LMFDB - Embedded newform 425.6.a.d.1.3

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			1.3
Root			$3.29890$ 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)

$f(q)$	$=$	\(q+4.29890 q^{2} +29.9706 q^{3} -13.5195 q^{4} +128.840 q^{6} -45.8012 q^{7} -195.684 q^{8} +655.235 q^{9} -504.686 q^{11} -405.186 q^{12} -513.142 q^{13} -196.895 q^{14} -408.601 q^{16} -289.000 q^{17} +2816.79 q^{18} -2084.02 q^{19} -1372.69 q^{21} -2169.60 q^{22} -4718.18 q^{23} -5864.75 q^{24} -2205.94 q^{26} +12354.9 q^{27} +619.207 q^{28} -3092.07 q^{29} +4157.17 q^{31} +4505.34 q^{32} -15125.7 q^{33} -1242.38 q^{34} -8858.42 q^{36} -4514.40 q^{37} -8958.99 q^{38} -15379.1 q^{39} +7375.94 q^{41} -5901.04 q^{42} +17592.2 q^{43} +6823.09 q^{44} -20283.0 q^{46} +14737.7 q^{47} -12246.0 q^{48} -14709.3 q^{49} -8661.49 q^{51} +6937.40 q^{52} +1008.07 q^{53} +53112.5 q^{54} +8962.53 q^{56} -62459.2 q^{57} -13292.5 q^{58} +496.132 q^{59} -13046.2 q^{61} +17871.2 q^{62} -30010.5 q^{63} +32443.2 q^{64} -65024.0 q^{66} -21128.0 q^{67} +3907.12 q^{68} -141406. q^{69} -41206.2 q^{71} -128219. q^{72} -53173.8 q^{73} -19406.9 q^{74} +28174.8 q^{76} +23115.2 q^{77} -66113.4 q^{78} +22083.6 q^{79} +211061. q^{81} +31708.4 q^{82} +63967.7 q^{83} +18558.0 q^{84} +75627.1 q^{86} -92671.1 q^{87} +98758.8 q^{88} -145034. q^{89} +23502.5 q^{91} +63787.2 q^{92} +124593. q^{93} +63356.1 q^{94} +135028. q^{96} +131574. q^{97} -63233.6 q^{98} -330688. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.29890	0.759945	0.379973	−	0.924998i	$-0.375933\pi$
$2$	4.29890	0.759945	0.379973	+	0.924998i	$0.375933\pi$
$3$	29.9706	1.92261	0.961306	−	0.275482i	$-0.0888374\pi$
$3$	29.9706	1.92261	0.961306	+	0.275482i	$0.0888374\pi$
$4$	−13.5195	−0.422483
$5$	0	0
$5$	0	0
$6$	128.840	1.46108
$7$	−45.8012	−0.353290	−0.176645	−	0.984275i	$-0.556524\pi$
$7$	−45.8012	−0.353290	−0.176645	+	0.984275i	$0.556524\pi$
$8$	−195.684	−1.08101
$9$	655.235	2.69644
$10$	0	0
$11$	−504.686	−1.25759	−0.628796	−	0.777570i	$-0.716452\pi$
$11$	−504.686	−1.25759	−0.628796	+	0.777570i	$0.716452\pi$
$12$	−405.186	−0.812271
$13$	−513.142	−0.842130	−0.421065	−	0.907031i	$-0.638344\pi$
$13$	−513.142	−0.842130	−0.421065	+	0.907031i	$0.638344\pi$
$14$	−196.895	−0.268481
$15$	0	0
$16$	−408.601	−0.399025
$17$	−289.000	−0.242536
$17$	−289.000	−0.242536
$18$	2816.79	2.04915
$19$	−2084.02	−1.32440	−0.662198	−	0.749329i	$-0.730376\pi$
$19$	−2084.02	−1.32440	−0.662198	+	0.749329i	$0.730376\pi$
$20$	0	0
$21$	−1372.69	−0.679240
$22$	−2169.60	−0.955701
$23$	−4718.18	−1.85975	−0.929875	−	0.367876i	$-0.880085\pi$
$23$	−4718.18	−1.85975	−0.929875	+	0.367876i	$0.880085\pi$
$24$	−5864.75	−2.07836
$25$	0	0
$26$	−2205.94	−0.639972
$27$	12354.9	3.26159
$28$	619.207	0.149259
$29$	−3092.07	−0.682739	−0.341369	−	0.939929i	$-0.610891\pi$
$29$	−3092.07	−0.682739	−0.341369	+	0.939929i	$0.610891\pi$
$30$	0	0
$31$	4157.17	0.776950	0.388475	−	0.921459i	$-0.373002\pi$
$31$	4157.17	0.776950	0.388475	+	0.921459i	$0.373002\pi$
$32$	4505.34	0.777772
$33$	−15125.7	−2.41786
$34$	−1242.38	−0.184314
$35$	0	0
$36$	−8858.42	−1.13920
$37$	−4514.40	−0.542120	−0.271060	−	0.962562i	$-0.587374\pi$
$37$	−4514.40	−0.542120	−0.271060	+	0.962562i	$0.587374\pi$
$38$	−8958.99	−1.00647
$39$	−15379.1	−1.61909
$40$	0	0
$41$	7375.94	0.685264	0.342632	−	0.939470i	$-0.388682\pi$
$41$	7375.94	0.685264	0.342632	+	0.939470i	$0.388682\pi$
$42$	−5901.04	−0.516185
$43$	17592.2	1.45094	0.725469	−	0.688255i	$-0.241623\pi$
$43$	17592.2	1.45094	0.725469	+	0.688255i	$0.241623\pi$
$44$	6823.09	0.531311
$45$	0	0
$46$	−20283.0	−1.41331
$47$	14737.7	0.973164	0.486582	−	0.873635i	$-0.338243\pi$
$47$	14737.7	0.973164	0.486582	+	0.873635i	$0.338243\pi$
$48$	−12246.0	−0.767170
$49$	−14709.3	−0.875186
$50$	0	0
$51$	−8661.49	−0.466302
$52$	6937.40	0.355786
$53$	1008.07	0.0492946	0.0246473	−	0.999696i	$-0.492154\pi$
$53$	1008.07	0.0492946	0.0246473	+	0.999696i	$0.492154\pi$
$54$	53112.5	2.47863
$55$	0	0
$56$	8962.53	0.381910
$57$	−62459.2	−2.54630
$58$	−13292.5	−0.518844
$59$	496.132	0.0185553	0.00927764	−	0.999957i	$-0.497047\pi$
$59$	496.132	0.0185553	0.00927764	+	0.999957i	$0.497047\pi$
$60$	0	0
$61$	−13046.2	−0.448910	−0.224455	−	0.974484i	$-0.572060\pi$
$61$	−13046.2	−0.448910	−0.224455	+	0.974484i	$0.572060\pi$
$62$	17871.2	0.590439
$63$	−30010.5	−0.952625
$64$	32443.2	0.990089
$65$	0	0
$66$	−65024.0	−1.83744
$67$	−21128.0	−0.575004	−0.287502	−	0.957780i	$-0.592825\pi$
$67$	−21128.0	−0.575004	−0.287502	+	0.957780i	$0.592825\pi$
$68$	3907.12	0.102467
$69$	−141406.	−3.57558
$70$	0	0
$71$	−41206.2	−0.970100	−0.485050	−	0.874487i	$-0.661199\pi$
$71$	−41206.2	−0.970100	−0.485050	+	0.874487i	$0.661199\pi$
$72$	−128219.	−2.91488
$73$	−53173.8	−1.16786	−0.583929	−	0.811805i	$-0.698485\pi$
$73$	−53173.8	−1.16786	−0.583929	+	0.811805i	$0.698485\pi$
$74$	−19406.9	−0.411981
$75$	0	0
$76$	28174.8	0.559535
$77$	23115.2	0.444295
$78$	−66113.4	−1.23042
$79$	22083.6	0.398110	0.199055	−	0.979988i	$-0.436213\pi$
$79$	22083.6	0.398110	0.199055	+	0.979988i	$0.436213\pi$
$80$	0	0
$81$	211061.	3.57434
$82$	31708.4	0.520763
$83$	63967.7	1.01922	0.509608	−	0.860407i	$-0.329791\pi$
$83$	63967.7	1.01922	0.509608	+	0.860407i	$0.329791\pi$
$84$	18558.0	0.286967
$85$	0	0
$86$	75627.1	1.10263
$87$	−92671.1	−1.31264
$88$	98758.8	1.35947
$89$	−145034.	−1.94087	−0.970433	−	0.241369i	$-0.922404\pi$
$89$	−145034.	−1.94087	−0.970433	+	0.241369i	$0.922404\pi$
$90$	0	0
$91$	23502.5	0.297516
$92$	63787.2	0.785713
$93$	124593.	1.49377
$94$	63356.1	0.739552
$95$	0	0
$96$	135028.	1.49535
$97$	131574.	1.41984	0.709921	−	0.704282i	$-0.248731\pi$
$97$	131574.	1.41984	0.709921	+	0.704282i	$0.248731\pi$
$98$	−63233.6	−0.665094
$99$	−330688.	−3.39102
$100$	0	0
$101$	−101611.	−0.991149	−0.495575	−	0.868565i	$-0.665043\pi$
$101$	−101611.	−0.991149	−0.495575	+	0.868565i	$0.665043\pi$
$102$	−37234.9	−0.354364
$103$	−72080.6	−0.669461	−0.334731	−	0.942314i	$-0.608645\pi$
$103$	−72080.6	−0.669461	−0.334731	+	0.942314i	$0.608645\pi$
$104$	100413.	0.910350
$105$	0	0
$106$	4333.57	0.0374612
$107$	175657.	1.48322	0.741610	−	0.670831i	$-0.234062\pi$
$107$	175657.	1.48322	0.741610	+	0.670831i	$0.234062\pi$
$108$	−167032.	−1.37797
$109$	−16726.2	−0.134844	−0.0674220	−	0.997725i	$-0.521477\pi$
$109$	−16726.2	−0.134844	−0.0674220	+	0.997725i	$0.521477\pi$
$110$	0	0
$111$	−135299.	−1.04229
$112$	18714.4	0.140971
$113$	106978.	0.788134	0.394067	−	0.919082i	$-0.371068\pi$
$113$	106978.	0.788134	0.394067	+	0.919082i	$0.371068\pi$
$114$	−268506.	−1.93505
$115$	0	0
$116$	41803.2	0.288446
$117$	−336228.	−2.27075
$118$	2132.82	0.0141010
$119$	13236.5	0.0856854
$120$	0	0
$121$	93657.2	0.581537
$122$	−56084.3	−0.341147
$123$	221061.	1.31750
$124$	−56202.6	−0.328248
$125$	0	0
$126$	−129012.	−0.723943
$127$	−156823.	−0.862781	−0.431390	−	0.902165i	$-0.641977\pi$
$127$	−156823.	−0.862781	−0.431390	+	0.902165i	$0.641977\pi$
$128$	−4700.61	−0.0253588
$129$	527248.	2.78959
$130$	0	0
$131$	124664.	0.634690	0.317345	−	0.948310i	$-0.397209\pi$
$131$	124664.	0.634690	0.317345	+	0.948310i	$0.397209\pi$
$132$	204492.	1.02151
$133$	95450.5	0.467896
$134$	−90827.1	−0.436972
$135$	0	0
$136$	56552.6	0.262183
$137$	49378.2	0.224767	0.112384	−	0.993665i	$-0.464151\pi$
$137$	49378.2	0.224767	0.112384	+	0.993665i	$0.464151\pi$
$138$	−607892.	−2.71724
$139$	41751.8	0.183290	0.0916449	−	0.995792i	$-0.470788\pi$
$139$	41751.8	0.183290	0.0916449	+	0.995792i	$0.470788\pi$
$140$	0	0
$141$	441698.	1.87102
$142$	−177141.	−0.737223
$143$	258976.	1.05906
$144$	−267730.	−1.07595
$145$	0	0
$146$	−228589.	−0.887509
$147$	−440845.	−1.68264
$148$	61032.2	0.229037
$149$	−46782.8	−0.172632	−0.0863159	−	0.996268i	$-0.527509\pi$
$149$	−46782.8	−0.172632	−0.0863159	+	0.996268i	$0.527509\pi$
$150$	0	0
$151$	−55545.1	−0.198246	−0.0991228	−	0.995075i	$-0.531604\pi$
$151$	−55545.1	−0.198246	−0.0991228	+	0.995075i	$0.531604\pi$
$152$	407808.	1.43168
$153$	−189363.	−0.653983
$154$	99370.0	0.337640
$155$	0	0
$156$	207918.	0.684038
$157$	−64212.8	−0.207909	−0.103954	−	0.994582i	$-0.533150\pi$
$157$	−64212.8	−0.207909	−0.103954	+	0.994582i	$0.533150\pi$
$158$	94935.4	0.302542
$159$	30212.3	0.0947744
$160$	0	0
$161$	216098.	0.657031
$162$	907332.	2.71631
$163$	−595853.	−1.75659	−0.878294	−	0.478121i	$-0.841318\pi$
$163$	−595853.	−1.75659	−0.878294	+	0.478121i	$0.841318\pi$
$164$	−99718.8	−0.289513
$165$	0	0
$166$	274991.	0.774548
$167$	533223.	1.47951	0.739755	−	0.672876i	$-0.234941\pi$
$167$	533223.	1.47951	0.739755	+	0.672876i	$0.234941\pi$
$168$	268612.	0.734265
$169$	−107979.	−0.290818
$170$	0	0
$171$	−1.36552e6	−3.57115
$172$	−237837.	−0.612997
$173$	−743259.	−1.88810	−0.944050	−	0.329804i	$-0.893017\pi$
$173$	−743259.	−1.88810	−0.944050	+	0.329804i	$0.893017\pi$
$174$	−398384.	−0.997536
$175$	0	0
$176$	206215.	0.501810
$177$	14869.4	0.0356746
$178$	−623488.	−1.47495
$179$	−369567.	−0.862105	−0.431053	−	0.902327i	$-0.641858\pi$
$179$	−369567.	−0.862105	−0.431053	+	0.902327i	$0.641858\pi$
$180$	0	0
$181$	490167.	1.11211	0.556055	−	0.831145i	$-0.312314\pi$
$181$	490167.	1.11211	0.556055	+	0.831145i	$0.312314\pi$
$182$	101035.	0.226096
$183$	−391002.	−0.863080
$184$	923270.	2.01041
$185$	0	0
$186$	535611.	1.13519
$187$	145854.	0.305011
$188$	−199246.	−0.411146
$189$	−565869.	−1.15229
$190$	0	0
$191$	−7647.56	−0.0151684	−0.00758420	−	0.999971i	$-0.502414\pi$
$191$	−7647.56	−0.0151684	−0.00758420	+	0.999971i	$0.502414\pi$
$192$	972342.	1.90356
$193$	195606.	0.377998	0.188999	−	0.981977i	$-0.439476\pi$
$193$	195606.	0.377998	0.188999	+	0.981977i	$0.439476\pi$
$194$	565622.	1.07900
$195$	0	0
$196$	198861.	0.369751
$197$	885254.	1.62518	0.812591	−	0.582834i	$-0.198056\pi$
$197$	885254.	1.62518	0.812591	+	0.582834i	$0.198056\pi$
$198$	−1.42159e6	−2.57699
$199$	−877413.	−1.57062	−0.785310	−	0.619102i	$-0.787497\pi$
$199$	−877413.	−1.57062	−0.785310	+	0.619102i	$0.787497\pi$
$200$	0	0
$201$	−633218.	−1.10551
$202$	−436817.	−0.753219
$203$	141620.	0.241205
$204$	117099.	0.197005
$205$	0	0
$206$	−309867.	−0.508754
$207$	−3.09151e6	−5.01470
$208$	209670.	0.336031
$209$	1.05178e6	1.66555
$210$	0	0
$211$	−325844.	−0.503852	−0.251926	−	0.967746i	$-0.581064\pi$
$211$	−325844.	−0.503852	−0.251926	+	0.967746i	$0.581064\pi$
$212$	−13628.5	−0.0208261
$213$	−1.23497e6	−1.86513
$214$	755131.	1.12717
$215$	0	0
$216$	−2.41765e6	−3.52581
$217$	−190403.	−0.274489
$218$	−71904.4	−0.102474
$219$	−1.59365e6	−2.24534
$220$	0	0
$221$	148298.	0.204246
$222$	−581637.	−0.792081
$223$	758258.	1.02107	0.510534	−	0.859858i	$-0.329448\pi$
$223$	758258.	1.02107	0.510534	+	0.859858i	$0.329448\pi$
$224$	−206350.	−0.274779
$225$	0	0
$226$	459890.	0.598939
$227$	−487718.	−0.628209	−0.314105	−	0.949388i	$-0.601704\pi$
$227$	−487718.	−0.628209	−0.314105	+	0.949388i	$0.601704\pi$
$228$	844415.	1.07577
$229$	−413467.	−0.521017	−0.260509	−	0.965472i	$-0.583890\pi$
$229$	−413467.	−0.521017	−0.260509	+	0.965472i	$0.583890\pi$
$230$	0	0
$231$	692776.	0.854206
$232$	605068.	0.738047
$233$	−750445.	−0.905584	−0.452792	−	0.891616i	$-0.649572\pi$
$233$	−750445.	−0.905584	−0.452792	+	0.891616i	$0.649572\pi$
$234$	−1.44541e6	−1.72565
$235$	0	0
$236$	−6707.44	−0.00783929
$237$	661859.	0.765411
$238$	56902.5	0.0651162
$239$	−421206.	−0.476980	−0.238490	−	0.971145i	$-0.576652\pi$
$239$	−421206.	−0.476980	−0.238490	+	0.971145i	$0.576652\pi$
$240$	0	0
$241$	212219.	0.235365	0.117682	−	0.993051i	$-0.462454\pi$
$241$	212219.	0.235365	0.117682	+	0.993051i	$0.462454\pi$
$242$	402623.	0.441937
$243$	3.32339e6	3.61048
$244$	176377.	0.189657
$245$	0	0
$246$	950320.	1.00123
$247$	1.06940e6	1.11531
$248$	−813489.	−0.839890
$249$	1.91715e6	1.95956
$250$	0	0
$251$	−937463.	−0.939226	−0.469613	−	0.882872i	$-0.655606\pi$
$251$	−937463.	−0.939226	−0.469613	+	0.882872i	$0.655606\pi$
$252$	405726.	0.402468
$253$	2.38120e6	2.33881
$254$	−674166.	−0.655666
$255$	0	0
$256$	−1.05839e6	−1.00936
$257$	1.48510e6	1.40256	0.701281	−	0.712885i	$-0.252612\pi$
$257$	1.48510e6	1.40256	0.701281	+	0.712885i	$0.252612\pi$
$258$	2.26659e6	2.11994
$259$	206765.	0.191526
$260$	0	0
$261$	−2.02603e6	−1.84096
$262$	535917.	0.482330
$263$	420914.	0.375235	0.187618	−	0.982242i	$-0.439923\pi$
$263$	420914.	0.375235	0.187618	+	0.982242i	$0.439923\pi$
$264$	2.95986e6	2.61373
$265$	0	0
$266$	410332.	0.355575
$267$	−4.34676e6	−3.73153
$268$	285639.	0.242930
$269$	−1.19320e6	−1.00538	−0.502691	−	0.864466i	$-0.667657\pi$
$269$	−1.19320e6	−1.00538	−0.502691	+	0.864466i	$0.667657\pi$
$270$	0	0
$271$	2.07801e6	1.71880	0.859398	−	0.511308i	$-0.170839\pi$
$271$	2.07801e6	1.71880	0.859398	+	0.511308i	$0.170839\pi$
$272$	118086.	0.0967777
$273$	704383.	0.572008
$274$	212272.	0.170811
$275$	0	0
$276$	1.91174e6	1.51062
$277$	−883691.	−0.691992	−0.345996	−	0.938236i	$-0.612459\pi$
$277$	−883691.	−0.691992	−0.345996	+	0.938236i	$0.612459\pi$
$278$	179487.	0.139290
$279$	2.72392e6	2.09500
$280$	0	0
$281$	353265.	0.266892	0.133446	−	0.991056i	$-0.457396\pi$
$281$	353265.	0.266892	0.133446	+	0.991056i	$0.457396\pi$
$282$	1.89882e6	1.42187
$283$	78060.8	0.0579384	0.0289692	−	0.999580i	$-0.490778\pi$
$283$	78060.8	0.0579384	0.0289692	+	0.999580i	$0.490778\pi$
$284$	557085.	0.409851
$285$	0	0
$286$	1.11331e6	0.804824
$287$	−337827.	−0.242097
$288$	2.95205e6	2.09722
$289$	83521.0	0.0588235
$290$	0	0
$291$	3.94334e6	2.72981
$292$	718881.	0.493401
$293$	1.79083e6	1.21867	0.609335	−	0.792913i	$-0.291436\pi$
$293$	1.79083e6	1.21867	0.609335	+	0.792913i	$0.291436\pi$
$294$	−1.89515e6	−1.27872
$295$	0	0
$296$	883393.	0.586037
$297$	−6.23535e6	−4.10176
$298$	−201115.	−0.131191
$299$	2.42109e6	1.56615
$300$	0	0
$301$	−805743.	−0.512602
$302$	−238783.	−0.150656
$303$	−3.04535e6	−1.90560
$304$	851533.	0.528466
$305$	0	0
$306$	−814052.	−0.496991
$307$	1.87896e6	1.13781	0.568907	−	0.822402i	$-0.307366\pi$
$307$	1.87896e6	1.13781	0.568907	+	0.822402i	$0.307366\pi$
$308$	−312505.	−0.187707
$309$	−2.16030e6	−1.28711
$310$	0	0
$311$	−2.65990e6	−1.55943	−0.779714	−	0.626136i	$-0.784635\pi$
$311$	−2.65990e6	−1.55943	−0.779714	+	0.626136i	$0.784635\pi$
$312$	3.00945e6	1.75025
$313$	−576430.	−0.332572	−0.166286	−	0.986078i	$-0.553178\pi$
$313$	−576430.	−0.332572	−0.166286	+	0.986078i	$0.553178\pi$
$314$	−276044.	−0.157999
$315$	0	0
$316$	−298559.	−0.168195
$317$	760504.	0.425063	0.212532	−	0.977154i	$-0.431829\pi$
$317$	760504.	0.425063	0.212532	+	0.977154i	$0.431829\pi$
$318$	129880.	0.0720233
$319$	1.56053e6	0.858607
$320$	0	0
$321$	5.26454e6	2.85166
$322$	928983.	0.499308
$323$	602281.	0.321213
$324$	−2.85344e6	−1.51010
$325$	0	0
$326$	−2.56151e6	−1.33491
$327$	−501295.	−0.259253
$328$	−1.44335e6	−0.740777
$329$	−675005.	−0.343809
$330$	0	0
$331$	−915317.	−0.459200	−0.229600	−	0.973285i	$-0.573742\pi$
$331$	−915317.	−0.459200	−0.229600	+	0.973285i	$0.573742\pi$
$332$	−864810.	−0.430601
$333$	−2.95799e6	−1.46179
$334$	2.29227e6	1.12435
$335$	0	0
$336$	560881.	0.271033
$337$	1.58665e6	0.761040	0.380520	−	0.924773i	$-0.375745\pi$
$337$	1.58665e6	0.761040	0.380520	+	0.924773i	$0.375745\pi$
$338$	−464189.	−0.221006
$339$	3.20620e6	1.51528
$340$	0	0
$341$	−2.09806e6	−0.977086
$342$	−5.87024e6	−2.71388
$343$	1.44348e6	0.662484
$344$	−3.44250e6	−1.56848
$345$	0	0
$346$	−3.19519e6	−1.43485
$347$	−3.71862e6	−1.65790	−0.828950	−	0.559322i	$-0.811061\pi$
$347$	−3.71862e6	−1.65790	−0.828950	+	0.559322i	$0.811061\pi$
$348$	1.25286e6	0.554569
$349$	2.20157e6	0.967540	0.483770	−	0.875195i	$-0.339267\pi$
$349$	2.20157e6	0.967540	0.483770	+	0.875195i	$0.339267\pi$
$350$	0	0
$351$	−6.33982e6	−2.74669
$352$	−2.27378e6	−0.978120
$353$	2.24412e6	0.958539	0.479269	−	0.877668i	$-0.340902\pi$
$353$	2.24412e6	0.958539	0.479269	+	0.877668i	$0.340902\pi$
$354$	63921.9	0.0271107
$355$	0	0
$356$	1.96079e6	0.819984
$357$	396706.	0.164740
$358$	−1.58873e6	−0.655153
$359$	−2.48856e6	−1.01909	−0.509544	−	0.860444i	$-0.670186\pi$
$359$	−2.48856e6	−1.01909	−0.509544	+	0.860444i	$0.670186\pi$
$360$	0	0
$361$	1.86703e6	0.754023
$362$	2.10718e6	0.845143
$363$	2.80696e6	1.11807
$364$	−317741.	−0.125695
$365$	0	0
$366$	−1.68088e6	−0.655893
$367$	4.14031e6	1.60460	0.802302	−	0.596919i	$-0.203609\pi$
$367$	4.14031e6	1.60460	0.802302	+	0.596919i	$0.203609\pi$
$368$	1.92785e6	0.742086
$369$	4.83297e6	1.84777
$370$	0	0
$371$	−46170.6	−0.0174153
$372$	−1.68442e6	−0.631094
$373$	−1.79738e6	−0.668910	−0.334455	−	0.942412i	$-0.608552\pi$
$373$	−1.79738e6	−0.668910	−0.334455	+	0.942412i	$0.608552\pi$
$374$	627013.	0.231792
$375$	0	0
$376$	−2.88393e6	−1.05200
$377$	1.58667e6	0.574955
$378$	−2.43261e6	−0.875676
$379$	90472.6	0.0323533	0.0161767	−	0.999869i	$-0.494851\pi$
$379$	90472.6	0.0323533	0.0161767	+	0.999869i	$0.494851\pi$
$380$	0	0
$381$	−4.70007e6	−1.65879
$382$	−32876.1	−0.0115272
$383$	−2.92745e6	−1.01975	−0.509873	−	0.860250i	$-0.670308\pi$
$383$	−2.92745e6	−1.01975	−0.509873	+	0.860250i	$0.670308\pi$
$384$	−140880.	−0.0487552
$385$	0	0
$386$	840892.	0.287258
$387$	1.15270e7	3.91237
$388$	−1.77881e6	−0.599859
$389$	1.35743e6	0.454823	0.227411	−	0.973799i	$-0.426974\pi$
$389$	1.35743e6	0.454823	0.227411	+	0.973799i	$0.426974\pi$
$390$	0	0
$391$	1.36355e6	0.451056
$392$	2.87836e6	0.946084
$393$	3.73624e6	1.22026
$394$	3.80562e6	1.23505
$395$	0	0
$396$	4.47072e6	1.43265
$397$	223344.	0.0711210	0.0355605	−	0.999368i	$-0.488678\pi$
$397$	223344.	0.0711210	0.0355605	+	0.999368i	$0.488678\pi$
$398$	−3.77191e6	−1.19359
$399$	2.86070e6	0.899582
$400$	0	0
$401$	3.43768e6	1.06759	0.533796	−	0.845613i	$-0.320765\pi$
$401$	3.43768e6	1.06759	0.533796	+	0.845613i	$0.320765\pi$
$402$	−2.72214e6	−0.840127
$403$	−2.13322e6	−0.654293
$404$	1.37373e6	0.418744
$405$	0	0
$406$	608812.	0.183302
$407$	2.27835e6	0.681766
$408$	1.69491e6	0.504077
$409$	−2.01964e6	−0.596987	−0.298494	−	0.954412i	$-0.596484\pi$
$409$	−2.01964e6	−0.596987	−0.298494	+	0.954412i	$0.596484\pi$
$410$	0	0
$411$	1.47989e6	0.432141
$412$	974491.	0.282836
$413$	−22723.4	−0.00655539
$414$	−1.32901e7	−3.81090
$415$	0	0
$416$	−2.31188e6	−0.654985
$417$	1.25133e6	0.352395
$418$	4.52148e6	1.26573
$419$	2.77100e6	0.771085	0.385542	−	0.922690i	$-0.374014\pi$
$419$	2.77100e6	0.771085	0.385542	+	0.922690i	$0.374014\pi$
$420$	0	0
$421$	−1.08269e6	−0.297714	−0.148857	−	0.988859i	$-0.547559\pi$
$421$	−1.08269e6	−0.297714	−0.148857	+	0.988859i	$0.547559\pi$
$422$	−1.40077e6	−0.382900
$423$	9.65668e6	2.62408
$424$	−197262.	−0.0532879
$425$	0	0
$426$	−5.30902e6	−1.41739
$427$	597531.	0.158595
$428$	−2.37479e6	−0.626636
$429$	7.76164e6	2.03615
$430$	0	0
$431$	−3.08443e6	−0.799800	−0.399900	−	0.916559i	$-0.630955\pi$
$431$	−3.08443e6	−0.799800	−0.399900	+	0.916559i	$0.630955\pi$
$432$	−5.04823e6	−1.30146
$433$	−3.83729e6	−0.983569	−0.491784	−	0.870717i	$-0.663655\pi$
$433$	−3.83729e6	−0.983569	−0.491784	+	0.870717i	$0.663655\pi$
$434$	−818523.	−0.208596
$435$	0	0
$436$	226130.	0.0569694
$437$	9.83277e6	2.46304
$438$	−6.85093e6	−1.70634
$439$	−1.77903e6	−0.440577	−0.220289	−	0.975435i	$-0.570700\pi$
$439$	−1.77903e6	−0.440577	−0.220289	+	0.975435i	$0.570700\pi$
$440$	0	0
$441$	−9.63801e6	−2.35989
$442$	637518.	0.155216
$443$	−2.29093e6	−0.554629	−0.277315	−	0.960779i	$-0.589444\pi$
$443$	−2.29093e6	−0.554629	−0.277315	+	0.960779i	$0.589444\pi$
$444$	1.82917e6	0.440349
$445$	0	0
$446$	3.25967e6	0.775956
$447$	−1.40211e6	−0.331904
$448$	−1.48594e6	−0.349789
$449$	6.34594e6	1.48552	0.742762	−	0.669555i	$-0.233515\pi$
$449$	6.34594e6	1.48552	0.742762	+	0.669555i	$0.233515\pi$
$450$	0	0
$451$	−3.72254e6	−0.861782
$452$	−1.44629e6	−0.332973
$453$	−1.66472e6	−0.381150
$454$	−2.09665e6	−0.477405
$455$	0	0
$456$	1.22222e7	2.75257
$457$	−1.39090e6	−0.311535	−0.155767	−	0.987794i	$-0.549785\pi$
$457$	−1.39090e6	−0.311535	−0.155767	+	0.987794i	$0.549785\pi$
$458$	−1.77745e6	−0.395944
$459$	−3.57057e6	−0.791053
$460$	0	0
$461$	−4.36882e6	−0.957441	−0.478721	−	0.877967i	$-0.658899\pi$
$461$	−4.36882e6	−0.957441	−0.478721	+	0.877967i	$0.658899\pi$
$462$	2.97817e6	0.649150
$463$	−737398.	−0.159864	−0.0799318	−	0.996800i	$-0.525470\pi$
$463$	−737398.	−0.159864	−0.0799318	+	0.996800i	$0.525470\pi$
$464$	1.26342e6	0.272430
$465$	0	0
$466$	−3.22609e6	−0.688194
$467$	−1.95716e6	−0.415273	−0.207636	−	0.978206i	$-0.566577\pi$
$467$	−1.95716e6	−0.415273	−0.207636	+	0.978206i	$0.566577\pi$
$468$	4.54563e6	0.959354
$469$	967686.	0.203143
$470$	0	0
$471$	−1.92449e6	−0.399728
$472$	−97084.9	−0.0200584
$473$	−8.87854e6	−1.82469
$474$	2.84527e6	0.581670
$475$	0	0
$476$	−178951.	−0.0362006
$477$	660520.	0.132920
$478$	−1.81072e6	−0.362478
$479$	−3.22574e6	−0.642377	−0.321189	−	0.947015i	$-0.604082\pi$
$479$	−3.22574e6	−0.642377	−0.321189	+	0.947015i	$0.604082\pi$
$480$	0	0
$481$	2.31652e6	0.456535
$482$	912308.	0.178864
$483$	6.47658e6	1.26322
$484$	−1.26619e6	−0.245690
$485$	0	0
$486$	1.42869e7	2.74377
$487$	−5.66306e6	−1.08200	−0.541001	−	0.841022i	$-0.681955\pi$
$487$	−5.66306e6	−1.08200	−0.541001	+	0.841022i	$0.681955\pi$
$488$	2.55293e6	0.485276
$489$	−1.78580e7	−3.37724
$490$	0	0
$491$	290042.	0.0542947	0.0271474	−	0.999631i	$-0.491358\pi$
$491$	290042.	0.0542947	0.0271474	+	0.999631i	$0.491358\pi$
$492$	−2.98863e6	−0.556620
$493$	893609.	0.165588
$494$	4.59723e6	0.847576
$495$	0	0
$496$	−1.69862e6	−0.310022
$497$	1.88729e6	0.342726
$498$	8.24163e6	1.48916
$499$	7.87433e6	1.41567	0.707835	−	0.706377i	$-0.249672\pi$
$499$	7.87433e6	1.41567	0.707835	+	0.706377i	$0.249672\pi$
$500$	0	0
$501$	1.59810e7	2.84452
$502$	−4.03006e6	−0.713760
$503$	−7.91006e6	−1.39399	−0.696995	−	0.717076i	$-0.745480\pi$
$503$	−7.91006e6	−1.39399	−0.696995	+	0.717076i	$0.745480\pi$
$504$	5.87256e6	1.02980
$505$	0	0
$506$	1.02365e7	1.77736
$507$	−3.23618e6	−0.559130
$508$	2.12016e6	0.364510
$509$	−4.88210e6	−0.835241	−0.417621	−	0.908621i	$-0.637136\pi$
$509$	−4.88210e6	−0.835241	−0.417621	+	0.908621i	$0.637136\pi$
$510$	0	0
$511$	2.43542e6	0.412593
$512$	−4.39950e6	−0.741700
$513$	−2.57479e7	−4.31964
$514$	6.38428e6	1.06587
$515$	0	0
$516$	−7.12811e6	−1.17856
$517$	−7.43793e6	−1.22384
$518$	888860.	0.145549
$519$	−2.22759e7	−3.63008
$520$	0	0
$521$	−4.67038e6	−0.753803	−0.376901	−	0.926253i	$-0.623010\pi$
$521$	−4.67038e6	−0.753803	−0.376901	+	0.926253i	$0.623010\pi$
$522$	−8.70971e6	−1.39903
$523$	8.87695e6	1.41909	0.709544	−	0.704661i	$-0.248901\pi$
$523$	8.87695e6	1.41909	0.709544	+	0.704661i	$0.248901\pi$
$524$	−1.68539e6	−0.268146
$525$	0	0
$526$	1.80947e6	0.285158
$527$	−1.20142e6	−0.188438
$528$	6.18039e6	0.964787
$529$	1.58248e7	2.45867
$530$	0	0
$531$	325083.	0.0500332
$532$	−1.29044e6	−0.197678
$533$	−3.78490e6	−0.577081
$534$	−1.86863e7	−2.83576
$535$	0	0
$536$	4.13440e6	0.621585
$537$	−1.10761e7	−1.65749
$538$	−5.12943e6	−0.764035
$539$	7.42356e6	1.10063
$540$	0	0
$541$	−9.68582e6	−1.42280	−0.711399	−	0.702789i	$-0.751938\pi$
$541$	−9.68582e6	−1.42280	−0.711399	+	0.702789i	$0.751938\pi$
$542$	8.93315e6	1.30619
$543$	1.46906e7	2.13816
$544$	−1.30204e6	−0.188638
$545$	0	0
$546$	3.02807e6	0.434695
$547$	1.98238e6	0.283282	0.141641	−	0.989918i	$-0.454762\pi$
$547$	1.98238e6	0.283282	0.141641	+	0.989918i	$0.454762\pi$
$548$	−667566.	−0.0949605
$549$	−8.54832e6	−1.21046
$550$	0	0
$551$	6.44394e6	0.904216
$552$	2.76709e7	3.86523
$553$	−1.01146e6	−0.140648
$554$	−3.79890e6	−0.525876
$555$	0	0
$556$	−564462.	−0.0774369
$557$	7.73137e6	1.05589	0.527945	−	0.849279i	$-0.322963\pi$
$557$	7.73137e6	1.05589	0.527945	+	0.849279i	$0.322963\pi$
$558$	1.17099e7	1.59208
$559$	−9.02729e6	−1.22188
$560$	0	0
$561$	4.37134e6	0.586418
$562$	1.51865e6	0.202823
$563$	6.73841e6	0.895955	0.447978	−	0.894045i	$-0.352144\pi$
$563$	6.73841e6	0.895955	0.447978	+	0.894045i	$0.352144\pi$
$564$	−5.97152e6	−0.790474
$565$	0	0
$566$	335575.	0.0440300
$567$	−9.66686e6	−1.26278
$568$	8.06337e6	1.04869
$569$	1.38027e6	0.178724	0.0893622	−	0.995999i	$-0.471517\pi$
$569$	1.38027e6	0.178724	0.0893622	+	0.995999i	$0.471517\pi$
$570$	0	0
$571$	7.93779e6	1.01885	0.509424	−	0.860516i	$-0.329859\pi$
$571$	7.93779e6	1.01885	0.509424	+	0.860516i	$0.329859\pi$
$572$	−3.50121e6	−0.447433
$573$	−229202.	−0.0291629
$574$	−1.45228e6	−0.183980
$575$	0	0
$576$	2.12579e7	2.66972
$577$	−8.26007e6	−1.03287	−0.516433	−	0.856327i	$-0.672741\pi$
$577$	−8.26007e6	−1.03287	−0.516433	+	0.856327i	$0.672741\pi$
$578$	359048.	0.0447027
$579$	5.86243e6	0.726744
$580$	0	0
$581$	−2.92980e6	−0.360078
$582$	1.69520e7	2.07450
$583$	−508757.	−0.0619925
$584$	1.04052e7	1.26247
$585$	0	0
$586$	7.69862e6	0.926123
$587$	−1.42684e7	−1.70915	−0.854573	−	0.519331i	$-0.826181\pi$
$587$	−1.42684e7	−1.70915	−0.854573	+	0.519331i	$0.826181\pi$
$588$	5.95998e6	0.710889
$589$	−8.66361e6	−1.02899
$590$	0	0
$591$	2.65316e7	3.12460
$592$	1.84459e6	0.216319
$593$	−1.26496e7	−1.47720	−0.738601	−	0.674143i	$-0.764513\pi$
$593$	−1.26496e7	−1.47720	−0.738601	+	0.674143i	$0.764513\pi$
$594$	−2.68051e7	−3.11711
$595$	0	0
$596$	632478.	0.0729340
$597$	−2.62966e7	−3.01970
$598$	1.04080e7	1.19019
$599$	−4.60166e6	−0.524019	−0.262010	−	0.965065i	$-0.584385\pi$
$599$	−4.60166e6	−0.524019	−0.262010	+	0.965065i	$0.584385\pi$
$600$	0	0
$601$	−8.69808e6	−0.982284	−0.491142	−	0.871080i	$-0.663420\pi$
$601$	−8.69808e6	−0.982284	−0.491142	+	0.871080i	$0.663420\pi$
$602$	−3.46381e6	−0.389549
$603$	−1.38438e7	−1.55046
$604$	750940.	0.0837555
$605$	0	0
$606$	−1.30917e7	−1.44815
$607$	−8.58642e6	−0.945889	−0.472945	−	0.881092i	$-0.656809\pi$
$607$	−8.58642e6	−0.945889	−0.472945	+	0.881092i	$0.656809\pi$
$608$	−9.38921e6	−1.03008
$609$	4.24445e6	0.463743
$610$	0	0
$611$	−7.56255e6	−0.819531
$612$	2.56008e6	0.276297
$613$	−1.80976e6	−0.194522	−0.0972611	−	0.995259i	$-0.531008\pi$
$613$	−1.80976e6	−0.194522	−0.0972611	+	0.995259i	$0.531008\pi$
$614$	8.07745e6	0.864676
$615$	0	0
$616$	−4.52327e6	−0.480287
$617$	−428692.	−0.0453348	−0.0226674	−	0.999743i	$-0.507216\pi$
$617$	−428692.	−0.0453348	−0.0226674	+	0.999743i	$0.507216\pi$
$618$	−9.28690e6	−0.978136
$619$	1.35257e7	1.41883	0.709417	−	0.704789i	$-0.248958\pi$
$619$	1.35257e7	1.41883	0.709417	+	0.704789i	$0.248958\pi$
$620$	0	0
$621$	−5.82926e7	−6.06575
$622$	−1.14347e7	−1.18508
$623$	6.64274e6	0.685689
$624$	6.28394e6	0.646057
$625$	0	0
$626$	−2.47802e6	−0.252737
$627$	3.15223e7	3.20220
$628$	868123.	0.0878379
$629$	1.30466e6	0.131483
$630$	0	0
$631$	943645.	0.0943486	0.0471743	−	0.998887i	$-0.484978\pi$
$631$	943645.	0.0943486	0.0471743	+	0.998887i	$0.484978\pi$
$632$	−4.32141e6	−0.430360
$633$	−9.76572e6	−0.968712
$634$	3.26933e6	0.323025
$635$	0	0
$636$	−408454.	−0.0400406
$637$	7.54793e6	0.737020
$638$	6.70855e6	0.652494
$639$	−2.69997e7	−2.61581
$640$	0	0
$641$	−1.91645e7	−1.84227	−0.921134	−	0.389247i	$-0.872735\pi$
$641$	−1.91645e7	−1.84227	−0.921134	+	0.389247i	$0.872735\pi$
$642$	2.26317e7	2.16710
$643$	5.39663e6	0.514748	0.257374	−	0.966312i	$-0.417143\pi$
$643$	5.39663e6	0.514748	0.257374	+	0.966312i	$0.417143\pi$
$644$	−2.92153e6	−0.277585
$645$	0	0
$646$	2.58915e6	0.244104
$647$	7.22529e6	0.678570	0.339285	−	0.940684i	$-0.389815\pi$
$647$	7.22529e6	0.678570	0.339285	+	0.940684i	$0.389815\pi$
$648$	−4.13013e7	−3.86390
$649$	−250391.	−0.0233350
$650$	0	0
$651$	−5.70648e6	−0.527735
$652$	8.05561e6	0.742129
$653$	−1.67534e7	−1.53752	−0.768758	−	0.639540i	$-0.779125\pi$
$653$	−1.67534e7	−1.53752	−0.768758	+	0.639540i	$0.779125\pi$
$654$	−2.15502e6	−0.197018
$655$	0	0
$656$	−3.01382e6	−0.273437
$657$	−3.48413e7	−3.14906
$658$	−2.90178e6	−0.261276
$659$	1.70752e7	1.53162	0.765810	−	0.643067i	$-0.222338\pi$
$659$	1.70752e7	1.53162	0.765810	+	0.643067i	$0.222338\pi$
$660$	0	0
$661$	−1.45834e7	−1.29824	−0.649122	−	0.760685i	$-0.724863\pi$
$661$	−1.45834e7	−1.29824	−0.649122	+	0.760685i	$0.724863\pi$
$662$	−3.93486e6	−0.348967
$663$	4.44457e6	0.392687
$664$	−1.25174e7	−1.10178
$665$	0	0
$666$	−1.27161e7	−1.11088
$667$	1.45889e7	1.26972
$668$	−7.20889e6	−0.625068
$669$	2.27254e7	1.96312
$670$	0	0
$671$	6.58423e6	0.564545
$672$	−6.18442e6	−0.528294
$673$	−9.19877e6	−0.782874	−0.391437	−	0.920205i	$-0.628022\pi$
$673$	−9.19877e6	−0.782874	−0.391437	+	0.920205i	$0.628022\pi$
$674$	6.82086e6	0.578348
$675$	0	0
$676$	1.45981e6	0.122866
$677$	1.16519e7	0.977068	0.488534	−	0.872545i	$-0.337532\pi$
$677$	1.16519e7	0.977068	0.488534	+	0.872545i	$0.337532\pi$
$678$	1.37831e7	1.15153
$679$	−6.02623e6	−0.501616
$680$	0	0
$681$	−1.46172e7	−1.20780
$682$	−9.01937e6	−0.742532
$683$	1.79480e7	1.47219	0.736094	−	0.676879i	$-0.236668\pi$
$683$	1.79480e7	1.47219	0.736094	+	0.676879i	$0.236668\pi$
$684$	1.84611e7	1.50875
$685$	0	0
$686$	6.20538e6	0.503452
$687$	−1.23918e7	−1.00171
$688$	−7.18820e6	−0.578960
$689$	−517281.	−0.0415124
$690$	0	0
$691$	−1.29794e7	−1.03410	−0.517048	−	0.855956i	$-0.672969\pi$
$691$	−1.29794e7	−1.03410	−0.517048	+	0.855956i	$0.672969\pi$
$692$	1.00485e7	0.797690
$693$	1.51459e7	1.19801
$694$	−1.59860e7	−1.25991
$695$	0	0
$696$	1.81342e7	1.41898
$697$	−2.13165e6	−0.166201
$698$	9.46433e6	0.735278
$699$	−2.24912e7	−1.74109
$700$	0	0
$701$	1.99415e7	1.53272	0.766358	−	0.642413i	$-0.222067\pi$
$701$	1.99415e7	1.53272	0.766358	+	0.642413i	$0.222067\pi$
$702$	−2.72542e7	−2.08733
$703$	9.40808e6	0.717981
$704$	−1.63737e7	−1.24513
$705$	0	0
$706$	9.64726e6	0.728437
$707$	4.65392e6	0.350163
$708$	−201026.	−0.0150719
$709$	−1.53176e7	−1.14439	−0.572196	−	0.820117i	$-0.693908\pi$
$709$	−1.53176e7	−1.14439	−0.572196	+	0.820117i	$0.693908\pi$
$710$	0	0
$711$	1.44700e7	1.07348
$712$	2.83808e7	2.09810
$713$	−1.96142e7	−1.44493
$714$	1.70540e6	0.125193
$715$	0	0
$716$	4.99634e6	0.364225
$717$	−1.26238e7	−0.917047
$718$	−1.06981e7	−0.774452
$719$	−1.24260e7	−0.896413	−0.448207	−	0.893930i	$-0.647937\pi$
$719$	−1.24260e7	−0.896413	−0.448207	+	0.893930i	$0.647937\pi$
$720$	0	0
$721$	3.30137e6	0.236514
$722$	8.02619e6	0.573016
$723$	6.36032e6	0.452515
$724$	−6.62680e6	−0.469848
$725$	0	0
$726$	1.20668e7	0.849673
$727$	−1.09417e7	−0.767803	−0.383902	−	0.923374i	$-0.625420\pi$
$727$	−1.09417e7	−0.767803	−0.383902	+	0.923374i	$0.625420\pi$
$728$	−4.59905e6	−0.321618
$729$	4.83159e7	3.36722
$730$	0	0
$731$	−5.08415e6	−0.351904
$732$	5.28613e6	0.364637
$733$	8.63822e6	0.593833	0.296916	−	0.954903i	$-0.404042\pi$
$733$	8.63822e6	0.593833	0.296916	+	0.954903i	$0.404042\pi$
$734$	1.77988e7	1.21941
$735$	0	0
$736$	−2.12570e7	−1.44646
$737$	1.06630e7	0.723121
$738$	2.07765e7	1.40421
$739$	−1.28244e7	−0.863825	−0.431913	−	0.901915i	$-0.642161\pi$
$739$	−1.28244e7	−0.863825	−0.431913	+	0.901915i	$0.642161\pi$
$740$	0	0
$741$	3.20504e7	2.14431
$742$	−198483.	−0.0132347
$743$	−1.16623e7	−0.775020	−0.387510	−	0.921865i	$-0.626665\pi$
$743$	−1.16623e7	−0.775020	−0.387510	+	0.921865i	$0.626665\pi$
$744$	−2.43807e7	−1.61478
$745$	0	0
$746$	−7.72675e6	−0.508335
$747$	4.19139e7	2.74825
$748$	−1.97187e6	−0.128862
$749$	−8.04529e6	−0.524007
$750$	0	0
$751$	2.02785e7	1.31200	0.656002	−	0.754759i	$-0.272246\pi$
$751$	2.02785e7	1.31200	0.656002	+	0.754759i	$0.272246\pi$
$752$	−6.02186e6	−0.388317
$753$	−2.80963e7	−1.80577
$754$	6.82094e6	0.436934
$755$	0	0
$756$	7.65024e6	0.486823
$757$	−4.98405e6	−0.316114	−0.158057	−	0.987430i	$-0.550523\pi$
$757$	−4.98405e6	−0.316114	−0.158057	+	0.987430i	$0.550523\pi$
$758$	388933.	0.0245868
$759$	7.13659e7	4.49662
$760$	0	0
$761$	1.63795e7	1.02527	0.512635	−	0.858607i	$-0.328669\pi$
$761$	1.63795e7	1.02527	0.512635	+	0.858607i	$0.328669\pi$
$762$	−2.02051e7	−1.26059
$763$	766081.	0.0476391
$764$	103391.	0.00640839
$765$	0	0
$766$	−1.25848e7	−0.774951
$767$	−254586.	−0.0156259
$768$	−3.17206e7	−1.94061
$769$	−2.09430e7	−1.27709	−0.638547	−	0.769582i	$-0.720464\pi$
$769$	−2.09430e7	−1.27709	−0.638547	+	0.769582i	$0.720464\pi$
$770$	0	0
$771$	4.45092e7	2.69658
$772$	−2.64449e6	−0.159698
$773$	1.56215e7	0.940318	0.470159	−	0.882582i	$-0.344197\pi$
$773$	1.56215e7	0.940318	0.470159	+	0.882582i	$0.344197\pi$
$774$	4.95535e7	2.97318
$775$	0	0
$776$	−2.57468e7	−1.53486
$777$	6.19685e6	0.368229
$778$	5.83544e6	0.345640
$779$	−1.53716e7	−0.907560
$780$	0	0
$781$	2.07962e7	1.21999
$782$	5.86178e6	0.342778
$783$	−3.82023e7	−2.22682
$784$	6.01022e6	0.349221
$785$	0	0
$786$	1.60617e7	0.927333
$787$	−2.34939e7	−1.35213	−0.676065	−	0.736842i	$-0.736316\pi$
$787$	−2.34939e7	−1.35213	−0.676065	+	0.736842i	$0.736316\pi$
$788$	−1.19682e7	−0.686613
$789$	1.26150e7	0.721432
$790$	0	0
$791$	−4.89974e6	−0.278440
$792$	6.47102e7	3.66572
$793$	6.69454e6	0.378040
$794$	960133.	0.0540480
$795$	0	0
$796$	1.18622e7	0.663561
$797$	2.03038e6	0.113222	0.0566111	−	0.998396i	$-0.481970\pi$
$797$	2.03038e6	0.113222	0.0566111	+	0.998396i	$0.481970\pi$
$798$	1.22979e7	0.683633
$799$	−4.25921e6	−0.236027
$800$	0	0
$801$	−9.50315e7	−5.23343
$802$	1.47783e7	0.811311
$803$	2.68361e7	1.46869
$804$	8.56076e6	0.467060
$805$	0	0
$806$	−9.17048e6	−0.497227
$807$	−3.57607e7	−1.93296
$808$	1.98837e7	1.07144
$809$	−1.53040e7	−0.822120	−0.411060	−	0.911608i	$-0.634841\pi$
$809$	−1.53040e7	−0.822120	−0.411060	+	0.911608i	$0.634841\pi$
$810$	0	0
$811$	−1.82768e6	−0.0975769	−0.0487884	−	0.998809i	$-0.515536\pi$
$811$	−1.82768e6	−0.0975769	−0.0487884	+	0.998809i	$0.515536\pi$
$812$	−1.91463e6	−0.101905
$813$	6.22791e7	3.30458
$814$	9.79441e6	0.518104
$815$	0	0
$816$	3.53910e6	0.186066
$817$	−3.66625e7	−1.92162
$818$	−8.68222e6	−0.453678
$819$	1.53996e7	0.802234
$820$	0	0
$821$	3.84838e6	0.199260	0.0996301	−	0.995025i	$-0.468234\pi$
$821$	3.84838e6	0.199260	0.0996301	+	0.995025i	$0.468234\pi$
$822$	6.36190e6	0.328403
$823$	−8.89074e6	−0.457550	−0.228775	−	0.973479i	$-0.573472\pi$
$823$	−8.89074e6	−0.457550	−0.228775	+	0.973479i	$0.573472\pi$
$824$	1.41050e7	0.723694
$825$	0	0
$826$	−97685.7	−0.00498174
$827$	−2.28605e7	−1.16231	−0.581155	−	0.813793i	$-0.697399\pi$
$827$	−2.28605e7	−1.16231	−0.581155	+	0.813793i	$0.697399\pi$
$828$	4.17956e7	2.11863
$829$	1.27813e7	0.645934	0.322967	−	0.946410i	$-0.395320\pi$
$829$	1.27813e7	0.645934	0.322967	+	0.946410i	$0.395320\pi$
$830$	0	0
$831$	−2.64847e7	−1.33043
$832$	−1.66480e7	−0.833783
$833$	4.25097e6	0.212264
$834$	5.37932e6	0.267801
$835$	0	0
$836$	−1.42194e7	−0.703666
$837$	5.13614e7	2.53410
$838$	1.19123e7	0.585982
$839$	−2.44397e7	−1.19865	−0.599323	−	0.800508i	$-0.704563\pi$
$839$	−2.44397e7	−1.19865	−0.599323	+	0.800508i	$0.704563\pi$
$840$	0	0
$841$	−1.09502e7	−0.533868
$842$	−4.65438e6	−0.226247
$843$	1.05876e7	0.513130
$844$	4.40523e6	0.212869
$845$	0	0
$846$	4.15131e7	1.99416
$847$	−4.28961e6	−0.205451
$848$	−411897.	−0.0196698
$849$	2.33953e6	0.111393
$850$	0	0
$851$	2.12997e7	1.00821
$852$	1.66962e7	0.787984
$853$	3.88293e7	1.82720	0.913601	−	0.406612i	$-0.133290\pi$
$853$	3.88293e7	1.82720	0.913601	+	0.406612i	$0.133290\pi$
$854$	2.56872e6	0.120524
$855$	0	0
$856$	−3.43732e7	−1.60338
$857$	2.51074e7	1.16775	0.583875	−	0.811843i	$-0.301536\pi$
$857$	2.51074e7	1.16775	0.583875	+	0.811843i	$0.301536\pi$
$858$	3.33665e7	1.54736
$859$	−1.46587e6	−0.0677817	−0.0338909	−	0.999426i	$-0.510790\pi$
$859$	−1.46587e6	−0.0677817	−0.0338909	+	0.999426i	$0.510790\pi$
$860$	0	0
$861$	−1.01249e7	−0.465458
$862$	−1.32596e7	−0.607804
$863$	1.85564e7	0.848138	0.424069	−	0.905630i	$-0.360601\pi$
$863$	1.85564e7	0.848138	0.424069	+	0.905630i	$0.360601\pi$
$864$	5.56630e7	2.53678
$865$	0	0
$866$	−1.64961e7	−0.747459
$867$	2.50317e6	0.113095
$868$	2.57415e6	0.115967
$869$	−1.11453e7	−0.500660
$870$	0	0
$871$	1.08417e7	0.484228
$872$	3.27305e6	0.145768
$873$	8.62117e7	3.82852
$874$	4.22701e7	1.87178
$875$	0	0
$876$	2.15453e7	0.948618
$877$	4.32523e7	1.89894	0.949468	−	0.313864i	$-0.101623\pi$
$877$	4.32523e7	1.89894	0.949468	+	0.313864i	$0.101623\pi$
$878$	−7.64787e6	−0.334814
$879$	5.36723e7	2.34303
$880$	0	0
$881$	2.75182e7	1.19448	0.597241	−	0.802062i	$-0.296264\pi$
$881$	2.75182e7	1.19448	0.597241	+	0.802062i	$0.296264\pi$
$882$	−4.14329e7	−1.79338
$883$	−3.80483e7	−1.64223	−0.821115	−	0.570763i	$-0.806648\pi$
$883$	−3.80483e7	−1.64223	−0.821115	+	0.570763i	$0.806648\pi$
$884$	−2.00491e6	−0.0862907
$885$	0	0
$886$	−9.84849e6	−0.421488
$887$	−1.82142e7	−0.777322	−0.388661	−	0.921381i	$-0.627062\pi$
$887$	−1.82142e7	−0.777322	−0.388661	+	0.921381i	$0.627062\pi$
$888$	2.64758e7	1.12672
$889$	7.18267e6	0.304812
$890$	0	0
$891$	−1.06520e8	−4.49507
$892$	−1.02512e7	−0.431384
$893$	−3.07137e7	−1.28885
$894$	−6.02752e6	−0.252229
$895$	0	0
$896$	215293.	0.00895902
$897$	7.25615e7	3.01110
$898$	2.72805e7	1.12892
$899$	−1.28543e7	−0.530454
$900$	0	0
$901$	−291331.	−0.0119557
$902$	−1.60028e7	−0.654907
$903$	−2.41486e7	−0.985535
$904$	−2.09339e7	−0.851980
$905$	0	0
$906$	−7.15646e6	−0.289653
$907$	−3.24189e7	−1.30852	−0.654259	−	0.756271i	$-0.727019\pi$
$907$	−3.24189e7	−1.30852	−0.654259	+	0.756271i	$0.727019\pi$
$908$	6.59369e6	0.265408
$909$	−6.65793e7	−2.67257
$910$	0	0
$911$	4.17461e7	1.66655	0.833277	−	0.552855i	$-0.186462\pi$
$911$	4.17461e7	1.66655	0.833277	+	0.552855i	$0.186462\pi$
$912$	2.55209e7	1.01604
$913$	−3.22836e7	−1.28176
$914$	−5.97935e6	−0.236749
$915$	0	0
$916$	5.58985e6	0.220121
$917$	−5.70974e6	−0.224230
$918$	−1.53495e7	−0.601157
$919$	−5.75989e6	−0.224970	−0.112485	−	0.993653i	$-0.535881\pi$
$919$	−5.75989e6	−0.224970	−0.112485	+	0.993653i	$0.535881\pi$
$920$	0	0
$921$	5.63134e7	2.18757
$922$	−1.87811e7	−0.727603
$923$	2.11446e7	0.816950
$924$	−9.36596e6	−0.360888
$925$	0	0
$926$	−3.17000e6	−0.121488
$927$	−4.72297e7	−1.80516
$928$	−1.39308e7	−0.531015
$929$	−3.14157e6	−0.119428	−0.0597142	−	0.998216i	$-0.519019\pi$
$929$	−3.14157e6	−0.119428	−0.0597142	+	0.998216i	$0.519019\pi$
$930$	0	0
$931$	3.06544e7	1.15909
$932$	1.01456e7	0.382594
$933$	−7.97188e7	−2.99817
$934$	−8.41362e6	−0.315585
$935$	0	0
$936$	6.57944e7	2.45470
$937$	−2.19517e7	−0.816808	−0.408404	−	0.912801i	$-0.633915\pi$
$937$	−2.19517e7	−0.816808	−0.408404	+	0.912801i	$0.633915\pi$
$938$	4.15999e6	0.154378
$939$	−1.72759e7	−0.639408
$940$	0	0
$941$	580958.	0.0213880	0.0106940	−	0.999943i	$-0.496596\pi$
$941$	580958.	0.0213880	0.0106940	+	0.999943i	$0.496596\pi$
$942$	−8.27321e6	−0.303771
$943$	−3.48010e7	−1.27442
$944$	−202720.	−0.00740401
$945$	0	0
$946$	−3.81680e7	−1.38666
$947$	−1.94273e7	−0.703945	−0.351972	−	0.936010i	$-0.614489\pi$
$947$	−1.94273e7	−0.703945	−0.351972	+	0.936010i	$0.614489\pi$
$948$	−8.94798e6	−0.323373
$949$	2.72857e7	0.983488
$950$	0	0
$951$	2.27927e7	0.817232
$952$	−2.59017e6	−0.0926267
$953$	5.06290e7	1.80579	0.902895	−	0.429860i	$-0.141437\pi$
$953$	5.06290e7	1.80579	0.902895	+	0.429860i	$0.141437\pi$
$954$	2.83951e6	0.101012
$955$	0	0
$956$	5.69448e6	0.201516
$957$	4.67698e7	1.65077
$958$	−1.38671e7	−0.488172
$959$	−2.26158e6	−0.0794081
$960$	0	0
$961$	−1.13471e7	−0.396349
$962$	9.95851e6	0.346942
$963$	1.15096e8	3.99941
$964$	−2.86909e6	−0.0994377
$965$	0	0
$966$	2.78422e7	0.959975
$967$	−2.24825e7	−0.773175	−0.386587	−	0.922253i	$-0.626346\pi$
$967$	−2.24825e7	−0.773175	−0.386587	+	0.922253i	$0.626346\pi$
$968$	−1.83272e7	−0.628647
$969$	1.80507e7	0.617568
$970$	0	0
$971$	−2.03554e7	−0.692836	−0.346418	−	0.938080i	$-0.612602\pi$
$971$	−2.03554e7	−0.692836	−0.346418	+	0.938080i	$0.612602\pi$
$972$	−4.49304e7	−1.52537
$973$	−1.91228e6	−0.0647545
$974$	−2.43449e7	−0.822263
$975$	0	0
$976$	5.33069e6	0.179126
$977$	1.70028e7	0.569880	0.284940	−	0.958545i	$-0.408026\pi$
$977$	1.70028e7	0.569880	0.284940	+	0.958545i	$0.408026\pi$
$978$	−7.67699e7	−2.56652
$979$	7.31968e7	2.44082
$980$	0	0
$981$	−1.09596e7	−0.363599
$982$	1.24686e6	0.0412610
$983$	−4.26423e7	−1.40753	−0.703763	−	0.710435i	$-0.748498\pi$
$983$	−4.26423e7	−1.40753	−0.703763	+	0.710435i	$0.748498\pi$
$984$	−4.32580e7	−1.42423
$985$	0	0
$986$	3.84153e6	0.125838
$987$	−2.02303e7	−0.661012
$988$	−1.44577e7	−0.471201
$989$	−8.30031e7	−2.69838
$990$	0	0
$991$	−2.07849e7	−0.672302	−0.336151	−	0.941808i	$-0.609125\pi$
$991$	−2.07849e7	−0.672302	−0.336151	+	0.941808i	$0.609125\pi$
$992$	1.87294e7	0.604290
$993$	−2.74326e7	−0.882863
$994$	8.11327e6	0.260453
$995$	0	0
$996$	−2.59188e7	−0.827879
$997$	2.54002e6	0.0809282	0.0404641	−	0.999181i	$-0.487116\pi$
$997$	2.54002e6	0.0809282	0.0404641	+	0.999181i	$0.487116\pi$
$998$	3.38510e7	1.07583
$999$	−5.57749e7	−1.76818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.6.a.d.1.3		5
5.4	even	2		85.6.a.a.1.3	✓	5
15.14	odd	2		765.6.a.g.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.6.a.a.1.3	✓	5	5.4	even	2
425.6.a.d.1.3		5	1.1	even	1	trivial
765.6.a.g.1.3		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)

\(f(q)\)	\(=\)	\(q+4.29890 q^{2} +29.9706 q^{3} -13.5195 q^{4} +128.840 q^{6} -45.8012 q^{7} -195.684 q^{8} +655.235 q^{9} -504.686 q^{11} -405.186 q^{12} -513.142 q^{13} -196.895 q^{14} -408.601 q^{16} -289.000 q^{17} +2816.79 q^{18} -2084.02 q^{19} -1372.69 q^{21} -2169.60 q^{22} -4718.18 q^{23} -5864.75 q^{24} -2205.94 q^{26} +12354.9 q^{27} +619.207 q^{28} -3092.07 q^{29} +4157.17 q^{31} +4505.34 q^{32} -15125.7 q^{33} -1242.38 q^{34} -8858.42 q^{36} -4514.40 q^{37} -8958.99 q^{38} -15379.1 q^{39} +7375.94 q^{41} -5901.04 q^{42} +17592.2 q^{43} +6823.09 q^{44} -20283.0 q^{46} +14737.7 q^{47} -12246.0 q^{48} -14709.3 q^{49} -8661.49 q^{51} +6937.40 q^{52} +1008.07 q^{53} +53112.5 q^{54} +8962.53 q^{56} -62459.2 q^{57} -13292.5 q^{58} +496.132 q^{59} -13046.2 q^{61} +17871.2 q^{62} -30010.5 q^{63} +32443.2 q^{64} -65024.0 q^{66} -21128.0 q^{67} +3907.12 q^{68} -141406. q^{69} -41206.2 q^{71} -128219. q^{72} -53173.8 q^{73} -19406.9 q^{74} +28174.8 q^{76} +23115.2 q^{77} -66113.4 q^{78} +22083.6 q^{79} +211061. q^{81} +31708.4 q^{82} +63967.7 q^{83} +18558.0 q^{84} +75627.1 q^{86} -92671.1 q^{87} +98758.8 q^{88} -145034. q^{89} +23502.5 q^{91} +63787.2 q^{92} +124593. q^{93} +63356.1 q^{94} +135028. q^{96} +131574. q^{97} -63233.6 q^{98} -330688. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{2} + 36 q^{3} + 43 q^{4} - 105 q^{6} + 204 q^{7} + 63 q^{8} + 531 q^{9} - 792 q^{11} - 785 q^{12} - 88 q^{13} + 860 q^{14} - 2365 q^{16} - 1445 q^{17} + 2052 q^{18} - 5160 q^{19} - 6428 q^{21}+ \cdots - 535112 q^{99}+O(q^{100}) \) 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

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