LMFDB - Embedded newform 528.6.o.b.175.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(175,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("528.175");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.o (of order $2$, degree $1$, 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:	$84.6826568613$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			175.8
Character	$\chi$	$=$	528.175
Dual form			528.6.o.b.175.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +48.1562 q^{5} +32.4843 q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} +48.1562 q^{5} +32.4843 q^{7} -81.0000 q^{9} +(-397.087 + 58.0781i) q^{11} -711.675i q^{13} +433.406i q^{15} +528.373i q^{17} +1617.57 q^{19} +292.359i q^{21} +2485.53i q^{23} -805.982 q^{25} -729.000i q^{27} -3537.99i q^{29} -9754.04i q^{31} +(-522.703 - 3573.78i) q^{33} +1564.32 q^{35} +13944.4 q^{37} +6405.08 q^{39} -11131.2i q^{41} +9935.97 q^{43} -3900.65 q^{45} +12926.4i q^{47} -15751.8 q^{49} -4755.36 q^{51} +9903.65 q^{53} +(-19122.2 + 2796.82i) q^{55} +14558.1i q^{57} +27977.6i q^{59} -53686.7i q^{61} -2631.23 q^{63} -34271.6i q^{65} +21925.2i q^{67} -22369.8 q^{69} -73681.3i q^{71} +54644.3i q^{73} -7253.84i q^{75} +(-12899.1 + 1886.63i) q^{77} +67869.0 q^{79} +6561.00 q^{81} -104286. q^{83} +25444.4i q^{85} +31841.9 q^{87} +59830.5 q^{89} -23118.3i q^{91} +87786.3 q^{93} +77896.0 q^{95} -98341.7 q^{97} +(32164.0 - 4704.33i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) $ 40 * q + 88 * q^5 - 3240 * q^9	$ 40 q + 88 q^{5} - 3240 q^{9} + 5464 q^{25} + 6840 q^{33} + 4576 q^{37} - 7128 q^{45} + 58760 q^{49} - 165336 q^{53} + 16632 q^{69} + 28136 q^{77} + 262440 q^{81} - 304608 q^{89} + 102672 q^{93} + 15664 q^{97}+O(q^{100}) $ 40 * q + 88 * q^5 - 3240 * q^9 + 5464 * q^25 + 6840 * q^33 + 4576 * q^37 - 7128 * q^45 + 58760 * q^49 - 165336 * q^53 + 16632 * q^69 + 28136 * q^77 + 262440 * q^81 - 304608 * q^89 + 102672 * q^93 + 15664 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$.

$n$	$133$	$145$	$353$	$463$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$			9.00000i			0.577350i
$3$			9.00000i			0.577350i
$4$		0			0
$5$	48.1562			0.861444			0.430722	−	0.902485i	$-0.358259\pi$
$5$	48.1562			0.861444			0.430722	+	0.902485i	$0.358259\pi$
$6$		0			0
$7$	32.4843			0.250570			0.125285	−	0.992121i	$-0.460016\pi$
$7$	32.4843			0.250570			0.125285	+	0.992121i	$0.460016\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	−397.087	+	58.0781i	−0.989473	+	0.144721i
$11$	−397.087	+	58.0781i	−0.989473	+	0.144721i
$12$		0			0
$13$		−	711.675i		−	1.16795i	−0.811772	−	0.583974i	$-0.801497\pi$
$13$		−	711.675i		−	1.16795i	0.811772	−	0.583974i	$-0.198503\pi$
$14$		0			0
$15$			433.406i			0.497355i
$16$		0			0
$17$			528.373i			0.443423i	0.975112	+	0.221712i	$0.0711643\pi$
$17$			528.373i			0.443423i	−0.975112	+	0.221712i	$0.928836\pi$
$18$		0			0
$19$	1617.57			1.02797			0.513984	−	0.857800i	$-0.328169\pi$
$19$	1617.57			1.02797			0.513984	+	0.857800i	$0.328169\pi$
$20$		0			0
$21$			292.359i			0.144666i
$22$		0			0
$23$			2485.53i			0.979715i	0.871803	+	0.489858i	$0.162951\pi$
$23$			2485.53i			0.979715i	−0.871803	+	0.489858i	$0.837049\pi$
$24$		0			0
$25$	−805.982			−0.257914
$26$		0			0
$27$		−	729.000i		−	0.192450i
$28$		0			0
$29$		−	3537.99i		−	0.781199i	−0.920561	−	0.390599i	$-0.872268\pi$
$29$		−	3537.99i		−	0.781199i	0.920561	−	0.390599i	$-0.127732\pi$
$30$		0			0
$31$		−	9754.04i		−	1.82297i	−0.411330	−	0.911486i	$-0.634936\pi$
$31$		−	9754.04i		−	1.82297i	0.411330	−	0.911486i	$-0.365064\pi$
$32$		0			0
$33$	−522.703	−	3573.78i	−0.0835546	−	0.571272i
$34$		0			0
$35$	1564.32			0.215852
$36$		0			0
$37$	13944.4			1.67454			0.837272	−	0.546787i	$-0.184149\pi$
$37$	13944.4			1.67454			0.837272	+	0.546787i	$0.184149\pi$
$38$		0			0
$39$	6405.08			0.674315
$40$		0			0
$41$		−	11131.2i		−	1.03415i	−0.855941	−	0.517074i	$-0.827021\pi$
$41$		−	11131.2i		−	1.03415i	0.855941	−	0.517074i	$-0.172979\pi$
$42$		0			0
$43$	9935.97			0.819481			0.409741	−	0.912202i	$-0.365619\pi$
$43$	9935.97			0.819481			0.409741	+	0.912202i	$0.365619\pi$
$44$		0			0
$45$	−3900.65			−0.287148
$46$		0			0
$47$			12926.4i			0.853559i	0.904356	+	0.426779i	$0.140352\pi$
$47$			12926.4i			0.853559i	−0.904356	+	0.426779i	$0.859648\pi$
$48$		0			0
$49$	−15751.8			−0.937215
$50$		0			0
$51$	−4755.36			−0.256010
$52$		0			0
$53$	9903.65			0.484290			0.242145	−	0.970240i	$-0.422149\pi$
$53$	9903.65			0.484290			0.242145	+	0.970240i	$0.422149\pi$
$54$		0			0
$55$	−19122.2	+	2796.82i	−0.852375	+	0.124669i
$56$		0			0
$57$			14558.1i			0.593497i
$58$		0			0
$59$			27977.6i			1.04636i	0.852223	+	0.523179i	$0.175254\pi$
$59$			27977.6i			1.04636i	−0.852223	+	0.523179i	$0.824746\pi$
$60$		0			0
$61$		−	53686.7i		−	1.84732i	−0.383213	−	0.923660i	$-0.625182\pi$
$61$		−	53686.7i		−	1.84732i	0.383213	−	0.923660i	$-0.374818\pi$
$62$		0			0
$63$	−2631.23			−0.0835232
$64$		0			0
$65$		−	34271.6i		−	1.00612i
$66$		0			0
$67$			21925.2i			0.596700i	0.954457	+	0.298350i	$0.0964363\pi$
$67$			21925.2i			0.596700i	−0.954457	+	0.298350i	$0.903564\pi$
$68$		0			0
$69$	−22369.8			−0.565639
$70$		0			0
$71$		−	73681.3i		−	1.73465i	−0.497744	−	0.867324i	$-0.665838\pi$
$71$		−	73681.3i		−	1.73465i	0.497744	−	0.867324i	$-0.334162\pi$
$72$		0			0
$73$			54644.3i			1.20016i	0.799942	+	0.600078i	$0.204864\pi$
$73$			54644.3i			1.20016i	−0.799942	+	0.600078i	$0.795136\pi$
$74$		0			0
$75$		−	7253.84i		−	0.148907i
$76$		0			0
$77$	−12899.1	+	1886.63i	−0.247932	+	0.0362626i
$78$		0			0
$79$	67869.0			1.22350			0.611750	−	0.791051i	$-0.290466\pi$
$79$	67869.0			1.22350			0.611750	+	0.791051i	$0.290466\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$	−104286.			−1.66162			−0.830810	−	0.556556i	$-0.812123\pi$
$83$	−104286.			−1.66162			−0.830810	+	0.556556i	$0.812123\pi$
$84$		0			0
$85$			25444.4i			0.381984i
$86$		0			0
$87$	31841.9			0.451025
$88$		0			0
$89$	59830.5			0.800659			0.400329	−	0.916371i	$-0.368896\pi$
$89$	59830.5			0.800659			0.400329	+	0.916371i	$0.368896\pi$
$90$		0			0
$91$		−	23118.3i		−	0.292652i
$92$		0			0
$93$	87786.3			1.05249
$94$		0			0
$95$	77896.0			0.885536
$96$		0			0
$97$	−98341.7			−1.06123			−0.530614	−	0.847614i	$-0.678038\pi$
$97$	−98341.7			−1.06123			−0.530614	+	0.847614i	$0.678038\pi$
$98$		0			0
$99$	32164.0	−	4704.33i	0.329824	−	0.0482402i
$100$		0			0
$101$		−	149509.i		−	1.45836i	−0.684324	−	0.729178i	$-0.739903\pi$
$101$		−	149509.i		−	1.45836i	0.684324	−	0.729178i	$-0.260097\pi$
$102$		0			0
$103$		−	25340.5i		−	0.235354i	−0.993052	−	0.117677i	$-0.962455\pi$
$103$		−	25340.5i		−	0.235354i	0.993052	−	0.117677i	$-0.0375448\pi$
$104$		0			0
$105$			14078.9i			0.124622i
$106$		0			0
$107$	142907.			1.20668			0.603341	−	0.797483i	$-0.293836\pi$
$107$	142907.			1.20668			0.603341	+	0.797483i	$0.293836\pi$
$108$		0			0
$109$		−	76546.4i		−	0.617104i	−0.951207	−	0.308552i	$-0.900156\pi$
$109$		−	76546.4i		−	0.617104i	0.951207	−	0.308552i	$-0.0998444\pi$
$110$		0			0
$111$			125500.i			0.966798i
$112$		0			0
$113$	74314.3			0.547490			0.273745	−	0.961802i	$-0.411738\pi$
$113$	74314.3			0.547490			0.273745	+	0.961802i	$0.411738\pi$
$114$		0			0
$115$			119694.i			0.843970i
$116$		0			0
$117$			57645.7i			0.389316i
$118$		0			0
$119$			17163.8i			0.111108i
$120$		0			0
$121$	154305.	−	46124.1i	0.958112	−	0.286394i
$122$		0			0
$123$	100181.			0.597065
$124$		0			0
$125$	−189301.			−1.08362
$126$		0			0
$127$	284222.			1.56368			0.781841	−	0.623477i	$-0.214281\pi$
$127$	284222.			1.56368			0.781841	+	0.623477i	$0.214281\pi$
$128$		0			0
$129$			89423.7i			0.473128i
$130$		0			0
$131$	348659.			1.77510			0.887550	−	0.460711i	$-0.152406\pi$
$131$	348659.			1.77510			0.887550	+	0.460711i	$0.152406\pi$
$132$		0			0
$133$	52545.6			0.257577
$134$		0			0
$135$		−	35105.9i		−	0.165785i
$136$		0			0
$137$	−82327.1			−0.374750			−0.187375	−	0.982288i	$-0.559998\pi$
$137$	−82327.1			−0.374750			−0.187375	+	0.982288i	$0.559998\pi$
$138$		0			0
$139$	193464.			0.849306			0.424653	−	0.905356i	$-0.360396\pi$
$139$	193464.			0.849306			0.424653	+	0.905356i	$0.360396\pi$
$140$		0			0
$141$	−116338.			−0.492802
$142$		0			0
$143$	41332.7	+	282597.i	0.169026	+	1.15565i
$144$		0			0
$145$		−	170376.i		−	0.672959i
$146$		0			0
$147$		−	141766.i		−	0.541101i
$148$		0			0
$149$		−	416084.i		−	1.53538i	−0.640823	−	0.767689i	$-0.721407\pi$
$149$		−	416084.i		−	1.53538i	0.640823	−	0.767689i	$-0.278593\pi$
$150$		0			0
$151$	298962.			1.06702			0.533511	−	0.845793i	$-0.320872\pi$
$151$	298962.			1.06702			0.533511	+	0.845793i	$0.320872\pi$
$152$		0			0
$153$		−	42798.2i		−	0.147808i
$154$		0			0
$155$		−	469717.i		−	1.57039i
$156$		0			0
$157$	−91170.3			−0.295192			−0.147596	−	0.989048i	$-0.547154\pi$
$157$	−91170.3			−0.295192			−0.147596	+	0.989048i	$0.547154\pi$
$158$		0			0
$159$			89132.8i			0.279605i
$160$		0			0
$161$			80740.8i			0.245487i
$162$		0			0
$163$			127405.i			0.375593i	0.982208	+	0.187796i	$0.0601345\pi$
$163$			127405.i			0.375593i	−0.982208	+	0.187796i	$0.939865\pi$
$164$		0			0
$165$	−25171.4	−	172100.i	−0.0719776	−	0.492119i
$166$		0			0
$167$	413434.			1.14714			0.573568	−	0.819158i	$-0.305559\pi$
$167$	413434.			1.14714			0.573568	+	0.819158i	$0.305559\pi$
$168$		0			0
$169$	−135188.			−0.364102
$170$		0			0
$171$	−131023.			−0.342656
$172$		0			0
$173$			105055.i			0.266870i	0.991058	+	0.133435i	$0.0426008\pi$
$173$			105055.i			0.266870i	−0.991058	+	0.133435i	$0.957399\pi$
$174$		0			0
$175$	−26181.8			−0.0646255
$176$		0			0
$177$	−251798.			−0.604115
$178$		0			0
$179$			578134.i			1.34864i	0.738439	+	0.674320i	$0.235563\pi$
$179$			578134.i			1.34864i	−0.738439	+	0.674320i	$0.764437\pi$
$180$		0			0
$181$	802152.			1.81995			0.909976	−	0.414660i	$-0.136099\pi$
$181$	802152.			1.81995			0.909976	+	0.414660i	$0.136099\pi$
$182$		0			0
$183$	483180.			1.06655
$184$		0			0
$185$	671510.			1.44253
$186$		0			0
$187$	−30686.9	−	209810.i	−0.0641725	−	0.438755i
$188$		0			0
$189$		−	23681.1i		−	0.0482221i
$190$		0			0
$191$			177279.i			0.351621i	0.984424	+	0.175810i	$0.0562545\pi$
$191$			177279.i			0.351621i	−0.984424	+	0.175810i	$0.943745\pi$
$192$		0			0
$193$		−	663345.i		−	1.28188i	−0.767593	−	0.640938i	$-0.778546\pi$
$193$		−	663345.i		−	1.28188i	0.767593	−	0.640938i	$-0.221454\pi$
$194$		0			0
$195$	308444.			0.580885
$196$		0			0
$197$			465544.i			0.854664i	0.904095	+	0.427332i	$0.140546\pi$
$197$			465544.i			0.854664i	−0.904095	+	0.427332i	$0.859454\pi$
$198$		0			0
$199$		−	44657.8i		−	0.0799401i	−0.999201	−	0.0399701i	$-0.987274\pi$
$199$		−	44657.8i		−	0.0799401i	0.999201	−	0.0399701i	$-0.0127263\pi$
$200$		0			0
$201$	−197327.			−0.344505
$202$		0			0
$203$		−	114929.i		−	0.195745i
$204$		0			0
$205$		−	536036.i		−	0.890860i
$206$		0			0
$207$		−	201328.i		−	0.326572i
$208$		0			0
$209$	−642316.	+	93945.4i	−1.01715	+	0.148768i
$210$		0			0
$211$	−438001.			−0.677281			−0.338640	−	0.940916i	$-0.609967\pi$
$211$	−438001.			−0.677281			−0.338640	+	0.940916i	$0.609967\pi$
$212$		0			0
$213$	663132.			1.00150
$214$		0			0
$215$	478478.			0.705937
$216$		0			0
$217$		−	316853.i		−	0.456782i
$218$		0			0
$219$	−491799.			−0.692910
$220$		0			0
$221$	376030.			0.517895
$222$		0			0
$223$			136109.i			0.183284i	0.995792	+	0.0916418i	$0.0292115\pi$
$223$			136109.i			0.183284i	−0.995792	+	0.0916418i	$0.970789\pi$
$224$		0			0
$225$	65284.5			0.0859714
$226$		0			0
$227$	530350.			0.683121			0.341561	−	0.939860i	$-0.389045\pi$
$227$	530350.			0.683121			0.341561	+	0.939860i	$0.389045\pi$
$228$		0			0
$229$	−1.31920e6			−1.66234			−0.831171	−	0.556017i	$-0.812329\pi$
$229$	−1.31920e6			−1.66234			−0.831171	+	0.556017i	$0.812329\pi$
$230$		0			0
$231$	−16979.6	−	116092.i	−0.0209362	−	0.143143i
$232$		0			0
$233$		−	177642.i		−	0.214366i	−0.994239	−	0.107183i	$-0.965817\pi$
$233$		−	177642.i		−	0.214366i	0.994239	−	0.107183i	$-0.0341832\pi$
$234$		0			0
$235$			622487.i			0.735293i
$236$		0			0
$237$			610821.i			0.706388i
$238$		0			0
$239$	−436163.			−0.493917			−0.246958	−	0.969026i	$-0.579431\pi$
$239$	−436163.			−0.493917			−0.246958	+	0.969026i	$0.579431\pi$
$240$		0			0
$241$		−	982694.i		−	1.08987i	−0.838477	−	0.544936i	$-0.816554\pi$
$241$		−	982694.i		−	1.08987i	0.838477	−	0.544936i	$-0.183446\pi$
$242$		0			0
$243$			59049.0i			0.0641500i
$244$		0			0
$245$	−758545.			−0.807358
$246$		0			0
$247$		−	1.15118e6i		−	1.20061i
$248$		0			0
$249$		−	938576.i		−	0.959337i
$250$		0			0
$251$		−	110185.i		−	0.110392i	−0.998476	−	0.0551962i	$-0.982422\pi$
$251$		−	110185.i		−	0.110392i	0.998476	−	0.0551962i	$-0.0175784\pi$
$252$		0			0
$253$	−144355.	−	986972.i	−0.141785	−	0.969401i
$254$		0			0
$255$	−229000.			−0.220539
$256$		0			0
$257$	−46235.0			−0.0436654			−0.0218327	−	0.999762i	$-0.506950\pi$
$257$	−46235.0			−0.0436654			−0.0218327	+	0.999762i	$0.506950\pi$
$258$		0			0
$259$	452975.			0.419590
$260$		0			0
$261$			286577.i			0.260400i
$262$		0			0
$263$	691192.			0.616182			0.308091	−	0.951357i	$-0.400310\pi$
$263$	691192.			0.616182			0.308091	+	0.951357i	$0.400310\pi$
$264$		0			0
$265$	476922.			0.417189
$266$		0			0
$267$			538474.i			0.462261i
$268$		0			0
$269$	1.24322e6			1.04753			0.523767	−	0.851862i	$-0.324526\pi$
$269$	1.24322e6			1.04753			0.523767	+	0.851862i	$0.324526\pi$
$270$		0			0
$271$	80838.9			0.0668648			0.0334324	−	0.999441i	$-0.489356\pi$
$271$	80838.9			0.0668648			0.0334324	+	0.999441i	$0.489356\pi$
$272$		0			0
$273$	208064.			0.168963
$274$		0			0
$275$	320045.	−	46809.9i	0.255199	−	0.0373255i
$276$		0			0
$277$		−	553359.i		−	0.433319i	−0.976247	−	0.216659i	$-0.930484\pi$
$277$		−	553359.i		−	0.433319i	0.976247	−	0.216659i	$-0.0695161\pi$
$278$		0			0
$279$			790077.i			0.607658i
$280$		0			0
$281$			1.66967e6i			1.26143i	0.776013	+	0.630717i	$0.217239\pi$
$281$			1.66967e6i			1.26143i	−0.776013	+	0.630717i	$0.782761\pi$
$282$		0			0
$283$	−2.04349e6			−1.51672			−0.758361	−	0.651835i	$-0.773999\pi$
$283$	−2.04349e6			−1.51672			−0.758361	+	0.651835i	$0.773999\pi$
$284$		0			0
$285$			701064.i			0.511264i
$286$		0			0
$287$		−	361589.i		−	0.259126i
$288$		0			0
$289$	1.14068e6			0.803376
$290$		0			0
$291$		−	885075.i		−	0.612700i
$292$		0			0
$293$			2.01875e6i			1.37377i	0.726766	+	0.686885i	$0.241023\pi$
$293$			2.01875e6i			1.37377i	−0.726766	+	0.686885i	$0.758977\pi$
$294$		0			0
$295$			1.34729e6i			0.901379i
$296$		0			0
$297$	42338.9	+	289476.i	0.0278515	+	0.190424i
$298$		0			0
$299$	1.76889e6			1.14426
$300$		0			0
$301$	322763.			0.205337
$302$		0			0
$303$	1.34558e6			0.841982
$304$		0			0
$305$		−	2.58535e6i		−	1.59136i
$306$		0			0
$307$	−1.98811e6			−1.20391			−0.601955	−	0.798530i	$-0.705611\pi$
$307$	−1.98811e6			−1.20391			−0.601955	+	0.798530i	$0.705611\pi$
$308$		0			0
$309$	228065.			0.135882
$310$		0			0
$311$		−	1.54009e6i		−	0.902912i	−0.892293	−	0.451456i	$-0.850905\pi$
$311$		−	1.54009e6i		−	0.902912i	0.892293	−	0.451456i	$-0.149095\pi$
$312$		0			0
$313$	583984.			0.336930			0.168465	−	0.985708i	$-0.446119\pi$
$313$	583984.			0.336930			0.168465	+	0.985708i	$0.446119\pi$
$314$		0			0
$315$	−126710.			−0.0719506
$316$		0			0
$317$	−3.30701e6			−1.84836			−0.924181	−	0.381955i	$-0.875251\pi$
$317$	−3.30701e6			−1.84836			−0.924181	+	0.381955i	$0.875251\pi$
$318$		0			0
$319$	205480.	+	1.40489e6i	0.113056	+	0.772975i
$320$		0			0
$321$			1.28616e6i			0.696679i
$322$		0			0
$323$			854681.i			0.455824i
$324$		0			0
$325$			573597.i			0.301230i
$326$		0			0
$327$	688917.			0.356285
$328$		0			0
$329$			419905.i			0.213876i
$330$		0			0
$331$		−	2.06333e6i		−	1.03514i	−0.855641	−	0.517570i	$-0.826837\pi$
$331$		−	2.06333e6i		−	1.03514i	0.855641	−	0.517570i	$-0.173163\pi$
$332$		0			0
$333$	−1.12950e6			−0.558181
$334$		0			0
$335$			1.05583e6i			0.514024i
$336$		0			0
$337$		−	3.45335e6i		−	1.65640i	−0.560430	−	0.828202i	$-0.689364\pi$
$337$		−	3.45335e6i		−	1.65640i	0.560430	−	0.828202i	$-0.310636\pi$
$338$		0			0
$339$			668828.i			0.316093i
$340$		0			0
$341$	566496.	+	3.87320e6i	0.263822	+	1.80378i
$342$		0			0
$343$	−1.05765e6			−0.485407
$344$		0			0
$345$	−1.07724e6			−0.487266
$346$		0			0
$347$	−3.67437e6			−1.63817			−0.819085	−	0.573672i	$-0.805519\pi$
$347$	−3.67437e6			−1.63817			−0.819085	+	0.573672i	$0.805519\pi$
$348$		0			0
$349$			1.63575e6i			0.718874i	0.933169	+	0.359437i	$0.117031\pi$
$349$			1.63575e6i			0.718874i	−0.933169	+	0.359437i	$0.882969\pi$
$350$		0			0
$351$	−518811.			−0.224772
$352$		0			0
$353$	−412235.			−0.176079			−0.0880396	−	0.996117i	$-0.528060\pi$
$353$	−412235.			−0.176079			−0.0880396	+	0.996117i	$0.528060\pi$
$354$		0			0
$355$		−	3.54821e6i		−	1.49430i
$356$		0			0
$357$	−154474.			−0.0641484
$358$		0			0
$359$	−3.60192e6			−1.47502			−0.737511	−	0.675336i	$-0.763999\pi$
$359$	−3.60192e6			−1.47502			−0.737511	+	0.675336i	$0.763999\pi$
$360$		0			0
$361$	140435.			0.0567163
$362$		0			0
$363$	415117.	+	1.38874e6i	0.165350	+	0.553166i
$364$		0			0
$365$			2.63146e6i			1.03387i
$366$		0			0
$367$		−	719389.i		−	0.278804i	−0.990236	−	0.139402i	$-0.955482\pi$
$367$		−	719389.i		−	0.278804i	0.990236	−	0.139402i	$-0.0445180\pi$
$368$		0			0
$369$			901627.i			0.344716i
$370$		0			0
$371$	321713.			0.121348
$372$		0			0
$373$			935613.i			0.348196i	0.984728	+	0.174098i	$0.0557010\pi$
$373$			935613.i			0.348196i	−0.984728	+	0.174098i	$0.944299\pi$
$374$		0			0
$375$		−	1.70371e6i		−	0.625630i
$376$		0			0
$377$	−2.51790e6			−0.912399
$378$		0			0
$379$			1.14559e6i			0.409668i	0.978797	+	0.204834i	$0.0656655\pi$
$379$			1.14559e6i			0.409668i	−0.978797	+	0.204834i	$0.934335\pi$
$380$		0			0
$381$			2.55800e6i			0.902793i
$382$		0			0
$383$		−	537149.i		−	0.187111i	−0.995614	−	0.0935553i	$-0.970177\pi$
$383$		−	537149.i		−	0.187111i	0.995614	−	0.0935553i	$-0.0298232\pi$
$384$		0			0
$385$	−621171.	+	90852.8i	−0.213579	+	0.0312382i
$386$		0			0
$387$	−804813.			−0.273160
$388$		0			0
$389$	3.59780e6			1.20549			0.602745	−	0.797934i	$-0.294074\pi$
$389$	3.59780e6			1.20549			0.602745	+	0.797934i	$0.294074\pi$
$390$		0			0
$391$	−1.31329e6			−0.434428
$392$		0			0
$393$			3.13793e6i			1.02485i
$394$		0			0
$395$	3.26831e6			1.05398
$396$		0			0
$397$	318165.			0.101315			0.0506577	−	0.998716i	$-0.483868\pi$
$397$	318165.			0.101315			0.0506577	+	0.998716i	$0.483868\pi$
$398$		0			0
$399$			472911.i			0.148712i
$400$		0			0
$401$	5.67129e6			1.76125			0.880625	−	0.473814i	$-0.157123\pi$
$401$	5.67129e6			1.76125			0.880625	+	0.473814i	$0.157123\pi$
$402$		0			0
$403$	−6.94170e6			−2.12914
$404$		0			0
$405$	315953.			0.0957160
$406$		0			0
$407$	−5.53715e6	+	809866.i	−1.65691	+	0.242341i
$408$		0			0
$409$		−	2.08152e6i		−	0.615280i	−0.951503	−	0.307640i	$-0.900461\pi$
$409$		−	2.08152e6i		−	0.615280i	0.951503	−	0.307640i	$-0.0995392\pi$
$410$		0			0
$411$		−	740943.i		−	0.216362i
$412$		0			0
$413$			908832.i			0.262185i
$414$		0			0
$415$	−5.02203e6			−1.43139
$416$		0			0
$417$			1.74118e6i			0.490347i
$418$		0			0
$419$			1.44847e6i			0.403064i	0.979482	+	0.201532i	$0.0645920\pi$
$419$			1.44847e6i			0.403064i	−0.979482	+	0.201532i	$0.935408\pi$
$420$		0			0
$421$	−3.36497e6			−0.925285			−0.462643	−	0.886545i	$-0.653099\pi$
$421$	−3.36497e6			−0.925285			−0.462643	+	0.886545i	$0.653099\pi$
$422$		0			0
$423$		−	1.04704e6i		−	0.284520i
$424$		0			0
$425$		−	425859.i		−	0.114365i
$426$		0			0
$427$		−	1.74397e6i		−	0.462882i
$428$		0			0
$429$	−2.54337e6	+	371995.i	−0.667216	+	0.0975874i
$430$		0			0
$431$	−1.68984e6			−0.438180			−0.219090	−	0.975705i	$-0.570309\pi$
$431$	−1.68984e6			−0.438180			−0.219090	+	0.975705i	$0.570309\pi$
$432$		0			0
$433$	−903413.			−0.231562			−0.115781	−	0.993275i	$-0.536937\pi$
$433$	−903413.			−0.231562			−0.115781	+	0.993275i	$0.536937\pi$
$434$		0			0
$435$	1.53338e6			0.388533
$436$		0			0
$437$			4.02052e6i			1.00711i
$438$		0			0
$439$	−4.57404e6			−1.13276			−0.566381	−	0.824143i	$-0.691657\pi$
$439$	−4.57404e6			−1.13276			−0.566381	+	0.824143i	$0.691657\pi$
$440$		0			0
$441$	1.27589e6			0.312405
$442$		0			0
$443$			2.93208e6i			0.709851i	0.934895	+	0.354925i	$0.115494\pi$
$443$			2.93208e6i			0.709851i	−0.934895	+	0.354925i	$0.884506\pi$
$444$		0			0
$445$	2.88121e6			0.689723
$446$		0			0
$447$	3.74475e6			0.886451
$448$		0			0
$449$	−5.59729e6			−1.31027			−0.655137	−	0.755510i	$-0.727389\pi$
$449$	−5.59729e6			−1.31027			−0.655137	+	0.755510i	$0.727389\pi$
$450$		0			0
$451$	646479.	+	4.42005e6i	0.149663	+	1.02326i
$452$		0			0
$453$			2.69066e6i			0.616045i
$454$		0			0
$455$		−	1.11329e6i		−	0.252103i
$456$		0			0
$457$		−	1.93233e6i		−	0.432804i	−0.976304	−	0.216402i	$-0.930568\pi$
$457$		−	1.93233e6i		−	0.432804i	0.976304	−	0.216402i	$-0.0694322\pi$
$458$		0			0
$459$	385184.			0.0853368
$460$		0			0
$461$		−	1.39027e6i		−	0.304683i	−0.988328	−	0.152341i	$-0.951319\pi$
$461$		−	1.39027e6i		−	0.304683i	0.988328	−	0.152341i	$-0.0486813\pi$
$462$		0			0
$463$		−	6.25968e6i		−	1.35706i	−0.734571	−	0.678531i	$-0.762617\pi$
$463$		−	6.25968e6i		−	1.35706i	0.734571	−	0.678531i	$-0.237383\pi$
$464$		0			0
$465$	4.22745e6			0.906664
$466$		0			0
$467$		−	5.92837e6i		−	1.25789i	−0.777449	−	0.628946i	$-0.783487\pi$
$467$		−	5.92837e6i		−	1.25789i	0.777449	−	0.628946i	$-0.216513\pi$
$468$		0			0
$469$			712224.i			0.149515i
$470$		0			0
$471$		−	820533.i		−	0.170429i
$472$		0			0
$473$	−3.94544e6	+	577062.i	−0.810854	+	0.118596i
$474$		0			0
$475$	−1.30373e6			−0.265127
$476$		0			0
$477$	−802196.			−0.161430
$478$		0			0
$479$	−553393.			−0.110203			−0.0551017	−	0.998481i	$-0.517548\pi$
$479$	−553393.			−0.110203			−0.0551017	+	0.998481i	$0.517548\pi$
$480$		0			0
$481$		−	9.92390e6i		−	1.95578i
$482$		0			0
$483$	−726667.			−0.141732
$484$		0			0
$485$	−4.73576e6			−0.914188
$486$		0			0
$487$			7.84047e6i			1.49803i	0.662555	+	0.749013i	$0.269472\pi$
$487$			7.84047e6i			1.49803i	−0.662555	+	0.749013i	$0.730528\pi$
$488$		0			0
$489$	−1.14664e6			−0.216849
$490$		0			0
$491$	−1.53171e6			−0.286729			−0.143365	−	0.989670i	$-0.545792\pi$
$491$	−1.53171e6			−0.286729			−0.143365	+	0.989670i	$0.545792\pi$
$492$		0			0
$493$	1.86938e6			0.346402
$494$		0			0
$495$	1.54890e6	−	226542.i	0.284125	−	0.0415563i
$496$		0			0
$497$		−	2.39349e6i		−	0.434650i
$498$		0			0
$499$			931235.i			0.167420i	0.996490	+	0.0837101i	$0.0266770\pi$
$499$			931235.i			0.167420i	−0.996490	+	0.0837101i	$0.973323\pi$
$500$		0			0
$501$			3.72091e6i			0.662299i
$502$		0			0
$503$	−967582.			−0.170517			−0.0852585	−	0.996359i	$-0.527172\pi$
$503$	−967582.			−0.170517			−0.0852585	+	0.996359i	$0.527172\pi$
$504$		0			0
$505$		−	7.19978e6i		−	1.25629i
$506$		0			0
$507$		−	1.21670e6i		−	0.210214i
$508$		0			0
$509$	9.75276e6			1.66853			0.834264	−	0.551366i	$-0.185893\pi$
$509$	9.75276e6			1.66853			0.834264	+	0.551366i	$0.185893\pi$
$510$		0			0
$511$			1.77508e6i			0.300723i
$512$		0			0
$513$		−	1.17921e6i		−	0.197832i
$514$		0			0
$515$		−	1.22030e6i		−	0.202745i
$516$		0			0
$517$	−750742.	−	5.13291e6i	−0.123528	−	0.844573i
$518$		0			0
$519$	−945492.			−0.154078
$520$		0			0
$521$	5.44205e6			0.878352			0.439176	−	0.898401i	$-0.355270\pi$
$521$	5.44205e6			0.878352			0.439176	+	0.898401i	$0.355270\pi$
$522$		0			0
$523$	−9.83685e6			−1.57254			−0.786270	−	0.617883i	$-0.787991\pi$
$523$	−9.83685e6			−1.57254			−0.786270	+	0.617883i	$0.787991\pi$
$524$		0			0
$525$		−	235636.i		−	0.0373115i
$526$		0			0
$527$	5.15377e6			0.808348
$528$		0			0
$529$	258474.			0.0401584
$530$		0			0
$531$		−	2.26618e6i		−	0.348786i
$532$		0			0
$533$	−7.92180e6			−1.20783
$534$		0			0
$535$	6.88184e6			1.03949
$536$		0			0
$537$	−5.20321e6			−0.778638
$538$		0			0
$539$	6.25482e6	−	914833.i	0.927348	−	0.135634i
$540$		0			0
$541$		−	2.23584e6i		−	0.328433i	−0.986424	−	0.164216i	$-0.947490\pi$
$541$		−	2.23584e6i		−	0.328433i	0.986424	−	0.164216i	$-0.0525096\pi$
$542$		0			0
$543$			7.21937e6i			1.05075i
$544$		0			0
$545$		−	3.68618e6i		−	0.531601i
$546$		0			0
$547$	9.72315e6			1.38944			0.694719	−	0.719282i	$-0.255529\pi$
$547$	9.72315e6			1.38944			0.694719	+	0.719282i	$0.255529\pi$
$548$		0			0
$549$			4.34862e6i			0.615773i
$550$		0			0
$551$		−	5.72295e6i		−	0.803047i
$552$		0			0
$553$	2.20468e6			0.306572
$554$		0			0
$555$			6.04359e6i			0.832842i
$556$		0			0
$557$			1.21423e7i			1.65829i	0.559031	+	0.829147i	$0.311173\pi$
$557$			1.21423e7i			1.65829i	−0.559031	+	0.829147i	$0.688827\pi$
$558$		0			0
$559$		−	7.07118e6i		−	0.957111i
$560$		0			0
$561$	1.88829e6	−	276182.i	0.253315	−	0.0370500i
$562$		0			0
$563$	6.96065e6			0.925505			0.462753	−	0.886487i	$-0.346862\pi$
$563$	6.96065e6			0.925505			0.462753	+	0.886487i	$0.346862\pi$
$564$		0			0
$565$	3.57869e6			0.471632
$566$		0			0
$567$	213129.			0.0278411
$568$		0			0
$569$			1.20347e7i			1.55831i	0.626830	+	0.779156i	$0.284352\pi$
$569$			1.20347e7i			1.55831i	−0.626830	+	0.779156i	$0.715648\pi$
$570$		0			0
$571$	−9.76231e6			−1.25303			−0.626516	−	0.779408i	$-0.715520\pi$
$571$	−9.76231e6			−1.25303			−0.626516	+	0.779408i	$0.715520\pi$
$572$		0			0
$573$	−1.59551e6			−0.203008
$574$		0			0
$575$		−	2.00329e6i		−	0.252682i
$576$		0			0
$577$	1.12858e7			1.41121			0.705605	−	0.708606i	$-0.250676\pi$
$577$	1.12858e7			1.41121			0.705605	+	0.708606i	$0.250676\pi$
$578$		0			0
$579$	5.97010e6			0.740091
$580$		0			0
$581$	−3.38767e6			−0.416352
$582$		0			0
$583$	−3.93261e6	+	575185.i	−0.479192	+	0.0700868i
$584$		0			0
$585$			2.77600e6i			0.335374i
$586$		0			0
$587$			1.48939e7i			1.78407i	0.451961	+	0.892037i	$0.350724\pi$
$587$			1.48939e7i			1.78407i	−0.451961	+	0.892037i	$0.649276\pi$
$588$		0			0
$589$		−	1.57778e7i		−	1.87396i
$590$		0			0
$591$	−4.18990e6			−0.493440
$592$		0			0
$593$		−	3.65515e6i		−	0.426843i	−0.976960	−	0.213421i	$-0.931539\pi$
$593$		−	3.65515e6i		−	0.426843i	0.976960	−	0.213421i	$-0.0684607\pi$
$594$		0			0
$595$			826545.i			0.0957136i
$596$		0			0
$597$	401920.			0.0461534
$598$		0			0
$599$		−	354755.i		−	0.0403982i	−0.999796	−	0.0201991i	$-0.993570\pi$
$599$		−	354755.i		−	0.0403982i	0.999796	−	0.0201991i	$-0.00643001\pi$
$600$		0			0
$601$		−	3.44883e6i		−	0.389480i	−0.980855	−	0.194740i	$-0.937614\pi$
$601$		−	3.44883e6i		−	0.389480i	0.980855	−	0.194740i	$-0.0623863\pi$
$602$		0			0
$603$		−	1.77594e6i		−	0.198900i
$604$		0			0
$605$	7.43073e6	−	2.22116e6i	0.825360	−	0.246713i
$606$		0			0
$607$	−1.35180e7			−1.48915			−0.744577	−	0.667537i	$-0.767349\pi$
$607$	−1.35180e7			−1.48915			−0.744577	+	0.667537i	$0.767349\pi$
$608$		0			0
$609$	1.03436e6			0.113013
$610$		0			0
$611$	9.19940e6			0.996912
$612$		0			0
$613$		−	1.04462e6i		−	0.112281i	−0.998423	−	0.0561406i	$-0.982120\pi$
$613$		−	1.04462e6i		−	0.112281i	0.998423	−	0.0561406i	$-0.0178795\pi$
$614$		0			0
$615$	4.82433e6			0.514338
$616$		0			0
$617$	−4.73327e6			−0.500551			−0.250275	−	0.968175i	$-0.580521\pi$
$617$	−4.73327e6			−0.500551			−0.250275	+	0.968175i	$0.580521\pi$
$618$		0			0
$619$		−	1.64161e7i		−	1.72204i	−0.508572	−	0.861019i	$-0.669827\pi$
$619$		−	1.64161e7i		−	1.72204i	0.508572	−	0.861019i	$-0.330173\pi$
$620$		0			0
$621$	1.81195e6			0.188546
$622$		0			0
$623$	1.94355e6			0.200621
$624$		0			0
$625$	−6.59732e6			−0.675566
$626$		0			0
$627$	−845509.	−	5.78084e6i	−0.0858913	−	0.587249i
$628$		0			0
$629$			7.36786e6i			0.742531i
$630$		0			0
$631$		−	7.75206e6i		−	0.775075i	−0.921854	−	0.387538i	$-0.873326\pi$
$631$		−	7.75206e6i		−	0.775075i	0.921854	−	0.387538i	$-0.126674\pi$
$632$		0			0
$633$		−	3.94201e6i		−	0.391028i
$634$		0			0
$635$	1.36871e7			1.34703
$636$		0			0
$637$			1.12101e7i			1.09462i
$638$		0			0
$639$			5.96819e6i			0.578216i
$640$		0			0
$641$	−19889.3			−0.00191194			−0.000955968	−	1.00000i	$-0.500304\pi$
$641$	−19889.3			−0.00191194			−0.000955968		1.00000i	$0.500304\pi$
$642$		0			0
$643$			9.98658e6i			0.952554i	0.879295	+	0.476277i	$0.158014\pi$
$643$			9.98658e6i			0.952554i	−0.879295	+	0.476277i	$0.841986\pi$
$644$		0			0
$645$			4.30630e6i			0.407573i
$646$		0			0
$647$			2.18818e6i			0.205505i	0.994707	+	0.102753i	$0.0327650\pi$
$647$			2.18818e6i			0.205505i	−0.994707	+	0.102753i	$0.967235\pi$
$648$		0			0
$649$	−1.62489e6	−	1.11095e7i	−0.151430	−	1.03534i
$650$		0			0
$651$	2.85168e6			0.263723
$652$		0			0
$653$	−1.54653e7			−1.41930			−0.709652	−	0.704552i	$-0.751148\pi$
$653$	−1.54653e7			−1.41930			−0.709652	+	0.704552i	$0.751148\pi$
$654$		0			0
$655$	1.67901e7			1.52915
$656$		0			0
$657$		−	4.42619e6i		−	0.400052i
$658$		0			0
$659$	−2.75320e6			−0.246958			−0.123479	−	0.992347i	$-0.539405\pi$
$659$	−2.75320e6			−0.246958			−0.123479	+	0.992347i	$0.539405\pi$
$660$		0			0
$661$	−1.18368e7			−1.05374			−0.526868	−	0.849947i	$-0.676634\pi$
$661$	−1.18368e7			−1.05374			−0.526868	+	0.849947i	$0.676634\pi$
$662$		0			0
$663$			3.38427e6i			0.299007i
$664$		0			0
$665$	2.53040e6			0.221888
$666$		0			0
$667$	8.79379e6			0.765352
$668$		0			0
$669$	−1.22498e6			−0.105819
$670$		0			0
$671$	3.11802e6	+	2.13183e7i	0.267346	+	1.82787i
$672$		0			0
$673$		−	1.55633e7i		−	1.32454i	−0.749266	−	0.662269i	$-0.769594\pi$
$673$		−	1.55633e7i		−	1.32454i	0.749266	−	0.662269i	$-0.230406\pi$
$674$		0			0
$675$			587561.i			0.0496356i
$676$		0			0
$677$			1.72331e7i			1.44508i	0.691328	+	0.722541i	$0.257026\pi$
$677$			1.72331e7i			1.44508i	−0.691328	+	0.722541i	$0.742974\pi$
$678$		0			0
$679$	−3.19456e6			−0.265911
$680$		0			0
$681$			4.77315e6i			0.394400i
$682$		0			0
$683$		−	8.30400e6i		−	0.681139i	−0.940219	−	0.340569i	$-0.889380\pi$
$683$		−	8.30400e6i		−	0.681139i	0.940219	−	0.340569i	$-0.110620\pi$
$684$		0			0
$685$	−3.96456e6			−0.322826
$686$		0			0
$687$		−	1.18728e7i		−	0.959754i
$688$		0			0
$689$		−	7.04818e6i		−	0.565625i
$690$		0			0
$691$			5.99380e6i			0.477537i	0.971076	+	0.238769i	$0.0767437\pi$
$691$			5.99380e6i			0.477537i	−0.971076	+	0.238769i	$0.923256\pi$
$692$		0			0
$693$	1.04483e6	−	152817.i	0.0826439	−	0.0120875i
$694$		0			0
$695$	9.31651e6			0.731629
$696$		0			0
$697$	5.88143e6			0.458565
$698$		0			0
$699$	1.59878e6			0.123765
$700$		0			0
$701$			558458.i			0.0429235i	0.999770	+	0.0214618i	$0.00683202\pi$
$701$			558458.i			0.0429235i	−0.999770	+	0.0214618i	$0.993168\pi$
$702$		0			0
$703$	2.25561e7			1.72138
$704$		0			0
$705$	−5.60238e6			−0.424522
$706$		0			0
$707$		−	4.85669e6i		−	0.365420i
$708$		0			0
$709$	1.20958e6			0.0903692			0.0451846	−	0.998979i	$-0.485612\pi$
$709$	1.20958e6			0.0903692			0.0451846	+	0.998979i	$0.485612\pi$
$710$		0			0
$711$	−5.49739e6			−0.407833
$712$		0			0
$713$	2.42440e7			1.78599
$714$		0			0
$715$	1.99043e6	+	1.36088e7i	0.145607	+	0.995530i
$716$		0			0
$717$		−	3.92546e6i		−	0.285163i
$718$		0			0
$719$			3.55666e6i			0.256578i	0.991737	+	0.128289i	$0.0409486\pi$
$719$			3.55666e6i			0.256578i	−0.991737	+	0.128289i	$0.959051\pi$
$720$		0			0
$721$		−	823169.i		−	0.0589727i
$722$		0			0
$723$	8.84425e6			0.629238
$724$		0			0
$725$			2.85156e6i			0.201482i
$726$		0			0
$727$			1.54104e7i			1.08138i	0.841223	+	0.540688i	$0.181836\pi$
$727$			1.54104e7i			1.08138i	−0.841223	+	0.540688i	$0.818164\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$			5.24990e6i			0.363377i
$732$		0			0
$733$		−	1.23525e7i		−	0.849171i	−0.905388	−	0.424586i	$-0.860420\pi$
$733$		−	1.23525e7i		−	0.849171i	0.905388	−	0.424586i	$-0.139580\pi$
$734$		0			0
$735$		−	6.82691e6i		−	0.466128i
$736$		0			0
$737$	−1.27337e6	−	8.70620e6i	−0.0863549	−	0.590418i
$738$		0			0
$739$	7.08140e6			0.476988			0.238494	−	0.971144i	$-0.423346\pi$
$739$	7.08140e6			0.476988			0.238494	+	0.971144i	$0.423346\pi$
$740$		0			0
$741$	1.03607e7			0.693174
$742$		0			0
$743$	561007.			0.0372817			0.0186409	−	0.999826i	$-0.494066\pi$
$743$	561007.			0.0372817			0.0186409	+	0.999826i	$0.494066\pi$
$744$		0			0
$745$		−	2.00370e7i		−	1.32264i
$746$		0			0
$747$	8.44719e6			0.553873
$748$		0			0
$749$	4.64222e6			0.302358
$750$		0			0
$751$		−	1.68693e7i		−	1.09143i	−0.837970	−	0.545716i	$-0.816258\pi$
$751$		−	1.68693e7i		−	1.09143i	0.837970	−	0.545716i	$-0.183742\pi$
$752$		0			0
$753$	991667.			0.0637350
$754$		0			0
$755$	1.43969e7			0.919179
$756$		0			0
$757$	2.65993e7			1.68706			0.843530	−	0.537082i	$-0.180473\pi$
$757$	2.65993e7			1.68706			0.843530	+	0.537082i	$0.180473\pi$
$758$		0			0
$759$	8.88275e6	−	1.29920e6i	0.559684	−	0.0818597i
$760$		0			0
$761$		−	3.44533e6i		−	0.215660i	−0.994169	−	0.107830i	$-0.965610\pi$
$761$		−	3.44533e6i		−	0.215660i	0.994169	−	0.107830i	$-0.0343902\pi$
$762$		0			0
$763$		−	2.48656e6i		−	0.154628i
$764$		0			0
$765$		−	2.06100e6i		−	0.127328i
$766$		0			0
$767$	1.99109e7			1.22209
$768$		0			0
$769$		−	2.54020e7i		−	1.54900i	−0.632572	−	0.774502i	$-0.718001\pi$
$769$		−	2.54020e7i		−	1.54900i	0.632572	−	0.774502i	$-0.281999\pi$
$770$		0			0
$771$		−	416115.i		−	0.0252103i
$772$		0			0
$773$	6.22317e6			0.374596			0.187298	−	0.982303i	$-0.440027\pi$
$773$	6.22317e6			0.374596			0.187298	+	0.982303i	$0.440027\pi$
$774$		0			0
$775$			7.86158e6i			0.470171i
$776$		0			0
$777$			4.07677e6i			0.242250i
$778$		0			0
$779$		−	1.80055e7i		−	1.06307i
$780$		0			0
$781$	4.27927e6	+	2.92579e7i	0.251040	+	1.71639i
$782$		0			0
$783$	−2.57919e6			−0.150342
$784$		0			0
$785$	−4.39041e6			−0.254291
$786$		0			0
$787$	−6.96316e6			−0.400746			−0.200373	−	0.979720i	$-0.564215\pi$
$787$	−6.96316e6			−0.400746			−0.200373	+	0.979720i	$0.564215\pi$
$788$		0			0
$789$			6.22073e6i			0.355753i
$790$		0			0
$791$	2.41405e6			0.137184
$792$		0			0
$793$	−3.82075e7			−2.15757
$794$		0			0
$795$			4.29230e6i			0.240864i
$796$		0			0
$797$	−3.14412e6			−0.175329			−0.0876645	−	0.996150i	$-0.527940\pi$
$797$	−3.14412e6			−0.175329			−0.0876645	+	0.996150i	$0.527940\pi$
$798$		0			0
$799$	−6.82997e6			−0.378488
$800$		0			0
$801$	−4.84627e6			−0.266886
$802$		0			0
$803$	−3.17364e6	−	2.16985e7i	−0.173687	−	1.18752i
$804$		0			0
$805$			3.88817e6i			0.211473i
$806$		0			0
$807$			1.11890e7i			0.604794i
$808$		0			0
$809$		−	2.84615e7i		−	1.52893i	−0.644666	−	0.764464i	$-0.723004\pi$
$809$		−	2.84615e7i		−	1.52893i	0.644666	−	0.764464i	$-0.276996\pi$
$810$		0			0
$811$	−2.06361e7			−1.10173			−0.550866	−	0.834594i	$-0.685703\pi$
$811$	−2.06361e7			−1.10173			−0.550866	+	0.834594i	$0.685703\pi$
$812$		0			0
$813$			727550.i			0.0386044i
$814$		0			0
$815$			6.13534e6i			0.323552i
$816$		0			0
$817$	1.60721e7			0.842400
$818$		0			0
$819$			1.87258e6i			0.0975507i
$820$		0			0
$821$		−	2.10176e7i		−	1.08824i	−0.839006	−	0.544122i	$-0.816863\pi$
$821$		−	2.10176e7i		−	1.08824i	0.839006	−	0.544122i	$-0.183137\pi$
$822$		0			0
$823$		−	4.95892e6i		−	0.255204i	−0.991825	−	0.127602i	$-0.959272\pi$
$823$		−	4.95892e6i		−	0.255204i	0.991825	−	0.127602i	$-0.0407281\pi$
$824$		0			0
$825$	421289.	+	2.88040e6i	0.0215499	+	0.147339i
$826$		0			0
$827$	−9.79265e6			−0.497893			−0.248947	−	0.968517i	$-0.580084\pi$
$827$	−9.79265e6			−0.497893			−0.248947	+	0.968517i	$0.580084\pi$
$828$		0			0
$829$	−1.64249e7			−0.830075			−0.415038	−	0.909804i	$-0.636232\pi$
$829$	−1.64249e7			−0.830075			−0.415038	+	0.909804i	$0.636232\pi$
$830$		0			0
$831$	4.98023e6			0.250177
$832$		0			0
$833$		−	8.32281e6i		−	0.415583i
$834$		0			0
$835$	1.99094e7			0.988194
$836$		0			0
$837$	−7.11069e6			−0.350831
$838$		0			0
$839$		−	5.03038e6i		−	0.246715i	−0.992362	−	0.123357i	$-0.960634\pi$
$839$		−	5.03038e6i		−	0.246715i	0.992362	−	0.123357i	$-0.0393662\pi$
$840$		0			0
$841$	7.99377e6			0.389728
$842$		0			0
$843$	−1.50270e7			−0.728289
$844$		0			0
$845$	−6.51016e6			−0.313653
$846$		0			0
$847$	5.01249e6	−	1.49831e6i	0.240074	−	0.0717617i
$848$		0			0
$849$		−	1.83914e7i		−	0.875679i
$850$		0			0
$851$			3.46593e7i			1.64058i
$852$		0			0
$853$			2.71742e7i			1.27875i	0.768897	+	0.639373i	$0.220806\pi$
$853$			2.71742e7i			1.27875i	−0.768897	+	0.639373i	$0.779194\pi$
$854$		0			0
$855$	−6.30958e6			−0.295179
$856$		0			0
$857$		−	2.59095e7i		−	1.20506i	−0.798097	−	0.602529i	$-0.794160\pi$
$857$		−	2.59095e7i		−	1.20506i	0.798097	−	0.602529i	$-0.205840\pi$
$858$		0			0
$859$		−	1.01361e7i		−	0.468692i	−0.972153	−	0.234346i	$-0.924705\pi$
$859$		−	1.01361e7i		−	0.468692i	0.972153	−	0.234346i	$-0.0752949\pi$
$860$		0			0
$861$	3.25430e6			0.149606
$862$		0			0
$863$		−	3.29021e6i		−	0.150382i	−0.997169	−	0.0751912i	$-0.976043\pi$
$863$		−	3.29021e6i		−	0.150382i	0.997169	−	0.0751912i	$-0.0239567\pi$
$864$		0			0
$865$			5.05903e6i			0.229894i
$866$		0			0
$867$			1.02661e7i			0.463829i
$868$		0			0
$869$	−2.69499e7	+	3.94170e6i	−1.21062	+	0.177066i
$870$		0			0
$871$	1.56036e7			0.696914
$872$		0			0
$873$	7.96568e6			0.353742
$874$		0			0
$875$	−6.14931e6			−0.271523
$876$		0			0
$877$			1.58064e7i			0.693958i	0.937873	+	0.346979i	$0.112793\pi$
$877$			1.58064e7i			0.693958i	−0.937873	+	0.346979i	$0.887207\pi$
$878$		0			0
$879$	−1.81688e7			−0.793147
$880$		0			0
$881$	−2.13995e7			−0.928890			−0.464445	−	0.885602i	$-0.653746\pi$
$881$	−2.13995e7			−0.928890			−0.464445	+	0.885602i	$0.653746\pi$
$882$		0			0
$883$		−	1.42409e7i		−	0.614660i	−0.951603	−	0.307330i	$-0.900564\pi$
$883$		−	1.42409e7i		−	0.614660i	0.951603	−	0.307330i	$-0.0994356\pi$
$884$		0			0
$885$	−1.21256e7			−0.520411
$886$		0			0
$887$	−1.93942e7			−0.827681			−0.413841	−	0.910349i	$-0.635813\pi$
$887$	−1.93942e7			−0.827681			−0.413841	+	0.910349i	$0.635813\pi$
$888$		0			0
$889$	9.23276e6			0.391811
$890$		0			0
$891$	−2.60529e6	+	381051.i	−0.109941	+	0.0160801i
$892$		0			0
$893$			2.09094e7i			0.877430i
$894$		0			0
$895$			2.78407e7i			1.16178i
$896$		0			0
$897$			1.59200e7i			0.660636i
$898$		0			0
$899$	−3.45097e7			−1.42410
$900$		0			0
$901$			5.23282e6i			0.214745i
$902$		0			0
$903$			2.90487e6i			0.118551i
$904$		0			0
$905$	3.86286e7			1.56779
$906$		0			0
$907$			5.23632e6i			0.211353i	0.994401	+	0.105676i	$0.0337008\pi$
$907$			5.23632e6i			0.211353i	−0.994401	+	0.105676i	$0.966299\pi$
$908$		0			0
$909$			1.21102e7i			0.486119i
$910$		0			0
$911$		−	3.56564e7i		−	1.42345i	−0.702458	−	0.711725i	$-0.747914\pi$
$911$		−	3.56564e7i		−	1.42345i	0.702458	−	0.711725i	$-0.252086\pi$
$912$		0			0
$913$	4.14107e7	−	6.05675e6i	1.64413	−	0.240471i
$914$		0			0
$915$	2.32681e7			0.918774
$916$		0			0
$917$	1.13260e7			0.444786
$918$		0			0
$919$	−1.30378e7			−0.509230			−0.254615	−	0.967042i	$-0.581949\pi$
$919$	−1.30378e7			−0.509230			−0.254615	+	0.967042i	$0.581949\pi$
$920$		0			0
$921$		−	1.78930e7i		−	0.695078i
$922$		0			0
$923$	−5.24371e7			−2.02598
$924$		0			0
$925$	−1.12390e7			−0.431888
$926$		0			0
$927$			2.05258e6i			0.0784515i
$928$		0			0
$929$	−1.11224e7			−0.422822			−0.211411	−	0.977397i	$-0.567806\pi$
$929$	−1.11224e7			−0.422822			−0.211411	+	0.977397i	$0.567806\pi$
$930$		0			0
$931$	−2.54796e7			−0.963426
$932$		0			0
$933$	1.38608e7			0.521296
$934$		0			0
$935$	−1.47776e6	−	1.01036e7i	−0.0552810	−	0.377963i
$936$		0			0
$937$			2.25921e7i			0.840635i	0.907377	+	0.420318i	$0.138081\pi$
$937$			2.25921e7i			0.840635i	−0.907377	+	0.420318i	$0.861919\pi$
$938$		0			0
$939$			5.25585e6i			0.194527i
$940$		0			0
$941$			2.49240e7i			0.917581i	0.888545	+	0.458790i	$0.151717\pi$
$941$			2.49240e7i			0.917581i	−0.888545	+	0.458790i	$0.848283\pi$
$942$		0			0
$943$	2.76670e7			1.01317
$944$		0			0
$945$		−	1.14039e6i		−	0.0415407i
$946$		0			0
$947$			2.25156e7i			0.815848i	0.913016	+	0.407924i	$0.133747\pi$
$947$			2.25156e7i			0.815848i	−0.913016	+	0.407924i	$0.866253\pi$
$948$		0			0
$949$	3.88890e7			1.40172
$950$		0			0
$951$		−	2.97631e7i		−	1.06715i
$952$		0			0
$953$		−	6.33086e6i		−	0.225803i	−0.993606	−	0.112902i	$-0.963985\pi$
$953$		−	6.33086e6i		−	0.225803i	0.993606	−	0.112902i	$-0.0360145\pi$
$954$		0			0
$955$			8.53709e6i			0.302901i
$956$		0			0
$957$	−1.26440e7	+	1.84932e6i	−0.446277	+	0.0652727i
$958$		0			0
$959$	−2.67434e6			−0.0939009
$960$		0			0
$961$	−6.65121e7			−2.32323
$962$		0			0
$963$	−1.15754e7			−0.402228
$964$		0			0
$965$		−	3.19441e7i		−	1.10426i
$966$		0			0
$967$	−9.30487e6			−0.319996			−0.159998	−	0.987117i	$-0.551149\pi$
$967$	−9.30487e6			−0.319996			−0.159998	+	0.987117i	$0.551149\pi$
$968$		0			0
$969$	−7.69213e6			−0.263170
$970$		0			0
$971$			9.33670e6i			0.317794i	0.987295	+	0.158897i	$0.0507937\pi$
$971$			9.33670e6i			0.317794i	−0.987295	+	0.158897i	$0.949206\pi$
$972$		0			0
$973$	6.28456e6			0.212810
$974$		0			0
$975$	−5.16238e6			−0.173915
$976$		0			0
$977$	5.33392e6			0.178776			0.0893881	−	0.995997i	$-0.471509\pi$
$977$	5.33392e6			0.178776			0.0893881	+	0.995997i	$0.471509\pi$
$978$		0			0
$979$	−2.37579e7	+	3.47484e6i	−0.792230	+	0.115872i
$980$		0			0
$981$			6.20026e6i			0.205701i
$982$		0			0
$983$		−	1.61253e7i		−	0.532260i	−0.963937	−	0.266130i	$-0.914255\pi$
$983$		−	1.61253e7i		−	0.532260i	0.963937	−	0.266130i	$-0.0857450\pi$
$984$		0			0
$985$			2.24188e7i			0.736245i
$986$		0			0
$987$	−3.77915e6			−0.123481
$988$		0			0
$989$			2.46962e7i			0.802858i
$990$		0			0
$991$		−	2.46911e7i		−	0.798651i	−0.916809	−	0.399326i	$-0.869244\pi$
$991$		−	2.46911e7i		−	0.798651i	0.916809	−	0.399326i	$-0.130756\pi$
$992$		0			0
$993$	1.85700e7			0.597639
$994$		0			0
$995$		−	2.15055e6i		−	0.0688639i
$996$		0			0
$997$			3.12188e7i			0.994667i	0.867559	+	0.497334i	$0.165687\pi$
$997$			3.12188e7i			0.994667i	−0.867559	+	0.497334i	$0.834313\pi$
$998$		0			0
$999$		−	1.01655e7i		−	0.322266i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.o.b.175.8	yes	40
4.3	odd	2	inner	528.6.o.b.175.35	yes	40
11.10	odd	2	inner	528.6.o.b.175.36	yes	40
44.43	even	2	inner	528.6.o.b.175.7	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
528.6.o.b.175.7	✓	40	44.43	even	2	inner
528.6.o.b.175.8	yes	40	1.1	even	1	trivial
528.6.o.b.175.35	yes	40	4.3	odd	2	inner
528.6.o.b.175.36	yes	40	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +48.1562 q^{5} +32.4843 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} +48.1562 q^{5} +32.4843 q^{7} -81.0000 q^{9} +(-397.087 + 58.0781i) q^{11} -711.675i q^{13} +433.406i q^{15} +528.373i q^{17} +1617.57 q^{19} +292.359i q^{21} +2485.53i q^{23} -805.982 q^{25} -729.000i q^{27} -3537.99i q^{29} -9754.04i q^{31} +(-522.703 - 3573.78i) q^{33} +1564.32 q^{35} +13944.4 q^{37} +6405.08 q^{39} -11131.2i q^{41} +9935.97 q^{43} -3900.65 q^{45} +12926.4i q^{47} -15751.8 q^{49} -4755.36 q^{51} +9903.65 q^{53} +(-19122.2 + 2796.82i) q^{55} +14558.1i q^{57} +27977.6i q^{59} -53686.7i q^{61} -2631.23 q^{63} -34271.6i q^{65} +21925.2i q^{67} -22369.8 q^{69} -73681.3i q^{71} +54644.3i q^{73} -7253.84i q^{75} +(-12899.1 + 1886.63i) q^{77} +67869.0 q^{79} +6561.00 q^{81} -104286. q^{83} +25444.4i q^{85} +31841.9 q^{87} +59830.5 q^{89} -23118.3i q^{91} +87786.3 q^{93} +77896.0 q^{95} -98341.7 q^{97} +(32164.0 - 4704.33i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) \) 40 * q + 88 * q^5 - 3240 * q^9	\( 40 q + 88 q^{5} - 3240 q^{9} + 5464 q^{25} + 6840 q^{33} + 4576 q^{37} - 7128 q^{45} + 58760 q^{49} - 165336 q^{53} + 16632 q^{69} + 28136 q^{77} + 262440 q^{81} - 304608 q^{89} + 102672 q^{93} + 15664 q^{97}+O(q^{100}) \) 40 * q + 88 * q^5 - 3240 * q^9 + 5464 * q^25 + 6840 * q^33 + 4576 * q^37 - 7128 * q^45 + 58760 * q^49 - 165336 * q^53 + 16632 * q^69 + 28136 * q^77 + 262440 * q^81 - 304608 * q^89 + 102672 * q^93 + 15664 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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