3651.02	0.366473	0.183237	−	0.983069i	$-0.441343\pi$
$463$	3651.02	0.366473	0.183237	+	0.983069i	$0.441343\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9230.06	−0.914596	−0.457298	−	0.889313i	$-0.651183\pi$
$467$	−9230.06	−0.914596	−0.457298	+	0.889313i	$0.651183\pi$
$468$	0	0
$469$	9771.73	0.962082
$470$	0	0
$471$	−7947.56	−0.777504
$472$	0	0
$473$	−225.219	−0.0218934
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2309.14	−0.221653
$478$	0	0
$479$	8663.36	0.826387	0.413193	−	0.910643i	$-0.364413\pi$
$479$	8663.36	0.826387	0.413193	+	0.910643i	$0.364413\pi$
$480$	0	0
$481$	−847.637	−0.0803512
$482$	0	0
$483$	10285.9	0.968992
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12154.2	1.13093	0.565463	−	0.824773i	$-0.308697\pi$
$487$	12154.2	1.13093	0.565463	+	0.824773i	$0.308697\pi$
$488$	0	0
$489$	−5362.21	−0.495884
$490$	0	0
$491$	5635.54	0.517980	0.258990	−	0.965880i	$-0.416610\pi$
$491$	5635.54	0.517980	0.258990	+	0.965880i	$0.416610\pi$
$492$	0	0
$493$	−146.211	−0.0133570
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14970.9	1.35118
$498$	0	0
$499$	−6749.53	−0.605512	−0.302756	−	0.953068i	$-0.597907\pi$
$499$	−6749.53	−0.605512	−0.302756	+	0.953068i	$0.597907\pi$
$500$	0	0
$501$	3670.79	0.327343
$502$	0	0
$503$	−17215.5	−1.52605	−0.763025	−	0.646369i	$-0.776287\pi$
$503$	−17215.5	−1.52605	−0.763025	+	0.646369i	$0.776287\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−10082.3	−0.883181
$508$	0	0
$509$	−17375.6	−1.51308	−0.756541	−	0.653946i	$-0.773112\pi$
$509$	−17375.6	−1.51308	−0.756541	+	0.653946i	$0.773112\pi$
$510$	0	0
$511$	−18190.8	−1.57478
$512$	0	0
$513$	−2424.57	−0.208669
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2286.86	0.194538
$518$	0	0
$519$	−2335.14	−0.197497
$520$	0	0
$521$	−10140.8	−0.852736	−0.426368	−	0.904550i	$-0.640207\pi$
$521$	−10140.8	−0.852736	−0.426368	+	0.904550i	$0.640207\pi$
$522$	0	0
$523$	−7367.10	−0.615948	−0.307974	−	0.951395i	$-0.599651\pi$
$523$	−7367.10	−0.615948	−0.307974	+	0.951395i	$0.599651\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	141.694	0.0117121
$528$	0	0
$529$	18378.5	1.51052
$530$	0	0
$531$	1568.41	0.128180
$532$	0	0
$533$	−28693.1	−2.33178
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3948.74	−0.317320
$538$	0	0
$539$	330.587	0.0264182
$540$	0	0
$541$	456.217	0.0362556	0.0181278	−	0.999836i	$-0.494229\pi$
$541$	456.217	0.0362556	0.0181278	+	0.999836i	$0.494229\pi$
$542$	0	0
$543$	−5467.16	−0.432078
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16435.7	1.28472	0.642359	−	0.766404i	$-0.277956\pi$
$547$	16435.7	1.28472	0.642359	+	0.766404i	$0.277956\pi$
$548$	0	0
$549$	−6596.22	−0.512787
$550$	0	0
$551$	−24779.2	−1.91584
$552$	0	0
$553$	−4260.93	−0.327655
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20749.6	−1.57844	−0.789218	−	0.614113i	$-0.789514\pi$
$557$	−20749.6	−1.57844	−0.789218	+	0.614113i	$0.789514\pi$
$558$	0	0
$559$	−2125.55	−0.160825
$560$	0	0
$561$	−12.5565	−0.000944987 0
$562$	0	0
$563$	24580.2	1.84002	0.920009	−	0.391898i	$-0.128181\pi$
$563$	24580.2	1.84002	0.920009	+	0.391898i	$0.128181\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1589.03	−0.117695
$568$	0	0
$569$	−10755.4	−0.792423	−0.396212	−	0.918159i	$-0.629675\pi$
$569$	−10755.4	−0.792423	−0.396212	+	0.918159i	$0.629675\pi$
$570$	0	0
$571$	9370.97	0.686800	0.343400	−	0.939189i	$-0.388421\pi$
$571$	9370.97	0.686800	0.343400	+	0.939189i	$0.388421\pi$
$572$	0	0
$573$	4253.24	0.310090
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14631.2	1.05564	0.527819	−	0.849357i	$-0.323010\pi$
$577$	14631.2	1.05564	0.527819	+	0.849357i	$0.323010\pi$
$578$	0	0
$579$	8394.72	0.602543
$580$	0	0
$581$	−427.922	−0.0305563
$582$	0	0
$583$	−2026.72	−0.143976
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20163.0	1.41774	0.708872	−	0.705337i	$-0.249204\pi$
$587$	20163.0	1.41774	0.708872	+	0.705337i	$0.249204\pi$
$588$	0	0
$589$	24013.7	1.67991
$590$	0	0
$591$	−2072.46	−0.144246
$592$	0	0
$593$	11427.2	0.791334	0.395667	−	0.918394i	$-0.370514\pi$
$593$	11427.2	0.791334	0.395667	+	0.918394i	$0.370514\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15001.7	−1.02844
$598$	0	0
$599$	−13907.2	−0.948637	−0.474318	−	0.880353i	$-0.657305\pi$
$599$	−13907.2	−0.948637	−0.474318	+	0.880353i	$0.657305\pi$
$600$	0	0
$601$	2823.38	0.191627	0.0958136	−	0.995399i	$-0.469455\pi$
$601$	2823.38	0.191627	0.0958136	+	0.995399i	$0.469455\pi$
$602$	0	0
$603$	−4482.99	−0.302755
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10782.2	0.720984	0.360492	−	0.932762i	$-0.382609\pi$
$607$	10782.2	0.720984	0.360492	+	0.932762i	$0.382609\pi$
$608$	0	0
$609$	−16239.9	−1.08058
$610$	0	0
$611$	21582.7	1.42904
$612$	0	0
$613$	−15504.4	−1.02156	−0.510781	−	0.859711i	$-0.670644\pi$
$613$	−15504.4	−1.02156	−0.510781	+	0.859711i	$0.670644\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1293.49	0.0843987	0.0421994	−	0.999109i	$-0.486564\pi$
$617$	1293.49	0.0843987	0.0421994	+	0.999109i	$0.486564\pi$
$618$	0	0
$619$	−4077.37	−0.264755	−0.132378	−	0.991199i	$-0.542261\pi$
$619$	−4077.37	−0.264755	−0.132378	+	0.991199i	$0.542261\pi$
$620$	0	0
$621$	−4718.86	−0.304930
$622$	0	0
$623$	3760.72	0.241846
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2128.02	−0.135542
$628$	0	0
$629$	−6.02453	−0.000381898 0
$630$	0	0
$631$	−4926.34	−0.310799	−0.155400	−	0.987852i	$-0.549667\pi$
$631$	−4926.34	−0.310799	−0.155400	+	0.987852i	$0.549667\pi$
$632$	0	0
$633$	3306.21	0.207599
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3119.98	0.194063
$638$	0	0
$639$	−6868.22	−0.425200
$640$	0	0
$641$	−2412.43	−0.148651	−0.0743255	−	0.997234i	$-0.523680\pi$
$641$	−2412.43	−0.148651	−0.0743255	+	0.997234i	$0.523680\pi$
$642$	0	0
$643$	9005.77	0.552337	0.276169	−	0.961109i	$-0.410935\pi$
$643$	9005.77	0.552337	0.276169	+	0.961109i	$0.410935\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7452.03	0.452812	0.226406	−	0.974033i	$-0.427302\pi$
$647$	7452.03	0.452812	0.226406	+	0.974033i	$0.427302\pi$
$648$	0	0
$649$	1376.59	0.0832600
$650$	0	0
$651$	15738.2	0.947511
$652$	0	0
$653$	19389.2	1.16195	0.580977	−	0.813920i	$-0.302671\pi$
$653$	19389.2	1.16195	0.580977	+	0.813920i	$0.302671\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8345.41	0.495564
$658$	0	0
$659$	29605.2	1.75001	0.875004	−	0.484116i	$-0.160859\pi$
$659$	29605.2	1.75001	0.875004	+	0.484116i	$0.160859\pi$
$660$	0	0
$661$	8840.38	0.520198	0.260099	−	0.965582i	$-0.416245\pi$
$661$	8840.38	0.520198	0.260099	+	0.965582i	$0.416245\pi$
$662$	0	0
$663$	−118.505	−0.00694170
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−48227.0	−2.79963
$668$	0	0
$669$	4914.69	0.284025
$670$	0	0
$671$	−5789.46	−0.333084
$672$	0	0
$673$	−16914.1	−0.968785	−0.484393	−	0.874851i	$-0.660959\pi$
$673$	−16914.1	−0.968785	−0.484393	+	0.874851i	$0.660959\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19762.2	−1.12190	−0.560949	−	0.827851i	$-0.689564\pi$
$677$	−19762.2	−1.12190	−0.560949	+	0.827851i	$0.689564\pi$
$678$	0	0
$679$	−17368.5	−0.981653
$680$	0	0
$681$	−11047.6	−0.621654
$682$	0	0
$683$	15403.9	0.862976	0.431488	−	0.902119i	$-0.357989\pi$
$683$	15403.9	0.862976	0.431488	+	0.902119i	$0.357989\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	11798.4	0.655222
$688$	0	0
$689$	−19127.5	−1.05762
$690$	0	0
$691$	11468.9	0.631399	0.315699	−	0.948859i	$-0.397761\pi$
$691$	11468.9	0.631399	0.315699	+	0.948859i	$0.397761\pi$
$692$	0	0
$693$	−1394.68	−0.0764493
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−203.935	−0.0110826
$698$	0	0
$699$	2491.18	0.134800
$700$	0	0
$701$	−22926.8	−1.23528	−0.617641	−	0.786460i	$-0.711911\pi$
$701$	−22926.8	−1.23528	−0.617641	+	0.786460i	$0.711911\pi$
$702$	0	0
$703$	−1021.01	−0.0547768
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−28104.0	−1.49499
$708$	0	0
$709$	−20023.3	−1.06064	−0.530319	−	0.847798i	$-0.677928\pi$
$709$	−20023.3	−1.06064	−0.530319	+	0.847798i	$0.677928\pi$
$710$	0	0
$711$	1954.79	0.103109
$712$	0	0
$713$	46737.1	2.45487
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5020.85	0.261516
$718$	0	0
$719$	23678.6	1.22818	0.614092	−	0.789234i	$-0.289522\pi$
$719$	23678.6	1.22818	0.614092	+	0.789234i	$0.289522\pi$
$720$	0	0
$721$	−12209.2	−0.630647
$722$	0	0
$723$	−13757.5	−0.707673
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−10302.0	−0.525559	−0.262779	−	0.964856i	$-0.584639\pi$
$727$	−10302.0	−0.525559	−0.262779	+	0.964856i	$0.584639\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−15.1072	−0.000764377 0
$732$	0	0
$733$	−23716.7	−1.19508	−0.597542	−	0.801838i	$-0.703856\pi$
$733$	−23716.7	−1.19508	−0.597542	+	0.801838i	$0.703856\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3934.69	−0.196657
$738$	0	0
$739$	13007.0	0.647459	0.323729	−	0.946150i	$-0.395063\pi$
$739$	13007.0	0.647459	0.323729	+	0.946150i	$0.395063\pi$
$740$	0	0
$741$	−20083.6	−0.995669
$742$	0	0
$743$	−33942.1	−1.67593	−0.837964	−	0.545725i	$-0.816254\pi$
$743$	−33942.1	−1.67593	−0.837964	+	0.545725i	$0.816254\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	196.319	0.00961569
$748$	0	0
$749$	−23636.1	−1.15307
$750$	0	0
$751$	2959.96	0.143822	0.0719112	−	0.997411i	$-0.477090\pi$
$751$	2959.96	0.143822	0.0719112	+	0.997411i	$0.477090\pi$
$752$	0	0
$753$	18418.1	0.891357
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29475.3	1.41519	0.707593	−	0.706620i	$-0.249781\pi$
$757$	29475.3	1.41519	0.707593	+	0.706620i	$0.249781\pi$
$758$	0	0
$759$	−4141.71	−0.198069
$760$	0	0
$761$	−7910.79	−0.376828	−0.188414	−	0.982090i	$-0.560335\pi$
$761$	−7910.79	−0.376828	−0.188414	+	0.982090i	$0.560335\pi$
$762$	0	0
$763$	−2149.57	−0.101992
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12991.8	0.611612
$768$	0	0
$769$	−8273.59	−0.387976	−0.193988	−	0.981004i	$-0.562142\pi$
$769$	−8273.59	−0.387976	−0.193988	+	0.981004i	$0.562142\pi$
$770$	0	0
$771$	−6654.03	−0.310816
$772$	0	0
$773$	−21985.0	−1.02296	−0.511479	−	0.859296i	$-0.670902\pi$
$773$	−21985.0	−1.02296	−0.511479	+	0.859296i	$0.670902\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−669.154	−0.0308955
$778$	0	0
$779$	−34561.9	−1.58961
$780$	0	0
$781$	−6028.19	−0.276191
$782$	0	0
$783$	7450.41	0.340046
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−25485.2	−1.15432	−0.577160	−	0.816631i	$-0.695839\pi$
$787$	−25485.2	−1.15432	−0.577160	+	0.816631i	$0.695839\pi$
$788$	0	0
$789$	−22438.5	−1.01246
$790$	0	0
$791$	18135.6	0.815206
$792$	0	0
$793$	−54639.1	−2.44677
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9611.91	0.427191	0.213596	−	0.976922i	$-0.431483\pi$
$797$	9611.91	0.427191	0.213596	+	0.976922i	$0.431483\pi$
$798$	0	0
$799$	153.398	0.00679201
$800$	0	0
$801$	−1725.31	−0.0761059
$802$	0	0
$803$	7324.71	0.321897
$804$	0	0
$805$	0	0
$806$	0	0
$807$	884.101	0.0385648
$808$	0	0
$809$	−1918.02	−0.0833547	−0.0416774	−	0.999131i	$-0.513270\pi$
$809$	−1918.02	−0.0833547	−0.0416774	+	0.999131i	$0.513270\pi$
$810$	0	0
$811$	38027.6	1.64652	0.823261	−	0.567663i	$-0.192152\pi$
$811$	38027.6	1.64652	0.823261	+	0.567663i	$0.192152\pi$
$812$	0	0
$813$	20884.2	0.900910
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2560.29	−0.109637
$818$	0	0
$819$	−13162.5	−0.561582
$820$	0	0
$821$	23150.3	0.984105	0.492052	−	0.870566i	$-0.336247\pi$
$821$	23150.3	0.984105	0.492052	+	0.870566i	$0.336247\pi$
$822$	0	0
$823$	−26584.1	−1.12596	−0.562978	−	0.826472i	$-0.690344\pi$
$823$	−26584.1	−1.12596	−0.562978	+	0.826472i	$0.690344\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19219.6	−0.808137	−0.404069	−	0.914729i	$-0.632404\pi$
$827$	−19219.6	−0.808137	−0.404069	+	0.914729i	$0.632404\pi$
$828$	0	0
$829$	38859.1	1.62803	0.814013	−	0.580846i	$-0.197278\pi$
$829$	38859.1	1.62803	0.814013	+	0.580846i	$0.197278\pi$
$830$	0	0
$831$	−6107.17	−0.254940
$832$	0	0
$833$	22.1751	0.000922354 0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7220.25	−0.298170
$838$	0	0
$839$	−32060.3	−1.31924	−0.659621	−	0.751598i	$-0.729283\pi$
$839$	−32060.3	−1.31924	−0.659621	+	0.751598i	$0.729283\pi$
$840$	0	0
$841$	51754.5	2.12204
$842$	0	0
$843$	3174.01	0.129678
$844$	0	0
$845$	0	0
$846$	0	0
$847$	24886.9	1.00959
$848$	0	0
$849$	−27758.9	−1.12213
$850$	0	0
$851$	−1987.16	−0.0800458
$852$	0	0
$853$	8889.57	0.356827	0.178413	−	0.983956i	$-0.442904\pi$
$853$	8889.57	0.356827	0.178413	+	0.983956i	$0.442904\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16548.6	0.659614	0.329807	−	0.944048i	$-0.393016\pi$
$857$	16548.6	0.659614	0.329807	+	0.944048i	$0.393016\pi$
$858$	0	0
$859$	46480.6	1.84621	0.923107	−	0.384543i	$-0.125641\pi$
$859$	46480.6	1.84621	0.923107	+	0.384543i	$0.125641\pi$
$860$	0	0
$861$	−22651.4	−0.896581
$862$	0	0
$863$	−40478.2	−1.59663	−0.798316	−	0.602239i	$-0.794275\pi$
$863$	−40478.2	−1.59663	−0.798316	+	0.602239i	$0.794275\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14738.2	0.577317
$868$	0	0
$869$	1715.71	0.0669751
$870$	0	0
$871$	−37134.4	−1.44460
$872$	0	0
$873$	7968.18	0.308914
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16898.6	0.650657	0.325329	−	0.945601i	$-0.394525\pi$
$877$	16898.6	0.650657	0.325329	+	0.945601i	$0.394525\pi$
$878$	0	0
$879$	−21029.5	−0.806949
$880$	0	0
$881$	−50678.1	−1.93801	−0.969006	−	0.247037i	$-0.920543\pi$
$881$	−50678.1	−1.93801	−0.969006	+	0.247037i	$0.920543\pi$
$882$	0	0
$883$	−19464.8	−0.741839	−0.370920	−	0.928665i	$-0.620957\pi$
$883$	−19464.8	−0.741839	−0.370920	+	0.928665i	$0.620957\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39554.0	1.49729	0.748644	−	0.662973i	$-0.230705\pi$
$887$	39554.0	1.49729	0.748644	+	0.662973i	$0.230705\pi$
$888$	0	0
$889$	24544.3	0.925974
$890$	0	0
$891$	639.838	0.0240577
$892$	0	0
$893$	25997.1	0.974200
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−39088.2	−1.45498
$898$	0	0
$899$	−73791.3	−2.73757
$900$	0	0
$901$	−135.948	−0.00502672
$902$	0	0
$903$	−1677.98	−0.0618380
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−12617.9	−0.461929	−0.230965	−	0.972962i	$-0.574188\pi$
$907$	−12617.9	−0.461929	−0.230965	+	0.972962i	$0.574188\pi$
$908$	0	0
$909$	12893.3	0.470456
$910$	0	0
$911$	2570.05	0.0934683	0.0467341	−	0.998907i	$-0.485119\pi$
$911$	2570.05	0.0934683	0.0467341	+	0.998907i	$0.485119\pi$
$912$	0	0
$913$	172.307	0.00624594
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−46908.5	−1.68927
$918$	0	0
$919$	−29738.3	−1.06744	−0.533719	−	0.845662i	$-0.679206\pi$
$919$	−29738.3	−1.06744	−0.533719	+	0.845662i	$0.679206\pi$
$920$	0	0
$921$	23918.2	0.855734
$922$	0	0
$923$	−56892.2	−2.02885
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5601.26	0.198457
$928$	0	0
$929$	17702.9	0.625203	0.312601	−	0.949884i	$-0.398800\pi$
$929$	17702.9	0.625203	0.312601	+	0.949884i	$0.398800\pi$
$930$	0	0
$931$	3758.13	0.132296
$932$	0	0
$933$	26325.2	0.923738
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17692.0	0.616833	0.308416	−	0.951251i	$-0.400201\pi$
$937$	17692.0	0.616833	0.308416	+	0.951251i	$0.400201\pi$
$938$	0	0
$939$	22786.0	0.791898
$940$	0	0
$941$	6445.10	0.223278	0.111639	−	0.993749i	$-0.464390\pi$
$941$	6445.10	0.223278	0.111639	+	0.993749i	$0.464390\pi$
$942$	0	0
$943$	−67266.8	−2.32291
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22687.6	−0.778510	−0.389255	−	0.921130i	$-0.627268\pi$
$947$	−22687.6	−0.778510	−0.389255	+	0.921130i	$0.627268\pi$
$948$	0	0
$949$	69128.3	2.36460
$950$	0	0
$951$	16326.9	0.556713
$952$	0	0
$953$	−21132.5	−0.718310	−0.359155	−	0.933278i	$-0.616935\pi$
$953$	−21132.5	−0.718310	−0.359155	+	0.933278i	$0.616935\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6539.17	0.220879
$958$	0	0
$959$	−11152.9	−0.375545
$960$	0	0
$961$	41720.7	1.40045
$962$	0	0
$963$	10843.6	0.362855
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−54081.6	−1.79850	−0.899248	−	0.437438i	$-0.855886\pi$
$967$	−54081.6	−1.79850	−0.899248	+	0.437438i	$0.855886\pi$
$968$	0	0
$969$	−142.743	−0.00473227
$970$	0	0
$971$	59224.0	1.95735	0.978676	−	0.205409i	$-0.0658524\pi$
$971$	59224.0	1.95735	0.978676	+	0.205409i	$0.0658524\pi$
$972$	0	0
$973$	−31700.5	−1.04447
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31672.4	1.03714	0.518572	−	0.855034i	$-0.326464\pi$
$977$	31672.4	1.03714	0.518572	+	0.855034i	$0.326464\pi$
$978$	0	0
$979$	−1514.29	−0.0494351
$980$	0	0
$981$	986.161	0.0320955
$982$	0	0
$983$	16842.2	0.546474	0.273237	−	0.961947i	$-0.411906\pi$
$983$	16842.2	0.546474	0.273237	+	0.961947i	$0.411906\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	17038.1	0.549473
$988$	0	0
$989$	−4983.03	−0.160213
$990$	0	0
$991$	−27973.5	−0.896678	−0.448339	−	0.893864i	$-0.647984\pi$
$991$	−27973.5	−0.896678	−0.448339	+	0.893864i	$0.647984\pi$
$992$	0	0
$993$	−19625.0	−0.627170
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−21935.4	−0.696792	−0.348396	−	0.937347i	$-0.613274\pi$
$997$	−21935.4	−0.696792	−0.348396	+	0.937347i	$0.613274\pi$
$998$	0	0
$999$	306.989	0.00972243

Twists

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

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

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -19.6176 q^{7} +9.00000 q^{9} +7.89924 q^{11} +74.5505 q^{13} +0.529863 q^{17} +89.7988 q^{19} +58.8528 q^{21} +174.773 q^{23} -27.0000 q^{27} -275.941 q^{29} +267.417 q^{31} -23.6977 q^{33} -11.3700 q^{37} -223.652 q^{39} -384.881 q^{41} -28.5115 q^{43} +289.504 q^{47} +41.8506 q^{49} -1.58959 q^{51} -256.571 q^{53} -269.396 q^{57} +174.268 q^{59} -732.913 q^{61} -176.558 q^{63} -498.110 q^{67} -524.318 q^{69} -763.136 q^{71} +927.268 q^{73} -154.964 q^{77} +217.199 q^{79} +81.0000 q^{81} +21.8132 q^{83} +827.823 q^{87} -191.701 q^{89} -1462.50 q^{91} -802.250 q^{93} +885.353 q^{97} +71.0931 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 19 q^{7} + 27 q^{9} + 48 q^{11} + 63 q^{13} + 16 q^{17} + 21 q^{19} - 57 q^{21} + 140 q^{23} - 81 q^{27} + 92 q^{29} + 179 q^{31} - 144 q^{33} + 550 q^{37} - 189 q^{39} - 206 q^{41} + 279 q^{43}+ \cdots + 432 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more