LMFDB - Embedded newform 3360.2.d.a.2911.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(2911,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.2911"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,-16,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 238x^{12} + 1262x^{10} + 3861x^{8} + 6834x^{6} + 6589x^{4} + 2916x^{2} + 324 $ x^16 + 24x^14 + 238x^12 + 1262x^10 + 3861x^8 + 6834x^6 + 6589x^4 + 2916*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2911.6
Root			$1.22519i$ of defining polynomial
Character	$\chi$	$=$	3360.2911
Dual form			3360.2.d.a.2911.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000i q^{5} +(1.22519 + 2.34498i) q^{7} +1.00000 q^{9} -3.01352i q^{11} -6.64828i q^{13} +1.00000i q^{15} +2.48852i q^{17} -7.51485 q^{19} +(-1.22519 - 2.34498i) q^{21} +2.99784i q^{23} -1.00000 q^{25} -1.00000 q^{27} -5.25959 q^{29} +0.732553 q^{31} +3.01352i q^{33} +(2.34498 - 1.22519i) q^{35} +4.92077 q^{37} +6.64828i q^{39} -1.88370i q^{41} +4.97184i q^{43} -1.00000i q^{45} -1.84555 q^{47} +(-3.99784 + 5.74607i) q^{49} -2.48852i q^{51} -9.99214 q^{53} -3.01352 q^{55} +7.51485 q^{57} +3.67590 q^{59} +3.07737i q^{61} +(1.22519 + 2.34498i) q^{63} -6.64828 q^{65} +6.80922i q^{67} -2.99784i q^{69} +9.38640i q^{71} -13.3201i q^{73} +1.00000 q^{75} +(7.06662 - 3.69212i) q^{77} -2.85035i q^{79} +1.00000 q^{81} +2.76948 q^{83} +2.48852 q^{85} +5.25959 q^{87} +10.1510i q^{89} +(15.5901 - 8.14538i) q^{91} -0.732553 q^{93} +7.51485i q^{95} -18.6360i q^{97} -3.01352i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 16 q^{9} + 16 q^{19} - 16 q^{25} - 16 q^{27} + 8 q^{29} - 8 q^{31} - 8 q^{37} + 16 q^{47} - 16 q^{49} - 48 q^{53} + 8 q^{55} - 16 q^{57} + 16 q^{59} - 8 q^{65} + 16 q^{75} + 16 q^{77}+ \cdots + 8 q^{93}+O(q^{100}) $ 16 * q - 16 * q^3 + 16 * q^9 + 16 * q^19 - 16 * q^25 - 16 * q^27 + 8 * q^29 - 8 * q^31 - 8 * q^37 + 16 * q^47 - 16 * q^49 - 48 * q^53 + 8 * q^55 - 16 * q^57 + 16 * q^59 - 8 * q^65 + 16 * q^75 + 16 * q^77 + 16 * q^81 - 24 * q^83 + 8 * q^85 - 8 * q^87 + 24 * q^91 + 8 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$1$	$1$	$-1$	$-1$	$1$

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
$7$	1.22519	+	2.34498i	0.463077	+	0.886318i
$7$	1.22519	+	2.34498i	0.463077	+	0.886318i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	3.01352i		−	0.908609i	−0.890846	−	0.454305i	$-0.849888\pi$
$11$		−	3.01352i		−	0.908609i	0.890846	−	0.454305i	$-0.150112\pi$
$12$		0			0
$13$		−	6.64828i		−	1.84390i	−0.387309	−	0.921950i	$-0.626595\pi$
$13$		−	6.64828i		−	1.84390i	0.387309	−	0.921950i	$-0.373405\pi$
$14$		0			0
$15$			1.00000i			0.258199i
$16$		0			0
$17$			2.48852i			0.603555i	0.953378	+	0.301778i	$0.0975800\pi$
$17$			2.48852i			0.603555i	−0.953378	+	0.301778i	$0.902420\pi$
$18$		0			0
$19$	−7.51485			−1.72403			−0.862013	−	0.506887i	$-0.830796\pi$
$19$	−7.51485			−1.72403			−0.862013	+	0.506887i	$0.830796\pi$
$20$		0			0
$21$	−1.22519	−	2.34498i	−0.267358	−	0.511716i
$22$		0			0
$23$			2.99784i			0.625092i	0.949903	+	0.312546i	$0.101182\pi$
$23$			2.99784i			0.625092i	−0.949903	+	0.312546i	$0.898818\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−5.25959			−0.976682			−0.488341	−	0.872653i	$-0.662398\pi$
$29$	−5.25959			−0.976682			−0.488341	+	0.872653i	$0.662398\pi$
$30$		0			0
$31$	0.732553			0.131570			0.0657852	−	0.997834i	$-0.479045\pi$
$31$	0.732553			0.131570			0.0657852	+	0.997834i	$0.479045\pi$
$32$		0			0
$33$			3.01352i			0.524586i
$34$		0			0
$35$	2.34498	−	1.22519i	0.396374	−	0.207094i
$36$		0			0
$37$	4.92077			0.808970			0.404485	−	0.914545i	$-0.367451\pi$
$37$	4.92077			0.808970			0.404485	+	0.914545i	$0.367451\pi$
$38$		0			0
$39$			6.64828i			1.06458i
$40$		0			0
$41$		−	1.88370i		−	0.294184i	−0.989123	−	0.147092i	$-0.953009\pi$
$41$		−	1.88370i		−	0.294184i	0.989123	−	0.147092i	$-0.0469914\pi$
$42$		0			0
$43$			4.97184i			0.758198i	0.925356	+	0.379099i	$0.123766\pi$
$43$			4.97184i			0.758198i	−0.925356	+	0.379099i	$0.876234\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$	−1.84555			−0.269201			−0.134601	−	0.990900i	$-0.542975\pi$
$47$	−1.84555			−0.269201			−0.134601	+	0.990900i	$0.542975\pi$
$48$		0			0
$49$	−3.99784	+	5.74607i	−0.571120	+	0.820867i
$50$		0			0
$51$		−	2.48852i		−	0.348463i
$52$		0			0
$53$	−9.99214			−1.37253			−0.686263	−	0.727353i	$-0.740750\pi$
$53$	−9.99214			−1.37253			−0.686263	+	0.727353i	$0.740750\pi$
$54$		0			0
$55$	−3.01352			−0.406342
$56$		0			0
$57$	7.51485			0.995366
$58$		0			0
$59$	3.67590			0.478561			0.239280	−	0.970950i	$-0.423088\pi$
$59$	3.67590			0.478561			0.239280	+	0.970950i	$0.423088\pi$
$60$		0			0
$61$			3.07737i			0.394016i	0.980402	+	0.197008i	$0.0631226\pi$
$61$			3.07737i			0.394016i	−0.980402	+	0.197008i	$0.936877\pi$
$62$		0			0
$63$	1.22519	+	2.34498i	0.154359	+	0.295439i
$64$		0			0
$65$	−6.64828			−0.824617
$66$		0			0
$67$			6.80922i			0.831878i	0.909392	+	0.415939i	$0.136547\pi$
$67$			6.80922i			0.831878i	−0.909392	+	0.415939i	$0.863453\pi$
$68$		0			0
$69$		−	2.99784i		−	0.360897i
$70$		0			0
$71$			9.38640i			1.11396i	0.830526	+	0.556981i	$0.188040\pi$
$71$			9.38640i			1.11396i	−0.830526	+	0.556981i	$0.811960\pi$
$72$		0			0
$73$		−	13.3201i		−	1.55900i	−0.626402	−	0.779500i	$-0.715473\pi$
$73$		−	13.3201i		−	1.55900i	0.626402	−	0.779500i	$-0.284527\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	7.06662	−	3.69212i	0.805317	−	0.420756i
$78$		0			0
$79$		−	2.85035i		−	0.320690i	−0.987061	−	0.160345i	$-0.948739\pi$
$79$		−	2.85035i		−	0.320690i	0.987061	−	0.160345i	$-0.0512607\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	2.76948			0.303990			0.151995	−	0.988381i	$-0.451430\pi$
$83$	2.76948			0.303990			0.151995	+	0.988381i	$0.451430\pi$
$84$		0			0
$85$	2.48852			0.269918
$86$		0			0
$87$	5.25959			0.563887
$88$		0			0
$89$			10.1510i			1.07600i	0.842944	+	0.538001i	$0.180820\pi$
$89$			10.1510i			1.07600i	−0.842944	+	0.538001i	$0.819180\pi$
$90$		0			0
$91$	15.5901	−	8.14538i	1.63428	−	0.853867i
$92$		0			0
$93$	−0.732553			−0.0759622
$94$		0			0
$95$			7.51485i			0.771007i
$96$		0			0
$97$		−	18.6360i		−	1.89220i	−0.323880	−	0.946098i	$-0.604988\pi$
$97$		−	18.6360i		−	1.89220i	0.323880	−	0.946098i	$-0.395012\pi$
$98$		0			0
$99$		−	3.01352i		−	0.302870i
$100$		0			0
$101$		−	1.52559i		−	0.151802i	−0.997115	−	0.0759011i	$-0.975817\pi$
$101$		−	1.52559i		−	0.151802i	0.997115	−	0.0759011i	$-0.0241833\pi$
$102$		0			0
$103$	−6.45037			−0.635574			−0.317787	−	0.948162i	$-0.602940\pi$
$103$	−6.45037			−0.635574			−0.317787	+	0.948162i	$0.602940\pi$
$104$		0			0
$105$	−2.34498	+	1.22519i	−0.228846	+	0.119566i
$106$		0			0
$107$			16.7156i			1.61596i	0.589210	+	0.807980i	$0.299439\pi$
$107$			16.7156i			1.61596i	−0.589210	+	0.807980i	$0.700561\pi$
$108$		0			0
$109$	−11.6688			−1.11767			−0.558836	−	0.829279i	$-0.688752\pi$
$109$	−11.6688			−1.11767			−0.558836	+	0.829279i	$0.688752\pi$
$110$		0			0
$111$	−4.92077			−0.467059
$112$		0			0
$113$	−15.0785			−1.41847			−0.709234	−	0.704973i	$-0.750959\pi$
$113$	−15.0785			−1.41847			−0.709234	+	0.704973i	$0.750959\pi$
$114$		0			0
$115$	2.99784			0.279550
$116$		0			0
$117$		−	6.64828i		−	0.614633i
$118$		0			0
$119$	−5.83553	+	3.04890i	−0.534942	+	0.279492i
$120$		0			0
$121$	1.91873			0.174430
$122$		0			0
$123$			1.88370i			0.169847i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	8.10656i		−	0.719341i	−0.933079	−	0.359670i	$-0.882889\pi$
$127$		−	8.10656i		−	0.719341i	0.933079	−	0.359670i	$-0.117111\pi$
$128$		0			0
$129$		−	4.97184i		−	0.437746i
$130$		0			0
$131$	−15.0858			−1.31805			−0.659024	−	0.752121i	$-0.729031\pi$
$131$	−15.0858			−1.31805			−0.659024	+	0.752121i	$0.729031\pi$
$132$		0			0
$133$	−9.20709	−	17.6222i	−0.798356	−	1.52803i
$134$		0			0
$135$			1.00000i			0.0860663i
$136$		0			0
$137$	−10.9952			−0.939380			−0.469690	−	0.882831i	$-0.655634\pi$
$137$	−10.9952			−0.939380			−0.469690	+	0.882831i	$0.655634\pi$
$138$		0			0
$139$	2.72739			0.231334			0.115667	−	0.993288i	$-0.463099\pi$
$139$	2.72739			0.231334			0.115667	+	0.993288i	$0.463099\pi$
$140$		0			0
$141$	1.84555			0.155423
$142$		0			0
$143$	−20.0347			−1.67538
$144$		0			0
$145$			5.25959i			0.436785i
$146$		0			0
$147$	3.99784	−	5.74607i	0.329736	−	0.473928i
$148$		0			0
$149$	8.35592			0.684544			0.342272	−	0.939601i	$-0.388804\pi$
$149$	8.35592			0.684544			0.342272	+	0.939601i	$0.388804\pi$
$150$		0			0
$151$			22.9354i			1.86646i	0.359282	+	0.933229i	$0.383022\pi$
$151$			22.9354i			1.86646i	−0.359282	+	0.933229i	$0.616978\pi$
$152$		0			0
$153$			2.48852i			0.201185i
$154$		0			0
$155$		−	0.732553i		−	0.0588401i
$156$		0			0
$157$			7.01845i			0.560133i	0.959981	+	0.280067i	$0.0903566\pi$
$157$			7.01845i			0.560133i	−0.959981	+	0.280067i	$0.909643\pi$
$158$		0			0
$159$	9.99214			0.792429
$160$		0			0
$161$	−7.02986	+	3.67291i	−0.554031	+	0.289466i
$162$		0			0
$163$		−	13.2184i		−	1.03534i	−0.855579	−	0.517672i	$-0.826799\pi$
$163$		−	13.2184i		−	1.03534i	0.855579	−	0.517672i	$-0.173201\pi$
$164$		0			0
$165$	3.01352			0.234602
$166$		0			0
$167$	−12.6407			−0.978164			−0.489082	−	0.872238i	$-0.662668\pi$
$167$	−12.6407			−0.978164			−0.489082	+	0.872238i	$0.662668\pi$
$168$		0			0
$169$	−31.1996			−2.39997
$170$		0			0
$171$	−7.51485			−0.574675
$172$		0			0
$173$			4.72615i			0.359323i	0.983729	+	0.179661i	$0.0575002\pi$
$173$			4.72615i			0.359323i	−0.983729	+	0.179661i	$0.942500\pi$
$174$		0			0
$175$	−1.22519	−	2.34498i	−0.0926154	−	0.177264i
$176$		0			0
$177$	−3.67590			−0.276297
$178$		0			0
$179$		−	13.3705i		−	0.999356i	−0.866211	−	0.499678i	$-0.833452\pi$
$179$		−	13.3705i		−	0.999356i	0.866211	−	0.499678i	$-0.166548\pi$
$180$		0			0
$181$			9.92238i			0.737525i	0.929524	+	0.368762i	$0.120218\pi$
$181$			9.92238i			0.737525i	−0.929524	+	0.368762i	$0.879782\pi$
$182$		0			0
$183$		−	3.07737i		−	0.227485i
$184$		0			0
$185$		−	4.92077i		−	0.361782i
$186$		0			0
$187$	7.49920			0.548396
$188$		0			0
$189$	−1.22519	−	2.34498i	−0.0891192	−	0.170572i
$190$		0			0
$191$			13.1111i			0.948689i	0.880339	+	0.474344i	$0.157315\pi$
$191$			13.1111i			0.948689i	−0.880339	+	0.474344i	$0.842685\pi$
$192$		0			0
$193$	6.03136			0.434146			0.217073	−	0.976155i	$-0.430349\pi$
$193$	6.03136			0.434146			0.217073	+	0.976155i	$0.430349\pi$
$194$		0			0
$195$	6.64828			0.476093
$196$		0			0
$197$	−21.2078			−1.51099			−0.755497	−	0.655152i	$-0.772605\pi$
$197$	−21.2078			−1.51099			−0.755497	+	0.655152i	$0.772605\pi$
$198$		0			0
$199$	−1.91024			−0.135414			−0.0677068	−	0.997705i	$-0.521568\pi$
$199$	−1.91024			−0.135414			−0.0677068	+	0.997705i	$0.521568\pi$
$200$		0			0
$201$		−	6.80922i		−	0.480285i
$202$		0			0
$203$	−6.44398	−	12.3336i	−0.452279	−	0.865651i
$204$		0			0
$205$	−1.88370			−0.131563
$206$		0			0
$207$			2.99784i			0.208364i
$208$		0			0
$209$			22.6461i			1.56646i
$210$		0			0
$211$			11.2462i			0.774218i	0.922034	+	0.387109i	$0.126526\pi$
$211$			11.2462i			0.774218i	−0.922034	+	0.387109i	$0.873474\pi$
$212$		0			0
$213$		−	9.38640i		−	0.643146i
$214$		0			0
$215$	4.97184			0.339076
$216$		0			0
$217$	0.897514	+	1.71782i	0.0609272	+	0.116613i
$218$		0			0
$219$			13.3201i			0.900089i
$220$		0			0
$221$	16.5444			1.11290
$222$		0			0
$223$	5.06340			0.339070			0.169535	−	0.985524i	$-0.445773\pi$
$223$	5.06340			0.339070			0.169535	+	0.985524i	$0.445773\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−19.3059			−1.28138			−0.640688	−	0.767801i	$-0.721351\pi$
$227$	−19.3059			−1.28138			−0.640688	+	0.767801i	$0.721351\pi$
$228$		0			0
$229$			12.6515i			0.836032i	0.908440	+	0.418016i	$0.137274\pi$
$229$			12.6515i			0.836032i	−0.908440	+	0.418016i	$0.862726\pi$
$230$		0			0
$231$	−7.06662	+	3.69212i	−0.464950	+	0.242923i
$232$		0			0
$233$	19.4858			1.27656			0.638279	−	0.769805i	$-0.279646\pi$
$233$	19.4858			1.27656			0.638279	+	0.769805i	$0.279646\pi$
$234$		0			0
$235$			1.84555i			0.120390i
$236$		0			0
$237$			2.85035i			0.185150i
$238$		0			0
$239$			15.4526i			0.999546i	0.866156	+	0.499773i	$0.166583\pi$
$239$			15.4526i			0.999546i	−0.866156	+	0.499773i	$0.833417\pi$
$240$		0			0
$241$		−	18.0126i		−	1.16029i	−0.814512	−	0.580147i	$-0.802995\pi$
$241$		−	18.0126i		−	1.16029i	0.814512	−	0.580147i	$-0.197005\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	5.74607	+	3.99784i	0.367103	+	0.255412i
$246$		0			0
$247$			49.9608i			3.17893i
$248$		0			0
$249$	−2.76948			−0.175509
$250$		0			0
$251$	−22.1128			−1.39575			−0.697874	−	0.716221i	$-0.745870\pi$
$251$	−22.1128			−1.39575			−0.697874	+	0.716221i	$0.745870\pi$
$252$		0			0
$253$	9.03403			0.567964
$254$		0			0
$255$	−2.48852			−0.155837
$256$		0			0
$257$		−	2.14970i		−	0.134095i	−0.997750	−	0.0670473i	$-0.978642\pi$
$257$		−	2.14970i		−	0.134095i	0.997750	−	0.0670473i	$-0.0213579\pi$
$258$		0			0
$259$	6.02886	+	11.5391i	0.374615	+	0.717005i
$260$		0			0
$261$	−5.25959			−0.325561
$262$		0			0
$263$		−	23.2694i		−	1.43485i	−0.696633	−	0.717427i	$-0.745320\pi$
$263$		−	23.2694i		−	1.43485i	0.696633	−	0.717427i	$-0.254680\pi$
$264$		0			0
$265$			9.99214i			0.613813i
$266$		0			0
$267$		−	10.1510i		−	0.621230i
$268$		0			0
$269$			3.36237i			0.205007i	0.994733	+	0.102504i	$0.0326853\pi$
$269$			3.36237i			0.205007i	−0.994733	+	0.102504i	$0.967315\pi$
$270$		0			0
$271$	−4.88011			−0.296446			−0.148223	−	0.988954i	$-0.547355\pi$
$271$	−4.88011			−0.296446			−0.148223	+	0.988954i	$0.547355\pi$
$272$		0			0
$273$	−15.5901	+	8.14538i	−0.943553	+	0.492981i
$274$		0			0
$275$			3.01352i			0.181722i
$276$		0			0
$277$	−26.2978			−1.58008			−0.790039	−	0.613056i	$-0.789940\pi$
$277$	−26.2978			−1.58008			−0.790039	+	0.613056i	$0.789940\pi$
$278$		0			0
$279$	0.732553			0.0438568
$280$		0			0
$281$	−8.18742			−0.488421			−0.244210	−	0.969722i	$-0.578529\pi$
$281$	−8.18742			−0.488421			−0.244210	+	0.969722i	$0.578529\pi$
$282$		0			0
$283$	13.4183			0.797637			0.398818	−	0.917030i	$-0.369420\pi$
$283$	13.4183			0.797637			0.398818	+	0.917030i	$0.369420\pi$
$284$		0			0
$285$		−	7.51485i		−	0.445141i
$286$		0			0
$287$	4.41723	−	2.30788i	0.260741	−	0.136230i
$288$		0			0
$289$	10.8073			0.635721
$290$		0			0
$291$			18.6360i			1.09246i
$292$		0			0
$293$			23.0697i			1.34775i	0.738847	+	0.673874i	$0.235371\pi$
$293$			23.0697i			1.34775i	−0.738847	+	0.673874i	$0.764629\pi$
$294$		0			0
$295$		−	3.67590i		−	0.214019i
$296$		0			0
$297$			3.01352i			0.174862i
$298$		0			0
$299$	19.9304			1.15261
$300$		0			0
$301$	−11.6588	+	6.09142i	−0.672005	+	0.351104i
$302$		0			0
$303$			1.52559i			0.0876430i
$304$		0			0
$305$	3.07737			0.176209
$306$		0			0
$307$	−13.4270			−0.766321			−0.383160	−	0.923682i	$-0.625164\pi$
$307$	−13.4270			−0.766321			−0.383160	+	0.923682i	$0.625164\pi$
$308$		0			0
$309$	6.45037			0.366949
$310$		0			0
$311$	−23.9720			−1.35933			−0.679665	−	0.733523i	$-0.737875\pi$
$311$	−23.9720			−1.35933			−0.679665	+	0.733523i	$0.737875\pi$
$312$		0			0
$313$			2.20964i			0.124897i	0.998048	+	0.0624483i	$0.0198908\pi$
$313$			2.20964i			0.124897i	−0.998048	+	0.0624483i	$0.980109\pi$
$314$		0			0
$315$	2.34498	−	1.22519i	0.132125	−	0.0690314i
$316$		0			0
$317$	−0.251931			−0.0141499			−0.00707494	−	0.999975i	$-0.502252\pi$
$317$	−0.251931			−0.0141499			−0.00707494	+	0.999975i	$0.502252\pi$
$318$		0			0
$319$			15.8499i			0.887422i
$320$		0			0
$321$		−	16.7156i		−	0.932975i
$322$		0			0
$323$		−	18.7009i		−	1.04054i
$324$		0			0
$325$			6.64828i			0.368780i
$326$		0			0
$327$	11.6688			0.645288
$328$		0			0
$329$	−2.26114	−	4.32777i	−0.124661	−	0.238598i
$330$		0			0
$331$		−	23.2542i		−	1.27817i	−0.769138	−	0.639083i	$-0.779314\pi$
$331$		−	23.2542i		−	1.27817i	0.769138	−	0.639083i	$-0.220686\pi$
$332$		0			0
$333$	4.92077			0.269657
$334$		0			0
$335$	6.80922			0.372027
$336$		0			0
$337$	33.7237			1.83705			0.918523	−	0.395368i	$-0.129383\pi$
$337$	33.7237			1.83705			0.918523	+	0.395368i	$0.129383\pi$
$338$		0			0
$339$	15.0785			0.818953
$340$		0			0
$341$		−	2.20756i		−	0.119546i
$342$		0			0
$343$	−18.3725	−	2.33483i	−0.992021	−	0.126069i
$344$		0			0
$345$	−2.99784			−0.161398
$346$		0			0
$347$		−	34.8813i		−	1.87253i	−0.351298	−	0.936264i	$-0.614260\pi$
$347$		−	34.8813i		−	1.87253i	0.351298	−	0.936264i	$-0.385740\pi$
$348$		0			0
$349$		−	7.45624i		−	0.399123i	−0.979885	−	0.199562i	$-0.936048\pi$
$349$		−	7.45624i		−	0.399123i	0.979885	−	0.199562i	$-0.0639518\pi$
$350$		0			0
$351$			6.64828i			0.354859i
$352$		0			0
$353$		−	30.8563i		−	1.64232i	−0.570701	−	0.821158i	$-0.693329\pi$
$353$		−	30.8563i		−	1.64232i	0.570701	−	0.821158i	$-0.306671\pi$
$354$		0			0
$355$	9.38640			0.498179
$356$		0			0
$357$	5.83553	−	3.04890i	0.308849	−	0.161365i
$358$		0			0
$359$			0.961413i			0.0507414i	0.999678	+	0.0253707i	$0.00807662\pi$
$359$			0.961413i			0.0507414i	−0.999678	+	0.0253707i	$0.991923\pi$
$360$		0			0
$361$	37.4730			1.97226
$362$		0			0
$363$	−1.91873			−0.100707
$364$		0			0
$365$	−13.3201			−0.697206
$366$		0			0
$367$	−24.3851			−1.27289			−0.636447	−	0.771320i	$-0.719597\pi$
$367$	−24.3851			−1.27289			−0.636447	+	0.771320i	$0.719597\pi$
$368$		0			0
$369$		−	1.88370i		−	0.0980615i
$370$		0			0
$371$	−12.2422	−	23.4314i	−0.635585	−	1.21650i
$372$		0			0
$373$	0.0929709			0.00481385			0.00240693	−	0.999997i	$-0.499234\pi$
$373$	0.0929709			0.00481385			0.00240693	+	0.999997i	$0.499234\pi$
$374$		0			0
$375$		−	1.00000i		−	0.0516398i
$376$		0			0
$377$			34.9672i			1.80090i
$378$		0			0
$379$		−	4.61620i		−	0.237118i	−0.992947	−	0.118559i	$-0.962173\pi$
$379$		−	4.61620i		−	0.237118i	0.992947	−	0.118559i	$-0.0378275\pi$
$380$		0			0
$381$			8.10656i			0.415312i
$382$		0			0
$383$	−30.2544			−1.54593			−0.772964	−	0.634450i	$-0.781227\pi$
$383$	−30.2544			−1.54593			−0.772964	+	0.634450i	$0.781227\pi$
$384$		0			0
$385$	−3.69212	−	7.06662i	−0.188168	−	0.360149i
$386$		0			0
$387$			4.97184i			0.252733i
$388$		0			0
$389$	−3.40555			−0.172668			−0.0863341	−	0.996266i	$-0.527515\pi$
$389$	−3.40555			−0.172668			−0.0863341	+	0.996266i	$0.527515\pi$
$390$		0			0
$391$	−7.46018			−0.377278
$392$		0			0
$393$	15.0858			0.760976
$394$		0			0
$395$	−2.85035			−0.143417
$396$		0			0
$397$		−	12.2638i		−	0.615502i	−0.951467	−	0.307751i	$-0.900424\pi$
$397$		−	12.2638i		−	0.615502i	0.951467	−	0.307751i	$-0.0995764\pi$
$398$		0			0
$399$	9.20709	+	17.6222i	0.460931	+	0.882211i
$400$		0			0
$401$	−34.6619			−1.73093			−0.865466	−	0.500967i	$-0.832978\pi$
$401$	−34.6619			−1.73093			−0.865466	+	0.500967i	$0.832978\pi$
$402$		0			0
$403$		−	4.87021i		−	0.242603i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$		−	14.8288i		−	0.735037i
$408$		0			0
$409$		−	2.77424i		−	0.137177i	−0.997645	−	0.0685887i	$-0.978150\pi$
$409$		−	2.77424i		−	0.137177i	0.997645	−	0.0685887i	$-0.0218496\pi$
$410$		0			0
$411$	10.9952			0.542351
$412$		0			0
$413$	4.50366	+	8.61989i	0.221610	+	0.424157i
$414$		0			0
$415$		−	2.76948i		−	0.135949i
$416$		0			0
$417$	−2.72739			−0.133561
$418$		0			0
$419$	7.36977			0.360037			0.180018	−	0.983663i	$-0.442384\pi$
$419$	7.36977			0.360037			0.180018	+	0.983663i	$0.442384\pi$
$420$		0			0
$421$	−8.01423			−0.390590			−0.195295	−	0.980745i	$-0.562566\pi$
$421$	−8.01423			−0.390590			−0.195295	+	0.980745i	$0.562566\pi$
$422$		0			0
$423$	−1.84555			−0.0897337
$424$		0			0
$425$		−	2.48852i		−	0.120711i
$426$		0			0
$427$	−7.21635	+	3.77035i	−0.349224	+	0.182460i
$428$		0			0
$429$	20.0347			0.967283
$430$		0			0
$431$		−	7.91884i		−	0.381437i	−0.981645	−	0.190719i	$-0.938918\pi$
$431$		−	7.91884i		−	0.381437i	0.981645	−	0.190719i	$-0.0610818\pi$
$432$		0			0
$433$		−	29.5651i		−	1.42081i	−0.703795	−	0.710404i	$-0.748512\pi$
$433$		−	29.5651i		−	1.42081i	0.703795	−	0.710404i	$-0.251488\pi$
$434$		0			0
$435$		−	5.25959i		−	0.252178i
$436$		0			0
$437$		−	22.5283i		−	1.07767i
$438$		0			0
$439$	25.6701			1.22517			0.612584	−	0.790405i	$-0.290130\pi$
$439$	25.6701			1.22517			0.612584	+	0.790405i	$0.290130\pi$
$440$		0			0
$441$	−3.99784	+	5.74607i	−0.190373	+	0.273622i
$442$		0			0
$443$			3.16301i			0.150279i	0.997173	+	0.0751396i	$0.0239403\pi$
$443$			3.16301i			0.150279i	−0.997173	+	0.0751396i	$0.976060\pi$
$444$		0			0
$445$	10.1510			0.481203
$446$		0			0
$447$	−8.35592			−0.395222
$448$		0			0
$449$	−14.1823			−0.669305			−0.334652	−	0.942342i	$-0.608619\pi$
$449$	−14.1823			−0.669305			−0.334652	+	0.942342i	$0.608619\pi$
$450$		0			0
$451$	−5.67656			−0.267299
$452$		0			0
$453$		−	22.9354i		−	1.07760i
$454$		0			0
$455$	−8.14538	−	15.5901i	−0.381861	−	0.730873i
$456$		0			0
$457$	26.4174			1.23575			0.617877	−	0.786275i	$-0.287993\pi$
$457$	26.4174			1.23575			0.617877	+	0.786275i	$0.287993\pi$
$458$		0			0
$459$		−	2.48852i		−	0.116154i
$460$		0			0
$461$		−	22.4308i		−	1.04470i	−0.852730	−	0.522352i	$-0.825055\pi$
$461$		−	22.4308i		−	1.04470i	0.852730	−	0.522352i	$-0.174945\pi$
$462$		0			0
$463$		−	15.6005i		−	0.725018i	−0.931980	−	0.362509i	$-0.881920\pi$
$463$		−	15.6005i		−	0.725018i	0.931980	−	0.362509i	$-0.118080\pi$
$464$		0			0
$465$			0.732553i			0.0339713i
$466$		0			0
$467$	30.2072			1.39782			0.698912	−	0.715208i	$-0.253668\pi$
$467$	30.2072			1.39782			0.698912	+	0.715208i	$0.253668\pi$
$468$		0			0
$469$	−15.9675	+	8.34256i	−0.737309	+	0.385224i
$470$		0			0
$471$		−	7.01845i		−	0.323393i
$472$		0			0
$473$	14.9827			0.688905
$474$		0			0
$475$	7.51485			0.344805
$476$		0			0
$477$	−9.99214			−0.457509
$478$		0			0
$479$	−1.37576			−0.0628602			−0.0314301	−	0.999506i	$-0.510006\pi$
$479$	−1.37576			−0.0628602			−0.0314301	+	0.999506i	$0.510006\pi$
$480$		0			0
$481$		−	32.7146i		−	1.49166i
$482$		0			0
$483$	7.02986	−	3.67291i	0.319870	−	0.167123i
$484$		0			0
$485$	−18.6360			−0.846216
$486$		0			0
$487$			11.0089i			0.498860i	0.968393	+	0.249430i	$0.0802432\pi$
$487$			11.0089i			0.498860i	−0.968393	+	0.249430i	$0.919757\pi$
$488$		0			0
$489$			13.2184i			0.597757i
$490$		0			0
$491$		−	18.8450i		−	0.850461i	−0.905085	−	0.425231i	$-0.860193\pi$
$491$		−	18.8450i		−	0.850461i	0.905085	−	0.425231i	$-0.139807\pi$
$492$		0			0
$493$		−	13.0886i		−	0.589481i
$494$		0			0
$495$	−3.01352			−0.135447
$496$		0			0
$497$	−22.0109	+	11.5001i	−0.987324	+	0.515850i
$498$		0			0
$499$			22.2338i			0.995321i	0.867372	+	0.497661i	$0.165807\pi$
$499$			22.2338i			0.995321i	−0.867372	+	0.497661i	$0.834193\pi$
$500$		0			0
$501$	12.6407			0.564743
$502$		0			0
$503$	−9.58715			−0.427470			−0.213735	−	0.976892i	$-0.568563\pi$
$503$	−9.58715			−0.427470			−0.213735	+	0.976892i	$0.568563\pi$
$504$		0			0
$505$	−1.52559			−0.0678880
$506$		0			0
$507$	31.1996			1.38562
$508$		0			0
$509$			18.9857i			0.841525i	0.907171	+	0.420762i	$0.138237\pi$
$509$			18.9857i			0.841525i	−0.907171	+	0.420762i	$0.861763\pi$
$510$		0			0
$511$	31.2353	−	16.3196i	1.38177	−	0.721937i
$512$		0			0
$513$	7.51485			0.331789
$514$		0			0
$515$			6.45037i			0.284237i
$516$		0			0
$517$			5.56159i			0.244599i
$518$		0			0
$519$		−	4.72615i		−	0.207455i
$520$		0			0
$521$			31.3876i			1.37512i	0.726129	+	0.687559i	$0.241318\pi$
$521$			31.3876i			1.37512i	−0.726129	+	0.687559i	$0.758682\pi$
$522$		0			0
$523$	39.3671			1.72140			0.860701	−	0.509111i	$-0.170026\pi$
$523$	39.3671			1.72140			0.860701	+	0.509111i	$0.170026\pi$
$524$		0			0
$525$	1.22519	+	2.34498i	0.0534715	+	0.102343i
$526$		0			0
$527$			1.82297i			0.0794100i
$528$		0			0
$529$	14.0130			0.609260
$530$		0			0
$531$	3.67590			0.159520
$532$		0			0
$533$	−12.5234			−0.542447
$534$		0			0
$535$	16.7156			0.722679
$536$		0			0
$537$			13.3705i			0.576978i
$538$		0			0
$539$	17.3159	+	12.0475i	0.745847	+	0.518924i
$540$		0			0
$541$	45.7683			1.96773			0.983866	−	0.178909i	$-0.0572567\pi$
$541$	45.7683			1.96773			0.983866	+	0.178909i	$0.0572567\pi$
$542$		0			0
$543$		−	9.92238i		−	0.425810i
$544$		0			0
$545$			11.6688i			0.499838i
$546$		0			0
$547$		−	33.1241i		−	1.41628i	−0.706070	−	0.708142i	$-0.749534\pi$
$547$		−	33.1241i		−	1.41628i	0.706070	−	0.708142i	$-0.250466\pi$
$548$		0			0
$549$			3.07737i			0.131339i
$550$		0			0
$551$	39.5250			1.68382
$552$		0			0
$553$	6.68402	−	3.49222i	0.284233	−	0.148504i
$554$		0			0
$555$			4.92077i			0.208875i
$556$		0			0
$557$	2.31738			0.0981905			0.0490952	−	0.998794i	$-0.484366\pi$
$557$	2.31738			0.0981905			0.0490952	+	0.998794i	$0.484366\pi$
$558$		0			0
$559$	33.0541			1.39804
$560$		0			0
$561$	−7.49920			−0.316616
$562$		0			0
$563$	9.60252			0.404698			0.202349	−	0.979313i	$-0.435142\pi$
$563$	9.60252			0.404698			0.202349	+	0.979313i	$0.435142\pi$
$564$		0			0
$565$			15.0785i			0.634358i
$566$		0			0
$567$	1.22519	+	2.34498i	0.0514530	+	0.0984798i
$568$		0			0
$569$	2.96246			0.124193			0.0620965	−	0.998070i	$-0.480221\pi$
$569$	2.96246			0.124193			0.0620965	+	0.998070i	$0.480221\pi$
$570$		0			0
$571$		−	7.86321i		−	0.329065i	−0.986372	−	0.164533i	$-0.947388\pi$
$571$		−	7.86321i		−	0.329065i	0.986372	−	0.164533i	$-0.0526116\pi$
$572$		0			0
$573$		−	13.1111i		−	0.547726i
$574$		0			0
$575$		−	2.99784i		−	0.125018i
$576$		0			0
$577$			1.82981i			0.0761759i	0.999274	+	0.0380880i	$0.0121267\pi$
$577$			1.82981i			0.0761759i	−0.999274	+	0.0380880i	$0.987873\pi$
$578$		0			0
$579$	−6.03136			−0.250655
$580$		0			0
$581$	3.39313	+	6.49438i	0.140771	+	0.269432i
$582$		0			0
$583$			30.1115i			1.24709i
$584$		0			0
$585$	−6.64828			−0.274872
$586$		0			0
$587$	−40.4704			−1.67039			−0.835196	−	0.549952i	$-0.814646\pi$
$587$	−40.4704			−1.67039			−0.835196	+	0.549952i	$0.814646\pi$
$588$		0			0
$589$	−5.50503			−0.226831
$590$		0			0
$591$	21.2078			0.872373
$592$		0			0
$593$		−	12.3851i		−	0.508594i	−0.967126	−	0.254297i	$-0.918156\pi$
$593$		−	12.3851i		−	0.508594i	0.967126	−	0.254297i	$-0.0818441\pi$
$594$		0			0
$595$	3.04890	+	5.83553i	0.124993	+	0.239233i
$596$		0			0
$597$	1.91024			0.0781810
$598$		0			0
$599$			48.6164i			1.98641i	0.116362	+	0.993207i	$0.462877\pi$
$599$			48.6164i			1.98641i	−0.116362	+	0.993207i	$0.537123\pi$
$600$		0			0
$601$			8.86805i			0.361735i	0.983507	+	0.180868i	$0.0578906\pi$
$601$			8.86805i			0.361735i	−0.983507	+	0.180868i	$0.942109\pi$
$602$		0			0
$603$			6.80922i			0.277293i
$604$		0			0
$605$		−	1.91873i		−	0.0780073i
$606$		0			0
$607$	−37.4194			−1.51881			−0.759404	−	0.650620i	$-0.774509\pi$
$607$	−37.4194			−1.51881			−0.759404	+	0.650620i	$0.774509\pi$
$608$		0			0
$609$	6.44398	+	12.3336i	0.261123	+	0.499784i
$610$		0			0
$611$			12.2697i			0.496380i
$612$		0			0
$613$	−30.5402			−1.23351			−0.616754	−	0.787156i	$-0.711553\pi$
$613$	−30.5402			−1.23351			−0.616754	+	0.787156i	$0.711553\pi$
$614$		0			0
$615$	1.88370			0.0759581
$616$		0			0
$617$	18.2966			0.736595			0.368297	−	0.929708i	$-0.379941\pi$
$617$	18.2966			0.736595			0.368297	+	0.929708i	$0.379941\pi$
$618$		0			0
$619$	25.5761			1.02799			0.513994	−	0.857794i	$-0.328165\pi$
$619$	25.5761			1.02799			0.513994	+	0.857794i	$0.328165\pi$
$620$		0			0
$621$		−	2.99784i		−	0.120299i
$622$		0			0
$623$	−23.8038	+	12.4368i	−0.953680	+	0.498271i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	22.6461i		−	0.904399i
$628$		0			0
$629$			12.2454i			0.488258i
$630$		0			0
$631$		−	26.9779i		−	1.07397i	−0.843591	−	0.536987i	$-0.819562\pi$
$631$		−	26.9779i		−	1.07397i	0.843591	−	0.536987i	$-0.180438\pi$
$632$		0			0
$633$		−	11.2462i		−	0.446995i
$634$		0			0
$635$	−8.10656			−0.321699
$636$		0			0
$637$	38.2014	+	26.5787i	1.51360	+	1.05309i
$638$		0			0
$639$			9.38640i			0.371320i
$640$		0			0
$641$	32.6905			1.29120			0.645599	−	0.763676i	$-0.276608\pi$
$641$	32.6905			1.29120			0.645599	+	0.763676i	$0.276608\pi$
$642$		0			0
$643$	−15.2025			−0.599530			−0.299765	−	0.954013i	$-0.596908\pi$
$643$	−15.2025			−0.599530			−0.299765	+	0.954013i	$0.596908\pi$
$644$		0			0
$645$	−4.97184			−0.195766
$646$		0			0
$647$	23.9264			0.940643			0.470322	−	0.882495i	$-0.344138\pi$
$647$	23.9264			0.940643			0.470322	+	0.882495i	$0.344138\pi$
$648$		0			0
$649$		−	11.0774i		−	0.434825i
$650$		0			0
$651$	−0.897514	−	1.71782i	−0.0351763	−	0.0673267i
$652$		0			0
$653$	51.0823			1.99901			0.999503	−	0.0315391i	$-0.0100409\pi$
$653$	51.0823			1.99901			0.999503	+	0.0315391i	$0.0100409\pi$
$654$		0			0
$655$			15.0858i			0.589449i
$656$		0			0
$657$		−	13.3201i		−	0.519667i
$658$		0			0
$659$		−	34.1652i		−	1.33089i	−0.746447	−	0.665444i	$-0.768242\pi$
$659$		−	34.1652i		−	1.33089i	0.746447	−	0.665444i	$-0.231758\pi$
$660$		0			0
$661$			14.5921i			0.567566i	0.958889	+	0.283783i	$0.0915895\pi$
$661$			14.5921i			0.567566i	−0.958889	+	0.283783i	$0.908410\pi$
$662$		0			0
$663$	−16.5444			−0.642530
$664$		0			0
$665$	−17.6222	+	9.20709i	−0.683358	+	0.357036i
$666$		0			0
$667$		−	15.7674i		−	0.610516i
$668$		0			0
$669$	−5.06340			−0.195762
$670$		0			0
$671$	9.27369			0.358007
$672$		0			0
$673$	−15.2224			−0.586779			−0.293389	−	0.955993i	$-0.594783\pi$
$673$	−15.2224			−0.586779			−0.293389	+	0.955993i	$0.594783\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$			49.4639i			1.90105i	0.310647	+	0.950525i	$0.399454\pi$
$677$			49.4639i			1.90105i	−0.310647	+	0.950525i	$0.600546\pi$
$678$		0			0
$679$	43.7009	−	22.8325i	1.67709	−	0.876232i
$680$		0			0
$681$	19.3059			0.739803
$682$		0			0
$683$			28.3295i			1.08400i	0.840379	+	0.542000i	$0.182333\pi$
$683$			28.3295i			1.08400i	−0.840379	+	0.542000i	$0.817667\pi$
$684$		0			0
$685$			10.9952i			0.420104i
$686$		0			0
$687$		−	12.6515i		−	0.482683i
$688$		0			0
$689$			66.4305i			2.53080i
$690$		0			0
$691$	−39.8336			−1.51534			−0.757670	−	0.652638i	$-0.773662\pi$
$691$	−39.8336			−1.51534			−0.757670	+	0.652638i	$0.773662\pi$
$692$		0			0
$693$	7.06662	−	3.69212i	0.268439	−	0.140252i
$694$		0			0
$695$		−	2.72739i		−	0.103456i
$696$		0			0
$697$	4.68763			0.177557
$698$		0			0
$699$	−19.4858			−0.737022
$700$		0			0
$701$	22.1877			0.838019			0.419010	−	0.907982i	$-0.362377\pi$
$701$	22.1877			0.838019			0.419010	+	0.907982i	$0.362377\pi$
$702$		0			0
$703$	−36.9789			−1.39468
$704$		0			0
$705$		−	1.84555i		−	0.0695075i
$706$		0			0
$707$	3.57748	−	1.86914i	0.134545	−	0.0702961i
$708$		0			0
$709$	32.3649			1.21549			0.607745	−	0.794133i	$-0.292074\pi$
$709$	32.3649			1.21549			0.607745	+	0.794133i	$0.292074\pi$
$710$		0			0
$711$		−	2.85035i		−	0.106897i
$712$		0			0
$713$			2.19607i			0.0822436i
$714$		0			0
$715$			20.0347i			0.749254i
$716$		0			0
$717$		−	15.4526i		−	0.577088i
$718$		0			0
$719$	−23.4628			−0.875016			−0.437508	−	0.899214i	$-0.644139\pi$
$719$	−23.4628			−0.875016			−0.437508	+	0.899214i	$0.644139\pi$
$720$		0			0
$721$	−7.90291	−	15.1260i	−0.294320	−	0.563321i
$722$		0			0
$723$			18.0126i			0.669896i
$724$		0			0
$725$	5.25959			0.195336
$726$		0			0
$727$	51.1744			1.89795			0.948977	−	0.315346i	$-0.102121\pi$
$727$	51.1744			1.89795			0.948977	+	0.315346i	$0.102121\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−12.3725			−0.457614
$732$		0			0
$733$			50.4754i			1.86435i	0.362007	+	0.932175i	$0.382092\pi$
$733$			50.4754i			1.86435i	−0.362007	+	0.932175i	$0.617908\pi$
$734$		0			0
$735$	−5.74607	−	3.99784i	−0.211947	−	0.147462i
$736$		0			0
$737$	20.5197			0.755852
$738$		0			0
$739$		−	39.2426i		−	1.44356i	−0.692122	−	0.721781i	$-0.743324\pi$
$739$		−	39.2426i		−	1.44356i	0.692122	−	0.721781i	$-0.256676\pi$
$740$		0			0
$741$		−	49.9608i		−	1.83536i
$742$		0			0
$743$		−	16.7484i		−	0.614440i	−0.951639	−	0.307220i	$-0.900601\pi$
$743$		−	16.7484i		−	0.614440i	0.951639	−	0.307220i	$-0.0993987\pi$
$744$		0			0
$745$		−	8.35592i		−	0.306137i
$746$		0			0
$747$	2.76948			0.101330
$748$		0			0
$749$	−39.1978	+	20.4798i	−1.43225	+	0.748314i
$750$		0			0
$751$			28.5845i			1.04306i	0.853232	+	0.521531i	$0.174639\pi$
$751$			28.5845i			1.04306i	−0.853232	+	0.521531i	$0.825361\pi$
$752$		0			0
$753$	22.1128			0.805835
$754$		0			0
$755$	22.9354			0.834706
$756$		0			0
$757$	−33.8836			−1.23152			−0.615760	−	0.787933i	$-0.711151\pi$
$757$	−33.8836			−1.23152			−0.615760	+	0.787933i	$0.711151\pi$
$758$		0			0
$759$	−9.03403			−0.327914
$760$		0			0
$761$			12.4539i			0.451452i	0.974191	+	0.225726i	$0.0724755\pi$
$761$			12.4539i			0.451452i	−0.974191	+	0.225726i	$0.927525\pi$
$762$		0			0
$763$	−14.2965	−	27.3631i	−0.517568	−	0.990612i
$764$		0			0
$765$	2.48852			0.0899727
$766$		0			0
$767$		−	24.4384i		−	0.882418i
$768$		0			0
$769$		−	32.4983i		−	1.17192i	−0.810340	−	0.585960i	$-0.800717\pi$
$769$		−	32.4983i		−	1.17192i	0.810340	−	0.585960i	$-0.199283\pi$
$770$		0			0
$771$			2.14970i			0.0774196i
$772$		0			0
$773$		−	48.4978i		−	1.74434i	−0.489199	−	0.872172i	$-0.662711\pi$
$773$		−	48.4978i		−	1.74434i	0.489199	−	0.872172i	$-0.337289\pi$
$774$		0			0
$775$	−0.732553			−0.0263141
$776$		0			0
$777$	−6.02886	−	11.5391i	−0.216284	−	0.413963i
$778$		0			0
$779$			14.1557i			0.507181i
$780$		0			0
$781$	28.2861			1.01216
$782$		0			0
$783$	5.25959			0.187962
$784$		0			0
$785$	7.01845			0.250499
$786$		0			0
$787$	−10.4860			−0.373785			−0.186893	−	0.982380i	$-0.559842\pi$
$787$	−10.4860			−0.373785			−0.186893	+	0.982380i	$0.559842\pi$
$788$		0			0
$789$			23.2694i			0.828414i
$790$		0			0
$791$	−18.4740	−	35.3588i	−0.656860	−	1.25721i
$792$		0			0
$793$	20.4592			0.726527
$794$		0			0
$795$		−	9.99214i		−	0.354385i
$796$		0			0
$797$		−	16.1299i		−	0.571351i	−0.958326	−	0.285676i	$-0.907782\pi$
$797$		−	16.1299i		−	0.571351i	0.958326	−	0.285676i	$-0.0922180\pi$
$798$		0			0
$799$		−	4.59269i		−	0.162478i
$800$		0			0
$801$			10.1510i			0.358667i
$802$		0			0
$803$	−40.1403			−1.41652
$804$		0			0
$805$	3.67291	+	7.02986i	0.129453	+	0.247770i
$806$		0			0
$807$		−	3.36237i		−	0.118361i
$808$		0			0
$809$	53.0829			1.86630			0.933148	−	0.359492i	$-0.117050\pi$
$809$	53.0829			1.86630			0.933148	+	0.359492i	$0.117050\pi$
$810$		0			0
$811$	−49.3864			−1.73419			−0.867096	−	0.498141i	$-0.834016\pi$
$811$	−49.3864			−1.73419			−0.867096	+	0.498141i	$0.834016\pi$
$812$		0			0
$813$	4.88011			0.171153
$814$		0			0
$815$	−13.2184			−0.463020
$816$		0			0
$817$		−	37.3626i		−	1.30715i
$818$		0			0
$819$	15.5901	−	8.14538i	0.544761	−	0.284622i
$820$		0			0
$821$	36.7324			1.28197			0.640984	−	0.767554i	$-0.278526\pi$
$821$	36.7324			1.28197			0.640984	+	0.767554i	$0.278526\pi$
$822$		0			0
$823$		−	29.2450i		−	1.01942i	−0.860347	−	0.509709i	$-0.829753\pi$
$823$		−	29.2450i		−	1.01942i	0.860347	−	0.509709i	$-0.170247\pi$
$824$		0			0
$825$		−	3.01352i		−	0.104917i
$826$		0			0
$827$		−	30.0895i		−	1.04632i	−0.852236	−	0.523158i	$-0.824754\pi$
$827$		−	30.0895i		−	1.04632i	0.852236	−	0.523158i	$-0.175246\pi$
$828$		0			0
$829$			33.7433i			1.17195i	0.810327	+	0.585977i	$0.199289\pi$
$829$			33.7433i			1.17195i	−0.810327	+	0.585977i	$0.800711\pi$
$830$		0			0
$831$	26.2978			0.912259
$832$		0			0
$833$	−14.2992	−	9.94870i	−0.495438	−	0.344702i
$834$		0			0
$835$			12.6407i			0.437448i
$836$		0			0
$837$	−0.732553			−0.0253207
$838$		0			0
$839$	20.4511			0.706052			0.353026	−	0.935614i	$-0.385153\pi$
$839$	20.4511			0.706052			0.353026	+	0.935614i	$0.385153\pi$
$840$		0			0
$841$	−1.33670			−0.0460931
$842$		0			0
$843$	8.18742			0.281990
$844$		0			0
$845$			31.1996i			1.07330i
$846$		0			0
$847$	2.35080	+	4.49937i	0.0807743	+	0.154600i
$848$		0			0
$849$	−13.4183			−0.460516
$850$		0			0
$851$			14.7517i			0.505681i
$852$		0			0
$853$		−	27.5322i		−	0.942685i	−0.881950	−	0.471342i	$-0.843770\pi$
$853$		−	27.5322i		−	0.942685i	0.881950	−	0.471342i	$-0.156230\pi$
$854$		0			0
$855$			7.51485i			0.257002i
$856$		0			0
$857$			2.62457i			0.0896537i	0.998995	+	0.0448269i	$0.0142736\pi$
$857$			2.62457i			0.0896537i	−0.998995	+	0.0448269i	$0.985726\pi$
$858$		0			0
$859$	40.7537			1.39050			0.695249	−	0.718769i	$-0.255294\pi$
$859$	40.7537			1.39050			0.695249	+	0.718769i	$0.255294\pi$
$860$		0			0
$861$	−4.41723	+	2.30788i	−0.150539	+	0.0786524i
$862$		0			0
$863$		−	25.6679i		−	0.873746i	−0.899523	−	0.436873i	$-0.856086\pi$
$863$		−	25.6679i		−	0.873746i	0.899523	−	0.436873i	$-0.143914\pi$
$864$		0			0
$865$	4.72615			0.160694
$866$		0			0
$867$	−10.8073			−0.367034
$868$		0			0
$869$	−8.58959			−0.291382
$870$		0			0
$871$	45.2696			1.53390
$872$		0			0
$873$		−	18.6360i		−	0.630732i
$874$		0			0
$875$	−2.34498	+	1.22519i	−0.0792747	+	0.0414189i
$876$		0			0
$877$	30.8851			1.04292			0.521458	−	0.853277i	$-0.325388\pi$
$877$	30.8851			1.04292			0.521458	+	0.853277i	$0.325388\pi$
$878$		0			0
$879$		−	23.0697i		−	0.778122i
$880$		0			0
$881$			24.6362i			0.830013i	0.909818	+	0.415007i	$0.136221\pi$
$881$			24.6362i			0.830013i	−0.909818	+	0.415007i	$0.863779\pi$
$882$		0			0
$883$		−	17.5506i		−	0.590624i	−0.955401	−	0.295312i	$-0.904576\pi$
$883$		−	17.5506i		−	0.590624i	0.955401	−	0.295312i	$-0.0954237\pi$
$884$		0			0
$885$			3.67590i			0.123564i
$886$		0			0
$887$	17.4618			0.586311			0.293155	−	0.956065i	$-0.405295\pi$
$887$	17.4618			0.586311			0.293155	+	0.956065i	$0.405295\pi$
$888$		0			0
$889$	19.0097	−	9.93205i	0.637565	−	0.333110i
$890$		0			0
$891$		−	3.01352i		−	0.100957i
$892$		0			0
$893$	13.8690			0.464110
$894$		0			0
$895$	−13.3705			−0.446925
$896$		0			0
$897$	−19.9304			−0.665458
$898$		0			0
$899$	−3.85293			−0.128502
$900$		0			0
$901$		−	24.8657i		−	0.828395i
$902$		0			0
$903$	11.6588	−	6.09142i	0.387982	−	0.202710i
$904$		0			0
$905$	9.92238			0.329831
$906$		0			0
$907$			46.7175i			1.55123i	0.631207	+	0.775614i	$0.282560\pi$
$907$			46.7175i			1.55123i	−0.631207	+	0.775614i	$0.717440\pi$
$908$		0			0
$909$		−	1.52559i		−	0.0506007i
$910$		0			0
$911$			20.2675i			0.671493i	0.941952	+	0.335747i	$0.108989\pi$
$911$			20.2675i			0.671493i	−0.941952	+	0.335747i	$0.891011\pi$
$912$		0			0
$913$		−	8.34588i		−	0.276208i
$914$		0			0
$915$	−3.07737			−0.101735
$916$		0			0
$917$	−18.4829	−	35.3758i	−0.610358	−	1.16821i
$918$		0			0
$919$		−	11.5972i		−	0.382557i	−0.981536	−	0.191278i	$-0.938737\pi$
$919$		−	11.5972i		−	0.382557i	0.981536	−	0.191278i	$-0.0612633\pi$
$920$		0			0
$921$	13.4270			0.442435
$922$		0			0
$923$	62.4034			2.05403
$924$		0			0
$925$	−4.92077			−0.161794
$926$		0			0
$927$	−6.45037			−0.211858
$928$		0			0
$929$			57.8656i			1.89851i	0.314511	+	0.949254i	$0.398159\pi$
$929$			57.8656i			1.89851i	−0.314511	+	0.949254i	$0.601841\pi$
$930$		0			0
$931$	30.0432	−	43.1808i	0.984625	−	1.41520i
$932$		0			0
$933$	23.9720			0.784809
$934$		0			0
$935$		−	7.49920i		−	0.245250i
$936$		0			0
$937$		−	0.668351i		−	0.0218341i	−0.999940	−	0.0109170i	$-0.996525\pi$
$937$		−	0.668351i		−	0.0218341i	0.999940	−	0.0109170i	$-0.00347507\pi$
$938$		0			0
$939$		−	2.20964i		−	0.0721090i
$940$		0			0
$941$			17.4343i			0.568344i	0.958773	+	0.284172i	$0.0917186\pi$
$941$			17.4343i			0.568344i	−0.958773	+	0.284172i	$0.908281\pi$
$942$		0			0
$943$	5.64702			0.183892
$944$		0			0
$945$	−2.34498	+	1.22519i	−0.0762821	+	0.0398553i
$946$		0			0
$947$		−	34.7083i		−	1.12787i	−0.825819	−	0.563935i	$-0.809287\pi$
$947$		−	34.7083i		−	1.12787i	0.825819	−	0.563935i	$-0.190713\pi$
$948$		0			0
$949$	−88.5557			−2.87464
$950$		0			0
$951$	0.251931			0.00816943
$952$		0			0
$953$	−1.76769			−0.0572612			−0.0286306	−	0.999590i	$-0.509115\pi$
$953$	−1.76769			−0.0572612			−0.0286306	+	0.999590i	$0.509115\pi$
$954$		0			0
$955$	13.1111			0.424267
$956$		0			0
$957$		−	15.8499i		−	0.512353i
$958$		0			0
$959$	−13.4711	−	25.7834i	−0.435005	−	0.832590i
$960$		0			0
$961$	−30.4634			−0.982689
$962$		0			0
$963$			16.7156i			0.538653i
$964$		0			0
$965$		−	6.03136i		−	0.194156i
$966$		0			0
$967$			3.46815i			0.111528i	0.998444	+	0.0557642i	$0.0177595\pi$
$967$			3.46815i			0.111528i	−0.998444	+	0.0557642i	$0.982241\pi$
$968$		0			0
$969$			18.7009i			0.600758i
$970$		0			0
$971$	−26.9721			−0.865575			−0.432787	−	0.901496i	$-0.642470\pi$
$971$	−26.9721			−0.865575			−0.432787	+	0.901496i	$0.642470\pi$
$972$		0			0
$973$	3.34156	+	6.39566i	0.107125	+	0.205035i
$974$		0			0
$975$		−	6.64828i		−	0.212915i
$976$		0			0
$977$	−21.9486			−0.702197			−0.351098	−	0.936339i	$-0.614192\pi$
$977$	−21.9486			−0.702197			−0.351098	+	0.936339i	$0.614192\pi$
$978$		0			0
$979$	30.5901			0.977665
$980$		0			0
$981$	−11.6688			−0.372557
$982$		0			0
$983$	−28.6929			−0.915161			−0.457580	−	0.889168i	$-0.651284\pi$
$983$	−28.6929			−0.915161			−0.457580	+	0.889168i	$0.651284\pi$
$984$		0			0
$985$			21.2078i			0.675737i
$986$		0			0
$987$	2.26114	+	4.32777i	0.0719730	+	0.137755i
$988$		0			0
$989$	−14.9048			−0.473944
$990$		0			0
$991$		−	20.0886i		−	0.638136i	−0.947732	−	0.319068i	$-0.896630\pi$
$991$		−	20.0886i		−	0.638136i	0.947732	−	0.319068i	$-0.103370\pi$
$992$		0			0
$993$			23.2542i			0.737950i
$994$		0			0
$995$			1.91024i			0.0605588i
$996$		0			0
$997$			36.9212i			1.16930i	0.811284	+	0.584652i	$0.198769\pi$
$997$			36.9212i			1.16930i	−0.811284	+	0.584652i	$0.801231\pi$
$998$		0			0
$999$	−4.92077			−0.155686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.d.a.2911.6	✓	16
4.3	odd	2		3360.2.d.d.2911.3	yes	16
7.6	odd	2		3360.2.d.d.2911.11	yes	16
28.27	even	2	inner	3360.2.d.a.2911.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.d.a.2911.6	✓	16	1.1	even	1	trivial
3360.2.d.a.2911.14	yes	16	28.27	even	2	inner
3360.2.d.d.2911.3	yes	16	4.3	odd	2
3360.2.d.d.2911.11	yes	16	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000i q^{5} +(1.22519 + 2.34498i) q^{7} +1.00000 q^{9} -3.01352i q^{11} -6.64828i q^{13} +1.00000i q^{15} +2.48852i q^{17} -7.51485 q^{19} +(-1.22519 - 2.34498i) q^{21} +2.99784i q^{23} -1.00000 q^{25} -1.00000 q^{27} -5.25959 q^{29} +0.732553 q^{31} +3.01352i q^{33} +(2.34498 - 1.22519i) q^{35} +4.92077 q^{37} +6.64828i q^{39} -1.88370i q^{41} +4.97184i q^{43} -1.00000i q^{45} -1.84555 q^{47} +(-3.99784 + 5.74607i) q^{49} -2.48852i q^{51} -9.99214 q^{53} -3.01352 q^{55} +7.51485 q^{57} +3.67590 q^{59} +3.07737i q^{61} +(1.22519 + 2.34498i) q^{63} -6.64828 q^{65} +6.80922i q^{67} -2.99784i q^{69} +9.38640i q^{71} -13.3201i q^{73} +1.00000 q^{75} +(7.06662 - 3.69212i) q^{77} -2.85035i q^{79} +1.00000 q^{81} +2.76948 q^{83} +2.48852 q^{85} +5.25959 q^{87} +10.1510i q^{89} +(15.5901 - 8.14538i) q^{91} -0.732553 q^{93} +7.51485i q^{95} -18.6360i q^{97} -3.01352i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 16 q^{9} + 16 q^{19} - 16 q^{25} - 16 q^{27} + 8 q^{29} - 8 q^{31} - 8 q^{37} + 16 q^{47} - 16 q^{49} - 48 q^{53} + 8 q^{55} - 16 q^{57} + 16 q^{59} - 8 q^{65} + 16 q^{75} + 16 q^{77}+ \cdots + 8 q^{93}+O(q^{100}) \) 16 * q - 16 * q^3 + 16 * q^9 + 16 * q^19 - 16 * q^25 - 16 * q^27 + 8 * q^29 - 8 * q^31 - 8 * q^37 + 16 * q^47 - 16 * q^49 - 48 * q^53 + 8 * q^55 - 16 * q^57 + 16 * q^59 - 8 * q^65 + 16 * q^75 + 16 * q^77 + 16 * q^81 - 24 * q^83 + 8 * q^85 - 8 * q^87 + 24 * q^91 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more