LMFDB - Embedded newform 650.6.b.o.599.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [650,6,Mod(599,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("650.599"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$650 = 2 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	650.b (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-192,0,72,0,0,-1258] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$104.249482878$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + 2075 x^{10} + 1596139 x^{8} + 560320345 x^{6} + 89526738200 x^{4} + 5863900282000 x^{2} + 131166627840000$ x^12 + 2075x^10 + 1596139x^8 + 560320345x^6 + 89526738200x^4 + 5863900282000*x^2 + 131166627840000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{8}\cdot 3^{4}\cdot 5^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.11
Root			$20.7411i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.6.b.o.599.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} +18.7411i q^{3} -16.0000 q^{4} -74.9644 q^{6} -230.695i q^{7} -64.0000i q^{8} -108.229 q^{9} -544.656 q^{11} -299.858i q^{12} -169.000i q^{13} +922.779 q^{14} +256.000 q^{16} -554.057i q^{17} -432.917i q^{18} +2228.18 q^{19} +4323.47 q^{21} -2178.63i q^{22} +2915.01i q^{23} +1199.43 q^{24} +676.000 q^{26} +2525.75i q^{27} +3691.11i q^{28} +6864.22 q^{29} -6257.84 q^{31} +1024.00i q^{32} -10207.5i q^{33} +2216.23 q^{34} +1731.67 q^{36} +5695.55i q^{37} +8912.71i q^{38} +3167.25 q^{39} -11997.2 q^{41} +17293.9i q^{42} -15451.4i q^{43} +8714.50 q^{44} -11660.0 q^{46} -16278.3i q^{47} +4797.72i q^{48} -36413.0 q^{49} +10383.6 q^{51} +2704.00i q^{52} +11052.3i q^{53} -10103.0 q^{54} -14764.5 q^{56} +41758.5i q^{57} +27456.9i q^{58} +24145.9 q^{59} -10681.8 q^{61} -25031.4i q^{62} +24967.9i q^{63} -4096.00 q^{64} +40829.9 q^{66} +52777.4i q^{67} +8864.91i q^{68} -54630.5 q^{69} -44162.3 q^{71} +6926.67i q^{72} +41761.3i q^{73} -22782.2 q^{74} -35650.8 q^{76} +125649. i q^{77} +12669.0i q^{78} +59573.9 q^{79} -73635.1 q^{81} -47988.7i q^{82} -22640.6i q^{83} -69175.6 q^{84} +61805.8 q^{86} +128643. i q^{87} +34858.0i q^{88} -129864. q^{89} -38987.4 q^{91} -46640.1i q^{92} -117279. i q^{93} +65113.1 q^{94} -19190.9 q^{96} +63287.9i q^{97} -145652. i q^{98} +58947.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 192 q^{4} + 72 q^{6} - 1258 q^{9} + 740 q^{11} + 504 q^{14} + 3072 q^{16} + 3318 q^{19} + 8644 q^{21} - 1152 q^{24} + 8112 q^{26} + 11822 q^{29} + 2806 q^{31} - 6704 q^{34} + 20128 q^{36} - 3042 q^{39}+ \cdots - 772238 q^{99}+O(q^{100})$ 12 * q - 192 * q^4 + 72 * q^6 - 1258 * q^9 + 740 * q^11 + 504 * q^14 + 3072 * q^16 + 3318 * q^19 + 8644 * q^21 - 1152 * q^24 + 8112 * q^26 + 11822 * q^29 + 2806 * q^31 - 6704 * q^34 + 20128 * q^36 - 3042 * q^39 + 2974 * q^41 - 11840 * q^44 + 12528 * q^46 - 57954 * q^49 + 14106 * q^51 - 60552 * q^54 - 8064 * q^56 - 44734 * q^59 + 13422 * q^61 - 49152 * q^64 + 71944 * q^66 - 268676 * q^69 + 30440 * q^71 - 84784 * q^74 - 53088 * q^76 - 190856 * q^79 - 16268 * q^81 - 138304 * q^84 + 128624 * q^86 - 584622 * q^89 - 21294 * q^91 - 57080 * q^94 + 18432 * q^96 - 772238 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$ .

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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

)

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

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	18.7411i	1.20224i	0.799158	+	0.601121i	$0.205279\pi$
$3$	18.7411i	1.20224i	−0.799158	+	0.601121i	$0.794721\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−74.9644	−0.850114
$7$	− 230.695i	− 1.77948i	−0.456470	−	0.889739i	$-0.650887\pi$
$7$	− 230.695i	− 1.77948i	0.456470	−	0.889739i	$-0.349113\pi$
$8$	− 64.0000i	− 0.353553i
$9$	−108.229	−0.445388
$10$	0	0
$11$	−544.656	−1.35719	−0.678595	−	0.734512i	$-0.737411\pi$
$11$	−544.656	−1.35719	−0.678595	+	0.734512i	$0.737411\pi$
$12$	− 299.858i	− 0.601121i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	922.779	1.25828
$15$	0	0
$16$	256.000	0.250000
$17$	− 554.057i	− 0.464977i	−0.972599	−	0.232489i	$-0.925313\pi$
$17$	− 554.057i	− 0.464977i	0.972599	−	0.232489i	$-0.0746869\pi$
$18$	− 432.917i	− 0.314937i
$19$	2228.18	1.41601	0.708004	−	0.706208i	$-0.249596\pi$
$19$	2228.18	1.41601	0.708004	+	0.706208i	$0.249596\pi$
$20$	0	0
$21$	4323.47	2.13936
$22$	− 2178.63i	− 0.959679i
$23$	2915.01i	1.14900i	0.818505	+	0.574500i	$0.194803\pi$
$23$	2915.01i	1.14900i	−0.818505	+	0.574500i	$0.805197\pi$
$24$	1199.43	0.425057
$25$	0	0
$26$	676.000	0.196116
$27$	2525.75i	0.666779i
$28$	3691.11i	0.889739i
$29$	6864.22	1.51564	0.757820	−	0.652464i	$-0.226264\pi$
$29$	6864.22	1.51564	0.757820	+	0.652464i	$0.226264\pi$
$30$	0	0
$31$	−6257.84	−1.16955	−0.584777	−	0.811194i	$-0.698818\pi$
$31$	−6257.84	−1.16955	−0.584777	+	0.811194i	$0.698818\pi$
$32$	1024.00i	0.176777i
$33$	− 10207.5i	− 1.63167i
$34$	2216.23	0.328789
$35$	0	0
$36$	1731.67	0.222694
$37$	5695.55i	0.683961i	0.939707	+	0.341980i	$0.111098\pi$
$37$	5695.55i	0.683961i	−0.939707	+	0.341980i	$0.888902\pi$
$38$	8912.71i	1.00127i
$39$	3167.25	0.333442
$40$	0	0
$41$	−11997.2	−1.11460	−0.557300	−	0.830311i	$-0.688163\pi$
$41$	−11997.2	−1.11460	−0.557300	+	0.830311i	$0.688163\pi$
$42$	17293.9i	1.51276i
$43$	− 15451.4i	− 1.27438i	−0.770708	−	0.637189i	$-0.780097\pi$
$43$	− 15451.4i	− 1.27438i	0.770708	−	0.637189i	$-0.219903\pi$
$44$	8714.50	0.678595
$45$	0	0
$46$	−11660.0	−0.812466
$47$	− 16278.3i	− 1.07489i	−0.843299	−	0.537445i	$-0.819390\pi$
$47$	− 16278.3i	− 1.07489i	0.843299	−	0.537445i	$-0.180610\pi$
$48$	4797.72i	0.300561i
$49$	−36413.0	−2.16654
$50$	0	0
$51$	10383.6	0.559016
$52$	2704.00i	0.138675i
$53$	11052.3i	0.540462i	0.962796	+	0.270231i	$0.0871000\pi$
$53$	11052.3i	0.540462i	−0.962796	+	0.270231i	$0.912900\pi$
$54$	−10103.0	−0.471484
$55$	0	0
$56$	−14764.5	−0.629140
$57$	41758.5i	1.70239i
$58$	27456.9i	1.07172i
$59$	24145.9	0.903054	0.451527	−	0.892257i	$-0.350879\pi$
$59$	24145.9	0.903054	0.451527	+	0.892257i	$0.350879\pi$
$60$	0	0
$61$	−10681.8	−0.367552	−0.183776	−	0.982968i	$-0.558832\pi$
$61$	−10681.8	−0.367552	−0.183776	+	0.982968i	$0.558832\pi$
$62$	− 25031.4i	− 0.827000i
$63$	24967.9i	0.792557i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	40829.9	1.15377
$67$	52777.4i	1.43635i	0.695861	+	0.718176i	$0.255023\pi$
$67$	52777.4i	1.43635i	−0.695861	+	0.718176i	$0.744977\pi$
$68$	8864.91i	0.232489i
$69$	−54630.5	−1.38138
$70$	0	0
$71$	−44162.3	−1.03970	−0.519848	−	0.854259i	$-0.674011\pi$
$71$	−44162.3	−1.03970	−0.519848	+	0.854259i	$0.674011\pi$
$72$	6926.67i	0.157468i
$73$	41761.3i	0.917205i	0.888642	+	0.458602i	$0.151650\pi$
$73$	41761.3i	0.917205i	−0.888642	+	0.458602i	$0.848350\pi$
$74$	−22782.2	−0.483633
$75$	0	0
$76$	−35650.8	−0.708004
$77$	125649.i	2.41509i
$78$	12669.0i	0.235779i
$79$	59573.9	1.07396	0.536980	−	0.843595i	$-0.319565\pi$
$79$	59573.9	1.07396	0.536980	+	0.843595i	$0.319565\pi$
$80$	0	0
$81$	−73635.1	−1.24702
$82$	− 47988.7i	− 0.788141i
$83$	− 22640.6i	− 0.360738i	−0.983599	−	0.180369i	$-0.942271\pi$
$83$	− 22640.6i	− 0.360738i	0.983599	−	0.180369i	$-0.0577292\pi$
$84$	−69175.6	−1.06968
$85$	0	0
$86$	61805.8	0.901121
$87$	128643.i	1.82217i
$88$	34858.0i	0.479839i
$89$	−129864.	−1.73786	−0.868930	−	0.494935i	$-0.835192\pi$
$89$	−129864.	−1.73786	−0.868930	+	0.494935i	$0.835192\pi$
$90$	0	0
$91$	−38987.4	−0.493538
$92$	− 46640.1i	− 0.574500i
$93$	− 117279.i	− 1.40609i
$94$	65113.1	0.760061
$95$	0	0
$96$	−19190.9	−0.212529
$97$	63287.9i	0.682953i	0.939890	+	0.341477i	$0.110927\pi$
$97$	63287.9i	0.682953i	−0.939890	+	0.341477i	$0.889073\pi$
$98$	− 145652.i	− 1.53197i
$99$	58947.7	0.604476
$100$	0	0
$101$	116458.	1.13596	0.567982	−	0.823041i	$-0.307724\pi$
$101$	116458.	1.13596	0.567982	+	0.823041i	$0.307724\pi$
$102$	41534.6i	0.395284i
$103$	48487.1i	0.450333i	0.974320	+	0.225166i	$0.0722926\pi$
$103$	48487.1i	0.450333i	−0.974320	+	0.225166i	$0.927707\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	−44209.4	−0.382164
$107$	− 189220.i	− 1.59775i	−0.601499	−	0.798874i	$-0.705430\pi$
$107$	− 189220.i	− 1.59775i	0.601499	−	0.798874i	$-0.294570\pi$
$108$	− 40412.1i	− 0.333389i
$109$	−157628.	−1.27077	−0.635384	−	0.772197i	$-0.719158\pi$
$109$	−157628.	−1.27077	−0.635384	+	0.772197i	$0.719158\pi$
$110$	0	0
$111$	−106741.	−0.822287
$112$	− 59057.8i	− 0.444869i
$113$	200949.i	1.48044i	0.672365	+	0.740219i	$0.265278\pi$
$113$	200949.i	1.48044i	−0.672365	+	0.740219i	$0.734722\pi$
$114$	−167034.	−1.20377
$115$	0	0
$116$	−109827.	−0.757820
$117$	18290.7i	0.123528i
$118$	96583.7i	0.638556i
$119$	−127818.	−0.827417
$120$	0	0
$121$	135600.	0.841966
$122$	− 42727.1i	− 0.259898i
$123$	− 224840.i	− 1.34002i
$124$	100125.	0.584777
$125$	0	0
$126$	−99871.6	−0.560423
$127$	283395.i	1.55913i	0.626321	+	0.779565i	$0.284560\pi$
$127$	283395.i	1.55913i	−0.626321	+	0.779565i	$0.715440\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	289577.	1.53211
$130$	0	0
$131$	−199352.	−1.01494	−0.507472	−	0.861668i	$-0.669420\pi$
$131$	−199352.	−1.01494	−0.507472	+	0.861668i	$0.669420\pi$
$132$	163319.i	0.815836i
$133$	− 514029.i	− 2.51975i
$134$	−211110.	−1.01565
$135$	0	0
$136$	−35459.6	−0.164394
$137$	− 265825.i	− 1.21003i	−0.796216	−	0.605013i	$-0.793168\pi$
$137$	− 265825.i	− 1.21003i	0.796216	−	0.605013i	$-0.206832\pi$
$138$	− 218522.i	− 0.976781i
$139$	−985.292	−0.00432541	−0.00216271	−	0.999998i	$-0.500688\pi$
$139$	−985.292	−0.00432541	−0.00216271	+	0.999998i	$0.500688\pi$
$140$	0	0
$141$	305073.	1.29228
$142$	− 176649.i	− 0.735176i
$143$	92046.9i	0.376417i
$144$	−27706.7	−0.111347
$145$	0	0
$146$	−167045.	−0.648562
$147$	− 682420.i	− 2.60471i
$148$	− 91128.7i	− 0.341980i
$149$	20636.5	0.0761502	0.0380751	−	0.999275i	$-0.487877\pi$
$149$	20636.5	0.0761502	0.0380751	+	0.999275i	$0.487877\pi$
$150$	0	0
$151$	31649.1	0.112959	0.0564793	−	0.998404i	$-0.482013\pi$
$151$	31649.1	0.112959	0.0564793	+	0.998404i	$0.482013\pi$
$152$	− 142603.i	− 0.500635i
$153$	59965.1i	0.207095i
$154$	−502597.	−1.70773
$155$	0	0
$156$	−50676.0	−0.166721
$157$	475091.i	1.53825i	0.639097	+	0.769126i	$0.279308\pi$
$157$	475091.i	1.53825i	−0.639097	+	0.769126i	$0.720692\pi$
$158$	238296.i	0.759405i
$159$	−207133.	−0.649766
$160$	0	0
$161$	672477.	2.04462
$162$	− 294541.i	− 0.881775i
$163$	368397.i	1.08604i	0.839719	+	0.543022i	$0.182720\pi$
$163$	368397.i	1.08604i	−0.839719	+	0.543022i	$0.817280\pi$
$164$	191955.	0.557300
$165$	0	0
$166$	90562.2	0.255080
$167$	446876.i	1.23993i	0.784631	+	0.619963i	$0.212852\pi$
$167$	446876.i	1.23993i	−0.784631	+	0.619963i	$0.787148\pi$
$168$	− 276702.i	− 0.756379i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−241154.	−0.630673
$172$	247223.i	0.637189i
$173$	− 218403.i	− 0.554810i	−0.960753	−	0.277405i	$-0.910526\pi$
$173$	− 218403.i	− 0.554810i	0.960753	−	0.277405i	$-0.0894743\pi$
$174$	−514572.	−1.28847
$175$	0	0
$176$	−139432.	−0.339298
$177$	452521.i	1.08569i
$178$	− 519457.i	− 1.22885i
$179$	−338950.	−0.790684	−0.395342	−	0.918534i	$-0.629374\pi$
$179$	−338950.	−0.790684	−0.395342	+	0.918534i	$0.629374\pi$
$180$	0	0
$181$	−440716.	−0.999915	−0.499957	−	0.866050i	$-0.666651\pi$
$181$	−440716.	−0.999915	−0.499957	+	0.866050i	$0.666651\pi$
$182$	− 155950.i	− 0.348984i
$183$	− 200188.i	− 0.441887i
$184$	186560.	0.406233
$185$	0	0
$186$	469116.	0.994255
$187$	301771.i	0.631063i
$188$	260452.i	0.537445i
$189$	582678.	1.18652
$190$	0	0
$191$	643313.	1.27596	0.637982	−	0.770051i	$-0.279769\pi$
$191$	643313.	1.27596	0.637982	+	0.770051i	$0.279769\pi$
$192$	− 76763.6i	− 0.150280i
$193$	− 389679.i	− 0.753033i	−0.926410	−	0.376517i	$-0.877122\pi$
$193$	− 389679.i	− 0.753033i	0.926410	−	0.376517i	$-0.122878\pi$
$194$	−253151.	−0.482921
$195$	0	0
$196$	582608.	1.08327
$197$	326458.i	0.599323i	0.954046	+	0.299662i	$0.0968738\pi$
$197$	326458.i	0.599323i	−0.954046	+	0.299662i	$0.903126\pi$
$198$	235791.i	0.427429i
$199$	−433426.	−0.775858	−0.387929	−	0.921689i	$-0.626809\pi$
$199$	−433426.	−0.775858	−0.387929	+	0.921689i	$0.626809\pi$
$200$	0	0
$201$	−989107.	−1.72684
$202$	465830.i	0.803247i
$203$	− 1.58354e6i	− 2.69705i
$204$	−166138.	−0.279508
$205$	0	0
$206$	−193949.	−0.318433
$207$	− 315489.i	− 0.511751i
$208$	− 43264.0i	− 0.0693375i
$209$	−1.21359e6	−1.92179
$210$	0	0
$211$	−76110.5	−0.117690	−0.0588448	−	0.998267i	$-0.518742\pi$
$211$	−76110.5	−0.117690	−0.0588448	+	0.998267i	$0.518742\pi$
$212$	− 176838.i	− 0.270231i
$213$	− 827651.i	− 1.24997i
$214$	756881.	1.12978
$215$	0	0
$216$	161648.	0.235742
$217$	1.44365e6i	2.08120i
$218$	− 630510.i	− 0.898568i
$219$	−782652.	−1.10270
$220$	0	0
$221$	−93635.6	−0.128962
$222$	− 426963.i	− 0.581444i
$223$	141163.i	0.190090i	0.995473	+	0.0950450i	$0.0302995\pi$
$223$	141163.i	0.190090i	−0.995473	+	0.0950450i	$0.969700\pi$
$224$	236231.	0.314570
$225$	0	0
$226$	−803797.	−1.04683
$227$	− 968458.i	− 1.24743i	−0.781652	−	0.623715i	$-0.785623\pi$
$227$	− 968458.i	− 1.24743i	0.781652	−	0.623715i	$-0.214377\pi$
$228$	− 668136.i	− 0.851193i
$229$	−72887.3	−0.0918466	−0.0459233	−	0.998945i	$-0.514623\pi$
$229$	−72887.3	−0.0918466	−0.0459233	+	0.998945i	$0.514623\pi$
$230$	0	0
$231$	−2.35481e6	−2.90352
$232$	− 439310.i	− 0.535860i
$233$	− 613692.i	− 0.740561i	−0.928920	−	0.370280i	$-0.879262\pi$
$233$	− 613692.i	− 0.740561i	0.928920	−	0.370280i	$-0.120738\pi$
$234$	−73163.0	−0.0873477
$235$	0	0
$236$	−386335.	−0.451527
$237$	1.11648e6i	1.29116i
$238$	− 511272.i	− 0.585072i
$239$	1.21444e6	1.37525	0.687626	−	0.726065i	$-0.258653\pi$
$239$	1.21444e6	1.37525	0.687626	+	0.726065i	$0.258653\pi$
$240$	0	0
$241$	−1.78850e6	−1.98356	−0.991780	−	0.127957i	$-0.959158\pi$
$241$	−1.78850e6	−1.98356	−0.991780	+	0.127957i	$0.959158\pi$
$242$	542398.i	0.595360i
$243$	− 766246.i	− 0.832439i
$244$	170908.	0.183776
$245$	0	0
$246$	899361.	0.947537
$247$	− 376562.i	− 0.392730i
$248$	400502.i	0.413500i
$249$	424309.	0.433695
$250$	0	0
$251$	427715.	0.428519	0.214259	−	0.976777i	$-0.431266\pi$
$251$	427715.	0.428519	0.214259	+	0.976777i	$0.431266\pi$
$252$	− 399486.i	− 0.396279i
$253$	− 1.58768e6i	− 1.55941i
$254$	−1.13358e6	−1.10247
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 271022.i	− 0.255960i	−0.991777	−	0.127980i	$-0.959151\pi$
$257$	− 271022.i	− 0.255960i	0.991777	−	0.127980i	$-0.0408493\pi$
$258$	1.15831e6i	1.08337i
$259$	1.31393e6	1.21709
$260$	0	0
$261$	−742909.	−0.675048
$262$	− 797407.i	− 0.717673i
$263$	1.31591e6i	1.17310i	0.809912	+	0.586552i	$0.199515\pi$
$263$	1.31591e6i	1.17310i	−0.809912	+	0.586552i	$0.800485\pi$
$264$	−653278.	−0.576883
$265$	0	0
$266$	2.05611e6	1.78174
$267$	− 2.43380e6i	− 2.08933i
$268$	− 844438.i	− 0.718176i
$269$	1.63605e6	1.37853	0.689266	−	0.724508i	$-0.257933\pi$
$269$	1.63605e6	1.37853	0.689266	+	0.724508i	$0.257933\pi$
$270$	0	0
$271$	−1.41289e6	−1.16865	−0.584326	−	0.811519i	$-0.698641\pi$
$271$	−1.41289e6	−1.16865	−0.584326	+	0.811519i	$0.698641\pi$
$272$	− 141839.i	− 0.116244i
$273$	− 730667.i	− 0.593353i
$274$	1.06330e6	0.855617
$275$	0	0
$276$	874088.	0.690689
$277$	755228.i	0.591397i	0.955281	+	0.295698i	$0.0955523\pi$
$277$	755228.i	0.591397i	−0.955281	+	0.295698i	$0.904448\pi$
$278$	− 3941.17i	− 0.00305853i
$279$	677282.	0.520905
$280$	0	0
$281$	−550427.	−0.415848	−0.207924	−	0.978145i	$-0.566671\pi$
$281$	−550427.	−0.415848	−0.207924	+	0.978145i	$0.566671\pi$
$282$	1.22029e6i	0.913778i
$283$	1.12011e6i	0.831367i	0.909509	+	0.415683i	$0.136458\pi$
$283$	1.12011e6i	0.831367i	−0.909509	+	0.415683i	$0.863542\pi$
$284$	706597.	0.519848
$285$	0	0
$286$	−368188.	−0.266167
$287$	2.76768e6i	1.98341i
$288$	− 110827.i	− 0.0787342i
$289$	1.11288e6	0.783796
$290$	0	0
$291$	−1.18608e6	−0.821076
$292$	− 668180.i	− 0.458602i
$293$	1.47320e6i	1.00252i	0.865297	+	0.501260i	$0.167130\pi$
$293$	1.47320e6i	1.00252i	−0.865297	+	0.501260i	$0.832870\pi$
$294$	2.72968e6	1.84181
$295$	0	0
$296$	364515.	0.241817
$297$	− 1.37567e6i	− 0.904946i
$298$	82546.1i	0.0538463i
$299$	492636.	0.318675
$300$	0	0
$301$	−3.56457e6	−2.26772
$302$	126596.i	0.0798737i
$303$	2.18254e6i	1.36570i
$304$	570414.	0.354002
$305$	0	0
$306$	−239861.	−0.146438
$307$	1.42452e6i	0.862623i	0.902203	+	0.431312i	$0.141949\pi$
$307$	1.42452e6i	0.862623i	−0.902203	+	0.431312i	$0.858051\pi$
$308$	− 2.01039e6i	− 1.20754i
$309$	−908703.	−0.541409
$310$	0	0
$311$	−1.79196e6	−1.05058	−0.525288	−	0.850924i	$-0.676043\pi$
$311$	−1.79196e6	−1.05058	−0.525288	+	0.850924i	$0.676043\pi$
$312$	− 202704.i	− 0.117890i
$313$	2.93685e6i	1.69442i	0.531260	+	0.847209i	$0.321719\pi$
$313$	2.93685e6i	1.69442i	−0.531260	+	0.847209i	$0.678281\pi$
$314$	−1.90036e6	−1.08771
$315$	0	0
$316$	−953182.	−0.536980
$317$	390977.i	0.218526i	0.994013	+	0.109263i	$0.0348491\pi$
$317$	390977.i	0.218526i	−0.994013	+	0.109263i	$0.965151\pi$
$318$	− 828533.i	− 0.459454i
$319$	−3.73864e6	−2.05701
$320$	0	0
$321$	3.54620e6	1.92088
$322$	2.68991e6i	1.44576i
$323$	− 1.23454e6i	− 0.658412i
$324$	1.17816e6	0.623509
$325$	0	0
$326$	−1.47359e6	−0.767949
$327$	− 2.95412e6i	− 1.52777i
$328$	767819.i	0.394071i
$329$	−3.75531e6	−1.91274
$330$	0	0
$331$	−2.78981e6	−1.39960	−0.699800	−	0.714339i	$-0.746728\pi$
$331$	−2.78981e6	−1.39960	−0.699800	+	0.714339i	$0.746728\pi$
$332$	362249.i	0.180369i
$333$	− 616425.i	− 0.304628i
$334$	−1.78750e6	−0.876760
$335$	0	0
$336$	1.10681e6	0.534841
$337$	2.51413e6i	1.20591i	0.797776	+	0.602953i	$0.206009\pi$
$337$	2.51413e6i	1.20591i	−0.797776	+	0.602953i	$0.793991\pi$
$338$	− 114244.i	− 0.0543928i
$339$	−3.76601e6	−1.77985
$340$	0	0
$341$	3.40837e6	1.58731
$342$	− 964616.i	− 0.445953i
$343$	4.52300e6i	2.07583i
$344$	−988893.	−0.450560
$345$	0	0
$346$	873614.	0.392310
$347$	310242.i	0.138317i	0.997606	+	0.0691587i	$0.0220315\pi$
$347$	310242.i	0.138317i	−0.997606	+	0.0691587i	$0.977969\pi$
$348$	− 2.05829e6i	− 0.911084i
$349$	1.17835e6	0.517859	0.258929	−	0.965896i	$-0.416630\pi$
$349$	1.17835e6	0.517859	0.258929	+	0.965896i	$0.416630\pi$
$350$	0	0
$351$	426852.	0.184931
$352$	− 557728.i	− 0.239920i
$353$	− 1.34357e6i	− 0.573885i	−0.957948	−	0.286942i	$-0.907361\pi$
$353$	− 1.34357e6i	− 0.573885i	0.957948	−	0.286942i	$-0.0926388\pi$
$354$	−1.81009e6	−0.767699
$355$	0	0
$356$	2.07783e6	0.868930
$357$	− 2.39545e6i	− 0.994756i
$358$	− 1.35580e6i	− 0.559098i
$359$	2.61851e6	1.07230	0.536152	−	0.844122i	$-0.319877\pi$
$359$	2.61851e6	1.07230	0.536152	+	0.844122i	$0.319877\pi$
$360$	0	0
$361$	2.48868e6	1.00508
$362$	− 1.76287e6i	− 0.707046i
$363$	2.54129e6i	1.01225i
$364$	623798.	0.246769
$365$	0	0
$366$	800753.	0.312461
$367$	119818.i	0.0464363i	0.999730	+	0.0232182i	$0.00739124\pi$
$367$	119818.i	0.0464363i	−0.999730	+	0.0232182i	$0.992609\pi$
$368$	746242.i	0.287250i
$369$	1.29844e6	0.496429
$370$	0	0
$371$	2.54972e6	0.961739
$372$	1.87646e6i	0.703044i
$373$	2.68734e6i	1.00012i	0.865992	+	0.500058i	$0.166688\pi$
$373$	2.68734e6i	1.00012i	−0.865992	+	0.500058i	$0.833312\pi$
$374$	−1.20708e6	−0.446229
$375$	0	0
$376$	−1.04181e6	−0.380031
$377$	− 1.16005e6i	− 0.420363i
$378$	2.33071e6i	0.838994i
$379$	−1.16326e6	−0.415988	−0.207994	−	0.978130i	$-0.566693\pi$
$379$	−1.16326e6	−0.415988	−0.207994	+	0.978130i	$0.566693\pi$
$380$	0	0
$381$	−5.31113e6	−1.87445
$382$	2.57325e6i	0.902243i
$383$	1.35360e6i	0.471514i	0.971812	+	0.235757i	$0.0757569\pi$
$383$	1.35360e6i	0.471514i	−0.971812	+	0.235757i	$0.924243\pi$
$384$	307054.	0.106264
$385$	0	0
$386$	1.55872e6	0.532475
$387$	1.67230e6i	0.567592i
$388$	− 1.01261e6i	− 0.341477i
$389$	−1.64100e6	−0.549839	−0.274920	−	0.961467i	$-0.588651\pi$
$389$	−1.64100e6	−0.549839	−0.274920	+	0.961467i	$0.588651\pi$
$390$	0	0
$391$	1.61508e6	0.534259
$392$	2.33043e6i	0.765987i
$393$	− 3.73607e6i	− 1.22021i
$394$	−1.30583e6	−0.423786
$395$	0	0
$396$	−943164.	−0.302238
$397$	− 1.51737e6i	− 0.483187i	−0.970378	−	0.241593i	$-0.922330\pi$
$397$	− 1.51737e6i	− 0.483187i	0.970378	−	0.241593i	$-0.0776700\pi$
$398$	− 1.73370e6i	− 0.548614i
$399$	9.63347e6	3.02936
$400$	0	0
$401$	−4.16517e6	−1.29352	−0.646758	−	0.762696i	$-0.723875\pi$
$401$	−4.16517e6	−1.29352	−0.646758	+	0.762696i	$0.723875\pi$
$402$	− 3.95643e6i	− 1.22106i
$403$	1.05758e6i	0.324376i
$404$	−1.86332e6	−0.567982
$405$	0	0
$406$	6.33415e6	1.90710
$407$	− 3.10212e6i	− 0.928265i
$408$	− 664553.i	− 0.197642i
$409$	−2.16258e6	−0.639240	−0.319620	−	0.947546i	$-0.603555\pi$
$409$	−2.16258e6	−0.639240	−0.319620	+	0.947546i	$0.603555\pi$
$410$	0	0
$411$	4.98186e6	1.45474
$412$	− 775794.i	− 0.225166i
$413$	− 5.57034e6i	− 1.60696i
$414$	1.26196e6	0.361862
$415$	0	0
$416$	173056.	0.0490290
$417$	− 18465.5i	− 0.00520020i
$418$	− 4.85436e6i	− 1.35891i
$419$	5.87037e6	1.63354	0.816772	−	0.576961i	$-0.195762\pi$
$419$	5.87037e6	1.63354	0.816772	+	0.576961i	$0.195762\pi$
$420$	0	0
$421$	−2.14426e6	−0.589621	−0.294810	−	0.955556i	$-0.595256\pi$
$421$	−2.14426e6	−0.589621	−0.294810	+	0.955556i	$0.595256\pi$
$422$	− 304442.i	− 0.0832192i
$423$	1.76178e6i	0.478742i
$424$	707350.	0.191082
$425$	0	0
$426$	3.31061e6	0.883860
$427$	2.46423e6i	0.654050i
$428$	3.02752e6i	0.798874i
$429$	−1.72506e6	−0.452545
$430$	0	0
$431$	2.56530e6	0.665189	0.332594	−	0.943070i	$-0.392076\pi$
$431$	2.56530e6	0.665189	0.332594	+	0.943070i	$0.392076\pi$
$432$	646593.i	0.166695i
$433$	6.07499e6i	1.55713i	0.627561	+	0.778567i	$0.284053\pi$
$433$	6.07499e6i	1.55713i	−0.627561	+	0.778567i	$0.715947\pi$
$434$	−5.77460e6	−1.47163
$435$	0	0
$436$	2.52204e6	0.635384
$437$	6.49515e6i	1.62699i
$438$	− 3.13061e6i	− 0.779729i
$439$	−7.17264e6	−1.77631	−0.888154	−	0.459547i	$-0.848012\pi$
$439$	−7.17264e6	−1.77631	−0.888154	+	0.459547i	$0.848012\pi$
$440$	0	0
$441$	3.94095e6	0.964950
$442$	− 374542.i	− 0.0911896i
$443$	1.55350e6i	0.376099i	0.982160	+	0.188049i	$0.0602165\pi$
$443$	1.55350e6i	0.376099i	−0.982160	+	0.188049i	$0.939783\pi$
$444$	1.70785e6	0.411143
$445$	0	0
$446$	−564653.	−0.134414
$447$	386751.i	0.0915510i
$448$	944925.i	0.222435i
$449$	3.57322e6	0.836456	0.418228	−	0.908342i	$-0.362651\pi$
$449$	3.57322e6	0.836456	0.418228	+	0.908342i	$0.362651\pi$
$450$	0	0
$451$	6.53433e6	1.51272
$452$	− 3.21519e6i	− 0.740219i
$453$	593139.i	0.135804i
$454$	3.87383e6	0.882066
$455$	0	0
$456$	2.67255e6	0.601884
$457$	− 8.13345e6i	− 1.82173i	−0.412703	−	0.910866i	$-0.635415\pi$
$457$	− 8.13345e6i	− 1.82173i	0.412703	−	0.910866i	$-0.364585\pi$
$458$	− 291549.i	− 0.0649454i
$459$	1.39941e6	0.310037
$460$	0	0
$461$	−3.69923e6	−0.810697	−0.405349	−	0.914162i	$-0.632850\pi$
$461$	−3.69923e6	−0.810697	−0.405349	+	0.914162i	$0.632850\pi$
$462$	− 9.41923e6i	− 2.05310i
$463$	1.34127e6i	0.290779i	0.989375	+	0.145389i	$0.0464435\pi$
$463$	1.34127e6i	0.290779i	−0.989375	+	0.145389i	$0.953557\pi$
$464$	1.75724e6	0.378910
$465$	0	0
$466$	2.45477e6	0.523655
$467$	2.41068e6i	0.511501i	0.966743	+	0.255751i	$0.0823226\pi$
$467$	2.41068e6i	0.511501i	−0.966743	+	0.255751i	$0.917677\pi$
$468$	− 292652.i	− 0.0617642i
$469$	1.21755e7	2.55596
$470$	0	0
$471$	−8.90373e6	−1.84935
$472$	− 1.54534e6i	− 0.319278i
$473$	8.41573e6i	1.72957i
$474$	−4.46592e6	−0.912989
$475$	0	0
$476$	2.04509e6	0.413708
$477$	− 1.19619e6i	− 0.240715i
$478$	4.85777e6i	0.972450i
$479$	−18669.1	−0.00371779	−0.00185890	−	0.999998i	$-0.500592\pi$
$479$	−18669.1	−0.00371779	−0.00185890	+	0.999998i	$0.500592\pi$
$480$	0	0
$481$	962547.	0.189697
$482$	− 7.15398e6i	− 1.40259i
$483$	1.26030e7i	2.45813i
$484$	−2.16959e6	−0.420983
$485$	0	0
$486$	3.06498e6	0.588623
$487$	− 4.53165e6i	− 0.865832i	−0.901434	−	0.432916i	$-0.857485\pi$
$487$	− 4.53165e6i	− 0.865832i	0.901434	−	0.432916i	$-0.142515\pi$
$488$	683633.i	0.129949i
$489$	−6.90417e6	−1.30569
$490$	0	0
$491$	3.84989e6	0.720684	0.360342	−	0.932820i	$-0.382660\pi$
$491$	3.84989e6	0.720684	0.360342	+	0.932820i	$0.382660\pi$
$492$	3.59744e6i	0.670010i
$493$	− 3.80317e6i	− 0.704739i
$494$	1.50625e6	0.277702
$495$	0	0
$496$	−1.60201e6	−0.292389
$497$	1.01880e7i	1.85011i
$498$	1.69724e6i	0.306668i
$499$	−2.18845e6	−0.393445	−0.196723	−	0.980459i	$-0.563030\pi$
$499$	−2.18845e6	−0.393445	−0.196723	+	0.980459i	$0.563030\pi$
$500$	0	0
$501$	−8.37495e6	−1.49069
$502$	1.71086e6i	0.303009i
$503$	− 1.37502e6i	− 0.242319i	−0.992633	−	0.121160i	$-0.961339\pi$
$503$	− 1.37502e6i	− 0.242319i	0.992633	−	0.121160i	$-0.0386613\pi$
$504$	1.59795e6	0.280211
$505$	0	0
$506$	6.35071e6	1.10267
$507$	− 535265.i	− 0.0924802i
$508$	− 4.53431e6i	− 0.779565i
$509$	−7.64261e6	−1.30752	−0.653759	−	0.756703i	$-0.726809\pi$
$509$	−7.64261e6	−1.30752	−0.653759	+	0.756703i	$0.726809\pi$
$510$	0	0
$511$	9.63410e6	1.63215
$512$	262144.i	0.0441942i
$513$	5.62783e6i	0.944164i
$514$	1.08409e6	0.180991
$515$	0	0
$516$	−4.63324e6	−0.766055
$517$	8.86606e6i	1.45883i
$518$	5.25573e6i	0.860614i
$519$	4.09312e6	0.667016
$520$	0	0
$521$	1.17487e7	1.89625	0.948123	−	0.317903i	$-0.102979\pi$
$521$	1.17487e7	1.89625	0.948123	+	0.317903i	$0.102979\pi$
$522$	− 2.97164e6i	− 0.477331i
$523$	− 5.40309e6i	− 0.863750i	−0.901933	−	0.431875i	$-0.857852\pi$
$523$	− 5.40309e6i	− 0.863750i	0.901933	−	0.431875i	$-0.142148\pi$
$524$	3.18963e6	0.507472
$525$	0	0
$526$	−5.26363e6	−0.829510
$527$	3.46720e6i	0.543817i
$528$	− 2.61311e6i	− 0.407918i
$529$	−2.06092e6	−0.320201
$530$	0	0
$531$	−2.61330e6	−0.402209
$532$	8.22446e6i	1.25988i
$533$	2.02752e6i	0.309134i
$534$	9.73520e6	1.47738
$535$	0	0
$536$	3.37775e6	0.507827
$537$	− 6.35230e6i	− 0.950594i
$538$	6.54422e6i	0.974769i
$539$	1.98326e7	2.94041
$540$	0	0
$541$	−1.33517e7	−1.96129	−0.980646	−	0.195789i	$-0.937273\pi$
$541$	−1.33517e7	−1.96129	−0.980646	+	0.195789i	$0.937273\pi$
$542$	− 5.65156e6i	− 0.826362i
$543$	− 8.25952e6i	− 1.20214i
$544$	567354.	0.0821972
$545$	0	0
$546$	2.92267e6	0.419564
$547$	2.40834e6i	0.344152i	0.985084	+	0.172076i	$0.0550475\pi$
$547$	2.40834e6i	0.344152i	−0.985084	+	0.172076i	$0.944953\pi$
$548$	4.25320e6i	0.605013i
$549$	1.15608e6	0.163703
$550$	0	0
$551$	1.52947e7	2.14616
$552$	3.49635e6i	0.488391i
$553$	− 1.37434e7i	− 1.91109i
$554$	−3.02091e6	−0.418180
$555$	0	0
$556$	15764.7	0.00216271
$557$	− 1.52042e6i	− 0.207647i	−0.994596	−	0.103823i	$-0.966892\pi$
$557$	− 1.52042e6i	− 0.207647i	0.994596	−	0.103823i	$-0.0331076\pi$
$558$	2.70913e6i	0.368336i
$559$	−2.61129e6	−0.353449
$560$	0	0
$561$	−5.65551e6	−0.758691
$562$	− 2.20171e6i	− 0.294049i
$563$	9.15496e6i	1.21727i	0.793452	+	0.608633i	$0.208282\pi$
$563$	9.15496e6i	1.21727i	−0.793452	+	0.608633i	$0.791718\pi$
$564$	−4.88117e6	−0.646139
$565$	0	0
$566$	−4.48042e6	−0.587865
$567$	1.69872e7i	2.21904i
$568$	2.82639e6i	0.367588i
$569$	−1.18973e7	−1.54052	−0.770262	−	0.637727i	$-0.779875\pi$
$569$	−1.18973e7	−1.54052	−0.770262	+	0.637727i	$0.779875\pi$
$570$	0	0
$571$	1.20642e7	1.54849	0.774245	−	0.632886i	$-0.218130\pi$
$571$	1.20642e7	1.54849	0.774245	+	0.632886i	$0.218130\pi$
$572$	− 1.47275e6i	− 0.188208i
$573$	1.20564e7i	1.53402i
$574$	−1.10707e7	−1.40248
$575$	0	0
$576$	443307.	0.0556735
$577$	− 1.38845e6i	− 0.173616i	−0.996225	−	0.0868081i	$-0.972333\pi$
$577$	− 1.38845e6i	− 0.173616i	0.996225	−	0.0868081i	$-0.0276667\pi$
$578$	4.45151e6i	0.554227i
$579$	7.30303e6	0.905329
$580$	0	0
$581$	−5.22305e6	−0.641925
$582$	− 4.74434e6i	− 0.580588i
$583$	− 6.01973e6i	− 0.733509i
$584$	2.67272e6	0.324281
$585$	0	0
$586$	−5.89280e6	−0.708888
$587$	5.80067e6i	0.694836i	0.937710	+	0.347418i	$0.112942\pi$
$587$	5.80067e6i	0.694836i	−0.937710	+	0.347418i	$0.887058\pi$
$588$	1.09187e7i	1.30235i
$589$	−1.39436e7	−1.65610
$590$	0	0
$591$	−6.11818e6	−0.720532
$592$	1.45806e6i	0.170990i
$593$	1.06130e7i	1.23937i	0.784852	+	0.619684i	$0.212739\pi$
$593$	1.06130e7i	1.23937i	−0.784852	+	0.619684i	$0.787261\pi$
$594$	5.50267e6	0.639893
$595$	0	0
$596$	−330184.	−0.0380751
$597$	− 8.12288e6i	− 0.932770i
$598$	1.97054e6i	0.225337i
$599$	4.68069e6	0.533020	0.266510	−	0.963832i	$-0.414130\pi$
$599$	4.68069e6	0.533020	0.266510	+	0.963832i	$0.414130\pi$
$600$	0	0
$601$	2.65345e6	0.299657	0.149828	−	0.988712i	$-0.452128\pi$
$601$	2.65345e6	0.299657	0.149828	+	0.988712i	$0.452128\pi$
$602$	− 1.42583e7i	− 1.60352i
$603$	− 5.71206e6i	− 0.639734i
$604$	−506386.	−0.0564793
$605$	0	0
$606$	−8.73018e6	−0.965698
$607$	− 6.58092e6i	− 0.724962i	−0.931991	−	0.362481i	$-0.881930\pi$
$607$	− 6.58092e6i	− 0.724962i	0.931991	−	0.362481i	$-0.118070\pi$
$608$	2.28165e6i	0.250317i
$609$	2.96773e7	3.24251
$610$	0	0
$611$	−2.75103e6	−0.298121
$612$	− 959442.i	− 0.103548i
$613$	8.28523e6i	0.890540i	0.895396	+	0.445270i	$0.146892\pi$
$613$	8.28523e6i	0.890540i	−0.895396	+	0.445270i	$0.853108\pi$
$614$	−5.69806e6	−0.609967
$615$	0	0
$616$	8.04155e6	0.853863
$617$	5.42747e6i	0.573964i	0.957936	+	0.286982i	$0.0926520\pi$
$617$	5.42747e6i	0.573964i	−0.957936	+	0.286982i	$0.907348\pi$
$618$	− 3.63481e6i	− 0.382834i
$619$	8.73793e6	0.916604	0.458302	−	0.888796i	$-0.348458\pi$
$619$	8.73793e6	0.916604	0.458302	+	0.888796i	$0.348458\pi$
$620$	0	0
$621$	−7.36259e6	−0.766129
$622$	− 7.16785e6i	− 0.742870i
$623$	2.99590e7i	3.09248i
$624$	810815.	0.0833605
$625$	0	0
$626$	−1.17474e7	−1.19813
$627$	− 2.27440e7i	− 2.31046i
$628$	− 7.60145e6i	− 0.769126i
$629$	3.15566e6	0.318026
$630$	0	0
$631$	−6.93247e6	−0.693130	−0.346565	−	0.938026i	$-0.612652\pi$
$631$	−6.93247e6	−0.693130	−0.346565	+	0.938026i	$0.612652\pi$
$632$	− 3.81273e6i	− 0.379702i
$633$	− 1.42639e6i	− 0.141492i
$634$	−1.56391e6	−0.154521
$635$	0	0
$636$	3.31413e6	0.324883
$637$	6.15380e6i	0.600890i
$638$	− 1.49546e7i	− 1.45453i
$639$	4.77966e6	0.463068
$640$	0	0
$641$	−9.65973e6	−0.928581	−0.464291	−	0.885683i	$-0.653691\pi$
$641$	−9.65973e6	−0.928581	−0.464291	+	0.885683i	$0.653691\pi$
$642$	1.41848e7i	1.35827i
$643$	− 588572.i	− 0.0561399i	−0.999606	−	0.0280700i	$-0.991064\pi$
$643$	− 588572.i	− 0.0561399i	0.999606	−	0.0280700i	$-0.00893612\pi$
$644$	−1.07596e7	−1.02231
$645$	0	0
$646$	4.93815e6	0.465568
$647$	− 7.53056e6i	− 0.707240i	−0.935389	−	0.353620i	$-0.884951\pi$
$647$	− 7.53056e6i	− 0.707240i	0.935389	−	0.353620i	$-0.115049\pi$
$648$	4.71265e6i	0.440887i
$649$	−1.31512e7	−1.22562
$650$	0	0
$651$	−2.70556e7	−2.50210
$652$	− 5.89436e6i	− 0.543022i
$653$	− 9.16834e6i	− 0.841410i	−0.907197	−	0.420705i	$-0.861783\pi$
$653$	− 9.16834e6i	− 0.841410i	0.907197	−	0.420705i	$-0.138217\pi$
$654$	1.18165e7	1.08030
$655$	0	0
$656$	−3.07127e6	−0.278650
$657$	− 4.51979e6i	− 0.408512i
$658$	− 1.50212e7i	− 1.35251i
$659$	1.25416e7	1.12496	0.562482	−	0.826809i	$-0.309847\pi$
$659$	1.25416e7	1.12496	0.562482	+	0.826809i	$0.309847\pi$
$660$	0	0
$661$	−2.80510e6	−0.249715	−0.124858	−	0.992175i	$-0.539847\pi$
$661$	−2.80510e6	−0.249715	−0.124858	+	0.992175i	$0.539847\pi$
$662$	− 1.11592e7i	− 0.989667i
$663$	− 1.75483e6i	− 0.155043i
$664$	−1.44900e6	−0.127540
$665$	0	0
$666$	2.46570e6	0.215404
$667$	2.00092e7i	1.74147i
$668$	− 7.15001e6i	− 0.619963i
$669$	−2.64556e6	−0.228534
$670$	0	0
$671$	5.81789e6	0.498838
$672$	4.42724e6i	0.378190i
$673$	− 1.88930e7i	− 1.60792i	−0.594686	−	0.803958i	$-0.702723\pi$
$673$	− 1.88930e7i	− 1.60792i	0.594686	−	0.803958i	$-0.297277\pi$
$674$	−1.00565e7	−0.852705
$675$	0	0
$676$	456976.	0.0384615
$677$	7.43678e6i	0.623610i	0.950146	+	0.311805i	$0.100934\pi$
$677$	7.43678e6i	0.623610i	−0.950146	+	0.311805i	$0.899066\pi$
$678$	− 1.50641e7i	− 1.25854i
$679$	1.46002e7	1.21530
$680$	0	0
$681$	1.81500e7	1.49971
$682$	1.36335e7i	1.12240i
$683$	− 1.01428e7i	− 0.831967i	−0.909372	−	0.415984i	$-0.863437\pi$
$683$	− 1.01428e7i	− 0.831967i	0.909372	−	0.415984i	$-0.136563\pi$
$684$	3.85846e6	0.315336
$685$	0	0
$686$	−1.80920e7	−1.46783
$687$	− 1.36599e6i	− 0.110422i
$688$	− 3.95557e6i	− 0.318594i
$689$	1.86785e6	0.149897
$690$	0	0
$691$	−2.15408e6	−0.171620	−0.0858098	−	0.996312i	$-0.527348\pi$
$691$	−2.15408e6	−0.171620	−0.0858098	+	0.996312i	$0.527348\pi$
$692$	3.49445e6i	0.277405i
$693$	− 1.35989e7i	− 1.07565i
$694$	−1.24097e6	−0.0978051
$695$	0	0
$696$	8.23316e6	0.644234
$697$	6.64711e6i	0.518264i
$698$	4.71341e6i	0.366181i
$699$	1.15013e7	0.890334
$700$	0	0
$701$	−6.90173e6	−0.530472	−0.265236	−	0.964183i	$-0.585450\pi$
$701$	−6.90173e6	−0.530472	−0.265236	+	0.964183i	$0.585450\pi$
$702$	1.70741e6i	0.130766i
$703$	1.26907e7i	0.968494i
$704$	2.23091e6	0.169649
$705$	0	0
$706$	5.37429e6	0.405798
$707$	− 2.68661e7i	− 2.02142i
$708$	− 7.24034e6i	− 0.542845i
$709$	−1.62641e7	−1.21511	−0.607555	−	0.794278i	$-0.707850\pi$
$709$	−1.62641e7	−1.21511	−0.607555	+	0.794278i	$0.707850\pi$
$710$	0	0
$711$	−6.44764e6	−0.478329
$712$	8.31131e6i	0.614426i
$713$	− 1.82417e7i	− 1.34382i
$714$	9.58180e6	0.703399
$715$	0	0
$716$	5.42320e6	0.395342
$717$	2.27600e7i	1.65339i
$718$	1.04740e7i	0.758233i
$719$	6.59643e6	0.475868	0.237934	−	0.971281i	$-0.423530\pi$
$719$	6.59643e6	0.475868	0.237934	+	0.971281i	$0.423530\pi$
$720$	0	0
$721$	1.11857e7	0.801357
$722$	9.95471e6i	0.710699i
$723$	− 3.35184e7i	− 2.38472i
$724$	7.05146e6	0.499957
$725$	0	0
$726$	−1.01651e7	−0.715767
$727$	− 2.19325e7i	− 1.53905i	−0.638620	−	0.769523i	$-0.720494\pi$
$727$	− 2.19325e7i	− 1.53905i	0.638620	−	0.769523i	$-0.279506\pi$
$728$	2.49519e6i	0.174492i
$729$	−3.53304e6	−0.246223
$730$	0	0
$731$	−8.56098e6	−0.592557
$732$	3.20301e6i	0.220943i
$733$	1.67003e7i	1.14806i	0.818836	+	0.574028i	$0.194620\pi$
$733$	1.67003e7i	1.14806i	−0.818836	+	0.574028i	$0.805380\pi$
$734$	−479273.	−0.0328355
$735$	0	0
$736$	−2.98497e6	−0.203116
$737$	− 2.87455e7i	− 1.94940i
$738$	5.19378e6i	0.351028i
$739$	1.69875e6	0.114425	0.0572123	−	0.998362i	$-0.481779\pi$
$739$	1.69875e6	0.114425	0.0572123	+	0.998362i	$0.481779\pi$
$740$	0	0
$741$	7.05719e6	0.472157
$742$	1.01989e7i	0.680052i
$743$	− 1.82334e7i	− 1.21170i	−0.795578	−	0.605851i	$-0.792833\pi$
$743$	− 1.82334e7i	− 1.21170i	0.795578	−	0.605851i	$-0.207167\pi$
$744$	−7.50585e6	−0.497127
$745$	0	0
$746$	−1.07493e7	−0.707188
$747$	2.45037e6i	0.160668i
$748$	− 4.82833e6i	− 0.315532i
$749$	−4.36521e7	−2.84315
$750$	0	0
$751$	−1.47109e6	−0.0951784	−0.0475892	−	0.998867i	$-0.515154\pi$
$751$	−1.47109e6	−0.0951784	−0.0475892	+	0.998867i	$0.515154\pi$
$752$	− 4.16724e6i	− 0.268722i
$753$	8.01585e6i	0.515184i
$754$	4.64021e6	0.297241
$755$	0	0
$756$	−9.32285e6	−0.593259
$757$	9.68714e6i	0.614407i	0.951644	+	0.307203i	$0.0993932\pi$
$757$	9.68714e6i	0.614407i	−0.951644	+	0.307203i	$0.900607\pi$
$758$	− 4.65306e6i	− 0.294148i
$759$	2.97548e7	1.87479
$760$	0	0
$761$	1.09169e7	0.683338	0.341669	−	0.939820i	$-0.389008\pi$
$761$	1.09169e7	0.683338	0.341669	+	0.939820i	$0.389008\pi$
$762$	− 2.12445e7i	− 1.32544i
$763$	3.63638e7i	2.26130i
$764$	−1.02930e7	−0.637982
$765$	0	0
$766$	−5.41441e6	−0.333411
$767$	− 4.08066e6i	− 0.250462i
$768$	1.22822e6i	0.0751402i
$769$	1.70886e7	1.04205	0.521027	−	0.853540i	$-0.325549\pi$
$769$	1.70886e7	1.04205	0.521027	+	0.853540i	$0.325549\pi$
$770$	0	0
$771$	5.07925e6	0.307726
$772$	6.23487e6i	0.376517i
$773$	− 5.92288e6i	− 0.356520i	−0.983983	−	0.178260i	$-0.942953\pi$
$773$	− 5.92288e6i	− 0.356520i	0.983983	−	0.178260i	$-0.0570468\pi$
$774$	−6.68919e6	−0.401348
$775$	0	0
$776$	4.05042e6	0.241461
$777$	2.46245e7i	1.46324i
$778$	− 6.56402e6i	− 0.388795i
$779$	−2.67318e7	−1.57828
$780$	0	0
$781$	2.40533e7	1.41107
$782$	6.46032e6i	0.377778i
$783$	1.73373e7i	1.01060i
$784$	−9.32173e6	−0.541635
$785$	0	0
$786$	1.49443e7	0.862818
$787$	2.75022e7i	1.58282i	0.611288	+	0.791409i	$0.290652\pi$
$787$	2.75022e7i	1.58282i	−0.611288	+	0.791409i	$0.709348\pi$
$788$	− 5.22332e6i	− 0.299662i
$789$	−2.46616e7	−1.41036
$790$	0	0
$791$	4.63579e7	2.63441
$792$	− 3.77265e6i	− 0.213715i
$793$	1.80522e6i	0.101941i
$794$	6.06947e6	0.341664
$795$	0	0
$796$	6.93481e6	0.387929
$797$	− 1.29045e7i	− 0.719606i	−0.933028	−	0.359803i	$-0.882844\pi$
$797$	− 1.29045e7i	− 0.719606i	0.933028	−	0.359803i	$-0.117156\pi$
$798$	3.85339e7i	2.14208i
$799$	−9.01909e6	−0.499799
$800$	0	0
$801$	1.40551e7	0.774021
$802$	− 1.66607e7i	− 0.914653i
$803$	− 2.27455e7i	− 1.24482i
$804$	1.58257e7	0.863422
$805$	0	0
$806$	−4.23030e6	−0.229369
$807$	3.06615e7i	1.65733i
$808$	− 7.45329e6i	− 0.401624i
$809$	−2.16172e7	−1.16126	−0.580628	−	0.814169i	$-0.697193\pi$
$809$	−2.16172e7	−1.16126	−0.580628	+	0.814169i	$0.697193\pi$
$810$	0	0
$811$	−1.68933e7	−0.901906	−0.450953	−	0.892548i	$-0.648916\pi$
$811$	−1.68933e7	−0.901906	−0.450953	+	0.892548i	$0.648916\pi$
$812$	2.53366e7i	1.34852i
$813$	− 2.64791e7i	− 1.40500i
$814$	1.24085e7	0.656382
$815$	0	0
$816$	2.65821e6	0.139754
$817$	− 3.44286e7i	− 1.80453i
$818$	− 8.65032e6i	− 0.452011i
$819$	4.21958e6	0.219816
$820$	0	0
$821$	−7.06381e6	−0.365747	−0.182874	−	0.983136i	$-0.558540\pi$
$821$	−7.06381e6	−0.365747	−0.182874	+	0.983136i	$0.558540\pi$
$822$	1.99274e7i	1.02866i
$823$	− 3.32953e7i	− 1.71350i	−0.515735	−	0.856748i	$-0.672481\pi$
$823$	− 3.32953e7i	− 1.71350i	0.515735	−	0.856748i	$-0.327519\pi$
$824$	3.10318e6	0.159217
$825$	0	0
$826$	2.22813e7	1.13630
$827$	− 1.49348e7i	− 0.759341i	−0.925122	−	0.379670i	$-0.876037\pi$
$827$	− 1.49348e7i	− 0.759341i	0.925122	−	0.379670i	$-0.123963\pi$
$828$	5.04782e6i	0.255875i
$829$	−1.10426e7	−0.558065	−0.279032	−	0.960282i	$-0.590014\pi$
$829$	−1.10426e7	−0.558065	−0.279032	+	0.960282i	$0.590014\pi$
$830$	0	0
$831$	−1.41538e7	−0.711002
$832$	692224.i	0.0346688i
$833$	2.01749e7i	1.00739i
$834$	73861.8	0.00367710
$835$	0	0
$836$	1.94175e7	0.960897
$837$	− 1.58058e7i	− 0.779834i
$838$	2.34815e7i	1.15509i
$839$	−2.86322e7	−1.40427	−0.702133	−	0.712046i	$-0.747769\pi$
$839$	−2.86322e7	−1.40427	−0.702133	+	0.712046i	$0.747769\pi$
$840$	0	0
$841$	2.66063e7	1.29717
$842$	− 8.57705e6i	− 0.416925i
$843$	− 1.03156e7i	− 0.499950i
$844$	1.21777e6	0.0588448
$845$	0	0
$846$	−7.04714e6	−0.338522
$847$	− 3.12821e7i	− 1.49826i
$848$	2.82940e6i	0.135115i
$849$	−2.09920e7	−0.999505
$850$	0	0
$851$	−1.66026e7	−0.785871
$852$	1.32424e7i	0.624983i
$853$	− 1.71429e7i	− 0.806700i	−0.915046	−	0.403350i	$-0.867846\pi$
$853$	− 1.71429e7i	− 0.806700i	0.915046	−	0.403350i	$-0.132154\pi$
$854$	−9.85691e6	−0.462483
$855$	0	0
$856$	−1.21101e7	−0.564889
$857$	132094.i	0.00614373i	0.999995	+	0.00307186i	$0.000977806\pi$
$857$	132094.i	0.00614373i	−0.999995	+	0.00307186i	$0.999022\pi$
$858$	− 6.90025e6i	− 0.319997i
$859$	−3.33727e7	−1.54315	−0.771577	−	0.636136i	$-0.780532\pi$
$859$	−3.33727e7	−1.54315	−0.771577	+	0.636136i	$0.780532\pi$
$860$	0	0
$861$	−5.18694e7	−2.38453
$862$	1.02612e7i	0.470359i
$863$	− 1.01919e7i	− 0.465829i	−0.972497	−	0.232915i	$-0.925174\pi$
$863$	− 1.01919e7i	− 0.465829i	0.972497	−	0.232915i	$-0.0748263\pi$
$864$	−2.58637e6	−0.117871
$865$	0	0
$866$	−2.43000e7	−1.10106
$867$	2.08566e7i	0.942313i
$868$	− 2.30984e7i	− 1.04060i
$869$	−3.24473e7	−1.45757
$870$	0	0
$871$	8.91938e6	0.398372
$872$	1.00882e7i	0.449284i
$873$	− 6.84960e6i	− 0.304179i
$874$	−2.59806e7	−1.15046
$875$	0	0
$876$	1.25224e7	0.551352
$877$	1.53857e6i	0.0675490i	0.999429	+	0.0337745i	$0.0107528\pi$
$877$	1.53857e6i	0.0675490i	−0.999429	+	0.0337745i	$0.989247\pi$
$878$	− 2.86906e7i	− 1.25604i
$879$	−2.76094e7	−1.20527
$880$	0	0
$881$	−5.38590e6	−0.233786	−0.116893	−	0.993145i	$-0.537293\pi$
$881$	−5.38590e6	−0.233786	−0.116893	+	0.993145i	$0.537293\pi$
$882$	1.57638e7i	0.682323i
$883$	3.24054e7i	1.39867i	0.714793	+	0.699336i	$0.246521\pi$
$883$	3.24054e7i	1.39867i	−0.714793	+	0.699336i	$0.753479\pi$
$884$	1.49817e6	0.0644808
$885$	0	0
$886$	−6.21400e6	−0.265942
$887$	− 3.98506e7i	− 1.70069i	−0.526224	−	0.850346i	$-0.676393\pi$
$887$	− 3.98506e7i	− 1.70069i	0.526224	−	0.850346i	$-0.323607\pi$
$888$	6.83142e6i	0.290722i
$889$	6.53776e7	2.77444
$890$	0	0
$891$	4.01058e7	1.69244
$892$	− 2.25861e6i	− 0.0950450i
$893$	− 3.62709e7i	− 1.52205i
$894$	−1.54701e6	−0.0647363
$895$	0	0
$896$	−3.77970e6	−0.157285
$897$	9.23255e6i	0.383125i
$898$	1.42929e7i	0.591464i
$899$	−4.29552e7	−1.77262
$900$	0	0
$901$	6.12363e6	0.251303
$902$	2.61373e7i	1.06966i
$903$	− 6.68039e7i	− 2.72636i
$904$	1.28608e7	0.523414
$905$	0	0
$906$	−2.37256e6	−0.0960276
$907$	− 494733.i	− 0.0199688i	−0.999950	−	0.00998441i	$-0.996822\pi$
$907$	− 494733.i	− 0.0199688i	0.999950	−	0.00998441i	$-0.00317819\pi$
$908$	1.54953e7i	0.623715i
$909$	−1.26041e7	−0.505944
$910$	0	0
$911$	1.03084e7	0.411526	0.205763	−	0.978602i	$-0.434032\pi$
$911$	1.03084e7	0.411526	0.205763	+	0.978602i	$0.434032\pi$
$912$	1.06902e7i	0.425597i
$913$	1.23313e7i	0.489590i
$914$	3.25338e7	1.28816
$915$	0	0
$916$	1.16620e6	0.0459233
$917$	4.59894e7i	1.80607i
$918$	5.59764e6i	0.219229i
$919$	−3.06006e7	−1.19520	−0.597602	−	0.801793i	$-0.703880\pi$
$919$	−3.06006e7	−1.19520	−0.597602	+	0.801793i	$0.703880\pi$
$920$	0	0
$921$	−2.66970e7	−1.03708
$922$	− 1.47969e7i	− 0.573250i
$923$	7.46344e6i	0.288360i
$924$	3.76769e7	1.45176
$925$	0	0
$926$	−5.36507e6	−0.205612
$927$	− 5.24772e6i	− 0.200573i
$928$	7.02896e6i	0.267930i
$929$	4.45717e7	1.69441	0.847207	−	0.531263i	$-0.178282\pi$
$929$	4.45717e7	1.69441	0.847207	+	0.531263i	$0.178282\pi$
$930$	0	0
$931$	−8.11347e7	−3.06784
$932$	9.81907e6i	0.370280i
$933$	− 3.35833e7i	− 1.26305i
$934$	−9.64271e6	−0.361686
$935$	0	0
$936$	1.17061e6	0.0436739
$937$	4.43114e7i	1.64879i	0.566012	+	0.824397i	$0.308485\pi$
$937$	4.43114e7i	1.64879i	−0.566012	+	0.824397i	$0.691515\pi$
$938$	4.87019e7i	1.80733i
$939$	−5.50398e7	−2.03710
$940$	0	0
$941$	1.48172e7	0.545498	0.272749	−	0.962085i	$-0.412067\pi$
$941$	1.48172e7	0.545498	0.272749	+	0.962085i	$0.412067\pi$
$942$	− 3.56149e7i	− 1.30769i
$943$	− 3.49718e7i	− 1.28068i
$944$	6.18136e6	0.225764
$945$	0	0
$946$	−3.36629e7	−1.22299
$947$	− 1.14497e7i	− 0.414876i	−0.978248	−	0.207438i	$-0.933487\pi$
$947$	− 1.14497e7i	− 0.414876i	0.978248	−	0.207438i	$-0.0665125\pi$
$948$	− 1.78637e7i	− 0.645581i
$949$	7.05765e6	0.254387
$950$	0	0
$951$	−7.32734e6	−0.262721
$952$	8.18035e6i	0.292536i
$953$	3.21943e7i	1.14828i	0.818758	+	0.574139i	$0.194663\pi$
$953$	3.21943e7i	1.14828i	−0.818758	+	0.574139i	$0.805337\pi$
$954$	4.78475e6	0.170211
$955$	0	0
$956$	−1.94311e7	−0.687626
$957$	− 7.00663e7i	− 2.47303i
$958$	− 74676.5i	− 0.00262888i
$959$	−6.13244e7	−2.15321
$960$	0	0
$961$	1.05315e7	0.367858
$962$	3.85019e6i	0.134136i
$963$	2.04792e7i	0.711617i
$964$	2.86159e7	0.991780
$965$	0	0
$966$	−5.04118e7	−1.73816
$967$	4.07255e7i	1.40055i	0.713871	+	0.700277i	$0.246940\pi$
$967$	4.07255e7i	1.40055i	−0.713871	+	0.700277i	$0.753060\pi$
$968$	− 8.67837e6i	− 0.297680i
$969$	2.31366e7	0.791571
$970$	0	0
$971$	1.63928e7	0.557961	0.278980	−	0.960297i	$-0.410004\pi$
$971$	1.63928e7	0.557961	0.278980	+	0.960297i	$0.410004\pi$
$972$	1.22599e7i	0.416220i
$973$	227301.i	0.00769698i
$974$	1.81266e7	0.612236
$975$	0	0
$976$	−2.73453e6	−0.0918880
$977$	4.03891e7i	1.35372i	0.736114	+	0.676858i	$0.236659\pi$
$977$	4.03891e7i	1.35372i	−0.736114	+	0.676858i	$0.763341\pi$
$978$	− 2.76167e7i	− 0.923261i
$979$	7.07314e7	2.35861
$980$	0	0
$981$	1.70599e7	0.565984
$982$	1.53996e7i	0.509601i
$983$	− 4.46121e7i	− 1.47254i	−0.676685	−	0.736272i	$-0.736584\pi$
$983$	− 4.46121e7i	− 1.47254i	0.676685	−	0.736272i	$-0.263416\pi$
$984$	−1.43898e7	−0.473769
$985$	0	0
$986$	1.52127e7	0.498325
$987$	− 7.03787e7i	− 2.29958i
$988$	6.02499e6i	0.196365i
$989$	4.50411e7	1.46426
$990$	0	0
$991$	−2.35070e7	−0.760351	−0.380175	−	0.924914i	$-0.624136\pi$
$991$	−2.35070e7	−0.760351	−0.380175	+	0.924914i	$0.624136\pi$
$992$	− 6.40803e6i	− 0.206750i
$993$	− 5.22841e7i	− 1.68266i
$994$	−4.07521e7	−1.30823
$995$	0	0
$996$	−6.78895e6	−0.216847
$997$	1.84154e6i	0.0586737i	0.999570	+	0.0293369i	$0.00933955\pi$
$997$	1.84154e6i	0.0586737i	−0.999570	+	0.0293369i	$0.990660\pi$
$998$	− 8.75378e6i	− 0.278208i
$999$	−1.43855e7	−0.456050

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.b.o.599.11		12
5.2	odd	4		650.6.a.s.1.5	✓	6
5.3	odd	4		650.6.a.t.1.2	yes	6
5.4	even	2	inner	650.6.b.o.599.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.6.a.s.1.5	✓	6	5.2	odd	4
650.6.a.t.1.2	yes	6	5.3	odd	4
650.6.b.o.599.2		12	5.4	even	2	inner
650.6.b.o.599.11		12	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	650.6.b.o.599.11

Level	$650$
Weight	$6$
Character	650.599
Analytic conductor	$104.249$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$