LMFDB - Embedded newform 8024.2.a.x.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.27679 q^{3} +2.13098 q^{5} +3.12134 q^{7} -1.36980 q^{9} +O(q^{10})$	$q+1.27679 q^{3} +2.13098 q^{5} +3.12134 q^{7} -1.36980 q^{9} -5.49989 q^{11} -1.81845 q^{13} +2.72082 q^{15} -1.00000 q^{17} +4.02030 q^{19} +3.98531 q^{21} -3.32139 q^{23} -0.458932 q^{25} -5.57933 q^{27} -3.90925 q^{29} -7.32580 q^{31} -7.02223 q^{33} +6.65151 q^{35} -1.91312 q^{37} -2.32179 q^{39} -3.34599 q^{41} -7.15978 q^{43} -2.91901 q^{45} +6.32070 q^{47} +2.74276 q^{49} -1.27679 q^{51} +3.32080 q^{53} -11.7201 q^{55} +5.13309 q^{57} +1.00000 q^{59} +6.06244 q^{61} -4.27560 q^{63} -3.87508 q^{65} +3.47840 q^{67} -4.24073 q^{69} -0.456612 q^{71} -7.62502 q^{73} -0.585961 q^{75} -17.1670 q^{77} -2.26166 q^{79} -3.01427 q^{81} -8.46810 q^{83} -2.13098 q^{85} -4.99131 q^{87} -3.75693 q^{89} -5.67601 q^{91} -9.35354 q^{93} +8.56717 q^{95} -4.48443 q^{97} +7.53373 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * 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
$3$	1.27679	0.737158	0.368579	−	0.929597i	$-0.379844\pi$
$3$	1.27679	0.737158	0.368579	+	0.929597i	$0.379844\pi$
$4$	0	0
$5$	2.13098	0.953002	0.476501	−	0.879174i	$-0.341905\pi$
$5$	2.13098	0.953002	0.476501	+	0.879174i	$0.341905\pi$
$6$	0	0
$7$	3.12134	1.17976	0.589878	−	0.807493i	$-0.299176\pi$
$7$	3.12134	1.17976	0.589878	+	0.807493i	$0.299176\pi$
$8$	0	0
$9$	−1.36980	−0.456599
$10$	0	0
$11$	−5.49989	−1.65828	−0.829140	−	0.559041i	$-0.811169\pi$
$11$	−5.49989	−1.65828	−0.829140	+	0.559041i	$0.811169\pi$
$12$	0	0
$13$	−1.81845	−0.504348	−0.252174	−	0.967682i	$-0.581146\pi$
$13$	−1.81845	−0.504348	−0.252174	+	0.967682i	$0.581146\pi$
$14$	0	0
$15$	2.72082	0.702513
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	4.02030	0.922320	0.461160	−	0.887317i	$-0.347433\pi$
$19$	4.02030	0.922320	0.461160	+	0.887317i	$0.347433\pi$
$20$	0	0
$21$	3.98531	0.869666
$22$	0	0
$23$	−3.32139	−0.692558	−0.346279	−	0.938132i	$-0.612555\pi$
$23$	−3.32139	−0.692558	−0.346279	+	0.938132i	$0.612555\pi$
$24$	0	0
$25$	−0.458932	−0.0917863
$26$	0	0
$27$	−5.57933	−1.07374
$28$	0	0
$29$	−3.90925	−0.725930	−0.362965	−	0.931803i	$-0.618236\pi$
$29$	−3.90925	−0.725930	−0.362965	+	0.931803i	$0.618236\pi$
$30$	0	0
$31$	−7.32580	−1.31575	−0.657876	−	0.753126i	$-0.728545\pi$
$31$	−7.32580	−1.31575	−0.657876	+	0.753126i	$0.728545\pi$
$32$	0	0
$33$	−7.02223	−1.22241
$34$	0	0
$35$	6.65151	1.12431
$36$	0	0
$37$	−1.91312	−0.314514	−0.157257	−	0.987558i	$-0.550265\pi$
$37$	−1.91312	−0.314514	−0.157257	+	0.987558i	$0.550265\pi$
$38$	0	0
$39$	−2.32179	−0.371784
$40$	0	0
$41$	−3.34599	−0.522556	−0.261278	−	0.965264i	$-0.584144\pi$
$41$	−3.34599	−0.522556	−0.261278	+	0.965264i	$0.584144\pi$
$42$	0	0
$43$	−7.15978	−1.09186	−0.545928	−	0.837832i	$-0.683823\pi$
$43$	−7.15978	−1.09186	−0.545928	+	0.837832i	$0.683823\pi$
$44$	0	0
$45$	−2.91901	−0.435140
$46$	0	0
$47$	6.32070	0.921969	0.460985	−	0.887408i	$-0.347496\pi$
$47$	6.32070	0.921969	0.460985	+	0.887408i	$0.347496\pi$
$48$	0	0
$49$	2.74276	0.391823
$50$	0	0
$51$	−1.27679	−0.178787
$52$	0	0
$53$	3.32080	0.456147	0.228073	−	0.973644i	$-0.426757\pi$
$53$	3.32080	0.456147	0.228073	+	0.973644i	$0.426757\pi$
$54$	0	0
$55$	−11.7201	−1.58034
$56$	0	0
$57$	5.13309	0.679895
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	6.06244	0.776216	0.388108	−	0.921614i	$-0.373129\pi$
$61$	6.06244	0.776216	0.388108	+	0.921614i	$0.373129\pi$
$62$	0	0
$63$	−4.27560	−0.538675
$64$	0	0
$65$	−3.87508	−0.480645
$66$	0	0
$67$	3.47840	0.424954	0.212477	−	0.977166i	$-0.431847\pi$
$67$	3.47840	0.424954	0.212477	+	0.977166i	$0.431847\pi$
$68$	0	0
$69$	−4.24073	−0.510524
$70$	0	0
$71$	−0.456612	−0.0541899	−0.0270950	−	0.999633i	$-0.508626\pi$
$71$	−0.456612	−0.0541899	−0.0270950	+	0.999633i	$0.508626\pi$
$72$	0	0
$73$	−7.62502	−0.892441	−0.446220	−	0.894923i	$-0.647230\pi$
$73$	−7.62502	−0.892441	−0.446220	+	0.894923i	$0.647230\pi$
$74$	0	0
$75$	−0.585961	−0.0676610
$76$	0	0
$77$	−17.1670	−1.95636
$78$	0	0
$79$	−2.26166	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$79$	−2.26166	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$80$	0	0
$81$	−3.01427	−0.334919
$82$	0	0
$83$	−8.46810	−0.929494	−0.464747	−	0.885443i	$-0.653855\pi$
$83$	−8.46810	−0.929494	−0.464747	+	0.885443i	$0.653855\pi$
$84$	0	0
$85$	−2.13098	−0.231137
$86$	0	0
$87$	−4.99131	−0.535125
$88$	0	0
$89$	−3.75693	−0.398233	−0.199117	−	0.979976i	$-0.563807\pi$
$89$	−3.75693	−0.398233	−0.199117	+	0.979976i	$0.563807\pi$
$90$	0	0
$91$	−5.67601	−0.595008
$92$	0	0
$93$	−9.35354	−0.969917
$94$	0	0
$95$	8.56717	0.878973
$96$	0	0
$97$	−4.48443	−0.455325	−0.227663	−	0.973740i	$-0.573108\pi$
$97$	−4.48443	−0.455325	−0.227663	+	0.973740i	$0.573108\pi$
$98$	0	0
$99$	7.53373	0.757169
$100$	0	0
$101$	−4.60208	−0.457924	−0.228962	−	0.973435i	$-0.573533\pi$
$101$	−4.60208	−0.457924	−0.228962	+	0.973435i	$0.573533\pi$
$102$	0	0
$103$	0.0280627	0.00276510	0.00138255	−	0.999999i	$-0.499560\pi$
$103$	0.0280627	0.00276510	0.00138255	+	0.999999i	$0.499560\pi$
$104$	0	0
$105$	8.49261	0.828794
$106$	0	0
$107$	−9.80103	−0.947501	−0.473751	−	0.880659i	$-0.657100\pi$
$107$	−9.80103	−0.947501	−0.473751	+	0.880659i	$0.657100\pi$
$108$	0	0
$109$	−8.63098	−0.826698	−0.413349	−	0.910573i	$-0.635641\pi$
$109$	−8.63098	−0.826698	−0.413349	+	0.910573i	$0.635641\pi$
$110$	0	0
$111$	−2.44266	−0.231847
$112$	0	0
$113$	15.0900	1.41955	0.709776	−	0.704428i	$-0.248796\pi$
$113$	15.0900	1.41955	0.709776	+	0.704428i	$0.248796\pi$
$114$	0	0
$115$	−7.07781	−0.660009
$116$	0	0
$117$	2.49091	0.230285
$118$	0	0
$119$	−3.12134	−0.286133
$120$	0	0
$121$	19.2488	1.74989
$122$	0	0
$123$	−4.27214	−0.385206
$124$	0	0
$125$	−11.6329	−1.04048
$126$	0	0
$127$	−4.67935	−0.415225	−0.207612	−	0.978211i	$-0.566569\pi$
$127$	−4.67935	−0.415225	−0.207612	+	0.978211i	$0.566569\pi$
$128$	0	0
$129$	−9.14156	−0.804870
$130$	0	0
$131$	−2.62999	−0.229783	−0.114892	−	0.993378i	$-0.536652\pi$
$131$	−2.62999	−0.229783	−0.114892	+	0.993378i	$0.536652\pi$
$132$	0	0
$133$	12.5487	1.08811
$134$	0	0
$135$	−11.8894	−1.02328
$136$	0	0
$137$	20.2189	1.72742	0.863708	−	0.503993i	$-0.168136\pi$
$137$	20.2189	1.72742	0.863708	+	0.503993i	$0.168136\pi$
$138$	0	0
$139$	11.9717	1.01542	0.507712	−	0.861527i	$-0.330491\pi$
$139$	11.9717	1.01542	0.507712	+	0.861527i	$0.330491\pi$
$140$	0	0
$141$	8.07024	0.679636
$142$	0	0
$143$	10.0013	0.836350
$144$	0	0
$145$	−8.33054	−0.691813
$146$	0	0
$147$	3.50195	0.288836
$148$	0	0
$149$	7.74134	0.634195	0.317098	−	0.948393i	$-0.397292\pi$
$149$	7.74134	0.634195	0.317098	+	0.948393i	$0.397292\pi$
$150$	0	0
$151$	−9.91952	−0.807239	−0.403620	−	0.914927i	$-0.632248\pi$
$151$	−9.91952	−0.807239	−0.403620	+	0.914927i	$0.632248\pi$
$152$	0	0
$153$	1.36980	0.110741
$154$	0	0
$155$	−15.6111	−1.25392
$156$	0	0
$157$	−22.9532	−1.83186	−0.915932	−	0.401334i	$-0.868547\pi$
$157$	−22.9532	−1.83186	−0.915932	+	0.401334i	$0.868547\pi$
$158$	0	0
$159$	4.23998	0.336252
$160$	0	0
$161$	−10.3672	−0.817049
$162$	0	0
$163$	6.98008	0.546722	0.273361	−	0.961911i	$-0.411865\pi$
$163$	6.98008	0.546722	0.273361	+	0.961911i	$0.411865\pi$
$164$	0	0
$165$	−14.9642	−1.16496
$166$	0	0
$167$	17.9950	1.39249	0.696247	−	0.717802i	$-0.254852\pi$
$167$	17.9950	1.39249	0.696247	+	0.717802i	$0.254852\pi$
$168$	0	0
$169$	−9.69323	−0.745633
$170$	0	0
$171$	−5.50699	−0.421130
$172$	0	0
$173$	−13.2207	−1.00515	−0.502577	−	0.864533i	$-0.667615\pi$
$173$	−13.2207	−1.00515	−0.502577	+	0.864533i	$0.667615\pi$
$174$	0	0
$175$	−1.43248	−0.108285
$176$	0	0
$177$	1.27679	0.0959697
$178$	0	0
$179$	21.5498	1.61071	0.805356	−	0.592792i	$-0.201974\pi$
$179$	21.5498	1.61071	0.805356	+	0.592792i	$0.201974\pi$
$180$	0	0
$181$	−18.5979	−1.38237	−0.691184	−	0.722679i	$-0.742911\pi$
$181$	−18.5979	−1.38237	−0.691184	+	0.722679i	$0.742911\pi$
$182$	0	0
$183$	7.74049	0.572194
$184$	0	0
$185$	−4.07681	−0.299733
$186$	0	0
$187$	5.49989	0.402192
$188$	0	0
$189$	−17.4150	−1.26675
$190$	0	0
$191$	−14.0736	−1.01833	−0.509166	−	0.860668i	$-0.670046\pi$
$191$	−14.0736	−1.01833	−0.509166	+	0.860668i	$0.670046\pi$
$192$	0	0
$193$	2.86803	0.206446	0.103223	−	0.994658i	$-0.467085\pi$
$193$	2.86803	0.206446	0.103223	+	0.994658i	$0.467085\pi$
$194$	0	0
$195$	−4.94769	−0.354311
$196$	0	0
$197$	18.9892	1.35293	0.676464	−	0.736476i	$-0.263512\pi$
$197$	18.9892	1.35293	0.676464	+	0.736476i	$0.263512\pi$
$198$	0	0
$199$	−1.18261	−0.0838332	−0.0419166	−	0.999121i	$-0.513346\pi$
$199$	−1.18261	−0.0838332	−0.0419166	+	0.999121i	$0.513346\pi$
$200$	0	0
$201$	4.44120	0.313258
$202$	0	0
$203$	−12.2021	−0.856420
$204$	0	0
$205$	−7.13023	−0.497997
$206$	0	0
$207$	4.54963	0.316221
$208$	0	0
$209$	−22.1112	−1.52946
$210$	0	0
$211$	−15.2591	−1.05048	−0.525240	−	0.850954i	$-0.676025\pi$
$211$	−15.2591	−1.05048	−0.525240	+	0.850954i	$0.676025\pi$
$212$	0	0
$213$	−0.583000	−0.0399465
$214$	0	0
$215$	−15.2573	−1.04054
$216$	0	0
$217$	−22.8663	−1.55227
$218$	0	0
$219$	−9.73558	−0.657869
$220$	0	0
$221$	1.81845	0.122322
$222$	0	0
$223$	−13.6233	−0.912286	−0.456143	−	0.889906i	$-0.650770\pi$
$223$	−13.6233	−0.912286	−0.456143	+	0.889906i	$0.650770\pi$
$224$	0	0
$225$	0.628643	0.0419095
$226$	0	0
$227$	24.6934	1.63896	0.819481	−	0.573107i	$-0.194262\pi$
$227$	24.6934	1.63896	0.819481	+	0.573107i	$0.194262\pi$
$228$	0	0
$229$	2.15629	0.142492	0.0712459	−	0.997459i	$-0.477302\pi$
$229$	2.15629	0.142492	0.0712459	+	0.997459i	$0.477302\pi$
$230$	0	0
$231$	−21.9188	−1.44215
$232$	0	0
$233$	−11.5363	−0.755766	−0.377883	−	0.925853i	$-0.623348\pi$
$233$	−11.5363	−0.755766	−0.377883	+	0.925853i	$0.623348\pi$
$234$	0	0
$235$	13.4693	0.878639
$236$	0	0
$237$	−2.88767	−0.187575
$238$	0	0
$239$	−18.0236	−1.16585	−0.582926	−	0.812525i	$-0.698092\pi$
$239$	−18.0236	−1.16585	−0.582926	+	0.812525i	$0.698092\pi$
$240$	0	0
$241$	−3.91143	−0.251957	−0.125979	−	0.992033i	$-0.540207\pi$
$241$	−3.91143	−0.251957	−0.125979	+	0.992033i	$0.540207\pi$
$242$	0	0
$243$	12.8894	0.826855
$244$	0	0
$245$	5.84477	0.373409
$246$	0	0
$247$	−7.31073	−0.465170
$248$	0	0
$249$	−10.8120	−0.685184
$250$	0	0
$251$	13.8047	0.871344	0.435672	−	0.900105i	$-0.356511\pi$
$251$	13.8047	0.871344	0.435672	+	0.900105i	$0.356511\pi$
$252$	0	0
$253$	18.2673	1.14845
$254$	0	0
$255$	−2.72082	−0.170384
$256$	0	0
$257$	0.965035	0.0601972	0.0300986	−	0.999547i	$-0.490418\pi$
$257$	0.965035	0.0601972	0.0300986	+	0.999547i	$0.490418\pi$
$258$	0	0
$259$	−5.97149	−0.371050
$260$	0	0
$261$	5.35488	0.331459
$262$	0	0
$263$	−0.933549	−0.0575651	−0.0287825	−	0.999586i	$-0.509163\pi$
$263$	−0.933549	−0.0575651	−0.0287825	+	0.999586i	$0.509163\pi$
$264$	0	0
$265$	7.07655	0.434709
$266$	0	0
$267$	−4.79682	−0.293561
$268$	0	0
$269$	9.47611	0.577768	0.288884	−	0.957364i	$-0.406716\pi$
$269$	9.47611	0.577768	0.288884	+	0.957364i	$0.406716\pi$
$270$	0	0
$271$	−20.0357	−1.21708	−0.608542	−	0.793522i	$-0.708245\pi$
$271$	−20.0357	−1.21708	−0.608542	+	0.793522i	$0.708245\pi$
$272$	0	0
$273$	−7.24710	−0.438614
$274$	0	0
$275$	2.52407	0.152207
$276$	0	0
$277$	25.8177	1.55123	0.775617	−	0.631204i	$-0.217439\pi$
$277$	25.8177	1.55123	0.775617	+	0.631204i	$0.217439\pi$
$278$	0	0
$279$	10.0349	0.600771
$280$	0	0
$281$	2.54610	0.151888	0.0759438	−	0.997112i	$-0.475803\pi$
$281$	2.54610	0.151888	0.0759438	+	0.997112i	$0.475803\pi$
$282$	0	0
$283$	11.7313	0.697354	0.348677	−	0.937243i	$-0.386631\pi$
$283$	11.7313	0.697354	0.348677	+	0.937243i	$0.386631\pi$
$284$	0	0
$285$	10.9385	0.647942
$286$	0	0
$287$	−10.4440	−0.616488
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−5.72570	−0.335646
$292$	0	0
$293$	1.59567	0.0932198	0.0466099	−	0.998913i	$-0.485158\pi$
$293$	1.59567	0.0932198	0.0466099	+	0.998913i	$0.485158\pi$
$294$	0	0
$295$	2.13098	0.124070
$296$	0	0
$297$	30.6857	1.78057
$298$	0	0
$299$	6.03979	0.349290
$300$	0	0
$301$	−22.3481	−1.28812
$302$	0	0
$303$	−5.87591	−0.337562
$304$	0	0
$305$	12.9189	0.739736
$306$	0	0
$307$	−5.92763	−0.338308	−0.169154	−	0.985590i	$-0.554103\pi$
$307$	−5.92763	−0.338308	−0.169154	+	0.985590i	$0.554103\pi$
$308$	0	0
$309$	0.0358302	0.00203831
$310$	0	0
$311$	8.19098	0.464468	0.232234	−	0.972660i	$-0.425397\pi$
$311$	8.19098	0.464468	0.232234	+	0.972660i	$0.425397\pi$
$312$	0	0
$313$	−12.5302	−0.708248	−0.354124	−	0.935198i	$-0.615221\pi$
$313$	−12.5302	−0.708248	−0.354124	+	0.935198i	$0.615221\pi$
$314$	0	0
$315$	−9.11121	−0.513359
$316$	0	0
$317$	−18.3358	−1.02984	−0.514922	−	0.857237i	$-0.672179\pi$
$317$	−18.3358	−1.02984	−0.514922	+	0.857237i	$0.672179\pi$
$318$	0	0
$319$	21.5005	1.20380
$320$	0	0
$321$	−12.5139	−0.698458
$322$	0	0
$323$	−4.02030	−0.223695
$324$	0	0
$325$	0.834546	0.0462923
$326$	0	0
$327$	−11.0200	−0.609406
$328$	0	0
$329$	19.7291	1.08770
$330$	0	0
$331$	2.73953	0.150578	0.0752892	−	0.997162i	$-0.476012\pi$
$331$	2.73953	0.150578	0.0752892	+	0.997162i	$0.476012\pi$
$332$	0	0
$333$	2.62058	0.143607
$334$	0	0
$335$	7.41240	0.404982
$336$	0	0
$337$	15.6041	0.850009	0.425005	−	0.905191i	$-0.360272\pi$
$337$	15.6041	0.850009	0.425005	+	0.905191i	$0.360272\pi$
$338$	0	0
$339$	19.2669	1.04643
$340$	0	0
$341$	40.2911	2.18189
$342$	0	0
$343$	−13.2883	−0.717500
$344$	0	0
$345$	−9.03691	−0.486531
$346$	0	0
$347$	30.4236	1.63323	0.816613	−	0.577185i	$-0.195849\pi$
$347$	30.4236	1.63323	0.816613	+	0.577185i	$0.195849\pi$
$348$	0	0
$349$	1.61925	0.0866765	0.0433382	−	0.999060i	$-0.486201\pi$
$349$	1.61925	0.0866765	0.0433382	+	0.999060i	$0.486201\pi$
$350$	0	0
$351$	10.1458	0.541540
$352$	0	0
$353$	−11.9980	−0.638590	−0.319295	−	0.947655i	$-0.603446\pi$
$353$	−11.9980	−0.638590	−0.319295	+	0.947655i	$0.603446\pi$
$354$	0	0
$355$	−0.973031	−0.0516431
$356$	0	0
$357$	−3.98531	−0.210925
$358$	0	0
$359$	21.4346	1.13127	0.565636	−	0.824655i	$-0.308631\pi$
$359$	21.4346	1.13127	0.565636	+	0.824655i	$0.308631\pi$
$360$	0	0
$361$	−2.83719	−0.149326
$362$	0	0
$363$	24.5768	1.28995
$364$	0	0
$365$	−16.2487	−0.850498
$366$	0	0
$367$	21.7985	1.13787	0.568935	−	0.822383i	$-0.307356\pi$
$367$	21.7985	1.13787	0.568935	+	0.822383i	$0.307356\pi$
$368$	0	0
$369$	4.58333	0.238598
$370$	0	0
$371$	10.3653	0.538142
$372$	0	0
$373$	−33.9262	−1.75663	−0.878317	−	0.478080i	$-0.841333\pi$
$373$	−33.9262	−1.75663	−0.878317	+	0.478080i	$0.841333\pi$
$374$	0	0
$375$	−14.8528	−0.766994
$376$	0	0
$377$	7.10880	0.366122
$378$	0	0
$379$	13.5769	0.697399	0.348700	−	0.937235i	$-0.386623\pi$
$379$	13.5769	0.697399	0.348700	+	0.937235i	$0.386623\pi$
$380$	0	0
$381$	−5.97456	−0.306086
$382$	0	0
$383$	30.6923	1.56830	0.784152	−	0.620569i	$-0.213098\pi$
$383$	30.6923	1.56830	0.784152	+	0.620569i	$0.213098\pi$
$384$	0	0
$385$	−36.5826	−1.86442
$386$	0	0
$387$	9.80744	0.498540
$388$	0	0
$389$	26.9283	1.36532	0.682659	−	0.730737i	$-0.260824\pi$
$389$	26.9283	1.36532	0.682659	+	0.730737i	$0.260824\pi$
$390$	0	0
$391$	3.32139	0.167970
$392$	0	0
$393$	−3.35796	−0.169386
$394$	0	0
$395$	−4.81955	−0.242498
$396$	0	0
$397$	−22.9160	−1.15012	−0.575062	−	0.818110i	$-0.695022\pi$
$397$	−22.9160	−1.15012	−0.575062	+	0.818110i	$0.695022\pi$
$398$	0	0
$399$	16.0221	0.802110
$400$	0	0
$401$	25.5970	1.27825	0.639126	−	0.769102i	$-0.279296\pi$
$401$	25.5970	1.27825	0.639126	+	0.769102i	$0.279296\pi$
$402$	0	0
$403$	13.3216	0.663598
$404$	0	0
$405$	−6.42334	−0.319178
$406$	0	0
$407$	10.5219	0.521553
$408$	0	0
$409$	−20.9730	−1.03705	−0.518524	−	0.855063i	$-0.673518\pi$
$409$	−20.9730	−1.03705	−0.518524	+	0.855063i	$0.673518\pi$
$410$	0	0
$411$	25.8154	1.27338
$412$	0	0
$413$	3.12134	0.153591
$414$	0	0
$415$	−18.0453	−0.885810
$416$	0	0
$417$	15.2854	0.748528
$418$	0	0
$419$	−25.3591	−1.23887	−0.619437	−	0.785047i	$-0.712639\pi$
$419$	−25.3591	−1.23887	−0.619437	+	0.785047i	$0.712639\pi$
$420$	0	0
$421$	−14.3714	−0.700421	−0.350210	−	0.936671i	$-0.613890\pi$
$421$	−14.3714	−0.700421	−0.350210	+	0.936671i	$0.613890\pi$
$422$	0	0
$423$	−8.65808	−0.420970
$424$	0	0
$425$	0.458932	0.0222615
$426$	0	0
$427$	18.9229	0.915746
$428$	0	0
$429$	12.7696	0.616522
$430$	0	0
$431$	3.55825	0.171395	0.0856974	−	0.996321i	$-0.472688\pi$
$431$	3.55825	0.171395	0.0856974	+	0.996321i	$0.472688\pi$
$432$	0	0
$433$	−11.6428	−0.559518	−0.279759	−	0.960070i	$-0.590255\pi$
$433$	−11.6428	−0.559518	−0.279759	+	0.960070i	$0.590255\pi$
$434$	0	0
$435$	−10.6364	−0.509975
$436$	0	0
$437$	−13.3530	−0.638760
$438$	0	0
$439$	8.65982	0.413311	0.206655	−	0.978414i	$-0.433742\pi$
$439$	8.65982	0.413311	0.206655	+	0.978414i	$0.433742\pi$
$440$	0	0
$441$	−3.75703	−0.178906
$442$	0	0
$443$	−30.9818	−1.47199	−0.735995	−	0.676987i	$-0.763285\pi$
$443$	−30.9818	−1.47199	−0.735995	+	0.676987i	$0.763285\pi$
$444$	0	0
$445$	−8.00593	−0.379517
$446$	0	0
$447$	9.88410	0.467502
$448$	0	0
$449$	7.60926	0.359103	0.179552	−	0.983749i	$-0.442535\pi$
$449$	7.60926	0.359103	0.179552	+	0.983749i	$0.442535\pi$
$450$	0	0
$451$	18.4026	0.866544
$452$	0	0
$453$	−12.6652	−0.595063
$454$	0	0
$455$	−12.0955	−0.567044
$456$	0	0
$457$	5.63394	0.263544	0.131772	−	0.991280i	$-0.457933\pi$
$457$	5.63394	0.263544	0.131772	+	0.991280i	$0.457933\pi$
$458$	0	0
$459$	5.57933	0.260421
$460$	0	0
$461$	33.3111	1.55145	0.775726	−	0.631070i	$-0.217384\pi$
$461$	33.3111	1.55145	0.775726	+	0.631070i	$0.217384\pi$
$462$	0	0
$463$	15.7275	0.730918	0.365459	−	0.930827i	$-0.380912\pi$
$463$	15.7275	0.730918	0.365459	+	0.930827i	$0.380912\pi$
$464$	0	0
$465$	−19.9322	−0.924333
$466$	0	0
$467$	−12.7969	−0.592170	−0.296085	−	0.955162i	$-0.595681\pi$
$467$	−12.7969	−0.592170	−0.296085	+	0.955162i	$0.595681\pi$
$468$	0	0
$469$	10.8573	0.501342
$470$	0	0
$471$	−29.3065	−1.35037
$472$	0	0
$473$	39.3780	1.81060
$474$	0	0
$475$	−1.84504	−0.0846564
$476$	0	0
$477$	−4.54882	−0.208276
$478$	0	0
$479$	−6.11944	−0.279604	−0.139802	−	0.990179i	$-0.544647\pi$
$479$	−6.11944	−0.279604	−0.139802	+	0.990179i	$0.544647\pi$
$480$	0	0
$481$	3.47891	0.158625
$482$	0	0
$483$	−13.2368	−0.602294
$484$	0	0
$485$	−9.55623	−0.433926
$486$	0	0
$487$	15.0479	0.681888	0.340944	−	0.940084i	$-0.389253\pi$
$487$	15.0479	0.681888	0.340944	+	0.940084i	$0.389253\pi$
$488$	0	0
$489$	8.91213	0.403020
$490$	0	0
$491$	25.4110	1.14678	0.573391	−	0.819282i	$-0.305627\pi$
$491$	25.4110	1.14678	0.573391	+	0.819282i	$0.305627\pi$
$492$	0	0
$493$	3.90925	0.176064
$494$	0	0
$495$	16.0542	0.721583
$496$	0	0
$497$	−1.42524	−0.0639309
$498$	0	0
$499$	32.4832	1.45415	0.727073	−	0.686560i	$-0.240880\pi$
$499$	32.4832	1.45415	0.727073	+	0.686560i	$0.240880\pi$
$500$	0	0
$501$	22.9759	1.02649
$502$	0	0
$503$	13.1109	0.584585	0.292293	−	0.956329i	$-0.405582\pi$
$503$	13.1109	0.584585	0.292293	+	0.956329i	$0.405582\pi$
$504$	0	0
$505$	−9.80693	−0.436403
$506$	0	0
$507$	−12.3763	−0.549649
$508$	0	0
$509$	17.2232	0.763407	0.381703	−	0.924285i	$-0.375338\pi$
$509$	17.2232	0.763407	0.381703	+	0.924285i	$0.375338\pi$
$510$	0	0
$511$	−23.8003	−1.05286
$512$	0	0
$513$	−22.4306	−0.990334
$514$	0	0
$515$	0.0598009	0.00263514
$516$	0	0
$517$	−34.7632	−1.52888
$518$	0	0
$519$	−16.8801	−0.740956
$520$	0	0
$521$	24.5405	1.07514	0.537570	−	0.843219i	$-0.319343\pi$
$521$	24.5405	1.07514	0.537570	+	0.843219i	$0.319343\pi$
$522$	0	0
$523$	−1.87930	−0.0821759	−0.0410879	−	0.999156i	$-0.513082\pi$
$523$	−1.87930	−0.0821759	−0.0410879	+	0.999156i	$0.513082\pi$
$524$	0	0
$525$	−1.82898	−0.0798234
$526$	0	0
$527$	7.32580	0.319117
$528$	0	0
$529$	−11.9684	−0.520364
$530$	0	0
$531$	−1.36980	−0.0594441
$532$	0	0
$533$	6.08453	0.263550
$534$	0	0
$535$	−20.8858	−0.902971
$536$	0	0
$537$	27.5147	1.18735
$538$	0	0
$539$	−15.0849	−0.649753
$540$	0	0
$541$	−17.6861	−0.760383	−0.380192	−	0.924908i	$-0.624142\pi$
$541$	−17.6861	−0.760383	−0.380192	+	0.924908i	$0.624142\pi$
$542$	0	0
$543$	−23.7456	−1.01902
$544$	0	0
$545$	−18.3924	−0.787845
$546$	0	0
$547$	33.3881	1.42757	0.713786	−	0.700363i	$-0.246979\pi$
$547$	33.3881	1.42757	0.713786	+	0.700363i	$0.246979\pi$
$548$	0	0
$549$	−8.30431	−0.354419
$550$	0	0
$551$	−15.7164	−0.669540
$552$	0	0
$553$	−7.05941	−0.300197
$554$	0	0
$555$	−5.20525	−0.220950
$556$	0	0
$557$	−23.2193	−0.983833	−0.491917	−	0.870642i	$-0.663704\pi$
$557$	−23.2193	−0.983833	−0.491917	+	0.870642i	$0.663704\pi$
$558$	0	0
$559$	13.0197	0.550676
$560$	0	0
$561$	7.02223	0.296479
$562$	0	0
$563$	−4.82783	−0.203469	−0.101734	−	0.994812i	$-0.532439\pi$
$563$	−4.82783	−0.203469	−0.101734	+	0.994812i	$0.532439\pi$
$564$	0	0
$565$	32.1566	1.35284
$566$	0	0
$567$	−9.40856	−0.395122
$568$	0	0
$569$	29.8564	1.25164	0.625822	−	0.779966i	$-0.284763\pi$
$569$	29.8564	1.25164	0.625822	+	0.779966i	$0.284763\pi$
$570$	0	0
$571$	−8.72493	−0.365127	−0.182563	−	0.983194i	$-0.558440\pi$
$571$	−8.72493	−0.365127	−0.182563	+	0.983194i	$0.558440\pi$
$572$	0	0
$573$	−17.9691	−0.750671
$574$	0	0
$575$	1.52429	0.0635674
$576$	0	0
$577$	−43.6513	−1.81723	−0.908613	−	0.417640i	$-0.862857\pi$
$577$	−43.6513	−1.81723	−0.908613	+	0.417640i	$0.862857\pi$
$578$	0	0
$579$	3.66189	0.152183
$580$	0	0
$581$	−26.4318	−1.09658
$582$	0	0
$583$	−18.2640	−0.756419
$584$	0	0
$585$	5.30808	0.219462
$586$	0	0
$587$	−42.4925	−1.75385	−0.876927	−	0.480623i	$-0.840411\pi$
$587$	−42.4925	−1.75385	−0.876927	+	0.480623i	$0.840411\pi$
$588$	0	0
$589$	−29.4519	−1.21354
$590$	0	0
$591$	24.2454	0.997321
$592$	0	0
$593$	−21.5713	−0.885827	−0.442913	−	0.896564i	$-0.646055\pi$
$593$	−21.5713	−0.885827	−0.442913	+	0.896564i	$0.646055\pi$
$594$	0	0
$595$	−6.65151	−0.272685
$596$	0	0
$597$	−1.50995	−0.0617982
$598$	0	0
$599$	−39.5851	−1.61740	−0.808702	−	0.588218i	$-0.799830\pi$
$599$	−39.5851	−1.61740	−0.808702	+	0.588218i	$0.799830\pi$
$600$	0	0
$601$	−8.22179	−0.335374	−0.167687	−	0.985840i	$-0.553630\pi$
$601$	−8.22179	−0.335374	−0.167687	+	0.985840i	$0.553630\pi$
$602$	0	0
$603$	−4.76470	−0.194034
$604$	0	0
$605$	41.0188	1.66765
$606$	0	0
$607$	0.897870	0.0364434	0.0182217	−	0.999834i	$-0.494200\pi$
$607$	0.897870	0.0364434	0.0182217	+	0.999834i	$0.494200\pi$
$608$	0	0
$609$	−15.5796	−0.631317
$610$	0	0
$611$	−11.4939	−0.464994
$612$	0	0
$613$	−2.29476	−0.0926846	−0.0463423	−	0.998926i	$-0.514756\pi$
$613$	−2.29476	−0.0926846	−0.0463423	+	0.998926i	$0.514756\pi$
$614$	0	0
$615$	−9.10384	−0.367102
$616$	0	0
$617$	−42.7497	−1.72104	−0.860519	−	0.509418i	$-0.829861\pi$
$617$	−42.7497	−1.72104	−0.860519	+	0.509418i	$0.829861\pi$
$618$	0	0
$619$	6.38217	0.256521	0.128261	−	0.991741i	$-0.459061\pi$
$619$	6.38217	0.256521	0.128261	+	0.991741i	$0.459061\pi$
$620$	0	0
$621$	18.5311	0.743629
$622$	0	0
$623$	−11.7266	−0.469818
$624$	0	0
$625$	−22.4947	−0.899789
$626$	0	0
$627$	−28.2315	−1.12746
$628$	0	0
$629$	1.91312	0.0762810
$630$	0	0
$631$	30.4826	1.21349	0.606747	−	0.794895i	$-0.292474\pi$
$631$	30.4826	1.21349	0.606747	+	0.794895i	$0.292474\pi$
$632$	0	0
$633$	−19.4827	−0.774370
$634$	0	0
$635$	−9.97158	−0.395710
$636$	0	0
$637$	−4.98759	−0.197615
$638$	0	0
$639$	0.625466	0.0247431
$640$	0	0
$641$	−37.2818	−1.47254	−0.736271	−	0.676687i	$-0.763415\pi$
$641$	−37.2818	−1.47254	−0.736271	+	0.676687i	$0.763415\pi$
$642$	0	0
$643$	7.41337	0.292355	0.146177	−	0.989258i	$-0.453303\pi$
$643$	7.41337	0.292355	0.146177	+	0.989258i	$0.453303\pi$
$644$	0	0
$645$	−19.4805	−0.767043
$646$	0	0
$647$	−10.8220	−0.425457	−0.212729	−	0.977111i	$-0.568235\pi$
$647$	−10.8220	−0.425457	−0.212729	+	0.977111i	$0.568235\pi$
$648$	0	0
$649$	−5.49989	−0.215890
$650$	0	0
$651$	−29.1956	−1.14427
$652$	0	0
$653$	−24.4160	−0.955472	−0.477736	−	0.878503i	$-0.658543\pi$
$653$	−24.4160	−0.955472	−0.477736	+	0.878503i	$0.658543\pi$
$654$	0	0
$655$	−5.60445	−0.218984
$656$	0	0
$657$	10.4447	0.407487
$658$	0	0
$659$	7.39335	0.288004	0.144002	−	0.989577i	$-0.454003\pi$
$659$	7.39335	0.288004	0.144002	+	0.989577i	$0.454003\pi$
$660$	0	0
$661$	26.9231	1.04719	0.523594	−	0.851968i	$-0.324591\pi$
$661$	26.9231	1.04719	0.523594	+	0.851968i	$0.324591\pi$
$662$	0	0
$663$	2.32179	0.0901709
$664$	0	0
$665$	26.7411	1.03697
$666$	0	0
$667$	12.9842	0.502749
$668$	0	0
$669$	−17.3942	−0.672499
$670$	0	0
$671$	−33.3428	−1.28718
$672$	0	0
$673$	−25.8654	−0.997039	−0.498519	−	0.866879i	$-0.666123\pi$
$673$	−25.8654	−0.997039	−0.498519	+	0.866879i	$0.666123\pi$
$674$	0	0
$675$	2.56053	0.0985549
$676$	0	0
$677$	22.9306	0.881294	0.440647	−	0.897680i	$-0.354749\pi$
$677$	22.9306	0.881294	0.440647	+	0.897680i	$0.354749\pi$
$678$	0	0
$679$	−13.9974	−0.537173
$680$	0	0
$681$	31.5284	1.20817
$682$	0	0
$683$	−1.43638	−0.0549615	−0.0274807	−	0.999622i	$-0.508748\pi$
$683$	−1.43638	−0.0549615	−0.0274807	+	0.999622i	$0.508748\pi$
$684$	0	0
$685$	43.0860	1.64623
$686$	0	0
$687$	2.75314	0.105039
$688$	0	0
$689$	−6.03872	−0.230057
$690$	0	0
$691$	−3.50892	−0.133486	−0.0667429	−	0.997770i	$-0.521261\pi$
$691$	−3.50892	−0.133486	−0.0667429	+	0.997770i	$0.521261\pi$
$692$	0	0
$693$	23.5153	0.893274
$694$	0	0
$695$	25.5114	0.967702
$696$	0	0
$697$	3.34599	0.126738
$698$	0	0
$699$	−14.7294	−0.557119
$700$	0	0
$701$	−39.1949	−1.48037	−0.740186	−	0.672402i	$-0.765263\pi$
$701$	−39.1949	−1.48037	−0.740186	+	0.672402i	$0.765263\pi$
$702$	0	0
$703$	−7.69130	−0.290083
$704$	0	0
$705$	17.1975	0.647695
$706$	0	0
$707$	−14.3647	−0.540239
$708$	0	0
$709$	−11.1504	−0.418762	−0.209381	−	0.977834i	$-0.567145\pi$
$709$	−11.1504	−0.418762	−0.209381	+	0.977834i	$0.567145\pi$
$710$	0	0
$711$	3.09801	0.116185
$712$	0	0
$713$	24.3318	0.911235
$714$	0	0
$715$	21.3125	0.797044
$716$	0	0
$717$	−23.0125	−0.859416
$718$	0	0
$719$	−6.17486	−0.230283	−0.115142	−	0.993349i	$-0.536732\pi$
$719$	−6.17486	−0.230283	−0.115142	+	0.993349i	$0.536732\pi$
$720$	0	0
$721$	0.0875931	0.00326214
$722$	0	0
$723$	−4.99409	−0.185732
$724$	0	0
$725$	1.79408	0.0666305
$726$	0	0
$727$	37.8527	1.40388	0.701940	−	0.712236i	$-0.252317\pi$
$727$	37.8527	1.40388	0.701940	+	0.712236i	$0.252317\pi$
$728$	0	0
$729$	25.4999	0.944441
$730$	0	0
$731$	7.15978	0.264814
$732$	0	0
$733$	−34.0886	−1.25909	−0.629545	−	0.776964i	$-0.716759\pi$
$733$	−34.0886	−1.25909	−0.629545	+	0.776964i	$0.716759\pi$
$734$	0	0
$735$	7.46257	0.275261
$736$	0	0
$737$	−19.1308	−0.704693
$738$	0	0
$739$	−21.8644	−0.804294	−0.402147	−	0.915575i	$-0.631736\pi$
$739$	−21.8644	−0.804294	−0.402147	+	0.915575i	$0.631736\pi$
$740$	0	0
$741$	−9.33429	−0.342904
$742$	0	0
$743$	−33.5435	−1.23059	−0.615297	−	0.788296i	$-0.710964\pi$
$743$	−33.5435	−1.23059	−0.615297	+	0.788296i	$0.710964\pi$
$744$	0	0
$745$	16.4966	0.604390
$746$	0	0
$747$	11.5996	0.424406
$748$	0	0
$749$	−30.5923	−1.11782
$750$	0	0
$751$	−37.5689	−1.37091	−0.685454	−	0.728116i	$-0.740396\pi$
$751$	−37.5689	−1.37091	−0.685454	+	0.728116i	$0.740396\pi$
$752$	0	0
$753$	17.6257	0.642318
$754$	0	0
$755$	−21.1383	−0.769301
$756$	0	0
$757$	−26.7254	−0.971351	−0.485676	−	0.874139i	$-0.661426\pi$
$757$	−26.7254	−0.971351	−0.485676	+	0.874139i	$0.661426\pi$
$758$	0	0
$759$	23.3236	0.846592
$760$	0	0
$761$	−5.95836	−0.215991	−0.107995	−	0.994151i	$-0.534443\pi$
$761$	−5.95836	−0.215991	−0.107995	+	0.994151i	$0.534443\pi$
$762$	0	0
$763$	−26.9402	−0.975301
$764$	0	0
$765$	2.91901	0.105537
$766$	0	0
$767$	−1.81845	−0.0656605
$768$	0	0
$769$	−5.25153	−0.189375	−0.0946875	−	0.995507i	$-0.530185\pi$
$769$	−5.25153	−0.189375	−0.0946875	+	0.995507i	$0.530185\pi$
$770$	0	0
$771$	1.23215	0.0443748
$772$	0	0
$773$	14.8480	0.534044	0.267022	−	0.963690i	$-0.413960\pi$
$773$	14.8480	0.534044	0.267022	+	0.963690i	$0.413960\pi$
$774$	0	0
$775$	3.36204	0.120768
$776$	0	0
$777$	−7.62436	−0.273522
$778$	0	0
$779$	−13.4519	−0.481964
$780$	0	0
$781$	2.51132	0.0898621
$782$	0	0
$783$	21.8110	0.779462
$784$	0	0
$785$	−48.9127	−1.74577
$786$	0	0
$787$	−37.7189	−1.34453	−0.672267	−	0.740309i	$-0.734679\pi$
$787$	−37.7189	−1.34453	−0.672267	+	0.740309i	$0.734679\pi$
$788$	0	0
$789$	−1.19195	−0.0424345
$790$	0	0
$791$	47.1012	1.67472
$792$	0	0
$793$	−11.0243	−0.391483
$794$	0	0
$795$	9.03530	0.320449
$796$	0	0
$797$	16.2719	0.576380	0.288190	−	0.957573i	$-0.406947\pi$
$797$	16.2719	0.576380	0.288190	+	0.957573i	$0.406947\pi$
$798$	0	0
$799$	−6.32070	−0.223610
$800$	0	0
$801$	5.14622	0.181833
$802$	0	0
$803$	41.9368	1.47992
$804$	0	0
$805$	−22.0923	−0.778650
$806$	0	0
$807$	12.0990	0.425906
$808$	0	0
$809$	−5.72768	−0.201374	−0.100687	−	0.994918i	$-0.532104\pi$
$809$	−5.72768	−0.201374	−0.100687	+	0.994918i	$0.532104\pi$
$810$	0	0
$811$	9.19240	0.322789	0.161394	−	0.986890i	$-0.448401\pi$
$811$	9.19240	0.322789	0.161394	+	0.986890i	$0.448401\pi$
$812$	0	0
$813$	−25.5815	−0.897182
$814$	0	0
$815$	14.8744	0.521028
$816$	0	0
$817$	−28.7844	−1.00704
$818$	0	0
$819$	7.77498	0.271680
$820$	0	0
$821$	12.2336	0.426955	0.213478	−	0.976948i	$-0.431521\pi$
$821$	12.2336	0.426955	0.213478	+	0.976948i	$0.431521\pi$
$822$	0	0
$823$	44.2138	1.54119	0.770597	−	0.637322i	$-0.219958\pi$
$823$	44.2138	1.54119	0.770597	+	0.637322i	$0.219958\pi$
$824$	0	0
$825$	3.22272	0.112201
$826$	0	0
$827$	−13.7428	−0.477882	−0.238941	−	0.971034i	$-0.576800\pi$
$827$	−13.7428	−0.477882	−0.238941	+	0.971034i	$0.576800\pi$
$828$	0	0
$829$	−36.9048	−1.28176	−0.640879	−	0.767642i	$-0.721430\pi$
$829$	−36.9048	−1.28176	−0.640879	+	0.767642i	$0.721430\pi$
$830$	0	0
$831$	32.9639	1.14350
$832$	0	0
$833$	−2.74276	−0.0950311
$834$	0	0
$835$	38.3469	1.32705
$836$	0	0
$837$	40.8731	1.41278
$838$	0	0
$839$	−16.1459	−0.557417	−0.278708	−	0.960376i	$-0.589906\pi$
$839$	−16.1459	−0.557417	−0.278708	+	0.960376i	$0.589906\pi$
$840$	0	0
$841$	−13.7177	−0.473025
$842$	0	0
$843$	3.25085	0.111965
$844$	0	0
$845$	−20.6561	−0.710590
$846$	0	0
$847$	60.0821	2.06444
$848$	0	0
$849$	14.9785	0.514060
$850$	0	0
$851$	6.35421	0.217819
$852$	0	0
$853$	12.0261	0.411766	0.205883	−	0.978577i	$-0.433993\pi$
$853$	12.0261	0.411766	0.205883	+	0.978577i	$0.433993\pi$
$854$	0	0
$855$	−11.7353	−0.401338
$856$	0	0
$857$	−4.81228	−0.164384	−0.0821922	−	0.996616i	$-0.526192\pi$
$857$	−4.81228	−0.164384	−0.0821922	+	0.996616i	$0.526192\pi$
$858$	0	0
$859$	47.9365	1.63557	0.817786	−	0.575522i	$-0.195201\pi$
$859$	47.9365	1.63557	0.817786	+	0.575522i	$0.195201\pi$
$860$	0	0
$861$	−13.3348	−0.454449
$862$	0	0
$863$	3.38212	0.115129	0.0575643	−	0.998342i	$-0.481667\pi$
$863$	3.38212	0.115129	0.0575643	+	0.998342i	$0.481667\pi$
$864$	0	0
$865$	−28.1731	−0.957914
$866$	0	0
$867$	1.27679	0.0433622
$868$	0	0
$869$	12.4389	0.421960
$870$	0	0
$871$	−6.32531	−0.214325
$872$	0	0
$873$	6.14276	0.207901
$874$	0	0
$875$	−36.3101	−1.22751
$876$	0	0
$877$	−18.9515	−0.639946	−0.319973	−	0.947427i	$-0.603674\pi$
$877$	−18.9515	−0.639946	−0.319973	+	0.947427i	$0.603674\pi$
$878$	0	0
$879$	2.03734	0.0687177
$880$	0	0
$881$	25.2408	0.850384	0.425192	−	0.905103i	$-0.360207\pi$
$881$	25.2408	0.850384	0.425192	+	0.905103i	$0.360207\pi$
$882$	0	0
$883$	−33.1723	−1.11634	−0.558168	−	0.829728i	$-0.688495\pi$
$883$	−33.1723	−1.11634	−0.558168	+	0.829728i	$0.688495\pi$
$884$	0	0
$885$	2.72082	0.0914594
$886$	0	0
$887$	−7.92844	−0.266211	−0.133105	−	0.991102i	$-0.542495\pi$
$887$	−7.92844	−0.266211	−0.133105	+	0.991102i	$0.542495\pi$
$888$	0	0
$889$	−14.6058	−0.489864
$890$	0	0
$891$	16.5781	0.555389
$892$	0	0
$893$	25.4111	0.850351
$894$	0	0
$895$	45.9223	1.53501
$896$	0	0
$897$	7.71157	0.257482
$898$	0	0
$899$	28.6384	0.955145
$900$	0	0
$901$	−3.32080	−0.110632
$902$	0	0
$903$	−28.5339	−0.949550
$904$	0	0
$905$	−39.6316	−1.31740
$906$	0	0
$907$	25.9498	0.861650	0.430825	−	0.902435i	$-0.358223\pi$
$907$	25.9498	0.861650	0.430825	+	0.902435i	$0.358223\pi$
$908$	0	0
$909$	6.30391	0.209088
$910$	0	0
$911$	3.60988	0.119601	0.0598003	−	0.998210i	$-0.480954\pi$
$911$	3.60988	0.119601	0.0598003	+	0.998210i	$0.480954\pi$
$912$	0	0
$913$	46.5736	1.54136
$914$	0	0
$915$	16.4948	0.545302
$916$	0	0
$917$	−8.20909	−0.271088
$918$	0	0
$919$	16.0338	0.528906	0.264453	−	0.964399i	$-0.414809\pi$
$919$	16.0338	0.528906	0.264453	+	0.964399i	$0.414809\pi$
$920$	0	0
$921$	−7.56836	−0.249386
$922$	0	0
$923$	0.830328	0.0273306
$924$	0	0
$925$	0.877990	0.0288681
$926$	0	0
$927$	−0.0384401	−0.00126254
$928$	0	0
$929$	−44.4502	−1.45836	−0.729182	−	0.684320i	$-0.760099\pi$
$929$	−44.4502	−1.45836	−0.729182	+	0.684320i	$0.760099\pi$
$930$	0	0
$931$	11.0267	0.361387
$932$	0	0
$933$	10.4582	0.342386
$934$	0	0
$935$	11.7201	0.383290
$936$	0	0
$937$	3.04242	0.0993914	0.0496957	−	0.998764i	$-0.484175\pi$
$937$	3.04242	0.0993914	0.0496957	+	0.998764i	$0.484175\pi$
$938$	0	0
$939$	−15.9985	−0.522090
$940$	0	0
$941$	25.9199	0.844965	0.422482	−	0.906371i	$-0.361159\pi$
$941$	25.9199	0.844965	0.422482	+	0.906371i	$0.361159\pi$
$942$	0	0
$943$	11.1133	0.361900
$944$	0	0
$945$	−37.1110	−1.20722
$946$	0	0
$947$	−4.34374	−0.141152	−0.0705762	−	0.997506i	$-0.522484\pi$
$947$	−4.34374	−0.141152	−0.0705762	+	0.997506i	$0.522484\pi$
$948$	0	0
$949$	13.8657	0.450101
$950$	0	0
$951$	−23.4111	−0.759157
$952$	0	0
$953$	−47.4786	−1.53798	−0.768991	−	0.639259i	$-0.779241\pi$
$953$	−47.4786	−1.53798	−0.768991	+	0.639259i	$0.779241\pi$
$954$	0	0
$955$	−29.9906	−0.970473
$956$	0	0
$957$	27.4517	0.887387
$958$	0	0
$959$	63.1100	2.03793
$960$	0	0
$961$	22.6674	0.731205
$962$	0	0
$963$	13.4254	0.432628
$964$	0	0
$965$	6.11172	0.196743
$966$	0	0
$967$	43.9658	1.41384	0.706922	−	0.707292i	$-0.250083\pi$
$967$	43.9658	1.41384	0.706922	+	0.707292i	$0.250083\pi$
$968$	0	0
$969$	−5.13309	−0.164899
$970$	0	0
$971$	−32.4585	−1.04164	−0.520821	−	0.853666i	$-0.674374\pi$
$971$	−32.4585	−1.04164	−0.520821	+	0.853666i	$0.674374\pi$
$972$	0	0
$973$	37.3677	1.19795
$974$	0	0
$975$	1.06554	0.0341247
$976$	0	0
$977$	−27.9625	−0.894599	−0.447300	−	0.894384i	$-0.647614\pi$
$977$	−27.9625	−0.894599	−0.447300	+	0.894384i	$0.647614\pi$
$978$	0	0
$979$	20.6627	0.660382
$980$	0	0
$981$	11.8227	0.377469
$982$	0	0
$983$	−37.4231	−1.19361	−0.596806	−	0.802386i	$-0.703564\pi$
$983$	−37.4231	−1.19361	−0.596806	+	0.802386i	$0.703564\pi$
$984$	0	0
$985$	40.4657	1.28934
$986$	0	0
$987$	25.1900	0.801805
$988$	0	0
$989$	23.7804	0.756173
$990$	0	0
$991$	−26.4222	−0.839328	−0.419664	−	0.907680i	$-0.637852\pi$
$991$	−26.4222	−0.839328	−0.419664	+	0.907680i	$0.637852\pi$
$992$	0	0
$993$	3.49782	0.111000
$994$	0	0
$995$	−2.52012	−0.0798932
$996$	0	0
$997$	−36.8242	−1.16624	−0.583118	−	0.812388i	$-0.698167\pi$
$997$	−36.8242	−1.16624	−0.583118	+	0.812388i	$0.698167\pi$
$998$	0	0
$999$	10.6739	0.337708

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.16	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.16	✓	22	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.27679 q^{3} +2.13098 q^{5} +3.12134 q^{7} -1.36980 q^{9} +O(q^{10})\)	\(q+1.27679 q^{3} +2.13098 q^{5} +3.12134 q^{7} -1.36980 q^{9} -5.49989 q^{11} -1.81845 q^{13} +2.72082 q^{15} -1.00000 q^{17} +4.02030 q^{19} +3.98531 q^{21} -3.32139 q^{23} -0.458932 q^{25} -5.57933 q^{27} -3.90925 q^{29} -7.32580 q^{31} -7.02223 q^{33} +6.65151 q^{35} -1.91312 q^{37} -2.32179 q^{39} -3.34599 q^{41} -7.15978 q^{43} -2.91901 q^{45} +6.32070 q^{47} +2.74276 q^{49} -1.27679 q^{51} +3.32080 q^{53} -11.7201 q^{55} +5.13309 q^{57} +1.00000 q^{59} +6.06244 q^{61} -4.27560 q^{63} -3.87508 q^{65} +3.47840 q^{67} -4.24073 q^{69} -0.456612 q^{71} -7.62502 q^{73} -0.585961 q^{75} -17.1670 q^{77} -2.26166 q^{79} -3.01427 q^{81} -8.46810 q^{83} -2.13098 q^{85} -4.99131 q^{87} -3.75693 q^{89} -5.67601 q^{91} -9.35354 q^{93} +8.56717 q^{95} -4.48443 q^{97} +7.53373 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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