LMFDB - Embedded newform 6028.2.a.c.1.18

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:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q+0.615083 q^{3} +0.179847 q^{5} +1.85704 q^{7} -2.62167 q^{9} +O(q^{10})$	$q+0.615083 q^{3} +0.179847 q^{5} +1.85704 q^{7} -2.62167 q^{9} +1.00000 q^{11} +0.370526 q^{13} +0.110621 q^{15} -1.41973 q^{17} +3.88780 q^{19} +1.14223 q^{21} -4.56140 q^{23} -4.96766 q^{25} -3.45779 q^{27} -8.96500 q^{29} +4.49396 q^{31} +0.615083 q^{33} +0.333983 q^{35} -0.121802 q^{37} +0.227904 q^{39} -11.6263 q^{41} +6.05931 q^{43} -0.471500 q^{45} +5.76166 q^{47} -3.55142 q^{49} -0.873251 q^{51} -10.1525 q^{53} +0.179847 q^{55} +2.39132 q^{57} +0.171687 q^{59} -3.59673 q^{61} -4.86854 q^{63} +0.0666380 q^{65} -1.63848 q^{67} -2.80564 q^{69} -0.291170 q^{71} +2.58192 q^{73} -3.05552 q^{75} +1.85704 q^{77} -9.34249 q^{79} +5.73819 q^{81} +6.87708 q^{83} -0.255334 q^{85} -5.51422 q^{87} -10.5150 q^{89} +0.688080 q^{91} +2.76415 q^{93} +0.699210 q^{95} +7.94096 q^{97} -2.62167 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) $ 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9	$ 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) $ 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9 + 25 * q^11 - 4 * q^13 - 10 * q^15 - 19 * q^17 - 12 * q^19 + 8 * q^21 - 31 * q^23 - q^25 - 44 * q^27 - q^29 - 8 * q^31 - 11 * q^33 - 16 * q^35 - 14 * q^37 - 18 * q^39 - 5 * q^41 - 15 * q^43 - 15 * q^45 - 41 * q^47 + 2 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 - 3 * q^57 - 35 * q^59 - 4 * q^61 - 45 * q^63 - 28 * q^65 - 30 * q^67 - 3 * q^69 + 4 * q^71 - 7 * q^73 - 18 * q^75 - 9 * q^77 - 9 * q^79 + 29 * q^81 - 72 * q^83 - 33 * q^87 - 30 * q^89 - 10 * q^91 + 7 * q^93 + 9 * q^95 - 37 * q^97 + 18 * 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
$3$	0.615083	0.355118	0.177559	−	0.984110i	$-0.443180\pi$
$3$	0.615083	0.355118	0.177559	+	0.984110i	$0.443180\pi$
$4$	0	0
$5$	0.179847	0.0804301	0.0402150	−	0.999191i	$-0.487196\pi$
$5$	0.179847	0.0804301	0.0402150	+	0.999191i	$0.487196\pi$
$6$	0	0
$7$	1.85704	0.701894	0.350947	−	0.936395i	$-0.385860\pi$
$7$	1.85704	0.701894	0.350947	+	0.936395i	$0.385860\pi$
$8$	0	0
$9$	−2.62167	−0.873891
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.370526	0.102765	0.0513827	−	0.998679i	$-0.483637\pi$
$13$	0.370526	0.102765	0.0513827	+	0.998679i	$0.483637\pi$
$14$	0	0
$15$	0.110621	0.0285622
$16$	0	0
$17$	−1.41973	−0.344335	−0.172168	−	0.985068i	$-0.555077\pi$
$17$	−1.41973	−0.344335	−0.172168	+	0.985068i	$0.555077\pi$
$18$	0	0
$19$	3.88780	0.891923	0.445961	−	0.895052i	$-0.352862\pi$
$19$	3.88780	0.891923	0.445961	+	0.895052i	$0.352862\pi$
$20$	0	0
$21$	1.14223	0.249255
$22$	0	0
$23$	−4.56140	−0.951118	−0.475559	−	0.879684i	$-0.657754\pi$
$23$	−4.56140	−0.951118	−0.475559	+	0.879684i	$0.657754\pi$
$24$	0	0
$25$	−4.96766	−0.993531
$26$	0	0
$27$	−3.45779	−0.665453
$28$	0	0
$29$	−8.96500	−1.66476	−0.832380	−	0.554206i	$-0.813022\pi$
$29$	−8.96500	−1.66476	−0.832380	+	0.554206i	$0.813022\pi$
$30$	0	0
$31$	4.49396	0.807138	0.403569	−	0.914949i	$-0.367769\pi$
$31$	4.49396	0.807138	0.403569	+	0.914949i	$0.367769\pi$
$32$	0	0
$33$	0.615083	0.107072
$34$	0	0
$35$	0.333983	0.0564534
$36$	0	0
$37$	−0.121802	−0.0200241	−0.0100120	−	0.999950i	$-0.503187\pi$
$37$	−0.121802	−0.0200241	−0.0100120	+	0.999950i	$0.503187\pi$
$38$	0	0
$39$	0.227904	0.0364939
$40$	0	0
$41$	−11.6263	−1.81572	−0.907860	−	0.419274i	$-0.862285\pi$
$41$	−11.6263	−1.81572	−0.907860	+	0.419274i	$0.862285\pi$
$42$	0	0
$43$	6.05931	0.924036	0.462018	−	0.886870i	$-0.347125\pi$
$43$	6.05931	0.924036	0.462018	+	0.886870i	$0.347125\pi$
$44$	0	0
$45$	−0.471500	−0.0702871
$46$	0	0
$47$	5.76166	0.840425	0.420212	−	0.907426i	$-0.361956\pi$
$47$	5.76166	0.840425	0.420212	+	0.907426i	$0.361956\pi$
$48$	0	0
$49$	−3.55142	−0.507345
$50$	0	0
$51$	−0.873251	−0.122280
$52$	0	0
$53$	−10.1525	−1.39455	−0.697273	−	0.716805i	$-0.745604\pi$
$53$	−10.1525	−1.39455	−0.697273	+	0.716805i	$0.745604\pi$
$54$	0	0
$55$	0.179847	0.0242506
$56$	0	0
$57$	2.39132	0.316738
$58$	0	0
$59$	0.171687	0.0223518	0.0111759	−	0.999938i	$-0.496443\pi$
$59$	0.171687	0.0223518	0.0111759	+	0.999938i	$0.496443\pi$
$60$	0	0
$61$	−3.59673	−0.460515	−0.230257	−	0.973130i	$-0.573957\pi$
$61$	−3.59673	−0.460515	−0.230257	+	0.973130i	$0.573957\pi$
$62$	0	0
$63$	−4.86854	−0.613379
$64$	0	0
$65$	0.0666380	0.00826543
$66$	0	0
$67$	−1.63848	−0.200172	−0.100086	−	0.994979i	$-0.531912\pi$
$67$	−1.63848	−0.200172	−0.100086	+	0.994979i	$0.531912\pi$
$68$	0	0
$69$	−2.80564	−0.337759
$70$	0	0
$71$	−0.291170	−0.0345555	−0.0172777	−	0.999851i	$-0.505500\pi$
$71$	−0.291170	−0.0345555	−0.0172777	+	0.999851i	$0.505500\pi$
$72$	0	0
$73$	2.58192	0.302191	0.151095	−	0.988519i	$-0.451720\pi$
$73$	2.58192	0.302191	0.151095	+	0.988519i	$0.451720\pi$
$74$	0	0
$75$	−3.05552	−0.352821
$76$	0	0
$77$	1.85704	0.211629
$78$	0	0
$79$	−9.34249	−1.05111	−0.525556	−	0.850759i	$-0.676143\pi$
$79$	−9.34249	−1.05111	−0.525556	+	0.850759i	$0.676143\pi$
$80$	0	0
$81$	5.73819	0.637577
$82$	0	0
$83$	6.87708	0.754857	0.377429	−	0.926039i	$-0.376808\pi$
$83$	6.87708	0.754857	0.377429	+	0.926039i	$0.376808\pi$
$84$	0	0
$85$	−0.255334	−0.0276949
$86$	0	0
$87$	−5.51422	−0.591186
$88$	0	0
$89$	−10.5150	−1.11459	−0.557295	−	0.830314i	$-0.688161\pi$
$89$	−10.5150	−1.11459	−0.557295	+	0.830314i	$0.688161\pi$
$90$	0	0
$91$	0.688080	0.0721304
$92$	0	0
$93$	2.76415	0.286629
$94$	0	0
$95$	0.699210	0.0717374
$96$	0	0
$97$	7.94096	0.806282	0.403141	−	0.915138i	$-0.367918\pi$
$97$	7.94096	0.806282	0.403141	+	0.915138i	$0.367918\pi$
$98$	0	0
$99$	−2.62167	−0.263488
$100$	0	0
$101$	−2.46988	−0.245762	−0.122881	−	0.992421i	$-0.539213\pi$
$101$	−2.46988	−0.245762	−0.122881	+	0.992421i	$0.539213\pi$
$102$	0	0
$103$	12.2613	1.20814	0.604069	−	0.796932i	$-0.293545\pi$
$103$	12.2613	1.20814	0.604069	+	0.796932i	$0.293545\pi$
$104$	0	0
$105$	0.205427	0.0200476
$106$	0	0
$107$	−13.9200	−1.34569	−0.672847	−	0.739782i	$-0.734929\pi$
$107$	−13.9200	−1.34569	−0.672847	+	0.739782i	$0.734929\pi$
$108$	0	0
$109$	−2.02096	−0.193573	−0.0967864	−	0.995305i	$-0.530856\pi$
$109$	−2.02096	−0.193573	−0.0967864	+	0.995305i	$0.530856\pi$
$110$	0	0
$111$	−0.0749181	−0.00711091
$112$	0	0
$113$	7.13134	0.670860	0.335430	−	0.942065i	$-0.391118\pi$
$113$	7.13134	0.670860	0.335430	+	0.942065i	$0.391118\pi$
$114$	0	0
$115$	−0.820355	−0.0764985
$116$	0	0
$117$	−0.971398	−0.0898058
$118$	0	0
$119$	−2.63649	−0.241687
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.15112	−0.644795
$124$	0	0
$125$	−1.79265	−0.160340
$126$	0	0
$127$	−8.19278	−0.726991	−0.363496	−	0.931596i	$-0.618417\pi$
$127$	−8.19278	−0.726991	−0.363496	+	0.931596i	$0.618417\pi$
$128$	0	0
$129$	3.72698	0.328142
$130$	0	0
$131$	−19.8306	−1.73261	−0.866303	−	0.499519i	$-0.833510\pi$
$131$	−19.8306	−1.73261	−0.866303	+	0.499519i	$0.833510\pi$
$132$	0	0
$133$	7.21979	0.626035
$134$	0	0
$135$	−0.621874	−0.0535224
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	−0.0556195	−0.00471759	−0.00235879	−	0.999997i	$-0.500751\pi$
$139$	−0.0556195	−0.00471759	−0.00235879	+	0.999997i	$0.500751\pi$
$140$	0	0
$141$	3.54390	0.298450
$142$	0	0
$143$	0.370526	0.0309849
$144$	0	0
$145$	−1.61233	−0.133897
$146$	0	0
$147$	−2.18441	−0.180167
$148$	0	0
$149$	21.1768	1.73487	0.867436	−	0.497548i	$-0.165766\pi$
$149$	21.1768	1.73487	0.867436	+	0.497548i	$0.165766\pi$
$150$	0	0
$151$	10.8189	0.880427	0.440214	−	0.897893i	$-0.354903\pi$
$151$	10.8189	0.880427	0.440214	+	0.897893i	$0.354903\pi$
$152$	0	0
$153$	3.72207	0.300911
$154$	0	0
$155$	0.808225	0.0649182
$156$	0	0
$157$	−0.643742	−0.0513762	−0.0256881	−	0.999670i	$-0.508178\pi$
$157$	−0.643742	−0.0513762	−0.0256881	+	0.999670i	$0.508178\pi$
$158$	0	0
$159$	−6.24460	−0.495229
$160$	0	0
$161$	−8.47069	−0.667584
$162$	0	0
$163$	11.5821	0.907179	0.453590	−	0.891211i	$-0.350143\pi$
$163$	11.5821	0.907179	0.453590	+	0.891211i	$0.350143\pi$
$164$	0	0
$165$	0.110621	0.00861182
$166$	0	0
$167$	18.0713	1.39840	0.699199	−	0.714928i	$-0.253540\pi$
$167$	18.0713	1.39840	0.699199	+	0.714928i	$0.253540\pi$
$168$	0	0
$169$	−12.8627	−0.989439
$170$	0	0
$171$	−10.1925	−0.779443
$172$	0	0
$173$	−19.4257	−1.47691	−0.738454	−	0.674304i	$-0.764444\pi$
$173$	−19.4257	−1.47691	−0.738454	+	0.674304i	$0.764444\pi$
$174$	0	0
$175$	−9.22512	−0.697353
$176$	0	0
$177$	0.105602	0.00793752
$178$	0	0
$179$	−9.56002	−0.714549	−0.357275	−	0.933999i	$-0.616294\pi$
$179$	−9.56002	−0.714549	−0.357275	+	0.933999i	$0.616294\pi$
$180$	0	0
$181$	0.263107	0.0195566	0.00977830	−	0.999952i	$-0.496887\pi$
$181$	0.263107	0.0195566	0.00977830	+	0.999952i	$0.496887\pi$
$182$	0	0
$183$	−2.21229	−0.163537
$184$	0	0
$185$	−0.0219057	−0.00161054
$186$	0	0
$187$	−1.41973	−0.103821
$188$	0	0
$189$	−6.42125	−0.467077
$190$	0	0
$191$	−11.0424	−0.798997	−0.399498	−	0.916734i	$-0.630816\pi$
$191$	−11.0424	−0.798997	−0.399498	+	0.916734i	$0.630816\pi$
$192$	0	0
$193$	−13.1618	−0.947410	−0.473705	−	0.880684i	$-0.657084\pi$
$193$	−13.1618	−0.947410	−0.473705	+	0.880684i	$0.657084\pi$
$194$	0	0
$195$	0.0409879	0.00293520
$196$	0	0
$197$	−27.2670	−1.94269	−0.971347	−	0.237665i	$-0.923618\pi$
$197$	−27.2670	−1.94269	−0.971347	+	0.237665i	$0.923618\pi$
$198$	0	0
$199$	3.19318	0.226358	0.113179	−	0.993575i	$-0.463897\pi$
$199$	3.19318	0.226358	0.113179	+	0.993575i	$0.463897\pi$
$200$	0	0
$201$	−1.00780	−0.0710848
$202$	0	0
$203$	−16.6483	−1.16848
$204$	0	0
$205$	−2.09095	−0.146038
$206$	0	0
$207$	11.9585	0.831174
$208$	0	0
$209$	3.88780	0.268925
$210$	0	0
$211$	8.92180	0.614202	0.307101	−	0.951677i	$-0.400641\pi$
$211$	8.92180	0.614202	0.307101	+	0.951677i	$0.400641\pi$
$212$	0	0
$213$	−0.179093	−0.0122713
$214$	0	0
$215$	1.08975	0.0743203
$216$	0	0
$217$	8.34544	0.566525
$218$	0	0
$219$	1.58809	0.107313
$220$	0	0
$221$	−0.526047	−0.0353858
$222$	0	0
$223$	−11.4225	−0.764904	−0.382452	−	0.923975i	$-0.624920\pi$
$223$	−11.4225	−0.764904	−0.382452	+	0.923975i	$0.624920\pi$
$224$	0	0
$225$	13.0236	0.868238
$226$	0	0
$227$	−4.55876	−0.302576	−0.151288	−	0.988490i	$-0.548342\pi$
$227$	−4.55876	−0.302576	−0.151288	+	0.988490i	$0.548342\pi$
$228$	0	0
$229$	−27.5220	−1.81871	−0.909353	−	0.416026i	$-0.863422\pi$
$229$	−27.5220	−1.81871	−0.909353	+	0.416026i	$0.863422\pi$
$230$	0	0
$231$	1.14223	0.0751533
$232$	0	0
$233$	6.04970	0.396329	0.198165	−	0.980169i	$-0.436502\pi$
$233$	6.04970	0.396329	0.198165	+	0.980169i	$0.436502\pi$
$234$	0	0
$235$	1.03622	0.0675954
$236$	0	0
$237$	−5.74640	−0.373269
$238$	0	0
$239$	−12.7794	−0.826633	−0.413317	−	0.910587i	$-0.635630\pi$
$239$	−12.7794	−0.826633	−0.413317	+	0.910587i	$0.635630\pi$
$240$	0	0
$241$	−4.25920	−0.274359	−0.137180	−	0.990546i	$-0.543804\pi$
$241$	−4.25920	−0.274359	−0.137180	+	0.990546i	$0.543804\pi$
$242$	0	0
$243$	13.9028	0.891868
$244$	0	0
$245$	−0.638712	−0.0408058
$246$	0	0
$247$	1.44053	0.0916588
$248$	0	0
$249$	4.22997	0.268064
$250$	0	0
$251$	−0.425910	−0.0268832	−0.0134416	−	0.999910i	$-0.504279\pi$
$251$	−0.425910	−0.0268832	−0.0134416	+	0.999910i	$0.504279\pi$
$252$	0	0
$253$	−4.56140	−0.286773
$254$	0	0
$255$	−0.157052	−0.00983496
$256$	0	0
$257$	−2.87781	−0.179513	−0.0897564	−	0.995964i	$-0.528609\pi$
$257$	−2.87781	−0.179513	−0.0897564	+	0.995964i	$0.528609\pi$
$258$	0	0
$259$	−0.226190	−0.0140548
$260$	0	0
$261$	23.5033	1.45482
$262$	0	0
$263$	7.48694	0.461665	0.230832	−	0.972994i	$-0.425855\pi$
$263$	7.48694	0.461665	0.230832	+	0.972994i	$0.425855\pi$
$264$	0	0
$265$	−1.82589	−0.112163
$266$	0	0
$267$	−6.46761	−0.395811
$268$	0	0
$269$	25.0262	1.52587	0.762937	−	0.646473i	$-0.223757\pi$
$269$	25.0262	1.52587	0.762937	+	0.646473i	$0.223757\pi$
$270$	0	0
$271$	5.73766	0.348538	0.174269	−	0.984698i	$-0.444244\pi$
$271$	5.73766	0.348538	0.174269	+	0.984698i	$0.444244\pi$
$272$	0	0
$273$	0.423226	0.0256148
$274$	0	0
$275$	−4.96766	−0.299561
$276$	0	0
$277$	−23.5129	−1.41275	−0.706376	−	0.707837i	$-0.749671\pi$
$277$	−23.5129	−1.41275	−0.706376	+	0.707837i	$0.749671\pi$
$278$	0	0
$279$	−11.7817	−0.705351
$280$	0	0
$281$	−3.24496	−0.193578	−0.0967891	−	0.995305i	$-0.530857\pi$
$281$	−3.24496	−0.193578	−0.0967891	+	0.995305i	$0.530857\pi$
$282$	0	0
$283$	−5.33969	−0.317412	−0.158706	−	0.987326i	$-0.550732\pi$
$283$	−5.33969	−0.317412	−0.158706	+	0.987326i	$0.550732\pi$
$284$	0	0
$285$	0.430072	0.0254752
$286$	0	0
$287$	−21.5904	−1.27444
$288$	0	0
$289$	−14.9844	−0.881433
$290$	0	0
$291$	4.88434	0.286325
$292$	0	0
$293$	2.26292	0.132201	0.0661005	−	0.997813i	$-0.478944\pi$
$293$	2.26292	0.132201	0.0661005	+	0.997813i	$0.478944\pi$
$294$	0	0
$295$	0.0308775	0.00179776
$296$	0	0
$297$	−3.45779	−0.200642
$298$	0	0
$299$	−1.69012	−0.0977421
$300$	0	0
$301$	11.2524	0.648575
$302$	0	0
$303$	−1.51918	−0.0872747
$304$	0	0
$305$	−0.646862	−0.0370392
$306$	0	0
$307$	13.9012	0.793382	0.396691	−	0.917952i	$-0.370158\pi$
$307$	13.9012	0.793382	0.396691	+	0.917952i	$0.370158\pi$
$308$	0	0
$309$	7.54169	0.429032
$310$	0	0
$311$	18.2042	1.03227	0.516134	−	0.856508i	$-0.327371\pi$
$311$	18.2042	1.03227	0.516134	+	0.856508i	$0.327371\pi$
$312$	0	0
$313$	21.9713	1.24189	0.620945	−	0.783854i	$-0.286749\pi$
$313$	21.9713	1.24189	0.620945	+	0.783854i	$0.286749\pi$
$314$	0	0
$315$	−0.875593	−0.0493341
$316$	0	0
$317$	−28.2600	−1.58724	−0.793620	−	0.608413i	$-0.791806\pi$
$317$	−28.2600	−1.58724	−0.793620	+	0.608413i	$0.791806\pi$
$318$	0	0
$319$	−8.96500	−0.501944
$320$	0	0
$321$	−8.56193	−0.477880
$322$	0	0
$323$	−5.51963	−0.307120
$324$	0	0
$325$	−1.84065	−0.102101
$326$	0	0
$327$	−1.24306	−0.0687412
$328$	0	0
$329$	10.6996	0.589889
$330$	0	0
$331$	28.9039	1.58870	0.794352	−	0.607458i	$-0.207811\pi$
$331$	28.9039	1.58870	0.794352	+	0.607458i	$0.207811\pi$
$332$	0	0
$333$	0.319324	0.0174989
$334$	0	0
$335$	−0.294676	−0.0160999
$336$	0	0
$337$	−21.5890	−1.17603	−0.588015	−	0.808850i	$-0.700090\pi$
$337$	−21.5890	−1.17603	−0.588015	+	0.808850i	$0.700090\pi$
$338$	0	0
$339$	4.38636	0.238234
$340$	0	0
$341$	4.49396	0.243361
$342$	0	0
$343$	−19.5944	−1.05800
$344$	0	0
$345$	−0.504586	−0.0271660
$346$	0	0
$347$	35.2318	1.89134	0.945670	−	0.325128i	$-0.105407\pi$
$347$	35.2318	1.89134	0.945670	+	0.325128i	$0.105407\pi$
$348$	0	0
$349$	34.1170	1.82624	0.913120	−	0.407690i	$-0.133666\pi$
$349$	34.1170	1.82624	0.913120	+	0.407690i	$0.133666\pi$
$350$	0	0
$351$	−1.28120	−0.0683855
$352$	0	0
$353$	15.7256	0.836989	0.418495	−	0.908219i	$-0.362558\pi$
$353$	15.7256	0.836989	0.418495	+	0.908219i	$0.362558\pi$
$354$	0	0
$355$	−0.0523660	−0.00277930
$356$	0	0
$357$	−1.62166	−0.0858273
$358$	0	0
$359$	31.2715	1.65045	0.825223	−	0.564807i	$-0.191049\pi$
$359$	31.2715	1.65045	0.825223	+	0.564807i	$0.191049\pi$
$360$	0	0
$361$	−3.88501	−0.204474
$362$	0	0
$363$	0.615083	0.0322835
$364$	0	0
$365$	0.464351	0.0243052
$366$	0	0
$367$	−5.64324	−0.294575	−0.147287	−	0.989094i	$-0.547054\pi$
$367$	−5.64324	−0.294575	−0.147287	+	0.989094i	$0.547054\pi$
$368$	0	0
$369$	30.4803	1.58674
$370$	0	0
$371$	−18.8535	−0.978824
$372$	0	0
$373$	−10.2006	−0.528168	−0.264084	−	0.964500i	$-0.585070\pi$
$373$	−10.2006	−0.528168	−0.264084	+	0.964500i	$0.585070\pi$
$374$	0	0
$375$	−1.10263	−0.0569396
$376$	0	0
$377$	−3.32177	−0.171080
$378$	0	0
$379$	−26.8498	−1.37918	−0.689590	−	0.724200i	$-0.742209\pi$
$379$	−26.8498	−1.37918	−0.689590	+	0.724200i	$0.742209\pi$
$380$	0	0
$381$	−5.03923	−0.258168
$382$	0	0
$383$	12.9990	0.664217	0.332108	−	0.943241i	$-0.392240\pi$
$383$	12.9990	0.664217	0.332108	+	0.943241i	$0.392240\pi$
$384$	0	0
$385$	0.333983	0.0170213
$386$	0	0
$387$	−15.8855	−0.807507
$388$	0	0
$389$	−25.2721	−1.28134	−0.640672	−	0.767815i	$-0.721344\pi$
$389$	−25.2721	−1.28134	−0.640672	+	0.767815i	$0.721344\pi$
$390$	0	0
$391$	6.47596	0.327503
$392$	0	0
$393$	−12.1974	−0.615280
$394$	0	0
$395$	−1.68022	−0.0845410
$396$	0	0
$397$	−9.85489	−0.494603	−0.247301	−	0.968939i	$-0.579544\pi$
$397$	−9.85489	−0.494603	−0.247301	+	0.968939i	$0.579544\pi$
$398$	0	0
$399$	4.44077	0.222316
$400$	0	0
$401$	−1.95637	−0.0976967	−0.0488483	−	0.998806i	$-0.515555\pi$
$401$	−1.95637	−0.0976967	−0.0488483	+	0.998806i	$0.515555\pi$
$402$	0	0
$403$	1.66513	0.0829459
$404$	0	0
$405$	1.03200	0.0512804
$406$	0	0
$407$	−0.121802	−0.00603748
$408$	0	0
$409$	3.72847	0.184361	0.0921806	−	0.995742i	$-0.470616\pi$
$409$	3.72847	0.184361	0.0921806	+	0.995742i	$0.470616\pi$
$410$	0	0
$411$	−0.615083	−0.0303398
$412$	0	0
$413$	0.318830	0.0156886
$414$	0	0
$415$	1.23682	0.0607132
$416$	0	0
$417$	−0.0342106	−0.00167530
$418$	0	0
$419$	20.9720	1.02455	0.512275	−	0.858822i	$-0.328803\pi$
$419$	20.9720	1.02455	0.512275	+	0.858822i	$0.328803\pi$
$420$	0	0
$421$	−32.9609	−1.60642	−0.803208	−	0.595698i	$-0.796875\pi$
$421$	−32.9609	−1.60642	−0.803208	+	0.595698i	$0.796875\pi$
$422$	0	0
$423$	−15.1052	−0.734440
$424$	0	0
$425$	7.05273	0.342108
$426$	0	0
$427$	−6.67927	−0.323232
$428$	0	0
$429$	0.227904	0.0110033
$430$	0	0
$431$	−37.6481	−1.81345	−0.906723	−	0.421726i	$-0.861425\pi$
$431$	−37.6481	−1.81345	−0.906723	+	0.421726i	$0.861425\pi$
$432$	0	0
$433$	36.8779	1.77224	0.886119	−	0.463457i	$-0.153391\pi$
$433$	36.8779	1.77224	0.886119	+	0.463457i	$0.153391\pi$
$434$	0	0
$435$	−0.991716	−0.0475491
$436$	0	0
$437$	−17.7338	−0.848324
$438$	0	0
$439$	−23.6171	−1.12718	−0.563591	−	0.826054i	$-0.690581\pi$
$439$	−23.6171	−1.12718	−0.563591	+	0.826054i	$0.690581\pi$
$440$	0	0
$441$	9.31065	0.443364
$442$	0	0
$443$	−15.9771	−0.759095	−0.379547	−	0.925172i	$-0.623920\pi$
$443$	−15.9771	−0.759095	−0.379547	+	0.925172i	$0.623920\pi$
$444$	0	0
$445$	−1.89110	−0.0896466
$446$	0	0
$447$	13.0255	0.616085
$448$	0	0
$449$	−31.6608	−1.49417	−0.747084	−	0.664730i	$-0.768547\pi$
$449$	−31.6608	−1.49417	−0.747084	+	0.664730i	$0.768547\pi$
$450$	0	0
$451$	−11.6263	−0.547460
$452$	0	0
$453$	6.65450	0.312656
$454$	0	0
$455$	0.123749	0.00580145
$456$	0	0
$457$	6.09461	0.285094	0.142547	−	0.989788i	$-0.454471\pi$
$457$	6.09461	0.285094	0.142547	+	0.989788i	$0.454471\pi$
$458$	0	0
$459$	4.90913	0.229139
$460$	0	0
$461$	40.0618	1.86587	0.932933	−	0.360050i	$-0.117240\pi$
$461$	40.0618	1.86587	0.932933	+	0.360050i	$0.117240\pi$
$462$	0	0
$463$	16.0213	0.744573	0.372287	−	0.928118i	$-0.378574\pi$
$463$	16.0213	0.744573	0.372287	+	0.928118i	$0.378574\pi$
$464$	0	0
$465$	0.497125	0.0230536
$466$	0	0
$467$	34.6176	1.60191	0.800957	−	0.598722i	$-0.204325\pi$
$467$	34.6176	1.60191	0.800957	+	0.598722i	$0.204325\pi$
$468$	0	0
$469$	−3.04272	−0.140500
$470$	0	0
$471$	−0.395954	−0.0182446
$472$	0	0
$473$	6.05931	0.278607
$474$	0	0
$475$	−19.3133	−0.886153
$476$	0	0
$477$	26.6164	1.21868
$478$	0	0
$479$	−42.5071	−1.94220	−0.971099	−	0.238678i	$-0.923286\pi$
$479$	−42.5071	−1.94220	−0.971099	+	0.238678i	$0.923286\pi$
$480$	0	0
$481$	−0.0451307	−0.00205778
$482$	0	0
$483$	−5.21017	−0.237071
$484$	0	0
$485$	1.42816	0.0648493
$486$	0	0
$487$	−37.8341	−1.71443	−0.857214	−	0.514960i	$-0.827807\pi$
$487$	−37.8341	−1.71443	−0.857214	+	0.514960i	$0.827807\pi$
$488$	0	0
$489$	7.12394	0.322156
$490$	0	0
$491$	−33.0524	−1.49164	−0.745818	−	0.666150i	$-0.767941\pi$
$491$	−33.0524	−1.49164	−0.745818	+	0.666150i	$0.767941\pi$
$492$	0	0
$493$	12.7279	0.573235
$494$	0	0
$495$	−0.471500	−0.0211924
$496$	0	0
$497$	−0.540713	−0.0242543
$498$	0	0
$499$	−9.13002	−0.408716	−0.204358	−	0.978896i	$-0.565511\pi$
$499$	−9.13002	−0.408716	−0.204358	+	0.978896i	$0.565511\pi$
$500$	0	0
$501$	11.1153	0.496596
$502$	0	0
$503$	40.3114	1.79739	0.898697	−	0.438569i	$-0.144515\pi$
$503$	40.3114	1.79739	0.898697	+	0.438569i	$0.144515\pi$
$504$	0	0
$505$	−0.444201	−0.0197667
$506$	0	0
$507$	−7.91163	−0.351368
$508$	0	0
$509$	−38.5450	−1.70848	−0.854238	−	0.519883i	$-0.825976\pi$
$509$	−38.5450	−1.70848	−0.854238	+	0.519883i	$0.825976\pi$
$510$	0	0
$511$	4.79472	0.212106
$512$	0	0
$513$	−13.4432	−0.593532
$514$	0	0
$515$	2.20515	0.0971706
$516$	0	0
$517$	5.76166	0.253398
$518$	0	0
$519$	−11.9484	−0.524477
$520$	0	0
$521$	20.3230	0.890367	0.445183	−	0.895439i	$-0.353139\pi$
$521$	20.3230	0.890367	0.445183	+	0.895439i	$0.353139\pi$
$522$	0	0
$523$	−35.0409	−1.53223	−0.766116	−	0.642703i	$-0.777813\pi$
$523$	−35.0409	−1.53223	−0.766116	+	0.642703i	$0.777813\pi$
$524$	0	0
$525$	−5.67421	−0.247643
$526$	0	0
$527$	−6.38021	−0.277926
$528$	0	0
$529$	−2.19361	−0.0953743
$530$	0	0
$531$	−0.450108	−0.0195330
$532$	0	0
$533$	−4.30784	−0.186593
$534$	0	0
$535$	−2.50347	−0.108234
$536$	0	0
$537$	−5.88020	−0.253749
$538$	0	0
$539$	−3.55142	−0.152970
$540$	0	0
$541$	29.1005	1.25113	0.625565	−	0.780172i	$-0.284869\pi$
$541$	29.1005	1.25113	0.625565	+	0.780172i	$0.284869\pi$
$542$	0	0
$543$	0.161833	0.00694490
$544$	0	0
$545$	−0.363464	−0.0155691
$546$	0	0
$547$	9.35953	0.400185	0.200092	−	0.979777i	$-0.435876\pi$
$547$	9.35953	0.400185	0.200092	+	0.979777i	$0.435876\pi$
$548$	0	0
$549$	9.42946	0.402440
$550$	0	0
$551$	−34.8541	−1.48484
$552$	0	0
$553$	−17.3493	−0.737769
$554$	0	0
$555$	−0.0134738	−0.000571931 0
$556$	0	0
$557$	−6.74836	−0.285937	−0.142969	−	0.989727i	$-0.545665\pi$
$557$	−6.74836	−0.285937	−0.142969	+	0.989727i	$0.545665\pi$
$558$	0	0
$559$	2.24513	0.0949590
$560$	0	0
$561$	−0.873251	−0.0368687
$562$	0	0
$563$	−26.2717	−1.10722	−0.553611	−	0.832775i	$-0.686751\pi$
$563$	−26.2717	−1.10722	−0.553611	+	0.832775i	$0.686751\pi$
$564$	0	0
$565$	1.28255	0.0539573
$566$	0	0
$567$	10.6560	0.447511
$568$	0	0
$569$	42.4010	1.77754	0.888771	−	0.458351i	$-0.151560\pi$
$569$	42.4010	1.77754	0.888771	+	0.458351i	$0.151560\pi$
$570$	0	0
$571$	−43.8568	−1.83535	−0.917676	−	0.397330i	$-0.869937\pi$
$571$	−43.8568	−1.83535	−0.917676	+	0.397330i	$0.869937\pi$
$572$	0	0
$573$	−6.79196	−0.283738
$574$	0	0
$575$	22.6595	0.944965
$576$	0	0
$577$	3.14902	0.131095	0.0655476	−	0.997849i	$-0.479121\pi$
$577$	3.14902	0.131095	0.0655476	+	0.997849i	$0.479121\pi$
$578$	0	0
$579$	−8.09562	−0.336442
$580$	0	0
$581$	12.7710	0.529830
$582$	0	0
$583$	−10.1525	−0.420472
$584$	0	0
$585$	−0.174703	−0.00722309
$586$	0	0
$587$	0.361845	0.0149349	0.00746746	−	0.999972i	$-0.497623\pi$
$587$	0.361845	0.0149349	0.00746746	+	0.999972i	$0.497623\pi$
$588$	0	0
$589$	17.4716	0.719905
$590$	0	0
$591$	−16.7715	−0.689886
$592$	0	0
$593$	−7.94688	−0.326339	−0.163170	−	0.986598i	$-0.552172\pi$
$593$	−7.94688	−0.326339	−0.163170	+	0.986598i	$0.552172\pi$
$594$	0	0
$595$	−0.474165	−0.0194389
$596$	0	0
$597$	1.96407	0.0803839
$598$	0	0
$599$	11.4266	0.466878	0.233439	−	0.972371i	$-0.425002\pi$
$599$	11.4266	0.466878	0.233439	+	0.972371i	$0.425002\pi$
$600$	0	0
$601$	22.6253	0.922905	0.461453	−	0.887165i	$-0.347328\pi$
$601$	22.6253	0.922905	0.461453	+	0.887165i	$0.347328\pi$
$602$	0	0
$603$	4.29556	0.174929
$604$	0	0
$605$	0.179847	0.00731182
$606$	0	0
$607$	6.30557	0.255935	0.127968	−	0.991778i	$-0.459155\pi$
$607$	6.30557	0.255935	0.127968	+	0.991778i	$0.459155\pi$
$608$	0	0
$609$	−10.2401	−0.414950
$610$	0	0
$611$	2.13485	0.0863666
$612$	0	0
$613$	6.46983	0.261314	0.130657	−	0.991428i	$-0.458291\pi$
$613$	6.46983	0.261314	0.130657	+	0.991428i	$0.458291\pi$
$614$	0	0
$615$	−1.28611	−0.0518609
$616$	0	0
$617$	−18.7992	−0.756828	−0.378414	−	0.925636i	$-0.623530\pi$
$617$	−18.7992	−0.756828	−0.378414	+	0.925636i	$0.623530\pi$
$618$	0	0
$619$	4.16499	0.167405	0.0837026	−	0.996491i	$-0.473325\pi$
$619$	4.16499	0.167405	0.0837026	+	0.996491i	$0.473325\pi$
$620$	0	0
$621$	15.7724	0.632924
$622$	0	0
$623$	−19.5268	−0.782324
$624$	0	0
$625$	24.5159	0.980635
$626$	0	0
$627$	2.39132	0.0955000
$628$	0	0
$629$	0.172925	0.00689499
$630$	0	0
$631$	34.8427	1.38707	0.693533	−	0.720425i	$-0.256053\pi$
$631$	34.8427	1.38707	0.693533	+	0.720425i	$0.256053\pi$
$632$	0	0
$633$	5.48764	0.218114
$634$	0	0
$635$	−1.47345	−0.0584720
$636$	0	0
$637$	−1.31589	−0.0521375
$638$	0	0
$639$	0.763352	0.0301977
$640$	0	0
$641$	−44.9883	−1.77693	−0.888465	−	0.458944i	$-0.848228\pi$
$641$	−44.9883	−1.77693	−0.888465	+	0.458944i	$0.848228\pi$
$642$	0	0
$643$	−28.6736	−1.13078	−0.565389	−	0.824825i	$-0.691274\pi$
$643$	−28.6736	−1.13078	−0.565389	+	0.824825i	$0.691274\pi$
$644$	0	0
$645$	0.670286	0.0263925
$646$	0	0
$647$	−15.6519	−0.615340	−0.307670	−	0.951493i	$-0.599549\pi$
$647$	−15.6519	−0.615340	−0.307670	+	0.951493i	$0.599549\pi$
$648$	0	0
$649$	0.171687	0.00673932
$650$	0	0
$651$	5.13313	0.201183
$652$	0	0
$653$	11.2343	0.439630	0.219815	−	0.975542i	$-0.429455\pi$
$653$	11.2343	0.439630	0.219815	+	0.975542i	$0.429455\pi$
$654$	0	0
$655$	−3.56647	−0.139354
$656$	0	0
$657$	−6.76895	−0.264082
$658$	0	0
$659$	7.17064	0.279328	0.139664	−	0.990199i	$-0.455398\pi$
$659$	7.17064	0.279328	0.139664	+	0.990199i	$0.455398\pi$
$660$	0	0
$661$	13.1431	0.511207	0.255603	−	0.966782i	$-0.417726\pi$
$661$	13.1431	0.511207	0.255603	+	0.966782i	$0.417726\pi$
$662$	0	0
$663$	−0.323562	−0.0125661
$664$	0	0
$665$	1.29846	0.0503520
$666$	0	0
$667$	40.8930	1.58338
$668$	0	0
$669$	−7.02575	−0.271631
$670$	0	0
$671$	−3.59673	−0.138850
$672$	0	0
$673$	−24.5170	−0.945061	−0.472531	−	0.881314i	$-0.656659\pi$
$673$	−24.5170	−0.945061	−0.472531	+	0.881314i	$0.656659\pi$
$674$	0	0
$675$	17.1771	0.661148
$676$	0	0
$677$	26.9454	1.03559	0.517797	−	0.855504i	$-0.326752\pi$
$677$	26.9454	1.03559	0.517797	+	0.855504i	$0.326752\pi$
$678$	0	0
$679$	14.7466	0.565924
$680$	0	0
$681$	−2.80402	−0.107450
$682$	0	0
$683$	3.75695	0.143756	0.0718778	−	0.997413i	$-0.477101\pi$
$683$	3.75695	0.143756	0.0718778	+	0.997413i	$0.477101\pi$
$684$	0	0
$685$	−0.179847	−0.00687160
$686$	0	0
$687$	−16.9283	−0.645855
$688$	0	0
$689$	−3.76175	−0.143311
$690$	0	0
$691$	17.4724	0.664681	0.332340	−	0.943160i	$-0.392162\pi$
$691$	17.4724	0.664681	0.332340	+	0.943160i	$0.392162\pi$
$692$	0	0
$693$	−4.86854	−0.184941
$694$	0	0
$695$	−0.0100030	−0.000379436 0
$696$	0	0
$697$	16.5062	0.625216
$698$	0	0
$699$	3.72107	0.140744
$700$	0	0
$701$	23.3970	0.883692	0.441846	−	0.897091i	$-0.354324\pi$
$701$	23.3970	0.883692	0.441846	+	0.897091i	$0.354324\pi$
$702$	0	0
$703$	−0.473540	−0.0178599
$704$	0	0
$705$	0.637359	0.0240043
$706$	0	0
$707$	−4.58666	−0.172499
$708$	0	0
$709$	38.4371	1.44353	0.721767	−	0.692136i	$-0.243330\pi$
$709$	38.4371	1.44353	0.721767	+	0.692136i	$0.243330\pi$
$710$	0	0
$711$	24.4930	0.918558
$712$	0	0
$713$	−20.4987	−0.767684
$714$	0	0
$715$	0.0666380	0.00249212
$716$	0	0
$717$	−7.86041	−0.293552
$718$	0	0
$719$	41.7065	1.55539	0.777695	−	0.628641i	$-0.216389\pi$
$719$	41.7065	1.55539	0.777695	+	0.628641i	$0.216389\pi$
$720$	0	0
$721$	22.7696	0.847985
$722$	0	0
$723$	−2.61976	−0.0974299
$724$	0	0
$725$	44.5350	1.65399
$726$	0	0
$727$	−12.8439	−0.476355	−0.238178	−	0.971222i	$-0.576550\pi$
$727$	−12.8439	−0.476355	−0.238178	+	0.971222i	$0.576550\pi$
$728$	0	0
$729$	−8.66318	−0.320859
$730$	0	0
$731$	−8.60259	−0.318178
$732$	0	0
$733$	16.7819	0.619855	0.309927	−	0.950760i	$-0.399695\pi$
$733$	16.7819	0.619855	0.309927	+	0.950760i	$0.399695\pi$
$734$	0	0
$735$	−0.392860	−0.0144909
$736$	0	0
$737$	−1.63848	−0.0603542
$738$	0	0
$739$	−15.1981	−0.559070	−0.279535	−	0.960136i	$-0.590180\pi$
$739$	−15.1981	−0.559070	−0.279535	+	0.960136i	$0.590180\pi$
$740$	0	0
$741$	0.886046	0.0325497
$742$	0	0
$743$	8.30584	0.304712	0.152356	−	0.988326i	$-0.451314\pi$
$743$	8.30584	0.304712	0.152356	+	0.988326i	$0.451314\pi$
$744$	0	0
$745$	3.80859	0.139536
$746$	0	0
$747$	−18.0295	−0.659663
$748$	0	0
$749$	−25.8499	−0.944534
$750$	0	0
$751$	−2.09857	−0.0765780	−0.0382890	−	0.999267i	$-0.512191\pi$
$751$	−2.09857	−0.0765780	−0.0382890	+	0.999267i	$0.512191\pi$
$752$	0	0
$753$	−0.261970	−0.00954671
$754$	0	0
$755$	1.94574	0.0708128
$756$	0	0
$757$	−4.66895	−0.169696	−0.0848479	−	0.996394i	$-0.527040\pi$
$757$	−4.66895	−0.169696	−0.0848479	+	0.996394i	$0.527040\pi$
$758$	0	0
$759$	−2.80564	−0.101838
$760$	0	0
$761$	−10.0030	−0.362607	−0.181304	−	0.983427i	$-0.558032\pi$
$761$	−10.0030	−0.362607	−0.181304	+	0.983427i	$0.558032\pi$
$762$	0	0
$763$	−3.75300	−0.135868
$764$	0	0
$765$	0.669403	0.0242023
$766$	0	0
$767$	0.0636146	0.00229699
$768$	0	0
$769$	−9.75632	−0.351822	−0.175911	−	0.984406i	$-0.556287\pi$
$769$	−9.75632	−0.351822	−0.175911	+	0.984406i	$0.556287\pi$
$770$	0	0
$771$	−1.77009	−0.0637482
$772$	0	0
$773$	−40.2531	−1.44780	−0.723902	−	0.689903i	$-0.757653\pi$
$773$	−40.2531	−1.44780	−0.723902	+	0.689903i	$0.757653\pi$
$774$	0	0
$775$	−22.3244	−0.801917
$776$	0	0
$777$	−0.139126	−0.00499110
$778$	0	0
$779$	−45.2007	−1.61948
$780$	0	0
$781$	−0.291170	−0.0104189
$782$	0	0
$783$	30.9991	1.10782
$784$	0	0
$785$	−0.115775	−0.00413219
$786$	0	0
$787$	22.3976	0.798388	0.399194	−	0.916866i	$-0.369290\pi$
$787$	22.3976	0.798388	0.399194	+	0.916866i	$0.369290\pi$
$788$	0	0
$789$	4.60509	0.163945
$790$	0	0
$791$	13.2432	0.470872
$792$	0	0
$793$	−1.33268	−0.0473250
$794$	0	0
$795$	−1.12307	−0.0398313
$796$	0	0
$797$	5.88026	0.208289	0.104145	−	0.994562i	$-0.466789\pi$
$797$	5.88026	0.208289	0.104145	+	0.994562i	$0.466789\pi$
$798$	0	0
$799$	−8.18000	−0.289388
$800$	0	0
$801$	27.5670	0.974031
$802$	0	0
$803$	2.58192	0.0911139
$804$	0	0
$805$	−1.52343	−0.0536938
$806$	0	0
$807$	15.3932	0.541865
$808$	0	0
$809$	−49.5525	−1.74217	−0.871086	−	0.491131i	$-0.836584\pi$
$809$	−49.5525	−1.74217	−0.871086	+	0.491131i	$0.836584\pi$
$810$	0	0
$811$	51.9979	1.82589	0.912946	−	0.408080i	$-0.133802\pi$
$811$	51.9979	1.82589	0.912946	+	0.408080i	$0.133802\pi$
$812$	0	0
$813$	3.52913	0.123772
$814$	0	0
$815$	2.08300	0.0729645
$816$	0	0
$817$	23.5574	0.824169
$818$	0	0
$819$	−1.80392	−0.0630341
$820$	0	0
$821$	12.9777	0.452925	0.226463	−	0.974020i	$-0.427284\pi$
$821$	12.9777	0.452925	0.226463	+	0.974020i	$0.427284\pi$
$822$	0	0
$823$	50.4267	1.75776	0.878882	−	0.477040i	$-0.158290\pi$
$823$	50.4267	1.75776	0.878882	+	0.477040i	$0.158290\pi$
$824$	0	0
$825$	−3.05552	−0.106379
$826$	0	0
$827$	50.7493	1.76473	0.882363	−	0.470568i	$-0.155951\pi$
$827$	50.7493	1.76473	0.882363	+	0.470568i	$0.155951\pi$
$828$	0	0
$829$	30.5901	1.06244	0.531219	−	0.847235i	$-0.321734\pi$
$829$	30.5901	1.06244	0.531219	+	0.847235i	$0.321734\pi$
$830$	0	0
$831$	−14.4624	−0.501694
$832$	0	0
$833$	5.04205	0.174697
$834$	0	0
$835$	3.25007	0.112473
$836$	0	0
$837$	−15.5392	−0.537112
$838$	0	0
$839$	−28.9075	−0.997997	−0.498998	−	0.866603i	$-0.666299\pi$
$839$	−28.9075	−0.997997	−0.498998	+	0.866603i	$0.666299\pi$
$840$	0	0
$841$	51.3713	1.77142
$842$	0	0
$843$	−1.99592	−0.0687431
$844$	0	0
$845$	−2.31332	−0.0795807
$846$	0	0
$847$	1.85704	0.0638085
$848$	0	0
$849$	−3.28435	−0.112719
$850$	0	0
$851$	0.555586	0.0190453
$852$	0	0
$853$	−20.2337	−0.692788	−0.346394	−	0.938089i	$-0.612594\pi$
$853$	−20.2337	−0.692788	−0.346394	+	0.938089i	$0.612594\pi$
$854$	0	0
$855$	−1.83310	−0.0626907
$856$	0	0
$857$	16.4777	0.562868	0.281434	−	0.959581i	$-0.409190\pi$
$857$	16.4777	0.562868	0.281434	+	0.959581i	$0.409190\pi$
$858$	0	0
$859$	−33.9369	−1.15791	−0.578956	−	0.815359i	$-0.696540\pi$
$859$	−33.9369	−1.15791	−0.578956	+	0.815359i	$0.696540\pi$
$860$	0	0
$861$	−13.2799	−0.452578
$862$	0	0
$863$	−47.9386	−1.63185	−0.815925	−	0.578158i	$-0.803772\pi$
$863$	−47.9386	−1.63185	−0.815925	+	0.578158i	$0.803772\pi$
$864$	0	0
$865$	−3.49365	−0.118788
$866$	0	0
$867$	−9.21662	−0.313013
$868$	0	0
$869$	−9.34249	−0.316922
$870$	0	0
$871$	−0.607100	−0.0205708
$872$	0	0
$873$	−20.8186	−0.704603
$874$	0	0
$875$	−3.32902	−0.112542
$876$	0	0
$877$	14.1827	0.478915	0.239457	−	0.970907i	$-0.423030\pi$
$877$	14.1827	0.478915	0.239457	+	0.970907i	$0.423030\pi$
$878$	0	0
$879$	1.39188	0.0469470
$880$	0	0
$881$	7.54145	0.254078	0.127039	−	0.991898i	$-0.459453\pi$
$881$	7.54145	0.254078	0.127039	+	0.991898i	$0.459453\pi$
$882$	0	0
$883$	−25.9023	−0.871681	−0.435841	−	0.900024i	$-0.643549\pi$
$883$	−25.9023	−0.871681	−0.435841	+	0.900024i	$0.643549\pi$
$884$	0	0
$885$	0.0189922	0.000638416 0
$886$	0	0
$887$	−43.6378	−1.46521	−0.732607	−	0.680652i	$-0.761697\pi$
$887$	−43.6378	−1.46521	−0.732607	+	0.680652i	$0.761697\pi$
$888$	0	0
$889$	−15.2143	−0.510271
$890$	0	0
$891$	5.73819	0.192237
$892$	0	0
$893$	22.4002	0.749594
$894$	0	0
$895$	−1.71934	−0.0574712
$896$	0	0
$897$	−1.03956	−0.0347100
$898$	0	0
$899$	−40.2883	−1.34369
$900$	0	0
$901$	14.4137	0.480192
$902$	0	0
$903$	6.92113	0.230321
$904$	0	0
$905$	0.0473190	0.00157294
$906$	0	0
$907$	18.6232	0.618373	0.309187	−	0.951001i	$-0.399943\pi$
$907$	18.6232	0.618373	0.309187	+	0.951001i	$0.399943\pi$
$908$	0	0
$909$	6.47522	0.214770
$910$	0	0
$911$	−29.9848	−0.993441	−0.496720	−	0.867911i	$-0.665463\pi$
$911$	−29.9848	−0.993441	−0.496720	+	0.867911i	$0.665463\pi$
$912$	0	0
$913$	6.87708	0.227598
$914$	0	0
$915$	−0.397874	−0.0131533
$916$	0	0
$917$	−36.8261	−1.21611
$918$	0	0
$919$	−29.0913	−0.959633	−0.479816	−	0.877369i	$-0.659297\pi$
$919$	−29.0913	−0.959633	−0.479816	+	0.877369i	$0.659297\pi$
$920$	0	0
$921$	8.55037	0.281744
$922$	0	0
$923$	−0.107886	−0.00355111
$924$	0	0
$925$	0.605068	0.0198945
$926$	0	0
$927$	−32.1450	−1.05578
$928$	0	0
$929$	18.0470	0.592104	0.296052	−	0.955172i	$-0.404330\pi$
$929$	18.0470	0.592104	0.296052	+	0.955172i	$0.404330\pi$
$930$	0	0
$931$	−13.8072	−0.452513
$932$	0	0
$933$	11.1971	0.366577
$934$	0	0
$935$	−0.255334	−0.00835033
$936$	0	0
$937$	22.5348	0.736178	0.368089	−	0.929790i	$-0.380012\pi$
$937$	22.5348	0.736178	0.368089	+	0.929790i	$0.380012\pi$
$938$	0	0
$939$	13.5141	0.441017
$940$	0	0
$941$	57.4608	1.87317	0.936585	−	0.350441i	$-0.113968\pi$
$941$	57.4608	1.87317	0.936585	+	0.350441i	$0.113968\pi$
$942$	0	0
$943$	53.0321	1.72696
$944$	0	0
$945$	−1.15484	−0.0375670
$946$	0	0
$947$	−49.4612	−1.60727	−0.803636	−	0.595121i	$-0.797104\pi$
$947$	−49.4612	−1.60727	−0.803636	+	0.595121i	$0.797104\pi$
$948$	0	0
$949$	0.956668	0.0310548
$950$	0	0
$951$	−17.3822	−0.563658
$952$	0	0
$953$	35.9403	1.16422	0.582111	−	0.813109i	$-0.302227\pi$
$953$	35.9403	1.16422	0.582111	+	0.813109i	$0.302227\pi$
$954$	0	0
$955$	−1.98594	−0.0642634
$956$	0	0
$957$	−5.51422	−0.178249
$958$	0	0
$959$	−1.85704	−0.0599668
$960$	0	0
$961$	−10.8044	−0.348528
$962$	0	0
$963$	36.4936	1.17599
$964$	0	0
$965$	−2.36712	−0.0762002
$966$	0	0
$967$	10.8990	0.350489	0.175244	−	0.984525i	$-0.443928\pi$
$967$	10.8990	0.350489	0.175244	+	0.984525i	$0.443928\pi$
$968$	0	0
$969$	−3.39503	−0.109064
$970$	0	0
$971$	48.5630	1.55846	0.779231	−	0.626737i	$-0.215610\pi$
$971$	48.5630	1.55846	0.779231	+	0.626737i	$0.215610\pi$
$972$	0	0
$973$	−0.103287	−0.00331124
$974$	0	0
$975$	−1.13215	−0.0362578
$976$	0	0
$977$	−11.5492	−0.369491	−0.184746	−	0.982786i	$-0.559146\pi$
$977$	−11.5492	−0.369491	−0.184746	+	0.982786i	$0.559146\pi$
$978$	0	0
$979$	−10.5150	−0.336062
$980$	0	0
$981$	5.29830	0.169162
$982$	0	0
$983$	54.2267	1.72956	0.864781	−	0.502149i	$-0.167457\pi$
$983$	54.2267	1.72956	0.864781	+	0.502149i	$0.167457\pi$
$984$	0	0
$985$	−4.90389	−0.156251
$986$	0	0
$987$	6.58115	0.209480
$988$	0	0
$989$	−27.6390	−0.878868
$990$	0	0
$991$	−2.94396	−0.0935181	−0.0467590	−	0.998906i	$-0.514889\pi$
$991$	−2.94396	−0.0935181	−0.0467590	+	0.998906i	$0.514889\pi$
$992$	0	0
$993$	17.7783	0.564178
$994$	0	0
$995$	0.574284	0.0182060
$996$	0	0
$997$	20.1848	0.639259	0.319630	−	0.947543i	$-0.396441\pi$
$997$	20.1848	0.639259	0.319630	+	0.947543i	$0.396441\pi$
$998$	0	0
$999$	0.421165	0.0133251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.c.1.18	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.c.1.18	✓	25	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)

\(f(q)\)	\(=\)	\(q+0.615083 q^{3} +0.179847 q^{5} +1.85704 q^{7} -2.62167 q^{9} +O(q^{10})\)	\(q+0.615083 q^{3} +0.179847 q^{5} +1.85704 q^{7} -2.62167 q^{9} +1.00000 q^{11} +0.370526 q^{13} +0.110621 q^{15} -1.41973 q^{17} +3.88780 q^{19} +1.14223 q^{21} -4.56140 q^{23} -4.96766 q^{25} -3.45779 q^{27} -8.96500 q^{29} +4.49396 q^{31} +0.615083 q^{33} +0.333983 q^{35} -0.121802 q^{37} +0.227904 q^{39} -11.6263 q^{41} +6.05931 q^{43} -0.471500 q^{45} +5.76166 q^{47} -3.55142 q^{49} -0.873251 q^{51} -10.1525 q^{53} +0.179847 q^{55} +2.39132 q^{57} +0.171687 q^{59} -3.59673 q^{61} -4.86854 q^{63} +0.0666380 q^{65} -1.63848 q^{67} -2.80564 q^{69} -0.291170 q^{71} +2.58192 q^{73} -3.05552 q^{75} +1.85704 q^{77} -9.34249 q^{79} +5.73819 q^{81} +6.87708 q^{83} -0.255334 q^{85} -5.51422 q^{87} -10.5150 q^{89} +0.688080 q^{91} +2.76415 q^{93} +0.699210 q^{95} +7.94096 q^{97} -2.62167 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) \) 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) \) 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9 + 25 * q^11 - 4 * q^13 - 10 * q^15 - 19 * q^17 - 12 * q^19 + 8 * q^21 - 31 * q^23 - q^25 - 44 * q^27 - q^29 - 8 * q^31 - 11 * q^33 - 16 * q^35 - 14 * q^37 - 18 * q^39 - 5 * q^41 - 15 * q^43 - 15 * q^45 - 41 * q^47 + 2 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 - 3 * q^57 - 35 * q^59 - 4 * q^61 - 45 * q^63 - 28 * q^65 - 30 * q^67 - 3 * q^69 + 4 * q^71 - 7 * q^73 - 18 * q^75 - 9 * q^77 - 9 * q^79 + 29 * q^81 - 72 * q^83 - 33 * q^87 - 30 * q^89 - 10 * q^91 + 7 * q^93 + 9 * q^95 - 37 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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