LMFDB - Embedded newform 2736.3.o.q.721.13

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 264 x^{18} + 28274 x^{16} - 1545308 x^{14} + 45358441 x^{12} - 637328868 x^{10} + \cdots + 194396337216 $ x^20 - 264x^18 + 28274x^16 - 1545308x^14 + 45358441x^12 - 637328868x^10 + 1825819356x^8 + 32794262368x^6 + 135580415344x^4 + 245217530816*x^2 + 194396337216
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32} $
Twist minimal:	no (minimal twist has level 1368)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.13
Root			$0.585690 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.q.721.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.585690 q^{5} +5.15901 q^{7} +O(q^{10})$	$q+0.585690 q^{5} +5.15901 q^{7} +11.8174 q^{11} -14.9048i q^{13} -8.91933 q^{17} +(-18.4806 - 4.41209i) q^{19} -35.6518 q^{23} -24.6570 q^{25} -7.12786i q^{29} -10.8064i q^{31} +3.02158 q^{35} +51.3985i q^{37} +33.2389i q^{41} -54.2006 q^{43} -32.6303 q^{47} -22.3846 q^{49} +91.2159i q^{53} +6.92130 q^{55} -107.615i q^{59} -23.6223 q^{61} -8.72959i q^{65} +107.283i q^{67} -72.7743i q^{71} -5.25863 q^{73} +60.9659 q^{77} +45.5813i q^{79} +132.488 q^{83} -5.22396 q^{85} +42.1195i q^{89} -76.8940i q^{91} +(-10.8239 - 2.58412i) q^{95} -143.255i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{7}+O(q^{10}) $ 20 * q + 16 * q^7	$ 20 q + 16 q^{7} + 8 q^{19} + 68 q^{25} + 128 q^{43} + 116 q^{49} - 144 q^{55} - 104 q^{61} - 88 q^{73} - 280 q^{85}+O(q^{100}) $ 20 * q + 16 * q^7 + 8 * q^19 + 68 * q^25 + 128 * q^43 + 116 * q^49 - 144 * q^55 - 104 * q^61 - 88 * q^73 - 280 * q^85

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
$5$	0.585690			0.117138			0.0585690	−	0.998283i	$-0.481346\pi$
$5$	0.585690			0.117138			0.0585690	+	0.998283i	$0.481346\pi$
$6$		0			0
$7$	5.15901			0.737002			0.368501	−	0.929627i	$-0.379871\pi$
$7$	5.15901			0.737002			0.368501	+	0.929627i	$0.379871\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	11.8174			1.07430			0.537152	−	0.843485i	$-0.319500\pi$
$11$	11.8174			1.07430			0.537152	+	0.843485i	$0.319500\pi$
$12$		0			0
$13$		−	14.9048i		−	1.14652i	−0.819372	−	0.573261i	$-0.805678\pi$
$13$		−	14.9048i		−	1.14652i	0.819372	−	0.573261i	$-0.194322\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−8.91933			−0.524667			−0.262333	−	0.964977i	$-0.584492\pi$
$17$	−8.91933			−0.524667			−0.262333	+	0.964977i	$0.584492\pi$
$18$		0			0
$19$	−18.4806	−	4.41209i	−0.972664	−	0.232215i
$19$	−18.4806	−	4.41209i	−0.972664	−	0.232215i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−35.6518			−1.55008			−0.775040	−	0.631912i	$-0.782270\pi$
$23$	−35.6518			−1.55008			−0.775040	+	0.631912i	$0.782270\pi$
$24$		0			0
$25$	−24.6570			−0.986279
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	7.12786i		−	0.245788i	−0.992420	−	0.122894i	$-0.960782\pi$
$29$		−	7.12786i		−	0.245788i	0.992420	−	0.122894i	$-0.0392176\pi$
$30$		0			0
$31$		−	10.8064i		−	0.348593i	−0.984693	−	0.174296i	$-0.944235\pi$
$31$		−	10.8064i		−	0.348593i	0.984693	−	0.174296i	$-0.0557651\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.02158			0.0863309
$36$		0			0
$37$			51.3985i			1.38915i	0.719421	+	0.694574i	$0.244407\pi$
$37$			51.3985i			1.38915i	−0.719421	+	0.694574i	$0.755593\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			33.2389i			0.810705i	0.914160	+	0.405352i	$0.132851\pi$
$41$			33.2389i			0.810705i	−0.914160	+	0.405352i	$0.867149\pi$
$42$		0			0
$43$	−54.2006			−1.26048			−0.630239	−	0.776401i	$-0.717043\pi$
$43$	−54.2006			−1.26048			−0.630239	+	0.776401i	$0.717043\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−32.6303			−0.694261			−0.347130	−	0.937817i	$-0.612844\pi$
$47$	−32.6303			−0.694261			−0.347130	+	0.937817i	$0.612844\pi$
$48$		0			0
$49$	−22.3846			−0.456829
$50$		0			0
$51$		0			0
$52$		0			0
$53$			91.2159i			1.72105i	0.509405	+	0.860527i	$0.329866\pi$
$53$			91.2159i			1.72105i	−0.509405	+	0.860527i	$0.670134\pi$
$54$		0			0
$55$	6.92130			0.125842
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	107.615i		−	1.82398i	−0.410210	−	0.911991i	$-0.634545\pi$
$59$		−	107.615i		−	1.82398i	0.410210	−	0.911991i	$-0.365455\pi$
$60$		0			0
$61$	−23.6223			−0.387251			−0.193625	−	0.981076i	$-0.562025\pi$
$61$	−23.6223			−0.387251			−0.193625	+	0.981076i	$0.562025\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	8.72959i		−	0.134301i
$66$		0			0
$67$			107.283i			1.60124i	0.599171	+	0.800621i	$0.295497\pi$
$67$			107.283i			1.60124i	−0.599171	+	0.800621i	$0.704503\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	72.7743i		−	1.02499i	−0.858690	−	0.512495i	$-0.828721\pi$
$71$		−	72.7743i		−	1.02499i	0.858690	−	0.512495i	$-0.171279\pi$
$72$		0			0
$73$	−5.25863			−0.0720360			−0.0360180	−	0.999351i	$-0.511467\pi$
$73$	−5.25863			−0.0720360			−0.0360180	+	0.999351i	$0.511467\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	60.9659			0.791764
$78$		0			0
$79$			45.5813i			0.576978i	0.957483	+	0.288489i	$0.0931529\pi$
$79$			45.5813i			0.576978i	−0.957483	+	0.288489i	$0.906847\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	132.488			1.59624			0.798119	−	0.602500i	$-0.205829\pi$
$83$	132.488			1.59624			0.798119	+	0.602500i	$0.205829\pi$
$84$		0			0
$85$	−5.22396			−0.0614584
$86$		0			0
$87$		0			0
$88$		0			0
$89$			42.1195i			0.473253i	0.971601	+	0.236627i	$0.0760418\pi$
$89$			42.1195i			0.473253i	−0.971601	+	0.236627i	$0.923958\pi$
$90$		0			0
$91$		−	76.8940i		−	0.844989i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.8239	−	2.58412i	−0.113936	−	0.0272012i
$96$		0			0
$97$		−	143.255i		−	1.47686i	−0.674332	−	0.738428i	$-0.735568\pi$
$97$		−	143.255i		−	1.47686i	0.674332	−	0.738428i	$-0.264432\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	89.3471			0.884625			0.442312	−	0.896861i	$-0.354158\pi$
$101$	89.3471			0.884625			0.442312	+	0.896861i	$0.354158\pi$
$102$		0			0
$103$		−	103.525i		−	1.00509i	−0.864550	−	0.502546i	$-0.832397\pi$
$103$		−	103.525i		−	1.00509i	0.864550	−	0.502546i	$-0.167603\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	64.4353i		−	0.602199i	−0.953593	−	0.301100i	$-0.902646\pi$
$107$		−	64.4353i		−	0.602199i	0.953593	−	0.301100i	$-0.0973536\pi$
$108$		0			0
$109$			173.357i			1.59043i	0.606329	+	0.795214i	$0.292641\pi$
$109$			173.357i			1.59043i	−0.606329	+	0.795214i	$0.707359\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			172.752i			1.52878i	0.644753	+	0.764391i	$0.276960\pi$
$113$			172.752i			1.52878i	−0.644753	+	0.764391i	$0.723040\pi$
$114$		0			0
$115$	−20.8809			−0.181573
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−46.0149			−0.386680
$120$		0			0
$121$	18.6498			0.154130
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−29.0836			−0.232669
$126$		0			0
$127$			177.397i			1.39683i	0.715694	+	0.698414i	$0.246111\pi$
$127$			177.397i			1.39683i	−0.715694	+	0.698414i	$0.753889\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−225.339			−1.72014			−0.860072	−	0.510173i	$-0.829581\pi$
$131$	−225.339			−1.72014			−0.860072	+	0.510173i	$0.829581\pi$
$132$		0			0
$133$	−95.3418	−	22.7620i	−0.716855	−	0.171143i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	185.839			1.35649			0.678245	−	0.734836i	$-0.262741\pi$
$137$	185.839			1.35649			0.678245	+	0.734836i	$0.262741\pi$
$138$		0			0
$139$	−188.027			−1.35272			−0.676358	−	0.736573i	$-0.736443\pi$
$139$	−188.027			−1.35272			−0.676358	+	0.736573i	$0.736443\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	176.135i		−	1.23171i
$144$		0			0
$145$		−	4.17472i		−	0.0287911i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−210.535			−1.41299			−0.706494	−	0.707719i	$-0.749724\pi$
$149$	−210.535			−1.41299			−0.706494	+	0.707719i	$0.749724\pi$
$150$		0			0
$151$		−	180.534i		−	1.19559i	−0.801649	−	0.597795i	$-0.796044\pi$
$151$		−	180.534i		−	1.19559i	0.801649	−	0.597795i	$-0.203956\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	6.32919i		−	0.0408335i
$156$		0			0
$157$	−250.193			−1.59359			−0.796794	−	0.604251i	$-0.793472\pi$
$157$	−250.193			−1.59359			−0.796794	+	0.604251i	$0.793472\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−183.928			−1.14241
$162$		0			0
$163$	−32.6614			−0.200377			−0.100188	−	0.994968i	$-0.531945\pi$
$163$	−32.6614			−0.200377			−0.100188	+	0.994968i	$0.531945\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	19.8364i		−	0.118781i	−0.998235	−	0.0593905i	$-0.981084\pi$
$167$		−	19.8364i		−	0.118781i	0.998235	−	0.0593905i	$-0.0189157\pi$
$168$		0			0
$169$	−53.1530			−0.314515
$170$		0			0
$171$		0			0
$172$		0			0
$173$			185.324i			1.07123i	0.844461	+	0.535617i	$0.179921\pi$
$173$			185.324i			1.07123i	−0.844461	+	0.535617i	$0.820079\pi$
$174$		0			0
$175$	−127.206			−0.726889
$176$		0			0
$177$		0			0
$178$		0			0
$179$			66.5882i			0.372001i	0.982550	+	0.186001i	$0.0595526\pi$
$179$			66.5882i			0.372001i	−0.982550	+	0.186001i	$0.940447\pi$
$180$		0			0
$181$		−	203.453i		−	1.12405i	−0.827121	−	0.562024i	$-0.810023\pi$
$181$		−	203.453i		−	1.12405i	0.827121	−	0.562024i	$-0.189977\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			30.1036i			0.162722i
$186$		0			0
$187$	−105.403			−0.563652
$188$		0			0
$189$		0			0
$190$		0			0
$191$	105.755			0.553693			0.276847	−	0.960914i	$-0.410711\pi$
$191$	105.755			0.553693			0.276847	+	0.960914i	$0.410711\pi$
$192$		0			0
$193$			352.487i			1.82636i	0.407560	+	0.913179i	$0.366380\pi$
$193$			352.487i			1.82636i	−0.407560	+	0.913179i	$0.633620\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	203.181			1.03138			0.515688	−	0.856776i	$-0.327536\pi$
$197$	203.181			1.03138			0.515688	+	0.856776i	$0.327536\pi$
$198$		0			0
$199$	−53.7401			−0.270051			−0.135025	−	0.990842i	$-0.543112\pi$
$199$	−53.7401			−0.270051			−0.135025	+	0.990842i	$0.543112\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	36.7727i		−	0.181146i
$204$		0			0
$205$			19.4677i			0.0949644i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−218.392	−	52.1392i	−1.04494	−	0.249470i
$210$		0			0
$211$			60.2566i			0.285576i	0.989753	+	0.142788i	$0.0456067\pi$
$211$			60.2566i			0.285576i	−0.989753	+	0.142788i	$0.954393\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−31.7447			−0.147650
$216$		0			0
$217$		−	55.7503i		−	0.256914i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			132.941i			0.601542i
$222$		0			0
$223$			114.546i			0.513661i	0.966456	+	0.256831i	$0.0826783\pi$
$223$			114.546i			0.513661i	−0.966456	+	0.256831i	$0.917322\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	355.584i		−	1.56645i	−0.621738	−	0.783225i	$-0.713573\pi$
$227$		−	355.584i		−	1.56645i	0.621738	−	0.783225i	$-0.286427\pi$
$228$		0			0
$229$	−186.383			−0.813899			−0.406950	−	0.913451i	$-0.633408\pi$
$229$	−186.383			−0.813899			−0.406950	+	0.913451i	$0.633408\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	427.065			1.83290			0.916449	−	0.400152i	$-0.131043\pi$
$233$	427.065			1.83290			0.916449	+	0.400152i	$0.131043\pi$
$234$		0			0
$235$	−19.1112			−0.0813243
$236$		0			0
$237$		0			0
$238$		0			0
$239$	37.5322			0.157038			0.0785192	−	0.996913i	$-0.474981\pi$
$239$	37.5322			0.157038			0.0785192	+	0.996913i	$0.474981\pi$
$240$		0			0
$241$			94.3513i			0.391499i	0.980654	+	0.195749i	$0.0627139\pi$
$241$			94.3513i			0.391499i	−0.980654	+	0.195749i	$0.937286\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−13.1104			−0.0535120
$246$		0			0
$247$	−65.7613	+	275.450i	−0.266240	+	1.11518i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	73.3775			0.292341			0.146170	−	0.989259i	$-0.453305\pi$
$251$	73.3775			0.292341			0.146170	+	0.989259i	$0.453305\pi$
$252$		0			0
$253$	−421.310			−1.66526
$254$		0			0
$255$		0			0
$256$		0			0
$257$			157.056i			0.611111i	0.952174	+	0.305556i	$0.0988422\pi$
$257$			157.056i			0.611111i	−0.952174	+	0.305556i	$0.901158\pi$
$258$		0			0
$259$			265.165i			1.02380i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−31.9429			−0.121456			−0.0607280	−	0.998154i	$-0.519342\pi$
$263$	−31.9429			−0.121456			−0.0607280	+	0.998154i	$0.519342\pi$
$264$		0			0
$265$			53.4242i			0.201601i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			178.356i			0.663035i	0.943449	+	0.331517i	$0.107561\pi$
$269$			178.356i			0.663035i	−0.943449	+	0.331517i	$0.892439\pi$
$270$		0			0
$271$	−333.321			−1.22997			−0.614983	−	0.788540i	$-0.710837\pi$
$271$	−333.321			−1.22997			−0.614983	+	0.788540i	$0.710837\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−291.380			−1.05956
$276$		0			0
$277$	153.669			0.554762			0.277381	−	0.960760i	$-0.410534\pi$
$277$	153.669			0.554762			0.277381	+	0.960760i	$0.410534\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	376.392i		−	1.33947i	−0.742598	−	0.669737i	$-0.766407\pi$
$281$		−	376.392i		−	1.33947i	0.742598	−	0.669737i	$-0.233593\pi$
$282$		0			0
$283$	−347.064			−1.22637			−0.613187	−	0.789937i	$-0.710113\pi$
$283$	−347.064			−1.22637			−0.613187	+	0.789937i	$0.710113\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			171.480i			0.597491i
$288$		0			0
$289$	−209.445			−0.724725
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	229.561i		−	0.783483i	−0.920075	−	0.391742i	$-0.871873\pi$
$293$		−	229.561i		−	0.783483i	0.920075	−	0.391742i	$-0.128127\pi$
$294$		0			0
$295$		−	63.0290i		−	0.213658i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			531.384i			1.77720i
$300$		0			0
$301$	−279.621			−0.928975
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−13.8353			−0.0453618
$306$		0			0
$307$		−	191.210i		−	0.622834i	−0.950273	−	0.311417i	$-0.899196\pi$
$307$		−	191.210i		−	0.622834i	0.950273	−	0.311417i	$-0.100804\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	419.281			1.34817			0.674084	−	0.738654i	$-0.264538\pi$
$311$	419.281			1.34817			0.674084	+	0.738654i	$0.264538\pi$
$312$		0			0
$313$	−103.826			−0.331711			−0.165856	−	0.986150i	$-0.553039\pi$
$313$	−103.826			−0.331711			−0.165856	+	0.986150i	$0.553039\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	507.269i		−	1.60022i	−0.599855	−	0.800109i	$-0.704775\pi$
$317$		−	507.269i		−	1.60022i	0.599855	−	0.800109i	$-0.295225\pi$
$318$		0			0
$319$		−	84.2324i		−	0.264051i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	164.835	+	39.3529i	0.510325	+	0.121836i
$324$		0			0
$325$			367.507i			1.13079i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−168.340			−0.511672
$330$		0			0
$331$		−	462.916i		−	1.39854i	−0.714858	−	0.699269i	$-0.753509\pi$
$331$		−	462.916i		−	1.39854i	0.714858	−	0.699269i	$-0.246491\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			62.8347i			0.187566i
$336$		0			0
$337$		−	68.7649i		−	0.204050i	−0.994782	−	0.102025i	$-0.967468\pi$
$337$		−	68.7649i		−	0.204050i	0.994782	−	0.102025i	$-0.0325322\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	127.703i		−	0.374495i
$342$		0			0
$343$	−368.274			−1.07369
$344$		0			0
$345$		0			0
$346$		0			0
$347$	277.920			0.800923			0.400461	−	0.916314i	$-0.368850\pi$
$347$	277.920			0.800923			0.400461	+	0.916314i	$0.368850\pi$
$348$		0			0
$349$	−146.447			−0.419620			−0.209810	−	0.977742i	$-0.567284\pi$
$349$	−146.447			−0.419620			−0.209810	+	0.977742i	$0.567284\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−127.776			−0.361972			−0.180986	−	0.983486i	$-0.557929\pi$
$353$	−127.776			−0.361972			−0.180986	+	0.983486i	$0.557929\pi$
$354$		0			0
$355$		−	42.6232i		−	0.120065i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−252.692			−0.703877			−0.351938	−	0.936023i	$-0.614477\pi$
$359$	−252.692			−0.703877			−0.351938	+	0.936023i	$0.614477\pi$
$360$		0			0
$361$	322.067	+	163.076i	0.892152	+	0.451735i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.07993			−0.00843815
$366$		0			0
$367$	−13.2138			−0.0360048			−0.0180024	−	0.999838i	$-0.505731\pi$
$367$	−13.2138			−0.0360048			−0.0180024	+	0.999838i	$0.505731\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			470.584i			1.26842i
$372$		0			0
$373$		−	542.028i		−	1.45316i	−0.687083	−	0.726579i	$-0.741109\pi$
$373$		−	542.028i		−	1.45316i	0.687083	−	0.726579i	$-0.258891\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−106.239			−0.281802
$378$		0			0
$379$			674.599i			1.77995i	0.456013	+	0.889973i	$0.349277\pi$
$379$			674.599i			1.77995i	−0.456013	+	0.889973i	$0.650723\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	415.700i		−	1.08538i	−0.839933	−	0.542690i	$-0.817406\pi$
$383$		−	415.700i		−	1.08538i	0.839933	−	0.542690i	$-0.182594\pi$
$384$		0			0
$385$	35.7071			0.0927457
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−408.921			−1.05121			−0.525605	−	0.850729i	$-0.676161\pi$
$389$	−408.921			−1.05121			−0.525605	+	0.850729i	$0.676161\pi$
$390$		0			0
$391$	317.991			0.813275
$392$		0			0
$393$		0			0
$394$		0			0
$395$			26.6965i			0.0675860i
$396$		0			0
$397$	−551.518			−1.38921			−0.694607	−	0.719389i	$-0.744422\pi$
$397$	−551.518			−1.38921			−0.694607	+	0.719389i	$0.744422\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	100.491i		−	0.250601i	−0.992119	−	0.125301i	$-0.960010\pi$
$401$		−	100.491i		−	0.250601i	0.992119	−	0.125301i	$-0.0399895\pi$
$402$		0			0
$403$	−161.067			−0.399670
$404$		0			0
$405$		0			0
$406$		0			0
$407$			607.394i			1.49237i
$408$		0			0
$409$		−	205.947i		−	0.503537i	−0.967787	−	0.251769i	$-0.918988\pi$
$409$		−	205.947i		−	0.503537i	0.967787	−	0.251769i	$-0.0810122\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	555.187i		−	1.34428i
$414$		0			0
$415$	77.5968			0.186980
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−32.1565			−0.0767459			−0.0383730	−	0.999263i	$-0.512217\pi$
$419$	−32.1565			−0.0767459			−0.0383730	+	0.999263i	$0.512217\pi$
$420$		0			0
$421$			216.724i			0.514785i	0.966307	+	0.257392i	$0.0828633\pi$
$421$			216.724i			0.514785i	−0.966307	+	0.257392i	$0.917137\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	219.924			0.517468
$426$		0			0
$427$	−121.868			−0.285405
$428$		0			0
$429$		0			0
$430$		0			0
$431$			721.789i			1.67469i	0.546678	+	0.837343i	$0.315892\pi$
$431$			721.789i			1.67469i	−0.546678	+	0.837343i	$0.684108\pi$
$432$		0			0
$433$		−	544.331i		−	1.25712i	−0.777763	−	0.628558i	$-0.783645\pi$
$433$		−	544.331i		−	1.25712i	0.777763	−	0.628558i	$-0.216355\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	658.868	+	157.299i	1.50771	+	0.359952i
$438$		0			0
$439$		−	213.302i		−	0.485882i	−0.970041	−	0.242941i	$-0.921888\pi$
$439$		−	213.302i		−	0.485882i	0.970041	−	0.242941i	$-0.0781121\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−297.473			−0.671497			−0.335748	−	0.941952i	$-0.608989\pi$
$443$	−297.473			−0.671497			−0.335748	+	0.941952i	$0.608989\pi$
$444$		0			0
$445$			24.6690i			0.0554359i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	297.207i		−	0.661931i	−0.943643	−	0.330966i	$-0.892626\pi$
$449$		−	297.207i		−	0.661931i	0.943643	−	0.330966i	$-0.107374\pi$
$450$		0			0
$451$			392.796i			0.870944i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	45.0361i		−	0.0989804i
$456$		0			0
$457$	96.6018			0.211383			0.105691	−	0.994399i	$-0.466294\pi$
$457$	96.6018			0.211383			0.105691	+	0.994399i	$0.466294\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−550.736			−1.19466			−0.597328	−	0.801997i	$-0.703771\pi$
$461$	−550.736			−1.19466			−0.597328	+	0.801997i	$0.703771\pi$
$462$		0			0
$463$	−45.2351			−0.0976999			−0.0488500	−	0.998806i	$-0.515556\pi$
$463$	−45.2351			−0.0976999			−0.0488500	+	0.998806i	$0.515556\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	118.446			0.253631			0.126815	−	0.991926i	$-0.459524\pi$
$467$	118.446			0.253631			0.126815	+	0.991926i	$0.459524\pi$
$468$		0			0
$469$			553.475i			1.18012i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−640.507			−1.35414
$474$		0			0
$475$	455.676	+	108.789i	0.959318	+	0.229029i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.9570			0.0374885			0.0187442	−	0.999824i	$-0.494033\pi$
$479$	17.9570			0.0374885			0.0187442	+	0.999824i	$0.494033\pi$
$480$		0			0
$481$	766.084			1.59269
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	83.9031i		−	0.172996i
$486$		0			0
$487$		−	475.493i		−	0.976372i	−0.872740	−	0.488186i	$-0.837659\pi$
$487$		−	475.493i		−	0.976372i	0.872740	−	0.488186i	$-0.162341\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−360.637			−0.734496			−0.367248	−	0.930123i	$-0.619700\pi$
$491$	−360.637			−0.734496			−0.367248	+	0.930123i	$0.619700\pi$
$492$		0			0
$493$			63.5758i			0.128957i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	375.443i		−	0.755419i
$498$		0			0
$499$	115.211			0.230884			0.115442	−	0.993314i	$-0.463172\pi$
$499$	115.211			0.230884			0.115442	+	0.993314i	$0.463172\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17.4753			0.0347422			0.0173711	−	0.999849i	$-0.494470\pi$
$503$	17.4753			0.0347422			0.0173711	+	0.999849i	$0.494470\pi$
$504$		0			0
$505$	52.3297			0.103623
$506$		0			0
$507$		0			0
$508$		0			0
$509$			292.333i			0.574328i	0.957881	+	0.287164i	$0.0927125\pi$
$509$			292.333i			0.574328i	−0.957881	+	0.287164i	$0.907287\pi$
$510$		0			0
$511$	−27.1293			−0.0530906
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	60.6333i		−	0.117735i
$516$		0			0
$517$	−385.603			−0.745848
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	687.193i		−	1.31899i	−0.751710	−	0.659494i	$-0.770771\pi$
$521$		−	687.193i		−	1.31899i	0.751710	−	0.659494i	$-0.229229\pi$
$522$		0			0
$523$		−	788.760i		−	1.50815i	−0.656791	−	0.754073i	$-0.728087\pi$
$523$		−	788.760i		−	1.50815i	0.656791	−	0.754073i	$-0.271913\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			96.3857i			0.182895i
$528$		0			0
$529$	742.054			1.40275
$530$		0			0
$531$		0			0
$532$		0			0
$533$	495.419			0.929492
$534$		0			0
$535$		−	37.7391i		−	0.0705404i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−264.527			−0.490773
$540$		0			0
$541$	−385.878			−0.713268			−0.356634	−	0.934244i	$-0.616076\pi$
$541$	−385.878			−0.713268			−0.356634	+	0.934244i	$0.616076\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			101.533i			0.186299i
$546$		0			0
$547$		−	432.569i		−	0.790802i	−0.918509	−	0.395401i	$-0.870606\pi$
$547$		−	432.569i		−	0.790802i	0.918509	−	0.395401i	$-0.129394\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−31.4488	+	131.727i	−0.0570758	+	0.239070i
$552$		0			0
$553$			235.154i			0.425234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−768.404			−1.37954			−0.689771	−	0.724028i	$-0.742289\pi$
$557$	−768.404			−1.37954			−0.689771	+	0.724028i	$0.742289\pi$
$558$		0			0
$559$			807.849i			1.44517i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			367.219i			0.652254i	0.945326	+	0.326127i	$0.105744\pi$
$563$			367.219i			0.652254i	−0.945326	+	0.326127i	$0.894256\pi$
$564$		0			0
$565$			101.179i			0.179079i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			691.754i			1.21574i	0.794038	+	0.607868i	$0.207975\pi$
$569$			691.754i			1.21574i	−0.794038	+	0.607868i	$0.792025\pi$
$570$		0			0
$571$	673.186			1.17896			0.589480	−	0.807783i	$-0.299333\pi$
$571$	673.186			1.17896			0.589480	+	0.807783i	$0.299333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	879.066			1.52881
$576$		0			0
$577$	571.057			0.989700			0.494850	−	0.868979i	$-0.335223\pi$
$577$	571.057			0.989700			0.494850	+	0.868979i	$0.335223\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	683.506			1.17643
$582$		0			0
$583$			1077.93i			1.84894i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	839.596			1.43032			0.715158	−	0.698962i	$-0.246355\pi$
$587$	839.596			1.43032			0.715158	+	0.698962i	$0.246355\pi$
$588$		0			0
$589$	−47.6787	+	199.709i	−0.0809486	+	0.339064i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−463.028			−0.780823			−0.390411	−	0.920641i	$-0.627667\pi$
$593$	−463.028			−0.780823			−0.390411	+	0.920641i	$0.627667\pi$
$594$		0			0
$595$	−26.9505			−0.0452950
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	656.487i		−	1.09597i	−0.836488	−	0.547986i	$-0.815395\pi$
$599$		−	656.487i		−	1.09597i	0.836488	−	0.547986i	$-0.184605\pi$
$600$		0			0
$601$			311.588i			0.518449i	0.965817	+	0.259224i	$0.0834669\pi$
$601$			311.588i			0.518449i	−0.965817	+	0.259224i	$0.916533\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.9230			0.0180545
$606$		0			0
$607$		−	186.898i		−	0.307904i	−0.988078	−	0.153952i	$-0.950800\pi$
$607$		−	186.898i		−	0.307904i	0.988078	−	0.153952i	$-0.0492001\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			486.348i			0.795986i
$612$		0			0
$613$	−626.130			−1.02142			−0.510710	−	0.859753i	$-0.670617\pi$
$613$	−626.130			−1.02142			−0.510710	+	0.859753i	$0.670617\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1028.16			1.66638			0.833190	−	0.552988i	$-0.186512\pi$
$617$	1028.16			1.66638			0.833190	+	0.552988i	$0.186512\pi$
$618$		0			0
$619$	35.3612			0.0571263			0.0285631	−	0.999592i	$-0.490907\pi$
$619$	35.3612			0.0571263			0.0285631	+	0.999592i	$0.490907\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			217.295i			0.348788i
$624$		0			0
$625$	599.390			0.959024
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	458.440i		−	0.728840i
$630$		0			0
$631$	−626.825			−0.993383			−0.496692	−	0.867927i	$-0.665452\pi$
$631$	−626.825			−0.993383			−0.496692	+	0.867927i	$0.665452\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			103.900i			0.163622i
$636$		0			0
$637$			333.638i			0.523764i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	749.678i		−	1.16954i	−0.811198	−	0.584772i	$-0.801184\pi$
$641$		−	749.678i		−	1.16954i	0.811198	−	0.584772i	$-0.198816\pi$
$642$		0			0
$643$	298.692			0.464528			0.232264	−	0.972653i	$-0.425387\pi$
$643$	298.692			0.464528			0.232264	+	0.972653i	$0.425387\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	375.883			0.580963			0.290481	−	0.956881i	$-0.406185\pi$
$647$	375.883			0.580963			0.290481	+	0.956881i	$0.406185\pi$
$648$		0			0
$649$		−	1271.72i		−	1.95951i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−641.912			−0.983020			−0.491510	−	0.870872i	$-0.663555\pi$
$653$	−641.912			−0.983020			−0.491510	+	0.870872i	$0.663555\pi$
$654$		0			0
$655$	−131.979			−0.201494
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	209.302i		−	0.317606i	−0.987310	−	0.158803i	$-0.949237\pi$
$659$		−	209.302i		−	0.317606i	0.987310	−	0.158803i	$-0.0507634\pi$
$660$		0			0
$661$			135.224i			0.204576i	0.994755	+	0.102288i	$0.0326163\pi$
$661$			135.224i			0.204576i	−0.994755	+	0.102288i	$0.967384\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−55.8407	−	13.3315i	−0.0839710	−	0.0200473i
$666$		0			0
$667$			254.121i			0.380992i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−279.153			−0.416025
$672$		0			0
$673$		−	589.529i		−	0.875971i	−0.898982	−	0.437986i	$-0.855692\pi$
$673$		−	589.529i		−	0.875971i	0.898982	−	0.437986i	$-0.144308\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	732.131i		−	1.08143i	−0.841205	−	0.540717i	$-0.818153\pi$
$677$		−	732.131i		−	1.08143i	0.841205	−	0.540717i	$-0.181847\pi$
$678$		0			0
$679$		−	739.055i		−	1.08845i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	192.642i		−	0.282053i	−0.990006	−	0.141026i	$-0.954960\pi$
$683$		−	192.642i		−	0.282053i	0.990006	−	0.141026i	$-0.0450402\pi$
$684$		0			0
$685$	108.844			0.158897
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1359.55			1.97323
$690$		0			0
$691$	794.271			1.14945			0.574726	−	0.818346i	$-0.305109\pi$
$691$	794.271			1.14945			0.574726	+	0.818346i	$0.305109\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−110.126			−0.158454
$696$		0			0
$697$		−	296.469i		−	0.425350i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	245.346			0.349994			0.174997	−	0.984569i	$-0.444008\pi$
$701$	245.346			0.349994			0.174997	+	0.984569i	$0.444008\pi$
$702$		0			0
$703$	226.775	−	949.876i	0.322581	−	1.35118i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	460.943			0.651970
$708$		0			0
$709$	−572.022			−0.806802			−0.403401	−	0.915023i	$-0.632172\pi$
$709$	−572.022			−0.806802			−0.403401	+	0.915023i	$0.632172\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			385.268i			0.540347i
$714$		0			0
$715$		−	103.161i		−	0.144281i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1142.49			−1.58900			−0.794501	−	0.607263i	$-0.792268\pi$
$719$	−1142.49			−1.58900			−0.794501	+	0.607263i	$0.792268\pi$
$720$		0			0
$721$		−	534.084i		−	0.740755i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			175.751i			0.242416i
$726$		0			0
$727$	−417.958			−0.574907			−0.287454	−	0.957795i	$-0.592809\pi$
$727$	−417.958			−0.574907			−0.287454	+	0.957795i	$0.592809\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	483.433			0.661331
$732$		0			0
$733$	280.328			0.382439			0.191220	−	0.981547i	$-0.438756\pi$
$733$	280.328			0.382439			0.191220	+	0.981547i	$0.438756\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1267.80i			1.72022i
$738$		0			0
$739$	309.436			0.418723			0.209361	−	0.977838i	$-0.432861\pi$
$739$	309.436			0.418723			0.209361	+	0.977838i	$0.432861\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	502.112i		−	0.675791i	−0.941184	−	0.337895i	$-0.890285\pi$
$743$		−	502.112i		−	0.675791i	0.941184	−	0.337895i	$-0.109715\pi$
$744$		0			0
$745$	−123.308			−0.165515
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	332.422i		−	0.443822i
$750$		0			0
$751$			365.970i			0.487310i	0.969862	+	0.243655i	$0.0783465\pi$
$751$			365.970i			0.487310i	−0.969862	+	0.243655i	$0.921654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	105.737i		−	0.140049i
$756$		0			0
$757$	321.740			0.425019			0.212510	−	0.977159i	$-0.431836\pi$
$757$	321.740			0.425019			0.212510	+	0.977159i	$0.431836\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1232.10			−1.61905			−0.809524	−	0.587087i	$-0.800275\pi$
$761$	−1232.10			−1.61905			−0.809524	+	0.587087i	$0.800275\pi$
$762$		0			0
$763$			894.349i			1.17215i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1603.98			−2.09124
$768$		0			0
$769$	916.235			1.19146			0.595732	−	0.803184i	$-0.296862\pi$
$769$	916.235			1.19146			0.595732	+	0.803184i	$0.296862\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			957.087i			1.23815i	0.785333	+	0.619073i	$0.212492\pi$
$773$			957.087i			1.23815i	−0.785333	+	0.619073i	$0.787508\pi$
$774$		0			0
$775$			266.453i			0.343810i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	146.653	−	614.276i	0.188258	−	0.788544i
$780$		0			0
$781$		−	860.000i		−	1.10115i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−146.536			−0.186670
$786$		0			0
$787$		−	378.091i		−	0.480421i	−0.970721	−	0.240210i	$-0.922784\pi$
$787$		−	378.091i		−	0.480421i	0.970721	−	0.240210i	$-0.0772164\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			891.232i			1.12672i
$792$		0			0
$793$			352.086i			0.443992i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1501.04i			1.88336i	0.336511	+	0.941679i	$0.390753\pi$
$797$			1501.04i			1.88336i	−0.336511	+	0.941679i	$0.609247\pi$
$798$		0			0
$799$	291.040			0.364256
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−62.1430			−0.0773886
$804$		0			0
$805$	−107.725			−0.133820
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−295.849			−0.365697			−0.182848	−	0.983141i	$-0.558532\pi$
$809$	−295.849			−0.365697			−0.182848	+	0.983141i	$0.558532\pi$
$810$		0			0
$811$		−	1520.23i		−	1.87452i	−0.348635	−	0.937259i	$-0.613355\pi$
$811$		−	1520.23i		−	1.87452i	0.348635	−	0.937259i	$-0.386645\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.1295			−0.0234717
$816$		0			0
$817$	1001.66	+	239.138i	1.22602	+	0.292702i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−538.450			−0.655847			−0.327923	−	0.944704i	$-0.606349\pi$
$821$	−538.450			−0.655847			−0.327923	+	0.944704i	$0.606349\pi$
$822$		0			0
$823$	1437.93			1.74718			0.873589	−	0.486664i	$-0.161786\pi$
$823$	1437.93			1.74718			0.873589	+	0.486664i	$0.161786\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	840.691i		−	1.01655i	−0.861193	−	0.508277i	$-0.830282\pi$
$827$		−	840.691i		−	1.01655i	0.861193	−	0.508277i	$-0.169718\pi$
$828$		0			0
$829$			1542.91i			1.86117i	0.366081	+	0.930583i	$0.380699\pi$
$829$			1542.91i			1.86117i	−0.366081	+	0.930583i	$0.619301\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	199.656			0.239683
$834$		0			0
$835$		−	11.6180i		−	0.0139138i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			163.302i			0.194639i	0.995253	+	0.0973194i	$0.0310268\pi$
$839$			163.302i			0.194639i	−0.995253	+	0.0973194i	$0.968973\pi$
$840$		0			0
$841$	790.194			0.939588
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−31.1312			−0.0368417
$846$		0			0
$847$	96.2145			0.113594
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1832.45i		−	2.15329i
$852$		0			0
$853$	257.115			0.301424			0.150712	−	0.988578i	$-0.451843\pi$
$853$	257.115			0.301424			0.150712	+	0.988578i	$0.451843\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			3.96935i			0.00463168i	0.999997	+	0.00231584i	$0.000737155\pi$
$857$			3.96935i			0.00463168i	−0.999997	+	0.00231584i	$0.999263\pi$
$858$		0			0
$859$	−252.099			−0.293480			−0.146740	−	0.989175i	$-0.546878\pi$
$859$	−252.099			−0.293480			−0.146740	+	0.989175i	$0.546878\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	304.185i		−	0.352474i	−0.984348	−	0.176237i	$-0.943607\pi$
$863$		−	304.185i		−	0.352474i	0.984348	−	0.176237i	$-0.0563925\pi$
$864$		0			0
$865$			108.542i			0.125482i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			538.650i			0.619850i
$870$		0			0
$871$	1599.04			1.83586
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−150.043			−0.171477
$876$		0			0
$877$			165.717i			0.188959i	0.995527	+	0.0944793i	$0.0301186\pi$
$877$			165.717i			0.188959i	−0.995527	+	0.0944793i	$0.969881\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	860.822			0.977096			0.488548	−	0.872537i	$-0.337527\pi$
$881$	860.822			0.977096			0.488548	+	0.872537i	$0.337527\pi$
$882$		0			0
$883$	−201.036			−0.227674			−0.113837	−	0.993499i	$-0.536314\pi$
$883$	−201.036			−0.227674			−0.113837	+	0.993499i	$0.536314\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1099.43i		−	1.23949i	−0.784803	−	0.619745i	$-0.787236\pi$
$887$		−	1099.43i		−	1.23949i	0.784803	−	0.619745i	$-0.212764\pi$
$888$		0			0
$889$			915.194i			1.02946i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	603.028	+	143.968i	0.675283	+	0.161218i
$894$		0			0
$895$			39.0001i			0.0435755i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−77.0264			−0.0856801
$900$		0			0
$901$		−	813.585i		−	0.902980i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	119.160i		−	0.131669i
$906$		0			0
$907$			552.953i			0.609650i	0.952408	+	0.304825i	$0.0985980\pi$
$907$			552.953i			0.609650i	−0.952408	+	0.304825i	$0.901402\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1140.54i		−	1.25196i	−0.779838	−	0.625981i	$-0.784699\pi$
$911$		−	1140.54i		−	1.25196i	0.779838	−	0.625981i	$-0.215301\pi$
$912$		0			0
$913$	1565.65			1.71485
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1162.53			−1.26775
$918$		0			0
$919$	138.162			0.150340			0.0751699	−	0.997171i	$-0.476050\pi$
$919$	138.162			0.150340			0.0751699	+	0.997171i	$0.476050\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1084.69			−1.17517
$924$		0			0
$925$		−	1267.33i		−	1.37009i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	46.7799			0.0503551			0.0251775	−	0.999683i	$-0.491985\pi$
$929$	46.7799			0.0503551			0.0251775	+	0.999683i	$0.491985\pi$
$930$		0			0
$931$	413.681	+	98.7628i	0.444341	+	0.106083i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−61.7334			−0.0660251
$936$		0			0
$937$	751.420			0.801943			0.400971	−	0.916091i	$-0.368673\pi$
$937$	751.420			0.801943			0.400971	+	0.916091i	$0.368673\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			786.319i			0.835621i	0.908534	+	0.417811i	$0.137202\pi$
$941$			786.319i			0.835621i	−0.908534	+	0.417811i	$0.862798\pi$
$942$		0			0
$943$		−	1185.03i		−	1.25666i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1061.09			1.12048			0.560240	−	0.828330i	$-0.310709\pi$
$947$	1061.09			1.12048			0.560240	+	0.828330i	$0.310709\pi$
$948$		0			0
$949$			78.3788i			0.0825909i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1356.96i			1.42388i	0.702241	+	0.711940i	$0.252183\pi$
$953$			1356.96i			1.42388i	−0.702241	+	0.711940i	$0.747817\pi$
$954$		0			0
$955$	61.9399			0.0648585
$956$		0			0
$957$		0			0
$958$		0			0
$959$	958.746			0.999735
$960$		0			0
$961$	844.222			0.878483
$962$		0			0
$963$		0			0
$964$		0			0
$965$			206.448i			0.213936i
$966$		0			0
$967$	1286.80			1.33071			0.665355	−	0.746527i	$-0.268280\pi$
$967$	1286.80			1.33071			0.665355	+	0.746527i	$0.268280\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1194.60i			1.23028i	0.788418	+	0.615140i	$0.210901\pi$
$971$			1194.60i			1.23028i	−0.788418	+	0.615140i	$0.789099\pi$
$972$		0			0
$973$	−970.036			−0.996953
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1162.09i			1.18944i	0.803932	+	0.594722i	$0.202738\pi$
$977$			1162.09i			1.18944i	−0.803932	+	0.594722i	$0.797262\pi$
$978$		0			0
$979$			497.741i			0.508418i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1512.55i			1.53871i	0.638821	+	0.769355i	$0.279422\pi$
$983$			1512.55i			1.53871i	−0.638821	+	0.769355i	$0.720578\pi$
$984$		0			0
$985$	119.001			0.120813
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1932.35			1.95384
$990$		0			0
$991$			1307.34i			1.31921i	0.751612	+	0.659606i	$0.229277\pi$
$991$			1307.34i			1.31921i	−0.751612	+	0.659606i	$0.770723\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−31.4750			−0.0316332
$996$		0			0
$997$	−1298.73			−1.30263			−0.651317	−	0.758806i	$-0.725783\pi$
$997$	−1298.73			−1.30263			−0.651317	+	0.758806i	$0.725783\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.q.721.13		20
3.2	odd	2	inner	2736.3.o.q.721.7		20
4.3	odd	2		1368.3.o.d.721.13	yes	20
12.11	even	2		1368.3.o.d.721.7	✓	20
19.18	odd	2	inner	2736.3.o.q.721.14		20
57.56	even	2	inner	2736.3.o.q.721.8		20
76.75	even	2		1368.3.o.d.721.14	yes	20
228.227	odd	2		1368.3.o.d.721.8	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.3.o.d.721.7	✓	20	12.11	even	2
1368.3.o.d.721.8	yes	20	228.227	odd	2
1368.3.o.d.721.13	yes	20	4.3	odd	2
1368.3.o.d.721.14	yes	20	76.75	even	2
2736.3.o.q.721.7		20	3.2	odd	2	inner
2736.3.o.q.721.8		20	57.56	even	2	inner
2736.3.o.q.721.13		20	1.1	even	1	trivial
2736.3.o.q.721.14		20	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)

\(f(q)\)	\(=\)	\(q+0.585690 q^{5} +5.15901 q^{7} +O(q^{10})\)	\(q+0.585690 q^{5} +5.15901 q^{7} +11.8174 q^{11} -14.9048i q^{13} -8.91933 q^{17} +(-18.4806 - 4.41209i) q^{19} -35.6518 q^{23} -24.6570 q^{25} -7.12786i q^{29} -10.8064i q^{31} +3.02158 q^{35} +51.3985i q^{37} +33.2389i q^{41} -54.2006 q^{43} -32.6303 q^{47} -22.3846 q^{49} +91.2159i q^{53} +6.92130 q^{55} -107.615i q^{59} -23.6223 q^{61} -8.72959i q^{65} +107.283i q^{67} -72.7743i q^{71} -5.25863 q^{73} +60.9659 q^{77} +45.5813i q^{79} +132.488 q^{83} -5.22396 q^{85} +42.1195i q^{89} -76.8940i q^{91} +(-10.8239 - 2.58412i) q^{95} -143.255i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7}+O(q^{10}) \) 20 * q + 16 * q^7	\( 20 q + 16 q^{7} + 8 q^{19} + 68 q^{25} + 128 q^{43} + 116 q^{49} - 144 q^{55} - 104 q^{61} - 88 q^{73} - 280 q^{85}+O(q^{100}) \) 20 * q + 16 * q^7 + 8 * q^19 + 68 * q^25 + 128 * q^43 + 116 * q^49 - 144 * q^55 - 104 * q^61 - 88 * q^73 - 280 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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