LMFDB - Embedded newform 816.2.e.c.239.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,2,Mod(239,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("816.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	816.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.51579280494$
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} - 2 x^{19} + 2 x^{18} - 2 x^{17} + 11 x^{16} - 28 x^{15} + 36 x^{14} - 38 x^{13} + 61 x^{12} + \cdots + 1024 $ x^20 - 2x^19 + 2x^18 - 2x^17 + 11x^16 - 28x^15 + 36x^14 - 38x^13 + 61x^12 - 138x^11 + 226x^10 - 276x^9 + 244x^8 - 304x^7 + 576x^6 - 896x^5 + 704x^4 - 256x^3 + 512x^2 - 1024*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.18
Root			$0.689587 - 1.23469i$ of defining polynomial
Character	$\chi$	$=$	816.239
Dual form			816.2.e.c.239.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.39392 + 1.02809i) q^{3} +3.10097i q^{5} -1.09021i q^{7} +(0.886042 + 2.86617i) q^{9} +O(q^{10})$	\(q+(1.39392 + 1.02809i) q^{3} +3.10097i q^{5} -1.09021i q^{7} +(0.886042 + 2.86617i) q^{9} -0.168447 q^{11} -4.08110 q^{13} +(-3.18809 + 4.32252i) q^{15} +1.00000i q^{17} +7.15205i q^{19} +(1.12084 - 1.51967i) q^{21} +7.51061 q^{23} -4.61603 q^{25} +(-1.71162 + 4.90615i) q^{27} -4.05922i q^{29} -5.53949i q^{31} +(-0.234802 - 0.173179i) q^{33} +3.38072 q^{35} -10.4493 q^{37} +(-5.68874 - 4.19576i) q^{39} +2.08110i q^{41} +1.33573i q^{43} +(-8.88791 + 2.74759i) q^{45} +2.73245 q^{47} +5.81143 q^{49} +(-1.02809 + 1.39392i) q^{51} +5.96479i q^{53} -0.522349i q^{55} +(-7.35299 + 9.96941i) q^{57} +1.28367 q^{59} -8.71247 q^{61} +(3.12474 - 0.965975i) q^{63} -12.6554i q^{65} -5.24977i q^{67} +(10.4692 + 7.72162i) q^{69} +11.1023 q^{71} +2.69752 q^{73} +(-6.43438 - 4.74571i) q^{75} +0.183643i q^{77} +1.09021i q^{79} +(-7.42986 + 5.07909i) q^{81} +12.6611 q^{83} -3.10097 q^{85} +(4.17326 - 5.65824i) q^{87} -8.18817i q^{89} +4.44927i q^{91} +(5.69512 - 7.72162i) q^{93} -22.1783 q^{95} +19.4432 q^{97} +(-0.149251 - 0.482797i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{9}+O(q^{10}) $ 20 * q - 4 * q^9	$ 20 q - 4 q^{9} + 12 q^{13} - 16 q^{21} - 48 q^{25} + 12 q^{33} - 16 q^{45} - 20 q^{49} + 64 q^{57} - 48 q^{61} + 28 q^{69} + 64 q^{73} - 60 q^{81} + 4 q^{85} + 32 q^{93} + 32 q^{97}+O(q^{100}) $ 20 * q - 4 * q^9 + 12 * q^13 - 16 * q^21 - 48 * q^25 + 12 * q^33 - 16 * q^45 - 20 * q^49 + 64 * q^57 - 48 * q^61 + 28 * q^69 + 64 * q^73 - 60 * q^81 + 4 * q^85 + 32 * q^93 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/816\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$545$	$613$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	1.39392	+	1.02809i	0.804782	+	0.593571i
$3$	1.39392	+	1.02809i	0.804782	+	0.593571i
$4$		0			0
$5$			3.10097i			1.38680i	0.720554	+	0.693398i	$0.243887\pi$
$5$			3.10097i			1.38680i	−0.720554	+	0.693398i	$0.756113\pi$
$6$		0			0
$7$		−	1.09021i		−	0.412062i	−0.978545	−	0.206031i	$-0.933945\pi$
$7$		−	1.09021i		−	0.412062i	0.978545	−	0.206031i	$-0.0660548\pi$
$8$		0			0
$9$	0.886042	+	2.86617i	0.295347	+	0.955390i
$10$		0			0
$11$	−0.168447			−0.0507886			−0.0253943	−	0.999678i	$-0.508084\pi$
$11$	−0.168447			−0.0507886			−0.0253943	+	0.999678i	$0.508084\pi$
$12$		0			0
$13$	−4.08110			−1.13189			−0.565947	−	0.824442i	$-0.691489\pi$
$13$	−4.08110			−1.13189			−0.565947	+	0.824442i	$0.691489\pi$
$14$		0			0
$15$	−3.18809	+	4.32252i	−0.823162	+	1.11607i
$16$		0			0
$17$			1.00000i			0.242536i
$17$			1.00000i			0.242536i
$18$		0			0
$19$			7.15205i			1.64079i	0.571795	+	0.820396i	$0.306247\pi$
$19$			7.15205i			1.64079i	−0.571795	+	0.820396i	$0.693753\pi$
$20$		0			0
$21$	1.12084	−	1.51967i	0.244588	−	0.331620i
$22$		0			0
$23$	7.51061			1.56607			0.783035	−	0.621978i	$-0.213671\pi$
$23$	7.51061			1.56607			0.783035	+	0.621978i	$0.213671\pi$
$24$		0			0
$25$	−4.61603			−0.923205
$26$		0			0
$27$	−1.71162	+	4.90615i	−0.329402	+	0.944190i
$28$		0			0
$29$		−	4.05922i		−	0.753778i	−0.926258	−	0.376889i	$-0.876994\pi$
$29$		−	4.05922i		−	0.753778i	0.926258	−	0.376889i	$-0.123006\pi$
$30$		0			0
$31$		−	5.53949i		−	0.994921i	−0.867487	−	0.497461i	$-0.834266\pi$
$31$		−	5.53949i		−	0.994921i	0.867487	−	0.497461i	$-0.165734\pi$
$32$		0			0
$33$	−0.234802	−	0.173179i	−0.0408737	−	0.0301466i
$34$		0			0
$35$	3.38072			0.571447
$36$		0			0
$37$	−10.4493			−1.71786			−0.858930	−	0.512093i	$-0.828870\pi$
$37$	−10.4493			−1.71786			−0.858930	+	0.512093i	$0.828870\pi$
$38$		0			0
$39$	−5.68874	−	4.19576i	−0.910927	−	0.671859i
$40$		0			0
$41$			2.08110i			0.325013i	0.986707	+	0.162507i	$0.0519579\pi$
$41$			2.08110i			0.325013i	−0.986707	+	0.162507i	$0.948042\pi$
$42$		0			0
$43$			1.33573i			0.203696i	0.994800	+	0.101848i	$0.0324756\pi$
$43$			1.33573i			0.203696i	−0.994800	+	0.101848i	$0.967524\pi$
$44$		0			0
$45$	−8.88791	+	2.74759i	−1.32493	+	0.409587i
$46$		0			0
$47$	2.73245			0.398568			0.199284	−	0.979942i	$-0.436138\pi$
$47$	2.73245			0.398568			0.199284	+	0.979942i	$0.436138\pi$
$48$		0			0
$49$	5.81143			0.830205
$50$		0			0
$51$	−1.02809	+	1.39392i	−0.143962	+	0.195188i
$52$		0			0
$53$			5.96479i			0.819326i	0.912237	+	0.409663i	$0.134354\pi$
$53$			5.96479i			0.819326i	−0.912237	+	0.409663i	$0.865646\pi$
$54$		0			0
$55$		−	0.522349i		−	0.0704335i
$56$		0			0
$57$	−7.35299	+	9.96941i	−0.973927	+	1.32048i
$58$		0			0
$59$	1.28367			0.167120			0.0835600	−	0.996503i	$-0.473371\pi$
$59$	1.28367			0.167120			0.0835600	+	0.996503i	$0.473371\pi$
$60$		0			0
$61$	−8.71247			−1.11552			−0.557759	−	0.830003i	$-0.688339\pi$
$61$	−8.71247			−1.11552			−0.557759	+	0.830003i	$0.688339\pi$
$62$		0			0
$63$	3.12474	−	0.965975i	0.393680	−	0.121701i
$64$		0			0
$65$		−	12.6554i		−	1.56971i
$66$		0			0
$67$		−	5.24977i		−	0.641361i	−0.947187	−	0.320681i	$-0.896088\pi$
$67$		−	5.24977i		−	0.641361i	0.947187	−	0.320681i	$-0.103912\pi$
$68$		0			0
$69$	10.4692	+	7.72162i	1.26034	+	0.929574i
$70$		0			0
$71$	11.1023			1.31760			0.658800	−	0.752318i	$-0.271064\pi$
$71$	11.1023			1.31760			0.658800	+	0.752318i	$0.271064\pi$
$72$		0			0
$73$	2.69752			0.315721			0.157860	−	0.987461i	$-0.449540\pi$
$73$	2.69752			0.315721			0.157860	+	0.987461i	$0.449540\pi$
$74$		0			0
$75$	−6.43438	−	4.74571i	−0.742979	−	0.547988i
$76$		0			0
$77$			0.183643i			0.0209281i
$78$		0			0
$79$			1.09021i			0.122659i	0.998118	+	0.0613293i	$0.0195340\pi$
$79$			1.09021i			0.122659i	−0.998118	+	0.0613293i	$0.980466\pi$
$80$		0			0
$81$	−7.42986	+	5.07909i	−0.825540	+	0.564344i
$82$		0			0
$83$	12.6611			1.38974			0.694871	−	0.719135i	$-0.255462\pi$
$83$	12.6611			1.38974			0.694871	+	0.719135i	$0.255462\pi$
$84$		0			0
$85$	−3.10097			−0.336348
$86$		0			0
$87$	4.17326	−	5.65824i	0.447421	−	0.606627i
$88$		0			0
$89$		−	8.18817i		−	0.867945i	−0.900926	−	0.433972i	$-0.857112\pi$
$89$		−	8.18817i		−	0.867945i	0.900926	−	0.433972i	$-0.142888\pi$
$90$		0			0
$91$			4.44927i			0.466411i
$92$		0			0
$93$	5.69512	−	7.72162i	0.590556	−	0.800694i
$94$		0			0
$95$	−22.1783			−2.27545
$96$		0			0
$97$	19.4432			1.97416			0.987081	−	0.160224i	$-0.0512215\pi$
$97$	19.4432			1.97416			0.987081	+	0.160224i	$0.0512215\pi$
$98$		0			0
$99$	−0.149251	−	0.482797i	−0.0150003	−	0.0485229i
$100$		0			0
$101$		−	2.92544i		−	0.291092i	−0.989352	−	0.145546i	$-0.953506\pi$
$101$		−	2.92544i		−	0.291092i	0.989352	−	0.145546i	$-0.0464938\pi$
$102$		0			0
$103$		−	7.46343i		−	0.735394i	−0.929946	−	0.367697i	$-0.880146\pi$
$103$		−	7.46343i		−	0.735394i	0.929946	−	0.367697i	$-0.119854\pi$
$104$		0			0
$105$	4.71247	+	3.47570i	0.459890	+	0.339194i
$106$		0			0
$107$	8.64168			0.835423			0.417712	−	0.908580i	$-0.362832\pi$
$107$	8.64168			0.835423			0.417712	+	0.908580i	$0.362832\pi$
$108$		0			0
$109$	11.3702			1.08907			0.544536	−	0.838738i	$-0.316706\pi$
$109$	11.3702			1.08907			0.544536	+	0.838738i	$0.316706\pi$
$110$		0			0
$111$	−14.5656	−	10.7429i	−1.38250	−	1.01967i
$112$		0			0
$113$			19.8569i			1.86798i	0.357296	+	0.933991i	$0.383699\pi$
$113$			19.8569i			1.86798i	−0.357296	+	0.933991i	$0.616301\pi$
$114$		0			0
$115$			23.2902i			2.17182i
$116$		0			0
$117$	−3.61603	−	11.6971i	−0.334302	−	1.08140i
$118$		0			0
$119$	1.09021			0.0999398
$120$		0			0
$121$	−10.9716			−0.997421
$122$		0			0
$123$	−2.13957	+	2.90089i	−0.192918	+	0.261565i
$124$		0			0
$125$			1.19069i			0.106499i
$126$		0			0
$127$			15.7825i			1.40047i	0.713910	+	0.700237i	$0.246922\pi$
$127$			15.7825i			1.40047i	−0.713910	+	0.700237i	$0.753078\pi$
$128$		0			0
$129$	−1.37325	+	1.86190i	−0.120908	+	0.163931i
$130$		0			0
$131$	8.31148			0.726178			0.363089	−	0.931755i	$-0.381722\pi$
$131$	8.31148			0.726178			0.363089	+	0.931755i	$0.381722\pi$
$132$		0			0
$133$	7.79727			0.676109
$134$		0			0
$135$	−15.2138	−	5.30769i	−1.30940	−	0.456813i
$136$		0			0
$137$		−	13.9740i		−	1.19388i	−0.802285	−	0.596941i	$-0.796383\pi$
$137$		−	13.9740i		−	1.19388i	0.802285	−	0.596941i	$-0.203617\pi$
$138$		0			0
$139$		−	1.08390i		−	0.0919353i	−0.998943	−	0.0459676i	$-0.985363\pi$
$139$		−	1.08390i		−	0.0919353i	0.998943	−	0.0459676i	$-0.0146371\pi$
$140$		0			0
$141$	3.80882	+	2.80921i	0.320760	+	0.236578i
$142$		0			0
$143$	0.687448			0.0574873
$144$		0			0
$145$	12.5875			1.04534
$146$		0			0
$147$	8.10069	+	5.97470i	0.668134	+	0.492785i
$148$		0			0
$149$		−	19.5519i		−	1.60176i	−0.598828	−	0.800878i	$-0.704367\pi$
$149$		−	19.5519i		−	1.60176i	0.598828	−	0.800878i	$-0.295633\pi$
$150$		0			0
$151$		−	11.8830i		−	0.967028i	−0.875337	−	0.483514i	$-0.839360\pi$
$151$		−	11.8830i		−	0.967028i	0.875337	−	0.483514i	$-0.160640\pi$
$152$		0			0
$153$	−2.86617	+	0.886042i	−0.231716	+	0.0716322i
$154$		0			0
$155$	17.1778			1.37975
$156$		0			0
$157$	6.19954			0.494777			0.247389	−	0.968916i	$-0.420428\pi$
$157$	6.19954			0.494777			0.247389	+	0.968916i	$0.420428\pi$
$158$		0			0
$159$	−6.13236	+	8.31445i	−0.486328	+	0.659379i
$160$		0			0
$161$		−	8.18817i		−	0.645318i
$162$		0			0
$163$		−	12.4740i		−	0.977039i	−0.872553	−	0.488520i	$-0.837537\pi$
$163$		−	12.4740i		−	0.977039i	0.872553	−	0.488520i	$-0.162463\pi$
$164$		0			0
$165$	0.537024	−	0.728114i	0.0418073	−	0.0566836i
$166$		0			0
$167$	−23.4915			−1.81783			−0.908913	−	0.416986i	$-0.863086\pi$
$167$	−23.4915			−1.81783			−0.908913	+	0.416986i	$0.863086\pi$
$168$		0			0
$169$	3.65537			0.281183
$170$		0			0
$171$	−20.4990	+	6.33702i	−1.56760	+	0.484604i
$172$		0			0
$173$		−	2.28501i		−	0.173726i	−0.996220	−	0.0868630i	$-0.972316\pi$
$173$		−	2.28501i		−	0.173726i	0.996220	−	0.0868630i	$-0.0276843\pi$
$174$		0			0
$175$			5.03246i			0.380418i
$176$		0			0
$177$	1.78934	+	1.31974i	0.134495	+	0.0991976i
$178$		0			0
$179$	−9.61732			−0.718832			−0.359416	−	0.933177i	$-0.617024\pi$
$179$	−9.61732			−0.718832			−0.359416	+	0.933177i	$0.617024\pi$
$180$		0			0
$181$	16.0678			1.19431			0.597154	−	0.802127i	$-0.296298\pi$
$181$	16.0678			1.19431			0.597154	+	0.802127i	$0.296298\pi$
$182$		0			0
$183$	−12.1445	−	8.95724i	−0.897748	−	0.662138i
$184$		0			0
$185$		−	32.4031i		−	2.38232i
$186$		0			0
$187$		−	0.168447i		−	0.0123180i
$188$		0			0
$189$	5.34876	+	1.86603i	0.389065	+	0.135734i
$190$		0			0
$191$	0.0609939			0.00441337			0.00220668	−	0.999998i	$-0.499298\pi$
$191$	0.0609939			0.00441337			0.00220668	+	0.999998i	$0.499298\pi$
$192$		0			0
$193$	−14.3601			−1.03366			−0.516832	−	0.856087i	$-0.672889\pi$
$193$	−14.3601			−1.03366			−0.516832	+	0.856087i	$0.672889\pi$
$194$		0			0
$195$	13.0109	−	17.6406i	0.931732	−	1.26327i
$196$		0			0
$197$			13.1500i			0.936896i	0.883491	+	0.468448i	$0.155187\pi$
$197$			13.1500i			0.936896i	−0.883491	+	0.468448i	$0.844813\pi$
$198$		0			0
$199$		−	9.33736i		−	0.661908i	−0.943647	−	0.330954i	$-0.892630\pi$
$199$		−	9.33736i		−	0.661908i	0.943647	−	0.330954i	$-0.107370\pi$
$200$		0			0
$201$	5.39726	−	7.31777i	0.380693	−	0.516156i
$202$		0			0
$203$	−4.42542			−0.310604
$204$		0			0
$205$	−6.45343			−0.450727
$206$		0			0
$207$	6.65471	+	21.5267i	0.462535	+	1.49621i
$208$		0			0
$209$		−	1.20474i		−	0.0833336i
$210$		0			0
$211$		−	13.7706i		−	0.948004i	−0.880524	−	0.474002i	$-0.842809\pi$
$211$		−	13.7706i		−	0.948004i	0.880524	−	0.474002i	$-0.157191\pi$
$212$		0			0
$213$	15.4757	+	11.4142i	1.06038	+	0.782089i
$214$		0			0
$215$	−4.14205			−0.282485
$216$		0			0
$217$	−6.03923			−0.409970
$218$		0			0
$219$	3.76013	+	2.77331i	0.254086	+	0.187403i
$220$		0			0
$221$		−	4.08110i		−	0.274524i
$222$		0			0
$223$		−	24.5000i		−	1.64064i	−0.571905	−	0.820320i	$-0.693795\pi$
$223$		−	24.5000i		−	1.64064i	0.571905	−	0.820320i	$-0.306205\pi$
$224$		0			0
$225$	−4.08999	−	13.2303i	−0.272666	−	0.882021i
$226$		0			0
$227$	21.9511			1.45695			0.728473	−	0.685074i	$-0.240230\pi$
$227$	21.9511			1.45695			0.728473	+	0.685074i	$0.240230\pi$
$228$		0			0
$229$	7.22299			0.477309			0.238654	−	0.971105i	$-0.423294\pi$
$229$	7.22299			0.477309			0.238654	+	0.971105i	$0.423294\pi$
$230$		0			0
$231$	−0.188802	+	0.255984i	−0.0124223	+	0.0168425i
$232$		0			0
$233$		−	5.92814i		−	0.388366i	−0.980965	−	0.194183i	$-0.937795\pi$
$233$		−	5.92814i		−	0.388366i	0.980965	−	0.194183i	$-0.0622055\pi$
$234$		0			0
$235$			8.47324i			0.552733i
$236$		0			0
$237$	−1.12084	+	1.51967i	−0.0728066	+	0.0987135i
$238$		0			0
$239$	28.6065			1.85040			0.925202	−	0.379476i	$-0.123896\pi$
$239$	28.6065			1.85040			0.925202	+	0.379476i	$0.123896\pi$
$240$		0			0
$241$	13.4737			0.867920			0.433960	−	0.900932i	$-0.357116\pi$
$241$	13.4737			0.867920			0.433960	+	0.900932i	$0.357116\pi$
$242$		0			0
$243$	−15.5784	−	0.558736i	−0.999357	−	0.0358430i
$244$		0			0
$245$			18.0211i			1.15133i
$246$		0			0
$247$		−	29.1882i		−	1.85720i
$248$		0			0
$249$	17.6487	+	13.0169i	1.11844	+	0.824910i
$250$		0			0
$251$	−4.43830			−0.280143			−0.140072	−	0.990141i	$-0.544733\pi$
$251$	−4.43830			−0.280143			−0.140072	+	0.990141i	$0.544733\pi$
$252$		0			0
$253$	−1.26514			−0.0795385
$254$		0			0
$255$	−4.32252	−	3.18809i	−0.270686	−	0.199646i
$256$		0			0
$257$			27.2101i			1.69732i	0.528940	+	0.848659i	$0.322590\pi$
$257$			27.2101i			1.69732i	−0.528940	+	0.848659i	$0.677410\pi$
$258$		0			0
$259$			11.3920i			0.707865i
$260$		0			0
$261$	11.6344	−	3.59664i	0.720152	−	0.222626i
$262$		0			0
$263$	−0.324267			−0.0199952			−0.00999758	−	0.999950i	$-0.503182\pi$
$263$	−0.324267			−0.0199952			−0.00999758	+	0.999950i	$0.503182\pi$
$264$		0			0
$265$	−18.4966			−1.13624
$266$		0			0
$267$	8.41822	−	11.4137i	0.515187	−	0.698506i
$268$		0			0
$269$			13.0267i			0.794253i	0.917764	+	0.397126i	$0.129993\pi$
$269$			13.0267i			0.794253i	−0.917764	+	0.397126i	$0.870007\pi$
$270$		0			0
$271$			4.43249i			0.269254i	0.990896	+	0.134627i	$0.0429837\pi$
$271$			4.43249i			0.269254i	−0.990896	+	0.134627i	$0.957016\pi$
$272$		0			0
$273$	−4.57427	+	6.20194i	−0.276848	+	0.375359i
$274$		0			0
$275$	0.777554			0.0468883
$276$		0			0
$277$	−12.4279			−0.746720			−0.373360	−	0.927687i	$-0.621794\pi$
$277$	−12.4279			−0.746720			−0.373360	+	0.927687i	$0.621794\pi$
$278$		0			0
$279$	15.8771	−	4.90822i	0.950538	−	0.293847i
$280$		0			0
$281$			5.39425i			0.321794i	0.986971	+	0.160897i	$0.0514387\pi$
$281$			5.39425i			0.321794i	−0.986971	+	0.160897i	$0.948561\pi$
$282$		0			0
$283$		−	22.4863i		−	1.33667i	−0.743859	−	0.668337i	$-0.767006\pi$
$283$		−	22.4863i		−	1.33667i	0.743859	−	0.668337i	$-0.232994\pi$
$284$		0			0
$285$	−30.9148	−	22.8014i	−1.83124	−	1.35064i
$286$		0			0
$287$	2.26884			0.133926
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	27.1024	+	19.9895i	1.58877	+	1.17180i
$292$		0			0
$293$		−	8.08724i		−	0.472462i	−0.971697	−	0.236231i	$-0.924088\pi$
$293$		−	8.08724i		−	0.472462i	0.971697	−	0.236231i	$-0.0759121\pi$
$294$		0			0
$295$			3.98063i			0.231761i
$296$		0			0
$297$	0.288317	−	0.826426i	0.0167298	−	0.0479541i
$298$		0			0
$299$	−30.6515			−1.77262
$300$		0			0
$301$	1.45623			0.0839355
$302$		0			0
$303$	3.00763	−	4.07783i	0.172784	−	0.234265i
$304$		0			0
$305$		−	27.0171i		−	1.54700i
$306$		0			0
$307$			1.68628i			0.0962411i	0.998842	+	0.0481206i	$0.0153232\pi$
$307$			1.68628i			0.0962411i	−0.998842	+	0.0481206i	$0.984677\pi$
$308$		0			0
$309$	7.67312	−	10.4035i	0.436508	−	0.591832i
$310$		0			0
$311$	−13.9998			−0.793858			−0.396929	−	0.917849i	$-0.629924\pi$
$311$	−13.9998			−0.793858			−0.396929	+	0.917849i	$0.629924\pi$
$312$		0			0
$313$	7.40428			0.418515			0.209257	−	0.977861i	$-0.432895\pi$
$313$	7.40428			0.418515			0.209257	+	0.977861i	$0.432895\pi$
$314$		0			0
$315$	2.99546	+	9.68973i	0.168775	+	0.545954i
$316$		0			0
$317$			21.5807i			1.21209i	0.795429	+	0.606047i	$0.207246\pi$
$317$			21.5807i			1.21209i	−0.795429	+	0.606047i	$0.792754\pi$
$318$		0			0
$319$			0.683763i			0.0382834i
$320$		0			0
$321$	12.0458	+	8.88447i	0.672333	+	0.495883i
$322$		0			0
$323$	−7.15205			−0.397951
$324$		0			0
$325$	18.8385			1.04497
$326$		0			0
$327$	15.8492	+	11.6897i	0.876465	+	0.646441i
$328$		0			0
$329$		−	2.97895i		−	0.164235i
$330$		0			0
$331$		−	1.42414i		−	0.0782779i	−0.999234	−	0.0391390i	$-0.987538\pi$
$331$		−	1.42414i		−	0.0782779i	0.999234	−	0.0391390i	$-0.0124615\pi$
$332$		0			0
$333$	−9.25855	−	29.9496i	−0.507365	−	1.64123i
$334$		0			0
$335$	16.2794			0.889438
$336$		0			0
$337$	−33.0119			−1.79827			−0.899136	−	0.437669i	$-0.855804\pi$
$337$	−33.0119			−1.79827			−0.899136	+	0.437669i	$0.855804\pi$
$338$		0			0
$339$	−20.4148	+	27.6790i	−1.10878	+	1.50332i
$340$		0			0
$341$			0.933109i			0.0505307i
$342$		0			0
$343$		−	13.9672i		−	0.754158i
$344$		0			0
$345$	−23.9445	+	32.4647i	−1.28913	+	1.74784i
$346$		0			0
$347$	−27.1438			−1.45715			−0.728577	−	0.684963i	$-0.759818\pi$
$347$	−27.1438			−1.45715			−0.728577	+	0.684963i	$0.759818\pi$
$348$		0			0
$349$	1.07627			0.0576112			0.0288056	−	0.999585i	$-0.490830\pi$
$349$	1.07627			0.0576112			0.0288056	+	0.999585i	$0.490830\pi$
$350$		0			0
$351$	6.98529	−	20.0225i	0.372847	−	1.06872i
$352$		0			0
$353$		−	21.7706i		−	1.15873i	−0.815068	−	0.579366i	$-0.803300\pi$
$353$		−	21.7706i		−	1.15873i	0.815068	−	0.579366i	$-0.196700\pi$
$354$		0			0
$355$			34.4279i			1.82724i
$356$		0			0
$357$	1.51967	+	1.12084i	0.0804297	+	0.0593213i
$358$		0			0
$359$	−30.5060			−1.61005			−0.805023	−	0.593243i	$-0.797847\pi$
$359$	−30.5060			−1.61005			−0.805023	+	0.593243i	$0.797847\pi$
$360$		0			0
$361$	−32.1518			−1.69220
$362$		0			0
$363$	−15.2936	−	11.2799i	−0.802706	−	0.592040i
$364$		0			0
$365$			8.36493i			0.437841i
$366$		0			0
$367$			23.1445i			1.20813i	0.796934	+	0.604067i	$0.206454\pi$
$367$			23.1445i			1.20813i	−0.796934	+	0.604067i	$0.793546\pi$
$368$		0			0
$369$	−5.96479	+	1.84394i	−0.310514	+	0.0959918i
$370$		0			0
$371$	6.50289			0.337613
$372$		0			0
$373$	−17.1832			−0.889715			−0.444857	−	0.895601i	$-0.646746\pi$
$373$	−17.1832			−0.889715			−0.444857	+	0.895601i	$0.646746\pi$
$374$		0			0
$375$	−1.22415	+	1.65974i	−0.0632147	+	0.0857084i
$376$		0			0
$377$			16.5661i			0.853197i
$378$		0			0
$379$			12.3573i			0.634750i	0.948300	+	0.317375i	$0.102801\pi$
$379$			12.3573i			0.634750i	−0.948300	+	0.317375i	$0.897199\pi$
$380$		0			0
$381$	−16.2259	+	21.9996i	−0.831281	+	1.12708i
$382$		0			0
$383$	20.9596			1.07098			0.535492	−	0.844540i	$-0.320126\pi$
$383$	20.9596			1.07098			0.535492	+	0.844540i	$0.320126\pi$
$384$		0			0
$385$	−0.569472			−0.0290230
$386$		0			0
$387$	−3.82842	+	1.18351i	−0.194609	+	0.0601611i
$388$		0			0
$389$			1.27730i			0.0647615i	0.999476	+	0.0323808i	$0.0103089\pi$
$389$			1.27730i			0.0647615i	−0.999476	+	0.0323808i	$0.989691\pi$
$390$		0			0
$391$			7.51061i			0.379828i
$392$		0			0
$393$	11.5856	+	8.54499i	0.584415	+	0.431038i
$394$		0			0
$395$	−3.38072			−0.170103
$396$		0			0
$397$	2.84094			0.142583			0.0712914	−	0.997456i	$-0.477288\pi$
$397$	2.84094			0.142583			0.0712914	+	0.997456i	$0.477288\pi$
$398$		0			0
$399$	10.8688	+	8.01633i	0.544120	+	0.401318i
$400$		0			0
$401$		−	11.8477i		−	0.591645i	−0.955243	−	0.295822i	$-0.904406\pi$
$401$		−	11.8477i		−	0.591645i	0.955243	−	0.295822i	$-0.0955937\pi$
$402$		0			0
$403$			22.6072i			1.12614i
$404$		0			0
$405$	−15.7501	−	23.0398i	−0.782630	−	1.14486i
$406$		0			0
$407$	1.76016			0.0872477
$408$		0			0
$409$	6.67682			0.330147			0.165074	−	0.986281i	$-0.447214\pi$
$409$	6.67682			0.330147			0.165074	+	0.986281i	$0.447214\pi$
$410$		0			0
$411$	14.3666	−	19.4787i	0.708653	−	0.960814i
$412$		0			0
$413$		−	1.39948i		−	0.0688638i
$414$		0			0
$415$			39.2619i			1.92729i
$416$		0			0
$417$	1.11435	−	1.51087i	0.0545701	−	0.0739878i
$418$		0			0
$419$	12.1181			0.592007			0.296003	−	0.955187i	$-0.404346\pi$
$419$	12.1181			0.592007			0.296003	+	0.955187i	$0.404346\pi$
$420$		0			0
$421$	3.29607			0.160640			0.0803202	−	0.996769i	$-0.474406\pi$
$421$	3.29607			0.160640			0.0803202	+	0.996769i	$0.474406\pi$
$422$		0			0
$423$	2.42106	+	7.83165i	0.117716	+	0.380788i
$424$		0			0
$425$		−	4.61603i		−	0.223910i
$426$		0			0
$427$			9.49846i			0.459663i
$428$		0			0
$429$	0.958249	+	0.706762i	0.0462647	+	0.0341228i
$430$		0			0
$431$	18.6357			0.897650			0.448825	−	0.893620i	$-0.351843\pi$
$431$	18.6357			0.897650			0.448825	+	0.893620i	$0.351843\pi$
$432$		0			0
$433$	5.76044			0.276829			0.138415	−	0.990374i	$-0.455799\pi$
$433$	5.76044			0.276829			0.138415	+	0.990374i	$0.455799\pi$
$434$		0			0
$435$	17.5460	+	12.9412i	0.841268	+	0.620482i
$436$		0			0
$437$			53.7162i			2.56960i
$438$		0			0
$439$			22.7061i			1.08370i	0.840474	+	0.541852i	$0.182277\pi$
$439$			22.7061i			1.08370i	−0.840474	+	0.541852i	$0.817723\pi$
$440$		0			0
$441$	5.14917	+	16.6566i	0.245199	+	0.793169i
$442$		0			0
$443$	25.5258			1.21277			0.606383	−	0.795172i	$-0.292620\pi$
$443$	25.5258			1.21277			0.606383	+	0.795172i	$0.292620\pi$
$444$		0			0
$445$	25.3913			1.20366
$446$		0			0
$447$	20.1012	−	27.2539i	0.950755	−	1.28906i
$448$		0			0
$449$			30.5958i			1.44390i	0.691943	+	0.721952i	$0.256755\pi$
$449$			30.5958i			1.44390i	−0.691943	+	0.721952i	$0.743245\pi$
$450$		0			0
$451$		−	0.350554i		−	0.0165070i
$452$		0			0
$453$	12.2169	−	16.5640i	0.574000	−	0.778246i
$454$		0			0
$455$	−13.7971			−0.646817
$456$		0			0
$457$	−39.1068			−1.82934			−0.914668	−	0.404205i	$-0.867548\pi$
$457$	−39.1068			−1.82934			−0.914668	+	0.404205i	$0.867548\pi$
$458$		0			0
$459$	−4.90615	−	1.71162i	−0.229000	−	0.0798916i
$460$		0			0
$461$		−	6.35639i		−	0.296047i	−0.988984	−	0.148023i	$-0.952709\pi$
$461$		−	6.35639i		−	0.296047i	0.988984	−	0.148023i	$-0.0472911\pi$
$462$		0			0
$463$		−	7.75421i		−	0.360369i	−0.983633	−	0.180184i	$-0.942331\pi$
$463$		−	7.75421i		−	0.360369i	0.983633	−	0.180184i	$-0.0576694\pi$
$464$		0			0
$465$	23.9445	+	17.6604i	1.11040	+	0.818981i
$466$		0			0
$467$	−36.6447			−1.69572			−0.847858	−	0.530224i	$-0.822108\pi$
$467$	−36.6447			−1.69572			−0.847858	+	0.530224i	$0.822108\pi$
$468$		0			0
$469$	−5.72337			−0.264281
$470$		0			0
$471$	8.64168	+	6.37372i	0.398188	+	0.293685i
$472$		0			0
$473$		−	0.224999i		−	0.0103455i
$474$		0			0
$475$		−	33.0140i		−	1.51479i
$476$		0			0
$477$	−17.0961	+	5.28505i	−0.782776	+	0.241986i
$478$		0			0
$479$	−3.90526			−0.178436			−0.0892179	−	0.996012i	$-0.528437\pi$
$479$	−3.90526			−0.178436			−0.0892179	+	0.996012i	$0.528437\pi$
$480$		0			0
$481$	42.6448			1.94443
$482$		0			0
$483$	8.41822	−	11.4137i	0.383042	−	0.519340i
$484$		0			0
$485$			60.2929i			2.73776i
$486$		0			0
$487$			14.9757i			0.678614i	0.940676	+	0.339307i	$0.110193\pi$
$487$			14.9757i			0.678614i	−0.940676	+	0.339307i	$0.889807\pi$
$488$		0			0
$489$	12.8245	−	17.3878i	0.579942	−	0.786303i
$490$		0			0
$491$	−20.0949			−0.906871			−0.453435	−	0.891289i	$-0.649802\pi$
$491$	−20.0949			−0.906871			−0.453435	+	0.891289i	$0.649802\pi$
$492$		0			0
$493$	4.05922			0.182818
$494$		0			0
$495$	1.49714	−	0.462823i	0.0672914	−	0.0208023i
$496$		0			0
$497$		−	12.1039i		−	0.542933i
$498$		0			0
$499$		−	24.7294i		−	1.10704i	−0.832836	−	0.553520i	$-0.813284\pi$
$499$		−	24.7294i		−	1.10704i	0.832836	−	0.553520i	$-0.186716\pi$
$500$		0			0
$501$	−32.7453	−	24.1515i	−1.46295	−	1.07901i
$502$		0			0
$503$	−36.2379			−1.61577			−0.807885	−	0.589341i	$-0.799388\pi$
$503$	−36.2379			−1.61577			−0.807885	+	0.589341i	$0.799388\pi$
$504$		0			0
$505$	9.07169			0.403685
$506$		0			0
$507$	5.09531	+	3.75807i	0.226291	+	0.166902i
$508$		0			0
$509$			35.7061i			1.58264i	0.611399	+	0.791322i	$0.290607\pi$
$509$			35.7061i			1.58264i	−0.611399	+	0.791322i	$0.709393\pi$
$510$		0			0
$511$		−	2.94087i		−	0.130097i
$512$		0			0
$513$	−35.0891	−	12.2416i	−1.54922	−	0.540480i
$514$		0			0
$515$	23.1439			1.01984
$516$		0			0
$517$	−0.460272			−0.0202427
$518$		0			0
$519$	2.34921	−	3.18513i	0.103119	−	0.139812i
$520$		0			0
$521$			19.1237i			0.837823i	0.908027	+	0.418912i	$0.137588\pi$
$521$			19.1237i			0.837823i	−0.908027	+	0.418912i	$0.862412\pi$
$522$		0			0
$523$			18.4324i			0.805992i	0.915202	+	0.402996i	$0.132031\pi$
$523$			18.4324i			0.805992i	−0.915202	+	0.402996i	$0.867969\pi$
$524$		0			0
$525$	−5.17384	+	7.01486i	−0.225805	+	0.306153i
$526$		0			0
$527$	5.53949			0.241304
$528$		0			0
$529$	33.4092			1.45258
$530$		0			0
$531$	1.13739	+	3.67923i	0.0493584	+	0.159665i
$532$		0			0
$533$		−	8.49318i		−	0.367880i
$534$		0			0
$535$			26.7976i			1.15856i
$536$		0			0
$537$	−13.4058	−	9.88752i	−0.578503	−	0.426678i
$538$		0			0
$539$	−0.978917			−0.0421649
$540$		0			0
$541$	−42.6597			−1.83409			−0.917043	−	0.398789i	$-0.869430\pi$
$541$	−42.6597			−1.83409			−0.917043	+	0.398789i	$0.869430\pi$
$542$		0			0
$543$	22.3972	+	16.5192i	0.961157	+	0.708906i
$544$		0			0
$545$			35.2588i			1.51032i
$546$		0			0
$547$		−	18.3482i		−	0.784512i	−0.919856	−	0.392256i	$-0.871695\pi$
$547$		−	18.3482i		−	0.784512i	0.919856	−	0.392256i	$-0.128305\pi$
$548$		0			0
$549$	−7.71961	−	24.9714i	−0.329465	−	1.06575i
$550$		0			0
$551$	29.0318			1.23679
$552$		0			0
$553$	1.18857			0.0505430
$554$		0			0
$555$	33.3135	−	45.1674i	1.41408	−	1.91725i
$556$		0			0
$557$		−	38.5598i		−	1.63383i	−0.576757	−	0.816915i	$-0.695682\pi$
$557$		−	38.5598i		−	1.63383i	0.576757	−	0.816915i	$-0.304318\pi$
$558$		0			0
$559$		−	5.45123i		−	0.230562i
$560$		0			0
$561$	0.173179	−	0.234802i	0.00731163	−	0.00991334i
$562$		0			0
$563$	−4.40731			−0.185746			−0.0928730	−	0.995678i	$-0.529605\pi$
$563$	−4.40731			−0.185746			−0.0928730	+	0.995678i	$0.529605\pi$
$564$		0			0
$565$	−61.5758			−2.59051
$566$		0			0
$567$	5.53730	+	8.10014i	0.232545	+	0.340174i
$568$		0			0
$569$		−	23.8686i		−	1.00062i	−0.865846	−	0.500311i	$-0.833219\pi$
$569$		−	23.8686i		−	1.00062i	0.865846	−	0.500311i	$-0.166781\pi$
$570$		0			0
$571$		−	43.0015i		−	1.79956i	−0.436346	−	0.899779i	$-0.643728\pi$
$571$		−	43.0015i		−	1.79956i	0.436346	−	0.899779i	$-0.356272\pi$
$572$		0			0
$573$	0.0850208	+	0.0627076i	0.00355180	+	0.00261965i
$574$		0			0
$575$	−34.6692			−1.44580
$576$		0			0
$577$	−36.8354			−1.53348			−0.766738	−	0.641960i	$-0.778121\pi$
$577$	−36.8354			−1.53348			−0.766738	+	0.641960i	$0.778121\pi$
$578$		0			0
$579$	−20.0169	−	14.7636i	−0.831875	−	0.613553i
$580$		0			0
$581$		−	13.8034i		−	0.572660i
$582$		0			0
$583$		−	1.00475i		−	0.0416124i
$584$		0			0
$585$	36.2725	−	11.2132i	1.49968	−	0.463608i
$586$		0			0
$587$	32.2245			1.33005			0.665024	−	0.746822i	$-0.268421\pi$
$587$	32.2245			1.33005			0.665024	+	0.746822i	$0.268421\pi$
$588$		0			0
$589$	39.6187			1.63246
$590$		0			0
$591$	−13.5194	+	18.3300i	−0.556114	+	0.753997i
$592$		0			0
$593$		−	13.7104i		−	0.563018i	−0.959559	−	0.281509i	$-0.909165\pi$
$593$		−	13.7104i		−	0.563018i	0.959559	−	0.281509i	$-0.0908349\pi$
$594$		0			0
$595$			3.38072i			0.138596i
$596$		0			0
$597$	9.59969	−	13.0156i	0.392889	−	0.532691i
$598$		0			0
$599$	−34.7552			−1.42006			−0.710029	−	0.704172i	$-0.751318\pi$
$599$	−34.7552			−1.42006			−0.710029	+	0.704172i	$0.751318\pi$
$600$		0			0
$601$	26.9814			1.10059			0.550297	−	0.834969i	$-0.314515\pi$
$601$	26.9814			1.10059			0.550297	+	0.834969i	$0.314515\pi$
$602$		0			0
$603$	15.0467	−	4.65151i	0.612750	−	0.189424i
$604$		0			0
$605$		−	34.0227i		−	1.38322i
$606$		0			0
$607$			30.2488i			1.22776i	0.789399	+	0.613880i	$0.210392\pi$
$607$			30.2488i			1.22776i	−0.789399	+	0.613880i	$0.789608\pi$
$608$		0			0
$609$	−6.16870	−	4.54975i	−0.249968	−	0.184365i
$610$		0			0
$611$	−11.1514			−0.451137
$612$		0			0
$613$	27.3561			1.10490			0.552452	−	0.833545i	$-0.313692\pi$
$613$	27.3561			1.10490			0.552452	+	0.833545i	$0.313692\pi$
$614$		0			0
$615$	−8.99559	−	6.63474i	−0.362737	−	0.267539i
$616$		0			0
$617$			6.50883i			0.262036i	0.991380	+	0.131018i	$0.0418245\pi$
$617$			6.50883i			0.262036i	−0.991380	+	0.131018i	$0.958175\pi$
$618$		0			0
$619$		−	45.0376i		−	1.81021i	−0.425185	−	0.905107i	$-0.639791\pi$
$619$		−	45.0376i		−	1.81021i	0.425185	−	0.905107i	$-0.360209\pi$
$620$		0			0
$621$	−12.8553	+	36.8482i	−0.515866	+	1.47867i
$622$		0			0
$623$	−8.92686			−0.357647
$624$		0			0
$625$	−26.7724			−1.07090
$626$		0			0
$627$	1.23859	−	1.67931i	0.0494644	−	0.0670653i
$628$		0			0
$629$		−	10.4493i		−	0.416642i
$630$		0			0
$631$		−	33.7130i		−	1.34209i	−0.741414	−	0.671047i	$-0.765845\pi$
$631$		−	33.7130i		−	1.34209i	0.741414	−	0.671047i	$-0.234155\pi$
$632$		0			0
$633$	14.1574	−	19.1951i	0.562708	−	0.762937i
$634$		0			0
$635$	−48.9412			−1.94217
$636$		0			0
$637$	−23.7170			−0.939703
$638$		0			0
$639$	9.83710	+	31.8211i	0.389150	+	1.25882i
$640$		0			0
$641$			14.5942i			0.576436i	0.957565	+	0.288218i	$0.0930627\pi$
$641$			14.5942i			0.576436i	−0.957565	+	0.288218i	$0.906937\pi$
$642$		0			0
$643$			31.9445i			1.25977i	0.776689	+	0.629884i	$0.216898\pi$
$643$			31.9445i			1.25977i	−0.776689	+	0.629884i	$0.783102\pi$
$644$		0			0
$645$	−5.77370	−	4.25842i	−0.227339	−	0.167675i
$646$		0			0
$647$	19.1484			0.752801			0.376400	−	0.926457i	$-0.377162\pi$
$647$	19.1484			0.752801			0.376400	+	0.926457i	$0.377162\pi$
$648$		0			0
$649$	−0.216231			−0.00848779
$650$		0			0
$651$	−8.41822	−	6.20890i	−0.329936	−	0.243346i
$652$		0			0
$653$		−	11.6937i		−	0.457611i	−0.973472	−	0.228806i	$-0.926518\pi$
$653$		−	11.6937i		−	0.457611i	0.973472	−	0.228806i	$-0.0734820\pi$
$654$		0			0
$655$			25.7737i			1.00706i
$656$		0			0
$657$	2.39012	+	7.73155i	0.0932473	+	0.301636i
$658$		0			0
$659$	−26.5948			−1.03599			−0.517993	−	0.855385i	$-0.673321\pi$
$659$	−26.5948			−1.03599			−0.517993	+	0.855385i	$0.673321\pi$
$660$		0			0
$661$	−17.2489			−0.670905			−0.335453	−	0.942057i	$-0.608889\pi$
$661$	−17.2489			−0.670905			−0.335453	+	0.942057i	$0.608889\pi$
$662$		0			0
$663$	4.19576	−	5.68874i	0.162950	−	0.220932i
$664$		0			0
$665$			24.1791i			0.937625i
$666$		0			0
$667$		−	30.4872i		−	1.18047i
$668$		0			0
$669$	25.1883	−	34.1511i	0.973836	−	1.32036i
$670$		0			0
$671$	1.46759			0.0566556
$672$		0			0
$673$	−13.9908			−0.539304			−0.269652	−	0.962958i	$-0.586909\pi$
$673$	−13.9908			−0.539304			−0.269652	+	0.962958i	$0.586909\pi$
$674$		0			0
$675$	7.90088	−	22.6469i	0.304105	−	0.871681i
$676$		0			0
$677$			1.10176i			0.0423441i	0.999776	+	0.0211721i	$0.00673978\pi$
$677$			1.10176i			0.0423441i	−0.999776	+	0.0211721i	$0.993260\pi$
$678$		0			0
$679$		−	21.1973i		−	0.813477i
$680$		0			0
$681$	30.5982	+	22.5678i	1.17252	+	0.864801i
$682$		0			0
$683$	−29.3480			−1.12297			−0.561484	−	0.827488i	$-0.689769\pi$
$683$	−29.3480			−1.12297			−0.561484	+	0.827488i	$0.689769\pi$
$684$		0			0
$685$	43.3331			1.65567
$686$		0			0
$687$	10.0683	+	7.42592i	0.384129	+	0.283317i
$688$		0			0
$689$		−	24.3429i		−	0.927390i
$690$		0			0
$691$			25.1447i			0.956548i	0.878211	+	0.478274i	$0.158737\pi$
$691$			25.1447i			0.956548i	−0.878211	+	0.478274i	$0.841263\pi$
$692$		0			0
$693$	−0.526352	+	0.162715i	−0.0199945	+	0.00618105i
$694$		0			0
$695$	3.36115			0.127496
$696$		0			0
$697$	−2.08110			−0.0788273
$698$		0			0
$699$	6.09469	−	8.26337i	0.230522	−	0.312549i
$700$		0			0
$701$		−	23.6537i		−	0.893389i	−0.894687	−	0.446694i	$-0.852601\pi$
$701$		−	23.6537i		−	0.893389i	0.894687	−	0.446694i	$-0.147399\pi$
$702$		0			0
$703$		−	74.7342i		−	2.81865i
$704$		0			0
$705$	−8.71129	+	11.8110i	−0.328086	+	0.444829i
$706$		0			0
$707$	−3.18935			−0.119948
$708$		0			0
$709$	5.61489			0.210872			0.105436	−	0.994426i	$-0.466376\pi$
$709$	5.61489			0.210872			0.105436	+	0.994426i	$0.466376\pi$
$710$		0			0
$711$	−3.12474	+	0.965975i	−0.117187	+	0.0362269i
$712$		0			0
$713$		−	41.6049i		−	1.55812i
$714$		0			0
$715$			2.13176i			0.0797232i
$716$		0			0
$717$	39.8753	+	29.4102i	1.48917	+	1.09835i
$718$		0			0
$719$	32.6326			1.21699			0.608495	−	0.793558i	$-0.291774\pi$
$719$	32.6326			1.21699			0.608495	+	0.793558i	$0.291774\pi$
$720$		0			0
$721$	−8.13674			−0.303028
$722$		0			0
$723$	18.7814	+	13.8523i	0.698486	+	0.515172i
$724$		0			0
$725$			18.7375i			0.695892i
$726$		0			0
$727$		−	4.33041i		−	0.160606i	−0.996770	−	0.0803030i	$-0.974411\pi$
$727$		−	4.33041i		−	0.160606i	0.996770	−	0.0803030i	$-0.0255888\pi$
$728$		0			0
$729$	−21.1407	−	16.7950i	−0.782989	−	0.622035i
$730$		0			0
$731$	−1.33573			−0.0494036
$732$		0			0
$733$	10.7413			0.396738			0.198369	−	0.980127i	$-0.436436\pi$
$733$	10.7413			0.396738			0.198369	+	0.980127i	$0.436436\pi$
$734$		0			0
$735$	−18.5274	+	25.1200i	−0.683393	+	0.926565i
$736$		0			0
$737$			0.884306i			0.0325738i
$738$		0			0
$739$		−	31.8147i		−	1.17032i	−0.810917	−	0.585161i	$-0.801031\pi$
$739$		−	31.8147i		−	1.17032i	0.810917	−	0.585161i	$-0.198969\pi$
$740$		0			0
$741$	30.0083	−	40.6861i	1.10238	−	1.49464i
$742$		0			0
$743$	−28.5017			−1.04563			−0.522813	−	0.852447i	$-0.675118\pi$
$743$	−28.5017			−1.04563			−0.522813	+	0.852447i	$0.675118\pi$
$744$		0			0
$745$	60.6299			2.22131
$746$		0			0
$747$	11.2183	+	36.2890i	0.410456	+	1.32775i
$748$		0			0
$749$		−	9.42129i		−	0.344246i
$750$		0			0
$751$			41.6427i			1.51956i	0.650178	+	0.759782i	$0.274694\pi$
$751$			41.6427i			1.51956i	−0.650178	+	0.759782i	$0.725306\pi$
$752$		0			0
$753$	−6.18665	−	4.56300i	−0.225454	−	0.166285i
$754$		0			0
$755$	36.8490			1.34107
$756$		0			0
$757$	−19.4316			−0.706253			−0.353127	−	0.935576i	$-0.614882\pi$
$757$	−19.4316			−0.706253			−0.353127	+	0.935576i	$0.614882\pi$
$758$		0			0
$759$	−1.76350	−	1.30068i	−0.0640111	−	0.0472117i
$760$		0			0
$761$		−	14.0473i		−	0.509215i	−0.967045	−	0.254607i	$-0.918054\pi$
$761$		−	14.0473i		−	0.509215i	0.967045	−	0.254607i	$-0.0819462\pi$
$762$		0			0
$763$		−	12.3960i		−	0.448765i
$764$		0			0
$765$	−2.74759	−	8.88791i	−0.0993394	−	0.321343i
$766$		0			0
$767$	−5.23880			−0.189162
$768$		0			0
$769$	40.2339			1.45087			0.725435	−	0.688291i	$-0.241639\pi$
$769$	40.2339			1.45087			0.725435	+	0.688291i	$0.241639\pi$
$770$		0			0
$771$	−27.9746	+	37.9288i	−1.00748	+	1.36597i
$772$		0			0
$773$			44.6057i			1.60436i	0.597084	+	0.802178i	$0.296326\pi$
$773$			44.6057i			1.60436i	−0.597084	+	0.802178i	$0.703674\pi$
$774$		0			0
$775$			25.5704i			0.918516i
$776$		0			0
$777$	−11.7121	+	15.8796i	−0.420168	+	0.569677i
$778$		0			0
$779$	−14.8841			−0.533279
$780$		0			0
$781$	−1.87015			−0.0669191
$782$		0			0
$783$	19.9152	+	6.94785i	0.711710	+	0.248296i
$784$		0			0
$785$			19.2246i			0.686156i
$786$		0			0
$787$			32.5848i			1.16152i	0.814074	+	0.580762i	$0.197245\pi$
$787$			32.5848i			1.16152i	−0.814074	+	0.580762i	$0.802755\pi$
$788$		0			0
$789$	−0.452003	−	0.333377i	−0.0160917	−	0.0118685i
$790$		0			0
$791$	21.6483			0.769725
$792$		0			0
$793$	35.5564			1.26265
$794$		0			0
$795$	−25.7829	−	19.0163i	−0.914424	−	0.674438i
$796$		0			0
$797$			33.2015i			1.17606i	0.808841	+	0.588028i	$0.200096\pi$
$797$			33.2015i			1.17606i	−0.808841	+	0.588028i	$0.799904\pi$
$798$		0			0
$799$			2.73245i			0.0966670i
$800$		0			0
$801$	23.4687	−	7.25506i	0.829225	−	0.256345i
$802$		0			0
$803$	−0.454388			−0.0160350
$804$		0			0
$805$	25.3913			0.894925
$806$		0			0
$807$	−13.3927	+	18.1582i	−0.471445	+	0.639200i
$808$		0			0
$809$		−	51.6323i		−	1.81530i	−0.419733	−	0.907648i	$-0.637876\pi$
$809$		−	51.6323i		−	1.81530i	0.419733	−	0.907648i	$-0.362124\pi$
$810$		0			0
$811$		−	48.7697i		−	1.71254i	−0.516531	−	0.856269i	$-0.672777\pi$
$811$		−	48.7697i		−	1.71254i	0.516531	−	0.856269i	$-0.327223\pi$
$812$		0			0
$813$	−4.55702	+	6.17855i	−0.159822	+	0.216691i
$814$		0			0
$815$	38.6815			1.35495
$816$		0			0
$817$	−9.55318			−0.334223
$818$		0			0
$819$	−12.7524	+	3.94224i	−0.445604	+	0.137753i
$820$		0			0
$821$		−	14.8502i		−	0.518276i	−0.965840	−	0.259138i	$-0.916561\pi$
$821$		−	14.8502i		−	0.518276i	0.965840	−	0.259138i	$-0.0834385\pi$
$822$		0			0
$823$			3.59778i			0.125411i	0.998032	+	0.0627053i	$0.0199728\pi$
$823$			3.59778i			0.125411i	−0.998032	+	0.0627053i	$0.980027\pi$
$824$		0			0
$825$	1.08385	+	0.799400i	0.0377348	+	0.0278315i
$826$		0			0
$827$	−4.66698			−0.162287			−0.0811434	−	0.996702i	$-0.525857\pi$
$827$	−4.66698			−0.162287			−0.0811434	+	0.996702i	$0.525857\pi$
$828$		0			0
$829$	2.73560			0.0950115			0.0475057	−	0.998871i	$-0.484873\pi$
$829$	2.73560			0.0950115			0.0475057	+	0.998871i	$0.484873\pi$
$830$		0			0
$831$	−17.3235	−	12.7771i	−0.600946	−	0.443231i
$832$		0			0
$833$			5.81143i			0.201354i
$834$		0			0
$835$		−	72.8464i		−	2.52095i
$836$		0			0
$837$	27.1776	+	9.48150i	0.939395	+	0.327729i
$838$		0			0
$839$	32.7235			1.12974			0.564871	−	0.825179i	$-0.308926\pi$
$839$	32.7235			1.12974			0.564871	+	0.825179i	$0.308926\pi$
$840$		0			0
$841$	12.5227			0.431818
$842$		0			0
$843$	−5.54580	+	7.51917i	−0.191008	+	0.258974i
$844$		0			0
$845$			11.3352i			0.389943i
$846$		0			0
$847$			11.9614i			0.410999i
$848$		0			0
$849$	23.1181	−	31.3442i	0.793411	−	1.07573i
$850$		0			0
$851$	−78.4809			−2.69029
$852$		0			0
$853$	−9.45976			−0.323896			−0.161948	−	0.986799i	$-0.551778\pi$
$853$	−9.45976			−0.323896			−0.161948	+	0.986799i	$0.551778\pi$
$854$		0			0
$855$	−19.6509	−	63.5668i	−0.672047	−	2.17394i
$856$		0			0
$857$			7.27266i			0.248429i	0.992255	+	0.124215i	$0.0396412\pi$
$857$			7.27266i			0.248429i	−0.992255	+	0.124215i	$0.960359\pi$
$858$		0			0
$859$			4.35729i			0.148669i	0.997233	+	0.0743344i	$0.0236832\pi$
$859$			4.35729i			0.148669i	−0.997233	+	0.0743344i	$0.976317\pi$
$860$		0			0
$861$	3.16259	+	2.33259i	0.107781	+	0.0794944i
$862$		0			0
$863$	−27.3967			−0.932596			−0.466298	−	0.884628i	$-0.654413\pi$
$863$	−27.3967			−0.932596			−0.466298	+	0.884628i	$0.654413\pi$
$864$		0			0
$865$	7.08575			0.240923
$866$		0			0
$867$	−1.39392	−	1.02809i	−0.0473401	−	0.0349159i
$868$		0			0
$869$		−	0.183643i		−	0.00622966i
$870$		0			0
$871$			21.4248i			0.725952i
$872$		0			0
$873$	17.2275	+	55.7276i	0.583063	+	1.88609i
$874$		0			0
$875$	1.29811			0.0438842
$876$		0			0
$877$	−15.3046			−0.516800			−0.258400	−	0.966038i	$-0.583195\pi$
$877$	−15.3046			−0.516800			−0.258400	+	0.966038i	$0.583195\pi$
$878$		0			0
$879$	8.31445	−	11.2730i	0.280439	−	0.380228i
$880$		0			0
$881$		−	52.8324i		−	1.77997i	−0.455990	−	0.889985i	$-0.650715\pi$
$881$		−	52.8324i		−	1.77997i	0.455990	−	0.889985i	$-0.349285\pi$
$882$		0			0
$883$			38.6903i			1.30203i	0.759065	+	0.651015i	$0.225657\pi$
$883$			38.6903i			1.30203i	−0.759065	+	0.651015i	$0.774343\pi$
$884$		0			0
$885$	−4.09247	+	5.54870i	−0.137567	+	0.186517i
$886$		0			0
$887$	−4.58730			−0.154026			−0.0770132	−	0.997030i	$-0.524538\pi$
$887$	−4.58730			−0.154026			−0.0770132	+	0.997030i	$0.524538\pi$
$888$		0			0
$889$	17.2064			0.577082
$890$		0			0
$891$	1.25154	−	0.855557i	0.0419280	−	0.0286622i
$892$		0			0
$893$			19.5426i			0.653968i
$894$		0			0
$895$		−	29.8230i		−	0.996874i
$896$		0			0
$897$	−42.7259	−	31.5127i	−1.42658	−	1.05218i
$898$		0			0
$899$	−22.4860			−0.749950
$900$		0			0
$901$	−5.96479			−0.198716
$902$		0			0
$903$	2.02987	+	1.49714i	0.0675498	+	0.0498217i
$904$		0			0
$905$			49.8257i			1.65626i
$906$		0			0
$907$			4.05385i			0.134606i	0.997733	+	0.0673030i	$0.0214394\pi$
$907$			4.05385i			0.134606i	−0.997733	+	0.0673030i	$0.978561\pi$
$908$		0			0
$909$	8.38480	−	2.59206i	0.278106	−	0.0859732i
$910$		0			0
$911$	49.2276			1.63098			0.815491	−	0.578769i	$-0.196467\pi$
$911$	49.2276			1.63098			0.815491	+	0.578769i	$0.196467\pi$
$912$		0			0
$913$	−2.13273			−0.0705830
$914$		0			0
$915$	27.7762	−	37.6598i	0.918251	−	1.24499i
$916$		0			0
$917$		−	9.06130i		−	0.299230i
$918$		0			0
$919$		−	23.6754i		−	0.780981i	−0.920607	−	0.390490i	$-0.872305\pi$
$919$		−	23.6754i		−	0.780981i	0.920607	−	0.390490i	$-0.127695\pi$
$920$		0			0
$921$	−1.73366	+	2.35054i	−0.0571259	+	0.0774531i
$922$		0			0
$923$	−45.3096			−1.49138
$924$		0			0
$925$	48.2344			1.58594
$926$		0			0
$927$	21.3915	−	6.61292i	0.702588	−	0.217197i
$928$		0			0
$929$			8.29926i			0.272290i	0.990689	+	0.136145i	$0.0434713\pi$
$929$			8.29926i			0.272290i	−0.990689	+	0.136145i	$0.956529\pi$
$930$		0			0
$931$			41.5637i			1.36219i
$932$		0			0
$933$	−19.5147	−	14.3932i	−0.638883	−	0.471211i
$934$		0			0
$935$	0.522349			0.0170826
$936$		0			0
$937$	−18.2859			−0.597373			−0.298687	−	0.954351i	$-0.596549\pi$
$937$	−18.2859			−0.597373			−0.298687	+	0.954351i	$0.596549\pi$
$938$		0			0
$939$	10.3210	+	7.61230i	0.336813	+	0.248418i
$940$		0			0
$941$		−	27.3824i		−	0.892639i	−0.894874	−	0.446320i	$-0.852734\pi$
$941$		−	27.3824i		−	0.892639i	0.894874	−	0.446320i	$-0.147266\pi$
$942$		0			0
$943$			15.6303i			0.508993i
$944$		0			0
$945$	−5.78652	+	16.5864i	−0.188235	+	0.539554i
$946$		0			0
$947$	−2.16508			−0.0703556			−0.0351778	−	0.999381i	$-0.511200\pi$
$947$	−2.16508			−0.0703556			−0.0351778	+	0.999381i	$0.511200\pi$
$948$		0			0
$949$	−11.0088			−0.357362
$950$		0			0
$951$	−22.1870	+	30.0819i	−0.719464	+	0.975472i
$952$		0			0
$953$		−	45.2985i		−	1.46736i	−0.679495	−	0.733680i	$-0.737801\pi$
$953$		−	45.2985i		−	1.46736i	0.679495	−	0.733680i	$-0.262199\pi$
$954$		0			0
$955$			0.189140i			0.00612044i
$956$		0			0
$957$	−0.702973	+	0.953112i	−0.0227239	+	0.0308097i
$958$		0			0
$959$	−15.2347			−0.491954
$960$		0			0
$961$	0.314081			0.0101317
$962$		0			0
$963$	7.65689	+	24.7685i	0.246740	+	0.798155i
$964$		0			0
$965$		−	44.5304i		−	1.43348i
$966$		0			0
$967$			23.6736i			0.761291i	0.924721	+	0.380645i	$0.124298\pi$
$967$			23.6736i			0.761291i	−0.924721	+	0.380645i	$0.875702\pi$
$968$		0			0
$969$	−9.96941	−	7.35299i	−0.320263	−	0.236212i
$970$		0			0
$971$	59.8717			1.92137			0.960686	−	0.277636i	$-0.0895508\pi$
$971$	59.8717			1.92137			0.960686	+	0.277636i	$0.0895508\pi$
$972$		0			0
$973$	−1.18168			−0.0378831
$974$		0			0
$975$	26.2594	+	19.3677i	0.840973	+	0.620264i
$976$		0			0
$977$			11.2749i			0.360717i	0.983601	+	0.180359i	$0.0577258\pi$
$977$			11.2749i			0.360717i	−0.983601	+	0.180359i	$0.942274\pi$
$978$		0			0
$979$			1.37927i			0.0440817i
$980$		0			0
$981$	10.0745	+	32.5890i	0.321654	+	1.04049i
$982$		0			0
$983$	40.9095			1.30481			0.652405	−	0.757871i	$-0.273760\pi$
$983$	40.9095			1.30481			0.652405	+	0.757871i	$0.273760\pi$
$984$		0			0
$985$	−40.7776			−1.29928
$986$		0			0
$987$	3.06264	−	4.15243i	0.0974850	−	0.132173i
$988$		0			0
$989$			10.0321i			0.319003i
$990$		0			0
$991$			42.1635i			1.33937i	0.742647	+	0.669683i	$0.233570\pi$
$991$			42.1635i			1.33937i	−0.742647	+	0.669683i	$0.766430\pi$
$992$		0			0
$993$	1.46415	−	1.98514i	0.0464635	−	0.0629966i
$994$		0			0
$995$	28.9549			0.917931
$996$		0			0
$997$	10.2484			0.324569			0.162285	−	0.986744i	$-0.448114\pi$
$997$	10.2484			0.324569			0.162285	+	0.986744i	$0.448114\pi$
$998$		0			0
$999$	17.8853	−	51.2661i	0.565866	−	1.62199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.2.e.c.239.18	yes	20
3.2	odd	2	inner	816.2.e.c.239.4	yes	20
4.3	odd	2	inner	816.2.e.c.239.3	✓	20
12.11	even	2	inner	816.2.e.c.239.17	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
816.2.e.c.239.3	✓	20	4.3	odd	2	inner
816.2.e.c.239.4	yes	20	3.2	odd	2	inner
816.2.e.c.239.17	yes	20	12.11	even	2	inner
816.2.e.c.239.18	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.39392 + 1.02809i) q^{3} +3.10097i q^{5} -1.09021i q^{7} +(0.886042 + 2.86617i) q^{9} +O(q^{10})\)	\(q+(1.39392 + 1.02809i) q^{3} +3.10097i q^{5} -1.09021i q^{7} +(0.886042 + 2.86617i) q^{9} -0.168447 q^{11} -4.08110 q^{13} +(-3.18809 + 4.32252i) q^{15} +1.00000i q^{17} +7.15205i q^{19} +(1.12084 - 1.51967i) q^{21} +7.51061 q^{23} -4.61603 q^{25} +(-1.71162 + 4.90615i) q^{27} -4.05922i q^{29} -5.53949i q^{31} +(-0.234802 - 0.173179i) q^{33} +3.38072 q^{35} -10.4493 q^{37} +(-5.68874 - 4.19576i) q^{39} +2.08110i q^{41} +1.33573i q^{43} +(-8.88791 + 2.74759i) q^{45} +2.73245 q^{47} +5.81143 q^{49} +(-1.02809 + 1.39392i) q^{51} +5.96479i q^{53} -0.522349i q^{55} +(-7.35299 + 9.96941i) q^{57} +1.28367 q^{59} -8.71247 q^{61} +(3.12474 - 0.965975i) q^{63} -12.6554i q^{65} -5.24977i q^{67} +(10.4692 + 7.72162i) q^{69} +11.1023 q^{71} +2.69752 q^{73} +(-6.43438 - 4.74571i) q^{75} +0.183643i q^{77} +1.09021i q^{79} +(-7.42986 + 5.07909i) q^{81} +12.6611 q^{83} -3.10097 q^{85} +(4.17326 - 5.65824i) q^{87} -8.18817i q^{89} +4.44927i q^{91} +(5.69512 - 7.72162i) q^{93} -22.1783 q^{95} +19.4432 q^{97} +(-0.149251 - 0.482797i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{9}+O(q^{10}) \) 20 * q - 4 * q^9	\( 20 q - 4 q^{9} + 12 q^{13} - 16 q^{21} - 48 q^{25} + 12 q^{33} - 16 q^{45} - 20 q^{49} + 64 q^{57} - 48 q^{61} + 28 q^{69} + 64 q^{73} - 60 q^{81} + 4 q^{85} + 32 q^{93} + 32 q^{97}+O(q^{100}) \) 20 * q - 4 * q^9 + 12 * q^13 - 16 * q^21 - 48 * q^25 + 12 * q^33 - 16 * q^45 - 20 * q^49 + 64 * q^57 - 48 * q^61 + 28 * q^69 + 64 * q^73 - 60 * q^81 + 4 * q^85 + 32 * q^93 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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