LMFDB - Embedded newform 1728.3.b.i.1567.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(1567,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1728.1567");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.b (of order $2$, degree $1$, 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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.116304318664704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64 $ x^12 - 2x^11 + x^10 + 6x^9 - 9x^8 - 2x^7 + 18x^6 - 4x^5 - 36x^4 + 48x^3 + 16x^2 - 64x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.2
Root			$0.742163 + 1.20382i$ of defining polynomial
Character	$\chi$	$=$	1728.1567
Dual form			1728.3.b.i.1567.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.36964i q^{5} +2.43910i q^{7} +O(q^{10})$	$q-8.36964i q^{5} +2.43910i q^{7} -2.41295 q^{11} -20.8843i q^{13} -13.1793 q^{17} -32.7976 q^{19} +33.8139i q^{23} -45.0508 q^{25} +34.7407i q^{29} +33.7335i q^{31} +20.4143 q^{35} -30.8546i q^{37} +35.1659 q^{41} -27.9121 q^{43} -21.9865i q^{47} +43.0508 q^{49} +26.3710i q^{53} +20.1955i q^{55} +93.3081 q^{59} -27.7128i q^{61} -174.794 q^{65} +9.97035 q^{67} +64.5448i q^{71} -45.5636 q^{73} -5.88541i q^{77} -82.8072i q^{79} -59.4956 q^{83} +110.306i q^{85} -139.628 q^{89} +50.9387 q^{91} +274.504i q^{95} +150.716 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 48 q^{17} - 72 q^{25} - 336 q^{41} + 48 q^{49} - 912 q^{65} + 60 q^{73} - 1248 q^{89} - 204 q^{97}+O(q^{100}) $ 12 * q - 48 * q^17 - 72 * q^25 - 336 * q^41 + 48 * q^49 - 912 * q^65 + 60 * q^73 - 1248 * q^89 - 204 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 8.36964i	− 1.67393i	−0.547259	−	0.836964i	$-0.684329\pi$
$5$	− 8.36964i	− 1.67393i	0.547259	−	0.836964i	$-0.315671\pi$
$6$	0	0
$7$	2.43910i	0.348442i	0.984707	+	0.174221i	$0.0557408\pi$
$7$	2.43910i	0.348442i	−0.984707	+	0.174221i	$0.944259\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.41295	−0.219359	−0.109679	−	0.993967i	$-0.534982\pi$
$11$	−2.41295	−0.219359	−0.109679	+	0.993967i	$0.534982\pi$
$12$	0	0
$13$	− 20.8843i	− 1.60648i	−0.595654	−	0.803241i	$-0.703107\pi$
$13$	− 20.8843i	− 1.60648i	0.595654	−	0.803241i	$-0.296893\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−13.1793	−0.775256	−0.387628	−	0.921816i	$-0.626705\pi$
$17$	−13.1793	−0.775256	−0.387628	+	0.921816i	$0.626705\pi$
$18$	0	0
$19$	−32.7976	−1.72619	−0.863096	−	0.505040i	$-0.831478\pi$
$19$	−32.7976	−1.72619	−0.863096	+	0.505040i	$0.831478\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	33.8139i	1.47017i	0.677975	+	0.735085i	$0.262858\pi$
$23$	33.8139i	1.47017i	−0.677975	+	0.735085i	$0.737142\pi$
$24$	0	0
$25$	−45.0508	−1.80203
$26$	0	0
$27$	0	0
$28$	0	0
$29$	34.7407i	1.19795i	0.800766	+	0.598977i	$0.204426\pi$
$29$	34.7407i	1.19795i	−0.800766	+	0.598977i	$0.795574\pi$
$30$	0	0
$31$	33.7335i	1.08818i	0.839027	+	0.544089i	$0.183125\pi$
$31$	33.7335i	1.08818i	−0.839027	+	0.544089i	$0.816875\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	20.4143	0.583267
$36$	0	0
$37$	− 30.8546i	− 0.833909i	−0.908927	−	0.416954i	$-0.863097\pi$
$37$	− 30.8546i	− 0.833909i	0.908927	−	0.416954i	$-0.136903\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	35.1659	0.857705	0.428852	−	0.903375i	$-0.358918\pi$
$41$	35.1659	0.857705	0.428852	+	0.903375i	$0.358918\pi$
$42$	0	0
$43$	−27.9121	−0.649119	−0.324560	−	0.945865i	$-0.605216\pi$
$43$	−27.9121	−0.649119	−0.324560	+	0.945865i	$0.605216\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 21.9865i	− 0.467799i	−0.972261	−	0.233899i	$-0.924851\pi$
$47$	− 21.9865i	− 0.467799i	0.972261	−	0.233899i	$-0.0751486\pi$
$48$	0	0
$49$	43.0508	0.878588
$50$	0	0
$51$	0	0
$52$	0	0
$53$	26.3710i	0.497567i	0.968559	+	0.248783i	$0.0800307\pi$
$53$	26.3710i	0.497567i	−0.968559	+	0.248783i	$0.919969\pi$
$54$	0	0
$55$	20.1955i	0.367191i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	93.3081	1.58149	0.790747	−	0.612143i	$-0.209693\pi$
$59$	93.3081	1.58149	0.790747	+	0.612143i	$0.209693\pi$
$60$	0	0
$61$	− 27.7128i	− 0.454308i	−0.973859	−	0.227154i	$-0.927058\pi$
$61$	− 27.7128i	− 0.454308i	0.973859	−	0.227154i	$-0.0729421\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−174.794	−2.68913
$66$	0	0
$67$	9.97035	0.148811	0.0744056	−	0.997228i	$-0.476294\pi$
$67$	9.97035	0.148811	0.0744056	+	0.997228i	$0.476294\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	64.5448i	0.909081i	0.890726	+	0.454541i	$0.150197\pi$
$71$	64.5448i	0.909081i	−0.890726	+	0.454541i	$0.849803\pi$
$72$	0	0
$73$	−45.5636	−0.624159	−0.312079	−	0.950056i	$-0.601025\pi$
$73$	−45.5636	−0.624159	−0.312079	+	0.950056i	$0.601025\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.88541i	− 0.0764339i
$78$	0	0
$79$	− 82.8072i	− 1.04819i	−0.851659	−	0.524096i	$-0.824403\pi$
$79$	− 82.8072i	− 1.04819i	0.851659	−	0.524096i	$-0.175597\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−59.4956	−0.716815	−0.358407	−	0.933565i	$-0.616680\pi$
$83$	−59.4956	−0.716815	−0.358407	+	0.933565i	$0.616680\pi$
$84$	0	0
$85$	110.306i	1.29772i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−139.628	−1.56885	−0.784426	−	0.620222i	$-0.787042\pi$
$89$	−139.628	−1.56885	−0.784426	+	0.620222i	$0.787042\pi$
$90$	0	0
$91$	50.9387	0.559766
$92$	0	0
$93$	0	0
$94$	0	0
$95$	274.504i	2.88952i
$96$	0	0
$97$	150.716	1.55377	0.776887	−	0.629641i	$-0.216798\pi$
$97$	150.716	1.55377	0.776887	+	0.629641i	$0.216798\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	74.0246i	0.732916i	0.930435	+	0.366458i	$0.119430\pi$
$101$	74.0246i	0.732916i	−0.930435	+	0.366458i	$0.880570\pi$
$102$	0	0
$103$	− 107.152i	− 1.04031i	−0.854070	−	0.520157i	$-0.825873\pi$
$103$	− 107.152i	− 1.04031i	0.854070	−	0.520157i	$-0.174127\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−34.6694	−0.324013	−0.162007	−	0.986790i	$-0.551797\pi$
$107$	−34.6694	−0.324013	−0.162007	+	0.986790i	$0.551797\pi$
$108$	0	0
$109$	197.530i	1.81220i	0.423060	+	0.906102i	$0.360956\pi$
$109$	197.530i	1.81220i	−0.423060	+	0.906102i	$0.639044\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	52.7803	0.467082	0.233541	−	0.972347i	$-0.424969\pi$
$113$	52.7803	0.467082	0.233541	+	0.972347i	$0.424969\pi$
$114$	0	0
$115$	283.010	2.46096
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 32.1457i	− 0.270132i
$120$	0	0
$121$	−115.178	−0.951882
$122$	0	0
$123$	0	0
$124$	0	0
$125$	167.818i	1.34254i
$126$	0	0
$127$	74.4391i	0.586135i	0.956092	+	0.293067i	$0.0946760\pi$
$127$	74.4391i	0.586135i	−0.956092	+	0.293067i	$0.905324\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	159.377	1.21662	0.608309	−	0.793700i	$-0.291848\pi$
$131$	159.377	1.21662	0.608309	+	0.793700i	$0.291848\pi$
$132$	0	0
$133$	− 79.9966i	− 0.601478i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−218.704	−1.59638	−0.798189	−	0.602406i	$-0.794209\pi$
$137$	−218.704	−1.59638	−0.798189	+	0.602406i	$0.794209\pi$
$138$	0	0
$139$	37.8825	0.272536	0.136268	−	0.990672i	$-0.456489\pi$
$139$	37.8825	0.272536	0.136268	+	0.990672i	$0.456489\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	50.3926i	0.352396i
$144$	0	0
$145$	290.767	2.00529
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 52.1967i	− 0.350314i	−0.984541	−	0.175157i	$-0.943957\pi$
$149$	− 52.1967i	− 0.350314i	0.984541	−	0.175157i	$-0.0560432\pi$
$150$	0	0
$151$	38.9569i	0.257993i	0.991645	+	0.128996i	$0.0411756\pi$
$151$	38.9569i	0.257993i	−0.991645	+	0.128996i	$0.958824\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	282.337	1.82153
$156$	0	0
$157$	21.8278i	0.139031i	0.997581	+	0.0695154i	$0.0221453\pi$
$157$	21.8278i	0.139031i	−0.997581	+	0.0695154i	$0.977855\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−82.4754	−0.512270
$162$	0	0
$163$	−262.870	−1.61270	−0.806350	−	0.591438i	$-0.798560\pi$
$163$	−262.870	−1.61270	−0.806350	+	0.591438i	$0.798560\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	233.614i	1.39889i	0.714687	+	0.699444i	$0.246569\pi$
$167$	233.614i	1.39889i	−0.714687	+	0.699444i	$0.753431\pi$
$168$	0	0
$169$	−267.153	−1.58079
$170$	0	0
$171$	0	0
$172$	0	0
$173$	293.868i	1.69866i	0.527864	+	0.849329i	$0.322993\pi$
$173$	293.868i	1.69866i	−0.527864	+	0.849329i	$0.677007\pi$
$174$	0	0
$175$	− 109.883i	− 0.627904i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	305.165	1.70483	0.852415	−	0.522866i	$-0.175137\pi$
$179$	305.165	1.70483	0.852415	+	0.522866i	$0.175137\pi$
$180$	0	0
$181$	55.0270i	0.304017i	0.988379	+	0.152008i	$0.0485740\pi$
$181$	55.0270i	0.304017i	−0.988379	+	0.152008i	$0.951426\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−258.242	−1.39590
$186$	0	0
$187$	31.8011	0.170059
$188$	0	0
$189$	0	0
$190$	0	0
$191$	252.518i	1.32208i	0.750349	+	0.661041i	$0.229885\pi$
$191$	252.518i	1.32208i	−0.750349	+	0.661041i	$0.770115\pi$
$192$	0	0
$193$	66.1013	0.342494	0.171247	−	0.985228i	$-0.445220\pi$
$193$	66.1013	0.342494	0.171247	+	0.985228i	$0.445220\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 152.421i	− 0.773710i	−0.922141	−	0.386855i	$-0.873561\pi$
$197$	− 152.421i	− 0.773710i	0.922141	−	0.386855i	$-0.126439\pi$
$198$	0	0
$199$	− 36.9774i	− 0.185816i	−0.995675	−	0.0929081i	$-0.970384\pi$
$199$	− 36.9774i	− 0.185816i	0.995675	−	0.0929081i	$-0.0296163\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−84.7359	−0.417418
$204$	0	0
$205$	− 294.326i	− 1.43573i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	79.1390	0.378655
$210$	0	0
$211$	−277.237	−1.31392	−0.656960	−	0.753926i	$-0.728158\pi$
$211$	−277.237	−1.31392	−0.656960	+	0.753926i	$0.728158\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	233.614i	1.08658i
$216$	0	0
$217$	−82.2793	−0.379167
$218$	0	0
$219$	0	0
$220$	0	0
$221$	275.241i	1.24543i
$222$	0	0
$223$	45.6907i	0.204891i	0.994739	+	0.102446i	$0.0326668\pi$
$223$	45.6907i	0.204891i	−0.994739	+	0.102446i	$0.967333\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−399.744	−1.76099	−0.880493	−	0.474060i	$-0.842788\pi$
$227$	−399.744	−1.76099	−0.880493	+	0.474060i	$0.842788\pi$
$228$	0	0
$229$	− 189.213i	− 0.826258i	−0.910672	−	0.413129i	$-0.864436\pi$
$229$	− 189.213i	− 0.826258i	0.910672	−	0.413129i	$-0.135564\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−57.0895	−0.245019	−0.122510	−	0.992467i	$-0.539094\pi$
$233$	−57.0895	−0.245019	−0.122510	+	0.992467i	$0.539094\pi$
$234$	0	0
$235$	−184.019	−0.783061
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 309.733i	− 1.29595i	−0.761660	−	0.647977i	$-0.775615\pi$
$239$	− 309.733i	− 1.29595i	0.761660	−	0.647977i	$-0.224385\pi$
$240$	0	0
$241$	−400.843	−1.66325	−0.831624	−	0.555340i	$-0.812588\pi$
$241$	−400.843	−1.66325	−0.831624	+	0.555340i	$0.812588\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 360.320i	− 1.47069i
$246$	0	0
$247$	684.955i	2.77310i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−264.779	−1.05490	−0.527448	−	0.849587i	$-0.676851\pi$
$251$	−264.779	−1.05490	−0.527448	+	0.849587i	$0.676851\pi$
$252$	0	0
$253$	− 81.5912i	− 0.322495i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−354.023	−1.37752	−0.688760	−	0.724990i	$-0.741845\pi$
$257$	−354.023	−1.37752	−0.688760	+	0.724990i	$0.741845\pi$
$258$	0	0
$259$	75.2574	0.290569
$260$	0	0
$261$	0	0
$262$	0	0
$263$	386.105i	1.46808i	0.679106	+	0.734041i	$0.262368\pi$
$263$	386.105i	1.46808i	−0.679106	+	0.734041i	$0.737632\pi$
$264$	0	0
$265$	220.716	0.832891
$266$	0	0
$267$	0	0
$268$	0	0
$269$	167.021i	0.620896i	0.950590	+	0.310448i	$0.100479\pi$
$269$	167.021i	0.620896i	−0.950590	+	0.310448i	$0.899521\pi$
$270$	0	0
$271$	254.266i	0.938253i	0.883131	+	0.469126i	$0.155431\pi$
$271$	254.266i	0.938253i	−0.883131	+	0.469126i	$0.844569\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	108.705	0.395292
$276$	0	0
$277$	− 414.204i	− 1.49532i	−0.664082	−	0.747660i	$-0.731177\pi$
$277$	− 414.204i	− 1.49532i	0.664082	−	0.747660i	$-0.268823\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	53.6902	0.191068	0.0955341	−	0.995426i	$-0.469544\pi$
$281$	53.6902	0.191068	0.0955341	+	0.995426i	$0.469544\pi$
$282$	0	0
$283$	−183.530	−0.648517	−0.324259	−	0.945969i	$-0.605115\pi$
$283$	−183.530	−0.648517	−0.324259	+	0.945969i	$0.605115\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	85.7730i	0.298861i
$288$	0	0
$289$	−115.305	−0.398979
$290$	0	0
$291$	0	0
$292$	0	0
$293$	165.240i	0.563959i	0.959420	+	0.281980i	$0.0909911\pi$
$293$	165.240i	0.563959i	−0.959420	+	0.281980i	$0.909009\pi$
$294$	0	0
$295$	− 780.955i	− 2.64730i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	706.179	2.36180
$300$	0	0
$301$	− 68.0804i	− 0.226181i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−231.946	−0.760479
$306$	0	0
$307$	−168.765	−0.549723	−0.274861	−	0.961484i	$-0.588632\pi$
$307$	−168.765	−0.549723	−0.274861	+	0.961484i	$0.588632\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 444.989i	− 1.43083i	−0.698699	−	0.715416i	$-0.746237\pi$
$311$	− 444.989i	− 1.43083i	0.698699	−	0.715416i	$-0.253763\pi$
$312$	0	0
$313$	20.7413	0.0662660	0.0331330	−	0.999451i	$-0.489452\pi$
$313$	20.7413	0.0662660	0.0331330	+	0.999451i	$0.489452\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 69.1362i	− 0.218095i	−0.994037	−	0.109048i	$-0.965220\pi$
$317$	− 69.1362i	− 0.218095i	0.994037	−	0.109048i	$-0.0347801\pi$
$318$	0	0
$319$	− 83.8274i	− 0.262782i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	432.252	1.33824
$324$	0	0
$325$	940.853i	2.89493i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	53.6273	0.163001
$330$	0	0
$331$	386.180	1.16671	0.583353	−	0.812219i	$-0.301740\pi$
$331$	386.180	1.16671	0.583353	+	0.812219i	$0.301740\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 83.4482i	− 0.249099i
$336$	0	0
$337$	−211.767	−0.628389	−0.314194	−	0.949359i	$-0.601734\pi$
$337$	−211.767	−0.628389	−0.314194	+	0.949359i	$0.601734\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 81.3972i	− 0.238701i
$342$	0	0
$343$	224.521i	0.654580i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−540.291	−1.55704	−0.778518	−	0.627623i	$-0.784028\pi$
$347$	−540.291	−1.55704	−0.778518	+	0.627623i	$0.784028\pi$
$348$	0	0
$349$	− 183.767i	− 0.526553i	−0.964720	−	0.263276i	$-0.915197\pi$
$349$	− 183.767i	− 0.526553i	0.964720	−	0.263276i	$-0.0848031\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	549.830	1.55759	0.778796	−	0.627277i	$-0.215831\pi$
$353$	549.830	1.55759	0.778796	+	0.627277i	$0.215831\pi$
$354$	0	0
$355$	540.216	1.52174
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 406.677i	− 1.13281i	−0.824129	−	0.566403i	$-0.808335\pi$
$359$	− 406.677i	− 1.13281i	0.824129	−	0.566403i	$-0.191665\pi$
$360$	0	0
$361$	714.686	1.97974
$362$	0	0
$363$	0	0
$364$	0	0
$365$	381.351i	1.04480i
$366$	0	0
$367$	− 443.688i	− 1.20896i	−0.796620	−	0.604480i	$-0.793381\pi$
$367$	− 443.688i	− 1.20896i	0.796620	−	0.604480i	$-0.206619\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−64.3215	−0.173373
$372$	0	0
$373$	98.5364i	0.264173i	0.991238	+	0.132086i	$0.0421676\pi$
$373$	98.5364i	0.264173i	−0.991238	+	0.132086i	$0.957832\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	725.534	1.92449
$378$	0	0
$379$	−431.635	−1.13888	−0.569439	−	0.822033i	$-0.692840\pi$
$379$	−431.635	−1.13888	−0.569439	+	0.822033i	$0.692840\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 260.504i	− 0.680167i	−0.940395	−	0.340083i	$-0.889545\pi$
$383$	− 260.504i	− 0.680167i	0.940395	−	0.340083i	$-0.110455\pi$
$384$	0	0
$385$	−49.2587	−0.127945
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 360.413i	− 0.926512i	−0.886224	−	0.463256i	$-0.846681\pi$
$389$	− 360.413i	− 0.926512i	0.886224	−	0.463256i	$-0.153319\pi$
$390$	0	0
$391$	− 445.645i	− 1.13976i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−693.066	−1.75460
$396$	0	0
$397$	524.685i	1.32162i	0.750551	+	0.660812i	$0.229788\pi$
$397$	524.685i	1.32162i	−0.750551	+	0.660812i	$0.770212\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−469.300	−1.17032	−0.585162	−	0.810916i	$-0.698969\pi$
$401$	−469.300	−1.17032	−0.585162	+	0.810916i	$0.698969\pi$
$402$	0	0
$403$	704.500	1.74814
$404$	0	0
$405$	0	0
$406$	0	0
$407$	74.4505i	0.182925i
$408$	0	0
$409$	493.509	1.20662	0.603311	−	0.797506i	$-0.293848\pi$
$409$	493.509	1.20662	0.603311	+	0.797506i	$0.293848\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	227.587i	0.551059i
$414$	0	0
$415$	497.957i	1.19990i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−527.341	−1.25857	−0.629285	−	0.777175i	$-0.716652\pi$
$419$	−527.341	−1.25857	−0.629285	+	0.777175i	$0.716652\pi$
$420$	0	0
$421$	− 417.579i	− 0.991875i	−0.868358	−	0.495937i	$-0.834825\pi$
$421$	− 417.579i	− 0.991875i	0.868358	−	0.495937i	$-0.165175\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	593.740	1.39704
$426$	0	0
$427$	67.5942	0.158300
$428$	0	0
$429$	0	0
$430$	0	0
$431$	462.224i	1.07245i	0.844076	+	0.536223i	$0.180149\pi$
$431$	462.224i	1.07245i	−0.844076	+	0.536223i	$0.819851\pi$
$432$	0	0
$433$	58.2079	0.134429	0.0672147	−	0.997739i	$-0.478589\pi$
$433$	58.2079	0.134429	0.0672147	+	0.997739i	$0.478589\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1109.02i	− 2.53780i
$438$	0	0
$439$	621.745i	1.41627i	0.706075	+	0.708137i	$0.250464\pi$
$439$	621.745i	1.41627i	−0.706075	+	0.708137i	$0.749536\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	127.815	0.288521	0.144261	−	0.989540i	$-0.453920\pi$
$443$	127.815	0.288521	0.144261	+	0.989540i	$0.453920\pi$
$444$	0	0
$445$	1168.63i	2.62614i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	335.372	0.746932	0.373466	−	0.927644i	$-0.378169\pi$
$449$	335.372	0.746932	0.373466	+	0.927644i	$0.378169\pi$
$450$	0	0
$451$	−84.8534	−0.188145
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 426.339i	− 0.937008i
$456$	0	0
$457$	−121.182	−0.265169	−0.132585	−	0.991172i	$-0.542328\pi$
$457$	−121.182	−0.265169	−0.132585	+	0.991172i	$0.542328\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	144.396i	0.313224i	0.987660	+	0.156612i	$0.0500573\pi$
$461$	144.396i	0.313224i	−0.987660	+	0.156612i	$0.949943\pi$
$462$	0	0
$463$	− 487.501i	− 1.05292i	−0.850201	−	0.526459i	$-0.823520\pi$
$463$	− 487.501i	− 1.05292i	0.850201	−	0.526459i	$-0.176480\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	298.591	0.639382	0.319691	−	0.947522i	$-0.396421\pi$
$467$	298.591	0.639382	0.319691	+	0.947522i	$0.396421\pi$
$468$	0	0
$469$	24.3187i	0.0518521i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	67.3505	0.142390
$474$	0	0
$475$	1477.56	3.11065
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 86.7847i	− 0.181179i	−0.995888	−	0.0905894i	$-0.971125\pi$
$479$	− 86.7847i	− 0.181179i	0.995888	−	0.0905894i	$-0.0288751\pi$
$480$	0	0
$481$	−644.376	−1.33966
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1261.44i	− 2.60090i
$486$	0	0
$487$	− 781.377i	− 1.60447i	−0.597009	−	0.802235i	$-0.703644\pi$
$487$	− 781.377i	− 1.60447i	0.597009	−	0.802235i	$-0.296356\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	158.487	0.322783	0.161392	−	0.986890i	$-0.448402\pi$
$491$	158.487	0.322783	0.161392	+	0.986890i	$0.448402\pi$
$492$	0	0
$493$	− 457.859i	− 0.928721i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−157.431	−0.316762
$498$	0	0
$499$	−780.549	−1.56423	−0.782113	−	0.623137i	$-0.785858\pi$
$499$	−780.549	−1.56423	−0.782113	+	0.623137i	$0.785858\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 517.773i	− 1.02937i	−0.857379	−	0.514685i	$-0.827909\pi$
$503$	− 517.773i	− 1.02937i	0.857379	−	0.514685i	$-0.172091\pi$
$504$	0	0
$505$	619.559	1.22685
$506$	0	0
$507$	0	0
$508$	0	0
$509$	653.045i	1.28300i	0.767124	+	0.641498i	$0.221687\pi$
$509$	653.045i	1.28300i	−0.767124	+	0.641498i	$0.778313\pi$
$510$	0	0
$511$	− 111.134i	− 0.217483i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−896.827	−1.74141
$516$	0	0
$517$	53.0523i	0.102616i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−310.175	−0.595346	−0.297673	−	0.954668i	$-0.596210\pi$
$521$	−310.175	−0.595346	−0.297673	+	0.954668i	$0.596210\pi$
$522$	0	0
$523$	−779.063	−1.48960	−0.744802	−	0.667285i	$-0.767456\pi$
$523$	−779.063	−1.48960	−0.744802	+	0.667285i	$0.767456\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 444.586i	− 0.843616i
$528$	0	0
$529$	−614.381	−1.16140
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 734.414i	− 1.37789i
$534$	0	0
$535$	290.170i	0.542374i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−103.879	−0.192726
$540$	0	0
$541$	− 410.019i	− 0.757892i	−0.925419	−	0.378946i	$-0.876287\pi$
$541$	− 410.019i	− 0.757892i	0.925419	−	0.378946i	$-0.123713\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1653.26	3.03350
$546$	0	0
$547$	−561.353	−1.02624	−0.513120	−	0.858317i	$-0.671510\pi$
$547$	−561.353	−1.02624	−0.513120	+	0.858317i	$0.671510\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1139.41i	− 2.06790i
$552$	0	0
$553$	201.975	0.365235
$554$	0	0
$555$	0	0
$556$	0	0
$557$	363.521i	0.652640i	0.945259	+	0.326320i	$0.105809\pi$
$557$	363.521i	0.652640i	−0.945259	+	0.326320i	$0.894191\pi$
$558$	0	0
$559$	582.924i	1.04280i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	287.769	0.511134	0.255567	−	0.966791i	$-0.417738\pi$
$563$	287.769	0.511134	0.255567	+	0.966791i	$0.417738\pi$
$564$	0	0
$565$	− 441.752i	− 0.781861i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	414.386	0.728270	0.364135	−	0.931346i	$-0.381365\pi$
$569$	414.386	0.728270	0.364135	+	0.931346i	$0.381365\pi$
$570$	0	0
$571$	−529.424	−0.927188	−0.463594	−	0.886048i	$-0.653440\pi$
$571$	−529.424	−0.927188	−0.463594	+	0.886048i	$0.653440\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1523.34i	− 2.64930i
$576$	0	0
$577$	−409.148	−0.709095	−0.354548	−	0.935038i	$-0.615365\pi$
$577$	−409.148	−0.709095	−0.354548	+	0.935038i	$0.615365\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 145.116i	− 0.249769i
$582$	0	0
$583$	− 63.6319i	− 0.109146i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−368.026	−0.626961	−0.313481	−	0.949595i	$-0.601495\pi$
$587$	−368.026	−0.626961	−0.313481	+	0.949595i	$0.601495\pi$
$588$	0	0
$589$	− 1106.38i	− 1.87840i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1055.08	1.77922	0.889612	−	0.456717i	$-0.150975\pi$
$593$	1055.08	1.77922	0.889612	+	0.456717i	$0.150975\pi$
$594$	0	0
$595$	−269.048	−0.452181
$596$	0	0
$597$	0	0
$598$	0	0
$599$	863.896i	1.44223i	0.692815	+	0.721116i	$0.256370\pi$
$599$	863.896i	1.44223i	−0.692815	+	0.721116i	$0.743630\pi$
$600$	0	0
$601$	−2.43642	−0.00405394	−0.00202697	−	0.999998i	$-0.500645\pi$
$601$	−2.43642	−0.00405394	−0.00202697	+	0.999998i	$0.500645\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	963.995i	1.59338i
$606$	0	0
$607$	755.681i	1.24494i	0.782642	+	0.622472i	$0.213872\pi$
$607$	755.681i	1.24494i	−0.782642	+	0.622472i	$0.786128\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−459.173	−0.751510
$612$	0	0
$613$	− 183.950i	− 0.300082i	−0.988680	−	0.150041i	$-0.952059\pi$
$613$	− 183.950i	− 0.300082i	0.988680	−	0.150041i	$-0.0479406\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	502.269	0.814050	0.407025	−	0.913417i	$-0.366566\pi$
$617$	502.269	0.814050	0.407025	+	0.913417i	$0.366566\pi$
$618$	0	0
$619$	222.995	0.360250	0.180125	−	0.983644i	$-0.442350\pi$
$619$	222.995	0.360250	0.180125	+	0.983644i	$0.442350\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 340.566i	− 0.546655i
$624$	0	0
$625$	278.305	0.445288
$626$	0	0
$627$	0	0
$628$	0	0
$629$	406.644i	0.646492i
$630$	0	0
$631$	1058.12i	1.67689i	0.544988	+	0.838444i	$0.316534\pi$
$631$	1058.12i	1.67689i	−0.544988	+	0.838444i	$0.683466\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	623.028	0.981147
$636$	0	0
$637$	− 899.085i	− 1.41144i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−353.113	−0.550878	−0.275439	−	0.961319i	$-0.588823\pi$
$641$	−353.113	−0.550878	−0.275439	+	0.961319i	$0.588823\pi$
$642$	0	0
$643$	404.720	0.629424	0.314712	−	0.949187i	$-0.398092\pi$
$643$	404.720	0.629424	0.314712	+	0.949187i	$0.398092\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	257.672i	0.398257i	0.979973	+	0.199129i	$0.0638111\pi$
$647$	257.672i	0.398257i	−0.979973	+	0.199129i	$0.936189\pi$
$648$	0	0
$649$	−225.147	−0.346914
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 841.878i	− 1.28925i	−0.764500	−	0.644624i	$-0.777014\pi$
$653$	− 841.878i	− 1.28925i	0.764500	−	0.644624i	$-0.222986\pi$
$654$	0	0
$655$	− 1333.93i	− 2.03653i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−673.789	−1.02244	−0.511221	−	0.859449i	$-0.670807\pi$
$659$	−673.789	−1.02244	−0.511221	+	0.859449i	$0.670807\pi$
$660$	0	0
$661$	116.970i	0.176959i	0.996078	+	0.0884796i	$0.0282008\pi$
$661$	116.970i	0.176959i	−0.996078	+	0.0884796i	$0.971799\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−669.543	−1.00683
$666$	0	0
$667$	−1174.72	−1.76120
$668$	0	0
$669$	0	0
$670$	0	0
$671$	66.8695i	0.0996565i
$672$	0	0
$673$	−772.635	−1.14805	−0.574023	−	0.818839i	$-0.694618\pi$
$673$	−772.635	−1.14805	−0.574023	+	0.818839i	$0.694618\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 172.800i	− 0.255244i	−0.991823	−	0.127622i	$-0.959266\pi$
$677$	− 172.800i	− 0.255244i	0.991823	−	0.127622i	$-0.0407344\pi$
$678$	0	0
$679$	367.611i	0.541400i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−808.636	−1.18395	−0.591974	−	0.805957i	$-0.701651\pi$
$683$	−808.636	−1.18395	−0.591974	+	0.805957i	$0.701651\pi$
$684$	0	0
$685$	1830.47i	2.67222i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	550.740	0.799332
$690$	0	0
$691$	−315.802	−0.457022	−0.228511	−	0.973541i	$-0.573386\pi$
$691$	−315.802	−0.457022	−0.228511	+	0.973541i	$0.573386\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 317.063i	− 0.456205i
$696$	0	0
$697$	−463.463	−0.664940
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 276.941i	− 0.395066i	−0.980296	−	0.197533i	$-0.936707\pi$
$701$	− 276.941i	− 0.395066i	0.980296	−	0.197533i	$-0.0632929\pi$
$702$	0	0
$703$	1011.96i	1.43949i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−180.553	−0.255379
$708$	0	0
$709$	− 512.419i	− 0.722734i	−0.932424	−	0.361367i	$-0.882310\pi$
$709$	− 512.419i	− 0.722734i	0.932424	−	0.361367i	$-0.117690\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1140.66	−1.59981
$714$	0	0
$715$	421.768	0.589885
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 129.848i	− 0.180595i	−0.995915	−	0.0902975i	$-0.971218\pi$
$719$	− 129.848i	− 0.180595i	0.995915	−	0.0902975i	$-0.0287818\pi$
$720$	0	0
$721$	261.355	0.362490
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1565.10i	− 2.15875i
$726$	0	0
$727$	− 581.866i	− 0.800366i	−0.916435	−	0.400183i	$-0.868947\pi$
$727$	− 581.866i	− 0.800366i	0.916435	−	0.400183i	$-0.131053\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	367.864	0.503233
$732$	0	0
$733$	− 452.139i	− 0.616834i	−0.951251	−	0.308417i	$-0.900201\pi$
$733$	− 452.139i	− 0.616834i	0.951251	−	0.308417i	$-0.0997992\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.0579	−0.0326430
$738$	0	0
$739$	42.9916	0.0581754	0.0290877	−	0.999577i	$-0.490740\pi$
$739$	42.9916	0.0581754	0.0290877	+	0.999577i	$0.490740\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 51.4541i	− 0.0692518i	−0.999400	−	0.0346259i	$-0.988976\pi$
$743$	− 51.4541i	− 0.0692518i	0.999400	−	0.0346259i	$-0.0110240\pi$
$744$	0	0
$745$	−436.868	−0.586399
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 84.5620i	− 0.112900i
$750$	0	0
$751$	− 390.652i	− 0.520175i	−0.965585	−	0.260088i	$-0.916249\pi$
$751$	− 390.652i	− 0.520175i	0.965585	−	0.260088i	$-0.0837514\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	326.055	0.431862
$756$	0	0
$757$	310.240i	0.409828i	0.978780	+	0.204914i	$0.0656915\pi$
$757$	310.240i	0.409828i	−0.978780	+	0.204914i	$0.934309\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	859.524	1.12947	0.564733	−	0.825273i	$-0.308979\pi$
$761$	859.524	1.12947	0.564733	+	0.825273i	$0.308979\pi$
$762$	0	0
$763$	−481.795	−0.631448
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1948.67i	− 2.54064i
$768$	0	0
$769$	1408.40	1.83147	0.915733	−	0.401787i	$-0.131611\pi$
$769$	1408.40	1.83147	0.915733	+	0.401787i	$0.131611\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1297.40i	− 1.67839i	−0.543828	−	0.839197i	$-0.683025\pi$
$773$	− 1297.40i	− 1.67839i	0.543828	−	0.839197i	$-0.316975\pi$
$774$	0	0
$775$	− 1519.72i	− 1.96093i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1153.36	−1.48056
$780$	0	0
$781$	− 155.743i	− 0.199415i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	182.691	0.232727
$786$	0	0
$787$	390.371	0.496024	0.248012	−	0.968757i	$-0.420223\pi$
$787$	390.371	0.496024	0.248012	+	0.968757i	$0.420223\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	128.736i	0.162751i
$792$	0	0
$793$	−578.762	−0.729838
$794$	0	0
$795$	0	0
$796$	0	0
$797$	480.484i	0.602866i	0.953487	+	0.301433i	$0.0974649\pi$
$797$	480.484i	0.602866i	−0.953487	+	0.301433i	$0.902535\pi$
$798$	0	0
$799$	289.768i	0.362664i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	109.942	0.136915
$804$	0	0
$805$	690.289i	0.857502i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1074.34	1.32798	0.663991	−	0.747741i	$-0.268861\pi$
$809$	1074.34	1.32798	0.663991	+	0.747741i	$0.268861\pi$
$810$	0	0
$811$	740.795	0.913434	0.456717	−	0.889612i	$-0.349025\pi$
$811$	740.795	0.913434	0.456717	+	0.889612i	$0.349025\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2200.13i	2.69954i
$816$	0	0
$817$	915.452	1.12050
$818$	0	0
$819$	0	0
$820$	0	0
$821$	216.891i	0.264180i	0.991238	+	0.132090i	$0.0421687\pi$
$821$	216.891i	0.264180i	−0.991238	+	0.132090i	$0.957831\pi$
$822$	0	0
$823$	1368.30i	1.66257i	0.555844	+	0.831287i	$0.312395\pi$
$823$	1368.30i	1.66257i	−0.555844	+	0.831287i	$0.687605\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	936.451	1.13235	0.566174	−	0.824286i	$-0.308423\pi$
$827$	936.451	1.13235	0.566174	+	0.824286i	$0.308423\pi$
$828$	0	0
$829$	− 17.5189i	− 0.0211325i	−0.999944	−	0.0105663i	$-0.996637\pi$
$829$	− 17.5189i	− 0.0211325i	0.999944	−	0.0105663i	$-0.00336341\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−567.381	−0.681130
$834$	0	0
$835$	1955.27	2.34164
$836$	0	0
$837$	0	0
$838$	0	0
$839$	115.442i	0.137595i	0.997631	+	0.0687976i	$0.0219163\pi$
$839$	115.442i	0.137595i	−0.997631	+	0.0687976i	$0.978084\pi$
$840$	0	0
$841$	−365.915	−0.435095
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2235.97i	2.64612i
$846$	0	0
$847$	− 280.929i	− 0.331676i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1043.32	1.22599
$852$	0	0
$853$	1273.82i	1.49335i	0.665191	+	0.746673i	$0.268350\pi$
$853$	1273.82i	1.49335i	−0.665191	+	0.746673i	$0.731650\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	517.583	0.603947	0.301973	−	0.953316i	$-0.402355\pi$
$857$	517.583	0.603947	0.301973	+	0.953316i	$0.402355\pi$
$858$	0	0
$859$	354.668	0.412885	0.206443	−	0.978459i	$-0.433811\pi$
$859$	354.668	0.412885	0.206443	+	0.978459i	$0.433811\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1069.96i	1.23982i	0.784673	+	0.619910i	$0.212831\pi$
$863$	1069.96i	1.23982i	−0.784673	+	0.619910i	$0.787169\pi$
$864$	0	0
$865$	2459.57	2.84343
$866$	0	0
$867$	0	0
$868$	0	0
$869$	199.809i	0.229930i
$870$	0	0
$871$	− 208.224i	− 0.239063i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−409.324	−0.467799
$876$	0	0
$877$	101.572i	0.115818i	0.998322	+	0.0579088i	$0.0184433\pi$
$877$	101.572i	0.115818i	−0.998322	+	0.0579088i	$0.981557\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−858.174	−0.974091	−0.487045	−	0.873377i	$-0.661925\pi$
$881$	−858.174	−0.974091	−0.487045	+	0.873377i	$0.661925\pi$
$882$	0	0
$883$	437.517	0.495489	0.247745	−	0.968825i	$-0.420311\pi$
$883$	437.517	0.495489	0.247745	+	0.968825i	$0.420311\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 938.095i	− 1.05760i	−0.848745	−	0.528802i	$-0.822641\pi$
$887$	− 938.095i	− 1.05760i	0.848745	−	0.528802i	$-0.177359\pi$
$888$	0	0
$889$	−181.564	−0.204234
$890$	0	0
$891$	0	0
$892$	0	0
$893$	721.107i	0.807510i
$894$	0	0
$895$	− 2554.12i	− 2.85376i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1171.93	−1.30359
$900$	0	0
$901$	− 347.553i	− 0.385741i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	460.556	0.508902
$906$	0	0
$907$	409.225	0.451185	0.225593	−	0.974222i	$-0.427568\pi$
$907$	409.225	0.451185	0.225593	+	0.974222i	$0.427568\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1544.02i	1.69486i	0.530907	+	0.847430i	$0.321852\pi$
$911$	1544.02i	1.69486i	−0.530907	+	0.847430i	$0.678148\pi$
$912$	0	0
$913$	143.560	0.157240
$914$	0	0
$915$	0	0
$916$	0	0
$917$	388.736i	0.423921i
$918$	0	0
$919$	− 1204.58i	− 1.31075i	−0.755305	−	0.655374i	$-0.772511\pi$
$919$	− 1204.58i	− 1.31075i	0.755305	−	0.655374i	$-0.227489\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1347.97	1.46042
$924$	0	0
$925$	1390.03i	1.50273i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1317.76	−1.41847	−0.709234	−	0.704973i	$-0.750959\pi$
$929$	−1317.76	−1.41847	−0.709234	+	0.704973i	$0.750959\pi$
$930$	0	0
$931$	−1411.97	−1.51661
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 266.163i	− 0.284667i
$936$	0	0
$937$	−649.172	−0.692820	−0.346410	−	0.938083i	$-0.612599\pi$
$937$	−649.172	−0.692820	−0.346410	+	0.938083i	$0.612599\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1352.72i	1.43753i	0.695252	+	0.718766i	$0.255293\pi$
$941$	1352.72i	1.43753i	−0.695252	+	0.718766i	$0.744707\pi$
$942$	0	0
$943$	1189.10i	1.26097i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1194.59	1.26145	0.630723	−	0.776008i	$-0.282758\pi$
$947$	1194.59	1.26145	0.630723	+	0.776008i	$0.282758\pi$
$948$	0	0
$949$	951.562i	1.00270i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−681.561	−0.715174	−0.357587	−	0.933880i	$-0.616400\pi$
$953$	−681.561	−0.715174	−0.357587	+	0.933880i	$0.616400\pi$
$954$	0	0
$955$	2113.48	2.21307
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 533.440i	− 0.556246i
$960$	0	0
$961$	−176.950	−0.184131
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 553.244i	− 0.573310i
$966$	0	0
$967$	1050.38i	1.08623i	0.839658	+	0.543115i	$0.182755\pi$
$967$	1050.38i	1.08623i	−0.839658	+	0.543115i	$0.817245\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1307.27	1.34631	0.673157	−	0.739500i	$-0.264938\pi$
$971$	1307.27	1.34631	0.673157	+	0.739500i	$0.264938\pi$
$972$	0	0
$973$	92.3990i	0.0949630i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.7045	0.0170977	0.00854886	−	0.999963i	$-0.497279\pi$
$977$	16.7045	0.0170977	0.00854886	+	0.999963i	$0.497279\pi$
$978$	0	0
$979$	336.914	0.344141
$980$	0	0
$981$	0	0
$982$	0	0
$983$	373.268i	0.379723i	0.981811	+	0.189862i	$0.0608040\pi$
$983$	373.268i	0.379723i	−0.981811	+	0.189862i	$0.939196\pi$
$984$	0	0
$985$	−1275.71	−1.29513
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 943.819i	− 0.954316i
$990$	0	0
$991$	1028.67i	1.03801i	0.854772	+	0.519004i	$0.173697\pi$
$991$	1028.67i	1.03801i	−0.854772	+	0.519004i	$0.826303\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−309.487	−0.311043
$996$	0	0
$997$	− 968.595i	− 0.971509i	−0.874095	−	0.485755i	$-0.838545\pi$
$997$	− 968.595i	− 0.971509i	0.874095	−	0.485755i	$-0.161455\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.b.i.1567.2	yes	12
3.2	odd	2		1728.3.b.j.1567.12	yes	12
4.3	odd	2	inner	1728.3.b.i.1567.1	✓	12
8.3	odd	2	inner	1728.3.b.i.1567.11	yes	12
8.5	even	2	inner	1728.3.b.i.1567.12	yes	12
12.11	even	2		1728.3.b.j.1567.11	yes	12
24.5	odd	2		1728.3.b.j.1567.2	yes	12
24.11	even	2		1728.3.b.j.1567.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.b.i.1567.1	✓	12	4.3	odd	2	inner
1728.3.b.i.1567.2	yes	12	1.1	even	1	trivial
1728.3.b.i.1567.11	yes	12	8.3	odd	2	inner
1728.3.b.i.1567.12	yes	12	8.5	even	2	inner
1728.3.b.j.1567.1	yes	12	24.11	even	2
1728.3.b.j.1567.2	yes	12	24.5	odd	2
1728.3.b.j.1567.11	yes	12	12.11	even	2
1728.3.b.j.1567.12	yes	12	3.2	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.36964i q^{5} +2.43910i q^{7} +O(q^{10})\)	\(q-8.36964i q^{5} +2.43910i q^{7} -2.41295 q^{11} -20.8843i q^{13} -13.1793 q^{17} -32.7976 q^{19} +33.8139i q^{23} -45.0508 q^{25} +34.7407i q^{29} +33.7335i q^{31} +20.4143 q^{35} -30.8546i q^{37} +35.1659 q^{41} -27.9121 q^{43} -21.9865i q^{47} +43.0508 q^{49} +26.3710i q^{53} +20.1955i q^{55} +93.3081 q^{59} -27.7128i q^{61} -174.794 q^{65} +9.97035 q^{67} +64.5448i q^{71} -45.5636 q^{73} -5.88541i q^{77} -82.8072i q^{79} -59.4956 q^{83} +110.306i q^{85} -139.628 q^{89} +50.9387 q^{91} +274.504i q^{95} +150.716 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 48 q^{17} - 72 q^{25} - 336 q^{41} + 48 q^{49} - 912 q^{65} + 60 q^{73} - 1248 q^{89} - 204 q^{97}+O(q^{100}) \) 12 * q - 48 * q^17 - 72 * q^25 - 336 * q^41 + 48 * q^49 - 912 * q^65 + 60 * q^73 - 1248 * q^89 - 204 * q^97

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