LMFDB - Embedded newform 1840.2.i.b.1471.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1471,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1840.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.i (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:	$14.6924739719$
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} - 8x^{14} + 33x^{12} - 98x^{10} + 272x^{8} - 882x^{6} + 2673x^{4} - 5832x^{2} + 6561 $ x^16 - 8x^14 + 33x^12 - 98x^10 + 272x^8 - 882x^6 + 2673x^4 - 5832*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.9
Root			$1.72432 + 0.163515i$ of defining polynomial
Character	$\chi$	$=$	1840.1471
Dual form			1840.2.i.b.1471.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.327030i q^{3} -1.00000i q^{5} -2.33999 q^{7} +2.89305 q^{9} +O(q^{10})$	$q+0.327030i q^{3} -1.00000i q^{5} -2.33999 q^{7} +2.89305 q^{9} -3.77566 q^{11} +1.12781 q^{13} +0.327030 q^{15} +2.52445i q^{17} -3.12160 q^{19} -0.765246i q^{21} +(1.12712 - 4.66150i) q^{23} -1.00000 q^{25} +1.92720i q^{27} -4.41750 q^{29} +2.01296i q^{31} -1.23475i q^{33} +2.33999i q^{35} +7.54531i q^{37} +0.368826i q^{39} -4.88701 q^{41} -6.20140 q^{43} -2.89305i q^{45} -7.22429i q^{47} -1.52445 q^{49} -0.825571 q^{51} -4.78006i q^{53} +3.77566i q^{55} -1.02086i q^{57} +9.02394i q^{59} +3.02086i q^{61} -6.76971 q^{63} -1.12781i q^{65} -8.58319 q^{67} +(1.52445 + 0.368601i) q^{69} +9.56428i q^{71} -10.1767 q^{73} -0.327030i q^{75} +8.83500 q^{77} -3.12160 q^{79} +8.04890 q^{81} -4.80753 q^{83} +2.52445 q^{85} -1.44466i q^{87} +17.8559i q^{89} -2.63905 q^{91} -0.658298 q^{93} +3.12160i q^{95} -8.55135i q^{97} -10.9232 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 8 q^{13} - 16 q^{25} + 36 q^{29} - 44 q^{41} + 20 q^{49} - 20 q^{69} - 48 q^{73} - 72 q^{77} + 40 q^{81} - 4 q^{85} + 88 q^{93}+O(q^{100}) $ 16 * q - 16 * q^9 - 8 * q^13 - 16 * q^25 + 36 * q^29 - 44 * q^41 + 20 * q^49 - 20 * q^69 - 48 * q^73 - 72 * q^77 + 40 * q^81 - 4 * q^85 + 88 * q^93

Display 100 coefficients 10 coefficients

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
$3$			0.327030i			0.188811i	0.995534	+	0.0944054i	$0.0300950\pi$
$3$			0.327030i			0.188811i	−0.995534	+	0.0944054i	$0.969905\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	−2.33999			−0.884433			−0.442216	−	0.896908i	$-0.645808\pi$
$7$	−2.33999			−0.884433			−0.442216	+	0.896908i	$0.645808\pi$
$8$		0			0
$9$	2.89305			0.964350
$10$		0			0
$11$	−3.77566			−1.13840			−0.569202	−	0.822198i	$-0.692748\pi$
$11$	−3.77566			−1.13840			−0.569202	+	0.822198i	$0.692748\pi$
$12$		0			0
$13$	1.12781			0.312797			0.156398	−	0.987694i	$-0.450012\pi$
$13$	1.12781			0.312797			0.156398	+	0.987694i	$0.450012\pi$
$14$		0			0
$15$	0.327030			0.0844387
$16$		0			0
$17$			2.52445i			0.612269i	0.951988	+	0.306135i	$0.0990358\pi$
$17$			2.52445i			0.612269i	−0.951988	+	0.306135i	$0.900964\pi$
$18$		0			0
$19$	−3.12160			−0.716144			−0.358072	−	0.933694i	$-0.616566\pi$
$19$	−3.12160			−0.716144			−0.358072	+	0.933694i	$0.616566\pi$
$20$		0			0
$21$		−	0.765246i		−	0.166990i
$22$		0			0
$23$	1.12712	−	4.66150i	0.235020	−	0.971991i
$23$	1.12712	−	4.66150i	0.235020	−	0.971991i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.92720i			0.370891i
$28$		0			0
$29$	−4.41750			−0.820310			−0.410155	−	0.912016i	$-0.634525\pi$
$29$	−4.41750			−0.820310			−0.410155	+	0.912016i	$0.634525\pi$
$30$		0			0
$31$			2.01296i			0.361538i	0.983526	+	0.180769i	$0.0578587\pi$
$31$			2.01296i			0.361538i	−0.983526	+	0.180769i	$0.942141\pi$
$32$		0			0
$33$		−	1.23475i		−	0.214943i
$34$		0			0
$35$			2.33999i			0.395530i
$36$		0			0
$37$			7.54531i			1.24044i	0.784428	+	0.620220i	$0.212957\pi$
$37$			7.54531i			1.24044i	−0.784428	+	0.620220i	$0.787043\pi$
$38$		0			0
$39$			0.368826i			0.0590594i
$40$		0			0
$41$	−4.88701			−0.763223			−0.381611	−	0.924323i	$-0.624631\pi$
$41$	−4.88701			−0.763223			−0.381611	+	0.924323i	$0.624631\pi$
$42$		0			0
$43$	−6.20140			−0.945705			−0.472853	−	0.881141i	$-0.656776\pi$
$43$	−6.20140			−0.945705			−0.472853	+	0.881141i	$0.656776\pi$
$44$		0			0
$45$		−	2.89305i		−	0.431271i
$46$		0			0
$47$		−	7.22429i		−	1.05377i	−0.849936	−	0.526886i	$-0.823360\pi$
$47$		−	7.22429i		−	1.05377i	0.849936	−	0.526886i	$-0.176640\pi$
$48$		0			0
$49$	−1.52445			−0.217779
$50$		0			0
$51$	−0.825571			−0.115603
$52$		0			0
$53$		−	4.78006i		−	0.656592i	−0.944575	−	0.328296i	$-0.893526\pi$
$53$		−	4.78006i		−	0.656592i	0.944575	−	0.328296i	$-0.106474\pi$
$54$		0			0
$55$			3.77566i			0.509110i
$56$		0			0
$57$		−	1.02086i		−	0.135216i
$58$		0			0
$59$			9.02394i			1.17482i	0.809290	+	0.587409i	$0.199852\pi$
$59$			9.02394i			1.17482i	−0.809290	+	0.587409i	$0.800148\pi$
$60$		0			0
$61$			3.02086i			0.386781i	0.981122	+	0.193391i	$0.0619484\pi$
$61$			3.02086i			0.386781i	−0.981122	+	0.193391i	$0.938052\pi$
$62$		0			0
$63$	−6.76971			−0.852903
$64$		0			0
$65$		−	1.12781i		−	0.139887i
$66$		0			0
$67$	−8.58319			−1.04860			−0.524302	−	0.851533i	$-0.675674\pi$
$67$	−8.58319			−1.04860			−0.524302	+	0.851533i	$0.675674\pi$
$68$		0			0
$69$	1.52445	+	0.368601i	0.183522	+	0.0443743i
$70$		0			0
$71$			9.56428i			1.13507i	0.823349	+	0.567536i	$0.192103\pi$
$71$			9.56428i			1.13507i	−0.823349	+	0.567536i	$0.807897\pi$
$72$		0			0
$73$	−10.1767			−1.19109			−0.595547	−	0.803321i	$-0.703065\pi$
$73$	−10.1767			−1.19109			−0.595547	+	0.803321i	$0.703065\pi$
$74$		0			0
$75$		−	0.327030i		−	0.0377622i
$76$		0			0
$77$	8.83500			1.00684
$78$		0			0
$79$	−3.12160			−0.351207			−0.175604	−	0.984461i	$-0.556188\pi$
$79$	−3.12160			−0.351207			−0.175604	+	0.984461i	$0.556188\pi$
$80$		0			0
$81$	8.04890			0.894322
$82$		0			0
$83$	−4.80753			−0.527695			−0.263847	−	0.964564i	$-0.584992\pi$
$83$	−4.80753			−0.527695			−0.263847	+	0.964564i	$0.584992\pi$
$84$		0			0
$85$	2.52445			0.273815
$86$		0			0
$87$		−	1.44466i		−	0.154883i
$88$		0			0
$89$			17.8559i			1.89272i	0.323118	+	0.946359i	$0.395269\pi$
$89$			17.8559i			1.89272i	−0.323118	+	0.946359i	$0.604731\pi$
$90$		0			0
$91$	−2.63905			−0.276648
$92$		0			0
$93$	−0.658298			−0.0682623
$94$		0			0
$95$			3.12160i			0.320269i
$96$		0			0
$97$		−	8.55135i		−	0.868258i	−0.900851	−	0.434129i	$-0.857056\pi$
$97$		−	8.55135i		−	0.868258i	0.900851	−	0.434129i	$-0.142944\pi$
$98$		0			0
$99$	−10.9232			−1.09782
$100$		0			0
$101$	−4.78006			−0.475634			−0.237817	−	0.971310i	$-0.576432\pi$
$101$	−4.78006			−0.475634			−0.237817	+	0.971310i	$0.576432\pi$
$102$		0			0
$103$	1.60017			0.157670			0.0788349	−	0.996888i	$-0.474880\pi$
$103$	1.60017			0.157670			0.0788349	+	0.996888i	$0.474880\pi$
$104$		0			0
$105$	−0.765246			−0.0746804
$106$		0			0
$107$	−10.5454			−1.01946			−0.509730	−	0.860335i	$-0.670255\pi$
$107$	−10.5454			−1.01946			−0.509730	+	0.860335i	$0.670255\pi$
$108$		0			0
$109$		−	12.0698i		−	1.15607i	−0.816011	−	0.578037i	$-0.803819\pi$
$109$		−	12.0698i		−	1.15607i	0.816011	−	0.578037i	$-0.196181\pi$
$110$		0			0
$111$	−2.46754			−0.234209
$112$		0			0
$113$		−	9.07580i		−	0.853779i	−0.904304	−	0.426890i	$-0.859609\pi$
$113$		−	9.07580i		−	0.853779i	0.904304	−	0.426890i	$-0.140391\pi$
$114$		0			0
$115$	−4.66150	−	1.12712i	−0.434687	−	0.105104i
$116$		0			0
$117$	3.26280			0.301646
$118$		0			0
$119$		−	5.90719i		−	0.541511i
$120$		0			0
$121$	3.25561			0.295965
$122$		0			0
$123$		−	1.59820i		−	0.144105i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	0.582133i		−	0.0516560i	−0.999666	−	0.0258280i	$-0.991778\pi$
$127$		−	0.582133i		−	0.0516560i	0.999666	−	0.0258280i	$-0.00822222\pi$
$128$		0			0
$129$		−	2.02804i		−	0.178559i
$130$		0			0
$131$			8.74572i			0.764117i	0.924138	+	0.382058i	$0.124785\pi$
$131$			8.74572i			0.764117i	−0.924138	+	0.382058i	$0.875215\pi$
$132$		0			0
$133$	7.30451			0.633381
$134$		0			0
$135$	1.92720			0.165867
$136$		0			0
$137$		−	8.25561i		−	0.705324i	−0.935751	−	0.352662i	$-0.885276\pi$
$137$		−	8.25561i		−	0.705324i	0.935751	−	0.352662i	$-0.114724\pi$
$138$		0			0
$139$		−	5.05581i		−	0.428828i	−0.976743	−	0.214414i	$-0.931216\pi$
$139$		−	5.05581i		−	0.428828i	0.976743	−	0.214414i	$-0.0687842\pi$
$140$		0			0
$141$	2.36256			0.198963
$142$		0			0
$143$	−4.25821			−0.356089
$144$		0			0
$145$			4.41750i			0.366854i
$146$		0			0
$147$		−	0.498541i		−	0.0411190i
$148$		0			0
$149$		−	1.30451i		−	0.106870i	−0.998571	−	0.0534349i	$-0.982983\pi$
$149$		−	1.30451i		−	0.106870i	0.998571	−	0.0534349i	$-0.0170170\pi$
$150$		0			0
$151$			18.1057i			1.47342i	0.676209	+	0.736709i	$0.263621\pi$
$151$			18.1057i			1.47342i	−0.676209	+	0.736709i	$0.736379\pi$
$152$		0			0
$153$			7.30337i			0.590442i
$154$		0			0
$155$	2.01296			0.161685
$156$		0			0
$157$			2.22871i			0.177871i	0.996037	+	0.0889353i	$0.0283464\pi$
$157$			2.22871i			0.177871i	−0.996037	+	0.0889353i	$0.971654\pi$
$158$		0			0
$159$	1.56322			0.123972
$160$		0			0
$161$	−2.63744	+	10.9079i	−0.207859	+	0.859660i
$162$		0			0
$163$			18.6300i			1.45922i	0.683866	+	0.729608i	$0.260297\pi$
$163$			18.6300i			1.45922i	−0.683866	+	0.729608i	$0.739703\pi$
$164$		0			0
$165$	−1.23475			−0.0961254
$166$		0			0
$167$		−	20.6451i		−	1.59757i	−0.601617	−	0.798785i	$-0.705477\pi$
$167$		−	20.6451i		−	1.59757i	0.601617	−	0.798785i	$-0.294523\pi$
$168$		0			0
$169$	−11.7281			−0.902158
$170$		0			0
$171$	−9.03095			−0.690614
$172$		0			0
$173$	−0.469507			−0.0356960			−0.0178480	−	0.999841i	$-0.505681\pi$
$173$	−0.469507			−0.0356960			−0.0178480	+	0.999841i	$0.505681\pi$
$174$		0			0
$175$	2.33999			0.176887
$176$		0			0
$177$	−2.95110			−0.221818
$178$		0			0
$179$		−	21.3060i		−	1.59249i	−0.604976	−	0.796243i	$-0.706817\pi$
$179$		−	21.3060i		−	1.59249i	0.604976	−	0.796243i	$-0.293183\pi$
$180$		0			0
$181$		−	0.539265i		−	0.0400833i	−0.999799	−	0.0200416i	$-0.993620\pi$
$181$		−	0.539265i		−	0.0400833i	0.999799	−	0.0200416i	$-0.00637988\pi$
$182$		0			0
$183$	−0.987910			−0.0730284
$184$		0			0
$185$	7.54531			0.554742
$186$		0			0
$187$		−	9.53147i		−	0.697010i
$188$		0			0
$189$		−	4.50964i		−	0.328028i
$190$		0			0
$191$	25.1167			1.81738			0.908689	−	0.417474i	$-0.137085\pi$
$191$	25.1167			1.81738			0.908689	+	0.417474i	$0.137085\pi$
$192$		0			0
$193$	9.12781			0.657034			0.328517	−	0.944498i	$-0.393451\pi$
$193$	9.12781			0.657034			0.328517	+	0.944498i	$0.393451\pi$
$194$		0			0
$195$	0.368826			0.0264122
$196$		0			0
$197$	−24.0045			−1.71025			−0.855126	−	0.518421i	$-0.826520\pi$
$197$	−24.0045			−1.71025			−0.855126	+	0.518421i	$0.826520\pi$
$198$		0			0
$199$	−3.52056			−0.249566			−0.124783	−	0.992184i	$-0.539823\pi$
$199$	−3.52056			−0.249566			−0.124783	+	0.992184i	$0.539823\pi$
$200$		0			0
$201$		−	2.80696i		−	0.197988i
$202$		0			0
$203$	10.3369			0.725509
$204$		0			0
$205$			4.88701i			0.341324i
$206$		0			0
$207$	3.26081	−	13.4860i	0.226642	−	0.937340i
$208$		0			0
$209$	11.7861			0.815262
$210$		0			0
$211$			24.1446i			1.66218i	0.556138	+	0.831090i	$0.312282\pi$
$211$			24.1446i			1.66218i	−0.556138	+	0.831090i	$0.687718\pi$
$212$		0			0
$213$	−3.12781			−0.214314
$214$		0			0
$215$			6.20140i			0.422932i
$216$		0			0
$217$		−	4.71030i		−	0.319756i
$218$		0			0
$219$		−	3.32809i		−	0.224891i
$220$		0			0
$221$			2.84709i			0.191516i
$222$		0			0
$223$		−	16.2942i		−	1.09114i	−0.838066	−	0.545569i	$-0.816314\pi$
$223$		−	16.2942i		−	1.09114i	0.838066	−	0.545569i	$-0.183686\pi$
$224$		0			0
$225$	−2.89305			−0.192870
$226$		0			0
$227$	1.03888			0.0689527			0.0344764	−	0.999406i	$-0.489024\pi$
$227$	1.03888			0.0689527			0.0344764	+	0.999406i	$0.489024\pi$
$228$		0			0
$229$		−	15.2765i		−	1.00950i	−0.863266	−	0.504749i	$-0.831585\pi$
$229$		−	15.2765i		−	1.00950i	0.863266	−	0.504749i	$-0.168415\pi$
$230$		0			0
$231$			2.88931i			0.190103i
$232$		0			0
$233$	−0.646214			−0.0423349			−0.0211674	−	0.999776i	$-0.506738\pi$
$233$	−0.646214			−0.0423349			−0.0211674	+	0.999776i	$0.506738\pi$
$234$		0			0
$235$	−7.22429			−0.471261
$236$		0			0
$237$		−	1.02086i		−	0.0663118i
$238$		0			0
$239$		−	6.52143i		−	0.421836i	−0.977504	−	0.210918i	$-0.932355\pi$
$239$		−	6.52143i		−	0.421836i	0.977504	−	0.210918i	$-0.0676453\pi$
$240$		0			0
$241$			8.57939i			0.552647i	0.961065	+	0.276324i	$0.0891162\pi$
$241$			8.57939i			0.552647i	−0.961065	+	0.276324i	$0.910884\pi$
$242$		0			0
$243$			8.41384i			0.539748i
$244$		0			0
$245$			1.52445i			0.0973936i
$246$		0			0
$247$	−3.52056			−0.224008
$248$		0			0
$249$		−	1.57221i		−	0.0996345i
$250$		0			0
$251$	10.0140			0.632079			0.316040	−	0.948746i	$-0.397647\pi$
$251$	10.0140			0.632079			0.316040	+	0.948746i	$0.397647\pi$
$252$		0			0
$253$	−4.25561	+	17.6003i	−0.267548	+	1.10652i
$254$		0			0
$255$			0.825571i			0.0516992i
$256$		0			0
$257$	12.2184			0.762164			0.381082	−	0.924541i	$-0.375552\pi$
$257$	12.2184			0.762164			0.381082	+	0.924541i	$0.375552\pi$
$258$		0			0
$259$		−	17.6559i		−	1.09709i
$260$		0			0
$261$	−12.7801			−0.791066
$262$		0			0
$263$	13.5882			0.837886			0.418943	−	0.908013i	$-0.362401\pi$
$263$	13.5882			0.837886			0.418943	+	0.908013i	$0.362401\pi$
$264$		0			0
$265$	−4.78006			−0.293637
$266$		0			0
$267$	−5.83940			−0.357365
$268$		0			0
$269$	−7.68030			−0.468276			−0.234138	−	0.972203i	$-0.575227\pi$
$269$	−7.68030			−0.468276			−0.234138	+	0.972203i	$0.575227\pi$
$270$		0			0
$271$			11.6678i			0.708771i	0.935099	+	0.354385i	$0.115310\pi$
$271$			11.6678i			0.708771i	−0.935099	+	0.354385i	$0.884690\pi$
$272$		0			0
$273$		−	0.863049i		−	0.0522341i
$274$		0			0
$275$	3.77566			0.227681
$276$		0			0
$277$	15.9749			0.959838			0.479919	−	0.877313i	$-0.340666\pi$
$277$	15.9749			0.959838			0.479919	+	0.877313i	$0.340666\pi$
$278$		0			0
$279$			5.82359i			0.348650i
$280$		0			0
$281$		−	22.8070i		−	1.36055i	−0.732957	−	0.680275i	$-0.761861\pi$
$281$		−	22.8070i		−	1.36055i	0.732957	−	0.680275i	$-0.238139\pi$
$282$		0			0
$283$	−10.5502			−0.627145			−0.313573	−	0.949564i	$-0.601526\pi$
$283$	−10.5502			−0.627145			−0.313573	+	0.949564i	$0.601526\pi$
$284$		0			0
$285$	−1.02086			−0.0604703
$286$		0			0
$287$	11.4355			0.675019
$288$		0			0
$289$	10.6271			0.625126
$290$		0			0
$291$	2.79655			0.163936
$292$		0			0
$293$			13.8290i			0.807897i	0.914782	+	0.403948i	$0.132362\pi$
$293$			13.8290i			0.807897i	−0.914782	+	0.403948i	$0.867638\pi$
$294$		0			0
$295$	9.02394			0.525394
$296$		0			0
$297$		−	7.27647i		−	0.422223i
$298$		0			0
$299$	1.27117	−	5.25727i	0.0735136	−	0.304036i
$300$		0			0
$301$	14.5112			0.836413
$302$		0			0
$303$		−	1.56322i		−	0.0898048i
$304$		0			0
$305$	3.02086			0.172974
$306$		0			0
$307$		−	14.4116i		−	0.822515i	−0.911519	−	0.411258i	$-0.865090\pi$
$307$		−	14.4116i		−	0.822515i	0.911519	−	0.411258i	$-0.134910\pi$
$308$		0			0
$309$			0.523305i			0.0297698i
$310$		0			0
$311$			12.9431i			0.733939i	0.930233	+	0.366969i	$0.119605\pi$
$311$			12.9431i			0.733939i	−0.930233	+	0.366969i	$0.880395\pi$
$312$		0			0
$313$			8.29847i			0.469057i	0.972109	+	0.234529i	$0.0753547\pi$
$313$			8.29847i			0.469057i	−0.972109	+	0.234529i	$0.924645\pi$
$314$		0			0
$315$			6.76971i			0.381430i
$316$		0			0
$317$	−8.57939			−0.481867			−0.240933	−	0.970542i	$-0.577454\pi$
$317$	−8.57939			−0.481867			−0.240933	+	0.970542i	$0.577454\pi$
$318$		0			0
$319$	16.6790			0.933844
$320$		0			0
$321$		−	3.44865i		−	0.192485i
$322$		0			0
$323$		−	7.88033i		−	0.438473i
$324$		0			0
$325$	−1.12781			−0.0625594
$326$		0			0
$327$	3.94717			0.218279
$328$		0			0
$329$			16.9048i			0.931990i
$330$		0			0
$331$			15.0698i			0.828313i	0.910206	+	0.414156i	$0.135923\pi$
$331$			15.0698i			0.828313i	−0.910206	+	0.414156i	$0.864077\pi$
$332$		0			0
$333$			21.8290i			1.19622i
$334$		0			0
$335$			8.58319i			0.468950i
$336$		0			0
$337$			2.80696i			0.152905i	0.997073	+	0.0764524i	$0.0243593\pi$
$337$			2.80696i			0.152905i	−0.997073	+	0.0764524i	$0.975641\pi$
$338$		0			0
$339$	2.96806			0.161203
$340$		0			0
$341$		−	7.60025i		−	0.411577i
$342$		0			0
$343$	19.9471			1.07704
$344$		0			0
$345$	0.368601	−	1.52445i	0.0198448	−	0.0820737i
$346$		0			0
$347$		−	11.3688i		−	0.610308i	−0.952303	−	0.305154i	$-0.901292\pi$
$347$		−	11.3688i		−	0.610308i	0.952303	−	0.305154i	$-0.0987079\pi$
$348$		0			0
$349$	−13.9480			−0.746619			−0.373310	−	0.927707i	$-0.621777\pi$
$349$	−13.9480			−0.746619			−0.373310	+	0.927707i	$0.621777\pi$
$350$		0			0
$351$			2.17351i			0.116013i
$352$		0			0
$353$	3.81121			0.202850			0.101425	−	0.994843i	$-0.467660\pi$
$353$	3.81121			0.202850			0.101425	+	0.994843i	$0.467660\pi$
$354$		0			0
$355$	9.56428			0.507619
$356$		0			0
$357$	1.93183			0.102243
$358$		0			0
$359$	21.3410			1.12633			0.563167	−	0.826343i	$-0.309583\pi$
$359$	21.3410			1.12633			0.563167	+	0.826343i	$0.309583\pi$
$360$		0			0
$361$	−9.25561			−0.487137
$362$		0			0
$363$			1.06468i			0.0558813i
$364$		0			0
$365$			10.1767i			0.532673i
$366$		0			0
$367$	16.7098			0.872246			0.436123	−	0.899887i	$-0.356351\pi$
$367$	16.7098			0.872246			0.436123	+	0.899887i	$0.356351\pi$
$368$		0			0
$369$	−14.1384			−0.736014
$370$		0			0
$371$			11.1853i			0.580711i
$372$		0			0
$373$		−	24.0417i		−	1.24483i	−0.782686	−	0.622416i	$-0.786151\pi$
$373$		−	24.0417i		−	1.24483i	0.782686	−	0.622416i	$-0.213849\pi$
$374$		0			0
$375$	−0.327030			−0.0168877
$376$		0			0
$377$	−4.98208			−0.256590
$378$		0			0
$379$	−33.6510			−1.72854			−0.864269	−	0.503030i	$-0.832219\pi$
$379$	−33.6510			−1.72854			−0.864269	+	0.503030i	$0.832219\pi$
$380$		0			0
$381$	0.190375			0.00975320
$382$		0			0
$383$	32.8183			1.67694			0.838469	−	0.544949i	$-0.183451\pi$
$383$	32.8183			1.67694			0.838469	+	0.544949i	$0.183451\pi$
$384$		0			0
$385$		−	8.83500i		−	0.450274i
$386$		0			0
$387$	−17.9410			−0.911991
$388$		0			0
$389$			17.3045i			0.877374i	0.898640	+	0.438687i	$0.144556\pi$
$389$			17.3045i			0.877374i	−0.898640	+	0.438687i	$0.855444\pi$
$390$		0			0
$391$	11.7677	+	2.84535i	0.595120	+	0.143896i
$392$		0			0
$393$	−2.86011			−0.144273
$394$		0			0
$395$			3.12160i			0.157065i
$396$		0			0
$397$	13.7193			0.688551			0.344276	−	0.938869i	$-0.388125\pi$
$397$	13.7193			0.688551			0.344276	+	0.938869i	$0.388125\pi$
$398$		0			0
$399$			2.38879i			0.119589i
$400$		0			0
$401$		−	14.8471i		−	0.741428i	−0.928747	−	0.370714i	$-0.879113\pi$
$401$		−	14.8471i		−	0.741428i	0.928747	−	0.370714i	$-0.120887\pi$
$402$		0			0
$403$			2.27023i			0.113088i
$404$		0			0
$405$		−	8.04890i		−	0.399953i
$406$		0			0
$407$		−	28.4885i		−	1.41212i
$408$		0			0
$409$	8.45922			0.418281			0.209141	−	0.977886i	$-0.432933\pi$
$409$	8.45922			0.418281			0.209141	+	0.977886i	$0.432933\pi$
$410$		0			0
$411$	2.69983			0.133173
$412$		0			0
$413$		−	21.1159i		−	1.03905i
$414$		0			0
$415$			4.80753i			0.235992i
$416$		0			0
$417$	1.65340			0.0809674
$418$		0			0
$419$	14.4714			0.706974			0.353487	−	0.935439i	$-0.384996\pi$
$419$	14.4714			0.706974			0.353487	+	0.935439i	$0.384996\pi$
$420$		0			0
$421$		−	11.3446i		−	0.552904i	−0.961028	−	0.276452i	$-0.910841\pi$
$421$		−	11.3446i		−	0.552904i	0.961028	−	0.276452i	$-0.0891587\pi$
$422$		0			0
$423$		−	20.9002i		−	1.01620i
$424$		0			0
$425$		−	2.52445i		−	0.122454i
$426$		0			0
$427$		−	7.06877i		−	0.342082i
$428$		0			0
$429$		−	1.39256i		−	0.0672335i
$430$		0			0
$431$	26.5312			1.27796			0.638981	−	0.769222i	$-0.279356\pi$
$431$	26.5312			1.27796			0.638981	+	0.769222i	$0.279356\pi$
$432$		0			0
$433$			12.5365i			0.602467i	0.953550	+	0.301234i	$0.0973984\pi$
$433$			12.5365i			0.602467i	−0.953550	+	0.301234i	$0.902602\pi$
$434$		0			0
$435$	−1.44466			−0.0692659
$436$		0			0
$437$	−3.51841	+	14.5513i	−0.168308	+	0.696085i
$438$		0			0
$439$		−	15.0628i		−	0.718909i	−0.933163	−	0.359455i	$-0.882963\pi$
$439$		−	15.0628i		−	0.718909i	0.933163	−	0.359455i	$-0.117037\pi$
$440$		0			0
$441$	−4.41031			−0.210015
$442$		0			0
$443$		−	7.70684i		−	0.366163i	−0.983098	−	0.183081i	$-0.941393\pi$
$443$		−	7.70684i		−	0.366163i	0.983098	−	0.183081i	$-0.0586072\pi$
$444$		0			0
$445$	17.8559			0.846449
$446$		0			0
$447$	0.426614			0.0201782
$448$		0			0
$449$	14.0846			0.664692			0.332346	−	0.943158i	$-0.392160\pi$
$449$	14.0846			0.664692			0.332346	+	0.943158i	$0.392160\pi$
$450$		0			0
$451$	18.4517			0.868856
$452$		0			0
$453$	−5.92110			−0.278197
$454$		0			0
$455$			2.63905i			0.123721i
$456$		0			0
$457$			34.0967i			1.59497i	0.603336	+	0.797487i	$0.293838\pi$
$457$			34.0967i			1.59497i	−0.603336	+	0.797487i	$0.706162\pi$
$458$		0			0
$459$	−4.86513			−0.227085
$460$		0			0
$461$	2.84415			0.132465			0.0662326	−	0.997804i	$-0.478902\pi$
$461$	2.84415			0.132465			0.0662326	+	0.997804i	$0.478902\pi$
$462$		0			0
$463$		−	9.80555i		−	0.455703i	−0.973696	−	0.227851i	$-0.926830\pi$
$463$		−	9.80555i		−	0.455703i	0.973696	−	0.227851i	$-0.0731700\pi$
$464$		0			0
$465$			0.658298i			0.0305278i
$466$		0			0
$467$	33.0506			1.52940			0.764700	−	0.644386i	$-0.222887\pi$
$467$	33.0506			1.52940			0.764700	+	0.644386i	$0.222887\pi$
$468$		0			0
$469$	20.0846			0.927419
$470$		0			0
$471$	−0.728855			−0.0335839
$472$		0			0
$473$	23.4144			1.07660
$474$		0			0
$475$	3.12160			0.143229
$476$		0			0
$477$		−	13.8290i		−	0.633185i
$478$		0			0
$479$	40.6362			1.85672			0.928358	−	0.371687i	$-0.121220\pi$
$479$	40.6362			1.85672			0.928358	+	0.371687i	$0.121220\pi$
$480$		0			0
$481$			8.50964i			0.388006i
$482$		0			0
$483$	−3.56720	−	0.862522i	−0.162313	−	0.0392461i
$484$		0			0
$485$	−8.55135			−0.388297
$486$		0			0
$487$		−	8.04986i		−	0.364774i	−0.983227	−	0.182387i	$-0.941618\pi$
$487$		−	8.04986i		−	0.364774i	0.983227	−	0.182387i	$-0.0583824\pi$
$488$		0			0
$489$	−6.09257			−0.275516
$490$		0			0
$491$			23.9293i			1.07991i	0.841693	+	0.539956i	$0.181559\pi$
$491$			23.9293i			1.07991i	−0.841693	+	0.539956i	$0.818441\pi$
$492$		0			0
$493$		−	11.1518i		−	0.502250i
$494$		0			0
$495$			10.9232i			0.490960i
$496$		0			0
$497$		−	22.3803i		−	1.00389i
$498$		0			0
$499$		−	18.1127i		−	0.810835i	−0.914132	−	0.405418i	$-0.867126\pi$
$499$		−	18.1127i		−	0.810835i	0.914132	−	0.405418i	$-0.132874\pi$
$500$		0			0
$501$	6.75158			0.301638
$502$		0			0
$503$	−23.2544			−1.03686			−0.518431	−	0.855120i	$-0.673484\pi$
$503$	−23.2544			−1.03686			−0.518431	+	0.855120i	$0.673484\pi$
$504$		0			0
$505$			4.78006i			0.212710i
$506$		0			0
$507$		−	3.83542i		−	0.170337i
$508$		0			0
$509$	20.7281			0.918755			0.459377	−	0.888241i	$-0.348073\pi$
$509$	20.7281			0.918755			0.459377	+	0.888241i	$0.348073\pi$
$510$		0			0
$511$	23.8134			1.05344
$512$		0			0
$513$		−	6.01596i		−	0.265611i
$514$		0			0
$515$		−	1.60017i		−	0.0705121i
$516$		0			0
$517$			27.2765i			1.19962i
$518$		0			0
$519$		−	0.153543i		−	0.00673979i
$520$		0			0
$521$		−	17.0489i		−	0.746926i	−0.927645	−	0.373463i	$-0.878170\pi$
$521$		−	17.0489i		−	0.746926i	0.927645	−	0.373463i	$-0.121830\pi$
$522$		0			0
$523$	−33.9343			−1.48384			−0.741922	−	0.670487i	$-0.766085\pi$
$523$	−33.9343			−1.48384			−0.741922	+	0.670487i	$0.766085\pi$
$524$		0			0
$525$			0.765246i			0.0333981i
$526$		0			0
$527$	−5.08162			−0.221359
$528$		0			0
$529$	−20.4592	−	10.5081i	−0.889531	−	0.456875i
$530$		0			0
$531$			26.1067i			1.13294i
$532$		0			0
$533$	−5.51159			−0.238734
$534$		0			0
$535$			10.5454i			0.455916i
$536$		0			0
$537$	6.96771			0.300679
$538$		0			0
$539$	5.75581			0.247920
$540$		0			0
$541$	−29.5927			−1.27229			−0.636145	−	0.771570i	$-0.719472\pi$
$541$	−29.5927			−1.27229			−0.636145	+	0.771570i	$0.719472\pi$
$542$		0			0
$543$	0.176356			0.00756815
$544$		0			0
$545$	−12.0698			−0.517012
$546$		0			0
$547$		−	26.1993i		−	1.12020i	−0.828425	−	0.560101i	$-0.810762\pi$
$547$		−	26.1993i		−	1.12020i	0.828425	−	0.560101i	$-0.189238\pi$
$548$		0			0
$549$			8.73949i			0.372992i
$550$		0			0
$551$	13.7897			0.587460
$552$		0			0
$553$	7.30451			0.310619
$554$		0			0
$555$			2.46754i			0.104741i
$556$		0			0
$557$		−	4.66859i		−	0.197815i	−0.995097	−	0.0989073i	$-0.968465\pi$
$557$		−	4.66859i		−	0.197815i	0.995097	−	0.0989073i	$-0.0315347\pi$
$558$		0			0
$559$	−6.99398			−0.295814
$560$		0			0
$561$	3.11707			0.131603
$562$		0			0
$563$	39.4557			1.66286			0.831429	−	0.555631i	$-0.187523\pi$
$563$	39.4557			1.66286			0.831429	+	0.555631i	$0.187523\pi$
$564$		0			0
$565$	−9.07580			−0.381822
$566$		0			0
$567$	−18.8343			−0.790968
$568$		0			0
$569$			10.0000i			0.419222i	0.977785	+	0.209611i	$0.0672197\pi$
$569$			10.0000i			0.419222i	−0.977785	+	0.209611i	$0.932780\pi$
$570$		0			0
$571$	−22.1714			−0.927845			−0.463922	−	0.885876i	$-0.653558\pi$
$571$	−22.1714			−0.927845			−0.463922	+	0.885876i	$0.653558\pi$
$572$		0			0
$573$			8.21390i			0.343140i
$574$		0			0
$575$	−1.12712	+	4.66150i	−0.0470040	+	0.194398i
$576$		0			0
$577$	−38.3996			−1.59860			−0.799299	−	0.600933i	$-0.794796\pi$
$577$	−38.3996			−1.59860			−0.799299	+	0.600933i	$0.794796\pi$
$578$		0			0
$579$			2.98506i			0.124055i
$580$		0			0
$581$	11.2496			0.466711
$582$		0			0
$583$			18.0479i			0.747467i
$584$		0			0
$585$		−	3.26280i		−	0.134900i
$586$		0			0
$587$		−	16.5842i		−	0.684505i	−0.939608	−	0.342253i	$-0.888810\pi$
$587$		−	16.5842i		−	0.684505i	0.939608	−	0.342253i	$-0.111190\pi$
$588$		0			0
$589$		−	6.28365i		−	0.258913i
$590$		0			0
$591$		−	7.85020i		−	0.322914i
$592$		0			0
$593$	−35.8839			−1.47358			−0.736788	−	0.676124i	$-0.763658\pi$
$593$	−35.8839			−1.47358			−0.736788	+	0.676124i	$0.763658\pi$
$594$		0			0
$595$	−5.90719			−0.242171
$596$		0			0
$597$		−	1.15133i		−	0.0471207i
$598$		0			0
$599$			10.9232i			0.446309i	0.974783	+	0.223155i	$0.0716354\pi$
$599$			10.9232i			0.446309i	−0.974783	+	0.223155i	$0.928365\pi$
$600$		0			0
$601$	7.25251			0.295836			0.147918	−	0.989000i	$-0.452743\pi$
$601$	7.25251			0.295836			0.147918	+	0.989000i	$0.452743\pi$
$602$		0			0
$603$	−24.8316			−1.01122
$604$		0			0
$605$		−	3.25561i		−	0.132359i
$606$		0			0
$607$		−	36.8844i		−	1.49709i	−0.663083	−	0.748546i	$-0.730752\pi$
$607$		−	36.8844i		−	1.49709i	0.663083	−	0.748546i	$-0.269248\pi$
$608$		0			0
$609$			3.38048i			0.136984i
$610$		0			0
$611$		−	8.14759i		−	0.329616i
$612$		0			0
$613$		−	42.2229i		−	1.70537i	−0.522427	−	0.852684i	$-0.674973\pi$
$613$		−	42.2229i		−	1.70537i	0.522427	−	0.852684i	$-0.325027\pi$
$614$		0			0
$615$	−1.59820			−0.0644456
$616$		0			0
$617$			16.7224i			0.673218i	0.941645	+	0.336609i	$0.109280\pi$
$617$			16.7224i			0.673218i	−0.941645	+	0.336609i	$0.890720\pi$
$618$		0			0
$619$	−26.6002			−1.06915			−0.534577	−	0.845120i	$-0.679529\pi$
$619$	−26.6002			−1.06915			−0.534577	+	0.845120i	$0.679529\pi$
$620$		0			0
$621$	8.98367	+	2.17218i	0.360502	+	0.0871667i
$622$		0			0
$623$		−	41.7825i		−	1.67398i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			3.85441i			0.153930i
$628$		0			0
$629$	−19.0478			−0.759484
$630$		0			0
$631$	−17.1712			−0.683576			−0.341788	−	0.939777i	$-0.611032\pi$
$631$	−17.1712			−0.683576			−0.341788	+	0.939777i	$0.611032\pi$
$632$		0			0
$633$	−7.89599			−0.313837
$634$		0			0
$635$	−0.582133			−0.0231012
$636$		0			0
$637$	−1.71928			−0.0681205
$638$		0			0
$639$			27.6700i			1.09461i
$640$		0			0
$641$			33.0626i			1.30589i	0.757404	+	0.652946i	$0.226467\pi$
$641$			33.0626i			1.30589i	−0.757404	+	0.652946i	$0.773533\pi$
$642$		0			0
$643$	13.0688			0.515385			0.257692	−	0.966227i	$-0.417038\pi$
$643$	13.0688			0.515385			0.257692	+	0.966227i	$0.417038\pi$
$644$		0			0
$645$	−2.02804			−0.0798542
$646$		0			0
$647$			1.38005i			0.0542552i	0.999632	+	0.0271276i	$0.00863604\pi$
$647$			1.38005i			0.0542552i	−0.999632	+	0.0271276i	$0.991364\pi$
$648$		0			0
$649$		−	34.0713i		−	1.33742i
$650$		0			0
$651$	1.54041			0.0603734
$652$		0			0
$653$	27.0117			1.05705			0.528525	−	0.848918i	$-0.322745\pi$
$653$	27.0117			1.05705			0.528525	+	0.848918i	$0.322745\pi$
$654$		0			0
$655$	8.74572			0.341723
$656$		0			0
$657$	−29.4417			−1.14863
$658$		0			0
$659$	−10.8955			−0.424429			−0.212215	−	0.977223i	$-0.568068\pi$
$659$	−10.8955			−0.424429			−0.212215	+	0.977223i	$0.568068\pi$
$660$		0			0
$661$			36.4786i			1.41885i	0.704780	+	0.709426i	$0.251046\pi$
$661$			36.4786i			1.41885i	−0.704780	+	0.709426i	$0.748954\pi$
$662$		0			0
$663$	−0.931083			−0.0361603
$664$		0			0
$665$		−	7.30451i		−	0.283257i
$666$		0			0
$667$	−4.97904	+	20.5922i	−0.192789	+	0.797333i
$668$		0			0
$669$	5.32868			0.206019
$670$		0			0
$671$		−	11.4057i		−	0.440313i
$672$		0			0
$673$	−30.2305			−1.16530			−0.582650	−	0.812723i	$-0.697984\pi$
$673$	−30.2305			−1.16530			−0.582650	+	0.812723i	$0.697984\pi$
$674$		0			0
$675$		−	1.92720i		−	0.0741781i
$676$		0			0
$677$		−	21.3177i		−	0.819307i	−0.912241	−	0.409654i	$-0.865650\pi$
$677$		−	21.3177i		−	0.819307i	0.912241	−	0.409654i	$-0.134350\pi$
$678$		0			0
$679$			20.0101i			0.767916i
$680$		0			0
$681$			0.339744i			0.0130190i
$682$		0			0
$683$			16.5007i			0.631380i	0.948862	+	0.315690i	$0.102236\pi$
$683$			16.5007i			0.631380i	−0.948862	+	0.315690i	$0.897764\pi$
$684$		0			0
$685$	−8.25561			−0.315431
$686$		0			0
$687$	4.99586			0.190604
$688$		0			0
$689$		−	5.39098i		−	0.205380i
$690$		0			0
$691$			45.4095i			1.72746i	0.503956	+	0.863729i	$0.331877\pi$
$691$			45.4095i			1.72746i	−0.503956	+	0.863729i	$0.668123\pi$
$692$		0			0
$693$	25.5601			0.970949
$694$		0			0
$695$	−5.05581			−0.191778
$696$		0			0
$697$		−	12.3370i		−	0.467298i
$698$		0			0
$699$		−	0.211331i		−	0.00799328i
$700$		0			0
$701$			29.7581i			1.12395i	0.827155	+	0.561973i	$0.189958\pi$
$701$			29.7581i			1.12395i	−0.827155	+	0.561973i	$0.810042\pi$
$702$		0			0
$703$		−	23.5534i		−	0.888335i
$704$		0			0
$705$		−	2.36256i		−	0.0889791i
$706$		0			0
$707$	11.1853			0.420666
$708$		0			0
$709$			26.6909i			1.00240i	0.865332	+	0.501198i	$0.167107\pi$
$709$			26.6909i			1.00240i	−0.865332	+	0.501198i	$0.832893\pi$
$710$		0			0
$711$	−9.03095			−0.338687
$712$		0			0
$713$	9.38342	+	2.26884i	0.351412	+	0.0849687i
$714$		0			0
$715$			4.25821i			0.159248i
$716$		0			0
$717$	2.13270			0.0796472
$718$		0			0
$719$			13.2311i			0.493435i	0.969087	+	0.246718i	$0.0793520\pi$
$719$			13.2311i			0.493435i	−0.969087	+	0.246718i	$0.920648\pi$
$720$		0			0
$721$	−3.74439			−0.139448
$722$		0			0
$723$	−2.80572			−0.104346
$724$		0			0
$725$	4.41750			0.164062
$726$		0			0
$727$	−18.4305			−0.683551			−0.341775	−	0.939782i	$-0.611028\pi$
$727$	−18.4305			−0.683551			−0.341775	+	0.939782i	$0.611028\pi$
$728$		0			0
$729$	21.3951			0.792412
$730$		0			0
$731$		−	15.6551i		−	0.579026i
$732$		0			0
$733$			1.90334i			0.0703016i	0.999382	+	0.0351508i	$0.0111912\pi$
$733$			1.90334i			0.0703016i	−0.999382	+	0.0351508i	$0.988809\pi$
$734$		0			0
$735$	−0.498541			−0.0183890
$736$		0			0
$737$	32.4072			1.19373
$738$		0			0
$739$		−	41.2672i		−	1.51804i	−0.651069	−	0.759019i	$-0.725679\pi$
$739$		−	41.2672i		−	1.51804i	0.651069	−	0.759019i	$-0.274321\pi$
$740$		0			0
$741$		−	1.15133i		−	0.0422951i
$742$		0			0
$743$	−26.0725			−0.956506			−0.478253	−	0.878222i	$-0.658730\pi$
$743$	−26.0725			−0.956506			−0.478253	+	0.878222i	$0.658730\pi$
$744$		0			0
$745$	−1.30451			−0.0477936
$746$		0			0
$747$	−13.9084			−0.508883
$748$		0			0
$749$	24.6761			0.901643
$750$		0			0
$751$	−40.6231			−1.48236			−0.741179	−	0.671307i	$-0.765733\pi$
$751$	−40.6231			−1.48236			−0.741179	+	0.671307i	$0.765733\pi$
$752$		0			0
$753$			3.27488i			0.119343i
$754$		0			0
$755$	18.1057			0.658933
$756$		0			0
$757$			25.0197i			0.909357i	0.890656	+	0.454678i	$0.150246\pi$
$757$			25.0197i			0.909357i	−0.890656	+	0.454678i	$0.849754\pi$
$758$		0			0
$759$	−5.75581	−	1.39171i	−0.208923	−	0.0505159i
$760$		0			0
$761$	28.9287			1.04867			0.524333	−	0.851513i	$-0.324315\pi$
$761$	28.9287			1.04867			0.524333	+	0.851513i	$0.324315\pi$
$762$		0			0
$763$			28.2431i			1.02247i
$764$		0			0
$765$	7.30337			0.264054
$766$		0			0
$767$			10.1772i			0.367479i
$768$		0			0
$769$		−	43.8960i		−	1.58293i	−0.611215	−	0.791465i	$-0.709319\pi$
$769$		−	43.8960i		−	1.58293i	0.611215	−	0.791465i	$-0.290681\pi$
$770$		0			0
$771$			3.99579i			0.143905i
$772$		0			0
$773$		−	12.7092i		−	0.457117i	−0.973530	−	0.228558i	$-0.926599\pi$
$773$		−	12.7092i		−	0.457117i	0.973530	−	0.228558i	$-0.0734012\pi$
$774$		0			0
$775$		−	2.01296i		−	0.0723076i
$776$		0			0
$777$	5.77402			0.207142
$778$		0			0
$779$	15.2553			0.546578
$780$		0			0
$781$		−	36.1115i		−	1.29217i
$782$		0			0
$783$		−	8.51343i		−	0.304245i
$784$		0			0
$785$	2.22871			0.0795461
$786$		0			0
$787$	−26.5523			−0.946488			−0.473244	−	0.880931i	$-0.656917\pi$
$787$	−26.5523			−0.946488			−0.473244	+	0.880931i	$0.656917\pi$
$788$		0			0
$789$			4.44375i			0.158202i
$790$		0			0
$791$			21.2373i			0.755111i
$792$		0			0
$793$			3.40694i			0.120984i
$794$		0			0
$795$		−	1.56322i		−	0.0554418i
$796$		0			0
$797$		−	45.9942i		−	1.62920i	−0.580024	−	0.814600i	$-0.696957\pi$
$797$		−	45.9942i		−	1.62920i	0.580024	−	0.814600i	$-0.303043\pi$
$798$		0			0
$799$	18.2374			0.645192
$800$		0			0
$801$			51.6579i			1.82524i
$802$		0			0
$803$	38.4238			1.35595
$804$		0			0
$805$	10.9079	+	2.63744i	0.384452	+	0.0929576i
$806$		0			0
$807$		−	2.51169i		−	0.0884156i
$808$		0			0
$809$	−7.30337			−0.256773			−0.128386	−	0.991724i	$-0.540980\pi$
$809$	−7.30337			−0.256773			−0.128386	+	0.991724i	$0.540980\pi$
$810$		0			0
$811$			36.0630i			1.26634i	0.774012	+	0.633171i	$0.218247\pi$
$811$			36.0630i			1.26634i	−0.774012	+	0.633171i	$0.781753\pi$
$812$		0			0
$813$	−3.81573			−0.133824
$814$		0			0
$815$	18.6300			0.652581
$816$		0			0
$817$	19.3583			0.677261
$818$		0			0
$819$	−7.63491			−0.266785
$820$		0			0
$821$	−3.51841			−0.122793			−0.0613967	−	0.998113i	$-0.519555\pi$
$821$	−3.51841			−0.122793			−0.0613967	+	0.998113i	$0.519555\pi$
$822$		0			0
$823$			7.86866i			0.274284i	0.990551	+	0.137142i	$0.0437917\pi$
$823$			7.86866i			0.274284i	−0.990551	+	0.137142i	$0.956208\pi$
$824$		0			0
$825$			1.23475i			0.0429886i
$826$		0			0
$827$	−4.40857			−0.153301			−0.0766506	−	0.997058i	$-0.524423\pi$
$827$	−4.40857			−0.153301			−0.0766506	+	0.997058i	$0.524423\pi$
$828$		0			0
$829$	−17.6271			−0.612216			−0.306108	−	0.951997i	$-0.599027\pi$
$829$	−17.6271			−0.612216			−0.306108	+	0.951997i	$0.599027\pi$
$830$		0			0
$831$			5.22427i			0.181228i
$832$		0			0
$833$		−	3.84840i		−	0.133339i
$834$		0			0
$835$	−20.6451			−0.714455
$836$		0			0
$837$	−3.87938			−0.134091
$838$		0			0
$839$	−36.3780			−1.25591			−0.627954	−	0.778250i	$-0.716108\pi$
$839$	−36.3780			−1.25591			−0.627954	+	0.778250i	$0.716108\pi$
$840$		0			0
$841$	−9.48568			−0.327092
$842$		0			0
$843$	7.45856			0.256886
$844$		0			0
$845$			11.7281i			0.403457i
$846$		0			0
$847$	−7.61809			−0.261761
$848$		0			0
$849$		−	3.45024i		−	0.118412i
$850$		0			0
$851$	35.1725	+	8.50444i	1.20570	+	0.291529i
$852$		0			0
$853$	−3.39098			−0.116105			−0.0580524	−	0.998314i	$-0.518489\pi$
$853$	−3.39098			−0.116105			−0.0580524	+	0.998314i	$0.518489\pi$
$854$		0			0
$855$			9.03095i			0.308852i
$856$		0			0
$857$	28.9677			0.989518			0.494759	−	0.869030i	$-0.335256\pi$
$857$	28.9677			0.989518			0.494759	+	0.869030i	$0.335256\pi$
$858$		0			0
$859$		−	31.3367i		−	1.06920i	−0.845107	−	0.534598i	$-0.820463\pi$
$859$		−	31.3367i		−	1.06920i	0.845107	−	0.534598i	$-0.179537\pi$
$860$		0			0
$861$			3.73977i			0.127451i
$862$		0			0
$863$		−	4.69597i		−	0.159853i	−0.996801	−	0.0799263i	$-0.974531\pi$
$863$		−	4.69597i		−	0.159853i	0.996801	−	0.0799263i	$-0.0254685\pi$
$864$		0			0
$865$			0.469507i			0.0159637i
$866$		0			0
$867$			3.47540i			0.118031i
$868$		0			0
$869$	11.7861			0.399816
$870$		0			0
$871$	−9.68017			−0.328000
$872$		0			0
$873$		−	24.7395i		−	0.837305i
$874$		0			0
$875$		−	2.33999i		−	0.0791061i
$876$		0			0
$877$	−42.2229			−1.42577			−0.712884	−	0.701282i	$-0.752611\pi$
$877$	−42.2229			−1.42577			−0.712884	+	0.701282i	$0.752611\pi$
$878$		0			0
$879$	−4.52248			−0.152540
$880$		0			0
$881$			2.00000i			0.0673817i	0.999432	+	0.0336909i	$0.0107262\pi$
$881$			2.00000i			0.0673817i	−0.999432	+	0.0336909i	$0.989274\pi$
$882$		0			0
$883$			42.5751i			1.43277i	0.697707	+	0.716383i	$0.254204\pi$
$883$			42.5751i			1.43277i	−0.697707	+	0.716383i	$0.745796\pi$
$884$		0			0
$885$			2.95110i			0.0992001i
$886$		0			0
$887$		−	6.82533i		−	0.229172i	−0.993413	−	0.114586i	$-0.963446\pi$
$887$		−	6.82533i		−	0.229172i	0.993413	−	0.114586i	$-0.0365542\pi$
$888$		0			0
$889$			1.36219i			0.0456862i
$890$		0			0
$891$	−30.3899			−1.01810
$892$		0			0
$893$			22.5513i			0.754652i
$894$		0			0
$895$	−21.3060			−0.712182
$896$		0			0
$897$	1.71928	+	0.415710i	0.0574052	+	0.0138802i
$898$		0			0
$899$		−	8.89225i		−	0.296573i
$900$		0			0
$901$	12.0670			0.402011
$902$		0			0
$903$			4.74560i			0.157924i
$904$		0			0
$905$	−0.539265			−0.0179258
$906$		0			0
$907$	−26.9561			−0.895064			−0.447532	−	0.894268i	$-0.647697\pi$
$907$	−26.9561			−0.895064			−0.447532	+	0.894268i	$0.647697\pi$
$908$		0			0
$909$	−13.8290			−0.458678
$910$		0			0
$911$	−15.8534			−0.525247			−0.262624	−	0.964898i	$-0.584588\pi$
$911$	−15.8534			−0.525247			−0.262624	+	0.964898i	$0.584588\pi$
$912$		0			0
$913$	18.1516			0.600730
$914$		0			0
$915$			0.987910i			0.0326593i
$916$		0			0
$917$		−	20.4649i		−	0.675810i
$918$		0			0
$919$	−47.0558			−1.55223			−0.776113	−	0.630593i	$-0.782812\pi$
$919$	−47.0558			−1.55223			−0.776113	+	0.630593i	$0.782812\pi$
$920$		0			0
$921$	4.71303			0.155300
$922$		0			0
$923$			10.7866i			0.355047i
$924$		0			0
$925$		−	7.54531i		−	0.248088i
$926$		0			0
$927$	4.62939			0.152049
$928$		0			0
$929$	−25.9359			−0.850930			−0.425465	−	0.904975i	$-0.639889\pi$
$929$	−25.9359			−0.850930			−0.425465	+	0.904975i	$0.639889\pi$
$930$		0			0
$931$	4.75873			0.155961
$932$		0			0
$933$	−4.23280			−0.138576
$934$		0			0
$935$	−9.53147			−0.311712
$936$		0			0
$937$			39.1604i			1.27931i	0.768661	+	0.639657i	$0.220924\pi$
$937$			39.1604i			1.27931i	−0.768661	+	0.639657i	$0.779076\pi$
$938$		0			0
$939$	−2.71385			−0.0885631
$940$		0			0
$941$			13.2870i			0.433143i	0.976267	+	0.216571i	$0.0694874\pi$
$941$			13.2870i			0.433143i	−0.976267	+	0.216571i	$0.930513\pi$
$942$		0			0
$943$	−5.50823	+	22.7808i	−0.179373	+	0.741845i
$944$		0			0
$945$	−4.50964			−0.146698
$946$		0			0
$947$		−	21.8304i		−	0.709392i	−0.934982	−	0.354696i	$-0.884584\pi$
$947$		−	21.8304i		−	0.709392i	0.934982	−	0.354696i	$-0.115416\pi$
$948$		0			0
$949$	−11.4773			−0.372570
$950$		0			0
$951$		−	2.80572i		−	0.0909816i
$952$		0			0
$953$		−	6.74929i		−	0.218631i	−0.994007	−	0.109315i	$-0.965134\pi$
$953$		−	6.74929i		−	0.218631i	0.994007	−	0.109315i	$-0.0348659\pi$
$954$		0			0
$955$		−	25.1167i		−	0.812756i
$956$		0			0
$957$			5.45453i			0.176320i
$958$		0			0
$959$			19.3180i			0.623812i
$960$		0			0
$961$	26.9480			0.869290
$962$		0			0
$963$	−30.5083			−0.983116
$964$		0			0
$965$		−	9.12781i		−	0.293834i
$966$		0			0
$967$			33.0689i			1.06342i	0.846925	+	0.531712i	$0.178451\pi$
$967$			33.0689i			1.06342i	−0.846925	+	0.531712i	$0.821549\pi$
$968$		0			0
$969$	2.57710			0.0827884
$970$		0			0
$971$	24.5505			0.787864			0.393932	−	0.919140i	$-0.371115\pi$
$971$	24.5505			0.787864			0.393932	+	0.919140i	$0.371115\pi$
$972$		0			0
$973$			11.8305i			0.379270i
$974$		0			0
$975$		−	0.368826i		−	0.0118119i
$976$		0			0
$977$			25.5870i			0.818601i	0.912400	+	0.409301i	$0.134227\pi$
$977$			25.5870i			0.818601i	−0.912400	+	0.409301i	$0.865773\pi$
$978$		0			0
$979$		−	67.4177i		−	2.15468i
$980$		0			0
$981$		−	34.9184i		−	1.11486i
$982$		0			0
$983$	−46.4924			−1.48288			−0.741439	−	0.671020i	$-0.765856\pi$
$983$	−46.4924			−1.48288			−0.741439	+	0.671020i	$0.765856\pi$
$984$		0			0
$985$			24.0045i			0.764848i
$986$		0			0
$987$	−5.52836			−0.175970
$988$		0			0
$989$	−6.98971	+	28.9079i	−0.222260	+	0.919217i
$990$		0			0
$991$		−	38.2681i		−	1.21563i	−0.794080	−	0.607813i	$-0.792047\pi$
$991$		−	38.2681i		−	1.21563i	0.794080	−	0.607813i	$-0.207953\pi$
$992$		0			0
$993$	−4.92828			−0.156394
$994$		0			0
$995$			3.52056i			0.111609i
$996$		0			0
$997$	1.46232			0.0463121			0.0231561	−	0.999732i	$-0.492629\pi$
$997$	1.46232			0.0463121			0.0231561	+	0.999732i	$0.492629\pi$
$998$		0			0
$999$	−14.5413			−0.460068

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.i.b.1471.9	yes	16
4.3	odd	2	inner	1840.2.i.b.1471.7	✓	16
23.22	odd	2	inner	1840.2.i.b.1471.10	yes	16
92.91	even	2	inner	1840.2.i.b.1471.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.i.b.1471.7	✓	16	4.3	odd	2	inner
1840.2.i.b.1471.8	yes	16	92.91	even	2	inner
1840.2.i.b.1471.9	yes	16	1.1	even	1	trivial
1840.2.i.b.1471.10	yes	16	23.22	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.327030i q^{3} -1.00000i q^{5} -2.33999 q^{7} +2.89305 q^{9} +O(q^{10})\)	\(q+0.327030i q^{3} -1.00000i q^{5} -2.33999 q^{7} +2.89305 q^{9} -3.77566 q^{11} +1.12781 q^{13} +0.327030 q^{15} +2.52445i q^{17} -3.12160 q^{19} -0.765246i q^{21} +(1.12712 - 4.66150i) q^{23} -1.00000 q^{25} +1.92720i q^{27} -4.41750 q^{29} +2.01296i q^{31} -1.23475i q^{33} +2.33999i q^{35} +7.54531i q^{37} +0.368826i q^{39} -4.88701 q^{41} -6.20140 q^{43} -2.89305i q^{45} -7.22429i q^{47} -1.52445 q^{49} -0.825571 q^{51} -4.78006i q^{53} +3.77566i q^{55} -1.02086i q^{57} +9.02394i q^{59} +3.02086i q^{61} -6.76971 q^{63} -1.12781i q^{65} -8.58319 q^{67} +(1.52445 + 0.368601i) q^{69} +9.56428i q^{71} -10.1767 q^{73} -0.327030i q^{75} +8.83500 q^{77} -3.12160 q^{79} +8.04890 q^{81} -4.80753 q^{83} +2.52445 q^{85} -1.44466i q^{87} +17.8559i q^{89} -2.63905 q^{91} -0.658298 q^{93} +3.12160i q^{95} -8.55135i q^{97} -10.9232 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 8 q^{13} - 16 q^{25} + 36 q^{29} - 44 q^{41} + 20 q^{49} - 20 q^{69} - 48 q^{73} - 72 q^{77} + 40 q^{81} - 4 q^{85} + 88 q^{93}+O(q^{100}) \) 16 * q - 16 * q^9 - 8 * q^13 - 16 * q^25 + 36 * q^29 - 44 * q^41 + 20 * q^49 - 20 * q^69 - 48 * q^73 - 72 * q^77 + 40 * q^81 - 4 * q^85 + 88 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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