LMFDB - Embedded newform 1840.3.k.e.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:	$48$
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.8
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.e.321.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.51681 q^{3} +2.23607i q^{5} -1.75483i q^{7} +11.4015 q^{9} +O(q^{10})$	$q-4.51681 q^{3} +2.23607i q^{5} -1.75483i q^{7} +11.4015 q^{9} -18.1298i q^{11} +15.0976 q^{13} -10.0999i q^{15} -14.5217i q^{17} +3.31102i q^{19} +7.92620i q^{21} +(-19.3405 - 12.4476i) q^{23} -5.00000 q^{25} -10.8473 q^{27} -10.7774 q^{29} +52.4830 q^{31} +81.8888i q^{33} +3.92391 q^{35} +40.4741i q^{37} -68.1931 q^{39} +0.449717 q^{41} -10.8456i q^{43} +25.4946i q^{45} -72.1590 q^{47} +45.9206 q^{49} +65.5917i q^{51} -11.0746i q^{53} +40.5395 q^{55} -14.9552i q^{57} +39.2246 q^{59} +87.8633i q^{61} -20.0077i q^{63} +33.7594i q^{65} -59.9394i q^{67} +(87.3574 + 56.2236i) q^{69} -54.5999 q^{71} +94.8773 q^{73} +22.5840 q^{75} -31.8146 q^{77} -30.4889i q^{79} -53.6188 q^{81} +98.2989i q^{83} +32.4715 q^{85} +48.6795 q^{87} -133.212i q^{89} -26.4937i q^{91} -237.055 q^{93} -7.40366 q^{95} -60.8967i q^{97} -206.708i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 128 q^{9}+O(q^{10}) $ 48 * q + 128 * q^9	$ 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) $ 48 * q + 128 * q^9 - 8 * q^23 - 240 * q^25 + 72 * q^29 - 32 * q^31 + 40 * q^35 + 96 * q^39 - 104 * q^41 - 128 * q^47 - 344 * q^49 - 80 * q^55 - 248 * q^59 + 292 * q^69 - 208 * q^71 + 224 * q^73 - 288 * q^77 + 184 * q^81 + 48 * q^87 - 672 * 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
$3$	−4.51681			−1.50560			−0.752801	−	0.658248i	$-0.771298\pi$
$3$	−4.51681			−1.50560			−0.752801	+	0.658248i	$0.771298\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	1.75483i		−	0.250689i	−0.992113	−	0.125345i	$-0.959996\pi$
$7$		−	1.75483i		−	0.250689i	0.992113	−	0.125345i	$-0.0400037\pi$
$8$		0			0
$9$	11.4015			1.26684
$10$		0			0
$11$		−	18.1298i		−	1.64816i	−0.566470	−	0.824082i	$-0.691691\pi$
$11$		−	18.1298i		−	1.64816i	0.566470	−	0.824082i	$-0.308309\pi$
$12$		0			0
$13$	15.0976			1.16136			0.580679	−	0.814133i	$-0.302787\pi$
$13$	15.0976			1.16136			0.580679	+	0.814133i	$0.302787\pi$
$14$		0			0
$15$		−	10.0999i		−	0.673326i
$16$		0			0
$17$		−	14.5217i		−	0.854218i	−0.904200	−	0.427109i	$-0.859532\pi$
$17$		−	14.5217i		−	0.854218i	0.904200	−	0.427109i	$-0.140468\pi$
$18$		0			0
$19$			3.31102i			0.174264i	0.996197	+	0.0871320i	$0.0277702\pi$
$19$			3.31102i			0.174264i	−0.996197	+	0.0871320i	$0.972230\pi$
$20$		0			0
$21$			7.92620i			0.377438i
$22$		0			0
$23$	−19.3405	−	12.4476i	−0.840893	−	0.541202i
$23$	−19.3405	−	12.4476i	−0.840893	−	0.541202i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−10.8473			−0.401751
$28$		0			0
$29$	−10.7774			−0.371635			−0.185817	−	0.982584i	$-0.559493\pi$
$29$	−10.7774			−0.371635			−0.185817	+	0.982584i	$0.559493\pi$
$30$		0			0
$31$	52.4830			1.69300			0.846500	−	0.532389i	$-0.178706\pi$
$31$	52.4830			1.69300			0.846500	+	0.532389i	$0.178706\pi$
$32$		0			0
$33$			81.8888i			2.48148i
$34$		0			0
$35$	3.92391			0.112112
$36$		0			0
$37$			40.4741i			1.09389i	0.837167	+	0.546947i	$0.184210\pi$
$37$			40.4741i			1.09389i	−0.837167	+	0.546947i	$0.815790\pi$
$38$		0			0
$39$	−68.1931			−1.74854
$40$		0			0
$41$	0.449717			0.0109687			0.00548436	−	0.999985i	$-0.498254\pi$
$41$	0.449717			0.0109687			0.00548436	+	0.999985i	$0.498254\pi$
$42$		0			0
$43$		−	10.8456i		−	0.252224i	−0.992016	−	0.126112i	$-0.959750\pi$
$43$		−	10.8456i		−	0.252224i	0.992016	−	0.126112i	$-0.0402499\pi$
$44$		0			0
$45$			25.4946i			0.566547i
$46$		0			0
$47$	−72.1590			−1.53530			−0.767649	−	0.640871i	$-0.778573\pi$
$47$	−72.1590			−1.53530			−0.767649	+	0.640871i	$0.778573\pi$
$48$		0			0
$49$	45.9206			0.937155
$50$		0			0
$51$			65.5917i			1.28611i
$52$		0			0
$53$		−	11.0746i		−	0.208954i	−0.994527	−	0.104477i	$-0.966683\pi$
$53$		−	11.0746i		−	0.208954i	0.994527	−	0.104477i	$-0.0333169\pi$
$54$		0			0
$55$	40.5395			0.737082
$56$		0			0
$57$		−	14.9552i		−	0.262372i
$58$		0			0
$59$	39.2246			0.664824			0.332412	−	0.943134i	$-0.392137\pi$
$59$	39.2246			0.664824			0.332412	+	0.943134i	$0.392137\pi$
$60$		0			0
$61$			87.8633i			1.44038i	0.693776	+	0.720191i	$0.255946\pi$
$61$			87.8633i			1.44038i	−0.693776	+	0.720191i	$0.744054\pi$
$62$		0			0
$63$		−	20.0077i		−	0.317583i
$64$		0			0
$65$			33.7594i			0.519375i
$66$		0			0
$67$		−	59.9394i		−	0.894617i	−0.894380	−	0.447309i	$-0.852383\pi$
$67$		−	59.9394i		−	0.894617i	0.894380	−	0.447309i	$-0.147617\pi$
$68$		0			0
$69$	87.3574	+	56.2236i	1.26605	+	0.814835i
$70$		0			0
$71$	−54.5999			−0.769013			−0.384507	−	0.923122i	$-0.625628\pi$
$71$	−54.5999			−0.769013			−0.384507	+	0.923122i	$0.625628\pi$
$72$		0			0
$73$	94.8773			1.29969			0.649845	−	0.760067i	$-0.274834\pi$
$73$	94.8773			1.29969			0.649845	+	0.760067i	$0.274834\pi$
$74$		0			0
$75$	22.5840			0.301120
$76$		0			0
$77$	−31.8146			−0.413177
$78$		0			0
$79$		−	30.4889i		−	0.385936i	−0.981205	−	0.192968i	$-0.938189\pi$
$79$		−	30.4889i		−	0.385936i	0.981205	−	0.192968i	$-0.0618114\pi$
$80$		0			0
$81$	−53.6188			−0.661961
$82$		0			0
$83$			98.2989i			1.18432i	0.805819	+	0.592162i	$0.201726\pi$
$83$			98.2989i			1.18432i	−0.805819	+	0.592162i	$0.798274\pi$
$84$		0			0
$85$	32.4715			0.382018
$86$		0			0
$87$	48.6795			0.559534
$88$		0			0
$89$		−	133.212i		−	1.49677i	−0.663266	−	0.748383i	$-0.730830\pi$
$89$		−	133.212i		−	1.49677i	0.663266	−	0.748383i	$-0.269170\pi$
$90$		0			0
$91$		−	26.4937i		−	0.291140i
$92$		0			0
$93$	−237.055			−2.54898
$94$		0			0
$95$	−7.40366			−0.0779332
$96$		0			0
$97$		−	60.8967i		−	0.627802i	−0.949456	−	0.313901i	$-0.898364\pi$
$97$		−	60.8967i		−	0.627802i	0.949456	−	0.313901i	$-0.101636\pi$
$98$		0			0
$99$		−	206.708i		−	2.08796i
$100$		0			0
$101$	10.3919			0.102891			0.0514453	−	0.998676i	$-0.483617\pi$
$101$	10.3919			0.102891			0.0514453	+	0.998676i	$0.483617\pi$
$102$		0			0
$103$			52.8464i			0.513072i	0.966535	+	0.256536i	$0.0825812\pi$
$103$			52.8464i			0.513072i	−0.966535	+	0.256536i	$0.917419\pi$
$104$		0			0
$105$	−17.7235			−0.168796
$106$		0			0
$107$			99.3988i			0.928961i	0.885583	+	0.464481i	$0.153759\pi$
$107$			99.3988i			0.928961i	−0.885583	+	0.464481i	$0.846241\pi$
$108$		0			0
$109$		−	187.793i		−	1.72287i	−0.507869	−	0.861434i	$-0.669567\pi$
$109$		−	187.793i		−	1.72287i	0.507869	−	0.861434i	$-0.330433\pi$
$110$		0			0
$111$		−	182.814i		−	1.64697i
$112$		0			0
$113$		−	34.4195i		−	0.304597i	−0.988335	−	0.152299i	$-0.951332\pi$
$113$		−	34.4195i		−	0.304597i	0.988335	−	0.152299i	$-0.0486676\pi$
$114$		0			0
$115$	27.8338	−	43.2467i	0.242033	−	0.376059i
$116$		0			0
$117$	172.136			1.47125
$118$		0			0
$119$	−25.4831			−0.214143
$120$		0			0
$121$	−207.690			−1.71645
$122$		0			0
$123$	−2.03129			−0.0165145
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−209.214			−1.64736			−0.823679	−	0.567057i	$-0.808082\pi$
$127$	−209.214			−1.64736			−0.823679	+	0.567057i	$0.808082\pi$
$128$		0			0
$129$			48.9877i			0.379750i
$130$		0			0
$131$	39.5453			0.301872			0.150936	−	0.988544i	$-0.451771\pi$
$131$	39.5453			0.301872			0.150936	+	0.988544i	$0.451771\pi$
$132$		0			0
$133$	5.81025			0.0436861
$134$		0			0
$135$		−	24.2552i		−	0.179668i
$136$		0			0
$137$		−	240.733i		−	1.75717i	−0.477582	−	0.878587i	$-0.658487\pi$
$137$		−	240.733i		−	1.75717i	0.477582	−	0.878587i	$-0.341513\pi$
$138$		0			0
$139$	85.3750			0.614209			0.307104	−	0.951676i	$-0.400640\pi$
$139$	85.3750			0.614209			0.307104	+	0.951676i	$0.400640\pi$
$140$		0			0
$141$	325.928			2.31155
$142$		0			0
$143$		−	273.717i		−	1.91411i
$144$		0			0
$145$		−	24.0990i		−	0.166200i
$146$		0			0
$147$	−207.414			−1.41098
$148$		0			0
$149$			41.8939i			0.281167i	0.990069	+	0.140583i	$0.0448978\pi$
$149$			41.8939i			0.281167i	−0.990069	+	0.140583i	$0.955102\pi$
$150$		0			0
$151$	26.6204			0.176294			0.0881470	−	0.996107i	$-0.471905\pi$
$151$	26.6204			0.176294			0.0881470	+	0.996107i	$0.471905\pi$
$152$		0			0
$153$		−	165.570i		−	1.08216i
$154$		0			0
$155$			117.355i			0.757132i
$156$		0			0
$157$			145.171i			0.924654i	0.886709	+	0.462327i	$0.152985\pi$
$157$			145.171i			0.924654i	−0.886709	+	0.462327i	$0.847015\pi$
$158$		0			0
$159$			50.0217i			0.314602i
$160$		0			0
$161$	−21.8434	+	33.9393i	−0.135674	+	0.210803i
$162$		0			0
$163$	103.941			0.637673			0.318836	−	0.947810i	$-0.396708\pi$
$163$	103.941			0.637673			0.318836	+	0.947810i	$0.396708\pi$
$164$		0			0
$165$	−183.109			−1.10975
$166$		0			0
$167$	−307.768			−1.84292			−0.921460	−	0.388473i	$-0.873003\pi$
$167$	−307.768			−1.84292			−0.921460	+	0.388473i	$0.873003\pi$
$168$		0			0
$169$	58.9389			0.348751
$170$		0			0
$171$			37.7507i			0.220764i
$172$		0			0
$173$	−192.783			−1.11435			−0.557177	−	0.830393i	$-0.688116\pi$
$173$	−192.783			−1.11435			−0.557177	+	0.830393i	$0.688116\pi$
$174$		0			0
$175$			8.77413i			0.0501379i
$176$		0			0
$177$	−177.170			−1.00096
$178$		0			0
$179$	94.9974			0.530712			0.265356	−	0.964151i	$-0.414511\pi$
$179$	94.9974			0.530712			0.265356	+	0.964151i	$0.414511\pi$
$180$		0			0
$181$		−	209.951i		−	1.15995i	−0.814635	−	0.579974i	$-0.803063\pi$
$181$		−	209.951i		−	1.15995i	0.814635	−	0.579974i	$-0.196937\pi$
$182$		0			0
$183$		−	396.862i		−	2.16864i
$184$		0			0
$185$	−90.5028			−0.489205
$186$		0			0
$187$	−263.276			−1.40789
$188$		0			0
$189$			19.0351i			0.100715i
$190$		0			0
$191$		−	353.455i		−	1.85055i	−0.379295	−	0.925276i	$-0.623834\pi$
$191$		−	353.455i		−	1.85055i	0.379295	−	0.925276i	$-0.376166\pi$
$192$		0			0
$193$	−163.217			−0.845684			−0.422842	−	0.906204i	$-0.638967\pi$
$193$	−163.217			−0.845684			−0.422842	+	0.906204i	$0.638967\pi$
$194$		0			0
$195$		−	152.484i		−	0.781972i
$196$		0			0
$197$	−213.811			−1.08533			−0.542667	−	0.839948i	$-0.682585\pi$
$197$	−213.811			−1.08533			−0.542667	+	0.839948i	$0.682585\pi$
$198$		0			0
$199$		−	71.4604i		−	0.359098i	−0.983749	−	0.179549i	$-0.942536\pi$
$199$		−	71.4604i		−	0.359098i	0.983749	−	0.179549i	$-0.0574638\pi$
$200$		0			0
$201$			270.734i			1.34694i
$202$		0			0
$203$			18.9125i			0.0931649i
$204$		0			0
$205$			1.00560i			0.00490536i
$206$		0			0
$207$	−220.512	−	141.922i	−1.06527	−	0.685615i
$208$		0			0
$209$	60.0281			0.287216
$210$		0			0
$211$	94.7516			0.449060			0.224530	−	0.974467i	$-0.427915\pi$
$211$	94.7516			0.449060			0.224530	+	0.974467i	$0.427915\pi$
$212$		0			0
$213$	246.617			1.15783
$214$		0			0
$215$	24.2516			0.112798
$216$		0			0
$217$		−	92.0984i		−	0.424417i
$218$		0			0
$219$	−428.542			−1.95681
$220$		0			0
$221$		−	219.244i		−	0.992053i
$222$		0			0
$223$	49.8370			0.223484			0.111742	−	0.993737i	$-0.464357\pi$
$223$	49.8370			0.223484			0.111742	+	0.993737i	$0.464357\pi$
$224$		0			0
$225$	−57.0077			−0.253367
$226$		0			0
$227$			392.399i			1.72863i	0.502952	+	0.864314i	$0.332247\pi$
$227$			392.399i			1.72863i	−0.502952	+	0.864314i	$0.667753\pi$
$228$		0			0
$229$		−	78.6363i		−	0.343390i	−0.985150	−	0.171695i	$-0.945076\pi$
$229$		−	78.6363i		−	0.343390i	0.985150	−	0.171695i	$-0.0549244\pi$
$230$		0			0
$231$	143.701			0.622080
$232$		0			0
$233$	147.754			0.634138			0.317069	−	0.948402i	$-0.397301\pi$
$233$	147.754			0.634138			0.317069	+	0.948402i	$0.397301\pi$
$234$		0			0
$235$		−	161.352i		−	0.686606i
$236$		0			0
$237$			137.713i			0.581066i
$238$		0			0
$239$	189.645			0.793495			0.396747	−	0.917928i	$-0.370139\pi$
$239$	189.645			0.793495			0.396747	+	0.917928i	$0.370139\pi$
$240$		0			0
$241$		−	388.492i		−	1.61200i	−0.591916	−	0.806000i	$-0.701628\pi$
$241$		−	388.492i		−	1.61200i	0.591916	−	0.806000i	$-0.298372\pi$
$242$		0			0
$243$	339.811			1.39840
$244$		0			0
$245$			102.682i			0.419108i
$246$		0			0
$247$			49.9885i			0.202383i
$248$		0			0
$249$		−	443.997i		−	1.78312i
$250$		0			0
$251$		−	236.403i		−	0.941846i	−0.882174	−	0.470923i	$-0.843921\pi$
$251$		−	236.403i		−	0.941846i	0.882174	−	0.470923i	$-0.156079\pi$
$252$		0			0
$253$	−225.673	+	350.640i	−0.891990	+	1.38593i
$254$		0			0
$255$	−146.668			−0.575167
$256$		0			0
$257$	6.77816			0.0263741			0.0131871	−	0.999913i	$-0.495802\pi$
$257$	6.77816			0.0263741			0.0131871	+	0.999913i	$0.495802\pi$
$258$		0			0
$259$	71.0250			0.274228
$260$		0			0
$261$	−122.879			−0.470801
$262$		0			0
$263$		−	128.891i		−	0.490080i	−0.969513	−	0.245040i	$-0.921199\pi$
$263$		−	128.891i		−	0.490080i	0.969513	−	0.245040i	$-0.0788011\pi$
$264$		0			0
$265$	24.7635			0.0934471
$266$		0			0
$267$			601.694i			2.25353i
$268$		0			0
$269$	25.2705			0.0939425			0.0469713	−	0.998896i	$-0.485043\pi$
$269$	25.2705			0.0939425			0.0469713	+	0.998896i	$0.485043\pi$
$270$		0			0
$271$	−418.799			−1.54538			−0.772692	−	0.634782i	$-0.781090\pi$
$271$	−418.799			−1.54538			−0.772692	+	0.634782i	$0.781090\pi$
$272$		0			0
$273$			119.667i			0.438341i
$274$		0			0
$275$			90.6490i			0.329633i
$276$		0			0
$277$	−275.940			−0.996173			−0.498087	−	0.867127i	$-0.665964\pi$
$277$	−275.940			−0.996173			−0.498087	+	0.867127i	$0.665964\pi$
$278$		0			0
$279$	598.386			2.14475
$280$		0			0
$281$		−	254.704i		−	0.906419i	−0.891404	−	0.453209i	$-0.850279\pi$
$281$		−	254.704i		−	0.906419i	0.891404	−	0.453209i	$-0.149721\pi$
$282$		0			0
$283$			12.5577i			0.0443737i	0.999754	+	0.0221868i	$0.00706287\pi$
$283$			12.5577i			0.0443737i	−0.999754	+	0.0221868i	$0.992937\pi$
$284$		0			0
$285$	33.4409			0.117336
$286$		0			0
$287$		−	0.789175i		−	0.00274974i
$288$		0			0
$289$	78.1200			0.270311
$290$		0			0
$291$			275.059i			0.945219i
$292$		0			0
$293$		−	391.986i		−	1.33784i	−0.743336	−	0.668918i	$-0.766758\pi$
$293$		−	391.986i		−	1.33784i	0.743336	−	0.668918i	$-0.233242\pi$
$294$		0			0
$295$			87.7090i			0.297319i
$296$		0			0
$297$			196.659i			0.662151i
$298$		0			0
$299$	−291.996	−	187.930i	−0.976577	−	0.628529i
$300$		0			0
$301$	−19.0322			−0.0632300
$302$		0			0
$303$	−46.9384			−0.154912
$304$		0			0
$305$	−196.468			−0.644159
$306$		0			0
$307$	−360.205			−1.17331			−0.586654	−	0.809838i	$-0.699555\pi$
$307$	−360.205			−1.17331			−0.586654	+	0.809838i	$0.699555\pi$
$308$		0			0
$309$		−	238.697i		−	0.772482i
$310$		0			0
$311$	242.371			0.779329			0.389664	−	0.920957i	$-0.372591\pi$
$311$	242.371			0.779329			0.389664	+	0.920957i	$0.372591\pi$
$312$		0			0
$313$		−	608.734i		−	1.94484i	−0.233242	−	0.972419i	$-0.574933\pi$
$313$		−	608.734i		−	1.94484i	0.233242	−	0.972419i	$-0.425067\pi$
$314$		0			0
$315$	44.7386			0.142027
$316$		0			0
$317$	−292.715			−0.923393			−0.461696	−	0.887038i	$-0.652759\pi$
$317$	−292.715			−0.923393			−0.461696	+	0.887038i	$0.652759\pi$
$318$		0			0
$319$			195.392i			0.612516i
$320$		0			0
$321$		−	448.965i		−	1.39865i
$322$		0			0
$323$	48.0816			0.148859
$324$		0			0
$325$	−75.4882			−0.232271
$326$		0			0
$327$			848.223i			2.59395i
$328$		0			0
$329$			126.626i			0.384883i
$330$		0			0
$331$	46.4347			0.140286			0.0701430	−	0.997537i	$-0.477654\pi$
$331$	46.4347			0.140286			0.0701430	+	0.997537i	$0.477654\pi$
$332$		0			0
$333$			461.467i			1.38579i
$334$		0			0
$335$	134.029			0.400085
$336$		0			0
$337$			591.810i			1.75611i	0.478558	+	0.878056i	$0.341160\pi$
$337$			591.810i			1.75611i	−0.478558	+	0.878056i	$0.658840\pi$
$338$		0			0
$339$			155.466i			0.458603i
$340$		0			0
$341$		−	951.506i		−	2.79034i
$342$		0			0
$343$		−	166.569i		−	0.485624i
$344$		0			0
$345$	−125.720	+	195.337i	−0.364405	+	0.566195i
$346$		0			0
$347$	437.242			1.26006			0.630032	−	0.776569i	$-0.283042\pi$
$347$	437.242			1.26006			0.630032	+	0.776569i	$0.283042\pi$
$348$		0			0
$349$	167.732			0.480607			0.240304	−	0.970698i	$-0.422753\pi$
$349$	167.732			0.480607			0.240304	+	0.970698i	$0.422753\pi$
$350$		0			0
$351$	−163.768			−0.466576
$352$		0			0
$353$	−162.657			−0.460785			−0.230392	−	0.973098i	$-0.574001\pi$
$353$	−162.657			−0.460785			−0.230392	+	0.973098i	$0.574001\pi$
$354$		0			0
$355$		−	122.089i		−	0.343913i
$356$		0			0
$357$	115.102			0.322415
$358$		0			0
$359$		−	56.4103i		−	0.157132i	−0.996909	−	0.0785659i	$-0.974966\pi$
$359$		−	56.4103i		−	0.157132i	0.996909	−	0.0785659i	$-0.0250341\pi$
$360$		0			0
$361$	350.037			0.969632
$362$		0			0
$363$	938.095			2.58428
$364$		0			0
$365$			212.152i			0.581239i
$366$		0			0
$367$		−	369.803i		−	1.00764i	−0.863809	−	0.503819i	$-0.831928\pi$
$367$		−	369.803i		−	1.00764i	0.863809	−	0.503819i	$-0.168072\pi$
$368$		0			0
$369$	5.12747			0.0138956
$370$		0			0
$371$	−19.4339			−0.0523825
$372$		0			0
$373$			413.529i			1.10866i	0.832298	+	0.554329i	$0.187025\pi$
$373$			413.529i			1.10866i	−0.832298	+	0.554329i	$0.812975\pi$
$374$		0			0
$375$			50.4994i			0.134665i
$376$		0			0
$377$	−162.714			−0.431601
$378$		0			0
$379$			298.865i			0.788561i	0.918990	+	0.394280i	$0.129006\pi$
$379$			298.865i			0.788561i	−0.918990	+	0.394280i	$0.870994\pi$
$380$		0			0
$381$	944.981			2.48026
$382$		0			0
$383$			225.641i			0.589140i	0.955630	+	0.294570i	$0.0951765\pi$
$383$			225.641i			0.589140i	−0.955630	+	0.294570i	$0.904824\pi$
$384$		0			0
$385$		−	71.1397i		−	0.184778i
$386$		0			0
$387$		−	123.657i		−	0.319527i
$388$		0			0
$389$		−	399.347i		−	1.02660i	−0.858210	−	0.513299i	$-0.828423\pi$
$389$		−	399.347i		−	1.02660i	0.858210	−	0.513299i	$-0.171577\pi$
$390$		0			0
$391$	−180.761	+	280.858i	−0.462304	+	0.718306i
$392$		0			0
$393$	−178.618			−0.454500
$394$		0			0
$395$	68.1753			0.172596
$396$		0			0
$397$	−319.785			−0.805504			−0.402752	−	0.915309i	$-0.631946\pi$
$397$	−319.785			−0.805504			−0.402752	+	0.915309i	$0.631946\pi$
$398$		0			0
$399$	−26.2438			−0.0657739
$400$		0			0
$401$			119.737i			0.298597i	0.988792	+	0.149298i	$0.0477015\pi$
$401$			119.737i			0.298597i	−0.988792	+	0.149298i	$0.952299\pi$
$402$		0			0
$403$	792.369			1.96618
$404$		0			0
$405$		−	119.895i		−	0.296038i
$406$		0			0
$407$	733.788			1.80292
$408$		0			0
$409$	−210.260			−0.514084			−0.257042	−	0.966400i	$-0.582748\pi$
$409$	−210.260			−0.514084			−0.257042	+	0.966400i	$0.582748\pi$
$410$		0			0
$411$			1087.34i			2.64561i
$412$		0			0
$413$		−	68.8324i		−	0.166664i
$414$		0			0
$415$	−219.803			−0.529646
$416$		0			0
$417$	−385.622			−0.924754
$418$		0			0
$419$			103.101i			0.246065i	0.992403	+	0.123033i	$0.0392620\pi$
$419$			103.101i			0.246065i	−0.992403	+	0.123033i	$0.960738\pi$
$420$		0			0
$421$			215.818i			0.512632i	0.966593	+	0.256316i	$0.0825088\pi$
$421$			215.818i			0.512632i	−0.966593	+	0.256316i	$0.917491\pi$
$422$		0			0
$423$	−822.723			−1.94497
$424$		0			0
$425$			72.6085i			0.170844i
$426$		0			0
$427$	154.185			0.361089
$428$		0			0
$429$			1236.33i			2.88188i
$430$		0			0
$431$			430.884i			0.999731i	0.866103	+	0.499866i	$0.166617\pi$
$431$			430.884i			0.999731i	−0.866103	+	0.499866i	$0.833383\pi$
$432$		0			0
$433$			104.542i			0.241437i	0.992687	+	0.120718i	$0.0385198\pi$
$433$			104.542i			0.241437i	−0.992687	+	0.120718i	$0.961480\pi$
$434$		0			0
$435$			108.851i			0.250231i
$436$		0			0
$437$	41.2143	−	64.0368i	0.0943120	−	0.146537i
$438$		0			0
$439$	−579.783			−1.32069			−0.660346	−	0.750962i	$-0.729590\pi$
$439$	−579.783			−1.32069			−0.660346	+	0.750962i	$0.729590\pi$
$440$		0			0
$441$	523.565			1.18722
$442$		0			0
$443$	405.291			0.914879			0.457439	−	0.889241i	$-0.348767\pi$
$443$	405.291			0.914879			0.457439	+	0.889241i	$0.348767\pi$
$444$		0			0
$445$	297.872			0.669374
$446$		0			0
$447$		−	189.227i		−	0.423326i
$448$		0			0
$449$	−340.772			−0.758959			−0.379479	−	0.925200i	$-0.623897\pi$
$449$	−340.772			−0.758959			−0.379479	+	0.925200i	$0.623897\pi$
$450$		0			0
$451$		−	8.15329i		−	0.0180782i
$452$		0			0
$453$	−120.239			−0.265428
$454$		0			0
$455$	59.2418			0.130202
$456$		0			0
$457$		−	274.333i		−	0.600291i	−0.953893	−	0.300145i	$-0.902965\pi$
$457$		−	274.333i		−	0.600291i	0.953893	−	0.300145i	$-0.0970352\pi$
$458$		0			0
$459$			157.521i			0.343183i
$460$		0			0
$461$	667.069			1.44700			0.723502	−	0.690322i	$-0.242531\pi$
$461$	667.069			1.44700			0.723502	+	0.690322i	$0.242531\pi$
$462$		0			0
$463$	536.570			1.15890			0.579449	−	0.815009i	$-0.303268\pi$
$463$	536.570			1.15890			0.579449	+	0.815009i	$0.303268\pi$
$464$		0			0
$465$		−	530.072i		−	1.13994i
$466$		0			0
$467$			99.3575i			0.212757i	0.994326	+	0.106378i	$0.0339255\pi$
$467$			99.3575i			0.212757i	−0.994326	+	0.106378i	$0.966075\pi$
$468$		0			0
$469$	−105.183			−0.224271
$470$		0			0
$471$		−	655.708i		−	1.39216i
$472$		0			0
$473$	−196.630			−0.415707
$474$		0			0
$475$		−	16.5551i		−	0.0348528i
$476$		0			0
$477$		−	126.267i		−	0.264711i
$478$		0			0
$479$			736.373i			1.53731i	0.639661	+	0.768657i	$0.279075\pi$
$479$			736.373i			1.53731i	−0.639661	+	0.768657i	$0.720925\pi$
$480$		0			0
$481$			611.064i			1.27040i
$482$		0			0
$483$	98.6626	−	153.297i	0.204270	−	0.317385i
$484$		0			0
$485$	136.169			0.280761
$486$		0			0
$487$	−802.901			−1.64867			−0.824333	−	0.566105i	$-0.808450\pi$
$487$	−802.901			−1.64867			−0.824333	+	0.566105i	$0.808450\pi$
$488$		0			0
$489$	−469.480			−0.960081
$490$		0			0
$491$	−750.640			−1.52880			−0.764399	−	0.644743i	$-0.776964\pi$
$491$	−750.640			−1.52880			−0.764399	+	0.644743i	$0.776964\pi$
$492$		0			0
$493$			156.506i			0.317457i
$494$		0			0
$495$	462.212			0.933762
$496$		0			0
$497$			95.8133i			0.192783i
$498$		0			0
$499$	82.6836			0.165699			0.0828493	−	0.996562i	$-0.473598\pi$
$499$	82.6836			0.165699			0.0828493	+	0.996562i	$0.473598\pi$
$500$		0			0
$501$	1390.13			2.77470
$502$		0			0
$503$			450.367i			0.895361i	0.894193	+	0.447681i	$0.147750\pi$
$503$			450.367i			0.895361i	−0.894193	+	0.447681i	$0.852250\pi$
$504$		0			0
$505$			23.2371i			0.0460140i
$506$		0			0
$507$	−266.215			−0.525080
$508$		0			0
$509$	383.319			0.753082			0.376541	−	0.926400i	$-0.377113\pi$
$509$	383.319			0.753082			0.376541	+	0.926400i	$0.377113\pi$
$510$		0			0
$511$		−	166.493i		−	0.325818i
$512$		0			0
$513$		−	35.9155i		−	0.0700107i
$514$		0			0
$515$	−118.168			−0.229453
$516$		0			0
$517$			1308.23i			2.53042i
$518$		0			0
$519$	870.765			1.67777
$520$		0			0
$521$			49.8457i			0.0956732i	0.998855	+	0.0478366i	$0.0152327\pi$
$521$			49.8457i			0.0956732i	−0.998855	+	0.0478366i	$0.984767\pi$
$522$		0			0
$523$		−	499.893i		−	0.955818i	−0.878409	−	0.477909i	$-0.841395\pi$
$523$		−	499.893i		−	0.955818i	0.878409	−	0.477909i	$-0.158605\pi$
$524$		0			0
$525$		−	39.6310i		−	0.0754877i
$526$		0			0
$527$		−	762.142i		−	1.44619i
$528$		0			0
$529$	219.112	+	481.488i	0.414201	+	0.910185i
$530$		0			0
$531$	447.221			0.842224
$532$		0			0
$533$	6.78967			0.0127386
$534$		0			0
$535$	−222.263			−0.415444
$536$		0			0
$537$	−429.085			−0.799040
$538$		0			0
$539$		−	832.532i		−	1.54459i
$540$		0			0
$541$	495.234			0.915404			0.457702	−	0.889106i	$-0.348673\pi$
$541$	495.234			0.915404			0.457702	+	0.889106i	$0.348673\pi$
$542$		0			0
$543$			948.306i			1.74642i
$544$		0			0
$545$	419.917			0.770490
$546$		0			0
$547$	−768.027			−1.40407			−0.702036	−	0.712142i	$-0.747725\pi$
$547$	−768.027			−1.40407			−0.702036	+	0.712142i	$0.747725\pi$
$548$		0			0
$549$			1001.78i			1.82473i
$550$		0			0
$551$		−	35.6842i		−	0.0647626i
$552$		0			0
$553$	−53.5028			−0.0967500
$554$		0			0
$555$	408.784			0.736547
$556$		0			0
$557$			491.942i			0.883200i	0.897212	+	0.441600i	$0.145589\pi$
$557$			491.942i			0.883200i	−0.897212	+	0.441600i	$0.854411\pi$
$558$		0			0
$559$		−	163.744i		−	0.292923i
$560$		0			0
$561$	1189.17			2.11973
$562$		0			0
$563$		−	666.364i		−	1.18359i	−0.806087	−	0.591797i	$-0.798419\pi$
$563$		−	666.364i		−	1.18359i	0.806087	−	0.591797i	$-0.201581\pi$
$564$		0			0
$565$	76.9644			0.136220
$566$		0			0
$567$			94.0917i			0.165946i
$568$		0			0
$569$			289.443i			0.508687i	0.967114	+	0.254344i	$0.0818594\pi$
$569$			289.443i			0.508687i	−0.967114	+	0.254344i	$0.918141\pi$
$570$		0			0
$571$		−	166.007i		−	0.290731i	−0.989378	−	0.145365i	$-0.953564\pi$
$571$		−	166.007i		−	0.290731i	0.989378	−	0.145365i	$-0.0464358\pi$
$572$		0			0
$573$			1596.49i			2.78619i
$574$		0			0
$575$	96.7027	+	62.2382i	0.168179	+	0.108240i
$576$		0			0
$577$	284.223			0.492588			0.246294	−	0.969195i	$-0.420787\pi$
$577$	284.223			0.492588			0.246294	+	0.969195i	$0.420787\pi$
$578$		0			0
$579$	737.219			1.27326
$580$		0			0
$581$	172.497			0.296898
$582$		0			0
$583$	−200.780			−0.344391
$584$		0			0
$585$			384.909i			0.657963i
$586$		0			0
$587$	−349.003			−0.594553			−0.297277	−	0.954791i	$-0.596078\pi$
$587$	−349.003			−0.594553			−0.297277	+	0.954791i	$0.596078\pi$
$588$		0			0
$589$			173.772i			0.295029i
$590$		0			0
$591$	965.742			1.63408
$592$		0			0
$593$	−200.503			−0.338116			−0.169058	−	0.985606i	$-0.554072\pi$
$593$	−200.503			−0.338116			−0.169058	+	0.985606i	$0.554072\pi$
$594$		0			0
$595$		−	56.9819i		−	0.0957678i
$596$		0			0
$597$			322.773i			0.540658i
$598$		0			0
$599$	−616.207			−1.02873			−0.514363	−	0.857572i	$-0.671972\pi$
$599$	−616.207			−1.02873			−0.514363	+	0.857572i	$0.671972\pi$
$600$		0			0
$601$	−730.896			−1.21613			−0.608067	−	0.793886i	$-0.708055\pi$
$601$	−730.896			−1.21613			−0.608067	+	0.793886i	$0.708055\pi$
$602$		0			0
$603$		−	683.401i		−	1.13333i
$604$		0			0
$605$		−	464.409i		−	0.767618i
$606$		0			0
$607$	651.970			1.07409			0.537043	−	0.843555i	$-0.319541\pi$
$607$	651.970			1.07409			0.537043	+	0.843555i	$0.319541\pi$
$608$		0			0
$609$		−	85.4240i		−	0.140269i
$610$		0			0
$611$	−1089.43			−1.78303
$612$		0			0
$613$			451.170i			0.736004i	0.929825	+	0.368002i	$0.119958\pi$
$613$			451.170i			0.736004i	−0.929825	+	0.368002i	$0.880042\pi$
$614$		0			0
$615$		−	4.54209i		−	0.00738552i
$616$		0			0
$617$		−	1170.11i		−	1.89646i	−0.317588	−	0.948229i	$-0.602873\pi$
$617$		−	1170.11i		−	1.89646i	0.317588	−	0.948229i	$-0.397127\pi$
$618$		0			0
$619$		−	1209.14i		−	1.95338i	−0.214656	−	0.976690i	$-0.568863\pi$
$619$		−	1209.14i		−	1.95338i	0.214656	−	0.976690i	$-0.431137\pi$
$620$		0			0
$621$	209.792	+	135.023i	0.337829	+	0.217428i
$622$		0			0
$623$	−233.764			−0.375223
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−271.135			−0.432432
$628$		0			0
$629$	587.753			0.934425
$630$		0			0
$631$			910.772i			1.44338i	0.692217	+	0.721689i	$0.256634\pi$
$631$			910.772i			1.44338i	−0.692217	+	0.721689i	$0.743366\pi$
$632$		0			0
$633$	−427.975			−0.676105
$634$		0			0
$635$		−	467.818i		−	0.736721i
$636$		0			0
$637$	693.293			1.08837
$638$		0			0
$639$	−622.523			−0.974214
$640$		0			0
$641$		−	147.213i		−	0.229662i	−0.993385	−	0.114831i	$-0.963367\pi$
$641$		−	147.213i		−	0.229662i	0.993385	−	0.114831i	$-0.0366326\pi$
$642$		0			0
$643$		−	1278.44i		−	1.98825i	−0.108247	−	0.994124i	$-0.534524\pi$
$643$		−	1278.44i		−	1.98825i	0.108247	−	0.994124i	$-0.465476\pi$
$644$		0			0
$645$	−109.540			−0.169829
$646$		0			0
$647$	−909.162			−1.40520			−0.702598	−	0.711587i	$-0.747977\pi$
$647$	−909.162			−1.40520			−0.702598	+	0.711587i	$0.747977\pi$
$648$		0			0
$649$		−	711.135i		−	1.09574i
$650$		0			0
$651$			415.991i			0.639003i
$652$		0			0
$653$	−812.029			−1.24354			−0.621768	−	0.783201i	$-0.713585\pi$
$653$	−812.029			−1.24354			−0.621768	+	0.783201i	$0.713585\pi$
$654$		0			0
$655$			88.4260i			0.135001i
$656$		0			0
$657$	1081.75			1.64649
$658$		0			0
$659$			862.069i			1.30815i	0.756431	+	0.654074i	$0.226941\pi$
$659$			862.069i			1.30815i	−0.756431	+	0.654074i	$0.773059\pi$
$660$		0			0
$661$			760.345i			1.15029i	0.818050	+	0.575147i	$0.195055\pi$
$661$			760.345i			1.15029i	−0.818050	+	0.575147i	$0.804945\pi$
$662$		0			0
$663$			990.281i			1.49364i
$664$		0			0
$665$			12.9921i			0.0195370i
$666$		0			0
$667$	208.441	+	134.153i	0.312505	+	0.201130i
$668$		0			0
$669$	−225.104			−0.336478
$670$		0			0
$671$	1592.95			2.37399
$672$		0			0
$673$	222.871			0.331161			0.165580	−	0.986196i	$-0.447050\pi$
$673$	222.871			0.331161			0.165580	+	0.986196i	$0.447050\pi$
$674$		0			0
$675$	54.2363			0.0803501
$676$		0			0
$677$			142.940i			0.211138i	0.994412	+	0.105569i	$0.0336664\pi$
$677$			142.940i			0.211138i	−0.994412	+	0.105569i	$0.966334\pi$
$678$		0			0
$679$	−106.863			−0.157383
$680$		0			0
$681$		−	1772.39i		−	2.60263i
$682$		0			0
$683$	−984.719			−1.44176			−0.720878	−	0.693062i	$-0.756261\pi$
$683$	−984.719			−1.44176			−0.720878	+	0.693062i	$0.756261\pi$
$684$		0			0
$685$	538.295			0.785832
$686$		0			0
$687$			355.185i			0.517009i
$688$		0			0
$689$		−	167.200i		−	0.242670i
$690$		0			0
$691$	−881.364			−1.27549			−0.637745	−	0.770247i	$-0.720133\pi$
$691$	−881.364			−1.27549			−0.637745	+	0.770247i	$0.720133\pi$
$692$		0			0
$693$	−362.736			−0.523428
$694$		0			0
$695$			190.904i			0.274683i
$696$		0			0
$697$		−	6.53066i		−	0.00936967i
$698$		0			0
$699$	−667.377			−0.954759
$700$		0			0
$701$			496.389i			0.708116i	0.935223	+	0.354058i	$0.115198\pi$
$701$			496.389i			0.708116i	−0.935223	+	0.354058i	$0.884802\pi$
$702$		0			0
$703$	−134.010			−0.190626
$704$		0			0
$705$			728.798i			1.03376i
$706$		0			0
$707$		−	18.2360i		−	0.0257936i
$708$		0			0
$709$		−	1231.04i		−	1.73631i	−0.496297	−	0.868153i	$-0.665307\pi$
$709$		−	1231.04i		−	1.73631i	0.496297	−	0.868153i	$-0.334693\pi$
$710$		0			0
$711$		−	347.621i		−	0.488918i
$712$		0			0
$713$	−1015.05	−	653.289i	−1.42363	−	0.916254i
$714$		0			0
$715$	612.051			0.856015
$716$		0			0
$717$	−856.591			−1.19469
$718$		0			0
$719$	85.1410			0.118416			0.0592079	−	0.998246i	$-0.481143\pi$
$719$	85.1410			0.118416			0.0592079	+	0.998246i	$0.481143\pi$
$720$		0			0
$721$	92.7362			0.128622
$722$		0			0
$723$			1754.74i			2.42703i
$724$		0			0
$725$	53.8871			0.0743270
$726$		0			0
$727$			366.107i			0.503585i	0.967781	+	0.251793i	$0.0810201\pi$
$727$			366.107i			0.503585i	−0.967781	+	0.251793i	$0.918980\pi$
$728$		0			0
$729$	−1052.29			−1.44347
$730$		0			0
$731$	−157.497			−0.215455
$732$		0			0
$733$		−	879.122i		−	1.19935i	−0.800245	−	0.599674i	$-0.795297\pi$
$733$		−	879.122i		−	1.19935i	0.800245	−	0.599674i	$-0.204703\pi$
$734$		0			0
$735$		−	463.793i		−	0.631010i
$736$		0			0
$737$	−1086.69			−1.47448
$738$		0			0
$739$	−1127.51			−1.52573			−0.762864	−	0.646559i	$-0.776208\pi$
$739$	−1127.51			−1.52573			−0.762864	+	0.646559i	$0.776208\pi$
$740$		0			0
$741$		−	225.788i		−	0.304708i
$742$		0			0
$743$		−	1365.53i		−	1.83786i	−0.394423	−	0.918929i	$-0.629055\pi$
$743$		−	1365.53i		−	1.83786i	0.394423	−	0.918929i	$-0.370945\pi$
$744$		0			0
$745$	−93.6776			−0.125742
$746$		0			0
$747$			1120.76i			1.50035i
$748$		0			0
$749$	174.428			0.232881
$750$		0			0
$751$		−	848.643i		−	1.13002i	−0.825085	−	0.565009i	$-0.808873\pi$
$751$		−	848.643i		−	1.13002i	0.825085	−	0.565009i	$-0.191127\pi$
$752$		0			0
$753$			1067.79i			1.41805i
$754$		0			0
$755$			59.5250i			0.0788410i
$756$		0			0
$757$			98.5863i			0.130233i	0.997878	+	0.0651164i	$0.0207419\pi$
$757$			98.5863i			0.130233i	−0.997878	+	0.0651164i	$0.979258\pi$
$758$		0			0
$759$	1019.32	−	1583.77i	1.34298	−	2.08666i
$760$		0			0
$761$	−126.165			−0.165788			−0.0828941	−	0.996558i	$-0.526416\pi$
$761$	−126.165			−0.165788			−0.0828941	+	0.996558i	$0.526416\pi$
$762$		0			0
$763$	−329.543			−0.431905
$764$		0			0
$765$	370.225			0.483955
$766$		0			0
$767$	592.200			0.772099
$768$		0			0
$769$		−	44.7341i		−	0.0581718i	−0.999577	−	0.0290859i	$-0.990740\pi$
$769$		−	44.7341i		−	0.0581718i	0.999577	−	0.0290859i	$-0.00925964\pi$
$770$		0			0
$771$	−30.6156			−0.0397090
$772$		0			0
$773$			153.135i			0.198105i	0.995082	+	0.0990524i	$0.0315811\pi$
$773$			153.135i			0.198105i	−0.995082	+	0.0990524i	$0.968419\pi$
$774$		0			0
$775$	−262.415			−0.338600
$776$		0			0
$777$	−320.806			−0.412878
$778$		0			0
$779$			1.48902i			0.00191145i
$780$		0			0
$781$			989.886i			1.26746i
$782$		0			0
$783$	116.905			0.149305
$784$		0			0
$785$	−324.612			−0.413518
$786$		0			0
$787$			304.484i			0.386891i	0.981111	+	0.193446i	$0.0619663\pi$
$787$			304.484i			0.386891i	−0.981111	+	0.193446i	$0.938034\pi$
$788$		0			0
$789$			582.176i			0.737865i
$790$		0			0
$791$	−60.4002			−0.0763593
$792$		0			0
$793$			1326.53i			1.67280i
$794$		0			0
$795$	−111.852			−0.140694
$796$		0			0
$797$			14.8282i			0.0186051i	0.999957	+	0.00930253i	$0.00296113\pi$
$797$			14.8282i			0.0186051i	−0.999957	+	0.00930253i	$0.997039\pi$
$798$		0			0
$799$			1047.87i			1.31148i
$800$		0			0
$801$		−	1518.82i		−	1.89616i
$802$		0			0
$803$		−	1720.11i		−	2.14210i
$804$		0			0
$805$	−75.8905	−	48.8434i	−0.0942739	−	0.0606750i
$806$		0			0
$807$	−114.142			−0.141440
$808$		0			0
$809$	−227.218			−0.280863			−0.140432	−	0.990090i	$-0.544849\pi$
$809$	−227.218			−0.280863			−0.140432	+	0.990090i	$0.544849\pi$
$810$		0			0
$811$	−457.765			−0.564446			−0.282223	−	0.959349i	$-0.591072\pi$
$811$	−457.765			−0.564446			−0.282223	+	0.959349i	$0.591072\pi$
$812$		0			0
$813$	1891.63			2.32673
$814$		0			0
$815$			232.418i			0.285176i
$816$		0			0
$817$	35.9101			0.0439536
$818$		0			0
$819$		−	302.069i		−	0.368827i
$820$		0			0
$821$	−1162.44			−1.41588			−0.707940	−	0.706272i	$-0.750375\pi$
$821$	−1162.44			−1.41588			−0.707940	+	0.706272i	$0.750375\pi$
$822$		0			0
$823$	−679.249			−0.825333			−0.412666	−	0.910882i	$-0.635403\pi$
$823$	−679.249			−0.825333			−0.412666	+	0.910882i	$0.635403\pi$
$824$		0			0
$825$		−	409.444i		−	0.496296i
$826$		0			0
$827$		−	725.358i		−	0.877095i	−0.898708	−	0.438548i	$-0.855493\pi$
$827$		−	725.358i		−	0.877095i	0.898708	−	0.438548i	$-0.144507\pi$
$828$		0			0
$829$	−372.426			−0.449248			−0.224624	−	0.974446i	$-0.572115\pi$
$829$	−372.426			−0.449248			−0.224624	+	0.974446i	$0.572115\pi$
$830$		0			0
$831$	1246.37			1.49984
$832$		0			0
$833$		−	666.845i		−	0.800535i
$834$		0			0
$835$		−	688.189i		−	0.824179i
$836$		0			0
$837$	−569.297			−0.680163
$838$		0			0
$839$			806.190i			0.960893i	0.877024	+	0.480447i	$0.159525\pi$
$839$			806.190i			0.960893i	−0.877024	+	0.480447i	$0.840475\pi$
$840$		0			0
$841$	−724.847			−0.861887
$842$		0			0
$843$			1150.45i			1.36471i
$844$		0			0
$845$			131.791i			0.155966i
$846$		0			0
$847$			364.460i			0.430295i
$848$		0			0
$849$		−	56.7209i		−	0.0668091i
$850$		0			0
$851$	503.807	−	782.791i	0.592018	−	0.919848i
$852$		0			0
$853$	391.264			0.458691			0.229346	−	0.973345i	$-0.426341\pi$
$853$	391.264			0.458691			0.229346	+	0.973345i	$0.426341\pi$
$854$		0			0
$855$	−84.4130			−0.0987287
$856$		0			0
$857$	−946.034			−1.10389			−0.551945	−	0.833880i	$-0.686114\pi$
$857$	−946.034			−1.10389			−0.551945	+	0.833880i	$0.686114\pi$
$858$		0			0
$859$	1265.11			1.47277			0.736383	−	0.676565i	$-0.236532\pi$
$859$	1265.11			1.47277			0.736383	+	0.676565i	$0.236532\pi$
$860$		0			0
$861$			3.56455i			0.00414001i
$862$		0			0
$863$	187.283			0.217013			0.108507	−	0.994096i	$-0.465393\pi$
$863$	187.283			0.217013			0.108507	+	0.994096i	$0.465393\pi$
$864$		0			0
$865$		−	431.077i		−	0.498355i
$866$		0			0
$867$	−352.853			−0.406981
$868$		0			0
$869$	−552.759			−0.636086
$870$		0			0
$871$		−	904.943i		−	1.03897i
$872$		0			0
$873$		−	694.316i		−	0.795322i
$874$		0			0
$875$	−19.6195			−0.0224223
$876$		0			0
$877$	1128.15			1.28638			0.643189	−	0.765707i	$-0.277611\pi$
$877$	1128.15			1.28638			0.643189	+	0.765707i	$0.277611\pi$
$878$		0			0
$879$			1770.52i			2.01425i
$880$		0			0
$881$		−	984.930i		−	1.11797i	−0.829178	−	0.558984i	$-0.811191\pi$
$881$		−	984.930i		−	1.11797i	0.829178	−	0.558984i	$-0.188809\pi$
$882$		0			0
$883$	1605.85			1.81863			0.909316	−	0.416106i	$-0.136605\pi$
$883$	1605.85			1.81863			0.909316	+	0.416106i	$0.136605\pi$
$884$		0			0
$885$		−	396.164i		−	0.447643i
$886$		0			0
$887$	871.151			0.982132			0.491066	−	0.871122i	$-0.336607\pi$
$887$	871.151			0.982132			0.491066	+	0.871122i	$0.336607\pi$
$888$		0			0
$889$			367.135i			0.412975i
$890$		0			0
$891$			972.099i			1.09102i
$892$		0			0
$893$		−	238.920i		−	0.267547i
$894$		0			0
$895$			212.421i			0.237341i
$896$		0			0
$897$	1318.89	+	848.844i	1.47034	+	0.946314i
$898$		0			0
$899$	−565.631			−0.629178
$900$		0			0
$901$	−160.822			−0.178492
$902$		0			0
$903$	85.9648			0.0951992
$904$		0			0
$905$	469.464			0.518745
$906$		0			0
$907$		−	1280.61i		−	1.41192i	−0.708252	−	0.705960i	$-0.750516\pi$
$907$		−	1280.61i		−	1.41192i	0.708252	−	0.705960i	$-0.249484\pi$
$908$		0			0
$909$	118.484			0.130346
$910$		0			0
$911$			514.004i			0.564220i	0.959382	+	0.282110i	$0.0910342\pi$
$911$			514.004i			0.564220i	−0.959382	+	0.282110i	$0.908966\pi$
$912$		0			0
$913$	1782.14			1.95196
$914$		0			0
$915$	887.410			0.969846
$916$		0			0
$917$		−	69.3951i		−	0.0756762i
$918$		0			0
$919$		−	936.617i		−	1.01917i	−0.860420	−	0.509585i	$-0.829799\pi$
$919$		−	936.617i		−	1.01917i	0.860420	−	0.509585i	$-0.170201\pi$
$920$		0			0
$921$	1626.98			1.76653
$922$		0			0
$923$	−824.330			−0.893099
$924$		0			0
$925$		−	202.370i		−	0.218779i
$926$		0			0
$927$			602.530i			0.649978i
$928$		0			0
$929$	788.720			0.848999			0.424500	−	0.905428i	$-0.360450\pi$
$929$	788.720			0.848999			0.424500	+	0.905428i	$0.360450\pi$
$930$		0			0
$931$			152.044i			0.163312i
$932$		0			0
$933$	−1094.74			−1.17336
$934$		0			0
$935$		−	588.703i		−	0.629628i
$936$		0			0
$937$		−	127.970i		−	0.136575i	−0.997666	−	0.0682873i	$-0.978247\pi$
$937$		−	127.970i		−	0.136575i	0.997666	−	0.0682873i	$-0.0217535\pi$
$938$		0			0
$939$			2749.53i			2.92815i
$940$		0			0
$941$			953.695i			1.01349i	0.862096	+	0.506746i	$0.169152\pi$
$941$			953.695i			1.01349i	−0.862096	+	0.506746i	$0.830848\pi$
$942$		0			0
$943$	−8.69777	−	5.59792i	−0.00922351	−	0.00593629i
$944$		0			0
$945$	−42.5637			−0.0450409
$946$		0			0
$947$	−487.630			−0.514920			−0.257460	−	0.966289i	$-0.582886\pi$
$947$	−487.630			−0.514920			−0.257460	+	0.966289i	$0.582886\pi$
$948$		0			0
$949$	1432.42			1.50940
$950$		0			0
$951$	1322.14			1.39026
$952$		0			0
$953$			763.062i			0.800695i	0.916363	+	0.400347i	$0.131110\pi$
$953$			763.062i			0.800695i	−0.916363	+	0.400347i	$0.868890\pi$
$954$		0			0
$955$	790.350			0.827592
$956$		0			0
$957$		−	882.550i		−	0.922205i
$958$		0			0
$959$	−422.444			−0.440505
$960$		0			0
$961$	1793.46			1.86625
$962$		0			0
$963$			1133.30i			1.17684i
$964$		0			0
$965$		−	364.964i		−	0.378201i
$966$		0			0
$967$	1093.33			1.13064			0.565321	−	0.824871i	$-0.308752\pi$
$967$	1093.33			1.13064			0.565321	+	0.824871i	$0.308752\pi$
$968$		0			0
$969$	−217.175			−0.224123
$970$		0			0
$971$		−	379.535i		−	0.390870i	−0.980717	−	0.195435i	$-0.937388\pi$
$971$		−	379.535i		−	0.390870i	0.980717	−	0.195435i	$-0.0626118\pi$
$972$		0			0
$973$		−	149.818i		−	0.153976i
$974$		0			0
$975$	340.966			0.349708
$976$		0			0
$977$			549.384i			0.562317i	0.959661	+	0.281159i	$0.0907187\pi$
$977$			549.384i			0.562317i	−0.959661	+	0.281159i	$0.909281\pi$
$978$		0			0
$979$	−2415.11			−2.46692
$980$		0			0
$981$		−	2141.13i		−	2.18259i
$982$		0			0
$983$			522.297i			0.531330i	0.964065	+	0.265665i	$0.0855915\pi$
$983$			522.297i			0.531330i	−0.964065	+	0.265665i	$0.914409\pi$
$984$		0			0
$985$		−	478.096i		−	0.485376i
$986$		0			0
$987$		−	571.947i		−	0.579480i
$988$		0			0
$989$	−135.003	+	209.761i	−0.136504	+	0.212094i
$990$		0			0
$991$	798.166			0.805414			0.402707	−	0.915329i	$-0.368069\pi$
$991$	798.166			0.805414			0.402707	+	0.915329i	$0.368069\pi$
$992$		0			0
$993$	−209.737			−0.211215
$994$		0			0
$995$	159.790			0.160593
$996$		0			0
$997$	−793.287			−0.795674			−0.397837	−	0.917456i	$-0.630239\pi$
$997$	−793.287			−0.795674			−0.397837	+	0.917456i	$0.630239\pi$
$998$		0			0
$999$		−	439.033i		−	0.439473i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.e.321.8		48
4.3	odd	2		920.3.k.a.321.42	yes	48
23.22	odd	2	inner	1840.3.k.e.321.7		48
92.91	even	2		920.3.k.a.321.41	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.3.k.a.321.41	✓	48	92.91	even	2
920.3.k.a.321.42	yes	48	4.3	odd	2
1840.3.k.e.321.7		48	23.22	odd	2	inner
1840.3.k.e.321.8		48	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)

\(f(q)\)	\(=\)	\(q-4.51681 q^{3} +2.23607i q^{5} -1.75483i q^{7} +11.4015 q^{9} +O(q^{10})\)	\(q-4.51681 q^{3} +2.23607i q^{5} -1.75483i q^{7} +11.4015 q^{9} -18.1298i q^{11} +15.0976 q^{13} -10.0999i q^{15} -14.5217i q^{17} +3.31102i q^{19} +7.92620i q^{21} +(-19.3405 - 12.4476i) q^{23} -5.00000 q^{25} -10.8473 q^{27} -10.7774 q^{29} +52.4830 q^{31} +81.8888i q^{33} +3.92391 q^{35} +40.4741i q^{37} -68.1931 q^{39} +0.449717 q^{41} -10.8456i q^{43} +25.4946i q^{45} -72.1590 q^{47} +45.9206 q^{49} +65.5917i q^{51} -11.0746i q^{53} +40.5395 q^{55} -14.9552i q^{57} +39.2246 q^{59} +87.8633i q^{61} -20.0077i q^{63} +33.7594i q^{65} -59.9394i q^{67} +(87.3574 + 56.2236i) q^{69} -54.5999 q^{71} +94.8773 q^{73} +22.5840 q^{75} -31.8146 q^{77} -30.4889i q^{79} -53.6188 q^{81} +98.2989i q^{83} +32.4715 q^{85} +48.6795 q^{87} -133.212i q^{89} -26.4937i q^{91} -237.055 q^{93} -7.40366 q^{95} -60.8967i q^{97} -206.708i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 128 q^{9}+O(q^{10}) \) 48 * q + 128 * q^9	\( 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) \) 48 * q + 128 * q^9 - 8 * q^23 - 240 * q^25 + 72 * q^29 - 32 * q^31 + 40 * q^35 + 96 * q^39 - 104 * q^41 - 128 * q^47 - 344 * q^49 - 80 * q^55 - 248 * q^59 + 292 * q^69 - 208 * q^71 + 224 * q^73 - 288 * q^77 + 184 * q^81 + 48 * q^87 - 672 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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