LMFDB - Embedded newform 528.6.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(1,528)]

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([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.a (trivial)

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:	yes
Analytic conductor:	$84.6826568613$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 2527x - 22146 $ x^3 - 2527*x - 22146
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 132)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-9.05783$ of defining polynomial
Character	$\chi$	$=$	528.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +30.1157 q^{5} +135.024 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +30.1157 q^{5} +135.024 q^{7} +81.0000 q^{9} +121.000 q^{11} +609.415 q^{13} +271.041 q^{15} +641.139 q^{17} +1937.40 q^{19} +1215.21 q^{21} -2154.29 q^{23} -2218.05 q^{25} +729.000 q^{27} +7102.77 q^{29} -9719.13 q^{31} +1089.00 q^{33} +4066.33 q^{35} +13994.2 q^{37} +5484.73 q^{39} +15491.3 q^{41} -17002.8 q^{43} +2439.37 q^{45} +4864.22 q^{47} +1424.41 q^{49} +5770.25 q^{51} -37681.0 q^{53} +3644.00 q^{55} +17436.6 q^{57} -13334.3 q^{59} +9242.38 q^{61} +10936.9 q^{63} +18352.9 q^{65} +15555.7 q^{67} -19388.6 q^{69} -3996.51 q^{71} +28992.7 q^{73} -19962.4 q^{75} +16337.9 q^{77} -48761.3 q^{79} +6561.00 q^{81} +98969.0 q^{83} +19308.3 q^{85} +63925.0 q^{87} -79242.9 q^{89} +82285.5 q^{91} -87472.1 q^{93} +58346.2 q^{95} +38374.4 q^{97} +9801.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 27 q^{3} + 36 q^{5} + 52 q^{7} + 243 q^{9}+O(q^{10}) $ 3 * q + 27 * q^3 + 36 * q^5 + 52 * q^7 + 243 * q^9	$ 3 q + 27 q^{3} + 36 q^{5} + 52 q^{7} + 243 q^{9} + 363 q^{11} - 888 q^{13} + 324 q^{15} + 1516 q^{17} - 1386 q^{19} + 468 q^{21} - 1030 q^{23} + 11273 q^{25} + 2187 q^{27} + 10904 q^{29} + 2568 q^{31} + 3267 q^{33} - 33596 q^{35} + 32910 q^{37} - 7992 q^{39} - 164 q^{41} - 25018 q^{43} + 2916 q^{45} + 26542 q^{47} + 42243 q^{49} + 13644 q^{51} + 24184 q^{53} + 4356 q^{55} - 12474 q^{57} + 39528 q^{59} + 17128 q^{61} + 4212 q^{63} + 6400 q^{65} - 49596 q^{67} - 9270 q^{69} + 50714 q^{71} + 64970 q^{73} + 101457 q^{75} + 6292 q^{77} - 64808 q^{79} + 19683 q^{81} + 117912 q^{83} + 4188 q^{85} + 98136 q^{87} - 122442 q^{89} + 158764 q^{91} + 23112 q^{93} + 177628 q^{95} + 150190 q^{97} + 29403 q^{99}+O(q^{100}) $ 3 * q + 27 * q^3 + 36 * q^5 + 52 * q^7 + 243 * q^9 + 363 * q^11 - 888 * q^13 + 324 * q^15 + 1516 * q^17 - 1386 * q^19 + 468 * q^21 - 1030 * q^23 + 11273 * q^25 + 2187 * q^27 + 10904 * q^29 + 2568 * q^31 + 3267 * q^33 - 33596 * q^35 + 32910 * q^37 - 7992 * q^39 - 164 * q^41 - 25018 * q^43 + 2916 * q^45 + 26542 * q^47 + 42243 * q^49 + 13644 * q^51 + 24184 * q^53 + 4356 * q^55 - 12474 * q^57 + 39528 * q^59 + 17128 * q^61 + 4212 * q^63 + 6400 * q^65 - 49596 * q^67 - 9270 * q^69 + 50714 * q^71 + 64970 * q^73 + 101457 * q^75 + 6292 * q^77 - 64808 * q^79 + 19683 * q^81 + 117912 * q^83 + 4188 * q^85 + 98136 * q^87 - 122442 * q^89 + 158764 * q^91 + 23112 * q^93 + 177628 * q^95 + 150190 * q^97 + 29403 * q^99

Display 100 coefficients 10 coefficients

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	30.1157	0.538725	0.269363	−	0.963039i	$-0.413187\pi$
$5$	30.1157	0.538725	0.269363	+	0.963039i	$0.413187\pi$
$6$	0	0
$7$	135.024	1.04151	0.520757	−	0.853705i	$-0.325650\pi$
$7$	135.024	1.04151	0.520757	+	0.853705i	$0.325650\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	609.415	1.00013	0.500063	−	0.865989i	$-0.333310\pi$
$13$	609.415	1.00013	0.500063	+	0.865989i	$0.333310\pi$
$14$	0	0
$15$	271.041	0.311033
$16$	0	0
$17$	641.139	0.538059	0.269030	−	0.963132i	$-0.413297\pi$
$17$	641.139	0.538059	0.269030	+	0.963132i	$0.413297\pi$
$18$	0	0
$19$	1937.40	1.23122	0.615611	−	0.788050i	$-0.288909\pi$
$19$	1937.40	1.23122	0.615611	+	0.788050i	$0.288909\pi$
$20$	0	0
$21$	1215.21	0.601318
$22$	0	0
$23$	−2154.29	−0.849149	−0.424574	−	0.905393i	$-0.639576\pi$
$23$	−2154.29	−0.849149	−0.424574	+	0.905393i	$0.639576\pi$
$24$	0	0
$25$	−2218.05	−0.709775
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	7102.77	1.56831	0.784157	−	0.620563i	$-0.213096\pi$
$29$	7102.77	1.56831	0.784157	+	0.620563i	$0.213096\pi$
$30$	0	0
$31$	−9719.13	−1.81645	−0.908224	−	0.418484i	$-0.862562\pi$
$31$	−9719.13	−1.81645	−0.908224	+	0.418484i	$0.862562\pi$
$32$	0	0
$33$	1089.00	0.174078
$34$	0	0
$35$	4066.33	0.561090
$36$	0	0
$37$	13994.2	1.68052	0.840261	−	0.542182i	$-0.182402\pi$
$37$	13994.2	1.68052	0.840261	+	0.542182i	$0.182402\pi$
$38$	0	0
$39$	5484.73	0.577423
$40$	0	0
$41$	15491.3	1.43922	0.719612	−	0.694376i	$-0.244320\pi$
$41$	15491.3	1.43922	0.719612	+	0.694376i	$0.244320\pi$
$42$	0	0
$43$	−17002.8	−1.40233	−0.701165	−	0.712999i	$-0.747336\pi$
$43$	−17002.8	−1.40233	−0.701165	+	0.712999i	$0.747336\pi$
$44$	0	0
$45$	2439.37	0.179575
$46$	0	0
$47$	4864.22	0.321195	0.160597	−	0.987020i	$-0.448658\pi$
$47$	4864.22	0.321195	0.160597	+	0.987020i	$0.448658\pi$
$48$	0	0
$49$	1424.41	0.0847512
$50$	0	0
$51$	5770.25	0.310649
$52$	0	0
$53$	−37681.0	−1.84261	−0.921304	−	0.388842i	$-0.872875\pi$
$53$	−37681.0	−1.84261	−0.921304	+	0.388842i	$0.872875\pi$
$54$	0	0
$55$	3644.00	0.162432
$56$	0	0
$57$	17436.6	0.710846
$58$	0	0
$59$	−13334.3	−0.498700	−0.249350	−	0.968413i	$-0.580217\pi$
$59$	−13334.3	−0.498700	−0.249350	+	0.968413i	$0.580217\pi$
$60$	0	0
$61$	9242.38	0.318023	0.159012	−	0.987277i	$-0.449169\pi$
$61$	9242.38	0.318023	0.159012	+	0.987277i	$0.449169\pi$
$62$	0	0
$63$	10936.9	0.347171
$64$	0	0
$65$	18352.9	0.538793
$66$	0	0
$67$	15555.7	0.423353	0.211676	−	0.977340i	$-0.432108\pi$
$67$	15555.7	0.423353	0.211676	+	0.977340i	$0.432108\pi$
$68$	0	0
$69$	−19388.6	−0.490256
$70$	0	0
$71$	−3996.51	−0.0940882	−0.0470441	−	0.998893i	$-0.514980\pi$
$71$	−3996.51	−0.0940882	−0.0470441	+	0.998893i	$0.514980\pi$
$72$	0	0
$73$	28992.7	0.636767	0.318384	−	0.947962i	$-0.396860\pi$
$73$	28992.7	0.636767	0.318384	+	0.947962i	$0.396860\pi$
$74$	0	0
$75$	−19962.4	−0.409789
$76$	0	0
$77$	16337.9	0.314028
$78$	0	0
$79$	−48761.3	−0.879037	−0.439519	−	0.898233i	$-0.644851\pi$
$79$	−48761.3	−0.879037	−0.439519	+	0.898233i	$0.644851\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	98969.0	1.57690	0.788449	−	0.615100i	$-0.210884\pi$
$83$	98969.0	1.57690	0.788449	+	0.615100i	$0.210884\pi$
$84$	0	0
$85$	19308.3	0.289866
$86$	0	0
$87$	63925.0	0.905466
$88$	0	0
$89$	−79242.9	−1.06044	−0.530219	−	0.847861i	$-0.677890\pi$
$89$	−79242.9	−1.06044	−0.530219	+	0.847861i	$0.677890\pi$
$90$	0	0
$91$	82285.5	1.04164
$92$	0	0
$93$	−87472.1	−1.04873
$94$	0	0
$95$	58346.2	0.663291
$96$	0	0
$97$	38374.4	0.414107	0.207054	−	0.978330i	$-0.433613\pi$
$97$	38374.4	0.414107	0.207054	+	0.978330i	$0.433613\pi$
$98$	0	0
$99$	9801.00	0.100504
$100$	0	0
$101$	60777.3	0.592841	0.296420	−	0.955058i	$-0.404207\pi$
$101$	60777.3	0.592841	0.296420	+	0.955058i	$0.404207\pi$
$102$	0	0
$103$	−61670.9	−0.572779	−0.286389	−	0.958113i	$-0.592455\pi$
$103$	−61670.9	−0.572779	−0.286389	+	0.958113i	$0.592455\pi$
$104$	0	0
$105$	36597.0	0.323945
$106$	0	0
$107$	42379.9	0.357849	0.178925	−	0.983863i	$-0.442738\pi$
$107$	42379.9	0.357849	0.178925	+	0.983863i	$0.442738\pi$
$108$	0	0
$109$	−3129.27	−0.0252277	−0.0126138	−	0.999920i	$-0.504015\pi$
$109$	−3129.27	−0.0252277	−0.0126138	+	0.999920i	$0.504015\pi$
$110$	0	0
$111$	125948.	0.970250
$112$	0	0
$113$	−78137.8	−0.575658	−0.287829	−	0.957682i	$-0.592934\pi$
$113$	−78137.8	−0.575658	−0.287829	+	0.957682i	$0.592934\pi$
$114$	0	0
$115$	−64877.8	−0.457458
$116$	0	0
$117$	49362.6	0.333375
$118$	0	0
$119$	86569.0	0.560396
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	139422.	0.830936
$124$	0	0
$125$	−160909.	−0.921099
$126$	0	0
$127$	−25587.1	−0.140771	−0.0703853	−	0.997520i	$-0.522423\pi$
$127$	−25587.1	−0.140771	−0.0703853	+	0.997520i	$0.522423\pi$
$128$	0	0
$129$	−153025.	−0.809635
$130$	0	0
$131$	183068.	0.932037	0.466018	−	0.884775i	$-0.345688\pi$
$131$	183068.	0.932037	0.466018	+	0.884775i	$0.345688\pi$
$132$	0	0
$133$	261596.	1.28233
$134$	0	0
$135$	21954.3	0.103678
$136$	0	0
$137$	380792.	1.73335	0.866675	−	0.498873i	$-0.166253\pi$
$137$	380792.	1.73335	0.866675	+	0.498873i	$0.166253\pi$
$138$	0	0
$139$	−140270.	−0.615784	−0.307892	−	0.951421i	$-0.599624\pi$
$139$	−140270.	−0.615784	−0.307892	+	0.951421i	$0.599624\pi$
$140$	0	0
$141$	43778.0	0.185442
$142$	0	0
$143$	73739.2	0.301549
$144$	0	0
$145$	213905.	0.844890
$146$	0	0
$147$	12819.7	0.0489311
$148$	0	0
$149$	−314102.	−1.15906	−0.579528	−	0.814952i	$-0.696763\pi$
$149$	−314102.	−1.15906	−0.579528	+	0.814952i	$0.696763\pi$
$150$	0	0
$151$	304345.	1.08623	0.543117	−	0.839657i	$-0.317244\pi$
$151$	304345.	1.08623	0.543117	+	0.839657i	$0.317244\pi$
$152$	0	0
$153$	51932.3	0.179353
$154$	0	0
$155$	−292698.	−0.978567
$156$	0	0
$157$	364821.	1.18122	0.590610	−	0.806957i	$-0.298887\pi$
$157$	364821.	1.18122	0.590610	+	0.806957i	$0.298887\pi$
$158$	0	0
$159$	−339129.	−1.06383
$160$	0	0
$161$	−290880.	−0.884400
$162$	0	0
$163$	−254691.	−0.750836	−0.375418	−	0.926856i	$-0.622501\pi$
$163$	−254691.	−0.750836	−0.375418	+	0.926856i	$0.622501\pi$
$164$	0	0
$165$	32796.0	0.0937801
$166$	0	0
$167$	−413103.	−1.14622	−0.573109	−	0.819479i	$-0.694263\pi$
$167$	−413103.	−1.14622	−0.573109	+	0.819479i	$0.694263\pi$
$168$	0	0
$169$	93.4384	0.000251657 0
$170$	0	0
$171$	156930.	0.410407
$172$	0	0
$173$	618993.	1.57243	0.786213	−	0.617956i	$-0.212039\pi$
$173$	618993.	1.57243	0.786213	+	0.617956i	$0.212039\pi$
$174$	0	0
$175$	−299489.	−0.739240
$176$	0	0
$177$	−120009.	−0.287925
$178$	0	0
$179$	226527.	0.528431	0.264215	−	0.964464i	$-0.414887\pi$
$179$	226527.	0.528431	0.264215	+	0.964464i	$0.414887\pi$
$180$	0	0
$181$	508200.	1.15302	0.576512	−	0.817089i	$-0.304413\pi$
$181$	508200.	1.15302	0.576512	+	0.817089i	$0.304413\pi$
$182$	0	0
$183$	83181.4	0.183611
$184$	0	0
$185$	421445.	0.905340
$186$	0	0
$187$	77577.9	0.162231
$188$	0	0
$189$	98432.3	0.200439
$190$	0	0
$191$	615746.	1.22129	0.610644	−	0.791905i	$-0.290911\pi$
$191$	615746.	1.22129	0.610644	+	0.791905i	$0.290911\pi$
$192$	0	0
$193$	172575.	0.333491	0.166745	−	0.986000i	$-0.446674\pi$
$193$	172575.	0.333491	0.166745	+	0.986000i	$0.446674\pi$
$194$	0	0
$195$	165176.	0.311072
$196$	0	0
$197$	395249.	0.725614	0.362807	−	0.931864i	$-0.381818\pi$
$197$	395249.	0.725614	0.362807	+	0.931864i	$0.381818\pi$
$198$	0	0
$199$	768927.	1.37642	0.688212	−	0.725509i	$-0.258396\pi$
$199$	768927.	1.37642	0.688212	+	0.725509i	$0.258396\pi$
$200$	0	0
$201$	140001.	0.244423
$202$	0	0
$203$	959043.	1.63342
$204$	0	0
$205$	466531.	0.775347
$206$	0	0
$207$	−174497.	−0.283050
$208$	0	0
$209$	234426.	0.371227
$210$	0	0
$211$	−775068.	−1.19849	−0.599244	−	0.800566i	$-0.704532\pi$
$211$	−775068.	−1.19849	−0.599244	+	0.800566i	$0.704532\pi$
$212$	0	0
$213$	−35968.6	−0.0543219
$214$	0	0
$215$	−512051.	−0.755470
$216$	0	0
$217$	−1.31231e6	−1.89186
$218$	0	0
$219$	260934.	0.367638
$220$	0	0
$221$	390720.	0.538127
$222$	0	0
$223$	−281949.	−0.379671	−0.189836	−	0.981816i	$-0.560796\pi$
$223$	−281949.	−0.379671	−0.189836	+	0.981816i	$0.560796\pi$
$224$	0	0
$225$	−179662.	−0.236592
$226$	0	0
$227$	513124.	0.660933	0.330467	−	0.943818i	$-0.392794\pi$
$227$	513124.	0.660933	0.330467	+	0.943818i	$0.392794\pi$
$228$	0	0
$229$	−821140.	−1.03473	−0.517367	−	0.855764i	$-0.673088\pi$
$229$	−821140.	−1.03473	−0.517367	+	0.855764i	$0.673088\pi$
$230$	0	0
$231$	147041.	0.181304
$232$	0	0
$233$	148419.	0.179101	0.0895506	−	0.995982i	$-0.471457\pi$
$233$	148419.	0.179101	0.0895506	+	0.995982i	$0.471457\pi$
$234$	0	0
$235$	146489.	0.173036
$236$	0	0
$237$	−438851.	−0.507512
$238$	0	0
$239$	374455.	0.424038	0.212019	−	0.977266i	$-0.431996\pi$
$239$	374455.	0.424038	0.212019	+	0.977266i	$0.431996\pi$
$240$	0	0
$241$	559182.	0.620170	0.310085	−	0.950709i	$-0.399642\pi$
$241$	559182.	0.620170	0.310085	+	0.950709i	$0.399642\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	42897.2	0.0456576
$246$	0	0
$247$	1.18068e6	1.23138
$248$	0	0
$249$	890721.	0.910423
$250$	0	0
$251$	−482470.	−0.483377	−0.241689	−	0.970354i	$-0.577701\pi$
$251$	−482470.	−0.483377	−0.241689	+	0.970354i	$0.577701\pi$
$252$	0	0
$253$	−260669.	−0.256028
$254$	0	0
$255$	173775.	0.167354
$256$	0	0
$257$	−1.28531e6	−1.21387	−0.606937	−	0.794750i	$-0.707602\pi$
$257$	−1.28531e6	−1.21387	−0.606937	+	0.794750i	$0.707602\pi$
$258$	0	0
$259$	1.88955e6	1.75029
$260$	0	0
$261$	575325.	0.522771
$262$	0	0
$263$	−755923.	−0.673889	−0.336944	−	0.941525i	$-0.609393\pi$
$263$	−755923.	−0.673889	−0.336944	+	0.941525i	$0.609393\pi$
$264$	0	0
$265$	−1.13479e6	−0.992660
$266$	0	0
$267$	−713186.	−0.612244
$268$	0	0
$269$	1.28255e6	1.08067	0.540337	−	0.841448i	$-0.318297\pi$
$269$	1.28255e6	1.08067	0.540337	+	0.841448i	$0.318297\pi$
$270$	0	0
$271$	1.37974e6	1.14123	0.570614	−	0.821218i	$-0.306705\pi$
$271$	1.37974e6	1.14123	0.570614	+	0.821218i	$0.306705\pi$
$272$	0	0
$273$	740569.	0.601394
$274$	0	0
$275$	−268384.	−0.214005
$276$	0	0
$277$	−805344.	−0.630641	−0.315321	−	0.948985i	$-0.602112\pi$
$277$	−805344.	−0.630641	−0.315321	+	0.948985i	$0.602112\pi$
$278$	0	0
$279$	−787249.	−0.605483
$280$	0	0
$281$	−2.15481e6	−1.62795	−0.813977	−	0.580897i	$-0.802702\pi$
$281$	−2.15481e6	−1.62795	−0.813977	+	0.580897i	$0.802702\pi$
$282$	0	0
$283$	−1.84636e6	−1.37041	−0.685203	−	0.728352i	$-0.740287\pi$
$283$	−1.84636e6	−1.37041	−0.685203	+	0.728352i	$0.740287\pi$
$284$	0	0
$285$	525116.	0.382951
$286$	0	0
$287$	2.09169e6	1.49897
$288$	0	0
$289$	−1.00880e6	−0.710492
$290$	0	0
$291$	345370.	0.239085
$292$	0	0
$293$	2.79556e6	1.90239	0.951195	−	0.308590i	$-0.0998570\pi$
$293$	2.79556e6	1.90239	0.951195	+	0.308590i	$0.0998570\pi$
$294$	0	0
$295$	−401571.	−0.268662
$296$	0	0
$297$	88209.0	0.0580259
$298$	0	0
$299$	−1.31285e6	−0.849256
$300$	0	0
$301$	−2.29579e6	−1.46055
$302$	0	0
$303$	546996.	0.342277
$304$	0	0
$305$	278340.	0.171327
$306$	0	0
$307$	−1.86860e6	−1.13154	−0.565772	−	0.824562i	$-0.691422\pi$
$307$	−1.86860e6	−1.13154	−0.565772	+	0.824562i	$0.691422\pi$
$308$	0	0
$309$	−555038.	−0.330694
$310$	0	0
$311$	1.45038e6	0.850319	0.425159	−	0.905119i	$-0.360218\pi$
$311$	1.45038e6	0.850319	0.425159	+	0.905119i	$0.360218\pi$
$312$	0	0
$313$	−3.31192e6	−1.91082	−0.955408	−	0.295288i	$-0.904585\pi$
$313$	−3.31192e6	−1.91082	−0.955408	+	0.295288i	$0.904585\pi$
$314$	0	0
$315$	329373.	0.187030
$316$	0	0
$317$	1.06722e6	0.596492	0.298246	−	0.954489i	$-0.403598\pi$
$317$	1.06722e6	0.596492	0.298246	+	0.954489i	$0.403598\pi$
$318$	0	0
$319$	859435.	0.472864
$320$	0	0
$321$	381419.	0.206604
$322$	0	0
$323$	1.24215e6	0.662470
$324$	0	0
$325$	−1.35171e6	−0.709864
$326$	0	0
$327$	−28163.5	−0.0145652
$328$	0	0
$329$	656785.	0.334529
$330$	0	0
$331$	−3.58124e6	−1.79665	−0.898324	−	0.439333i	$-0.855215\pi$
$331$	−3.58124e6	−1.79665	−0.898324	+	0.439333i	$0.855215\pi$
$332$	0	0
$333$	1.13353e6	0.560174
$334$	0	0
$335$	468470.	0.228071
$336$	0	0
$337$	2.06974e6	0.992752	0.496376	−	0.868108i	$-0.334664\pi$
$337$	2.06974e6	0.992752	0.496376	+	0.868108i	$0.334664\pi$
$338$	0	0
$339$	−703240.	−0.332356
$340$	0	0
$341$	−1.17601e6	−0.547680
$342$	0	0
$343$	−2.07701e6	−0.953244
$344$	0	0
$345$	−583900.	−0.264113
$346$	0	0
$347$	−1.59487e6	−0.711053	−0.355526	−	0.934666i	$-0.615698\pi$
$347$	−1.59487e6	−0.711053	−0.355526	+	0.934666i	$0.615698\pi$
$348$	0	0
$349$	−1.58011e6	−0.694424	−0.347212	−	0.937787i	$-0.612872\pi$
$349$	−1.58011e6	−0.694424	−0.347212	+	0.937787i	$0.612872\pi$
$350$	0	0
$351$	444263.	0.192474
$352$	0	0
$353$	−445483.	−0.190281	−0.0951403	−	0.995464i	$-0.530330\pi$
$353$	−445483.	−0.190281	−0.0951403	+	0.995464i	$0.530330\pi$
$354$	0	0
$355$	−120358.	−0.0506877
$356$	0	0
$357$	779121.	0.323545
$358$	0	0
$359$	−2.80695e6	−1.14947	−0.574737	−	0.818338i	$-0.694896\pi$
$359$	−2.80695e6	−1.14947	−0.574737	+	0.818338i	$0.694896\pi$
$360$	0	0
$361$	1.27744e6	0.515907
$362$	0	0
$363$	131769.	0.0524864
$364$	0	0
$365$	873133.	0.343043
$366$	0	0
$367$	2.95709e6	1.14604	0.573020	−	0.819542i	$-0.305772\pi$
$367$	2.95709e6	1.14604	0.573020	+	0.819542i	$0.305772\pi$
$368$	0	0
$369$	1.25480e6	0.479741
$370$	0	0
$371$	−5.08783e6	−1.91910
$372$	0	0
$373$	−2.68309e6	−0.998534	−0.499267	−	0.866448i	$-0.666397\pi$
$373$	−2.68309e6	−0.998534	−0.499267	+	0.866448i	$0.666397\pi$
$374$	0	0
$375$	−1.44818e6	−0.531797
$376$	0	0
$377$	4.32854e6	1.56851
$378$	0	0
$379$	−3.70416e6	−1.32462	−0.662310	−	0.749230i	$-0.730424\pi$
$379$	−3.70416e6	−1.32462	−0.662310	+	0.749230i	$0.730424\pi$
$380$	0	0
$381$	−230284.	−0.0812740
$382$	0	0
$383$	−2.67522e6	−0.931884	−0.465942	−	0.884815i	$-0.654284\pi$
$383$	−2.67522e6	−0.931884	−0.465942	+	0.884815i	$0.654284\pi$
$384$	0	0
$385$	492026.	0.169175
$386$	0	0
$387$	−1.37723e6	−0.467443
$388$	0	0
$389$	−2.24051e6	−0.750713	−0.375356	−	0.926881i	$-0.622480\pi$
$389$	−2.24051e6	−0.750713	−0.375356	+	0.926881i	$0.622480\pi$
$390$	0	0
$391$	−1.38120e6	−0.456892
$392$	0	0
$393$	1.64761e6	0.538112
$394$	0	0
$395$	−1.46848e6	−0.473560
$396$	0	0
$397$	−4.86544e6	−1.54934	−0.774669	−	0.632367i	$-0.782083\pi$
$397$	−4.86544e6	−1.54934	−0.774669	+	0.632367i	$0.782083\pi$
$398$	0	0
$399$	2.35436e6	0.740356
$400$	0	0
$401$	−1.64816e6	−0.511846	−0.255923	−	0.966697i	$-0.582379\pi$
$401$	−1.64816e6	−0.511846	−0.255923	+	0.966697i	$0.582379\pi$
$402$	0	0
$403$	−5.92298e6	−1.81668
$404$	0	0
$405$	197589.	0.0598584
$406$	0	0
$407$	1.69330e6	0.506696
$408$	0	0
$409$	3.77916e6	1.11709	0.558543	−	0.829476i	$-0.311361\pi$
$409$	3.77916e6	1.11709	0.558543	+	0.829476i	$0.311361\pi$
$410$	0	0
$411$	3.42713e6	1.00075
$412$	0	0
$413$	−1.80044e6	−0.519403
$414$	0	0
$415$	2.98052e6	0.849515
$416$	0	0
$417$	−1.26243e6	−0.355523
$418$	0	0
$419$	5.40002e6	1.50266	0.751329	−	0.659927i	$-0.229413\pi$
$419$	5.40002e6	1.50266	0.751329	+	0.659927i	$0.229413\pi$
$420$	0	0
$421$	742544.	0.204182	0.102091	−	0.994775i	$-0.467447\pi$
$421$	742544.	0.204182	0.102091	+	0.994775i	$0.467447\pi$
$422$	0	0
$423$	394002.	0.107065
$424$	0	0
$425$	−1.42208e6	−0.381901
$426$	0	0
$427$	1.24794e6	0.331226
$428$	0	0
$429$	663653.	0.174100
$430$	0	0
$431$	−2.99668e6	−0.777046	−0.388523	−	0.921439i	$-0.627015\pi$
$431$	−2.99668e6	−0.777046	−0.388523	+	0.921439i	$0.627015\pi$
$432$	0	0
$433$	−2.47753e6	−0.635036	−0.317518	−	0.948252i	$-0.602849\pi$
$433$	−2.47753e6	−0.635036	−0.317518	+	0.948252i	$0.602849\pi$
$434$	0	0
$435$	1.92514e6	0.487798
$436$	0	0
$437$	−4.17372e6	−1.04549
$438$	0	0
$439$	−3.51673e6	−0.870919	−0.435460	−	0.900208i	$-0.643414\pi$
$439$	−3.51673e6	−0.870919	−0.435460	+	0.900208i	$0.643414\pi$
$440$	0	0
$441$	115377.	0.0282504
$442$	0	0
$443$	−5.01627e6	−1.21443	−0.607213	−	0.794539i	$-0.707713\pi$
$443$	−5.01627e6	−1.21443	−0.607213	+	0.794539i	$0.707713\pi$
$444$	0	0
$445$	−2.38645e6	−0.571285
$446$	0	0
$447$	−2.82691e6	−0.669181
$448$	0	0
$449$	142480.	0.0333532	0.0166766	−	0.999861i	$-0.494691\pi$
$449$	142480.	0.0333532	0.0166766	+	0.999861i	$0.494691\pi$
$450$	0	0
$451$	1.87445e6	0.433942
$452$	0	0
$453$	2.73910e6	0.627137
$454$	0	0
$455$	2.47808e6	0.561161
$456$	0	0
$457$	−7.64696e6	−1.71277	−0.856384	−	0.516340i	$-0.827294\pi$
$457$	−7.64696e6	−1.71277	−0.856384	+	0.516340i	$0.827294\pi$
$458$	0	0
$459$	467391.	0.103550
$460$	0	0
$461$	−2.41362e6	−0.528953	−0.264477	−	0.964392i	$-0.585199\pi$
$461$	−2.41362e6	−0.528953	−0.264477	+	0.964392i	$0.585199\pi$
$462$	0	0
$463$	−3.96363e6	−0.859291	−0.429646	−	0.902998i	$-0.641361\pi$
$463$	−3.96363e6	−0.859291	−0.429646	+	0.902998i	$0.641361\pi$
$464$	0	0
$465$	−2.63428e6	−0.564976
$466$	0	0
$467$	1.24552e6	0.264276	0.132138	−	0.991231i	$-0.457816\pi$
$467$	1.24552e6	0.264276	0.132138	+	0.991231i	$0.457816\pi$
$468$	0	0
$469$	2.10039e6	0.440928
$470$	0	0
$471$	3.28339e6	0.681977
$472$	0	0
$473$	−2.05734e6	−0.422818
$474$	0	0
$475$	−4.29725e6	−0.873890
$476$	0	0
$477$	−3.05216e6	−0.614203
$478$	0	0
$479$	7.42922e6	1.47946	0.739732	−	0.672901i	$-0.234952\pi$
$479$	7.42922e6	1.47946	0.739732	+	0.672901i	$0.234952\pi$
$480$	0	0
$481$	8.52828e6	1.68073
$482$	0	0
$483$	−2.61792e6	−0.510609
$484$	0	0
$485$	1.15567e6	0.223090
$486$	0	0
$487$	−4.84866e6	−0.926403	−0.463201	−	0.886253i	$-0.653299\pi$
$487$	−4.84866e6	−0.926403	−0.463201	+	0.886253i	$0.653299\pi$
$488$	0	0
$489$	−2.29222e6	−0.433495
$490$	0	0
$491$	4.85108e6	0.908102	0.454051	−	0.890976i	$-0.349978\pi$
$491$	4.85108e6	0.908102	0.454051	+	0.890976i	$0.349978\pi$
$492$	0	0
$493$	4.55387e6	0.843846
$494$	0	0
$495$	295164.	0.0541439
$496$	0	0
$497$	−539624.	−0.0979942
$498$	0	0
$499$	9.18412e6	1.65115	0.825575	−	0.564293i	$-0.190851\pi$
$499$	9.18412e6	1.65115	0.825575	+	0.564293i	$0.190851\pi$
$500$	0	0
$501$	−3.71793e6	−0.661769
$502$	0	0
$503$	9.43777e6	1.66322	0.831609	−	0.555361i	$-0.187420\pi$
$503$	9.43777e6	1.66322	0.831609	+	0.555361i	$0.187420\pi$
$504$	0	0
$505$	1.83035e6	0.319378
$506$	0	0
$507$	840.945	0.000145294 0
$508$	0	0
$509$	−7.98520e6	−1.36613	−0.683064	−	0.730359i	$-0.739353\pi$
$509$	−7.98520e6	−1.36613	−0.683064	+	0.730359i	$0.739353\pi$
$510$	0	0
$511$	3.91470e6	0.663202
$512$	0	0
$513$	1.41237e6	0.236949
$514$	0	0
$515$	−1.85726e6	−0.308570
$516$	0	0
$517$	588571.	0.0968439
$518$	0	0
$519$	5.57093e6	0.907841
$520$	0	0
$521$	−1.55745e6	−0.251374	−0.125687	−	0.992070i	$-0.540113\pi$
$521$	−1.55745e6	−0.251374	−0.125687	+	0.992070i	$0.540113\pi$
$522$	0	0
$523$	−6.11756e6	−0.977967	−0.488983	−	0.872293i	$-0.662632\pi$
$523$	−6.11756e6	−0.977967	−0.488983	+	0.872293i	$0.662632\pi$
$524$	0	0
$525$	−2.69540e6	−0.426801
$526$	0	0
$527$	−6.23132e6	−0.977357
$528$	0	0
$529$	−1.79540e6	−0.278947
$530$	0	0
$531$	−1.08008e6	−0.166233
$532$	0	0
$533$	9.44063e6	1.43941
$534$	0	0
$535$	1.27630e6	0.192783
$536$	0	0
$537$	2.03875e6	0.305090
$538$	0	0
$539$	172354.	0.0255534
$540$	0	0
$541$	5.15367e6	0.757048	0.378524	−	0.925592i	$-0.376432\pi$
$541$	5.15367e6	0.757048	0.378524	+	0.925592i	$0.376432\pi$
$542$	0	0
$543$	4.57380e6	0.665699
$544$	0	0
$545$	−94240.2	−0.0135908
$546$	0	0
$547$	−3.56570e6	−0.509538	−0.254769	−	0.967002i	$-0.581999\pi$
$547$	−3.56570e6	−0.509538	−0.254769	+	0.967002i	$0.581999\pi$
$548$	0	0
$549$	748633.	0.106008
$550$	0	0
$551$	1.37609e7	1.93094
$552$	0	0
$553$	−6.58393e6	−0.915530
$554$	0	0
$555$	3.79300e6	0.522698
$556$	0	0
$557$	5.77351e6	0.788501	0.394250	−	0.919003i	$-0.371004\pi$
$557$	5.77351e6	0.788501	0.394250	+	0.919003i	$0.371004\pi$
$558$	0	0
$559$	−1.03618e7	−1.40251
$560$	0	0
$561$	698201.	0.0936641
$562$	0	0
$563$	−2.62555e6	−0.349100	−0.174550	−	0.984648i	$-0.555847\pi$
$563$	−2.62555e6	−0.349100	−0.174550	+	0.984648i	$0.555847\pi$
$564$	0	0
$565$	−2.35317e6	−0.310122
$566$	0	0
$567$	885891.	0.115724
$568$	0	0
$569$	1.95960e6	0.253738	0.126869	−	0.991919i	$-0.459507\pi$
$569$	1.95960e6	0.253738	0.126869	+	0.991919i	$0.459507\pi$
$570$	0	0
$571$	−8.84615e6	−1.13544	−0.567719	−	0.823222i	$-0.692174\pi$
$571$	−8.84615e6	−1.13544	−0.567719	+	0.823222i	$0.692174\pi$
$572$	0	0
$573$	5.54171e6	0.705111
$574$	0	0
$575$	4.77831e6	0.602704
$576$	0	0
$577$	4.55927e6	0.570106	0.285053	−	0.958512i	$-0.407989\pi$
$577$	4.55927e6	0.570106	0.285053	+	0.958512i	$0.407989\pi$
$578$	0	0
$579$	1.55317e6	0.192541
$580$	0	0
$581$	1.33632e7	1.64236
$582$	0	0
$583$	−4.55940e6	−0.555567
$584$	0	0
$585$	1.48659e6	0.179598
$586$	0	0
$587$	−6.48858e6	−0.777239	−0.388620	−	0.921398i	$-0.627048\pi$
$587$	−6.48858e6	−0.777239	−0.388620	+	0.921398i	$0.627048\pi$
$588$	0	0
$589$	−1.88299e7	−2.23645
$590$	0	0
$591$	3.55725e6	0.418933
$592$	0	0
$593$	1.10585e7	1.29140	0.645700	−	0.763591i	$-0.276566\pi$
$593$	1.10585e7	1.29140	0.645700	+	0.763591i	$0.276566\pi$
$594$	0	0
$595$	2.60708e6	0.301900
$596$	0	0
$597$	6.92034e6	0.794679
$598$	0	0
$599$	1.06958e7	1.21799	0.608997	−	0.793172i	$-0.291572\pi$
$599$	1.06958e7	1.21799	0.608997	+	0.793172i	$0.291572\pi$
$600$	0	0
$601$	−2.98886e6	−0.337535	−0.168768	−	0.985656i	$-0.553979\pi$
$601$	−2.98886e6	−0.337535	−0.168768	+	0.985656i	$0.553979\pi$
$602$	0	0
$603$	1.26001e6	0.141118
$604$	0	0
$605$	440923.	0.0489750
$606$	0	0
$607$	−3.88941e6	−0.428462	−0.214231	−	0.976783i	$-0.568724\pi$
$607$	−3.88941e6	−0.428462	−0.214231	+	0.976783i	$0.568724\pi$
$608$	0	0
$609$	8.63139e6	0.943056
$610$	0	0
$611$	2.96433e6	0.321235
$612$	0	0
$613$	−4.38126e6	−0.470921	−0.235461	−	0.971884i	$-0.575660\pi$
$613$	−4.38126e6	−0.470921	−0.235461	+	0.971884i	$0.575660\pi$
$614$	0	0
$615$	4.19878e6	0.447647
$616$	0	0
$617$	−1.12342e7	−1.18804	−0.594019	−	0.804451i	$-0.702460\pi$
$617$	−1.12342e7	−1.18804	−0.594019	+	0.804451i	$0.702460\pi$
$618$	0	0
$619$	−7.30226e6	−0.766004	−0.383002	−	0.923748i	$-0.625110\pi$
$619$	−7.30226e6	−0.766004	−0.383002	+	0.923748i	$0.625110\pi$
$620$	0	0
$621$	−1.57047e6	−0.163419
$622$	0	0
$623$	−1.06997e7	−1.10446
$624$	0	0
$625$	2.08550e6	0.213555
$626$	0	0
$627$	2.10983e6	0.214328
$628$	0	0
$629$	8.97224e6	0.904220
$630$	0	0
$631$	−4.41706e6	−0.441631	−0.220816	−	0.975316i	$-0.570872\pi$
$631$	−4.41706e6	−0.441631	−0.220816	+	0.975316i	$0.570872\pi$
$632$	0	0
$633$	−6.97562e6	−0.691948
$634$	0	0
$635$	−770573.	−0.0758367
$636$	0	0
$637$	868059.	0.0847619
$638$	0	0
$639$	−323718.	−0.0313627
$640$	0	0
$641$	7.82517e6	0.752227	0.376113	−	0.926574i	$-0.377260\pi$
$641$	7.82517e6	0.752227	0.376113	+	0.926574i	$0.377260\pi$
$642$	0	0
$643$	−1.23411e7	−1.17713	−0.588567	−	0.808448i	$-0.700308\pi$
$643$	−1.23411e7	−1.17713	−0.588567	+	0.808448i	$0.700308\pi$
$644$	0	0
$645$	−4.60846e6	−0.436171
$646$	0	0
$647$	1.62995e7	1.53078	0.765390	−	0.643567i	$-0.222546\pi$
$647$	1.62995e7	1.53078	0.765390	+	0.643567i	$0.222546\pi$
$648$	0	0
$649$	−1.61345e6	−0.150364
$650$	0	0
$651$	−1.18108e7	−1.09226
$652$	0	0
$653$	−1.99859e7	−1.83417	−0.917087	−	0.398686i	$-0.869466\pi$
$653$	−1.99859e7	−1.83417	−0.917087	+	0.398686i	$0.869466\pi$
$654$	0	0
$655$	5.51320e6	0.502112
$656$	0	0
$657$	2.34840e6	0.212256
$658$	0	0
$659$	−1.61200e7	−1.44594	−0.722970	−	0.690879i	$-0.757224\pi$
$659$	−1.61200e7	−1.44594	−0.722970	+	0.690879i	$0.757224\pi$
$660$	0	0
$661$	7.23532e6	0.644101	0.322051	−	0.946722i	$-0.395628\pi$
$661$	7.23532e6	0.644101	0.322051	+	0.946722i	$0.395628\pi$
$662$	0	0
$663$	3.51648e6	0.310688
$664$	0	0
$665$	7.87813e6	0.690826
$666$	0	0
$667$	−1.53014e7	−1.33173
$668$	0	0
$669$	−2.53754e6	−0.219203
$670$	0	0
$671$	1.11833e6	0.0958877
$672$	0	0
$673$	6.76614e6	0.575842	0.287921	−	0.957654i	$-0.407036\pi$
$673$	6.76614e6	0.575842	0.287921	+	0.957654i	$0.407036\pi$
$674$	0	0
$675$	−1.61696e6	−0.136596
$676$	0	0
$677$	2.24883e7	1.88575	0.942876	−	0.333143i	$-0.108109\pi$
$677$	2.24883e7	1.88575	0.942876	+	0.333143i	$0.108109\pi$
$678$	0	0
$679$	5.18146e6	0.431298
$680$	0	0
$681$	4.61812e6	0.381590
$682$	0	0
$683$	−2.06615e6	−0.169477	−0.0847385	−	0.996403i	$-0.527005\pi$
$683$	−2.06615e6	−0.169477	−0.0847385	+	0.996403i	$0.527005\pi$
$684$	0	0
$685$	1.14678e7	0.933800
$686$	0	0
$687$	−7.39026e6	−0.597404
$688$	0	0
$689$	−2.29634e7	−1.84284
$690$	0	0
$691$	65054.6	0.00518302	0.00259151	−	0.999997i	$-0.499175\pi$
$691$	65054.6	0.00518302	0.00259151	+	0.999997i	$0.499175\pi$
$692$	0	0
$693$	1.32337e6	0.104676
$694$	0	0
$695$	−4.22433e6	−0.331738
$696$	0	0
$697$	9.93209e6	0.774388
$698$	0	0
$699$	1.33577e6	0.103404
$700$	0	0
$701$	8.75462e6	0.672887	0.336444	−	0.941704i	$-0.390776\pi$
$701$	8.75462e6	0.672887	0.336444	+	0.941704i	$0.390776\pi$
$702$	0	0
$703$	2.71124e7	2.06909
$704$	0	0
$705$	1.31840e6	0.0999023
$706$	0	0
$707$	8.20638e6	0.617452
$708$	0	0
$709$	−9.53225e6	−0.712164	−0.356082	−	0.934455i	$-0.615888\pi$
$709$	−9.53225e6	−0.712164	−0.356082	+	0.934455i	$0.615888\pi$
$710$	0	0
$711$	−3.94966e6	−0.293012
$712$	0	0
$713$	2.09378e7	1.54243
$714$	0	0
$715$	2.22070e6	0.162452
$716$	0	0
$717$	3.37009e6	0.244818
$718$	0	0
$719$	2.23471e7	1.61213	0.806064	−	0.591829i	$-0.201594\pi$
$719$	2.23471e7	1.61213	0.806064	+	0.591829i	$0.201594\pi$
$720$	0	0
$721$	−8.32703e6	−0.596557
$722$	0	0
$723$	5.03264e6	0.358055
$724$	0	0
$725$	−1.57543e7	−1.11315
$726$	0	0
$727$	9.26653e6	0.650251	0.325125	−	0.945671i	$-0.394593\pi$
$727$	9.26653e6	0.650251	0.325125	+	0.945671i	$0.394593\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.09012e7	−0.754536
$732$	0	0
$733$	−4.71371e6	−0.324043	−0.162022	−	0.986787i	$-0.551801\pi$
$733$	−4.71371e6	−0.324043	−0.162022	+	0.986787i	$0.551801\pi$
$734$	0	0
$735$	386074.	0.0263604
$736$	0	0
$737$	1.88224e6	0.127646
$738$	0	0
$739$	2.35284e7	1.58482	0.792412	−	0.609986i	$-0.208825\pi$
$739$	2.35284e7	1.58482	0.792412	+	0.609986i	$0.208825\pi$
$740$	0	0
$741$	1.06261e7	0.710936
$742$	0	0
$743$	−9.33917e6	−0.620635	−0.310318	−	0.950633i	$-0.600435\pi$
$743$	−9.33917e6	−0.620635	−0.310318	+	0.950633i	$0.600435\pi$
$744$	0	0
$745$	−9.45938e6	−0.624413
$746$	0	0
$747$	8.01649e6	0.525633
$748$	0	0
$749$	5.72229e6	0.372705
$750$	0	0
$751$	6.81588e6	0.440983	0.220492	−	0.975389i	$-0.429234\pi$
$751$	6.81588e6	0.440983	0.220492	+	0.975389i	$0.429234\pi$
$752$	0	0
$753$	−4.34223e6	−0.279078
$754$	0	0
$755$	9.16554e6	0.585182
$756$	0	0
$757$	2.37462e6	0.150610	0.0753052	−	0.997161i	$-0.476007\pi$
$757$	2.37462e6	0.150610	0.0753052	+	0.997161i	$0.476007\pi$
$758$	0	0
$759$	−2.34602e6	−0.147818
$760$	0	0
$761$	7.50939e6	0.470049	0.235024	−	0.971989i	$-0.424483\pi$
$761$	7.50939e6	0.470049	0.235024	+	0.971989i	$0.424483\pi$
$762$	0	0
$763$	−422526.	−0.0262750
$764$	0	0
$765$	1.56398e6	0.0966221
$766$	0	0
$767$	−8.12611e6	−0.498763
$768$	0	0
$769$	−1.85630e7	−1.13197	−0.565983	−	0.824417i	$-0.691503\pi$
$769$	−1.85630e7	−1.13197	−0.565983	+	0.824417i	$0.691503\pi$
$770$	0	0
$771$	−1.15678e7	−0.700831
$772$	0	0
$773$	−2.01490e7	−1.21284	−0.606421	−	0.795144i	$-0.707395\pi$
$773$	−2.01490e7	−1.21284	−0.606421	+	0.795144i	$0.707395\pi$
$774$	0	0
$775$	2.15575e7	1.28927
$776$	0	0
$777$	1.70060e7	1.01053
$778$	0	0
$779$	3.00129e7	1.77200
$780$	0	0
$781$	−483578.	−0.0283687
$782$	0	0
$783$	5.17792e6	0.301822
$784$	0	0
$785$	1.09868e7	0.636353
$786$	0	0
$787$	2.10592e7	1.21201	0.606004	−	0.795462i	$-0.292772\pi$
$787$	2.10592e7	1.21201	0.606004	+	0.795462i	$0.292772\pi$
$788$	0	0
$789$	−6.80331e6	−0.389070
$790$	0	0
$791$	−1.05505e7	−0.599556
$792$	0	0
$793$	5.63244e6	0.318064
$794$	0	0
$795$	−1.02131e7	−0.573113
$796$	0	0
$797$	1.92818e7	1.07523	0.537617	−	0.843189i	$-0.319325\pi$
$797$	1.92818e7	1.07523	0.537617	+	0.843189i	$0.319325\pi$
$798$	0	0
$799$	3.11864e6	0.172822
$800$	0	0
$801$	−6.41868e6	−0.353479
$802$	0	0
$803$	3.50811e6	0.191993
$804$	0	0
$805$	−8.76004e6	−0.476449
$806$	0	0
$807$	1.15430e7	0.623928
$808$	0	0
$809$	1.68866e7	0.907131	0.453566	−	0.891223i	$-0.350152\pi$
$809$	1.68866e7	0.907131	0.453566	+	0.891223i	$0.350152\pi$
$810$	0	0
$811$	2.54646e7	1.35952	0.679758	−	0.733437i	$-0.262085\pi$
$811$	2.54646e7	1.35952	0.679758	+	0.733437i	$0.262085\pi$
$812$	0	0
$813$	1.24176e7	0.658889
$814$	0	0
$815$	−7.67019e6	−0.404494
$816$	0	0
$817$	−3.29414e7	−1.72658
$818$	0	0
$819$	6.66512e6	0.347215
$820$	0	0
$821$	−1.60389e7	−0.830459	−0.415229	−	0.909717i	$-0.636299\pi$
$821$	−1.60389e7	−0.830459	−0.415229	+	0.909717i	$0.636299\pi$
$822$	0	0
$823$	1.73085e7	0.890760	0.445380	−	0.895342i	$-0.353069\pi$
$823$	1.73085e7	0.890760	0.445380	+	0.895342i	$0.353069\pi$
$824$	0	0
$825$	−2.41545e6	−0.123556
$826$	0	0
$827$	7.03900e6	0.357888	0.178944	−	0.983859i	$-0.442732\pi$
$827$	7.03900e6	0.357888	0.178944	+	0.983859i	$0.442732\pi$
$828$	0	0
$829$	−1.65892e7	−0.838378	−0.419189	−	0.907899i	$-0.637685\pi$
$829$	−1.65892e7	−0.838378	−0.419189	+	0.907899i	$0.637685\pi$
$830$	0	0
$831$	−7.24810e6	−0.364101
$832$	0	0
$833$	913248.	0.0456012
$834$	0	0
$835$	−1.24409e7	−0.617497
$836$	0	0
$837$	−7.08524e6	−0.349576
$838$	0	0
$839$	2.62360e7	1.28674	0.643372	−	0.765554i	$-0.277535\pi$
$839$	2.62360e7	1.28674	0.643372	+	0.765554i	$0.277535\pi$
$840$	0	0
$841$	2.99382e7	1.45961
$842$	0	0
$843$	−1.93933e7	−0.939900
$844$	0	0
$845$	2813.96	0.000135574 0
$846$	0	0
$847$	1.97688e6	0.0946831
$848$	0	0
$849$	−1.66172e7	−0.791205
$850$	0	0
$851$	−3.01475e7	−1.42701
$852$	0	0
$853$	2.66631e7	1.25469	0.627347	−	0.778740i	$-0.284141\pi$
$853$	2.66631e7	1.25469	0.627347	+	0.778740i	$0.284141\pi$
$854$	0	0
$855$	4.72604e6	0.221097
$856$	0	0
$857$	−9.20684e6	−0.428212	−0.214106	−	0.976810i	$-0.568684\pi$
$857$	−9.20684e6	−0.428212	−0.214106	+	0.976810i	$0.568684\pi$
$858$	0	0
$859$	−1.99138e7	−0.920811	−0.460405	−	0.887709i	$-0.652296\pi$
$859$	−1.99138e7	−0.920811	−0.460405	+	0.887709i	$0.652296\pi$
$860$	0	0
$861$	1.88252e7	0.865432
$862$	0	0
$863$	140963.	0.00644285	0.00322143	−	0.999995i	$-0.498975\pi$
$863$	140963.	0.00644285	0.00322143	+	0.999995i	$0.498975\pi$
$864$	0	0
$865$	1.86414e7	0.847106
$866$	0	0
$867$	−9.07918e6	−0.410203
$868$	0	0
$869$	−5.90011e6	−0.265040
$870$	0	0
$871$	9.47987e6	0.423406
$872$	0	0
$873$	3.10833e6	0.138036
$874$	0	0
$875$	−2.17266e7	−0.959338
$876$	0	0
$877$	1.37863e7	0.605268	0.302634	−	0.953107i	$-0.402134\pi$
$877$	1.37863e7	0.605268	0.302634	+	0.953107i	$0.402134\pi$
$878$	0	0
$879$	2.51600e7	1.09835
$880$	0	0
$881$	−2.76950e7	−1.20216	−0.601079	−	0.799190i	$-0.705262\pi$
$881$	−2.76950e7	−1.20216	−0.601079	+	0.799190i	$0.705262\pi$
$882$	0	0
$883$	−3.95137e7	−1.70548	−0.852739	−	0.522337i	$-0.825060\pi$
$883$	−3.95137e7	−1.70548	−0.852739	+	0.522337i	$0.825060\pi$
$884$	0	0
$885$	−3.61414e6	−0.155112
$886$	0	0
$887$	2.61806e7	1.11730	0.558652	−	0.829402i	$-0.311319\pi$
$887$	2.61806e7	1.11730	0.558652	+	0.829402i	$0.311319\pi$
$888$	0	0
$889$	−3.45487e6	−0.146615
$890$	0	0
$891$	793881.	0.0335013
$892$	0	0
$893$	9.42396e6	0.395462
$894$	0	0
$895$	6.82202e6	0.284679
$896$	0	0
$897$	−1.18157e7	−0.490318
$898$	0	0
$899$	−6.90328e7	−2.84876
$900$	0	0
$901$	−2.41588e7	−0.991433
$902$	0	0
$903$	−2.06621e7	−0.843246
$904$	0	0
$905$	1.53048e7	0.621163
$906$	0	0
$907$	2.93574e7	1.18495	0.592474	−	0.805589i	$-0.298151\pi$
$907$	2.93574e7	1.18495	0.592474	+	0.805589i	$0.298151\pi$
$908$	0	0
$909$	4.92296e6	0.197614
$910$	0	0
$911$	−1.77251e7	−0.707608	−0.353804	−	0.935320i	$-0.615112\pi$
$911$	−1.77251e7	−0.707608	−0.353804	+	0.935320i	$0.615112\pi$
$912$	0	0
$913$	1.19752e7	0.475453
$914$	0	0
$915$	2.50506e6	0.0989159
$916$	0	0
$917$	2.47185e7	0.970729
$918$	0	0
$919$	−2.64714e7	−1.03392	−0.516962	−	0.856008i	$-0.672937\pi$
$919$	−2.64714e7	−1.03392	−0.516962	+	0.856008i	$0.672937\pi$
$920$	0	0
$921$	−1.68174e7	−0.653297
$922$	0	0
$923$	−2.43553e6	−0.0941001
$924$	0	0
$925$	−3.10398e7	−1.19279
$926$	0	0
$927$	−4.99534e6	−0.190926
$928$	0	0
$929$	4.44247e6	0.168883	0.0844413	−	0.996428i	$-0.473089\pi$
$929$	4.44247e6	0.168883	0.0844413	+	0.996428i	$0.473089\pi$
$930$	0	0
$931$	2.75967e6	0.104348
$932$	0	0
$933$	1.30534e7	0.490932
$934$	0	0
$935$	2.33631e6	0.0873980
$936$	0	0
$937$	−1.68366e7	−0.626479	−0.313240	−	0.949674i	$-0.601414\pi$
$937$	−1.68366e7	−0.626479	−0.313240	+	0.949674i	$0.601414\pi$
$938$	0	0
$939$	−2.98073e7	−1.10321
$940$	0	0
$941$	−3.16888e7	−1.16663	−0.583314	−	0.812247i	$-0.698244\pi$
$941$	−3.16888e7	−1.16663	−0.583314	+	0.812247i	$0.698244\pi$
$942$	0	0
$943$	−3.33727e7	−1.22212
$944$	0	0
$945$	2.96435e6	0.107982
$946$	0	0
$947$	−1.26628e7	−0.458833	−0.229416	−	0.973328i	$-0.573682\pi$
$947$	−1.26628e7	−0.458833	−0.229416	+	0.973328i	$0.573682\pi$
$948$	0	0
$949$	1.76686e7	0.636847
$950$	0	0
$951$	9.60495e6	0.344385
$952$	0	0
$953$	−3.31163e7	−1.18116	−0.590581	−	0.806978i	$-0.701102\pi$
$953$	−3.31163e7	−1.18116	−0.590581	+	0.806978i	$0.701102\pi$
$954$	0	0
$955$	1.85436e7	0.657939
$956$	0	0
$957$	7.73492e6	0.273008
$958$	0	0
$959$	5.14159e7	1.80531
$960$	0	0
$961$	6.58323e7	2.29948
$962$	0	0
$963$	3.43277e6	0.119283
$964$	0	0
$965$	5.19720e6	0.179660
$966$	0	0
$967$	4.65490e6	0.160083	0.0800413	−	0.996792i	$-0.474495\pi$
$967$	4.65490e6	0.160083	0.0800413	+	0.996792i	$0.474495\pi$
$968$	0	0
$969$	1.11793e7	0.382477
$970$	0	0
$971$	−1.90251e7	−0.647558	−0.323779	−	0.946133i	$-0.604953\pi$
$971$	−1.90251e7	−0.647558	−0.323779	+	0.946133i	$0.604953\pi$
$972$	0	0
$973$	−1.89398e7	−0.641347
$974$	0	0
$975$	−1.21654e7	−0.409840
$976$	0	0
$977$	−5.32853e7	−1.78596	−0.892978	−	0.450100i	$-0.851388\pi$
$977$	−5.32853e7	−1.78596	−0.892978	+	0.450100i	$0.851388\pi$
$978$	0	0
$979$	−9.58839e6	−0.319734
$980$	0	0
$981$	−253471.	−0.00840923
$982$	0	0
$983$	−1.39336e7	−0.459918	−0.229959	−	0.973200i	$-0.573859\pi$
$983$	−1.39336e7	−0.459918	−0.229959	+	0.973200i	$0.573859\pi$
$984$	0	0
$985$	1.19032e7	0.390907
$986$	0	0
$987$	5.91107e6	0.193140
$988$	0	0
$989$	3.66290e7	1.19079
$990$	0	0
$991$	−5.74985e7	−1.85983	−0.929913	−	0.367780i	$-0.880118\pi$
$991$	−5.74985e7	−1.85983	−0.929913	+	0.367780i	$0.880118\pi$
$992$	0	0
$993$	−3.22311e7	−1.03730
$994$	0	0
$995$	2.31567e7	0.741515
$996$	0	0
$997$	1.20706e7	0.384584	0.192292	−	0.981338i	$-0.438408\pi$
$997$	1.20706e7	0.384584	0.192292	+	0.981338i	$0.438408\pi$
$998$	0	0
$999$	1.02018e7	0.323417

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.a.y.1.2		3
4.3	odd	2		132.6.a.e.1.2	✓	3
12.11	even	2		396.6.a.g.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.6.a.e.1.2	✓	3	4.3	odd	2
396.6.a.g.1.2		3	12.11	even	2
528.6.a.y.1.2		3	1.1	even	1	trivial

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.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +30.1157 q^{5} +135.024 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +30.1157 q^{5} +135.024 q^{7} +81.0000 q^{9} +121.000 q^{11} +609.415 q^{13} +271.041 q^{15} +641.139 q^{17} +1937.40 q^{19} +1215.21 q^{21} -2154.29 q^{23} -2218.05 q^{25} +729.000 q^{27} +7102.77 q^{29} -9719.13 q^{31} +1089.00 q^{33} +4066.33 q^{35} +13994.2 q^{37} +5484.73 q^{39} +15491.3 q^{41} -17002.8 q^{43} +2439.37 q^{45} +4864.22 q^{47} +1424.41 q^{49} +5770.25 q^{51} -37681.0 q^{53} +3644.00 q^{55} +17436.6 q^{57} -13334.3 q^{59} +9242.38 q^{61} +10936.9 q^{63} +18352.9 q^{65} +15555.7 q^{67} -19388.6 q^{69} -3996.51 q^{71} +28992.7 q^{73} -19962.4 q^{75} +16337.9 q^{77} -48761.3 q^{79} +6561.00 q^{81} +98969.0 q^{83} +19308.3 q^{85} +63925.0 q^{87} -79242.9 q^{89} +82285.5 q^{91} -87472.1 q^{93} +58346.2 q^{95} +38374.4 q^{97} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 27 q^{3} + 36 q^{5} + 52 q^{7} + 243 q^{9}+O(q^{10}) \) 3 * q + 27 * q^3 + 36 * q^5 + 52 * q^7 + 243 * q^9	\( 3 q + 27 q^{3} + 36 q^{5} + 52 q^{7} + 243 q^{9} + 363 q^{11} - 888 q^{13} + 324 q^{15} + 1516 q^{17} - 1386 q^{19} + 468 q^{21} - 1030 q^{23} + 11273 q^{25} + 2187 q^{27} + 10904 q^{29} + 2568 q^{31} + 3267 q^{33} - 33596 q^{35} + 32910 q^{37} - 7992 q^{39} - 164 q^{41} - 25018 q^{43} + 2916 q^{45} + 26542 q^{47} + 42243 q^{49} + 13644 q^{51} + 24184 q^{53} + 4356 q^{55} - 12474 q^{57} + 39528 q^{59} + 17128 q^{61} + 4212 q^{63} + 6400 q^{65} - 49596 q^{67} - 9270 q^{69} + 50714 q^{71} + 64970 q^{73} + 101457 q^{75} + 6292 q^{77} - 64808 q^{79} + 19683 q^{81} + 117912 q^{83} + 4188 q^{85} + 98136 q^{87} - 122442 q^{89} + 158764 q^{91} + 23112 q^{93} + 177628 q^{95} + 150190 q^{97} + 29403 q^{99}+O(q^{100}) \) 3 * q + 27 * q^3 + 36 * q^5 + 52 * q^7 + 243 * q^9 + 363 * q^11 - 888 * q^13 + 324 * q^15 + 1516 * q^17 - 1386 * q^19 + 468 * q^21 - 1030 * q^23 + 11273 * q^25 + 2187 * q^27 + 10904 * q^29 + 2568 * q^31 + 3267 * q^33 - 33596 * q^35 + 32910 * q^37 - 7992 * q^39 - 164 * q^41 - 25018 * q^43 + 2916 * q^45 + 26542 * q^47 + 42243 * q^49 + 13644 * q^51 + 24184 * q^53 + 4356 * q^55 - 12474 * q^57 + 39528 * q^59 + 17128 * q^61 + 4212 * q^63 + 6400 * q^65 - 49596 * q^67 - 9270 * q^69 + 50714 * q^71 + 64970 * q^73 + 101457 * q^75 + 6292 * q^77 - 64808 * q^79 + 19683 * q^81 + 117912 * q^83 + 4188 * q^85 + 98136 * q^87 - 122442 * q^89 + 158764 * q^91 + 23112 * q^93 + 177628 * q^95 + 150190 * q^97 + 29403 * q^99

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