LMFDB - Embedded newform 2736.3.o.n.721.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.o (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:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.184143974400.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 215x^{4} - 568x^{3} - 1022x^{2} + 1320x + 2628 $ x^8 - 4x^7 - 22x^6 + 80x^5 + 215x^4 - 568x^3 - 1022x^2 + 1320*x + 2628
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.7
Root			$3.87998 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.n.721.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.84873 q^{5} +3.94975 q^{7} +O(q^{10})$	$q+6.84873 q^{5} +3.94975 q^{7} -15.1751 q^{11} -1.55708i q^{13} -2.94923 q^{17} +(-16.0741 - 10.1303i) q^{19} -20.2249 q^{23} +21.9051 q^{25} -4.45583i q^{29} +3.63817i q^{31} +27.0508 q^{35} +17.6705i q^{37} +76.0775i q^{41} -26.7041 q^{43} -48.9731 q^{47} -33.3995 q^{49} +49.0610i q^{53} -103.930 q^{55} +97.9347i q^{59} -105.291 q^{61} -10.6640i q^{65} -129.286i q^{67} +130.161i q^{71} -15.6005 q^{73} -59.9380 q^{77} +113.344i q^{79} +62.5629 q^{83} -20.1985 q^{85} -134.617i q^{89} -6.15007i q^{91} +(-110.087 - 69.3797i) q^{95} -116.736i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{5} + 12 q^{7}+O(q^{10}) $ 8 * q - 4 * q^5 + 12 * q^7	$ 8 q - 4 q^{5} + 12 q^{7} + 4 q^{11} - 4 q^{17} + 36 q^{19} - 56 q^{23} + 140 q^{25} + 236 q^{35} - 100 q^{43} - 188 q^{47} - 36 q^{49} - 28 q^{55} - 180 q^{61} - 356 q^{73} - 68 q^{77} + 136 q^{83} + 148 q^{85} - 140 q^{95}+O(q^{100}) $ 8 * q - 4 * q^5 + 12 * q^7 + 4 * q^11 - 4 * q^17 + 36 * q^19 - 56 * q^23 + 140 * q^25 + 236 * q^35 - 100 * q^43 - 188 * q^47 - 36 * q^49 - 28 * q^55 - 180 * q^61 - 356 * q^73 - 68 * q^77 + 136 * q^83 + 148 * q^85 - 140 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	6.84873			1.36975			0.684873	−	0.728662i	$-0.259858\pi$
$5$	6.84873			1.36975			0.684873	+	0.728662i	$0.259858\pi$
$6$		0			0
$7$	3.94975			0.564250			0.282125	−	0.959378i	$-0.408961\pi$
$7$	3.94975			0.564250			0.282125	+	0.959378i	$0.408961\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−15.1751			−1.37956			−0.689779	−	0.724020i	$-0.742292\pi$
$11$	−15.1751			−1.37956			−0.689779	+	0.724020i	$0.742292\pi$
$12$		0			0
$13$		−	1.55708i		−	0.119775i	−0.998205	−	0.0598876i	$-0.980926\pi$
$13$		−	1.55708i		−	0.119775i	0.998205	−	0.0598876i	$-0.0190742\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.94923			−0.173484			−0.0867420	−	0.996231i	$-0.527646\pi$
$17$	−2.94923			−0.173484			−0.0867420	+	0.996231i	$0.527646\pi$
$18$		0			0
$19$	−16.0741	−	10.1303i	−0.846006	−	0.533174i
$19$	−16.0741	−	10.1303i	−0.846006	−	0.533174i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−20.2249			−0.879342			−0.439671	−	0.898159i	$-0.644905\pi$
$23$	−20.2249			−0.879342			−0.439671	+	0.898159i	$0.644905\pi$
$24$		0			0
$25$	21.9051			0.876204
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.45583i		−	0.153649i	−0.997045	−	0.0768247i	$-0.975522\pi$
$29$		−	4.45583i		−	0.153649i	0.997045	−	0.0768247i	$-0.0244782\pi$
$30$		0			0
$31$			3.63817i			0.117360i	0.998277	+	0.0586802i	$0.0186892\pi$
$31$			3.63817i			0.117360i	−0.998277	+	0.0586802i	$0.981311\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	27.0508			0.772879
$36$		0			0
$37$			17.6705i			0.477580i	0.971071	+	0.238790i	$0.0767507\pi$
$37$			17.6705i			0.477580i	−0.971071	+	0.238790i	$0.923249\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			76.0775i			1.85555i	0.373142	+	0.927774i	$0.378281\pi$
$41$			76.0775i			1.85555i	−0.373142	+	0.927774i	$0.621719\pi$
$42$		0			0
$43$	−26.7041			−0.621026			−0.310513	−	0.950569i	$-0.600501\pi$
$43$	−26.7041			−0.621026			−0.310513	+	0.950569i	$0.600501\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−48.9731			−1.04198			−0.520990	−	0.853563i	$-0.674437\pi$
$47$	−48.9731			−1.04198			−0.520990	+	0.853563i	$0.674437\pi$
$48$		0			0
$49$	−33.3995			−0.681622
$50$		0			0
$51$		0			0
$52$		0			0
$53$			49.0610i			0.925679i	0.886442	+	0.462839i	$0.153169\pi$
$53$			49.0610i			0.925679i	−0.886442	+	0.462839i	$0.846831\pi$
$54$		0			0
$55$	−103.930			−1.88964
$56$		0			0
$57$		0			0
$58$		0			0
$59$			97.9347i			1.65991i	0.557830	+	0.829955i	$0.311634\pi$
$59$			97.9347i			1.65991i	−0.557830	+	0.829955i	$0.688366\pi$
$60$		0			0
$61$	−105.291			−1.72608			−0.863040	−	0.505136i	$-0.831442\pi$
$61$	−105.291			−1.72608			−0.863040	+	0.505136i	$0.831442\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	10.6640i		−	0.164062i
$66$		0			0
$67$		−	129.286i		−	1.92964i	−0.262917	−	0.964819i	$-0.584684\pi$
$67$		−	129.286i		−	1.92964i	0.262917	−	0.964819i	$-0.415316\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			130.161i			1.83326i	0.399737	+	0.916630i	$0.369101\pi$
$71$			130.161i			1.83326i	−0.399737	+	0.916630i	$0.630899\pi$
$72$		0			0
$73$	−15.6005			−0.213706			−0.106853	−	0.994275i	$-0.534077\pi$
$73$	−15.6005			−0.213706			−0.106853	+	0.994275i	$0.534077\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−59.9380			−0.778415
$78$		0			0
$79$			113.344i			1.43473i	0.696697	+	0.717366i	$0.254652\pi$
$79$			113.344i			1.43473i	−0.696697	+	0.717366i	$0.745348\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	62.5629			0.753770			0.376885	−	0.926260i	$-0.376995\pi$
$83$	62.5629			0.753770			0.376885	+	0.926260i	$0.376995\pi$
$84$		0			0
$85$	−20.1985			−0.237629
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	134.617i		−	1.51255i	−0.654252	−	0.756277i	$-0.727016\pi$
$89$		−	134.617i		−	1.51255i	0.654252	−	0.756277i	$-0.272984\pi$
$90$		0			0
$91$		−	6.15007i		−	0.0675832i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−110.087	−	69.3797i	−1.15881	−	0.730312i
$96$		0			0
$97$		−	116.736i		−	1.20346i	−0.798699	−	0.601731i	$-0.794478\pi$
$97$		−	116.736i		−	1.20346i	0.798699	−	0.601731i	$-0.205522\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	181.084			1.79291			0.896456	−	0.443132i	$-0.146133\pi$
$101$	181.084			1.79291			0.896456	+	0.443132i	$0.146133\pi$
$102$		0			0
$103$			64.2671i			0.623952i	0.950090	+	0.311976i	$0.100991\pi$
$103$			64.2671i			0.623952i	−0.950090	+	0.311976i	$0.899009\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			50.4107i			0.471128i	0.971859	+	0.235564i	$0.0756937\pi$
$107$			50.4107i			0.471128i	−0.971859	+	0.235564i	$0.924306\pi$
$108$		0			0
$109$		−	8.83703i		−	0.0810737i	−0.999178	−	0.0405368i	$-0.987093\pi$
$109$		−	8.83703i		−	0.0810737i	0.999178	−	0.0405368i	$-0.0129068\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			104.143i			0.921618i	0.887499	+	0.460809i	$0.152441\pi$
$113$			104.143i			0.921618i	−0.887499	+	0.460809i	$0.847559\pi$
$114$		0			0
$115$	−138.515			−1.20447
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−11.6487			−0.0978884
$120$		0			0
$121$	109.285			0.903179
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−21.1962			−0.169569
$126$		0			0
$127$			144.856i			1.14060i	0.821436	+	0.570301i	$0.193173\pi$
$127$			144.856i			1.14060i	−0.821436	+	0.570301i	$0.806827\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	68.6024			0.523683			0.261841	−	0.965111i	$-0.415670\pi$
$131$	68.6024			0.523683			0.261841	+	0.965111i	$0.415670\pi$
$132$		0			0
$133$	−63.4887	−	40.0121i	−0.477359	−	0.300843i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−55.9677			−0.408523			−0.204262	−	0.978916i	$-0.565479\pi$
$137$	−55.9677			−0.408523			−0.204262	+	0.978916i	$0.565479\pi$
$138$		0			0
$139$	171.838			1.23624			0.618122	−	0.786082i	$-0.287894\pi$
$139$	171.838			1.23624			0.618122	+	0.786082i	$0.287894\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			23.6289i			0.165237i
$144$		0			0
$145$		−	30.5168i		−	0.210461i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−166.212			−1.11552			−0.557759	−	0.830003i	$-0.688338\pi$
$149$	−166.212			−1.11552			−0.557759	+	0.830003i	$0.688338\pi$
$150$		0			0
$151$		−	141.248i		−	0.935419i	−0.883882	−	0.467709i	$-0.845079\pi$
$151$		−	141.248i		−	0.935419i	0.883882	−	0.467709i	$-0.154921\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			24.9169i			0.160754i
$156$		0			0
$157$	−182.180			−1.16038			−0.580190	−	0.814481i	$-0.697022\pi$
$157$	−182.180			−1.16038			−0.580190	+	0.814481i	$0.697022\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−79.8832			−0.496169
$162$		0			0
$163$	144.215			0.884755			0.442377	−	0.896829i	$-0.354135\pi$
$163$	144.215			0.884755			0.442377	+	0.896829i	$0.354135\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	205.489i		−	1.23047i	−0.788343	−	0.615236i	$-0.789061\pi$
$167$		−	205.489i		−	1.23047i	0.788343	−	0.615236i	$-0.210939\pi$
$168$		0			0
$169$	166.576			0.985654
$170$		0			0
$171$		0			0
$172$		0			0
$173$			76.7474i			0.443627i	0.975089	+	0.221813i	$0.0711976\pi$
$173$			76.7474i			0.443627i	−0.975089	+	0.221813i	$0.928802\pi$
$174$		0			0
$175$	86.5197			0.494398
$176$		0			0
$177$		0			0
$178$		0			0
$179$			102.452i			0.572359i	0.958176	+	0.286179i	$0.0923853\pi$
$179$			102.452i			0.572359i	−0.958176	+	0.286179i	$0.907615\pi$
$180$		0			0
$181$		−	65.0036i		−	0.359136i	−0.983746	−	0.179568i	$-0.942530\pi$
$181$		−	65.0036i		−	0.359136i	0.983746	−	0.179568i	$-0.0574700\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			121.020i			0.654163i
$186$		0			0
$187$	44.7549			0.239331
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.68904			−0.0140787			−0.00703936	−	0.999975i	$-0.502241\pi$
$191$	−2.68904			−0.0140787			−0.00703936	+	0.999975i	$0.502241\pi$
$192$		0			0
$193$		−	219.173i		−	1.13561i	−0.823163	−	0.567805i	$-0.807793\pi$
$193$		−	219.173i		−	1.13561i	0.823163	−	0.567805i	$-0.192207\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−152.168			−0.772428			−0.386214	−	0.922409i	$-0.626217\pi$
$197$	−152.168			−0.772428			−0.386214	+	0.922409i	$0.626217\pi$
$198$		0			0
$199$	90.2747			0.453642			0.226821	−	0.973937i	$-0.427167\pi$
$199$	90.2747			0.453642			0.226821	+	0.973937i	$0.427167\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	17.5994i		−	0.0866967i
$204$		0			0
$205$			521.034i			2.54163i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	243.927	+	153.729i	1.16711	+	0.735544i
$210$		0			0
$211$		−	205.002i		−	0.971574i	−0.874077	−	0.485787i	$-0.838533\pi$
$211$		−	205.002i		−	0.971574i	0.874077	−	0.485787i	$-0.161467\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−182.889			−0.850647
$216$		0			0
$217$			14.3699i			0.0662206i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			4.59218i			0.0207791i
$222$		0			0
$223$		−	277.603i		−	1.24486i	−0.782677	−	0.622428i	$-0.786146\pi$
$223$		−	277.603i		−	1.24486i	0.782677	−	0.622428i	$-0.213854\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	65.1981i		−	0.287216i	−0.989635	−	0.143608i	$-0.954130\pi$
$227$		−	65.1981i		−	0.287216i	0.989635	−	0.143608i	$-0.0458705\pi$
$228$		0			0
$229$	87.0612			0.380180			0.190090	−	0.981767i	$-0.439122\pi$
$229$	87.0612			0.380180			0.190090	+	0.981767i	$0.439122\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−312.923			−1.34302			−0.671508	−	0.740997i	$-0.734353\pi$
$233$	−312.923			−1.34302			−0.671508	+	0.740997i	$0.734353\pi$
$234$		0			0
$235$	−335.403			−1.42725
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−263.792			−1.10373			−0.551867	−	0.833932i	$-0.686084\pi$
$239$	−263.792			−1.10373			−0.551867	+	0.833932i	$0.686084\pi$
$240$		0			0
$241$			168.628i			0.699702i	0.936805	+	0.349851i	$0.113768\pi$
$241$			168.628i			0.699702i	−0.936805	+	0.349851i	$0.886232\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−228.744			−0.933649
$246$		0			0
$247$	−15.7737	+	25.0286i	−0.0638610	+	0.101331i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−88.4188			−0.352266			−0.176133	−	0.984366i	$-0.556359\pi$
$251$	−88.4188			−0.352266			−0.176133	+	0.984366i	$0.556359\pi$
$252$		0			0
$253$	306.915			1.21310
$254$		0			0
$255$		0			0
$256$		0			0
$257$			339.207i			1.31987i	0.751323	+	0.659935i	$0.229416\pi$
$257$			339.207i			1.31987i	−0.751323	+	0.659935i	$0.770584\pi$
$258$		0			0
$259$			69.7939i			0.269474i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	464.614			1.76659			0.883296	−	0.468815i	$-0.155319\pi$
$263$	464.614			1.76659			0.883296	+	0.468815i	$0.155319\pi$
$264$		0			0
$265$			336.005i			1.26794i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	163.779i		−	0.608844i	−0.952537	−	0.304422i	$-0.901537\pi$
$269$		−	163.779i		−	0.608844i	0.952537	−	0.304422i	$-0.0984633\pi$
$270$		0			0
$271$	44.7542			0.165145			0.0825723	−	0.996585i	$-0.473686\pi$
$271$	44.7542			0.165145			0.0825723	+	0.996585i	$0.473686\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−332.413			−1.20877
$276$		0			0
$277$	−79.1110			−0.285599			−0.142800	−	0.989752i	$-0.545610\pi$
$277$	−79.1110			−0.285599			−0.142800	+	0.989752i	$0.545610\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	491.314i		−	1.74845i	−0.485522	−	0.874225i	$-0.661370\pi$
$281$		−	491.314i		−	1.74845i	0.485522	−	0.874225i	$-0.338630\pi$
$282$		0			0
$283$	−425.727			−1.50434			−0.752168	−	0.658971i	$-0.770992\pi$
$283$	−425.727			−1.50434			−0.752168	+	0.658971i	$0.770992\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			300.487i			1.04699i
$288$		0			0
$289$	−280.302			−0.969903
$290$		0			0
$291$		0			0
$292$		0			0
$293$			336.003i			1.14677i	0.819287	+	0.573384i	$0.194370\pi$
$293$			336.003i			1.14677i	−0.819287	+	0.573384i	$0.805630\pi$
$294$		0			0
$295$			670.728i			2.27366i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			31.4917i			0.105323i
$300$		0			0
$301$	−105.475			−0.350414
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−721.109			−2.36429
$306$		0			0
$307$			228.329i			0.743742i	0.928284	+	0.371871i	$0.121284\pi$
$307$			228.329i			0.743742i	−0.928284	+	0.371871i	$0.878716\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−374.720			−1.20489			−0.602444	−	0.798161i	$-0.705806\pi$
$311$	−374.720			−1.20489			−0.602444	+	0.798161i	$0.705806\pi$
$312$		0			0
$313$	−12.9612			−0.0414096			−0.0207048	−	0.999786i	$-0.506591\pi$
$313$	−12.9612			−0.0414096			−0.0207048	+	0.999786i	$0.506591\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	287.779i		−	0.907819i	−0.891048	−	0.453910i	$-0.850029\pi$
$317$		−	287.779i		−	0.907819i	0.891048	−	0.453910i	$-0.149971\pi$
$318$		0			0
$319$			67.6179i			0.211968i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	47.4062	+	29.8766i	0.146769	+	0.0924971i
$324$		0			0
$325$		−	34.1079i		−	0.104948i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−193.431			−0.587938
$330$		0			0
$331$		−	303.001i		−	0.915411i	−0.889104	−	0.457705i	$-0.848671\pi$
$331$		−	303.001i		−	0.915411i	0.889104	−	0.457705i	$-0.151329\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	885.443i		−	2.64311i
$336$		0			0
$337$			289.383i			0.858704i	0.903137	+	0.429352i	$0.141258\pi$
$337$			289.383i			0.858704i	−0.903137	+	0.429352i	$0.858742\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	55.2097i		−	0.161905i
$342$		0			0
$343$	−325.457			−0.948855
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−294.427			−0.848493			−0.424246	−	0.905547i	$-0.639461\pi$
$347$	−294.427			−0.848493			−0.424246	+	0.905547i	$0.639461\pi$
$348$		0			0
$349$	−320.982			−0.919720			−0.459860	−	0.887991i	$-0.652100\pi$
$349$	−320.982			−0.919720			−0.459860	+	0.887991i	$0.652100\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	134.912			0.382187			0.191093	−	0.981572i	$-0.438797\pi$
$353$	134.912			0.382187			0.191093	+	0.981572i	$0.438797\pi$
$354$		0			0
$355$			891.441i			2.51110i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−400.755			−1.11631			−0.558154	−	0.829737i	$-0.688490\pi$
$359$	−400.755			−1.11631			−0.558154	+	0.829737i	$0.688490\pi$
$360$		0			0
$361$	155.754	+	325.671i	0.431452	+	0.902136i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−106.844			−0.292723
$366$		0			0
$367$	638.456			1.73966			0.869831	−	0.493350i	$-0.164228\pi$
$367$	638.456			1.73966			0.869831	+	0.493350i	$0.164228\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			193.779i			0.522314i
$372$		0			0
$373$			608.583i			1.63159i	0.578342	+	0.815795i	$0.303700\pi$
$373$			608.583i			1.63159i	−0.578342	+	0.815795i	$0.696300\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.93808			−0.0184034
$378$		0			0
$379$			152.474i			0.402305i	0.979560	+	0.201153i	$0.0644687\pi$
$379$			152.474i			0.402305i	−0.979560	+	0.201153i	$0.935531\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			236.214i			0.616748i	0.951265	+	0.308374i	$0.0997848\pi$
$383$			236.214i			0.616748i	−0.951265	+	0.308374i	$0.900215\pi$
$384$		0			0
$385$	−410.499			−1.06623
$386$		0			0
$387$		0			0
$388$		0			0
$389$	305.806			0.786133			0.393067	−	0.919510i	$-0.371414\pi$
$389$	305.806			0.786133			0.393067	+	0.919510i	$0.371414\pi$
$390$		0			0
$391$	59.6478			0.152552
$392$		0			0
$393$		0			0
$394$		0			0
$395$			776.261i			1.96522i
$396$		0			0
$397$	−46.6168			−0.117423			−0.0587114	−	0.998275i	$-0.518699\pi$
$397$	−46.6168			−0.117423			−0.0587114	+	0.998275i	$0.518699\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	373.096i		−	0.930414i	−0.885202	−	0.465207i	$-0.845980\pi$
$401$		−	373.096i		−	0.930414i	0.885202	−	0.465207i	$-0.154020\pi$
$402$		0			0
$403$	5.66492			0.0140569
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	268.151i		−	0.658849i
$408$		0			0
$409$		−	408.783i		−	0.999470i	−0.866178	−	0.499735i	$-0.833431\pi$
$409$		−	408.783i		−	0.999470i	0.866178	−	0.499735i	$-0.166569\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			386.818i			0.936604i
$414$		0			0
$415$	428.476			1.03247
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−100.128			−0.238970			−0.119485	−	0.992836i	$-0.538124\pi$
$419$	−100.128			−0.238970			−0.119485	+	0.992836i	$0.538124\pi$
$420$		0			0
$421$		−	511.871i		−	1.21585i	−0.793996	−	0.607923i	$-0.792003\pi$
$421$		−	511.871i		−	1.21585i	0.793996	−	0.607923i	$-0.207997\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−64.6032			−0.152007
$426$		0			0
$427$	−415.873			−0.973940
$428$		0			0
$429$		0			0
$430$		0			0
$431$			66.9213i			0.155270i	0.996982	+	0.0776349i	$0.0247369\pi$
$431$			66.9213i			0.155270i	−0.996982	+	0.0776349i	$0.975263\pi$
$432$		0			0
$433$		−	235.456i		−	0.543778i	−0.962329	−	0.271889i	$-0.912352\pi$
$433$		−	235.456i		−	0.543778i	0.962329	−	0.271889i	$-0.0876484\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	325.097	+	204.884i	0.743928	+	0.468842i
$438$		0			0
$439$			490.771i			1.11793i	0.829191	+	0.558965i	$0.188801\pi$
$439$			490.771i			1.11793i	−0.829191	+	0.558965i	$0.811199\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	132.347			0.298751			0.149375	−	0.988781i	$-0.452274\pi$
$443$	132.347			0.298751			0.149375	+	0.988781i	$0.452274\pi$
$444$		0			0
$445$		−	921.957i		−	2.07181i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			207.049i			0.461133i	0.973057	+	0.230566i	$0.0740579\pi$
$449$			207.049i			0.461133i	−0.973057	+	0.230566i	$0.925942\pi$
$450$		0			0
$451$		−	1154.49i		−	2.55984i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	42.1202i		−	0.0925718i
$456$		0			0
$457$	−813.901			−1.78096			−0.890482	−	0.455018i	$-0.849633\pi$
$457$	−813.901			−1.78096			−0.890482	+	0.455018i	$0.849633\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.43527			0.0182978			0.00914889	−	0.999958i	$-0.497088\pi$
$461$	8.43527			0.0182978			0.00914889	+	0.999958i	$0.497088\pi$
$462$		0			0
$463$	−165.437			−0.357316			−0.178658	−	0.983911i	$-0.557176\pi$
$463$	−165.437			−0.357316			−0.178658	+	0.983911i	$0.557176\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	701.232			1.50157			0.750784	−	0.660548i	$-0.229676\pi$
$467$	701.232			1.50157			0.750784	+	0.660548i	$0.229676\pi$
$468$		0			0
$469$		−	510.646i		−	1.08880i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	405.238			0.856740
$474$		0			0
$475$	−352.105	−	221.905i	−0.741274	−	0.467169i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	306.927			0.640766			0.320383	−	0.947288i	$-0.396188\pi$
$479$	306.927			0.640766			0.320383	+	0.947288i	$0.396188\pi$
$480$		0			0
$481$	27.5143			0.0572022
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	799.492i		−	1.64844i
$486$		0			0
$487$			649.613i			1.33391i	0.745099	+	0.666954i	$0.232402\pi$
$487$			649.613i			1.33391i	−0.745099	+	0.666954i	$0.767598\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	21.9061			0.0446153			0.0223077	−	0.999751i	$-0.492899\pi$
$491$	21.9061			0.0446153			0.0223077	+	0.999751i	$0.492899\pi$
$492$		0			0
$493$			13.1413i			0.0266557i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			514.105i			1.03442i
$498$		0			0
$499$	−133.896			−0.268329			−0.134164	−	0.990959i	$-0.542835\pi$
$499$	−133.896			−0.268329			−0.134164	+	0.990959i	$0.542835\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−382.719			−0.760873			−0.380436	−	0.924807i	$-0.624226\pi$
$503$	−382.719			−0.760873			−0.380436	+	0.924807i	$0.624226\pi$
$504$		0			0
$505$	1240.20			2.45584
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	379.671i		−	0.745915i	−0.927848	−	0.372957i	$-0.878344\pi$
$509$		−	379.671i		−	0.745915i	0.927848	−	0.372957i	$-0.121656\pi$
$510$		0			0
$511$	−61.6182			−0.120584
$512$		0			0
$513$		0			0
$514$		0			0
$515$			440.148i			0.854656i
$516$		0			0
$517$	743.173			1.43747
$518$		0			0
$519$		0			0
$520$		0			0
$521$			419.577i			0.805330i	0.915347	+	0.402665i	$0.131916\pi$
$521$			419.577i			0.805330i	−0.915347	+	0.402665i	$0.868084\pi$
$522$		0			0
$523$		−	363.239i		−	0.694529i	−0.937767	−	0.347264i	$-0.887111\pi$
$523$		−	363.239i		−	0.694529i	0.937767	−	0.347264i	$-0.112889\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	10.7298i		−	0.0203602i
$528$		0			0
$529$	−119.955			−0.226758
$530$		0			0
$531$		0			0
$532$		0			0
$533$	118.459			0.222249
$534$		0			0
$535$			345.249i			0.645325i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	506.841			0.940337
$540$		0			0
$541$	829.374			1.53304			0.766519	−	0.642221i	$-0.221987\pi$
$541$	829.374			1.53304			0.766519	+	0.642221i	$0.221987\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	60.5224i		−	0.111050i
$546$		0			0
$547$		−	706.759i		−	1.29206i	−0.763311	−	0.646032i	$-0.776427\pi$
$547$		−	706.759i		−	1.29206i	0.763311	−	0.646032i	$-0.223573\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−45.1389	+	71.6236i	−0.0819218	+	0.129988i
$552$		0			0
$553$			447.680i			0.809547i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	53.1087			0.0953478			0.0476739	−	0.998863i	$-0.484819\pi$
$557$	53.1087			0.0953478			0.0476739	+	0.998863i	$0.484819\pi$
$558$		0			0
$559$			41.5804i			0.0743835i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	411.166i		−	0.730312i	−0.930946	−	0.365156i	$-0.881016\pi$
$563$		−	411.166i		−	0.730312i	0.930946	−	0.365156i	$-0.118984\pi$
$564$		0			0
$565$			713.246i			1.26238i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	2.95079i		−	0.00518592i	−0.999997	−	0.00259296i	$-0.999175\pi$
$569$		−	2.95079i		−	0.00518592i	0.999997	−	0.00259296i	$-0.000825366\pi$
$570$		0			0
$571$	227.769			0.398895			0.199448	−	0.979908i	$-0.436085\pi$
$571$	227.769			0.398895			0.199448	+	0.979908i	$0.436085\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−443.028			−0.770483
$576$		0			0
$577$	−908.173			−1.57396			−0.786979	−	0.616980i	$-0.788356\pi$
$577$	−908.173			−1.57396			−0.786979	+	0.616980i	$0.788356\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	247.108			0.425315
$582$		0			0
$583$		−	744.507i		−	1.27703i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−787.659			−1.34184			−0.670919	−	0.741530i	$-0.734100\pi$
$587$	−787.659			−1.34184			−0.670919	+	0.741530i	$0.734100\pi$
$588$		0			0
$589$	36.8558	−	58.4804i	0.0625735	−	0.0992876i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−183.636			−0.309673			−0.154836	−	0.987940i	$-0.549485\pi$
$593$	−183.636			−0.309673			−0.154836	+	0.987940i	$0.549485\pi$
$594$		0			0
$595$	−79.7789			−0.134082
$596$		0			0
$597$		0			0
$598$		0			0
$599$			51.2977i			0.0856389i	0.999083	+	0.0428194i	$0.0136340\pi$
$599$			51.2977i			0.0856389i	−0.999083	+	0.0428194i	$0.986366\pi$
$600$		0			0
$601$			1118.43i			1.86095i	0.366356	+	0.930475i	$0.380605\pi$
$601$			1118.43i			1.86095i	−0.366356	+	0.930475i	$0.619395\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	748.461			1.23713
$606$		0			0
$607$		−	926.291i		−	1.52601i	−0.646390	−	0.763007i	$-0.723722\pi$
$607$		−	926.291i		−	1.52601i	0.646390	−	0.763007i	$-0.276278\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			76.2549i			0.124803i
$612$		0			0
$613$	425.005			0.693320			0.346660	−	0.937991i	$-0.387316\pi$
$613$	425.005			0.693320			0.346660	+	0.937991i	$0.387316\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−390.471			−0.632854			−0.316427	−	0.948617i	$-0.602483\pi$
$617$	−390.471			−0.632854			−0.316427	+	0.948617i	$0.602483\pi$
$618$		0			0
$619$	1019.60			1.64717			0.823587	−	0.567190i	$-0.191970\pi$
$619$	1019.60			1.64717			0.823587	+	0.567190i	$0.191970\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	531.705i		−	0.853458i
$624$		0			0
$625$	−692.794			−1.10847
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	52.1142i		−	0.0828525i
$630$		0			0
$631$	−740.158			−1.17299			−0.586496	−	0.809952i	$-0.699493\pi$
$631$	−740.158			−1.17299			−0.586496	+	0.809952i	$0.699493\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			992.083i			1.56234i
$636$		0			0
$637$			52.0056i			0.0816414i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			735.330i			1.14716i	0.819149	+	0.573580i	$0.194446\pi$
$641$			735.330i			1.14716i	−0.819149	+	0.573580i	$0.805554\pi$
$642$		0			0
$643$	−698.990			−1.08708			−0.543538	−	0.839384i	$-0.682916\pi$
$643$	−698.990			−1.08708			−0.543538	+	0.839384i	$0.682916\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−708.552			−1.09513			−0.547567	−	0.836762i	$-0.684446\pi$
$647$	−708.552			−1.09513			−0.547567	+	0.836762i	$0.684446\pi$
$648$		0			0
$649$		−	1486.17i		−	2.28994i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	656.636			1.00557			0.502784	−	0.864412i	$-0.332309\pi$
$653$	656.636			1.00557			0.502784	+	0.864412i	$0.332309\pi$
$654$		0			0
$655$	469.839			0.717312
$656$		0			0
$657$		0			0
$658$		0			0
$659$			404.014i			0.613071i	0.951859	+	0.306536i	$0.0991699\pi$
$659$			404.014i			0.613071i	−0.951859	+	0.306536i	$0.900830\pi$
$660$		0			0
$661$		−	204.858i		−	0.309922i	−0.987921	−	0.154961i	$-0.950475\pi$
$661$		−	204.858i		−	0.309922i	0.987921	−	0.154961i	$-0.0495252\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−434.817	−	274.032i	−0.653860	−	0.412079i
$666$		0			0
$667$			90.1186i			0.135110i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1597.80			2.38123
$672$		0			0
$673$		−	127.110i		−	0.188871i	−0.995531	−	0.0944357i	$-0.969895\pi$
$673$		−	127.110i		−	0.188871i	0.995531	−	0.0944357i	$-0.0301047\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	675.862i		−	0.998320i	−0.866510	−	0.499160i	$-0.833642\pi$
$677$		−	675.862i		−	0.998320i	0.866510	−	0.499160i	$-0.166358\pi$
$678$		0			0
$679$		−	461.077i		−	0.679054i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	378.282i		−	0.553854i	−0.960891	−	0.276927i	$-0.910684\pi$
$683$		−	378.282i		−	0.553854i	0.960891	−	0.276927i	$-0.0893161\pi$
$684$		0			0
$685$	−383.307			−0.559573
$686$		0			0
$687$		0			0
$688$		0			0
$689$	76.3918			0.110873
$690$		0			0
$691$	352.626			0.510312			0.255156	−	0.966900i	$-0.417873\pi$
$691$	352.626			0.510312			0.255156	+	0.966900i	$0.417873\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1176.87			1.69334
$696$		0			0
$697$		−	224.370i		−	0.321908i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1227.13			1.75054			0.875271	−	0.483632i	$-0.160683\pi$
$701$	1227.13			1.75054			0.875271	+	0.483632i	$0.160683\pi$
$702$		0			0
$703$	179.007	−	284.037i	0.254633	−	0.404035i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	715.237			1.01165
$708$		0			0
$709$	1299.03			1.83220			0.916102	−	0.400946i	$-0.131319\pi$
$709$	1299.03			1.83220			0.916102	+	0.400946i	$0.131319\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	73.5815i		−	0.103200i
$714$		0			0
$715$			161.828i			0.226332i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−79.6223			−0.110740			−0.0553702	−	0.998466i	$-0.517634\pi$
$719$	−79.6223			−0.110740			−0.0553702	+	0.998466i	$0.517634\pi$
$720$		0			0
$721$			253.839i			0.352065i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	97.6055i		−	0.134628i
$726$		0			0
$727$	−1113.22			−1.53125			−0.765626	−	0.643286i	$-0.777571\pi$
$727$	−1113.22			−1.53125			−0.765626	+	0.643286i	$0.777571\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	78.7565			0.107738
$732$		0			0
$733$	349.881			0.477327			0.238663	−	0.971102i	$-0.423291\pi$
$733$	349.881			0.477327			0.238663	+	0.971102i	$0.423291\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1961.93i			2.66205i
$738$		0			0
$739$	731.257			0.989522			0.494761	−	0.869029i	$-0.335256\pi$
$739$	731.257			0.989522			0.494761	+	0.869029i	$0.335256\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	513.328i		−	0.690885i	−0.938440	−	0.345443i	$-0.887729\pi$
$743$		−	513.328i		−	0.690885i	0.938440	−	0.345443i	$-0.112271\pi$
$744$		0			0
$745$	−1138.34			−1.52797
$746$		0			0
$747$		0			0
$748$		0			0
$749$			199.110i			0.265834i
$750$		0			0
$751$			120.844i			0.160911i	0.996758	+	0.0804554i	$0.0256375\pi$
$751$			120.844i			0.160911i	−0.996758	+	0.0804554i	$0.974363\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	967.371i		−	1.28129i
$756$		0			0
$757$	460.093			0.607785			0.303892	−	0.952706i	$-0.401714\pi$
$757$	460.093			0.607785			0.303892	+	0.952706i	$0.401714\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−285.484			−0.375143			−0.187572	−	0.982251i	$-0.560062\pi$
$761$	−285.484			−0.375143			−0.187572	+	0.982251i	$0.560062\pi$
$762$		0			0
$763$		−	34.9041i		−	0.0457458i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	152.492			0.198816
$768$		0			0
$769$	−304.690			−0.396216			−0.198108	−	0.980180i	$-0.563480\pi$
$769$	−304.690			−0.396216			−0.198108	+	0.980180i	$0.563480\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	975.225i		−	1.26161i	−0.775941	−	0.630805i	$-0.782725\pi$
$773$		−	975.225i		−	1.26161i	0.775941	−	0.630805i	$-0.217275\pi$
$774$		0			0
$775$			79.6945i			0.102832i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	770.688	−	1222.88i	0.989329	−	1.56981i
$780$		0			0
$781$		−	1975.22i		−	2.52909i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1247.70			−1.58943
$786$		0			0
$787$		−	5.90262i		−	0.00750016i	−0.999993	−	0.00375008i	$-0.998806\pi$
$787$		−	5.90262i		−	0.00750016i	0.999993	−	0.00375008i	$-0.00119369\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			411.338i			0.520023i
$792$		0			0
$793$			163.946i			0.206742i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			985.089i			1.23600i	0.786180	+	0.617998i	$0.212056\pi$
$797$			985.089i			1.23600i	−0.786180	+	0.617998i	$0.787944\pi$
$798$		0			0
$799$	144.433			0.180767
$800$		0			0
$801$		0			0
$802$		0			0
$803$	236.740			0.294819
$804$		0			0
$805$	−547.098			−0.679625
$806$		0			0
$807$		0			0
$808$		0			0
$809$	536.353			0.662983			0.331491	−	0.943458i	$-0.392448\pi$
$809$	536.353			0.662983			0.331491	+	0.943458i	$0.392448\pi$
$810$		0			0
$811$			1244.74i			1.53482i	0.641160	+	0.767408i	$0.278454\pi$
$811$			1244.74i			1.53482i	−0.641160	+	0.767408i	$0.721546\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	987.690			1.21189
$816$		0			0
$817$	429.245	+	270.520i	0.525391	+	0.331114i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	426.595			0.519604			0.259802	−	0.965662i	$-0.416343\pi$
$821$	426.595			0.519604			0.259802	+	0.965662i	$0.416343\pi$
$822$		0			0
$823$	616.996			0.749691			0.374845	−	0.927087i	$-0.377696\pi$
$823$	616.996			0.749691			0.374845	+	0.927087i	$0.377696\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	814.804i		−	0.985252i	−0.870241	−	0.492626i	$-0.836037\pi$
$827$		−	814.804i		−	0.985252i	0.870241	−	0.492626i	$-0.163963\pi$
$828$		0			0
$829$		−	949.783i		−	1.14570i	−0.819661	−	0.572849i	$-0.805838\pi$
$829$		−	949.783i		−	1.14570i	0.819661	−	0.572849i	$-0.194162\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	98.5027			0.118251
$834$		0			0
$835$		−	1407.34i		−	1.68544i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			567.492i			0.676391i	0.941076	+	0.338196i	$0.109817\pi$
$839$			567.492i			0.676391i	−0.941076	+	0.338196i	$0.890183\pi$
$840$		0			0
$841$	821.146			0.976392
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1140.83			1.35010
$846$		0			0
$847$	431.647			0.509619
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	357.382i		−	0.419956i
$852$		0			0
$853$	−1537.39			−1.80233			−0.901164	−	0.433477i	$-0.857286\pi$
$853$	−1537.39			−1.80233			−0.901164	+	0.433477i	$0.857286\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1220.29i		−	1.42391i	−0.702223	−	0.711957i	$-0.747809\pi$
$857$		−	1220.29i		−	1.42391i	0.702223	−	0.711957i	$-0.252191\pi$
$858$		0			0
$859$	494.606			0.575793			0.287896	−	0.957662i	$-0.407044\pi$
$859$	494.606			0.575793			0.287896	+	0.957662i	$0.407044\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			251.028i			0.290878i	0.989367	+	0.145439i	$0.0464595\pi$
$863$			251.028i			0.290878i	−0.989367	+	0.145439i	$0.953541\pi$
$864$		0			0
$865$			525.623i			0.607656i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1720.01i		−	1.97929i
$870$		0			0
$871$	−201.308			−0.231123
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−83.7195			−0.0956794
$876$		0			0
$877$		−	39.6379i		−	0.0451972i	−0.999745	−	0.0225986i	$-0.992806\pi$
$877$		−	39.6379i		−	0.0451972i	0.999745	−	0.0225986i	$-0.00719397\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1464.98			−1.66286			−0.831430	−	0.555629i	$-0.812478\pi$
$881$	−1464.98			−1.66286			−0.831430	+	0.555629i	$0.812478\pi$
$882$		0			0
$883$	1026.22			1.16219			0.581096	−	0.813835i	$-0.302624\pi$
$883$	1026.22			1.16219			0.581096	+	0.813835i	$0.302624\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			467.059i			0.526560i	0.964719	+	0.263280i	$0.0848043\pi$
$887$			467.059i			0.526560i	−0.964719	+	0.263280i	$0.915196\pi$
$888$		0			0
$889$			572.147i			0.643585i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	787.199	+	496.112i	0.881522	+	0.555557i
$894$		0			0
$895$			701.668i			0.783986i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	16.2111			0.0180324
$900$		0			0
$901$		−	144.692i		−	0.160590i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	445.192i		−	0.491925i
$906$		0			0
$907$			1437.13i			1.58449i	0.610201	+	0.792246i	$0.291089\pi$
$907$			1437.13i			1.58449i	−0.610201	+	0.792246i	$0.708911\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1114.61i			1.22350i	0.791052	+	0.611748i	$0.209534\pi$
$911$			1114.61i			1.22350i	−0.791052	+	0.611748i	$0.790466\pi$
$912$		0			0
$913$	−949.400			−1.03987
$914$		0			0
$915$		0			0
$916$		0			0
$917$	270.962			0.295488
$918$		0			0
$919$	78.0758			0.0849574			0.0424787	−	0.999097i	$-0.486475\pi$
$919$	78.0758			0.0849574			0.0424787	+	0.999097i	$0.486475\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	202.672			0.219579
$924$		0			0
$925$			387.073i			0.418457i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−396.926			−0.427262			−0.213631	−	0.976914i	$-0.568529\pi$
$929$	−396.926			−0.427262			−0.213631	+	0.976914i	$0.568529\pi$
$930$		0			0
$931$	536.867	+	338.347i	0.576656	+	0.363423i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	306.515			0.327823
$936$		0			0
$937$	163.341			0.174323			0.0871615	−	0.996194i	$-0.472220\pi$
$937$	163.341			0.174323			0.0871615	+	0.996194i	$0.472220\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	498.935i		−	0.530218i	−0.964218	−	0.265109i	$-0.914592\pi$
$941$		−	498.935i		−	0.530218i	0.964218	−	0.265109i	$-0.0854080\pi$
$942$		0			0
$943$		−	1538.66i		−	1.63166i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1226.45			−1.29509			−0.647545	−	0.762027i	$-0.724204\pi$
$947$	−1226.45			−1.29509			−0.647545	+	0.762027i	$0.724204\pi$
$948$		0			0
$949$			24.2912i			0.0255967i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			791.987i			0.831047i	0.909583	+	0.415523i	$0.136402\pi$
$953$			791.987i			0.831047i	−0.909583	+	0.415523i	$0.863598\pi$
$954$		0			0
$955$	−18.4165			−0.0192843
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−221.058			−0.230509
$960$		0			0
$961$	947.764			0.986227
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1501.06i		−	1.55550i
$966$		0			0
$967$	−521.534			−0.539332			−0.269666	−	0.962954i	$-0.586913\pi$
$967$	−521.534			−0.539332			−0.269666	+	0.962954i	$0.586913\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1118.43i		−	1.15183i	−0.817508	−	0.575917i	$-0.804645\pi$
$971$		−	1118.43i		−	1.15183i	0.817508	−	0.575917i	$-0.195355\pi$
$972$		0			0
$973$	678.717			0.697551
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1389.31i		−	1.42202i	−0.703184	−	0.711008i	$-0.748239\pi$
$977$		−	1389.31i		−	1.42202i	0.703184	−	0.711008i	$-0.251761\pi$
$978$		0			0
$979$			2042.84i			2.08665i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			924.244i			0.940227i	0.882606	+	0.470114i	$0.155787\pi$
$983$			924.244i			0.940227i	−0.882606	+	0.470114i	$0.844213\pi$
$984$		0			0
$985$	−1042.16			−1.05803
$986$		0			0
$987$		0			0
$988$		0			0
$989$	540.087			0.546094
$990$		0			0
$991$		−	775.227i		−	0.782268i	−0.920334	−	0.391134i	$-0.872083\pi$
$991$		−	775.227i		−	0.782268i	0.920334	−	0.391134i	$-0.127917\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	618.267			0.621374
$996$		0			0
$997$	1227.41			1.23111			0.615553	−	0.788095i	$-0.288933\pi$
$997$	1227.41			1.23111			0.615553	+	0.788095i	$0.288933\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.n.721.7		8
3.2	odd	2		912.3.o.d.721.1		8
4.3	odd	2		342.3.d.b.37.8		8
12.11	even	2		114.3.d.a.37.3	✓	8
19.18	odd	2	inner	2736.3.o.n.721.8		8
57.56	even	2		912.3.o.d.721.5		8
76.75	even	2		342.3.d.b.37.4		8
228.227	odd	2		114.3.d.a.37.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.3.d.a.37.3	✓	8	12.11	even	2
114.3.d.a.37.5	yes	8	228.227	odd	2
342.3.d.b.37.4		8	76.75	even	2
342.3.d.b.37.8		8	4.3	odd	2
912.3.o.d.721.1		8	3.2	odd	2
912.3.o.d.721.5		8	57.56	even	2
2736.3.o.n.721.7		8	1.1	even	1	trivial
2736.3.o.n.721.8		8	19.18	odd	2	inner

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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.84873 q^{5} +3.94975 q^{7} +O(q^{10})\)	\(q+6.84873 q^{5} +3.94975 q^{7} -15.1751 q^{11} -1.55708i q^{13} -2.94923 q^{17} +(-16.0741 - 10.1303i) q^{19} -20.2249 q^{23} +21.9051 q^{25} -4.45583i q^{29} +3.63817i q^{31} +27.0508 q^{35} +17.6705i q^{37} +76.0775i q^{41} -26.7041 q^{43} -48.9731 q^{47} -33.3995 q^{49} +49.0610i q^{53} -103.930 q^{55} +97.9347i q^{59} -105.291 q^{61} -10.6640i q^{65} -129.286i q^{67} +130.161i q^{71} -15.6005 q^{73} -59.9380 q^{77} +113.344i q^{79} +62.5629 q^{83} -20.1985 q^{85} -134.617i q^{89} -6.15007i q^{91} +(-110.087 - 69.3797i) q^{95} -116.736i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{5} + 12 q^{7}+O(q^{10}) \) 8 * q - 4 * q^5 + 12 * q^7	\( 8 q - 4 q^{5} + 12 q^{7} + 4 q^{11} - 4 q^{17} + 36 q^{19} - 56 q^{23} + 140 q^{25} + 236 q^{35} - 100 q^{43} - 188 q^{47} - 36 q^{49} - 28 q^{55} - 180 q^{61} - 356 q^{73} - 68 q^{77} + 136 q^{83} + 148 q^{85} - 140 q^{95}+O(q^{100}) \) 8 * q - 4 * q^5 + 12 * q^7 + 4 * q^11 - 4 * q^17 + 36 * q^19 - 56 * q^23 + 140 * q^25 + 236 * q^35 - 100 * q^43 - 188 * q^47 - 36 * q^49 - 28 * q^55 - 180 * q^61 - 356 * q^73 - 68 * q^77 + 136 * q^83 + 148 * q^85 - 140 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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