LMFDB - Embedded newform 5648.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5648,2,Mod(1,5648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5648, 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("5648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5648 = 2^{4} \cdot 353 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5648.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:	$45.0995070616$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} - 14 x^{12} + 71 x^{11} + 47 x^{10} - 452 x^{9} + 101 x^{8} + 1251 x^{7} - 740 x^{6} + \cdots - 1 $ x^14 - 4x^13 - 14x^12 + 71x^11 + 47x^10 - 452x^9 + 101x^8 + 1251x^7 - 740x^6 - 1488x^5 + 1096x^4 + 600x^3 - 410x^2 - 42*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 353)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41172$ of defining polynomial
Character	$\chi$	$=$	5648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61874 q^{3} +3.04446 q^{5} -5.11343 q^{7} +3.85779 q^{9} +O(q^{10})$	$q-2.61874 q^{3} +3.04446 q^{5} -5.11343 q^{7} +3.85779 q^{9} +5.60579 q^{11} -4.54347 q^{13} -7.97265 q^{15} -0.673860 q^{17} +1.47041 q^{19} +13.3907 q^{21} +3.58108 q^{23} +4.26875 q^{25} -2.24633 q^{27} -3.94782 q^{29} -3.77024 q^{31} -14.6801 q^{33} -15.5677 q^{35} -8.44910 q^{37} +11.8982 q^{39} +7.82680 q^{41} -6.68167 q^{43} +11.7449 q^{45} +6.32711 q^{47} +19.1472 q^{49} +1.76466 q^{51} +1.77362 q^{53} +17.0666 q^{55} -3.85061 q^{57} +6.66991 q^{59} -4.03506 q^{61} -19.7266 q^{63} -13.8324 q^{65} +0.538541 q^{67} -9.37790 q^{69} +1.60224 q^{71} +10.8613 q^{73} -11.1787 q^{75} -28.6649 q^{77} -16.9961 q^{79} -5.69082 q^{81} +4.48784 q^{83} -2.05154 q^{85} +10.3383 q^{87} -8.48296 q^{89} +23.2327 q^{91} +9.87327 q^{93} +4.47659 q^{95} +1.99439 q^{97} +21.6260 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{3} - 4 q^{5} - 29 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q + 2 * q^3 - 4 * q^5 - 29 * q^7 + 14 * q^9	$ 14 q + 2 q^{3} - 4 q^{5} - 29 q^{7} + 14 q^{9} + 7 q^{11} + 2 q^{13} - 16 q^{15} + 8 q^{17} - 14 q^{19} + 6 q^{21} - 16 q^{23} + 24 q^{25} + 8 q^{27} - 9 q^{29} - 28 q^{31} - 10 q^{33} + 16 q^{35} + 8 q^{37} + 23 q^{39} + 14 q^{41} - 11 q^{43} - 37 q^{45} - 14 q^{47} + 39 q^{49} + 6 q^{51} - 27 q^{53} - 3 q^{55} + 7 q^{57} + 25 q^{59} - 3 q^{61} - 56 q^{63} - 41 q^{65} - 37 q^{67} - 32 q^{69} - 3 q^{71} + 14 q^{73} - 15 q^{75} - 43 q^{77} - 39 q^{79} - 26 q^{81} - 9 q^{83} - 12 q^{85} - 26 q^{87} - 12 q^{91} - 7 q^{93} + 24 q^{97} + 14 q^{99}+O(q^{100}) $ 14 * q + 2 * q^3 - 4 * q^5 - 29 * q^7 + 14 * q^9 + 7 * q^11 + 2 * q^13 - 16 * q^15 + 8 * q^17 - 14 * q^19 + 6 * q^21 - 16 * q^23 + 24 * q^25 + 8 * q^27 - 9 * q^29 - 28 * q^31 - 10 * q^33 + 16 * q^35 + 8 * q^37 + 23 * q^39 + 14 * q^41 - 11 * q^43 - 37 * q^45 - 14 * q^47 + 39 * q^49 + 6 * q^51 - 27 * q^53 - 3 * q^55 + 7 * q^57 + 25 * q^59 - 3 * q^61 - 56 * q^63 - 41 * q^65 - 37 * q^67 - 32 * q^69 - 3 * q^71 + 14 * q^73 - 15 * q^75 - 43 * q^77 - 39 * q^79 - 26 * q^81 - 9 * q^83 - 12 * q^85 - 26 * q^87 - 12 * q^91 - 7 * q^93 + 24 * q^97 + 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.61874	−1.51193	−0.755965	−	0.654612i	$-0.772832\pi$
$3$	−2.61874	−1.51193	−0.755965	+	0.654612i	$0.772832\pi$
$4$	0	0
$5$	3.04446	1.36152	0.680762	−	0.732504i	$-0.261649\pi$
$5$	3.04446	1.36152	0.680762	+	0.732504i	$0.261649\pi$
$6$	0	0
$7$	−5.11343	−1.93270	−0.966348	−	0.257238i	$-0.917188\pi$
$7$	−5.11343	−1.93270	−0.966348	+	0.257238i	$0.917188\pi$
$8$	0	0
$9$	3.85779	1.28593
$10$	0	0
$11$	5.60579	1.69021	0.845105	−	0.534600i	$-0.179538\pi$
$11$	5.60579	1.69021	0.845105	+	0.534600i	$0.179538\pi$
$12$	0	0
$13$	−4.54347	−1.26013	−0.630066	−	0.776541i	$-0.716972\pi$
$13$	−4.54347	−1.26013	−0.630066	+	0.776541i	$0.716972\pi$
$14$	0	0
$15$	−7.97265	−2.05853
$16$	0	0
$17$	−0.673860	−0.163435	−0.0817175	−	0.996656i	$-0.526041\pi$
$17$	−0.673860	−0.163435	−0.0817175	+	0.996656i	$0.526041\pi$
$18$	0	0
$19$	1.47041	0.337334	0.168667	−	0.985673i	$-0.446054\pi$
$19$	1.47041	0.337334	0.168667	+	0.985673i	$0.446054\pi$
$20$	0	0
$21$	13.3907	2.92210
$22$	0	0
$23$	3.58108	0.746706	0.373353	−	0.927689i	$-0.378208\pi$
$23$	3.58108	0.746706	0.373353	+	0.927689i	$0.378208\pi$
$24$	0	0
$25$	4.26875	0.853750
$26$	0	0
$27$	−2.24633	−0.432306
$28$	0	0
$29$	−3.94782	−0.733091	−0.366546	−	0.930400i	$-0.619460\pi$
$29$	−3.94782	−0.733091	−0.366546	+	0.930400i	$0.619460\pi$
$30$	0	0
$31$	−3.77024	−0.677155	−0.338577	−	0.940939i	$-0.609946\pi$
$31$	−3.77024	−0.677155	−0.338577	+	0.940939i	$0.609946\pi$
$32$	0	0
$33$	−14.6801	−2.55548
$34$	0	0
$35$	−15.5677	−2.63141
$36$	0	0
$37$	−8.44910	−1.38902	−0.694512	−	0.719481i	$-0.744380\pi$
$37$	−8.44910	−1.38902	−0.694512	+	0.719481i	$0.744380\pi$
$38$	0	0
$39$	11.8982	1.90523
$40$	0	0
$41$	7.82680	1.22234	0.611171	−	0.791499i	$-0.290699\pi$
$41$	7.82680	1.22234	0.611171	+	0.791499i	$0.290699\pi$
$42$	0	0
$43$	−6.68167	−1.01894	−0.509472	−	0.860487i	$-0.670159\pi$
$43$	−6.68167	−1.01894	−0.509472	+	0.860487i	$0.670159\pi$
$44$	0	0
$45$	11.7449	1.75083
$46$	0	0
$47$	6.32711	0.922904	0.461452	−	0.887165i	$-0.347329\pi$
$47$	6.32711	0.922904	0.461452	+	0.887165i	$0.347329\pi$
$48$	0	0
$49$	19.1472	2.73532
$50$	0	0
$51$	1.76466	0.247102
$52$	0	0
$53$	1.77362	0.243626	0.121813	−	0.992553i	$-0.461129\pi$
$53$	1.77362	0.243626	0.121813	+	0.992553i	$0.461129\pi$
$54$	0	0
$55$	17.0666	2.30126
$56$	0	0
$57$	−3.85061	−0.510025
$58$	0	0
$59$	6.66991	0.868348	0.434174	−	0.900829i	$-0.357040\pi$
$59$	6.66991	0.868348	0.434174	+	0.900829i	$0.357040\pi$
$60$	0	0
$61$	−4.03506	−0.516637	−0.258319	−	0.966060i	$-0.583168\pi$
$61$	−4.03506	−0.516637	−0.258319	+	0.966060i	$0.583168\pi$
$62$	0	0
$63$	−19.7266	−2.48531
$64$	0	0
$65$	−13.8324	−1.71570
$66$	0	0
$67$	0.538541	0.0657933	0.0328966	−	0.999459i	$-0.489527\pi$
$67$	0.538541	0.0657933	0.0328966	+	0.999459i	$0.489527\pi$
$68$	0	0
$69$	−9.37790	−1.12897
$70$	0	0
$71$	1.60224	0.190151	0.0950755	−	0.995470i	$-0.469691\pi$
$71$	1.60224	0.190151	0.0950755	+	0.995470i	$0.469691\pi$
$72$	0	0
$73$	10.8613	1.27122	0.635610	−	0.772011i	$-0.280749\pi$
$73$	10.8613	1.27122	0.635610	+	0.772011i	$0.280749\pi$
$74$	0	0
$75$	−11.1787	−1.29081
$76$	0	0
$77$	−28.6649	−3.26666
$78$	0	0
$79$	−16.9961	−1.91221	−0.956106	−	0.293022i	$-0.905339\pi$
$79$	−16.9961	−1.91221	−0.956106	+	0.293022i	$0.905339\pi$
$80$	0	0
$81$	−5.69082	−0.632314
$82$	0	0
$83$	4.48784	0.492605	0.246302	−	0.969193i	$-0.420784\pi$
$83$	4.48784	0.492605	0.246302	+	0.969193i	$0.420784\pi$
$84$	0	0
$85$	−2.05154	−0.222521
$86$	0	0
$87$	10.3383	1.10838
$88$	0	0
$89$	−8.48296	−0.899192	−0.449596	−	0.893232i	$-0.648432\pi$
$89$	−8.48296	−0.899192	−0.449596	+	0.893232i	$0.648432\pi$
$90$	0	0
$91$	23.2327	2.43545
$92$	0	0
$93$	9.87327	1.02381
$94$	0	0
$95$	4.47659	0.459289
$96$	0	0
$97$	1.99439	0.202500	0.101250	−	0.994861i	$-0.467716\pi$
$97$	1.99439	0.202500	0.101250	+	0.994861i	$0.467716\pi$
$98$	0	0
$99$	21.6260	2.17349
$100$	0	0
$101$	−4.10575	−0.408538	−0.204269	−	0.978915i	$-0.565482\pi$
$101$	−4.10575	−0.408538	−0.204269	+	0.978915i	$0.565482\pi$
$102$	0	0
$103$	−12.9859	−1.27954	−0.639771	−	0.768566i	$-0.720971\pi$
$103$	−12.9859	−1.27954	−0.639771	+	0.768566i	$0.720971\pi$
$104$	0	0
$105$	40.7676	3.97851
$106$	0	0
$107$	6.07667	0.587454	0.293727	−	0.955889i	$-0.405104\pi$
$107$	6.07667	0.587454	0.293727	+	0.955889i	$0.405104\pi$
$108$	0	0
$109$	10.0360	0.961279	0.480639	−	0.876918i	$-0.340405\pi$
$109$	10.0360	0.961279	0.480639	+	0.876918i	$0.340405\pi$
$110$	0	0
$111$	22.1260	2.10011
$112$	0	0
$113$	1.11671	0.105052	0.0525258	−	0.998620i	$-0.483273\pi$
$113$	1.11671	0.105052	0.0525258	+	0.998620i	$0.483273\pi$
$114$	0	0
$115$	10.9024	1.01666
$116$	0	0
$117$	−17.5278	−1.62044
$118$	0	0
$119$	3.44574	0.315870
$120$	0	0
$121$	20.4249	1.85681
$122$	0	0
$123$	−20.4964	−1.84809
$124$	0	0
$125$	−2.22627	−0.199123
$126$	0	0
$127$	−3.76595	−0.334174	−0.167087	−	0.985942i	$-0.553436\pi$
$127$	−3.76595	−0.334174	−0.167087	+	0.985942i	$0.553436\pi$
$128$	0	0
$129$	17.4975	1.54057
$130$	0	0
$131$	−16.6771	−1.45708	−0.728542	−	0.685001i	$-0.759802\pi$
$131$	−16.6771	−1.45708	−0.728542	+	0.685001i	$0.759802\pi$
$132$	0	0
$133$	−7.51882	−0.651964
$134$	0	0
$135$	−6.83886	−0.588596
$136$	0	0
$137$	8.32520	0.711270	0.355635	−	0.934625i	$-0.384265\pi$
$137$	8.32520	0.711270	0.355635	+	0.934625i	$0.384265\pi$
$138$	0	0
$139$	18.2291	1.54617	0.773085	−	0.634302i	$-0.218712\pi$
$139$	18.2291	1.54617	0.773085	+	0.634302i	$0.218712\pi$
$140$	0	0
$141$	−16.5690	−1.39536
$142$	0	0
$143$	−25.4698	−2.12989
$144$	0	0
$145$	−12.0190	−0.998122
$146$	0	0
$147$	−50.1415	−4.13560
$148$	0	0
$149$	1.00216	0.0821003	0.0410502	−	0.999157i	$-0.486930\pi$
$149$	1.00216	0.0821003	0.0410502	+	0.999157i	$0.486930\pi$
$150$	0	0
$151$	−1.13674	−0.0925062	−0.0462531	−	0.998930i	$-0.514728\pi$
$151$	−1.13674	−0.0925062	−0.0462531	+	0.998930i	$0.514728\pi$
$152$	0	0
$153$	−2.59961	−0.210166
$154$	0	0
$155$	−11.4783	−0.921963
$156$	0	0
$157$	−17.0879	−1.36376	−0.681881	−	0.731463i	$-0.738838\pi$
$157$	−17.0879	−1.36376	−0.681881	+	0.731463i	$0.738838\pi$
$158$	0	0
$159$	−4.64466	−0.368345
$160$	0	0
$161$	−18.3116	−1.44316
$162$	0	0
$163$	7.00795	0.548905	0.274453	−	0.961601i	$-0.411503\pi$
$163$	7.00795	0.548905	0.274453	+	0.961601i	$0.411503\pi$
$164$	0	0
$165$	−44.6930	−3.47935
$166$	0	0
$167$	−15.4504	−1.19559	−0.597795	−	0.801649i	$-0.703956\pi$
$167$	−15.4504	−1.19559	−0.597795	+	0.801649i	$0.703956\pi$
$168$	0	0
$169$	7.64314	0.587934
$170$	0	0
$171$	5.67252	0.433788
$172$	0	0
$173$	0.806790	0.0613391	0.0306696	−	0.999530i	$-0.490236\pi$
$173$	0.806790	0.0613391	0.0306696	+	0.999530i	$0.490236\pi$
$174$	0	0
$175$	−21.8280	−1.65004
$176$	0	0
$177$	−17.4667	−1.31288
$178$	0	0
$179$	−5.40379	−0.403898	−0.201949	−	0.979396i	$-0.564728\pi$
$179$	−5.40379	−0.403898	−0.201949	+	0.979396i	$0.564728\pi$
$180$	0	0
$181$	19.3386	1.43743	0.718713	−	0.695307i	$-0.244732\pi$
$181$	19.3386	1.43743	0.718713	+	0.695307i	$0.244732\pi$
$182$	0	0
$183$	10.5668	0.781119
$184$	0	0
$185$	−25.7230	−1.89119
$186$	0	0
$187$	−3.77752	−0.276240
$188$	0	0
$189$	11.4865	0.835516
$190$	0	0
$191$	10.2501	0.741670	0.370835	−	0.928699i	$-0.379072\pi$
$191$	10.2501	0.741670	0.370835	+	0.928699i	$0.379072\pi$
$192$	0	0
$193$	6.11743	0.440342	0.220171	−	0.975461i	$-0.429338\pi$
$193$	6.11743	0.440342	0.220171	+	0.975461i	$0.429338\pi$
$194$	0	0
$195$	36.2235	2.59402
$196$	0	0
$197$	−9.92518	−0.707140	−0.353570	−	0.935408i	$-0.615032\pi$
$197$	−9.92518	−0.707140	−0.353570	+	0.935408i	$0.615032\pi$
$198$	0	0
$199$	−2.18701	−0.155033	−0.0775165	−	0.996991i	$-0.524699\pi$
$199$	−2.18701	−0.155033	−0.0775165	+	0.996991i	$0.524699\pi$
$200$	0	0
$201$	−1.41030	−0.0994747
$202$	0	0
$203$	20.1869	1.41684
$204$	0	0
$205$	23.8284	1.66425
$206$	0	0
$207$	13.8150	0.960212
$208$	0	0
$209$	8.24279	0.570166
$210$	0	0
$211$	3.56421	0.245370	0.122685	−	0.992446i	$-0.460849\pi$
$211$	3.56421	0.245370	0.122685	+	0.992446i	$0.460849\pi$
$212$	0	0
$213$	−4.19585	−0.287495
$214$	0	0
$215$	−20.3421	−1.38732
$216$	0	0
$217$	19.2789	1.30873
$218$	0	0
$219$	−28.4429	−1.92199
$220$	0	0
$221$	3.06166	0.205950
$222$	0	0
$223$	−21.3448	−1.42935	−0.714675	−	0.699456i	$-0.753426\pi$
$223$	−21.3448	−1.42935	−0.714675	+	0.699456i	$0.753426\pi$
$224$	0	0
$225$	16.4679	1.09786
$226$	0	0
$227$	−10.9439	−0.726375	−0.363187	−	0.931716i	$-0.618311\pi$
$227$	−10.9439	−0.726375	−0.363187	+	0.931716i	$0.618311\pi$
$228$	0	0
$229$	2.66608	0.176179	0.0880897	−	0.996113i	$-0.471924\pi$
$229$	2.66608	0.176179	0.0880897	+	0.996113i	$0.471924\pi$
$230$	0	0
$231$	75.0658	4.93896
$232$	0	0
$233$	0.210570	0.0137949	0.00689744	−	0.999976i	$-0.497804\pi$
$233$	0.210570	0.0137949	0.00689744	+	0.999976i	$0.497804\pi$
$234$	0	0
$235$	19.2626	1.25656
$236$	0	0
$237$	44.5083	2.89113
$238$	0	0
$239$	−24.4705	−1.58286	−0.791432	−	0.611257i	$-0.790664\pi$
$239$	−24.4705	−1.58286	−0.791432	+	0.611257i	$0.790664\pi$
$240$	0	0
$241$	−24.2611	−1.56280	−0.781399	−	0.624032i	$-0.785493\pi$
$241$	−24.2611	−1.56280	−0.781399	+	0.624032i	$0.785493\pi$
$242$	0	0
$243$	21.6418	1.38832
$244$	0	0
$245$	58.2929	3.72420
$246$	0	0
$247$	−6.68075	−0.425086
$248$	0	0
$249$	−11.7525	−0.744783
$250$	0	0
$251$	−13.9457	−0.880242	−0.440121	−	0.897939i	$-0.645064\pi$
$251$	−13.9457	−0.880242	−0.440121	+	0.897939i	$0.645064\pi$
$252$	0	0
$253$	20.0748	1.26209
$254$	0	0
$255$	5.37245	0.336436
$256$	0	0
$257$	−18.5108	−1.15468	−0.577338	−	0.816506i	$-0.695908\pi$
$257$	−18.5108	−1.15468	−0.577338	+	0.816506i	$0.695908\pi$
$258$	0	0
$259$	43.2039	2.68456
$260$	0	0
$261$	−15.2299	−0.942704
$262$	0	0
$263$	19.3081	1.19059	0.595295	−	0.803507i	$-0.297035\pi$
$263$	19.3081	1.19059	0.595295	+	0.803507i	$0.297035\pi$
$264$	0	0
$265$	5.39973	0.331703
$266$	0	0
$267$	22.2147	1.35952
$268$	0	0
$269$	−2.22226	−0.135493	−0.0677467	−	0.997703i	$-0.521581\pi$
$269$	−2.22226	−0.135493	−0.0677467	+	0.997703i	$0.521581\pi$
$270$	0	0
$271$	19.8279	1.20446	0.602229	−	0.798323i	$-0.294279\pi$
$271$	19.8279	1.20446	0.602229	+	0.798323i	$0.294279\pi$
$272$	0	0
$273$	−60.8405	−3.68223
$274$	0	0
$275$	23.9297	1.44302
$276$	0	0
$277$	−18.5125	−1.11231	−0.556155	−	0.831078i	$-0.687724\pi$
$277$	−18.5125	−1.11231	−0.556155	+	0.831078i	$0.687724\pi$
$278$	0	0
$279$	−14.5448	−0.870774
$280$	0	0
$281$	−15.3494	−0.915671	−0.457835	−	0.889037i	$-0.651375\pi$
$281$	−15.3494	−0.915671	−0.457835	+	0.889037i	$0.651375\pi$
$282$	0	0
$283$	−24.9708	−1.48436	−0.742181	−	0.670200i	$-0.766209\pi$
$283$	−24.9708	−1.48436	−0.742181	+	0.670200i	$0.766209\pi$
$284$	0	0
$285$	−11.7230	−0.694412
$286$	0	0
$287$	−40.0218	−2.36241
$288$	0	0
$289$	−16.5459	−0.973289
$290$	0	0
$291$	−5.22280	−0.306166
$292$	0	0
$293$	25.3414	1.48046	0.740230	−	0.672354i	$-0.234717\pi$
$293$	25.3414	1.48046	0.740230	+	0.672354i	$0.234717\pi$
$294$	0	0
$295$	20.3063	1.18228
$296$	0	0
$297$	−12.5925	−0.730688
$298$	0	0
$299$	−16.2705	−0.940948
$300$	0	0
$301$	34.1663	1.96931
$302$	0	0
$303$	10.7519	0.617680
$304$	0	0
$305$	−12.2846	−0.703414
$306$	0	0
$307$	−1.31592	−0.0751035	−0.0375517	−	0.999295i	$-0.511956\pi$
$307$	−1.31592	−0.0751035	−0.0375517	+	0.999295i	$0.511956\pi$
$308$	0	0
$309$	34.0068	1.93458
$310$	0	0
$311$	−14.9157	−0.845792	−0.422896	−	0.906178i	$-0.638987\pi$
$311$	−14.9157	−0.845792	−0.422896	+	0.906178i	$0.638987\pi$
$312$	0	0
$313$	−0.556997	−0.0314833	−0.0157417	−	0.999876i	$-0.505011\pi$
$313$	−0.556997	−0.0314833	−0.0157417	+	0.999876i	$0.505011\pi$
$314$	0	0
$315$	−60.0568	−3.38381
$316$	0	0
$317$	−33.4087	−1.87642	−0.938209	−	0.346070i	$-0.887516\pi$
$317$	−33.4087	−1.87642	−0.938209	+	0.346070i	$0.887516\pi$
$318$	0	0
$319$	−22.1306	−1.23908
$320$	0	0
$321$	−15.9132	−0.888189
$322$	0	0
$323$	−0.990848	−0.0551322
$324$	0	0
$325$	−19.3949	−1.07584
$326$	0	0
$327$	−26.2818	−1.45339
$328$	0	0
$329$	−32.3532	−1.78369
$330$	0	0
$331$	6.34767	0.348900	0.174450	−	0.984666i	$-0.444185\pi$
$331$	6.34767	0.348900	0.174450	+	0.984666i	$0.444185\pi$
$332$	0	0
$333$	−32.5949	−1.78619
$334$	0	0
$335$	1.63957	0.0895791
$336$	0	0
$337$	−10.1315	−0.551899	−0.275950	−	0.961172i	$-0.588992\pi$
$337$	−10.1315	−0.551899	−0.275950	+	0.961172i	$0.588992\pi$
$338$	0	0
$339$	−2.92438	−0.158831
$340$	0	0
$341$	−21.1352	−1.14453
$342$	0	0
$343$	−62.1139	−3.35384
$344$	0	0
$345$	−28.5507	−1.53712
$346$	0	0
$347$	5.85162	0.314132	0.157066	−	0.987588i	$-0.449797\pi$
$347$	5.85162	0.314132	0.157066	+	0.987588i	$0.449797\pi$
$348$	0	0
$349$	15.0385	0.804994	0.402497	−	0.915421i	$-0.368142\pi$
$349$	15.0385	0.804994	0.402497	+	0.915421i	$0.368142\pi$
$350$	0	0
$351$	10.2061	0.544763
$352$	0	0
$353$	1.00000	0.0532246
$353$	1.00000	0.0532246
$354$	0	0
$355$	4.87796	0.258895
$356$	0	0
$357$	−9.02349	−0.477574
$358$	0	0
$359$	−13.9758	−0.737614	−0.368807	−	0.929506i	$-0.620234\pi$
$359$	−13.9758	−0.737614	−0.368807	+	0.929506i	$0.620234\pi$
$360$	0	0
$361$	−16.8379	−0.886206
$362$	0	0
$363$	−53.4875	−2.80737
$364$	0	0
$365$	33.0668	1.73080
$366$	0	0
$367$	33.8149	1.76512	0.882562	−	0.470197i	$-0.155817\pi$
$367$	33.8149	1.76512	0.882562	+	0.470197i	$0.155817\pi$
$368$	0	0
$369$	30.1942	1.57185
$370$	0	0
$371$	−9.06931	−0.470855
$372$	0	0
$373$	19.5910	1.01438	0.507192	−	0.861833i	$-0.330684\pi$
$373$	19.5910	1.01438	0.507192	+	0.861833i	$0.330684\pi$
$374$	0	0
$375$	5.83001	0.301060
$376$	0	0
$377$	17.9368	0.923792
$378$	0	0
$379$	−12.4984	−0.641998	−0.320999	−	0.947079i	$-0.604019\pi$
$379$	−12.4984	−0.641998	−0.320999	+	0.947079i	$0.604019\pi$
$380$	0	0
$381$	9.86204	0.505247
$382$	0	0
$383$	11.7008	0.597881	0.298940	−	0.954272i	$-0.403367\pi$
$383$	11.7008	0.597881	0.298940	+	0.954272i	$0.403367\pi$
$384$	0	0
$385$	−87.2691	−4.44764
$386$	0	0
$387$	−25.7765	−1.31029
$388$	0	0
$389$	−29.4539	−1.49337	−0.746687	−	0.665175i	$-0.768357\pi$
$389$	−29.4539	−1.49337	−0.746687	+	0.665175i	$0.768357\pi$
$390$	0	0
$391$	−2.41314	−0.122038
$392$	0	0
$393$	43.6729	2.20301
$394$	0	0
$395$	−51.7440	−2.60352
$396$	0	0
$397$	−17.6528	−0.885969	−0.442985	−	0.896529i	$-0.646080\pi$
$397$	−17.6528	−0.885969	−0.442985	+	0.896529i	$0.646080\pi$
$398$	0	0
$399$	19.6898	0.985724
$400$	0	0
$401$	−11.2075	−0.559676	−0.279838	−	0.960047i	$-0.590281\pi$
$401$	−11.2075	−0.559676	−0.279838	+	0.960047i	$0.590281\pi$
$402$	0	0
$403$	17.1300	0.853305
$404$	0	0
$405$	−17.3255	−0.860911
$406$	0	0
$407$	−47.3639	−2.34774
$408$	0	0
$409$	−16.2593	−0.803971	−0.401986	−	0.915646i	$-0.631680\pi$
$409$	−16.2593	−0.803971	−0.401986	+	0.915646i	$0.631680\pi$
$410$	0	0
$411$	−21.8015	−1.07539
$412$	0	0
$413$	−34.1061	−1.67825
$414$	0	0
$415$	13.6631	0.670693
$416$	0	0
$417$	−47.7372	−2.33770
$418$	0	0
$419$	−12.3354	−0.602623	−0.301312	−	0.953526i	$-0.597424\pi$
$419$	−12.3354	−0.602623	−0.301312	+	0.953526i	$0.597424\pi$
$420$	0	0
$421$	−4.11586	−0.200595	−0.100297	−	0.994958i	$-0.531979\pi$
$421$	−4.11586	−0.200595	−0.100297	+	0.994958i	$0.531979\pi$
$422$	0	0
$423$	24.4087	1.18679
$424$	0	0
$425$	−2.87654	−0.139533
$426$	0	0
$427$	20.6330	0.998502
$428$	0	0
$429$	66.6987	3.22024
$430$	0	0
$431$	−24.2345	−1.16734	−0.583668	−	0.811993i	$-0.698383\pi$
$431$	−24.2345	−1.16734	−0.583668	+	0.811993i	$0.698383\pi$
$432$	0	0
$433$	−36.8169	−1.76931	−0.884653	−	0.466249i	$-0.845605\pi$
$433$	−36.8169	−1.76931	−0.884653	+	0.466249i	$0.845605\pi$
$434$	0	0
$435$	31.4746	1.50909
$436$	0	0
$437$	5.26563	0.251889
$438$	0	0
$439$	−13.2117	−0.630561	−0.315281	−	0.948999i	$-0.602099\pi$
$439$	−13.2117	−0.630561	−0.315281	+	0.948999i	$0.602099\pi$
$440$	0	0
$441$	73.8659	3.51742
$442$	0	0
$443$	−18.2696	−0.868016	−0.434008	−	0.900909i	$-0.642901\pi$
$443$	−18.2696	−0.868016	−0.434008	+	0.900909i	$0.642901\pi$
$444$	0	0
$445$	−25.8261	−1.22427
$446$	0	0
$447$	−2.62440	−0.124130
$448$	0	0
$449$	−10.0609	−0.474804	−0.237402	−	0.971411i	$-0.576296\pi$
$449$	−10.0609	−0.474804	−0.237402	+	0.971411i	$0.576296\pi$
$450$	0	0
$451$	43.8755	2.06601
$452$	0	0
$453$	2.97681	0.139863
$454$	0	0
$455$	70.7312	3.31593
$456$	0	0
$457$	−14.6259	−0.684171	−0.342085	−	0.939669i	$-0.611133\pi$
$457$	−14.6259	−0.684171	−0.342085	+	0.939669i	$0.611133\pi$
$458$	0	0
$459$	1.51371	0.0706540
$460$	0	0
$461$	37.3827	1.74109	0.870543	−	0.492092i	$-0.163768\pi$
$461$	37.3827	1.74109	0.870543	+	0.492092i	$0.163768\pi$
$462$	0	0
$463$	−22.0440	−1.02447	−0.512235	−	0.858845i	$-0.671182\pi$
$463$	−22.0440	−1.02447	−0.512235	+	0.858845i	$0.671182\pi$
$464$	0	0
$465$	30.0588	1.39394
$466$	0	0
$467$	−8.10108	−0.374873	−0.187437	−	0.982277i	$-0.560018\pi$
$467$	−8.10108	−0.374873	−0.187437	+	0.982277i	$0.560018\pi$
$468$	0	0
$469$	−2.75379	−0.127158
$470$	0	0
$471$	44.7487	2.06191
$472$	0	0
$473$	−37.4561	−1.72223
$474$	0	0
$475$	6.27679	0.287999
$476$	0	0
$477$	6.84227	0.313286
$478$	0	0
$479$	24.8904	1.13727	0.568635	−	0.822590i	$-0.307472\pi$
$479$	24.8904	1.13727	0.568635	+	0.822590i	$0.307472\pi$
$480$	0	0
$481$	38.3883	1.75035
$482$	0	0
$483$	47.9533	2.18195
$484$	0	0
$485$	6.07186	0.275709
$486$	0	0
$487$	−19.8035	−0.897384	−0.448692	−	0.893686i	$-0.648110\pi$
$487$	−19.8035	−0.897384	−0.448692	+	0.893686i	$0.648110\pi$
$488$	0	0
$489$	−18.3520	−0.829906
$490$	0	0
$491$	−11.2839	−0.509234	−0.254617	−	0.967042i	$-0.581949\pi$
$491$	−11.2839	−0.509234	−0.254617	+	0.967042i	$0.581949\pi$
$492$	0	0
$493$	2.66028	0.119813
$494$	0	0
$495$	65.8395	2.95926
$496$	0	0
$497$	−8.19295	−0.367504
$498$	0	0
$499$	4.26523	0.190938	0.0954689	−	0.995432i	$-0.469565\pi$
$499$	4.26523	0.190938	0.0954689	+	0.995432i	$0.469565\pi$
$500$	0	0
$501$	40.4606	1.80765
$502$	0	0
$503$	−37.7929	−1.68510	−0.842551	−	0.538617i	$-0.818947\pi$
$503$	−37.7929	−1.68510	−0.842551	+	0.538617i	$0.818947\pi$
$504$	0	0
$505$	−12.4998	−0.556234
$506$	0	0
$507$	−20.0154	−0.888915
$508$	0	0
$509$	28.0857	1.24488	0.622438	−	0.782669i	$-0.286142\pi$
$509$	28.0857	1.24488	0.622438	+	0.782669i	$0.286142\pi$
$510$	0	0
$511$	−55.5385	−2.45688
$512$	0	0
$513$	−3.30301	−0.145832
$514$	0	0
$515$	−39.5352	−1.74213
$516$	0	0
$517$	35.4685	1.55990
$518$	0	0
$519$	−2.11277	−0.0927404
$520$	0	0
$521$	10.1363	0.444080	0.222040	−	0.975038i	$-0.428728\pi$
$521$	10.1363	0.444080	0.222040	+	0.975038i	$0.428728\pi$
$522$	0	0
$523$	−41.3038	−1.80609	−0.903045	−	0.429547i	$-0.858673\pi$
$523$	−41.3038	−1.80609	−0.903045	+	0.429547i	$0.858673\pi$
$524$	0	0
$525$	57.1617	2.49474
$526$	0	0
$527$	2.54061	0.110671
$528$	0	0
$529$	−10.1759	−0.442430
$530$	0	0
$531$	25.7311	1.11663
$532$	0	0
$533$	−35.5609	−1.54031
$534$	0	0
$535$	18.5002	0.799833
$536$	0	0
$537$	14.1511	0.610665
$538$	0	0
$539$	107.335	4.62326
$540$	0	0
$541$	−6.76395	−0.290805	−0.145402	−	0.989373i	$-0.546448\pi$
$541$	−6.76395	−0.290805	−0.145402	+	0.989373i	$0.546448\pi$
$542$	0	0
$543$	−50.6427	−2.17329
$544$	0	0
$545$	30.5544	1.30880
$546$	0	0
$547$	−12.9228	−0.552541	−0.276271	−	0.961080i	$-0.589099\pi$
$547$	−12.9228	−0.552541	−0.276271	+	0.961080i	$0.589099\pi$
$548$	0	0
$549$	−15.5664	−0.664359
$550$	0	0
$551$	−5.80489	−0.247297
$552$	0	0
$553$	86.9085	3.69572
$554$	0	0
$555$	67.3617	2.85935
$556$	0	0
$557$	−2.96140	−0.125478	−0.0627392	−	0.998030i	$-0.519984\pi$
$557$	−2.96140	−0.125478	−0.0627392	+	0.998030i	$0.519984\pi$
$558$	0	0
$559$	30.3580	1.28401
$560$	0	0
$561$	9.89234	0.417655
$562$	0	0
$563$	−31.2075	−1.31524	−0.657619	−	0.753351i	$-0.728436\pi$
$563$	−31.2075	−1.31524	−0.657619	+	0.753351i	$0.728436\pi$
$564$	0	0
$565$	3.39979	0.143030
$566$	0	0
$567$	29.0997	1.22207
$568$	0	0
$569$	−40.0483	−1.67891	−0.839456	−	0.543428i	$-0.817126\pi$
$569$	−40.0483	−1.67891	−0.839456	+	0.543428i	$0.817126\pi$
$570$	0	0
$571$	19.9104	0.833223	0.416612	−	0.909085i	$-0.363217\pi$
$571$	19.9104	0.833223	0.416612	+	0.909085i	$0.363217\pi$
$572$	0	0
$573$	−26.8423	−1.12135
$574$	0	0
$575$	15.2867	0.637500
$576$	0	0
$577$	7.62218	0.317315	0.158658	−	0.987334i	$-0.449283\pi$
$577$	7.62218	0.317315	0.158658	+	0.987334i	$0.449283\pi$
$578$	0	0
$579$	−16.0199	−0.665766
$580$	0	0
$581$	−22.9483	−0.952055
$582$	0	0
$583$	9.94258	0.411779
$584$	0	0
$585$	−53.3626	−2.20627
$586$	0	0
$587$	−6.63324	−0.273783	−0.136891	−	0.990586i	$-0.543711\pi$
$587$	−6.63324	−0.273783	−0.136891	+	0.990586i	$0.543711\pi$
$588$	0	0
$589$	−5.54378	−0.228427
$590$	0	0
$591$	25.9914	1.06915
$592$	0	0
$593$	3.29488	0.135304	0.0676522	−	0.997709i	$-0.478449\pi$
$593$	3.29488	0.135304	0.0676522	+	0.997709i	$0.478449\pi$
$594$	0	0
$595$	10.4904	0.430065
$596$	0	0
$597$	5.72721	0.234399
$598$	0	0
$599$	−15.0380	−0.614434	−0.307217	−	0.951639i	$-0.599398\pi$
$599$	−15.0380	−0.614434	−0.307217	+	0.951639i	$0.599398\pi$
$600$	0	0
$601$	−13.3474	−0.544450	−0.272225	−	0.962234i	$-0.587759\pi$
$601$	−13.3474	−0.544450	−0.272225	+	0.962234i	$0.587759\pi$
$602$	0	0
$603$	2.07758	0.0846055
$604$	0	0
$605$	62.1829	2.52810
$606$	0	0
$607$	−3.16195	−0.128340	−0.0641698	−	0.997939i	$-0.520440\pi$
$607$	−3.16195	−0.128340	−0.0641698	+	0.997939i	$0.520440\pi$
$608$	0	0
$609$	−52.8642	−2.14217
$610$	0	0
$611$	−28.7470	−1.16298
$612$	0	0
$613$	−24.1355	−0.974823	−0.487411	−	0.873173i	$-0.662059\pi$
$613$	−24.1355	−0.974823	−0.487411	+	0.873173i	$0.662059\pi$
$614$	0	0
$615$	−62.4004	−2.51623
$616$	0	0
$617$	12.6797	0.510465	0.255233	−	0.966880i	$-0.417848\pi$
$617$	12.6797	0.510465	0.255233	+	0.966880i	$0.417848\pi$
$618$	0	0
$619$	0.839105	0.0337265	0.0168632	−	0.999858i	$-0.494632\pi$
$619$	0.839105	0.0337265	0.0168632	+	0.999858i	$0.494632\pi$
$620$	0	0
$621$	−8.04427	−0.322806
$622$	0	0
$623$	43.3771	1.73787
$624$	0	0
$625$	−28.1215	−1.12486
$626$	0	0
$627$	−21.5857	−0.862050
$628$	0	0
$629$	5.69351	0.227015
$630$	0	0
$631$	21.6456	0.861698	0.430849	−	0.902424i	$-0.358214\pi$
$631$	21.6456	0.861698	0.430849	+	0.902424i	$0.358214\pi$
$632$	0	0
$633$	−9.33373	−0.370983
$634$	0	0
$635$	−11.4653	−0.454986
$636$	0	0
$637$	−86.9948	−3.44686
$638$	0	0
$639$	6.18111	0.244521
$640$	0	0
$641$	17.8646	0.705608	0.352804	−	0.935697i	$-0.385228\pi$
$641$	17.8646	0.705608	0.352804	+	0.935697i	$0.385228\pi$
$642$	0	0
$643$	26.4832	1.04440	0.522198	−	0.852824i	$-0.325112\pi$
$643$	26.4832	1.04440	0.522198	+	0.852824i	$0.325112\pi$
$644$	0	0
$645$	53.2706	2.09753
$646$	0	0
$647$	14.5194	0.570816	0.285408	−	0.958406i	$-0.407871\pi$
$647$	14.5194	0.570816	0.285408	+	0.958406i	$0.407871\pi$
$648$	0	0
$649$	37.3901	1.46769
$650$	0	0
$651$	−50.4863	−1.97871
$652$	0	0
$653$	−13.3955	−0.524206	−0.262103	−	0.965040i	$-0.584416\pi$
$653$	−13.3955	−0.524206	−0.262103	+	0.965040i	$0.584416\pi$
$654$	0	0
$655$	−50.7728	−1.98386
$656$	0	0
$657$	41.9006	1.63470
$658$	0	0
$659$	−4.84784	−0.188845	−0.0944225	−	0.995532i	$-0.530100\pi$
$659$	−4.84784	−0.188845	−0.0944225	+	0.995532i	$0.530100\pi$
$660$	0	0
$661$	27.5713	1.07240	0.536199	−	0.844092i	$-0.319860\pi$
$661$	27.5713	1.07240	0.536199	+	0.844092i	$0.319860\pi$
$662$	0	0
$663$	−8.01770	−0.311382
$664$	0	0
$665$	−22.8908	−0.887666
$666$	0	0
$667$	−14.1374	−0.547404
$668$	0	0
$669$	55.8963	2.16108
$670$	0	0
$671$	−22.6197	−0.873225
$672$	0	0
$673$	43.9898	1.69568	0.847840	−	0.530252i	$-0.177903\pi$
$673$	43.9898	1.69568	0.847840	+	0.530252i	$0.177903\pi$
$674$	0	0
$675$	−9.58901	−0.369081
$676$	0	0
$677$	−42.1977	−1.62179	−0.810895	−	0.585191i	$-0.801019\pi$
$677$	−42.1977	−1.62179	−0.810895	+	0.585191i	$0.801019\pi$
$678$	0	0
$679$	−10.1982	−0.391371
$680$	0	0
$681$	28.6593	1.09823
$682$	0	0
$683$	0.993542	0.0380168	0.0190084	−	0.999819i	$-0.493949\pi$
$683$	0.993542	0.0380168	0.0190084	+	0.999819i	$0.493949\pi$
$684$	0	0
$685$	25.3458	0.968411
$686$	0	0
$687$	−6.98176	−0.266371
$688$	0	0
$689$	−8.05842	−0.307001
$690$	0	0
$691$	−16.5834	−0.630861	−0.315431	−	0.948949i	$-0.602149\pi$
$691$	−16.5834	−0.630861	−0.315431	+	0.948949i	$0.602149\pi$
$692$	0	0
$693$	−110.583	−4.20070
$694$	0	0
$695$	55.4978	2.10515
$696$	0	0
$697$	−5.27417	−0.199773
$698$	0	0
$699$	−0.551427	−0.0208569
$700$	0	0
$701$	−32.7630	−1.23744	−0.618721	−	0.785611i	$-0.712349\pi$
$701$	−32.7630	−1.23744	−0.618721	+	0.785611i	$0.712349\pi$
$702$	0	0
$703$	−12.4236	−0.468565
$704$	0	0
$705$	−50.4438	−1.89982
$706$	0	0
$707$	20.9945	0.789580
$708$	0	0
$709$	37.9061	1.42359	0.711797	−	0.702385i	$-0.247882\pi$
$709$	37.9061	1.42359	0.711797	+	0.702385i	$0.247882\pi$
$710$	0	0
$711$	−65.5674	−2.45897
$712$	0	0
$713$	−13.5015	−0.505635
$714$	0	0
$715$	−77.5418	−2.89990
$716$	0	0
$717$	64.0818	2.39318
$718$	0	0
$719$	28.7241	1.07123	0.535614	−	0.844463i	$-0.320080\pi$
$719$	28.7241	1.07123	0.535614	+	0.844463i	$0.320080\pi$
$720$	0	0
$721$	66.4027	2.47297
$722$	0	0
$723$	63.5336	2.36284
$724$	0	0
$725$	−16.8522	−0.625876
$726$	0	0
$727$	−15.4970	−0.574752	−0.287376	−	0.957818i	$-0.592783\pi$
$727$	−15.4970	−0.574752	−0.287376	+	0.957818i	$0.592783\pi$
$728$	0	0
$729$	−39.6016	−1.46673
$730$	0	0
$731$	4.50251	0.166531
$732$	0	0
$733$	19.2671	0.711646	0.355823	−	0.934553i	$-0.384201\pi$
$733$	19.2671	0.711646	0.355823	+	0.934553i	$0.384201\pi$
$734$	0	0
$735$	−152.654	−5.63073
$736$	0	0
$737$	3.01895	0.111204
$738$	0	0
$739$	4.55240	0.167463	0.0837313	−	0.996488i	$-0.473316\pi$
$739$	4.55240	0.167463	0.0837313	+	0.996488i	$0.473316\pi$
$740$	0	0
$741$	17.4951	0.642700
$742$	0	0
$743$	−0.520108	−0.0190809	−0.00954045	−	0.999954i	$-0.503037\pi$
$743$	−0.520108	−0.0190809	−0.00954045	+	0.999954i	$0.503037\pi$
$744$	0	0
$745$	3.05104	0.111782
$746$	0	0
$747$	17.3132	0.633455
$748$	0	0
$749$	−31.0727	−1.13537
$750$	0	0
$751$	44.3885	1.61976	0.809879	−	0.586597i	$-0.199533\pi$
$751$	44.3885	1.61976	0.809879	+	0.586597i	$0.199533\pi$
$752$	0	0
$753$	36.5200	1.33086
$754$	0	0
$755$	−3.46075	−0.125950
$756$	0	0
$757$	25.5363	0.928133	0.464066	−	0.885800i	$-0.346390\pi$
$757$	25.5363	0.928133	0.464066	+	0.885800i	$0.346390\pi$
$758$	0	0
$759$	−52.5706	−1.90819
$760$	0	0
$761$	50.7385	1.83927	0.919634	−	0.392776i	$-0.128485\pi$
$761$	50.7385	1.83927	0.919634	+	0.392776i	$0.128485\pi$
$762$	0	0
$763$	−51.3186	−1.85786
$764$	0	0
$765$	−7.91442	−0.286146
$766$	0	0
$767$	−30.3045	−1.09423
$768$	0	0
$769$	3.90315	0.140751	0.0703756	−	0.997521i	$-0.477580\pi$
$769$	3.90315	0.140751	0.0703756	+	0.997521i	$0.477580\pi$
$770$	0	0
$771$	48.4751	1.74579
$772$	0	0
$773$	45.3052	1.62951	0.814757	−	0.579803i	$-0.196870\pi$
$773$	45.3052	1.62951	0.814757	+	0.579803i	$0.196870\pi$
$774$	0	0
$775$	−16.0942	−0.578121
$776$	0	0
$777$	−113.140	−4.05887
$778$	0	0
$779$	11.5086	0.412338
$780$	0	0
$781$	8.98183	0.321395
$782$	0	0
$783$	8.86809	0.316920
$784$	0	0
$785$	−52.0234	−1.85680
$786$	0	0
$787$	−21.2385	−0.757069	−0.378534	−	0.925587i	$-0.623572\pi$
$787$	−21.2385	−0.757069	−0.378534	+	0.925587i	$0.623572\pi$
$788$	0	0
$789$	−50.5629	−1.80009
$790$	0	0
$791$	−5.71024	−0.203033
$792$	0	0
$793$	18.3332	0.651031
$794$	0	0
$795$	−14.1405	−0.501511
$796$	0	0
$797$	37.9180	1.34312	0.671562	−	0.740949i	$-0.265624\pi$
$797$	37.9180	1.34312	0.671562	+	0.740949i	$0.265624\pi$
$798$	0	0
$799$	−4.26359	−0.150835
$800$	0	0
$801$	−32.7255	−1.15630
$802$	0	0
$803$	60.8862	2.14863
$804$	0	0
$805$	−55.7490	−1.96489
$806$	0	0
$807$	5.81951	0.204856
$808$	0	0
$809$	−31.3642	−1.10271	−0.551353	−	0.834272i	$-0.685888\pi$
$809$	−31.3642	−1.10271	−0.551353	+	0.834272i	$0.685888\pi$
$810$	0	0
$811$	−8.51099	−0.298861	−0.149431	−	0.988772i	$-0.547744\pi$
$811$	−8.51099	−0.298861	−0.149431	+	0.988772i	$0.547744\pi$
$812$	0	0
$813$	−51.9240	−1.82106
$814$	0	0
$815$	21.3355	0.747348
$816$	0	0
$817$	−9.82476	−0.343725
$818$	0	0
$819$	89.6271	3.13182
$820$	0	0
$821$	25.3744	0.885571	0.442786	−	0.896627i	$-0.353990\pi$
$821$	25.3744	0.885571	0.442786	+	0.896627i	$0.353990\pi$
$822$	0	0
$823$	40.2080	1.40156	0.700782	−	0.713376i	$-0.252835\pi$
$823$	40.2080	1.40156	0.700782	+	0.713376i	$0.252835\pi$
$824$	0	0
$825$	−62.6657	−2.18174
$826$	0	0
$827$	35.4154	1.23151	0.615757	−	0.787936i	$-0.288850\pi$
$827$	35.4154	1.23151	0.615757	+	0.787936i	$0.288850\pi$
$828$	0	0
$829$	−31.3464	−1.08870	−0.544352	−	0.838857i	$-0.683225\pi$
$829$	−31.3464	−1.08870	−0.544352	+	0.838857i	$0.683225\pi$
$830$	0	0
$831$	48.4795	1.68174
$832$	0	0
$833$	−12.9025	−0.447046
$834$	0	0
$835$	−47.0382	−1.62783
$836$	0	0
$837$	8.46919	0.292738
$838$	0	0
$839$	5.45174	0.188215	0.0941076	−	0.995562i	$-0.470000\pi$
$839$	5.45174	0.188215	0.0941076	+	0.995562i	$0.470000\pi$
$840$	0	0
$841$	−13.4147	−0.462577
$842$	0	0
$843$	40.1962	1.38443
$844$	0	0
$845$	23.2693	0.800487
$846$	0	0
$847$	−104.442	−3.58865
$848$	0	0
$849$	65.3921	2.24425
$850$	0	0
$851$	−30.2569	−1.03719
$852$	0	0
$853$	−13.5404	−0.463614	−0.231807	−	0.972762i	$-0.574464\pi$
$853$	−13.5404	−0.463614	−0.231807	+	0.972762i	$0.574464\pi$
$854$	0	0
$855$	17.2698	0.590613
$856$	0	0
$857$	−47.6033	−1.62610	−0.813049	−	0.582196i	$-0.802194\pi$
$857$	−47.6033	−1.62610	−0.813049	+	0.582196i	$0.802194\pi$
$858$	0	0
$859$	20.5547	0.701318	0.350659	−	0.936503i	$-0.385958\pi$
$859$	20.5547	0.701318	0.350659	+	0.936503i	$0.385958\pi$
$860$	0	0
$861$	104.807	3.57180
$862$	0	0
$863$	30.6136	1.04210	0.521049	−	0.853527i	$-0.325541\pi$
$863$	30.6136	1.04210	0.521049	+	0.853527i	$0.325541\pi$
$864$	0	0
$865$	2.45624	0.0835147
$866$	0	0
$867$	43.3294	1.47154
$868$	0	0
$869$	−95.2767	−3.23204
$870$	0	0
$871$	−2.44685	−0.0829082
$872$	0	0
$873$	7.69396	0.260401
$874$	0	0
$875$	11.3839	0.384845
$876$	0	0
$877$	42.5599	1.43715	0.718574	−	0.695451i	$-0.244795\pi$
$877$	42.5599	1.43715	0.718574	+	0.695451i	$0.244795\pi$
$878$	0	0
$879$	−66.3625	−2.23835
$880$	0	0
$881$	12.9370	0.435860	0.217930	−	0.975964i	$-0.430070\pi$
$881$	12.9370	0.435860	0.217930	+	0.975964i	$0.430070\pi$
$882$	0	0
$883$	55.0299	1.85190	0.925951	−	0.377643i	$-0.123265\pi$
$883$	55.0299	1.85190	0.925951	+	0.377643i	$0.123265\pi$
$884$	0	0
$885$	−53.1768	−1.78752
$886$	0	0
$887$	12.3418	0.414395	0.207198	−	0.978299i	$-0.433566\pi$
$887$	12.3418	0.414395	0.207198	+	0.978299i	$0.433566\pi$
$888$	0	0
$889$	19.2569	0.645857
$890$	0	0
$891$	−31.9016	−1.06874
$892$	0	0
$893$	9.30341	0.311327
$894$	0	0
$895$	−16.4516	−0.549917
$896$	0	0
$897$	42.6082	1.42265
$898$	0	0
$899$	14.8842	0.496416
$900$	0	0
$901$	−1.19517	−0.0398170
$902$	0	0
$903$	−89.4725	−2.97746
$904$	0	0
$905$	58.8756	1.95709
$906$	0	0
$907$	−7.11084	−0.236111	−0.118056	−	0.993007i	$-0.537666\pi$
$907$	−7.11084	−0.236111	−0.118056	+	0.993007i	$0.537666\pi$
$908$	0	0
$909$	−15.8391	−0.525351
$910$	0	0
$911$	−18.8259	−0.623731	−0.311865	−	0.950126i	$-0.600954\pi$
$911$	−18.8259	−0.623731	−0.311865	+	0.950126i	$0.600954\pi$
$912$	0	0
$913$	25.1579	0.832605
$914$	0	0
$915$	32.1702	1.06351
$916$	0	0
$917$	85.2772	2.81610
$918$	0	0
$919$	6.66602	0.219892	0.109946	−	0.993938i	$-0.464932\pi$
$919$	6.66602	0.219892	0.109946	+	0.993938i	$0.464932\pi$
$920$	0	0
$921$	3.44605	0.113551
$922$	0	0
$923$	−7.27973	−0.239615
$924$	0	0
$925$	−36.0671	−1.18588
$926$	0	0
$927$	−50.0970	−1.64540
$928$	0	0
$929$	27.0434	0.887265	0.443633	−	0.896209i	$-0.353690\pi$
$929$	27.0434	0.887265	0.443633	+	0.896209i	$0.353690\pi$
$930$	0	0
$931$	28.1542	0.922715
$932$	0	0
$933$	39.0603	1.27878
$934$	0	0
$935$	−11.5005	−0.376107
$936$	0	0
$937$	−10.2554	−0.335029	−0.167515	−	0.985870i	$-0.553574\pi$
$937$	−10.2554	−0.335029	−0.167515	+	0.985870i	$0.553574\pi$
$938$	0	0
$939$	1.45863	0.0476005
$940$	0	0
$941$	−25.7686	−0.840031	−0.420015	−	0.907517i	$-0.637975\pi$
$941$	−25.7686	−0.840031	−0.420015	+	0.907517i	$0.637975\pi$
$942$	0	0
$943$	28.0284	0.912730
$944$	0	0
$945$	34.9701	1.13758
$946$	0	0
$947$	12.4435	0.404360	0.202180	−	0.979348i	$-0.435197\pi$
$947$	12.4435	0.404360	0.202180	+	0.979348i	$0.435197\pi$
$948$	0	0
$949$	−49.3480	−1.60190
$950$	0	0
$951$	87.4885	2.83701
$952$	0	0
$953$	19.3343	0.626301	0.313150	−	0.949704i	$-0.398616\pi$
$953$	19.3343	0.626301	0.313150	+	0.949704i	$0.398616\pi$
$954$	0	0
$955$	31.2060	1.00980
$956$	0	0
$957$	57.9544	1.87340
$958$	0	0
$959$	−42.5704	−1.37467
$960$	0	0
$961$	−16.7853	−0.541462
$962$	0	0
$963$	23.4425	0.755425
$964$	0	0
$965$	18.6243	0.599537
$966$	0	0
$967$	−54.8580	−1.76411	−0.882057	−	0.471142i	$-0.843842\pi$
$967$	−54.8580	−1.76411	−0.882057	+	0.471142i	$0.843842\pi$
$968$	0	0
$969$	2.59477	0.0833560
$970$	0	0
$971$	21.2781	0.682846	0.341423	−	0.939910i	$-0.389091\pi$
$971$	21.2781	0.682846	0.341423	+	0.939910i	$0.389091\pi$
$972$	0	0
$973$	−93.2132	−2.98828
$974$	0	0
$975$	50.7903	1.62659
$976$	0	0
$977$	16.2306	0.519261	0.259631	−	0.965708i	$-0.416399\pi$
$977$	16.2306	0.519261	0.259631	+	0.965708i	$0.416399\pi$
$978$	0	0
$979$	−47.5537	−1.51982
$980$	0	0
$981$	38.7170	1.23614
$982$	0	0
$983$	7.43350	0.237092	0.118546	−	0.992949i	$-0.462177\pi$
$983$	7.43350	0.237092	0.118546	+	0.992949i	$0.462177\pi$
$984$	0	0
$985$	−30.2168	−0.962788
$986$	0	0
$987$	84.7247	2.69682
$988$	0	0
$989$	−23.9276	−0.760852
$990$	0	0
$991$	−17.4924	−0.555665	−0.277832	−	0.960630i	$-0.589616\pi$
$991$	−17.4924	−0.555665	−0.277832	+	0.960630i	$0.589616\pi$
$992$	0	0
$993$	−16.6229	−0.527512
$994$	0	0
$995$	−6.65827	−0.211081
$996$	0	0
$997$	−24.6866	−0.781833	−0.390916	−	0.920426i	$-0.627842\pi$
$997$	−24.6866	−0.781833	−0.390916	+	0.920426i	$0.627842\pi$
$998$	0	0
$999$	18.9795	0.600484

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5648.2.a.p.1.1		14
4.3	odd	2		353.2.a.d.1.4	✓	14
12.11	even	2		3177.2.a.h.1.11		14
20.19	odd	2		8825.2.a.i.1.11		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
353.2.a.d.1.4	✓	14	4.3	odd	2
3177.2.a.h.1.11		14	12.11	even	2
5648.2.a.p.1.1		14	1.1	even	1	trivial
8825.2.a.i.1.11		14	20.19	odd	2

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:	\( N \)	\(=\)	\( 5648 = 2^{4} \cdot 353 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5648.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61874 q^{3} +3.04446 q^{5} -5.11343 q^{7} +3.85779 q^{9} +O(q^{10})\)	\(q-2.61874 q^{3} +3.04446 q^{5} -5.11343 q^{7} +3.85779 q^{9} +5.60579 q^{11} -4.54347 q^{13} -7.97265 q^{15} -0.673860 q^{17} +1.47041 q^{19} +13.3907 q^{21} +3.58108 q^{23} +4.26875 q^{25} -2.24633 q^{27} -3.94782 q^{29} -3.77024 q^{31} -14.6801 q^{33} -15.5677 q^{35} -8.44910 q^{37} +11.8982 q^{39} +7.82680 q^{41} -6.68167 q^{43} +11.7449 q^{45} +6.32711 q^{47} +19.1472 q^{49} +1.76466 q^{51} +1.77362 q^{53} +17.0666 q^{55} -3.85061 q^{57} +6.66991 q^{59} -4.03506 q^{61} -19.7266 q^{63} -13.8324 q^{65} +0.538541 q^{67} -9.37790 q^{69} +1.60224 q^{71} +10.8613 q^{73} -11.1787 q^{75} -28.6649 q^{77} -16.9961 q^{79} -5.69082 q^{81} +4.48784 q^{83} -2.05154 q^{85} +10.3383 q^{87} -8.48296 q^{89} +23.2327 q^{91} +9.87327 q^{93} +4.47659 q^{95} +1.99439 q^{97} +21.6260 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{3} - 4 q^{5} - 29 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q + 2 * q^3 - 4 * q^5 - 29 * q^7 + 14 * q^9	\( 14 q + 2 q^{3} - 4 q^{5} - 29 q^{7} + 14 q^{9} + 7 q^{11} + 2 q^{13} - 16 q^{15} + 8 q^{17} - 14 q^{19} + 6 q^{21} - 16 q^{23} + 24 q^{25} + 8 q^{27} - 9 q^{29} - 28 q^{31} - 10 q^{33} + 16 q^{35} + 8 q^{37} + 23 q^{39} + 14 q^{41} - 11 q^{43} - 37 q^{45} - 14 q^{47} + 39 q^{49} + 6 q^{51} - 27 q^{53} - 3 q^{55} + 7 q^{57} + 25 q^{59} - 3 q^{61} - 56 q^{63} - 41 q^{65} - 37 q^{67} - 32 q^{69} - 3 q^{71} + 14 q^{73} - 15 q^{75} - 43 q^{77} - 39 q^{79} - 26 q^{81} - 9 q^{83} - 12 q^{85} - 26 q^{87} - 12 q^{91} - 7 q^{93} + 24 q^{97} + 14 q^{99}+O(q^{100}) \) 14 * q + 2 * q^3 - 4 * q^5 - 29 * q^7 + 14 * q^9 + 7 * q^11 + 2 * q^13 - 16 * q^15 + 8 * q^17 - 14 * q^19 + 6 * q^21 - 16 * q^23 + 24 * q^25 + 8 * q^27 - 9 * q^29 - 28 * q^31 - 10 * q^33 + 16 * q^35 + 8 * q^37 + 23 * q^39 + 14 * q^41 - 11 * q^43 - 37 * q^45 - 14 * q^47 + 39 * q^49 + 6 * q^51 - 27 * q^53 - 3 * q^55 + 7 * q^57 + 25 * q^59 - 3 * q^61 - 56 * q^63 - 41 * q^65 - 37 * q^67 - 32 * q^69 - 3 * q^71 + 14 * q^73 - 15 * q^75 - 43 * q^77 - 39 * q^79 - 26 * q^81 - 9 * q^83 - 12 * q^85 - 26 * q^87 - 12 * q^91 - 7 * q^93 + 24 * q^97 + 14 * q^99

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