LMFDB - Embedded newform 6032.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 10x^{8} + 21x^{7} + 28x^{6} - 67x^{5} - 20x^{4} + 76x^{3} - 8x^{2} - 26x + 8 $ x^10 - 2x^9 - 10x^8 + 21x^7 + 28x^6 - 67x^5 - 20x^4 + 76x^3 - 8x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.418899$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56651 q^{3} +0.728348 q^{5} +3.51001 q^{7} +3.58699 q^{9} +O(q^{10})$	$q-2.56651 q^{3} +0.728348 q^{5} +3.51001 q^{7} +3.58699 q^{9} +2.48266 q^{11} -1.00000 q^{13} -1.86931 q^{15} -2.68325 q^{17} +2.82721 q^{19} -9.00848 q^{21} +1.17126 q^{23} -4.46951 q^{25} -1.50653 q^{27} +1.00000 q^{29} +7.55568 q^{31} -6.37179 q^{33} +2.55651 q^{35} -1.38963 q^{37} +2.56651 q^{39} +1.18173 q^{41} +7.86515 q^{43} +2.61258 q^{45} -4.29483 q^{47} +5.32015 q^{49} +6.88661 q^{51} +8.82523 q^{53} +1.80824 q^{55} -7.25607 q^{57} +5.41513 q^{59} +9.31337 q^{61} +12.5904 q^{63} -0.728348 q^{65} +13.8215 q^{67} -3.00605 q^{69} -1.88224 q^{71} +9.81188 q^{73} +11.4711 q^{75} +8.71417 q^{77} -15.8465 q^{79} -6.89446 q^{81} -12.2292 q^{83} -1.95434 q^{85} -2.56651 q^{87} -17.9021 q^{89} -3.51001 q^{91} -19.3918 q^{93} +2.05919 q^{95} -14.8074 q^{97} +8.90530 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9}+O(q^{10}) $ 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9	$ 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9} + 13 q^{11} - 10 q^{13} - 3 q^{15} + 9 q^{17} + 12 q^{19} - 10 q^{21} + 5 q^{23} + 7 q^{25} + 3 q^{27} + 10 q^{29} - 5 q^{31} - 9 q^{33} + 23 q^{35} - 4 q^{37} - 3 q^{39} - 3 q^{41} + 27 q^{43} - 20 q^{45} + 16 q^{47} + 6 q^{49} + 34 q^{51} + 11 q^{53} + q^{55} + 27 q^{59} - 7 q^{61} + 6 q^{63} + 5 q^{65} + 35 q^{67} - 22 q^{69} + 21 q^{71} + 7 q^{75} - 18 q^{77} + 12 q^{79} + 6 q^{81} + 24 q^{83} - 2 q^{85} + 3 q^{87} - 23 q^{89} - 3 q^{93} + 17 q^{95} + 2 q^{97} + 65 q^{99}+O(q^{100}) $ 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9 + 13 * q^11 - 10 * q^13 - 3 * q^15 + 9 * q^17 + 12 * q^19 - 10 * q^21 + 5 * q^23 + 7 * q^25 + 3 * q^27 + 10 * q^29 - 5 * q^31 - 9 * q^33 + 23 * q^35 - 4 * q^37 - 3 * q^39 - 3 * q^41 + 27 * q^43 - 20 * q^45 + 16 * q^47 + 6 * q^49 + 34 * q^51 + 11 * q^53 + q^55 + 27 * q^59 - 7 * q^61 + 6 * q^63 + 5 * q^65 + 35 * q^67 - 22 * q^69 + 21 * q^71 + 7 * q^75 - 18 * q^77 + 12 * q^79 + 6 * q^81 + 24 * q^83 - 2 * q^85 + 3 * q^87 - 23 * q^89 - 3 * q^93 + 17 * q^95 + 2 * q^97 + 65 * 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
$3$	−2.56651	−1.48178	−0.740889	−	0.671628i	$-0.765595\pi$
$3$	−2.56651	−1.48178	−0.740889	+	0.671628i	$0.765595\pi$
$4$	0	0
$5$	0.728348	0.325727	0.162863	−	0.986649i	$-0.447927\pi$
$5$	0.728348	0.325727	0.162863	+	0.986649i	$0.447927\pi$
$6$	0	0
$7$	3.51001	1.32666	0.663329	−	0.748328i	$-0.269143\pi$
$7$	3.51001	1.32666	0.663329	+	0.748328i	$0.269143\pi$
$8$	0	0
$9$	3.58699	1.19566
$10$	0	0
$11$	2.48266	0.748551	0.374276	−	0.927318i	$-0.377891\pi$
$11$	2.48266	0.748551	0.374276	+	0.927318i	$0.377891\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.86931	−0.482655
$16$	0	0
$17$	−2.68325	−0.650785	−0.325392	−	0.945579i	$-0.605496\pi$
$17$	−2.68325	−0.650785	−0.325392	+	0.945579i	$0.605496\pi$
$18$	0	0
$19$	2.82721	0.648606	0.324303	−	0.945953i	$-0.394870\pi$
$19$	2.82721	0.648606	0.324303	+	0.945953i	$0.394870\pi$
$20$	0	0
$21$	−9.00848	−1.96581
$22$	0	0
$23$	1.17126	0.244224	0.122112	−	0.992516i	$-0.461033\pi$
$23$	1.17126	0.244224	0.122112	+	0.992516i	$0.461033\pi$
$24$	0	0
$25$	−4.46951	−0.893902
$26$	0	0
$27$	−1.50653	−0.289931
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	7.55568	1.35704	0.678520	−	0.734582i	$-0.262621\pi$
$31$	7.55568	1.35704	0.678520	+	0.734582i	$0.262621\pi$
$32$	0	0
$33$	−6.37179	−1.10919
$34$	0	0
$35$	2.55651	0.432128
$36$	0	0
$37$	−1.38963	−0.228453	−0.114227	−	0.993455i	$-0.536439\pi$
$37$	−1.38963	−0.228453	−0.114227	+	0.993455i	$0.536439\pi$
$38$	0	0
$39$	2.56651	0.410971
$40$	0	0
$41$	1.18173	0.184555	0.0922777	−	0.995733i	$-0.470585\pi$
$41$	1.18173	0.184555	0.0922777	+	0.995733i	$0.470585\pi$
$42$	0	0
$43$	7.86515	1.19942	0.599712	−	0.800216i	$-0.295282\pi$
$43$	7.86515	1.19942	0.599712	+	0.800216i	$0.295282\pi$
$44$	0	0
$45$	2.61258	0.389460
$46$	0	0
$47$	−4.29483	−0.626466	−0.313233	−	0.949676i	$-0.601412\pi$
$47$	−4.29483	−0.626466	−0.313233	+	0.949676i	$0.601412\pi$
$48$	0	0
$49$	5.32015	0.760022
$50$	0	0
$51$	6.88661	0.964318
$52$	0	0
$53$	8.82523	1.21224	0.606120	−	0.795373i	$-0.292725\pi$
$53$	8.82523	1.21224	0.606120	+	0.795373i	$0.292725\pi$
$54$	0	0
$55$	1.80824	0.243823
$56$	0	0
$57$	−7.25607	−0.961090
$58$	0	0
$59$	5.41513	0.704989	0.352495	−	0.935814i	$-0.385333\pi$
$59$	5.41513	0.704989	0.352495	+	0.935814i	$0.385333\pi$
$60$	0	0
$61$	9.31337	1.19245	0.596227	−	0.802816i	$-0.296666\pi$
$61$	9.31337	1.19245	0.596227	+	0.802816i	$0.296666\pi$
$62$	0	0
$63$	12.5904	1.58624
$64$	0	0
$65$	−0.728348	−0.0903404
$66$	0	0
$67$	13.8215	1.68856	0.844280	−	0.535903i	$-0.180029\pi$
$67$	13.8215	1.68856	0.844280	+	0.535903i	$0.180029\pi$
$68$	0	0
$69$	−3.00605	−0.361886
$70$	0	0
$71$	−1.88224	−0.223381	−0.111691	−	0.993743i	$-0.535627\pi$
$71$	−1.88224	−0.223381	−0.111691	+	0.993743i	$0.535627\pi$
$72$	0	0
$73$	9.81188	1.14839	0.574197	−	0.818717i	$-0.305314\pi$
$73$	9.81188	1.14839	0.574197	+	0.818717i	$0.305314\pi$
$74$	0	0
$75$	11.4711	1.32456
$76$	0	0
$77$	8.71417	0.993072
$78$	0	0
$79$	−15.8465	−1.78287	−0.891435	−	0.453148i	$-0.850301\pi$
$79$	−15.8465	−1.78287	−0.891435	+	0.453148i	$0.850301\pi$
$80$	0	0
$81$	−6.89446	−0.766051
$82$	0	0
$83$	−12.2292	−1.34233	−0.671166	−	0.741307i	$-0.734206\pi$
$83$	−12.2292	−1.34233	−0.671166	+	0.741307i	$0.734206\pi$
$84$	0	0
$85$	−1.95434	−0.211978
$86$	0	0
$87$	−2.56651	−0.275159
$88$	0	0
$89$	−17.9021	−1.89762	−0.948811	−	0.315846i	$-0.897712\pi$
$89$	−17.9021	−1.89762	−0.948811	+	0.315846i	$0.897712\pi$
$90$	0	0
$91$	−3.51001	−0.367949
$92$	0	0
$93$	−19.3918	−2.01083
$94$	0	0
$95$	2.05919	0.211268
$96$	0	0
$97$	−14.8074	−1.50346	−0.751732	−	0.659468i	$-0.770781\pi$
$97$	−14.8074	−1.50346	−0.751732	+	0.659468i	$0.770781\pi$
$98$	0	0
$99$	8.90530	0.895016
$100$	0	0
$101$	−4.86651	−0.484236	−0.242118	−	0.970247i	$-0.577842\pi$
$101$	−4.86651	−0.484236	−0.242118	+	0.970247i	$0.577842\pi$
$102$	0	0
$103$	−11.7278	−1.15557	−0.577786	−	0.816188i	$-0.696083\pi$
$103$	−11.7278	−1.15557	−0.577786	+	0.816188i	$0.696083\pi$
$104$	0	0
$105$	−6.56131	−0.640318
$106$	0	0
$107$	−1.81904	−0.175853	−0.0879265	−	0.996127i	$-0.528024\pi$
$107$	−1.81904	−0.175853	−0.0879265	+	0.996127i	$0.528024\pi$
$108$	0	0
$109$	7.14656	0.684516	0.342258	−	0.939606i	$-0.388808\pi$
$109$	7.14656	0.684516	0.342258	+	0.939606i	$0.388808\pi$
$110$	0	0
$111$	3.56649	0.338517
$112$	0	0
$113$	5.95500	0.560199	0.280099	−	0.959971i	$-0.409633\pi$
$113$	5.95500	0.560199	0.280099	+	0.959971i	$0.409633\pi$
$114$	0	0
$115$	0.853083	0.0795504
$116$	0	0
$117$	−3.58699	−0.331618
$118$	0	0
$119$	−9.41824	−0.863369
$120$	0	0
$121$	−4.83638	−0.439671
$122$	0	0
$123$	−3.03293	−0.273470
$124$	0	0
$125$	−6.89709	−0.616895
$126$	0	0
$127$	21.3738	1.89662	0.948310	−	0.317346i	$-0.102792\pi$
$127$	21.3738	1.89662	0.948310	+	0.317346i	$0.102792\pi$
$128$	0	0
$129$	−20.1860	−1.77728
$130$	0	0
$131$	4.82929	0.421937	0.210969	−	0.977493i	$-0.432338\pi$
$131$	4.82929	0.421937	0.210969	+	0.977493i	$0.432338\pi$
$132$	0	0
$133$	9.92352	0.860479
$134$	0	0
$135$	−1.09728	−0.0944384
$136$	0	0
$137$	12.6792	1.08326	0.541628	−	0.840618i	$-0.317808\pi$
$137$	12.6792	1.08326	0.541628	+	0.840618i	$0.317808\pi$
$138$	0	0
$139$	−2.90698	−0.246567	−0.123283	−	0.992372i	$-0.539342\pi$
$139$	−2.90698	−0.246567	−0.123283	+	0.992372i	$0.539342\pi$
$140$	0	0
$141$	11.0227	0.928283
$142$	0	0
$143$	−2.48266	−0.207611
$144$	0	0
$145$	0.728348	0.0604860
$146$	0	0
$147$	−13.6542	−1.12618
$148$	0	0
$149$	0.739629	0.0605928	0.0302964	−	0.999541i	$-0.490355\pi$
$149$	0.739629	0.0605928	0.0302964	+	0.999541i	$0.490355\pi$
$150$	0	0
$151$	−8.09329	−0.658623	−0.329311	−	0.944221i	$-0.606817\pi$
$151$	−8.09329	−0.658623	−0.329311	+	0.944221i	$0.606817\pi$
$152$	0	0
$153$	−9.62481	−0.778120
$154$	0	0
$155$	5.50316	0.442025
$156$	0	0
$157$	−5.12512	−0.409029	−0.204515	−	0.978864i	$-0.565562\pi$
$157$	−5.12512	−0.409029	−0.204515	+	0.978864i	$0.565562\pi$
$158$	0	0
$159$	−22.6501	−1.79627
$160$	0	0
$161$	4.11112	0.324002
$162$	0	0
$163$	9.30469	0.728800	0.364400	−	0.931243i	$-0.381274\pi$
$163$	9.30469	0.728800	0.364400	+	0.931243i	$0.381274\pi$
$164$	0	0
$165$	−4.64088	−0.361292
$166$	0	0
$167$	22.4694	1.73873	0.869366	−	0.494168i	$-0.164527\pi$
$167$	22.4694	1.73873	0.869366	+	0.494168i	$0.164527\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	10.1412	0.775515
$172$	0	0
$173$	9.17685	0.697703	0.348851	−	0.937178i	$-0.386572\pi$
$173$	9.17685	0.697703	0.348851	+	0.937178i	$0.386572\pi$
$174$	0	0
$175$	−15.6880	−1.18590
$176$	0	0
$177$	−13.8980	−1.04464
$178$	0	0
$179$	3.87343	0.289514	0.144757	−	0.989467i	$-0.453760\pi$
$179$	3.87343	0.289514	0.144757	+	0.989467i	$0.453760\pi$
$180$	0	0
$181$	−9.91365	−0.736876	−0.368438	−	0.929652i	$-0.620107\pi$
$181$	−9.91365	−0.736876	−0.368438	+	0.929652i	$0.620107\pi$
$182$	0	0
$183$	−23.9029	−1.76695
$184$	0	0
$185$	−1.01213	−0.0744133
$186$	0	0
$187$	−6.66162	−0.487146
$188$	0	0
$189$	−5.28792	−0.384640
$190$	0	0
$191$	3.59430	0.260075	0.130037	−	0.991509i	$-0.458490\pi$
$191$	3.59430	0.260075	0.130037	+	0.991509i	$0.458490\pi$
$192$	0	0
$193$	20.2009	1.45409	0.727047	−	0.686588i	$-0.240892\pi$
$193$	20.2009	1.45409	0.727047	+	0.686588i	$0.240892\pi$
$194$	0	0
$195$	1.86931	0.133864
$196$	0	0
$197$	−17.2036	−1.22571	−0.612854	−	0.790196i	$-0.709979\pi$
$197$	−17.2036	−1.22571	−0.612854	+	0.790196i	$0.709979\pi$
$198$	0	0
$199$	0.427816	0.0303271	0.0151635	−	0.999885i	$-0.495173\pi$
$199$	0.427816	0.0303271	0.0151635	+	0.999885i	$0.495173\pi$
$200$	0	0
$201$	−35.4729	−2.50207
$202$	0	0
$203$	3.51001	0.246354
$204$	0	0
$205$	0.860711	0.0601146
$206$	0	0
$207$	4.20129	0.292010
$208$	0	0
$209$	7.01901	0.485515
$210$	0	0
$211$	−11.6415	−0.801432	−0.400716	−	0.916202i	$-0.631239\pi$
$211$	−11.6415	−0.801432	−0.400716	+	0.916202i	$0.631239\pi$
$212$	0	0
$213$	4.83080	0.331001
$214$	0	0
$215$	5.72856	0.390685
$216$	0	0
$217$	26.5205	1.80033
$218$	0	0
$219$	−25.1823	−1.70166
$220$	0	0
$221$	2.68325	0.180495
$222$	0	0
$223$	7.65395	0.512547	0.256273	−	0.966604i	$-0.417505\pi$
$223$	7.65395	0.512547	0.256273	+	0.966604i	$0.417505\pi$
$224$	0	0
$225$	−16.0321	−1.06881
$226$	0	0
$227$	−1.97830	−0.131304	−0.0656521	−	0.997843i	$-0.520913\pi$
$227$	−1.97830	−0.131304	−0.0656521	+	0.997843i	$0.520913\pi$
$228$	0	0
$229$	24.8466	1.64191	0.820955	−	0.570993i	$-0.193442\pi$
$229$	24.8466	1.64191	0.820955	+	0.570993i	$0.193442\pi$
$230$	0	0
$231$	−22.3650	−1.47151
$232$	0	0
$233$	18.1124	1.18659	0.593293	−	0.804987i	$-0.297828\pi$
$233$	18.1124	1.18659	0.593293	+	0.804987i	$0.297828\pi$
$234$	0	0
$235$	−3.12813	−0.204057
$236$	0	0
$237$	40.6703	2.64182
$238$	0	0
$239$	11.6834	0.755738	0.377869	−	0.925859i	$-0.376657\pi$
$239$	11.6834	0.755738	0.377869	+	0.925859i	$0.376657\pi$
$240$	0	0
$241$	−16.4729	−1.06111	−0.530557	−	0.847650i	$-0.678017\pi$
$241$	−16.4729	−1.06111	−0.530557	+	0.847650i	$0.678017\pi$
$242$	0	0
$243$	22.2143	1.42505
$244$	0	0
$245$	3.87492	0.247560
$246$	0	0
$247$	−2.82721	−0.179891
$248$	0	0
$249$	31.3865	1.98904
$250$	0	0
$251$	−29.3494	−1.85252	−0.926259	−	0.376889i	$-0.876994\pi$
$251$	−29.3494	−1.85252	−0.926259	+	0.376889i	$0.876994\pi$
$252$	0	0
$253$	2.90784	0.182814
$254$	0	0
$255$	5.01584	0.314104
$256$	0	0
$257$	12.6851	0.791274	0.395637	−	0.918407i	$-0.370524\pi$
$257$	12.6851	0.791274	0.395637	+	0.918407i	$0.370524\pi$
$258$	0	0
$259$	−4.87760	−0.303079
$260$	0	0
$261$	3.58699	0.222029
$262$	0	0
$263$	−10.7938	−0.665571	−0.332786	−	0.943002i	$-0.607989\pi$
$263$	−10.7938	−0.665571	−0.332786	+	0.943002i	$0.607989\pi$
$264$	0	0
$265$	6.42784	0.394859
$266$	0	0
$267$	45.9460	2.81185
$268$	0	0
$269$	24.9444	1.52089	0.760443	−	0.649404i	$-0.224982\pi$
$269$	24.9444	1.52089	0.760443	+	0.649404i	$0.224982\pi$
$270$	0	0
$271$	20.3686	1.23730	0.618652	−	0.785665i	$-0.287679\pi$
$271$	20.3686	1.23730	0.618652	+	0.785665i	$0.287679\pi$
$272$	0	0
$273$	9.00848	0.545218
$274$	0	0
$275$	−11.0963	−0.669131
$276$	0	0
$277$	−5.43692	−0.326673	−0.163336	−	0.986570i	$-0.552226\pi$
$277$	−5.43692	−0.326673	−0.163336	+	0.986570i	$0.552226\pi$
$278$	0	0
$279$	27.1022	1.62257
$280$	0	0
$281$	−32.5405	−1.94120	−0.970600	−	0.240697i	$-0.922624\pi$
$281$	−32.5405	−1.94120	−0.970600	+	0.240697i	$0.922624\pi$
$282$	0	0
$283$	13.4632	0.800305	0.400153	−	0.916449i	$-0.368957\pi$
$283$	13.4632	0.800305	0.400153	+	0.916449i	$0.368957\pi$
$284$	0	0
$285$	−5.28494	−0.313053
$286$	0	0
$287$	4.14788	0.244842
$288$	0	0
$289$	−9.80015	−0.576479
$290$	0	0
$291$	38.0034	2.22780
$292$	0	0
$293$	−13.5828	−0.793515	−0.396757	−	0.917924i	$-0.629865\pi$
$293$	−13.5828	−0.793515	−0.396757	+	0.917924i	$0.629865\pi$
$294$	0	0
$295$	3.94409	0.229634
$296$	0	0
$297$	−3.74020	−0.217028
$298$	0	0
$299$	−1.17126	−0.0677356
$300$	0	0
$301$	27.6067	1.59122
$302$	0	0
$303$	12.4900	0.717529
$304$	0	0
$305$	6.78337	0.388415
$306$	0	0
$307$	−1.61049	−0.0919155	−0.0459577	−	0.998943i	$-0.514634\pi$
$307$	−1.61049	−0.0919155	−0.0459577	+	0.998943i	$0.514634\pi$
$308$	0	0
$309$	30.0995	1.71230
$310$	0	0
$311$	−15.3634	−0.871181	−0.435591	−	0.900145i	$-0.643460\pi$
$311$	−15.3634	−0.871181	−0.435591	+	0.900145i	$0.643460\pi$
$312$	0	0
$313$	−25.0467	−1.41572	−0.707860	−	0.706353i	$-0.750339\pi$
$313$	−25.0467	−1.41572	−0.707860	+	0.706353i	$0.750339\pi$
$314$	0	0
$315$	9.17017	0.516680
$316$	0	0
$317$	13.7656	0.773151	0.386575	−	0.922258i	$-0.373658\pi$
$317$	13.7656	0.773151	0.386575	+	0.922258i	$0.373658\pi$
$318$	0	0
$319$	2.48266	0.139002
$320$	0	0
$321$	4.66859	0.260575
$322$	0	0
$323$	−7.58612	−0.422103
$324$	0	0
$325$	4.46951	0.247924
$326$	0	0
$327$	−18.3417	−1.01430
$328$	0	0
$329$	−15.0749	−0.831106
$330$	0	0
$331$	10.8233	0.594902	0.297451	−	0.954737i	$-0.403863\pi$
$331$	10.8233	0.594902	0.297451	+	0.954737i	$0.403863\pi$
$332$	0	0
$333$	−4.98458	−0.273153
$334$	0	0
$335$	10.0668	0.550009
$336$	0	0
$337$	−1.69801	−0.0924965	−0.0462482	−	0.998930i	$-0.514727\pi$
$337$	−1.69801	−0.0924965	−0.0462482	+	0.998930i	$0.514727\pi$
$338$	0	0
$339$	−15.2836	−0.830090
$340$	0	0
$341$	18.7582	1.01581
$342$	0	0
$343$	−5.89628	−0.318369
$344$	0	0
$345$	−2.18945	−0.117876
$346$	0	0
$347$	−31.2885	−1.67966	−0.839828	−	0.542853i	$-0.817344\pi$
$347$	−31.2885	−1.67966	−0.839828	+	0.542853i	$0.817344\pi$
$348$	0	0
$349$	20.5463	1.09982	0.549909	−	0.835224i	$-0.314662\pi$
$349$	20.5463	1.09982	0.549909	+	0.835224i	$0.314662\pi$
$350$	0	0
$351$	1.50653	0.0804124
$352$	0	0
$353$	3.64010	0.193743	0.0968714	−	0.995297i	$-0.469116\pi$
$353$	3.64010	0.193743	0.0968714	+	0.995297i	$0.469116\pi$
$354$	0	0
$355$	−1.37093	−0.0727612
$356$	0	0
$357$	24.1720	1.27932
$358$	0	0
$359$	26.1317	1.37918	0.689588	−	0.724202i	$-0.257792\pi$
$359$	26.1317	1.37918	0.689588	+	0.724202i	$0.257792\pi$
$360$	0	0
$361$	−11.0069	−0.579310
$362$	0	0
$363$	12.4126	0.651495
$364$	0	0
$365$	7.14646	0.374063
$366$	0	0
$367$	17.0537	0.890197	0.445099	−	0.895481i	$-0.353169\pi$
$367$	17.0537	0.890197	0.445099	+	0.895481i	$0.353169\pi$
$368$	0	0
$369$	4.23886	0.220666
$370$	0	0
$371$	30.9766	1.60823
$372$	0	0
$373$	11.9159	0.616980	0.308490	−	0.951228i	$-0.400176\pi$
$373$	11.9159	0.616980	0.308490	+	0.951228i	$0.400176\pi$
$374$	0	0
$375$	17.7015	0.914101
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	30.3423	1.55858	0.779289	−	0.626665i	$-0.215581\pi$
$379$	30.3423	1.55858	0.779289	+	0.626665i	$0.215581\pi$
$380$	0	0
$381$	−54.8562	−2.81037
$382$	0	0
$383$	23.0655	1.17859	0.589296	−	0.807917i	$-0.299405\pi$
$383$	23.0655	1.17859	0.589296	+	0.807917i	$0.299405\pi$
$384$	0	0
$385$	6.34694	0.323470
$386$	0	0
$387$	28.2122	1.43411
$388$	0	0
$389$	3.14384	0.159399	0.0796995	−	0.996819i	$-0.474604\pi$
$389$	3.14384	0.159399	0.0796995	+	0.996819i	$0.474604\pi$
$390$	0	0
$391$	−3.14278	−0.158937
$392$	0	0
$393$	−12.3945	−0.625217
$394$	0	0
$395$	−11.5418	−0.580729
$396$	0	0
$397$	17.6654	0.886601	0.443301	−	0.896373i	$-0.353807\pi$
$397$	17.6654	0.886601	0.443301	+	0.896373i	$0.353807\pi$
$398$	0	0
$399$	−25.4689	−1.27504
$400$	0	0
$401$	10.9704	0.547837	0.273919	−	0.961753i	$-0.411680\pi$
$401$	10.9704	0.547837	0.273919	+	0.961753i	$0.411680\pi$
$402$	0	0
$403$	−7.55568	−0.376375
$404$	0	0
$405$	−5.02156	−0.249523
$406$	0	0
$407$	−3.44997	−0.171009
$408$	0	0
$409$	22.0817	1.09187	0.545934	−	0.837828i	$-0.316175\pi$
$409$	22.0817	1.09187	0.545934	+	0.837828i	$0.316175\pi$
$410$	0	0
$411$	−32.5413	−1.60514
$412$	0	0
$413$	19.0071	0.935280
$414$	0	0
$415$	−8.90712	−0.437233
$416$	0	0
$417$	7.46080	0.365357
$418$	0	0
$419$	16.5507	0.808555	0.404278	−	0.914636i	$-0.367523\pi$
$419$	16.5507	0.808555	0.404278	+	0.914636i	$0.367523\pi$
$420$	0	0
$421$	11.7075	0.570587	0.285293	−	0.958440i	$-0.407909\pi$
$421$	11.7075	0.570587	0.285293	+	0.958440i	$0.407909\pi$
$422$	0	0
$423$	−15.4055	−0.749043
$424$	0	0
$425$	11.9928	0.581738
$426$	0	0
$427$	32.6900	1.58198
$428$	0	0
$429$	6.37179	0.307633
$430$	0	0
$431$	15.2149	0.732876	0.366438	−	0.930443i	$-0.380577\pi$
$431$	15.2149	0.732876	0.366438	+	0.930443i	$0.380577\pi$
$432$	0	0
$433$	19.3326	0.929064	0.464532	−	0.885556i	$-0.346223\pi$
$433$	19.3326	0.929064	0.464532	+	0.885556i	$0.346223\pi$
$434$	0	0
$435$	−1.86931	−0.0896267
$436$	0	0
$437$	3.31139	0.158405
$438$	0	0
$439$	−0.948963	−0.0452915	−0.0226458	−	0.999744i	$-0.507209\pi$
$439$	−0.948963	−0.0452915	−0.0226458	+	0.999744i	$0.507209\pi$
$440$	0	0
$441$	19.0834	0.908731
$442$	0	0
$443$	39.6157	1.88220	0.941100	−	0.338127i	$-0.109793\pi$
$443$	39.6157	1.88220	0.941100	+	0.338127i	$0.109793\pi$
$444$	0	0
$445$	−13.0390	−0.618106
$446$	0	0
$447$	−1.89827	−0.0897850
$448$	0	0
$449$	−23.7227	−1.11954	−0.559772	−	0.828646i	$-0.689111\pi$
$449$	−23.7227	−1.11954	−0.559772	+	0.828646i	$0.689111\pi$
$450$	0	0
$451$	2.93384	0.138149
$452$	0	0
$453$	20.7715	0.975932
$454$	0	0
$455$	−2.55651	−0.119851
$456$	0	0
$457$	−17.7488	−0.830254	−0.415127	−	0.909764i	$-0.636263\pi$
$457$	−17.7488	−0.830254	−0.415127	+	0.909764i	$0.636263\pi$
$458$	0	0
$459$	4.04239	0.188683
$460$	0	0
$461$	−20.6844	−0.963366	−0.481683	−	0.876345i	$-0.659974\pi$
$461$	−20.6844	−0.963366	−0.481683	+	0.876345i	$0.659974\pi$
$462$	0	0
$463$	25.0208	1.16281	0.581407	−	0.813613i	$-0.302502\pi$
$463$	25.0208	1.16281	0.581407	+	0.813613i	$0.302502\pi$
$464$	0	0
$465$	−14.1239	−0.654982
$466$	0	0
$467$	−20.1775	−0.933706	−0.466853	−	0.884335i	$-0.654612\pi$
$467$	−20.1775	−0.933706	−0.466853	+	0.884335i	$0.654612\pi$
$468$	0	0
$469$	48.5134	2.24014
$470$	0	0
$471$	13.1537	0.606090
$472$	0	0
$473$	19.5265	0.897830
$474$	0	0
$475$	−12.6362	−0.579790
$476$	0	0
$477$	31.6561	1.44943
$478$	0	0
$479$	−2.60245	−0.118909	−0.0594545	−	0.998231i	$-0.518936\pi$
$479$	−2.60245	−0.118909	−0.0594545	+	0.998231i	$0.518936\pi$
$480$	0	0
$481$	1.38963	0.0633615
$482$	0	0
$483$	−10.5513	−0.480099
$484$	0	0
$485$	−10.7849	−0.489719
$486$	0	0
$487$	−26.7031	−1.21003	−0.605016	−	0.796213i	$-0.706833\pi$
$487$	−26.7031	−1.21003	−0.605016	+	0.796213i	$0.706833\pi$
$488$	0	0
$489$	−23.8806	−1.07992
$490$	0	0
$491$	7.37582	0.332866	0.166433	−	0.986053i	$-0.446775\pi$
$491$	7.37582	0.332866	0.166433	+	0.986053i	$0.446775\pi$
$492$	0	0
$493$	−2.68325	−0.120848
$494$	0	0
$495$	6.48615	0.291531
$496$	0	0
$497$	−6.60668	−0.296350
$498$	0	0
$499$	5.36217	0.240044	0.120022	−	0.992771i	$-0.461704\pi$
$499$	5.36217	0.240044	0.120022	+	0.992771i	$0.461704\pi$
$500$	0	0
$501$	−57.6680	−2.57641
$502$	0	0
$503$	19.7440	0.880342	0.440171	−	0.897914i	$-0.354918\pi$
$503$	19.7440	0.880342	0.440171	+	0.897914i	$0.354918\pi$
$504$	0	0
$505$	−3.54451	−0.157729
$506$	0	0
$507$	−2.56651	−0.113983
$508$	0	0
$509$	6.55866	0.290708	0.145354	−	0.989380i	$-0.453568\pi$
$509$	6.55866	0.290708	0.145354	+	0.989380i	$0.453568\pi$
$510$	0	0
$511$	34.4398	1.52353
$512$	0	0
$513$	−4.25927	−0.188051
$514$	0	0
$515$	−8.54190	−0.376401
$516$	0	0
$517$	−10.6626	−0.468942
$518$	0	0
$519$	−23.5525	−1.03384
$520$	0	0
$521$	−2.12784	−0.0932226	−0.0466113	−	0.998913i	$-0.514842\pi$
$521$	−2.12784	−0.0932226	−0.0466113	+	0.998913i	$0.514842\pi$
$522$	0	0
$523$	12.2324	0.534884	0.267442	−	0.963574i	$-0.413822\pi$
$523$	12.2324	0.534884	0.267442	+	0.963574i	$0.413822\pi$
$524$	0	0
$525$	40.2635	1.75724
$526$	0	0
$527$	−20.2738	−0.883141
$528$	0	0
$529$	−21.6282	−0.940355
$530$	0	0
$531$	19.4240	0.842931
$532$	0	0
$533$	−1.18173	−0.0511864
$534$	0	0
$535$	−1.32489	−0.0572801
$536$	0	0
$537$	−9.94121	−0.428995
$538$	0	0
$539$	13.2082	0.568915
$540$	0	0
$541$	8.48247	0.364690	0.182345	−	0.983235i	$-0.441631\pi$
$541$	8.48247	0.364690	0.182345	+	0.983235i	$0.441631\pi$
$542$	0	0
$543$	25.4435	1.09189
$544$	0	0
$545$	5.20518	0.222965
$546$	0	0
$547$	0.541072	0.0231346	0.0115673	−	0.999933i	$-0.496318\pi$
$547$	0.541072	0.0231346	0.0115673	+	0.999933i	$0.496318\pi$
$548$	0	0
$549$	33.4070	1.42578
$550$	0	0
$551$	2.82721	0.120443
$552$	0	0
$553$	−55.6213	−2.36526
$554$	0	0
$555$	2.59765	0.110264
$556$	0	0
$557$	−11.3815	−0.482248	−0.241124	−	0.970494i	$-0.577516\pi$
$557$	−11.3815	−0.482248	−0.241124	+	0.970494i	$0.577516\pi$
$558$	0	0
$559$	−7.86515	−0.332660
$560$	0	0
$561$	17.0971	0.721841
$562$	0	0
$563$	−1.34275	−0.0565901	−0.0282951	−	0.999600i	$-0.509008\pi$
$563$	−1.34275	−0.0565901	−0.0282951	+	0.999600i	$0.509008\pi$
$564$	0	0
$565$	4.33731	0.182472
$566$	0	0
$567$	−24.1996	−1.01629
$568$	0	0
$569$	−9.62395	−0.403457	−0.201728	−	0.979441i	$-0.564656\pi$
$569$	−9.62395	−0.403457	−0.201728	+	0.979441i	$0.564656\pi$
$570$	0	0
$571$	−24.5756	−1.02846	−0.514228	−	0.857654i	$-0.671921\pi$
$571$	−24.5756	−1.02846	−0.514228	+	0.857654i	$0.671921\pi$
$572$	0	0
$573$	−9.22483	−0.385373
$574$	0	0
$575$	−5.23495	−0.218312
$576$	0	0
$577$	28.1374	1.17137	0.585687	−	0.810537i	$-0.300825\pi$
$577$	28.1374	1.17137	0.585687	+	0.810537i	$0.300825\pi$
$578$	0	0
$579$	−51.8459	−2.15464
$580$	0	0
$581$	−42.9247	−1.78081
$582$	0	0
$583$	21.9101	0.907423
$584$	0	0
$585$	−2.61258	−0.108017
$586$	0	0
$587$	19.5404	0.806520	0.403260	−	0.915085i	$-0.367877\pi$
$587$	19.5404	0.806520	0.403260	+	0.915085i	$0.367877\pi$
$588$	0	0
$589$	21.3615	0.880185
$590$	0	0
$591$	44.1533	1.81623
$592$	0	0
$593$	0.466301	0.0191487	0.00957434	−	0.999954i	$-0.496952\pi$
$593$	0.466301	0.0191487	0.00957434	+	0.999954i	$0.496952\pi$
$594$	0	0
$595$	−6.85975	−0.281222
$596$	0	0
$597$	−1.09800	−0.0449379
$598$	0	0
$599$	2.10980	0.0862040	0.0431020	−	0.999071i	$-0.486276\pi$
$599$	2.10980	0.0862040	0.0431020	+	0.999071i	$0.486276\pi$
$600$	0	0
$601$	14.0944	0.574922	0.287461	−	0.957792i	$-0.407189\pi$
$601$	14.0944	0.574922	0.287461	+	0.957792i	$0.407189\pi$
$602$	0	0
$603$	49.5775	2.01895
$604$	0	0
$605$	−3.52257	−0.143213
$606$	0	0
$607$	−8.83303	−0.358521	−0.179261	−	0.983802i	$-0.557371\pi$
$607$	−8.83303	−0.358521	−0.179261	+	0.983802i	$0.557371\pi$
$608$	0	0
$609$	−9.00848	−0.365042
$610$	0	0
$611$	4.29483	0.173750
$612$	0	0
$613$	27.5885	1.11429	0.557144	−	0.830416i	$-0.311897\pi$
$613$	27.5885	1.11429	0.557144	+	0.830416i	$0.311897\pi$
$614$	0	0
$615$	−2.20903	−0.0890765
$616$	0	0
$617$	31.7426	1.27791	0.638954	−	0.769245i	$-0.279367\pi$
$617$	31.7426	1.27791	0.638954	+	0.769245i	$0.279367\pi$
$618$	0	0
$619$	35.1817	1.41407	0.707036	−	0.707178i	$-0.250032\pi$
$619$	35.1817	1.41407	0.707036	+	0.707178i	$0.250032\pi$
$620$	0	0
$621$	−1.76453	−0.0708082
$622$	0	0
$623$	−62.8366	−2.51749
$624$	0	0
$625$	17.3241	0.692963
$626$	0	0
$627$	−18.0144	−0.719425
$628$	0	0
$629$	3.72872	0.148674
$630$	0	0
$631$	−9.59417	−0.381938	−0.190969	−	0.981596i	$-0.561163\pi$
$631$	−9.59417	−0.381938	−0.190969	+	0.981596i	$0.561163\pi$
$632$	0	0
$633$	29.8780	1.18754
$634$	0	0
$635$	15.5676	0.617780
$636$	0	0
$637$	−5.32015	−0.210792
$638$	0	0
$639$	−6.75159	−0.267089
$640$	0	0
$641$	−24.9356	−0.984899	−0.492449	−	0.870341i	$-0.663898\pi$
$641$	−24.9356	−0.984899	−0.492449	+	0.870341i	$0.663898\pi$
$642$	0	0
$643$	27.8899	1.09987	0.549934	−	0.835208i	$-0.314653\pi$
$643$	27.8899	1.09987	0.549934	+	0.835208i	$0.314653\pi$
$644$	0	0
$645$	−14.7024	−0.578908
$646$	0	0
$647$	−5.94879	−0.233871	−0.116936	−	0.993140i	$-0.537307\pi$
$647$	−5.94879	−0.233871	−0.116936	+	0.993140i	$0.537307\pi$
$648$	0	0
$649$	13.4439	0.527721
$650$	0	0
$651$	−68.0652	−2.66769
$652$	0	0
$653$	−16.2642	−0.636468	−0.318234	−	0.948012i	$-0.603090\pi$
$653$	−16.2642	−0.636468	−0.318234	+	0.948012i	$0.603090\pi$
$654$	0	0
$655$	3.51740	0.137436
$656$	0	0
$657$	35.1951	1.37309
$658$	0	0
$659$	30.9334	1.20500	0.602498	−	0.798120i	$-0.294172\pi$
$659$	30.9334	1.20500	0.602498	+	0.798120i	$0.294172\pi$
$660$	0	0
$661$	11.0905	0.431369	0.215685	−	0.976463i	$-0.430802\pi$
$661$	11.0905	0.431369	0.215685	+	0.976463i	$0.430802\pi$
$662$	0	0
$663$	−6.88661	−0.267454
$664$	0	0
$665$	7.22777	0.280281
$666$	0	0
$667$	1.17126	0.0453513
$668$	0	0
$669$	−19.6440	−0.759480
$670$	0	0
$671$	23.1220	0.892613
$672$	0	0
$673$	32.9902	1.27168	0.635839	−	0.771822i	$-0.280654\pi$
$673$	32.9902	1.27168	0.635839	+	0.771822i	$0.280654\pi$
$674$	0	0
$675$	6.73344	0.259170
$676$	0	0
$677$	−20.8467	−0.801206	−0.400603	−	0.916252i	$-0.631199\pi$
$677$	−20.8467	−0.801206	−0.400603	+	0.916252i	$0.631199\pi$
$678$	0	0
$679$	−51.9741	−1.99458
$680$	0	0
$681$	5.07733	0.194564
$682$	0	0
$683$	−24.2783	−0.928984	−0.464492	−	0.885577i	$-0.653763\pi$
$683$	−24.2783	−0.928984	−0.464492	+	0.885577i	$0.653763\pi$
$684$	0	0
$685$	9.23485	0.352846
$686$	0	0
$687$	−63.7692	−2.43294
$688$	0	0
$689$	−8.82523	−0.336215
$690$	0	0
$691$	−6.73295	−0.256133	−0.128067	−	0.991766i	$-0.540877\pi$
$691$	−6.73295	−0.256133	−0.128067	+	0.991766i	$0.540877\pi$
$692$	0	0
$693$	31.2577	1.18738
$694$	0	0
$695$	−2.11729	−0.0803134
$696$	0	0
$697$	−3.17088	−0.120106
$698$	0	0
$699$	−46.4858	−1.75826
$700$	0	0
$701$	−20.4043	−0.770660	−0.385330	−	0.922779i	$-0.625912\pi$
$701$	−20.4043	−0.770660	−0.385330	+	0.922779i	$0.625912\pi$
$702$	0	0
$703$	−3.92876	−0.148176
$704$	0	0
$705$	8.02839	0.302367
$706$	0	0
$707$	−17.0815	−0.642415
$708$	0	0
$709$	−13.2613	−0.498037	−0.249019	−	0.968499i	$-0.580108\pi$
$709$	−13.2613	−0.498037	−0.249019	+	0.968499i	$0.580108\pi$
$710$	0	0
$711$	−56.8413	−2.13172
$712$	0	0
$713$	8.84966	0.331422
$714$	0	0
$715$	−1.80824	−0.0676244
$716$	0	0
$717$	−29.9857	−1.11984
$718$	0	0
$719$	35.1164	1.30962	0.654811	−	0.755792i	$-0.272748\pi$
$719$	35.1164	1.30962	0.654811	+	0.755792i	$0.272748\pi$
$720$	0	0
$721$	−41.1646	−1.53305
$722$	0	0
$723$	42.2779	1.57233
$724$	0	0
$725$	−4.46951	−0.165993
$726$	0	0
$727$	−34.4720	−1.27850	−0.639248	−	0.769001i	$-0.720754\pi$
$727$	−34.4720	−1.27850	−0.639248	+	0.769001i	$0.720754\pi$
$728$	0	0
$729$	−36.3299	−1.34555
$730$	0	0
$731$	−21.1042	−0.780566
$732$	0	0
$733$	41.5666	1.53530	0.767649	−	0.640870i	$-0.221426\pi$
$733$	41.5666	1.53530	0.767649	+	0.640870i	$0.221426\pi$
$734$	0	0
$735$	−9.94504	−0.366828
$736$	0	0
$737$	34.3140	1.26397
$738$	0	0
$739$	−19.4474	−0.715383	−0.357691	−	0.933840i	$-0.616436\pi$
$739$	−19.4474	−0.715383	−0.357691	+	0.933840i	$0.616436\pi$
$740$	0	0
$741$	7.25607	0.266558
$742$	0	0
$743$	−10.9495	−0.401699	−0.200850	−	0.979622i	$-0.564370\pi$
$743$	−10.9495	−0.401699	−0.200850	+	0.979622i	$0.564370\pi$
$744$	0	0
$745$	0.538707	0.0197367
$746$	0	0
$747$	−43.8661	−1.60498
$748$	0	0
$749$	−6.38484	−0.233297
$750$	0	0
$751$	31.9806	1.16699	0.583494	−	0.812117i	$-0.301685\pi$
$751$	31.9806	1.16699	0.583494	+	0.812117i	$0.301685\pi$
$752$	0	0
$753$	75.3256	2.74502
$754$	0	0
$755$	−5.89473	−0.214531
$756$	0	0
$757$	−13.5092	−0.490999	−0.245499	−	0.969397i	$-0.578952\pi$
$757$	−13.5092	−0.490999	−0.245499	+	0.969397i	$0.578952\pi$
$758$	0	0
$759$	−7.46301	−0.270890
$760$	0	0
$761$	6.92504	0.251032	0.125516	−	0.992092i	$-0.459941\pi$
$761$	6.92504	0.251032	0.125516	+	0.992092i	$0.459941\pi$
$762$	0	0
$763$	25.0845	0.908119
$764$	0	0
$765$	−7.01021	−0.253455
$766$	0	0
$767$	−5.41513	−0.195529
$768$	0	0
$769$	39.9602	1.44100	0.720500	−	0.693455i	$-0.243912\pi$
$769$	39.9602	1.44100	0.720500	+	0.693455i	$0.243912\pi$
$770$	0	0
$771$	−32.5564	−1.17249
$772$	0	0
$773$	−7.21535	−0.259518	−0.129759	−	0.991546i	$-0.541420\pi$
$773$	−7.21535	−0.259518	−0.129759	+	0.991546i	$0.541420\pi$
$774$	0	0
$775$	−33.7702	−1.21306
$776$	0	0
$777$	12.5184	0.449096
$778$	0	0
$779$	3.34100	0.119704
$780$	0	0
$781$	−4.67297	−0.167212
$782$	0	0
$783$	−1.50653	−0.0538389
$784$	0	0
$785$	−3.73287	−0.133232
$786$	0	0
$787$	−32.0483	−1.14240	−0.571200	−	0.820811i	$-0.693522\pi$
$787$	−32.0483	−1.14240	−0.571200	+	0.820811i	$0.693522\pi$
$788$	0	0
$789$	27.7023	0.986229
$790$	0	0
$791$	20.9021	0.743192
$792$	0	0
$793$	−9.31337	−0.330727
$794$	0	0
$795$	−16.4971	−0.585093
$796$	0	0
$797$	−50.2933	−1.78148	−0.890739	−	0.454515i	$-0.849813\pi$
$797$	−50.2933	−1.78148	−0.890739	+	0.454515i	$0.849813\pi$
$798$	0	0
$799$	11.5241	0.407694
$800$	0	0
$801$	−64.2148	−2.26892
$802$	0	0
$803$	24.3596	0.859632
$804$	0	0
$805$	2.99433	0.105536
$806$	0	0
$807$	−64.0201	−2.25362
$808$	0	0
$809$	38.9810	1.37050	0.685250	−	0.728308i	$-0.259693\pi$
$809$	38.9810	1.37050	0.685250	+	0.728308i	$0.259693\pi$
$810$	0	0
$811$	−31.9849	−1.12314	−0.561571	−	0.827429i	$-0.689803\pi$
$811$	−31.9849	−1.12314	−0.561571	+	0.827429i	$0.689803\pi$
$812$	0	0
$813$	−52.2763	−1.83341
$814$	0	0
$815$	6.77705	0.237390
$816$	0	0
$817$	22.2364	0.777953
$818$	0	0
$819$	−12.5904	−0.439943
$820$	0	0
$821$	25.7153	0.897468	0.448734	−	0.893665i	$-0.351875\pi$
$821$	25.7153	0.897468	0.448734	+	0.893665i	$0.351875\pi$
$822$	0	0
$823$	5.93723	0.206959	0.103479	−	0.994632i	$-0.467002\pi$
$823$	5.93723	0.206959	0.103479	+	0.994632i	$0.467002\pi$
$824$	0	0
$825$	28.4788	0.991504
$826$	0	0
$827$	39.3162	1.36716	0.683580	−	0.729876i	$-0.260422\pi$
$827$	39.3162	1.36716	0.683580	+	0.729876i	$0.260422\pi$
$828$	0	0
$829$	−21.4547	−0.745152	−0.372576	−	0.928002i	$-0.621525\pi$
$829$	−21.4547	−0.745152	−0.372576	+	0.928002i	$0.621525\pi$
$830$	0	0
$831$	13.9539	0.484057
$832$	0	0
$833$	−14.2753	−0.494610
$834$	0	0
$835$	16.3655	0.566352
$836$	0	0
$837$	−11.3828	−0.393448
$838$	0	0
$839$	−45.9976	−1.58801	−0.794006	−	0.607910i	$-0.792008\pi$
$839$	−45.9976	−1.58801	−0.794006	+	0.607910i	$0.792008\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	83.5155	2.87643
$844$	0	0
$845$	0.728348	0.0250559
$846$	0	0
$847$	−16.9757	−0.583293
$848$	0	0
$849$	−34.5535	−1.18587
$850$	0	0
$851$	−1.62761	−0.0557938
$852$	0	0
$853$	24.4883	0.838465	0.419232	−	0.907879i	$-0.362299\pi$
$853$	24.4883	0.838465	0.419232	+	0.907879i	$0.362299\pi$
$854$	0	0
$855$	7.38630	0.252606
$856$	0	0
$857$	−3.12152	−0.106629	−0.0533145	−	0.998578i	$-0.516979\pi$
$857$	−3.12152	−0.106629	−0.0533145	+	0.998578i	$0.516979\pi$
$858$	0	0
$859$	8.38573	0.286117	0.143059	−	0.989714i	$-0.454306\pi$
$859$	8.38573	0.286117	0.143059	+	0.989714i	$0.454306\pi$
$860$	0	0
$861$	−10.6456	−0.362801
$862$	0	0
$863$	1.54208	0.0524930	0.0262465	−	0.999656i	$-0.491645\pi$
$863$	1.54208	0.0524930	0.0262465	+	0.999656i	$0.491645\pi$
$864$	0	0
$865$	6.68394	0.227261
$866$	0	0
$867$	25.1522	0.854214
$868$	0	0
$869$	−39.3415	−1.33457
$870$	0	0
$871$	−13.8215	−0.468322
$872$	0	0
$873$	−53.1141	−1.79764
$874$	0	0
$875$	−24.2089	−0.818409
$876$	0	0
$877$	15.1006	0.509912	0.254956	−	0.966953i	$-0.417939\pi$
$877$	15.1006	0.509912	0.254956	+	0.966953i	$0.417939\pi$
$878$	0	0
$879$	34.8604	1.17581
$880$	0	0
$881$	−18.4583	−0.621876	−0.310938	−	0.950430i	$-0.600643\pi$
$881$	−18.4583	−0.621876	−0.310938	+	0.950430i	$0.600643\pi$
$882$	0	0
$883$	−41.2974	−1.38977	−0.694884	−	0.719122i	$-0.744544\pi$
$883$	−41.2974	−1.38977	−0.694884	+	0.719122i	$0.744544\pi$
$884$	0	0
$885$	−10.1226	−0.340266
$886$	0	0
$887$	−30.0375	−1.00856	−0.504280	−	0.863540i	$-0.668242\pi$
$887$	−30.0375	−1.00856	−0.504280	+	0.863540i	$0.668242\pi$
$888$	0	0
$889$	75.0223	2.51617
$890$	0	0
$891$	−17.1166	−0.573428
$892$	0	0
$893$	−12.1424	−0.406330
$894$	0	0
$895$	2.82120	0.0943024
$896$	0	0
$897$	3.00605	0.100369
$898$	0	0
$899$	7.55568	0.251996
$900$	0	0
$901$	−23.6803	−0.788907
$902$	0	0
$903$	−70.8530	−2.35784
$904$	0	0
$905$	−7.22058	−0.240020
$906$	0	0
$907$	−33.5376	−1.11360	−0.556799	−	0.830647i	$-0.687971\pi$
$907$	−33.5376	−1.11360	−0.556799	+	0.830647i	$0.687971\pi$
$908$	0	0
$909$	−17.4561	−0.578983
$910$	0	0
$911$	−4.41916	−0.146413	−0.0732067	−	0.997317i	$-0.523323\pi$
$911$	−4.41916	−0.146413	−0.0732067	+	0.997317i	$0.523323\pi$
$912$	0	0
$913$	−30.3610	−1.00480
$914$	0	0
$915$	−17.4096	−0.575544
$916$	0	0
$917$	16.9509	0.559767
$918$	0	0
$919$	24.8134	0.818520	0.409260	−	0.912418i	$-0.365787\pi$
$919$	24.8134	0.818520	0.409260	+	0.912418i	$0.365787\pi$
$920$	0	0
$921$	4.13334	0.136198
$922$	0	0
$923$	1.88224	0.0619548
$924$	0	0
$925$	6.21095	0.204215
$926$	0	0
$927$	−42.0675	−1.38168
$928$	0	0
$929$	−45.4085	−1.48981	−0.744903	−	0.667173i	$-0.767504\pi$
$929$	−45.4085	−1.48981	−0.744903	+	0.667173i	$0.767504\pi$
$930$	0	0
$931$	15.0412	0.492955
$932$	0	0
$933$	39.4305	1.29090
$934$	0	0
$935$	−4.85197	−0.158676
$936$	0	0
$937$	−4.30821	−0.140743	−0.0703716	−	0.997521i	$-0.522419\pi$
$937$	−4.30821	−0.140743	−0.0703716	+	0.997521i	$0.522419\pi$
$938$	0	0
$939$	64.2826	2.09778
$940$	0	0
$941$	−36.9980	−1.20610	−0.603051	−	0.797703i	$-0.706048\pi$
$941$	−36.9980	−1.20610	−0.603051	+	0.797703i	$0.706048\pi$
$942$	0	0
$943$	1.38411	0.0450729
$944$	0	0
$945$	−3.85144	−0.125287
$946$	0	0
$947$	35.9693	1.16885	0.584423	−	0.811449i	$-0.301321\pi$
$947$	35.9693	1.16885	0.584423	+	0.811449i	$0.301321\pi$
$948$	0	0
$949$	−9.81188	−0.318507
$950$	0	0
$951$	−35.3295	−1.14564
$952$	0	0
$953$	30.2012	0.978312	0.489156	−	0.872196i	$-0.337305\pi$
$953$	30.2012	0.978312	0.489156	+	0.872196i	$0.337305\pi$
$954$	0	0
$955$	2.61790	0.0847133
$956$	0	0
$957$	−6.37179	−0.205971
$958$	0	0
$959$	44.5040	1.43711
$960$	0	0
$961$	26.0884	0.841560
$962$	0	0
$963$	−6.52488	−0.210261
$964$	0	0
$965$	14.7133	0.473637
$966$	0	0
$967$	1.58796	0.0510653	0.0255326	−	0.999674i	$-0.491872\pi$
$967$	1.58796	0.0510653	0.0255326	+	0.999674i	$0.491872\pi$
$968$	0	0
$969$	19.4699	0.625463
$970$	0	0
$971$	19.8208	0.636080	0.318040	−	0.948077i	$-0.396975\pi$
$971$	19.8208	0.636080	0.318040	+	0.948077i	$0.396975\pi$
$972$	0	0
$973$	−10.2035	−0.327110
$974$	0	0
$975$	−11.4711	−0.367368
$976$	0	0
$977$	16.9761	0.543115	0.271557	−	0.962422i	$-0.412461\pi$
$977$	16.9761	0.543115	0.271557	+	0.962422i	$0.412461\pi$
$978$	0	0
$979$	−44.4449	−1.42047
$980$	0	0
$981$	25.6347	0.818452
$982$	0	0
$983$	32.0012	1.02068	0.510341	−	0.859972i	$-0.329519\pi$
$983$	32.0012	1.02068	0.510341	+	0.859972i	$0.329519\pi$
$984$	0	0
$985$	−12.5302	−0.399246
$986$	0	0
$987$	38.6899	1.23151
$988$	0	0
$989$	9.21212	0.292928
$990$	0	0
$991$	−17.8555	−0.567198	−0.283599	−	0.958943i	$-0.591528\pi$
$991$	−17.8555	−0.567198	−0.283599	+	0.958943i	$0.591528\pi$
$992$	0	0
$993$	−27.7782	−0.881513
$994$	0	0
$995$	0.311599	0.00987834
$996$	0	0
$997$	−48.2502	−1.52810	−0.764050	−	0.645157i	$-0.776792\pi$
$997$	−48.2502	−1.52810	−0.764050	+	0.645157i	$0.776792\pi$
$998$	0	0
$999$	2.09351	0.0662357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bb.1.2		10
4.3	odd	2		3016.2.a.f.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.f.1.9	✓	10	4.3	odd	2
6032.2.a.bb.1.2		10	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56651 q^{3} +0.728348 q^{5} +3.51001 q^{7} +3.58699 q^{9} +O(q^{10})\)	\(q-2.56651 q^{3} +0.728348 q^{5} +3.51001 q^{7} +3.58699 q^{9} +2.48266 q^{11} -1.00000 q^{13} -1.86931 q^{15} -2.68325 q^{17} +2.82721 q^{19} -9.00848 q^{21} +1.17126 q^{23} -4.46951 q^{25} -1.50653 q^{27} +1.00000 q^{29} +7.55568 q^{31} -6.37179 q^{33} +2.55651 q^{35} -1.38963 q^{37} +2.56651 q^{39} +1.18173 q^{41} +7.86515 q^{43} +2.61258 q^{45} -4.29483 q^{47} +5.32015 q^{49} +6.88661 q^{51} +8.82523 q^{53} +1.80824 q^{55} -7.25607 q^{57} +5.41513 q^{59} +9.31337 q^{61} +12.5904 q^{63} -0.728348 q^{65} +13.8215 q^{67} -3.00605 q^{69} -1.88224 q^{71} +9.81188 q^{73} +11.4711 q^{75} +8.71417 q^{77} -15.8465 q^{79} -6.89446 q^{81} -12.2292 q^{83} -1.95434 q^{85} -2.56651 q^{87} -17.9021 q^{89} -3.51001 q^{91} -19.3918 q^{93} +2.05919 q^{95} -14.8074 q^{97} +8.90530 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9}+O(q^{10}) \) 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9	\( 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9} + 13 q^{11} - 10 q^{13} - 3 q^{15} + 9 q^{17} + 12 q^{19} - 10 q^{21} + 5 q^{23} + 7 q^{25} + 3 q^{27} + 10 q^{29} - 5 q^{31} - 9 q^{33} + 23 q^{35} - 4 q^{37} - 3 q^{39} - 3 q^{41} + 27 q^{43} - 20 q^{45} + 16 q^{47} + 6 q^{49} + 34 q^{51} + 11 q^{53} + q^{55} + 27 q^{59} - 7 q^{61} + 6 q^{63} + 5 q^{65} + 35 q^{67} - 22 q^{69} + 21 q^{71} + 7 q^{75} - 18 q^{77} + 12 q^{79} + 6 q^{81} + 24 q^{83} - 2 q^{85} + 3 q^{87} - 23 q^{89} - 3 q^{93} + 17 q^{95} + 2 q^{97} + 65 q^{99}+O(q^{100}) \) 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9 + 13 * q^11 - 10 * q^13 - 3 * q^15 + 9 * q^17 + 12 * q^19 - 10 * q^21 + 5 * q^23 + 7 * q^25 + 3 * q^27 + 10 * q^29 - 5 * q^31 - 9 * q^33 + 23 * q^35 - 4 * q^37 - 3 * q^39 - 3 * q^41 + 27 * q^43 - 20 * q^45 + 16 * q^47 + 6 * q^49 + 34 * q^51 + 11 * q^53 + q^55 + 27 * q^59 - 7 * q^61 + 6 * q^63 + 5 * q^65 + 35 * q^67 - 22 * q^69 + 21 * q^71 + 7 * q^75 - 18 * q^77 + 12 * q^79 + 6 * q^81 + 24 * q^83 - 2 * q^85 + 3 * q^87 - 23 * q^89 - 3 * q^93 + 17 * q^95 + 2 * q^97 + 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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