LMFDB - Embedded newform 1840.2.e.h.369.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1840,2,Mod(369,1840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1840.369"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,-2,0,0,0,-22,0,-14,0,0,0,-6,0,0,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} + \cdots + 1024 $ x^16 - 2x^15 + 2x^14 + 6x^13 + 100x^12 - 196x^11 + 210x^10 + 702x^9 + 1572x^8 - 1594x^7 + 2464x^6 + 9568x^5 + 15457x^4 + 4336x^3 + 128x^2 - 512*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.11
Root			$0.303680 - 0.303680i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.h.369.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.540724i q^{3} +(2.21535 + 0.303680i) q^{5} +1.15693i q^{7} +2.70762 q^{9} -2.10688 q^{11} +5.59296i q^{13} +(-0.164207 + 1.19789i) q^{15} -0.244199i q^{17} +1.45043 q^{19} -0.625580 q^{21} +1.00000i q^{23} +(4.81556 + 1.34552i) q^{25} +3.08625i q^{27} -4.29653 q^{29} -3.19717 q^{31} -1.13924i q^{33} +(-0.351337 + 2.56301i) q^{35} +0.807940i q^{37} -3.02425 q^{39} -1.59546 q^{41} +5.98219i q^{43} +(5.99832 + 0.822250i) q^{45} -0.624940i q^{47} +5.66151 q^{49} +0.132044 q^{51} -0.536087i q^{53} +(-4.66747 - 0.639817i) q^{55} +0.784283i q^{57} -1.03806 q^{59} +3.77846 q^{61} +3.13253i q^{63} +(-1.69847 + 12.3904i) q^{65} -4.61571i q^{67} -0.540724 q^{69} -6.77489 q^{71} -8.87290i q^{73} +(-0.727553 + 2.60389i) q^{75} -2.43751i q^{77} +14.8521 q^{79} +6.45405 q^{81} -6.11896i q^{83} +(0.0741583 - 0.540985i) q^{85} -2.32323i q^{87} -5.79942 q^{89} -6.47066 q^{91} -1.72879i q^{93} +(3.21321 + 0.440467i) q^{95} +2.38582i q^{97} -5.70462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} - 22 q^{9} - 14 q^{11} - 6 q^{15} + 22 q^{19} + 12 q^{25} - 44 q^{29} - 18 q^{31} - 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} + 38 q^{51} - 30 q^{55} + 64 q^{59} + 34 q^{61} + 6 q^{65}+ \cdots + 122 q^{99}+O(q^{100}) $ 16 * q - 2 * q^5 - 22 * q^9 - 14 * q^11 - 6 * q^15 + 22 * q^19 + 12 * q^25 - 44 * q^29 - 18 * q^31 - 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 + 38 * q^51 - 30 * q^55 + 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 - 30 * q^71 - 56 * q^75 - 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 + 70 * q^91 - 38 * q^95 + 122 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$			0.540724i			0.312187i	0.987742	+	0.156094i	$0.0498901\pi$
$3$			0.540724i			0.312187i	−0.987742	+	0.156094i	$0.950110\pi$
$4$		0			0
$5$	2.21535	+	0.303680i	0.990735	+	0.135810i
$5$	2.21535	+	0.303680i	0.990735	+	0.135810i
$6$		0			0
$7$			1.15693i			0.437279i	0.975806	+	0.218639i	$0.0701618\pi$
$7$			1.15693i			0.437279i	−0.975806	+	0.218639i	$0.929838\pi$
$8$		0			0
$9$	2.70762			0.902539
$10$		0			0
$11$	−2.10688			−0.635247			−0.317624	−	0.948217i	$-0.602885\pi$
$11$	−2.10688			−0.635247			−0.317624	+	0.948217i	$0.602885\pi$
$12$		0			0
$13$			5.59296i			1.55121i	0.631220	+	0.775604i	$0.282554\pi$
$13$			5.59296i			1.55121i	−0.631220	+	0.775604i	$0.717446\pi$
$14$		0			0
$15$	−0.164207	+	1.19789i	−0.0423981	+	0.309295i
$16$		0			0
$17$		−	0.244199i		−	0.0592268i	−0.999561	−	0.0296134i	$-0.990572\pi$
$17$		−	0.244199i		−	0.0592268i	0.999561	−	0.0296134i	$-0.00942762\pi$
$18$		0			0
$19$	1.45043			0.332752			0.166376	−	0.986062i	$-0.446794\pi$
$19$	1.45043			0.332752			0.166376	+	0.986062i	$0.446794\pi$
$20$		0			0
$21$	−0.625580			−0.136513
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	4.81556	+	1.34552i	0.963111	+	0.269103i
$26$		0			0
$27$			3.08625i			0.593948i
$28$		0			0
$29$	−4.29653			−0.797845			−0.398922	−	0.916985i	$-0.630616\pi$
$29$	−4.29653			−0.797845			−0.398922	+	0.916985i	$0.630616\pi$
$30$		0			0
$31$	−3.19717			−0.574229			−0.287114	−	0.957896i	$-0.592696\pi$
$31$	−3.19717			−0.574229			−0.287114	+	0.957896i	$0.592696\pi$
$32$		0			0
$33$		−	1.13924i		−	0.198316i
$34$		0			0
$35$	−0.351337	+	2.56301i	−0.0593868	+	0.433227i
$36$		0			0
$37$			0.807940i			0.132824i	0.997792	+	0.0664122i	$0.0211552\pi$
$37$			0.807940i			0.132824i	−0.997792	+	0.0664122i	$0.978845\pi$
$38$		0			0
$39$	−3.02425			−0.484267
$40$		0			0
$41$	−1.59546			−0.249169			−0.124585	−	0.992209i	$-0.539760\pi$
$41$	−1.59546			−0.249169			−0.124585	+	0.992209i	$0.539760\pi$
$42$		0			0
$43$			5.98219i			0.912275i	0.889909	+	0.456138i	$0.150768\pi$
$43$			5.98219i			0.912275i	−0.889909	+	0.456138i	$0.849232\pi$
$44$		0			0
$45$	5.99832	+	0.822250i	0.894177	+	0.122574i
$46$		0			0
$47$		−	0.624940i		−	0.0911569i	−0.998961	−	0.0455784i	$-0.985487\pi$
$47$		−	0.624940i		−	0.0911569i	0.998961	−	0.0455784i	$-0.0145131\pi$
$48$		0			0
$49$	5.66151			0.808787
$50$		0			0
$51$	0.132044			0.0184899
$52$		0			0
$53$		−	0.536087i		−	0.0736372i	−0.999322	−	0.0368186i	$-0.988278\pi$
$53$		−	0.536087i		−	0.0736372i	0.999322	−	0.0368186i	$-0.0117224\pi$
$54$		0			0
$55$	−4.66747	−	0.639817i	−0.629362	−	0.0862729i
$56$		0			0
$57$			0.784283i			0.103881i
$58$		0			0
$59$	−1.03806			−0.135144			−0.0675721	−	0.997714i	$-0.521525\pi$
$59$	−1.03806			−0.135144			−0.0675721	+	0.997714i	$0.521525\pi$
$60$		0			0
$61$	3.77846			0.483783			0.241891	−	0.970303i	$-0.422232\pi$
$61$	3.77846			0.483783			0.241891	+	0.970303i	$0.422232\pi$
$62$		0			0
$63$			3.13253i			0.394661i
$64$		0			0
$65$	−1.69847	+	12.3904i	−0.210669	+	1.53684i
$66$		0			0
$67$		−	4.61571i		−	0.563899i	−0.959429	−	0.281949i	$-0.909019\pi$
$67$		−	4.61571i		−	0.563899i	0.959429	−	0.281949i	$-0.0909811\pi$
$68$		0			0
$69$	−0.540724			−0.0650955
$70$		0			0
$71$	−6.77489			−0.804032			−0.402016	−	0.915633i	$-0.631690\pi$
$71$	−6.77489			−0.804032			−0.402016	+	0.915633i	$0.631690\pi$
$72$		0			0
$73$		−	8.87290i		−	1.03849i	−0.854624	−	0.519247i	$-0.826212\pi$
$73$		−	8.87290i		−	1.03849i	0.854624	−	0.519247i	$-0.173788\pi$
$74$		0			0
$75$	−0.727553	+	2.60389i	−0.0840106	+	0.300671i
$76$		0			0
$77$		−	2.43751i		−	0.277780i
$78$		0			0
$79$	14.8521			1.67099			0.835494	−	0.549500i	$-0.185182\pi$
$79$	14.8521			1.67099			0.835494	+	0.549500i	$0.185182\pi$
$80$		0			0
$81$	6.45405			0.717116
$82$		0			0
$83$		−	6.11896i		−	0.671644i	−0.941926	−	0.335822i	$-0.890986\pi$
$83$		−	6.11896i		−	0.671644i	0.941926	−	0.335822i	$-0.109014\pi$
$84$		0			0
$85$	0.0741583	−	0.540985i	0.00804360	−	0.0586781i
$86$		0			0
$87$		−	2.32323i		−	0.249077i
$88$		0			0
$89$	−5.79942			−0.614737			−0.307369	−	0.951591i	$-0.599448\pi$
$89$	−5.79942			−0.614737			−0.307369	+	0.951591i	$0.599448\pi$
$90$		0			0
$91$	−6.47066			−0.678310
$92$		0			0
$93$		−	1.72879i		−	0.179267i
$94$		0			0
$95$	3.21321	+	0.440467i	0.329669	+	0.0451910i
$96$		0			0
$97$			2.38582i			0.242244i	0.992638	+	0.121122i	$0.0386492\pi$
$97$			2.38582i			0.242244i	−0.992638	+	0.121122i	$0.961351\pi$
$98$		0			0
$99$	−5.70462			−0.573336
$100$		0			0
$101$	8.33379			0.829243			0.414621	−	0.909994i	$-0.363914\pi$
$101$	8.33379			0.829243			0.414621	+	0.909994i	$0.363914\pi$
$102$		0			0
$103$			12.4250i			1.22427i	0.790754	+	0.612135i	$0.209689\pi$
$103$			12.4250i			1.22427i	−0.790754	+	0.612135i	$0.790311\pi$
$104$		0			0
$105$	−1.38588	−	0.189976i	−0.135248	−	0.0185398i
$106$		0			0
$107$			15.5804i			1.50621i	0.657899	+	0.753106i	$0.271445\pi$
$107$			15.5804i			1.50621i	−0.657899	+	0.753106i	$0.728555\pi$
$108$		0			0
$109$	−9.71396			−0.930428			−0.465214	−	0.885198i	$-0.654023\pi$
$109$	−9.71396			−0.930428			−0.465214	+	0.885198i	$0.654023\pi$
$110$		0			0
$111$	−0.436872			−0.0414661
$112$		0			0
$113$		−	2.69724i		−	0.253735i	−0.991920	−	0.126867i	$-0.959508\pi$
$113$		−	2.69724i		−	0.253735i	0.991920	−	0.126867i	$-0.0404922\pi$
$114$		0			0
$115$	−0.303680	+	2.21535i	−0.0283183	+	0.206583i
$116$		0			0
$117$			15.1436i			1.40003i
$118$		0			0
$119$	0.282521			0.0258986
$120$		0			0
$121$	−6.56107			−0.596461
$122$		0			0
$123$		−	0.862705i		−	0.0777875i
$124$		0			0
$125$	10.2595	+	4.44318i	0.917641	+	0.397410i
$126$		0			0
$127$			4.40424i			0.390813i	0.980722	+	0.195406i	$0.0626026\pi$
$127$			4.40424i			0.390813i	−0.980722	+	0.195406i	$0.937397\pi$
$128$		0			0
$129$	−3.23471			−0.284801
$130$		0			0
$131$	−1.65815			−0.144873			−0.0724367	−	0.997373i	$-0.523078\pi$
$131$	−1.65815			−0.144873			−0.0724367	+	0.997373i	$0.523078\pi$
$132$		0			0
$133$			1.67805i			0.145505i
$134$		0			0
$135$	−0.937232	+	6.83712i	−0.0806641	+	0.588445i
$136$		0			0
$137$			2.70894i			0.231440i	0.993282	+	0.115720i	$0.0369176\pi$
$137$			2.70894i			0.231440i	−0.993282	+	0.115720i	$0.963082\pi$
$138$		0			0
$139$	8.99503			0.762949			0.381474	−	0.924379i	$-0.375416\pi$
$139$	8.99503			0.762949			0.381474	+	0.924379i	$0.375416\pi$
$140$		0			0
$141$	0.337920			0.0284580
$142$		0			0
$143$		−	11.7837i		−	0.985400i
$144$		0			0
$145$	−9.51831	−	1.30477i	−0.790453	−	0.108355i
$146$		0			0
$147$			3.06132i			0.252493i
$148$		0			0
$149$	−9.54172			−0.781688			−0.390844	−	0.920457i	$-0.627817\pi$
$149$	−9.54172			−0.781688			−0.390844	+	0.920457i	$0.627817\pi$
$150$		0			0
$151$	12.1471			0.988514			0.494257	−	0.869316i	$-0.335440\pi$
$151$	12.1471			0.988514			0.494257	+	0.869316i	$0.335440\pi$
$152$		0			0
$153$		−	0.661196i		−	0.0534545i
$154$		0			0
$155$	−7.08285	−	0.970918i	−0.568909	−	0.0779860i
$156$		0			0
$157$			19.1756i			1.53038i	0.643806	+	0.765189i	$0.277354\pi$
$157$			19.1756i			1.53038i	−0.643806	+	0.765189i	$0.722646\pi$
$158$		0			0
$159$	0.289875			0.0229886
$160$		0			0
$161$	−1.15693			−0.0911789
$162$		0			0
$163$		−	17.3397i		−	1.35815i	−0.734071	−	0.679073i	$-0.762382\pi$
$163$		−	17.3397i		−	1.35815i	0.734071	−	0.679073i	$-0.237618\pi$
$164$		0			0
$165$	0.345965	−	2.52381i	0.0269333	−	0.196479i
$166$		0			0
$167$		−	0.582642i		−	0.0450862i	−0.999746	−	0.0225431i	$-0.992824\pi$
$167$		−	0.582642i		−	0.0450862i	0.999746	−	0.0225431i	$-0.00717629\pi$
$168$		0			0
$169$	−18.2812			−1.40624
$170$		0			0
$171$	3.92721			0.300321
$172$		0			0
$173$			12.0591i			0.916840i	0.888736	+	0.458420i	$0.151584\pi$
$173$			12.0591i			0.916840i	−0.888736	+	0.458420i	$0.848416\pi$
$174$		0			0
$175$	−1.55667	+	5.57126i	−0.117673	+	0.421148i
$176$		0			0
$177$		−	0.561305i		−	0.0421903i
$178$		0			0
$179$	18.0142			1.34645			0.673223	−	0.739439i	$-0.264909\pi$
$179$	18.0142			1.34645			0.673223	+	0.739439i	$0.264909\pi$
$180$		0			0
$181$	3.33577			0.247946			0.123973	−	0.992286i	$-0.460436\pi$
$181$	3.33577			0.247946			0.123973	+	0.992286i	$0.460436\pi$
$182$		0			0
$183$			2.04311i			0.151031i
$184$		0			0
$185$	−0.245355	+	1.78987i	−0.0180389	+	0.131594i
$186$		0			0
$187$			0.514496i			0.0376237i
$188$		0			0
$189$	−3.57057			−0.259721
$190$		0			0
$191$	2.90013			0.209846			0.104923	−	0.994480i	$-0.466540\pi$
$191$	2.90013			0.209846			0.104923	+	0.994480i	$0.466540\pi$
$192$		0			0
$193$		−	11.6127i		−	0.835898i	−0.908470	−	0.417949i	$-0.862749\pi$
$193$		−	11.6127i		−	0.835898i	0.908470	−	0.417949i	$-0.137251\pi$
$194$		0			0
$195$	−6.69977	−	0.918404i	−0.479780	−	0.0657683i
$196$		0			0
$197$		−	0.156603i		−	0.0111575i	−0.999984	−	0.00557876i	$-0.998224\pi$
$197$		−	0.156603i		−	0.0111575i	0.999984	−	0.00557876i	$-0.00177579\pi$
$198$		0			0
$199$	23.3474			1.65506			0.827528	−	0.561425i	$-0.189747\pi$
$199$	23.3474			1.65506			0.827528	+	0.561425i	$0.189747\pi$
$200$		0			0
$201$	2.49583			0.176042
$202$		0			0
$203$		−	4.97078i		−	0.348880i
$204$		0			0
$205$	−3.53451	−	0.484511i	−0.246861	−	0.0338397i
$206$		0			0
$207$			2.70762i			0.188192i
$208$		0			0
$209$	−3.05588			−0.211380
$210$		0			0
$211$	21.2620			1.46374			0.731870	−	0.681445i	$-0.238648\pi$
$211$	21.2620			1.46374			0.731870	+	0.681445i	$0.238648\pi$
$212$		0			0
$213$		−	3.66335i		−	0.251008i
$214$		0			0
$215$	−1.81667	+	13.2526i	−0.123896	+	0.903823i
$216$		0			0
$217$		−	3.69890i		−	0.251098i
$218$		0			0
$219$	4.79779			0.324205
$220$		0			0
$221$	1.36579			0.0918731
$222$		0			0
$223$		−	28.7628i		−	1.92610i	−0.269328	−	0.963049i	$-0.586801\pi$
$223$		−	28.7628i		−	1.92610i	0.269328	−	0.963049i	$-0.413199\pi$
$224$		0			0
$225$	13.0387	+	3.64315i	0.869246	+	0.242876i
$226$		0			0
$227$			6.08990i			0.404201i	0.979365	+	0.202100i	$0.0647768\pi$
$227$			6.08990i			0.404201i	−0.979365	+	0.202100i	$0.935223\pi$
$228$		0			0
$229$	−19.9294			−1.31697			−0.658486	−	0.752593i	$-0.728803\pi$
$229$	−19.9294			−1.31697			−0.658486	+	0.752593i	$0.728803\pi$
$230$		0			0
$231$	1.31802			0.0867194
$232$		0			0
$233$		−	9.63207i		−	0.631018i	−0.948923	−	0.315509i	$-0.897825\pi$
$233$		−	9.63207i		−	0.631018i	0.948923	−	0.315509i	$-0.102175\pi$
$234$		0			0
$235$	0.189782	−	1.38446i	0.0123800	−	0.0903123i
$236$		0			0
$237$			8.03087i			0.521661i
$238$		0			0
$239$	13.2583			0.857607			0.428803	−	0.903398i	$-0.358935\pi$
$239$	13.2583			0.857607			0.428803	+	0.903398i	$0.358935\pi$
$240$		0			0
$241$	−14.4250			−0.929195			−0.464597	−	0.885522i	$-0.653801\pi$
$241$	−14.4250			−0.929195			−0.464597	+	0.885522i	$0.653801\pi$
$242$		0			0
$243$			12.7486i			0.817823i
$244$		0			0
$245$	12.5422	+	1.71929i	0.801294	+	0.109841i
$246$		0			0
$247$			8.11220i			0.516167i
$248$		0			0
$249$	3.30867			0.209678
$250$		0			0
$251$	−18.6789			−1.17900			−0.589500	−	0.807768i	$-0.700675\pi$
$251$	−18.6789			−1.17900			−0.589500	+	0.807768i	$0.700675\pi$
$252$		0			0
$253$		−	2.10688i		−	0.132458i
$254$		0			0
$255$	0.292524	+	0.0400992i	0.0183185	+	0.00251111i
$256$		0			0
$257$		−	25.5035i		−	1.59087i	−0.606042	−	0.795433i	$-0.707244\pi$
$257$		−	25.5035i		−	1.59087i	0.606042	−	0.795433i	$-0.292756\pi$
$258$		0			0
$259$	−0.934730			−0.0580813
$260$		0			0
$261$	−11.6333			−0.720086
$262$		0			0
$263$		−	18.4521i		−	1.13780i	−0.822405	−	0.568902i	$-0.807368\pi$
$263$		−	18.4521i		−	1.13780i	0.822405	−	0.568902i	$-0.192632\pi$
$264$		0			0
$265$	0.162799	−	1.18762i	0.0100007	−	0.0729549i
$266$		0			0
$267$		−	3.13589i		−	0.191913i
$268$		0			0
$269$	−2.02723			−0.123602			−0.0618012	−	0.998088i	$-0.519684\pi$
$269$	−2.02723			−0.123602			−0.0618012	+	0.998088i	$0.519684\pi$
$270$		0			0
$271$	−26.7402			−1.62435			−0.812175	−	0.583414i	$-0.801716\pi$
$271$	−26.7402			−1.62435			−0.812175	+	0.583414i	$0.801716\pi$
$272$		0			0
$273$		−	3.49884i		−	0.211760i
$274$		0			0
$275$	−10.1458	−	2.83484i	−0.611814	−	0.170947i
$276$		0			0
$277$			18.8601i			1.13320i	0.823995	+	0.566598i	$0.191741\pi$
$277$			18.8601i			1.13320i	−0.823995	+	0.566598i	$0.808259\pi$
$278$		0			0
$279$	−8.65672			−0.518264
$280$		0			0
$281$	6.09240			0.363442			0.181721	−	0.983350i	$-0.441833\pi$
$281$	6.09240			0.363442			0.181721	+	0.983350i	$0.441833\pi$
$282$		0			0
$283$		−	15.6971i		−	0.933095i	−0.884496	−	0.466548i	$-0.845498\pi$
$283$		−	15.6971i		−	0.933095i	0.884496	−	0.466548i	$-0.154502\pi$
$284$		0			0
$285$	−0.238171	+	1.73746i	−0.0141080	+	0.102918i
$286$		0			0
$287$		−	1.84584i		−	0.108956i
$288$		0			0
$289$	16.9404			0.996492
$290$		0			0
$291$	−1.29007			−0.0756254
$292$		0			0
$293$		−	20.3629i		−	1.18962i	−0.803868	−	0.594808i	$-0.797228\pi$
$293$		−	20.3629i		−	1.18962i	0.803868	−	0.594808i	$-0.202772\pi$
$294$		0			0
$295$	−2.29967	−	0.315239i	−0.133892	−	0.0183539i
$296$		0			0
$297$		−	6.50234i		−	0.377304i
$298$		0			0
$299$	−5.59296			−0.323449
$300$		0			0
$301$	−6.92098			−0.398918
$302$		0			0
$303$			4.50628i			0.258879i
$304$		0			0
$305$	8.37062	+	1.14744i	0.479300	+	0.0657025i
$306$		0			0
$307$			6.31386i			0.360351i	0.983634	+	0.180176i	$0.0576666\pi$
$307$			6.31386i			0.360351i	−0.983634	+	0.180176i	$0.942333\pi$
$308$		0			0
$309$	−6.71848			−0.382201
$310$		0			0
$311$	2.26901			0.128664			0.0643318	−	0.997929i	$-0.479508\pi$
$311$	2.26901			0.128664			0.0643318	+	0.997929i	$0.479508\pi$
$312$		0			0
$313$		−	25.8886i		−	1.46331i	−0.681676	−	0.731654i	$-0.738749\pi$
$313$		−	25.8886i		−	1.46331i	0.681676	−	0.731654i	$-0.261251\pi$
$314$		0			0
$315$	−0.951286	+	6.93964i	−0.0535989	+	0.391004i
$316$		0			0
$317$		−	0.952783i		−	0.0535136i	−0.999642	−	0.0267568i	$-0.991482\pi$
$317$		−	0.952783i		−	0.0535136i	0.999642	−	0.0267568i	$-0.00851798\pi$
$318$		0			0
$319$	9.05225			0.506829
$320$		0			0
$321$	−8.42468			−0.470220
$322$		0			0
$323$		−	0.354193i		−	0.0197078i
$324$		0			0
$325$	−7.52542	+	26.9332i	−0.417435	+	1.49399i
$326$		0			0
$327$		−	5.25257i		−	0.290468i
$328$		0			0
$329$	0.723012			0.0398609
$330$		0			0
$331$	14.7188			0.809017			0.404509	−	0.914534i	$-0.367443\pi$
$331$	14.7188			0.809017			0.404509	+	0.914534i	$0.367443\pi$
$332$		0			0
$333$			2.18759i			0.119879i
$334$		0			0
$335$	1.40170	−	10.2254i	0.0765831	−	0.558674i
$336$		0			0
$337$		−	23.7550i		−	1.29402i	−0.762482	−	0.647010i	$-0.776019\pi$
$337$		−	23.7550i		−	1.29402i	0.762482	−	0.647010i	$-0.223981\pi$
$338$		0			0
$339$	1.45846			0.0792127
$340$		0			0
$341$	6.73605			0.364777
$342$		0			0
$343$			14.6485i			0.790944i
$344$		0			0
$345$	−1.19789	−	0.164207i	−0.0644924	−	0.00884062i
$346$		0			0
$347$		−	20.8557i		−	1.11959i	−0.828630	−	0.559797i	$-0.810879\pi$
$347$		−	20.8557i		−	1.11959i	0.828630	−	0.559797i	$-0.189121\pi$
$348$		0			0
$349$	12.8328			0.686922			0.343461	−	0.939167i	$-0.388401\pi$
$349$	12.8328			0.686922			0.343461	+	0.939167i	$0.388401\pi$
$350$		0			0
$351$	−17.2612			−0.921337
$352$		0			0
$353$		−	30.4888i		−	1.62275i	−0.584523	−	0.811377i	$-0.698718\pi$
$353$		−	30.4888i		−	1.62275i	0.584523	−	0.811377i	$-0.301282\pi$
$354$		0			0
$355$	−15.0088	−	2.05740i	−0.796583	−	0.109196i
$356$		0			0
$357$			0.152766i			0.00808522i
$358$		0			0
$359$	11.0247			0.581864			0.290932	−	0.956744i	$-0.406035\pi$
$359$	11.0247			0.581864			0.290932	+	0.956744i	$0.406035\pi$
$360$		0			0
$361$	−16.8963			−0.889276
$362$		0			0
$363$		−	3.54773i		−	0.186207i
$364$		0			0
$365$	2.69453	−	19.6566i	0.141038	−	1.02887i
$366$		0			0
$367$		−	18.5124i		−	0.966340i	−0.875527	−	0.483170i	$-0.839485\pi$
$367$		−	18.5124i		−	0.966340i	0.875527	−	0.483170i	$-0.160515\pi$
$368$		0			0
$369$	−4.31990			−0.224885
$370$		0			0
$371$	0.620215			0.0322000
$372$		0			0
$373$			26.4707i			1.37060i	0.728260	+	0.685301i	$0.240330\pi$
$373$			26.4707i			1.37060i	−0.728260	+	0.685301i	$0.759670\pi$
$374$		0			0
$375$	−2.40254	+	5.54758i	−0.124066	+	0.286476i
$376$		0			0
$377$		−	24.0303i		−	1.23762i
$378$		0			0
$379$	10.8785			0.558791			0.279395	−	0.960176i	$-0.409866\pi$
$379$	10.8785			0.558791			0.279395	+	0.960176i	$0.409866\pi$
$380$		0			0
$381$	−2.38148			−0.122007
$382$		0			0
$383$		−	21.6026i		−	1.10384i	−0.833897	−	0.551920i	$-0.813895\pi$
$383$		−	21.6026i		−	1.10384i	0.833897	−	0.551920i	$-0.186105\pi$
$384$		0			0
$385$	0.740224	−	5.39994i	0.0377253	−	0.275206i
$386$		0			0
$387$			16.1975i			0.823364i
$388$		0			0
$389$	29.4117			1.49123			0.745617	−	0.666374i	$-0.232155\pi$
$389$	29.4117			1.49123			0.745617	+	0.666374i	$0.232155\pi$
$390$		0			0
$391$	0.244199			0.0123497
$392$		0			0
$393$		−	0.896603i		−	0.0452276i
$394$		0			0
$395$	32.9025	+	4.51028i	1.65551	+	0.226937i
$396$		0			0
$397$			33.2753i			1.67004i	0.550219	+	0.835021i	$0.314544\pi$
$397$			33.2753i			1.67004i	−0.550219	+	0.835021i	$0.685456\pi$
$398$		0			0
$399$	−0.907360			−0.0454248
$400$		0			0
$401$	33.7280			1.68429			0.842147	−	0.539248i	$-0.181292\pi$
$401$	33.7280			1.68429			0.842147	+	0.539248i	$0.181292\pi$
$402$		0			0
$403$		−	17.8816i		−	0.890748i
$404$		0			0
$405$	14.2980	+	1.95997i	0.710472	+	0.0973915i
$406$		0			0
$407$		−	1.70223i		−	0.0843764i
$408$		0			0
$409$	−22.6282			−1.11889			−0.559445	−	0.828867i	$-0.688986\pi$
$409$	−22.6282			−1.11889			−0.559445	+	0.828867i	$0.688986\pi$
$410$		0			0
$411$	−1.46479			−0.0722527
$412$		0			0
$413$		−	1.20097i		−	0.0590956i
$414$		0			0
$415$	1.85821	−	13.5556i	0.0912159	−	0.665421i
$416$		0			0
$417$			4.86383i			0.238183i
$418$		0			0
$419$	−34.6846			−1.69446			−0.847228	−	0.531230i	$-0.821730\pi$
$419$	−34.6846			−1.69446			−0.847228	+	0.531230i	$0.821730\pi$
$420$		0			0
$421$	−0.348044			−0.0169626			−0.00848131	−	0.999964i	$-0.502700\pi$
$421$	−0.348044			−0.0169626			−0.00848131	+	0.999964i	$0.502700\pi$
$422$		0			0
$423$		−	1.69210i		−	0.0822726i
$424$		0			0
$425$	0.328573	−	1.17595i	0.0159381	−	0.0570420i
$426$		0			0
$427$			4.37142i			0.211548i
$428$		0			0
$429$	6.37172			0.307629
$430$		0			0
$431$	−21.2559			−1.02386			−0.511930	−	0.859027i	$-0.671069\pi$
$431$	−21.2559			−1.02386			−0.511930	+	0.859027i	$0.671069\pi$
$432$		0			0
$433$		−	29.3363i		−	1.40981i	−0.709301	−	0.704906i	$-0.750989\pi$
$433$		−	29.3363i		−	1.40981i	0.709301	−	0.704906i	$-0.249011\pi$
$434$		0			0
$435$	0.705521	−	5.14678i	0.0338271	−	0.246769i
$436$		0			0
$437$			1.45043i			0.0693835i
$438$		0			0
$439$	20.2210			0.965096			0.482548	−	0.875870i	$-0.339711\pi$
$439$	20.2210			0.965096			0.482548	+	0.875870i	$0.339711\pi$
$440$		0			0
$441$	15.3292			0.729962
$442$		0			0
$443$		−	33.8124i		−	1.60648i	−0.595657	−	0.803239i	$-0.703108\pi$
$443$		−	33.8124i		−	1.60648i	0.595657	−	0.803239i	$-0.296892\pi$
$444$		0			0
$445$	−12.8478	−	1.76117i	−0.609042	−	0.0834875i
$446$		0			0
$447$		−	5.15944i		−	0.244033i
$448$		0			0
$449$	−17.7383			−0.837120			−0.418560	−	0.908189i	$-0.637465\pi$
$449$	−17.7383			−0.837120			−0.418560	+	0.908189i	$0.637465\pi$
$450$		0			0
$451$	3.36144			0.158284
$452$		0			0
$453$			6.56821i			0.308601i
$454$		0			0
$455$	−14.3348	−	1.96501i	−0.672025	−	0.0921212i
$456$		0			0
$457$		−	31.7861i		−	1.48689i	−0.668797	−	0.743445i	$-0.733190\pi$
$457$		−	31.7861i		−	1.48689i	0.668797	−	0.743445i	$-0.266810\pi$
$458$		0			0
$459$	0.753657			0.0351777
$460$		0			0
$461$	17.2206			0.802044			0.401022	−	0.916068i	$-0.368655\pi$
$461$	17.2206			0.802044			0.401022	+	0.916068i	$0.368655\pi$
$462$		0			0
$463$		−	40.0085i		−	1.85935i	−0.368376	−	0.929677i	$-0.620086\pi$
$463$		−	40.0085i		−	1.85935i	0.368376	−	0.929677i	$-0.379914\pi$
$464$		0			0
$465$	0.524999	−	3.82987i	0.0243462	−	0.177606i
$466$		0			0
$467$		−	11.9947i		−	0.555049i	−0.960719	−	0.277524i	$-0.910486\pi$
$467$		−	11.9947i		−	0.555049i	0.960719	−	0.277524i	$-0.0895140\pi$
$468$		0			0
$469$	5.34006			0.246581
$470$		0			0
$471$	−10.3687			−0.477764
$472$		0			0
$473$		−	12.6037i		−	0.579520i
$474$		0			0
$475$	6.98463	+	1.95158i	0.320477	+	0.0895446i
$476$		0			0
$477$		−	1.45152i		−	0.0664604i
$478$		0			0
$479$	−20.8249			−0.951514			−0.475757	−	0.879577i	$-0.657826\pi$
$479$	−20.8249			−0.951514			−0.475757	+	0.879577i	$0.657826\pi$
$480$		0			0
$481$	−4.51877			−0.206038
$482$		0			0
$483$		−	0.625580i		−	0.0284649i
$484$		0			0
$485$	−0.724528	+	5.28544i	−0.0328991	+	0.239999i
$486$		0			0
$487$			16.5640i			0.750584i	0.926907	+	0.375292i	$0.122458\pi$
$487$			16.5640i			0.750584i	−0.926907	+	0.375292i	$0.877542\pi$
$488$		0			0
$489$	9.37597			0.423996
$490$		0			0
$491$	−28.8494			−1.30196			−0.650978	−	0.759097i	$-0.725641\pi$
$491$	−28.8494			−1.30196			−0.650978	+	0.759097i	$0.725641\pi$
$492$		0			0
$493$			1.04921i			0.0472538i
$494$		0			0
$495$	−12.6377	−	1.73238i	−0.568024	−	0.0778647i
$496$		0			0
$497$		−	7.83808i		−	0.351586i
$498$		0			0
$499$	−14.0289			−0.628021			−0.314011	−	0.949419i	$-0.601673\pi$
$499$	−14.0289			−0.628021			−0.314011	+	0.949419i	$0.601673\pi$
$500$		0			0
$501$	0.315048			0.0140753
$502$		0			0
$503$			23.2157i			1.03514i	0.855642	+	0.517568i	$0.173162\pi$
$503$			23.2157i			1.03514i	−0.855642	+	0.517568i	$0.826838\pi$
$504$		0			0
$505$	18.4623	+	2.53081i	0.821560	+	0.112619i
$506$		0			0
$507$		−	9.88507i		−	0.439011i
$508$		0			0
$509$	17.6290			0.781394			0.390697	−	0.920519i	$-0.372234\pi$
$509$	17.6290			0.781394			0.390697	+	0.920519i	$0.372234\pi$
$510$		0			0
$511$	10.2653			0.454111
$512$		0			0
$513$			4.47638i			0.197637i
$514$		0			0
$515$	−3.77322	+	27.5257i	−0.166268	+	1.21293i
$516$		0			0
$517$			1.31667i			0.0579072i
$518$		0			0
$519$	−6.52067			−0.286226
$520$		0			0
$521$	−1.64734			−0.0721711			−0.0360855	−	0.999349i	$-0.511489\pi$
$521$	−1.64734			−0.0721711			−0.0360855	+	0.999349i	$0.511489\pi$
$522$		0			0
$523$			40.1998i			1.75781i	0.476994	+	0.878907i	$0.341726\pi$
$523$			40.1998i			1.75781i	−0.476994	+	0.878907i	$0.658274\pi$
$524$		0			0
$525$	−3.01252	−	0.841729i	−0.131477	−	0.0367360i
$526$		0			0
$527$			0.780744i			0.0340098i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−2.81067			−0.121973
$532$		0			0
$533$		−	8.92336i		−	0.386513i
$534$		0			0
$535$	−4.73145	+	34.5160i	−0.204559	+	1.49226i
$536$		0			0
$537$			9.74072i			0.420343i
$538$		0			0
$539$	−11.9281			−0.513780
$540$		0			0
$541$	26.8419			1.15403			0.577013	−	0.816735i	$-0.304218\pi$
$541$	26.8419			1.15403			0.577013	+	0.816735i	$0.304218\pi$
$542$		0			0
$543$			1.80373i			0.0774055i
$544$		0			0
$545$	−21.5198	−	2.94994i	−0.921808	−	0.126361i
$546$		0			0
$547$			20.7472i			0.887087i	0.896253	+	0.443544i	$0.146279\pi$
$547$			20.7472i			0.887087i	−0.896253	+	0.443544i	$0.853721\pi$
$548$		0			0
$549$	10.2306			0.436633
$550$		0			0
$551$	−6.23181			−0.265484
$552$		0			0
$553$			17.1828i			0.730687i
$554$		0			0
$555$	−0.967826	−	0.132670i	−0.0410819	−	0.00563151i
$556$		0			0
$557$		−	18.1711i		−	0.769933i	−0.922931	−	0.384966i	$-0.874213\pi$
$557$		−	18.1711i		−	0.769933i	0.922931	−	0.384966i	$-0.125787\pi$
$558$		0			0
$559$	−33.4581			−1.41513
$560$		0			0
$561$	−0.278201			−0.0117456
$562$		0			0
$563$		−	22.9947i		−	0.969110i	−0.874761	−	0.484555i	$-0.838982\pi$
$563$		−	22.9947i		−	0.969110i	0.874761	−	0.484555i	$-0.161018\pi$
$564$		0			0
$565$	0.819098	−	5.97532i	0.0344597	−	0.251384i
$566$		0			0
$567$			7.46688i			0.313580i
$568$		0			0
$569$	−27.0829			−1.13537			−0.567687	−	0.823245i	$-0.692162\pi$
$569$	−27.0829			−1.13537			−0.567687	+	0.823245i	$0.692162\pi$
$570$		0			0
$571$	10.5631			0.442053			0.221027	−	0.975268i	$-0.429059\pi$
$571$	10.5631			0.442053			0.221027	+	0.975268i	$0.429059\pi$
$572$		0			0
$573$			1.56817i			0.0655112i
$574$		0			0
$575$	−1.34552	+	4.81556i	−0.0561119	+	0.200823i
$576$		0			0
$577$			2.51016i			0.104499i	0.998634	+	0.0522496i	$0.0166392\pi$
$577$			2.51016i			0.104499i	−0.998634	+	0.0522496i	$0.983361\pi$
$578$		0			0
$579$	6.27925			0.260957
$580$		0			0
$581$	7.07922			0.293695
$582$		0			0
$583$			1.12947i			0.0467778i
$584$		0			0
$585$	−4.59881	+	33.5484i	−0.190137	+	1.38705i
$586$		0			0
$587$		−	43.0923i		−	1.77861i	−0.457315	−	0.889305i	$-0.651189\pi$
$587$		−	43.0923i		−	1.77861i	0.457315	−	0.889305i	$-0.348811\pi$
$588$		0			0
$589$	−4.63727			−0.191076
$590$		0			0
$591$	0.0846792			0.00348324
$592$		0			0
$593$			29.7306i			1.22089i	0.792059	+	0.610444i	$0.209009\pi$
$593$			29.7306i			1.22089i	−0.792059	+	0.610444i	$0.790991\pi$
$594$		0			0
$595$	0.625882	+	0.0857960i	0.0256587	+	0.00351729i
$596$		0			0
$597$			12.6245i			0.516687i
$598$		0			0
$599$	−22.9491			−0.937674			−0.468837	−	0.883285i	$-0.655327\pi$
$599$	−22.9491			−0.937674			−0.468837	+	0.883285i	$0.655327\pi$
$600$		0			0
$601$	3.49920			0.142735			0.0713677	−	0.997450i	$-0.477264\pi$
$601$	3.49920			0.142735			0.0713677	+	0.997450i	$0.477264\pi$
$602$		0			0
$603$		−	12.4976i		−	0.508941i
$604$		0			0
$605$	−14.5351	−	1.99247i	−0.590934	−	0.0810053i
$606$		0			0
$607$			42.5670i			1.72774i	0.503714	+	0.863871i	$0.331967\pi$
$607$			42.5670i			1.72774i	−0.503714	+	0.863871i	$0.668033\pi$
$608$		0			0
$609$	2.68782			0.108916
$610$		0			0
$611$	3.49526			0.141403
$612$		0			0
$613$		−	28.0072i		−	1.13120i	−0.824679	−	0.565600i	$-0.808644\pi$
$613$		−	28.0072i		−	1.13120i	0.824679	−	0.565600i	$-0.191356\pi$
$614$		0			0
$615$	0.261987	−	1.91119i	0.0105643	−	0.0770668i
$616$		0			0
$617$		−	3.14856i		−	0.126756i	−0.997990	−	0.0633781i	$-0.979813\pi$
$617$		−	3.14856i		−	0.126756i	0.997990	−	0.0633781i	$-0.0201874\pi$
$618$		0			0
$619$	13.1433			0.528275			0.264137	−	0.964485i	$-0.414913\pi$
$619$	13.1433			0.528275			0.264137	+	0.964485i	$0.414913\pi$
$620$		0			0
$621$	−3.08625			−0.123847
$622$		0			0
$623$		−	6.70953i		−	0.268812i
$624$		0			0
$625$	21.3792	+	12.9588i	0.855167	+	0.518353i
$626$		0			0
$627$		−	1.65239i		−	0.0659900i
$628$		0			0
$629$	0.197298			0.00786677
$630$		0			0
$631$	35.4545			1.41142			0.705711	−	0.708500i	$-0.250628\pi$
$631$	35.4545			1.41142			0.705711	+	0.708500i	$0.250628\pi$
$632$		0			0
$633$			11.4969i			0.456961i
$634$		0			0
$635$	−1.33748	+	9.75693i	−0.0530763	+	0.387192i
$636$		0			0
$637$			31.6646i			1.25460i
$638$		0			0
$639$	−18.3438			−0.725670
$640$		0			0
$641$	9.79611			0.386923			0.193462	−	0.981108i	$-0.438029\pi$
$641$	9.79611			0.386923			0.193462	+	0.981108i	$0.438029\pi$
$642$		0			0
$643$			15.6613i			0.617621i	0.951124	+	0.308810i	$0.0999308\pi$
$643$			15.6613i			0.617621i	−0.951124	+	0.308810i	$0.900069\pi$
$644$		0			0
$645$	−7.16602	−	0.982319i	−0.282162	−	0.0386788i
$646$		0			0
$647$		−	38.6953i		−	1.52127i	−0.649180	−	0.760634i	$-0.724888\pi$
$647$		−	38.6953i		−	1.52127i	0.649180	−	0.760634i	$-0.275112\pi$
$648$		0			0
$649$	2.18707			0.0858500
$650$		0			0
$651$	2.00009			0.0783896
$652$		0			0
$653$		−	37.8261i		−	1.48025i	−0.672469	−	0.740125i	$-0.734766\pi$
$653$		−	37.8261i		−	1.48025i	0.672469	−	0.740125i	$-0.265234\pi$
$654$		0			0
$655$	−3.67339	−	0.503548i	−0.143531	−	0.0196753i
$656$		0			0
$657$		−	24.0244i		−	0.937282i
$658$		0			0
$659$	9.00852			0.350922			0.175461	−	0.984486i	$-0.443858\pi$
$659$	9.00852			0.350922			0.175461	+	0.984486i	$0.443858\pi$
$660$		0			0
$661$	−26.0490			−1.01319			−0.506594	−	0.862185i	$-0.669096\pi$
$661$	−26.0490			−1.01319			−0.506594	+	0.862185i	$0.669096\pi$
$662$		0			0
$663$			0.738517i			0.0286816i
$664$		0			0
$665$	−0.509590	+	3.71746i	−0.0197611	+	0.144157i
$666$		0			0
$667$		−	4.29653i		−	0.166362i
$668$		0			0
$669$	15.5527			0.601303
$670$		0			0
$671$	−7.96076			−0.307322
$672$		0			0
$673$			4.68835i			0.180722i	0.995909	+	0.0903612i	$0.0288022\pi$
$673$			4.68835i			0.180722i	−0.995909	+	0.0903612i	$0.971198\pi$
$674$		0			0
$675$	−4.15260	+	14.8620i	−0.159834	+	0.572038i
$676$		0			0
$677$			31.1370i			1.19669i	0.801238	+	0.598346i	$0.204175\pi$
$677$			31.1370i			1.19669i	−0.801238	+	0.598346i	$0.795825\pi$
$678$		0			0
$679$	−2.76023			−0.105928
$680$		0			0
$681$	−3.29296			−0.126186
$682$		0			0
$683$			15.2671i			0.584180i	0.956391	+	0.292090i	$0.0943507\pi$
$683$			15.2671i			0.584180i	−0.956391	+	0.292090i	$0.905649\pi$
$684$		0			0
$685$	−0.822652	+	6.00125i	−0.0314319	+	0.229296i
$686$		0			0
$687$		−	10.7763i		−	0.411142i
$688$		0			0
$689$	2.99831			0.114227
$690$		0			0
$691$	30.3229			1.15354			0.576768	−	0.816908i	$-0.304314\pi$
$691$	30.3229			1.15354			0.576768	+	0.816908i	$0.304314\pi$
$692$		0			0
$693$		−	6.59985i		−	0.250707i
$694$		0			0
$695$	19.9271	+	2.73161i	0.755880	+	0.103616i
$696$		0			0
$697$			0.389610i			0.0147575i
$698$		0			0
$699$	5.20829			0.196996
$700$		0			0
$701$	−19.7545			−0.746118			−0.373059	−	0.927808i	$-0.621691\pi$
$701$	−19.7545			−0.746118			−0.373059	+	0.927808i	$0.621691\pi$
$702$		0			0
$703$			1.17186i			0.0441975i
$704$		0			0
$705$	0.748611	+	0.102620i	0.0281943	+	0.00386488i
$706$		0			0
$707$			9.64161i			0.362610i
$708$		0			0
$709$	−7.45728			−0.280064			−0.140032	−	0.990147i	$-0.544721\pi$
$709$	−7.45728			−0.280064			−0.140032	+	0.990147i	$0.544721\pi$
$710$		0			0
$711$	40.2137			1.50813
$712$		0			0
$713$		−	3.19717i		−	0.119735i
$714$		0			0
$715$	3.57847	−	26.1050i	0.133827	−	0.976271i
$716$		0			0
$717$			7.16907i			0.267734i
$718$		0			0
$719$	48.4683			1.80756			0.903781	−	0.427995i	$-0.140780\pi$
$719$	48.4683			1.80756			0.903781	+	0.427995i	$0.140780\pi$
$720$		0			0
$721$	−14.3748			−0.535347
$722$		0			0
$723$		−	7.79993i		−	0.290083i
$724$		0			0
$725$	−20.6902	−	5.78105i	−0.768413	−	0.214703i
$726$		0			0
$727$			49.7498i			1.84512i	0.385856	+	0.922559i	$0.373906\pi$
$727$			49.7498i			1.84512i	−0.385856	+	0.922559i	$0.626094\pi$
$728$		0			0
$729$	12.4687			0.461802
$730$		0			0
$731$	1.46084			0.0540312
$732$		0			0
$733$		−	12.1579i		−	0.449063i	−0.974467	−	0.224531i	$-0.927915\pi$
$733$		−	12.1579i		−	0.449063i	0.974467	−	0.224531i	$-0.0720851\pi$
$734$		0			0
$735$	−0.929661	+	6.78189i	−0.0342911	+	0.250154i
$736$		0			0
$737$			9.72474i			0.358215i
$738$		0			0
$739$	−20.5807			−0.757075			−0.378537	−	0.925586i	$-0.623573\pi$
$739$	−20.5807			−0.757075			−0.378537	+	0.925586i	$0.623573\pi$
$740$		0			0
$741$	−4.38646			−0.161141
$742$		0			0
$743$			22.7862i			0.835944i	0.908460	+	0.417972i	$0.137259\pi$
$743$			22.7862i			0.835944i	−0.908460	+	0.417972i	$0.862741\pi$
$744$		0			0
$745$	−21.1383	−	2.89763i	−0.774446	−	0.106161i
$746$		0			0
$747$		−	16.5678i		−	0.606185i
$748$		0			0
$749$	−18.0254			−0.658634
$750$		0			0
$751$	−1.69147			−0.0617226			−0.0308613	−	0.999524i	$-0.509825\pi$
$751$	−1.69147			−0.0617226			−0.0308613	+	0.999524i	$0.509825\pi$
$752$		0			0
$753$		−	10.1001i		−	0.368069i
$754$		0			0
$755$	26.9100	+	3.68883i	0.979355	+	0.134250i
$756$		0			0
$757$			25.7514i			0.935952i	0.883741	+	0.467976i	$0.155017\pi$
$757$			25.7514i			0.935952i	−0.883741	+	0.467976i	$0.844983\pi$
$758$		0			0
$759$	1.13924			0.0413518
$760$		0			0
$761$	41.5286			1.50541			0.752706	−	0.658357i	$-0.228748\pi$
$761$	41.5286			1.50541			0.752706	+	0.658357i	$0.228748\pi$
$762$		0			0
$763$		−	11.2384i		−	0.406856i
$764$		0			0
$765$	0.200792	−	1.46478i	0.00725966	−	0.0529593i
$766$		0			0
$767$		−	5.80584i		−	0.209637i
$768$		0			0
$769$	−0.962676			−0.0347150			−0.0173575	−	0.999849i	$-0.505525\pi$
$769$	−0.962676			−0.0347150			−0.0173575	+	0.999849i	$0.505525\pi$
$770$		0			0
$771$	13.7904			0.496648
$772$		0			0
$773$		−	42.5570i		−	1.53067i	−0.643633	−	0.765334i	$-0.722574\pi$
$773$		−	42.5570i		−	1.53067i	0.643633	−	0.765334i	$-0.277426\pi$
$774$		0			0
$775$	−15.3962	−	4.30185i	−0.553046	−	0.154527i
$776$		0			0
$777$		−	0.505431i		−	0.0181322i
$778$		0			0
$779$	−2.31411			−0.0829115
$780$		0			0
$781$	14.2739			0.510759
$782$		0			0
$783$		−	13.2601i		−	0.473879i
$784$		0			0
$785$	−5.82325	+	42.4806i	−0.207841	+	1.51620i
$786$		0			0
$787$			32.0542i			1.14261i	0.820738	+	0.571304i	$0.193562\pi$
$787$			32.0542i			1.14261i	−0.820738	+	0.571304i	$0.806438\pi$
$788$		0			0
$789$	9.97749			0.355208
$790$		0			0
$791$	3.12051			0.110953
$792$		0			0
$793$			21.1328i			0.750447i
$794$		0			0
$795$	0.642175	+	0.0880293i	0.0227756	+	0.00312208i
$796$		0			0
$797$		−	42.6339i		−	1.51017i	−0.655628	−	0.755084i	$-0.727596\pi$
$797$		−	42.6339i		−	1.51017i	0.655628	−	0.755084i	$-0.272404\pi$
$798$		0			0
$799$	−0.152609			−0.00539893
$800$		0			0
$801$	−15.7026			−0.554825
$802$		0			0
$803$			18.6941i			0.659701i
$804$		0			0
$805$	−2.56301	−	0.351337i	−0.0903341	−	0.0123830i
$806$		0			0
$807$		−	1.09617i		−	0.0385871i
$808$		0			0
$809$	5.90921			0.207757			0.103878	−	0.994590i	$-0.466875\pi$
$809$	5.90921			0.207757			0.103878	+	0.994590i	$0.466875\pi$
$810$		0			0
$811$	38.1150			1.33840			0.669199	−	0.743084i	$-0.266638\pi$
$811$	38.1150			1.33840			0.669199	+	0.743084i	$0.266638\pi$
$812$		0			0
$813$		−	14.4591i		−	0.507101i
$814$		0			0
$815$	5.26571	−	38.4134i	0.184450	−	1.34556i
$816$		0			0
$817$			8.67675i			0.303561i
$818$		0			0
$819$	−17.5201			−0.612201
$820$		0			0
$821$	−31.4861			−1.09887			−0.549437	−	0.835535i	$-0.685158\pi$
$821$	−31.4861			−1.09887			−0.549437	+	0.835535i	$0.685158\pi$
$822$		0			0
$823$		−	2.04415i		−	0.0712546i	−0.999365	−	0.0356273i	$-0.988657\pi$
$823$		−	2.04415i		−	0.0712546i	0.999365	−	0.0356273i	$-0.0113429\pi$
$824$		0			0
$825$	1.53287	−	5.48607i	0.0533675	−	0.191000i
$826$		0			0
$827$		−	34.9513i		−	1.21538i	−0.794175	−	0.607689i	$-0.792097\pi$
$827$		−	34.9513i		−	1.21538i	0.794175	−	0.607689i	$-0.207903\pi$
$828$		0			0
$829$	2.48455			0.0862919			0.0431460	−	0.999069i	$-0.486262\pi$
$829$	2.48455			0.0862919			0.0431460	+	0.999069i	$0.486262\pi$
$830$		0			0
$831$	−10.1981			−0.353769
$832$		0			0
$833$		−	1.38253i		−	0.0479019i
$834$		0			0
$835$	0.176937	−	1.29076i	0.00612315	−	0.0446684i
$836$		0			0
$837$		−	9.86726i		−	0.341062i
$838$		0			0
$839$	−46.5747			−1.60794			−0.803969	−	0.594672i	$-0.797282\pi$
$839$	−46.5747			−1.60794			−0.803969	+	0.594672i	$0.797282\pi$
$840$		0			0
$841$	−10.5399			−0.363444
$842$		0			0
$843$			3.29431i			0.113462i
$844$		0			0
$845$	−40.4992	−	5.55163i	−1.39322	−	0.190982i
$846$		0			0
$847$		−	7.59070i		−	0.260820i
$848$		0			0
$849$	8.48780			0.291300
$850$		0			0
$851$	−0.807940			−0.0276958
$852$		0			0
$853$		−	1.74956i		−	0.0599036i	−0.999551	−	0.0299518i	$-0.990465\pi$
$853$		−	1.74956i		−	0.0599036i	0.999551	−	0.0299518i	$-0.00953538\pi$
$854$		0			0
$855$	8.70015	+	1.19262i	0.297539	+	0.0407866i
$856$		0			0
$857$			5.88603i			0.201063i	0.994934	+	0.100532i	$0.0320543\pi$
$857$			5.88603i			0.201063i	−0.994934	+	0.100532i	$0.967946\pi$
$858$		0			0
$859$	−38.4929			−1.31336			−0.656680	−	0.754169i	$-0.728040\pi$
$859$	−38.4929			−1.31336			−0.656680	+	0.754169i	$0.728040\pi$
$860$		0			0
$861$	0.998090			0.0340148
$862$		0			0
$863$		−	36.8687i		−	1.25503i	−0.778606	−	0.627513i	$-0.784073\pi$
$863$		−	36.8687i		−	1.25503i	0.778606	−	0.627513i	$-0.215927\pi$
$864$		0			0
$865$	−3.66213	+	26.7152i	−0.124516	+	0.908345i
$866$		0			0
$867$			9.16006i			0.311092i
$868$		0			0
$869$	−31.2915			−1.06149
$870$		0			0
$871$	25.8155			0.874724
$872$		0			0
$873$			6.45990i			0.218634i
$874$		0			0
$875$	−5.14045	+	11.8696i	−0.173779	+	0.401265i
$876$		0			0
$877$			46.7182i			1.57756i	0.614676	+	0.788780i	$0.289287\pi$
$877$			46.7182i			1.57756i	−0.614676	+	0.788780i	$0.710713\pi$
$878$		0			0
$879$	11.0107			0.371383
$880$		0			0
$881$	−13.8791			−0.467600			−0.233800	−	0.972285i	$-0.575116\pi$
$881$	−13.8791			−0.467600			−0.233800	+	0.972285i	$0.575116\pi$
$882$		0			0
$883$			23.8941i			0.804100i	0.915618	+	0.402050i	$0.131702\pi$
$883$			23.8941i			0.804100i	−0.915618	+	0.402050i	$0.868298\pi$
$884$		0			0
$885$	0.170457	−	1.24349i	0.00572986	−	0.0417994i
$886$		0			0
$887$			2.36292i			0.0793391i	0.999213	+	0.0396695i	$0.0126305\pi$
$887$			2.36292i			0.0793391i	−0.999213	+	0.0396695i	$0.987369\pi$
$888$		0			0
$889$	−5.09540			−0.170894
$890$		0			0
$891$	−13.5979			−0.455546
$892$		0			0
$893$		−	0.906432i		−	0.0303326i
$894$		0			0
$895$	39.9078	+	5.47057i	1.33397	+	0.182861i
$896$		0			0
$897$		−	3.02425i		−	0.100977i
$898$		0			0
$899$	13.7367			0.458145
$900$		0			0
$901$	−0.130912			−0.00436130
$902$		0			0
$903$		−	3.74234i		−	0.124537i
$904$		0			0
$905$	7.38990	+	1.01301i	0.245649	+	0.0336735i
$906$		0			0
$907$		−	23.1649i		−	0.769179i	−0.923088	−	0.384590i	$-0.874343\pi$
$907$		−	23.1649i		−	0.769179i	0.923088	−	0.384590i	$-0.125657\pi$
$908$		0			0
$909$	22.5647			0.748424
$910$		0			0
$911$	43.6423			1.44593			0.722967	−	0.690882i	$-0.242778\pi$
$911$	43.6423			1.44593			0.722967	+	0.690882i	$0.242778\pi$
$912$		0			0
$913$			12.8919i			0.426660i
$914$		0			0
$915$	−0.620451	+	4.52619i	−0.0205115	+	0.149631i
$916$		0			0
$917$		−	1.91837i		−	0.0633500i
$918$		0			0
$919$	−10.3644			−0.341890			−0.170945	−	0.985281i	$-0.554682\pi$
$919$	−10.3644			−0.341890			−0.170945	+	0.985281i	$0.554682\pi$
$920$		0			0
$921$	−3.41406			−0.112497
$922$		0			0
$923$		−	37.8917i		−	1.24722i
$924$		0			0
$925$	−1.08710	+	3.89068i	−0.0357435	+	0.127925i
$926$		0			0
$927$			33.6421i			1.10495i
$928$		0			0
$929$	38.1805			1.25266			0.626331	−	0.779557i	$-0.284556\pi$
$929$	38.1805			1.25266			0.626331	+	0.779557i	$0.284556\pi$
$930$		0			0
$931$	8.21163			0.269125
$932$		0			0
$933$			1.22691i			0.0401671i
$934$		0			0
$935$	−0.156242	+	1.13979i	−0.00510967	+	0.0372751i
$936$		0			0
$937$		−	12.1405i		−	0.396613i	−0.980140	−	0.198306i	$-0.936456\pi$
$937$		−	12.1405i		−	0.396613i	0.980140	−	0.198306i	$-0.0635441\pi$
$938$		0			0
$939$	13.9986			0.456826
$940$		0			0
$941$	−18.2871			−0.596141			−0.298071	−	0.954544i	$-0.596343\pi$
$941$	−18.2871			−0.596141			−0.298071	+	0.954544i	$0.596343\pi$
$942$		0			0
$943$		−	1.59546i		−	0.0519554i
$944$		0			0
$945$	−7.91007	−	1.08431i	−0.257315	−	0.0352727i
$946$		0			0
$947$			51.7376i			1.68125i	0.541621	+	0.840623i	$0.317811\pi$
$947$			51.7376i			1.68125i	−0.541621	+	0.840623i	$0.682189\pi$
$948$		0			0
$949$	49.6257			1.61092
$950$		0			0
$951$	0.515193			0.0167063
$952$		0			0
$953$			58.1675i			1.88423i	0.335292	+	0.942114i	$0.391165\pi$
$953$			58.1675i			1.88423i	−0.335292	+	0.942114i	$0.608835\pi$
$954$		0			0
$955$	6.42480	+	0.880712i	0.207902	+	0.0284992i
$956$		0			0
$957$			4.89477i			0.158225i
$958$		0			0
$959$	−3.13406			−0.101204
$960$		0			0
$961$	−20.7781			−0.670261
$962$		0			0
$963$			42.1857i			1.35942i
$964$		0			0
$965$	3.52654	−	25.7261i	0.113523	−	0.828153i
$966$		0			0
$967$			12.1138i			0.389555i	0.980847	+	0.194777i	$0.0623984\pi$
$967$			12.1138i			0.389555i	−0.980847	+	0.194777i	$0.937602\pi$
$968$		0			0
$969$	0.191521			0.00615253
$970$		0			0
$971$	−36.5922			−1.17430			−0.587149	−	0.809479i	$-0.699750\pi$
$971$	−36.5922			−1.17430			−0.587149	+	0.809479i	$0.699750\pi$
$972$		0			0
$973$			10.4066i			0.333621i
$974$		0			0
$975$	−14.5634	−	4.06917i	−0.466403	−	0.130318i
$976$		0			0
$977$			22.3327i			0.714487i	0.934011	+	0.357243i	$0.116283\pi$
$977$			22.3327i			0.714487i	−0.934011	+	0.357243i	$0.883717\pi$
$978$		0			0
$979$	12.2187			0.390510
$980$		0			0
$981$	−26.3017			−0.839748
$982$		0			0
$983$			49.8139i			1.58882i	0.607384	+	0.794408i	$0.292219\pi$
$983$			49.8139i			1.58882i	−0.607384	+	0.794408i	$0.707781\pi$
$984$		0			0
$985$	0.0475574	−	0.346931i	0.00151530	−	0.0110542i
$986$		0			0
$987$			0.390950i			0.0124441i
$988$		0			0
$989$	−5.98219			−0.190223
$990$		0			0
$991$	14.3340			0.455334			0.227667	−	0.973739i	$-0.426890\pi$
$991$	14.3340			0.455334			0.227667	+	0.973739i	$0.426890\pi$
$992$		0			0
$993$			7.95880i			0.252565i
$994$		0			0
$995$	51.7227	+	7.09016i	1.63972	+	0.224773i
$996$		0			0
$997$			51.7021i			1.63742i	0.574205	+	0.818711i	$0.305311\pi$
$997$			51.7021i			1.63742i	−0.574205	+	0.818711i	$0.694689\pi$
$998$		0			0
$999$	−2.49350			−0.0788909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.h.369.11		16
4.3	odd	2		920.2.e.c.369.6	✓	16
5.2	odd	4		9200.2.a.dd.1.6		8
5.3	odd	4		9200.2.a.de.1.3		8
5.4	even	2	inner	1840.2.e.h.369.6		16
20.3	even	4		4600.2.a.bj.1.6		8
20.7	even	4		4600.2.a.bk.1.3		8
20.19	odd	2		920.2.e.c.369.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.6	✓	16	4.3	odd	2
920.2.e.c.369.11	yes	16	20.19	odd	2
1840.2.e.h.369.6		16	5.4	even	2	inner
1840.2.e.h.369.11		16	1.1	even	1	trivial
4600.2.a.bj.1.6		8	20.3	even	4
4600.2.a.bk.1.3		8	20.7	even	4
9200.2.a.dd.1.6		8	5.2	odd	4
9200.2.a.de.1.3		8	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.540724i q^{3} +(2.21535 + 0.303680i) q^{5} +1.15693i q^{7} +2.70762 q^{9} -2.10688 q^{11} +5.59296i q^{13} +(-0.164207 + 1.19789i) q^{15} -0.244199i q^{17} +1.45043 q^{19} -0.625580 q^{21} +1.00000i q^{23} +(4.81556 + 1.34552i) q^{25} +3.08625i q^{27} -4.29653 q^{29} -3.19717 q^{31} -1.13924i q^{33} +(-0.351337 + 2.56301i) q^{35} +0.807940i q^{37} -3.02425 q^{39} -1.59546 q^{41} +5.98219i q^{43} +(5.99832 + 0.822250i) q^{45} -0.624940i q^{47} +5.66151 q^{49} +0.132044 q^{51} -0.536087i q^{53} +(-4.66747 - 0.639817i) q^{55} +0.784283i q^{57} -1.03806 q^{59} +3.77846 q^{61} +3.13253i q^{63} +(-1.69847 + 12.3904i) q^{65} -4.61571i q^{67} -0.540724 q^{69} -6.77489 q^{71} -8.87290i q^{73} +(-0.727553 + 2.60389i) q^{75} -2.43751i q^{77} +14.8521 q^{79} +6.45405 q^{81} -6.11896i q^{83} +(0.0741583 - 0.540985i) q^{85} -2.32323i q^{87} -5.79942 q^{89} -6.47066 q^{91} -1.72879i q^{93} +(3.21321 + 0.440467i) q^{95} +2.38582i q^{97} -5.70462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} - 22 q^{9} - 14 q^{11} - 6 q^{15} + 22 q^{19} + 12 q^{25} - 44 q^{29} - 18 q^{31} - 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} + 38 q^{51} - 30 q^{55} + 64 q^{59} + 34 q^{61} + 6 q^{65}+ \cdots + 122 q^{99}+O(q^{100}) \) 16 * q - 2 * q^5 - 22 * q^9 - 14 * q^11 - 6 * q^15 + 22 * q^19 + 12 * q^25 - 44 * q^29 - 18 * q^31 - 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 + 38 * q^51 - 30 * q^55 + 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 - 30 * q^71 - 56 * q^75 - 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 + 70 * q^91 - 38 * q^95 + 122 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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