LMFDB - Embedded newform 2736.3.o.q.721.3

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			721.3
Root			$-6.04300 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.q.721.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.04300 q^{5} +0.880524 q^{7} +O(q^{10})$	$q-6.04300 q^{5} +0.880524 q^{7} -12.2988 q^{11} -7.16565i q^{13} -15.7504 q^{17} +(18.2477 + 5.29345i) q^{19} -19.1457 q^{23} +11.5179 q^{25} +51.7618i q^{29} +32.5020i q^{31} -5.32101 q^{35} -25.3948i q^{37} -19.0976i q^{41} -39.5873 q^{43} -24.4667 q^{47} -48.2247 q^{49} +13.1419i q^{53} +74.3217 q^{55} -65.5796i q^{59} +22.2165 q^{61} +43.3020i q^{65} -18.7255i q^{67} -76.2174i q^{71} +85.4812 q^{73} -10.8294 q^{77} -150.744i q^{79} +29.0189 q^{83} +95.1798 q^{85} +127.303i q^{89} -6.30952i q^{91} +(-110.271 - 31.9883i) q^{95} +127.180i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−6.04300			−1.20860			−0.604300	−	0.796757i	$-0.706547\pi$
$5$	−6.04300			−1.20860			−0.604300	+	0.796757i	$0.706547\pi$
$6$		0			0
$7$	0.880524			0.125789			0.0628946	−	0.998020i	$-0.479967\pi$
$7$	0.880524			0.125789			0.0628946	+	0.998020i	$0.479967\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−12.2988			−1.11807			−0.559037	−	0.829143i	$-0.688829\pi$
$11$	−12.2988			−1.11807			−0.559037	+	0.829143i	$0.688829\pi$
$12$		0			0
$13$		−	7.16565i		−	0.551204i	−0.961272	−	0.275602i	$-0.911123\pi$
$13$		−	7.16565i		−	0.551204i	0.961272	−	0.275602i	$-0.0888771\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.7504			−0.926495			−0.463248	−	0.886229i	$-0.653316\pi$
$17$	−15.7504			−0.926495			−0.463248	+	0.886229i	$0.653316\pi$
$18$		0			0
$19$	18.2477	+	5.29345i	0.960407	+	0.278602i
$19$	18.2477	+	5.29345i	0.960407	+	0.278602i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−19.1457			−0.832423			−0.416212	−	0.909268i	$-0.636642\pi$
$23$	−19.1457			−0.832423			−0.416212	+	0.909268i	$0.636642\pi$
$24$		0			0
$25$	11.5179			0.460715
$26$		0			0
$27$		0			0
$28$		0			0
$29$			51.7618i			1.78489i	0.451156	+	0.892445i	$0.351012\pi$
$29$			51.7618i			1.78489i	−0.451156	+	0.892445i	$0.648988\pi$
$30$		0			0
$31$			32.5020i			1.04845i	0.851580	+	0.524225i	$0.175645\pi$
$31$			32.5020i			1.04845i	−0.851580	+	0.524225i	$0.824355\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.32101			−0.152029
$36$		0			0
$37$		−	25.3948i		−	0.686345i	−0.939272	−	0.343172i	$-0.888499\pi$
$37$		−	25.3948i		−	0.686345i	0.939272	−	0.343172i	$-0.111501\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	19.0976i		−	0.465796i	−0.972501	−	0.232898i	$-0.925179\pi$
$41$		−	19.0976i		−	0.465796i	0.972501	−	0.232898i	$-0.0748209\pi$
$42$		0			0
$43$	−39.5873			−0.920636			−0.460318	−	0.887754i	$-0.652265\pi$
$43$	−39.5873			−0.920636			−0.460318	+	0.887754i	$0.652265\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−24.4667			−0.520569			−0.260284	−	0.965532i	$-0.583816\pi$
$47$	−24.4667			−0.520569			−0.260284	+	0.965532i	$0.583816\pi$
$48$		0			0
$49$	−48.2247			−0.984177
$50$		0			0
$51$		0			0
$52$		0			0
$53$			13.1419i			0.247960i	0.992285	+	0.123980i	$0.0395659\pi$
$53$			13.1419i			0.247960i	−0.992285	+	0.123980i	$0.960434\pi$
$54$		0			0
$55$	74.3217			1.35130
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	65.5796i		−	1.11152i	−0.831344	−	0.555759i	$-0.812428\pi$
$59$		−	65.5796i		−	1.11152i	0.831344	−	0.555759i	$-0.187572\pi$
$60$		0			0
$61$	22.2165			0.364205			0.182103	−	0.983280i	$-0.441710\pi$
$61$	22.2165			0.364205			0.182103	+	0.983280i	$0.441710\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			43.3020i			0.666185i
$66$		0			0
$67$		−	18.7255i		−	0.279486i	−0.990188	−	0.139743i	$-0.955372\pi$
$67$		−	18.7255i		−	0.279486i	0.990188	−	0.139743i	$-0.0446276\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	76.2174i		−	1.07348i	−0.843746	−	0.536742i	$-0.819655\pi$
$71$		−	76.2174i		−	1.07348i	0.843746	−	0.536742i	$-0.180345\pi$
$72$		0			0
$73$	85.4812			1.17097			0.585487	−	0.810681i	$-0.300903\pi$
$73$	85.4812			1.17097			0.585487	+	0.810681i	$0.300903\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.8294			−0.140642
$78$		0			0
$79$		−	150.744i		−	1.90815i	−0.299572	−	0.954074i	$-0.596844\pi$
$79$		−	150.744i		−	1.90815i	0.299572	−	0.954074i	$-0.403156\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	29.0189			0.349625			0.174812	−	0.984602i	$-0.444068\pi$
$83$	29.0189			0.349625			0.174812	+	0.984602i	$0.444068\pi$
$84$		0			0
$85$	95.1798			1.11976
$86$		0			0
$87$		0			0
$88$		0			0
$89$			127.303i			1.43037i	0.698933	+	0.715187i	$0.253659\pi$
$89$			127.303i			1.43037i	−0.698933	+	0.715187i	$0.746341\pi$
$90$		0			0
$91$		−	6.30952i		−	0.0693354i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−110.271	−	31.9883i	−1.16075	−	0.336719i
$96$		0			0
$97$			127.180i			1.31113i	0.755137	+	0.655567i	$0.227570\pi$
$97$			127.180i			1.31113i	−0.755137	+	0.655567i	$0.772430\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	79.8127			0.790225			0.395113	−	0.918633i	$-0.370706\pi$
$101$	79.8127			0.790225			0.395113	+	0.918633i	$0.370706\pi$
$102$		0			0
$103$		−	107.140i		−	1.04019i	−0.854108	−	0.520095i	$-0.825896\pi$
$103$		−	107.140i		−	1.04019i	0.854108	−	0.520095i	$-0.174104\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	77.3179i		−	0.722597i	−0.932450	−	0.361299i	$-0.882334\pi$
$107$		−	77.3179i		−	0.722597i	0.932450	−	0.361299i	$-0.117666\pi$
$108$		0			0
$109$			28.8689i			0.264852i	0.991193	+	0.132426i	$0.0422767\pi$
$109$			28.8689i			0.264852i	−0.991193	+	0.132426i	$0.957723\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			128.054i			1.13322i	0.823986	+	0.566610i	$0.191746\pi$
$113$			128.054i			1.13322i	−0.823986	+	0.566610i	$0.808254\pi$
$114$		0			0
$115$	115.698			1.00607
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−13.8686			−0.116543
$120$		0			0
$121$	30.2607			0.250089
$122$		0			0
$123$		0			0
$124$		0			0
$125$	81.4725			0.651780
$126$		0			0
$127$			183.797i			1.44722i	0.690209	+	0.723610i	$0.257519\pi$
$127$			183.797i			1.44722i	−0.690209	+	0.723610i	$0.742481\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	101.552			0.775208			0.387604	−	0.921826i	$-0.373303\pi$
$131$	101.552			0.775208			0.387604	+	0.921826i	$0.373303\pi$
$132$		0			0
$133$	16.0676	+	4.66101i	0.120809	+	0.0350452i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−78.9622			−0.576366			−0.288183	−	0.957575i	$-0.593051\pi$
$137$	−78.9622			−0.576366			−0.288183	+	0.957575i	$0.593051\pi$
$138$		0			0
$139$	116.421			0.837560			0.418780	−	0.908088i	$-0.362458\pi$
$139$	116.421			0.837560			0.418780	+	0.908088i	$0.362458\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			88.1289i			0.616286i
$144$		0			0
$145$		−	312.797i		−	2.15722i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	146.838			0.985488			0.492744	−	0.870174i	$-0.335994\pi$
$149$	146.838			0.985488			0.492744	+	0.870174i	$0.335994\pi$
$150$		0			0
$151$			117.072i			0.775310i	0.921805	+	0.387655i	$0.126715\pi$
$151$			117.072i			0.775310i	−0.921805	+	0.387655i	$0.873285\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	196.409i		−	1.26716i
$156$		0			0
$157$	78.5321			0.500205			0.250102	−	0.968219i	$-0.419536\pi$
$157$	78.5321			0.500205			0.250102	+	0.968219i	$0.419536\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16.8583			−0.104710
$162$		0			0
$163$	36.2301			0.222271			0.111135	−	0.993805i	$-0.464551\pi$
$163$	36.2301			0.222271			0.111135	+	0.993805i	$0.464551\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	259.286i		−	1.55261i	−0.630359	−	0.776304i	$-0.717092\pi$
$167$		−	259.286i		−	1.55261i	0.630359	−	0.776304i	$-0.282908\pi$
$168$		0			0
$169$	117.654			0.696175
$170$		0			0
$171$		0			0
$172$		0			0
$173$			15.8453i			0.0915914i	0.998951	+	0.0457957i	$0.0145823\pi$
$173$			15.8453i			0.0915914i	−0.998951	+	0.0457957i	$0.985418\pi$
$174$		0			0
$175$	10.1418			0.0579529
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	118.769i		−	0.663513i	−0.943365	−	0.331756i	$-0.892359\pi$
$179$		−	118.769i		−	0.663513i	0.943365	−	0.331756i	$-0.107641\pi$
$180$		0			0
$181$		−	331.705i		−	1.83263i	−0.400462	−	0.916313i	$-0.631150\pi$
$181$		−	331.705i		−	1.83263i	0.400462	−	0.916313i	$-0.368850\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			153.461i			0.829516i
$186$		0			0
$187$	193.711			1.03589
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−64.8350			−0.339450			−0.169725	−	0.985491i	$-0.554288\pi$
$191$	−64.8350			−0.339450			−0.169725	+	0.985491i	$0.554288\pi$
$192$		0			0
$193$			289.564i			1.50033i	0.661249	+	0.750166i	$0.270026\pi$
$193$			289.564i			1.50033i	−0.661249	+	0.750166i	$0.729974\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	318.180			1.61513			0.807563	−	0.589781i	$-0.200786\pi$
$197$	318.180			1.61513			0.807563	+	0.589781i	$0.200786\pi$
$198$		0			0
$199$	82.5519			0.414834			0.207417	−	0.978253i	$-0.433494\pi$
$199$	82.5519			0.414834			0.207417	+	0.978253i	$0.433494\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			45.5775i			0.224520i
$204$		0			0
$205$			115.407i			0.562961i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−224.425	−	65.1031i	−1.07381	−	0.311498i
$210$		0			0
$211$		−	125.964i		−	0.596987i	−0.954412	−	0.298493i	$-0.903516\pi$
$211$		−	125.964i		−	0.596987i	0.954412	−	0.298493i	$-0.0964841\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	239.226			1.11268
$216$		0			0
$217$			28.6188i			0.131884i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			112.862i			0.510688i
$222$		0			0
$223$		−	56.7738i		−	0.254591i	−0.991865	−	0.127296i	$-0.959370\pi$
$223$		−	56.7738i		−	0.254591i	0.991865	−	0.127296i	$-0.0406297\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			239.912i			1.05688i	0.848971	+	0.528440i	$0.177223\pi$
$227$			239.912i			1.05688i	−0.848971	+	0.528440i	$0.822777\pi$
$228$		0			0
$229$	280.197			1.22357			0.611785	−	0.791024i	$-0.290452\pi$
$229$	280.197			1.22357			0.611785	+	0.791024i	$0.290452\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−126.899			−0.544632			−0.272316	−	0.962208i	$-0.587790\pi$
$233$	−126.899			−0.544632			−0.272316	+	0.962208i	$0.587790\pi$
$234$		0			0
$235$	147.853			0.629160
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−53.7118			−0.224736			−0.112368	−	0.993667i	$-0.535843\pi$
$239$	−53.7118			−0.224736			−0.112368	+	0.993667i	$0.535843\pi$
$240$		0			0
$241$		−	330.110i		−	1.36975i	−0.728660	−	0.684876i	$-0.759856\pi$
$241$		−	330.110i		−	1.36975i	0.728660	−	0.684876i	$-0.240144\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	291.422			1.18948
$246$		0			0
$247$	37.9310	−	130.757i	0.153567	−	0.529380i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	23.1047			0.0920505			0.0460253	−	0.998940i	$-0.485345\pi$
$251$	23.1047			0.0920505			0.0460253	+	0.998940i	$0.485345\pi$
$252$		0			0
$253$	235.470			0.930710
$254$		0			0
$255$		0			0
$256$		0			0
$257$			60.7728i			0.236470i	0.992986	+	0.118235i	$0.0377236\pi$
$257$			60.7728i			0.236470i	−0.992986	+	0.118235i	$0.962276\pi$
$258$		0			0
$259$		−	22.3607i		−	0.0863347i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	116.322			0.442287			0.221144	−	0.975241i	$-0.429021\pi$
$263$	116.322			0.442287			0.221144	+	0.975241i	$0.429021\pi$
$264$		0			0
$265$		−	79.4164i		−	0.299685i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	94.7378i		−	0.352185i	−0.984374	−	0.176093i	$-0.943654\pi$
$269$		−	94.7378i		−	0.352185i	0.984374	−	0.176093i	$-0.0563458\pi$
$270$		0			0
$271$	487.057			1.79726			0.898629	−	0.438710i	$-0.144564\pi$
$271$	487.057			1.79726			0.898629	+	0.438710i	$0.144564\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−141.656			−0.515113
$276$		0			0
$277$	495.554			1.78901			0.894503	−	0.447063i	$-0.147530\pi$
$277$	495.554			1.78901			0.894503	+	0.447063i	$0.147530\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	350.650i		−	1.24786i	−0.781479	−	0.623932i	$-0.785534\pi$
$281$		−	350.650i		−	1.24786i	0.781479	−	0.623932i	$-0.214466\pi$
$282$		0			0
$283$	−51.8494			−0.183214			−0.0916068	−	0.995795i	$-0.529200\pi$
$283$	−51.8494			−0.183214			−0.0916068	+	0.995795i	$0.529200\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	16.8159i		−	0.0585921i
$288$		0			0
$289$	−40.9242			−0.141606
$290$		0			0
$291$		0			0
$292$		0			0
$293$			160.419i			0.547506i	0.961800	+	0.273753i	$0.0882650\pi$
$293$			160.419i			0.547506i	−0.961800	+	0.273753i	$0.911735\pi$
$294$		0			0
$295$			396.297i			1.34338i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			137.192i			0.458835i
$300$		0			0
$301$	−34.8576			−0.115806
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−134.255			−0.440179
$306$		0			0
$307$		−	436.739i		−	1.42260i	−0.702888	−	0.711301i	$-0.748106\pi$
$307$		−	436.739i		−	1.42260i	0.702888	−	0.711301i	$-0.251894\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	107.433			0.345443			0.172721	−	0.984971i	$-0.444744\pi$
$311$	107.433			0.345443			0.172721	+	0.984971i	$0.444744\pi$
$312$		0			0
$313$	−394.436			−1.26018			−0.630089	−	0.776523i	$-0.716982\pi$
$313$	−394.436			−1.26018			−0.630089	+	0.776523i	$0.716982\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	189.281i		−	0.597100i	−0.954394	−	0.298550i	$-0.903497\pi$
$317$		−	189.281i		−	0.597100i	0.954394	−	0.298550i	$-0.0965030\pi$
$318$		0			0
$319$		−	636.609i		−	1.99564i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−287.409	−	83.3740i	−0.889812	−	0.258124i
$324$		0			0
$325$		−	82.5330i		−	0.253948i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−21.5436			−0.0654819
$330$		0			0
$331$		−	76.0556i		−	0.229775i	−0.993378	−	0.114888i	$-0.963349\pi$
$331$		−	76.0556i		−	0.229775i	0.993378	−	0.114888i	$-0.0366508\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			113.159i			0.337787i
$336$		0			0
$337$			327.362i			0.971400i	0.874126	+	0.485700i	$0.161435\pi$
$337$			327.362i			0.971400i	−0.874126	+	0.485700i	$0.838565\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	399.735i		−	1.17224i
$342$		0			0
$343$	−85.6087			−0.249588
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−193.294			−0.557044			−0.278522	−	0.960430i	$-0.589845\pi$
$347$	−193.294			−0.557044			−0.278522	+	0.960430i	$0.589845\pi$
$348$		0			0
$349$	326.241			0.934788			0.467394	−	0.884049i	$-0.345193\pi$
$349$	326.241			0.934788			0.467394	+	0.884049i	$0.345193\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−437.916			−1.24056			−0.620278	−	0.784382i	$-0.712980\pi$
$353$	−437.916			−1.24056			−0.620278	+	0.784382i	$0.712980\pi$
$354$		0			0
$355$			460.582i			1.29741i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	412.477			1.14896			0.574481	−	0.818518i	$-0.305204\pi$
$359$	412.477			1.14896			0.574481	+	0.818518i	$0.305204\pi$
$360$		0			0
$361$	304.959	+	193.187i	0.844761	+	0.535143i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−516.563			−1.41524
$366$		0			0
$367$	−3.45182			−0.00940550			−0.00470275	−	0.999989i	$-0.501497\pi$
$367$	−3.45182			−0.00940550			−0.00470275	+	0.999989i	$0.501497\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			11.5717i			0.0311907i
$372$		0			0
$373$			419.041i			1.12343i	0.827329	+	0.561717i	$0.189859\pi$
$373$			419.041i			1.12343i	−0.827329	+	0.561717i	$0.810141\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	370.907			0.983838
$378$		0			0
$379$			99.6300i			0.262876i	0.991324	+	0.131438i	$0.0419594\pi$
$379$			99.6300i			0.262876i	−0.991324	+	0.131438i	$0.958041\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			556.513i			1.45304i	0.687147	+	0.726518i	$0.258863\pi$
$383$			556.513i			1.45304i	−0.687147	+	0.726518i	$0.741137\pi$
$384$		0			0
$385$	65.4421			0.169979
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−217.930			−0.560231			−0.280115	−	0.959966i	$-0.590373\pi$
$389$	−217.930			−0.560231			−0.280115	+	0.959966i	$0.590373\pi$
$390$		0			0
$391$	301.553			0.771236
$392$		0			0
$393$		0			0
$394$		0			0
$395$			910.944i			2.30619i
$396$		0			0
$397$	202.104			0.509079			0.254539	−	0.967062i	$-0.418076\pi$
$397$	202.104			0.509079			0.254539	+	0.967062i	$0.418076\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	187.126i		−	0.466649i	−0.972399	−	0.233324i	$-0.925040\pi$
$401$		−	187.126i		−	0.466649i	0.972399	−	0.233324i	$-0.0749604\pi$
$402$		0			0
$403$	232.898			0.577910
$404$		0			0
$405$		0			0
$406$		0			0
$407$			312.325i			0.767384i
$408$		0			0
$409$		−	642.898i		−	1.57188i	−0.618304	−	0.785939i	$-0.712180\pi$
$409$		−	642.898i		−	1.57188i	0.618304	−	0.785939i	$-0.287820\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	57.7444i		−	0.139817i
$414$		0			0
$415$	−175.361			−0.422557
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−317.320			−0.757328			−0.378664	−	0.925534i	$-0.623616\pi$
$419$	−317.320			−0.757328			−0.378664	+	0.925534i	$0.623616\pi$
$420$		0			0
$421$			286.711i			0.681024i	0.940240	+	0.340512i	$0.110600\pi$
$421$			286.711i			0.681024i	−0.940240	+	0.340512i	$0.889400\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−181.411			−0.426850
$426$		0			0
$427$	19.5622			0.0458131
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	195.003i		−	0.452444i	−0.974076	−	0.226222i	$-0.927363\pi$
$431$		−	195.003i		−	0.452444i	0.974076	−	0.226222i	$-0.0726375\pi$
$432$		0			0
$433$			445.143i			1.02804i	0.857777	+	0.514022i	$0.171845\pi$
$433$			445.143i			1.02804i	−0.857777	+	0.514022i	$0.828155\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−349.366	−	101.347i	−0.799465	−	0.231915i
$438$		0			0
$439$		−	84.5171i		−	0.192522i	−0.995356	−	0.0962610i	$-0.969312\pi$
$439$		−	84.5171i		−	0.192522i	0.995356	−	0.0962610i	$-0.0306883\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−222.354			−0.501928			−0.250964	−	0.967996i	$-0.580748\pi$
$443$	−222.354			−0.501928			−0.250964	+	0.967996i	$0.580748\pi$
$444$		0			0
$445$		−	769.294i		−	1.72875i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	73.3880i		−	0.163448i	−0.996655	−	0.0817238i	$-0.973957\pi$
$449$		−	73.3880i		−	0.163448i	0.996655	−	0.0817238i	$-0.0260425\pi$
$450$		0			0
$451$			234.878i			0.520794i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			38.1285i			0.0837988i
$456$		0			0
$457$	−427.096			−0.934564			−0.467282	−	0.884108i	$-0.654767\pi$
$457$	−427.096			−0.934564			−0.467282	+	0.884108i	$0.654767\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−406.262			−0.881263			−0.440631	−	0.897688i	$-0.645245\pi$
$461$	−406.262			−0.881263			−0.440631	+	0.897688i	$0.645245\pi$
$462$		0			0
$463$	218.446			0.471806			0.235903	−	0.971777i	$-0.424195\pi$
$463$	218.446			0.471806			0.235903	+	0.971777i	$0.424195\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−194.669			−0.416849			−0.208425	−	0.978038i	$-0.566834\pi$
$467$	−194.669			−0.416849			−0.208425	+	0.978038i	$0.566834\pi$
$468$		0			0
$469$		−	16.4883i		−	0.0351563i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	486.877			1.02934
$474$		0			0
$475$	210.175	+	60.9692i	0.442474	+	0.128356i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−829.575			−1.73189			−0.865944	−	0.500141i	$-0.833282\pi$
$479$	−829.575			−1.73189			−0.865944	+	0.500141i	$0.833282\pi$
$480$		0			0
$481$	−181.970			−0.378316
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	768.549i		−	1.58464i
$486$		0			0
$487$		−	327.110i		−	0.671683i	−0.941918	−	0.335842i	$-0.890979\pi$
$487$		−	327.110i		−	0.671683i	0.941918	−	0.335842i	$-0.109021\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	756.825			1.54139			0.770697	−	0.637202i	$-0.219908\pi$
$491$	756.825			1.54139			0.770697	+	0.637202i	$0.219908\pi$
$492$		0			0
$493$		−	815.270i		−	1.65369i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	67.1113i		−	0.135033i
$498$		0			0
$499$	−700.692			−1.40419			−0.702096	−	0.712083i	$-0.747752\pi$
$499$	−700.692			−1.40419			−0.702096	+	0.712083i	$0.747752\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	433.003			0.860841			0.430421	−	0.902628i	$-0.358365\pi$
$503$	433.003			0.860841			0.430421	+	0.902628i	$0.358365\pi$
$504$		0			0
$505$	−482.309			−0.955066
$506$		0			0
$507$		0			0
$508$		0			0
$509$			200.538i			0.393983i	0.980405	+	0.196992i	$0.0631172\pi$
$509$			200.538i			0.393983i	−0.980405	+	0.196992i	$0.936883\pi$
$510$		0			0
$511$	75.2682			0.147296
$512$		0			0
$513$		0			0
$514$		0			0
$515$			647.445i			1.25717i
$516$		0			0
$517$	300.912			0.582034
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	39.8220i		−	0.0764337i	−0.999269	−	0.0382169i	$-0.987832\pi$
$521$		−	39.8220i		−	0.0764337i	0.999269	−	0.0382169i	$-0.0121678\pi$
$522$		0			0
$523$			735.534i			1.40637i	0.711005	+	0.703187i	$0.248241\pi$
$523$			735.534i			1.40637i	−0.711005	+	0.703187i	$0.751759\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	511.920i		−	0.971384i
$528$		0			0
$529$	−162.441			−0.307072
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−136.847			−0.256749
$534$		0			0
$535$			467.232i			0.873331i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	593.106			1.10038
$540$		0			0
$541$	836.284			1.54581			0.772906	−	0.634521i	$-0.218803\pi$
$541$	836.284			1.54581			0.772906	+	0.634521i	$0.218803\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	174.455i		−	0.320100i
$546$		0			0
$547$		−	806.768i		−	1.47490i	−0.675404	−	0.737448i	$-0.736031\pi$
$547$		−	806.768i		−	1.47490i	0.675404	−	0.737448i	$-0.263969\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−273.998	+	944.535i	−0.497275	+	1.71422i
$552$		0			0
$553$		−	132.733i		−	0.240024i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−31.2562			−0.0561152			−0.0280576	−	0.999606i	$-0.508932\pi$
$557$	−31.2562			−0.0561152			−0.0280576	+	0.999606i	$0.508932\pi$
$558$		0			0
$559$			283.669i			0.507458i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	624.626i		−	1.10946i	−0.832031	−	0.554730i	$-0.812822\pi$
$563$		−	624.626i		−	1.10946i	0.832031	−	0.554730i	$-0.187178\pi$
$564$		0			0
$565$		−	773.830i		−	1.36961i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	841.035i		−	1.47809i	−0.673655	−	0.739046i	$-0.735276\pi$
$569$		−	841.035i		−	1.47809i	0.673655	−	0.739046i	$-0.264724\pi$
$570$		0			0
$571$	154.965			0.271392			0.135696	−	0.990751i	$-0.456673\pi$
$571$	154.965			0.271392			0.135696	+	0.990751i	$0.456673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−220.518			−0.383510
$576$		0			0
$577$	−569.953			−0.987787			−0.493893	−	0.869522i	$-0.664427\pi$
$577$	−569.953			−0.987787			−0.493893	+	0.869522i	$0.664427\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	25.5518			0.0439790
$582$		0			0
$583$		−	161.630i		−	0.277238i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1118.55			1.90554			0.952768	−	0.303698i	$-0.0982214\pi$
$587$	1118.55			1.90554			0.952768	+	0.303698i	$0.0982214\pi$
$588$		0			0
$589$	−172.047	+	593.087i	−0.292101	+	1.00694i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	481.241			0.811536			0.405768	−	0.913976i	$-0.367004\pi$
$593$	481.241			0.811536			0.405768	+	0.913976i	$0.367004\pi$
$594$		0			0
$595$	83.8081			0.140854
$596$		0			0
$597$		0			0
$598$		0			0
$599$			262.338i			0.437960i	0.975729	+	0.218980i	$0.0702729\pi$
$599$			262.338i			0.437960i	−0.975729	+	0.218980i	$0.929727\pi$
$600$		0			0
$601$			767.853i			1.27763i	0.769362	+	0.638813i	$0.220574\pi$
$601$			767.853i			1.27763i	−0.769362	+	0.638813i	$0.779426\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−182.866			−0.302257
$606$		0			0
$607$			661.285i			1.08943i	0.838621	+	0.544716i	$0.183362\pi$
$607$			661.285i			1.08943i	−0.838621	+	0.544716i	$0.816638\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			175.320i			0.286939i
$612$		0			0
$613$	−244.930			−0.399559			−0.199780	−	0.979841i	$-0.564023\pi$
$613$	−244.930			−0.399559			−0.199780	+	0.979841i	$0.564023\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1072.05			1.73753			0.868764	−	0.495227i	$-0.164915\pi$
$617$	1072.05			1.73753			0.868764	+	0.495227i	$0.164915\pi$
$618$		0			0
$619$	22.3756			0.0361480			0.0180740	−	0.999837i	$-0.494247\pi$
$619$	22.3756			0.0361480			0.0180740	+	0.999837i	$0.494247\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			112.094i			0.179926i
$624$		0			0
$625$	−780.285			−1.24846
$626$		0			0
$627$		0			0
$628$		0			0
$629$			399.978i			0.635895i
$630$		0			0
$631$	−465.941			−0.738417			−0.369208	−	0.929347i	$-0.620371\pi$
$631$	−465.941			−0.738417			−0.369208	+	0.929347i	$0.620371\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	1110.69i		−	1.74911i
$636$		0			0
$637$			345.561i			0.542482i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			167.880i			0.261903i	0.991389	+	0.130952i	$0.0418032\pi$
$641$			167.880i			0.261903i	−0.991389	+	0.130952i	$0.958197\pi$
$642$		0			0
$643$	−386.222			−0.600656			−0.300328	−	0.953836i	$-0.597096\pi$
$643$	−386.222			−0.600656			−0.300328	+	0.953836i	$0.597096\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1225.10			1.89350			0.946752	−	0.321963i	$-0.104343\pi$
$647$	1225.10			1.89350			0.946752	+	0.321963i	$0.104343\pi$
$648$		0			0
$649$			806.550i			1.24276i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−71.4949			−0.109487			−0.0547434	−	0.998500i	$-0.517434\pi$
$653$	−71.4949			−0.109487			−0.0547434	+	0.998500i	$0.517434\pi$
$654$		0			0
$655$	−613.680			−0.936916
$656$		0			0
$657$		0			0
$658$		0			0
$659$			366.489i			0.556129i	0.960562	+	0.278065i	$0.0896929\pi$
$659$			366.489i			0.556129i	−0.960562	+	0.278065i	$0.910307\pi$
$660$		0			0
$661$			156.057i			0.236092i	0.993008	+	0.118046i	$0.0376631\pi$
$661$			156.057i			0.236092i	−0.993008	+	0.118046i	$0.962337\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−97.0963	−	28.1665i	−0.146009	−	0.0423556i
$666$		0			0
$667$		−	991.018i		−	1.48578i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−273.237			−0.407208
$672$		0			0
$673$			189.795i			0.282014i	0.990009	+	0.141007i	$0.0450339\pi$
$673$			189.795i			0.282014i	−0.990009	+	0.141007i	$0.954966\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			244.297i			0.360852i	0.983589	+	0.180426i	$0.0577476\pi$
$677$			244.297i			0.360852i	−0.983589	+	0.180426i	$0.942252\pi$
$678$		0			0
$679$			111.985i			0.164926i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			239.170i			0.350175i	0.984553	+	0.175088i	$0.0560209\pi$
$683$			239.170i			0.350175i	−0.984553	+	0.175088i	$0.943979\pi$
$684$		0			0
$685$	477.169			0.696597
$686$		0			0
$687$		0			0
$688$		0			0
$689$	94.1701			0.136676
$690$		0			0
$691$	−974.552			−1.41035			−0.705175	−	0.709033i	$-0.749132\pi$
$691$	−974.552			−1.41035			−0.705175	+	0.709033i	$0.749132\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−703.531			−1.01228
$696$		0			0
$697$			300.796i			0.431558i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1200.32			1.71229			0.856145	−	0.516736i	$-0.172853\pi$
$701$	1200.32			1.71229			0.856145	+	0.516736i	$0.172853\pi$
$702$		0			0
$703$	134.426	−	463.396i	0.191217	−	0.659170i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	70.2770			0.0994018
$708$		0			0
$709$	−211.764			−0.298679			−0.149340	−	0.988786i	$-0.547715\pi$
$709$	−211.764			−0.298679			−0.149340	+	0.988786i	$0.547715\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	622.274i		−	0.872754i
$714$		0			0
$715$		−	532.563i		−	0.744844i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	253.686			0.352831			0.176416	−	0.984316i	$-0.443550\pi$
$719$	253.686			0.352831			0.176416	+	0.984316i	$0.443550\pi$
$720$		0			0
$721$		−	94.3390i		−	0.130845i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			596.186i			0.822325i
$726$		0			0
$727$	−1191.34			−1.63871			−0.819355	−	0.573287i	$-0.805668\pi$
$727$	−1191.34			−1.63871			−0.819355	+	0.573287i	$0.805668\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	623.517			0.852965
$732$		0			0
$733$	336.376			0.458903			0.229451	−	0.973320i	$-0.426307\pi$
$733$	336.376			0.458903			0.229451	+	0.973320i	$0.426307\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			230.302i			0.312486i
$738$		0			0
$739$	818.392			1.10743			0.553716	−	0.832706i	$-0.313209\pi$
$739$	818.392			1.10743			0.553716	+	0.832706i	$0.313209\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1451.72i		−	1.95387i	−0.213539	−	0.976934i	$-0.568499\pi$
$743$		−	1451.72i		−	1.95387i	0.213539	−	0.976934i	$-0.431501\pi$
$744$		0			0
$745$	−887.340			−1.19106
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	68.0803i		−	0.0908949i
$750$		0			0
$751$		−	503.999i		−	0.671104i	−0.942022	−	0.335552i	$-0.891077\pi$
$751$		−	503.999i		−	0.671104i	0.942022	−	0.335552i	$-0.108923\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	707.465i		−	0.937040i
$756$		0			0
$757$	434.031			0.573357			0.286679	−	0.958027i	$-0.407449\pi$
$757$	434.031			0.573357			0.286679	+	0.958027i	$0.407449\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1313.70			1.72628			0.863140	−	0.504964i	$-0.168494\pi$
$761$	1313.70			1.72628			0.863140	+	0.504964i	$0.168494\pi$
$762$		0			0
$763$			25.4197i			0.0333155i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−469.920			−0.612673
$768$		0			0
$769$	−822.085			−1.06903			−0.534516	−	0.845159i	$-0.679506\pi$
$769$	−822.085			−1.06903			−0.534516	+	0.845159i	$0.679506\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			248.059i			0.320904i	0.987044	+	0.160452i	$0.0512951\pi$
$773$			248.059i			0.320904i	−0.987044	+	0.160452i	$0.948705\pi$
$774$		0			0
$775$			374.353i			0.483037i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	101.092	−	348.489i	0.129772	−	0.447354i
$780$		0			0
$781$			937.383i			1.20023i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−474.570			−0.604548
$786$		0			0
$787$			1402.06i			1.78152i	0.454473	+	0.890761i	$0.349828\pi$
$787$			1402.06i			1.78152i	−0.454473	+	0.890761i	$0.650172\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			112.755i			0.142547i
$792$		0			0
$793$		−	159.196i		−	0.200751i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	217.355i		−	0.272716i	−0.990660	−	0.136358i	$-0.956460\pi$
$797$		−	217.355i		−	0.272716i	0.990660	−	0.136358i	$-0.0435397\pi$
$798$		0			0
$799$	385.362			0.482305
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1051.32			−1.30924
$804$		0			0
$805$	101.875			0.126552
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1405.64			−1.73751			−0.868753	−	0.495246i	$-0.835078\pi$
$809$	−1405.64			−1.73751			−0.868753	+	0.495246i	$0.835078\pi$
$810$		0			0
$811$		−	653.317i		−	0.805570i	−0.915295	−	0.402785i	$-0.868042\pi$
$811$		−	653.317i		−	0.805570i	0.915295	−	0.402785i	$-0.131958\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−218.939			−0.268637
$816$		0			0
$817$	−722.379	−	209.553i	−0.884184	−	0.256491i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1188.57			−1.44771			−0.723854	−	0.689954i	$-0.757631\pi$
$821$	−1188.57			−1.44771			−0.723854	+	0.689954i	$0.757631\pi$
$822$		0			0
$823$	−582.737			−0.708064			−0.354032	−	0.935233i	$-0.615190\pi$
$823$	−582.737			−0.708064			−0.354032	+	0.935233i	$0.615190\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	277.658i		−	0.335742i	−0.985809	−	0.167871i	$-0.946311\pi$
$827$		−	277.658i		−	0.335742i	0.985809	−	0.167871i	$-0.0536891\pi$
$828$		0			0
$829$		−	404.811i		−	0.488313i	−0.969736	−	0.244156i	$-0.921489\pi$
$829$		−	404.811i		−	0.488313i	0.969736	−	0.244156i	$-0.0785110\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	759.559			0.911836
$834$		0			0
$835$			1566.86i			1.87648i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			305.667i			0.364323i	0.983269	+	0.182162i	$0.0583094\pi$
$839$			305.667i			0.364323i	−0.983269	+	0.182162i	$0.941691\pi$
$840$		0			0
$841$	−1838.28			−2.18583
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−710.980			−0.841397
$846$		0			0
$847$	26.6453			0.0314584
$848$		0			0
$849$		0			0
$850$		0			0
$851$			486.201i			0.571329i
$852$		0			0
$853$	1034.38			1.21263			0.606317	−	0.795223i	$-0.292646\pi$
$853$	1034.38			1.21263			0.606317	+	0.795223i	$0.292646\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1227.91i			1.43280i	0.697690	+	0.716400i	$0.254211\pi$
$857$			1227.91i			1.43280i	−0.697690	+	0.716400i	$0.745789\pi$
$858$		0			0
$859$	−949.077			−1.10486			−0.552431	−	0.833559i	$-0.686300\pi$
$859$	−949.077			−1.10486			−0.552431	+	0.833559i	$0.686300\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	543.866i		−	0.630204i	−0.949058	−	0.315102i	$-0.897961\pi$
$863$		−	543.866i		−	0.630204i	0.949058	−	0.315102i	$-0.102039\pi$
$864$		0			0
$865$		−	95.7533i		−	0.110697i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1853.97i			2.13345i
$870$		0			0
$871$	−134.181			−0.154054
$872$		0			0
$873$		0			0
$874$		0			0
$875$	71.7385			0.0819869
$876$		0			0
$877$		−	1216.65i		−	1.38728i	−0.720321	−	0.693641i	$-0.756006\pi$
$877$		−	1216.65i		−	1.38728i	0.720321	−	0.693641i	$-0.243994\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−154.334			−0.175180			−0.0875900	−	0.996157i	$-0.527917\pi$
$881$	−154.334			−0.175180			−0.0875900	+	0.996157i	$0.527917\pi$
$882$		0			0
$883$	−133.831			−0.151564			−0.0757818	−	0.997124i	$-0.524145\pi$
$883$	−133.831			−0.151564			−0.0757818	+	0.997124i	$0.524145\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1333.43i		−	1.50331i	−0.659557	−	0.751654i	$-0.729256\pi$
$887$		−	1333.43i		−	1.50331i	0.659557	−	0.751654i	$-0.270744\pi$
$888$		0			0
$889$			161.838i			0.182045i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−446.462	−	129.513i	−0.499958	−	0.145032i
$894$		0			0
$895$			717.720i			0.801922i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1682.36			−1.87137
$900$		0			0
$901$		−	206.990i		−	0.229734i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			2004.50i			2.21491i
$906$		0			0
$907$			824.747i			0.909314i	0.890667	+	0.454657i	$0.150238\pi$
$907$			824.747i			0.909314i	−0.890667	+	0.454657i	$0.849762\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			846.919i			0.929658i	0.885400	+	0.464829i	$0.153884\pi$
$911$			846.919i			0.929658i	−0.885400	+	0.464829i	$0.846116\pi$
$912$		0			0
$913$	−356.897			−0.390906
$914$		0			0
$915$		0			0
$916$		0			0
$917$	89.4192			0.0975127
$918$		0			0
$919$	291.763			0.317479			0.158739	−	0.987321i	$-0.449257\pi$
$919$	291.763			0.317479			0.158739	+	0.987321i	$0.449257\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−546.147			−0.591709
$924$		0			0
$925$		−	292.494i		−	0.316209i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	274.096			0.295044			0.147522	−	0.989059i	$-0.452870\pi$
$929$	274.096			0.295044			0.147522	+	0.989059i	$0.452870\pi$
$930$		0			0
$931$	−879.991	−	255.275i	−0.945210	−	0.274194i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1170.60			−1.25198
$936$		0			0
$937$	−654.761			−0.698785			−0.349392	−	0.936977i	$-0.613612\pi$
$937$	−654.761			−0.698785			−0.349392	+	0.936977i	$0.613612\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1206.52i			1.28216i	0.767473	+	0.641081i	$0.221514\pi$
$941$			1206.52i			1.28216i	−0.767473	+	0.641081i	$0.778486\pi$
$942$		0			0
$943$			365.638i			0.387740i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−50.5232			−0.0533508			−0.0266754	−	0.999644i	$-0.508492\pi$
$947$	−50.5232			−0.0533508			−0.0266754	+	0.999644i	$0.508492\pi$
$948$		0			0
$949$		−	612.528i		−	0.645446i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	617.380i		−	0.647828i	−0.946086	−	0.323914i	$-0.895001\pi$
$953$		−	617.380i		−	0.647828i	0.946086	−	0.323914i	$-0.104999\pi$
$954$		0			0
$955$	391.798			0.410260
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−69.5281			−0.0725007
$960$		0			0
$961$	−95.3773			−0.0992479
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1749.84i		−	1.81330i
$966$		0			0
$967$	−1287.02			−1.33094			−0.665470	−	0.746425i	$-0.731769\pi$
$967$	−1287.02			−1.33094			−0.665470	+	0.746425i	$0.731769\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			44.5046i			0.0458337i	0.999737	+	0.0229169i	$0.00729531\pi$
$971$			44.5046i			0.0458337i	−0.999737	+	0.0229169i	$0.992705\pi$
$972$		0			0
$973$	102.511			0.105356
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1416.29i			1.44963i	0.688941	+	0.724817i	$0.258076\pi$
$977$			1416.29i			1.44963i	−0.688941	+	0.724817i	$0.741924\pi$
$978$		0			0
$979$		−	1565.68i		−	1.59926i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1387.41i		−	1.41141i	−0.708507	−	0.705704i	$-0.750631\pi$
$983$		−	1387.41i		−	1.41141i	0.708507	−	0.705704i	$-0.249369\pi$
$984$		0			0
$985$	−1922.76			−1.95204
$986$		0			0
$987$		0			0
$988$		0			0
$989$	757.928			0.766358
$990$		0			0
$991$		−	281.318i		−	0.283873i	−0.989876	−	0.141937i	$-0.954667\pi$
$991$		−	281.318i		−	0.283873i	0.989876	−	0.141937i	$-0.0453329\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−498.861			−0.501368
$996$		0			0
$997$	−1808.55			−1.81399			−0.906996	−	0.421138i	$-0.861631\pi$
$997$	−1808.55			−1.81399			−0.906996	+	0.421138i	$0.861631\pi$
$998$		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.721.3		20
3.2	odd	2	inner	2736.3.o.q.721.17		20
4.3	odd	2		1368.3.o.d.721.3	✓	20
12.11	even	2		1368.3.o.d.721.17	yes	20
19.18	odd	2	inner	2736.3.o.q.721.4		20
57.56	even	2	inner	2736.3.o.q.721.18		20
76.75	even	2		1368.3.o.d.721.4	yes	20
228.227	odd	2		1368.3.o.d.721.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.3.o.d.721.3	✓	20	4.3	odd	2
1368.3.o.d.721.4	yes	20	76.75	even	2
1368.3.o.d.721.17	yes	20	12.11	even	2
1368.3.o.d.721.18	yes	20	228.227	odd	2
2736.3.o.q.721.3		20	1.1	even	1	trivial
2736.3.o.q.721.4		20	19.18	odd	2	inner
2736.3.o.q.721.17		20	3.2	odd	2	inner
2736.3.o.q.721.18		20	57.56	even	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)

\(f(q)\)	\(=\)	\(q-6.04300 q^{5} +0.880524 q^{7} +O(q^{10})\)	\(q-6.04300 q^{5} +0.880524 q^{7} -12.2988 q^{11} -7.16565i q^{13} -15.7504 q^{17} +(18.2477 + 5.29345i) q^{19} -19.1457 q^{23} +11.5179 q^{25} +51.7618i q^{29} +32.5020i q^{31} -5.32101 q^{35} -25.3948i q^{37} -19.0976i q^{41} -39.5873 q^{43} -24.4667 q^{47} -48.2247 q^{49} +13.1419i q^{53} +74.3217 q^{55} -65.5796i q^{59} +22.2165 q^{61} +43.3020i q^{65} -18.7255i q^{67} -76.2174i q^{71} +85.4812 q^{73} -10.8294 q^{77} -150.744i q^{79} +29.0189 q^{83} +95.1798 q^{85} +127.303i q^{89} -6.30952i q^{91} +(-110.271 - 31.9883i) q^{95} +127.180i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7}+O(q^{10}) \) 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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