LMFDB - Embedded newform 1680.2.ba.d.911.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(911,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.911"); 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.ba (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-10,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4148675396$
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} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 640 x^{12} - 1656 x^{11} + 3522 x^{10} - 6148 x^{9} + \cdots + 13 $ x^16 - 8x^15 + 48x^14 - 196x^13 + 640x^12 - 1656x^11 + 3522x^10 - 6148x^9 + 8856x^8 - 10460x^7 + 10032x^6 - 7680x^5 + 4586x^4 - 2062x^3 + 659x^2 - 134*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			911.4
Root			$0.500000 + 2.12118i$ of defining polynomial
Character	$\chi$	$=$	1680.911
Dual form			1680.2.ba.d.911.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.18129 + 1.26671i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(-0.209100 - 2.99270i) q^{9} +5.47573 q^{11} -3.61183 q^{13} +(-1.26671 - 1.18129i) q^{15} +4.22648i q^{17} -7.43110i q^{19} +(1.26671 + 1.18129i) q^{21} -0.817252 q^{23} -1.00000 q^{25} +(4.03789 + 3.27039i) q^{27} -6.76900i q^{29} -7.87478i q^{31} +(-6.46843 + 6.93615i) q^{33} +1.00000 q^{35} +1.59343 q^{37} +(4.26663 - 4.57514i) q^{39} +7.57715i q^{41} -10.1778i q^{43} +(2.99270 - 0.209100i) q^{45} -0.833284 q^{47} -1.00000 q^{49} +(-5.35372 - 4.99270i) q^{51} -4.92079i q^{53} +5.47573i q^{55} +(9.41304 + 8.77830i) q^{57} +5.74788 q^{59} -2.84267 q^{61} +(-2.99270 + 0.209100i) q^{63} -3.61183i q^{65} +0.159787i q^{67} +(0.965413 - 1.03522i) q^{69} +9.22757 q^{71} +15.8610 q^{73} +(1.18129 - 1.26671i) q^{75} -5.47573i q^{77} +11.3527i q^{79} +(-8.91255 + 1.25155i) q^{81} +11.3286 q^{83} -4.22648 q^{85} +(8.57434 + 7.99616i) q^{87} -9.93325i q^{89} +3.61183i q^{91} +(9.97505 + 9.30241i) q^{93} +7.43110 q^{95} -8.44229 q^{97} +(-1.14497 - 16.3872i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 10 q^{9} + 12 q^{11} - 2 q^{15} + 2 q^{21} + 8 q^{23} - 16 q^{25} - 12 q^{27} + 12 q^{33} + 16 q^{35} - 8 q^{37} - 14 q^{39} + 8 q^{45} + 16 q^{47} - 16 q^{49} - 34 q^{51} + 12 q^{57} + 32 q^{59}+ \cdots - 50 q^{99}+O(q^{100}) $ 16 * q - 10 * q^9 + 12 * q^11 - 2 * q^15 + 2 * q^21 + 8 * q^23 - 16 * q^25 - 12 * q^27 + 12 * q^33 + 16 * q^35 - 8 * q^37 - 14 * q^39 + 8 * q^45 + 16 * q^47 - 16 * q^49 - 34 * q^51 + 12 * q^57 + 32 * q^59 + 8 * q^61 - 8 * q^63 - 20 * q^69 - 32 * q^71 + 48 * q^73 - 22 * q^81 + 72 * q^83 - 12 * q^85 - 44 * q^87 + 28 * q^93 + 8 * q^95 - 8 * q^97 - 50 * q^99

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
$3$	−1.18129	+	1.26671i	−0.682019	+	0.731334i
$3$	−1.18129	+	1.26671i	−0.682019	+	0.731334i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−0.209100	−	2.99270i	−0.0696999	−	0.997568i
$10$		0			0
$11$	5.47573			1.65099			0.825497	−	0.564407i	$-0.190895\pi$
$11$	5.47573			1.65099			0.825497	+	0.564407i	$0.190895\pi$
$12$		0			0
$13$	−3.61183			−1.00174			−0.500871	−	0.865522i	$-0.666987\pi$
$13$	−3.61183			−1.00174			−0.500871	+	0.865522i	$0.666987\pi$
$14$		0			0
$15$	−1.26671	−	1.18129i	−0.327063	−	0.305008i
$16$		0			0
$17$			4.22648i			1.02507i	0.858666	+	0.512536i	$0.171294\pi$
$17$			4.22648i			1.02507i	−0.858666	+	0.512536i	$0.828706\pi$
$18$		0			0
$19$		−	7.43110i		−	1.70481i	−0.522880	−	0.852406i	$-0.675143\pi$
$19$		−	7.43110i		−	1.70481i	0.522880	−	0.852406i	$-0.324857\pi$
$20$		0			0
$21$	1.26671	+	1.18129i	0.276418	+	0.257779i
$22$		0			0
$23$	−0.817252			−0.170409			−0.0852044	−	0.996363i	$-0.527154\pi$
$23$	−0.817252			−0.170409			−0.0852044	+	0.996363i	$0.527154\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	4.03789	+	3.27039i	0.777092	+	0.629386i
$28$		0			0
$29$		−	6.76900i		−	1.25697i	−0.777821	−	0.628485i	$-0.783675\pi$
$29$		−	6.76900i		−	1.25697i	0.777821	−	0.628485i	$-0.216325\pi$
$30$		0			0
$31$		−	7.87478i		−	1.41435i	−0.707037	−	0.707176i	$-0.749969\pi$
$31$		−	7.87478i		−	1.41435i	0.707037	−	0.707176i	$-0.250031\pi$
$32$		0			0
$33$	−6.46843	+	6.93615i	−1.12601	+	1.20743i
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	1.59343			0.261958			0.130979	−	0.991385i	$-0.458188\pi$
$37$	1.59343			0.261958			0.130979	+	0.991385i	$0.458188\pi$
$38$		0			0
$39$	4.26663	−	4.57514i	0.683207	−	0.732608i
$40$		0			0
$41$			7.57715i			1.18335i	0.806176	+	0.591676i	$0.201533\pi$
$41$			7.57715i			1.18335i	−0.806176	+	0.591676i	$0.798467\pi$
$42$		0			0
$43$		−	10.1778i		−	1.55210i	−0.630672	−	0.776049i	$-0.717221\pi$
$43$		−	10.1778i		−	1.55210i	0.630672	−	0.776049i	$-0.282779\pi$
$44$		0			0
$45$	2.99270	−	0.209100i	0.446126	−	0.0311707i
$46$		0			0
$47$	−0.833284			−0.121547			−0.0607735	−	0.998152i	$-0.519357\pi$
$47$	−0.833284			−0.121547			−0.0607735	+	0.998152i	$0.519357\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−5.35372	−	4.99270i	−0.749670	−	0.699118i
$52$		0			0
$53$		−	4.92079i		−	0.675922i	−0.941160	−	0.337961i	$-0.890263\pi$
$53$		−	4.92079i		−	0.675922i	0.941160	−	0.337961i	$-0.109737\pi$
$54$		0			0
$55$			5.47573i			0.738347i
$56$		0			0
$57$	9.41304	+	8.77830i	1.24679	+	1.16271i
$58$		0			0
$59$	5.74788			0.748311			0.374155	−	0.927366i	$-0.377933\pi$
$59$	5.74788			0.748311			0.374155	+	0.927366i	$0.377933\pi$
$60$		0			0
$61$	−2.84267			−0.363967			−0.181984	−	0.983302i	$-0.558252\pi$
$61$	−2.84267			−0.363967			−0.181984	+	0.983302i	$0.558252\pi$
$62$		0			0
$63$	−2.99270	+	0.209100i	−0.377045	+	0.0263441i
$64$		0			0
$65$		−	3.61183i		−	0.447993i
$66$		0			0
$67$			0.159787i			0.0195210i	0.999952	+	0.00976052i	$0.00310692\pi$
$67$			0.159787i			0.0195210i	−0.999952	+	0.00976052i	$0.996893\pi$
$68$		0			0
$69$	0.965413	−	1.03522i	0.116222	−	0.124626i
$70$		0			0
$71$	9.22757			1.09511			0.547555	−	0.836769i	$-0.315559\pi$
$71$	9.22757			1.09511			0.547555	+	0.836769i	$0.315559\pi$
$72$		0			0
$73$	15.8610			1.85639			0.928196	−	0.372093i	$-0.121360\pi$
$73$	15.8610			1.85639			0.928196	+	0.372093i	$0.121360\pi$
$74$		0			0
$75$	1.18129	−	1.26671i	0.136404	−	0.146267i
$76$		0			0
$77$		−	5.47573i		−	0.624017i
$78$		0			0
$79$			11.3527i			1.27728i	0.769504	+	0.638642i	$0.220503\pi$
$79$			11.3527i			1.27728i	−0.769504	+	0.638642i	$0.779497\pi$
$80$		0			0
$81$	−8.91255	+	1.25155i	−0.990284	+	0.139061i
$82$		0			0
$83$	11.3286			1.24348			0.621739	−	0.783224i	$-0.286426\pi$
$83$	11.3286			1.24348			0.621739	+	0.783224i	$0.286426\pi$
$84$		0			0
$85$	−4.22648			−0.458426
$86$		0			0
$87$	8.57434	+	7.99616i	0.919266	+	0.857278i
$88$		0			0
$89$		−	9.93325i		−	1.05292i	−0.850199	−	0.526461i	$-0.823519\pi$
$89$		−	9.93325i		−	1.05292i	0.850199	−	0.526461i	$-0.176481\pi$
$90$		0			0
$91$			3.61183i			0.378623i
$92$		0			0
$93$	9.97505	+	9.30241i	1.03436	+	0.964615i
$94$		0			0
$95$	7.43110			0.762415
$96$		0			0
$97$	−8.44229			−0.857185			−0.428592	−	0.903498i	$-0.640990\pi$
$97$	−8.44229			−0.857185			−0.428592	+	0.903498i	$0.640990\pi$
$98$		0			0
$99$	−1.14497	−	16.3872i	−0.115074	−	1.64698i
$100$		0			0
$101$			7.07147i			0.703638i	0.936068	+	0.351819i	$0.114437\pi$
$101$			7.07147i			0.703638i	−0.936068	+	0.351819i	$0.885563\pi$
$102$		0			0
$103$			13.3172i			1.31218i	0.754681	+	0.656092i	$0.227792\pi$
$103$			13.3172i			1.31218i	−0.754681	+	0.656092i	$0.772208\pi$
$104$		0			0
$105$	−1.18129	+	1.26671i	−0.115282	+	0.123618i
$106$		0			0
$107$	18.5906			1.79723			0.898613	−	0.438743i	$-0.144576\pi$
$107$	18.5906			1.79723			0.898613	+	0.438743i	$0.144576\pi$
$108$		0			0
$109$	15.6589			1.49985			0.749925	−	0.661523i	$-0.230090\pi$
$109$	15.6589			1.49985			0.749925	+	0.661523i	$0.230090\pi$
$110$		0			0
$111$	−1.88230	+	2.01841i	−0.178660	+	0.191579i
$112$		0			0
$113$			3.93263i			0.369951i	0.982743	+	0.184975i	$0.0592205\pi$
$113$			3.93263i			0.369951i	−0.982743	+	0.184975i	$0.940779\pi$
$114$		0			0
$115$		−	0.817252i		−	0.0762092i
$116$		0			0
$117$	0.755233	+	10.8091i	0.0698213	+	0.999306i
$118$		0			0
$119$	4.22648			0.387441
$120$		0			0
$121$	18.9836			1.72578
$122$		0			0
$123$	−9.59804	−	8.95082i	−0.865426	−	0.807068i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	14.1243i		−	1.25333i	−0.779290	−	0.626663i	$-0.784420\pi$
$127$		−	14.1243i		−	1.25333i	0.779290	−	0.626663i	$-0.215580\pi$
$128$		0			0
$129$	12.8923	+	12.0229i	1.13510	+	1.05856i
$130$		0			0
$131$	−15.0358			−1.31368			−0.656840	−	0.754030i	$-0.728107\pi$
$131$	−15.0358			−1.31368			−0.656840	+	0.754030i	$0.728107\pi$
$132$		0			0
$133$	−7.43110			−0.644358
$134$		0			0
$135$	−3.27039	+	4.03789i	−0.281470	+	0.347526i
$136$		0			0
$137$			5.41909i			0.462984i	0.972837	+	0.231492i	$0.0743607\pi$
$137$			5.41909i			0.462984i	−0.972837	+	0.231492i	$0.925639\pi$
$138$		0			0
$139$		−	4.50851i		−	0.382407i	−0.981550	−	0.191203i	$-0.938761\pi$
$139$		−	4.50851i		−	0.382407i	0.981550	−	0.191203i	$-0.0612390\pi$
$140$		0			0
$141$	0.984351	−	1.05553i	0.0828973	−	0.0888915i
$142$		0			0
$143$	−19.7774			−1.65387
$144$		0			0
$145$	6.76900			0.562134
$146$		0			0
$147$	1.18129	−	1.26671i	0.0974313	−	0.104476i
$148$		0			0
$149$		−	5.25864i		−	0.430804i	−0.976525	−	0.215402i	$-0.930894\pi$
$149$		−	5.25864i		−	0.430804i	0.976525	−	0.215402i	$-0.0691062\pi$
$150$		0			0
$151$		−	1.58998i		−	0.129391i	−0.997905	−	0.0646955i	$-0.979392\pi$
$151$		−	1.58998i		−	0.129391i	0.997905	−	0.0646955i	$-0.0206076\pi$
$152$		0			0
$153$	12.6486	−	0.883756i	1.02258	−	0.0714474i
$154$		0			0
$155$	7.87478			0.632518
$156$		0			0
$157$	−17.1358			−1.36759			−0.683794	−	0.729675i	$-0.739671\pi$
$157$	−17.1358			−1.36759			−0.683794	+	0.729675i	$0.739671\pi$
$158$		0			0
$159$	6.23320	+	5.81289i	0.494325	+	0.460992i
$160$		0			0
$161$			0.817252i			0.0644085i
$162$		0			0
$163$		−	11.9313i		−	0.934534i	−0.884116	−	0.467267i	$-0.845239\pi$
$163$		−	11.9313i		−	0.934534i	0.884116	−	0.467267i	$-0.154761\pi$
$164$		0			0
$165$	−6.93615	−	6.46843i	−0.539978	−	0.503567i
$166$		0			0
$167$	−1.99274			−0.154203			−0.0771014	−	0.997023i	$-0.524567\pi$
$167$	−1.99274			−0.154203			−0.0771014	+	0.997023i	$0.524567\pi$
$168$		0			0
$169$	0.0453320			0.00348707
$170$		0			0
$171$	−22.2391	+	1.55384i	−1.70067	+	0.118825i
$172$		0			0
$173$			4.01054i			0.304915i	0.988310	+	0.152458i	$0.0487188\pi$
$173$			4.01054i			0.304915i	−0.988310	+	0.152458i	$0.951281\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$	−6.78993	+	7.28089i	−0.510362	+	0.547265i
$178$		0			0
$179$	−7.24176			−0.541274			−0.270637	−	0.962681i	$-0.587234\pi$
$179$	−7.24176			−0.541274			−0.270637	+	0.962681i	$0.587234\pi$
$180$		0			0
$181$	−0.749663			−0.0557220			−0.0278610	−	0.999612i	$-0.508870\pi$
$181$	−0.749663			−0.0557220			−0.0278610	+	0.999612i	$0.508870\pi$
$182$		0			0
$183$	3.35803	−	3.60084i	0.248233	−	0.266182i
$184$		0			0
$185$			1.59343i			0.117151i
$186$		0			0
$187$			23.1430i			1.69239i
$188$		0			0
$189$	3.27039	−	4.03789i	0.237886	−	0.293713i
$190$		0			0
$191$	18.7456			1.35639			0.678194	−	0.734883i	$-0.262763\pi$
$191$	18.7456			1.35639			0.678194	+	0.734883i	$0.262763\pi$
$192$		0			0
$193$	7.33556			0.528025			0.264013	−	0.964519i	$-0.414954\pi$
$193$	7.33556			0.528025			0.264013	+	0.964519i	$0.414954\pi$
$194$		0			0
$195$	4.57514	+	4.26663i	0.327632	+	0.305540i
$196$		0			0
$197$		−	27.0541i		−	1.92753i	−0.266758	−	0.963764i	$-0.585952\pi$
$197$		−	27.0541i		−	1.92753i	0.266758	−	0.963764i	$-0.414048\pi$
$198$		0			0
$199$		−	20.1432i		−	1.42791i	−0.700190	−	0.713956i	$-0.746902\pi$
$199$		−	20.1432i		−	1.42791i	0.700190	−	0.713956i	$-0.253098\pi$
$200$		0			0
$201$	−0.202403	−	0.188755i	−0.0142764	−	0.0133137i
$202$		0			0
$203$	−6.76900			−0.475090
$204$		0			0
$205$	−7.57715			−0.529211
$206$		0			0
$207$	0.170887	+	2.44579i	0.0118775	+	0.169994i
$208$		0			0
$209$		−	40.6907i		−	2.81463i
$210$		0			0
$211$		−	1.94424i		−	0.133847i	−0.997758	−	0.0669236i	$-0.978682\pi$
$211$		−	1.94424i		−	0.133847i	0.997758	−	0.0669236i	$-0.0213184\pi$
$212$		0			0
$213$	−10.9004	+	11.6886i	−0.746886	+	0.800892i
$214$		0			0
$215$	10.1778			0.694120
$216$		0			0
$217$	−7.87478			−0.534575
$218$		0			0
$219$	−18.7365	+	20.0913i	−1.26609	+	1.35764i
$220$		0			0
$221$		−	15.2653i		−	1.02686i
$222$		0			0
$223$		−	17.8213i		−	1.19341i	−0.802462	−	0.596703i	$-0.796477\pi$
$223$		−	17.8213i		−	1.19341i	0.802462	−	0.596703i	$-0.203523\pi$
$224$		0			0
$225$	0.209100	+	2.99270i	0.0139400	+	0.199514i
$226$		0			0
$227$	6.61057			0.438759			0.219380	−	0.975640i	$-0.429597\pi$
$227$	6.61057			0.438759			0.219380	+	0.975640i	$0.429597\pi$
$228$		0			0
$229$	6.37119			0.421020			0.210510	−	0.977592i	$-0.432487\pi$
$229$	6.37119			0.421020			0.210510	+	0.977592i	$0.432487\pi$
$230$		0			0
$231$	6.93615	+	6.46843i	0.456365	+	0.425592i
$232$		0			0
$233$			16.7228i			1.09555i	0.836627	+	0.547773i	$0.184524\pi$
$233$			16.7228i			1.09555i	−0.836627	+	0.547773i	$0.815476\pi$
$234$		0			0
$235$		−	0.833284i		−	0.0543574i
$236$		0			0
$237$	−14.3806	−	13.4109i	−0.934121	−	0.871132i
$238$		0			0
$239$	−14.0396			−0.908149			−0.454075	−	0.890964i	$-0.650030\pi$
$239$	−14.0396			−0.908149			−0.454075	+	0.890964i	$0.650030\pi$
$240$		0			0
$241$	27.5217			1.77283			0.886414	−	0.462894i	$-0.153189\pi$
$241$	27.5217			1.77283			0.886414	+	0.462894i	$0.153189\pi$
$242$		0			0
$243$	8.94298	−	12.7680i	0.573693	−	0.819071i
$244$		0			0
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$			26.8399i			1.70778i
$248$		0			0
$249$	−13.3824	+	14.3501i	−0.848076	+	0.909399i
$250$		0			0
$251$	−7.29441			−0.460419			−0.230209	−	0.973141i	$-0.573941\pi$
$251$	−7.29441			−0.460419			−0.230209	+	0.973141i	$0.573941\pi$
$252$		0			0
$253$	−4.47505			−0.281344
$254$		0			0
$255$	4.99270	−	5.35372i	0.312655	−	0.335263i
$256$		0			0
$257$			10.1930i			0.635825i	0.948120	+	0.317912i	$0.102982\pi$
$257$			10.1930i			0.635825i	−0.948120	+	0.317912i	$0.897018\pi$
$258$		0			0
$259$		−	1.59343i		−	0.0990107i
$260$		0			0
$261$	−20.2576	+	1.41540i	−1.25391	+	0.0876108i
$262$		0			0
$263$	−12.6745			−0.781545			−0.390773	−	0.920487i	$-0.627792\pi$
$263$	−12.6745			−0.781545			−0.390773	+	0.920487i	$0.627792\pi$
$264$		0			0
$265$	4.92079			0.302282
$266$		0			0
$267$	12.5825	+	11.7341i	0.770038	+	0.718113i
$268$		0			0
$269$			8.89603i			0.542401i	0.962523	+	0.271200i	$0.0874206\pi$
$269$			8.89603i			0.542401i	−0.962523	+	0.271200i	$0.912579\pi$
$270$		0			0
$271$			12.7128i			0.772249i	0.922447	+	0.386125i	$0.126187\pi$
$271$			12.7128i			0.772249i	−0.922447	+	0.386125i	$0.873813\pi$
$272$		0			0
$273$	−4.57514	−	4.26663i	−0.276900	−	0.258228i
$274$		0			0
$275$	−5.47573			−0.330199
$276$		0			0
$277$	−11.7469			−0.705805			−0.352903	−	0.935660i	$-0.614805\pi$
$277$	−11.7469			−0.705805			−0.352903	+	0.935660i	$0.614805\pi$
$278$		0			0
$279$	−23.5669	+	1.64661i	−1.41091	+	0.0985802i
$280$		0			0
$281$		−	16.4003i		−	0.978358i	−0.872183	−	0.489179i	$-0.837296\pi$
$281$		−	16.4003i		−	0.978358i	0.872183	−	0.489179i	$-0.162704\pi$
$282$		0			0
$283$			18.3066i			1.08821i	0.839016	+	0.544107i	$0.183132\pi$
$283$			18.3066i			1.08821i	−0.839016	+	0.544107i	$0.816868\pi$
$284$		0			0
$285$	−8.77830	+	9.41304i	−0.519982	+	0.557580i
$286$		0			0
$287$	7.57715			0.447265
$288$		0			0
$289$	−0.863120			−0.0507718
$290$		0			0
$291$	9.97281	−	10.6939i	0.584616	−	0.626889i
$292$		0			0
$293$		−	11.1436i		−	0.651019i	−0.945539	−	0.325509i	$-0.894464\pi$
$293$		−	11.1436i		−	0.651019i	0.945539	−	0.325509i	$-0.105536\pi$
$294$		0			0
$295$			5.74788i			0.334655i
$296$		0			0
$297$	22.1104	+	17.9078i	1.28297	+	1.03911i
$298$		0			0
$299$	2.95178			0.170706
$300$		0			0
$301$	−10.1778			−0.586638
$302$		0			0
$303$	−8.95749	−	8.35347i	−0.514595	−	0.479894i
$304$		0			0
$305$		−	2.84267i		−	0.162771i
$306$		0			0
$307$		−	1.22655i		−	0.0700026i	−0.999387	−	0.0350013i	$-0.988856\pi$
$307$		−	1.22655i		−	0.0700026i	0.999387	−	0.0350013i	$-0.0111435\pi$
$308$		0			0
$309$	−16.8690	−	15.7315i	−0.959646	−	0.894935i
$310$		0			0
$311$	−10.0883			−0.572056			−0.286028	−	0.958221i	$-0.592335\pi$
$311$	−10.0883			−0.572056			−0.286028	+	0.958221i	$0.592335\pi$
$312$		0			0
$313$	−11.9069			−0.673017			−0.336509	−	0.941680i	$-0.609246\pi$
$313$	−11.9069			−0.673017			−0.336509	+	0.941680i	$0.609246\pi$
$314$		0			0
$315$	−0.209100	−	2.99270i	−0.0117814	−	0.168620i
$316$		0			0
$317$			13.8364i			0.777131i	0.921421	+	0.388565i	$0.127029\pi$
$317$			13.8364i			0.777131i	−0.921421	+	0.388565i	$0.872971\pi$
$318$		0			0
$319$		−	37.0652i		−	2.07525i
$320$		0			0
$321$	−21.9610	+	23.5489i	−1.22574	+	1.31437i
$322$		0			0
$323$	31.4074			1.74755
$324$		0			0
$325$	3.61183			0.200348
$326$		0			0
$327$	−18.4977	+	19.8352i	−1.02293	+	1.09689i
$328$		0			0
$329$			0.833284i			0.0459404i
$330$		0			0
$331$			18.2569i			1.00349i	0.865016	+	0.501744i	$0.167308\pi$
$331$			18.2569i			1.00349i	−0.865016	+	0.501744i	$0.832692\pi$
$332$		0			0
$333$	−0.333185	−	4.76865i	−0.0182584	−	0.261321i
$334$		0			0
$335$	−0.159787			−0.00873008
$336$		0			0
$337$	−26.9506			−1.46809			−0.734047	−	0.679098i	$-0.762371\pi$
$337$	−26.9506			−1.46809			−0.734047	+	0.679098i	$0.762371\pi$
$338$		0			0
$339$	−4.98150	−	4.64558i	−0.270558	−	0.252313i
$340$		0			0
$341$		−	43.1201i		−	2.33509i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	1.03522	+	0.965413i	0.0557344	+	0.0519761i
$346$		0			0
$347$	13.9989			0.751498			0.375749	−	0.926721i	$-0.377385\pi$
$347$	13.9989			0.751498			0.375749	+	0.926721i	$0.377385\pi$
$348$		0			0
$349$	−23.5755			−1.26197			−0.630983	−	0.775797i	$-0.717348\pi$
$349$	−23.5755			−1.26197			−0.630983	+	0.775797i	$0.717348\pi$
$350$		0			0
$351$	−14.5842	−	11.8121i	−0.778446	−	0.630483i
$352$		0			0
$353$		−	25.4111i		−	1.35249i	−0.736675	−	0.676247i	$-0.763605\pi$
$353$		−	25.4111i		−	1.35249i	0.736675	−	0.676247i	$-0.236395\pi$
$354$		0			0
$355$			9.22757i			0.489748i
$356$		0			0
$357$	−4.99270	+	5.35372i	−0.264242	+	0.283349i
$358$		0			0
$359$	−35.4716			−1.87212			−0.936059	−	0.351844i	$-0.885555\pi$
$359$	−35.4716			−1.87212			−0.936059	+	0.351844i	$0.885555\pi$
$360$		0			0
$361$	−36.2213			−1.90638
$362$		0			0
$363$	−22.4252	+	24.0467i	−1.17702	+	1.26212i
$364$		0			0
$365$			15.8610i			0.830203i
$366$		0			0
$367$			21.9232i			1.14438i	0.820120	+	0.572191i	$0.193906\pi$
$367$			21.9232i			1.14438i	−0.820120	+	0.572191i	$0.806094\pi$
$368$		0			0
$369$	22.6762	−	1.58438i	1.18047	−	0.0824795i
$370$		0			0
$371$	−4.92079			−0.255475
$372$		0			0
$373$	−25.4056			−1.31545			−0.657725	−	0.753258i	$-0.728481\pi$
$373$	−25.4056			−1.31545			−0.657725	+	0.753258i	$0.728481\pi$
$374$		0			0
$375$	1.26671	+	1.18129i	0.0654125	+	0.0610016i
$376$		0			0
$377$			24.4485i			1.25916i
$378$		0			0
$379$		−	18.8097i		−	0.966190i	−0.875568	−	0.483095i	$-0.839513\pi$
$379$		−	18.8097i		−	0.966190i	0.875568	−	0.483095i	$-0.160487\pi$
$380$		0			0
$381$	17.8913	+	16.6849i	0.916601	+	0.854793i
$382$		0			0
$383$	−16.3421			−0.835044			−0.417522	−	0.908667i	$-0.637101\pi$
$383$	−16.3421			−0.835044			−0.417522	+	0.908667i	$0.637101\pi$
$384$		0			0
$385$	5.47573			0.279069
$386$		0			0
$387$	−30.4591	+	2.12817i	−1.54832	+	0.108181i
$388$		0			0
$389$		−	29.0878i		−	1.47481i	−0.675452	−	0.737404i	$-0.736052\pi$
$389$		−	29.0878i		−	1.47481i	0.675452	−	0.737404i	$-0.263948\pi$
$390$		0			0
$391$		−	3.45410i		−	0.174681i
$392$		0			0
$393$	17.7616	−	19.0459i	0.895955	−	0.960740i
$394$		0			0
$395$	−11.3527			−0.571218
$396$		0			0
$397$	7.31540			0.367149			0.183575	−	0.983006i	$-0.441233\pi$
$397$	7.31540			0.367149			0.183575	+	0.983006i	$0.441233\pi$
$398$		0			0
$399$	8.77830	−	9.41304i	0.439465	−	0.471241i
$400$		0			0
$401$			22.1972i			1.10848i	0.832358	+	0.554238i	$0.186990\pi$
$401$			22.1972i			1.10848i	−0.832358	+	0.554238i	$0.813010\pi$
$402$		0			0
$403$			28.4424i			1.41682i
$404$		0			0
$405$	−1.25155	−	8.91255i	−0.0621899	−	0.442868i
$406$		0			0
$407$	8.72517			0.432490
$408$		0			0
$409$	15.1172			0.747497			0.373749	−	0.927530i	$-0.378072\pi$
$409$	15.1172			0.747497			0.373749	+	0.927530i	$0.378072\pi$
$410$		0			0
$411$	−6.86441	−	6.40153i	−0.338596	−	0.315764i
$412$		0			0
$413$		−	5.74788i		−	0.282835i
$414$		0			0
$415$			11.3286i			0.556101i
$416$		0			0
$417$	5.71097	+	5.32587i	0.279667	+	0.260809i
$418$		0			0
$419$	20.7130			1.01190			0.505948	−	0.862564i	$-0.331143\pi$
$419$	20.7130			1.01190			0.505948	+	0.862564i	$0.331143\pi$
$420$		0			0
$421$	35.7381			1.74177			0.870884	−	0.491488i	$-0.163547\pi$
$421$	35.7381			1.74177			0.870884	+	0.491488i	$0.163547\pi$
$422$		0			0
$423$	0.174239	+	2.49377i	0.00847181	+	0.121251i
$424$		0			0
$425$		−	4.22648i		−	0.205014i
$426$		0			0
$427$			2.84267i			0.137567i
$428$		0			0
$429$	23.3629	−	25.0522i	1.12797	−	1.20953i
$430$		0			0
$431$	19.1560			0.922710			0.461355	−	0.887216i	$-0.347363\pi$
$431$	19.1560			0.922710			0.461355	+	0.887216i	$0.347363\pi$
$432$		0			0
$433$	−2.53254			−0.121706			−0.0608531	−	0.998147i	$-0.519382\pi$
$433$	−2.53254			−0.121706			−0.0608531	+	0.998147i	$0.519382\pi$
$434$		0			0
$435$	−7.99616	+	8.57434i	−0.383386	+	0.411108i
$436$		0			0
$437$			6.07309i			0.290515i
$438$		0			0
$439$		−	5.39558i		−	0.257517i	−0.991676	−	0.128759i	$-0.958901\pi$
$439$		−	5.39558i		−	0.257517i	0.991676	−	0.128759i	$-0.0410992\pi$
$440$		0			0
$441$	0.209100	+	2.99270i	0.00995713	+	0.142510i
$442$		0			0
$443$	−16.1061			−0.765223			−0.382612	−	0.923909i	$-0.624975\pi$
$443$	−16.1061			−0.765223			−0.382612	+	0.923909i	$0.624975\pi$
$444$		0			0
$445$	9.93325			0.470881
$446$		0			0
$447$	6.66116	+	6.21198i	0.315062	+	0.293817i
$448$		0			0
$449$			12.6902i			0.598889i	0.954114	+	0.299444i	$0.0968013\pi$
$449$			12.6902i			0.598889i	−0.954114	+	0.299444i	$0.903199\pi$
$450$		0			0
$451$			41.4904i			1.95371i
$452$		0			0
$453$	2.01405	+	1.87824i	0.0946281	+	0.0882472i
$454$		0			0
$455$	−3.61183			−0.169325
$456$		0			0
$457$	0.748135			0.0349963			0.0174982	−	0.999847i	$-0.494430\pi$
$457$	0.748135			0.0349963			0.0174982	+	0.999847i	$0.494430\pi$
$458$		0			0
$459$	−13.8222	+	17.0661i	−0.645166	+	0.796575i
$460$		0			0
$461$		−	29.5987i		−	1.37855i	−0.724500	−	0.689275i	$-0.757929\pi$
$461$		−	29.5987i		−	1.37855i	0.724500	−	0.689275i	$-0.242071\pi$
$462$		0			0
$463$			1.17658i			0.0546805i	0.999626	+	0.0273402i	$0.00870375\pi$
$463$			1.17658i			0.0546805i	−0.999626	+	0.0273402i	$0.991296\pi$
$464$		0			0
$465$	−9.30241	+	9.97505i	−0.431389	+	0.462582i
$466$		0			0
$467$	12.2127			0.565136			0.282568	−	0.959247i	$-0.408814\pi$
$467$	12.2127			0.565136			0.282568	+	0.959247i	$0.408814\pi$
$468$		0			0
$469$	0.159787			0.00737826
$470$		0			0
$471$	20.2424	−	21.7061i	0.932722	−	1.00016i
$472$		0			0
$473$		−	55.7308i		−	2.56251i
$474$		0			0
$475$			7.43110i			0.340962i
$476$		0			0
$477$	−14.7265	+	1.02894i	−0.674279	+	0.0471117i
$478$		0			0
$479$	−35.0924			−1.60341			−0.801707	−	0.597717i	$-0.796075\pi$
$479$	−35.0924			−1.60341			−0.801707	+	0.597717i	$0.796075\pi$
$480$		0			0
$481$	−5.75519			−0.262414
$482$		0			0
$483$	−1.03522	−	0.965413i	−0.0471041	−	0.0439278i
$484$		0			0
$485$		−	8.44229i		−	0.383345i
$486$		0			0
$487$		−	1.77034i		−	0.0802217i	−0.999195	−	0.0401108i	$-0.987229\pi$
$487$		−	1.77034i		−	0.0802217i	0.999195	−	0.0401108i	$-0.0127711\pi$
$488$		0			0
$489$	15.1135	+	14.0944i	0.683457	+	0.637370i
$490$		0			0
$491$	−12.9841			−0.585964			−0.292982	−	0.956118i	$-0.594648\pi$
$491$	−12.9841			−0.585964			−0.292982	+	0.956118i	$0.594648\pi$
$492$		0			0
$493$	28.6090			1.28849
$494$		0			0
$495$	16.3872	−	1.14497i	0.736551	−	0.0514627i
$496$		0			0
$497$		−	9.22757i		−	0.413913i
$498$		0			0
$499$		−	19.7417i		−	0.883759i	−0.897074	−	0.441880i	$-0.854312\pi$
$499$		−	19.7417i		−	0.883759i	0.897074	−	0.441880i	$-0.145688\pi$
$500$		0			0
$501$	2.35401	−	2.52422i	0.105169	−	0.112774i
$502$		0			0
$503$	34.7008			1.54723			0.773616	−	0.633655i	$-0.218446\pi$
$503$	34.7008			1.54723			0.773616	+	0.633655i	$0.218446\pi$
$504$		0			0
$505$	−7.07147			−0.314676
$506$		0			0
$507$	−0.0535503	+	0.0574224i	−0.00237825	+	0.00255022i
$508$		0			0
$509$			23.0266i			1.02064i	0.859985	+	0.510319i	$0.170473\pi$
$509$			23.0266i			1.02064i	−0.859985	+	0.510319i	$0.829527\pi$
$510$		0			0
$511$		−	15.8610i		−	0.701650i
$512$		0			0
$513$	24.3026	−	30.0060i	1.07299	−	1.32480i
$514$		0			0
$515$	−13.3172			−0.586827
$516$		0			0
$517$	−4.56284			−0.200673
$518$		0			0
$519$	−5.08018	−	4.73761i	−0.222995	−	0.207958i
$520$		0			0
$521$			33.7984i			1.48074i	0.672202	+	0.740368i	$0.265349\pi$
$521$			33.7984i			1.48074i	−0.672202	+	0.740368i	$0.734651\pi$
$522$		0			0
$523$		−	15.8227i		−	0.691877i	−0.938257	−	0.345939i	$-0.887561\pi$
$523$		−	15.8227i		−	0.691877i	0.938257	−	0.345939i	$-0.112439\pi$
$524$		0			0
$525$	−1.26671	−	1.18129i	−0.0552837	−	0.0515558i
$526$		0			0
$527$	33.2826			1.44981
$528$		0			0
$529$	−22.3321			−0.970961
$530$		0			0
$531$	−1.20188	−	17.2017i	−0.0521572	−	0.746491i
$532$		0			0
$533$		−	27.3674i		−	1.18541i
$534$		0			0
$535$			18.5906i			0.803744i
$536$		0			0
$537$	8.55463	−	9.17319i	0.369159	−	0.395853i
$538$		0			0
$539$	−5.47573			−0.235856
$540$		0			0
$541$	22.7126			0.976492			0.488246	−	0.872706i	$-0.337637\pi$
$541$	22.7126			0.976492			0.488246	+	0.872706i	$0.337637\pi$
$542$		0			0
$543$	0.885570	−	0.949604i	0.0380035	−	0.0407514i
$544$		0			0
$545$			15.6589i			0.670753i
$546$		0			0
$547$			11.6802i			0.499408i	0.968322	+	0.249704i	$0.0803334\pi$
$547$			11.6802i			0.499408i	−0.968322	+	0.249704i	$0.919667\pi$
$548$		0			0
$549$	0.594403	+	8.50728i	0.0253685	+	0.363082i
$550$		0			0
$551$	−50.3011			−2.14290
$552$		0			0
$553$	11.3527			0.482768
$554$		0			0
$555$	−2.01841	−	1.88230i	−0.0856766	−	0.0798992i
$556$		0			0
$557$			19.6461i			0.832430i	0.909266	+	0.416215i	$0.136644\pi$
$557$			19.6461i			0.832430i	−0.909266	+	0.416215i	$0.863356\pi$
$558$		0			0
$559$			36.7605i			1.55480i
$560$		0			0
$561$	−29.3155	−	27.3387i	−1.23770	−	1.15424i
$562$		0			0
$563$	−39.4517			−1.66269			−0.831346	−	0.555756i	$-0.812429\pi$
$563$	−39.4517			−1.66269			−0.831346	+	0.555756i	$0.812429\pi$
$564$		0			0
$565$	−3.93263			−0.165447
$566$		0			0
$567$	1.25155	+	8.91255i	0.0525600	+	0.374292i
$568$		0			0
$569$			42.8676i			1.79710i	0.438867	+	0.898552i	$0.355380\pi$
$569$			42.8676i			1.79710i	−0.438867	+	0.898552i	$0.644620\pi$
$570$		0			0
$571$		−	9.10927i		−	0.381211i	−0.981667	−	0.190605i	$-0.938955\pi$
$571$		−	9.10927i		−	0.381211i	0.981667	−	0.190605i	$-0.0610451\pi$
$572$		0			0
$573$	−22.1441	+	23.7453i	−0.925082	+	0.991973i
$574$		0			0
$575$	0.817252			0.0340818
$576$		0			0
$577$	11.9698			0.498310			0.249155	−	0.968464i	$-0.419847\pi$
$577$	11.9698			0.498310			0.249155	+	0.968464i	$0.419847\pi$
$578$		0			0
$579$	−8.66544	+	9.29202i	−0.360123	+	0.386163i
$580$		0			0
$581$		−	11.3286i		−	0.469991i
$582$		0			0
$583$		−	26.9449i		−	1.11594i
$584$		0			0
$585$	−10.8091	+	0.755233i	−0.446903	+	0.0312250i
$586$		0			0
$587$	34.8252			1.43739			0.718695	−	0.695325i	$-0.244740\pi$
$587$	34.8252			1.43739			0.718695	+	0.695325i	$0.244740\pi$
$588$		0			0
$589$	−58.5183			−2.41120
$590$		0			0
$591$	34.2697	+	31.9588i	1.40967	+	1.31461i
$592$		0			0
$593$			43.0067i			1.76607i	0.469304	+	0.883036i	$0.344505\pi$
$593$			43.0067i			1.76607i	−0.469304	+	0.883036i	$0.655495\pi$
$594$		0			0
$595$			4.22648i			0.173269i
$596$		0			0
$597$	25.5155	+	23.7950i	1.04428	+	0.973864i
$598$		0			0
$599$	12.4879			0.510240			0.255120	−	0.966909i	$-0.417885\pi$
$599$	12.4879			0.510240			0.255120	+	0.966909i	$0.417885\pi$
$600$		0			0
$601$	34.1728			1.39394			0.696968	−	0.717102i	$-0.254532\pi$
$601$	34.1728			1.39394			0.696968	+	0.717102i	$0.254532\pi$
$602$		0			0
$603$	0.478194	−	0.0334114i	0.0194736	−	0.00136062i
$604$		0			0
$605$			18.9836i			0.771793i
$606$		0			0
$607$		−	19.0849i		−	0.774634i	−0.921947	−	0.387317i	$-0.873402\pi$
$607$		−	19.0849i		−	0.774634i	0.921947	−	0.387317i	$-0.126598\pi$
$608$		0			0
$609$	7.99616	−	8.57434i	0.324021	−	0.347450i
$610$		0			0
$611$	3.00968			0.121759
$612$		0			0
$613$	−23.3572			−0.943389			−0.471694	−	0.881762i	$-0.656357\pi$
$613$	−23.3572			−0.943389			−0.471694	+	0.881762i	$0.656357\pi$
$614$		0			0
$615$	8.95082	−	9.59804i	0.360932	−	0.387030i
$616$		0			0
$617$		−	13.1699i		−	0.530202i	−0.964221	−	0.265101i	$-0.914595\pi$
$617$		−	13.1699i		−	0.530202i	0.964221	−	0.265101i	$-0.0854053\pi$
$618$		0			0
$619$			17.6170i			0.708086i	0.935229	+	0.354043i	$0.115193\pi$
$619$			17.6170i			0.708086i	−0.935229	+	0.354043i	$0.884807\pi$
$620$		0			0
$621$	−3.29997	−	2.67273i	−0.132423	−	0.107253i
$622$		0			0
$623$	−9.93325			−0.397967
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	51.5432	+	48.0676i	2.05844	+	1.91963i
$628$		0			0
$629$			6.73458i			0.268525i
$630$		0			0
$631$			15.9879i			0.636470i	0.948012	+	0.318235i	$0.103090\pi$
$631$			15.9879i			0.636470i	−0.948012	+	0.318235i	$0.896910\pi$
$632$		0			0
$633$	2.46279	+	2.29672i	0.0978870	+	0.0912863i
$634$		0			0
$635$	14.1243			0.560505
$636$		0			0
$637$	3.61183			0.143106
$638$		0			0
$639$	−1.92948	−	27.6154i	−0.0763291	−	1.09245i
$640$		0			0
$641$		−	26.7669i		−	1.05723i	−0.848862	−	0.528615i	$-0.822712\pi$
$641$		−	26.7669i		−	1.05723i	0.848862	−	0.528615i	$-0.177288\pi$
$642$		0			0
$643$		−	36.4167i		−	1.43613i	−0.695974	−	0.718067i	$-0.745027\pi$
$643$		−	36.4167i		−	1.43613i	0.695974	−	0.718067i	$-0.254973\pi$
$644$		0			0
$645$	−12.0229	+	12.8923i	−0.473403	+	0.507634i
$646$		0			0
$647$	−28.3136			−1.11312			−0.556561	−	0.830807i	$-0.687879\pi$
$647$	−28.3136			−1.11312			−0.556561	+	0.830807i	$0.687879\pi$
$648$		0			0
$649$	31.4738			1.23546
$650$		0			0
$651$	9.30241	−	9.97505i	0.364590	−	0.390953i
$652$		0			0
$653$			7.20424i			0.281924i	0.990015	+	0.140962i	$0.0450195\pi$
$653$			7.20424i			0.281924i	−0.990015	+	0.140962i	$0.954981\pi$
$654$		0			0
$655$		−	15.0358i		−	0.587496i
$656$		0			0
$657$	−3.31653	−	47.4673i	−0.129390	−	1.85188i
$658$		0			0
$659$	−17.0537			−0.664318			−0.332159	−	0.943223i	$-0.607777\pi$
$659$	−17.0537			−0.664318			−0.332159	+	0.943223i	$0.607777\pi$
$660$		0			0
$661$	4.65316			0.180987			0.0904935	−	0.995897i	$-0.471156\pi$
$661$	4.65316			0.180987			0.0904935	+	0.995897i	$0.471156\pi$
$662$		0			0
$663$	19.3367	+	18.0328i	0.750976	+	0.700336i
$664$		0			0
$665$		−	7.43110i		−	0.288166i
$666$		0			0
$667$			5.53198i			0.214199i
$668$		0			0
$669$	22.5744	+	21.0522i	0.872778	+	0.813925i
$670$		0			0
$671$	−15.5657			−0.600908
$672$		0			0
$673$	26.0720			1.00500			0.502500	−	0.864577i	$-0.332414\pi$
$673$	26.0720			1.00500			0.502500	+	0.864577i	$0.332414\pi$
$674$		0			0
$675$	−4.03789	−	3.27039i	−0.155418	−	0.125877i
$676$		0			0
$677$		−	30.7182i		−	1.18060i	−0.807186	−	0.590298i	$-0.799010\pi$
$677$		−	30.7182i		−	1.18060i	0.807186	−	0.590298i	$-0.200990\pi$
$678$		0			0
$679$			8.44229i			0.323985i
$680$		0			0
$681$	−7.80902	+	8.37367i	−0.299242	+	0.320880i
$682$		0			0
$683$	−9.84648			−0.376765			−0.188382	−	0.982096i	$-0.560324\pi$
$683$	−9.84648			−0.376765			−0.188382	+	0.982096i	$0.560324\pi$
$684$		0			0
$685$	−5.41909			−0.207053
$686$		0			0
$687$	−7.52624	+	8.07044i	−0.287144	+	0.307907i
$688$		0			0
$689$			17.7731i			0.677100i
$690$		0			0
$691$		−	35.0457i		−	1.33320i	−0.745416	−	0.666600i	$-0.767749\pi$
$691$		−	35.0457i		−	1.33320i	0.745416	−	0.666600i	$-0.232251\pi$
$692$		0			0
$693$	−16.3872	+	1.14497i	−0.622499	+	0.0434939i
$694$		0			0
$695$	4.50851			0.171018
$696$		0			0
$697$	−32.0246			−1.21302
$698$		0			0
$699$	−21.1829	−	19.7545i	−0.801210	−	0.747183i
$700$		0			0
$701$			8.08916i			0.305523i	0.988263	+	0.152762i	$0.0488167\pi$
$701$			8.08916i			0.305523i	−0.988263	+	0.152762i	$0.951183\pi$
$702$		0			0
$703$		−	11.8409i		−	0.446589i
$704$		0			0
$705$	1.05553	+	0.984351i	0.0397535	+	0.0370728i
$706$		0			0
$707$	7.07147			0.265950
$708$		0			0
$709$	35.8946			1.34805			0.674026	−	0.738708i	$-0.264564\pi$
$709$	35.8946			1.34805			0.674026	+	0.738708i	$0.264564\pi$
$710$		0			0
$711$	33.9754	−	2.37386i	1.27418	−	0.0890265i
$712$		0			0
$713$			6.43568i			0.241018i
$714$		0			0
$715$		−	19.7774i		−	0.739633i
$716$		0			0
$717$	16.5849	−	17.7841i	0.619375	−	0.664161i
$718$		0			0
$719$	−41.4403			−1.54546			−0.772732	−	0.634732i	$-0.781110\pi$
$719$	−41.4403			−1.54546			−0.772732	+	0.634732i	$0.781110\pi$
$720$		0			0
$721$	13.3172			0.495959
$722$		0			0
$723$	−32.5111	+	34.8619i	−1.20910	+	1.29653i
$724$		0			0
$725$			6.76900i			0.251394i
$726$		0			0
$727$		−	15.8794i		−	0.588933i	−0.955662	−	0.294467i	$-0.904858\pi$
$727$		−	15.8794i		−	0.588933i	0.955662	−	0.294467i	$-0.0951420\pi$
$728$		0			0
$729$	5.60912	+	26.4109i	0.207745	+	0.978183i
$730$		0			0
$731$	43.0162			1.59101
$732$		0			0
$733$	22.2284			0.821023			0.410512	−	0.911855i	$-0.365350\pi$
$733$	22.2284			0.821023			0.410512	+	0.911855i	$0.365350\pi$
$734$		0			0
$735$	1.26671	+	1.18129i	0.0467232	+	0.0435726i
$736$		0			0
$737$			0.874948i			0.0322291i
$738$		0			0
$739$		−	10.9792i		−	0.403877i	−0.979398	−	0.201939i	$-0.935276\pi$
$739$		−	10.9792i		−	0.403877i	0.979398	−	0.201939i	$-0.0647242\pi$
$740$		0			0
$741$	−33.9983	−	31.7058i	−1.24896	−	1.16474i
$742$		0			0
$743$	23.8119			0.873573			0.436786	−	0.899565i	$-0.356117\pi$
$743$	23.8119			0.873573			0.436786	+	0.899565i	$0.356117\pi$
$744$		0			0
$745$	5.25864			0.192662
$746$		0			0
$747$	−2.36881	−	33.9032i	−0.0866704	−	1.24045i
$748$		0			0
$749$		−	18.5906i		−	0.679287i
$750$		0			0
$751$			7.92816i			0.289303i	0.989483	+	0.144651i	$0.0462060\pi$
$751$			7.92816i			0.289303i	−0.989483	+	0.144651i	$0.953794\pi$
$752$		0			0
$753$	8.61682	−	9.23989i	0.314014	−	0.336720i
$754$		0			0
$755$	1.58998			0.0578655
$756$		0			0
$757$	−39.6358			−1.44059			−0.720293	−	0.693670i	$-0.755993\pi$
$757$	−39.6358			−1.44059			−0.720293	+	0.693670i	$0.755993\pi$
$758$		0			0
$759$	5.28634	−	5.66858i	0.191882	−	0.205757i
$760$		0			0
$761$			42.6209i			1.54500i	0.635012	+	0.772502i	$0.280995\pi$
$761$			42.6209i			1.54500i	−0.635012	+	0.772502i	$0.719005\pi$
$762$		0			0
$763$		−	15.6589i		−	0.566890i
$764$		0			0
$765$	0.883756	+	12.6486i	0.0319522	+	0.457311i
$766$		0			0
$767$	−20.7604			−0.749614
$768$		0			0
$769$	37.7181			1.36015			0.680074	−	0.733144i	$-0.261948\pi$
$769$	37.7181			1.36015			0.680074	+	0.733144i	$0.261948\pi$
$770$		0			0
$771$	−12.9116	−	12.0410i	−0.465000	−	0.433645i
$772$		0			0
$773$		−	3.47876i		−	0.125122i	−0.998041	−	0.0625611i	$-0.980073\pi$
$773$		−	3.47876i		−	0.125122i	0.998041	−	0.0625611i	$-0.0199268\pi$
$774$		0			0
$775$			7.87478i			0.282870i
$776$		0			0
$777$	2.01841	+	1.88230i	0.0724099	+	0.0675272i
$778$		0			0
$779$	56.3066			2.01739
$780$		0			0
$781$	50.5276			1.80802
$782$		0			0
$783$	22.1372	−	27.3325i	0.791121	−	0.976783i
$784$		0			0
$785$		−	17.1358i		−	0.611604i
$786$		0			0
$787$		−	7.84882i		−	0.279780i	−0.990167	−	0.139890i	$-0.955325\pi$
$787$		−	7.84882i		−	0.279780i	0.990167	−	0.139890i	$-0.0446749\pi$
$788$		0			0
$789$	14.9723	−	16.0549i	0.533029	−	0.571571i
$790$		0			0
$791$	3.93263			0.139828
$792$		0			0
$793$	10.2673			0.364601
$794$		0			0
$795$	−5.81289	+	6.23320i	−0.206162	+	0.221069i
$796$		0			0
$797$			30.3207i			1.07401i	0.843578	+	0.537007i	$0.180445\pi$
$797$			30.3207i			1.07401i	−0.843578	+	0.537007i	$0.819555\pi$
$798$		0			0
$799$		−	3.52186i		−	0.124594i
$800$		0			0
$801$	−29.7273	+	2.07704i	−1.05036	+	0.0733886i
$802$		0			0
$803$	86.8506			3.06489
$804$		0			0
$805$	−0.817252			−0.0288044
$806$		0			0
$807$	−11.2687	−	10.5088i	−0.396676	−	0.369928i
$808$		0			0
$809$			14.9144i			0.524364i	0.965018	+	0.262182i	$0.0844421\pi$
$809$			14.9144i			0.524364i	−0.965018	+	0.262182i	$0.915558\pi$
$810$		0			0
$811$		−	10.2714i		−	0.360676i	−0.983605	−	0.180338i	$-0.942281\pi$
$811$		−	10.2714i		−	0.360676i	0.983605	−	0.180338i	$-0.0577192\pi$
$812$		0			0
$813$	−16.1034	−	15.0176i	−0.564772	−	0.526689i
$814$		0			0
$815$	11.9313			0.417936
$816$		0			0
$817$	−75.6322			−2.64604
$818$		0			0
$819$	10.8091	−	0.755233i	0.377702	−	0.0263900i
$820$		0			0
$821$			22.1123i			0.771725i	0.922556	+	0.385863i	$0.126096\pi$
$821$			22.1123i			0.771725i	−0.922556	+	0.385863i	$0.873904\pi$
$822$		0			0
$823$		−	32.9990i		−	1.15027i	−0.818057	−	0.575137i	$-0.804949\pi$
$823$		−	32.9990i		−	1.15027i	0.818057	−	0.575137i	$-0.195051\pi$
$824$		0			0
$825$	6.46843	−	6.93615i	0.225202	−	0.241486i
$826$		0			0
$827$	52.3962			1.82199			0.910997	−	0.412413i	$-0.135314\pi$
$827$	52.3962			1.82199			0.910997	+	0.412413i	$0.135314\pi$
$828$		0			0
$829$	−1.95264			−0.0678179			−0.0339090	−	0.999425i	$-0.510796\pi$
$829$	−1.95264			−0.0678179			−0.0339090	+	0.999425i	$0.510796\pi$
$830$		0			0
$831$	13.8766	−	14.8799i	0.481373	−	0.516180i
$832$		0			0
$833$		−	4.22648i		−	0.146439i
$834$		0			0
$835$		−	1.99274i		−	0.0689616i
$836$		0			0
$837$	25.7536	−	31.7975i	0.890174	−	1.09908i
$838$		0			0
$839$	26.3162			0.908537			0.454269	−	0.890865i	$-0.349901\pi$
$839$	26.3162			0.908537			0.454269	+	0.890865i	$0.349901\pi$
$840$		0			0
$841$	−16.8193			−0.579976
$842$		0			0
$843$	20.7744	+	19.3735i	0.715507	+	0.667259i
$844$		0			0
$845$			0.0453320i			0.00155947i
$846$		0			0
$847$		−	18.9836i		−	0.652284i
$848$		0			0
$849$	−23.1891	−	21.6254i	−0.795849	−	0.742183i
$850$		0			0
$851$	−1.30223			−0.0446399
$852$		0			0
$853$	−47.5515			−1.62813			−0.814066	−	0.580773i	$-0.802750\pi$
$853$	−47.5515			−1.62813			−0.814066	+	0.580773i	$0.802750\pi$
$854$		0			0
$855$	−1.55384	−	22.2391i	−0.0531403	−	0.760561i
$856$		0			0
$857$			31.7562i			1.08477i	0.840130	+	0.542386i	$0.182479\pi$
$857$			31.7562i			1.08477i	−0.840130	+	0.542386i	$0.817521\pi$
$858$		0			0
$859$		−	12.0161i		−	0.409984i	−0.978764	−	0.204992i	$-0.934283\pi$
$859$		−	12.0161i		−	0.409984i	0.978764	−	0.204992i	$-0.0657168\pi$
$860$		0			0
$861$	−8.95082	+	9.59804i	−0.305043	+	0.327100i
$862$		0			0
$863$	−3.85760			−0.131314			−0.0656572	−	0.997842i	$-0.520914\pi$
$863$	−3.85760			−0.131314			−0.0656572	+	0.997842i	$0.520914\pi$
$864$		0			0
$865$	−4.01054			−0.136362
$866$		0			0
$867$	1.01960	−	1.09332i	0.0346273	−	0.0371311i
$868$		0			0
$869$			62.1645i			2.10879i
$870$		0			0
$871$		−	0.577123i		−	0.0195551i
$872$		0			0
$873$	1.76528	+	25.2653i	0.0597457	+	0.855100i
$874$		0			0
$875$	−1.00000			−0.0338062
$876$		0			0
$877$	−9.44601			−0.318969			−0.159485	−	0.987200i	$-0.550983\pi$
$877$	−9.44601			−0.318969			−0.159485	+	0.987200i	$0.550983\pi$
$878$		0			0
$879$	14.1158	+	13.1639i	0.476113	+	0.444007i
$880$		0			0
$881$			7.98637i			0.269068i	0.990909	+	0.134534i	$0.0429537\pi$
$881$			7.98637i			0.269068i	−0.990909	+	0.134534i	$0.957046\pi$
$882$		0			0
$883$			36.5457i			1.22986i	0.788582	+	0.614930i	$0.210816\pi$
$883$			36.5457i			1.22986i	−0.788582	+	0.614930i	$0.789184\pi$
$884$		0			0
$885$	−7.28089	−	6.78993i	−0.244744	−	0.228241i
$886$		0			0
$887$	−41.2225			−1.38412			−0.692058	−	0.721842i	$-0.743296\pi$
$887$	−41.2225			−1.38412			−0.692058	+	0.721842i	$0.743296\pi$
$888$		0			0
$889$	−14.1243			−0.473713
$890$		0			0
$891$	−48.8027	+	6.85313i	−1.63495	+	0.229589i
$892$		0			0
$893$			6.19222i			0.207215i
$894$		0			0
$895$		−	7.24176i		−	0.242065i
$896$		0			0
$897$	−3.48691	+	3.73904i	−0.116425	+	0.124843i
$898$		0			0
$899$	−53.3044			−1.77780
$900$		0			0
$901$	20.7976			0.692869
$902$		0			0
$903$	12.0229	−	12.8923i	0.400098	−	0.429029i
$904$		0			0
$905$		−	0.749663i		−	0.0249196i
$906$		0			0
$907$		−	9.40601i		−	0.312322i	−0.987732	−	0.156161i	$-0.950088\pi$
$907$		−	9.40601i		−	0.312322i	0.987732	−	0.156161i	$-0.0499118\pi$
$908$		0			0
$909$	21.1628	−	1.47864i	0.701927	−	0.0490435i
$910$		0			0
$911$	11.4174			0.378276			0.189138	−	0.981951i	$-0.439431\pi$
$911$	11.4174			0.378276			0.189138	+	0.981951i	$0.439431\pi$
$912$		0			0
$913$	62.0325			2.05298
$914$		0			0
$915$	3.60084	+	3.35803i	0.119040	+	0.111013i
$916$		0			0
$917$			15.0358i			0.496524i
$918$		0			0
$919$		−	7.24065i		−	0.238847i	−0.992843	−	0.119423i	$-0.961895\pi$
$919$		−	7.24065i		−	0.238847i	0.992843	−	0.119423i	$-0.0381046\pi$
$920$		0			0
$921$	1.55367	+	1.44891i	0.0511953	+	0.0477431i
$922$		0			0
$923$	−33.3284			−1.09702
$924$		0			0
$925$	−1.59343			−0.0523915
$926$		0			0
$927$	39.8545	−	2.78463i	1.30899	−	0.0914592i
$928$		0			0
$929$		−	33.4931i		−	1.09887i	−0.835536	−	0.549436i	$-0.814842\pi$
$929$		−	33.4931i		−	1.09887i	0.835536	−	0.549436i	$-0.185158\pi$
$930$		0			0
$931$			7.43110i			0.243545i
$932$		0			0
$933$	11.9173	−	12.7790i	0.390153	−	0.418364i
$934$		0			0
$935$	−23.1430			−0.756858
$936$		0			0
$937$	−16.1097			−0.526280			−0.263140	−	0.964758i	$-0.584758\pi$
$937$	−16.1097			−0.526280			−0.263140	+	0.964758i	$0.584758\pi$
$938$		0			0
$939$	14.0655	−	15.0826i	0.459011	−	0.492201i
$940$		0			0
$941$		−	14.5124i		−	0.473092i	−0.971620	−	0.236546i	$-0.923985\pi$
$941$		−	14.5124i		−	0.473092i	0.971620	−	0.236546i	$-0.0760154\pi$
$942$		0			0
$943$		−	6.19244i		−	0.201654i
$944$		0			0
$945$	4.03789	+	3.27039i	0.131353	+	0.106386i
$946$		0			0
$947$	9.36786			0.304415			0.152207	−	0.988349i	$-0.451362\pi$
$947$	9.36786			0.304415			0.152207	+	0.988349i	$0.451362\pi$
$948$		0			0
$949$	−57.2873			−1.85962
$950$		0			0
$951$	−17.5267	−	16.3448i	−0.568342	−	0.530018i
$952$		0			0
$953$		−	17.0284i		−	0.551604i	−0.961214	−	0.275802i	$-0.911057\pi$
$953$		−	17.0284i		−	0.551604i	0.961214	−	0.275802i	$-0.0889434\pi$
$954$		0			0
$955$			18.7456i			0.606595i
$956$		0			0
$957$	46.9508	+	43.7848i	1.51770	+	1.41536i
$958$		0			0
$959$	5.41909			0.174992
$960$		0			0
$961$	−31.0122			−1.00039
$962$		0			0
$963$	−3.88730	−	55.6363i	−0.125266	−	1.79285i
$964$		0			0
$965$			7.33556i			0.236140i
$966$		0			0
$967$			29.9365i			0.962692i	0.876531	+	0.481346i	$0.159852\pi$
$967$			29.9365i			0.962692i	−0.876531	+	0.481346i	$0.840148\pi$
$968$		0			0
$969$	−37.1013	+	39.7840i	−1.19187	+	1.27805i
$970$		0			0
$971$	−23.7701			−0.762819			−0.381410	−	0.924406i	$-0.624561\pi$
$971$	−23.7701			−0.762819			−0.381410	+	0.924406i	$0.624561\pi$
$972$		0			0
$973$	−4.50851			−0.144536
$974$		0			0
$975$	−4.26663	+	4.57514i	−0.136641	+	0.146522i
$976$		0			0
$977$			13.9813i			0.447302i	0.974669	+	0.223651i	$0.0717975\pi$
$977$			13.9813i			0.447302i	−0.974669	+	0.223651i	$0.928202\pi$
$978$		0			0
$979$		−	54.3918i		−	1.73837i
$980$		0			0
$981$	−3.27427	−	46.8624i	−0.104539	−	1.49620i
$982$		0			0
$983$	23.1193			0.737392			0.368696	−	0.929550i	$-0.379804\pi$
$983$	23.1193			0.737392			0.368696	+	0.929550i	$0.379804\pi$
$984$		0			0
$985$	27.0541			0.862016
$986$		0			0
$987$	−1.05553	−	0.984351i	−0.0335978	−	0.0313322i
$988$		0			0
$989$			8.31782i			0.264491i
$990$		0			0
$991$			34.1504i			1.08482i	0.840113	+	0.542411i	$0.182489\pi$
$991$			34.1504i			1.08482i	−0.840113	+	0.542411i	$0.817511\pi$
$992$		0			0
$993$	−23.1261	−	21.5667i	−0.733885	−	0.684398i
$994$		0			0
$995$	20.1432			0.638582
$996$		0			0
$997$	12.3151			0.390023			0.195012	−	0.980801i	$-0.437526\pi$
$997$	12.3151			0.390023			0.195012	+	0.980801i	$0.437526\pi$
$998$		0			0
$999$	6.43408	+	5.21112i	0.203565	+	0.164873i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.ba.d.911.4	yes	16
3.2	odd	2		1680.2.ba.c.911.14	yes	16
4.3	odd	2		1680.2.ba.c.911.13	✓	16
12.11	even	2	inner	1680.2.ba.d.911.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.ba.c.911.13	✓	16	4.3	odd	2
1680.2.ba.c.911.14	yes	16	3.2	odd	2
1680.2.ba.d.911.3	yes	16	12.11	even	2	inner
1680.2.ba.d.911.4	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.ba (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.18129 + 1.26671i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(-0.209100 - 2.99270i) q^{9} +5.47573 q^{11} -3.61183 q^{13} +(-1.26671 - 1.18129i) q^{15} +4.22648i q^{17} -7.43110i q^{19} +(1.26671 + 1.18129i) q^{21} -0.817252 q^{23} -1.00000 q^{25} +(4.03789 + 3.27039i) q^{27} -6.76900i q^{29} -7.87478i q^{31} +(-6.46843 + 6.93615i) q^{33} +1.00000 q^{35} +1.59343 q^{37} +(4.26663 - 4.57514i) q^{39} +7.57715i q^{41} -10.1778i q^{43} +(2.99270 - 0.209100i) q^{45} -0.833284 q^{47} -1.00000 q^{49} +(-5.35372 - 4.99270i) q^{51} -4.92079i q^{53} +5.47573i q^{55} +(9.41304 + 8.77830i) q^{57} +5.74788 q^{59} -2.84267 q^{61} +(-2.99270 + 0.209100i) q^{63} -3.61183i q^{65} +0.159787i q^{67} +(0.965413 - 1.03522i) q^{69} +9.22757 q^{71} +15.8610 q^{73} +(1.18129 - 1.26671i) q^{75} -5.47573i q^{77} +11.3527i q^{79} +(-8.91255 + 1.25155i) q^{81} +11.3286 q^{83} -4.22648 q^{85} +(8.57434 + 7.99616i) q^{87} -9.93325i q^{89} +3.61183i q^{91} +(9.97505 + 9.30241i) q^{93} +7.43110 q^{95} -8.44229 q^{97} +(-1.14497 - 16.3872i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 10 q^{9} + 12 q^{11} - 2 q^{15} + 2 q^{21} + 8 q^{23} - 16 q^{25} - 12 q^{27} + 12 q^{33} + 16 q^{35} - 8 q^{37} - 14 q^{39} + 8 q^{45} + 16 q^{47} - 16 q^{49} - 34 q^{51} + 12 q^{57} + 32 q^{59}+ \cdots - 50 q^{99}+O(q^{100}) \) 16 * q - 10 * q^9 + 12 * q^11 - 2 * q^15 + 2 * q^21 + 8 * q^23 - 16 * q^25 - 12 * q^27 + 12 * q^33 + 16 * q^35 - 8 * q^37 - 14 * q^39 + 8 * q^45 + 16 * q^47 - 16 * q^49 - 34 * q^51 + 12 * q^57 + 32 * q^59 + 8 * q^61 - 8 * q^63 - 20 * q^69 - 32 * q^71 + 48 * q^73 - 22 * q^81 + 72 * q^83 - 12 * q^85 - 44 * q^87 + 28 * q^93 + 8 * q^95 - 8 * q^97 - 50 * q^99

Introduction

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