LMFDB - Embedded newform 2288.2.j.k.1585.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2288,2,Mod(1585,2288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2288, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2288.1585");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2288 = 2^{4} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2288.j (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$18.2697719825$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 19x^{10} + 133x^{8} + 423x^{6} + 601x^{4} + 312x^{2} + 36 $ x^12 + 19x^10 + 133x^8 + 423x^6 + 601x^4 + 312*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 143)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.10
Root			$2.66546i$ of defining polynomial
Character	$\chi$	$=$	2288.1585
Dual form			2288.2.j.k.1585.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.17072 q^{3} +0.458686i q^{5} +1.17072i q^{7} +1.71204 q^{9} +O(q^{10})$	$q+2.17072 q^{3} +0.458686i q^{5} +1.17072i q^{7} +1.71204 q^{9} +1.00000i q^{11} +(-3.50921 + 0.827915i) q^{13} +0.995680i q^{15} -6.55353 q^{17} +5.85202i q^{19} +2.54131i q^{21} +3.16452 q^{23} +4.78961 q^{25} -2.79581 q^{27} -2.13998 q^{29} +8.66805i q^{31} +2.17072i q^{33} -0.536994 q^{35} +8.23483i q^{37} +(-7.61752 + 1.79717i) q^{39} -5.38281i q^{41} +3.12946 q^{43} +0.785287i q^{45} +10.5508i q^{47} +5.62941 q^{49} -14.2259 q^{51} -1.26682 q^{53} -0.458686 q^{55} +12.7031i q^{57} +0.884643i q^{59} +5.03768 q^{61} +2.00432i q^{63} +(-0.379753 - 1.60963i) q^{65} -3.84383i q^{67} +6.86929 q^{69} -5.11327i q^{71} -5.92140i q^{73} +10.3969 q^{75} -1.17072 q^{77} -2.48143 q^{79} -11.2050 q^{81} -7.40248i q^{83} -3.00601i q^{85} -4.64531 q^{87} +7.39861i q^{89} +(-0.969259 - 4.10831i) q^{91} +18.8159i q^{93} -2.68424 q^{95} -14.6680i q^{97} +1.71204i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 12 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 12 * q^9	$ 12 q + 4 q^{3} + 12 q^{9} - 8 q^{13} - 12 q^{17} + 4 q^{23} - 16 q^{25} + 28 q^{27} + 8 q^{29} - 16 q^{35} - 4 q^{39} - 12 q^{43} + 32 q^{49} - 20 q^{53} + 8 q^{55} - 12 q^{61} - 20 q^{65} + 52 q^{69} + 8 q^{75} + 8 q^{77} + 48 q^{79} - 36 q^{81} - 12 q^{87} + 4 q^{95}+O(q^{100}) $ 12 * q + 4 * q^3 + 12 * q^9 - 8 * q^13 - 12 * q^17 + 4 * q^23 - 16 * q^25 + 28 * q^27 + 8 * q^29 - 16 * q^35 - 4 * q^39 - 12 * q^43 + 32 * q^49 - 20 * q^53 + 8 * q^55 - 12 * q^61 - 20 * q^65 + 52 * q^69 + 8 * q^75 + 8 * q^77 + 48 * q^79 - 36 * q^81 - 12 * q^87 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2288\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$1717$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	2.17072			1.25327			0.626634	−	0.779314i	$-0.284432\pi$
$3$	2.17072			1.25327			0.626634	+	0.779314i	$0.284432\pi$
$4$		0			0
$5$			0.458686i			0.205131i	0.994726	+	0.102565i	$0.0327051\pi$
$5$			0.458686i			0.205131i	−0.994726	+	0.102565i	$0.967295\pi$
$6$		0			0
$7$			1.17072i			0.442492i	0.975218	+	0.221246i	$0.0710123\pi$
$7$			1.17072i			0.442492i	−0.975218	+	0.221246i	$0.928988\pi$
$8$		0			0
$9$	1.71204			0.570679
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−3.50921	+	0.827915i	−0.973280	+	0.229622i
$13$	−3.50921	+	0.827915i	−0.973280	+	0.229622i
$14$		0			0
$15$			0.995680i			0.257083i
$16$		0			0
$17$	−6.55353			−1.58946			−0.794732	−	0.606960i	$-0.792389\pi$
$17$	−6.55353			−1.58946			−0.794732	+	0.606960i	$0.792389\pi$
$18$		0			0
$19$			5.85202i			1.34254i	0.741211	+	0.671272i	$0.234252\pi$
$19$			5.85202i			1.34254i	−0.741211	+	0.671272i	$0.765748\pi$
$20$		0			0
$21$			2.54131i			0.554560i
$22$		0			0
$23$	3.16452			0.659848			0.329924	−	0.944008i	$-0.392977\pi$
$23$	3.16452			0.659848			0.329924	+	0.944008i	$0.392977\pi$
$24$		0			0
$25$	4.78961			0.957921
$26$		0			0
$27$	−2.79581			−0.538054
$28$		0			0
$29$	−2.13998			−0.397385			−0.198692	−	0.980062i	$-0.563669\pi$
$29$	−2.13998			−0.397385			−0.198692	+	0.980062i	$0.563669\pi$
$30$		0			0
$31$			8.66805i			1.55683i	0.627752	+	0.778414i	$0.283975\pi$
$31$			8.66805i			1.55683i	−0.627752	+	0.778414i	$0.716025\pi$
$32$		0			0
$33$			2.17072i			0.377874i
$34$		0			0
$35$	−0.536994			−0.0907686
$36$		0			0
$37$			8.23483i			1.35380i	0.736076	+	0.676899i	$0.236676\pi$
$37$			8.23483i			1.35380i	−0.736076	+	0.676899i	$0.763324\pi$
$38$		0			0
$39$	−7.61752	+	1.79717i	−1.21978	+	0.287778i
$40$		0			0
$41$		−	5.38281i		−	0.840653i	−0.907373	−	0.420327i	$-0.861915\pi$
$41$		−	5.38281i		−	0.840653i	0.907373	−	0.420327i	$-0.138085\pi$
$42$		0			0
$43$	3.12946			0.477238			0.238619	−	0.971113i	$-0.423305\pi$
$43$	3.12946			0.477238			0.238619	+	0.971113i	$0.423305\pi$
$44$		0			0
$45$			0.785287i			0.117064i
$46$		0			0
$47$			10.5508i			1.53899i	0.638651	+	0.769497i	$0.279493\pi$
$47$			10.5508i			1.53899i	−0.638651	+	0.769497i	$0.720507\pi$
$48$		0			0
$49$	5.62941			0.804201
$50$		0			0
$51$	−14.2259			−1.99202
$52$		0			0
$53$	−1.26682			−0.174011			−0.0870054	−	0.996208i	$-0.527730\pi$
$53$	−1.26682			−0.174011			−0.0870054	+	0.996208i	$0.527730\pi$
$54$		0			0
$55$	−0.458686			−0.0618492
$56$		0			0
$57$			12.7031i			1.68257i
$58$		0			0
$59$			0.884643i			0.115171i	0.998341	+	0.0575854i	$0.0183401\pi$
$59$			0.884643i			0.115171i	−0.998341	+	0.0575854i	$0.981660\pi$
$60$		0			0
$61$	5.03768			0.645009			0.322505	−	0.946568i	$-0.395475\pi$
$61$	5.03768			0.645009			0.322505	+	0.946568i	$0.395475\pi$
$62$		0			0
$63$			2.00432i			0.252521i
$64$		0			0
$65$	−0.379753	−	1.60963i	−0.0471026	−	0.199649i
$66$		0			0
$67$		−	3.84383i		−	0.469598i	−0.972044	−	0.234799i	$-0.924557\pi$
$67$		−	3.84383i		−	0.469598i	0.972044	−	0.234799i	$-0.0754432\pi$
$68$		0			0
$69$	6.86929			0.826966
$70$		0			0
$71$		−	5.11327i		−	0.606833i	−0.952858	−	0.303416i	$-0.901873\pi$
$71$		−	5.11327i		−	0.606833i	0.952858	−	0.303416i	$-0.0981273\pi$
$72$		0			0
$73$		−	5.92140i		−	0.693047i	−0.938041	−	0.346524i	$-0.887362\pi$
$73$		−	5.92140i		−	0.693047i	0.938041	−	0.346524i	$-0.112638\pi$
$74$		0			0
$75$	10.3969			1.20053
$76$		0			0
$77$	−1.17072			−0.133416
$78$		0			0
$79$	−2.48143			−0.279182			−0.139591	−	0.990209i	$-0.544579\pi$
$79$	−2.48143			−0.279182			−0.139591	+	0.990209i	$0.544579\pi$
$80$		0			0
$81$	−11.2050			−1.24500
$82$		0			0
$83$		−	7.40248i		−	0.812528i	−0.913756	−	0.406264i	$-0.866831\pi$
$83$		−	7.40248i		−	0.812528i	0.913756	−	0.406264i	$-0.133169\pi$
$84$		0			0
$85$		−	3.00601i		−	0.326048i
$86$		0			0
$87$	−4.64531			−0.498029
$88$		0			0
$89$			7.39861i			0.784251i	0.919912	+	0.392125i	$0.128260\pi$
$89$			7.39861i			0.784251i	−0.919912	+	0.392125i	$0.871740\pi$
$90$		0			0
$91$	−0.969259	−	4.10831i	−0.101606	−	0.430668i
$92$		0			0
$93$			18.8159i			1.95112i
$94$		0			0
$95$	−2.68424			−0.275397
$96$		0			0
$97$		−	14.6680i		−	1.48931i	−0.667447	−	0.744657i	$-0.732613\pi$
$97$		−	14.6680i		−	1.48931i	0.667447	−	0.744657i	$-0.267387\pi$
$98$		0			0
$99$			1.71204i			0.172066i
$100$		0			0
$101$	−11.6018			−1.15442			−0.577209	−	0.816596i	$-0.695858\pi$
$101$	−11.6018			−1.15442			−0.577209	+	0.816596i	$0.695858\pi$
$102$		0			0
$103$	3.98768			0.392918			0.196459	−	0.980512i	$-0.437056\pi$
$103$	3.98768			0.392918			0.196459	+	0.980512i	$0.437056\pi$
$104$		0			0
$105$	−1.16567			−0.113757
$106$		0			0
$107$	13.7192			1.32628			0.663142	−	0.748493i	$-0.269222\pi$
$107$	13.7192			1.32628			0.663142	+	0.748493i	$0.269222\pi$
$108$		0			0
$109$			7.12146i			0.682112i	0.940043	+	0.341056i	$0.110785\pi$
$109$			7.12146i			0.682112i	−0.940043	+	0.341056i	$0.889215\pi$
$110$		0			0
$111$			17.8755i			1.69667i
$112$		0			0
$113$	8.48050			0.797778			0.398889	−	0.916999i	$-0.369396\pi$
$113$	8.48050			0.797778			0.398889	+	0.916999i	$0.369396\pi$
$114$		0			0
$115$			1.45152i			0.135355i
$116$		0			0
$117$	−6.00790	+	1.41742i	−0.555430	+	0.131041i
$118$		0			0
$119$		−	7.67237i		−	0.703325i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	11.6846i		−	1.05356i
$124$		0			0
$125$			4.49036i			0.401630i
$126$		0			0
$127$	−14.9557			−1.32711			−0.663553	−	0.748130i	$-0.730952\pi$
$127$	−14.9557			−1.32711			−0.663553	+	0.748130i	$0.730952\pi$
$128$		0			0
$129$	6.79318			0.598106
$130$		0			0
$131$	12.1089			1.05796			0.528982	−	0.848633i	$-0.322574\pi$
$131$	12.1089			1.05796			0.528982	+	0.848633i	$0.322574\pi$
$132$		0			0
$133$	−6.85109			−0.594065
$134$		0			0
$135$		−	1.28240i		−	0.110371i
$136$		0			0
$137$			7.58680i			0.648184i	0.946026	+	0.324092i	$0.105059\pi$
$137$			7.58680i			0.648184i	−0.946026	+	0.324092i	$0.894941\pi$
$138$		0			0
$139$	−10.7464			−0.911494			−0.455747	−	0.890109i	$-0.650628\pi$
$139$	−10.7464			−0.911494			−0.455747	+	0.890109i	$0.650628\pi$
$140$		0			0
$141$			22.9029i			1.92877i
$142$		0			0
$143$	−0.827915	−	3.50921i	−0.0692337	−	0.293455i
$144$		0			0
$145$		−	0.981579i		−	0.0815157i
$146$		0			0
$147$	12.2199			1.00788
$148$		0			0
$149$		−	11.4579i		−	0.938672i	−0.883020	−	0.469336i	$-0.844493\pi$
$149$		−	11.4579i		−	0.938672i	0.883020	−	0.469336i	$-0.155507\pi$
$150$		0			0
$151$			16.6187i			1.35241i	0.736714	+	0.676205i	$0.236376\pi$
$151$			16.6187i			1.35241i	−0.736714	+	0.676205i	$0.763624\pi$
$152$		0			0
$153$	−11.2199			−0.907074
$154$		0			0
$155$	−3.97591			−0.319353
$156$		0			0
$157$	−14.1605			−1.13013			−0.565067	−	0.825045i	$-0.691150\pi$
$157$	−14.1605			−1.13013			−0.565067	+	0.825045i	$0.691150\pi$
$158$		0			0
$159$	−2.74991			−0.218082
$160$		0			0
$161$			3.70477i			0.291977i
$162$		0			0
$163$		−	5.44196i		−	0.426247i	−0.977025	−	0.213124i	$-0.931636\pi$
$163$		−	5.44196i		−	0.426247i	0.977025	−	0.213124i	$-0.0683637\pi$
$164$		0			0
$165$	−0.995680			−0.0775136
$166$		0			0
$167$			0.886440i			0.0685948i	0.999412	+	0.0342974i	$0.0109193\pi$
$167$			0.886440i			0.0685948i	−0.999412	+	0.0342974i	$0.989081\pi$
$168$		0			0
$169$	11.6291	−	5.81066i	0.894547	−	0.446973i
$170$		0			0
$171$			10.0189i			0.766162i
$172$		0			0
$173$	6.77560			0.515140			0.257570	−	0.966260i	$-0.417078\pi$
$173$	6.77560			0.515140			0.257570	+	0.966260i	$0.417078\pi$
$174$		0			0
$175$			5.60730i			0.423872i
$176$		0			0
$177$			1.92032i			0.144340i
$178$		0			0
$179$	−1.31745			−0.0984711			−0.0492356	−	0.998787i	$-0.515679\pi$
$179$	−1.31745			−0.0984711			−0.0492356	+	0.998787i	$0.515679\pi$
$180$		0			0
$181$	11.0987			0.824961			0.412481	−	0.910966i	$-0.364662\pi$
$181$	11.0987			0.824961			0.412481	+	0.910966i	$0.364662\pi$
$182$		0			0
$183$	10.9354			0.808369
$184$		0			0
$185$	−3.77720			−0.277705
$186$		0			0
$187$		−	6.55353i		−	0.479242i
$188$		0			0
$189$		−	3.27312i		−	0.238084i
$190$		0			0
$191$	19.2393			1.39210			0.696052	−	0.717991i	$-0.254938\pi$
$191$	19.2393			1.39210			0.696052	+	0.717991i	$0.254938\pi$
$192$		0			0
$193$		−	19.7460i		−	1.42135i	−0.703523	−	0.710673i	$-0.748391\pi$
$193$		−	19.7460i		−	1.42135i	0.703523	−	0.710673i	$-0.251609\pi$
$194$		0			0
$195$	−0.824338	−	3.49405i	−0.0590321	−	0.250214i
$196$		0			0
$197$		−	21.0198i		−	1.49760i	−0.662796	−	0.748800i	$-0.730630\pi$
$197$		−	21.0198i		−	1.49760i	0.662796	−	0.748800i	$-0.269370\pi$
$198$		0			0
$199$	3.72266			0.263892			0.131946	−	0.991257i	$-0.457877\pi$
$199$	3.72266			0.263892			0.131946	+	0.991257i	$0.457877\pi$
$200$		0			0
$201$		−	8.34388i		−	0.588532i
$202$		0			0
$203$		−	2.50532i		−	0.175839i
$204$		0			0
$205$	2.46902			0.172444
$206$		0			0
$207$	5.41777			0.376561
$208$		0			0
$209$	−5.85202			−0.404793
$210$		0			0
$211$	11.9327			0.821478			0.410739	−	0.911753i	$-0.365271\pi$
$211$	11.9327			0.821478			0.410739	+	0.911753i	$0.365271\pi$
$212$		0			0
$213$		−	11.0995i		−	0.760524i
$214$		0			0
$215$			1.43544i			0.0978961i
$216$		0			0
$217$	−10.1479			−0.688883
$218$		0			0
$219$		−	12.8537i		−	0.868573i
$220$		0			0
$221$	22.9977	−	5.42577i	1.54699	−	0.364977i
$222$		0			0
$223$		−	2.88605i		−	0.193264i	−0.995320	−	0.0966320i	$-0.969193\pi$
$223$		−	2.88605i		−	0.193264i	0.995320	−	0.0966320i	$-0.0308070\pi$
$224$		0			0
$225$	8.19998			0.546666
$226$		0			0
$227$			8.00487i			0.531302i	0.964069	+	0.265651i	$0.0855868\pi$
$227$			8.00487i			0.531302i	−0.964069	+	0.265651i	$0.914413\pi$
$228$		0			0
$229$			26.0987i			1.72465i	0.506356	+	0.862324i	$0.330992\pi$
$229$			26.0987i			1.72465i	−0.506356	+	0.862324i	$0.669008\pi$
$230$		0			0
$231$	−2.54131			−0.167206
$232$		0			0
$233$	0.160396			0.0105079			0.00525394	−	0.999986i	$-0.498328\pi$
$233$	0.160396			0.0105079			0.00525394	+	0.999986i	$0.498328\pi$
$234$		0			0
$235$	−4.83951			−0.315695
$236$		0			0
$237$	−5.38649			−0.349890
$238$		0			0
$239$		−	9.08010i		−	0.587343i	−0.955906	−	0.293672i	$-0.905123\pi$
$239$		−	9.08010i		−	0.587343i	0.955906	−	0.293672i	$-0.0948772\pi$
$240$		0			0
$241$			0.132631i			0.00854350i	0.999991	+	0.00427175i	$0.00135974\pi$
$241$			0.132631i			0.00854350i	−0.999991	+	0.00427175i	$0.998640\pi$
$242$		0			0
$243$	−15.9356			−1.02227
$244$		0			0
$245$			2.58213i			0.164966i
$246$		0			0
$247$	−4.84497	−	20.5360i	−0.308278	−	1.30667i
$248$		0			0
$249$		−	16.0687i		−	1.01831i
$250$		0			0
$251$	−27.5007			−1.73583			−0.867915	−	0.496713i	$-0.834540\pi$
$251$	−27.5007			−1.73583			−0.867915	+	0.496713i	$0.834540\pi$
$252$		0			0
$253$			3.16452i			0.198952i
$254$		0			0
$255$		−	6.52522i		−	0.408625i
$256$		0			0
$257$	7.78887			0.485856			0.242928	−	0.970044i	$-0.421892\pi$
$257$	7.78887			0.485856			0.242928	+	0.970044i	$0.421892\pi$
$258$		0			0
$259$	−9.64070			−0.599044
$260$		0			0
$261$	−3.66373			−0.226779
$262$		0			0
$263$	−7.74585			−0.477629			−0.238815	−	0.971065i	$-0.576759\pi$
$263$	−7.74585			−0.477629			−0.238815	+	0.971065i	$0.576759\pi$
$264$		0			0
$265$		−	0.581072i		−	0.0356949i
$266$		0			0
$267$			16.0603i			0.982876i
$268$		0			0
$269$	−3.34660			−0.204046			−0.102023	−	0.994782i	$-0.532532\pi$
$269$	−3.34660			−0.204046			−0.102023	+	0.994782i	$0.532532\pi$
$270$		0			0
$271$			10.5534i			0.641075i	0.947236	+	0.320538i	$0.103864\pi$
$271$			10.5534i			0.641075i	−0.947236	+	0.320538i	$0.896136\pi$
$272$		0			0
$273$	−2.10399	−	8.91801i	−0.127339	−	0.539742i
$274$		0			0
$275$			4.78961i			0.288824i
$276$		0			0
$277$	10.9571			0.658348			0.329174	−	0.944269i	$-0.393230\pi$
$277$	10.9571			0.658348			0.329174	+	0.944269i	$0.393230\pi$
$278$		0			0
$279$			14.8400i			0.888448i
$280$		0			0
$281$			16.5005i			0.984338i	0.870500	+	0.492169i	$0.163796\pi$
$281$			16.5005i			0.984338i	−0.870500	+	0.492169i	$0.836204\pi$
$282$		0			0
$283$	1.15458			0.0686326			0.0343163	−	0.999411i	$-0.489075\pi$
$283$	1.15458			0.0686326			0.0343163	+	0.999411i	$0.489075\pi$
$284$		0			0
$285$	−5.82674			−0.345146
$286$		0			0
$287$	6.30177			0.371982
$288$		0			0
$289$	25.9488			1.52640
$290$		0			0
$291$		−	31.8403i		−	1.86651i
$292$		0			0
$293$			13.4247i			0.784281i	0.919905	+	0.392141i	$0.128265\pi$
$293$			13.4247i			0.784281i	−0.919905	+	0.392141i	$0.871735\pi$
$294$		0			0
$295$	−0.405773			−0.0236250
$296$		0			0
$297$		−	2.79581i		−	0.162229i
$298$		0			0
$299$	−11.1050	+	2.61995i	−0.642216	+	0.151516i
$300$		0			0
$301$			3.66373i			0.211174i
$302$		0			0
$303$	−25.1842			−1.44680
$304$		0			0
$305$			2.31071i			0.132311i
$306$		0			0
$307$			30.2535i			1.72666i	0.504643	+	0.863328i	$0.331624\pi$
$307$			30.2535i			1.72666i	−0.504643	+	0.863328i	$0.668376\pi$
$308$		0			0
$309$	8.65615			0.492431
$310$		0			0
$311$	8.48341			0.481050			0.240525	−	0.970643i	$-0.422680\pi$
$311$	8.48341			0.481050			0.240525	+	0.970643i	$0.422680\pi$
$312$		0			0
$313$	−10.7899			−0.609881			−0.304941	−	0.952371i	$-0.598637\pi$
$313$	−10.7899			−0.609881			−0.304941	+	0.952371i	$0.598637\pi$
$314$		0			0
$315$	−0.919353			−0.0517997
$316$		0			0
$317$		−	19.8347i		−	1.11403i	−0.830504	−	0.557013i	$-0.811947\pi$
$317$		−	19.8347i		−	1.11403i	0.830504	−	0.557013i	$-0.188053\pi$
$318$		0			0
$319$		−	2.13998i		−	0.119816i
$320$		0			0
$321$	29.7806			1.66219
$322$		0			0
$323$		−	38.3514i		−	2.13393i
$324$		0			0
$325$	−16.8077	+	3.96539i	−0.932326	+	0.219960i
$326$		0			0
$327$			15.4587i			0.854868i
$328$		0			0
$329$	−12.3521			−0.680991
$330$		0			0
$331$			3.41102i			0.187486i	0.995596	+	0.0937432i	$0.0298833\pi$
$331$			3.41102i			0.187486i	−0.995596	+	0.0937432i	$0.970117\pi$
$332$		0			0
$333$			14.0983i			0.772583i
$334$		0			0
$335$	1.76311			0.0963290
$336$		0			0
$337$	−7.64192			−0.416282			−0.208141	−	0.978099i	$-0.566741\pi$
$337$	−7.64192			−0.416282			−0.208141	+	0.978099i	$0.566741\pi$
$338$		0			0
$339$	18.4088			0.999829
$340$		0			0
$341$	−8.66805			−0.469401
$342$		0			0
$343$			14.7855i			0.798344i
$344$		0			0
$345$			3.15085i			0.169636i
$346$		0			0
$347$	23.9328			1.28478			0.642390	−	0.766378i	$-0.277943\pi$
$347$	23.9328			1.28478			0.642390	+	0.766378i	$0.277943\pi$
$348$		0			0
$349$			13.6173i			0.728917i	0.931220	+	0.364458i	$0.118746\pi$
$349$			13.6173i			0.728917i	−0.931220	+	0.364458i	$0.881254\pi$
$350$		0			0
$351$	9.81109	−	2.31469i	0.523677	−	0.123549i
$352$		0			0
$353$		−	3.44816i		−	0.183527i	−0.995781	−	0.0917636i	$-0.970750\pi$
$353$		−	3.44816i		−	0.183527i	0.995781	−	0.0917636i	$-0.0292504\pi$
$354$		0			0
$355$	2.34538			0.124480
$356$		0			0
$357$		−	16.6546i		−	0.881454i
$358$		0			0
$359$			20.5605i			1.08514i	0.840011	+	0.542570i	$0.182549\pi$
$359$			20.5605i			1.08514i	−0.840011	+	0.542570i	$0.817451\pi$
$360$		0			0
$361$	−15.2461			−0.802427
$362$		0			0
$363$	−2.17072			−0.113933
$364$		0			0
$365$	2.71606			0.142165
$366$		0			0
$367$	33.1416			1.72998			0.864989	−	0.501790i	$-0.167325\pi$
$367$	33.1416			1.72998			0.864989	+	0.501790i	$0.167325\pi$
$368$		0			0
$369$		−	9.21556i		−	0.479743i
$370$		0			0
$371$		−	1.48309i		−	0.0769983i
$372$		0			0
$373$	7.24503			0.375134			0.187567	−	0.982252i	$-0.439940\pi$
$373$	7.24503			0.375134			0.187567	+	0.982252i	$0.439940\pi$
$374$		0			0
$375$			9.74732i			0.503349i
$376$		0			0
$377$	7.50964	−	1.77172i	0.386766	−	0.0912484i
$378$		0			0
$379$			26.9600i			1.38484i	0.721493	+	0.692421i	$0.243456\pi$
$379$			26.9600i			1.38484i	−0.721493	+	0.692421i	$0.756544\pi$
$380$		0			0
$381$	−32.4647			−1.66322
$382$		0			0
$383$		−	16.9498i		−	0.866094i	−0.901371	−	0.433047i	$-0.857438\pi$
$383$		−	16.9498i		−	0.866094i	0.901371	−	0.433047i	$-0.142562\pi$
$384$		0			0
$385$		−	0.536994i		−	0.0273678i
$386$		0			0
$387$	5.35775			0.272350
$388$		0			0
$389$	2.61306			0.132488			0.0662438	−	0.997803i	$-0.478899\pi$
$389$	2.61306			0.132488			0.0662438	+	0.997803i	$0.478899\pi$
$390$		0			0
$391$	−20.7388			−1.04880
$392$		0			0
$393$	26.2852			1.32591
$394$		0			0
$395$		−	1.13820i		−	0.0572688i
$396$		0			0
$397$			20.0083i			1.00419i	0.864813	+	0.502095i	$0.167437\pi$
$397$			20.0083i			1.00419i	−0.864813	+	0.502095i	$0.832563\pi$
$398$		0			0
$399$	−14.8718			−0.744522
$400$		0			0
$401$		−	28.1352i		−	1.40500i	−0.711682	−	0.702502i	$-0.752066\pi$
$401$		−	28.1352i		−	1.40500i	0.711682	−	0.702502i	$-0.247934\pi$
$402$		0			0
$403$	−7.17641	−	30.4180i	−0.357482	−	1.51523i
$404$		0			0
$405$		−	5.13959i		−	0.255389i
$406$		0			0
$407$	−8.23483			−0.408185
$408$		0			0
$409$		−	32.3950i		−	1.60183i	−0.598779	−	0.800914i	$-0.704347\pi$
$409$		−	32.3950i		−	1.60183i	0.598779	−	0.800914i	$-0.295653\pi$
$410$		0			0
$411$			16.4688i			0.812347i
$412$		0			0
$413$	−1.03567			−0.0509621
$414$		0			0
$415$	3.39541			0.166674
$416$		0			0
$417$	−23.3274			−1.14235
$418$		0			0
$419$	−7.23409			−0.353408			−0.176704	−	0.984264i	$-0.556544\pi$
$419$	−7.23409			−0.353408			−0.176704	+	0.984264i	$0.556544\pi$
$420$		0			0
$421$		−	6.86086i		−	0.334378i	−0.985925	−	0.167189i	$-0.946531\pi$
$421$		−	6.86086i		−	0.334378i	0.985925	−	0.167189i	$-0.0534690\pi$
$422$		0			0
$423$			18.0634i			0.878271i
$424$		0			0
$425$	−31.3888			−1.52258
$426$		0			0
$427$			5.89773i			0.285411i
$428$		0			0
$429$	−1.79717	−	7.61752i	−0.0867684	−	0.367777i
$430$		0			0
$431$		−	13.2631i		−	0.638863i	−0.947609	−	0.319431i	$-0.896508\pi$
$431$		−	13.2631i		−	0.638863i	0.947609	−	0.319431i	$-0.103492\pi$
$432$		0			0
$433$	9.73932			0.468042			0.234021	−	0.972232i	$-0.424812\pi$
$433$	9.73932			0.468042			0.234021	+	0.972232i	$0.424812\pi$
$434$		0			0
$435$		−	2.13074i		−	0.102161i
$436$		0			0
$437$			18.5188i			0.885875i
$438$		0			0
$439$	11.7838			0.562412			0.281206	−	0.959647i	$-0.409266\pi$
$439$	11.7838			0.562412			0.281206	+	0.959647i	$0.409266\pi$
$440$		0			0
$441$	9.63775			0.458941
$442$		0			0
$443$	8.14169			0.386823			0.193412	−	0.981118i	$-0.438045\pi$
$443$	8.14169			0.386823			0.193412	+	0.981118i	$0.438045\pi$
$444$		0			0
$445$	−3.39364			−0.160874
$446$		0			0
$447$		−	24.8720i		−	1.17641i
$448$		0			0
$449$		−	35.3317i		−	1.66741i	−0.552211	−	0.833704i	$-0.686216\pi$
$449$		−	35.3317i		−	1.66741i	0.552211	−	0.833704i	$-0.313784\pi$
$450$		0			0
$451$	5.38281			0.253467
$452$		0			0
$453$			36.0746i			1.69493i
$454$		0			0
$455$	1.88443	−	0.444585i	0.0883432	−	0.0208425i
$456$		0			0
$457$			19.6535i			0.919350i	0.888087	+	0.459675i	$0.152034\pi$
$457$			19.6535i			0.919350i	−0.888087	+	0.459675i	$0.847966\pi$
$458$		0			0
$459$	18.3224			0.855218
$460$		0			0
$461$			11.8871i			0.553636i	0.960922	+	0.276818i	$0.0892799\pi$
$461$			11.8871i			0.553636i	−0.960922	+	0.276818i	$0.910720\pi$
$462$		0			0
$463$		−	33.4104i		−	1.55271i	−0.630293	−	0.776357i	$-0.717065\pi$
$463$		−	33.4104i		−	1.55271i	0.630293	−	0.776357i	$-0.282935\pi$
$464$		0			0
$465$	−8.63060			−0.400235
$466$		0			0
$467$	−7.45002			−0.344746			−0.172373	−	0.985032i	$-0.555143\pi$
$467$	−7.45002			−0.344746			−0.172373	+	0.985032i	$0.555143\pi$
$468$		0			0
$469$	4.50005			0.207793
$470$		0			0
$471$	−30.7386			−1.41636
$472$		0			0
$473$			3.12946i			0.143893i
$474$		0			0
$475$			28.0289i			1.28605i
$476$		0			0
$477$	−2.16884			−0.0993043
$478$		0			0
$479$		−	22.1756i		−	1.01323i	−0.862173	−	0.506614i	$-0.830897\pi$
$479$		−	22.1756i		−	1.01323i	0.862173	−	0.506614i	$-0.169103\pi$
$480$		0			0
$481$	−6.81774	−	28.8977i	−0.310862	−	1.31762i
$482$		0			0
$483$			8.04204i			0.365925i
$484$		0			0
$485$	6.72803			0.305504
$486$		0			0
$487$		−	8.84223i		−	0.400680i	−0.979726	−	0.200340i	$-0.935795\pi$
$487$		−	8.84223i		−	0.400680i	0.979726	−	0.200340i	$-0.0642046\pi$
$488$		0			0
$489$		−	11.8130i		−	0.534201i
$490$		0			0
$491$	−27.1206			−1.22393			−0.611967	−	0.790883i	$-0.709621\pi$
$491$	−27.1206			−1.22393			−0.611967	+	0.790883i	$0.709621\pi$
$492$		0			0
$493$	14.0244			0.631629
$494$		0			0
$495$	−0.785287			−0.0352960
$496$		0			0
$497$	5.98622			0.268518
$498$		0			0
$499$			27.8865i			1.24837i	0.781276	+	0.624185i	$0.214569\pi$
$499$			27.8865i			1.24837i	−0.781276	+	0.624185i	$0.785431\pi$
$500$		0			0
$501$			1.92422i			0.0859676i
$502$		0			0
$503$	18.4384			0.822127			0.411064	−	0.911607i	$-0.365157\pi$
$503$	18.4384			0.822127			0.411064	+	0.911607i	$0.365157\pi$
$504$		0			0
$505$		−	5.32157i		−	0.236807i
$506$		0			0
$507$	25.2436	−	12.6133i	1.12111	−	0.560177i
$508$		0			0
$509$			23.5533i			1.04398i	0.852951	+	0.521992i	$0.174811\pi$
$509$			23.5533i			1.04398i	−0.852951	+	0.521992i	$0.825189\pi$
$510$		0			0
$511$	6.93231			0.306667
$512$		0			0
$513$		−	16.3611i		−	0.722362i
$514$		0			0
$515$			1.82909i			0.0805994i
$516$		0			0
$517$	−10.5508			−0.464024
$518$		0			0
$519$	14.7080			0.645608
$520$		0			0
$521$	22.7100			0.994942			0.497471	−	0.867481i	$-0.334262\pi$
$521$	22.7100			0.994942			0.497471	+	0.867481i	$0.334262\pi$
$522$		0			0
$523$	−10.4633			−0.457526			−0.228763	−	0.973482i	$-0.573468\pi$
$523$	−10.4633			−0.457526			−0.228763	+	0.973482i	$0.573468\pi$
$524$		0			0
$525$			12.1719i			0.531225i
$526$		0			0
$527$		−	56.8063i		−	2.47452i
$528$		0			0
$529$	−12.9858			−0.564601
$530$		0			0
$531$			1.51454i			0.0657255i
$532$		0			0
$533$	4.45651	+	18.8894i	0.193033	+	0.818191i
$534$		0			0
$535$			6.29280i			0.272062i
$536$		0			0
$537$	−2.85983			−0.123411
$538$		0			0
$539$			5.62941i			0.242476i
$540$		0			0
$541$			42.1990i			1.81428i	0.420832	+	0.907139i	$0.361738\pi$
$541$			42.1990i			1.81428i	−0.420832	+	0.907139i	$0.638262\pi$
$542$		0			0
$543$	24.0922			1.03390
$544$		0			0
$545$	−3.26651			−0.139922
$546$		0			0
$547$	−9.87841			−0.422370			−0.211185	−	0.977446i	$-0.567732\pi$
$547$	−9.87841			−0.422370			−0.211185	+	0.977446i	$0.567732\pi$
$548$		0			0
$549$	8.62470			0.368093
$550$		0			0
$551$		−	12.5232i		−	0.533507i
$552$		0			0
$553$		−	2.90506i		−	0.123536i
$554$		0			0
$555$	−8.19925			−0.348039
$556$		0			0
$557$		−	27.7902i		−	1.17751i	−0.808312	−	0.588754i	$-0.799619\pi$
$557$		−	27.7902i		−	1.17751i	0.808312	−	0.588754i	$-0.200381\pi$
$558$		0			0
$559$	−10.9819	+	2.59092i	−0.464486	+	0.109584i
$560$		0			0
$561$		−	14.2259i		−	0.600618i
$562$		0			0
$563$	27.1946			1.14612			0.573058	−	0.819515i	$-0.305757\pi$
$563$	27.1946			1.14612			0.573058	+	0.819515i	$0.305757\pi$
$564$		0			0
$565$			3.88989i			0.163649i
$566$		0			0
$567$		−	13.1180i		−	0.550904i
$568$		0			0
$569$	33.0535			1.38567			0.692837	−	0.721095i	$-0.256361\pi$
$569$	33.0535			1.38567			0.692837	+	0.721095i	$0.256361\pi$
$570$		0			0
$571$	30.0787			1.25875			0.629377	−	0.777100i	$-0.283310\pi$
$571$	30.0787			1.25875			0.629377	+	0.777100i	$0.283310\pi$
$572$		0			0
$573$	41.7631			1.74468
$574$		0			0
$575$	15.1568			0.632082
$576$		0			0
$577$			6.67291i			0.277797i	0.990307	+	0.138898i	$0.0443561\pi$
$577$			6.67291i			0.277797i	−0.990307	+	0.138898i	$0.955644\pi$
$578$		0			0
$579$		−	42.8630i		−	1.78133i
$580$		0			0
$581$	8.66625			0.359537
$582$		0			0
$583$		−	1.26682i		−	0.0524662i
$584$		0			0
$585$	−0.650151	−	2.75574i	−0.0268804	−	0.113936i
$586$		0			0
$587$			16.5282i			0.682191i	0.940029	+	0.341096i	$0.110798\pi$
$587$			16.5282i			0.682191i	−0.940029	+	0.341096i	$0.889202\pi$
$588$		0			0
$589$	−50.7256			−2.09011
$590$		0			0
$591$		−	45.6282i		−	1.87689i
$592$		0			0
$593$		−	17.3007i		−	0.710456i	−0.934780	−	0.355228i	$-0.884403\pi$
$593$		−	17.3007i		−	0.710456i	0.934780	−	0.355228i	$-0.115597\pi$
$594$		0			0
$595$	3.51921			0.144273
$596$		0			0
$597$	8.08086			0.330728
$598$		0			0
$599$	−35.7952			−1.46255			−0.731276	−	0.682082i	$-0.761075\pi$
$599$	−35.7952			−1.46255			−0.731276	+	0.682082i	$0.761075\pi$
$600$		0			0
$601$	−45.7772			−1.86729			−0.933646	−	0.358196i	$-0.883392\pi$
$601$	−45.7772			−1.86729			−0.933646	+	0.358196i	$0.883392\pi$
$602$		0			0
$603$		−	6.58077i		−	0.267990i
$604$		0			0
$605$		−	0.458686i		−	0.0186482i
$606$		0			0
$607$	7.60403			0.308638			0.154319	−	0.988021i	$-0.450682\pi$
$607$	7.60403			0.308638			0.154319	+	0.988021i	$0.450682\pi$
$608$		0			0
$609$		−	5.43836i		−	0.220374i
$610$		0			0
$611$	−8.73517	−	37.0250i	−0.353387	−	1.49787i
$612$		0			0
$613$		−	5.27304i		−	0.212976i	−0.994314	−	0.106488i	$-0.966039\pi$
$613$		−	5.27304i		−	0.212976i	0.994314	−	0.106488i	$-0.0339606\pi$
$614$		0			0
$615$	5.35955			0.216118
$616$		0			0
$617$		−	15.9614i		−	0.642584i	−0.946980	−	0.321292i	$-0.895883\pi$
$617$		−	15.9614i		−	0.642584i	0.946980	−	0.321292i	$-0.104117\pi$
$618$		0			0
$619$		−	20.0005i		−	0.803887i	−0.915665	−	0.401943i	$-0.868335\pi$
$619$		−	20.0005i		−	0.803887i	0.915665	−	0.401943i	$-0.131665\pi$
$620$		0			0
$621$	−8.84740			−0.355034
$622$		0			0
$623$	−8.66172			−0.347024
$624$		0			0
$625$	21.8884			0.875535
$626$		0			0
$627$	−12.7031			−0.507313
$628$		0			0
$629$		−	53.9672i		−	2.15181i
$630$		0			0
$631$		−	28.7753i		−	1.14553i	−0.819721	−	0.572764i	$-0.805871\pi$
$631$		−	28.7753i		−	1.14553i	0.819721	−	0.572764i	$-0.194129\pi$
$632$		0			0
$633$	25.9025			1.02953
$634$		0			0
$635$		−	6.85998i		−	0.272230i
$636$		0			0
$637$	−19.7548	+	4.66067i	−0.782713	+	0.184663i
$638$		0			0
$639$		−	8.75410i		−	0.346307i
$640$		0			0
$641$	39.5621			1.56261			0.781304	−	0.624150i	$-0.214555\pi$
$641$	39.5621			1.56261			0.781304	+	0.624150i	$0.214555\pi$
$642$		0			0
$643$			8.36156i			0.329748i	0.986315	+	0.164874i	$0.0527217\pi$
$643$			8.36156i			0.329748i	−0.986315	+	0.164874i	$0.947278\pi$
$644$		0			0
$645$			3.11594i			0.122690i
$646$		0			0
$647$	−9.64297			−0.379104			−0.189552	−	0.981871i	$-0.560704\pi$
$647$	−9.64297			−0.379104			−0.189552	+	0.981871i	$0.560704\pi$
$648$		0			0
$649$	−0.884643			−0.0347253
$650$		0			0
$651$	−22.0282			−0.863354
$652$		0			0
$653$	25.0322			0.979587			0.489794	−	0.871838i	$-0.337072\pi$
$653$	25.0322			0.979587			0.489794	+	0.871838i	$0.337072\pi$
$654$		0			0
$655$			5.55420i			0.217021i
$656$		0			0
$657$		−	10.1376i		−	0.395507i
$658$		0			0
$659$	−27.2819			−1.06275			−0.531375	−	0.847136i	$-0.678325\pi$
$659$	−27.2819			−1.06275			−0.531375	+	0.847136i	$0.678325\pi$
$660$		0			0
$661$			1.09069i			0.0424229i	0.999775	+	0.0212114i	$0.00675232\pi$
$661$			1.09069i			0.0424229i	−0.999775	+	0.0212114i	$0.993248\pi$
$662$		0			0
$663$	49.9217	−	11.7778i	1.93880	−	0.457413i
$664$		0			0
$665$		−	3.14250i		−	0.121861i
$666$		0			0
$667$	−6.77201			−0.262213
$668$		0			0
$669$		−	6.26481i		−	0.242211i
$670$		0			0
$671$			5.03768i			0.194478i
$672$		0			0
$673$	48.0360			1.85165			0.925826	−	0.377950i	$-0.123371\pi$
$673$	48.0360			1.85165			0.925826	+	0.377950i	$0.123371\pi$
$674$		0			0
$675$	−13.3908			−0.515414
$676$		0			0
$677$	2.28541			0.0878355			0.0439178	−	0.999035i	$-0.486016\pi$
$677$	2.28541			0.0878355			0.0439178	+	0.999035i	$0.486016\pi$
$678$		0			0
$679$	17.1722			0.659009
$680$		0			0
$681$			17.3763i			0.665863i
$682$		0			0
$683$			35.3074i			1.35100i	0.737360	+	0.675500i	$0.236072\pi$
$683$			35.3074i			1.35100i	−0.737360	+	0.675500i	$0.763928\pi$
$684$		0			0
$685$	−3.47996			−0.132962
$686$		0			0
$687$			56.6530i			2.16145i
$688$		0			0
$689$	4.44553	−	1.04882i	0.169361	−	0.0399568i
$690$		0			0
$691$			31.3666i			1.19324i	0.802523	+	0.596621i	$0.203491\pi$
$691$			31.3666i			1.19324i	−0.802523	+	0.596621i	$0.796509\pi$
$692$		0			0
$693$	−2.00432			−0.0761378
$694$		0			0
$695$		−	4.92920i		−	0.186975i
$696$		0			0
$697$			35.2764i			1.33619i
$698$		0			0
$699$	0.348175			0.0131692
$700$		0			0
$701$	−21.8220			−0.824206			−0.412103	−	0.911137i	$-0.635206\pi$
$701$	−21.8220			−0.824206			−0.412103	+	0.911137i	$0.635206\pi$
$702$		0			0
$703$	−48.1903			−1.81753
$704$		0			0
$705$	−10.5052			−0.395650
$706$		0			0
$707$		−	13.5824i		−	0.510821i
$708$		0			0
$709$		−	36.9873i		−	1.38909i	−0.719451	−	0.694543i	$-0.755607\pi$
$709$		−	36.9873i		−	1.38909i	0.719451	−	0.694543i	$-0.244393\pi$
$710$		0			0
$711$	−4.24829			−0.159323
$712$		0			0
$713$			27.4302i			1.02727i
$714$		0			0
$715$	1.60963	−	0.379753i	0.0601966	−	0.0142020i
$716$		0			0
$717$		−	19.7104i		−	0.736098i
$718$		0			0
$719$	11.0686			0.412788			0.206394	−	0.978469i	$-0.433827\pi$
$719$	11.0686			0.412788			0.206394	+	0.978469i	$0.433827\pi$
$720$		0			0
$721$			4.66847i			0.173863i
$722$		0			0
$723$			0.287905i			0.0107073i
$724$		0			0
$725$	−10.2497			−0.380663
$726$		0			0
$727$	−20.0495			−0.743595			−0.371798	−	0.928314i	$-0.621258\pi$
$727$	−20.0495			−0.743595			−0.371798	+	0.928314i	$0.621258\pi$
$728$		0			0
$729$	−0.976648			−0.0361722
$730$		0			0
$731$	−20.5090			−0.758553
$732$		0			0
$733$			17.6554i			0.652117i	0.945350	+	0.326059i	$0.105721\pi$
$733$			17.6554i			0.652117i	−0.945350	+	0.326059i	$0.894279\pi$
$734$		0			0
$735$			5.60509i			0.206747i
$736$		0			0
$737$	3.84383			0.141589
$738$		0			0
$739$		−	20.1704i		−	0.741981i	−0.928637	−	0.370990i	$-0.879018\pi$
$739$		−	20.1704i		−	0.741981i	0.928637	−	0.370990i	$-0.120982\pi$
$740$		0			0
$741$	−10.5171	−	44.5779i	−0.386355	−	1.63761i
$742$		0			0
$743$		−	2.45850i		−	0.0901934i	−0.998983	−	0.0450967i	$-0.985640\pi$
$743$		−	2.45850i		−	0.0901934i	0.998983	−	0.0450967i	$-0.0143596\pi$
$744$		0			0
$745$	5.25560			0.192550
$746$		0			0
$747$		−	12.6733i		−	0.463692i
$748$		0			0
$749$			16.0614i			0.586870i
$750$		0			0
$751$	17.0807			0.623285			0.311642	−	0.950199i	$-0.399121\pi$
$751$	17.0807			0.623285			0.311642	+	0.950199i	$0.399121\pi$
$752$		0			0
$753$	−59.6964			−2.17546
$754$		0			0
$755$	−7.62276			−0.277421
$756$		0			0
$757$	22.8517			0.830560			0.415280	−	0.909694i	$-0.363684\pi$
$757$	22.8517			0.830560			0.415280	+	0.909694i	$0.363684\pi$
$758$		0			0
$759$			6.86929i			0.249340i
$760$		0			0
$761$		−	11.6590i		−	0.422639i	−0.977417	−	0.211319i	$-0.932224\pi$
$761$		−	11.6590i		−	0.422639i	0.977417	−	0.211319i	$-0.0677760\pi$
$762$		0			0
$763$	−8.33725			−0.301829
$764$		0			0
$765$		−	5.14640i		−	0.186069i
$766$		0			0
$767$	−0.732409	−	3.10440i	−0.0264458	−	0.112093i
$768$		0			0
$769$		−	43.9248i		−	1.58397i	−0.610540	−	0.791985i	$-0.709048\pi$
$769$		−	43.9248i		−	1.58397i	0.610540	−	0.791985i	$-0.290952\pi$
$770$		0			0
$771$	16.9075			0.608908
$772$		0			0
$773$		−	44.5121i		−	1.60099i	−0.599341	−	0.800494i	$-0.704571\pi$
$773$		−	44.5121i		−	1.60099i	0.599341	−	0.800494i	$-0.295429\pi$
$774$		0			0
$775$			41.5165i			1.49132i
$776$		0			0
$777$	−20.9273			−0.750762
$778$		0			0
$779$	31.5003			1.12861
$780$		0			0
$781$	5.11327			0.182967
$782$		0			0
$783$	5.98298			0.213814
$784$		0			0
$785$		−	6.49524i		−	0.231825i
$786$		0			0
$787$			26.3410i			0.938954i	0.882945	+	0.469477i	$0.155558\pi$
$787$			26.3410i			0.938954i	−0.882945	+	0.469477i	$0.844442\pi$
$788$		0			0
$789$	−16.8141			−0.598597
$790$		0			0
$791$			9.92831i			0.353010i
$792$		0			0
$793$	−17.6783	+	4.17077i	−0.627774	+	0.148108i
$794$		0			0
$795$		−	1.26135i		−	0.0447353i
$796$		0			0
$797$	20.1094			0.712312			0.356156	−	0.934427i	$-0.384087\pi$
$797$	20.1094			0.712312			0.356156	+	0.934427i	$0.384087\pi$
$798$		0			0
$799$		−	69.1450i		−	2.44618i
$800$		0			0
$801$			12.6667i			0.447555i
$802$		0			0
$803$	5.92140			0.208962
$804$		0			0
$805$	−1.69933			−0.0598934
$806$		0			0
$807$	−7.26454			−0.255724
$808$		0			0
$809$	7.67428			0.269813			0.134907	−	0.990858i	$-0.456927\pi$
$809$	7.67428			0.269813			0.134907	+	0.990858i	$0.456927\pi$
$810$		0			0
$811$			4.39093i			0.154187i	0.997024	+	0.0770933i	$0.0245639\pi$
$811$			4.39093i			0.154187i	−0.997024	+	0.0770933i	$0.975436\pi$
$812$		0			0
$813$			22.9086i			0.803438i
$814$		0			0
$815$	2.49615			0.0874363
$816$		0			0
$817$			18.3136i			0.640713i
$818$		0			0
$819$	−1.65941	−	7.03358i	−0.0579844	−	0.245773i
$820$		0			0
$821$		−	4.48612i		−	0.156567i	−0.996931	−	0.0782833i	$-0.975056\pi$
$821$		−	4.48612i		−	0.156567i	0.996931	−	0.0782833i	$-0.0249439\pi$
$822$		0			0
$823$	20.7536			0.723426			0.361713	−	0.932289i	$-0.382192\pi$
$823$	20.7536			0.723426			0.361713	+	0.932289i	$0.382192\pi$
$824$		0			0
$825$			10.3969i			0.361974i
$826$		0			0
$827$			35.0088i			1.21737i	0.793410	+	0.608687i	$0.208303\pi$
$827$			35.0088i			1.21737i	−0.793410	+	0.608687i	$0.791697\pi$
$828$		0			0
$829$	48.1248			1.67144			0.835721	−	0.549154i	$-0.185050\pi$
$829$	48.1248			1.67144			0.835721	+	0.549154i	$0.185050\pi$
$830$		0			0
$831$	23.7848			0.825086
$832$		0			0
$833$	−36.8925			−1.27825
$834$		0			0
$835$	−0.406598			−0.0140709
$836$		0			0
$837$		−	24.2342i		−	0.837657i
$838$		0			0
$839$		−	15.3315i		−	0.529304i	−0.964344	−	0.264652i	$-0.914743\pi$
$839$		−	15.3315i		−	0.529304i	0.964344	−	0.264652i	$-0.0852570\pi$
$840$		0			0
$841$	−24.4205			−0.842086
$842$		0			0
$843$			35.8180i			1.23364i
$844$		0			0
$845$	2.66527	+	5.33411i	0.0916879	+	0.183499i
$846$		0			0
$847$		−	1.17072i		−	0.0402265i
$848$		0			0
$849$	2.50627			0.0860149
$850$		0			0
$851$			26.0593i			0.893300i
$852$		0			0
$853$		−	10.0332i		−	0.343531i	−0.985138	−	0.171765i	$-0.945053\pi$
$853$		−	10.0332i		−	0.343531i	0.985138	−	0.171765i	$-0.0549471\pi$
$854$		0			0
$855$	−4.59551			−0.157163
$856$		0			0
$857$	27.6616			0.944903			0.472452	−	0.881357i	$-0.343369\pi$
$857$	27.6616			0.944903			0.472452	+	0.881357i	$0.343369\pi$
$858$		0			0
$859$	43.3633			1.47954			0.739768	−	0.672862i	$-0.234935\pi$
$859$	43.3633			1.47954			0.739768	+	0.672862i	$0.234935\pi$
$860$		0			0
$861$	13.6794			0.466193
$862$		0			0
$863$		−	38.2865i		−	1.30329i	−0.758525	−	0.651644i	$-0.774080\pi$
$863$		−	38.2865i		−	1.30329i	0.758525	−	0.651644i	$-0.225920\pi$
$864$		0			0
$865$			3.10787i			0.105671i
$866$		0			0
$867$	56.3276			1.91298
$868$		0			0
$869$		−	2.48143i		−	0.0841766i
$870$		0			0
$871$	3.18236	+	13.4888i	0.107830	+	0.457050i
$872$		0			0
$873$		−	25.1122i		−	0.849920i
$874$		0			0
$875$	−5.25696			−0.177718
$876$		0			0
$877$		−	4.41266i		−	0.149005i	−0.997221	−	0.0745024i	$-0.976263\pi$
$877$		−	4.41266i		−	0.149005i	0.997221	−	0.0745024i	$-0.0237369\pi$
$878$		0			0
$879$			29.1414i			0.982914i
$880$		0			0
$881$	−9.20818			−0.310232			−0.155116	−	0.987896i	$-0.549575\pi$
$881$	−9.20818			−0.310232			−0.155116	+	0.987896i	$0.549575\pi$
$882$		0			0
$883$	−4.27713			−0.143937			−0.0719685	−	0.997407i	$-0.522928\pi$
$883$	−4.27713			−0.143937			−0.0719685	+	0.997407i	$0.522928\pi$
$884$		0			0
$885$	−0.880822			−0.0296085
$886$		0			0
$887$	−46.4860			−1.56085			−0.780424	−	0.625251i	$-0.784997\pi$
$887$	−46.4860			−1.56085			−0.780424	+	0.625251i	$0.784997\pi$
$888$		0			0
$889$		−	17.5090i		−	0.587233i
$890$		0			0
$891$		−	11.2050i		−	0.375383i
$892$		0			0
$893$	−61.7435			−2.06617
$894$		0			0
$895$		−	0.604298i		−	0.0201994i
$896$		0			0
$897$	−24.1058	+	5.68719i	−0.804869	+	0.189890i
$898$		0			0
$899$		−	18.5495i		−	0.618659i
$900$		0			0
$901$	8.30213			0.276584
$902$		0			0
$903$			7.95293i			0.264657i
$904$		0			0
$905$			5.09083i			0.169225i
$906$		0			0
$907$	−1.17192			−0.0389129			−0.0194564	−	0.999811i	$-0.506194\pi$
$907$	−1.17192			−0.0389129			−0.0194564	+	0.999811i	$0.506194\pi$
$908$		0			0
$909$	−19.8626			−0.658802
$910$		0			0
$911$	42.8661			1.42022			0.710109	−	0.704091i	$-0.248645\pi$
$911$	42.8661			1.42022			0.710109	+	0.704091i	$0.248645\pi$
$912$		0			0
$913$	7.40248			0.244986
$914$		0			0
$915$			5.01592i			0.165821i
$916$		0			0
$917$			14.1762i			0.468140i
$918$		0			0
$919$	−24.0196			−0.792333			−0.396167	−	0.918179i	$-0.629660\pi$
$919$	−24.0196			−0.792333			−0.396167	+	0.918179i	$0.629660\pi$
$920$		0			0
$921$			65.6719i			2.16396i
$922$		0			0
$923$	4.23335	+	17.9435i	0.139342	+	0.590618i
$924$		0			0
$925$			39.4416i			1.29683i
$926$		0			0
$927$	6.82705			0.224230
$928$		0			0
$929$		−	19.2035i		−	0.630046i	−0.949084	−	0.315023i	$-0.897988\pi$
$929$		−	19.2035i		−	0.630046i	0.949084	−	0.315023i	$-0.102012\pi$
$930$		0			0
$931$			32.9434i			1.07968i
$932$		0			0
$933$	18.4151			0.602885
$934$		0			0
$935$	3.00601			0.0983071
$936$		0			0
$937$	51.2628			1.67468			0.837342	−	0.546680i	$-0.184109\pi$
$937$	51.2628			1.67468			0.837342	+	0.546680i	$0.184109\pi$
$938$		0			0
$939$	−23.4219			−0.764344
$940$		0			0
$941$			1.61030i			0.0524943i	0.999655	+	0.0262471i	$0.00835568\pi$
$941$			1.61030i			0.0524943i	−0.999655	+	0.0262471i	$0.991644\pi$
$942$		0			0
$943$		−	17.0340i		−	0.554703i
$944$		0			0
$945$	1.50133			0.0488384
$946$		0			0
$947$		−	11.4406i		−	0.371770i	−0.982571	−	0.185885i	$-0.940485\pi$
$947$		−	11.4406i		−	0.371770i	0.982571	−	0.185885i	$-0.0595152\pi$
$948$		0			0
$949$	4.90241	+	20.7794i	0.159139	+	0.674529i
$950$		0			0
$951$		−	43.0556i		−	1.39617i
$952$		0			0
$953$	16.5093			0.534789			0.267394	−	0.963587i	$-0.413837\pi$
$953$	16.5093			0.534789			0.267394	+	0.963587i	$0.413837\pi$
$954$		0			0
$955$			8.82478i			0.285563i
$956$		0			0
$957$		−	4.64531i		−	0.150161i
$958$		0			0
$959$	−8.88203			−0.286816
$960$		0			0
$961$	−44.1350			−1.42371
$962$		0			0
$963$	23.4878			0.756883
$964$		0			0
$965$	9.05720			0.291561
$966$		0			0
$967$			30.8852i			0.993202i	0.867979	+	0.496601i	$0.165419\pi$
$967$			30.8852i			0.993202i	−0.867979	+	0.496601i	$0.834581\pi$
$968$		0			0
$969$		−	83.2502i		−	2.67438i
$970$		0			0
$971$	−23.7146			−0.761038			−0.380519	−	0.924773i	$-0.624255\pi$
$971$	−23.7146			−0.761038			−0.380519	+	0.924773i	$0.624255\pi$
$972$		0			0
$973$		−	12.5810i		−	0.403328i
$974$		0			0
$975$	−36.4849	+	8.60776i	−1.16845	+	0.275669i
$976$		0			0
$977$			52.6096i			1.68313i	0.540156	+	0.841565i	$0.318365\pi$
$977$			52.6096i			1.68313i	−0.540156	+	0.841565i	$0.681635\pi$
$978$		0			0
$979$	−7.39861			−0.236461
$980$		0			0
$981$			12.1922i			0.389267i
$982$		0			0
$983$			44.4857i			1.41887i	0.704769	+	0.709437i	$0.251051\pi$
$983$			44.4857i			1.41887i	−0.704769	+	0.709437i	$0.748949\pi$
$984$		0			0
$985$	9.64150			0.307204
$986$		0			0
$987$	−26.8129			−0.853464
$988$		0			0
$989$	9.90323			0.314904
$990$		0			0
$991$	−8.68365			−0.275845			−0.137923	−	0.990443i	$-0.544043\pi$
$991$	−8.68365			−0.275845			−0.137923	+	0.990443i	$0.544043\pi$
$992$		0			0
$993$			7.40437i			0.234971i
$994$		0			0
$995$			1.70753i			0.0541324i
$996$		0			0
$997$	−25.3265			−0.802100			−0.401050	−	0.916056i	$-0.631355\pi$
$997$	−25.3265			−0.802100			−0.401050	+	0.916056i	$0.631355\pi$
$998$		0			0
$999$		−	23.0230i		−	0.728416i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2288.2.j.k.1585.10		12
4.3	odd	2		143.2.b.a.12.1	✓	12
12.11	even	2		1287.2.b.b.298.12		12
13.12	even	2	inner	2288.2.j.k.1585.9		12
52.31	even	4		1859.2.a.j.1.1		6
52.47	even	4		1859.2.a.n.1.6		6
52.51	odd	2		143.2.b.a.12.12	yes	12
156.155	even	2		1287.2.b.b.298.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.b.a.12.1	✓	12	4.3	odd	2
143.2.b.a.12.12	yes	12	52.51	odd	2
1287.2.b.b.298.1		12	156.155	even	2
1287.2.b.b.298.12		12	12.11	even	2
1859.2.a.j.1.1		6	52.31	even	4
1859.2.a.n.1.6		6	52.47	even	4
2288.2.j.k.1585.9		12	13.12	even	2	inner
2288.2.j.k.1585.10		12	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:	\( N \)	\(=\)	\( 2288 = 2^{4} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2288.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.17072 q^{3} +0.458686i q^{5} +1.17072i q^{7} +1.71204 q^{9} +O(q^{10})\)	\(q+2.17072 q^{3} +0.458686i q^{5} +1.17072i q^{7} +1.71204 q^{9} +1.00000i q^{11} +(-3.50921 + 0.827915i) q^{13} +0.995680i q^{15} -6.55353 q^{17} +5.85202i q^{19} +2.54131i q^{21} +3.16452 q^{23} +4.78961 q^{25} -2.79581 q^{27} -2.13998 q^{29} +8.66805i q^{31} +2.17072i q^{33} -0.536994 q^{35} +8.23483i q^{37} +(-7.61752 + 1.79717i) q^{39} -5.38281i q^{41} +3.12946 q^{43} +0.785287i q^{45} +10.5508i q^{47} +5.62941 q^{49} -14.2259 q^{51} -1.26682 q^{53} -0.458686 q^{55} +12.7031i q^{57} +0.884643i q^{59} +5.03768 q^{61} +2.00432i q^{63} +(-0.379753 - 1.60963i) q^{65} -3.84383i q^{67} +6.86929 q^{69} -5.11327i q^{71} -5.92140i q^{73} +10.3969 q^{75} -1.17072 q^{77} -2.48143 q^{79} -11.2050 q^{81} -7.40248i q^{83} -3.00601i q^{85} -4.64531 q^{87} +7.39861i q^{89} +(-0.969259 - 4.10831i) q^{91} +18.8159i q^{93} -2.68424 q^{95} -14.6680i q^{97} +1.71204i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 12 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 12 * q^9	\( 12 q + 4 q^{3} + 12 q^{9} - 8 q^{13} - 12 q^{17} + 4 q^{23} - 16 q^{25} + 28 q^{27} + 8 q^{29} - 16 q^{35} - 4 q^{39} - 12 q^{43} + 32 q^{49} - 20 q^{53} + 8 q^{55} - 12 q^{61} - 20 q^{65} + 52 q^{69} + 8 q^{75} + 8 q^{77} + 48 q^{79} - 36 q^{81} - 12 q^{87} + 4 q^{95}+O(q^{100}) \) 12 * q + 4 * q^3 + 12 * q^9 - 8 * q^13 - 12 * q^17 + 4 * q^23 - 16 * q^25 + 28 * q^27 + 8 * q^29 - 16 * q^35 - 4 * q^39 - 12 * q^43 + 32 * q^49 - 20 * q^53 + 8 * q^55 - 12 * q^61 - 20 * q^65 + 52 * q^69 + 8 * q^75 + 8 * q^77 + 48 * q^79 - 36 * q^81 - 12 * q^87 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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