LMFDB - Embedded newform 6008.2.a.b.1.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.30
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.09745 q^{3} -0.0446663 q^{5} +4.14700 q^{7} -1.79560 q^{9} +O(q^{10})$	$q+1.09745 q^{3} -0.0446663 q^{5} +4.14700 q^{7} -1.79560 q^{9} -6.33765 q^{11} +2.84157 q^{13} -0.0490192 q^{15} +3.68590 q^{17} -2.76271 q^{19} +4.55115 q^{21} -6.82373 q^{23} -4.99800 q^{25} -5.26294 q^{27} +9.76715 q^{29} -9.04708 q^{31} -6.95528 q^{33} -0.185232 q^{35} -7.68047 q^{37} +3.11850 q^{39} +4.70273 q^{41} -0.928314 q^{43} +0.0802027 q^{45} -1.75621 q^{47} +10.1976 q^{49} +4.04511 q^{51} -6.88021 q^{53} +0.283080 q^{55} -3.03195 q^{57} +5.45578 q^{59} -1.13711 q^{61} -7.44634 q^{63} -0.126923 q^{65} -10.4060 q^{67} -7.48873 q^{69} -9.49858 q^{71} -12.8208 q^{73} -5.48508 q^{75} -26.2823 q^{77} +6.76255 q^{79} -0.389050 q^{81} +10.3424 q^{83} -0.164636 q^{85} +10.7190 q^{87} +4.86666 q^{89} +11.7840 q^{91} -9.92875 q^{93} +0.123400 q^{95} -7.30869 q^{97} +11.3799 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * 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
$3$	1.09745	0.633615	0.316808	−	0.948490i	$-0.397389\pi$
$3$	1.09745	0.633615	0.316808	+	0.948490i	$0.397389\pi$
$4$	0	0
$5$	−0.0446663	−0.0199754	−0.00998770	−	0.999950i	$-0.503179\pi$
$5$	−0.0446663	−0.0199754	−0.00998770	+	0.999950i	$0.503179\pi$
$6$	0	0
$7$	4.14700	1.56742	0.783710	−	0.621127i	$-0.213325\pi$
$7$	4.14700	1.56742	0.783710	+	0.621127i	$0.213325\pi$
$8$	0	0
$9$	−1.79560	−0.598532
$10$	0	0
$11$	−6.33765	−1.91087	−0.955437	−	0.295194i	$-0.904616\pi$
$11$	−6.33765	−1.91087	−0.955437	+	0.295194i	$0.904616\pi$
$12$	0	0
$13$	2.84157	0.788111	0.394055	−	0.919087i	$-0.371072\pi$
$13$	2.84157	0.788111	0.394055	+	0.919087i	$0.371072\pi$
$14$	0	0
$15$	−0.0490192	−0.0126567
$16$	0	0
$17$	3.68590	0.893962	0.446981	−	0.894543i	$-0.352499\pi$
$17$	3.68590	0.893962	0.446981	+	0.894543i	$0.352499\pi$
$18$	0	0
$19$	−2.76271	−0.633810	−0.316905	−	0.948457i	$-0.602644\pi$
$19$	−2.76271	−0.633810	−0.316905	+	0.948457i	$0.602644\pi$
$20$	0	0
$21$	4.55115	0.993141
$22$	0	0
$23$	−6.82373	−1.42285	−0.711423	−	0.702764i	$-0.751949\pi$
$23$	−6.82373	−1.42285	−0.711423	+	0.702764i	$0.751949\pi$
$24$	0	0
$25$	−4.99800	−0.999601
$26$	0	0
$27$	−5.26294	−1.01285
$28$	0	0
$29$	9.76715	1.81371	0.906857	−	0.421439i	$-0.138475\pi$
$29$	9.76715	1.81371	0.906857	+	0.421439i	$0.138475\pi$
$30$	0	0
$31$	−9.04708	−1.62490	−0.812452	−	0.583028i	$-0.801868\pi$
$31$	−9.04708	−1.62490	−0.812452	+	0.583028i	$0.801868\pi$
$32$	0	0
$33$	−6.95528	−1.21076
$34$	0	0
$35$	−0.185232	−0.0313098
$36$	0	0
$37$	−7.68047	−1.26266	−0.631331	−	0.775513i	$-0.717491\pi$
$37$	−7.68047	−1.26266	−0.631331	+	0.775513i	$0.717491\pi$
$38$	0	0
$39$	3.11850	0.499359
$40$	0	0
$41$	4.70273	0.734443	0.367222	−	0.930133i	$-0.380309\pi$
$41$	4.70273	0.734443	0.367222	+	0.930133i	$0.380309\pi$
$42$	0	0
$43$	−0.928314	−0.141566	−0.0707832	−	0.997492i	$-0.522550\pi$
$43$	−0.928314	−0.141566	−0.0707832	+	0.997492i	$0.522550\pi$
$44$	0	0
$45$	0.0802027	0.0119559
$46$	0	0
$47$	−1.75621	−0.256170	−0.128085	−	0.991763i	$-0.540883\pi$
$47$	−1.75621	−0.256170	−0.128085	+	0.991763i	$0.540883\pi$
$48$	0	0
$49$	10.1976	1.45681
$50$	0	0
$51$	4.04511	0.566428
$52$	0	0
$53$	−6.88021	−0.945070	−0.472535	−	0.881312i	$-0.656661\pi$
$53$	−6.88021	−0.945070	−0.472535	+	0.881312i	$0.656661\pi$
$54$	0	0
$55$	0.283080	0.0381705
$56$	0	0
$57$	−3.03195	−0.401592
$58$	0	0
$59$	5.45578	0.710282	0.355141	−	0.934813i	$-0.384433\pi$
$59$	5.45578	0.710282	0.355141	+	0.934813i	$0.384433\pi$
$60$	0	0
$61$	−1.13711	−0.145592	−0.0727961	−	0.997347i	$-0.523192\pi$
$61$	−1.13711	−0.145592	−0.0727961	+	0.997347i	$0.523192\pi$
$62$	0	0
$63$	−7.44634	−0.938151
$64$	0	0
$65$	−0.126923	−0.0157428
$66$	0	0
$67$	−10.4060	−1.27130	−0.635648	−	0.771979i	$-0.719267\pi$
$67$	−10.4060	−1.27130	−0.635648	+	0.771979i	$0.719267\pi$
$68$	0	0
$69$	−7.48873	−0.901537
$70$	0	0
$71$	−9.49858	−1.12727	−0.563637	−	0.826022i	$-0.690598\pi$
$71$	−9.49858	−1.12727	−0.563637	+	0.826022i	$0.690598\pi$
$72$	0	0
$73$	−12.8208	−1.50057	−0.750283	−	0.661117i	$-0.770083\pi$
$73$	−12.8208	−1.50057	−0.750283	+	0.661117i	$0.770083\pi$
$74$	0	0
$75$	−5.48508	−0.633362
$76$	0	0
$77$	−26.2823	−2.99514
$78$	0	0
$79$	6.76255	0.760847	0.380423	−	0.924812i	$-0.375778\pi$
$79$	6.76255	0.760847	0.380423	+	0.924812i	$0.375778\pi$
$80$	0	0
$81$	−0.389050	−0.0432278
$82$	0	0
$83$	10.3424	1.13522	0.567611	−	0.823297i	$-0.307868\pi$
$83$	10.3424	1.13522	0.567611	+	0.823297i	$0.307868\pi$
$84$	0	0
$85$	−0.164636	−0.0178572
$86$	0	0
$87$	10.7190	1.14920
$88$	0	0
$89$	4.86666	0.515865	0.257933	−	0.966163i	$-0.416959\pi$
$89$	4.86666	0.515865	0.257933	+	0.966163i	$0.416959\pi$
$90$	0	0
$91$	11.7840	1.23530
$92$	0	0
$93$	−9.92875	−1.02956
$94$	0	0
$95$	0.123400	0.0126606
$96$	0	0
$97$	−7.30869	−0.742085	−0.371043	−	0.928616i	$-0.621000\pi$
$97$	−7.30869	−0.742085	−0.371043	+	0.928616i	$0.621000\pi$
$98$	0	0
$99$	11.3799	1.14372
$100$	0	0
$101$	−14.3674	−1.42961	−0.714803	−	0.699325i	$-0.753484\pi$
$101$	−14.3674	−1.42961	−0.714803	+	0.699325i	$0.753484\pi$
$102$	0	0
$103$	−14.3615	−1.41508	−0.707542	−	0.706671i	$-0.750196\pi$
$103$	−14.3615	−1.41508	−0.707542	+	0.706671i	$0.750196\pi$
$104$	0	0
$105$	−0.203283	−0.0198384
$106$	0	0
$107$	6.61529	0.639525	0.319762	−	0.947498i	$-0.396397\pi$
$107$	6.61529	0.639525	0.319762	+	0.947498i	$0.396397\pi$
$108$	0	0
$109$	15.6887	1.50270	0.751351	−	0.659903i	$-0.229403\pi$
$109$	15.6887	1.50270	0.751351	+	0.659903i	$0.229403\pi$
$110$	0	0
$111$	−8.42896	−0.800042
$112$	0	0
$113$	−1.45766	−0.137125	−0.0685627	−	0.997647i	$-0.521841\pi$
$113$	−1.45766	−0.137125	−0.0685627	+	0.997647i	$0.521841\pi$
$114$	0	0
$115$	0.304791	0.0284219
$116$	0	0
$117$	−5.10232	−0.471709
$118$	0	0
$119$	15.2854	1.40121
$120$	0	0
$121$	29.1659	2.65144
$122$	0	0
$123$	5.16103	0.465354
$124$	0	0
$125$	0.446574	0.0399428
$126$	0	0
$127$	0.340971	0.0302563	0.0151281	−	0.999886i	$-0.495184\pi$
$127$	0.340971	0.0302563	0.0151281	+	0.999886i	$0.495184\pi$
$128$	0	0
$129$	−1.01878	−0.0896987
$130$	0	0
$131$	−0.930386	−0.0812882	−0.0406441	−	0.999174i	$-0.512941\pi$
$131$	−0.930386	−0.0812882	−0.0406441	+	0.999174i	$0.512941\pi$
$132$	0	0
$133$	−11.4570	−0.993447
$134$	0	0
$135$	0.235076	0.0202322
$136$	0	0
$137$	5.82807	0.497925	0.248963	−	0.968513i	$-0.419910\pi$
$137$	5.82807	0.497925	0.248963	+	0.968513i	$0.419910\pi$
$138$	0	0
$139$	−8.12159	−0.688865	−0.344432	−	0.938811i	$-0.611929\pi$
$139$	−8.12159	−0.688865	−0.344432	+	0.938811i	$0.611929\pi$
$140$	0	0
$141$	−1.92736	−0.162313
$142$	0	0
$143$	−18.0089	−1.50598
$144$	0	0
$145$	−0.436263	−0.0362297
$146$	0	0
$147$	11.1914	0.923055
$148$	0	0
$149$	0.144482	0.0118364	0.00591821	−	0.999982i	$-0.498116\pi$
$149$	0.144482	0.0118364	0.00591821	+	0.999982i	$0.498116\pi$
$150$	0	0
$151$	−2.52023	−0.205094	−0.102547	−	0.994728i	$-0.532699\pi$
$151$	−2.52023	−0.205094	−0.102547	+	0.994728i	$0.532699\pi$
$152$	0	0
$153$	−6.61839	−0.535065
$154$	0	0
$155$	0.404100	0.0324581
$156$	0	0
$157$	−9.89356	−0.789592	−0.394796	−	0.918769i	$-0.629185\pi$
$157$	−9.89356	−0.789592	−0.394796	+	0.918769i	$0.629185\pi$
$158$	0	0
$159$	−7.55071	−0.598810
$160$	0	0
$161$	−28.2980	−2.23020
$162$	0	0
$163$	−3.02561	−0.236984	−0.118492	−	0.992955i	$-0.537806\pi$
$163$	−3.02561	−0.236984	−0.118492	+	0.992955i	$0.537806\pi$
$164$	0	0
$165$	0.310667	0.0241854
$166$	0	0
$167$	−2.58138	−0.199753	−0.0998765	−	0.995000i	$-0.531845\pi$
$167$	−2.58138	−0.199753	−0.0998765	+	0.995000i	$0.531845\pi$
$168$	0	0
$169$	−4.92546	−0.378881
$170$	0	0
$171$	4.96072	0.379355
$172$	0	0
$173$	2.26721	0.172373	0.0861864	−	0.996279i	$-0.472532\pi$
$173$	2.26721	0.172373	0.0861864	+	0.996279i	$0.472532\pi$
$174$	0	0
$175$	−20.7268	−1.56680
$176$	0	0
$177$	5.98746	0.450045
$178$	0	0
$179$	1.23146	0.0920433	0.0460217	−	0.998940i	$-0.485346\pi$
$179$	1.23146	0.0920433	0.0460217	+	0.998940i	$0.485346\pi$
$180$	0	0
$181$	−4.51384	−0.335511	−0.167756	−	0.985829i	$-0.553652\pi$
$181$	−4.51384	−0.335511	−0.167756	+	0.985829i	$0.553652\pi$
$182$	0	0
$183$	−1.24793	−0.0922494
$184$	0	0
$185$	0.343059	0.0252222
$186$	0	0
$187$	−23.3600	−1.70825
$188$	0	0
$189$	−21.8255	−1.58757
$190$	0	0
$191$	9.98607	0.722567	0.361283	−	0.932456i	$-0.382339\pi$
$191$	9.98607	0.722567	0.361283	+	0.932456i	$0.382339\pi$
$192$	0	0
$193$	5.39231	0.388147	0.194074	−	0.980987i	$-0.437830\pi$
$193$	5.39231	0.388147	0.194074	+	0.980987i	$0.437830\pi$
$194$	0	0
$195$	−0.139292	−0.00997489
$196$	0	0
$197$	−20.8726	−1.48711	−0.743556	−	0.668674i	$-0.766862\pi$
$197$	−20.8726	−1.48711	−0.743556	+	0.668674i	$0.766862\pi$
$198$	0	0
$199$	10.7561	0.762478	0.381239	−	0.924476i	$-0.375497\pi$
$199$	10.7561	0.762478	0.381239	+	0.924476i	$0.375497\pi$
$200$	0	0
$201$	−11.4201	−0.805512
$202$	0	0
$203$	40.5044	2.84285
$204$	0	0
$205$	−0.210054	−0.0146708
$206$	0	0
$207$	12.2527	0.851618
$208$	0	0
$209$	17.5091	1.21113
$210$	0	0
$211$	−11.1041	−0.764438	−0.382219	−	0.924072i	$-0.624840\pi$
$211$	−11.1041	−0.764438	−0.382219	+	0.924072i	$0.624840\pi$
$212$	0	0
$213$	−10.4243	−0.714258
$214$	0	0
$215$	0.0414644	0.00282785
$216$	0	0
$217$	−37.5183	−2.54691
$218$	0	0
$219$	−14.0703	−0.950781
$220$	0	0
$221$	10.4738	0.704541
$222$	0	0
$223$	−16.3917	−1.09767	−0.548836	−	0.835930i	$-0.684929\pi$
$223$	−16.3917	−1.09767	−0.548836	+	0.835930i	$0.684929\pi$
$224$	0	0
$225$	8.97440	0.598293
$226$	0	0
$227$	−7.66340	−0.508637	−0.254319	−	0.967120i	$-0.581851\pi$
$227$	−7.66340	−0.508637	−0.254319	+	0.967120i	$0.581851\pi$
$228$	0	0
$229$	16.8035	1.11040	0.555202	−	0.831715i	$-0.312641\pi$
$229$	16.8035	1.11040	0.555202	+	0.831715i	$0.312641\pi$
$230$	0	0
$231$	−28.8436	−1.89777
$232$	0	0
$233$	8.43463	0.552571	0.276285	−	0.961076i	$-0.410897\pi$
$233$	8.43463	0.552571	0.276285	+	0.961076i	$0.410897\pi$
$234$	0	0
$235$	0.0784437	0.00511710
$236$	0	0
$237$	7.42159	0.482084
$238$	0	0
$239$	−6.66677	−0.431237	−0.215619	−	0.976478i	$-0.569177\pi$
$239$	−6.66677	−0.431237	−0.215619	+	0.976478i	$0.569177\pi$
$240$	0	0
$241$	−4.64555	−0.299246	−0.149623	−	0.988743i	$-0.547806\pi$
$241$	−4.64555	−0.299246	−0.149623	+	0.988743i	$0.547806\pi$
$242$	0	0
$243$	15.3619	0.985464
$244$	0	0
$245$	−0.455492	−0.0291003
$246$	0	0
$247$	−7.85046	−0.499513
$248$	0	0
$249$	11.3503	0.719294
$250$	0	0
$251$	26.8129	1.69242	0.846208	−	0.532853i	$-0.178880\pi$
$251$	26.8129	1.69242	0.846208	+	0.532853i	$0.178880\pi$
$252$	0	0
$253$	43.2464	2.71888
$254$	0	0
$255$	−0.180680	−0.0113146
$256$	0	0
$257$	8.30438	0.518013	0.259006	−	0.965876i	$-0.416605\pi$
$257$	8.30438	0.518013	0.259006	+	0.965876i	$0.416605\pi$
$258$	0	0
$259$	−31.8510	−1.97912
$260$	0	0
$261$	−17.5378	−1.08557
$262$	0	0
$263$	7.37953	0.455041	0.227521	−	0.973773i	$-0.426938\pi$
$263$	7.37953	0.455041	0.227521	+	0.973773i	$0.426938\pi$
$264$	0	0
$265$	0.307314	0.0188781
$266$	0	0
$267$	5.34094	0.326860
$268$	0	0
$269$	19.9288	1.21508	0.607539	−	0.794290i	$-0.292157\pi$
$269$	19.9288	1.21508	0.607539	+	0.794290i	$0.292157\pi$
$270$	0	0
$271$	−3.17593	−0.192924	−0.0964619	−	0.995337i	$-0.530753\pi$
$271$	−3.17593	−0.192924	−0.0964619	+	0.995337i	$0.530753\pi$
$272$	0	0
$273$	12.9324	0.782706
$274$	0	0
$275$	31.6756	1.91011
$276$	0	0
$277$	24.1463	1.45081	0.725405	−	0.688322i	$-0.241652\pi$
$277$	24.1463	1.45081	0.725405	+	0.688322i	$0.241652\pi$
$278$	0	0
$279$	16.2449	0.972557
$280$	0	0
$281$	6.22309	0.371239	0.185619	−	0.982622i	$-0.440571\pi$
$281$	6.22309	0.371239	0.185619	+	0.982622i	$0.440571\pi$
$282$	0	0
$283$	−28.8669	−1.71596	−0.857981	−	0.513682i	$-0.828281\pi$
$283$	−28.8669	−1.71596	−0.857981	+	0.513682i	$0.828281\pi$
$284$	0	0
$285$	0.135426	0.00802195
$286$	0	0
$287$	19.5023	1.15118
$288$	0	0
$289$	−3.41414	−0.200832
$290$	0	0
$291$	−8.02095	−0.470197
$292$	0	0
$293$	27.6136	1.61321	0.806603	−	0.591093i	$-0.201304\pi$
$293$	27.6136	1.61321	0.806603	+	0.591093i	$0.201304\pi$
$294$	0	0
$295$	−0.243690	−0.0141882
$296$	0	0
$297$	33.3547	1.93544
$298$	0	0
$299$	−19.3901	−1.12136
$300$	0	0
$301$	−3.84972	−0.221894
$302$	0	0
$303$	−15.7675	−0.905821
$304$	0	0
$305$	0.0507906	0.00290826
$306$	0	0
$307$	31.3405	1.78869	0.894347	−	0.447373i	$-0.147641\pi$
$307$	31.3405	1.78869	0.894347	+	0.447373i	$0.147641\pi$
$308$	0	0
$309$	−15.7611	−0.896619
$310$	0	0
$311$	−23.4732	−1.33104	−0.665521	−	0.746379i	$-0.731791\pi$
$311$	−23.4732	−1.33104	−0.665521	+	0.746379i	$0.731791\pi$
$312$	0	0
$313$	−6.27594	−0.354737	−0.177368	−	0.984145i	$-0.556758\pi$
$313$	−6.27594	−0.354737	−0.177368	+	0.984145i	$0.556758\pi$
$314$	0	0
$315$	0.332601	0.0187399
$316$	0	0
$317$	−12.0524	−0.676929	−0.338464	−	0.940979i	$-0.609907\pi$
$317$	−12.0524	−0.676929	−0.338464	+	0.940979i	$0.609907\pi$
$318$	0	0
$319$	−61.9008	−3.46578
$320$	0	0
$321$	7.25998	0.405212
$322$	0	0
$323$	−10.1831	−0.566602
$324$	0	0
$325$	−14.2022	−0.787796
$326$	0	0
$327$	17.2176	0.952135
$328$	0	0
$329$	−7.28303	−0.401526
$330$	0	0
$331$	−15.1868	−0.834745	−0.417372	−	0.908736i	$-0.637049\pi$
$331$	−15.1868	−0.834745	−0.417372	+	0.908736i	$0.637049\pi$
$332$	0	0
$333$	13.7910	0.755743
$334$	0	0
$335$	0.464798	0.0253946
$336$	0	0
$337$	−29.2744	−1.59468	−0.797339	−	0.603531i	$-0.793760\pi$
$337$	−29.2744	−1.59468	−0.797339	+	0.603531i	$0.793760\pi$
$338$	0	0
$339$	−1.59972	−0.0868847
$340$	0	0
$341$	57.3373	3.10499
$342$	0	0
$343$	13.2607	0.716008
$344$	0	0
$345$	0.334494	0.0180086
$346$	0	0
$347$	−1.13669	−0.0610208	−0.0305104	−	0.999534i	$-0.509713\pi$
$347$	−1.13669	−0.0610208	−0.0305104	+	0.999534i	$0.509713\pi$
$348$	0	0
$349$	30.4800	1.63156	0.815779	−	0.578364i	$-0.196309\pi$
$349$	30.4800	1.63156	0.815779	+	0.578364i	$0.196309\pi$
$350$	0	0
$351$	−14.9550	−0.798241
$352$	0	0
$353$	−16.1432	−0.859218	−0.429609	−	0.903015i	$-0.641349\pi$
$353$	−16.1432	−0.859218	−0.429609	+	0.903015i	$0.641349\pi$
$354$	0	0
$355$	0.424267	0.0225178
$356$	0	0
$357$	16.7751	0.887831
$358$	0	0
$359$	−24.0899	−1.27142	−0.635708	−	0.771929i	$-0.719292\pi$
$359$	−24.0899	−1.27142	−0.635708	+	0.771929i	$0.719292\pi$
$360$	0	0
$361$	−11.3674	−0.598285
$362$	0	0
$363$	32.0082	1.67999
$364$	0	0
$365$	0.572660	0.0299744
$366$	0	0
$367$	−20.3603	−1.06280	−0.531399	−	0.847122i	$-0.678334\pi$
$367$	−20.3603	−1.06280	−0.531399	+	0.847122i	$0.678334\pi$
$368$	0	0
$369$	−8.44421	−0.439588
$370$	0	0
$371$	−28.5323	−1.48132
$372$	0	0
$373$	−37.4220	−1.93764	−0.968818	−	0.247772i	$-0.920302\pi$
$373$	−37.4220	−1.93764	−0.968818	+	0.247772i	$0.920302\pi$
$374$	0	0
$375$	0.490095	0.0253084
$376$	0	0
$377$	27.7541	1.42941
$378$	0	0
$379$	11.4790	0.589635	0.294817	−	0.955554i	$-0.404741\pi$
$379$	11.4790	0.589635	0.294817	+	0.955554i	$0.404741\pi$
$380$	0	0
$381$	0.374200	0.0191709
$382$	0	0
$383$	−6.17583	−0.315570	−0.157785	−	0.987473i	$-0.550435\pi$
$383$	−6.17583	−0.315570	−0.157785	+	0.987473i	$0.550435\pi$
$384$	0	0
$385$	1.17393	0.0598292
$386$	0	0
$387$	1.66688	0.0847320
$388$	0	0
$389$	−28.8214	−1.46130	−0.730650	−	0.682752i	$-0.760783\pi$
$389$	−28.8214	−1.46130	−0.730650	+	0.682752i	$0.760783\pi$
$390$	0	0
$391$	−25.1516	−1.27197
$392$	0	0
$393$	−1.02106	−0.0515054
$394$	0	0
$395$	−0.302058	−0.0151982
$396$	0	0
$397$	10.1865	0.511248	0.255624	−	0.966776i	$-0.417719\pi$
$397$	10.1865	0.511248	0.255624	+	0.966776i	$0.417719\pi$
$398$	0	0
$399$	−12.5735	−0.629463
$400$	0	0
$401$	−28.1703	−1.40676	−0.703378	−	0.710816i	$-0.748326\pi$
$401$	−28.1703	−1.40676	−0.703378	+	0.710816i	$0.748326\pi$
$402$	0	0
$403$	−25.7080	−1.28060
$404$	0	0
$405$	0.0173775	0.000863493 0
$406$	0	0
$407$	48.6762	2.41279
$408$	0	0
$409$	35.7516	1.76780	0.883900	−	0.467675i	$-0.154908\pi$
$409$	35.7516	1.76780	0.883900	+	0.467675i	$0.154908\pi$
$410$	0	0
$411$	6.39603	0.315493
$412$	0	0
$413$	22.6251	1.11331
$414$	0	0
$415$	−0.461956	−0.0226765
$416$	0	0
$417$	−8.91307	−0.436475
$418$	0	0
$419$	18.5723	0.907317	0.453658	−	0.891176i	$-0.350119\pi$
$419$	18.5723	0.907317	0.453658	+	0.891176i	$0.350119\pi$
$420$	0	0
$421$	29.5659	1.44095	0.720476	−	0.693480i	$-0.243923\pi$
$421$	29.5659	1.44095	0.720476	+	0.693480i	$0.243923\pi$
$422$	0	0
$423$	3.15345	0.153326
$424$	0	0
$425$	−18.4221	−0.893606
$426$	0	0
$427$	−4.71561	−0.228204
$428$	0	0
$429$	−19.7640	−0.954213
$430$	0	0
$431$	36.9474	1.77969	0.889846	−	0.456261i	$-0.150811\pi$
$431$	36.9474	1.77969	0.889846	+	0.456261i	$0.150811\pi$
$432$	0	0
$433$	10.6218	0.510452	0.255226	−	0.966881i	$-0.417850\pi$
$433$	10.6218	0.510452	0.255226	+	0.966881i	$0.417850\pi$
$434$	0	0
$435$	−0.478778	−0.0229557
$436$	0	0
$437$	18.8520	0.901814
$438$	0	0
$439$	22.8405	1.09012	0.545058	−	0.838398i	$-0.316507\pi$
$439$	22.8405	1.09012	0.545058	+	0.838398i	$0.316507\pi$
$440$	0	0
$441$	−18.3109	−0.871945
$442$	0	0
$443$	11.6837	0.555109	0.277554	−	0.960710i	$-0.410476\pi$
$443$	11.6837	0.555109	0.277554	+	0.960710i	$0.410476\pi$
$444$	0	0
$445$	−0.217376	−0.0103046
$446$	0	0
$447$	0.158562	0.00749973
$448$	0	0
$449$	13.5886	0.641287	0.320644	−	0.947200i	$-0.396101\pi$
$449$	13.5886	0.641287	0.320644	+	0.947200i	$0.396101\pi$
$450$	0	0
$451$	−29.8043	−1.40343
$452$	0	0
$453$	−2.76584	−0.129950
$454$	0	0
$455$	−0.526349	−0.0246756
$456$	0	0
$457$	−34.8883	−1.63201	−0.816004	−	0.578046i	$-0.803815\pi$
$457$	−34.8883	−1.63201	−0.816004	+	0.578046i	$0.803815\pi$
$458$	0	0
$459$	−19.3987	−0.905453
$460$	0	0
$461$	31.9313	1.48719	0.743595	−	0.668630i	$-0.233119\pi$
$461$	31.9313	1.48719	0.743595	+	0.668630i	$0.233119\pi$
$462$	0	0
$463$	−32.0806	−1.49091	−0.745456	−	0.666555i	$-0.767768\pi$
$463$	−32.0806	−1.49091	−0.745456	+	0.666555i	$0.767768\pi$
$464$	0	0
$465$	0.443481	0.0205659
$466$	0	0
$467$	7.22053	0.334126	0.167063	−	0.985946i	$-0.446572\pi$
$467$	7.22053	0.334126	0.167063	+	0.985946i	$0.446572\pi$
$468$	0	0
$469$	−43.1538	−1.99266
$470$	0	0
$471$	−10.8577	−0.500297
$472$	0	0
$473$	5.88333	0.270516
$474$	0	0
$475$	13.8081	0.633557
$476$	0	0
$477$	12.3541	0.565654
$478$	0	0
$479$	−37.2220	−1.70072	−0.850358	−	0.526205i	$-0.823614\pi$
$479$	−37.2220	−1.70072	−0.850358	+	0.526205i	$0.823614\pi$
$480$	0	0
$481$	−21.8246	−0.995118
$482$	0	0
$483$	−31.0558	−1.41309
$484$	0	0
$485$	0.326453	0.0148234
$486$	0	0
$487$	−14.6015	−0.661657	−0.330829	−	0.943691i	$-0.607328\pi$
$487$	−14.6015	−0.661657	−0.330829	+	0.943691i	$0.607328\pi$
$488$	0	0
$489$	−3.32047	−0.150157
$490$	0	0
$491$	22.7655	1.02739	0.513696	−	0.857972i	$-0.328276\pi$
$491$	22.7655	1.02739	0.513696	+	0.857972i	$0.328276\pi$
$492$	0	0
$493$	36.0007	1.62139
$494$	0	0
$495$	−0.508297	−0.0228462
$496$	0	0
$497$	−39.3907	−1.76691
$498$	0	0
$499$	−13.5960	−0.608642	−0.304321	−	0.952570i	$-0.598430\pi$
$499$	−13.5960	−0.608642	−0.304321	+	0.952570i	$0.598430\pi$
$500$	0	0
$501$	−2.83294	−0.126567
$502$	0	0
$503$	20.8709	0.930586	0.465293	−	0.885157i	$-0.345949\pi$
$503$	20.8709	0.930586	0.465293	+	0.885157i	$0.345949\pi$
$504$	0	0
$505$	0.641738	0.0285570
$506$	0	0
$507$	−5.40546	−0.240065
$508$	0	0
$509$	11.8806	0.526598	0.263299	−	0.964714i	$-0.415189\pi$
$509$	11.8806	0.526598	0.263299	+	0.964714i	$0.415189\pi$
$510$	0	0
$511$	−53.1681	−2.35202
$512$	0	0
$513$	14.5400	0.641957
$514$	0	0
$515$	0.641477	0.0282669
$516$	0	0
$517$	11.1303	0.489509
$518$	0	0
$519$	2.48816	0.109218
$520$	0	0
$521$	−1.23770	−0.0542247	−0.0271123	−	0.999632i	$-0.508631\pi$
$521$	−1.23770	−0.0542247	−0.0271123	+	0.999632i	$0.508631\pi$
$522$	0	0
$523$	−17.2331	−0.753553	−0.376776	−	0.926304i	$-0.622967\pi$
$523$	−17.2331	−0.753553	−0.376776	+	0.926304i	$0.622967\pi$
$524$	0	0
$525$	−22.7466	−0.992745
$526$	0	0
$527$	−33.3466	−1.45260
$528$	0	0
$529$	23.5633	1.02449
$530$	0	0
$531$	−9.79637	−0.425126
$532$	0	0
$533$	13.3632	0.578823
$534$	0	0
$535$	−0.295481	−0.0127748
$536$	0	0
$537$	1.35147	0.0583200
$538$	0	0
$539$	−64.6292	−2.78378
$540$	0	0
$541$	−42.9788	−1.84780	−0.923901	−	0.382631i	$-0.875018\pi$
$541$	−42.9788	−1.84780	−0.923901	+	0.382631i	$0.875018\pi$
$542$	0	0
$543$	−4.95373	−0.212585
$544$	0	0
$545$	−0.700755	−0.0300171
$546$	0	0
$547$	−44.2130	−1.89041	−0.945206	−	0.326475i	$-0.894139\pi$
$547$	−44.2130	−1.89041	−0.945206	+	0.326475i	$0.894139\pi$
$548$	0	0
$549$	2.04179	0.0871416
$550$	0	0
$551$	−26.9838	−1.14955
$552$	0	0
$553$	28.0443	1.19257
$554$	0	0
$555$	0.376491	0.0159811
$556$	0	0
$557$	−13.7776	−0.583777	−0.291888	−	0.956452i	$-0.594284\pi$
$557$	−13.7776	−0.583777	−0.291888	+	0.956452i	$0.594284\pi$
$558$	0	0
$559$	−2.63787	−0.111570
$560$	0	0
$561$	−25.6365	−1.08237
$562$	0	0
$563$	32.2279	1.35824	0.679121	−	0.734026i	$-0.262361\pi$
$563$	32.2279	1.35824	0.679121	+	0.734026i	$0.262361\pi$
$564$	0	0
$565$	0.0651085	0.00273913
$566$	0	0
$567$	−1.61339	−0.0677562
$568$	0	0
$569$	−0.774502	−0.0324688	−0.0162344	−	0.999868i	$-0.505168\pi$
$569$	−0.774502	−0.0324688	−0.0162344	+	0.999868i	$0.505168\pi$
$570$	0	0
$571$	12.2585	0.513001	0.256500	−	0.966544i	$-0.417431\pi$
$571$	12.2585	0.513001	0.256500	+	0.966544i	$0.417431\pi$
$572$	0	0
$573$	10.9593	0.457829
$574$	0	0
$575$	34.1050	1.42228
$576$	0	0
$577$	−33.1886	−1.38166	−0.690830	−	0.723017i	$-0.742755\pi$
$577$	−33.1886	−1.38166	−0.690830	+	0.723017i	$0.742755\pi$
$578$	0	0
$579$	5.91781	0.245936
$580$	0	0
$581$	42.8899	1.77937
$582$	0	0
$583$	43.6044	1.80591
$584$	0	0
$585$	0.227902	0.00942258
$586$	0	0
$587$	−14.6862	−0.606163	−0.303082	−	0.952965i	$-0.598015\pi$
$587$	−14.6862	−0.606163	−0.303082	+	0.952965i	$0.598015\pi$
$588$	0	0
$589$	24.9945	1.02988
$590$	0	0
$591$	−22.9067	−0.942256
$592$	0	0
$593$	10.4206	0.427922	0.213961	−	0.976842i	$-0.431364\pi$
$593$	10.4206	0.427922	0.213961	+	0.976842i	$0.431364\pi$
$594$	0	0
$595$	−0.682745	−0.0279898
$596$	0	0
$597$	11.8043	0.483118
$598$	0	0
$599$	−36.7275	−1.50064	−0.750322	−	0.661073i	$-0.770101\pi$
$599$	−36.7275	−1.50064	−0.750322	+	0.661073i	$0.770101\pi$
$600$	0	0
$601$	4.09296	0.166955	0.0834777	−	0.996510i	$-0.473397\pi$
$601$	4.09296	0.166955	0.0834777	+	0.996510i	$0.473397\pi$
$602$	0	0
$603$	18.6850	0.760911
$604$	0	0
$605$	−1.30273	−0.0529636
$606$	0	0
$607$	42.9951	1.74512	0.872559	−	0.488509i	$-0.162459\pi$
$607$	42.9951	1.74512	0.872559	+	0.488509i	$0.162459\pi$
$608$	0	0
$609$	44.4517	1.80127
$610$	0	0
$611$	−4.99041	−0.201890
$612$	0	0
$613$	−19.4209	−0.784404	−0.392202	−	0.919879i	$-0.628287\pi$
$613$	−19.4209	−0.784404	−0.392202	+	0.919879i	$0.628287\pi$
$614$	0	0
$615$	−0.230524	−0.00929564
$616$	0	0
$617$	−39.8522	−1.60439	−0.802195	−	0.597062i	$-0.796335\pi$
$617$	−39.8522	−1.60439	−0.802195	+	0.597062i	$0.796335\pi$
$618$	0	0
$619$	26.8560	1.07943	0.539716	−	0.841847i	$-0.318532\pi$
$619$	26.8560	1.07943	0.539716	+	0.841847i	$0.318532\pi$
$620$	0	0
$621$	35.9129	1.44114
$622$	0	0
$623$	20.1821	0.808578
$624$	0	0
$625$	24.9701	0.998803
$626$	0	0
$627$	19.2155	0.767391
$628$	0	0
$629$	−28.3095	−1.12877
$630$	0	0
$631$	−9.92711	−0.395192	−0.197596	−	0.980284i	$-0.563313\pi$
$631$	−9.92711	−0.395192	−0.197596	+	0.980284i	$0.563313\pi$
$632$	0	0
$633$	−12.1862	−0.484359
$634$	0	0
$635$	−0.0152299	−0.000604382 0
$636$	0	0
$637$	28.9774	1.14813
$638$	0	0
$639$	17.0556	0.674710
$640$	0	0
$641$	1.94948	0.0769998	0.0384999	−	0.999259i	$-0.487742\pi$
$641$	1.94948	0.0769998	0.0384999	+	0.999259i	$0.487742\pi$
$642$	0	0
$643$	9.35437	0.368900	0.184450	−	0.982842i	$-0.440950\pi$
$643$	9.35437	0.368900	0.184450	+	0.982842i	$0.440950\pi$
$644$	0	0
$645$	0.0455052	0.00179177
$646$	0	0
$647$	28.7731	1.13119	0.565594	−	0.824684i	$-0.308647\pi$
$647$	28.7731	1.13119	0.565594	+	0.824684i	$0.308647\pi$
$648$	0	0
$649$	−34.5768	−1.35726
$650$	0	0
$651$	−41.1746	−1.61376
$652$	0	0
$653$	38.7694	1.51717	0.758583	−	0.651577i	$-0.225892\pi$
$653$	38.7694	1.51717	0.758583	+	0.651577i	$0.225892\pi$
$654$	0	0
$655$	0.0415569	0.00162376
$656$	0	0
$657$	23.0210	0.898136
$658$	0	0
$659$	−38.9003	−1.51534	−0.757671	−	0.652637i	$-0.773663\pi$
$659$	−38.9003	−1.51534	−0.757671	+	0.652637i	$0.773663\pi$
$660$	0	0
$661$	−47.4785	−1.84670	−0.923349	−	0.383961i	$-0.874560\pi$
$661$	−47.4785	−1.84670	−0.923349	+	0.383961i	$0.874560\pi$
$662$	0	0
$663$	11.4945	0.446408
$664$	0	0
$665$	0.511742	0.0198445
$666$	0	0
$667$	−66.6484	−2.58064
$668$	0	0
$669$	−17.9892	−0.695502
$670$	0	0
$671$	7.20662	0.278209
$672$	0	0
$673$	−32.4504	−1.25087	−0.625436	−	0.780275i	$-0.715079\pi$
$673$	−32.4504	−1.25087	−0.625436	+	0.780275i	$0.715079\pi$
$674$	0	0
$675$	26.3042	1.01245
$676$	0	0
$677$	−9.73894	−0.374298	−0.187149	−	0.982332i	$-0.559925\pi$
$677$	−9.73894	−0.374298	−0.187149	+	0.982332i	$0.559925\pi$
$678$	0	0
$679$	−30.3092	−1.16316
$680$	0	0
$681$	−8.41022	−0.322280
$682$	0	0
$683$	−6.64784	−0.254373	−0.127186	−	0.991879i	$-0.540595\pi$
$683$	−6.64784	−0.254373	−0.127186	+	0.991879i	$0.540595\pi$
$684$	0	0
$685$	−0.260318	−0.00994625
$686$	0	0
$687$	18.4410	0.703569
$688$	0	0
$689$	−19.5506	−0.744820
$690$	0	0
$691$	5.08544	0.193459	0.0967296	−	0.995311i	$-0.469162\pi$
$691$	5.08544	0.193459	0.0967296	+	0.995311i	$0.469162\pi$
$692$	0	0
$693$	47.1924	1.79269
$694$	0	0
$695$	0.362762	0.0137603
$696$	0	0
$697$	17.3338	0.656565
$698$	0	0
$699$	9.25661	0.350117
$700$	0	0
$701$	35.2295	1.33060	0.665299	−	0.746577i	$-0.268304\pi$
$701$	35.2295	1.33060	0.665299	+	0.746577i	$0.268304\pi$
$702$	0	0
$703$	21.2189	0.800288
$704$	0	0
$705$	0.0860883	0.00324227
$706$	0	0
$707$	−59.5816	−2.24080
$708$	0	0
$709$	−0.699610	−0.0262744	−0.0131372	−	0.999914i	$-0.504182\pi$
$709$	−0.699610	−0.0262744	−0.0131372	+	0.999914i	$0.504182\pi$
$710$	0	0
$711$	−12.1428	−0.455391
$712$	0	0
$713$	61.7348	2.31199
$714$	0	0
$715$	0.804392	0.0300826
$716$	0	0
$717$	−7.31647	−0.273238
$718$	0	0
$719$	17.7467	0.661840	0.330920	−	0.943659i	$-0.392641\pi$
$719$	17.7467	0.661840	0.330920	+	0.943659i	$0.392641\pi$
$720$	0	0
$721$	−59.5574	−2.21803
$722$	0	0
$723$	−5.09827	−0.189607
$724$	0	0
$725$	−48.8163	−1.81299
$726$	0	0
$727$	11.5641	0.428889	0.214444	−	0.976736i	$-0.431206\pi$
$727$	11.5641	0.428889	0.214444	+	0.976736i	$0.431206\pi$
$728$	0	0
$729$	18.0261	0.667633
$730$	0	0
$731$	−3.42167	−0.126555
$732$	0	0
$733$	−4.43798	−0.163920	−0.0819602	−	0.996636i	$-0.526118\pi$
$733$	−4.43798	−0.163920	−0.0819602	+	0.996636i	$0.526118\pi$
$734$	0	0
$735$	−0.499881	−0.0184384
$736$	0	0
$737$	65.9497	2.42929
$738$	0	0
$739$	−3.07315	−0.113048	−0.0565238	−	0.998401i	$-0.518002\pi$
$739$	−3.07315	−0.113048	−0.0565238	+	0.998401i	$0.518002\pi$
$740$	0	0
$741$	−8.61551	−0.316499
$742$	0	0
$743$	−40.4436	−1.48373	−0.741867	−	0.670547i	$-0.766059\pi$
$743$	−40.4436	−1.48373	−0.741867	+	0.670547i	$0.766059\pi$
$744$	0	0
$745$	−0.00645348	−0.000236437 0
$746$	0	0
$747$	−18.5707	−0.679467
$748$	0	0
$749$	27.4337	1.00240
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	29.4259	1.07234
$754$	0	0
$755$	0.112570	0.00409682
$756$	0	0
$757$	43.4929	1.58078	0.790388	−	0.612606i	$-0.209879\pi$
$757$	43.4929	1.58078	0.790388	+	0.612606i	$0.209879\pi$
$758$	0	0
$759$	47.4610	1.72272
$760$	0	0
$761$	−27.5143	−0.997393	−0.498697	−	0.866777i	$-0.666188\pi$
$761$	−27.5143	−0.997393	−0.498697	+	0.866777i	$0.666188\pi$
$762$	0	0
$763$	65.0610	2.35537
$764$	0	0
$765$	0.295619	0.0106881
$766$	0	0
$767$	15.5030	0.559781
$768$	0	0
$769$	30.5849	1.10292	0.551459	−	0.834202i	$-0.314071\pi$
$769$	30.5849	1.10292	0.551459	+	0.834202i	$0.314071\pi$
$770$	0	0
$771$	9.11367	0.328221
$772$	0	0
$773$	−25.4624	−0.915818	−0.457909	−	0.888999i	$-0.651402\pi$
$773$	−25.4624	−0.915818	−0.457909	+	0.888999i	$0.651402\pi$
$774$	0	0
$775$	45.2174	1.62426
$776$	0	0
$777$	−34.9549	−1.25400
$778$	0	0
$779$	−12.9923	−0.465498
$780$	0	0
$781$	60.1987	2.15408
$782$	0	0
$783$	−51.4040	−1.83703
$784$	0	0
$785$	0.441909	0.0157724
$786$	0	0
$787$	11.1056	0.395873	0.197937	−	0.980215i	$-0.436576\pi$
$787$	11.1056	0.395873	0.197937	+	0.980215i	$0.436576\pi$
$788$	0	0
$789$	8.09870	0.288321
$790$	0	0
$791$	−6.04493	−0.214933
$792$	0	0
$793$	−3.23119	−0.114743
$794$	0	0
$795$	0.337263	0.0119615
$796$	0	0
$797$	−29.8227	−1.05637	−0.528187	−	0.849128i	$-0.677128\pi$
$797$	−29.8227	−1.05637	−0.528187	+	0.849128i	$0.677128\pi$
$798$	0	0
$799$	−6.47323	−0.229006
$800$	0	0
$801$	−8.73856	−0.308762
$802$	0	0
$803$	81.2540	2.86739
$804$	0	0
$805$	1.26397	0.0445491
$806$	0	0
$807$	21.8709	0.769892
$808$	0	0
$809$	−21.6261	−0.760334	−0.380167	−	0.924918i	$-0.624133\pi$
$809$	−21.6261	−0.760334	−0.380167	+	0.924918i	$0.624133\pi$
$810$	0	0
$811$	10.6998	0.375720	0.187860	−	0.982196i	$-0.439845\pi$
$811$	10.6998	0.375720	0.187860	+	0.982196i	$0.439845\pi$
$812$	0	0
$813$	−3.48543	−0.122239
$814$	0	0
$815$	0.135143	0.00473385
$816$	0	0
$817$	2.56466	0.0897262
$818$	0	0
$819$	−21.1593	−0.739367
$820$	0	0
$821$	−23.9459	−0.835718	−0.417859	−	0.908512i	$-0.637219\pi$
$821$	−23.9459	−0.835718	−0.417859	+	0.908512i	$0.637219\pi$
$822$	0	0
$823$	−7.12071	−0.248212	−0.124106	−	0.992269i	$-0.539606\pi$
$823$	−7.12071	−0.248212	−0.124106	+	0.992269i	$0.539606\pi$
$824$	0	0
$825$	34.7625	1.21028
$826$	0	0
$827$	−28.7219	−0.998758	−0.499379	−	0.866384i	$-0.666438\pi$
$827$	−28.7219	−0.998758	−0.499379	+	0.866384i	$0.666438\pi$
$828$	0	0
$829$	−32.9632	−1.14486	−0.572430	−	0.819954i	$-0.693999\pi$
$829$	−32.9632	−1.14486	−0.572430	+	0.819954i	$0.693999\pi$
$830$	0	0
$831$	26.4994	0.919255
$832$	0	0
$833$	37.5875	1.30233
$834$	0	0
$835$	0.115301	0.00399015
$836$	0	0
$837$	47.6143	1.64579
$838$	0	0
$839$	39.1866	1.35287	0.676437	−	0.736501i	$-0.263523\pi$
$839$	39.1866	1.35287	0.676437	+	0.736501i	$0.263523\pi$
$840$	0	0
$841$	66.3972	2.28956
$842$	0	0
$843$	6.82956	0.235222
$844$	0	0
$845$	0.220002	0.00756830
$846$	0	0
$847$	120.951	4.15593
$848$	0	0
$849$	−31.6801	−1.08726
$850$	0	0
$851$	52.4095	1.79657
$852$	0	0
$853$	49.7163	1.70226	0.851128	−	0.524959i	$-0.175919\pi$
$853$	49.7163	1.70226	0.851128	+	0.524959i	$0.175919\pi$
$854$	0	0
$855$	−0.221577	−0.00757777
$856$	0	0
$857$	27.0587	0.924309	0.462154	−	0.886799i	$-0.347077\pi$
$857$	27.0587	0.924309	0.462154	+	0.886799i	$0.347077\pi$
$858$	0	0
$859$	−9.56499	−0.326353	−0.163177	−	0.986597i	$-0.552174\pi$
$859$	−9.56499	−0.326353	−0.163177	+	0.986597i	$0.552174\pi$
$860$	0	0
$861$	21.4028	0.729406
$862$	0	0
$863$	−6.13783	−0.208934	−0.104467	−	0.994528i	$-0.533314\pi$
$863$	−6.13783	−0.208934	−0.104467	+	0.994528i	$0.533314\pi$
$864$	0	0
$865$	−0.101268	−0.00344322
$866$	0	0
$867$	−3.74686	−0.127250
$868$	0	0
$869$	−42.8587	−1.45388
$870$	0	0
$871$	−29.5694	−1.00192
$872$	0	0
$873$	13.1235	0.444162
$874$	0	0
$875$	1.85195	0.0626072
$876$	0	0
$877$	57.8520	1.95352	0.976762	−	0.214327i	$-0.0687558\pi$
$877$	57.8520	1.95352	0.976762	+	0.214327i	$0.0687558\pi$
$878$	0	0
$879$	30.3047	1.02215
$880$	0	0
$881$	−3.36736	−0.113449	−0.0567246	−	0.998390i	$-0.518066\pi$
$881$	−3.36736	−0.113449	−0.0567246	+	0.998390i	$0.518066\pi$
$882$	0	0
$883$	22.7789	0.766572	0.383286	−	0.923630i	$-0.374792\pi$
$883$	22.7789	0.766572	0.383286	+	0.923630i	$0.374792\pi$
$884$	0	0
$885$	−0.267438	−0.00898983
$886$	0	0
$887$	−5.41195	−0.181715	−0.0908577	−	0.995864i	$-0.528961\pi$
$887$	−5.41195	−0.181715	−0.0908577	+	0.995864i	$0.528961\pi$
$888$	0	0
$889$	1.41401	0.0474243
$890$	0	0
$891$	2.46567	0.0826030
$892$	0	0
$893$	4.85192	0.162363
$894$	0	0
$895$	−0.0550046	−0.00183860
$896$	0	0
$897$	−21.2798	−0.710511
$898$	0	0
$899$	−88.3642	−2.94711
$900$	0	0
$901$	−25.3598	−0.844857
$902$	0	0
$903$	−4.22489	−0.140596
$904$	0	0
$905$	0.201617	0.00670197
$906$	0	0
$907$	12.1356	0.402957	0.201479	−	0.979493i	$-0.435425\pi$
$907$	12.1356	0.402957	0.201479	+	0.979493i	$0.435425\pi$
$908$	0	0
$909$	25.7980	0.855665
$910$	0	0
$911$	−16.8848	−0.559418	−0.279709	−	0.960085i	$-0.590238\pi$
$911$	−16.8848	−0.559418	−0.279709	+	0.960085i	$0.590238\pi$
$912$	0	0
$913$	−65.5464	−2.16927
$914$	0	0
$915$	0.0557403	0.00184272
$916$	0	0
$917$	−3.85831	−0.127413
$918$	0	0
$919$	30.3203	1.00017	0.500087	−	0.865975i	$-0.333301\pi$
$919$	30.3203	1.00017	0.500087	+	0.865975i	$0.333301\pi$
$920$	0	0
$921$	34.3947	1.13334
$922$	0	0
$923$	−26.9909	−0.888417
$924$	0	0
$925$	38.3870	1.26216
$926$	0	0
$927$	25.7875	0.846973
$928$	0	0
$929$	−4.34012	−0.142395	−0.0711974	−	0.997462i	$-0.522682\pi$
$929$	−4.34012	−0.142395	−0.0711974	+	0.997462i	$0.522682\pi$
$930$	0	0
$931$	−28.1732	−0.923339
$932$	0	0
$933$	−25.7607	−0.843368
$934$	0	0
$935$	1.04340	0.0341230
$936$	0	0
$937$	−18.1328	−0.592372	−0.296186	−	0.955130i	$-0.595715\pi$
$937$	−18.1328	−0.592372	−0.296186	+	0.955130i	$0.595715\pi$
$938$	0	0
$939$	−6.88755	−0.224767
$940$	0	0
$941$	8.34741	0.272118	0.136059	−	0.990701i	$-0.456556\pi$
$941$	8.34741	0.272118	0.136059	+	0.990701i	$0.456556\pi$
$942$	0	0
$943$	−32.0902	−1.04500
$944$	0	0
$945$	0.974863	0.0317123
$946$	0	0
$947$	50.1004	1.62804	0.814022	−	0.580833i	$-0.197273\pi$
$947$	50.1004	1.62804	0.814022	+	0.580833i	$0.197273\pi$
$948$	0	0
$949$	−36.4314	−1.18261
$950$	0	0
$951$	−13.2269	−0.428912
$952$	0	0
$953$	31.2503	1.01230	0.506148	−	0.862446i	$-0.331069\pi$
$953$	31.2503	1.01230	0.506148	+	0.862446i	$0.331069\pi$
$954$	0	0
$955$	−0.446041	−0.0144336
$956$	0	0
$957$	−67.9333	−2.19597
$958$	0	0
$959$	24.1690	0.780458
$960$	0	0
$961$	50.8497	1.64031
$962$	0	0
$963$	−11.8784	−0.382776
$964$	0	0
$965$	−0.240855	−0.00775339
$966$	0	0
$967$	−25.6787	−0.825771	−0.412886	−	0.910783i	$-0.635479\pi$
$967$	−25.6787	−0.825771	−0.412886	+	0.910783i	$0.635479\pi$
$968$	0	0
$969$	−11.1755	−0.359008
$970$	0	0
$971$	8.83558	0.283547	0.141774	−	0.989899i	$-0.454720\pi$
$971$	8.83558	0.283547	0.141774	+	0.989899i	$0.454720\pi$
$972$	0	0
$973$	−33.6803	−1.07974
$974$	0	0
$975$	−15.5863	−0.499160
$976$	0	0
$977$	−1.63013	−0.0521525	−0.0260763	−	0.999660i	$-0.508301\pi$
$977$	−1.63013	−0.0521525	−0.0260763	+	0.999660i	$0.508301\pi$
$978$	0	0
$979$	−30.8432	−0.985754
$980$	0	0
$981$	−28.1705	−0.899415
$982$	0	0
$983$	−15.8500	−0.505537	−0.252768	−	0.967527i	$-0.581341\pi$
$983$	−15.8500	−0.505537	−0.252768	+	0.967527i	$0.581341\pi$
$984$	0	0
$985$	0.932303	0.0297056
$986$	0	0
$987$	−7.99279	−0.254413
$988$	0	0
$989$	6.33456	0.201427
$990$	0	0
$991$	25.5922	0.812962	0.406481	−	0.913659i	$-0.366756\pi$
$991$	25.5922	0.812962	0.406481	+	0.913659i	$0.366756\pi$
$992$	0	0
$993$	−16.6669	−0.528907
$994$	0	0
$995$	−0.480435	−0.0152308
$996$	0	0
$997$	−4.34006	−0.137451	−0.0687256	−	0.997636i	$-0.521893\pi$
$997$	−4.34006	−0.137451	−0.0687256	+	0.997636i	$0.521893\pi$
$998$	0	0
$999$	40.4219	1.27889

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.30	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.30	✓	44	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.09745 q^{3} -0.0446663 q^{5} +4.14700 q^{7} -1.79560 q^{9} +O(q^{10})\)	\(q+1.09745 q^{3} -0.0446663 q^{5} +4.14700 q^{7} -1.79560 q^{9} -6.33765 q^{11} +2.84157 q^{13} -0.0490192 q^{15} +3.68590 q^{17} -2.76271 q^{19} +4.55115 q^{21} -6.82373 q^{23} -4.99800 q^{25} -5.26294 q^{27} +9.76715 q^{29} -9.04708 q^{31} -6.95528 q^{33} -0.185232 q^{35} -7.68047 q^{37} +3.11850 q^{39} +4.70273 q^{41} -0.928314 q^{43} +0.0802027 q^{45} -1.75621 q^{47} +10.1976 q^{49} +4.04511 q^{51} -6.88021 q^{53} +0.283080 q^{55} -3.03195 q^{57} +5.45578 q^{59} -1.13711 q^{61} -7.44634 q^{63} -0.126923 q^{65} -10.4060 q^{67} -7.48873 q^{69} -9.49858 q^{71} -12.8208 q^{73} -5.48508 q^{75} -26.2823 q^{77} +6.76255 q^{79} -0.389050 q^{81} +10.3424 q^{83} -0.164636 q^{85} +10.7190 q^{87} +4.86666 q^{89} +11.7840 q^{91} -9.92875 q^{93} +0.123400 q^{95} -7.30869 q^{97} +11.3799 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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