LMFDB - Embedded newform 425.6.a.d.1.5

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			1.5
Root			$8.31248$ 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)

$f(q)$	$=$	\(q+9.31248 q^{2} -17.0184 q^{3} +54.7222 q^{4} -158.484 q^{6} +147.923 q^{7} +211.600 q^{8} +46.6264 q^{9} -351.681 q^{11} -931.285 q^{12} -334.042 q^{13} +1377.53 q^{14} +219.408 q^{16} -289.000 q^{17} +434.208 q^{18} +272.021 q^{19} -2517.42 q^{21} -3275.02 q^{22} +2934.77 q^{23} -3601.09 q^{24} -3110.76 q^{26} +3341.97 q^{27} +8094.70 q^{28} -3709.94 q^{29} -8038.46 q^{31} -4727.96 q^{32} +5985.05 q^{33} -2691.31 q^{34} +2551.50 q^{36} -12788.2 q^{37} +2533.19 q^{38} +5684.86 q^{39} -8814.73 q^{41} -23443.4 q^{42} +3996.06 q^{43} -19244.7 q^{44} +27330.0 q^{46} +17425.9 q^{47} -3733.98 q^{48} +5074.35 q^{49} +4918.32 q^{51} -18279.5 q^{52} -2247.33 q^{53} +31122.0 q^{54} +31300.6 q^{56} -4629.37 q^{57} -34548.7 q^{58} -44941.1 q^{59} -5.42801 q^{61} -74857.9 q^{62} +6897.15 q^{63} -51050.1 q^{64} +55735.6 q^{66} -12642.5 q^{67} -15814.7 q^{68} -49945.2 q^{69} -20430.2 q^{71} +9866.15 q^{72} -35431.0 q^{73} -119090. q^{74} +14885.6 q^{76} -52021.8 q^{77} +52940.1 q^{78} +51363.2 q^{79} -68205.2 q^{81} -82086.9 q^{82} +28251.5 q^{83} -137759. q^{84} +37213.3 q^{86} +63137.3 q^{87} -74415.6 q^{88} -84586.2 q^{89} -49412.6 q^{91} +160597. q^{92} +136802. q^{93} +162278. q^{94} +80462.4 q^{96} +33426.9 q^{97} +47254.7 q^{98} -16397.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	9.31248	1.64623	0.823114	−	0.567876i	$-0.192235\pi$
$2$	9.31248	1.64623	0.823114	+	0.567876i	$0.192235\pi$
$3$	−17.0184	−1.09173	−0.545866	−	0.837873i	$-0.683799\pi$
$3$	−17.0184	−1.09173	−0.545866	+	0.837873i	$0.683799\pi$
$4$	54.7222	1.71007
$5$	0	0
$5$	0	0
$6$	−158.484	−1.79724
$7$	147.923	1.14102	0.570508	−	0.821292i	$-0.306746\pi$
$7$	147.923	1.14102	0.570508	+	0.821292i	$0.306746\pi$
$8$	211.600	1.16894
$9$	46.6264	0.191878
$10$	0	0
$11$	−351.681	−0.876328	−0.438164	−	0.898895i	$-0.644371\pi$
$11$	−351.681	−0.876328	−0.438164	+	0.898895i	$0.644371\pi$
$12$	−931.285	−1.86694
$13$	−334.042	−0.548204	−0.274102	−	0.961701i	$-0.588381\pi$
$13$	−334.042	−0.548204	−0.274102	+	0.961701i	$0.588381\pi$
$14$	1377.53	1.87837
$15$	0	0
$16$	219.408	0.214266
$17$	−289.000	−0.242536
$17$	−289.000	−0.242536
$18$	434.208	0.315876
$19$	272.021	0.172870	0.0864348	−	0.996258i	$-0.472453\pi$
$19$	272.021	0.172870	0.0864348	+	0.996258i	$0.472453\pi$
$20$	0	0
$21$	−2517.42	−1.24568
$22$	−3275.02	−1.44264
$23$	2934.77	1.15679	0.578395	−	0.815757i	$-0.303679\pi$
$23$	2934.77	1.15679	0.578395	+	0.815757i	$0.303679\pi$
$24$	−3601.09	−1.27616
$25$	0	0
$26$	−3110.76	−0.902469
$27$	3341.97	0.882252
$28$	8094.70	1.95122
$29$	−3709.94	−0.819167	−0.409583	−	0.912273i	$-0.634326\pi$
$29$	−3709.94	−0.819167	−0.409583	+	0.912273i	$0.634326\pi$
$30$	0	0
$31$	−8038.46	−1.50234	−0.751170	−	0.660108i	$-0.770510\pi$
$31$	−8038.46	−1.50234	−0.751170	+	0.660108i	$0.770510\pi$
$32$	−4727.96	−0.816205
$33$	5985.05	0.956715
$34$	−2691.31	−0.399269
$35$	0	0
$36$	2551.50	0.328125
$37$	−12788.2	−1.53569	−0.767847	−	0.640634i	$-0.778672\pi$
$37$	−12788.2	−1.53569	−0.767847	+	0.640634i	$0.778672\pi$
$38$	2533.19	0.284583
$39$	5684.86	0.598492
$40$	0	0
$41$	−8814.73	−0.818935	−0.409467	−	0.912325i	$-0.634285\pi$
$41$	−8814.73	−0.818935	−0.409467	+	0.912325i	$0.634285\pi$
$42$	−23443.4	−2.05068
$43$	3996.06	0.329580	0.164790	−	0.986329i	$-0.447305\pi$
$43$	3996.06	0.329580	0.164790	+	0.986329i	$0.447305\pi$
$44$	−19244.7	−1.49858
$45$	0	0
$46$	27330.0	1.90434
$47$	17425.9	1.15067	0.575334	−	0.817919i	$-0.304872\pi$
$47$	17425.9	1.15067	0.575334	+	0.817919i	$0.304872\pi$
$48$	−3733.98	−0.233921
$49$	5074.35	0.301919
$50$	0	0
$51$	4918.32	0.264784
$52$	−18279.5	−0.937467
$53$	−2247.33	−0.109895	−0.0549474	−	0.998489i	$-0.517499\pi$
$53$	−2247.33	−0.109895	−0.0549474	+	0.998489i	$0.517499\pi$
$54$	31122.0	1.45239
$55$	0	0
$56$	31300.6	1.33377
$57$	−4629.37	−0.188727
$58$	−34548.7	−1.34854
$59$	−44941.1	−1.68079	−0.840395	−	0.541974i	$-0.817677\pi$
$59$	−44941.1	−1.68079	−0.840395	+	0.541974i	$0.817677\pi$
$60$	0	0
$61$	−5.42801	−0.000186774 0	−9.33869e−5	−	1.00000i	$-0.500030\pi$
$61$	−5.42801	−0.000186774 0	−9.33869e−5		1.00000i	$0.500030\pi$
$62$	−74857.9	−2.47320
$63$	6897.15	0.218936
$64$	−51050.1	−1.55793
$65$	0	0
$66$	55735.6	1.57497
$67$	−12642.5	−0.344071	−0.172035	−	0.985091i	$-0.555034\pi$
$67$	−12642.5	−0.344071	−0.172035	+	0.985091i	$0.555034\pi$
$68$	−15814.7	−0.414753
$69$	−49945.2	−1.26290
$70$	0	0
$71$	−20430.2	−0.480980	−0.240490	−	0.970652i	$-0.577308\pi$
$71$	−20430.2	−0.480980	−0.240490	+	0.970652i	$0.577308\pi$
$72$	9866.15	0.224293
$73$	−35431.0	−0.778174	−0.389087	−	0.921201i	$-0.627209\pi$
$73$	−35431.0	−0.778174	−0.389087	+	0.921201i	$0.627209\pi$
$74$	−119090.	−2.52810
$75$	0	0
$76$	14885.6	0.295619
$77$	−52021.8	−0.999905
$78$	52940.1	0.985255
$79$	51363.2	0.925943	0.462972	−	0.886373i	$-0.346783\pi$
$79$	51363.2	0.925943	0.462972	+	0.886373i	$0.346783\pi$
$80$	0	0
$81$	−68205.2	−1.15506
$82$	−82086.9	−1.34815
$83$	28251.5	0.450139	0.225070	−	0.974343i	$-0.427739\pi$
$83$	28251.5	0.450139	0.225070	+	0.974343i	$0.427739\pi$
$84$	−137759.	−2.13021
$85$	0	0
$86$	37213.3	0.542565
$87$	63137.3	0.894310
$88$	−74415.6	−1.02437
$89$	−84586.2	−1.13194	−0.565972	−	0.824425i	$-0.691499\pi$
$89$	−84586.2	−1.13194	−0.565972	+	0.824425i	$0.691499\pi$
$90$	0	0
$91$	−49412.6	−0.625510
$92$	160597.	1.97819
$93$	136802.	1.64015
$94$	162278.	1.89426
$95$	0	0
$96$	80462.4	0.891077
$97$	33426.9	0.360717	0.180358	−	0.983601i	$-0.442274\pi$
$97$	33426.9	0.360717	0.180358	+	0.983601i	$0.442274\pi$
$98$	47254.7	0.497027
$99$	−16397.6	−0.168148
$100$	0	0
$101$	−75053.8	−0.732098	−0.366049	−	0.930596i	$-0.619290\pi$
$101$	−75053.8	−0.732098	−0.366049	+	0.930596i	$0.619290\pi$
$102$	45801.8	0.435895
$103$	−182702.	−1.69687	−0.848437	−	0.529297i	$-0.822456\pi$
$103$	−182702.	−1.69687	−0.848437	+	0.529297i	$0.822456\pi$
$104$	−70683.2	−0.640815
$105$	0	0
$106$	−20928.2	−0.180912
$107$	−70251.2	−0.593191	−0.296595	−	0.955003i	$-0.595851\pi$
$107$	−70251.2	−0.593191	−0.296595	+	0.955003i	$0.595851\pi$
$108$	182880.	1.50871
$109$	182369.	1.47023	0.735114	−	0.677944i	$-0.237129\pi$
$109$	182369.	1.47023	0.735114	+	0.677944i	$0.237129\pi$
$110$	0	0
$111$	217635.	1.67657
$112$	32455.6	0.244481
$113$	−198318.	−1.46106	−0.730528	−	0.682883i	$-0.760726\pi$
$113$	−198318.	−1.46106	−0.730528	+	0.682883i	$0.760726\pi$
$114$	−43110.9	−0.310688
$115$	0	0
$116$	−203016.	−1.40083
$117$	−15575.2	−0.105189
$118$	−418513.	−2.76697
$119$	−42749.9	−0.276737
$120$	0	0
$121$	−37371.8	−0.232049
$122$	−50.5482	−0.000307473 0
$123$	150013.	0.894057
$124$	−439882.	−2.56911
$125$	0	0
$126$	64229.5	0.360419
$127$	−131329.	−0.722522	−0.361261	−	0.932465i	$-0.617654\pi$
$127$	−131329.	−0.722522	−0.361261	+	0.932465i	$0.617654\pi$
$128$	−324108.	−1.74850
$129$	−68006.7	−0.359813
$130$	0	0
$131$	146339.	0.745043	0.372522	−	0.928024i	$-0.378493\pi$
$131$	146339.	0.745043	0.372522	+	0.928024i	$0.378493\pi$
$132$	327515.	1.63605
$133$	40238.3	0.197247
$134$	−117733.	−0.566419
$135$	0	0
$136$	−61152.4	−0.283508
$137$	−37245.9	−0.169542	−0.0847708	−	0.996400i	$-0.527016\pi$
$137$	−37245.9	−0.169542	−0.0847708	+	0.996400i	$0.527016\pi$
$138$	−465113.	−2.07903
$139$	407618.	1.78944	0.894718	−	0.446632i	$-0.147377\pi$
$139$	407618.	1.78944	0.894718	+	0.446632i	$0.147377\pi$
$140$	0	0
$141$	−296561.	−1.25622
$142$	−190256.	−0.791802
$143$	117476.	0.480407
$144$	10230.2	0.0411130
$145$	0	0
$146$	−329950.	−1.28105
$147$	−86357.4	−0.329614
$148$	−699797.	−2.62614
$149$	183343.	0.676548	0.338274	−	0.941048i	$-0.390157\pi$
$149$	183343.	0.676548	0.338274	+	0.941048i	$0.390157\pi$
$150$	0	0
$151$	−372353.	−1.32896	−0.664480	−	0.747306i	$-0.731347\pi$
$151$	−372353.	−1.32896	−0.664480	+	0.747306i	$0.731347\pi$
$152$	57559.6	0.202073
$153$	−13475.0	−0.0465373
$154$	−484452.	−1.64607
$155$	0	0
$156$	311088.	1.02346
$157$	469749.	1.52096	0.760478	−	0.649364i	$-0.224965\pi$
$157$	469749.	1.52096	0.760478	+	0.649364i	$0.224965\pi$
$158$	478319.	1.52431
$159$	38246.0	0.119976
$160$	0	0
$161$	434122.	1.31992
$162$	−635159.	−1.90149
$163$	261451.	0.770763	0.385382	−	0.922757i	$-0.374070\pi$
$163$	261451.	0.770763	0.385382	+	0.922757i	$0.374070\pi$
$164$	−482361.	−1.40043
$165$	0	0
$166$	263092.	0.741032
$167$	201294.	0.558522	0.279261	−	0.960215i	$-0.409911\pi$
$167$	201294.	0.558522	0.279261	+	0.960215i	$0.409911\pi$
$168$	−532686.	−1.45612
$169$	−259709.	−0.699472
$170$	0	0
$171$	12683.4	0.0331699
$172$	218673.	0.563605
$173$	258569.	0.656844	0.328422	−	0.944531i	$-0.393483\pi$
$173$	258569.	0.656844	0.328422	+	0.944531i	$0.393483\pi$
$174$	587965.	1.47224
$175$	0	0
$176$	−77161.6	−0.187767
$177$	764826.	1.83497
$178$	−787707.	−1.86344
$179$	843550.	1.96779	0.983894	−	0.178751i	$-0.0572057\pi$
$179$	843550.	1.96779	0.983894	+	0.178751i	$0.0572057\pi$
$180$	0	0
$181$	652818.	1.48114	0.740569	−	0.671980i	$-0.234556\pi$
$181$	652818.	1.48114	0.740569	+	0.671980i	$0.234556\pi$
$182$	−460154.	−1.02973
$183$	92.3762	0.000203907 0
$184$	620997.	1.35221
$185$	0	0
$186$	1.27396e6	2.70007
$187$	101636.	0.212541
$188$	953582.	1.96772
$189$	494355.	1.00666
$190$	0	0
$191$	306162.	0.607251	0.303625	−	0.952791i	$-0.401803\pi$
$191$	306162.	0.607251	0.303625	+	0.952791i	$0.401803\pi$
$192$	868792.	1.70084
$193$	764545.	1.47744	0.738720	−	0.674013i	$-0.235431\pi$
$193$	764545.	1.47744	0.738720	+	0.674013i	$0.235431\pi$
$194$	311287.	0.593822
$195$	0	0
$196$	277679.	0.516302
$197$	−304807.	−0.559576	−0.279788	−	0.960062i	$-0.590264\pi$
$197$	−304807.	−0.559576	−0.279788	+	0.960062i	$0.590264\pi$
$198$	−152702.	−0.276811
$199$	−1.01589e6	−1.81850	−0.909248	−	0.416254i	$-0.863343\pi$
$199$	−1.01589e6	−1.81850	−0.909248	+	0.416254i	$0.863343\pi$
$200$	0	0
$201$	215156.	0.375633
$202$	−698936.	−1.20520
$203$	−548788.	−0.934683
$204$	269141.	0.452799
$205$	0	0
$206$	−1.70140e6	−2.79344
$207$	136838.	0.221963
$208$	−73291.5	−0.117461
$209$	−95664.5	−0.151490
$210$	0	0
$211$	903449.	1.39700	0.698502	−	0.715608i	$-0.253850\pi$
$211$	903449.	1.39700	0.698502	+	0.715608i	$0.253850\pi$
$212$	−122979.	−0.187928
$213$	347690.	0.525101
$214$	−654212.	−0.976527
$215$	0	0
$216$	707160.	1.03130
$217$	−1.18908e6	−1.71420
$218$	1.69831e6	2.42033
$219$	602980.	0.849557
$220$	0	0
$221$	96538.1	0.132959
$222$	2.02672e6	2.76001
$223$	66712.7	0.0898351	0.0449176	−	0.998991i	$-0.485697\pi$
$223$	66712.7	0.0898351	0.0449176	+	0.998991i	$0.485697\pi$
$224$	−699377.	−0.931303
$225$	0	0
$226$	−1.84683e6	−2.40523
$227$	1.33859e6	1.72418	0.862090	−	0.506756i	$-0.169155\pi$
$227$	1.33859e6	1.72418	0.862090	+	0.506756i	$0.169155\pi$
$228$	−253329.	−0.322736
$229$	−268798.	−0.338717	−0.169358	−	0.985555i	$-0.554170\pi$
$229$	−268798.	−0.338717	−0.169358	+	0.985555i	$0.554170\pi$
$230$	0	0
$231$	885329.	1.09163
$232$	−785023.	−0.957552
$233$	883834.	1.06655	0.533274	−	0.845942i	$-0.320961\pi$
$233$	883834.	1.06655	0.533274	+	0.845942i	$0.320961\pi$
$234$	−145043.	−0.173164
$235$	0	0
$236$	−2.45927e6	−2.87427
$237$	−874120.	−1.01088
$238$	−398107.	−0.455573
$239$	−188258.	−0.213186	−0.106593	−	0.994303i	$-0.533994\pi$
$239$	−188258.	−0.213186	−0.106593	+	0.994303i	$0.533994\pi$
$240$	0	0
$241$	1.33170e6	1.47694	0.738472	−	0.674284i	$-0.235548\pi$
$241$	1.33170e6	1.47694	0.738472	+	0.674284i	$0.235548\pi$
$242$	−348024.	−0.382006
$243$	348647.	0.378765
$244$	−297.033	−0.000319396 0
$245$	0	0
$246$	1.39699e6	1.47182
$247$	−90866.4	−0.0947678
$248$	−1.70094e6	−1.75614
$249$	−480796.	−0.491431
$250$	0	0
$251$	456339.	0.457197	0.228598	−	0.973521i	$-0.426586\pi$
$251$	456339.	0.457197	0.228598	+	0.973521i	$0.426586\pi$
$252$	377427.	0.374396
$253$	−1.03210e6	−1.01373
$254$	−1.22300e6	−1.18944
$255$	0	0
$256$	−1.38464e6	−1.32050
$257$	444163.	0.419479	0.209739	−	0.977757i	$-0.432738\pi$
$257$	444163.	0.419479	0.209739	+	0.977757i	$0.432738\pi$
$258$	−633311.	−0.592335
$259$	−1.89167e6	−1.75225
$260$	0	0
$261$	−172981.	−0.157180
$262$	1.36278e6	1.22651
$263$	−1.10033e6	−0.980918	−0.490459	−	0.871464i	$-0.663171\pi$
$263$	−1.10033e6	−0.980918	−0.490459	+	0.871464i	$0.663171\pi$
$264$	1.26643e6	1.11834
$265$	0	0
$266$	374718.	0.324714
$267$	1.43952e6	1.23578
$268$	−691828.	−0.588384
$269$	1.50044e6	1.26427	0.632133	−	0.774860i	$-0.282180\pi$
$269$	1.50044e6	1.26427	0.632133	+	0.774860i	$0.282180\pi$
$270$	0	0
$271$	−1.63387e6	−1.35143	−0.675714	−	0.737164i	$-0.736165\pi$
$271$	−1.63387e6	−1.35143	−0.675714	+	0.737164i	$0.736165\pi$
$272$	−63409.0	−0.0519671
$273$	840924.	0.682889
$274$	−346851.	−0.279104
$275$	0	0
$276$	−2.73311e6	−2.15965
$277$	−2.07998e6	−1.62877	−0.814387	−	0.580323i	$-0.802926\pi$
$277$	−2.07998e6	−1.62877	−0.814387	+	0.580323i	$0.802926\pi$
$278$	3.79593e6	2.94582
$279$	−374805.	−0.288267
$280$	0	0
$281$	−1.52771e6	−1.15418	−0.577091	−	0.816680i	$-0.695812\pi$
$281$	−1.52771e6	−1.15418	−0.577091	+	0.816680i	$0.695812\pi$
$282$	−2.76171e6	−2.06803
$283$	293520.	0.217857	0.108929	−	0.994050i	$-0.465258\pi$
$283$	293520.	0.217857	0.108929	+	0.994050i	$0.465258\pi$
$284$	−1.11798e6	−0.822508
$285$	0	0
$286$	1.09399e6	0.790859
$287$	−1.30391e6	−0.934418
$288$	−220448.	−0.156612
$289$	83521.0	0.0588235
$290$	0	0
$291$	−568872.	−0.393806
$292$	−1.93886e6	−1.33073
$293$	−143236.	−0.0974728	−0.0487364	−	0.998812i	$-0.515519\pi$
$293$	−143236.	−0.0974728	−0.0487364	+	0.998812i	$0.515519\pi$
$294$	−804201.	−0.542621
$295$	0	0
$296$	−2.70598e6	−1.79513
$297$	−1.17530e6	−0.773142
$298$	1.70738e6	1.11375
$299$	−980336.	−0.634157
$300$	0	0
$301$	591112.	0.376057
$302$	−3.46753e6	−2.18777
$303$	1.27730e6	0.799254
$304$	59683.6	0.0370400
$305$	0	0
$306$	−125486.	−0.0766111
$307$	−2.44074e6	−1.47801	−0.739003	−	0.673702i	$-0.764703\pi$
$307$	−2.44074e6	−1.47801	−0.739003	+	0.673702i	$0.764703\pi$
$308$	−2.84675e6	−1.70991
$309$	3.10929e6	1.85253
$310$	0	0
$311$	−2.52347e6	−1.47944	−0.739721	−	0.672914i	$-0.765043\pi$
$311$	−2.52347e6	−1.47944	−0.739721	+	0.672914i	$0.765043\pi$
$312$	1.20292e6	0.699598
$313$	2.73286e6	1.57673	0.788364	−	0.615210i	$-0.210929\pi$
$313$	2.73286e6	1.57673	0.788364	+	0.615210i	$0.210929\pi$
$314$	4.37452e6	2.50384
$315$	0	0
$316$	2.81071e6	1.58343
$317$	−3.03021e6	−1.69366	−0.846828	−	0.531867i	$-0.821491\pi$
$317$	−3.03021e6	−1.69366	−0.846828	+	0.531867i	$0.821491\pi$
$318$	356165.	0.197507
$319$	1.30471e6	0.717859
$320$	0	0
$321$	1.19556e6	0.647605
$322$	4.04275e6	2.17289
$323$	−78614.1	−0.0419270
$324$	−3.73234e6	−1.97523
$325$	0	0
$326$	2.43475e6	1.26885
$327$	−3.10363e6	−1.60509
$328$	−1.86520e6	−0.957282
$329$	2.57769e6	1.31293
$330$	0	0
$331$	−2.45111e6	−1.22968	−0.614841	−	0.788651i	$-0.710780\pi$
$331$	−2.45111e6	−1.22968	−0.614841	+	0.788651i	$0.710780\pi$
$332$	1.54599e6	0.769769
$333$	−596267.	−0.294666
$334$	1.87455e6	0.919455
$335$	0	0
$336$	−552343.	−0.266907
$337$	−3.43996e6	−1.64998	−0.824990	−	0.565148i	$-0.808819\pi$
$337$	−3.43996e6	−1.64998	−0.824990	+	0.565148i	$0.808819\pi$
$338$	−2.41853e6	−1.15149
$339$	3.37506e6	1.59508
$340$	0	0
$341$	2.82697e6	1.31654
$342$	118114.	0.0546053
$343$	−1.73553e6	−0.796522
$344$	845567.	0.385258
$345$	0	0
$346$	2.40792e6	1.08131
$347$	3.25247e6	1.45007	0.725036	−	0.688711i	$-0.241823\pi$
$347$	3.25247e6	1.45007	0.725036	+	0.688711i	$0.241823\pi$
$348$	3.45501e6	1.52933
$349$	−1.81765e6	−0.798817	−0.399408	−	0.916773i	$-0.630784\pi$
$349$	−1.81765e6	−0.798817	−0.399408	+	0.916773i	$0.630784\pi$
$350$	0	0
$351$	−1.11636e6	−0.483654
$352$	1.66273e6	0.715263
$353$	1.73961e6	0.743047	0.371523	−	0.928424i	$-0.378836\pi$
$353$	1.73961e6	0.743047	0.371523	+	0.928424i	$0.378836\pi$
$354$	7.12242e6	3.02078
$355$	0	0
$356$	−4.62874e6	−1.93570
$357$	727535.	0.302123
$358$	7.85554e6	3.23943
$359$	270669.	0.110841	0.0554207	−	0.998463i	$-0.482350\pi$
$359$	270669.	0.110841	0.0554207	+	0.998463i	$0.482350\pi$
$360$	0	0
$361$	−2.40210e6	−0.970116
$362$	6.07935e6	2.43829
$363$	636008.	0.253336
$364$	−2.70397e6	−1.06967
$365$	0	0
$366$	860.251	0.000335678 0
$367$	3.44973e6	1.33697	0.668483	−	0.743728i	$-0.266944\pi$
$367$	3.44973e6	1.33697	0.668483	+	0.743728i	$0.266944\pi$
$368$	643913.	0.247861
$369$	−411000.	−0.157136
$370$	0	0
$371$	−332433.	−0.125392
$372$	7.48609e6	2.80477
$373$	−2.14205e6	−0.797182	−0.398591	−	0.917129i	$-0.630501\pi$
$373$	−2.14205e6	−0.797182	−0.398591	+	0.917129i	$0.630501\pi$
$374$	946480.	0.349891
$375$	0	0
$376$	3.68731e6	1.34506
$377$	1.23928e6	0.449071
$378$	4.60367e6	1.65720
$379$	−4.13431e6	−1.47844	−0.739222	−	0.673462i	$-0.764807\pi$
$379$	−4.13431e6	−1.47844	−0.739222	+	0.673462i	$0.764807\pi$
$380$	0	0
$381$	2.23501e6	0.788800
$382$	2.85113e6	0.999674
$383$	3.48669e6	1.21455	0.607276	−	0.794491i	$-0.292262\pi$
$383$	3.48669e6	1.21455	0.607276	+	0.794491i	$0.292262\pi$
$384$	5.51580e6	1.90889
$385$	0	0
$386$	7.11980e6	2.43220
$387$	186322.	0.0632394
$388$	1.82919e6	0.616850
$389$	241098.	0.0807830	0.0403915	−	0.999184i	$-0.487139\pi$
$389$	241098.	0.0807830	0.0403915	+	0.999184i	$0.487139\pi$
$390$	0	0
$391$	−848149.	−0.280563
$392$	1.07373e6	0.352923
$393$	−2.49046e6	−0.813387
$394$	−2.83850e6	−0.921189
$395$	0	0
$396$	−897314.	−0.287545
$397$	−1.48978e6	−0.474401	−0.237201	−	0.971461i	$-0.576230\pi$
$397$	−1.48978e6	−0.474401	−0.237201	+	0.971461i	$0.576230\pi$
$398$	−9.46042e6	−2.99366
$399$	−684792.	−0.215341
$400$	0	0
$401$	−1.32010e6	−0.409964	−0.204982	−	0.978766i	$-0.565714\pi$
$401$	−1.32010e6	−0.409964	−0.204982	+	0.978766i	$0.565714\pi$
$402$	2.00364e6	0.618377
$403$	2.68518e6	0.823589
$404$	−4.10711e6	−1.25194
$405$	0	0
$406$	−5.11057e6	−1.53870
$407$	4.49735e6	1.34577
$408$	1.04072e6	0.309515
$409$	−3.51850e6	−1.04004	−0.520020	−	0.854154i	$-0.674075\pi$
$409$	−3.51850e6	−1.04004	−0.520020	+	0.854154i	$0.674075\pi$
$410$	0	0
$411$	633865.	0.185094
$412$	−9.99784e6	−2.90177
$413$	−6.64784e6	−1.91781
$414$	1.27430e6	0.365402
$415$	0	0
$416$	1.57934e6	0.447447
$417$	−6.93701e6	−1.95358
$418$	−890873.	−0.249388
$419$	5.14824e6	1.43260	0.716298	−	0.697795i	$-0.245835\pi$
$419$	5.14824e6	1.43260	0.716298	+	0.697795i	$0.245835\pi$
$420$	0	0
$421$	−774383.	−0.212937	−0.106468	−	0.994316i	$-0.533954\pi$
$421$	−774383.	−0.212937	−0.106468	+	0.994316i	$0.533954\pi$
$422$	8.41335e6	2.29979
$423$	812506.	0.220788
$424$	−475534.	−0.128460
$425$	0	0
$426$	3.23785e6	0.864436
$427$	−802.930	−0.000213112 0
$428$	−3.84430e6	−1.01440
$429$	−1.99926e6	−0.524475
$430$	0	0
$431$	−1.27633e6	−0.330955	−0.165478	−	0.986214i	$-0.552917\pi$
$431$	−1.27633e6	−0.330955	−0.165478	+	0.986214i	$0.552917\pi$
$432$	733255.	0.189036
$433$	953054.	0.244286	0.122143	−	0.992513i	$-0.461023\pi$
$433$	953054.	0.244286	0.122143	+	0.992513i	$0.461023\pi$
$434$	−1.10732e7	−2.82196
$435$	0	0
$436$	9.97963e6	2.51419
$437$	798320.	0.199974
$438$	5.61523e6	1.39856
$439$	4.69271e6	1.16215	0.581075	−	0.813850i	$-0.302632\pi$
$439$	4.69271e6	1.16215	0.581075	+	0.813850i	$0.302632\pi$
$440$	0	0
$441$	236599.	0.0579317
$442$	899008.	0.218881
$443$	2.78578e6	0.674432	0.337216	−	0.941427i	$-0.390515\pi$
$443$	2.78578e6	0.674432	0.337216	+	0.941427i	$0.390515\pi$
$444$	1.19094e7	2.86704
$445$	0	0
$446$	621260.	0.147889
$447$	−3.12021e6	−0.738609
$448$	−7.55151e6	−1.77762
$449$	−1.99708e6	−0.467498	−0.233749	−	0.972297i	$-0.575099\pi$
$449$	−1.99708e6	−0.467498	−0.233749	+	0.972297i	$0.575099\pi$
$450$	0	0
$451$	3.09997e6	0.717655
$452$	−1.08524e7	−2.49850
$453$	6.33685e6	1.45087
$454$	1.24656e7	2.83839
$455$	0	0
$456$	−979573.	−0.220610
$457$	4.17496e6	0.935109	0.467555	−	0.883964i	$-0.345135\pi$
$457$	4.17496e6	0.935109	0.467555	+	0.883964i	$0.345135\pi$
$458$	−2.50317e6	−0.557605
$459$	−965828.	−0.213978
$460$	0	0
$461$	1.69931e6	0.372408	0.186204	−	0.982511i	$-0.440381\pi$
$461$	1.69931e6	0.372408	0.186204	+	0.982511i	$0.440381\pi$
$462$	8.24460e6	1.79707
$463$	2.09596e6	0.454392	0.227196	−	0.973849i	$-0.427044\pi$
$463$	2.09596e6	0.454392	0.227196	+	0.973849i	$0.427044\pi$
$464$	−813992.	−0.175519
$465$	0	0
$466$	8.23068e6	1.75578
$467$	−622138.	−0.132006	−0.0660032	−	0.997819i	$-0.521025\pi$
$467$	−622138.	−0.132006	−0.0660032	+	0.997819i	$0.521025\pi$
$468$	−852308.	−0.179880
$469$	−1.87013e6	−0.392590
$470$	0	0
$471$	−7.99438e6	−1.66048
$472$	−9.50953e6	−1.96473
$473$	−1.40534e6	−0.288821
$474$	−8.14022e6	−1.66414
$475$	0	0
$476$	−2.33937e6	−0.473240
$477$	−104785.	−0.0210864
$478$	−1.75315e6	−0.350953
$479$	−5.60503e6	−1.11619	−0.558096	−	0.829776i	$-0.688468\pi$
$479$	−5.60503e6	−1.11619	−0.558096	+	0.829776i	$0.688468\pi$
$480$	0	0
$481$	4.27179e6	0.841873
$482$	1.24014e7	2.43139
$483$	−7.38806e6	−1.44100
$484$	−2.04506e6	−0.396820
$485$	0	0
$486$	3.24676e6	0.623534
$487$	−4.17152e6	−0.797025	−0.398512	−	0.917163i	$-0.630473\pi$
$487$	−4.17152e6	−0.797025	−0.398512	+	0.917163i	$0.630473\pi$
$488$	−1148.57	−0.000218327 0
$489$	−4.44948e6	−0.841467
$490$	0	0
$491$	−9.70533e6	−1.81680	−0.908398	−	0.418106i	$-0.862694\pi$
$491$	−9.70533e6	−1.81680	−0.908398	+	0.418106i	$0.862694\pi$
$492$	8.20902e6	1.52890
$493$	1.07217e6	0.198677
$494$	−846191.	−0.156009
$495$	0	0
$496$	−1.76370e6	−0.321900
$497$	−3.02210e6	−0.548806
$498$	−4.47740e6	−0.809008
$499$	874420.	0.157206	0.0786029	−	0.996906i	$-0.474954\pi$
$499$	874420.	0.157206	0.0786029	+	0.996906i	$0.474954\pi$
$500$	0	0
$501$	−3.42571e6	−0.609756
$502$	4.24964e6	0.752650
$503$	2.98048e6	0.525250	0.262625	−	0.964898i	$-0.415412\pi$
$503$	2.98048e6	0.525250	0.262625	+	0.964898i	$0.415412\pi$
$504$	1.45943e6	0.255922
$505$	0	0
$506$	−9.61143e6	−1.66883
$507$	4.41984e6	0.763636
$508$	−7.18660e6	−1.23556
$509$	1.01876e7	1.74291	0.871457	−	0.490471i	$-0.163175\pi$
$509$	1.01876e7	1.74291	0.871457	+	0.490471i	$0.163175\pi$
$510$	0	0
$511$	−5.24108e6	−0.887909
$512$	−2.52301e6	−0.425347
$513$	909085.	0.152515
$514$	4.13626e6	0.690558
$515$	0	0
$516$	−3.72147e6	−0.615306
$517$	−6.12834e6	−1.00836
$518$	−1.76161e7	−2.88461
$519$	−4.40044e6	−0.717097
$520$	0	0
$521$	−7.80248e6	−1.25933	−0.629663	−	0.776868i	$-0.716807\pi$
$521$	−7.80248e6	−1.25933	−0.629663	+	0.776868i	$0.716807\pi$
$522$	−1.61089e6	−0.258755
$523$	3.39460e6	0.542668	0.271334	−	0.962485i	$-0.412535\pi$
$523$	3.39460e6	0.542668	0.271334	+	0.962485i	$0.412535\pi$
$524$	8.00798e6	1.27407
$525$	0	0
$526$	−1.02468e7	−1.61481
$527$	2.32311e6	0.364371
$528$	1.31317e6	0.204991
$529$	2.17654e6	0.338164
$530$	0	0
$531$	−2.09544e6	−0.322507
$532$	2.20193e6	0.337306
$533$	2.94449e6	0.448943
$534$	1.34055e7	2.03437
$535$	0	0
$536$	−2.67516e6	−0.402196
$537$	−1.43559e7	−2.14830
$538$	1.39728e7	2.08127
$539$	−1.78455e6	−0.264580
$540$	0	0
$541$	2.63412e6	0.386938	0.193469	−	0.981106i	$-0.438026\pi$
$541$	2.63412e6	0.386938	0.193469	+	0.981106i	$0.438026\pi$
$542$	−1.52153e7	−2.22476
$543$	−1.11099e7	−1.61701
$544$	1.36638e6	0.197959
$545$	0	0
$546$	7.83109e6	1.12419
$547$	5.43351e6	0.776447	0.388224	−	0.921565i	$-0.373089\pi$
$547$	5.43351e6	0.776447	0.388224	+	0.921565i	$0.373089\pi$
$548$	−2.03817e6	−0.289928
$549$	−253.089	−3.58379e−5 0
$550$	0	0
$551$	−1.00918e6	−0.141609
$552$	−1.05684e7	−1.47625
$553$	7.59782e6	1.05652
$554$	−1.93698e7	−2.68133
$555$	0	0
$556$	2.23057e7	3.06006
$557$	356279.	0.0486578	0.0243289	−	0.999704i	$-0.492255\pi$
$557$	356279.	0.0486578	0.0243289	+	0.999704i	$0.492255\pi$
$558$	−3.49036e6	−0.474553
$559$	−1.33485e6	−0.180677
$560$	0	0
$561$	−1.72968e6	−0.232038
$562$	−1.42267e7	−1.90005
$563$	−1.08214e7	−1.43884	−0.719422	−	0.694573i	$-0.755593\pi$
$563$	−1.08214e7	−1.43884	−0.719422	+	0.694573i	$0.755593\pi$
$564$	−1.62284e7	−2.14822
$565$	0	0
$566$	2.73340e6	0.358643
$567$	−1.00891e7	−1.31794
$568$	−4.32303e6	−0.562234
$569$	−2.53077e6	−0.327697	−0.163848	−	0.986486i	$-0.552391\pi$
$569$	−2.53077e6	−0.327697	−0.163848	+	0.986486i	$0.552391\pi$
$570$	0	0
$571$	−1.09119e7	−1.40059	−0.700296	−	0.713852i	$-0.746949\pi$
$571$	−1.09119e7	−1.40059	−0.700296	+	0.713852i	$0.746949\pi$
$572$	6.42854e6	0.821528
$573$	−5.21040e6	−0.662955
$574$	−1.21426e7	−1.53827
$575$	0	0
$576$	−2.38028e6	−0.298932
$577$	1.36386e7	1.70541	0.852707	−	0.522389i	$-0.174959\pi$
$577$	1.36386e7	1.70541	0.852707	+	0.522389i	$0.174959\pi$
$578$	777787.	0.0968370
$579$	−1.30113e7	−1.61297
$580$	0	0
$581$	4.17907e6	0.513616
$582$	−5.29761e6	−0.648295
$583$	790342.	0.0963038
$584$	−7.49720e6	−0.909634
$585$	0	0
$586$	−1.33388e6	−0.160462
$587$	6.57942e6	0.788121	0.394060	−	0.919085i	$-0.371070\pi$
$587$	6.57942e6	0.788121	0.394060	+	0.919085i	$0.371070\pi$
$588$	−4.72566e6	−0.563663
$589$	−2.18663e6	−0.259709
$590$	0	0
$591$	5.18732e6	0.610906
$592$	−2.80583e6	−0.329046
$593$	4.75136e6	0.554858	0.277429	−	0.960746i	$-0.410518\pi$
$593$	4.75136e6	0.554858	0.277429	+	0.960746i	$0.410518\pi$
$594$	−1.09450e7	−1.27277
$595$	0	0
$596$	1.00329e7	1.15694
$597$	1.72888e7	1.98531
$598$	−9.12936e6	−1.04397
$599$	−1.79172e6	−0.204034	−0.102017	−	0.994783i	$-0.532530\pi$
$599$	−1.79172e6	−0.204034	−0.102017	+	0.994783i	$0.532530\pi$
$600$	0	0
$601$	4.80312e6	0.542422	0.271211	−	0.962520i	$-0.412576\pi$
$601$	4.80312e6	0.542422	0.271211	+	0.962520i	$0.412576\pi$
$602$	5.50471e6	0.619075
$603$	−589477.	−0.0660197
$604$	−2.03760e7	−2.27261
$605$	0	0
$606$	1.18948e7	1.31576
$607$	−7.35442e6	−0.810171	−0.405086	−	0.914279i	$-0.632758\pi$
$607$	−7.35442e6	−0.810171	−0.405086	+	0.914279i	$0.632758\pi$
$608$	−1.28611e6	−0.141097
$609$	9.33949e6	1.02042
$610$	0	0
$611$	−5.82097e6	−0.630801
$612$	−737384.	−0.0795820
$613$	−2.75267e6	−0.295872	−0.147936	−	0.988997i	$-0.547263\pi$
$613$	−2.75267e6	−0.295872	−0.147936	+	0.988997i	$0.547263\pi$
$614$	−2.27294e7	−2.43314
$615$	0	0
$616$	−1.10078e7	−1.16882
$617$	−1.49396e7	−1.57988	−0.789941	−	0.613183i	$-0.789889\pi$
$617$	−1.49396e7	−1.57988	−0.789941	+	0.613183i	$0.789889\pi$
$618$	2.89552e7	3.04969
$619$	5.61975e6	0.589509	0.294754	−	0.955573i	$-0.404762\pi$
$619$	5.61975e6	0.589509	0.294754	+	0.955573i	$0.404762\pi$
$620$	0	0
$621$	9.80791e6	1.02058
$622$	−2.34998e7	−2.43550
$623$	−1.25123e7	−1.29157
$624$	1.24730e6	0.128236
$625$	0	0
$626$	2.54497e7	2.59565
$627$	1.62806e6	0.165387
$628$	2.57057e7	2.60094
$629$	3.69578e6	0.372460
$630$	0	0
$631$	2.04112e6	0.204077	0.102039	−	0.994780i	$-0.467463\pi$
$631$	2.04112e6	0.204077	0.102039	+	0.994780i	$0.467463\pi$
$632$	1.08684e7	1.08237
$633$	−1.53753e7	−1.52515
$634$	−2.82188e7	−2.78815
$635$	0	0
$636$	2.09290e6	0.205166
$637$	−1.69504e6	−0.165513
$638$	1.21501e7	1.18176
$639$	−952587.	−0.0922896
$640$	0	0
$641$	−5.16052e6	−0.496076	−0.248038	−	0.968750i	$-0.579786\pi$
$641$	−5.16052e6	−0.496076	−0.248038	+	0.968750i	$0.579786\pi$
$642$	1.11337e7	1.06611
$643$	−1.73411e7	−1.65406	−0.827028	−	0.562161i	$-0.809970\pi$
$643$	−1.73411e7	−1.65406	−0.827028	+	0.562161i	$0.809970\pi$
$644$	2.37561e7	2.25715
$645$	0	0
$646$	−732092.	−0.0690215
$647$	2.82554e6	0.265363	0.132682	−	0.991159i	$-0.457641\pi$
$647$	2.82554e6	0.265363	0.132682	+	0.991159i	$0.457641\pi$
$648$	−1.44322e7	−1.35019
$649$	1.58049e7	1.47292
$650$	0	0
$651$	2.02362e7	1.87144
$652$	1.43072e7	1.31806
$653$	1.88867e7	1.73329	0.866647	−	0.498921i	$-0.166270\pi$
$653$	1.88867e7	1.73329	0.866647	+	0.498921i	$0.166270\pi$
$654$	−2.89025e7	−2.64235
$655$	0	0
$656$	−1.93402e6	−0.175470
$657$	−1.65202e6	−0.149315
$658$	2.40047e7	2.16138
$659$	8.56581e6	0.768343	0.384171	−	0.923262i	$-0.374487\pi$
$659$	8.56581e6	0.768343	0.384171	+	0.923262i	$0.374487\pi$
$660$	0	0
$661$	−386924.	−0.0344446	−0.0172223	−	0.999852i	$-0.505482\pi$
$661$	−386924.	−0.0344446	−0.0172223	+	0.999852i	$0.505482\pi$
$662$	−2.28259e7	−2.02434
$663$	−1.64292e6	−0.145156
$664$	5.97802e6	0.526184
$665$	0	0
$666$	−5.55273e6	−0.485088
$667$	−1.08878e7	−0.947604
$668$	1.10153e7	0.955111
$669$	−1.13534e6	−0.0980759
$670$	0	0
$671$	1908.93	0.000163675 0
$672$	1.19023e7	1.01673
$673$	1.47130e6	0.125217	0.0626087	−	0.998038i	$-0.480058\pi$
$673$	1.47130e6	0.125217	0.0626087	+	0.998038i	$0.480058\pi$
$674$	−3.20345e7	−2.71624
$675$	0	0
$676$	−1.42119e7	−1.19615
$677$	−2.82123e6	−0.236574	−0.118287	−	0.992979i	$-0.537740\pi$
$677$	−2.82123e6	−0.236574	−0.118287	+	0.992979i	$0.537740\pi$
$678$	3.14302e7	2.62587
$679$	4.94462e6	0.411584
$680$	0	0
$681$	−2.27807e7	−1.88234
$682$	2.63261e7	2.16733
$683$	−1.50304e7	−1.23287	−0.616435	−	0.787405i	$-0.711424\pi$
$683$	−1.50304e7	−1.23287	−0.616435	+	0.787405i	$0.711424\pi$
$684$	694062.	0.0567229
$685$	0	0
$686$	−1.61621e7	−1.31126
$687$	4.57451e6	0.369788
$688$	876769.	0.0706178
$689$	750701.	0.0602447
$690$	0	0
$691$	−1.59452e7	−1.27038	−0.635192	−	0.772354i	$-0.719079\pi$
$691$	−1.59452e7	−1.27038	−0.635192	+	0.772354i	$0.719079\pi$
$692$	1.41495e7	1.12325
$693$	−2.42559e6	−0.191860
$694$	3.02885e7	2.38715
$695$	0	0
$696$	1.33599e7	1.04539
$697$	2.54746e6	0.198621
$698$	−1.69268e7	−1.31504
$699$	−1.50414e7	−1.16439
$700$	0	0
$701$	1.06412e7	0.817893	0.408946	−	0.912558i	$-0.365896\pi$
$701$	1.06412e7	0.817893	0.408946	+	0.912558i	$0.365896\pi$
$702$	−1.03960e7	−0.796206
$703$	−3.47865e6	−0.265475
$704$	1.79533e7	1.36525
$705$	0	0
$706$	1.62001e7	1.22323
$707$	−1.11022e7	−0.835336
$708$	4.18530e7	3.13793
$709$	1.45758e7	1.08898	0.544488	−	0.838769i	$-0.316724\pi$
$709$	1.45758e7	1.08898	0.544488	+	0.838769i	$0.316724\pi$
$710$	0	0
$711$	2.39488e6	0.177668
$712$	−1.78984e7	−1.32317
$713$	−2.35910e7	−1.73789
$714$	6.77515e6	0.497363
$715$	0	0
$716$	4.61609e7	3.36505
$717$	3.20385e6	0.232742
$718$	2.52060e6	0.182470
$719$	−333021.	−0.0240242	−0.0120121	−	0.999928i	$-0.503824\pi$
$719$	−333021.	−0.0240242	−0.0120121	+	0.999928i	$0.503824\pi$
$720$	0	0
$721$	−2.70259e7	−1.93616
$722$	−2.23695e7	−1.59703
$723$	−2.26634e7	−1.61243
$724$	3.57236e7	2.53285
$725$	0	0
$726$	5.92281e6	0.417048
$727$	−7.40443e6	−0.519584	−0.259792	−	0.965665i	$-0.583654\pi$
$727$	−7.40443e6	−0.519584	−0.259792	+	0.965665i	$0.583654\pi$
$728$	−1.04557e7	−0.731181
$729$	1.06405e7	0.741551
$730$	0	0
$731$	−1.15486e6	−0.0799350
$732$	5055.03	0.000348695 0
$733$	2.45275e7	1.68614	0.843070	−	0.537804i	$-0.180746\pi$
$733$	2.45275e7	1.68614	0.843070	+	0.537804i	$0.180746\pi$
$734$	3.21255e7	2.20095
$735$	0	0
$736$	−1.38755e7	−0.944178
$737$	4.44614e6	0.301519
$738$	−3.82742e6	−0.258682
$739$	−1.59675e7	−1.07554	−0.537771	−	0.843091i	$-0.680733\pi$
$739$	−1.59675e7	−1.07554	−0.537771	+	0.843091i	$0.680733\pi$
$740$	0	0
$741$	1.54640e6	0.103461
$742$	−3.09577e6	−0.206423
$743$	−1.86157e7	−1.23711	−0.618554	−	0.785742i	$-0.712281\pi$
$743$	−1.86157e7	−1.23711	−0.618554	+	0.785742i	$0.712281\pi$
$744$	2.89472e7	1.91723
$745$	0	0
$746$	−1.99478e7	−1.31234
$747$	1.31727e6	0.0863720
$748$	5.56173e6	0.363459
$749$	−1.03918e7	−0.676840
$750$	0	0
$751$	−4.16383e6	−0.269397	−0.134699	−	0.990887i	$-0.543007\pi$
$751$	−4.16383e6	−0.269397	−0.134699	+	0.990887i	$0.543007\pi$
$752$	3.82338e6	0.246549
$753$	−7.76616e6	−0.499136
$754$	1.15407e7	0.739273
$755$	0	0
$756$	2.70522e7	1.72146
$757$	4.45014e6	0.282250	0.141125	−	0.989992i	$-0.454928\pi$
$757$	4.45014e6	0.282250	0.141125	+	0.989992i	$0.454928\pi$
$758$	−3.85006e7	−2.43386
$759$	1.75647e7	1.10672
$760$	0	0
$761$	−1.93876e7	−1.21356	−0.606782	−	0.794868i	$-0.707540\pi$
$761$	−1.93876e7	−1.21356	−0.606782	+	0.794868i	$0.707540\pi$
$762$	2.08135e7	1.29854
$763$	2.69766e7	1.67755
$764$	1.67539e7	1.03844
$765$	0	0
$766$	3.24697e7	1.99943
$767$	1.50122e7	0.921416
$768$	2.35644e7	1.44163
$769$	2.29489e7	1.39941	0.699706	−	0.714431i	$-0.253314\pi$
$769$	2.29489e7	1.39941	0.699706	+	0.714431i	$0.253314\pi$
$770$	0	0
$771$	−7.55895e6	−0.457958
$772$	4.18376e7	2.52652
$773$	−2.30148e7	−1.38535	−0.692675	−	0.721250i	$-0.743568\pi$
$773$	−2.30148e7	−1.38535	−0.692675	+	0.721250i	$0.743568\pi$
$774$	1.73512e6	0.104106
$775$	0	0
$776$	7.07312e6	0.421654
$777$	3.21933e7	1.91299
$778$	2.24522e6	0.132987
$779$	−2.39779e6	−0.141569
$780$	0	0
$781$	7.18490e6	0.421496
$782$	−7.89837e6	−0.461871
$783$	−1.23985e7	−0.722711
$784$	1.11335e6	0.0646909
$785$	0	0
$786$	−2.31923e7	−1.33902
$787$	4.90366e6	0.282217	0.141109	−	0.989994i	$-0.454933\pi$
$787$	4.90366e6	0.282217	0.141109	+	0.989994i	$0.454933\pi$
$788$	−1.66797e7	−0.956912
$789$	1.87258e7	1.07090
$790$	0	0
$791$	−2.93359e7	−1.66709
$792$	−3.46973e6	−0.196555
$793$	1813.18	0.000102390 0
$794$	−1.38735e7	−0.780973
$795$	0	0
$796$	−5.55915e7	−3.10975
$797$	−2.67952e6	−0.149421	−0.0747104	−	0.997205i	$-0.523803\pi$
$797$	−2.67952e6	−0.149421	−0.0747104	+	0.997205i	$0.523803\pi$
$798$	−6.37711e6	−0.354500
$799$	−5.03608e6	−0.279078
$800$	0	0
$801$	−3.94395e6	−0.217195
$802$	−1.22934e7	−0.674894
$803$	1.24604e7	0.681935
$804$	1.17738e7	0.642358
$805$	0	0
$806$	2.50057e7	1.35582
$807$	−2.55351e7	−1.38024
$808$	−1.58814e7	−0.855775
$809$	−2.30277e7	−1.23703	−0.618514	−	0.785774i	$-0.712265\pi$
$809$	−2.30277e7	−1.23703	−0.618514	+	0.785774i	$0.712265\pi$
$810$	0	0
$811$	−1.91486e7	−1.02232	−0.511158	−	0.859487i	$-0.670783\pi$
$811$	−1.91486e7	−1.02232	−0.511158	+	0.859487i	$0.670783\pi$
$812$	−3.00309e7	−1.59837
$813$	2.78058e7	1.47540
$814$	4.18815e7	2.21545
$815$	0	0
$816$	1.07912e6	0.0567341
$817$	1.08701e6	0.0569744
$818$	−3.27660e7	−1.71214
$819$	−2.30393e6	−0.120022
$820$	0	0
$821$	2.06817e7	1.07085	0.535424	−	0.844583i	$-0.320152\pi$
$821$	2.06817e7	1.07085	0.535424	+	0.844583i	$0.320152\pi$
$822$	5.90286e6	0.304707
$823$	3.00687e7	1.54745	0.773723	−	0.633524i	$-0.218392\pi$
$823$	3.00687e7	1.54745	0.773723	+	0.633524i	$0.218392\pi$
$824$	−3.86596e7	−1.98353
$825$	0	0
$826$	−6.19079e7	−3.15715
$827$	3.14519e7	1.59913	0.799563	−	0.600582i	$-0.205064\pi$
$827$	3.14519e7	1.59913	0.799563	+	0.600582i	$0.205064\pi$
$828$	7.48807e6	0.379572
$829$	1.23366e7	0.623459	0.311729	−	0.950171i	$-0.399092\pi$
$829$	1.23366e7	0.623459	0.311729	+	0.950171i	$0.399092\pi$
$830$	0	0
$831$	3.53980e7	1.77818
$832$	1.70529e7	0.854061
$833$	−1.46649e6	−0.0732261
$834$	−6.46007e7	−3.21604
$835$	0	0
$836$	−5.23497e6	−0.259059
$837$	−2.68643e7	−1.32544
$838$	4.79428e7	2.35838
$839$	1.83838e7	0.901636	0.450818	−	0.892616i	$-0.351132\pi$
$839$	1.83838e7	0.901636	0.450818	+	0.892616i	$0.351132\pi$
$840$	0	0
$841$	−6.74747e6	−0.328966
$842$	−7.21143e6	−0.350543
$843$	2.59992e7	1.26006
$844$	4.94387e7	2.38897
$845$	0	0
$846$	7.56645e6	0.363468
$847$	−5.52816e6	−0.264772
$848$	−493082.	−0.0235467
$849$	−4.99525e6	−0.237842
$850$	0	0
$851$	−3.75304e7	−1.77648
$852$	1.90263e7	0.897958
$853$	−1.05623e7	−0.497036	−0.248518	−	0.968627i	$-0.579943\pi$
$853$	−1.05623e7	−0.497036	−0.248518	+	0.968627i	$0.579943\pi$
$854$	−7477.27	−0.000350831 0
$855$	0	0
$856$	−1.48651e7	−0.693401
$857$	1.39462e7	0.648642	0.324321	−	0.945947i	$-0.394864\pi$
$857$	1.39462e7	0.648642	0.324321	+	0.945947i	$0.394864\pi$
$858$	−1.86180e7	−0.863406
$859$	−1.54908e6	−0.0716295	−0.0358148	−	0.999358i	$-0.511403\pi$
$859$	−1.54908e6	−0.0716295	−0.0358148	+	0.999358i	$0.511403\pi$
$860$	0	0
$861$	2.21904e7	1.02013
$862$	−1.18858e7	−0.544828
$863$	−3.85664e6	−0.176271	−0.0881357	−	0.996108i	$-0.528091\pi$
$863$	−3.85664e6	−0.176271	−0.0881357	+	0.996108i	$0.528091\pi$
$864$	−1.58007e7	−0.720098
$865$	0	0
$866$	8.87529e6	0.402150
$867$	−1.42140e6	−0.0642195
$868$	−6.50689e7	−2.93139
$869$	−1.80634e7	−0.811430
$870$	0	0
$871$	4.22314e6	0.188621
$872$	3.85892e7	1.71860
$873$	1.55858e6	0.0692138
$874$	7.43433e6	0.329203
$875$	0	0
$876$	3.29964e7	1.45280
$877$	1.12606e7	0.494380	0.247190	−	0.968967i	$-0.420493\pi$
$877$	1.12606e7	0.494380	0.247190	+	0.968967i	$0.420493\pi$
$878$	4.37007e7	1.91316
$879$	2.43765e6	0.106414
$880$	0	0
$881$	−3.47489e7	−1.50835	−0.754173	−	0.656675i	$-0.771962\pi$
$881$	−3.47489e7	−1.50835	−0.754173	+	0.656675i	$0.771962\pi$
$882$	2.20332e6	0.0953688
$883$	−2.66893e7	−1.15196	−0.575978	−	0.817465i	$-0.695379\pi$
$883$	−2.66893e7	−1.15196	−0.575978	+	0.817465i	$0.695379\pi$
$884$	5.28277e6	0.227369
$885$	0	0
$886$	2.59425e7	1.11027
$887$	4.99833e6	0.213312	0.106656	−	0.994296i	$-0.465986\pi$
$887$	4.99833e6	0.213312	0.106656	+	0.994296i	$0.465986\pi$
$888$	4.60514e7	1.95980
$889$	−1.94266e7	−0.824409
$890$	0	0
$891$	2.39864e7	1.01221
$892$	3.65066e6	0.153624
$893$	4.74020e6	0.198915
$894$	−2.90569e7	−1.21592
$895$	0	0
$896$	−4.79432e7	−1.99506
$897$	1.66838e7	0.692330
$898$	−1.85978e7	−0.769609
$899$	2.98222e7	1.23067
$900$	0	0
$901$	649478.	0.0266534
$902$	2.88684e7	1.18142
$903$	−1.00598e7	−0.410553
$904$	−4.19641e7	−1.70788
$905$	0	0
$906$	5.90118e7	2.38846
$907$	−3.01530e7	−1.21706	−0.608531	−	0.793530i	$-0.708241\pi$
$907$	−3.01530e7	−1.21706	−0.608531	+	0.793530i	$0.708241\pi$
$908$	7.32505e7	2.94846
$909$	−3.49949e6	−0.140474
$910$	0	0
$911$	−3.32353e7	−1.32680	−0.663398	−	0.748267i	$-0.730886\pi$
$911$	−3.32353e7	−1.32680	−0.663398	+	0.748267i	$0.730886\pi$
$912$	−1.01572e6	−0.0404378
$913$	−9.93552e6	−0.394470
$914$	3.88792e7	1.53940
$915$	0	0
$916$	−1.47092e7	−0.579229
$917$	2.16470e7	0.850107
$918$	−8.99425e6	−0.352256
$919$	−1.87194e7	−0.731146	−0.365573	−	0.930783i	$-0.619127\pi$
$919$	−1.87194e7	−0.731146	−0.365573	+	0.930783i	$0.619127\pi$
$920$	0	0
$921$	4.15376e7	1.61359
$922$	1.58247e7	0.613069
$923$	6.82454e6	0.263675
$924$	4.84471e7	1.86676
$925$	0	0
$926$	1.95186e7	0.748034
$927$	−8.51873e6	−0.325593
$928$	1.75405e7	0.668608
$929$	4.27881e7	1.62661	0.813306	−	0.581836i	$-0.197665\pi$
$929$	4.27881e7	1.62661	0.813306	+	0.581836i	$0.197665\pi$
$930$	0	0
$931$	1.38033e6	0.0521926
$932$	4.83653e7	1.82387
$933$	4.29455e7	1.61515
$934$	−5.79365e6	−0.217313
$935$	0	0
$936$	−3.29571e6	−0.122959
$937$	−3.24593e7	−1.20779	−0.603893	−	0.797065i	$-0.706385\pi$
$937$	−3.24593e7	−1.20779	−0.603893	+	0.797065i	$0.706385\pi$
$938$	−1.74155e7	−0.646293
$939$	−4.65089e7	−1.72136
$940$	0	0
$941$	7.45310e6	0.274387	0.137193	−	0.990544i	$-0.456192\pi$
$941$	7.45310e6	0.274387	0.137193	+	0.990544i	$0.456192\pi$
$942$	−7.44474e7	−2.73352
$943$	−2.58692e7	−0.947336
$944$	−9.86044e6	−0.360136
$945$	0	0
$946$	−1.30872e7	−0.475465
$947$	3.76947e7	1.36586	0.682929	−	0.730485i	$-0.260706\pi$
$947$	3.76947e7	1.36586	0.682929	+	0.730485i	$0.260706\pi$
$948$	−4.78338e7	−1.72868
$949$	1.18354e7	0.426598
$950$	0	0
$951$	5.15694e7	1.84902
$952$	−9.04587e6	−0.323488
$953$	3.57243e7	1.27418	0.637091	−	0.770789i	$-0.280138\pi$
$953$	3.57243e7	1.27418	0.637091	+	0.770789i	$0.280138\pi$
$954$	−975807.	−0.0347131
$955$	0	0
$956$	−1.03019e7	−0.364562
$957$	−2.22042e7	−0.783709
$958$	−5.21967e7	−1.83751
$959$	−5.50954e6	−0.193450
$960$	0	0
$961$	3.59876e7	1.25703
$962$	3.97809e7	1.38592
$963$	−3.27556e6	−0.113820
$964$	7.28736e7	2.52568
$965$	0	0
$966$	−6.88011e7	−2.37221
$967$	2.18028e7	0.749800	0.374900	−	0.927065i	$-0.377677\pi$
$967$	2.18028e7	0.749800	0.374900	+	0.927065i	$0.377677\pi$
$968$	−7.90786e6	−0.271251
$969$	1.33789e6	0.0457731
$970$	0	0
$971$	−4.30522e7	−1.46537	−0.732685	−	0.680568i	$-0.761733\pi$
$971$	−4.30522e7	−1.46537	−0.732685	+	0.680568i	$0.761733\pi$
$972$	1.90787e7	0.647714
$973$	6.02962e7	2.04177
$974$	−3.88472e7	−1.31209
$975$	0	0
$976$	−1190.95	−4.00193e−5 0
$977$	2.33233e7	0.781725	0.390862	−	0.920449i	$-0.372177\pi$
$977$	2.33233e7	0.781725	0.390862	+	0.920449i	$0.372177\pi$
$978$	−4.14357e7	−1.38525
$979$	2.97473e7	0.991953
$980$	0	0
$981$	8.50322e6	0.282105
$982$	−9.03806e7	−2.99086
$983$	1.04232e7	0.344048	0.172024	−	0.985093i	$-0.444969\pi$
$983$	1.04232e7	0.344048	0.172024	+	0.985093i	$0.444969\pi$
$984$	3.17427e7	1.04509
$985$	0	0
$986$	9.98459e6	0.327068
$987$	−4.38683e7	−1.43337
$988$	−4.97241e6	−0.162059
$989$	1.17275e7	0.381255
$990$	0	0
$991$	8.54735e6	0.276470	0.138235	−	0.990399i	$-0.455857\pi$
$991$	8.54735e6	0.276470	0.138235	+	0.990399i	$0.455857\pi$
$992$	3.80055e7	1.22622
$993$	4.17140e7	1.34248
$994$	−2.81433e7	−0.903459
$995$	0	0
$996$	−2.63102e7	−0.840381
$997$	−9.53203e6	−0.303702	−0.151851	−	0.988403i	$-0.548523\pi$
$997$	−9.53203e6	−0.303702	−0.151851	+	0.988403i	$0.548523\pi$
$998$	8.14301e6	0.258797
$999$	−4.27377e7	−1.35487

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.6.a.a.1.1	✓	5	5.4	even	2
425.6.a.d.1.5		5	1.1	even	1	trivial
765.6.a.g.1.5		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)

\(f(q)\)	\(=\)	\(q+9.31248 q^{2} -17.0184 q^{3} +54.7222 q^{4} -158.484 q^{6} +147.923 q^{7} +211.600 q^{8} +46.6264 q^{9} -351.681 q^{11} -931.285 q^{12} -334.042 q^{13} +1377.53 q^{14} +219.408 q^{16} -289.000 q^{17} +434.208 q^{18} +272.021 q^{19} -2517.42 q^{21} -3275.02 q^{22} +2934.77 q^{23} -3601.09 q^{24} -3110.76 q^{26} +3341.97 q^{27} +8094.70 q^{28} -3709.94 q^{29} -8038.46 q^{31} -4727.96 q^{32} +5985.05 q^{33} -2691.31 q^{34} +2551.50 q^{36} -12788.2 q^{37} +2533.19 q^{38} +5684.86 q^{39} -8814.73 q^{41} -23443.4 q^{42} +3996.06 q^{43} -19244.7 q^{44} +27330.0 q^{46} +17425.9 q^{47} -3733.98 q^{48} +5074.35 q^{49} +4918.32 q^{51} -18279.5 q^{52} -2247.33 q^{53} +31122.0 q^{54} +31300.6 q^{56} -4629.37 q^{57} -34548.7 q^{58} -44941.1 q^{59} -5.42801 q^{61} -74857.9 q^{62} +6897.15 q^{63} -51050.1 q^{64} +55735.6 q^{66} -12642.5 q^{67} -15814.7 q^{68} -49945.2 q^{69} -20430.2 q^{71} +9866.15 q^{72} -35431.0 q^{73} -119090. q^{74} +14885.6 q^{76} -52021.8 q^{77} +52940.1 q^{78} +51363.2 q^{79} -68205.2 q^{81} -82086.9 q^{82} +28251.5 q^{83} -137759. q^{84} +37213.3 q^{86} +63137.3 q^{87} -74415.6 q^{88} -84586.2 q^{89} -49412.6 q^{91} +160597. q^{92} +136802. q^{93} +162278. q^{94} +80462.4 q^{96} +33426.9 q^{97} +47254.7 q^{98} -16397.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{2} + 36 q^{3} + 43 q^{4} - 105 q^{6} + 204 q^{7} + 63 q^{8} + 531 q^{9} - 792 q^{11} - 785 q^{12} - 88 q^{13} + 860 q^{14} - 2365 q^{16} - 1445 q^{17} + 2052 q^{18} - 5160 q^{19} - 6428 q^{21}+ \cdots - 535112 q^{99}+O(q^{100}) \) 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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