LMFDB - Embedded newform 960.3.e.d.511.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(511,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, 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("960.511");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.e (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:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.5
Root			$1.40126 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	960.511
Dual form			960.3.e.d.511.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -2.23607 q^{5} +4.45607i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -2.23607 q^{5} +4.45607i q^{7} -3.00000 q^{9} -12.1543i q^{11} -0.265796 q^{13} -3.87298i q^{15} +8.82953 q^{17} -28.7701i q^{19} -7.71813 q^{21} +15.3177i q^{23} +5.00000 q^{25} -5.19615i q^{27} +25.4164 q^{29} -21.4990i q^{31} +21.0519 q^{33} -9.96407i q^{35} -32.9872 q^{37} -0.460373i q^{39} +72.3161 q^{41} +13.1039i q^{43} +6.70820 q^{45} +56.9909i q^{47} +29.1435 q^{49} +15.2932i q^{51} +17.6608 q^{53} +27.1779i q^{55} +49.8312 q^{57} -28.4094i q^{59} +22.2444 q^{61} -13.3682i q^{63} +0.594339 q^{65} -51.3161i q^{67} -26.5311 q^{69} +49.9422i q^{71} +128.973 q^{73} +8.66025i q^{75} +54.1606 q^{77} -88.6084i q^{79} +9.00000 q^{81} +43.3955i q^{83} -19.7434 q^{85} +44.0225i q^{87} -148.097 q^{89} -1.18441i q^{91} +37.2374 q^{93} +64.3318i q^{95} +121.426 q^{97} +36.4630i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 16 q^{13} - 16 q^{17} - 96 q^{21} + 40 q^{25} + 96 q^{29} + 48 q^{33} + 112 q^{37} + 112 q^{41} - 184 q^{49} - 224 q^{53} - 144 q^{57} - 80 q^{61} - 160 q^{65} + 144 q^{69} + 272 q^{73} - 256 q^{77} + 72 q^{81} + 160 q^{85} - 48 q^{89} + 240 q^{93} + 528 q^{97}+O(q^{100}) $ 8 * q - 24 * q^9 + 16 * q^13 - 16 * q^17 - 96 * q^21 + 40 * q^25 + 96 * q^29 + 48 * q^33 + 112 * q^37 + 112 * q^41 - 184 * q^49 - 224 * q^53 - 144 * q^57 - 80 * q^61 - 160 * q^65 + 144 * q^69 + 272 * q^73 - 256 * q^77 + 72 * q^81 + 160 * q^85 - 48 * q^89 + 240 * q^93 + 528 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	4.45607i	0.636581i	0.947993	+	0.318291i	$0.103109\pi$
$7$	4.45607i	0.636581i	−0.947993	+	0.318291i	$0.896891\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 12.1543i	− 1.10494i	−0.833533	−	0.552470i	$-0.813685\pi$
$11$	− 12.1543i	− 1.10494i	0.833533	−	0.552470i	$-0.186315\pi$
$12$	0	0
$13$	−0.265796	−0.0204459	−0.0102229	−	0.999948i	$-0.503254\pi$
$13$	−0.265796	−0.0204459	−0.0102229	+	0.999948i	$0.503254\pi$
$14$	0	0
$15$	− 3.87298i	− 0.258199i
$16$	0	0
$17$	8.82953	0.519384	0.259692	−	0.965692i	$-0.416379\pi$
$17$	8.82953	0.519384	0.259692	+	0.965692i	$0.416379\pi$
$18$	0	0
$19$	− 28.7701i	− 1.51421i	−0.653291	−	0.757107i	$-0.726612\pi$
$19$	− 28.7701i	− 1.51421i	0.653291	−	0.757107i	$-0.273388\pi$
$20$	0	0
$21$	−7.71813	−0.367530
$22$	0	0
$23$	15.3177i	0.665988i	0.942929	+	0.332994i	$0.108059\pi$
$23$	15.3177i	0.665988i	−0.942929	+	0.332994i	$0.891941\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	25.4164	0.876428	0.438214	−	0.898871i	$-0.355611\pi$
$29$	25.4164	0.876428	0.438214	+	0.898871i	$0.355611\pi$
$30$	0	0
$31$	− 21.4990i	− 0.693517i	−0.937955	−	0.346758i	$-0.887282\pi$
$31$	− 21.4990i	− 0.693517i	0.937955	−	0.346758i	$-0.112718\pi$
$32$	0	0
$33$	21.0519	0.637937
$34$	0	0
$35$	− 9.96407i	− 0.284688i
$36$	0	0
$37$	−32.9872	−0.891545	−0.445772	−	0.895146i	$-0.647071\pi$
$37$	−32.9872	−0.891545	−0.445772	+	0.895146i	$0.647071\pi$
$38$	0	0
$39$	− 0.460373i	− 0.0118044i
$40$	0	0
$41$	72.3161	1.76381	0.881904	−	0.471429i	$-0.156262\pi$
$41$	72.3161	1.76381	0.881904	+	0.471429i	$0.156262\pi$
$42$	0	0
$43$	13.1039i	0.304743i	0.988323	+	0.152371i	$0.0486910\pi$
$43$	13.1039i	0.304743i	−0.988323	+	0.152371i	$0.951309\pi$
$44$	0	0
$45$	6.70820	0.149071
$46$	0	0
$47$	56.9909i	1.21257i	0.795246	+	0.606287i	$0.207342\pi$
$47$	56.9909i	1.21257i	−0.795246	+	0.606287i	$0.792658\pi$
$48$	0	0
$49$	29.1435	0.594765
$50$	0	0
$51$	15.2932i	0.299866i
$52$	0	0
$53$	17.6608	0.333223	0.166611	−	0.986023i	$-0.446717\pi$
$53$	17.6608	0.333223	0.166611	+	0.986023i	$0.446717\pi$
$54$	0	0
$55$	27.1779i	0.494144i
$56$	0	0
$57$	49.8312	0.874232
$58$	0	0
$59$	− 28.4094i	− 0.481515i	−0.970585	−	0.240758i	$-0.922604\pi$
$59$	− 28.4094i	− 0.481515i	0.970585	−	0.240758i	$-0.0773959\pi$
$60$	0	0
$61$	22.2444	0.364662	0.182331	−	0.983237i	$-0.441636\pi$
$61$	22.2444	0.364662	0.182331	+	0.983237i	$0.441636\pi$
$62$	0	0
$63$	− 13.3682i	− 0.212194i
$64$	0	0
$65$	0.594339	0.00914367
$66$	0	0
$67$	− 51.3161i	− 0.765912i	−0.923767	−	0.382956i	$-0.874906\pi$
$67$	− 51.3161i	− 0.765912i	0.923767	−	0.382956i	$-0.125094\pi$
$68$	0	0
$69$	−26.5311	−0.384509
$70$	0	0
$71$	49.9422i	0.703411i	0.936111	+	0.351705i	$0.114398\pi$
$71$	49.9422i	0.703411i	−0.936111	+	0.351705i	$0.885602\pi$
$72$	0	0
$73$	128.973	1.76676	0.883378	−	0.468661i	$-0.155263\pi$
$73$	128.973	1.76676	0.883378	+	0.468661i	$0.155263\pi$
$74$	0	0
$75$	8.66025i	0.115470i
$76$	0	0
$77$	54.1606	0.703384
$78$	0	0
$79$	− 88.6084i	− 1.12162i	−0.827943	−	0.560812i	$-0.810489\pi$
$79$	− 88.6084i	− 1.12162i	0.827943	−	0.560812i	$-0.189511\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	43.3955i	0.522838i	0.965225	+	0.261419i	$0.0841904\pi$
$83$	43.3955i	0.522838i	−0.965225	+	0.261419i	$0.915810\pi$
$84$	0	0
$85$	−19.7434	−0.232276
$86$	0	0
$87$	44.0225i	0.506006i
$88$	0	0
$89$	−148.097	−1.66401	−0.832004	−	0.554769i	$-0.812807\pi$
$89$	−148.097	−1.66401	−0.832004	+	0.554769i	$0.812807\pi$
$90$	0	0
$91$	− 1.18441i	− 0.0130155i
$92$	0	0
$93$	37.2374	0.400402
$94$	0	0
$95$	64.3318i	0.677177i
$96$	0	0
$97$	121.426	1.25181	0.625905	−	0.779899i	$-0.284730\pi$
$97$	121.426	1.25181	0.625905	+	0.779899i	$0.284730\pi$
$98$	0	0
$99$	36.4630i	0.368313i
$100$	0	0
$101$	155.858	1.54315	0.771574	−	0.636139i	$-0.219470\pi$
$101$	155.858	1.54315	0.771574	+	0.636139i	$0.219470\pi$
$102$	0	0
$103$	− 29.2417i	− 0.283900i	−0.989874	−	0.141950i	$-0.954663\pi$
$103$	− 29.2417i	− 0.283900i	0.989874	−	0.141950i	$-0.0453372\pi$
$104$	0	0
$105$	17.2583	0.164365
$106$	0	0
$107$	− 52.0592i	− 0.486534i	−0.969959	−	0.243267i	$-0.921781\pi$
$107$	− 52.0592i	− 0.486534i	0.969959	−	0.243267i	$-0.0782192\pi$
$108$	0	0
$109$	146.593	1.34489	0.672443	−	0.740149i	$-0.265245\pi$
$109$	146.593	1.34489	0.672443	+	0.740149i	$0.265245\pi$
$110$	0	0
$111$	− 57.1354i	− 0.514734i
$112$	0	0
$113$	−74.5415	−0.659659	−0.329829	−	0.944041i	$-0.606991\pi$
$113$	−74.5415	−0.659659	−0.329829	+	0.944041i	$0.606991\pi$
$114$	0	0
$115$	− 34.2515i	− 0.297839i
$116$	0	0
$117$	0.797389	0.00681529
$118$	0	0
$119$	39.3450i	0.330630i
$120$	0	0
$121$	−26.7280	−0.220892
$122$	0	0
$123$	125.255i	1.01834i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 142.991i	− 1.12591i	−0.826487	−	0.562955i	$-0.809664\pi$
$127$	− 142.991i	− 1.12591i	0.826487	−	0.562955i	$-0.190336\pi$
$128$	0	0
$129$	−22.6967	−0.175943
$130$	0	0
$131$	− 71.2101i	− 0.543588i	−0.962355	−	0.271794i	$-0.912383\pi$
$131$	− 71.2101i	− 0.543588i	0.962355	−	0.271794i	$-0.0876170\pi$
$132$	0	0
$133$	128.201	0.963920
$134$	0	0
$135$	11.6190i	0.0860663i
$136$	0	0
$137$	82.0860	0.599168	0.299584	−	0.954070i	$-0.403152\pi$
$137$	82.0860	0.599168	0.299584	+	0.954070i	$0.403152\pi$
$138$	0	0
$139$	218.408i	1.57128i	0.618684	+	0.785640i	$0.287666\pi$
$139$	218.408i	1.57128i	−0.618684	+	0.785640i	$0.712334\pi$
$140$	0	0
$141$	−98.7112	−0.700080
$142$	0	0
$143$	3.23058i	0.0225915i
$144$	0	0
$145$	−56.8328	−0.391950
$146$	0	0
$147$	50.4780i	0.343387i
$148$	0	0
$149$	54.7913	0.367727	0.183864	−	0.982952i	$-0.441140\pi$
$149$	54.7913	0.367727	0.183864	+	0.982952i	$0.441140\pi$
$150$	0	0
$151$	− 159.042i	− 1.05326i	−0.850094	−	0.526631i	$-0.823455\pi$
$151$	− 159.042i	− 1.05326i	0.850094	−	0.526631i	$-0.176545\pi$
$152$	0	0
$153$	−26.4886	−0.173128
$154$	0	0
$155$	48.0733i	0.310150i
$156$	0	0
$157$	−196.132	−1.24925	−0.624625	−	0.780925i	$-0.714748\pi$
$157$	−196.132	−1.24925	−0.624625	+	0.780925i	$0.714748\pi$
$158$	0	0
$159$	30.5894i	0.192386i
$160$	0	0
$161$	−68.2568	−0.423956
$162$	0	0
$163$	9.14060i	0.0560773i	0.999607	+	0.0280387i	$0.00892615\pi$
$163$	9.14060i	0.0560773i	−0.999607	+	0.0280387i	$0.991074\pi$
$164$	0	0
$165$	−47.0736	−0.285294
$166$	0	0
$167$	255.845i	1.53201i	0.642835	+	0.766004i	$0.277758\pi$
$167$	255.845i	1.53201i	−0.642835	+	0.766004i	$0.722242\pi$
$168$	0	0
$169$	−168.929	−0.999582
$170$	0	0
$171$	86.3102i	0.504738i
$172$	0	0
$173$	328.479	1.89872	0.949361	−	0.314187i	$-0.101732\pi$
$173$	328.479	1.89872	0.949361	+	0.314187i	$0.101732\pi$
$174$	0	0
$175$	22.2803i	0.127316i
$176$	0	0
$177$	49.2065	0.278003
$178$	0	0
$179$	136.205i	0.760921i	0.924797	+	0.380461i	$0.124235\pi$
$179$	136.205i	0.760921i	−0.924797	+	0.380461i	$0.875765\pi$
$180$	0	0
$181$	183.088	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$181$	183.088	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$182$	0	0
$183$	38.5284i	0.210538i
$184$	0	0
$185$	73.7615	0.398711
$186$	0	0
$187$	− 107.317i	− 0.573888i
$188$	0	0
$189$	23.1544	0.122510
$190$	0	0
$191$	− 215.085i	− 1.12610i	−0.826424	−	0.563049i	$-0.809628\pi$
$191$	− 215.085i	− 1.12610i	0.826424	−	0.563049i	$-0.190372\pi$
$192$	0	0
$193$	1.49648	0.00775378	0.00387689	−	0.999992i	$-0.498766\pi$
$193$	1.49648	0.00775378	0.00387689	+	0.999992i	$0.498766\pi$
$194$	0	0
$195$	1.02943i	0.00527910i
$196$	0	0
$197$	−108.472	−0.550621	−0.275311	−	0.961355i	$-0.588781\pi$
$197$	−108.472	−0.550621	−0.275311	+	0.961355i	$0.588781\pi$
$198$	0	0
$199$	− 244.087i	− 1.22657i	−0.789863	−	0.613283i	$-0.789849\pi$
$199$	− 244.087i	− 1.22657i	0.789863	−	0.613283i	$-0.210151\pi$
$200$	0	0
$201$	88.8820	0.442199
$202$	0	0
$203$	113.257i	0.557917i
$204$	0	0
$205$	−161.704	−0.788799
$206$	0	0
$207$	− 45.9532i	− 0.221996i
$208$	0	0
$209$	−349.681	−1.67312
$210$	0	0
$211$	− 347.613i	− 1.64746i	−0.566984	−	0.823729i	$-0.691890\pi$
$211$	− 347.613i	− 1.64746i	0.566984	−	0.823729i	$-0.308110\pi$
$212$	0	0
$213$	−86.5024	−0.406114
$214$	0	0
$215$	− 29.3013i	− 0.136285i
$216$	0	0
$217$	95.8011	0.441480
$218$	0	0
$219$	223.388i	1.02004i
$220$	0	0
$221$	−2.34686	−0.0106193
$222$	0	0
$223$	− 305.374i	− 1.36939i	−0.728830	−	0.684694i	$-0.759936\pi$
$223$	− 305.374i	− 1.36939i	0.728830	−	0.684694i	$-0.240064\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	− 404.300i	− 1.78106i	−0.454926	−	0.890529i	$-0.650334\pi$
$227$	− 404.300i	− 1.78106i	0.454926	−	0.890529i	$-0.349666\pi$
$228$	0	0
$229$	−354.858	−1.54960	−0.774799	−	0.632208i	$-0.782149\pi$
$229$	−354.858	−1.54960	−0.774799	+	0.632208i	$0.782149\pi$
$230$	0	0
$231$	93.8088i	0.406099i
$232$	0	0
$233$	111.633	0.479111	0.239556	−	0.970883i	$-0.422998\pi$
$233$	111.633	0.479111	0.239556	+	0.970883i	$0.422998\pi$
$234$	0	0
$235$	− 127.436i	− 0.542279i
$236$	0	0
$237$	153.474	0.647570
$238$	0	0
$239$	− 248.183i	− 1.03842i	−0.854646	−	0.519211i	$-0.826226\pi$
$239$	− 248.183i	− 1.03842i	0.854646	−	0.519211i	$-0.173774\pi$
$240$	0	0
$241$	−13.4186	−0.0556788	−0.0278394	−	0.999612i	$-0.508863\pi$
$241$	−13.4186	−0.0556788	−0.0278394	+	0.999612i	$0.508863\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−65.1668	−0.265987
$246$	0	0
$247$	7.64698i	0.0309594i
$248$	0	0
$249$	−75.1632	−0.301860
$250$	0	0
$251$	338.305i	1.34783i	0.738809	+	0.673915i	$0.235389\pi$
$251$	338.305i	1.34783i	−0.738809	+	0.673915i	$0.764611\pi$
$252$	0	0
$253$	186.177	0.735877
$254$	0	0
$255$	− 34.1966i	− 0.134104i
$256$	0	0
$257$	−122.247	−0.475671	−0.237836	−	0.971305i	$-0.576438\pi$
$257$	−122.247	−0.475671	−0.237836	+	0.971305i	$0.576438\pi$
$258$	0	0
$259$	− 146.993i	− 0.567540i
$260$	0	0
$261$	−76.2492	−0.292143
$262$	0	0
$263$	289.777i	1.10181i	0.834566	+	0.550907i	$0.185718\pi$
$263$	289.777i	1.10181i	−0.834566	+	0.550907i	$0.814282\pi$
$264$	0	0
$265$	−39.4908	−0.149022
$266$	0	0
$267$	− 256.511i	− 0.960716i
$268$	0	0
$269$	−245.321	−0.911974	−0.455987	−	0.889986i	$-0.650714\pi$
$269$	−245.321	−0.911974	−0.455987	+	0.889986i	$0.650714\pi$
$270$	0	0
$271$	262.071i	0.967050i	0.875331	+	0.483525i	$0.160644\pi$
$271$	262.071i	0.967050i	−0.875331	+	0.483525i	$0.839356\pi$
$272$	0	0
$273$	2.05145	0.00751448
$274$	0	0
$275$	− 60.7717i	− 0.220988i
$276$	0	0
$277$	−100.002	−0.361018	−0.180509	−	0.983573i	$-0.557774\pi$
$277$	−100.002	−0.361018	−0.180509	+	0.983573i	$0.557774\pi$
$278$	0	0
$279$	64.4970i	0.231172i
$280$	0	0
$281$	29.0676	0.103443	0.0517217	−	0.998662i	$-0.483529\pi$
$281$	29.0676	0.103443	0.0517217	+	0.998662i	$0.483529\pi$
$282$	0	0
$283$	63.6053i	0.224754i	0.993666	+	0.112377i	$0.0358464\pi$
$283$	63.6053i	0.224754i	−0.993666	+	0.112377i	$0.964154\pi$
$284$	0	0
$285$	−111.426	−0.390968
$286$	0	0
$287$	322.246i	1.12281i
$288$	0	0
$289$	−211.039	−0.730240
$290$	0	0
$291$	210.315i	0.722733i
$292$	0	0
$293$	−8.96048	−0.0305819	−0.0152909	−	0.999883i	$-0.504867\pi$
$293$	−8.96048	−0.0305819	−0.0152909	+	0.999883i	$0.504867\pi$
$294$	0	0
$295$	63.5253i	0.215340i
$296$	0	0
$297$	−63.1558	−0.212646
$298$	0	0
$299$	− 4.07140i	− 0.0136167i
$300$	0	0
$301$	−58.3920	−0.193993
$302$	0	0
$303$	269.954i	0.890937i
$304$	0	0
$305$	−49.7400	−0.163082
$306$	0	0
$307$	− 347.678i	− 1.13250i	−0.824233	−	0.566250i	$-0.808394\pi$
$307$	− 347.678i	− 1.13250i	0.824233	−	0.566250i	$-0.191606\pi$
$308$	0	0
$309$	50.6481	0.163910
$310$	0	0
$311$	192.789i	0.619901i	0.950753	+	0.309950i	$0.100312\pi$
$311$	192.789i	0.619901i	−0.950753	+	0.309950i	$0.899688\pi$
$312$	0	0
$313$	−235.513	−0.752439	−0.376219	−	0.926531i	$-0.622776\pi$
$313$	−235.513	−0.752439	−0.376219	+	0.926531i	$0.622776\pi$
$314$	0	0
$315$	29.8922i	0.0948959i
$316$	0	0
$317$	−546.577	−1.72422	−0.862109	−	0.506723i	$-0.830857\pi$
$317$	−546.577	−1.72422	−0.862109	+	0.506723i	$0.830857\pi$
$318$	0	0
$319$	− 308.920i	− 0.968400i
$320$	0	0
$321$	90.1692	0.280901
$322$	0	0
$323$	− 254.026i	− 0.786458i
$324$	0	0
$325$	−1.32898	−0.00408918
$326$	0	0
$327$	253.906i	0.776470i
$328$	0	0
$329$	−253.956	−0.771901
$330$	0	0
$331$	− 244.297i	− 0.738058i	−0.929418	−	0.369029i	$-0.879690\pi$
$331$	− 244.297i	− 0.738058i	0.929418	−	0.369029i	$-0.120310\pi$
$332$	0	0
$333$	98.9615	0.297182
$334$	0	0
$335$	114.746i	0.342526i
$336$	0	0
$337$	581.581	1.72576	0.862879	−	0.505410i	$-0.168659\pi$
$337$	581.581	1.72576	0.862879	+	0.505410i	$0.168659\pi$
$338$	0	0
$339$	− 129.110i	− 0.380854i
$340$	0	0
$341$	−261.306	−0.766294
$342$	0	0
$343$	348.213i	1.01520i
$344$	0	0
$345$	59.3253	0.171957
$346$	0	0
$347$	532.168i	1.53363i	0.641871	+	0.766813i	$0.278159\pi$
$347$	532.168i	1.53363i	−0.641871	+	0.766813i	$0.721841\pi$
$348$	0	0
$349$	−636.743	−1.82448	−0.912239	−	0.409659i	$-0.865648\pi$
$349$	−636.743	−1.82448	−0.912239	+	0.409659i	$0.865648\pi$
$350$	0	0
$351$	1.38112i	0.00393481i
$352$	0	0
$353$	350.211	0.992100	0.496050	−	0.868294i	$-0.334783\pi$
$353$	350.211	0.992100	0.496050	+	0.868294i	$0.334783\pi$
$354$	0	0
$355$	− 111.674i	− 0.314575i
$356$	0	0
$357$	−68.1475	−0.190889
$358$	0	0
$359$	76.6399i	0.213482i	0.994287	+	0.106741i	$0.0340415\pi$
$359$	76.6399i	0.213482i	−0.994287	+	0.106741i	$0.965958\pi$
$360$	0	0
$361$	−466.717	−1.29284
$362$	0	0
$363$	− 46.2942i	− 0.127532i
$364$	0	0
$365$	−288.393	−0.790118
$366$	0	0
$367$	− 410.325i	− 1.11805i	−0.829150	−	0.559026i	$-0.811175\pi$
$367$	− 410.325i	− 1.11805i	0.829150	−	0.559026i	$-0.188825\pi$
$368$	0	0
$369$	−216.948	−0.587936
$370$	0	0
$371$	78.6978i	0.212123i
$372$	0	0
$373$	−56.1032	−0.150411	−0.0752053	−	0.997168i	$-0.523961\pi$
$373$	−56.1032	−0.150411	−0.0752053	+	0.997168i	$0.523961\pi$
$374$	0	0
$375$	− 19.3649i	− 0.0516398i
$376$	0	0
$377$	−6.75559	−0.0179193
$378$	0	0
$379$	197.508i	0.521130i	0.965456	+	0.260565i	$0.0839088\pi$
$379$	197.508i	0.521130i	−0.965456	+	0.260565i	$0.916091\pi$
$380$	0	0
$381$	247.667	0.650045
$382$	0	0
$383$	− 388.780i	− 1.01509i	−0.861625	−	0.507546i	$-0.830553\pi$
$383$	− 388.780i	− 1.01509i	0.861625	−	0.507546i	$-0.169447\pi$
$384$	0	0
$385$	−121.107	−0.314563
$386$	0	0
$387$	− 39.3118i	− 0.101581i
$388$	0	0
$389$	−537.091	−1.38070	−0.690348	−	0.723477i	$-0.742543\pi$
$389$	−537.091	−1.38070	−0.690348	+	0.723477i	$0.742543\pi$
$390$	0	0
$391$	135.248i	0.345904i
$392$	0	0
$393$	123.339	0.313841
$394$	0	0
$395$	198.134i	0.501606i
$396$	0	0
$397$	−536.944	−1.35250	−0.676252	−	0.736670i	$-0.736397\pi$
$397$	−536.944	−1.35250	−0.676252	+	0.736670i	$0.736397\pi$
$398$	0	0
$399$	222.051i	0.556519i
$400$	0	0
$401$	658.617	1.64244	0.821218	−	0.570615i	$-0.193295\pi$
$401$	658.617	1.64244	0.821218	+	0.570615i	$0.193295\pi$
$402$	0	0
$403$	5.71436i	0.0141796i
$404$	0	0
$405$	−20.1246	−0.0496904
$406$	0	0
$407$	400.937i	0.985103i
$408$	0	0
$409$	−695.505	−1.70050	−0.850251	−	0.526377i	$-0.823550\pi$
$409$	−695.505	−1.70050	−0.850251	+	0.526377i	$0.823550\pi$
$410$	0	0
$411$	142.177i	0.345930i
$412$	0	0
$413$	126.594	0.306523
$414$	0	0
$415$	− 97.0353i	− 0.233820i
$416$	0	0
$417$	−378.293	−0.907179
$418$	0	0
$419$	− 95.8396i	− 0.228734i	−0.993439	−	0.114367i	$-0.963516\pi$
$419$	− 95.8396i	− 0.228734i	0.993439	−	0.114367i	$-0.0364840\pi$
$420$	0	0
$421$	601.155	1.42792	0.713961	−	0.700185i	$-0.246899\pi$
$421$	601.155	1.42792	0.713961	+	0.700185i	$0.246899\pi$
$422$	0	0
$423$	− 170.973i	− 0.404191i
$424$	0	0
$425$	44.1476	0.103877
$426$	0	0
$427$	99.1226i	0.232137i
$428$	0	0
$429$	−5.59553	−0.0130432
$430$	0	0
$431$	− 74.1664i	− 0.172080i	−0.996292	−	0.0860399i	$-0.972579\pi$
$431$	− 74.1664i	− 0.172080i	0.996292	−	0.0860399i	$-0.0274213\pi$
$432$	0	0
$433$	436.312	1.00765	0.503824	−	0.863806i	$-0.331926\pi$
$433$	436.312	1.00765	0.503824	+	0.863806i	$0.331926\pi$
$434$	0	0
$435$	− 98.4373i	− 0.226293i
$436$	0	0
$437$	440.692	1.00845
$438$	0	0
$439$	416.602i	0.948979i	0.880261	+	0.474489i	$0.157367\pi$
$439$	416.602i	0.948979i	−0.880261	+	0.474489i	$0.842633\pi$
$440$	0	0
$441$	−87.4304	−0.198255
$442$	0	0
$443$	283.661i	0.640319i	0.947364	+	0.320159i	$0.103736\pi$
$443$	283.661i	0.640319i	−0.947364	+	0.320159i	$0.896264\pi$
$444$	0	0
$445$	331.154	0.744167
$446$	0	0
$447$	94.9014i	0.212307i
$448$	0	0
$449$	−81.2571	−0.180973	−0.0904867	−	0.995898i	$-0.528842\pi$
$449$	−81.2571	−0.180973	−0.0904867	+	0.995898i	$0.528842\pi$
$450$	0	0
$451$	− 878.955i	− 1.94890i
$452$	0	0
$453$	275.470	0.608101
$454$	0	0
$455$	2.64841i	0.00582069i
$456$	0	0
$457$	386.747	0.846274	0.423137	−	0.906066i	$-0.360929\pi$
$457$	386.747	0.846274	0.423137	+	0.906066i	$0.360929\pi$
$458$	0	0
$459$	− 45.8796i	− 0.0999555i
$460$	0	0
$461$	−23.0382	−0.0499744	−0.0249872	−	0.999688i	$-0.507955\pi$
$461$	−23.0382	−0.0499744	−0.0249872	+	0.999688i	$0.507955\pi$
$462$	0	0
$463$	− 606.274i	− 1.30945i	−0.755868	−	0.654724i	$-0.772785\pi$
$463$	− 606.274i	− 1.30945i	0.755868	−	0.654724i	$-0.227215\pi$
$464$	0	0
$465$	−83.2653	−0.179065
$466$	0	0
$467$	816.731i	1.74889i	0.485125	+	0.874445i	$0.338774\pi$
$467$	816.731i	1.74889i	−0.485125	+	0.874445i	$0.661226\pi$
$468$	0	0
$469$	228.668	0.487565
$470$	0	0
$471$	− 339.711i	− 0.721255i
$472$	0	0
$473$	159.270	0.336722
$474$	0	0
$475$	− 143.850i	− 0.302843i
$476$	0	0
$477$	−52.9824	−0.111074
$478$	0	0
$479$	− 528.270i	− 1.10286i	−0.834221	−	0.551430i	$-0.814082\pi$
$479$	− 528.270i	− 1.10286i	0.834221	−	0.551430i	$-0.185918\pi$
$480$	0	0
$481$	8.76787	0.0182284
$482$	0	0
$483$	− 118.224i	− 0.244771i
$484$	0	0
$485$	−271.516	−0.559827
$486$	0	0
$487$	57.5041i	0.118078i	0.998256	+	0.0590391i	$0.0188037\pi$
$487$	57.5041i	0.118078i	−0.998256	+	0.0590391i	$0.981196\pi$
$488$	0	0
$489$	−15.8320	−0.0323762
$490$	0	0
$491$	466.970i	0.951058i	0.879700	+	0.475529i	$0.157743\pi$
$491$	466.970i	0.951058i	−0.879700	+	0.475529i	$0.842257\pi$
$492$	0	0
$493$	224.415	0.455203
$494$	0	0
$495$	− 81.5338i	− 0.164715i
$496$	0	0
$497$	−222.546	−0.447778
$498$	0	0
$499$	246.634i	0.494257i	0.968983	+	0.247129i	$0.0794870\pi$
$499$	246.634i	0.494257i	−0.968983	+	0.247129i	$0.920513\pi$
$500$	0	0
$501$	−443.137	−0.884506
$502$	0	0
$503$	− 569.336i	− 1.13188i	−0.824446	−	0.565940i	$-0.808513\pi$
$503$	− 569.336i	− 1.13188i	0.824446	−	0.565940i	$-0.191487\pi$
$504$	0	0
$505$	−348.509	−0.690117
$506$	0	0
$507$	− 292.594i	− 0.577109i
$508$	0	0
$509$	276.352	0.542931	0.271466	−	0.962448i	$-0.412492\pi$
$509$	276.352	0.542931	0.271466	+	0.962448i	$0.412492\pi$
$510$	0	0
$511$	574.713i	1.12468i
$512$	0	0
$513$	−149.494	−0.291411
$514$	0	0
$515$	65.3864i	0.126964i
$516$	0	0
$517$	692.687	1.33982
$518$	0	0
$519$	568.942i	1.09623i
$520$	0	0
$521$	487.826	0.936326	0.468163	−	0.883642i	$-0.344916\pi$
$521$	487.826	0.936326	0.468163	+	0.883642i	$0.344916\pi$
$522$	0	0
$523$	− 800.276i	− 1.53016i	−0.643933	−	0.765082i	$-0.722698\pi$
$523$	− 800.276i	− 1.53016i	0.643933	−	0.765082i	$-0.277302\pi$
$524$	0	0
$525$	−38.5907	−0.0735060
$526$	0	0
$527$	− 189.826i	− 0.360201i
$528$	0	0
$529$	294.367	0.556460
$530$	0	0
$531$	85.2282i	0.160505i
$532$	0	0
$533$	−19.2214	−0.0360626
$534$	0	0
$535$	116.408i	0.217585i
$536$	0	0
$537$	−235.914	−0.439318
$538$	0	0
$539$	− 354.220i	− 0.657179i
$540$	0	0
$541$	667.426	1.23369	0.616845	−	0.787085i	$-0.288410\pi$
$541$	667.426	1.23369	0.616845	+	0.787085i	$0.288410\pi$
$542$	0	0
$543$	317.117i	0.584010i
$544$	0	0
$545$	−327.791	−0.601451
$546$	0	0
$547$	674.633i	1.23333i	0.787224	+	0.616667i	$0.211517\pi$
$547$	674.633i	1.23333i	−0.787224	+	0.616667i	$0.788483\pi$
$548$	0	0
$549$	−66.7332	−0.121554
$550$	0	0
$551$	− 731.232i	− 1.32710i
$552$	0	0
$553$	394.845	0.714005
$554$	0	0
$555$	127.759i	0.230196i
$556$	0	0
$557$	861.502	1.54668	0.773341	−	0.633990i	$-0.218584\pi$
$557$	861.502	1.54668	0.773341	+	0.633990i	$0.218584\pi$
$558$	0	0
$559$	− 3.48298i	− 0.00623073i
$560$	0	0
$561$	185.879	0.331334
$562$	0	0
$563$	− 319.214i	− 0.566987i	−0.958974	−	0.283494i	$-0.908507\pi$
$563$	− 319.214i	− 0.566987i	0.958974	−	0.283494i	$-0.0914935\pi$
$564$	0	0
$565$	166.680	0.295008
$566$	0	0
$567$	40.1046i	0.0707312i
$568$	0	0
$569$	−116.908	−0.205463	−0.102731	−	0.994709i	$-0.532758\pi$
$569$	−116.908	−0.205463	−0.102731	+	0.994709i	$0.532758\pi$
$570$	0	0
$571$	1104.69i	1.93467i	0.253508	+	0.967333i	$0.418416\pi$
$571$	1104.69i	1.93467i	−0.253508	+	0.967333i	$0.581584\pi$
$572$	0	0
$573$	372.537	0.650152
$574$	0	0
$575$	76.5887i	0.133198i
$576$	0	0
$577$	−802.968	−1.39163	−0.695813	−	0.718223i	$-0.744956\pi$
$577$	−802.968	−1.39163	−0.695813	+	0.718223i	$0.744956\pi$
$578$	0	0
$579$	2.59198i	0.00447665i
$580$	0	0
$581$	−193.373	−0.332829
$582$	0	0
$583$	− 214.656i	− 0.368191i
$584$	0	0
$585$	−1.78302	−0.00304789
$586$	0	0
$587$	− 164.619i	− 0.280442i	−0.990120	−	0.140221i	$-0.955219\pi$
$587$	− 164.619i	− 0.280442i	0.990120	−	0.140221i	$-0.0447813\pi$
$588$	0	0
$589$	−618.528	−1.05013
$590$	0	0
$591$	− 187.880i	− 0.317901i
$592$	0	0
$593$	378.471	0.638231	0.319116	−	0.947716i	$-0.396614\pi$
$593$	378.471	0.638231	0.319116	+	0.947716i	$0.396614\pi$
$594$	0	0
$595$	− 87.9780i	− 0.147862i
$596$	0	0
$597$	422.770	0.708158
$598$	0	0
$599$	− 86.5006i	− 0.144408i	−0.997390	−	0.0722042i	$-0.976997\pi$
$599$	− 86.5006i	− 0.144408i	0.997390	−	0.0722042i	$-0.0230033\pi$
$600$	0	0
$601$	−416.690	−0.693327	−0.346664	−	0.937990i	$-0.612685\pi$
$601$	−416.690	−0.693327	−0.346664	+	0.937990i	$0.612685\pi$
$602$	0	0
$603$	153.948i	0.255304i
$604$	0	0
$605$	59.7656	0.0987861
$606$	0	0
$607$	− 1056.06i	− 1.73980i	−0.493226	−	0.869901i	$-0.664183\pi$
$607$	− 1056.06i	− 1.73980i	0.493226	−	0.869901i	$-0.335817\pi$
$608$	0	0
$609$	−196.167	−0.322114
$610$	0	0
$611$	− 15.1480i	− 0.0247921i
$612$	0	0
$613$	1015.11	1.65596	0.827982	−	0.560754i	$-0.189489\pi$
$613$	1015.11	1.65596	0.827982	+	0.560754i	$0.189489\pi$
$614$	0	0
$615$	− 280.079i	− 0.455413i
$616$	0	0
$617$	575.077	0.932053	0.466027	−	0.884771i	$-0.345685\pi$
$617$	575.077	0.932053	0.466027	+	0.884771i	$0.345685\pi$
$618$	0	0
$619$	− 339.933i	− 0.549165i	−0.961564	−	0.274583i	$-0.911460\pi$
$619$	− 339.933i	− 0.549165i	0.961564	−	0.274583i	$-0.0885397\pi$
$620$	0	0
$621$	79.5933	0.128170
$622$	0	0
$623$	− 659.929i	− 1.05928i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 605.666i	− 0.965974i
$628$	0	0
$629$	−291.261	−0.463054
$630$	0	0
$631$	497.309i	0.788128i	0.919083	+	0.394064i	$0.128931\pi$
$631$	497.309i	0.788128i	−0.919083	+	0.394064i	$0.871069\pi$
$632$	0	0
$633$	602.084	0.951160
$634$	0	0
$635$	319.737i	0.503522i
$636$	0	0
$637$	−7.74623	−0.0121605
$638$	0	0
$639$	− 149.827i	− 0.234470i
$640$	0	0
$641$	17.7231	0.0276491	0.0138246	−	0.999904i	$-0.495599\pi$
$641$	17.7231	0.0276491	0.0138246	+	0.999904i	$0.495599\pi$
$642$	0	0
$643$	551.144i	0.857145i	0.903507	+	0.428573i	$0.140983\pi$
$643$	551.144i	0.857145i	−0.903507	+	0.428573i	$0.859017\pi$
$644$	0	0
$645$	50.7513	0.0786842
$646$	0	0
$647$	433.954i	0.670717i	0.942091	+	0.335358i	$0.108857\pi$
$647$	433.954i	0.670717i	−0.942091	+	0.335358i	$0.891143\pi$
$648$	0	0
$649$	−345.297	−0.532045
$650$	0	0
$651$	165.932i	0.254888i
$652$	0	0
$653$	−49.9181	−0.0764443	−0.0382221	−	0.999269i	$-0.512169\pi$
$653$	−49.9181	−0.0764443	−0.0382221	+	0.999269i	$0.512169\pi$
$654$	0	0
$655$	159.231i	0.243100i
$656$	0	0
$657$	−386.920	−0.588919
$658$	0	0
$659$	− 410.479i	− 0.622882i	−0.950266	−	0.311441i	$-0.899188\pi$
$659$	− 410.479i	− 0.622882i	0.950266	−	0.311441i	$-0.100812\pi$
$660$	0	0
$661$	565.667	0.855774	0.427887	−	0.903832i	$-0.359258\pi$
$661$	565.667	0.855774	0.427887	+	0.903832i	$0.359258\pi$
$662$	0	0
$663$	− 4.06487i	− 0.00613103i
$664$	0	0
$665$	−286.667	−0.431078
$666$	0	0
$667$	389.322i	0.583691i
$668$	0	0
$669$	528.923	0.790617
$670$	0	0
$671$	− 270.366i	− 0.402930i
$672$	0	0
$673$	853.392	1.26804	0.634021	−	0.773316i	$-0.281403\pi$
$673$	853.392	1.26804	0.634021	+	0.773316i	$0.281403\pi$
$674$	0	0
$675$	− 25.9808i	− 0.0384900i
$676$	0	0
$677$	570.348	0.842464	0.421232	−	0.906953i	$-0.361598\pi$
$677$	570.348	0.842464	0.421232	+	0.906953i	$0.361598\pi$
$678$	0	0
$679$	541.081i	0.796879i
$680$	0	0
$681$	700.269	1.02829
$682$	0	0
$683$	− 138.032i	− 0.202096i	−0.994882	−	0.101048i	$-0.967780\pi$
$683$	− 138.032i	− 0.202096i	0.994882	−	0.101048i	$-0.0322196\pi$
$684$	0	0
$685$	−183.550	−0.267956
$686$	0	0
$687$	− 614.632i	− 0.894661i
$688$	0	0
$689$	−4.69418	−0.00681303
$690$	0	0
$691$	− 200.350i	− 0.289943i	−0.989436	−	0.144971i	$-0.953691\pi$
$691$	− 200.350i	− 0.289943i	0.989436	−	0.144971i	$-0.0463090\pi$
$692$	0	0
$693$	−162.482	−0.234461
$694$	0	0
$695$	− 488.375i	− 0.702698i
$696$	0	0
$697$	638.517	0.916094
$698$	0	0
$699$	193.354i	0.276615i
$700$	0	0
$701$	−148.485	−0.211819	−0.105909	−	0.994376i	$-0.533775\pi$
$701$	−148.485	−0.211819	−0.105909	+	0.994376i	$0.533775\pi$
$702$	0	0
$703$	949.043i	1.34999i
$704$	0	0
$705$	220.725	0.313085
$706$	0	0
$707$	694.514i	0.982339i
$708$	0	0
$709$	−808.403	−1.14020	−0.570101	−	0.821575i	$-0.693096\pi$
$709$	−808.403	−1.14020	−0.570101	+	0.821575i	$0.693096\pi$
$710$	0	0
$711$	265.825i	0.373875i
$712$	0	0
$713$	329.316	0.461874
$714$	0	0
$715$	− 7.22380i	− 0.0101032i
$716$	0	0
$717$	429.865	0.599533
$718$	0	0
$719$	796.673i	1.10803i	0.832507	+	0.554015i	$0.186905\pi$
$719$	796.673i	1.10803i	−0.832507	+	0.554015i	$0.813095\pi$
$720$	0	0
$721$	130.303	0.180725
$722$	0	0
$723$	− 23.2417i	− 0.0321462i
$724$	0	0
$725$	127.082	0.175286
$726$	0	0
$727$	− 1059.29i	− 1.45707i	−0.685010	−	0.728534i	$-0.740202\pi$
$727$	− 1059.29i	− 1.45707i	0.685010	−	0.728534i	$-0.259798\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	115.702i	0.158278i
$732$	0	0
$733$	−1145.22	−1.56237	−0.781187	−	0.624298i	$-0.785385\pi$
$733$	−1145.22	−1.56237	−0.781187	+	0.624298i	$0.785385\pi$
$734$	0	0
$735$	− 112.872i	− 0.153568i
$736$	0	0
$737$	−623.713	−0.846286
$738$	0	0
$739$	155.577i	0.210523i	0.994445	+	0.105262i	$0.0335680\pi$
$739$	155.577i	0.210523i	−0.994445	+	0.105262i	$0.966432\pi$
$740$	0	0
$741$	−13.2450	−0.0178744
$742$	0	0
$743$	733.962i	0.987836i	0.869508	+	0.493918i	$0.164436\pi$
$743$	733.962i	0.987836i	−0.869508	+	0.493918i	$0.835564\pi$
$744$	0	0
$745$	−122.517	−0.164453
$746$	0	0
$747$	− 130.187i	− 0.174279i
$748$	0	0
$749$	231.979	0.309719
$750$	0	0
$751$	1032.25i	1.37450i	0.726423	+	0.687248i	$0.241182\pi$
$751$	1032.25i	1.37450i	−0.726423	+	0.687248i	$0.758818\pi$
$752$	0	0
$753$	−585.962	−0.778170
$754$	0	0
$755$	355.630i	0.471033i
$756$	0	0
$757$	70.8915	0.0936480	0.0468240	−	0.998903i	$-0.485090\pi$
$757$	70.8915	0.0936480	0.0468240	+	0.998903i	$0.485090\pi$
$758$	0	0
$759$	322.468i	0.424859i
$760$	0	0
$761$	446.601	0.586861	0.293430	−	0.955980i	$-0.405203\pi$
$761$	446.601	0.586861	0.293430	+	0.955980i	$0.405203\pi$
$762$	0	0
$763$	653.226i	0.856129i
$764$	0	0
$765$	59.2303	0.0774252
$766$	0	0
$767$	7.55111i	0.00984500i
$768$	0	0
$769$	1477.88	1.92183	0.960913	−	0.276850i	$-0.0892904\pi$
$769$	1477.88	1.92183	0.960913	+	0.276850i	$0.0892904\pi$
$770$	0	0
$771$	− 211.739i	− 0.274629i
$772$	0	0
$773$	396.016	0.512310	0.256155	−	0.966636i	$-0.417544\pi$
$773$	396.016	0.512310	0.256155	+	0.966636i	$0.417544\pi$
$774$	0	0
$775$	− 107.495i	− 0.138703i
$776$	0	0
$777$	254.599	0.327670
$778$	0	0
$779$	− 2080.54i	− 2.67078i
$780$	0	0
$781$	607.014	0.777227
$782$	0	0
$783$	− 132.068i	− 0.168669i
$784$	0	0
$785$	438.565	0.558681
$786$	0	0
$787$	1221.57i	1.55219i	0.630616	+	0.776095i	$0.282802\pi$
$787$	1221.57i	1.55219i	−0.630616	+	0.776095i	$0.717198\pi$
$788$	0	0
$789$	−501.909	−0.636133
$790$	0	0
$791$	− 332.162i	− 0.419926i
$792$	0	0
$793$	−5.91248	−0.00745584
$794$	0	0
$795$	− 68.4000i	− 0.0860378i
$796$	0	0
$797$	430.044	0.539579	0.269789	−	0.962919i	$-0.413046\pi$
$797$	430.044	0.539579	0.269789	+	0.962919i	$0.413046\pi$
$798$	0	0
$799$	503.203i	0.629791i
$800$	0	0
$801$	444.290	0.554669
$802$	0	0
$803$	− 1567.58i	− 1.95216i
$804$	0	0
$805$	152.627	0.189599
$806$	0	0
$807$	− 424.908i	− 0.526528i
$808$	0	0
$809$	980.456	1.21194	0.605968	−	0.795489i	$-0.292786\pi$
$809$	980.456	1.21194	0.605968	+	0.795489i	$0.292786\pi$
$810$	0	0
$811$	− 712.890i	− 0.879025i	−0.898236	−	0.439513i	$-0.855151\pi$
$811$	− 712.890i	− 0.879025i	0.898236	−	0.439513i	$-0.144849\pi$
$812$	0	0
$813$	−453.919	−0.558327
$814$	0	0
$815$	− 20.4390i	− 0.0250785i
$816$	0	0
$817$	377.001	0.461446
$818$	0	0
$819$	3.55322i	0.00433849i
$820$	0	0
$821$	−644.594	−0.785133	−0.392566	−	0.919724i	$-0.628413\pi$
$821$	−644.594	−0.785133	−0.392566	+	0.919724i	$0.628413\pi$
$822$	0	0
$823$	1005.30i	1.22151i	0.791819	+	0.610756i	$0.209134\pi$
$823$	1005.30i	1.22151i	−0.791819	+	0.610756i	$0.790866\pi$
$824$	0	0
$825$	105.260	0.127587
$826$	0	0
$827$	4.18687i	0.00506272i	0.999997	+	0.00253136i	$0.000805757\pi$
$827$	4.18687i	0.00506272i	−0.999997	+	0.00253136i	$0.999194\pi$
$828$	0	0
$829$	−25.8104	−0.0311343	−0.0155672	−	0.999879i	$-0.504955\pi$
$829$	−25.8104	−0.0311343	−0.0155672	+	0.999879i	$0.504955\pi$
$830$	0	0
$831$	− 173.208i	− 0.208434i
$832$	0	0
$833$	257.323	0.308911
$834$	0	0
$835$	− 572.088i	− 0.685135i
$836$	0	0
$837$	−111.712	−0.133467
$838$	0	0
$839$	− 1299.25i	− 1.54857i	−0.632838	−	0.774284i	$-0.718110\pi$
$839$	− 1299.25i	− 1.54857i	0.632838	−	0.774284i	$-0.281890\pi$
$840$	0	0
$841$	−195.006	−0.231874
$842$	0	0
$843$	50.3465i	0.0597231i
$844$	0	0
$845$	377.738	0.447027
$846$	0	0
$847$	− 119.102i	− 0.140616i
$848$	0	0
$849$	−110.168	−0.129762
$850$	0	0
$851$	− 505.288i	− 0.593758i
$852$	0	0
$853$	16.3846	0.0192082	0.00960410	−	0.999954i	$-0.496943\pi$
$853$	16.3846	0.0192082	0.00960410	+	0.999954i	$0.496943\pi$
$854$	0	0
$855$	− 192.995i	− 0.225726i
$856$	0	0
$857$	−146.244	−0.170646	−0.0853231	−	0.996353i	$-0.527192\pi$
$857$	−146.244	−0.170646	−0.0853231	+	0.996353i	$0.527192\pi$
$858$	0	0
$859$	1277.30i	1.48696i	0.668756	+	0.743482i	$0.266827\pi$
$859$	1277.30i	1.48696i	−0.668756	+	0.743482i	$0.733173\pi$
$860$	0	0
$861$	−558.146	−0.648253
$862$	0	0
$863$	1209.38i	1.40137i	0.713471	+	0.700685i	$0.247122\pi$
$863$	1209.38i	1.40137i	−0.713471	+	0.700685i	$0.752878\pi$
$864$	0	0
$865$	−734.501	−0.849134
$866$	0	0
$867$	− 365.531i	− 0.421604i
$868$	0	0
$869$	−1076.98	−1.23933
$870$	0	0
$871$	13.6396i	0.0156597i
$872$	0	0
$873$	−364.277	−0.417270
$874$	0	0
$875$	− 49.8203i	− 0.0569375i
$876$	0	0
$877$	−923.114	−1.05258	−0.526291	−	0.850305i	$-0.676418\pi$
$877$	−923.114	−1.05258	−0.526291	+	0.850305i	$0.676418\pi$
$878$	0	0
$879$	− 15.5200i	− 0.0176564i
$880$	0	0
$881$	−1243.05	−1.41095	−0.705476	−	0.708734i	$-0.749267\pi$
$881$	−1243.05	−1.41095	−0.705476	+	0.708734i	$0.749267\pi$
$882$	0	0
$883$	542.111i	0.613943i	0.951719	+	0.306971i	$0.0993156\pi$
$883$	542.111i	0.613943i	−0.951719	+	0.306971i	$0.900684\pi$
$884$	0	0
$885$	−110.029	−0.124327
$886$	0	0
$887$	1719.74i	1.93883i	0.245434	+	0.969413i	$0.421070\pi$
$887$	1719.74i	1.93883i	−0.245434	+	0.969413i	$0.578930\pi$
$888$	0	0
$889$	637.176	0.716733
$890$	0	0
$891$	− 109.389i	− 0.122771i
$892$	0	0
$893$	1639.63	1.83610
$894$	0	0
$895$	− 304.563i	− 0.340294i
$896$	0	0
$897$	7.05187	0.00786161
$898$	0	0
$899$	− 546.428i	− 0.607817i
$900$	0	0
$901$	155.937	0.173071
$902$	0	0
$903$	− 101.138i	− 0.112002i
$904$	0	0
$905$	−409.397	−0.452372
$906$	0	0
$907$	− 496.545i	− 0.547458i	−0.961807	−	0.273729i	$-0.911743\pi$
$907$	− 496.545i	− 0.547458i	0.961807	−	0.273729i	$-0.0882572\pi$
$908$	0	0
$909$	−467.574	−0.514383
$910$	0	0
$911$	1135.10i	1.24600i	0.782223	+	0.622999i	$0.214086\pi$
$911$	1135.10i	1.24600i	−0.782223	+	0.622999i	$0.785914\pi$
$912$	0	0
$913$	527.444	0.577704
$914$	0	0
$915$	− 86.1522i	− 0.0941554i
$916$	0	0
$917$	317.317	0.346038
$918$	0	0
$919$	1378.92i	1.50045i	0.661180	+	0.750227i	$0.270056\pi$
$919$	1378.92i	1.50045i	−0.661180	+	0.750227i	$0.729944\pi$
$920$	0	0
$921$	602.195	0.653849
$922$	0	0
$923$	− 13.2745i	− 0.0143819i
$924$	0	0
$925$	−164.936	−0.178309
$926$	0	0
$927$	87.7251i	0.0946333i
$928$	0	0
$929$	317.155	0.341394	0.170697	−	0.985324i	$-0.445398\pi$
$929$	317.155	0.341394	0.170697	+	0.985324i	$0.445398\pi$
$930$	0	0
$931$	− 838.460i	− 0.900601i
$932$	0	0
$933$	−333.921	−0.357900
$934$	0	0
$935$	239.968i	0.256651i
$936$	0	0
$937$	−1718.74	−1.83430	−0.917150	−	0.398542i	$-0.869516\pi$
$937$	−1718.74	−1.83430	−0.917150	+	0.398542i	$0.869516\pi$
$938$	0	0
$939$	− 407.921i	− 0.434421i
$940$	0	0
$941$	703.030	0.747109	0.373555	−	0.927608i	$-0.378139\pi$
$941$	703.030	0.747109	0.373555	+	0.927608i	$0.378139\pi$
$942$	0	0
$943$	1107.72i	1.17468i
$944$	0	0
$945$	−51.7748	−0.0547882
$946$	0	0
$947$	838.953i	0.885906i	0.896545	+	0.442953i	$0.146069\pi$
$947$	838.953i	0.885906i	−0.896545	+	0.442953i	$0.853931\pi$
$948$	0	0
$949$	−34.2806	−0.0361229
$950$	0	0
$951$	− 946.699i	− 0.995478i
$952$	0	0
$953$	−404.555	−0.424507	−0.212253	−	0.977215i	$-0.568080\pi$
$953$	−404.555	−0.424507	−0.212253	+	0.977215i	$0.568080\pi$
$954$	0	0
$955$	480.944i	0.503606i
$956$	0	0
$957$	535.065	0.559106
$958$	0	0
$959$	365.781i	0.381419i
$960$	0	0
$961$	498.792	0.519035
$962$	0	0
$963$	156.178i	0.162178i
$964$	0	0
$965$	−3.34623	−0.00346759
$966$	0	0
$967$	− 912.447i	− 0.943585i	−0.881710	−	0.471793i	$-0.843607\pi$
$967$	− 912.447i	− 0.943585i	0.881710	−	0.471793i	$-0.156393\pi$
$968$	0	0
$969$	439.986	0.454062
$970$	0	0
$971$	− 357.552i	− 0.368231i	−0.982905	−	0.184115i	$-0.941058\pi$
$971$	− 357.552i	− 0.368231i	0.982905	−	0.184115i	$-0.0589420\pi$
$972$	0	0
$973$	−973.240	−1.00025
$974$	0	0
$975$	− 2.30186i	− 0.00236089i
$976$	0	0
$977$	−1028.81	−1.05303	−0.526513	−	0.850167i	$-0.676501\pi$
$977$	−1028.81	−1.05303	−0.526513	+	0.850167i	$0.676501\pi$
$978$	0	0
$979$	1800.02i	1.83863i
$980$	0	0
$981$	−439.778	−0.448295
$982$	0	0
$983$	− 1434.99i	− 1.45980i	−0.683552	−	0.729902i	$-0.739566\pi$
$983$	− 1434.99i	− 1.45980i	0.683552	−	0.729902i	$-0.260434\pi$
$984$	0	0
$985$	242.552	0.246245
$986$	0	0
$987$	− 439.864i	− 0.445657i
$988$	0	0
$989$	−200.723	−0.202955
$990$	0	0
$991$	1312.26i	1.32418i	0.749426	+	0.662088i	$0.230329\pi$
$991$	1312.26i	1.32418i	−0.749426	+	0.662088i	$0.769671\pi$
$992$	0	0
$993$	423.135	0.426118
$994$	0	0
$995$	545.794i	0.548537i
$996$	0	0
$997$	−655.906	−0.657879	−0.328940	−	0.944351i	$-0.606691\pi$
$997$	−655.906	−0.657879	−0.328940	+	0.944351i	$0.606691\pi$
$998$	0	0
$999$	171.406i	0.171578i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.e.d.511.5		8
3.2	odd	2		2880.3.e.n.2431.7		8
4.3	odd	2	inner	960.3.e.d.511.2		8
8.3	odd	2		480.3.e.a.31.8	yes	8
8.5	even	2		480.3.e.a.31.3	✓	8
12.11	even	2		2880.3.e.n.2431.6		8
24.5	odd	2		1440.3.e.e.991.3		8
24.11	even	2		1440.3.e.e.991.2		8
40.3	even	4		2400.3.j.d.799.3		8
40.13	odd	4		2400.3.j.j.799.6		8
40.19	odd	2		2400.3.e.g.1951.2		8
40.27	even	4		2400.3.j.j.799.5		8
40.29	even	2		2400.3.e.g.1951.7		8
40.37	odd	4		2400.3.j.d.799.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.e.a.31.3	✓	8	8.5	even	2
480.3.e.a.31.8	yes	8	8.3	odd	2
960.3.e.d.511.2		8	4.3	odd	2	inner
960.3.e.d.511.5		8	1.1	even	1	trivial
1440.3.e.e.991.2		8	24.11	even	2
1440.3.e.e.991.3		8	24.5	odd	2
2400.3.e.g.1951.2		8	40.19	odd	2
2400.3.e.g.1951.7		8	40.29	even	2
2400.3.j.d.799.3		8	40.3	even	4
2400.3.j.d.799.4		8	40.37	odd	4
2400.3.j.j.799.5		8	40.27	even	4
2400.3.j.j.799.6		8	40.13	odd	4
2880.3.e.n.2431.6		8	12.11	even	2
2880.3.e.n.2431.7		8	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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -2.23607 q^{5} +4.45607i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -2.23607 q^{5} +4.45607i q^{7} -3.00000 q^{9} -12.1543i q^{11} -0.265796 q^{13} -3.87298i q^{15} +8.82953 q^{17} -28.7701i q^{19} -7.71813 q^{21} +15.3177i q^{23} +5.00000 q^{25} -5.19615i q^{27} +25.4164 q^{29} -21.4990i q^{31} +21.0519 q^{33} -9.96407i q^{35} -32.9872 q^{37} -0.460373i q^{39} +72.3161 q^{41} +13.1039i q^{43} +6.70820 q^{45} +56.9909i q^{47} +29.1435 q^{49} +15.2932i q^{51} +17.6608 q^{53} +27.1779i q^{55} +49.8312 q^{57} -28.4094i q^{59} +22.2444 q^{61} -13.3682i q^{63} +0.594339 q^{65} -51.3161i q^{67} -26.5311 q^{69} +49.9422i q^{71} +128.973 q^{73} +8.66025i q^{75} +54.1606 q^{77} -88.6084i q^{79} +9.00000 q^{81} +43.3955i q^{83} -19.7434 q^{85} +44.0225i q^{87} -148.097 q^{89} -1.18441i q^{91} +37.2374 q^{93} +64.3318i q^{95} +121.426 q^{97} +36.4630i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 16 q^{13} - 16 q^{17} - 96 q^{21} + 40 q^{25} + 96 q^{29} + 48 q^{33} + 112 q^{37} + 112 q^{41} - 184 q^{49} - 224 q^{53} - 144 q^{57} - 80 q^{61} - 160 q^{65} + 144 q^{69} + 272 q^{73} - 256 q^{77} + 72 q^{81} + 160 q^{85} - 48 q^{89} + 240 q^{93} + 528 q^{97}+O(q^{100}) \) 8 * q - 24 * q^9 + 16 * q^13 - 16 * q^17 - 96 * q^21 + 40 * q^25 + 96 * q^29 + 48 * q^33 + 112 * q^37 + 112 * q^41 - 184 * q^49 - 224 * q^53 - 144 * q^57 - 80 * q^61 - 160 * q^65 + 144 * q^69 + 272 * q^73 - 256 * q^77 + 72 * q^81 + 160 * q^85 - 48 * q^89 + 240 * q^93 + 528 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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