LMFDB - Embedded newform 2400.4.a.cb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$141.604584014$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1965645.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 43x^{2} + 100x - 32 $ x^4 - x^3 - 43x^2 + 100x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-7.11205$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +11.4568 q^{7} +9.00000 q^{9} +16.0092 q^{11} -73.9390 q^{13} +103.922 q^{17} +27.9470 q^{19} +34.3705 q^{21} -112.741 q^{23} +27.0000 q^{27} -241.111 q^{29} -303.899 q^{31} +48.0277 q^{33} +177.584 q^{37} -221.817 q^{39} +4.91366 q^{41} -171.309 q^{43} -373.482 q^{47} -211.741 q^{49} +311.765 q^{51} -45.4391 q^{53} +83.8411 q^{57} +711.166 q^{59} +567.928 q^{61} +103.111 q^{63} -390.568 q^{67} -338.223 q^{69} +687.567 q^{71} -431.420 q^{73} +183.415 q^{77} -1086.42 q^{79} +81.0000 q^{81} -720.446 q^{83} -723.334 q^{87} -633.410 q^{89} -847.106 q^{91} -911.696 q^{93} +219.505 q^{97} +144.083 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} - 12 q^{7} + 36 q^{9} - 36 q^{21} - 104 q^{23} + 108 q^{27} - 444 q^{29} - 96 q^{41} + 240 q^{43} - 800 q^{47} - 500 q^{49} - 504 q^{61} - 108 q^{63} - 984 q^{67} - 312 q^{69} + 324 q^{81}+ \cdots + 936 q^{89}+O(q^{100}) $ 4 * q + 12 * q^3 - 12 * q^7 + 36 * q^9 - 36 * q^21 - 104 * q^23 + 108 * q^27 - 444 * q^29 - 96 * q^41 + 240 * q^43 - 800 * q^47 - 500 * q^49 - 504 * q^61 - 108 * q^63 - 984 * q^67 - 312 * q^69 + 324 * q^81 - 800 * q^83 - 1332 * q^87 + 936 * q^89

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
$7$	11.4568	0.618611	0.309305	−	0.950963i	$-0.399903\pi$
$7$	11.4568	0.618611	0.309305	+	0.950963i	$0.399903\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	16.0092	0.438815	0.219407	−	0.975633i	$-0.429588\pi$
$11$	16.0092	0.438815	0.219407	+	0.975633i	$0.429588\pi$
$12$	0	0
$13$	−73.9390	−1.57746	−0.788730	−	0.614740i	$-0.789261\pi$
$13$	−73.9390	−1.57746	−0.788730	+	0.614740i	$0.789261\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	103.922	1.48263	0.741315	−	0.671157i	$-0.234202\pi$
$17$	103.922	1.48263	0.741315	+	0.671157i	$0.234202\pi$
$18$	0	0
$19$	27.9470	0.337447	0.168723	−	0.985663i	$-0.446036\pi$
$19$	27.9470	0.337447	0.168723	+	0.985663i	$0.446036\pi$
$20$	0	0
$21$	34.3705	0.357155
$22$	0	0
$23$	−112.741	−1.02209	−0.511046	−	0.859553i	$-0.670742\pi$
$23$	−112.741	−1.02209	−0.511046	+	0.859553i	$0.670742\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−241.111	−1.54391	−0.771953	−	0.635679i	$-0.780720\pi$
$29$	−241.111	−1.54391	−0.771953	+	0.635679i	$0.780720\pi$
$30$	0	0
$31$	−303.899	−1.76070	−0.880352	−	0.474321i	$-0.842694\pi$
$31$	−303.899	−1.76070	−0.880352	+	0.474321i	$0.842694\pi$
$32$	0	0
$33$	48.0277	0.253350
$34$	0	0
$35$	0	0
$36$	0	0
$37$	177.584	0.789045	0.394523	−	0.918886i	$-0.370910\pi$
$37$	177.584	0.789045	0.394523	+	0.918886i	$0.370910\pi$
$38$	0	0
$39$	−221.817	−0.910747
$40$	0	0
$41$	4.91366	0.0187167	0.00935836	−	0.999956i	$-0.497021\pi$
$41$	4.91366	0.0187167	0.00935836	+	0.999956i	$0.497021\pi$
$42$	0	0
$43$	−171.309	−0.607545	−0.303772	−	0.952745i	$-0.598246\pi$
$43$	−171.309	−0.607545	−0.303772	+	0.952745i	$0.598246\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−373.482	−1.15911	−0.579553	−	0.814935i	$-0.696773\pi$
$47$	−373.482	−1.15911	−0.579553	+	0.814935i	$0.696773\pi$
$48$	0	0
$49$	−211.741	−0.617321
$50$	0	0
$51$	311.765	0.855997
$52$	0	0
$53$	−45.4391	−0.117765	−0.0588824	−	0.998265i	$-0.518754\pi$
$53$	−45.4391	−0.117765	−0.0588824	+	0.998265i	$0.518754\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	83.8411	0.194825
$58$	0	0
$59$	711.166	1.56925	0.784627	−	0.619968i	$-0.212855\pi$
$59$	711.166	1.56925	0.784627	+	0.619968i	$0.212855\pi$
$60$	0	0
$61$	567.928	1.19206	0.596031	−	0.802962i	$-0.296744\pi$
$61$	567.928	1.19206	0.596031	+	0.802962i	$0.296744\pi$
$62$	0	0
$63$	103.111	0.206204
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−390.568	−0.712172	−0.356086	−	0.934453i	$-0.615889\pi$
$67$	−390.568	−0.712172	−0.356086	+	0.934453i	$0.615889\pi$
$68$	0	0
$69$	−338.223	−0.590105
$70$	0	0
$71$	687.567	1.14928	0.574642	−	0.818405i	$-0.305141\pi$
$71$	687.567	1.14928	0.574642	+	0.818405i	$0.305141\pi$
$72$	0	0
$73$	−431.420	−0.691696	−0.345848	−	0.938290i	$-0.612409\pi$
$73$	−431.420	−0.691696	−0.345848	+	0.938290i	$0.612409\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	183.415	0.271455
$78$	0	0
$79$	−1086.42	−1.54723	−0.773616	−	0.633655i	$-0.781554\pi$
$79$	−1086.42	−1.54723	−0.773616	+	0.633655i	$0.781554\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−720.446	−0.952762	−0.476381	−	0.879239i	$-0.658052\pi$
$83$	−720.446	−0.952762	−0.476381	+	0.879239i	$0.658052\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−723.334	−0.891375
$88$	0	0
$89$	−633.410	−0.754397	−0.377198	−	0.926132i	$-0.623113\pi$
$89$	−633.410	−0.754397	−0.377198	+	0.926132i	$0.623113\pi$
$90$	0	0
$91$	−847.106	−0.975834
$92$	0	0
$93$	−911.696	−1.01654
$94$	0	0
$95$	0	0
$96$	0	0
$97$	219.505	0.229766	0.114883	−	0.993379i	$-0.463351\pi$
$97$	219.505	0.229766	0.114883	+	0.993379i	$0.463351\pi$
$98$	0	0
$99$	144.083	0.146272
$100$	0	0
$101$	−131.925	−0.129970	−0.0649851	−	0.997886i	$-0.520700\pi$
$101$	−131.925	−0.129970	−0.0649851	+	0.997886i	$0.520700\pi$
$102$	0	0
$103$	812.349	0.777117	0.388559	−	0.921424i	$-0.372973\pi$
$103$	812.349	0.777117	0.388559	+	0.921424i	$0.372973\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1730.82	1.56378	0.781891	−	0.623415i	$-0.214255\pi$
$107$	1730.82	1.56378	0.781891	+	0.623415i	$0.214255\pi$
$108$	0	0
$109$	−226.295	−0.198854	−0.0994272	−	0.995045i	$-0.531701\pi$
$109$	−226.295	−0.198854	−0.0994272	+	0.995045i	$0.531701\pi$
$110$	0	0
$111$	532.753	0.455555
$112$	0	0
$113$	−650.648	−0.541662	−0.270831	−	0.962627i	$-0.587298\pi$
$113$	−650.648	−0.541662	−0.270831	+	0.962627i	$0.587298\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−665.451	−0.525820
$118$	0	0
$119$	1190.61	0.917171
$120$	0	0
$121$	−1074.70	−0.807442
$122$	0	0
$123$	14.7410	0.0108061
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2255.61	−1.57601	−0.788003	−	0.615671i	$-0.788885\pi$
$127$	−2255.61	−1.57601	−0.788003	+	0.615671i	$0.788885\pi$
$128$	0	0
$129$	−513.928	−0.350766
$130$	0	0
$131$	−2028.19	−1.35270	−0.676352	−	0.736579i	$-0.736440\pi$
$131$	−2028.19	−1.35270	−0.676352	+	0.736579i	$0.736440\pi$
$132$	0	0
$133$	320.184	0.208748
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−215.710	−0.134521	−0.0672603	−	0.997735i	$-0.521426\pi$
$137$	−215.710	−0.134521	−0.0672603	+	0.997735i	$0.521426\pi$
$138$	0	0
$139$	−532.652	−0.325028	−0.162514	−	0.986706i	$-0.551960\pi$
$139$	−532.652	−0.325028	−0.162514	+	0.986706i	$0.551960\pi$
$140$	0	0
$141$	−1120.45	−0.669210
$142$	0	0
$143$	−1183.70	−0.692212
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−635.223	−0.356410
$148$	0	0
$149$	−469.011	−0.257872	−0.128936	−	0.991653i	$-0.541156\pi$
$149$	−469.011	−0.257872	−0.128936	+	0.991653i	$0.541156\pi$
$150$	0	0
$151$	3108.96	1.67552	0.837759	−	0.546040i	$-0.183865\pi$
$151$	3108.96	1.67552	0.837759	+	0.546040i	$0.183865\pi$
$152$	0	0
$153$	935.295	0.494210
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.5264	0.00738427	0.00369214	−	0.999993i	$-0.498825\pi$
$157$	14.5264	0.00738427	0.00369214	+	0.999993i	$0.498825\pi$
$158$	0	0
$159$	−136.317	−0.0679916
$160$	0	0
$161$	−1291.65	−0.632277
$162$	0	0
$163$	432.050	0.207612	0.103806	−	0.994598i	$-0.466898\pi$
$163$	432.050	0.207612	0.103806	+	0.994598i	$0.466898\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−463.849	−0.214932	−0.107466	−	0.994209i	$-0.534274\pi$
$167$	−463.849	−0.214932	−0.107466	+	0.994209i	$0.534274\pi$
$168$	0	0
$169$	3269.97	1.48838
$170$	0	0
$171$	251.523	0.112482
$172$	0	0
$173$	2565.85	1.12762	0.563809	−	0.825905i	$-0.309336\pi$
$173$	2565.85	1.12762	0.563809	+	0.825905i	$0.309336\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2133.50	0.906009
$178$	0	0
$179$	−194.598	−0.0812568	−0.0406284	−	0.999174i	$-0.512936\pi$
$179$	−194.598	−0.0812568	−0.0406284	+	0.999174i	$0.512936\pi$
$180$	0	0
$181$	−3437.26	−1.41154	−0.705772	−	0.708439i	$-0.749400\pi$
$181$	−3437.26	−1.41154	−0.705772	+	0.708439i	$0.749400\pi$
$182$	0	0
$183$	1703.78	0.688237
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1663.70	0.650600
$188$	0	0
$189$	309.334	0.119052
$190$	0	0
$191$	−4117.81	−1.55997	−0.779985	−	0.625798i	$-0.784774\pi$
$191$	−4117.81	−1.55997	−0.779985	+	0.625798i	$0.784774\pi$
$192$	0	0
$193$	1199.51	0.447371	0.223686	−	0.974661i	$-0.428191\pi$
$193$	1199.51	0.447371	0.223686	+	0.974661i	$0.428191\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2167.35	0.783845	0.391922	−	0.919998i	$-0.371810\pi$
$197$	2167.35	0.783845	0.391922	+	0.919998i	$0.371810\pi$
$198$	0	0
$199$	1816.36	0.647025	0.323513	−	0.946224i	$-0.395136\pi$
$199$	1816.36	0.647025	0.323513	+	0.946224i	$0.395136\pi$
$200$	0	0
$201$	−1171.70	−0.411173
$202$	0	0
$203$	−2762.37	−0.955077
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1014.67	−0.340697
$208$	0	0
$209$	447.410	0.148077
$210$	0	0
$211$	536.372	0.175002	0.0875009	−	0.996164i	$-0.472112\pi$
$211$	536.372	0.175002	0.0875009	+	0.996164i	$0.472112\pi$
$212$	0	0
$213$	2062.70	0.663540
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3481.72	−1.08919
$218$	0	0
$219$	−1294.26	−0.399351
$220$	0	0
$221$	−7683.86	−2.33879
$222$	0	0
$223$	5669.44	1.70248	0.851242	−	0.524773i	$-0.175850\pi$
$223$	5669.44	1.70248	0.851242	+	0.524773i	$0.175850\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2681.34	0.783994	0.391997	−	0.919966i	$-0.371784\pi$
$227$	2681.34	0.783994	0.391997	+	0.919966i	$0.371784\pi$
$228$	0	0
$229$	−4182.30	−1.20687	−0.603436	−	0.797411i	$-0.706202\pi$
$229$	−4182.30	−1.20687	−0.603436	+	0.797411i	$0.706202\pi$
$230$	0	0
$231$	550.245	0.156725
$232$	0	0
$233$	1625.28	0.456976	0.228488	−	0.973547i	$-0.426622\pi$
$233$	1625.28	0.456976	0.228488	+	0.973547i	$0.426622\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3259.25	−0.893294
$238$	0	0
$239$	−4102.63	−1.11036	−0.555182	−	0.831729i	$-0.687352\pi$
$239$	−4102.63	−1.11036	−0.555182	+	0.831729i	$0.687352\pi$
$240$	0	0
$241$	−5815.34	−1.55435	−0.777176	−	0.629283i	$-0.783349\pi$
$241$	−5815.34	−1.55435	−0.777176	+	0.629283i	$0.783349\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2066.37	−0.532309
$248$	0	0
$249$	−2161.34	−0.550077
$250$	0	0
$251$	60.5934	0.0152375	0.00761877	−	0.999971i	$-0.497575\pi$
$251$	60.5934	0.0152375	0.00761877	+	0.999971i	$0.497575\pi$
$252$	0	0
$253$	−1804.90	−0.448509
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7615.89	−1.84851	−0.924253	−	0.381780i	$-0.875311\pi$
$257$	−7615.89	−1.84851	−0.924253	+	0.381780i	$0.875311\pi$
$258$	0	0
$259$	2034.55	0.488112
$260$	0	0
$261$	−2170.00	−0.514635
$262$	0	0
$263$	−6838.09	−1.60325	−0.801625	−	0.597827i	$-0.796031\pi$
$263$	−6838.09	−1.60325	−0.801625	+	0.597827i	$0.796031\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1900.23	−0.435551
$268$	0	0
$269$	−1783.58	−0.404263	−0.202131	−	0.979358i	$-0.564787\pi$
$269$	−1783.58	−0.404263	−0.202131	+	0.979358i	$0.564787\pi$
$270$	0	0
$271$	3660.86	0.820595	0.410298	−	0.911952i	$-0.365425\pi$
$271$	3660.86	0.820595	0.410298	+	0.911952i	$0.365425\pi$
$272$	0	0
$273$	−2541.32	−0.563398
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1767.95	−0.383487	−0.191743	−	0.981445i	$-0.561414\pi$
$277$	−1767.95	−0.383487	−0.191743	+	0.981445i	$0.561414\pi$
$278$	0	0
$279$	−2735.09	−0.586901
$280$	0	0
$281$	−7877.06	−1.67226	−0.836132	−	0.548528i	$-0.815188\pi$
$281$	−7877.06	−1.67226	−0.836132	+	0.548528i	$0.815188\pi$
$282$	0	0
$283$	1186.41	0.249204	0.124602	−	0.992207i	$-0.460235\pi$
$283$	1186.41	0.249204	0.124602	+	0.992207i	$0.460235\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	56.2950	0.0115784
$288$	0	0
$289$	5886.72	1.19819
$290$	0	0
$291$	658.514	0.132656
$292$	0	0
$293$	−5000.35	−0.997010	−0.498505	−	0.866887i	$-0.666117\pi$
$293$	−5000.35	−0.997010	−0.498505	+	0.866887i	$0.666117\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	432.249	0.0844499
$298$	0	0
$299$	8335.95	1.61231
$300$	0	0
$301$	−1962.66	−0.375834
$302$	0	0
$303$	−395.774	−0.0750383
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3101.08	0.576508	0.288254	−	0.957554i	$-0.406925\pi$
$307$	3101.08	0.576508	0.288254	+	0.957554i	$0.406925\pi$
$308$	0	0
$309$	2437.05	0.448669
$310$	0	0
$311$	−3781.74	−0.689528	−0.344764	−	0.938689i	$-0.612041\pi$
$311$	−3781.74	−0.689528	−0.344764	+	0.938689i	$0.612041\pi$
$312$	0	0
$313$	5791.67	1.04589	0.522946	−	0.852366i	$-0.324833\pi$
$313$	5791.67	1.04589	0.522946	+	0.852366i	$0.324833\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3540.27	−0.627261	−0.313630	−	0.949545i	$-0.601545\pi$
$317$	−3540.27	−0.627261	−0.313630	+	0.949545i	$0.601545\pi$
$318$	0	0
$319$	−3860.01	−0.677489
$320$	0	0
$321$	5192.46	0.902850
$322$	0	0
$323$	2904.30	0.500309
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−678.885	−0.114809
$328$	0	0
$329$	−4278.92	−0.717035
$330$	0	0
$331$	5480.73	0.910116	0.455058	−	0.890462i	$-0.349619\pi$
$331$	5480.73	0.910116	0.455058	+	0.890462i	$0.349619\pi$
$332$	0	0
$333$	1598.26	0.263015
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6231.03	1.00720	0.503599	−	0.863937i	$-0.332009\pi$
$337$	6231.03	1.00720	0.503599	+	0.863937i	$0.332009\pi$
$338$	0	0
$339$	−1951.94	−0.312729
$340$	0	0
$341$	−4865.18	−0.772623
$342$	0	0
$343$	−6355.57	−1.00049
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11243.1	1.73937	0.869686	−	0.493605i	$-0.164321\pi$
$347$	11243.1	1.73937	0.869686	+	0.493605i	$0.164321\pi$
$348$	0	0
$349$	−12464.4	−1.91176	−0.955880	−	0.293758i	$-0.905094\pi$
$349$	−12464.4	−1.91176	−0.955880	+	0.293758i	$0.905094\pi$
$350$	0	0
$351$	−1996.35	−0.303582
$352$	0	0
$353$	−6940.19	−1.04643	−0.523214	−	0.852202i	$-0.675267\pi$
$353$	−6940.19	−1.04643	−0.523214	+	0.852202i	$0.675267\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3571.84	0.529529
$358$	0	0
$359$	11994.6	1.76337	0.881686	−	0.471836i	$-0.156409\pi$
$359$	11994.6	1.76337	0.881686	+	0.471836i	$0.156409\pi$
$360$	0	0
$361$	−6077.96	−0.886130
$362$	0	0
$363$	−3224.11	−0.466177
$364$	0	0
$365$	0	0
$366$	0	0
$367$	292.939	0.0416656	0.0208328	−	0.999783i	$-0.493368\pi$
$367$	292.939	0.0416656	0.0208328	+	0.999783i	$0.493368\pi$
$368$	0	0
$369$	44.2230	0.00623891
$370$	0	0
$371$	−520.588	−0.0728506
$372$	0	0
$373$	−3680.41	−0.510897	−0.255448	−	0.966823i	$-0.582223\pi$
$373$	−3680.41	−0.510897	−0.255448	+	0.966823i	$0.582223\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17827.5	2.43545
$378$	0	0
$379$	−5684.86	−0.770479	−0.385239	−	0.922817i	$-0.625881\pi$
$379$	−5684.86	−0.770479	−0.385239	+	0.922817i	$0.625881\pi$
$380$	0	0
$381$	−6766.82	−0.909908
$382$	0	0
$383$	−6737.64	−0.898896	−0.449448	−	0.893306i	$-0.648379\pi$
$383$	−6737.64	−0.898896	−0.449448	+	0.893306i	$0.648379\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1541.78	−0.202515
$388$	0	0
$389$	−2460.91	−0.320753	−0.160377	−	0.987056i	$-0.551271\pi$
$389$	−2460.91	−0.320753	−0.160377	+	0.987056i	$0.551271\pi$
$390$	0	0
$391$	−11716.2	−1.51538
$392$	0	0
$393$	−6084.58	−0.780984
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15437.4	1.95158	0.975792	−	0.218700i	$-0.0701816\pi$
$397$	15437.4	1.95158	0.975792	+	0.218700i	$0.0701816\pi$
$398$	0	0
$399$	960.553	0.120521
$400$	0	0
$401$	8128.55	1.01227	0.506135	−	0.862454i	$-0.331074\pi$
$401$	8128.55	1.01227	0.506135	+	0.862454i	$0.331074\pi$
$402$	0	0
$403$	22470.0	2.77744
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2842.98	0.346244
$408$	0	0
$409$	−12735.2	−1.53965	−0.769824	−	0.638257i	$-0.779656\pi$
$409$	−12735.2	−1.53965	−0.769824	+	0.638257i	$0.779656\pi$
$410$	0	0
$411$	−647.129	−0.0776655
$412$	0	0
$413$	8147.71	0.970757
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1597.96	−0.187655
$418$	0	0
$419$	−11559.0	−1.34772	−0.673861	−	0.738858i	$-0.735365\pi$
$419$	−11559.0	−1.34772	−0.673861	+	0.738858i	$0.735365\pi$
$420$	0	0
$421$	−5096.66	−0.590014	−0.295007	−	0.955495i	$-0.595322\pi$
$421$	−5096.66	−0.590014	−0.295007	+	0.955495i	$0.595322\pi$
$422$	0	0
$423$	−3361.34	−0.386368
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6506.66	0.737422
$428$	0	0
$429$	−3551.11	−0.399649
$430$	0	0
$431$	13427.8	1.50069	0.750344	−	0.661048i	$-0.229888\pi$
$431$	13427.8	1.50069	0.750344	+	0.661048i	$0.229888\pi$
$432$	0	0
$433$	7029.48	0.780174	0.390087	−	0.920778i	$-0.372445\pi$
$433$	7029.48	0.780174	0.390087	+	0.920778i	$0.372445\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3150.78	−0.344902
$438$	0	0
$439$	2124.38	0.230959	0.115479	−	0.993310i	$-0.463160\pi$
$439$	2124.38	0.230959	0.115479	+	0.993310i	$0.463160\pi$
$440$	0	0
$441$	−1905.67	−0.205774
$442$	0	0
$443$	8144.17	0.873457	0.436728	−	0.899593i	$-0.356137\pi$
$443$	8144.17	0.873457	0.436728	+	0.899593i	$0.356137\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1407.03	−0.148882
$448$	0	0
$449$	−4584.57	−0.481869	−0.240934	−	0.970541i	$-0.577454\pi$
$449$	−4584.57	−0.481869	−0.240934	+	0.970541i	$0.577454\pi$
$450$	0	0
$451$	78.6639	0.00821317
$452$	0	0
$453$	9326.87	0.967361
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14742.6	−1.50904	−0.754519	−	0.656279i	$-0.772130\pi$
$457$	−14742.6	−1.50904	−0.754519	+	0.656279i	$0.772130\pi$
$458$	0	0
$459$	2805.89	0.285332
$460$	0	0
$461$	−163.931	−0.0165619	−0.00828096	−	0.999966i	$-0.502636\pi$
$461$	−163.931	−0.0165619	−0.00828096	+	0.999966i	$0.502636\pi$
$462$	0	0
$463$	9995.68	1.00332	0.501662	−	0.865064i	$-0.332722\pi$
$463$	9995.68	1.00332	0.501662	+	0.865064i	$0.332722\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10863.0	−1.07640	−0.538201	−	0.842816i	$-0.680896\pi$
$467$	−10863.0	−1.07640	−0.538201	+	0.842816i	$0.680896\pi$
$468$	0	0
$469$	−4474.68	−0.440557
$470$	0	0
$471$	43.5791	0.00426331
$472$	0	0
$473$	−2742.53	−0.266600
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−408.952	−0.0392550
$478$	0	0
$479$	2014.55	0.192165	0.0960825	−	0.995373i	$-0.469369\pi$
$479$	2014.55	0.192165	0.0960825	+	0.995373i	$0.469369\pi$
$480$	0	0
$481$	−13130.4	−1.24469
$482$	0	0
$483$	−3874.96	−0.365045
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9012.28	0.838574	0.419287	−	0.907854i	$-0.362280\pi$
$487$	9012.28	0.838574	0.419287	+	0.907854i	$0.362280\pi$
$488$	0	0
$489$	1296.15	0.119865
$490$	0	0
$491$	1584.21	0.145610	0.0728049	−	0.997346i	$-0.476805\pi$
$491$	1584.21	0.145610	0.0728049	+	0.997346i	$0.476805\pi$
$492$	0	0
$493$	−25056.7	−2.28904
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7877.34	0.710960
$498$	0	0
$499$	−12988.3	−1.16521	−0.582603	−	0.812757i	$-0.697966\pi$
$499$	−12988.3	−1.16521	−0.582603	+	0.812757i	$0.697966\pi$
$500$	0	0
$501$	−1391.55	−0.124091
$502$	0	0
$503$	−5645.47	−0.500435	−0.250218	−	0.968190i	$-0.580502\pi$
$503$	−5645.47	−0.500435	−0.250218	+	0.968190i	$0.580502\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9809.91	0.859317
$508$	0	0
$509$	4786.46	0.416809	0.208405	−	0.978043i	$-0.433173\pi$
$509$	4786.46	0.416809	0.208405	+	0.978043i	$0.433173\pi$
$510$	0	0
$511$	−4942.70	−0.427891
$512$	0	0
$513$	754.570	0.0649416
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5979.15	−0.508632
$518$	0	0
$519$	7697.54	0.651030
$520$	0	0
$521$	−8508.80	−0.715503	−0.357752	−	0.933817i	$-0.616457\pi$
$521$	−8508.80	−0.715503	−0.357752	+	0.933817i	$0.616457\pi$
$522$	0	0
$523$	−2414.60	−0.201879	−0.100940	−	0.994893i	$-0.532185\pi$
$523$	−2414.60	−0.201879	−0.100940	+	0.994893i	$0.532185\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31581.7	−2.61047
$528$	0	0
$529$	543.532	0.0446726
$530$	0	0
$531$	6400.50	0.523084
$532$	0	0
$533$	−363.311	−0.0295249
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−583.795	−0.0469137
$538$	0	0
$539$	−3389.81	−0.270889
$540$	0	0
$541$	10550.6	0.838460	0.419230	−	0.907880i	$-0.362300\pi$
$541$	10550.6	0.838460	0.419230	+	0.907880i	$0.362300\pi$
$542$	0	0
$543$	−10311.8	−0.814955
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12254.3	0.957870	0.478935	−	0.877850i	$-0.341023\pi$
$547$	12254.3	0.957870	0.478935	+	0.877850i	$0.341023\pi$
$548$	0	0
$549$	5111.35	0.397354
$550$	0	0
$551$	−6738.35	−0.520986
$552$	0	0
$553$	−12446.9	−0.957134
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20611.8	−1.56795	−0.783976	−	0.620791i	$-0.786811\pi$
$557$	−20611.8	−1.56795	−0.783976	+	0.620791i	$0.786811\pi$
$558$	0	0
$559$	12666.4	0.958378
$560$	0	0
$561$	4991.11	0.375624
$562$	0	0
$563$	14083.1	1.05423	0.527117	−	0.849793i	$-0.323273\pi$
$563$	14083.1	1.05423	0.527117	+	0.849793i	$0.323273\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	928.003	0.0687345
$568$	0	0
$569$	−13836.4	−1.01943	−0.509713	−	0.860344i	$-0.670248\pi$
$569$	−13836.4	−1.01943	−0.509713	+	0.860344i	$0.670248\pi$
$570$	0	0
$571$	−23484.8	−1.72121	−0.860604	−	0.509274i	$-0.829914\pi$
$571$	−23484.8	−1.72121	−0.860604	+	0.509274i	$0.829914\pi$
$572$	0	0
$573$	−12353.4	−0.900650
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10828.7	−0.781288	−0.390644	−	0.920542i	$-0.627748\pi$
$577$	−10828.7	−0.781288	−0.390644	+	0.920542i	$0.627748\pi$
$578$	0	0
$579$	3598.53	0.258290
$580$	0	0
$581$	−8254.03	−0.589389
$582$	0	0
$583$	−727.444	−0.0516769
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6065.64	0.426500	0.213250	−	0.976998i	$-0.431595\pi$
$587$	6065.64	0.426500	0.213250	+	0.976998i	$0.431595\pi$
$588$	0	0
$589$	−8493.06	−0.594144
$590$	0	0
$591$	6502.05	0.452553
$592$	0	0
$593$	−2166.82	−0.150052	−0.0750260	−	0.997182i	$-0.523904\pi$
$593$	−2166.82	−0.150052	−0.0750260	+	0.997182i	$0.523904\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5449.07	0.373560
$598$	0	0
$599$	−26728.3	−1.82319	−0.911594	−	0.411093i	$-0.865147\pi$
$599$	−26728.3	−1.82319	−0.911594	+	0.411093i	$0.865147\pi$
$600$	0	0
$601$	−14709.0	−0.998324	−0.499162	−	0.866509i	$-0.666359\pi$
$601$	−14709.0	−0.998324	−0.499162	+	0.866509i	$0.666359\pi$
$602$	0	0
$603$	−3515.11	−0.237391
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9621.82	−0.643390	−0.321695	−	0.946843i	$-0.604253\pi$
$607$	−9621.82	−0.643390	−0.321695	+	0.946843i	$0.604253\pi$
$608$	0	0
$609$	−8287.12	−0.551414
$610$	0	0
$611$	27614.9	1.82844
$612$	0	0
$613$	21971.7	1.44768	0.723842	−	0.689966i	$-0.242375\pi$
$613$	21971.7	1.44768	0.723842	+	0.689966i	$0.242375\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−516.493	−0.0337005	−0.0168503	−	0.999858i	$-0.505364\pi$
$617$	−516.493	−0.0337005	−0.0168503	+	0.999858i	$0.505364\pi$
$618$	0	0
$619$	30127.1	1.95624	0.978119	−	0.208044i	$-0.0667097\pi$
$619$	30127.1	1.95624	0.978119	+	0.208044i	$0.0667097\pi$
$620$	0	0
$621$	−3044.01	−0.196702
$622$	0	0
$623$	−7256.87	−0.466678
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1342.23	0.0854920
$628$	0	0
$629$	18454.9	1.16986
$630$	0	0
$631$	−9949.54	−0.627710	−0.313855	−	0.949471i	$-0.601621\pi$
$631$	−9949.54	−0.627710	−0.313855	+	0.949471i	$0.601621\pi$
$632$	0	0
$633$	1609.12	0.101037
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15655.9	0.973799
$638$	0	0
$639$	6188.10	0.383095
$640$	0	0
$641$	−30158.4	−1.85832	−0.929162	−	0.369672i	$-0.879470\pi$
$641$	−30158.4	−1.85832	−0.929162	+	0.369672i	$0.879470\pi$
$642$	0	0
$643$	16915.2	1.03743	0.518717	−	0.854946i	$-0.326410\pi$
$643$	16915.2	1.03743	0.518717	+	0.854946i	$0.326410\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14552.4	−0.884259	−0.442130	−	0.896951i	$-0.645777\pi$
$647$	−14552.4	−0.884259	−0.442130	+	0.896951i	$0.645777\pi$
$648$	0	0
$649$	11385.2	0.688611
$650$	0	0
$651$	−10445.1	−0.628844
$652$	0	0
$653$	15894.1	0.952503	0.476252	−	0.879309i	$-0.341995\pi$
$653$	15894.1	0.952503	0.476252	+	0.879309i	$0.341995\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3882.78	−0.230565
$658$	0	0
$659$	−20281.8	−1.19889	−0.599444	−	0.800416i	$-0.704612\pi$
$659$	−20281.8	−1.19889	−0.599444	+	0.800416i	$0.704612\pi$
$660$	0	0
$661$	−10123.9	−0.595727	−0.297864	−	0.954608i	$-0.596274\pi$
$661$	−10123.9	−0.595727	−0.297864	+	0.954608i	$0.596274\pi$
$662$	0	0
$663$	−23051.6	−1.35030
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27183.1	1.57801
$668$	0	0
$669$	17008.3	0.982930
$670$	0	0
$671$	9092.08	0.523094
$672$	0	0
$673$	29893.1	1.71217	0.856087	−	0.516832i	$-0.172889\pi$
$673$	29893.1	1.71217	0.856087	+	0.516832i	$0.172889\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6612.39	0.375383	0.187692	−	0.982228i	$-0.439899\pi$
$677$	6612.39	0.375383	0.187692	+	0.982228i	$0.439899\pi$
$678$	0	0
$679$	2514.83	0.142136
$680$	0	0
$681$	8044.01	0.452639
$682$	0	0
$683$	−19096.8	−1.06986	−0.534932	−	0.844895i	$-0.679663\pi$
$683$	−19096.8	−1.06986	−0.534932	+	0.844895i	$0.679663\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−12546.9	−0.696788
$688$	0	0
$689$	3359.72	0.185769
$690$	0	0
$691$	31232.2	1.71943	0.859717	−	0.510771i	$-0.170640\pi$
$691$	31232.2	1.71943	0.859717	+	0.510771i	$0.170640\pi$
$692$	0	0
$693$	1650.73	0.0904851
$694$	0	0
$695$	0	0
$696$	0	0
$697$	510.636	0.0277500
$698$	0	0
$699$	4875.83	0.263835
$700$	0	0
$701$	20678.4	1.11414	0.557070	−	0.830465i	$-0.311925\pi$
$701$	20678.4	1.11414	0.557070	+	0.830465i	$0.311925\pi$
$702$	0	0
$703$	4962.95	0.266261
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1511.44	−0.0804009
$708$	0	0
$709$	15125.0	0.801172	0.400586	−	0.916259i	$-0.368807\pi$
$709$	15125.0	0.801172	0.400586	+	0.916259i	$0.368807\pi$
$710$	0	0
$711$	−9777.74	−0.515744
$712$	0	0
$713$	34261.8	1.79960
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12307.9	−0.641070
$718$	0	0
$719$	2310.05	0.119820	0.0599098	−	0.998204i	$-0.480919\pi$
$719$	2310.05	0.119820	0.0599098	+	0.998204i	$0.480919\pi$
$720$	0	0
$721$	9306.94	0.480733
$722$	0	0
$723$	−17446.0	−0.897406
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−21890.3	−1.11674	−0.558368	−	0.829594i	$-0.688572\pi$
$727$	−21890.3	−1.11674	−0.558368	+	0.829594i	$0.688572\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−17802.8	−0.900764
$732$	0	0
$733$	31535.0	1.58905	0.794525	−	0.607232i	$-0.207720\pi$
$733$	31535.0	1.58905	0.794525	+	0.607232i	$0.207720\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6252.69	−0.312511
$738$	0	0
$739$	17924.8	0.892252	0.446126	−	0.894970i	$-0.352803\pi$
$739$	17924.8	0.892252	0.446126	+	0.894970i	$0.352803\pi$
$740$	0	0
$741$	−6199.12	−0.307329
$742$	0	0
$743$	8987.45	0.443765	0.221883	−	0.975073i	$-0.428780\pi$
$743$	8987.45	0.443765	0.221883	+	0.975073i	$0.428780\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−6484.01	−0.317587
$748$	0	0
$749$	19829.7	0.967372
$750$	0	0
$751$	23280.1	1.13116	0.565580	−	0.824693i	$-0.308652\pi$
$751$	23280.1	1.13116	0.565580	+	0.824693i	$0.308652\pi$
$752$	0	0
$753$	181.780	0.00879739
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−607.949	−0.0291893	−0.0145946	−	0.999893i	$-0.504646\pi$
$757$	−607.949	−0.0291893	−0.0145946	+	0.999893i	$0.504646\pi$
$758$	0	0
$759$	−5414.69	−0.258947
$760$	0	0
$761$	−8024.09	−0.382225	−0.191112	−	0.981568i	$-0.561210\pi$
$761$	−8024.09	−0.382225	−0.191112	+	0.981568i	$0.561210\pi$
$762$	0	0
$763$	−2592.62	−0.123013
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−52582.9	−2.47543
$768$	0	0
$769$	32499.1	1.52399	0.761994	−	0.647584i	$-0.224220\pi$
$769$	32499.1	1.52399	0.761994	+	0.647584i	$0.224220\pi$
$770$	0	0
$771$	−22847.7	−1.06724
$772$	0	0
$773$	−31904.3	−1.48450	−0.742249	−	0.670124i	$-0.766241\pi$
$773$	−31904.3	−1.48450	−0.742249	+	0.670124i	$0.766241\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6103.66	0.281811
$778$	0	0
$779$	137.322	0.00631589
$780$	0	0
$781$	11007.4	0.504323
$782$	0	0
$783$	−6510.01	−0.297125
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9472.32	0.429037	0.214518	−	0.976720i	$-0.431182\pi$
$787$	9472.32	0.429037	0.214518	+	0.976720i	$0.431182\pi$
$788$	0	0
$789$	−20514.3	−0.925637
$790$	0	0
$791$	−7454.36	−0.335078
$792$	0	0
$793$	−41992.0	−1.88043
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25280.8	1.12358	0.561790	−	0.827280i	$-0.310113\pi$
$797$	25280.8	1.12358	0.561790	+	0.827280i	$0.310113\pi$
$798$	0	0
$799$	−38812.9	−1.71852
$800$	0	0
$801$	−5700.69	−0.251466
$802$	0	0
$803$	−6906.69	−0.303526
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5350.74	−0.233401
$808$	0	0
$809$	−2732.05	−0.118732	−0.0593658	−	0.998236i	$-0.518908\pi$
$809$	−2732.05	−0.118732	−0.0593658	+	0.998236i	$0.518908\pi$
$810$	0	0
$811$	−10478.9	−0.453715	−0.226858	−	0.973928i	$-0.572845\pi$
$811$	−10478.9	−0.453715	−0.226858	+	0.973928i	$0.572845\pi$
$812$	0	0
$813$	10982.6	0.473771
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4787.59	−0.205014
$818$	0	0
$819$	−7623.96	−0.325278
$820$	0	0
$821$	23402.2	0.994815	0.497408	−	0.867517i	$-0.334285\pi$
$821$	23402.2	0.994815	0.497408	+	0.867517i	$0.334285\pi$
$822$	0	0
$823$	32871.9	1.39228	0.696138	−	0.717908i	$-0.254900\pi$
$823$	32871.9	1.39228	0.696138	+	0.717908i	$0.254900\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10385.7	0.436693	0.218346	−	0.975871i	$-0.429934\pi$
$827$	10385.7	0.436693	0.218346	+	0.975871i	$0.429934\pi$
$828$	0	0
$829$	8669.72	0.363223	0.181611	−	0.983370i	$-0.441869\pi$
$829$	8669.72	0.363223	0.181611	+	0.983370i	$0.441869\pi$
$830$	0	0
$831$	−5303.85	−0.221406
$832$	0	0
$833$	−22004.5	−0.915258
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8205.26	−0.338848
$838$	0	0
$839$	−11290.2	−0.464578	−0.232289	−	0.972647i	$-0.574621\pi$
$839$	−11290.2	−0.464578	−0.232289	+	0.972647i	$0.574621\pi$
$840$	0	0
$841$	33745.8	1.38365
$842$	0	0
$843$	−23631.2	−0.965482
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−12312.7	−0.499492
$848$	0	0
$849$	3559.23	0.143878
$850$	0	0
$851$	−20021.0	−0.806477
$852$	0	0
$853$	−27372.7	−1.09874	−0.549370	−	0.835579i	$-0.685132\pi$
$853$	−27372.7	−1.09874	−0.549370	+	0.835579i	$0.685132\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−32578.4	−1.29855	−0.649275	−	0.760554i	$-0.724928\pi$
$857$	−32578.4	−1.29855	−0.649275	+	0.760554i	$0.724928\pi$
$858$	0	0
$859$	−21054.5	−0.836287	−0.418144	−	0.908381i	$-0.637319\pi$
$859$	−21054.5	−0.836287	−0.418144	+	0.908381i	$0.637319\pi$
$860$	0	0
$861$	168.885	0.00668477
$862$	0	0
$863$	9800.23	0.386563	0.193281	−	0.981143i	$-0.438087\pi$
$863$	9800.23	0.386563	0.193281	+	0.981143i	$0.438087\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17660.2	0.691777
$868$	0	0
$869$	−17392.7	−0.678948
$870$	0	0
$871$	28878.2	1.12342
$872$	0	0
$873$	1975.54	0.0765888
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−25843.1	−0.995051	−0.497525	−	0.867449i	$-0.665758\pi$
$877$	−25843.1	−0.995051	−0.497525	+	0.867449i	$0.665758\pi$
$878$	0	0
$879$	−15001.1	−0.575624
$880$	0	0
$881$	45926.2	1.75629	0.878146	−	0.478393i	$-0.158780\pi$
$881$	45926.2	1.75629	0.878146	+	0.478393i	$0.158780\pi$
$882$	0	0
$883$	44707.6	1.70389	0.851943	−	0.523635i	$-0.175424\pi$
$883$	44707.6	1.70389	0.851943	+	0.523635i	$0.175424\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−187.047	−0.00708051	−0.00354025	−	0.999994i	$-0.501127\pi$
$887$	−187.047	−0.00708051	−0.00354025	+	0.999994i	$0.501127\pi$
$888$	0	0
$889$	−25842.1	−0.974935
$890$	0	0
$891$	1296.75	0.0487572
$892$	0	0
$893$	−10437.7	−0.391136
$894$	0	0
$895$	0	0
$896$	0	0
$897$	25007.9	0.930867
$898$	0	0
$899$	73273.5	2.71836
$900$	0	0
$901$	−4722.11	−0.174602
$902$	0	0
$903$	−5887.99	−0.216988
$904$	0	0
$905$	0	0
$906$	0	0
$907$	48905.8	1.79040	0.895199	−	0.445667i	$-0.147034\pi$
$907$	48905.8	1.79040	0.895199	+	0.445667i	$0.147034\pi$
$908$	0	0
$909$	−1187.32	−0.0433234
$910$	0	0
$911$	38028.5	1.38303	0.691514	−	0.722363i	$-0.256944\pi$
$911$	38028.5	1.38303	0.691514	+	0.722363i	$0.256944\pi$
$912$	0	0
$913$	−11533.8	−0.418086
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−23236.7	−0.836797
$918$	0	0
$919$	30750.5	1.10377	0.551885	−	0.833920i	$-0.313909\pi$
$919$	30750.5	1.10377	0.551885	+	0.833920i	$0.313909\pi$
$920$	0	0
$921$	9303.24	0.332847
$922$	0	0
$923$	−50838.0	−1.81295
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7311.14	0.259039
$928$	0	0
$929$	−19376.5	−0.684309	−0.342155	−	0.939644i	$-0.611157\pi$
$929$	−19376.5	−0.684309	−0.342155	+	0.939644i	$0.611157\pi$
$930$	0	0
$931$	−5917.53	−0.208313
$932$	0	0
$933$	−11345.2	−0.398099
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38450.9	1.34059	0.670296	−	0.742094i	$-0.266167\pi$
$937$	38450.9	1.34059	0.670296	+	0.742094i	$0.266167\pi$
$938$	0	0
$939$	17375.0	0.603846
$940$	0	0
$941$	43296.2	1.49991	0.749956	−	0.661488i	$-0.230075\pi$
$941$	43296.2	1.49991	0.749956	+	0.661488i	$0.230075\pi$
$942$	0	0
$943$	−553.971	−0.0191302
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12241.0	0.420041	0.210020	−	0.977697i	$-0.432647\pi$
$947$	12241.0	0.420041	0.210020	+	0.977697i	$0.432647\pi$
$948$	0	0
$949$	31898.7	1.09112
$950$	0	0
$951$	−10620.8	−0.362149
$952$	0	0
$953$	7604.93	0.258497	0.129249	−	0.991612i	$-0.458743\pi$
$953$	7604.93	0.258497	0.129249	+	0.991612i	$0.458743\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−11580.0	−0.391148
$958$	0	0
$959$	−2471.35	−0.0832159
$960$	0	0
$961$	62563.4	2.10008
$962$	0	0
$963$	15577.4	0.521261
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4640.44	−0.154319	−0.0771595	−	0.997019i	$-0.524585\pi$
$967$	−4640.44	−0.154319	−0.0771595	+	0.997019i	$0.524585\pi$
$968$	0	0
$969$	8712.91	0.288853
$970$	0	0
$971$	−8635.15	−0.285392	−0.142696	−	0.989767i	$-0.545577\pi$
$971$	−8635.15	−0.285392	−0.142696	+	0.989767i	$0.545577\pi$
$972$	0	0
$973$	−6102.50	−0.201066
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9333.88	0.305647	0.152824	−	0.988253i	$-0.451163\pi$
$977$	9333.88	0.305647	0.152824	+	0.988253i	$0.451163\pi$
$978$	0	0
$979$	−10140.4	−0.331040
$980$	0	0
$981$	−2036.66	−0.0662848
$982$	0	0
$983$	−16819.4	−0.545733	−0.272867	−	0.962052i	$-0.587972\pi$
$983$	−16819.4	−0.545733	−0.272867	+	0.962052i	$0.587972\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−12836.8	−0.413980
$988$	0	0
$989$	19313.6	0.620967
$990$	0	0
$991$	−33375.0	−1.06982	−0.534910	−	0.844909i	$-0.679654\pi$
$991$	−33375.0	−1.06982	−0.534910	+	0.844909i	$0.679654\pi$
$992$	0	0
$993$	16442.2	0.525456
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22267.5	−0.707341	−0.353670	−	0.935370i	$-0.615067\pi$
$997$	−22267.5	−0.707341	−0.353670	+	0.935370i	$0.615067\pi$
$998$	0	0
$999$	4794.77	0.151852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.cb.1.4		4
4.3	odd	2		2400.4.a.by.1.1		4
5.2	odd	4		480.4.f.g.289.3	✓	8
5.3	odd	4		480.4.f.g.289.8	yes	8
5.4	even	2		2400.4.a.by.1.2		4
15.2	even	4		1440.4.f.l.289.4		8
15.8	even	4		1440.4.f.l.289.1		8
20.3	even	4		480.4.f.g.289.4	yes	8
20.7	even	4		480.4.f.g.289.7	yes	8
20.19	odd	2	inner	2400.4.a.cb.1.3		4
40.3	even	4		960.4.f.t.769.5		8
40.13	odd	4		960.4.f.t.769.1		8
40.27	even	4		960.4.f.t.769.2		8
40.37	odd	4		960.4.f.t.769.6		8
60.23	odd	4		1440.4.f.l.289.2		8
60.47	odd	4		1440.4.f.l.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.f.g.289.3	✓	8	5.2	odd	4
480.4.f.g.289.4	yes	8	20.3	even	4
480.4.f.g.289.7	yes	8	20.7	even	4
480.4.f.g.289.8	yes	8	5.3	odd	4
960.4.f.t.769.1		8	40.13	odd	4
960.4.f.t.769.2		8	40.27	even	4
960.4.f.t.769.5		8	40.3	even	4
960.4.f.t.769.6		8	40.37	odd	4
1440.4.f.l.289.1		8	15.8	even	4
1440.4.f.l.289.2		8	60.23	odd	4
1440.4.f.l.289.3		8	60.47	odd	4
1440.4.f.l.289.4		8	15.2	even	4
2400.4.a.by.1.1		4	4.3	odd	2
2400.4.a.by.1.2		4	5.4	even	2
2400.4.a.cb.1.3		4	20.19	odd	2	inner
2400.4.a.cb.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +11.4568 q^{7} +9.00000 q^{9} +16.0092 q^{11} -73.9390 q^{13} +103.922 q^{17} +27.9470 q^{19} +34.3705 q^{21} -112.741 q^{23} +27.0000 q^{27} -241.111 q^{29} -303.899 q^{31} +48.0277 q^{33} +177.584 q^{37} -221.817 q^{39} +4.91366 q^{41} -171.309 q^{43} -373.482 q^{47} -211.741 q^{49} +311.765 q^{51} -45.4391 q^{53} +83.8411 q^{57} +711.166 q^{59} +567.928 q^{61} +103.111 q^{63} -390.568 q^{67} -338.223 q^{69} +687.567 q^{71} -431.420 q^{73} +183.415 q^{77} -1086.42 q^{79} +81.0000 q^{81} -720.446 q^{83} -723.334 q^{87} -633.410 q^{89} -847.106 q^{91} -911.696 q^{93} +219.505 q^{97} +144.083 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} - 12 q^{7} + 36 q^{9} - 36 q^{21} - 104 q^{23} + 108 q^{27} - 444 q^{29} - 96 q^{41} + 240 q^{43} - 800 q^{47} - 500 q^{49} - 504 q^{61} - 108 q^{63} - 984 q^{67} - 312 q^{69} + 324 q^{81}+ \cdots + 936 q^{89}+O(q^{100}) \) 4 * q + 12 * q^3 - 12 * q^7 + 36 * q^9 - 36 * q^21 - 104 * q^23 + 108 * q^27 - 444 * q^29 - 96 * q^41 + 240 * q^43 - 800 * q^47 - 500 * q^49 - 504 * q^61 - 108 * q^63 - 984 * q^67 - 312 * q^69 + 324 * q^81 - 800 * q^83 - 1332 * q^87 + 936 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more