LMFDB - Embedded newform 75.9.f.a.7.2

Newspace parameters

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.f (of order $4$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$30.5533957546$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.7
Dual form			75.9.f.a.43.2

$q$-expansion

$f(q)$	$=$	$q+(7.34847 + 7.34847i) q^{2} +(33.0681 - 33.0681i) q^{3} -148.000i q^{4} +486.000 q^{6} +(2872.03 + 2872.03i) q^{7} +(2968.78 - 2968.78i) q^{8} -2187.00i q^{9} +O(q^{10})$	\(q+(7.34847 + 7.34847i) q^{2} +(33.0681 - 33.0681i) q^{3} -148.000i q^{4} +486.000 q^{6} +(2872.03 + 2872.03i) q^{7} +(2968.78 - 2968.78i) q^{8} -2187.00i q^{9} -234.000 q^{11} +(-4894.08 - 4894.08i) q^{12} +(-11930.2 + 11930.2i) q^{13} +42210.0i q^{14} +5744.00 q^{16} +(89012.0 + 89012.0i) q^{17} +(16071.1 - 16071.1i) q^{18} -181693. i q^{19} +189945. q^{21} +(-1719.54 - 1719.54i) q^{22} +(269432. - 269432. i) q^{23} -196344. i q^{24} -175338. q^{26} +(-72320.0 - 72320.0i) q^{27} +(425060. - 425060. i) q^{28} +240174. i q^{29} +836725. q^{31} +(-717798. - 717798. i) q^{32} +(-7737.94 + 7737.94i) q^{33} +1.30820e6i q^{34} -323676. q^{36} +(608282. + 608282. i) q^{37} +(1.33517e6 - 1.33517e6i) q^{38} +789021. i q^{39} +2.82222e6 q^{41} +(1.39580e6 + 1.39580e6i) q^{42} +(2.80202e6 - 2.80202e6i) q^{43} +34632.0i q^{44} +3.95982e6 q^{46} +(-5.39321e6 - 5.39321e6i) q^{47} +(189943. - 189943. i) q^{48} +1.07323e7i q^{49} +5.88692e6 q^{51} +(1.76568e6 + 1.76568e6i) q^{52} +(-1.19258e6 + 1.19258e6i) q^{53} -1.06288e6i q^{54} +1.70528e7 q^{56} +(-6.00824e6 - 6.00824e6i) q^{57} +(-1.76491e6 + 1.76491e6i) q^{58} +1.27860e7i q^{59} +517403. q^{61} +(6.14865e6 + 6.14865e6i) q^{62} +(6.28112e6 - 6.28112e6i) q^{63} -1.20199e7i q^{64} -113724. q^{66} +(2.06617e6 + 2.06617e6i) q^{67} +(1.31738e7 - 1.31738e7i) q^{68} -1.78192e7i q^{69} -2.08286e7 q^{71} +(-6.49273e6 - 6.49273e6i) q^{72} +(2.88730e7 - 2.88730e7i) q^{73} +8.93988e6i q^{74} -2.68906e7 q^{76} +(-672054. - 672054. i) q^{77} +(-5.79810e6 + 5.79810e6i) q^{78} +4.21879e7i q^{79} -4.78297e6 q^{81} +(2.07390e7 + 2.07390e7i) q^{82} +(-6.66763e7 + 6.66763e7i) q^{83} -2.81119e7i q^{84} +4.11811e7 q^{86} +(7.94210e6 + 7.94210e6i) q^{87} +(-694695. + 694695. i) q^{88} -9.51614e7i q^{89} -6.85279e7 q^{91} +(-3.98759e7 - 3.98759e7i) q^{92} +(2.76689e7 - 2.76689e7i) q^{93} -7.92637e7i q^{94} -4.74725e7 q^{96} +(7.08678e7 + 7.08678e7i) q^{97} +(-7.88658e7 + 7.88658e7i) q^{98} +511758. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 1944 q^{6}+O(q^{10}) $ 4 * q + 1944 * q^6	$ 4 q + 1944 q^{6} - 936 q^{11} + 22976 q^{16} + 759780 q^{21} - 701352 q^{26} + 3346900 q^{31} - 1294704 q^{36} + 11288880 q^{41} + 15839280 q^{46} + 23547672 q^{51} + 68211360 q^{56} + 2069612 q^{61} - 454896 q^{66} - 83314512 q^{71} - 107562256 q^{76} - 19131876 q^{81} + 164724264 q^{86} - 274111740 q^{91} - 189889920 q^{96}+O(q^{100}) $ 4 * q + 1944 * q^6 - 936 * q^11 + 22976 * q^16 + 759780 * q^21 - 701352 * q^26 + 3346900 * q^31 - 1294704 * q^36 + 11288880 * q^41 + 15839280 * q^46 + 23547672 * q^51 + 68211360 * q^56 + 2069612 * q^61 - 454896 * q^66 - 83314512 * q^71 - 107562256 * q^76 - 19131876 * q^81 + 164724264 * q^86 - 274111740 * q^91 - 189889920 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	7.34847	+	7.34847i	0.459279	+	0.459279i	0.898419	−	0.439140i	$-0.144717\pi$
$2$	7.34847	+	7.34847i	0.459279	+	0.459279i	−0.439140	+	0.898419i	$0.644717\pi$
$3$	33.0681	−	33.0681i	0.408248	−	0.408248i
$3$	33.0681	−	33.0681i	0.408248	−	0.408248i
$4$		−	148.000i		−	0.578125i
$5$		0			0
$5$		0			0
$6$	486.000			0.375000
$7$	2872.03	+	2872.03i	1.19618	+	1.19618i	0.975301	+	0.220878i	$0.0708922\pi$
$7$	2872.03	+	2872.03i	1.19618	+	1.19618i	0.220878	+	0.975301i	$0.429108\pi$
$8$	2968.78	−	2968.78i	0.724800	−	0.724800i
$9$		−	2187.00i		−	0.333333i
$10$		0			0
$11$	−234.000			−0.0159825			−0.00799126	−	0.999968i	$-0.502544\pi$
$11$	−234.000			−0.0159825			−0.00799126	+	0.999968i	$0.502544\pi$
$12$	−4894.08	−	4894.08i	−0.236019	−	0.236019i
$13$	−11930.2	+	11930.2i	−0.417711	+	0.417711i	−0.884414	−	0.466703i	$-0.845442\pi$
$13$	−11930.2	+	11930.2i	−0.417711	+	0.417711i	0.466703	+	0.884414i	$0.345442\pi$
$14$			42210.0i			1.09876i
$15$		0			0
$16$	5744.00			0.0876465
$17$	89012.0	+	89012.0i	1.06574	+	1.06574i	0.997681	+	0.0680630i	$0.0216819\pi$
$17$	89012.0	+	89012.0i	1.06574	+	1.06574i	0.0680630	+	0.997681i	$0.478318\pi$
$18$	16071.1	−	16071.1i	0.153093	−	0.153093i
$19$		−	181693.i		−	1.39420i	−0.716976	−	0.697098i	$-0.754474\pi$
$19$		−	181693.i		−	1.39420i	0.716976	−	0.697098i	$-0.245526\pi$
$20$		0			0
$21$	189945.			0.976676
$22$	−1719.54	−	1719.54i	−0.00734044	−	0.00734044i
$23$	269432.	−	269432.i	0.962803	−	0.962803i	−0.0365300	−	0.999333i	$-0.511630\pi$
$23$	269432.	−	269432.i	0.962803	−	0.962803i	0.999333	+	0.0365300i	$0.0116305\pi$
$24$		−	196344.i		−	0.591797i
$25$		0			0
$26$	−175338.			−0.383692
$27$	−72320.0	−	72320.0i	−0.136083	−	0.136083i
$28$	425060.	−	425060.i	0.691541	−	0.691541i
$29$			240174.i			0.339574i	0.985481	+	0.169787i	$0.0543079\pi$
$29$			240174.i			0.339574i	−0.985481	+	0.169787i	$0.945692\pi$
$30$		0			0
$31$	836725.			0.906016			0.453008	−	0.891506i	$-0.350351\pi$
$31$	836725.			0.906016			0.453008	+	0.891506i	$0.350351\pi$
$32$	−717798.	−	717798.i	−0.684546	−	0.684546i
$33$	−7737.94	+	7737.94i	−0.00652483	+	0.00652483i
$34$			1.30820e6i			0.978948i
$35$		0			0
$36$	−323676.			−0.192708
$37$	608282.	+	608282.i	0.324562	+	0.324562i	0.850514	−	0.525952i	$-0.176291\pi$
$37$	608282.	+	608282.i	0.324562	+	0.324562i	−0.525952	+	0.850514i	$0.676291\pi$
$38$	1.33517e6	−	1.33517e6i	0.640325	−	0.640325i
$39$			789021.i			0.341059i
$40$		0			0
$41$	2.82222e6			0.998747			0.499373	−	0.866387i	$-0.333564\pi$
$41$	2.82222e6			0.998747			0.499373	+	0.866387i	$0.333564\pi$
$42$	1.39580e6	+	1.39580e6i	0.448567	+	0.448567i
$43$	2.80202e6	−	2.80202e6i	0.819590	−	0.819590i	−0.166458	−	0.986049i	$-0.553233\pi$
$43$	2.80202e6	−	2.80202e6i	0.819590	−	0.819590i	0.986049	+	0.166458i	$0.0532330\pi$
$44$			34632.0i			0.00923989i
$45$		0			0
$46$	3.95982e6			0.884391
$47$	−5.39321e6	−	5.39321e6i	−1.10524	−	1.10524i	−0.993768	−	0.111471i	$-0.964444\pi$
$47$	−5.39321e6	−	5.39321e6i	−1.10524	−	1.10524i	−0.111471	−	0.993768i	$-0.535556\pi$
$48$	189943.	−	189943.i	0.0357815	−	0.0357815i
$49$			1.07323e7i			1.86169i
$50$		0			0
$51$	5.88692e6			0.870176
$52$	1.76568e6	+	1.76568e6i	0.241489	+	0.241489i
$53$	−1.19258e6	+	1.19258e6i	−0.151142	+	0.151142i	−0.778628	−	0.627486i	$-0.784084\pi$
$53$	−1.19258e6	+	1.19258e6i	−0.151142	+	0.151142i	0.627486	+	0.778628i	$0.284084\pi$
$54$		−	1.06288e6i		−	0.125000i
$55$		0			0
$56$	1.70528e7			1.73398
$57$	−6.00824e6	−	6.00824e6i	−0.569178	−	0.569178i
$58$	−1.76491e6	+	1.76491e6i	−0.155959	+	0.155959i
$59$			1.27860e7i			1.05518i	0.849500	+	0.527588i	$0.176904\pi$
$59$			1.27860e7i			1.05518i	−0.849500	+	0.527588i	$0.823096\pi$
$60$		0			0
$61$	517403.			0.0373688			0.0186844	−	0.999825i	$-0.494052\pi$
$61$	517403.			0.0373688			0.0186844	+	0.999825i	$0.494052\pi$
$62$	6.14865e6	+	6.14865e6i	0.416115	+	0.416115i
$63$	6.28112e6	−	6.28112e6i	0.398726	−	0.398726i
$64$		−	1.20199e7i		−	0.716442i
$65$		0			0
$66$	−113724.			−0.00599344
$67$	2.06617e6	+	2.06617e6i	0.102534	+	0.102534i	0.756513	−	0.653979i	$-0.226902\pi$
$67$	2.06617e6	+	2.06617e6i	0.102534	+	0.102534i	−0.653979	+	0.756513i	$0.726902\pi$
$68$	1.31738e7	−	1.31738e7i	0.616133	−	0.616133i
$69$		−	1.78192e7i		−	0.786125i
$70$		0			0
$71$	−2.08286e7			−0.819648			−0.409824	−	0.912165i	$-0.634410\pi$
$71$	−2.08286e7			−0.819648			−0.409824	+	0.912165i	$0.634410\pi$
$72$	−6.49273e6	−	6.49273e6i	−0.241600	−	0.241600i
$73$	2.88730e7	−	2.88730e7i	1.01672	−	1.01672i	0.0168598	−	0.999858i	$-0.494633\pi$
$73$	2.88730e7	−	2.88730e7i	1.01672	−	1.01672i	0.999858	−	0.0168598i	$-0.00536689\pi$
$74$			8.93988e6i			0.298129i
$75$		0			0
$76$	−2.68906e7			−0.806019
$77$	−672054.	−	672054.i	−0.0191180	−	0.0191180i
$78$	−5.79810e6	+	5.79810e6i	−0.156642	+	0.156642i
$79$			4.21879e7i			1.08313i	0.840659	+	0.541564i	$0.182168\pi$
$79$			4.21879e7i			1.08313i	−0.840659	+	0.541564i	$0.817832\pi$
$80$		0			0
$81$	−4.78297e6			−0.111111
$82$	2.07390e7	+	2.07390e7i	0.458704	+	0.458704i
$83$	−6.66763e7	+	6.66763e7i	−1.40494	+	1.40494i	−0.621642	+	0.783302i	$0.713534\pi$
$83$	−6.66763e7	+	6.66763e7i	−1.40494	+	1.40494i	−0.783302	+	0.621642i	$0.786466\pi$
$84$		−	2.81119e7i		−	0.564641i
$85$		0			0
$86$	4.11811e7			0.752842
$87$	7.94210e6	+	7.94210e6i	0.138630	+	0.138630i
$88$	−694695.	+	694695.i	−0.0115841	+	0.0115841i
$89$		−	9.51614e7i		−	1.51670i	−0.651845	−	0.758352i	$-0.726005\pi$
$89$		−	9.51614e7i		−	1.51670i	0.651845	−	0.758352i	$-0.273995\pi$
$90$		0			0
$91$	−6.85279e7			−0.999314
$92$	−3.98759e7	−	3.98759e7i	−0.556620	−	0.556620i
$93$	2.76689e7	−	2.76689e7i	0.369880	−	0.369880i
$94$		−	7.92637e7i		−	1.01523i
$95$		0			0
$96$	−4.74725e7			−0.558929
$97$	7.08678e7	+	7.08678e7i	0.800501	+	0.800501i	0.983174	−	0.182672i	$-0.0584748\pi$
$97$	7.08678e7	+	7.08678e7i	0.800501	+	0.800501i	−0.182672	+	0.983174i	$0.558475\pi$
$98$	−7.88658e7	+	7.88658e7i	−0.855036	+	0.855036i
$99$			511758.i			0.00532750i
$100$		0			0
$101$	−8.63379e7			−0.829690			−0.414845	−	0.909892i	$-0.636164\pi$
$101$	−8.63379e7			−0.829690			−0.414845	+	0.909892i	$0.636164\pi$
$102$	4.32598e7	+	4.32598e7i	0.399654	+	0.399654i
$103$	−7.61161e7	+	7.61161e7i	−0.676282	+	0.676282i	−0.959157	−	0.282875i	$-0.908712\pi$
$103$	−7.61161e7	+	7.61161e7i	−0.676282	+	0.676282i	0.282875	+	0.959157i	$0.408712\pi$
$104$			7.08366e7i			0.605514i
$105$		0			0
$106$	−1.75273e7			−0.138833
$107$	−1.09281e8	−	1.09281e8i	−0.833701	−	0.833701i	0.154320	−	0.988021i	$-0.450681\pi$
$107$	−1.09281e8	−	1.09281e8i	−0.833701	−	0.833701i	−0.988021	+	0.154320i	$0.950681\pi$
$108$	−1.07034e7	+	1.07034e7i	−0.0786728	+	0.0786728i
$109$		−	5.04631e7i		−	0.357494i	−0.983895	−	0.178747i	$-0.942796\pi$
$109$		−	5.04631e7i		−	0.357494i	0.983895	−	0.178747i	$-0.0572043\pi$
$110$		0			0
$111$	4.02295e7			0.265004
$112$	1.64969e7	+	1.64969e7i	0.104841	+	0.104841i
$113$	−1.21244e8	+	1.21244e8i	−0.743611	+	0.743611i	−0.973271	−	0.229660i	$-0.926238\pi$
$113$	−1.21244e8	+	1.21244e8i	−0.743611	+	0.743611i	0.229660	+	0.973271i	$0.426238\pi$
$114$		−	8.83028e7i		−	0.522823i
$115$		0			0
$116$	3.55458e7			0.196316
$117$	2.60914e7	+	2.60914e7i	0.139237	+	0.139237i
$118$	−9.39572e7	+	9.39572e7i	−0.484621	+	0.484621i
$119$			5.11290e8i			2.54964i
$120$		0			0
$121$	−2.14304e8			−0.999745
$122$	3.80212e6	+	3.80212e6i	0.0171627	+	0.0171627i
$123$	9.33255e7	−	9.33255e7i	0.407737	−	0.407737i
$124$		−	1.23835e8i		−	0.523791i
$125$		0			0
$126$	9.23133e7			0.366254
$127$	−9.55780e7	−	9.55780e7i	−0.367403	−	0.367403i	0.499126	−	0.866529i	$-0.333654\pi$
$127$	−9.55780e7	−	9.55780e7i	−0.367403	−	0.367403i	−0.866529	+	0.499126i	$0.833654\pi$
$128$	−9.54285e7	+	9.54285e7i	−0.355499	+	0.355499i
$129$		−	1.85315e8i		−	0.669193i
$130$		0			0
$131$	−3.13838e8			−1.06566			−0.532832	−	0.846221i	$-0.678872\pi$
$131$	−3.13838e8			−1.06566			−0.532832	+	0.846221i	$0.678872\pi$
$132$	1.14521e6	+	1.14521e6i	0.00377217	+	0.00377217i
$133$	5.21827e8	−	5.21827e8i	1.66771	−	1.66771i
$134$			3.03664e7i			0.0941834i
$135$		0			0
$136$	5.28514e8			1.54490
$137$	5.04691e7	+	5.04691e7i	0.143266	+	0.143266i	0.775102	−	0.631836i	$-0.217698\pi$
$137$	5.04691e7	+	5.04691e7i	0.143266	+	0.143266i	−0.631836	+	0.775102i	$0.717698\pi$
$138$	1.30944e8	−	1.30944e8i	0.361051	−	0.361051i
$139$		−	5.26688e8i		−	1.41089i	−0.708763	−	0.705446i	$-0.750747\pi$
$139$		−	5.26688e8i		−	1.41089i	0.708763	−	0.705446i	$-0.249253\pi$
$140$		0			0
$141$	−3.56687e8			−0.902423
$142$	−1.53059e8	−	1.53059e8i	−0.376447	−	0.376447i
$143$	2.79168e6	−	2.79168e6i	0.00667607	−	0.00667607i
$144$		−	1.25621e7i		−	0.0292155i
$145$		0			0
$146$	4.24345e8			0.933915
$147$	3.54896e8	+	3.54896e8i	0.760032	+	0.760032i
$148$	9.00257e7	−	9.00257e7i	0.187638	−	0.187638i
$149$			8.99246e8i			1.82446i	0.409683	+	0.912228i	$0.365639\pi$
$149$			8.99246e8i			1.82446i	−0.409683	+	0.912228i	$0.634361\pi$
$150$		0			0
$151$	8.05999e8			1.55034			0.775170	−	0.631753i	$-0.217664\pi$
$151$	8.05999e8			1.55034			0.775170	+	0.631753i	$0.217664\pi$
$152$	−5.39407e8	−	5.39407e8i	−1.01051	−	1.01051i
$153$	1.94669e8	−	1.94669e8i	0.355248	−	0.355248i
$154$		−	9.87714e6i		−	0.0175610i
$155$		0			0
$156$	1.16775e8			0.197175
$157$	4.78600e8	+	4.78600e8i	0.787724	+	0.787724i	0.981121	−	0.193397i	$-0.0619504\pi$
$157$	4.78600e8	+	4.78600e8i	0.787724	+	0.787724i	−0.193397	+	0.981121i	$0.561950\pi$
$158$	−3.10017e8	+	3.10017e8i	−0.497458	+	0.497458i
$159$			7.88729e7i			0.123407i
$160$		0			0
$161$	1.54763e9			2.30337
$162$	−3.51475e7	−	3.51475e7i	−0.0510310	−	0.0510310i
$163$	4.56533e8	−	4.56533e8i	0.646729	−	0.646729i	−0.305472	−	0.952201i	$-0.598814\pi$
$163$	4.56533e8	−	4.56533e8i	0.646729	−	0.646729i	0.952201	+	0.305472i	$0.0988144\pi$
$164$		−	4.17689e8i		−	0.577401i
$165$		0			0
$166$	−9.79937e8			−1.29052
$167$	2.74258e8	+	2.74258e8i	0.352609	+	0.352609i	0.861079	−	0.508470i	$-0.169789\pi$
$167$	2.74258e8	+	2.74258e8i	0.352609	+	0.352609i	−0.508470	+	0.861079i	$0.669789\pi$
$168$	5.63905e8	−	5.63905e8i	0.707895	−	0.707895i
$169$			5.31069e8i			0.651035i
$170$		0			0
$171$	−3.97363e8			−0.464732
$172$	−4.14698e8	−	4.14698e8i	−0.473826	−	0.473826i
$173$	−1.75277e8	+	1.75277e8i	−0.195677	+	0.195677i	−0.798144	−	0.602467i	$-0.794185\pi$
$173$	−1.75277e8	+	1.75277e8i	−0.195677	+	0.195677i	0.602467	+	0.798144i	$0.294185\pi$
$174$			1.16725e8i			0.127340i
$175$		0			0
$176$	−1.34410e6			−0.00140081
$177$	4.22807e8	+	4.22807e8i	0.430774	+	0.430774i
$178$	6.99291e8	−	6.99291e8i	0.696591	−	0.696591i
$179$			1.65138e9i			1.60855i	0.594255	+	0.804277i	$0.297447\pi$
$179$			1.65138e9i			1.60855i	−0.594255	+	0.804277i	$0.702553\pi$
$180$		0			0
$181$	−3.19153e8			−0.297362			−0.148681	−	0.988885i	$-0.547503\pi$
$181$	−3.19153e8			−0.297362			−0.148681	+	0.988885i	$0.547503\pi$
$182$	−5.03575e8	−	5.03575e8i	−0.458964	−	0.458964i
$183$	1.71095e7	−	1.71095e7i	0.0152558	−	0.0152558i
$184$		−	1.59977e9i		−	1.39568i
$185$		0			0
$186$	4.06648e8			0.339756
$187$	−2.08288e7	−	2.08288e7i	−0.0170333	−	0.0170333i
$188$	−7.98195e8	+	7.98195e8i	−0.638966	+	0.638966i
$189$		−	4.15410e8i		−	0.325559i
$190$		0			0
$191$	−1.13627e9			−0.853785			−0.426893	−	0.904302i	$-0.640392\pi$
$191$	−1.13627e9			−0.853785			−0.426893	+	0.904302i	$0.640392\pi$
$192$	−3.97476e8	−	3.97476e8i	−0.292486	−	0.292486i
$193$	−1.71674e9	+	1.71674e9i	−1.23730	+	1.23730i	−0.276197	+	0.961101i	$0.589074\pi$
$193$	−1.71674e9	+	1.71674e9i	−1.23730	+	1.23730i	−0.961101	+	0.276197i	$0.910926\pi$
$194$			1.04154e9i			0.735307i
$195$		0			0
$196$	1.58838e9			1.07629
$197$	−4.17614e8	−	4.17614e8i	−0.277275	−	0.277275i	0.554745	−	0.832020i	$-0.312816\pi$
$197$	−4.17614e8	−	4.17614e8i	−0.277275	−	0.277275i	−0.832020	+	0.554745i	$0.812816\pi$
$198$	−3.76064e6	+	3.76064e6i	−0.00244681	+	0.00244681i
$199$			1.06994e9i			0.682258i	0.940017	+	0.341129i	$0.110809\pi$
$199$			1.06994e9i			0.682258i	−0.940017	+	0.341129i	$0.889191\pi$
$200$		0			0
$201$	1.36649e8			0.0837186
$202$	−6.34451e8	−	6.34451e8i	−0.381059	−	0.381059i
$203$	−6.89786e8	+	6.89786e8i	−0.406191	+	0.406191i
$204$		−	8.71264e8i		−	0.503071i
$205$		0			0
$206$	−1.11867e9			−0.621205
$207$	−5.89247e8	−	5.89247e8i	−0.320934	−	0.320934i
$208$	−6.85273e7	+	6.85273e7i	−0.0366109	+	0.0366109i
$209$			4.25162e7i			0.0222828i
$210$		0			0
$211$	−2.26915e9			−1.14481			−0.572406	−	0.819971i	$-0.693990\pi$
$211$	−2.26915e9			−1.14481			−0.572406	+	0.819971i	$0.693990\pi$
$212$	1.76502e8	+	1.76502e8i	0.0873790	+	0.0873790i
$213$	−6.88763e8	+	6.88763e8i	−0.334620	+	0.334620i
$214$		−	1.60610e9i		−	0.765803i
$215$		0			0
$216$	−4.29404e8			−0.197266
$217$	2.40310e9	+	2.40310e9i	1.08376	+	1.08376i
$218$	3.70827e8	−	3.70827e8i	0.164189	−	0.164189i
$219$		−	1.90955e9i		−	0.830146i
$220$		0			0
$221$	−2.12387e9			−0.890346
$222$	2.95625e8	+	2.95625e8i	0.121711	+	0.121711i
$223$	1.99573e9	−	1.99573e9i	0.807015	−	0.807015i	−0.177166	−	0.984181i	$-0.556693\pi$
$223$	1.99573e9	−	1.99573e9i	0.807015	−	0.807015i	0.984181	+	0.177166i	$0.0566928\pi$
$224$		−	4.12307e9i		−	1.63768i
$225$		0			0
$226$	−1.78191e9			−0.683050
$227$	1.32121e9	+	1.32121e9i	0.497588	+	0.497588i	0.910686	−	0.413099i	$-0.135554\pi$
$227$	1.32121e9	+	1.32121e9i	0.497588	+	0.497588i	−0.413099	+	0.910686i	$0.635554\pi$
$228$	−8.89220e8	+	8.89220e8i	−0.329056	+	0.329056i
$229$			2.03118e9i			0.738595i	0.929311	+	0.369297i	$0.120402\pi$
$229$			2.03118e9i			0.738595i	−0.929311	+	0.369297i	$0.879598\pi$
$230$		0			0
$231$	−4.44471e7			−0.0156097
$232$	7.13024e8	+	7.13024e8i	0.246123	+	0.246123i
$233$	−1.51062e9	+	1.51062e9i	−0.512545	+	0.512545i	−0.915305	−	0.402760i	$-0.868051\pi$
$233$	−1.51062e9	+	1.51062e9i	−0.512545	+	0.512545i	0.402760	+	0.915305i	$0.368051\pi$
$234$			3.83464e8i			0.127897i
$235$		0			0
$236$	1.89232e9			0.610024
$237$	1.39508e9	+	1.39508e9i	0.442185	+	0.442185i
$238$	−3.75720e9	+	3.75720e9i	−1.17100	+	1.17100i
$239$		−	3.44530e9i		−	1.05593i	−0.849266	−	0.527965i	$-0.822955\pi$
$239$		−	3.44530e9i		−	1.05593i	0.849266	−	0.527965i	$-0.177045\pi$
$240$		0			0
$241$	2.45959e7			0.00729112			0.00364556	−	0.999993i	$-0.498840\pi$
$241$	2.45959e7			0.00729112			0.00364556	+	0.999993i	$0.498840\pi$
$242$	−1.57481e9	−	1.57481e9i	−0.459162	−	0.459162i
$243$	−1.58164e8	+	1.58164e8i	−0.0453609	+	0.0453609i
$244$		−	7.65756e7i		−	0.0216039i
$245$		0			0
$246$	1.37160e9			0.374530
$247$	2.16764e9	+	2.16764e9i	0.582371	+	0.582371i
$248$	2.48405e9	−	2.48405e9i	0.656681	−	0.656681i
$249$			4.40972e9i			1.14713i
$250$		0			0
$251$	−3.62820e9			−0.914106			−0.457053	−	0.889440i	$-0.651095\pi$
$251$	−3.62820e9			−0.914106			−0.457053	+	0.889440i	$0.651095\pi$
$252$	−9.29606e8	−	9.29606e8i	−0.230514	−	0.230514i
$253$	−6.30470e7	+	6.30470e7i	−0.0153880	+	0.0153880i
$254$		−	1.40470e9i		−	0.337482i
$255$		0			0
$256$	−4.47960e9			−1.04299
$257$	−1.05639e9	−	1.05639e9i	−0.242154	−	0.242154i	0.575586	−	0.817741i	$-0.304774\pi$
$257$	−1.05639e9	−	1.05639e9i	−0.242154	−	0.242154i	−0.817741	+	0.575586i	$0.804774\pi$
$258$	1.36178e9	−	1.36178e9i	0.307346	−	0.307346i
$259$			3.49400e9i			0.776469i
$260$		0			0
$261$	5.25261e8			0.113191
$262$	−2.30623e9	−	2.30623e9i	−0.489438	−	0.489438i
$263$	2.89677e9	−	2.89677e9i	0.605468	−	0.605468i	−0.336290	−	0.941758i	$-0.609172\pi$
$263$	2.89677e9	−	2.89677e9i	0.605468	−	0.605468i	0.941758	+	0.336290i	$0.109172\pi$
$264$			4.59445e7i			0.00945840i
$265$		0			0
$266$	7.66926e9			1.53189
$267$	−3.14681e9	−	3.14681e9i	−0.619192	−	0.619192i
$268$	3.05794e8	−	3.05794e8i	0.0592774	−	0.0592774i
$269$		−	4.75635e9i		−	0.908374i	−0.890906	−	0.454187i	$-0.849930\pi$
$269$		−	4.75635e9i		−	0.908374i	0.890906	−	0.454187i	$-0.150070\pi$
$270$		0			0
$271$	4.98291e9			0.923860			0.461930	−	0.886916i	$-0.347157\pi$
$271$	4.98291e9			0.923860			0.461930	+	0.886916i	$0.347157\pi$
$272$	5.11285e8	+	5.11285e8i	0.0934087	+	0.0934087i
$273$	−2.26609e9	+	2.26609e9i	−0.407968	+	0.407968i
$274$			7.41741e8i			0.131598i
$275$		0			0
$276$	−2.63724e9			−0.454479
$277$	−2.27910e9	−	2.27910e9i	−0.387119	−	0.387119i	0.486539	−	0.873659i	$-0.338259\pi$
$277$	−2.27910e9	−	2.27910e9i	−0.387119	−	0.387119i	−0.873659	+	0.486539i	$0.838259\pi$
$278$	3.87035e9	−	3.87035e9i	0.647994	−	0.647994i
$279$		−	1.82992e9i		−	0.302005i
$280$		0			0
$281$	−1.00667e9			−0.161459			−0.0807294	−	0.996736i	$-0.525725\pi$
$281$	−1.00667e9			−0.161459			−0.0807294	+	0.996736i	$0.525725\pi$
$282$	−2.62110e9	−	2.62110e9i	−0.414464	−	0.414464i
$283$	3.14496e9	−	3.14496e9i	0.490308	−	0.490308i	−0.418095	−	0.908403i	$-0.637302\pi$
$283$	3.14496e9	−	3.14496e9i	0.490308	−	0.490308i	0.908403	+	0.418095i	$0.137302\pi$
$284$			3.08264e9i			0.473859i
$285$		0			0
$286$	4.10291e7			0.00613236
$287$	8.10549e9	+	8.10549e9i	1.19468	+	1.19468i
$288$	−1.56983e9	+	1.56983e9i	−0.228182	+	0.228182i
$289$			8.87052e9i			1.27162i
$290$		0			0
$291$	4.68693e9			0.653607
$292$	−4.27320e9	−	4.27320e9i	−0.587790	−	0.587790i
$293$	3.60077e9	−	3.60077e9i	0.488568	−	0.488568i	−0.419286	−	0.907854i	$-0.637720\pi$
$293$	3.60077e9	−	3.60077e9i	0.488568	−	0.488568i	0.907854	+	0.419286i	$0.137720\pi$
$294$			5.21589e9i			0.698134i
$295$		0			0
$296$	3.61171e9			0.470485
$297$	1.69229e7	+	1.69229e7i	0.00217494	+	0.00217494i
$298$	−6.60808e9	+	6.60808e9i	−0.837935	+	0.837935i
$299$			6.42877e9i			0.804346i
$300$		0			0
$301$	1.60949e10			1.96075
$302$	5.92286e9	+	5.92286e9i	0.712039	+	0.712039i
$303$	−2.85503e9	+	2.85503e9i	−0.338719	+	0.338719i
$304$		−	1.04364e9i		−	0.122196i
$305$		0			0
$306$	2.86104e9			0.326316
$307$	−1.23211e10	−	1.23211e10i	−1.38706	−	1.38706i	−0.831420	−	0.555644i	$-0.812472\pi$
$307$	−1.23211e10	−	1.23211e10i	−1.38706	−	1.38706i	−0.555644	−	0.831420i	$-0.687528\pi$
$308$	−9.94640e7	+	9.94640e7i	−0.0110526	+	0.0110526i
$309$			5.03403e9i			0.552182i
$310$		0			0
$311$	−8.62846e9			−0.922341			−0.461170	−	0.887312i	$-0.652570\pi$
$311$	−8.62846e9			−0.922341			−0.461170	+	0.887312i	$0.652570\pi$
$312$	2.34243e9	+	2.34243e9i	0.247200	+	0.247200i
$313$	2.96428e9	−	2.96428e9i	0.308846	−	0.308846i	−0.535616	−	0.844462i	$-0.679921\pi$
$313$	2.96428e9	−	2.96428e9i	0.308846	−	0.308846i	0.844462	+	0.535616i	$0.179921\pi$
$314$			7.03396e9i			0.723571i
$315$		0			0
$316$	6.24381e9			0.626183
$317$	−1.98335e9	−	1.98335e9i	−0.196409	−	0.196409i	0.602050	−	0.798459i	$-0.294351\pi$
$317$	−1.98335e9	−	1.98335e9i	−0.196409	−	0.196409i	−0.798459	+	0.602050i	$0.794351\pi$
$318$	−5.79595e8	+	5.79595e8i	−0.0566782	+	0.0566782i
$319$		−	5.62007e7i		−	0.00542724i
$320$		0			0
$321$	−7.22745e9			−0.680714
$322$	1.13727e10	+	1.13727e10i	1.05789	+	1.05789i
$323$	1.61729e10	−	1.61729e10i	1.48586	−	1.48586i
$324$			7.07879e8i			0.0642361i
$325$		0			0
$326$	6.70964e9			0.594058
$327$	−1.66872e9	−	1.66872e9i	−0.145946	−	0.145946i
$328$	8.37855e9	−	8.37855e9i	0.723892	−	0.723892i
$329$		−	3.09789e10i		−	2.64413i
$330$		0			0
$331$	−4.30793e9			−0.358886			−0.179443	−	0.983768i	$-0.557430\pi$
$331$	−4.30793e9			−0.358886			−0.179443	+	0.983768i	$0.557430\pi$
$332$	9.86809e9	+	9.86809e9i	0.812233	+	0.812233i
$333$	1.33031e9	−	1.33031e9i	0.108187	−	0.108187i
$334$			4.03075e9i			0.323892i
$335$		0			0
$336$	1.09104e9			0.0856023
$337$	−1.18877e8	−	1.18877e8i	−0.00921677	−	0.00921677i	0.702483	−	0.711700i	$-0.252075\pi$
$337$	−1.18877e8	−	1.18877e8i	−0.00921677	−	0.00921677i	−0.711700	+	0.702483i	$0.752075\pi$
$338$	−3.90255e9	+	3.90255e9i	−0.299007	+	0.299007i
$339$			8.01860e9i			0.607156i
$340$		0			0
$341$	−1.95794e8			−0.0144804
$342$	−2.92001e9	−	2.92001e9i	−0.213442	−	0.213442i
$343$	−1.42667e10	+	1.42667e10i	−1.03074	+	1.03074i
$344$		−	1.66372e10i		−	1.18808i
$345$		0			0
$346$	−2.57603e9			−0.179741
$347$	−1.63550e10	−	1.63550e10i	−1.12806	−	1.12806i	−0.990492	−	0.137569i	$-0.956071\pi$
$347$	−1.63550e10	−	1.63550e10i	−1.12806	−	1.12806i	−0.137569	−	0.990492i	$-0.543929\pi$
$348$	1.17543e9	−	1.17543e9i	0.0801457	−	0.0801457i
$349$		−	2.63406e10i		−	1.77552i	−0.460310	−	0.887758i	$-0.652262\pi$
$349$		−	2.63406e10i		−	1.77552i	0.460310	−	0.887758i	$-0.347738\pi$
$350$		0			0
$351$	1.72559e9			0.113686
$352$	1.67965e8	+	1.67965e8i	0.0109408	+	0.0109408i
$353$	4.48404e9	−	4.48404e9i	0.288783	−	0.288783i	−0.547816	−	0.836599i	$-0.684541\pi$
$353$	4.48404e9	−	4.48404e9i	0.288783	−	0.288783i	0.836599	+	0.547816i	$0.184541\pi$
$354$			6.21398e9i			0.395691i
$355$		0			0
$356$	−1.40839e10			−0.876845
$357$	1.69074e10	+	1.69074e10i	1.04089	+	1.04089i
$358$	−1.21351e10	+	1.21351e10i	−0.738775	+	0.738775i
$359$			1.76017e10i			1.05968i	0.848097	+	0.529841i	$0.177749\pi$
$359$			1.76017e10i			1.05968i	−0.848097	+	0.529841i	$0.822251\pi$
$360$		0			0
$361$	−1.60288e10			−0.943782
$362$	−2.34529e9	−	2.34529e9i	−0.136572	−	0.136572i
$363$	−7.08663e9	+	7.08663e9i	−0.408144	+	0.408144i
$364$			1.01421e10i			0.577729i
$365$		0			0
$366$	2.51458e8			0.0140133
$367$	−5.58417e9	−	5.58417e9i	−0.307818	−	0.307818i	0.536245	−	0.844063i	$-0.319842\pi$
$367$	−5.58417e9	−	5.58417e9i	−0.307818	−	0.307818i	−0.844063	+	0.536245i	$0.819842\pi$
$368$	1.54762e9	−	1.54762e9i	0.0843863	−	0.0843863i
$369$		−	6.17220e9i		−	0.332916i
$370$		0			0
$371$	−6.85026e9			−0.361586
$372$	−4.09500e9	−	4.09500e9i	−0.213837	−	0.213837i
$373$	−1.94503e9	+	1.94503e9i	−0.100483	+	0.100483i	−0.755561	−	0.655078i	$-0.772636\pi$
$373$	−1.94503e9	+	1.94503e9i	−0.100483	+	0.100483i	0.655078	+	0.755561i	$0.272636\pi$
$374$		−	3.06120e8i		−	0.0156461i
$375$		0			0
$376$	−3.20225e10			−1.60215
$377$	−2.86533e9	−	2.86533e9i	−0.141844	−	0.141844i
$378$	3.05263e9	−	3.05263e9i	0.149522	−	0.149522i
$379$		−	1.29779e10i		−	0.628996i	−0.949258	−	0.314498i	$-0.898164\pi$
$379$		−	1.29779e10i		−	0.628996i	0.949258	−	0.314498i	$-0.101836\pi$
$380$		0			0
$381$	−6.32117e9			−0.299984
$382$	−8.34986e9	−	8.34986e9i	−0.392126	−	0.392126i
$383$	−1.84005e10	+	1.84005e10i	−0.855135	+	0.855135i	−0.990760	−	0.135625i	$-0.956696\pi$
$383$	−1.84005e10	+	1.84005e10i	−0.855135	+	0.855135i	0.135625	+	0.990760i	$0.456696\pi$
$384$			6.31128e9i			0.290264i
$385$		0			0
$386$	−2.52308e10			−1.13653
$387$	−6.12801e9	−	6.12801e9i	−0.273197	−	0.273197i
$388$	1.04884e10	−	1.04884e10i	0.462790	−	0.462790i
$389$		−	1.69662e9i		−	0.0740944i	−0.999314	−	0.0370472i	$-0.988205\pi$
$389$		−	1.69662e9i		−	0.0740944i	0.999314	−	0.0370472i	$-0.0117952\pi$
$390$		0			0
$391$	4.79653e10			2.05220
$392$	3.18618e10	+	3.18618e10i	1.34935	+	1.34935i
$393$	−1.03780e10	+	1.03780e10i	−0.435056	+	0.435056i
$394$		−	6.13765e9i		−	0.254693i
$395$		0			0
$396$	7.57402e7			0.00307996
$397$	1.61839e10	+	1.61839e10i	0.651510	+	0.651510i	0.953357	−	0.301846i	$-0.0976029\pi$
$397$	1.61839e10	+	1.61839e10i	0.651510	+	0.651510i	−0.301846	+	0.953357i	$0.597603\pi$
$398$	−7.86245e9	+	7.86245e9i	−0.313347	+	0.313347i
$399$		−	3.45117e10i		−	1.36168i
$400$		0			0
$401$	−3.67977e9			−0.142313			−0.0711563	−	0.997465i	$-0.522669\pi$
$401$	−3.67977e9			−0.142313			−0.0711563	+	0.997465i	$0.522669\pi$
$402$	1.00416e9	+	1.00416e9i	0.0384502	+	0.0384502i
$403$	−9.98233e9	+	9.98233e9i	−0.378453	+	0.378453i
$404$			1.27780e10i			0.479664i
$405$		0			0
$406$	−1.01377e10			−0.373110
$407$	−1.42338e8	−	1.42338e8i	−0.00518732	−	0.00518732i
$408$	1.74770e10	−	1.74770e10i	0.630704	−	0.630704i
$409$			2.44824e10i			0.874905i	0.899242	+	0.437452i	$0.144119\pi$
$409$			2.44824e10i			0.874905i	−0.899242	+	0.437452i	$0.855881\pi$
$410$		0			0
$411$	3.33783e9			0.116976
$412$	1.12652e10	+	1.12652e10i	0.390976	+	0.390976i
$413$	−3.67216e10	+	3.67216e10i	−1.26218	+	1.26218i
$414$		−	8.66013e9i		−	0.294797i
$415$		0			0
$416$	1.71270e10			0.571885
$417$	−1.74166e10	−	1.74166e10i	−0.575995	−	0.575995i
$418$	−3.12429e8	+	3.12429e8i	−0.0102340	+	0.0102340i
$419$			2.07283e10i			0.672522i	0.941769	+	0.336261i	$0.109162\pi$
$419$			2.07283e10i			0.672522i	−0.941769	+	0.336261i	$0.890838\pi$
$420$		0			0
$421$	−1.53035e10			−0.487150			−0.243575	−	0.969882i	$-0.578320\pi$
$421$	−1.53035e10			−0.487150			−0.243575	+	0.969882i	$0.578320\pi$
$422$	−1.66748e10	−	1.66748e10i	−0.525788	−	0.525788i
$423$	−1.17950e10	+	1.17950e10i	−0.368413	+	0.368413i
$424$			7.08104e9i			0.219095i
$425$		0			0
$426$	−1.01227e10			−0.307368
$427$	1.48600e9	+	1.48600e9i	0.0446998	+	0.0446998i
$428$	−1.61736e10	+	1.61736e10i	−0.481983	+	0.481983i
$429$		−	1.84631e8i		−	0.00545099i
$430$		0			0
$431$	3.04251e10			0.881705			0.440852	−	0.897580i	$-0.354676\pi$
$431$	3.04251e10			0.881705			0.440852	+	0.897580i	$0.354676\pi$
$432$	−4.15406e8	−	4.15406e8i	−0.0119272	−	0.0119272i
$433$	3.30421e9	−	3.30421e9i	0.0939974	−	0.0939974i	−0.658544	−	0.752542i	$-0.728828\pi$
$433$	3.30421e9	−	3.30421e9i	0.0939974	−	0.0939974i	0.752542	+	0.658544i	$0.228828\pi$
$434$			3.53182e10i			0.995495i
$435$		0			0
$436$	−7.46855e9			−0.206676
$437$	−4.89538e10	−	4.89538e10i	−1.34234	−	1.34234i
$438$	1.40323e10	−	1.40323e10i	0.381269	−	0.381269i
$439$		−	3.09771e10i		−	0.834032i	−0.908899	−	0.417016i	$-0.863076\pi$
$439$		−	3.09771e10i		−	0.834032i	0.908899	−	0.417016i	$-0.136924\pi$
$440$		0			0
$441$	2.34715e10			0.620563
$442$	−1.56072e10	−	1.56072e10i	−0.408917	−	0.408917i
$443$	3.59125e10	−	3.59125e10i	0.932460	−	0.932460i	−0.0653990	−	0.997859i	$-0.520832\pi$
$443$	3.59125e10	−	3.59125e10i	0.932460	−	0.932460i	0.997859	+	0.0653990i	$0.0208320\pi$
$444$		−	5.95396e9i		−	0.153205i
$445$		0			0
$446$	2.93311e10			0.741291
$447$	2.97364e10	+	2.97364e10i	0.744831	+	0.744831i
$448$	3.45215e10	−	3.45215e10i	0.856993	−	0.856993i
$449$		−	3.25449e10i		−	0.800751i	−0.916351	−	0.400376i	$-0.868880\pi$
$449$		−	3.25449e10i		−	0.800751i	0.916351	−	0.400376i	$-0.131120\pi$
$450$		0			0
$451$	−6.60399e8			−0.0159625
$452$	1.79441e10	+	1.79441e10i	0.429900	+	0.429900i
$453$	2.66529e10	−	2.66529e10i	0.632924	−	0.632924i
$454$			1.94178e10i			0.457064i
$455$		0			0
$456$	−3.56743e10			−0.825081
$457$	1.89124e10	+	1.89124e10i	0.433593	+	0.433593i	0.889849	−	0.456256i	$-0.150810\pi$
$457$	1.89124e10	+	1.89124e10i	0.433593	+	0.433593i	−0.456256	+	0.889849i	$0.650810\pi$
$458$	−1.49261e10	+	1.49261e10i	−0.339221	+	0.339221i
$459$		−	1.28747e10i		−	0.290059i
$460$		0			0
$461$	3.64502e10			0.807042			0.403521	−	0.914970i	$-0.367786\pi$
$461$	3.64502e10			0.807042			0.403521	+	0.914970i	$0.367786\pi$
$462$	−3.26618e8	−	3.26618e8i	−0.00716923	−	0.00716923i
$463$	5.92927e10	−	5.92927e10i	1.29026	−	1.29026i	0.355636	−	0.934625i	$-0.384264\pi$
$463$	5.92927e10	−	5.92927e10i	1.29026	−	1.29026i	0.934625	−	0.355636i	$-0.115736\pi$
$464$			1.37956e9i			0.0297624i
$465$		0			0
$466$	−2.22015e10			−0.470803
$467$	−3.42203e10	−	3.42203e10i	−0.719476	−	0.719476i	0.249021	−	0.968498i	$-0.419891\pi$
$467$	−3.42203e10	−	3.42203e10i	−0.719476	−	0.719476i	−0.968498	+	0.249021i	$0.919891\pi$
$468$	3.86153e9	−	3.86153e9i	0.0804964	−	0.0804964i
$469$			1.18682e10i			0.245298i
$470$		0			0
$471$	3.16528e10			0.643174
$472$	3.79587e10	+	3.79587e10i	0.764792	+	0.764792i
$473$	−6.55672e8	+	6.55672e8i	−0.0130991	+	0.0130991i
$474$			2.05033e10i			0.406173i
$475$		0			0
$476$	7.56709e10			1.47401
$477$	2.60818e9	+	2.60818e9i	0.0503807	+	0.0503807i
$478$	2.53177e10	−	2.53177e10i	0.484967	−	0.484967i
$479$			4.05336e10i			0.769970i	0.922923	+	0.384985i	$0.125793\pi$
$479$			4.05336e10i			0.769970i	−0.922923	+	0.384985i	$0.874207\pi$
$480$		0			0
$481$	−1.45139e10			−0.271146
$482$	1.80742e8	+	1.80742e8i	0.00334866	+	0.00334866i
$483$	5.11772e10	−	5.11772e10i	0.940346	−	0.940346i
$484$			3.17170e10i			0.577977i
$485$		0			0
$486$	−2.32452e9			−0.0416667
$487$	3.57760e10	+	3.57760e10i	0.636028	+	0.636028i	0.949573	−	0.313545i	$-0.101517\pi$
$487$	3.57760e10	+	3.57760e10i	0.636028	+	0.636028i	−0.313545	+	0.949573i	$0.601517\pi$
$488$	1.53606e9	−	1.53606e9i	0.0270849	−	0.0270849i
$489$		−	3.01934e10i		−	0.528052i
$490$		0			0
$491$	−5.52678e10			−0.950925			−0.475462	−	0.879736i	$-0.657719\pi$
$491$	−5.52678e10			−0.950925			−0.475462	+	0.879736i	$0.657719\pi$
$492$	−1.38122e10	−	1.38122e10i	−0.235723	−	0.235723i
$493$	−2.13784e10	+	2.13784e10i	−0.361899	+	0.361899i
$494$			3.18577e10i			0.534942i
$495$		0			0
$496$	4.80615e9			0.0794091
$497$	−5.98204e10	−	5.98204e10i	−0.980446	−	0.980446i
$498$	−3.24047e10	+	3.24047e10i	−0.526854	+	0.526854i
$499$		−	2.17632e10i		−	0.351011i	−0.984478	−	0.175506i	$-0.943844\pi$
$499$		−	2.17632e10i		−	0.351011i	0.984478	−	0.175506i	$-0.0561560\pi$
$500$		0			0
$501$	1.81384e10			0.287904
$502$	−2.66617e10	−	2.66617e10i	−0.419830	−	0.419830i
$503$	−1.36083e10	+	1.36083e10i	−0.212585	+	0.212585i	−0.805365	−	0.592779i	$-0.798031\pi$
$503$	−1.36083e10	+	1.36083e10i	−0.212585	+	0.212585i	0.592779	+	0.805365i	$0.298031\pi$
$504$		−	3.72946e10i		−	0.577994i
$505$		0			0
$506$	−9.26598e8			−0.0141348
$507$	1.75615e10	+	1.75615e10i	0.265784	+	0.265784i
$508$	−1.41455e10	+	1.41455e10i	−0.212405	+	0.212405i
$509$			8.50274e10i			1.26674i	0.773849	+	0.633370i	$0.218329\pi$
$509$			8.50274e10i			1.26674i	−0.773849	+	0.633370i	$0.781671\pi$
$510$		0			0
$511$	1.65848e11			2.43235
$512$	−8.48852e9	−	8.48852e9i	−0.123524	−	0.123524i
$513$	−1.31400e10	+	1.31400e10i	−0.189726	+	0.189726i
$514$		−	1.55257e10i		−	0.222433i
$515$		0			0
$516$	−2.74266e10			−0.386877
$517$	1.26201e9	+	1.26201e9i	0.0176645	+	0.0176645i
$518$	−2.56756e10	+	2.56756e10i	−0.356616	+	0.356616i
$519$			1.15921e10i			0.159770i
$520$		0			0
$521$	−1.04749e11			−1.42167			−0.710836	−	0.703358i	$-0.751683\pi$
$521$	−1.04749e11			−1.42167			−0.710836	+	0.703358i	$0.751683\pi$
$522$	3.85986e9	+	3.85986e9i	0.0519864	+	0.0519864i
$523$	1.81952e10	−	1.81952e10i	0.243192	−	0.243192i	−0.574977	−	0.818169i	$-0.694989\pi$
$523$	1.81952e10	−	1.81952e10i	0.243192	−	0.243192i	0.818169	+	0.574977i	$0.194989\pi$
$524$			4.64480e10i			0.616087i
$525$		0			0
$526$	4.25737e10			0.556158
$527$	7.44786e10	+	7.44786e10i	0.965581	+	0.965581i
$528$	−4.44467e7	+	4.44467e7i	−0.000571879	+	0.000571879i
$529$		−	6.68758e10i		−	0.853977i
$530$		0			0
$531$	2.79629e10			0.351726
$532$	−7.72304e10	−	7.72304e10i	−0.964144	−	0.964144i
$533$	−3.36698e10	+	3.36698e10i	−0.417187	+	0.417187i
$534$		−	4.62485e10i		−	0.568764i
$535$		0			0
$536$	1.22680e10			0.148633
$537$	5.46081e10	+	5.46081e10i	0.656689	+	0.656689i
$538$	3.49519e10	−	3.49519e10i	0.417198	−	0.417198i
$539$		−	2.51135e9i		−	0.0297545i
$540$		0			0
$541$	−6.68793e10			−0.780733			−0.390367	−	0.920660i	$-0.627652\pi$
$541$	−6.68793e10			−0.780733			−0.390367	+	0.920660i	$0.627652\pi$
$542$	3.66168e10	+	3.66168e10i	0.424310	+	0.424310i
$543$	−1.05538e10	+	1.05538e10i	−0.121397	+	0.121397i
$544$		−	1.27785e11i		−	1.45910i
$545$		0			0
$546$	−3.33046e10			−0.374743
$547$	1.80208e9	+	1.80208e9i	0.0201291	+	0.0201291i	0.717100	−	0.696971i	$-0.245469\pi$
$547$	1.80208e9	+	1.80208e9i	0.0201291	+	0.0201291i	−0.696971	+	0.717100i	$0.745469\pi$
$548$	7.46942e9	−	7.46942e9i	0.0828256	−	0.0828256i
$549$		−	1.13156e9i		−	0.0124563i
$550$		0			0
$551$	4.36379e10			0.473432
$552$	−5.29013e10	−	5.29013e10i	−0.569784	−	0.569784i
$553$	−1.21165e11	+	1.21165e11i	−1.29562	+	1.29562i
$554$		−	3.34958e10i		−	0.355592i
$555$		0			0
$556$	−7.79498e10			−0.815672
$557$	−2.12113e10	−	2.12113e10i	−0.220367	−	0.220367i	0.588286	−	0.808653i	$-0.299803\pi$
$557$	−2.12113e10	−	2.12113e10i	−0.220367	−	0.220367i	−0.808653	+	0.588286i	$0.799803\pi$
$558$	1.34471e10	−	1.34471e10i	0.138705	−	0.138705i
$559$			6.68575e10i			0.684704i
$560$		0			0
$561$	−1.37754e9			−0.0139076
$562$	−7.39748e9	−	7.39748e9i	−0.0741547	−	0.0741547i
$563$	−2.95427e10	+	2.95427e10i	−0.294047	+	0.294047i	−0.838677	−	0.544629i	$-0.816670\pi$
$563$	−2.95427e10	+	2.95427e10i	−0.294047	+	0.294047i	0.544629	+	0.838677i	$0.316670\pi$
$564$			5.27896e10i			0.521713i
$565$		0			0
$566$	4.62212e10			0.450377
$567$	−1.37368e10	−	1.37368e10i	−0.132909	−	0.132909i
$568$	−6.18356e10	+	6.18356e10i	−0.594081	+	0.594081i
$569$			6.53964e10i			0.623886i	0.950101	+	0.311943i	$0.100980\pi$
$569$			6.53964e10i			0.623886i	−0.950101	+	0.311943i	$0.899020\pi$
$570$		0			0
$571$	−1.87778e11			−1.76644			−0.883221	−	0.468957i	$-0.844630\pi$
$571$	−1.87778e11			−1.76644			−0.883221	+	0.468957i	$0.844630\pi$
$572$	−4.13168e8	−	4.13168e8i	−0.00385960	−	0.00385960i
$573$	−3.75744e10	+	3.75744e10i	−0.348556	+	0.348556i
$574$			1.19126e11i			1.09738i
$575$		0			0
$576$	−2.62875e10			−0.238814
$577$	−9.25087e7	−	9.25087e7i	−0.000834602	−	0.000834602i	0.706689	−	0.707524i	$-0.250188\pi$
$577$	−9.25087e7	−	9.25087e7i	−0.000834602	−	0.000834602i	−0.707524	+	0.706689i	$0.750188\pi$
$578$	−6.51847e10	+	6.51847e10i	−0.584029	+	0.584029i
$579$			1.13538e11i			1.01025i
$580$		0			0
$581$	−3.82992e11			−3.36113
$582$	3.44418e10	+	3.44418e10i	0.300188	+	0.300188i
$583$	2.79064e8	−	2.79064e8i	0.00241563	−	0.00241563i
$584$		−	1.71435e11i		−	1.47383i
$585$		0			0
$586$	5.29204e10			0.448779
$587$	−1.12620e11	−	1.12620e11i	−0.948560	−	0.948560i	0.0501803	−	0.998740i	$-0.484020\pi$
$587$	−1.12620e11	−	1.12620e11i	−0.948560	−	0.948560i	−0.998740	+	0.0501803i	$0.984020\pi$
$588$	5.25246e10	−	5.25246e10i	0.439393	−	0.439393i
$589$		−	1.52027e11i		−	1.26316i
$590$		0			0
$591$	−2.76194e10			−0.226394
$592$	3.49397e9	+	3.49397e9i	0.0284467	+	0.0284467i
$593$	1.37020e11	−	1.37020e11i	1.10807	−	1.10807i	0.114661	−	0.993405i	$-0.463422\pi$
$593$	1.37020e11	−	1.37020e11i	1.10807	−	1.10807i	0.993405	−	0.114661i	$-0.0365782\pi$
$594$			2.48714e8i			0.00199781i
$595$		0			0
$596$	1.33088e11			1.05476
$597$	3.53810e10	+	3.53810e10i	0.278531	+	0.278531i
$598$	−4.72416e10	+	4.72416e10i	−0.369420	+	0.369420i
$599$		−	1.62549e11i		−	1.26263i	−0.775525	−	0.631317i	$-0.782515\pi$
$599$		−	1.62549e11i		−	1.26263i	0.775525	−	0.631317i	$-0.217485\pi$
$600$		0			0
$601$	1.32519e10			0.101574			0.0507869	−	0.998710i	$-0.483827\pi$
$601$	1.32519e10			0.101574			0.0507869	+	0.998710i	$0.483827\pi$
$602$	1.18273e11	+	1.18273e11i	0.900534	+	0.900534i
$603$	4.51872e9	−	4.51872e9i	0.0341780	−	0.0341780i
$604$		−	1.19288e11i		−	0.896290i
$605$		0			0
$606$	−4.19602e10			−0.311134
$607$	8.11866e10	+	8.11866e10i	0.598039	+	0.598039i	0.939791	−	0.341751i	$-0.111020\pi$
$607$	8.11866e10	+	8.11866e10i	0.598039	+	0.598039i	−0.341751	+	0.939791i	$0.611020\pi$
$608$	−1.30419e11	+	1.30419e11i	−0.954391	+	0.954391i
$609$			4.56199e10i			0.331654i
$610$		0			0
$611$	1.28685e11			0.923340
$612$	−2.88111e10	−	2.88111e10i	−0.205378	−	0.205378i
$613$	4.13613e10	−	4.13613e10i	0.292922	−	0.292922i	−0.545311	−	0.838234i	$-0.683589\pi$
$613$	4.13613e10	−	4.13613e10i	0.292922	−	0.292922i	0.838234	+	0.545311i	$0.183589\pi$
$614$		−	1.81083e11i		−	1.27410i
$615$		0			0
$616$	−3.99036e9			−0.0277134
$617$	1.45292e11	+	1.45292e11i	1.00254	+	1.00254i	0.999997	+	0.00254243i	$0.000809283\pi$
$617$	1.45292e11	+	1.45292e11i	1.00254	+	1.00254i	0.00254243	+	0.999997i	$0.499191\pi$
$618$	−3.69924e10	+	3.69924e10i	−0.253606	+	0.253606i
$619$			1.15947e11i			0.789765i	0.918732	+	0.394882i	$0.129215\pi$
$619$			1.15947e11i			0.789765i	−0.918732	+	0.394882i	$0.870785\pi$
$620$		0			0
$621$	−3.89706e10			−0.262042
$622$	−6.34059e10	−	6.34059e10i	−0.423612	−	0.423612i
$623$	2.73306e11	−	2.73306e11i	1.81425	−	1.81425i
$624$			4.53214e9i			0.0298927i
$625$		0			0
$626$	4.35658e10			0.283693
$627$	1.40593e9	+	1.40593e9i	0.00909690	+	0.00909690i
$628$	7.08328e10	−	7.08328e10i	0.455403	−	0.455403i
$629$			1.08289e11i			0.691800i
$630$		0			0
$631$	−3.77923e10			−0.238389			−0.119194	−	0.992871i	$-0.538031\pi$
$631$	−3.77923e10			−0.238389			−0.119194	+	0.992871i	$0.538031\pi$
$632$	1.25247e11	+	1.25247e11i	0.785052	+	0.785052i
$633$	−7.50366e10	+	7.50366e10i	−0.467367	+	0.467367i
$634$		−	2.91491e10i		−	0.180413i
$635$		0			0
$636$	1.16732e10			0.0713446
$637$	−1.28039e11	−	1.28039e11i	−0.777648	−	0.777648i
$638$	4.12989e8	−	4.12989e8i	0.00249262	−	0.00249262i
$639$			4.55522e10i			0.273216i
$640$		0			0
$641$	−1.49659e10			−0.0886484			−0.0443242	−	0.999017i	$-0.514113\pi$
$641$	−1.49659e10			−0.0886484			−0.0443242	+	0.999017i	$0.514113\pi$
$642$	−5.31107e10	−	5.31107e10i	−0.312638	−	0.312638i
$643$	−1.62422e11	+	1.62422e11i	−0.950170	+	0.950170i	−0.998816	−	0.0486457i	$-0.984509\pi$
$643$	−1.62422e11	+	1.62422e11i	−0.950170	+	0.950170i	0.0486457	+	0.998816i	$0.484509\pi$
$644$		−	2.29049e11i		−	1.33164i
$645$		0			0
$646$	2.37692e11			1.36485
$647$	−7.96349e10	−	7.96349e10i	−0.454450	−	0.454450i	0.442378	−	0.896829i	$-0.354135\pi$
$647$	−7.96349e10	−	7.96349e10i	−0.454450	−	0.454450i	−0.896829	+	0.442378i	$0.854135\pi$
$648$	−1.41996e10	+	1.41996e10i	−0.0805334	+	0.0805334i
$649$		−	2.99191e9i		−	0.0168644i
$650$		0			0
$651$	1.58932e11			0.884885
$652$	−6.75669e10	−	6.75669e10i	−0.373890	−	0.373890i
$653$	−3.49865e9	+	3.49865e9i	−0.0192419	+	0.0192419i	−0.716662	−	0.697420i	$-0.754331\pi$
$653$	−3.49865e9	+	3.49865e9i	−0.0192419	+	0.0192419i	0.697420	+	0.716662i	$0.254331\pi$
$654$		−	2.45251e10i		−	0.134060i
$655$		0			0
$656$	1.62108e10			0.0875367
$657$	−6.31452e10	−	6.31452e10i	−0.338906	−	0.338906i
$658$	2.27647e11	−	2.27647e11i	1.21439	−	1.21439i
$659$			2.49093e11i			1.32075i	0.750938	+	0.660373i	$0.229602\pi$
$659$			2.49093e11i			1.32075i	−0.750938	+	0.660373i	$0.770398\pi$
$660$		0			0
$661$	8.01902e9			0.0420064			0.0210032	−	0.999779i	$-0.493314\pi$
$661$	8.01902e9			0.0420064			0.0210032	+	0.999779i	$0.493314\pi$
$662$	−3.16567e10	−	3.16567e10i	−0.164829	−	0.164829i
$663$	−7.02323e10	+	7.02323e10i	−0.363482	+	0.363482i
$664$			3.95894e11i			2.03661i
$665$		0			0
$666$	1.95515e10			0.0993765
$667$	6.47105e10	+	6.47105e10i	0.326942	+	0.326942i
$668$	4.05902e10	−	4.05902e10i	0.203852	−	0.203852i
$669$		−	1.31990e11i		−	0.658925i
$670$		0			0
$671$	−1.21072e8			−0.000597248
$672$	−1.36342e11	−	1.36342e11i	−0.668580	−	0.668580i
$673$	−1.75658e11	+	1.75658e11i	−0.856264	+	0.856264i	−0.990896	−	0.134632i	$-0.957015\pi$
$673$	−1.75658e11	+	1.75658e11i	−0.856264	+	0.856264i	0.134632	+	0.990896i	$0.457015\pi$
$674$		−	1.74713e9i		−	0.00846614i
$675$		0			0
$676$	7.85983e10			0.376380
$677$	1.49210e11	+	1.49210e11i	0.710301	+	0.710301i	0.966598	−	0.256297i	$-0.0825025\pi$
$677$	1.49210e11	+	1.49210e11i	0.710301	+	0.710301i	−0.256297	+	0.966598i	$0.582502\pi$
$678$	−5.89245e10	+	5.89245e10i	−0.278854	+	0.278854i
$679$			4.07068e11i			1.91509i
$680$		0			0
$681$	8.73801e10			0.406279
$682$	−1.43878e9	−	1.43878e9i	−0.00665056	−	0.00665056i
$683$	2.32934e11	−	2.32934e11i	1.07041	−	1.07041i	0.0730842	−	0.997326i	$-0.476716\pi$
$683$	2.32934e11	−	2.32934e11i	1.07041	−	1.07041i	0.997326	−	0.0730842i	$-0.0232842\pi$
$684$			5.88097e10i			0.268673i
$685$		0			0
$686$	−2.09677e11			−0.946792
$687$	6.71672e10	+	6.71672e10i	0.301530	+	0.301530i
$688$	1.60948e10	−	1.60948e10i	0.0718342	−	0.0718342i
$689$		−	2.84556e10i		−	0.126267i
$690$		0			0
$691$	2.85720e11			1.25322			0.626611	−	0.779332i	$-0.284441\pi$
$691$	2.85720e11			1.25322			0.626611	+	0.779332i	$0.284441\pi$
$692$	2.59410e10	+	2.59410e10i	0.113126	+	0.113126i
$693$	−1.46978e9	+	1.46978e9i	−0.00637265	+	0.00637265i
$694$		−	2.40368e11i		−	1.03619i
$695$		0			0
$696$	4.71567e10			0.200959
$697$	2.51211e11	+	2.51211e11i	1.06441	+	1.06441i
$698$	1.93563e11	−	1.93563e11i	0.815458	−	0.815458i
$699$			9.99068e10i			0.418491i
$700$		0			0
$701$	−6.05393e10			−0.250706			−0.125353	−	0.992112i	$-0.540006\pi$
$701$	−6.05393e10			−0.250706			−0.125353	+	0.992112i	$0.540006\pi$
$702$	1.26804e10	+	1.26804e10i	0.0522139	+	0.0522139i
$703$	1.10521e11	−	1.10521e11i	0.452503	−	0.452503i
$704$			2.81266e9i			0.0114505i
$705$		0			0
$706$	6.59017e10			0.265264
$707$	−2.47965e11	−	2.47965e11i	−0.992458	−	0.992458i
$708$	6.25755e10	−	6.25755e10i	0.249041	−	0.249041i
$709$			1.05416e10i			0.0417178i	0.999782	+	0.0208589i	$0.00664007\pi$
$709$			1.05416e10i			0.0417178i	−0.999782	+	0.0208589i	$0.993360\pi$
$710$		0			0
$711$	9.22650e10			0.361043
$712$	−2.82513e11	−	2.82513e11i	−1.09931	−	1.09931i
$713$	2.25440e11	−	2.25440e11i	0.872315	−	0.872315i
$714$			2.48487e11i			0.956116i
$715$		0			0
$716$	2.44405e11			0.929945
$717$	−1.13929e11	−	1.13929e11i	−0.431082	−	0.431082i
$718$	−1.29345e11	+	1.29345e11i	−0.486690	+	0.486690i
$719$		−	4.26585e10i		−	0.159621i	−0.996810	−	0.0798104i	$-0.974569\pi$
$719$		−	4.26585e10i		−	0.159621i	0.996810	−	0.0798104i	$-0.0254315\pi$
$720$		0			0
$721$	−4.37215e11			−1.61791
$722$	−1.17787e11	−	1.17787e11i	−0.433460	−	0.433460i
$723$	8.13339e8	−	8.13339e8i	0.00297659	−	0.00297659i
$724$			4.72347e10i			0.171912i
$725$		0			0
$726$	−1.04152e11			−0.374904
$727$	5.57133e10	+	5.57133e10i	0.199444	+	0.199444i	0.799762	−	0.600318i	$-0.204959\pi$
$727$	5.57133e10	+	5.57133e10i	0.199444	+	0.199444i	−0.600318	+	0.799762i	$0.704959\pi$
$728$	−2.03444e11	+	2.03444e11i	−0.724303	+	0.724303i
$729$			1.04604e10i			0.0370370i
$730$		0			0
$731$	4.98826e11			1.74695
$732$	−2.53221e9	−	2.53221e9i	−0.00881974	−	0.00881974i
$733$	−6.92078e10	+	6.92078e10i	−0.239739	+	0.239739i	−0.816742	−	0.577003i	$-0.804222\pi$
$733$	−6.92078e10	+	6.92078e10i	−0.239739	+	0.239739i	0.577003	+	0.816742i	$0.304222\pi$
$734$		−	8.20702e10i		−	0.282749i
$735$		0			0
$736$	−3.86795e11			−1.31817
$737$	−4.83484e8	−	4.83484e8i	−0.00163875	−	0.00163875i
$738$	4.53562e10	−	4.53562e10i	0.152901	−	0.152901i
$739$			5.42380e11i			1.81855i	0.416193	+	0.909276i	$0.363364\pi$
$739$			5.42380e11i			1.81855i	−0.416193	+	0.909276i	$0.636636\pi$
$740$		0			0
$741$	1.43360e11			0.475504
$742$	−5.03389e10	−	5.03389e10i	−0.166069	−	0.166069i
$743$	−1.05763e11	+	1.05763e11i	−0.347038	+	0.347038i	−0.859005	−	0.511967i	$-0.828917\pi$
$743$	−1.05763e11	+	1.05763e11i	−0.347038	+	0.347038i	0.511967	+	0.859005i	$0.328917\pi$
$744$		−	1.64286e11i		−	0.536178i
$745$		0			0
$746$	−2.85860e10			−0.0922992
$747$	1.45821e11	+	1.45821e11i	0.468314	+	0.468314i
$748$	−3.08266e9	+	3.08266e9i	−0.00984736	+	0.00984736i
$749$		−	6.27717e11i		−	1.99451i
$750$		0			0
$751$	4.27682e11			1.34450			0.672251	−	0.740323i	$-0.265328\pi$
$751$	4.27682e11			1.34450			0.672251	+	0.740323i	$0.265328\pi$
$752$	−3.09786e10	−	3.09786e10i	−0.0968703	−	0.0968703i
$753$	−1.19978e11	+	1.19978e11i	−0.373182	+	0.373182i
$754$		−	4.21116e10i		−	0.130292i
$755$		0			0
$756$	−6.14806e10			−0.188214
$757$	2.41193e11	+	2.41193e11i	0.734482	+	0.734482i	0.971504	−	0.237022i	$-0.0761715\pi$
$757$	2.41193e11	+	2.41193e11i	0.734482	+	0.734482i	−0.237022	+	0.971504i	$0.576171\pi$
$758$	9.53677e10	−	9.53677e10i	0.288885	−	0.288885i
$759$			4.16969e9i			0.0125643i
$760$		0			0
$761$	−4.31324e11			−1.28607			−0.643035	−	0.765837i	$-0.722325\pi$
$761$	−4.31324e11			−1.28607			−0.643035	+	0.765837i	$0.722325\pi$
$762$	−4.64509e10	−	4.64509e10i	−0.137776	−	0.137776i
$763$	1.44931e11	−	1.44931e11i	0.427627	−	0.427627i
$764$			1.68168e11i			0.493595i
$765$		0			0
$766$	−2.70431e11			−0.785491
$767$	−1.52540e11	−	1.52540e11i	−0.440759	−	0.440759i
$768$	−1.48132e11	+	1.48132e11i	−0.425798	+	0.425798i
$769$		−	3.88788e11i		−	1.11175i	−0.831265	−	0.555876i	$-0.812383\pi$
$769$		−	3.88788e11i		−	1.11175i	0.831265	−	0.555876i	$-0.187617\pi$
$770$		0			0
$771$	−6.98658e10			−0.197718
$772$	2.54077e11	+	2.54077e11i	0.715313	+	0.715313i
$773$	−2.51696e11	+	2.51696e11i	−0.704951	+	0.704951i	−0.965469	−	0.260518i	$-0.916107\pi$
$773$	−2.51696e11	+	2.51696e11i	−0.704951	+	0.704951i	0.260518	+	0.965469i	$0.416107\pi$

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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(7.34847 + 7.34847i) q^{2} +(33.0681 - 33.0681i) q^{3} -148.000i q^{4} +486.000 q^{6} +(2872.03 + 2872.03i) q^{7} +(2968.78 - 2968.78i) q^{8} -2187.00i q^{9} +O(q^{10})\)	\(q+(7.34847 + 7.34847i) q^{2} +(33.0681 - 33.0681i) q^{3} -148.000i q^{4} +486.000 q^{6} +(2872.03 + 2872.03i) q^{7} +(2968.78 - 2968.78i) q^{8} -2187.00i q^{9} -234.000 q^{11} +(-4894.08 - 4894.08i) q^{12} +(-11930.2 + 11930.2i) q^{13} +42210.0i q^{14} +5744.00 q^{16} +(89012.0 + 89012.0i) q^{17} +(16071.1 - 16071.1i) q^{18} -181693. i q^{19} +189945. q^{21} +(-1719.54 - 1719.54i) q^{22} +(269432. - 269432. i) q^{23} -196344. i q^{24} -175338. q^{26} +(-72320.0 - 72320.0i) q^{27} +(425060. - 425060. i) q^{28} +240174. i q^{29} +836725. q^{31} +(-717798. - 717798. i) q^{32} +(-7737.94 + 7737.94i) q^{33} +1.30820e6i q^{34} -323676. q^{36} +(608282. + 608282. i) q^{37} +(1.33517e6 - 1.33517e6i) q^{38} +789021. i q^{39} +2.82222e6 q^{41} +(1.39580e6 + 1.39580e6i) q^{42} +(2.80202e6 - 2.80202e6i) q^{43} +34632.0i q^{44} +3.95982e6 q^{46} +(-5.39321e6 - 5.39321e6i) q^{47} +(189943. - 189943. i) q^{48} +1.07323e7i q^{49} +5.88692e6 q^{51} +(1.76568e6 + 1.76568e6i) q^{52} +(-1.19258e6 + 1.19258e6i) q^{53} -1.06288e6i q^{54} +1.70528e7 q^{56} +(-6.00824e6 - 6.00824e6i) q^{57} +(-1.76491e6 + 1.76491e6i) q^{58} +1.27860e7i q^{59} +517403. q^{61} +(6.14865e6 + 6.14865e6i) q^{62} +(6.28112e6 - 6.28112e6i) q^{63} -1.20199e7i q^{64} -113724. q^{66} +(2.06617e6 + 2.06617e6i) q^{67} +(1.31738e7 - 1.31738e7i) q^{68} -1.78192e7i q^{69} -2.08286e7 q^{71} +(-6.49273e6 - 6.49273e6i) q^{72} +(2.88730e7 - 2.88730e7i) q^{73} +8.93988e6i q^{74} -2.68906e7 q^{76} +(-672054. - 672054. i) q^{77} +(-5.79810e6 + 5.79810e6i) q^{78} +4.21879e7i q^{79} -4.78297e6 q^{81} +(2.07390e7 + 2.07390e7i) q^{82} +(-6.66763e7 + 6.66763e7i) q^{83} -2.81119e7i q^{84} +4.11811e7 q^{86} +(7.94210e6 + 7.94210e6i) q^{87} +(-694695. + 694695. i) q^{88} -9.51614e7i q^{89} -6.85279e7 q^{91} +(-3.98759e7 - 3.98759e7i) q^{92} +(2.76689e7 - 2.76689e7i) q^{93} -7.92637e7i q^{94} -4.74725e7 q^{96} +(7.08678e7 + 7.08678e7i) q^{97} +(-7.88658e7 + 7.88658e7i) q^{98} +511758. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 1944 q^{6}+O(q^{10}) \) 4 * q + 1944 * q^6	\( 4 q + 1944 q^{6} - 936 q^{11} + 22976 q^{16} + 759780 q^{21} - 701352 q^{26} + 3346900 q^{31} - 1294704 q^{36} + 11288880 q^{41} + 15839280 q^{46} + 23547672 q^{51} + 68211360 q^{56} + 2069612 q^{61} - 454896 q^{66} - 83314512 q^{71} - 107562256 q^{76} - 19131876 q^{81} + 164724264 q^{86} - 274111740 q^{91} - 189889920 q^{96}+O(q^{100}) \) 4 * q + 1944 * q^6 - 936 * q^11 + 22976 * q^16 + 759780 * q^21 - 701352 * q^26 + 3346900 * q^31 - 1294704 * q^36 + 11288880 * q^41 + 15839280 * q^46 + 23547672 * q^51 + 68211360 * q^56 + 2069612 * q^61 - 454896 * q^66 - 83314512 * q^71 - 107562256 * q^76 - 19131876 * q^81 + 164724264 * q^86 - 274111740 * q^91 - 189889920 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more