LMFDB - Embedded newform 4608.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, 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("4608.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.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:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1536)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4608.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41421 q^{5} +1.41421 q^{7} +O(q^{10})$	$q-3.41421 q^{5} +1.41421 q^{7} -4.82843 q^{11} -0.828427 q^{13} -4.82843 q^{17} -2.82843 q^{19} +1.17157 q^{23} +6.65685 q^{25} -7.41421 q^{29} -7.07107 q^{31} -4.82843 q^{35} +11.6569 q^{37} +10.4853 q^{41} -6.82843 q^{43} +12.4853 q^{47} -5.00000 q^{49} -1.75736 q^{53} +16.4853 q^{55} +1.65685 q^{59} +0.343146 q^{61} +2.82843 q^{65} +5.65685 q^{67} -8.48528 q^{71} -11.3137 q^{73} -6.82843 q^{77} +17.4142 q^{79} -8.82843 q^{83} +16.4853 q^{85} -5.31371 q^{89} -1.17157 q^{91} +9.65685 q^{95} -7.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} - 4 q^{11} + 4 q^{13} - 4 q^{17} + 8 q^{23} + 2 q^{25} - 12 q^{29} - 4 q^{35} + 12 q^{37} + 4 q^{41} - 8 q^{43} + 8 q^{47} - 10 q^{49} - 12 q^{53} + 16 q^{55} - 8 q^{59} + 12 q^{61} - 8 q^{77} + 32 q^{79} - 12 q^{83} + 16 q^{85} + 12 q^{89} - 8 q^{91} + 8 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 4 * q^11 + 4 * q^13 - 4 * q^17 + 8 * q^23 + 2 * q^25 - 12 * q^29 - 4 * q^35 + 12 * q^37 + 4 * q^41 - 8 * q^43 + 8 * q^47 - 10 * q^49 - 12 * q^53 + 16 * q^55 - 8 * q^59 + 12 * q^61 - 8 * q^77 + 32 * q^79 - 12 * q^83 + 16 * q^85 + 12 * q^89 - 8 * q^91 + 8 * q^95 - 4 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	0	0
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.17157	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$23$	1.17157	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.41421	−1.37678	−0.688392	−	0.725338i	$-0.741683\pi$
$29$	−7.41421	−1.37678	−0.688392	+	0.725338i	$0.741683\pi$
$30$	0	0
$31$	−7.07107	−1.27000	−0.635001	−	0.772512i	$-0.719000\pi$
$31$	−7.07107	−1.27000	−0.635001	+	0.772512i	$0.719000\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.82843	−0.816153
$36$	0	0
$37$	11.6569	1.91638	0.958188	−	0.286141i	$-0.0923726\pi$
$37$	11.6569	1.91638	0.958188	+	0.286141i	$0.0923726\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.4853	1.63753	0.818763	−	0.574132i	$-0.194660\pi$
$41$	10.4853	1.63753	0.818763	+	0.574132i	$0.194660\pi$
$42$	0	0
$43$	−6.82843	−1.04133	−0.520663	−	0.853762i	$-0.674315\pi$
$43$	−6.82843	−1.04133	−0.520663	+	0.853762i	$0.674315\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.4853	1.82117	0.910583	−	0.413327i	$-0.135633\pi$
$47$	12.4853	1.82117	0.910583	+	0.413327i	$0.135633\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.75736	−0.241392	−0.120696	−	0.992690i	$-0.538513\pi$
$53$	−1.75736	−0.241392	−0.120696	+	0.992690i	$0.538513\pi$
$54$	0	0
$55$	16.4853	2.22287
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.65685	0.215704	0.107852	−	0.994167i	$-0.465603\pi$
$59$	1.65685	0.215704	0.107852	+	0.994167i	$0.465603\pi$
$60$	0	0
$61$	0.343146	0.0439353	0.0219677	−	0.999759i	$-0.493007\pi$
$61$	0.343146	0.0439353	0.0219677	+	0.999759i	$0.493007\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.82843	0.350823
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.48528	−1.00702	−0.503509	−	0.863990i	$-0.667958\pi$
$71$	−8.48528	−1.00702	−0.503509	+	0.863990i	$0.667958\pi$
$72$	0	0
$73$	−11.3137	−1.32417	−0.662085	−	0.749429i	$-0.730328\pi$
$73$	−11.3137	−1.32417	−0.662085	+	0.749429i	$0.730328\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.82843	−0.778171
$78$	0	0
$79$	17.4142	1.95925	0.979626	−	0.200830i	$-0.0643640\pi$
$79$	17.4142	1.95925	0.979626	+	0.200830i	$0.0643640\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.82843	−0.969046	−0.484523	−	0.874779i	$-0.661007\pi$
$83$	−8.82843	−0.969046	−0.484523	+	0.874779i	$0.661007\pi$
$84$	0	0
$85$	16.4853	1.78808
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.31371	−0.563252	−0.281626	−	0.959524i	$-0.590874\pi$
$89$	−5.31371	−0.563252	−0.281626	+	0.959524i	$0.590874\pi$
$90$	0	0
$91$	−1.17157	−0.122814
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9.65685	0.990772
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.7279	1.06747	0.533734	−	0.845652i	$-0.320788\pi$
$101$	10.7279	1.06747	0.533734	+	0.845652i	$0.320788\pi$
$102$	0	0
$103$	9.41421	0.927610	0.463805	−	0.885937i	$-0.346484\pi$
$103$	9.41421	0.927610	0.463805	+	0.885937i	$0.346484\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.31371	−0.707043	−0.353521	−	0.935426i	$-0.615016\pi$
$107$	−7.31371	−0.707043	−0.353521	+	0.935426i	$0.615016\pi$
$108$	0	0
$109$	7.17157	0.686912	0.343456	−	0.939169i	$-0.388402\pi$
$109$	7.17157	0.686912	0.343456	+	0.939169i	$0.388402\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.82843	−0.625961
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−1.89949	−0.168553	−0.0842765	−	0.996442i	$-0.526858\pi$
$127$	−1.89949	−0.168553	−0.0842765	+	0.996442i	$0.526858\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.3137	1.33796	0.668982	−	0.743278i	$-0.266730\pi$
$131$	15.3137	1.33796	0.668982	+	0.743278i	$0.266730\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.48528	0.212332	0.106166	−	0.994348i	$-0.466143\pi$
$137$	2.48528	0.212332	0.106166	+	0.994348i	$0.466143\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	25.3137	2.10219
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.41421	−0.279703	−0.139852	−	0.990172i	$-0.544663\pi$
$149$	−3.41421	−0.279703	−0.139852	+	0.990172i	$0.544663\pi$
$150$	0	0
$151$	6.58579	0.535944	0.267972	−	0.963427i	$-0.413647\pi$
$151$	6.58579	0.535944	0.267972	+	0.963427i	$0.413647\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	24.1421	1.93914
$156$	0	0
$157$	−17.3137	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$157$	−17.3137	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.65685	0.130578
$162$	0	0
$163$	−12.4853	−0.977923	−0.488961	−	0.872305i	$-0.662624\pi$
$163$	−12.4853	−0.977923	−0.488961	+	0.872305i	$0.662624\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.89949	−0.600587	−0.300294	−	0.953847i	$-0.597085\pi$
$173$	−7.89949	−0.600587	−0.300294	+	0.953847i	$0.597085\pi$
$174$	0	0
$175$	9.41421	0.711648
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.3431	1.07206	0.536029	−	0.844200i	$-0.319924\pi$
$179$	14.3431	1.07206	0.536029	+	0.844200i	$0.319924\pi$
$180$	0	0
$181$	12.1421	0.902518	0.451259	−	0.892393i	$-0.350975\pi$
$181$	12.1421	0.902518	0.451259	+	0.892393i	$0.350975\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−39.7990	−2.92608
$186$	0	0
$187$	23.3137	1.70487
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.3137	−1.10806	−0.554031	−	0.832496i	$-0.686911\pi$
$191$	−15.3137	−1.10806	−0.554031	+	0.832496i	$0.686911\pi$
$192$	0	0
$193$	−11.3137	−0.814379	−0.407189	−	0.913344i	$-0.633491\pi$
$193$	−11.3137	−0.814379	−0.407189	+	0.913344i	$0.633491\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.585786	0.0417356	0.0208678	−	0.999782i	$-0.493357\pi$
$197$	0.585786	0.0417356	0.0208678	+	0.999782i	$0.493357\pi$
$198$	0	0
$199$	11.7574	0.833457	0.416729	−	0.909031i	$-0.363177\pi$
$199$	11.7574	0.833457	0.416729	+	0.909031i	$0.363177\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.4853	−0.735922
$204$	0	0
$205$	−35.7990	−2.50031
$206$	0	0
$207$	0	0
$208$	0	0
$209$	13.6569	0.944664
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	23.3137	1.58998
$216$	0	0
$217$	−10.0000	−0.678844
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	23.5563	1.57745	0.788725	−	0.614746i	$-0.210742\pi$
$223$	23.5563	1.57745	0.788725	+	0.614746i	$0.210742\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.1421	0.805902	0.402951	−	0.915222i	$-0.367985\pi$
$227$	12.1421	0.805902	0.402951	+	0.915222i	$0.367985\pi$
$228$	0	0
$229$	17.7990	1.17619	0.588095	−	0.808792i	$-0.299878\pi$
$229$	17.7990	1.17619	0.588095	+	0.808792i	$0.299878\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	−42.6274	−2.78071
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.686292	0.0443925	0.0221963	−	0.999754i	$-0.492934\pi$
$239$	0.686292	0.0443925	0.0221963	+	0.999754i	$0.492934\pi$
$240$	0	0
$241$	1.65685	0.106727	0.0533637	−	0.998575i	$-0.483006\pi$
$241$	1.65685	0.106727	0.0533637	+	0.998575i	$0.483006\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	17.0711	1.09063
$246$	0	0
$247$	2.34315	0.149091
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.7990	1.12346	0.561731	−	0.827320i	$-0.310136\pi$
$251$	17.7990	1.12346	0.561731	+	0.827320i	$0.310136\pi$
$252$	0	0
$253$	−5.65685	−0.355643
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.6569	1.22616	0.613080	−	0.790020i	$-0.289930\pi$
$257$	19.6569	1.22616	0.613080	+	0.790020i	$0.289930\pi$
$258$	0	0
$259$	16.4853	1.02435
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.9706	1.53975	0.769875	−	0.638194i	$-0.220318\pi$
$263$	24.9706	1.53975	0.769875	+	0.638194i	$0.220318\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.24264	−0.380621	−0.190310	−	0.981724i	$-0.560949\pi$
$269$	−6.24264	−0.380621	−0.190310	+	0.981724i	$0.560949\pi$
$270$	0	0
$271$	−3.75736	−0.228243	−0.114122	−	0.993467i	$-0.536405\pi$
$271$	−3.75736	−0.228243	−0.114122	+	0.993467i	$0.536405\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−32.1421	−1.93824
$276$	0	0
$277$	−15.1716	−0.911571	−0.455786	−	0.890090i	$-0.650642\pi$
$277$	−15.1716	−0.911571	−0.455786	+	0.890090i	$0.650642\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	23.6569	1.41125	0.705625	−	0.708586i	$-0.250666\pi$
$281$	23.6569	1.41125	0.705625	+	0.708586i	$0.250666\pi$
$282$	0	0
$283$	12.9706	0.771020	0.385510	−	0.922704i	$-0.374026\pi$
$283$	12.9706	0.771020	0.385510	+	0.922704i	$0.374026\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.8284	0.875294
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.92893	0.171110	0.0855550	−	0.996333i	$-0.472734\pi$
$293$	2.92893	0.171110	0.0855550	+	0.996333i	$0.472734\pi$
$294$	0	0
$295$	−5.65685	−0.329355
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.970563	−0.0561291
$300$	0	0
$301$	−9.65685	−0.556612
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.17157	−0.0670841
$306$	0	0
$307$	−18.3431	−1.04690	−0.523449	−	0.852057i	$-0.675355\pi$
$307$	−18.3431	−1.04690	−0.523449	+	0.852057i	$0.675355\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.3137	1.54882	0.774409	−	0.632685i	$-0.218047\pi$
$311$	27.3137	1.54882	0.774409	+	0.632685i	$0.218047\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.1005	0.679632	0.339816	−	0.940492i	$-0.389635\pi$
$317$	12.1005	0.679632	0.339816	+	0.940492i	$0.389635\pi$
$318$	0	0
$319$	35.7990	2.00436
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.6569	0.759888
$324$	0	0
$325$	−5.51472	−0.305902
$326$	0	0
$327$	0	0
$328$	0	0
$329$	17.6569	0.973454
$330$	0	0
$331$	2.34315	0.128791	0.0643955	−	0.997924i	$-0.479488\pi$
$331$	2.34315	0.128791	0.0643955	+	0.997924i	$0.479488\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−19.3137	−1.05522
$336$	0	0
$337$	−19.3137	−1.05208	−0.526042	−	0.850458i	$-0.676325\pi$
$337$	−19.3137	−1.05208	−0.526042	+	0.850458i	$0.676325\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	34.1421	1.84890
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.7990	1.17023	0.585116	−	0.810950i	$-0.301049\pi$
$347$	21.7990	1.17023	0.585116	+	0.810950i	$0.301049\pi$
$348$	0	0
$349$	−21.3137	−1.14090	−0.570448	−	0.821333i	$-0.693231\pi$
$349$	−21.3137	−1.14090	−0.570448	+	0.821333i	$0.693231\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	28.9706	1.53760
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.51472	−0.396612	−0.198306	−	0.980140i	$-0.563544\pi$
$359$	−7.51472	−0.396612	−0.198306	+	0.980140i	$0.563544\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	38.6274	2.02185
$366$	0	0
$367$	0.928932	0.0484899	0.0242449	−	0.999706i	$-0.492282\pi$
$367$	0.928932	0.0484899	0.0242449	+	0.999706i	$0.492282\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.48528	−0.129029
$372$	0	0
$373$	−11.6569	−0.603569	−0.301785	−	0.953376i	$-0.597582\pi$
$373$	−11.6569	−0.603569	−0.301785	+	0.953376i	$0.597582\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.14214	0.316336
$378$	0	0
$379$	−5.17157	−0.265646	−0.132823	−	0.991140i	$-0.542404\pi$
$379$	−5.17157	−0.265646	−0.132823	+	0.991140i	$0.542404\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.3431	1.14168	0.570841	−	0.821061i	$-0.306617\pi$
$383$	22.3431	1.14168	0.570841	+	0.821061i	$0.306617\pi$
$384$	0	0
$385$	23.3137	1.18818
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.07107	0.0543053	0.0271526	−	0.999631i	$-0.491356\pi$
$389$	1.07107	0.0543053	0.0271526	+	0.999631i	$0.491356\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−59.4558	−2.99155
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.4853	−0.923111	−0.461555	−	0.887111i	$-0.652708\pi$
$401$	−18.4853	−0.923111	−0.461555	+	0.887111i	$0.652708\pi$
$402$	0	0
$403$	5.85786	0.291801
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−56.2843	−2.78991
$408$	0	0
$409$	31.6569	1.56533	0.782665	−	0.622443i	$-0.213860\pi$
$409$	31.6569	1.56533	0.782665	+	0.622443i	$0.213860\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.34315	0.115299
$414$	0	0
$415$	30.1421	1.47962
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.828427	0.0404713	0.0202357	−	0.999795i	$-0.493558\pi$
$419$	0.828427	0.0404713	0.0202357	+	0.999795i	$0.493558\pi$
$420$	0	0
$421$	22.4853	1.09587	0.547933	−	0.836522i	$-0.315415\pi$
$421$	22.4853	1.09587	0.547933	+	0.836522i	$0.315415\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−32.1421	−1.55912
$426$	0	0
$427$	0.485281	0.0234844
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.1421	−1.64457	−0.822284	−	0.569077i	$-0.807301\pi$
$431$	−34.1421	−1.64457	−0.822284	+	0.569077i	$0.807301\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.31371	−0.158516
$438$	0	0
$439$	−21.2132	−1.01245	−0.506225	−	0.862401i	$-0.668960\pi$
$439$	−21.2132	−1.01245	−0.506225	+	0.862401i	$0.668960\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.5147	−0.642104	−0.321052	−	0.947062i	$-0.604036\pi$
$443$	−13.5147	−0.642104	−0.321052	+	0.947062i	$0.604036\pi$
$444$	0	0
$445$	18.1421	0.860020
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.7990	−0.651215	−0.325607	−	0.945505i	$-0.605569\pi$
$449$	−13.7990	−0.651215	−0.325607	+	0.945505i	$0.605569\pi$
$450$	0	0
$451$	−50.6274	−2.38395
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	34.2843	1.60375	0.801875	−	0.597491i	$-0.203836\pi$
$457$	34.2843	1.60375	0.801875	+	0.597491i	$0.203836\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−19.2132	−0.894848	−0.447424	−	0.894322i	$-0.647659\pi$
$461$	−19.2132	−0.894848	−0.447424	+	0.894322i	$0.647659\pi$
$462$	0	0
$463$	−12.2426	−0.568964	−0.284482	−	0.958681i	$-0.591822\pi$
$463$	−12.2426	−0.568964	−0.284482	+	0.958681i	$0.591822\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.1716	−0.702057	−0.351028	−	0.936365i	$-0.614168\pi$
$467$	−15.1716	−0.702057	−0.351028	+	0.936365i	$0.614168\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	32.9706	1.51599
$474$	0	0
$475$	−18.8284	−0.863907
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.8284	−1.22582	−0.612911	−	0.790152i	$-0.710002\pi$
$479$	−26.8284	−1.22582	−0.612911	+	0.790152i	$0.710002\pi$
$480$	0	0
$481$	−9.65685	−0.440315
$482$	0	0
$483$	0	0
$484$	0	0
$485$	26.1421	1.18705
$486$	0	0
$487$	20.7279	0.939272	0.469636	−	0.882860i	$-0.344385\pi$
$487$	20.7279	0.939272	0.469636	+	0.882860i	$0.344385\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.2843	−0.734899	−0.367449	−	0.930043i	$-0.619769\pi$
$491$	−16.2843	−0.734899	−0.367449	+	0.930043i	$0.619769\pi$
$492$	0	0
$493$	35.7990	1.61231
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	−40.2843	−1.80337	−0.901686	−	0.432392i	$-0.857670\pi$
$499$	−40.2843	−1.80337	−0.901686	+	0.432392i	$0.857670\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	27.7990	1.23950	0.619748	−	0.784801i	$-0.287235\pi$
$503$	27.7990	1.23950	0.619748	+	0.784801i	$0.287235\pi$
$504$	0	0
$505$	−36.6274	−1.62990
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12.8701	0.570455	0.285228	−	0.958460i	$-0.407931\pi$
$509$	12.8701	0.570455	0.285228	+	0.958460i	$0.407931\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−32.1421	−1.41635
$516$	0	0
$517$	−60.2843	−2.65130
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.1421	−0.707200	−0.353600	−	0.935397i	$-0.615043\pi$
$521$	−16.1421	−0.707200	−0.353600	+	0.935397i	$0.615043\pi$
$522$	0	0
$523$	−9.45584	−0.413475	−0.206738	−	0.978396i	$-0.566285\pi$
$523$	−9.45584	−0.413475	−0.206738	+	0.978396i	$0.566285\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	34.1421	1.48725
$528$	0	0
$529$	−21.6274	−0.940322
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.68629	−0.376245
$534$	0	0
$535$	24.9706	1.07957
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24.1421	1.03988
$540$	0	0
$541$	−4.82843	−0.207590	−0.103795	−	0.994599i	$-0.533099\pi$
$541$	−4.82843	−0.207590	−0.103795	+	0.994599i	$0.533099\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−24.4853	−1.04883
$546$	0	0
$547$	10.1421	0.433646	0.216823	−	0.976211i	$-0.430430\pi$
$547$	10.1421	0.433646	0.216823	+	0.976211i	$0.430430\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.9706	0.893376
$552$	0	0
$553$	24.6274	1.04726
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.7279	−0.624042	−0.312021	−	0.950075i	$-0.601006\pi$
$557$	−14.7279	−0.624042	−0.312021	+	0.950075i	$0.601006\pi$
$558$	0	0
$559$	5.65685	0.239259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	27.1716	1.14515	0.572573	−	0.819854i	$-0.305945\pi$
$563$	27.1716	1.14515	0.572573	+	0.819854i	$0.305945\pi$
$564$	0	0
$565$	12.4853	0.525260
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.1127	1.05278	0.526390	−	0.850244i	$-0.323545\pi$
$569$	25.1127	1.05278	0.526390	+	0.850244i	$0.323545\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.79899	0.325240
$576$	0	0
$577$	31.3137	1.30361	0.651803	−	0.758388i	$-0.274013\pi$
$577$	31.3137	1.30361	0.651803	+	0.758388i	$0.274013\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.4853	−0.517977
$582$	0	0
$583$	8.48528	0.351424
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.9706	−0.865548	−0.432774	−	0.901503i	$-0.642465\pi$
$587$	−20.9706	−0.865548	−0.432774	+	0.901503i	$0.642465\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20.3431	0.835393	0.417696	−	0.908587i	$-0.362838\pi$
$593$	20.3431	0.835393	0.417696	+	0.908587i	$0.362838\pi$
$594$	0	0
$595$	23.3137	0.955769
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.4853	0.837006	0.418503	−	0.908215i	$-0.362555\pi$
$599$	20.4853	0.837006	0.418503	+	0.908215i	$0.362555\pi$
$600$	0	0
$601$	−18.6274	−0.759828	−0.379914	−	0.925022i	$-0.624046\pi$
$601$	−18.6274	−0.759828	−0.379914	+	0.925022i	$0.624046\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−42.0416	−1.70924
$606$	0	0
$607$	−9.89949	−0.401808	−0.200904	−	0.979611i	$-0.564388\pi$
$607$	−9.89949	−0.401808	−0.200904	+	0.979611i	$0.564388\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.3431	−0.418439
$612$	0	0
$613$	−4.62742	−0.186900	−0.0934498	−	0.995624i	$-0.529789\pi$
$613$	−4.62742	−0.186900	−0.0934498	+	0.995624i	$0.529789\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.68629	0.269180	0.134590	−	0.990901i	$-0.457028\pi$
$617$	6.68629	0.269180	0.134590	+	0.990901i	$0.457028\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.51472	−0.301071
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−56.2843	−2.24420
$630$	0	0
$631$	−19.2721	−0.767209	−0.383605	−	0.923497i	$-0.625317\pi$
$631$	−19.2721	−0.767209	−0.383605	+	0.923497i	$0.625317\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.48528	0.257361
$636$	0	0
$637$	4.14214	0.164117
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.1716	0.915222	0.457611	−	0.889152i	$-0.348705\pi$
$641$	23.1716	0.915222	0.457611	+	0.889152i	$0.348705\pi$
$642$	0	0
$643$	43.1127	1.70020	0.850099	−	0.526622i	$-0.176542\pi$
$643$	43.1127	1.70020	0.850099	+	0.526622i	$0.176542\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−46.4264	−1.82521	−0.912605	−	0.408842i	$-0.865933\pi$
$647$	−46.4264	−1.82521	−0.912605	+	0.408842i	$0.865933\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.38478	0.171590	0.0857948	−	0.996313i	$-0.472657\pi$
$653$	4.38478	0.171590	0.0857948	+	0.996313i	$0.472657\pi$
$654$	0	0
$655$	−52.2843	−2.04292
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−25.6569	−0.999449	−0.499725	−	0.866184i	$-0.666565\pi$
$659$	−25.6569	−0.999449	−0.499725	+	0.866184i	$0.666565\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13.6569	0.529590
$666$	0	0
$667$	−8.68629	−0.336335
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.65685	−0.0639621
$672$	0	0
$673$	−41.3137	−1.59253	−0.796263	−	0.604950i	$-0.793193\pi$
$673$	−41.3137	−1.59253	−0.796263	+	0.604950i	$0.793193\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.92893	0.266300	0.133150	−	0.991096i	$-0.457491\pi$
$677$	6.92893	0.266300	0.133150	+	0.991096i	$0.457491\pi$
$678$	0	0
$679$	−10.8284	−0.415557
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.1421	1.68905	0.844526	−	0.535515i	$-0.179882\pi$
$683$	44.1421	1.68905	0.844526	+	0.535515i	$0.179882\pi$
$684$	0	0
$685$	−8.48528	−0.324206
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.45584	0.0554632
$690$	0	0
$691$	−25.1716	−0.957572	−0.478786	−	0.877932i	$-0.658923\pi$
$691$	−25.1716	−0.957572	−0.478786	+	0.877932i	$0.658923\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−54.6274	−2.07214
$696$	0	0
$697$	−50.6274	−1.91765
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.27208	−0.199124	−0.0995618	−	0.995031i	$-0.531744\pi$
$701$	−5.27208	−0.199124	−0.0995618	+	0.995031i	$0.531744\pi$
$702$	0	0
$703$	−32.9706	−1.24351
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.1716	0.570586
$708$	0	0
$709$	40.8284	1.53334	0.766672	−	0.642039i	$-0.221911\pi$
$709$	40.8284	1.53334	0.766672	+	0.642039i	$0.221911\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.28427	−0.310248
$714$	0	0
$715$	−13.6569	−0.510737
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.48528	−0.316448	−0.158224	−	0.987403i	$-0.550577\pi$
$719$	−8.48528	−0.316448	−0.158224	+	0.987403i	$0.550577\pi$
$720$	0	0
$721$	13.3137	0.495828
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−49.3553	−1.83301
$726$	0	0
$727$	24.5269	0.909653	0.454826	−	0.890580i	$-0.349701\pi$
$727$	24.5269	0.909653	0.454826	+	0.890580i	$0.349701\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	32.9706	1.21946
$732$	0	0
$733$	8.82843	0.326085	0.163043	−	0.986619i	$-0.447869\pi$
$733$	8.82843	0.326085	0.163043	+	0.986619i	$0.447869\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27.3137	−1.00611
$738$	0	0
$739$	8.68629	0.319530	0.159765	−	0.987155i	$-0.448926\pi$
$739$	8.68629	0.319530	0.159765	+	0.987155i	$0.448926\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.28427	−0.303920	−0.151960	−	0.988387i	$-0.548559\pi$
$743$	−8.28427	−0.303920	−0.151960	+	0.988387i	$0.548559\pi$
$744$	0	0
$745$	11.6569	0.427074
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.3431	−0.377930
$750$	0	0
$751$	−14.5858	−0.532243	−0.266121	−	0.963940i	$-0.585742\pi$
$751$	−14.5858	−0.532243	−0.266121	+	0.963940i	$0.585742\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−22.4853	−0.818323
$756$	0	0
$757$	−46.7696	−1.69987	−0.849934	−	0.526889i	$-0.823358\pi$
$757$	−46.7696	−1.69987	−0.849934	+	0.526889i	$0.823358\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.1716	−0.839969	−0.419984	−	0.907531i	$-0.637964\pi$
$761$	−23.1716	−0.839969	−0.419984	+	0.907531i	$0.637964\pi$
$762$	0	0
$763$	10.1421	0.367170
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.37258	−0.0495611
$768$	0	0
$769$	−4.68629	−0.168992	−0.0844960	−	0.996424i	$-0.526928\pi$
$769$	−4.68629	−0.168992	−0.0844960	+	0.996424i	$0.526928\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−38.2426	−1.37549	−0.687746	−	0.725951i	$-0.741400\pi$
$773$	−38.2426	−1.37549	−0.687746	+	0.725951i	$0.741400\pi$
$774$	0	0
$775$	−47.0711	−1.69084
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−29.6569	−1.06257
$780$	0	0
$781$	40.9706	1.46604
$782$	0	0
$783$	0	0
$784$	0	0
$785$	59.1127	2.10982
$786$	0	0
$787$	13.1716	0.469516	0.234758	−	0.972054i	$-0.424570\pi$
$787$	13.1716	0.469516	0.234758	+	0.972054i	$0.424570\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.17157	−0.183880
$792$	0	0
$793$	−0.284271	−0.0100948
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.5269	1.36469	0.682347	−	0.731029i	$-0.260960\pi$
$797$	38.5269	1.36469	0.682347	+	0.731029i	$0.260960\pi$
$798$	0	0
$799$	−60.2843	−2.13270
$800$	0	0
$801$	0	0
$802$	0	0
$803$	54.6274	1.92776
$804$	0	0
$805$	−5.65685	−0.199378
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.4853	−1.35307	−0.676535	−	0.736410i	$-0.736519\pi$
$809$	−38.4853	−1.35307	−0.676535	+	0.736410i	$0.736519\pi$
$810$	0	0
$811$	0.485281	0.0170405	0.00852027	−	0.999964i	$-0.497288\pi$
$811$	0.485281	0.0170405	0.00852027	+	0.999964i	$0.497288\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	42.6274	1.49317
$816$	0	0
$817$	19.3137	0.675701
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.0122	1.64074	0.820368	−	0.571835i	$-0.193768\pi$
$821$	47.0122	1.64074	0.820368	+	0.571835i	$0.193768\pi$
$822$	0	0
$823$	−16.9289	−0.590105	−0.295053	−	0.955481i	$-0.595337\pi$
$823$	−16.9289	−0.590105	−0.295053	+	0.955481i	$0.595337\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.6274	0.369551	0.184776	−	0.982781i	$-0.440844\pi$
$827$	10.6274	0.369551	0.184776	+	0.982781i	$0.440844\pi$
$828$	0	0
$829$	42.4853	1.47557	0.737787	−	0.675033i	$-0.235871\pi$
$829$	42.4853	1.47557	0.737787	+	0.675033i	$0.235871\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	24.1421	0.836475
$834$	0	0
$835$	−40.9706	−1.41784
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.11270	0.245558	0.122779	−	0.992434i	$-0.460819\pi$
$839$	7.11270	0.245558	0.122779	+	0.992434i	$0.460819\pi$
$840$	0	0
$841$	25.9706	0.895537
$842$	0	0
$843$	0	0
$844$	0	0
$845$	42.0416	1.44628
$846$	0	0
$847$	17.4142	0.598359
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13.6569	0.468151
$852$	0	0
$853$	−1.31371	−0.0449805	−0.0224903	−	0.999747i	$-0.507159\pi$
$853$	−1.31371	−0.0449805	−0.0224903	+	0.999747i	$0.507159\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.48528	−0.221533	−0.110766	−	0.993846i	$-0.535331\pi$
$857$	−6.48528	−0.221533	−0.110766	+	0.993846i	$0.535331\pi$
$858$	0	0
$859$	9.17157	0.312930	0.156465	−	0.987684i	$-0.449990\pi$
$859$	9.17157	0.312930	0.156465	+	0.987684i	$0.449990\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	26.9706	0.917027
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−84.0833	−2.85233
$870$	0	0
$871$	−4.68629	−0.158789
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.00000	−0.270449
$876$	0	0
$877$	31.9411	1.07858	0.539288	−	0.842122i	$-0.318694\pi$
$877$	31.9411	1.07858	0.539288	+	0.842122i	$0.318694\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.34315	−0.146324	−0.0731621	−	0.997320i	$-0.523309\pi$
$881$	−4.34315	−0.146324	−0.0731621	+	0.997320i	$0.523309\pi$
$882$	0	0
$883$	23.7990	0.800900	0.400450	−	0.916319i	$-0.368854\pi$
$883$	23.7990	0.800900	0.400450	+	0.916319i	$0.368854\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.28427	−0.143852	−0.0719259	−	0.997410i	$-0.522915\pi$
$887$	−4.28427	−0.143852	−0.0719259	+	0.997410i	$0.522915\pi$
$888$	0	0
$889$	−2.68629	−0.0900953
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−35.3137	−1.18173
$894$	0	0
$895$	−48.9706	−1.63691
$896$	0	0
$897$	0	0
$898$	0	0
$899$	52.4264	1.74852
$900$	0	0
$901$	8.48528	0.282686
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−41.4558	−1.37804
$906$	0	0
$907$	25.1716	0.835808	0.417904	−	0.908491i	$-0.362765\pi$
$907$	25.1716	0.835808	0.417904	+	0.908491i	$0.362765\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19.5980	0.649310	0.324655	−	0.945833i	$-0.394752\pi$
$911$	19.5980	0.649310	0.324655	+	0.945833i	$0.394752\pi$
$912$	0	0
$913$	42.6274	1.41076
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.6569	0.715172
$918$	0	0
$919$	−15.5563	−0.513157	−0.256578	−	0.966523i	$-0.582595\pi$
$919$	−15.5563	−0.513157	−0.256578	+	0.966523i	$0.582595\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.02944	0.231377
$924$	0	0
$925$	77.5980	2.55141
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−48.1421	−1.57949	−0.789746	−	0.613434i	$-0.789788\pi$
$929$	−48.1421	−1.57949	−0.789746	+	0.613434i	$0.789788\pi$
$930$	0	0
$931$	14.1421	0.463490
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−79.5980	−2.60313
$936$	0	0
$937$	36.6274	1.19657	0.598283	−	0.801285i	$-0.295850\pi$
$937$	36.6274	1.19657	0.598283	+	0.801285i	$0.295850\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	18.0416	0.588140	0.294070	−	0.955784i	$-0.404990\pi$
$941$	18.0416	0.588140	0.294070	+	0.955784i	$0.404990\pi$
$942$	0	0
$943$	12.2843	0.400031
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49.6569	1.61363	0.806815	−	0.590804i	$-0.201189\pi$
$947$	49.6569	1.61363	0.806815	+	0.590804i	$0.201189\pi$
$948$	0	0
$949$	9.37258	0.304247
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17.5147	−0.567357	−0.283679	−	0.958919i	$-0.591555\pi$
$953$	−17.5147	−0.567357	−0.283679	+	0.958919i	$0.591555\pi$
$954$	0	0
$955$	52.2843	1.69188
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.51472	0.113496
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	38.6274	1.24346
$966$	0	0
$967$	16.5269	0.531470	0.265735	−	0.964046i	$-0.414385\pi$
$967$	16.5269	0.531470	0.265735	+	0.964046i	$0.414385\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.8284	0.796782	0.398391	−	0.917216i	$-0.369569\pi$
$971$	24.8284	0.796782	0.398391	+	0.917216i	$0.369569\pi$
$972$	0	0
$973$	22.6274	0.725402
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.1716	1.12524	0.562619	−	0.826716i	$-0.309794\pi$
$977$	35.1716	1.12524	0.562619	+	0.826716i	$0.309794\pi$
$978$	0	0
$979$	25.6569	0.819997
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−37.6569	−1.20107	−0.600534	−	0.799600i	$-0.705045\pi$
$983$	−37.6569	−1.20107	−0.600534	+	0.799600i	$0.705045\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−41.8995	−1.33098	−0.665491	−	0.746406i	$-0.731778\pi$
$991$	−41.8995	−1.33098	−0.665491	+	0.746406i	$0.731778\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−40.1421	−1.27259
$996$	0	0
$997$	−26.9706	−0.854166	−0.427083	−	0.904212i	$-0.640459\pi$
$997$	−26.9706	−0.854166	−0.427083	+	0.904212i	$0.640459\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.a.b.1.1		2
3.2	odd	2		1536.2.a.f.1.2	yes	2
4.3	odd	2		4608.2.a.d.1.1		2
8.3	odd	2		4608.2.a.o.1.2		2
8.5	even	2		4608.2.a.q.1.2		2
12.11	even	2		1536.2.a.k.1.2	yes	2
16.3	odd	4		4608.2.d.h.2305.4		4
16.5	even	4		4608.2.d.f.2305.1		4
16.11	odd	4		4608.2.d.h.2305.1		4
16.13	even	4		4608.2.d.f.2305.4		4
24.5	odd	2		1536.2.a.h.1.1	yes	2
24.11	even	2		1536.2.a.a.1.1	✓	2
48.5	odd	4		1536.2.d.b.769.4		4
48.11	even	4		1536.2.d.e.769.2		4
48.29	odd	4		1536.2.d.b.769.1		4
48.35	even	4		1536.2.d.e.769.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.a.1.1	✓	2	24.11	even	2
1536.2.a.f.1.2	yes	2	3.2	odd	2
1536.2.a.h.1.1	yes	2	24.5	odd	2
1536.2.a.k.1.2	yes	2	12.11	even	2
1536.2.d.b.769.1		4	48.29	odd	4
1536.2.d.b.769.4		4	48.5	odd	4
1536.2.d.e.769.2		4	48.11	even	4
1536.2.d.e.769.3		4	48.35	even	4
4608.2.a.b.1.1		2	1.1	even	1	trivial
4608.2.a.d.1.1		2	4.3	odd	2
4608.2.a.o.1.2		2	8.3	odd	2
4608.2.a.q.1.2		2	8.5	even	2
4608.2.d.f.2305.1		4	16.5	even	4
4608.2.d.f.2305.4		4	16.13	even	4
4608.2.d.h.2305.1		4	16.11	odd	4
4608.2.d.h.2305.4		4	16.3	odd	4

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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.a (trivial)

\(f(q)\)	\(=\)	\(q-3.41421 q^{5} +1.41421 q^{7} +O(q^{10})\)	\(q-3.41421 q^{5} +1.41421 q^{7} -4.82843 q^{11} -0.828427 q^{13} -4.82843 q^{17} -2.82843 q^{19} +1.17157 q^{23} +6.65685 q^{25} -7.41421 q^{29} -7.07107 q^{31} -4.82843 q^{35} +11.6569 q^{37} +10.4853 q^{41} -6.82843 q^{43} +12.4853 q^{47} -5.00000 q^{49} -1.75736 q^{53} +16.4853 q^{55} +1.65685 q^{59} +0.343146 q^{61} +2.82843 q^{65} +5.65685 q^{67} -8.48528 q^{71} -11.3137 q^{73} -6.82843 q^{77} +17.4142 q^{79} -8.82843 q^{83} +16.4853 q^{85} -5.31371 q^{89} -1.17157 q^{91} +9.65685 q^{95} -7.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} - 4 q^{11} + 4 q^{13} - 4 q^{17} + 8 q^{23} + 2 q^{25} - 12 q^{29} - 4 q^{35} + 12 q^{37} + 4 q^{41} - 8 q^{43} + 8 q^{47} - 10 q^{49} - 12 q^{53} + 16 q^{55} - 8 q^{59} + 12 q^{61} - 8 q^{77} + 32 q^{79} - 12 q^{83} + 16 q^{85} + 12 q^{89} - 8 q^{91} + 8 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 4 * q^11 + 4 * q^13 - 4 * q^17 + 8 * q^23 + 2 * q^25 - 12 * q^29 - 4 * q^35 + 12 * q^37 + 4 * q^41 - 8 * q^43 + 8 * q^47 - 10 * q^49 - 12 * q^53 + 16 * q^55 - 8 * q^59 + 12 * q^61 - 8 * q^77 + 32 * q^79 - 12 * q^83 + 16 * q^85 + 12 * q^89 - 8 * q^91 + 8 * q^95 - 4 * q^97

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