LMFDB - Embedded newform 531.6.a.c.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,6,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("531.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	531.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:	$85.1638083207$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + \cdots + 49172480 $ x^12 - 2x^11 - 269x^10 + 143x^9 + 25384x^8 + 8539x^7 - 1009736x^6 - 720516x^5 + 15565376x^4 + 6775664x^3 - 75006848x^2 + 21512960*x + 49172480
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.70991$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.290087 q^{2} -31.9158 q^{4} -87.1048 q^{5} +167.610 q^{7} +18.5412 q^{8} +O(q^{10})$	\(q-0.290087 q^{2} -31.9158 q^{4} -87.1048 q^{5} +167.610 q^{7} +18.5412 q^{8} +25.2680 q^{10} -252.804 q^{11} -289.403 q^{13} -48.6216 q^{14} +1015.93 q^{16} +227.568 q^{17} -1162.71 q^{19} +2780.02 q^{20} +73.3353 q^{22} +1231.89 q^{23} +4462.24 q^{25} +83.9520 q^{26} -5349.42 q^{28} -1975.28 q^{29} +7791.87 q^{31} -888.026 q^{32} -66.0147 q^{34} -14599.6 q^{35} +7046.70 q^{37} +337.287 q^{38} -1615.03 q^{40} +11251.3 q^{41} +18519.7 q^{43} +8068.46 q^{44} -357.355 q^{46} +20101.5 q^{47} +11286.1 q^{49} -1294.44 q^{50} +9236.53 q^{52} -37357.8 q^{53} +22020.5 q^{55} +3107.69 q^{56} +573.003 q^{58} +3481.00 q^{59} -998.231 q^{61} -2260.32 q^{62} -32252.1 q^{64} +25208.3 q^{65} -19664.7 q^{67} -7263.03 q^{68} +4235.17 q^{70} -21610.0 q^{71} +46265.2 q^{73} -2044.16 q^{74} +37108.8 q^{76} -42372.5 q^{77} -85014.5 q^{79} -88492.2 q^{80} -3263.86 q^{82} -75468.7 q^{83} -19822.3 q^{85} -5372.32 q^{86} -4687.29 q^{88} -105603. q^{89} -48506.8 q^{91} -39316.7 q^{92} -5831.19 q^{94} +101277. q^{95} -109794. q^{97} -3273.97 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) $ 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8	$ 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) $ 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8 + 601 * q^10 - 1480 * q^11 + 472 * q^13 - 1065 * q^14 + 6370 * q^16 - 1565 * q^17 + 3939 * q^19 - 8033 * q^20 - 1738 * q^22 - 7245 * q^23 + 9690 * q^25 - 3764 * q^26 + 12154 * q^28 - 10003 * q^29 + 7295 * q^31 - 11628 * q^32 - 16344 * q^34 - 11015 * q^35 + 6741 * q^37 - 3035 * q^38 + 5572 * q^40 - 34025 * q^41 - 6336 * q^43 - 41168 * q^44 + 2345 * q^46 - 66167 * q^47 + 28319 * q^49 - 31173 * q^50 + 16440 * q^52 - 62290 * q^53 + 55764 * q^55 - 107306 * q^56 + 37952 * q^58 + 41772 * q^59 + 68469 * q^61 - 99190 * q^62 + 68525 * q^64 - 80156 * q^65 + 113310 * q^67 - 33887 * q^68 + 32034 * q^70 - 84520 * q^71 + 135895 * q^73 + 31962 * q^74 - 61848 * q^76 + 3799 * q^77 + 14122 * q^79 - 77609 * q^80 - 1501 * q^82 - 114463 * q^83 - 101097 * q^85 + 203536 * q^86 - 244967 * q^88 - 189109 * q^89 - 168249 * q^91 + 71946 * q^92 - 472284 * q^94 - 21923 * q^95 - 76192 * q^97 + 17544 * q^98

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 $

$2$	−0.290087	−0.0512807	−0.0256403	−	0.999671i	$-0.508162\pi$
$2$	−0.290087	−0.0512807	−0.0256403	+	0.999671i	$0.508162\pi$
$3$	0	0
$3$	0	0
$4$	−31.9158	−0.997370
$5$	−87.1048	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$5$	−87.1048	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$6$	0	0
$7$	167.610	1.29287	0.646435	−	0.762969i	$-0.276259\pi$
$7$	167.610	1.29287	0.646435	+	0.762969i	$0.276259\pi$
$8$	18.5412	0.102427
$9$	0	0
$10$	25.2680	0.0799044
$11$	−252.804	−0.629945	−0.314972	−	0.949101i	$-0.601995\pi$
$11$	−252.804	−0.629945	−0.314972	+	0.949101i	$0.601995\pi$
$12$	0	0
$13$	−289.403	−0.474946	−0.237473	−	0.971394i	$-0.576319\pi$
$13$	−289.403	−0.474946	−0.237473	+	0.971394i	$0.576319\pi$
$14$	−48.6216	−0.0662993
$15$	0	0
$16$	1015.93	0.992118
$17$	227.568	0.190981	0.0954903	−	0.995430i	$-0.469558\pi$
$17$	227.568	0.190981	0.0954903	+	0.995430i	$0.469558\pi$
$18$	0	0
$19$	−1162.71	−0.738902	−0.369451	−	0.929250i	$-0.620454\pi$
$19$	−1162.71	−0.738902	−0.369451	+	0.929250i	$0.620454\pi$
$20$	2780.02	1.55408
$21$	0	0
$22$	73.3353	0.0323040
$23$	1231.89	0.485570	0.242785	−	0.970080i	$-0.421939\pi$
$23$	1231.89	0.485570	0.242785	+	0.970080i	$0.421939\pi$
$24$	0	0
$25$	4462.24	1.42792
$26$	83.9520	0.0243555
$27$	0	0
$28$	−5349.42	−1.28947
$29$	−1975.28	−0.436147	−0.218073	−	0.975932i	$-0.569977\pi$
$29$	−1975.28	−0.436147	−0.218073	+	0.975932i	$0.569977\pi$
$30$	0	0
$31$	7791.87	1.45626	0.728128	−	0.685441i	$-0.240391\pi$
$31$	7791.87	1.45626	0.728128	+	0.685441i	$0.240391\pi$
$32$	−888.026	−0.153303
$33$	0	0
$34$	−66.0147	−0.00979362
$35$	−14599.6	−2.01452
$36$	0	0
$37$	7046.70	0.846216	0.423108	−	0.906079i	$-0.360939\pi$
$37$	7046.70	0.846216	0.423108	+	0.906079i	$0.360939\pi$
$38$	337.287	0.0378914
$39$	0	0
$40$	−1615.03	−0.159599
$41$	11251.3	1.04530	0.522652	−	0.852546i	$-0.324943\pi$
$41$	11251.3	1.04530	0.522652	+	0.852546i	$0.324943\pi$
$42$	0	0
$43$	18519.7	1.52743	0.763716	−	0.645552i	$-0.223373\pi$
$43$	18519.7	1.52743	0.763716	+	0.645552i	$0.223373\pi$
$44$	8068.46	0.628288
$45$	0	0
$46$	−357.355	−0.0249003
$47$	20101.5	1.32735	0.663673	−	0.748023i	$-0.268997\pi$
$47$	20101.5	1.32735	0.663673	+	0.748023i	$0.268997\pi$
$48$	0	0
$49$	11286.1	0.671514
$50$	−1294.44	−0.0732246
$51$	0	0
$52$	9236.53	0.473697
$53$	−37357.8	−1.82681	−0.913403	−	0.407058i	$-0.866555\pi$
$53$	−37357.8	−1.82681	−0.913403	+	0.407058i	$0.866555\pi$
$54$	0	0
$55$	22020.5	0.981566
$56$	3107.69	0.132424
$57$	0	0
$58$	573.003	0.0223659
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	−998.231	−0.0343484	−0.0171742	−	0.999853i	$-0.505467\pi$
$61$	−998.231	−0.0343484	−0.0171742	+	0.999853i	$0.505467\pi$
$62$	−2260.32	−0.0746778
$63$	0	0
$64$	−32252.1	−0.984256
$65$	25208.3	0.740050
$66$	0	0
$67$	−19664.7	−0.535181	−0.267590	−	0.963533i	$-0.586227\pi$
$67$	−19664.7	−0.535181	−0.267590	+	0.963533i	$0.586227\pi$
$68$	−7263.03	−0.190478
$69$	0	0
$70$	4235.17	0.103306
$71$	−21610.0	−0.508755	−0.254378	−	0.967105i	$-0.581871\pi$
$71$	−21610.0	−0.508755	−0.254378	+	0.967105i	$0.581871\pi$
$72$	0	0
$73$	46265.2	1.01613	0.508063	−	0.861320i	$-0.330362\pi$
$73$	46265.2	1.01613	0.508063	+	0.861320i	$0.330362\pi$
$74$	−2044.16	−0.0433945
$75$	0	0
$76$	37108.8	0.736958
$77$	−42372.5	−0.814437
$78$	0	0
$79$	−85014.5	−1.53259	−0.766294	−	0.642490i	$-0.777901\pi$
$79$	−85014.5	−1.53259	−0.766294	+	0.642490i	$0.777901\pi$
$80$	−88492.2	−1.54590
$81$	0	0
$82$	−3263.86	−0.0536039
$83$	−75468.7	−1.20246	−0.601231	−	0.799075i	$-0.705323\pi$
$83$	−75468.7	−1.20246	−0.601231	+	0.799075i	$0.705323\pi$
$84$	0	0
$85$	−19822.3	−0.297582
$86$	−5372.32	−0.0783278
$87$	0	0
$88$	−4687.29	−0.0645231
$89$	−105603.	−1.41319	−0.706596	−	0.707617i	$-0.749770\pi$
$89$	−105603.	−1.41319	−0.706596	+	0.707617i	$0.749770\pi$
$90$	0	0
$91$	−48506.8	−0.614043
$92$	−39316.7	−0.484293
$93$	0	0
$94$	−5831.19	−0.0680672
$95$	101277.	1.15134
$96$	0	0
$97$	−109794.	−1.18481	−0.592404	−	0.805641i	$-0.701821\pi$
$97$	−109794.	−1.18481	−0.592404	+	0.805641i	$0.701821\pi$
$98$	−3273.97	−0.0344357
$99$	0	0
$100$	−142416.	−1.42416
$101$	−96278.1	−0.939126	−0.469563	−	0.882899i	$-0.655589\pi$
$101$	−96278.1	−0.939126	−0.469563	+	0.882899i	$0.655589\pi$
$102$	0	0
$103$	−120043.	−1.11492	−0.557459	−	0.830205i	$-0.688223\pi$
$103$	−120043.	−1.11492	−0.557459	+	0.830205i	$0.688223\pi$
$104$	−5365.87	−0.0486470
$105$	0	0
$106$	10837.0	0.0936798
$107$	37527.7	0.316878	0.158439	−	0.987369i	$-0.449354\pi$
$107$	37527.7	0.316878	0.158439	+	0.987369i	$0.449354\pi$
$108$	0	0
$109$	−100498.	−0.810201	−0.405101	−	0.914272i	$-0.632763\pi$
$109$	−100498.	−0.810201	−0.405101	+	0.914272i	$0.632763\pi$
$110$	−6387.85	−0.0503354
$111$	0	0
$112$	170280.	1.28268
$113$	96992.3	0.714564	0.357282	−	0.933997i	$-0.383703\pi$
$113$	96992.3	0.714564	0.357282	+	0.933997i	$0.383703\pi$
$114$	0	0
$115$	−107303.	−0.756604
$116$	63042.6	0.435000
$117$	0	0
$118$	−1009.79	−0.00667618
$119$	38142.7	0.246913
$120$	0	0
$121$	−97141.1	−0.603170
$122$	289.574	0.00176141
$123$	0	0
$124$	−248684.	−1.45243
$125$	−116480.	−0.666772
$126$	0	0
$127$	22759.5	0.125214	0.0626071	−	0.998038i	$-0.480058\pi$
$127$	22759.5	0.125214	0.0626071	+	0.998038i	$0.480058\pi$
$128$	37772.8	0.203776
$129$	0	0
$130$	−7312.62	−0.0379503
$131$	218805.	1.11398	0.556992	−	0.830518i	$-0.311955\pi$
$131$	218805.	1.11398	0.556992	+	0.830518i	$0.311955\pi$
$132$	0	0
$133$	−194882.	−0.955304
$134$	5704.48	0.0274444
$135$	0	0
$136$	4219.38	0.0195615
$137$	377313.	1.71752	0.858758	−	0.512382i	$-0.171237\pi$
$137$	377313.	1.71752	0.858758	+	0.512382i	$0.171237\pi$
$138$	0	0
$139$	−5026.98	−0.0220683	−0.0110342	−	0.999939i	$-0.503512\pi$
$139$	−5026.98	−0.0220683	−0.0110342	+	0.999939i	$0.503512\pi$
$140$	465960.	2.00922
$141$	0	0
$142$	6268.79	0.0260893
$143$	73162.2	0.299190
$144$	0	0
$145$	172056.	0.679594
$146$	−13421.0	−0.0521076
$147$	0	0
$148$	−224901.	−0.843991
$149$	165819.	0.611884	0.305942	−	0.952050i	$-0.401029\pi$
$149$	165819.	0.611884	0.305942	+	0.952050i	$0.401029\pi$
$150$	0	0
$151$	305526.	1.09045	0.545225	−	0.838290i	$-0.316444\pi$
$151$	305526.	1.09045	0.545225	+	0.838290i	$0.316444\pi$
$152$	−21558.0	−0.0756831
$153$	0	0
$154$	12291.7	0.0417649
$155$	−678709.	−2.26910
$156$	0	0
$157$	−412996.	−1.33720	−0.668600	−	0.743622i	$-0.733106\pi$
$157$	−412996.	−1.33720	−0.668600	+	0.743622i	$0.733106\pi$
$158$	24661.6	0.0785921
$159$	0	0
$160$	77351.3	0.238873
$161$	206477.	0.627779
$162$	0	0
$163$	234544.	0.691441	0.345721	−	0.938337i	$-0.387634\pi$
$163$	234544.	0.691441	0.345721	+	0.938337i	$0.387634\pi$
$164$	−359094.	−1.04255
$165$	0	0
$166$	21892.5	0.0616631
$167$	−138449.	−0.384148	−0.192074	−	0.981380i	$-0.561521\pi$
$167$	−138449.	−0.384148	−0.192074	+	0.981380i	$0.561521\pi$
$168$	0	0
$169$	−287539.	−0.774427
$170$	5750.19	0.0152602
$171$	0	0
$172$	−591071.	−1.52342
$173$	−150940.	−0.383433	−0.191717	−	0.981450i	$-0.561405\pi$
$173$	−150940.	−0.383433	−0.191717	+	0.981450i	$0.561405\pi$
$174$	0	0
$175$	747917.	1.84611
$176$	−256831.	−0.624979
$177$	0	0
$178$	30634.1	0.0724694
$179$	−2851.21	−0.00665114	−0.00332557	−	0.999994i	$-0.501059\pi$
$179$	−2851.21	−0.00665114	−0.00332557	+	0.999994i	$0.501059\pi$
$180$	0	0
$181$	340696.	0.772985	0.386492	−	0.922293i	$-0.373687\pi$
$181$	340696.	0.772985	0.386492	+	0.922293i	$0.373687\pi$
$182$	14071.2	0.0314886
$183$	0	0
$184$	22840.7	0.0497352
$185$	−613801.	−1.31856
$186$	0	0
$187$	−57530.2	−0.120307
$188$	−641557.	−1.32386
$189$	0	0
$190$	−29379.3	−0.0590415
$191$	101124.	0.200571	0.100286	−	0.994959i	$-0.468024\pi$
$191$	101124.	0.200571	0.100286	+	0.994959i	$0.468024\pi$
$192$	0	0
$193$	344995.	0.666683	0.333342	−	0.942806i	$-0.391824\pi$
$193$	344995.	0.666683	0.333342	+	0.942806i	$0.391824\pi$
$194$	31849.8	0.0607578
$195$	0	0
$196$	−360207.	−0.669748
$197$	506599.	0.930034	0.465017	−	0.885302i	$-0.346048\pi$
$197$	506599.	0.930034	0.465017	+	0.885302i	$0.346048\pi$
$198$	0	0
$199$	−221500.	−0.396498	−0.198249	−	0.980152i	$-0.563525\pi$
$199$	−221500.	−0.396498	−0.198249	+	0.980152i	$0.563525\pi$
$200$	82735.3	0.146257
$201$	0	0
$202$	27929.1	0.0481591
$203$	−331076.	−0.563882
$204$	0	0
$205$	−980041.	−1.62877
$206$	34822.9	0.0571737
$207$	0	0
$208$	−294012.	−0.471202
$209$	293937.	0.465467
$210$	0	0
$211$	−969455.	−1.49907	−0.749534	−	0.661966i	$-0.769723\pi$
$211$	−969455.	−1.49907	−0.749534	+	0.661966i	$0.769723\pi$
$212$	1.19231e6	1.82200
$213$	0	0
$214$	−10886.3	−0.0162497
$215$	−1.61315e6	−2.38001
$216$	0	0
$217$	1.30600e6	1.88275
$218$	29153.3	0.0415477
$219$	0	0
$220$	−702801.	−0.978985
$221$	−65858.8	−0.0907054
$222$	0	0
$223$	−1.33305e6	−1.79509	−0.897544	−	0.440925i	$-0.854650\pi$
$223$	−1.33305e6	−1.79509	−0.897544	+	0.440925i	$0.854650\pi$
$224$	−148842.	−0.198201
$225$	0	0
$226$	−28136.2	−0.0366433
$227$	−734891.	−0.946582	−0.473291	−	0.880906i	$-0.656934\pi$
$227$	−734891.	−0.946582	−0.473291	+	0.880906i	$0.656934\pi$
$228$	0	0
$229$	315332.	0.397355	0.198677	−	0.980065i	$-0.436335\pi$
$229$	315332.	0.397355	0.198677	+	0.980065i	$0.436335\pi$
$230$	31127.3	0.0387992
$231$	0	0
$232$	−36624.0	−0.0446730
$233$	−447622.	−0.540159	−0.270079	−	0.962838i	$-0.587050\pi$
$233$	−447622.	−0.540159	−0.270079	+	0.962838i	$0.587050\pi$
$234$	0	0
$235$	−1.75094e6	−2.06824
$236$	−111099.	−0.129847
$237$	0	0
$238$	−11064.7	−0.0126619
$239$	−220701.	−0.249925	−0.124962	−	0.992161i	$-0.539881\pi$
$239$	−220701.	−0.249925	−0.124962	+	0.992161i	$0.539881\pi$
$240$	0	0
$241$	643890.	0.714117	0.357058	−	0.934082i	$-0.383780\pi$
$241$	643890.	0.714117	0.357058	+	0.934082i	$0.383780\pi$
$242$	28179.4	0.0309310
$243$	0	0
$244$	31859.4	0.0342581
$245$	−983077.	−1.04634
$246$	0	0
$247$	336491.	0.350938
$248$	144471.	0.149159
$249$	0	0
$250$	33789.5	0.0341925
$251$	−1.13484e6	−1.13697	−0.568485	−	0.822693i	$-0.692470\pi$
$251$	−1.13484e6	−1.13697	−0.568485	+	0.822693i	$0.692470\pi$
$252$	0	0
$253$	−311426.	−0.305882
$254$	−6602.25	−0.00642107
$255$	0	0
$256$	1.02111e6	0.973807
$257$	−157266.	−0.148526	−0.0742630	−	0.997239i	$-0.523660\pi$
$257$	−157266.	−0.148526	−0.0742630	+	0.997239i	$0.523660\pi$
$258$	0	0
$259$	1.18110e6	1.09405
$260$	−804546.	−0.738104
$261$	0	0
$262$	−63472.6	−0.0571259
$263$	943147.	0.840794	0.420397	−	0.907340i	$-0.361891\pi$
$263$	943147.	0.840794	0.420397	+	0.907340i	$0.361891\pi$
$264$	0	0
$265$	3.25405e6	2.84649
$266$	56532.7	0.0489886
$267$	0	0
$268$	627616.	0.533773
$269$	−528620.	−0.445413	−0.222707	−	0.974885i	$-0.571489\pi$
$269$	−528620.	−0.445413	−0.222707	+	0.974885i	$0.571489\pi$
$270$	0	0
$271$	1.81630e6	1.50233	0.751163	−	0.660117i	$-0.229493\pi$
$271$	1.81630e6	1.50233	0.751163	+	0.660117i	$0.229493\pi$
$272$	231193.	0.189475
$273$	0	0
$274$	−109454.	−0.0880754
$275$	−1.12807e6	−0.899509
$276$	0	0
$277$	−1.67266e6	−1.30981	−0.654905	−	0.755711i	$-0.727291\pi$
$277$	−1.67266e6	−1.30981	−0.654905	+	0.755711i	$0.727291\pi$
$278$	1458.26	0.00113168
$279$	0	0
$280$	−270695.	−0.206340
$281$	−461678.	−0.348798	−0.174399	−	0.984675i	$-0.555798\pi$
$281$	−461678.	−0.348798	−0.174399	+	0.984675i	$0.555798\pi$
$282$	0	0
$283$	880352.	0.653417	0.326708	−	0.945125i	$-0.394061\pi$
$283$	880352.	0.653417	0.326708	+	0.945125i	$0.394061\pi$
$284$	689702.	0.507418
$285$	0	0
$286$	−21223.4	−0.0153426
$287$	1.88583e6	1.35144
$288$	0	0
$289$	−1.36807e6	−0.963526
$290$	−49911.3	−0.0348501
$291$	0	0
$292$	−1.47659e6	−1.01345
$293$	2.64738e6	1.80155	0.900777	−	0.434281i	$-0.142998\pi$
$293$	2.64738e6	1.80155	0.900777	+	0.434281i	$0.142998\pi$
$294$	0	0
$295$	−303212.	−0.202857
$296$	130654.	0.0866750
$297$	0	0
$298$	−48102.0	−0.0313778
$299$	−356512.	−0.230619
$300$	0	0
$301$	3.10408e6	1.97477
$302$	−88629.2	−0.0559190
$303$	0	0
$304$	−1.18123e6	−0.733077
$305$	86950.7	0.0535209
$306$	0	0
$307$	−2.30973e6	−1.39867	−0.699336	−	0.714794i	$-0.746521\pi$
$307$	−2.30973e6	−1.39867	−0.699336	+	0.714794i	$0.746521\pi$
$308$	1.35236e6	0.812295
$309$	0	0
$310$	196885.	0.116361
$311$	2.74862e6	1.61144	0.805719	−	0.592298i	$-0.201779\pi$
$311$	2.74862e6	1.61144	0.805719	+	0.592298i	$0.201779\pi$
$312$	0	0
$313$	−2.11318e6	−1.21920	−0.609601	−	0.792708i	$-0.708670\pi$
$313$	−2.11318e6	−1.21920	−0.609601	+	0.792708i	$0.708670\pi$
$314$	119805.	0.0685725
$315$	0	0
$316$	2.71331e6	1.52856
$317$	−1.75231e6	−0.979407	−0.489703	−	0.871889i	$-0.662895\pi$
$317$	−1.75231e6	−0.979407	−0.489703	+	0.871889i	$0.662895\pi$
$318$	0	0
$319$	499358.	0.274748
$320$	2.80931e6	1.53365
$321$	0	0
$322$	−59896.3	−0.0321929
$323$	−264595.	−0.141116
$324$	0	0
$325$	−1.29138e6	−0.678184
$326$	−68038.2	−0.0354576
$327$	0	0
$328$	208612.	0.107067
$329$	3.36922e6	1.71609
$330$	0	0
$331$	−971356.	−0.487313	−0.243657	−	0.969862i	$-0.578347\pi$
$331$	−971356.	−0.487313	−0.243657	+	0.969862i	$0.578347\pi$
$332$	2.40865e6	1.19930
$333$	0	0
$334$	40162.3	0.0196994
$335$	1.71289e6	0.833907
$336$	0	0
$337$	1.02817e6	0.493164	0.246582	−	0.969122i	$-0.420693\pi$
$337$	1.02817e6	0.493164	0.246582	+	0.969122i	$0.420693\pi$
$338$	83411.5	0.0397131
$339$	0	0
$340$	632645.	0.296799
$341$	−1.96982e6	−0.917360
$342$	0	0
$343$	−925352.	−0.424690
$344$	343376.	0.156450
$345$	0	0
$346$	43785.9	0.0196627
$347$	−2.24381e6	−1.00038	−0.500188	−	0.865917i	$-0.666736\pi$
$347$	−2.24381e6	−1.00038	−0.500188	+	0.865917i	$0.666736\pi$
$348$	0	0
$349$	2.31221e6	1.01616	0.508081	−	0.861309i	$-0.330355\pi$
$349$	2.31221e6	1.01616	0.508081	+	0.861309i	$0.330355\pi$
$350$	−216961.	−0.0946700
$351$	0	0
$352$	224497.	0.0965724
$353$	749517.	0.320144	0.160072	−	0.987105i	$-0.448827\pi$
$353$	749517.	0.320144	0.160072	+	0.987105i	$0.448827\pi$
$354$	0	0
$355$	1.88234e6	0.792731
$356$	3.37041e6	1.40948
$357$	0	0
$358$	827.099	0.000341075 0
$359$	−2.49839e6	−1.02312	−0.511558	−	0.859249i	$-0.670931\pi$
$359$	−2.49839e6	−1.02312	−0.511558	+	0.859249i	$0.670931\pi$
$360$	0	0
$361$	−1.12421e6	−0.454024
$362$	−98831.7	−0.0396392
$363$	0	0
$364$	1.54814e6	0.612429
$365$	−4.02992e6	−1.58330
$366$	0	0
$367$	−2.57748e6	−0.998920	−0.499460	−	0.866337i	$-0.666468\pi$
$367$	−2.57748e6	−0.998920	−0.499460	+	0.866337i	$0.666468\pi$
$368$	1.25151e6	0.481742
$369$	0	0
$370$	178056.	0.0676164
$371$	−6.26155e6	−2.36182
$372$	0	0
$373$	−999728.	−0.372057	−0.186029	−	0.982544i	$-0.559562\pi$
$373$	−999728.	−0.372057	−0.186029	+	0.982544i	$0.559562\pi$
$374$	16688.8	0.00616944
$375$	0	0
$376$	372706.	0.135955
$377$	571650.	0.207146
$378$	0	0
$379$	−3.57477e6	−1.27835	−0.639175	−	0.769062i	$-0.720724\pi$
$379$	−3.57477e6	−1.27835	−0.639175	+	0.769062i	$0.720724\pi$
$380$	−3.23235e6	−1.14831
$381$	0	0
$382$	−29334.7	−0.0102854
$383$	47574.2	0.0165720	0.00828599	−	0.999966i	$-0.497362\pi$
$383$	47574.2	0.0165720	0.00828599	+	0.999966i	$0.497362\pi$
$384$	0	0
$385$	3.69085e6	1.26904
$386$	−100079.	−0.0341880
$387$	0	0
$388$	3.50416e6	1.18169
$389$	4.27552e6	1.43257	0.716284	−	0.697809i	$-0.245842\pi$
$389$	4.27552e6	1.43257	0.716284	+	0.697809i	$0.245842\pi$
$390$	0	0
$391$	280339.	0.0927344
$392$	209258.	0.0687809
$393$	0	0
$394$	−146958.	−0.0476928
$395$	7.40517e6	2.38804
$396$	0	0
$397$	−4.92533e6	−1.56841	−0.784204	−	0.620504i	$-0.786928\pi$
$397$	−4.92533e6	−1.56841	−0.784204	+	0.620504i	$0.786928\pi$
$398$	64254.3	0.0203327
$399$	0	0
$400$	4.53332e6	1.41666
$401$	3.08195e6	0.957117	0.478559	−	0.878056i	$-0.341159\pi$
$401$	3.08195e6	0.957117	0.478559	+	0.878056i	$0.341159\pi$
$402$	0	0
$403$	−2.25499e6	−0.691642
$404$	3.07280e6	0.936657
$405$	0	0
$406$	96041.0	0.0289162
$407$	−1.78143e6	−0.533069
$408$	0	0
$409$	−4.03756e6	−1.19347	−0.596734	−	0.802439i	$-0.703535\pi$
$409$	−4.03756e6	−1.19347	−0.596734	+	0.802439i	$0.703535\pi$
$410$	284297.	0.0835244
$411$	0	0
$412$	3.83126e6	1.11199
$413$	583451.	0.168317
$414$	0	0
$415$	6.57369e6	1.87365
$416$	256997.	0.0728106
$417$	0	0
$418$	−85267.5	−0.0238695
$419$	−3.20507e6	−0.891873	−0.445936	−	0.895065i	$-0.647129\pi$
$419$	−3.20507e6	−0.891873	−0.445936	+	0.895065i	$0.647129\pi$
$420$	0	0
$421$	3.86174e6	1.06189	0.530943	−	0.847408i	$-0.321838\pi$
$421$	3.86174e6	1.06189	0.530943	+	0.847408i	$0.321838\pi$
$422$	281227.	0.0768733
$423$	0	0
$424$	−692659.	−0.187113
$425$	1.01546e6	0.272705
$426$	0	0
$427$	−167314.	−0.0444080
$428$	−1.19773e6	−0.316045
$429$	0	0
$430$	467955.	0.122049
$431$	−4.57174e6	−1.18547	−0.592733	−	0.805399i	$-0.701951\pi$
$431$	−4.57174e6	−1.18547	−0.592733	+	0.805399i	$0.701951\pi$
$432$	0	0
$433$	6.35435e6	1.62874	0.814370	−	0.580346i	$-0.197083\pi$
$433$	6.35435e6	1.62874	0.814370	+	0.580346i	$0.197083\pi$
$434$	−378853.	−0.0965487
$435$	0	0
$436$	3.20749e6	0.808070
$437$	−1.43233e6	−0.358788
$438$	0	0
$439$	−5.50122e6	−1.36238	−0.681189	−	0.732108i	$-0.738537\pi$
$439$	−5.50122e6	−1.36238	−0.681189	+	0.732108i	$0.738537\pi$
$440$	408285.	0.100538
$441$	0	0
$442$	19104.8	0.00465144
$443$	−3.19936e6	−0.774558	−0.387279	−	0.921963i	$-0.626585\pi$
$443$	−3.19936e6	−0.774558	−0.387279	+	0.921963i	$0.626585\pi$
$444$	0	0
$445$	9.19852e6	2.20200
$446$	386702.	0.0920533
$447$	0	0
$448$	−5.40578e6	−1.27252
$449$	4.16986e6	0.976124	0.488062	−	0.872809i	$-0.337704\pi$
$449$	4.16986e6	0.976124	0.488062	+	0.872809i	$0.337704\pi$
$450$	0	0
$451$	−2.84437e6	−0.658484
$452$	−3.09559e6	−0.712685
$453$	0	0
$454$	213183.	0.0485414
$455$	4.22517e6	0.956789
$456$	0	0
$457$	5.33516e6	1.19497	0.597485	−	0.801880i	$-0.296167\pi$
$457$	5.33516e6	1.19497	0.597485	+	0.801880i	$0.296167\pi$
$458$	−91473.7	−0.0203766
$459$	0	0
$460$	3.42468e6	0.754614
$461$	5.91111e6	1.29544	0.647719	−	0.761879i	$-0.275723\pi$
$461$	5.91111e6	1.29544	0.647719	+	0.761879i	$0.275723\pi$
$462$	0	0
$463$	−7.25730e6	−1.57334	−0.786670	−	0.617373i	$-0.788197\pi$
$463$	−7.25730e6	−1.57334	−0.786670	+	0.617373i	$0.788197\pi$
$464$	−2.00674e6	−0.432709
$465$	0	0
$466$	129849.	0.0276997
$467$	−8.09273e6	−1.71713	−0.858565	−	0.512706i	$-0.828643\pi$
$467$	−8.09273e6	−1.71713	−0.858565	+	0.512706i	$0.828643\pi$
$468$	0	0
$469$	−3.29600e6	−0.691919
$470$	507925.	0.106061
$471$	0	0
$472$	64541.9	0.0133348
$473$	−4.68185e6	−0.962198
$474$	0	0
$475$	−5.18829e6	−1.05509
$476$	−1.21736e6	−0.246264
$477$	0	0
$478$	64022.6	0.0128163
$479$	−4.31869e6	−0.860030	−0.430015	−	0.902822i	$-0.641492\pi$
$479$	−4.31869e6	−0.860030	−0.430015	+	0.902822i	$0.641492\pi$
$480$	0	0
$481$	−2.03933e6	−0.401907
$482$	−186784.	−0.0366204
$483$	0	0
$484$	3.10034e6	0.601583
$485$	9.56356e6	1.84614
$486$	0	0
$487$	−134182.	−0.0256372	−0.0128186	−	0.999918i	$-0.504080\pi$
$487$	−134182.	−0.0256372	−0.0128186	+	0.999918i	$0.504080\pi$
$488$	−18508.4	−0.00351819
$489$	0	0
$490$	285178.	0.0536570
$491$	−3.28974e6	−0.615825	−0.307913	−	0.951415i	$-0.599630\pi$
$491$	−3.28974e6	−0.615825	−0.307913	+	0.951415i	$0.599630\pi$
$492$	0	0
$493$	−449510.	−0.0832956
$494$	−97611.7	−0.0179964
$495$	0	0
$496$	7.91599e6	1.44478
$497$	−3.62205e6	−0.657755
$498$	0	0
$499$	9.24373e6	1.66187	0.830933	−	0.556373i	$-0.187807\pi$
$499$	9.24373e6	1.66187	0.830933	+	0.556373i	$0.187807\pi$
$500$	3.71757e6	0.665019
$501$	0	0
$502$	329202.	0.0583046
$503$	5.84762e6	1.03053	0.515263	−	0.857032i	$-0.327694\pi$
$503$	5.84762e6	1.03053	0.515263	+	0.857032i	$0.327694\pi$
$504$	0	0
$505$	8.38628e6	1.46333
$506$	90340.9	0.0156858
$507$	0	0
$508$	−726389.	−0.124885
$509$	9.97745e6	1.70697	0.853483	−	0.521120i	$-0.174486\pi$
$509$	9.97745e6	1.70697	0.853483	+	0.521120i	$0.174486\pi$
$510$	0	0
$511$	7.75451e6	1.31372
$512$	−1.50494e6	−0.253714
$513$	0	0
$514$	45620.9	0.00761652
$515$	1.04563e7	1.73724
$516$	0	0
$517$	−5.08174e6	−0.836154
$518$	−342621.	−0.0561035
$519$	0	0
$520$	467393.	0.0758007
$521$	−6.29007e6	−1.01522	−0.507611	−	0.861586i	$-0.669471\pi$
$521$	−6.29007e6	−1.01522	−0.507611	+	0.861586i	$0.669471\pi$
$522$	0	0
$523$	777756.	0.124334	0.0621669	−	0.998066i	$-0.480199\pi$
$523$	777756.	0.124334	0.0621669	+	0.998066i	$0.480199\pi$
$524$	−6.98335e6	−1.11106
$525$	0	0
$526$	−273595.	−0.0431165
$527$	1.77318e6	0.278117
$528$	0	0
$529$	−4.91880e6	−0.764222
$530$	−943958.	−0.145970
$531$	0	0
$532$	6.21981e6	0.952792
$533$	−3.25615e6	−0.496463
$534$	0	0
$535$	−3.26884e6	−0.493753
$536$	−364607.	−0.0548167
$537$	0	0
$538$	153346.	0.0228411
$539$	−2.85318e6	−0.423017
$540$	0	0
$541$	3.53668e6	0.519520	0.259760	−	0.965673i	$-0.416357\pi$
$541$	3.53668e6	0.519520	0.259760	+	0.965673i	$0.416357\pi$
$542$	−526885.	−0.0770403
$543$	0	0
$544$	−202086.	−0.0292779
$545$	8.75389e6	1.26244
$546$	0	0
$547$	−7.93013e6	−1.13321	−0.566607	−	0.823988i	$-0.691744\pi$
$547$	−7.93013e6	−1.13321	−0.566607	+	0.823988i	$0.691744\pi$
$548$	−1.20423e7	−1.71300
$549$	0	0
$550$	327240.	0.0461275
$551$	2.29667e6	0.322270
$552$	0	0
$553$	−1.42493e7	−1.98144
$554$	485217.	0.0671679
$555$	0	0
$556$	160440.	0.0220103
$557$	1.08864e7	1.48678	0.743392	−	0.668857i	$-0.233216\pi$
$557$	1.08864e7	1.48678	0.743392	+	0.668857i	$0.233216\pi$
$558$	0	0
$559$	−5.35964e6	−0.725447
$560$	−1.48322e7	−1.99864
$561$	0	0
$562$	133927.	0.0178866
$563$	−3.05550e6	−0.406267	−0.203133	−	0.979151i	$-0.565113\pi$
$563$	−3.05550e6	−0.406267	−0.203133	+	0.979151i	$0.565113\pi$
$564$	0	0
$565$	−8.44850e6	−1.11342
$566$	−255379.	−0.0335077
$567$	0	0
$568$	−400675.	−0.0521100
$569$	−7.58838e6	−0.982581	−0.491291	−	0.870996i	$-0.663475\pi$
$569$	−7.58838e6	−0.982581	−0.491291	+	0.870996i	$0.663475\pi$
$570$	0	0
$571$	−6.32584e6	−0.811948	−0.405974	−	0.913885i	$-0.633068\pi$
$571$	−6.32584e6	−0.811948	−0.405974	+	0.913885i	$0.633068\pi$
$572$	−2.33503e6	−0.298403
$573$	0	0
$574$	−547055.	−0.0693029
$575$	5.49698e6	0.693354
$576$	0	0
$577$	−2.66897e6	−0.333737	−0.166868	−	0.985979i	$-0.553365\pi$
$577$	−2.66897e6	−0.333737	−0.166868	+	0.985979i	$0.553365\pi$
$578$	396860.	0.0494103
$579$	0	0
$580$	−5.49131e6	−0.677807
$581$	−1.26493e7	−1.55463
$582$	0	0
$583$	9.44422e6	1.15079
$584$	857811.	0.104078
$585$	0	0
$586$	−767972.	−0.0923850
$587$	−5.82526e6	−0.697782	−0.348891	−	0.937163i	$-0.613442\pi$
$587$	−5.82526e6	−0.697782	−0.348891	+	0.937163i	$0.613442\pi$
$588$	0	0
$589$	−9.05967e6	−1.07603
$590$	87957.9	0.0104027
$591$	0	0
$592$	7.15894e6	0.839546
$593$	1.39791e7	1.63246	0.816232	−	0.577724i	$-0.196059\pi$
$593$	1.39791e7	1.63246	0.816232	+	0.577724i	$0.196059\pi$
$594$	0	0
$595$	−3.32241e6	−0.384735
$596$	−5.29226e6	−0.610275
$597$	0	0
$598$	103419.	0.0118263
$599$	1.42274e7	1.62016	0.810081	−	0.586319i	$-0.199423\pi$
$599$	1.42274e7	1.62016	0.810081	+	0.586319i	$0.199423\pi$
$600$	0	0
$601$	8.29425e6	0.936679	0.468340	−	0.883549i	$-0.344852\pi$
$601$	8.29425e6	0.936679	0.468340	+	0.883549i	$0.344852\pi$
$602$	−900455.	−0.101268
$603$	0	0
$604$	−9.75111e6	−1.08758
$605$	8.46145e6	0.939845
$606$	0	0
$607$	6.09959e6	0.671938	0.335969	−	0.941873i	$-0.390936\pi$
$607$	6.09959e6	0.671938	0.335969	+	0.941873i	$0.390936\pi$
$608$	1.03251e6	0.113276
$609$	0	0
$610$	−25223.3	−0.00274459
$611$	−5.81743e6	−0.630417
$612$	0	0
$613$	−9.06279e6	−0.974116	−0.487058	−	0.873370i	$-0.661930\pi$
$613$	−9.06279e6	−0.974116	−0.487058	+	0.873370i	$0.661930\pi$
$614$	670024.	0.0717248
$615$	0	0
$616$	−785637.	−0.0834200
$617$	−1.70732e7	−1.80552	−0.902760	−	0.430145i	$-0.858463\pi$
$617$	−1.70732e7	−1.80552	−0.902760	+	0.430145i	$0.858463\pi$
$618$	0	0
$619$	8.30455e6	0.871143	0.435572	−	0.900154i	$-0.356546\pi$
$619$	8.30455e6	0.871143	0.435572	+	0.900154i	$0.356546\pi$
$620$	2.16616e7	2.26314
$621$	0	0
$622$	−797340.	−0.0826357
$623$	−1.77001e7	−1.82707
$624$	0	0
$625$	−3.79852e6	−0.388968
$626$	613007.	0.0625216
$627$	0	0
$628$	1.31811e7	1.33368
$629$	1.60360e6	0.161611
$630$	0	0
$631$	1.07615e7	1.07597	0.537983	−	0.842956i	$-0.319186\pi$
$631$	1.07615e7	1.07597	0.537983	+	0.842956i	$0.319186\pi$
$632$	−1.57627e6	−0.156978
$633$	0	0
$634$	508323.	0.0502246
$635$	−1.98246e6	−0.195106
$636$	0	0
$637$	−3.26624e6	−0.318933
$638$	−144857.	−0.0140893
$639$	0	0
$640$	−3.29019e6	−0.317520
$641$	1.33893e7	1.28711	0.643553	−	0.765402i	$-0.277460\pi$
$641$	1.33893e7	1.28711	0.643553	+	0.765402i	$0.277460\pi$
$642$	0	0
$643$	−2.78332e6	−0.265482	−0.132741	−	0.991151i	$-0.542378\pi$
$643$	−2.78332e6	−0.265482	−0.132741	+	0.991151i	$0.542378\pi$
$644$	−6.58988e6	−0.626128
$645$	0	0
$646$	76755.8	0.00723652
$647$	−2.39129e6	−0.224580	−0.112290	−	0.993675i	$-0.535819\pi$
$647$	−2.39129e6	−0.224580	−0.112290	+	0.993675i	$0.535819\pi$
$648$	0	0
$649$	−880011.	−0.0820118
$650$	374614.	0.0347777
$651$	0	0
$652$	−7.48567e6	−0.689623
$653$	−1.02862e7	−0.943997	−0.471998	−	0.881599i	$-0.656467\pi$
$653$	−1.02862e7	−0.943997	−0.471998	+	0.881599i	$0.656467\pi$
$654$	0	0
$655$	−1.90590e7	−1.73579
$656$	1.14305e7	1.03706
$657$	0	0
$658$	−977367.	−0.0880021
$659$	−1.52742e7	−1.37007	−0.685036	−	0.728509i	$-0.740214\pi$
$659$	−1.52742e7	−1.37007	−0.685036	+	0.728509i	$0.740214\pi$
$660$	0	0
$661$	−2.48992e6	−0.221657	−0.110829	−	0.993840i	$-0.535350\pi$
$661$	−2.48992e6	−0.221657	−0.110829	+	0.993840i	$0.535350\pi$
$662$	281778.	0.0249898
$663$	0	0
$664$	−1.39928e6	−0.123164
$665$	1.69751e7	1.48853
$666$	0	0
$667$	−2.43332e6	−0.211780
$668$	4.41872e6	0.383138
$669$	0	0
$670$	−496888.	−0.0427633
$671$	252357.	0.0216376
$672$	0	0
$673$	1.62042e7	1.37908	0.689542	−	0.724246i	$-0.257812\pi$
$673$	1.62042e7	1.37908	0.689542	+	0.724246i	$0.257812\pi$
$674$	−298260.	−0.0252898
$675$	0	0
$676$	9.17706e6	0.772390
$677$	−8.11683e6	−0.680636	−0.340318	−	0.940310i	$-0.610535\pi$
$677$	−8.11683e6	−0.680636	−0.340318	+	0.940310i	$0.610535\pi$
$678$	0	0
$679$	−1.84025e7	−1.53180
$680$	−367529.	−0.0304803
$681$	0	0
$682$	571419.	0.0470429
$683$	1.39257e7	1.14226	0.571131	−	0.820859i	$-0.306505\pi$
$683$	1.39257e7	1.14226	0.571131	+	0.820859i	$0.306505\pi$
$684$	0	0
$685$	−3.28658e7	−2.67619
$686$	268433.	0.0217784
$687$	0	0
$688$	1.88147e7	1.51539
$689$	1.08115e7	0.867633
$690$	0	0
$691$	2.03394e7	1.62048	0.810238	−	0.586101i	$-0.199338\pi$
$691$	2.03394e7	1.62048	0.810238	+	0.586101i	$0.199338\pi$
$692$	4.81739e6	0.382425
$693$	0	0
$694$	650902.	0.0513000
$695$	437874.	0.0343864
$696$	0	0
$697$	2.56043e6	0.199633
$698$	−670742.	−0.0521095
$699$	0	0
$700$	−2.38704e7	−1.84126
$701$	−2.46892e6	−0.189763	−0.0948815	−	0.995489i	$-0.530247\pi$
$701$	−2.46892e6	−0.189763	−0.0948815	+	0.995489i	$0.530247\pi$
$702$	0	0
$703$	−8.19325e6	−0.625270
$704$	8.15347e6	0.620027
$705$	0	0
$706$	−217426.	−0.0164172
$707$	−1.61372e7	−1.21417
$708$	0	0
$709$	1.05339e7	0.786994	0.393497	−	0.919326i	$-0.371265\pi$
$709$	1.05339e7	0.786994	0.393497	+	0.919326i	$0.371265\pi$
$710$	−546042.	−0.0406518
$711$	0	0
$712$	−1.95800e6	−0.144748
$713$	9.59871e6	0.707113
$714$	0	0
$715$	−6.37278e6	−0.466191
$716$	90998.7	0.00663365
$717$	0	0
$718$	724752.	0.0524660
$719$	−1.89639e7	−1.36806	−0.684029	−	0.729455i	$-0.739774\pi$
$719$	−1.89639e7	−1.36806	−0.684029	+	0.729455i	$0.739774\pi$
$720$	0	0
$721$	−2.01204e7	−1.44144
$722$	326119.	0.0232827
$723$	0	0
$724$	−1.08736e7	−0.770952
$725$	−8.81416e6	−0.622782
$726$	0	0
$727$	−579638.	−0.0406744	−0.0203372	−	0.999793i	$-0.506474\pi$
$727$	−579638.	−0.0406744	−0.0203372	+	0.999793i	$0.506474\pi$
$728$	−899373.	−0.0628943
$729$	0	0
$730$	1.16903e6	0.0811929
$731$	4.21449e6	0.291710
$732$	0	0
$733$	−1.22334e7	−0.840980	−0.420490	−	0.907297i	$-0.638142\pi$
$733$	−1.22334e7	−0.840980	−0.420490	+	0.907297i	$0.638142\pi$
$734$	747696.	0.0512253
$735$	0	0
$736$	−1.09395e6	−0.0744393
$737$	4.97132e6	0.337134
$738$	0	0
$739$	1.16099e7	0.782017	0.391008	−	0.920387i	$-0.372126\pi$
$739$	1.16099e7	0.782017	0.391008	+	0.920387i	$0.372126\pi$
$740$	1.95900e7	1.31509
$741$	0	0
$742$	1.81640e6	0.121116
$743$	−2.39450e7	−1.59127	−0.795634	−	0.605777i	$-0.792862\pi$
$743$	−2.39450e7	−1.59127	−0.795634	+	0.605777i	$0.792862\pi$
$744$	0	0
$745$	−1.44436e7	−0.953424
$746$	290008.	0.0190793
$747$	0	0
$748$	1.83612e6	0.119991
$749$	6.29002e6	0.409683
$750$	0	0
$751$	1.54209e6	0.0997720	0.0498860	−	0.998755i	$-0.484114\pi$
$751$	1.54209e6	0.0997720	0.0498860	+	0.998755i	$0.484114\pi$
$752$	2.04217e7	1.31688
$753$	0	0
$754$	−165828.	−0.0106226
$755$	−2.66128e7	−1.69911
$756$	0	0
$757$	−2.03724e7	−1.29212	−0.646059	−	0.763288i	$-0.723584\pi$
$757$	−2.03724e7	−1.29212	−0.646059	+	0.763288i	$0.723584\pi$
$758$	1.03699e6	0.0655546
$759$	0	0
$760$	1.87780e6	0.117928
$761$	6.63887e6	0.415559	0.207779	−	0.978176i	$-0.433376\pi$
$761$	6.63887e6	0.415559	0.207779	+	0.978176i	$0.433376\pi$
$762$	0	0
$763$	−1.68445e7	−1.04749
$764$	−3.22744e6	−0.200044
$765$	0	0
$766$	−13800.7	−0.000849822 0
$767$	−1.00741e6	−0.0618327
$768$	0	0
$769$	−1.86945e7	−1.13998	−0.569992	−	0.821650i	$-0.693054\pi$
$769$	−1.86945e7	−1.13998	−0.569992	+	0.821650i	$0.693054\pi$
$770$	−1.07067e6	−0.0650771
$771$	0	0
$772$	−1.10108e7	−0.664930
$773$	−2.02459e7	−1.21867	−0.609337	−	0.792911i	$-0.708564\pi$
$773$	−2.02459e7	−1.21867	−0.609337	+	0.792911i	$0.708564\pi$
$774$	0	0
$775$	3.47692e7	2.07941
$776$	−2.03570e6	−0.121356
$777$	0	0
$778$	−1.24027e6	−0.0734630
$779$	−1.30820e7	−0.772377
$780$	0	0
$781$	5.46310e6	0.320488
$782$	−81322.7	−0.00475548
$783$	0	0
$784$	1.14659e7	0.666221
$785$	3.59739e7	2.08359
$786$	0	0
$787$	1.82754e7	1.05179	0.525895	−	0.850550i	$-0.323730\pi$
$787$	1.82754e7	1.05179	0.525895	+	0.850550i	$0.323730\pi$
$788$	−1.61685e7	−0.927588
$789$	0	0
$790$	−2.14815e6	−0.122461
$791$	1.62569e7	0.923839
$792$	0	0
$793$	288891.	0.0163136
$794$	1.42878e6	0.0804290
$795$	0	0
$796$	7.06935e6	0.395455
$797$	2.12442e7	1.18466	0.592331	−	0.805695i	$-0.298208\pi$
$797$	2.12442e7	1.18466	0.592331	+	0.805695i	$0.298208\pi$
$798$	0	0
$799$	4.57446e6	0.253497
$800$	−3.96259e6	−0.218904
$801$	0	0
$802$	−894036.	−0.0490816
$803$	−1.16960e7	−0.640103
$804$	0	0
$805$	−1.79851e7	−0.978191
$806$	654144.	0.0354679
$807$	0	0
$808$	−1.78511e6	−0.0961915
$809$	−1.25471e7	−0.674020	−0.337010	−	0.941501i	$-0.609416\pi$
$809$	−1.25471e7	−0.674020	−0.337010	+	0.941501i	$0.609416\pi$
$810$	0	0
$811$	727251.	0.0388269	0.0194134	−	0.999812i	$-0.493820\pi$
$811$	727251.	0.0388269	0.0194134	+	0.999812i	$0.493820\pi$
$812$	1.05666e7	0.562399
$813$	0	0
$814$	516772.	0.0273362
$815$	−2.04299e7	−1.07739
$816$	0	0
$817$	−2.15330e7	−1.12862
$818$	1.17125e6	0.0612019
$819$	0	0
$820$	3.12788e7	1.62449
$821$	−1.33577e7	−0.691629	−0.345815	−	0.938303i	$-0.612397\pi$
$821$	−1.33577e7	−0.691629	−0.345815	+	0.938303i	$0.612397\pi$
$822$	0	0
$823$	−1.38719e7	−0.713896	−0.356948	−	0.934124i	$-0.616183\pi$
$823$	−1.38719e7	−0.713896	−0.356948	+	0.934124i	$0.616183\pi$
$824$	−2.22573e6	−0.114197
$825$	0	0
$826$	−169252.	−0.00863143
$827$	−3.82559e7	−1.94507	−0.972533	−	0.232765i	$-0.925223\pi$
$827$	−3.82559e7	−1.94507	−0.972533	+	0.232765i	$0.925223\pi$
$828$	0	0
$829$	1.78067e7	0.899907	0.449954	−	0.893052i	$-0.351441\pi$
$829$	1.78067e7	0.899907	0.449954	+	0.893052i	$0.351441\pi$
$830$	−1.90694e6	−0.0960821
$831$	0	0
$832$	9.33384e6	0.467468
$833$	2.56837e6	0.128246
$834$	0	0
$835$	1.20596e7	0.598571
$836$	−9.38126e6	−0.464243
$837$	0	0
$838$	929751.	0.0457359
$839$	−9.51395e6	−0.466612	−0.233306	−	0.972403i	$-0.574954\pi$
$839$	−9.51395e6	−0.466612	−0.233306	+	0.972403i	$0.574954\pi$
$840$	0	0
$841$	−1.66094e7	−0.809776
$842$	−1.12024e6	−0.0544542
$843$	0	0
$844$	3.09410e7	1.49513
$845$	2.50460e7	1.20669
$846$	0	0
$847$	−1.62818e7	−0.779820
$848$	−3.79529e7	−1.81241
$849$	0	0
$850$	−294574.	−0.0139845
$851$	8.68074e6	0.410897
$852$	0	0
$853$	−1.04920e7	−0.493727	−0.246864	−	0.969050i	$-0.579400\pi$
$853$	−1.04920e7	−0.493727	−0.246864	+	0.969050i	$0.579400\pi$
$854$	48535.6	0.00227727
$855$	0	0
$856$	695808.	0.0324567
$857$	−1.41436e7	−0.657821	−0.328911	−	0.944361i	$-0.606682\pi$
$857$	−1.41436e7	−0.657821	−0.328911	+	0.944361i	$0.606682\pi$
$858$	0	0
$859$	3.56902e7	1.65031	0.825155	−	0.564906i	$-0.191088\pi$
$859$	3.56902e7	1.65031	0.825155	+	0.564906i	$0.191088\pi$
$860$	5.14851e7	2.37375
$861$	0	0
$862$	1.32621e6	0.0607915
$863$	−1.25188e7	−0.572183	−0.286091	−	0.958202i	$-0.592356\pi$
$863$	−1.25188e7	−0.572183	−0.286091	+	0.958202i	$0.592356\pi$
$864$	0	0
$865$	1.31476e7	0.597457
$866$	−1.84332e6	−0.0835229
$867$	0	0
$868$	−4.16820e7	−1.87780
$869$	2.14920e7	0.965445
$870$	0	0
$871$	5.69102e6	0.254182
$872$	−1.86336e6	−0.0829861
$873$	0	0
$874$	415500.	0.0183989
$875$	−1.95233e7	−0.862050
$876$	0	0
$877$	−3.08425e7	−1.35410	−0.677050	−	0.735937i	$-0.736742\pi$
$877$	−3.08425e7	−1.35410	−0.677050	+	0.735937i	$0.736742\pi$
$878$	1.59583e6	0.0698637
$879$	0	0
$880$	2.23712e7	0.973829
$881$	1.52828e6	0.0663379	0.0331690	−	0.999450i	$-0.489440\pi$
$881$	1.52828e6	0.0663379	0.0331690	+	0.999450i	$0.489440\pi$
$882$	0	0
$883$	1.20855e7	0.521632	0.260816	−	0.965389i	$-0.416008\pi$
$883$	1.20855e7	0.521632	0.260816	+	0.965389i	$0.416008\pi$
$884$	2.10194e6	0.0904669
$885$	0	0
$886$	928094.	0.0397198
$887$	−1.34386e7	−0.573516	−0.286758	−	0.958003i	$-0.592578\pi$
$887$	−1.34386e7	−0.573516	−0.286758	+	0.958003i	$0.592578\pi$
$888$	0	0
$889$	3.81473e6	0.161886
$890$	−2.66838e6	−0.112920
$891$	0	0
$892$	4.25456e7	1.79037
$893$	−2.33722e7	−0.980778
$894$	0	0
$895$	248354.	0.0103637
$896$	6.33110e6	0.263456
$897$	0	0
$898$	−1.20962e6	−0.0500563
$899$	−1.53911e7	−0.635141
$900$	0	0
$901$	−8.50146e6	−0.348884
$902$	825116.	0.0337675
$903$	0	0
$904$	1.79835e6	0.0731903
$905$	−2.96763e7	−1.20445
$906$	0	0
$907$	4.92075e6	0.198616	0.0993078	−	0.995057i	$-0.468337\pi$
$907$	4.92075e6	0.198616	0.0993078	+	0.995057i	$0.468337\pi$
$908$	2.34547e7	0.944093
$909$	0	0
$910$	−1.22567e6	−0.0490648
$911$	4.20882e7	1.68021	0.840106	−	0.542422i	$-0.182493\pi$
$911$	4.20882e7	1.68021	0.840106	+	0.542422i	$0.182493\pi$
$912$	0	0
$913$	1.90788e7	0.757485
$914$	−1.54766e6	−0.0612789
$915$	0	0
$916$	−1.00641e7	−0.396310
$917$	3.66739e7	1.44024
$918$	0	0
$919$	−4.32245e7	−1.68827	−0.844134	−	0.536133i	$-0.819885\pi$
$919$	−4.32245e7	−1.68827	−0.844134	+	0.536133i	$0.819885\pi$
$920$	−1.98953e6	−0.0774963
$921$	0	0
$922$	−1.71474e6	−0.0664310
$923$	6.25399e6	0.241631
$924$	0	0
$925$	3.14441e7	1.20833
$926$	2.10525e6	0.0806820
$927$	0	0
$928$	1.75410e6	0.0668626
$929$	−4.09390e6	−0.155632	−0.0778158	−	0.996968i	$-0.524795\pi$
$929$	−4.09390e6	−0.155632	−0.0778158	+	0.996968i	$0.524795\pi$
$930$	0	0
$931$	−1.31225e7	−0.496183
$932$	1.42862e7	0.538738
$933$	0	0
$934$	2.34760e6	0.0880556
$935$	5.01115e6	0.187460
$936$	0	0
$937$	1.35185e6	0.0503015	0.0251507	−	0.999684i	$-0.491993\pi$
$937$	1.35185e6	0.0503015	0.0251507	+	0.999684i	$0.491993\pi$
$938$	956129.	0.0354821
$939$	0	0
$940$	5.58827e7	2.06280
$941$	−3.76790e7	−1.38715	−0.693577	−	0.720382i	$-0.743966\pi$
$941$	−3.76790e7	−1.38715	−0.693577	+	0.720382i	$0.743966\pi$
$942$	0	0
$943$	1.38603e7	0.507568
$944$	3.53645e6	0.129163
$945$	0	0
$946$	1.35814e6	0.0493422
$947$	−2.29344e7	−0.831024	−0.415512	−	0.909588i	$-0.636397\pi$
$947$	−2.29344e7	−0.831024	−0.415512	+	0.909588i	$0.636397\pi$
$948$	0	0
$949$	−1.33893e7	−0.482604
$950$	1.50506e6	0.0541058
$951$	0	0
$952$	707211.	0.0252905
$953$	−3.44029e7	−1.22705	−0.613526	−	0.789675i	$-0.710249\pi$
$953$	−3.44029e7	−1.22705	−0.613526	+	0.789675i	$0.710249\pi$
$954$	0	0
$955$	−8.80834e6	−0.312526
$956$	7.04386e6	0.249268
$957$	0	0
$958$	1.25280e6	0.0441029
$959$	6.32415e7	2.22052
$960$	0	0
$961$	3.20841e7	1.12068
$962$	591585.	0.0206101
$963$	0	0
$964$	−2.05503e7	−0.712239
$965$	−3.00507e7	−1.03881
$966$	0	0
$967$	4.00865e7	1.37858	0.689289	−	0.724486i	$-0.257923\pi$
$967$	4.00865e7	1.37858	0.689289	+	0.724486i	$0.257923\pi$
$968$	−1.80111e6	−0.0617806
$969$	0	0
$970$	−2.77427e6	−0.0946714
$971$	−3.52454e7	−1.19965	−0.599825	−	0.800131i	$-0.704763\pi$
$971$	−3.52454e7	−1.19965	−0.599825	+	0.800131i	$0.704763\pi$
$972$	0	0
$973$	−842572.	−0.0285315
$974$	38924.5	0.00131470
$975$	0	0
$976$	−1.01413e6	−0.0340777
$977$	1.35507e7	0.454178	0.227089	−	0.973874i	$-0.427079\pi$
$977$	1.35507e7	0.454178	0.227089	+	0.973874i	$0.427079\pi$
$978$	0	0
$979$	2.66969e7	0.890233
$980$	3.13757e7	1.04359
$981$	0	0
$982$	954311.	0.0315799
$983$	−4.74375e7	−1.56580	−0.782902	−	0.622145i	$-0.786262\pi$
$983$	−4.74375e7	−1.56580	−0.782902	+	0.622145i	$0.786262\pi$
$984$	0	0
$985$	−4.41272e7	−1.44916
$986$	130397.	0.00427146
$987$	0	0
$988$	−1.07394e7	−0.350015
$989$	2.28141e7	0.741675
$990$	0	0
$991$	4.59032e7	1.48477	0.742385	−	0.669974i	$-0.233695\pi$
$991$	4.59032e7	1.48477	0.742385	+	0.669974i	$0.233695\pi$
$992$	−6.91938e6	−0.223248
$993$	0	0
$994$	1.05071e6	0.0337301
$995$	1.92937e7	0.617814
$996$	0	0
$997$	2.51295e6	0.0800655	0.0400327	−	0.999198i	$-0.487254\pi$
$997$	2.51295e6	0.0800655	0.0400327	+	0.999198i	$0.487254\pi$
$998$	−2.68149e6	−0.0852216
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.c.1.8		12
3.2	odd	2		177.6.a.c.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.c.1.5	✓	12	3.2	odd	2
531.6.a.c.1.8		12	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-0.290087 q^{2} -31.9158 q^{4} -87.1048 q^{5} +167.610 q^{7} +18.5412 q^{8} +O(q^{10})\)	\(q-0.290087 q^{2} -31.9158 q^{4} -87.1048 q^{5} +167.610 q^{7} +18.5412 q^{8} +25.2680 q^{10} -252.804 q^{11} -289.403 q^{13} -48.6216 q^{14} +1015.93 q^{16} +227.568 q^{17} -1162.71 q^{19} +2780.02 q^{20} +73.3353 q^{22} +1231.89 q^{23} +4462.24 q^{25} +83.9520 q^{26} -5349.42 q^{28} -1975.28 q^{29} +7791.87 q^{31} -888.026 q^{32} -66.0147 q^{34} -14599.6 q^{35} +7046.70 q^{37} +337.287 q^{38} -1615.03 q^{40} +11251.3 q^{41} +18519.7 q^{43} +8068.46 q^{44} -357.355 q^{46} +20101.5 q^{47} +11286.1 q^{49} -1294.44 q^{50} +9236.53 q^{52} -37357.8 q^{53} +22020.5 q^{55} +3107.69 q^{56} +573.003 q^{58} +3481.00 q^{59} -998.231 q^{61} -2260.32 q^{62} -32252.1 q^{64} +25208.3 q^{65} -19664.7 q^{67} -7263.03 q^{68} +4235.17 q^{70} -21610.0 q^{71} +46265.2 q^{73} -2044.16 q^{74} +37108.8 q^{76} -42372.5 q^{77} -85014.5 q^{79} -88492.2 q^{80} -3263.86 q^{82} -75468.7 q^{83} -19822.3 q^{85} -5372.32 q^{86} -4687.29 q^{88} -105603. q^{89} -48506.8 q^{91} -39316.7 q^{92} -5831.19 q^{94} +101277. q^{95} -109794. q^{97} -3273.97 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) \) 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) \) 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8 + 601 * q^10 - 1480 * q^11 + 472 * q^13 - 1065 * q^14 + 6370 * q^16 - 1565 * q^17 + 3939 * q^19 - 8033 * q^20 - 1738 * q^22 - 7245 * q^23 + 9690 * q^25 - 3764 * q^26 + 12154 * q^28 - 10003 * q^29 + 7295 * q^31 - 11628 * q^32 - 16344 * q^34 - 11015 * q^35 + 6741 * q^37 - 3035 * q^38 + 5572 * q^40 - 34025 * q^41 - 6336 * q^43 - 41168 * q^44 + 2345 * q^46 - 66167 * q^47 + 28319 * q^49 - 31173 * q^50 + 16440 * q^52 - 62290 * q^53 + 55764 * q^55 - 107306 * q^56 + 37952 * q^58 + 41772 * q^59 + 68469 * q^61 - 99190 * q^62 + 68525 * q^64 - 80156 * q^65 + 113310 * q^67 - 33887 * q^68 + 32034 * q^70 - 84520 * q^71 + 135895 * q^73 + 31962 * q^74 - 61848 * q^76 + 3799 * q^77 + 14122 * q^79 - 77609 * q^80 - 1501 * q^82 - 114463 * q^83 - 101097 * q^85 + 203536 * q^86 - 244967 * q^88 - 189109 * q^89 - 168249 * q^91 + 71946 * q^92 - 472284 * q^94 - 21923 * q^95 - 76192 * q^97 + 17544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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