LMFDB - Embedded newform 2160.3.e.d.271.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(271,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2160.271");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.e (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:	$58.8557371018$
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} - 3 x^{11} + 17 x^{10} - 22 x^{9} + 127 x^{8} - 157 x^{7} + 552 x^{6} - 9 x^{5} + 251 x^{4} + \cdots + 1 $ x^12 - 3x^11 + 17x^10 - 22x^9 + 127x^8 - 157x^7 + 552x^6 - 9x^5 + 251x^4 - 100x^3 + 75x^2 - 9*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.6
Root			$1.46938 + 2.54504i$ of defining polynomial
Character	$\chi$	$=$	2160.271
Dual form			2160.3.e.d.271.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{5} +8.95014i q^{7} -1.50831i q^{11} +17.5052 q^{13} -26.4713 q^{17} -26.5986i q^{19} -11.2211i q^{23} +5.00000 q^{25} +15.1795 q^{29} -10.6288i q^{31} -20.0131i q^{35} +68.0363 q^{37} +62.8912 q^{41} -28.8278i q^{43} +84.6303i q^{47} -31.1050 q^{49} -66.7081 q^{53} +3.37267i q^{55} -27.2925i q^{59} +8.24504 q^{61} -39.1429 q^{65} +57.3201i q^{67} -69.9696i q^{71} -67.3962 q^{73} +13.4995 q^{77} +84.1060i q^{79} -90.0589i q^{83} +59.1916 q^{85} -46.6933 q^{89} +156.674i q^{91} +59.4762i q^{95} +104.470 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{13} - 84 q^{17} + 60 q^{25} - 132 q^{29} + 72 q^{37} - 84 q^{41} - 204 q^{49} - 12 q^{53} + 12 q^{61} - 60 q^{65} - 84 q^{73} + 144 q^{77} - 300 q^{89} + 240 q^{97}+O(q^{100}) $ 12 * q - 12 * q^13 - 84 * q^17 + 60 * q^25 - 132 * q^29 + 72 * q^37 - 84 * q^41 - 204 * q^49 - 12 * q^53 + 12 * q^61 - 60 * q^65 - 84 * q^73 + 144 * q^77 - 300 * q^89 + 240 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	8.95014i	1.27859i	0.768961	+	0.639296i	$0.220774\pi$
$7$	8.95014i	1.27859i	−0.768961	+	0.639296i	$0.779226\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.50831i	− 0.137119i	−0.997647	−	0.0685594i	$-0.978160\pi$
$11$	− 1.50831i	− 0.137119i	0.997647	−	0.0685594i	$-0.0218403\pi$
$12$	0	0
$13$	17.5052	1.34656	0.673278	−	0.739390i	$-0.264886\pi$
$13$	17.5052	1.34656	0.673278	+	0.739390i	$0.264886\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−26.4713	−1.55713	−0.778567	−	0.627562i	$-0.784053\pi$
$17$	−26.4713	−1.55713	−0.778567	+	0.627562i	$0.784053\pi$
$18$	0	0
$19$	− 26.5986i	− 1.39992i	−0.714180	−	0.699962i	$-0.753200\pi$
$19$	− 26.5986i	− 1.39992i	0.714180	−	0.699962i	$-0.246800\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 11.2211i	− 0.487874i	−0.969791	−	0.243937i	$-0.921561\pi$
$23$	− 11.2211i	− 0.487874i	0.969791	−	0.243937i	$-0.0784390\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	15.1795	0.523430	0.261715	−	0.965145i	$-0.415712\pi$
$29$	15.1795	0.523430	0.261715	+	0.965145i	$0.415712\pi$
$30$	0	0
$31$	− 10.6288i	− 0.342865i	−0.985196	−	0.171432i	$-0.945160\pi$
$31$	− 10.6288i	− 0.342865i	0.985196	−	0.171432i	$-0.0548395\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 20.0131i	− 0.571803i
$36$	0	0
$37$	68.0363	1.83882	0.919410	−	0.393301i	$-0.128667\pi$
$37$	68.0363	1.83882	0.919410	+	0.393301i	$0.128667\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	62.8912	1.53393	0.766966	−	0.641688i	$-0.221766\pi$
$41$	62.8912	1.53393	0.766966	+	0.641688i	$0.221766\pi$
$42$	0	0
$43$	− 28.8278i	− 0.670413i	−0.942145	−	0.335207i	$-0.891194\pi$
$43$	− 28.8278i	− 0.670413i	0.942145	−	0.335207i	$-0.108806\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	84.6303i	1.80065i	0.435223	+	0.900323i	$0.356670\pi$
$47$	84.6303i	1.80065i	−0.435223	+	0.900323i	$0.643330\pi$
$48$	0	0
$49$	−31.1050	−0.634795
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−66.7081	−1.25864	−0.629322	−	0.777145i	$-0.716667\pi$
$53$	−66.7081	−1.25864	−0.629322	+	0.777145i	$0.716667\pi$
$54$	0	0
$55$	3.37267i	0.0613214i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 27.2925i	− 0.462584i	−0.972884	−	0.231292i	$-0.925705\pi$
$59$	− 27.2925i	− 0.462584i	0.972884	−	0.231292i	$-0.0742953\pi$
$60$	0	0
$61$	8.24504	0.135165	0.0675823	−	0.997714i	$-0.478471\pi$
$61$	8.24504	0.135165	0.0675823	+	0.997714i	$0.478471\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−39.1429	−0.602198
$66$	0	0
$67$	57.3201i	0.855524i	0.903891	+	0.427762i	$0.140698\pi$
$67$	57.3201i	0.855524i	−0.903891	+	0.427762i	$0.859302\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 69.9696i	− 0.985487i	−0.870175	−	0.492744i	$-0.835994\pi$
$71$	− 69.9696i	− 0.985487i	0.870175	−	0.492744i	$-0.164006\pi$
$72$	0	0
$73$	−67.3962	−0.923235	−0.461618	−	0.887079i	$-0.652731\pi$
$73$	−67.3962	−0.923235	−0.461618	+	0.887079i	$0.652731\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.4995	0.175319
$78$	0	0
$79$	84.1060i	1.06463i	0.846545	+	0.532316i	$0.178678\pi$
$79$	84.1060i	1.06463i	−0.846545	+	0.532316i	$0.821322\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 90.0589i	− 1.08505i	−0.840041	−	0.542523i	$-0.817469\pi$
$83$	− 90.0589i	− 1.08505i	0.840041	−	0.542523i	$-0.182531\pi$
$84$	0	0
$85$	59.1916	0.696371
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−46.6933	−0.524644	−0.262322	−	0.964980i	$-0.584488\pi$
$89$	−46.6933	−0.524644	−0.262322	+	0.964980i	$0.584488\pi$
$90$	0	0
$91$	156.674i	1.72169i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	59.4762i	0.626065i
$96$	0	0
$97$	104.470	1.07702	0.538508	−	0.842621i	$-0.318988\pi$
$97$	104.470	1.07702	0.538508	+	0.842621i	$0.318988\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	52.3771	0.518585	0.259292	−	0.965799i	$-0.416511\pi$
$101$	52.3771	0.518585	0.259292	+	0.965799i	$0.416511\pi$
$102$	0	0
$103$	149.385i	1.45034i	0.688570	+	0.725170i	$0.258239\pi$
$103$	149.385i	1.45034i	−0.688570	+	0.725170i	$0.741761\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	194.426i	1.81706i	0.417816	+	0.908532i	$0.362796\pi$
$107$	194.426i	1.81706i	−0.417816	+	0.908532i	$0.637204\pi$
$108$	0	0
$109$	148.169	1.35935	0.679676	−	0.733513i	$-0.262120\pi$
$109$	148.169	1.35935	0.679676	+	0.733513i	$0.262120\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	21.9566	0.194306	0.0971530	−	0.995269i	$-0.469026\pi$
$113$	21.9566	0.194306	0.0971530	+	0.995269i	$0.469026\pi$
$114$	0	0
$115$	25.0912i	0.218184i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 236.922i	− 1.99094i
$120$	0	0
$121$	118.725	0.981198
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	61.5469i	0.484621i	0.970199	+	0.242310i	$0.0779052\pi$
$127$	61.5469i	0.484621i	−0.970199	+	0.242310i	$0.922095\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	211.233i	1.61247i	0.591597	+	0.806234i	$0.298498\pi$
$131$	211.233i	1.61247i	−0.591597	+	0.806234i	$0.701502\pi$
$132$	0	0
$133$	238.061	1.78993
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.8182	0.130060	0.0650301	−	0.997883i	$-0.479286\pi$
$137$	17.8182	0.130060	0.0650301	+	0.997883i	$0.479286\pi$
$138$	0	0
$139$	35.9588i	0.258696i	0.991599	+	0.129348i	$0.0412885\pi$
$139$	35.9588i	0.258696i	−0.991599	+	0.129348i	$0.958712\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 26.4032i	− 0.184638i
$144$	0	0
$145$	−33.9423	−0.234085
$146$	0	0
$147$	0	0
$148$	0	0
$149$	143.778	0.964952	0.482476	−	0.875909i	$-0.339738\pi$
$149$	143.778	0.964952	0.482476	+	0.875909i	$0.339738\pi$
$150$	0	0
$151$	− 177.059i	− 1.17258i	−0.810102	−	0.586289i	$-0.800588\pi$
$151$	− 177.059i	− 1.17258i	0.810102	−	0.586289i	$-0.199412\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	23.7667i	0.153334i
$156$	0	0
$157$	−122.908	−0.782855	−0.391427	−	0.920209i	$-0.628019\pi$
$157$	−122.908	−0.782855	−0.391427	+	0.920209i	$0.628019\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	100.430	0.623791
$162$	0	0
$163$	90.2474i	0.553665i	0.960918	+	0.276832i	$0.0892847\pi$
$163$	90.2474i	0.553665i	−0.960918	+	0.276832i	$0.910715\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.8896i	0.143052i	0.997439	+	0.0715258i	$0.0227868\pi$
$167$	23.8896i	0.143052i	−0.997439	+	0.0715258i	$0.977213\pi$
$168$	0	0
$169$	137.433	0.813212
$170$	0	0
$171$	0	0
$172$	0	0
$173$	286.987	1.65888	0.829442	−	0.558592i	$-0.188658\pi$
$173$	286.987	1.65888	0.829442	+	0.558592i	$0.188658\pi$
$174$	0	0
$175$	44.7507i	0.255718i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	26.3002i	0.146928i	0.997298	+	0.0734642i	$0.0234055\pi$
$179$	26.3002i	0.146928i	−0.997298	+	0.0734642i	$0.976595\pi$
$180$	0	0
$181$	272.919	1.50784	0.753920	−	0.656967i	$-0.228161\pi$
$181$	272.919	1.50784	0.753920	+	0.656967i	$0.228161\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−152.134	−0.822345
$186$	0	0
$187$	39.9268i	0.213512i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.4853i	0.0810749i	0.999178	+	0.0405374i	$0.0129070\pi$
$191$	15.4853i	0.0810749i	−0.999178	+	0.0405374i	$0.987093\pi$
$192$	0	0
$193$	260.783	1.35121	0.675604	−	0.737264i	$-0.263883\pi$
$193$	260.783	1.35121	0.675604	+	0.737264i	$0.263883\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−288.494	−1.46444	−0.732219	−	0.681069i	$-0.761515\pi$
$197$	−288.494	−1.46444	−0.732219	+	0.681069i	$0.761515\pi$
$198$	0	0
$199$	− 125.417i	− 0.630238i	−0.949052	−	0.315119i	$-0.897956\pi$
$199$	− 125.417i	− 0.630238i	0.949052	−	0.315119i	$-0.102044\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	135.858i	0.669253i
$204$	0	0
$205$	−140.629	−0.685995
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−40.1188	−0.191956
$210$	0	0
$211$	320.712i	1.51996i	0.649944	+	0.759982i	$0.274792\pi$
$211$	320.712i	1.51996i	−0.649944	+	0.759982i	$0.725208\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	64.4608i	0.299818i
$216$	0	0
$217$	95.1293	0.438384
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−463.385	−2.09677
$222$	0	0
$223$	− 38.2653i	− 0.171593i	−0.996313	−	0.0857966i	$-0.972656\pi$
$223$	− 38.2653i	− 0.171593i	0.996313	−	0.0857966i	$-0.0273435\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 415.130i	− 1.82877i	−0.404851	−	0.914383i	$-0.632677\pi$
$227$	− 415.130i	− 1.82877i	0.404851	−	0.914383i	$-0.367323\pi$
$228$	0	0
$229$	255.259	1.11467	0.557334	−	0.830289i	$-0.311824\pi$
$229$	255.259	1.11467	0.557334	+	0.830289i	$0.311824\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	260.047	1.11608	0.558040	−	0.829814i	$-0.311554\pi$
$233$	260.047	1.11608	0.558040	+	0.829814i	$0.311554\pi$
$234$	0	0
$235$	− 189.239i	− 0.805273i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 367.472i	− 1.53754i	−0.639526	−	0.768770i	$-0.720869\pi$
$239$	− 367.472i	− 1.53754i	0.639526	−	0.768770i	$-0.279131\pi$
$240$	0	0
$241$	−43.2834	−0.179599	−0.0897996	−	0.995960i	$-0.528623\pi$
$241$	−43.2834	−0.179599	−0.0897996	+	0.995960i	$0.528623\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	69.5528	0.283889
$246$	0	0
$247$	− 465.614i	− 1.88508i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 17.4517i	− 0.0695287i	−0.999396	−	0.0347644i	$-0.988932\pi$
$251$	− 17.4517i	− 0.0695287i	0.999396	−	0.0347644i	$-0.0110681\pi$
$252$	0	0
$253$	−16.9249	−0.0668967
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.18461	−0.00460939	−0.00230469	−	0.999997i	$-0.500734\pi$
$257$	−1.18461	−0.00460939	−0.00230469	+	0.999997i	$0.500734\pi$
$258$	0	0
$259$	608.935i	2.35110i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	57.3419i	0.218030i	0.994040	+	0.109015i	$0.0347697\pi$
$263$	57.3419i	0.218030i	−0.994040	+	0.109015i	$0.965230\pi$
$264$	0	0
$265$	149.164	0.562883
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−194.567	−0.723296	−0.361648	−	0.932315i	$-0.617786\pi$
$269$	−194.567	−0.723296	−0.361648	+	0.932315i	$0.617786\pi$
$270$	0	0
$271$	− 128.007i	− 0.472352i	−0.971710	−	0.236176i	$-0.924106\pi$
$271$	− 128.007i	− 0.472352i	0.971710	−	0.236176i	$-0.0758942\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 7.54153i	− 0.0274237i
$276$	0	0
$277$	380.773	1.37463	0.687317	−	0.726358i	$-0.258788\pi$
$277$	380.773	1.37463	0.687317	+	0.726358i	$0.258788\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	136.748	0.486649	0.243325	−	0.969945i	$-0.421762\pi$
$281$	136.748	0.486649	0.243325	+	0.969945i	$0.421762\pi$
$282$	0	0
$283$	42.2842i	0.149414i	0.997206	+	0.0747071i	$0.0238022\pi$
$283$	42.2842i	0.149414i	−0.997206	+	0.0747071i	$0.976198\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	562.885i	1.96127i
$288$	0	0
$289$	411.728	1.42467
$290$	0	0
$291$	0	0
$292$	0	0
$293$	339.835	1.15985	0.579923	−	0.814672i	$-0.303083\pi$
$293$	339.835	1.15985	0.579923	+	0.814672i	$0.303083\pi$
$294$	0	0
$295$	61.0278i	0.206874i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 196.428i	− 0.656950i
$300$	0	0
$301$	258.012	0.857184
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−18.4365	−0.0604474
$306$	0	0
$307$	558.162i	1.81812i	0.416669	+	0.909058i	$0.363198\pi$
$307$	558.162i	1.81812i	−0.416669	+	0.909058i	$0.636802\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 195.480i	− 0.628555i	−0.949331	−	0.314277i	$-0.898238\pi$
$311$	− 195.480i	− 0.628555i	0.949331	−	0.314277i	$-0.101762\pi$
$312$	0	0
$313$	178.786	0.571202	0.285601	−	0.958349i	$-0.407807\pi$
$313$	178.786	0.571202	0.285601	+	0.958349i	$0.407807\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	202.112	0.637578	0.318789	−	0.947826i	$-0.396724\pi$
$317$	202.112	0.637578	0.318789	+	0.947826i	$0.396724\pi$
$318$	0	0
$319$	− 22.8953i	− 0.0717721i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	704.098i	2.17987i
$324$	0	0
$325$	87.5261	0.269311
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−757.453	−2.30229
$330$	0	0
$331$	54.2631i	0.163937i	0.996635	+	0.0819685i	$0.0261207\pi$
$331$	54.2631i	0.163937i	−0.996635	+	0.0819685i	$0.973879\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 128.172i	− 0.382602i
$336$	0	0
$337$	−121.814	−0.361467	−0.180733	−	0.983532i	$-0.557847\pi$
$337$	−121.814	−0.361467	−0.180733	+	0.983532i	$0.557847\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16.0315	−0.0470132
$342$	0	0
$343$	160.163i	0.466948i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 201.620i	− 0.581037i	−0.956869	−	0.290518i	$-0.906172\pi$
$347$	− 201.620i	− 0.581037i	0.956869	−	0.290518i	$-0.0938278\pi$
$348$	0	0
$349$	−303.158	−0.868646	−0.434323	−	0.900757i	$-0.643012\pi$
$349$	−303.158	−0.868646	−0.434323	+	0.900757i	$0.643012\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.3302	−0.0859213	−0.0429607	−	0.999077i	$-0.513679\pi$
$353$	−30.3302	−0.0859213	−0.0429607	+	0.999077i	$0.513679\pi$
$354$	0	0
$355$	156.457i	0.440723i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	222.648i	0.620189i	0.950706	+	0.310095i	$0.100361\pi$
$359$	222.648i	0.620189i	−0.950706	+	0.310095i	$0.899639\pi$
$360$	0	0
$361$	−346.484	−0.959790
$362$	0	0
$363$	0	0
$364$	0	0
$365$	150.702	0.412883
$366$	0	0
$367$	348.939i	0.950788i	0.879773	+	0.475394i	$0.157694\pi$
$367$	348.939i	0.950788i	−0.879773	+	0.475394i	$0.842306\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 597.047i	− 1.60929i
$372$	0	0
$373$	−354.702	−0.950944	−0.475472	−	0.879731i	$-0.657723\pi$
$373$	−354.702	−0.950944	−0.475472	+	0.879731i	$0.657723\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	265.720	0.704828
$378$	0	0
$379$	− 59.4308i	− 0.156809i	−0.996922	−	0.0784047i	$-0.975017\pi$
$379$	− 59.4308i	− 0.156809i	0.996922	−	0.0784047i	$-0.0249826\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 140.672i	− 0.367289i	−0.982993	−	0.183644i	$-0.941211\pi$
$383$	− 140.672i	− 0.367289i	0.982993	−	0.183644i	$-0.0587894\pi$
$384$	0	0
$385$	−30.1859	−0.0784049
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−521.150	−1.33972	−0.669858	−	0.742489i	$-0.733645\pi$
$389$	−521.150	−1.33972	−0.669858	+	0.742489i	$0.733645\pi$
$390$	0	0
$391$	297.037i	0.759685i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 188.067i	− 0.476118i
$396$	0	0
$397$	−293.333	−0.738873	−0.369437	−	0.929256i	$-0.620449\pi$
$397$	−293.333	−0.738873	−0.369437	+	0.929256i	$0.620449\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	674.542	1.68215	0.841075	−	0.540918i	$-0.181923\pi$
$401$	674.542	1.68215	0.841075	+	0.540918i	$0.181923\pi$
$402$	0	0
$403$	− 186.060i	− 0.461686i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 102.620i	− 0.252137i
$408$	0	0
$409$	−775.003	−1.89487	−0.947436	−	0.319946i	$-0.896335\pi$
$409$	−775.003	−1.89487	−0.947436	+	0.319946i	$0.896335\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	244.271	0.591456
$414$	0	0
$415$	201.378i	0.485248i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	526.375i	1.25627i	0.778106	+	0.628133i	$0.216181\pi$
$419$	526.375i	1.25627i	−0.778106	+	0.628133i	$0.783819\pi$
$420$	0	0
$421$	−594.600	−1.41235	−0.706176	−	0.708036i	$-0.749581\pi$
$421$	−594.600	−1.41235	−0.706176	+	0.708036i	$0.749581\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−132.356	−0.311427
$426$	0	0
$427$	73.7942i	0.172820i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	760.198i	1.76380i	0.471436	+	0.881900i	$0.343736\pi$
$431$	760.198i	1.76380i	−0.471436	+	0.881900i	$0.656264\pi$
$432$	0	0
$433$	229.928	0.531011	0.265505	−	0.964109i	$-0.414461\pi$
$433$	229.928	0.531011	0.265505	+	0.964109i	$0.414461\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−298.465	−0.682987
$438$	0	0
$439$	− 567.326i	− 1.29231i	−0.763205	−	0.646157i	$-0.776375\pi$
$439$	− 567.326i	− 1.29231i	0.763205	−	0.646157i	$-0.223625\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 260.009i	− 0.586928i	−0.955970	−	0.293464i	$-0.905192\pi$
$443$	− 260.009i	− 0.586928i	0.955970	−	0.293464i	$-0.0948081\pi$
$444$	0	0
$445$	104.409	0.234628
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−681.439	−1.51768	−0.758840	−	0.651277i	$-0.774234\pi$
$449$	−681.439	−1.51768	−0.758840	+	0.651277i	$0.774234\pi$
$450$	0	0
$451$	− 94.8592i	− 0.210331i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 350.334i	− 0.769965i
$456$	0	0
$457$	479.067	1.04829	0.524144	−	0.851630i	$-0.324385\pi$
$457$	479.067	1.04829	0.524144	+	0.851630i	$0.324385\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	296.009	0.642101	0.321051	−	0.947062i	$-0.395964\pi$
$461$	296.009	0.642101	0.321051	+	0.947062i	$0.395964\pi$
$462$	0	0
$463$	− 478.325i	− 1.03310i	−0.856258	−	0.516549i	$-0.827216\pi$
$463$	− 478.325i	− 1.03310i	0.856258	−	0.516549i	$-0.172784\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 638.914i	− 1.36812i	−0.729424	−	0.684062i	$-0.760212\pi$
$467$	− 638.914i	− 1.36812i	0.729424	−	0.684062i	$-0.239788\pi$
$468$	0	0
$469$	−513.023	−1.09387
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−43.4811	−0.0919262
$474$	0	0
$475$	− 132.993i	− 0.279985i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 321.667i	− 0.671539i	−0.941944	−	0.335770i	$-0.891004\pi$
$479$	− 321.667i	− 0.671539i	0.941944	−	0.335770i	$-0.108996\pi$
$480$	0	0
$481$	1190.99	2.47607
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−233.603	−0.481656
$486$	0	0
$487$	− 365.324i	− 0.750151i	−0.926994	−	0.375076i	$-0.877617\pi$
$487$	− 365.324i	− 0.750151i	0.926994	−	0.375076i	$-0.122383\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	461.476i	0.939869i	0.882701	+	0.469935i	$0.155722\pi$
$491$	461.476i	0.939869i	−0.882701	+	0.469935i	$0.844278\pi$
$492$	0	0
$493$	−401.820	−0.815051
$494$	0	0
$495$	0	0
$496$	0	0
$497$	626.237	1.26003
$498$	0	0
$499$	− 438.091i	− 0.877937i	−0.898502	−	0.438969i	$-0.855344\pi$
$499$	− 438.091i	− 0.877937i	0.898502	−	0.438969i	$-0.144656\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	739.951i	1.47108i	0.677483	+	0.735538i	$0.263071\pi$
$503$	739.951i	1.47108i	−0.677483	+	0.735538i	$0.736929\pi$
$504$	0	0
$505$	−117.119	−0.231918
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−291.485	−0.572663	−0.286331	−	0.958131i	$-0.592436\pi$
$509$	−291.485	−0.572663	−0.286331	+	0.958131i	$0.592436\pi$
$510$	0	0
$511$	− 603.205i	− 1.18044i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 334.035i	− 0.648612i
$516$	0	0
$517$	127.648	0.246902
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−122.192	−0.234533	−0.117267	−	0.993100i	$-0.537413\pi$
$521$	−122.192	−0.234533	−0.117267	+	0.993100i	$0.537413\pi$
$522$	0	0
$523$	188.835i	0.361060i	0.983569	+	0.180530i	$0.0577814\pi$
$523$	188.835i	0.361060i	−0.983569	+	0.180530i	$0.942219\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	281.358i	0.533886i
$528$	0	0
$529$	403.087	0.761979
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1100.92	2.06552
$534$	0	0
$535$	− 434.749i	− 0.812616i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	46.9158i	0.0870423i
$540$	0	0
$541$	−447.677	−0.827500	−0.413750	−	0.910391i	$-0.635781\pi$
$541$	−447.677	−0.827500	−0.413750	+	0.910391i	$0.635781\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−331.317	−0.607921
$546$	0	0
$547$	− 34.5743i	− 0.0632071i	−0.999500	−	0.0316036i	$-0.989939\pi$
$547$	− 34.5743i	− 0.0632071i	0.999500	−	0.0316036i	$-0.0100614\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 403.752i	− 0.732763i
$552$	0	0
$553$	−752.760	−1.36123
$554$	0	0
$555$	0	0
$556$	0	0
$557$	359.449	0.645331	0.322665	−	0.946513i	$-0.395421\pi$
$557$	359.449	0.645331	0.322665	+	0.946513i	$0.395421\pi$
$558$	0	0
$559$	− 504.636i	− 0.902748i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	967.640i	1.71872i	0.511370	+	0.859361i	$0.329138\pi$
$563$	967.640i	1.71872i	−0.511370	+	0.859361i	$0.670862\pi$
$564$	0	0
$565$	−49.0964	−0.0868963
$566$	0	0
$567$	0	0
$568$	0	0
$569$	879.916	1.54643	0.773213	−	0.634146i	$-0.218648\pi$
$569$	879.916	1.54643	0.773213	+	0.634146i	$0.218648\pi$
$570$	0	0
$571$	− 10.9579i	− 0.0191907i	−0.999954	−	0.00959534i	$-0.996946\pi$
$571$	− 10.9579i	− 0.0191907i	0.999954	−	0.00959534i	$-0.00305434\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 56.1055i	− 0.0975748i
$576$	0	0
$577$	116.051	0.201128	0.100564	−	0.994931i	$-0.467935\pi$
$577$	116.051	0.201128	0.100564	+	0.994931i	$0.467935\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	806.039	1.38733
$582$	0	0
$583$	100.616i	0.172584i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 623.516i	− 1.06221i	−0.847307	−	0.531104i	$-0.821777\pi$
$587$	− 623.516i	− 1.06221i	0.847307	−	0.531104i	$-0.178223\pi$
$588$	0	0
$589$	−282.711	−0.479985
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−940.297	−1.58566	−0.792830	−	0.609442i	$-0.791393\pi$
$593$	−940.297	−1.58566	−0.792830	+	0.609442i	$0.791393\pi$
$594$	0	0
$595$	529.773i	0.890374i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	358.102i	0.597833i	0.954279	+	0.298917i	$0.0966253\pi$
$599$	358.102i	0.597833i	−0.954279	+	0.298917i	$0.903375\pi$
$600$	0	0
$601$	−713.065	−1.18646	−0.593232	−	0.805031i	$-0.702149\pi$
$601$	−713.065	−1.18646	−0.593232	+	0.805031i	$0.702149\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−265.477	−0.438805
$606$	0	0
$607$	− 1088.78i	− 1.79370i	−0.442331	−	0.896852i	$-0.645848\pi$
$607$	− 1088.78i	− 1.79370i	0.442331	−	0.896852i	$-0.354152\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1481.47i	2.42467i
$612$	0	0
$613$	465.516	0.759407	0.379703	−	0.925108i	$-0.376026\pi$
$613$	465.516	0.759407	0.379703	+	0.925108i	$0.376026\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	493.257	0.799444	0.399722	−	0.916636i	$-0.369107\pi$
$617$	493.257	0.799444	0.399722	+	0.916636i	$0.369107\pi$
$618$	0	0
$619$	714.889i	1.15491i	0.816423	+	0.577455i	$0.195954\pi$
$619$	714.889i	1.15491i	−0.816423	+	0.577455i	$0.804046\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 417.912i	− 0.670806i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1801.01	−2.86329
$630$	0	0
$631$	− 594.089i	− 0.941504i	−0.882266	−	0.470752i	$-0.843983\pi$
$631$	− 594.089i	− 0.941504i	0.882266	−	0.470752i	$-0.156017\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 137.623i	− 0.216729i
$636$	0	0
$637$	−544.499	−0.854787
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1077.54	1.68103	0.840516	−	0.541787i	$-0.182252\pi$
$641$	1077.54	1.68103	0.840516	+	0.541787i	$0.182252\pi$
$642$	0	0
$643$	326.166i	0.507257i	0.967302	+	0.253629i	$0.0816241\pi$
$643$	326.166i	0.507257i	−0.967302	+	0.253629i	$0.918376\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 779.108i	− 1.20419i	−0.798426	−	0.602093i	$-0.794334\pi$
$647$	− 779.108i	− 1.20419i	0.798426	−	0.602093i	$-0.205666\pi$
$648$	0	0
$649$	−41.1654	−0.0634290
$650$	0	0
$651$	0	0
$652$	0	0
$653$	398.239	0.609861	0.304931	−	0.952375i	$-0.401367\pi$
$653$	398.239	0.609861	0.304931	+	0.952375i	$0.401367\pi$
$654$	0	0
$655$	− 472.332i	− 0.721118i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1232.75i	− 1.87063i	−0.353814	−	0.935316i	$-0.615115\pi$
$659$	− 1232.75i	− 1.87063i	0.353814	−	0.935316i	$-0.384885\pi$
$660$	0	0
$661$	−226.636	−0.342869	−0.171435	−	0.985196i	$-0.554840\pi$
$661$	−226.636	−0.342869	−0.171435	+	0.985196i	$0.554840\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−532.320	−0.800482
$666$	0	0
$667$	− 170.331i	− 0.255368i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 12.4360i	− 0.0185336i
$672$	0	0
$673$	−884.153	−1.31375	−0.656874	−	0.754000i	$-0.728122\pi$
$673$	−884.153	−1.31375	−0.656874	+	0.754000i	$0.728122\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	491.587	0.726125	0.363063	−	0.931765i	$-0.381731\pi$
$677$	491.587	0.726125	0.363063	+	0.931765i	$0.381731\pi$
$678$	0	0
$679$	935.025i	1.37706i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 612.020i	− 0.896076i	−0.894015	−	0.448038i	$-0.852123\pi$
$683$	− 612.020i	− 0.896076i	0.894015	−	0.448038i	$-0.147877\pi$
$684$	0	0
$685$	−39.8428	−0.0581647
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1167.74	−1.69483
$690$	0	0
$691$	39.5679i	0.0572618i	0.999590	+	0.0286309i	$0.00911474\pi$
$691$	39.5679i	0.0572618i	−0.999590	+	0.0286309i	$0.990885\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 80.4063i	− 0.115692i
$696$	0	0
$697$	−1664.81	−2.38854
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1089.38	−1.55404	−0.777021	−	0.629474i	$-0.783270\pi$
$701$	−1089.38	−1.55404	−0.777021	+	0.629474i	$0.783270\pi$
$702$	0	0
$703$	− 1809.67i	− 2.57421i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	468.782i	0.663058i
$708$	0	0
$709$	353.103	0.498029	0.249015	−	0.968500i	$-0.419893\pi$
$709$	353.103	0.498029	0.249015	+	0.968500i	$0.419893\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−119.267	−0.167275
$714$	0	0
$715$	59.0394i	0.0825726i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 312.773i	− 0.435011i	−0.976059	−	0.217505i	$-0.930208\pi$
$719$	− 312.773i	− 0.435011i	0.976059	−	0.217505i	$-0.0697920\pi$
$720$	0	0
$721$	−1337.02	−1.85439
$722$	0	0
$723$	0	0
$724$	0	0
$725$	75.8974	0.104686
$726$	0	0
$727$	− 47.8900i	− 0.0658735i	−0.999457	−	0.0329367i	$-0.989514\pi$
$727$	− 47.8900i	− 0.0658735i	0.999457	−	0.0329367i	$-0.0104860\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	763.107i	1.04392i
$732$	0	0
$733$	−172.445	−0.235259	−0.117630	−	0.993058i	$-0.537530\pi$
$733$	−172.445	−0.235259	−0.117630	+	0.993058i	$0.537530\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	86.4563	0.117308
$738$	0	0
$739$	− 282.710i	− 0.382558i	−0.981536	−	0.191279i	$-0.938736\pi$
$739$	− 282.710i	− 0.382558i	0.981536	−	0.191279i	$-0.0612635\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1065.64i	1.43423i	0.696953	+	0.717117i	$0.254539\pi$
$743$	1065.64i	1.43423i	−0.696953	+	0.717117i	$0.745461\pi$
$744$	0	0
$745$	−321.497	−0.431540
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1740.14	−2.32328
$750$	0	0
$751$	− 459.318i	− 0.611608i	−0.952094	−	0.305804i	$-0.901075\pi$
$751$	− 459.318i	− 0.611608i	0.952094	−	0.305804i	$-0.0989252\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	395.916i	0.524392i
$756$	0	0
$757$	−1429.77	−1.88873	−0.944366	−	0.328896i	$-0.893324\pi$
$757$	−1429.77	−1.88873	−0.944366	+	0.328896i	$0.893324\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−312.340	−0.410433	−0.205217	−	0.978717i	$-0.565790\pi$
$761$	−312.340	−0.410433	−0.205217	+	0.978717i	$0.565790\pi$
$762$	0	0
$763$	1326.14i	1.73805i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 477.761i	− 0.622896i
$768$	0	0
$769$	−264.329	−0.343731	−0.171865	−	0.985120i	$-0.554979\pi$
$769$	−264.329	−0.343731	−0.171865	+	0.985120i	$0.554979\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1338.95	1.73214	0.866072	−	0.499920i	$-0.166637\pi$
$773$	1338.95	1.73214	0.866072	+	0.499920i	$0.166637\pi$
$774$	0	0
$775$	− 53.1440i	− 0.0685730i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1672.82i	− 2.14739i
$780$	0	0
$781$	−105.536	−0.135129
$782$	0	0
$783$	0	0
$784$	0	0
$785$	274.831	0.350103
$786$	0	0
$787$	243.443i	0.309330i	0.987967	+	0.154665i	$0.0494299\pi$
$787$	243.443i	0.309330i	−0.987967	+	0.154665i	$0.950570\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	196.514i	0.248438i
$792$	0	0
$793$	144.331	0.182007
$794$	0	0
$795$	0	0
$796$	0	0
$797$	690.771	0.866714	0.433357	−	0.901222i	$-0.357329\pi$
$797$	690.771	0.866714	0.433357	+	0.901222i	$0.357329\pi$
$798$	0	0
$799$	− 2240.27i	− 2.80385i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	101.654i	0.126593i
$804$	0	0
$805$	−224.569	−0.278968
$806$	0	0
$807$	0	0
$808$	0	0
$809$	355.995	0.440043	0.220022	−	0.975495i	$-0.429387\pi$
$809$	355.995	0.440043	0.220022	+	0.975495i	$0.429387\pi$
$810$	0	0
$811$	484.000i	0.596794i	0.954442	+	0.298397i	$0.0964519\pi$
$811$	484.000i	0.596794i	−0.954442	+	0.298397i	$0.903548\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 201.799i	− 0.247606i
$816$	0	0
$817$	−766.777	−0.938528
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.1422	0.0476763	0.0238382	−	0.999716i	$-0.492411\pi$
$821$	39.1422	0.0476763	0.0238382	+	0.999716i	$0.492411\pi$
$822$	0	0
$823$	− 516.197i	− 0.627214i	−0.949553	−	0.313607i	$-0.898463\pi$
$823$	− 516.197i	− 0.627214i	0.949553	−	0.313607i	$-0.101537\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1278.13i	1.54550i	0.634711	+	0.772750i	$0.281119\pi$
$827$	1278.13i	1.54550i	−0.634711	+	0.772750i	$0.718881\pi$
$828$	0	0
$829$	401.121	0.483862	0.241931	−	0.970294i	$-0.422219\pi$
$829$	401.121	0.483862	0.241931	+	0.970294i	$0.422219\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	823.388	0.988461
$834$	0	0
$835$	− 53.4188i	− 0.0639746i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 481.868i	− 0.574336i	−0.957880	−	0.287168i	$-0.907286\pi$
$839$	− 481.868i	− 0.574336i	0.957880	−	0.287168i	$-0.0927137\pi$
$840$	0	0
$841$	−610.583	−0.726021
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−307.309	−0.363679
$846$	0	0
$847$	1062.61i	1.25455i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 763.443i	− 0.897113i
$852$	0	0
$853$	79.7999	0.0935521	0.0467760	−	0.998905i	$-0.485105\pi$
$853$	79.7999	0.0935521	0.0467760	+	0.998905i	$0.485105\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−899.082	−1.04910	−0.524552	−	0.851379i	$-0.675767\pi$
$857$	−899.082	−1.04910	−0.524552	+	0.851379i	$0.675767\pi$
$858$	0	0
$859$	868.736i	1.01133i	0.862729	+	0.505667i	$0.168754\pi$
$859$	868.736i	1.01133i	−0.862729	+	0.505667i	$0.831246\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1331.10i	1.54241i	0.636587	+	0.771205i	$0.280346\pi$
$863$	1331.10i	1.54241i	−0.636587	+	0.771205i	$0.719654\pi$
$864$	0	0
$865$	−641.723	−0.741876
$866$	0	0
$867$	0	0
$868$	0	0
$869$	126.858	0.145981
$870$	0	0
$871$	1003.40i	1.15201i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 100.066i	− 0.114361i
$876$	0	0
$877$	1359.62	1.55031	0.775155	−	0.631770i	$-0.217671\pi$
$877$	1359.62	1.55031	0.775155	+	0.631770i	$0.217671\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1144.56	1.29916	0.649580	−	0.760293i	$-0.274945\pi$
$881$	1144.56	1.29916	0.649580	+	0.760293i	$0.274945\pi$
$882$	0	0
$883$	611.883i	0.692959i	0.938057	+	0.346480i	$0.112623\pi$
$883$	611.883i	0.692959i	−0.938057	+	0.346480i	$0.887377\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	791.204i	0.892000i	0.895033	+	0.446000i	$0.147152\pi$
$887$	791.204i	0.892000i	−0.895033	+	0.446000i	$0.852848\pi$
$888$	0	0
$889$	−550.853	−0.619632
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2251.05	2.52077
$894$	0	0
$895$	− 58.8090i	− 0.0657084i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 161.340i	− 0.179466i
$900$	0	0
$901$	1765.85	1.95988
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−610.265	−0.674326
$906$	0	0
$907$	− 1214.75i	− 1.33931i	−0.742674	−	0.669654i	$-0.766443\pi$
$907$	− 1214.75i	− 1.33931i	0.742674	−	0.669654i	$-0.233557\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 959.599i	− 1.05335i	−0.850068	−	0.526673i	$-0.823439\pi$
$911$	− 959.599i	− 1.05335i	0.850068	−	0.526673i	$-0.176561\pi$
$912$	0	0
$913$	−135.836	−0.148780
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1890.57	−2.06169
$918$	0	0
$919$	791.713i	0.861494i	0.902473	+	0.430747i	$0.141750\pi$
$919$	791.713i	0.861494i	−0.902473	+	0.430747i	$0.858250\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1224.83i	− 1.32701i
$924$	0	0
$925$	340.182	0.367764
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1224.69	−1.31829	−0.659146	−	0.752015i	$-0.729082\pi$
$929$	−1224.69	−1.31829	−0.659146	+	0.752015i	$0.729082\pi$
$930$	0	0
$931$	827.347i	0.888665i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 89.2790i	− 0.0954855i
$936$	0	0
$937$	−448.821	−0.478998	−0.239499	−	0.970897i	$-0.576983\pi$
$937$	−448.821	−0.478998	−0.239499	+	0.970897i	$0.576983\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	60.9083	0.0647272	0.0323636	−	0.999476i	$-0.489697\pi$
$941$	60.9083	0.0647272	0.0323636	+	0.999476i	$0.489697\pi$
$942$	0	0
$943$	− 705.709i	− 0.748366i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	197.672i	0.208735i	0.994539	+	0.104368i	$0.0332818\pi$
$947$	197.672i	0.208735i	−0.994539	+	0.104368i	$0.966718\pi$
$948$	0	0
$949$	−1179.78	−1.24319
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1495.61	1.56937	0.784684	−	0.619896i	$-0.212825\pi$
$953$	1495.61	1.56937	0.784684	+	0.619896i	$0.212825\pi$
$954$	0	0
$955$	− 34.6262i	− 0.0362578i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	159.476i	0.166294i
$960$	0	0
$961$	848.028	0.882444
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−583.129	−0.604279
$966$	0	0
$967$	− 1427.58i	− 1.47630i	−0.674637	−	0.738150i	$-0.735700\pi$
$967$	− 1427.58i	− 1.47630i	0.674637	−	0.738150i	$-0.264300\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 293.000i	− 0.301751i	−0.988553	−	0.150875i	$-0.951791\pi$
$971$	− 293.000i	− 0.301751i	0.988553	−	0.150875i	$-0.0482092\pi$
$972$	0	0
$973$	−321.836	−0.330767
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1245.62	−1.27494	−0.637470	−	0.770475i	$-0.720019\pi$
$977$	−1245.62	−1.27494	−0.637470	+	0.770475i	$0.720019\pi$
$978$	0	0
$979$	70.4279i	0.0719386i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	671.026i	0.682631i	0.939949	+	0.341315i	$0.110872\pi$
$983$	671.026i	0.682631i	−0.939949	+	0.341315i	$0.889128\pi$
$984$	0	0
$985$	645.093	0.654917
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−323.479	−0.327077
$990$	0	0
$991$	− 1646.58i	− 1.66153i	−0.556621	−	0.830767i	$-0.687902\pi$
$991$	− 1646.58i	− 1.66153i	0.556621	−	0.830767i	$-0.312098\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	280.442i	0.281851i
$996$	0	0
$997$	715.806	0.717960	0.358980	−	0.933345i	$-0.383125\pi$
$997$	715.806	0.717960	0.358980	+	0.933345i	$0.383125\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.e.d.271.6	yes	12
3.2	odd	2		2160.3.e.e.271.12	yes	12
4.3	odd	2	inner	2160.3.e.d.271.1	✓	12
12.11	even	2		2160.3.e.e.271.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.3.e.d.271.1	✓	12	4.3	odd	2	inner
2160.3.e.d.271.6	yes	12	1.1	even	1	trivial
2160.3.e.e.271.7	yes	12	12.11	even	2
2160.3.e.e.271.12	yes	12	3.2	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607 q^{5} +8.95014i q^{7} -1.50831i q^{11} +17.5052 q^{13} -26.4713 q^{17} -26.5986i q^{19} -11.2211i q^{23} +5.00000 q^{25} +15.1795 q^{29} -10.6288i q^{31} -20.0131i q^{35} +68.0363 q^{37} +62.8912 q^{41} -28.8278i q^{43} +84.6303i q^{47} -31.1050 q^{49} -66.7081 q^{53} +3.37267i q^{55} -27.2925i q^{59} +8.24504 q^{61} -39.1429 q^{65} +57.3201i q^{67} -69.9696i q^{71} -67.3962 q^{73} +13.4995 q^{77} +84.1060i q^{79} -90.0589i q^{83} +59.1916 q^{85} -46.6933 q^{89} +156.674i q^{91} +59.4762i q^{95} +104.470 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{13} - 84 q^{17} + 60 q^{25} - 132 q^{29} + 72 q^{37} - 84 q^{41} - 204 q^{49} - 12 q^{53} + 12 q^{61} - 60 q^{65} - 84 q^{73} + 144 q^{77} - 300 q^{89} + 240 q^{97}+O(q^{100}) \) 12 * q - 12 * q^13 - 84 * q^17 + 60 * q^25 - 132 * q^29 + 72 * q^37 - 84 * q^41 - 204 * q^49 - 12 * q^53 + 12 * q^61 - 60 * q^65 - 84 * q^73 + 144 * q^77 - 300 * q^89 + 240 * q^97

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