LMFDB - Embedded newform 6044.2.a.a.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$1$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.57704 q^{3} +3.78585 q^{5} +1.51605 q^{7} -0.512930 q^{9} +O(q^{10})$	$q-1.57704 q^{3} +3.78585 q^{5} +1.51605 q^{7} -0.512930 q^{9} -2.76818 q^{11} +4.80806 q^{13} -5.97045 q^{15} -2.88524 q^{17} -8.31022 q^{19} -2.39087 q^{21} -7.23282 q^{23} +9.33266 q^{25} +5.54005 q^{27} +7.44970 q^{29} -1.37858 q^{31} +4.36555 q^{33} +5.73953 q^{35} +2.27773 q^{37} -7.58252 q^{39} +2.20921 q^{41} +3.44247 q^{43} -1.94187 q^{45} -10.0775 q^{47} -4.70160 q^{49} +4.55016 q^{51} -11.8321 q^{53} -10.4799 q^{55} +13.1056 q^{57} -6.48510 q^{59} -9.02274 q^{61} -0.777625 q^{63} +18.2026 q^{65} -5.98905 q^{67} +11.4065 q^{69} -4.84111 q^{71} -15.8749 q^{73} -14.7180 q^{75} -4.19669 q^{77} -1.72548 q^{79} -7.19811 q^{81} +9.91848 q^{83} -10.9231 q^{85} -11.7485 q^{87} -3.24839 q^{89} +7.28924 q^{91} +2.17408 q^{93} -31.4612 q^{95} +10.5422 q^{97} +1.41988 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * 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
$3$	−1.57704	−0.910507	−0.455254	−	0.890362i	$-0.650451\pi$
$3$	−1.57704	−0.910507	−0.455254	+	0.890362i	$0.650451\pi$
$4$	0	0
$5$	3.78585	1.69308	0.846542	−	0.532322i	$-0.178681\pi$
$5$	3.78585	1.69308	0.846542	+	0.532322i	$0.178681\pi$
$6$	0	0
$7$	1.51605	0.573012	0.286506	−	0.958078i	$-0.407506\pi$
$7$	1.51605	0.573012	0.286506	+	0.958078i	$0.407506\pi$
$8$	0	0
$9$	−0.512930	−0.170977
$10$	0	0
$11$	−2.76818	−0.834638	−0.417319	−	0.908760i	$-0.637030\pi$
$11$	−2.76818	−0.834638	−0.417319	+	0.908760i	$0.637030\pi$
$12$	0	0
$13$	4.80806	1.33352	0.666758	−	0.745275i	$-0.267682\pi$
$13$	4.80806	1.33352	0.666758	+	0.745275i	$0.267682\pi$
$14$	0	0
$15$	−5.97045	−1.54156
$16$	0	0
$17$	−2.88524	−0.699775	−0.349887	−	0.936792i	$-0.613780\pi$
$17$	−2.88524	−0.699775	−0.349887	+	0.936792i	$0.613780\pi$
$18$	0	0
$19$	−8.31022	−1.90649	−0.953247	−	0.302192i	$-0.902282\pi$
$19$	−8.31022	−1.90649	−0.953247	+	0.302192i	$0.902282\pi$
$20$	0	0
$21$	−2.39087	−0.521731
$22$	0	0
$23$	−7.23282	−1.50815	−0.754074	−	0.656790i	$-0.771914\pi$
$23$	−7.23282	−1.50815	−0.754074	+	0.656790i	$0.771914\pi$
$24$	0	0
$25$	9.33266	1.86653
$26$	0	0
$27$	5.54005	1.06618
$28$	0	0
$29$	7.44970	1.38338	0.691688	−	0.722197i	$-0.256867\pi$
$29$	7.44970	1.38338	0.691688	+	0.722197i	$0.256867\pi$
$30$	0	0
$31$	−1.37858	−0.247600	−0.123800	−	0.992307i	$-0.539508\pi$
$31$	−1.37858	−0.247600	−0.123800	+	0.992307i	$0.539508\pi$
$32$	0	0
$33$	4.36555	0.759944
$34$	0	0
$35$	5.73953	0.970157
$36$	0	0
$37$	2.27773	0.374457	0.187229	−	0.982316i	$-0.440049\pi$
$37$	2.27773	0.374457	0.187229	+	0.982316i	$0.440049\pi$
$38$	0	0
$39$	−7.58252	−1.21418
$40$	0	0
$41$	2.20921	0.345020	0.172510	−	0.985008i	$-0.444812\pi$
$41$	2.20921	0.345020	0.172510	+	0.985008i	$0.444812\pi$
$42$	0	0
$43$	3.44247	0.524972	0.262486	−	0.964936i	$-0.415458\pi$
$43$	3.44247	0.524972	0.262486	+	0.964936i	$0.415458\pi$
$44$	0	0
$45$	−1.94187	−0.289478
$46$	0	0
$47$	−10.0775	−1.46996	−0.734979	−	0.678090i	$-0.762808\pi$
$47$	−10.0775	−1.46996	−0.734979	+	0.678090i	$0.762808\pi$
$48$	0	0
$49$	−4.70160	−0.671657
$50$	0	0
$51$	4.55016	0.637150
$52$	0	0
$53$	−11.8321	−1.62527	−0.812633	−	0.582776i	$-0.801966\pi$
$53$	−11.8321	−1.62527	−0.812633	+	0.582776i	$0.801966\pi$
$54$	0	0
$55$	−10.4799	−1.41311
$56$	0	0
$57$	13.1056	1.73588
$58$	0	0
$59$	−6.48510	−0.844288	−0.422144	−	0.906529i	$-0.638722\pi$
$59$	−6.48510	−0.844288	−0.422144	+	0.906529i	$0.638722\pi$
$60$	0	0
$61$	−9.02274	−1.15524	−0.577622	−	0.816305i	$-0.696019\pi$
$61$	−9.02274	−1.15524	−0.577622	+	0.816305i	$0.696019\pi$
$62$	0	0
$63$	−0.777625	−0.0979716
$64$	0	0
$65$	18.2026	2.25775
$66$	0	0
$67$	−5.98905	−0.731679	−0.365840	−	0.930678i	$-0.619218\pi$
$67$	−5.98905	−0.731679	−0.365840	+	0.930678i	$0.619218\pi$
$68$	0	0
$69$	11.4065	1.37318
$70$	0	0
$71$	−4.84111	−0.574534	−0.287267	−	0.957851i	$-0.592747\pi$
$71$	−4.84111	−0.574534	−0.287267	+	0.957851i	$0.592747\pi$
$72$	0	0
$73$	−15.8749	−1.85802	−0.929008	−	0.370059i	$-0.879337\pi$
$73$	−15.8749	−1.85802	−0.929008	+	0.370059i	$0.879337\pi$
$74$	0	0
$75$	−14.7180	−1.69949
$76$	0	0
$77$	−4.19669	−0.478258
$78$	0	0
$79$	−1.72548	−0.194132	−0.0970660	−	0.995278i	$-0.530946\pi$
$79$	−1.72548	−0.194132	−0.0970660	+	0.995278i	$0.530946\pi$
$80$	0	0
$81$	−7.19811	−0.799790
$82$	0	0
$83$	9.91848	1.08869	0.544347	−	0.838860i	$-0.316777\pi$
$83$	9.91848	1.08869	0.544347	+	0.838860i	$0.316777\pi$
$84$	0	0
$85$	−10.9231	−1.18478
$86$	0	0
$87$	−11.7485	−1.25957
$88$	0	0
$89$	−3.24839	−0.344328	−0.172164	−	0.985068i	$-0.555076\pi$
$89$	−3.24839	−0.344328	−0.172164	+	0.985068i	$0.555076\pi$
$90$	0	0
$91$	7.28924	0.764120
$92$	0	0
$93$	2.17408	0.225442
$94$	0	0
$95$	−31.4612	−3.22785
$96$	0	0
$97$	10.5422	1.07040	0.535202	−	0.844724i	$-0.320236\pi$
$97$	10.5422	1.07040	0.535202	+	0.844724i	$0.320236\pi$
$98$	0	0
$99$	1.41988	0.142704
$100$	0	0
$101$	11.4340	1.13773	0.568865	−	0.822431i	$-0.307383\pi$
$101$	11.4340	1.13773	0.568865	+	0.822431i	$0.307383\pi$
$102$	0	0
$103$	12.6198	1.24347	0.621734	−	0.783228i	$-0.286428\pi$
$103$	12.6198	1.24347	0.621734	+	0.783228i	$0.286428\pi$
$104$	0	0
$105$	−9.05149	−0.883335
$106$	0	0
$107$	0.568604	0.0549690	0.0274845	−	0.999622i	$-0.491250\pi$
$107$	0.568604	0.0549690	0.0274845	+	0.999622i	$0.491250\pi$
$108$	0	0
$109$	−11.8795	−1.13785	−0.568923	−	0.822391i	$-0.692640\pi$
$109$	−11.8795	−1.13785	−0.568923	+	0.822391i	$0.692640\pi$
$110$	0	0
$111$	−3.59209	−0.340946
$112$	0	0
$113$	9.26960	0.872011	0.436005	−	0.899944i	$-0.356393\pi$
$113$	9.26960	0.872011	0.436005	+	0.899944i	$0.356393\pi$
$114$	0	0
$115$	−27.3824	−2.55342
$116$	0	0
$117$	−2.46620	−0.228000
$118$	0	0
$119$	−4.37417	−0.400979
$120$	0	0
$121$	−3.33717	−0.303379
$122$	0	0
$123$	−3.48402	−0.314143
$124$	0	0
$125$	16.4028	1.46711
$126$	0	0
$127$	−15.4249	−1.36874	−0.684368	−	0.729137i	$-0.739922\pi$
$127$	−15.4249	−1.36874	−0.684368	+	0.729137i	$0.739922\pi$
$128$	0	0
$129$	−5.42893	−0.477990
$130$	0	0
$131$	13.8267	1.20804	0.604021	−	0.796969i	$-0.293564\pi$
$131$	13.8267	1.20804	0.604021	+	0.796969i	$0.293564\pi$
$132$	0	0
$133$	−12.5987	−1.09244
$134$	0	0
$135$	20.9738	1.80514
$136$	0	0
$137$	−17.3044	−1.47841	−0.739206	−	0.673480i	$-0.764799\pi$
$137$	−17.3044	−1.47841	−0.739206	+	0.673480i	$0.764799\pi$
$138$	0	0
$139$	1.25780	0.106685	0.0533427	−	0.998576i	$-0.483012\pi$
$139$	1.25780	0.106685	0.0533427	+	0.998576i	$0.483012\pi$
$140$	0	0
$141$	15.8927	1.33841
$142$	0	0
$143$	−13.3096	−1.11300
$144$	0	0
$145$	28.2035	2.34217
$146$	0	0
$147$	7.41464	0.611549
$148$	0	0
$149$	19.6083	1.60637	0.803186	−	0.595728i	$-0.203136\pi$
$149$	19.6083	1.60637	0.803186	+	0.595728i	$0.203136\pi$
$150$	0	0
$151$	1.89147	0.153925	0.0769627	−	0.997034i	$-0.475478\pi$
$151$	1.89147	0.153925	0.0769627	+	0.997034i	$0.475478\pi$
$152$	0	0
$153$	1.47993	0.119645
$154$	0	0
$155$	−5.21910	−0.419208
$156$	0	0
$157$	−7.09861	−0.566531	−0.283266	−	0.959042i	$-0.591418\pi$
$157$	−7.09861	−0.566531	−0.283266	+	0.959042i	$0.591418\pi$
$158$	0	0
$159$	18.6598	1.47982
$160$	0	0
$161$	−10.9653	−0.864187
$162$	0	0
$163$	5.23785	0.410260	0.205130	−	0.978735i	$-0.434238\pi$
$163$	5.23785	0.410260	0.205130	+	0.978735i	$0.434238\pi$
$164$	0	0
$165$	16.5273	1.28665
$166$	0	0
$167$	−9.94206	−0.769340	−0.384670	−	0.923054i	$-0.625685\pi$
$167$	−9.94206	−0.769340	−0.384670	+	0.923054i	$0.625685\pi$
$168$	0	0
$169$	10.1174	0.778263
$170$	0	0
$171$	4.26256	0.325966
$172$	0	0
$173$	8.04892	0.611948	0.305974	−	0.952040i	$-0.401018\pi$
$173$	8.04892	0.611948	0.305974	+	0.952040i	$0.401018\pi$
$174$	0	0
$175$	14.1487	1.06954
$176$	0	0
$177$	10.2273	0.768731
$178$	0	0
$179$	13.2693	0.991791	0.495895	−	0.868382i	$-0.334840\pi$
$179$	13.2693	0.991791	0.495895	+	0.868382i	$0.334840\pi$
$180$	0	0
$181$	−20.5512	−1.52756	−0.763781	−	0.645476i	$-0.776659\pi$
$181$	−20.5512	−1.52756	−0.763781	+	0.645476i	$0.776659\pi$
$182$	0	0
$183$	14.2293	1.05186
$184$	0	0
$185$	8.62316	0.633987
$186$	0	0
$187$	7.98688	0.584059
$188$	0	0
$189$	8.39897	0.610935
$190$	0	0
$191$	−25.2182	−1.82473	−0.912363	−	0.409383i	$-0.865744\pi$
$191$	−25.2182	−1.82473	−0.912363	+	0.409383i	$0.865744\pi$
$192$	0	0
$193$	−22.1052	−1.59117	−0.795583	−	0.605845i	$-0.792835\pi$
$193$	−22.1052	−1.59117	−0.795583	+	0.605845i	$0.792835\pi$
$194$	0	0
$195$	−28.7063	−2.05570
$196$	0	0
$197$	4.09878	0.292026	0.146013	−	0.989283i	$-0.453356\pi$
$197$	4.09878	0.292026	0.146013	+	0.989283i	$0.453356\pi$
$198$	0	0
$199$	25.0743	1.77747	0.888734	−	0.458423i	$-0.151586\pi$
$199$	25.0743	1.77747	0.888734	+	0.458423i	$0.151586\pi$
$200$	0	0
$201$	9.44501	0.666199
$202$	0	0
$203$	11.2941	0.792690
$204$	0	0
$205$	8.36372	0.584148
$206$	0	0
$207$	3.70993	0.257858
$208$	0	0
$209$	23.0042	1.59123
$210$	0	0
$211$	−2.09829	−0.144452	−0.0722261	−	0.997388i	$-0.523010\pi$
$211$	−2.09829	−0.144452	−0.0722261	+	0.997388i	$0.523010\pi$
$212$	0	0
$213$	7.63464	0.523117
$214$	0	0
$215$	13.0327	0.888821
$216$	0	0
$217$	−2.08999	−0.141878
$218$	0	0
$219$	25.0354	1.69174
$220$	0	0
$221$	−13.8724	−0.933160
$222$	0	0
$223$	−19.1754	−1.28408	−0.642041	−	0.766671i	$-0.721912\pi$
$223$	−19.1754	−1.28408	−0.642041	+	0.766671i	$0.721912\pi$
$224$	0	0
$225$	−4.78700	−0.319133
$226$	0	0
$227$	−7.20019	−0.477894	−0.238947	−	0.971033i	$-0.576802\pi$
$227$	−7.20019	−0.477894	−0.238947	+	0.971033i	$0.576802\pi$
$228$	0	0
$229$	1.16013	0.0766634	0.0383317	−	0.999265i	$-0.487796\pi$
$229$	1.16013	0.0766634	0.0383317	+	0.999265i	$0.487796\pi$
$230$	0	0
$231$	6.61837	0.435457
$232$	0	0
$233$	28.1387	1.84343	0.921715	−	0.387868i	$-0.126788\pi$
$233$	28.1387	1.84343	0.921715	+	0.387868i	$0.126788\pi$
$234$	0	0
$235$	−38.1520	−2.48876
$236$	0	0
$237$	2.72116	0.176759
$238$	0	0
$239$	30.8660	1.99656	0.998278	−	0.0586618i	$-0.0186833\pi$
$239$	30.8660	1.99656	0.998278	+	0.0586618i	$0.0186833\pi$
$240$	0	0
$241$	9.40037	0.605531	0.302765	−	0.953065i	$-0.402090\pi$
$241$	9.40037	0.605531	0.302765	+	0.953065i	$0.402090\pi$
$242$	0	0
$243$	−5.26839	−0.337968
$244$	0	0
$245$	−17.7996	−1.13717
$246$	0	0
$247$	−39.9560	−2.54234
$248$	0	0
$249$	−15.6419	−0.991264
$250$	0	0
$251$	6.39159	0.403434	0.201717	−	0.979444i	$-0.435348\pi$
$251$	6.39159	0.403434	0.201717	+	0.979444i	$0.435348\pi$
$252$	0	0
$253$	20.0218	1.25876
$254$	0	0
$255$	17.2262	1.07875
$256$	0	0
$257$	−14.3501	−0.895137	−0.447568	−	0.894250i	$-0.647710\pi$
$257$	−14.3501	−0.895137	−0.447568	+	0.894250i	$0.647710\pi$
$258$	0	0
$259$	3.45315	0.214568
$260$	0	0
$261$	−3.82117	−0.236525
$262$	0	0
$263$	−4.39454	−0.270979	−0.135489	−	0.990779i	$-0.543261\pi$
$263$	−4.39454	−0.270979	−0.135489	+	0.990779i	$0.543261\pi$
$264$	0	0
$265$	−44.7946	−2.75171
$266$	0	0
$267$	5.12285	0.313513
$268$	0	0
$269$	5.86634	0.357677	0.178839	−	0.983878i	$-0.442766\pi$
$269$	5.86634	0.357677	0.178839	+	0.983878i	$0.442766\pi$
$270$	0	0
$271$	16.8266	1.02214	0.511071	−	0.859538i	$-0.329249\pi$
$271$	16.8266	1.02214	0.511071	+	0.859538i	$0.329249\pi$
$272$	0	0
$273$	−11.4955	−0.695737
$274$	0	0
$275$	−25.8345	−1.55788
$276$	0	0
$277$	−8.36949	−0.502874	−0.251437	−	0.967874i	$-0.580903\pi$
$277$	−8.36949	−0.502874	−0.251437	+	0.967874i	$0.580903\pi$
$278$	0	0
$279$	0.707115	0.0423339
$280$	0	0
$281$	−26.7817	−1.59766	−0.798831	−	0.601555i	$-0.794548\pi$
$281$	−26.7817	−1.59766	−0.798831	+	0.601555i	$0.794548\pi$
$282$	0	0
$283$	−10.2333	−0.608307	−0.304153	−	0.952623i	$-0.598374\pi$
$283$	−10.2333	−0.608307	−0.304153	+	0.952623i	$0.598374\pi$
$284$	0	0
$285$	49.6158	2.93898
$286$	0	0
$287$	3.34926	0.197701
$288$	0	0
$289$	−8.67536	−0.510316
$290$	0	0
$291$	−16.6256	−0.974610
$292$	0	0
$293$	−15.6834	−0.916235	−0.458117	−	0.888892i	$-0.651476\pi$
$293$	−15.6834	−0.916235	−0.458117	+	0.888892i	$0.651476\pi$
$294$	0	0
$295$	−24.5516	−1.42945
$296$	0	0
$297$	−15.3359	−0.889877
$298$	0	0
$299$	−34.7758	−2.01114
$300$	0	0
$301$	5.21894	0.300815
$302$	0	0
$303$	−18.0320	−1.03591
$304$	0	0
$305$	−34.1587	−1.95592
$306$	0	0
$307$	19.8572	1.13331	0.566655	−	0.823955i	$-0.308237\pi$
$307$	19.8572	1.13331	0.566655	+	0.823955i	$0.308237\pi$
$308$	0	0
$309$	−19.9020	−1.13219
$310$	0	0
$311$	−3.23643	−0.183521	−0.0917607	−	0.995781i	$-0.529249\pi$
$311$	−3.23643	−0.183521	−0.0917607	+	0.995781i	$0.529249\pi$
$312$	0	0
$313$	−22.9067	−1.29477	−0.647383	−	0.762165i	$-0.724136\pi$
$313$	−22.9067	−1.29477	−0.647383	+	0.762165i	$0.724136\pi$
$314$	0	0
$315$	−2.94397	−0.165874
$316$	0	0
$317$	−23.4166	−1.31521	−0.657605	−	0.753363i	$-0.728430\pi$
$317$	−23.4166	−1.31521	−0.657605	+	0.753363i	$0.728430\pi$
$318$	0	0
$319$	−20.6221	−1.15462
$320$	0	0
$321$	−0.896714	−0.0500497
$322$	0	0
$323$	23.9770	1.33412
$324$	0	0
$325$	44.8720	2.48905
$326$	0	0
$327$	18.7345	1.03602
$328$	0	0
$329$	−15.2780	−0.842303
$330$	0	0
$331$	−22.3361	−1.22771	−0.613853	−	0.789421i	$-0.710381\pi$
$331$	−22.3361	−1.22771	−0.613853	+	0.789421i	$0.710381\pi$
$332$	0	0
$333$	−1.16832	−0.0640234
$334$	0	0
$335$	−22.6737	−1.23879
$336$	0	0
$337$	20.3474	1.10839	0.554196	−	0.832386i	$-0.313026\pi$
$337$	20.3474	1.10839	0.554196	+	0.832386i	$0.313026\pi$
$338$	0	0
$339$	−14.6186	−0.793972
$340$	0	0
$341$	3.81616	0.206657
$342$	0	0
$343$	−17.7402	−0.957880
$344$	0	0
$345$	43.1832	2.32491
$346$	0	0
$347$	17.8187	0.956557	0.478279	−	0.878208i	$-0.341261\pi$
$347$	17.8187	0.956557	0.478279	+	0.878208i	$0.341261\pi$
$348$	0	0
$349$	5.12008	0.274072	0.137036	−	0.990566i	$-0.456242\pi$
$349$	5.12008	0.274072	0.137036	+	0.990566i	$0.456242\pi$
$350$	0	0
$351$	26.6369	1.42177
$352$	0	0
$353$	−18.4006	−0.979366	−0.489683	−	0.871901i	$-0.662887\pi$
$353$	−18.4006	−0.979366	−0.489683	+	0.871901i	$0.662887\pi$
$354$	0	0
$355$	−18.3277	−0.972734
$356$	0	0
$357$	6.89826	0.365094
$358$	0	0
$359$	10.1192	0.534073	0.267037	−	0.963686i	$-0.413956\pi$
$359$	10.1192	0.534073	0.267037	+	0.963686i	$0.413956\pi$
$360$	0	0
$361$	50.0597	2.63472
$362$	0	0
$363$	5.26286	0.276229
$364$	0	0
$365$	−60.1000	−3.14578
$366$	0	0
$367$	21.6090	1.12798	0.563990	−	0.825781i	$-0.309266\pi$
$367$	21.6090	1.12798	0.563990	+	0.825781i	$0.309266\pi$
$368$	0	0
$369$	−1.13317	−0.0589903
$370$	0	0
$371$	−17.9380	−0.931296
$372$	0	0
$373$	−1.98606	−0.102834	−0.0514172	−	0.998677i	$-0.516374\pi$
$373$	−1.98606	−0.102834	−0.0514172	+	0.998677i	$0.516374\pi$
$374$	0	0
$375$	−25.8679	−1.33581
$376$	0	0
$377$	35.8186	1.84475
$378$	0	0
$379$	−10.7470	−0.552034	−0.276017	−	0.961153i	$-0.589015\pi$
$379$	−10.7470	−0.552034	−0.276017	+	0.961153i	$0.589015\pi$
$380$	0	0
$381$	24.3257	1.24624
$382$	0	0
$383$	−5.32153	−0.271918	−0.135959	−	0.990714i	$-0.543411\pi$
$383$	−5.32153	−0.271918	−0.135959	+	0.990714i	$0.543411\pi$
$384$	0	0
$385$	−15.8881	−0.809730
$386$	0	0
$387$	−1.76574	−0.0897578
$388$	0	0
$389$	14.9437	0.757678	0.378839	−	0.925463i	$-0.376323\pi$
$389$	14.9437	0.757678	0.378839	+	0.925463i	$0.376323\pi$
$390$	0	0
$391$	20.8685	1.05536
$392$	0	0
$393$	−21.8053	−1.09993
$394$	0	0
$395$	−6.53242	−0.328682
$396$	0	0
$397$	15.9887	0.802451	0.401226	−	0.915979i	$-0.368584\pi$
$397$	15.9887	0.802451	0.401226	+	0.915979i	$0.368584\pi$
$398$	0	0
$399$	19.8687	0.994678
$400$	0	0
$401$	−7.07824	−0.353470	−0.176735	−	0.984258i	$-0.556554\pi$
$401$	−7.07824	−0.353470	−0.176735	+	0.984258i	$0.556554\pi$
$402$	0	0
$403$	−6.62830	−0.330179
$404$	0	0
$405$	−27.2510	−1.35411
$406$	0	0
$407$	−6.30518	−0.312536
$408$	0	0
$409$	−28.4022	−1.40440	−0.702199	−	0.711981i	$-0.747798\pi$
$409$	−28.4022	−1.40440	−0.702199	+	0.711981i	$0.747798\pi$
$410$	0	0
$411$	27.2898	1.34610
$412$	0	0
$413$	−9.83172	−0.483787
$414$	0	0
$415$	37.5499	1.84325
$416$	0	0
$417$	−1.98361	−0.0971379
$418$	0	0
$419$	−23.7692	−1.16120	−0.580601	−	0.814188i	$-0.697182\pi$
$419$	−23.7692	−1.16120	−0.580601	+	0.814188i	$0.697182\pi$
$420$	0	0
$421$	−6.22336	−0.303308	−0.151654	−	0.988434i	$-0.548460\pi$
$421$	−6.22336	−0.303308	−0.151654	+	0.988434i	$0.548460\pi$
$422$	0	0
$423$	5.16906	0.251328
$424$	0	0
$425$	−26.9270	−1.30615
$426$	0	0
$427$	−13.6789	−0.661968
$428$	0	0
$429$	20.9898	1.01340
$430$	0	0
$431$	27.9045	1.34411	0.672056	−	0.740500i	$-0.265411\pi$
$431$	27.9045	1.34411	0.672056	+	0.740500i	$0.265411\pi$
$432$	0	0
$433$	−6.86541	−0.329931	−0.164965	−	0.986299i	$-0.552751\pi$
$433$	−6.86541	−0.329931	−0.164965	+	0.986299i	$0.552751\pi$
$434$	0	0
$435$	−44.4781	−2.13256
$436$	0	0
$437$	60.1063	2.87528
$438$	0	0
$439$	−1.64942	−0.0787224	−0.0393612	−	0.999225i	$-0.512532\pi$
$439$	−1.64942	−0.0787224	−0.0393612	+	0.999225i	$0.512532\pi$
$440$	0	0
$441$	2.41159	0.114838
$442$	0	0
$443$	23.2318	1.10378	0.551888	−	0.833918i	$-0.313908\pi$
$443$	23.2318	1.10378	0.551888	+	0.833918i	$0.313908\pi$
$444$	0	0
$445$	−12.2979	−0.582976
$446$	0	0
$447$	−30.9231	−1.46261
$448$	0	0
$449$	−11.8110	−0.557395	−0.278697	−	0.960379i	$-0.589903\pi$
$449$	−11.8110	−0.557395	−0.278697	+	0.960379i	$0.589903\pi$
$450$	0	0
$451$	−6.11549	−0.287967
$452$	0	0
$453$	−2.98293	−0.140150
$454$	0	0
$455$	27.5960	1.29372
$456$	0	0
$457$	−7.28605	−0.340827	−0.170413	−	0.985373i	$-0.554510\pi$
$457$	−7.28605	−0.340827	−0.170413	+	0.985373i	$0.554510\pi$
$458$	0	0
$459$	−15.9844	−0.746087
$460$	0	0
$461$	−14.7215	−0.685649	−0.342824	−	0.939400i	$-0.611384\pi$
$461$	−14.7215	−0.685649	−0.342824	+	0.939400i	$0.611384\pi$
$462$	0	0
$463$	26.7308	1.24229	0.621143	−	0.783697i	$-0.286668\pi$
$463$	26.7308	1.24229	0.621143	+	0.783697i	$0.286668\pi$
$464$	0	0
$465$	8.23075	0.381692
$466$	0	0
$467$	8.01772	0.371016	0.185508	−	0.982643i	$-0.440607\pi$
$467$	8.01772	0.371016	0.185508	+	0.982643i	$0.440607\pi$
$468$	0	0
$469$	−9.07969	−0.419261
$470$	0	0
$471$	11.1948	0.515831
$472$	0	0
$473$	−9.52938	−0.438161
$474$	0	0
$475$	−77.5564	−3.55853
$476$	0	0
$477$	6.06904	0.277882
$478$	0	0
$479$	−40.7213	−1.86060	−0.930302	−	0.366795i	$-0.880455\pi$
$479$	−40.7213	−1.86060	−0.930302	+	0.366795i	$0.880455\pi$
$480$	0	0
$481$	10.9515	0.499345
$482$	0	0
$483$	17.2928	0.786848
$484$	0	0
$485$	39.9114	1.81228
$486$	0	0
$487$	−42.7036	−1.93508	−0.967542	−	0.252708i	$-0.918679\pi$
$487$	−42.7036	−1.93508	−0.967542	+	0.252708i	$0.918679\pi$
$488$	0	0
$489$	−8.26032	−0.373544
$490$	0	0
$491$	−9.20637	−0.415478	−0.207739	−	0.978184i	$-0.566610\pi$
$491$	−9.20637	−0.415478	−0.207739	+	0.978184i	$0.566610\pi$
$492$	0	0
$493$	−21.4942	−0.968051
$494$	0	0
$495$	5.37546	0.241609
$496$	0	0
$497$	−7.33935	−0.329215
$498$	0	0
$499$	−7.01677	−0.314114	−0.157057	−	0.987590i	$-0.550201\pi$
$499$	−7.01677	−0.314114	−0.157057	+	0.987590i	$0.550201\pi$
$500$	0	0
$501$	15.6791	0.700489
$502$	0	0
$503$	−12.8225	−0.571726	−0.285863	−	0.958271i	$-0.592280\pi$
$503$	−12.8225	−0.571726	−0.285863	+	0.958271i	$0.592280\pi$
$504$	0	0
$505$	43.2876	1.92627
$506$	0	0
$507$	−15.9556	−0.708614
$508$	0	0
$509$	44.1939	1.95886	0.979430	−	0.201783i	$-0.0646736\pi$
$509$	44.1939	1.95886	0.979430	+	0.201783i	$0.0646736\pi$
$510$	0	0
$511$	−24.0671	−1.06467
$512$	0	0
$513$	−46.0390	−2.03267
$514$	0	0
$515$	47.7768	2.10530
$516$	0	0
$517$	27.8964	1.22688
$518$	0	0
$519$	−12.6935	−0.557183
$520$	0	0
$521$	−14.8631	−0.651165	−0.325583	−	0.945514i	$-0.605560\pi$
$521$	−14.8631	−0.651165	−0.325583	+	0.945514i	$0.605560\pi$
$522$	0	0
$523$	−0.645786	−0.0282382	−0.0141191	−	0.999900i	$-0.504494\pi$
$523$	−0.645786	−0.0282382	−0.0141191	+	0.999900i	$0.504494\pi$
$524$	0	0
$525$	−22.3132	−0.973828
$526$	0	0
$527$	3.97754	0.173264
$528$	0	0
$529$	29.3137	1.27451
$530$	0	0
$531$	3.32640	0.144353
$532$	0	0
$533$	10.6220	0.460090
$534$	0	0
$535$	2.15265	0.0930672
$536$	0	0
$537$	−20.9262	−0.903033
$538$	0	0
$539$	13.0149	0.560591
$540$	0	0
$541$	29.1688	1.25407	0.627033	−	0.778992i	$-0.284269\pi$
$541$	29.1688	1.25407	0.627033	+	0.778992i	$0.284269\pi$
$542$	0	0
$543$	32.4102	1.39086
$544$	0	0
$545$	−44.9739	−1.92647
$546$	0	0
$547$	10.4569	0.447105	0.223552	−	0.974692i	$-0.428235\pi$
$547$	10.4569	0.447105	0.223552	+	0.974692i	$0.428235\pi$
$548$	0	0
$549$	4.62803	0.197520
$550$	0	0
$551$	−61.9086	−2.63740
$552$	0	0
$553$	−2.61591	−0.111240
$554$	0	0
$555$	−13.5991	−0.577250
$556$	0	0
$557$	25.2114	1.06824	0.534121	−	0.845408i	$-0.320643\pi$
$557$	25.2114	1.06824	0.534121	+	0.845408i	$0.320643\pi$
$558$	0	0
$559$	16.5516	0.700058
$560$	0	0
$561$	−12.5957	−0.531790
$562$	0	0
$563$	0.302101	0.0127320	0.00636602	−	0.999980i	$-0.497974\pi$
$563$	0.302101	0.0127320	0.00636602	+	0.999980i	$0.497974\pi$
$564$	0	0
$565$	35.0933	1.47639
$566$	0	0
$567$	−10.9127	−0.458289
$568$	0	0
$569$	20.4100	0.855631	0.427815	−	0.903866i	$-0.359283\pi$
$569$	20.4100	0.855631	0.427815	+	0.903866i	$0.359283\pi$
$570$	0	0
$571$	−30.1984	−1.26376	−0.631882	−	0.775065i	$-0.717717\pi$
$571$	−30.1984	−1.26376	−0.631882	+	0.775065i	$0.717717\pi$
$572$	0	0
$573$	39.7702	1.66143
$574$	0	0
$575$	−67.5015	−2.81501
$576$	0	0
$577$	−11.5788	−0.482031	−0.241016	−	0.970521i	$-0.577481\pi$
$577$	−11.5788	−0.482031	−0.241016	+	0.970521i	$0.577481\pi$
$578$	0	0
$579$	34.8609	1.44877
$580$	0	0
$581$	15.0369	0.623835
$582$	0	0
$583$	32.7534	1.35651
$584$	0	0
$585$	−9.33665	−0.386023
$586$	0	0
$587$	−29.6532	−1.22392	−0.611959	−	0.790889i	$-0.709618\pi$
$587$	−29.6532	−1.22392	−0.611959	+	0.790889i	$0.709618\pi$
$588$	0	0
$589$	11.4563	0.472049
$590$	0	0
$591$	−6.46396	−0.265892
$592$	0	0
$593$	21.9519	0.901457	0.450728	−	0.892661i	$-0.351164\pi$
$593$	21.9519	0.901457	0.450728	+	0.892661i	$0.351164\pi$
$594$	0	0
$595$	−16.5599	−0.678891
$596$	0	0
$597$	−39.5433	−1.61840
$598$	0	0
$599$	31.8377	1.30085	0.650427	−	0.759569i	$-0.274590\pi$
$599$	31.8377	1.30085	0.650427	+	0.759569i	$0.274590\pi$
$600$	0	0
$601$	−31.4382	−1.28239	−0.641196	−	0.767377i	$-0.721562\pi$
$601$	−31.4382	−1.28239	−0.641196	+	0.767377i	$0.721562\pi$
$602$	0	0
$603$	3.07196	0.125100
$604$	0	0
$605$	−12.6340	−0.513646
$606$	0	0
$607$	−23.9956	−0.973953	−0.486976	−	0.873415i	$-0.661900\pi$
$607$	−23.9956	−0.973953	−0.486976	+	0.873415i	$0.661900\pi$
$608$	0	0
$609$	−17.8113	−0.721750
$610$	0	0
$611$	−48.4533	−1.96021
$612$	0	0
$613$	−0.0456290	−0.00184294	−0.000921469	−	1.00000i	$-0.500293\pi$
$613$	−0.0456290	−0.00184294	−0.000921469		1.00000i	$0.500293\pi$
$614$	0	0
$615$	−13.1900	−0.531871
$616$	0	0
$617$	−1.92190	−0.0773728	−0.0386864	−	0.999251i	$-0.512317\pi$
$617$	−1.92190	−0.0773728	−0.0386864	+	0.999251i	$0.512317\pi$
$618$	0	0
$619$	0.0670238	0.00269391	0.00134696	−	0.999999i	$-0.499571\pi$
$619$	0.0670238	0.00269391	0.00134696	+	0.999999i	$0.499571\pi$
$620$	0	0
$621$	−40.0702	−1.60796
$622$	0	0
$623$	−4.92471	−0.197304
$624$	0	0
$625$	15.4352	0.617408
$626$	0	0
$627$	−36.2786	−1.44883
$628$	0	0
$629$	−6.57182	−0.262036
$630$	0	0
$631$	−8.32788	−0.331528	−0.165764	−	0.986165i	$-0.553009\pi$
$631$	−8.32788	−0.331528	−0.165764	+	0.986165i	$0.553009\pi$
$632$	0	0
$633$	3.30909	0.131525
$634$	0	0
$635$	−58.3962	−2.31738
$636$	0	0
$637$	−22.6056	−0.895665
$638$	0	0
$639$	2.48315	0.0982318
$640$	0	0
$641$	24.1119	0.952361	0.476181	−	0.879347i	$-0.342021\pi$
$641$	24.1119	0.952361	0.476181	+	0.879347i	$0.342021\pi$
$642$	0	0
$643$	4.51549	0.178074	0.0890368	−	0.996028i	$-0.471621\pi$
$643$	4.51549	0.178074	0.0890368	+	0.996028i	$0.471621\pi$
$644$	0	0
$645$	−20.5531	−0.809278
$646$	0	0
$647$	20.4825	0.805252	0.402626	−	0.915365i	$-0.368098\pi$
$647$	20.4825	0.805252	0.402626	+	0.915365i	$0.368098\pi$
$648$	0	0
$649$	17.9519	0.704675
$650$	0	0
$651$	3.29601	0.129181
$652$	0	0
$653$	−19.2806	−0.754506	−0.377253	−	0.926110i	$-0.623131\pi$
$653$	−19.2806	−0.754506	−0.377253	+	0.926110i	$0.623131\pi$
$654$	0	0
$655$	52.3457	2.04531
$656$	0	0
$657$	8.14271	0.317677
$658$	0	0
$659$	−27.4770	−1.07035	−0.535175	−	0.844741i	$-0.679754\pi$
$659$	−27.4770	−1.07035	−0.535175	+	0.844741i	$0.679754\pi$
$660$	0	0
$661$	−15.6154	−0.607369	−0.303684	−	0.952773i	$-0.598217\pi$
$661$	−15.6154	−0.607369	−0.303684	+	0.952773i	$0.598217\pi$
$662$	0	0
$663$	21.8774	0.849649
$664$	0	0
$665$	−47.6967	−1.84960
$666$	0	0
$667$	−53.8824	−2.08633
$668$	0	0
$669$	30.2405	1.16917
$670$	0	0
$671$	24.9766	0.964210
$672$	0	0
$673$	37.4618	1.44405	0.722023	−	0.691869i	$-0.243212\pi$
$673$	37.4618	1.44405	0.722023	+	0.691869i	$0.243212\pi$
$674$	0	0
$675$	51.7034	1.99006
$676$	0	0
$677$	14.4348	0.554776	0.277388	−	0.960758i	$-0.410531\pi$
$677$	14.4348	0.554776	0.277388	+	0.960758i	$0.410531\pi$
$678$	0	0
$679$	15.9825	0.613354
$680$	0	0
$681$	11.3550	0.435126
$682$	0	0
$683$	−36.1162	−1.38195	−0.690975	−	0.722879i	$-0.742818\pi$
$683$	−36.1162	−1.38195	−0.690975	+	0.722879i	$0.742818\pi$
$684$	0	0
$685$	−65.5117	−2.50307
$686$	0	0
$687$	−1.82957	−0.0698026
$688$	0	0
$689$	−56.8895	−2.16732
$690$	0	0
$691$	−17.0280	−0.647776	−0.323888	−	0.946095i	$-0.604990\pi$
$691$	−17.0280	−0.647776	−0.323888	+	0.946095i	$0.604990\pi$
$692$	0	0
$693$	2.15261	0.0817708
$694$	0	0
$695$	4.76185	0.180627
$696$	0	0
$697$	−6.37410	−0.241436
$698$	0	0
$699$	−44.3760	−1.67846
$700$	0	0
$701$	−48.7873	−1.84267	−0.921336	−	0.388766i	$-0.872901\pi$
$701$	−48.7873	−1.84267	−0.921336	+	0.388766i	$0.872901\pi$
$702$	0	0
$703$	−18.9285	−0.713901
$704$	0	0
$705$	60.1674	2.26604
$706$	0	0
$707$	17.3345	0.651933
$708$	0	0
$709$	5.38974	0.202416	0.101208	−	0.994865i	$-0.467729\pi$
$709$	5.38974	0.202416	0.101208	+	0.994865i	$0.467729\pi$
$710$	0	0
$711$	0.885052	0.0331920
$712$	0	0
$713$	9.97103	0.373418
$714$	0	0
$715$	−50.3881	−1.88441
$716$	0	0
$717$	−48.6771	−1.81788
$718$	0	0
$719$	−19.9439	−0.743781	−0.371890	−	0.928277i	$-0.621290\pi$
$719$	−19.9439	−0.743781	−0.371890	+	0.928277i	$0.621290\pi$
$720$	0	0
$721$	19.1322	0.712522
$722$	0	0
$723$	−14.8248	−0.551340
$724$	0	0
$725$	69.5255	2.58211
$726$	0	0
$727$	34.4816	1.27885	0.639427	−	0.768852i	$-0.279172\pi$
$727$	34.4816	1.27885	0.639427	+	0.768852i	$0.279172\pi$
$728$	0	0
$729$	29.9028	1.10751
$730$	0	0
$731$	−9.93236	−0.367362
$732$	0	0
$733$	−25.9131	−0.957124	−0.478562	−	0.878054i	$-0.658842\pi$
$733$	−25.9131	−0.957124	−0.478562	+	0.878054i	$0.658842\pi$
$734$	0	0
$735$	28.0707	1.03540
$736$	0	0
$737$	16.5788	0.610688
$738$	0	0
$739$	−22.4408	−0.825499	−0.412750	−	0.910844i	$-0.635432\pi$
$739$	−22.4408	−0.825499	−0.412750	+	0.910844i	$0.635432\pi$
$740$	0	0
$741$	63.0124	2.31482
$742$	0	0
$743$	−27.7674	−1.01869	−0.509344	−	0.860563i	$-0.670112\pi$
$743$	−27.7674	−1.01869	−0.509344	+	0.860563i	$0.670112\pi$
$744$	0	0
$745$	74.2340	2.71972
$746$	0	0
$747$	−5.08748	−0.186141
$748$	0	0
$749$	0.862031	0.0314979
$750$	0	0
$751$	−3.43312	−0.125276	−0.0626381	−	0.998036i	$-0.519951\pi$
$751$	−3.43312	−0.125276	−0.0626381	+	0.998036i	$0.519951\pi$
$752$	0	0
$753$	−10.0798	−0.367329
$754$	0	0
$755$	7.16081	0.260609
$756$	0	0
$757$	−1.47518	−0.0536165	−0.0268083	−	0.999641i	$-0.508534\pi$
$757$	−1.47518	−0.0536165	−0.0268083	+	0.999641i	$0.508534\pi$
$758$	0	0
$759$	−31.5752	−1.14611
$760$	0	0
$761$	4.56538	0.165495	0.0827474	−	0.996571i	$-0.473631\pi$
$761$	4.56538	0.165495	0.0827474	+	0.996571i	$0.473631\pi$
$762$	0	0
$763$	−18.0098	−0.652000
$764$	0	0
$765$	5.60278	0.202569
$766$	0	0
$767$	−31.1807	−1.12587
$768$	0	0
$769$	−21.0100	−0.757640	−0.378820	−	0.925470i	$-0.623670\pi$
$769$	−21.0100	−0.757640	−0.378820	+	0.925470i	$0.623670\pi$
$770$	0	0
$771$	22.6308	0.815029
$772$	0	0
$773$	−36.1671	−1.30084	−0.650419	−	0.759575i	$-0.725407\pi$
$773$	−36.1671	−1.30084	−0.650419	+	0.759575i	$0.725407\pi$
$774$	0	0
$775$	−12.8658	−0.462154
$776$	0	0
$777$	−5.44578	−0.195366
$778$	0	0
$779$	−18.3590	−0.657779
$780$	0	0
$781$	13.4011	0.479528
$782$	0	0
$783$	41.2717	1.47493
$784$	0	0
$785$	−26.8743	−0.959184
$786$	0	0
$787$	25.4570	0.907444	0.453722	−	0.891143i	$-0.350096\pi$
$787$	25.4570	0.907444	0.453722	+	0.891143i	$0.350096\pi$
$788$	0	0
$789$	6.93038	0.246728
$790$	0	0
$791$	14.0531	0.499672
$792$	0	0
$793$	−43.3819	−1.54053
$794$	0	0
$795$	70.6431	2.50545
$796$	0	0
$797$	−55.4682	−1.96478	−0.982391	−	0.186834i	$-0.940177\pi$
$797$	−55.4682	−1.96478	−0.982391	+	0.186834i	$0.940177\pi$
$798$	0	0
$799$	29.0761	1.02864
$800$	0	0
$801$	1.66619	0.0588720
$802$	0	0
$803$	43.9446	1.55077
$804$	0	0
$805$	−41.5130	−1.46314
$806$	0	0
$807$	−9.25148	−0.325668
$808$	0	0
$809$	50.0849	1.76089	0.880445	−	0.474148i	$-0.157244\pi$
$809$	50.0849	1.76089	0.880445	+	0.474148i	$0.157244\pi$
$810$	0	0
$811$	27.9536	0.981583	0.490791	−	0.871277i	$-0.336708\pi$
$811$	27.9536	0.981583	0.490791	+	0.871277i	$0.336708\pi$
$812$	0	0
$813$	−26.5363	−0.930668
$814$	0	0
$815$	19.8297	0.694604
$816$	0	0
$817$	−28.6077	−1.00086
$818$	0	0
$819$	−3.73887	−0.130647
$820$	0	0
$821$	14.1166	0.492674	0.246337	−	0.969184i	$-0.420773\pi$
$821$	14.1166	0.492674	0.246337	+	0.969184i	$0.420773\pi$
$822$	0	0
$823$	6.33452	0.220808	0.110404	−	0.993887i	$-0.464786\pi$
$823$	6.33452	0.220808	0.110404	+	0.993887i	$0.464786\pi$
$824$	0	0
$825$	40.7422	1.41846
$826$	0	0
$827$	23.5085	0.817469	0.408735	−	0.912653i	$-0.365970\pi$
$827$	23.5085	0.817469	0.408735	+	0.912653i	$0.365970\pi$
$828$	0	0
$829$	−6.62623	−0.230138	−0.115069	−	0.993357i	$-0.536709\pi$
$829$	−6.62623	−0.230138	−0.115069	+	0.993357i	$0.536709\pi$
$830$	0	0
$831$	13.1991	0.457870
$832$	0	0
$833$	13.5653	0.470009
$834$	0	0
$835$	−37.6391	−1.30256
$836$	0	0
$837$	−7.63740	−0.263987
$838$	0	0
$839$	27.3109	0.942878	0.471439	−	0.881899i	$-0.343735\pi$
$839$	27.3109	0.942878	0.471439	+	0.881899i	$0.343735\pi$
$840$	0	0
$841$	26.4981	0.913727
$842$	0	0
$843$	42.2359	1.45468
$844$	0	0
$845$	38.3030	1.31766
$846$	0	0
$847$	−5.05930	−0.173840
$848$	0	0
$849$	16.1384	0.553868
$850$	0	0
$851$	−16.4745	−0.564737
$852$	0	0
$853$	45.9827	1.57442	0.787209	−	0.616686i	$-0.211525\pi$
$853$	45.9827	1.57442	0.787209	+	0.616686i	$0.211525\pi$
$854$	0	0
$855$	16.1374	0.551887
$856$	0	0
$857$	0.623246	0.0212897	0.0106448	−	0.999943i	$-0.496612\pi$
$857$	0.623246	0.0212897	0.0106448	+	0.999943i	$0.496612\pi$
$858$	0	0
$859$	13.9961	0.477540	0.238770	−	0.971076i	$-0.423256\pi$
$859$	13.9961	0.477540	0.238770	+	0.971076i	$0.423256\pi$
$860$	0	0
$861$	−5.28193	−0.180008
$862$	0	0
$863$	42.1531	1.43491	0.717455	−	0.696605i	$-0.245307\pi$
$863$	42.1531	1.43491	0.717455	+	0.696605i	$0.245307\pi$
$864$	0	0
$865$	30.4720	1.03608
$866$	0	0
$867$	13.6814	0.464646
$868$	0	0
$869$	4.77645	0.162030
$870$	0	0
$871$	−28.7957	−0.975706
$872$	0	0
$873$	−5.40743	−0.183014
$874$	0	0
$875$	24.8674	0.840671
$876$	0	0
$877$	21.1705	0.714879	0.357439	−	0.933936i	$-0.383650\pi$
$877$	21.1705	0.714879	0.357439	+	0.933936i	$0.383650\pi$
$878$	0	0
$879$	24.7334	0.834238
$880$	0	0
$881$	0.961507	0.0323940	0.0161970	−	0.999869i	$-0.494844\pi$
$881$	0.961507	0.0323940	0.0161970	+	0.999869i	$0.494844\pi$
$882$	0	0
$883$	−23.2063	−0.780953	−0.390477	−	0.920613i	$-0.627690\pi$
$883$	−23.2063	−0.780953	−0.390477	+	0.920613i	$0.627690\pi$
$884$	0	0
$885$	38.7190	1.30153
$886$	0	0
$887$	−56.0808	−1.88301	−0.941504	−	0.337002i	$-0.890587\pi$
$887$	−56.0808	−1.88301	−0.941504	+	0.337002i	$0.890587\pi$
$888$	0	0
$889$	−23.3848	−0.784302
$890$	0	0
$891$	19.9257	0.667536
$892$	0	0
$893$	83.7464	2.80247
$894$	0	0
$895$	50.2354	1.67918
$896$	0	0
$897$	54.8430	1.83116
$898$	0	0
$899$	−10.2700	−0.342524
$900$	0	0
$901$	34.1385	1.13732
$902$	0	0
$903$	−8.23051	−0.273894
$904$	0	0
$905$	−77.8039	−2.58629
$906$	0	0
$907$	21.6648	0.719369	0.359684	−	0.933074i	$-0.382884\pi$
$907$	21.6648	0.719369	0.359684	+	0.933074i	$0.382884\pi$
$908$	0	0
$909$	−5.86486	−0.194525
$910$	0	0
$911$	−32.8995	−1.09001	−0.545005	−	0.838433i	$-0.683472\pi$
$911$	−32.8995	−1.09001	−0.545005	+	0.838433i	$0.683472\pi$
$912$	0	0
$913$	−27.4562	−0.908666
$914$	0	0
$915$	53.8699	1.78088
$916$	0	0
$917$	20.9619	0.692222
$918$	0	0
$919$	6.47140	0.213472	0.106736	−	0.994287i	$-0.465960\pi$
$919$	6.47140	0.213472	0.106736	+	0.994287i	$0.465960\pi$
$920$	0	0
$921$	−31.3157	−1.03189
$922$	0	0
$923$	−23.2763	−0.766150
$924$	0	0
$925$	21.2573	0.698936
$926$	0	0
$927$	−6.47308	−0.212604
$928$	0	0
$929$	2.78382	0.0913341	0.0456670	−	0.998957i	$-0.485459\pi$
$929$	2.78382	0.0913341	0.0456670	+	0.998957i	$0.485459\pi$
$930$	0	0
$931$	39.0713	1.28051
$932$	0	0
$933$	5.10400	0.167098
$934$	0	0
$935$	30.2371	0.988860
$936$	0	0
$937$	19.9679	0.652322	0.326161	−	0.945314i	$-0.394245\pi$
$937$	19.9679	0.652322	0.326161	+	0.945314i	$0.394245\pi$
$938$	0	0
$939$	36.1250	1.17889
$940$	0	0
$941$	56.2061	1.83227	0.916133	−	0.400874i	$-0.131294\pi$
$941$	56.2061	1.83227	0.916133	+	0.400874i	$0.131294\pi$
$942$	0	0
$943$	−15.9788	−0.520341
$944$	0	0
$945$	31.7972	1.03436
$946$	0	0
$947$	−4.22309	−0.137232	−0.0686160	−	0.997643i	$-0.521858\pi$
$947$	−4.22309	−0.137232	−0.0686160	+	0.997643i	$0.521858\pi$
$948$	0	0
$949$	−76.3274	−2.47769
$950$	0	0
$951$	36.9291	1.19751
$952$	0	0
$953$	30.1774	0.977541	0.488770	−	0.872412i	$-0.337446\pi$
$953$	30.1774	0.977541	0.488770	+	0.872412i	$0.337446\pi$
$954$	0	0
$955$	−95.4723	−3.08941
$956$	0	0
$957$	32.5220	1.05129
$958$	0	0
$959$	−26.2342	−0.847147
$960$	0	0
$961$	−29.0995	−0.938694
$962$	0	0
$963$	−0.291654	−0.00939842
$964$	0	0
$965$	−83.6869	−2.69398
$966$	0	0
$967$	10.0212	0.322260	0.161130	−	0.986933i	$-0.448486\pi$
$967$	10.0212	0.322260	0.161130	+	0.986933i	$0.448486\pi$
$968$	0	0
$969$	−37.8128	−1.21472
$970$	0	0
$971$	38.7895	1.24481	0.622406	−	0.782694i	$-0.286155\pi$
$971$	38.7895	1.24481	0.622406	+	0.782694i	$0.286155\pi$
$972$	0	0
$973$	1.90689	0.0611320
$974$	0	0
$975$	−70.7651	−2.26630
$976$	0	0
$977$	55.4004	1.77242	0.886209	−	0.463287i	$-0.153330\pi$
$977$	55.4004	1.77242	0.886209	+	0.463287i	$0.153330\pi$
$978$	0	0
$979$	8.99212	0.287390
$980$	0	0
$981$	6.09333	0.194545
$982$	0	0
$983$	52.0063	1.65874	0.829372	−	0.558696i	$-0.188698\pi$
$983$	52.0063	1.65874	0.829372	+	0.558696i	$0.188698\pi$
$984$	0	0
$985$	15.5174	0.494424
$986$	0	0
$987$	24.0941	0.766923
$988$	0	0
$989$	−24.8988	−0.791735
$990$	0	0
$991$	−9.92605	−0.315312	−0.157656	−	0.987494i	$-0.550394\pi$
$991$	−9.92605	−0.315312	−0.157656	+	0.987494i	$0.550394\pi$
$992$	0	0
$993$	35.2251	1.11783
$994$	0	0
$995$	94.9274	3.00940
$996$	0	0
$997$	−57.8563	−1.83233	−0.916164	−	0.400804i	$-0.868731\pi$
$997$	−57.8563	−1.83233	−0.916164	+	0.400804i	$0.868731\pi$
$998$	0	0
$999$	12.6188	0.399240

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.20	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.20	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-1.57704 q^{3} +3.78585 q^{5} +1.51605 q^{7} -0.512930 q^{9} +O(q^{10})\)	\(q-1.57704 q^{3} +3.78585 q^{5} +1.51605 q^{7} -0.512930 q^{9} -2.76818 q^{11} +4.80806 q^{13} -5.97045 q^{15} -2.88524 q^{17} -8.31022 q^{19} -2.39087 q^{21} -7.23282 q^{23} +9.33266 q^{25} +5.54005 q^{27} +7.44970 q^{29} -1.37858 q^{31} +4.36555 q^{33} +5.73953 q^{35} +2.27773 q^{37} -7.58252 q^{39} +2.20921 q^{41} +3.44247 q^{43} -1.94187 q^{45} -10.0775 q^{47} -4.70160 q^{49} +4.55016 q^{51} -11.8321 q^{53} -10.4799 q^{55} +13.1056 q^{57} -6.48510 q^{59} -9.02274 q^{61} -0.777625 q^{63} +18.2026 q^{65} -5.98905 q^{67} +11.4065 q^{69} -4.84111 q^{71} -15.8749 q^{73} -14.7180 q^{75} -4.19669 q^{77} -1.72548 q^{79} -7.19811 q^{81} +9.91848 q^{83} -10.9231 q^{85} -11.7485 q^{87} -3.24839 q^{89} +7.28924 q^{91} +2.17408 q^{93} -31.4612 q^{95} +10.5422 q^{97} +1.41988 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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