LMFDB - Embedded newform 425.6.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,6,Mod(1,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("425.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.1631234205$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 $ x^5 - 2x^4 - 95x^3 + 220x^2 + 1668x - 4640
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.57023$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-7.57023 q^{2} +20.2147 q^{3} +25.3084 q^{4} -153.030 q^{6} -57.9243 q^{7} +50.6569 q^{8} +165.633 q^{9} -590.831 q^{11} +511.602 q^{12} +810.340 q^{13} +438.500 q^{14} -1193.35 q^{16} -289.000 q^{17} -1253.88 q^{18} +270.112 q^{19} -1170.92 q^{21} +4472.73 q^{22} +4208.47 q^{23} +1024.01 q^{24} -6134.46 q^{26} -1563.94 q^{27} -1465.97 q^{28} -404.699 q^{29} +2000.47 q^{31} +7412.94 q^{32} -11943.5 q^{33} +2187.80 q^{34} +4191.92 q^{36} -10822.6 q^{37} -2044.81 q^{38} +16380.8 q^{39} +2216.89 q^{41} +8864.14 q^{42} -10418.0 q^{43} -14953.0 q^{44} -31859.1 q^{46} +7914.78 q^{47} -24123.3 q^{48} -13451.8 q^{49} -5842.04 q^{51} +20508.4 q^{52} -19868.5 q^{53} +11839.4 q^{54} -2934.26 q^{56} +5460.22 q^{57} +3063.66 q^{58} -46845.0 q^{59} -11533.8 q^{61} -15144.0 q^{62} -9594.19 q^{63} -17930.4 q^{64} +90414.8 q^{66} +60931.7 q^{67} -7314.13 q^{68} +85072.8 q^{69} +70722.9 q^{71} +8390.47 q^{72} -42807.6 q^{73} +81929.8 q^{74} +6836.10 q^{76} +34223.5 q^{77} -124006. q^{78} +21266.0 q^{79} -71863.5 q^{81} -16782.4 q^{82} +29097.1 q^{83} -29634.1 q^{84} +78866.7 q^{86} -8180.86 q^{87} -29929.7 q^{88} -20515.1 q^{89} -46938.4 q^{91} +106510. q^{92} +40438.8 q^{93} -59916.7 q^{94} +149850. q^{96} -100926. q^{97} +101833. q^{98} -97861.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 7 q^{2} + 36 q^{3} + 43 q^{4} - 105 q^{6} + 204 q^{7} + 63 q^{8} + 531 q^{9} - 792 q^{11} - 785 q^{12} - 88 q^{13} + 860 q^{14} - 2365 q^{16} - 1445 q^{17} + 2052 q^{18} - 5160 q^{19} - 6428 q^{21}+ \cdots - 535112 q^{99}+O(q^{100}) $ 5 * q + 7 * q^2 + 36 * q^3 + 43 * q^4 - 105 * q^6 + 204 * q^7 + 63 * q^8 + 531 * q^9 - 792 * q^11 - 785 * q^12 - 88 * q^13 + 860 * q^14 - 2365 * q^16 - 1445 * q^17 + 2052 * q^18 - 5160 * q^19 - 6428 * q^21 + 3058 * q^22 + 6140 * q^23 - 11751 * q^24 - 9643 * q^26 + 13380 * q^27 + 4568 * q^28 - 7328 * q^29 - 8968 * q^31 + 4703 * q^32 - 13072 * q^33 - 2023 * q^34 + 2048 * q^36 - 9834 * q^37 - 10625 * q^38 + 10108 * q^39 - 5302 * q^41 - 16222 * q^42 + 4916 * q^43 - 31514 * q^44 - 16768 * q^46 + 20364 * q^47 - 45557 * q^48 - 26483 * q^49 - 10404 * q^51 + 11525 * q^52 - 56492 * q^53 + 64635 * q^54 + 73228 * q^56 - 67554 * q^57 - 54213 * q^58 - 96200 * q^59 - 7284 * q^61 - 122335 * q^62 - 64108 * q^63 + 28483 * q^64 + 115232 * q^66 + 60308 * q^67 - 12427 * q^68 - 89076 * q^69 - 25304 * q^71 - 118722 * q^72 - 163328 * q^73 - 82768 * q^74 + 92027 * q^76 - 19912 * q^77 - 134895 * q^78 - 50940 * q^79 + 92445 * q^81 - 191390 * q^82 + 179060 * q^83 - 128054 * q^84 + 87086 * q^86 - 55764 * q^87 - 167634 * q^88 - 387356 * q^89 - 115200 * q^91 + 293228 * q^92 + 202558 * q^93 + 100515 * q^94 + 428113 * q^96 + 106076 * q^97 - 49329 * q^98 - 535112 * q^99

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 $

$2$	−7.57023	−1.33824	−0.669120	−	0.743154i	$-0.733329\pi$
$2$	−7.57023	−1.33824	−0.669120	+	0.743154i	$0.733329\pi$
$3$	20.2147	1.29677	0.648386	−	0.761312i	$-0.275444\pi$
$3$	20.2147	1.29677	0.648386	+	0.761312i	$0.275444\pi$
$4$	25.3084	0.790888
$5$	0	0
$5$	0	0
$6$	−153.030	−1.73539
$7$	−57.9243	−0.446802	−0.223401	−	0.974727i	$-0.571716\pi$
$7$	−57.9243	−0.446802	−0.223401	+	0.974727i	$0.571716\pi$
$8$	50.6569	0.279842
$9$	165.633	0.681619
$10$	0	0
$11$	−590.831	−1.47225	−0.736126	−	0.676845i	$-0.763347\pi$
$11$	−590.831	−1.47225	−0.736126	+	0.676845i	$0.763347\pi$
$12$	511.602	1.02560
$13$	810.340	1.32987	0.664935	−	0.746901i	$-0.268459\pi$
$13$	810.340	1.32987	0.664935	+	0.746901i	$0.268459\pi$
$14$	438.500	0.597929
$15$	0	0
$16$	−1193.35	−1.16538
$17$	−289.000	−0.242536
$17$	−289.000	−0.242536
$18$	−1253.88	−0.912170
$19$	270.112	0.171656	0.0858281	−	0.996310i	$-0.472646\pi$
$19$	270.112	0.171656	0.0858281	+	0.996310i	$0.472646\pi$
$20$	0	0
$21$	−1170.92	−0.579401
$22$	4472.73	1.97023
$23$	4208.47	1.65884	0.829419	−	0.558626i	$-0.188671\pi$
$23$	4208.47	1.65884	0.829419	+	0.558626i	$0.188671\pi$
$24$	1024.01	0.362892
$25$	0	0
$26$	−6134.46	−1.77969
$27$	−1563.94	−0.412868
$28$	−1465.97	−0.353371
$29$	−404.699	−0.0893587	−0.0446794	−	0.999001i	$-0.514227\pi$
$29$	−404.699	−0.0893587	−0.0446794	+	0.999001i	$0.514227\pi$
$30$	0	0
$31$	2000.47	0.373875	0.186938	−	0.982372i	$-0.440144\pi$
$31$	2000.47	0.373875	0.186938	+	0.982372i	$0.440144\pi$
$32$	7412.94	1.27972
$33$	−11943.5	−1.90917
$34$	2187.80	0.324571
$35$	0	0
$36$	4191.92	0.539084
$37$	−10822.6	−1.29966	−0.649828	−	0.760081i	$-0.725159\pi$
$37$	−10822.6	−1.29966	−0.649828	+	0.760081i	$0.725159\pi$
$38$	−2044.81	−0.229717
$39$	16380.8	1.72454
$40$	0	0
$41$	2216.89	0.205961	0.102980	−	0.994683i	$-0.467162\pi$
$41$	2216.89	0.205961	0.102980	+	0.994683i	$0.467162\pi$
$42$	8864.14	0.775378
$43$	−10418.0	−0.859238	−0.429619	−	0.903010i	$-0.641352\pi$
$43$	−10418.0	−0.859238	−0.429619	+	0.903010i	$0.641352\pi$
$44$	−14953.0	−1.16439
$45$	0	0
$46$	−31859.1	−2.21993
$47$	7914.78	0.522630	0.261315	−	0.965254i	$-0.415844\pi$
$47$	7914.78	0.522630	0.261315	+	0.965254i	$0.415844\pi$
$48$	−24123.3	−1.51124
$49$	−13451.8	−0.800368
$50$	0	0
$51$	−5842.04	−0.314514
$52$	20508.4	1.05178
$53$	−19868.5	−0.971574	−0.485787	−	0.874077i	$-0.661467\pi$
$53$	−19868.5	−0.971574	−0.485787	+	0.874077i	$0.661467\pi$
$54$	11839.4	0.552517
$55$	0	0
$56$	−2934.26	−0.125034
$57$	5460.22	0.222599
$58$	3063.66	0.119583
$59$	−46845.0	−1.75200	−0.875998	−	0.482315i	$-0.839796\pi$
$59$	−46845.0	−1.75200	−0.875998	+	0.482315i	$0.839796\pi$
$60$	0	0
$61$	−11533.8	−0.396869	−0.198434	−	0.980114i	$-0.563586\pi$
$61$	−11533.8	−0.396869	−0.198434	+	0.980114i	$0.563586\pi$
$62$	−15144.0	−0.500335
$63$	−9594.19	−0.304549
$64$	−17930.4	−0.547192
$65$	0	0
$66$	90414.8	2.55494
$67$	60931.7	1.65827	0.829137	−	0.559045i	$-0.188832\pi$
$67$	60931.7	1.65827	0.829137	+	0.559045i	$0.188832\pi$
$68$	−7314.13	−0.191819
$69$	85072.8	2.15114
$70$	0	0
$71$	70722.9	1.66500	0.832500	−	0.554026i	$-0.186909\pi$
$71$	70722.9	1.66500	0.832500	+	0.554026i	$0.186909\pi$
$72$	8390.47	0.190746
$73$	−42807.6	−0.940186	−0.470093	−	0.882617i	$-0.655780\pi$
$73$	−42807.6	−0.940186	−0.470093	+	0.882617i	$0.655780\pi$
$74$	81929.8	1.73925
$75$	0	0
$76$	6836.10	0.135761
$77$	34223.5	0.657805
$78$	−124006.	−2.30785
$79$	21266.0	0.383370	0.191685	−	0.981456i	$-0.438605\pi$
$79$	21266.0	0.383370	0.191685	+	0.981456i	$0.438605\pi$
$80$	0	0
$81$	−71863.5	−1.21701
$82$	−16782.4	−0.275625
$83$	29097.1	0.463611	0.231806	−	0.972762i	$-0.425537\pi$
$83$	29097.1	0.463611	0.231806	+	0.972762i	$0.425537\pi$
$84$	−29634.1	−0.458241
$85$	0	0
$86$	78866.7	1.14987
$87$	−8180.86	−0.115878
$88$	−29929.7	−0.411998
$89$	−20515.1	−0.274536	−0.137268	−	0.990534i	$-0.543832\pi$
$89$	−20515.1	−0.274536	−0.137268	+	0.990534i	$0.543832\pi$
$90$	0	0
$91$	−46938.4	−0.594189
$92$	106510.	1.31196
$93$	40438.8	0.484831
$94$	−59916.7	−0.699405
$95$	0	0
$96$	149850.	1.65951
$97$	−100926.	−1.08911	−0.544557	−	0.838724i	$-0.683302\pi$
$97$	−100926.	−1.08911	−0.544557	+	0.838724i	$0.683302\pi$
$98$	101833.	1.07108
$99$	−97861.4	−1.00351
$100$	0	0
$101$	−45296.3	−0.441834	−0.220917	−	0.975293i	$-0.570905\pi$
$101$	−45296.3	−0.441834	−0.220917	+	0.975293i	$0.570905\pi$
$102$	44225.6	0.420895
$103$	168235.	1.56251	0.781255	−	0.624212i	$-0.214580\pi$
$103$	168235.	1.56251	0.781255	+	0.624212i	$0.214580\pi$
$104$	41049.3	0.372154
$105$	0	0
$106$	150409.	1.30020
$107$	−95188.3	−0.803756	−0.401878	−	0.915693i	$-0.631642\pi$
$107$	−95188.3	−0.803756	−0.401878	+	0.915693i	$0.631642\pi$
$108$	−39580.9	−0.326532
$109$	−133637.	−1.07736	−0.538678	−	0.842512i	$-0.681076\pi$
$109$	−133637.	−1.07736	−0.538678	+	0.842512i	$0.681076\pi$
$110$	0	0
$111$	−218776.	−1.68536
$112$	69124.1	0.520696
$113$	−8986.10	−0.0662026	−0.0331013	−	0.999452i	$-0.510538\pi$
$113$	−8986.10	−0.0662026	−0.0331013	+	0.999452i	$0.510538\pi$
$114$	−41335.1	−0.297891
$115$	0	0
$116$	−10242.3	−0.0706728
$117$	134219.	0.906464
$118$	354627.	2.34459
$119$	16740.1	0.108365
$120$	0	0
$121$	188031.	1.16752
$122$	87313.3	0.531106
$123$	44813.7	0.267084
$124$	50628.6	0.295693
$125$	0	0
$126$	72630.3	0.407560
$127$	−106730.	−0.587189	−0.293594	−	0.955930i	$-0.594851\pi$
$127$	−106730.	−0.587189	−0.293594	+	0.955930i	$0.594851\pi$
$128$	−101477.	−0.547448
$129$	−210597.	−1.11424
$130$	0	0
$131$	−272062.	−1.38513	−0.692564	−	0.721356i	$-0.743519\pi$
$131$	−272062.	−1.38513	−0.692564	+	0.721356i	$0.743519\pi$
$132$	−302270.	−1.50994
$133$	−15646.0	−0.0766964
$134$	−461267.	−2.21917
$135$	0	0
$136$	−14639.8	−0.0678717
$137$	−281000.	−1.27910	−0.639551	−	0.768749i	$-0.720880\pi$
$137$	−281000.	−1.27910	−0.639551	+	0.768749i	$0.720880\pi$
$138$	−644021.	−2.87874
$139$	−367694.	−1.61417	−0.807085	−	0.590435i	$-0.798956\pi$
$139$	−367694.	−1.61417	−0.807085	+	0.590435i	$0.798956\pi$
$140$	0	0
$141$	159995.	0.677732
$142$	−535389.	−2.22817
$143$	−478775.	−1.95790
$144$	−197659.	−0.794348
$145$	0	0
$146$	324063.	1.25819
$147$	−271923.	−1.03789
$148$	−273904.	−1.02788
$149$	249934.	0.922274	0.461137	−	0.887329i	$-0.347442\pi$
$149$	249934.	0.922274	0.461137	+	0.887329i	$0.347442\pi$
$150$	0	0
$151$	−469965.	−1.67735	−0.838674	−	0.544634i	$-0.816669\pi$
$151$	−469965.	−1.67735	−0.838674	+	0.544634i	$0.816669\pi$
$152$	13683.0	0.0480366
$153$	−47868.0	−0.165317
$154$	−259080.	−0.880302
$155$	0	0
$156$	414571.	1.36392
$157$	55643.6	0.180163	0.0900817	−	0.995934i	$-0.471287\pi$
$157$	55643.6	0.180163	0.0900817	+	0.995934i	$0.471287\pi$
$158$	−160989.	−0.513042
$159$	−401636.	−1.25991
$160$	0	0
$161$	−243772.	−0.741173
$162$	544023.	1.62866
$163$	−129793.	−0.382632	−0.191316	−	0.981528i	$-0.561276\pi$
$163$	−129793.	−0.382632	−0.191316	+	0.981528i	$0.561276\pi$
$164$	56106.0	0.162892
$165$	0	0
$166$	−220272.	−0.620424
$167$	42946.0	0.119160	0.0595801	−	0.998224i	$-0.481024\pi$
$167$	42946.0	0.119160	0.0595801	+	0.998224i	$0.481024\pi$
$168$	−59315.2	−0.162141
$169$	285358.	0.768553
$170$	0	0
$171$	44739.5	0.117004
$172$	−263663.	−0.679561
$173$	485194.	1.23254	0.616268	−	0.787536i	$-0.288644\pi$
$173$	485194.	1.23254	0.616268	+	0.787536i	$0.288644\pi$
$174$	61931.0	0.155073
$175$	0	0
$176$	705071.	1.71574
$177$	−946956.	−2.27194
$178$	155304.	0.367395
$179$	−845266.	−1.97179	−0.985895	−	0.167364i	$-0.946474\pi$
$179$	−845266.	−1.97179	−0.985895	+	0.167364i	$0.946474\pi$
$180$	0	0
$181$	−121793.	−0.276328	−0.138164	−	0.990409i	$-0.544120\pi$
$181$	−121793.	−0.276328	−0.138164	+	0.990409i	$0.544120\pi$
$182$	355334.	0.795168
$183$	−233152.	−0.514648
$184$	213188.	0.464213
$185$	0	0
$186$	−306131.	−0.648821
$187$	170750.	0.357073
$188$	200311.	0.413342
$189$	90590.2	0.184470
$190$	0	0
$191$	−337069.	−0.668553	−0.334277	−	0.942475i	$-0.608492\pi$
$191$	−337069.	−0.668553	−0.334277	+	0.942475i	$0.608492\pi$
$192$	−362457.	−0.709584
$193$	−828463.	−1.60096	−0.800478	−	0.599362i	$-0.795421\pi$
$193$	−828463.	−1.60096	−0.800478	+	0.599362i	$0.795421\pi$
$194$	764033.	1.45750
$195$	0	0
$196$	−340443.	−0.633001
$197$	289241.	0.531000	0.265500	−	0.964111i	$-0.414463\pi$
$197$	289241.	0.531000	0.265500	+	0.964111i	$0.414463\pi$
$198$	740833.	1.34294
$199$	74484.2	0.133331	0.0666656	−	0.997775i	$-0.478764\pi$
$199$	74484.2	0.133331	0.0666656	+	0.997775i	$0.478764\pi$
$200$	0	0
$201$	1.23172e6	2.15041
$202$	342903.	0.591280
$203$	23441.9	0.0399257
$204$	−147853.	−0.248745
$205$	0	0
$206$	−1.27358e6	−2.09102
$207$	697062.	1.13070
$208$	−967022.	−1.54981
$209$	−159590.	−0.252721
$210$	0	0
$211$	81694.1	0.126324	0.0631618	−	0.998003i	$-0.479882\pi$
$211$	81694.1	0.126324	0.0631618	+	0.998003i	$0.479882\pi$
$212$	−502841.	−0.768406
$213$	1.42964e6	2.15912
$214$	720598.	1.07562
$215$	0	0
$216$	−79224.4	−0.115538
$217$	−115875.	−0.167048
$218$	1.01166e6	1.44176
$219$	−865342.	−1.21921
$220$	0	0
$221$	−234188.	−0.322541
$222$	1.65619e6	2.25541
$223$	−1.10167e6	−1.48350	−0.741751	−	0.670675i	$-0.766005\pi$
$223$	−1.10167e6	−1.48350	−0.741751	+	0.670675i	$0.766005\pi$
$224$	−429389.	−0.571783
$225$	0	0
$226$	68026.9	0.0885950
$227$	10758.5	0.0138576	0.00692878	−	0.999976i	$-0.497794\pi$
$227$	10758.5	0.0138576	0.00692878	+	0.999976i	$0.497794\pi$
$228$	138190.	0.176051
$229$	328783.	0.414305	0.207153	−	0.978309i	$-0.433580\pi$
$229$	328783.	0.414305	0.207153	+	0.978309i	$0.433580\pi$
$230$	0	0
$231$	691817.	0.853024
$232$	−20500.8	−0.0250064
$233$	319913.	0.386049	0.193024	−	0.981194i	$-0.438170\pi$
$233$	319913.	0.386049	0.193024	+	0.981194i	$0.438170\pi$
$234$	−1.01607e6	−1.21307
$235$	0	0
$236$	−1.18557e6	−1.38563
$237$	429886.	0.497144
$238$	−126727.	−0.145019
$239$	5271.90	0.00596998	0.00298499	−	0.999996i	$-0.499050\pi$
$239$	5271.90	0.00596998	0.00298499	+	0.999996i	$0.499050\pi$
$240$	0	0
$241$	−163951.	−0.181833	−0.0909165	−	0.995859i	$-0.528980\pi$
$241$	−163951.	−0.181833	−0.0909165	+	0.995859i	$0.528980\pi$
$242$	−1.42344e6	−1.56243
$243$	−1.07266e6	−1.16532
$244$	−291902.	−0.313879
$245$	0	0
$246$	−339250.	−0.357423
$247$	218882.	0.228280
$248$	101337.	0.104626
$249$	588188.	0.601199
$250$	0	0
$251$	1.05403e6	1.05601	0.528004	−	0.849242i	$-0.322940\pi$
$251$	1.05403e6	1.05601	0.528004	+	0.849242i	$0.322940\pi$
$252$	−242814.	−0.240864
$253$	−2.48649e6	−2.44223
$254$	807972.	0.785800
$255$	0	0
$256$	1.34198e6	1.27981
$257$	203668.	0.192349	0.0961746	−	0.995364i	$-0.469339\pi$
$257$	203668.	0.192349	0.0961746	+	0.995364i	$0.469339\pi$
$258$	1.59427e6	1.49112
$259$	626893.	0.580689
$260$	0	0
$261$	−67031.6	−0.0609086
$262$	2.05957e6	1.85363
$263$	−178668.	−0.159279	−0.0796393	−	0.996824i	$-0.525377\pi$
$263$	−178668.	−0.159279	−0.0796393	+	0.996824i	$0.525377\pi$
$264$	−605019.	−0.534268
$265$	0	0
$266$	118444.	0.102638
$267$	−414707.	−0.356010
$268$	1.54209e6	1.31151
$269$	−171993.	−0.144920	−0.0724602	−	0.997371i	$-0.523085\pi$
$269$	−171993.	−0.144920	−0.0724602	+	0.997371i	$0.523085\pi$
$270$	0	0
$271$	−953905.	−0.789009	−0.394504	−	0.918894i	$-0.629084\pi$
$271$	−953905.	−0.789009	−0.394504	+	0.918894i	$0.629084\pi$
$272$	344879.	0.282647
$273$	−948844.	−0.770528
$274$	2.12724e6	1.71175
$275$	0	0
$276$	2.15306e6	1.70131
$277$	739945.	0.579429	0.289714	−	0.957113i	$-0.406440\pi$
$277$	739945.	0.579429	0.289714	+	0.957113i	$0.406440\pi$
$278$	2.78353e6	2.16015
$279$	331344.	0.254840
$280$	0	0
$281$	−1.79135e6	−1.35336	−0.676682	−	0.736275i	$-0.736583\pi$
$281$	−1.79135e6	−1.35336	−0.676682	+	0.736275i	$0.736583\pi$
$282$	−1.21120e6	−0.906969
$283$	2.17360e6	1.61329	0.806646	−	0.591035i	$-0.201280\pi$
$283$	2.17360e6	1.61329	0.806646	+	0.591035i	$0.201280\pi$
$284$	1.78988e6	1.31683
$285$	0	0
$286$	3.62443e6	2.62014
$287$	−128412.	−0.0920237
$288$	1.22783e6	0.872283
$289$	83521.0	0.0588235
$290$	0	0
$291$	−2.04019e6	−1.41233
$292$	−1.08339e6	−0.743582
$293$	1.39396e6	0.948593	0.474296	−	0.880365i	$-0.342702\pi$
$293$	1.39396e6	0.948593	0.474296	+	0.880365i	$0.342702\pi$
$294$	2.05852e6	1.38895
$295$	0	0
$296$	−548240.	−0.363699
$297$	924026.	0.607845
$298$	−1.89206e6	−1.23422
$299$	3.41029e6	2.20604
$300$	0	0
$301$	603455.	0.383909
$302$	3.55774e6	2.24469
$303$	−915650.	−0.572958
$304$	−322339.	−0.200045
$305$	0	0
$306$	362372.	0.221234
$307$	−2.99450e6	−1.81334	−0.906668	−	0.421845i	$-0.861383\pi$
$307$	−2.99450e6	−1.81334	−0.906668	+	0.421845i	$0.861383\pi$
$308$	866142.	0.520250
$309$	3.40081e6	2.02622
$310$	0	0
$311$	−2.22950e6	−1.30709	−0.653546	−	0.756886i	$-0.726720\pi$
$311$	−2.22950e6	−1.30709	−0.653546	+	0.756886i	$0.726720\pi$
$312$	829798.	0.482599
$313$	−3.11891e6	−1.79946	−0.899730	−	0.436446i	$-0.856237\pi$
$313$	−3.11891e6	−1.79946	−0.899730	+	0.436446i	$0.856237\pi$
$314$	−421235.	−0.241102
$315$	0	0
$316$	538209.	0.303203
$317$	1.58091e6	0.883604	0.441802	−	0.897113i	$-0.354339\pi$
$317$	1.58091e6	0.883604	0.441802	+	0.897113i	$0.354339\pi$
$318$	3.04048e6	1.68606
$319$	239109.	0.131559
$320$	0	0
$321$	−1.92420e6	−1.04229
$322$	1.84541e6	0.991868
$323$	−78062.3	−0.0416327
$324$	−1.81875e6	−0.962522
$325$	0	0
$326$	982562.	0.512054
$327$	−2.70142e6	−1.39709
$328$	112301.	0.0576365
$329$	−458458.	−0.233512
$330$	0	0
$331$	714667.	0.358537	0.179268	−	0.983800i	$-0.442627\pi$
$331$	714667.	0.358537	0.179268	+	0.983800i	$0.442627\pi$
$332$	736401.	0.366665
$333$	−1.79259e6	−0.885870
$334$	−325111.	−0.159465
$335$	0	0
$336$	1.39732e6	0.675225
$337$	−1.32736e6	−0.636671	−0.318335	−	0.947978i	$-0.603124\pi$
$337$	−1.32736e6	−0.636671	−0.318335	+	0.947978i	$0.603124\pi$
$338$	−2.16023e6	−1.02851
$339$	−181651.	−0.0858497
$340$	0	0
$341$	−1.18194e6	−0.550438
$342$	−338688.	−0.156580
$343$	1.75272e6	0.804409
$344$	−527743.	−0.240451
$345$	0	0
$346$	−3.67303e6	−1.64943
$347$	−293121.	−0.130684	−0.0653420	−	0.997863i	$-0.520814\pi$
$347$	−293121.	−0.130684	−0.0653420	+	0.997863i	$0.520814\pi$
$348$	−207045.	−0.0916465
$349$	1.33136e6	0.585102	0.292551	−	0.956250i	$-0.405496\pi$
$349$	1.33136e6	0.585102	0.292551	+	0.956250i	$0.405496\pi$
$350$	0	0
$351$	−1.26733e6	−0.549061
$352$	−4.37980e6	−1.88407
$353$	4.50680e6	1.92501	0.962503	−	0.271272i	$-0.0874444\pi$
$353$	4.50680e6	1.92501	0.962503	+	0.271272i	$0.0874444\pi$
$354$	7.16868e6	3.04040
$355$	0	0
$356$	−519205.	−0.217127
$357$	338396.	0.140525
$358$	6.39886e6	2.63873
$359$	846073.	0.346475	0.173237	−	0.984880i	$-0.444577\pi$
$359$	846073.	0.346475	0.173237	+	0.984880i	$0.444577\pi$
$360$	0	0
$361$	−2.40314e6	−0.970534
$362$	922000.	0.369794
$363$	3.80098e6	1.51401
$364$	−1.18794e6	−0.469937
$365$	0	0
$366$	1.76501e6	0.688723
$367$	1.07689e6	0.417357	0.208678	−	0.977984i	$-0.433084\pi$
$367$	1.07689e6	0.417357	0.208678	+	0.977984i	$0.433084\pi$
$368$	−5.02219e6	−1.93318
$369$	367191.	0.140387
$370$	0	0
$371$	1.15087e6	0.434101
$372$	1.02344e6	0.383447
$373$	−4.96823e6	−1.84897	−0.924485	−	0.381218i	$-0.875505\pi$
$373$	−4.96823e6	−1.84897	−0.924485	+	0.381218i	$0.875505\pi$
$374$	−1.29262e6	−0.477850
$375$	0	0
$376$	400938.	0.146254
$377$	−327944.	−0.118835
$378$	−685789.	−0.246866
$379$	−2.76952e6	−0.990390	−0.495195	−	0.868782i	$-0.664903\pi$
$379$	−2.76952e6	−0.990390	−0.495195	+	0.868782i	$0.664903\pi$
$380$	0	0
$381$	−2.15751e6	−0.761450
$382$	2.55169e6	0.894685
$383$	4.29830e6	1.49727	0.748634	−	0.662983i	$-0.230710\pi$
$383$	4.29830e6	1.49727	0.748634	+	0.662983i	$0.230710\pi$
$384$	−2.05132e6	−0.709915
$385$	0	0
$386$	6.27165e6	2.14247
$387$	−1.72557e6	−0.585673
$388$	−2.55428e6	−0.861368
$389$	1.72218e6	0.577037	0.288519	−	0.957474i	$-0.406837\pi$
$389$	1.72218e6	0.577037	0.288519	+	0.957474i	$0.406837\pi$
$390$	0	0
$391$	−1.21625e6	−0.402327
$392$	−681425.	−0.223977
$393$	−5.49965e6	−1.79620
$394$	−2.18962e6	−0.710606
$395$	0	0
$396$	−2.47672e6	−0.793667
$397$	3.88896e6	1.23839	0.619195	−	0.785238i	$-0.287459\pi$
$397$	3.88896e6	1.23839	0.619195	+	0.785238i	$0.287459\pi$
$398$	−563863.	−0.178429
$399$	−316279.	−0.0994577
$400$	0	0
$401$	983117.	0.305312	0.152656	−	0.988279i	$-0.451217\pi$
$401$	983117.	0.305312	0.152656	+	0.988279i	$0.451217\pi$
$402$	−9.32437e6	−2.87776
$403$	1.62106e6	0.497205
$404$	−1.14638e6	−0.349441
$405$	0	0
$406$	−177461.	−0.0534302
$407$	6.39435e6	1.91342
$408$	−295940.	−0.0880142
$409$	1.76631e6	0.522106	0.261053	−	0.965324i	$-0.415930\pi$
$409$	1.76631e6	0.522106	0.261053	+	0.965324i	$0.415930\pi$
$410$	0	0
$411$	−5.68033e6	−1.65870
$412$	4.25776e6	1.23577
$413$	2.71346e6	0.782796
$414$	−5.27692e6	−1.51314
$415$	0	0
$416$	6.00701e6	1.70186
$417$	−7.43282e6	−2.09321
$418$	1.20814e6	0.338201
$419$	−4.06790e6	−1.13197	−0.565986	−	0.824415i	$-0.691504\pi$
$419$	−4.06790e6	−1.13197	−0.565986	+	0.824415i	$0.691504\pi$
$420$	0	0
$421$	−782501.	−0.215169	−0.107584	−	0.994196i	$-0.534312\pi$
$421$	−782501.	−0.215169	−0.107584	+	0.994196i	$0.534312\pi$
$422$	−618443.	−0.169051
$423$	1.31095e6	0.356234
$424$	−1.00648e6	−0.271887
$425$	0	0
$426$	−1.08227e7	−2.88943
$427$	668085.	0.177322
$428$	−2.40907e6	−0.635681
$429$	−9.67828e6	−2.53895
$430$	0	0
$431$	5.81089e6	1.50678	0.753389	−	0.657575i	$-0.228418\pi$
$431$	5.81089e6	1.50678	0.753389	+	0.657575i	$0.228418\pi$
$432$	1.86634e6	0.481150
$433$	−6.52867e6	−1.67342	−0.836710	−	0.547646i	$-0.815524\pi$
$433$	−6.52867e6	−1.67342	−0.836710	+	0.547646i	$0.815524\pi$
$434$	877204.	0.223551
$435$	0	0
$436$	−3.38213e6	−0.852068
$437$	1.13676e6	0.284750
$438$	6.55084e6	1.63159
$439$	−3.73384e6	−0.924686	−0.462343	−	0.886701i	$-0.652991\pi$
$439$	−3.73384e6	−0.924686	−0.462343	+	0.886701i	$0.652991\pi$
$440$	0	0
$441$	−2.22806e6	−0.545546
$442$	1.77286e6	0.431637
$443$	5.07062e6	1.22759	0.613793	−	0.789467i	$-0.289643\pi$
$443$	5.07062e6	1.22759	0.613793	+	0.789467i	$0.289643\pi$
$444$	−5.53687e6	−1.33293
$445$	0	0
$446$	8.33988e6	1.98528
$447$	5.05234e6	1.19598
$448$	1.03860e6	0.244487
$449$	4.32682e6	1.01287	0.506434	−	0.862279i	$-0.330964\pi$
$449$	4.32682e6	1.01287	0.506434	+	0.862279i	$0.330964\pi$
$450$	0	0
$451$	−1.30981e6	−0.303226
$452$	−227424.	−0.0523588
$453$	−9.50019e6	−2.17514
$454$	−81444.3	−0.0185448
$455$	0	0
$456$	276598.	0.0622926
$457$	2.96675e6	0.664494	0.332247	−	0.943192i	$-0.392193\pi$
$457$	2.96675e6	0.664494	0.332247	+	0.943192i	$0.392193\pi$
$458$	−2.48896e6	−0.554440
$459$	451979.	0.100135
$460$	0	0
$461$	−250988.	−0.0550049	−0.0275024	−	0.999622i	$-0.508755\pi$
$461$	−250988.	−0.0550049	−0.0275024	+	0.999622i	$0.508755\pi$
$462$	−5.23721e6	−1.14155
$463$	−4.74104e6	−1.02783	−0.513915	−	0.857841i	$-0.671805\pi$
$463$	−4.74104e6	−1.02783	−0.513915	+	0.857841i	$0.671805\pi$
$464$	482949.	0.104137
$465$	0	0
$466$	−2.42182e6	−0.516626
$467$	−8.11532e6	−1.72192	−0.860961	−	0.508672i	$-0.830137\pi$
$467$	−8.11532e6	−1.72192	−0.860961	+	0.508672i	$0.830137\pi$
$468$	3.39688e6	0.716911
$469$	−3.52943e6	−0.740921
$470$	0	0
$471$	1.12482e6	0.233631
$472$	−2.37302e6	−0.490282
$473$	6.15528e6	1.26501
$474$	−3.25434e6	−0.665298
$475$	0	0
$476$	423666.	0.0857050
$477$	−3.29089e6	−0.662243
$478$	−39909.5	−0.00798926
$479$	9.23044e6	1.83816	0.919081	−	0.394069i	$-0.128933\pi$
$479$	9.23044e6	1.83816	0.919081	+	0.394069i	$0.128933\pi$
$480$	0	0
$481$	−8.77001e6	−1.72837
$482$	1.24115e6	0.243336
$483$	−4.92778e6	−0.961133
$484$	4.75876e6	0.923380
$485$	0	0
$486$	8.12028e6	1.55948
$487$	−7.83485e6	−1.49695	−0.748477	−	0.663161i	$-0.769215\pi$
$487$	−7.83485e6	−1.49695	−0.748477	+	0.663161i	$0.769215\pi$
$488$	−584265.	−0.111061
$489$	−2.62372e6	−0.496187
$490$	0	0
$491$	1.68201e6	0.314865	0.157433	−	0.987530i	$-0.449678\pi$
$491$	1.68201e6	0.314865	0.157433	+	0.987530i	$0.449678\pi$
$492$	1.13416e6	0.211234
$493$	116958.	0.0216727
$494$	−1.65699e6	−0.305494
$495$	0	0
$496$	−2.38726e6	−0.435708
$497$	−4.09657e6	−0.743926
$498$	−4.45272e6	−0.804548
$499$	6.47769e6	1.16458	0.582289	−	0.812982i	$-0.302157\pi$
$499$	6.47769e6	1.16458	0.582289	+	0.812982i	$0.302157\pi$
$500$	0	0
$501$	868139.	0.154524
$502$	−7.97923e6	−1.41319
$503$	1.10035e7	1.93915	0.969575	−	0.244793i	$-0.0787201\pi$
$503$	1.10035e7	1.93915	0.969575	+	0.244793i	$0.0787201\pi$
$504$	−486012.	−0.0852256
$505$	0	0
$506$	1.88233e7	3.26829
$507$	5.76843e6	0.996639
$508$	−2.70117e6	−0.464400
$509$	−914828.	−0.156511	−0.0782555	−	0.996933i	$-0.524935\pi$
$509$	−914828.	−0.156511	−0.0782555	+	0.996933i	$0.524935\pi$
$510$	0	0
$511$	2.47960e6	0.420077
$512$	−6.91181e6	−1.16524
$513$	−422439.	−0.0708713
$514$	−1.54181e6	−0.257409
$515$	0	0
$516$	−5.32987e6	−0.881236
$517$	−4.67630e6	−0.769443
$518$	−4.74572e6	−0.777102
$519$	9.80804e6	1.59832
$520$	0	0
$521$	7.06411e6	1.14015	0.570077	−	0.821592i	$-0.306914\pi$
$521$	7.06411e6	1.14015	0.570077	+	0.821592i	$0.306914\pi$
$522$	507445.	0.0815104
$523$	−3.13073e6	−0.500485	−0.250242	−	0.968183i	$-0.580510\pi$
$523$	−3.13073e6	−0.500485	−0.250242	+	0.968183i	$0.580510\pi$
$524$	−6.88546e6	−1.09548
$525$	0	0
$526$	1.35256e6	0.213153
$527$	−578134.	−0.0906781
$528$	1.42528e7	2.22492
$529$	1.12748e7	1.75175
$530$	0	0
$531$	−7.75909e6	−1.19419
$532$	−395976.	−0.0606582
$533$	1.79644e6	0.273901
$534$	3.13943e6	0.476428
$535$	0	0
$536$	3.08661e6	0.464055
$537$	−1.70868e7	−2.55696
$538$	1.30202e6	0.193938
$539$	7.94773e6	1.17834
$540$	0	0
$541$	2.46824e6	0.362571	0.181286	−	0.983430i	$-0.441974\pi$
$541$	2.46824e6	0.362571	0.181286	+	0.983430i	$0.441974\pi$
$542$	7.22128e6	1.05588
$543$	−2.46200e6	−0.358335
$544$	−2.14234e6	−0.310378
$545$	0	0
$546$	7.18297e6	1.03115
$547$	1.03370e7	1.47715	0.738576	−	0.674170i	$-0.235499\pi$
$547$	1.03370e7	1.47715	0.738576	+	0.674170i	$0.235499\pi$
$548$	−7.11167e6	−1.01163
$549$	−1.91038e6	−0.270513
$550$	0	0
$551$	−109314.	−0.0153390
$552$	4.30952e6	0.601979
$553$	−1.23182e6	−0.171291
$554$	−5.60156e6	−0.775415
$555$	0	0
$556$	−9.30575e6	−1.27663
$557$	1.26155e7	1.72292	0.861462	−	0.507823i	$-0.169550\pi$
$557$	1.26155e7	1.72292	0.861462	+	0.507823i	$0.169550\pi$
$558$	−2.50835e6	−0.341038
$559$	−8.44213e6	−1.14267
$560$	0	0
$561$	3.45166e6	0.463043
$562$	1.35609e7	1.81113
$563$	4.42925e6	0.588924	0.294462	−	0.955663i	$-0.404860\pi$
$563$	4.42925e6	0.588924	0.294462	+	0.955663i	$0.404860\pi$
$564$	4.04921e6	0.536010
$565$	0	0
$566$	−1.64546e7	−2.15897
$567$	4.16264e6	0.543765
$568$	3.58260e6	0.465937
$569$	1.75417e6	0.227139	0.113569	−	0.993530i	$-0.463772\pi$
$569$	1.75417e6	0.227139	0.113569	+	0.993530i	$0.463772\pi$
$570$	0	0
$571$	1.46164e7	1.87607	0.938037	−	0.346535i	$-0.112642\pi$
$571$	1.46164e7	1.87607	0.938037	+	0.346535i	$0.112642\pi$
$572$	−1.21170e7	−1.54848
$573$	−6.81375e6	−0.866961
$574$	972107.	0.123150
$575$	0	0
$576$	−2.96987e6	−0.372976
$577$	1.34051e7	1.67621	0.838107	−	0.545506i	$-0.183662\pi$
$577$	1.34051e7	1.67621	0.838107	+	0.545506i	$0.183662\pi$
$578$	−632273.	−0.0787200
$579$	−1.67471e7	−2.07608
$580$	0	0
$581$	−1.68543e6	−0.207143
$582$	1.54447e7	1.89004
$583$	1.17389e7	1.43040
$584$	−2.16850e6	−0.263104
$585$	0	0
$586$	−1.05526e7	−1.26945
$587$	4.77829e6	0.572371	0.286185	−	0.958174i	$-0.407613\pi$
$587$	4.77829e6	0.572371	0.286185	+	0.958174i	$0.407613\pi$
$588$	−6.88195e6	−0.820858
$589$	540349.	0.0641780
$590$	0	0
$591$	5.84692e6	0.688586
$592$	1.29152e7	1.51460
$593$	−1.35929e7	−1.58736	−0.793680	−	0.608335i	$-0.791838\pi$
$593$	−1.35929e7	−1.58736	−0.793680	+	0.608335i	$0.791838\pi$
$594$	−6.99509e6	−0.813444
$595$	0	0
$596$	6.32544e6	0.729415
$597$	1.50567e6	0.172900
$598$	−2.58167e7	−2.95221
$599$	−1.55453e7	−1.77024	−0.885120	−	0.465362i	$-0.845924\pi$
$599$	−1.55453e7	−1.77024	−0.885120	+	0.465362i	$0.845924\pi$
$600$	0	0
$601$	136540.	0.0154197	0.00770983	−	0.999970i	$-0.497546\pi$
$601$	136540.	0.0154197	0.00770983	+	0.999970i	$0.497546\pi$
$602$	−4.56830e6	−0.513763
$603$	1.00923e7	1.13031
$604$	−1.18941e7	−1.32659
$605$	0	0
$606$	6.93168e6	0.766756
$607$	−7.83964e6	−0.863623	−0.431812	−	0.901964i	$-0.642125\pi$
$607$	−7.83964e6	−0.863623	−0.431812	+	0.901964i	$0.642125\pi$
$608$	2.00232e6	0.219672
$609$	473870.	0.0517745
$610$	0	0
$611$	6.41367e6	0.695030
$612$	−1.21146e6	−0.130747
$613$	149717.	0.0160923	0.00804616	−	0.999968i	$-0.497439\pi$
$613$	149717.	0.0160923	0.00804616	+	0.999968i	$0.497439\pi$
$614$	2.26691e7	2.42668
$615$	0	0
$616$	1.73365e6	0.184082
$617$	−1.00603e7	−1.06389	−0.531944	−	0.846779i	$-0.678538\pi$
$617$	−1.00603e7	−1.06389	−0.531944	+	0.846779i	$0.678538\pi$
$618$	−2.57450e7	−2.71157
$619$	−1.36299e7	−1.42977	−0.714885	−	0.699242i	$-0.753521\pi$
$619$	−1.36299e7	−1.42977	−0.714885	+	0.699242i	$0.753521\pi$
$620$	0	0
$621$	−6.58180e6	−0.684882
$622$	1.68778e7	1.74920
$623$	1.18832e6	0.122663
$624$	−1.95481e7	−2.00975
$625$	0	0
$626$	2.36109e7	2.40811
$627$	−3.22607e6	−0.327722
$628$	1.40825e6	0.142489
$629$	3.12774e6	0.315213
$630$	0	0
$631$	−1.23119e7	−1.23098	−0.615489	−	0.788145i	$-0.711042\pi$
$631$	−1.23119e7	−1.23098	−0.615489	+	0.788145i	$0.711042\pi$
$632$	1.07727e6	0.107283
$633$	1.65142e6	0.163813
$634$	−1.19678e7	−1.18248
$635$	0	0
$636$	−1.01648e7	−0.996448
$637$	−1.09005e7	−1.06438
$638$	−1.81011e6	−0.176057
$639$	1.17141e7	1.13489
$640$	0	0
$641$	−1.26659e6	−0.121756	−0.0608781	−	0.998145i	$-0.519390\pi$
$641$	−1.26659e6	−0.121756	−0.0608781	+	0.998145i	$0.519390\pi$
$642$	1.45667e7	1.39483
$643$	−151975.	−0.0144959	−0.00724796	−	0.999974i	$-0.502307\pi$
$643$	−151975.	−0.0144959	−0.00724796	+	0.999974i	$0.502307\pi$
$644$	−6.16949e6	−0.586185
$645$	0	0
$646$	590949.	0.0557146
$647$	−5.22174e6	−0.490405	−0.245202	−	0.969472i	$-0.578854\pi$
$647$	−5.22174e6	−0.490405	−0.245202	+	0.969472i	$0.578854\pi$
$648$	−3.64038e6	−0.340572
$649$	2.76775e7	2.57938
$650$	0	0
$651$	−2.34239e6	−0.216624
$652$	−3.28485e6	−0.302619
$653$	3.34884e6	0.307335	0.153667	−	0.988123i	$-0.450892\pi$
$653$	3.34884e6	0.307335	0.153667	+	0.988123i	$0.450892\pi$
$654$	2.04504e7	1.86964
$655$	0	0
$656$	−2.64553e6	−0.240023
$657$	−7.09036e6	−0.640848
$658$	3.47063e6	0.312496
$659$	−4.45618e6	−0.399714	−0.199857	−	0.979825i	$-0.564048\pi$
$659$	−4.45618e6	−0.399714	−0.199857	+	0.979825i	$0.564048\pi$
$660$	0	0
$661$	1.22926e7	1.09431	0.547156	−	0.837030i	$-0.315710\pi$
$661$	1.22926e7	1.09431	0.547156	+	0.837030i	$0.315710\pi$
$662$	−5.41019e6	−0.479808
$663$	−4.73404e6	−0.418262
$664$	1.47397e6	0.129738
$665$	0	0
$666$	1.35703e7	1.18551
$667$	−1.70316e6	−0.148232
$668$	1.08689e6	0.0942423
$669$	−2.22699e7	−1.92377
$670$	0	0
$671$	6.81452e6	0.584290
$672$	−8.67997e6	−0.741472
$673$	−9.46310e6	−0.805370	−0.402685	−	0.915339i	$-0.631923\pi$
$673$	−9.46310e6	−0.805370	−0.402685	+	0.915339i	$0.631923\pi$
$674$	1.00484e7	0.852019
$675$	0	0
$676$	7.22197e6	0.607840
$677$	1.30040e7	1.09045	0.545225	−	0.838290i	$-0.316444\pi$
$677$	1.30040e7	1.09045	0.545225	+	0.838290i	$0.316444\pi$
$678$	1.37514e6	0.114888
$679$	5.84607e6	0.486619
$680$	0	0
$681$	217480.	0.0179701
$682$	8.94754e6	0.736619
$683$	1.02124e7	0.837672	0.418836	−	0.908062i	$-0.362438\pi$
$683$	1.02124e7	0.837672	0.418836	+	0.908062i	$0.362438\pi$
$684$	1.13229e6	0.0925371
$685$	0	0
$686$	−1.32685e7	−1.07649
$687$	6.64624e6	0.537260
$688$	1.24324e7	1.00134
$689$	−1.61003e7	−1.29207
$690$	0	0
$691$	−1.99799e7	−1.59183	−0.795917	−	0.605405i	$-0.793011\pi$
$691$	−1.99799e7	−1.59183	−0.795917	+	0.605405i	$0.793011\pi$
$692$	1.22795e7	0.974798
$693$	5.66855e6	0.448372
$694$	2.21899e6	0.174887
$695$	0	0
$696$	−414417.	−0.0324275
$697$	−640681.	−0.0499528
$698$	−1.00787e7	−0.783007
$699$	6.46694e6	0.500617
$700$	0	0
$701$	−1.32804e7	−1.02074	−0.510372	−	0.859954i	$-0.670492\pi$
$701$	−1.32804e7	−1.02074	−0.510372	+	0.859954i	$0.670492\pi$
$702$	9.59395e6	0.734775
$703$	−2.92332e6	−0.223094
$704$	1.05938e7	0.805604
$705$	0	0
$706$	−3.41176e7	−2.57612
$707$	2.62375e6	0.197412
$708$	−2.39660e7	−1.79685
$709$	2.49313e7	1.86264	0.931321	−	0.364198i	$-0.118657\pi$
$709$	2.49313e7	1.86264	0.931321	+	0.364198i	$0.118657\pi$
$710$	0	0
$711$	3.52236e6	0.261312
$712$	−1.03923e6	−0.0768267
$713$	8.41889e6	0.620199
$714$	−2.56174e6	−0.188057
$715$	0	0
$716$	−2.13923e7	−1.55947
$717$	106570.	0.00774170
$718$	−6.40497e6	−0.463667
$719$	1.08777e7	0.784718	0.392359	−	0.919812i	$-0.371659\pi$
$719$	1.08777e7	0.784718	0.392359	+	0.919812i	$0.371659\pi$
$720$	0	0
$721$	−9.74488e6	−0.698133
$722$	1.81923e7	1.29881
$723$	−3.31423e6	−0.235796
$724$	−3.08238e6	−0.218545
$725$	0	0
$726$	−2.87743e7	−2.02611
$727$	−1.86033e7	−1.30543	−0.652715	−	0.757604i	$-0.726370\pi$
$727$	−1.86033e7	−1.30543	−0.652715	+	0.757604i	$0.726370\pi$
$728$	−2.37775e6	−0.166279
$729$	−4.22065e6	−0.294144
$730$	0	0
$731$	3.01080e6	0.208396
$732$	−5.90070e6	−0.407029
$733$	1.98387e7	1.36381	0.681905	−	0.731441i	$-0.261152\pi$
$733$	1.98387e7	1.36381	0.681905	+	0.731441i	$0.261152\pi$
$734$	−8.15233e6	−0.558524
$735$	0	0
$736$	3.11971e7	2.12285
$737$	−3.60004e7	−2.44140
$738$	−2.77972e6	−0.187871
$739$	2.02803e7	1.36604	0.683019	−	0.730400i	$-0.260667\pi$
$739$	2.02803e7	1.36604	0.683019	+	0.730400i	$0.260667\pi$
$740$	0	0
$741$	4.42464e6	0.296028
$742$	−8.71235e6	−0.580932
$743$	−1.90469e7	−1.26576	−0.632882	−	0.774248i	$-0.718128\pi$
$743$	−1.90469e7	−1.26576	−0.632882	+	0.774248i	$0.718128\pi$
$744$	2.04850e6	0.135676
$745$	0	0
$746$	3.76107e7	2.47437
$747$	4.81945e6	0.316006
$748$	4.32142e6	0.282405
$749$	5.51371e6	0.359120
$750$	0	0
$751$	1.31465e7	0.850568	0.425284	−	0.905060i	$-0.360174\pi$
$751$	1.31465e7	0.850568	0.425284	+	0.905060i	$0.360174\pi$
$752$	−9.44513e6	−0.609065
$753$	2.13068e7	1.36940
$754$	2.48261e6	0.159030
$755$	0	0
$756$	2.29269e6	0.145895
$757$	−1.50517e7	−0.954652	−0.477326	−	0.878726i	$-0.658394\pi$
$757$	−1.50517e7	−0.954652	−0.477326	+	0.878726i	$0.658394\pi$
$758$	2.09659e7	1.32538
$759$	−5.02637e7	−3.16701
$760$	0	0
$761$	1.67383e7	1.04773	0.523865	−	0.851801i	$-0.324490\pi$
$761$	1.67383e7	1.04773	0.523865	+	0.851801i	$0.324490\pi$
$762$	1.63329e7	1.01900
$763$	7.74080e6	0.481365
$764$	−8.53069e6	−0.528751
$765$	0	0
$766$	−3.25391e7	−2.00371
$767$	−3.79604e7	−2.32993
$768$	2.71276e7	1.65962
$769$	1.27095e7	0.775018	0.387509	−	0.921866i	$-0.373336\pi$
$769$	1.27095e7	0.775018	0.387509	+	0.921866i	$0.373336\pi$
$770$	0	0
$771$	4.11709e6	0.249433
$772$	−2.09671e7	−1.26618
$773$	2.38129e7	1.43339	0.716693	−	0.697389i	$-0.245655\pi$
$773$	2.38129e7	1.43339	0.716693	+	0.697389i	$0.245655\pi$
$774$	1.30630e7	0.783771
$775$	0	0
$776$	−5.11259e6	−0.304780
$777$	1.26724e7	0.753022
$778$	−1.30373e7	−0.772215
$779$	598808.	0.0353544
$780$	0	0
$781$	−4.17853e7	−2.45130
$782$	9.20727e6	0.538411
$783$	632926.	0.0368934
$784$	1.60527e7	0.932736
$785$	0	0
$786$	4.16336e7	2.40374
$787$	−2.29342e7	−1.31992	−0.659959	−	0.751302i	$-0.729426\pi$
$787$	−2.29342e7	−1.31992	−0.659959	+	0.751302i	$0.729426\pi$
$788$	7.32024e6	0.419962
$789$	−3.61172e6	−0.206548
$790$	0	0
$791$	520513.	0.0295795
$792$	−4.95735e6	−0.280826
$793$	−9.34628e6	−0.527784
$794$	−2.94403e7	−1.65726
$795$	0	0
$796$	1.88508e6	0.105450
$797$	−6.16887e6	−0.344001	−0.172001	−	0.985097i	$-0.555023\pi$
$797$	−6.16887e6	−0.344001	−0.172001	+	0.985097i	$0.555023\pi$
$798$	2.39431e6	0.133098
$799$	−2.28737e6	−0.126756
$800$	0	0
$801$	−3.39799e6	−0.187129
$802$	−7.44243e6	−0.408581
$803$	2.52921e7	1.38419
$804$	3.11728e7	1.70073
$805$	0	0
$806$	−1.22718e7	−0.665380
$807$	−3.47678e6	−0.187929
$808$	−2.29457e6	−0.123644
$809$	−1.32716e7	−0.712936	−0.356468	−	0.934307i	$-0.616019\pi$
$809$	−1.32716e7	−0.712936	−0.356468	+	0.934307i	$0.616019\pi$
$810$	0	0
$811$	1.18752e7	0.634002	0.317001	−	0.948425i	$-0.397324\pi$
$811$	1.18752e7	0.634002	0.317001	+	0.948425i	$0.397324\pi$
$812$	593277.	0.0315768
$813$	−1.92829e7	−1.02316
$814$	−4.84067e7	−2.56062
$815$	0	0
$816$	6.97162e6	0.366529
$817$	−2.81402e6	−0.147493
$818$	−1.33714e7	−0.698704
$819$	−7.77456e6	−0.405010
$820$	0	0
$821$	1.94701e7	1.00812	0.504058	−	0.863670i	$-0.331840\pi$
$821$	1.94701e7	1.00812	0.504058	+	0.863670i	$0.331840\pi$
$822$	4.30014e7	2.21975
$823$	−1.25032e7	−0.643462	−0.321731	−	0.946831i	$-0.604265\pi$
$823$	−1.25032e7	−0.643462	−0.321731	+	0.946831i	$0.604265\pi$
$824$	8.52225e6	0.437257
$825$	0	0
$826$	−2.05415e7	−1.04757
$827$	−1.73208e7	−0.880654	−0.440327	−	0.897837i	$-0.645137\pi$
$827$	−1.73208e7	−0.880654	−0.440327	+	0.897837i	$0.645137\pi$
$828$	1.76415e7	0.894253
$829$	−6.47806e6	−0.327385	−0.163693	−	0.986511i	$-0.552341\pi$
$829$	−6.47806e6	−0.327385	−0.163693	+	0.986511i	$0.552341\pi$
$830$	0	0
$831$	1.49578e7	0.751387
$832$	−1.45297e7	−0.727694
$833$	3.88756e6	0.194118
$834$	5.62681e7	2.80122
$835$	0	0
$836$	−4.03898e6	−0.199874
$837$	−3.12861e6	−0.154361
$838$	3.07950e7	1.51485
$839$	3.25114e6	0.159452	0.0797262	−	0.996817i	$-0.474595\pi$
$839$	3.25114e6	0.159452	0.0797262	+	0.996817i	$0.474595\pi$
$840$	0	0
$841$	−2.03474e7	−0.992015
$842$	5.92371e6	0.287948
$843$	−3.62116e7	−1.75501
$844$	2.06755e6	0.0999078
$845$	0	0
$846$	−9.92421e6	−0.476727
$847$	−1.08915e7	−0.521652
$848$	2.37102e7	1.13226
$849$	4.39386e7	2.09207
$850$	0	0
$851$	−4.55467e7	−2.15592
$852$	3.61819e7	1.70763
$853$	2.84698e7	1.33971	0.669856	−	0.742491i	$-0.266356\pi$
$853$	2.84698e7	1.33971	0.669856	+	0.742491i	$0.266356\pi$
$854$	−5.05756e6	−0.237299
$855$	0	0
$856$	−4.82194e6	−0.224925
$857$	−2.54857e7	−1.18534	−0.592672	−	0.805444i	$-0.701927\pi$
$857$	−2.54857e7	−1.18534	−0.592672	+	0.805444i	$0.701927\pi$
$858$	7.32668e7	3.39773
$859$	1.08818e7	0.503174	0.251587	−	0.967835i	$-0.419048\pi$
$859$	1.08818e7	0.503174	0.251587	+	0.967835i	$0.419048\pi$
$860$	0	0
$861$	−2.59580e6	−0.119334
$862$	−4.39898e7	−2.01643
$863$	−1.54637e7	−0.706785	−0.353393	−	0.935475i	$-0.614972\pi$
$863$	−1.54637e7	−0.706785	−0.353393	+	0.935475i	$0.614972\pi$
$864$	−1.15934e7	−0.528356
$865$	0	0
$866$	4.94235e7	2.23944
$867$	1.68835e6	0.0762807
$868$	−2.93262e6	−0.132117
$869$	−1.25646e7	−0.564417
$870$	0	0
$871$	4.93754e7	2.20529
$872$	−6.76961e6	−0.301490
$873$	−1.67167e7	−0.742361
$874$	−8.60550e6	−0.381064
$875$	0	0
$876$	−2.19004e7	−0.964256
$877$	1.94708e7	0.854839	0.427419	−	0.904054i	$-0.359423\pi$
$877$	1.94708e7	0.854839	0.427419	+	0.904054i	$0.359423\pi$
$878$	2.82660e7	1.23745
$879$	2.81784e7	1.23011
$880$	0	0
$881$	−3.33714e7	−1.44855	−0.724276	−	0.689510i	$-0.757826\pi$
$881$	−3.33714e7	−1.44855	−0.724276	+	0.689510i	$0.757826\pi$
$882$	1.68670e7	0.730071
$883$	2.68149e7	1.15737	0.578687	−	0.815550i	$-0.303565\pi$
$883$	2.68149e7	1.15737	0.578687	+	0.815550i	$0.303565\pi$
$884$	−5.92694e6	−0.255094
$885$	0	0
$886$	−3.83858e7	−1.64281
$887$	4.47933e7	1.91163	0.955815	−	0.293970i	$-0.0949765\pi$
$887$	4.47933e7	1.91163	0.955815	+	0.293970i	$0.0949765\pi$
$888$	−1.10825e7	−0.471634
$889$	6.18226e6	0.262357
$890$	0	0
$891$	4.24592e7	1.79175
$892$	−2.78815e7	−1.17328
$893$	2.13787e6	0.0897126
$894$	−3.82474e7	−1.60051
$895$	0	0
$896$	5.87798e6	0.244601
$897$	6.89379e7	2.86073
$898$	−3.27550e7	−1.35546
$899$	−809586.	−0.0334090
$900$	0	0
$901$	5.74200e6	0.235641
$902$	9.91555e6	0.405789
$903$	1.21987e7	0.497843
$904$	−455208.	−0.0185263
$905$	0	0
$906$	7.19187e7	2.91086
$907$	−3.91153e7	−1.57880	−0.789402	−	0.613876i	$-0.789609\pi$
$907$	−3.91153e7	−1.57880	−0.789402	+	0.613876i	$0.789609\pi$
$908$	272281.	0.0109598
$909$	−7.50257e6	−0.301162
$910$	0	0
$911$	−1.01380e7	−0.404723	−0.202362	−	0.979311i	$-0.564862\pi$
$911$	−1.01380e7	−0.404723	−0.202362	+	0.979311i	$0.564862\pi$
$912$	−6.51597e6	−0.259413
$913$	−1.71915e7	−0.682553
$914$	−2.24590e7	−0.889253
$915$	0	0
$916$	8.32097e6	0.327669
$917$	1.57590e7	0.618878
$918$	−3.42159e6	−0.134005
$919$	3.15062e7	1.23057	0.615285	−	0.788304i	$-0.289041\pi$
$919$	3.15062e7	1.23057	0.615285	+	0.788304i	$0.289041\pi$
$920$	0	0
$921$	−6.05329e7	−2.35148
$922$	1.90004e6	0.0736098
$923$	5.73096e7	2.21423
$924$	1.75088e7	0.674646
$925$	0	0
$926$	3.58908e7	1.37548
$927$	2.78653e7	1.06504
$928$	−3.00001e6	−0.114354
$929$	−2.88637e7	−1.09727	−0.548634	−	0.836063i	$-0.684852\pi$
$929$	−2.88637e7	−1.09727	−0.548634	+	0.836063i	$0.684852\pi$
$930$	0	0
$931$	−3.63348e6	−0.137388
$932$	8.09649e6	0.305321
$933$	−4.50686e7	−1.69500
$934$	6.14348e7	2.30434
$935$	0	0
$936$	6.79913e6	0.253667
$937$	3.52072e6	0.131003	0.0655016	−	0.997852i	$-0.479135\pi$
$937$	3.52072e6	0.131003	0.0655016	+	0.997852i	$0.479135\pi$
$938$	2.67186e7	0.991531
$939$	−6.30478e7	−2.33349
$940$	0	0
$941$	−3.30676e7	−1.21739	−0.608694	−	0.793405i	$-0.708306\pi$
$941$	−3.30676e7	−1.21739	−0.608694	+	0.793405i	$0.708306\pi$
$942$	−8.51514e6	−0.312654
$943$	9.32970e6	0.341656
$944$	5.59026e7	2.04175
$945$	0	0
$946$	−4.65969e7	−1.69289
$947$	−2.20160e7	−0.797744	−0.398872	−	0.917007i	$-0.630598\pi$
$947$	−2.20160e7	−0.797744	−0.398872	+	0.917007i	$0.630598\pi$
$948$	1.08797e7	0.393185
$949$	−3.46887e7	−1.25032
$950$	0	0
$951$	3.19575e7	1.14583
$952$	848002.	0.0303252
$953$	−1.20561e7	−0.430005	−0.215002	−	0.976614i	$-0.568976\pi$
$953$	−1.20561e7	−0.430005	−0.215002	+	0.976614i	$0.568976\pi$
$954$	2.49128e7	0.886240
$955$	0	0
$956$	133423.	0.00472158
$957$	4.83351e6	0.170601
$958$	−6.98766e7	−2.45990
$959$	1.62767e7	0.571506
$960$	0	0
$961$	−2.46273e7	−0.860217
$962$	6.63910e7	2.31298
$963$	−1.57664e7	−0.547855
$964$	−4.14935e6	−0.143810
$965$	0	0
$966$	3.73044e7	1.28623
$967$	1.22744e6	0.0422118	0.0211059	−	0.999777i	$-0.493281\pi$
$967$	1.22744e6	0.0422118	0.0211059	+	0.999777i	$0.493281\pi$
$968$	9.52505e6	0.326722
$969$	−1.57800e6	−0.0539882
$970$	0	0
$971$	2.52894e7	0.860775	0.430388	−	0.902644i	$-0.358377\pi$
$971$	2.52894e7	0.860775	0.430388	+	0.902644i	$0.358377\pi$
$972$	−2.71473e7	−0.921640
$973$	2.12984e7	0.721215
$974$	5.93117e7	2.00328
$975$	0	0
$976$	1.37639e7	0.462505
$977$	5.63871e7	1.88992	0.944960	−	0.327186i	$-0.106100\pi$
$977$	5.63871e7	1.88992	0.944960	+	0.327186i	$0.106100\pi$
$978$	1.98622e7	0.664017
$979$	1.21210e7	0.404186
$980$	0	0
$981$	−2.21347e7	−0.734346
$982$	−1.27332e7	−0.421365
$983$	−3.41291e7	−1.12652	−0.563262	−	0.826278i	$-0.690454\pi$
$983$	−3.41291e7	−1.12652	−0.563262	+	0.826278i	$0.690454\pi$
$984$	2.27012e6	0.0747414
$985$	0	0
$986$	−885399.	−0.0290033
$987$	−9.26758e6	−0.302812
$988$	5.53957e6	0.180544
$989$	−4.38438e7	−1.42534
$990$	0	0
$991$	−3.81751e7	−1.23480	−0.617400	−	0.786649i	$-0.711814\pi$
$991$	−3.81751e7	−1.23480	−0.617400	+	0.786649i	$0.711814\pi$
$992$	1.48293e7	0.478457
$993$	1.44468e7	0.464940
$994$	3.10120e7	0.995551
$995$	0	0
$996$	1.48861e7	0.475481
$997$	4.52990e7	1.44328	0.721640	−	0.692269i	$-0.243389\pi$
$997$	4.52990e7	1.44328	0.721640	+	0.692269i	$0.243389\pi$
$998$	−4.90376e7	−1.55849
$999$	1.69260e7	0.536586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.6.a.d.1.1		5
5.4	even	2		85.6.a.a.1.5	✓	5
15.14	odd	2		765.6.a.g.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.6.a.a.1.5	✓	5	5.4	even	2
425.6.a.d.1.1		5	1.1	even	1	trivial
765.6.a.g.1.1		5	15.14	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q-7.57023 q^{2} +20.2147 q^{3} +25.3084 q^{4} -153.030 q^{6} -57.9243 q^{7} +50.6569 q^{8} +165.633 q^{9} -590.831 q^{11} +511.602 q^{12} +810.340 q^{13} +438.500 q^{14} -1193.35 q^{16} -289.000 q^{17} -1253.88 q^{18} +270.112 q^{19} -1170.92 q^{21} +4472.73 q^{22} +4208.47 q^{23} +1024.01 q^{24} -6134.46 q^{26} -1563.94 q^{27} -1465.97 q^{28} -404.699 q^{29} +2000.47 q^{31} +7412.94 q^{32} -11943.5 q^{33} +2187.80 q^{34} +4191.92 q^{36} -10822.6 q^{37} -2044.81 q^{38} +16380.8 q^{39} +2216.89 q^{41} +8864.14 q^{42} -10418.0 q^{43} -14953.0 q^{44} -31859.1 q^{46} +7914.78 q^{47} -24123.3 q^{48} -13451.8 q^{49} -5842.04 q^{51} +20508.4 q^{52} -19868.5 q^{53} +11839.4 q^{54} -2934.26 q^{56} +5460.22 q^{57} +3063.66 q^{58} -46845.0 q^{59} -11533.8 q^{61} -15144.0 q^{62} -9594.19 q^{63} -17930.4 q^{64} +90414.8 q^{66} +60931.7 q^{67} -7314.13 q^{68} +85072.8 q^{69} +70722.9 q^{71} +8390.47 q^{72} -42807.6 q^{73} +81929.8 q^{74} +6836.10 q^{76} +34223.5 q^{77} -124006. q^{78} +21266.0 q^{79} -71863.5 q^{81} -16782.4 q^{82} +29097.1 q^{83} -29634.1 q^{84} +78866.7 q^{86} -8180.86 q^{87} -29929.7 q^{88} -20515.1 q^{89} -46938.4 q^{91} +106510. q^{92} +40438.8 q^{93} -59916.7 q^{94} +149850. q^{96} -100926. q^{97} +101833. q^{98} -97861.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{2} + 36 q^{3} + 43 q^{4} - 105 q^{6} + 204 q^{7} + 63 q^{8} + 531 q^{9} - 792 q^{11} - 785 q^{12} - 88 q^{13} + 860 q^{14} - 2365 q^{16} - 1445 q^{17} + 2052 q^{18} - 5160 q^{19} - 6428 q^{21}+ \cdots - 535112 q^{99}+O(q^{100}) \) 5 * q + 7 * q^2 + 36 * q^3 + 43 * q^4 - 105 * q^6 + 204 * q^7 + 63 * q^8 + 531 * q^9 - 792 * q^11 - 785 * q^12 - 88 * q^13 + 860 * q^14 - 2365 * q^16 - 1445 * q^17 + 2052 * q^18 - 5160 * q^19 - 6428 * q^21 + 3058 * q^22 + 6140 * q^23 - 11751 * q^24 - 9643 * q^26 + 13380 * q^27 + 4568 * q^28 - 7328 * q^29 - 8968 * q^31 + 4703 * q^32 - 13072 * q^33 - 2023 * q^34 + 2048 * q^36 - 9834 * q^37 - 10625 * q^38 + 10108 * q^39 - 5302 * q^41 - 16222 * q^42 + 4916 * q^43 - 31514 * q^44 - 16768 * q^46 + 20364 * q^47 - 45557 * q^48 - 26483 * q^49 - 10404 * q^51 + 11525 * q^52 - 56492 * q^53 + 64635 * q^54 + 73228 * q^56 - 67554 * q^57 - 54213 * q^58 - 96200 * q^59 - 7284 * q^61 - 122335 * q^62 - 64108 * q^63 + 28483 * q^64 + 115232 * q^66 + 60308 * q^67 - 12427 * q^68 - 89076 * q^69 - 25304 * q^71 - 118722 * q^72 - 163328 * q^73 - 82768 * q^74 + 92027 * q^76 - 19912 * q^77 - 134895 * q^78 - 50940 * q^79 + 92445 * q^81 - 191390 * q^82 + 179060 * q^83 - 128054 * q^84 + 87086 * q^86 - 55764 * q^87 - 167634 * q^88 - 387356 * q^89 - 115200 * q^91 + 293228 * q^92 + 202558 * q^93 + 100515 * q^94 + 428113 * q^96 + 106076 * q^97 - 49329 * q^98 - 535112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more