LMFDB - Embedded newform 8024.2.a.ba.1.8

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:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.91871 q^{3} +0.891187 q^{5} +4.37465 q^{7} +0.681439 q^{9} +O(q^{10})$	$q-1.91871 q^{3} +0.891187 q^{5} +4.37465 q^{7} +0.681439 q^{9} -0.506311 q^{11} -2.73007 q^{13} -1.70993 q^{15} -1.00000 q^{17} -4.67019 q^{19} -8.39367 q^{21} -5.20075 q^{23} -4.20579 q^{25} +4.44864 q^{27} +7.35756 q^{29} -5.22692 q^{31} +0.971462 q^{33} +3.89863 q^{35} -5.81887 q^{37} +5.23822 q^{39} -5.15217 q^{41} +0.273467 q^{43} +0.607289 q^{45} -1.21300 q^{47} +12.1375 q^{49} +1.91871 q^{51} +1.84954 q^{53} -0.451217 q^{55} +8.96072 q^{57} +1.00000 q^{59} +13.1058 q^{61} +2.98106 q^{63} -2.43301 q^{65} -7.57977 q^{67} +9.97871 q^{69} +5.53991 q^{71} +10.2546 q^{73} +8.06967 q^{75} -2.21493 q^{77} +16.8880 q^{79} -10.5800 q^{81} +15.1838 q^{83} -0.891187 q^{85} -14.1170 q^{87} +1.69020 q^{89} -11.9431 q^{91} +10.0289 q^{93} -4.16201 q^{95} -6.14463 q^{97} -0.345020 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * 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
$3$	−1.91871	−1.10777	−0.553883	−	0.832594i	$-0.686855\pi$
$3$	−1.91871	−1.10777	−0.553883	+	0.832594i	$0.686855\pi$
$4$	0	0
$5$	0.891187	0.398551	0.199275	−	0.979944i	$-0.436141\pi$
$5$	0.891187	0.398551	0.199275	+	0.979944i	$0.436141\pi$
$6$	0	0
$7$	4.37465	1.65346	0.826731	−	0.562598i	$-0.190198\pi$
$7$	4.37465	1.65346	0.826731	+	0.562598i	$0.190198\pi$
$8$	0	0
$9$	0.681439	0.227146
$10$	0	0
$11$	−0.506311	−0.152658	−0.0763292	−	0.997083i	$-0.524320\pi$
$11$	−0.506311	−0.152658	−0.0763292	+	0.997083i	$0.524320\pi$
$12$	0	0
$13$	−2.73007	−0.757187	−0.378593	−	0.925563i	$-0.623592\pi$
$13$	−2.73007	−0.757187	−0.378593	+	0.925563i	$0.623592\pi$
$14$	0	0
$15$	−1.70993	−0.441501
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.67019	−1.07141	−0.535707	−	0.844404i	$-0.679955\pi$
$19$	−4.67019	−1.07141	−0.535707	+	0.844404i	$0.679955\pi$
$20$	0	0
$21$	−8.39367	−1.83165
$22$	0	0
$23$	−5.20075	−1.08443	−0.542215	−	0.840240i	$-0.682414\pi$
$23$	−5.20075	−1.08443	−0.542215	+	0.840240i	$0.682414\pi$
$24$	0	0
$25$	−4.20579	−0.841157
$26$	0	0
$27$	4.44864	0.856141
$28$	0	0
$29$	7.35756	1.36626	0.683132	−	0.730295i	$-0.260617\pi$
$29$	7.35756	1.36626	0.683132	+	0.730295i	$0.260617\pi$
$30$	0	0
$31$	−5.22692	−0.938783	−0.469392	−	0.882990i	$-0.655527\pi$
$31$	−5.22692	−0.938783	−0.469392	+	0.882990i	$0.655527\pi$
$32$	0	0
$33$	0.971462	0.169110
$34$	0	0
$35$	3.89863	0.658989
$36$	0	0
$37$	−5.81887	−0.956617	−0.478308	−	0.878192i	$-0.658750\pi$
$37$	−5.81887	−0.956617	−0.478308	+	0.878192i	$0.658750\pi$
$38$	0	0
$39$	5.23822	0.838786
$40$	0	0
$41$	−5.15217	−0.804633	−0.402317	−	0.915501i	$-0.631795\pi$
$41$	−5.15217	−0.804633	−0.402317	+	0.915501i	$0.631795\pi$
$42$	0	0
$43$	0.273467	0.0417033	0.0208516	−	0.999783i	$-0.493362\pi$
$43$	0.273467	0.0417033	0.0208516	+	0.999783i	$0.493362\pi$
$44$	0	0
$45$	0.607289	0.0905294
$46$	0	0
$47$	−1.21300	−0.176935	−0.0884675	−	0.996079i	$-0.528197\pi$
$47$	−1.21300	−0.176935	−0.0884675	+	0.996079i	$0.528197\pi$
$48$	0	0
$49$	12.1375	1.73394
$50$	0	0
$51$	1.91871	0.268673
$52$	0	0
$53$	1.84954	0.254053	0.127027	−	0.991899i	$-0.459457\pi$
$53$	1.84954	0.254053	0.127027	+	0.991899i	$0.459457\pi$
$54$	0	0
$55$	−0.451217	−0.0608421
$56$	0	0
$57$	8.96072	1.18688
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	13.1058	1.67803	0.839015	−	0.544108i	$-0.183132\pi$
$61$	13.1058	1.67803	0.839015	+	0.544108i	$0.183132\pi$
$62$	0	0
$63$	2.98106	0.375578
$64$	0	0
$65$	−2.43301	−0.301777
$66$	0	0
$67$	−7.57977	−0.926016	−0.463008	−	0.886354i	$-0.653230\pi$
$67$	−7.57977	−0.926016	−0.463008	+	0.886354i	$0.653230\pi$
$68$	0	0
$69$	9.97871	1.20130
$70$	0	0
$71$	5.53991	0.657466	0.328733	−	0.944423i	$-0.393378\pi$
$71$	5.53991	0.657466	0.328733	+	0.944423i	$0.393378\pi$
$72$	0	0
$73$	10.2546	1.20021	0.600106	−	0.799920i	$-0.295125\pi$
$73$	10.2546	1.20021	0.600106	+	0.799920i	$0.295125\pi$
$74$	0	0
$75$	8.06967	0.931806
$76$	0	0
$77$	−2.21493	−0.252415
$78$	0	0
$79$	16.8880	1.90005	0.950024	−	0.312178i	$-0.101059\pi$
$79$	16.8880	1.90005	0.950024	+	0.312178i	$0.101059\pi$
$80$	0	0
$81$	−10.5800	−1.17555
$82$	0	0
$83$	15.1838	1.66664	0.833321	−	0.552789i	$-0.186436\pi$
$83$	15.1838	1.66664	0.833321	+	0.552789i	$0.186436\pi$
$84$	0	0
$85$	−0.891187	−0.0966628
$86$	0	0
$87$	−14.1170	−1.51350
$88$	0	0
$89$	1.69020	0.179161	0.0895805	−	0.995980i	$-0.471447\pi$
$89$	1.69020	0.179161	0.0895805	+	0.995980i	$0.471447\pi$
$90$	0	0
$91$	−11.9431	−1.25198
$92$	0	0
$93$	10.0289	1.03995
$94$	0	0
$95$	−4.16201	−0.427013
$96$	0	0
$97$	−6.14463	−0.623893	−0.311947	−	0.950100i	$-0.600981\pi$
$97$	−6.14463	−0.623893	−0.311947	+	0.950100i	$0.600981\pi$
$98$	0	0
$99$	−0.345020	−0.0346758
$100$	0	0
$101$	−1.22153	−0.121547	−0.0607733	−	0.998152i	$-0.519357\pi$
$101$	−1.22153	−0.121547	−0.0607733	+	0.998152i	$0.519357\pi$
$102$	0	0
$103$	14.2740	1.40646	0.703231	−	0.710961i	$-0.251740\pi$
$103$	14.2740	1.40646	0.703231	+	0.710961i	$0.251740\pi$
$104$	0	0
$105$	−7.48033	−0.730005
$106$	0	0
$107$	15.2176	1.47114	0.735571	−	0.677447i	$-0.236914\pi$
$107$	15.2176	1.47114	0.735571	+	0.677447i	$0.236914\pi$
$108$	0	0
$109$	13.8179	1.32352	0.661758	−	0.749718i	$-0.269811\pi$
$109$	13.8179	1.32352	0.661758	+	0.749718i	$0.269811\pi$
$110$	0	0
$111$	11.1647	1.05971
$112$	0	0
$113$	3.04359	0.286317	0.143159	−	0.989700i	$-0.454274\pi$
$113$	3.04359	0.286317	0.143159	+	0.989700i	$0.454274\pi$
$114$	0	0
$115$	−4.63484	−0.432201
$116$	0	0
$117$	−1.86038	−0.171992
$118$	0	0
$119$	−4.37465	−0.401023
$120$	0	0
$121$	−10.7436	−0.976695
$122$	0	0
$123$	9.88550	0.891346
$124$	0	0
$125$	−8.20408	−0.733795
$126$	0	0
$127$	−15.0969	−1.33963	−0.669816	−	0.742527i	$-0.733627\pi$
$127$	−15.0969	−1.33963	−0.669816	+	0.742527i	$0.733627\pi$
$128$	0	0
$129$	−0.524703	−0.0461975
$130$	0	0
$131$	7.67015	0.670144	0.335072	−	0.942192i	$-0.391239\pi$
$131$	7.67015	0.670144	0.335072	+	0.942192i	$0.391239\pi$
$132$	0	0
$133$	−20.4304	−1.77154
$134$	0	0
$135$	3.96457	0.341216
$136$	0	0
$137$	−6.88030	−0.587823	−0.293912	−	0.955833i	$-0.594957\pi$
$137$	−6.88030	−0.587823	−0.293912	+	0.955833i	$0.594957\pi$
$138$	0	0
$139$	11.7745	0.998701	0.499350	−	0.866400i	$-0.333572\pi$
$139$	11.7745	0.998701	0.499350	+	0.866400i	$0.333572\pi$
$140$	0	0
$141$	2.32740	0.196003
$142$	0	0
$143$	1.38227	0.115591
$144$	0	0
$145$	6.55696	0.544526
$146$	0	0
$147$	−23.2884	−1.92080
$148$	0	0
$149$	15.2962	1.25311	0.626556	−	0.779376i	$-0.284464\pi$
$149$	15.2962	1.25311	0.626556	+	0.779376i	$0.284464\pi$
$150$	0	0
$151$	20.7971	1.69244	0.846222	−	0.532830i	$-0.178871\pi$
$151$	20.7971	1.69244	0.846222	+	0.532830i	$0.178871\pi$
$152$	0	0
$153$	−0.681439	−0.0550911
$154$	0	0
$155$	−4.65817	−0.374153
$156$	0	0
$157$	−12.0561	−0.962179	−0.481089	−	0.876672i	$-0.659759\pi$
$157$	−12.0561	−0.962179	−0.481089	+	0.876672i	$0.659759\pi$
$158$	0	0
$159$	−3.54872	−0.281432
$160$	0	0
$161$	−22.7514	−1.79306
$162$	0	0
$163$	14.4341	1.13057	0.565283	−	0.824897i	$-0.308767\pi$
$163$	14.4341	1.13057	0.565283	+	0.824897i	$0.308767\pi$
$164$	0	0
$165$	0.865754	0.0673989
$166$	0	0
$167$	14.7854	1.14413	0.572065	−	0.820208i	$-0.306142\pi$
$167$	14.7854	1.14413	0.572065	+	0.820208i	$0.306142\pi$
$168$	0	0
$169$	−5.54669	−0.426669
$170$	0	0
$171$	−3.18245	−0.243368
$172$	0	0
$173$	−16.2849	−1.23812	−0.619060	−	0.785344i	$-0.712486\pi$
$173$	−16.2849	−1.23812	−0.619060	+	0.785344i	$0.712486\pi$
$174$	0	0
$175$	−18.3988	−1.39082
$176$	0	0
$177$	−1.91871	−0.144219
$178$	0	0
$179$	−2.41399	−0.180430	−0.0902150	−	0.995922i	$-0.528755\pi$
$179$	−2.41399	−0.180430	−0.0902150	+	0.995922i	$0.528755\pi$
$180$	0	0
$181$	2.64416	0.196539	0.0982694	−	0.995160i	$-0.468669\pi$
$181$	2.64416	0.196539	0.0982694	+	0.995160i	$0.468669\pi$
$182$	0	0
$183$	−25.1463	−1.85887
$184$	0	0
$185$	−5.18570	−0.381261
$186$	0	0
$187$	0.506311	0.0370251
$188$	0	0
$189$	19.4612	1.41560
$190$	0	0
$191$	−6.13723	−0.444075	−0.222037	−	0.975038i	$-0.571271\pi$
$191$	−6.13723	−0.444075	−0.222037	+	0.975038i	$0.571271\pi$
$192$	0	0
$193$	1.78683	0.128619	0.0643094	−	0.997930i	$-0.479516\pi$
$193$	1.78683	0.128619	0.0643094	+	0.997930i	$0.479516\pi$
$194$	0	0
$195$	4.66823	0.334299
$196$	0	0
$197$	7.07222	0.503875	0.251937	−	0.967744i	$-0.418932\pi$
$197$	7.07222	0.503875	0.251937	+	0.967744i	$0.418932\pi$
$198$	0	0
$199$	0.953217	0.0675718	0.0337859	−	0.999429i	$-0.489244\pi$
$199$	0.953217	0.0675718	0.0337859	+	0.999429i	$0.489244\pi$
$200$	0	0
$201$	14.5434	1.02581
$202$	0	0
$203$	32.1867	2.25907
$204$	0	0
$205$	−4.59154	−0.320687
$206$	0	0
$207$	−3.54399	−0.246324
$208$	0	0
$209$	2.36456	0.163560
$210$	0	0
$211$	−2.34137	−0.161187	−0.0805933	−	0.996747i	$-0.525681\pi$
$211$	−2.34137	−0.161187	−0.0805933	+	0.996747i	$0.525681\pi$
$212$	0	0
$213$	−10.6295	−0.728319
$214$	0	0
$215$	0.243710	0.0166209
$216$	0	0
$217$	−22.8660	−1.55224
$218$	0	0
$219$	−19.6756	−1.32956
$220$	0	0
$221$	2.73007	0.183645
$222$	0	0
$223$	8.83577	0.591687	0.295843	−	0.955236i	$-0.404399\pi$
$223$	8.83577	0.591687	0.295843	+	0.955236i	$0.404399\pi$
$224$	0	0
$225$	−2.86599	−0.191066
$226$	0	0
$227$	−1.51152	−0.100323	−0.0501615	−	0.998741i	$-0.515974\pi$
$227$	−1.51152	−0.100323	−0.0501615	+	0.998741i	$0.515974\pi$
$228$	0	0
$229$	−3.96747	−0.262178	−0.131089	−	0.991371i	$-0.541847\pi$
$229$	−3.96747	−0.262178	−0.131089	+	0.991371i	$0.541847\pi$
$230$	0	0
$231$	4.24980	0.279617
$232$	0	0
$233$	18.9144	1.23913	0.619563	−	0.784947i	$-0.287310\pi$
$233$	18.9144	1.23913	0.619563	+	0.784947i	$0.287310\pi$
$234$	0	0
$235$	−1.08101	−0.0705176
$236$	0	0
$237$	−32.4031	−2.10481
$238$	0	0
$239$	−26.8177	−1.73469	−0.867346	−	0.497705i	$-0.834176\pi$
$239$	−26.8177	−1.73469	−0.867346	+	0.497705i	$0.834176\pi$
$240$	0	0
$241$	−25.4420	−1.63886	−0.819432	−	0.573176i	$-0.805711\pi$
$241$	−25.4420	−1.63886	−0.819432	+	0.573176i	$0.805711\pi$
$242$	0	0
$243$	6.95392	0.446094
$244$	0	0
$245$	10.8168	0.691062
$246$	0	0
$247$	12.7500	0.811260
$248$	0	0
$249$	−29.1333	−1.84625
$250$	0	0
$251$	−3.29213	−0.207797	−0.103899	−	0.994588i	$-0.533132\pi$
$251$	−3.29213	−0.207797	−0.103899	+	0.994588i	$0.533132\pi$
$252$	0	0
$253$	2.63319	0.165547
$254$	0	0
$255$	1.70993	0.107080
$256$	0	0
$257$	−8.31472	−0.518658	−0.259329	−	0.965789i	$-0.583501\pi$
$257$	−8.31472	−0.518658	−0.259329	+	0.965789i	$0.583501\pi$
$258$	0	0
$259$	−25.4555	−1.58173
$260$	0	0
$261$	5.01373	0.310342
$262$	0	0
$263$	−10.9801	−0.677064	−0.338532	−	0.940955i	$-0.609930\pi$
$263$	−10.9801	−0.677064	−0.338532	+	0.940955i	$0.609930\pi$
$264$	0	0
$265$	1.64828	0.101253
$266$	0	0
$267$	−3.24300	−0.198468
$268$	0	0
$269$	−5.41788	−0.330334	−0.165167	−	0.986266i	$-0.552816\pi$
$269$	−5.41788	−0.330334	−0.165167	+	0.986266i	$0.552816\pi$
$270$	0	0
$271$	19.2800	1.17118	0.585588	−	0.810609i	$-0.300864\pi$
$271$	19.2800	1.17118	0.585588	+	0.810609i	$0.300864\pi$
$272$	0	0
$273$	22.9154	1.38690
$274$	0	0
$275$	2.12943	0.128410
$276$	0	0
$277$	1.52593	0.0916845	0.0458423	−	0.998949i	$-0.485403\pi$
$277$	1.52593	0.0916845	0.0458423	+	0.998949i	$0.485403\pi$
$278$	0	0
$279$	−3.56183	−0.213241
$280$	0	0
$281$	−1.71328	−0.102206	−0.0511028	−	0.998693i	$-0.516274\pi$
$281$	−1.71328	−0.102206	−0.0511028	+	0.998693i	$0.516274\pi$
$282$	0	0
$283$	22.6161	1.34439	0.672194	−	0.740375i	$-0.265352\pi$
$283$	22.6161	1.34439	0.672194	+	0.740375i	$0.265352\pi$
$284$	0	0
$285$	7.98568	0.473031
$286$	0	0
$287$	−22.5389	−1.33043
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	11.7898	0.691128
$292$	0	0
$293$	−16.7508	−0.978595	−0.489298	−	0.872117i	$-0.662747\pi$
$293$	−16.7508	−0.978595	−0.489298	+	0.872117i	$0.662747\pi$
$294$	0	0
$295$	0.891187	0.0518869
$296$	0	0
$297$	−2.25239	−0.130697
$298$	0	0
$299$	14.1984	0.821116
$300$	0	0
$301$	1.19632	0.0689548
$302$	0	0
$303$	2.34376	0.134645
$304$	0	0
$305$	11.6798	0.668781
$306$	0	0
$307$	13.7158	0.782805	0.391402	−	0.920220i	$-0.371990\pi$
$307$	13.7158	0.782805	0.391402	+	0.920220i	$0.371990\pi$
$308$	0	0
$309$	−27.3877	−1.55803
$310$	0	0
$311$	8.48219	0.480981	0.240491	−	0.970651i	$-0.422692\pi$
$311$	8.48219	0.480981	0.240491	+	0.970651i	$0.422692\pi$
$312$	0	0
$313$	−6.80602	−0.384699	−0.192349	−	0.981327i	$-0.561611\pi$
$313$	−6.80602	−0.384699	−0.192349	+	0.981327i	$0.561611\pi$
$314$	0	0
$315$	2.65668	0.149687
$316$	0	0
$317$	21.3926	1.20153	0.600764	−	0.799426i	$-0.294863\pi$
$317$	21.3926	1.20153	0.600764	+	0.799426i	$0.294863\pi$
$318$	0	0
$319$	−3.72521	−0.208572
$320$	0	0
$321$	−29.1982	−1.62968
$322$	0	0
$323$	4.67019	0.259856
$324$	0	0
$325$	11.4821	0.636913
$326$	0	0
$327$	−26.5125	−1.46615
$328$	0	0
$329$	−5.30647	−0.292555
$330$	0	0
$331$	23.9165	1.31457	0.657285	−	0.753642i	$-0.271705\pi$
$331$	23.9165	1.31457	0.657285	+	0.753642i	$0.271705\pi$
$332$	0	0
$333$	−3.96521	−0.217292
$334$	0	0
$335$	−6.75499	−0.369064
$336$	0	0
$337$	23.3972	1.27452	0.637262	−	0.770647i	$-0.280067\pi$
$337$	23.3972	1.27452	0.637262	+	0.770647i	$0.280067\pi$
$338$	0	0
$339$	−5.83977	−0.317173
$340$	0	0
$341$	2.64645	0.143313
$342$	0	0
$343$	22.4750	1.21353
$344$	0	0
$345$	8.89290	0.478778
$346$	0	0
$347$	12.9698	0.696255	0.348128	−	0.937447i	$-0.386818\pi$
$347$	12.9698	0.696255	0.348128	+	0.937447i	$0.386818\pi$
$348$	0	0
$349$	−22.9330	−1.22757	−0.613787	−	0.789472i	$-0.710355\pi$
$349$	−22.9330	−1.22757	−0.613787	+	0.789472i	$0.710355\pi$
$350$	0	0
$351$	−12.1451	−0.648259
$352$	0	0
$353$	29.6313	1.57712	0.788558	−	0.614960i	$-0.210828\pi$
$353$	29.6313	1.57712	0.788558	+	0.614960i	$0.210828\pi$
$354$	0	0
$355$	4.93709	0.262034
$356$	0	0
$357$	8.39367	0.444240
$358$	0	0
$359$	−5.27058	−0.278171	−0.139085	−	0.990280i	$-0.544416\pi$
$359$	−5.27058	−0.278171	−0.139085	+	0.990280i	$0.544416\pi$
$360$	0	0
$361$	2.81063	0.147928
$362$	0	0
$363$	20.6139	1.08195
$364$	0	0
$365$	9.13879	0.478346
$366$	0	0
$367$	5.36334	0.279964	0.139982	−	0.990154i	$-0.455296\pi$
$367$	5.36334	0.279964	0.139982	+	0.990154i	$0.455296\pi$
$368$	0	0
$369$	−3.51089	−0.182769
$370$	0	0
$371$	8.09107	0.420068
$372$	0	0
$373$	−18.4446	−0.955026	−0.477513	−	0.878625i	$-0.658462\pi$
$373$	−18.4446	−0.955026	−0.477513	+	0.878625i	$0.658462\pi$
$374$	0	0
$375$	15.7412	0.812873
$376$	0	0
$377$	−20.0867	−1.03452
$378$	0	0
$379$	32.2143	1.65474	0.827369	−	0.561659i	$-0.189837\pi$
$379$	32.2143	1.65474	0.827369	+	0.561659i	$0.189837\pi$
$380$	0	0
$381$	28.9665	1.48400
$382$	0	0
$383$	20.9701	1.07152	0.535761	−	0.844369i	$-0.320025\pi$
$383$	20.9701	1.07152	0.535761	+	0.844369i	$0.320025\pi$
$384$	0	0
$385$	−1.97392	−0.100600
$386$	0	0
$387$	0.186351	0.00947274
$388$	0	0
$389$	−10.3921	−0.526901	−0.263451	−	0.964673i	$-0.584861\pi$
$389$	−10.3921	−0.526901	−0.263451	+	0.964673i	$0.584861\pi$
$390$	0	0
$391$	5.20075	0.263013
$392$	0	0
$393$	−14.7168	−0.742363
$394$	0	0
$395$	15.0504	0.757265
$396$	0	0
$397$	10.6607	0.535044	0.267522	−	0.963552i	$-0.413795\pi$
$397$	10.6607	0.535044	0.267522	+	0.963552i	$0.413795\pi$
$398$	0	0
$399$	39.2000	1.96245
$400$	0	0
$401$	2.39510	0.119606	0.0598029	−	0.998210i	$-0.480953\pi$
$401$	2.39510	0.119606	0.0598029	+	0.998210i	$0.480953\pi$
$402$	0	0
$403$	14.2699	0.710834
$404$	0	0
$405$	−9.42872	−0.468517
$406$	0	0
$407$	2.94616	0.146036
$408$	0	0
$409$	19.7965	0.978876	0.489438	−	0.872038i	$-0.337202\pi$
$409$	19.7965	0.978876	0.489438	+	0.872038i	$0.337202\pi$
$410$	0	0
$411$	13.2013	0.651171
$412$	0	0
$413$	4.37465	0.215262
$414$	0	0
$415$	13.5316	0.664242
$416$	0	0
$417$	−22.5918	−1.10633
$418$	0	0
$419$	20.0469	0.979357	0.489679	−	0.871903i	$-0.337114\pi$
$419$	20.0469	0.979357	0.489679	+	0.871903i	$0.337114\pi$
$420$	0	0
$421$	16.1388	0.786557	0.393278	−	0.919419i	$-0.371341\pi$
$421$	16.1388	0.786557	0.393278	+	0.919419i	$0.371341\pi$
$422$	0	0
$423$	−0.826589	−0.0401901
$424$	0	0
$425$	4.20579	0.204011
$426$	0	0
$427$	57.3334	2.77456
$428$	0	0
$429$	−2.65216	−0.128048
$430$	0	0
$431$	−20.2522	−0.975514	−0.487757	−	0.872980i	$-0.662185\pi$
$431$	−20.2522	−0.975514	−0.487757	+	0.872980i	$0.662185\pi$
$432$	0	0
$433$	2.82232	0.135632	0.0678161	−	0.997698i	$-0.478397\pi$
$433$	2.82232	0.135632	0.0678161	+	0.997698i	$0.478397\pi$
$434$	0	0
$435$	−12.5809	−0.603208
$436$	0	0
$437$	24.2885	1.16187
$438$	0	0
$439$	−10.3494	−0.493950	−0.246975	−	0.969022i	$-0.579436\pi$
$439$	−10.3494	−0.493950	−0.246975	+	0.969022i	$0.579436\pi$
$440$	0	0
$441$	8.27100	0.393857
$442$	0	0
$443$	21.1701	1.00582	0.502910	−	0.864339i	$-0.332263\pi$
$443$	21.1701	1.00582	0.502910	+	0.864339i	$0.332263\pi$
$444$	0	0
$445$	1.50629	0.0714048
$446$	0	0
$447$	−29.3489	−1.38816
$448$	0	0
$449$	−34.5453	−1.63029	−0.815147	−	0.579254i	$-0.803344\pi$
$449$	−34.5453	−1.63029	−0.815147	+	0.579254i	$0.803344\pi$
$450$	0	0
$451$	2.60860	0.122834
$452$	0	0
$453$	−39.9036	−1.87483
$454$	0	0
$455$	−10.6436	−0.498977
$456$	0	0
$457$	5.60786	0.262325	0.131162	−	0.991361i	$-0.458129\pi$
$457$	5.60786	0.262325	0.131162	+	0.991361i	$0.458129\pi$
$458$	0	0
$459$	−4.44864	−0.207645
$460$	0	0
$461$	13.4004	0.624117	0.312058	−	0.950063i	$-0.398982\pi$
$461$	13.4004	0.624117	0.312058	+	0.950063i	$0.398982\pi$
$462$	0	0
$463$	−19.9721	−0.928183	−0.464091	−	0.885787i	$-0.653619\pi$
$463$	−19.9721	−0.928183	−0.464091	+	0.885787i	$0.653619\pi$
$464$	0	0
$465$	8.93766	0.414474
$466$	0	0
$467$	−8.16633	−0.377892	−0.188946	−	0.981987i	$-0.560507\pi$
$467$	−8.16633	−0.377892	−0.188946	+	0.981987i	$0.560507\pi$
$468$	0	0
$469$	−33.1588	−1.53113
$470$	0	0
$471$	23.1321	1.06587
$472$	0	0
$473$	−0.138459	−0.00636635
$474$	0	0
$475$	19.6418	0.901228
$476$	0	0
$477$	1.26035	0.0577073
$478$	0	0
$479$	−2.77149	−0.126633	−0.0633163	−	0.997994i	$-0.520168\pi$
$479$	−2.77149	−0.126633	−0.0633163	+	0.997994i	$0.520168\pi$
$480$	0	0
$481$	15.8860	0.724337
$482$	0	0
$483$	43.6534	1.98630
$484$	0	0
$485$	−5.47602	−0.248653
$486$	0	0
$487$	31.9703	1.44871	0.724357	−	0.689426i	$-0.242137\pi$
$487$	31.9703	1.44871	0.724357	+	0.689426i	$0.242137\pi$
$488$	0	0
$489$	−27.6948	−1.25240
$490$	0	0
$491$	−10.5423	−0.475766	−0.237883	−	0.971294i	$-0.576453\pi$
$491$	−10.5423	−0.475766	−0.237883	+	0.971294i	$0.576453\pi$
$492$	0	0
$493$	−7.35756	−0.331368
$494$	0	0
$495$	−0.307477	−0.0138201
$496$	0	0
$497$	24.2352	1.08710
$498$	0	0
$499$	−9.56453	−0.428167	−0.214084	−	0.976815i	$-0.568676\pi$
$499$	−9.56453	−0.428167	−0.214084	+	0.976815i	$0.568676\pi$
$500$	0	0
$501$	−28.3689	−1.26743
$502$	0	0
$503$	−32.8938	−1.46666	−0.733332	−	0.679871i	$-0.762036\pi$
$503$	−32.8938	−1.46666	−0.733332	+	0.679871i	$0.762036\pi$
$504$	0	0
$505$	−1.08861	−0.0484425
$506$	0	0
$507$	10.6425	0.472649
$508$	0	0
$509$	21.9496	0.972899	0.486449	−	0.873709i	$-0.338292\pi$
$509$	21.9496	0.972899	0.486449	+	0.873709i	$0.338292\pi$
$510$	0	0
$511$	44.8604	1.98451
$512$	0	0
$513$	−20.7760	−0.917282
$514$	0	0
$515$	12.7208	0.560547
$516$	0	0
$517$	0.614157	0.0270106
$518$	0	0
$519$	31.2460	1.37155
$520$	0	0
$521$	29.2359	1.28085	0.640423	−	0.768022i	$-0.278759\pi$
$521$	29.2359	1.28085	0.640423	+	0.768022i	$0.278759\pi$
$522$	0	0
$523$	−17.1644	−0.750548	−0.375274	−	0.926914i	$-0.622451\pi$
$523$	−17.1644	−0.750548	−0.375274	+	0.926914i	$0.622451\pi$
$524$	0	0
$525$	35.3020	1.54070
$526$	0	0
$527$	5.22692	0.227688
$528$	0	0
$529$	4.04777	0.175990
$530$	0	0
$531$	0.681439	0.0295719
$532$	0	0
$533$	14.0658	0.609258
$534$	0	0
$535$	13.5617	0.586325
$536$	0	0
$537$	4.63174	0.199874
$538$	0	0
$539$	−6.14537	−0.264700
$540$	0	0
$541$	5.02029	0.215839	0.107919	−	0.994160i	$-0.465581\pi$
$541$	5.02029	0.215839	0.107919	+	0.994160i	$0.465581\pi$
$542$	0	0
$543$	−5.07337	−0.217719
$544$	0	0
$545$	12.3143	0.527488
$546$	0	0
$547$	21.1668	0.905028	0.452514	−	0.891757i	$-0.350527\pi$
$547$	21.1668	0.905028	0.452514	+	0.891757i	$0.350527\pi$
$548$	0	0
$549$	8.93083	0.381158
$550$	0	0
$551$	−34.3612	−1.46384
$552$	0	0
$553$	73.8790	3.14165
$554$	0	0
$555$	9.94985	0.422348
$556$	0	0
$557$	−20.9660	−0.888356	−0.444178	−	0.895938i	$-0.646504\pi$
$557$	−20.9660	−0.888356	−0.444178	+	0.895938i	$0.646504\pi$
$558$	0	0
$559$	−0.746585	−0.0315772
$560$	0	0
$561$	−0.971462	−0.0410152
$562$	0	0
$563$	22.9259	0.966210	0.483105	−	0.875562i	$-0.339509\pi$
$563$	22.9259	0.966210	0.483105	+	0.875562i	$0.339509\pi$
$564$	0	0
$565$	2.71241	0.114112
$566$	0	0
$567$	−46.2836	−1.94373
$568$	0	0
$569$	−39.3266	−1.64866	−0.824328	−	0.566112i	$-0.808447\pi$
$569$	−39.3266	−1.64866	−0.824328	+	0.566112i	$0.808447\pi$
$570$	0	0
$571$	12.8966	0.539707	0.269853	−	0.962901i	$-0.413025\pi$
$571$	12.8966	0.539707	0.269853	+	0.962901i	$0.413025\pi$
$572$	0	0
$573$	11.7756	0.491931
$574$	0	0
$575$	21.8732	0.912177
$576$	0	0
$577$	−0.736128	−0.0306454	−0.0153227	−	0.999883i	$-0.504878\pi$
$577$	−0.736128	−0.0306454	−0.0153227	+	0.999883i	$0.504878\pi$
$578$	0	0
$579$	−3.42840	−0.142480
$580$	0	0
$581$	66.4240	2.75573
$582$	0	0
$583$	−0.936440	−0.0387834
$584$	0	0
$585$	−1.65795	−0.0685476
$586$	0	0
$587$	9.22823	0.380890	0.190445	−	0.981698i	$-0.439007\pi$
$587$	9.22823	0.380890	0.190445	+	0.981698i	$0.439007\pi$
$588$	0	0
$589$	24.4107	1.00583
$590$	0	0
$591$	−13.5695	−0.558175
$592$	0	0
$593$	−20.4643	−0.840370	−0.420185	−	0.907439i	$-0.638035\pi$
$593$	−20.4643	−0.840370	−0.420185	+	0.907439i	$0.638035\pi$
$594$	0	0
$595$	−3.89863	−0.159828
$596$	0	0
$597$	−1.82894	−0.0748537
$598$	0	0
$599$	−44.7628	−1.82896	−0.914478	−	0.404635i	$-0.867399\pi$
$599$	−44.7628	−1.82896	−0.914478	+	0.404635i	$0.867399\pi$
$600$	0	0
$601$	13.6617	0.557274	0.278637	−	0.960396i	$-0.410117\pi$
$601$	13.6617	0.557274	0.278637	+	0.960396i	$0.410117\pi$
$602$	0	0
$603$	−5.16515	−0.210341
$604$	0	0
$605$	−9.57460	−0.389263
$606$	0	0
$607$	−0.786702	−0.0319312	−0.0159656	−	0.999873i	$-0.505082\pi$
$607$	−0.786702	−0.0319312	−0.0159656	+	0.999873i	$0.505082\pi$
$608$	0	0
$609$	−61.7570	−2.50252
$610$	0	0
$611$	3.31159	0.133973
$612$	0	0
$613$	2.12708	0.0859121	0.0429561	−	0.999077i	$-0.486322\pi$
$613$	2.12708	0.0859121	0.0429561	+	0.999077i	$0.486322\pi$
$614$	0	0
$615$	8.80983	0.355247
$616$	0	0
$617$	−37.5497	−1.51170	−0.755848	−	0.654747i	$-0.772775\pi$
$617$	−37.5497	−1.51170	−0.755848	+	0.654747i	$0.772775\pi$
$618$	0	0
$619$	5.46438	0.219632	0.109816	−	0.993952i	$-0.464974\pi$
$619$	5.46438	0.219632	0.109816	+	0.993952i	$0.464974\pi$
$620$	0	0
$621$	−23.1363	−0.928426
$622$	0	0
$623$	7.39404	0.296236
$624$	0	0
$625$	13.7176	0.548703
$626$	0	0
$627$	−4.53691	−0.181187
$628$	0	0
$629$	5.81887	0.232014
$630$	0	0
$631$	15.2830	0.608405	0.304202	−	0.952607i	$-0.401610\pi$
$631$	15.2830	0.608405	0.304202	+	0.952607i	$0.401610\pi$
$632$	0	0
$633$	4.49241	0.178557
$634$	0	0
$635$	−13.4541	−0.533911
$636$	0	0
$637$	−33.1364	−1.31291
$638$	0	0
$639$	3.77511	0.149341
$640$	0	0
$641$	5.71256	0.225632	0.112816	−	0.993616i	$-0.464013\pi$
$641$	5.71256	0.225632	0.112816	+	0.993616i	$0.464013\pi$
$642$	0	0
$643$	32.2523	1.27191	0.635953	−	0.771728i	$-0.280607\pi$
$643$	32.2523	1.27191	0.635953	+	0.771728i	$0.280607\pi$
$644$	0	0
$645$	−0.467608	−0.0184121
$646$	0	0
$647$	−35.9587	−1.41368	−0.706842	−	0.707372i	$-0.749881\pi$
$647$	−35.9587	−1.41368	−0.706842	+	0.707372i	$0.749881\pi$
$648$	0	0
$649$	−0.506311	−0.0198744
$650$	0	0
$651$	43.8731	1.71952
$652$	0	0
$653$	21.5251	0.842340	0.421170	−	0.906982i	$-0.361620\pi$
$653$	21.5251	0.842340	0.421170	+	0.906982i	$0.361620\pi$
$654$	0	0
$655$	6.83554	0.267087
$656$	0	0
$657$	6.98790	0.272624
$658$	0	0
$659$	7.83260	0.305115	0.152557	−	0.988295i	$-0.451249\pi$
$659$	7.83260	0.305115	0.152557	+	0.988295i	$0.451249\pi$
$660$	0	0
$661$	34.3038	1.33426	0.667132	−	0.744939i	$-0.267522\pi$
$661$	34.3038	1.33426	0.667132	+	0.744939i	$0.267522\pi$
$662$	0	0
$663$	−5.23822	−0.203435
$664$	0	0
$665$	−18.2073	−0.706050
$666$	0	0
$667$	−38.2648	−1.48162
$668$	0	0
$669$	−16.9533	−0.655451
$670$	0	0
$671$	−6.63562	−0.256165
$672$	0	0
$673$	−31.4791	−1.21343	−0.606716	−	0.794919i	$-0.707513\pi$
$673$	−31.4791	−1.21343	−0.606716	+	0.794919i	$0.707513\pi$
$674$	0	0
$675$	−18.7100	−0.720149
$676$	0	0
$677$	−13.8367	−0.531788	−0.265894	−	0.964002i	$-0.585667\pi$
$677$	−13.8367	−0.531788	−0.265894	+	0.964002i	$0.585667\pi$
$678$	0	0
$679$	−26.8806	−1.03158
$680$	0	0
$681$	2.90016	0.111134
$682$	0	0
$683$	−2.84027	−0.108680	−0.0543400	−	0.998522i	$-0.517305\pi$
$683$	−2.84027	−0.108680	−0.0543400	+	0.998522i	$0.517305\pi$
$684$	0	0
$685$	−6.13163	−0.234278
$686$	0	0
$687$	7.61241	0.290432
$688$	0	0
$689$	−5.04937	−0.192366
$690$	0	0
$691$	24.4888	0.931597	0.465798	−	0.884891i	$-0.345767\pi$
$691$	24.4888	0.931597	0.465798	+	0.884891i	$0.345767\pi$
$692$	0	0
$693$	−1.50934	−0.0573351
$694$	0	0
$695$	10.4933	0.398033
$696$	0	0
$697$	5.15217	0.195152
$698$	0	0
$699$	−36.2913	−1.37266
$700$	0	0
$701$	7.92906	0.299477	0.149738	−	0.988726i	$-0.452157\pi$
$701$	7.92906	0.299477	0.149738	+	0.988726i	$0.452157\pi$
$702$	0	0
$703$	27.1752	1.02493
$704$	0	0
$705$	2.07415	0.0781170
$706$	0	0
$707$	−5.34376	−0.200973
$708$	0	0
$709$	21.5002	0.807455	0.403728	−	0.914879i	$-0.367714\pi$
$709$	21.5002	0.807455	0.403728	+	0.914879i	$0.367714\pi$
$710$	0	0
$711$	11.5081	0.431589
$712$	0	0
$713$	27.1839	1.01805
$714$	0	0
$715$	1.23186	0.0460688
$716$	0	0
$717$	51.4553	1.92163
$718$	0	0
$719$	−34.3561	−1.28127	−0.640633	−	0.767847i	$-0.721328\pi$
$719$	−34.3561	−1.28127	−0.640633	+	0.767847i	$0.721328\pi$
$720$	0	0
$721$	62.4439	2.32553
$722$	0	0
$723$	48.8158	1.81548
$724$	0	0
$725$	−30.9443	−1.14924
$726$	0	0
$727$	2.18050	0.0808703	0.0404352	−	0.999182i	$-0.487126\pi$
$727$	2.18050	0.0808703	0.0404352	+	0.999182i	$0.487126\pi$
$728$	0	0
$729$	18.3973	0.681383
$730$	0	0
$731$	−0.273467	−0.0101145
$732$	0	0
$733$	47.8726	1.76821	0.884107	−	0.467284i	$-0.154768\pi$
$733$	47.8726	1.76821	0.884107	+	0.467284i	$0.154768\pi$
$734$	0	0
$735$	−20.7543	−0.765535
$736$	0	0
$737$	3.83772	0.141364
$738$	0	0
$739$	13.5291	0.497675	0.248837	−	0.968545i	$-0.419952\pi$
$739$	13.5291	0.497675	0.248837	+	0.968545i	$0.419952\pi$
$740$	0	0
$741$	−24.4634	−0.898687
$742$	0	0
$743$	−8.30478	−0.304673	−0.152336	−	0.988329i	$-0.548680\pi$
$743$	−8.30478	−0.304673	−0.152336	+	0.988329i	$0.548680\pi$
$744$	0	0
$745$	13.6318	0.499429
$746$	0	0
$747$	10.3469	0.378572
$748$	0	0
$749$	66.5717	2.43248
$750$	0	0
$751$	49.6673	1.81239	0.906194	−	0.422863i	$-0.138975\pi$
$751$	49.6673	1.81239	0.906194	+	0.422863i	$0.138975\pi$
$752$	0	0
$753$	6.31663	0.230191
$754$	0	0
$755$	18.5341	0.674525
$756$	0	0
$757$	−10.1952	−0.370549	−0.185275	−	0.982687i	$-0.559317\pi$
$757$	−10.1952	−0.370549	−0.185275	+	0.982687i	$0.559317\pi$
$758$	0	0
$759$	−5.05233	−0.183388
$760$	0	0
$761$	−39.0239	−1.41461	−0.707307	−	0.706906i	$-0.750090\pi$
$761$	−39.0239	−1.41461	−0.707307	+	0.706906i	$0.750090\pi$
$762$	0	0
$763$	60.4485	2.18838
$764$	0	0
$765$	−0.607289	−0.0219566
$766$	0	0
$767$	−2.73007	−0.0985773
$768$	0	0
$769$	−3.36338	−0.121287	−0.0606434	−	0.998159i	$-0.519315\pi$
$769$	−3.36338	−0.121287	−0.0606434	+	0.998159i	$0.519315\pi$
$770$	0	0
$771$	15.9535	0.574552
$772$	0	0
$773$	−27.3653	−0.984261	−0.492130	−	0.870521i	$-0.663782\pi$
$773$	−27.3653	−0.984261	−0.492130	+	0.870521i	$0.663782\pi$
$774$	0	0
$775$	21.9833	0.789664
$776$	0	0
$777$	48.8417	1.75219
$778$	0	0
$779$	24.0616	0.862095
$780$	0	0
$781$	−2.80491	−0.100368
$782$	0	0
$783$	32.7311	1.16972
$784$	0	0
$785$	−10.7442	−0.383477
$786$	0	0
$787$	−19.2197	−0.685110	−0.342555	−	0.939498i	$-0.611292\pi$
$787$	−19.2197	−0.685110	−0.342555	+	0.939498i	$0.611292\pi$
$788$	0	0
$789$	21.0677	0.750029
$790$	0	0
$791$	13.3146	0.473414
$792$	0	0
$793$	−35.7799	−1.27058
$794$	0	0
$795$	−3.16257	−0.112165
$796$	0	0
$797$	−1.85119	−0.0655725	−0.0327863	−	0.999462i	$-0.510438\pi$
$797$	−1.85119	−0.0655725	−0.0327863	+	0.999462i	$0.510438\pi$
$798$	0	0
$799$	1.21300	0.0429130
$800$	0	0
$801$	1.15177	0.0406957
$802$	0	0
$803$	−5.19202	−0.183223
$804$	0	0
$805$	−20.2758	−0.714628
$806$	0	0
$807$	10.3953	0.365933
$808$	0	0
$809$	20.0979	0.706606	0.353303	−	0.935509i	$-0.385059\pi$
$809$	20.0979	0.706606	0.353303	+	0.935509i	$0.385059\pi$
$810$	0	0
$811$	29.7843	1.04587	0.522935	−	0.852373i	$-0.324837\pi$
$811$	29.7843	1.04587	0.522935	+	0.852373i	$0.324837\pi$
$812$	0	0
$813$	−36.9927	−1.29739
$814$	0	0
$815$	12.8635	0.450588
$816$	0	0
$817$	−1.27714	−0.0446815
$818$	0	0
$819$	−8.13850	−0.284382
$820$	0	0
$821$	−33.4952	−1.16899	−0.584496	−	0.811397i	$-0.698708\pi$
$821$	−33.4952	−1.16899	−0.584496	+	0.811397i	$0.698708\pi$
$822$	0	0
$823$	6.37741	0.222302	0.111151	−	0.993804i	$-0.464546\pi$
$823$	6.37741	0.222302	0.111151	+	0.993804i	$0.464546\pi$
$824$	0	0
$825$	−4.08576	−0.142248
$826$	0	0
$827$	−7.84146	−0.272674	−0.136337	−	0.990662i	$-0.543533\pi$
$827$	−7.84146	−0.272674	−0.136337	+	0.990662i	$0.543533\pi$
$828$	0	0
$829$	15.3962	0.534731	0.267366	−	0.963595i	$-0.413847\pi$
$829$	15.3962	0.534731	0.267366	+	0.963595i	$0.413847\pi$
$830$	0	0
$831$	−2.92782	−0.101565
$832$	0	0
$833$	−12.1375	−0.420541
$834$	0	0
$835$	13.1766	0.455994
$836$	0	0
$837$	−23.2527	−0.803731
$838$	0	0
$839$	6.32392	0.218326	0.109163	−	0.994024i	$-0.465183\pi$
$839$	6.32392	0.218326	0.109163	+	0.994024i	$0.465183\pi$
$840$	0	0
$841$	25.1337	0.866680
$842$	0	0
$843$	3.28728	0.113220
$844$	0	0
$845$	−4.94314	−0.170049
$846$	0	0
$847$	−46.9997	−1.61493
$848$	0	0
$849$	−43.3937	−1.48927
$850$	0	0
$851$	30.2625	1.03738
$852$	0	0
$853$	−58.0300	−1.98691	−0.993454	−	0.114231i	$-0.963560\pi$
$853$	−58.0300	−1.98691	−0.993454	+	0.114231i	$0.963560\pi$
$854$	0	0
$855$	−2.83615	−0.0969944
$856$	0	0
$857$	30.8791	1.05481	0.527406	−	0.849614i	$-0.323165\pi$
$857$	30.8791	1.05481	0.527406	+	0.849614i	$0.323165\pi$
$858$	0	0
$859$	53.0599	1.81038	0.905190	−	0.425008i	$-0.139729\pi$
$859$	53.0599	1.81038	0.905190	+	0.425008i	$0.139729\pi$
$860$	0	0
$861$	43.2456	1.47381
$862$	0	0
$863$	−2.27488	−0.0774379	−0.0387190	−	0.999250i	$-0.512328\pi$
$863$	−2.27488	−0.0774379	−0.0387190	+	0.999250i	$0.512328\pi$
$864$	0	0
$865$	−14.5129	−0.493454
$866$	0	0
$867$	−1.91871	−0.0651627
$868$	0	0
$869$	−8.55057	−0.290058
$870$	0	0
$871$	20.6933	0.701167
$872$	0	0
$873$	−4.18719	−0.141715
$874$	0	0
$875$	−35.8899	−1.21330
$876$	0	0
$877$	−1.67794	−0.0566600	−0.0283300	−	0.999599i	$-0.509019\pi$
$877$	−1.67794	−0.0566600	−0.0283300	+	0.999599i	$0.509019\pi$
$878$	0	0
$879$	32.1400	1.08405
$880$	0	0
$881$	−5.92444	−0.199600	−0.0997998	−	0.995008i	$-0.531820\pi$
$881$	−5.92444	−0.199600	−0.0997998	+	0.995008i	$0.531820\pi$
$882$	0	0
$883$	−38.6090	−1.29930	−0.649648	−	0.760235i	$-0.725084\pi$
$883$	−38.6090	−1.29930	−0.649648	+	0.760235i	$0.725084\pi$
$884$	0	0
$885$	−1.70993	−0.0574786
$886$	0	0
$887$	37.1978	1.24898	0.624490	−	0.781033i	$-0.285307\pi$
$887$	37.1978	1.24898	0.624490	+	0.781033i	$0.285307\pi$
$888$	0	0
$889$	−66.0435	−2.21503
$890$	0	0
$891$	5.35674	0.179458
$892$	0	0
$893$	5.66496	0.189571
$894$	0	0
$895$	−2.15132	−0.0719105
$896$	0	0
$897$	−27.2426	−0.909605
$898$	0	0
$899$	−38.4574	−1.28263
$900$	0	0
$901$	−1.84954	−0.0616170
$902$	0	0
$903$	−2.29539	−0.0763858
$904$	0	0
$905$	2.35644	0.0783307
$906$	0	0
$907$	−9.00494	−0.299004	−0.149502	−	0.988761i	$-0.547767\pi$
$907$	−9.00494	−0.299004	−0.149502	+	0.988761i	$0.547767\pi$
$908$	0	0
$909$	−0.832397	−0.0276089
$910$	0	0
$911$	−31.0898	−1.03005	−0.515026	−	0.857174i	$-0.672218\pi$
$911$	−31.0898	−1.03005	−0.515026	+	0.857174i	$0.672218\pi$
$912$	0	0
$913$	−7.68774	−0.254427
$914$	0	0
$915$	−22.4100	−0.740853
$916$	0	0
$917$	33.5542	1.10806
$918$	0	0
$919$	−33.7614	−1.11369	−0.556843	−	0.830618i	$-0.687988\pi$
$919$	−33.7614	−1.11369	−0.556843	+	0.830618i	$0.687988\pi$
$920$	0	0
$921$	−26.3167	−0.867165
$922$	0	0
$923$	−15.1244	−0.497825
$924$	0	0
$925$	24.4729	0.804665
$926$	0	0
$927$	9.72688	0.319473
$928$	0	0
$929$	27.6999	0.908805	0.454403	−	0.890796i	$-0.349853\pi$
$929$	27.6999	0.908805	0.454403	+	0.890796i	$0.349853\pi$
$930$	0	0
$931$	−56.6846	−1.85776
$932$	0	0
$933$	−16.2748	−0.532815
$934$	0	0
$935$	0.451217	0.0147564
$936$	0	0
$937$	47.5003	1.55177	0.775883	−	0.630877i	$-0.217305\pi$
$937$	47.5003	1.55177	0.775883	+	0.630877i	$0.217305\pi$
$938$	0	0
$939$	13.0588	0.426156
$940$	0	0
$941$	−11.4133	−0.372064	−0.186032	−	0.982544i	$-0.559563\pi$
$941$	−11.4133	−0.372064	−0.186032	+	0.982544i	$0.559563\pi$
$942$	0	0
$943$	26.7951	0.872569
$944$	0	0
$945$	17.3436	0.564187
$946$	0	0
$947$	20.1587	0.655071	0.327536	−	0.944839i	$-0.393782\pi$
$947$	20.1587	0.655071	0.327536	+	0.944839i	$0.393782\pi$
$948$	0	0
$949$	−27.9959	−0.908785
$950$	0	0
$951$	−41.0462	−1.33101
$952$	0	0
$953$	−54.9545	−1.78015	−0.890076	−	0.455812i	$-0.849349\pi$
$953$	−54.9545	−1.78015	−0.890076	+	0.455812i	$0.849349\pi$
$954$	0	0
$955$	−5.46942	−0.176986
$956$	0	0
$957$	7.14759	0.231049
$958$	0	0
$959$	−30.0989	−0.971943
$960$	0	0
$961$	−3.67926	−0.118686
$962$	0	0
$963$	10.3699	0.334165
$964$	0	0
$965$	1.59240	0.0512612
$966$	0	0
$967$	−29.4955	−0.948512	−0.474256	−	0.880387i	$-0.657283\pi$
$967$	−29.4955	−0.948512	−0.474256	+	0.880387i	$0.657283\pi$
$968$	0	0
$969$	−8.96072	−0.287860
$970$	0	0
$971$	−43.1642	−1.38521	−0.692603	−	0.721319i	$-0.743536\pi$
$971$	−43.1642	−1.38521	−0.692603	+	0.721319i	$0.743536\pi$
$972$	0	0
$973$	51.5093	1.65131
$974$	0	0
$975$	−22.0308	−0.705551
$976$	0	0
$977$	42.3620	1.35528	0.677640	−	0.735394i	$-0.263003\pi$
$977$	42.3620	1.35528	0.677640	+	0.735394i	$0.263003\pi$
$978$	0	0
$979$	−0.855767	−0.0273504
$980$	0	0
$981$	9.41606	0.300632
$982$	0	0
$983$	45.2811	1.44424	0.722121	−	0.691767i	$-0.243167\pi$
$983$	45.2811	1.44424	0.722121	+	0.691767i	$0.243167\pi$
$984$	0	0
$985$	6.30267	0.200820
$986$	0	0
$987$	10.1816	0.324083
$988$	0	0
$989$	−1.42223	−0.0452243
$990$	0	0
$991$	−47.9689	−1.52378	−0.761892	−	0.647704i	$-0.775729\pi$
$991$	−47.9689	−1.52378	−0.761892	+	0.647704i	$0.775729\pi$
$992$	0	0
$993$	−45.8888	−1.45624
$994$	0	0
$995$	0.849495	0.0269308
$996$	0	0
$997$	−51.8972	−1.64360	−0.821800	−	0.569776i	$-0.807030\pi$
$997$	−51.8972	−1.64360	−0.821800	+	0.569776i	$0.807030\pi$
$998$	0	0
$999$	−25.8861	−0.818999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.ba.1.8	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.ba.1.8	✓	30	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)

\(f(q)\)	\(=\)	\(q-1.91871 q^{3} +0.891187 q^{5} +4.37465 q^{7} +0.681439 q^{9} +O(q^{10})\)	\(q-1.91871 q^{3} +0.891187 q^{5} +4.37465 q^{7} +0.681439 q^{9} -0.506311 q^{11} -2.73007 q^{13} -1.70993 q^{15} -1.00000 q^{17} -4.67019 q^{19} -8.39367 q^{21} -5.20075 q^{23} -4.20579 q^{25} +4.44864 q^{27} +7.35756 q^{29} -5.22692 q^{31} +0.971462 q^{33} +3.89863 q^{35} -5.81887 q^{37} +5.23822 q^{39} -5.15217 q^{41} +0.273467 q^{43} +0.607289 q^{45} -1.21300 q^{47} +12.1375 q^{49} +1.91871 q^{51} +1.84954 q^{53} -0.451217 q^{55} +8.96072 q^{57} +1.00000 q^{59} +13.1058 q^{61} +2.98106 q^{63} -2.43301 q^{65} -7.57977 q^{67} +9.97871 q^{69} +5.53991 q^{71} +10.2546 q^{73} +8.06967 q^{75} -2.21493 q^{77} +16.8880 q^{79} -10.5800 q^{81} +15.1838 q^{83} -0.891187 q^{85} -14.1170 q^{87} +1.69020 q^{89} -11.9431 q^{91} +10.0289 q^{93} -4.16201 q^{95} -6.14463 q^{97} -0.345020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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