LMFDB - Embedded newform 768.3.h.g.641.12

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			641.12
Root			$-2.18318 - 0.904303i$ of defining polynomial
Character	$\chi$	$=$	768.641
Dual form			768.3.h.g.641.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.32750 + 2.69031i) q^{3} -0.640013 q^{5} -2.72077 q^{7} +(-5.47550 + 7.14275i) q^{9} +O(q^{10})$	\(q+(1.32750 + 2.69031i) q^{3} -0.640013 q^{5} -2.72077 q^{7} +(-5.47550 + 7.14275i) q^{9} -11.2836 q^{11} -5.25176i q^{13} +(-0.849616 - 1.72183i) q^{15} -14.8718i q^{17} +15.0798i q^{19} +(-3.61181 - 7.31969i) q^{21} -36.4411i q^{23} -24.5904 q^{25} +(-26.4849 - 5.24877i) q^{27} +51.7310 q^{29} -36.5009 q^{31} +(-14.9790 - 30.3565i) q^{33} +1.74133 q^{35} -63.6951i q^{37} +(14.1289 - 6.97170i) q^{39} -12.1500i q^{41} +11.8032i q^{43} +(3.50439 - 4.57146i) q^{45} +61.1247i q^{47} -41.5974 q^{49} +(40.0097 - 19.7423i) q^{51} -59.1695 q^{53} +7.22168 q^{55} +(-40.5694 + 20.0185i) q^{57} +37.2898 q^{59} -58.1987i q^{61} +(14.8975 - 19.4338i) q^{63} +3.36120i q^{65} +23.0991i q^{67} +(98.0376 - 48.3754i) q^{69} -7.29656i q^{71} -73.4504 q^{73} +(-32.6437 - 66.1557i) q^{75} +30.7002 q^{77} +58.5098 q^{79} +(-21.0379 - 78.2203i) q^{81} -32.3939 q^{83} +9.51815i q^{85} +(68.6728 + 139.172i) q^{87} -112.260i q^{89} +14.2888i q^{91} +(-48.4549 - 98.1987i) q^{93} -9.65130i q^{95} -80.0338 q^{97} +(61.7836 - 80.5963i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.32750	+	2.69031i	0.442499	+	0.896769i
$3$	1.32750	+	2.69031i	0.442499	+	0.896769i
$4$		0			0
$5$	−0.640013			−0.128003			−0.0640013	−	0.997950i	$-0.520386\pi$
$5$	−0.640013			−0.128003			−0.0640013	+	0.997950i	$0.520386\pi$
$6$		0			0
$7$	−2.72077			−0.388681			−0.194340	−	0.980934i	$-0.562257\pi$
$7$	−2.72077			−0.388681			−0.194340	+	0.980934i	$0.562257\pi$
$8$		0			0
$9$	−5.47550	+	7.14275i	−0.608389	+	0.793639i
$10$		0			0
$11$	−11.2836			−1.02579			−0.512893	−	0.858452i	$-0.671426\pi$
$11$	−11.2836			−1.02579			−0.512893	+	0.858452i	$0.671426\pi$
$12$		0			0
$13$		−	5.25176i		−	0.403982i	−0.979387	−	0.201991i	$-0.935259\pi$
$13$		−	5.25176i		−	0.403982i	0.979387	−	0.201991i	$-0.0647411\pi$
$14$		0			0
$15$	−0.849616	−	1.72183i	−0.0566411	−	0.114789i
$16$		0			0
$17$		−	14.8718i		−	0.874812i	−0.899264	−	0.437406i	$-0.855897\pi$
$17$		−	14.8718i		−	0.874812i	0.899264	−	0.437406i	$-0.144103\pi$
$18$		0			0
$19$			15.0798i			0.793676i	0.917889	+	0.396838i	$0.129893\pi$
$19$			15.0798i			0.793676i	−0.917889	+	0.396838i	$0.870107\pi$
$20$		0			0
$21$	−3.61181	−	7.31969i	−0.171991	−	0.348557i
$22$		0			0
$23$		−	36.4411i		−	1.58439i	−0.610266	−	0.792197i	$-0.708937\pi$
$23$		−	36.4411i		−	1.58439i	0.610266	−	0.792197i	$-0.291063\pi$
$24$		0			0
$25$	−24.5904			−0.983615
$26$		0			0
$27$	−26.4849	−	5.24877i	−0.980923	−	0.194399i
$28$		0			0
$29$	51.7310			1.78383			0.891914	−	0.452205i	$-0.149362\pi$
$29$	51.7310			1.78383			0.891914	+	0.452205i	$0.149362\pi$
$30$		0			0
$31$	−36.5009			−1.17745			−0.588724	−	0.808334i	$-0.700370\pi$
$31$	−36.5009			−1.17745			−0.588724	+	0.808334i	$0.700370\pi$
$32$		0			0
$33$	−14.9790	−	30.3565i	−0.453910	−	0.919893i
$34$		0			0
$35$	1.74133			0.0497522
$36$		0			0
$37$		−	63.6951i		−	1.72149i	−0.509036	−	0.860745i	$-0.669998\pi$
$37$		−	63.6951i		−	1.72149i	0.509036	−	0.860745i	$-0.330002\pi$
$38$		0			0
$39$	14.1289	−	6.97170i	0.362278	−	0.178762i
$40$		0			0
$41$		−	12.1500i		−	0.296342i	−0.988962	−	0.148171i	$-0.952661\pi$
$41$		−	12.1500i		−	0.296342i	0.988962	−	0.148171i	$-0.0473386\pi$
$42$		0			0
$43$			11.8032i			0.274493i	0.990537	+	0.137247i	$0.0438253\pi$
$43$			11.8032i			0.274493i	−0.990537	+	0.137247i	$0.956175\pi$
$44$		0			0
$45$	3.50439	−	4.57146i	0.0778753	−	0.101588i
$46$		0			0
$47$			61.1247i			1.30053i	0.759709	+	0.650263i	$0.225341\pi$
$47$			61.1247i			1.30053i	−0.759709	+	0.650263i	$0.774659\pi$
$48$		0			0
$49$	−41.5974			−0.848927
$50$		0			0
$51$	40.0097	−	19.7423i	0.784504	−	0.387104i
$52$		0			0
$53$	−59.1695			−1.11641			−0.558203	−	0.829705i	$-0.688509\pi$
$53$	−59.1695			−1.11641			−0.558203	+	0.829705i	$0.688509\pi$
$54$		0			0
$55$	7.22168			0.131303
$56$		0			0
$57$	−40.5694	+	20.0185i	−0.711744	+	0.351201i
$58$		0			0
$59$	37.2898			0.632031			0.316016	−	0.948754i	$-0.397655\pi$
$59$	37.2898			0.632031			0.316016	+	0.948754i	$0.397655\pi$
$60$		0			0
$61$		−	58.1987i		−	0.954076i	−0.878883	−	0.477038i	$-0.841710\pi$
$61$		−	58.1987i		−	0.954076i	0.878883	−	0.477038i	$-0.158290\pi$
$62$		0			0
$63$	14.8975	−	19.4338i	0.236469	−	0.308472i
$64$		0			0
$65$			3.36120i			0.0517107i
$66$		0			0
$67$			23.0991i			0.344762i	0.985030	+	0.172381i	$0.0551461\pi$
$67$			23.0991i			0.344762i	−0.985030	+	0.172381i	$0.944854\pi$
$68$		0			0
$69$	98.0376	−	48.3754i	1.42084	−	0.701093i
$70$		0			0
$71$		−	7.29656i		−	0.102768i	−0.998679	−	0.0513842i	$-0.983637\pi$
$71$		−	7.29656i		−	0.102768i	0.998679	−	0.0513842i	$-0.0163633\pi$
$72$		0			0
$73$	−73.4504			−1.00617			−0.503085	−	0.864237i	$-0.667802\pi$
$73$	−73.4504			−1.00617			−0.503085	+	0.864237i	$0.667802\pi$
$74$		0			0
$75$	−32.6437	−	66.1557i	−0.435249	−	0.882076i
$76$		0			0
$77$	30.7002			0.398703
$78$		0			0
$79$	58.5098			0.740630			0.370315	−	0.928906i	$-0.379250\pi$
$79$	58.5098			0.740630			0.370315	+	0.928906i	$0.379250\pi$
$80$		0			0
$81$	−21.0379	−	78.2203i	−0.259727	−	0.965682i
$82$		0			0
$83$	−32.3939			−0.390287			−0.195144	−	0.980775i	$-0.562517\pi$
$83$	−32.3939			−0.390287			−0.195144	+	0.980775i	$0.562517\pi$
$84$		0			0
$85$			9.51815i			0.111978i
$86$		0			0
$87$	68.6728	+	139.172i	0.789343	+	1.59968i
$88$		0			0
$89$		−	112.260i		−	1.26135i	−0.776046	−	0.630677i	$-0.782777\pi$
$89$		−	112.260i		−	1.26135i	0.776046	−	0.630677i	$-0.217223\pi$
$90$		0			0
$91$			14.2888i			0.157020i
$92$		0			0
$93$	−48.4549	−	98.1987i	−0.521020	−	1.05590i
$94$		0			0
$95$		−	9.65130i		−	0.101593i
$96$		0			0
$97$	−80.0338			−0.825090			−0.412545	−	0.910937i	$-0.635360\pi$
$97$	−80.0338			−0.825090			−0.412545	+	0.910937i	$0.635360\pi$
$98$		0			0
$99$	61.7836	−	80.5963i	0.624077	−	0.814104i
$100$		0			0
$101$	119.373			1.18191			0.590954	−	0.806705i	$-0.298751\pi$
$101$	119.373			1.18191			0.590954	+	0.806705i	$0.298751\pi$
$102$		0			0
$103$	−125.522			−1.21866			−0.609330	−	0.792917i	$-0.708562\pi$
$103$	−125.522			−1.21866			−0.609330	+	0.792917i	$0.708562\pi$
$104$		0			0
$105$	2.31161	+	4.68470i	0.0220153	+	0.0446162i
$106$		0			0
$107$	148.669			1.38943			0.694716	−	0.719284i	$-0.255530\pi$
$107$	148.669			1.38943			0.694716	+	0.719284i	$0.255530\pi$
$108$		0			0
$109$		−	70.6664i		−	0.648316i	−0.946003	−	0.324158i	$-0.894919\pi$
$109$		−	70.6664i		−	0.648316i	0.946003	−	0.324158i	$-0.105081\pi$
$110$		0			0
$111$	171.359	−	84.5552i	1.54378	−	0.761758i
$112$		0			0
$113$			130.968i			1.15901i	0.814969	+	0.579504i	$0.196754\pi$
$113$			130.968i			1.15901i	−0.814969	+	0.579504i	$0.803246\pi$
$114$		0			0
$115$			23.3228i			0.202807i
$116$		0			0
$117$	37.5120	+	28.7560i	0.320616	+	0.245778i
$118$		0			0
$119$			40.4627i			0.340023i
$120$		0			0
$121$	6.32073			0.0522374
$122$		0			0
$123$	32.6873	−	16.1291i	0.265750	−	0.131131i
$124$		0			0
$125$	31.7385			0.253908
$126$		0			0
$127$	−209.206			−1.64729			−0.823647	−	0.567103i	$-0.808064\pi$
$127$	−209.206			−1.64729			−0.823647	+	0.567103i	$0.808064\pi$
$128$		0			0
$129$	−31.7543	+	15.6687i	−0.246157	+	0.121463i
$130$		0			0
$131$	−174.839			−1.33465			−0.667326	−	0.744766i	$-0.732561\pi$
$131$	−174.839			−1.33465			−0.667326	+	0.744766i	$0.732561\pi$
$132$		0			0
$133$		−	41.0287i		−	0.308487i
$134$		0			0
$135$	16.9507	+	3.35929i	0.125561	+	0.0248836i
$136$		0			0
$137$		−	162.945i		−	1.18938i	−0.803956	−	0.594689i	$-0.797275\pi$
$137$		−	162.945i		−	1.18938i	0.803956	−	0.594689i	$-0.202725\pi$
$138$		0			0
$139$			194.227i			1.39732i	0.715454	+	0.698660i	$0.246220\pi$
$139$			194.227i			1.39732i	−0.715454	+	0.698660i	$0.753780\pi$
$140$		0			0
$141$	−164.444	+	81.1429i	−1.16627	+	0.575482i
$142$		0			0
$143$			59.2590i			0.414399i
$144$		0			0
$145$	−33.1085			−0.228335
$146$		0			0
$147$	−55.2205	−	111.910i	−0.375650	−	0.761291i
$148$		0			0
$149$	−83.4695			−0.560198			−0.280099	−	0.959971i	$-0.590367\pi$
$149$	−83.4695			−0.560198			−0.280099	+	0.959971i	$0.590367\pi$
$150$		0			0
$151$	−60.3488			−0.399661			−0.199830	−	0.979830i	$-0.564039\pi$
$151$	−60.3488			−0.399661			−0.199830	+	0.979830i	$0.564039\pi$
$152$		0			0
$153$	106.226	+	81.4306i	0.694285	+	0.532226i
$154$		0			0
$155$	23.3611			0.150717
$156$		0			0
$157$			100.762i			0.641798i	0.947113	+	0.320899i	$0.103985\pi$
$157$			100.762i			0.641798i	−0.947113	+	0.320899i	$0.896015\pi$
$158$		0			0
$159$	−78.5474	−	159.184i	−0.494009	−	1.00116i
$160$		0			0
$161$			99.1476i			0.615824i
$162$		0			0
$163$			69.7149i			0.427699i	0.976867	+	0.213849i	$0.0686002\pi$
$163$			69.7149i			0.427699i	−0.976867	+	0.213849i	$0.931400\pi$
$164$		0			0
$165$	9.58677	+	19.4285i	0.0581016	+	0.117749i
$166$		0			0
$167$		−	120.823i		−	0.723491i	−0.932277	−	0.361745i	$-0.882181\pi$
$167$		−	120.823i		−	0.723491i	0.932277	−	0.361745i	$-0.117819\pi$
$168$		0			0
$169$	141.419			0.836799
$170$		0			0
$171$	−107.712	−	82.5697i	−0.629893	−	0.482863i
$172$		0			0
$173$	−30.0602			−0.173758			−0.0868791	−	0.996219i	$-0.527689\pi$
$173$	−30.0602			−0.173758			−0.0868791	+	0.996219i	$0.527689\pi$
$174$		0			0
$175$	66.9047			0.382312
$176$		0			0
$177$	49.5022	+	100.321i	0.279673	+	0.566786i
$178$		0			0
$179$	−114.784			−0.641250			−0.320625	−	0.947206i	$-0.603893\pi$
$179$	−114.784			−0.641250			−0.320625	+	0.947206i	$0.603893\pi$
$180$		0			0
$181$			181.627i			1.00347i	0.865022	+	0.501733i	$0.167304\pi$
$181$			181.627i			1.00347i	−0.865022	+	0.501733i	$0.832696\pi$
$182$		0			0
$183$	156.572	−	77.2586i	0.855586	−	0.422178i
$184$		0			0
$185$			40.7657i			0.220355i
$186$		0			0
$187$			167.808i			0.897370i
$188$		0			0
$189$	72.0592	+	14.2807i	0.381266	+	0.0755592i
$190$		0			0
$191$		−	209.226i		−	1.09543i	−0.836666	−	0.547713i	$-0.815498\pi$
$191$		−	209.226i		−	1.09543i	0.836666	−	0.547713i	$-0.184502\pi$
$192$		0			0
$193$	161.330			0.835907			0.417954	−	0.908468i	$-0.362747\pi$
$193$	161.330			0.835907			0.417954	+	0.908468i	$0.362747\pi$
$194$		0			0
$195$	−9.04265	+	4.46198i	−0.0463726	+	0.0228820i
$196$		0			0
$197$	−144.953			−0.735801			−0.367901	−	0.929865i	$-0.619923\pi$
$197$	−144.953			−0.735801			−0.367901	+	0.929865i	$0.619923\pi$
$198$		0			0
$199$	−237.264			−1.19228			−0.596140	−	0.802880i	$-0.703300\pi$
$199$	−237.264			−1.19228			−0.596140	+	0.802880i	$0.703300\pi$
$200$		0			0
$201$	−62.1436	+	30.6640i	−0.309172	+	0.152557i
$202$		0			0
$203$	−140.748			−0.693340
$204$		0			0
$205$			7.77617i			0.0379325i
$206$		0			0
$207$	260.289	+	199.533i	1.25744	+	0.963927i
$208$		0			0
$209$		−	170.156i		−	0.814142i
$210$		0			0
$211$			307.117i			1.45553i	0.685827	+	0.727765i	$0.259441\pi$
$211$			307.117i			1.45553i	−0.685827	+	0.727765i	$0.740559\pi$
$212$		0			0
$213$	19.6300	−	9.68617i	0.0921595	−	0.0454750i
$214$		0			0
$215$		−	7.55422i		−	0.0351359i
$216$		0			0
$217$	99.3105			0.457652
$218$		0			0
$219$	−97.5053	−	197.604i	−0.445230	−	0.902302i
$220$		0			0
$221$	−78.1032			−0.353408
$222$		0			0
$223$	−438.001			−1.96413			−0.982065	−	0.188544i	$-0.939623\pi$
$223$	−438.001			−1.96413			−0.982065	+	0.188544i	$0.939623\pi$
$224$		0			0
$225$	134.645	−	175.643i	0.598420	−	0.780636i
$226$		0			0
$227$	270.755			1.19275			0.596376	−	0.802705i	$-0.296607\pi$
$227$	270.755			1.19275			0.596376	+	0.802705i	$0.296607\pi$
$228$		0			0
$229$			98.6778i			0.430907i	0.976514	+	0.215454i	$0.0691230\pi$
$229$			98.6778i			0.430907i	−0.976514	+	0.215454i	$0.930877\pi$
$230$		0			0
$231$	40.7544	+	82.5929i	0.176426	+	0.357545i
$232$		0			0
$233$			432.405i			1.85581i	0.372812	+	0.927907i	$0.378394\pi$
$233$			432.405i			1.85581i	−0.372812	+	0.927907i	$0.621606\pi$
$234$		0			0
$235$		−	39.1206i		−	0.166471i
$236$		0			0
$237$	77.6717	+	157.409i	0.327729	+	0.664174i
$238$		0			0
$239$			275.799i			1.15397i	0.816754	+	0.576986i	$0.195771\pi$
$239$			275.799i			1.15397i	−0.816754	+	0.576986i	$0.804229\pi$
$240$		0			0
$241$	416.770			1.72934			0.864669	−	0.502343i	$-0.167528\pi$
$241$	416.770			1.72934			0.864669	+	0.502343i	$0.167528\pi$
$242$		0			0
$243$	182.509	−	160.436i	0.751065	−	0.660228i
$244$		0			0
$245$	26.6229			0.108665
$246$		0			0
$247$	79.1958			0.320631
$248$		0			0
$249$	−43.0028	−	87.1494i	−0.172702	−	0.349998i
$250$		0			0
$251$	−165.903			−0.660966			−0.330483	−	0.943812i	$-0.607212\pi$
$251$	−165.903			−0.660966			−0.330483	+	0.943812i	$0.607212\pi$
$252$		0			0
$253$			411.188i			1.62525i
$254$		0			0
$255$	−25.6068	+	12.6353i	−0.100419	+	0.0495503i
$256$		0			0
$257$			59.6381i			0.232055i	0.993246	+	0.116028i	$0.0370161\pi$
$257$			59.6381i			0.232055i	−0.993246	+	0.116028i	$0.962984\pi$
$258$		0			0
$259$			173.300i			0.669110i
$260$		0			0
$261$	−283.253	+	369.502i	−1.08526	+	1.41572i
$262$		0			0
$263$		−	67.2606i		−	0.255744i	−0.991791	−	0.127872i	$-0.959185\pi$
$263$		−	67.2606i		−	0.255744i	0.991791	−	0.127872i	$-0.0408146\pi$
$264$		0			0
$265$	37.8693			0.142903
$266$		0			0
$267$	302.015	−	149.026i	1.13114	−	0.558148i
$268$		0			0
$269$	36.1495			0.134385			0.0671923	−	0.997740i	$-0.478596\pi$
$269$	36.1495			0.134385			0.0671923	+	0.997740i	$0.478596\pi$
$270$		0			0
$271$	328.311			1.21148			0.605741	−	0.795662i	$-0.292877\pi$
$271$	328.311			1.21148			0.605741	+	0.795662i	$0.292877\pi$
$272$		0			0
$273$	−38.4413	+	18.9684i	−0.140811	+	0.0694812i
$274$		0			0
$275$	277.469			1.00898
$276$		0			0
$277$		−	60.7455i		−	0.219298i	−0.993970	−	0.109649i	$-0.965027\pi$
$277$		−	60.7455i		−	0.219298i	0.993970	−	0.109649i	$-0.0349727\pi$
$278$		0			0
$279$	199.861	−	260.717i	0.716347	−	0.934470i
$280$		0			0
$281$			366.827i			1.30543i	0.757602	+	0.652717i	$0.226371\pi$
$281$			366.827i			1.30543i	−0.757602	+	0.652717i	$0.773629\pi$
$282$		0			0
$283$		−	104.032i		−	0.367603i	−0.982963	−	0.183801i	$-0.941160\pi$
$283$		−	104.032i		−	0.367603i	0.982963	−	0.183801i	$-0.0588404\pi$
$284$		0			0
$285$	25.9650	−	12.8121i	0.0911051	−	0.0449547i
$286$		0			0
$287$			33.0574i			0.115182i
$288$		0			0
$289$	67.8293			0.234703
$290$		0			0
$291$	−106.245	−	215.315i	−0.365102	−	0.739915i
$292$		0			0
$293$	−410.579			−1.40129			−0.700647	−	0.713508i	$-0.747105\pi$
$293$	−410.579			−1.40129			−0.700647	+	0.713508i	$0.747105\pi$
$294$		0			0
$295$	−23.8660			−0.0809017
$296$		0			0
$297$	298.846	+	59.2253i	1.00622	+	0.199412i
$298$		0			0
$299$	−191.380			−0.640066
$300$		0			0
$301$		−	32.1138i		−	0.106690i
$302$		0			0
$303$	158.467	+	321.149i	0.522994	+	1.05990i
$304$		0			0
$305$			37.2479i			0.122124i
$306$		0			0
$307$		−	270.025i		−	0.879561i	−0.898105	−	0.439781i	$-0.855056\pi$
$307$		−	270.025i		−	0.879561i	0.898105	−	0.439781i	$-0.144944\pi$
$308$		0			0
$309$	−166.630	−	337.693i	−0.539256	−	1.09286i
$310$		0			0
$311$			251.555i			0.808858i	0.914569	+	0.404429i	$0.132530\pi$
$311$			251.555i			0.808858i	−0.914569	+	0.404429i	$0.867470\pi$
$312$		0			0
$313$	83.9560			0.268230			0.134115	−	0.990966i	$-0.457181\pi$
$313$	83.9560			0.268230			0.134115	+	0.990966i	$0.457181\pi$
$314$		0			0
$315$	−9.53463	+	12.4379i	−0.0302687	+	0.0394853i
$316$		0			0
$317$	−270.737			−0.854060			−0.427030	−	0.904238i	$-0.640440\pi$
$317$	−270.737			−0.854060			−0.427030	+	0.904238i	$0.640440\pi$
$318$		0			0
$319$	−583.715			−1.82983
$320$		0			0
$321$	197.358	+	399.966i	0.614823	+	1.24600i
$322$		0			0
$323$	224.265			0.694318
$324$		0			0
$325$			129.143i			0.397363i
$326$		0			0
$327$	190.114	−	93.8095i	0.581389	−	0.286879i
$328$		0			0
$329$		−	166.306i		−	0.505489i
$330$		0			0
$331$		−	9.91562i		−	0.0299566i	−0.999888	−	0.0149783i	$-0.995232\pi$
$331$		−	9.91562i		−	0.0299566i	0.999888	−	0.0149783i	$-0.00476791\pi$
$332$		0			0
$333$	454.959	+	348.763i	1.36624	+	1.04733i
$334$		0			0
$335$		−	14.7837i		−	0.0441305i
$336$		0			0
$337$	−289.542			−0.859174			−0.429587	−	0.903025i	$-0.641341\pi$
$337$	−289.542			−0.859174			−0.429587	+	0.903025i	$0.641341\pi$
$338$		0			0
$339$	−352.344	+	173.860i	−1.03936	+	0.512860i
$340$		0			0
$341$	411.864			1.20781
$342$		0			0
$343$	246.494			0.718643
$344$		0			0
$345$	−62.7454	+	30.9609i	−0.181871	+	0.0897418i
$346$		0			0
$347$	331.079			0.954119			0.477060	−	0.878871i	$-0.341703\pi$
$347$	331.079			0.954119			0.477060	+	0.878871i	$0.341703\pi$
$348$		0			0
$349$			302.280i			0.866133i	0.901362	+	0.433067i	$0.142569\pi$
$349$			302.280i			0.866133i	−0.901362	+	0.433067i	$0.857431\pi$
$350$		0			0
$351$	−27.5653	+	139.092i	−0.0785337	+	0.396275i
$352$		0			0
$353$		−	528.345i		−	1.49673i	−0.663289	−	0.748364i	$-0.730840\pi$
$353$		−	528.345i		−	1.49673i	0.663289	−	0.748364i	$-0.269160\pi$
$354$		0			0
$355$			4.66989i			0.0131546i
$356$		0			0
$357$	−108.857	+	53.7142i	−0.304922	+	0.150460i
$358$		0			0
$359$		−	59.5167i		−	0.165785i	−0.996559	−	0.0828923i	$-0.973584\pi$
$359$		−	59.5167i		−	0.165785i	0.996559	−	0.0828923i	$-0.0264158\pi$
$360$		0			0
$361$	133.598			0.370078
$362$		0			0
$363$	8.39076	+	17.0047i	0.0231150	+	0.0468449i
$364$		0			0
$365$	47.0092			0.128792
$366$		0			0
$367$	423.762			1.15466			0.577332	−	0.816509i	$-0.304094\pi$
$367$	423.762			1.15466			0.577332	+	0.816509i	$0.304094\pi$
$368$		0			0
$369$	86.7846	+	66.5274i	0.235189	+	0.180291i
$370$		0			0
$371$	160.986			0.433925
$372$		0			0
$373$		−	26.5076i		−	0.0710660i	−0.999369	−	0.0355330i	$-0.988687\pi$
$373$		−	26.5076i		−	0.0710660i	0.999369	−	0.0355330i	$-0.0113129\pi$
$374$		0			0
$375$	42.1328	+	85.3863i	0.112354	+	0.227697i
$376$		0			0
$377$		−	271.679i		−	0.720634i
$378$		0			0
$379$			383.169i			1.01100i	0.862827	+	0.505500i	$0.168692\pi$
$379$			383.169i			1.01100i	−0.862827	+	0.505500i	$0.831308\pi$
$380$		0			0
$381$	−277.721	−	562.829i	−0.728927	−	1.47724i
$382$		0			0
$383$		−	299.496i		−	0.781973i	−0.920396	−	0.390987i	$-0.872134\pi$
$383$		−	299.496i		−	0.781973i	0.920396	−	0.390987i	$-0.127866\pi$
$384$		0			0
$385$	−19.6485			−0.0510351
$386$		0			0
$387$	−84.3075	−	64.6285i	−0.217849	−	0.166999i
$388$		0			0
$389$	437.687			1.12516			0.562580	−	0.826743i	$-0.309809\pi$
$389$	437.687			1.12516			0.562580	+	0.826743i	$0.309809\pi$
$390$		0			0
$391$	−541.944			−1.38605
$392$		0			0
$393$	−232.099	−	470.372i	−0.590583	−	1.19687i
$394$		0			0
$395$	−37.4470			−0.0948027
$396$		0			0
$397$		−	669.283i		−	1.68585i	−0.538029	−	0.842926i	$-0.680831\pi$
$397$		−	669.283i		−	1.68585i	0.538029	−	0.842926i	$-0.319169\pi$
$398$		0			0
$399$	110.380	−	54.4656i	0.276641	−	0.136505i
$400$		0			0
$401$		−	56.1062i		−	0.139916i	−0.997550	−	0.0699578i	$-0.977714\pi$
$401$		−	56.1062i		−	0.139916i	0.997550	−	0.0699578i	$-0.0222865\pi$
$402$		0			0
$403$			191.694i			0.475668i
$404$		0			0
$405$	13.4645	+	50.0620i	0.0332457	+	0.123610i
$406$		0			0
$407$			718.713i			1.76588i
$408$		0			0
$409$	−248.582			−0.607779			−0.303889	−	0.952707i	$-0.598285\pi$
$409$	−248.582			−0.607779			−0.303889	+	0.952707i	$0.598285\pi$
$410$		0			0
$411$	438.371	−	216.309i	1.06660	−	0.526299i
$412$		0			0
$413$	−101.457			−0.245658
$414$		0			0
$415$	20.7325			0.0499578
$416$		0			0
$417$	−522.531	+	257.836i	−1.25307	+	0.618313i
$418$		0			0
$419$	−459.935			−1.09770			−0.548849	−	0.835922i	$-0.684934\pi$
$419$	−459.935			−1.09770			−0.548849	+	0.835922i	$0.684934\pi$
$420$		0			0
$421$			97.4789i			0.231541i	0.993276	+	0.115771i	$0.0369338\pi$
$421$			97.4789i			0.231541i	−0.993276	+	0.115771i	$0.963066\pi$
$422$		0			0
$423$	−436.599	−	334.688i	−1.03215	−	0.791225i
$424$		0			0
$425$			365.703i			0.860479i
$426$		0			0
$427$			158.345i			0.370831i
$428$		0			0
$429$	−159.425	+	78.6663i	−0.371620	+	0.183371i
$430$		0			0
$431$		−	545.207i		−	1.26498i	−0.774568	−	0.632490i	$-0.782033\pi$
$431$		−	545.207i		−	1.26498i	0.774568	−	0.632490i	$-0.217967\pi$
$432$		0			0
$433$	−24.5297			−0.0566506			−0.0283253	−	0.999599i	$-0.509017\pi$
$433$	−24.5297			−0.0566506			−0.0283253	+	0.999599i	$0.509017\pi$
$434$		0			0
$435$	−43.9515	−	89.0721i	−0.101038	−	0.204763i
$436$		0			0
$437$	549.526			1.25750
$438$		0			0
$439$	−81.8743			−0.186502			−0.0932509	−	0.995643i	$-0.529726\pi$
$439$	−81.8743			−0.186502			−0.0932509	+	0.995643i	$0.529726\pi$
$440$		0			0
$441$	227.767	−	297.120i	0.516478	−	0.673742i
$442$		0			0
$443$	−831.356			−1.87665			−0.938325	−	0.345753i	$-0.887623\pi$
$443$	−831.356			−1.87665			−0.938325	+	0.345753i	$0.887623\pi$
$444$		0			0
$445$			71.8482i			0.161457i
$446$		0			0
$447$	−110.806	−	224.559i	−0.247887	−	0.502368i
$448$		0			0
$449$			259.553i			0.578070i	0.957319	+	0.289035i	$0.0933344\pi$
$449$			259.553i			0.578070i	−0.957319	+	0.289035i	$0.906666\pi$
$450$		0			0
$451$			137.097i			0.303983i
$452$		0			0
$453$	−80.1129	−	162.357i	−0.176850	−	0.358403i
$454$		0			0
$455$		−	9.14503i		−	0.0200990i
$456$		0			0
$457$	−373.184			−0.816596			−0.408298	−	0.912849i	$-0.633878\pi$
$457$	−373.184			−0.816596			−0.408298	+	0.912849i	$0.633878\pi$
$458$		0			0
$459$	−78.0588	+	393.879i	−0.170063	+	0.858123i
$460$		0			0
$461$	−672.996			−1.45986			−0.729930	−	0.683522i	$-0.760447\pi$
$461$	−672.996			−1.45986			−0.729930	+	0.683522i	$0.760447\pi$
$462$		0			0
$463$	−28.0435			−0.0605690			−0.0302845	−	0.999541i	$-0.509641\pi$
$463$	−28.0435			−0.0605690			−0.0302845	+	0.999541i	$0.509641\pi$
$464$		0			0
$465$	31.0118	+	62.8484i	0.0666920	+	0.135158i
$466$		0			0
$467$	−373.758			−0.800339			−0.400169	−	0.916441i	$-0.631049\pi$
$467$	−373.758			−0.800339			−0.400169	+	0.916441i	$0.631049\pi$
$468$		0			0
$469$		−	62.8472i		−	0.134003i
$470$		0			0
$471$	−271.082	+	133.762i	−0.575545	+	0.283995i
$472$		0			0
$473$		−	133.183i		−	0.281572i
$474$		0			0
$475$		−	370.819i		−	0.780672i
$476$		0			0
$477$	323.982	−	422.633i	0.679208	−	0.886023i
$478$		0			0
$479$		−	915.154i		−	1.91055i	−0.295718	−	0.955275i	$-0.595559\pi$
$479$		−	915.154i		−	1.91055i	0.295718	−	0.955275i	$-0.404441\pi$
$480$		0			0
$481$	−334.512			−0.695451
$482$		0			0
$483$	−266.737	+	131.618i	−0.552251	+	0.272502i
$484$		0			0
$485$	51.2227			0.105614
$486$		0			0
$487$	238.560			0.489856			0.244928	−	0.969541i	$-0.421236\pi$
$487$	238.560			0.489856			0.244928	+	0.969541i	$0.421236\pi$
$488$		0			0
$489$	−187.555	+	92.5464i	−0.383547	+	0.189257i
$490$		0			0
$491$	435.657			0.887286			0.443643	−	0.896204i	$-0.353686\pi$
$491$	435.657			0.887286			0.443643	+	0.896204i	$0.353686\pi$
$492$		0			0
$493$		−	769.334i		−	1.56052i
$494$		0			0
$495$	−39.5423	+	51.5827i	−0.0798835	+	0.104207i
$496$		0			0
$497$			19.8522i			0.0399441i
$498$		0			0
$499$		−	209.499i		−	0.419838i	−0.977719	−	0.209919i	$-0.932680\pi$
$499$		−	209.499i		−	0.419838i	0.977719	−	0.209919i	$-0.0673200\pi$
$500$		0			0
$501$	325.051	−	160.392i	0.648804	−	0.320144i
$502$		0			0
$503$		−	558.314i		−	1.10997i	−0.831861	−	0.554984i	$-0.812724\pi$
$503$		−	558.314i		−	1.10997i	0.831861	−	0.554984i	$-0.187276\pi$
$504$		0			0
$505$	−76.4002			−0.151287
$506$		0			0
$507$	187.733	+	380.460i	0.370283	+	0.750415i
$508$		0			0
$509$	319.621			0.627939			0.313969	−	0.949433i	$-0.398341\pi$
$509$	319.621			0.627939			0.313969	+	0.949433i	$0.398341\pi$
$510$		0			0
$511$	199.841			0.391079
$512$		0			0
$513$	79.1507	−	399.388i	0.154290	−	0.778535i
$514$		0			0
$515$	80.3357			0.155992
$516$		0			0
$517$		−	689.710i		−	1.33406i
$518$		0			0
$519$	−39.9048	−	80.8710i	−0.0768879	−	0.155821i
$520$		0			0
$521$		−	771.602i		−	1.48100i	−0.672055	−	0.740501i	$-0.734588\pi$
$521$		−	771.602i		−	1.48100i	0.672055	−	0.740501i	$-0.265412\pi$
$522$		0			0
$523$		−	288.856i		−	0.552306i	−0.961114	−	0.276153i	$-0.910940\pi$
$523$		−	288.856i		−	0.552306i	0.961114	−	0.276153i	$-0.0890597\pi$
$524$		0			0
$525$	88.8158	+	179.994i	0.169173	+	0.342846i
$526$		0			0
$527$			542.835i			1.03005i
$528$		0			0
$529$	−798.951			−1.51030
$530$		0			0
$531$	−204.180	+	266.352i	−0.384521	+	0.501605i
$532$		0			0
$533$	−63.8090			−0.119717
$534$		0			0
$535$	−95.1503			−0.177851
$536$		0			0
$537$	−152.375	−	308.803i	−0.283753	−	0.575053i
$538$		0			0
$539$	469.371			0.870818
$540$		0			0
$541$			232.871i			0.430446i	0.976565	+	0.215223i	$0.0690478\pi$
$541$			232.871i			0.430446i	−0.976565	+	0.215223i	$0.930952\pi$
$542$		0			0
$543$	−488.633	+	241.110i	−0.899877	+	0.444033i
$544$		0			0
$545$			45.2274i			0.0829861i
$546$		0			0
$547$		−	910.003i		−	1.66363i	−0.555056	−	0.831813i	$-0.687303\pi$
$547$		−	910.003i		−	1.66363i	0.555056	−	0.831813i	$-0.312697\pi$
$548$		0			0
$549$	415.699	+	318.667i	0.757192	+	0.580449i
$550$		0			0
$551$			780.096i			1.41578i
$552$		0			0
$553$	−159.191			−0.287869
$554$		0			0
$555$	−109.672	+	54.1164i	−0.197608	+	0.0975071i
$556$		0			0
$557$	297.809			0.534666			0.267333	−	0.963604i	$-0.413858\pi$
$557$	297.809			0.534666			0.267333	+	0.963604i	$0.413858\pi$
$558$		0			0
$559$	61.9877			0.110890
$560$		0			0
$561$	−451.456	+	222.765i	−0.804734	+	0.397086i
$562$		0			0
$563$	184.465			0.327647			0.163824	−	0.986490i	$-0.447617\pi$
$563$	184.465			0.327647			0.163824	+	0.986490i	$0.447617\pi$
$564$		0			0
$565$		−	83.8212i		−	0.148356i
$566$		0			0
$567$	57.2391	+	212.819i	0.100951	+	0.375342i
$568$		0			0
$569$			404.137i			0.710258i	0.934817	+	0.355129i	$0.115563\pi$
$569$			404.137i			0.710258i	−0.934817	+	0.355129i	$0.884437\pi$
$570$		0			0
$571$			762.365i			1.33514i	0.744547	+	0.667570i	$0.232666\pi$
$571$			762.365i			1.33514i	−0.744547	+	0.667570i	$0.767334\pi$
$572$		0			0
$573$	562.883	−	277.748i	0.982344	−	0.484726i
$574$		0			0
$575$			896.100i			1.55843i
$576$		0			0
$577$	−290.766			−0.503927			−0.251964	−	0.967737i	$-0.581076\pi$
$577$	−290.766			−0.503927			−0.251964	+	0.967737i	$0.581076\pi$
$578$		0			0
$579$	214.165	+	434.027i	0.369888	+	0.749616i
$580$		0			0
$581$	88.1361			0.151697
$582$		0			0
$583$	667.648			1.14519
$584$		0			0
$585$	−24.0082	−	18.4042i	−0.0410397	−	0.0314602i
$586$		0			0
$587$	−224.506			−0.382464			−0.191232	−	0.981545i	$-0.561248\pi$
$587$	−224.506			−0.382464			−0.191232	+	0.981545i	$0.561248\pi$
$588$		0			0
$589$		−	550.428i		−	0.934513i
$590$		0			0
$591$	−192.425	−	389.968i	−0.325592	−	0.659844i
$592$		0			0
$593$		−	482.620i		−	0.813862i	−0.913459	−	0.406931i	$-0.866599\pi$
$593$		−	482.620i		−	0.813862i	0.913459	−	0.406931i	$-0.133401\pi$
$594$		0			0
$595$		−	25.8967i		−	0.0435238i
$596$		0			0
$597$	−314.967	−	638.312i	−0.527583	−	1.06920i
$598$		0			0
$599$		−	803.277i		−	1.34103i	−0.741896	−	0.670515i	$-0.766073\pi$
$599$		−	803.277i		−	1.34103i	0.741896	−	0.670515i	$-0.233927\pi$
$600$		0			0
$601$	126.365			0.210258			0.105129	−	0.994459i	$-0.466474\pi$
$601$	126.365			0.210258			0.105129	+	0.994459i	$0.466474\pi$
$602$		0			0
$603$	−164.991	−	126.479i	−0.273617	−	0.209750i
$604$		0			0
$605$	−4.04535			−0.00668653
$606$		0			0
$607$	397.356			0.654623			0.327312	−	0.944916i	$-0.393857\pi$
$607$	397.356			0.654623			0.327312	+	0.944916i	$0.393857\pi$
$608$		0			0
$609$	−186.843	−	378.655i	−0.306802	−	0.621766i
$610$		0			0
$611$	321.012			0.525388
$612$		0			0
$613$		−	1056.55i		−	1.72357i	−0.507275	−	0.861785i	$-0.669347\pi$
$613$		−	1056.55i		−	1.72357i	0.507275	−	0.861785i	$-0.330653\pi$
$614$		0			0
$615$	−20.9203	+	10.3229i	−0.0340167	+	0.0167851i
$616$		0			0
$617$		−	411.284i		−	0.666586i	−0.942823	−	0.333293i	$-0.891840\pi$
$617$		−	411.284i		−	0.666586i	0.942823	−	0.333293i	$-0.108160\pi$
$618$		0			0
$619$		−	691.383i		−	1.11693i	−0.829526	−	0.558467i	$-0.811390\pi$
$619$		−	691.383i		−	1.11693i	0.829526	−	0.558467i	$-0.188610\pi$
$620$		0			0
$621$	−191.271	+	965.138i	−0.308005	+	1.55417i
$622$		0			0
$623$			305.435i			0.490264i
$624$		0			0
$625$	594.447			0.951114
$626$		0			0
$627$	457.771	−	225.881i	0.730097	−	0.360257i
$628$		0			0
$629$	−947.262			−1.50598
$630$		0			0
$631$	528.057			0.836858			0.418429	−	0.908250i	$-0.362581\pi$
$631$	528.057			0.836858			0.418429	+	0.908250i	$0.362581\pi$
$632$		0			0
$633$	−826.238	+	407.697i	−1.30527	+	0.644071i
$634$		0			0
$635$	133.895			0.210858
$636$		0			0
$637$			218.460i			0.342951i
$638$		0			0
$639$	52.1175	+	39.9523i	0.0815611	+	0.0625232i
$640$		0			0
$641$		−	134.165i		−	0.209306i	−0.994509	−	0.104653i	$-0.966627\pi$
$641$		−	134.165i		−	0.209306i	0.994509	−	0.104653i	$-0.0333731\pi$
$642$		0			0
$643$			633.985i			0.985979i	0.870035	+	0.492990i	$0.164096\pi$
$643$			633.985i			0.985979i	−0.870035	+	0.492990i	$0.835904\pi$
$644$		0			0
$645$	20.3232	−	10.0282i	0.0315088	−	0.0155476i
$646$		0			0
$647$			227.758i			0.352021i	0.984388	+	0.176011i	$0.0563193\pi$
$647$			227.758i			0.352021i	−0.984388	+	0.176011i	$0.943681\pi$
$648$		0			0
$649$	−420.766			−0.648329
$650$		0			0
$651$	131.834	+	267.176i	0.202511	+	0.410408i
$652$		0			0
$653$	1190.65			1.82336			0.911678	−	0.410906i	$-0.134788\pi$
$653$	1190.65			1.82336			0.911678	+	0.410906i	$0.134788\pi$
$654$		0			0
$655$	111.900			0.170839
$656$		0			0
$657$	402.178	−	524.638i	0.612142	−	0.798536i
$658$		0			0
$659$	266.873			0.404967			0.202484	−	0.979286i	$-0.435099\pi$
$659$	266.873			0.404967			0.202484	+	0.979286i	$0.435099\pi$
$660$		0			0
$661$			1045.10i			1.58109i	0.612405	+	0.790544i	$0.290202\pi$
$661$			1045.10i			1.58109i	−0.612405	+	0.790544i	$0.709798\pi$
$662$		0			0
$663$	−103.682	−	210.122i	−0.156383	−	0.316925i
$664$		0			0
$665$			26.2589i			0.0394871i
$666$		0			0
$667$		−	1885.13i		−	2.82629i
$668$		0			0
$669$	−581.445	−	1178.36i	−0.869126	−	1.76137i
$670$		0			0
$671$			656.693i			0.978678i
$672$		0			0
$673$	408.300			0.606687			0.303343	−	0.952881i	$-0.401897\pi$
$673$	408.300			0.606687			0.303343	+	0.952881i	$0.401897\pi$
$674$		0			0
$675$	651.274	+	129.069i	0.964850	+	0.191214i
$676$		0			0
$677$	749.557			1.10717			0.553587	−	0.832791i	$-0.313259\pi$
$677$	749.557			1.10717			0.553587	+	0.832791i	$0.313259\pi$
$678$		0			0
$679$	217.753			0.320697
$680$		0			0
$681$	359.426	+	728.413i	0.527792	+	1.06962i
$682$		0			0
$683$	−258.242			−0.378099			−0.189050	−	0.981968i	$-0.560541\pi$
$683$	−258.242			−0.378099			−0.189050	+	0.981968i	$0.560541\pi$
$684$		0			0
$685$			104.287i			0.152244i
$686$		0			0
$687$	−265.474	+	130.995i	−0.386424	+	0.190676i
$688$		0			0
$689$			310.744i			0.451007i
$690$		0			0
$691$		−	929.714i		−	1.34546i	−0.739887	−	0.672731i	$-0.765121\pi$
$691$		−	929.714i		−	1.34546i	0.739887	−	0.672731i	$-0.234879\pi$
$692$		0			0
$693$	−168.099	+	219.284i	−0.242567	+	0.316427i
$694$		0			0
$695$		−	124.308i		−	0.178861i
$696$		0			0
$697$	−180.693			−0.259244
$698$		0			0
$699$	−1163.30	+	574.016i	−1.66424	+	0.821197i
$700$		0			0
$701$	335.731			0.478932			0.239466	−	0.970905i	$-0.423028\pi$
$701$	335.731			0.478932			0.239466	+	0.970905i	$0.423028\pi$
$702$		0			0
$703$	960.513			1.36631
$704$		0			0
$705$	105.246	−	51.9325i	0.149286	−	0.0736632i
$706$		0			0
$707$	−324.785			−0.459385
$708$		0			0
$709$		−	356.121i		−	0.502287i	−0.967950	−	0.251143i	$-0.919193\pi$
$709$		−	356.121i		−	0.502287i	0.967950	−	0.251143i	$-0.0808065\pi$
$710$		0			0
$711$	−320.370	+	417.921i	−0.450591	+	0.587793i
$712$		0			0
$713$			1330.13i			1.86554i
$714$		0			0
$715$		−	37.9266i		−	0.0530442i
$716$		0			0
$717$	−741.985	+	366.123i	−1.03485	+	0.510632i
$718$		0			0
$719$			31.7234i			0.0441216i	0.999757	+	0.0220608i	$0.00702274\pi$
$719$			31.7234i			0.0441216i	−0.999757	+	0.0220608i	$0.992977\pi$
$720$		0			0
$721$	341.516			0.473670
$722$		0			0
$723$	553.262	+	1121.24i	0.765231	+	1.55082i
$724$		0			0
$725$	−1272.09			−1.75460
$726$		0			0
$727$	234.202			0.322148			0.161074	−	0.986942i	$-0.448504\pi$
$727$	234.202			0.322148			0.161074	+	0.986942i	$0.448504\pi$
$728$		0			0
$729$	673.901	+	278.027i	0.924418	+	0.381381i
$730$		0			0
$731$	175.535			0.240130
$732$		0			0
$733$			616.813i			0.841491i	0.907179	+	0.420745i	$0.138231\pi$
$733$			616.813i			0.841491i	−0.907179	+	0.420745i	$0.861769\pi$
$734$		0			0
$735$	35.3419	+	71.6238i	0.0480842	+	0.0974473i
$736$		0			0
$737$		−	260.642i		−	0.353653i
$738$		0			0
$739$		−	605.806i		−	0.819764i	−0.912138	−	0.409882i	$-0.865570\pi$
$739$		−	605.806i		−	0.819764i	0.912138	−	0.409882i	$-0.134430\pi$
$740$		0			0
$741$	105.132	+	213.061i	0.141879	+	0.287532i
$742$		0			0
$743$		−	1033.33i		−	1.39076i	−0.718642	−	0.695380i	$-0.755236\pi$
$743$		−	1033.33i		−	1.39076i	0.718642	−	0.695380i	$-0.244764\pi$
$744$		0			0
$745$	53.4216			0.0717068
$746$		0			0
$747$	177.372	−	231.381i	0.237446	−	0.309747i
$748$		0			0
$749$	−404.494			−0.540046
$750$		0			0
$751$	−969.301			−1.29068			−0.645340	−	0.763896i	$-0.723284\pi$
$751$	−969.301			−1.29068			−0.645340	+	0.763896i	$0.723284\pi$
$752$		0			0
$753$	−220.235	−	446.329i	−0.292477	−	0.592734i
$754$		0			0
$755$	38.6240			0.0511577
$756$		0			0
$757$			533.594i			0.704880i	0.935835	+	0.352440i	$0.114648\pi$
$757$			533.594i			0.704880i	−0.935835	+	0.352440i	$0.885352\pi$
$758$		0			0
$759$	−1106.22	+	545.851i	−1.45747	+	0.719172i
$760$		0			0
$761$		−	867.717i		−	1.14023i	−0.821564	−	0.570116i	$-0.806898\pi$
$761$		−	867.717i		−	1.14023i	0.821564	−	0.570116i	$-0.193102\pi$
$762$		0			0
$763$			192.267i			0.251988i
$764$		0			0
$765$	−67.9858	−	52.1166i	−0.0888704	−	0.0681263i
$766$		0			0
$767$		−	195.837i		−	0.255329i
$768$		0			0
$769$	194.555			0.252997			0.126498	−	0.991967i	$-0.459626\pi$
$769$	194.555			0.252997			0.126498	+	0.991967i	$0.459626\pi$
$770$		0			0
$771$	−160.445	+	79.1695i	−0.208100	+	0.102684i
$772$		0			0
$773$	−420.140			−0.543519			−0.271760	−	0.962365i	$-0.587606\pi$
$773$	−420.140			−0.543519			−0.271760	+	0.962365i	$0.587606\pi$
$774$		0			0
$775$	897.572			1.15816
$776$		0			0
$777$	−466.229	+	230.055i	−0.600037	+	0.296081i
$778$		0			0
$779$	183.220			0.235199
$780$		0			0
$781$			82.3318i			0.105418i
$782$		0			0
$783$	−1370.09	−	271.524i	−1.74980	−	0.346775i
$784$		0			0
$785$		−	64.4892i		−	0.0821519i
$786$		0			0
$787$			849.081i			1.07888i	0.842023	+	0.539441i	$0.181365\pi$
$787$			849.081i			1.07888i	−0.842023	+	0.539441i	$0.818635\pi$
$788$		0			0
$789$	180.952	−	89.2883i	0.229343	−	0.113166i
$790$		0			0
$791$		−	356.333i		−	0.450484i
$792$		0			0
$793$	−305.646			−0.385429
$794$		0			0
$795$	50.2714	+	101.880i	0.0632344	+	0.128151i
$796$		0			0
$797$	779.186			0.977648			0.488824	−	0.872382i	$-0.337426\pi$
$797$	779.186			0.977648			0.488824	+	0.872382i	$0.337426\pi$
$798$		0			0
$799$	909.035			1.13772
$800$		0			0
$801$	801.849	+	614.682i	1.00106	+	0.767393i
$802$		0			0
$803$	828.789			1.03212
$804$		0			0
$805$		−	63.4558i		−	0.0788270i
$806$		0			0
$807$	47.9884	+	97.2532i	0.0594651	+	0.120512i
$808$		0			0
$809$		−	865.140i		−	1.06939i	−0.845044	−	0.534697i	$-0.820426\pi$
$809$		−	865.140i		−	1.06939i	0.845044	−	0.534697i	$-0.179574\pi$
$810$		0			0
$811$			251.360i			0.309938i	0.987919	+	0.154969i	$0.0495278\pi$
$811$			251.360i			0.309938i	−0.987919	+	0.154969i	$0.950472\pi$
$812$		0			0
$813$	435.833	+	883.258i	0.536080	+	1.08642i
$814$		0			0
$815$		−	44.6185i		−	0.0547466i
$816$		0			0
$817$	−177.991			−0.217859
$818$		0			0
$819$	−102.061	−	78.2384i	−0.124617	−	0.0955292i
$820$		0			0
$821$	−103.359			−0.125894			−0.0629470	−	0.998017i	$-0.520050\pi$
$821$	−103.359			−0.125894			−0.0629470	+	0.998017i	$0.520050\pi$
$822$		0			0
$823$	−474.728			−0.576826			−0.288413	−	0.957506i	$-0.593128\pi$
$823$	−474.728			−0.576826			−0.288413	+	0.957506i	$0.593128\pi$
$824$		0			0
$825$	368.340	+	746.477i	0.446473	+	0.904821i
$826$		0			0
$827$	471.776			0.570467			0.285234	−	0.958458i	$-0.407929\pi$
$827$	471.776			0.570467			0.285234	+	0.958458i	$0.407929\pi$
$828$		0			0
$829$			771.883i			0.931102i	0.885021	+	0.465551i	$0.154144\pi$
$829$			771.883i			0.931102i	−0.885021	+	0.465551i	$0.845856\pi$
$830$		0			0
$831$	163.424	−	80.6396i	0.196660	−	0.0970392i
$832$		0			0
$833$			618.629i			0.742652i
$834$		0			0
$835$			77.3283i			0.0926087i
$836$		0			0
$837$	966.723	+	191.585i	1.15499	+	0.228895i
$838$		0			0
$839$		−	510.944i		−	0.608992i	−0.952514	−	0.304496i	$-0.901512\pi$
$839$		−	510.944i		−	0.608992i	0.952514	−	0.304496i	$-0.0984881\pi$
$840$		0			0
$841$	1835.10			2.18204
$842$		0			0
$843$	−986.877	+	486.962i	−1.17067	+	0.577654i
$844$		0			0
$845$	−90.5100			−0.107112
$846$		0			0
$847$	−17.1972			−0.0203037
$848$		0			0
$849$	279.877	−	138.102i	0.329655	−	0.162664i
$850$		0			0
$851$	−2321.12			−2.72752
$852$		0			0
$853$			581.090i			0.681231i	0.940203	+	0.340616i	$0.110635\pi$
$853$			581.090i			0.681231i	−0.940203	+	0.340616i	$0.889365\pi$
$854$		0			0
$855$	68.9369	+	52.8457i	0.0806279	+	0.0618078i
$856$		0			0
$857$		−	7.21760i		−	0.00842193i	−0.999991	−	0.00421097i	$-0.998660\pi$
$857$		−	7.21760i		−	0.00842193i	0.999991	−	0.00421097i	$-0.00134040\pi$
$858$		0			0
$859$			104.856i			0.122068i	0.998136	+	0.0610339i	$0.0194398\pi$
$859$			104.856i			0.122068i	−0.998136	+	0.0610339i	$0.980560\pi$
$860$		0			0
$861$	−88.9344	+	43.8836i	−0.103292	+	0.0509681i
$862$		0			0
$863$			1213.72i			1.40640i	0.710992	+	0.703200i	$0.248246\pi$
$863$			1213.72i			1.40640i	−0.710992	+	0.703200i	$0.751754\pi$
$864$		0			0
$865$	19.2389			0.0222415
$866$		0			0
$867$	90.0432	+	182.482i	0.103856	+	0.210475i
$868$		0			0
$869$	−660.204			−0.759729
$870$		0			0
$871$	121.311			0.139278
$872$		0			0
$873$	438.225	−	571.661i	0.501975	−	0.654824i
$874$		0			0
$875$	−86.3530			−0.0986892
$876$		0			0
$877$		−	706.591i		−	0.805691i	−0.915268	−	0.402846i	$-0.868021\pi$
$877$		−	706.591i		−	0.805691i	0.915268	−	0.402846i	$-0.131979\pi$
$878$		0			0
$879$	−545.043	−	1104.58i	−0.620072	−	1.25664i
$880$		0			0
$881$			1715.95i			1.94773i	0.227134	+	0.973864i	$0.427065\pi$
$881$			1715.95i			1.94773i	−0.227134	+	0.973864i	$0.572935\pi$
$882$		0			0
$883$		−	1472.11i		−	1.66717i	−0.552390	−	0.833586i	$-0.686284\pi$
$883$		−	1472.11i		−	1.66717i	0.552390	−	0.833586i	$-0.313716\pi$
$884$		0			0
$885$	−31.6821	−	64.2068i	−0.0357989	−	0.0725501i
$886$		0			0
$887$			277.658i			0.313030i	0.987676	+	0.156515i	$0.0500259\pi$
$887$			277.658i			0.313030i	−0.987676	+	0.156515i	$0.949974\pi$
$888$		0			0
$889$	569.202			0.640272
$890$		0			0
$891$	237.384	+	882.610i	0.266424	+	0.990584i
$892$		0			0
$893$	−921.751			−1.03220
$894$		0			0
$895$	73.4631			0.0820817
$896$		0			0
$897$	−254.056	−	514.870i	−0.283229	−	0.573991i
$898$		0			0
$899$	−1888.23			−2.10037
$900$		0			0
$901$			879.957i			0.976645i
$902$		0			0
$903$	86.3959	−	42.6310i	0.0956766	−	0.0472104i
$904$		0			0
$905$		−	116.244i		−	0.128446i
$906$		0			0
$907$			365.591i			0.403078i	0.979481	+	0.201539i	$0.0645942\pi$
$907$			365.591i			0.403078i	−0.979481	+	0.201539i	$0.935406\pi$
$908$		0			0
$909$	−653.625	+	852.650i	−0.719060	+	0.938009i
$910$		0			0
$911$		−	1664.90i		−	1.82755i	−0.406223	−	0.913774i	$-0.633154\pi$
$911$		−	1664.90i		−	1.82755i	0.406223	−	0.913774i	$-0.366846\pi$
$912$		0			0
$913$	365.521			0.400351
$914$		0			0
$915$	−100.208	+	49.4465i	−0.109517	+	0.0540399i
$916$		0			0
$917$	475.697			0.518754
$918$		0			0
$919$	−195.745			−0.212998			−0.106499	−	0.994313i	$-0.533964\pi$
$919$	−195.745			−0.212998			−0.106499	+	0.994313i	$0.533964\pi$
$920$		0			0
$921$	726.451	−	358.458i	0.788763	−	0.389205i
$922$		0			0
$923$	−38.3198			−0.0415166
$924$		0			0
$925$			1566.29i			1.69328i
$926$		0			0
$927$	687.295	−	896.573i	0.741419	−	0.967177i
$928$		0			0
$929$			769.046i			0.827822i	0.910317	+	0.413911i	$0.135837\pi$
$929$			769.046i			0.827822i	−0.910317	+	0.413911i	$0.864163\pi$
$930$		0			0
$931$		−	627.283i		−	0.673773i
$932$		0			0
$933$	−676.760	+	333.939i	−0.725359	+	0.357919i
$934$		0			0
$935$		−	107.400i		−	0.114866i
$936$		0			0
$937$	0.0296961			3.16928e−5			1.58464e−5	−	1.00000i	$-0.499995\pi$
$937$	0.0296961			3.16928e−5			1.58464e−5		1.00000i	$0.499995\pi$
$938$		0			0
$939$	111.451	+	225.867i	0.118692	+	0.240540i
$940$		0			0
$941$	−1283.68			−1.36416			−0.682081	−	0.731277i	$-0.738925\pi$
$941$	−1283.68			−1.36416			−0.682081	+	0.731277i	$0.738925\pi$
$942$		0			0
$943$	−442.760			−0.469522
$944$		0			0
$945$	−46.1189	−	9.13983i	−0.0488030	−	0.00967178i
$946$		0			0
$947$	1514.62			1.59938			0.799692	−	0.600410i	$-0.204996\pi$
$947$	1514.62			1.59938			0.799692	+	0.600410i	$0.204996\pi$
$948$		0			0
$949$			385.744i			0.406474i
$950$		0			0
$951$	−359.403	−	728.365i	−0.377921	−	0.765894i
$952$		0			0
$953$			937.870i			0.984124i	0.870560	+	0.492062i	$0.163757\pi$
$953$			937.870i			0.984124i	−0.870560	+	0.492062i	$0.836243\pi$
$954$		0			0
$955$			133.908i			0.140218i
$956$		0			0
$957$	−774.880	−	1570.37i	−0.809697	−	1.64093i
$958$		0			0
$959$			443.335i			0.462288i
$960$		0			0
$961$	371.317			0.386386
$962$		0			0
$963$	−814.038	+	1061.91i	−0.845315	+	1.10271i
$964$		0			0
$965$	−103.253			−0.106998
$966$		0			0
$967$	−461.673			−0.477428			−0.238714	−	0.971090i	$-0.576726\pi$
$967$	−461.673			−0.477428			−0.238714	+	0.971090i	$0.576726\pi$
$968$		0			0
$969$	297.711	+	603.340i	0.307235	+	0.622642i
$970$		0			0
$971$	1360.95			1.40160			0.700798	−	0.713360i	$-0.252827\pi$
$971$	1360.95			1.40160			0.700798	+	0.713360i	$0.252827\pi$
$972$		0			0
$973$		−	528.447i		−	0.543111i
$974$		0			0
$975$	−347.434	+	171.437i	−0.356342	+	0.175833i
$976$		0			0
$977$			1162.17i			1.18953i	0.803901	+	0.594763i	$0.202754\pi$
$977$			1162.17i			1.18953i	−0.803901	+	0.594763i	$0.797246\pi$
$978$		0			0
$979$			1266.71i			1.29388i
$980$		0			0
$981$	504.753	+	386.934i	0.514529	+	0.394428i
$982$		0			0
$983$			969.961i			0.986735i	0.869821	+	0.493368i	$0.164234\pi$
$983$			969.961i			0.986735i	−0.869821	+	0.493368i	$0.835766\pi$
$984$		0			0
$985$	92.7717			0.0941845
$986$		0			0
$987$	447.414	−	220.771i	0.453307	−	0.223679i
$988$		0			0
$989$	430.122			0.434906
$990$		0			0
$991$	1602.81			1.61737			0.808684	−	0.588243i	$-0.200180\pi$
$991$	1602.81			1.61737			0.808684	+	0.588243i	$0.200180\pi$
$992$		0			0
$993$	26.6761	−	13.1630i	0.0268641	−	0.0132558i
$994$		0			0
$995$	151.852			0.152615
$996$		0			0
$997$			742.855i			0.745090i	0.928014	+	0.372545i	$0.121515\pi$
$997$			742.855i			0.745090i	−0.928014	+	0.372545i	$0.878485\pi$
$998$		0			0
$999$	−334.321	+	1686.96i	−0.334656	+	1.68865i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.h.g.641.12		16
3.2	odd	2	inner	768.3.h.g.641.6		16
4.3	odd	2		768.3.h.h.641.5		16
8.3	odd	2		768.3.h.h.641.12		16
8.5	even	2	inner	768.3.h.g.641.5		16
12.11	even	2		768.3.h.h.641.11		16
16.3	odd	4		384.3.e.a.257.7	✓	8
16.5	even	4		384.3.e.b.257.7	yes	8
16.11	odd	4		384.3.e.c.257.2	yes	8
16.13	even	4		384.3.e.d.257.2	yes	8
24.5	odd	2	inner	768.3.h.g.641.11		16
24.11	even	2		768.3.h.h.641.6		16
48.5	odd	4		384.3.e.b.257.8	yes	8
48.11	even	4		384.3.e.c.257.1	yes	8
48.29	odd	4		384.3.e.d.257.1	yes	8
48.35	even	4		384.3.e.a.257.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.e.a.257.7	✓	8	16.3	odd	4
384.3.e.a.257.8	yes	8	48.35	even	4
384.3.e.b.257.7	yes	8	16.5	even	4
384.3.e.b.257.8	yes	8	48.5	odd	4
384.3.e.c.257.1	yes	8	48.11	even	4
384.3.e.c.257.2	yes	8	16.11	odd	4
384.3.e.d.257.1	yes	8	48.29	odd	4
384.3.e.d.257.2	yes	8	16.13	even	4
768.3.h.g.641.5		16	8.5	even	2	inner
768.3.h.g.641.6		16	3.2	odd	2	inner
768.3.h.g.641.11		16	24.5	odd	2	inner
768.3.h.g.641.12		16	1.1	even	1	trivial
768.3.h.h.641.5		16	4.3	odd	2
768.3.h.h.641.6		16	24.11	even	2
768.3.h.h.641.11		16	12.11	even	2
768.3.h.h.641.12		16	8.3	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)

\(f(q)\)	\(=\)	\(q+(1.32750 + 2.69031i) q^{3} -0.640013 q^{5} -2.72077 q^{7} +(-5.47550 + 7.14275i) q^{9} +O(q^{10})\)	\(q+(1.32750 + 2.69031i) q^{3} -0.640013 q^{5} -2.72077 q^{7} +(-5.47550 + 7.14275i) q^{9} -11.2836 q^{11} -5.25176i q^{13} +(-0.849616 - 1.72183i) q^{15} -14.8718i q^{17} +15.0798i q^{19} +(-3.61181 - 7.31969i) q^{21} -36.4411i q^{23} -24.5904 q^{25} +(-26.4849 - 5.24877i) q^{27} +51.7310 q^{29} -36.5009 q^{31} +(-14.9790 - 30.3565i) q^{33} +1.74133 q^{35} -63.6951i q^{37} +(14.1289 - 6.97170i) q^{39} -12.1500i q^{41} +11.8032i q^{43} +(3.50439 - 4.57146i) q^{45} +61.1247i q^{47} -41.5974 q^{49} +(40.0097 - 19.7423i) q^{51} -59.1695 q^{53} +7.22168 q^{55} +(-40.5694 + 20.0185i) q^{57} +37.2898 q^{59} -58.1987i q^{61} +(14.8975 - 19.4338i) q^{63} +3.36120i q^{65} +23.0991i q^{67} +(98.0376 - 48.3754i) q^{69} -7.29656i q^{71} -73.4504 q^{73} +(-32.6437 - 66.1557i) q^{75} +30.7002 q^{77} +58.5098 q^{79} +(-21.0379 - 78.2203i) q^{81} -32.3939 q^{83} +9.51815i q^{85} +(68.6728 + 139.172i) q^{87} -112.260i q^{89} +14.2888i q^{91} +(-48.4549 - 98.1987i) q^{93} -9.65130i q^{95} -80.0338 q^{97} +(61.7836 - 80.5963i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{7}+O(q^{10}) \) 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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