LMFDB - Embedded newform 528.6.a.z.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 2117x^{2} + 1518x + 1092672 $ x^4 - 2x^3 - 2117x^2 + 1518*x + 1092672
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 264)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$29.5436$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} +47.0872 q^{5} +114.978 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +47.0872 q^{5} +114.978 q^{7} +81.0000 q^{9} -121.000 q^{11} +362.277 q^{13} -423.785 q^{15} -1289.15 q^{17} -1491.20 q^{19} -1034.80 q^{21} -3936.18 q^{23} -907.798 q^{25} -729.000 q^{27} +8955.63 q^{29} -3652.38 q^{31} +1089.00 q^{33} +5413.97 q^{35} -13689.2 q^{37} -3260.49 q^{39} +17810.5 q^{41} +7212.26 q^{43} +3814.06 q^{45} -6378.07 q^{47} -3587.17 q^{49} +11602.3 q^{51} -30639.4 q^{53} -5697.55 q^{55} +13420.8 q^{57} -45573.3 q^{59} +49238.1 q^{61} +9313.18 q^{63} +17058.6 q^{65} -9147.39 q^{67} +35425.7 q^{69} +61058.6 q^{71} +33357.0 q^{73} +8170.18 q^{75} -13912.3 q^{77} -84149.4 q^{79} +6561.00 q^{81} -64944.5 q^{83} -60702.4 q^{85} -80600.7 q^{87} +99595.7 q^{89} +41653.7 q^{91} +32871.4 q^{93} -70216.2 q^{95} -5373.67 q^{97} -9801.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{3} - 44 q^{5} - 38 q^{7} + 324 q^{9}+O(q^{10}) $ 4 * q - 36 * q^3 - 44 * q^5 - 38 * q^7 + 324 * q^9	$ 4 q - 36 q^{3} - 44 q^{5} - 38 q^{7} + 324 q^{9} - 484 q^{11} + 288 q^{13} + 396 q^{15} + 418 q^{17} + 10 q^{19} + 342 q^{21} - 576 q^{23} + 4932 q^{25} - 2916 q^{27} + 11930 q^{29} - 9968 q^{31} + 4356 q^{33} + 428 q^{35} + 13740 q^{37} - 2592 q^{39} + 29766 q^{41} - 25650 q^{43} - 3564 q^{45} + 5776 q^{47} + 39888 q^{49} - 3762 q^{51} - 7840 q^{53} + 5324 q^{55} - 90 q^{57} - 28800 q^{59} + 33932 q^{61} - 3078 q^{63} + 33216 q^{65} - 83056 q^{67} + 5184 q^{69} - 21336 q^{71} + 27044 q^{73} - 44388 q^{75} + 4598 q^{77} - 102542 q^{79} + 26244 q^{81} - 64996 q^{83} - 12132 q^{85} - 107370 q^{87} + 37888 q^{89} - 273612 q^{91} + 89712 q^{93} - 254380 q^{95} - 20996 q^{97} - 39204 q^{99}+O(q^{100}) $ 4 * q - 36 * q^3 - 44 * q^5 - 38 * q^7 + 324 * q^9 - 484 * q^11 + 288 * q^13 + 396 * q^15 + 418 * q^17 + 10 * q^19 + 342 * q^21 - 576 * q^23 + 4932 * q^25 - 2916 * q^27 + 11930 * q^29 - 9968 * q^31 + 4356 * q^33 + 428 * q^35 + 13740 * q^37 - 2592 * q^39 + 29766 * q^41 - 25650 * q^43 - 3564 * q^45 + 5776 * q^47 + 39888 * q^49 - 3762 * q^51 - 7840 * q^53 + 5324 * q^55 - 90 * q^57 - 28800 * q^59 + 33932 * q^61 - 3078 * q^63 + 33216 * q^65 - 83056 * q^67 + 5184 * q^69 - 21336 * q^71 + 27044 * q^73 - 44388 * q^75 + 4598 * q^77 - 102542 * q^79 + 26244 * q^81 - 64996 * q^83 - 12132 * q^85 - 107370 * q^87 + 37888 * q^89 - 273612 * q^91 + 89712 * q^93 - 254380 * q^95 - 20996 * q^97 - 39204 * 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
$5$	47.0872	0.842321	0.421160	−	0.906986i	$-0.361623\pi$
$5$	47.0872	0.842321	0.421160	+	0.906986i	$0.361623\pi$
$6$	0	0
$7$	114.978	0.886886	0.443443	−	0.896303i	$-0.353757\pi$
$7$	114.978	0.886886	0.443443	+	0.896303i	$0.353757\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	362.277	0.594541	0.297271	−	0.954793i	$-0.403924\pi$
$13$	362.277	0.594541	0.297271	+	0.954793i	$0.403924\pi$
$14$	0	0
$15$	−423.785	−0.486314
$16$	0	0
$17$	−1289.15	−1.08188	−0.540942	−	0.841060i	$-0.681932\pi$
$17$	−1289.15	−1.08188	−0.540942	+	0.841060i	$0.681932\pi$
$18$	0	0
$19$	−1491.20	−0.947656	−0.473828	−	0.880618i	$-0.657128\pi$
$19$	−1491.20	−0.947656	−0.473828	+	0.880618i	$0.657128\pi$
$20$	0	0
$21$	−1034.80	−0.512044
$22$	0	0
$23$	−3936.18	−1.55151	−0.775757	−	0.631032i	$-0.782632\pi$
$23$	−3936.18	−1.55151	−0.775757	+	0.631032i	$0.782632\pi$
$24$	0	0
$25$	−907.798	−0.290495
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	8955.63	1.97743	0.988716	−	0.149805i	$-0.0478645\pi$
$29$	8955.63	1.97743	0.988716	+	0.149805i	$0.0478645\pi$
$30$	0	0
$31$	−3652.38	−0.682609	−0.341304	−	0.939953i	$-0.610869\pi$
$31$	−3652.38	−0.682609	−0.341304	+	0.939953i	$0.610869\pi$
$32$	0	0
$33$	1089.00	0.174078
$34$	0	0
$35$	5413.97	0.747043
$36$	0	0
$37$	−13689.2	−1.64390	−0.821948	−	0.569562i	$-0.807113\pi$
$37$	−13689.2	−1.64390	−0.821948	+	0.569562i	$0.807113\pi$
$38$	0	0
$39$	−3260.49	−0.343259
$40$	0	0
$41$	17810.5	1.65469	0.827344	−	0.561696i	$-0.189851\pi$
$41$	17810.5	1.65469	0.827344	+	0.561696i	$0.189851\pi$
$42$	0	0
$43$	7212.26	0.594840	0.297420	−	0.954747i	$-0.403874\pi$
$43$	7212.26	0.594840	0.297420	+	0.954747i	$0.403874\pi$
$44$	0	0
$45$	3814.06	0.280774
$46$	0	0
$47$	−6378.07	−0.421157	−0.210579	−	0.977577i	$-0.567535\pi$
$47$	−6378.07	−0.421157	−0.210579	+	0.977577i	$0.567535\pi$
$48$	0	0
$49$	−3587.17	−0.213433
$50$	0	0
$51$	11602.3	0.624626
$52$	0	0
$53$	−30639.4	−1.49827	−0.749135	−	0.662418i	$-0.769531\pi$
$53$	−30639.4	−1.49827	−0.749135	+	0.662418i	$0.769531\pi$
$54$	0	0
$55$	−5697.55	−0.253969
$56$	0	0
$57$	13420.8	0.547129
$58$	0	0
$59$	−45573.3	−1.70444	−0.852218	−	0.523187i	$-0.824743\pi$
$59$	−45573.3	−1.70444	−0.852218	+	0.523187i	$0.824743\pi$
$60$	0	0
$61$	49238.1	1.69425	0.847124	−	0.531395i	$-0.178332\pi$
$61$	49238.1	1.69425	0.847124	+	0.531395i	$0.178332\pi$
$62$	0	0
$63$	9313.18	0.295629
$64$	0	0
$65$	17058.6	0.500795
$66$	0	0
$67$	−9147.39	−0.248949	−0.124475	−	0.992223i	$-0.539724\pi$
$67$	−9147.39	−0.248949	−0.124475	+	0.992223i	$0.539724\pi$
$68$	0	0
$69$	35425.7	0.895767
$70$	0	0
$71$	61058.6	1.43748	0.718738	−	0.695281i	$-0.244720\pi$
$71$	61058.6	1.43748	0.718738	+	0.695281i	$0.244720\pi$
$72$	0	0
$73$	33357.0	0.732622	0.366311	−	0.930492i	$-0.380621\pi$
$73$	33357.0	0.732622	0.366311	+	0.930492i	$0.380621\pi$
$74$	0	0
$75$	8170.18	0.167718
$76$	0	0
$77$	−13912.3	−0.267406
$78$	0	0
$79$	−84149.4	−1.51699	−0.758496	−	0.651678i	$-0.774066\pi$
$79$	−84149.4	−1.51699	−0.758496	+	0.651678i	$0.774066\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−64944.5	−1.03478	−0.517389	−	0.855750i	$-0.673096\pi$
$83$	−64944.5	−1.03478	−0.517389	+	0.855750i	$0.673096\pi$
$84$	0	0
$85$	−60702.4	−0.911294
$86$	0	0
$87$	−80600.7	−1.14167
$88$	0	0
$89$	99595.7	1.33280	0.666401	−	0.745593i	$-0.267834\pi$
$89$	99595.7	1.33280	0.666401	+	0.745593i	$0.267834\pi$
$90$	0	0
$91$	41653.7	0.527290
$92$	0	0
$93$	32871.4	0.394104
$94$	0	0
$95$	−70216.2	−0.798230
$96$	0	0
$97$	−5373.67	−0.0579885	−0.0289942	−	0.999580i	$-0.509230\pi$
$97$	−5373.67	−0.0579885	−0.0289942	+	0.999580i	$0.509230\pi$
$98$	0	0
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−200358.	−1.95435	−0.977175	−	0.212435i	$-0.931860\pi$
$101$	−200358.	−1.95435	−0.977175	+	0.212435i	$0.931860\pi$
$102$	0	0
$103$	−98843.5	−0.918026	−0.459013	−	0.888430i	$-0.651797\pi$
$103$	−98843.5	−0.918026	−0.459013	+	0.888430i	$0.651797\pi$
$104$	0	0
$105$	−48725.7	−0.431305
$106$	0	0
$107$	82703.7	0.698337	0.349169	−	0.937060i	$-0.386464\pi$
$107$	82703.7	0.698337	0.349169	+	0.937060i	$0.386464\pi$
$108$	0	0
$109$	116860.	0.942107	0.471054	−	0.882105i	$-0.343874\pi$
$109$	116860.	0.942107	0.471054	+	0.882105i	$0.343874\pi$
$110$	0	0
$111$	123203.	0.949104
$112$	0	0
$113$	−92535.0	−0.681726	−0.340863	−	0.940113i	$-0.610719\pi$
$113$	−92535.0	−0.681726	−0.340863	+	0.940113i	$0.610719\pi$
$114$	0	0
$115$	−185344.	−1.30687
$116$	0	0
$117$	29344.4	0.198180
$118$	0	0
$119$	−148223.	−0.959509
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−160294.	−0.955334
$124$	0	0
$125$	−189893.	−1.08701
$126$	0	0
$127$	−184391.	−1.01445	−0.507223	−	0.861815i	$-0.669328\pi$
$127$	−184391.	−1.01445	−0.507223	+	0.861815i	$0.669328\pi$
$128$	0	0
$129$	−64910.4	−0.343431
$130$	0	0
$131$	−139520.	−0.710327	−0.355164	−	0.934804i	$-0.615575\pi$
$131$	−139520.	−0.710327	−0.355164	+	0.934804i	$0.615575\pi$
$132$	0	0
$133$	−171454.	−0.840463
$134$	0	0
$135$	−34326.5	−0.162105
$136$	0	0
$137$	−45180.2	−0.205659	−0.102829	−	0.994699i	$-0.532790\pi$
$137$	−45180.2	−0.205659	−0.102829	+	0.994699i	$0.532790\pi$
$138$	0	0
$139$	12218.0	0.0536369	0.0268184	−	0.999640i	$-0.491462\pi$
$139$	12218.0	0.0536369	0.0268184	+	0.999640i	$0.491462\pi$
$140$	0	0
$141$	57402.6	0.243155
$142$	0	0
$143$	−43835.5	−0.179261
$144$	0	0
$145$	421696.	1.66563
$146$	0	0
$147$	32284.5	0.123226
$148$	0	0
$149$	140013.	0.516659	0.258329	−	0.966057i	$-0.416828\pi$
$149$	140013.	0.516659	0.258329	+	0.966057i	$0.416828\pi$
$150$	0	0
$151$	−193028.	−0.688933	−0.344467	−	0.938799i	$-0.611940\pi$
$151$	−193028.	−0.688933	−0.344467	+	0.938799i	$0.611940\pi$
$152$	0	0
$153$	−104421.	−0.360628
$154$	0	0
$155$	−171980.	−0.574976
$156$	0	0
$157$	−488732.	−1.58242	−0.791210	−	0.611544i	$-0.790549\pi$
$157$	−488732.	−1.58242	−0.791210	+	0.611544i	$0.790549\pi$
$158$	0	0
$159$	275754.	0.865026
$160$	0	0
$161$	−452573.	−1.37602
$162$	0	0
$163$	173619.	0.511833	0.255916	−	0.966699i	$-0.417623\pi$
$163$	173619.	0.511833	0.255916	+	0.966699i	$0.417623\pi$
$164$	0	0
$165$	51277.9	0.146629
$166$	0	0
$167$	253937.	0.704586	0.352293	−	0.935890i	$-0.385402\pi$
$167$	253937.	0.704586	0.352293	+	0.935890i	$0.385402\pi$
$168$	0	0
$169$	−240049.	−0.646521
$170$	0	0
$171$	−120787.	−0.315885
$172$	0	0
$173$	−488693.	−1.24143	−0.620713	−	0.784038i	$-0.713157\pi$
$173$	−488693.	−1.24143	−0.620713	+	0.784038i	$0.713157\pi$
$174$	0	0
$175$	−104376.	−0.257636
$176$	0	0
$177$	410160.	0.984057
$178$	0	0
$179$	615409.	1.43559	0.717796	−	0.696253i	$-0.245151\pi$
$179$	615409.	1.43559	0.717796	+	0.696253i	$0.245151\pi$
$180$	0	0
$181$	−664790.	−1.50830	−0.754151	−	0.656701i	$-0.771951\pi$
$181$	−664790.	−1.50830	−0.754151	+	0.656701i	$0.771951\pi$
$182$	0	0
$183$	−443143.	−0.978175
$184$	0	0
$185$	−644587.	−1.38469
$186$	0	0
$187$	155987.	0.326201
$188$	0	0
$189$	−83818.6	−0.170681
$190$	0	0
$191$	−63718.7	−0.126381	−0.0631907	−	0.998001i	$-0.520128\pi$
$191$	−63718.7	−0.126381	−0.0631907	+	0.998001i	$0.520128\pi$
$192$	0	0
$193$	−760331.	−1.46930	−0.734649	−	0.678448i	$-0.762653\pi$
$193$	−760331.	−1.46930	−0.734649	+	0.678448i	$0.762653\pi$
$194$	0	0
$195$	−153527.	−0.289134
$196$	0	0
$197$	−73889.2	−0.135649	−0.0678243	−	0.997697i	$-0.521606\pi$
$197$	−73889.2	−0.135649	−0.0678243	+	0.997697i	$0.521606\pi$
$198$	0	0
$199$	−140823.	−0.252082	−0.126041	−	0.992025i	$-0.540227\pi$
$199$	−140823.	−0.252082	−0.126041	+	0.992025i	$0.540227\pi$
$200$	0	0
$201$	82326.5	0.143731
$202$	0	0
$203$	1.02970e6	1.75376
$204$	0	0
$205$	838645.	1.39378
$206$	0	0
$207$	−318831.	−0.517171
$208$	0	0
$209$	180435.	0.285729
$210$	0	0
$211$	−70030.6	−0.108288	−0.0541441	−	0.998533i	$-0.517243\pi$
$211$	−70030.6	−0.108288	−0.0541441	+	0.998533i	$0.517243\pi$
$212$	0	0
$213$	−549527.	−0.829927
$214$	0	0
$215$	339605.	0.501046
$216$	0	0
$217$	−419942.	−0.605396
$218$	0	0
$219$	−300213.	−0.422980
$220$	0	0
$221$	−467029.	−0.643225
$222$	0	0
$223$	235914.	0.317681	0.158841	−	0.987304i	$-0.449224\pi$
$223$	235914.	0.317681	0.158841	+	0.987304i	$0.449224\pi$
$224$	0	0
$225$	−73531.6	−0.0968318
$226$	0	0
$227$	758879.	0.977480	0.488740	−	0.872430i	$-0.337457\pi$
$227$	758879.	0.977480	0.488740	+	0.872430i	$0.337457\pi$
$228$	0	0
$229$	428227.	0.539616	0.269808	−	0.962914i	$-0.413040\pi$
$229$	428227.	0.539616	0.269808	+	0.962914i	$0.413040\pi$
$230$	0	0
$231$	125211.	0.154387
$232$	0	0
$233$	132082.	0.159388	0.0796939	−	0.996819i	$-0.474606\pi$
$233$	132082.	0.159388	0.0796939	+	0.996819i	$0.474606\pi$
$234$	0	0
$235$	−300325.	−0.354750
$236$	0	0
$237$	757345.	0.875836
$238$	0	0
$239$	127997.	0.144946	0.0724729	−	0.997370i	$-0.476911\pi$
$239$	127997.	0.144946	0.0724729	+	0.997370i	$0.476911\pi$
$240$	0	0
$241$	732055.	0.811897	0.405949	−	0.913896i	$-0.366941\pi$
$241$	732055.	0.811897	0.405949	+	0.913896i	$0.366941\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−168910.	−0.179779
$246$	0	0
$247$	−540225.	−0.563420
$248$	0	0
$249$	584501.	0.597430
$250$	0	0
$251$	−1.18468e6	−1.18691	−0.593455	−	0.804867i	$-0.702237\pi$
$251$	−1.18468e6	−1.18691	−0.593455	+	0.804867i	$0.702237\pi$
$252$	0	0
$253$	476278.	0.467799
$254$	0	0
$255$	546322.	0.526136
$256$	0	0
$257$	873545.	0.824997	0.412499	−	0.910958i	$-0.364656\pi$
$257$	873545.	0.824997	0.412499	+	0.910958i	$0.364656\pi$
$258$	0	0
$259$	−1.57395e6	−1.45795
$260$	0	0
$261$	725406.	0.659144
$262$	0	0
$263$	1.81086e6	1.61434	0.807172	−	0.590317i	$-0.200997\pi$
$263$	1.81086e6	1.61434	0.807172	+	0.590317i	$0.200997\pi$
$264$	0	0
$265$	−1.44272e6	−1.26202
$266$	0	0
$267$	−896362.	−0.769494
$268$	0	0
$269$	−1.50052e6	−1.26433	−0.632167	−	0.774832i	$-0.717834\pi$
$269$	−1.50052e6	−1.26433	−0.632167	+	0.774832i	$0.717834\pi$
$270$	0	0
$271$	1.76632e6	1.46098	0.730491	−	0.682922i	$-0.239291\pi$
$271$	1.76632e6	1.46098	0.730491	+	0.682922i	$0.239291\pi$
$272$	0	0
$273$	−374883.	−0.304431
$274$	0	0
$275$	109844.	0.0875877
$276$	0	0
$277$	−1.67769e6	−1.31375	−0.656875	−	0.754000i	$-0.728122\pi$
$277$	−1.67769e6	−1.31375	−0.656875	+	0.754000i	$0.728122\pi$
$278$	0	0
$279$	−295843.	−0.227536
$280$	0	0
$281$	−812722.	−0.614011	−0.307005	−	0.951708i	$-0.599327\pi$
$281$	−812722.	−0.614011	−0.307005	+	0.951708i	$0.599327\pi$
$282$	0	0
$283$	−1.83530e6	−1.36220	−0.681101	−	0.732189i	$-0.738499\pi$
$283$	−1.83530e6	−1.36220	−0.681101	+	0.732189i	$0.738499\pi$
$284$	0	0
$285$	631946.	0.460858
$286$	0	0
$287$	2.04780e6	1.46752
$288$	0	0
$289$	242050.	0.170475
$290$	0	0
$291$	48363.0	0.0334797
$292$	0	0
$293$	880729.	0.599340	0.299670	−	0.954043i	$-0.403123\pi$
$293$	880729.	0.599340	0.299670	+	0.954043i	$0.403123\pi$
$294$	0	0
$295$	−2.14592e6	−1.43568
$296$	0	0
$297$	88209.0	0.0580259
$298$	0	0
$299$	−1.42599e6	−0.922439
$300$	0	0
$301$	829248.	0.527556
$302$	0	0
$303$	1.80322e6	1.12834
$304$	0	0
$305$	2.31848e6	1.42710
$306$	0	0
$307$	−948307.	−0.574252	−0.287126	−	0.957893i	$-0.592700\pi$
$307$	−948307.	−0.574252	−0.287126	+	0.957893i	$0.592700\pi$
$308$	0	0
$309$	889591.	0.530022
$310$	0	0
$311$	1.10358e6	0.646997	0.323499	−	0.946229i	$-0.395141\pi$
$311$	1.10358e6	0.646997	0.323499	+	0.946229i	$0.395141\pi$
$312$	0	0
$313$	2.41603e6	1.39393	0.696966	−	0.717104i	$-0.254533\pi$
$313$	2.41603e6	1.39393	0.696966	+	0.717104i	$0.254533\pi$
$314$	0	0
$315$	438531.	0.249014
$316$	0	0
$317$	−1.38660e6	−0.775001	−0.387501	−	0.921869i	$-0.626662\pi$
$317$	−1.38660e6	−0.775001	−0.387501	+	0.921869i	$0.626662\pi$
$318$	0	0
$319$	−1.08363e6	−0.596218
$320$	0	0
$321$	−744333.	−0.403185
$322$	0	0
$323$	1.92237e6	1.02525
$324$	0	0
$325$	−328874.	−0.172711
$326$	0	0
$327$	−1.05174e6	−0.543926
$328$	0	0
$329$	−733334.	−0.373519
$330$	0	0
$331$	−1.59482e6	−0.800097	−0.400048	−	0.916494i	$-0.631007\pi$
$331$	−1.59482e6	−0.800097	−0.400048	+	0.916494i	$0.631007\pi$
$332$	0	0
$333$	−1.10883e6	−0.547966
$334$	0	0
$335$	−430725.	−0.209695
$336$	0	0
$337$	−908191.	−0.435614	−0.217807	−	0.975992i	$-0.569890\pi$
$337$	−908191.	−0.435614	−0.217807	+	0.975992i	$0.569890\pi$
$338$	0	0
$339$	832815.	0.393595
$340$	0	0
$341$	441938.	0.205814
$342$	0	0
$343$	−2.34487e6	−1.07618
$344$	0	0
$345$	1.66809e6	0.754523
$346$	0	0
$347$	166693.	0.0743181	0.0371590	−	0.999309i	$-0.488169\pi$
$347$	166693.	0.0743181	0.0371590	+	0.999309i	$0.488169\pi$
$348$	0	0
$349$	−2.64737e6	−1.16346	−0.581730	−	0.813382i	$-0.697624\pi$
$349$	−2.64737e6	−1.16346	−0.581730	+	0.813382i	$0.697624\pi$
$350$	0	0
$351$	−264100.	−0.114420
$352$	0	0
$353$	3.51059e6	1.49949	0.749745	−	0.661727i	$-0.230176\pi$
$353$	3.51059e6	1.49949	0.749745	+	0.661727i	$0.230176\pi$
$354$	0	0
$355$	2.87507e6	1.21082
$356$	0	0
$357$	1.33401e6	0.553973
$358$	0	0
$359$	2.18715e6	0.895660	0.447830	−	0.894119i	$-0.352197\pi$
$359$	2.18715e6	0.895660	0.447830	+	0.894119i	$0.352197\pi$
$360$	0	0
$361$	−252436.	−0.101949
$362$	0	0
$363$	−131769.	−0.0524864
$364$	0	0
$365$	1.57069e6	0.617103
$366$	0	0
$367$	−965069.	−0.374018	−0.187009	−	0.982358i	$-0.559879\pi$
$367$	−965069.	−0.374018	−0.187009	+	0.982358i	$0.559879\pi$
$368$	0	0
$369$	1.44265e6	0.551562
$370$	0	0
$371$	−3.52284e6	−1.32879
$372$	0	0
$373$	2.59031e6	0.964004	0.482002	−	0.876170i	$-0.339910\pi$
$373$	2.59031e6	0.964004	0.482002	+	0.876170i	$0.339910\pi$
$374$	0	0
$375$	1.70904e6	0.627586
$376$	0	0
$377$	3.24442e6	1.17566
$378$	0	0
$379$	−2.46520e6	−0.881563	−0.440782	−	0.897614i	$-0.645299\pi$
$379$	−2.46520e6	−0.881563	−0.440782	+	0.897614i	$0.645299\pi$
$380$	0	0
$381$	1.65951e6	0.585691
$382$	0	0
$383$	110495.	0.0384899	0.0192450	−	0.999815i	$-0.493874\pi$
$383$	110495.	0.0384899	0.0192450	+	0.999815i	$0.493874\pi$
$384$	0	0
$385$	−655090.	−0.225242
$386$	0	0
$387$	584193.	0.198280
$388$	0	0
$389$	433701.	0.145317	0.0726584	−	0.997357i	$-0.476852\pi$
$389$	433701.	0.145317	0.0726584	+	0.997357i	$0.476852\pi$
$390$	0	0
$391$	5.07433e6	1.67856
$392$	0	0
$393$	1.25568e6	0.410108
$394$	0	0
$395$	−3.96236e6	−1.27779
$396$	0	0
$397$	4.24027e6	1.35026	0.675130	−	0.737699i	$-0.264087\pi$
$397$	4.24027e6	1.35026	0.675130	+	0.737699i	$0.264087\pi$
$398$	0	0
$399$	1.54309e6	0.485241
$400$	0	0
$401$	−5.03680e6	−1.56420	−0.782102	−	0.623150i	$-0.785852\pi$
$401$	−5.03680e6	−1.56420	−0.782102	+	0.623150i	$0.785852\pi$
$402$	0	0
$403$	−1.32317e6	−0.405839
$404$	0	0
$405$	308939.	0.0935912
$406$	0	0
$407$	1.65640e6	0.495654
$408$	0	0
$409$	2.82671e6	0.835550	0.417775	−	0.908551i	$-0.362810\pi$
$409$	2.82671e6	0.835550	0.417775	+	0.908551i	$0.362810\pi$
$410$	0	0
$411$	406622.	0.118737
$412$	0	0
$413$	−5.23991e6	−1.51164
$414$	0	0
$415$	−3.05806e6	−0.871616
$416$	0	0
$417$	−109962.	−0.0309673
$418$	0	0
$419$	−641023.	−0.178377	−0.0891885	−	0.996015i	$-0.528427\pi$
$419$	−641023.	−0.178377	−0.0891885	+	0.996015i	$0.528427\pi$
$420$	0	0
$421$	857213.	0.235713	0.117856	−	0.993031i	$-0.462398\pi$
$421$	857213.	0.235713	0.117856	+	0.993031i	$0.462398\pi$
$422$	0	0
$423$	−516623.	−0.140386
$424$	0	0
$425$	1.17029e6	0.314283
$426$	0	0
$427$	5.66128e6	1.50261
$428$	0	0
$429$	394519.	0.103496
$430$	0	0
$431$	7.30262e6	1.89359	0.946794	−	0.321839i	$-0.104301\pi$
$431$	7.30262e6	1.89359	0.946794	+	0.321839i	$0.104301\pi$
$432$	0	0
$433$	−2.74346e6	−0.703199	−0.351600	−	0.936150i	$-0.614362\pi$
$433$	−2.74346e6	−0.703199	−0.351600	+	0.936150i	$0.614362\pi$
$434$	0	0
$435$	−3.79526e6	−0.961653
$436$	0	0
$437$	5.86962e6	1.47030
$438$	0	0
$439$	−63420.0	−0.0157060	−0.00785299	−	0.999969i	$-0.502500\pi$
$439$	−63420.0	−0.0157060	−0.00785299	+	0.999969i	$0.502500\pi$
$440$	0	0
$441$	−290561.	−0.0711443
$442$	0	0
$443$	−1.62690e6	−0.393868	−0.196934	−	0.980417i	$-0.563099\pi$
$443$	−1.62690e6	−0.393868	−0.196934	+	0.980417i	$0.563099\pi$
$444$	0	0
$445$	4.68968e6	1.12265
$446$	0	0
$447$	−1.26012e6	−0.298293
$448$	0	0
$449$	3.89663e6	0.912165	0.456082	−	0.889938i	$-0.349252\pi$
$449$	3.89663e6	0.912165	0.456082	+	0.889938i	$0.349252\pi$
$450$	0	0
$451$	−2.15507e6	−0.498907
$452$	0	0
$453$	1.73725e6	0.397756
$454$	0	0
$455$	1.96135e6	0.444148
$456$	0	0
$457$	−4.13511e6	−0.926182	−0.463091	−	0.886311i	$-0.653260\pi$
$457$	−4.13511e6	−0.926182	−0.463091	+	0.886311i	$0.653260\pi$
$458$	0	0
$459$	939790.	0.208209
$460$	0	0
$461$	4.75990e6	1.04315	0.521573	−	0.853206i	$-0.325345\pi$
$461$	4.75990e6	1.04315	0.521573	+	0.853206i	$0.325345\pi$
$462$	0	0
$463$	8.00193e6	1.73477	0.867385	−	0.497637i	$-0.165799\pi$
$463$	8.00193e6	1.73477	0.867385	+	0.497637i	$0.165799\pi$
$464$	0	0
$465$	1.54782e6	0.331962
$466$	0	0
$467$	5.17904e6	1.09890	0.549449	−	0.835527i	$-0.314838\pi$
$467$	5.17904e6	1.09890	0.549449	+	0.835527i	$0.314838\pi$
$468$	0	0
$469$	−1.05174e6	−0.220789
$470$	0	0
$471$	4.39859e6	0.913611
$472$	0	0
$473$	−872684.	−0.179351
$474$	0	0
$475$	1.35370e6	0.275290
$476$	0	0
$477$	−2.48179e6	−0.499423
$478$	0	0
$479$	−2.15259e6	−0.428670	−0.214335	−	0.976760i	$-0.568758\pi$
$479$	−2.15259e6	−0.428670	−0.214335	+	0.976760i	$0.568758\pi$
$480$	0	0
$481$	−4.95929e6	−0.977364
$482$	0	0
$483$	4.07315e6	0.794444
$484$	0	0
$485$	−253031.	−0.0488449
$486$	0	0
$487$	−5.13188e6	−0.980515	−0.490257	−	0.871578i	$-0.663097\pi$
$487$	−5.13188e6	−0.980515	−0.490257	+	0.871578i	$0.663097\pi$
$488$	0	0
$489$	−1.56257e6	−0.295507
$490$	0	0
$491$	4.42325e6	0.828014	0.414007	−	0.910274i	$-0.364129\pi$
$491$	4.42325e6	0.828014	0.414007	+	0.910274i	$0.364129\pi$
$492$	0	0
$493$	−1.15452e7	−2.13935
$494$	0	0
$495$	−461501.	−0.0846564
$496$	0	0
$497$	7.02036e6	1.27488
$498$	0	0
$499$	237726.	0.0427392	0.0213696	−	0.999772i	$-0.493197\pi$
$499$	237726.	0.0427392	0.0213696	+	0.999772i	$0.493197\pi$
$500$	0	0
$501$	−2.28543e6	−0.406793
$502$	0	0
$503$	−699663.	−0.123302	−0.0616508	−	0.998098i	$-0.519637\pi$
$503$	−699663.	−0.123302	−0.0616508	+	0.998098i	$0.519637\pi$
$504$	0	0
$505$	−9.43427e6	−1.64619
$506$	0	0
$507$	2.16044e6	0.373269
$508$	0	0
$509$	−6.58799e6	−1.12709	−0.563545	−	0.826086i	$-0.690563\pi$
$509$	−6.58799e6	−1.12709	−0.563545	+	0.826086i	$0.690563\pi$
$510$	0	0
$511$	3.83531e6	0.649753
$512$	0	0
$513$	1.08708e6	0.182376
$514$	0	0
$515$	−4.65426e6	−0.773272
$516$	0	0
$517$	771746.	0.126984
$518$	0	0
$519$	4.39823e6	0.716737
$520$	0	0
$521$	731910.	0.118131	0.0590655	−	0.998254i	$-0.481188\pi$
$521$	731910.	0.118131	0.0590655	+	0.998254i	$0.481188\pi$
$522$	0	0
$523$	4.12489e6	0.659414	0.329707	−	0.944083i	$-0.393050\pi$
$523$	4.12489e6	0.659414	0.329707	+	0.944083i	$0.393050\pi$
$524$	0	0
$525$	939387.	0.148746
$526$	0	0
$527$	4.70847e6	0.738504
$528$	0	0
$529$	9.05720e6	1.40720
$530$	0	0
$531$	−3.69144e6	−0.568145
$532$	0	0
$533$	6.45232e6	0.983780
$534$	0	0
$535$	3.89428e6	0.588224
$536$	0	0
$537$	−5.53868e6	−0.828840
$538$	0	0
$539$	434047.	0.0643524
$540$	0	0
$541$	−2.61793e6	−0.384561	−0.192281	−	0.981340i	$-0.561588\pi$
$541$	−2.61793e6	−0.384561	−0.192281	+	0.981340i	$0.561588\pi$
$542$	0	0
$543$	5.98311e6	0.870818
$544$	0	0
$545$	5.50262e6	0.793557
$546$	0	0
$547$	−1.04121e7	−1.48789	−0.743946	−	0.668240i	$-0.767048\pi$
$547$	−1.04121e7	−1.48789	−0.743946	+	0.668240i	$0.767048\pi$
$548$	0	0
$549$	3.98829e6	0.564749
$550$	0	0
$551$	−1.33546e7	−1.87392
$552$	0	0
$553$	−9.67529e6	−1.34540
$554$	0	0
$555$	5.80128e6	0.799450
$556$	0	0
$557$	−9.81719e6	−1.34075	−0.670377	−	0.742020i	$-0.733868\pi$
$557$	−9.81719e6	−1.34075	−0.670377	+	0.742020i	$0.733868\pi$
$558$	0	0
$559$	2.61283e6	0.353657
$560$	0	0
$561$	−1.40388e6	−0.188332
$562$	0	0
$563$	−6.78261e6	−0.901833	−0.450916	−	0.892566i	$-0.648903\pi$
$563$	−6.78261e6	−0.901833	−0.450916	+	0.892566i	$0.648903\pi$
$564$	0	0
$565$	−4.35721e6	−0.574232
$566$	0	0
$567$	754368.	0.0985429
$568$	0	0
$569$	−4.33100e6	−0.560799	−0.280400	−	0.959883i	$-0.590467\pi$
$569$	−4.33100e6	−0.560799	−0.280400	+	0.959883i	$0.590467\pi$
$570$	0	0
$571$	−1.34607e7	−1.72774	−0.863869	−	0.503717i	$-0.831965\pi$
$571$	−1.34607e7	−1.72774	−0.863869	+	0.503717i	$0.831965\pi$
$572$	0	0
$573$	573468.	0.0729663
$574$	0	0
$575$	3.57326e6	0.450708
$576$	0	0
$577$	−9.14330e6	−1.14331	−0.571654	−	0.820495i	$-0.693698\pi$
$577$	−9.14330e6	−1.14331	−0.571654	+	0.820495i	$0.693698\pi$
$578$	0	0
$579$	6.84298e6	0.848299
$580$	0	0
$581$	−7.46716e6	−0.917731
$582$	0	0
$583$	3.70736e6	0.451745
$584$	0	0
$585$	1.38175e6	0.166932
$586$	0	0
$587$	7.05129e6	0.844644	0.422322	−	0.906446i	$-0.361215\pi$
$587$	7.05129e6	0.844644	0.422322	+	0.906446i	$0.361215\pi$
$588$	0	0
$589$	5.44641e6	0.646878
$590$	0	0
$591$	665003.	0.0783168
$592$	0	0
$593$	8.24383e6	0.962703	0.481352	−	0.876528i	$-0.340146\pi$
$593$	8.24383e6	0.962703	0.481352	+	0.876528i	$0.340146\pi$
$594$	0	0
$595$	−6.97941e6	−0.808214
$596$	0	0
$597$	1.26741e6	0.145540
$598$	0	0
$599$	−1.22700e7	−1.39726	−0.698632	−	0.715481i	$-0.746207\pi$
$599$	−1.22700e7	−1.39726	−0.698632	+	0.715481i	$0.746207\pi$
$600$	0	0
$601$	−1.18308e7	−1.33607	−0.668035	−	0.744130i	$-0.732864\pi$
$601$	−1.18308e7	−1.33607	−0.668035	+	0.744130i	$0.732864\pi$
$602$	0	0
$603$	−740939.	−0.0829830
$604$	0	0
$605$	689403.	0.0765746
$606$	0	0
$607$	−7.85107e6	−0.864883	−0.432441	−	0.901662i	$-0.642348\pi$
$607$	−7.85107e6	−0.864883	−0.432441	+	0.901662i	$0.642348\pi$
$608$	0	0
$609$	−9.26727e6	−1.01253
$610$	0	0
$611$	−2.31062e6	−0.250395
$612$	0	0
$613$	1.01897e6	0.109524	0.0547619	−	0.998499i	$-0.482560\pi$
$613$	1.01897e6	0.109524	0.0547619	+	0.998499i	$0.482560\pi$
$614$	0	0
$615$	−7.54780e6	−0.804698
$616$	0	0
$617$	9.89015e6	1.04590	0.522950	−	0.852363i	$-0.324831\pi$
$617$	9.89015e6	1.04590	0.522950	+	0.852363i	$0.324831\pi$
$618$	0	0
$619$	1.47138e7	1.54347	0.771737	−	0.635942i	$-0.219388\pi$
$619$	1.47138e7	1.54347	0.771737	+	0.635942i	$0.219388\pi$
$620$	0	0
$621$	2.86948e6	0.298589
$622$	0	0
$623$	1.14513e7	1.18204
$624$	0	0
$625$	−6.10466e6	−0.625117
$626$	0	0
$627$	−1.62391e6	−0.164966
$628$	0	0
$629$	1.76475e7	1.77851
$630$	0	0
$631$	9.56270e6	0.956109	0.478054	−	0.878330i	$-0.341342\pi$
$631$	9.56270e6	0.956109	0.478054	+	0.878330i	$0.341342\pi$
$632$	0	0
$633$	630275.	0.0625203
$634$	0	0
$635$	−8.68243e6	−0.854490
$636$	0	0
$637$	−1.29955e6	−0.126895
$638$	0	0
$639$	4.94574e6	0.479159
$640$	0	0
$641$	−9.03092e6	−0.868134	−0.434067	−	0.900880i	$-0.642922\pi$
$641$	−9.03092e6	−0.868134	−0.434067	+	0.900880i	$0.642922\pi$
$642$	0	0
$643$	−1.78314e7	−1.70082	−0.850410	−	0.526120i	$-0.823646\pi$
$643$	−1.78314e7	−1.70082	−0.850410	+	0.526120i	$0.823646\pi$
$644$	0	0
$645$	−3.05644e6	−0.289279
$646$	0	0
$647$	1.33600e7	1.25472	0.627360	−	0.778729i	$-0.284135\pi$
$647$	1.33600e7	1.25472	0.627360	+	0.778729i	$0.284135\pi$
$648$	0	0
$649$	5.51437e6	0.513907
$650$	0	0
$651$	3.77948e6	0.349526
$652$	0	0
$653$	1.53459e7	1.40835	0.704175	−	0.710027i	$-0.251317\pi$
$653$	1.53459e7	1.40835	0.704175	+	0.710027i	$0.251317\pi$
$654$	0	0
$655$	−6.56961e6	−0.598324
$656$	0	0
$657$	2.70192e6	0.244207
$658$	0	0
$659$	2.08123e7	1.86684	0.933420	−	0.358786i	$-0.116809\pi$
$659$	2.08123e7	1.86684	0.933420	+	0.358786i	$0.116809\pi$
$660$	0	0
$661$	−2.66977e6	−0.237667	−0.118834	−	0.992914i	$-0.537916\pi$
$661$	−2.66977e6	−0.237667	−0.118834	+	0.992914i	$0.537916\pi$
$662$	0	0
$663$	4.20326e6	0.371366
$664$	0	0
$665$	−8.07328e6	−0.707939
$666$	0	0
$667$	−3.52510e7	−3.06801
$668$	0	0
$669$	−2.12323e6	−0.183413
$670$	0	0
$671$	−5.95781e6	−0.510835
$672$	0	0
$673$	8.80823e6	0.749637	0.374818	−	0.927098i	$-0.377705\pi$
$673$	8.80823e6	0.749637	0.374818	+	0.927098i	$0.377705\pi$
$674$	0	0
$675$	661785.	0.0559059
$676$	0	0
$677$	8.93254e6	0.749037	0.374519	−	0.927219i	$-0.377808\pi$
$677$	8.93254e6	0.749037	0.374519	+	0.927219i	$0.377808\pi$
$678$	0	0
$679$	−617851.	−0.0514292
$680$	0	0
$681$	−6.82991e6	−0.564348
$682$	0	0
$683$	−1.74989e7	−1.43536	−0.717678	−	0.696375i	$-0.754795\pi$
$683$	−1.74989e7	−1.43536	−0.717678	+	0.696375i	$0.754795\pi$
$684$	0	0
$685$	−2.12741e6	−0.173231
$686$	0	0
$687$	−3.85404e6	−0.311548
$688$	0	0
$689$	−1.10999e7	−0.890783
$690$	0	0
$691$	−1.75267e7	−1.39638	−0.698192	−	0.715911i	$-0.746012\pi$
$691$	−1.75267e7	−1.39638	−0.698192	+	0.715911i	$0.746012\pi$
$692$	0	0
$693$	−1.12689e6	−0.0891354
$694$	0	0
$695$	575312.	0.0451795
$696$	0	0
$697$	−2.29604e7	−1.79018
$698$	0	0
$699$	−1.18874e6	−0.0920226
$700$	0	0
$701$	1.55468e7	1.19494	0.597470	−	0.801891i	$-0.296173\pi$
$701$	1.55468e7	1.19494	0.597470	+	0.801891i	$0.296173\pi$
$702$	0	0
$703$	2.04133e7	1.55785
$704$	0	0
$705$	2.70293e6	0.204815
$706$	0	0
$707$	−2.30366e7	−1.73329
$708$	0	0
$709$	−1.03223e7	−0.771188	−0.385594	−	0.922669i	$-0.626003\pi$
$709$	−1.03223e7	−0.771188	−0.385594	+	0.922669i	$0.626003\pi$
$710$	0	0
$711$	−6.81610e6	−0.505664
$712$	0	0
$713$	1.43764e7	1.05908
$714$	0	0
$715$	−2.06409e6	−0.150995
$716$	0	0
$717$	−1.15197e6	−0.0836845
$718$	0	0
$719$	−3.56995e6	−0.257537	−0.128769	−	0.991675i	$-0.541102\pi$
$719$	−3.56995e6	−0.257537	−0.128769	+	0.991675i	$0.541102\pi$
$720$	0	0
$721$	−1.13648e7	−0.814184
$722$	0	0
$723$	−6.58849e6	−0.468749
$724$	0	0
$725$	−8.12991e6	−0.574435
$726$	0	0
$727$	−1.93880e7	−1.36049	−0.680246	−	0.732984i	$-0.738127\pi$
$727$	−1.93880e7	−1.36049	−0.680246	+	0.732984i	$0.738127\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−9.29768e6	−0.643549
$732$	0	0
$733$	−1.50900e7	−1.03736	−0.518681	−	0.854968i	$-0.673577\pi$
$733$	−1.50900e7	−1.03736	−0.518681	+	0.854968i	$0.673577\pi$
$734$	0	0
$735$	1.52019e6	0.103795
$736$	0	0
$737$	1.10683e6	0.0750609
$738$	0	0
$739$	1.13824e7	0.766694	0.383347	−	0.923604i	$-0.374771\pi$
$739$	1.13824e7	0.766694	0.383347	+	0.923604i	$0.374771\pi$
$740$	0	0
$741$	4.86203e6	0.325291
$742$	0	0
$743$	5.62900e6	0.374076	0.187038	−	0.982353i	$-0.440111\pi$
$743$	5.62900e6	0.374076	0.187038	+	0.982353i	$0.440111\pi$
$744$	0	0
$745$	6.59283e6	0.435193
$746$	0	0
$747$	−5.26051e6	−0.344926
$748$	0	0
$749$	9.50906e6	0.619346
$750$	0	0
$751$	9.53159e6	0.616688	0.308344	−	0.951275i	$-0.400225\pi$
$751$	9.53159e6	0.616688	0.308344	+	0.951275i	$0.400225\pi$
$752$	0	0
$753$	1.06622e7	0.685263
$754$	0	0
$755$	−9.08913e6	−0.580303
$756$	0	0
$757$	1.29356e7	0.820438	0.410219	−	0.911987i	$-0.365452\pi$
$757$	1.29356e7	0.820438	0.410219	+	0.911987i	$0.365452\pi$
$758$	0	0
$759$	−4.28650e6	−0.270084
$760$	0	0
$761$	2.42242e6	0.151631	0.0758156	−	0.997122i	$-0.475844\pi$
$761$	2.42242e6	0.151631	0.0758156	+	0.997122i	$0.475844\pi$
$762$	0	0
$763$	1.34363e7	0.835542
$764$	0	0
$765$	−4.91690e6	−0.303765
$766$	0	0
$767$	−1.65102e7	−1.01336
$768$	0	0
$769$	1.71630e7	1.04659	0.523297	−	0.852151i	$-0.324702\pi$
$769$	1.71630e7	1.04659	0.523297	+	0.852151i	$0.324702\pi$
$770$	0	0
$771$	−7.86190e6	−0.476312
$772$	0	0
$773$	−4.62806e6	−0.278580	−0.139290	−	0.990252i	$-0.544482\pi$
$773$	−4.62806e6	−0.278580	−0.139290	+	0.990252i	$0.544482\pi$
$774$	0	0
$775$	3.31562e6	0.198295
$776$	0	0
$777$	1.41656e7	0.841748
$778$	0	0
$779$	−2.65589e7	−1.56807
$780$	0	0
$781$	−7.38809e6	−0.433415
$782$	0	0
$783$	−6.52866e6	−0.380557
$784$	0	0
$785$	−2.30130e7	−1.33291
$786$	0	0
$787$	−2.65435e7	−1.52764	−0.763822	−	0.645427i	$-0.776679\pi$
$787$	−2.65435e7	−1.52764	−0.763822	+	0.645427i	$0.776679\pi$
$788$	0	0
$789$	−1.62978e7	−0.932042
$790$	0	0
$791$	−1.06394e7	−0.604613
$792$	0	0
$793$	1.78378e7	1.00730
$794$	0	0
$795$	1.29845e7	0.728630
$796$	0	0
$797$	−2.51872e7	−1.40454	−0.702270	−	0.711911i	$-0.747830\pi$
$797$	−2.51872e7	−1.40454	−0.702270	+	0.711911i	$0.747830\pi$
$798$	0	0
$799$	8.22228e6	0.455644
$800$	0	0
$801$	8.06726e6	0.444267
$802$	0	0
$803$	−4.03620e6	−0.220894
$804$	0	0
$805$	−2.13104e7	−1.15905
$806$	0	0
$807$	1.35047e7	0.729963
$808$	0	0
$809$	5.76153e6	0.309504	0.154752	−	0.987953i	$-0.450542\pi$
$809$	5.76153e6	0.309504	0.154752	+	0.987953i	$0.450542\pi$
$810$	0	0
$811$	−8.85867e6	−0.472951	−0.236476	−	0.971637i	$-0.575992\pi$
$811$	−8.85867e6	−0.472951	−0.236476	+	0.971637i	$0.575992\pi$
$812$	0	0
$813$	−1.58968e7	−0.843499
$814$	0	0
$815$	8.17522e6	0.431127
$816$	0	0
$817$	−1.07549e7	−0.563704
$818$	0	0
$819$	3.37395e6	0.175763
$820$	0	0
$821$	−8.76450e6	−0.453805	−0.226903	−	0.973917i	$-0.572860\pi$
$821$	−8.76450e6	−0.453805	−0.226903	+	0.973917i	$0.572860\pi$
$822$	0	0
$823$	1.91042e7	0.983172	0.491586	−	0.870829i	$-0.336417\pi$
$823$	1.91042e7	0.983172	0.491586	+	0.870829i	$0.336417\pi$
$824$	0	0
$825$	−988592.	−0.0505688
$826$	0	0
$827$	−1.40989e7	−0.716840	−0.358420	−	0.933560i	$-0.616684\pi$
$827$	−1.40989e7	−0.716840	−0.358420	+	0.933560i	$0.616684\pi$
$828$	0	0
$829$	−1.36584e7	−0.690262	−0.345131	−	0.938554i	$-0.612166\pi$
$829$	−1.36584e7	−0.690262	−0.345131	+	0.938554i	$0.612166\pi$
$830$	0	0
$831$	1.50992e7	0.758494
$832$	0	0
$833$	4.62439e6	0.230910
$834$	0	0
$835$	1.19572e7	0.593488
$836$	0	0
$837$	2.66259e6	0.131368
$838$	0	0
$839$	−5.23597e6	−0.256799	−0.128399	−	0.991723i	$-0.540984\pi$
$839$	−5.23597e6	−0.256799	−0.128399	+	0.991723i	$0.540984\pi$
$840$	0	0
$841$	5.96922e7	2.91023
$842$	0	0
$843$	7.31449e6	0.354499
$844$	0	0
$845$	−1.13032e7	−0.544578
$846$	0	0
$847$	1.68339e6	0.0806260
$848$	0	0
$849$	1.65177e7	0.786468
$850$	0	0
$851$	5.38833e7	2.55053
$852$	0	0
$853$	1.29513e7	0.609456	0.304728	−	0.952439i	$-0.401434\pi$
$853$	1.29513e7	0.609456	0.304728	+	0.952439i	$0.401434\pi$
$854$	0	0
$855$	−5.68751e6	−0.266077
$856$	0	0
$857$	949475.	0.0441602	0.0220801	−	0.999756i	$-0.492971\pi$
$857$	949475.	0.0441602	0.0220801	+	0.999756i	$0.492971\pi$
$858$	0	0
$859$	2.60502e6	0.120456	0.0602279	−	0.998185i	$-0.480817\pi$
$859$	2.60502e6	0.120456	0.0602279	+	0.998185i	$0.480817\pi$
$860$	0	0
$861$	−1.84302e7	−0.847273
$862$	0	0
$863$	3.46650e7	1.58440	0.792200	−	0.610262i	$-0.208936\pi$
$863$	3.46650e7	1.58440	0.792200	+	0.610262i	$0.208936\pi$
$864$	0	0
$865$	−2.30112e7	−1.04568
$866$	0	0
$867$	−2.17845e6	−0.0984236
$868$	0	0
$869$	1.01821e7	0.457390
$870$	0	0
$871$	−3.31389e6	−0.148010
$872$	0	0
$873$	−435267.	−0.0193295
$874$	0	0
$875$	−2.18334e7	−0.964055
$876$	0	0
$877$	4.18909e7	1.83916	0.919582	−	0.392898i	$-0.128528\pi$
$877$	4.18909e7	1.83916	0.919582	+	0.392898i	$0.128528\pi$
$878$	0	0
$879$	−7.92656e6	−0.346029
$880$	0	0
$881$	−3.78090e6	−0.164118	−0.0820589	−	0.996627i	$-0.526150\pi$
$881$	−3.78090e6	−0.164118	−0.0820589	+	0.996627i	$0.526150\pi$
$882$	0	0
$883$	−1.90970e7	−0.824256	−0.412128	−	0.911126i	$-0.635214\pi$
$883$	−1.90970e7	−0.824256	−0.412128	+	0.911126i	$0.635214\pi$
$884$	0	0
$885$	1.93133e7	0.828892
$886$	0	0
$887$	123945.	0.00528958	0.00264479	−	0.999997i	$-0.499158\pi$
$887$	123945.	0.00528958	0.00264479	+	0.999997i	$0.499158\pi$
$888$	0	0
$889$	−2.12008e7	−0.899699
$890$	0	0
$891$	−793881.	−0.0335013
$892$	0	0
$893$	9.51094e6	0.399112
$894$	0	0
$895$	2.89779e7	1.20923
$896$	0	0
$897$	1.28339e7	0.532570
$898$	0	0
$899$	−3.27094e7	−1.34981
$900$	0	0
$901$	3.94987e7	1.62096
$902$	0	0
$903$	−7.46323e6	−0.304584
$904$	0	0
$905$	−3.13031e7	−1.27047
$906$	0	0
$907$	4.54693e7	1.83527	0.917634	−	0.397426i	$-0.130096\pi$
$907$	4.54693e7	1.83527	0.917634	+	0.397426i	$0.130096\pi$
$908$	0	0
$909$	−1.62290e7	−0.651450
$910$	0	0
$911$	4.08634e7	1.63132	0.815659	−	0.578533i	$-0.196375\pi$
$911$	4.08634e7	1.63132	0.815659	+	0.578533i	$0.196375\pi$
$912$	0	0
$913$	7.85829e6	0.311997
$914$	0	0
$915$	−2.08664e7	−0.823937
$916$	0	0
$917$	−1.60417e7	−0.629979
$918$	0	0
$919$	3.55765e7	1.38955	0.694776	−	0.719226i	$-0.255503\pi$
$919$	3.55765e7	1.38955	0.694776	+	0.719226i	$0.255503\pi$
$920$	0	0
$921$	8.53476e6	0.331545
$922$	0	0
$923$	2.21201e7	0.854639
$924$	0	0
$925$	1.24271e7	0.477544
$926$	0	0
$927$	−8.00632e6	−0.306009
$928$	0	0
$929$	4.28980e7	1.63079	0.815395	−	0.578905i	$-0.196520\pi$
$929$	4.28980e7	1.63079	0.815395	+	0.578905i	$0.196520\pi$
$930$	0	0
$931$	5.34917e6	0.202261
$932$	0	0
$933$	−9.93221e6	−0.373544
$934$	0	0
$935$	7.34499e6	0.274766
$936$	0	0
$937$	2.62652e7	0.977308	0.488654	−	0.872478i	$-0.337488\pi$
$937$	2.62652e7	0.977308	0.488654	+	0.872478i	$0.337488\pi$
$938$	0	0
$939$	−2.17443e7	−0.804787
$940$	0	0
$941$	−2.11153e6	−0.0777362	−0.0388681	−	0.999244i	$-0.512375\pi$
$941$	−2.11153e6	−0.0777362	−0.0388681	+	0.999244i	$0.512375\pi$
$942$	0	0
$943$	−7.01053e7	−2.56727
$944$	0	0
$945$	−3.94678e6	−0.143768
$946$	0	0
$947$	−9.04218e6	−0.327641	−0.163820	−	0.986490i	$-0.552382\pi$
$947$	−9.04218e6	−0.327641	−0.163820	+	0.986490i	$0.552382\pi$
$948$	0	0
$949$	1.20845e7	0.435574
$950$	0	0
$951$	1.24794e7	0.447447
$952$	0	0
$953$	1.10750e7	0.395014	0.197507	−	0.980301i	$-0.436715\pi$
$953$	1.10750e7	0.395014	0.197507	+	0.980301i	$0.436715\pi$
$954$	0	0
$955$	−3.00033e6	−0.106454
$956$	0	0
$957$	9.75269e6	0.344227
$958$	0	0
$959$	−5.19471e6	−0.182396
$960$	0	0
$961$	−1.52893e7	−0.534045
$962$	0	0
$963$	6.69900e6	0.232779
$964$	0	0
$965$	−3.58019e7	−1.23762
$966$	0	0
$967$	2.51054e7	0.863379	0.431689	−	0.902022i	$-0.357918\pi$
$967$	2.51054e7	0.863379	0.431689	+	0.902022i	$0.357918\pi$
$968$	0	0
$969$	−1.73014e7	−0.591931
$970$	0	0
$971$	4.72855e7	1.60946	0.804729	−	0.593642i	$-0.202311\pi$
$971$	4.72855e7	1.60946	0.804729	+	0.593642i	$0.202311\pi$
$972$	0	0
$973$	1.40480e6	0.0475698
$974$	0	0
$975$	2.95987e6	0.0997150
$976$	0	0
$977$	−2.10706e7	−0.706222	−0.353111	−	0.935581i	$-0.614876\pi$
$977$	−2.10706e7	−0.706222	−0.353111	+	0.935581i	$0.614876\pi$
$978$	0	0
$979$	−1.20511e7	−0.401855
$980$	0	0
$981$	9.46567e6	0.314036
$982$	0	0
$983$	2.84949e7	0.940553	0.470277	−	0.882519i	$-0.344154\pi$
$983$	2.84949e7	0.940553	0.470277	+	0.882519i	$0.344154\pi$
$984$	0	0
$985$	−3.47923e6	−0.114260
$986$	0	0
$987$	6.60001e6	0.215651
$988$	0	0
$989$	−2.83888e7	−0.922903
$990$	0	0
$991$	−5.56340e7	−1.79952	−0.899759	−	0.436387i	$-0.856258\pi$
$991$	−5.56340e7	−1.79952	−0.899759	+	0.436387i	$0.856258\pi$
$992$	0	0
$993$	1.43534e7	0.461936
$994$	0	0
$995$	−6.63098e6	−0.212334
$996$	0	0
$997$	−2.01671e7	−0.642549	−0.321275	−	0.946986i	$-0.604111\pi$
$997$	−2.01671e7	−0.642549	−0.321275	+	0.946986i	$0.604111\pi$
$998$	0	0
$999$	9.97945e6	0.316368

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.a.z.1.3		4
4.3	odd	2		264.6.a.i.1.3	✓	4
12.11	even	2		792.6.a.m.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.6.a.i.1.3	✓	4	4.3	odd	2
528.6.a.z.1.3		4	1.1	even	1	trivial
792.6.a.m.1.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +47.0872 q^{5} +114.978 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +47.0872 q^{5} +114.978 q^{7} +81.0000 q^{9} -121.000 q^{11} +362.277 q^{13} -423.785 q^{15} -1289.15 q^{17} -1491.20 q^{19} -1034.80 q^{21} -3936.18 q^{23} -907.798 q^{25} -729.000 q^{27} +8955.63 q^{29} -3652.38 q^{31} +1089.00 q^{33} +5413.97 q^{35} -13689.2 q^{37} -3260.49 q^{39} +17810.5 q^{41} +7212.26 q^{43} +3814.06 q^{45} -6378.07 q^{47} -3587.17 q^{49} +11602.3 q^{51} -30639.4 q^{53} -5697.55 q^{55} +13420.8 q^{57} -45573.3 q^{59} +49238.1 q^{61} +9313.18 q^{63} +17058.6 q^{65} -9147.39 q^{67} +35425.7 q^{69} +61058.6 q^{71} +33357.0 q^{73} +8170.18 q^{75} -13912.3 q^{77} -84149.4 q^{79} +6561.00 q^{81} -64944.5 q^{83} -60702.4 q^{85} -80600.7 q^{87} +99595.7 q^{89} +41653.7 q^{91} +32871.4 q^{93} -70216.2 q^{95} -5373.67 q^{97} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{3} - 44 q^{5} - 38 q^{7} + 324 q^{9}+O(q^{10}) \) 4 * q - 36 * q^3 - 44 * q^5 - 38 * q^7 + 324 * q^9	\( 4 q - 36 q^{3} - 44 q^{5} - 38 q^{7} + 324 q^{9} - 484 q^{11} + 288 q^{13} + 396 q^{15} + 418 q^{17} + 10 q^{19} + 342 q^{21} - 576 q^{23} + 4932 q^{25} - 2916 q^{27} + 11930 q^{29} - 9968 q^{31} + 4356 q^{33} + 428 q^{35} + 13740 q^{37} - 2592 q^{39} + 29766 q^{41} - 25650 q^{43} - 3564 q^{45} + 5776 q^{47} + 39888 q^{49} - 3762 q^{51} - 7840 q^{53} + 5324 q^{55} - 90 q^{57} - 28800 q^{59} + 33932 q^{61} - 3078 q^{63} + 33216 q^{65} - 83056 q^{67} + 5184 q^{69} - 21336 q^{71} + 27044 q^{73} - 44388 q^{75} + 4598 q^{77} - 102542 q^{79} + 26244 q^{81} - 64996 q^{83} - 12132 q^{85} - 107370 q^{87} + 37888 q^{89} - 273612 q^{91} + 89712 q^{93} - 254380 q^{95} - 20996 q^{97} - 39204 q^{99}+O(q^{100}) \) 4 * q - 36 * q^3 - 44 * q^5 - 38 * q^7 + 324 * q^9 - 484 * q^11 + 288 * q^13 + 396 * q^15 + 418 * q^17 + 10 * q^19 + 342 * q^21 - 576 * q^23 + 4932 * q^25 - 2916 * q^27 + 11930 * q^29 - 9968 * q^31 + 4356 * q^33 + 428 * q^35 + 13740 * q^37 - 2592 * q^39 + 29766 * q^41 - 25650 * q^43 - 3564 * q^45 + 5776 * q^47 + 39888 * q^49 - 3762 * q^51 - 7840 * q^53 + 5324 * q^55 - 90 * q^57 - 28800 * q^59 + 33932 * q^61 - 3078 * q^63 + 33216 * q^65 - 83056 * q^67 + 5184 * q^69 - 21336 * q^71 + 27044 * q^73 - 44388 * q^75 + 4598 * q^77 - 102542 * q^79 + 26244 * q^81 - 64996 * q^83 - 12132 * q^85 - 107370 * q^87 + 37888 * q^89 - 273612 * q^91 + 89712 * q^93 - 254380 * q^95 - 20996 * q^97 - 39204 * q^99

Introduction

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