LMFDB - Embedded newform 10.4.b.a.9.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10,4,Mod(9,10)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("10.9");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 10 = 2 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	10.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.590019100057$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	10.9
Dual form			10.4.b.a.9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -2.00000i q^{3} -4.00000 q^{4} +(-5.00000 - 10.0000i) q^{5} +4.00000 q^{6} +26.0000i q^{7} -8.00000i q^{8} +23.0000 q^{9} +O(q^{10})$	\(q+2.00000i q^{2} -2.00000i q^{3} -4.00000 q^{4} +(-5.00000 - 10.0000i) q^{5} +4.00000 q^{6} +26.0000i q^{7} -8.00000i q^{8} +23.0000 q^{9} +(20.0000 - 10.0000i) q^{10} -28.0000 q^{11} +8.00000i q^{12} -12.0000i q^{13} -52.0000 q^{14} +(-20.0000 + 10.0000i) q^{15} +16.0000 q^{16} -64.0000i q^{17} +46.0000i q^{18} +60.0000 q^{19} +(20.0000 + 40.0000i) q^{20} +52.0000 q^{21} -56.0000i q^{22} +58.0000i q^{23} -16.0000 q^{24} +(-75.0000 + 100.000i) q^{25} +24.0000 q^{26} -100.000i q^{27} -104.000i q^{28} -90.0000 q^{29} +(-20.0000 - 40.0000i) q^{30} -128.000 q^{31} +32.0000i q^{32} +56.0000i q^{33} +128.000 q^{34} +(260.000 - 130.000i) q^{35} -92.0000 q^{36} +236.000i q^{37} +120.000i q^{38} -24.0000 q^{39} +(-80.0000 + 40.0000i) q^{40} +242.000 q^{41} +104.000i q^{42} -362.000i q^{43} +112.000 q^{44} +(-115.000 - 230.000i) q^{45} -116.000 q^{46} +226.000i q^{47} -32.0000i q^{48} -333.000 q^{49} +(-200.000 - 150.000i) q^{50} -128.000 q^{51} +48.0000i q^{52} +108.000i q^{53} +200.000 q^{54} +(140.000 + 280.000i) q^{55} +208.000 q^{56} -120.000i q^{57} -180.000i q^{58} +20.0000 q^{59} +(80.0000 - 40.0000i) q^{60} +542.000 q^{61} -256.000i q^{62} +598.000i q^{63} -64.0000 q^{64} +(-120.000 + 60.0000i) q^{65} -112.000 q^{66} -434.000i q^{67} +256.000i q^{68} +116.000 q^{69} +(260.000 + 520.000i) q^{70} -1128.00 q^{71} -184.000i q^{72} -632.000i q^{73} -472.000 q^{74} +(200.000 + 150.000i) q^{75} -240.000 q^{76} -728.000i q^{77} -48.0000i q^{78} +720.000 q^{79} +(-80.0000 - 160.000i) q^{80} +421.000 q^{81} +484.000i q^{82} +478.000i q^{83} -208.000 q^{84} +(-640.000 + 320.000i) q^{85} +724.000 q^{86} +180.000i q^{87} +224.000i q^{88} +490.000 q^{89} +(460.000 - 230.000i) q^{90} +312.000 q^{91} -232.000i q^{92} +256.000i q^{93} -452.000 q^{94} +(-300.000 - 600.000i) q^{95} +64.0000 q^{96} +1456.00i q^{97} -666.000i q^{98} -644.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 10 q^{5} + 8 q^{6} + 46 q^{9}+O(q^{10}) $ 2 * q - 8 * q^4 - 10 * q^5 + 8 * q^6 + 46 * q^9	$ 2 q - 8 q^{4} - 10 q^{5} + 8 q^{6} + 46 q^{9} + 40 q^{10} - 56 q^{11} - 104 q^{14} - 40 q^{15} + 32 q^{16} + 120 q^{19} + 40 q^{20} + 104 q^{21} - 32 q^{24} - 150 q^{25} + 48 q^{26} - 180 q^{29} - 40 q^{30} - 256 q^{31} + 256 q^{34} + 520 q^{35} - 184 q^{36} - 48 q^{39} - 160 q^{40} + 484 q^{41} + 224 q^{44} - 230 q^{45} - 232 q^{46} - 666 q^{49} - 400 q^{50} - 256 q^{51} + 400 q^{54} + 280 q^{55} + 416 q^{56} + 40 q^{59} + 160 q^{60} + 1084 q^{61} - 128 q^{64} - 240 q^{65} - 224 q^{66} + 232 q^{69} + 520 q^{70} - 2256 q^{71} - 944 q^{74} + 400 q^{75} - 480 q^{76} + 1440 q^{79} - 160 q^{80} + 842 q^{81} - 416 q^{84} - 1280 q^{85} + 1448 q^{86} + 980 q^{89} + 920 q^{90} + 624 q^{91} - 904 q^{94} - 600 q^{95} + 128 q^{96} - 1288 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 10 * q^5 + 8 * q^6 + 46 * q^9 + 40 * q^10 - 56 * q^11 - 104 * q^14 - 40 * q^15 + 32 * q^16 + 120 * q^19 + 40 * q^20 + 104 * q^21 - 32 * q^24 - 150 * q^25 + 48 * q^26 - 180 * q^29 - 40 * q^30 - 256 * q^31 + 256 * q^34 + 520 * q^35 - 184 * q^36 - 48 * q^39 - 160 * q^40 + 484 * q^41 + 224 * q^44 - 230 * q^45 - 232 * q^46 - 666 * q^49 - 400 * q^50 - 256 * q^51 + 400 * q^54 + 280 * q^55 + 416 * q^56 + 40 * q^59 + 160 * q^60 + 1084 * q^61 - 128 * q^64 - 240 * q^65 - 224 * q^66 + 232 * q^69 + 520 * q^70 - 2256 * q^71 - 944 * q^74 + 400 * q^75 - 480 * q^76 + 1440 * q^79 - 160 * q^80 + 842 * q^81 - 416 * q^84 - 1280 * q^85 + 1448 * q^86 + 980 * q^89 + 920 * q^90 + 624 * q^91 - 904 * q^94 - 600 * q^95 + 128 * q^96 - 1288 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$.

$n$	$7$
$\chi(n)$	$-1$

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 $

$2$			2.00000i			0.707107i
$2$			2.00000i			0.707107i
$3$		−	2.00000i		−	0.384900i	−0.981307	−	0.192450i	$-0.938357\pi$
$3$		−	2.00000i		−	0.384900i	0.981307	−	0.192450i	$-0.0616434\pi$
$4$	−4.00000			−0.500000
$5$	−5.00000	−	10.0000i	−0.447214	−	0.894427i
$5$	−5.00000	−	10.0000i	−0.447214	−	0.894427i
$6$	4.00000			0.272166
$7$			26.0000i			1.40387i	0.712242	+	0.701934i	$0.247680\pi$
$7$			26.0000i			1.40387i	−0.712242	+	0.701934i	$0.752320\pi$
$8$		−	8.00000i		−	0.353553i
$9$	23.0000			0.851852
$10$	20.0000	−	10.0000i	0.632456	−	0.316228i
$11$	−28.0000			−0.767483			−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−28.0000			−0.767483			−0.383742	+	0.923440i	$0.625365\pi$
$12$			8.00000i			0.192450i
$13$		−	12.0000i		−	0.256015i	−0.991773	−	0.128008i	$-0.959142\pi$
$13$		−	12.0000i		−	0.256015i	0.991773	−	0.128008i	$-0.0408582\pi$
$14$	−52.0000			−0.992685
$15$	−20.0000	+	10.0000i	−0.344265	+	0.172133i
$16$	16.0000			0.250000
$17$		−	64.0000i		−	0.913075i	−0.889704	−	0.456538i	$-0.849089\pi$
$17$		−	64.0000i		−	0.913075i	0.889704	−	0.456538i	$-0.150911\pi$
$18$			46.0000i			0.602350i
$19$	60.0000			0.724471			0.362235	−	0.932087i	$-0.382014\pi$
$19$	60.0000			0.724471			0.362235	+	0.932087i	$0.382014\pi$
$20$	20.0000	+	40.0000i	0.223607	+	0.447214i
$21$	52.0000			0.540349
$22$		−	56.0000i		−	0.542693i
$23$			58.0000i			0.525819i	0.964821	+	0.262909i	$0.0846821\pi$
$23$			58.0000i			0.525819i	−0.964821	+	0.262909i	$0.915318\pi$
$24$	−16.0000			−0.136083
$25$	−75.0000	+	100.000i	−0.600000	+	0.800000i
$26$	24.0000			0.181030
$27$		−	100.000i		−	0.712778i
$28$		−	104.000i		−	0.701934i
$29$	−90.0000			−0.576296			−0.288148	−	0.957586i	$-0.593039\pi$
$29$	−90.0000			−0.576296			−0.288148	+	0.957586i	$0.593039\pi$
$30$	−20.0000	−	40.0000i	−0.121716	−	0.243432i
$31$	−128.000			−0.741596			−0.370798	−	0.928714i	$-0.620916\pi$
$31$	−128.000			−0.741596			−0.370798	+	0.928714i	$0.620916\pi$
$32$			32.0000i			0.176777i
$33$			56.0000i			0.295405i
$34$	128.000			0.645642
$35$	260.000	−	130.000i	1.25566	−	0.627829i
$36$	−92.0000			−0.425926
$37$			236.000i			1.04860i	0.851534	+	0.524299i	$0.175673\pi$
$37$			236.000i			1.04860i	−0.851534	+	0.524299i	$0.824327\pi$
$38$			120.000i			0.512278i
$39$	−24.0000			−0.0985404
$40$	−80.0000	+	40.0000i	−0.316228	+	0.158114i
$41$	242.000			0.921806			0.460903	−	0.887450i	$-0.347526\pi$
$41$	242.000			0.921806			0.460903	+	0.887450i	$0.347526\pi$
$42$			104.000i			0.382084i
$43$		−	362.000i		−	1.28383i	−0.766778	−	0.641913i	$-0.778141\pi$
$43$		−	362.000i		−	1.28383i	0.766778	−	0.641913i	$-0.221859\pi$
$44$	112.000			0.383742
$45$	−115.000	−	230.000i	−0.380960	−	0.761919i
$46$	−116.000			−0.371810
$47$			226.000i			0.701393i	0.936489	+	0.350697i	$0.114055\pi$
$47$			226.000i			0.701393i	−0.936489	+	0.350697i	$0.885945\pi$
$48$		−	32.0000i		−	0.0962250i
$49$	−333.000			−0.970845
$50$	−200.000	−	150.000i	−0.565685	−	0.424264i
$51$	−128.000			−0.351443
$52$			48.0000i			0.128008i
$53$			108.000i			0.279905i	0.990158	+	0.139952i	$0.0446949\pi$
$53$			108.000i			0.279905i	−0.990158	+	0.139952i	$0.955305\pi$
$54$	200.000			0.504010
$55$	140.000	+	280.000i	0.343229	+	0.686458i
$56$	208.000			0.496342
$57$		−	120.000i		−	0.278849i
$58$		−	180.000i		−	0.407503i
$59$	20.0000			0.0441318			0.0220659	−	0.999757i	$-0.492976\pi$
$59$	20.0000			0.0441318			0.0220659	+	0.999757i	$0.492976\pi$
$60$	80.0000	−	40.0000i	0.172133	−	0.0860663i
$61$	542.000			1.13764			0.568820	−	0.822462i	$-0.307400\pi$
$61$	542.000			1.13764			0.568820	+	0.822462i	$0.307400\pi$
$62$		−	256.000i		−	0.524388i
$63$			598.000i			1.19589i
$64$	−64.0000			−0.125000
$65$	−120.000	+	60.0000i	−0.228987	+	0.114494i
$66$	−112.000			−0.208883
$67$		−	434.000i		−	0.791366i	−0.918387	−	0.395683i	$-0.870508\pi$
$67$		−	434.000i		−	0.791366i	0.918387	−	0.395683i	$-0.129492\pi$
$68$			256.000i			0.456538i
$69$	116.000			0.202388
$70$	260.000	+	520.000i	0.443942	+	0.887884i
$71$	−1128.00			−1.88548			−0.942739	−	0.333531i	$-0.891760\pi$
$71$	−1128.00			−1.88548			−0.942739	+	0.333531i	$0.891760\pi$
$72$		−	184.000i		−	0.301175i
$73$		−	632.000i		−	1.01329i	−0.862155	−	0.506644i	$-0.830886\pi$
$73$		−	632.000i		−	1.01329i	0.862155	−	0.506644i	$-0.169114\pi$
$74$	−472.000			−0.741471
$75$	200.000	+	150.000i	0.307920	+	0.230940i
$76$	−240.000			−0.362235
$77$		−	728.000i		−	1.07745i
$78$		−	48.0000i		−	0.0696786i
$79$	720.000			1.02540			0.512698	−	0.858569i	$-0.328646\pi$
$79$	720.000			1.02540			0.512698	+	0.858569i	$0.328646\pi$
$80$	−80.0000	−	160.000i	−0.111803	−	0.223607i
$81$	421.000			0.577503
$82$			484.000i			0.651815i
$83$			478.000i			0.632136i	0.948736	+	0.316068i	$0.102363\pi$
$83$			478.000i			0.632136i	−0.948736	+	0.316068i	$0.897637\pi$
$84$	−208.000			−0.270175
$85$	−640.000	+	320.000i	−0.816679	+	0.408340i
$86$	724.000			0.907801
$87$			180.000i			0.221816i
$88$			224.000i			0.271346i
$89$	490.000			0.583594			0.291797	−	0.956480i	$-0.405747\pi$
$89$	490.000			0.583594			0.291797	+	0.956480i	$0.405747\pi$
$90$	460.000	−	230.000i	0.538758	−	0.269379i
$91$	312.000			0.359412
$92$		−	232.000i		−	0.262909i
$93$			256.000i			0.285440i
$94$	−452.000			−0.495960
$95$	−300.000	−	600.000i	−0.323993	−	0.647986i
$96$	64.0000			0.0680414
$97$			1456.00i			1.52407i	0.647538	+	0.762033i	$0.275799\pi$
$97$			1456.00i			1.52407i	−0.647538	+	0.762033i	$0.724201\pi$
$98$		−	666.000i		−	0.686491i
$99$	−644.000			−0.653782
$100$	300.000	−	400.000i	0.300000	−	0.400000i
$101$	−578.000			−0.569437			−0.284719	−	0.958611i	$-0.591900\pi$
$101$	−578.000			−0.569437			−0.284719	+	0.958611i	$0.591900\pi$
$102$		−	256.000i		−	0.248508i
$103$		−	1462.00i		−	1.39859i	−0.714831	−	0.699297i	$-0.753497\pi$
$103$		−	1462.00i		−	1.39859i	0.714831	−	0.699297i	$-0.246503\pi$
$104$	−96.0000			−0.0905151
$105$	−260.000	−	520.000i	−0.241651	−	0.483303i
$106$	−216.000			−0.197922
$107$			966.000i			0.872773i	0.899759	+	0.436387i	$0.143742\pi$
$107$			966.000i			0.872773i	−0.899759	+	0.436387i	$0.856258\pi$
$108$			400.000i			0.356389i
$109$	−370.000			−0.325134			−0.162567	−	0.986698i	$-0.551977\pi$
$109$	−370.000			−0.325134			−0.162567	+	0.986698i	$0.551977\pi$
$110$	−560.000	+	280.000i	−0.485399	+	0.242700i
$111$	472.000			0.403606
$112$			416.000i			0.350967i
$113$			528.000i			0.439558i	0.975550	+	0.219779i	$0.0705336\pi$
$113$			528.000i			0.439558i	−0.975550	+	0.219779i	$0.929466\pi$
$114$	240.000			0.197176
$115$	580.000	−	290.000i	0.470307	−	0.235153i
$116$	360.000			0.288148
$117$		−	276.000i		−	0.218087i
$118$			40.0000i			0.0312059i
$119$	1664.00			1.28184
$120$	80.0000	+	160.000i	0.0608581	+	0.121716i
$121$	−547.000			−0.410969
$122$			1084.00i			0.804432i
$123$		−	484.000i		−	0.354803i
$124$	512.000			0.370798
$125$	1375.00	+	250.000i	0.983870	+	0.178885i
$126$	−1196.00			−0.845620
$127$		−	1534.00i		−	1.07181i	−0.844277	−	0.535907i	$-0.819970\pi$
$127$		−	1534.00i		−	1.07181i	0.844277	−	0.535907i	$-0.180030\pi$
$128$		−	128.000i		−	0.0883883i
$129$	−724.000			−0.494145
$130$	−120.000	−	240.000i	−0.0809592	−	0.161918i
$131$	12.0000			0.00800340			0.00400170	−	0.999992i	$-0.498726\pi$
$131$	12.0000			0.00800340			0.00400170	+	0.999992i	$0.498726\pi$
$132$		−	224.000i		−	0.147702i
$133$			1560.00i			1.01706i
$134$	868.000			0.559580
$135$	−1000.00	+	500.000i	−0.637528	+	0.318764i
$136$	−512.000			−0.322821
$137$		−	1224.00i		−	0.763309i	−0.924305	−	0.381655i	$-0.875354\pi$
$137$		−	1224.00i		−	0.763309i	0.924305	−	0.381655i	$-0.124646\pi$
$138$			232.000i			0.143110i
$139$	−3100.00			−1.89164			−0.945822	−	0.324685i	$-0.894742\pi$
$139$	−3100.00			−1.89164			−0.945822	+	0.324685i	$0.894742\pi$
$140$	−1040.00	+	520.000i	−0.627829	+	0.313914i
$141$	452.000			0.269966
$142$		−	2256.00i		−	1.33323i
$143$			336.000i			0.196488i
$144$	368.000			0.212963
$145$	450.000	+	900.000i	0.257727	+	0.515455i
$146$	1264.00			0.716503
$147$			666.000i			0.373679i
$148$		−	944.000i		−	0.524299i
$149$	−250.000			−0.137455			−0.0687275	−	0.997635i	$-0.521894\pi$
$149$	−250.000			−0.137455			−0.0687275	+	0.997635i	$0.521894\pi$
$150$	−300.000	+	400.000i	−0.163299	+	0.217732i
$151$	2152.00			1.15978			0.579892	−	0.814694i	$-0.303095\pi$
$151$	2152.00			1.15978			0.579892	+	0.814694i	$0.303095\pi$
$152$		−	480.000i		−	0.256139i
$153$		−	1472.00i		−	0.777805i
$154$	1456.00			0.761869
$155$	640.000	+	1280.00i	0.331652	+	0.663304i
$156$	96.0000			0.0492702
$157$		−	524.000i		−	0.266368i	−0.991091	−	0.133184i	$-0.957480\pi$
$157$		−	524.000i		−	0.266368i	0.991091	−	0.133184i	$-0.0425201\pi$
$158$			1440.00i			0.725065i
$159$	216.000			0.107735
$160$	320.000	−	160.000i	0.158114	−	0.0790569i
$161$	−1508.00			−0.738180
$162$			842.000i			0.408357i
$163$			3518.00i			1.69050i	0.534373	+	0.845249i	$0.320548\pi$
$163$			3518.00i			1.69050i	−0.534373	+	0.845249i	$0.679452\pi$
$164$	−968.000			−0.460903
$165$	560.000	−	280.000i	0.264218	−	0.132109i
$166$	−956.000			−0.446988
$167$		−	534.000i		−	0.247438i	−0.992317	−	0.123719i	$-0.960518\pi$
$167$		−	534.000i		−	0.247438i	0.992317	−	0.123719i	$-0.0394822\pi$
$168$		−	416.000i		−	0.191042i
$169$	2053.00			0.934456
$170$	−640.000	−	1280.00i	−0.288740	−	0.577480i
$171$	1380.00			0.617142
$172$			1448.00i			0.641913i
$173$		−	4252.00i		−	1.86863i	−0.356444	−	0.934317i	$-0.616011\pi$
$173$		−	4252.00i		−	1.86863i	0.356444	−	0.934317i	$-0.383989\pi$
$174$	−360.000			−0.156848
$175$	−2600.00	−	1950.00i	−1.12309	−	0.842321i
$176$	−448.000			−0.191871
$177$		−	40.0000i		−	0.0169864i
$178$			980.000i			0.412664i
$179$	−2500.00			−1.04390			−0.521952	−	0.852975i	$-0.674796\pi$
$179$	−2500.00			−1.04390			−0.521952	+	0.852975i	$0.674796\pi$
$180$	460.000	+	920.000i	0.190480	+	0.380960i
$181$	−2578.00			−1.05868			−0.529340	−	0.848410i	$-0.677561\pi$
$181$	−2578.00			−1.05868			−0.529340	+	0.848410i	$0.677561\pi$
$182$			624.000i			0.254143i
$183$		−	1084.00i		−	0.437878i
$184$	464.000			0.185905
$185$	2360.00	−	1180.00i	0.937895	−	0.468948i
$186$	−512.000			−0.201837
$187$			1792.00i			0.700770i
$188$		−	904.000i		−	0.350697i
$189$	2600.00			1.00065
$190$	1200.00	−	600.000i	0.458196	−	0.229098i
$191$	−768.000			−0.290945			−0.145473	−	0.989362i	$-0.546470\pi$
$191$	−768.000			−0.290945			−0.145473	+	0.989362i	$0.546470\pi$
$192$			128.000i			0.0481125i
$193$			2608.00i			0.972684i	0.873769	+	0.486342i	$0.161669\pi$
$193$			2608.00i			0.972684i	−0.873769	+	0.486342i	$0.838331\pi$
$194$	−2912.00			−1.07768
$195$	120.000	+	240.000i	0.0440686	+	0.0881372i
$196$	1332.00			0.485423
$197$			5116.00i			1.85025i	0.379659	+	0.925127i	$0.376041\pi$
$197$			5116.00i			1.85025i	−0.379659	+	0.925127i	$0.623959\pi$
$198$		−	1288.00i		−	0.462294i
$199$	3480.00			1.23965			0.619826	−	0.784739i	$-0.287203\pi$
$199$	3480.00			1.23965			0.619826	+	0.784739i	$0.287203\pi$
$200$	800.000	+	600.000i	0.282843	+	0.212132i
$201$	−868.000			−0.304597
$202$		−	1156.00i		−	0.402653i
$203$		−	2340.00i		−	0.809043i
$204$	512.000			0.175721
$205$	−1210.00	−	2420.00i	−0.412244	−	0.824488i
$206$	2924.00			0.988955
$207$			1334.00i			0.447920i
$208$		−	192.000i		−	0.0640039i
$209$	−1680.00			−0.556019
$210$	1040.00	−	520.000i	0.341747	−	0.170873i
$211$	3132.00			1.02188			0.510938	−	0.859618i	$-0.329298\pi$
$211$	3132.00			1.02188			0.510938	+	0.859618i	$0.329298\pi$
$212$		−	432.000i		−	0.139952i
$213$			2256.00i			0.725721i
$214$	−1932.00			−0.617144
$215$	−3620.00	+	1810.00i	−1.14829	+	0.574144i
$216$	−800.000			−0.252005
$217$		−	3328.00i		−	1.04110i
$218$		−	740.000i		−	0.229904i
$219$	−1264.00			−0.390015
$220$	−560.000	−	1120.00i	−0.171615	−	0.343229i
$221$	−768.000			−0.233761
$222$			944.000i			0.285392i
$223$		−	62.0000i		−	0.0186181i	−0.999957	−	0.00930903i	$-0.997037\pi$
$223$		−	62.0000i		−	0.0186181i	0.999957	−	0.00930903i	$-0.00296320\pi$
$224$	−832.000			−0.248171
$225$	−1725.00	+	2300.00i	−0.511111	+	0.681481i
$226$	−1056.00			−0.310814
$227$		−	5314.00i		−	1.55376i	−0.629651	−	0.776878i	$-0.716802\pi$
$227$		−	5314.00i		−	1.55376i	0.629651	−	0.776878i	$-0.283198\pi$
$228$			480.000i			0.139424i
$229$	190.000			0.0548277			0.0274139	−	0.999624i	$-0.491273\pi$
$229$	190.000			0.0548277			0.0274139	+	0.999624i	$0.491273\pi$
$230$	580.000	+	1160.00i	0.166279	+	0.332557i
$231$	−1456.00			−0.414709
$232$			720.000i			0.203751i
$233$			2408.00i			0.677053i	0.940957	+	0.338526i	$0.109928\pi$
$233$			2408.00i			0.677053i	−0.940957	+	0.338526i	$0.890072\pi$
$234$	552.000			0.154211
$235$	2260.00	−	1130.00i	0.627345	−	0.313673i
$236$	−80.0000			−0.0220659
$237$		−	1440.00i		−	0.394675i
$238$			3328.00i			0.906396i
$239$	5680.00			1.53727			0.768637	−	0.639685i	$-0.220935\pi$
$239$	5680.00			1.53727			0.768637	+	0.639685i	$0.220935\pi$
$240$	−320.000	+	160.000i	−0.0860663	+	0.0430331i
$241$	−278.000			−0.0743052			−0.0371526	−	0.999310i	$-0.511829\pi$
$241$	−278.000			−0.0743052			−0.0371526	+	0.999310i	$0.511829\pi$
$242$		−	1094.00i		−	0.290599i
$243$		−	3542.00i		−	0.935059i
$244$	−2168.00			−0.568820
$245$	1665.00	+	3330.00i	0.434175	+	0.868351i
$246$	968.000			0.250884
$247$		−	720.000i		−	0.185476i
$248$			1024.00i			0.262194i
$249$	956.000			0.243309
$250$	−500.000	+	2750.00i	−0.126491	+	0.695701i
$251$	3252.00			0.817787			0.408893	−	0.912582i	$-0.365915\pi$
$251$	3252.00			0.817787			0.408893	+	0.912582i	$0.365915\pi$
$252$		−	2392.00i		−	0.597944i
$253$		−	1624.00i		−	0.403557i
$254$	3068.00			0.757888
$255$	640.000	+	1280.00i	0.157170	+	0.314340i
$256$	256.000			0.0625000
$257$			1536.00i			0.372813i	0.982473	+	0.186407i	$0.0596842\pi$
$257$			1536.00i			0.372813i	−0.982473	+	0.186407i	$0.940316\pi$
$258$		−	1448.00i		−	0.349413i
$259$	−6136.00			−1.47209
$260$	480.000	−	240.000i	0.114494	−	0.0572468i
$261$	−2070.00			−0.490919
$262$			24.0000i			0.00565926i
$263$			4858.00i			1.13900i	0.821991	+	0.569500i	$0.192863\pi$
$263$			4858.00i			1.13900i	−0.821991	+	0.569500i	$0.807137\pi$
$264$	448.000			0.104441
$265$	1080.00	−	540.000i	0.250354	−	0.125177i
$266$	−3120.00			−0.719171
$267$		−	980.000i		−	0.224626i
$268$			1736.00i			0.395683i
$269$	−2610.00			−0.591578			−0.295789	−	0.955253i	$-0.595583\pi$
$269$	−2610.00			−0.591578			−0.295789	+	0.955253i	$0.595583\pi$
$270$	−1000.00	−	2000.00i	−0.225400	−	0.450800i
$271$	−5168.00			−1.15843			−0.579213	−	0.815176i	$-0.696640\pi$
$271$	−5168.00			−1.15843			−0.579213	+	0.815176i	$0.696640\pi$
$272$		−	1024.00i		−	0.228269i
$273$		−	624.000i		−	0.138338i
$274$	2448.00			0.539741
$275$	2100.00	−	2800.00i	0.460490	−	0.613987i
$276$	−464.000			−0.101194
$277$		−	1924.00i		−	0.417336i	−0.977987	−	0.208668i	$-0.933087\pi$
$277$		−	1924.00i		−	0.417336i	0.977987	−	0.208668i	$-0.0669127\pi$
$278$		−	6200.00i		−	1.33759i
$279$	−2944.00			−0.631730
$280$	−1040.00	−	2080.00i	−0.221971	−	0.443942i
$281$	3042.00			0.645803			0.322901	−	0.946433i	$-0.395342\pi$
$281$	3042.00			0.645803			0.322901	+	0.946433i	$0.395342\pi$
$282$			904.000i			0.190895i
$283$			1718.00i			0.360864i	0.983587	+	0.180432i	$0.0577496\pi$
$283$			1718.00i			0.360864i	−0.983587	+	0.180432i	$0.942250\pi$
$284$	4512.00			0.942739
$285$	−1200.00	+	600.000i	−0.249410	+	0.124705i
$286$	−672.000			−0.138938
$287$			6292.00i			1.29409i
$288$			736.000i			0.150588i
$289$	817.000			0.166294
$290$	−1800.00	+	900.000i	−0.364482	+	0.182241i
$291$	2912.00			0.586613
$292$			2528.00i			0.506644i
$293$		−	2292.00i		−	0.456997i	−0.973544	−	0.228498i	$-0.926618\pi$
$293$		−	2292.00i		−	0.456997i	0.973544	−	0.228498i	$-0.0733816\pi$
$294$	−1332.00			−0.264231
$295$	−100.000	−	200.000i	−0.0197364	−	0.0394727i
$296$	1888.00			0.370736
$297$			2800.00i			0.547045i
$298$		−	500.000i		−	0.0971954i
$299$	696.000			0.134618
$300$	−800.000	−	600.000i	−0.153960	−	0.115470i
$301$	9412.00			1.80232
$302$			4304.00i			0.820091i
$303$			1156.00i			0.219176i
$304$	960.000			0.181118
$305$	−2710.00	−	5420.00i	−0.508768	−	1.01754i
$306$	2944.00			0.549991
$307$			5406.00i			1.00501i	0.864576	+	0.502503i	$0.167587\pi$
$307$			5406.00i			1.00501i	−0.864576	+	0.502503i	$0.832413\pi$
$308$			2912.00i			0.538723i
$309$	−2924.00			−0.538319
$310$	−2560.00	+	1280.00i	−0.469027	+	0.234513i
$311$	−5688.00			−1.03710			−0.518548	−	0.855048i	$-0.673527\pi$
$311$	−5688.00			−1.03710			−0.518548	+	0.855048i	$0.673527\pi$
$312$			192.000i			0.0348393i
$313$		−	7352.00i		−	1.32767i	−0.747881	−	0.663833i	$-0.768928\pi$
$313$		−	7352.00i		−	1.32767i	0.747881	−	0.663833i	$-0.231072\pi$
$314$	1048.00			0.188351
$315$	5980.00	−	2990.00i	1.06963	−	0.534817i
$316$	−2880.00			−0.512698
$317$		−	3484.00i		−	0.617290i	−0.951177	−	0.308645i	$-0.900124\pi$
$317$		−	3484.00i		−	0.617290i	0.951177	−	0.308645i	$-0.0998755\pi$
$318$			432.000i			0.0761804i
$319$	2520.00			0.442298
$320$	320.000	+	640.000i	0.0559017	+	0.111803i
$321$	1932.00			0.335931
$322$		−	3016.00i		−	0.521972i
$323$		−	3840.00i		−	0.661496i
$324$	−1684.00			−0.288752
$325$	1200.00	+	900.000i	0.204812	+	0.153609i
$326$	−7036.00			−1.19536
$327$			740.000i			0.125144i
$328$		−	1936.00i		−	0.325908i
$329$	−5876.00			−0.984664
$330$	560.000	+	1120.00i	0.0934151	+	0.186830i
$331$	−7868.00			−1.30654			−0.653269	−	0.757125i	$-0.726603\pi$
$331$	−7868.00			−1.30654			−0.653269	+	0.757125i	$0.726603\pi$
$332$		−	1912.00i		−	0.316068i
$333$			5428.00i			0.893251i
$334$	1068.00			0.174965
$335$	−4340.00	+	2170.00i	−0.707819	+	0.353910i
$336$	832.000			0.135087
$337$			656.000i			0.106037i	0.998594	+	0.0530187i	$0.0168843\pi$
$337$			656.000i			0.106037i	−0.998594	+	0.0530187i	$0.983116\pi$
$338$			4106.00i			0.660760i
$339$	1056.00			0.169186
$340$	2560.00	−	1280.00i	0.408340	−	0.204170i
$341$	3584.00			0.569163
$342$			2760.00i			0.436385i
$343$			260.000i			0.0409291i
$344$	−2896.00			−0.453901
$345$	−580.000	−	1160.00i	−0.0905106	−	0.181021i
$346$	8504.00			1.32132
$347$		−	5754.00i		−	0.890176i	−0.895487	−	0.445088i	$-0.853172\pi$
$347$		−	5754.00i		−	0.890176i	0.895487	−	0.445088i	$-0.146828\pi$
$348$		−	720.000i		−	0.110908i
$349$	3110.00			0.477004			0.238502	−	0.971142i	$-0.423344\pi$
$349$	3110.00			0.477004			0.238502	+	0.971142i	$0.423344\pi$
$350$	3900.00	−	5200.00i	0.595611	−	0.794148i
$351$	−1200.00			−0.182482
$352$		−	896.000i		−	0.135673i
$353$			7808.00i			1.17727i	0.808397	+	0.588637i	$0.200335\pi$
$353$			7808.00i			1.17727i	−0.808397	+	0.588637i	$0.799665\pi$
$354$	80.0000			0.0120112
$355$	5640.00	+	11280.0i	0.843212	+	1.68642i
$356$	−1960.00			−0.291797
$357$		−	3328.00i		−	0.493379i
$358$		−	5000.00i		−	0.738151i
$359$	9240.00			1.35841			0.679204	−	0.733949i	$-0.262325\pi$
$359$	9240.00			1.35841			0.679204	+	0.733949i	$0.262325\pi$
$360$	−1840.00	+	920.000i	−0.269379	+	0.134690i
$361$	−3259.00			−0.475142
$362$		−	5156.00i		−	0.748600i
$363$			1094.00i			0.158182i
$364$	−1248.00			−0.179706
$365$	−6320.00	+	3160.00i	−0.906312	+	0.453156i
$366$	2168.00			0.309626
$367$		−	3214.00i		−	0.457137i	−0.973528	−	0.228569i	$-0.926595\pi$
$367$		−	3214.00i		−	0.457137i	0.973528	−	0.228569i	$-0.0734046\pi$
$368$			928.000i			0.131455i
$369$	5566.00			0.785242
$370$	2360.00	+	4720.00i	0.331596	+	0.663192i
$371$	−2808.00			−0.392949
$372$		−	1024.00i		−	0.142720i
$373$			348.000i			0.0483077i	0.999708	+	0.0241538i	$0.00768915\pi$
$373$			348.000i			0.0483077i	−0.999708	+	0.0241538i	$0.992311\pi$
$374$	−3584.00			−0.495519
$375$	500.000	−	2750.00i	0.0688530	−	0.378692i
$376$	1808.00			0.247980
$377$			1080.00i			0.147541i
$378$			5200.00i			0.707564i
$379$	−4940.00			−0.669527			−0.334764	−	0.942302i	$-0.608656\pi$
$379$	−4940.00			−0.669527			−0.334764	+	0.942302i	$0.608656\pi$
$380$	1200.00	+	2400.00i	0.161997	+	0.323993i
$381$	−3068.00			−0.412542
$382$		−	1536.00i		−	0.205729i
$383$		−	6142.00i		−	0.819430i	−0.912214	−	0.409715i	$-0.865628\pi$
$383$		−	6142.00i		−	0.819430i	0.912214	−	0.409715i	$-0.134372\pi$
$384$	−256.000			−0.0340207
$385$	−7280.00	+	3640.00i	−0.963697	+	0.481848i
$386$	−5216.00			−0.687791
$387$		−	8326.00i		−	1.09363i
$388$		−	5824.00i		−	0.762033i
$389$	−3050.00			−0.397535			−0.198768	−	0.980047i	$-0.563694\pi$
$389$	−3050.00			−0.397535			−0.198768	+	0.980047i	$0.563694\pi$
$390$	−480.000	+	240.000i	−0.0623224	+	0.0311612i
$391$	3712.00			0.480112
$392$			2664.00i			0.343246i
$393$		−	24.0000i		−	0.00308051i
$394$	−10232.0			−1.30833
$395$	−3600.00	−	7200.00i	−0.458571	−	0.917143i
$396$	2576.00			0.326891
$397$			5396.00i			0.682160i	0.940034	+	0.341080i	$0.110793\pi$
$397$			5396.00i			0.682160i	−0.940034	+	0.341080i	$0.889207\pi$
$398$			6960.00i			0.876566i
$399$	3120.00			0.391467
$400$	−1200.00	+	1600.00i	−0.150000	+	0.200000i
$401$	14482.0			1.80348			0.901741	−	0.432276i	$-0.142289\pi$
$401$	14482.0			1.80348			0.901741	+	0.432276i	$0.142289\pi$
$402$		−	1736.00i		−	0.215383i
$403$			1536.00i			0.189860i
$404$	2312.00			0.284719
$405$	−2105.00	−	4210.00i	−0.258267	−	0.516535i
$406$	4680.00			0.572080
$407$		−	6608.00i		−	0.804782i
$408$			1024.00i			0.124254i
$409$	1090.00			0.131778			0.0658888	−	0.997827i	$-0.479012\pi$
$409$	1090.00			0.131778			0.0658888	+	0.997827i	$0.479012\pi$
$410$	4840.00	−	2420.00i	0.583001	−	0.291501i
$411$	−2448.00			−0.293798
$412$			5848.00i			0.699297i
$413$			520.000i			0.0619553i
$414$	−2668.00			−0.316727
$415$	4780.00	−	2390.00i	0.565400	−	0.282700i
$416$	384.000			0.0452576
$417$			6200.00i			0.728094i
$418$		−	3360.00i		−	0.393165i
$419$	7180.00			0.837150			0.418575	−	0.908182i	$-0.362530\pi$
$419$	7180.00			0.837150			0.418575	+	0.908182i	$0.362530\pi$
$420$	1040.00	+	2080.00i	0.120826	+	0.241651i
$421$	−8138.00			−0.942095			−0.471047	−	0.882108i	$-0.656124\pi$
$421$	−8138.00			−0.942095			−0.471047	+	0.882108i	$0.656124\pi$
$422$			6264.00i			0.722575i
$423$			5198.00i			0.597483i
$424$	864.000			0.0989612
$425$	6400.00	+	4800.00i	0.730460	+	0.547845i
$426$	−4512.00			−0.513162
$427$			14092.0i			1.59710i
$428$		−	3864.00i		−	0.436387i
$429$	672.000			0.0756281
$430$	−3620.00	−	7240.00i	−0.405981	−	0.811962i
$431$	−208.000			−0.0232460			−0.0116230	−	0.999932i	$-0.503700\pi$
$431$	−208.000			−0.0232460			−0.0116230	+	0.999932i	$0.503700\pi$
$432$		−	1600.00i		−	0.178195i
$433$		−	12992.0i		−	1.44193i	−0.692971	−	0.720965i	$-0.743699\pi$
$433$		−	12992.0i		−	1.44193i	0.692971	−	0.720965i	$-0.256301\pi$
$434$	6656.00			0.736171
$435$	1800.00	−	900.000i	0.198399	−	0.0991993i
$436$	1480.00			0.162567
$437$			3480.00i			0.380940i
$438$		−	2528.00i		−	0.275782i
$439$	−1080.00			−0.117416			−0.0587080	−	0.998275i	$-0.518698\pi$
$439$	−1080.00			−0.117416			−0.0587080	+	0.998275i	$0.518698\pi$
$440$	2240.00	−	1120.00i	0.242700	−	0.121350i
$441$	−7659.00			−0.827017
$442$		−	1536.00i		−	0.165294i
$443$			9078.00i			0.973609i	0.873511	+	0.486805i	$0.161838\pi$
$443$			9078.00i			0.973609i	−0.873511	+	0.486805i	$0.838162\pi$
$444$	−1888.00			−0.201803
$445$	−2450.00	−	4900.00i	−0.260991	−	0.521983i
$446$	124.000			0.0131650
$447$			500.000i			0.0529065i
$448$		−	1664.00i		−	0.175484i
$449$	−14310.0			−1.50408			−0.752039	−	0.659119i	$-0.770929\pi$
$449$	−14310.0			−1.50408			−0.752039	+	0.659119i	$0.770929\pi$
$450$	−4600.00	−	3450.00i	−0.481880	−	0.361410i
$451$	−6776.00			−0.707471
$452$		−	2112.00i		−	0.219779i
$453$		−	4304.00i		−	0.446401i
$454$	10628.0			1.09867
$455$	−1560.00	−	3120.00i	−0.160734	−	0.321468i
$456$	−960.000			−0.0985880
$457$		−	2344.00i		−	0.239929i	−0.992778	−	0.119965i	$-0.961722\pi$
$457$		−	2344.00i		−	0.239929i	0.992778	−	0.119965i	$-0.0382781\pi$
$458$			380.000i			0.0387691i
$459$	−6400.00			−0.650820
$460$	−2320.00	+	1160.00i	−0.235153	+	0.117577i
$461$	11382.0			1.14992			0.574959	−	0.818182i	$-0.305018\pi$
$461$	11382.0			1.14992			0.574959	+	0.818182i	$0.305018\pi$
$462$		−	2912.00i		−	0.293244i
$463$		−	16062.0i		−	1.61223i	−0.591756	−	0.806117i	$-0.701565\pi$
$463$		−	16062.0i		−	1.61223i	0.591756	−	0.806117i	$-0.298435\pi$
$464$	−1440.00			−0.144074
$465$	2560.00	−	1280.00i	0.255306	−	0.127653i
$466$	−4816.00			−0.478749
$467$			17166.0i			1.70096i	0.526008	+	0.850479i	$0.323688\pi$
$467$			17166.0i			1.70096i	−0.526008	+	0.850479i	$0.676312\pi$
$468$			1104.00i			0.109044i
$469$	11284.0			1.11097
$470$	2260.00	+	4520.00i	0.221800	+	0.443600i
$471$	−1048.00			−0.102525
$472$		−	160.000i		−	0.0156030i
$473$			10136.0i			0.985315i
$474$	2880.00			0.279078
$475$	−4500.00	+	6000.00i	−0.434682	+	0.579577i
$476$	−6656.00			−0.640919
$477$			2484.00i			0.238437i
$478$			11360.0i			1.08702i
$479$	−7520.00			−0.717323			−0.358661	−	0.933468i	$-0.616767\pi$
$479$	−7520.00			−0.717323			−0.358661	+	0.933468i	$0.616767\pi$
$480$	−320.000	−	640.000i	−0.0304290	−	0.0608581i
$481$	2832.00			0.268458
$482$		−	556.000i		−	0.0525417i
$483$			3016.00i			0.284126i
$484$	2188.00			0.205485
$485$	14560.0	−	7280.00i	1.36317	−	0.681583i
$486$	7084.00			0.661187
$487$		−	11814.0i		−	1.09927i	−0.835406	−	0.549634i	$-0.814767\pi$
$487$		−	11814.0i		−	1.09927i	0.835406	−	0.549634i	$-0.185233\pi$
$488$		−	4336.00i		−	0.402216i
$489$	7036.00			0.650673
$490$	−6660.00	+	3330.00i	−0.614017	+	0.307008i
$491$	14052.0			1.29156			0.645782	−	0.763522i	$-0.276532\pi$
$491$	14052.0			1.29156			0.645782	+	0.763522i	$0.276532\pi$
$492$			1936.00i			0.177402i
$493$			5760.00i			0.526202i
$494$	1440.00			0.131151
$495$	3220.00	+	6440.00i	0.292380	+	0.584761i
$496$	−2048.00			−0.185399
$497$		−	29328.0i		−	2.64696i
$498$			1912.00i			0.172046i
$499$	−7620.00			−0.683603			−0.341802	−	0.939772i	$-0.611037\pi$
$499$	−7620.00			−0.683603			−0.341802	+	0.939772i	$0.611037\pi$
$500$	−5500.00	−	1000.00i	−0.491935	−	0.0894427i
$501$	−1068.00			−0.0952390
$502$			6504.00i			0.578262i
$503$			1818.00i			0.161154i	0.996748	+	0.0805772i	$0.0256763\pi$
$503$			1818.00i			0.161154i	−0.996748	+	0.0805772i	$0.974324\pi$
$504$	4784.00			0.422810
$505$	2890.00	+	5780.00i	0.254660	+	0.509320i
$506$	3248.00			0.285358
$507$		−	4106.00i		−	0.359672i
$508$			6136.00i			0.535907i
$509$	−17850.0			−1.55440			−0.777198	−	0.629256i	$-0.783360\pi$
$509$	−17850.0			−1.55440			−0.777198	+	0.629256i	$0.783360\pi$
$510$	−2560.00	+	1280.00i	−0.222272	+	0.111136i
$511$	16432.0			1.42252
$512$			512.000i			0.0441942i
$513$		−	6000.00i		−	0.516387i
$514$	−3072.00			−0.263619
$515$	−14620.0	+	7310.00i	−1.25094	+	0.625470i
$516$	2896.00			0.247072
$517$		−	6328.00i		−	0.538308i
$518$		−	12272.0i		−	1.04093i
$519$	−8504.00			−0.719237
$520$	480.000	+	960.000i	0.0404796	+	0.0809592i
$521$	−19238.0			−1.61772			−0.808860	−	0.588001i	$-0.799915\pi$
$521$	−19238.0			−1.61772			−0.808860	+	0.588001i	$0.799915\pi$
$522$		−	4140.00i		−	0.347132i
$523$			6278.00i			0.524891i	0.964947	+	0.262445i	$0.0845289\pi$
$523$			6278.00i			0.524891i	−0.964947	+	0.262445i	$0.915471\pi$
$524$	−48.0000			−0.00400170
$525$	−3900.00	+	5200.00i	−0.324209	+	0.432279i
$526$	−9716.00			−0.805395
$527$			8192.00i			0.677133i
$528$			896.000i			0.0738511i
$529$	8803.00			0.723514
$530$	1080.00	+	2160.00i	0.0885136	+	0.177027i
$531$	460.000			0.0375938
$532$		−	6240.00i		−	0.508531i
$533$		−	2904.00i		−	0.235997i
$534$	1960.00			0.158834
$535$	9660.00	−	4830.00i	0.780632	−	0.390316i
$536$	−3472.00			−0.279790
$537$			5000.00i			0.401799i
$538$		−	5220.00i		−	0.418309i
$539$	9324.00			0.745108
$540$	4000.00	−	2000.00i	0.318764	−	0.159382i
$541$	−9818.00			−0.780238			−0.390119	−	0.920764i	$-0.627566\pi$
$541$	−9818.00			−0.780238			−0.390119	+	0.920764i	$0.627566\pi$
$542$		−	10336.0i		−	0.819131i
$543$			5156.00i			0.407486i
$544$	2048.00			0.161410
$545$	1850.00	+	3700.00i	0.145404	+	0.290808i
$546$	1248.00			0.0978195
$547$		−	12514.0i		−	0.978172i	−0.872236	−	0.489086i	$-0.837330\pi$
$547$		−	12514.0i		−	0.978172i	0.872236	−	0.489086i	$-0.162670\pi$
$548$			4896.00i			0.381655i
$549$	12466.0			0.969100
$550$	5600.00	+	4200.00i	0.434154	+	0.325616i
$551$	−5400.00			−0.417509
$552$		−	928.000i		−	0.0715549i
$553$			18720.0i			1.43952i
$554$	3848.00			0.295101
$555$	−2360.00	−	4720.00i	−0.180498	−	0.360996i
$556$	12400.0			0.945822
$557$			10596.0i			0.806045i	0.915190	+	0.403022i	$0.132040\pi$
$557$			10596.0i			0.806045i	−0.915190	+	0.403022i	$0.867960\pi$
$558$		−	5888.00i		−	0.446701i
$559$	−4344.00			−0.328679
$560$	4160.00	−	2080.00i	0.313914	−	0.156957i
$561$	3584.00			0.269727
$562$			6084.00i			0.456651i
$563$		−	14002.0i		−	1.04816i	−0.851669	−	0.524080i	$-0.824409\pi$
$563$		−	14002.0i		−	1.04816i	0.851669	−	0.524080i	$-0.175591\pi$
$564$	−1808.00			−0.134983
$565$	5280.00	−	2640.00i	0.393153	−	0.196576i
$566$	−3436.00			−0.255169
$567$			10946.0i			0.810739i
$568$			9024.00i			0.666617i
$569$	7330.00			0.540052			0.270026	−	0.962853i	$-0.412968\pi$
$569$	7330.00			0.540052			0.270026	+	0.962853i	$0.412968\pi$
$570$	−1200.00	−	2400.00i	−0.0881798	−	0.176360i
$571$	5812.00			0.425963			0.212981	−	0.977056i	$-0.431683\pi$
$571$	5812.00			0.425963			0.212981	+	0.977056i	$0.431683\pi$
$572$		−	1344.00i		−	0.0982438i
$573$			1536.00i			0.111985i
$574$	−12584.0			−0.915063
$575$	−5800.00	−	4350.00i	−0.420655	−	0.315491i
$576$	−1472.00			−0.106481
$577$			16736.0i			1.20750i	0.797173	+	0.603751i	$0.206328\pi$
$577$			16736.0i			1.20750i	−0.797173	+	0.603751i	$0.793672\pi$
$578$			1634.00i			0.117587i
$579$	5216.00			0.374386
$580$	−1800.00	−	3600.00i	−0.128864	−	0.257727i
$581$	−12428.0			−0.887436
$582$			5824.00i			0.414798i
$583$		−	3024.00i		−	0.214822i
$584$	−5056.00			−0.358251
$585$	−2760.00	+	1380.00i	−0.195063	+	0.0975316i
$586$	4584.00			0.323146
$587$		−	7434.00i		−	0.522716i	−0.965242	−	0.261358i	$-0.915830\pi$
$587$		−	7434.00i		−	0.522716i	0.965242	−	0.261358i	$-0.0841702\pi$
$588$		−	2664.00i		−	0.186839i
$589$	−7680.00			−0.537265
$590$	400.000	−	200.000i	0.0279114	−	0.0139557i
$591$	10232.0			0.712163
$592$			3776.00i			0.262150i
$593$		−	25872.0i		−	1.79163i	−0.444429	−	0.895814i	$-0.646593\pi$
$593$		−	25872.0i		−	1.79163i	0.444429	−	0.895814i	$-0.353407\pi$
$594$	−5600.00			−0.386820
$595$	−8320.00	−	16640.0i	−0.573255	−	1.14651i
$596$	1000.00			0.0687275
$597$		−	6960.00i		−	0.477142i
$598$			1392.00i			0.0951892i
$599$	3720.00			0.253748			0.126874	−	0.991919i	$-0.459506\pi$
$599$	3720.00			0.253748			0.126874	+	0.991919i	$0.459506\pi$
$600$	1200.00	−	1600.00i	0.0816497	−	0.108866i
$601$	−12958.0			−0.879481			−0.439740	−	0.898125i	$-0.644930\pi$
$601$	−12958.0			−0.879481			−0.439740	+	0.898125i	$0.644930\pi$
$602$			18824.0i			1.27443i
$603$		−	9982.00i		−	0.674127i
$604$	−8608.00			−0.579892
$605$	2735.00	+	5470.00i	0.183791	+	0.367582i
$606$	−2312.00			−0.154981
$607$		−	7214.00i		−	0.482384i	−0.970477	−	0.241192i	$-0.922462\pi$
$607$		−	7214.00i		−	0.482384i	0.970477	−	0.241192i	$-0.0775384\pi$
$608$			1920.00i			0.128070i
$609$	−4680.00			−0.311401
$610$	10840.0	−	5420.00i	0.719506	−	0.359753i
$611$	2712.00			0.179568
$612$			5888.00i			0.388902i
$613$			4828.00i			0.318109i	0.987270	+	0.159055i	$0.0508446\pi$
$613$			4828.00i			0.318109i	−0.987270	+	0.159055i	$0.949155\pi$
$614$	−10812.0			−0.710646
$615$	−4840.00	+	2420.00i	−0.317346	+	0.158673i
$616$	−5824.00			−0.380934
$617$			27656.0i			1.80452i	0.431193	+	0.902260i	$0.358093\pi$
$617$			27656.0i			1.80452i	−0.431193	+	0.902260i	$0.641907\pi$
$618$		−	5848.00i		−	0.380649i
$619$	21220.0			1.37787			0.688937	−	0.724821i	$-0.258078\pi$
$619$	21220.0			1.37787			0.688937	+	0.724821i	$0.258078\pi$
$620$	−2560.00	−	5120.00i	−0.165826	−	0.331652i
$621$	5800.00			0.374792
$622$		−	11376.0i		−	0.733338i
$623$			12740.0i			0.819289i
$624$	−384.000			−0.0246351
$625$	−4375.00	−	15000.0i	−0.280000	−	0.960000i
$626$	14704.0			0.938802
$627$			3360.00i			0.214012i
$628$			2096.00i			0.133184i
$629$	15104.0			0.957450
$630$	5980.00	+	11960.0i	0.378173	+	0.756346i
$631$	17672.0			1.11491			0.557457	−	0.830206i	$-0.311777\pi$
$631$	17672.0			1.11491			0.557457	+	0.830206i	$0.311777\pi$
$632$		−	5760.00i		−	0.362532i
$633$		−	6264.00i		−	0.393320i
$634$	6968.00			0.436490
$635$	−15340.0	+	7670.00i	−0.958660	+	0.479330i
$636$	−864.000			−0.0538677
$637$			3996.00i			0.248551i
$638$			5040.00i			0.312752i
$639$	−25944.0			−1.60615
$640$	−1280.00	+	640.000i	−0.0790569	+	0.0395285i
$641$	7322.00			0.451173			0.225586	−	0.974223i	$-0.427570\pi$
$641$	7322.00			0.451173			0.225586	+	0.974223i	$0.427570\pi$
$642$			3864.00i			0.237539i
$643$			8238.00i			0.505249i	0.967564	+	0.252624i	$0.0812937\pi$
$643$			8238.00i			0.505249i	−0.967564	+	0.252624i	$0.918706\pi$
$644$	6032.00			0.369090
$645$	3620.00	+	7240.00i	0.220988	+	0.441976i
$646$	7680.00			0.467749
$647$			6426.00i			0.390467i	0.980757	+	0.195233i	$0.0625465\pi$
$647$			6426.00i			0.390467i	−0.980757	+	0.195233i	$0.937454\pi$
$648$		−	3368.00i		−	0.204178i
$649$	−560.000			−0.0338705
$650$	−1800.00	+	2400.00i	−0.108618	+	0.144824i
$651$	−6656.00			−0.400721
$652$		−	14072.0i		−	0.845249i
$653$			5908.00i			0.354055i	0.984206	+	0.177027i	$0.0566482\pi$
$653$			5908.00i			0.354055i	−0.984206	+	0.177027i	$0.943352\pi$
$654$	−1480.00			−0.0884902
$655$	−60.0000	−	120.000i	−0.00357923	−	0.00715845i
$656$	3872.00			0.230452
$657$		−	14536.0i		−	0.863171i
$658$		−	11752.0i		−	0.696262i
$659$	26780.0			1.58301			0.791503	−	0.611166i	$-0.209299\pi$
$659$	26780.0			1.58301			0.791503	+	0.611166i	$0.209299\pi$
$660$	−2240.00	+	1120.00i	−0.132109	+	0.0660545i
$661$	−24538.0			−1.44390			−0.721950	−	0.691945i	$-0.756754\pi$
$661$	−24538.0			−1.44390			−0.721950	+	0.691945i	$0.756754\pi$
$662$		−	15736.0i		−	0.923863i
$663$			1536.00i			0.0899748i
$664$	3824.00			0.223494
$665$	15600.0	−	7800.00i	0.909687	−	0.454844i
$666$	−10856.0			−0.631624
$667$		−	5220.00i		−	0.303027i
$668$			2136.00i			0.123719i
$669$	−124.000			−0.00716609
$670$	−4340.00	−	8680.00i	−0.250252	−	0.500504i
$671$	−15176.0			−0.873119
$672$			1664.00i			0.0955211i
$673$			28848.0i			1.65232i	0.563439	+	0.826158i	$0.309478\pi$
$673$			28848.0i			1.65232i	−0.563439	+	0.826158i	$0.690522\pi$
$674$	−1312.00			−0.0749798
$675$	10000.0	+	7500.00i	0.570222	+	0.427667i
$676$	−8212.00			−0.467228
$677$		−	26884.0i		−	1.52620i	−0.646282	−	0.763099i	$-0.723677\pi$
$677$		−	26884.0i		−	1.52620i	0.646282	−	0.763099i	$-0.276323\pi$
$678$			2112.00i			0.119633i
$679$	−37856.0			−2.13959
$680$	2560.00	+	5120.00i	0.144370	+	0.288740i
$681$	−10628.0			−0.598041
$682$			7168.00i			0.402459i
$683$		−	14282.0i		−	0.800125i	−0.916488	−	0.400063i	$-0.868988\pi$
$683$		−	14282.0i		−	0.800125i	0.916488	−	0.400063i	$-0.131012\pi$
$684$	−5520.00			−0.308571
$685$	−12240.0	+	6120.00i	−0.682725	+	0.341362i
$686$	−520.000			−0.0289412
$687$		−	380.000i		−	0.0211032i
$688$		−	5792.00i		−	0.320956i
$689$	1296.00			0.0716599
$690$	2320.00	−	1160.00i	0.128001	−	0.0640006i
$691$	−3428.00			−0.188723			−0.0943613	−	0.995538i	$-0.530081\pi$
$691$	−3428.00			−0.188723			−0.0943613	+	0.995538i	$0.530081\pi$
$692$			17008.0i			0.934317i
$693$		−	16744.0i		−	0.917824i
$694$	11508.0			0.629449
$695$	15500.0	+	31000.0i	0.845969	+	1.69194i
$696$	1440.00			0.0784239
$697$		−	15488.0i		−	0.841678i
$698$			6220.00i			0.337293i
$699$	4816.00			0.260598
$700$	10400.0	+	7800.00i	0.561547	+	0.421160i
$701$	26942.0			1.45162			0.725810	−	0.687895i	$-0.241465\pi$
$701$	26942.0			1.45162			0.725810	+	0.687895i	$0.241465\pi$
$702$		−	2400.00i		−	0.129034i
$703$			14160.0i			0.759679i
$704$	1792.00			0.0959354
$705$	−2260.00	−	4520.00i	−0.120733	−	0.241465i
$706$	−15616.0			−0.832459
$707$		−	15028.0i		−	0.799415i
$708$			160.000i			0.00849318i
$709$	1950.00			0.103292			0.0516458	−	0.998665i	$-0.483553\pi$
$709$	1950.00			0.103292			0.0516458	+	0.998665i	$0.483553\pi$
$710$	−22560.0	+	11280.0i	−1.19248	+	0.596241i
$711$	16560.0			0.873486
$712$		−	3920.00i		−	0.206332i
$713$		−	7424.00i		−	0.389945i
$714$	6656.00			0.348872
$715$	3360.00	−	1680.00i	0.175744	−	0.0878719i
$716$	10000.0			0.521952
$717$		−	11360.0i		−	0.591697i
$718$			18480.0i			0.960540i
$719$	−12080.0			−0.626576			−0.313288	−	0.949658i	$-0.601430\pi$
$719$	−12080.0			−0.626576			−0.313288	+	0.949658i	$0.601430\pi$
$720$	−1840.00	−	3680.00i	−0.0952399	−	0.190480i
$721$	38012.0			1.96344
$722$		−	6518.00i		−	0.335976i
$723$			556.000i			0.0286001i
$724$	10312.0			0.529340
$725$	6750.00	−	9000.00i	0.345778	−	0.461037i
$726$	−2188.00			−0.111852
$727$			17226.0i			0.878785i	0.898295	+	0.439393i	$0.144806\pi$
$727$			17226.0i			0.878785i	−0.898295	+	0.439393i	$0.855194\pi$
$728$		−	2496.00i		−	0.127071i
$729$	4283.00			0.217599
$730$	−6320.00	−	12640.0i	−0.320430	−	0.640859i
$731$	−23168.0			−1.17223
$732$			4336.00i			0.218939i
$733$			788.000i			0.0397073i	0.999803	+	0.0198536i	$0.00632003\pi$
$733$			788.000i			0.0397073i	−0.999803	+	0.0198536i	$0.993680\pi$
$734$	6428.00			0.323245
$735$	6660.00	−	3330.00i	0.334228	−	0.167114i
$736$	−1856.00			−0.0929525
$737$			12152.0i			0.607360i
$738$			11132.0i			0.555250i
$739$	2060.00			0.102542			0.0512709	−	0.998685i	$-0.483673\pi$
$739$	2060.00			0.102542			0.0512709	+	0.998685i	$0.483673\pi$
$740$	−9440.00	+	4720.00i	−0.468948	+	0.234474i
$741$	−1440.00			−0.0713896
$742$		−	5616.00i		−	0.277857i
$743$			3258.00i			0.160867i	0.996760	+	0.0804337i	$0.0256305\pi$
$743$			3258.00i			0.160867i	−0.996760	+	0.0804337i	$0.974369\pi$
$744$	2048.00			0.100918
$745$	1250.00	+	2500.00i	0.0614718	+	0.122944i
$746$	−696.000			−0.0341587
$747$			10994.0i			0.538487i
$748$		−	7168.00i		−	0.350385i
$749$	−25116.0			−1.22526
$750$	5500.00	+	1000.00i	0.267775	+	0.0486864i
$751$	−4528.00			−0.220012			−0.110006	−	0.993931i	$-0.535087\pi$
$751$	−4528.00			−0.220012			−0.110006	+	0.993931i	$0.535087\pi$
$752$			3616.00i			0.175348i
$753$		−	6504.00i		−	0.314766i
$754$	−2160.00			−0.104327
$755$	−10760.0	−	21520.0i	−0.518671	−	1.03734i
$756$	−10400.0			−0.500323
$757$			18236.0i			0.875560i	0.899082	+	0.437780i	$0.144235\pi$
$757$			18236.0i			0.875560i	−0.899082	+	0.437780i	$0.855765\pi$
$758$		−	9880.00i		−	0.473427i
$759$	−3248.00			−0.155329
$760$	−4800.00	+	2400.00i	−0.229098	+	0.114549i
$761$	−18678.0			−0.889720			−0.444860	−	0.895600i	$-0.646747\pi$
$761$	−18678.0			−0.889720			−0.444860	+	0.895600i	$0.646747\pi$
$762$		−	6136.00i		−	0.291711i
$763$		−	9620.00i		−	0.456445i
$764$	3072.00			0.145473
$765$	−14720.0	+	7360.00i	−0.695690	+	0.347845i
$766$	12284.0			0.579424
$767$		−	240.000i		−	0.0112984i
$768$		−	512.000i		−	0.0240563i
$769$	−27390.0			−1.28441			−0.642203	−	0.766534i	$-0.721980\pi$
$769$	−27390.0			−1.28441			−0.642203	+	0.766534i	$0.721980\pi$
$770$	−7280.00	−	14560.0i	−0.340718	−	0.681436i
$771$	3072.00			0.143496
$772$		−	10432.0i		−	0.486342i
$773$		−	9252.00i		−	0.430493i	−0.976560	−	0.215247i	$-0.930944\pi$
$773$		−	9252.00i		−	0.430493i	0.976560	−	0.215247i	$-0.0690555\pi$
$774$	16652.0			0.773312
$775$	9600.00	−	12800.0i	0.444958	−	0.593277i
$776$	11648.0			0.538839

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	10.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -2.00000i q^{3} -4.00000 q^{4} +(-5.00000 - 10.0000i) q^{5} +4.00000 q^{6} +26.0000i q^{7} -8.00000i q^{8} +23.0000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} -2.00000i q^{3} -4.00000 q^{4} +(-5.00000 - 10.0000i) q^{5} +4.00000 q^{6} +26.0000i q^{7} -8.00000i q^{8} +23.0000 q^{9} +(20.0000 - 10.0000i) q^{10} -28.0000 q^{11} +8.00000i q^{12} -12.0000i q^{13} -52.0000 q^{14} +(-20.0000 + 10.0000i) q^{15} +16.0000 q^{16} -64.0000i q^{17} +46.0000i q^{18} +60.0000 q^{19} +(20.0000 + 40.0000i) q^{20} +52.0000 q^{21} -56.0000i q^{22} +58.0000i q^{23} -16.0000 q^{24} +(-75.0000 + 100.000i) q^{25} +24.0000 q^{26} -100.000i q^{27} -104.000i q^{28} -90.0000 q^{29} +(-20.0000 - 40.0000i) q^{30} -128.000 q^{31} +32.0000i q^{32} +56.0000i q^{33} +128.000 q^{34} +(260.000 - 130.000i) q^{35} -92.0000 q^{36} +236.000i q^{37} +120.000i q^{38} -24.0000 q^{39} +(-80.0000 + 40.0000i) q^{40} +242.000 q^{41} +104.000i q^{42} -362.000i q^{43} +112.000 q^{44} +(-115.000 - 230.000i) q^{45} -116.000 q^{46} +226.000i q^{47} -32.0000i q^{48} -333.000 q^{49} +(-200.000 - 150.000i) q^{50} -128.000 q^{51} +48.0000i q^{52} +108.000i q^{53} +200.000 q^{54} +(140.000 + 280.000i) q^{55} +208.000 q^{56} -120.000i q^{57} -180.000i q^{58} +20.0000 q^{59} +(80.0000 - 40.0000i) q^{60} +542.000 q^{61} -256.000i q^{62} +598.000i q^{63} -64.0000 q^{64} +(-120.000 + 60.0000i) q^{65} -112.000 q^{66} -434.000i q^{67} +256.000i q^{68} +116.000 q^{69} +(260.000 + 520.000i) q^{70} -1128.00 q^{71} -184.000i q^{72} -632.000i q^{73} -472.000 q^{74} +(200.000 + 150.000i) q^{75} -240.000 q^{76} -728.000i q^{77} -48.0000i q^{78} +720.000 q^{79} +(-80.0000 - 160.000i) q^{80} +421.000 q^{81} +484.000i q^{82} +478.000i q^{83} -208.000 q^{84} +(-640.000 + 320.000i) q^{85} +724.000 q^{86} +180.000i q^{87} +224.000i q^{88} +490.000 q^{89} +(460.000 - 230.000i) q^{90} +312.000 q^{91} -232.000i q^{92} +256.000i q^{93} -452.000 q^{94} +(-300.000 - 600.000i) q^{95} +64.0000 q^{96} +1456.00i q^{97} -666.000i q^{98} -644.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 10 q^{5} + 8 q^{6} + 46 q^{9}+O(q^{10}) \) 2 * q - 8 * q^4 - 10 * q^5 + 8 * q^6 + 46 * q^9	\( 2 q - 8 q^{4} - 10 q^{5} + 8 q^{6} + 46 q^{9} + 40 q^{10} - 56 q^{11} - 104 q^{14} - 40 q^{15} + 32 q^{16} + 120 q^{19} + 40 q^{20} + 104 q^{21} - 32 q^{24} - 150 q^{25} + 48 q^{26} - 180 q^{29} - 40 q^{30} - 256 q^{31} + 256 q^{34} + 520 q^{35} - 184 q^{36} - 48 q^{39} - 160 q^{40} + 484 q^{41} + 224 q^{44} - 230 q^{45} - 232 q^{46} - 666 q^{49} - 400 q^{50} - 256 q^{51} + 400 q^{54} + 280 q^{55} + 416 q^{56} + 40 q^{59} + 160 q^{60} + 1084 q^{61} - 128 q^{64} - 240 q^{65} - 224 q^{66} + 232 q^{69} + 520 q^{70} - 2256 q^{71} - 944 q^{74} + 400 q^{75} - 480 q^{76} + 1440 q^{79} - 160 q^{80} + 842 q^{81} - 416 q^{84} - 1280 q^{85} + 1448 q^{86} + 980 q^{89} + 920 q^{90} + 624 q^{91} - 904 q^{94} - 600 q^{95} + 128 q^{96} - 1288 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 10 * q^5 + 8 * q^6 + 46 * q^9 + 40 * q^10 - 56 * q^11 - 104 * q^14 - 40 * q^15 + 32 * q^16 + 120 * q^19 + 40 * q^20 + 104 * q^21 - 32 * q^24 - 150 * q^25 + 48 * q^26 - 180 * q^29 - 40 * q^30 - 256 * q^31 + 256 * q^34 + 520 * q^35 - 184 * q^36 - 48 * q^39 - 160 * q^40 + 484 * q^41 + 224 * q^44 - 230 * q^45 - 232 * q^46 - 666 * q^49 - 400 * q^50 - 256 * q^51 + 400 * q^54 + 280 * q^55 + 416 * q^56 + 40 * q^59 + 160 * q^60 + 1084 * q^61 - 128 * q^64 - 240 * q^65 - 224 * q^66 + 232 * q^69 + 520 * q^70 - 2256 * q^71 - 944 * q^74 + 400 * q^75 - 480 * q^76 + 1440 * q^79 - 160 * q^80 + 842 * q^81 - 416 * q^84 - 1280 * q^85 + 1448 * q^86 + 980 * q^89 + 920 * q^90 + 624 * q^91 - 904 * q^94 - 600 * q^95 + 128 * q^96 - 1288 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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