LMFDB - Embedded newform 1680.2.d.b.1231.1

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			1231.1
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1680.1231
Dual form			1680.2.d.b.1231.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000i q^{5} +(-2.44949 - 1.00000i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000i q^{5} +(-2.44949 - 1.00000i) q^{7} +1.00000 q^{9} +2.44949i q^{11} +4.44949i q^{13} -1.00000i q^{15} +6.89898i q^{17} +4.44949 q^{19} +(-2.44949 - 1.00000i) q^{21} -2.00000i q^{23} -1.00000 q^{25} +1.00000 q^{27} -6.89898 q^{29} -8.44949 q^{31} +2.44949i q^{33} +(-1.00000 + 2.44949i) q^{35} +6.89898 q^{37} +4.44949i q^{39} +0.898979i q^{41} +8.00000i q^{43} -1.00000i q^{45} +10.8990 q^{47} +(5.00000 + 4.89898i) q^{49} +6.89898i q^{51} +10.4495 q^{53} +2.44949 q^{55} +4.44949 q^{57} +4.89898 q^{59} +12.8990i q^{61} +(-2.44949 - 1.00000i) q^{63} +4.44949 q^{65} -15.7980i q^{67} -2.00000i q^{69} +6.44949i q^{71} +5.34847i q^{73} -1.00000 q^{75} +(2.44949 - 6.00000i) q^{77} -9.79796i q^{79} +1.00000 q^{81} -2.89898 q^{83} +6.89898 q^{85} -6.89898 q^{87} +15.7980i q^{89} +(4.44949 - 10.8990i) q^{91} -8.44949 q^{93} -4.44949i q^{95} -2.65153i q^{97} +2.44949i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.120394\pi$
$11$			2.44949i			0.738549i	−0.929320	+	0.369274i	$0.879606\pi$
$12$		0			0
$13$			4.44949i			1.23407i	0.786937	+	0.617033i	$0.211666\pi$
$13$			4.44949i			1.23407i	−0.786937	+	0.617033i	$0.788334\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$			6.89898i			1.67325i	0.547777	+	0.836624i	$0.315474\pi$
$17$			6.89898i			1.67325i	−0.547777	+	0.836624i	$0.684526\pi$
$18$		0			0
$19$	4.44949			1.02078			0.510391	−	0.859942i	$-0.329501\pi$
$19$	4.44949			1.02078			0.510391	+	0.859942i	$0.329501\pi$
$20$		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
$29$	−6.89898			−1.28111			−0.640554	−	0.767913i	$-0.721295\pi$
$29$	−6.89898			−1.28111			−0.640554	+	0.767913i	$0.721295\pi$
$30$		0			0
$31$	−8.44949			−1.51757			−0.758787	−	0.651339i	$-0.774207\pi$
$31$	−8.44949			−1.51757			−0.758787	+	0.651339i	$0.774207\pi$
$32$		0			0
$33$			2.44949i			0.426401i
$34$		0			0
$35$	−1.00000	+	2.44949i	−0.169031	+	0.414039i
$36$		0			0
$37$	6.89898			1.13419			0.567093	−	0.823654i	$-0.308068\pi$
$37$	6.89898			1.13419			0.567093	+	0.823654i	$0.308068\pi$
$38$		0			0
$39$			4.44949i			0.712489i
$40$		0			0
$41$			0.898979i			0.140397i	0.997533	+	0.0701985i	$0.0223633\pi$
$41$			0.898979i			0.140397i	−0.997533	+	0.0701985i	$0.977637\pi$
$42$		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
$47$	10.8990			1.58978			0.794890	−	0.606754i	$-0.207529\pi$
$47$	10.8990			1.58978			0.794890	+	0.606754i	$0.207529\pi$
$48$		0			0
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$		0			0
$51$			6.89898i			0.966050i
$52$		0			0
$53$	10.4495			1.43535			0.717674	−	0.696379i	$-0.245207\pi$
$53$	10.4495			1.43535			0.717674	+	0.696379i	$0.245207\pi$
$54$		0			0
$55$	2.44949			0.330289
$56$		0			0
$57$	4.44949			0.589349
$58$		0			0
$59$	4.89898			0.637793			0.318896	−	0.947790i	$-0.396688\pi$
$59$	4.89898			0.637793			0.318896	+	0.947790i	$0.396688\pi$
$60$		0			0
$61$			12.8990i			1.65155i	0.564003	+	0.825773i	$0.309261\pi$
$61$			12.8990i			1.65155i	−0.564003	+	0.825773i	$0.690739\pi$
$62$		0			0
$63$	−2.44949	−	1.00000i	−0.308607	−	0.125988i
$64$		0			0
$65$	4.44949			0.551891
$66$		0			0
$67$		−	15.7980i		−	1.93003i	−0.262199	−	0.965014i	$-0.584448\pi$
$67$		−	15.7980i		−	1.93003i	0.262199	−	0.965014i	$-0.415552\pi$
$68$		0			0
$69$		−	2.00000i		−	0.240772i
$70$		0			0
$71$			6.44949i			0.765414i	0.923870	+	0.382707i	$0.125008\pi$
$71$			6.44949i			0.765414i	−0.923870	+	0.382707i	$0.874992\pi$
$72$		0			0
$73$			5.34847i			0.625991i	0.949755	+	0.312995i	$0.101332\pi$
$73$			5.34847i			0.625991i	−0.949755	+	0.312995i	$0.898668\pi$
$74$		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	$-0.814178\pi$
$79$		−	9.79796i		−	1.10236i	0.834388	−	0.551178i	$-0.185822\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−2.89898			−0.318204			−0.159102	−	0.987262i	$-0.550860\pi$
$83$	−2.89898			−0.318204			−0.159102	+	0.987262i	$0.550860\pi$
$84$		0			0
$85$	6.89898			0.748299
$86$		0			0
$87$	−6.89898			−0.739648
$88$		0			0
$89$			15.7980i			1.67458i	0.546759	+	0.837290i	$0.315861\pi$
$89$			15.7980i			1.67458i	−0.546759	+	0.837290i	$0.684139\pi$
$90$		0			0
$91$	4.44949	−	10.8990i	0.466433	−	1.14252i
$92$		0			0
$93$	−8.44949			−0.876171
$94$		0			0
$95$		−	4.44949i		−	0.456508i
$96$		0			0
$97$		−	2.65153i		−	0.269222i	−0.990899	−	0.134611i	$-0.957021\pi$
$97$		−	2.65153i		−	0.269222i	0.990899	−	0.134611i	$-0.0429785\pi$
$98$		0			0
$99$			2.44949i			0.246183i
$100$		0			0
$101$			8.89898i			0.885482i	0.896650	+	0.442741i	$0.145994\pi$
$101$			8.89898i			0.885482i	−0.896650	+	0.442741i	$0.854006\pi$
$102$		0			0
$103$	−8.89898			−0.876843			−0.438421	−	0.898770i	$-0.644462\pi$
$103$	−8.89898			−0.876843			−0.438421	+	0.898770i	$0.644462\pi$
$104$		0			0
$105$	−1.00000	+	2.44949i	−0.0975900	+	0.239046i
$106$		0			0
$107$			15.7980i			1.52725i	0.645662	+	0.763623i	$0.276581\pi$
$107$			15.7980i			1.52725i	−0.645662	+	0.763623i	$0.723419\pi$
$108$		0			0
$109$	−13.7980			−1.32160			−0.660802	−	0.750560i	$-0.729784\pi$
$109$	−13.7980			−1.32160			−0.660802	+	0.750560i	$0.729784\pi$
$110$		0			0
$111$	6.89898			0.654822
$112$		0			0
$113$	−3.34847			−0.314997			−0.157499	−	0.987519i	$-0.550343\pi$
$113$	−3.34847			−0.314997			−0.157499	+	0.987519i	$0.550343\pi$
$114$		0			0
$115$	−2.00000			−0.186501
$116$		0			0
$117$			4.44949i			0.411355i
$118$		0			0
$119$	6.89898	−	16.8990i	0.632428	−	1.54913i
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$			0.898979i			0.0810583i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	1.79796i		−	0.159543i	−0.996813	−	0.0797715i	$-0.974581\pi$
$127$		−	1.79796i		−	0.159543i	0.996813	−	0.0797715i	$-0.0254191\pi$
$128$		0			0
$129$			8.00000i			0.704361i
$130$		0			0
$131$	−18.6969			−1.63356			−0.816780	−	0.576950i	$-0.804243\pi$
$131$	−18.6969			−1.63356			−0.816780	+	0.576950i	$0.804243\pi$
$132$		0			0
$133$	−10.8990	−	4.44949i	−0.945061	−	0.385820i
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$	21.1464			1.80666			0.903331	−	0.428945i	$-0.141115\pi$
$137$	21.1464			1.80666			0.903331	+	0.428945i	$0.141115\pi$
$138$		0			0
$139$	−7.55051			−0.640426			−0.320213	−	0.947346i	$-0.603754\pi$
$139$	−7.55051			−0.640426			−0.320213	+	0.947346i	$0.603754\pi$
$140$		0			0
$141$	10.8990			0.917860
$142$		0			0
$143$	−10.8990			−0.911418
$144$		0			0
$145$			6.89898i			0.572929i
$146$		0			0
$147$	5.00000	+	4.89898i	0.412393	+	0.404061i
$148$		0			0
$149$	−3.79796			−0.311141			−0.155570	−	0.987825i	$-0.549722\pi$
$149$	−3.79796			−0.311141			−0.155570	+	0.987825i	$0.549722\pi$
$150$		0			0
$151$		−	8.89898i		−	0.724189i	−0.932141	−	0.362094i	$-0.882062\pi$
$151$		−	8.89898i		−	0.724189i	0.932141	−	0.362094i	$-0.117938\pi$
$152$		0			0
$153$			6.89898i			0.557749i
$154$		0			0
$155$			8.44949i			0.678679i
$156$		0			0
$157$		−	17.3485i		−	1.38456i	−0.721630	−	0.692279i	$-0.756607\pi$
$157$		−	17.3485i		−	1.38456i	0.721630	−	0.692279i	$-0.243393\pi$
$158$		0			0
$159$	10.4495			0.828698
$160$		0			0
$161$	−2.00000	+	4.89898i	−0.157622	+	0.386094i
$162$		0			0
$163$		−	23.5959i		−	1.84817i	−0.382181	−	0.924087i	$-0.624827\pi$
$163$		−	23.5959i		−	1.84817i	0.382181	−	0.924087i	$-0.375173\pi$
$164$		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
$169$	−6.79796			−0.522920
$170$		0			0
$171$	4.44949			0.340261
$172$		0			0
$173$		−	3.79796i		−	0.288753i	−0.989523	−	0.144377i	$-0.953882\pi$
$173$		−	3.79796i		−	0.288753i	0.989523	−	0.144377i	$-0.0461177\pi$
$174$		0			0
$175$	2.44949	+	1.00000i	0.185164	+	0.0755929i
$176$		0			0
$177$	4.89898			0.368230
$178$		0			0
$179$			0.247449i			0.0184952i	0.999957	+	0.00924759i	$0.00294364\pi$
$179$			0.247449i			0.0184952i	−0.999957	+	0.00924759i	$0.997056\pi$
$180$		0			0
$181$		−	13.7980i		−	1.02559i	−0.858510	−	0.512797i	$-0.828609\pi$
$181$		−	13.7980i		−	1.02559i	0.858510	−	0.512797i	$-0.171391\pi$
$182$		0			0
$183$			12.8990i			0.953520i
$184$		0			0
$185$		−	6.89898i		−	0.507223i
$186$		0			0
$187$	−16.8990			−1.23578
$188$		0			0
$189$	−2.44949	−	1.00000i	−0.178174	−	0.0727393i
$190$		0			0
$191$			10.4495i			0.756099i	0.925785	+	0.378049i	$0.123405\pi$
$191$			10.4495i			0.756099i	−0.925785	+	0.378049i	$0.876595\pi$
$192$		0			0
$193$	−12.6969			−0.913946			−0.456973	−	0.889481i	$-0.651066\pi$
$193$	−12.6969			−0.913946			−0.456973	+	0.889481i	$0.651066\pi$
$194$		0			0
$195$	4.44949			0.318635
$196$		0			0
$197$	0.247449			0.0176300			0.00881500	−	0.999961i	$-0.497194\pi$
$197$	0.247449			0.0176300			0.00881500	+	0.999961i	$0.497194\pi$
$198$		0			0
$199$	4.44949			0.315416			0.157708	−	0.987486i	$-0.449590\pi$
$199$	4.44949			0.315416			0.157708	+	0.987486i	$0.449590\pi$
$200$		0			0
$201$		−	15.7980i		−	1.11430i
$202$		0			0
$203$	16.8990	+	6.89898i	1.18608	+	0.484213i
$204$		0			0
$205$	0.898979			0.0627875
$206$		0			0
$207$		−	2.00000i		−	0.139010i
$208$		0			0
$209$			10.8990i			0.753898i
$210$		0			0
$211$		−	6.69694i		−	0.461036i	−0.973068	−	0.230518i	$-0.925958\pi$
$211$		−	6.69694i		−	0.461036i	0.973068	−	0.230518i	$-0.0740421\pi$
$212$		0			0
$213$			6.44949i			0.441912i
$214$		0			0
$215$	8.00000			0.545595
$216$		0			0
$217$	20.6969	+	8.44949i	1.40500	+	0.573589i
$218$		0			0
$219$			5.34847i			0.361416i
$220$		0			0
$221$	−30.6969			−2.06490
$222$		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
$227$	20.6969			1.37370			0.686852	−	0.726797i	$-0.258992\pi$
$227$	20.6969			1.37370			0.686852	+	0.726797i	$0.258992\pi$
$228$		0			0
$229$			7.10102i			0.469249i	0.972086	+	0.234624i	$0.0753860\pi$
$229$			7.10102i			0.469249i	−0.972086	+	0.234624i	$0.924614\pi$
$230$		0			0
$231$	2.44949	−	6.00000i	0.161165	−	0.394771i
$232$		0			0
$233$	−12.2474			−0.802357			−0.401179	−	0.916000i	$-0.631399\pi$
$233$	−12.2474			−0.802357			−0.401179	+	0.916000i	$0.631399\pi$
$234$		0			0
$235$		−	10.8990i		−	0.710971i
$236$		0			0
$237$		−	9.79796i		−	0.636446i
$238$		0			0
$239$			26.4495i			1.71088i	0.517906	+	0.855438i	$0.326712\pi$
$239$			26.4495i			1.71088i	−0.517906	+	0.855438i	$0.673288\pi$
$240$		0			0
$241$			0.898979i			0.0579084i	0.999581	+	0.0289542i	$0.00921769\pi$
$241$			0.898979i			0.0579084i	−0.999581	+	0.0289542i	$0.990782\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	4.89898	−	5.00000i	0.312984	−	0.319438i
$246$		0			0
$247$			19.7980i			1.25971i
$248$		0			0
$249$	−2.89898			−0.183715
$250$		0			0
$251$	1.30306			0.0822485			0.0411243	−	0.999154i	$-0.486906\pi$
$251$	1.30306			0.0822485			0.0411243	+	0.999154i	$0.486906\pi$
$252$		0			0
$253$	4.89898			0.307996
$254$		0			0
$255$	6.89898			0.432031
$256$		0			0
$257$		−	1.10102i		−	0.0686798i	−0.999410	−	0.0343399i	$-0.989067\pi$
$257$		−	1.10102i		−	0.0686798i	0.999410	−	0.0343399i	$-0.0109329\pi$
$258$		0			0
$259$	−16.8990	−	6.89898i	−1.05005	−	0.428682i
$260$		0			0
$261$	−6.89898			−0.427036
$262$		0			0
$263$		−	1.10102i		−	0.0678918i	−0.999424	−	0.0339459i	$-0.989193\pi$
$263$		−	1.10102i		−	0.0678918i	0.999424	−	0.0339459i	$-0.0108074\pi$
$264$		0			0
$265$		−	10.4495i		−	0.641907i
$266$		0			0
$267$			15.7980i			0.966819i
$268$		0			0
$269$		−	20.4949i		−	1.24960i	−0.780786	−	0.624798i	$-0.785181\pi$
$269$		−	20.4949i		−	1.24960i	0.780786	−	0.624798i	$-0.214819\pi$
$270$		0			0
$271$	19.1464			1.16306			0.581531	−	0.813524i	$-0.302454\pi$
$271$	19.1464			1.16306			0.581531	+	0.813524i	$0.302454\pi$
$272$		0			0
$273$	4.44949	−	10.8990i	0.269295	−	0.659636i
$274$		0			0
$275$		−	2.44949i		−	0.147710i
$276$		0			0
$277$	13.5959			0.816900			0.408450	−	0.912781i	$-0.366070\pi$
$277$	13.5959			0.816900			0.408450	+	0.912781i	$0.366070\pi$
$278$		0			0
$279$	−8.44949			−0.505858
$280$		0			0
$281$	−1.10102			−0.0656814			−0.0328407	−	0.999461i	$-0.510455\pi$
$281$	−1.10102			−0.0656814			−0.0328407	+	0.999461i	$0.510455\pi$
$282$		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
$285$		−	4.44949i		−	0.263565i
$286$		0			0
$287$	0.898979	−	2.20204i	0.0530651	−	0.129982i
$288$		0			0
$289$	−30.5959			−1.79976
$290$		0			0
$291$		−	2.65153i		−	0.155435i
$292$		0			0
$293$		−	24.6969i		−	1.44281i	−0.692513	−	0.721405i	$-0.743497\pi$
$293$		−	24.6969i		−	1.44281i	0.692513	−	0.721405i	$-0.256503\pi$
$294$		0			0
$295$		−	4.89898i		−	0.285230i
$296$		0			0
$297$			2.44949i			0.142134i
$298$		0			0
$299$	8.89898			0.514641
$300$		0			0
$301$	8.00000	−	19.5959i	0.461112	−	1.12949i
$302$		0			0
$303$			8.89898i			0.511233i
$304$		0			0
$305$	12.8990			0.738593
$306$		0			0
$307$	−8.89898			−0.507892			−0.253946	−	0.967218i	$-0.581728\pi$
$307$	−8.89898			−0.507892			−0.253946	+	0.967218i	$0.581728\pi$
$308$		0			0
$309$	−8.89898			−0.506245
$310$		0			0
$311$	3.10102			0.175843			0.0879214	−	0.996127i	$-0.471978\pi$
$311$	3.10102			0.175843			0.0879214	+	0.996127i	$0.471978\pi$
$312$		0			0
$313$			14.6515i			0.828153i	0.910242	+	0.414077i	$0.135895\pi$
$313$			14.6515i			0.828153i	−0.910242	+	0.414077i	$0.864105\pi$
$314$		0			0
$315$	−1.00000	+	2.44949i	−0.0563436	+	0.138013i
$316$		0			0
$317$	17.1464			0.963039			0.481520	−	0.876435i	$-0.340085\pi$
$317$	17.1464			0.963039			0.481520	+	0.876435i	$0.340085\pi$
$318$		0			0
$319$		−	16.8990i		−	0.946161i
$320$		0			0
$321$			15.7980i			0.881756i
$322$		0			0
$323$			30.6969i			1.70802i
$324$		0			0
$325$		−	4.44949i		−	0.246813i
$326$		0			0
$327$	−13.7980			−0.763029
$328$		0			0
$329$	−26.6969	−	10.8990i	−1.47185	−	0.600880i
$330$		0			0
$331$			7.10102i			0.390307i	0.980773	+	0.195154i	$0.0625206\pi$
$331$			7.10102i			0.390307i	−0.980773	+	0.195154i	$0.937479\pi$
$332$		0			0
$333$	6.89898			0.378062
$334$		0			0
$335$	−15.7980			−0.863135
$336$		0			0
$337$	12.6969			0.691646			0.345823	−	0.938300i	$-0.387600\pi$
$337$	12.6969			0.691646			0.345823	+	0.938300i	$0.387600\pi$
$338$		0			0
$339$	−3.34847			−0.181864
$340$		0			0
$341$		−	20.6969i		−	1.12080i
$342$		0			0
$343$	−7.34847	−	17.0000i	−0.396780	−	0.917914i
$344$		0			0
$345$	−2.00000			−0.107676
$346$		0			0
$347$		−	34.8990i		−	1.87348i	−0.350032	−	0.936738i	$-0.613829\pi$
$347$		−	34.8990i		−	1.87348i	0.350032	−	0.936738i	$-0.386171\pi$
$348$		0			0
$349$		−	10.6969i		−	0.572594i	−0.958141	−	0.286297i	$-0.907576\pi$
$349$		−	10.6969i		−	0.572594i	0.958141	−	0.286297i	$-0.0924244\pi$
$350$		0			0
$351$			4.44949i			0.237496i
$352$		0			0
$353$			31.7980i			1.69243i	0.532838	+	0.846217i	$0.321126\pi$
$353$			31.7980i			1.69243i	−0.532838	+	0.846217i	$0.678874\pi$
$354$		0			0
$355$	6.44949			0.342303
$356$		0			0
$357$	6.89898	−	16.8990i	0.365133	−	0.894389i
$358$		0			0
$359$			12.2474i			0.646396i	0.946331	+	0.323198i	$0.104758\pi$
$359$			12.2474i			0.646396i	−0.946331	+	0.323198i	$0.895242\pi$
$360$		0			0
$361$	0.797959			0.0419978
$362$		0			0
$363$	5.00000			0.262432
$364$		0			0
$365$	5.34847			0.279952
$366$		0			0
$367$	−9.79796			−0.511449			−0.255725	−	0.966750i	$-0.582314\pi$
$367$	−9.79796			−0.511449			−0.255725	+	0.966750i	$0.582314\pi$
$368$		0			0
$369$			0.898979i			0.0467990i
$370$		0			0
$371$	−25.5959	−	10.4495i	−1.32887	−	0.542510i
$372$		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
$377$		−	30.6969i		−	1.58097i
$378$		0			0
$379$			11.1010i			0.570221i	0.958495	+	0.285111i	$0.0920303\pi$
$379$			11.1010i			0.570221i	−0.958495	+	0.285111i	$0.907970\pi$
$380$		0			0
$381$		−	1.79796i		−	0.0921122i
$382$		0			0
$383$	−6.20204			−0.316909			−0.158455	−	0.987366i	$-0.550651\pi$
$383$	−6.20204			−0.316909			−0.158455	+	0.987366i	$0.550651\pi$
$384$		0			0
$385$	−6.00000	−	2.44949i	−0.305788	−	0.124838i
$386$		0			0
$387$			8.00000i			0.406663i
$388$		0			0
$389$	−21.5959			−1.09496			−0.547478	−	0.836820i	$-0.684412\pi$
$389$	−21.5959			−1.09496			−0.547478	+	0.836820i	$0.684412\pi$
$390$		0			0
$391$	13.7980			0.697793
$392$		0			0
$393$	−18.6969			−0.943136
$394$		0			0
$395$	−9.79796			−0.492989
$396$		0			0
$397$		−	37.8434i		−	1.89930i	−0.313306	−	0.949652i	$-0.601437\pi$
$397$		−	37.8434i		−	1.89930i	0.313306	−	0.949652i	$-0.398563\pi$
$398$		0			0
$399$	−10.8990	−	4.44949i	−0.545631	−	0.222753i
$400$		0			0
$401$	27.7980			1.38816			0.694082	−	0.719896i	$-0.255811\pi$
$401$	27.7980			1.38816			0.694082	+	0.719896i	$0.255811\pi$
$402$		0			0
$403$		−	37.5959i		−	1.87279i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$			16.8990i			0.837651i
$408$		0			0
$409$		−	17.7980i		−	0.880052i	−0.897985	−	0.440026i	$-0.854969\pi$
$409$		−	17.7980i		−	0.880052i	0.897985	−	0.440026i	$-0.145031\pi$
$410$		0			0
$411$	21.1464			1.04308
$412$		0			0
$413$	−12.0000	−	4.89898i	−0.590481	−	0.241063i
$414$		0			0
$415$			2.89898i			0.142305i
$416$		0			0
$417$	−7.55051			−0.369750
$418$		0			0
$419$	15.5959			0.761910			0.380955	−	0.924593i	$-0.375595\pi$
$419$	15.5959			0.761910			0.380955	+	0.924593i	$0.375595\pi$
$420$		0			0
$421$	−29.5959			−1.44242			−0.721208	−	0.692718i	$-0.756413\pi$
$421$	−29.5959			−1.44242			−0.721208	+	0.692718i	$0.756413\pi$
$422$		0			0
$423$	10.8990			0.529927
$424$		0			0
$425$		−	6.89898i		−	0.334650i
$426$		0			0
$427$	12.8990	−	31.5959i	0.624225	−	1.52903i
$428$		0			0
$429$	−10.8990			−0.526208
$430$		0			0
$431$		−	1.55051i		−	0.0746855i	−0.999303	−	0.0373427i	$-0.988111\pi$
$431$		−	1.55051i		−	0.0746855i	0.999303	−	0.0373427i	$-0.0118893\pi$
$432$		0			0
$433$			0.449490i			0.0216011i	0.999942	+	0.0108005i	$0.00343799\pi$
$433$			0.449490i			0.0216011i	−0.999942	+	0.0108005i	$0.996562\pi$
$434$		0			0
$435$			6.89898i			0.330781i
$436$		0			0
$437$		−	8.89898i		−	0.425696i
$438$		0			0
$439$	9.75255			0.465464			0.232732	−	0.972541i	$-0.425234\pi$
$439$	9.75255			0.465464			0.232732	+	0.972541i	$0.425234\pi$
$440$		0			0
$441$	5.00000	+	4.89898i	0.238095	+	0.233285i
$442$		0			0
$443$			1.10102i			0.0523111i	0.999658	+	0.0261555i	$0.00832651\pi$
$443$			1.10102i			0.0523111i	−0.999658	+	0.0261555i	$0.991673\pi$
$444$		0			0
$445$	15.7980			0.748895
$446$		0			0
$447$	−3.79796			−0.179637
$448$		0			0
$449$	0.202041			0.00953491			0.00476745	−	0.999989i	$-0.498482\pi$
$449$	0.202041			0.00953491			0.00476745	+	0.999989i	$0.498482\pi$
$450$		0			0
$451$	−2.20204			−0.103690
$452$		0			0
$453$		−	8.89898i		−	0.418111i
$454$		0			0
$455$	−10.8990	−	4.44949i	−0.510952	−	0.208595i
$456$		0			0
$457$	22.4949			1.05227			0.526133	−	0.850402i	$-0.323641\pi$
$457$	22.4949			1.05227			0.526133	+	0.850402i	$0.323641\pi$
$458$		0			0
$459$			6.89898i			0.322017i
$460$		0			0
$461$			8.20204i			0.382007i	0.981589	+	0.191004i	$0.0611742\pi$
$461$			8.20204i			0.382007i	−0.981589	+	0.191004i	$0.938826\pi$
$462$		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
$465$			8.44949i			0.391836i
$466$		0			0
$467$	12.6969			0.587544			0.293772	−	0.955875i	$-0.405089\pi$
$467$	12.6969			0.587544			0.293772	+	0.955875i	$0.405089\pi$
$468$		0			0
$469$	−15.7980	+	38.6969i	−0.729482	+	1.78686i
$470$		0			0
$471$		−	17.3485i		−	0.799375i
$472$		0			0
$473$	−19.5959			−0.901021
$474$		0			0
$475$	−4.44949			−0.204157
$476$		0			0
$477$	10.4495			0.478449
$478$		0			0
$479$	3.10102			0.141689			0.0708446	−	0.997487i	$-0.477431\pi$
$479$	3.10102			0.141689			0.0708446	+	0.997487i	$0.477431\pi$
$480$		0			0
$481$			30.6969i			1.39966i
$482$		0			0
$483$	−2.00000	+	4.89898i	−0.0910032	+	0.222911i
$484$		0			0
$485$	−2.65153			−0.120400
$486$		0			0
$487$			23.5959i			1.06923i	0.845095	+	0.534617i	$0.179544\pi$
$487$			23.5959i			1.06923i	−0.845095	+	0.534617i	$0.820456\pi$
$488$		0			0
$489$		−	23.5959i		−	1.06704i
$490$		0			0
$491$			20.6515i			0.931991i	0.884787	+	0.465995i	$0.154304\pi$
$491$			20.6515i			0.931991i	−0.884787	+	0.465995i	$0.845696\pi$
$492$		0			0
$493$		−	47.5959i		−	2.14361i
$494$		0			0
$495$	2.44949			0.110096
$496$		0			0
$497$	6.44949	−	15.7980i	0.289299	−	0.708635i
$498$		0			0
$499$			5.30306i			0.237398i	0.992930	+	0.118699i	$0.0378723\pi$
$499$			5.30306i			0.237398i	−0.992930	+	0.118699i	$0.962128\pi$
$500$		0			0
$501$	−8.00000			−0.357414
$502$		0			0
$503$	43.5959			1.94385			0.971923	−	0.235299i	$-0.0756070\pi$
$503$	43.5959			1.94385			0.971923	+	0.235299i	$0.0756070\pi$
$504$		0			0
$505$	8.89898			0.395999
$506$		0			0
$507$	−6.79796			−0.301908
$508$		0			0
$509$		−	4.89898i		−	0.217143i	−0.994089	−	0.108572i	$-0.965372\pi$
$509$		−	4.89898i		−	0.217143i	0.994089	−	0.108572i	$-0.0346277\pi$
$510$		0			0
$511$	5.34847	−	13.1010i	0.236602	−	0.579555i
$512$		0			0
$513$	4.44949			0.196450
$514$		0			0
$515$			8.89898i			0.392136i
$516$		0			0
$517$			26.6969i			1.17413i
$518$		0			0
$519$		−	3.79796i		−	0.166712i
$520$		0			0
$521$			3.10102i			0.135858i	0.997690	+	0.0679291i	$0.0216392\pi$
$521$			3.10102i			0.135858i	−0.997690	+	0.0679291i	$0.978361\pi$
$522$		0			0
$523$	−27.1010			−1.18504			−0.592522	−	0.805554i	$-0.701868\pi$
$523$	−27.1010			−1.18504			−0.592522	+	0.805554i	$0.701868\pi$
$524$		0			0
$525$	2.44949	+	1.00000i	0.106904	+	0.0436436i
$526$		0			0
$527$		−	58.2929i		−	2.53928i
$528$		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
$535$	15.7980			0.683005
$536$		0			0
$537$			0.247449i			0.0106782i
$538$		0			0
$539$	−12.0000	+	12.2474i	−0.516877	+	0.527535i
$540$		0			0
$541$	−25.5959			−1.10045			−0.550227	−	0.835015i	$-0.685459\pi$
$541$	−25.5959			−1.10045			−0.550227	+	0.835015i	$0.685459\pi$
$542$		0			0
$543$		−	13.7980i		−	0.592127i
$544$		0			0
$545$			13.7980i			0.591040i
$546$		0			0
$547$			6.40408i			0.273819i	0.990584	+	0.136909i	$0.0437169\pi$
$547$			6.40408i			0.273819i	−0.990584	+	0.136909i	$0.956283\pi$
$548$		0			0
$549$			12.8990i			0.550515i
$550$		0			0
$551$	−30.6969			−1.30773
$552$		0			0
$553$	−9.79796	+	24.0000i	−0.416652	+	1.02058i
$554$		0			0
$555$		−	6.89898i		−	0.292845i
$556$		0			0
$557$	−10.4495			−0.442759			−0.221380	−	0.975188i	$-0.571056\pi$
$557$	−10.4495			−0.442759			−0.221380	+	0.975188i	$0.571056\pi$
$558$		0			0
$559$	−35.5959			−1.50555
$560$		0			0
$561$	−16.8990			−0.713475
$562$		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
$565$			3.34847i			0.140871i
$566$		0			0
$567$	−2.44949	−	1.00000i	−0.102869	−	0.0419961i
$568$		0			0
$569$	−11.7980			−0.494596			−0.247298	−	0.968939i	$-0.579543\pi$
$569$	−11.7980			−0.494596			−0.247298	+	0.968939i	$0.579543\pi$
$570$		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
$573$			10.4495i			0.436534i
$574$		0			0
$575$			2.00000i			0.0834058i
$576$		0			0
$577$		−	23.5505i		−	0.980421i	−0.871604	−	0.490210i	$-0.836920\pi$
$577$		−	23.5505i		−	0.980421i	0.871604	−	0.490210i	$-0.163080\pi$
$578$		0			0
$579$	−12.6969			−0.527667
$580$		0			0
$581$	7.10102	+	2.89898i	0.294600	+	0.120270i
$582$		0			0
$583$			25.5959i			1.06007i
$584$		0			0
$585$	4.44949			0.183964
$586$		0			0
$587$	43.1918			1.78272			0.891359	−	0.453298i	$-0.149753\pi$
$587$	43.1918			1.78272			0.891359	+	0.453298i	$0.149753\pi$
$588$		0			0
$589$	−37.5959			−1.54911
$590$		0			0
$591$	0.247449			0.0101787
$592$		0			0
$593$		−	20.2020i		−	0.829598i	−0.909913	−	0.414799i	$-0.863852\pi$
$593$		−	20.2020i		−	0.829598i	0.909913	−	0.414799i	$-0.136148\pi$
$594$		0			0
$595$	−16.8990	−	6.89898i	−0.692791	−	0.282831i
$596$		0			0
$597$	4.44949			0.182105
$598$		0			0
$599$		−	22.0454i		−	0.900751i	−0.892839	−	0.450375i	$-0.851290\pi$
$599$		−	22.0454i		−	0.900751i	0.892839	−	0.450375i	$-0.148710\pi$
$600$		0			0
$601$		−	20.8990i		−	0.852487i	−0.904608	−	0.426244i	$-0.859837\pi$
$601$		−	20.8990i		−	0.852487i	0.904608	−	0.426244i	$-0.140163\pi$
$602$		0			0
$603$		−	15.7980i		−	0.643343i
$604$		0			0
$605$		−	5.00000i		−	0.203279i
$606$		0			0
$607$	−15.5959			−0.633019			−0.316509	−	0.948589i	$-0.602511\pi$
$607$	−15.5959			−0.633019			−0.316509	+	0.948589i	$0.602511\pi$
$608$		0			0
$609$	16.8990	+	6.89898i	0.684781	+	0.279561i
$610$		0			0
$611$			48.4949i			1.96189i
$612$		0			0
$613$	31.7980			1.28431			0.642154	−	0.766576i	$-0.278041\pi$
$613$	31.7980			1.28431			0.642154	+	0.766576i	$0.278041\pi$
$614$		0			0
$615$	0.898979			0.0362504
$616$		0			0
$617$	31.8434			1.28197			0.640983	−	0.767555i	$-0.278527\pi$
$617$	31.8434			1.28197			0.640983	+	0.767555i	$0.278527\pi$
$618$		0			0
$619$	−12.0454			−0.484146			−0.242073	−	0.970258i	$-0.577827\pi$
$619$	−12.0454			−0.484146			−0.242073	+	0.970258i	$0.577827\pi$
$620$		0			0
$621$		−	2.00000i		−	0.0802572i
$622$		0			0
$623$	15.7980	−	38.6969i	0.632932	−	1.55036i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			10.8990i			0.435263i
$628$		0			0
$629$			47.5959i			1.89777i
$630$		0			0
$631$			49.3939i			1.96634i	0.182695	+	0.983170i	$0.441518\pi$
$631$			49.3939i			1.96634i	−0.182695	+	0.983170i	$0.558482\pi$
$632$		0			0
$633$		−	6.69694i		−	0.266179i
$634$		0			0
$635$	−1.79796			−0.0713498
$636$		0			0
$637$	−21.7980	+	22.2474i	−0.863667	+	0.881476i
$638$		0			0
$639$			6.44949i			0.255138i
$640$		0			0
$641$	12.6969			0.501499			0.250749	−	0.968052i	$-0.419323\pi$
$641$	12.6969			0.501499			0.250749	+	0.968052i	$0.419323\pi$
$642$		0			0
$643$	28.8990			1.13966			0.569832	−	0.821761i	$-0.307008\pi$
$643$	28.8990			1.13966			0.569832	+	0.821761i	$0.307008\pi$
$644$		0			0
$645$	8.00000			0.315000
$646$		0			0
$647$	27.5959			1.08491			0.542454	−	0.840086i	$-0.317495\pi$
$647$	27.5959			1.08491			0.542454	+	0.840086i	$0.317495\pi$
$648$		0			0
$649$			12.0000i			0.471041i
$650$		0			0
$651$	20.6969	+	8.44949i	0.811177	+	0.331162i
$652$		0			0
$653$	9.14643			0.357927			0.178964	−	0.983856i	$-0.442726\pi$
$653$	9.14643			0.357927			0.178964	+	0.983856i	$0.442726\pi$
$654$		0			0
$655$			18.6969i			0.730550i
$656$		0			0
$657$			5.34847i			0.208664i
$658$		0			0
$659$		−	24.2474i		−	0.944546i	−0.881452	−	0.472273i	$-0.843434\pi$
$659$		−	24.2474i		−	0.944546i	0.881452	−	0.472273i	$-0.156566\pi$
$660$		0			0
$661$		−	35.1010i		−	1.36527i	−0.730759	−	0.682636i	$-0.760834\pi$
$661$		−	35.1010i		−	1.36527i	0.730759	−	0.682636i	$-0.239166\pi$
$662$		0			0
$663$	−30.6969			−1.19217
$664$		0			0
$665$	−4.44949	+	10.8990i	−0.172544	+	0.422644i
$666$		0			0
$667$			13.7980i			0.534259i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−31.5959			−1.21975
$672$		0			0
$673$	3.30306			0.127324			0.0636618	−	0.997972i	$-0.479722\pi$
$673$	3.30306			0.127324			0.0636618	+	0.997972i	$0.479722\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$			8.69694i			0.334250i	0.985936	+	0.167125i	$0.0534484\pi$
$677$			8.69694i			0.334250i	−0.985936	+	0.167125i	$0.946552\pi$
$678$		0			0
$679$	−2.65153	+	6.49490i	−0.101756	+	0.249251i
$680$		0			0
$681$	20.6969			0.793108
$682$		0			0
$683$		−	29.1010i		−	1.11352i	−0.830674	−	0.556760i	$-0.812044\pi$
$683$		−	29.1010i		−	1.11352i	0.830674	−	0.556760i	$-0.187956\pi$
$684$		0			0
$685$		−	21.1464i		−	0.807963i
$686$		0			0
$687$			7.10102i			0.270921i
$688$		0			0
$689$			46.4949i			1.77131i
$690$		0			0
$691$	−23.5505			−0.895904			−0.447952	−	0.894058i	$-0.647846\pi$
$691$	−23.5505			−0.895904			−0.447952	+	0.894058i	$0.647846\pi$
$692$		0			0
$693$	2.44949	−	6.00000i	0.0930484	−	0.227921i
$694$		0			0
$695$			7.55051i			0.286407i
$696$		0			0
$697$	−6.20204			−0.234919
$698$		0			0
$699$	−12.2474			−0.463241
$700$		0			0
$701$	−12.6969			−0.479557			−0.239778	−	0.970828i	$-0.577075\pi$
$701$	−12.6969			−0.479557			−0.239778	+	0.970828i	$0.577075\pi$
$702$		0			0
$703$	30.6969			1.15776
$704$		0			0
$705$		−	10.8990i		−	0.410479i
$706$		0			0
$707$	8.89898	−	21.7980i	0.334681	−	0.819797i
$708$		0			0
$709$	35.7980			1.34442			0.672210	−	0.740360i	$-0.265345\pi$
$709$	35.7980			1.34442			0.672210	+	0.740360i	$0.265345\pi$
$710$		0			0
$711$		−	9.79796i		−	0.367452i
$712$		0			0
$713$			16.8990i			0.632872i
$714$		0			0
$715$			10.8990i			0.407599i
$716$		0			0
$717$			26.4495i			0.987774i
$718$		0			0
$719$	31.1010			1.15987			0.579936	−	0.814662i	$-0.303077\pi$
$719$	31.1010			1.15987			0.579936	+	0.814662i	$0.303077\pi$
$720$		0			0
$721$	21.7980	+	8.89898i	0.811798	+	0.331415i
$722$		0			0
$723$			0.898979i			0.0334334i
$724$		0			0
$725$	6.89898			0.256222
$726$		0			0
$727$	6.20204			0.230021			0.115010	−	0.993364i	$-0.463310\pi$
$727$	6.20204			0.230021			0.115010	+	0.993364i	$0.463310\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−55.1918			−2.04134
$732$		0			0
$733$		−	2.65153i		−	0.0979365i	−0.998800	−	0.0489683i	$-0.984407\pi$
$733$		−	2.65153i		−	0.0979365i	0.998800	−	0.0489683i	$-0.0155933\pi$
$734$		0			0
$735$	4.89898	−	5.00000i	0.180702	−	0.184428i
$736$		0			0
$737$	38.6969			1.42542
$738$		0			0
$739$		−	47.5959i		−	1.75084i	−0.483359	−	0.875422i	$-0.660584\pi$
$739$		−	47.5959i		−	1.75084i	0.483359	−	0.875422i	$-0.339416\pi$
$740$		0			0
$741$			19.7980i			0.727296i
$742$		0			0
$743$		−	20.2929i		−	0.744473i	−0.928138	−	0.372236i	$-0.878591\pi$
$743$		−	20.2929i		−	0.744473i	0.928138	−	0.372236i	$-0.121409\pi$
$744$		0			0
$745$			3.79796i			0.139146i
$746$		0			0
$747$	−2.89898			−0.106068
$748$		0			0
$749$	15.7980	−	38.6969i	0.577245	−	1.41396i
$750$		0			0
$751$		−	48.8990i		−	1.78435i	−0.451691	−	0.892175i	$-0.649179\pi$
$751$		−	48.8990i		−	1.78435i	0.451691	−	0.892175i	$-0.350821\pi$
$752$		0			0
$753$	1.30306			0.0474862
$754$		0			0
$755$	−8.89898			−0.323867
$756$		0			0
$757$	48.6969			1.76992			0.884960	−	0.465667i	$-0.154185\pi$
$757$	48.6969			1.76992			0.884960	+	0.465667i	$0.154185\pi$
$758$		0			0
$759$	4.89898			0.177822
$760$		0			0
$761$		−	27.7980i		−	1.00768i	−0.863798	−	0.503838i	$-0.831921\pi$
$761$		−	27.7980i		−	1.00768i	0.863798	−	0.503838i	$-0.168079\pi$
$762$		0			0
$763$	33.7980	+	13.7980i	1.22357	+	0.499520i
$764$		0			0
$765$	6.89898			0.249433
$766$		0			0
$767$			21.7980i			0.787079i
$768$		0			0
$769$			8.49490i			0.306334i	0.988200	+	0.153167i	$0.0489472\pi$
$769$			8.49490i			0.306334i	−0.988200	+	0.153167i	$0.951053\pi$
$770$		0			0
$771$		−	1.10102i		−	0.0396523i
$772$		0			0
$773$			43.3939i			1.56077i	0.625300	+	0.780385i	$0.284977\pi$
$773$			43.3939i			1.56077i	−0.625300	+	0.780385i	$0.715023\pi$
$774$		0			0
$775$	8.44949			0.303515
$776$		0			0
$777$	−16.8990	−	6.89898i	−0.606248	−	0.247500i
$778$		0			0
$779$			4.00000i			0.143315i
$780$		0			0
$781$	−15.7980			−0.565295
$782$		0			0
$783$	−6.89898			−0.246549
$784$		0			0
$785$	−17.3485			−0.619193
$786$		0			0
$787$	−18.6969			−0.666474			−0.333237	−	0.942843i	$-0.608141\pi$
$787$	−18.6969			−0.666474			−0.333237	+	0.942843i	$0.608141\pi$
$788$		0			0
$789$		−	1.10102i		−	0.0391974i
$790$		0			0
$791$	8.20204	+	3.34847i	0.291631	+	0.119058i
$792$		0			0
$793$	−57.3939			−2.03812
$794$		0			0
$795$		−	10.4495i		−	0.370605i
$796$		0			0
$797$			15.7980i			0.559592i	0.960059	+	0.279796i	$0.0902669\pi$
$797$			15.7980i			0.559592i	−0.960059	+	0.279796i	$0.909733\pi$
$798$		0			0
$799$			75.1918i			2.66010i
$800$		0			0
$801$			15.7980i			0.558193i
$802$		0			0
$803$	−13.1010			−0.462325
$804$		0			0
$805$	4.89898	+	2.00000i	0.172666	+	0.0704907i
$806$		0			0
$807$		−	20.4949i		−	0.721455i
$808$		0			0
$809$	−34.8990			−1.22698			−0.613491	−	0.789701i	$-0.710235\pi$
$809$	−34.8990			−1.22698			−0.613491	+	0.789701i	$0.710235\pi$
$810$		0			0
$811$	35.1464			1.23416			0.617079	−	0.786901i	$-0.288316\pi$
$811$	35.1464			1.23416			0.617079	+	0.786901i	$0.288316\pi$
$812$		0			0
$813$	19.1464			0.671495
$814$		0			0
$815$	−23.5959			−0.826529
$816$		0			0
$817$			35.5959i			1.24534i
$818$		0			0
$819$	4.44949	−	10.8990i	0.155478	−	0.380841i
$820$		0			0
$821$	−6.49490			−0.226673			−0.113337	−	0.993557i	$-0.536154\pi$
$821$	−6.49490			−0.226673			−0.113337	+	0.993557i	$0.536154\pi$
$822$		0			0
$823$			45.3939i			1.58233i	0.611602	+	0.791166i	$0.290525\pi$
$823$			45.3939i			1.58233i	−0.611602	+	0.791166i	$0.709475\pi$
$824$		0			0
$825$		−	2.44949i		−	0.0852803i
$826$		0			0
$827$			43.3939i			1.50895i	0.656327	+	0.754476i	$0.272109\pi$
$827$			43.3939i			1.50895i	−0.656327	+	0.754476i	$0.727891\pi$
$828$		0			0
$829$			5.39388i			0.187337i	0.995603	+	0.0936685i	$0.0298594\pi$
$829$			5.39388i			0.187337i	−0.995603	+	0.0936685i	$0.970141\pi$
$830$		0			0
$831$	13.5959			0.471637
$832$		0			0
$833$	−33.7980	+	34.4949i	−1.17103	+	1.19518i
$834$		0			0
$835$			8.00000i			0.276851i
$836$		0			0
$837$	−8.44949			−0.292057
$838$		0			0
$839$	−53.7980			−1.85731			−0.928656	−	0.370942i	$-0.879035\pi$
$839$	−53.7980			−1.85731			−0.928656	+	0.370942i	$0.879035\pi$
$840$		0			0
$841$	18.5959			0.641239
$842$		0			0
$843$	−1.10102			−0.0379212
$844$		0			0
$845$			6.79796i			0.233857i
$846$		0			0
$847$	−12.2474	−	5.00000i	−0.420827	−	0.171802i
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$		−	13.7980i		−	0.472988i
$852$		0			0
$853$			36.0454i			1.23417i	0.786896	+	0.617086i	$0.211687\pi$
$853$			36.0454i			1.23417i	−0.786896	+	0.617086i	$0.788313\pi$
$854$		0			0
$855$		−	4.44949i		−	0.152169i
$856$		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
$859$	35.5505			1.21297			0.606484	−	0.795096i	$-0.292579\pi$
$859$	35.5505			1.21297			0.606484	+	0.795096i	$0.292579\pi$
$860$		0			0
$861$	0.898979	−	2.20204i	0.0306371	−	0.0750454i
$862$		0			0
$863$			31.3939i			1.06866i	0.845276	+	0.534330i	$0.179436\pi$
$863$			31.3939i			1.06866i	−0.845276	+	0.534330i	$0.820564\pi$
$864$		0			0
$865$	−3.79796			−0.129134
$866$		0			0
$867$	−30.5959			−1.03909
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$	70.2929			2.38178
$872$		0			0
$873$		−	2.65153i		−	0.0897407i
$874$		0			0
$875$	1.00000	−	2.44949i	0.0338062	−	0.0828079i
$876$		0			0
$877$	−13.5959			−0.459102			−0.229551	−	0.973297i	$-0.573726\pi$
$877$	−13.5959			−0.459102			−0.229551	+	0.973297i	$0.573726\pi$
$878$		0			0
$879$		−	24.6969i		−	0.833007i
$880$		0			0
$881$		−	8.20204i		−	0.276334i	−0.990409	−	0.138167i	$-0.955879\pi$
$881$		−	8.20204i		−	0.276334i	0.990409	−	0.138167i	$-0.0441210\pi$
$882$		0			0
$883$		−	0.202041i		−	0.00679922i	−0.999994	−	0.00339961i	$-0.998918\pi$
$883$		−	0.202041i		−	0.00679922i	0.999994	−	0.00339961i	$-0.00108213\pi$
$884$		0			0
$885$		−	4.89898i		−	0.164677i
$886$		0			0
$887$	12.2929			0.412754			0.206377	−	0.978473i	$-0.433833\pi$
$887$	12.2929			0.412754			0.206377	+	0.978473i	$0.433833\pi$
$888$		0			0
$889$	−1.79796	+	4.40408i	−0.0603016	+	0.147708i
$890$		0			0
$891$			2.44949i			0.0820610i
$892$		0			0
$893$	48.4949			1.62282
$894$		0			0
$895$	0.247449			0.00827130
$896$		0			0
$897$	8.89898			0.297128
$898$		0			0
$899$	58.2929			1.94418
$900$		0			0
$901$			72.0908i			2.40169i
$902$		0			0
$903$	8.00000	−	19.5959i	0.266223	−	0.652111i
$904$		0			0
$905$	−13.7980			−0.458660
$906$		0			0
$907$			1.59592i			0.0529916i	0.999649	+	0.0264958i	$0.00843486\pi$
$907$			1.59592i			0.0529916i	−0.999649	+	0.0264958i	$0.991565\pi$
$908$		0			0
$909$			8.89898i			0.295161i
$910$		0			0
$911$			32.6515i			1.08179i	0.841089	+	0.540897i	$0.181915\pi$
$911$			32.6515i			1.08179i	−0.841089	+	0.540897i	$0.818085\pi$
$912$		0			0
$913$		−	7.10102i		−	0.235009i
$914$		0			0
$915$	12.8990			0.426427
$916$		0			0
$917$	45.7980	+	18.6969i	1.51238	+	0.617427i
$918$		0			0
$919$			28.4949i			0.939960i	0.882677	+	0.469980i	$0.155739\pi$
$919$			28.4949i			0.939960i	−0.882677	+	0.469980i	$0.844261\pi$
$920$		0			0
$921$	−8.89898			−0.293231
$922$		0			0
$923$	−28.6969			−0.944571
$924$		0			0
$925$	−6.89898			−0.226837
$926$		0			0
$927$	−8.89898			−0.292281
$928$		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
$931$	22.2474	+	21.7980i	0.729131	+	0.714399i
$932$		0			0
$933$	3.10102			0.101523
$934$		0			0
$935$			16.8990i			0.552656i
$936$		0			0
$937$		−	18.2474i		−	0.596118i	−0.954547	−	0.298059i	$-0.903661\pi$
$937$		−	18.2474i		−	0.596118i	0.954547	−	0.298059i	$-0.0963392\pi$
$938$		0			0
$939$			14.6515i			0.478135i
$940$		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
$943$	1.79796			0.0585496
$944$		0			0
$945$	−1.00000	+	2.44949i	−0.0325300	+	0.0796819i
$946$		0			0
$947$		−	4.69694i		−	0.152630i	−0.997084	−	0.0763150i	$-0.975685\pi$
$947$		−	4.69694i		−	0.152630i	0.997084	−	0.0763150i	$-0.0243155\pi$
$948$		0			0
$949$	−23.7980			−0.772514
$950$		0			0
$951$	17.1464			0.556011
$952$		0			0
$953$	−0.247449			−0.00801565			−0.00400783	−	0.999992i	$-0.501276\pi$
$953$	−0.247449			−0.00801565			−0.00400783	+	0.999992i	$0.501276\pi$
$954$		0			0
$955$	10.4495			0.338138
$956$		0			0
$957$		−	16.8990i		−	0.546266i
$958$		0			0
$959$	−51.7980	−	21.1464i	−1.67264	−	0.682854i
$960$		0			0
$961$	40.3939			1.30303
$962$		0			0
$963$			15.7980i			0.509082i
$964$		0			0
$965$			12.6969i			0.408729i
$966$		0			0
$967$			31.3939i			1.00956i	0.863248	+	0.504780i	$0.168426\pi$
$967$			31.3939i			1.00956i	−0.863248	+	0.504780i	$0.831574\pi$
$968$		0			0
$969$			30.6969i			0.986128i
$970$		0			0
$971$	−39.1918			−1.25773			−0.628863	−	0.777516i	$-0.716479\pi$
$971$	−39.1918			−1.25773			−0.628863	+	0.777516i	$0.716479\pi$
$972$		0			0
$973$	18.4949	+	7.55051i	0.592919	+	0.242058i
$974$		0			0
$975$		−	4.44949i		−	0.142498i
$976$		0			0
$977$	16.6515			0.532730			0.266365	−	0.963872i	$-0.414177\pi$
$977$	16.6515			0.532730			0.266365	+	0.963872i	$0.414177\pi$
$978$		0			0
$979$	−38.6969			−1.23676
$980$		0			0
$981$	−13.7980			−0.440535
$982$		0			0
$983$	−30.8990			−0.985524			−0.492762	−	0.870164i	$-0.664013\pi$
$983$	−30.8990			−0.985524			−0.492762	+	0.870164i	$0.664013\pi$
$984$		0			0
$985$		−	0.247449i		−	0.00788437i
$986$		0			0
$987$	−26.6969	−	10.8990i	−0.849773	−	0.346918i
$988$		0			0
$989$	16.0000			0.508770
$990$		0			0
$991$			22.6969i			0.720992i	0.932761	+	0.360496i	$0.117393\pi$
$991$			22.6969i			0.720992i	−0.932761	+	0.360496i	$0.882607\pi$
$992$		0			0
$993$			7.10102i			0.225344i
$994$		0			0
$995$		−	4.44949i		−	0.141058i
$996$		0			0
$997$		−	11.9546i		−	0.378606i	−0.981919	−	0.189303i	$-0.939377\pi$
$997$		−	11.9546i		−	0.378606i	0.981919	−	0.189303i	$-0.0606228\pi$
$998$		0			0
$999$	6.89898			0.218274

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000i q^{5} +(-2.44949 - 1.00000i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000i q^{5} +(-2.44949 - 1.00000i) q^{7} +1.00000 q^{9} +2.44949i q^{11} +4.44949i q^{13} -1.00000i q^{15} +6.89898i q^{17} +4.44949 q^{19} +(-2.44949 - 1.00000i) q^{21} -2.00000i q^{23} -1.00000 q^{25} +1.00000 q^{27} -6.89898 q^{29} -8.44949 q^{31} +2.44949i q^{33} +(-1.00000 + 2.44949i) q^{35} +6.89898 q^{37} +4.44949i q^{39} +0.898979i q^{41} +8.00000i q^{43} -1.00000i q^{45} +10.8990 q^{47} +(5.00000 + 4.89898i) q^{49} +6.89898i q^{51} +10.4495 q^{53} +2.44949 q^{55} +4.44949 q^{57} +4.89898 q^{59} +12.8990i q^{61} +(-2.44949 - 1.00000i) q^{63} +4.44949 q^{65} -15.7980i q^{67} -2.00000i q^{69} +6.44949i q^{71} +5.34847i q^{73} -1.00000 q^{75} +(2.44949 - 6.00000i) q^{77} -9.79796i q^{79} +1.00000 q^{81} -2.89898 q^{83} +6.89898 q^{85} -6.89898 q^{87} +15.7980i q^{89} +(4.44949 - 10.8990i) q^{91} -8.44949 q^{93} -4.44949i q^{95} -2.65153i q^{97} +2.44949i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more