LMFDB - Embedded newform 6024.2.a.q.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 57 x^{16} + 51 x^{15} + 1328 x^{14} - 1116 x^{13} - 16275 x^{12} + 13699 x^{11} + \cdots + 44032 $ x^18 - x^17 - 57x^16 + 51x^15 + 1328x^14 - 1116x^13 - 16275x^12 + 13699x^11 + 112394x^10 - 101250x^9 - 432956x^8 + 439806x^7 + 844117x^6 - 1010785x^5 - 593552x^4 + 965980x^3 - 88040x^2 - 184992x + 44032
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.75663$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.75663 q^{5} +3.11001 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.75663 q^{5} +3.11001 q^{7} +1.00000 q^{9} +5.38988 q^{11} +0.687759 q^{13} +1.75663 q^{15} -0.0345783 q^{17} -0.408316 q^{19} +3.11001 q^{21} -2.49563 q^{23} -1.91425 q^{25} +1.00000 q^{27} +6.93496 q^{29} -2.74102 q^{31} +5.38988 q^{33} +5.46313 q^{35} -0.936365 q^{37} +0.687759 q^{39} +6.76236 q^{41} -5.49266 q^{43} +1.75663 q^{45} +12.3248 q^{47} +2.67215 q^{49} -0.0345783 q^{51} -7.33060 q^{53} +9.46802 q^{55} -0.408316 q^{57} -12.4764 q^{59} +13.2145 q^{61} +3.11001 q^{63} +1.20814 q^{65} +12.2685 q^{67} -2.49563 q^{69} -1.95641 q^{71} -0.242780 q^{73} -1.91425 q^{75} +16.7626 q^{77} -17.7036 q^{79} +1.00000 q^{81} +3.41055 q^{83} -0.0607412 q^{85} +6.93496 q^{87} +5.78283 q^{89} +2.13894 q^{91} -2.74102 q^{93} -0.717260 q^{95} -1.72490 q^{97} +5.38988 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.75663	0.785589	0.392794	−	0.919626i	$-0.371508\pi$
$5$	1.75663	0.785589	0.392794	+	0.919626i	$0.371508\pi$
$6$	0	0
$7$	3.11001	1.17547	0.587736	−	0.809053i	$-0.300019\pi$
$7$	3.11001	1.17547	0.587736	+	0.809053i	$0.300019\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.38988	1.62511	0.812555	−	0.582885i	$-0.198076\pi$
$11$	5.38988	1.62511	0.812555	+	0.582885i	$0.198076\pi$
$12$	0	0
$13$	0.687759	0.190750	0.0953750	−	0.995441i	$-0.469595\pi$
$13$	0.687759	0.190750	0.0953750	+	0.995441i	$0.469595\pi$
$14$	0	0
$15$	1.75663	0.453560
$16$	0	0
$17$	−0.0345783	−0.00838647	−0.00419323	−	0.999991i	$-0.501335\pi$
$17$	−0.0345783	−0.00838647	−0.00419323	+	0.999991i	$0.501335\pi$
$18$	0	0
$19$	−0.408316	−0.0936741	−0.0468370	−	0.998903i	$-0.514914\pi$
$19$	−0.408316	−0.0936741	−0.0468370	+	0.998903i	$0.514914\pi$
$20$	0	0
$21$	3.11001	0.678659
$22$	0	0
$23$	−2.49563	−0.520374	−0.260187	−	0.965558i	$-0.583784\pi$
$23$	−2.49563	−0.520374	−0.260187	+	0.965558i	$0.583784\pi$
$24$	0	0
$25$	−1.91425	−0.382851
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.93496	1.28779	0.643895	−	0.765114i	$-0.277317\pi$
$29$	6.93496	1.28779	0.643895	+	0.765114i	$0.277317\pi$
$30$	0	0
$31$	−2.74102	−0.492301	−0.246150	−	0.969232i	$-0.579166\pi$
$31$	−2.74102	−0.492301	−0.246150	+	0.969232i	$0.579166\pi$
$32$	0	0
$33$	5.38988	0.938257
$34$	0	0
$35$	5.46313	0.923438
$36$	0	0
$37$	−0.936365	−0.153938	−0.0769688	−	0.997034i	$-0.524524\pi$
$37$	−0.936365	−0.153938	−0.0769688	+	0.997034i	$0.524524\pi$
$38$	0	0
$39$	0.687759	0.110130
$40$	0	0
$41$	6.76236	1.05610	0.528052	−	0.849212i	$-0.322923\pi$
$41$	6.76236	1.05610	0.528052	+	0.849212i	$0.322923\pi$
$42$	0	0
$43$	−5.49266	−0.837623	−0.418811	−	0.908073i	$-0.637553\pi$
$43$	−5.49266	−0.837623	−0.418811	+	0.908073i	$0.637553\pi$
$44$	0	0
$45$	1.75663	0.261863
$46$	0	0
$47$	12.3248	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$47$	12.3248	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$48$	0	0
$49$	2.67215	0.381736
$50$	0	0
$51$	−0.0345783	−0.00484193
$52$	0	0
$53$	−7.33060	−1.00694	−0.503468	−	0.864014i	$-0.667943\pi$
$53$	−7.33060	−1.00694	−0.503468	+	0.864014i	$0.667943\pi$
$54$	0	0
$55$	9.46802	1.27667
$56$	0	0
$57$	−0.408316	−0.0540828
$58$	0	0
$59$	−12.4764	−1.62429	−0.812146	−	0.583454i	$-0.801701\pi$
$59$	−12.4764	−1.62429	−0.812146	+	0.583454i	$0.801701\pi$
$60$	0	0
$61$	13.2145	1.69194	0.845972	−	0.533228i	$-0.179021\pi$
$61$	13.2145	1.69194	0.845972	+	0.533228i	$0.179021\pi$
$62$	0	0
$63$	3.11001	0.391824
$64$	0	0
$65$	1.20814	0.149851
$66$	0	0
$67$	12.2685	1.49884	0.749419	−	0.662096i	$-0.230333\pi$
$67$	12.2685	1.49884	0.749419	+	0.662096i	$0.230333\pi$
$68$	0	0
$69$	−2.49563	−0.300438
$70$	0	0
$71$	−1.95641	−0.232183	−0.116092	−	0.993239i	$-0.537037\pi$
$71$	−1.95641	−0.232183	−0.116092	+	0.993239i	$0.537037\pi$
$72$	0	0
$73$	−0.242780	−0.0284152	−0.0142076	−	0.999899i	$-0.504523\pi$
$73$	−0.242780	−0.0284152	−0.0142076	+	0.999899i	$0.504523\pi$
$74$	0	0
$75$	−1.91425	−0.221039
$76$	0	0
$77$	16.7626	1.91027
$78$	0	0
$79$	−17.7036	−1.99181	−0.995907	−	0.0903799i	$-0.971192\pi$
$79$	−17.7036	−1.99181	−0.995907	+	0.0903799i	$0.971192\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.41055	0.374356	0.187178	−	0.982326i	$-0.440066\pi$
$83$	3.41055	0.374356	0.187178	+	0.982326i	$0.440066\pi$
$84$	0	0
$85$	−0.0607412	−0.00658831
$86$	0	0
$87$	6.93496	0.743506
$88$	0	0
$89$	5.78283	0.612978	0.306489	−	0.951874i	$-0.400846\pi$
$89$	5.78283	0.612978	0.306489	+	0.951874i	$0.400846\pi$
$90$	0	0
$91$	2.13894	0.224222
$92$	0	0
$93$	−2.74102	−0.284230
$94$	0	0
$95$	−0.717260	−0.0735893
$96$	0	0
$97$	−1.72490	−0.175137	−0.0875683	−	0.996159i	$-0.527910\pi$
$97$	−1.72490	−0.175137	−0.0875683	+	0.996159i	$0.527910\pi$
$98$	0	0
$99$	5.38988	0.541703
$100$	0	0
$101$	−8.54817	−0.850575	−0.425287	−	0.905058i	$-0.639827\pi$
$101$	−8.54817	−0.850575	−0.425287	+	0.905058i	$0.639827\pi$
$102$	0	0
$103$	4.28183	0.421902	0.210951	−	0.977497i	$-0.432344\pi$
$103$	4.28183	0.421902	0.210951	+	0.977497i	$0.432344\pi$
$104$	0	0
$105$	5.46313	0.533147
$106$	0	0
$107$	−13.6691	−1.32144	−0.660722	−	0.750631i	$-0.729750\pi$
$107$	−13.6691	−1.32144	−0.660722	+	0.750631i	$0.729750\pi$
$108$	0	0
$109$	−4.45221	−0.426444	−0.213222	−	0.977004i	$-0.568396\pi$
$109$	−4.45221	−0.426444	−0.213222	+	0.977004i	$0.568396\pi$
$110$	0	0
$111$	−0.936365	−0.0888759
$112$	0	0
$113$	−12.3401	−1.16086	−0.580428	−	0.814312i	$-0.697115\pi$
$113$	−12.3401	−1.16086	−0.580428	+	0.814312i	$0.697115\pi$
$114$	0	0
$115$	−4.38389	−0.408800
$116$	0	0
$117$	0.687759	0.0635834
$118$	0	0
$119$	−0.107539	−0.00985806
$120$	0	0
$121$	18.0508	1.64098
$122$	0	0
$123$	6.76236	0.609741
$124$	0	0
$125$	−12.1458	−1.08635
$126$	0	0
$127$	7.82051	0.693958	0.346979	−	0.937873i	$-0.387207\pi$
$127$	7.82051	0.693958	0.346979	+	0.937873i	$0.387207\pi$
$128$	0	0
$129$	−5.49266	−0.483602
$130$	0	0
$131$	−2.44352	−0.213492	−0.106746	−	0.994286i	$-0.534043\pi$
$131$	−2.44352	−0.213492	−0.106746	+	0.994286i	$0.534043\pi$
$132$	0	0
$133$	−1.26987	−0.110111
$134$	0	0
$135$	1.75663	0.151187
$136$	0	0
$137$	10.2615	0.876698	0.438349	−	0.898805i	$-0.355563\pi$
$137$	10.2615	0.876698	0.438349	+	0.898805i	$0.355563\pi$
$138$	0	0
$139$	−10.4628	−0.887446	−0.443723	−	0.896164i	$-0.646343\pi$
$139$	−10.4628	−0.887446	−0.443723	+	0.896164i	$0.646343\pi$
$140$	0	0
$141$	12.3248	1.03793
$142$	0	0
$143$	3.70694	0.309990
$144$	0	0
$145$	12.1822	1.01167
$146$	0	0
$147$	2.67215	0.220395
$148$	0	0
$149$	−10.7365	−0.879570	−0.439785	−	0.898103i	$-0.644945\pi$
$149$	−10.7365	−0.879570	−0.439785	+	0.898103i	$0.644945\pi$
$150$	0	0
$151$	8.77194	0.713850	0.356925	−	0.934133i	$-0.383825\pi$
$151$	8.77194	0.713850	0.356925	+	0.934133i	$0.383825\pi$
$152$	0	0
$153$	−0.0345783	−0.00279549
$154$	0	0
$155$	−4.81495	−0.386746
$156$	0	0
$157$	−21.8656	−1.74506	−0.872532	−	0.488557i	$-0.837523\pi$
$157$	−21.8656	−1.74506	−0.872532	+	0.488557i	$0.837523\pi$
$158$	0	0
$159$	−7.33060	−0.581354
$160$	0	0
$161$	−7.76142	−0.611686
$162$	0	0
$163$	3.56939	0.279577	0.139788	−	0.990181i	$-0.455358\pi$
$163$	3.56939	0.279577	0.139788	+	0.990181i	$0.455358\pi$
$164$	0	0
$165$	9.46802	0.737084
$166$	0	0
$167$	−18.7800	−1.45324	−0.726620	−	0.687039i	$-0.758910\pi$
$167$	−18.7800	−1.45324	−0.726620	+	0.687039i	$0.758910\pi$
$168$	0	0
$169$	−12.5270	−0.963614
$170$	0	0
$171$	−0.408316	−0.0312247
$172$	0	0
$173$	−0.210485	−0.0160028	−0.00800142	−	0.999968i	$-0.502547\pi$
$173$	−0.210485	−0.0160028	−0.00800142	+	0.999968i	$0.502547\pi$
$174$	0	0
$175$	−5.95334	−0.450030
$176$	0	0
$177$	−12.4764	−0.937786
$178$	0	0
$179$	6.74736	0.504322	0.252161	−	0.967685i	$-0.418859\pi$
$179$	6.74736	0.504322	0.252161	+	0.967685i	$0.418859\pi$
$180$	0	0
$181$	−0.331031	−0.0246053	−0.0123027	−	0.999924i	$-0.503916\pi$
$181$	−0.331031	−0.0246053	−0.0123027	+	0.999924i	$0.503916\pi$
$182$	0	0
$183$	13.2145	0.976844
$184$	0	0
$185$	−1.64485	−0.120932
$186$	0	0
$187$	−0.186373	−0.0136289
$188$	0	0
$189$	3.11001	0.226220
$190$	0	0
$191$	1.47940	0.107046	0.0535230	−	0.998567i	$-0.482955\pi$
$191$	1.47940	0.107046	0.0535230	+	0.998567i	$0.482955\pi$
$192$	0	0
$193$	1.83550	0.132122	0.0660612	−	0.997816i	$-0.478957\pi$
$193$	1.83550	0.132122	0.0660612	+	0.997816i	$0.478957\pi$
$194$	0	0
$195$	1.20814	0.0865166
$196$	0	0
$197$	−17.1774	−1.22384	−0.611920	−	0.790919i	$-0.709603\pi$
$197$	−17.1774	−1.22384	−0.611920	+	0.790919i	$0.709603\pi$
$198$	0	0
$199$	10.3576	0.734230	0.367115	−	0.930176i	$-0.380346\pi$
$199$	10.3576	0.734230	0.367115	+	0.930176i	$0.380346\pi$
$200$	0	0
$201$	12.2685	0.865355
$202$	0	0
$203$	21.5678	1.51376
$204$	0	0
$205$	11.8790	0.829663
$206$	0	0
$207$	−2.49563	−0.173458
$208$	0	0
$209$	−2.20077	−0.152231
$210$	0	0
$211$	18.5958	1.28019	0.640094	−	0.768297i	$-0.278895\pi$
$211$	18.5958	1.28019	0.640094	+	0.768297i	$0.278895\pi$
$212$	0	0
$213$	−1.95641	−0.134051
$214$	0	0
$215$	−9.64857	−0.658027
$216$	0	0
$217$	−8.52458	−0.578686
$218$	0	0
$219$	−0.242780	−0.0164055
$220$	0	0
$221$	−0.0237815	−0.00159972
$222$	0	0
$223$	7.94122	0.531783	0.265892	−	0.964003i	$-0.414334\pi$
$223$	7.94122	0.531783	0.265892	+	0.964003i	$0.414334\pi$
$224$	0	0
$225$	−1.91425	−0.127617
$226$	0	0
$227$	3.84808	0.255406	0.127703	−	0.991812i	$-0.459240\pi$
$227$	3.84808	0.255406	0.127703	+	0.991812i	$0.459240\pi$
$228$	0	0
$229$	−11.3397	−0.749347	−0.374674	−	0.927157i	$-0.622245\pi$
$229$	−11.3397	−0.749347	−0.374674	+	0.927157i	$0.622245\pi$
$230$	0	0
$231$	16.7626	1.10290
$232$	0	0
$233$	−6.28565	−0.411787	−0.205893	−	0.978574i	$-0.566010\pi$
$233$	−6.28565	−0.411787	−0.205893	+	0.978574i	$0.566010\pi$
$234$	0	0
$235$	21.6501	1.41229
$236$	0	0
$237$	−17.7036	−1.14997
$238$	0	0
$239$	15.4887	1.00188	0.500941	−	0.865481i	$-0.332987\pi$
$239$	15.4887	1.00188	0.500941	+	0.865481i	$0.332987\pi$
$240$	0	0
$241$	18.0763	1.16439	0.582197	−	0.813047i	$-0.302193\pi$
$241$	18.0763	1.16439	0.582197	+	0.813047i	$0.302193\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.69398	0.299888
$246$	0	0
$247$	−0.280823	−0.0178683
$248$	0	0
$249$	3.41055	0.216135
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−13.4511	−0.845665
$254$	0	0
$255$	−0.0607412	−0.00380376
$256$	0	0
$257$	−11.9863	−0.747685	−0.373843	−	0.927492i	$-0.621960\pi$
$257$	−11.9863	−0.747685	−0.373843	+	0.927492i	$0.621960\pi$
$258$	0	0
$259$	−2.91210	−0.180949
$260$	0	0
$261$	6.93496	0.429263
$262$	0	0
$263$	30.4317	1.87650	0.938248	−	0.345963i	$-0.112448\pi$
$263$	30.4317	1.87650	0.938248	+	0.345963i	$0.112448\pi$
$264$	0	0
$265$	−12.8771	−0.791037
$266$	0	0
$267$	5.78283	0.353903
$268$	0	0
$269$	3.69836	0.225493	0.112746	−	0.993624i	$-0.464035\pi$
$269$	3.69836	0.225493	0.112746	+	0.993624i	$0.464035\pi$
$270$	0	0
$271$	−15.8124	−0.960536	−0.480268	−	0.877122i	$-0.659461\pi$
$271$	−15.8124	−0.960536	−0.480268	+	0.877122i	$0.659461\pi$
$272$	0	0
$273$	2.13894	0.129454
$274$	0	0
$275$	−10.3176	−0.622174
$276$	0	0
$277$	26.6349	1.60033	0.800167	−	0.599777i	$-0.204744\pi$
$277$	26.6349	1.60033	0.800167	+	0.599777i	$0.204744\pi$
$278$	0	0
$279$	−2.74102	−0.164100
$280$	0	0
$281$	2.74502	0.163754	0.0818770	−	0.996642i	$-0.473909\pi$
$281$	2.74502	0.163754	0.0818770	+	0.996642i	$0.473909\pi$
$282$	0	0
$283$	−4.92418	−0.292712	−0.146356	−	0.989232i	$-0.546755\pi$
$283$	−4.92418	−0.292712	−0.146356	+	0.989232i	$0.546755\pi$
$284$	0	0
$285$	−0.717260	−0.0424868
$286$	0	0
$287$	21.0310	1.24142
$288$	0	0
$289$	−16.9988	−0.999930
$290$	0	0
$291$	−1.72490	−0.101115
$292$	0	0
$293$	−25.7079	−1.50187	−0.750937	−	0.660374i	$-0.770398\pi$
$293$	−25.7079	−1.50187	−0.750937	+	0.660374i	$0.770398\pi$
$294$	0	0
$295$	−21.9165	−1.27603
$296$	0	0
$297$	5.38988	0.312752
$298$	0	0
$299$	−1.71639	−0.0992615
$300$	0	0
$301$	−17.0822	−0.984603
$302$	0	0
$303$	−8.54817	−0.491079
$304$	0	0
$305$	23.2130	1.32917
$306$	0	0
$307$	9.98223	0.569716	0.284858	−	0.958570i	$-0.408054\pi$
$307$	9.98223	0.569716	0.284858	+	0.958570i	$0.408054\pi$
$308$	0	0
$309$	4.28183	0.243585
$310$	0	0
$311$	20.6531	1.17113	0.585565	−	0.810626i	$-0.300873\pi$
$311$	20.6531	1.17113	0.585565	+	0.810626i	$0.300873\pi$
$312$	0	0
$313$	7.82577	0.442339	0.221169	−	0.975235i	$-0.429013\pi$
$313$	7.82577	0.442339	0.221169	+	0.975235i	$0.429013\pi$
$314$	0	0
$315$	5.46313	0.307813
$316$	0	0
$317$	27.4243	1.54030	0.770150	−	0.637863i	$-0.220181\pi$
$317$	27.4243	1.54030	0.770150	+	0.637863i	$0.220181\pi$
$318$	0	0
$319$	37.3786	2.09280
$320$	0	0
$321$	−13.6691	−0.762936
$322$	0	0
$323$	0.0141189	0.000785595 0
$324$	0	0
$325$	−1.31655	−0.0730288
$326$	0	0
$327$	−4.45221	−0.246208
$328$	0	0
$329$	38.3302	2.11321
$330$	0	0
$331$	4.96028	0.272642	0.136321	−	0.990665i	$-0.456472\pi$
$331$	4.96028	0.272642	0.136321	+	0.990665i	$0.456472\pi$
$332$	0	0
$333$	−0.936365	−0.0513125
$334$	0	0
$335$	21.5512	1.17747
$336$	0	0
$337$	11.7235	0.638622	0.319311	−	0.947650i	$-0.396549\pi$
$337$	11.7235	0.638622	0.319311	+	0.947650i	$0.396549\pi$
$338$	0	0
$339$	−12.3401	−0.670220
$340$	0	0
$341$	−14.7737	−0.800043
$342$	0	0
$343$	−13.4596	−0.726752
$344$	0	0
$345$	−4.38389	−0.236021
$346$	0	0
$347$	3.31167	0.177780	0.0888900	−	0.996041i	$-0.471668\pi$
$347$	3.31167	0.177780	0.0888900	+	0.996041i	$0.471668\pi$
$348$	0	0
$349$	8.71412	0.466456	0.233228	−	0.972422i	$-0.425071\pi$
$349$	8.71412	0.466456	0.233228	+	0.972422i	$0.425071\pi$
$350$	0	0
$351$	0.687759	0.0367099
$352$	0	0
$353$	−26.9502	−1.43442	−0.717208	−	0.696859i	$-0.754580\pi$
$353$	−26.9502	−1.43442	−0.717208	+	0.696859i	$0.754580\pi$
$354$	0	0
$355$	−3.43669	−0.182401
$356$	0	0
$357$	−0.107539	−0.00569156
$358$	0	0
$359$	−15.1232	−0.798172	−0.399086	−	0.916914i	$-0.630672\pi$
$359$	−15.1232	−0.798172	−0.399086	+	0.916914i	$0.630672\pi$
$360$	0	0
$361$	−18.8333	−0.991225
$362$	0	0
$363$	18.0508	0.947420
$364$	0	0
$365$	−0.426474	−0.0223227
$366$	0	0
$367$	−16.6145	−0.867271	−0.433635	−	0.901088i	$-0.642769\pi$
$367$	−16.6145	−0.867271	−0.433635	+	0.901088i	$0.642769\pi$
$368$	0	0
$369$	6.76236	0.352034
$370$	0	0
$371$	−22.7982	−1.18362
$372$	0	0
$373$	3.24717	0.168132	0.0840661	−	0.996460i	$-0.473209\pi$
$373$	3.24717	0.168132	0.0840661	+	0.996460i	$0.473209\pi$
$374$	0	0
$375$	−12.1458	−0.627205
$376$	0	0
$377$	4.76959	0.245646
$378$	0	0
$379$	22.5328	1.15743	0.578716	−	0.815530i	$-0.303554\pi$
$379$	22.5328	1.15743	0.578716	+	0.815530i	$0.303554\pi$
$380$	0	0
$381$	7.82051	0.400657
$382$	0	0
$383$	−10.0422	−0.513132	−0.256566	−	0.966527i	$-0.582591\pi$
$383$	−10.0422	−0.513132	−0.256566	+	0.966527i	$0.582591\pi$
$384$	0	0
$385$	29.4456	1.50069
$386$	0	0
$387$	−5.49266	−0.279208
$388$	0	0
$389$	15.6571	0.793845	0.396923	−	0.917852i	$-0.370078\pi$
$389$	15.6571	0.793845	0.396923	+	0.917852i	$0.370078\pi$
$390$	0	0
$391$	0.0862946	0.00436410
$392$	0	0
$393$	−2.44352	−0.123259
$394$	0	0
$395$	−31.0987	−1.56475
$396$	0	0
$397$	−26.5211	−1.33106	−0.665528	−	0.746373i	$-0.731794\pi$
$397$	−26.5211	−1.33106	−0.665528	+	0.746373i	$0.731794\pi$
$398$	0	0
$399$	−1.26987	−0.0635728
$400$	0	0
$401$	−2.99478	−0.149552	−0.0747762	−	0.997200i	$-0.523824\pi$
$401$	−2.99478	−0.149552	−0.0747762	+	0.997200i	$0.523824\pi$
$402$	0	0
$403$	−1.88516	−0.0939064
$404$	0	0
$405$	1.75663	0.0872876
$406$	0	0
$407$	−5.04689	−0.250165
$408$	0	0
$409$	19.5837	0.968351	0.484176	−	0.874971i	$-0.339120\pi$
$409$	19.5837	0.968351	0.484176	+	0.874971i	$0.339120\pi$
$410$	0	0
$411$	10.2615	0.506162
$412$	0	0
$413$	−38.8018	−1.90931
$414$	0	0
$415$	5.99107	0.294090
$416$	0	0
$417$	−10.4628	−0.512367
$418$	0	0
$419$	−26.2003	−1.27997	−0.639983	−	0.768389i	$-0.721059\pi$
$419$	−26.2003	−1.27997	−0.639983	+	0.768389i	$0.721059\pi$
$420$	0	0
$421$	25.2319	1.22973	0.614865	−	0.788633i	$-0.289211\pi$
$421$	25.2319	1.22973	0.614865	+	0.788633i	$0.289211\pi$
$422$	0	0
$423$	12.3248	0.599251
$424$	0	0
$425$	0.0661916	0.00321076
$426$	0	0
$427$	41.0972	1.98883
$428$	0	0
$429$	3.70694	0.178973
$430$	0	0
$431$	12.6390	0.608798	0.304399	−	0.952545i	$-0.401544\pi$
$431$	12.6390	0.608798	0.304399	+	0.952545i	$0.401544\pi$
$432$	0	0
$433$	−36.1709	−1.73826	−0.869132	−	0.494581i	$-0.835322\pi$
$433$	−36.1709	−1.73826	−0.869132	+	0.494581i	$0.835322\pi$
$434$	0	0
$435$	12.1822	0.584090
$436$	0	0
$437$	1.01900	0.0487456
$438$	0	0
$439$	41.2446	1.96850	0.984248	−	0.176796i	$-0.0565732\pi$
$439$	41.2446	1.96850	0.984248	+	0.176796i	$0.0565732\pi$
$440$	0	0
$441$	2.67215	0.127245
$442$	0	0
$443$	−0.452572	−0.0215024	−0.0107512	−	0.999942i	$-0.503422\pi$
$443$	−0.452572	−0.0215024	−0.0107512	+	0.999942i	$0.503422\pi$
$444$	0	0
$445$	10.1583	0.481549
$446$	0	0
$447$	−10.7365	−0.507820
$448$	0	0
$449$	−22.4944	−1.06158	−0.530789	−	0.847504i	$-0.678104\pi$
$449$	−22.4944	−1.06158	−0.530789	+	0.847504i	$0.678104\pi$
$450$	0	0
$451$	36.4483	1.71628
$452$	0	0
$453$	8.77194	0.412142
$454$	0	0
$455$	3.75732	0.176146
$456$	0	0
$457$	5.02989	0.235289	0.117644	−	0.993056i	$-0.462466\pi$
$457$	5.02989	0.235289	0.117644	+	0.993056i	$0.462466\pi$
$458$	0	0
$459$	−0.0345783	−0.00161398
$460$	0	0
$461$	−9.57608	−0.446003	−0.223001	−	0.974818i	$-0.571585\pi$
$461$	−9.57608	−0.446003	−0.223001	+	0.974818i	$0.571585\pi$
$462$	0	0
$463$	3.74899	0.174230	0.0871151	−	0.996198i	$-0.472235\pi$
$463$	3.74899	0.174230	0.0871151	+	0.996198i	$0.472235\pi$
$464$	0	0
$465$	−4.81495	−0.223288
$466$	0	0
$467$	−11.9874	−0.554711	−0.277355	−	0.960767i	$-0.589458\pi$
$467$	−11.9874	−0.554711	−0.277355	+	0.960767i	$0.589458\pi$
$468$	0	0
$469$	38.1552	1.76184
$470$	0	0
$471$	−21.8656	−1.00751
$472$	0	0
$473$	−29.6048	−1.36123
$474$	0	0
$475$	0.781620	0.0358632
$476$	0	0
$477$	−7.33060	−0.335645
$478$	0	0
$479$	−14.2720	−0.652105	−0.326053	−	0.945352i	$-0.605719\pi$
$479$	−14.2720	−0.652105	−0.326053	+	0.945352i	$0.605719\pi$
$480$	0	0
$481$	−0.643994	−0.0293636
$482$	0	0
$483$	−7.76142	−0.353157
$484$	0	0
$485$	−3.03000	−0.137585
$486$	0	0
$487$	−25.8986	−1.17358	−0.586788	−	0.809741i	$-0.699608\pi$
$487$	−25.8986	−1.17358	−0.586788	+	0.809741i	$0.699608\pi$
$488$	0	0
$489$	3.56939	0.161414
$490$	0	0
$491$	−18.0885	−0.816321	−0.408161	−	0.912910i	$-0.633830\pi$
$491$	−18.0885	−0.816321	−0.408161	+	0.912910i	$0.633830\pi$
$492$	0	0
$493$	−0.239799	−0.0108000
$494$	0	0
$495$	9.46802	0.425556
$496$	0	0
$497$	−6.08446	−0.272925
$498$	0	0
$499$	37.0236	1.65740	0.828701	−	0.559691i	$-0.189080\pi$
$499$	37.0236	1.65740	0.828701	+	0.559691i	$0.189080\pi$
$500$	0	0
$501$	−18.7800	−0.839029
$502$	0	0
$503$	15.3737	0.685481	0.342740	−	0.939430i	$-0.388645\pi$
$503$	15.3737	0.685481	0.342740	+	0.939430i	$0.388645\pi$
$504$	0	0
$505$	−15.0160	−0.668202
$506$	0	0
$507$	−12.5270	−0.556343
$508$	0	0
$509$	25.5275	1.13149	0.565744	−	0.824581i	$-0.308589\pi$
$509$	25.5275	1.13149	0.565744	+	0.824581i	$0.308589\pi$
$510$	0	0
$511$	−0.755047	−0.0334013
$512$	0	0
$513$	−0.408316	−0.0180276
$514$	0	0
$515$	7.52159	0.331441
$516$	0	0
$517$	66.4290	2.92155
$518$	0	0
$519$	−0.210485	−0.00923925
$520$	0	0
$521$	14.3423	0.628346	0.314173	−	0.949366i	$-0.398273\pi$
$521$	14.3423	0.628346	0.314173	+	0.949366i	$0.398273\pi$
$522$	0	0
$523$	19.7669	0.864347	0.432174	−	0.901790i	$-0.357747\pi$
$523$	19.7669	0.864347	0.432174	+	0.901790i	$0.357747\pi$
$524$	0	0
$525$	−5.95334	−0.259825
$526$	0	0
$527$	0.0947796	0.00412867
$528$	0	0
$529$	−16.7718	−0.729210
$530$	0	0
$531$	−12.4764	−0.541431
$532$	0	0
$533$	4.65088	0.201452
$534$	0	0
$535$	−24.0116	−1.03811
$536$	0	0
$537$	6.74736	0.291170
$538$	0	0
$539$	14.4026	0.620363
$540$	0	0
$541$	11.5200	0.495284	0.247642	−	0.968852i	$-0.420344\pi$
$541$	11.5200	0.495284	0.247642	+	0.968852i	$0.420344\pi$
$542$	0	0
$543$	−0.331031	−0.0142059
$544$	0	0
$545$	−7.82088	−0.335010
$546$	0	0
$547$	25.4412	1.08779	0.543894	−	0.839154i	$-0.316949\pi$
$547$	25.4412	1.08779	0.543894	+	0.839154i	$0.316949\pi$
$548$	0	0
$549$	13.2145	0.563981
$550$	0	0
$551$	−2.83166	−0.120633
$552$	0	0
$553$	−55.0585	−2.34132
$554$	0	0
$555$	−1.64485	−0.0698199
$556$	0	0
$557$	−42.4648	−1.79929	−0.899646	−	0.436620i	$-0.856175\pi$
$557$	−42.4648	−1.79929	−0.899646	+	0.436620i	$0.856175\pi$
$558$	0	0
$559$	−3.77763	−0.159777
$560$	0	0
$561$	−0.186373	−0.00786866
$562$	0	0
$563$	−21.3944	−0.901667	−0.450833	−	0.892608i	$-0.648873\pi$
$563$	−21.3944	−0.901667	−0.450833	+	0.892608i	$0.648873\pi$
$564$	0	0
$565$	−21.6769	−0.911955
$566$	0	0
$567$	3.11001	0.130608
$568$	0	0
$569$	−37.0593	−1.55361	−0.776803	−	0.629743i	$-0.783160\pi$
$569$	−37.0593	−1.55361	−0.776803	+	0.629743i	$0.783160\pi$
$570$	0	0
$571$	12.7892	0.535211	0.267606	−	0.963529i	$-0.413768\pi$
$571$	12.7892	0.535211	0.267606	+	0.963529i	$0.413768\pi$
$572$	0	0
$573$	1.47940	0.0618030
$574$	0	0
$575$	4.77726	0.199226
$576$	0	0
$577$	28.6737	1.19370	0.596851	−	0.802352i	$-0.296418\pi$
$577$	28.6737	1.19370	0.596851	+	0.802352i	$0.296418\pi$
$578$	0	0
$579$	1.83550	0.0762809
$580$	0	0
$581$	10.6068	0.440046
$582$	0	0
$583$	−39.5110	−1.63638
$584$	0	0
$585$	1.20814	0.0499504
$586$	0	0
$587$	25.5534	1.05470	0.527352	−	0.849647i	$-0.323185\pi$
$587$	25.5534	1.05470	0.527352	+	0.849647i	$0.323185\pi$
$588$	0	0
$589$	1.11920	0.0461158
$590$	0	0
$591$	−17.1774	−0.706585
$592$	0	0
$593$	28.6640	1.17709	0.588545	−	0.808464i	$-0.299701\pi$
$593$	28.6640	1.17709	0.588545	+	0.808464i	$0.299701\pi$
$594$	0	0
$595$	−0.188906	−0.00774438
$596$	0	0
$597$	10.3576	0.423908
$598$	0	0
$599$	−7.42582	−0.303411	−0.151705	−	0.988426i	$-0.548477\pi$
$599$	−7.42582	−0.303411	−0.151705	+	0.988426i	$0.548477\pi$
$600$	0	0
$601$	−5.19014	−0.211710	−0.105855	−	0.994382i	$-0.533758\pi$
$601$	−5.19014	−0.211710	−0.105855	+	0.994382i	$0.533758\pi$
$602$	0	0
$603$	12.2685	0.499613
$604$	0	0
$605$	31.7085	1.28913
$606$	0	0
$607$	−10.9578	−0.444764	−0.222382	−	0.974960i	$-0.571383\pi$
$607$	−10.9578	−0.444764	−0.222382	+	0.974960i	$0.571383\pi$
$608$	0	0
$609$	21.5678	0.873971
$610$	0	0
$611$	8.47648	0.342922
$612$	0	0
$613$	17.2927	0.698448	0.349224	−	0.937039i	$-0.386445\pi$
$613$	17.2927	0.698448	0.349224	+	0.937039i	$0.386445\pi$
$614$	0	0
$615$	11.8790	0.479006
$616$	0	0
$617$	10.4358	0.420128	0.210064	−	0.977688i	$-0.432633\pi$
$617$	10.4358	0.420128	0.210064	+	0.977688i	$0.432633\pi$
$618$	0	0
$619$	16.5660	0.665844	0.332922	−	0.942954i	$-0.391965\pi$
$619$	16.5660	0.665844	0.332922	+	0.942954i	$0.391965\pi$
$620$	0	0
$621$	−2.49563	−0.100146
$622$	0	0
$623$	17.9846	0.720539
$624$	0	0
$625$	−11.7644	−0.470575
$626$	0	0
$627$	−2.20077	−0.0878904
$628$	0	0
$629$	0.0323779	0.00129099
$630$	0	0
$631$	−36.3907	−1.44869	−0.724346	−	0.689437i	$-0.757858\pi$
$631$	−36.3907	−1.44869	−0.724346	+	0.689437i	$0.757858\pi$
$632$	0	0
$633$	18.5958	0.739117
$634$	0	0
$635$	13.7377	0.545166
$636$	0	0
$637$	1.83780	0.0728162
$638$	0	0
$639$	−1.95641	−0.0773945
$640$	0	0
$641$	0.820408	0.0324042	0.0162021	−	0.999869i	$-0.494842\pi$
$641$	0.820408	0.0324042	0.0162021	+	0.999869i	$0.494842\pi$
$642$	0	0
$643$	−22.7221	−0.896072	−0.448036	−	0.894016i	$-0.647876\pi$
$643$	−22.7221	−0.896072	−0.448036	+	0.894016i	$0.647876\pi$
$644$	0	0
$645$	−9.64857	−0.379912
$646$	0	0
$647$	−27.5128	−1.08164	−0.540820	−	0.841138i	$-0.681886\pi$
$647$	−27.5128	−1.08164	−0.540820	+	0.841138i	$0.681886\pi$
$648$	0	0
$649$	−67.2464	−2.63965
$650$	0	0
$651$	−8.52458	−0.334105
$652$	0	0
$653$	−25.8017	−1.00970	−0.504850	−	0.863207i	$-0.668452\pi$
$653$	−25.8017	−1.00970	−0.504850	+	0.863207i	$0.668452\pi$
$654$	0	0
$655$	−4.29236	−0.167717
$656$	0	0
$657$	−0.242780	−0.00947174
$658$	0	0
$659$	39.1124	1.52360	0.761801	−	0.647811i	$-0.224315\pi$
$659$	39.1124	1.52360	0.761801	+	0.647811i	$0.224315\pi$
$660$	0	0
$661$	−17.5398	−0.682219	−0.341109	−	0.940024i	$-0.610803\pi$
$661$	−17.5398	−0.682219	−0.341109	+	0.940024i	$0.610803\pi$
$662$	0	0
$663$	−0.0237815	−0.000923598 0
$664$	0	0
$665$	−2.23068	−0.0865022
$666$	0	0
$667$	−17.3071	−0.670133
$668$	0	0
$669$	7.94122	0.307025
$670$	0	0
$671$	71.2245	2.74959
$672$	0	0
$673$	33.1631	1.27834	0.639172	−	0.769064i	$-0.279277\pi$
$673$	33.1631	1.27834	0.639172	+	0.769064i	$0.279277\pi$
$674$	0	0
$675$	−1.91425	−0.0736796
$676$	0	0
$677$	−21.7508	−0.835952	−0.417976	−	0.908458i	$-0.637260\pi$
$677$	−21.7508	−0.835952	−0.417976	+	0.908458i	$0.637260\pi$
$678$	0	0
$679$	−5.36444	−0.205868
$680$	0	0
$681$	3.84808	0.147459
$682$	0	0
$683$	30.0751	1.15079	0.575397	−	0.817874i	$-0.304848\pi$
$683$	30.0751	1.15079	0.575397	+	0.817874i	$0.304848\pi$
$684$	0	0
$685$	18.0256	0.688724
$686$	0	0
$687$	−11.3397	−0.432636
$688$	0	0
$689$	−5.04169	−0.192073
$690$	0	0
$691$	19.0590	0.725038	0.362519	−	0.931976i	$-0.381917\pi$
$691$	19.0590	0.725038	0.362519	+	0.931976i	$0.381917\pi$
$692$	0	0
$693$	16.7626	0.636757
$694$	0	0
$695$	−18.3793	−0.697167
$696$	0	0
$697$	−0.233831	−0.00885697
$698$	0	0
$699$	−6.28565	−0.237745
$700$	0	0
$701$	12.0744	0.456044	0.228022	−	0.973656i	$-0.426774\pi$
$701$	12.0744	0.456044	0.228022	+	0.973656i	$0.426774\pi$
$702$	0	0
$703$	0.382333	0.0144200
$704$	0	0
$705$	21.6501	0.815389
$706$	0	0
$707$	−26.5849	−0.999827
$708$	0	0
$709$	−10.6380	−0.399517	−0.199758	−	0.979845i	$-0.564016\pi$
$709$	−10.6380	−0.399517	−0.199758	+	0.979845i	$0.564016\pi$
$710$	0	0
$711$	−17.7036	−0.663938
$712$	0	0
$713$	6.84056	0.256181
$714$	0	0
$715$	6.51172	0.243524
$716$	0	0
$717$	15.4887	0.578437
$718$	0	0
$719$	−26.3695	−0.983416	−0.491708	−	0.870760i	$-0.663627\pi$
$719$	−26.3695	−0.983416	−0.491708	+	0.870760i	$0.663627\pi$
$720$	0	0
$721$	13.3165	0.495934
$722$	0	0
$723$	18.0763	0.672264
$724$	0	0
$725$	−13.2753	−0.493031
$726$	0	0
$727$	18.0208	0.668355	0.334178	−	0.942510i	$-0.391541\pi$
$727$	18.0208	0.668355	0.334178	+	0.942510i	$0.391541\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.189927	0.00702470
$732$	0	0
$733$	−39.2770	−1.45073	−0.725364	−	0.688366i	$-0.758329\pi$
$733$	−39.2770	−1.45073	−0.725364	+	0.688366i	$0.758329\pi$
$734$	0	0
$735$	4.69398	0.173140
$736$	0	0
$737$	66.1258	2.43578
$738$	0	0
$739$	−20.5567	−0.756191	−0.378096	−	0.925767i	$-0.623421\pi$
$739$	−20.5567	−0.756191	−0.378096	+	0.925767i	$0.623421\pi$
$740$	0	0
$741$	−0.280823	−0.0103163
$742$	0	0
$743$	40.9074	1.50075	0.750373	−	0.661015i	$-0.229874\pi$
$743$	40.9074	1.50075	0.750373	+	0.661015i	$0.229874\pi$
$744$	0	0
$745$	−18.8601	−0.690980
$746$	0	0
$747$	3.41055	0.124785
$748$	0	0
$749$	−42.5111	−1.55332
$750$	0	0
$751$	−0.126191	−0.00460478	−0.00230239	−	0.999997i	$-0.500733\pi$
$751$	−0.126191	−0.00460478	−0.00230239	+	0.999997i	$0.500733\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	15.4091	0.560793
$756$	0	0
$757$	−32.5325	−1.18241	−0.591207	−	0.806520i	$-0.701348\pi$
$757$	−32.5325	−1.18241	−0.591207	+	0.806520i	$0.701348\pi$
$758$	0	0
$759$	−13.4511	−0.488245
$760$	0	0
$761$	−32.0966	−1.16350	−0.581750	−	0.813368i	$-0.697632\pi$
$761$	−32.0966	−1.16350	−0.581750	+	0.813368i	$0.697632\pi$
$762$	0	0
$763$	−13.8464	−0.501273
$764$	0	0
$765$	−0.0607412	−0.00219610
$766$	0	0
$767$	−8.58078	−0.309834
$768$	0	0
$769$	16.1559	0.582596	0.291298	−	0.956632i	$-0.405913\pi$
$769$	16.1559	0.582596	0.291298	+	0.956632i	$0.405913\pi$
$770$	0	0
$771$	−11.9863	−0.431676
$772$	0	0
$773$	−12.4905	−0.449251	−0.224625	−	0.974445i	$-0.572116\pi$
$773$	−12.4905	−0.449251	−0.224625	+	0.974445i	$0.572116\pi$
$774$	0	0
$775$	5.24700	0.188478
$776$	0	0
$777$	−2.91210	−0.104471
$778$	0	0
$779$	−2.76118	−0.0989295
$780$	0	0
$781$	−10.5448	−0.377323
$782$	0	0
$783$	6.93496	0.247835
$784$	0	0
$785$	−38.4097	−1.37090
$786$	0	0
$787$	−4.56278	−0.162646	−0.0813228	−	0.996688i	$-0.525914\pi$
$787$	−4.56278	−0.162646	−0.0813228	+	0.996688i	$0.525914\pi$
$788$	0	0
$789$	30.4317	1.08340
$790$	0	0
$791$	−38.3777	−1.36455
$792$	0	0
$793$	9.08840	0.322738
$794$	0	0
$795$	−12.8771	−0.456705
$796$	0	0
$797$	−44.7969	−1.58679	−0.793393	−	0.608710i	$-0.791687\pi$
$797$	−44.7969	−1.58679	−0.793393	+	0.608710i	$0.791687\pi$
$798$	0	0
$799$	−0.426170	−0.0150768
$800$	0	0
$801$	5.78283	0.204326
$802$	0	0
$803$	−1.30855	−0.0461778
$804$	0	0
$805$	−13.6339	−0.480533
$806$	0	0
$807$	3.69836	0.130188
$808$	0	0
$809$	−41.3954	−1.45539	−0.727693	−	0.685903i	$-0.759407\pi$
$809$	−41.3954	−1.45539	−0.727693	+	0.685903i	$0.759407\pi$
$810$	0	0
$811$	13.0752	0.459131	0.229565	−	0.973293i	$-0.426270\pi$
$811$	13.0752	0.459131	0.229565	+	0.973293i	$0.426270\pi$
$812$	0	0
$813$	−15.8124	−0.554566
$814$	0	0
$815$	6.27010	0.219632
$816$	0	0
$817$	2.24274	0.0784636
$818$	0	0
$819$	2.13894	0.0747405
$820$	0	0
$821$	48.5860	1.69566	0.847832	−	0.530265i	$-0.177908\pi$
$821$	48.5860	1.69566	0.847832	+	0.530265i	$0.177908\pi$
$822$	0	0
$823$	26.5197	0.924418	0.462209	−	0.886771i	$-0.347057\pi$
$823$	26.5197	0.924418	0.462209	+	0.886771i	$0.347057\pi$
$824$	0	0
$825$	−10.3176	−0.359212
$826$	0	0
$827$	6.23383	0.216771	0.108386	−	0.994109i	$-0.465432\pi$
$827$	6.23383	0.216771	0.108386	+	0.994109i	$0.465432\pi$
$828$	0	0
$829$	−12.0676	−0.419124	−0.209562	−	0.977795i	$-0.567204\pi$
$829$	−12.0676	−0.419124	−0.209562	+	0.977795i	$0.567204\pi$
$830$	0	0
$831$	26.6349	0.923954
$832$	0	0
$833$	−0.0923985	−0.00320142
$834$	0	0
$835$	−32.9895	−1.14165
$836$	0	0
$837$	−2.74102	−0.0947434
$838$	0	0
$839$	−39.6770	−1.36980	−0.684900	−	0.728637i	$-0.740154\pi$
$839$	−39.6770	−1.36980	−0.684900	+	0.728637i	$0.740154\pi$
$840$	0	0
$841$	19.0937	0.658404
$842$	0	0
$843$	2.74502	0.0945434
$844$	0	0
$845$	−22.0053	−0.757004
$846$	0	0
$847$	56.1381	1.92893
$848$	0	0
$849$	−4.92418	−0.168998
$850$	0	0
$851$	2.33682	0.0801052
$852$	0	0
$853$	23.1604	0.792998	0.396499	−	0.918035i	$-0.370225\pi$
$853$	23.1604	0.792998	0.396499	+	0.918035i	$0.370225\pi$
$854$	0	0
$855$	−0.717260	−0.0245298
$856$	0	0
$857$	57.5245	1.96500	0.982499	−	0.186267i	$-0.0596390\pi$
$857$	57.5245	1.96500	0.982499	+	0.186267i	$0.0596390\pi$
$858$	0	0
$859$	34.8670	1.18965	0.594823	−	0.803857i	$-0.297222\pi$
$859$	34.8670	1.18965	0.594823	+	0.803857i	$0.297222\pi$
$860$	0	0
$861$	21.0310	0.716734
$862$	0	0
$863$	9.51509	0.323897	0.161949	−	0.986799i	$-0.448222\pi$
$863$	9.51509	0.323897	0.161949	+	0.986799i	$0.448222\pi$
$864$	0	0
$865$	−0.369743	−0.0125717
$866$	0	0
$867$	−16.9988	−0.577310
$868$	0	0
$869$	−95.4204	−3.23692
$870$	0	0
$871$	8.43779	0.285903
$872$	0	0
$873$	−1.72490	−0.0583789
$874$	0	0
$875$	−37.7735	−1.27698
$876$	0	0
$877$	−14.0202	−0.473429	−0.236715	−	0.971579i	$-0.576071\pi$
$877$	−14.0202	−0.473429	−0.236715	+	0.971579i	$0.576071\pi$
$878$	0	0
$879$	−25.7079	−0.867107
$880$	0	0
$881$	−4.23216	−0.142585	−0.0712926	−	0.997455i	$-0.522712\pi$
$881$	−4.23216	−0.142585	−0.0712926	+	0.997455i	$0.522712\pi$
$882$	0	0
$883$	2.40104	0.0808015	0.0404007	−	0.999184i	$-0.487137\pi$
$883$	2.40104	0.0808015	0.0404007	+	0.999184i	$0.487137\pi$
$884$	0	0
$885$	−21.9165	−0.736714
$886$	0	0
$887$	41.7605	1.40218	0.701091	−	0.713072i	$-0.252697\pi$
$887$	41.7605	1.40218	0.701091	+	0.713072i	$0.252697\pi$
$888$	0	0
$889$	24.3219	0.815729
$890$	0	0
$891$	5.38988	0.180568
$892$	0	0
$893$	−5.03240	−0.168403
$894$	0	0
$895$	11.8526	0.396189
$896$	0	0
$897$	−1.71639	−0.0573086
$898$	0	0
$899$	−19.0088	−0.633980
$900$	0	0
$901$	0.253480	0.00844463
$902$	0	0
$903$	−17.0822	−0.568461
$904$	0	0
$905$	−0.581498	−0.0193296
$906$	0	0
$907$	24.9057	0.826981	0.413491	−	0.910508i	$-0.364309\pi$
$907$	24.9057	0.826981	0.413491	+	0.910508i	$0.364309\pi$
$908$	0	0
$909$	−8.54817	−0.283525
$910$	0	0
$911$	−39.7529	−1.31707	−0.658536	−	0.752549i	$-0.728824\pi$
$911$	−39.7529	−1.31707	−0.658536	+	0.752549i	$0.728824\pi$
$912$	0	0
$913$	18.3824	0.608370
$914$	0	0
$915$	23.2130	0.767398
$916$	0	0
$917$	−7.59938	−0.250953
$918$	0	0
$919$	38.6434	1.27473	0.637364	−	0.770563i	$-0.280025\pi$
$919$	38.6434	1.27473	0.637364	+	0.770563i	$0.280025\pi$
$920$	0	0
$921$	9.98223	0.328926
$922$	0	0
$923$	−1.34554	−0.0442890
$924$	0	0
$925$	1.79244	0.0589351
$926$	0	0
$927$	4.28183	0.140634
$928$	0	0
$929$	20.0921	0.659200	0.329600	−	0.944121i	$-0.393086\pi$
$929$	20.0921	0.659200	0.329600	+	0.944121i	$0.393086\pi$
$930$	0	0
$931$	−1.09108	−0.0357588
$932$	0	0
$933$	20.6531	0.676152
$934$	0	0
$935$	−0.327388	−0.0107067
$936$	0	0
$937$	16.3280	0.533412	0.266706	−	0.963778i	$-0.414065\pi$
$937$	16.3280	0.533412	0.266706	+	0.963778i	$0.414065\pi$
$938$	0	0
$939$	7.82577	0.255384
$940$	0	0
$941$	−22.3189	−0.727576	−0.363788	−	0.931482i	$-0.618517\pi$
$941$	−22.3189	−0.727576	−0.363788	+	0.931482i	$0.618517\pi$
$942$	0	0
$943$	−16.8763	−0.549569
$944$	0	0
$945$	5.46313	0.177716
$946$	0	0
$947$	−34.9263	−1.13495	−0.567476	−	0.823390i	$-0.692080\pi$
$947$	−34.9263	−1.13495	−0.567476	+	0.823390i	$0.692080\pi$
$948$	0	0
$949$	−0.166974	−0.00542020
$950$	0	0
$951$	27.4243	0.889293
$952$	0	0
$953$	28.5344	0.924320	0.462160	−	0.886796i	$-0.347075\pi$
$953$	28.5344	0.924320	0.462160	+	0.886796i	$0.347075\pi$
$954$	0	0
$955$	2.59877	0.0840941
$956$	0	0
$957$	37.3786	1.20828
$958$	0	0
$959$	31.9133	1.03054
$960$	0	0
$961$	−23.4868	−0.757640
$962$	0	0
$963$	−13.6691	−0.440481
$964$	0	0
$965$	3.22430	0.103794
$966$	0	0
$967$	−46.7518	−1.50344	−0.751718	−	0.659485i	$-0.770774\pi$
$967$	−46.7518	−1.50344	−0.751718	+	0.659485i	$0.770774\pi$
$968$	0	0
$969$	0.0141189	0.000453563 0
$970$	0	0
$971$	1.38476	0.0444391	0.0222195	−	0.999753i	$-0.492927\pi$
$971$	1.38476	0.0444391	0.0222195	+	0.999753i	$0.492927\pi$
$972$	0	0
$973$	−32.5395	−1.04317
$974$	0	0
$975$	−1.31655	−0.0421632
$976$	0	0
$977$	19.0942	0.610876	0.305438	−	0.952212i	$-0.401197\pi$
$977$	19.0942	0.610876	0.305438	+	0.952212i	$0.401197\pi$
$978$	0	0
$979$	31.1687	0.996157
$980$	0	0
$981$	−4.45221	−0.142148
$982$	0	0
$983$	−42.3115	−1.34953	−0.674764	−	0.738033i	$-0.735755\pi$
$983$	−42.3115	−1.34953	−0.674764	+	0.738033i	$0.735755\pi$
$984$	0	0
$985$	−30.1744	−0.961435
$986$	0	0
$987$	38.3302	1.22006
$988$	0	0
$989$	13.7076	0.435878
$990$	0	0
$991$	58.2927	1.85173	0.925864	−	0.377857i	$-0.123339\pi$
$991$	58.2927	1.85173	0.925864	+	0.377857i	$0.123339\pi$
$992$	0	0
$993$	4.96028	0.157410
$994$	0	0
$995$	18.1944	0.576803
$996$	0	0
$997$	−25.7621	−0.815895	−0.407947	−	0.913005i	$-0.633755\pi$
$997$	−25.7621	−0.815895	−0.407947	+	0.913005i	$0.633755\pi$
$998$	0	0
$999$	−0.936365	−0.0296253

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.q.1.13	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.q.1.13	✓	18	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.75663 q^{5} +3.11001 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.75663 q^{5} +3.11001 q^{7} +1.00000 q^{9} +5.38988 q^{11} +0.687759 q^{13} +1.75663 q^{15} -0.0345783 q^{17} -0.408316 q^{19} +3.11001 q^{21} -2.49563 q^{23} -1.91425 q^{25} +1.00000 q^{27} +6.93496 q^{29} -2.74102 q^{31} +5.38988 q^{33} +5.46313 q^{35} -0.936365 q^{37} +0.687759 q^{39} +6.76236 q^{41} -5.49266 q^{43} +1.75663 q^{45} +12.3248 q^{47} +2.67215 q^{49} -0.0345783 q^{51} -7.33060 q^{53} +9.46802 q^{55} -0.408316 q^{57} -12.4764 q^{59} +13.2145 q^{61} +3.11001 q^{63} +1.20814 q^{65} +12.2685 q^{67} -2.49563 q^{69} -1.95641 q^{71} -0.242780 q^{73} -1.91425 q^{75} +16.7626 q^{77} -17.7036 q^{79} +1.00000 q^{81} +3.41055 q^{83} -0.0607412 q^{85} +6.93496 q^{87} +5.78283 q^{89} +2.13894 q^{91} -2.74102 q^{93} -0.717260 q^{95} -1.72490 q^{97} +5.38988 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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