LMFDB - Embedded newform 6028.2.a.f.1.22

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:	$0$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
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)

$f(q)$	$=$	$q+2.24948 q^{3} -3.83763 q^{5} -2.94590 q^{7} +2.06016 q^{9} +O(q^{10})$	$q+2.24948 q^{3} -3.83763 q^{5} -2.94590 q^{7} +2.06016 q^{9} +1.00000 q^{11} +3.80735 q^{13} -8.63266 q^{15} +5.40273 q^{17} -7.07548 q^{19} -6.62674 q^{21} -0.626715 q^{23} +9.72738 q^{25} -2.11416 q^{27} +2.65894 q^{29} -6.13893 q^{31} +2.24948 q^{33} +11.3053 q^{35} -1.01540 q^{37} +8.56455 q^{39} -7.39086 q^{41} +6.47851 q^{43} -7.90611 q^{45} -3.03981 q^{47} +1.67833 q^{49} +12.1533 q^{51} -13.2026 q^{53} -3.83763 q^{55} -15.9162 q^{57} +1.35796 q^{59} +5.33316 q^{61} -6.06902 q^{63} -14.6112 q^{65} +14.5073 q^{67} -1.40978 q^{69} +16.4771 q^{71} +9.05649 q^{73} +21.8815 q^{75} -2.94590 q^{77} +9.24911 q^{79} -10.9362 q^{81} +9.94844 q^{83} -20.7337 q^{85} +5.98123 q^{87} +6.11750 q^{89} -11.2161 q^{91} -13.8094 q^{93} +27.1531 q^{95} -0.140681 q^{97} +2.06016 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * 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
$3$	2.24948	1.29874	0.649369	−	0.760474i	$-0.275033\pi$
$3$	2.24948	1.29874	0.649369	+	0.760474i	$0.275033\pi$
$4$	0	0
$5$	−3.83763	−1.71624	−0.858119	−	0.513450i	$-0.828367\pi$
$5$	−3.83763	−1.71624	−0.858119	+	0.513450i	$0.828367\pi$
$6$	0	0
$7$	−2.94590	−1.11345	−0.556723	−	0.830698i	$-0.687941\pi$
$7$	−2.94590	−1.11345	−0.556723	+	0.830698i	$0.687941\pi$
$8$	0	0
$9$	2.06016	0.686719
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.80735	1.05597	0.527984	−	0.849254i	$-0.322948\pi$
$13$	3.80735	1.05597	0.527984	+	0.849254i	$0.322948\pi$
$14$	0	0
$15$	−8.63266	−2.22894
$16$	0	0
$17$	5.40273	1.31035	0.655177	−	0.755475i	$-0.272594\pi$
$17$	5.40273	1.31035	0.655177	+	0.755475i	$0.272594\pi$
$18$	0	0
$19$	−7.07548	−1.62323	−0.811613	−	0.584195i	$-0.801410\pi$
$19$	−7.07548	−1.62323	−0.811613	+	0.584195i	$0.801410\pi$
$20$	0	0
$21$	−6.62674	−1.44607
$22$	0	0
$23$	−0.626715	−0.130679	−0.0653395	−	0.997863i	$-0.520813\pi$
$23$	−0.626715	−0.130679	−0.0653395	+	0.997863i	$0.520813\pi$
$24$	0	0
$25$	9.72738	1.94548
$26$	0	0
$27$	−2.11416	−0.406870
$28$	0	0
$29$	2.65894	0.493753	0.246876	−	0.969047i	$-0.420596\pi$
$29$	2.65894	0.493753	0.246876	+	0.969047i	$0.420596\pi$
$30$	0	0
$31$	−6.13893	−1.10258	−0.551292	−	0.834312i	$-0.685865\pi$
$31$	−6.13893	−1.10258	−0.551292	+	0.834312i	$0.685865\pi$
$32$	0	0
$33$	2.24948	0.391584
$34$	0	0
$35$	11.3053	1.91094
$36$	0	0
$37$	−1.01540	−0.166931	−0.0834655	−	0.996511i	$-0.526599\pi$
$37$	−1.01540	−0.166931	−0.0834655	+	0.996511i	$0.526599\pi$
$38$	0	0
$39$	8.56455	1.37143
$40$	0	0
$41$	−7.39086	−1.15426	−0.577129	−	0.816653i	$-0.695827\pi$
$41$	−7.39086	−1.15426	−0.577129	+	0.816653i	$0.695827\pi$
$42$	0	0
$43$	6.47851	0.987964	0.493982	−	0.869472i	$-0.335541\pi$
$43$	6.47851	0.987964	0.493982	+	0.869472i	$0.335541\pi$
$44$	0	0
$45$	−7.90611	−1.17857
$46$	0	0
$47$	−3.03981	−0.443402	−0.221701	−	0.975115i	$-0.571161\pi$
$47$	−3.03981	−0.443402	−0.221701	+	0.975115i	$0.571161\pi$
$48$	0	0
$49$	1.67833	0.239762
$50$	0	0
$51$	12.1533	1.70181
$52$	0	0
$53$	−13.2026	−1.81352	−0.906758	−	0.421651i	$-0.861451\pi$
$53$	−13.2026	−1.81352	−0.906758	+	0.421651i	$0.861451\pi$
$54$	0	0
$55$	−3.83763	−0.517465
$56$	0	0
$57$	−15.9162	−2.10815
$58$	0	0
$59$	1.35796	0.176792	0.0883958	−	0.996085i	$-0.471826\pi$
$59$	1.35796	0.176792	0.0883958	+	0.996085i	$0.471826\pi$
$60$	0	0
$61$	5.33316	0.682842	0.341421	−	0.939911i	$-0.389092\pi$
$61$	5.33316	0.682842	0.341421	+	0.939911i	$0.389092\pi$
$62$	0	0
$63$	−6.06902	−0.764624
$64$	0	0
$65$	−14.6112	−1.81229
$66$	0	0
$67$	14.5073	1.77235	0.886173	−	0.463354i	$-0.153354\pi$
$67$	14.5073	1.77235	0.886173	+	0.463354i	$0.153354\pi$
$68$	0	0
$69$	−1.40978	−0.169718
$70$	0	0
$71$	16.4771	1.95547	0.977735	−	0.209845i	$-0.0672959\pi$
$71$	16.4771	1.95547	0.977735	+	0.209845i	$0.0672959\pi$
$72$	0	0
$73$	9.05649	1.05998	0.529991	−	0.848003i	$-0.322195\pi$
$73$	9.05649	1.05998	0.529991	+	0.848003i	$0.322195\pi$
$74$	0	0
$75$	21.8815	2.52666
$76$	0	0
$77$	−2.94590	−0.335717
$78$	0	0
$79$	9.24911	1.04061	0.520303	−	0.853982i	$-0.325819\pi$
$79$	9.24911	1.04061	0.520303	+	0.853982i	$0.325819\pi$
$80$	0	0
$81$	−10.9362	−1.21514
$82$	0	0
$83$	9.94844	1.09198	0.545992	−	0.837791i	$-0.316153\pi$
$83$	9.94844	1.09198	0.545992	+	0.837791i	$0.316153\pi$
$84$	0	0
$85$	−20.7337	−2.24888
$86$	0	0
$87$	5.98123	0.641255
$88$	0	0
$89$	6.11750	0.648454	0.324227	−	0.945979i	$-0.394896\pi$
$89$	6.11750	0.648454	0.324227	+	0.945979i	$0.394896\pi$
$90$	0	0
$91$	−11.2161	−1.17576
$92$	0	0
$93$	−13.8094	−1.43197
$94$	0	0
$95$	27.1531	2.78585
$96$	0	0
$97$	−0.140681	−0.0142840	−0.00714200	−	0.999974i	$-0.502273\pi$
$97$	−0.140681	−0.0142840	−0.00714200	+	0.999974i	$0.502273\pi$
$98$	0	0
$99$	2.06016	0.207054
$100$	0	0
$101$	16.5449	1.64628	0.823140	−	0.567839i	$-0.192220\pi$
$101$	16.5449	1.64628	0.823140	+	0.567839i	$0.192220\pi$
$102$	0	0
$103$	6.88400	0.678301	0.339151	−	0.940732i	$-0.389860\pi$
$103$	6.88400	0.678301	0.339151	+	0.940732i	$0.389860\pi$
$104$	0	0
$105$	25.4310	2.48181
$106$	0	0
$107$	4.52640	0.437583	0.218792	−	0.975772i	$-0.429789\pi$
$107$	4.52640	0.437583	0.218792	+	0.975772i	$0.429789\pi$
$108$	0	0
$109$	0.168687	0.0161573	0.00807865	−	0.999967i	$-0.497428\pi$
$109$	0.168687	0.0161573	0.00807865	+	0.999967i	$0.497428\pi$
$110$	0	0
$111$	−2.28412	−0.216799
$112$	0	0
$113$	5.91098	0.556059	0.278029	−	0.960573i	$-0.410319\pi$
$113$	5.91098	0.556059	0.278029	+	0.960573i	$0.410319\pi$
$114$	0	0
$115$	2.40510	0.224276
$116$	0	0
$117$	7.84374	0.725154
$118$	0	0
$119$	−15.9159	−1.45901
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−16.6256	−1.49908
$124$	0	0
$125$	−18.1419	−1.62266
$126$	0	0
$127$	−17.8799	−1.58659	−0.793294	−	0.608839i	$-0.791635\pi$
$127$	−17.8799	−1.58659	−0.793294	+	0.608839i	$0.791635\pi$
$128$	0	0
$129$	14.5733	1.28311
$130$	0	0
$131$	13.7978	1.20552	0.602758	−	0.797924i	$-0.294068\pi$
$131$	13.7978	1.20552	0.602758	+	0.797924i	$0.294068\pi$
$132$	0	0
$133$	20.8437	1.80738
$134$	0	0
$135$	8.11335	0.698286
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	−2.90083	−0.246046	−0.123023	−	0.992404i	$-0.539259\pi$
$139$	−2.90083	−0.246046	−0.123023	+	0.992404i	$0.539259\pi$
$140$	0	0
$141$	−6.83799	−0.575862
$142$	0	0
$143$	3.80735	0.318387
$144$	0	0
$145$	−10.2040	−0.847397
$146$	0	0
$147$	3.77537	0.311388
$148$	0	0
$149$	−10.8238	−0.886724	−0.443362	−	0.896343i	$-0.646214\pi$
$149$	−10.8238	−0.886724	−0.443362	+	0.896343i	$0.646214\pi$
$150$	0	0
$151$	0.911671	0.0741907	0.0370954	−	0.999312i	$-0.488189\pi$
$151$	0.911671	0.0741907	0.0370954	+	0.999312i	$0.488189\pi$
$152$	0	0
$153$	11.1305	0.899845
$154$	0	0
$155$	23.5589	1.89230
$156$	0	0
$157$	0.600984	0.0479638	0.0239819	−	0.999712i	$-0.492366\pi$
$157$	0.600984	0.0479638	0.0239819	+	0.999712i	$0.492366\pi$
$158$	0	0
$159$	−29.6990	−2.35528
$160$	0	0
$161$	1.84624	0.145504
$162$	0	0
$163$	18.5169	1.45036	0.725179	−	0.688561i	$-0.241757\pi$
$163$	18.5169	1.45036	0.725179	+	0.688561i	$0.241757\pi$
$164$	0	0
$165$	−8.63266	−0.672052
$166$	0	0
$167$	−9.23804	−0.714861	−0.357431	−	0.933940i	$-0.616347\pi$
$167$	−9.23804	−0.714861	−0.357431	+	0.933940i	$0.616347\pi$
$168$	0	0
$169$	1.49592	0.115070
$170$	0	0
$171$	−14.5766	−1.11470
$172$	0	0
$173$	−6.40093	−0.486654	−0.243327	−	0.969944i	$-0.578239\pi$
$173$	−6.40093	−0.486654	−0.243327	+	0.969944i	$0.578239\pi$
$174$	0	0
$175$	−28.6559	−2.16618
$176$	0	0
$177$	3.05471	0.229606
$178$	0	0
$179$	0.540809	0.0404219	0.0202110	−	0.999796i	$-0.493566\pi$
$179$	0.540809	0.0404219	0.0202110	+	0.999796i	$0.493566\pi$
$180$	0	0
$181$	21.3673	1.58822	0.794108	−	0.607776i	$-0.207938\pi$
$181$	21.3673	1.58822	0.794108	+	0.607776i	$0.207938\pi$
$182$	0	0
$183$	11.9968	0.886832
$184$	0	0
$185$	3.89673	0.286493
$186$	0	0
$187$	5.40273	0.395087
$188$	0	0
$189$	6.22810	0.453028
$190$	0	0
$191$	−7.62110	−0.551443	−0.275722	−	0.961238i	$-0.588917\pi$
$191$	−7.62110	−0.551443	−0.275722	+	0.961238i	$0.588917\pi$
$192$	0	0
$193$	11.5837	0.833812	0.416906	−	0.908950i	$-0.363114\pi$
$193$	11.5837	0.833812	0.416906	+	0.908950i	$0.363114\pi$
$194$	0	0
$195$	−32.8676	−2.35370
$196$	0	0
$197$	−8.07523	−0.575336	−0.287668	−	0.957730i	$-0.592880\pi$
$197$	−8.07523	−0.575336	−0.287668	+	0.957730i	$0.592880\pi$
$198$	0	0
$199$	8.73682	0.619337	0.309668	−	0.950845i	$-0.399782\pi$
$199$	8.73682	0.619337	0.309668	+	0.950845i	$0.399782\pi$
$200$	0	0
$201$	32.6338	2.30181
$202$	0	0
$203$	−7.83297	−0.549767
$204$	0	0
$205$	28.3633	1.98098
$206$	0	0
$207$	−1.29113	−0.0897398
$208$	0	0
$209$	−7.07548	−0.489421
$210$	0	0
$211$	−11.1317	−0.766337	−0.383169	−	0.923678i	$-0.625167\pi$
$211$	−11.1317	−0.766337	−0.383169	+	0.923678i	$0.625167\pi$
$212$	0	0
$213$	37.0648	2.53964
$214$	0	0
$215$	−24.8621	−1.69558
$216$	0	0
$217$	18.0847	1.22767
$218$	0	0
$219$	20.3724	1.37664
$220$	0	0
$221$	20.5701	1.38369
$222$	0	0
$223$	11.0199	0.737950	0.368975	−	0.929439i	$-0.379709\pi$
$223$	11.0199	0.737950	0.368975	+	0.929439i	$0.379709\pi$
$224$	0	0
$225$	20.0399	1.33599
$226$	0	0
$227$	9.83107	0.652511	0.326256	−	0.945282i	$-0.394213\pi$
$227$	9.83107	0.652511	0.326256	+	0.945282i	$0.394213\pi$
$228$	0	0
$229$	−25.2737	−1.67013	−0.835067	−	0.550149i	$-0.814571\pi$
$229$	−25.2737	−1.67013	−0.835067	+	0.550149i	$0.814571\pi$
$230$	0	0
$231$	−6.62674	−0.436008
$232$	0	0
$233$	−4.07427	−0.266914	−0.133457	−	0.991055i	$-0.542608\pi$
$233$	−4.07427	−0.266914	−0.133457	+	0.991055i	$0.542608\pi$
$234$	0	0
$235$	11.6657	0.760983
$236$	0	0
$237$	20.8057	1.35147
$238$	0	0
$239$	16.8766	1.09166	0.545829	−	0.837896i	$-0.316215\pi$
$239$	16.8766	1.09166	0.545829	+	0.837896i	$0.316215\pi$
$240$	0	0
$241$	12.1156	0.780436	0.390218	−	0.920722i	$-0.372400\pi$
$241$	12.1156	0.780436	0.390218	+	0.920722i	$0.372400\pi$
$242$	0	0
$243$	−18.2583	−1.17127
$244$	0	0
$245$	−6.44081	−0.411488
$246$	0	0
$247$	−26.9388	−1.71408
$248$	0	0
$249$	22.3788	1.41820
$250$	0	0
$251$	14.5984	0.921444	0.460722	−	0.887545i	$-0.347591\pi$
$251$	14.5984	0.921444	0.460722	+	0.887545i	$0.347591\pi$
$252$	0	0
$253$	−0.626715	−0.0394012
$254$	0	0
$255$	−46.6399	−2.92070
$256$	0	0
$257$	22.0087	1.37287	0.686433	−	0.727193i	$-0.259176\pi$
$257$	22.0087	1.37287	0.686433	+	0.727193i	$0.259176\pi$
$258$	0	0
$259$	2.99127	0.185869
$260$	0	0
$261$	5.47783	0.339069
$262$	0	0
$263$	28.7096	1.77031	0.885155	−	0.465297i	$-0.154052\pi$
$263$	28.7096	1.77031	0.885155	+	0.465297i	$0.154052\pi$
$264$	0	0
$265$	50.6666	3.11243
$266$	0	0
$267$	13.7612	0.842172
$268$	0	0
$269$	9.67990	0.590194	0.295097	−	0.955467i	$-0.404648\pi$
$269$	9.67990	0.590194	0.295097	+	0.955467i	$0.404648\pi$
$270$	0	0
$271$	15.0417	0.913718	0.456859	−	0.889539i	$-0.348974\pi$
$271$	15.0417	0.913718	0.456859	+	0.889539i	$0.348974\pi$
$272$	0	0
$273$	−25.2303	−1.52701
$274$	0	0
$275$	9.72738	0.586583
$276$	0	0
$277$	−23.5846	−1.41706	−0.708529	−	0.705681i	$-0.750641\pi$
$277$	−23.5846	−1.41706	−0.708529	+	0.705681i	$0.750641\pi$
$278$	0	0
$279$	−12.6472	−0.757165
$280$	0	0
$281$	4.82432	0.287795	0.143897	−	0.989593i	$-0.454037\pi$
$281$	4.82432	0.287795	0.143897	+	0.989593i	$0.454037\pi$
$282$	0	0
$283$	15.0305	0.893471	0.446736	−	0.894666i	$-0.352586\pi$
$283$	15.0305	0.893471	0.446736	+	0.894666i	$0.352586\pi$
$284$	0	0
$285$	61.0802	3.61808
$286$	0	0
$287$	21.7727	1.28520
$288$	0	0
$289$	12.1895	0.717027
$290$	0	0
$291$	−0.316459	−0.0185512
$292$	0	0
$293$	−27.0950	−1.58291	−0.791453	−	0.611230i	$-0.790675\pi$
$293$	−27.0950	−1.58291	−0.791453	+	0.611230i	$0.790675\pi$
$294$	0	0
$295$	−5.21135	−0.303416
$296$	0	0
$297$	−2.11416	−0.122676
$298$	0	0
$299$	−2.38612	−0.137993
$300$	0	0
$301$	−19.0851	−1.10004
$302$	0	0
$303$	37.2174	2.13808
$304$	0	0
$305$	−20.4667	−1.17192
$306$	0	0
$307$	−4.57015	−0.260833	−0.130416	−	0.991459i	$-0.541631\pi$
$307$	−4.57015	−0.260833	−0.130416	+	0.991459i	$0.541631\pi$
$308$	0	0
$309$	15.4854	0.880935
$310$	0	0
$311$	−10.4317	−0.591527	−0.295763	−	0.955261i	$-0.595574\pi$
$311$	−10.4317	−0.591527	−0.295763	+	0.955261i	$0.595574\pi$
$312$	0	0
$313$	8.03289	0.454046	0.227023	−	0.973889i	$-0.427101\pi$
$313$	8.03289	0.454046	0.227023	+	0.973889i	$0.427101\pi$
$314$	0	0
$315$	23.2906	1.31228
$316$	0	0
$317$	−15.0444	−0.844980	−0.422490	−	0.906368i	$-0.638844\pi$
$317$	−15.0444	−0.844980	−0.422490	+	0.906368i	$0.638844\pi$
$318$	0	0
$319$	2.65894	0.148872
$320$	0	0
$321$	10.1820	0.568306
$322$	0	0
$323$	−38.2269	−2.12700
$324$	0	0
$325$	37.0355	2.05436
$326$	0	0
$327$	0.379458	0.0209841
$328$	0	0
$329$	8.95498	0.493704
$330$	0	0
$331$	−1.84639	−0.101487	−0.0507434	−	0.998712i	$-0.516159\pi$
$331$	−1.84639	−0.101487	−0.0507434	+	0.998712i	$0.516159\pi$
$332$	0	0
$333$	−2.09189	−0.114635
$334$	0	0
$335$	−55.6735	−3.04177
$336$	0	0
$337$	−30.9774	−1.68745	−0.843724	−	0.536778i	$-0.819641\pi$
$337$	−30.9774	−1.68745	−0.843724	+	0.536778i	$0.819641\pi$
$338$	0	0
$339$	13.2966	0.722174
$340$	0	0
$341$	−6.13893	−0.332442
$342$	0	0
$343$	15.6771	0.846484
$344$	0	0
$345$	5.41022	0.291276
$346$	0	0
$347$	21.4803	1.15312	0.576561	−	0.817054i	$-0.304394\pi$
$347$	21.4803	1.15312	0.576561	+	0.817054i	$0.304394\pi$
$348$	0	0
$349$	−33.7580	−1.80702	−0.903511	−	0.428564i	$-0.859020\pi$
$349$	−33.7580	−1.80702	−0.903511	+	0.428564i	$0.859020\pi$
$350$	0	0
$351$	−8.04934	−0.429642
$352$	0	0
$353$	18.4554	0.982280	0.491140	−	0.871081i	$-0.336580\pi$
$353$	18.4554	0.982280	0.491140	+	0.871081i	$0.336580\pi$
$354$	0	0
$355$	−63.2329	−3.35605
$356$	0	0
$357$	−35.8025	−1.89487
$358$	0	0
$359$	20.4378	1.07867	0.539334	−	0.842092i	$-0.318676\pi$
$359$	20.4378	1.07867	0.539334	+	0.842092i	$0.318676\pi$
$360$	0	0
$361$	31.0625	1.63487
$362$	0	0
$363$	2.24948	0.118067
$364$	0	0
$365$	−34.7554	−1.81918
$366$	0	0
$367$	1.52424	0.0795646	0.0397823	−	0.999208i	$-0.487334\pi$
$367$	1.52424	0.0795646	0.0397823	+	0.999208i	$0.487334\pi$
$368$	0	0
$369$	−15.2263	−0.792651
$370$	0	0
$371$	38.8936	2.01925
$372$	0	0
$373$	−2.40222	−0.124382	−0.0621912	−	0.998064i	$-0.519809\pi$
$373$	−2.40222	−0.124382	−0.0621912	+	0.998064i	$0.519809\pi$
$374$	0	0
$375$	−40.8099	−2.10741
$376$	0	0
$377$	10.1235	0.521387
$378$	0	0
$379$	31.0799	1.59647	0.798234	−	0.602348i	$-0.205768\pi$
$379$	31.0799	1.59647	0.798234	+	0.602348i	$0.205768\pi$
$380$	0	0
$381$	−40.2205	−2.06056
$382$	0	0
$383$	−21.2770	−1.08720	−0.543601	−	0.839344i	$-0.682940\pi$
$383$	−21.2770	−1.08720	−0.543601	+	0.839344i	$0.682940\pi$
$384$	0	0
$385$	11.3053	0.576170
$386$	0	0
$387$	13.3468	0.678454
$388$	0	0
$389$	13.4769	0.683304	0.341652	−	0.939827i	$-0.389014\pi$
$389$	13.4769	0.683304	0.341652	+	0.939827i	$0.389014\pi$
$390$	0	0
$391$	−3.38597	−0.171236
$392$	0	0
$393$	31.0378	1.56565
$394$	0	0
$395$	−35.4946	−1.78593
$396$	0	0
$397$	5.48507	0.275288	0.137644	−	0.990482i	$-0.456047\pi$
$397$	5.48507	0.275288	0.137644	+	0.990482i	$0.456047\pi$
$398$	0	0
$399$	46.8874	2.34731
$400$	0	0
$401$	19.6587	0.981708	0.490854	−	0.871242i	$-0.336685\pi$
$401$	19.6587	0.981708	0.490854	+	0.871242i	$0.336685\pi$
$402$	0	0
$403$	−23.3730	−1.16429
$404$	0	0
$405$	41.9691	2.08546
$406$	0	0
$407$	−1.01540	−0.0503316
$408$	0	0
$409$	−6.71708	−0.332138	−0.166069	−	0.986114i	$-0.553107\pi$
$409$	−6.71708	−0.332138	−0.166069	+	0.986114i	$0.553107\pi$
$410$	0	0
$411$	2.24948	0.110959
$412$	0	0
$413$	−4.00042	−0.196848
$414$	0	0
$415$	−38.1784	−1.87410
$416$	0	0
$417$	−6.52537	−0.319549
$418$	0	0
$419$	36.0692	1.76210	0.881048	−	0.473027i	$-0.156839\pi$
$419$	36.0692	1.76210	0.881048	+	0.473027i	$0.156839\pi$
$420$	0	0
$421$	13.2313	0.644853	0.322427	−	0.946594i	$-0.395501\pi$
$421$	13.2313	0.644853	0.322427	+	0.946594i	$0.395501\pi$
$422$	0	0
$423$	−6.26248	−0.304492
$424$	0	0
$425$	52.5544	2.54926
$426$	0	0
$427$	−15.7110	−0.760307
$428$	0	0
$429$	8.56455	0.413501
$430$	0	0
$431$	16.9887	0.818315	0.409158	−	0.912464i	$-0.365823\pi$
$431$	16.9887	0.818315	0.409158	+	0.912464i	$0.365823\pi$
$432$	0	0
$433$	1.65604	0.0795844	0.0397922	−	0.999208i	$-0.487330\pi$
$433$	1.65604	0.0795844	0.0397922	+	0.999208i	$0.487330\pi$
$434$	0	0
$435$	−22.9537	−1.10055
$436$	0	0
$437$	4.43431	0.212122
$438$	0	0
$439$	−12.7768	−0.609803	−0.304901	−	0.952384i	$-0.598624\pi$
$439$	−12.7768	−0.609803	−0.304901	+	0.952384i	$0.598624\pi$
$440$	0	0
$441$	3.45763	0.164649
$442$	0	0
$443$	2.76446	0.131343	0.0656717	−	0.997841i	$-0.479081\pi$
$443$	2.76446	0.131343	0.0656717	+	0.997841i	$0.479081\pi$
$444$	0	0
$445$	−23.4767	−1.11290
$446$	0	0
$447$	−24.3480	−1.15162
$448$	0	0
$449$	−40.7552	−1.92336	−0.961678	−	0.274182i	$-0.911593\pi$
$449$	−40.7552	−1.92336	−0.961678	+	0.274182i	$0.911593\pi$
$450$	0	0
$451$	−7.39086	−0.348022
$452$	0	0
$453$	2.05079	0.0963543
$454$	0	0
$455$	43.0431	2.01789
$456$	0	0
$457$	−28.0289	−1.31114	−0.655568	−	0.755136i	$-0.727571\pi$
$457$	−28.0289	−1.31114	−0.655568	+	0.755136i	$0.727571\pi$
$458$	0	0
$459$	−11.4222	−0.533144
$460$	0	0
$461$	22.7470	1.05944	0.529718	−	0.848174i	$-0.322298\pi$
$461$	22.7470	1.05944	0.529718	+	0.848174i	$0.322298\pi$
$462$	0	0
$463$	−34.8745	−1.62076	−0.810379	−	0.585906i	$-0.800739\pi$
$463$	−34.8745	−1.62076	−0.810379	+	0.585906i	$0.800739\pi$
$464$	0	0
$465$	52.9953	2.45760
$466$	0	0
$467$	−10.9383	−0.506165	−0.253083	−	0.967445i	$-0.581444\pi$
$467$	−10.9383	−0.506165	−0.253083	+	0.967445i	$0.581444\pi$
$468$	0	0
$469$	−42.7370	−1.97341
$470$	0	0
$471$	1.35190	0.0622923
$472$	0	0
$473$	6.47851	0.297882
$474$	0	0
$475$	−68.8259	−3.15795
$476$	0	0
$477$	−27.1994	−1.24538
$478$	0	0
$479$	33.0610	1.51060	0.755298	−	0.655382i	$-0.227492\pi$
$479$	33.0610	1.51060	0.755298	+	0.655382i	$0.227492\pi$
$480$	0	0
$481$	−3.86599	−0.176274
$482$	0	0
$483$	4.15308	0.188972
$484$	0	0
$485$	0.539881	0.0245148
$486$	0	0
$487$	−40.3325	−1.82764	−0.913819	−	0.406122i	$-0.866881\pi$
$487$	−40.3325	−1.82764	−0.913819	+	0.406122i	$0.866881\pi$
$488$	0	0
$489$	41.6534	1.88363
$490$	0	0
$491$	−13.2722	−0.598966	−0.299483	−	0.954102i	$-0.596814\pi$
$491$	−13.2722	−0.598966	−0.299483	+	0.954102i	$0.596814\pi$
$492$	0	0
$493$	14.3655	0.646991
$494$	0	0
$495$	−7.90611	−0.355353
$496$	0	0
$497$	−48.5398	−2.17731
$498$	0	0
$499$	−4.78404	−0.214163	−0.107082	−	0.994250i	$-0.534151\pi$
$499$	−4.78404	−0.214163	−0.107082	+	0.994250i	$0.534151\pi$
$500$	0	0
$501$	−20.7808	−0.928417
$502$	0	0
$503$	17.6142	0.785378	0.392689	−	0.919671i	$-0.371545\pi$
$503$	17.6142	0.785378	0.392689	+	0.919671i	$0.371545\pi$
$504$	0	0
$505$	−63.4931	−2.82541
$506$	0	0
$507$	3.36503	0.149446
$508$	0	0
$509$	8.19782	0.363362	0.181681	−	0.983358i	$-0.441846\pi$
$509$	8.19782	0.363362	0.181681	+	0.983358i	$0.441846\pi$
$510$	0	0
$511$	−26.6795	−1.18023
$512$	0	0
$513$	14.9587	0.660442
$514$	0	0
$515$	−26.4182	−1.16413
$516$	0	0
$517$	−3.03981	−0.133691
$518$	0	0
$519$	−14.3988	−0.632036
$520$	0	0
$521$	−12.7528	−0.558712	−0.279356	−	0.960188i	$-0.590121\pi$
$521$	−12.7528	−0.558712	−0.279356	+	0.960188i	$0.590121\pi$
$522$	0	0
$523$	−14.3034	−0.625445	−0.312723	−	0.949845i	$-0.601241\pi$
$523$	−14.3034	−0.625445	−0.312723	+	0.949845i	$0.601241\pi$
$524$	0	0
$525$	−64.4608	−2.81330
$526$	0	0
$527$	−33.1669	−1.44478
$528$	0	0
$529$	−22.6072	−0.982923
$530$	0	0
$531$	2.79761	0.121406
$532$	0	0
$533$	−28.1396	−1.21886
$534$	0	0
$535$	−17.3706	−0.750997
$536$	0	0
$537$	1.21654	0.0524975
$538$	0	0
$539$	1.67833	0.0722909
$540$	0	0
$541$	−18.6149	−0.800317	−0.400158	−	0.916446i	$-0.631045\pi$
$541$	−18.6149	−0.800317	−0.400158	+	0.916446i	$0.631045\pi$
$542$	0	0
$543$	48.0652	2.06268
$544$	0	0
$545$	−0.647358	−0.0277298
$546$	0	0
$547$	−1.93536	−0.0827502	−0.0413751	−	0.999144i	$-0.513174\pi$
$547$	−1.93536	−0.0827502	−0.0413751	+	0.999144i	$0.513174\pi$
$548$	0	0
$549$	10.9872	0.468920
$550$	0	0
$551$	−18.8133	−0.801472
$552$	0	0
$553$	−27.2470	−1.15866
$554$	0	0
$555$	8.76562	0.372080
$556$	0	0
$557$	−13.1610	−0.557651	−0.278826	−	0.960342i	$-0.589945\pi$
$557$	−13.1610	−0.557651	−0.278826	+	0.960342i	$0.589945\pi$
$558$	0	0
$559$	24.6660	1.04326
$560$	0	0
$561$	12.1533	0.513114
$562$	0	0
$563$	−10.6535	−0.448990	−0.224495	−	0.974475i	$-0.572073\pi$
$563$	−10.6535	−0.448990	−0.224495	+	0.974475i	$0.572073\pi$
$564$	0	0
$565$	−22.6842	−0.954329
$566$	0	0
$567$	32.2170	1.35299
$568$	0	0
$569$	17.5420	0.735398	0.367699	−	0.929945i	$-0.380146\pi$
$569$	17.5420	0.735398	0.367699	+	0.929945i	$0.380146\pi$
$570$	0	0
$571$	−14.6568	−0.613368	−0.306684	−	0.951811i	$-0.599219\pi$
$571$	−14.6568	−0.613368	−0.306684	+	0.951811i	$0.599219\pi$
$572$	0	0
$573$	−17.1435	−0.716180
$574$	0	0
$575$	−6.09629	−0.254233
$576$	0	0
$577$	10.2681	0.427467	0.213734	−	0.976892i	$-0.431438\pi$
$577$	10.2681	0.427467	0.213734	+	0.976892i	$0.431438\pi$
$578$	0	0
$579$	26.0572	1.08290
$580$	0	0
$581$	−29.3071	−1.21586
$582$	0	0
$583$	−13.2026	−0.546796
$584$	0	0
$585$	−30.1013	−1.24454
$586$	0	0
$587$	43.1974	1.78295	0.891474	−	0.453071i	$-0.149672\pi$
$587$	43.1974	1.78295	0.891474	+	0.453071i	$0.149672\pi$
$588$	0	0
$589$	43.4359	1.78974
$590$	0	0
$591$	−18.1651	−0.747211
$592$	0	0
$593$	4.59496	0.188692	0.0943462	−	0.995539i	$-0.469924\pi$
$593$	4.59496	0.188692	0.0943462	+	0.995539i	$0.469924\pi$
$594$	0	0
$595$	61.0793	2.50401
$596$	0	0
$597$	19.6533	0.804356
$598$	0	0
$599$	39.4072	1.61014	0.805068	−	0.593183i	$-0.202129\pi$
$599$	39.4072	1.61014	0.805068	+	0.593183i	$0.202129\pi$
$600$	0	0
$601$	−29.8336	−1.21694	−0.608470	−	0.793577i	$-0.708216\pi$
$601$	−29.8336	−1.21694	−0.608470	+	0.793577i	$0.708216\pi$
$602$	0	0
$603$	29.8873	1.21710
$604$	0	0
$605$	−3.83763	−0.156022
$606$	0	0
$607$	1.35689	0.0550745	0.0275372	−	0.999621i	$-0.491234\pi$
$607$	1.35689	0.0550745	0.0275372	+	0.999621i	$0.491234\pi$
$608$	0	0
$609$	−17.6201	−0.714003
$610$	0	0
$611$	−11.5736	−0.468218
$612$	0	0
$613$	−9.77532	−0.394821	−0.197411	−	0.980321i	$-0.563253\pi$
$613$	−9.77532	−0.394821	−0.197411	+	0.980321i	$0.563253\pi$
$614$	0	0
$615$	63.8028	2.57278
$616$	0	0
$617$	5.11668	0.205990	0.102995	−	0.994682i	$-0.467157\pi$
$617$	5.11668	0.205990	0.102995	+	0.994682i	$0.467157\pi$
$618$	0	0
$619$	−6.41978	−0.258033	−0.129016	−	0.991642i	$-0.541182\pi$
$619$	−6.41978	−0.258033	−0.129016	+	0.991642i	$0.541182\pi$
$620$	0	0
$621$	1.32497	0.0531694
$622$	0	0
$623$	−18.0216	−0.722019
$624$	0	0
$625$	20.9850	0.839400
$626$	0	0
$627$	−15.9162	−0.635630
$628$	0	0
$629$	−5.48594	−0.218739
$630$	0	0
$631$	−15.2910	−0.608723	−0.304362	−	0.952557i	$-0.598443\pi$
$631$	−15.2910	−0.608723	−0.304362	+	0.952557i	$0.598443\pi$
$632$	0	0
$633$	−25.0405	−0.995271
$634$	0	0
$635$	68.6165	2.72296
$636$	0	0
$637$	6.39000	0.253181
$638$	0	0
$639$	33.9454	1.34286
$640$	0	0
$641$	24.4737	0.966653	0.483326	−	0.875440i	$-0.339428\pi$
$641$	24.4737	0.966653	0.483326	+	0.875440i	$0.339428\pi$
$642$	0	0
$643$	−22.0855	−0.870968	−0.435484	−	0.900196i	$-0.643423\pi$
$643$	−22.0855	−0.870968	−0.435484	+	0.900196i	$0.643423\pi$
$644$	0	0
$645$	−55.9268	−2.20212
$646$	0	0
$647$	5.69847	0.224030	0.112015	−	0.993707i	$-0.464270\pi$
$647$	5.69847	0.224030	0.112015	+	0.993707i	$0.464270\pi$
$648$	0	0
$649$	1.35796	0.0533046
$650$	0	0
$651$	40.6811	1.59442
$652$	0	0
$653$	−5.09485	−0.199377	−0.0996885	−	0.995019i	$-0.531785\pi$
$653$	−5.09485	−0.199377	−0.0996885	+	0.995019i	$0.531785\pi$
$654$	0	0
$655$	−52.9507	−2.06895
$656$	0	0
$657$	18.6578	0.727910
$658$	0	0
$659$	−6.38594	−0.248761	−0.124381	−	0.992235i	$-0.539694\pi$
$659$	−6.38594	−0.248761	−0.124381	+	0.992235i	$0.539694\pi$
$660$	0	0
$661$	13.0654	0.508186	0.254093	−	0.967180i	$-0.418223\pi$
$661$	13.0654	0.508186	0.254093	+	0.967180i	$0.418223\pi$
$662$	0	0
$663$	46.2720	1.79705
$664$	0	0
$665$	−79.9902	−3.10189
$666$	0	0
$667$	−1.66640	−0.0645231
$668$	0	0
$669$	24.7891	0.958403
$670$	0	0
$671$	5.33316	0.205884
$672$	0	0
$673$	−5.54559	−0.213767	−0.106883	−	0.994272i	$-0.534087\pi$
$673$	−5.54559	−0.213767	−0.106883	+	0.994272i	$0.534087\pi$
$674$	0	0
$675$	−20.5652	−0.791556
$676$	0	0
$677$	−27.1444	−1.04325	−0.521623	−	0.853176i	$-0.674673\pi$
$677$	−27.1444	−1.04325	−0.521623	+	0.853176i	$0.674673\pi$
$678$	0	0
$679$	0.414432	0.0159045
$680$	0	0
$681$	22.1148	0.847441
$682$	0	0
$683$	−28.3602	−1.08517	−0.542586	−	0.840000i	$-0.682555\pi$
$683$	−28.3602	−1.08517	−0.542586	+	0.840000i	$0.682555\pi$
$684$	0	0
$685$	−3.83763	−0.146628
$686$	0	0
$687$	−56.8527	−2.16906
$688$	0	0
$689$	−50.2669	−1.91502
$690$	0	0
$691$	6.59593	0.250921	0.125460	−	0.992099i	$-0.459959\pi$
$691$	6.59593	0.250921	0.125460	+	0.992099i	$0.459959\pi$
$692$	0	0
$693$	−6.06902	−0.230543
$694$	0	0
$695$	11.1323	0.422273
$696$	0	0
$697$	−39.9308	−1.51249
$698$	0	0
$699$	−9.16498	−0.346651
$700$	0	0
$701$	9.16590	0.346191	0.173096	−	0.984905i	$-0.444623\pi$
$701$	9.16590	0.346191	0.173096	+	0.984905i	$0.444623\pi$
$702$	0	0
$703$	7.18445	0.270967
$704$	0	0
$705$	26.2416	0.988317
$706$	0	0
$707$	−48.7396	−1.83304
$708$	0	0
$709$	−46.6749	−1.75291	−0.876456	−	0.481481i	$-0.840099\pi$
$709$	−46.6749	−1.75291	−0.876456	+	0.481481i	$0.840099\pi$
$710$	0	0
$711$	19.0546	0.714604
$712$	0	0
$713$	3.84736	0.144085
$714$	0	0
$715$	−14.6112	−0.546427
$716$	0	0
$717$	37.9636	1.41778
$718$	0	0
$719$	30.6225	1.14203	0.571013	−	0.820941i	$-0.306551\pi$
$719$	30.6225	1.14203	0.571013	+	0.820941i	$0.306551\pi$
$720$	0	0
$721$	−20.2796	−0.755252
$722$	0	0
$723$	27.2539	1.01358
$724$	0	0
$725$	25.8645	0.960584
$726$	0	0
$727$	42.6408	1.58146	0.790730	−	0.612164i	$-0.209701\pi$
$727$	42.6408	1.58146	0.790730	+	0.612164i	$0.209701\pi$
$728$	0	0
$729$	−8.26307	−0.306040
$730$	0	0
$731$	35.0016	1.29458
$732$	0	0
$733$	48.0468	1.77465	0.887324	−	0.461146i	$-0.152562\pi$
$733$	48.0468	1.77465	0.887324	+	0.461146i	$0.152562\pi$
$734$	0	0
$735$	−14.4885	−0.534415
$736$	0	0
$737$	14.5073	0.534383
$738$	0	0
$739$	−52.5106	−1.93163	−0.965817	−	0.259226i	$-0.916533\pi$
$739$	−52.5106	−1.93163	−0.965817	+	0.259226i	$0.916533\pi$
$740$	0	0
$741$	−60.5984	−2.22614
$742$	0	0
$743$	−48.9390	−1.79540	−0.897698	−	0.440611i	$-0.854762\pi$
$743$	−48.9390	−1.79540	−0.897698	+	0.440611i	$0.854762\pi$
$744$	0	0
$745$	41.5379	1.52183
$746$	0	0
$747$	20.4953	0.749886
$748$	0	0
$749$	−13.3343	−0.487225
$750$	0	0
$751$	−7.23486	−0.264004	−0.132002	−	0.991249i	$-0.542140\pi$
$751$	−7.23486	−0.264004	−0.132002	+	0.991249i	$0.542140\pi$
$752$	0	0
$753$	32.8388	1.19671
$754$	0	0
$755$	−3.49865	−0.127329
$756$	0	0
$757$	−11.6174	−0.422242	−0.211121	−	0.977460i	$-0.567711\pi$
$757$	−11.6174	−0.422242	−0.211121	+	0.977460i	$0.567711\pi$
$758$	0	0
$759$	−1.40978	−0.0511718
$760$	0	0
$761$	18.9920	0.688460	0.344230	−	0.938885i	$-0.388140\pi$
$761$	18.9920	0.688460	0.344230	+	0.938885i	$0.388140\pi$
$762$	0	0
$763$	−0.496936	−0.0179903
$764$	0	0
$765$	−42.7146	−1.54435
$766$	0	0
$767$	5.17023	0.186686
$768$	0	0
$769$	−14.7983	−0.533639	−0.266820	−	0.963746i	$-0.585973\pi$
$769$	−14.7983	−0.533639	−0.266820	+	0.963746i	$0.585973\pi$
$770$	0	0
$771$	49.5081	1.78299
$772$	0	0
$773$	−28.3258	−1.01881	−0.509405	−	0.860527i	$-0.670134\pi$
$773$	−28.3258	−1.01881	−0.509405	+	0.860527i	$0.670134\pi$
$774$	0	0
$775$	−59.7157	−2.14505
$776$	0	0
$777$	6.72880	0.241394
$778$	0	0
$779$	52.2939	1.87362
$780$	0	0
$781$	16.4771	0.589596
$782$	0	0
$783$	−5.62142	−0.200893
$784$	0	0
$785$	−2.30635	−0.0823173
$786$	0	0
$787$	−37.7855	−1.34691	−0.673454	−	0.739229i	$-0.735190\pi$
$787$	−37.7855	−1.34691	−0.673454	+	0.739229i	$0.735190\pi$
$788$	0	0
$789$	64.5816	2.29917
$790$	0	0
$791$	−17.4132	−0.619141
$792$	0	0
$793$	20.3052	0.721060
$794$	0	0
$795$	113.974	4.04223
$796$	0	0
$797$	42.5770	1.50815	0.754077	−	0.656786i	$-0.228085\pi$
$797$	42.5770	1.50815	0.754077	+	0.656786i	$0.228085\pi$
$798$	0	0
$799$	−16.4233	−0.581013
$800$	0	0
$801$	12.6030	0.445306
$802$	0	0
$803$	9.05649	0.319597
$804$	0	0
$805$	−7.08518	−0.249720
$806$	0	0
$807$	21.7747	0.766507
$808$	0	0
$809$	−35.1438	−1.23559	−0.617795	−	0.786339i	$-0.711974\pi$
$809$	−35.1438	−1.23559	−0.617795	+	0.786339i	$0.711974\pi$
$810$	0	0
$811$	40.5282	1.42314	0.711569	−	0.702616i	$-0.247985\pi$
$811$	40.5282	1.42314	0.711569	+	0.702616i	$0.247985\pi$
$812$	0	0
$813$	33.8360	1.18668
$814$	0	0
$815$	−71.0611	−2.48916
$816$	0	0
$817$	−45.8386	−1.60369
$818$	0	0
$819$	−23.1069	−0.807419
$820$	0	0
$821$	41.8584	1.46087	0.730434	−	0.682984i	$-0.239318\pi$
$821$	41.8584	1.46087	0.730434	+	0.682984i	$0.239318\pi$
$822$	0	0
$823$	18.0127	0.627882	0.313941	−	0.949442i	$-0.398351\pi$
$823$	18.0127	0.627882	0.313941	+	0.949442i	$0.398351\pi$
$824$	0	0
$825$	21.8815	0.761817
$826$	0	0
$827$	−15.1114	−0.525475	−0.262737	−	0.964867i	$-0.584625\pi$
$827$	−15.1114	−0.525475	−0.262737	+	0.964867i	$0.584625\pi$
$828$	0	0
$829$	−4.76672	−0.165555	−0.0827776	−	0.996568i	$-0.526379\pi$
$829$	−4.76672	−0.165555	−0.0827776	+	0.996568i	$0.526379\pi$
$830$	0	0
$831$	−53.0530	−1.84039
$832$	0	0
$833$	9.06757	0.314173
$834$	0	0
$835$	35.4522	1.22687
$836$	0	0
$837$	12.9787	0.448608
$838$	0	0
$839$	26.2254	0.905401	0.452700	−	0.891663i	$-0.350461\pi$
$839$	26.2254	0.905401	0.452700	+	0.891663i	$0.350461\pi$
$840$	0	0
$841$	−21.9300	−0.756208
$842$	0	0
$843$	10.8522	0.373770
$844$	0	0
$845$	−5.74077	−0.197488
$846$	0	0
$847$	−2.94590	−0.101222
$848$	0	0
$849$	33.8108	1.16038
$850$	0	0
$851$	0.636367	0.0218144
$852$	0	0
$853$	−12.1506	−0.416029	−0.208015	−	0.978126i	$-0.566700\pi$
$853$	−12.1506	−0.416029	−0.208015	+	0.978126i	$0.566700\pi$
$854$	0	0
$855$	55.9396	1.91309
$856$	0	0
$857$	19.7074	0.673194	0.336597	−	0.941649i	$-0.390724\pi$
$857$	19.7074	0.673194	0.336597	+	0.941649i	$0.390724\pi$
$858$	0	0
$859$	4.76593	0.162611	0.0813057	−	0.996689i	$-0.474091\pi$
$859$	4.76593	0.162611	0.0813057	+	0.996689i	$0.474091\pi$
$860$	0	0
$861$	48.9773	1.66914
$862$	0	0
$863$	−32.9566	−1.12186	−0.560929	−	0.827864i	$-0.689556\pi$
$863$	−32.9566	−1.12186	−0.560929	+	0.827864i	$0.689556\pi$
$864$	0	0
$865$	24.5644	0.835215
$866$	0	0
$867$	27.4199	0.931230
$868$	0	0
$869$	9.24911	0.313755
$870$	0	0
$871$	55.2343	1.87154
$872$	0	0
$873$	−0.289825	−0.00980909
$874$	0	0
$875$	53.4443	1.80675
$876$	0	0
$877$	38.3589	1.29529	0.647644	−	0.761943i	$-0.275754\pi$
$877$	38.3589	1.29529	0.647644	+	0.761943i	$0.275754\pi$
$878$	0	0
$879$	−60.9496	−2.05578
$880$	0	0
$881$	−11.1615	−0.376040	−0.188020	−	0.982165i	$-0.560207\pi$
$881$	−11.1615	−0.376040	−0.188020	+	0.982165i	$0.560207\pi$
$882$	0	0
$883$	−37.4535	−1.26041	−0.630206	−	0.776428i	$-0.717030\pi$
$883$	−37.4535	−1.26041	−0.630206	+	0.776428i	$0.717030\pi$
$884$	0	0
$885$	−11.7228	−0.394058
$886$	0	0
$887$	16.2814	0.546675	0.273337	−	0.961918i	$-0.411872\pi$
$887$	16.2814	0.546675	0.273337	+	0.961918i	$0.411872\pi$
$888$	0	0
$889$	52.6725	1.76658
$890$	0	0
$891$	−10.9362	−0.366377
$892$	0	0
$893$	21.5081	0.719742
$894$	0	0
$895$	−2.07542	−0.0693737
$896$	0	0
$897$	−5.36753	−0.179217
$898$	0	0
$899$	−16.3230	−0.544404
$900$	0	0
$901$	−71.3300	−2.37635
$902$	0	0
$903$	−42.9314	−1.42867
$904$	0	0
$905$	−81.9996	−2.72576
$906$	0	0
$907$	53.3539	1.77159	0.885794	−	0.464079i	$-0.153615\pi$
$907$	53.3539	1.77159	0.885794	+	0.464079i	$0.153615\pi$
$908$	0	0
$909$	34.0851	1.13053
$910$	0	0
$911$	−56.3547	−1.86711	−0.933557	−	0.358429i	$-0.883313\pi$
$911$	−56.3547	−1.86711	−0.933557	+	0.358429i	$0.883313\pi$
$912$	0	0
$913$	9.94844	0.329245
$914$	0	0
$915$	−46.0394	−1.52202
$916$	0	0
$917$	−40.6468	−1.34228
$918$	0	0
$919$	27.5086	0.907425	0.453713	−	0.891148i	$-0.350099\pi$
$919$	27.5086	0.907425	0.453713	+	0.891148i	$0.350099\pi$
$920$	0	0
$921$	−10.2805	−0.338753
$922$	0	0
$923$	62.7340	2.06491
$924$	0	0
$925$	−9.87719	−0.324760
$926$	0	0
$927$	14.1821	0.465802
$928$	0	0
$929$	1.07666	0.0353240	0.0176620	−	0.999844i	$-0.494378\pi$
$929$	1.07666	0.0353240	0.0176620	+	0.999844i	$0.494378\pi$
$930$	0	0
$931$	−11.8750	−0.389188
$932$	0	0
$933$	−23.4659	−0.768238
$934$	0	0
$935$	−20.7337	−0.678063
$936$	0	0
$937$	10.4120	0.340145	0.170072	−	0.985432i	$-0.445600\pi$
$937$	10.4120	0.340145	0.170072	+	0.985432i	$0.445600\pi$
$938$	0	0
$939$	18.0698	0.589686
$940$	0	0
$941$	−29.4078	−0.958666	−0.479333	−	0.877633i	$-0.659121\pi$
$941$	−29.4078	−0.958666	−0.479333	+	0.877633i	$0.659121\pi$
$942$	0	0
$943$	4.63196	0.150837
$944$	0	0
$945$	−23.9011	−0.777504
$946$	0	0
$947$	27.4359	0.891546	0.445773	−	0.895146i	$-0.352929\pi$
$947$	27.4359	0.891546	0.445773	+	0.895146i	$0.352929\pi$
$948$	0	0
$949$	34.4812	1.11931
$950$	0	0
$951$	−33.8422	−1.09741
$952$	0	0
$953$	34.0961	1.10448	0.552240	−	0.833685i	$-0.313773\pi$
$953$	34.0961	1.10448	0.552240	+	0.833685i	$0.313773\pi$
$954$	0	0
$955$	29.2469	0.946408
$956$	0	0
$957$	5.98123	0.193346
$958$	0	0
$959$	−2.94590	−0.0951281
$960$	0	0
$961$	6.68643	0.215691
$962$	0	0
$963$	9.32508	0.300497
$964$	0	0
$965$	−44.4538	−1.43102
$966$	0	0
$967$	−60.3239	−1.93989	−0.969943	−	0.243333i	$-0.921759\pi$
$967$	−60.3239	−1.93989	−0.969943	+	0.243333i	$0.921759\pi$
$968$	0	0
$969$	−85.9906	−2.76242
$970$	0	0
$971$	12.4501	0.399544	0.199772	−	0.979842i	$-0.435980\pi$
$971$	12.4501	0.399544	0.199772	+	0.979842i	$0.435980\pi$
$972$	0	0
$973$	8.54557	0.273958
$974$	0	0
$975$	83.3107	2.66808
$976$	0	0
$977$	−29.3470	−0.938893	−0.469446	−	0.882961i	$-0.655546\pi$
$977$	−29.3470	−0.938893	−0.469446	+	0.882961i	$0.655546\pi$
$978$	0	0
$979$	6.11750	0.195516
$980$	0	0
$981$	0.347522	0.0110955
$982$	0	0
$983$	−35.4937	−1.13207	−0.566036	−	0.824380i	$-0.691524\pi$
$983$	−35.4937	−1.13207	−0.566036	+	0.824380i	$0.691524\pi$
$984$	0	0
$985$	30.9897	0.987414
$986$	0	0
$987$	20.1440	0.641192
$988$	0	0
$989$	−4.06018	−0.129106
$990$	0	0
$991$	−5.86447	−0.186291	−0.0931455	−	0.995653i	$-0.529692\pi$
$991$	−5.86447	−0.186291	−0.0931455	+	0.995653i	$0.529692\pi$
$992$	0	0
$993$	−4.15342	−0.131805
$994$	0	0
$995$	−33.5287	−1.06293
$996$	0	0
$997$	−6.63908	−0.210262	−0.105131	−	0.994458i	$-0.533526\pi$
$997$	−6.63908	−0.210262	−0.105131	+	0.994458i	$0.533526\pi$
$998$	0	0
$999$	2.14672	0.0679192

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.f.1.22	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.f.1.22	✓	29	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)

\(f(q)\)	\(=\)	\(q+2.24948 q^{3} -3.83763 q^{5} -2.94590 q^{7} +2.06016 q^{9} +O(q^{10})\)	\(q+2.24948 q^{3} -3.83763 q^{5} -2.94590 q^{7} +2.06016 q^{9} +1.00000 q^{11} +3.80735 q^{13} -8.63266 q^{15} +5.40273 q^{17} -7.07548 q^{19} -6.62674 q^{21} -0.626715 q^{23} +9.72738 q^{25} -2.11416 q^{27} +2.65894 q^{29} -6.13893 q^{31} +2.24948 q^{33} +11.3053 q^{35} -1.01540 q^{37} +8.56455 q^{39} -7.39086 q^{41} +6.47851 q^{43} -7.90611 q^{45} -3.03981 q^{47} +1.67833 q^{49} +12.1533 q^{51} -13.2026 q^{53} -3.83763 q^{55} -15.9162 q^{57} +1.35796 q^{59} +5.33316 q^{61} -6.06902 q^{63} -14.6112 q^{65} +14.5073 q^{67} -1.40978 q^{69} +16.4771 q^{71} +9.05649 q^{73} +21.8815 q^{75} -2.94590 q^{77} +9.24911 q^{79} -10.9362 q^{81} +9.94844 q^{83} -20.7337 q^{85} +5.98123 q^{87} +6.11750 q^{89} -11.2161 q^{91} -13.8094 q^{93} +27.1531 q^{95} -0.140681 q^{97} +2.06016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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