LMFDB - Embedded newform 6028.2.a.d.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.900414 q^{3} +4.31358 q^{5} +4.18187 q^{7} -2.18926 q^{9} +O(q^{10})$	$q-0.900414 q^{3} +4.31358 q^{5} +4.18187 q^{7} -2.18926 q^{9} -1.00000 q^{11} -6.12366 q^{13} -3.88401 q^{15} -7.84422 q^{17} +0.784023 q^{19} -3.76542 q^{21} -6.75064 q^{23} +13.6070 q^{25} +4.67248 q^{27} +1.15882 q^{29} +2.81537 q^{31} +0.900414 q^{33} +18.0389 q^{35} -3.36007 q^{37} +5.51383 q^{39} -10.1413 q^{41} -3.48329 q^{43} -9.44354 q^{45} -6.79750 q^{47} +10.4881 q^{49} +7.06305 q^{51} -9.22830 q^{53} -4.31358 q^{55} -0.705945 q^{57} -4.10449 q^{59} +9.94756 q^{61} -9.15519 q^{63} -26.4149 q^{65} -6.67646 q^{67} +6.07837 q^{69} -0.268097 q^{71} -7.49879 q^{73} -12.2519 q^{75} -4.18187 q^{77} +8.20310 q^{79} +2.36060 q^{81} -10.3886 q^{83} -33.8367 q^{85} -1.04341 q^{87} +1.21472 q^{89} -25.6084 q^{91} -2.53499 q^{93} +3.38195 q^{95} +2.51963 q^{97} +2.18926 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * 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
$3$	−0.900414	−0.519854	−0.259927	−	0.965628i	$-0.583699\pi$
$3$	−0.900414	−0.519854	−0.259927	+	0.965628i	$0.583699\pi$
$4$	0	0
$5$	4.31358	1.92909	0.964547	−	0.263912i	$-0.0850128\pi$
$5$	4.31358	1.92909	0.964547	+	0.263912i	$0.0850128\pi$
$6$	0	0
$7$	4.18187	1.58060	0.790300	−	0.612720i	$-0.209925\pi$
$7$	4.18187	1.58060	0.790300	+	0.612720i	$0.209925\pi$
$8$	0	0
$9$	−2.18926	−0.729752
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.12366	−1.69840	−0.849199	−	0.528073i	$-0.822915\pi$
$13$	−6.12366	−1.69840	−0.849199	+	0.528073i	$0.822915\pi$
$14$	0	0
$15$	−3.88401	−1.00285
$16$	0	0
$17$	−7.84422	−1.90250	−0.951252	−	0.308416i	$-0.900201\pi$
$17$	−7.84422	−1.90250	−0.951252	+	0.308416i	$0.900201\pi$
$18$	0	0
$19$	0.784023	0.179867	0.0899336	−	0.995948i	$-0.471335\pi$
$19$	0.784023	0.179867	0.0899336	+	0.995948i	$0.471335\pi$
$20$	0	0
$21$	−3.76542	−0.821681
$22$	0	0
$23$	−6.75064	−1.40761	−0.703803	−	0.710395i	$-0.748516\pi$
$23$	−6.75064	−1.40761	−0.703803	+	0.710395i	$0.748516\pi$
$24$	0	0
$25$	13.6070	2.72140
$26$	0	0
$27$	4.67248	0.899219
$28$	0	0
$29$	1.15882	0.215187	0.107593	−	0.994195i	$-0.465686\pi$
$29$	1.15882	0.215187	0.107593	+	0.994195i	$0.465686\pi$
$30$	0	0
$31$	2.81537	0.505655	0.252827	−	0.967511i	$-0.418640\pi$
$31$	2.81537	0.505655	0.252827	+	0.967511i	$0.418640\pi$
$32$	0	0
$33$	0.900414	0.156742
$34$	0	0
$35$	18.0389	3.04912
$36$	0	0
$37$	−3.36007	−0.552393	−0.276196	−	0.961101i	$-0.589074\pi$
$37$	−3.36007	−0.552393	−0.276196	+	0.961101i	$0.589074\pi$
$38$	0	0
$39$	5.51383	0.882919
$40$	0	0
$41$	−10.1413	−1.58381	−0.791904	−	0.610645i	$-0.790910\pi$
$41$	−10.1413	−1.58381	−0.791904	+	0.610645i	$0.790910\pi$
$42$	0	0
$43$	−3.48329	−0.531196	−0.265598	−	0.964084i	$-0.585569\pi$
$43$	−3.48329	−0.531196	−0.265598	+	0.964084i	$0.585569\pi$
$44$	0	0
$45$	−9.44354	−1.40776
$46$	0	0
$47$	−6.79750	−0.991517	−0.495759	−	0.868460i	$-0.665110\pi$
$47$	−6.79750	−0.991517	−0.495759	+	0.868460i	$0.665110\pi$
$48$	0	0
$49$	10.4881	1.49829
$50$	0	0
$51$	7.06305	0.989024
$52$	0	0
$53$	−9.22830	−1.26760	−0.633802	−	0.773495i	$-0.718507\pi$
$53$	−9.22830	−1.26760	−0.633802	+	0.773495i	$0.718507\pi$
$54$	0	0
$55$	−4.31358	−0.581644
$56$	0	0
$57$	−0.705945	−0.0935048
$58$	0	0
$59$	−4.10449	−0.534359	−0.267180	−	0.963647i	$-0.586092\pi$
$59$	−4.10449	−0.534359	−0.267180	+	0.963647i	$0.586092\pi$
$60$	0	0
$61$	9.94756	1.27366	0.636828	−	0.771006i	$-0.280246\pi$
$61$	9.94756	1.27366	0.636828	+	0.771006i	$0.280246\pi$
$62$	0	0
$63$	−9.15519	−1.15345
$64$	0	0
$65$	−26.4149	−3.27637
$66$	0	0
$67$	−6.67646	−0.815660	−0.407830	−	0.913058i	$-0.633714\pi$
$67$	−6.67646	−0.815660	−0.407830	+	0.913058i	$0.633714\pi$
$68$	0	0
$69$	6.07837	0.731750
$70$	0	0
$71$	−0.268097	−0.0318172	−0.0159086	−	0.999873i	$-0.505064\pi$
$71$	−0.268097	−0.0318172	−0.0159086	+	0.999873i	$0.505064\pi$
$72$	0	0
$73$	−7.49879	−0.877667	−0.438834	−	0.898568i	$-0.644608\pi$
$73$	−7.49879	−0.877667	−0.438834	+	0.898568i	$0.644608\pi$
$74$	0	0
$75$	−12.2519	−1.41473
$76$	0	0
$77$	−4.18187	−0.476569
$78$	0	0
$79$	8.20310	0.922921	0.461460	−	0.887161i	$-0.347326\pi$
$79$	8.20310	0.922921	0.461460	+	0.887161i	$0.347326\pi$
$80$	0	0
$81$	2.36060	0.262289
$82$	0	0
$83$	−10.3886	−1.14030	−0.570151	−	0.821540i	$-0.693115\pi$
$83$	−10.3886	−1.14030	−0.570151	+	0.821540i	$0.693115\pi$
$84$	0	0
$85$	−33.8367	−3.67011
$86$	0	0
$87$	−1.04341	−0.111866
$88$	0	0
$89$	1.21472	0.128760	0.0643799	−	0.997925i	$-0.479493\pi$
$89$	1.21472	0.128760	0.0643799	+	0.997925i	$0.479493\pi$
$90$	0	0
$91$	−25.6084	−2.68449
$92$	0	0
$93$	−2.53499	−0.262867
$94$	0	0
$95$	3.38195	0.346981
$96$	0	0
$97$	2.51963	0.255830	0.127915	−	0.991785i	$-0.459172\pi$
$97$	2.51963	0.255830	0.127915	+	0.991785i	$0.459172\pi$
$98$	0	0
$99$	2.18926	0.220028
$100$	0	0
$101$	12.5771	1.25147	0.625735	−	0.780036i	$-0.284799\pi$
$101$	12.5771	1.25147	0.625735	+	0.780036i	$0.284799\pi$
$102$	0	0
$103$	8.82461	0.869514	0.434757	−	0.900548i	$-0.356834\pi$
$103$	8.82461	0.869514	0.434757	+	0.900548i	$0.356834\pi$
$104$	0	0
$105$	−16.2424	−1.58510
$106$	0	0
$107$	−17.5215	−1.69387	−0.846935	−	0.531697i	$-0.821555\pi$
$107$	−17.5215	−1.69387	−0.846935	+	0.531697i	$0.821555\pi$
$108$	0	0
$109$	3.85236	0.368989	0.184495	−	0.982834i	$-0.440935\pi$
$109$	3.85236	0.368989	0.184495	+	0.982834i	$0.440935\pi$
$110$	0	0
$111$	3.02546	0.287164
$112$	0	0
$113$	−3.69830	−0.347907	−0.173953	−	0.984754i	$-0.555654\pi$
$113$	−3.69830	−0.347907	−0.173953	+	0.984754i	$0.555654\pi$
$114$	0	0
$115$	−29.1195	−2.71540
$116$	0	0
$117$	13.4063	1.23941
$118$	0	0
$119$	−32.8035	−3.00710
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	9.13139	0.823350
$124$	0	0
$125$	37.1271	3.32075
$126$	0	0
$127$	−14.4939	−1.28612	−0.643061	−	0.765815i	$-0.722336\pi$
$127$	−14.4939	−1.28612	−0.643061	+	0.765815i	$0.722336\pi$
$128$	0	0
$129$	3.13640	0.276145
$130$	0	0
$131$	−19.5230	−1.70573	−0.852864	−	0.522133i	$-0.825136\pi$
$131$	−19.5230	−1.70573	−0.852864	+	0.522133i	$0.825136\pi$
$132$	0	0
$133$	3.27869	0.284298
$134$	0	0
$135$	20.1551	1.73468
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	18.0512	1.53109	0.765543	−	0.643384i	$-0.222470\pi$
$139$	18.0512	1.53109	0.765543	+	0.643384i	$0.222470\pi$
$140$	0	0
$141$	6.12056	0.515444
$142$	0	0
$143$	6.12366	0.512086
$144$	0	0
$145$	4.99865	0.415115
$146$	0	0
$147$	−9.44360	−0.778895
$148$	0	0
$149$	−14.2607	−1.16828	−0.584140	−	0.811653i	$-0.698568\pi$
$149$	−14.2607	−1.16828	−0.584140	+	0.811653i	$0.698568\pi$
$150$	0	0
$151$	−3.48475	−0.283585	−0.141793	−	0.989896i	$-0.545287\pi$
$151$	−3.48475	−0.283585	−0.141793	+	0.989896i	$0.545287\pi$
$152$	0	0
$153$	17.1730	1.38835
$154$	0	0
$155$	12.1443	0.975455
$156$	0	0
$157$	−19.7801	−1.57863	−0.789314	−	0.613990i	$-0.789564\pi$
$157$	−19.7801	−1.57863	−0.789314	+	0.613990i	$0.789564\pi$
$158$	0	0
$159$	8.30929	0.658969
$160$	0	0
$161$	−28.2303	−2.22486
$162$	0	0
$163$	10.5236	0.824270	0.412135	−	0.911123i	$-0.364783\pi$
$163$	10.5236	0.824270	0.412135	+	0.911123i	$0.364783\pi$
$164$	0	0
$165$	3.88401	0.302370
$166$	0	0
$167$	−18.4233	−1.42564	−0.712820	−	0.701348i	$-0.752582\pi$
$167$	−18.4233	−1.42564	−0.712820	+	0.701348i	$0.752582\pi$
$168$	0	0
$169$	24.4992	1.88455
$170$	0	0
$171$	−1.71643	−0.131258
$172$	0	0
$173$	20.6890	1.57295	0.786476	−	0.617621i	$-0.211903\pi$
$173$	20.6890	1.57295	0.786476	+	0.617621i	$0.211903\pi$
$174$	0	0
$175$	56.9028	4.30145
$176$	0	0
$177$	3.69574	0.277789
$178$	0	0
$179$	−10.0825	−0.753603	−0.376802	−	0.926294i	$-0.622976\pi$
$179$	−10.0825	−0.753603	−0.376802	+	0.926294i	$0.622976\pi$
$180$	0	0
$181$	6.69666	0.497759	0.248879	−	0.968535i	$-0.419938\pi$
$181$	6.69666	0.497759	0.248879	+	0.968535i	$0.419938\pi$
$182$	0	0
$183$	−8.95692	−0.662115
$184$	0	0
$185$	−14.4940	−1.06562
$186$	0	0
$187$	7.84422	0.573626
$188$	0	0
$189$	19.5397	1.42130
$190$	0	0
$191$	15.3249	1.10887	0.554434	−	0.832228i	$-0.312935\pi$
$191$	15.3249	1.10887	0.554434	+	0.832228i	$0.312935\pi$
$192$	0	0
$193$	18.3914	1.32384	0.661920	−	0.749574i	$-0.269742\pi$
$193$	18.3914	1.32384	0.661920	+	0.749574i	$0.269742\pi$
$194$	0	0
$195$	23.7844	1.70323
$196$	0	0
$197$	2.61996	0.186664	0.0933322	−	0.995635i	$-0.470248\pi$
$197$	2.61996	0.186664	0.0933322	+	0.995635i	$0.470248\pi$
$198$	0	0
$199$	12.7705	0.905279	0.452639	−	0.891694i	$-0.350483\pi$
$199$	12.7705	0.905279	0.452639	+	0.891694i	$0.350483\pi$
$200$	0	0
$201$	6.01158	0.424024
$202$	0	0
$203$	4.84602	0.340124
$204$	0	0
$205$	−43.7455	−3.05532
$206$	0	0
$207$	14.7789	1.02720
$208$	0	0
$209$	−0.784023	−0.0542320
$210$	0	0
$211$	25.4953	1.75517	0.877585	−	0.479420i	$-0.159153\pi$
$211$	25.4953	1.75517	0.877585	+	0.479420i	$0.159153\pi$
$212$	0	0
$213$	0.241398	0.0165403
$214$	0	0
$215$	−15.0254	−1.02473
$216$	0	0
$217$	11.7735	0.799237
$218$	0	0
$219$	6.75201	0.456259
$220$	0	0
$221$	48.0353	3.23121
$222$	0	0
$223$	3.04579	0.203961	0.101981	−	0.994786i	$-0.467482\pi$
$223$	3.04579	0.203961	0.101981	+	0.994786i	$0.467482\pi$
$224$	0	0
$225$	−29.7892	−1.98595
$226$	0	0
$227$	−17.5350	−1.16384	−0.581920	−	0.813246i	$-0.697698\pi$
$227$	−17.5350	−1.16384	−0.581920	+	0.813246i	$0.697698\pi$
$228$	0	0
$229$	14.1686	0.936289	0.468145	−	0.883652i	$-0.344923\pi$
$229$	14.1686	0.936289	0.468145	+	0.883652i	$0.344923\pi$
$230$	0	0
$231$	3.76542	0.247746
$232$	0	0
$233$	−0.169324	−0.0110928	−0.00554640	−	0.999985i	$-0.501765\pi$
$233$	−0.169324	−0.0110928	−0.00554640	+	0.999985i	$0.501765\pi$
$234$	0	0
$235$	−29.3216	−1.91273
$236$	0	0
$237$	−7.38619	−0.479784
$238$	0	0
$239$	12.2905	0.795008	0.397504	−	0.917600i	$-0.369876\pi$
$239$	12.2905	0.795008	0.397504	+	0.917600i	$0.369876\pi$
$240$	0	0
$241$	−22.1824	−1.42890	−0.714448	−	0.699689i	$-0.753322\pi$
$241$	−22.1824	−1.42890	−0.714448	+	0.699689i	$0.753322\pi$
$242$	0	0
$243$	−16.1429	−1.03557
$244$	0	0
$245$	45.2411	2.89035
$246$	0	0
$247$	−4.80109	−0.305486
$248$	0	0
$249$	9.35408	0.592791
$250$	0	0
$251$	4.47856	0.282684	0.141342	−	0.989961i	$-0.454858\pi$
$251$	4.47856	0.282684	0.141342	+	0.989961i	$0.454858\pi$
$252$	0	0
$253$	6.75064	0.424409
$254$	0	0
$255$	30.4670	1.90792
$256$	0	0
$257$	16.3132	1.01759	0.508796	−	0.860887i	$-0.330091\pi$
$257$	16.3132	1.01759	0.508796	+	0.860887i	$0.330091\pi$
$258$	0	0
$259$	−14.0514	−0.873111
$260$	0	0
$261$	−2.53694	−0.157033
$262$	0	0
$263$	−3.10918	−0.191720	−0.0958600	−	0.995395i	$-0.530560\pi$
$263$	−3.10918	−0.191720	−0.0958600	+	0.995395i	$0.530560\pi$
$264$	0	0
$265$	−39.8070	−2.44533
$266$	0	0
$267$	−1.09375	−0.0669363
$268$	0	0
$269$	29.8737	1.82143	0.910715	−	0.413036i	$-0.135532\pi$
$269$	29.8737	1.82143	0.910715	+	0.413036i	$0.135532\pi$
$270$	0	0
$271$	−19.2134	−1.16713	−0.583565	−	0.812067i	$-0.698343\pi$
$271$	−19.2134	−1.16713	−0.583565	+	0.812067i	$0.698343\pi$
$272$	0	0
$273$	23.0581	1.39554
$274$	0	0
$275$	−13.6070	−0.820534
$276$	0	0
$277$	−31.7805	−1.90951	−0.954753	−	0.297400i	$-0.903881\pi$
$277$	−31.7805	−1.90951	−0.954753	+	0.297400i	$0.903881\pi$
$278$	0	0
$279$	−6.16355	−0.369002
$280$	0	0
$281$	8.58057	0.511874	0.255937	−	0.966693i	$-0.417616\pi$
$281$	8.58057	0.511874	0.255937	+	0.966693i	$0.417616\pi$
$282$	0	0
$283$	−13.3349	−0.792675	−0.396337	−	0.918105i	$-0.629719\pi$
$283$	−13.3349	−0.792675	−0.396337	+	0.918105i	$0.629719\pi$
$284$	0	0
$285$	−3.04516	−0.180379
$286$	0	0
$287$	−42.4097	−2.50337
$288$	0	0
$289$	44.5318	2.61952
$290$	0	0
$291$	−2.26871	−0.132994
$292$	0	0
$293$	−9.12688	−0.533198	−0.266599	−	0.963808i	$-0.585900\pi$
$293$	−9.12688	−0.533198	−0.266599	+	0.963808i	$0.585900\pi$
$294$	0	0
$295$	−17.7051	−1.03083
$296$	0	0
$297$	−4.67248	−0.271125
$298$	0	0
$299$	41.3386	2.39067
$300$	0	0
$301$	−14.5667	−0.839608
$302$	0	0
$303$	−11.3246	−0.650582
$304$	0	0
$305$	42.9097	2.45700
$306$	0	0
$307$	−12.0155	−0.685759	−0.342879	−	0.939379i	$-0.611402\pi$
$307$	−12.0155	−0.685759	−0.342879	+	0.939379i	$0.611402\pi$
$308$	0	0
$309$	−7.94580	−0.452021
$310$	0	0
$311$	10.8527	0.615399	0.307699	−	0.951484i	$-0.400441\pi$
$311$	10.8527	0.615399	0.307699	+	0.951484i	$0.400441\pi$
$312$	0	0
$313$	−13.5492	−0.765847	−0.382923	−	0.923780i	$-0.625083\pi$
$313$	−13.5492	−0.765847	−0.382923	+	0.923780i	$0.625083\pi$
$314$	0	0
$315$	−39.4917	−2.22510
$316$	0	0
$317$	−27.6706	−1.55414	−0.777068	−	0.629417i	$-0.783294\pi$
$317$	−27.6706	−1.55414	−0.777068	+	0.629417i	$0.783294\pi$
$318$	0	0
$319$	−1.15882	−0.0648812
$320$	0	0
$321$	15.7766	0.880565
$322$	0	0
$323$	−6.15005	−0.342198
$324$	0	0
$325$	−83.3247	−4.62202
$326$	0	0
$327$	−3.46872	−0.191821
$328$	0	0
$329$	−28.4263	−1.56719
$330$	0	0
$331$	−17.0629	−0.937861	−0.468931	−	0.883235i	$-0.655361\pi$
$331$	−17.0629	−0.937861	−0.468931	+	0.883235i	$0.655361\pi$
$332$	0	0
$333$	7.35606	0.403109
$334$	0	0
$335$	−28.7995	−1.57348
$336$	0	0
$337$	21.1636	1.15286	0.576428	−	0.817148i	$-0.304446\pi$
$337$	21.1636	1.15286	0.576428	+	0.817148i	$0.304446\pi$
$338$	0	0
$339$	3.33000	0.180861
$340$	0	0
$341$	−2.81537	−0.152461
$342$	0	0
$343$	14.5866	0.787604
$344$	0	0
$345$	26.2196	1.41161
$346$	0	0
$347$	18.4294	0.989340	0.494670	−	0.869081i	$-0.335289\pi$
$347$	18.4294	0.989340	0.494670	+	0.869081i	$0.335289\pi$
$348$	0	0
$349$	−18.2484	−0.976815	−0.488407	−	0.872616i	$-0.662422\pi$
$349$	−18.2484	−0.976815	−0.488407	+	0.872616i	$0.662422\pi$
$350$	0	0
$351$	−28.6127	−1.52723
$352$	0	0
$353$	6.10052	0.324698	0.162349	−	0.986733i	$-0.448093\pi$
$353$	6.10052	0.324698	0.162349	+	0.986733i	$0.448093\pi$
$354$	0	0
$355$	−1.15646	−0.0613784
$356$	0	0
$357$	29.5368	1.56325
$358$	0	0
$359$	12.2207	0.644982	0.322491	−	0.946573i	$-0.395480\pi$
$359$	12.2207	0.644982	0.322491	+	0.946573i	$0.395480\pi$
$360$	0	0
$361$	−18.3853	−0.967648
$362$	0	0
$363$	−0.900414	−0.0472595
$364$	0	0
$365$	−32.3467	−1.69310
$366$	0	0
$367$	−1.67687	−0.0875320	−0.0437660	−	0.999042i	$-0.513936\pi$
$367$	−1.67687	−0.0875320	−0.0437660	+	0.999042i	$0.513936\pi$
$368$	0	0
$369$	22.2019	1.15579
$370$	0	0
$371$	−38.5916	−2.00357
$372$	0	0
$373$	18.8184	0.974381	0.487190	−	0.873296i	$-0.338022\pi$
$373$	18.8184	0.974381	0.487190	+	0.873296i	$0.338022\pi$
$374$	0	0
$375$	−33.4297	−1.72630
$376$	0	0
$377$	−7.09620	−0.365473
$378$	0	0
$379$	−9.21354	−0.473268	−0.236634	−	0.971599i	$-0.576044\pi$
$379$	−9.21354	−0.473268	−0.236634	+	0.971599i	$0.576044\pi$
$380$	0	0
$381$	13.0505	0.668596
$382$	0	0
$383$	−13.4819	−0.688895	−0.344448	−	0.938806i	$-0.611934\pi$
$383$	−13.4819	−0.688895	−0.344448	+	0.938806i	$0.611934\pi$
$384$	0	0
$385$	−18.0389	−0.919346
$386$	0	0
$387$	7.62580	0.387641
$388$	0	0
$389$	−5.29836	−0.268637	−0.134319	−	0.990938i	$-0.542885\pi$
$389$	−5.29836	−0.268637	−0.134319	+	0.990938i	$0.542885\pi$
$390$	0	0
$391$	52.9535	2.67797
$392$	0	0
$393$	17.5787	0.886730
$394$	0	0
$395$	35.3848	1.78040
$396$	0	0
$397$	36.3185	1.82277	0.911385	−	0.411554i	$-0.135014\pi$
$397$	36.3185	1.82277	0.911385	+	0.411554i	$0.135014\pi$
$398$	0	0
$399$	−2.95217	−0.147794
$400$	0	0
$401$	1.48176	0.0739956	0.0369978	−	0.999315i	$-0.488221\pi$
$401$	1.48176	0.0739956	0.0369978	+	0.999315i	$0.488221\pi$
$402$	0	0
$403$	−17.2403	−0.858802
$404$	0	0
$405$	10.1827	0.505980
$406$	0	0
$407$	3.36007	0.166553
$408$	0	0
$409$	−26.1469	−1.29288	−0.646441	−	0.762964i	$-0.723743\pi$
$409$	−26.1469	−1.29288	−0.646441	+	0.762964i	$0.723743\pi$
$410$	0	0
$411$	−0.900414	−0.0444141
$412$	0	0
$413$	−17.1645	−0.844608
$414$	0	0
$415$	−44.8123	−2.19975
$416$	0	0
$417$	−16.2536	−0.795942
$418$	0	0
$419$	21.0576	1.02873	0.514366	−	0.857571i	$-0.328027\pi$
$419$	21.0576	1.02873	0.514366	+	0.857571i	$0.328027\pi$
$420$	0	0
$421$	14.9573	0.728973	0.364486	−	0.931209i	$-0.381245\pi$
$421$	14.9573	0.728973	0.364486	+	0.931209i	$0.381245\pi$
$422$	0	0
$423$	14.8815	0.723561
$424$	0	0
$425$	−106.736	−5.17748
$426$	0	0
$427$	41.5995	2.01314
$428$	0	0
$429$	−5.51383	−0.266210
$430$	0	0
$431$	27.0308	1.30203	0.651013	−	0.759066i	$-0.274344\pi$
$431$	27.0308	1.30203	0.651013	+	0.759066i	$0.274344\pi$
$432$	0	0
$433$	34.3854	1.65246	0.826229	−	0.563335i	$-0.190482\pi$
$433$	34.3854	1.65246	0.826229	+	0.563335i	$0.190482\pi$
$434$	0	0
$435$	−4.50085	−0.215799
$436$	0	0
$437$	−5.29266	−0.253182
$438$	0	0
$439$	−16.3785	−0.781704	−0.390852	−	0.920454i	$-0.627820\pi$
$439$	−16.3785	−0.781704	−0.390852	+	0.920454i	$0.627820\pi$
$440$	0	0
$441$	−22.9610	−1.09338
$442$	0	0
$443$	−4.48733	−0.213200	−0.106600	−	0.994302i	$-0.533996\pi$
$443$	−4.48733	−0.213200	−0.106600	+	0.994302i	$0.533996\pi$
$444$	0	0
$445$	5.23979	0.248390
$446$	0	0
$447$	12.8405	0.607335
$448$	0	0
$449$	10.8903	0.513945	0.256973	−	0.966419i	$-0.417275\pi$
$449$	10.8903	0.513945	0.256973	+	0.966419i	$0.417275\pi$
$450$	0	0
$451$	10.1413	0.477536
$452$	0	0
$453$	3.13772	0.147423
$454$	0	0
$455$	−110.464	−5.17863
$456$	0	0
$457$	28.6460	1.34000	0.670001	−	0.742360i	$-0.266294\pi$
$457$	28.6460	1.34000	0.670001	+	0.742360i	$0.266294\pi$
$458$	0	0
$459$	−36.6519	−1.71077
$460$	0	0
$461$	27.0293	1.25888	0.629439	−	0.777050i	$-0.283285\pi$
$461$	27.0293	1.25888	0.629439	+	0.777050i	$0.283285\pi$
$462$	0	0
$463$	−40.9957	−1.90523	−0.952615	−	0.304177i	$-0.901618\pi$
$463$	−40.9957	−1.90523	−0.952615	+	0.304177i	$0.901618\pi$
$464$	0	0
$465$	−10.9349	−0.507094
$466$	0	0
$467$	−6.57903	−0.304441	−0.152221	−	0.988347i	$-0.548642\pi$
$467$	−6.57903	−0.304441	−0.152221	+	0.988347i	$0.548642\pi$
$468$	0	0
$469$	−27.9201	−1.28923
$470$	0	0
$471$	17.8103	0.820656
$472$	0	0
$473$	3.48329	0.160162
$474$	0	0
$475$	10.6682	0.489491
$476$	0	0
$477$	20.2031	0.925036
$478$	0	0
$479$	−41.2547	−1.88498	−0.942488	−	0.334240i	$-0.891520\pi$
$479$	−41.2547	−1.88498	−0.942488	+	0.334240i	$0.891520\pi$
$480$	0	0
$481$	20.5759	0.938182
$482$	0	0
$483$	25.4190	1.15660
$484$	0	0
$485$	10.8686	0.493519
$486$	0	0
$487$	−41.9295	−1.90001	−0.950004	−	0.312237i	$-0.898922\pi$
$487$	−41.9295	−1.90001	−0.950004	+	0.312237i	$0.898922\pi$
$488$	0	0
$489$	−9.47558	−0.428500
$490$	0	0
$491$	−13.8700	−0.625946	−0.312973	−	0.949762i	$-0.601325\pi$
$491$	−13.8700	−0.625946	−0.312973	+	0.949762i	$0.601325\pi$
$492$	0	0
$493$	−9.09001	−0.409393
$494$	0	0
$495$	9.44354	0.424455
$496$	0	0
$497$	−1.12115	−0.0502903
$498$	0	0
$499$	15.4539	0.691810	0.345905	−	0.938270i	$-0.387572\pi$
$499$	15.4539	0.691810	0.345905	+	0.938270i	$0.387572\pi$
$500$	0	0
$501$	16.5886	0.741124
$502$	0	0
$503$	23.7376	1.05841	0.529204	−	0.848495i	$-0.322491\pi$
$503$	23.7376	1.05841	0.529204	+	0.848495i	$0.322491\pi$
$504$	0	0
$505$	54.2525	2.41420
$506$	0	0
$507$	−22.0594	−0.979693
$508$	0	0
$509$	−20.8186	−0.922768	−0.461384	−	0.887201i	$-0.652647\pi$
$509$	−20.8186	−0.922768	−0.461384	+	0.887201i	$0.652647\pi$
$510$	0	0
$511$	−31.3590	−1.38724
$512$	0	0
$513$	3.66333	0.161740
$514$	0	0
$515$	38.0657	1.67737
$516$	0	0
$517$	6.79750	0.298954
$518$	0	0
$519$	−18.6286	−0.817706
$520$	0	0
$521$	−33.4225	−1.46427	−0.732134	−	0.681161i	$-0.761475\pi$
$521$	−33.4225	−1.46427	−0.732134	+	0.681161i	$0.761475\pi$
$522$	0	0
$523$	17.3507	0.758694	0.379347	−	0.925255i	$-0.376149\pi$
$523$	17.3507	0.758694	0.379347	+	0.925255i	$0.376149\pi$
$524$	0	0
$525$	−51.2361	−2.23613
$526$	0	0
$527$	−22.0843	−0.962009
$528$	0	0
$529$	22.5712	0.981354
$530$	0	0
$531$	8.98578	0.389949
$532$	0	0
$533$	62.1020	2.68994
$534$	0	0
$535$	−75.5806	−3.26763
$536$	0	0
$537$	9.07844	0.391764
$538$	0	0
$539$	−10.4881	−0.451753
$540$	0	0
$541$	5.87717	0.252679	0.126340	−	0.991987i	$-0.459677\pi$
$541$	5.87717	0.252679	0.126340	+	0.991987i	$0.459677\pi$
$542$	0	0
$543$	−6.02976	−0.258762
$544$	0	0
$545$	16.6175	0.711815
$546$	0	0
$547$	−23.4158	−1.00119	−0.500593	−	0.865683i	$-0.666885\pi$
$547$	−23.4158	−1.00119	−0.500593	+	0.865683i	$0.666885\pi$
$548$	0	0
$549$	−21.7778	−0.929452
$550$	0	0
$551$	0.908539	0.0387051
$552$	0	0
$553$	34.3043	1.45877
$554$	0	0
$555$	13.0506	0.553965
$556$	0	0
$557$	9.33349	0.395473	0.197736	−	0.980255i	$-0.436641\pi$
$557$	9.33349	0.395473	0.197736	+	0.980255i	$0.436641\pi$
$558$	0	0
$559$	21.3305	0.902182
$560$	0	0
$561$	−7.06305	−0.298202
$562$	0	0
$563$	2.43230	0.102509	0.0512546	−	0.998686i	$-0.483678\pi$
$563$	2.43230	0.102509	0.0512546	+	0.998686i	$0.483678\pi$
$564$	0	0
$565$	−15.9529	−0.671144
$566$	0	0
$567$	9.87174	0.414574
$568$	0	0
$569$	−7.18174	−0.301074	−0.150537	−	0.988604i	$-0.548100\pi$
$569$	−7.18174	−0.301074	−0.150537	+	0.988604i	$0.548100\pi$
$570$	0	0
$571$	−2.74533	−0.114888	−0.0574442	−	0.998349i	$-0.518295\pi$
$571$	−2.74533	−0.114888	−0.0574442	+	0.998349i	$0.518295\pi$
$572$	0	0
$573$	−13.7987	−0.576450
$574$	0	0
$575$	−91.8561	−3.83066
$576$	0	0
$577$	3.44450	0.143396	0.0716982	−	0.997426i	$-0.477158\pi$
$577$	3.44450	0.143396	0.0716982	+	0.997426i	$0.477158\pi$
$578$	0	0
$579$	−16.5599	−0.688204
$580$	0	0
$581$	−43.4440	−1.80236
$582$	0	0
$583$	9.22830	0.382197
$584$	0	0
$585$	57.8290	2.39093
$586$	0	0
$587$	12.8689	0.531157	0.265579	−	0.964089i	$-0.414437\pi$
$587$	12.8689	0.531157	0.265579	+	0.964089i	$0.414437\pi$
$588$	0	0
$589$	2.20731	0.0909507
$590$	0	0
$591$	−2.35905	−0.0970382
$592$	0	0
$593$	15.6045	0.640800	0.320400	−	0.947282i	$-0.396183\pi$
$593$	15.6045	0.640800	0.320400	+	0.947282i	$0.396183\pi$
$594$	0	0
$595$	−141.501	−5.80097
$596$	0	0
$597$	−11.4988	−0.470613
$598$	0	0
$599$	−34.9506	−1.42804	−0.714021	−	0.700124i	$-0.753128\pi$
$599$	−34.9506	−1.42804	−0.714021	+	0.700124i	$0.753128\pi$
$600$	0	0
$601$	−32.2958	−1.31737	−0.658686	−	0.752418i	$-0.728888\pi$
$601$	−32.2958	−1.31737	−0.658686	+	0.752418i	$0.728888\pi$
$602$	0	0
$603$	14.6165	0.595229
$604$	0	0
$605$	4.31358	0.175372
$606$	0	0
$607$	21.0203	0.853188	0.426594	−	0.904443i	$-0.359713\pi$
$607$	21.0203	0.853188	0.426594	+	0.904443i	$0.359713\pi$
$608$	0	0
$609$	−4.36343	−0.176815
$610$	0	0
$611$	41.6256	1.68399
$612$	0	0
$613$	7.04814	0.284672	0.142336	−	0.989818i	$-0.454539\pi$
$613$	7.04814	0.284672	0.142336	+	0.989818i	$0.454539\pi$
$614$	0	0
$615$	39.3890	1.58832
$616$	0	0
$617$	25.6721	1.03352	0.516760	−	0.856130i	$-0.327138\pi$
$617$	25.6721	1.03352	0.516760	+	0.856130i	$0.327138\pi$
$618$	0	0
$619$	−9.14798	−0.367688	−0.183844	−	0.982955i	$-0.558854\pi$
$619$	−9.14798	−0.367688	−0.183844	+	0.982955i	$0.558854\pi$
$620$	0	0
$621$	−31.5422	−1.26575
$622$	0	0
$623$	5.07980	0.203518
$624$	0	0
$625$	92.1157	3.68463
$626$	0	0
$627$	0.705945	0.0281927
$628$	0	0
$629$	26.3572	1.05093
$630$	0	0
$631$	−45.3273	−1.80445	−0.902225	−	0.431265i	$-0.858068\pi$
$631$	−45.3273	−1.80445	−0.902225	+	0.431265i	$0.858068\pi$
$632$	0	0
$633$	−22.9563	−0.912433
$634$	0	0
$635$	−62.5205	−2.48105
$636$	0	0
$637$	−64.2253	−2.54470
$638$	0	0
$639$	0.586932	0.0232187
$640$	0	0
$641$	13.3956	0.529095	0.264547	−	0.964373i	$-0.414777\pi$
$641$	13.3956	0.529095	0.264547	+	0.964373i	$0.414777\pi$
$642$	0	0
$643$	40.7740	1.60797	0.803984	−	0.594651i	$-0.202710\pi$
$643$	40.7740	1.60797	0.803984	+	0.594651i	$0.202710\pi$
$644$	0	0
$645$	13.5291	0.532709
$646$	0	0
$647$	−14.8296	−0.583011	−0.291506	−	0.956569i	$-0.594156\pi$
$647$	−14.8296	−0.583011	−0.291506	+	0.956569i	$0.594156\pi$
$648$	0	0
$649$	4.10449	0.161115
$650$	0	0
$651$	−10.6010	−0.415487
$652$	0	0
$653$	−14.3435	−0.561303	−0.280651	−	0.959810i	$-0.590550\pi$
$653$	−14.3435	−0.561303	−0.280651	+	0.959810i	$0.590550\pi$
$654$	0	0
$655$	−84.2139	−3.29051
$656$	0	0
$657$	16.4168	0.640479
$658$	0	0
$659$	20.4726	0.797497	0.398749	−	0.917060i	$-0.369445\pi$
$659$	20.4726	0.797497	0.398749	+	0.917060i	$0.369445\pi$
$660$	0	0
$661$	−20.6821	−0.804442	−0.402221	−	0.915543i	$-0.631762\pi$
$661$	−20.6821	−0.804442	−0.402221	+	0.915543i	$0.631762\pi$
$662$	0	0
$663$	−43.2517	−1.67976
$664$	0	0
$665$	14.1429	0.548438
$666$	0	0
$667$	−7.82275	−0.302898
$668$	0	0
$669$	−2.74247	−0.106030
$670$	0	0
$671$	−9.94756	−0.384021
$672$	0	0
$673$	−13.3429	−0.514330	−0.257165	−	0.966368i	$-0.582788\pi$
$673$	−13.3429	−0.514330	−0.257165	+	0.966368i	$0.582788\pi$
$674$	0	0
$675$	63.5784	2.44714
$676$	0	0
$677$	−1.84401	−0.0708710	−0.0354355	−	0.999372i	$-0.511282\pi$
$677$	−1.84401	−0.0708710	−0.0354355	+	0.999372i	$0.511282\pi$
$678$	0	0
$679$	10.5368	0.404364
$680$	0	0
$681$	15.7888	0.605027
$682$	0	0
$683$	42.3592	1.62083	0.810415	−	0.585856i	$-0.199242\pi$
$683$	42.3592	1.62083	0.810415	+	0.585856i	$0.199242\pi$
$684$	0	0
$685$	4.31358	0.164814
$686$	0	0
$687$	−12.7576	−0.486734
$688$	0	0
$689$	56.5109	2.15290
$690$	0	0
$691$	−23.0688	−0.877580	−0.438790	−	0.898590i	$-0.644593\pi$
$691$	−23.0688	−0.877580	−0.438790	+	0.898590i	$0.644593\pi$
$692$	0	0
$693$	9.15519	0.347777
$694$	0	0
$695$	77.8656	2.95361
$696$	0	0
$697$	79.5508	3.01320
$698$	0	0
$699$	0.152462	0.00576664
$700$	0	0
$701$	24.5836	0.928509	0.464254	−	0.885702i	$-0.346322\pi$
$701$	24.5836	0.928509	0.464254	+	0.885702i	$0.346322\pi$
$702$	0	0
$703$	−2.63438	−0.0993573
$704$	0	0
$705$	26.4016	0.994341
$706$	0	0
$707$	52.5959	1.97807
$708$	0	0
$709$	20.9306	0.786067	0.393033	−	0.919524i	$-0.371426\pi$
$709$	20.9306	0.786067	0.393033	+	0.919524i	$0.371426\pi$
$710$	0	0
$711$	−17.9587	−0.673503
$712$	0	0
$713$	−19.0055	−0.711762
$714$	0	0
$715$	26.4149	0.987862
$716$	0	0
$717$	−11.0666	−0.413288
$718$	0	0
$719$	−41.9177	−1.56327	−0.781633	−	0.623739i	$-0.785613\pi$
$719$	−41.9177	−1.56327	−0.781633	+	0.623739i	$0.785613\pi$
$720$	0	0
$721$	36.9034	1.37435
$722$	0	0
$723$	19.9734	0.742817
$724$	0	0
$725$	15.7680	0.585610
$726$	0	0
$727$	−8.67668	−0.321800	−0.160900	−	0.986971i	$-0.551440\pi$
$727$	−8.67668	−0.321800	−0.160900	+	0.986971i	$0.551440\pi$
$728$	0	0
$729$	7.45353	0.276057
$730$	0	0
$731$	27.3237	1.01060
$732$	0	0
$733$	1.91494	0.0707298	0.0353649	−	0.999374i	$-0.488741\pi$
$733$	1.91494	0.0707298	0.0353649	+	0.999374i	$0.488741\pi$
$734$	0	0
$735$	−40.7358	−1.50256
$736$	0	0
$737$	6.67646	0.245931
$738$	0	0
$739$	23.4819	0.863795	0.431898	−	0.901923i	$-0.357844\pi$
$739$	23.4819	0.863795	0.431898	+	0.901923i	$0.357844\pi$
$740$	0	0
$741$	4.32297	0.158808
$742$	0	0
$743$	46.8571	1.71902	0.859511	−	0.511118i	$-0.170768\pi$
$743$	46.8571	1.71902	0.859511	+	0.511118i	$0.170768\pi$
$744$	0	0
$745$	−61.5146	−2.25372
$746$	0	0
$747$	22.7434	0.832137
$748$	0	0
$749$	−73.2728	−2.67733
$750$	0	0
$751$	−19.0550	−0.695328	−0.347664	−	0.937619i	$-0.613025\pi$
$751$	−19.0550	−0.695328	−0.347664	+	0.937619i	$0.613025\pi$
$752$	0	0
$753$	−4.03256	−0.146955
$754$	0	0
$755$	−15.0318	−0.547063
$756$	0	0
$757$	11.9166	0.433115	0.216557	−	0.976270i	$-0.430517\pi$
$757$	11.9166	0.433115	0.216557	+	0.976270i	$0.430517\pi$
$758$	0	0
$759$	−6.07837	−0.220631
$760$	0	0
$761$	24.8951	0.902446	0.451223	−	0.892411i	$-0.350988\pi$
$761$	24.8951	0.902446	0.451223	+	0.892411i	$0.350988\pi$
$762$	0	0
$763$	16.1101	0.583224
$764$	0	0
$765$	74.0772	2.67827
$766$	0	0
$767$	25.1345	0.907554
$768$	0	0
$769$	−28.5362	−1.02904	−0.514520	−	0.857478i	$-0.672030\pi$
$769$	−28.5362	−1.02904	−0.514520	+	0.857478i	$0.672030\pi$
$770$	0	0
$771$	−14.6887	−0.528999
$772$	0	0
$773$	−7.38912	−0.265768	−0.132884	−	0.991132i	$-0.542424\pi$
$773$	−7.38912	−0.265768	−0.132884	+	0.991132i	$0.542424\pi$
$774$	0	0
$775$	38.3087	1.37609
$776$	0	0
$777$	12.6521	0.453891
$778$	0	0
$779$	−7.95104	−0.284875
$780$	0	0
$781$	0.268097	0.00959325
$782$	0	0
$783$	5.41454	0.193500
$784$	0	0
$785$	−85.3233	−3.04532
$786$	0	0
$787$	−34.8311	−1.24159	−0.620797	−	0.783971i	$-0.713191\pi$
$787$	−34.8311	−1.24159	−0.620797	+	0.783971i	$0.713191\pi$
$788$	0	0
$789$	2.79955	0.0996665
$790$	0	0
$791$	−15.4658	−0.549901
$792$	0	0
$793$	−60.9155	−2.16317
$794$	0	0
$795$	35.8428	1.27121
$796$	0	0
$797$	−38.7729	−1.37341	−0.686703	−	0.726938i	$-0.740943\pi$
$797$	−38.7729	−1.37341	−0.686703	+	0.726938i	$0.740943\pi$
$798$	0	0
$799$	53.3211	1.88636
$800$	0	0
$801$	−2.65933	−0.0939627
$802$	0	0
$803$	7.49879	0.264627
$804$	0	0
$805$	−121.774	−4.29197
$806$	0	0
$807$	−26.8987	−0.946878
$808$	0	0
$809$	−43.4499	−1.52762	−0.763809	−	0.645442i	$-0.776673\pi$
$809$	−43.4499	−1.52762	−0.763809	+	0.645442i	$0.776673\pi$
$810$	0	0
$811$	−2.44749	−0.0859429	−0.0429715	−	0.999076i	$-0.513682\pi$
$811$	−2.44749	−0.0859429	−0.0429715	+	0.999076i	$0.513682\pi$
$812$	0	0
$813$	17.3000	0.606737
$814$	0	0
$815$	45.3944	1.59009
$816$	0	0
$817$	−2.73098	−0.0955448
$818$	0	0
$819$	56.0632	1.95901
$820$	0	0
$821$	−36.0811	−1.25924	−0.629619	−	0.776904i	$-0.716789\pi$
$821$	−36.0811	−1.25924	−0.629619	+	0.776904i	$0.716789\pi$
$822$	0	0
$823$	17.9930	0.627198	0.313599	−	0.949556i	$-0.398465\pi$
$823$	17.9930	0.627198	0.313599	+	0.949556i	$0.398465\pi$
$824$	0	0
$825$	12.2519	0.426558
$826$	0	0
$827$	−0.631391	−0.0219556	−0.0109778	−	0.999940i	$-0.503494\pi$
$827$	−0.631391	−0.0219556	−0.0109778	+	0.999940i	$0.503494\pi$
$828$	0	0
$829$	16.1825	0.562040	0.281020	−	0.959702i	$-0.409327\pi$
$829$	16.1825	0.562040	0.281020	+	0.959702i	$0.409327\pi$
$830$	0	0
$831$	28.6156	0.992665
$832$	0	0
$833$	−82.2707	−2.85051
$834$	0	0
$835$	−79.4705	−2.75019
$836$	0	0
$837$	13.1547	0.454694
$838$	0	0
$839$	25.9211	0.894895	0.447447	−	0.894310i	$-0.352333\pi$
$839$	25.9211	0.894895	0.447447	+	0.894310i	$0.352333\pi$
$840$	0	0
$841$	−27.6571	−0.953695
$842$	0	0
$843$	−7.72606	−0.266100
$844$	0	0
$845$	105.679	3.63548
$846$	0	0
$847$	4.18187	0.143691
$848$	0	0
$849$	12.0069	0.412075
$850$	0	0
$851$	22.6826	0.777551
$852$	0	0
$853$	17.4306	0.596813	0.298406	−	0.954439i	$-0.403545\pi$
$853$	17.4306	0.596813	0.298406	+	0.954439i	$0.403545\pi$
$854$	0	0
$855$	−7.40395	−0.253210
$856$	0	0
$857$	−8.31111	−0.283902	−0.141951	−	0.989874i	$-0.545338\pi$
$857$	−8.31111	−0.283902	−0.141951	+	0.989874i	$0.545338\pi$
$858$	0	0
$859$	7.00870	0.239134	0.119567	−	0.992826i	$-0.461849\pi$
$859$	7.00870	0.239134	0.119567	+	0.992826i	$0.461849\pi$
$860$	0	0
$861$	38.1863	1.30139
$862$	0	0
$863$	5.06617	0.172454	0.0862272	−	0.996275i	$-0.472519\pi$
$863$	5.06617	0.172454	0.0862272	+	0.996275i	$0.472519\pi$
$864$	0	0
$865$	89.2436	3.03437
$866$	0	0
$867$	−40.0971	−1.36177
$868$	0	0
$869$	−8.20310	−0.278271
$870$	0	0
$871$	40.8844	1.38531
$872$	0	0
$873$	−5.51611	−0.186692
$874$	0	0
$875$	155.261	5.24877
$876$	0	0
$877$	−26.9947	−0.911546	−0.455773	−	0.890096i	$-0.650637\pi$
$877$	−26.9947	−0.911546	−0.455773	+	0.890096i	$0.650637\pi$
$878$	0	0
$879$	8.21797	0.277185
$880$	0	0
$881$	−13.0330	−0.439092	−0.219546	−	0.975602i	$-0.570458\pi$
$881$	−13.0330	−0.439092	−0.219546	+	0.975602i	$0.570458\pi$
$882$	0	0
$883$	−14.3202	−0.481914	−0.240957	−	0.970536i	$-0.577461\pi$
$883$	−14.3202	−0.481914	−0.240957	+	0.970536i	$0.577461\pi$
$884$	0	0
$885$	15.9419	0.535881
$886$	0	0
$887$	−59.4074	−1.99471	−0.997353	−	0.0727086i	$-0.976836\pi$
$887$	−59.4074	−1.99471	−0.997353	+	0.0727086i	$0.976836\pi$
$888$	0	0
$889$	−60.6115	−2.03284
$890$	0	0
$891$	−2.36060	−0.0790832
$892$	0	0
$893$	−5.32940	−0.178342
$894$	0	0
$895$	−43.4918	−1.45377
$896$	0	0
$897$	−37.2219	−1.24280
$898$	0	0
$899$	3.26249	0.108810
$900$	0	0
$901$	72.3888	2.41162
$902$	0	0
$903$	13.1160	0.436474
$904$	0	0
$905$	28.8866	0.960223
$906$	0	0
$907$	22.6318	0.751477	0.375739	−	0.926726i	$-0.377389\pi$
$907$	22.6318	0.751477	0.375739	+	0.926726i	$0.377389\pi$
$908$	0	0
$909$	−27.5345	−0.913263
$910$	0	0
$911$	−12.4698	−0.413143	−0.206572	−	0.978431i	$-0.566231\pi$
$911$	−12.4698	−0.413143	−0.206572	+	0.978431i	$0.566231\pi$
$912$	0	0
$913$	10.3886	0.343814
$914$	0	0
$915$	−38.6364	−1.27728
$916$	0	0
$917$	−81.6425	−2.69607
$918$	0	0
$919$	−27.7438	−0.915183	−0.457592	−	0.889162i	$-0.651288\pi$
$919$	−27.7438	−0.915183	−0.457592	+	0.889162i	$0.651288\pi$
$920$	0	0
$921$	10.8189	0.356495
$922$	0	0
$923$	1.64173	0.0540383
$924$	0	0
$925$	−45.7205	−1.50328
$926$	0	0
$927$	−19.3193	−0.634529
$928$	0	0
$929$	1.07235	0.0351828	0.0175914	−	0.999845i	$-0.494400\pi$
$929$	1.07235	0.0351828	0.0175914	+	0.999845i	$0.494400\pi$
$930$	0	0
$931$	8.22289	0.269494
$932$	0	0
$933$	−9.77190	−0.319918
$934$	0	0
$935$	33.8367	1.10658
$936$	0	0
$937$	31.8148	1.03934	0.519672	−	0.854366i	$-0.326054\pi$
$937$	31.8148	1.03934	0.519672	+	0.854366i	$0.326054\pi$
$938$	0	0
$939$	12.1999	0.398129
$940$	0	0
$941$	2.03201	0.0662418	0.0331209	−	0.999451i	$-0.489455\pi$
$941$	2.03201	0.0662418	0.0331209	+	0.999451i	$0.489455\pi$
$942$	0	0
$943$	68.4604	2.22938
$944$	0	0
$945$	84.2862	2.74183
$946$	0	0
$947$	−43.0433	−1.39872	−0.699360	−	0.714770i	$-0.746531\pi$
$947$	−43.0433	−1.39872	−0.699360	+	0.714770i	$0.746531\pi$
$948$	0	0
$949$	45.9200	1.49063
$950$	0	0
$951$	24.9150	0.807924
$952$	0	0
$953$	16.8370	0.545403	0.272702	−	0.962099i	$-0.412083\pi$
$953$	16.8370	0.545403	0.272702	+	0.962099i	$0.412083\pi$
$954$	0	0
$955$	66.1051	2.13911
$956$	0	0
$957$	1.04341	0.0337288
$958$	0	0
$959$	4.18187	0.135040
$960$	0	0
$961$	−23.0737	−0.744313
$962$	0	0
$963$	38.3591	1.23610
$964$	0	0
$965$	79.3328	2.55381
$966$	0	0
$967$	19.1374	0.615416	0.307708	−	0.951481i	$-0.400438\pi$
$967$	19.1374	0.615416	0.307708	+	0.951481i	$0.400438\pi$
$968$	0	0
$969$	5.53759	0.177893
$970$	0	0
$971$	7.44092	0.238790	0.119395	−	0.992847i	$-0.461904\pi$
$971$	7.44092	0.238790	0.119395	+	0.992847i	$0.461904\pi$
$972$	0	0
$973$	75.4880	2.42004
$974$	0	0
$975$	75.0267	2.40278
$976$	0	0
$977$	21.3110	0.681800	0.340900	−	0.940100i	$-0.389268\pi$
$977$	21.3110	0.681800	0.340900	+	0.940100i	$0.389268\pi$
$978$	0	0
$979$	−1.21472	−0.0388226
$980$	0	0
$981$	−8.43380	−0.269270
$982$	0	0
$983$	−17.7686	−0.566731	−0.283366	−	0.959012i	$-0.591451\pi$
$983$	−17.7686	−0.566731	−0.283366	+	0.959012i	$0.591451\pi$
$984$	0	0
$985$	11.3014	0.360093
$986$	0	0
$987$	25.5954	0.814711
$988$	0	0
$989$	23.5144	0.747715
$990$	0	0
$991$	−16.5426	−0.525493	−0.262746	−	0.964865i	$-0.584628\pi$
$991$	−16.5426	−0.525493	−0.262746	+	0.964865i	$0.584628\pi$
$992$	0	0
$993$	15.3637	0.487551
$994$	0	0
$995$	55.0867	1.74637
$996$	0	0
$997$	1.42752	0.0452100	0.0226050	−	0.999744i	$-0.492804\pi$
$997$	1.42752	0.0452100	0.0226050	+	0.999744i	$0.492804\pi$
$998$	0	0
$999$	−15.6999	−0.496722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.d.1.13	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.d.1.13	✓	27	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-0.900414 q^{3} +4.31358 q^{5} +4.18187 q^{7} -2.18926 q^{9} +O(q^{10})\)	\(q-0.900414 q^{3} +4.31358 q^{5} +4.18187 q^{7} -2.18926 q^{9} -1.00000 q^{11} -6.12366 q^{13} -3.88401 q^{15} -7.84422 q^{17} +0.784023 q^{19} -3.76542 q^{21} -6.75064 q^{23} +13.6070 q^{25} +4.67248 q^{27} +1.15882 q^{29} +2.81537 q^{31} +0.900414 q^{33} +18.0389 q^{35} -3.36007 q^{37} +5.51383 q^{39} -10.1413 q^{41} -3.48329 q^{43} -9.44354 q^{45} -6.79750 q^{47} +10.4881 q^{49} +7.06305 q^{51} -9.22830 q^{53} -4.31358 q^{55} -0.705945 q^{57} -4.10449 q^{59} +9.94756 q^{61} -9.15519 q^{63} -26.4149 q^{65} -6.67646 q^{67} +6.07837 q^{69} -0.268097 q^{71} -7.49879 q^{73} -12.2519 q^{75} -4.18187 q^{77} +8.20310 q^{79} +2.36060 q^{81} -10.3886 q^{83} -33.8367 q^{85} -1.04341 q^{87} +1.21472 q^{89} -25.6084 q^{91} -2.53499 q^{93} +3.38195 q^{95} +2.51963 q^{97} +2.18926 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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