LMFDB - Embedded newform 1680.2.d.b.1231.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(1231,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1680.1231");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1231.2
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1680.1231
Dual form			1680.2.d.b.1231.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000i q^{5} +(2.44949 - 1.00000i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000i q^{5} +(2.44949 - 1.00000i) q^{7} +1.00000 q^{9} -2.44949i q^{11} -0.449490i q^{13} -1.00000i q^{15} -2.89898i q^{17} -0.449490 q^{19} +(2.44949 - 1.00000i) q^{21} -2.00000i q^{23} -1.00000 q^{25} +1.00000 q^{27} +2.89898 q^{29} -3.55051 q^{31} -2.44949i q^{33} +(-1.00000 - 2.44949i) q^{35} -2.89898 q^{37} -0.449490i q^{39} -8.89898i q^{41} +8.00000i q^{43} -1.00000i q^{45} +1.10102 q^{47} +(5.00000 - 4.89898i) q^{49} -2.89898i q^{51} +5.55051 q^{53} -2.44949 q^{55} -0.449490 q^{57} -4.89898 q^{59} +3.10102i q^{61} +(2.44949 - 1.00000i) q^{63} -0.449490 q^{65} +3.79796i q^{67} -2.00000i q^{69} +1.55051i q^{71} -9.34847i q^{73} -1.00000 q^{75} +(-2.44949 - 6.00000i) q^{77} +9.79796i q^{79} +1.00000 q^{81} +6.89898 q^{83} -2.89898 q^{85} +2.89898 q^{87} -3.79796i q^{89} +(-0.449490 - 1.10102i) q^{91} -3.55051 q^{93} +0.449490i q^{95} -17.3485i q^{97} -2.44949i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} + 8 q^{19} - 4 q^{25} + 4 q^{27} - 8 q^{29} - 24 q^{31} - 4 q^{35} + 8 q^{37} + 24 q^{47} + 20 q^{49} + 32 q^{53} + 8 q^{57} + 8 q^{65} - 4 q^{75} + 4 q^{81} + 8 q^{83} + 8 q^{85} - 8 q^{87} + 8 q^{91} - 24 q^{93}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 8 * q^19 - 4 * q^25 + 4 * q^27 - 8 * q^29 - 24 * q^31 - 4 * q^35 + 8 * q^37 + 24 * q^47 + 20 * q^49 + 32 * q^53 + 8 * q^57 + 8 * q^65 - 4 * q^75 + 4 * q^81 + 8 * q^83 + 8 * q^85 - 8 * q^87 + 8 * q^91 - 24 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	2.44949	−	1.00000i	0.925820	−	0.377964i
$7$	2.44949	−	1.00000i	0.925820	−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	2.44949i		−	0.738549i	−0.929320	−	0.369274i	$-0.879606\pi$
$11$		−	2.44949i		−	0.738549i	0.929320	−	0.369274i	$-0.120394\pi$
$12$		0			0
$13$		−	0.449490i		−	0.124666i	−0.998055	−	0.0623330i	$-0.980146\pi$
$13$		−	0.449490i		−	0.124666i	0.998055	−	0.0623330i	$-0.0198541\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$		−	2.89898i		−	0.703106i	−0.936168	−	0.351553i	$-0.885654\pi$
$17$		−	2.89898i		−	0.703106i	0.936168	−	0.351553i	$-0.114346\pi$
$18$		0			0
$19$	−0.449490			−0.103120			−0.0515600	−	0.998670i	$-0.516419\pi$
$19$	−0.449490			−0.103120			−0.0515600	+	0.998670i	$0.516419\pi$
$20$		0			0
$21$	2.44949	−	1.00000i	0.534522	−	0.218218i
$22$		0			0
$23$		−	2.00000i		−	0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$		−	2.00000i		−	0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	2.89898			0.538327			0.269163	−	0.963095i	$-0.413253\pi$
$29$	2.89898			0.538327			0.269163	+	0.963095i	$0.413253\pi$
$30$		0			0
$31$	−3.55051			−0.637690			−0.318845	−	0.947807i	$-0.603295\pi$
$31$	−3.55051			−0.637690			−0.318845	+	0.947807i	$0.603295\pi$
$32$		0			0
$33$		−	2.44949i		−	0.426401i
$34$		0			0
$35$	−1.00000	−	2.44949i	−0.169031	−	0.414039i
$36$		0			0
$37$	−2.89898			−0.476589			−0.238295	−	0.971193i	$-0.576588\pi$
$37$	−2.89898			−0.476589			−0.238295	+	0.971193i	$0.576588\pi$
$38$		0			0
$39$		−	0.449490i		−	0.0719760i
$40$		0			0
$41$		−	8.89898i		−	1.38979i	−0.719113	−	0.694894i	$-0.755451\pi$
$41$		−	8.89898i		−	1.38979i	0.719113	−	0.694894i	$-0.244549\pi$
$42$		0			0
$43$			8.00000i			1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$			8.00000i			1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$	1.10102			0.160600			0.0803002	−	0.996771i	$-0.474412\pi$
$47$	1.10102			0.160600			0.0803002	+	0.996771i	$0.474412\pi$
$48$		0			0
$49$	5.00000	−	4.89898i	0.714286	−	0.699854i
$50$		0			0
$51$		−	2.89898i		−	0.405938i
$52$		0			0
$53$	5.55051			0.762421			0.381211	−	0.924488i	$-0.375507\pi$
$53$	5.55051			0.762421			0.381211	+	0.924488i	$0.375507\pi$
$54$		0			0
$55$	−2.44949			−0.330289
$56$		0			0
$57$	−0.449490			−0.0595364
$58$		0			0
$59$	−4.89898			−0.637793			−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898			−0.637793			−0.318896	+	0.947790i	$0.603312\pi$
$60$		0			0
$61$			3.10102i			0.397045i	0.980096	+	0.198522i	$0.0636143\pi$
$61$			3.10102i			0.397045i	−0.980096	+	0.198522i	$0.936386\pi$
$62$		0			0
$63$	2.44949	−	1.00000i	0.308607	−	0.125988i
$64$		0			0
$65$	−0.449490			−0.0557523
$66$		0			0
$67$			3.79796i			0.463995i	0.972716	+	0.231997i	$0.0745260\pi$
$67$			3.79796i			0.463995i	−0.972716	+	0.231997i	$0.925474\pi$
$68$		0			0
$69$		−	2.00000i		−	0.240772i
$70$		0			0
$71$			1.55051i			0.184012i	0.995758	+	0.0920059i	$0.0293279\pi$
$71$			1.55051i			0.184012i	−0.995758	+	0.0920059i	$0.970672\pi$
$72$		0			0
$73$		−	9.34847i		−	1.09416i	−0.837082	−	0.547078i	$-0.815740\pi$
$73$		−	9.34847i		−	1.09416i	0.837082	−	0.547078i	$-0.184260\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	−2.44949	−	6.00000i	−0.279145	−	0.683763i
$78$		0			0
$79$			9.79796i			1.10236i	0.834388	+	0.551178i	$0.185822\pi$
$79$			9.79796i			1.10236i	−0.834388	+	0.551178i	$0.814178\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	6.89898			0.757261			0.378631	−	0.925548i	$-0.376395\pi$
$83$	6.89898			0.757261			0.378631	+	0.925548i	$0.376395\pi$
$84$		0			0
$85$	−2.89898			−0.314438
$86$		0			0
$87$	2.89898			0.310803
$88$		0			0
$89$		−	3.79796i		−	0.402583i	−0.979531	−	0.201291i	$-0.935486\pi$
$89$		−	3.79796i		−	0.402583i	0.979531	−	0.201291i	$-0.0645138\pi$
$90$		0			0
$91$	−0.449490	−	1.10102i	−0.0471193	−	0.115418i
$92$		0			0
$93$	−3.55051			−0.368171
$94$		0			0
$95$			0.449490i			0.0461167i
$96$		0			0
$97$		−	17.3485i		−	1.76147i	−0.473609	−	0.880735i	$-0.657049\pi$
$97$		−	17.3485i		−	1.76147i	0.473609	−	0.880735i	$-0.342951\pi$
$98$		0			0
$99$		−	2.44949i		−	0.246183i
$100$		0			0
$101$		−	0.898979i		−	0.0894518i	−0.998999	−	0.0447259i	$-0.985759\pi$
$101$		−	0.898979i		−	0.0894518i	0.998999	−	0.0447259i	$-0.0142414\pi$
$102$		0			0
$103$	0.898979			0.0885791			0.0442895	−	0.999019i	$-0.485898\pi$
$103$	0.898979			0.0885791			0.0442895	+	0.999019i	$0.485898\pi$
$104$		0			0
$105$	−1.00000	−	2.44949i	−0.0975900	−	0.239046i
$106$		0			0
$107$		−	3.79796i		−	0.367163i	−0.983005	−	0.183581i	$-0.941231\pi$
$107$		−	3.79796i		−	0.367163i	0.983005	−	0.183581i	$-0.0587690\pi$
$108$		0			0
$109$	5.79796			0.555344			0.277672	−	0.960676i	$-0.410437\pi$
$109$	5.79796			0.555344			0.277672	+	0.960676i	$0.410437\pi$
$110$		0			0
$111$	−2.89898			−0.275159
$112$		0			0
$113$	11.3485			1.06757			0.533787	−	0.845619i	$-0.320768\pi$
$113$	11.3485			1.06757			0.533787	+	0.845619i	$0.320768\pi$
$114$		0			0
$115$	−2.00000			−0.186501
$116$		0			0
$117$		−	0.449490i		−	0.0415553i
$118$		0			0
$119$	−2.89898	−	7.10102i	−0.265749	−	0.650949i
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		−	8.89898i		−	0.802394i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			17.7980i			1.57931i	0.613549	+	0.789657i	$0.289741\pi$
$127$			17.7980i			1.57931i	−0.613549	+	0.789657i	$0.710259\pi$
$128$		0			0
$129$			8.00000i			0.704361i
$130$		0			0
$131$	10.6969			0.934596			0.467298	−	0.884100i	$-0.345228\pi$
$131$	10.6969			0.934596			0.467298	+	0.884100i	$0.345228\pi$
$132$		0			0
$133$	−1.10102	+	0.449490i	−0.0954706	+	0.0389757i
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$	−13.1464			−1.12318			−0.561588	−	0.827417i	$-0.689809\pi$
$137$	−13.1464			−1.12318			−0.561588	+	0.827417i	$0.689809\pi$
$138$		0			0
$139$	−12.4495			−1.05595			−0.527976	−	0.849259i	$-0.677049\pi$
$139$	−12.4495			−1.05595			−0.527976	+	0.849259i	$0.677049\pi$
$140$		0			0
$141$	1.10102			0.0927227
$142$		0			0
$143$	−1.10102			−0.0920720
$144$		0			0
$145$		−	2.89898i		−	0.240747i
$146$		0			0
$147$	5.00000	−	4.89898i	0.412393	−	0.404061i
$148$		0			0
$149$	15.7980			1.29422			0.647110	−	0.762397i	$-0.275978\pi$
$149$	15.7980			1.29422			0.647110	+	0.762397i	$0.275978\pi$
$150$		0			0
$151$			0.898979i			0.0731579i	0.999331	+	0.0365790i	$0.0116460\pi$
$151$			0.898979i			0.0731579i	−0.999331	+	0.0365790i	$0.988354\pi$
$152$		0			0
$153$		−	2.89898i		−	0.234369i
$154$		0			0
$155$			3.55051i			0.285184i
$156$		0			0
$157$		−	2.65153i		−	0.211615i	−0.994387	−	0.105808i	$-0.966257\pi$
$157$		−	2.65153i		−	0.211615i	0.994387	−	0.105808i	$-0.0337428\pi$
$158$		0			0
$159$	5.55051			0.440184
$160$		0			0
$161$	−2.00000	−	4.89898i	−0.157622	−	0.386094i
$162$		0			0
$163$			15.5959i			1.22157i	0.791798	+	0.610783i	$0.209145\pi$
$163$			15.5959i			1.22157i	−0.791798	+	0.610783i	$0.790855\pi$
$164$		0			0
$165$	−2.44949			−0.190693
$166$		0			0
$167$	−8.00000			−0.619059			−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000			−0.619059			−0.309529	+	0.950890i	$0.600171\pi$
$168$		0			0
$169$	12.7980			0.984458
$170$		0			0
$171$	−0.449490			−0.0343733
$172$		0			0
$173$			15.7980i			1.20110i	0.799588	+	0.600548i	$0.205051\pi$
$173$			15.7980i			1.20110i	−0.799588	+	0.600548i	$0.794949\pi$
$174$		0			0
$175$	−2.44949	+	1.00000i	−0.185164	+	0.0755929i
$176$		0			0
$177$	−4.89898			−0.368230
$178$		0			0
$179$		−	24.2474i		−	1.81234i	−0.422914	−	0.906170i	$-0.638993\pi$
$179$		−	24.2474i		−	1.81234i	0.422914	−	0.906170i	$-0.361007\pi$
$180$		0			0
$181$			5.79796i			0.430959i	0.976508	+	0.215479i	$0.0691314\pi$
$181$			5.79796i			0.430959i	−0.976508	+	0.215479i	$0.930869\pi$
$182$		0			0
$183$			3.10102i			0.229234i
$184$		0			0
$185$			2.89898i			0.213137i
$186$		0			0
$187$	−7.10102			−0.519278
$188$		0			0
$189$	2.44949	−	1.00000i	0.178174	−	0.0727393i
$190$		0			0
$191$			5.55051i			0.401621i	0.979630	+	0.200810i	$0.0643575\pi$
$191$			5.55051i			0.401621i	−0.979630	+	0.200810i	$0.935642\pi$
$192$		0			0
$193$	16.6969			1.20187			0.600936	−	0.799297i	$-0.294795\pi$
$193$	16.6969			1.20187			0.600936	+	0.799297i	$0.294795\pi$
$194$		0			0
$195$	−0.449490			−0.0321886
$196$		0			0
$197$	−24.2474			−1.72756			−0.863780	−	0.503870i	$-0.831909\pi$
$197$	−24.2474			−1.72756			−0.863780	+	0.503870i	$0.831909\pi$
$198$		0			0
$199$	−0.449490			−0.0318635			−0.0159317	−	0.999873i	$-0.505071\pi$
$199$	−0.449490			−0.0318635			−0.0159317	+	0.999873i	$0.505071\pi$
$200$		0			0
$201$			3.79796i			0.267887i
$202$		0			0
$203$	7.10102	−	2.89898i	0.498394	−	0.203468i
$204$		0			0
$205$	−8.89898			−0.621532
$206$		0			0
$207$		−	2.00000i		−	0.139010i
$208$		0			0
$209$			1.10102i			0.0761592i
$210$		0			0
$211$			22.6969i			1.56252i	0.624205	+	0.781261i	$0.285423\pi$
$211$			22.6969i			1.56252i	−0.624205	+	0.781261i	$0.714577\pi$
$212$		0			0
$213$			1.55051i			0.106239i
$214$		0			0
$215$	8.00000			0.545595
$216$		0			0
$217$	−8.69694	+	3.55051i	−0.590387	+	0.241024i
$218$		0			0
$219$		−	9.34847i		−	0.631711i
$220$		0			0
$221$	−1.30306			−0.0876534
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−8.69694			−0.577236			−0.288618	−	0.957444i	$-0.593196\pi$
$227$	−8.69694			−0.577236			−0.288618	+	0.957444i	$0.593196\pi$
$228$		0			0
$229$			16.8990i			1.11672i	0.829600	+	0.558358i	$0.188568\pi$
$229$			16.8990i			1.11672i	−0.829600	+	0.558358i	$0.811432\pi$
$230$		0			0
$231$	−2.44949	−	6.00000i	−0.161165	−	0.394771i
$232$		0			0
$233$	12.2474			0.802357			0.401179	−	0.916000i	$-0.368601\pi$
$233$	12.2474			0.802357			0.401179	+	0.916000i	$0.368601\pi$
$234$		0			0
$235$		−	1.10102i		−	0.0718227i
$236$		0			0
$237$			9.79796i			0.636446i
$238$		0			0
$239$			21.5505i			1.39399i	0.717078	+	0.696993i	$0.245479\pi$
$239$			21.5505i			1.39399i	−0.717078	+	0.696993i	$0.754521\pi$
$240$		0			0
$241$		−	8.89898i		−	0.573234i	−0.958045	−	0.286617i	$-0.907469\pi$
$241$		−	8.89898i		−	0.573234i	0.958045	−	0.286617i	$-0.0925307\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−4.89898	−	5.00000i	−0.312984	−	0.319438i
$246$		0			0
$247$			0.202041i			0.0128556i
$248$		0			0
$249$	6.89898			0.437205
$250$		0			0
$251$	30.6969			1.93757			0.968787	−	0.247895i	$-0.0797390\pi$
$251$	30.6969			1.93757			0.968787	+	0.247895i	$0.0797390\pi$
$252$		0			0
$253$	−4.89898			−0.307996
$254$		0			0
$255$	−2.89898			−0.181541
$256$		0			0
$257$		−	10.8990i		−	0.679860i	−0.940451	−	0.339930i	$-0.889597\pi$
$257$		−	10.8990i		−	0.679860i	0.940451	−	0.339930i	$-0.110403\pi$
$258$		0			0
$259$	−7.10102	+	2.89898i	−0.441236	+	0.180134i
$260$		0			0
$261$	2.89898			0.179442
$262$		0			0
$263$		−	10.8990i		−	0.672060i	−0.941851	−	0.336030i	$-0.890916\pi$
$263$		−	10.8990i		−	0.672060i	0.941851	−	0.336030i	$-0.109084\pi$
$264$		0			0
$265$		−	5.55051i		−	0.340965i
$266$		0			0
$267$		−	3.79796i		−	0.232431i
$268$		0			0
$269$			28.4949i			1.73736i	0.495370	+	0.868682i	$0.335033\pi$
$269$			28.4949i			1.73736i	−0.495370	+	0.868682i	$0.664967\pi$
$270$		0			0
$271$	−15.1464			−0.920080			−0.460040	−	0.887898i	$-0.652165\pi$
$271$	−15.1464			−0.920080			−0.460040	+	0.887898i	$0.652165\pi$
$272$		0			0
$273$	−0.449490	−	1.10102i	−0.0272044	−	0.0666368i
$274$		0			0
$275$			2.44949i			0.147710i
$276$		0			0
$277$	−25.5959			−1.53791			−0.768955	−	0.639303i	$-0.779223\pi$
$277$	−25.5959			−1.53791			−0.768955	+	0.639303i	$0.779223\pi$
$278$		0			0
$279$	−3.55051			−0.212563
$280$		0			0
$281$	−10.8990			−0.650179			−0.325089	−	0.945683i	$-0.605394\pi$
$281$	−10.8990			−0.650179			−0.325089	+	0.945683i	$0.605394\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$			0.449490i			0.0266255i
$286$		0			0
$287$	−8.89898	−	21.7980i	−0.525290	−	1.28669i
$288$		0			0
$289$	8.59592			0.505642
$290$		0			0
$291$		−	17.3485i		−	1.01699i
$292$		0			0
$293$			4.69694i			0.274398i	0.990543	+	0.137199i	$0.0438100\pi$
$293$			4.69694i			0.274398i	−0.990543	+	0.137199i	$0.956190\pi$
$294$		0			0
$295$			4.89898i			0.285230i
$296$		0			0
$297$		−	2.44949i		−	0.142134i
$298$		0			0
$299$	−0.898979			−0.0519893
$300$		0			0
$301$	8.00000	+	19.5959i	0.461112	+	1.12949i
$302$		0			0
$303$		−	0.898979i		−	0.0516450i
$304$		0			0
$305$	3.10102			0.177564
$306$		0			0
$307$	0.898979			0.0513075			0.0256537	−	0.999671i	$-0.491833\pi$
$307$	0.898979			0.0513075			0.0256537	+	0.999671i	$0.491833\pi$
$308$		0			0
$309$	0.898979			0.0511412
$310$		0			0
$311$	12.8990			0.731434			0.365717	−	0.930726i	$-0.380824\pi$
$311$	12.8990			0.731434			0.365717	+	0.930726i	$0.380824\pi$
$312$		0			0
$313$			29.3485i			1.65887i	0.558601	+	0.829437i	$0.311338\pi$
$313$			29.3485i			1.65887i	−0.558601	+	0.829437i	$0.688662\pi$
$314$		0			0
$315$	−1.00000	−	2.44949i	−0.0563436	−	0.138013i
$316$		0			0
$317$	−17.1464			−0.963039			−0.481520	−	0.876435i	$-0.659915\pi$
$317$	−17.1464			−0.963039			−0.481520	+	0.876435i	$0.659915\pi$
$318$		0			0
$319$		−	7.10102i		−	0.397581i
$320$		0			0
$321$		−	3.79796i		−	0.211981i
$322$		0			0
$323$			1.30306i			0.0725043i
$324$		0			0
$325$			0.449490i			0.0249332i
$326$		0			0
$327$	5.79796			0.320628
$328$		0			0
$329$	2.69694	−	1.10102i	0.148687	−	0.0607012i
$330$		0			0
$331$			16.8990i			0.928852i	0.885612	+	0.464426i	$0.153739\pi$
$331$			16.8990i			0.928852i	−0.885612	+	0.464426i	$0.846261\pi$
$332$		0			0
$333$	−2.89898			−0.158863
$334$		0			0
$335$	3.79796			0.207505
$336$		0			0
$337$	−16.6969			−0.909540			−0.454770	−	0.890609i	$-0.650279\pi$
$337$	−16.6969			−0.909540			−0.454770	+	0.890609i	$0.650279\pi$
$338$		0			0
$339$	11.3485			0.616364
$340$		0			0
$341$			8.69694i			0.470966i
$342$		0			0
$343$	7.34847	−	17.0000i	0.396780	−	0.917914i
$344$		0			0
$345$	−2.00000			−0.107676
$346$		0			0
$347$		−	25.1010i		−	1.34749i	−0.738962	−	0.673747i	$-0.764684\pi$
$347$		−	25.1010i		−	1.34749i	0.738962	−	0.673747i	$-0.235316\pi$
$348$		0			0
$349$			18.6969i			1.00082i	0.865787	+	0.500412i	$0.166818\pi$
$349$			18.6969i			1.00082i	−0.865787	+	0.500412i	$0.833182\pi$
$350$		0			0
$351$		−	0.449490i		−	0.0239920i
$352$		0			0
$353$			12.2020i			0.649449i	0.945809	+	0.324725i	$0.105272\pi$
$353$			12.2020i			0.649449i	−0.945809	+	0.324725i	$0.894728\pi$
$354$		0			0
$355$	1.55051			0.0822925
$356$		0			0
$357$	−2.89898	−	7.10102i	−0.153430	−	0.375826i
$358$		0			0
$359$		−	12.2474i		−	0.646396i	−0.946331	−	0.323198i	$-0.895242\pi$
$359$		−	12.2474i		−	0.646396i	0.946331	−	0.323198i	$-0.104758\pi$
$360$		0			0
$361$	−18.7980			−0.989366
$362$		0			0
$363$	5.00000			0.262432
$364$		0			0
$365$	−9.34847			−0.489321
$366$		0			0
$367$	9.79796			0.511449			0.255725	−	0.966750i	$-0.417686\pi$
$367$	9.79796			0.511449			0.255725	+	0.966750i	$0.417686\pi$
$368$		0			0
$369$		−	8.89898i		−	0.463262i
$370$		0			0
$371$	13.5959	−	5.55051i	0.705865	−	0.288168i
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$		−	1.30306i		−	0.0671111i
$378$		0			0
$379$			20.8990i			1.07351i	0.843739	+	0.536754i	$0.180350\pi$
$379$			20.8990i			1.07351i	−0.843739	+	0.536754i	$0.819650\pi$
$380$		0			0
$381$			17.7980i			0.911817i
$382$		0			0
$383$	−25.7980			−1.31821			−0.659107	−	0.752049i	$-0.729066\pi$
$383$	−25.7980			−1.31821			−0.659107	+	0.752049i	$0.729066\pi$
$384$		0			0
$385$	−6.00000	+	2.44949i	−0.305788	+	0.124838i
$386$		0			0
$387$			8.00000i			0.406663i
$388$		0			0
$389$	17.5959			0.892148			0.446074	−	0.894996i	$-0.352822\pi$
$389$	17.5959			0.892148			0.446074	+	0.894996i	$0.352822\pi$
$390$		0			0
$391$	−5.79796			−0.293215
$392$		0			0
$393$	10.6969			0.539589
$394$		0			0
$395$	9.79796			0.492989
$396$		0			0
$397$			25.8434i			1.29704i	0.761197	+	0.648521i	$0.224612\pi$
$397$			25.8434i			1.29704i	−0.761197	+	0.648521i	$0.775388\pi$
$398$		0			0
$399$	−1.10102	+	0.449490i	−0.0551200	+	0.0225026i
$400$		0			0
$401$	8.20204			0.409590			0.204795	−	0.978805i	$-0.434347\pi$
$401$	8.20204			0.409590			0.204795	+	0.978805i	$0.434347\pi$
$402$		0			0
$403$			1.59592i			0.0794983i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$			7.10102i			0.351985i
$408$		0			0
$409$			1.79796i			0.0889033i	0.999012	+	0.0444517i	$0.0141541\pi$
$409$			1.79796i			0.0889033i	−0.999012	+	0.0444517i	$0.985846\pi$
$410$		0			0
$411$	−13.1464			−0.648465
$412$		0			0
$413$	−12.0000	+	4.89898i	−0.590481	+	0.241063i
$414$		0			0
$415$		−	6.89898i		−	0.338658i
$416$		0			0
$417$	−12.4495			−0.609654
$418$		0			0
$419$	−23.5959			−1.15274			−0.576368	−	0.817190i	$-0.695531\pi$
$419$	−23.5959			−1.15274			−0.576368	+	0.817190i	$0.695531\pi$
$420$		0			0
$421$	9.59592			0.467676			0.233838	−	0.972276i	$-0.424871\pi$
$421$	9.59592			0.467676			0.233838	+	0.972276i	$0.424871\pi$
$422$		0			0
$423$	1.10102			0.0535334
$424$		0			0
$425$			2.89898i			0.140621i
$426$		0			0
$427$	3.10102	+	7.59592i	0.150069	+	0.367592i
$428$		0			0
$429$	−1.10102			−0.0531578
$430$		0			0
$431$		−	6.44949i		−	0.310661i	−0.987863	−	0.155330i	$-0.950356\pi$
$431$		−	6.44949i		−	0.310661i	0.987863	−	0.155330i	$-0.0496442\pi$
$432$		0			0
$433$		−	4.44949i		−	0.213829i	−0.994268	−	0.106914i	$-0.965903\pi$
$433$		−	4.44949i		−	0.213829i	0.994268	−	0.106914i	$-0.0340971\pi$
$434$		0			0
$435$		−	2.89898i		−	0.138995i
$436$		0			0
$437$			0.898979i			0.0430040i
$438$		0			0
$439$	34.2474			1.63454			0.817271	−	0.576254i	$-0.195486\pi$
$439$	34.2474			1.63454			0.817271	+	0.576254i	$0.195486\pi$
$440$		0			0
$441$	5.00000	−	4.89898i	0.238095	−	0.233285i
$442$		0			0
$443$			10.8990i			0.517826i	0.965901	+	0.258913i	$0.0833643\pi$
$443$			10.8990i			0.517826i	−0.965901	+	0.258913i	$0.916636\pi$
$444$		0			0
$445$	−3.79796			−0.180041
$446$		0			0
$447$	15.7980			0.747218
$448$		0			0
$449$	19.7980			0.934323			0.467162	−	0.884172i	$-0.345277\pi$
$449$	19.7980			0.934323			0.467162	+	0.884172i	$0.345277\pi$
$450$		0			0
$451$	−21.7980			−1.02643
$452$		0			0
$453$			0.898979i			0.0422377i
$454$		0			0
$455$	−1.10102	+	0.449490i	−0.0516166	+	0.0210724i
$456$		0			0
$457$	−26.4949			−1.23938			−0.619690	−	0.784847i	$-0.712741\pi$
$457$	−26.4949			−1.23938			−0.619690	+	0.784847i	$0.712741\pi$
$458$		0			0
$459$		−	2.89898i		−	0.135313i
$460$		0			0
$461$			27.7980i			1.29468i	0.762201	+	0.647340i	$0.224119\pi$
$461$			27.7980i			1.29468i	−0.762201	+	0.647340i	$0.775881\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		0			0
$465$			3.55051i			0.164651i
$466$		0			0
$467$	−16.6969			−0.772642			−0.386321	−	0.922364i	$-0.626254\pi$
$467$	−16.6969			−0.772642			−0.386321	+	0.922364i	$0.626254\pi$
$468$		0			0
$469$	3.79796	+	9.30306i	0.175373	+	0.429575i
$470$		0			0
$471$		−	2.65153i		−	0.122176i
$472$		0			0
$473$	19.5959			0.901021
$474$		0			0
$475$	0.449490			0.0206240
$476$		0			0
$477$	5.55051			0.254140
$478$		0			0
$479$	12.8990			0.589369			0.294685	−	0.955595i	$-0.404785\pi$
$479$	12.8990			0.589369			0.294685	+	0.955595i	$0.404785\pi$
$480$		0			0
$481$			1.30306i			0.0594145i
$482$		0			0
$483$	−2.00000	−	4.89898i	−0.0910032	−	0.222911i
$484$		0			0
$485$	−17.3485			−0.787753
$486$		0			0
$487$		−	15.5959i		−	0.706719i	−0.935488	−	0.353359i	$-0.885039\pi$
$487$		−	15.5959i		−	0.706719i	0.935488	−	0.353359i	$-0.114961\pi$
$488$		0			0
$489$			15.5959i			0.705272i
$490$		0			0
$491$			35.3485i			1.59525i	0.603151	+	0.797627i	$0.293912\pi$
$491$			35.3485i			1.59525i	−0.603151	+	0.797627i	$0.706088\pi$
$492$		0			0
$493$		−	8.40408i		−	0.378501i
$494$		0			0
$495$	−2.44949			−0.110096
$496$		0			0
$497$	1.55051	+	3.79796i	0.0695499	+	0.170362i
$498$		0			0
$499$			34.6969i			1.55325i	0.629964	+	0.776624i	$0.283070\pi$
$499$			34.6969i			1.55325i	−0.629964	+	0.776624i	$0.716930\pi$
$500$		0			0
$501$	−8.00000			−0.357414
$502$		0			0
$503$	4.40408			0.196368			0.0981842	−	0.995168i	$-0.468697\pi$
$503$	4.40408			0.196368			0.0981842	+	0.995168i	$0.468697\pi$
$504$		0			0
$505$	−0.898979			−0.0400041
$506$		0			0
$507$	12.7980			0.568377
$508$		0			0
$509$			4.89898i			0.217143i	0.994089	+	0.108572i	$0.0346277\pi$
$509$			4.89898i			0.217143i	−0.994089	+	0.108572i	$0.965372\pi$
$510$		0			0
$511$	−9.34847	−	22.8990i	−0.413552	−	1.01299i
$512$		0			0
$513$	−0.449490			−0.0198455
$514$		0			0
$515$		−	0.898979i		−	0.0396138i
$516$		0			0
$517$		−	2.69694i		−	0.118611i
$518$		0			0
$519$			15.7980i			0.693453i
$520$		0			0
$521$			12.8990i			0.565115i	0.959250	+	0.282557i	$0.0911827\pi$
$521$			12.8990i			0.565115i	−0.959250	+	0.282557i	$0.908817\pi$
$522$		0			0
$523$	−36.8990			−1.61348			−0.806740	−	0.590907i	$-0.798770\pi$
$523$	−36.8990			−1.61348			−0.806740	+	0.590907i	$0.798770\pi$
$524$		0			0
$525$	−2.44949	+	1.00000i	−0.106904	+	0.0436436i
$526$		0			0
$527$			10.2929i			0.448364i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$	−4.89898			−0.212598
$532$		0			0
$533$	−4.00000			−0.173259
$534$		0			0
$535$	−3.79796			−0.164200
$536$		0			0
$537$		−	24.2474i		−	1.04635i
$538$		0			0
$539$	−12.0000	−	12.2474i	−0.516877	−	0.527535i
$540$		0			0
$541$	13.5959			0.584534			0.292267	−	0.956337i	$-0.405590\pi$
$541$	13.5959			0.584534			0.292267	+	0.956337i	$0.405590\pi$
$542$		0			0
$543$			5.79796i			0.248814i
$544$		0			0
$545$		−	5.79796i		−	0.248357i
$546$		0			0
$547$			45.5959i			1.94954i	0.223210	+	0.974770i	$0.428346\pi$
$547$			45.5959i			1.94954i	−0.223210	+	0.974770i	$0.571654\pi$
$548$		0			0
$549$			3.10102i			0.132348i
$550$		0			0
$551$	−1.30306			−0.0555123
$552$		0			0
$553$	9.79796	+	24.0000i	0.416652	+	1.02058i
$554$		0			0
$555$			2.89898i			0.123055i
$556$		0			0
$557$	−5.55051			−0.235183			−0.117591	−	0.993062i	$-0.537517\pi$
$557$	−5.55051			−0.235183			−0.117591	+	0.993062i	$0.537517\pi$
$558$		0			0
$559$	3.59592			0.152091
$560$		0			0
$561$	−7.10102			−0.299805
$562$		0			0
$563$	−4.00000			−0.168580			−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000			−0.168580			−0.0842900	+	0.996441i	$0.526862\pi$
$564$		0			0
$565$		−	11.3485i		−	0.477434i
$566$		0			0
$567$	2.44949	−	1.00000i	0.102869	−	0.0419961i
$568$		0			0
$569$	7.79796			0.326907			0.163454	−	0.986551i	$-0.447737\pi$
$569$	7.79796			0.326907			0.163454	+	0.986551i	$0.447737\pi$
$570$		0			0
$571$		−	28.0000i		−	1.17176i	−0.810397	−	0.585882i	$-0.800748\pi$
$571$		−	28.0000i		−	1.17176i	0.810397	−	0.585882i	$-0.199252\pi$
$572$		0			0
$573$			5.55051i			0.231876i
$574$		0			0
$575$			2.00000i			0.0834058i
$576$		0			0
$577$		−	28.4495i		−	1.18437i	−0.805803	−	0.592184i	$-0.798266\pi$
$577$		−	28.4495i		−	1.18437i	0.805803	−	0.592184i	$-0.201734\pi$
$578$		0			0
$579$	16.6969			0.693901
$580$		0			0
$581$	16.8990	−	6.89898i	0.701088	−	0.286218i
$582$		0			0
$583$		−	13.5959i		−	0.563085i
$584$		0			0
$585$	−0.449490			−0.0185841
$586$		0			0
$587$	−35.1918			−1.45252			−0.726261	−	0.687419i	$-0.758744\pi$
$587$	−35.1918			−1.45252			−0.726261	+	0.687419i	$0.758744\pi$
$588$		0			0
$589$	1.59592			0.0657587
$590$		0			0
$591$	−24.2474			−0.997407
$592$		0			0
$593$		−	39.7980i		−	1.63431i	−0.576421	−	0.817153i	$-0.695551\pi$
$593$		−	39.7980i		−	1.63431i	0.576421	−	0.817153i	$-0.304449\pi$
$594$		0			0
$595$	−7.10102	+	2.89898i	−0.291113	+	0.118847i
$596$		0			0
$597$	−0.449490			−0.0183964
$598$		0			0
$599$			22.0454i			0.900751i	0.892839	+	0.450375i	$0.148710\pi$
$599$			22.0454i			0.900751i	−0.892839	+	0.450375i	$0.851290\pi$
$600$		0			0
$601$		−	11.1010i		−	0.452820i	−0.974032	−	0.226410i	$-0.927301\pi$
$601$		−	11.1010i		−	0.452820i	0.974032	−	0.226410i	$-0.0726989\pi$
$602$		0			0
$603$			3.79796i			0.154665i
$604$		0			0
$605$		−	5.00000i		−	0.203279i
$606$		0			0
$607$	23.5959			0.957729			0.478864	−	0.877889i	$-0.341049\pi$
$607$	23.5959			0.957729			0.478864	+	0.877889i	$0.341049\pi$
$608$		0			0
$609$	7.10102	−	2.89898i	0.287748	−	0.117473i
$610$		0			0
$611$		−	0.494897i		−	0.0200214i
$612$		0			0
$613$	12.2020			0.492836			0.246418	−	0.969164i	$-0.420746\pi$
$613$	12.2020			0.492836			0.246418	+	0.969164i	$0.420746\pi$
$614$		0			0
$615$	−8.89898			−0.358841
$616$		0			0
$617$	−31.8434			−1.28197			−0.640983	−	0.767555i	$-0.721473\pi$
$617$	−31.8434			−1.28197			−0.640983	+	0.767555i	$0.721473\pi$
$618$		0			0
$619$	32.0454			1.28801			0.644007	−	0.765020i	$-0.277271\pi$
$619$	32.0454			1.28801			0.644007	+	0.765020i	$0.277271\pi$
$620$		0			0
$621$		−	2.00000i		−	0.0802572i
$622$		0			0
$623$	−3.79796	−	9.30306i	−0.152162	−	0.372719i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			1.10102i			0.0439705i
$628$		0			0
$629$			8.40408i			0.335093i
$630$		0			0
$631$		−	9.39388i		−	0.373964i	−0.982363	−	0.186982i	$-0.940129\pi$
$631$		−	9.39388i		−	0.373964i	0.982363	−	0.186982i	$-0.0598707\pi$
$632$		0			0
$633$			22.6969i			0.902122i
$634$		0			0
$635$	17.7980			0.706290
$636$		0			0
$637$	−2.20204	−	2.24745i	−0.0872480	−	0.0890472i
$638$		0			0
$639$			1.55051i			0.0613372i
$640$		0			0
$641$	−16.6969			−0.659489			−0.329745	−	0.944070i	$-0.606963\pi$
$641$	−16.6969			−0.659489			−0.329745	+	0.944070i	$0.606963\pi$
$642$		0			0
$643$	19.1010			0.753271			0.376635	−	0.926362i	$-0.377081\pi$
$643$	19.1010			0.753271			0.376635	+	0.926362i	$0.377081\pi$
$644$		0			0
$645$	8.00000			0.315000
$646$		0			0
$647$	−11.5959			−0.455883			−0.227941	−	0.973675i	$-0.573199\pi$
$647$	−11.5959			−0.455883			−0.227941	+	0.973675i	$0.573199\pi$
$648$		0			0
$649$			12.0000i			0.471041i
$650$		0			0
$651$	−8.69694	+	3.55051i	−0.340860	+	0.139155i
$652$		0			0
$653$	−25.1464			−0.984056			−0.492028	−	0.870579i	$-0.663744\pi$
$653$	−25.1464			−0.984056			−0.492028	+	0.870579i	$0.663744\pi$
$654$		0			0
$655$		−	10.6969i		−	0.417964i
$656$		0			0
$657$		−	9.34847i		−	0.364719i
$658$		0			0
$659$			0.247449i			0.00963923i	0.999988	+	0.00481962i	$0.00153414\pi$
$659$			0.247449i			0.00963923i	−0.999988	+	0.00481962i	$0.998466\pi$
$660$		0			0
$661$		−	44.8990i		−	1.74637i	−0.487391	−	0.873184i	$-0.662051\pi$
$661$		−	44.8990i		−	1.74637i	0.487391	−	0.873184i	$-0.337949\pi$
$662$		0			0
$663$	−1.30306			−0.0506067
$664$		0			0
$665$	0.449490	+	1.10102i	0.0174305	+	0.0426957i
$666$		0			0
$667$		−	5.79796i		−	0.224498i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.59592			0.293237
$672$		0			0
$673$	32.6969			1.26037			0.630187	−	0.776443i	$-0.282978\pi$
$673$	32.6969			1.26037			0.630187	+	0.776443i	$0.282978\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$		−	20.6969i		−	0.795448i	−0.917505	−	0.397724i	$-0.869800\pi$
$677$		−	20.6969i		−	0.795448i	0.917505	−	0.397724i	$-0.130200\pi$
$678$		0			0
$679$	−17.3485	−	42.4949i	−0.665773	−	1.63080i
$680$		0			0
$681$	−8.69694			−0.333267
$682$		0			0
$683$		−	38.8990i		−	1.48843i	−0.667941	−	0.744214i	$-0.732824\pi$
$683$		−	38.8990i		−	1.48843i	0.667941	−	0.744214i	$-0.267176\pi$
$684$		0			0
$685$			13.1464i			0.502299i
$686$		0			0
$687$			16.8990i			0.644736i
$688$		0			0
$689$		−	2.49490i		−	0.0950480i
$690$		0			0
$691$	−28.4495			−1.08227			−0.541135	−	0.840936i	$-0.682005\pi$
$691$	−28.4495			−1.08227			−0.541135	+	0.840936i	$0.682005\pi$
$692$		0			0
$693$	−2.44949	−	6.00000i	−0.0930484	−	0.227921i
$694$		0			0
$695$			12.4495i			0.472236i
$696$		0			0
$697$	−25.7980			−0.977167
$698$		0			0
$699$	12.2474			0.463241
$700$		0			0
$701$	16.6969			0.630635			0.315317	−	0.948986i	$-0.397889\pi$
$701$	16.6969			0.630635			0.315317	+	0.948986i	$0.397889\pi$
$702$		0			0
$703$	1.30306			0.0491459
$704$		0			0
$705$		−	1.10102i		−	0.0414668i
$706$		0			0
$707$	−0.898979	−	2.20204i	−0.0338096	−	0.0828163i
$708$		0			0
$709$	16.2020			0.608480			0.304240	−	0.952595i	$-0.401597\pi$
$709$	16.2020			0.608480			0.304240	+	0.952595i	$0.401597\pi$
$710$		0			0
$711$			9.79796i			0.367452i
$712$		0			0
$713$			7.10102i			0.265935i
$714$		0			0
$715$			1.10102i			0.0411758i
$716$		0			0
$717$			21.5505i			0.804819i
$718$		0			0
$719$	40.8990			1.52527			0.762637	−	0.646826i	$-0.223904\pi$
$719$	40.8990			1.52527			0.762637	+	0.646826i	$0.223904\pi$
$720$		0			0
$721$	2.20204	−	0.898979i	0.0820083	−	0.0334797i
$722$		0			0
$723$		−	8.89898i		−	0.330957i
$724$		0			0
$725$	−2.89898			−0.107665
$726$		0			0
$727$	25.7980			0.956793			0.478397	−	0.878144i	$-0.341218\pi$
$727$	25.7980			0.956793			0.478397	+	0.878144i	$0.341218\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	23.1918			0.857781
$732$		0			0
$733$		−	17.3485i		−	0.640780i	−0.947286	−	0.320390i	$-0.896186\pi$
$733$		−	17.3485i		−	0.640780i	0.947286	−	0.320390i	$-0.103814\pi$
$734$		0			0
$735$	−4.89898	−	5.00000i	−0.180702	−	0.184428i
$736$		0			0
$737$	9.30306			0.342683
$738$		0			0
$739$		−	8.40408i		−	0.309149i	−0.987981	−	0.154575i	$-0.950599\pi$
$739$		−	8.40408i		−	0.309149i	0.987981	−	0.154575i	$-0.0494007\pi$
$740$		0			0
$741$			0.202041i			0.00742216i
$742$		0			0
$743$			48.2929i			1.77169i	0.463978	+	0.885847i	$0.346422\pi$
$743$			48.2929i			1.77169i	−0.463978	+	0.885847i	$0.653578\pi$
$744$		0			0
$745$		−	15.7980i		−	0.578792i
$746$		0			0
$747$	6.89898			0.252420
$748$		0			0
$749$	−3.79796	−	9.30306i	−0.138774	−	0.339926i
$750$		0			0
$751$		−	39.1010i		−	1.42682i	−0.700749	−	0.713408i	$-0.747151\pi$
$751$		−	39.1010i		−	1.42682i	0.700749	−	0.713408i	$-0.252849\pi$
$752$		0			0
$753$	30.6969			1.11866
$754$		0			0
$755$	0.898979			0.0327172
$756$		0			0
$757$	19.3031			0.701582			0.350791	−	0.936454i	$-0.385913\pi$
$757$	19.3031			0.701582			0.350791	+	0.936454i	$0.385913\pi$
$758$		0			0
$759$	−4.89898			−0.177822
$760$		0			0
$761$		−	8.20204i		−	0.297324i	−0.988888	−	0.148662i	$-0.952503\pi$
$761$		−	8.20204i		−	0.297324i	0.988888	−	0.148662i	$-0.0474966\pi$
$762$		0			0
$763$	14.2020	−	5.79796i	0.514148	−	0.209900i
$764$		0			0
$765$	−2.89898			−0.104813
$766$		0			0
$767$			2.20204i			0.0795111i
$768$		0			0
$769$		−	40.4949i		−	1.46028i	−0.683296	−	0.730142i	$-0.739454\pi$
$769$		−	40.4949i		−	1.46028i	0.683296	−	0.730142i	$-0.260546\pi$
$770$		0			0
$771$		−	10.8990i		−	0.392517i
$772$		0			0
$773$		−	15.3939i		−	0.553679i	−0.960916	−	0.276840i	$-0.910713\pi$
$773$		−	15.3939i		−	0.553679i	0.960916	−	0.276840i	$-0.0892871\pi$
$774$		0			0
$775$	3.55051			0.127538
$776$		0			0
$777$	−7.10102	+	2.89898i	−0.254748	+	0.104000i
$778$		0			0
$779$			4.00000i			0.143315i
$780$		0			0
$781$	3.79796			0.135902
$782$		0			0
$783$	2.89898			0.103601
$784$		0			0
$785$	−2.65153			−0.0946372
$786$		0			0
$787$	10.6969			0.381305			0.190652	−	0.981658i	$-0.438940\pi$
$787$	10.6969			0.381305			0.190652	+	0.981658i	$0.438940\pi$
$788$		0			0
$789$		−	10.8990i		−	0.388014i
$790$		0			0
$791$	27.7980	−	11.3485i	0.988382	−	0.403505i
$792$		0			0
$793$	1.39388			0.0494980
$794$		0			0
$795$		−	5.55051i		−	0.196856i
$796$		0			0
$797$		−	3.79796i		−	0.134531i	−0.997735	−	0.0672653i	$-0.978573\pi$
$797$		−	3.79796i		−	0.134531i	0.997735	−	0.0672653i	$-0.0214274\pi$
$798$		0			0
$799$		−	3.19184i		−	0.112919i
$800$		0			0
$801$		−	3.79796i		−	0.134194i
$802$		0			0
$803$	−22.8990			−0.808087
$804$		0			0
$805$	−4.89898	+	2.00000i	−0.172666	+	0.0704907i
$806$		0			0
$807$			28.4949i			1.00307i
$808$		0			0
$809$	−25.1010			−0.882505			−0.441252	−	0.897383i	$-0.645466\pi$
$809$	−25.1010			−0.882505			−0.441252	+	0.897383i	$0.645466\pi$
$810$		0			0
$811$	0.853572			0.0299730			0.0149865	−	0.999888i	$-0.495229\pi$
$811$	0.853572			0.0299730			0.0149865	+	0.999888i	$0.495229\pi$
$812$		0			0
$813$	−15.1464			−0.531208
$814$		0			0
$815$	15.5959			0.546301
$816$		0			0
$817$		−	3.59592i		−	0.125805i
$818$		0			0
$819$	−0.449490	−	1.10102i	−0.0157064	−	0.0384728i
$820$		0			0
$821$	42.4949			1.48308			0.741541	−	0.670907i	$-0.234095\pi$
$821$	42.4949			1.48308			0.741541	+	0.670907i	$0.234095\pi$
$822$		0			0
$823$		−	13.3939i		−	0.466881i	−0.972371	−	0.233441i	$-0.925002\pi$
$823$		−	13.3939i		−	0.466881i	0.972371	−	0.233441i	$-0.0749984\pi$
$824$		0			0
$825$			2.44949i			0.0852803i
$826$		0			0
$827$		−	15.3939i		−	0.535298i	−0.963517	−	0.267649i	$-0.913753\pi$
$827$		−	15.3939i		−	0.535298i	0.963517	−	0.267649i	$-0.0862467\pi$
$828$		0			0
$829$		−	53.3939i		−	1.85445i	−0.374510	−	0.927223i	$-0.622189\pi$
$829$		−	53.3939i		−	1.85445i	0.374510	−	0.927223i	$-0.377811\pi$
$830$		0			0
$831$	−25.5959			−0.887913
$832$		0			0
$833$	−14.2020	−	14.4949i	−0.492072	−	0.502218i
$834$		0			0
$835$			8.00000i			0.276851i
$836$		0			0
$837$	−3.55051			−0.122724
$838$		0			0
$839$	−34.2020			−1.18079			−0.590393	−	0.807116i	$-0.701027\pi$
$839$	−34.2020			−1.18079			−0.590393	+	0.807116i	$0.701027\pi$
$840$		0			0
$841$	−20.5959			−0.710204
$842$		0			0
$843$	−10.8990			−0.375381
$844$		0			0
$845$		−	12.7980i		−	0.440263i
$846$		0			0
$847$	12.2474	−	5.00000i	0.420827	−	0.171802i
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$			5.79796i			0.198751i
$852$		0			0
$853$		−	8.04541i		−	0.275470i	−0.990469	−	0.137735i	$-0.956018\pi$
$853$		−	8.04541i		−	0.275470i	0.990469	−	0.137735i	$-0.0439822\pi$
$854$		0			0
$855$			0.449490i			0.0153722i
$856$		0			0
$857$			6.00000i			0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$			6.00000i			0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$		0			0
$859$	40.4495			1.38012			0.690059	−	0.723753i	$-0.257584\pi$
$859$	40.4495			1.38012			0.690059	+	0.723753i	$0.257584\pi$
$860$		0			0
$861$	−8.89898	−	21.7980i	−0.303276	−	0.742872i
$862$		0			0
$863$		−	27.3939i		−	0.932498i	−0.884653	−	0.466249i	$-0.845605\pi$
$863$		−	27.3939i		−	0.932498i	0.884653	−	0.466249i	$-0.154395\pi$
$864$		0			0
$865$	15.7980			0.537147
$866$		0			0
$867$	8.59592			0.291933
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$	1.70714			0.0578444
$872$		0			0
$873$		−	17.3485i		−	0.587157i
$874$		0			0
$875$	1.00000	+	2.44949i	0.0338062	+	0.0828079i
$876$		0			0
$877$	25.5959			0.864313			0.432156	−	0.901799i	$-0.357753\pi$
$877$	25.5959			0.864313			0.432156	+	0.901799i	$0.357753\pi$
$878$		0			0
$879$			4.69694i			0.158424i
$880$		0			0
$881$		−	27.7980i		−	0.936537i	−0.883586	−	0.468269i	$-0.844878\pi$
$881$		−	27.7980i		−	0.936537i	0.883586	−	0.468269i	$-0.155122\pi$
$882$		0			0
$883$		−	19.7980i		−	0.666254i	−0.942882	−	0.333127i	$-0.891896\pi$
$883$		−	19.7980i		−	0.666254i	0.942882	−	0.333127i	$-0.108104\pi$
$884$		0			0
$885$			4.89898i			0.164677i
$886$		0			0
$887$	−56.2929			−1.89013			−0.945065	−	0.326884i	$-0.894001\pi$
$887$	−56.2929			−1.89013			−0.945065	+	0.326884i	$0.894001\pi$
$888$		0			0
$889$	17.7980	+	43.5959i	0.596924	+	1.46216i
$890$		0			0
$891$		−	2.44949i		−	0.0820610i
$892$		0			0
$893$	−0.494897			−0.0165611
$894$		0			0
$895$	−24.2474			−0.810503
$896$		0			0
$897$	−0.898979			−0.0300161
$898$		0			0
$899$	−10.2929			−0.343286
$900$		0			0
$901$		−	16.0908i		−	0.536063i
$902$		0			0
$903$	8.00000	+	19.5959i	0.266223	+	0.652111i
$904$		0			0
$905$	5.79796			0.192731
$906$		0			0
$907$		−	37.5959i		−	1.24835i	−0.781284	−	0.624176i	$-0.785435\pi$
$907$		−	37.5959i		−	1.24835i	0.781284	−	0.624176i	$-0.214565\pi$
$908$		0			0
$909$		−	0.898979i		−	0.0298173i
$910$		0			0
$911$			47.3485i			1.56872i	0.620303	+	0.784362i	$0.287010\pi$
$911$			47.3485i			1.56872i	−0.620303	+	0.784362i	$0.712990\pi$
$912$		0			0
$913$		−	16.8990i		−	0.559275i
$914$		0			0
$915$	3.10102			0.102517
$916$		0			0
$917$	26.2020	−	10.6969i	0.865268	−	0.353244i
$918$		0			0
$919$		−	20.4949i		−	0.676064i	−0.941135	−	0.338032i	$-0.890239\pi$
$919$		−	20.4949i		−	0.676064i	0.941135	−	0.338032i	$-0.109761\pi$
$920$		0			0
$921$	0.898979			0.0296224
$922$		0			0
$923$	0.696938			0.0229400
$924$		0			0
$925$	2.89898			0.0953179
$926$		0			0
$927$	0.898979			0.0295264
$928$		0			0
$929$		−	30.0000i		−	0.984268i	−0.870519	−	0.492134i	$-0.836217\pi$
$929$		−	30.0000i		−	0.984268i	0.870519	−	0.492134i	$-0.163783\pi$
$930$		0			0
$931$	−2.24745	+	2.20204i	−0.0736572	+	0.0721690i
$932$		0			0
$933$	12.8990			0.422294
$934$		0			0
$935$			7.10102i			0.232228i
$936$		0			0
$937$			6.24745i			0.204095i	0.994780	+	0.102048i	$0.0325394\pi$
$937$			6.24745i			0.204095i	−0.994780	+	0.102048i	$0.967461\pi$
$938$		0			0
$939$			29.3485i			0.957751i
$940$		0			0
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$		0			0
$943$	−17.7980			−0.579581
$944$		0			0
$945$	−1.00000	−	2.44949i	−0.0325300	−	0.0796819i
$946$		0			0
$947$			24.6969i			0.802543i	0.915959	+	0.401271i	$0.131432\pi$
$947$			24.6969i			0.802543i	−0.915959	+	0.401271i	$0.868568\pi$
$948$		0			0
$949$	−4.20204			−0.136404
$950$		0			0
$951$	−17.1464			−0.556011
$952$		0			0
$953$	24.2474			0.785452			0.392726	−	0.919656i	$-0.371532\pi$
$953$	24.2474			0.785452			0.392726	+	0.919656i	$0.371532\pi$
$954$		0			0
$955$	5.55051			0.179610
$956$		0			0
$957$		−	7.10102i		−	0.229543i
$958$		0			0
$959$	−32.2020	+	13.1464i	−1.03986	+	0.424520i
$960$		0			0
$961$	−18.3939			−0.593351
$962$		0			0
$963$		−	3.79796i		−	0.122388i
$964$		0			0
$965$		−	16.6969i		−	0.537493i
$966$		0			0
$967$		−	27.3939i		−	0.880928i	−0.897770	−	0.440464i	$-0.854814\pi$
$967$		−	27.3939i		−	0.880928i	0.897770	−	0.440464i	$-0.145186\pi$
$968$		0			0
$969$			1.30306i			0.0418604i
$970$		0			0
$971$	39.1918			1.25773			0.628863	−	0.777516i	$-0.283521\pi$
$971$	39.1918			1.25773			0.628863	+	0.777516i	$0.283521\pi$
$972$		0			0
$973$	−30.4949	+	12.4495i	−0.977622	+	0.399112i
$974$		0			0
$975$			0.449490i			0.0143952i
$976$		0			0
$977$	31.3485			1.00293			0.501463	−	0.865179i	$-0.332795\pi$
$977$	31.3485			1.00293			0.501463	+	0.865179i	$0.332795\pi$
$978$		0			0
$979$	−9.30306			−0.297327
$980$		0			0
$981$	5.79796			0.185115
$982$		0			0
$983$	−21.1010			−0.673018			−0.336509	−	0.941680i	$-0.609246\pi$
$983$	−21.1010			−0.673018			−0.336509	+	0.941680i	$0.609246\pi$
$984$		0			0
$985$			24.2474i			0.772588i
$986$		0			0
$987$	2.69694	−	1.10102i	0.0858445	−	0.0350459i
$988$		0			0
$989$	16.0000			0.508770
$990$		0			0
$991$		−	6.69694i		−	0.212735i	−0.994327	−	0.106368i	$-0.966078\pi$
$991$		−	6.69694i		−	0.212735i	0.994327	−	0.106368i	$-0.0339220\pi$
$992$		0			0
$993$			16.8990i			0.536273i
$994$		0			0
$995$			0.449490i			0.0142498i
$996$		0			0
$997$		−	56.0454i		−	1.77498i	−0.460831	−	0.887488i	$-0.652449\pi$
$997$		−	56.0454i		−	1.77498i	0.460831	−	0.887488i	$-0.347551\pi$
$998$		0			0
$999$	−2.89898			−0.0917197

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.d.b.1231.2	yes	4
3.2	odd	2		5040.2.d.b.4591.4		4
4.3	odd	2		1680.2.d.a.1231.1	✓	4
7.6	odd	2		1680.2.d.a.1231.3	yes	4
12.11	even	2		5040.2.d.a.4591.3		4
21.20	even	2		5040.2.d.a.4591.1		4
28.27	even	2	inner	1680.2.d.b.1231.4	yes	4
84.83	odd	2		5040.2.d.b.4591.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.d.a.1231.1	✓	4	4.3	odd	2
1680.2.d.a.1231.3	yes	4	7.6	odd	2
1680.2.d.b.1231.2	yes	4	1.1	even	1	trivial
1680.2.d.b.1231.4	yes	4	28.27	even	2	inner
5040.2.d.a.4591.1		4	21.20	even	2
5040.2.d.a.4591.3		4	12.11	even	2
5040.2.d.b.4591.2		4	84.83	odd	2
5040.2.d.b.4591.4		4	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000i q^{5} +(2.44949 - 1.00000i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000i q^{5} +(2.44949 - 1.00000i) q^{7} +1.00000 q^{9} -2.44949i q^{11} -0.449490i q^{13} -1.00000i q^{15} -2.89898i q^{17} -0.449490 q^{19} +(2.44949 - 1.00000i) q^{21} -2.00000i q^{23} -1.00000 q^{25} +1.00000 q^{27} +2.89898 q^{29} -3.55051 q^{31} -2.44949i q^{33} +(-1.00000 - 2.44949i) q^{35} -2.89898 q^{37} -0.449490i q^{39} -8.89898i q^{41} +8.00000i q^{43} -1.00000i q^{45} +1.10102 q^{47} +(5.00000 - 4.89898i) q^{49} -2.89898i q^{51} +5.55051 q^{53} -2.44949 q^{55} -0.449490 q^{57} -4.89898 q^{59} +3.10102i q^{61} +(2.44949 - 1.00000i) q^{63} -0.449490 q^{65} +3.79796i q^{67} -2.00000i q^{69} +1.55051i q^{71} -9.34847i q^{73} -1.00000 q^{75} +(-2.44949 - 6.00000i) q^{77} +9.79796i q^{79} +1.00000 q^{81} +6.89898 q^{83} -2.89898 q^{85} +2.89898 q^{87} -3.79796i q^{89} +(-0.449490 - 1.10102i) q^{91} -3.55051 q^{93} +0.449490i q^{95} -17.3485i q^{97} -2.44949i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} + 8 q^{19} - 4 q^{25} + 4 q^{27} - 8 q^{29} - 24 q^{31} - 4 q^{35} + 8 q^{37} + 24 q^{47} + 20 q^{49} + 32 q^{53} + 8 q^{57} + 8 q^{65} - 4 q^{75} + 4 q^{81} + 8 q^{83} + 8 q^{85} - 8 q^{87} + 8 q^{91} - 24 q^{93}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 8 * q^19 - 4 * q^25 + 4 * q^27 - 8 * q^29 - 24 * q^31 - 4 * q^35 + 8 * q^37 + 24 * q^47 + 20 * q^49 + 32 * q^53 + 8 * q^57 + 8 * q^65 - 4 * q^75 + 4 * q^81 + 8 * q^83 + 8 * q^85 - 8 * q^87 + 8 * q^91 - 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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