LMFDB - Embedded newform 912.3.o.b.721.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,3,Mod(721,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("912.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	912.o (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:	$24.8502001097$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.4
Root			$-1.63746 + 1.52274i$ of defining polynomial
Character	$\chi$	$=$	912.721
Dual form			912.3.o.b.721.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +1.27492 q^{5} -4.72508 q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +1.27492 q^{5} -4.72508 q^{7} -3.00000 q^{9} +7.27492 q^{11} -4.30136i q^{13} +2.20822i q^{15} -20.3746 q^{17} +(7.54983 + 17.4356i) q^{19} -8.18408i q^{21} -5.45017 q^{23} -23.3746 q^{25} -5.19615i q^{27} +8.60271i q^{29} -20.0624i q^{31} +12.6005i q^{33} -6.02409 q^{35} +40.6169i q^{37} +7.45017 q^{39} -31.2920i q^{41} -65.1238 q^{43} -3.82475 q^{45} -55.4743 q^{47} -26.6736 q^{49} -35.2898i q^{51} -78.1149i q^{53} +9.27492 q^{55} +(-30.1993 + 13.0767i) q^{57} +69.5122i q^{59} -6.17525 q^{61} +14.1752 q^{63} -5.48387i q^{65} -123.724i q^{67} -9.43996i q^{69} -3.11884i q^{71} +33.8248 q^{73} -40.4860i q^{75} -34.3746 q^{77} +87.6700i q^{79} +9.00000 q^{81} -121.698 q^{83} -25.9759 q^{85} -14.9003 q^{87} +69.9726i q^{89} +20.3243i q^{91} +34.7492 q^{93} +(9.62541 + 22.2289i) q^{95} -111.081i q^{97} -21.8248 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{5} - 34 q^{7} - 12 q^{9}+O(q^{10}) $ 4 * q - 10 * q^5 - 34 * q^7 - 12 * q^9	$ 4 q - 10 q^{5} - 34 q^{7} - 12 q^{9} + 14 q^{11} - 6 q^{17} - 52 q^{23} - 18 q^{25} + 142 q^{35} + 60 q^{39} - 34 q^{43} + 30 q^{45} - 86 q^{47} + 150 q^{49} + 22 q^{55} - 70 q^{61} + 102 q^{63} + 90 q^{73} - 62 q^{77} + 36 q^{81} - 64 q^{83} - 270 q^{85} - 120 q^{87} - 12 q^{93} + 114 q^{95} - 42 q^{99}+O(q^{100}) $ 4 * q - 10 * q^5 - 34 * q^7 - 12 * q^9 + 14 * q^11 - 6 * q^17 - 52 * q^23 - 18 * q^25 + 142 * q^35 + 60 * q^39 - 34 * q^43 + 30 * q^45 - 86 * q^47 + 150 * q^49 + 22 * q^55 - 70 * q^61 + 102 * q^63 + 90 * q^73 - 62 * q^77 + 36 * q^81 - 64 * q^83 - 270 * q^85 - 120 * q^87 - 12 * q^93 + 114 * q^95 - 42 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.73205i			0.577350i
$3$			1.73205i			0.577350i
$4$		0			0
$5$	1.27492			0.254983			0.127492	−	0.991840i	$-0.459307\pi$
$5$	1.27492			0.254983			0.127492	+	0.991840i	$0.459307\pi$
$6$		0			0
$7$	−4.72508			−0.675012			−0.337506	−	0.941323i	$-0.609583\pi$
$7$	−4.72508			−0.675012			−0.337506	+	0.941323i	$0.609583\pi$
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	7.27492			0.661356			0.330678	−	0.943744i	$-0.392723\pi$
$11$	7.27492			0.661356			0.330678	+	0.943744i	$0.392723\pi$
$12$		0			0
$13$		−	4.30136i		−	0.330873i	−0.986220	−	0.165437i	$-0.947097\pi$
$13$		−	4.30136i		−	0.330873i	0.986220	−	0.165437i	$-0.0529034\pi$
$14$		0			0
$15$			2.20822i			0.147215i
$16$		0			0
$17$	−20.3746			−1.19851			−0.599253	−	0.800560i	$-0.704535\pi$
$17$	−20.3746			−1.19851			−0.599253	+	0.800560i	$0.704535\pi$
$18$		0			0
$19$	7.54983	+	17.4356i	0.397360	+	0.917663i
$19$	7.54983	+	17.4356i	0.397360	+	0.917663i
$20$		0			0
$21$		−	8.18408i		−	0.389718i
$22$		0			0
$23$	−5.45017			−0.236964			−0.118482	−	0.992956i	$-0.537803\pi$
$23$	−5.45017			−0.236964			−0.118482	+	0.992956i	$0.537803\pi$
$24$		0			0
$25$	−23.3746			−0.934983
$26$		0			0
$27$		−	5.19615i		−	0.192450i
$28$		0			0
$29$			8.60271i			0.296645i	0.988939	+	0.148323i	$0.0473874\pi$
$29$			8.60271i			0.296645i	−0.988939	+	0.148323i	$0.952613\pi$
$30$		0			0
$31$		−	20.0624i		−	0.647176i	−0.946198	−	0.323588i	$-0.895111\pi$
$31$		−	20.0624i		−	0.647176i	0.946198	−	0.323588i	$-0.104889\pi$
$32$		0			0
$33$			12.6005i			0.381834i
$34$		0			0
$35$	−6.02409			−0.172117
$36$		0			0
$37$			40.6169i			1.09775i	0.835903	+	0.548877i	$0.184944\pi$
$37$			40.6169i			1.09775i	−0.835903	+	0.548877i	$0.815056\pi$
$38$		0			0
$39$	7.45017			0.191030
$40$		0			0
$41$		−	31.2920i		−	0.763220i	−0.924324	−	0.381610i	$-0.875370\pi$
$41$		−	31.2920i		−	0.763220i	0.924324	−	0.381610i	$-0.124630\pi$
$42$		0			0
$43$	−65.1238			−1.51451			−0.757253	−	0.653122i	$-0.773459\pi$
$43$	−65.1238			−1.51451			−0.757253	+	0.653122i	$0.773459\pi$
$44$		0			0
$45$	−3.82475			−0.0849945
$46$		0			0
$47$	−55.4743			−1.18030			−0.590152	−	0.807292i	$-0.700932\pi$
$47$	−55.4743			−1.18030			−0.590152	+	0.807292i	$0.700932\pi$
$48$		0			0
$49$	−26.6736			−0.544359
$50$		0			0
$51$		−	35.2898i		−	0.691957i
$52$		0			0
$53$		−	78.1149i		−	1.47387i	−0.675966	−	0.736933i	$-0.736273\pi$
$53$		−	78.1149i		−	1.47387i	0.675966	−	0.736933i	$-0.263727\pi$
$54$		0			0
$55$	9.27492			0.168635
$56$		0			0
$57$	−30.1993	+	13.0767i	−0.529813	+	0.229416i
$58$		0			0
$59$			69.5122i			1.17817i	0.808070	+	0.589087i	$0.200512\pi$
$59$			69.5122i			1.17817i	−0.808070	+	0.589087i	$0.799488\pi$
$60$		0			0
$61$	−6.17525			−0.101234			−0.0506168	−	0.998718i	$-0.516119\pi$
$61$	−6.17525			−0.101234			−0.0506168	+	0.998718i	$0.516119\pi$
$62$		0			0
$63$	14.1752			0.225004
$64$		0			0
$65$		−	5.48387i		−	0.0843673i
$66$		0			0
$67$		−	123.724i		−	1.84662i	−0.384053	−	0.923311i	$-0.625472\pi$
$67$		−	123.724i		−	1.84662i	0.384053	−	0.923311i	$-0.374528\pi$
$68$		0			0
$69$		−	9.43996i		−	0.136811i
$70$		0			0
$71$		−	3.11884i		−	0.0439273i	−0.999759	−	0.0219637i	$-0.993008\pi$
$71$		−	3.11884i		−	0.0439273i	0.999759	−	0.0219637i	$-0.00699181\pi$
$72$		0			0
$73$	33.8248			0.463353			0.231676	−	0.972793i	$-0.425579\pi$
$73$	33.8248			0.463353			0.231676	+	0.972793i	$0.425579\pi$
$74$		0			0
$75$		−	40.4860i		−	0.539813i
$76$		0			0
$77$	−34.3746			−0.446423
$78$		0			0
$79$			87.6700i			1.10975i	0.831935	+	0.554873i	$0.187233\pi$
$79$			87.6700i			1.10975i	−0.831935	+	0.554873i	$0.812767\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	−121.698			−1.46624			−0.733119	−	0.680101i	$-0.761936\pi$
$83$	−121.698			−1.46624			−0.733119	+	0.680101i	$0.761936\pi$
$84$		0			0
$85$	−25.9759			−0.305599
$86$		0			0
$87$	−14.9003			−0.171268
$88$		0			0
$89$			69.9726i			0.786209i	0.919494	+	0.393104i	$0.128599\pi$
$89$			69.9726i			0.786209i	−0.919494	+	0.393104i	$0.871401\pi$
$90$		0			0
$91$			20.3243i			0.223344i
$92$		0			0
$93$	34.7492			0.373647
$94$		0			0
$95$	9.62541	+	22.2289i	0.101320	+	0.233989i
$96$		0			0
$97$		−	111.081i		−	1.14517i	−0.819846	−	0.572585i	$-0.805941\pi$
$97$		−	111.081i		−	1.14517i	0.819846	−	0.572585i	$-0.194059\pi$
$98$		0			0
$99$	−21.8248			−0.220452
$100$		0			0
$101$	75.2508			0.745058			0.372529	−	0.928021i	$-0.378491\pi$
$101$	75.2508			0.745058			0.372529	+	0.928021i	$0.378491\pi$
$102$		0			0
$103$		−	188.474i		−	1.82985i	−0.403628	−	0.914923i	$-0.632251\pi$
$103$		−	188.474i		−	1.82985i	0.403628	−	0.914923i	$-0.367749\pi$
$104$		0			0
$105$		−	10.4340i		−	0.0993717i
$106$		0			0
$107$		−	8.14236i		−	0.0760968i	−0.999276	−	0.0380484i	$-0.987886\pi$
$107$		−	8.14236i		−	0.0760968i	0.999276	−	0.0380484i	$-0.0121141\pi$
$108$		0			0
$109$		−	119.192i		−	1.09351i	−0.837294	−	0.546753i	$-0.815864\pi$
$109$		−	119.192i		−	1.09351i	0.837294	−	0.546753i	$-0.184136\pi$
$110$		0			0
$111$	−70.3505			−0.633788
$112$		0			0
$113$			196.355i			1.73765i	0.495117	+	0.868826i	$0.335125\pi$
$113$			196.355i			1.73765i	−0.495117	+	0.868826i	$0.664875\pi$
$114$		0			0
$115$	−6.94851			−0.0604218
$116$		0			0
$117$			12.9041i			0.110291i
$118$		0			0
$119$	96.2716			0.809005
$120$		0			0
$121$	−68.0756			−0.562608
$122$		0			0
$123$	54.1993			0.440645
$124$		0			0
$125$	−61.6736			−0.493389
$126$		0			0
$127$		−	16.0229i		−	0.126165i	−0.998008	−	0.0630823i	$-0.979907\pi$
$127$		−	16.0229i		−	0.126165i	0.998008	−	0.0630823i	$-0.0200931\pi$
$128$		0			0
$129$		−	112.798i		−	0.874400i
$130$		0			0
$131$	−165.921			−1.26657			−0.633287	−	0.773917i	$-0.718295\pi$
$131$	−165.921			−1.26657			−0.633287	+	0.773917i	$0.718295\pi$
$132$		0			0
$133$	−35.6736	−	82.3846i	−0.268223	−	0.619433i
$134$		0			0
$135$		−	6.62466i		−	0.0490716i
$136$		0			0
$137$	−199.674			−1.45747			−0.728736	−	0.684795i	$-0.759892\pi$
$137$	−199.674			−1.45747			−0.728736	+	0.684795i	$0.759892\pi$
$138$		0			0
$139$	54.5739			0.392618			0.196309	−	0.980542i	$-0.437104\pi$
$139$	54.5739			0.392618			0.196309	+	0.980542i	$0.437104\pi$
$140$		0			0
$141$		−	96.0842i		−	0.681448i
$142$		0			0
$143$		−	31.2920i		−	0.218825i
$144$		0			0
$145$			10.9677i			0.0756396i
$146$		0			0
$147$		−	46.2000i		−	0.314286i
$148$		0			0
$149$	−245.323			−1.64646			−0.823232	−	0.567705i	$-0.807831\pi$
$149$	−245.323			−1.64646			−0.823232	+	0.567705i	$0.807831\pi$
$150$		0			0
$151$			79.2974i			0.525149i	0.964912	+	0.262574i	$0.0845715\pi$
$151$			79.2974i			0.525149i	−0.964912	+	0.262574i	$0.915429\pi$
$152$		0			0
$153$	61.1238			0.399502
$154$		0			0
$155$		−	25.5780i		−	0.165019i
$156$		0			0
$157$	−53.6977			−0.342023			−0.171012	−	0.985269i	$-0.554704\pi$
$157$	−53.6977			−0.342023			−0.171012	+	0.985269i	$0.554704\pi$
$158$		0			0
$159$	135.299			0.850937
$160$		0			0
$161$	25.7525			0.159953
$162$		0			0
$163$	−78.3987			−0.480973			−0.240487	−	0.970652i	$-0.577307\pi$
$163$	−78.3987			−0.480973			−0.240487	+	0.970652i	$0.577307\pi$
$164$		0			0
$165$			16.0646i			0.0973614i
$166$		0			0
$167$			293.350i			1.75658i	0.478124	+	0.878292i	$0.341317\pi$
$167$			293.350i			1.75658i	−0.478124	+	0.878292i	$0.658683\pi$
$168$		0			0
$169$	150.498			0.890523
$170$		0			0
$171$	−22.6495	−	52.3068i	−0.132453	−	0.305888i
$172$		0			0
$173$			10.5074i			0.0607364i	0.999539	+	0.0303682i	$0.00966798\pi$
$173$			10.5074i			0.0607364i	−0.999539	+	0.0303682i	$0.990332\pi$
$174$		0			0
$175$	110.447			0.631125
$176$		0			0
$177$	−120.399			−0.680219
$178$		0			0
$179$			12.4121i			0.0693412i	0.999399	+	0.0346706i	$0.0110382\pi$
$179$			12.4121i			0.0693412i	−0.999399	+	0.0346706i	$0.988962\pi$
$180$		0			0
$181$			106.780i			0.589945i	0.955506	+	0.294973i	$0.0953105\pi$
$181$			106.780i			0.589945i	−0.955506	+	0.294973i	$0.904689\pi$
$182$		0			0
$183$		−	10.6958i		−	0.0584472i
$184$		0			0
$185$			51.7832i			0.279909i
$186$		0			0
$187$	−148.223			−0.792639
$188$		0			0
$189$			24.5523i			0.129906i
$190$		0			0
$191$	19.0274			0.0996199			0.0498099	−	0.998759i	$-0.484138\pi$
$191$	19.0274			0.0996199			0.0498099	+	0.998759i	$0.484138\pi$
$192$		0			0
$193$			110.391i			0.571974i	0.958234	+	0.285987i	$0.0923213\pi$
$193$			110.391i			0.571974i	−0.958234	+	0.285987i	$0.907679\pi$
$194$		0			0
$195$	9.49834			0.0487095
$196$		0			0
$197$	328.749			1.66878			0.834389	−	0.551176i	$-0.185821\pi$
$197$	328.749			1.66878			0.834389	+	0.551176i	$0.185821\pi$
$198$		0			0
$199$	266.973			1.34157			0.670785	−	0.741651i	$-0.265957\pi$
$199$	266.973			1.34157			0.670785	+	0.741651i	$0.265957\pi$
$200$		0			0
$201$	214.296			1.06615
$202$		0			0
$203$		−	40.6485i		−	0.200239i
$204$		0			0
$205$		−	39.8947i		−	0.194608i
$206$		0			0
$207$	16.3505			0.0789879
$208$		0			0
$209$	54.9244	+	126.843i	0.262796	+	0.606902i
$210$		0			0
$211$		−	70.4329i		−	0.333805i	−0.985973	−	0.166903i	$-0.946623\pi$
$211$		−	70.4329i		−	0.333805i	0.985973	−	0.166903i	$-0.0533766\pi$
$212$		0			0
$213$	5.40199			0.0253614
$214$		0			0
$215$	−83.0274			−0.386174
$216$		0			0
$217$			94.7967i			0.436851i
$218$		0			0
$219$			58.5862i			0.267517i
$220$		0			0
$221$			87.6383i			0.396554i
$222$		0			0
$223$			237.432i			1.06472i	0.846519	+	0.532359i	$0.178694\pi$
$223$			237.432i			1.06472i	−0.846519	+	0.532359i	$0.821306\pi$
$224$		0			0
$225$	70.1238			0.311661
$226$		0			0
$227$			269.740i			1.18828i	0.804362	+	0.594140i	$0.202508\pi$
$227$			269.740i			1.18828i	−0.804362	+	0.594140i	$0.797492\pi$
$228$		0			0
$229$	259.419			1.13284			0.566418	−	0.824118i	$-0.308329\pi$
$229$	259.419			1.13284			0.566418	+	0.824118i	$0.308329\pi$
$230$		0			0
$231$		−	59.5385i		−	0.257743i
$232$		0			0
$233$	−52.6769			−0.226081			−0.113041	−	0.993590i	$-0.536059\pi$
$233$	−52.6769			−0.226081			−0.113041	+	0.993590i	$0.536059\pi$
$234$		0			0
$235$	−70.7251			−0.300958
$236$		0			0
$237$	−151.849			−0.640712
$238$		0			0
$239$	346.120			1.44820			0.724101	−	0.689694i	$-0.242255\pi$
$239$	346.120			1.44820			0.724101	+	0.689694i	$0.242255\pi$
$240$		0			0
$241$		−	300.278i		−	1.24597i	−0.782235	−	0.622983i	$-0.785921\pi$
$241$		−	300.278i		−	1.24597i	0.782235	−	0.622983i	$-0.214079\pi$
$242$		0			0
$243$			15.5885i			0.0641500i
$244$		0			0
$245$	−34.0066			−0.138803
$246$		0			0
$247$	74.9967	−	32.4745i	0.303630	−	0.131476i
$248$		0			0
$249$		−	210.787i		−	0.846532i
$250$		0			0
$251$	−190.223			−0.757862			−0.378931	−	0.925425i	$-0.623708\pi$
$251$	−190.223			−0.757862			−0.378931	+	0.925425i	$0.623708\pi$
$252$		0			0
$253$	−39.6495			−0.156717
$254$		0			0
$255$		−	44.9916i		−	0.176438i
$256$		0			0
$257$			25.3478i			0.0986295i	0.998783	+	0.0493147i	$0.0157037\pi$
$257$			25.3478i			0.0986295i	−0.998783	+	0.0493147i	$0.984296\pi$
$258$		0			0
$259$		−	191.918i		−	0.740997i
$260$		0			0
$261$		−	25.8081i		−	0.0988817i
$262$		0			0
$263$	450.120			1.71148			0.855742	−	0.517402i	$-0.173101\pi$
$263$	450.120			1.71148			0.855742	+	0.517402i	$0.173101\pi$
$264$		0			0
$265$		−	99.5901i		−	0.375812i
$266$		0			0
$267$	−121.196			−0.453918
$268$		0			0
$269$			485.141i			1.80350i	0.432259	+	0.901749i	$0.357716\pi$
$269$			485.141i			1.80350i	−0.432259	+	0.901749i	$0.642284\pi$
$270$		0			0
$271$	−156.997			−0.579324			−0.289662	−	0.957129i	$-0.593543\pi$
$271$	−156.997			−0.579324			−0.289662	+	0.957129i	$0.593543\pi$
$272$		0			0
$273$	−35.2026			−0.128947
$274$		0			0
$275$	−170.048			−0.618357
$276$		0			0
$277$	156.629			0.565447			0.282723	−	0.959202i	$-0.408762\pi$
$277$	156.629			0.565447			0.282723	+	0.959202i	$0.408762\pi$
$278$		0			0
$279$			60.1873i			0.215725i
$280$		0			0
$281$			457.658i			1.62868i	0.580390	+	0.814339i	$0.302900\pi$
$281$			457.658i			1.62868i	−0.580390	+	0.814339i	$0.697100\pi$
$282$		0			0
$283$	46.5739			0.164572			0.0822861	−	0.996609i	$-0.473778\pi$
$283$	46.5739			0.164572			0.0822861	+	0.996609i	$0.473778\pi$
$284$		0			0
$285$	−38.5017	+	16.6717i	−0.135094	+	0.0584972i
$286$		0			0
$287$			147.857i			0.515182i
$288$		0			0
$289$	126.124			0.436414
$290$		0			0
$291$	192.399			0.661164
$292$		0			0
$293$			222.163i			0.758235i	0.925349	+	0.379117i	$0.123772\pi$
$293$			222.163i			0.758235i	−0.925349	+	0.379117i	$0.876228\pi$
$294$		0			0
$295$			88.6223i			0.300415i
$296$		0			0
$297$		−	37.8016i		−	0.127278i
$298$		0			0
$299$			23.4431i			0.0784050i
$300$		0			0
$301$	307.715			1.02231
$302$		0			0
$303$			130.338i			0.430159i
$304$		0			0
$305$	−7.87293			−0.0258129
$306$		0			0
$307$		−	492.069i		−	1.60283i	−0.598108	−	0.801416i	$-0.704080\pi$
$307$		−	492.069i		−	1.60283i	0.598108	−	0.801416i	$-0.295920\pi$
$308$		0			0
$309$	326.447			1.05646
$310$		0			0
$311$	−162.773			−0.523387			−0.261693	−	0.965151i	$-0.584281\pi$
$311$	−162.773			−0.523387			−0.261693	+	0.965151i	$0.584281\pi$
$312$		0			0
$313$	414.894			1.32554			0.662770	−	0.748823i	$-0.269381\pi$
$313$	414.894			1.32554			0.662770	+	0.748823i	$0.269381\pi$
$314$		0			0
$315$	18.0723			0.0573723
$316$		0			0
$317$		−	365.750i		−	1.15379i	−0.816819	−	0.576893i	$-0.804265\pi$
$317$		−	365.750i		−	1.15379i	0.816819	−	0.576893i	$-0.195735\pi$
$318$		0			0
$319$			62.5840i			0.196188i
$320$		0			0
$321$	14.1030			0.0439345
$322$		0			0
$323$	−153.825	−	355.243i	−0.476238	−	1.09982i
$324$		0			0
$325$			100.542i			0.309361i
$326$		0			0
$327$	206.447			0.631336
$328$		0			0
$329$	262.120			0.796719
$330$		0			0
$331$			220.258i			0.665433i	0.943027	+	0.332716i	$0.107965\pi$
$331$			220.258i			0.665433i	−0.943027	+	0.332716i	$0.892035\pi$
$332$		0			0
$333$		−	121.851i		−	0.365918i
$334$		0			0
$335$		−	157.737i		−	0.470858i
$336$		0			0
$337$		−	240.980i		−	0.715073i	−0.933899	−	0.357536i	$-0.883617\pi$
$337$		−	240.980i		−	0.715073i	0.933899	−	0.357536i	$-0.116383\pi$
$338$		0			0
$339$	−340.096			−1.00323
$340$		0			0
$341$		−	145.953i		−	0.428014i
$342$		0			0
$343$	357.564			1.04246
$344$		0			0
$345$		−	12.0352i		−	0.0348846i
$346$		0			0
$347$	234.876			0.676877			0.338438	−	0.940989i	$-0.390101\pi$
$347$	234.876			0.676877			0.338438	+	0.940989i	$0.390101\pi$
$348$		0			0
$349$	−440.677			−1.26268			−0.631342	−	0.775504i	$-0.717496\pi$
$349$	−440.677			−1.26268			−0.631342	+	0.775504i	$0.717496\pi$
$350$		0			0
$351$	−22.3505			−0.0636766
$352$		0			0
$353$	−543.292			−1.53907			−0.769536	−	0.638603i	$-0.779512\pi$
$353$	−543.292			−1.53907			−0.769536	+	0.638603i	$0.779512\pi$
$354$		0			0
$355$		−	3.97626i		−	0.0112007i
$356$		0			0
$357$			166.747i			0.467079i
$358$		0			0
$359$	38.5191			0.107296			0.0536478	−	0.998560i	$-0.482915\pi$
$359$	38.5191			0.107296			0.0536478	+	0.998560i	$0.482915\pi$
$360$		0			0
$361$	−247.000	+	263.272i	−0.684211	+	0.729285i
$362$		0			0
$363$		−	117.910i		−	0.324822i
$364$		0			0
$365$	43.1238			0.118147
$366$		0			0
$367$	568.997			1.55040			0.775200	−	0.631716i	$-0.217649\pi$
$367$	568.997			1.55040			0.775200	+	0.631716i	$0.217649\pi$
$368$		0			0
$369$			93.8760i			0.254407i
$370$		0			0
$371$			369.099i			0.994877i
$372$		0			0
$373$		−	61.8618i		−	0.165849i	−0.996556	−	0.0829247i	$-0.973574\pi$
$373$		−	61.8618i		−	0.165849i	0.996556	−	0.0829247i	$-0.0264261\pi$
$374$		0			0
$375$		−	106.822i		−	0.284858i
$376$		0			0
$377$	37.0033			0.0981520
$378$		0			0
$379$			342.601i			0.903960i	0.892028	+	0.451980i	$0.149282\pi$
$379$			342.601i			0.903960i	−0.892028	+	0.451980i	$0.850718\pi$
$380$		0			0
$381$	27.7525			0.0728412
$382$		0			0
$383$			77.1309i			0.201386i	0.994918	+	0.100693i	$0.0321060\pi$
$383$			77.1309i			0.201386i	−0.994918	+	0.100693i	$0.967894\pi$
$384$		0			0
$385$	−43.8248			−0.113831
$386$		0			0
$387$	195.371			0.504835
$388$		0			0
$389$	433.770			1.11509			0.557545	−	0.830147i	$-0.311743\pi$
$389$	433.770			1.11509			0.557545	+	0.830147i	$0.311743\pi$
$390$		0			0
$391$	111.045			0.284002
$392$		0			0
$393$		−	287.384i		−	0.731256i
$394$		0			0
$395$			111.772i			0.282967i
$396$		0			0
$397$	−283.069			−0.713020			−0.356510	−	0.934291i	$-0.616033\pi$
$397$	−283.069			−0.713020			−0.356510	+	0.934291i	$0.616033\pi$
$398$		0			0
$399$	142.694	−	61.7885i	0.357630	−	0.154858i
$400$		0			0
$401$		−	323.491i		−	0.806710i	−0.915044	−	0.403355i	$-0.867844\pi$
$401$		−	323.491i		−	0.806710i	0.915044	−	0.403355i	$-0.132156\pi$
$402$		0			0
$403$	−86.2957			−0.214133
$404$		0			0
$405$	11.4743			0.0283315
$406$		0			0
$407$			295.484i			0.726006i
$408$		0			0
$409$		−	546.741i		−	1.33678i	−0.743813	−	0.668388i	$-0.766985\pi$
$409$		−	546.741i		−	1.33678i	0.743813	−	0.668388i	$-0.233015\pi$
$410$		0			0
$411$		−	345.845i		−	0.841472i
$412$		0			0
$413$		−	328.451i		−	0.795281i
$414$		0			0
$415$	−155.154			−0.373866
$416$		0			0
$417$			94.5248i			0.226678i
$418$		0			0
$419$	352.997			0.842474			0.421237	−	0.906951i	$-0.361596\pi$
$419$	352.997			0.842474			0.421237	+	0.906951i	$0.361596\pi$
$420$		0			0
$421$			578.525i			1.37417i	0.726578	+	0.687084i	$0.241110\pi$
$421$			578.525i			1.37417i	−0.726578	+	0.687084i	$0.758890\pi$
$422$		0			0
$423$	166.423			0.393434
$424$		0			0
$425$	476.248			1.12058
$426$		0			0
$427$	29.1786			0.0683339
$428$		0			0
$429$	54.1993			0.126339
$430$		0			0
$431$			628.896i			1.45915i	0.683898	+	0.729577i	$0.260283\pi$
$431$			628.896i			1.45915i	−0.683898	+	0.729577i	$0.739717\pi$
$432$		0			0
$433$		−	122.803i		−	0.283610i	−0.989895	−	0.141805i	$-0.954709\pi$
$433$		−	122.803i		−	0.283610i	0.989895	−	0.141805i	$-0.0452905\pi$
$434$		0			0
$435$	−18.9967			−0.0436705
$436$		0			0
$437$	−41.1478	−	95.0269i	−0.0941598	−	0.217453i
$438$		0			0
$439$			63.3694i			0.144350i	0.997392	+	0.0721748i	$0.0229939\pi$
$439$			63.3694i			0.144350i	−0.997392	+	0.0721748i	$0.977006\pi$
$440$		0			0
$441$	80.0208			0.181453
$442$		0			0
$443$	−610.320			−1.37770			−0.688849	−	0.724905i	$-0.741883\pi$
$443$	−610.320			−1.37770			−0.688849	+	0.724905i	$0.741883\pi$
$444$		0			0
$445$			89.2092i			0.200470i
$446$		0			0
$447$		−	424.912i		−	0.950586i
$448$		0			0
$449$			410.772i			0.914860i	0.889246	+	0.457430i	$0.151230\pi$
$449$			410.772i			0.914860i	−0.889246	+	0.457430i	$0.848770\pi$
$450$		0			0
$451$		−	227.647i		−	0.504760i
$452$		0			0
$453$	−137.347			−0.303195
$454$		0			0
$455$			25.9117i			0.0569489i
$456$		0			0
$457$	251.722			0.550814			0.275407	−	0.961328i	$-0.411187\pi$
$457$	251.722			0.550814			0.275407	+	0.961328i	$0.411187\pi$
$458$		0			0
$459$			105.869i			0.230652i
$460$		0			0
$461$	−224.622			−0.487250			−0.243625	−	0.969870i	$-0.578337\pi$
$461$	−224.622			−0.487250			−0.243625	+	0.969870i	$0.578337\pi$
$462$		0			0
$463$	196.375			0.424135			0.212068	−	0.977255i	$-0.431980\pi$
$463$	196.375			0.424135			0.212068	+	0.977255i	$0.431980\pi$
$464$		0			0
$465$	44.3023			0.0952738
$466$		0			0
$467$	106.381			0.227797			0.113899	−	0.993492i	$-0.463666\pi$
$467$	106.381			0.227797			0.113899	+	0.993492i	$0.463666\pi$
$468$		0			0
$469$			584.605i			1.24649i
$470$		0			0
$471$		−	93.0071i		−	0.197467i
$472$		0			0
$473$	−473.770			−1.00163
$474$		0			0
$475$	−176.474	−	407.550i	−0.371525	−	0.858000i
$476$		0			0
$477$			234.345i			0.491289i
$478$		0			0
$479$	−463.154			−0.966920			−0.483460	−	0.875367i	$-0.660620\pi$
$479$	−463.154			−0.966920			−0.483460	+	0.875367i	$0.660620\pi$
$480$		0			0
$481$	174.708			0.363217
$482$		0			0
$483$			44.6046i			0.0923491i
$484$		0			0
$485$		−	141.620i		−	0.291999i
$486$		0			0
$487$		−	248.463i		−	0.510191i	−0.966916	−	0.255095i	$-0.917893\pi$
$487$		−	248.463i		−	0.510191i	0.966916	−	0.255095i	$-0.0821069\pi$
$488$		0			0
$489$		−	135.790i		−	0.277690i
$490$		0			0
$491$	−305.588			−0.622379			−0.311189	−	0.950348i	$-0.600727\pi$
$491$	−305.588			−0.622379			−0.311189	+	0.950348i	$0.600727\pi$
$492$		0			0
$493$		−	175.277i		−	0.355531i
$494$		0			0
$495$	−27.8248			−0.0562116
$496$		0			0
$497$			14.7368i			0.0296514i
$498$		0			0
$499$	−268.918			−0.538913			−0.269457	−	0.963013i	$-0.586844\pi$
$499$	−268.918			−0.538913			−0.269457	+	0.963013i	$0.586844\pi$
$500$		0			0
$501$	−508.096			−1.01416
$502$		0			0
$503$	507.154			1.00826			0.504130	−	0.863628i	$-0.331813\pi$
$503$	507.154			1.00826			0.504130	+	0.863628i	$0.331813\pi$
$504$		0			0
$505$	95.9386			0.189977
$506$		0			0
$507$			260.671i			0.514144i
$508$		0			0
$509$		−	407.256i		−	0.800111i	−0.916491	−	0.400055i	$-0.868991\pi$
$509$		−	407.256i		−	0.800111i	0.916491	−	0.400055i	$-0.131009\pi$
$510$		0			0
$511$	−159.825			−0.312769
$512$		0			0
$513$	90.5980	−	39.2301i	0.176604	−	0.0764719i
$514$		0			0
$515$		−	240.289i		−	0.466581i
$516$		0			0
$517$	−403.571			−0.780601
$518$		0			0
$519$	−18.1993			−0.0350662
$520$		0			0
$521$			607.190i			1.16543i	0.812676	+	0.582716i	$0.198010\pi$
$521$			607.190i			1.16543i	−0.812676	+	0.582716i	$0.801990\pi$
$522$		0			0
$523$		−	209.227i		−	0.400052i	−0.979791	−	0.200026i	$-0.935897\pi$
$523$		−	209.227i		−	0.400052i	0.979791	−	0.200026i	$-0.0641026\pi$
$524$		0			0
$525$			191.300i			0.364380i
$526$		0			0
$527$			408.764i			0.775643i
$528$		0			0
$529$	−499.296			−0.943848
$530$		0			0
$531$		−	208.537i		−	0.392724i
$532$		0			0
$533$	−134.598			−0.252529
$534$		0			0
$535$		−	10.3808i		−	0.0194034i
$536$		0			0
$537$	−21.4983			−0.0400342
$538$		0			0
$539$	−194.048			−0.360015
$540$		0			0
$541$	−44.6637			−0.0825576			−0.0412788	−	0.999148i	$-0.513143\pi$
$541$	−44.6637			−0.0825576			−0.0412788	+	0.999148i	$0.513143\pi$
$542$		0			0
$543$	−184.949			−0.340605
$544$		0			0
$545$		−	151.960i		−	0.278826i
$546$		0			0
$547$			595.405i			1.08849i	0.838925	+	0.544246i	$0.183184\pi$
$547$			595.405i			1.08849i	−0.838925	+	0.544246i	$0.816816\pi$
$548$		0			0
$549$	18.5257			0.0337445
$550$		0			0
$551$	−149.993	+	64.9490i	−0.272220	+	0.117875i
$552$		0			0
$553$		−	414.248i		−	0.749092i
$554$		0			0
$555$	−89.6911			−0.161606
$556$		0			0
$557$	−78.4228			−0.140795			−0.0703975	−	0.997519i	$-0.522427\pi$
$557$	−78.4228			−0.140795			−0.0703975	+	0.997519i	$0.522427\pi$
$558$		0			0
$559$			280.120i			0.501110i
$560$		0			0
$561$		−	256.731i		−	0.457630i
$562$		0			0
$563$		−	457.825i		−	0.813189i	−0.913609	−	0.406594i	$-0.866716\pi$
$563$		−	457.825i		−	0.813189i	0.913609	−	0.406594i	$-0.133284\pi$
$564$		0			0
$565$			250.336i			0.443073i
$566$		0			0
$567$	−42.5257			−0.0750013
$568$		0			0
$569$			307.833i			0.541007i	0.962719	+	0.270504i	$0.0871902\pi$
$569$			307.833i			0.541007i	−0.962719	+	0.270504i	$0.912810\pi$
$570$		0			0
$571$	−589.492			−1.03238			−0.516192	−	0.856473i	$-0.672651\pi$
$571$	−589.492			−1.03238			−0.516192	+	0.856473i	$0.672651\pi$
$572$		0			0
$573$			32.9564i			0.0575156i
$574$		0			0
$575$	127.395			0.221557
$576$		0			0
$577$	−363.262			−0.629570			−0.314785	−	0.949163i	$-0.601932\pi$
$577$	−363.262			−0.629570			−0.314785	+	0.949163i	$0.601932\pi$
$578$		0			0
$579$	−191.203			−0.330229
$580$		0			0
$581$	575.032			0.989727
$582$		0			0
$583$		−	568.280i		−	0.974751i
$584$		0			0
$585$			16.4516i			0.0281224i
$586$		0			0
$587$	−391.625			−0.667164			−0.333582	−	0.942721i	$-0.608257\pi$
$587$	−391.625			−0.667164			−0.333582	+	0.942721i	$0.608257\pi$
$588$		0			0
$589$	349.801	−	151.468i	0.593889	−	0.257162i
$590$		0			0
$591$			569.410i			0.963469i
$592$		0			0
$593$	−212.900			−0.359022			−0.179511	−	0.983756i	$-0.557452\pi$
$593$	−212.900			−0.359022			−0.179511	+	0.983756i	$0.557452\pi$
$594$		0			0
$595$	122.738			0.206283
$596$		0			0
$597$			462.410i			0.774556i
$598$		0			0
$599$		−	916.888i		−	1.53070i	−0.643616	−	0.765349i	$-0.722566\pi$
$599$		−	916.888i		−	1.53070i	0.643616	−	0.765349i	$-0.277434\pi$
$600$		0			0
$601$		−	485.078i		−	0.807118i	−0.914954	−	0.403559i	$-0.867773\pi$
$601$		−	485.078i		−	0.807118i	0.914954	−	0.403559i	$-0.132227\pi$
$602$		0			0
$603$			371.171i			0.615541i
$604$		0			0
$605$	−86.7907			−0.143456
$606$		0			0
$607$		−	193.101i		−	0.318123i	−0.987269	−	0.159061i	$-0.949153\pi$
$607$		−	193.101i		−	0.318123i	0.987269	−	0.159061i	$-0.0508468\pi$
$608$		0			0
$609$	70.4053			0.115608
$610$		0			0
$611$			238.614i			0.390531i
$612$		0			0
$613$	−142.368			−0.232248			−0.116124	−	0.993235i	$-0.537047\pi$
$613$	−142.368			−0.232248			−0.116124	+	0.993235i	$0.537047\pi$
$614$		0			0
$615$	69.0997			0.112357
$616$		0			0
$617$	−547.770			−0.887796			−0.443898	−	0.896077i	$-0.646405\pi$
$617$	−547.770			−0.887796			−0.443898	+	0.896077i	$0.646405\pi$
$618$		0			0
$619$	−814.482			−1.31580			−0.657901	−	0.753104i	$-0.728556\pi$
$619$	−814.482			−1.31580			−0.657901	+	0.753104i	$0.728556\pi$
$620$		0			0
$621$			28.3199i			0.0456037i
$622$		0			0
$623$		−	330.626i		−	0.530700i
$624$		0			0
$625$	505.736			0.809177
$626$		0			0
$627$	−219.698	+	95.1319i	−0.350395	+	0.151725i
$628$		0			0
$629$		−	827.552i		−	1.31566i
$630$		0			0
$631$	−641.509			−1.01665			−0.508327	−	0.861164i	$-0.669736\pi$
$631$	−641.509			−1.01665			−0.508327	+	0.861164i	$0.669736\pi$
$632$		0			0
$633$	121.993			0.192723
$634$		0			0
$635$		−	20.4279i		−	0.0321699i
$636$		0			0
$637$			114.733i			0.180114i
$638$		0			0
$639$			9.35652i			0.0146424i
$640$		0			0
$641$			352.878i			0.550512i	0.961371	+	0.275256i	$0.0887626\pi$
$641$			352.878i			0.550512i	−0.961371	+	0.275256i	$0.911237\pi$
$642$		0			0
$643$	455.757			0.708797			0.354399	−	0.935094i	$-0.384686\pi$
$643$	455.757			0.708797			0.354399	+	0.935094i	$0.384686\pi$
$644$		0			0
$645$		−	143.808i		−	0.222958i
$646$		0			0
$647$	−484.368			−0.748637			−0.374318	−	0.927300i	$-0.622123\pi$
$647$	−484.368			−0.748637			−0.374318	+	0.927300i	$0.622123\pi$
$648$		0			0
$649$			505.696i			0.779192i
$650$		0			0
$651$	−164.193			−0.252216
$652$		0			0
$653$	280.780			0.429985			0.214992	−	0.976616i	$-0.431027\pi$
$653$	280.780			0.429985			0.214992	+	0.976616i	$0.431027\pi$
$654$		0			0
$655$	−211.536			−0.322955
$656$		0			0
$657$	−101.474			−0.154451
$658$		0			0
$659$		−	800.259i		−	1.21435i	−0.794567	−	0.607177i	$-0.792302\pi$
$659$		−	800.259i		−	1.21435i	0.794567	−	0.607177i	$-0.207698\pi$
$660$		0			0
$661$		−	812.996i		−	1.22995i	−0.788547	−	0.614975i	$-0.789166\pi$
$661$		−	812.996i		−	1.22995i	0.788547	−	0.614975i	$-0.210834\pi$
$662$		0			0
$663$	−151.794			−0.228950
$664$		0			0
$665$	−45.4809	−	105.034i	−0.0683923	−	0.157945i
$666$		0			0
$667$		−	46.8862i		−	0.0702941i
$668$		0			0
$669$	−411.244			−0.614715
$670$		0			0
$671$	−44.9244			−0.0669514
$672$		0			0
$673$		−	79.8930i		−	0.118712i	−0.998237	−	0.0593559i	$-0.981095\pi$
$673$		−	79.8930i		−	0.118712i	0.998237	−	0.0593559i	$-0.0189047\pi$
$674$		0			0
$675$			121.458i			0.179938i
$676$		0			0
$677$		−	1153.64i		−	1.70404i	−0.523506	−	0.852022i	$-0.675376\pi$
$677$		−	1153.64i		−	1.70404i	0.523506	−	0.852022i	$-0.324624\pi$
$678$		0			0
$679$			524.869i			0.773003i
$680$		0			0
$681$	−467.203			−0.686054
$682$		0			0
$683$		−	382.766i		−	0.560419i	−0.959939	−	0.280209i	$-0.909596\pi$
$683$		−	382.766i		−	0.560419i	0.959939	−	0.280209i	$-0.0904039\pi$
$684$		0			0
$685$	−254.567			−0.371631
$686$		0			0
$687$			449.328i			0.654043i
$688$		0			0
$689$	−336.000			−0.487663
$690$		0			0
$691$	−133.330			−0.192952			−0.0964759	−	0.995335i	$-0.530757\pi$
$691$	−133.330			−0.192952			−0.0964759	+	0.995335i	$0.530757\pi$
$692$		0			0
$693$	103.124			0.148808
$694$		0			0
$695$	69.5772			0.100111
$696$		0			0
$697$			637.562i			0.914723i
$698$		0			0
$699$		−	91.2391i		−	0.130528i
$700$		0			0
$701$	−934.042			−1.33244			−0.666221	−	0.745755i	$-0.732089\pi$
$701$	−934.042			−1.33244			−0.666221	+	0.745755i	$0.732089\pi$
$702$		0			0
$703$	−708.179	+	306.651i	−1.00737	+	0.436203i
$704$		0			0
$705$		−	122.499i		−	0.173758i
$706$		0			0
$707$	−355.566			−0.502923
$708$		0			0
$709$	287.993			0.406197			0.203098	−	0.979158i	$-0.434899\pi$
$709$	287.993			0.406197			0.203098	+	0.979158i	$0.434899\pi$
$710$		0			0
$711$		−	263.010i		−	0.369915i
$712$		0			0
$713$			109.344i			0.153357i
$714$		0			0
$715$		−	39.8947i		−	0.0557968i
$716$		0			0
$717$			599.498i			0.836120i
$718$		0			0
$719$	−78.0723			−0.108585			−0.0542923	−	0.998525i	$-0.517290\pi$
$719$	−78.0723			−0.108585			−0.0542923	+	0.998525i	$0.517290\pi$
$720$		0			0
$721$			890.556i			1.23517i
$722$		0			0
$723$	520.096			0.719359
$724$		0			0
$725$		−	201.085i		−	0.277358i
$726$		0			0
$727$	320.169			0.440397			0.220199	−	0.975455i	$-0.429329\pi$
$727$	320.169			0.440397			0.220199	+	0.975455i	$0.429329\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$	1326.87			1.81514
$732$		0			0
$733$	−565.781			−0.771870			−0.385935	−	0.922526i	$-0.626121\pi$
$733$	−565.781			−0.771870			−0.385935	+	0.922526i	$0.626121\pi$
$734$		0			0
$735$		−	58.9012i		−	0.0801377i
$736$		0			0
$737$		−	900.080i		−	1.22127i
$738$		0			0
$739$	869.557			1.17667			0.588334	−	0.808618i	$-0.299784\pi$
$739$	869.557			1.17667			0.588334	+	0.808618i	$0.299784\pi$
$740$		0			0
$741$	56.2475	+	129.898i	0.0759076	+	0.175301i
$742$		0			0
$743$			7.74529i			0.0104243i	0.999986	+	0.00521217i	$0.00165909\pi$
$743$			7.74529i			0.0104243i	−0.999986	+	0.00521217i	$0.998341\pi$
$744$		0			0
$745$	−312.767			−0.419821
$746$		0			0
$747$	365.093			0.488746
$748$		0			0
$749$			38.4733i			0.0513663i
$750$		0			0
$751$		−	1281.23i		−	1.70603i	−0.521890	−	0.853013i	$-0.674773\pi$
$751$		−	1281.23i		−	1.70603i	0.521890	−	0.853013i	$-0.325227\pi$
$752$		0			0
$753$		−	329.477i		−	0.437552i
$754$		0			0
$755$			101.098i			0.133904i
$756$		0			0
$757$	−441.172			−0.582790			−0.291395	−	0.956603i	$-0.594119\pi$
$757$	−441.172			−0.582790			−0.291395	+	0.956603i	$0.594119\pi$
$758$		0			0
$759$		−	68.6750i		−	0.0904808i
$760$		0			0
$761$	476.532			0.626192			0.313096	−	0.949721i	$-0.398634\pi$
$761$	476.532			0.626192			0.313096	+	0.949721i	$0.398634\pi$
$762$		0			0
$763$			563.193i			0.738129i
$764$		0			0
$765$	77.9277			0.101866
$766$		0			0
$767$	298.997			0.389826
$768$		0			0
$769$	−624.567			−0.812181			−0.406091	−	0.913833i	$-0.633108\pi$
$769$	−624.567			−0.812181			−0.406091	+	0.913833i	$0.633108\pi$
$770$		0			0
$771$	−43.9036			−0.0569438
$772$		0			0
$773$		−	334.038i		−	0.432133i	−0.976379	−	0.216066i	$-0.930677\pi$
$773$		−	334.038i		−	0.432133i	0.976379	−	0.216066i	$-0.0693227\pi$
$774$		0			0
$775$			468.951i			0.605098i
$776$		0			0
$777$	332.412			0.427815
$778$		0			0
$779$	545.595	−	236.249i	0.700378	−	0.303273i
$780$		0			0
$781$		−	22.6893i		−	0.0290516i
$782$		0			0
$783$	44.7010			0.0570894
$784$		0			0
$785$	−68.4601			−0.0872103
$786$		0			0
$787$			479.323i			0.609051i	0.952504	+	0.304526i	$0.0984980\pi$
$787$			479.323i			0.609051i	−0.952504	+	0.304526i	$0.901502\pi$
$788$		0			0
$789$			779.631i			0.988126i
$790$		0			0
$791$		−	927.792i		−	1.17294i
$792$		0			0
$793$			26.5619i			0.0334955i
$794$		0			0
$795$	172.495			0.216975
$796$		0			0
$797$		−	126.025i		−	0.158125i	−0.996870	−	0.0790624i	$-0.974807\pi$
$797$		−	126.025i		−	0.158125i	0.996870	−	0.0790624i	$-0.0251926\pi$
$798$		0			0
$799$	1130.26			1.41460
$800$		0			0
$801$		−	209.918i		−	0.262070i
$802$		0			0
$803$	246.072			0.306441
$804$		0			0
$805$	32.8323			0.0407854
$806$		0			0
$807$	−840.289			−1.04125
$808$		0			0
$809$	901.825			1.11474			0.557370	−	0.830264i	$-0.311810\pi$
$809$	901.825			1.11474			0.557370	+	0.830264i	$0.311810\pi$
$810$		0			0
$811$			183.776i			0.226604i	0.993561	+	0.113302i	$0.0361427\pi$
$811$			183.776i			0.226604i	−0.993561	+	0.113302i	$0.963857\pi$
$812$		0			0
$813$		−	271.926i		−	0.334473i
$814$		0			0
$815$	−99.9518			−0.122640
$816$		0			0
$817$	−491.674	−	1135.47i	−0.601804	−	1.38981i
$818$		0			0
$819$		−	60.9728i		−	0.0744478i
$820$		0			0
$821$	1034.35			1.25986			0.629932	−	0.776650i	$-0.283083\pi$
$821$	1034.35			1.25986			0.629932	+	0.776650i	$0.283083\pi$
$822$		0			0
$823$	−952.107			−1.15687			−0.578437	−	0.815727i	$-0.696337\pi$
$823$	−952.107			−1.15687			−0.578437	+	0.815727i	$0.696337\pi$
$824$		0			0
$825$		−	294.532i		−	0.357009i
$826$		0			0
$827$		−	970.806i		−	1.17389i	−0.809627	−	0.586944i	$-0.800331\pi$
$827$		−	970.806i		−	1.17389i	0.809627	−	0.586944i	$-0.199669\pi$
$828$		0			0
$829$		−	374.908i		−	0.452242i	−0.974099	−	0.226121i	$-0.927396\pi$
$829$		−	374.908i		−	0.452242i	0.974099	−	0.226121i	$-0.0726044\pi$
$830$		0			0
$831$			271.289i			0.326461i
$832$		0			0
$833$	543.463			0.652417
$834$		0			0
$835$			373.996i			0.447900i
$836$		0			0
$837$	−104.248			−0.124549
$838$		0			0
$839$			1282.87i			1.52904i	0.644597	+	0.764522i	$0.277025\pi$
$839$			1282.87i			1.52904i	−0.644597	+	0.764522i	$0.722975\pi$
$840$		0			0
$841$	766.993			0.912002
$842$		0			0
$843$	−792.688			−0.940318
$844$		0			0
$845$	191.873			0.227069
$846$		0			0
$847$	321.663			0.379767
$848$		0			0
$849$			80.6684i			0.0950158i
$850$		0			0
$851$		−	221.369i		−	0.260128i
$852$		0			0
$853$	827.595			0.970217			0.485108	−	0.874454i	$-0.338780\pi$
$853$	827.595			0.970217			0.485108	+	0.874454i	$0.338780\pi$
$854$		0			0
$855$	−28.8762	−	66.6868i	−0.0337734	−	0.0779963i
$856$		0			0
$857$			986.671i			1.15131i	0.817694	+	0.575654i	$0.195252\pi$
$857$			986.671i			1.15131i	−0.817694	+	0.575654i	$0.804748\pi$
$858$		0			0
$859$	1382.27			1.60916			0.804582	−	0.593842i	$-0.202389\pi$
$859$	1382.27			1.60916			0.804582	+	0.593842i	$0.202389\pi$
$860$		0			0
$861$	−256.096			−0.297441
$862$		0			0
$863$		−	485.475i		−	0.562543i	−0.959628	−	0.281272i	$-0.909244\pi$
$863$		−	485.475i		−	0.562543i	0.959628	−	0.281272i	$-0.0907562\pi$
$864$		0			0
$865$			13.3961i			0.0154868i
$866$		0			0
$867$			218.453i			0.251964i
$868$		0			0
$869$			637.792i			0.733938i
$870$		0			0
$871$	−532.179			−0.610998
$872$		0			0
$873$			333.244i			0.381723i
$874$		0			0
$875$	291.413			0.333043
$876$		0			0
$877$			266.756i			0.304169i	0.988368	+	0.152084i	$0.0485985\pi$
$877$			266.756i			0.304169i	−0.988368	+	0.152084i	$0.951401\pi$
$878$		0			0
$879$	−384.797			−0.437767
$880$		0			0
$881$	−979.248			−1.11152			−0.555760	−	0.831343i	$-0.687573\pi$
$881$	−979.248			−1.11152			−0.555760	+	0.831343i	$0.687573\pi$
$882$		0			0
$883$	−1551.91			−1.75755			−0.878774	−	0.477239i	$-0.841638\pi$
$883$	−1551.91			−1.75755			−0.878774	+	0.477239i	$0.841638\pi$
$884$		0			0
$885$	−153.498			−0.173444
$886$		0			0
$887$		−	700.543i		−	0.789789i	−0.918727	−	0.394894i	$-0.870781\pi$
$887$		−	700.543i		−	0.789789i	0.918727	−	0.394894i	$-0.129219\pi$
$888$		0			0
$889$			75.7095i			0.0851626i
$890$		0			0
$891$	65.4743			0.0734840
$892$		0			0
$893$	−418.821	−	967.227i	−0.469005	−	1.08312i
$894$		0			0
$895$			15.8244i			0.0176809i
$896$		0			0
$897$	−40.6046			−0.0452672
$898$		0			0
$899$	172.591			0.191982
$900$		0			0
$901$			1591.56i			1.76644i
$902$		0			0
$903$			532.978i			0.590231i
$904$		0			0
$905$			136.136i			0.150426i
$906$		0			0
$907$			1228.07i			1.35399i	0.735987	+	0.676996i	$0.236718\pi$
$907$			1228.07i			1.35399i	−0.735987	+	0.676996i	$0.763282\pi$
$908$		0			0
$909$	−225.752			−0.248353
$910$		0			0
$911$		−	1077.26i		−	1.18250i	−0.806487	−	0.591252i	$-0.798634\pi$
$911$		−	1077.26i		−	1.18250i	0.806487	−	0.591252i	$-0.201366\pi$
$912$		0			0
$913$	−885.341			−0.969705
$914$		0			0
$915$		−	13.6363i		−	0.0149031i
$916$		0			0
$917$	783.991			0.854952
$918$		0			0
$919$	348.488			0.379204			0.189602	−	0.981861i	$-0.439280\pi$
$919$	348.488			0.379204			0.189602	+	0.981861i	$0.439280\pi$
$920$		0			0
$921$	852.289			0.925395
$922$		0			0
$923$	−13.4152			−0.0145344
$924$		0			0
$925$		−	949.403i		−	1.02638i
$926$		0			0
$927$			565.423i			0.609949i
$928$		0			0
$929$	−1506.48			−1.62162			−0.810808	−	0.585312i	$-0.800972\pi$
$929$	−1506.48			−1.62162			−0.810808	+	0.585312i	$0.800972\pi$
$930$		0			0
$931$	−201.381	−	465.070i	−0.216306	−	0.499538i
$932$		0			0
$933$		−	281.932i		−	0.302177i
$934$		0			0
$935$	−188.973			−0.202110
$936$		0			0
$937$	−145.777			−0.155578			−0.0777890	−	0.996970i	$-0.524786\pi$
$937$	−145.777			−0.155578			−0.0777890	+	0.996970i	$0.524786\pi$
$938$		0			0
$939$			718.617i			0.765300i
$940$		0			0
$941$		−	1501.75i		−	1.59590i	−0.602721	−	0.797952i	$-0.705917\pi$
$941$		−	1501.75i		−	1.59590i	0.602721	−	0.797952i	$-0.294083\pi$
$942$		0			0
$943$			170.547i			0.180855i
$944$		0			0
$945$			31.3021i			0.0331239i
$946$		0			0
$947$	−765.794			−0.808653			−0.404326	−	0.914615i	$-0.632494\pi$
$947$	−765.794			−0.808653			−0.404326	+	0.914615i	$0.632494\pi$
$948$		0			0
$949$		−	145.492i		−	0.153311i
$950$		0			0
$951$	633.498			0.666139
$952$		0			0
$953$			1308.88i			1.37344i	0.726924	+	0.686718i	$0.240949\pi$
$953$			1308.88i			1.37344i	−0.726924	+	0.686718i	$0.759051\pi$
$954$		0			0
$955$	24.2584			0.0254014
$956$		0			0
$957$	−108.399			−0.113269
$958$		0			0
$959$	943.474			0.983810
$960$		0			0
$961$	558.498			0.581164
$962$		0			0
$963$			24.4271i			0.0253656i
$964$		0			0
$965$			140.739i			0.145844i
$966$		0			0
$967$	−587.704			−0.607760			−0.303880	−	0.952710i	$-0.598282\pi$
$967$	−587.704			−0.607760			−0.303880	+	0.952710i	$0.598282\pi$
$968$		0			0
$969$	615.299	−	266.432i	0.634983	−	0.274956i
$970$		0			0
$971$			653.823i			0.673351i	0.941621	+	0.336675i	$0.109302\pi$
$971$			653.823i			0.673351i	−0.941621	+	0.336675i	$0.890698\pi$
$972$		0			0
$973$	−257.866			−0.265022
$974$		0			0
$975$	−174.145			−0.178610
$976$		0			0
$977$			101.034i			0.103413i	0.998662	+	0.0517064i	$0.0164660\pi$
$977$			101.034i			0.103413i	−0.998662	+	0.0517064i	$0.983534\pi$
$978$		0			0
$979$			509.045i			0.519964i
$980$		0			0
$981$			357.576i			0.364502i
$982$		0			0
$983$		−	888.065i		−	0.903423i	−0.892164	−	0.451711i	$-0.850814\pi$
$983$		−	888.065i		−	0.903423i	0.892164	−	0.451711i	$-0.149186\pi$
$984$		0			0
$985$	419.128			0.425511
$986$		0			0
$987$			454.006i			0.459986i
$988$		0			0
$989$	354.935			0.358883
$990$		0			0
$991$			1782.19i			1.79838i	0.437562	+	0.899188i	$0.355842\pi$
$991$			1782.19i			1.79838i	−0.437562	+	0.899188i	$0.644158\pi$
$992$		0			0
$993$	−381.498			−0.384188
$994$		0			0
$995$	340.368			0.342078
$996$		0			0
$997$	891.200			0.893882			0.446941	−	0.894563i	$-0.352513\pi$
$997$	891.200			0.893882			0.446941	+	0.894563i	$0.352513\pi$
$998$		0			0
$999$	211.051			0.211263

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.3.o.b.721.4		4
3.2	odd	2		2736.3.o.l.721.1		4
4.3	odd	2		57.3.c.b.37.3	yes	4
12.11	even	2		171.3.c.f.37.2		4
19.18	odd	2	inner	912.3.o.b.721.2		4
57.56	even	2		2736.3.o.l.721.2		4
76.75	even	2		57.3.c.b.37.2	✓	4
228.227	odd	2		171.3.c.f.37.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.c.b.37.2	✓	4	76.75	even	2
57.3.c.b.37.3	yes	4	4.3	odd	2
171.3.c.f.37.2		4	12.11	even	2
171.3.c.f.37.3		4	228.227	odd	2
912.3.o.b.721.2		4	19.18	odd	2	inner
912.3.o.b.721.4		4	1.1	even	1	trivial
2736.3.o.l.721.1		4	3.2	odd	2
2736.3.o.l.721.2		4	57.56	even	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	912.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +1.27492 q^{5} -4.72508 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +1.27492 q^{5} -4.72508 q^{7} -3.00000 q^{9} +7.27492 q^{11} -4.30136i q^{13} +2.20822i q^{15} -20.3746 q^{17} +(7.54983 + 17.4356i) q^{19} -8.18408i q^{21} -5.45017 q^{23} -23.3746 q^{25} -5.19615i q^{27} +8.60271i q^{29} -20.0624i q^{31} +12.6005i q^{33} -6.02409 q^{35} +40.6169i q^{37} +7.45017 q^{39} -31.2920i q^{41} -65.1238 q^{43} -3.82475 q^{45} -55.4743 q^{47} -26.6736 q^{49} -35.2898i q^{51} -78.1149i q^{53} +9.27492 q^{55} +(-30.1993 + 13.0767i) q^{57} +69.5122i q^{59} -6.17525 q^{61} +14.1752 q^{63} -5.48387i q^{65} -123.724i q^{67} -9.43996i q^{69} -3.11884i q^{71} +33.8248 q^{73} -40.4860i q^{75} -34.3746 q^{77} +87.6700i q^{79} +9.00000 q^{81} -121.698 q^{83} -25.9759 q^{85} -14.9003 q^{87} +69.9726i q^{89} +20.3243i q^{91} +34.7492 q^{93} +(9.62541 + 22.2289i) q^{95} -111.081i q^{97} -21.8248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{5} - 34 q^{7} - 12 q^{9}+O(q^{10}) \) 4 * q - 10 * q^5 - 34 * q^7 - 12 * q^9	\( 4 q - 10 q^{5} - 34 q^{7} - 12 q^{9} + 14 q^{11} - 6 q^{17} - 52 q^{23} - 18 q^{25} + 142 q^{35} + 60 q^{39} - 34 q^{43} + 30 q^{45} - 86 q^{47} + 150 q^{49} + 22 q^{55} - 70 q^{61} + 102 q^{63} + 90 q^{73} - 62 q^{77} + 36 q^{81} - 64 q^{83} - 270 q^{85} - 120 q^{87} - 12 q^{93} + 114 q^{95} - 42 q^{99}+O(q^{100}) \) 4 * q - 10 * q^5 - 34 * q^7 - 12 * q^9 + 14 * q^11 - 6 * q^17 - 52 * q^23 - 18 * q^25 + 142 * q^35 + 60 * q^39 - 34 * q^43 + 30 * q^45 - 86 * q^47 + 150 * q^49 + 22 * q^55 - 70 * q^61 + 102 * q^63 + 90 * q^73 - 62 * q^77 + 36 * q^81 - 64 * q^83 - 270 * q^85 - 120 * q^87 - 12 * q^93 + 114 * q^95 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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