LMFDB - Embedded newform 768.3.h.g.641.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(641,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("768.641");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$20.9264843029$
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} - 4 x^{15} + 14 x^{14} - 28 x^{13} + 50 x^{12} - 104 x^{11} - 66 x^{10} + 640 x^{9} + 555 x^{8} + \cdots + 2401 $ x^16 - 4x^15 + 14x^14 - 28x^13 + 50x^12 - 104x^11 - 66x^10 + 640x^9 + 555x^8 - 7060x^7 + 17714x^6 - 25496x^5 + 24840x^4 - 17932x^3 + 11724x^2 - 7056*x + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{34}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.1
Root			$1.38361 + 0.573111i$ of defining polynomial
Character	$\chi$	$=$	768.641
Dual form			768.3.h.g.641.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.98985 - 0.246559i) q^{3} -6.63641 q^{5} -0.578158 q^{7} +(8.87842 + 1.47435i) q^{9} +O(q^{10})$	\(q+(-2.98985 - 0.246559i) q^{3} -6.63641 q^{5} -0.578158 q^{7} +(8.87842 + 1.47435i) q^{9} +8.68786 q^{11} -17.9269i q^{13} +(19.8419 + 1.63626i) q^{15} +19.0110i q^{17} -32.1769i q^{19} +(1.72861 + 0.142550i) q^{21} -20.4836i q^{23} +19.0419 q^{25} +(-26.1816 - 6.59713i) q^{27} -22.0310 q^{29} +26.2344 q^{31} +(-25.9754 - 2.14207i) q^{33} +3.83689 q^{35} +53.3855i q^{37} +(-4.42003 + 53.5988i) q^{39} +35.6935i q^{41} +50.4895i q^{43} +(-58.9208 - 9.78437i) q^{45} -30.6265i q^{47} -48.6657 q^{49} +(4.68732 - 56.8401i) q^{51} -88.8962 q^{53} -57.6562 q^{55} +(-7.93348 + 96.2040i) q^{57} +63.1939 q^{59} +33.5317i q^{61} +(-5.13313 - 0.852405i) q^{63} +118.970i q^{65} +108.562i q^{67} +(-5.05041 + 61.2430i) q^{69} -59.3600i q^{71} +5.60477 q^{73} +(-56.9325 - 4.69495i) q^{75} -5.02296 q^{77} -78.9955 q^{79} +(76.6526 + 26.1797i) q^{81} -48.5283 q^{83} -126.165i q^{85} +(65.8694 + 5.43193i) q^{87} -58.7109i q^{89} +10.3646i q^{91} +(-78.4369 - 6.46831i) q^{93} +213.539i q^{95} +93.3544 q^{97} +(77.1345 + 12.8089i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{7}+O(q^{10}) $ 16 * q - 16 * q^7	$ 16 q - 16 q^{7} + 32 q^{15} + 80 q^{25} - 112 q^{31} + 16 q^{33} + 208 q^{39} + 144 q^{49} - 384 q^{55} + 80 q^{57} + 528 q^{63} + 160 q^{73} - 816 q^{79} + 144 q^{81} + 736 q^{87} + 192 q^{97}+O(q^{100}) $ 16 * q - 16 * q^7 + 32 * q^15 + 80 * q^25 - 112 * q^31 + 16 * q^33 + 208 * q^39 + 144 * q^49 - 384 * q^55 + 80 * q^57 + 528 * q^63 + 160 * q^73 - 816 * q^79 + 144 * q^81 + 736 * q^87 + 192 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−2.98985	−	0.246559i	−0.996617	−	0.0821862i
$3$	−2.98985	−	0.246559i	−0.996617	−	0.0821862i
$4$		0			0
$5$	−6.63641			−1.32728			−0.663641	−	0.748051i	$-0.730990\pi$
$5$	−6.63641			−1.32728			−0.663641	+	0.748051i	$0.730990\pi$
$6$		0			0
$7$	−0.578158			−0.0825940			−0.0412970	−	0.999147i	$-0.513149\pi$
$7$	−0.578158			−0.0825940			−0.0412970	+	0.999147i	$0.513149\pi$
$8$		0			0
$9$	8.87842	+	1.47435i	0.986491	+	0.163816i
$10$		0			0
$11$	8.68786			0.789806			0.394903	−	0.918723i	$-0.370778\pi$
$11$	8.68786			0.789806			0.394903	+	0.918723i	$0.370778\pi$
$12$		0			0
$13$		−	17.9269i		−	1.37899i	−0.724289	−	0.689497i	$-0.757832\pi$
$13$		−	17.9269i		−	1.37899i	0.724289	−	0.689497i	$-0.242168\pi$
$14$		0			0
$15$	19.8419	+	1.63626i	1.32279	+	0.109084i
$16$		0			0
$17$			19.0110i			1.11829i	0.829068	+	0.559147i	$0.188871\pi$
$17$			19.0110i			1.11829i	−0.829068	+	0.559147i	$0.811129\pi$
$18$		0			0
$19$		−	32.1769i		−	1.69352i	−0.531976	−	0.846760i	$-0.678550\pi$
$19$		−	32.1769i		−	1.69352i	0.531976	−	0.846760i	$-0.321450\pi$
$20$		0			0
$21$	1.72861	+	0.142550i	0.0823146	+	0.00678808i
$22$		0			0
$23$		−	20.4836i		−	0.890592i	−0.895383	−	0.445296i	$-0.853098\pi$
$23$		−	20.4836i		−	0.890592i	0.895383	−	0.445296i	$-0.146902\pi$
$24$		0			0
$25$	19.0419			0.761677
$26$		0			0
$27$	−26.1816	−	6.59713i	−0.969690	−	0.244338i
$28$		0			0
$29$	−22.0310			−0.759690			−0.379845	−	0.925050i	$-0.624023\pi$
$29$	−22.0310			−0.759690			−0.379845	+	0.925050i	$0.624023\pi$
$30$		0			0
$31$	26.2344			0.846270			0.423135	−	0.906067i	$-0.360930\pi$
$31$	26.2344			0.846270			0.423135	+	0.906067i	$0.360930\pi$
$32$		0			0
$33$	−25.9754	−	2.14207i	−0.787134	−	0.0649111i
$34$		0			0
$35$	3.83689			0.109625
$36$		0			0
$37$			53.3855i			1.44285i	0.692492	+	0.721426i	$0.256513\pi$
$37$			53.3855i			1.44285i	−0.692492	+	0.721426i	$0.743487\pi$
$38$		0			0
$39$	−4.42003	+	53.5988i	−0.113334	+	1.37433i
$40$		0			0
$41$			35.6935i			0.870574i	0.900292	+	0.435287i	$0.143353\pi$
$41$			35.6935i			0.870574i	−0.900292	+	0.435287i	$0.856647\pi$
$42$		0			0
$43$			50.4895i			1.17417i	0.809524	+	0.587087i	$0.199725\pi$
$43$			50.4895i			1.17417i	−0.809524	+	0.587087i	$0.800275\pi$
$44$		0			0
$45$	−58.9208	−	9.78437i	−1.30935	−	0.217430i
$46$		0			0
$47$		−	30.6265i		−	0.651628i	−0.945434	−	0.325814i	$-0.894362\pi$
$47$		−	30.6265i		−	0.651628i	0.945434	−	0.325814i	$-0.105638\pi$
$48$		0			0
$49$	−48.6657			−0.993178
$50$		0			0
$51$	4.68732	−	56.8401i	0.0919083	−	1.11451i
$52$		0			0
$53$	−88.8962			−1.67729			−0.838644	−	0.544681i	$-0.816651\pi$
$53$	−88.8962			−1.67729			−0.838644	+	0.544681i	$0.816651\pi$
$54$		0			0
$55$	−57.6562			−1.04829
$56$		0			0
$57$	−7.93348	+	96.2040i	−0.139184	+	1.68779i
$58$		0			0
$59$	63.1939			1.07108			0.535542	−	0.844509i	$-0.320107\pi$
$59$	63.1939			1.07108			0.535542	+	0.844509i	$0.320107\pi$
$60$		0			0
$61$			33.5317i			0.549700i	0.961487	+	0.274850i	$0.0886282\pi$
$61$			33.5317i			0.549700i	−0.961487	+	0.274850i	$0.911372\pi$
$62$		0			0
$63$	−5.13313	−	0.852405i	−0.0814782	−	0.0135302i
$64$		0			0
$65$			118.970i			1.83031i
$66$		0			0
$67$			108.562i			1.62032i	0.586206	+	0.810162i	$0.300621\pi$
$67$			108.562i			1.62032i	−0.586206	+	0.810162i	$0.699379\pi$
$68$		0			0
$69$	−5.05041	+	61.2430i	−0.0731944	+	0.887579i
$70$		0			0
$71$		−	59.3600i		−	0.836056i	−0.908434	−	0.418028i	$-0.862721\pi$
$71$		−	59.3600i		−	0.836056i	0.908434	−	0.418028i	$-0.137279\pi$
$72$		0			0
$73$	5.60477			0.0767777			0.0383889	−	0.999263i	$-0.487777\pi$
$73$	5.60477			0.0767777			0.0383889	+	0.999263i	$0.487777\pi$
$74$		0			0
$75$	−56.9325	−	4.69495i	−0.759101	−	0.0625994i
$76$		0			0
$77$	−5.02296			−0.0652332
$78$		0			0
$79$	−78.9955			−0.999943			−0.499971	−	0.866042i	$-0.666656\pi$
$79$	−78.9955			−0.999943			−0.499971	+	0.866042i	$0.666656\pi$
$80$		0			0
$81$	76.6526	+	26.1797i	0.946328	+	0.323207i
$82$		0			0
$83$	−48.5283			−0.584679			−0.292339	−	0.956315i	$-0.594434\pi$
$83$	−48.5283			−0.584679			−0.292339	+	0.956315i	$0.594434\pi$
$84$		0			0
$85$		−	126.165i		−	1.48429i
$86$		0			0
$87$	65.8694	+	5.43193i	0.757120	+	0.0624360i
$88$		0			0
$89$		−	58.7109i		−	0.659672i	−0.944038	−	0.329836i	$-0.893006\pi$
$89$		−	58.7109i		−	0.659672i	0.944038	−	0.329836i	$-0.106994\pi$
$90$		0			0
$91$			10.3646i			0.113897i
$92$		0			0
$93$	−78.4369	−	6.46831i	−0.843407	−	0.0695517i
$94$		0			0
$95$			213.539i			2.24778i
$96$		0			0
$97$	93.3544			0.962416			0.481208	−	0.876606i	$-0.340198\pi$
$97$	93.3544			0.962416			0.481208	+	0.876606i	$0.340198\pi$
$98$		0			0
$99$	77.1345	+	12.8089i	0.779136	+	0.129383i
$100$		0			0
$101$	−114.161			−1.13031			−0.565155	−	0.824985i	$-0.691184\pi$
$101$	−114.161			−1.13031			−0.565155	+	0.824985i	$0.691184\pi$
$102$		0			0
$103$	−70.0355			−0.679956			−0.339978	−	0.940433i	$-0.610420\pi$
$103$	−70.0355			−0.679956			−0.339978	+	0.940433i	$0.610420\pi$
$104$		0			0
$105$	−11.4717	−	0.946018i	−0.109255	−	0.00900970i
$106$		0			0
$107$	50.9072			0.475769			0.237884	−	0.971293i	$-0.423546\pi$
$107$	50.9072			0.475769			0.237884	+	0.971293i	$0.423546\pi$
$108$		0			0
$109$		−	13.2178i		−	0.121264i	−0.998160	−	0.0606320i	$-0.980688\pi$
$109$		−	13.2178i		−	0.121264i	0.998160	−	0.0606320i	$-0.0193116\pi$
$110$		0			0
$111$	13.1627	−	159.615i	0.118582	−	1.43797i
$112$		0			0
$113$			62.9470i			0.557053i	0.960429	+	0.278527i	$0.0898460\pi$
$113$			62.9470i			0.557053i	−0.960429	+	0.278527i	$0.910154\pi$
$114$		0			0
$115$			135.938i			1.18207i
$116$		0			0
$117$	26.4305	−	159.163i	0.225902	−	1.36036i
$118$		0			0
$119$		−	10.9914i		−	0.0923644i
$120$		0			0
$121$	−45.5210			−0.376207
$122$		0			0
$123$	8.80054	−	106.718i	0.0715491	−	0.867629i
$124$		0			0
$125$	39.5402			0.316321
$126$		0			0
$127$	−236.659			−1.86345			−0.931727	−	0.363160i	$-0.881698\pi$
$127$	−236.659			−1.86345			−0.931727	+	0.363160i	$0.881698\pi$
$128$		0			0
$129$	12.4486	−	150.956i	0.0965009	−	1.17020i
$130$		0			0
$131$	−121.212			−0.925279			−0.462640	−	0.886546i	$-0.653098\pi$
$131$	−121.212			−0.925279			−0.462640	+	0.886546i	$0.653098\pi$
$132$		0			0
$133$			18.6033i			0.139875i
$134$		0			0
$135$	173.752	+	43.7812i	1.28705	+	0.324305i
$136$		0			0
$137$			136.677i			0.997645i	0.866704	+	0.498822i	$0.166234\pi$
$137$			136.677i			0.997645i	−0.866704	+	0.498822i	$0.833766\pi$
$138$		0			0
$139$		−	29.5616i		−	0.212673i	−0.994330	−	0.106337i	$-0.966088\pi$
$139$		−	29.5616i		−	0.212673i	0.994330	−	0.106337i	$-0.0339121\pi$
$140$		0			0
$141$	−7.55123	+	91.5687i	−0.0535548	+	0.649424i
$142$		0			0
$143$		−	155.747i		−	1.08914i
$144$		0			0
$145$	146.207			1.00832
$146$		0			0
$147$	145.503	+	11.9990i	0.989818	+	0.0816255i
$148$		0			0
$149$	−17.5092			−0.117511			−0.0587556	−	0.998272i	$-0.518713\pi$
$149$	−17.5092			−0.117511			−0.0587556	+	0.998272i	$0.518713\pi$
$150$		0			0
$151$	119.607			0.792099			0.396050	−	0.918229i	$-0.370381\pi$
$151$	119.607			0.792099			0.396050	+	0.918229i	$0.370381\pi$
$152$		0			0
$153$	−28.0288	+	168.788i	−0.183195	+	1.10319i
$154$		0			0
$155$	−174.102			−1.12324
$156$		0			0
$157$			189.488i			1.20693i	0.797388	+	0.603466i	$0.206214\pi$
$157$			189.488i			1.20693i	−0.797388	+	0.603466i	$0.793786\pi$
$158$		0			0
$159$	265.786	+	21.9181i	1.67161	+	0.137850i
$160$		0			0
$161$			11.8428i			0.0735576i
$162$		0			0
$163$		−	104.654i		−	0.642051i	−0.947071	−	0.321026i	$-0.895972\pi$
$163$		−	104.654i		−	0.642051i	0.947071	−	0.321026i	$-0.104028\pi$
$164$		0			0
$165$	172.384	+	14.2156i	1.04475	+	0.0861553i
$166$		0			0
$167$			142.664i			0.854278i	0.904186	+	0.427139i	$0.140478\pi$
$167$			142.664i			0.854278i	−0.904186	+	0.427139i	$0.859522\pi$
$168$		0			0
$169$	−152.374			−0.901623
$170$		0			0
$171$	47.4399	−	285.680i	0.277426	−	1.67064i
$172$		0			0
$173$	11.6594			0.0673952			0.0336976	−	0.999432i	$-0.489272\pi$
$173$	11.6594			0.0673952			0.0336976	+	0.999432i	$0.489272\pi$
$174$		0			0
$175$	−11.0092			−0.0629100
$176$		0			0
$177$	−188.940	−	15.5810i	−1.06746	−	0.0880283i
$178$		0			0
$179$	−40.3979			−0.225687			−0.112843	−	0.993613i	$-0.535996\pi$
$179$	−40.3979			−0.225687			−0.112843	+	0.993613i	$0.535996\pi$
$180$		0			0
$181$			121.175i			0.669472i	0.942312	+	0.334736i	$0.108647\pi$
$181$			121.175i			0.669472i	−0.942312	+	0.334736i	$0.891353\pi$
$182$		0			0
$183$	8.26752	−	100.255i	0.0451777	−	0.547840i
$184$		0			0
$185$		−	354.288i		−	1.91507i
$186$		0			0
$187$			165.165i			0.883235i
$188$		0			0
$189$	15.1371	+	3.81418i	0.0800906	+	0.0201808i
$190$		0			0
$191$			337.287i			1.76590i	0.469465	+	0.882951i	$0.344447\pi$
$191$			337.287i			1.76590i	−0.469465	+	0.882951i	$0.655553\pi$
$192$		0			0
$193$	−227.937			−1.18102			−0.590511	−	0.807029i	$-0.701074\pi$
$193$	−227.937			−1.18102			−0.590511	+	0.807029i	$0.701074\pi$
$194$		0			0
$195$	29.3332	−	355.704i	0.150426	−	1.82412i
$196$		0			0
$197$	172.276			0.874495			0.437248	−	0.899341i	$-0.355953\pi$
$197$	172.276			0.874495			0.437248	+	0.899341i	$0.355953\pi$
$198$		0			0
$199$	−0.598837			−0.00300923			−0.00150461	−	0.999999i	$-0.500479\pi$
$199$	−0.598837			−0.00300923			−0.00150461	+	0.999999i	$0.500479\pi$
$200$		0			0
$201$	26.7668	−	324.583i	0.133168	−	1.61484i
$202$		0			0
$203$	12.7374			0.0627458
$204$		0			0
$205$		−	236.877i		−	1.15550i
$206$		0			0
$207$	30.2000	−	181.862i	0.145894	−	0.878561i
$208$		0			0
$209$		−	279.548i		−	1.33755i
$210$		0			0
$211$			243.040i			1.15185i	0.817504	+	0.575923i	$0.195357\pi$
$211$			243.040i			1.15185i	−0.817504	+	0.575923i	$0.804643\pi$
$212$		0			0
$213$	−14.6357	+	177.478i	−0.0687123	+	0.833228i
$214$		0			0
$215$		−	335.069i		−	1.55846i
$216$		0			0
$217$	−15.1676			−0.0698968
$218$		0			0
$219$	−16.7574	−	1.38190i	−0.0765180	−	0.00631007i
$220$		0			0
$221$	340.809			1.54212
$222$		0			0
$223$	191.174			0.857284			0.428642	−	0.903474i	$-0.358992\pi$
$223$	191.174			0.857284			0.428642	+	0.903474i	$0.358992\pi$
$224$		0			0
$225$	169.062	+	28.0744i	0.751388	+	0.124775i
$226$		0			0
$227$	−8.76834			−0.0386271			−0.0193135	−	0.999813i	$-0.506148\pi$
$227$	−8.76834			−0.0386271			−0.0193135	+	0.999813i	$0.506148\pi$
$228$		0			0
$229$		−	131.740i		−	0.575283i	−0.957738	−	0.287641i	$-0.907129\pi$
$229$		−	131.740i		−	0.575283i	0.957738	−	0.287641i	$-0.0928711\pi$
$230$		0			0
$231$	15.0179	+	1.23845i	0.0650125	+	0.00536127i
$232$		0			0
$233$		−	96.0149i		−	0.412081i	−0.978543	−	0.206040i	$-0.933942\pi$
$233$		−	96.0149i		−	0.412081i	0.978543	−	0.206040i	$-0.0660579\pi$
$234$		0			0
$235$			203.250i			0.864894i
$236$		0			0
$237$	236.185	+	19.4770i	0.996560	+	0.0821815i
$238$		0			0
$239$		−	317.978i		−	1.33045i	−0.746643	−	0.665225i	$-0.768336\pi$
$239$		−	317.978i		−	1.33045i	0.746643	−	0.665225i	$-0.231664\pi$
$240$		0			0
$241$	−114.189			−0.473812			−0.236906	−	0.971533i	$-0.576133\pi$
$241$	−114.189			−0.473812			−0.236906	+	0.971533i	$0.576133\pi$
$242$		0			0
$243$	−222.725	−	97.1728i	−0.916564	−	0.399888i
$244$		0			0
$245$	322.966			1.31823
$246$		0			0
$247$	−576.832			−2.33535
$248$		0			0
$249$	145.093	+	11.9651i	0.582701	+	0.0480525i
$250$		0			0
$251$	376.853			1.50140			0.750702	−	0.660641i	$-0.229715\pi$
$251$	376.853			1.50140			0.750702	+	0.660641i	$0.229715\pi$
$252$		0			0
$253$		−	177.959i		−	0.703395i
$254$		0			0
$255$	−31.1070	+	377.214i	−0.121988	+	1.47927i
$256$		0			0
$257$		−	152.041i		−	0.591599i	−0.955250	−	0.295800i	$-0.904414\pi$
$257$		−	152.041i		−	0.591599i	0.955250	−	0.295800i	$-0.0955861\pi$
$258$		0			0
$259$		−	30.8653i		−	0.119171i
$260$		0			0
$261$	−195.600	−	32.4813i	−0.749427	−	0.124450i
$262$		0			0
$263$			495.013i			1.88218i	0.338158	+	0.941089i	$0.390196\pi$
$263$			495.013i			1.88218i	−0.338158	+	0.941089i	$0.609804\pi$
$264$		0			0
$265$	589.952			2.22623
$266$		0			0
$267$	−14.4757	+	175.537i	−0.0542160	+	0.657441i
$268$		0			0
$269$	173.556			0.645190			0.322595	−	0.946537i	$-0.395445\pi$
$269$	173.556			0.645190			0.322595	+	0.946537i	$0.395445\pi$
$270$		0			0
$271$	−125.415			−0.462784			−0.231392	−	0.972861i	$-0.574328\pi$
$271$	−125.415			−0.462784			−0.231392	+	0.972861i	$0.574328\pi$
$272$		0			0
$273$	2.55548	−	30.9886i	0.00936072	−	0.113511i
$274$		0			0
$275$	165.434			0.601577
$276$		0			0
$277$			226.300i			0.816967i	0.912766	+	0.408483i	$0.133942\pi$
$277$			226.300i			0.816967i	−0.912766	+	0.408483i	$0.866058\pi$
$278$		0			0
$279$	232.920	+	38.6786i	0.834838	+	0.138633i
$280$		0			0
$281$		−	13.8834i		−	0.0494073i	−0.999695	−	0.0247036i	$-0.992136\pi$
$281$		−	13.8834i		−	0.0494073i	0.999695	−	0.0247036i	$-0.00786421\pi$
$282$		0			0
$283$			306.461i			1.08290i	0.840733	+	0.541450i	$0.182125\pi$
$283$			306.461i			1.08290i	−0.840733	+	0.541450i	$0.817875\pi$
$284$		0			0
$285$	52.6498	−	638.449i	0.184736	−	2.24017i
$286$		0			0
$287$		−	20.6365i		−	0.0719042i
$288$		0			0
$289$	−72.4183			−0.250582
$290$		0			0
$291$	−279.116	−	23.0173i	−0.959160	−	0.0790973i
$292$		0			0
$293$	121.869			0.415935			0.207968	−	0.978136i	$-0.433315\pi$
$293$	121.869			0.415935			0.207968	+	0.978136i	$0.433315\pi$
$294$		0			0
$295$	−419.381			−1.42163
$296$		0			0
$297$	−227.462	−	57.3149i	−0.765867	−	0.192980i
$298$		0			0
$299$	−367.208			−1.22812
$300$		0			0
$301$		−	29.1909i		−	0.0969797i
$302$		0			0
$303$	341.325	+	28.1474i	1.12649	+	0.0928959i
$304$		0			0
$305$		−	222.530i		−	0.729607i
$306$		0			0
$307$			19.0431i			0.0620298i	0.999519	+	0.0310149i	$0.00987393\pi$
$307$			19.0431i			0.0620298i	−0.999519	+	0.0310149i	$0.990126\pi$
$308$		0			0
$309$	209.396	+	17.2678i	0.677656	+	0.0558830i
$310$		0			0
$311$		−	223.159i		−	0.717552i	−0.933424	−	0.358776i	$-0.883194\pi$
$311$		−	223.159i		−	0.717552i	0.933424	−	0.358776i	$-0.116806\pi$
$312$		0			0
$313$	194.708			0.622070			0.311035	−	0.950399i	$-0.399324\pi$
$313$	194.708			0.622070			0.311035	+	0.950399i	$0.399324\pi$
$314$		0			0
$315$	34.0655	+	5.65691i	0.108145	+	0.0179584i
$316$		0			0
$317$	−532.192			−1.67884			−0.839420	−	0.543483i	$-0.817105\pi$
$317$	−532.192			−1.67884			−0.839420	+	0.543483i	$0.817105\pi$
$318$		0			0
$319$	−191.402			−0.600007
$320$		0			0
$321$	−152.205	−	12.5516i	−0.474159	−	0.0391016i
$322$		0			0
$323$	611.715			1.89385
$324$		0			0
$325$		−	341.363i		−	1.05035i
$326$		0			0
$327$	−3.25896	+	39.5192i	−0.00996623	+	0.120854i
$328$		0			0
$329$			17.7070i			0.0538206i
$330$		0			0
$331$			91.7273i			0.277122i	0.990354	+	0.138561i	$0.0442476\pi$
$331$			91.7273i			0.277122i	−0.990354	+	0.138561i	$0.955752\pi$
$332$		0			0
$333$	−78.7088	+	473.979i	−0.236363	+	1.42336i
$334$		0			0
$335$		−	720.460i		−	2.15063i
$336$		0			0
$337$	82.1658			0.243815			0.121908	−	0.992541i	$-0.461099\pi$
$337$	82.1658			0.243815			0.121908	+	0.992541i	$0.461099\pi$
$338$		0			0
$339$	15.5201	−	188.202i	0.0457821	−	0.555169i
$340$		0			0
$341$	227.921			0.668389
$342$		0			0
$343$	56.4662			0.164625
$344$		0			0
$345$	33.5166	−	406.433i	0.0971496	−	1.17807i
$346$		0			0
$347$	−323.122			−0.931189			−0.465594	−	0.884998i	$-0.654159\pi$
$347$	−323.122			−0.931189			−0.465594	+	0.884998i	$0.654159\pi$
$348$		0			0
$349$			255.324i			0.731586i	0.930696	+	0.365793i	$0.119202\pi$
$349$			255.324i			0.731586i	−0.930696	+	0.365793i	$0.880798\pi$
$350$		0			0
$351$	−118.266	+	469.356i	−0.336941	+	1.33720i
$352$		0			0
$353$		−	344.332i		−	0.975444i	−0.872999	−	0.487722i	$-0.837828\pi$
$353$		−	344.332i		−	0.975444i	0.872999	−	0.487722i	$-0.162172\pi$
$354$		0			0
$355$			393.937i			1.10968i
$356$		0			0
$357$	−2.71001	+	32.8625i	−0.00759107	+	0.0920519i
$358$		0			0
$359$		−	119.962i		−	0.334157i	−0.985944	−	0.167079i	$-0.946567\pi$
$359$		−	119.962i		−	0.334157i	0.985944	−	0.167079i	$-0.0534334\pi$
$360$		0			0
$361$	−674.351			−1.86801
$362$		0			0
$363$	136.101	+	11.2236i	0.374934	+	0.0309190i
$364$		0			0
$365$	−37.1956			−0.101906
$366$		0			0
$367$	−398.726			−1.08645			−0.543223	−	0.839589i	$-0.682796\pi$
$367$	−398.726			−1.08645			−0.543223	+	0.839589i	$0.682796\pi$
$368$		0			0
$369$	−52.6246	+	316.902i	−0.142614	+	0.858813i
$370$		0			0
$371$	51.3960			0.138534
$372$		0			0
$373$			606.592i			1.62625i	0.582088	+	0.813126i	$0.302236\pi$
$373$			606.592i			1.62625i	−0.582088	+	0.813126i	$0.697764\pi$
$374$		0			0
$375$	−118.219	−	9.74896i	−0.315251	−	0.0259972i
$376$		0			0
$377$			394.948i			1.04761i
$378$		0			0
$379$		−	266.492i		−	0.703146i	−0.936161	−	0.351573i	$-0.885647\pi$
$379$		−	266.492i		−	0.703146i	0.936161	−	0.351573i	$-0.114353\pi$
$380$		0			0
$381$	707.574	+	58.3502i	1.85715	+	0.153150i
$382$		0			0
$383$			406.790i			1.06212i	0.847336	+	0.531058i	$0.178205\pi$
$383$			406.790i			1.06212i	−0.847336	+	0.531058i	$0.821795\pi$
$384$		0			0
$385$	33.3344			0.0865828
$386$		0			0
$387$	−74.4390	+	448.267i	−0.192349	+	1.15831i
$388$		0			0
$389$	217.748			0.559763			0.279881	−	0.960035i	$-0.409705\pi$
$389$	217.748			0.559763			0.279881	+	0.960035i	$0.409705\pi$
$390$		0			0
$391$	389.414			0.995944
$392$		0			0
$393$	362.405	+	29.8858i	0.922149	+	0.0760452i
$394$		0			0
$395$	524.246			1.32721
$396$		0			0
$397$		−	53.9651i		−	0.135932i	−0.997688	−	0.0679661i	$-0.978349\pi$
$397$		−	53.9651i		−	0.135932i	0.997688	−	0.0679661i	$-0.0216510\pi$
$398$		0			0
$399$	4.58680	−	55.6211i	0.0114958	−	0.139401i
$400$		0			0
$401$			294.291i			0.733893i	0.930242	+	0.366946i	$0.119597\pi$
$401$			294.291i			0.733893i	−0.930242	+	0.366946i	$0.880403\pi$
$402$		0			0
$403$		−	470.302i		−	1.16700i
$404$		0			0
$405$	−508.698	−	173.739i	−1.25604	−	0.428986i
$406$		0			0
$407$			463.806i			1.13957i
$408$		0			0
$409$	−569.204			−1.39170			−0.695849	−	0.718188i	$-0.744972\pi$
$409$	−569.204			−1.39170			−0.695849	+	0.718188i	$0.744972\pi$
$410$		0			0
$411$	33.6990	−	408.645i	0.0819926	−	0.994270i
$412$		0			0
$413$	−36.5361			−0.0884651
$414$		0			0
$415$	322.054			0.776034
$416$		0			0
$417$	−7.28866	+	88.3847i	−0.0174788	+	0.211954i
$418$		0			0
$419$	−590.728			−1.40985			−0.704926	−	0.709281i	$-0.749020\pi$
$419$	−590.728			−1.40985			−0.704926	+	0.709281i	$0.749020\pi$
$420$		0			0
$421$		−	475.989i		−	1.13061i	−0.824880	−	0.565307i	$-0.808758\pi$
$421$		−	475.989i		−	1.13061i	0.824880	−	0.565307i	$-0.191242\pi$
$422$		0			0
$423$	45.1541	−	271.915i	0.106747	−	0.642825i
$424$		0			0
$425$			362.006i			0.851780i
$426$		0			0
$427$		−	19.3866i		−	0.0454019i
$428$		0			0
$429$	−38.4007	+	465.659i	−0.0895120	+	1.08545i
$430$		0			0
$431$		−	341.539i		−	0.792435i	−0.918157	−	0.396217i	$-0.870323\pi$
$431$		−	341.539i		−	0.792435i	0.918157	−	0.396217i	$-0.129677\pi$
$432$		0			0
$433$	−88.2258			−0.203755			−0.101877	−	0.994797i	$-0.532485\pi$
$433$	−88.2258			−0.203755			−0.101877	+	0.994797i	$0.532485\pi$
$434$		0			0
$435$	−437.136	−	36.0485i	−1.00491	−	0.0828702i
$436$		0			0
$437$	−659.099			−1.50824
$438$		0			0
$439$	735.647			1.67573			0.837867	−	0.545875i	$-0.183803\pi$
$439$	735.647			1.67573			0.837867	+	0.545875i	$0.183803\pi$
$440$		0			0
$441$	−432.075	−	71.7502i	−0.979761	−	0.162699i
$442$		0			0
$443$	−333.798			−0.753494			−0.376747	−	0.926316i	$-0.622957\pi$
$443$	−333.798			−0.753494			−0.376747	+	0.926316i	$0.622957\pi$
$444$		0			0
$445$			389.629i			0.875571i
$446$		0			0
$447$	52.3498	+	4.31703i	0.117114	+	0.00965779i
$448$		0			0
$449$			396.485i			0.883039i	0.897252	+	0.441520i	$0.145560\pi$
$449$			396.485i			0.883039i	−0.897252	+	0.441520i	$0.854440\pi$
$450$		0			0
$451$			310.100i			0.687584i
$452$		0			0
$453$	−357.607	−	29.4901i	−0.789419	−	0.0650996i
$454$		0			0
$455$		−	68.7836i		−	0.151173i
$456$		0			0
$457$	−34.5362			−0.0755715			−0.0377857	−	0.999286i	$-0.512030\pi$
$457$	−34.5362			−0.0755715			−0.0377857	+	0.999286i	$0.512030\pi$
$458$		0			0
$459$	125.418	−	497.739i	0.273242	−	1.08440i
$460$		0			0
$461$	−324.050			−0.702929			−0.351464	−	0.936201i	$-0.614316\pi$
$461$	−324.050			−0.702929			−0.351464	+	0.936201i	$0.614316\pi$
$462$		0			0
$463$	6.33727			0.0136874			0.00684371	−	0.999977i	$-0.497822\pi$
$463$	6.33727			0.0136874			0.00684371	+	0.999977i	$0.497822\pi$
$464$		0			0
$465$	520.539	+	42.9264i	1.11944	+	0.0923147i
$466$		0			0
$467$	450.706			0.965109			0.482554	−	0.875866i	$-0.339709\pi$
$467$	450.706			0.965109			0.482554	+	0.875866i	$0.339709\pi$
$468$		0			0
$469$		−	62.7658i		−	0.133829i
$470$		0			0
$471$	46.7200	−	566.542i	0.0991932	−	1.20285i
$472$		0			0
$473$			438.646i			0.927370i
$474$		0			0
$475$		−	612.710i		−	1.28992i
$476$		0			0
$477$	−789.258	−	131.064i	−1.65463	−	0.274767i
$478$		0			0
$479$			259.094i			0.540906i	0.962733	+	0.270453i	$0.0871735\pi$
$479$			259.094i			0.540906i	−0.962733	+	0.270453i	$0.912827\pi$
$480$		0			0
$481$	957.038			1.98968
$482$		0			0
$483$	2.91993	−	35.4081i	0.00604541	−	0.0733087i
$484$		0			0
$485$	−619.538			−1.27740
$486$		0			0
$487$	628.526			1.29061			0.645304	−	0.763926i	$-0.276731\pi$
$487$	628.526			1.29061			0.645304	+	0.763926i	$0.276731\pi$
$488$		0			0
$489$	−25.8034	+	312.901i	−0.0527677	+	0.639879i
$490$		0			0
$491$	−352.334			−0.717585			−0.358792	−	0.933417i	$-0.616811\pi$
$491$	−352.334			−0.717585			−0.358792	+	0.933417i	$0.616811\pi$
$492$		0			0
$493$		−	418.831i		−	0.849557i
$494$		0			0
$495$	−511.896	−	85.0052i	−1.03413	−	0.171728i
$496$		0			0
$497$			34.3195i			0.0690532i
$498$		0			0
$499$		−	240.576i		−	0.482116i	−0.970511	−	0.241058i	$-0.922506\pi$
$499$		−	240.576i		−	0.482116i	0.970511	−	0.241058i	$-0.0774943\pi$
$500$		0			0
$501$	35.1751	−	426.545i	0.0702098	−	0.851388i
$502$		0			0
$503$			13.1648i			0.0261725i	0.999914	+	0.0130862i	$0.00416560\pi$
$503$			13.1648i			0.0261725i	−0.999914	+	0.0130862i	$0.995834\pi$
$504$		0			0
$505$	757.621			1.50024
$506$		0			0
$507$	455.577	+	37.5692i	0.898573	+	0.0741010i
$508$		0			0
$509$	−468.752			−0.920927			−0.460464	−	0.887679i	$-0.652317\pi$
$509$	−468.752			−0.920927			−0.460464	+	0.887679i	$0.652317\pi$
$510$		0			0
$511$	−3.24044			−0.00634138
$512$		0			0
$513$	−212.275	+	842.443i	−0.413791	+	1.64219i
$514$		0			0
$515$	464.784			0.902494
$516$		0			0
$517$		−	266.079i		−	0.514660i
$518$		0			0
$519$	−34.8598	−	2.87472i	−0.0671672	−	0.00553895i
$520$		0			0
$521$		−	456.517i		−	0.876232i	−0.898918	−	0.438116i	$-0.855646\pi$
$521$		−	456.517i		−	0.876232i	0.898918	−	0.438116i	$-0.144354\pi$
$522$		0			0
$523$		−	287.095i		−	0.548939i	−0.961596	−	0.274469i	$-0.911498\pi$
$523$		−	287.095i		−	0.548939i	0.961596	−	0.274469i	$-0.0885022\pi$
$524$		0			0
$525$	32.9160	+	2.71442i	0.0626971	+	0.00517033i
$526$		0			0
$527$			498.742i			0.946379i
$528$		0			0
$529$	109.421			0.206845
$530$		0			0
$531$	561.062	+	93.1698i	1.05661	+	0.175461i
$532$		0			0
$533$	639.875			1.20052
$534$		0			0
$535$	−337.841			−0.631479
$536$		0			0
$537$	120.784	+	9.96046i	0.224923	+	0.0185483i
$538$		0			0
$539$	−422.801			−0.784418
$540$		0			0
$541$			804.779i			1.48758i	0.668416	+	0.743788i	$0.266973\pi$
$541$			804.779i			1.48758i	−0.668416	+	0.743788i	$0.733027\pi$
$542$		0			0
$543$	29.8766	−	362.294i	0.0550214	−	0.667208i
$544$		0			0
$545$			87.7186i			0.160952i
$546$		0			0
$547$			76.3034i			0.139494i	0.997565	+	0.0697471i	$0.0222192\pi$
$547$			76.3034i			0.139494i	−0.997565	+	0.0697471i	$0.977781\pi$
$548$		0			0
$549$	−49.4373	+	297.708i	−0.0900498	+	0.542274i
$550$		0			0
$551$			708.889i			1.28655i
$552$		0			0
$553$	45.6718			0.0825892
$554$		0			0
$555$	−87.3528	+	1059.27i	−0.157392	+	1.90859i
$556$		0			0
$557$	−540.544			−0.970455			−0.485228	−	0.874388i	$-0.661263\pi$
$557$	−540.544			−0.970455			−0.485228	+	0.874388i	$0.661263\pi$
$558$		0			0
$559$	905.121			1.61918
$560$		0			0
$561$	40.7228	−	493.819i	0.0725897	−	0.880247i
$562$		0			0
$563$	−1047.65			−1.86084			−0.930422	−	0.366491i	$-0.880559\pi$
$563$	−1047.65			−1.86084			−0.930422	+	0.366491i	$0.880559\pi$
$564$		0			0
$565$		−	417.742i		−	0.739366i
$566$		0			0
$567$	−44.3173	−	15.1360i	−0.0781610	−	0.0266949i
$568$		0			0
$569$			958.593i			1.68470i	0.538933	+	0.842348i	$0.318827\pi$
$569$			958.593i			1.68470i	−0.538933	+	0.842348i	$0.681173\pi$
$570$		0			0
$571$			118.951i			0.208321i	0.994560	+	0.104161i	$0.0332156\pi$
$571$			118.951i			0.208321i	−0.994560	+	0.104161i	$0.966784\pi$
$572$		0			0
$573$	83.1611	−	1008.44i	0.145133	−	1.75993i
$574$		0			0
$575$		−	390.048i		−	0.678344i
$576$		0			0
$577$	−355.825			−0.616681			−0.308340	−	0.951276i	$-0.599774\pi$
$577$	−355.825			−0.616681			−0.308340	+	0.951276i	$0.599774\pi$
$578$		0			0
$579$	681.499	+	56.1999i	1.17703	+	0.0970637i
$580$		0			0
$581$	28.0570			0.0482910
$582$		0			0
$583$	−772.318			−1.32473
$584$		0			0
$585$	−175.404	+	1056.27i	−0.299835	+	1.80559i
$586$		0			0
$587$	−571.017			−0.972771			−0.486386	−	0.873744i	$-0.661685\pi$
$587$	−571.017			−0.972771			−0.486386	+	0.873744i	$0.661685\pi$
$588$		0			0
$589$		−	844.140i		−	1.43318i
$590$		0			0
$591$	−515.078	−	42.4760i	−0.871537	−	0.0718714i
$592$		0			0
$593$			370.411i			0.624640i	0.949977	+	0.312320i	$0.101106\pi$
$593$			370.411i			0.624640i	−0.949977	+	0.312320i	$0.898894\pi$
$594$		0			0
$595$			72.9432i			0.122594i
$596$		0			0
$597$	1.79043	+	0.147648i	0.00299905	+	0.000247317i
$598$		0			0
$599$		−	68.6579i		−	0.114621i	−0.998356	−	0.0573104i	$-0.981748\pi$
$599$		−	68.6579i		−	0.114621i	0.998356	−	0.0573104i	$-0.0182525\pi$
$600$		0			0
$601$	−9.39898			−0.0156389			−0.00781945	−	0.999969i	$-0.502489\pi$
$601$	−9.39898			−0.0156389			−0.00781945	+	0.999969i	$0.502489\pi$
$602$		0			0
$603$	−160.058	+	963.857i	−0.265436	+	1.59844i
$604$		0			0
$605$	302.096			0.499333
$606$		0			0
$607$	−1064.78			−1.75417			−0.877085	−	0.480336i	$-0.840515\pi$
$607$	−1064.78			−1.75417			−0.877085	+	0.480336i	$0.840515\pi$
$608$		0			0
$609$	−38.0829	−	3.14051i	−0.0625335	−	0.00515684i
$610$		0			0
$611$	−549.039			−0.898591
$612$		0			0
$613$			651.884i			1.06343i	0.846922	+	0.531716i	$0.178453\pi$
$613$			651.884i			1.06343i	−0.846922	+	0.531716i	$0.821547\pi$
$614$		0			0
$615$	−58.4040	+	708.227i	−0.0949659	+	1.15159i
$616$		0			0
$617$		−	146.821i		−	0.237959i	−0.992897	−	0.118980i	$-0.962038\pi$
$617$		−	146.821i		−	0.237959i	0.992897	−	0.118980i	$-0.0379623\pi$
$618$		0			0
$619$		−	596.307i		−	0.963340i	−0.876353	−	0.481670i	$-0.840030\pi$
$619$		−	596.307i		−	0.963340i	0.876353	−	0.481670i	$-0.159970\pi$
$620$		0			0
$621$	−135.133	+	536.295i	−0.217606	+	0.863598i
$622$		0			0
$623$			33.9441i			0.0544850i
$624$		0			0
$625$	−738.453			−1.18152
$626$		0			0
$627$	−68.9250	+	835.808i	−0.109928	+	1.33303i
$628$		0			0
$629$	−1014.91			−1.61353
$630$		0			0
$631$	77.6556			0.123067			0.0615337	−	0.998105i	$-0.480401\pi$
$631$	77.6556			0.123067			0.0615337	+	0.998105i	$0.480401\pi$
$632$		0			0
$633$	59.9235	−	726.652i	0.0946658	−	1.14795i
$634$		0			0
$635$	1570.56			2.47333
$636$		0			0
$637$			872.427i			1.36959i
$638$		0			0
$639$	87.5172	−	527.023i	0.136960	−	0.824762i
$640$		0			0
$641$		−	574.386i		−	0.896078i	−0.894014	−	0.448039i	$-0.852123\pi$
$641$		−	574.386i		−	0.896078i	0.894014	−	0.448039i	$-0.147877\pi$
$642$		0			0
$643$		−	1217.20i		−	1.89299i	−0.322712	−	0.946497i	$-0.604594\pi$
$643$		−	1217.20i		−	1.89299i	0.322712	−	0.946497i	$-0.395406\pi$
$644$		0			0
$645$	−82.6141	+	1001.81i	−0.128084	+	1.55319i
$646$		0			0
$647$		−	802.675i		−	1.24061i	−0.784360	−	0.620305i	$-0.787009\pi$
$647$		−	802.675i		−	1.24061i	0.784360	−	0.620305i	$-0.212991\pi$
$648$		0			0
$649$	549.020			0.845948
$650$		0			0
$651$	45.3489	+	3.73970i	0.0696604	+	0.00574455i
$652$		0			0
$653$	216.282			0.331212			0.165606	−	0.986192i	$-0.447042\pi$
$653$	216.282			0.331212			0.165606	+	0.986192i	$0.447042\pi$
$654$		0			0
$655$	804.410			1.22811
$656$		0			0
$657$	49.7615	+	8.26338i	0.0757405	+	0.0125774i
$658$		0			0
$659$	611.696			0.928218			0.464109	−	0.885778i	$-0.346375\pi$
$659$	611.696			0.928218			0.464109	+	0.885778i	$0.346375\pi$
$660$		0			0
$661$			214.961i			0.325206i	0.986692	+	0.162603i	$0.0519890\pi$
$661$			214.961i			0.325206i	−0.986692	+	0.162603i	$0.948011\pi$
$662$		0			0
$663$	−1018.97	−	84.0293i	−1.53690	−	0.126741i
$664$		0			0
$665$		−	123.459i		−	0.185653i
$666$		0			0
$667$			451.275i			0.676574i
$668$		0			0
$669$	−571.583	−	47.1357i	−0.854384	−	0.0704569i
$670$		0			0
$671$			291.319i			0.434156i
$672$		0			0
$673$	−900.809			−1.33850			−0.669249	−	0.743038i	$-0.733384\pi$
$673$	−900.809			−1.33850			−0.669249	+	0.743038i	$0.733384\pi$
$674$		0			0
$675$	−498.549	−	125.622i	−0.738591	−	0.186107i
$676$		0			0
$677$	−322.312			−0.476088			−0.238044	−	0.971254i	$-0.576506\pi$
$677$	−322.312			−0.476088			−0.238044	+	0.971254i	$0.576506\pi$
$678$		0			0
$679$	−53.9736			−0.0794898
$680$		0			0
$681$	26.2160	+	2.16191i	0.0384964	+	0.00317461i
$682$		0			0
$683$	451.564			0.661148			0.330574	−	0.943780i	$-0.392758\pi$
$683$	451.564			0.661148			0.330574	+	0.943780i	$0.392758\pi$
$684$		0			0
$685$		−	907.047i		−	1.32416i
$686$		0			0
$687$	−32.4816	+	393.882i	−0.0472803	+	0.573337i
$688$		0			0
$689$			1593.64i			2.31297i
$690$		0			0
$691$			259.954i			0.376199i	0.982150	+	0.188100i	$0.0602328\pi$
$691$			259.954i			0.376199i	−0.982150	+	0.188100i	$0.939767\pi$
$692$		0			0
$693$	−44.5959	−	7.40558i	−0.0643520	−	0.0106863i
$694$		0			0
$695$			196.183i			0.282277i
$696$		0			0
$697$	−678.570			−0.973558
$698$		0			0
$699$	−23.6733	+	287.070i	−0.0338674	+	0.410687i
$700$		0			0
$701$	434.792			0.620246			0.310123	−	0.950696i	$-0.399630\pi$
$701$	434.792			0.620246			0.310123	+	0.950696i	$0.399630\pi$
$702$		0			0
$703$	1717.78			2.44350
$704$		0			0
$705$	50.1131	−	607.688i	0.0710823	−	0.861968i
$706$		0			0
$707$	66.0033			0.0933568
$708$		0			0
$709$			1136.05i			1.60232i	0.598448	+	0.801162i	$0.295784\pi$
$709$			1136.05i			1.60232i	−0.598448	+	0.801162i	$0.704216\pi$
$710$		0			0
$711$	−701.355	−	116.467i	−0.986434	−	0.163807i
$712$		0			0
$713$		−	537.375i		−	0.753682i
$714$		0			0
$715$			1033.60i			1.44559i
$716$		0			0
$717$	−78.4001	+	950.705i	−0.109345	+	1.32595i
$718$		0			0
$719$			1284.33i			1.78627i	0.449792	+	0.893133i	$0.351498\pi$
$719$			1284.33i			1.78627i	−0.449792	+	0.893133i	$0.648502\pi$
$720$		0			0
$721$	40.4916			0.0561603
$722$		0			0
$723$	341.407	+	28.1542i	0.472209	+	0.0389408i
$724$		0			0
$725$	−419.513			−0.578638
$726$		0			0
$727$	−31.6562			−0.0435437			−0.0217718	−	0.999763i	$-0.506931\pi$
$727$	−31.6562			−0.0435437			−0.0217718	+	0.999763i	$0.506931\pi$
$728$		0			0
$729$	641.956	+	345.447i	0.880598	+	0.473864i
$730$		0			0
$731$	−959.856			−1.31307
$732$		0			0
$733$		−	596.421i		−	0.813671i	−0.913501	−	0.406836i	$-0.866632\pi$
$733$		−	596.421i		−	0.813671i	0.913501	−	0.406836i	$-0.133368\pi$
$734$		0			0
$735$	−965.619	−	79.6300i	−1.31377	−	0.108340i
$736$		0			0
$737$			943.170i			1.27974i
$738$		0			0
$739$		−	1146.14i		−	1.55093i	−0.631391	−	0.775464i	$-0.717516\pi$
$739$		−	1146.14i		−	1.55093i	0.631391	−	0.775464i	$-0.282484\pi$
$740$		0			0
$741$	1724.64	+	142.223i	2.32745	+	0.191934i
$742$		0			0
$743$		−	842.850i		−	1.13439i	−0.823584	−	0.567194i	$-0.808029\pi$
$743$		−	842.850i		−	1.13439i	0.823584	−	0.567194i	$-0.191971\pi$
$744$		0			0
$745$	116.198			0.155970
$746$		0			0
$747$	−430.855	−	71.5476i	−0.576780	−	0.0957799i
$748$		0			0
$749$	−29.4324			−0.0392956
$750$		0			0
$751$	−797.793			−1.06231			−0.531154	−	0.847275i	$-0.678241\pi$
$751$	−797.793			−1.06231			−0.531154	+	0.847275i	$0.678241\pi$
$752$		0			0
$753$	−1126.73	−	92.9162i	−1.49633	−	0.123395i
$754$		0			0
$755$	−793.761			−1.05134
$756$		0			0
$757$		−	278.402i		−	0.367770i	−0.982948	−	0.183885i	$-0.941133\pi$
$757$		−	278.402i		−	0.367770i	0.982948	−	0.183885i	$-0.0588674\pi$
$758$		0			0
$759$	−43.8773	+	532.071i	−0.0578093	+	0.701015i
$760$		0			0
$761$		−	986.126i		−	1.29583i	−0.761713	−	0.647914i	$-0.775641\pi$
$761$		−	986.126i		−	1.29583i	0.761713	−	0.647914i	$-0.224359\pi$
$762$		0			0
$763$			7.64197i			0.0100157i
$764$		0			0
$765$	186.011	−	1120.14i	0.243151	−	1.46424i
$766$		0			0
$767$		−	1132.87i		−	1.47702i
$768$		0			0
$769$	242.053			0.314764			0.157382	−	0.987538i	$-0.449695\pi$
$769$	242.053			0.314764			0.157382	+	0.987538i	$0.449695\pi$
$770$		0			0
$771$	−37.4870	+	454.580i	−0.0486213	+	0.589598i
$772$		0			0
$773$	590.233			0.763561			0.381780	−	0.924253i	$-0.375311\pi$
$773$	590.233			0.763561			0.381780	+	0.924253i	$0.375311\pi$
$774$		0			0
$775$	499.553			0.644585
$776$		0			0
$777$	−7.61009	+	92.2825i	−0.00979420	+	0.118768i
$778$		0			0
$779$	1148.51			1.47433
$780$		0			0
$781$		−	515.712i		−	0.660322i
$782$		0			0
$783$	576.808	+	145.341i	0.736664	+	0.185621i
$784$		0			0
$785$		−	1257.52i		−	1.60194i
$786$		0			0
$787$			565.777i			0.718903i	0.933164	+	0.359451i	$0.117036\pi$
$787$			565.777i			0.718903i	−0.933164	+	0.359451i	$0.882964\pi$
$788$		0			0
$789$	122.050	−	1480.02i	0.154689	−	1.87581i
$790$		0			0
$791$		−	36.3933i		−	0.0460092i
$792$		0			0
$793$	601.120			0.758033
$794$		0			0
$795$	−1763.87	−	145.458i	−2.21870	−	0.182966i
$796$		0			0
$797$	1128.02			1.41533			0.707664	−	0.706549i	$-0.249749\pi$
$797$	1128.02			1.41533			0.707664	+	0.706549i	$0.249749\pi$
$798$		0			0
$799$	582.241			0.728712
$800$		0			0
$801$	86.5601	−	521.259i	0.108065	−	0.650761i
$802$		0			0
$803$	48.6935			0.0606395
$804$		0			0
$805$		−	78.5934i		−	0.0976316i
$806$		0			0
$807$	−518.907	−	42.7917i	−0.643007	−	0.0530257i
$808$		0			0
$809$		−	794.472i		−	0.982042i	−0.871148	−	0.491021i	$-0.836624\pi$
$809$		−	794.472i		−	0.982042i	0.871148	−	0.491021i	$-0.163376\pi$
$810$		0			0
$811$		−	1430.50i		−	1.76387i	−0.471367	−	0.881937i	$-0.656239\pi$
$811$		−	1430.50i		−	1.76387i	0.471367	−	0.881937i	$-0.343761\pi$
$812$		0			0
$813$	374.971	+	30.9220i	0.461219	+	0.0380345i
$814$		0			0
$815$			694.529i			0.852183i
$816$		0			0
$817$	1624.59			1.98849
$818$		0			0
$819$	−15.2810	+	92.0211i	−0.0186581	+	0.112358i
$820$		0			0
$821$	1289.62			1.57079			0.785397	−	0.618992i	$-0.212459\pi$
$821$	1289.62			1.57079			0.785397	+	0.618992i	$0.212459\pi$
$822$		0			0
$823$	1064.14			1.29300			0.646502	−	0.762913i	$-0.276231\pi$
$823$	1064.14			1.29300			0.646502	+	0.762913i	$0.276231\pi$
$824$		0			0
$825$	−494.622	−	40.7891i	−0.599542	−	0.0494413i
$826$		0			0
$827$	1258.48			1.52174			0.760868	−	0.648907i	$-0.224773\pi$
$827$	1258.48			1.52174			0.760868	+	0.648907i	$0.224773\pi$
$828$		0			0
$829$			1197.65i			1.44469i	0.691531	+	0.722347i	$0.256936\pi$
$829$			1197.65i			1.44469i	−0.691531	+	0.722347i	$0.743064\pi$
$830$		0			0
$831$	55.7961	−	676.603i	0.0671434	−	0.814203i
$832$		0			0
$833$		−	925.184i		−	1.11067i
$834$		0			0
$835$		−	946.779i		−	1.13387i
$836$		0			0
$837$	−686.859	−	173.071i	−0.820620	−	0.206776i
$838$		0			0
$839$		−	423.903i		−	0.505248i	−0.967565	−	0.252624i	$-0.918706\pi$
$839$		−	423.903i		−	0.505248i	0.967565	−	0.252624i	$-0.0812936\pi$
$840$		0			0
$841$	−355.635			−0.422872
$842$		0			0
$843$	−3.42308	+	41.5094i	−0.00406059	+	0.0492401i
$844$		0			0
$845$	1011.22			1.19671
$846$		0			0
$847$	26.3183			0.0310724
$848$		0			0
$849$	75.5605	−	916.272i	0.0889995	−	1.07924i
$850$		0			0
$851$	1093.53			1.28499
$852$		0			0
$853$			33.0080i			0.0386964i	0.999813	+	0.0193482i	$0.00615911\pi$
$853$			33.0080i			0.0386964i	−0.999813	+	0.0193482i	$0.993841\pi$
$854$		0			0
$855$	−314.830	+	1895.89i	−0.368223	+	2.21741i
$856$		0			0
$857$			315.995i			0.368722i	0.982859	+	0.184361i	$0.0590215\pi$
$857$			315.995i			0.368722i	−0.982859	+	0.184361i	$0.940978\pi$
$858$		0			0
$859$		−	41.3157i		−	0.0480975i	−0.999711	−	0.0240487i	$-0.992344\pi$
$859$		−	41.3157i		−	0.0480975i	0.999711	−	0.0240487i	$-0.00765569\pi$
$860$		0			0
$861$	−5.08810	+	61.7000i	−0.00590953	+	0.0716609i
$862$		0			0
$863$		−	1173.36i		−	1.35963i	−0.733382	−	0.679817i	$-0.762059\pi$
$863$		−	1173.36i		−	1.35963i	0.733382	−	0.679817i	$-0.237941\pi$
$864$		0			0
$865$	−77.3763			−0.0894524
$866$		0			0
$867$	216.520	+	17.8553i	0.249735	+	0.0205944i
$868$		0			0
$869$	−686.302			−0.789760
$870$		0			0
$871$	1946.18			2.23442
$872$		0			0
$873$	828.839	+	137.637i	0.949415	+	0.157659i
$874$		0			0
$875$	−22.8605			−0.0261262
$876$		0			0
$877$		−	917.504i		−	1.04618i	−0.852276	−	0.523092i	$-0.824778\pi$
$877$		−	917.504i		−	1.04618i	0.852276	−	0.523092i	$-0.175222\pi$
$878$		0			0
$879$	−364.370	−	30.0478i	−0.414528	−	0.0341841i
$880$		0			0
$881$		−	1662.48i		−	1.88704i	−0.331322	−	0.943518i	$-0.607495\pi$
$881$		−	1662.48i		−	1.88704i	0.331322	−	0.943518i	$-0.392505\pi$
$882$		0			0
$883$		−	1258.25i		−	1.42497i	−0.701686	−	0.712486i	$-0.747569\pi$
$883$		−	1258.25i		−	1.42497i	0.701686	−	0.712486i	$-0.252431\pi$
$884$		0			0
$885$	1253.89	+	103.402i	1.41682	+	0.116838i
$886$		0			0
$887$		−	237.781i		−	0.268073i	−0.990976	−	0.134037i	$-0.957206\pi$
$887$		−	237.781i		−	0.268073i	0.990976	−	0.134037i	$-0.0427940\pi$
$888$		0			0
$889$	136.826			0.153910
$890$		0			0
$891$	665.947	+	227.446i	0.747416	+	0.255270i
$892$		0			0
$893$	−985.466			−1.10354
$894$		0			0
$895$	268.097			0.299550
$896$		0			0
$897$	1097.90	+	90.5383i	1.22397	+	0.100935i
$898$		0			0
$899$	−577.970			−0.642903
$900$		0			0
$901$		−	1690.01i		−	1.87570i
$902$		0			0
$903$	−7.19727	+	87.2764i	−0.00797039	+	0.0966517i
$904$		0			0
$905$		−	804.164i		−	0.888579i
$906$		0			0
$907$		−	1118.51i		−	1.23320i	−0.787277	−	0.616599i	$-0.788510\pi$
$907$		−	1118.51i		−	1.23320i	0.787277	−	0.616599i	$-0.211490\pi$
$908$		0			0
$909$	−1013.57	−	168.313i	−1.11504	−	0.185163i
$910$		0			0
$911$		−	307.830i		−	0.337903i	−0.985624	−	0.168952i	$-0.945962\pi$
$911$		−	307.830i		−	0.337903i	0.985624	−	0.168952i	$-0.0540382\pi$
$912$		0			0
$913$	−421.608			−0.461783
$914$		0			0
$915$	−54.8667	+	665.332i	−0.0599636	+	0.727138i
$916$		0			0
$917$	70.0794			0.0764225
$918$		0			0
$919$	195.828			0.213088			0.106544	−	0.994308i	$-0.466022\pi$
$919$	195.828			0.213088			0.106544	+	0.994308i	$0.466022\pi$
$920$		0			0
$921$	4.69525	−	56.9362i	0.00509799	−	0.0618199i
$922$		0			0
$923$	−1064.14			−1.15292
$924$		0			0
$925$			1016.56i			1.09899i
$926$		0			0
$927$	−621.804	−	103.257i	−0.670771	−	0.111388i
$928$		0			0
$929$			80.3569i			0.0864982i	0.999064	+	0.0432491i	$0.0137709\pi$
$929$			80.3569i			0.0864982i	−0.999064	+	0.0432491i	$0.986229\pi$
$930$		0			0
$931$			1565.91i			1.68197i
$932$		0			0
$933$	−55.0217	+	667.211i	−0.0589728	+	0.715124i
$934$		0			0
$935$		−	1096.10i		−	1.17230i
$936$		0			0
$937$	−502.272			−0.536043			−0.268021	−	0.963413i	$-0.586370\pi$
$937$	−502.272			−0.536043			−0.268021	+	0.963413i	$0.586370\pi$
$938$		0			0
$939$	−582.147	−	48.0069i	−0.619965	−	0.0511255i
$940$		0			0
$941$	1752.05			1.86190			0.930951	−	0.365144i	$-0.118980\pi$
$941$	1752.05			1.86190			0.930951	+	0.365144i	$0.118980\pi$
$942$		0			0
$943$	731.133			0.775326
$944$		0			0
$945$	−100.456	−	25.3125i	−0.106303	−	0.0267857i
$946$		0			0
$947$	−634.493			−0.670003			−0.335001	−	0.942218i	$-0.608737\pi$
$947$	−634.493			−0.670003			−0.335001	+	0.942218i	$0.608737\pi$
$948$		0			0
$949$		−	100.476i		−	0.105876i
$950$		0			0
$951$	1591.18	+	131.217i	1.67316	+	0.137977i
$952$		0			0
$953$		−	590.937i		−	0.620081i	−0.950723	−	0.310040i	$-0.899657\pi$
$953$		−	590.937i		−	0.620081i	0.950723	−	0.310040i	$-0.100343\pi$
$954$		0			0
$955$		−	2238.38i		−	2.34385i
$956$		0			0
$957$	572.264	+	47.1919i	0.597977	+	0.0493123i
$958$		0			0
$959$		−	79.0211i		−	0.0823994i
$960$		0			0
$961$	−272.757			−0.283827
$962$		0			0
$963$	451.976	+	75.0549i	0.469341	+	0.0779386i
$964$		0			0
$965$	1512.69			1.56755
$966$		0			0
$967$	−1194.01			−1.23476			−0.617381	−	0.786665i	$-0.711806\pi$
$967$	−1194.01			−1.23476			−0.617381	+	0.786665i	$0.711806\pi$
$968$		0			0
$969$	−1828.94	−	150.823i	−1.88745	−	0.155649i
$970$		0			0
$971$	−827.456			−0.852169			−0.426084	−	0.904683i	$-0.640107\pi$
$971$	−827.456			−0.852169			−0.426084	+	0.904683i	$0.640107\pi$
$972$		0			0
$973$			17.0913i			0.0175655i
$974$		0			0
$975$	−84.1660	+	1020.63i	−0.0863241	+	1.04679i
$976$		0			0
$977$			1770.96i			1.81266i	0.422576	+	0.906328i	$0.361126\pi$
$977$			1770.96i			1.81266i	−0.422576	+	0.906328i	$0.638874\pi$
$978$		0			0
$979$		−	510.072i		−	0.521013i
$980$		0			0
$981$	19.4876	−	117.353i	0.0198650	−	0.119626i
$982$		0			0
$983$			406.666i			0.413699i	0.978373	+	0.206850i	$0.0663211\pi$
$983$			406.666i			0.413699i	−0.978373	+	0.206850i	$0.933679\pi$
$984$		0			0
$985$	−1143.29			−1.16070
$986$		0			0
$987$	4.36580	−	52.9412i	0.00442331	−	0.0536385i
$988$		0			0
$989$	1034.21			1.04571
$990$		0			0
$991$	−1364.81			−1.37720			−0.688602	−	0.725140i	$-0.741775\pi$
$991$	−1364.81			−1.37720			−0.688602	+	0.725140i	$0.741775\pi$
$992$		0			0
$993$	22.6161	−	274.251i	0.0227756	−	0.276184i
$994$		0			0
$995$	3.97413			0.00399410
$996$		0			0
$997$		−	1201.02i		−	1.20463i	−0.798257	−	0.602317i	$-0.794244\pi$
$997$		−	1201.02i		−	1.20463i	0.798257	−	0.602317i	$-0.205756\pi$
$998$		0			0
$999$	352.191	−	1397.72i	0.352544	−	1.39912i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.h.g.641.1		16
3.2	odd	2	inner	768.3.h.g.641.15		16
4.3	odd	2		768.3.h.h.641.16		16
8.3	odd	2		768.3.h.h.641.1		16
8.5	even	2	inner	768.3.h.g.641.16		16
12.11	even	2		768.3.h.h.641.2		16
16.3	odd	4		384.3.e.a.257.6	yes	8
16.5	even	4		384.3.e.b.257.6	yes	8
16.11	odd	4		384.3.e.c.257.3	yes	8
16.13	even	4		384.3.e.d.257.3	yes	8
24.5	odd	2	inner	768.3.h.g.641.2		16
24.11	even	2		768.3.h.h.641.15		16
48.5	odd	4		384.3.e.b.257.5	yes	8
48.11	even	4		384.3.e.c.257.4	yes	8
48.29	odd	4		384.3.e.d.257.4	yes	8
48.35	even	4		384.3.e.a.257.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.e.a.257.5	✓	8	48.35	even	4
384.3.e.a.257.6	yes	8	16.3	odd	4
384.3.e.b.257.5	yes	8	48.5	odd	4
384.3.e.b.257.6	yes	8	16.5	even	4
384.3.e.c.257.3	yes	8	16.11	odd	4
384.3.e.c.257.4	yes	8	48.11	even	4
384.3.e.d.257.3	yes	8	16.13	even	4
384.3.e.d.257.4	yes	8	48.29	odd	4
768.3.h.g.641.1		16	1.1	even	1	trivial
768.3.h.g.641.2		16	24.5	odd	2	inner
768.3.h.g.641.15		16	3.2	odd	2	inner
768.3.h.g.641.16		16	8.5	even	2	inner
768.3.h.h.641.1		16	8.3	odd	2
768.3.h.h.641.2		16	12.11	even	2
768.3.h.h.641.15		16	24.11	even	2
768.3.h.h.641.16		16	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.98985 - 0.246559i) q^{3} -6.63641 q^{5} -0.578158 q^{7} +(8.87842 + 1.47435i) q^{9} +O(q^{10})\)	\(q+(-2.98985 - 0.246559i) q^{3} -6.63641 q^{5} -0.578158 q^{7} +(8.87842 + 1.47435i) q^{9} +8.68786 q^{11} -17.9269i q^{13} +(19.8419 + 1.63626i) q^{15} +19.0110i q^{17} -32.1769i q^{19} +(1.72861 + 0.142550i) q^{21} -20.4836i q^{23} +19.0419 q^{25} +(-26.1816 - 6.59713i) q^{27} -22.0310 q^{29} +26.2344 q^{31} +(-25.9754 - 2.14207i) q^{33} +3.83689 q^{35} +53.3855i q^{37} +(-4.42003 + 53.5988i) q^{39} +35.6935i q^{41} +50.4895i q^{43} +(-58.9208 - 9.78437i) q^{45} -30.6265i q^{47} -48.6657 q^{49} +(4.68732 - 56.8401i) q^{51} -88.8962 q^{53} -57.6562 q^{55} +(-7.93348 + 96.2040i) q^{57} +63.1939 q^{59} +33.5317i q^{61} +(-5.13313 - 0.852405i) q^{63} +118.970i q^{65} +108.562i q^{67} +(-5.05041 + 61.2430i) q^{69} -59.3600i q^{71} +5.60477 q^{73} +(-56.9325 - 4.69495i) q^{75} -5.02296 q^{77} -78.9955 q^{79} +(76.6526 + 26.1797i) q^{81} -48.5283 q^{83} -126.165i q^{85} +(65.8694 + 5.43193i) q^{87} -58.7109i q^{89} +10.3646i q^{91} +(-78.4369 - 6.46831i) q^{93} +213.539i q^{95} +93.3544 q^{97} +(77.1345 + 12.8089i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{7}+O(q^{10}) \) 16 * q - 16 * q^7	\( 16 q - 16 q^{7} + 32 q^{15} + 80 q^{25} - 112 q^{31} + 16 q^{33} + 208 q^{39} + 144 q^{49} - 384 q^{55} + 80 q^{57} + 528 q^{63} + 160 q^{73} - 816 q^{79} + 144 q^{81} + 736 q^{87} + 192 q^{97}+O(q^{100}) \) 16 * q - 16 * q^7 + 32 * q^15 + 80 * q^25 - 112 * q^31 + 16 * q^33 + 208 * q^39 + 144 * q^49 - 384 * q^55 + 80 * q^57 + 528 * q^63 + 160 * q^73 - 816 * q^79 + 144 * q^81 + 736 * q^87 + 192 * q^97

Introduction

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