LMFDB - Embedded newform 8024.2.a.z.1.5

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:	$24$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-2.20476 q^{3} +3.30270 q^{5} -4.05648 q^{7} +1.86097 q^{9} +O(q^{10})$	$q-2.20476 q^{3} +3.30270 q^{5} -4.05648 q^{7} +1.86097 q^{9} -2.06533 q^{11} -0.105306 q^{13} -7.28168 q^{15} +1.00000 q^{17} -3.19402 q^{19} +8.94357 q^{21} +1.62662 q^{23} +5.90786 q^{25} +2.51128 q^{27} -0.938798 q^{29} +0.822775 q^{31} +4.55356 q^{33} -13.3974 q^{35} +7.92374 q^{37} +0.232175 q^{39} +1.99489 q^{41} -9.90354 q^{43} +6.14625 q^{45} +7.98841 q^{47} +9.45504 q^{49} -2.20476 q^{51} +3.19895 q^{53} -6.82117 q^{55} +7.04205 q^{57} +1.00000 q^{59} +5.05246 q^{61} -7.54901 q^{63} -0.347795 q^{65} +7.19464 q^{67} -3.58631 q^{69} +9.80318 q^{71} -9.52013 q^{73} -13.0254 q^{75} +8.37796 q^{77} +9.13416 q^{79} -11.1197 q^{81} -12.1826 q^{83} +3.30270 q^{85} +2.06983 q^{87} +1.21783 q^{89} +0.427173 q^{91} -1.81402 q^{93} -10.5489 q^{95} -13.2777 q^{97} -3.84352 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * 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
$3$	−2.20476	−1.27292	−0.636460	−	0.771310i	$-0.719602\pi$
$3$	−2.20476	−1.27292	−0.636460	+	0.771310i	$0.719602\pi$
$4$	0	0
$5$	3.30270	1.47701	0.738507	−	0.674246i	$-0.235531\pi$
$5$	3.30270	1.47701	0.738507	+	0.674246i	$0.235531\pi$
$6$	0	0
$7$	−4.05648	−1.53321	−0.766603	−	0.642122i	$-0.778054\pi$
$7$	−4.05648	−1.53321	−0.766603	+	0.642122i	$0.778054\pi$
$8$	0	0
$9$	1.86097	0.620325
$10$	0	0
$11$	−2.06533	−0.622720	−0.311360	−	0.950292i	$-0.600784\pi$
$11$	−2.06533	−0.622720	−0.311360	+	0.950292i	$0.600784\pi$
$12$	0	0
$13$	−0.105306	−0.0292067	−0.0146033	−	0.999893i	$-0.504649\pi$
$13$	−0.105306	−0.0292067	−0.0146033	+	0.999893i	$0.504649\pi$
$14$	0	0
$15$	−7.28168	−1.88012
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−3.19402	−0.732758	−0.366379	−	0.930466i	$-0.619403\pi$
$19$	−3.19402	−0.732758	−0.366379	+	0.930466i	$0.619403\pi$
$20$	0	0
$21$	8.94357	1.95165
$22$	0	0
$23$	1.62662	0.339174	0.169587	−	0.985515i	$-0.445757\pi$
$23$	1.62662	0.339174	0.169587	+	0.985515i	$0.445757\pi$
$24$	0	0
$25$	5.90786	1.18157
$26$	0	0
$27$	2.51128	0.483296
$28$	0	0
$29$	−0.938798	−0.174330	−0.0871652	−	0.996194i	$-0.527781\pi$
$29$	−0.938798	−0.174330	−0.0871652	+	0.996194i	$0.527781\pi$
$30$	0	0
$31$	0.822775	0.147775	0.0738873	−	0.997267i	$-0.476459\pi$
$31$	0.822775	0.147775	0.0738873	+	0.997267i	$0.476459\pi$
$32$	0	0
$33$	4.55356	0.792672
$34$	0	0
$35$	−13.3974	−2.26457
$36$	0	0
$37$	7.92374	1.30265	0.651327	−	0.758797i	$-0.274213\pi$
$37$	7.92374	1.30265	0.651327	+	0.758797i	$0.274213\pi$
$38$	0	0
$39$	0.232175	0.0371778
$40$	0	0
$41$	1.99489	0.311549	0.155775	−	0.987793i	$-0.450213\pi$
$41$	1.99489	0.311549	0.155775	+	0.987793i	$0.450213\pi$
$42$	0	0
$43$	−9.90354	−1.51028	−0.755138	−	0.655566i	$-0.772430\pi$
$43$	−9.90354	−1.51028	−0.755138	+	0.655566i	$0.772430\pi$
$44$	0	0
$45$	6.14625	0.916229
$46$	0	0
$47$	7.98841	1.16523	0.582615	−	0.812748i	$-0.302030\pi$
$47$	7.98841	1.16523	0.582615	+	0.812748i	$0.302030\pi$
$48$	0	0
$49$	9.45504	1.35072
$50$	0	0
$51$	−2.20476	−0.308728
$52$	0	0
$53$	3.19895	0.439410	0.219705	−	0.975566i	$-0.429491\pi$
$53$	3.19895	0.439410	0.219705	+	0.975566i	$0.429491\pi$
$54$	0	0
$55$	−6.82117	−0.919766
$56$	0	0
$57$	7.04205	0.932742
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	5.05246	0.646901	0.323451	−	0.946245i	$-0.395157\pi$
$61$	5.05246	0.646901	0.323451	+	0.946245i	$0.395157\pi$
$62$	0	0
$63$	−7.54901	−0.951086
$64$	0	0
$65$	−0.347795	−0.0431387
$66$	0	0
$67$	7.19464	0.878965	0.439482	−	0.898251i	$-0.355162\pi$
$67$	7.19464	0.878965	0.439482	+	0.898251i	$0.355162\pi$
$68$	0	0
$69$	−3.58631	−0.431741
$70$	0	0
$71$	9.80318	1.16342	0.581712	−	0.813395i	$-0.302383\pi$
$71$	9.80318	1.16342	0.581712	+	0.813395i	$0.302383\pi$
$72$	0	0
$73$	−9.52013	−1.11425	−0.557124	−	0.830430i	$-0.688095\pi$
$73$	−9.52013	−1.11425	−0.557124	+	0.830430i	$0.688095\pi$
$74$	0	0
$75$	−13.0254	−1.50405
$76$	0	0
$77$	8.37796	0.954758
$78$	0	0
$79$	9.13416	1.02767	0.513837	−	0.857888i	$-0.328224\pi$
$79$	9.13416	1.02767	0.513837	+	0.857888i	$0.328224\pi$
$80$	0	0
$81$	−11.1197	−1.23552
$82$	0	0
$83$	−12.1826	−1.33722	−0.668609	−	0.743614i	$-0.733110\pi$
$83$	−12.1826	−1.33722	−0.668609	+	0.743614i	$0.733110\pi$
$84$	0	0
$85$	3.30270	0.358229
$86$	0	0
$87$	2.06983	0.221909
$88$	0	0
$89$	1.21783	0.129089	0.0645446	−	0.997915i	$-0.479441\pi$
$89$	1.21783	0.129089	0.0645446	+	0.997915i	$0.479441\pi$
$90$	0	0
$91$	0.427173	0.0447799
$92$	0	0
$93$	−1.81402	−0.188105
$94$	0	0
$95$	−10.5489	−1.08229
$96$	0	0
$97$	−13.2777	−1.34815	−0.674075	−	0.738663i	$-0.735458\pi$
$97$	−13.2777	−1.34815	−0.674075	+	0.738663i	$0.735458\pi$
$98$	0	0
$99$	−3.84352	−0.386289
$100$	0	0
$101$	−4.40895	−0.438707	−0.219354	−	0.975645i	$-0.570395\pi$
$101$	−4.40895	−0.438707	−0.219354	+	0.975645i	$0.570395\pi$
$102$	0	0
$103$	−4.45137	−0.438606	−0.219303	−	0.975657i	$-0.570378\pi$
$103$	−4.45137	−0.438606	−0.219303	+	0.975657i	$0.570378\pi$
$104$	0	0
$105$	29.5380	2.88261
$106$	0	0
$107$	15.3245	1.48148	0.740740	−	0.671792i	$-0.234475\pi$
$107$	15.3245	1.48148	0.740740	+	0.671792i	$0.234475\pi$
$108$	0	0
$109$	−9.12840	−0.874342	−0.437171	−	0.899378i	$-0.644020\pi$
$109$	−9.12840	−0.874342	−0.437171	+	0.899378i	$0.644020\pi$
$110$	0	0
$111$	−17.4699	−1.65817
$112$	0	0
$113$	−10.8842	−1.02390	−0.511948	−	0.859016i	$-0.671076\pi$
$113$	−10.8842	−1.02390	−0.511948	+	0.859016i	$0.671076\pi$
$114$	0	0
$115$	5.37224	0.500964
$116$	0	0
$117$	−0.195972	−0.0181176
$118$	0	0
$119$	−4.05648	−0.371857
$120$	0	0
$121$	−6.73442	−0.612220
$122$	0	0
$123$	−4.39825	−0.396577
$124$	0	0
$125$	2.99838	0.268183
$126$	0	0
$127$	6.57910	0.583801	0.291900	−	0.956449i	$-0.405712\pi$
$127$	6.57910	0.583801	0.291900	+	0.956449i	$0.405712\pi$
$128$	0	0
$129$	21.8349	1.92246
$130$	0	0
$131$	−14.1559	−1.23681	−0.618405	−	0.785859i	$-0.712221\pi$
$131$	−14.1559	−1.23681	−0.618405	+	0.785859i	$0.712221\pi$
$132$	0	0
$133$	12.9565	1.12347
$134$	0	0
$135$	8.29401	0.713835
$136$	0	0
$137$	−5.90102	−0.504158	−0.252079	−	0.967707i	$-0.581114\pi$
$137$	−5.90102	−0.504158	−0.252079	+	0.967707i	$0.581114\pi$
$138$	0	0
$139$	0.199479	0.0169196	0.00845982	−	0.999964i	$-0.497307\pi$
$139$	0.199479	0.0169196	0.00845982	+	0.999964i	$0.497307\pi$
$140$	0	0
$141$	−17.6125	−1.48324
$142$	0	0
$143$	0.217492	0.0181876
$144$	0	0
$145$	−3.10057	−0.257489
$146$	0	0
$147$	−20.8461	−1.71936
$148$	0	0
$149$	18.0092	1.47537	0.737687	−	0.675143i	$-0.235918\pi$
$149$	18.0092	1.47537	0.737687	+	0.675143i	$0.235918\pi$
$150$	0	0
$151$	−1.07521	−0.0874990	−0.0437495	−	0.999043i	$-0.513930\pi$
$151$	−1.07521	−0.0874990	−0.0437495	+	0.999043i	$0.513930\pi$
$152$	0	0
$153$	1.86097	0.150451
$154$	0	0
$155$	2.71738	0.218265
$156$	0	0
$157$	−0.120080	−0.00958342	−0.00479171	−	0.999989i	$-0.501525\pi$
$157$	−0.120080	−0.00958342	−0.00479171	+	0.999989i	$0.501525\pi$
$158$	0	0
$159$	−7.05293	−0.559333
$160$	0	0
$161$	−6.59835	−0.520023
$162$	0	0
$163$	−1.03675	−0.0812047	−0.0406024	−	0.999175i	$-0.512928\pi$
$163$	−1.03675	−0.0812047	−0.0406024	+	0.999175i	$0.512928\pi$
$164$	0	0
$165$	15.0391	1.17079
$166$	0	0
$167$	−13.1676	−1.01894	−0.509469	−	0.860489i	$-0.670158\pi$
$167$	−13.1676	−1.01894	−0.509469	+	0.860489i	$0.670158\pi$
$168$	0	0
$169$	−12.9889	−0.999147
$170$	0	0
$171$	−5.94399	−0.454548
$172$	0	0
$173$	7.01126	0.533056	0.266528	−	0.963827i	$-0.414123\pi$
$173$	7.01126	0.533056	0.266528	+	0.963827i	$0.414123\pi$
$174$	0	0
$175$	−23.9651	−1.81159
$176$	0	0
$177$	−2.20476	−0.165720
$178$	0	0
$179$	−2.95870	−0.221144	−0.110572	−	0.993868i	$-0.535268\pi$
$179$	−2.95870	−0.221144	−0.110572	+	0.993868i	$0.535268\pi$
$180$	0	0
$181$	−8.40831	−0.624984	−0.312492	−	0.949920i	$-0.601164\pi$
$181$	−8.40831	−0.624984	−0.312492	+	0.949920i	$0.601164\pi$
$182$	0	0
$183$	−11.1395	−0.823454
$184$	0	0
$185$	26.1698	1.92404
$186$	0	0
$187$	−2.06533	−0.151032
$188$	0	0
$189$	−10.1870	−0.740992
$190$	0	0
$191$	−26.5344	−1.91996	−0.959979	−	0.280071i	$-0.909642\pi$
$191$	−26.5344	−1.91996	−0.959979	+	0.280071i	$0.909642\pi$
$192$	0	0
$193$	23.8918	1.71977	0.859883	−	0.510491i	$-0.170536\pi$
$193$	23.8918	1.71977	0.859883	+	0.510491i	$0.170536\pi$
$194$	0	0
$195$	0.766806	0.0549121
$196$	0	0
$197$	−14.7325	−1.04965	−0.524824	−	0.851210i	$-0.675869\pi$
$197$	−14.7325	−1.04965	−0.524824	+	0.851210i	$0.675869\pi$
$198$	0	0
$199$	12.8339	0.909768	0.454884	−	0.890551i	$-0.349681\pi$
$199$	12.8339	0.909768	0.454884	+	0.890551i	$0.349681\pi$
$200$	0	0
$201$	−15.8625	−1.11885
$202$	0	0
$203$	3.80822	0.267284
$204$	0	0
$205$	6.58852	0.460163
$206$	0	0
$207$	3.02710	0.210398
$208$	0	0
$209$	6.59670	0.456303
$210$	0	0
$211$	−12.7419	−0.877188	−0.438594	−	0.898685i	$-0.644523\pi$
$211$	−12.7419	−0.877188	−0.438594	+	0.898685i	$0.644523\pi$
$212$	0	0
$213$	−21.6137	−1.48094
$214$	0	0
$215$	−32.7085	−2.23070
$216$	0	0
$217$	−3.33757	−0.226569
$218$	0	0
$219$	20.9896	1.41835
$220$	0	0
$221$	−0.105306	−0.00708367
$222$	0	0
$223$	25.2121	1.68833	0.844163	−	0.536086i	$-0.180098\pi$
$223$	25.2121	1.68833	0.844163	+	0.536086i	$0.180098\pi$
$224$	0	0
$225$	10.9944	0.732958
$226$	0	0
$227$	−11.1965	−0.743138	−0.371569	−	0.928405i	$-0.621180\pi$
$227$	−11.1965	−0.743138	−0.371569	+	0.928405i	$0.621180\pi$
$228$	0	0
$229$	−6.57556	−0.434525	−0.217263	−	0.976113i	$-0.569713\pi$
$229$	−6.57556	−0.434525	−0.217263	+	0.976113i	$0.569713\pi$
$230$	0	0
$231$	−18.4714	−1.21533
$232$	0	0
$233$	0.242706	0.0159002	0.00795009	−	0.999968i	$-0.497469\pi$
$233$	0.242706	0.0159002	0.00795009	+	0.999968i	$0.497469\pi$
$234$	0	0
$235$	26.3834	1.72106
$236$	0	0
$237$	−20.1386	−1.30815
$238$	0	0
$239$	−2.39182	−0.154714	−0.0773569	−	0.997003i	$-0.524648\pi$
$239$	−2.39182	−0.154714	−0.0773569	+	0.997003i	$0.524648\pi$
$240$	0	0
$241$	−19.1550	−1.23388	−0.616942	−	0.787009i	$-0.711629\pi$
$241$	−19.1550	−1.23388	−0.616942	+	0.787009i	$0.711629\pi$
$242$	0	0
$243$	16.9824	1.08942
$244$	0	0
$245$	31.2272	1.99503
$246$	0	0
$247$	0.336350	0.0214014
$248$	0	0
$249$	26.8598	1.70217
$250$	0	0
$251$	12.4176	0.783794	0.391897	−	0.920009i	$-0.371819\pi$
$251$	12.4176	0.783794	0.391897	+	0.920009i	$0.371819\pi$
$252$	0	0
$253$	−3.35950	−0.211210
$254$	0	0
$255$	−7.28168	−0.455996
$256$	0	0
$257$	6.49042	0.404861	0.202431	−	0.979297i	$-0.435116\pi$
$257$	6.49042	0.404861	0.202431	+	0.979297i	$0.435116\pi$
$258$	0	0
$259$	−32.1425	−1.99724
$260$	0	0
$261$	−1.74708	−0.108141
$262$	0	0
$263$	−5.07891	−0.313179	−0.156590	−	0.987664i	$-0.550050\pi$
$263$	−5.07891	−0.313179	−0.156590	+	0.987664i	$0.550050\pi$
$264$	0	0
$265$	10.5652	0.649015
$266$	0	0
$267$	−2.68501	−0.164320
$268$	0	0
$269$	−8.80464	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$269$	−8.80464	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$270$	0	0
$271$	3.12874	0.190057	0.0950286	−	0.995475i	$-0.469706\pi$
$271$	3.12874	0.190057	0.0950286	+	0.995475i	$0.469706\pi$
$272$	0	0
$273$	−0.941814	−0.0570012
$274$	0	0
$275$	−12.2017	−0.735788
$276$	0	0
$277$	1.24892	0.0750404	0.0375202	−	0.999296i	$-0.488054\pi$
$277$	1.24892	0.0750404	0.0375202	+	0.999296i	$0.488054\pi$
$278$	0	0
$279$	1.53116	0.0916683
$280$	0	0
$281$	−17.3800	−1.03681	−0.518403	−	0.855136i	$-0.673473\pi$
$281$	−17.3800	−1.03681	−0.518403	+	0.855136i	$0.673473\pi$
$282$	0	0
$283$	−16.9031	−1.00478	−0.502392	−	0.864640i	$-0.667547\pi$
$283$	−16.9031	−1.00478	−0.502392	+	0.864640i	$0.667547\pi$
$284$	0	0
$285$	23.2578	1.37767
$286$	0	0
$287$	−8.09222	−0.477669
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	29.2743	1.71609
$292$	0	0
$293$	18.5181	1.08184	0.540920	−	0.841074i	$-0.318076\pi$
$293$	18.5181	1.08184	0.540920	+	0.841074i	$0.318076\pi$
$294$	0	0
$295$	3.30270	0.192291
$296$	0	0
$297$	−5.18662	−0.300958
$298$	0	0
$299$	−0.171293	−0.00990614
$300$	0	0
$301$	40.1735	2.31556
$302$	0	0
$303$	9.72069	0.558439
$304$	0	0
$305$	16.6868	0.955483
$306$	0	0
$307$	28.0391	1.60028	0.800138	−	0.599816i	$-0.204760\pi$
$307$	28.0391	1.60028	0.800138	+	0.599816i	$0.204760\pi$
$308$	0	0
$309$	9.81421	0.558311
$310$	0	0
$311$	−2.70130	−0.153177	−0.0765884	−	0.997063i	$-0.524403\pi$
$311$	−2.70130	−0.153177	−0.0765884	+	0.997063i	$0.524403\pi$
$312$	0	0
$313$	−20.3696	−1.15136	−0.575679	−	0.817676i	$-0.695262\pi$
$313$	−20.3696	−1.15136	−0.575679	+	0.817676i	$0.695262\pi$
$314$	0	0
$315$	−24.9321	−1.40477
$316$	0	0
$317$	0.376674	0.0211561	0.0105781	−	0.999944i	$-0.496633\pi$
$317$	0.376674	0.0211561	0.0105781	+	0.999944i	$0.496633\pi$
$318$	0	0
$319$	1.93893	0.108559
$320$	0	0
$321$	−33.7870	−1.88580
$322$	0	0
$323$	−3.19402	−0.177720
$324$	0	0
$325$	−0.622134	−0.0345098
$326$	0	0
$327$	20.1260	1.11297
$328$	0	0
$329$	−32.4048	−1.78654
$330$	0	0
$331$	−8.31478	−0.457022	−0.228511	−	0.973541i	$-0.573386\pi$
$331$	−8.31478	−0.457022	−0.228511	+	0.973541i	$0.573386\pi$
$332$	0	0
$333$	14.7459	0.808069
$334$	0	0
$335$	23.7618	1.29824
$336$	0	0
$337$	4.56579	0.248714	0.124357	−	0.992238i	$-0.460313\pi$
$337$	4.56579	0.248714	0.124357	+	0.992238i	$0.460313\pi$
$338$	0	0
$339$	23.9970	1.30334
$340$	0	0
$341$	−1.69930	−0.0920222
$342$	0	0
$343$	−9.95881	−0.537725
$344$	0	0
$345$	−11.8445	−0.637688
$346$	0	0
$347$	−5.58547	−0.299844	−0.149922	−	0.988698i	$-0.547902\pi$
$347$	−5.58547	−0.299844	−0.149922	+	0.988698i	$0.547902\pi$
$348$	0	0
$349$	21.0206	1.12521	0.562603	−	0.826727i	$-0.309800\pi$
$349$	21.0206	1.12521	0.562603	+	0.826727i	$0.309800\pi$
$350$	0	0
$351$	−0.264453	−0.0141155
$352$	0	0
$353$	−23.6268	−1.25753	−0.628764	−	0.777596i	$-0.716439\pi$
$353$	−23.6268	−1.25753	−0.628764	+	0.777596i	$0.716439\pi$
$354$	0	0
$355$	32.3770	1.71839
$356$	0	0
$357$	8.94357	0.473344
$358$	0	0
$359$	13.8028	0.728485	0.364243	−	0.931304i	$-0.381328\pi$
$359$	13.8028	0.728485	0.364243	+	0.931304i	$0.381328\pi$
$360$	0	0
$361$	−8.79825	−0.463066
$362$	0	0
$363$	14.8478	0.779307
$364$	0	0
$365$	−31.4422	−1.64576
$366$	0	0
$367$	13.3756	0.698200	0.349100	−	0.937086i	$-0.386487\pi$
$367$	13.3756	0.698200	0.349100	+	0.937086i	$0.386487\pi$
$368$	0	0
$369$	3.71244	0.193262
$370$	0	0
$371$	−12.9765	−0.673706
$372$	0	0
$373$	9.34422	0.483825	0.241913	−	0.970298i	$-0.422225\pi$
$373$	9.34422	0.483825	0.241913	+	0.970298i	$0.422225\pi$
$374$	0	0
$375$	−6.61072	−0.341376
$376$	0	0
$377$	0.0988613	0.00509162
$378$	0	0
$379$	−23.7703	−1.22100	−0.610500	−	0.792016i	$-0.709031\pi$
$379$	−23.7703	−1.22100	−0.610500	+	0.792016i	$0.709031\pi$
$380$	0	0
$381$	−14.5053	−0.743131
$382$	0	0
$383$	−8.20664	−0.419340	−0.209670	−	0.977772i	$-0.567239\pi$
$383$	−8.20664	−0.419340	−0.209670	+	0.977772i	$0.567239\pi$
$384$	0	0
$385$	27.6699	1.41019
$386$	0	0
$387$	−18.4302	−0.936861
$388$	0	0
$389$	4.72358	0.239495	0.119748	−	0.992804i	$-0.461791\pi$
$389$	4.72358	0.239495	0.119748	+	0.992804i	$0.461791\pi$
$390$	0	0
$391$	1.62662	0.0822617
$392$	0	0
$393$	31.2105	1.57436
$394$	0	0
$395$	30.1674	1.51789
$396$	0	0
$397$	−14.4813	−0.726794	−0.363397	−	0.931634i	$-0.618383\pi$
$397$	−14.4813	−0.726794	−0.363397	+	0.931634i	$0.618383\pi$
$398$	0	0
$399$	−28.5659	−1.43009
$400$	0	0
$401$	14.8633	0.742240	0.371120	−	0.928585i	$-0.378974\pi$
$401$	14.8633	0.742240	0.371120	+	0.928585i	$0.378974\pi$
$402$	0	0
$403$	−0.0866433	−0.00431601
$404$	0	0
$405$	−36.7251	−1.82488
$406$	0	0
$407$	−16.3651	−0.811189
$408$	0	0
$409$	12.5597	0.621036	0.310518	−	0.950567i	$-0.399497\pi$
$409$	12.5597	0.621036	0.310518	+	0.950567i	$0.399497\pi$
$410$	0	0
$411$	13.0103	0.641753
$412$	0	0
$413$	−4.05648	−0.199606
$414$	0	0
$415$	−40.2357	−1.97509
$416$	0	0
$417$	−0.439805	−0.0215373
$418$	0	0
$419$	−8.17320	−0.399287	−0.199643	−	0.979869i	$-0.563978\pi$
$419$	−8.17320	−0.399287	−0.199643	+	0.979869i	$0.563978\pi$
$420$	0	0
$421$	−15.6267	−0.761600	−0.380800	−	0.924657i	$-0.624351\pi$
$421$	−15.6267	−0.761600	−0.380800	+	0.924657i	$0.624351\pi$
$422$	0	0
$423$	14.8662	0.722821
$424$	0	0
$425$	5.90786	0.286573
$426$	0	0
$427$	−20.4952	−0.991833
$428$	0	0
$429$	−0.479518	−0.0231513
$430$	0	0
$431$	−32.5151	−1.56620	−0.783098	−	0.621898i	$-0.786362\pi$
$431$	−32.5151	−1.56620	−0.783098	+	0.621898i	$0.786362\pi$
$432$	0	0
$433$	8.04693	0.386711	0.193355	−	0.981129i	$-0.438063\pi$
$433$	8.04693	0.386711	0.193355	+	0.981129i	$0.438063\pi$
$434$	0	0
$435$	6.83602	0.327762
$436$	0	0
$437$	−5.19545	−0.248532
$438$	0	0
$439$	22.2531	1.06208	0.531041	−	0.847346i	$-0.321801\pi$
$439$	22.2531	1.06208	0.531041	+	0.847346i	$0.321801\pi$
$440$	0	0
$441$	17.5956	0.837885
$442$	0	0
$443$	19.9878	0.949650	0.474825	−	0.880080i	$-0.342511\pi$
$443$	19.9878	0.949650	0.474825	+	0.880080i	$0.342511\pi$
$444$	0	0
$445$	4.02212	0.190667
$446$	0	0
$447$	−39.7061	−1.87803
$448$	0	0
$449$	−10.4091	−0.491234	−0.245617	−	0.969367i	$-0.578991\pi$
$449$	−10.4091	−0.491234	−0.245617	+	0.969367i	$0.578991\pi$
$450$	0	0
$451$	−4.12010	−0.194008
$452$	0	0
$453$	2.37057	0.111379
$454$	0	0
$455$	1.41083	0.0661405
$456$	0	0
$457$	17.7705	0.831270	0.415635	−	0.909532i	$-0.363559\pi$
$457$	17.7705	0.831270	0.415635	+	0.909532i	$0.363559\pi$
$458$	0	0
$459$	2.51128	0.117217
$460$	0	0
$461$	−23.5914	−1.09876	−0.549380	−	0.835573i	$-0.685136\pi$
$461$	−23.5914	−1.09876	−0.549380	+	0.835573i	$0.685136\pi$
$462$	0	0
$463$	−22.9398	−1.06610	−0.533051	−	0.846083i	$-0.678955\pi$
$463$	−22.9398	−1.06610	−0.533051	+	0.846083i	$0.678955\pi$
$464$	0	0
$465$	−5.99118	−0.277834
$466$	0	0
$467$	−11.8952	−0.550442	−0.275221	−	0.961381i	$-0.588751\pi$
$467$	−11.8952	−0.550442	−0.275221	+	0.961381i	$0.588751\pi$
$468$	0	0
$469$	−29.1849	−1.34763
$470$	0	0
$471$	0.264748	0.0121989
$472$	0	0
$473$	20.4541	0.940478
$474$	0	0
$475$	−18.8698	−0.865806
$476$	0	0
$477$	5.95317	0.272577
$478$	0	0
$479$	−41.9225	−1.91549	−0.957745	−	0.287619i	$-0.907136\pi$
$479$	−41.9225	−1.91549	−0.957745	+	0.287619i	$0.907136\pi$
$480$	0	0
$481$	−0.834419	−0.0380462
$482$	0	0
$483$	14.5478	0.661948
$484$	0	0
$485$	−43.8525	−1.99124
$486$	0	0
$487$	−19.4017	−0.879173	−0.439587	−	0.898200i	$-0.644875\pi$
$487$	−19.4017	−0.879173	−0.439587	+	0.898200i	$0.644875\pi$
$488$	0	0
$489$	2.28579	0.103367
$490$	0	0
$491$	−3.27628	−0.147856	−0.0739281	−	0.997264i	$-0.523554\pi$
$491$	−3.27628	−0.147856	−0.0739281	+	0.997264i	$0.523554\pi$
$492$	0	0
$493$	−0.938798	−0.0422813
$494$	0	0
$495$	−12.6940	−0.570554
$496$	0	0
$497$	−39.7664	−1.78377
$498$	0	0
$499$	−30.9857	−1.38711	−0.693555	−	0.720404i	$-0.743956\pi$
$499$	−30.9857	−1.38711	−0.693555	+	0.720404i	$0.743956\pi$
$500$	0	0
$501$	29.0314	1.29703
$502$	0	0
$503$	−36.7866	−1.64023	−0.820116	−	0.572197i	$-0.806091\pi$
$503$	−36.7866	−1.64023	−0.820116	+	0.572197i	$0.806091\pi$
$504$	0	0
$505$	−14.5615	−0.647977
$506$	0	0
$507$	28.6375	1.27183
$508$	0	0
$509$	−3.30809	−0.146629	−0.0733143	−	0.997309i	$-0.523358\pi$
$509$	−3.30809	−0.146629	−0.0733143	+	0.997309i	$0.523358\pi$
$510$	0	0
$511$	38.6182	1.70837
$512$	0	0
$513$	−8.02107	−0.354139
$514$	0	0
$515$	−14.7016	−0.647828
$516$	0	0
$517$	−16.4987	−0.725612
$518$	0	0
$519$	−15.4582	−0.678538
$520$	0	0
$521$	−8.95942	−0.392519	−0.196260	−	0.980552i	$-0.562880\pi$
$521$	−8.95942	−0.392519	−0.196260	+	0.980552i	$0.562880\pi$
$522$	0	0
$523$	31.1519	1.36218	0.681088	−	0.732202i	$-0.261507\pi$
$523$	31.1519	1.36218	0.681088	+	0.732202i	$0.261507\pi$
$524$	0	0
$525$	52.8374	2.30601
$526$	0	0
$527$	0.822775	0.0358406
$528$	0	0
$529$	−20.3541	−0.884961
$530$	0	0
$531$	1.86097	0.0807594
$532$	0	0
$533$	−0.210074	−0.00909932
$534$	0	0
$535$	50.6124	2.18817
$536$	0	0
$537$	6.52324	0.281498
$538$	0	0
$539$	−19.5278	−0.841120
$540$	0	0
$541$	12.2251	0.525600	0.262800	−	0.964850i	$-0.415354\pi$
$541$	12.2251	0.525600	0.262800	+	0.964850i	$0.415354\pi$
$542$	0	0
$543$	18.5383	0.795555
$544$	0	0
$545$	−30.1484	−1.29142
$546$	0	0
$547$	−19.5277	−0.834946	−0.417473	−	0.908689i	$-0.637084\pi$
$547$	−19.5277	−0.834946	−0.417473	+	0.908689i	$0.637084\pi$
$548$	0	0
$549$	9.40250	0.401289
$550$	0	0
$551$	2.99854	0.127742
$552$	0	0
$553$	−37.0525	−1.57563
$554$	0	0
$555$	−57.6981	−2.44915
$556$	0	0
$557$	−14.7002	−0.622869	−0.311435	−	0.950268i	$-0.600809\pi$
$557$	−14.7002	−0.622869	−0.311435	+	0.950268i	$0.600809\pi$
$558$	0	0
$559$	1.04290	0.0441102
$560$	0	0
$561$	4.55356	0.192251
$562$	0	0
$563$	−24.6107	−1.03722	−0.518608	−	0.855012i	$-0.673550\pi$
$563$	−24.6107	−1.03722	−0.518608	+	0.855012i	$0.673550\pi$
$564$	0	0
$565$	−35.9472	−1.51231
$566$	0	0
$567$	45.1068	1.89431
$568$	0	0
$569$	−29.3593	−1.23081	−0.615403	−	0.788213i	$-0.711007\pi$
$569$	−29.3593	−1.23081	−0.615403	+	0.788213i	$0.711007\pi$
$570$	0	0
$571$	−14.0917	−0.589720	−0.294860	−	0.955540i	$-0.595273\pi$
$571$	−14.0917	−0.589720	−0.294860	+	0.955540i	$0.595273\pi$
$572$	0	0
$573$	58.5019	2.44395
$574$	0	0
$575$	9.60984	0.400758
$576$	0	0
$577$	29.2258	1.21669	0.608344	−	0.793674i	$-0.291834\pi$
$577$	29.2258	1.21669	0.608344	+	0.793674i	$0.291834\pi$
$578$	0	0
$579$	−52.6756	−2.18912
$580$	0	0
$581$	49.4187	2.05023
$582$	0	0
$583$	−6.60688	−0.273629
$584$	0	0
$585$	−0.647238	−0.0267600
$586$	0	0
$587$	−38.5888	−1.59273	−0.796364	−	0.604817i	$-0.793246\pi$
$587$	−38.5888	−1.59273	−0.796364	+	0.604817i	$0.793246\pi$
$588$	0	0
$589$	−2.62796	−0.108283
$590$	0	0
$591$	32.4817	1.33612
$592$	0	0
$593$	−12.0440	−0.494586	−0.247293	−	0.968941i	$-0.579541\pi$
$593$	−12.0440	−0.494586	−0.247293	+	0.968941i	$0.579541\pi$
$594$	0	0
$595$	−13.3974	−0.549238
$596$	0	0
$597$	−28.2956	−1.15806
$598$	0	0
$599$	−41.3038	−1.68763	−0.843814	−	0.536636i	$-0.819695\pi$
$599$	−41.3038	−1.68763	−0.843814	+	0.536636i	$0.819695\pi$
$600$	0	0
$601$	−44.3202	−1.80786	−0.903930	−	0.427681i	$-0.859330\pi$
$601$	−44.3202	−1.80786	−0.903930	+	0.427681i	$0.859330\pi$
$602$	0	0
$603$	13.3890	0.545244
$604$	0	0
$605$	−22.2418	−0.904258
$606$	0	0
$607$	7.96443	0.323266	0.161633	−	0.986851i	$-0.448324\pi$
$607$	7.96443	0.323266	0.161633	+	0.986851i	$0.448324\pi$
$608$	0	0
$609$	−8.39621	−0.340232
$610$	0	0
$611$	−0.841230	−0.0340325
$612$	0	0
$613$	5.83090	0.235508	0.117754	−	0.993043i	$-0.462431\pi$
$613$	5.83090	0.235508	0.117754	+	0.993043i	$0.462431\pi$
$614$	0	0
$615$	−14.5261	−0.585750
$616$	0	0
$617$	1.64933	0.0663994	0.0331997	−	0.999449i	$-0.489430\pi$
$617$	1.64933	0.0663994	0.0331997	+	0.999449i	$0.489430\pi$
$618$	0	0
$619$	22.0243	0.885230	0.442615	−	0.896712i	$-0.354051\pi$
$619$	22.0243	0.885230	0.442615	+	0.896712i	$0.354051\pi$
$620$	0	0
$621$	4.08490	0.163921
$622$	0	0
$623$	−4.94009	−0.197920
$624$	0	0
$625$	−19.6365	−0.785461
$626$	0	0
$627$	−14.5441	−0.580837
$628$	0	0
$629$	7.92374	0.315940
$630$	0	0
$631$	20.0057	0.796412	0.398206	−	0.917296i	$-0.369633\pi$
$631$	20.0057	0.796412	0.398206	+	0.917296i	$0.369633\pi$
$632$	0	0
$633$	28.0928	1.11659
$634$	0	0
$635$	21.7288	0.862282
$636$	0	0
$637$	−0.995675	−0.0394501
$638$	0	0
$639$	18.2435	0.721700
$640$	0	0
$641$	2.62620	0.103729	0.0518644	−	0.998654i	$-0.483484\pi$
$641$	2.62620	0.103729	0.0518644	+	0.998654i	$0.483484\pi$
$642$	0	0
$643$	−6.68996	−0.263826	−0.131913	−	0.991261i	$-0.542112\pi$
$643$	−6.68996	−0.263826	−0.131913	+	0.991261i	$0.542112\pi$
$644$	0	0
$645$	72.1143	2.83950
$646$	0	0
$647$	24.8291	0.976132	0.488066	−	0.872807i	$-0.337703\pi$
$647$	24.8291	0.976132	0.488066	+	0.872807i	$0.337703\pi$
$648$	0	0
$649$	−2.06533	−0.0810712
$650$	0	0
$651$	7.35855	0.288404
$652$	0	0
$653$	−47.0903	−1.84279	−0.921393	−	0.388632i	$-0.872948\pi$
$653$	−47.0903	−1.84279	−0.921393	+	0.388632i	$0.872948\pi$
$654$	0	0
$655$	−46.7529	−1.82679
$656$	0	0
$657$	−17.7167	−0.691195
$658$	0	0
$659$	45.9780	1.79105	0.895524	−	0.445013i	$-0.146801\pi$
$659$	45.9780	1.79105	0.895524	+	0.445013i	$0.146801\pi$
$660$	0	0
$661$	45.8321	1.78266	0.891331	−	0.453354i	$-0.149773\pi$
$661$	45.8321	1.78266	0.891331	+	0.453354i	$0.149773\pi$
$662$	0	0
$663$	0.232175	0.00901694
$664$	0	0
$665$	42.7914	1.65938
$666$	0	0
$667$	−1.52707	−0.0591283
$668$	0	0
$669$	−55.5867	−2.14910
$670$	0	0
$671$	−10.4350	−0.402838
$672$	0	0
$673$	−3.21120	−0.123783	−0.0618913	−	0.998083i	$-0.519713\pi$
$673$	−3.21120	−0.123783	−0.0618913	+	0.998083i	$0.519713\pi$
$674$	0	0
$675$	14.8363	0.571049
$676$	0	0
$677$	−10.6520	−0.409388	−0.204694	−	0.978826i	$-0.565620\pi$
$677$	−10.6520	−0.409388	−0.204694	+	0.978826i	$0.565620\pi$
$678$	0	0
$679$	53.8609	2.06699
$680$	0	0
$681$	24.6856	0.945955
$682$	0	0
$683$	4.85779	0.185878	0.0929391	−	0.995672i	$-0.470374\pi$
$683$	4.85779	0.185878	0.0929391	+	0.995672i	$0.470374\pi$
$684$	0	0
$685$	−19.4893	−0.744649
$686$	0	0
$687$	14.4975	0.553116
$688$	0	0
$689$	−0.336870	−0.0128337
$690$	0	0
$691$	−30.7923	−1.17140	−0.585698	−	0.810530i	$-0.699179\pi$
$691$	−30.7923	−1.17140	−0.585698	+	0.810530i	$0.699179\pi$
$692$	0	0
$693$	15.5912	0.592260
$694$	0	0
$695$	0.658822	0.0249905
$696$	0	0
$697$	1.99489	0.0755618
$698$	0	0
$699$	−0.535108	−0.0202396
$700$	0	0
$701$	−2.48829	−0.0939814	−0.0469907	−	0.998895i	$-0.514963\pi$
$701$	−2.48829	−0.0939814	−0.0469907	+	0.998895i	$0.514963\pi$
$702$	0	0
$703$	−25.3086	−0.954530
$704$	0	0
$705$	−58.1690	−2.19077
$706$	0	0
$707$	17.8848	0.672629
$708$	0	0
$709$	7.91022	0.297075	0.148537	−	0.988907i	$-0.452543\pi$
$709$	7.91022	0.297075	0.148537	+	0.988907i	$0.452543\pi$
$710$	0	0
$711$	16.9984	0.637491
$712$	0	0
$713$	1.33834	0.0501213
$714$	0	0
$715$	0.718312	0.0268633
$716$	0	0
$717$	5.27338	0.196938
$718$	0	0
$719$	−16.4893	−0.614947	−0.307474	−	0.951557i	$-0.599484\pi$
$719$	−16.4893	−0.614947	−0.307474	+	0.951557i	$0.599484\pi$
$720$	0	0
$721$	18.0569	0.672474
$722$	0	0
$723$	42.2322	1.57063
$724$	0	0
$725$	−5.54628	−0.205984
$726$	0	0
$727$	3.73245	0.138429	0.0692144	−	0.997602i	$-0.477951\pi$
$727$	3.73245	0.138429	0.0692144	+	0.997602i	$0.477951\pi$
$728$	0	0
$729$	−4.08315	−0.151228
$730$	0	0
$731$	−9.90354	−0.366296
$732$	0	0
$733$	−5.96447	−0.220303	−0.110151	−	0.993915i	$-0.535134\pi$
$733$	−5.96447	−0.220303	−0.110151	+	0.993915i	$0.535134\pi$
$734$	0	0
$735$	−68.8485	−2.53952
$736$	0	0
$737$	−14.8593	−0.547349
$738$	0	0
$739$	−27.4341	−1.00918	−0.504590	−	0.863359i	$-0.668356\pi$
$739$	−27.4341	−1.00918	−0.504590	+	0.863359i	$0.668356\pi$
$740$	0	0
$741$	−0.741572	−0.0272423
$742$	0	0
$743$	37.8593	1.38892	0.694461	−	0.719530i	$-0.255643\pi$
$743$	37.8593	1.38892	0.694461	+	0.719530i	$0.255643\pi$
$744$	0	0
$745$	59.4792	2.17915
$746$	0	0
$747$	−22.6716	−0.829510
$748$	0	0
$749$	−62.1637	−2.27141
$750$	0	0
$751$	39.6020	1.44510	0.722548	−	0.691321i	$-0.242971\pi$
$751$	39.6020	1.44510	0.722548	+	0.691321i	$0.242971\pi$
$752$	0	0
$753$	−27.3779	−0.997707
$754$	0	0
$755$	−3.55109	−0.129237
$756$	0	0
$757$	0.635794	0.0231083	0.0115542	−	0.999933i	$-0.496322\pi$
$757$	0.635794	0.0231083	0.0115542	+	0.999933i	$0.496322\pi$
$758$	0	0
$759$	7.40691	0.268854
$760$	0	0
$761$	22.8776	0.829311	0.414655	−	0.909979i	$-0.363902\pi$
$761$	22.8776	0.829311	0.414655	+	0.909979i	$0.363902\pi$
$762$	0	0
$763$	37.0292	1.34055
$764$	0	0
$765$	6.14625	0.222218
$766$	0	0
$767$	−0.105306	−0.00380239
$768$	0	0
$769$	−12.2310	−0.441062	−0.220531	−	0.975380i	$-0.570779\pi$
$769$	−12.2310	−0.441062	−0.220531	+	0.975380i	$0.570779\pi$
$770$	0	0
$771$	−14.3098	−0.515356
$772$	0	0
$773$	−30.3240	−1.09068	−0.545340	−	0.838215i	$-0.683599\pi$
$773$	−30.3240	−1.09068	−0.545340	+	0.838215i	$0.683599\pi$
$774$	0	0
$775$	4.86083	0.174606
$776$	0	0
$777$	70.8665	2.54232
$778$	0	0
$779$	−6.37171	−0.228290
$780$	0	0
$781$	−20.2468	−0.724487
$782$	0	0
$783$	−2.35758	−0.0842532
$784$	0	0
$785$	−0.396588	−0.0141549
$786$	0	0
$787$	−25.7381	−0.917464	−0.458732	−	0.888575i	$-0.651696\pi$
$787$	−25.7381	−0.917464	−0.458732	+	0.888575i	$0.651696\pi$
$788$	0	0
$789$	11.1978	0.398652
$790$	0	0
$791$	44.1514	1.56984
$792$	0	0
$793$	−0.532056	−0.0188939
$794$	0	0
$795$	−23.2937	−0.826143
$796$	0	0
$797$	−18.7726	−0.664959	−0.332479	−	0.943111i	$-0.607885\pi$
$797$	−18.7726	−0.664959	−0.332479	+	0.943111i	$0.607885\pi$
$798$	0	0
$799$	7.98841	0.282610
$800$	0	0
$801$	2.26634	0.0800773
$802$	0	0
$803$	19.6622	0.693864
$804$	0	0
$805$	−21.7924	−0.768081
$806$	0	0
$807$	19.4121	0.683339
$808$	0	0
$809$	−3.50241	−0.123138	−0.0615690	−	0.998103i	$-0.519610\pi$
$809$	−3.50241	−0.123138	−0.0615690	+	0.998103i	$0.519610\pi$
$810$	0	0
$811$	15.5786	0.547038	0.273519	−	0.961867i	$-0.411812\pi$
$811$	15.5786	0.547038	0.273519	+	0.961867i	$0.411812\pi$
$812$	0	0
$813$	−6.89812	−0.241928
$814$	0	0
$815$	−3.42409	−0.119941
$816$	0	0
$817$	31.6321	1.10667
$818$	0	0
$819$	0.794958	0.0277781
$820$	0	0
$821$	−41.7951	−1.45866	−0.729329	−	0.684163i	$-0.760168\pi$
$821$	−41.7951	−1.45866	−0.729329	+	0.684163i	$0.760168\pi$
$822$	0	0
$823$	30.2154	1.05324	0.526622	−	0.850100i	$-0.323458\pi$
$823$	30.2154	1.05324	0.526622	+	0.850100i	$0.323458\pi$
$824$	0	0
$825$	26.9018	0.936599
$826$	0	0
$827$	−37.1224	−1.29087	−0.645437	−	0.763814i	$-0.723325\pi$
$827$	−37.1224	−1.29087	−0.645437	+	0.763814i	$0.723325\pi$
$828$	0	0
$829$	14.0644	0.488476	0.244238	−	0.969715i	$-0.421462\pi$
$829$	14.0644	0.488476	0.244238	+	0.969715i	$0.421462\pi$
$830$	0	0
$831$	−2.75357	−0.0955204
$832$	0	0
$833$	9.45504	0.327598
$834$	0	0
$835$	−43.4886	−1.50499
$836$	0	0
$837$	2.06622	0.0714189
$838$	0	0
$839$	−22.3219	−0.770639	−0.385320	−	0.922783i	$-0.625909\pi$
$839$	−22.3219	−0.770639	−0.385320	+	0.922783i	$0.625909\pi$
$840$	0	0
$841$	−28.1187	−0.969609
$842$	0	0
$843$	38.3188	1.31977
$844$	0	0
$845$	−42.8985	−1.47575
$846$	0	0
$847$	27.3180	0.938659
$848$	0	0
$849$	37.2673	1.27901
$850$	0	0
$851$	12.8889	0.441826
$852$	0	0
$853$	−36.1826	−1.23887	−0.619434	−	0.785049i	$-0.712638\pi$
$853$	−36.1826	−1.23887	−0.619434	+	0.785049i	$0.712638\pi$
$854$	0	0
$855$	−19.6312	−0.671374
$856$	0	0
$857$	11.3114	0.386389	0.193195	−	0.981160i	$-0.438115\pi$
$857$	11.3114	0.386389	0.193195	+	0.981160i	$0.438115\pi$
$858$	0	0
$859$	−15.2076	−0.518875	−0.259438	−	0.965760i	$-0.583537\pi$
$859$	−15.2076	−0.518875	−0.259438	+	0.965760i	$0.583537\pi$
$860$	0	0
$861$	17.8414	0.608034
$862$	0	0
$863$	58.3243	1.98538	0.992690	−	0.120688i	$-0.0385102\pi$
$863$	58.3243	1.98538	0.992690	+	0.120688i	$0.0385102\pi$
$864$	0	0
$865$	23.1561	0.787332
$866$	0	0
$867$	−2.20476	−0.0748776
$868$	0	0
$869$	−18.8650	−0.639952
$870$	0	0
$871$	−0.757640	−0.0256717
$872$	0	0
$873$	−24.7095	−0.836291
$874$	0	0
$875$	−12.1629	−0.411180
$876$	0	0
$877$	20.3998	0.688853	0.344427	−	0.938813i	$-0.388073\pi$
$877$	20.3998	0.688853	0.344427	+	0.938813i	$0.388073\pi$
$878$	0	0
$879$	−40.8280	−1.37710
$880$	0	0
$881$	−23.9988	−0.808540	−0.404270	−	0.914640i	$-0.632474\pi$
$881$	−23.9988	−0.808540	−0.404270	+	0.914640i	$0.632474\pi$
$882$	0	0
$883$	−21.8577	−0.735571	−0.367785	−	0.929911i	$-0.619884\pi$
$883$	−21.8577	−0.735571	−0.367785	+	0.929911i	$0.619884\pi$
$884$	0	0
$885$	−7.28168	−0.244771
$886$	0	0
$887$	11.5906	0.389173	0.194586	−	0.980885i	$-0.437664\pi$
$887$	11.5906	0.389173	0.194586	+	0.980885i	$0.437664\pi$
$888$	0	0
$889$	−26.6880	−0.895086
$890$	0	0
$891$	22.9658	0.769384
$892$	0	0
$893$	−25.5151	−0.853832
$894$	0	0
$895$	−9.77172	−0.326633
$896$	0	0
$897$	0.377661	0.0126097
$898$	0	0
$899$	−0.772419	−0.0257616
$900$	0	0
$901$	3.19895	0.106573
$902$	0	0
$903$	−88.5730	−2.94753
$904$	0	0
$905$	−27.7701	−0.923111
$906$	0	0
$907$	−12.3443	−0.409887	−0.204944	−	0.978774i	$-0.565701\pi$
$907$	−12.3443	−0.409887	−0.204944	+	0.978774i	$0.565701\pi$
$908$	0	0
$909$	−8.20495	−0.272141
$910$	0	0
$911$	23.1318	0.766390	0.383195	−	0.923667i	$-0.374824\pi$
$911$	23.1318	0.766390	0.383195	+	0.923667i	$0.374824\pi$
$912$	0	0
$913$	25.1612	0.832713
$914$	0	0
$915$	−36.7904	−1.21625
$916$	0	0
$917$	57.4233	1.89629
$918$	0	0
$919$	5.09651	0.168118	0.0840592	−	0.996461i	$-0.473212\pi$
$919$	5.09651	0.168118	0.0840592	+	0.996461i	$0.473212\pi$
$920$	0	0
$921$	−61.8195	−2.03702
$922$	0	0
$923$	−1.03234	−0.0339798
$924$	0	0
$925$	46.8123	1.53918
$926$	0	0
$927$	−8.28388	−0.272078
$928$	0	0
$929$	46.3178	1.51964	0.759819	−	0.650135i	$-0.225288\pi$
$929$	46.3178	1.51964	0.759819	+	0.650135i	$0.225288\pi$
$930$	0	0
$931$	−30.1996	−0.989751
$932$	0	0
$933$	5.95573	0.194982
$934$	0	0
$935$	−6.82117	−0.223076
$936$	0	0
$937$	25.2331	0.824328	0.412164	−	0.911110i	$-0.364773\pi$
$937$	25.2331	0.824328	0.412164	+	0.911110i	$0.364773\pi$
$938$	0	0
$939$	44.9101	1.46559
$940$	0	0
$941$	−24.8563	−0.810292	−0.405146	−	0.914252i	$-0.632779\pi$
$941$	−24.8563	−0.810292	−0.405146	+	0.914252i	$0.632779\pi$
$942$	0	0
$943$	3.24492	0.105669
$944$	0	0
$945$	−33.6445	−1.09446
$946$	0	0
$947$	−27.4565	−0.892215	−0.446108	−	0.894979i	$-0.647190\pi$
$947$	−27.4565	−0.892215	−0.446108	+	0.894979i	$0.647190\pi$
$948$	0	0
$949$	1.00253	0.0325435
$950$	0	0
$951$	−0.830476	−0.0269300
$952$	0	0
$953$	2.79186	0.0904372	0.0452186	−	0.998977i	$-0.485602\pi$
$953$	2.79186	0.0904372	0.0452186	+	0.998977i	$0.485602\pi$
$954$	0	0
$955$	−87.6351	−2.83581
$956$	0	0
$957$	−4.27487	−0.138187
$958$	0	0
$959$	23.9374	0.772978
$960$	0	0
$961$	−30.3230	−0.978163
$962$	0	0
$963$	28.5186	0.918998
$964$	0	0
$965$	78.9074	2.54012
$966$	0	0
$967$	50.6629	1.62921	0.814604	−	0.580017i	$-0.196954\pi$
$967$	50.6629	1.62921	0.814604	+	0.580017i	$0.196954\pi$
$968$	0	0
$969$	7.04205	0.226223
$970$	0	0
$971$	43.4034	1.39288	0.696440	−	0.717615i	$-0.254766\pi$
$971$	43.4034	1.39288	0.696440	+	0.717615i	$0.254766\pi$
$972$	0	0
$973$	−0.809185	−0.0259413
$974$	0	0
$975$	1.37166	0.0439282
$976$	0	0
$977$	36.0684	1.15393	0.576965	−	0.816769i	$-0.304237\pi$
$977$	36.0684	1.15393	0.576965	+	0.816769i	$0.304237\pi$
$978$	0	0
$979$	−2.51521	−0.0803864
$980$	0	0
$981$	−16.9877	−0.542376
$982$	0	0
$983$	25.0963	0.800447	0.400224	−	0.916417i	$-0.368932\pi$
$983$	25.0963	0.800447	0.400224	+	0.916417i	$0.368932\pi$
$984$	0	0
$985$	−48.6572	−1.55035
$986$	0	0
$987$	71.4450	2.27412
$988$	0	0
$989$	−16.1093	−0.512246
$990$	0	0
$991$	15.0529	0.478171	0.239085	−	0.970999i	$-0.423152\pi$
$991$	15.0529	0.478171	0.239085	+	0.970999i	$0.423152\pi$
$992$	0	0
$993$	18.3321	0.581752
$994$	0	0
$995$	42.3865	1.34374
$996$	0	0
$997$	−52.1891	−1.65285	−0.826423	−	0.563049i	$-0.809628\pi$
$997$	−52.1891	−1.65285	−0.826423	+	0.563049i	$0.809628\pi$
$998$	0	0
$999$	19.8987	0.629568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.z.1.5	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.z.1.5	✓	24	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.20476 q^{3} +3.30270 q^{5} -4.05648 q^{7} +1.86097 q^{9} +O(q^{10})\)	\(q-2.20476 q^{3} +3.30270 q^{5} -4.05648 q^{7} +1.86097 q^{9} -2.06533 q^{11} -0.105306 q^{13} -7.28168 q^{15} +1.00000 q^{17} -3.19402 q^{19} +8.94357 q^{21} +1.62662 q^{23} +5.90786 q^{25} +2.51128 q^{27} -0.938798 q^{29} +0.822775 q^{31} +4.55356 q^{33} -13.3974 q^{35} +7.92374 q^{37} +0.232175 q^{39} +1.99489 q^{41} -9.90354 q^{43} +6.14625 q^{45} +7.98841 q^{47} +9.45504 q^{49} -2.20476 q^{51} +3.19895 q^{53} -6.82117 q^{55} +7.04205 q^{57} +1.00000 q^{59} +5.05246 q^{61} -7.54901 q^{63} -0.347795 q^{65} +7.19464 q^{67} -3.58631 q^{69} +9.80318 q^{71} -9.52013 q^{73} -13.0254 q^{75} +8.37796 q^{77} +9.13416 q^{79} -11.1197 q^{81} -12.1826 q^{83} +3.30270 q^{85} +2.06983 q^{87} +1.21783 q^{89} +0.427173 q^{91} -1.81402 q^{93} -10.5489 q^{95} -13.2777 q^{97} -3.84352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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