LMFDB - Embedded newform 6004.2.a.g.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56278 q^{3} -2.37537 q^{5} +2.65817 q^{7} -0.557733 q^{9} +O(q^{10})$	$q+1.56278 q^{3} -2.37537 q^{5} +2.65817 q^{7} -0.557733 q^{9} -2.81237 q^{11} +2.78992 q^{13} -3.71217 q^{15} -7.35621 q^{17} +1.00000 q^{19} +4.15412 q^{21} +5.90957 q^{23} +0.642390 q^{25} -5.55994 q^{27} +2.21464 q^{29} +1.03131 q^{31} -4.39511 q^{33} -6.31413 q^{35} +4.56380 q^{37} +4.36002 q^{39} +2.65150 q^{41} +6.23613 q^{43} +1.32482 q^{45} +0.283410 q^{47} +0.0658470 q^{49} -11.4961 q^{51} -8.74189 q^{53} +6.68043 q^{55} +1.56278 q^{57} -10.5307 q^{59} +2.64975 q^{61} -1.48255 q^{63} -6.62710 q^{65} -12.4441 q^{67} +9.23533 q^{69} +0.602756 q^{71} -4.37373 q^{73} +1.00391 q^{75} -7.47576 q^{77} -1.00000 q^{79} -7.01573 q^{81} -2.88505 q^{83} +17.4737 q^{85} +3.46098 q^{87} -17.4000 q^{89} +7.41607 q^{91} +1.61170 q^{93} -2.37537 q^{95} -10.5573 q^{97} +1.56855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	1.56278	0.902269	0.451134	−	0.892456i	$-0.351020\pi$
$3$	1.56278	0.902269	0.451134	+	0.892456i	$0.351020\pi$
$4$	0	0
$5$	−2.37537	−1.06230	−0.531149	−	0.847278i	$-0.678240\pi$
$5$	−2.37537	−1.06230	−0.531149	+	0.847278i	$0.678240\pi$
$6$	0	0
$7$	2.65817	1.00469	0.502346	−	0.864667i	$-0.332470\pi$
$7$	2.65817	1.00469	0.502346	+	0.864667i	$0.332470\pi$
$8$	0	0
$9$	−0.557733	−0.185911
$10$	0	0
$11$	−2.81237	−0.847963	−0.423981	−	0.905671i	$-0.639368\pi$
$11$	−2.81237	−0.847963	−0.423981	+	0.905671i	$0.639368\pi$
$12$	0	0
$13$	2.78992	0.773785	0.386892	−	0.922125i	$-0.373548\pi$
$13$	2.78992	0.773785	0.386892	+	0.922125i	$0.373548\pi$
$14$	0	0
$15$	−3.71217	−0.958479
$16$	0	0
$17$	−7.35621	−1.78414	−0.892071	−	0.451895i	$-0.850748\pi$
$17$	−7.35621	−1.78414	−0.892071	+	0.451895i	$0.850748\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.15412	0.906503
$22$	0	0
$23$	5.90957	1.23223	0.616115	−	0.787656i	$-0.288706\pi$
$23$	5.90957	1.23223	0.616115	+	0.787656i	$0.288706\pi$
$24$	0	0
$25$	0.642390	0.128478
$26$	0	0
$27$	−5.55994	−1.07001
$28$	0	0
$29$	2.21464	0.411248	0.205624	−	0.978631i	$-0.434078\pi$
$29$	2.21464	0.411248	0.205624	+	0.978631i	$0.434078\pi$
$30$	0	0
$31$	1.03131	0.185228	0.0926142	−	0.995702i	$-0.470478\pi$
$31$	1.03131	0.185228	0.0926142	+	0.995702i	$0.470478\pi$
$32$	0	0
$33$	−4.39511	−0.765090
$34$	0	0
$35$	−6.31413	−1.06728
$36$	0	0
$37$	4.56380	0.750283	0.375142	−	0.926967i	$-0.377594\pi$
$37$	4.56380	0.750283	0.375142	+	0.926967i	$0.377594\pi$
$38$	0	0
$39$	4.36002	0.698162
$40$	0	0
$41$	2.65150	0.414095	0.207048	−	0.978331i	$-0.433615\pi$
$41$	2.65150	0.414095	0.207048	+	0.978331i	$0.433615\pi$
$42$	0	0
$43$	6.23613	0.951001	0.475500	−	0.879715i	$-0.342267\pi$
$43$	6.23613	0.951001	0.475500	+	0.879715i	$0.342267\pi$
$44$	0	0
$45$	1.32482	0.197493
$46$	0	0
$47$	0.283410	0.0413395	0.0206698	−	0.999786i	$-0.493420\pi$
$47$	0.283410	0.0413395	0.0206698	+	0.999786i	$0.493420\pi$
$48$	0	0
$49$	0.0658470	0.00940672
$50$	0	0
$51$	−11.4961	−1.60978
$52$	0	0
$53$	−8.74189	−1.20079	−0.600395	−	0.799703i	$-0.704990\pi$
$53$	−8.74189	−1.20079	−0.600395	+	0.799703i	$0.704990\pi$
$54$	0	0
$55$	6.68043	0.900789
$56$	0	0
$57$	1.56278	0.206995
$58$	0	0
$59$	−10.5307	−1.37098	−0.685491	−	0.728081i	$-0.740413\pi$
$59$	−10.5307	−1.37098	−0.685491	+	0.728081i	$0.740413\pi$
$60$	0	0
$61$	2.64975	0.339266	0.169633	−	0.985507i	$-0.445742\pi$
$61$	2.64975	0.339266	0.169633	+	0.985507i	$0.445742\pi$
$62$	0	0
$63$	−1.48255	−0.186783
$64$	0	0
$65$	−6.62710	−0.821990
$66$	0	0
$67$	−12.4441	−1.52028	−0.760142	−	0.649757i	$-0.774871\pi$
$67$	−12.4441	−1.52028	−0.760142	+	0.649757i	$0.774871\pi$
$68$	0	0
$69$	9.23533	1.11180
$70$	0	0
$71$	0.602756	0.0715339	0.0357670	−	0.999360i	$-0.488613\pi$
$71$	0.602756	0.0715339	0.0357670	+	0.999360i	$0.488613\pi$
$72$	0	0
$73$	−4.37373	−0.511907	−0.255953	−	0.966689i	$-0.582389\pi$
$73$	−4.37373	−0.511907	−0.255953	+	0.966689i	$0.582389\pi$
$74$	0	0
$75$	1.00391	0.115922
$76$	0	0
$77$	−7.47576	−0.851942
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−7.01573	−0.779526
$82$	0	0
$83$	−2.88505	−0.316676	−0.158338	−	0.987385i	$-0.550614\pi$
$83$	−2.88505	−0.316676	−0.158338	+	0.987385i	$0.550614\pi$
$84$	0	0
$85$	17.4737	1.89529
$86$	0	0
$87$	3.46098	0.371056
$88$	0	0
$89$	−17.4000	−1.84439	−0.922196	−	0.386723i	$-0.873607\pi$
$89$	−17.4000	−1.84439	−0.922196	+	0.386723i	$0.873607\pi$
$90$	0	0
$91$	7.41607	0.777416
$92$	0	0
$93$	1.61170	0.167126
$94$	0	0
$95$	−2.37537	−0.243708
$96$	0	0
$97$	−10.5573	−1.07193	−0.535965	−	0.844240i	$-0.680052\pi$
$97$	−10.5573	−1.07193	−0.535965	+	0.844240i	$0.680052\pi$
$98$	0	0
$99$	1.56855	0.157646
$100$	0	0
$101$	−2.75201	−0.273835	−0.136917	−	0.990582i	$-0.543720\pi$
$101$	−2.75201	−0.273835	−0.136917	+	0.990582i	$0.543720\pi$
$102$	0	0
$103$	−5.66925	−0.558608	−0.279304	−	0.960203i	$-0.590104\pi$
$103$	−5.66925	−0.558608	−0.279304	+	0.960203i	$0.590104\pi$
$104$	0	0
$105$	−9.86757	−0.962976
$106$	0	0
$107$	−14.3653	−1.38875	−0.694375	−	0.719613i	$-0.744319\pi$
$107$	−14.3653	−1.38875	−0.694375	+	0.719613i	$0.744319\pi$
$108$	0	0
$109$	−0.523715	−0.0501628	−0.0250814	−	0.999685i	$-0.507984\pi$
$109$	−0.523715	−0.0501628	−0.0250814	+	0.999685i	$0.507984\pi$
$110$	0	0
$111$	7.13219	0.676957
$112$	0	0
$113$	−2.92082	−0.274767	−0.137384	−	0.990518i	$-0.543869\pi$
$113$	−2.92082	−0.274767	−0.137384	+	0.990518i	$0.543869\pi$
$114$	0	0
$115$	−14.0374	−1.30900
$116$	0	0
$117$	−1.55603	−0.143855
$118$	0	0
$119$	−19.5540	−1.79251
$120$	0	0
$121$	−3.09055	−0.280959
$122$	0	0
$123$	4.14370	0.373625
$124$	0	0
$125$	10.3509	0.925817
$126$	0	0
$127$	4.42976	0.393078	0.196539	−	0.980496i	$-0.437030\pi$
$127$	4.42976	0.393078	0.196539	+	0.980496i	$0.437030\pi$
$128$	0	0
$129$	9.74567	0.858058
$130$	0	0
$131$	−9.70591	−0.848009	−0.424005	−	0.905660i	$-0.639376\pi$
$131$	−9.70591	−0.848009	−0.424005	+	0.905660i	$0.639376\pi$
$132$	0	0
$133$	2.65817	0.230492
$134$	0	0
$135$	13.2069	1.13667
$136$	0	0
$137$	18.0611	1.54307	0.771533	−	0.636189i	$-0.219490\pi$
$137$	18.0611	1.54307	0.771533	+	0.636189i	$0.219490\pi$
$138$	0	0
$139$	−0.330382	−0.0280226	−0.0140113	−	0.999902i	$-0.504460\pi$
$139$	−0.330382	−0.0280226	−0.0140113	+	0.999902i	$0.504460\pi$
$140$	0	0
$141$	0.442906	0.0372994
$142$	0	0
$143$	−7.84630	−0.656141
$144$	0	0
$145$	−5.26058	−0.436868
$146$	0	0
$147$	0.102904	0.00848739
$148$	0	0
$149$	−19.6987	−1.61378	−0.806890	−	0.590702i	$-0.798851\pi$
$149$	−19.6987	−1.61378	−0.806890	+	0.590702i	$0.798851\pi$
$150$	0	0
$151$	−14.0623	−1.14437	−0.572186	−	0.820124i	$-0.693905\pi$
$151$	−14.0623	−1.14437	−0.572186	+	0.820124i	$0.693905\pi$
$152$	0	0
$153$	4.10280	0.331692
$154$	0	0
$155$	−2.44974	−0.196768
$156$	0	0
$157$	6.60714	0.527307	0.263653	−	0.964617i	$-0.415072\pi$
$157$	6.60714	0.527307	0.263653	+	0.964617i	$0.415072\pi$
$158$	0	0
$159$	−13.6616	−1.08344
$160$	0	0
$161$	15.7086	1.23801
$162$	0	0
$163$	−15.5967	−1.22163	−0.610815	−	0.791773i	$-0.709158\pi$
$163$	−15.5967	−1.22163	−0.610815	+	0.791773i	$0.709158\pi$
$164$	0	0
$165$	10.4400	0.812754
$166$	0	0
$167$	6.06399	0.469246	0.234623	−	0.972086i	$-0.424615\pi$
$167$	6.06399	0.469246	0.234623	+	0.972086i	$0.424615\pi$
$168$	0	0
$169$	−5.21634	−0.401257
$170$	0	0
$171$	−0.557733	−0.0426509
$172$	0	0
$173$	23.5265	1.78869	0.894344	−	0.447380i	$-0.147643\pi$
$173$	23.5265	1.78869	0.894344	+	0.447380i	$0.147643\pi$
$174$	0	0
$175$	1.70758	0.129081
$176$	0	0
$177$	−16.4571	−1.23700
$178$	0	0
$179$	−9.87404	−0.738021	−0.369010	−	0.929425i	$-0.620303\pi$
$179$	−9.87404	−0.738021	−0.369010	+	0.929425i	$0.620303\pi$
$180$	0	0
$181$	11.4674	0.852363	0.426181	−	0.904638i	$-0.359859\pi$
$181$	11.4674	0.852363	0.426181	+	0.904638i	$0.359859\pi$
$182$	0	0
$183$	4.14097	0.306109
$184$	0	0
$185$	−10.8407	−0.797025
$186$	0	0
$187$	20.6884	1.51289
$188$	0	0
$189$	−14.7792	−1.07503
$190$	0	0
$191$	2.72793	0.197386	0.0986932	−	0.995118i	$-0.468534\pi$
$191$	2.72793	0.197386	0.0986932	+	0.995118i	$0.468534\pi$
$192$	0	0
$193$	5.72730	0.412260	0.206130	−	0.978525i	$-0.433913\pi$
$193$	5.72730	0.412260	0.206130	+	0.978525i	$0.433913\pi$
$194$	0	0
$195$	−10.3567	−0.741656
$196$	0	0
$197$	9.81881	0.699561	0.349781	−	0.936832i	$-0.386256\pi$
$197$	9.81881	0.699561	0.349781	+	0.936832i	$0.386256\pi$
$198$	0	0
$199$	−5.27929	−0.374239	−0.187120	−	0.982337i	$-0.559915\pi$
$199$	−5.27929	−0.374239	−0.187120	+	0.982337i	$0.559915\pi$
$200$	0	0
$201$	−19.4473	−1.37170
$202$	0	0
$203$	5.88687	0.413177
$204$	0	0
$205$	−6.29830	−0.439893
$206$	0	0
$207$	−3.29596	−0.229085
$208$	0	0
$209$	−2.81237	−0.194536
$210$	0	0
$211$	18.3437	1.26283	0.631415	−	0.775445i	$-0.282475\pi$
$211$	18.3437	1.26283	0.631415	+	0.775445i	$0.282475\pi$
$212$	0	0
$213$	0.941972	0.0645428
$214$	0	0
$215$	−14.8131	−1.01025
$216$	0	0
$217$	2.74139	0.186098
$218$	0	0
$219$	−6.83516	−0.461878
$220$	0	0
$221$	−20.5232	−1.38054
$222$	0	0
$223$	−20.1720	−1.35081	−0.675407	−	0.737445i	$-0.736032\pi$
$223$	−20.1720	−1.35081	−0.675407	+	0.737445i	$0.736032\pi$
$224$	0	0
$225$	−0.358282	−0.0238855
$226$	0	0
$227$	12.4056	0.823391	0.411695	−	0.911322i	$-0.364937\pi$
$227$	12.4056	0.823391	0.411695	+	0.911322i	$0.364937\pi$
$228$	0	0
$229$	−13.6254	−0.900389	−0.450195	−	0.892930i	$-0.648645\pi$
$229$	−13.6254	−0.900389	−0.450195	+	0.892930i	$0.648645\pi$
$230$	0	0
$231$	−11.6829	−0.768680
$232$	0	0
$233$	−1.35532	−0.0887900	−0.0443950	−	0.999014i	$-0.514136\pi$
$233$	−1.35532	−0.0887900	−0.0443950	+	0.999014i	$0.514136\pi$
$234$	0	0
$235$	−0.673203	−0.0439149
$236$	0	0
$237$	−1.56278	−0.101513
$238$	0	0
$239$	−17.3282	−1.12087	−0.560434	−	0.828199i	$-0.689366\pi$
$239$	−17.3282	−1.12087	−0.560434	+	0.828199i	$0.689366\pi$
$240$	0	0
$241$	23.3441	1.50373	0.751864	−	0.659318i	$-0.229155\pi$
$241$	23.3441	1.50373	0.751864	+	0.659318i	$0.229155\pi$
$242$	0	0
$243$	5.71580	0.366669
$244$	0	0
$245$	−0.156411	−0.00999275
$246$	0	0
$247$	2.78992	0.177518
$248$	0	0
$249$	−4.50869	−0.285727
$250$	0	0
$251$	23.7545	1.49937	0.749686	−	0.661794i	$-0.230205\pi$
$251$	23.7545	1.49937	0.749686	+	0.661794i	$0.230205\pi$
$252$	0	0
$253$	−16.6199	−1.04489
$254$	0	0
$255$	27.3075	1.71006
$256$	0	0
$257$	−17.6146	−1.09877	−0.549385	−	0.835570i	$-0.685138\pi$
$257$	−17.6146	−1.09877	−0.549385	+	0.835570i	$0.685138\pi$
$258$	0	0
$259$	12.1313	0.753804
$260$	0	0
$261$	−1.23518	−0.0764555
$262$	0	0
$263$	−7.39406	−0.455937	−0.227969	−	0.973668i	$-0.573208\pi$
$263$	−7.39406	−0.455937	−0.227969	+	0.973668i	$0.573208\pi$
$264$	0	0
$265$	20.7652	1.27560
$266$	0	0
$267$	−27.1922	−1.66414
$268$	0	0
$269$	−23.5474	−1.43571	−0.717855	−	0.696193i	$-0.754876\pi$
$269$	−23.5474	−1.43571	−0.717855	+	0.696193i	$0.754876\pi$
$270$	0	0
$271$	28.8464	1.75229	0.876147	−	0.482044i	$-0.160105\pi$
$271$	28.8464	1.75229	0.876147	+	0.482044i	$0.160105\pi$
$272$	0	0
$273$	11.5897	0.701438
$274$	0	0
$275$	−1.80664	−0.108945
$276$	0	0
$277$	16.6207	0.998639	0.499320	−	0.866418i	$-0.333583\pi$
$277$	16.6207	0.998639	0.499320	+	0.866418i	$0.333583\pi$
$278$	0	0
$279$	−0.575195	−0.0344360
$280$	0	0
$281$	−8.39482	−0.500793	−0.250397	−	0.968143i	$-0.580561\pi$
$281$	−8.39482	−0.500793	−0.250397	+	0.968143i	$0.580561\pi$
$282$	0	0
$283$	−22.6926	−1.34893	−0.674467	−	0.738305i	$-0.735627\pi$
$283$	−22.6926	−1.34893	−0.674467	+	0.738305i	$0.735627\pi$
$284$	0	0
$285$	−3.71217	−0.219890
$286$	0	0
$287$	7.04813	0.416038
$288$	0	0
$289$	37.1138	2.18316
$290$	0	0
$291$	−16.4987	−0.967169
$292$	0	0
$293$	29.5373	1.72559	0.862795	−	0.505554i	$-0.168712\pi$
$293$	29.5373	1.72559	0.862795	+	0.505554i	$0.168712\pi$
$294$	0	0
$295$	25.0144	1.45639
$296$	0	0
$297$	15.6366	0.907329
$298$	0	0
$299$	16.4872	0.953481
$300$	0	0
$301$	16.5767	0.955463
$302$	0	0
$303$	−4.30077	−0.247073
$304$	0	0
$305$	−6.29415	−0.360402
$306$	0	0
$307$	−31.4187	−1.79316	−0.896579	−	0.442883i	$-0.853956\pi$
$307$	−31.4187	−1.79316	−0.896579	+	0.442883i	$0.853956\pi$
$308$	0	0
$309$	−8.85977	−0.504014
$310$	0	0
$311$	−13.9990	−0.793811	−0.396906	−	0.917859i	$-0.629916\pi$
$311$	−13.9990	−0.793811	−0.396906	+	0.917859i	$0.629916\pi$
$312$	0	0
$313$	−17.8802	−1.01065	−0.505324	−	0.862930i	$-0.668627\pi$
$313$	−17.8802	−1.01065	−0.505324	+	0.862930i	$0.668627\pi$
$314$	0	0
$315$	3.52160	0.198420
$316$	0	0
$317$	−33.4054	−1.87623	−0.938117	−	0.346318i	$-0.887432\pi$
$317$	−33.4054	−1.87623	−0.938117	+	0.346318i	$0.887432\pi$
$318$	0	0
$319$	−6.22839	−0.348723
$320$	0	0
$321$	−22.4498	−1.25303
$322$	0	0
$323$	−7.35621	−0.409310
$324$	0	0
$325$	1.79222	0.0994143
$326$	0	0
$327$	−0.818448	−0.0452603
$328$	0	0
$329$	0.753350	0.0415335
$330$	0	0
$331$	15.4450	0.848936	0.424468	−	0.905443i	$-0.360461\pi$
$331$	15.4450	0.848936	0.424468	+	0.905443i	$0.360461\pi$
$332$	0	0
$333$	−2.54538	−0.139486
$334$	0	0
$335$	29.5593	1.61500
$336$	0	0
$337$	21.9521	1.19581	0.597904	−	0.801567i	$-0.296000\pi$
$337$	21.9521	1.19581	0.597904	+	0.801567i	$0.296000\pi$
$338$	0	0
$339$	−4.56458	−0.247914
$340$	0	0
$341$	−2.90042	−0.157067
$342$	0	0
$343$	−18.4321	−0.995241
$344$	0	0
$345$	−21.9373	−1.18107
$346$	0	0
$347$	−35.1377	−1.88629	−0.943146	−	0.332377i	$-0.892149\pi$
$347$	−35.1377	−1.88629	−0.943146	+	0.332377i	$0.892149\pi$
$348$	0	0
$349$	4.90555	0.262588	0.131294	−	0.991343i	$-0.458087\pi$
$349$	4.90555	0.262588	0.131294	+	0.991343i	$0.458087\pi$
$350$	0	0
$351$	−15.5118	−0.827958
$352$	0	0
$353$	15.3524	0.817127	0.408564	−	0.912730i	$-0.366030\pi$
$353$	15.3524	0.817127	0.408564	+	0.912730i	$0.366030\pi$
$354$	0	0
$355$	−1.43177	−0.0759904
$356$	0	0
$357$	−30.5585	−1.61733
$358$	0	0
$359$	−32.8499	−1.73375	−0.866876	−	0.498523i	$-0.833876\pi$
$359$	−32.8499	−1.73375	−0.866876	+	0.498523i	$0.833876\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.82984	−0.253501
$364$	0	0
$365$	10.3892	0.543798
$366$	0	0
$367$	−23.4768	−1.22548	−0.612740	−	0.790285i	$-0.709933\pi$
$367$	−23.4768	−1.22548	−0.612740	+	0.790285i	$0.709933\pi$
$368$	0	0
$369$	−1.47883	−0.0769849
$370$	0	0
$371$	−23.2374	−1.20643
$372$	0	0
$373$	17.5148	0.906883	0.453442	−	0.891286i	$-0.350196\pi$
$373$	17.5148	0.906883	0.453442	+	0.891286i	$0.350196\pi$
$374$	0	0
$375$	16.1762	0.835335
$376$	0	0
$377$	6.17866	0.318217
$378$	0	0
$379$	21.3936	1.09892	0.549458	−	0.835521i	$-0.314834\pi$
$379$	21.3936	1.09892	0.549458	+	0.835521i	$0.314834\pi$
$380$	0	0
$381$	6.92272	0.354662
$382$	0	0
$383$	12.0971	0.618131	0.309065	−	0.951041i	$-0.399984\pi$
$383$	12.0971	0.618131	0.309065	+	0.951041i	$0.399984\pi$
$384$	0	0
$385$	17.7577	0.905016
$386$	0	0
$387$	−3.47810	−0.176802
$388$	0	0
$389$	−29.1021	−1.47553	−0.737767	−	0.675055i	$-0.764120\pi$
$389$	−29.1021	−1.47553	−0.737767	+	0.675055i	$0.764120\pi$
$390$	0	0
$391$	−43.4720	−2.19847
$392$	0	0
$393$	−15.1682	−0.765132
$394$	0	0
$395$	2.37537	0.119518
$396$	0	0
$397$	21.4051	1.07429	0.537145	−	0.843490i	$-0.319503\pi$
$397$	21.4051	1.07429	0.537145	+	0.843490i	$0.319503\pi$
$398$	0	0
$399$	4.15412	0.207966
$400$	0	0
$401$	−16.0342	−0.800711	−0.400355	−	0.916360i	$-0.631113\pi$
$401$	−16.0342	−0.800711	−0.400355	+	0.916360i	$0.631113\pi$
$402$	0	0
$403$	2.87727	0.143327
$404$	0	0
$405$	16.6650	0.828089
$406$	0	0
$407$	−12.8351	−0.636212
$408$	0	0
$409$	9.84131	0.486621	0.243311	−	0.969948i	$-0.421767\pi$
$409$	9.84131	0.486621	0.243311	+	0.969948i	$0.421767\pi$
$410$	0	0
$411$	28.2255	1.39226
$412$	0	0
$413$	−27.9924	−1.37742
$414$	0	0
$415$	6.85307	0.336404
$416$	0	0
$417$	−0.516312	−0.0252839
$418$	0	0
$419$	−8.50107	−0.415304	−0.207652	−	0.978203i	$-0.566582\pi$
$419$	−8.50107	−0.415304	−0.207652	+	0.978203i	$0.566582\pi$
$420$	0	0
$421$	−15.3125	−0.746285	−0.373143	−	0.927774i	$-0.621720\pi$
$421$	−15.3125	−0.746285	−0.373143	+	0.927774i	$0.621720\pi$
$422$	0	0
$423$	−0.158067	−0.00768548
$424$	0	0
$425$	−4.72555	−0.229223
$426$	0	0
$427$	7.04348	0.340858
$428$	0	0
$429$	−12.2620	−0.592015
$430$	0	0
$431$	34.0627	1.64074	0.820371	−	0.571831i	$-0.193767\pi$
$431$	34.0627	1.64074	0.820371	+	0.571831i	$0.193767\pi$
$432$	0	0
$433$	3.44618	0.165613	0.0828063	−	0.996566i	$-0.473612\pi$
$433$	3.44618	0.165613	0.0828063	+	0.996566i	$0.473612\pi$
$434$	0	0
$435$	−8.22111	−0.394172
$436$	0	0
$437$	5.90957	0.282693
$438$	0	0
$439$	−12.0948	−0.577252	−0.288626	−	0.957442i	$-0.593198\pi$
$439$	−12.0948	−0.577252	−0.288626	+	0.957442i	$0.593198\pi$
$440$	0	0
$441$	−0.0367251	−0.00174881
$442$	0	0
$443$	−19.9393	−0.947345	−0.473672	−	0.880701i	$-0.657072\pi$
$443$	−19.9393	−0.947345	−0.473672	+	0.880701i	$0.657072\pi$
$444$	0	0
$445$	41.3314	1.95929
$446$	0	0
$447$	−30.7846	−1.45606
$448$	0	0
$449$	−0.0515726	−0.00243386	−0.00121693	−	0.999999i	$-0.500387\pi$
$449$	−0.0515726	−0.00243386	−0.00121693	+	0.999999i	$0.500387\pi$
$450$	0	0
$451$	−7.45702	−0.351137
$452$	0	0
$453$	−21.9762	−1.03253
$454$	0	0
$455$	−17.6159	−0.825848
$456$	0	0
$457$	−17.3459	−0.811405	−0.405702	−	0.914005i	$-0.632973\pi$
$457$	−17.3459	−0.811405	−0.405702	+	0.914005i	$0.632973\pi$
$458$	0	0
$459$	40.9000	1.90905
$460$	0	0
$461$	−9.08283	−0.423030	−0.211515	−	0.977375i	$-0.567840\pi$
$461$	−9.08283	−0.423030	−0.211515	+	0.977375i	$0.567840\pi$
$462$	0	0
$463$	−2.23886	−0.104049	−0.0520243	−	0.998646i	$-0.516567\pi$
$463$	−2.23886	−0.104049	−0.0520243	+	0.998646i	$0.516567\pi$
$464$	0	0
$465$	−3.82839	−0.177537
$466$	0	0
$467$	−40.7948	−1.88776	−0.943879	−	0.330292i	$-0.892853\pi$
$467$	−40.7948	−1.88776	−0.943879	+	0.330292i	$0.892853\pi$
$468$	0	0
$469$	−33.0784	−1.52742
$470$	0	0
$471$	10.3255	0.475772
$472$	0	0
$473$	−17.5383	−0.806413
$474$	0	0
$475$	0.642390	0.0294749
$476$	0	0
$477$	4.87564	0.223240
$478$	0	0
$479$	23.9061	1.09230	0.546148	−	0.837688i	$-0.316093\pi$
$479$	23.9061	1.09230	0.546148	+	0.837688i	$0.316093\pi$
$480$	0	0
$481$	12.7326	0.580558
$482$	0	0
$483$	24.5490	1.11702
$484$	0	0
$485$	25.0775	1.13871
$486$	0	0
$487$	−9.03173	−0.409267	−0.204633	−	0.978839i	$-0.565600\pi$
$487$	−9.03173	−0.409267	−0.204633	+	0.978839i	$0.565600\pi$
$488$	0	0
$489$	−24.3742	−1.10224
$490$	0	0
$491$	6.96464	0.314310	0.157155	−	0.987574i	$-0.449768\pi$
$491$	6.96464	0.314310	0.157155	+	0.987574i	$0.449768\pi$
$492$	0	0
$493$	−16.2913	−0.733724
$494$	0	0
$495$	−3.72590	−0.167467
$496$	0	0
$497$	1.60222	0.0718696
$498$	0	0
$499$	4.05697	0.181615	0.0908075	−	0.995868i	$-0.471055\pi$
$499$	4.05697	0.181615	0.0908075	+	0.995868i	$0.471055\pi$
$500$	0	0
$501$	9.47665	0.423386
$502$	0	0
$503$	24.9372	1.11190	0.555948	−	0.831217i	$-0.312355\pi$
$503$	24.9372	1.11190	0.555948	+	0.831217i	$0.312355\pi$
$504$	0	0
$505$	6.53704	0.290894
$506$	0	0
$507$	−8.15197	−0.362042
$508$	0	0
$509$	2.36751	0.104938	0.0524690	−	0.998623i	$-0.483291\pi$
$509$	2.36751	0.104938	0.0524690	+	0.998623i	$0.483291\pi$
$510$	0	0
$511$	−11.6261	−0.514309
$512$	0	0
$513$	−5.55994	−0.245477
$514$	0	0
$515$	13.4666	0.593408
$516$	0	0
$517$	−0.797054	−0.0350544
$518$	0	0
$519$	36.7667	1.61388
$520$	0	0
$521$	36.6137	1.60407	0.802037	−	0.597274i	$-0.203749\pi$
$521$	36.6137	1.60407	0.802037	+	0.597274i	$0.203749\pi$
$522$	0	0
$523$	−9.84827	−0.430635	−0.215317	−	0.976544i	$-0.569079\pi$
$523$	−9.84827	−0.430635	−0.215317	+	0.976544i	$0.569079\pi$
$524$	0	0
$525$	2.66856	0.116466
$526$	0	0
$527$	−7.58652	−0.330474
$528$	0	0
$529$	11.9230	0.518391
$530$	0	0
$531$	5.87333	0.254881
$532$	0	0
$533$	7.39748	0.320421
$534$	0	0
$535$	34.1230	1.47527
$536$	0	0
$537$	−15.4309	−0.665893
$538$	0	0
$539$	−0.185187	−0.00797655
$540$	0	0
$541$	10.8869	0.468064	0.234032	−	0.972229i	$-0.424808\pi$
$541$	10.8869	0.468064	0.234032	+	0.972229i	$0.424808\pi$
$542$	0	0
$543$	17.9209	0.769060
$544$	0	0
$545$	1.24402	0.0532878
$546$	0	0
$547$	27.0875	1.15818	0.579088	−	0.815265i	$-0.303409\pi$
$547$	27.0875	1.15818	0.579088	+	0.815265i	$0.303409\pi$
$548$	0	0
$549$	−1.47786	−0.0630733
$550$	0	0
$551$	2.21464	0.0943467
$552$	0	0
$553$	−2.65817	−0.113037
$554$	0	0
$555$	−16.9416	−0.719131
$556$	0	0
$557$	30.8838	1.30859	0.654294	−	0.756240i	$-0.272966\pi$
$557$	30.8838	1.30859	0.654294	+	0.756240i	$0.272966\pi$
$558$	0	0
$559$	17.3983	0.735870
$560$	0	0
$561$	32.3313	1.36503
$562$	0	0
$563$	−0.864093	−0.0364172	−0.0182086	−	0.999834i	$-0.505796\pi$
$563$	−0.864093	−0.0364172	−0.0182086	+	0.999834i	$0.505796\pi$
$564$	0	0
$565$	6.93802	0.291885
$566$	0	0
$567$	−18.6490	−0.783184
$568$	0	0
$569$	−8.20239	−0.343862	−0.171931	−	0.985109i	$-0.555001\pi$
$569$	−8.20239	−0.343862	−0.171931	+	0.985109i	$0.555001\pi$
$570$	0	0
$571$	22.4774	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$571$	22.4774	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$572$	0	0
$573$	4.26315	0.178096
$574$	0	0
$575$	3.79625	0.158314
$576$	0	0
$577$	−24.5620	−1.02253	−0.511264	−	0.859423i	$-0.670823\pi$
$577$	−24.5620	−1.02253	−0.511264	+	0.859423i	$0.670823\pi$
$578$	0	0
$579$	8.95049	0.371970
$580$	0	0
$581$	−7.66895	−0.318162
$582$	0	0
$583$	24.5855	1.01823
$584$	0	0
$585$	3.69615	0.152817
$586$	0	0
$587$	13.3448	0.550801	0.275400	−	0.961330i	$-0.411190\pi$
$587$	13.3448	0.550801	0.275400	+	0.961330i	$0.411190\pi$
$588$	0	0
$589$	1.03131	0.0424943
$590$	0	0
$591$	15.3446	0.631192
$592$	0	0
$593$	38.9410	1.59911	0.799557	−	0.600591i	$-0.205068\pi$
$593$	38.9410	1.59911	0.799557	+	0.600591i	$0.205068\pi$
$594$	0	0
$595$	46.4481	1.90418
$596$	0	0
$597$	−8.25035	−0.337664
$598$	0	0
$599$	4.09820	0.167448	0.0837239	−	0.996489i	$-0.473319\pi$
$599$	4.09820	0.167448	0.0837239	+	0.996489i	$0.473319\pi$
$600$	0	0
$601$	12.7060	0.518288	0.259144	−	0.965839i	$-0.416560\pi$
$601$	12.7060	0.518288	0.259144	+	0.965839i	$0.416560\pi$
$602$	0	0
$603$	6.94047	0.282638
$604$	0	0
$605$	7.34121	0.298463
$606$	0	0
$607$	13.5034	0.548088	0.274044	−	0.961717i	$-0.411639\pi$
$607$	13.5034	0.548088	0.274044	+	0.961717i	$0.411639\pi$
$608$	0	0
$609$	9.19986	0.372797
$610$	0	0
$611$	0.790690	0.0319879
$612$	0	0
$613$	24.6511	0.995648	0.497824	−	0.867278i	$-0.334133\pi$
$613$	24.6511	0.995648	0.497824	+	0.867278i	$0.334133\pi$
$614$	0	0
$615$	−9.84283	−0.396901
$616$	0	0
$617$	−22.2078	−0.894052	−0.447026	−	0.894521i	$-0.647517\pi$
$617$	−22.2078	−0.894052	−0.447026	+	0.894521i	$0.647517\pi$
$618$	0	0
$619$	13.8966	0.558553	0.279276	−	0.960211i	$-0.409905\pi$
$619$	13.8966	0.558553	0.279276	+	0.960211i	$0.409905\pi$
$620$	0	0
$621$	−32.8568	−1.31850
$622$	0	0
$623$	−46.2520	−1.85305
$624$	0	0
$625$	−27.7993	−1.11197
$626$	0	0
$627$	−4.39511	−0.175524
$628$	0	0
$629$	−33.5722	−1.33861
$630$	0	0
$631$	−14.5999	−0.581212	−0.290606	−	0.956843i	$-0.593857\pi$
$631$	−14.5999	−0.581212	−0.290606	+	0.956843i	$0.593857\pi$
$632$	0	0
$633$	28.6670	1.13941
$634$	0	0
$635$	−10.5223	−0.417566
$636$	0	0
$637$	0.183708	0.00727878
$638$	0	0
$639$	−0.336177	−0.0132990
$640$	0	0
$641$	32.8320	1.29679	0.648394	−	0.761305i	$-0.275441\pi$
$641$	32.8320	1.29679	0.648394	+	0.761305i	$0.275441\pi$
$642$	0	0
$643$	−14.4574	−0.570143	−0.285071	−	0.958506i	$-0.592017\pi$
$643$	−14.4574	−0.570143	−0.285071	+	0.958506i	$0.592017\pi$
$644$	0	0
$645$	−23.1496	−0.911514
$646$	0	0
$647$	38.1091	1.49822	0.749111	−	0.662444i	$-0.230481\pi$
$647$	38.1091	1.49822	0.749111	+	0.662444i	$0.230481\pi$
$648$	0	0
$649$	29.6163	1.16254
$650$	0	0
$651$	4.28417	0.167910
$652$	0	0
$653$	−27.2350	−1.06579	−0.532895	−	0.846182i	$-0.678896\pi$
$653$	−27.2350	−1.06579	−0.532895	+	0.846182i	$0.678896\pi$
$654$	0	0
$655$	23.0551	0.900839
$656$	0	0
$657$	2.43938	0.0951692
$658$	0	0
$659$	31.0547	1.20972	0.604859	−	0.796333i	$-0.293230\pi$
$659$	31.0547	1.20972	0.604859	+	0.796333i	$0.293230\pi$
$660$	0	0
$661$	33.2533	1.29341	0.646703	−	0.762742i	$-0.276148\pi$
$661$	33.2533	1.29341	0.646703	+	0.762742i	$0.276148\pi$
$662$	0	0
$663$	−32.0732	−1.24562
$664$	0	0
$665$	−6.31413	−0.244852
$666$	0	0
$667$	13.0875	0.506752
$668$	0	0
$669$	−31.5242	−1.21880
$670$	0	0
$671$	−7.45210	−0.287685
$672$	0	0
$673$	−36.2614	−1.39777	−0.698886	−	0.715233i	$-0.746321\pi$
$673$	−36.2614	−1.39777	−0.698886	+	0.715233i	$0.746321\pi$
$674$	0	0
$675$	−3.57165	−0.137473
$676$	0	0
$677$	40.8989	1.57187	0.785937	−	0.618307i	$-0.212181\pi$
$677$	40.8989	1.57187	0.785937	+	0.618307i	$0.212181\pi$
$678$	0	0
$679$	−28.0630	−1.07696
$680$	0	0
$681$	19.3872	0.742920
$682$	0	0
$683$	−13.2518	−0.507065	−0.253533	−	0.967327i	$-0.581592\pi$
$683$	−13.2518	−0.507065	−0.253533	+	0.967327i	$0.581592\pi$
$684$	0	0
$685$	−42.9019	−1.63920
$686$	0	0
$687$	−21.2934	−0.812393
$688$	0	0
$689$	−24.3892	−0.929154
$690$	0	0
$691$	10.9408	0.416207	0.208104	−	0.978107i	$-0.433271\pi$
$691$	10.9408	0.416207	0.208104	+	0.978107i	$0.433271\pi$
$692$	0	0
$693$	4.16948	0.158385
$694$	0	0
$695$	0.784779	0.0297684
$696$	0	0
$697$	−19.5050	−0.738805
$698$	0	0
$699$	−2.11806	−0.0801125
$700$	0	0
$701$	−15.2890	−0.577456	−0.288728	−	0.957411i	$-0.593232\pi$
$701$	−15.2890	−0.577456	−0.288728	+	0.957411i	$0.593232\pi$
$702$	0	0
$703$	4.56380	0.172127
$704$	0	0
$705$	−1.05207	−0.0396231
$706$	0	0
$707$	−7.31529	−0.275120
$708$	0	0
$709$	−14.8840	−0.558980	−0.279490	−	0.960149i	$-0.590165\pi$
$709$	−14.8840	−0.558980	−0.279490	+	0.960149i	$0.590165\pi$
$710$	0	0
$711$	0.557733	0.0209166
$712$	0	0
$713$	6.09459	0.228244
$714$	0	0
$715$	18.6379	0.697017
$716$	0	0
$717$	−27.0801	−1.01132
$718$	0	0
$719$	50.5214	1.88413	0.942065	−	0.335430i	$-0.108882\pi$
$719$	50.5214	1.88413	0.942065	+	0.335430i	$0.108882\pi$
$720$	0	0
$721$	−15.0698	−0.561229
$722$	0	0
$723$	36.4816	1.35677
$724$	0	0
$725$	1.42266	0.0528363
$726$	0	0
$727$	25.2650	0.937026	0.468513	−	0.883457i	$-0.344790\pi$
$727$	25.2650	0.937026	0.468513	+	0.883457i	$0.344790\pi$
$728$	0	0
$729$	29.9797	1.11036
$730$	0	0
$731$	−45.8743	−1.69672
$732$	0	0
$733$	49.0305	1.81098	0.905491	−	0.424364i	$-0.139502\pi$
$733$	49.0305	1.81098	0.905491	+	0.424364i	$0.139502\pi$
$734$	0	0
$735$	−0.244436	−0.00901614
$736$	0	0
$737$	34.9973	1.28914
$738$	0	0
$739$	−23.6030	−0.868251	−0.434126	−	0.900852i	$-0.642943\pi$
$739$	−23.6030	−0.868251	−0.434126	+	0.900852i	$0.642943\pi$
$740$	0	0
$741$	4.36002	0.160169
$742$	0	0
$743$	−28.0798	−1.03015	−0.515074	−	0.857146i	$-0.672235\pi$
$743$	−28.0798	−1.03015	−0.515074	+	0.857146i	$0.672235\pi$
$744$	0	0
$745$	46.7917	1.71432
$746$	0	0
$747$	1.60909	0.0588735
$748$	0	0
$749$	−38.1855	−1.39527
$750$	0	0
$751$	24.5568	0.896092	0.448046	−	0.894011i	$-0.352120\pi$
$751$	24.5568	0.896092	0.448046	+	0.894011i	$0.352120\pi$
$752$	0	0
$753$	37.1230	1.35284
$754$	0	0
$755$	33.4032	1.21567
$756$	0	0
$757$	−3.33164	−0.121090	−0.0605452	−	0.998165i	$-0.519284\pi$
$757$	−3.33164	−0.121090	−0.0605452	+	0.998165i	$0.519284\pi$
$758$	0	0
$759$	−25.9732	−0.942767
$760$	0	0
$761$	−36.9530	−1.33954	−0.669772	−	0.742567i	$-0.733608\pi$
$761$	−36.9530	−1.33954	−0.669772	+	0.742567i	$0.733608\pi$
$762$	0	0
$763$	−1.39212	−0.0503981
$764$	0	0
$765$	−9.74568	−0.352356
$766$	0	0
$767$	−29.3799	−1.06085
$768$	0	0
$769$	−39.3221	−1.41799	−0.708995	−	0.705213i	$-0.750851\pi$
$769$	−39.3221	−1.41799	−0.708995	+	0.705213i	$0.750851\pi$
$770$	0	0
$771$	−27.5277	−0.991385
$772$	0	0
$773$	−36.9053	−1.32739	−0.663695	−	0.748003i	$-0.731013\pi$
$773$	−36.9053	−1.32739	−0.663695	+	0.748003i	$0.731013\pi$
$774$	0	0
$775$	0.662502	0.0237978
$776$	0	0
$777$	18.9585	0.680134
$778$	0	0
$779$	2.65150	0.0949999
$780$	0	0
$781$	−1.69517	−0.0606581
$782$	0	0
$783$	−12.3132	−0.440039
$784$	0	0
$785$	−15.6944	−0.560157
$786$	0	0
$787$	−35.7524	−1.27444	−0.637219	−	0.770683i	$-0.719915\pi$
$787$	−35.7524	−1.27444	−0.637219	+	0.770683i	$0.719915\pi$
$788$	0	0
$789$	−11.5553	−0.411378
$790$	0	0
$791$	−7.76402	−0.276057
$792$	0	0
$793$	7.39260	0.262519
$794$	0	0
$795$	32.4514	1.15093
$796$	0	0
$797$	−24.8500	−0.880231	−0.440115	−	0.897941i	$-0.645062\pi$
$797$	−24.8500	−0.880231	−0.440115	+	0.897941i	$0.645062\pi$
$798$	0	0
$799$	−2.08482	−0.0737556
$800$	0	0
$801$	9.70454	0.342893
$802$	0	0
$803$	12.3006	0.434078
$804$	0	0
$805$	−37.3138	−1.31514
$806$	0	0
$807$	−36.7993	−1.29540
$808$	0	0
$809$	−22.9418	−0.806589	−0.403295	−	0.915070i	$-0.632135\pi$
$809$	−22.9418	−0.806589	−0.403295	+	0.915070i	$0.632135\pi$
$810$	0	0
$811$	−7.06552	−0.248104	−0.124052	−	0.992276i	$-0.539589\pi$
$811$	−7.06552	−0.248104	−0.124052	+	0.992276i	$0.539589\pi$
$812$	0	0
$813$	45.0805	1.58104
$814$	0	0
$815$	37.0480	1.29774
$816$	0	0
$817$	6.23613	0.218175
$818$	0	0
$819$	−4.13619	−0.144530
$820$	0	0
$821$	54.7782	1.91177	0.955887	−	0.293735i	$-0.0948982\pi$
$821$	54.7782	1.91177	0.955887	+	0.293735i	$0.0948982\pi$
$822$	0	0
$823$	−40.8241	−1.42304	−0.711518	−	0.702667i	$-0.751992\pi$
$823$	−40.8241	−1.42304	−0.711518	+	0.702667i	$0.751992\pi$
$824$	0	0
$825$	−2.82337	−0.0982972
$826$	0	0
$827$	16.2035	0.563451	0.281726	−	0.959495i	$-0.409093\pi$
$827$	16.2035	0.563451	0.281726	+	0.959495i	$0.409093\pi$
$828$	0	0
$829$	−33.9941	−1.18066	−0.590332	−	0.807160i	$-0.701003\pi$
$829$	−33.9941	−1.18066	−0.590332	+	0.807160i	$0.701003\pi$
$830$	0	0
$831$	25.9744	0.901041
$832$	0	0
$833$	−0.484384	−0.0167829
$834$	0	0
$835$	−14.4042	−0.498479
$836$	0	0
$837$	−5.73401	−0.198196
$838$	0	0
$839$	25.8596	0.892771	0.446386	−	0.894841i	$-0.352711\pi$
$839$	25.8596	0.892771	0.446386	+	0.894841i	$0.352711\pi$
$840$	0	0
$841$	−24.0954	−0.830875
$842$	0	0
$843$	−13.1192	−0.451850
$844$	0	0
$845$	12.3907	0.426255
$846$	0	0
$847$	−8.21520	−0.282278
$848$	0	0
$849$	−35.4634	−1.21710
$850$	0	0
$851$	26.9701	0.924522
$852$	0	0
$853$	20.9860	0.718547	0.359273	−	0.933232i	$-0.383025\pi$
$853$	20.9860	0.718547	0.359273	+	0.933232i	$0.383025\pi$
$854$	0	0
$855$	1.32482	0.0453080
$856$	0	0
$857$	−25.1394	−0.858745	−0.429372	−	0.903128i	$-0.641265\pi$
$857$	−25.1394	−0.858745	−0.429372	+	0.903128i	$0.641265\pi$
$858$	0	0
$859$	38.0207	1.29725	0.648626	−	0.761108i	$-0.275344\pi$
$859$	38.0207	1.29725	0.648626	+	0.761108i	$0.275344\pi$
$860$	0	0
$861$	11.0147	0.375378
$862$	0	0
$863$	13.1640	0.448106	0.224053	−	0.974577i	$-0.428071\pi$
$863$	13.1640	0.448106	0.224053	+	0.974577i	$0.428071\pi$
$864$	0	0
$865$	−55.8842	−1.90012
$866$	0	0
$867$	58.0005	1.96980
$868$	0	0
$869$	2.81237	0.0954032
$870$	0	0
$871$	−34.7179	−1.17637
$872$	0	0
$873$	5.88815	0.199284
$874$	0	0
$875$	27.5145	0.930161
$876$	0	0
$877$	16.4665	0.556035	0.278018	−	0.960576i	$-0.410323\pi$
$877$	16.4665	0.556035	0.278018	+	0.960576i	$0.410323\pi$
$878$	0	0
$879$	46.1602	1.55695
$880$	0	0
$881$	9.27474	0.312474	0.156237	−	0.987720i	$-0.450064\pi$
$881$	9.27474	0.312474	0.156237	+	0.987720i	$0.450064\pi$
$882$	0	0
$883$	54.7385	1.84210	0.921048	−	0.389449i	$-0.127335\pi$
$883$	54.7385	1.84210	0.921048	+	0.389449i	$0.127335\pi$
$884$	0	0
$885$	39.0918	1.31406
$886$	0	0
$887$	−11.9704	−0.401926	−0.200963	−	0.979599i	$-0.564407\pi$
$887$	−11.9704	−0.401926	−0.200963	+	0.979599i	$0.564407\pi$
$888$	0	0
$889$	11.7750	0.394922
$890$	0	0
$891$	19.7309	0.661009
$892$	0	0
$893$	0.283410	0.00948394
$894$	0	0
$895$	23.4545	0.783998
$896$	0	0
$897$	25.7658	0.860296
$898$	0	0
$899$	2.28397	0.0761748
$900$	0	0
$901$	64.3071	2.14238
$902$	0	0
$903$	25.9056	0.862085
$904$	0	0
$905$	−27.2393	−0.905464
$906$	0	0
$907$	12.9993	0.431636	0.215818	−	0.976434i	$-0.430758\pi$
$907$	12.9993	0.431636	0.215818	+	0.976434i	$0.430758\pi$
$908$	0	0
$909$	1.53489	0.0509090
$910$	0	0
$911$	−23.2411	−0.770014	−0.385007	−	0.922914i	$-0.625801\pi$
$911$	−23.2411	−0.770014	−0.385007	+	0.922914i	$0.625801\pi$
$912$	0	0
$913$	8.11385	0.268529
$914$	0	0
$915$	−9.83634	−0.325179
$916$	0	0
$917$	−25.7999	−0.851988
$918$	0	0
$919$	−47.3640	−1.56239	−0.781197	−	0.624284i	$-0.785391\pi$
$919$	−47.3640	−1.56239	−0.781197	+	0.624284i	$0.785391\pi$
$920$	0	0
$921$	−49.1003	−1.61791
$922$	0	0
$923$	1.68164	0.0553519
$924$	0	0
$925$	2.93174	0.0963949
$926$	0	0
$927$	3.16193	0.103851
$928$	0	0
$929$	58.9374	1.93367	0.966837	−	0.255393i	$-0.0822047\pi$
$929$	58.9374	1.93367	0.966837	+	0.255393i	$0.0822047\pi$
$930$	0	0
$931$	0.0658470	0.00215805
$932$	0	0
$933$	−21.8773	−0.716231
$934$	0	0
$935$	−49.1426	−1.60714
$936$	0	0
$937$	−23.6257	−0.771820	−0.385910	−	0.922537i	$-0.626112\pi$
$937$	−23.6257	−0.771820	−0.385910	+	0.922537i	$0.626112\pi$
$938$	0	0
$939$	−27.9427	−0.911876
$940$	0	0
$941$	−49.6290	−1.61786	−0.808930	−	0.587906i	$-0.799953\pi$
$941$	−49.6290	−1.61786	−0.808930	+	0.587906i	$0.799953\pi$
$942$	0	0
$943$	15.6692	0.510261
$944$	0	0
$945$	35.1062	1.14200
$946$	0	0
$947$	36.6499	1.19096	0.595481	−	0.803370i	$-0.296962\pi$
$947$	36.6499	1.19096	0.595481	+	0.803370i	$0.296962\pi$
$948$	0	0
$949$	−12.2024	−0.396106
$950$	0	0
$951$	−52.2051	−1.69287
$952$	0	0
$953$	−35.4904	−1.14965	−0.574823	−	0.818278i	$-0.694929\pi$
$953$	−35.4904	−1.14965	−0.574823	+	0.818278i	$0.694929\pi$
$954$	0	0
$955$	−6.47986	−0.209683
$956$	0	0
$957$	−9.73357	−0.314642
$958$	0	0
$959$	48.0095	1.55031
$960$	0	0
$961$	−29.9364	−0.965690
$962$	0	0
$963$	8.01203	0.258184
$964$	0	0
$965$	−13.6045	−0.437944
$966$	0	0
$967$	−4.87977	−0.156923	−0.0784614	−	0.996917i	$-0.525001\pi$
$967$	−4.87977	−0.156923	−0.0784614	+	0.996917i	$0.525001\pi$
$968$	0	0
$969$	−11.4961	−0.369308
$970$	0	0
$971$	27.5433	0.883908	0.441954	−	0.897038i	$-0.354285\pi$
$971$	27.5433	0.883908	0.441954	+	0.897038i	$0.354285\pi$
$972$	0	0
$973$	−0.878210	−0.0281541
$974$	0	0
$975$	2.80083	0.0896984
$976$	0	0
$977$	−18.9025	−0.604746	−0.302373	−	0.953190i	$-0.597779\pi$
$977$	−18.9025	−0.604746	−0.302373	+	0.953190i	$0.597779\pi$
$978$	0	0
$979$	48.9352	1.56398
$980$	0	0
$981$	0.292093	0.00932581
$982$	0	0
$983$	23.0231	0.734323	0.367161	−	0.930157i	$-0.380330\pi$
$983$	23.0231	0.734323	0.367161	+	0.930157i	$0.380330\pi$
$984$	0	0
$985$	−23.3233	−0.743143
$986$	0	0
$987$	1.17732	0.0374744
$988$	0	0
$989$	36.8528	1.17185
$990$	0	0
$991$	10.7418	0.341224	0.170612	−	0.985338i	$-0.445426\pi$
$991$	10.7418	0.341224	0.170612	+	0.985338i	$0.445426\pi$
$992$	0	0
$993$	24.1371	0.765968
$994$	0	0
$995$	12.5403	0.397554
$996$	0	0
$997$	39.6665	1.25625	0.628126	−	0.778112i	$-0.283822\pi$
$997$	39.6665	1.25625	0.628126	+	0.778112i	$0.283822\pi$
$998$	0	0
$999$	−25.3744	−0.802811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.22	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.22	✓	27	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56278 q^{3} -2.37537 q^{5} +2.65817 q^{7} -0.557733 q^{9} +O(q^{10})\)	\(q+1.56278 q^{3} -2.37537 q^{5} +2.65817 q^{7} -0.557733 q^{9} -2.81237 q^{11} +2.78992 q^{13} -3.71217 q^{15} -7.35621 q^{17} +1.00000 q^{19} +4.15412 q^{21} +5.90957 q^{23} +0.642390 q^{25} -5.55994 q^{27} +2.21464 q^{29} +1.03131 q^{31} -4.39511 q^{33} -6.31413 q^{35} +4.56380 q^{37} +4.36002 q^{39} +2.65150 q^{41} +6.23613 q^{43} +1.32482 q^{45} +0.283410 q^{47} +0.0658470 q^{49} -11.4961 q^{51} -8.74189 q^{53} +6.68043 q^{55} +1.56278 q^{57} -10.5307 q^{59} +2.64975 q^{61} -1.48255 q^{63} -6.62710 q^{65} -12.4441 q^{67} +9.23533 q^{69} +0.602756 q^{71} -4.37373 q^{73} +1.00391 q^{75} -7.47576 q^{77} -1.00000 q^{79} -7.01573 q^{81} -2.88505 q^{83} +17.4737 q^{85} +3.46098 q^{87} -17.4000 q^{89} +7.41607 q^{91} +1.61170 q^{93} -2.37537 q^{95} -10.5573 q^{97} +1.56855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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