LMFDB - Embedded newform 1840.3.k.b.321.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

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([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 10 x^{8} + 34 x^{7} + 346 x^{6} - 968 x^{5} + 165 x^{4} + 6972 x^{3} + 19344 x^{2} + \cdots + 225444 $ x^10 - 2x^9 - 10x^8 + 34x^7 + 346x^6 - 968x^5 + 165x^4 + 6972x^3 + 19344x^2 - 89080*x + 225444
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.8
Root			$1.32878 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.b.321.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.60299 q^{3} +2.23607i q^{5} -8.05652i q^{7} -2.22446 q^{9} +O(q^{10})$	$q+2.60299 q^{3} +2.23607i q^{5} -8.05652i q^{7} -2.22446 q^{9} -5.14870i q^{11} -6.63907 q^{13} +5.82045i q^{15} +10.7993i q^{17} +9.70153i q^{19} -20.9710i q^{21} +(-16.6172 + 15.9019i) q^{23} -5.00000 q^{25} -29.2171 q^{27} -9.52776 q^{29} +10.0640 q^{31} -13.4020i q^{33} +18.0149 q^{35} +54.9767i q^{37} -17.2814 q^{39} -47.4339 q^{41} -29.8177i q^{43} -4.97405i q^{45} -10.2009 q^{47} -15.9075 q^{49} +28.1105i q^{51} +86.4254i q^{53} +11.5128 q^{55} +25.2529i q^{57} +4.11297 q^{59} +17.2778i q^{61} +17.9214i q^{63} -14.8454i q^{65} +49.4243i q^{67} +(-43.2543 + 41.3923i) q^{69} +42.0951 q^{71} +52.2218 q^{73} -13.0149 q^{75} -41.4806 q^{77} +110.634i q^{79} -56.0316 q^{81} -138.133i q^{83} -24.1480 q^{85} -24.8006 q^{87} +163.641i q^{89} +53.4878i q^{91} +26.1964 q^{93} -21.6933 q^{95} +90.4792i q^{97} +11.4531i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 16 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 16 * q^9	$ 10 q + 2 q^{3} - 16 q^{9} - 2 q^{13} - 44 q^{23} - 50 q^{25} - 40 q^{27} - 46 q^{29} - 16 q^{31} + 60 q^{35} - 72 q^{39} - 84 q^{41} - 112 q^{47} + 50 q^{49} + 10 q^{55} + 262 q^{59} + 124 q^{69} - 236 q^{71} + 168 q^{73} - 10 q^{75} + 300 q^{77} - 258 q^{81} - 540 q^{87} + 100 q^{93} + 90 q^{95}+O(q^{100}) $ 10 * q + 2 * q^3 - 16 * q^9 - 2 * q^13 - 44 * q^23 - 50 * q^25 - 40 * q^27 - 46 * q^29 - 16 * q^31 + 60 * q^35 - 72 * q^39 - 84 * q^41 - 112 * q^47 + 50 * q^49 + 10 * q^55 + 262 * q^59 + 124 * q^69 - 236 * q^71 + 168 * q^73 - 10 * q^75 + 300 * q^77 - 258 * q^81 - 540 * q^87 + 100 * q^93 + 90 * q^95

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
$3$	2.60299			0.867662			0.433831	−	0.900994i	$-0.357161\pi$
$3$	2.60299			0.867662			0.433831	+	0.900994i	$0.357161\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	8.05652i		−	1.15093i	−0.817826	−	0.575466i	$-0.804821\pi$
$7$		−	8.05652i		−	1.15093i	0.817826	−	0.575466i	$-0.195179\pi$
$8$		0			0
$9$	−2.22446			−0.247163
$10$		0			0
$11$		−	5.14870i		−	0.468064i	−0.972229	−	0.234032i	$-0.924808\pi$
$11$		−	5.14870i		−	0.468064i	0.972229	−	0.234032i	$-0.0751920\pi$
$12$		0			0
$13$	−6.63907			−0.510698			−0.255349	−	0.966849i	$-0.582190\pi$
$13$	−6.63907			−0.510698			−0.255349	+	0.966849i	$0.582190\pi$
$14$		0			0
$15$			5.82045i			0.388030i
$16$		0			0
$17$			10.7993i			0.635254i	0.948216	+	0.317627i	$0.102886\pi$
$17$			10.7993i			0.635254i	−0.948216	+	0.317627i	$0.897114\pi$
$18$		0			0
$19$			9.70153i			0.510607i	0.966861	+	0.255303i	$0.0821754\pi$
$19$			9.70153i			0.510607i	−0.966861	+	0.255303i	$0.917825\pi$
$20$		0			0
$21$		−	20.9710i		−	0.998620i
$22$		0			0
$23$	−16.6172	+	15.9019i	−0.722486	+	0.691385i
$23$	−16.6172	+	15.9019i	−0.722486	+	0.691385i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−29.2171			−1.08212
$28$		0			0
$29$	−9.52776			−0.328544			−0.164272	−	0.986415i	$-0.552527\pi$
$29$	−9.52776			−0.328544			−0.164272	+	0.986415i	$0.552527\pi$
$30$		0			0
$31$	10.0640			0.324644			0.162322	−	0.986738i	$-0.448102\pi$
$31$	10.0640			0.324644			0.162322	+	0.986738i	$0.448102\pi$
$32$		0			0
$33$		−	13.4020i		−	0.406121i
$34$		0			0
$35$	18.0149			0.514712
$36$		0			0
$37$			54.9767i			1.48586i	0.669371	+	0.742928i	$0.266564\pi$
$37$			54.9767i			1.48586i	−0.669371	+	0.742928i	$0.733436\pi$
$38$		0			0
$39$	−17.2814			−0.443113
$40$		0			0
$41$	−47.4339			−1.15692			−0.578462	−	0.815710i	$-0.696347\pi$
$41$	−47.4339			−1.15692			−0.578462	+	0.815710i	$0.696347\pi$
$42$		0			0
$43$		−	29.8177i		−	0.693435i	−0.937970	−	0.346717i	$-0.887296\pi$
$43$		−	29.8177i		−	0.693435i	0.937970	−	0.346717i	$-0.112704\pi$
$44$		0			0
$45$		−	4.97405i		−	0.110535i
$46$		0			0
$47$	−10.2009			−0.217041			−0.108521	−	0.994094i	$-0.534611\pi$
$47$	−10.2009			−0.217041			−0.108521	+	0.994094i	$0.534611\pi$
$48$		0			0
$49$	−15.9075			−0.324644
$50$		0			0
$51$			28.1105i			0.551186i
$52$		0			0
$53$			86.4254i			1.63067i	0.578991	+	0.815334i	$0.303447\pi$
$53$			86.4254i			1.63067i	−0.578991	+	0.815334i	$0.696553\pi$
$54$		0			0
$55$	11.5128			0.209324
$56$		0			0
$57$			25.2529i			0.443034i
$58$		0			0
$59$	4.11297			0.0697114			0.0348557	−	0.999392i	$-0.488903\pi$
$59$	4.11297			0.0697114			0.0348557	+	0.999392i	$0.488903\pi$
$60$		0			0
$61$			17.2778i			0.283242i	0.989921	+	0.141621i	$0.0452315\pi$
$61$			17.2778i			0.283242i	−0.989921	+	0.141621i	$0.954769\pi$
$62$		0			0
$63$			17.9214i			0.284467i
$64$		0			0
$65$		−	14.8454i		−	0.228391i
$66$		0			0
$67$			49.4243i			0.737676i	0.929494	+	0.368838i	$0.120244\pi$
$67$			49.4243i			0.737676i	−0.929494	+	0.368838i	$0.879756\pi$
$68$		0			0
$69$	−43.2543	+	41.3923i	−0.626874	+	0.599889i
$70$		0			0
$71$	42.0951			0.592889			0.296444	−	0.955050i	$-0.404199\pi$
$71$	42.0951			0.592889			0.296444	+	0.955050i	$0.404199\pi$
$72$		0			0
$73$	52.2218			0.715367			0.357683	−	0.933843i	$-0.383567\pi$
$73$	52.2218			0.715367			0.357683	+	0.933843i	$0.383567\pi$
$74$		0			0
$75$	−13.0149			−0.173532
$76$		0			0
$77$	−41.4806			−0.538709
$78$		0			0
$79$			110.634i			1.40043i	0.713930	+	0.700217i	$0.246914\pi$
$79$			110.634i			1.40043i	−0.713930	+	0.700217i	$0.753086\pi$
$80$		0			0
$81$	−56.0316			−0.691748
$82$		0			0
$83$		−	138.133i		−	1.66426i	−0.554584	−	0.832128i	$-0.687123\pi$
$83$		−	138.133i		−	1.66426i	0.554584	−	0.832128i	$-0.312877\pi$
$84$		0			0
$85$	−24.1480			−0.284094
$86$		0			0
$87$	−24.8006			−0.285065
$88$		0			0
$89$			163.641i			1.83866i	0.393489	+	0.919329i	$0.371268\pi$
$89$			163.641i			1.83866i	−0.393489	+	0.919329i	$0.628732\pi$
$90$		0			0
$91$			53.4878i			0.587778i
$92$		0			0
$93$	26.1964			0.281681
$94$		0			0
$95$	−21.6933			−0.228350
$96$		0			0
$97$			90.4792i			0.932775i	0.884581	+	0.466388i	$0.154445\pi$
$97$			90.4792i			0.932775i	−0.884581	+	0.466388i	$0.845555\pi$
$98$		0			0
$99$			11.4531i			0.115688i
$100$		0			0
$101$	−136.457			−1.35105			−0.675527	−	0.737335i	$-0.736084\pi$
$101$	−136.457			−1.35105			−0.675527	+	0.737335i	$0.736084\pi$
$102$		0			0
$103$			51.1083i			0.496197i	0.968735	+	0.248098i	$0.0798056\pi$
$103$			51.1083i			0.496197i	−0.968735	+	0.248098i	$0.920194\pi$
$104$		0			0
$105$	46.8926			0.446596
$106$		0			0
$107$			5.13765i			0.0480154i	0.999712	+	0.0240077i	$0.00764262\pi$
$107$			5.13765i			0.0480154i	−0.999712	+	0.0240077i	$0.992357\pi$
$108$		0			0
$109$		−	169.516i		−	1.55519i	−0.628765	−	0.777595i	$-0.716439\pi$
$109$		−	169.516i		−	1.55519i	0.628765	−	0.777595i	$-0.283561\pi$
$110$		0			0
$111$			143.104i			1.28922i
$112$		0			0
$113$		−	28.2160i		−	0.249699i	−0.992176	−	0.124849i	$-0.960155\pi$
$113$		−	28.2160i		−	0.249699i	0.992176	−	0.124849i	$-0.0398448\pi$
$114$		0			0
$115$	−35.5576	−	37.1572i	−0.309197	−	0.323106i
$116$		0			0
$117$	14.7684			0.126225
$118$		0			0
$119$	87.0050			0.731134
$120$		0			0
$121$	94.4909			0.780916
$122$		0			0
$123$	−123.470			−1.00382
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−238.395			−1.87713			−0.938563	−	0.345109i	$-0.887842\pi$
$127$	−238.395			−1.87713			−0.938563	+	0.345109i	$0.887842\pi$
$128$		0			0
$129$		−	77.6150i		−	0.601667i
$130$		0			0
$131$	−143.887			−1.09838			−0.549188	−	0.835699i	$-0.685063\pi$
$131$	−143.887			−1.09838			−0.549188	+	0.835699i	$0.685063\pi$
$132$		0			0
$133$	78.1606			0.587674
$134$		0			0
$135$		−	65.3315i		−	0.483937i
$136$		0			0
$137$		−	128.956i		−	0.941282i	−0.882325	−	0.470641i	$-0.844023\pi$
$137$		−	128.956i		−	0.941282i	0.882325	−	0.470641i	$-0.155977\pi$
$138$		0			0
$139$	133.982			0.963899			0.481949	−	0.876199i	$-0.339929\pi$
$139$	133.982			0.963899			0.481949	+	0.876199i	$0.339929\pi$
$140$		0			0
$141$	−26.5529			−0.188318
$142$		0			0
$143$			34.1826i			0.239039i
$144$		0			0
$145$		−	21.3047i		−	0.146929i
$146$		0			0
$147$	−41.4071			−0.281681
$148$		0			0
$149$		−	69.7376i		−	0.468038i	−0.972232	−	0.234019i	$-0.924812\pi$
$149$		−	69.7376i		−	0.468038i	0.972232	−	0.234019i	$-0.0751878\pi$
$150$		0			0
$151$	−276.903			−1.83379			−0.916897	−	0.399123i	$-0.869314\pi$
$151$	−276.903			−1.83379			−0.916897	+	0.399123i	$0.869314\pi$
$152$		0			0
$153$		−	24.0227i		−	0.157011i
$154$		0			0
$155$			22.5037i			0.145185i
$156$		0			0
$157$		−	164.782i		−	1.04957i	−0.851236	−	0.524783i	$-0.824146\pi$
$157$		−	164.782i		−	1.04957i	0.851236	−	0.524783i	$-0.175854\pi$
$158$		0			0
$159$			224.964i			1.41487i
$160$		0			0
$161$	128.114	+	133.877i	0.795737	+	0.831532i
$162$		0			0
$163$	40.7616			0.250071			0.125036	−	0.992152i	$-0.460095\pi$
$163$	40.7616			0.250071			0.125036	+	0.992152i	$0.460095\pi$
$164$		0			0
$165$	29.9678			0.181623
$166$		0			0
$167$	132.027			0.790578			0.395289	−	0.918557i	$-0.370644\pi$
$167$	132.027			0.790578			0.395289	+	0.918557i	$0.370644\pi$
$168$		0			0
$169$	−124.923			−0.739188
$170$		0			0
$171$		−	21.5807i		−	0.126203i
$172$		0			0
$173$	−262.356			−1.51651			−0.758255	−	0.651958i	$-0.773948\pi$
$173$	−262.356			−1.51651			−0.758255	+	0.651958i	$0.773948\pi$
$174$		0			0
$175$			40.2826i			0.230186i
$176$		0			0
$177$	10.7060			0.0604859
$178$		0			0
$179$	−50.6552			−0.282990			−0.141495	−	0.989939i	$-0.545191\pi$
$179$	−50.6552			−0.282990			−0.141495	+	0.989939i	$0.545191\pi$
$180$		0			0
$181$			255.326i			1.41064i	0.708890	+	0.705319i	$0.249196\pi$
$181$			255.326i			1.41064i	−0.708890	+	0.705319i	$0.750804\pi$
$182$		0			0
$183$			44.9738i			0.245758i
$184$		0			0
$185$	−122.932			−0.664495
$186$		0			0
$187$	55.6025			0.297339
$188$		0			0
$189$			235.388i			1.24544i
$190$		0			0
$191$		−	370.197i		−	1.93820i	−0.246665	−	0.969101i	$-0.579335\pi$
$191$		−	370.197i		−	1.93820i	0.246665	−	0.969101i	$-0.420665\pi$
$192$		0			0
$193$	171.758			0.889940			0.444970	−	0.895545i	$-0.353214\pi$
$193$	171.758			0.889940			0.444970	+	0.895545i	$0.353214\pi$
$194$		0			0
$195$		−	38.6424i		−	0.198166i
$196$		0			0
$197$	−265.313			−1.34677			−0.673384	−	0.739293i	$-0.735160\pi$
$197$	−265.313			−1.34677			−0.673384	+	0.739293i	$0.735160\pi$
$198$		0			0
$199$			289.040i			1.45246i	0.687451	+	0.726231i	$0.258730\pi$
$199$			289.040i			1.45246i	−0.687451	+	0.726231i	$0.741270\pi$
$200$		0			0
$201$			128.651i			0.640054i
$202$		0			0
$203$			76.7606i			0.378131i
$204$		0			0
$205$		−	106.065i		−	0.517392i
$206$		0			0
$207$	36.9643	−	35.3731i	0.178572	−	0.170885i
$208$		0			0
$209$	49.9503			0.238997
$210$		0			0
$211$	−235.268			−1.11501			−0.557507	−	0.830172i	$-0.688242\pi$
$211$	−235.268			−1.11501			−0.557507	+	0.830172i	$0.688242\pi$
$212$		0			0
$213$	109.573			0.514427
$214$		0			0
$215$	66.6744			0.310113
$216$		0			0
$217$		−	81.0805i		−	0.373643i
$218$		0			0
$219$	135.933			0.620697
$220$		0			0
$221$		−	71.6975i		−	0.324423i
$222$		0			0
$223$	221.630			0.993855			0.496928	−	0.867792i	$-0.334461\pi$
$223$	221.630			0.993855			0.496928	+	0.867792i	$0.334461\pi$
$224$		0			0
$225$	11.1223			0.0494325
$226$		0			0
$227$		−	223.695i		−	0.985441i	−0.870188	−	0.492720i	$-0.836003\pi$
$227$		−	223.695i		−	0.985441i	0.870188	−	0.492720i	$-0.163997\pi$
$228$		0			0
$229$			170.011i			0.742406i	0.928552	+	0.371203i	$0.121055\pi$
$229$			170.011i			0.742406i	−0.928552	+	0.371203i	$0.878945\pi$
$230$		0			0
$231$	−107.973			−0.467418
$232$		0			0
$233$	29.9648			0.128604			0.0643020	−	0.997930i	$-0.479518\pi$
$233$	29.9648			0.128604			0.0643020	+	0.997930i	$0.479518\pi$
$234$		0			0
$235$		−	22.8100i		−	0.0970638i
$236$		0			0
$237$			287.980i			1.21510i
$238$		0			0
$239$	−181.416			−0.759061			−0.379530	−	0.925179i	$-0.623914\pi$
$239$	−181.416			−0.759061			−0.379530	+	0.925179i	$0.623914\pi$
$240$		0			0
$241$		−	156.298i		−	0.648538i	−0.945965	−	0.324269i	$-0.894882\pi$
$241$		−	156.298i		−	0.648538i	0.945965	−	0.324269i	$-0.105118\pi$
$242$		0			0
$243$	117.105			0.481912
$244$		0			0
$245$		−	35.5703i		−	0.145185i
$246$		0			0
$247$		−	64.4092i		−	0.260766i
$248$		0			0
$249$		−	359.559i		−	1.44401i
$250$		0			0
$251$		−	250.907i		−	0.999628i	−0.866133	−	0.499814i	$-0.833402\pi$
$251$		−	250.907i		−	0.999628i	0.866133	−	0.499814i	$-0.166598\pi$
$252$		0			0
$253$	81.8739	+	85.5569i	0.323612	+	0.338170i
$254$		0			0
$255$	−62.8569			−0.246498
$256$		0			0
$257$	−134.881			−0.524831			−0.262415	−	0.964955i	$-0.584519\pi$
$257$	−134.881			−0.524831			−0.262415	+	0.964955i	$0.584519\pi$
$258$		0			0
$259$	442.921			1.71012
$260$		0			0
$261$	21.1942			0.0812037
$262$		0			0
$263$			191.244i			0.727164i	0.931562	+	0.363582i	$0.118446\pi$
$263$			191.244i			0.727164i	−0.931562	+	0.363582i	$0.881554\pi$
$264$		0			0
$265$	−193.253			−0.729257
$266$		0			0
$267$			425.954i			1.59533i
$268$		0			0
$269$	211.957			0.787946			0.393973	−	0.919122i	$-0.371100\pi$
$269$	211.957			0.787946			0.393973	+	0.919122i	$0.371100\pi$
$270$		0			0
$271$	−41.3602			−0.152621			−0.0763104	−	0.997084i	$-0.524314\pi$
$271$	−41.3602			−0.152621			−0.0763104	+	0.997084i	$0.524314\pi$
$272$		0			0
$273$			139.228i			0.509993i
$274$		0			0
$275$			25.7435i			0.0936128i
$276$		0			0
$277$	209.778			0.757323			0.378662	−	0.925535i	$-0.376384\pi$
$277$	209.778			0.757323			0.378662	+	0.925535i	$0.376384\pi$
$278$		0			0
$279$	−22.3869			−0.0802399
$280$		0			0
$281$		−	132.745i		−	0.472402i	−0.971704	−	0.236201i	$-0.924098\pi$
$281$		−	132.745i		−	0.472402i	0.971704	−	0.236201i	$-0.0759023\pi$
$282$		0			0
$283$		−	218.496i		−	0.772071i	−0.922484	−	0.386036i	$-0.873844\pi$
$283$		−	218.496i		−	0.772071i	0.922484	−	0.386036i	$-0.126156\pi$
$284$		0			0
$285$	−56.4673			−0.198131
$286$		0			0
$287$			382.152i			1.33154i
$288$		0			0
$289$	172.375			0.596452
$290$		0			0
$291$			235.516i			0.809333i
$292$		0			0
$293$			257.693i			0.879497i	0.898121	+	0.439748i	$0.144932\pi$
$293$			257.693i			0.879497i	−0.898121	+	0.439748i	$0.855068\pi$
$294$		0			0
$295$			9.19688i			0.0311759i
$296$		0			0
$297$			150.430i			0.506499i
$298$		0			0
$299$	110.323	−	105.574i	0.368972	−	0.353089i
$300$		0			0
$301$	−240.227			−0.798096
$302$		0			0
$303$	−355.194			−1.17226
$304$		0			0
$305$	−38.6343			−0.126670
$306$		0			0
$307$	12.9089			0.0420485			0.0210243	−	0.999779i	$-0.493307\pi$
$307$	12.9089			0.0420485			0.0210243	+	0.999779i	$0.493307\pi$
$308$		0			0
$309$			133.034i			0.430531i
$310$		0			0
$311$	−212.171			−0.682221			−0.341110	−	0.940023i	$-0.610803\pi$
$311$	−212.171			−0.682221			−0.341110	+	0.940023i	$0.610803\pi$
$312$		0			0
$313$		−	12.5955i		−	0.0402414i	−0.999798	−	0.0201207i	$-0.993595\pi$
$313$		−	12.5955i		−	0.0402414i	0.999798	−	0.0201207i	$-0.00640504\pi$
$314$		0			0
$315$	−40.0736			−0.127218
$316$		0			0
$317$	−313.064			−0.987584			−0.493792	−	0.869580i	$-0.664390\pi$
$317$	−313.064			−0.987584			−0.493792	+	0.869580i	$0.664390\pi$
$318$		0			0
$319$			49.0556i			0.153779i
$320$		0			0
$321$			13.3732i			0.0416611i
$322$		0			0
$323$	−104.770			−0.324365
$324$		0			0
$325$	33.1954			0.102140
$326$		0			0
$327$		−	441.247i		−	1.34938i
$328$		0			0
$329$			82.1841i			0.249800i
$330$		0			0
$331$	382.892			1.15677			0.578386	−	0.815763i	$-0.303683\pi$
$331$	382.892			1.15677			0.578386	+	0.815763i	$0.303683\pi$
$332$		0			0
$333$		−	122.294i		−	0.367248i
$334$		0			0
$335$	−110.516			−0.329899
$336$		0			0
$337$			61.1020i			0.181311i	0.995882	+	0.0906557i	$0.0288963\pi$
$337$			61.1020i			0.181311i	−0.995882	+	0.0906557i	$0.971104\pi$
$338$		0			0
$339$		−	73.4458i		−	0.216654i
$340$		0			0
$341$		−	51.8163i		−	0.151954i
$342$		0			0
$343$		−	266.610i		−	0.777289i
$344$		0			0
$345$	−92.5560	−	96.7196i	−0.268278	−	0.280347i
$346$		0			0
$347$	−287.399			−0.828238			−0.414119	−	0.910223i	$-0.635910\pi$
$347$	−287.399			−0.828238			−0.414119	+	0.910223i	$0.635910\pi$
$348$		0			0
$349$	−660.469			−1.89246			−0.946231	−	0.323492i	$-0.895143\pi$
$349$	−660.469			−1.89246			−0.946231	+	0.323492i	$0.895143\pi$
$350$		0			0
$351$	193.975			0.552634
$352$		0			0
$353$	386.196			1.09404			0.547020	−	0.837119i	$-0.315762\pi$
$353$	386.196			1.09404			0.547020	+	0.837119i	$0.315762\pi$
$354$		0			0
$355$			94.1275i			0.265148i
$356$		0			0
$357$	226.473			0.634377
$358$		0			0
$359$			91.2104i			0.254068i	0.991898	+	0.127034i	$0.0405457\pi$
$359$			91.2104i			0.254068i	−0.991898	+	0.127034i	$0.959454\pi$
$360$		0			0
$361$	266.880			0.739281
$362$		0			0
$363$	245.958			0.677571
$364$		0			0
$365$			116.771i			0.319922i
$366$		0			0
$367$		−	336.449i		−	0.916755i	−0.888758	−	0.458378i	$-0.848431\pi$
$367$		−	336.449i		−	0.916755i	0.888758	−	0.458378i	$-0.151569\pi$
$368$		0			0
$369$	105.515			0.285948
$370$		0			0
$371$	696.288			1.87679
$372$		0			0
$373$			706.033i			1.89285i	0.322926	+	0.946424i	$0.395334\pi$
$373$			706.033i			1.89285i	−0.322926	+	0.946424i	$0.604666\pi$
$374$		0			0
$375$		−	29.1023i		−	0.0776060i
$376$		0			0
$377$	63.2555			0.167786
$378$		0			0
$379$			293.798i			0.775193i	0.921829	+	0.387597i	$0.126695\pi$
$379$			293.798i			0.775193i	−0.921829	+	0.387597i	$0.873305\pi$
$380$		0			0
$381$	−620.539			−1.62871
$382$		0			0
$383$			405.545i			1.05886i	0.848352	+	0.529432i	$0.177595\pi$
$383$			405.545i			1.05886i	−0.848352	+	0.529432i	$0.822405\pi$
$384$		0			0
$385$		−	92.7535i		−	0.240918i
$386$		0			0
$387$			66.3284i			0.171391i
$388$		0			0
$389$			671.711i			1.72676i	0.504551	+	0.863382i	$0.331658\pi$
$389$			671.711i			1.72676i	−0.504551	+	0.863382i	$0.668342\pi$
$390$		0			0
$391$	−171.729	−	179.454i	−0.439205	−	0.458962i
$392$		0			0
$393$	−374.537			−0.953020
$394$		0			0
$395$	−247.386			−0.626293
$396$		0			0
$397$	461.838			1.16332			0.581661	−	0.813432i	$-0.302403\pi$
$397$	461.838			1.16332			0.581661	+	0.813432i	$0.302403\pi$
$398$		0			0
$399$	203.451			0.509902
$400$		0			0
$401$		−	422.085i		−	1.05258i	−0.850305	−	0.526290i	$-0.823583\pi$
$401$		−	422.085i		−	1.05258i	0.850305	−	0.526290i	$-0.176417\pi$
$402$		0			0
$403$	−66.8154			−0.165795
$404$		0			0
$405$		−	125.290i		−	0.309359i
$406$		0			0
$407$	283.059			0.695475
$408$		0			0
$409$	227.299			0.555743			0.277872	−	0.960618i	$-0.410371\pi$
$409$	227.299			0.555743			0.277872	+	0.960618i	$0.410371\pi$
$410$		0			0
$411$		−	335.670i		−	0.816715i
$412$		0			0
$413$		−	33.1362i		−	0.0802330i
$414$		0			0
$415$	308.875			0.744278
$416$		0			0
$417$	348.753			0.836338
$418$		0			0
$419$		−	135.807i		−	0.324121i	−0.986781	−	0.162060i	$-0.948186\pi$
$419$		−	135.807i		−	0.324121i	0.986781	−	0.162060i	$-0.0518139\pi$
$420$		0			0
$421$		−	600.720i		−	1.42689i	−0.700713	−	0.713444i	$-0.747134\pi$
$421$		−	600.720i		−	1.42689i	0.700713	−	0.713444i	$-0.252866\pi$
$422$		0			0
$423$	22.6916			0.0536445
$424$		0			0
$425$		−	53.9966i		−	0.127051i
$426$		0			0
$427$	139.199			0.325992
$428$		0			0
$429$			88.9768i			0.207405i
$430$		0			0
$431$			375.319i			0.870810i	0.900235	+	0.435405i	$0.143395\pi$
$431$			375.319i			0.870810i	−0.900235	+	0.435405i	$0.856605\pi$
$432$		0			0
$433$			33.4009i			0.0771382i	0.999256	+	0.0385691i	$0.0122800\pi$
$433$			33.4009i			0.0771382i	−0.999256	+	0.0385691i	$0.987720\pi$
$434$		0			0
$435$		−	55.4559i		−	0.127485i
$436$		0			0
$437$	−154.272	−	161.212i	−0.353026	−	0.368906i
$438$		0			0
$439$	−600.923			−1.36885			−0.684423	−	0.729085i	$-0.739946\pi$
$439$	−600.923			−1.36885			−0.684423	+	0.729085i	$0.739946\pi$
$440$		0			0
$441$	35.3857			0.0802398
$442$		0			0
$443$	−666.130			−1.50368			−0.751839	−	0.659347i	$-0.770833\pi$
$443$	−666.130			−1.50368			−0.751839	+	0.659347i	$0.770833\pi$
$444$		0			0
$445$	−365.912			−0.822273
$446$		0			0
$447$		−	181.526i		−	0.406099i
$448$		0			0
$449$	194.665			0.433551			0.216776	−	0.976221i	$-0.430446\pi$
$449$	194.665			0.433551			0.216776	+	0.976221i	$0.430446\pi$
$450$		0			0
$451$			244.223i			0.541514i
$452$		0			0
$453$	−720.775			−1.59111
$454$		0			0
$455$	−119.602			−0.262862
$456$		0			0
$457$		−	271.646i		−	0.594411i	−0.954814	−	0.297206i	$-0.903945\pi$
$457$		−	271.646i		−	0.594411i	0.954814	−	0.297206i	$-0.0960547\pi$
$458$		0			0
$459$		−	315.525i		−	0.687418i
$460$		0			0
$461$	−325.220			−0.705466			−0.352733	−	0.935724i	$-0.614748\pi$
$461$	−325.220			−0.705466			−0.352733	+	0.935724i	$0.614748\pi$
$462$		0			0
$463$	−403.699			−0.871920			−0.435960	−	0.899966i	$-0.643591\pi$
$463$	−403.699			−0.871920			−0.435960	+	0.899966i	$0.643591\pi$
$464$		0			0
$465$			58.5768i			0.125972i
$466$		0			0
$467$			284.878i			0.610017i	0.952350	+	0.305009i	$0.0986594\pi$
$467$			284.878i			0.610017i	−0.952350	+	0.305009i	$0.901341\pi$
$468$		0			0
$469$	398.188			0.849015
$470$		0			0
$471$		−	428.925i		−	0.910669i
$472$		0			0
$473$	−153.522			−0.324572
$474$		0			0
$475$		−	48.5076i		−	0.102121i
$476$		0			0
$477$		−	192.250i		−	0.403040i
$478$		0			0
$479$		−	296.818i		−	0.619662i	−0.950792	−	0.309831i	$-0.899727\pi$
$479$		−	296.818i		−	0.619662i	0.950792	−	0.309831i	$-0.100273\pi$
$480$		0			0
$481$		−	364.994i		−	0.758824i
$482$		0			0
$483$	333.478	+	348.479i	0.690431	+	0.721489i
$484$		0			0
$485$	−202.318			−0.417150
$486$		0			0
$487$	498.378			1.02336			0.511682	−	0.859175i	$-0.329023\pi$
$487$	498.378			1.02336			0.511682	+	0.859175i	$0.329023\pi$
$488$		0			0
$489$	106.102			0.216977
$490$		0			0
$491$	−756.668			−1.54108			−0.770538	−	0.637394i	$-0.780012\pi$
$491$	−756.668			−1.54108			−0.770538	+	0.637394i	$0.780012\pi$
$492$		0			0
$493$		−	102.893i		−	0.208709i
$494$		0			0
$495$	−25.6099			−0.0517372
$496$		0			0
$497$		−	339.140i		−	0.682374i
$498$		0			0
$499$	337.955			0.677264			0.338632	−	0.940919i	$-0.390036\pi$
$499$	337.955			0.677264			0.338632	+	0.940919i	$0.390036\pi$
$500$		0			0
$501$	343.663			0.685955
$502$		0			0
$503$		−	727.880i		−	1.44708i	−0.690283	−	0.723539i	$-0.742514\pi$
$503$		−	727.880i		−	1.44708i	0.690283	−	0.723539i	$-0.257486\pi$
$504$		0			0
$505$		−	305.126i		−	0.604210i
$506$		0			0
$507$	−325.172			−0.641365
$508$		0			0
$509$	811.883			1.59505			0.797527	−	0.603283i	$-0.206141\pi$
$509$	811.883			1.59505			0.797527	+	0.603283i	$0.206141\pi$
$510$		0			0
$511$		−	420.726i		−	0.823338i
$512$		0			0
$513$		−	283.451i		−	0.552536i
$514$		0			0
$515$	−114.282			−0.221906
$516$		0			0
$517$			52.5216i			0.101589i
$518$		0			0
$519$	−682.910			−1.31582
$520$		0			0
$521$		−	410.451i		−	0.787813i	−0.919150	−	0.393907i	$-0.871123\pi$
$521$		−	410.451i		−	0.787813i	0.919150	−	0.393907i	$-0.128877\pi$
$522$		0			0
$523$			1011.33i			1.93372i	0.255316	+	0.966858i	$0.417821\pi$
$523$			1011.33i			1.93372i	−0.255316	+	0.966858i	$0.582179\pi$
$524$		0			0
$525$			104.855i			0.199724i
$526$		0			0
$527$			108.684i			0.206231i
$528$		0			0
$529$	23.2618	−	528.488i	0.0439731	−	0.999033i
$530$		0			0
$531$	−9.14916			−0.0172300
$532$		0			0
$533$	314.917			0.590838
$534$		0			0
$535$	−11.4881			−0.0214731
$536$		0			0
$537$	−131.855			−0.245540
$538$		0			0
$539$			81.9032i			0.151954i
$540$		0			0
$541$	618.852			1.14390			0.571952	−	0.820287i	$-0.306186\pi$
$541$	618.852			1.14390			0.571952	+	0.820287i	$0.306186\pi$
$542$		0			0
$543$			664.609i			1.22396i
$544$		0			0
$545$	379.049			0.695503
$546$		0			0
$547$	423.226			0.773723			0.386861	−	0.922138i	$-0.373559\pi$
$547$	423.226			0.773723			0.386861	+	0.922138i	$0.373559\pi$
$548$		0			0
$549$		−	38.4338i		−	0.0700069i
$550$		0			0
$551$		−	92.4339i		−	0.167757i
$552$		0			0
$553$	891.328			1.61180
$554$		0			0
$555$	−319.989			−0.576557
$556$		0			0
$557$			461.268i			0.828130i	0.910247	+	0.414065i	$0.135891\pi$
$557$			461.268i			0.828130i	−0.910247	+	0.414065i	$0.864109\pi$
$558$		0			0
$559$			197.962i			0.354136i
$560$		0			0
$561$	144.732			0.257990
$562$		0			0
$563$		−	510.695i		−	0.907096i	−0.891232	−	0.453548i	$-0.850158\pi$
$563$		−	510.695i		−	0.907096i	0.891232	−	0.453548i	$-0.149842\pi$
$564$		0			0
$565$	63.0928			0.111669
$566$		0			0
$567$			451.420i			0.796155i
$568$		0			0
$569$		−	186.632i		−	0.328001i	−0.986460	−	0.164000i	$-0.947560\pi$
$569$		−	186.632i		−	0.328001i	0.986460	−	0.164000i	$-0.0524398\pi$
$570$		0			0
$571$		−	730.831i		−	1.27991i	−0.768411	−	0.639957i	$-0.778952\pi$
$571$		−	730.831i		−	1.27991i	0.768411	−	0.639957i	$-0.221048\pi$
$572$		0			0
$573$		−	963.616i		−	1.68170i
$574$		0			0
$575$	83.0859	−	79.5093i	0.144497	−	0.138277i
$576$		0			0
$577$	920.204			1.59481			0.797404	−	0.603446i	$-0.206206\pi$
$577$	920.204			1.59481			0.797404	+	0.603446i	$0.206206\pi$
$578$		0			0
$579$	447.085			0.772167
$580$		0			0
$581$	−1112.87			−1.91544
$582$		0			0
$583$	444.979			0.763257
$584$		0			0
$585$			33.0231i			0.0564497i
$586$		0			0
$587$	50.4562			0.0859561			0.0429781	−	0.999076i	$-0.486315\pi$
$587$	50.4562			0.0859561			0.0429781	+	0.999076i	$0.486315\pi$
$588$		0			0
$589$			97.6358i			0.165765i
$590$		0			0
$591$	−690.607			−1.16854
$592$		0			0
$593$	−192.419			−0.324485			−0.162242	−	0.986751i	$-0.551873\pi$
$593$	−192.419			−0.324485			−0.162242	+	0.986751i	$0.551873\pi$
$594$		0			0
$595$			194.549i			0.326973i
$596$		0			0
$597$			752.367i			1.26025i
$598$		0			0
$599$	769.203			1.28415			0.642073	−	0.766644i	$-0.278075\pi$
$599$	769.203			1.28415			0.642073	+	0.766644i	$0.278075\pi$
$600$		0			0
$601$	−965.858			−1.60709			−0.803543	−	0.595247i	$-0.797054\pi$
$601$	−965.858			−1.60709			−0.803543	+	0.595247i	$0.797054\pi$
$602$		0			0
$603$		−	109.943i		−	0.182326i
$604$		0			0
$605$			211.288i			0.349236i
$606$		0			0
$607$	609.221			1.00366			0.501830	−	0.864966i	$-0.332660\pi$
$607$	609.221			1.00366			0.501830	+	0.864966i	$0.332660\pi$
$608$		0			0
$609$			199.807i			0.328090i
$610$		0			0
$611$	67.7248			0.110842
$612$		0			0
$613$			686.597i			1.12006i	0.828472	+	0.560030i	$0.189211\pi$
$613$			686.597i			1.12006i	−0.828472	+	0.560030i	$0.810789\pi$
$614$		0			0
$615$		−	276.087i		−	0.448921i
$616$		0			0
$617$		−	53.0755i		−	0.0860219i	−0.999075	−	0.0430109i	$-0.986305\pi$
$617$		−	53.0755i		−	0.0860219i	0.999075	−	0.0430109i	$-0.0136950\pi$
$618$		0			0
$619$			263.514i			0.425709i	0.977084	+	0.212855i	$0.0682761\pi$
$619$			263.514i			0.425709i	−0.977084	+	0.212855i	$0.931724\pi$
$620$		0			0
$621$	485.506	−	464.607i	0.781814	−	0.748159i
$622$		0			0
$623$	1318.37			2.11617
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	130.020			0.207368
$628$		0			0
$629$	−593.711			−0.943896
$630$		0			0
$631$			124.292i			0.196976i	0.995138	+	0.0984879i	$0.0314006\pi$
$631$			124.292i			0.196976i	−0.995138	+	0.0984879i	$0.968599\pi$
$632$		0			0
$633$	−612.399			−0.967455
$634$		0			0
$635$		−	533.067i		−	0.839476i
$636$		0			0
$637$	105.611			0.165795
$638$		0			0
$639$	−93.6390			−0.146540
$640$		0			0
$641$			603.832i			0.942015i	0.882129	+	0.471008i	$0.156110\pi$
$641$			603.832i			0.942015i	−0.882129	+	0.471008i	$0.843890\pi$
$642$		0			0
$643$		−	1112.73i		−	1.73053i	−0.501319	−	0.865263i	$-0.667152\pi$
$643$		−	1112.73i		−	1.73053i	0.501319	−	0.865263i	$-0.332848\pi$
$644$		0			0
$645$	173.552			0.269074
$646$		0			0
$647$	695.666			1.07522			0.537609	−	0.843194i	$-0.319328\pi$
$647$	695.666			1.07522			0.537609	+	0.843194i	$0.319328\pi$
$648$		0			0
$649$		−	21.1765i		−	0.0326294i
$650$		0			0
$651$		−	211.051i		−	0.324196i
$652$		0			0
$653$	357.346			0.547237			0.273618	−	0.961838i	$-0.411779\pi$
$653$	357.346			0.547237			0.273618	+	0.961838i	$0.411779\pi$
$654$		0			0
$655$		−	321.742i		−	0.491209i
$656$		0			0
$657$	−116.165			−0.176812
$658$		0			0
$659$		−	367.276i		−	0.557324i	−0.960389	−	0.278662i	$-0.910109\pi$
$659$		−	367.276i		−	0.557324i	0.960389	−	0.278662i	$-0.0898908\pi$
$660$		0			0
$661$		−	443.045i		−	0.670265i	−0.942171	−	0.335132i	$-0.891219\pi$
$661$		−	443.045i		−	0.670265i	0.942171	−	0.335132i	$-0.108781\pi$
$662$		0			0
$663$		−	186.628i		−	0.281489i
$664$		0			0
$665$			174.772i			0.262816i
$666$		0			0
$667$	158.325	−	151.509i	0.237368	−	0.227150i
$668$		0			0
$669$	576.899			0.862331
$670$		0			0
$671$	88.9581			0.132575
$672$		0			0
$673$	1093.39			1.62465			0.812324	−	0.583206i	$-0.198202\pi$
$673$	1093.39			1.62465			0.812324	+	0.583206i	$0.198202\pi$
$674$		0			0
$675$	146.086			0.216423
$676$		0			0
$677$			842.558i			1.24455i	0.782800	+	0.622273i	$0.213791\pi$
$677$			842.558i			1.24455i	−0.782800	+	0.622273i	$0.786209\pi$
$678$		0			0
$679$	728.947			1.07356
$680$		0			0
$681$		−	582.275i		−	0.855029i
$682$		0			0
$683$	858.398			1.25681			0.628403	−	0.777888i	$-0.283709\pi$
$683$	858.398			1.25681			0.628403	+	0.777888i	$0.283709\pi$
$684$		0			0
$685$	288.354			0.420954
$686$		0			0
$687$			442.536i			0.644157i
$688$		0			0
$689$		−	573.784i		−	0.832779i
$690$		0			0
$691$	−206.282			−0.298527			−0.149264	−	0.988797i	$-0.547690\pi$
$691$	−206.282			−0.298527			−0.149264	+	0.988797i	$0.547690\pi$
$692$		0			0
$693$	92.2722			0.133149
$694$		0			0
$695$			299.593i			0.431069i
$696$		0			0
$697$		−	512.253i		−	0.734940i
$698$		0			0
$699$	77.9978			0.111585
$700$		0			0
$701$			374.399i			0.534093i	0.963684	+	0.267047i	$0.0860478\pi$
$701$			374.399i			0.534093i	−0.963684	+	0.267047i	$0.913952\pi$
$702$		0			0
$703$	−533.358			−0.758688
$704$		0			0
$705$		−	59.3741i		−	0.0842186i
$706$		0			0
$707$			1099.36i			1.55497i
$708$		0			0
$709$			179.348i			0.252959i	0.991969	+	0.126480i	$0.0403678\pi$
$709$			179.348i			0.252959i	−0.991969	+	0.126480i	$0.959632\pi$
$710$		0			0
$711$		−	246.102i		−	0.346135i
$712$		0			0
$713$	−167.235	+	160.036i	−0.234551	+	0.224454i
$714$		0			0
$715$	−76.4346			−0.106902
$716$		0			0
$717$	−472.222			−0.658608
$718$		0			0
$719$	324.519			0.451347			0.225674	−	0.974203i	$-0.427542\pi$
$719$	324.519			0.451347			0.225674	+	0.974203i	$0.427542\pi$
$720$		0			0
$721$	411.755			0.571088
$722$		0			0
$723$		−	406.841i		−	0.562712i
$724$		0			0
$725$	47.6388			0.0657087
$726$		0			0
$727$		−	666.947i		−	0.917396i	−0.888592	−	0.458698i	$-0.848316\pi$
$727$		−	666.947i		−	0.917396i	0.888592	−	0.458698i	$-0.151684\pi$
$728$		0			0
$729$	809.106			1.10988
$730$		0			0
$731$	322.011			0.440507
$732$		0			0
$733$		−	854.699i		−	1.16603i	−0.812462	−	0.583014i	$-0.801873\pi$
$733$		−	854.699i		−	1.16603i	0.812462	−	0.583014i	$-0.198127\pi$
$734$		0			0
$735$		−	92.5891i		−	0.125972i
$736$		0			0
$737$	254.471			0.345280
$738$		0			0
$739$	−713.025			−0.964851			−0.482426	−	0.875937i	$-0.660244\pi$
$739$	−713.025			−0.964851			−0.482426	+	0.875937i	$0.660244\pi$
$740$		0			0
$741$		−	167.656i		−	0.226257i
$742$		0			0
$743$			1115.52i			1.50137i	0.660662	+	0.750683i	$0.270276\pi$
$743$			1115.52i			1.50137i	−0.660662	+	0.750683i	$0.729724\pi$
$744$		0			0
$745$	155.938			0.209313
$746$		0			0
$747$			307.272i			0.411342i
$748$		0			0
$749$	41.3916			0.0552624
$750$		0			0
$751$		−	928.797i		−	1.23675i	−0.785885	−	0.618373i	$-0.787792\pi$
$751$		−	928.797i		−	1.23675i	0.785885	−	0.618373i	$-0.212208\pi$
$752$		0			0
$753$		−	653.107i		−	0.867339i
$754$		0			0
$755$		−	619.174i		−	0.820098i
$756$		0			0
$757$		−	930.052i		−	1.22860i	−0.789071	−	0.614301i	$-0.789438\pi$
$757$		−	930.052i		−	1.22860i	0.789071	−	0.614301i	$-0.210562\pi$
$758$		0			0
$759$	213.117	+	222.704i	0.280786	+	0.293417i
$760$		0			0
$761$	660.941			0.868517			0.434258	−	0.900788i	$-0.357011\pi$
$761$	660.941			0.868517			0.434258	+	0.900788i	$0.357011\pi$
$762$		0			0
$763$	−1365.71			−1.78992
$764$		0			0
$765$	53.7164			0.0702175
$766$		0			0
$767$	−27.3063			−0.0356014
$768$		0			0
$769$		−	15.1066i		−	0.0196445i	−0.999952	−	0.00982226i	$-0.996873\pi$
$769$		−	15.1066i		−	0.0196445i	0.999952	−	0.00982226i	$-0.00312657\pi$
$770$		0			0
$771$	−351.095			−0.455376
$772$		0			0
$773$		−	450.498i		−	0.582792i	−0.956602	−	0.291396i	$-0.905880\pi$
$773$		−	450.498i		−	0.582792i	0.956602	−	0.291396i	$-0.0941198\pi$
$774$		0			0
$775$	−50.3198			−0.0649288
$776$		0			0
$777$	1152.92			1.48381
$778$		0			0
$779$		−	460.181i		−	0.590733i
$780$		0			0
$781$		−	216.735i		−	0.277510i
$782$		0			0
$783$	278.374			0.355522
$784$		0			0
$785$	368.464			0.469381
$786$		0			0
$787$			602.812i			0.765963i	0.923756	+	0.382981i	$0.125103\pi$
$787$			602.812i			0.765963i	−0.923756	+	0.382981i	$0.874897\pi$
$788$		0			0
$789$			497.806i			0.630932i
$790$		0			0
$791$	−227.323			−0.287386
$792$		0			0
$793$		−	114.708i		−	0.144651i
$794$		0			0
$795$	−503.035			−0.632748
$796$		0			0
$797$			1169.49i			1.46736i	0.679494	+	0.733681i	$0.262199\pi$
$797$			1169.49i			1.46736i	−0.679494	+	0.733681i	$0.737801\pi$
$798$		0			0
$799$		−	110.163i		−	0.137876i
$800$		0			0
$801$		−	364.013i		−	0.454448i
$802$		0			0
$803$		−	268.874i		−	0.334837i
$804$		0			0
$805$	−299.357	+	286.471i	−0.371873	+	0.355864i
$806$		0			0
$807$	551.722			0.683670
$808$		0			0
$809$	−696.820			−0.861335			−0.430668	−	0.902511i	$-0.641722\pi$
$809$	−696.820			−0.861335			−0.430668	+	0.902511i	$0.641722\pi$
$810$		0			0
$811$	384.936			0.474643			0.237322	−	0.971431i	$-0.423731\pi$
$811$	384.936			0.474643			0.237322	+	0.971431i	$0.423731\pi$
$812$		0			0
$813$	−107.660			−0.132423
$814$		0			0
$815$			91.1458i			0.111835i
$816$		0			0
$817$	289.277			0.354072
$818$		0			0
$819$		−	118.982i		−	0.145277i
$820$		0			0
$821$	−506.372			−0.616775			−0.308388	−	0.951261i	$-0.599789\pi$
$821$	−506.372			−0.616775			−0.308388	+	0.951261i	$0.599789\pi$
$822$		0			0
$823$	1449.62			1.76139			0.880695	−	0.473684i	$-0.157076\pi$
$823$	1449.62			1.76139			0.880695	+	0.473684i	$0.157076\pi$
$824$		0			0
$825$			67.0100i			0.0812242i
$826$		0			0
$827$			801.629i			0.969322i	0.874702	+	0.484661i	$0.161057\pi$
$827$			801.629i			0.969322i	−0.874702	+	0.484661i	$0.838943\pi$
$828$		0			0
$829$	−643.431			−0.776153			−0.388076	−	0.921627i	$-0.626860\pi$
$829$	−643.431			−0.776153			−0.388076	+	0.921627i	$0.626860\pi$
$830$		0			0
$831$	546.050			0.657100
$832$		0			0
$833$		−	171.791i		−	0.206231i
$834$		0			0
$835$			295.220i			0.353557i
$836$		0			0
$837$	−294.040			−0.351302
$838$		0			0
$839$		−	460.853i		−	0.549288i	−0.961546	−	0.274644i	$-0.911440\pi$
$839$		−	460.853i		−	0.549288i	0.961546	−	0.274644i	$-0.0885600\pi$
$840$		0			0
$841$	−750.222			−0.892059
$842$		0			0
$843$		−	345.533i		−	0.409885i
$844$		0			0
$845$		−	279.336i		−	0.330575i
$846$		0			0
$847$		−	761.268i		−	0.898781i
$848$		0			0
$849$		−	568.742i		−	0.669897i
$850$		0			0
$851$	−874.231	−	913.558i	−1.02730	−	1.07351i
$852$		0			0
$853$	−1399.59			−1.64079			−0.820393	−	0.571799i	$-0.806246\pi$
$853$	−1399.59			−1.64079			−0.820393	+	0.571799i	$0.806246\pi$
$854$		0			0
$855$	48.2559			0.0564397
$856$		0			0
$857$	477.360			0.557012			0.278506	−	0.960434i	$-0.410161\pi$
$857$	477.360			0.557012			0.278506	+	0.960434i	$0.410161\pi$
$858$		0			0
$859$	−1176.53			−1.36965			−0.684826	−	0.728707i	$-0.740122\pi$
$859$	−1176.53			−1.36965			−0.684826	+	0.728707i	$0.740122\pi$
$860$		0			0
$861$			994.736i			1.15533i
$862$		0			0
$863$	687.885			0.797086			0.398543	−	0.917150i	$-0.369516\pi$
$863$	687.885			0.797086			0.398543	+	0.917150i	$0.369516\pi$
$864$		0			0
$865$		−	586.646i		−	0.678204i
$866$		0			0
$867$	448.689			0.517519
$868$		0			0
$869$	569.623			0.655493
$870$		0			0
$871$		−	328.132i		−	0.376730i
$872$		0			0
$873$		−	201.268i		−	0.230547i
$874$		0			0
$875$	−90.0746			−0.102942
$876$		0			0
$877$	51.0202			0.0581758			0.0290879	−	0.999577i	$-0.490740\pi$
$877$	51.0202			0.0581758			0.0290879	+	0.999577i	$0.490740\pi$
$878$		0			0
$879$			670.770i			0.763106i
$880$		0			0
$881$		−	1644.35i		−	1.86646i	−0.359276	−	0.933231i	$-0.616976\pi$
$881$		−	1644.35i		−	1.86646i	0.359276	−	0.933231i	$-0.383024\pi$
$882$		0			0
$883$	351.562			0.398145			0.199073	−	0.979985i	$-0.436207\pi$
$883$	351.562			0.398145			0.199073	+	0.979985i	$0.436207\pi$
$884$		0			0
$885$			23.9394i			0.0270501i
$886$		0			0
$887$	116.003			0.130781			0.0653906	−	0.997860i	$-0.479171\pi$
$887$	116.003			0.130781			0.0653906	+	0.997860i	$0.479171\pi$
$888$		0			0
$889$			1920.63i			2.16044i
$890$		0			0
$891$			288.490i			0.323782i
$892$		0			0
$893$		−	98.9647i		−	0.110823i
$894$		0			0
$895$		−	113.269i		−	0.126557i
$896$		0			0
$897$	287.168	−	274.807i	0.320143	−	0.306362i
$898$		0			0
$899$	−95.8870			−0.106660
$900$		0			0
$901$	−933.335			−1.03589
$902$		0			0
$903$	−625.307			−0.692477
$904$		0			0
$905$	−570.925			−0.630857
$906$		0			0
$907$		−	13.8397i		−	0.0152587i	−0.999971	−	0.00762936i	$-0.997571\pi$
$907$		−	13.8397i		−	0.0152587i	0.999971	−	0.00762936i	$-0.00242852\pi$
$908$		0			0
$909$	303.543			0.333930
$910$		0			0
$911$			446.848i			0.490503i	0.969460	+	0.245251i	$0.0788705\pi$
$911$			446.848i			0.490503i	−0.969460	+	0.245251i	$0.921129\pi$
$912$		0			0
$913$	−711.207			−0.778978
$914$		0			0
$915$	−100.564			−0.109907
$916$		0			0
$917$			1159.23i			1.26416i
$918$		0			0
$919$		−	1164.24i		−	1.26686i	−0.773800	−	0.633430i	$-0.781646\pi$
$919$		−	1164.24i		−	1.26686i	0.773800	−	0.633430i	$-0.218354\pi$
$920$		0			0
$921$	33.6017			0.0364839
$922$		0			0
$923$	−279.472			−0.302787
$924$		0			0
$925$		−	274.883i		−	0.297171i
$926$		0			0
$927$		−	113.688i		−	0.122641i
$928$		0			0
$929$	15.7379			0.0169407			0.00847035	−	0.999964i	$-0.497304\pi$
$929$	15.7379			0.0169407			0.00847035	+	0.999964i	$0.497304\pi$
$930$		0			0
$931$		−	154.327i		−	0.165765i
$932$		0			0
$933$	−552.277			−0.591937
$934$		0			0
$935$			124.331i			0.132974i
$936$		0			0
$937$			282.231i			0.301207i	0.988594	+	0.150604i	$0.0481217\pi$
$937$			282.231i			0.301207i	−0.988594	+	0.150604i	$0.951878\pi$
$938$		0			0
$939$		−	32.7860i		−	0.0349159i
$940$		0			0
$941$			241.059i			0.256174i	0.991763	+	0.128087i	$0.0408836\pi$
$941$			241.059i			0.256174i	−0.991763	+	0.128087i	$0.959116\pi$
$942$		0			0
$943$	788.217	−	754.287i	0.835861	−	0.799880i
$944$		0			0
$945$	−526.344			−0.556978
$946$		0			0
$947$	−568.623			−0.600447			−0.300223	−	0.953869i	$-0.597061\pi$
$947$	−568.623			−0.600447			−0.300223	+	0.953869i	$0.597061\pi$
$948$		0			0
$949$	−346.704			−0.365336
$950$		0			0
$951$	−814.901			−0.856889
$952$		0			0
$953$			1106.49i			1.16106i	0.814240	+	0.580528i	$0.197154\pi$
$953$			1106.49i			1.16106i	−0.814240	+	0.580528i	$0.802846\pi$
$954$		0			0
$955$	827.785			0.866790
$956$		0			0
$957$			127.691i			0.133428i
$958$		0			0
$959$	−1038.93			−1.08335
$960$		0			0
$961$	−859.717			−0.894606
$962$		0			0
$963$		−	11.4285i		−	0.0118676i
$964$		0			0
$965$			384.063i			0.397993i
$966$		0			0
$967$	409.424			0.423396			0.211698	−	0.977335i	$-0.432101\pi$
$967$	409.424			0.423396			0.211698	+	0.977335i	$0.432101\pi$
$968$		0			0
$969$	−272.715			−0.281439
$970$		0			0
$971$			1540.49i			1.58650i	0.608899	+	0.793248i	$0.291612\pi$
$971$			1540.49i			1.58650i	−0.608899	+	0.793248i	$0.708388\pi$
$972$		0			0
$973$		−	1079.43i		−	1.10938i
$974$		0			0
$975$	86.4071			0.0886226
$976$		0			0
$977$			1116.89i			1.14318i	0.820539	+	0.571590i	$0.193673\pi$
$977$			1116.89i			1.14318i	−0.820539	+	0.571590i	$0.806327\pi$
$978$		0			0
$979$	842.537			0.860610
$980$		0			0
$981$			377.082i			0.384385i
$982$		0			0
$983$			752.534i			0.765549i	0.923842	+	0.382774i	$0.125031\pi$
$983$			752.534i			0.765549i	−0.923842	+	0.382774i	$0.874969\pi$
$984$		0			0
$985$		−	593.259i		−	0.602293i
$986$		0			0
$987$			213.924i			0.216742i
$988$		0			0
$989$	474.157	+	495.486i	0.479430	+	0.500997i
$990$		0			0
$991$	−1678.60			−1.69385			−0.846923	−	0.531716i	$-0.821547\pi$
$991$	−1678.60			−1.69385			−0.846923	+	0.531716i	$0.821547\pi$
$992$		0			0
$993$	996.662			1.00369
$994$		0			0
$995$	−646.313			−0.649561
$996$		0			0
$997$	1062.89			1.06609			0.533045	−	0.846087i	$-0.321048\pi$
$997$	1062.89			1.06609			0.533045	+	0.846087i	$0.321048\pi$
$998$		0			0
$999$		−	1606.26i		−	1.60787i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.b.321.8		10
4.3	odd	2		115.3.d.b.91.6	yes	10
12.11	even	2		1035.3.g.b.91.5		10
20.3	even	4		575.3.c.d.574.10		20
20.7	even	4		575.3.c.d.574.11		20
20.19	odd	2		575.3.d.g.551.5		10
23.22	odd	2	inner	1840.3.k.b.321.7		10
92.91	even	2		115.3.d.b.91.5	✓	10
276.275	odd	2		1035.3.g.b.91.6		10
460.183	odd	4		575.3.c.d.574.9		20
460.367	odd	4		575.3.c.d.574.12		20
460.459	even	2		575.3.d.g.551.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.d.b.91.5	✓	10	92.91	even	2
115.3.d.b.91.6	yes	10	4.3	odd	2
575.3.c.d.574.9		20	460.183	odd	4
575.3.c.d.574.10		20	20.3	even	4
575.3.c.d.574.11		20	20.7	even	4
575.3.c.d.574.12		20	460.367	odd	4
575.3.d.g.551.5		10	20.19	odd	2
575.3.d.g.551.6		10	460.459	even	2
1035.3.g.b.91.5		10	12.11	even	2
1035.3.g.b.91.6		10	276.275	odd	2
1840.3.k.b.321.7		10	23.22	odd	2	inner
1840.3.k.b.321.8		10	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.60299 q^{3} +2.23607i q^{5} -8.05652i q^{7} -2.22446 q^{9} +O(q^{10})\)	\(q+2.60299 q^{3} +2.23607i q^{5} -8.05652i q^{7} -2.22446 q^{9} -5.14870i q^{11} -6.63907 q^{13} +5.82045i q^{15} +10.7993i q^{17} +9.70153i q^{19} -20.9710i q^{21} +(-16.6172 + 15.9019i) q^{23} -5.00000 q^{25} -29.2171 q^{27} -9.52776 q^{29} +10.0640 q^{31} -13.4020i q^{33} +18.0149 q^{35} +54.9767i q^{37} -17.2814 q^{39} -47.4339 q^{41} -29.8177i q^{43} -4.97405i q^{45} -10.2009 q^{47} -15.9075 q^{49} +28.1105i q^{51} +86.4254i q^{53} +11.5128 q^{55} +25.2529i q^{57} +4.11297 q^{59} +17.2778i q^{61} +17.9214i q^{63} -14.8454i q^{65} +49.4243i q^{67} +(-43.2543 + 41.3923i) q^{69} +42.0951 q^{71} +52.2218 q^{73} -13.0149 q^{75} -41.4806 q^{77} +110.634i q^{79} -56.0316 q^{81} -138.133i q^{83} -24.1480 q^{85} -24.8006 q^{87} +163.641i q^{89} +53.4878i q^{91} +26.1964 q^{93} -21.6933 q^{95} +90.4792i q^{97} +11.4531i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 16 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 16 * q^9	\( 10 q + 2 q^{3} - 16 q^{9} - 2 q^{13} - 44 q^{23} - 50 q^{25} - 40 q^{27} - 46 q^{29} - 16 q^{31} + 60 q^{35} - 72 q^{39} - 84 q^{41} - 112 q^{47} + 50 q^{49} + 10 q^{55} + 262 q^{59} + 124 q^{69} - 236 q^{71} + 168 q^{73} - 10 q^{75} + 300 q^{77} - 258 q^{81} - 540 q^{87} + 100 q^{93} + 90 q^{95}+O(q^{100}) \) 10 * q + 2 * q^3 - 16 * q^9 - 2 * q^13 - 44 * q^23 - 50 * q^25 - 40 * q^27 - 46 * q^29 - 16 * q^31 + 60 * q^35 - 72 * q^39 - 84 * q^41 - 112 * q^47 + 50 * q^49 + 10 * q^55 + 262 * q^59 + 124 * q^69 - 236 * q^71 + 168 * q^73 - 10 * q^75 + 300 * q^77 - 258 * q^81 - 540 * q^87 + 100 * q^93 + 90 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more