LMFDB - Embedded newform 320.6.d.c.161.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(161,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("320.161");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.d (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:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + \cdots + 653157349 $ x^12 - 343x^10 - 696x^9 + 44406x^8 + 179640x^7 - 2401691x^6 - 15554592x^5 + 26901210x^4 + 434775816x^3 + 1271335685x^2 + 1475231592x + 653157349
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.7
Root			$-3.41426 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	320.161
Dual form			320.6.d.c.161.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.09968i q^{3} -25.0000i q^{5} +118.821 q^{7} +233.392 q^{9} +O(q^{10})$	$q+3.09968i q^{3} -25.0000i q^{5} +118.821 q^{7} +233.392 q^{9} -355.467i q^{11} -370.735i q^{13} +77.4920 q^{15} -586.019 q^{17} -972.770i q^{19} +368.307i q^{21} +1298.93 q^{23} -625.000 q^{25} +1476.66i q^{27} +5226.75i q^{29} -4622.64 q^{31} +1101.83 q^{33} -2970.52i q^{35} -5296.01i q^{37} +1149.16 q^{39} +14042.0 q^{41} -4229.77i q^{43} -5834.80i q^{45} -18577.1 q^{47} -2688.58 q^{49} -1816.47i q^{51} -20963.9i q^{53} -8886.67 q^{55} +3015.28 q^{57} -48573.7i q^{59} -3620.88i q^{61} +27731.9 q^{63} -9268.37 q^{65} -1683.52i q^{67} +4026.27i q^{69} -5332.40 q^{71} +80560.7 q^{73} -1937.30i q^{75} -42236.9i q^{77} -39272.8 q^{79} +52137.1 q^{81} -18385.7i q^{83} +14650.5i q^{85} -16201.3 q^{87} +43137.3 q^{89} -44051.1i q^{91} -14328.7i q^{93} -24319.2 q^{95} +134744. q^{97} -82963.1i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 268 q^{7} - 428 q^{9}+O(q^{10}) $ 12 * q - 268 * q^7 - 428 * q^9	$ 12 q - 268 q^{7} - 428 q^{9} + 900 q^{15} + 2400 q^{17} - 8108 q^{23} - 7500 q^{25} - 7976 q^{31} - 20776 q^{33} - 40984 q^{39} - 56408 q^{41} - 21172 q^{47} - 5540 q^{49} - 18400 q^{55} + 39992 q^{57} - 179516 q^{63} + 44000 q^{65} - 367704 q^{71} + 58736 q^{73} - 26192 q^{79} + 411692 q^{81} - 183200 q^{87} + 87672 q^{89} - 121000 q^{95} - 172336 q^{97}+O(q^{100}) $ 12 * q - 268 * q^7 - 428 * q^9 + 900 * q^15 + 2400 * q^17 - 8108 * q^23 - 7500 * q^25 - 7976 * q^31 - 20776 * q^33 - 40984 * q^39 - 56408 * q^41 - 21172 * q^47 - 5540 * q^49 - 18400 * q^55 + 39992 * q^57 - 179516 * q^63 + 44000 * q^65 - 367704 * q^71 + 58736 * q^73 - 26192 * q^79 + 411692 * q^81 - 183200 * q^87 + 87672 * q^89 - 121000 * q^95 - 172336 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	3.09968i	0.198845i	0.995045	+	0.0994223i	$0.0316995\pi$
$3$	3.09968i	0.198845i	−0.995045	+	0.0994223i	$0.968301\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	118.821	0.916533	0.458266	−	0.888815i	$-0.348471\pi$
$7$	118.821	0.916533	0.458266	+	0.888815i	$0.348471\pi$
$8$	0	0
$9$	233.392	0.960461
$10$	0	0
$11$	− 355.467i	− 0.885763i	−0.896580	−	0.442881i	$-0.853956\pi$
$11$	− 355.467i	− 0.885763i	0.896580	−	0.442881i	$-0.146044\pi$
$12$	0	0
$13$	− 370.735i	− 0.608422i	−0.952605	−	0.304211i	$-0.901607\pi$
$13$	− 370.735i	− 0.608422i	0.952605	−	0.304211i	$-0.0983928\pi$
$14$	0	0
$15$	77.4920	0.0889260
$16$	0	0
$17$	−586.019	−0.491801	−0.245900	−	0.969295i	$-0.579084\pi$
$17$	−586.019	−0.491801	−0.245900	+	0.969295i	$0.579084\pi$
$18$	0	0
$19$	− 972.770i	− 0.618196i	−0.951030	−	0.309098i	$-0.899973\pi$
$19$	− 972.770i	− 0.618196i	0.951030	−	0.309098i	$-0.100027\pi$
$20$	0	0
$21$	368.307i	0.182248i
$22$	0	0
$23$	1298.93	0.511996	0.255998	−	0.966677i	$-0.417596\pi$
$23$	1298.93	0.511996	0.255998	+	0.966677i	$0.417596\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	1476.66i	0.389827i
$28$	0	0
$29$	5226.75i	1.15408i	0.816715	+	0.577041i	$0.195793\pi$
$29$	5226.75i	1.15408i	−0.816715	+	0.577041i	$0.804207\pi$
$30$	0	0
$31$	−4622.64	−0.863944	−0.431972	−	0.901887i	$-0.642182\pi$
$31$	−4622.64	−0.863944	−0.431972	+	0.901887i	$0.642182\pi$
$32$	0	0
$33$	1101.83	0.176129
$34$	0	0
$35$	− 2970.52i	− 0.409886i
$36$	0	0
$37$	− 5296.01i	− 0.635982i	−0.948094	−	0.317991i	$-0.896992\pi$
$37$	− 5296.01i	− 0.635982i	0.948094	−	0.317991i	$-0.103008\pi$
$38$	0	0
$39$	1149.16	0.120981
$40$	0	0
$41$	14042.0	1.30457	0.652286	−	0.757973i	$-0.273810\pi$
$41$	14042.0	1.30457	0.652286	+	0.757973i	$0.273810\pi$
$42$	0	0
$43$	− 4229.77i	− 0.348856i	−0.984670	−	0.174428i	$-0.944192\pi$
$43$	− 4229.77i	− 0.348856i	0.984670	−	0.174428i	$-0.0558076\pi$
$44$	0	0
$45$	− 5834.80i	− 0.429531i
$46$	0	0
$47$	−18577.1	−1.22668	−0.613341	−	0.789818i	$-0.710175\pi$
$47$	−18577.1	−1.22668	−0.613341	+	0.789818i	$0.710175\pi$
$48$	0	0
$49$	−2688.58	−0.159968
$50$	0	0
$51$	− 1816.47i	− 0.0977920i
$52$	0	0
$53$	− 20963.9i	− 1.02514i	−0.858645	−	0.512570i	$-0.828694\pi$
$53$	− 20963.9i	− 1.02514i	0.858645	−	0.512570i	$-0.171306\pi$
$54$	0	0
$55$	−8886.67	−0.396125
$56$	0	0
$57$	3015.28	0.122925
$58$	0	0
$59$	− 48573.7i	− 1.81665i	−0.418265	−	0.908325i	$-0.637362\pi$
$59$	− 48573.7i	− 1.81665i	0.418265	−	0.908325i	$-0.362638\pi$
$60$	0	0
$61$	− 3620.88i	− 0.124592i	−0.998058	−	0.0622958i	$-0.980158\pi$
$61$	− 3620.88i	− 0.124592i	0.998058	−	0.0622958i	$-0.0198422\pi$
$62$	0	0
$63$	27731.9	0.880294
$64$	0	0
$65$	−9268.37	−0.272095
$66$	0	0
$67$	− 1683.52i	− 0.0458174i	−0.999738	−	0.0229087i	$-0.992707\pi$
$67$	− 1683.52i	− 0.0458174i	0.999738	−	0.0229087i	$-0.00729270\pi$
$68$	0	0
$69$	4026.27i	0.101808i
$70$	0	0
$71$	−5332.40	−0.125538	−0.0627692	−	0.998028i	$-0.519993\pi$
$71$	−5332.40	−0.125538	−0.0627692	+	0.998028i	$0.519993\pi$
$72$	0	0
$73$	80560.7	1.76936	0.884680	−	0.466199i	$-0.154377\pi$
$73$	80560.7	1.76936	0.884680	+	0.466199i	$0.154377\pi$
$74$	0	0
$75$	− 1937.30i	− 0.0397689i
$76$	0	0
$77$	− 42236.9i	− 0.811831i
$78$	0	0
$79$	−39272.8	−0.707985	−0.353992	−	0.935248i	$-0.615176\pi$
$79$	−39272.8	−0.707985	−0.353992	+	0.935248i	$0.615176\pi$
$80$	0	0
$81$	52137.1	0.882946
$82$	0	0
$83$	− 18385.7i	− 0.292944i	−0.989215	−	0.146472i	$-0.953208\pi$
$83$	− 18385.7i	− 0.292944i	0.989215	−	0.146472i	$-0.0467918\pi$
$84$	0	0
$85$	14650.5i	0.219940i
$86$	0	0
$87$	−16201.3	−0.229483
$88$	0	0
$89$	43137.3	0.577269	0.288635	−	0.957439i	$-0.406799\pi$
$89$	43137.3	0.577269	0.288635	+	0.957439i	$0.406799\pi$
$90$	0	0
$91$	− 44051.1i	− 0.557639i
$92$	0	0
$93$	− 14328.7i	− 0.171791i
$94$	0	0
$95$	−24319.2	−0.276466
$96$	0	0
$97$	134744.	1.45405	0.727027	−	0.686609i	$-0.240902\pi$
$97$	134744.	1.45405	0.727027	+	0.686609i	$0.240902\pi$
$98$	0	0
$99$	− 82963.1i	− 0.850740i
$100$	0	0
$101$	− 39471.3i	− 0.385015i	−0.981296	−	0.192507i	$-0.938338\pi$
$101$	− 39471.3i	− 0.385015i	0.981296	−	0.192507i	$-0.0616620\pi$
$102$	0	0
$103$	−164437.	−1.52724	−0.763620	−	0.645666i	$-0.776580\pi$
$103$	−164437.	−1.52724	−0.763620	+	0.645666i	$0.776580\pi$
$104$	0	0
$105$	9207.68	0.0815036
$106$	0	0
$107$	− 203860.i	− 1.72137i	−0.509142	−	0.860683i	$-0.670037\pi$
$107$	− 203860.i	− 1.72137i	0.509142	−	0.860683i	$-0.329963\pi$
$108$	0	0
$109$	− 65561.7i	− 0.528548i	−0.964448	−	0.264274i	$-0.914868\pi$
$109$	− 65561.7i	− 0.528548i	0.964448	−	0.264274i	$-0.0851322\pi$
$110$	0	0
$111$	16415.9	0.126462
$112$	0	0
$113$	35057.0	0.258273	0.129136	−	0.991627i	$-0.458780\pi$
$113$	35057.0	0.258273	0.129136	+	0.991627i	$0.458780\pi$
$114$	0	0
$115$	− 32473.3i	− 0.228972i
$116$	0	0
$117$	− 86526.6i	− 0.584366i
$118$	0	0
$119$	−69631.3	−0.450752
$120$	0	0
$121$	34694.3	0.215424
$122$	0	0
$123$	43525.6i	0.259407i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	157228.	0.865007	0.432503	−	0.901632i	$-0.357630\pi$
$127$	157228.	0.865007	0.432503	+	0.901632i	$0.357630\pi$
$128$	0	0
$129$	13110.9	0.0693681
$130$	0	0
$131$	− 85680.9i	− 0.436220i	−0.975924	−	0.218110i	$-0.930011\pi$
$131$	− 85680.9i	− 0.436220i	0.975924	−	0.218110i	$-0.0699892\pi$
$132$	0	0
$133$	− 115585.i	− 0.566597i
$134$	0	0
$135$	36916.6	0.174336
$136$	0	0
$137$	−99340.8	−0.452196	−0.226098	−	0.974105i	$-0.572597\pi$
$137$	−99340.8	−0.452196	−0.226098	+	0.974105i	$0.572597\pi$
$138$	0	0
$139$	36370.0i	0.159664i	0.996808	+	0.0798319i	$0.0254384\pi$
$139$	36370.0i	0.159664i	−0.996808	+	0.0798319i	$0.974562\pi$
$140$	0	0
$141$	− 57582.9i	− 0.243919i
$142$	0	0
$143$	−131784.	−0.538918
$144$	0	0
$145$	130669.	0.516121
$146$	0	0
$147$	− 8333.74i	− 0.0318088i
$148$	0	0
$149$	− 241962.i	− 0.892857i	−0.894819	−	0.446428i	$-0.852696\pi$
$149$	− 241962.i	− 0.892857i	0.894819	−	0.446428i	$-0.147304\pi$
$150$	0	0
$151$	−241079.	−0.860433	−0.430217	−	0.902726i	$-0.641563\pi$
$151$	−241079.	−0.860433	−0.430217	+	0.902726i	$0.641563\pi$
$152$	0	0
$153$	−136772.	−0.472356
$154$	0	0
$155$	115566.i	0.386367i
$156$	0	0
$157$	450362.i	1.45818i	0.684416	+	0.729092i	$0.260057\pi$
$157$	450362.i	1.45818i	−0.684416	+	0.729092i	$0.739943\pi$
$158$	0	0
$159$	64981.5	0.203844
$160$	0	0
$161$	154340.	0.469261
$162$	0	0
$163$	− 388586.i	− 1.14556i	−0.819709	−	0.572780i	$-0.805865\pi$
$163$	− 388586.i	− 1.14556i	0.819709	−	0.572780i	$-0.194135\pi$
$164$	0	0
$165$	− 27545.8i	− 0.0787674i
$166$	0	0
$167$	−16419.7	−0.0455589	−0.0227794	−	0.999741i	$-0.507252\pi$
$167$	−16419.7	−0.0455589	−0.0227794	+	0.999741i	$0.507252\pi$
$168$	0	0
$169$	233849.	0.629822
$170$	0	0
$171$	− 227037.i	− 0.593753i
$172$	0	0
$173$	− 111836.i	− 0.284097i	−0.989860	−	0.142048i	$-0.954631\pi$
$173$	− 111836.i	− 0.284097i	0.989860	−	0.142048i	$-0.0453689\pi$
$174$	0	0
$175$	−74263.1	−0.183307
$176$	0	0
$177$	150563.	0.361231
$178$	0	0
$179$	432673.i	1.00932i	0.863319	+	0.504658i	$0.168381\pi$
$179$	432673.i	1.00932i	−0.863319	+	0.504658i	$0.831619\pi$
$180$	0	0
$181$	800005.i	1.81508i	0.419962	+	0.907542i	$0.362043\pi$
$181$	800005.i	1.81508i	−0.419962	+	0.907542i	$0.637957\pi$
$182$	0	0
$183$	11223.6	0.0247744
$184$	0	0
$185$	−132400.	−0.284420
$186$	0	0
$187$	208310.i	0.435619i
$188$	0	0
$189$	175459.i	0.357289i
$190$	0	0
$191$	372676.	0.739177	0.369588	−	0.929196i	$-0.379499\pi$
$191$	372676.	0.739177	0.369588	+	0.929196i	$0.379499\pi$
$192$	0	0
$193$	−784772.	−1.51653	−0.758263	−	0.651948i	$-0.773952\pi$
$193$	−784772.	−1.51653	−0.758263	+	0.651948i	$0.773952\pi$
$194$	0	0
$195$	− 28729.0i	− 0.0541046i
$196$	0	0
$197$	− 328945.i	− 0.603889i	−0.953325	−	0.301945i	$-0.902364\pi$
$197$	− 328945.i	− 0.603889i	0.953325	−	0.301945i	$-0.0976358\pi$
$198$	0	0
$199$	473709.	0.847968	0.423984	−	0.905670i	$-0.360631\pi$
$199$	473709.	0.847968	0.423984	+	0.905670i	$0.360631\pi$
$200$	0	0
$201$	5218.36	0.00911054
$202$	0	0
$203$	621047.i	1.05775i
$204$	0	0
$205$	− 351049.i	− 0.583423i
$206$	0	0
$207$	303160.	0.491752
$208$	0	0
$209$	−345787.	−0.547575
$210$	0	0
$211$	704656.i	1.08961i	0.838563	+	0.544805i	$0.183396\pi$
$211$	704656.i	1.08961i	−0.838563	+	0.544805i	$0.816604\pi$
$212$	0	0
$213$	− 16528.7i	− 0.0249626i
$214$	0	0
$215$	−105744.	−0.156013
$216$	0	0
$217$	−549266.	−0.791833
$218$	0	0
$219$	249712.i	0.351828i
$220$	0	0
$221$	217258.i	0.299223i
$222$	0	0
$223$	487789.	0.656855	0.328427	−	0.944529i	$-0.393481\pi$
$223$	487789.	0.656855	0.328427	+	0.944529i	$0.393481\pi$
$224$	0	0
$225$	−145870.	−0.192092
$226$	0	0
$227$	280630.i	0.361467i	0.983532	+	0.180734i	$0.0578472\pi$
$227$	280630.i	0.361467i	−0.983532	+	0.180734i	$0.942153\pi$
$228$	0	0
$229$	336464.i	0.423984i	0.977271	+	0.211992i	$0.0679950\pi$
$229$	336464.i	0.423984i	−0.977271	+	0.211992i	$0.932005\pi$
$230$	0	0
$231$	130921.	0.161428
$232$	0	0
$233$	−248441.	−0.299802	−0.149901	−	0.988701i	$-0.547895\pi$
$233$	−248441.	−0.299802	−0.149901	+	0.988701i	$0.547895\pi$
$234$	0	0
$235$	464426.i	0.548589i
$236$	0	0
$237$	− 121733.i	− 0.140779i
$238$	0	0
$239$	685682.	0.776476	0.388238	−	0.921559i	$-0.373084\pi$
$239$	685682.	0.776476	0.388238	+	0.921559i	$0.373084\pi$
$240$	0	0
$241$	−762667.	−0.845848	−0.422924	−	0.906165i	$-0.638996\pi$
$241$	−762667.	−0.845848	−0.422924	+	0.906165i	$0.638996\pi$
$242$	0	0
$243$	520437.i	0.565396i
$244$	0	0
$245$	67214.5i	0.0715398i
$246$	0	0
$247$	−360640.	−0.376124
$248$	0	0
$249$	56989.8	0.0582504
$250$	0	0
$251$	− 92462.4i	− 0.0926362i	−0.998927	−	0.0463181i	$-0.985251\pi$
$251$	− 92462.4i	− 0.0926362i	0.998927	−	0.0463181i	$-0.0147488\pi$
$252$	0	0
$253$	− 461727.i	− 0.453507i
$254$	0	0
$255$	−45411.8	−0.0437339
$256$	0	0
$257$	−715466.	−0.675704	−0.337852	−	0.941199i	$-0.609700\pi$
$257$	−715466.	−0.675704	−0.337852	+	0.941199i	$0.609700\pi$
$258$	0	0
$259$	− 629277.i	− 0.582898i
$260$	0	0
$261$	1.21988e6i	1.10845i
$262$	0	0
$263$	−2.07592e6	−1.85064	−0.925320	−	0.379187i	$-0.876204\pi$
$263$	−2.07592e6	−1.85064	−0.925320	+	0.379187i	$0.876204\pi$
$264$	0	0
$265$	−524098.	−0.458457
$266$	0	0
$267$	133712.i	0.114787i
$268$	0	0
$269$	1.61680e6i	1.36231i	0.732140	+	0.681154i	$0.238522\pi$
$269$	1.61680e6i	1.36231i	−0.732140	+	0.681154i	$0.761478\pi$
$270$	0	0
$271$	2.21414e6	1.83139	0.915695	−	0.401873i	$-0.131641\pi$
$271$	2.21414e6	1.83139	0.915695	+	0.401873i	$0.131641\pi$
$272$	0	0
$273$	136544.	0.110883
$274$	0	0
$275$	222167.i	0.177153i
$276$	0	0
$277$	568957.i	0.445533i	0.974872	+	0.222767i	$0.0715088\pi$
$277$	568957.i	0.445533i	−0.974872	+	0.222767i	$0.928491\pi$
$278$	0	0
$279$	−1.07889e6	−0.829784
$280$	0	0
$281$	786523.	0.594217	0.297109	−	0.954844i	$-0.403978\pi$
$281$	786523.	0.594217	0.297109	+	0.954844i	$0.403978\pi$
$282$	0	0
$283$	2.33703e6i	1.73460i	0.497789	+	0.867298i	$0.334145\pi$
$283$	2.33703e6i	1.73460i	−0.497789	+	0.867298i	$0.665855\pi$
$284$	0	0
$285$	− 75381.9i	− 0.0549737i
$286$	0	0
$287$	1.66848e6	1.19568
$288$	0	0
$289$	−1.07644e6	−0.758132
$290$	0	0
$291$	417664.i	0.289131i
$292$	0	0
$293$	− 1.06744e6i	− 0.726398i	−0.931712	−	0.363199i	$-0.881685\pi$
$293$	− 1.06744e6i	− 0.726398i	0.931712	−	0.363199i	$-0.118315\pi$
$294$	0	0
$295$	−1.21434e6	−0.812431
$296$	0	0
$297$	524905.	0.345294
$298$	0	0
$299$	− 481559.i	− 0.311510i
$300$	0	0
$301$	− 502585.i	− 0.319738i
$302$	0	0
$303$	122348.	0.0765582
$304$	0	0
$305$	−90521.9	−0.0557191
$306$	0	0
$307$	345396.i	0.209156i	0.994517	+	0.104578i	$0.0333492\pi$
$307$	345396.i	0.209156i	−0.994517	+	0.104578i	$0.966651\pi$
$308$	0	0
$309$	− 509703.i	− 0.303683i
$310$	0	0
$311$	1.38990e6	0.814860	0.407430	−	0.913236i	$-0.366425\pi$
$311$	1.38990e6	0.814860	0.407430	+	0.913236i	$0.366425\pi$
$312$	0	0
$313$	−1.35897e6	−0.784061	−0.392031	−	0.919952i	$-0.628227\pi$
$313$	−1.35897e6	−0.784061	−0.392031	+	0.919952i	$0.628227\pi$
$314$	0	0
$315$	− 693296.i	− 0.393679i
$316$	0	0
$317$	2.04882e6i	1.14513i	0.819859	+	0.572565i	$0.194052\pi$
$317$	2.04882e6i	1.14513i	−0.819859	+	0.572565i	$0.805948\pi$
$318$	0	0
$319$	1.85794e6	1.02224
$320$	0	0
$321$	631902.	0.342284
$322$	0	0
$323$	570061.i	0.304029i
$324$	0	0
$325$	231709.i	0.121684i
$326$	0	0
$327$	203220.	0.105099
$328$	0	0
$329$	−2.20734e6	−1.12429
$330$	0	0
$331$	− 2.75605e6i	− 1.38267i	−0.722536	−	0.691333i	$-0.757024\pi$
$331$	− 2.75605e6i	− 1.38267i	0.722536	−	0.691333i	$-0.242976\pi$
$332$	0	0
$333$	− 1.23605e6i	− 0.610836i
$334$	0	0
$335$	−42087.9	−0.0204902
$336$	0	0
$337$	1.96450e6	0.942275	0.471137	−	0.882060i	$-0.343844\pi$
$337$	1.96450e6	0.942275	0.471137	+	0.882060i	$0.343844\pi$
$338$	0	0
$339$	108665.i	0.0513561i
$340$	0	0
$341$	1.64319e6i	0.765249i
$342$	0	0
$343$	−2.31648e6	−1.06315
$344$	0	0
$345$	100657.	0.0455298
$346$	0	0
$347$	− 2.37477e6i	− 1.05876i	−0.848384	−	0.529381i	$-0.822424\pi$
$347$	− 2.37477e6i	− 1.05876i	0.848384	−	0.529381i	$-0.177576\pi$
$348$	0	0
$349$	− 1.89577e6i	− 0.833146i	−0.909102	−	0.416573i	$-0.863231\pi$
$349$	− 1.89577e6i	− 0.833146i	0.909102	−	0.416573i	$-0.136769\pi$
$350$	0	0
$351$	547451.	0.237179
$352$	0	0
$353$	89473.2	0.0382170	0.0191085	−	0.999817i	$-0.493917\pi$
$353$	89473.2	0.0382170	0.0191085	+	0.999817i	$0.493917\pi$
$354$	0	0
$355$	133310.i	0.0561425i
$356$	0	0
$357$	− 215835.i	− 0.0896295i
$358$	0	0
$359$	−4.27153e6	−1.74923	−0.874615	−	0.484818i	$-0.838886\pi$
$359$	−4.27153e6	−1.74923	−0.874615	+	0.484818i	$0.838886\pi$
$360$	0	0
$361$	1.52982e6	0.617834
$362$	0	0
$363$	107541.i	0.0428360i
$364$	0	0
$365$	− 2.01402e6i	− 0.791282i
$366$	0	0
$367$	2.20466e6	0.854429	0.427214	−	0.904150i	$-0.359495\pi$
$367$	2.20466e6	0.854429	0.427214	+	0.904150i	$0.359495\pi$
$368$	0	0
$369$	3.27728e6	1.25299
$370$	0	0
$371$	− 2.49095e6i	− 0.939574i
$372$	0	0
$373$	− 5.02519e6i	− 1.87017i	−0.354427	−	0.935084i	$-0.615324\pi$
$373$	− 5.02519e6i	− 1.87017i	0.354427	−	0.935084i	$-0.384676\pi$
$374$	0	0
$375$	−48432.5	−0.0177852
$376$	0	0
$377$	1.93774e6	0.702169
$378$	0	0
$379$	128535.i	0.0459647i	0.999736	+	0.0229823i	$0.00731615\pi$
$379$	128535.i	0.0459647i	−0.999736	+	0.0229823i	$0.992684\pi$
$380$	0	0
$381$	487355.i	0.172002i
$382$	0	0
$383$	−3.97607e6	−1.38502	−0.692512	−	0.721406i	$-0.743496\pi$
$383$	−3.97607e6	−1.38502	−0.692512	+	0.721406i	$0.743496\pi$
$384$	0	0
$385$	−1.05592e6	−0.363062
$386$	0	0
$387$	− 987195.i	− 0.335062i
$388$	0	0
$389$	724285.i	0.242681i	0.992611	+	0.121340i	$0.0387192\pi$
$389$	724285.i	0.242681i	−0.992611	+	0.121340i	$0.961281\pi$
$390$	0	0
$391$	−761198.	−0.251800
$392$	0	0
$393$	265584.	0.0867401
$394$	0	0
$395$	981819.i	0.316620i
$396$	0	0
$397$	3.92474e6i	1.24978i	0.780711	+	0.624892i	$0.214857\pi$
$397$	3.92474e6i	1.24978i	−0.780711	+	0.624892i	$0.785143\pi$
$398$	0	0
$399$	358278.	0.112665
$400$	0	0
$401$	5.85395e6	1.81798	0.908988	−	0.416822i	$-0.136856\pi$
$401$	5.85395e6	1.81798	0.908988	+	0.416822i	$0.136856\pi$
$402$	0	0
$403$	1.71377e6i	0.525643i
$404$	0	0
$405$	− 1.30343e6i	− 0.394865i
$406$	0	0
$407$	−1.88256e6	−0.563329
$408$	0	0
$409$	2.05857e6	0.608496	0.304248	−	0.952593i	$-0.401595\pi$
$409$	2.05857e6	0.608496	0.304248	+	0.952593i	$0.401595\pi$
$410$	0	0
$411$	− 307925.i	− 0.0899167i
$412$	0	0
$413$	− 5.77158e6i	− 1.66502i
$414$	0	0
$415$	−459642.	−0.131009
$416$	0	0
$417$	−112735.	−0.0317483
$418$	0	0
$419$	− 1.30873e6i	− 0.364180i	−0.983282	−	0.182090i	$-0.941714\pi$
$419$	− 1.30873e6i	− 0.364180i	0.983282	−	0.182090i	$-0.0582863\pi$
$420$	0	0
$421$	4.48649e6i	1.23368i	0.787089	+	0.616839i	$0.211587\pi$
$421$	4.48649e6i	1.23368i	−0.787089	+	0.616839i	$0.788413\pi$
$422$	0	0
$423$	−4.33574e6	−1.17818
$424$	0	0
$425$	366262.	0.0983602
$426$	0	0
$427$	− 430236.i	− 0.114192i
$428$	0	0
$429$	− 408488.i	− 0.107161i
$430$	0	0
$431$	6.60957e6	1.71388	0.856940	−	0.515417i	$-0.172363\pi$
$431$	6.60957e6	1.71388	0.856940	+	0.515417i	$0.172363\pi$
$432$	0	0
$433$	−6.04227e6	−1.54875	−0.774373	−	0.632729i	$-0.781935\pi$
$433$	−6.04227e6	−1.54875	−0.774373	+	0.632729i	$0.781935\pi$
$434$	0	0
$435$	405031.i	0.102628i
$436$	0	0
$437$	− 1.26356e6i	− 0.316514i
$438$	0	0
$439$	656087.	0.162480	0.0812401	−	0.996695i	$-0.474112\pi$
$439$	656087.	0.162480	0.0812401	+	0.996695i	$0.474112\pi$
$440$	0	0
$441$	−627493.	−0.153643
$442$	0	0
$443$	− 5.38448e6i	− 1.30357i	−0.758403	−	0.651786i	$-0.774020\pi$
$443$	− 5.38448e6i	− 1.30357i	0.758403	−	0.651786i	$-0.225980\pi$
$444$	0	0
$445$	− 1.07843e6i	− 0.258163i
$446$	0	0
$447$	750006.	0.177540
$448$	0	0
$449$	7.00212e6	1.63913	0.819565	−	0.572986i	$-0.194215\pi$
$449$	7.00212e6	1.63913	0.819565	+	0.572986i	$0.194215\pi$
$450$	0	0
$451$	− 4.99145e6i	− 1.15554i
$452$	0	0
$453$	− 747268.i	− 0.171093i
$454$	0	0
$455$	−1.10128e6	−0.249384
$456$	0	0
$457$	5.95736e6	1.33433	0.667165	−	0.744910i	$-0.267508\pi$
$457$	5.95736e6	1.33433	0.667165	+	0.744910i	$0.267508\pi$
$458$	0	0
$459$	− 865353.i	− 0.191717i
$460$	0	0
$461$	2.60215e6i	0.570269i	0.958487	+	0.285135i	$0.0920383\pi$
$461$	2.60215e6i	0.570269i	−0.958487	+	0.285135i	$0.907962\pi$
$462$	0	0
$463$	−2.68993e6	−0.583160	−0.291580	−	0.956546i	$-0.594181\pi$
$463$	−2.68993e6	−0.583160	−0.291580	+	0.956546i	$0.594181\pi$
$464$	0	0
$465$	−358217.	−0.0768271
$466$	0	0
$467$	− 6.49920e6i	− 1.37901i	−0.724280	−	0.689506i	$-0.757828\pi$
$467$	− 6.49920e6i	− 1.37901i	0.724280	−	0.689506i	$-0.242172\pi$
$468$	0	0
$469$	− 200037.i	− 0.0419931i
$470$	0	0
$471$	−1.39598e6	−0.289952
$472$	0	0
$473$	−1.50354e6	−0.309003
$474$	0	0
$475$	607981.i	0.123639i
$476$	0	0
$477$	− 4.89281e6i	− 0.984607i
$478$	0	0
$479$	7.30319e6	1.45437	0.727183	−	0.686443i	$-0.240829\pi$
$479$	7.30319e6	1.45437	0.727183	+	0.686443i	$0.240829\pi$
$480$	0	0
$481$	−1.96342e6	−0.386945
$482$	0	0
$483$	478406.i	0.0933100i
$484$	0	0
$485$	− 3.36860e6i	− 0.650272i
$486$	0	0
$487$	−4.25953e6	−0.813841	−0.406921	−	0.913464i	$-0.633397\pi$
$487$	−4.25953e6	−0.813841	−0.406921	+	0.913464i	$0.633397\pi$
$488$	0	0
$489$	1.20449e6	0.227789
$490$	0	0
$491$	− 888023.i	− 0.166234i	−0.996540	−	0.0831171i	$-0.973512\pi$
$491$	− 888023.i	− 0.166234i	0.996540	−	0.0831171i	$-0.0264876\pi$
$492$	0	0
$493$	− 3.06297e6i	− 0.567579i
$494$	0	0
$495$	−2.07408e6	−0.380463
$496$	0	0
$497$	−633601.	−0.115060
$498$	0	0
$499$	− 597315.i	− 0.107387i	−0.998557	−	0.0536935i	$-0.982901\pi$
$499$	− 597315.i	− 0.107387i	0.998557	−	0.0536935i	$-0.0170994\pi$
$500$	0	0
$501$	− 50895.7i	− 0.00905914i
$502$	0	0
$503$	−9.67283e6	−1.70464	−0.852322	−	0.523018i	$-0.824806\pi$
$503$	−9.67283e6	−1.70464	−0.852322	+	0.523018i	$0.824806\pi$
$504$	0	0
$505$	−986782.	−0.172184
$506$	0	0
$507$	724856.i	0.125237i
$508$	0	0
$509$	2.66674e6i	0.456232i	0.973634	+	0.228116i	$0.0732566\pi$
$509$	2.66674e6i	0.456232i	−0.973634	+	0.228116i	$0.926743\pi$
$510$	0	0
$511$	9.57230e6	1.62168
$512$	0	0
$513$	1.43645e6	0.240989
$514$	0	0
$515$	4.11093e6i	0.683002i
$516$	0	0
$517$	6.60353e6i	1.08655i
$518$	0	0
$519$	346656.	0.0564911
$520$	0	0
$521$	5.96858e6	0.963333	0.481667	−	0.876355i	$-0.340032\pi$
$521$	5.96858e6	0.963333	0.481667	+	0.876355i	$0.340032\pi$
$522$	0	0
$523$	9.24654e6i	1.47817i	0.673611	+	0.739086i	$0.264742\pi$
$523$	9.24654e6i	1.47817i	−0.673611	+	0.739086i	$0.735258\pi$
$524$	0	0
$525$	− 230192.i	− 0.0364495i
$526$	0	0
$527$	2.70895e6	0.424888
$528$	0	0
$529$	−4.74912e6	−0.737860
$530$	0	0
$531$	− 1.13367e7i	− 1.74482i
$532$	0	0
$533$	− 5.20585e6i	− 0.793731i
$534$	0	0
$535$	−5.09651e6	−0.769818
$536$	0	0
$537$	−1.34115e6	−0.200697
$538$	0	0
$539$	955702.i	0.141694i
$540$	0	0
$541$	1.19550e7i	1.75612i	0.478547	+	0.878062i	$0.341164\pi$
$541$	1.19550e7i	1.75612i	−0.478547	+	0.878062i	$0.658836\pi$
$542$	0	0
$543$	−2.47976e6	−0.360919
$544$	0	0
$545$	−1.63904e6	−0.236374
$546$	0	0
$547$	1.07515e7i	1.53638i	0.640221	+	0.768190i	$0.278843\pi$
$547$	1.07515e7i	1.53638i	−0.640221	+	0.768190i	$0.721157\pi$
$548$	0	0
$549$	− 845083.i	− 0.119665i
$550$	0	0
$551$	5.08442e6	0.713449
$552$	0	0
$553$	−4.66643e6	−0.648891
$554$	0	0
$555$	− 410399.i	− 0.0565553i
$556$	0	0
$557$	3.09710e6i	0.422978i	0.977380	+	0.211489i	$0.0678313\pi$
$557$	3.09710e6i	0.422978i	−0.977380	+	0.211489i	$0.932169\pi$
$558$	0	0
$559$	−1.56812e6	−0.212252
$560$	0	0
$561$	−645696.	−0.0866205
$562$	0	0
$563$	1.01607e7i	1.35100i	0.737361	+	0.675499i	$0.236072\pi$
$563$	1.01607e7i	1.35100i	−0.737361	+	0.675499i	$0.763928\pi$
$564$	0	0
$565$	− 876424.i	− 0.115503i
$566$	0	0
$567$	6.19498e6	0.809249
$568$	0	0
$569$	−2.86628e6	−0.371141	−0.185570	−	0.982631i	$-0.559413\pi$
$569$	−2.86628e6	−0.371141	−0.185570	+	0.982631i	$0.559413\pi$
$570$	0	0
$571$	6.30651e6i	0.809466i	0.914435	+	0.404733i	$0.132636\pi$
$571$	6.30651e6i	0.809466i	−0.914435	+	0.404733i	$0.867364\pi$
$572$	0	0
$573$	1.15518e6i	0.146981i
$574$	0	0
$575$	−811832.	−0.102399
$576$	0	0
$577$	−8.94056e6	−1.11796	−0.558979	−	0.829182i	$-0.688807\pi$
$577$	−8.94056e6	−1.11796	−0.558979	+	0.829182i	$0.688807\pi$
$578$	0	0
$579$	− 2.43254e6i	− 0.301553i
$580$	0	0
$581$	− 2.18461e6i	− 0.268493i
$582$	0	0
$583$	−7.45199e6	−0.908031
$584$	0	0
$585$	−2.16316e6	−0.261336
$586$	0	0
$587$	− 7.46374e6i	− 0.894049i	−0.894522	−	0.447024i	$-0.852484\pi$
$587$	− 7.46374e6i	− 0.894049i	0.894522	−	0.447024i	$-0.147516\pi$
$588$	0	0
$589$	4.49676e6i	0.534086i
$590$	0	0
$591$	1.01962e6	0.120080
$592$	0	0
$593$	−7.68119e6	−0.896999	−0.448500	−	0.893783i	$-0.648041\pi$
$593$	−7.68119e6	−0.896999	−0.448500	+	0.893783i	$0.648041\pi$
$594$	0	0
$595$	1.74078e6i	0.201582i
$596$	0	0
$597$	1.46835e6i	0.168614i
$598$	0	0
$599$	−4.04401e6	−0.460517	−0.230258	−	0.973130i	$-0.573957\pi$
$599$	−4.04401e6	−0.460517	−0.230258	+	0.973130i	$0.573957\pi$
$600$	0	0
$601$	733125.	0.0827927	0.0413963	−	0.999143i	$-0.486819\pi$
$601$	733125.	0.0827927	0.0413963	+	0.999143i	$0.486819\pi$
$602$	0	0
$603$	− 392919.i	− 0.0440058i
$604$	0	0
$605$	− 867357.i	− 0.0963407i
$606$	0	0
$607$	1.38418e7	1.52483	0.762413	−	0.647091i	$-0.224015\pi$
$607$	1.38418e7	1.52483	0.762413	+	0.647091i	$0.224015\pi$
$608$	0	0
$609$	−1.92505e6	−0.210329
$610$	0	0
$611$	6.88716e6i	0.746341i
$612$	0	0
$613$	− 4.82596e6i	− 0.518719i	−0.965781	−	0.259360i	$-0.916489\pi$
$613$	− 4.82596e6i	− 0.518719i	0.965781	−	0.259360i	$-0.0835115\pi$
$614$	0	0
$615$	1.08814e6	0.116010
$616$	0	0
$617$	−3.36275e6	−0.355616	−0.177808	−	0.984065i	$-0.556901\pi$
$617$	−3.36275e6	−0.355616	−0.177808	+	0.984065i	$0.556901\pi$
$618$	0	0
$619$	9.71613e6i	1.01922i	0.860406	+	0.509608i	$0.170210\pi$
$619$	9.71613e6i	1.01922i	−0.860406	+	0.509608i	$0.829790\pi$
$620$	0	0
$621$	1.91808e6i	0.199590i
$622$	0	0
$623$	5.12562e6	0.529086
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	− 1.07183e6i	− 0.108882i
$628$	0	0
$629$	3.10356e6i	0.312776i
$630$	0	0
$631$	1.51112e7	1.51086	0.755432	−	0.655227i	$-0.227427\pi$
$631$	1.51112e7	1.51086	0.755432	+	0.655227i	$0.227427\pi$
$632$	0	0
$633$	−2.18421e6	−0.216663
$634$	0	0
$635$	− 3.93069e6i	− 0.386843i
$636$	0	0
$637$	996751.i	0.0973281i
$638$	0	0
$639$	−1.24454e6	−0.120575
$640$	0	0
$641$	9.92719e6	0.954292	0.477146	−	0.878824i	$-0.341671\pi$
$641$	9.92719e6	0.954292	0.477146	+	0.878824i	$0.341671\pi$
$642$	0	0
$643$	1.34898e7i	1.28670i	0.765573	+	0.643349i	$0.222456\pi$
$643$	1.34898e7i	1.28670i	−0.765573	+	0.643349i	$0.777544\pi$
$644$	0	0
$645$	− 327773.i	− 0.0310223i
$646$	0	0
$647$	6.20615e6	0.582857	0.291428	−	0.956593i	$-0.405869\pi$
$647$	6.20615e6	0.582857	0.291428	+	0.956593i	$0.405869\pi$
$648$	0	0
$649$	−1.72664e7	−1.60912
$650$	0	0
$651$	− 1.70255e6i	− 0.157452i
$652$	0	0
$653$	− 1.72472e7i	− 1.58283i	−0.611277	−	0.791417i	$-0.709344\pi$
$653$	− 1.72472e7i	− 1.58283i	0.611277	−	0.791417i	$-0.290656\pi$
$654$	0	0
$655$	−2.14202e6	−0.195084
$656$	0	0
$657$	1.88022e7	1.69940
$658$	0	0
$659$	− 4.84307e6i	− 0.434418i	−0.976125	−	0.217209i	$-0.930305\pi$
$659$	− 4.84307e6i	− 0.434418i	0.976125	−	0.217209i	$-0.0696952\pi$
$660$	0	0
$661$	1.29176e7i	1.14995i	0.818172	+	0.574973i	$0.194987\pi$
$661$	1.29176e7i	1.14995i	−0.818172	+	0.574973i	$0.805013\pi$
$662$	0	0
$663$	−673429.	−0.0594988
$664$	0	0
$665$	−2.88964e6	−0.253390
$666$	0	0
$667$	6.78919e6i	0.590885i
$668$	0	0
$669$	1.51199e6i	0.130612i
$670$	0	0
$671$	−1.28710e6	−0.110359
$672$	0	0
$673$	−6.16137e6	−0.524372	−0.262186	−	0.965017i	$-0.584443\pi$
$673$	−6.16137e6	−0.524372	−0.262186	+	0.965017i	$0.584443\pi$
$674$	0	0
$675$	− 922914.i	− 0.0779654i
$676$	0	0
$677$	1.95790e7i	1.64179i	0.571077	+	0.820896i	$0.306526\pi$
$677$	1.95790e7i	1.64179i	−0.571077	+	0.820896i	$0.693474\pi$
$678$	0	0
$679$	1.60104e7	1.33269
$680$	0	0
$681$	−869862.	−0.0718758
$682$	0	0
$683$	1.33176e7i	1.09238i	0.837661	+	0.546190i	$0.183922\pi$
$683$	1.33176e7i	1.09238i	−0.837661	+	0.546190i	$0.816078\pi$
$684$	0	0
$685$	2.48352e6i	0.202228i
$686$	0	0
$687$	−1.04293e6	−0.0843069
$688$	0	0
$689$	−7.77206e6	−0.623718
$690$	0	0
$691$	8.17155e6i	0.651043i	0.945535	+	0.325521i	$0.105540\pi$
$691$	8.17155e6i	0.651043i	−0.945535	+	0.325521i	$0.894460\pi$
$692$	0	0
$693$	− 9.85776e6i	− 0.779731i
$694$	0	0
$695$	909250.	0.0714038
$696$	0	0
$697$	−8.22886e6	−0.641590
$698$	0	0
$699$	− 770089.i	− 0.0596140i
$700$	0	0
$701$	1.90780e7i	1.46635i	0.680041	+	0.733174i	$0.261962\pi$
$701$	1.90780e7i	1.46635i	−0.680041	+	0.733174i	$0.738038\pi$
$702$	0	0
$703$	−5.15180e6	−0.393161
$704$	0	0
$705$	−1.43957e6	−0.109084
$706$	0	0
$707$	− 4.69001e6i	− 0.352879i
$708$	0	0
$709$	1.13159e7i	0.845422i	0.906265	+	0.422711i	$0.138921\pi$
$709$	1.13159e7i	0.845422i	−0.906265	+	0.422711i	$0.861079\pi$
$710$	0	0
$711$	−9.16595e6	−0.679992
$712$	0	0
$713$	−6.00449e6	−0.442336
$714$	0	0
$715$	3.29460e6i	0.241011i
$716$	0	0
$717$	2.12540e6i	0.154398i
$718$	0	0
$719$	−1.63847e7	−1.18200	−0.590998	−	0.806673i	$-0.701266\pi$
$719$	−1.63847e7	−1.18200	−0.590998	+	0.806673i	$0.701266\pi$
$720$	0	0
$721$	−1.95386e7	−1.39976
$722$	0	0
$723$	− 2.36402e6i	− 0.168192i
$724$	0	0
$725$	− 3.26672e6i	− 0.230816i
$726$	0	0
$727$	8.06838e6	0.566175	0.283087	−	0.959094i	$-0.408641\pi$
$727$	8.06838e6	0.566175	0.283087	+	0.959094i	$0.408641\pi$
$728$	0	0
$729$	1.10561e7	0.770520
$730$	0	0
$731$	2.47873e6i	0.171568i
$732$	0	0
$733$	− 563282.i	− 0.0387227i	−0.999813	−	0.0193614i	$-0.993837\pi$
$733$	− 563282.i	− 0.0387227i	0.999813	−	0.0193614i	$-0.00616330\pi$
$734$	0	0
$735$	−208344.	−0.0142253
$736$	0	0
$737$	−598434.	−0.0405833
$738$	0	0
$739$	6.99287e6i	0.471026i	0.971871	+	0.235513i	$0.0756770\pi$
$739$	6.99287e6i	0.471026i	−0.971871	+	0.235513i	$0.924323\pi$
$740$	0	0
$741$	− 1.11787e6i	− 0.0747902i
$742$	0	0
$743$	−690902.	−0.0459139	−0.0229570	−	0.999736i	$-0.507308\pi$
$743$	−690902.	−0.0459139	−0.0229570	+	0.999736i	$0.507308\pi$
$744$	0	0
$745$	−6.04906e6	−0.399298
$746$	0	0
$747$	− 4.29107e6i	− 0.281361i
$748$	0	0
$749$	− 2.42229e7i	− 1.57769i
$750$	0	0
$751$	1.61091e6	0.104225	0.0521123	−	0.998641i	$-0.483405\pi$
$751$	1.61091e6	0.104225	0.0521123	+	0.998641i	$0.483405\pi$
$752$	0	0
$753$	286604.	0.0184202
$754$	0	0
$755$	6.02698e6i	0.384797i
$756$	0	0
$757$	− 9.75412e6i	− 0.618655i	−0.950956	−	0.309327i	$-0.899896\pi$
$757$	− 9.75412e6i	− 0.618655i	0.950956	−	0.309327i	$-0.100104\pi$
$758$	0	0
$759$	1.43121e6	0.0901774
$760$	0	0
$761$	1.42949e7	0.894786	0.447393	−	0.894337i	$-0.352352\pi$
$761$	1.42949e7	0.894786	0.447393	+	0.894337i	$0.352352\pi$
$762$	0	0
$763$	− 7.79011e6i	− 0.484431i
$764$	0	0
$765$	3.41930e6i	0.211244i
$766$	0	0
$767$	−1.80080e7	−1.10529
$768$	0	0
$769$	1.02217e7	0.623313	0.311656	−	0.950195i	$-0.399116\pi$
$769$	1.02217e7	0.623313	0.311656	+	0.950195i	$0.399116\pi$
$770$	0	0
$771$	− 2.21772e6i	− 0.134360i
$772$	0	0
$773$	6.53435e6i	0.393327i	0.980471	+	0.196663i	$0.0630106\pi$
$773$	6.53435e6i	0.393327i	−0.980471	+	0.196663i	$0.936989\pi$
$774$	0	0
$775$	2.88915e6	0.172789
$776$	0	0
$777$	1.95056e6	0.115906
$778$	0	0
$779$	− 1.36596e7i	− 0.806481i
$780$	0	0
$781$	1.89549e6i	0.111197i
$782$	0	0
$783$	−7.71815e6	−0.449892
$784$	0	0
$785$	1.12590e7	0.652120
$786$	0	0
$787$	1.32185e7i	0.760756i	0.924831	+	0.380378i	$0.124206\pi$
$787$	1.32185e7i	0.760756i	−0.924831	+	0.380378i	$0.875794\pi$
$788$	0	0
$789$	− 6.43470e6i	− 0.367990i
$790$	0	0
$791$	4.16550e6	0.236715
$792$	0	0
$793$	−1.34238e6	−0.0758043
$794$	0	0
$795$	− 1.62454e6i	− 0.0911616i
$796$	0	0
$797$	− 1.45286e7i	− 0.810173i	−0.914278	−	0.405087i	$-0.867241\pi$
$797$	− 1.45286e7i	− 0.810173i	0.914278	−	0.405087i	$-0.132759\pi$
$798$	0	0
$799$	1.08865e7	0.603284
$800$	0	0
$801$	1.00679e7	0.554444
$802$	0	0
$803$	− 2.86367e7i	− 1.56723i
$804$	0	0
$805$	− 3.85851e6i	− 0.209860i
$806$	0	0
$807$	−5.01156e6	−0.270888
$808$	0	0
$809$	1.59141e7	0.854891	0.427445	−	0.904041i	$-0.359414\pi$
$809$	1.59141e7	0.854891	0.427445	+	0.904041i	$0.359414\pi$
$810$	0	0
$811$	− 1.87197e7i	− 0.999418i	−0.866193	−	0.499709i	$-0.833440\pi$
$811$	− 1.87197e7i	− 0.999418i	0.866193	−	0.499709i	$-0.166560\pi$
$812$	0	0
$813$	6.86311e6i	0.364162i
$814$	0	0
$815$	−9.71465e6	−0.512310
$816$	0	0
$817$	−4.11459e6	−0.215661
$818$	0	0
$819$	− 1.02812e7i	− 0.535590i
$820$	0	0
$821$	− 3.44309e7i	− 1.78275i	−0.453267	−	0.891375i	$-0.649742\pi$
$821$	− 3.44309e7i	− 1.78275i	0.453267	−	0.891375i	$-0.350258\pi$
$822$	0	0
$823$	1.98856e7	1.02338	0.511692	−	0.859169i	$-0.329019\pi$
$823$	1.98856e7	1.02338	0.511692	+	0.859169i	$0.329019\pi$
$824$	0	0
$825$	−688646.	−0.0352258
$826$	0	0
$827$	− 2.39990e7i	− 1.22019i	−0.792327	−	0.610097i	$-0.791131\pi$
$827$	− 2.39990e7i	− 1.22019i	0.792327	−	0.610097i	$-0.208869\pi$
$828$	0	0
$829$	1.03074e7i	0.520908i	0.965486	+	0.260454i	$0.0838722\pi$
$829$	1.03074e7i	0.520908i	−0.965486	+	0.260454i	$0.916128\pi$
$830$	0	0
$831$	−1.76359e6	−0.0885919
$832$	0	0
$833$	1.57556e6	0.0786724
$834$	0	0
$835$	410492.i	0.0203746i
$836$	0	0
$837$	− 6.82608e6i	− 0.336789i
$838$	0	0
$839$	808698.	0.0396626	0.0198313	−	0.999803i	$-0.493687\pi$
$839$	808698.	0.0396626	0.0198313	+	0.999803i	$0.493687\pi$
$840$	0	0
$841$	−6.80776e6	−0.331905
$842$	0	0
$843$	2.43797e6i	0.118157i
$844$	0	0
$845$	− 5.84622e6i	− 0.281665i
$846$	0	0
$847$	4.12241e6	0.197443
$848$	0	0
$849$	−7.24405e6	−0.344915
$850$	0	0
$851$	− 6.87916e6i	− 0.325620i
$852$	0	0
$853$	1.48576e7i	0.699161i	0.936906	+	0.349580i	$0.113676\pi$
$853$	1.48576e7i	0.699161i	−0.936906	+	0.349580i	$0.886324\pi$
$854$	0	0
$855$	−5.67592e6	−0.265534
$856$	0	0
$857$	873459.	0.0406247	0.0203124	−	0.999794i	$-0.493534\pi$
$857$	873459.	0.0406247	0.0203124	+	0.999794i	$0.493534\pi$
$858$	0	0
$859$	3.79173e7i	1.75329i	0.481135	+	0.876647i	$0.340225\pi$
$859$	3.79173e7i	1.75329i	−0.481135	+	0.876647i	$0.659775\pi$
$860$	0	0
$861$	5.17175e6i	0.237755i
$862$	0	0
$863$	−3.76129e7	−1.71914	−0.859568	−	0.511022i	$-0.829267\pi$
$863$	−3.76129e7	−1.71914	−0.859568	+	0.511022i	$0.829267\pi$
$864$	0	0
$865$	−2.79590e6	−0.127052
$866$	0	0
$867$	− 3.33662e6i	− 0.150750i
$868$	0	0
$869$	1.39602e7i	0.627107i
$870$	0	0
$871$	−624138.	−0.0278763
$872$	0	0
$873$	3.14482e7	1.39656
$874$	0	0
$875$	1.85658e6i	0.0819772i
$876$	0	0
$877$	− 1.68861e7i	− 0.741360i	−0.928761	−	0.370680i	$-0.879125\pi$
$877$	− 1.68861e7i	− 0.741360i	0.928761	−	0.370680i	$-0.120875\pi$
$878$	0	0
$879$	3.30872e6	0.144440
$880$	0	0
$881$	1.09397e7	0.474858	0.237429	−	0.971405i	$-0.423695\pi$
$881$	1.09397e7	0.474858	0.237429	+	0.971405i	$0.423695\pi$
$882$	0	0
$883$	− 1.07162e7i	− 0.462530i	−0.972891	−	0.231265i	$-0.925714\pi$
$883$	− 1.07162e7i	− 0.462530i	0.972891	−	0.231265i	$-0.0742864\pi$
$884$	0	0
$885$	− 3.76408e6i	− 0.161547i
$886$	0	0
$887$	−3.53953e6	−0.151056	−0.0755278	−	0.997144i	$-0.524064\pi$
$887$	−3.53953e6	−0.151056	−0.0755278	+	0.997144i	$0.524064\pi$
$888$	0	0
$889$	1.86819e7	0.792807
$890$	0	0
$891$	− 1.85330e7i	− 0.782081i
$892$	0	0
$893$	1.80712e7i	0.758330i
$894$	0	0
$895$	1.08168e7	0.451380
$896$	0	0
$897$	1.49268e6	0.0619420
$898$	0	0
$899$	− 2.41614e7i	− 0.997062i
$900$	0	0
$901$	1.22853e7i	0.504165i
$902$	0	0
$903$	1.55785e6	0.0635781
$904$	0	0
$905$	2.00001e7	0.811730
$906$	0	0
$907$	− 1.70933e7i	− 0.689935i	−0.938615	−	0.344967i	$-0.887890\pi$
$907$	− 1.70933e7i	− 0.689935i	0.938615	−	0.344967i	$-0.112110\pi$
$908$	0	0
$909$	− 9.21228e6i	− 0.369792i
$910$	0	0
$911$	−2.54600e7	−1.01639	−0.508196	−	0.861241i	$-0.669688\pi$
$911$	−2.54600e7	−1.01639	−0.508196	+	0.861241i	$0.669688\pi$
$912$	0	0
$913$	−6.53550e6	−0.259479
$914$	0	0
$915$	− 280589.i	− 0.0110794i
$916$	0	0
$917$	− 1.01807e7i	− 0.399810i
$918$	0	0
$919$	3.02310e6	0.118077	0.0590383	−	0.998256i	$-0.481197\pi$
$919$	3.02310e6	0.118077	0.0590383	+	0.998256i	$0.481197\pi$
$920$	0	0
$921$	−1.07062e6	−0.0415896
$922$	0	0
$923$	1.97691e6i	0.0763804i
$924$	0	0
$925$	3.31001e6i	0.127196i
$926$	0	0
$927$	−3.83783e7	−1.46685
$928$	0	0
$929$	3.67045e7	1.39534	0.697669	−	0.716420i	$-0.254220\pi$
$929$	3.67045e7	1.39534	0.697669	+	0.716420i	$0.254220\pi$
$930$	0	0
$931$	2.61537e6i	0.0988915i
$932$	0	0
$933$	4.30825e6i	0.162031i
$934$	0	0
$935$	5.20776e6	0.194815
$936$	0	0
$937$	−2.27722e6	−0.0847338	−0.0423669	−	0.999102i	$-0.513490\pi$
$937$	−2.27722e6	−0.0847338	−0.0423669	+	0.999102i	$0.513490\pi$
$938$	0	0
$939$	− 4.21238e6i	− 0.155906i
$940$	0	0
$941$	− 4.68229e7i	− 1.72379i	−0.507087	−	0.861895i	$-0.669278\pi$
$941$	− 4.68229e7i	− 1.72379i	0.507087	−	0.861895i	$-0.330722\pi$
$942$	0	0
$943$	1.82395e7	0.667936
$944$	0	0
$945$	4.38646e6	0.159785
$946$	0	0
$947$	4.27905e7i	1.55050i	0.631654	+	0.775250i	$0.282376\pi$
$947$	4.27905e7i	1.55050i	−0.631654	+	0.775250i	$0.717624\pi$
$948$	0	0
$949$	− 2.98667e7i	− 1.07652i
$950$	0	0
$951$	−6.35067e6	−0.227703
$952$	0	0
$953$	3.95519e7	1.41070	0.705351	−	0.708858i	$-0.250789\pi$
$953$	3.95519e7	1.41070	0.705351	+	0.708858i	$0.250789\pi$
$954$	0	0
$955$	− 9.31691e6i	− 0.330570i
$956$	0	0
$957$	5.75901e6i	0.203267i
$958$	0	0
$959$	−1.18038e7	−0.414452
$960$	0	0
$961$	−7.26038e6	−0.253601
$962$	0	0
$963$	− 4.75793e7i	− 1.65330i
$964$	0	0
$965$	1.96193e7i	0.678211i
$966$	0	0
$967$	2.15339e7	0.740553	0.370277	−	0.928922i	$-0.379263\pi$
$967$	2.15339e7	0.740553	0.370277	+	0.928922i	$0.379263\pi$
$968$	0	0
$969$	−1.76701e6	−0.0604546
$970$	0	0
$971$	559354.i	0.0190388i	0.999955	+	0.00951939i	$0.00303016\pi$
$971$	559354.i	0.0190388i	−0.999955	+	0.00951939i	$0.996970\pi$
$972$	0	0
$973$	4.32152e6i	0.146337i
$974$	0	0
$975$	−718225.	−0.0241963
$976$	0	0
$977$	2.11889e7	0.710187	0.355094	−	0.934831i	$-0.384449\pi$
$977$	2.11889e7	0.710187	0.355094	+	0.934831i	$0.384449\pi$
$978$	0	0
$979$	− 1.53339e7i	− 0.511324i
$980$	0	0
$981$	− 1.53016e7i	− 0.507649i
$982$	0	0
$983$	4.00893e7	1.32326	0.661630	−	0.749830i	$-0.269865\pi$
$983$	4.00893e7	1.32326	0.661630	+	0.749830i	$0.269865\pi$
$984$	0	0
$985$	−8.22362e6	−0.270067
$986$	0	0
$987$	− 6.84206e6i	− 0.223560i
$988$	0	0
$989$	− 5.49418e6i	− 0.178613i
$990$	0	0
$991$	−5.50729e7	−1.78137	−0.890684	−	0.454623i	$-0.849774\pi$
$991$	−5.50729e7	−1.78137	−0.890684	+	0.454623i	$0.849774\pi$
$992$	0	0
$993$	8.54288e6	0.274936
$994$	0	0
$995$	− 1.18427e7i	− 0.379223i
$996$	0	0
$997$	1.44969e7i	0.461889i	0.972967	+	0.230944i	$0.0741816\pi$
$997$	1.44969e7i	0.461889i	−0.972967	+	0.230944i	$0.925818\pi$
$998$	0	0
$999$	7.82043e6	0.247923

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.d.c.161.7	yes	12
4.3	odd	2		320.6.d.d.161.6	yes	12
8.3	odd	2		320.6.d.d.161.7	yes	12
8.5	even	2	inner	320.6.d.c.161.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.d.c.161.6	✓	12	8.5	even	2	inner
320.6.d.c.161.7	yes	12	1.1	even	1	trivial
320.6.d.d.161.6	yes	12	4.3	odd	2
320.6.d.d.161.7	yes	12	8.3	odd	2

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.09968i q^{3} -25.0000i q^{5} +118.821 q^{7} +233.392 q^{9} +O(q^{10})\)	\(q+3.09968i q^{3} -25.0000i q^{5} +118.821 q^{7} +233.392 q^{9} -355.467i q^{11} -370.735i q^{13} +77.4920 q^{15} -586.019 q^{17} -972.770i q^{19} +368.307i q^{21} +1298.93 q^{23} -625.000 q^{25} +1476.66i q^{27} +5226.75i q^{29} -4622.64 q^{31} +1101.83 q^{33} -2970.52i q^{35} -5296.01i q^{37} +1149.16 q^{39} +14042.0 q^{41} -4229.77i q^{43} -5834.80i q^{45} -18577.1 q^{47} -2688.58 q^{49} -1816.47i q^{51} -20963.9i q^{53} -8886.67 q^{55} +3015.28 q^{57} -48573.7i q^{59} -3620.88i q^{61} +27731.9 q^{63} -9268.37 q^{65} -1683.52i q^{67} +4026.27i q^{69} -5332.40 q^{71} +80560.7 q^{73} -1937.30i q^{75} -42236.9i q^{77} -39272.8 q^{79} +52137.1 q^{81} -18385.7i q^{83} +14650.5i q^{85} -16201.3 q^{87} +43137.3 q^{89} -44051.1i q^{91} -14328.7i q^{93} -24319.2 q^{95} +134744. q^{97} -82963.1i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 268 q^{7} - 428 q^{9}+O(q^{10}) \) 12 * q - 268 * q^7 - 428 * q^9	\( 12 q - 268 q^{7} - 428 q^{9} + 900 q^{15} + 2400 q^{17} - 8108 q^{23} - 7500 q^{25} - 7976 q^{31} - 20776 q^{33} - 40984 q^{39} - 56408 q^{41} - 21172 q^{47} - 5540 q^{49} - 18400 q^{55} + 39992 q^{57} - 179516 q^{63} + 44000 q^{65} - 367704 q^{71} + 58736 q^{73} - 26192 q^{79} + 411692 q^{81} - 183200 q^{87} + 87672 q^{89} - 121000 q^{95} - 172336 q^{97}+O(q^{100}) \) 12 * q - 268 * q^7 - 428 * q^9 + 900 * q^15 + 2400 * q^17 - 8108 * q^23 - 7500 * q^25 - 7976 * q^31 - 20776 * q^33 - 40984 * q^39 - 56408 * q^41 - 21172 * q^47 - 5540 * q^49 - 18400 * q^55 + 39992 * q^57 - 179516 * q^63 + 44000 * q^65 - 367704 * q^71 + 58736 * q^73 - 26192 * q^79 + 411692 * q^81 - 183200 * q^87 + 87672 * q^89 - 121000 * q^95 - 172336 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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