LMFDB - Embedded newform 544.2.b.d.33.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [544,2,Mod(33,544)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("544.33"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 544 = 2^{5} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	544.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,0,0,8,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$4.34386186996$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.4
Root			$0.765367i$ of defining polynomial
Character	$\chi$	$=$	544.33
Dual form			544.2.b.d.33.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61313i q^{3} +3.69552i q^{5} -1.08239i q^{7} -3.82843 q^{9} +2.61313i q^{11} +4.82843 q^{13} -9.65685 q^{15} +(3.82843 + 1.53073i) q^{17} -5.65685 q^{19} +2.82843 q^{21} -1.08239i q^{23} -8.65685 q^{25} -2.16478i q^{27} -6.75699i q^{29} -8.47343i q^{31} -6.82843 q^{33} +4.00000 q^{35} -3.69552i q^{37} +12.6173i q^{39} +3.06147i q^{41} -9.65685 q^{43} -14.1480i q^{45} +9.65685 q^{47} +5.82843 q^{49} +(-4.00000 + 10.0042i) q^{51} -2.00000 q^{53} -9.65685 q^{55} -14.7821i q^{57} +1.65685 q^{59} +8.02509i q^{61} +4.14386i q^{63} +17.8435i q^{65} +13.6569 q^{67} +2.82843 q^{69} +10.6382i q^{71} +10.4525i q^{73} -22.6215i q^{75} +2.82843 q^{77} +3.24718i q^{79} -5.82843 q^{81} +1.65685 q^{83} +(-5.65685 + 14.1480i) q^{85} +17.6569 q^{87} -6.48528 q^{89} -5.22625i q^{91} +22.1421 q^{93} -20.9050i q^{95} -13.5140i q^{97} -10.0042i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} + 8 q^{13} - 16 q^{15} + 4 q^{17} - 12 q^{25} - 16 q^{33} + 16 q^{35} - 16 q^{43} + 16 q^{47} + 12 q^{49} - 16 q^{51} - 8 q^{53} - 16 q^{55} - 16 q^{59} + 32 q^{67} - 12 q^{81} - 16 q^{83}+ \cdots + 32 q^{93}+O(q^{100}) $ 4 * q - 4 * q^9 + 8 * q^13 - 16 * q^15 + 4 * q^17 - 12 * q^25 - 16 * q^33 + 16 * q^35 - 16 * q^43 + 16 * q^47 + 12 * q^49 - 16 * q^51 - 8 * q^53 - 16 * q^55 - 16 * q^59 + 32 * q^67 - 12 * q^81 - 16 * q^83 + 48 * q^87 + 8 * q^89 + 32 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/544\mathbb{Z}\right)^\times$.

$n$	$69$	$511$	$513$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			2.61313i			1.50869i	0.656479	+	0.754344i	$0.272045\pi$
$3$			2.61313i			1.50869i	−0.656479	+	0.754344i	$0.727955\pi$
$4$		0			0
$5$			3.69552i			1.65269i	0.563167	+	0.826343i	$0.309583\pi$
$5$			3.69552i			1.65269i	−0.563167	+	0.826343i	$0.690417\pi$
$6$		0			0
$7$		−	1.08239i		−	0.409106i	−0.978856	−	0.204553i	$-0.934426\pi$
$7$		−	1.08239i		−	0.409106i	0.978856	−	0.204553i	$-0.0655740\pi$
$8$		0			0
$9$	−3.82843			−1.27614
$10$		0			0
$11$			2.61313i			0.787887i	0.919135	+	0.393944i	$0.128889\pi$
$11$			2.61313i			0.787887i	−0.919135	+	0.393944i	$0.871111\pi$
$12$		0			0
$13$	4.82843			1.33916			0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843			1.33916			0.669582	+	0.742738i	$0.266473\pi$
$14$		0			0
$15$	−9.65685			−2.49339
$16$		0			0
$17$	3.82843	+	1.53073i	0.928530	+	0.371257i
$17$	3.82843	+	1.53073i	0.928530	+	0.371257i
$18$		0			0
$19$	−5.65685			−1.29777			−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685			−1.29777			−0.648886	+	0.760886i	$0.724765\pi$
$20$		0			0
$21$	2.82843			0.617213
$22$		0			0
$23$		−	1.08239i		−	0.225694i	−0.993612	−	0.112847i	$-0.964003\pi$
$23$		−	1.08239i		−	0.225694i	0.993612	−	0.112847i	$-0.0359971\pi$
$24$		0			0
$25$	−8.65685			−1.73137
$26$		0			0
$27$		−	2.16478i		−	0.416613i
$28$		0			0
$29$		−	6.75699i		−	1.25474i	−0.778721	−	0.627370i	$-0.784131\pi$
$29$		−	6.75699i		−	1.25474i	0.778721	−	0.627370i	$-0.215869\pi$
$30$		0			0
$31$		−	8.47343i		−	1.52187i	−0.648827	−	0.760936i	$-0.724740\pi$
$31$		−	8.47343i		−	1.52187i	0.648827	−	0.760936i	$-0.275260\pi$
$32$		0			0
$33$	−6.82843			−1.18868
$34$		0			0
$35$	4.00000			0.676123
$36$		0			0
$37$		−	3.69552i		−	0.607539i	−0.952745	−	0.303770i	$-0.901755\pi$
$37$		−	3.69552i		−	0.607539i	0.952745	−	0.303770i	$-0.0982453\pi$
$38$		0			0
$39$			12.6173i			2.02038i
$40$		0			0
$41$			3.06147i			0.478121i	0.971005	+	0.239060i	$0.0768394\pi$
$41$			3.06147i			0.478121i	−0.971005	+	0.239060i	$0.923161\pi$
$42$		0			0
$43$	−9.65685			−1.47266			−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685			−1.47266			−0.736328	+	0.676625i	$0.763442\pi$
$44$		0			0
$45$		−	14.1480i		−	2.10906i
$46$		0			0
$47$	9.65685			1.40860			0.704298	−	0.709904i	$-0.251262\pi$
$47$	9.65685			1.40860			0.704298	+	0.709904i	$0.251262\pi$
$48$		0			0
$49$	5.82843			0.832632
$50$		0			0
$51$	−4.00000	+	10.0042i	−0.560112	+	1.40086i
$52$		0			0
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	−9.65685			−1.30213
$56$		0			0
$57$		−	14.7821i		−	1.95793i
$58$		0			0
$59$	1.65685			0.215704			0.107852	−	0.994167i	$-0.465603\pi$
$59$	1.65685			0.215704			0.107852	+	0.994167i	$0.465603\pi$
$60$		0			0
$61$			8.02509i			1.02751i	0.857938	+	0.513754i	$0.171745\pi$
$61$			8.02509i			1.02751i	−0.857938	+	0.513754i	$0.828255\pi$
$62$		0			0
$63$			4.14386i			0.522077i
$64$		0			0
$65$			17.8435i			2.21322i
$66$		0			0
$67$	13.6569			1.66845			0.834225	−	0.551424i	$-0.185915\pi$
$67$	13.6569			1.66845			0.834225	+	0.551424i	$0.185915\pi$
$68$		0			0
$69$	2.82843			0.340503
$70$		0			0
$71$			10.6382i			1.26252i	0.775570	+	0.631262i	$0.217463\pi$
$71$			10.6382i			1.26252i	−0.775570	+	0.631262i	$0.782537\pi$
$72$		0			0
$73$			10.4525i			1.22337i	0.791100	+	0.611687i	$0.209509\pi$
$73$			10.4525i			1.22337i	−0.791100	+	0.611687i	$0.790491\pi$
$74$		0			0
$75$		−	22.6215i		−	2.61210i
$76$		0			0
$77$	2.82843			0.322329
$78$		0			0
$79$			3.24718i			0.365336i	0.983175	+	0.182668i	$0.0584733\pi$
$79$			3.24718i			0.365336i	−0.983175	+	0.182668i	$0.941527\pi$
$80$		0			0
$81$	−5.82843			−0.647603
$82$		0			0
$83$	1.65685			0.181863			0.0909317	−	0.995857i	$-0.471016\pi$
$83$	1.65685			0.181863			0.0909317	+	0.995857i	$0.471016\pi$
$84$		0			0
$85$	−5.65685	+	14.1480i	−0.613572	+	1.53457i
$86$		0			0
$87$	17.6569			1.89301
$88$		0			0
$89$	−6.48528			−0.687438			−0.343719	−	0.939072i	$-0.611687\pi$
$89$	−6.48528			−0.687438			−0.343719	+	0.939072i	$0.611687\pi$
$90$		0			0
$91$		−	5.22625i		−	0.547860i
$92$		0			0
$93$	22.1421			2.29603
$94$		0			0
$95$		−	20.9050i		−	2.14481i
$96$		0			0
$97$		−	13.5140i		−	1.37214i	−0.727537	−	0.686068i	$-0.759335\pi$
$97$		−	13.5140i		−	1.37214i	0.727537	−	0.686068i	$-0.240665\pi$
$98$		0			0
$99$		−	10.0042i		−	1.00546i
$100$		0			0
$101$	12.8284			1.27648			0.638238	−	0.769839i	$-0.279664\pi$
$101$	12.8284			1.27648			0.638238	+	0.769839i	$0.279664\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$			10.4525i			1.02006i
$106$		0			0
$107$			3.88123i			0.375212i	0.982244	+	0.187606i	$0.0600729\pi$
$107$			3.88123i			0.375212i	−0.982244	+	0.187606i	$0.939927\pi$
$108$		0			0
$109$			6.75699i			0.647202i	0.946194	+	0.323601i	$0.104894\pi$
$109$			6.75699i			0.647202i	−0.946194	+	0.323601i	$0.895106\pi$
$110$		0			0
$111$	9.65685			0.916588
$112$		0			0
$113$			4.32957i			0.407292i	0.979045	+	0.203646i	$0.0652791\pi$
$113$			4.32957i			0.407292i	−0.979045	+	0.203646i	$0.934721\pi$
$114$		0			0
$115$	4.00000			0.373002
$116$		0			0
$117$	−18.4853			−1.70896
$118$		0			0
$119$	1.65685	−	4.14386i	0.151884	−	0.379867i
$120$		0			0
$121$	4.17157			0.379234
$122$		0			0
$123$	−8.00000			−0.721336
$124$		0			0
$125$		−	13.5140i		−	1.20873i
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		0			0
$129$		−	25.2346i		−	2.22178i
$130$		0			0
$131$		−	6.57128i		−	0.574135i	−0.957910	−	0.287068i	$-0.907320\pi$
$131$		−	6.57128i		−	0.574135i	0.957910	−	0.287068i	$-0.0926804\pi$
$132$		0			0
$133$			6.12293i			0.530926i
$134$		0			0
$135$	8.00000			0.688530
$136$		0			0
$137$	−8.82843			−0.754263			−0.377132	−	0.926160i	$-0.623090\pi$
$137$	−8.82843			−0.754263			−0.377132	+	0.926160i	$0.623090\pi$
$138$		0			0
$139$			2.61313i			0.221642i	0.993840	+	0.110821i	$0.0353481\pi$
$139$			2.61313i			0.221642i	−0.993840	+	0.110821i	$0.964652\pi$
$140$		0			0
$141$			25.2346i			2.12513i
$142$		0			0
$143$			12.6173i			1.05511i
$144$		0			0
$145$	24.9706			2.07369
$146$		0			0
$147$			15.2304i			1.25618i
$148$		0			0
$149$	−13.3137			−1.09070			−0.545351	−	0.838208i	$-0.683604\pi$
$149$	−13.3137			−1.09070			−0.545351	+	0.838208i	$0.683604\pi$
$150$		0			0
$151$	3.31371			0.269666			0.134833	−	0.990868i	$-0.456950\pi$
$151$	3.31371			0.269666			0.134833	+	0.990868i	$0.456950\pi$
$152$		0			0
$153$	−14.6569	−	5.86030i	−1.18494	−	0.473777i
$154$		0			0
$155$	31.3137			2.51518
$156$		0			0
$157$	−10.0000			−0.798087			−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000			−0.798087			−0.399043	+	0.916932i	$0.630658\pi$
$158$		0			0
$159$		−	5.22625i		−	0.414469i
$160$		0			0
$161$	−1.17157			−0.0923329
$162$		0			0
$163$			16.1271i			1.26317i	0.775306	+	0.631586i	$0.217596\pi$
$163$			16.1271i			1.26317i	−0.775306	+	0.631586i	$0.782404\pi$
$164$		0			0
$165$		−	25.2346i		−	1.96451i
$166$		0			0
$167$		−	21.9874i		−	1.70144i	−0.525622	−	0.850718i	$-0.676167\pi$
$167$		−	21.9874i		−	1.70144i	0.525622	−	0.850718i	$-0.323833\pi$
$168$		0			0
$169$	10.3137			0.793362
$170$		0			0
$171$	21.6569			1.65614
$172$		0			0
$173$		−	11.0866i		−	0.842895i	−0.906853	−	0.421448i	$-0.861522\pi$
$173$		−	11.0866i		−	0.842895i	0.906853	−	0.421448i	$-0.138478\pi$
$174$		0			0
$175$			9.37011i			0.708314i
$176$		0			0
$177$			4.32957i			0.325430i
$178$		0			0
$179$	24.9706			1.86639			0.933194	−	0.359374i	$-0.117010\pi$
$179$	24.9706			1.86639			0.933194	+	0.359374i	$0.117010\pi$
$180$		0			0
$181$		−	15.4161i		−	1.14587i	−0.819600	−	0.572936i	$-0.805804\pi$
$181$		−	15.4161i		−	1.14587i	0.819600	−	0.572936i	$-0.194196\pi$
$182$		0			0
$183$	−20.9706			−1.55019
$184$		0			0
$185$	13.6569			1.00407
$186$		0			0
$187$	−4.00000	+	10.0042i	−0.292509	+	0.731577i
$188$		0			0
$189$	−2.34315			−0.170439
$190$		0			0
$191$	12.9706			0.938517			0.469258	−	0.883061i	$-0.344521\pi$
$191$	12.9706			0.938517			0.469258	+	0.883061i	$0.344521\pi$
$192$		0			0
$193$		−	23.9665i		−	1.72514i	−0.505934	−	0.862572i	$-0.668852\pi$
$193$		−	23.9665i		−	1.72514i	0.505934	−	0.862572i	$-0.331148\pi$
$194$		0			0
$195$	−46.6274			−3.33906
$196$		0			0
$197$		−	0.634051i		−	0.0451742i	−0.999745	−	0.0225871i	$-0.992810\pi$
$197$		−	0.634051i		−	0.0451742i	0.999745	−	0.0225871i	$-0.00719032\pi$
$198$		0			0
$199$			5.04054i			0.357315i	0.983911	+	0.178657i	$0.0571753\pi$
$199$			5.04054i			0.357315i	−0.983911	+	0.178657i	$0.942825\pi$
$200$		0			0
$201$			35.6871i			2.51717i
$202$		0			0
$203$	−7.31371			−0.513322
$204$		0			0
$205$	−11.3137			−0.790184
$206$		0			0
$207$			4.14386i			0.288018i
$208$		0			0
$209$		−	14.7821i		−	1.02250i
$210$		0			0
$211$		−	15.2304i		−	1.04850i	−0.851563	−	0.524252i	$-0.824345\pi$
$211$		−	15.2304i		−	1.04850i	0.851563	−	0.524252i	$-0.175655\pi$
$212$		0			0
$213$	−27.7990			−1.90476
$214$		0			0
$215$		−	35.6871i		−	2.43384i
$216$		0			0
$217$	−9.17157			−0.622607
$218$		0			0
$219$	−27.3137			−1.84569
$220$		0			0
$221$	18.4853	+	7.39104i	1.24345	+	0.497175i
$222$		0			0
$223$	−4.97056			−0.332854			−0.166427	−	0.986054i	$-0.553223\pi$
$223$	−4.97056			−0.332854			−0.166427	+	0.986054i	$0.553223\pi$
$224$		0			0
$225$	33.1421			2.20948
$226$		0			0
$227$			10.0042i			0.664000i	0.943279	+	0.332000i	$0.107723\pi$
$227$			10.0042i			0.664000i	−0.943279	+	0.332000i	$0.892277\pi$
$228$		0			0
$229$	−17.7990			−1.17619			−0.588095	−	0.808792i	$-0.700122\pi$
$229$	−17.7990			−1.17619			−0.588095	+	0.808792i	$0.700122\pi$
$230$		0			0
$231$			7.39104i			0.486294i
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$			35.6871i			2.32797i
$236$		0			0
$237$	−8.48528			−0.551178
$238$		0			0
$239$	14.3431			0.927781			0.463890	−	0.885893i	$-0.346453\pi$
$239$	14.3431			0.927781			0.463890	+	0.885893i	$0.346453\pi$
$240$		0			0
$241$			16.0502i			1.03388i	0.856021	+	0.516941i	$0.172929\pi$
$241$			16.0502i			1.03388i	−0.856021	+	0.516941i	$0.827071\pi$
$242$		0			0
$243$		−	21.7248i		−	1.39364i
$244$		0			0
$245$			21.5391i			1.37608i
$246$		0			0
$247$	−27.3137			−1.73793
$248$		0			0
$249$			4.32957i			0.274375i
$250$		0			0
$251$	−5.65685			−0.357057			−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685			−0.357057			−0.178529	+	0.983935i	$0.557134\pi$
$252$		0			0
$253$	2.82843			0.177822
$254$		0			0
$255$	−36.9706	−	14.7821i	−2.31519	−	0.925689i
$256$		0			0
$257$	18.4853			1.15308			0.576540	−	0.817069i	$-0.304402\pi$
$257$	18.4853			1.15308			0.576540	+	0.817069i	$0.304402\pi$
$258$		0			0
$259$	−4.00000			−0.248548
$260$		0			0
$261$			25.8686i			1.60123i
$262$		0			0
$263$	−16.0000			−0.986602			−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000			−0.986602			−0.493301	+	0.869859i	$0.664210\pi$
$264$		0			0
$265$		−	7.39104i		−	0.454028i
$266$		0			0
$267$		−	16.9469i		−	1.03713i
$268$		0			0
$269$		−	0.634051i		−	0.0386588i	−0.999813	−	0.0193294i	$-0.993847\pi$
$269$		−	0.634051i		−	0.0386588i	0.999813	−	0.0193294i	$-0.00615311\pi$
$270$		0			0
$271$	14.3431			0.871284			0.435642	−	0.900120i	$-0.356521\pi$
$271$	14.3431			0.871284			0.435642	+	0.900120i	$0.356521\pi$
$272$		0			0
$273$	13.6569			0.826550
$274$		0			0
$275$		−	22.6215i		−	1.36412i
$276$		0			0
$277$		−	9.29319i		−	0.558374i	−0.960237	−	0.279187i	$-0.909935\pi$
$277$		−	9.29319i		−	0.558374i	0.960237	−	0.279187i	$-0.0900649\pi$
$278$		0			0
$279$			32.4399i			1.94213i
$280$		0			0
$281$	−14.9706			−0.893069			−0.446534	−	0.894766i	$-0.647342\pi$
$281$	−14.9706			−0.893069			−0.446534	+	0.894766i	$0.647342\pi$
$282$		0			0
$283$			29.6411i			1.76198i	0.473136	+	0.880989i	$0.343122\pi$
$283$			29.6411i			1.76198i	−0.473136	+	0.880989i	$0.656878\pi$
$284$		0			0
$285$	54.6274			3.23585
$286$		0			0
$287$	3.31371			0.195602
$288$		0			0
$289$	12.3137	+	11.7206i	0.724336	+	0.689447i
$290$		0			0
$291$	35.3137			2.07013
$292$		0			0
$293$	−18.0000			−1.05157			−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000			−1.05157			−0.525786	+	0.850617i	$0.676229\pi$
$294$		0			0
$295$			6.12293i			0.356491i
$296$		0			0
$297$	5.65685			0.328244
$298$		0			0
$299$		−	5.22625i		−	0.302242i
$300$		0			0
$301$			10.4525i			0.602472i
$302$		0			0
$303$			33.5223i			1.92581i
$304$		0			0
$305$	−29.6569			−1.69815
$306$		0			0
$307$	−12.9706			−0.740269			−0.370135	−	0.928978i	$-0.620688\pi$
$307$	−12.9706			−0.740269			−0.370135	+	0.928978i	$0.620688\pi$
$308$		0			0
$309$			20.9050i			1.18924i
$310$		0			0
$311$		−	1.08239i		−	0.0613768i	−0.999529	−	0.0306884i	$-0.990230\pi$
$311$		−	1.08239i		−	0.0613768i	0.999529	−	0.0306884i	$-0.00976996\pi$
$312$		0			0
$313$		−	7.39104i		−	0.417766i	−0.977941	−	0.208883i	$-0.933017\pi$
$313$		−	7.39104i		−	0.417766i	0.977941	−	0.208883i	$-0.0669828\pi$
$314$		0			0
$315$	−15.3137			−0.862830
$316$		0			0
$317$		−	6.75699i		−	0.379510i	−0.981831	−	0.189755i	$-0.939231\pi$
$317$		−	6.75699i		−	0.379510i	0.981831	−	0.189755i	$-0.0607694\pi$
$318$		0			0
$319$	17.6569			0.988594
$320$		0			0
$321$	−10.1421			−0.566079
$322$		0			0
$323$	−21.6569	−	8.65914i	−1.20502	−	0.481807i
$324$		0			0
$325$	−41.7990			−2.31859
$326$		0			0
$327$	−17.6569			−0.976426
$328$		0			0
$329$		−	10.4525i		−	0.576265i
$330$		0			0
$331$	4.97056			0.273207			0.136603	−	0.990626i	$-0.456381\pi$
$331$	4.97056			0.273207			0.136603	+	0.990626i	$0.456381\pi$
$332$		0			0
$333$			14.1480i			0.775307i
$334$		0			0
$335$			50.4692i			2.75742i
$336$		0			0
$337$		−	31.3575i		−	1.70815i	−0.520148	−	0.854076i	$-0.674123\pi$
$337$		−	31.3575i		−	1.70815i	0.520148	−	0.854076i	$-0.325877\pi$
$338$		0			0
$339$	−11.3137			−0.614476
$340$		0			0
$341$	22.1421			1.19906
$342$		0			0
$343$		−	13.8854i		−	0.749741i
$344$		0			0
$345$			10.4525i			0.562744i
$346$		0			0
$347$			22.9929i			1.23432i	0.786837	+	0.617161i	$0.211717\pi$
$347$			22.9929i			1.23432i	−0.786837	+	0.617161i	$0.788283\pi$
$348$		0			0
$349$	1.31371			0.0703212			0.0351606	−	0.999382i	$-0.488806\pi$
$349$	1.31371			0.0703212			0.0351606	+	0.999382i	$0.488806\pi$
$350$		0			0
$351$		−	10.4525i		−	0.557913i
$352$		0			0
$353$	−0.343146			−0.0182638			−0.00913190	−	0.999958i	$-0.502907\pi$
$353$	−0.343146			−0.0182638			−0.00913190	+	0.999958i	$0.502907\pi$
$354$		0			0
$355$	−39.3137			−2.08655
$356$		0			0
$357$	10.8284	+	4.32957i	0.573101	+	0.229145i
$358$		0			0
$359$	25.6569			1.35412			0.677058	−	0.735929i	$-0.263254\pi$
$359$	25.6569			1.35412			0.677058	+	0.735929i	$0.263254\pi$
$360$		0			0
$361$	13.0000			0.684211
$362$		0			0
$363$			10.9008i			0.572146i
$364$		0			0
$365$	−38.6274			−2.02185
$366$		0			0
$367$			0.185709i			0.00969394i	0.999988	+	0.00484697i	$0.00154284\pi$
$367$			0.185709i			0.00969394i	−0.999988	+	0.00484697i	$0.998457\pi$
$368$		0			0
$369$		−	11.7206i		−	0.610150i
$370$		0			0
$371$			2.16478i			0.112390i
$372$		0			0
$373$	9.51472			0.492653			0.246327	−	0.969187i	$-0.420776\pi$
$373$	9.51472			0.492653			0.246327	+	0.969187i	$0.420776\pi$
$374$		0			0
$375$	35.3137			1.82359
$376$		0			0
$377$		−	32.6256i		−	1.68030i
$378$		0			0
$379$		−	37.4035i		−	1.92129i	−0.277779	−	0.960645i	$-0.589598\pi$
$379$		−	37.4035i		−	1.92129i	0.277779	−	0.960645i	$-0.410402\pi$
$380$		0			0
$381$		−	41.8100i		−	2.14199i
$382$		0			0
$383$	−11.3137			−0.578103			−0.289052	−	0.957313i	$-0.593340\pi$
$383$	−11.3137			−0.578103			−0.289052	+	0.957313i	$0.593340\pi$
$384$		0			0
$385$			10.4525i			0.532709i
$386$		0			0
$387$	36.9706			1.87932
$388$		0			0
$389$	19.4558			0.986450			0.493225	−	0.869902i	$-0.335818\pi$
$389$	19.4558			0.986450			0.493225	+	0.869902i	$0.335818\pi$
$390$		0			0
$391$	1.65685	−	4.14386i	0.0837907	−	0.209564i
$392$		0			0
$393$	17.1716			0.866191
$394$		0			0
$395$	−12.0000			−0.603786
$396$		0			0
$397$		−	4.96362i		−	0.249117i	−0.992212	−	0.124558i	$-0.960249\pi$
$397$		−	4.96362i		−	0.249117i	0.992212	−	0.124558i	$-0.0397514\pi$
$398$		0			0
$399$	−16.0000			−0.801002
$400$		0			0
$401$		−	1.26810i		−	0.0633260i	−0.999499	−	0.0316630i	$-0.989920\pi$
$401$		−	1.26810i		−	0.0633260i	0.999499	−	0.0316630i	$-0.0100803\pi$
$402$		0			0
$403$		−	40.9133i		−	2.03804i
$404$		0			0
$405$		−	21.5391i		−	1.07028i
$406$		0			0
$407$	9.65685			0.478672
$408$		0			0
$409$	−0.343146			−0.0169675			−0.00848373	−	0.999964i	$-0.502700\pi$
$409$	−0.343146			−0.0169675			−0.00848373	+	0.999964i	$0.502700\pi$
$410$		0			0
$411$		−	23.0698i		−	1.13795i
$412$		0			0
$413$		−	1.79337i		−	0.0882458i
$414$		0			0
$415$			6.12293i			0.300563i
$416$		0			0
$417$	−6.82843			−0.334390
$418$		0			0
$419$		−	33.0740i		−	1.61577i	−0.589341	−	0.807884i	$-0.700613\pi$
$419$		−	33.0740i		−	1.61577i	0.589341	−	0.807884i	$-0.299387\pi$
$420$		0			0
$421$	−6.48528			−0.316073			−0.158037	−	0.987433i	$-0.550516\pi$
$421$	−6.48528			−0.316073			−0.158037	+	0.987433i	$0.550516\pi$
$422$		0			0
$423$	−36.9706			−1.79757
$424$		0			0
$425$	−33.1421	−	13.2513i	−1.60763	−	0.642784i
$426$		0			0
$427$	8.68629			0.420359
$428$		0			0
$429$	−32.9706			−1.59183
$430$		0			0
$431$		−	8.99869i		−	0.433452i	−0.976232	−	0.216726i	$-0.930462\pi$
$431$		−	8.99869i		−	0.433452i	0.976232	−	0.216726i	$-0.0695378\pi$
$432$		0			0
$433$	0.142136			0.00683060			0.00341530	−	0.999994i	$-0.498913\pi$
$433$	0.142136			0.00683060			0.00341530	+	0.999994i	$0.498913\pi$
$434$		0			0
$435$			65.2512i			3.12856i
$436$		0			0
$437$			6.12293i			0.292900i
$438$		0			0
$439$		−	1.60766i		−	0.0767293i	−0.999264	−	0.0383646i	$-0.987785\pi$
$439$		−	1.60766i		−	0.0767293i	0.999264	−	0.0383646i	$-0.0122148\pi$
$440$		0			0
$441$	−22.3137			−1.06256
$442$		0			0
$443$	29.6569			1.40904			0.704520	−	0.709684i	$-0.251162\pi$
$443$	29.6569			1.40904			0.704520	+	0.709684i	$0.251162\pi$
$444$		0			0
$445$		−	23.9665i		−	1.13612i
$446$		0			0
$447$		−	34.7904i		−	1.64553i
$448$		0			0
$449$			32.6256i			1.53970i	0.638226	+	0.769849i	$0.279668\pi$
$449$			32.6256i			1.53970i	−0.638226	+	0.769849i	$0.720332\pi$
$450$		0			0
$451$	−8.00000			−0.376705
$452$		0			0
$453$			8.65914i			0.406842i
$454$		0			0
$455$	19.3137			0.905441
$456$		0			0
$457$	12.8284			0.600089			0.300044	−	0.953925i	$-0.402999\pi$
$457$	12.8284			0.600089			0.300044	+	0.953925i	$0.402999\pi$
$458$		0			0
$459$	3.31371	−	8.28772i	0.154671	−	0.386838i
$460$		0			0
$461$	1.31371			0.0611855			0.0305928	−	0.999532i	$-0.490261\pi$
$461$	1.31371			0.0611855			0.0305928	+	0.999532i	$0.490261\pi$
$462$		0			0
$463$	−11.3137			−0.525793			−0.262896	−	0.964824i	$-0.584678\pi$
$463$	−11.3137			−0.525793			−0.262896	+	0.964824i	$0.584678\pi$
$464$		0			0
$465$			81.8267i			3.79462i
$466$		0			0
$467$	20.2843			0.938644			0.469322	−	0.883027i	$-0.344498\pi$
$467$	20.2843			0.938644			0.469322	+	0.883027i	$0.344498\pi$
$468$		0			0
$469$		−	14.7821i		−	0.682573i
$470$		0			0
$471$		−	26.1313i		−	1.20406i
$472$		0			0
$473$		−	25.2346i		−	1.16029i
$474$		0			0
$475$	48.9706			2.24692
$476$		0			0
$477$	7.65685			0.350583
$478$		0			0
$479$			9.37011i			0.428131i	0.976819	+	0.214066i	$0.0686707\pi$
$479$			9.37011i			0.428131i	−0.976819	+	0.214066i	$0.931329\pi$
$480$		0			0
$481$		−	17.8435i		−	0.813595i
$482$		0			0
$483$		−	3.06147i		−	0.139302i
$484$		0			0
$485$	49.9411			2.26771
$486$		0			0
$487$		−	25.0489i		−	1.13507i	−0.823348	−	0.567536i	$-0.807897\pi$
$487$		−	25.0489i		−	1.13507i	0.823348	−	0.567536i	$-0.192103\pi$
$488$		0			0
$489$	−42.1421			−1.90573
$490$		0			0
$491$	8.97056			0.404836			0.202418	−	0.979299i	$-0.435120\pi$
$491$	8.97056			0.404836			0.202418	+	0.979299i	$0.435120\pi$
$492$		0			0
$493$	10.3431	−	25.8686i	0.465832	−	1.16506i
$494$		0			0
$495$	36.9706			1.66170
$496$		0			0
$497$	11.5147			0.516506
$498$		0			0
$499$			14.3337i			0.641666i	0.947136	+	0.320833i	$0.103963\pi$
$499$			14.3337i			0.641666i	−0.947136	+	0.320833i	$0.896037\pi$
$500$		0			0
$501$	57.4558			2.56694
$502$		0			0
$503$		−	6.68006i		−	0.297849i	−0.988849	−	0.148925i	$-0.952419\pi$
$503$		−	6.68006i		−	0.297849i	0.988849	−	0.148925i	$-0.0475812\pi$
$504$		0			0
$505$			47.4077i			2.10961i
$506$		0			0
$507$			26.9510i			1.19694i
$508$		0			0
$509$	33.3137			1.47660			0.738302	−	0.674470i	$-0.235628\pi$
$509$	33.3137			1.47660			0.738302	+	0.674470i	$0.235628\pi$
$510$		0			0
$511$	11.3137			0.500489
$512$		0			0
$513$			12.2459i			0.540668i
$514$		0			0
$515$			29.5641i			1.30275i
$516$		0			0
$517$			25.2346i			1.10982i
$518$		0			0
$519$	28.9706			1.27167
$520$		0			0
$521$		−	5.59767i		−	0.245238i	−0.992454	−	0.122619i	$-0.960871\pi$
$521$		−	5.59767i		−	0.245238i	0.992454	−	0.122619i	$-0.0391294\pi$
$522$		0			0
$523$	−5.65685			−0.247357			−0.123678	−	0.992322i	$-0.539469\pi$
$523$	−5.65685			−0.247357			−0.123678	+	0.992322i	$0.539469\pi$
$524$		0			0
$525$	−24.4853			−1.06863
$526$		0			0
$527$	12.9706	−	32.4399i	0.565007	−	1.41310i
$528$		0			0
$529$	21.8284			0.949062
$530$		0			0
$531$	−6.34315			−0.275269
$532$		0			0
$533$			14.7821i			0.640283i
$534$		0			0
$535$	−14.3431			−0.620108
$536$		0			0
$537$			65.2512i			2.81580i
$538$		0			0
$539$			15.2304i			0.656020i
$540$		0			0
$541$			3.17025i			0.136300i	0.997675	+	0.0681499i	$0.0217096\pi$
$541$			3.17025i			0.136300i	−0.997675	+	0.0681499i	$0.978290\pi$
$542$		0			0
$543$	40.2843			1.72876
$544$		0			0
$545$	−24.9706			−1.06962
$546$		0			0
$547$		−	6.57128i		−	0.280967i	−0.990083	−	0.140484i	$-0.955134\pi$
$547$		−	6.57128i		−	0.280967i	0.990083	−	0.140484i	$-0.0448658\pi$
$548$		0			0
$549$		−	30.7235i		−	1.31125i
$550$		0			0
$551$			38.2233i			1.62837i
$552$		0			0
$553$	3.51472			0.149461
$554$		0			0
$555$			35.6871i			1.51483i
$556$		0			0
$557$	−9.79899			−0.415197			−0.207598	−	0.978214i	$-0.566565\pi$
$557$	−9.79899			−0.415197			−0.207598	+	0.978214i	$0.566565\pi$
$558$		0			0
$559$	−46.6274			−1.97213
$560$		0			0
$561$	−26.1421	−	10.4525i	−1.10372	−	0.441305i
$562$		0			0
$563$	−43.5980			−1.83744			−0.918718	−	0.394914i	$-0.870774\pi$
$563$	−43.5980			−1.83744			−0.918718	+	0.394914i	$0.870774\pi$
$564$		0			0
$565$	−16.0000			−0.673125
$566$		0			0
$567$			6.30864i			0.264938i
$568$		0			0
$569$	29.3137			1.22889			0.614447	−	0.788958i	$-0.289379\pi$
$569$	29.3137			1.22889			0.614447	+	0.788958i	$0.289379\pi$
$570$		0			0
$571$		−	20.0852i		−	0.840541i	−0.907399	−	0.420271i	$-0.861935\pi$
$571$		−	20.0852i		−	0.840541i	0.907399	−	0.420271i	$-0.138065\pi$
$572$		0			0
$573$			33.8937i			1.41593i
$574$		0			0
$575$			9.37011i			0.390761i
$576$		0			0
$577$	−13.5147			−0.562625			−0.281313	−	0.959616i	$-0.590770\pi$
$577$	−13.5147			−0.562625			−0.281313	+	0.959616i	$0.590770\pi$
$578$		0			0
$579$	62.6274			2.60271
$580$		0			0
$581$		−	1.79337i		−	0.0744014i
$582$		0			0
$583$		−	5.22625i		−	0.216449i
$584$		0			0
$585$		−	68.3127i		−	2.82438i
$586$		0			0
$587$	5.65685			0.233483			0.116742	−	0.993162i	$-0.462755\pi$
$587$	5.65685			0.233483			0.116742	+	0.993162i	$0.462755\pi$
$588$		0			0
$589$			47.9329i			1.97504i
$590$		0			0
$591$	1.65685			0.0681539
$592$		0			0
$593$	−23.9411			−0.983144			−0.491572	−	0.870837i	$-0.663578\pi$
$593$	−23.9411			−0.983144			−0.491572	+	0.870837i	$0.663578\pi$
$594$		0			0
$595$	15.3137	+	6.12293i	0.627801	+	0.251016i
$596$		0			0
$597$	−13.1716			−0.539077
$598$		0			0
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$		−	17.3183i		−	0.706427i	−0.935543	−	0.353214i	$-0.885089\pi$
$601$		−	17.3183i		−	0.706427i	0.935543	−	0.353214i	$-0.114911\pi$
$602$		0			0
$603$	−52.2843			−2.12918
$604$		0			0
$605$			15.4161i			0.626755i
$606$		0			0
$607$			0.185709i			0.00753770i	0.999993	+	0.00376885i	$0.00119967\pi$
$607$			0.185709i			0.00753770i	−0.999993	+	0.00376885i	$0.998800\pi$
$608$		0			0
$609$		−	19.1116i		−	0.774443i
$610$		0			0
$611$	46.6274			1.88634
$612$		0			0
$613$	−29.3137			−1.18397			−0.591985	−	0.805949i	$-0.701655\pi$
$613$	−29.3137			−1.18397			−0.591985	+	0.805949i	$0.701655\pi$
$614$		0			0
$615$		−	29.5641i		−	1.19214i
$616$		0			0
$617$			2.53620i			0.102104i	0.998696	+	0.0510518i	$0.0162574\pi$
$617$			2.53620i			0.102104i	−0.998696	+	0.0510518i	$0.983743\pi$
$618$		0			0
$619$		−	15.2304i		−	0.612162i	−0.952005	−	0.306081i	$-0.900982\pi$
$619$		−	15.2304i		−	0.612162i	0.952005	−	0.306081i	$-0.0990178\pi$
$620$		0			0
$621$	−2.34315			−0.0940272
$622$		0			0
$623$			7.01962i			0.281235i
$624$		0			0
$625$	6.65685			0.266274
$626$		0			0
$627$	38.6274			1.54263
$628$		0			0
$629$	5.65685	−	14.1480i	0.225554	−	0.564119i
$630$		0			0
$631$	14.3431			0.570992			0.285496	−	0.958380i	$-0.407842\pi$
$631$	14.3431			0.570992			0.285496	+	0.958380i	$0.407842\pi$
$632$		0			0
$633$	39.7990			1.58187
$634$		0			0
$635$		−	59.1283i		−	2.34643i
$636$		0			0
$637$	28.1421			1.11503
$638$		0			0
$639$		−	40.7276i		−	1.61116i
$640$		0			0
$641$			25.7598i			1.01745i	0.860928	+	0.508726i	$0.169883\pi$
$641$			25.7598i			1.01745i	−0.860928	+	0.508726i	$0.830117\pi$
$642$		0			0
$643$		−	20.8281i		−	0.821379i	−0.911775	−	0.410690i	$-0.865288\pi$
$643$		−	20.8281i		−	0.821379i	0.911775	−	0.410690i	$-0.134712\pi$
$644$		0			0
$645$	93.2548			3.67191
$646$		0			0
$647$	−4.97056			−0.195413			−0.0977065	−	0.995215i	$-0.531151\pi$
$647$	−4.97056			−0.195413			−0.0977065	+	0.995215i	$0.531151\pi$
$648$		0			0
$649$			4.32957i			0.169950i
$650$		0			0
$651$		−	23.9665i		−	0.939320i
$652$		0			0
$653$		−	27.1367i		−	1.06194i	−0.847390	−	0.530971i	$-0.821827\pi$
$653$		−	27.1367i		−	1.06194i	0.847390	−	0.530971i	$-0.178173\pi$
$654$		0			0
$655$	24.2843			0.948865
$656$		0			0
$657$		−	40.0166i		−	1.56120i
$658$		0			0
$659$	−28.9706			−1.12853			−0.564266	−	0.825593i	$-0.690841\pi$
$659$	−28.9706			−1.12853			−0.564266	+	0.825593i	$0.690841\pi$
$660$		0			0
$661$	−18.0000			−0.700119			−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000			−0.700119			−0.350059	+	0.936727i	$0.613839\pi$
$662$		0			0
$663$	−19.3137	+	48.3044i	−0.750082	+	1.87599i
$664$		0			0
$665$	−22.6274			−0.877454
$666$		0			0
$667$	−7.31371			−0.283188
$668$		0			0
$669$		−	12.9887i		−	0.502172i
$670$		0			0
$671$	−20.9706			−0.809560
$672$		0			0
$673$		−	9.18440i		−	0.354033i	−0.984208	−	0.177016i	$-0.943355\pi$
$673$		−	9.18440i		−	0.354033i	0.984208	−	0.177016i	$-0.0566445\pi$
$674$		0			0
$675$			18.7402i			0.721312i
$676$		0			0
$677$		−	36.8464i		−	1.41612i	−0.706151	−	0.708061i	$-0.749570\pi$
$677$		−	36.8464i		−	1.41612i	0.706151	−	0.708061i	$-0.250430\pi$
$678$		0			0
$679$	−14.6274			−0.561349
$680$		0			0
$681$	−26.1421			−1.00177
$682$		0			0
$683$			42.6298i			1.63118i	0.578628	+	0.815591i	$0.303588\pi$
$683$			42.6298i			1.63118i	−0.578628	+	0.815591i	$0.696412\pi$
$684$		0			0
$685$		−	32.6256i		−	1.24656i
$686$		0			0
$687$		−	46.5110i		−	1.77451i
$688$		0			0
$689$	−9.65685			−0.367897
$690$		0			0
$691$		−	5.30318i		−	0.201742i	−0.994899	−	0.100871i	$-0.967837\pi$
$691$		−	5.30318i		−	0.201742i	0.994899	−	0.100871i	$-0.0321630\pi$
$692$		0			0
$693$	−10.8284			−0.411338
$694$		0			0
$695$	−9.65685			−0.366305
$696$		0			0
$697$	−4.68629	+	11.7206i	−0.177506	+	0.443950i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15.8579			−0.598943			−0.299472	−	0.954105i	$-0.596810\pi$
$701$	−15.8579			−0.598943			−0.299472	+	0.954105i	$0.596810\pi$
$702$		0			0
$703$			20.9050i			0.788447i
$704$		0			0
$705$	−93.2548			−3.51218
$706$		0			0
$707$		−	13.8854i		−	0.522214i
$708$		0			0
$709$			39.3826i			1.47904i	0.673132	+	0.739522i	$0.264949\pi$
$709$			39.3826i			1.47904i	−0.673132	+	0.739522i	$0.735051\pi$
$710$		0			0
$711$		−	12.4316i		−	0.466221i
$712$		0			0
$713$	−9.17157			−0.343478
$714$		0			0
$715$	−46.6274			−1.74377
$716$		0			0
$717$			37.4804i			1.39973i
$718$		0			0
$719$			27.2137i			1.01490i	0.861682	+	0.507449i	$0.169411\pi$
$719$			27.2137i			1.01490i	−0.861682	+	0.507449i	$0.830589\pi$
$720$		0			0
$721$		−	8.65914i		−	0.322483i
$722$		0			0
$723$	−41.9411			−1.55981
$724$		0			0
$725$			58.4942i			2.17242i
$726$		0			0
$727$	36.9706			1.37116			0.685581	−	0.727996i	$-0.259548\pi$
$727$	36.9706			1.37116			0.685581	+	0.727996i	$0.259548\pi$
$728$		0			0
$729$	39.2843			1.45497
$730$		0			0
$731$	−36.9706	−	14.7821i	−1.36741	−	0.546735i
$732$		0			0
$733$	23.9411			0.884286			0.442143	−	0.896945i	$-0.354218\pi$
$733$	23.9411			0.884286			0.442143	+	0.896945i	$0.354218\pi$
$734$		0			0
$735$	−56.2843			−2.07608
$736$		0			0
$737$			35.6871i			1.31455i
$738$		0			0
$739$	−29.6569			−1.09095			−0.545473	−	0.838129i	$-0.683650\pi$
$739$	−29.6569			−1.09095			−0.545473	+	0.838129i	$0.683650\pi$
$740$		0			0
$741$		−	71.3742i		−	2.62200i
$742$		0			0
$743$			22.8841i			0.839536i	0.907632	+	0.419768i	$0.137888\pi$
$743$			22.8841i			0.839536i	−0.907632	+	0.419768i	$0.862112\pi$
$744$		0			0
$745$		−	49.2011i		−	1.80259i
$746$		0			0
$747$	−6.34315			−0.232084
$748$		0			0
$749$	4.20101			0.153502
$750$		0			0
$751$			6.30864i			0.230206i	0.993354	+	0.115103i	$0.0367198\pi$
$751$			6.30864i			0.230206i	−0.993354	+	0.115103i	$0.963280\pi$
$752$		0			0
$753$		−	14.7821i		−	0.538689i
$754$		0			0
$755$			12.2459i			0.445673i
$756$		0			0
$757$	−48.4264			−1.76009			−0.880044	−	0.474892i	$-0.842487\pi$
$757$	−48.4264			−1.76009			−0.880044	+	0.474892i	$0.842487\pi$
$758$		0			0
$759$			7.39104i			0.268278i
$760$		0			0
$761$	−2.20101			−0.0797866			−0.0398933	−	0.999204i	$-0.512702\pi$
$761$	−2.20101			−0.0797866			−0.0398933	+	0.999204i	$0.512702\pi$
$762$		0			0
$763$	7.31371			0.264774
$764$		0			0
$765$	21.6569	−	54.1647i	0.783005	−	1.95833i
$766$		0			0
$767$	8.00000			0.288863
$768$		0			0
$769$	3.85786			0.139118			0.0695591	−	0.997578i	$-0.477841\pi$
$769$	3.85786			0.139118			0.0695591	+	0.997578i	$0.477841\pi$
$770$		0			0
$771$			48.3044i			1.73964i
$772$		0			0
$773$	20.8284			0.749146			0.374573	−	0.927197i	$-0.377789\pi$
$773$	20.8284			0.749146			0.374573	+	0.927197i	$0.377789\pi$
$774$		0			0
$775$			73.3532i			2.63493i
$776$		0			0
$777$		−	10.4525i		−	0.374981i
$778$		0			0
$779$		−	17.3183i		−	0.620492i
$780$		0			0
$781$	−27.7990			−0.994726
$782$		0			0
$783$	−14.6274			−0.522741
$784$		0			0
$785$		−	36.9552i		−	1.31899i
$786$		0			0
$787$			19.1886i			0.683998i	0.939700	+	0.341999i	$0.111104\pi$
$787$			19.1886i			0.683998i	−0.939700	+	0.341999i	$0.888896\pi$
$788$		0			0
$789$		−	41.8100i		−	1.48848i
$790$		0			0
$791$	4.68629			0.166625
$792$		0			0
$793$			38.7485i			1.37600i
$794$		0			0
$795$	19.3137			0.684987
$796$		0			0
$797$	−21.3137			−0.754970			−0.377485	−	0.926016i	$-0.623211\pi$
$797$	−21.3137			−0.754970			−0.377485	+	0.926016i	$0.623211\pi$
$798$		0			0
$799$	36.9706	+	14.7821i	1.30792	+	0.522952i
$800$		0			0
$801$	24.8284			0.877269
$802$		0			0
$803$	−27.3137			−0.963880
$804$		0			0
$805$		−	4.32957i		−	0.152597i
$806$		0			0
$807$	1.65685			0.0583240
$808$		0			0
$809$			45.6143i			1.60371i	0.597516	+	0.801857i	$0.296154\pi$
$809$			45.6143i			1.60371i	−0.597516	+	0.801857i	$0.703846\pi$
$810$		0			0
$811$		−	2.98454i		−	0.104802i	−0.998626	−	0.0524008i	$-0.983313\pi$
$811$		−	2.98454i		−	0.104802i	0.998626	−	0.0524008i	$-0.0166873\pi$
$812$		0			0
$813$			37.4804i			1.31450i
$814$		0			0
$815$	−59.5980			−2.08763
$816$		0			0
$817$	54.6274			1.91117
$818$		0			0
$819$			20.0083i			0.699147i
$820$		0			0
$821$		−	30.7235i		−	1.07226i	−0.844137	−	0.536128i	$-0.819886\pi$
$821$		−	30.7235i		−	1.07226i	0.844137	−	0.536128i	$-0.180114\pi$
$822$		0			0
$823$		−	45.4286i		−	1.58354i	−0.610818	−	0.791771i	$-0.709159\pi$
$823$		−	45.4286i		−	1.58354i	0.610818	−	0.791771i	$-0.290841\pi$
$824$		0			0
$825$	59.1127			2.05804
$826$		0			0
$827$		−	28.2191i		−	0.981275i	−0.871364	−	0.490637i	$-0.836764\pi$
$827$		−	28.2191i		−	0.981275i	0.871364	−	0.490637i	$-0.163236\pi$
$828$		0			0
$829$	26.6863			0.926853			0.463427	−	0.886135i	$-0.346620\pi$
$829$	26.6863			0.926853			0.463427	+	0.886135i	$0.346620\pi$
$830$		0			0
$831$	24.2843			0.842412
$832$		0			0
$833$	22.3137	+	8.92177i	0.773124	+	0.309121i
$834$		0			0
$835$	81.2548			2.81194
$836$		0			0
$837$	−18.3431			−0.634032
$838$		0			0
$839$			25.9456i			0.895740i	0.894099	+	0.447870i	$0.147817\pi$
$839$			25.9456i			0.895740i	−0.894099	+	0.447870i	$0.852183\pi$
$840$		0			0
$841$	−16.6569			−0.574374
$842$		0			0
$843$		−	39.1200i		−	1.34736i
$844$		0			0
$845$			38.1145i			1.31118i
$846$		0			0
$847$		−	4.51528i		−	0.155147i
$848$		0			0
$849$	−77.4558			−2.65828
$850$		0			0
$851$	−4.00000			−0.137118
$852$		0			0
$853$			3.69552i			0.126532i	0.997997	+	0.0632661i	$0.0201517\pi$
$853$			3.69552i			0.126532i	−0.997997	+	0.0632661i	$0.979848\pi$
$854$		0			0
$855$			80.0333i			2.73708i
$856$		0			0
$857$			36.9552i			1.26236i	0.775634	+	0.631182i	$0.217430\pi$
$857$			36.9552i			1.26236i	−0.775634	+	0.631182i	$0.782570\pi$
$858$		0			0
$859$	24.2843			0.828569			0.414284	−	0.910148i	$-0.364032\pi$
$859$	24.2843			0.828569			0.414284	+	0.910148i	$0.364032\pi$
$860$		0			0
$861$			8.65914i			0.295103i
$862$		0			0
$863$	−53.2548			−1.81282			−0.906408	−	0.422404i	$-0.861186\pi$
$863$	−53.2548			−1.81282			−0.906408	+	0.422404i	$0.861186\pi$
$864$		0			0
$865$	40.9706			1.39304
$866$		0			0
$867$	−30.6274	+	32.1773i	−1.04016	+	1.09280i
$868$		0			0
$869$	−8.48528			−0.287843
$870$		0			0
$871$	65.9411			2.23433
$872$		0			0
$873$			51.7373i			1.75104i
$874$		0			0
$875$	−14.6274			−0.494497
$876$		0			0
$877$			48.5670i			1.63999i	0.572370	+	0.819996i	$0.306024\pi$
$877$			48.5670i			1.63999i	−0.572370	+	0.819996i	$0.693976\pi$
$878$		0			0
$879$		−	47.0363i		−	1.58649i
$880$		0			0
$881$		−	10.4525i		−	0.352154i	−0.984376	−	0.176077i	$-0.943659\pi$
$881$		−	10.4525i		−	0.352154i	0.984376	−	0.176077i	$-0.0563407\pi$
$882$		0			0
$883$	−28.2843			−0.951842			−0.475921	−	0.879488i	$-0.657885\pi$
$883$	−28.2843			−0.951842			−0.475921	+	0.879488i	$0.657885\pi$
$884$		0			0
$885$	−16.0000			−0.537834
$886$		0			0
$887$			7.57675i			0.254402i	0.991877	+	0.127201i	$0.0405994\pi$
$887$			7.57675i			0.254402i	−0.991877	+	0.127201i	$0.959401\pi$
$888$		0			0
$889$			17.3183i			0.580836i
$890$		0			0
$891$		−	15.2304i		−	0.510238i
$892$		0			0
$893$	−54.6274			−1.82804
$894$		0			0
$895$			92.2792i			3.08455i
$896$		0			0
$897$	13.6569			0.455989
$898$		0			0
$899$	−57.2548			−1.90956
$900$		0			0
$901$	−7.65685	−	3.06147i	−0.255087	−	0.101992i
$902$		0			0
$903$	−27.3137			−0.908943
$904$		0			0
$905$	56.9706			1.89377
$906$		0			0
$907$		−	45.3198i		−	1.50482i	−0.658695	−	0.752410i	$-0.728891\pi$
$907$		−	45.3198i		−	1.50482i	0.658695	−	0.752410i	$-0.271109\pi$
$908$		0			0
$909$	−49.1127			−1.62897
$910$		0			0
$911$		−	43.6352i		−	1.44570i	−0.691005	−	0.722850i	$-0.742832\pi$
$911$		−	43.6352i		−	1.44570i	0.691005	−	0.722850i	$-0.257168\pi$
$912$		0			0
$913$			4.32957i			0.143288i
$914$		0			0
$915$		−	77.4971i		−	2.56197i
$916$		0			0
$917$	−7.11270			−0.234882
$918$		0			0
$919$	−33.6569			−1.11024			−0.555119	−	0.831771i	$-0.687327\pi$
$919$	−33.6569			−1.11024			−0.555119	+	0.831771i	$0.687327\pi$
$920$		0			0
$921$		−	33.8937i		−	1.11684i
$922$		0			0
$923$			51.3658i			1.69073i
$924$		0			0
$925$			31.9916i			1.05188i
$926$		0			0
$927$	−30.6274			−1.00594
$928$		0			0
$929$			34.4190i			1.12925i	0.825348	+	0.564625i	$0.190979\pi$
$929$			34.4190i			1.12925i	−0.825348	+	0.564625i	$0.809021\pi$
$930$		0			0
$931$	−32.9706			−1.08057
$932$		0			0
$933$	2.82843			0.0925985
$934$		0			0
$935$	−36.9706	−	14.7821i	−1.20907	−	0.483425i
$936$		0			0
$937$	42.9706			1.40379			0.701894	−	0.712282i	$-0.252338\pi$
$937$	42.9706			1.40379			0.701894	+	0.712282i	$0.252338\pi$
$938$		0			0
$939$	19.3137			0.630279
$940$		0			0
$941$		−	34.5278i		−	1.12557i	−0.826602	−	0.562786i	$-0.809729\pi$
$941$		−	34.5278i		−	1.12557i	0.826602	−	0.562786i	$-0.190271\pi$
$942$		0			0
$943$	3.31371			0.107909
$944$		0			0
$945$		−	8.65914i		−	0.281682i
$946$		0			0
$947$			36.5068i			1.18631i	0.805087	+	0.593156i	$0.202118\pi$
$947$			36.5068i			1.18631i	−0.805087	+	0.593156i	$0.797882\pi$
$948$		0			0
$949$			50.4692i			1.63830i
$950$		0			0
$951$	17.6569			0.572563
$952$		0			0
$953$	42.0833			1.36321			0.681605	−	0.731720i	$-0.261282\pi$
$953$	42.0833			1.36321			0.681605	+	0.731720i	$0.261282\pi$
$954$		0			0
$955$			47.9329i			1.55107i
$956$		0			0
$957$			46.1396i			1.49148i
$958$		0			0
$959$			9.55582i			0.308574i
$960$		0			0
$961$	−40.7990			−1.31610
$962$		0			0
$963$		−	14.8590i		−	0.478824i
$964$		0			0
$965$	88.5685			2.85112
$966$		0			0
$967$	28.9706			0.931630			0.465815	−	0.884882i	$-0.345761\pi$
$967$	28.9706			0.931630			0.465815	+	0.884882i	$0.345761\pi$
$968$		0			0
$969$	22.6274	−	56.5921i	0.726897	−	1.81800i
$970$		0			0
$971$	−14.3431			−0.460293			−0.230147	−	0.973156i	$-0.573921\pi$
$971$	−14.3431			−0.460293			−0.230147	+	0.973156i	$0.573921\pi$
$972$		0			0
$973$	2.82843			0.0906752
$974$		0			0
$975$		−	109.226i		−	3.49803i
$976$		0			0
$977$	5.31371			0.170001			0.0850003	−	0.996381i	$-0.472911\pi$
$977$	5.31371			0.170001			0.0850003	+	0.996381i	$0.472911\pi$
$978$		0			0
$979$		−	16.9469i		−	0.541624i
$980$		0			0
$981$		−	25.8686i		−	0.825922i
$982$		0			0
$983$		−	42.3671i		−	1.35130i	−0.737222	−	0.675651i	$-0.763863\pi$
$983$		−	42.3671i		−	1.35130i	0.737222	−	0.675651i	$-0.236137\pi$
$984$		0			0
$985$	2.34315			0.0746588
$986$		0			0
$987$	27.3137			0.869405
$988$		0			0
$989$			10.4525i			0.332370i
$990$		0			0
$991$		−	47.7472i		−	1.51674i	−0.651824	−	0.758371i	$-0.725996\pi$
$991$		−	47.7472i		−	1.51674i	0.651824	−	0.758371i	$-0.274004\pi$
$992$		0			0
$993$			12.9887i			0.412184i
$994$		0			0
$995$	−18.6274			−0.590529
$996$		0			0
$997$		−	15.4161i		−	0.488234i	−0.969746	−	0.244117i	$-0.921502\pi$
$997$		−	15.4161i		−	0.488234i	0.969746	−	0.244117i	$-0.0784980\pi$
$998$		0			0
$999$	−8.00000			−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	544.2.b.d.33.4	yes	4
3.2	odd	2		4896.2.c.l.577.1		4
4.3	odd	2		544.2.b.e.33.1	yes	4
8.3	odd	2		1088.2.b.m.577.4		4
8.5	even	2		1088.2.b.l.577.1		4
12.11	even	2		4896.2.c.m.577.1		4
17.4	even	4		9248.2.a.be.1.4		4
17.13	even	4		9248.2.a.be.1.1		4
17.16	even	2	inner	544.2.b.d.33.1	✓	4
51.50	odd	2		4896.2.c.l.577.4		4
68.47	odd	4		9248.2.a.bd.1.4		4
68.55	odd	4		9248.2.a.bd.1.1		4
68.67	odd	2		544.2.b.e.33.4	yes	4
136.67	odd	2		1088.2.b.m.577.1		4
136.101	even	2		1088.2.b.l.577.4		4
204.203	even	2		4896.2.c.m.577.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.b.d.33.1	✓	4	17.16	even	2	inner
544.2.b.d.33.4	yes	4	1.1	even	1	trivial
544.2.b.e.33.1	yes	4	4.3	odd	2
544.2.b.e.33.4	yes	4	68.67	odd	2
1088.2.b.l.577.1		4	8.5	even	2
1088.2.b.l.577.4		4	136.101	even	2
1088.2.b.m.577.1		4	136.67	odd	2
1088.2.b.m.577.4		4	8.3	odd	2
4896.2.c.l.577.1		4	3.2	odd	2
4896.2.c.l.577.4		4	51.50	odd	2
4896.2.c.m.577.1		4	12.11	even	2
4896.2.c.m.577.4		4	204.203	even	2
9248.2.a.bd.1.1		4	68.55	odd	4
9248.2.a.bd.1.4		4	68.47	odd	4
9248.2.a.be.1.1		4	17.13	even	4
9248.2.a.be.1.4		4	17.4	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 544 = 2^{5} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	544.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.61313i q^{3} +3.69552i q^{5} -1.08239i q^{7} -3.82843 q^{9} +2.61313i q^{11} +4.82843 q^{13} -9.65685 q^{15} +(3.82843 + 1.53073i) q^{17} -5.65685 q^{19} +2.82843 q^{21} -1.08239i q^{23} -8.65685 q^{25} -2.16478i q^{27} -6.75699i q^{29} -8.47343i q^{31} -6.82843 q^{33} +4.00000 q^{35} -3.69552i q^{37} +12.6173i q^{39} +3.06147i q^{41} -9.65685 q^{43} -14.1480i q^{45} +9.65685 q^{47} +5.82843 q^{49} +(-4.00000 + 10.0042i) q^{51} -2.00000 q^{53} -9.65685 q^{55} -14.7821i q^{57} +1.65685 q^{59} +8.02509i q^{61} +4.14386i q^{63} +17.8435i q^{65} +13.6569 q^{67} +2.82843 q^{69} +10.6382i q^{71} +10.4525i q^{73} -22.6215i q^{75} +2.82843 q^{77} +3.24718i q^{79} -5.82843 q^{81} +1.65685 q^{83} +(-5.65685 + 14.1480i) q^{85} +17.6569 q^{87} -6.48528 q^{89} -5.22625i q^{91} +22.1421 q^{93} -20.9050i q^{95} -13.5140i q^{97} -10.0042i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} + 8 q^{13} - 16 q^{15} + 4 q^{17} - 12 q^{25} - 16 q^{33} + 16 q^{35} - 16 q^{43} + 16 q^{47} + 12 q^{49} - 16 q^{51} - 8 q^{53} - 16 q^{55} - 16 q^{59} + 32 q^{67} - 12 q^{81} - 16 q^{83}+ \cdots + 32 q^{93}+O(q^{100}) \) 4 * q - 4 * q^9 + 8 * q^13 - 16 * q^15 + 4 * q^17 - 12 * q^25 - 16 * q^33 + 16 * q^35 - 16 * q^43 + 16 * q^47 + 12 * q^49 - 16 * q^51 - 8 * q^53 - 16 * q^55 - 16 * q^59 + 32 * q^67 - 12 * q^81 - 16 * q^83 + 48 * q^87 + 8 * q^89 + 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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