LMFDB - Embedded newform 6040.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 9x^{6} + 32x^{5} - 17x^{4} - 27x^{3} + 10x^{2} + 3x - 1 $ x^9 - x^8 - 11x^7 + 9x^6 + 32x^5 - 17x^4 - 27x^3 + 10x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.02556$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.75822 q^{3} +1.00000 q^{5} +1.53187 q^{7} +0.0913304 q^{9} +O(q^{10})$	$q-1.75822 q^{3} +1.00000 q^{5} +1.53187 q^{7} +0.0913304 q^{9} -5.86599 q^{11} +0.896819 q^{13} -1.75822 q^{15} -5.84925 q^{17} +3.50732 q^{19} -2.69336 q^{21} +2.74263 q^{23} +1.00000 q^{25} +5.11408 q^{27} +2.53080 q^{29} +5.38052 q^{31} +10.3137 q^{33} +1.53187 q^{35} +1.58441 q^{37} -1.57680 q^{39} +1.42627 q^{41} +8.60118 q^{43} +0.0913304 q^{45} +1.37054 q^{47} -4.65338 q^{49} +10.2843 q^{51} -10.7071 q^{53} -5.86599 q^{55} -6.16663 q^{57} -0.677690 q^{59} +8.86107 q^{61} +0.139906 q^{63} +0.896819 q^{65} -9.39938 q^{67} -4.82215 q^{69} -4.08725 q^{71} -4.48639 q^{73} -1.75822 q^{75} -8.98591 q^{77} +6.93088 q^{79} -9.26565 q^{81} -5.69567 q^{83} -5.84925 q^{85} -4.44971 q^{87} -3.48085 q^{89} +1.37381 q^{91} -9.46012 q^{93} +3.50732 q^{95} -10.1605 q^{97} -0.535743 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.75822	−1.01511	−0.507554	−	0.861620i	$-0.669450\pi$
$3$	−1.75822	−1.01511	−0.507554	+	0.861620i	$0.669450\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.53187	0.578991	0.289496	−	0.957179i	$-0.406512\pi$
$7$	1.53187	0.578991	0.289496	+	0.957179i	$0.406512\pi$
$8$	0	0
$9$	0.0913304	0.0304435
$10$	0	0
$11$	−5.86599	−1.76866	−0.884331	−	0.466860i	$-0.845385\pi$
$11$	−5.86599	−1.76866	−0.884331	+	0.466860i	$0.845385\pi$
$12$	0	0
$13$	0.896819	0.248733	0.124366	−	0.992236i	$-0.460310\pi$
$13$	0.896819	0.248733	0.124366	+	0.992236i	$0.460310\pi$
$14$	0	0
$15$	−1.75822	−0.453970
$16$	0	0
$17$	−5.84925	−1.41865	−0.709326	−	0.704880i	$-0.751001\pi$
$17$	−5.84925	−1.41865	−0.709326	+	0.704880i	$0.751001\pi$
$18$	0	0
$19$	3.50732	0.804634	0.402317	−	0.915500i	$-0.368205\pi$
$19$	3.50732	0.804634	0.402317	+	0.915500i	$0.368205\pi$
$20$	0	0
$21$	−2.69336	−0.587738
$22$	0	0
$23$	2.74263	0.571879	0.285939	−	0.958248i	$-0.407694\pi$
$23$	2.74263	0.571879	0.285939	+	0.958248i	$0.407694\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.11408	0.984204
$28$	0	0
$29$	2.53080	0.469959	0.234979	−	0.972000i	$-0.424498\pi$
$29$	2.53080	0.469959	0.234979	+	0.972000i	$0.424498\pi$
$30$	0	0
$31$	5.38052	0.966369	0.483185	−	0.875519i	$-0.339480\pi$
$31$	5.38052	0.966369	0.483185	+	0.875519i	$0.339480\pi$
$32$	0	0
$33$	10.3137	1.79538
$34$	0	0
$35$	1.53187	0.258933
$36$	0	0
$37$	1.58441	0.260476	0.130238	−	0.991483i	$-0.458426\pi$
$37$	1.58441	0.260476	0.130238	+	0.991483i	$0.458426\pi$
$38$	0	0
$39$	−1.57680	−0.252491
$40$	0	0
$41$	1.42627	0.222746	0.111373	−	0.993779i	$-0.464475\pi$
$41$	1.42627	0.222746	0.111373	+	0.993779i	$0.464475\pi$
$42$	0	0
$43$	8.60118	1.31167	0.655833	−	0.754906i	$-0.272317\pi$
$43$	8.60118	1.31167	0.655833	+	0.754906i	$0.272317\pi$
$44$	0	0
$45$	0.0913304	0.0136147
$46$	0	0
$47$	1.37054	0.199913	0.0999567	−	0.994992i	$-0.468130\pi$
$47$	1.37054	0.199913	0.0999567	+	0.994992i	$0.468130\pi$
$48$	0	0
$49$	−4.65338	−0.664769
$50$	0	0
$51$	10.2843	1.44008
$52$	0	0
$53$	−10.7071	−1.47074	−0.735369	−	0.677667i	$-0.762991\pi$
$53$	−10.7071	−1.47074	−0.735369	+	0.677667i	$0.762991\pi$
$54$	0	0
$55$	−5.86599	−0.790970
$56$	0	0
$57$	−6.16663	−0.816790
$58$	0	0
$59$	−0.677690	−0.0882278	−0.0441139	−	0.999027i	$-0.514046\pi$
$59$	−0.677690	−0.0882278	−0.0441139	+	0.999027i	$0.514046\pi$
$60$	0	0
$61$	8.86107	1.13454	0.567272	−	0.823531i	$-0.307999\pi$
$61$	8.86107	1.13454	0.567272	+	0.823531i	$0.307999\pi$
$62$	0	0
$63$	0.139906	0.0176265
$64$	0	0
$65$	0.896819	0.111237
$66$	0	0
$67$	−9.39938	−1.14832	−0.574158	−	0.818744i	$-0.694671\pi$
$67$	−9.39938	−1.14832	−0.574158	+	0.818744i	$0.694671\pi$
$68$	0	0
$69$	−4.82215	−0.580519
$70$	0	0
$71$	−4.08725	−0.485067	−0.242533	−	0.970143i	$-0.577978\pi$
$71$	−4.08725	−0.485067	−0.242533	+	0.970143i	$0.577978\pi$
$72$	0	0
$73$	−4.48639	−0.525093	−0.262546	−	0.964919i	$-0.584562\pi$
$73$	−4.48639	−0.525093	−0.262546	+	0.964919i	$0.584562\pi$
$74$	0	0
$75$	−1.75822	−0.203022
$76$	0	0
$77$	−8.98591	−1.02404
$78$	0	0
$79$	6.93088	0.779785	0.389892	−	0.920860i	$-0.372512\pi$
$79$	6.93088	0.779785	0.389892	+	0.920860i	$0.372512\pi$
$80$	0	0
$81$	−9.26565	−1.02952
$82$	0	0
$83$	−5.69567	−0.625181	−0.312590	−	0.949888i	$-0.601197\pi$
$83$	−5.69567	−0.625181	−0.312590	+	0.949888i	$0.601197\pi$
$84$	0	0
$85$	−5.84925	−0.634441
$86$	0	0
$87$	−4.44971	−0.477059
$88$	0	0
$89$	−3.48085	−0.368969	−0.184485	−	0.982835i	$-0.559062\pi$
$89$	−3.48085	−0.368969	−0.184485	+	0.982835i	$0.559062\pi$
$90$	0	0
$91$	1.37381	0.144014
$92$	0	0
$93$	−9.46012	−0.980969
$94$	0	0
$95$	3.50732	0.359843
$96$	0	0
$97$	−10.1605	−1.03164	−0.515821	−	0.856696i	$-0.672513\pi$
$97$	−10.1605	−1.03164	−0.515821	+	0.856696i	$0.672513\pi$
$98$	0	0
$99$	−0.535743	−0.0538442
$100$	0	0
$101$	−4.31990	−0.429846	−0.214923	−	0.976631i	$-0.568950\pi$
$101$	−4.31990	−0.429846	−0.214923	+	0.976631i	$0.568950\pi$
$102$	0	0
$103$	1.60268	0.157916	0.0789581	−	0.996878i	$-0.474841\pi$
$103$	1.60268	0.157916	0.0789581	+	0.996878i	$0.474841\pi$
$104$	0	0
$105$	−2.69336	−0.262845
$106$	0	0
$107$	3.00707	0.290704	0.145352	−	0.989380i	$-0.453568\pi$
$107$	3.00707	0.290704	0.145352	+	0.989380i	$0.453568\pi$
$108$	0	0
$109$	−3.79992	−0.363967	−0.181983	−	0.983302i	$-0.558252\pi$
$109$	−3.79992	−0.363967	−0.181983	+	0.983302i	$0.558252\pi$
$110$	0	0
$111$	−2.78574	−0.264411
$112$	0	0
$113$	−9.81560	−0.923374	−0.461687	−	0.887043i	$-0.652756\pi$
$113$	−9.81560	−0.923374	−0.461687	+	0.887043i	$0.652756\pi$
$114$	0	0
$115$	2.74263	0.255752
$116$	0	0
$117$	0.0819068	0.00757229
$118$	0	0
$119$	−8.96028	−0.821387
$120$	0	0
$121$	23.4098	2.12817
$122$	0	0
$123$	−2.50770	−0.226111
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.8098	1.22542	0.612712	−	0.790306i	$-0.290079\pi$
$127$	13.8098	1.22542	0.612712	+	0.790306i	$0.290079\pi$
$128$	0	0
$129$	−15.1227	−1.33148
$130$	0	0
$131$	4.30009	0.375700	0.187850	−	0.982198i	$-0.439848\pi$
$131$	4.30009	0.375700	0.187850	+	0.982198i	$0.439848\pi$
$132$	0	0
$133$	5.37275	0.465876
$134$	0	0
$135$	5.11408	0.440150
$136$	0	0
$137$	−8.88971	−0.759499	−0.379750	−	0.925089i	$-0.623990\pi$
$137$	−8.88971	−0.759499	−0.379750	+	0.925089i	$0.623990\pi$
$138$	0	0
$139$	−13.0154	−1.10395	−0.551974	−	0.833861i	$-0.686125\pi$
$139$	−13.0154	−1.10395	−0.551974	+	0.833861i	$0.686125\pi$
$140$	0	0
$141$	−2.40970	−0.202934
$142$	0	0
$143$	−5.26073	−0.439924
$144$	0	0
$145$	2.53080	0.210172
$146$	0	0
$147$	8.18166	0.674812
$148$	0	0
$149$	−16.0145	−1.31196	−0.655979	−	0.754780i	$-0.727744\pi$
$149$	−16.0145	−1.31196	−0.655979	+	0.754780i	$0.727744\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−0.534214	−0.0431887
$154$	0	0
$155$	5.38052	0.432173
$156$	0	0
$157$	−4.59585	−0.366789	−0.183395	−	0.983039i	$-0.558709\pi$
$157$	−4.59585	−0.366789	−0.183395	+	0.983039i	$0.558709\pi$
$158$	0	0
$159$	18.8255	1.49296
$160$	0	0
$161$	4.20135	0.331113
$162$	0	0
$163$	9.09815	0.712622	0.356311	−	0.934367i	$-0.384034\pi$
$163$	9.09815	0.712622	0.356311	+	0.934367i	$0.384034\pi$
$164$	0	0
$165$	10.3137	0.802919
$166$	0	0
$167$	−0.796650	−0.0616466	−0.0308233	−	0.999525i	$-0.509813\pi$
$167$	−0.796650	−0.0616466	−0.0308233	+	0.999525i	$0.509813\pi$
$168$	0	0
$169$	−12.1957	−0.938132
$170$	0	0
$171$	0.320325	0.0244958
$172$	0	0
$173$	7.12220	0.541491	0.270745	−	0.962651i	$-0.412730\pi$
$173$	7.12220	0.541491	0.270745	+	0.962651i	$0.412730\pi$
$174$	0	0
$175$	1.53187	0.115798
$176$	0	0
$177$	1.19153	0.0895607
$178$	0	0
$179$	17.2246	1.28743	0.643714	−	0.765266i	$-0.277393\pi$
$179$	17.2246	1.28743	0.643714	+	0.765266i	$0.277393\pi$
$180$	0	0
$181$	−14.0189	−1.04202	−0.521010	−	0.853551i	$-0.674445\pi$
$181$	−14.0189	−1.04202	−0.521010	+	0.853551i	$0.674445\pi$
$182$	0	0
$183$	−15.5797	−1.15168
$184$	0	0
$185$	1.58441	0.116488
$186$	0	0
$187$	34.3117	2.50912
$188$	0	0
$189$	7.83408	0.569846
$190$	0	0
$191$	−20.1365	−1.45703	−0.728515	−	0.685030i	$-0.759789\pi$
$191$	−20.1365	−1.45703	−0.728515	+	0.685030i	$0.759789\pi$
$192$	0	0
$193$	15.4161	1.10968	0.554839	−	0.831958i	$-0.312780\pi$
$193$	15.4161	1.10968	0.554839	+	0.831958i	$0.312780\pi$
$194$	0	0
$195$	−1.57680	−0.112917
$196$	0	0
$197$	−1.48952	−0.106124	−0.0530621	−	0.998591i	$-0.516898\pi$
$197$	−1.48952	−0.106124	−0.0530621	+	0.998591i	$0.516898\pi$
$198$	0	0
$199$	−19.6758	−1.39478	−0.697389	−	0.716693i	$-0.745655\pi$
$199$	−19.6758	−1.39478	−0.697389	+	0.716693i	$0.745655\pi$
$200$	0	0
$201$	16.5262	1.16567
$202$	0	0
$203$	3.87686	0.272102
$204$	0	0
$205$	1.42627	0.0996151
$206$	0	0
$207$	0.250486	0.0174100
$208$	0	0
$209$	−20.5739	−1.42313
$210$	0	0
$211$	7.42206	0.510956	0.255478	−	0.966815i	$-0.417767\pi$
$211$	7.42206	0.510956	0.255478	+	0.966815i	$0.417767\pi$
$212$	0	0
$213$	7.18627	0.492395
$214$	0	0
$215$	8.60118	0.586595
$216$	0	0
$217$	8.24223	0.559519
$218$	0	0
$219$	7.88806	0.533025
$220$	0	0
$221$	−5.24572	−0.352865
$222$	0	0
$223$	−16.3488	−1.09480	−0.547398	−	0.836873i	$-0.684381\pi$
$223$	−16.3488	−1.09480	−0.547398	+	0.836873i	$0.684381\pi$
$224$	0	0
$225$	0.0913304	0.00608869
$226$	0	0
$227$	17.1484	1.13818	0.569091	−	0.822275i	$-0.307295\pi$
$227$	17.1484	1.13818	0.569091	+	0.822275i	$0.307295\pi$
$228$	0	0
$229$	−3.25018	−0.214778	−0.107389	−	0.994217i	$-0.534249\pi$
$229$	−3.25018	−0.214778	−0.107389	+	0.994217i	$0.534249\pi$
$230$	0	0
$231$	15.7992	1.03951
$232$	0	0
$233$	5.62635	0.368594	0.184297	−	0.982871i	$-0.440999\pi$
$233$	5.62635	0.368594	0.184297	+	0.982871i	$0.440999\pi$
$234$	0	0
$235$	1.37054	0.0894040
$236$	0	0
$237$	−12.1860	−0.791565
$238$	0	0
$239$	−24.2274	−1.56714	−0.783569	−	0.621305i	$-0.786603\pi$
$239$	−24.2274	−1.56714	−0.783569	+	0.621305i	$0.786603\pi$
$240$	0	0
$241$	6.52347	0.420214	0.210107	−	0.977678i	$-0.432619\pi$
$241$	6.52347	0.420214	0.210107	+	0.977678i	$0.432619\pi$
$242$	0	0
$243$	0.948806	0.0608660
$244$	0	0
$245$	−4.65338	−0.297294
$246$	0	0
$247$	3.14543	0.200139
$248$	0	0
$249$	10.0142	0.634626
$250$	0	0
$251$	−18.6179	−1.17515	−0.587575	−	0.809170i	$-0.699917\pi$
$251$	−18.6179	−1.17515	−0.587575	+	0.809170i	$0.699917\pi$
$252$	0	0
$253$	−16.0883	−1.01146
$254$	0	0
$255$	10.2843	0.644025
$256$	0	0
$257$	−8.21540	−0.512462	−0.256231	−	0.966616i	$-0.582481\pi$
$257$	−8.21540	−0.512462	−0.256231	+	0.966616i	$0.582481\pi$
$258$	0	0
$259$	2.42711	0.150813
$260$	0	0
$261$	0.231139	0.0143072
$262$	0	0
$263$	−5.54055	−0.341645	−0.170822	−	0.985302i	$-0.554642\pi$
$263$	−5.54055	−0.341645	−0.170822	+	0.985302i	$0.554642\pi$
$264$	0	0
$265$	−10.7071	−0.657734
$266$	0	0
$267$	6.12009	0.374543
$268$	0	0
$269$	−11.4492	−0.698067	−0.349034	−	0.937110i	$-0.613490\pi$
$269$	−11.4492	−0.698067	−0.349034	+	0.937110i	$0.613490\pi$
$270$	0	0
$271$	−27.4858	−1.66965	−0.834823	−	0.550519i	$-0.814430\pi$
$271$	−27.4858	−1.66965	−0.834823	+	0.550519i	$0.814430\pi$
$272$	0	0
$273$	−2.41545	−0.146190
$274$	0	0
$275$	−5.86599	−0.353732
$276$	0	0
$277$	16.8535	1.01263	0.506316	−	0.862348i	$-0.331007\pi$
$277$	16.8535	1.01263	0.506316	+	0.862348i	$0.331007\pi$
$278$	0	0
$279$	0.491404	0.0294196
$280$	0	0
$281$	−21.7851	−1.29959	−0.649794	−	0.760110i	$-0.725145\pi$
$281$	−21.7851	−1.29959	−0.649794	+	0.760110i	$0.725145\pi$
$282$	0	0
$283$	0.920695	0.0547297	0.0273648	−	0.999626i	$-0.491288\pi$
$283$	0.920695	0.0547297	0.0273648	+	0.999626i	$0.491288\pi$
$284$	0	0
$285$	−6.16663	−0.365280
$286$	0	0
$287$	2.18486	0.128968
$288$	0	0
$289$	17.2138	1.01257
$290$	0	0
$291$	17.8644	1.04723
$292$	0	0
$293$	−20.7956	−1.21489	−0.607446	−	0.794361i	$-0.707806\pi$
$293$	−20.7956	−1.21489	−0.607446	+	0.794361i	$0.707806\pi$
$294$	0	0
$295$	−0.677690	−0.0394567
$296$	0	0
$297$	−29.9991	−1.74072
$298$	0	0
$299$	2.45965	0.142245
$300$	0	0
$301$	13.1759	0.759444
$302$	0	0
$303$	7.59533	0.436340
$304$	0	0
$305$	8.86107	0.507383
$306$	0	0
$307$	28.2546	1.61257	0.806287	−	0.591524i	$-0.201474\pi$
$307$	28.2546	1.61257	0.806287	+	0.591524i	$0.201474\pi$
$308$	0	0
$309$	−2.81785	−0.160302
$310$	0	0
$311$	−29.4020	−1.66723	−0.833617	−	0.552343i	$-0.813734\pi$
$311$	−29.4020	−1.66723	−0.833617	+	0.552343i	$0.813734\pi$
$312$	0	0
$313$	−10.4870	−0.592759	−0.296379	−	0.955070i	$-0.595779\pi$
$313$	−10.4870	−0.592759	−0.296379	+	0.955070i	$0.595779\pi$
$314$	0	0
$315$	0.139906	0.00788281
$316$	0	0
$317$	16.4809	0.925660	0.462830	−	0.886447i	$-0.346834\pi$
$317$	16.4809	0.925660	0.462830	+	0.886447i	$0.346834\pi$
$318$	0	0
$319$	−14.8457	−0.831198
$320$	0	0
$321$	−5.28708	−0.295096
$322$	0	0
$323$	−20.5152	−1.14150
$324$	0	0
$325$	0.896819	0.0497466
$326$	0	0
$327$	6.68109	0.369465
$328$	0	0
$329$	2.09948	0.115748
$330$	0	0
$331$	−21.4363	−1.17824	−0.589122	−	0.808044i	$-0.700526\pi$
$331$	−21.4363	−1.17824	−0.589122	+	0.808044i	$0.700526\pi$
$332$	0	0
$333$	0.144705	0.00792978
$334$	0	0
$335$	−9.39938	−0.513543
$336$	0	0
$337$	12.3688	0.673769	0.336884	−	0.941546i	$-0.390627\pi$
$337$	12.3688	0.673769	0.336884	+	0.941546i	$0.390627\pi$
$338$	0	0
$339$	17.2580	0.937324
$340$	0	0
$341$	−31.5620	−1.70918
$342$	0	0
$343$	−17.8514	−0.963887
$344$	0	0
$345$	−4.82215	−0.259616
$346$	0	0
$347$	18.5505	0.995844	0.497922	−	0.867222i	$-0.334097\pi$
$347$	18.5505	0.995844	0.497922	+	0.867222i	$0.334097\pi$
$348$	0	0
$349$	23.2080	1.24230	0.621148	−	0.783693i	$-0.286667\pi$
$349$	23.2080	1.24230	0.621148	+	0.783693i	$0.286667\pi$
$350$	0	0
$351$	4.58640	0.244804
$352$	0	0
$353$	−0.625977	−0.0333174	−0.0166587	−	0.999861i	$-0.505303\pi$
$353$	−0.625977	−0.0333174	−0.0166587	+	0.999861i	$0.505303\pi$
$354$	0	0
$355$	−4.08725	−0.216929
$356$	0	0
$357$	15.7541	0.833796
$358$	0	0
$359$	14.0173	0.739804	0.369902	−	0.929071i	$-0.379391\pi$
$359$	14.0173	0.739804	0.369902	+	0.929071i	$0.379391\pi$
$360$	0	0
$361$	−6.69872	−0.352564
$362$	0	0
$363$	−41.1596	−2.16032
$364$	0	0
$365$	−4.48639	−0.234829
$366$	0	0
$367$	−5.42961	−0.283423	−0.141712	−	0.989908i	$-0.545261\pi$
$367$	−5.42961	−0.283423	−0.141712	+	0.989908i	$0.545261\pi$
$368$	0	0
$369$	0.130262	0.00678116
$370$	0	0
$371$	−16.4019	−0.851544
$372$	0	0
$373$	−10.6591	−0.551909	−0.275954	−	0.961171i	$-0.588994\pi$
$373$	−10.6591	−0.551909	−0.275954	+	0.961171i	$0.588994\pi$
$374$	0	0
$375$	−1.75822	−0.0907940
$376$	0	0
$377$	2.26967	0.116894
$378$	0	0
$379$	−26.0060	−1.33584	−0.667919	−	0.744234i	$-0.732815\pi$
$379$	−26.0060	−1.33584	−0.667919	+	0.744234i	$0.732815\pi$
$380$	0	0
$381$	−24.2807	−1.24394
$382$	0	0
$383$	−6.36092	−0.325028	−0.162514	−	0.986706i	$-0.551960\pi$
$383$	−6.36092	−0.325028	−0.162514	+	0.986706i	$0.551960\pi$
$384$	0	0
$385$	−8.98591	−0.457965
$386$	0	0
$387$	0.785549	0.0399317
$388$	0	0
$389$	37.6073	1.90677	0.953384	−	0.301760i	$-0.0975741\pi$
$389$	37.6073	1.90677	0.953384	+	0.301760i	$0.0975741\pi$
$390$	0	0
$391$	−16.0424	−0.811297
$392$	0	0
$393$	−7.56049	−0.381376
$394$	0	0
$395$	6.93088	0.348730
$396$	0	0
$397$	−33.8051	−1.69663	−0.848315	−	0.529491i	$-0.822383\pi$
$397$	−33.8051	−1.69663	−0.848315	+	0.529491i	$0.822383\pi$
$398$	0	0
$399$	−9.44646	−0.472914
$400$	0	0
$401$	15.1960	0.758851	0.379425	−	0.925222i	$-0.376122\pi$
$401$	15.1960	0.758851	0.379425	+	0.925222i	$0.376122\pi$
$402$	0	0
$403$	4.82535	0.240368
$404$	0	0
$405$	−9.26565	−0.460414
$406$	0	0
$407$	−9.29415	−0.460694
$408$	0	0
$409$	−8.30582	−0.410696	−0.205348	−	0.978689i	$-0.565833\pi$
$409$	−8.30582	−0.410696	−0.205348	+	0.978689i	$0.565833\pi$
$410$	0	0
$411$	15.6300	0.770973
$412$	0	0
$413$	−1.03813	−0.0510831
$414$	0	0
$415$	−5.69567	−0.279589
$416$	0	0
$417$	22.8838	1.12063
$418$	0	0
$419$	15.1450	0.739880	0.369940	−	0.929056i	$-0.379378\pi$
$419$	15.1450	0.739880	0.369940	+	0.929056i	$0.379378\pi$
$420$	0	0
$421$	−9.05358	−0.441245	−0.220622	−	0.975359i	$-0.570809\pi$
$421$	−9.05358	−0.441245	−0.220622	+	0.975359i	$0.570809\pi$
$422$	0	0
$423$	0.125172	0.00608606
$424$	0	0
$425$	−5.84925	−0.283730
$426$	0	0
$427$	13.5740	0.656891
$428$	0	0
$429$	9.24951	0.446571
$430$	0	0
$431$	−7.77083	−0.374307	−0.187154	−	0.982331i	$-0.559926\pi$
$431$	−7.77083	−0.374307	−0.187154	+	0.982331i	$0.559926\pi$
$432$	0	0
$433$	13.7060	0.658669	0.329334	−	0.944213i	$-0.393176\pi$
$433$	13.7060	0.658669	0.329334	+	0.944213i	$0.393176\pi$
$434$	0	0
$435$	−4.44971	−0.213347
$436$	0	0
$437$	9.61929	0.460153
$438$	0	0
$439$	13.1755	0.628834	0.314417	−	0.949285i	$-0.398191\pi$
$439$	13.1755	0.628834	0.314417	+	0.949285i	$0.398191\pi$
$440$	0	0
$441$	−0.424995	−0.0202379
$442$	0	0
$443$	−18.4434	−0.876271	−0.438136	−	0.898909i	$-0.644361\pi$
$443$	−18.4434	−0.876271	−0.438136	+	0.898909i	$0.644361\pi$
$444$	0	0
$445$	−3.48085	−0.165008
$446$	0	0
$447$	28.1569	1.33178
$448$	0	0
$449$	−36.2875	−1.71251	−0.856257	−	0.516550i	$-0.827216\pi$
$449$	−36.2875	−1.71251	−0.856257	+	0.516550i	$0.827216\pi$
$450$	0	0
$451$	−8.36649	−0.393963
$452$	0	0
$453$	1.75822	0.0826083
$454$	0	0
$455$	1.37381	0.0644051
$456$	0	0
$457$	−21.3689	−0.999594	−0.499797	−	0.866143i	$-0.666592\pi$
$457$	−21.3689	−0.999594	−0.499797	+	0.866143i	$0.666592\pi$
$458$	0	0
$459$	−29.9135	−1.39624
$460$	0	0
$461$	−30.9613	−1.44201	−0.721006	−	0.692929i	$-0.756320\pi$
$461$	−30.9613	−1.44201	−0.721006	+	0.692929i	$0.756320\pi$
$462$	0	0
$463$	1.32893	0.0617606	0.0308803	−	0.999523i	$-0.490169\pi$
$463$	1.32893	0.0617606	0.0308803	+	0.999523i	$0.490169\pi$
$464$	0	0
$465$	−9.46012	−0.438703
$466$	0	0
$467$	19.4495	0.900017	0.450009	−	0.893024i	$-0.351421\pi$
$467$	19.4495	0.900017	0.450009	+	0.893024i	$0.351421\pi$
$468$	0	0
$469$	−14.3986	−0.664866
$470$	0	0
$471$	8.08051	0.372330
$472$	0	0
$473$	−50.4544	−2.31990
$474$	0	0
$475$	3.50732	0.160927
$476$	0	0
$477$	−0.977886	−0.0447743
$478$	0	0
$479$	29.0961	1.32944	0.664718	−	0.747095i	$-0.268552\pi$
$479$	29.0961	1.32944	0.664718	+	0.747095i	$0.268552\pi$
$480$	0	0
$481$	1.42093	0.0647889
$482$	0	0
$483$	−7.38689	−0.336115
$484$	0	0
$485$	−10.1605	−0.461365
$486$	0	0
$487$	−6.13509	−0.278007	−0.139004	−	0.990292i	$-0.544390\pi$
$487$	−6.13509	−0.278007	−0.139004	+	0.990292i	$0.544390\pi$
$488$	0	0
$489$	−15.9965	−0.723388
$490$	0	0
$491$	41.3118	1.86438	0.932188	−	0.361975i	$-0.117898\pi$
$491$	41.3118	1.86438	0.932188	+	0.361975i	$0.117898\pi$
$492$	0	0
$493$	−14.8033	−0.666708
$494$	0	0
$495$	−0.535743	−0.0240799
$496$	0	0
$497$	−6.26112	−0.280850
$498$	0	0
$499$	1.53948	0.0689165	0.0344582	−	0.999406i	$-0.489029\pi$
$499$	1.53948	0.0689165	0.0344582	+	0.999406i	$0.489029\pi$
$500$	0	0
$501$	1.40068	0.0625780
$502$	0	0
$503$	−13.6651	−0.609296	−0.304648	−	0.952465i	$-0.598539\pi$
$503$	−13.6651	−0.609296	−0.304648	+	0.952465i	$0.598539\pi$
$504$	0	0
$505$	−4.31990	−0.192233
$506$	0	0
$507$	21.4427	0.952305
$508$	0	0
$509$	−31.6725	−1.40386	−0.701929	−	0.712247i	$-0.747678\pi$
$509$	−31.6725	−1.40386	−0.701929	+	0.712247i	$0.747678\pi$
$510$	0	0
$511$	−6.87256	−0.304024
$512$	0	0
$513$	17.9367	0.791924
$514$	0	0
$515$	1.60268	0.0706223
$516$	0	0
$517$	−8.03956	−0.353579
$518$	0	0
$519$	−12.5224	−0.549671
$520$	0	0
$521$	16.7786	0.735084	0.367542	−	0.930007i	$-0.380199\pi$
$521$	16.7786	0.735084	0.367542	+	0.930007i	$0.380199\pi$
$522$	0	0
$523$	23.9418	1.04690	0.523452	−	0.852055i	$-0.324644\pi$
$523$	23.9418	1.04690	0.523452	+	0.852055i	$0.324644\pi$
$524$	0	0
$525$	−2.69336	−0.117548
$526$	0	0
$527$	−31.4720	−1.37094
$528$	0	0
$529$	−15.4780	−0.672955
$530$	0	0
$531$	−0.0618937	−0.00268596
$532$	0	0
$533$	1.27911	0.0554043
$534$	0	0
$535$	3.00707	0.130007
$536$	0	0
$537$	−30.2846	−1.30688
$538$	0	0
$539$	27.2967	1.17575
$540$	0	0
$541$	−20.2387	−0.870130	−0.435065	−	0.900399i	$-0.643275\pi$
$541$	−20.2387	−0.870130	−0.435065	+	0.900399i	$0.643275\pi$
$542$	0	0
$543$	24.6484	1.05776
$544$	0	0
$545$	−3.79992	−0.162771
$546$	0	0
$547$	7.96450	0.340537	0.170269	−	0.985398i	$-0.445536\pi$
$547$	7.96450	0.340537	0.170269	+	0.985398i	$0.445536\pi$
$548$	0	0
$549$	0.809284	0.0345394
$550$	0	0
$551$	8.87634	0.378145
$552$	0	0
$553$	10.6172	0.451489
$554$	0	0
$555$	−2.78574	−0.118248
$556$	0	0
$557$	−17.1594	−0.727066	−0.363533	−	0.931581i	$-0.618430\pi$
$557$	−17.1594	−0.727066	−0.363533	+	0.931581i	$0.618430\pi$
$558$	0	0
$559$	7.71370	0.326255
$560$	0	0
$561$	−60.3274	−2.54702
$562$	0	0
$563$	−26.2028	−1.10431	−0.552157	−	0.833740i	$-0.686195\pi$
$563$	−26.2028	−1.10431	−0.552157	+	0.833740i	$0.686195\pi$
$564$	0	0
$565$	−9.81560	−0.412945
$566$	0	0
$567$	−14.1937	−0.596081
$568$	0	0
$569$	0.963737	0.0404020	0.0202010	−	0.999796i	$-0.493569\pi$
$569$	0.963737	0.0404020	0.0202010	+	0.999796i	$0.493569\pi$
$570$	0	0
$571$	2.97981	0.124701	0.0623505	−	0.998054i	$-0.480140\pi$
$571$	2.97981	0.124701	0.0623505	+	0.998054i	$0.480140\pi$
$572$	0	0
$573$	35.4044	1.47904
$574$	0	0
$575$	2.74263	0.114376
$576$	0	0
$577$	−2.20361	−0.0917376	−0.0458688	−	0.998947i	$-0.514606\pi$
$577$	−2.20361	−0.0917376	−0.0458688	+	0.998947i	$0.514606\pi$
$578$	0	0
$579$	−27.1049	−1.12644
$580$	0	0
$581$	−8.72501	−0.361974
$582$	0	0
$583$	62.8079	2.60124
$584$	0	0
$585$	0.0819068	0.00338643
$586$	0	0
$587$	29.2899	1.20892	0.604462	−	0.796634i	$-0.293388\pi$
$587$	29.2899	1.20892	0.604462	+	0.796634i	$0.293388\pi$
$588$	0	0
$589$	18.8712	0.777573
$590$	0	0
$591$	2.61891	0.107727
$592$	0	0
$593$	16.2252	0.666291	0.333145	−	0.942875i	$-0.391890\pi$
$593$	16.2252	0.666291	0.333145	+	0.942875i	$0.391890\pi$
$594$	0	0
$595$	−8.96028	−0.367336
$596$	0	0
$597$	34.5943	1.41585
$598$	0	0
$599$	−47.2495	−1.93056	−0.965281	−	0.261215i	$-0.915877\pi$
$599$	−47.2495	−1.93056	−0.965281	+	0.261215i	$0.915877\pi$
$600$	0	0
$601$	−22.0922	−0.901159	−0.450580	−	0.892736i	$-0.648783\pi$
$601$	−22.0922	−0.901159	−0.450580	+	0.892736i	$0.648783\pi$
$602$	0	0
$603$	−0.858449	−0.0349587
$604$	0	0
$605$	23.4098	0.951745
$606$	0	0
$607$	4.45091	0.180657	0.0903285	−	0.995912i	$-0.471208\pi$
$607$	4.45091	0.180657	0.0903285	+	0.995912i	$0.471208\pi$
$608$	0	0
$609$	−6.81636	−0.276213
$610$	0	0
$611$	1.22912	0.0497250
$612$	0	0
$613$	−5.61033	−0.226599	−0.113299	−	0.993561i	$-0.536142\pi$
$613$	−5.61033	−0.226599	−0.113299	+	0.993561i	$0.536142\pi$
$614$	0	0
$615$	−2.50770	−0.101120
$616$	0	0
$617$	−9.51146	−0.382917	−0.191458	−	0.981501i	$-0.561322\pi$
$617$	−9.51146	−0.382917	−0.191458	+	0.981501i	$0.561322\pi$
$618$	0	0
$619$	0.189188	0.00760410	0.00380205	−	0.999993i	$-0.498790\pi$
$619$	0.189188	0.00760410	0.00380205	+	0.999993i	$0.498790\pi$
$620$	0	0
$621$	14.0260	0.562846
$622$	0	0
$623$	−5.33219	−0.213630
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	36.1734	1.44463
$628$	0	0
$629$	−9.26763	−0.369525
$630$	0	0
$631$	−9.21580	−0.366875	−0.183438	−	0.983031i	$-0.558723\pi$
$631$	−9.21580	−0.366875	−0.183438	+	0.983031i	$0.558723\pi$
$632$	0	0
$633$	−13.0496	−0.518675
$634$	0	0
$635$	13.8098	0.548026
$636$	0	0
$637$	−4.17324	−0.165350
$638$	0	0
$639$	−0.373290	−0.0147671
$640$	0	0
$641$	−34.7524	−1.37264	−0.686318	−	0.727301i	$-0.740774\pi$
$641$	−34.7524	−1.37264	−0.686318	+	0.727301i	$0.740774\pi$
$642$	0	0
$643$	−43.7367	−1.72481	−0.862403	−	0.506223i	$-0.831041\pi$
$643$	−43.7367	−1.72481	−0.862403	+	0.506223i	$0.831041\pi$
$644$	0	0
$645$	−15.1227	−0.595457
$646$	0	0
$647$	37.9083	1.49033	0.745165	−	0.666880i	$-0.232371\pi$
$647$	37.9083	1.49033	0.745165	+	0.666880i	$0.232371\pi$
$648$	0	0
$649$	3.97532	0.156045
$650$	0	0
$651$	−14.4916	−0.567972
$652$	0	0
$653$	−47.5355	−1.86021	−0.930104	−	0.367296i	$-0.880283\pi$
$653$	−47.5355	−1.86021	−0.930104	+	0.367296i	$0.880283\pi$
$654$	0	0
$655$	4.30009	0.168018
$656$	0	0
$657$	−0.409744	−0.0159856
$658$	0	0
$659$	30.6495	1.19394	0.596968	−	0.802265i	$-0.296372\pi$
$659$	30.6495	1.19394	0.596968	+	0.802265i	$0.296372\pi$
$660$	0	0
$661$	3.88104	0.150955	0.0754775	−	0.997148i	$-0.475952\pi$
$661$	3.88104	0.150955	0.0754775	+	0.997148i	$0.475952\pi$
$662$	0	0
$663$	9.22312	0.358196
$664$	0	0
$665$	5.37275	0.208346
$666$	0	0
$667$	6.94107	0.268759
$668$	0	0
$669$	28.7447	1.11134
$670$	0	0
$671$	−51.9789	−2.00662
$672$	0	0
$673$	7.29019	0.281016	0.140508	−	0.990080i	$-0.455126\pi$
$673$	7.29019	0.281016	0.140508	+	0.990080i	$0.455126\pi$
$674$	0	0
$675$	5.11408	0.196841
$676$	0	0
$677$	26.5940	1.02209	0.511045	−	0.859554i	$-0.329259\pi$
$677$	26.5940	1.02209	0.511045	+	0.859554i	$0.329259\pi$
$678$	0	0
$679$	−15.5645	−0.597312
$680$	0	0
$681$	−30.1507	−1.15538
$682$	0	0
$683$	−10.4189	−0.398669	−0.199335	−	0.979931i	$-0.563878\pi$
$683$	−10.4189	−0.398669	−0.199335	+	0.979931i	$0.563878\pi$
$684$	0	0
$685$	−8.88971	−0.339658
$686$	0	0
$687$	5.71453	0.218023
$688$	0	0
$689$	−9.60236	−0.365821
$690$	0	0
$691$	−12.3495	−0.469797	−0.234899	−	0.972020i	$-0.575476\pi$
$691$	−12.3495	−0.469797	−0.234899	+	0.972020i	$0.575476\pi$
$692$	0	0
$693$	−0.820687	−0.0311753
$694$	0	0
$695$	−13.0154	−0.493701
$696$	0	0
$697$	−8.34262	−0.315999
$698$	0	0
$699$	−9.89235	−0.374163
$700$	0	0
$701$	48.3114	1.82470	0.912348	−	0.409415i	$-0.134267\pi$
$701$	48.3114	1.82470	0.912348	+	0.409415i	$0.134267\pi$
$702$	0	0
$703$	5.55704	0.209588
$704$	0	0
$705$	−2.40970	−0.0907547
$706$	0	0
$707$	−6.61751	−0.248877
$708$	0	0
$709$	19.8506	0.745505	0.372752	−	0.927931i	$-0.378414\pi$
$709$	19.8506	0.745505	0.372752	+	0.927931i	$0.378414\pi$
$710$	0	0
$711$	0.633000	0.0237393
$712$	0	0
$713$	14.7568	0.552646
$714$	0	0
$715$	−5.26073	−0.196740
$716$	0	0
$717$	42.5970	1.59081
$718$	0	0
$719$	14.8568	0.554066	0.277033	−	0.960860i	$-0.410649\pi$
$719$	14.8568	0.554066	0.277033	+	0.960860i	$0.410649\pi$
$720$	0	0
$721$	2.45509	0.0914321
$722$	0	0
$723$	−11.4697	−0.426562
$724$	0	0
$725$	2.53080	0.0939917
$726$	0	0
$727$	−13.7074	−0.508378	−0.254189	−	0.967155i	$-0.581809\pi$
$727$	−13.7074	−0.508378	−0.254189	+	0.967155i	$0.581809\pi$
$728$	0	0
$729$	26.1287	0.967731
$730$	0	0
$731$	−50.3105	−1.86080
$732$	0	0
$733$	6.62488	0.244696	0.122348	−	0.992487i	$-0.460958\pi$
$733$	6.62488	0.244696	0.122348	+	0.992487i	$0.460958\pi$
$734$	0	0
$735$	8.18166	0.301785
$736$	0	0
$737$	55.1367	2.03098
$738$	0	0
$739$	−6.95622	−0.255889	−0.127944	−	0.991781i	$-0.540838\pi$
$739$	−6.95622	−0.255889	−0.127944	+	0.991781i	$0.540838\pi$
$740$	0	0
$741$	−5.53035	−0.203163
$742$	0	0
$743$	43.3827	1.59156	0.795778	−	0.605589i	$-0.207062\pi$
$743$	43.3827	1.59156	0.795778	+	0.605589i	$0.207062\pi$
$744$	0	0
$745$	−16.0145	−0.586725
$746$	0	0
$747$	−0.520188	−0.0190327
$748$	0	0
$749$	4.60643	0.168315
$750$	0	0
$751$	−0.760362	−0.0277460	−0.0138730	−	0.999904i	$-0.504416\pi$
$751$	−0.760362	−0.0277460	−0.0138730	+	0.999904i	$0.504416\pi$
$752$	0	0
$753$	32.7343	1.19290
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	3.55688	0.129277	0.0646385	−	0.997909i	$-0.479411\pi$
$757$	3.55688	0.129277	0.0646385	+	0.997909i	$0.479411\pi$
$758$	0	0
$759$	28.2867	1.02674
$760$	0	0
$761$	−18.5180	−0.671277	−0.335638	−	0.941991i	$-0.608952\pi$
$761$	−18.5180	−0.671277	−0.335638	+	0.941991i	$0.608952\pi$
$762$	0	0
$763$	−5.82098	−0.210733
$764$	0	0
$765$	−0.534214	−0.0193146
$766$	0	0
$767$	−0.607766	−0.0219452
$768$	0	0
$769$	12.4366	0.448474	0.224237	−	0.974535i	$-0.428011\pi$
$769$	12.4366	0.448474	0.224237	+	0.974535i	$0.428011\pi$
$770$	0	0
$771$	14.4445	0.520205
$772$	0	0
$773$	−17.0639	−0.613745	−0.306872	−	0.951751i	$-0.599283\pi$
$773$	−17.0639	−0.613745	−0.306872	+	0.951751i	$0.599283\pi$
$774$	0	0
$775$	5.38052	0.193274
$776$	0	0
$777$	−4.26739	−0.153092
$778$	0	0
$779$	5.00239	0.179229
$780$	0	0
$781$	23.9757	0.857920
$782$	0	0
$783$	12.9427	0.462535
$784$	0	0
$785$	−4.59585	−0.164033
$786$	0	0
$787$	−29.9387	−1.06720	−0.533599	−	0.845737i	$-0.679161\pi$
$787$	−29.9387	−1.06720	−0.533599	+	0.845737i	$0.679161\pi$
$788$	0	0
$789$	9.74149	0.346806
$790$	0	0
$791$	−15.0362	−0.534626
$792$	0	0
$793$	7.94677	0.282198
$794$	0	0
$795$	18.8255	0.667671
$796$	0	0
$797$	36.0516	1.27701	0.638506	−	0.769617i	$-0.279553\pi$
$797$	36.0516	1.27701	0.638506	+	0.769617i	$0.279553\pi$
$798$	0	0
$799$	−8.01662	−0.283608
$800$	0	0
$801$	−0.317907	−0.0112327
$802$	0	0
$803$	26.3171	0.928711
$804$	0	0
$805$	4.20135	0.148078
$806$	0	0
$807$	20.1301	0.708613
$808$	0	0
$809$	−22.5150	−0.791585	−0.395793	−	0.918340i	$-0.629530\pi$
$809$	−22.5150	−0.791585	−0.395793	+	0.918340i	$0.629530\pi$
$810$	0	0
$811$	−25.6944	−0.902252	−0.451126	−	0.892460i	$-0.648977\pi$
$811$	−25.6944	−0.902252	−0.451126	+	0.892460i	$0.648977\pi$
$812$	0	0
$813$	48.3261	1.69487
$814$	0	0
$815$	9.09815	0.318694
$816$	0	0
$817$	30.1671	1.05541
$818$	0	0
$819$	0.125470	0.00438429
$820$	0	0
$821$	35.0675	1.22387	0.611933	−	0.790910i	$-0.290392\pi$
$821$	35.0675	1.22387	0.611933	+	0.790910i	$0.290392\pi$
$822$	0	0
$823$	−3.25821	−0.113574	−0.0567870	−	0.998386i	$-0.518086\pi$
$823$	−3.25821	−0.113574	−0.0567870	+	0.998386i	$0.518086\pi$
$824$	0	0
$825$	10.3137	0.359076
$826$	0	0
$827$	31.2078	1.08520	0.542601	−	0.839990i	$-0.317440\pi$
$827$	31.2078	1.08520	0.542601	+	0.839990i	$0.317440\pi$
$828$	0	0
$829$	−17.1949	−0.597204	−0.298602	−	0.954378i	$-0.596520\pi$
$829$	−17.1949	−0.597204	−0.298602	+	0.954378i	$0.596520\pi$
$830$	0	0
$831$	−29.6322	−1.02793
$832$	0	0
$833$	27.2188	0.943076
$834$	0	0
$835$	−0.796650	−0.0275692
$836$	0	0
$837$	27.5164	0.951105
$838$	0	0
$839$	−33.0184	−1.13992	−0.569960	−	0.821672i	$-0.693041\pi$
$839$	−33.0184	−1.13992	−0.569960	+	0.821672i	$0.693041\pi$
$840$	0	0
$841$	−22.5950	−0.779139
$842$	0	0
$843$	38.3029	1.31922
$844$	0	0
$845$	−12.1957	−0.419545
$846$	0	0
$847$	35.8607	1.23219
$848$	0	0
$849$	−1.61878	−0.0555565
$850$	0	0
$851$	4.34546	0.148961
$852$	0	0
$853$	−22.9546	−0.785950	−0.392975	−	0.919549i	$-0.628554\pi$
$853$	−22.9546	−0.785950	−0.392975	+	0.919549i	$0.628554\pi$
$854$	0	0
$855$	0.320325	0.0109549
$856$	0	0
$857$	30.0367	1.02603	0.513017	−	0.858379i	$-0.328528\pi$
$857$	30.0367	1.02603	0.513017	+	0.858379i	$0.328528\pi$
$858$	0	0
$859$	−1.43708	−0.0490326	−0.0245163	−	0.999699i	$-0.507805\pi$
$859$	−1.43708	−0.0490326	−0.0245163	+	0.999699i	$0.507805\pi$
$860$	0	0
$861$	−3.84146	−0.130916
$862$	0	0
$863$	−33.2010	−1.13017	−0.565087	−	0.825031i	$-0.691158\pi$
$863$	−33.2010	−1.13017	−0.565087	+	0.825031i	$0.691158\pi$
$864$	0	0
$865$	7.12220	0.242162
$866$	0	0
$867$	−30.2655	−1.02787
$868$	0	0
$869$	−40.6565	−1.37918
$870$	0	0
$871$	−8.42954	−0.285624
$872$	0	0
$873$	−0.927962	−0.0314068
$874$	0	0
$875$	1.53187	0.0517866
$876$	0	0
$877$	20.6050	0.695780	0.347890	−	0.937535i	$-0.386898\pi$
$877$	20.6050	0.695780	0.347890	+	0.937535i	$0.386898\pi$
$878$	0	0
$879$	36.5632	1.23325
$880$	0	0
$881$	−7.46176	−0.251393	−0.125696	−	0.992069i	$-0.540117\pi$
$881$	−7.46176	−0.251393	−0.125696	+	0.992069i	$0.540117\pi$
$882$	0	0
$883$	21.2146	0.713927	0.356963	−	0.934118i	$-0.383812\pi$
$883$	21.2146	0.713927	0.356963	+	0.934118i	$0.383812\pi$
$884$	0	0
$885$	1.19153	0.0400528
$886$	0	0
$887$	−20.6074	−0.691930	−0.345965	−	0.938247i	$-0.612448\pi$
$887$	−20.6074	−0.691930	−0.345965	+	0.938247i	$0.612448\pi$
$888$	0	0
$889$	21.1548	0.709510
$890$	0	0
$891$	54.3522	1.82087
$892$	0	0
$893$	4.80691	0.160857
$894$	0	0
$895$	17.2246	0.575755
$896$	0	0
$897$	−4.32460	−0.144394
$898$	0	0
$899$	13.6170	0.454153
$900$	0	0
$901$	62.6287	2.08647
$902$	0	0
$903$	−23.1660	−0.770917
$904$	0	0
$905$	−14.0189	−0.466005
$906$	0	0
$907$	4.29099	0.142480	0.0712400	−	0.997459i	$-0.477304\pi$
$907$	4.29099	0.142480	0.0712400	+	0.997459i	$0.477304\pi$
$908$	0	0
$909$	−0.394538	−0.0130860
$910$	0	0
$911$	1.41804	0.0469817	0.0234908	−	0.999724i	$-0.492522\pi$
$911$	1.41804	0.0469817	0.0234908	+	0.999724i	$0.492522\pi$
$912$	0	0
$913$	33.4107	1.10573
$914$	0	0
$915$	−15.5797	−0.515048
$916$	0	0
$917$	6.58716	0.217527
$918$	0	0
$919$	−3.14868	−0.103865	−0.0519327	−	0.998651i	$-0.516538\pi$
$919$	−3.14868	−0.103865	−0.0519327	+	0.998651i	$0.516538\pi$
$920$	0	0
$921$	−49.6777	−1.63694
$922$	0	0
$923$	−3.66552	−0.120652
$924$	0	0
$925$	1.58441	0.0520952
$926$	0	0
$927$	0.146373	0.00480752
$928$	0	0
$929$	−33.5346	−1.10023	−0.550117	−	0.835087i	$-0.685417\pi$
$929$	−33.5346	−1.10023	−0.550117	+	0.835087i	$0.685417\pi$
$930$	0	0
$931$	−16.3209	−0.534896
$932$	0	0
$933$	51.6951	1.69242
$934$	0	0
$935$	34.3117	1.12211
$936$	0	0
$937$	−55.8247	−1.82371	−0.911857	−	0.410508i	$-0.865351\pi$
$937$	−55.8247	−1.82371	−0.911857	+	0.410508i	$0.865351\pi$
$938$	0	0
$939$	18.4384	0.601714
$940$	0	0
$941$	15.2251	0.496323	0.248162	−	0.968719i	$-0.420174\pi$
$941$	15.2251	0.496323	0.248162	+	0.968719i	$0.420174\pi$
$942$	0	0
$943$	3.91174	0.127384
$944$	0	0
$945$	7.83408	0.254843
$946$	0	0
$947$	46.7913	1.52051	0.760256	−	0.649623i	$-0.225073\pi$
$947$	46.7913	1.52051	0.760256	+	0.649623i	$0.225073\pi$
$948$	0	0
$949$	−4.02348	−0.130608
$950$	0	0
$951$	−28.9770	−0.939645
$952$	0	0
$953$	47.7062	1.54536	0.772678	−	0.634798i	$-0.218917\pi$
$953$	47.7062	1.54536	0.772678	+	0.634798i	$0.218917\pi$
$954$	0	0
$955$	−20.1365	−0.651603
$956$	0	0
$957$	26.1019	0.843755
$958$	0	0
$959$	−13.6179	−0.439743
$960$	0	0
$961$	−2.05005	−0.0661307
$962$	0	0
$963$	0.274637	0.00885004
$964$	0	0
$965$	15.4161	0.496263
$966$	0	0
$967$	−15.6087	−0.501942	−0.250971	−	0.967995i	$-0.580750\pi$
$967$	−15.6087	−0.501942	−0.250971	+	0.967995i	$0.580750\pi$
$968$	0	0
$969$	36.0702	1.15874
$970$	0	0
$971$	30.2597	0.971081	0.485541	−	0.874214i	$-0.338623\pi$
$971$	30.2597	0.971081	0.485541	+	0.874214i	$0.338623\pi$
$972$	0	0
$973$	−19.9378	−0.639176
$974$	0	0
$975$	−1.57680	−0.0504981
$976$	0	0
$977$	26.0786	0.834330	0.417165	−	0.908831i	$-0.363024\pi$
$977$	26.0786	0.834330	0.417165	+	0.908831i	$0.363024\pi$
$978$	0	0
$979$	20.4186	0.652582
$980$	0	0
$981$	−0.347048	−0.0110804
$982$	0	0
$983$	−6.67475	−0.212891	−0.106446	−	0.994319i	$-0.533947\pi$
$983$	−6.67475	−0.212891	−0.106446	+	0.994319i	$0.533947\pi$
$984$	0	0
$985$	−1.48952	−0.0474602
$986$	0	0
$987$	−3.69134	−0.117497
$988$	0	0
$989$	23.5899	0.750115
$990$	0	0
$991$	−39.8910	−1.26718	−0.633589	−	0.773669i	$-0.718419\pi$
$991$	−39.8910	−1.26718	−0.633589	+	0.773669i	$0.718419\pi$
$992$	0	0
$993$	37.6896	1.19604
$994$	0	0
$995$	−19.6758	−0.623763
$996$	0	0
$997$	23.0871	0.731177	0.365588	−	0.930777i	$-0.380868\pi$
$997$	23.0871	0.731177	0.365588	+	0.930777i	$0.380868\pi$
$998$	0	0
$999$	8.10281	0.256361

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.l.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.l.1.2	✓	9	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.75822 q^{3} +1.00000 q^{5} +1.53187 q^{7} +0.0913304 q^{9} +O(q^{10})\)	\(q-1.75822 q^{3} +1.00000 q^{5} +1.53187 q^{7} +0.0913304 q^{9} -5.86599 q^{11} +0.896819 q^{13} -1.75822 q^{15} -5.84925 q^{17} +3.50732 q^{19} -2.69336 q^{21} +2.74263 q^{23} +1.00000 q^{25} +5.11408 q^{27} +2.53080 q^{29} +5.38052 q^{31} +10.3137 q^{33} +1.53187 q^{35} +1.58441 q^{37} -1.57680 q^{39} +1.42627 q^{41} +8.60118 q^{43} +0.0913304 q^{45} +1.37054 q^{47} -4.65338 q^{49} +10.2843 q^{51} -10.7071 q^{53} -5.86599 q^{55} -6.16663 q^{57} -0.677690 q^{59} +8.86107 q^{61} +0.139906 q^{63} +0.896819 q^{65} -9.39938 q^{67} -4.82215 q^{69} -4.08725 q^{71} -4.48639 q^{73} -1.75822 q^{75} -8.98591 q^{77} +6.93088 q^{79} -9.26565 q^{81} -5.69567 q^{83} -5.84925 q^{85} -4.44971 q^{87} -3.48085 q^{89} +1.37381 q^{91} -9.46012 q^{93} +3.50732 q^{95} -10.1605 q^{97} -0.535743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) \) 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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