LMFDB - Embedded newform 2128.2.m.e.1329.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2128,2,Mod(1329,2128)] 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("2128.1329"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2128 = 2^{4} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2128.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,4] 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:	$16.9921655501$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.113164960000.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 17x^{6} - 30x^{5} + 174x^{4} - 208x^{3} + 962x^{2} - 382x + 2449 $ x^8 - 2x^7 + 17x^6 - 30x^5 + 174x^4 - 208x^3 + 962x^2 - 382*x + 2449
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 532)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1329.5
Root			$0.737640 + 2.57255i$ of defining polynomial
Character	$\chi$	$=$	2128.1329
Dual form			2128.2.m.e.1329.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.09331 q^{3} -1.58993i q^{5} +(-0.618034 - 2.57255i) q^{7} +1.38197 q^{9} -1.61803 q^{11} -3.38705 q^{13} -3.32821i q^{15} -2.57255i q^{17} +(-1.29374 - 4.16248i) q^{19} +(-1.29374 - 5.38516i) q^{21} -7.47214 q^{23} +2.47214 q^{25} -3.38705 q^{27} +8.71338i q^{29} -8.06785 q^{31} -3.38705 q^{33} +(-4.09017 + 0.982628i) q^{35} +5.38516i q^{37} -7.09017 q^{39} +8.06785 q^{41} +5.70820 q^{43} -2.19722i q^{45} -4.76978i q^{47} +(-6.23607 + 3.17985i) q^{49} -5.38516i q^{51} -14.0985i q^{53} +2.57255i q^{55} +(-2.70820 - 8.71338i) q^{57} +0.799575 q^{59} +6.73503i q^{61} +(-0.854102 - 3.55518i) q^{63} +5.38516i q^{65} -2.05695i q^{67} -15.6415 q^{69} -12.0416i q^{71} -9.30759i q^{73} +5.17496 q^{75} +(1.00000 + 4.16248i) q^{77} +10.7703i q^{79} -11.2361 q^{81} -2.94788i q^{83} -4.09017 q^{85} +18.2398i q^{87} +8.37326 q^{89} +(2.09331 + 8.71338i) q^{91} -16.8885 q^{93} +(-6.61803 + 2.05695i) q^{95} +10.1612 q^{97} -2.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7} + 20 q^{9} - 4 q^{11} - 24 q^{23} - 16 q^{25} + 12 q^{35} - 12 q^{39} - 8 q^{43} - 32 q^{49} + 32 q^{57} + 20 q^{63} + 8 q^{77} - 72 q^{81} + 12 q^{85} + 8 q^{93} - 44 q^{95}+O(q^{100}) $ 8 * q + 4 * q^7 + 20 * q^9 - 4 * q^11 - 24 * q^23 - 16 * q^25 + 12 * q^35 - 12 * q^39 - 8 * q^43 - 32 * q^49 + 32 * q^57 + 20 * q^63 + 8 * q^77 - 72 * q^81 + 12 * q^85 + 8 * q^93 - 44 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2128\mathbb{Z}\right)^\times$.

$n$	$533$	$799$	$913$	$1009$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	2.09331			1.20858			0.604288	−	0.796766i	$-0.293458\pi$
$3$	2.09331			1.20858			0.604288	+	0.796766i	$0.293458\pi$
$4$		0			0
$5$		−	1.58993i		−	0.711036i	−0.934669	−	0.355518i	$-0.884304\pi$
$5$		−	1.58993i		−	0.711036i	0.934669	−	0.355518i	$-0.115696\pi$
$6$		0			0
$7$	−0.618034	−	2.57255i	−0.233595	−	0.972334i
$7$	−0.618034	−	2.57255i	−0.233595	−	0.972334i
$8$		0			0
$9$	1.38197			0.460655
$10$		0			0
$11$	−1.61803			−0.487856			−0.243928	−	0.969793i	$-0.578436\pi$
$11$	−1.61803			−0.487856			−0.243928	+	0.969793i	$0.578436\pi$
$12$		0			0
$13$	−3.38705			−0.939400			−0.469700	−	0.882826i	$-0.655638\pi$
$13$	−3.38705			−0.939400			−0.469700	+	0.882826i	$0.655638\pi$
$14$		0			0
$15$		−	3.32821i		−	0.859341i
$16$		0			0
$17$		−	2.57255i		−	0.623936i	−0.950093	−	0.311968i	$-0.899012\pi$
$17$		−	2.57255i		−	0.623936i	0.950093	−	0.311968i	$-0.100988\pi$
$18$		0			0
$19$	−1.29374	−	4.16248i	−0.296804	−	0.954938i
$19$	−1.29374	−	4.16248i	−0.296804	−	0.954938i
$20$		0			0
$21$	−1.29374	−	5.38516i	−0.282317	−	1.17514i
$22$		0			0
$23$	−7.47214			−1.55805			−0.779024	−	0.626994i	$-0.784285\pi$
$23$	−7.47214			−1.55805			−0.779024	+	0.626994i	$0.784285\pi$
$24$		0			0
$25$	2.47214			0.494427
$26$		0			0
$27$	−3.38705			−0.651839
$28$		0			0
$29$			8.71338i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$29$			8.71338i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$30$		0			0
$31$	−8.06785			−1.44903			−0.724514	−	0.689260i	$-0.757936\pi$
$31$	−8.06785			−1.44903			−0.724514	+	0.689260i	$0.757936\pi$
$32$		0			0
$33$	−3.38705			−0.589610
$34$		0			0
$35$	−4.09017	+	0.982628i	−0.691365	+	0.166094i
$36$		0			0
$37$			5.38516i			0.885316i	0.896691	+	0.442658i	$0.145964\pi$
$37$			5.38516i			0.885316i	−0.896691	+	0.442658i	$0.854036\pi$
$38$		0			0
$39$	−7.09017			−1.13534
$40$		0			0
$41$	8.06785			1.25999			0.629993	−	0.776601i	$-0.283058\pi$
$41$	8.06785			1.25999			0.629993	+	0.776601i	$0.283058\pi$
$42$		0			0
$43$	5.70820			0.870493			0.435246	−	0.900311i	$-0.356661\pi$
$43$	5.70820			0.870493			0.435246	+	0.900311i	$0.356661\pi$
$44$		0			0
$45$		−	2.19722i		−	0.327543i
$46$		0			0
$47$		−	4.76978i		−	0.695744i	−0.937542	−	0.347872i	$-0.886904\pi$
$47$		−	4.76978i		−	0.695744i	0.937542	−	0.347872i	$-0.113096\pi$
$48$		0			0
$49$	−6.23607	+	3.17985i	−0.890867	+	0.454265i
$50$		0			0
$51$		−	5.38516i		−	0.754074i
$52$		0			0
$53$		−	14.0985i		−	1.93658i	−0.249821	−	0.968292i	$-0.580372\pi$
$53$		−	14.0985i		−	1.93658i	0.249821	−	0.968292i	$-0.419628\pi$
$54$		0			0
$55$			2.57255i			0.346883i
$56$		0			0
$57$	−2.70820	−	8.71338i	−0.358710	−	1.15412i
$58$		0			0
$59$	0.799575			0.104096			0.0520479	−	0.998645i	$-0.483425\pi$
$59$	0.799575			0.104096			0.0520479	+	0.998645i	$0.483425\pi$
$60$		0			0
$61$			6.73503i			0.862333i	0.902272	+	0.431166i	$0.141898\pi$
$61$			6.73503i			0.862333i	−0.902272	+	0.431166i	$0.858102\pi$
$62$		0			0
$63$	−0.854102	−	3.55518i	−0.107607	−	0.447911i
$64$		0			0
$65$			5.38516i			0.667947i
$66$		0			0
$67$		−	2.05695i		−	0.251296i	−0.992075	−	0.125648i	$-0.959899\pi$
$67$		−	2.05695i		−	0.251296i	0.992075	−	0.125648i	$-0.0401011\pi$
$68$		0			0
$69$	−15.6415			−1.88302
$70$		0			0
$71$		−	12.0416i		−	1.42907i	−0.699597	−	0.714537i	$-0.746637\pi$
$71$		−	12.0416i		−	1.42907i	0.699597	−	0.714537i	$-0.253363\pi$
$72$		0			0
$73$		−	9.30759i		−	1.08937i	−0.838640	−	0.544685i	$-0.816649\pi$
$73$		−	9.30759i		−	1.08937i	0.838640	−	0.544685i	$-0.183351\pi$
$74$		0			0
$75$	5.17496			0.597553
$76$		0			0
$77$	1.00000	+	4.16248i	0.113961	+	0.474359i
$78$		0			0
$79$			10.7703i			1.21176i	0.795557	+	0.605878i	$0.207178\pi$
$79$			10.7703i			1.21176i	−0.795557	+	0.605878i	$0.792822\pi$
$80$		0			0
$81$	−11.2361			−1.24845
$82$		0			0
$83$		−	2.94788i		−	0.323572i	−0.986826	−	0.161786i	$-0.948275\pi$
$83$		−	2.94788i		−	0.323572i	0.986826	−	0.161786i	$-0.0517255\pi$
$84$		0			0
$85$	−4.09017			−0.443641
$86$		0			0
$87$			18.2398i			1.95552i
$88$		0			0
$89$	8.37326			0.887564			0.443782	−	0.896135i	$-0.353637\pi$
$89$	8.37326			0.887564			0.443782	+	0.896135i	$0.353637\pi$
$90$		0			0
$91$	2.09331	+	8.71338i	0.219439	+	0.913410i
$92$		0			0
$93$	−16.8885			−1.75126
$94$		0			0
$95$	−6.61803	+	2.05695i	−0.678996	+	0.211039i
$96$		0			0
$97$	10.1612			1.03171			0.515855	−	0.856676i	$-0.327474\pi$
$97$	10.1612			1.03171			0.515855	+	0.856676i	$0.327474\pi$
$98$		0			0
$99$	−2.23607			−0.224733
$100$		0			0
$101$			1.58993i			0.158204i	0.996867	+	0.0791018i	$0.0252052\pi$
$101$			1.58993i			0.158204i	−0.996867	+	0.0791018i	$0.974795\pi$
$102$		0			0
$103$	12.7486			1.25616			0.628080	−	0.778148i	$-0.283841\pi$
$103$	12.7486			1.25616			0.628080	+	0.778148i	$0.283841\pi$
$104$		0			0
$105$	−8.56201	+	2.05695i	−0.835567	+	0.200738i
$106$		0			0
$107$			5.38516i			0.520604i	0.965527	+	0.260302i	$0.0838220\pi$
$107$			5.38516i			0.520604i	−0.965527	+	0.260302i	$0.916178\pi$
$108$		0			0
$109$		−	12.0416i		−	1.15338i	−0.816965	−	0.576688i	$-0.804345\pi$
$109$		−	12.0416i		−	1.15338i	0.816965	−	0.576688i	$-0.195655\pi$
$110$		0			0
$111$			11.2728i			1.06997i
$112$		0			0
$113$			7.44211i			0.700095i	0.936732	+	0.350048i	$0.113835\pi$
$113$			7.44211i			0.700095i	−0.936732	+	0.350048i	$0.886165\pi$
$114$		0			0
$115$			11.8801i			1.10783i
$116$		0			0
$117$	−4.68079			−0.432740
$118$		0			0
$119$	−6.61803	+	1.58993i	−0.606674	+	0.145748i
$120$		0			0
$121$	−8.38197			−0.761997
$122$		0			0
$123$	16.8885			1.52279
$124$		0			0
$125$		−	11.8801i		−	1.06259i
$126$		0			0
$127$		−	10.7703i		−	0.955712i	−0.878438	−	0.477856i	$-0.841414\pi$
$127$		−	10.7703i		−	0.955712i	0.878438	−	0.477856i	$-0.158586\pi$
$128$		0			0
$129$	11.9491			1.05206
$130$		0			0
$131$		−	10.8975i		−	0.952120i	−0.879413	−	0.476060i	$-0.842064\pi$
$131$		−	10.8975i		−	0.952120i	0.879413	−	0.476060i	$-0.157936\pi$
$132$		0			0
$133$	−9.90863	+	5.90077i	−0.859187	+	0.511662i
$134$		0			0
$135$			5.38516i			0.463481i
$136$		0			0
$137$	−6.29180			−0.537544			−0.268772	−	0.963204i	$-0.586618\pi$
$137$	−6.29180			−0.537544			−0.268772	+	0.963204i	$0.586618\pi$
$138$		0			0
$139$		−	8.70029i		−	0.737949i	−0.929440	−	0.368974i	$-0.879709\pi$
$139$		−	8.70029i		−	0.737949i	0.929440	−	0.368974i	$-0.120291\pi$
$140$		0			0
$141$		−	9.98464i		−	0.840859i
$142$		0			0
$143$	5.48037			0.458291
$144$		0			0
$145$	13.8536			1.15048
$146$		0			0
$147$	−13.0541	+	6.65643i	−1.07668	+	0.549013i
$148$		0			0
$149$	1.47214			0.120602			0.0603010	−	0.998180i	$-0.480794\pi$
$149$	1.47214			0.120602			0.0603010	+	0.998180i	$0.480794\pi$
$150$		0			0
$151$		−	19.4837i		−	1.58556i	−0.609507	−	0.792781i	$-0.708632\pi$
$151$		−	19.4837i		−	1.58556i	0.609507	−	0.792781i	$-0.291368\pi$
$152$		0			0
$153$		−	3.55518i		−	0.287419i
$154$		0			0
$155$			12.8273i			1.03031i
$156$		0			0
$157$		−	9.68292i		−	0.772781i	−0.922335	−	0.386391i	$-0.873722\pi$
$157$		−	9.68292i		−	0.772781i	0.922335	−	0.386391i	$-0.126278\pi$
$158$		0			0
$159$		−	29.5127i		−	2.34051i
$160$		0			0
$161$	4.61803	+	19.2225i	0.363952	+	1.51494i
$162$		0			0
$163$	9.38197			0.734852			0.367426	−	0.930053i	$-0.380239\pi$
$163$	9.38197			0.734852			0.367426	+	0.930053i	$0.380239\pi$
$164$		0			0
$165$			5.38516i			0.419235i
$166$		0			0
$167$	2.39873			0.185619			0.0928095	−	0.995684i	$-0.470415\pi$
$167$	2.39873			0.185619			0.0928095	+	0.995684i	$0.470415\pi$
$168$		0			0
$169$	−1.52786			−0.117528
$170$		0			0
$171$	−1.78790	−	5.75241i	−0.136724	−	0.439897i
$172$		0			0
$173$	−5.78578			−0.439885			−0.219942	−	0.975513i	$-0.570587\pi$
$173$	−5.78578			−0.439885			−0.219942	+	0.975513i	$0.570587\pi$
$174$		0			0
$175$	−1.52786	−	6.35970i	−0.115496	−	0.480748i
$176$		0			0
$177$	1.67376			0.125808
$178$		0			0
$179$		−	15.3698i		−	1.14879i	−0.818577	−	0.574397i	$-0.805237\pi$
$179$		−	15.3698i		−	1.14879i	0.818577	−	0.574397i	$-0.194763\pi$
$180$		0			0
$181$	18.0403			1.34092			0.670461	−	0.741945i	$-0.266096\pi$
$181$	18.0403			1.34092			0.670461	+	0.741945i	$0.266096\pi$
$182$		0			0
$183$			14.0985i			1.04219i
$184$		0			0
$185$	8.56201			0.629492
$186$		0			0
$187$			4.16248i			0.304391i
$188$		0			0
$189$	2.09331	+	8.71338i	0.152266	+	0.633805i
$190$		0			0
$191$	−6.03444			−0.436637			−0.218318	−	0.975878i	$-0.570057\pi$
$191$	−6.03444			−0.436637			−0.218318	+	0.975878i	$0.570057\pi$
$192$		0			0
$193$		−	8.71338i		−	0.627203i	−0.949555	−	0.313601i	$-0.898464\pi$
$193$		−	8.71338i		−	0.627203i	0.949555	−	0.313601i	$-0.101536\pi$
$194$		0			0
$195$			11.2728i			0.807265i
$196$		0			0
$197$	2.61803			0.186527			0.0932636	−	0.995641i	$-0.470270\pi$
$197$	2.61803			0.186527			0.0932636	+	0.995641i	$0.470270\pi$
$198$		0			0
$199$			11.1295i			0.788948i	0.918907	+	0.394474i	$0.129073\pi$
$199$			11.1295i			0.788948i	−0.918907	+	0.394474i	$0.870927\pi$
$200$		0			0
$201$		−	4.30584i		−	0.303711i
$202$		0			0
$203$	22.4156	−	5.38516i	1.57327	−	0.377964i
$204$		0			0
$205$		−	12.8273i		−	0.895896i
$206$		0			0
$207$	−10.3262			−0.717723
$208$		0			0
$209$	2.09331	+	6.73503i	0.144798	+	0.465872i
$210$		0			0
$211$			9.98464i			0.687371i	0.939085	+	0.343686i	$0.111675\pi$
$211$			9.98464i			0.687371i	−0.939085	+	0.343686i	$0.888325\pi$
$212$		0			0
$213$		−	25.2068i		−	1.72714i
$214$		0			0
$215$		−	9.07562i		−	0.618952i
$216$		0			0
$217$	4.98620	+	20.7550i	0.338486	+	1.40894i
$218$		0			0
$219$		−	19.4837i		−	1.31659i
$220$		0			0
$221$			8.71338i			0.586125i
$222$		0			0
$223$	12.7486			0.853712			0.426856	−	0.904320i	$-0.359621\pi$
$223$	12.7486			0.853712			0.426856	+	0.904320i	$0.359621\pi$
$224$		0			0
$225$	3.41641			0.227761
$226$		0			0
$227$	−5.48037			−0.363745			−0.181872	−	0.983322i	$-0.558216\pi$
$227$	−5.48037			−0.363745			−0.181872	+	0.983322i	$0.558216\pi$
$228$		0			0
$229$		−	20.5804i		−	1.35999i	−0.733215	−	0.679997i	$-0.761981\pi$
$229$		−	20.5804i		−	1.35999i	0.733215	−	0.679997i	$-0.238019\pi$
$230$		0			0
$231$	2.09331	+	8.71338i	0.137730	+	0.573298i
$232$		0			0
$233$	16.5623			1.08503			0.542516	−	0.840045i	$-0.317472\pi$
$233$	16.5623			1.08503			0.542516	+	0.840045i	$0.317472\pi$
$234$		0			0
$235$	−7.58359			−0.494699
$236$		0			0
$237$			22.5457i			1.46450i
$238$		0			0
$239$	−29.6525			−1.91806			−0.959030	−	0.283306i	$-0.908569\pi$
$239$	−29.6525			−1.91806			−0.959030	+	0.283306i	$0.908569\pi$
$240$		0			0
$241$	−6.77411			−0.436359			−0.218179	−	0.975909i	$-0.570012\pi$
$241$	−6.77411			−0.436359			−0.218179	+	0.975909i	$0.570012\pi$
$242$		0			0
$243$	−13.3595			−0.857010
$244$		0			0
$245$	5.05573	+	9.91489i	0.322999	+	0.633439i
$246$		0			0
$247$	4.38197	+	14.0985i	0.278818	+	0.897069i
$248$		0			0
$249$		−	6.17085i		−	0.391062i
$250$		0			0
$251$			19.2225i			1.21331i	0.794965	+	0.606656i	$0.207489\pi$
$251$			19.2225i			1.21331i	−0.794965	+	0.606656i	$0.792511\pi$
$252$		0			0
$253$	12.0902			0.760102
$254$		0			0
$255$	−8.56201			−0.536174
$256$		0			0
$257$	−21.4273			−1.33660			−0.668299	−	0.743892i	$-0.732978\pi$
$257$	−21.4273			−1.33660			−0.668299	+	0.743892i	$0.732978\pi$
$258$		0			0
$259$	13.8536	−	3.32821i	0.860823	−	0.206805i
$260$		0			0
$261$			12.0416i			0.745356i
$262$		0			0
$263$	−6.43769			−0.396965			−0.198483	−	0.980104i	$-0.563601\pi$
$263$	−6.43769			−0.396965			−0.198483	+	0.980104i	$0.563601\pi$
$264$		0			0
$265$	−22.4156			−1.37698
$266$		0			0
$267$	17.5279			1.07269
$268$		0			0
$269$	0.494165			0.0301297			0.0150649	−	0.999887i	$-0.495205\pi$
$269$	0.494165			0.0301297			0.0150649	+	0.999887i	$0.495205\pi$
$270$		0			0
$271$			4.53781i			0.275652i	0.990456	+	0.137826i	$0.0440115\pi$
$271$			4.53781i			0.275652i	−0.990456	+	0.137826i	$0.955988\pi$
$272$		0			0
$273$	4.38197	+	18.2398i	0.265209	+	1.10393i
$274$		0			0
$275$	−4.00000			−0.241209
$276$		0			0
$277$	14.9443			0.897914			0.448957	−	0.893553i	$-0.351796\pi$
$277$	14.9443			0.897914			0.448957	+	0.893553i	$0.351796\pi$
$278$		0			0
$279$	−11.1495			−0.667503
$280$		0			0
$281$		−	16.1555i		−	0.963756i	−0.876238	−	0.481878i	$-0.839955\pi$
$281$		−	16.1555i		−	0.963756i	0.876238	−	0.481878i	$-0.160045\pi$
$282$		0			0
$283$			0.231967i			0.0137890i	0.999976	+	0.00689450i	$0.00219461\pi$
$283$			0.231967i			0.0137890i	−0.999976	+	0.00689450i	$0.997805\pi$
$284$		0			0
$285$	−13.8536	+	4.30584i	−0.820618	+	0.255056i
$286$		0			0
$287$	−4.98620	−	20.7550i	−0.294326	−	1.22513i
$288$		0			0
$289$	10.3820			0.610704
$290$		0			0
$291$	21.2705			1.24690
$292$		0			0
$293$	−5.97453			−0.349036			−0.174518	−	0.984654i	$-0.555837\pi$
$293$	−5.97453			−0.349036			−0.174518	+	0.984654i	$0.555837\pi$
$294$		0			0
$295$		−	1.27126i		−	0.0740159i
$296$		0			0
$297$	5.48037			0.318003
$298$		0			0
$299$	25.3085			1.46363
$300$		0			0
$301$	−3.52786	−	14.6847i	−0.203343	−	0.846410i
$302$		0			0
$303$			3.32821i			0.191201i
$304$		0			0
$305$	10.7082			0.613150
$306$		0			0
$307$	−6.46870			−0.369188			−0.184594	−	0.982815i	$-0.559097\pi$
$307$	−6.46870			−0.369188			−0.184594	+	0.982815i	$0.559097\pi$
$308$		0			0
$309$	26.6869			1.51817
$310$		0			0
$311$		−	18.0079i		−	1.02113i	−0.859838	−	0.510567i	$-0.829436\pi$
$311$		−	18.0079i		−	1.02113i	0.859838	−	0.510567i	$-0.170564\pi$
$312$		0			0
$313$			25.7255i			1.45409i	0.686588	+	0.727047i	$0.259108\pi$
$313$			25.7255i			1.45409i	−0.686588	+	0.727047i	$0.740892\pi$
$314$		0			0
$315$	−5.65248	+	1.35796i	−0.318481	+	0.0765123i
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		−	14.0985i		−	0.789367i
$320$		0			0
$321$			11.2728i			0.629189i
$322$		0			0
$323$	−10.7082	+	3.32821i	−0.595820	+	0.185187i
$324$		0			0
$325$	−8.37326			−0.464465
$326$		0			0
$327$		−	25.2068i		−	1.39394i
$328$		0			0
$329$	−12.2705	+	2.94788i	−0.676495	+	0.162522i
$330$		0			0
$331$			20.7550i			1.14080i	0.821368	+	0.570398i	$0.193211\pi$
$331$			20.7550i			1.14080i	−0.821368	+	0.570398i	$0.806789\pi$
$332$		0			0
$333$			7.44211i			0.407825i
$334$		0			0
$335$	−3.27040			−0.178681
$336$		0			0
$337$		−	12.8273i		−	0.698746i	−0.936984	−	0.349373i	$-0.886395\pi$
$337$		−	12.8273i		−	0.698746i	0.936984	−	0.349373i	$-0.113605\pi$
$338$		0			0
$339$			15.5787i			0.846118i
$340$		0			0
$341$	13.0541			0.706917
$342$		0			0
$343$	12.0344	+	14.0774i	0.649799	+	0.760106i
$344$		0			0
$345$			24.8689i			1.33890i
$346$		0			0
$347$	0.437694			0.0234967			0.0117483	−	0.999931i	$-0.496260\pi$
$347$	0.437694			0.0234967			0.0117483	+	0.999931i	$0.496260\pi$
$348$		0			0
$349$			22.0270i			1.17908i	0.807740	+	0.589539i	$0.200691\pi$
$349$			22.0270i			1.17908i	−0.807740	+	0.589539i	$0.799309\pi$
$350$		0			0
$351$	11.4721			0.612337
$352$		0			0
$353$		−	32.6925i		−	1.74005i	−0.493009	−	0.870024i	$-0.664103\pi$
$353$		−	32.6925i		−	1.74005i	0.493009	−	0.870024i	$-0.335897\pi$
$354$		0			0
$355$	−19.1452			−1.01612
$356$		0			0
$357$	−13.8536	+	3.32821i	−0.733212	+	0.176148i
$358$		0			0
$359$	−32.7426			−1.72809			−0.864045	−	0.503414i	$-0.832077\pi$
$359$	−32.7426			−1.72809			−0.864045	+	0.503414i	$0.832077\pi$
$360$		0			0
$361$	−15.6525	+	10.7703i	−0.823815	+	0.566859i
$362$		0			0
$363$	−17.5461			−0.920931
$364$		0			0
$365$	−14.7984			−0.774582
$366$		0			0
$367$		−	29.7447i		−	1.55266i	−0.630328	−	0.776329i	$-0.717079\pi$
$367$		−	29.7447i		−	1.55266i	0.630328	−	0.776329i	$-0.282921\pi$
$368$		0			0
$369$	11.1495			0.580419
$370$		0			0
$371$	−36.2693	+	8.71338i	−1.88301	+	0.452376i
$372$		0			0
$373$			7.44211i			0.385338i	0.981264	+	0.192669i	$0.0617144\pi$
$373$			7.44211i			0.385338i	−0.981264	+	0.192669i	$0.938286\pi$
$374$		0			0
$375$		−	24.8689i		−	1.28422i
$376$		0			0
$377$		−	29.5127i		−	1.51998i
$378$		0			0
$379$			12.0416i			0.618535i	0.950975	+	0.309268i	$0.100084\pi$
$379$			12.0416i			0.618535i	−0.950975	+	0.309268i	$0.899916\pi$
$380$		0			0
$381$		−	22.5457i		−	1.15505i
$382$		0			0
$383$	−23.7094			−1.21149			−0.605746	−	0.795658i	$-0.707125\pi$
$383$	−23.7094			−1.21149			−0.605746	+	0.795658i	$0.707125\pi$
$384$		0			0
$385$	6.61803	−	1.58993i	0.337286	−	0.0810301i
$386$		0			0
$387$	7.88854			0.400997
$388$		0			0
$389$	27.5066			1.39464			0.697319	−	0.716761i	$-0.254376\pi$
$389$	27.5066			1.39464			0.697319	+	0.716761i	$0.254376\pi$
$390$		0			0
$391$			19.2225i			0.972122i
$392$		0			0
$393$		−	22.8119i		−	1.15071i
$394$		0			0
$395$	17.1240			0.861603
$396$		0			0
$397$			10.2902i			0.516451i	0.966085	+	0.258226i	$0.0831378\pi$
$397$			10.2902i			0.516451i	−0.966085	+	0.258226i	$0.916862\pi$
$398$		0			0
$399$	−20.7419	+	12.3522i	−1.03839	+	0.618382i
$400$		0			0
$401$		−	15.3698i		−	0.767532i	−0.923430	−	0.383766i	$-0.874627\pi$
$401$		−	15.3698i		−	0.767532i	0.923430	−	0.383766i	$-0.125373\pi$
$402$		0			0
$403$	27.3262			1.36122
$404$		0			0
$405$			17.8645i			0.887695i
$406$		0			0
$407$		−	8.71338i		−	0.431906i
$408$		0			0
$409$	−7.87909			−0.389596			−0.194798	−	0.980843i	$-0.562405\pi$
$409$	−7.87909			−0.389596			−0.194798	+	0.980843i	$0.562405\pi$
$410$		0			0
$411$	−13.1707			−0.649663
$412$		0			0
$413$	−0.494165	−	2.05695i	−0.0243162	−	0.101216i
$414$		0			0
$415$	−4.68692			−0.230072
$416$		0			0
$417$		−	18.2124i		−	0.891867i
$418$		0			0
$419$			26.5648i			1.29778i	0.760884	+	0.648888i	$0.224766\pi$
$419$			26.5648i			1.29778i	−0.760884	+	0.648888i	$0.775234\pi$
$420$		0			0
$421$			22.8119i			1.11179i	0.831254	+	0.555893i	$0.187623\pi$
$421$			22.8119i			1.11179i	−0.831254	+	0.555893i	$0.812377\pi$
$422$		0			0
$423$		−	6.59167i		−	0.320498i
$424$		0			0
$425$		−	6.35970i		−	0.308491i
$426$		0			0
$427$	17.3262	−	4.16248i	0.838475	−	0.201437i
$428$		0			0
$429$	11.4721			0.553880
$430$		0			0
$431$		−	6.65643i		−	0.320629i	−0.987066	−	0.160314i	$-0.948749\pi$
$431$		−	6.65643i		−	0.320629i	0.987066	−	0.160314i	$-0.0512508\pi$
$432$		0			0
$433$	−18.5344			−0.890707			−0.445354	−	0.895355i	$-0.646922\pi$
$433$	−18.5344			−0.890707			−0.445354	+	0.895355i	$0.646922\pi$
$434$		0			0
$435$	29.0000			1.39044
$436$		0			0
$437$	9.66700	+	31.1026i	0.462435	+	1.48784i
$438$		0			0
$439$	35.4697			1.69288			0.846438	−	0.532487i	$-0.178743\pi$
$439$	35.4697			1.69288			0.846438	+	0.532487i	$0.178743\pi$
$440$		0			0
$441$	−8.61803	+	4.39445i	−0.410383	+	0.209259i
$442$		0			0
$443$	19.7426			0.938001			0.469001	−	0.883198i	$-0.344614\pi$
$443$	19.7426			0.938001			0.469001	+	0.883198i	$0.344614\pi$
$444$		0			0
$445$		−	13.3129i		−	0.631090i
$446$		0			0
$447$	3.08164			0.145757
$448$		0			0
$449$			31.5253i			1.48777i	0.668307	+	0.743886i	$0.267019\pi$
$449$			31.5253i			1.48777i	−0.668307	+	0.743886i	$0.732981\pi$
$450$		0			0
$451$	−13.0541			−0.614691
$452$		0			0
$453$		−	40.7855i		−	1.91627i
$454$		0			0
$455$	13.8536	−	3.32821i	0.649468	−	0.156029i
$456$		0			0
$457$	2.90983			0.136116			0.0680581	−	0.997681i	$-0.478320\pi$
$457$	2.90983			0.136116			0.0680581	+	0.997681i	$0.478320\pi$
$458$		0			0
$459$			8.71338i			0.406706i
$460$		0			0
$461$			17.6325i			0.821230i	0.911809	+	0.410615i	$0.134686\pi$
$461$			17.6325i			0.821230i	−0.911809	+	0.410615i	$0.865314\pi$
$462$		0			0
$463$	42.0132			1.95252			0.976258	−	0.216609i	$-0.0694996\pi$
$463$	42.0132			1.95252			0.976258	+	0.216609i	$0.0694996\pi$
$464$		0			0
$465$			26.8515i			1.24521i
$466$		0			0
$467$			6.87840i			0.318294i	0.987255	+	0.159147i	$0.0508744\pi$
$467$			6.87840i			0.318294i	−0.987255	+	0.159147i	$0.949126\pi$
$468$		0			0
$469$	−5.29161	+	1.27126i	−0.244344	+	0.0587016i
$470$		0			0
$471$		−	20.2694i		−	0.933964i
$472$		0			0
$473$	−9.23607			−0.424675
$474$		0			0
$475$	−3.19830	−	10.2902i	−0.146748	−	0.472147i
$476$		0			0
$477$		−	19.4837i		−	0.892098i
$478$		0			0
$479$			1.82189i			0.0832444i	0.999133	+	0.0416222i	$0.0132526\pi$
$479$			1.82189i			0.0832444i	−0.999133	+	0.0416222i	$0.986747\pi$
$480$		0			0
$481$		−	18.2398i		−	0.831665i
$482$		0			0
$483$	9.66700	+	40.2387i	0.439864	+	1.83092i
$484$		0			0
$485$		−	16.1555i		−	0.733583i
$486$		0			0
$487$			40.2387i			1.82339i	0.410869	+	0.911694i	$0.365225\pi$
$487$			40.2387i			1.82339i	−0.410869	+	0.911694i	$0.634775\pi$
$488$		0			0
$489$	19.6394			0.888125
$490$		0			0
$491$	13.4164			0.605474			0.302737	−	0.953074i	$-0.402100\pi$
$491$	13.4164			0.605474			0.302737	+	0.953074i	$0.402100\pi$
$492$		0			0
$493$	22.4156			1.00955
$494$		0			0
$495$			3.55518i			0.159794i
$496$		0			0
$497$	−30.9777	+	7.44211i	−1.38954	+	0.333824i
$498$		0			0
$499$	31.4508			1.40793			0.703967	−	0.710233i	$-0.251410\pi$
$499$	31.4508			1.40793			0.703967	+	0.710233i	$0.251410\pi$
$500$		0			0
$501$	5.02129			0.224335
$502$		0			0
$503$			8.32496i			0.371192i	0.982626	+	0.185596i	$0.0594215\pi$
$503$			8.32496i			0.371192i	−0.982626	+	0.185596i	$0.940578\pi$
$504$		0			0
$505$	2.52786			0.112488
$506$		0			0
$507$	−3.19830			−0.142041
$508$		0			0
$509$	26.1081			1.15722			0.578611	−	0.815604i	$-0.303595\pi$
$509$	26.1081			1.15722			0.578611	+	0.815604i	$0.303595\pi$
$510$		0			0
$511$	−23.9443	+	5.75241i	−1.05923	+	0.254471i
$512$		0			0
$513$	4.38197	+	14.0985i	0.193469	+	0.622466i
$514$		0			0
$515$		−	20.2694i		−	0.893176i
$516$		0			0
$517$			7.71766i			0.339422i
$518$		0			0
$519$	−12.1115			−0.531634
$520$		0			0
$521$	−31.4718			−1.37881			−0.689403	−	0.724378i	$-0.742127\pi$
$521$	−31.4718			−1.37881			−0.689403	+	0.724378i	$0.742127\pi$
$522$		0			0
$523$	43.6542			1.90886			0.954432	−	0.298427i	$-0.0964621\pi$
$523$	43.6542			1.90886			0.954432	+	0.298427i	$0.0964621\pi$
$524$		0			0
$525$	−3.19830	−	13.3129i	−0.139585	−	0.581021i
$526$		0			0
$527$			20.7550i			0.904101i
$528$		0			0
$529$	32.8328			1.42751
$530$		0			0
$531$	1.10499			0.0479523
$532$		0			0
$533$	−27.3262			−1.18363
$534$		0			0
$535$	8.56201			0.370168
$536$		0			0
$537$		−	32.1738i		−	1.38840i
$538$		0			0
$539$	10.0902	−	5.14511i	0.434614	−	0.221615i
$540$		0			0
$541$	−14.7639			−0.634751			−0.317376	−	0.948300i	$-0.602802\pi$
$541$	−14.7639			−0.634751			−0.317376	+	0.948300i	$0.602802\pi$
$542$		0			0
$543$	37.7639			1.62061
$544$		0			0
$545$	−19.1452			−0.820092
$546$		0			0
$547$		−	18.2124i		−	0.778708i	−0.921088	−	0.389354i	$-0.872698\pi$
$547$		−	18.2124i		−	0.778708i	0.921088	−	0.389354i	$-0.127302\pi$
$548$		0			0
$549$			9.30759i			0.397238i
$550$		0			0
$551$	36.2693	−	11.2728i	1.54512	−	0.480239i
$552$		0			0
$553$	27.7073	−	6.65643i	1.17823	−	0.283060i
$554$		0			0
$555$	17.9230			0.760788
$556$		0			0
$557$	−27.6180			−1.17021			−0.585107	−	0.810956i	$-0.698947\pi$
$557$	−27.6180			−1.17021			−0.585107	+	0.810956i	$0.698947\pi$
$558$		0			0
$559$	−19.3340			−0.817741
$560$		0			0
$561$			8.71338i			0.367879i
$562$		0			0
$563$	9.55034			0.402499			0.201249	−	0.979540i	$-0.435500\pi$
$563$	9.55034			0.402499			0.201249	+	0.979540i	$0.435500\pi$
$564$		0			0
$565$	11.8324			0.497793
$566$		0			0
$567$	6.94427	+	28.9054i	0.291632	+	1.21391i
$568$		0			0
$569$		−	5.38516i		−	0.225758i	−0.993609	−	0.112879i	$-0.963993\pi$
$569$		−	5.38516i		−	0.225758i	0.993609	−	0.112879i	$-0.0360072\pi$
$570$		0			0
$571$	3.18034			0.133093			0.0665465	−	0.997783i	$-0.478802\pi$
$571$	3.18034			0.133093			0.0665465	+	0.997783i	$0.478802\pi$
$572$		0			0
$573$	−12.6320			−0.527709
$574$		0			0
$575$	−18.4721			−0.770341
$576$		0			0
$577$			4.91314i			0.204537i	0.994757	+	0.102268i	$0.0326100\pi$
$577$			4.91314i			0.204537i	−0.994757	+	0.102268i	$0.967390\pi$
$578$		0			0
$579$		−	18.2398i		−	0.758022i
$580$		0			0
$581$	−7.58359	+	1.82189i	−0.314620	+	0.0755849i
$582$		0			0
$583$			22.8119i			0.944773i
$584$		0			0
$585$			7.44211i			0.307694i
$586$		0			0
$587$			15.4353i			0.637084i	0.947909	+	0.318542i	$0.103193\pi$
$587$			15.4353i			0.637084i	−0.947909	+	0.318542i	$0.896807\pi$
$588$		0			0
$589$	10.4377	+	33.5823i	0.430078	+	1.38373i
$590$		0			0
$591$	5.48037			0.225432
$592$		0			0
$593$		−	37.6057i		−	1.54428i	−0.635452	−	0.772140i	$-0.719186\pi$
$593$		−	37.6057i		−	1.54428i	0.635452	−	0.772140i	$-0.280814\pi$
$594$		0			0
$595$	2.52786	+	10.5222i	0.103632	+	0.431367i
$596$		0			0
$597$			23.2975i			0.953503i
$598$		0			0
$599$			46.4095i			1.89624i	0.317910	+	0.948121i	$0.397019\pi$
$599$			46.4095i			1.89624i	−0.317910	+	0.948121i	$0.602981\pi$
$600$		0			0
$601$	−30.7889			−1.25591			−0.627953	−	0.778252i	$-0.716107\pi$
$601$	−30.7889			−1.25591			−0.627953	+	0.778252i	$0.716107\pi$
$602$		0			0
$603$		−	2.84263i		−	0.115761i
$604$		0			0
$605$			13.3267i			0.541808i
$606$		0			0
$607$	−32.0826			−1.30219			−0.651097	−	0.758994i	$-0.725691\pi$
$607$	−32.0826			−1.30219			−0.651097	+	0.758994i	$0.725691\pi$
$608$		0			0
$609$	46.9230	−	11.2728i	1.90142	−	0.456799i
$610$		0			0
$611$			16.1555i			0.653581i
$612$		0			0
$613$	12.0344			0.486067			0.243033	−	0.970018i	$-0.421858\pi$
$613$	12.0344			0.486067			0.243033	+	0.970018i	$0.421858\pi$
$614$		0			0
$615$		−	26.8515i		−	1.08276i
$616$		0			0
$617$	6.20163			0.249668			0.124834	−	0.992178i	$-0.460160\pi$
$617$	6.20163			0.249668			0.124834	+	0.992178i	$0.460160\pi$
$618$		0			0
$619$			20.0617i			0.806349i	0.915123	+	0.403175i	$0.132093\pi$
$619$			20.0617i			0.806349i	−0.915123	+	0.403175i	$0.867907\pi$
$620$		0			0
$621$	25.3085			1.01560
$622$		0			0
$623$	−5.17496	−	21.5407i	−0.207330	−	0.863008i
$624$		0			0
$625$	−6.52786			−0.261115
$626$		0			0
$627$	4.38197	+	14.0985i	0.174999	+	0.563042i
$628$		0			0
$629$	13.8536			0.552380
$630$		0			0
$631$	−32.7082			−1.30209			−0.651047	−	0.759038i	$-0.725670\pi$
$631$	−32.7082			−1.30209			−0.651047	+	0.759038i	$0.725670\pi$
$632$		0			0
$633$			20.9010i			0.830740i
$634$		0			0
$635$	−17.1240			−0.679546
$636$		0			0
$637$	21.1219	−	10.7703i	0.836880	−	0.426736i
$638$		0			0
$639$		−	16.6411i		−	0.658311i
$640$		0			0
$641$		−	12.8273i		−	0.506647i	−0.967382	−	0.253324i	$-0.918476\pi$
$641$		−	12.8273i		−	0.506647i	0.967382	−	0.253324i	$-0.0815237\pi$
$642$		0			0
$643$		−	12.3441i		−	0.486803i	−0.969926	−	0.243401i	$-0.921737\pi$
$643$		−	12.3441i		−	0.486803i	0.969926	−	0.243401i	$-0.0782632\pi$
$644$		0			0
$645$		−	18.9981i		−	0.748051i
$646$		0			0
$647$		−	34.8898i		−	1.37166i	−0.727763	−	0.685829i	$-0.759440\pi$
$647$		−	34.8898i		−	1.37166i	0.727763	−	0.685829i	$-0.240560\pi$
$648$		0			0
$649$	−1.29374			−0.0507837
$650$		0			0
$651$	10.4377	+	43.4467i	0.409085	+	1.70281i
$652$		0			0
$653$	−44.1246			−1.72673			−0.863365	−	0.504580i	$-0.831647\pi$
$653$	−44.1246			−1.72673			−0.863365	+	0.504580i	$0.831647\pi$
$654$		0			0
$655$	−17.3262			−0.676992
$656$		0			0
$657$		−	12.8628i		−	0.501824i
$658$		0			0
$659$			2.05695i			0.0801274i	0.999197	+	0.0400637i	$0.0127561\pi$
$659$			2.05695i			0.0801274i	−0.999197	+	0.0400637i	$0.987244\pi$
$660$		0			0
$661$	28.6956			1.11613			0.558064	−	0.829798i	$-0.311544\pi$
$661$	28.6956			1.11613			0.558064	+	0.829798i	$0.311544\pi$
$662$		0			0
$663$			18.2398i			0.708377i
$664$		0			0
$665$	9.38178	+	15.7540i	0.363810	+	0.610913i
$666$		0			0
$667$		−	65.1076i		−	2.52097i
$668$		0			0
$669$	26.6869			1.03178
$670$		0			0
$671$		−	10.8975i		−	0.420694i
$672$		0			0
$673$			27.4114i			1.05663i	0.849048	+	0.528316i	$0.177176\pi$
$673$			27.4114i			1.05663i	−0.849048	+	0.528316i	$0.822824\pi$
$674$		0			0
$675$	−8.37326			−0.322287
$676$		0			0
$677$	31.3997			1.20679			0.603395	−	0.797443i	$-0.293814\pi$
$677$	31.3997			1.20679			0.603395	+	0.797443i	$0.293814\pi$
$678$		0			0
$679$	−6.27994	−	26.1401i	−0.241002	−	1.00317i
$680$		0			0
$681$	−11.4721			−0.439613
$682$		0			0
$683$			40.2387i			1.53969i	0.638231	+	0.769845i	$0.279667\pi$
$683$			40.2387i			1.53969i	−0.638231	+	0.769845i	$0.720333\pi$
$684$		0			0
$685$			10.0035i			0.382214i
$686$		0			0
$687$		−	43.0813i		−	1.64365i
$688$		0			0
$689$			47.7525i			1.81923i
$690$		0			0
$691$		−	44.8046i		−	1.70445i	−0.523176	−	0.852225i	$-0.675253\pi$
$691$		−	44.8046i		−	1.70445i	0.523176	−	0.852225i	$-0.324747\pi$
$692$		0			0
$693$	1.38197	+	5.75241i	0.0524965	+	0.218516i
$694$		0			0
$695$	−13.8328			−0.524709
$696$		0			0
$697$		−	20.7550i		−	0.786151i
$698$		0			0
$699$	34.6701			1.31134
$700$		0			0
$701$	−29.1246			−1.10002			−0.550011	−	0.835158i	$-0.685376\pi$
$701$	−29.1246			−1.10002			−0.550011	+	0.835158i	$0.685376\pi$
$702$		0			0
$703$	22.4156	−	6.96700i	0.845422	−	0.262765i
$704$		0			0
$705$	−15.8748			−0.597881
$706$		0			0
$707$	4.09017	−	0.982628i	0.153827	−	0.0369555i
$708$		0			0
$709$	46.4853			1.74579			0.872896	−	0.487907i	$-0.162239\pi$
$709$	46.4853			1.74579			0.872896	+	0.487907i	$0.162239\pi$
$710$		0			0
$711$			14.8842i			0.558202i
$712$		0			0
$713$	60.2841			2.25766
$714$		0			0
$715$		−	8.71338i		−	0.325862i
$716$		0			0
$717$	−62.0720			−2.31812
$718$		0			0
$719$			36.4797i			1.36046i	0.732997	+	0.680231i	$0.238121\pi$
$719$			36.4797i			1.36046i	−0.732997	+	0.680231i	$0.761879\pi$
$720$		0			0
$721$	−7.87909	−	32.7966i	−0.293433	−	1.22141i
$722$		0			0
$723$	−14.1803			−0.527373
$724$		0			0
$725$			21.5407i			0.800000i
$726$		0			0
$727$			38.8203i			1.43976i	0.694096	+	0.719882i	$0.255804\pi$
$727$			38.8203i			1.43976i	−0.694096	+	0.719882i	$0.744196\pi$
$728$		0			0
$729$	5.74265			0.212691
$730$		0			0
$731$		−	14.6847i		−	0.543132i
$732$		0			0
$733$		−	13.4701i		−	0.497528i	−0.968564	−	0.248764i	$-0.919976\pi$
$733$		−	13.4701i		−	0.497528i	0.968564	−	0.248764i	$-0.0800244\pi$
$734$		0			0
$735$	10.5832	+	20.7550i	0.390368	+	0.765559i
$736$		0			0
$737$			3.32821i			0.122596i
$738$		0			0
$739$	−9.18034			−0.337704			−0.168852	−	0.985641i	$-0.554006\pi$
$739$	−9.18034			−0.337704			−0.168852	+	0.985641i	$0.554006\pi$
$740$		0			0
$741$	9.17283	+	29.5127i	0.336972	+	1.08418i
$742$		0			0
$743$		−	40.2387i		−	1.47621i	−0.674684	−	0.738107i	$-0.735720\pi$
$743$		−	40.2387i		−	1.47621i	0.674684	−	0.738107i	$-0.264280\pi$
$744$		0			0
$745$		−	2.34059i		−	0.0857525i
$746$		0			0
$747$		−	4.07388i		−	0.149055i
$748$		0			0
$749$	13.8536	−	3.32821i	0.506201	−	0.121610i
$750$		0			0
$751$			46.4095i			1.69351i	0.531985	+	0.846754i	$0.321446\pi$
$751$			46.4095i			1.69351i	−0.531985	+	0.846754i	$0.678554\pi$
$752$		0			0
$753$			40.2387i			1.46638i
$754$		0			0
$755$	−30.9777			−1.12739
$756$		0			0
$757$	9.88854			0.359405			0.179703	−	0.983721i	$-0.442486\pi$
$757$	9.88854			0.359405			0.179703	+	0.983721i	$0.442486\pi$
$758$		0			0
$759$	25.3085			0.918641
$760$		0			0
$761$		−	34.1391i		−	1.23754i	−0.785572	−	0.618771i	$-0.787631\pi$
$761$		−	34.1391i		−	1.23754i	0.785572	−	0.618771i	$-0.212369\pi$
$762$		0			0
$763$	−30.9777	+	7.44211i	−1.12147	+	0.269423i
$764$		0			0
$765$	−5.65248			−0.204366
$766$		0			0
$767$	−2.70820			−0.0977876
$768$		0			0
$769$			30.8706i			1.11322i	0.830773	+	0.556612i	$0.187899\pi$
$769$			30.8706i			1.11322i	−0.830773	+	0.556612i	$0.812101\pi$
$770$		0			0
$771$	−44.8541			−1.61538
$772$		0			0
$773$	38.0572			1.36882			0.684411	−	0.729097i	$-0.260060\pi$
$773$	38.0572			1.36882			0.684411	+	0.729097i	$0.260060\pi$
$774$		0			0
$775$	−19.9448			−0.716439
$776$		0			0
$777$	29.0000	−	6.96700i	1.04037	−	0.249940i
$778$		0			0
$779$	−10.4377	−	33.5823i	−0.373969	−	1.20321i
$780$		0			0
$781$			19.4837i			0.697182i
$782$		0			0
$783$		−	29.5127i		−	1.05470i
$784$		0			0
$785$	−15.3951			−0.549475
$786$		0			0
$787$	−7.76244			−0.276701			−0.138351	−	0.990383i	$-0.544180\pi$
$787$	−7.76244			−0.276701			−0.138351	+	0.990383i	$0.544180\pi$
$788$		0			0
$789$	−13.4761			−0.479763
$790$		0			0
$791$	19.1452	−	4.59948i	0.680726	−	0.163539i
$792$		0			0
$793$		−	22.8119i		−	0.810075i
$794$		0			0
$795$	−46.9230			−1.66419
$796$		0			0
$797$	−19.6394			−0.695663			−0.347832	−	0.937557i	$-0.613082\pi$
$797$	−19.6394			−0.695663			−0.347832	+	0.937557i	$0.613082\pi$
$798$		0			0
$799$	−12.2705			−0.434099
$800$		0			0
$801$	11.5716			0.408861
$802$		0			0
$803$			15.0600i			0.531456i
$804$		0			0
$805$	30.5623	−	7.34233i	1.07718	−	0.258783i
$806$		0			0
$807$	1.03444			0.0364141
$808$		0			0
$809$	−39.0689			−1.37359			−0.686794	−	0.726852i	$-0.740983\pi$
$809$	−39.0689			−1.37359			−0.686794	+	0.726852i	$0.740983\pi$
$810$		0			0
$811$	41.2555			1.44868			0.724338	−	0.689445i	$-0.242146\pi$
$811$	41.2555			1.44868			0.724338	+	0.689445i	$0.242146\pi$
$812$		0			0
$813$			9.49906i			0.333147i
$814$		0			0
$815$		−	14.9166i		−	0.522507i
$816$		0			0
$817$	−7.38493	−	23.7603i	−0.258366	−	0.831267i
$818$		0			0
$819$	2.89289	+	12.0416i	0.101086	+	0.420767i
$820$		0			0
$821$	26.5410			0.926288			0.463144	−	0.886283i	$-0.346721\pi$
$821$	26.5410			0.926288			0.463144	+	0.886283i	$0.346721\pi$
$822$		0			0
$823$	−6.65248			−0.231891			−0.115945	−	0.993256i	$-0.536990\pi$
$823$	−6.65248			−0.231891			−0.115945	+	0.993256i	$0.536990\pi$
$824$		0			0
$825$	−8.37326			−0.291519
$826$		0			0
$827$			36.1248i			1.25618i	0.778140	+	0.628091i	$0.216163\pi$
$827$			36.1248i			1.25618i	−0.778140	+	0.628091i	$0.783837\pi$
$828$		0			0
$829$	3.19830			0.111082			0.0555408	−	0.998456i	$-0.482312\pi$
$829$	3.19830			0.111082			0.0555408	+	0.998456i	$0.482312\pi$
$830$		0			0
$831$	31.2831			1.08520
$832$		0			0
$833$	8.18034	+	16.0426i	0.283432	+	0.555844i
$834$		0			0
$835$		−	3.81379i		−	0.131982i
$836$		0			0
$837$	27.3262			0.944533
$838$		0			0
$839$	38.6680			1.33497			0.667484	−	0.744624i	$-0.267371\pi$
$839$	38.6680			1.33497			0.667484	+	0.744624i	$0.267371\pi$
$840$		0			0
$841$	−46.9230			−1.61803
$842$		0			0
$843$		−	33.8185i		−	1.16477i
$844$		0			0
$845$			2.42919i			0.0835667i
$846$		0			0
$847$	5.18034	+	21.5631i	0.177999	+	0.740916i
$848$		0			0
$849$			0.485580i			0.0166651i
$850$		0			0
$851$		−	40.2387i		−	1.37936i
$852$		0			0
$853$			47.2886i			1.61913i	0.587030	+	0.809565i	$0.300297\pi$
$853$			47.2886i			1.61913i	−0.587030	+	0.809565i	$0.699703\pi$
$854$		0			0
$855$	−9.14590	+	2.84263i	−0.312783	+	0.0972161i
$856$		0			0
$857$	−23.5927			−0.805912			−0.402956	−	0.915219i	$-0.632017\pi$
$857$	−23.5927			−0.805912			−0.402956	+	0.915219i	$0.632017\pi$
$858$		0			0
$859$		−	16.0426i		−	0.547367i	−0.961820	−	0.273684i	$-0.911758\pi$
$859$		−	16.0426i		−	0.547367i	0.961820	−	0.273684i	$-0.0882421\pi$
$860$		0			0
$861$	−10.4377	−	43.4467i	−0.355716	−	1.48066i
$862$		0			0
$863$		−	0.785685i		−	0.0267450i	−0.999911	−	0.0133725i	$-0.995743\pi$
$863$		−	0.785685i		−	0.0267450i	0.999911	−	0.0133725i	$-0.00425673\pi$
$864$		0			0
$865$			9.19896i			0.312774i
$866$		0			0
$867$	21.7327			0.738082
$868$		0			0
$869$		−	17.4268i		−	0.591162i
$870$		0			0
$871$			6.96700i			0.236068i
$872$		0			0
$873$	14.0424			0.475263
$874$		0			0
$875$	−30.5623	+	7.34233i	−1.03319	+	0.248216i
$876$		0			0
$877$			14.8842i			0.502605i	0.967909	+	0.251302i	$0.0808588\pi$
$877$			14.8842i			0.502605i	−0.967909	+	0.251302i	$0.919141\pi$
$878$		0			0
$879$	−12.5066			−0.421836
$880$		0			0
$881$		−	38.6769i		−	1.30306i	−0.758624	−	0.651529i	$-0.774128\pi$
$881$		−	38.6769i		−	1.30306i	0.758624	−	0.651529i	$-0.225872\pi$
$882$		0			0
$883$	−24.5967			−0.827746			−0.413873	−	0.910335i	$-0.635824\pi$
$883$	−24.5967			−0.827746			−0.413873	+	0.910335i	$0.635824\pi$
$884$		0			0
$885$		−	2.66116i		−	0.0894538i
$886$		0			0
$887$	−34.2926			−1.15143			−0.575717	−	0.817649i	$-0.695277\pi$
$887$	−34.2926			−1.15143			−0.575717	+	0.817649i	$0.695277\pi$
$888$		0			0
$889$	−27.7073	+	6.65643i	−0.929271	+	0.223249i
$890$		0			0
$891$	18.1803			0.609064
$892$		0			0
$893$	−19.8541	+	6.17085i	−0.664392	+	0.206500i
$894$		0			0
$895$	−24.4369			−0.816834
$896$		0			0
$897$	52.9787			1.76891
$898$		0			0
$899$		−	70.2982i		−	2.34458i
$900$		0			0
$901$	−36.2693			−1.20830
$902$		0			0
$903$	−7.38493	−	30.7396i	−0.245755	−	1.02295i
$904$		0			0
$905$		−	28.6827i		−	0.953444i
$906$		0			0
$907$			20.2694i			0.673034i	0.941677	+	0.336517i	$0.109249\pi$
$907$			20.2694i			0.673034i	−0.941677	+	0.336517i	$0.890751\pi$
$908$		0			0
$909$			2.19722i			0.0728773i
$910$		0			0
$911$		−	17.4268i		−	0.577374i	−0.957423	−	0.288687i	$-0.906781\pi$
$911$		−	17.4268i		−	0.577374i	0.957423	−	0.288687i	$-0.0932187\pi$
$912$		0			0
$913$			4.76978i			0.157857i
$914$		0			0
$915$	22.4156			0.741038
$916$		0			0
$917$	−28.0344	+	6.73503i	−0.925779	+	0.222410i
$918$		0			0
$919$	−18.2361			−0.601552			−0.300776	−	0.953695i	$-0.597246\pi$
$919$	−18.2361			−0.601552			−0.300776	+	0.953695i	$0.597246\pi$
$920$		0			0
$921$	−13.5410			−0.446192
$922$		0			0
$923$			40.7855i			1.34247i
$924$		0			0
$925$			13.3129i			0.437724i
$926$		0			0
$927$	17.6182			0.578657
$928$		0			0
$929$		−	8.09299i		−	0.265522i	−0.991148	−	0.132761i	$-0.957616\pi$
$929$		−	8.09299i		−	0.265522i	0.991148	−	0.132761i	$-0.0423843\pi$
$930$		0			0
$931$	21.3039	+	21.8436i	0.698208	+	0.715895i
$932$		0			0
$933$		−	37.6962i		−	1.23412i
$934$		0			0
$935$	6.61803			0.216433
$936$		0			0
$937$		−	21.6517i		−	0.707329i	−0.935372	−	0.353665i	$-0.884935\pi$
$937$		−	21.6517i		−	0.707329i	0.935372	−	0.353665i	$-0.115065\pi$
$938$		0			0
$939$			53.8516i			1.75738i
$940$		0			0
$941$	−50.6171			−1.65007			−0.825034	−	0.565082i	$-0.808844\pi$
$941$	−50.6171			−1.65007			−0.825034	+	0.565082i	$0.808844\pi$
$942$		0			0
$943$	−60.2841			−1.96312
$944$		0			0
$945$	13.8536	−	3.32821i	0.450659	−	0.108267i
$946$		0			0
$947$	22.0344			0.716023			0.358012	−	0.933717i	$-0.383455\pi$
$947$	22.0344			0.716023			0.358012	+	0.933717i	$0.383455\pi$
$948$		0			0
$949$			31.5253i			1.02335i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	22.0262i		−	0.713500i	−0.934200	−	0.356750i	$-0.883885\pi$
$953$		−	22.0262i		−	0.713500i	0.934200	−	0.356750i	$-0.116115\pi$
$954$		0			0
$955$			9.59431i			0.310465i
$956$		0			0
$957$		−	29.5127i		−	0.954010i
$958$		0			0
$959$	3.88854	+	16.1860i	0.125568	+	0.522673i
$960$		0			0
$961$	34.0902			1.09968
$962$		0			0
$963$			7.44211i			0.239819i
$964$		0			0
$965$	−13.8536			−0.445964
$966$		0			0
$967$	48.0902			1.54648			0.773238	−	0.634116i	$-0.218636\pi$
$967$	48.0902			1.54648			0.773238	+	0.634116i	$0.218636\pi$
$968$		0			0
$969$	−22.4156	+	6.96700i	−0.720094	+	0.223812i
$970$		0			0
$971$	29.4952			0.946545			0.473272	−	0.880916i	$-0.343073\pi$
$971$	29.4952			0.946545			0.473272	+	0.880916i	$0.343073\pi$
$972$		0			0
$973$	−22.3820	+	5.37708i	−0.717533	+	0.172381i
$974$		0			0
$975$	−17.5279			−0.561341
$976$		0			0
$977$		−	19.9693i		−	0.638874i	−0.947607	−	0.319437i	$-0.896506\pi$
$977$		−	19.9693i		−	0.638874i	0.947607	−	0.319437i	$-0.103494\pi$
$978$		0			0
$979$	−13.5482			−0.433003
$980$		0			0
$981$		−	16.6411i		−	0.531309i
$982$		0			0
$983$	−21.7327			−0.693166			−0.346583	−	0.938019i	$-0.612658\pi$
$983$	−21.7327			−0.693166			−0.346583	+	0.938019i	$0.612658\pi$
$984$		0			0
$985$		−	4.16248i		−	0.132628i
$986$		0			0
$987$	−25.6860	+	6.17085i	−0.817596	+	0.196420i
$988$		0			0
$989$	−42.6525			−1.35627
$990$		0			0
$991$		−	24.0832i		−	0.765028i	−0.923950	−	0.382514i	$-0.875058\pi$
$991$		−	24.0832i		−	0.765028i	0.923950	−	0.382514i	$-0.124942\pi$
$992$		0			0
$993$			43.4467i			1.37874i
$994$		0			0
$995$	17.6950			0.560971
$996$		0			0
$997$		−	2.19722i		−	0.0695868i	−0.999395	−	0.0347934i	$-0.988923\pi$
$997$		−	2.19722i		−	0.0695868i	0.999395	−	0.0347934i	$-0.0110773\pi$
$998$		0			0
$999$		−	18.2398i		−	0.577083i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2128.2.m.e.1329.5		8
4.3	odd	2		532.2.g.b.265.3	✓	8
7.6	odd	2	inner	2128.2.m.e.1329.4		8
12.11	even	2		4788.2.i.e.3457.5		8
19.18	odd	2	inner	2128.2.m.e.1329.3		8
28.27	even	2		532.2.g.b.265.6	yes	8
76.75	even	2		532.2.g.b.265.5	yes	8
84.83	odd	2		4788.2.i.e.3457.4		8
133.132	even	2	inner	2128.2.m.e.1329.6		8
228.227	odd	2		4788.2.i.e.3457.6		8
532.531	odd	2		532.2.g.b.265.4	yes	8
1596.1595	even	2		4788.2.i.e.3457.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.g.b.265.3	✓	8	4.3	odd	2
532.2.g.b.265.4	yes	8	532.531	odd	2
532.2.g.b.265.5	yes	8	76.75	even	2
532.2.g.b.265.6	yes	8	28.27	even	2
2128.2.m.e.1329.3		8	19.18	odd	2	inner
2128.2.m.e.1329.4		8	7.6	odd	2	inner
2128.2.m.e.1329.5		8	1.1	even	1	trivial
2128.2.m.e.1329.6		8	133.132	even	2	inner
4788.2.i.e.3457.3		8	1596.1595	even	2
4788.2.i.e.3457.4		8	84.83	odd	2
4788.2.i.e.3457.5		8	12.11	even	2
4788.2.i.e.3457.6		8	228.227	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:	\( N \)	\(=\)	\( 2128 = 2^{4} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2128.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.09331 q^{3} -1.58993i q^{5} +(-0.618034 - 2.57255i) q^{7} +1.38197 q^{9} -1.61803 q^{11} -3.38705 q^{13} -3.32821i q^{15} -2.57255i q^{17} +(-1.29374 - 4.16248i) q^{19} +(-1.29374 - 5.38516i) q^{21} -7.47214 q^{23} +2.47214 q^{25} -3.38705 q^{27} +8.71338i q^{29} -8.06785 q^{31} -3.38705 q^{33} +(-4.09017 + 0.982628i) q^{35} +5.38516i q^{37} -7.09017 q^{39} +8.06785 q^{41} +5.70820 q^{43} -2.19722i q^{45} -4.76978i q^{47} +(-6.23607 + 3.17985i) q^{49} -5.38516i q^{51} -14.0985i q^{53} +2.57255i q^{55} +(-2.70820 - 8.71338i) q^{57} +0.799575 q^{59} +6.73503i q^{61} +(-0.854102 - 3.55518i) q^{63} +5.38516i q^{65} -2.05695i q^{67} -15.6415 q^{69} -12.0416i q^{71} -9.30759i q^{73} +5.17496 q^{75} +(1.00000 + 4.16248i) q^{77} +10.7703i q^{79} -11.2361 q^{81} -2.94788i q^{83} -4.09017 q^{85} +18.2398i q^{87} +8.37326 q^{89} +(2.09331 + 8.71338i) q^{91} -16.8885 q^{93} +(-6.61803 + 2.05695i) q^{95} +10.1612 q^{97} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7} + 20 q^{9} - 4 q^{11} - 24 q^{23} - 16 q^{25} + 12 q^{35} - 12 q^{39} - 8 q^{43} - 32 q^{49} + 32 q^{57} + 20 q^{63} + 8 q^{77} - 72 q^{81} + 12 q^{85} + 8 q^{93} - 44 q^{95}+O(q^{100}) \) 8 * q + 4 * q^7 + 20 * q^9 - 4 * q^11 - 24 * q^23 - 16 * q^25 + 12 * q^35 - 12 * q^39 - 8 * q^43 - 32 * q^49 + 32 * q^57 + 20 * q^63 + 8 * q^77 - 72 * q^81 + 12 * q^85 + 8 * q^93 - 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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