LMFDB - Embedded newform 5408.2.a.bn.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5408,2,Mod(1,5408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5408, 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("5408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5408 = 2^{5} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5408.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:	$43.1830974131$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.134509248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} + 60x^{2} - 48 $ x^6 - 15x^4 + 60x^2 - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 416)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.03053$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.03053 q^{3} -0.701519 q^{5} -2.93800 q^{7} -1.93800 q^{9} +2.93800 q^{11} -0.722938 q^{15} -1.00000 q^{17} +1.78493 q^{19} -3.02771 q^{21} -5.89767 q^{23} -4.50787 q^{25} -5.08877 q^{27} +5.72294 q^{29} -0.722938 q^{31} +3.02771 q^{33} +2.06106 q^{35} +7.85135 q^{37} +3.79312 q^{41} +7.14984 q^{43} +1.35955 q^{45} -5.15307 q^{47} +1.63186 q^{49} -1.03053 q^{51} -3.21507 q^{53} -2.06106 q^{55} +1.83943 q^{57} +6.50787 q^{59} -6.15307 q^{61} +5.69386 q^{63} +12.0911 q^{67} -6.07774 q^{69} +2.21507 q^{71} +10.9430 q^{73} -4.64551 q^{75} -8.63186 q^{77} +14.2997 q^{79} +0.569868 q^{81} +13.8760 q^{83} +0.701519 q^{85} +5.89767 q^{87} -7.89485 q^{89} -0.745010 q^{93} -1.25216 q^{95} +1.83943 q^{97} -5.69386 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{7} + 12 q^{9} - 6 q^{11} + 24 q^{15} - 6 q^{17} + 6 q^{19} + 6 q^{25} + 6 q^{29} + 24 q^{31} - 12 q^{47} + 24 q^{49} - 24 q^{53} + 6 q^{59} - 18 q^{61} + 72 q^{63} + 30 q^{67} + 6 q^{69} + 18 q^{71}+ \cdots - 72 q^{99}+O(q^{100}) $ 6 * q + 6 * q^7 + 12 * q^9 - 6 * q^11 + 24 * q^15 - 6 * q^17 + 6 * q^19 + 6 * q^25 + 6 * q^29 + 24 * q^31 - 12 * q^47 + 24 * q^49 - 24 * q^53 + 6 * q^59 - 18 * q^61 + 72 * q^63 + 30 * q^67 + 6 * q^69 + 18 * q^71 - 66 * q^77 - 6 * q^81 + 36 * q^83 - 72 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.03053	0.594978	0.297489	−	0.954725i	$-0.403851\pi$
$3$	1.03053	0.594978	0.297489	+	0.954725i	$0.403851\pi$
$4$	0	0
$5$	−0.701519	−0.313729	−0.156864	−	0.987620i	$-0.550139\pi$
$5$	−0.701519	−0.313729	−0.156864	+	0.987620i	$0.550139\pi$
$6$	0	0
$7$	−2.93800	−1.11046	−0.555230	−	0.831697i	$-0.687370\pi$
$7$	−2.93800	−1.11046	−0.555230	+	0.831697i	$0.687370\pi$
$8$	0	0
$9$	−1.93800	−0.646001
$10$	0	0
$11$	2.93800	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$11$	2.93800	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.722938	−0.186662
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	1.78493	0.409492	0.204746	−	0.978815i	$-0.434363\pi$
$19$	1.78493	0.409492	0.204746	+	0.978815i	$0.434363\pi$
$20$	0	0
$21$	−3.02771	−0.660700
$22$	0	0
$23$	−5.89767	−1.22975	−0.614875	−	0.788625i	$-0.710793\pi$
$23$	−5.89767	−1.22975	−0.614875	+	0.788625i	$0.710793\pi$
$24$	0	0
$25$	−4.50787	−0.901574
$26$	0	0
$27$	−5.08877	−0.979334
$28$	0	0
$29$	5.72294	1.06272	0.531361	−	0.847145i	$-0.321681\pi$
$29$	5.72294	1.06272	0.531361	+	0.847145i	$0.321681\pi$
$30$	0	0
$31$	−0.722938	−0.129843	−0.0649217	−	0.997890i	$-0.520680\pi$
$31$	−0.722938	−0.129843	−0.0649217	+	0.997890i	$0.520680\pi$
$32$	0	0
$33$	3.02771	0.527056
$34$	0	0
$35$	2.06106	0.348383
$36$	0	0
$37$	7.85135	1.29075	0.645377	−	0.763864i	$-0.276700\pi$
$37$	7.85135	1.29075	0.645377	+	0.763864i	$0.276700\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.79312	0.592385	0.296193	−	0.955128i	$-0.404283\pi$
$41$	3.79312	0.592385	0.296193	+	0.955128i	$0.404283\pi$
$42$	0	0
$43$	7.14984	1.09034	0.545170	−	0.838326i	$-0.316465\pi$
$43$	7.14984	1.09034	0.545170	+	0.838326i	$0.316465\pi$
$44$	0	0
$45$	1.35955	0.202669
$46$	0	0
$47$	−5.15307	−0.751652	−0.375826	−	0.926690i	$-0.622641\pi$
$47$	−5.15307	−0.751652	−0.375826	+	0.926690i	$0.622641\pi$
$48$	0	0
$49$	1.63186	0.233124
$50$	0	0
$51$	−1.03053	−0.144303
$52$	0	0
$53$	−3.21507	−0.441623	−0.220812	−	0.975316i	$-0.570871\pi$
$53$	−3.21507	−0.441623	−0.220812	+	0.975316i	$0.570871\pi$
$54$	0	0
$55$	−2.06106	−0.277914
$56$	0	0
$57$	1.83943	0.243639
$58$	0	0
$59$	6.50787	0.847253	0.423626	−	0.905837i	$-0.360757\pi$
$59$	6.50787	0.847253	0.423626	+	0.905837i	$0.360757\pi$
$60$	0	0
$61$	−6.15307	−0.787820	−0.393910	−	0.919149i	$-0.628878\pi$
$61$	−6.15307	−0.787820	−0.393910	+	0.919149i	$0.628878\pi$
$62$	0	0
$63$	5.69386	0.717359
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0911	1.47716	0.738580	−	0.674166i	$-0.235497\pi$
$67$	12.0911	1.47716	0.738580	+	0.674166i	$0.235497\pi$
$68$	0	0
$69$	−6.07774	−0.731674
$70$	0	0
$71$	2.21507	0.262880	0.131440	−	0.991324i	$-0.458040\pi$
$71$	2.21507	0.262880	0.131440	+	0.991324i	$0.458040\pi$
$72$	0	0
$73$	10.9430	1.28078	0.640388	−	0.768052i	$-0.278774\pi$
$73$	10.9430	1.28078	0.640388	+	0.768052i	$0.278774\pi$
$74$	0	0
$75$	−4.64551	−0.536417
$76$	0	0
$77$	−8.63186	−0.983692
$78$	0	0
$79$	14.2997	1.60884	0.804419	−	0.594062i	$-0.202477\pi$
$79$	14.2997	1.60884	0.804419	+	0.594062i	$0.202477\pi$
$80$	0	0
$81$	0.569868	0.0633187
$82$	0	0
$83$	13.8760	1.52309	0.761545	−	0.648112i	$-0.224441\pi$
$83$	13.8760	1.52309	0.761545	+	0.648112i	$0.224441\pi$
$84$	0	0
$85$	0.701519	0.0760904
$86$	0	0
$87$	5.89767	0.632297
$88$	0	0
$89$	−7.89485	−0.836852	−0.418426	−	0.908251i	$-0.637418\pi$
$89$	−7.89485	−0.836852	−0.418426	+	0.908251i	$0.637418\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.745010	−0.0772540
$94$	0	0
$95$	−1.25216	−0.128469
$96$	0	0
$97$	1.83943	0.186766	0.0933830	−	0.995630i	$-0.470232\pi$
$97$	1.83943	0.186766	0.0933830	+	0.995630i	$0.470232\pi$
$98$	0	0
$99$	−5.69386	−0.572255
$100$	0	0
$101$	1.72294	0.171439	0.0857193	−	0.996319i	$-0.472681\pi$
$101$	1.72294	0.171439	0.0857193	+	0.996319i	$0.472681\pi$
$102$	0	0
$103$	−17.1696	−1.69177	−0.845887	−	0.533362i	$-0.820928\pi$
$103$	−17.1696	−1.69177	−0.845887	+	0.533362i	$0.820928\pi$
$104$	0	0
$105$	2.12399	0.207280
$106$	0	0
$107$	5.15266	0.498127	0.249063	−	0.968487i	$-0.419877\pi$
$107$	5.15266	0.498127	0.249063	+	0.968487i	$0.419877\pi$
$108$	0	0
$109$	−4.86714	−0.466187	−0.233094	−	0.972454i	$-0.574885\pi$
$109$	−4.86714	−0.466187	−0.233094	+	0.972454i	$0.574885\pi$
$110$	0	0
$111$	8.09107	0.767971
$112$	0	0
$113$	20.7520	1.95219	0.976093	−	0.217355i	$-0.0697429\pi$
$113$	20.7520	1.95219	0.976093	+	0.217355i	$0.0697429\pi$
$114$	0	0
$115$	4.13733	0.385808
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.93800	0.269326
$120$	0	0
$121$	−2.36814	−0.215285
$122$	0	0
$123$	3.90893	0.352456
$124$	0	0
$125$	6.66995	0.596578
$126$	0	0
$127$	−20.1973	−1.79222	−0.896112	−	0.443828i	$-0.853620\pi$
$127$	−20.1973	−1.79222	−0.896112	+	0.443828i	$0.853620\pi$
$128$	0	0
$129$	7.36814	0.648728
$130$	0	0
$131$	1.99717	0.174494	0.0872470	−	0.996187i	$-0.472193\pi$
$131$	1.99717	0.174494	0.0872470	+	0.996187i	$0.472193\pi$
$132$	0	0
$133$	−5.24414	−0.454725
$134$	0	0
$135$	3.56987	0.307245
$136$	0	0
$137$	−22.9599	−1.96160	−0.980799	−	0.195019i	$-0.937523\pi$
$137$	−22.9599	−1.96160	−0.980799	+	0.195019i	$0.937523\pi$
$138$	0	0
$139$	8.76764	0.743661	0.371831	−	0.928301i	$-0.378730\pi$
$139$	8.76764	0.743661	0.371831	+	0.928301i	$0.378730\pi$
$140$	0	0
$141$	−5.31040	−0.447217
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.01475	−0.333407
$146$	0	0
$147$	1.68169	0.138703
$148$	0	0
$149$	−1.13791	−0.0932215	−0.0466107	−	0.998913i	$-0.514842\pi$
$149$	−1.13791	−0.0932215	−0.0466107	+	0.998913i	$0.514842\pi$
$150$	0	0
$151$	0.722938	0.0588318	0.0294159	−	0.999567i	$-0.490635\pi$
$151$	0.722938	0.0588318	0.0294159	+	0.999567i	$0.490635\pi$
$152$	0	0
$153$	1.93800	0.156678
$154$	0	0
$155$	0.507154	0.0407356
$156$	0	0
$157$	4.66094	0.371984	0.185992	−	0.982551i	$-0.440450\pi$
$157$	4.66094	0.371984	0.185992	+	0.982551i	$0.440450\pi$
$158$	0	0
$159$	−3.31323	−0.262756
$160$	0	0
$161$	17.3274	1.36559
$162$	0	0
$163$	10.9380	0.856731	0.428365	−	0.903606i	$-0.359090\pi$
$163$	10.9380	0.856731	0.428365	+	0.903606i	$0.359090\pi$
$164$	0	0
$165$	−2.12399	−0.165353
$166$	0	0
$167$	23.9537	1.85360	0.926798	−	0.375560i	$-0.122549\pi$
$167$	23.9537	1.85360	0.926798	+	0.375560i	$0.122549\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.45921	−0.264532
$172$	0	0
$173$	12.2599	0.932102	0.466051	−	0.884758i	$-0.345676\pi$
$173$	12.2599	0.932102	0.466051	+	0.884758i	$0.345676\pi$
$174$	0	0
$175$	13.2441	1.00116
$176$	0	0
$177$	6.70657	0.504097
$178$	0	0
$179$	−21.0062	−1.57008	−0.785040	−	0.619445i	$-0.787358\pi$
$179$	−21.0062	−1.57008	−0.785040	+	0.619445i	$0.787358\pi$
$180$	0	0
$181$	−8.66094	−0.643763	−0.321881	−	0.946780i	$-0.604315\pi$
$181$	−8.66094	−0.643763	−0.321881	+	0.946780i	$0.604315\pi$
$182$	0	0
$183$	−6.34094	−0.468735
$184$	0	0
$185$	−5.50787	−0.404947
$186$	0	0
$187$	−2.93800	−0.214848
$188$	0	0
$189$	14.9508	1.08751
$190$	0	0
$191$	16.0752	1.16316	0.581581	−	0.813489i	$-0.302434\pi$
$191$	16.0752	1.16316	0.581581	+	0.813489i	$0.302434\pi$
$192$	0	0
$193$	8.44549	0.607920	0.303960	−	0.952685i	$-0.401691\pi$
$193$	8.44549	0.607920	0.303960	+	0.952685i	$0.401691\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.89485	0.562484	0.281242	−	0.959637i	$-0.409254\pi$
$197$	7.89485	0.562484	0.281242	+	0.959637i	$0.409254\pi$
$198$	0	0
$199$	16.0752	1.13954	0.569771	−	0.821804i	$-0.307032\pi$
$199$	16.0752	1.13954	0.569771	+	0.821804i	$0.307032\pi$
$200$	0	0
$201$	12.4602	0.878878
$202$	0	0
$203$	−16.8140	−1.18011
$204$	0	0
$205$	−2.66094	−0.185848
$206$	0	0
$207$	11.4297	0.794420
$208$	0	0
$209$	5.24414	0.362745
$210$	0	0
$211$	23.0034	1.58362	0.791810	−	0.610767i	$-0.209139\pi$
$211$	23.0034	1.58362	0.791810	+	0.610767i	$0.209139\pi$
$212$	0	0
$213$	2.28270	0.156408
$214$	0	0
$215$	−5.01574	−0.342071
$216$	0	0
$217$	2.12399	0.144186
$218$	0	0
$219$	11.2771	0.762033
$220$	0	0
$221$	0	0
$222$	0	0
$223$	15.2308	1.01993	0.509965	−	0.860195i	$-0.329658\pi$
$223$	15.2308	1.01993	0.509965	+	0.860195i	$0.329658\pi$
$224$	0	0
$225$	8.73627	0.582418
$226$	0	0
$227$	29.5369	1.96044	0.980218	−	0.197921i	$-0.0634188\pi$
$227$	29.5369	1.96044	0.980218	+	0.197921i	$0.0634188\pi$
$228$	0	0
$229$	−13.1114	−0.866425	−0.433213	−	0.901292i	$-0.642620\pi$
$229$	−13.1114	−0.866425	−0.433213	+	0.901292i	$0.642620\pi$
$230$	0	0
$231$	−8.89541	−0.585275
$232$	0	0
$233$	25.3219	1.65889	0.829446	−	0.558587i	$-0.188656\pi$
$233$	25.3219	1.65889	0.829446	+	0.558587i	$0.188656\pi$
$234$	0	0
$235$	3.61497	0.235815
$236$	0	0
$237$	14.7363	0.957224
$238$	0	0
$239$	6.73627	0.435733	0.217867	−	0.975979i	$-0.430090\pi$
$239$	6.73627	0.435733	0.217867	+	0.975979i	$0.430090\pi$
$240$	0	0
$241$	−20.8989	−1.34621	−0.673107	−	0.739545i	$-0.735041\pi$
$241$	−20.8989	−1.34621	−0.673107	+	0.739545i	$0.735041\pi$
$242$	0	0
$243$	15.8536	1.01701
$244$	0	0
$245$	−1.14478	−0.0731375
$246$	0	0
$247$	0	0
$248$	0	0
$249$	14.2997	0.906205
$250$	0	0
$251$	18.8813	1.19178	0.595888	−	0.803067i	$-0.296800\pi$
$251$	18.8813	1.19178	0.595888	+	0.803067i	$0.296800\pi$
$252$	0	0
$253$	−17.3274	−1.08936
$254$	0	0
$255$	0.722938	0.0452721
$256$	0	0
$257$	21.1821	1.32131	0.660653	−	0.750691i	$-0.270279\pi$
$257$	21.1821	1.32131	0.660653	+	0.750691i	$0.270279\pi$
$258$	0	0
$259$	−23.0673	−1.43333
$260$	0	0
$261$	−11.0911	−0.686520
$262$	0	0
$263$	0.966643	0.0596057	0.0298029	−	0.999556i	$-0.490512\pi$
$263$	0.966643	0.0596057	0.0298029	+	0.999556i	$0.490512\pi$
$264$	0	0
$265$	2.25543	0.138550
$266$	0	0
$267$	−8.13589	−0.497909
$268$	0	0
$269$	−14.6319	−0.892121	−0.446060	−	0.895003i	$-0.647173\pi$
$269$	−14.6319	−0.892121	−0.446060	+	0.895003i	$0.647173\pi$
$270$	0	0
$271$	22.5079	1.36726	0.683628	−	0.729831i	$-0.260401\pi$
$271$	22.5079	1.36726	0.683628	+	0.729831i	$0.260401\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.2441	−0.798652
$276$	0	0
$277$	−16.3352	−0.981488	−0.490744	−	0.871304i	$-0.663275\pi$
$277$	−16.3352	−0.981488	−0.490744	+	0.871304i	$0.663275\pi$
$278$	0	0
$279$	1.40106	0.0838790
$280$	0	0
$281$	−16.2534	−0.969594	−0.484797	−	0.874627i	$-0.661106\pi$
$281$	−16.2534	−0.969594	−0.484797	+	0.874627i	$0.661106\pi$
$282$	0	0
$283$	−7.14984	−0.425014	−0.212507	−	0.977160i	$-0.568163\pi$
$283$	−7.14984	−0.425014	−0.212507	+	0.977160i	$0.568163\pi$
$284$	0	0
$285$	−1.29040	−0.0764365
$286$	0	0
$287$	−11.1442	−0.657820
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	1.89559	0.111122
$292$	0	0
$293$	17.5856	1.02736	0.513682	−	0.857981i	$-0.328281\pi$
$293$	17.5856	1.02736	0.513682	+	0.857981i	$0.328281\pi$
$294$	0	0
$295$	−4.56539	−0.265807
$296$	0	0
$297$	−14.9508	−0.867535
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−21.0062	−1.21078
$302$	0	0
$303$	1.77554	0.102002
$304$	0	0
$305$	4.31649	0.247162
$306$	0	0
$307$	3.27706	0.187032	0.0935159	−	0.995618i	$-0.470189\pi$
$307$	3.27706	0.187032	0.0935159	+	0.995618i	$0.470189\pi$
$308$	0	0
$309$	−17.6939	−1.00657
$310$	0	0
$311$	−21.1640	−1.20010	−0.600050	−	0.799963i	$-0.704853\pi$
$311$	−21.1640	−1.20010	−0.600050	+	0.799963i	$0.704853\pi$
$312$	0	0
$313$	10.1821	0.575529	0.287764	−	0.957701i	$-0.407088\pi$
$313$	10.1821	0.575529	0.287764	+	0.957701i	$0.407088\pi$
$314$	0	0
$315$	−3.99435	−0.225056
$316$	0	0
$317$	−28.7706	−1.61592	−0.807959	−	0.589238i	$-0.799428\pi$
$317$	−28.7706	−1.61592	−0.807959	+	0.589238i	$0.799428\pi$
$318$	0	0
$319$	16.8140	0.941404
$320$	0	0
$321$	5.30998	0.296374
$322$	0	0
$323$	−1.78493	−0.0993164
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.01574	−0.277371
$328$	0	0
$329$	15.1397	0.834681
$330$	0	0
$331$	−4.38388	−0.240960	−0.120480	−	0.992716i	$-0.538443\pi$
$331$	−4.38388	−0.240960	−0.120480	+	0.992716i	$0.538443\pi$
$332$	0	0
$333$	−15.2160	−0.833829
$334$	0	0
$335$	−8.48211	−0.463427
$336$	0	0
$337$	−1.22840	−0.0669152	−0.0334576	−	0.999440i	$-0.510652\pi$
$337$	−1.22840	−0.0669152	−0.0334576	+	0.999440i	$0.510652\pi$
$338$	0	0
$339$	21.3856	1.16151
$340$	0	0
$341$	−2.12399	−0.115021
$342$	0	0
$343$	15.7716	0.851586
$344$	0	0
$345$	4.26365	0.229547
$346$	0	0
$347$	7.21373	0.387253	0.193627	−	0.981075i	$-0.437975\pi$
$347$	7.21373	0.387253	0.193627	+	0.981075i	$0.437975\pi$
$348$	0	0
$349$	26.7599	1.43243	0.716213	−	0.697882i	$-0.245874\pi$
$349$	26.7599	1.43243	0.716213	+	0.697882i	$0.245874\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.8434	−0.790037	−0.395018	−	0.918673i	$-0.629262\pi$
$353$	−14.8434	−0.790037	−0.395018	+	0.918673i	$0.629262\pi$
$354$	0	0
$355$	−1.55391	−0.0824730
$356$	0	0
$357$	3.02771	0.160243
$358$	0	0
$359$	−19.5699	−1.03286	−0.516429	−	0.856330i	$-0.672739\pi$
$359$	−19.5699	−1.03286	−0.516429	+	0.856330i	$0.672739\pi$
$360$	0	0
$361$	−15.8140	−0.832316
$362$	0	0
$363$	−2.44044	−0.128090
$364$	0	0
$365$	−7.67668	−0.401816
$366$	0	0
$367$	10.0198	0.523029	0.261515	−	0.965200i	$-0.415778\pi$
$367$	10.0198	0.523029	0.261515	+	0.965200i	$0.415778\pi$
$368$	0	0
$369$	−7.35107	−0.382681
$370$	0	0
$371$	9.44588	0.490405
$372$	0	0
$373$	−25.5989	−1.32546	−0.662732	−	0.748857i	$-0.730603\pi$
$373$	−25.5989	−1.32546	−0.662732	+	0.748857i	$0.730603\pi$
$374$	0	0
$375$	6.87360	0.354951
$376$	0	0
$377$	0	0
$378$	0	0
$379$	23.6609	1.21538	0.607690	−	0.794174i	$-0.292096\pi$
$379$	23.6609	1.21538	0.607690	+	0.794174i	$0.292096\pi$
$380$	0	0
$381$	−20.8140	−1.06633
$382$	0	0
$383$	−19.7058	−1.00692	−0.503459	−	0.864019i	$-0.667939\pi$
$383$	−19.7058	−1.00692	−0.503459	+	0.864019i	$0.667939\pi$
$384$	0	0
$385$	6.05541	0.308612
$386$	0	0
$387$	−13.8564	−0.704361
$388$	0	0
$389$	4.35480	0.220797	0.110399	−	0.993887i	$-0.464787\pi$
$389$	4.35480	0.220797	0.110399	+	0.993887i	$0.464787\pi$
$390$	0	0
$391$	5.89767	0.298258
$392$	0	0
$393$	2.05815	0.103820
$394$	0	0
$395$	−10.0315	−0.504739
$396$	0	0
$397$	31.9288	1.60246	0.801230	−	0.598356i	$-0.204179\pi$
$397$	31.9288	1.60246	0.801230	+	0.598356i	$0.204179\pi$
$398$	0	0
$399$	−5.40426	−0.270551
$400$	0	0
$401$	−17.4348	−0.870650	−0.435325	−	0.900273i	$-0.643367\pi$
$401$	−17.4348	−0.870650	−0.435325	+	0.900273i	$0.643367\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−0.399773	−0.0198649
$406$	0	0
$407$	23.0673	1.14340
$408$	0	0
$409$	−27.0821	−1.33912	−0.669561	−	0.742757i	$-0.733518\pi$
$409$	−27.0821	−1.33912	−0.669561	+	0.742757i	$0.733518\pi$
$410$	0	0
$411$	−23.6609	−1.16711
$412$	0	0
$413$	−19.1201	−0.940841
$414$	0	0
$415$	−9.73428	−0.477837
$416$	0	0
$417$	9.03533	0.442462
$418$	0	0
$419$	−39.3642	−1.92306	−0.961532	−	0.274692i	$-0.911424\pi$
$419$	−39.3642	−1.92306	−0.961532	+	0.274692i	$0.911424\pi$
$420$	0	0
$421$	−18.4653	−0.899943	−0.449972	−	0.893043i	$-0.648566\pi$
$421$	−18.4653	−0.899943	−0.449972	+	0.893043i	$0.648566\pi$
$422$	0	0
$423$	9.98667	0.485568
$424$	0	0
$425$	4.50787	0.218664
$426$	0	0
$427$	18.0777	0.874843
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.95375	0.383118	0.191559	−	0.981481i	$-0.438646\pi$
$431$	7.95375	0.383118	0.191559	+	0.981481i	$0.438646\pi$
$432$	0	0
$433$	24.5699	1.18075	0.590376	−	0.807128i	$-0.298979\pi$
$433$	24.5699	1.18075	0.590376	+	0.807128i	$0.298979\pi$
$434$	0	0
$435$	−4.13733	−0.198370
$436$	0	0
$437$	−10.5270	−0.503572
$438$	0	0
$439$	13.3330	0.636351	0.318175	−	0.948032i	$-0.396930\pi$
$439$	13.3330	0.636351	0.318175	+	0.948032i	$0.396930\pi$
$440$	0	0
$441$	−3.16256	−0.150598
$442$	0	0
$443$	6.62646	0.314833	0.157416	−	0.987532i	$-0.449684\pi$
$443$	6.62646	0.314833	0.157416	+	0.987532i	$0.449684\pi$
$444$	0	0
$445$	5.53838	0.262544
$446$	0	0
$447$	−1.17266	−0.0554647
$448$	0	0
$449$	12.4602	0.588035	0.294018	−	0.955800i	$-0.405008\pi$
$449$	12.4602	0.588035	0.294018	+	0.955800i	$0.405008\pi$
$450$	0	0
$451$	11.1442	0.524759
$452$	0	0
$453$	0.745010	0.0350036
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.66025	0.405110	0.202555	−	0.979271i	$-0.435076\pi$
$457$	8.66025	0.405110	0.202555	+	0.979271i	$0.435076\pi$
$458$	0	0
$459$	5.08877	0.237524
$460$	0	0
$461$	11.8457	0.551709	0.275855	−	0.961199i	$-0.411039\pi$
$461$	11.8457	0.551709	0.275855	+	0.961199i	$0.411039\pi$
$462$	0	0
$463$	13.8760	0.644873	0.322436	−	0.946591i	$-0.395498\pi$
$463$	13.8760	0.644873	0.322436	+	0.946591i	$0.395498\pi$
$464$	0	0
$465$	0.522639	0.0242368
$466$	0	0
$467$	6.05541	0.280211	0.140106	−	0.990137i	$-0.455256\pi$
$467$	6.05541	0.280211	0.140106	+	0.990137i	$0.455256\pi$
$468$	0	0
$469$	−35.5236	−1.64033
$470$	0	0
$471$	4.80325	0.221322
$472$	0	0
$473$	21.0062	0.965868
$474$	0	0
$475$	−8.04625	−0.369187
$476$	0	0
$477$	6.23081	0.285289
$478$	0	0
$479$	−14.8273	−0.677479	−0.338739	−	0.940880i	$-0.610000\pi$
$479$	−14.8273	−0.677479	−0.338739	+	0.940880i	$0.610000\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	17.8564	0.812495
$484$	0	0
$485$	−1.29040	−0.0585939
$486$	0	0
$487$	23.0935	1.04647	0.523233	−	0.852190i	$-0.324726\pi$
$487$	23.0935	1.04647	0.523233	+	0.852190i	$0.324726\pi$
$488$	0	0
$489$	11.2720	0.509736
$490$	0	0
$491$	−14.8230	−0.668955	−0.334477	−	0.942404i	$-0.608560\pi$
$491$	−14.8230	−0.668955	−0.334477	+	0.942404i	$0.608560\pi$
$492$	0	0
$493$	−5.72294	−0.257748
$494$	0	0
$495$	3.99435	0.179533
$496$	0	0
$497$	−6.50787	−0.291918
$498$	0	0
$499$	23.6147	1.05714	0.528569	−	0.848890i	$-0.322729\pi$
$499$	23.6147	1.05714	0.528569	+	0.848890i	$0.322729\pi$
$500$	0	0
$501$	24.6851	1.10285
$502$	0	0
$503$	5.53204	0.246661	0.123331	−	0.992366i	$-0.460642\pi$
$503$	5.53204	0.246661	0.123331	+	0.992366i	$0.460642\pi$
$504$	0	0
$505$	−1.20867	−0.0537852
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.2329	−1.20708	−0.603539	−	0.797333i	$-0.706243\pi$
$509$	−27.2329	−1.20708	−0.603539	+	0.797333i	$0.706243\pi$
$510$	0	0
$511$	−32.1504	−1.42225
$512$	0	0
$513$	−9.08312	−0.401030
$514$	0	0
$515$	12.0448	0.530758
$516$	0	0
$517$	−15.1397	−0.665845
$518$	0	0
$519$	12.6342	0.554580
$520$	0	0
$521$	−27.1359	−1.18885	−0.594423	−	0.804153i	$-0.702619\pi$
$521$	−27.1359	−1.18885	−0.594423	+	0.804153i	$0.702619\pi$
$522$	0	0
$523$	14.8230	0.648167	0.324083	−	0.946029i	$-0.394944\pi$
$523$	14.8230	0.648167	0.324083	+	0.946029i	$0.394944\pi$
$524$	0	0
$525$	13.6485	0.595670
$526$	0	0
$527$	0.722938	0.0314917
$528$	0	0
$529$	11.7825	0.512284
$530$	0	0
$531$	−12.6123	−0.547326
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−3.61469	−0.156277
$536$	0	0
$537$	−21.6476	−0.934163
$538$	0	0
$539$	4.79442	0.206510
$540$	0	0
$541$	−12.1951	−0.524309	−0.262154	−	0.965026i	$-0.584433\pi$
$541$	−12.1951	−0.524309	−0.262154	+	0.965026i	$0.584433\pi$
$542$	0	0
$543$	−8.92538	−0.383025
$544$	0	0
$545$	3.41439	0.146256
$546$	0	0
$547$	−30.5965	−1.30821	−0.654106	−	0.756403i	$-0.726955\pi$
$547$	−30.5965	−1.30821	−0.654106	+	0.756403i	$0.726955\pi$
$548$	0	0
$549$	11.9247	0.508933
$550$	0	0
$551$	10.2151	0.435176
$552$	0	0
$553$	−42.0125	−1.78655
$554$	0	0
$555$	−5.67604	−0.240934
$556$	0	0
$557$	12.9333	0.548000	0.274000	−	0.961730i	$-0.411653\pi$
$557$	12.9333	0.548000	0.274000	+	0.961730i	$0.411653\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3.02771	−0.127830
$562$	0	0
$563$	−5.53204	−0.233148	−0.116574	−	0.993182i	$-0.537191\pi$
$563$	−5.53204	−0.233148	−0.116574	+	0.993182i	$0.537191\pi$
$564$	0	0
$565$	−14.5579	−0.612457
$566$	0	0
$567$	−1.67427	−0.0703129
$568$	0	0
$569$	17.9537	0.752660	0.376330	−	0.926486i	$-0.377186\pi$
$569$	17.9537	0.752660	0.376330	+	0.926486i	$0.377186\pi$
$570$	0	0
$571$	−1.61780	−0.0677028	−0.0338514	−	0.999427i	$-0.510777\pi$
$571$	−1.61780	−0.0677028	−0.0338514	+	0.999427i	$0.510777\pi$
$572$	0	0
$573$	16.5660	0.692056
$574$	0	0
$575$	26.5859	1.10871
$576$	0	0
$577$	−17.8712	−0.743986	−0.371993	−	0.928236i	$-0.621325\pi$
$577$	−17.8712	−0.743986	−0.371993	+	0.928236i	$0.621325\pi$
$578$	0	0
$579$	8.70335	0.361699
$580$	0	0
$581$	−40.7678	−1.69133
$582$	0	0
$583$	−9.44588	−0.391208
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.95375	−0.328286	−0.164143	−	0.986437i	$-0.552486\pi$
$587$	−7.95375	−0.328286	−0.164143	+	0.986437i	$0.552486\pi$
$588$	0	0
$589$	−1.29040	−0.0531698
$590$	0	0
$591$	8.13589	0.334666
$592$	0	0
$593$	24.6716	1.01314	0.506570	−	0.862199i	$-0.330913\pi$
$593$	24.6716	1.01314	0.506570	+	0.862199i	$0.330913\pi$
$594$	0	0
$595$	−2.06106	−0.0844954
$596$	0	0
$597$	16.5660	0.678002
$598$	0	0
$599$	−8.48211	−0.346570	−0.173285	−	0.984872i	$-0.555438\pi$
$599$	−8.48211	−0.346570	−0.173285	+	0.984872i	$0.555438\pi$
$600$	0	0
$601$	−39.0896	−1.59450	−0.797250	−	0.603650i	$-0.793713\pi$
$601$	−39.0896	−1.59450	−0.797250	+	0.603650i	$0.793713\pi$
$602$	0	0
$603$	−23.4325	−0.954247
$604$	0	0
$605$	1.66129	0.0675411
$606$	0	0
$607$	20.6406	0.837776	0.418888	−	0.908038i	$-0.362420\pi$
$607$	20.6406	0.837776	0.418888	+	0.908038i	$0.362420\pi$
$608$	0	0
$609$	−17.3274	−0.702141
$610$	0	0
$611$	0	0
$612$	0	0
$613$	25.1719	1.01668	0.508341	−	0.861156i	$-0.330259\pi$
$613$	25.1719	1.01668	0.508341	+	0.861156i	$0.330259\pi$
$614$	0	0
$615$	−2.74219	−0.110576
$616$	0	0
$617$	17.3478	0.698395	0.349198	−	0.937049i	$-0.386454\pi$
$617$	17.3478	0.698395	0.349198	+	0.937049i	$0.386454\pi$
$618$	0	0
$619$	37.0157	1.48779	0.743894	−	0.668297i	$-0.232977\pi$
$619$	37.0157	1.48779	0.743894	+	0.668297i	$0.232977\pi$
$620$	0	0
$621$	30.0119	1.20434
$622$	0	0
$623$	23.1951	0.929291
$624$	0	0
$625$	17.8603	0.714411
$626$	0	0
$627$	5.40426	0.215825
$628$	0	0
$629$	−7.85135	−0.313054
$630$	0	0
$631$	−39.9537	−1.59053	−0.795267	−	0.606260i	$-0.792669\pi$
$631$	−39.9537	−1.59053	−0.795267	+	0.606260i	$0.792669\pi$
$632$	0	0
$633$	23.7058	0.942219
$634$	0	0
$635$	14.1688	0.562272
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.29281	−0.169821
$640$	0	0
$641$	17.5541	0.693346	0.346673	−	0.937986i	$-0.387311\pi$
$641$	17.5541	0.693346	0.346673	+	0.937986i	$0.387311\pi$
$642$	0	0
$643$	1.30998	0.0516607	0.0258303	−	0.999666i	$-0.491777\pi$
$643$	1.30998	0.0516607	0.0258303	+	0.999666i	$0.491777\pi$
$644$	0	0
$645$	−5.16888	−0.203525
$646$	0	0
$647$	−26.6960	−1.04953	−0.524764	−	0.851247i	$-0.675847\pi$
$647$	−26.6960	−1.04953	−0.524764	+	0.851247i	$0.675847\pi$
$648$	0	0
$649$	19.1201	0.750532
$650$	0	0
$651$	2.18884	0.0857875
$652$	0	0
$653$	13.1201	0.513431	0.256716	−	0.966487i	$-0.417360\pi$
$653$	13.1201	0.513431	0.256716	+	0.966487i	$0.417360\pi$
$654$	0	0
$655$	−1.40106	−0.0547438
$656$	0	0
$657$	−21.2075	−0.827382
$658$	0	0
$659$	−43.8657	−1.70876	−0.854382	−	0.519646i	$-0.826064\pi$
$659$	−43.8657	−1.70876	−0.854382	+	0.519646i	$0.826064\pi$
$660$	0	0
$661$	−16.9276	−0.658408	−0.329204	−	0.944259i	$-0.606780\pi$
$661$	−16.9276	−0.658408	−0.329204	+	0.944259i	$0.606780\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.67886	0.142660
$666$	0	0
$667$	−33.7520	−1.30688
$668$	0	0
$669$	15.6958	0.606836
$670$	0	0
$671$	−18.0777	−0.697883
$672$	0	0
$673$	19.1555	0.738389	0.369195	−	0.929352i	$-0.379634\pi$
$673$	19.1555	0.738389	0.369195	+	0.929352i	$0.379634\pi$
$674$	0	0
$675$	22.9395	0.882943
$676$	0	0
$677$	36.2136	1.39180	0.695901	−	0.718137i	$-0.255005\pi$
$677$	36.2136	1.39180	0.695901	+	0.718137i	$0.255005\pi$
$678$	0	0
$679$	−5.40426	−0.207396
$680$	0	0
$681$	30.4388	1.16642
$682$	0	0
$683$	4.76919	0.182488	0.0912440	−	0.995829i	$-0.470916\pi$
$683$	4.76919	0.182488	0.0912440	+	0.995829i	$0.470916\pi$
$684$	0	0
$685$	16.1068	0.615410
$686$	0	0
$687$	−13.5117	−0.515504
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−28.9695	−1.10205	−0.551026	−	0.834488i	$-0.685763\pi$
$691$	−28.9695	−1.10205	−0.551026	+	0.834488i	$0.685763\pi$
$692$	0	0
$693$	16.7286	0.635466
$694$	0	0
$695$	−6.15066	−0.233308
$696$	0	0
$697$	−3.79312	−0.143674
$698$	0	0
$699$	26.0950	0.987004
$700$	0	0
$701$	−39.0157	−1.47360	−0.736802	−	0.676108i	$-0.763665\pi$
$701$	−39.0157	−1.47360	−0.736802	+	0.676108i	$0.763665\pi$
$702$	0	0
$703$	14.0141	0.528554
$704$	0	0
$705$	3.72535	0.140305
$706$	0	0
$707$	−5.06200	−0.190376
$708$	0	0
$709$	−28.2064	−1.05932	−0.529658	−	0.848212i	$-0.677680\pi$
$709$	−28.2064	−1.05932	−0.529658	+	0.848212i	$0.677680\pi$
$710$	0	0
$711$	−27.7128	−1.03931
$712$	0	0
$713$	4.26365	0.159675
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.94194	0.259252
$718$	0	0
$719$	9.21090	0.343509	0.171754	−	0.985140i	$-0.445056\pi$
$719$	9.21090	0.343509	0.171754	+	0.985140i	$0.445056\pi$
$720$	0	0
$721$	50.4444	1.87865
$722$	0	0
$723$	−21.5369	−0.800968
$724$	0	0
$725$	−25.7983	−0.958124
$726$	0	0
$727$	12.2386	0.453905	0.226952	−	0.973906i	$-0.427124\pi$
$727$	12.2386	0.453905	0.226952	+	0.973906i	$0.427124\pi$
$728$	0	0
$729$	14.6280	0.541779
$730$	0	0
$731$	−7.14984	−0.264446
$732$	0	0
$733$	6.66995	0.246360	0.123180	−	0.992384i	$-0.460691\pi$
$733$	6.66995	0.246360	0.123180	+	0.992384i	$0.460691\pi$
$734$	0	0
$735$	−1.17974	−0.0435152
$736$	0	0
$737$	35.5236	1.30853
$738$	0	0
$739$	20.5660	0.756533	0.378267	−	0.925697i	$-0.376520\pi$
$739$	20.5660	0.756533	0.378267	+	0.925697i	$0.376520\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−0.0462534	−0.00169687	−0.000848436	−	1.00000i	$-0.500270\pi$
$743$	−0.0462534	−0.00169687	−0.000848436		1.00000i	$0.500270\pi$
$744$	0	0
$745$	0.798267	0.0292463
$746$	0	0
$747$	−26.8918	−0.983918
$748$	0	0
$749$	−15.1385	−0.553150
$750$	0	0
$751$	38.9848	1.42257	0.711287	−	0.702901i	$-0.248112\pi$
$751$	38.9848	1.42257	0.711287	+	0.702901i	$0.248112\pi$
$752$	0	0
$753$	19.4578	0.709081
$754$	0	0
$755$	−0.507154	−0.0184572
$756$	0	0
$757$	0.259886	0.00944570	0.00472285	−	0.999989i	$-0.498497\pi$
$757$	0.259886	0.00944570	0.00472285	+	0.999989i	$0.498497\pi$
$758$	0	0
$759$	−17.8564	−0.648147
$760$	0	0
$761$	−34.1176	−1.23676	−0.618382	−	0.785878i	$-0.712211\pi$
$761$	−34.1176	−1.23676	−0.618382	+	0.785878i	$0.712211\pi$
$762$	0	0
$763$	14.2997	0.517683
$764$	0	0
$765$	−1.35955	−0.0491545
$766$	0	0
$767$	0	0
$768$	0	0
$769$	2.72596	0.0983007	0.0491503	−	0.998791i	$-0.484349\pi$
$769$	2.72596	0.0983007	0.0491503	+	0.998791i	$0.484349\pi$
$770$	0	0
$771$	21.8289	0.786148
$772$	0	0
$773$	−45.4835	−1.63593	−0.817963	−	0.575271i	$-0.804897\pi$
$773$	−45.4835	−1.63593	−0.817963	+	0.575271i	$0.804897\pi$
$774$	0	0
$775$	3.25891	0.117063
$776$	0	0
$777$	−23.7716	−0.852801
$778$	0	0
$779$	6.77046	0.242577
$780$	0	0
$781$	6.50787	0.232870
$782$	0	0
$783$	−29.1227	−1.04076
$784$	0	0
$785$	−3.26974	−0.116702
$786$	0	0
$787$	−36.8588	−1.31388	−0.656938	−	0.753945i	$-0.728149\pi$
$787$	−36.8588	−1.31388	−0.656938	+	0.753945i	$0.728149\pi$
$788$	0	0
$789$	0.996156	0.0354641
$790$	0	0
$791$	−60.9695	−2.16783
$792$	0	0
$793$	0	0
$794$	0	0
$795$	2.32429	0.0824341
$796$	0	0
$797$	−10.1359	−0.359032	−0.179516	−	0.983755i	$-0.557453\pi$
$797$	−10.1359	−0.359032	−0.179516	+	0.983755i	$0.557453\pi$
$798$	0	0
$799$	5.15307	0.182302
$800$	0	0
$801$	15.3002	0.540607
$802$	0	0
$803$	32.1504	1.13456
$804$	0	0
$805$	−12.1555	−0.428424
$806$	0	0
$807$	−15.0786	−0.530792
$808$	0	0
$809$	19.2795	0.677830	0.338915	−	0.940817i	$-0.389940\pi$
$809$	19.2795	0.677830	0.338915	+	0.940817i	$0.389940\pi$
$810$	0	0
$811$	21.7653	0.764285	0.382142	−	0.924103i	$-0.375186\pi$
$811$	21.7653	0.764285	0.382142	+	0.924103i	$0.375186\pi$
$812$	0	0
$813$	23.1951	0.813487
$814$	0	0
$815$	−7.67321	−0.268781
$816$	0	0
$817$	12.7620	0.446485
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44.7384	1.56138	0.780691	−	0.624917i	$-0.214867\pi$
$821$	44.7384	1.56138	0.780691	+	0.624917i	$0.214867\pi$
$822$	0	0
$823$	−12.3187	−0.429404	−0.214702	−	0.976680i	$-0.568878\pi$
$823$	−12.3187	−0.429404	−0.214702	+	0.976680i	$0.568878\pi$
$824$	0	0
$825$	−13.6485	−0.475180
$826$	0	0
$827$	43.6595	1.51819	0.759095	−	0.650980i	$-0.225642\pi$
$827$	43.6595	1.51819	0.759095	+	0.650980i	$0.225642\pi$
$828$	0	0
$829$	49.0448	1.70340	0.851698	−	0.524032i	$-0.175573\pi$
$829$	49.0448	1.70340	0.851698	+	0.524032i	$0.175573\pi$
$830$	0	0
$831$	−16.8340	−0.583964
$832$	0	0
$833$	−1.63186	−0.0565408
$834$	0	0
$835$	−16.8040	−0.581526
$836$	0	0
$837$	3.67886	0.127160
$838$	0	0
$839$	39.8164	1.37462	0.687308	−	0.726366i	$-0.258792\pi$
$839$	39.8164	1.37462	0.687308	+	0.726366i	$0.258792\pi$
$840$	0	0
$841$	3.75201	0.129380
$842$	0	0
$843$	−16.7496	−0.576887
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.95759	0.239066
$848$	0	0
$849$	−7.36814	−0.252874
$850$	0	0
$851$	−46.3047	−1.58730
$852$	0	0
$853$	33.4230	1.14438	0.572191	−	0.820121i	$-0.306094\pi$
$853$	33.4230	1.14438	0.572191	+	0.820121i	$0.306094\pi$
$854$	0	0
$855$	2.42670	0.0829914
$856$	0	0
$857$	11.5345	0.394012	0.197006	−	0.980402i	$-0.436878\pi$
$857$	11.5345	0.394012	0.197006	+	0.980402i	$0.436878\pi$
$858$	0	0
$859$	−3.61497	−0.123341	−0.0616707	−	0.998097i	$-0.519643\pi$
$859$	−3.61497	−0.123341	−0.0616707	+	0.998097i	$0.519643\pi$
$860$	0	0
$861$	−11.4844	−0.391389
$862$	0	0
$863$	−46.8259	−1.59397	−0.796986	−	0.603997i	$-0.793574\pi$
$863$	−46.8259	−1.59397	−0.796986	+	0.603997i	$0.793574\pi$
$864$	0	0
$865$	−8.60054	−0.292427
$866$	0	0
$867$	−16.4885	−0.559979
$868$	0	0
$869$	42.0125	1.42518
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−3.56483	−0.120651
$874$	0	0
$875$	−19.5963	−0.662477
$876$	0	0
$877$	22.0640	0.745050	0.372525	−	0.928022i	$-0.378492\pi$
$877$	22.0640	0.745050	0.372525	+	0.928022i	$0.378492\pi$
$878$	0	0
$879$	18.1226	0.611259
$880$	0	0
$881$	32.5699	1.09731	0.548653	−	0.836050i	$-0.315141\pi$
$881$	32.5699	1.09731	0.548653	+	0.836050i	$0.315141\pi$
$882$	0	0
$883$	−44.5807	−1.50026	−0.750130	−	0.661290i	$-0.770009\pi$
$883$	−44.5807	−1.50026	−0.750130	+	0.661290i	$0.770009\pi$
$884$	0	0
$885$	−4.70478	−0.158150
$886$	0	0
$887$	−39.4280	−1.32386	−0.661932	−	0.749564i	$-0.730263\pi$
$887$	−39.4280	−1.32386	−0.661932	+	0.749564i	$0.730263\pi$
$888$	0	0
$889$	59.3399	1.99020
$890$	0	0
$891$	1.67427	0.0560903
$892$	0	0
$893$	−9.19789	−0.307796
$894$	0	0
$895$	14.7363	0.492579
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.13733	−0.137988
$900$	0	0
$901$	3.21507	0.107109
$902$	0	0
$903$	−21.6476	−0.720387
$904$	0	0
$905$	6.07581	0.201967
$906$	0	0
$907$	58.7688	1.95139	0.975693	−	0.219141i	$-0.0703253\pi$
$907$	58.7688	1.95139	0.975693	+	0.219141i	$0.0703253\pi$
$908$	0	0
$909$	−3.33906	−0.110750
$910$	0	0
$911$	9.86206	0.326745	0.163372	−	0.986564i	$-0.447763\pi$
$911$	9.86206	0.326745	0.163372	+	0.986564i	$0.447763\pi$
$912$	0	0
$913$	40.7678	1.34922
$914$	0	0
$915$	4.44828	0.147056
$916$	0	0
$917$	−5.86771	−0.193769
$918$	0	0
$919$	13.6987	0.451877	0.225939	−	0.974142i	$-0.427455\pi$
$919$	13.6987	0.451877	0.225939	+	0.974142i	$0.427455\pi$
$920$	0	0
$921$	3.37712	0.111280
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−35.3929	−1.16371
$926$	0	0
$927$	33.2748	1.09289
$928$	0	0
$929$	8.44549	0.277088	0.138544	−	0.990356i	$-0.455758\pi$
$929$	8.44549	0.277088	0.138544	+	0.990356i	$0.455758\pi$
$930$	0	0
$931$	2.91277	0.0954622
$932$	0	0
$933$	−21.8102	−0.714033
$934$	0	0
$935$	2.06106	0.0674040
$936$	0	0
$937$	−10.6743	−0.348713	−0.174357	−	0.984683i	$-0.555785\pi$
$937$	−10.6743	−0.348713	−0.174357	+	0.984683i	$0.555785\pi$
$938$	0	0
$939$	10.4930	0.342427
$940$	0	0
$941$	−25.9672	−0.846508	−0.423254	−	0.906011i	$-0.639112\pi$
$941$	−25.9672	−0.846508	−0.423254	+	0.906011i	$0.639112\pi$
$942$	0	0
$943$	−22.3705	−0.728485
$944$	0	0
$945$	−10.4883	−0.341184
$946$	0	0
$947$	−33.9671	−1.10378	−0.551891	−	0.833916i	$-0.686094\pi$
$947$	−33.9671	−1.10378	−0.551891	+	0.833916i	$0.686094\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−29.6490	−0.961436
$952$	0	0
$953$	−4.35239	−0.140988	−0.0704939	−	0.997512i	$-0.522458\pi$
$953$	−4.35239	−0.140988	−0.0704939	+	0.997512i	$0.522458\pi$
$954$	0	0
$955$	−11.2771	−0.364917
$956$	0	0
$957$	17.3274	0.560115
$958$	0	0
$959$	67.4563	2.17828
$960$	0	0
$961$	−30.4774	−0.983141
$962$	0	0
$963$	−9.98587	−0.321790
$964$	0	0
$965$	−5.92467	−0.190722
$966$	0	0
$967$	−19.0291	−0.611934	−0.305967	−	0.952042i	$-0.598980\pi$
$967$	−19.0291	−0.611934	−0.305967	+	0.952042i	$0.598980\pi$
$968$	0	0
$969$	−1.83943	−0.0590911
$970$	0	0
$971$	−35.2420	−1.13097	−0.565485	−	0.824758i	$-0.691311\pi$
$971$	−35.2420	−1.13097	−0.565485	+	0.824758i	$0.691311\pi$
$972$	0	0
$973$	−25.7593	−0.825807
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.32393	−0.170328	−0.0851638	−	0.996367i	$-0.527141\pi$
$977$	−5.32393	−0.170328	−0.0851638	+	0.996367i	$0.527141\pi$
$978$	0	0
$979$	−23.1951	−0.741318
$980$	0	0
$981$	9.43253	0.301158
$982$	0	0
$983$	−12.4750	−0.397889	−0.198945	−	0.980011i	$-0.563751\pi$
$983$	−12.4750	−0.397889	−0.198945	+	0.980011i	$0.563751\pi$
$984$	0	0
$985$	−5.53838	−0.176468
$986$	0	0
$987$	15.6020	0.496617
$988$	0	0
$989$	−42.1674	−1.34084
$990$	0	0
$991$	45.4058	1.44236	0.721182	−	0.692746i	$-0.243599\pi$
$991$	45.4058	1.44236	0.721182	+	0.692746i	$0.243599\pi$
$992$	0	0
$993$	−4.51773	−0.143366
$994$	0	0
$995$	−11.2771	−0.357507
$996$	0	0
$997$	−19.0133	−0.602158	−0.301079	−	0.953599i	$-0.597347\pi$
$997$	−19.0133	−0.602158	−0.301079	+	0.953599i	$0.597347\pi$
$998$	0	0
$999$	−39.9537	−1.26408

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5408.2.a.bn.1.4		6
4.3	odd	2		5408.2.a.bm.1.3		6
13.2	odd	12		416.2.w.d.225.3	✓	12
13.7	odd	12		416.2.w.d.257.3	yes	12
13.12	even	2		5408.2.a.bm.1.4		6
52.7	even	12		416.2.w.d.257.4	yes	12
52.15	even	12		416.2.w.d.225.4	yes	12
52.51	odd	2	inner	5408.2.a.bn.1.3		6
104.59	even	12		832.2.w.j.257.3		12
104.67	even	12		832.2.w.j.641.3		12
104.85	odd	12		832.2.w.j.257.4		12
104.93	odd	12		832.2.w.j.641.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.w.d.225.3	✓	12	13.2	odd	12
416.2.w.d.225.4	yes	12	52.15	even	12
416.2.w.d.257.3	yes	12	13.7	odd	12
416.2.w.d.257.4	yes	12	52.7	even	12
832.2.w.j.257.3		12	104.59	even	12
832.2.w.j.257.4		12	104.85	odd	12
832.2.w.j.641.3		12	104.67	even	12
832.2.w.j.641.4		12	104.93	odd	12
5408.2.a.bm.1.3		6	4.3	odd	2
5408.2.a.bm.1.4		6	13.12	even	2
5408.2.a.bn.1.3		6	52.51	odd	2	inner
5408.2.a.bn.1.4		6	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:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

\(f(q)\)	\(=\)	\(q+1.03053 q^{3} -0.701519 q^{5} -2.93800 q^{7} -1.93800 q^{9} +2.93800 q^{11} -0.722938 q^{15} -1.00000 q^{17} +1.78493 q^{19} -3.02771 q^{21} -5.89767 q^{23} -4.50787 q^{25} -5.08877 q^{27} +5.72294 q^{29} -0.722938 q^{31} +3.02771 q^{33} +2.06106 q^{35} +7.85135 q^{37} +3.79312 q^{41} +7.14984 q^{43} +1.35955 q^{45} -5.15307 q^{47} +1.63186 q^{49} -1.03053 q^{51} -3.21507 q^{53} -2.06106 q^{55} +1.83943 q^{57} +6.50787 q^{59} -6.15307 q^{61} +5.69386 q^{63} +12.0911 q^{67} -6.07774 q^{69} +2.21507 q^{71} +10.9430 q^{73} -4.64551 q^{75} -8.63186 q^{77} +14.2997 q^{79} +0.569868 q^{81} +13.8760 q^{83} +0.701519 q^{85} +5.89767 q^{87} -7.89485 q^{89} -0.745010 q^{93} -1.25216 q^{95} +1.83943 q^{97} -5.69386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{7} + 12 q^{9} - 6 q^{11} + 24 q^{15} - 6 q^{17} + 6 q^{19} + 6 q^{25} + 6 q^{29} + 24 q^{31} - 12 q^{47} + 24 q^{49} - 24 q^{53} + 6 q^{59} - 18 q^{61} + 72 q^{63} + 30 q^{67} + 6 q^{69} + 18 q^{71}+ \cdots - 72 q^{99}+O(q^{100}) \) 6 * q + 6 * q^7 + 12 * q^9 - 6 * q^11 + 24 * q^15 - 6 * q^17 + 6 * q^19 + 6 * q^25 + 6 * q^29 + 24 * q^31 - 12 * q^47 + 24 * q^49 - 24 * q^53 + 6 * q^59 - 18 * q^61 + 72 * q^63 + 30 * q^67 + 6 * q^69 + 18 * q^71 - 66 * q^77 - 6 * q^81 + 36 * q^83 - 72 * q^99

Introduction

L-functions

Modular forms

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