LMFDB - Embedded newform 1050.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,4,Mod(1,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("1050.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1050.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$61.9520055060$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{8761}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 2190 $ x^2 - x - 2190
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$47.3001$ of defining polynomial
Character	$\chi$	$=$	1050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -50.3001 q^{11} -12.0000 q^{12} +45.3001 q^{13} +14.0000 q^{14} +16.0000 q^{16} -83.3001 q^{17} -18.0000 q^{18} +20.0000 q^{19} +21.0000 q^{21} +100.600 q^{22} -151.600 q^{23} +24.0000 q^{24} -90.6002 q^{26} -27.0000 q^{27} -28.0000 q^{28} +15.0000 q^{29} +328.501 q^{31} -32.0000 q^{32} +150.900 q^{33} +166.600 q^{34} +36.0000 q^{36} -307.501 q^{37} -40.0000 q^{38} -135.900 q^{39} +89.3001 q^{41} -42.0000 q^{42} -386.200 q^{43} -201.200 q^{44} +303.200 q^{46} +153.200 q^{47} -48.0000 q^{48} +49.0000 q^{49} +249.900 q^{51} +181.200 q^{52} -393.900 q^{53} +54.0000 q^{54} +56.0000 q^{56} -60.0000 q^{57} -30.0000 q^{58} +730.301 q^{59} -732.901 q^{61} -657.001 q^{62} -63.0000 q^{63} +64.0000 q^{64} -301.801 q^{66} +899.301 q^{67} -333.200 q^{68} +454.801 q^{69} +104.300 q^{71} -72.0000 q^{72} -172.800 q^{73} +615.001 q^{74} +80.0000 q^{76} +352.101 q^{77} +271.801 q^{78} -799.102 q^{79} +81.0000 q^{81} -178.600 q^{82} +157.502 q^{83} +84.0000 q^{84} +772.401 q^{86} -45.0000 q^{87} +402.401 q^{88} -962.201 q^{89} -317.101 q^{91} -606.401 q^{92} -985.502 q^{93} -306.401 q^{94} +96.0000 q^{96} -252.002 q^{97} -98.0000 q^{98} -452.701 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 + 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9	$ 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9} - 7 q^{11} - 24 q^{12} - 3 q^{13} + 28 q^{14} + 32 q^{16} - 73 q^{17} - 36 q^{18} + 40 q^{19} + 42 q^{21} + 14 q^{22} - 116 q^{23} + 48 q^{24} + 6 q^{26} - 54 q^{27} - 56 q^{28} + 30 q^{29} + 189 q^{31} - 64 q^{32} + 21 q^{33} + 146 q^{34} + 72 q^{36} - 147 q^{37} - 80 q^{38} + 9 q^{39} + 85 q^{41} - 84 q^{42} - 398 q^{43} - 28 q^{44} + 232 q^{46} - 68 q^{47} - 96 q^{48} + 98 q^{49} + 219 q^{51} - 12 q^{52} - 507 q^{53} + 108 q^{54} + 112 q^{56} - 120 q^{57} - 60 q^{58} + 431 q^{59} - 249 q^{61} - 378 q^{62} - 126 q^{63} + 128 q^{64} - 42 q^{66} + 769 q^{67} - 292 q^{68} + 348 q^{69} + 115 q^{71} - 144 q^{72} - 720 q^{73} + 294 q^{74} + 160 q^{76} + 49 q^{77} - 18 q^{78} - 7 q^{79} + 162 q^{81} - 170 q^{82} - 1089 q^{83} + 168 q^{84} + 796 q^{86} - 90 q^{87} + 56 q^{88} - 614 q^{89} + 21 q^{91} - 464 q^{92} - 567 q^{93} + 136 q^{94} + 192 q^{96} + 1368 q^{97} - 196 q^{98} - 63 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 + 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9 - 7 * q^11 - 24 * q^12 - 3 * q^13 + 28 * q^14 + 32 * q^16 - 73 * q^17 - 36 * q^18 + 40 * q^19 + 42 * q^21 + 14 * q^22 - 116 * q^23 + 48 * q^24 + 6 * q^26 - 54 * q^27 - 56 * q^28 + 30 * q^29 + 189 * q^31 - 64 * q^32 + 21 * q^33 + 146 * q^34 + 72 * q^36 - 147 * q^37 - 80 * q^38 + 9 * q^39 + 85 * q^41 - 84 * q^42 - 398 * q^43 - 28 * q^44 + 232 * q^46 - 68 * q^47 - 96 * q^48 + 98 * q^49 + 219 * q^51 - 12 * q^52 - 507 * q^53 + 108 * q^54 + 112 * q^56 - 120 * q^57 - 60 * q^58 + 431 * q^59 - 249 * q^61 - 378 * q^62 - 126 * q^63 + 128 * q^64 - 42 * q^66 + 769 * q^67 - 292 * q^68 + 348 * q^69 + 115 * q^71 - 144 * q^72 - 720 * q^73 + 294 * q^74 + 160 * q^76 + 49 * q^77 - 18 * q^78 - 7 * q^79 + 162 * q^81 - 170 * q^82 - 1089 * q^83 + 168 * q^84 + 796 * q^86 - 90 * q^87 + 56 * q^88 - 614 * q^89 + 21 * q^91 - 464 * q^92 - 567 * q^93 + 136 * q^94 + 192 * q^96 + 1368 * q^97 - 196 * q^98 - 63 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	−50.3001	−1.37873	−0.689366	−	0.724413i	$-0.742111\pi$
$11$	−50.3001	−1.37873	−0.689366	+	0.724413i	$0.742111\pi$
$12$	−12.0000	−0.288675
$13$	45.3001	0.966461	0.483230	−	0.875493i	$-0.339463\pi$
$13$	45.3001	0.966461	0.483230	+	0.875493i	$0.339463\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−83.3001	−1.18843	−0.594213	−	0.804308i	$-0.702536\pi$
$17$	−83.3001	−1.18843	−0.594213	+	0.804308i	$0.702536\pi$
$18$	−18.0000	−0.235702
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	100.600	0.974911
$23$	−151.600	−1.37438	−0.687192	−	0.726476i	$-0.741157\pi$
$23$	−151.600	−1.37438	−0.687192	+	0.726476i	$0.741157\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	−90.6002	−0.683391
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	15.0000	0.0960493	0.0480247	−	0.998846i	$-0.484707\pi$
$29$	15.0000	0.0960493	0.0480247	+	0.998846i	$0.484707\pi$
$30$	0	0
$31$	328.501	1.90324	0.951620	−	0.307277i	$-0.0994180\pi$
$31$	328.501	1.90324	0.951620	+	0.307277i	$0.0994180\pi$
$32$	−32.0000	−0.176777
$33$	150.900	0.796011
$34$	166.600	0.840344
$35$	0	0
$36$	36.0000	0.166667
$37$	−307.501	−1.36629	−0.683146	−	0.730282i	$-0.739389\pi$
$37$	−307.501	−1.36629	−0.683146	+	0.730282i	$0.739389\pi$
$38$	−40.0000	−0.170759
$39$	−135.900	−0.557986
$40$	0	0
$41$	89.3001	0.340154	0.170077	−	0.985431i	$-0.445598\pi$
$41$	89.3001	0.340154	0.170077	+	0.985431i	$0.445598\pi$
$42$	−42.0000	−0.154303
$43$	−386.200	−1.36965	−0.684826	−	0.728707i	$-0.740122\pi$
$43$	−386.200	−1.36965	−0.684826	+	0.728707i	$0.740122\pi$
$44$	−201.200	−0.689366
$45$	0	0
$46$	303.200	0.971836
$47$	153.200	0.475459	0.237730	−	0.971331i	$-0.423597\pi$
$47$	153.200	0.475459	0.237730	+	0.971331i	$0.423597\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	0	0
$51$	249.900	0.686138
$52$	181.200	0.483230
$53$	−393.900	−1.02087	−0.510437	−	0.859915i	$-0.670517\pi$
$53$	−393.900	−1.02087	−0.510437	+	0.859915i	$0.670517\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	56.0000	0.133631
$57$	−60.0000	−0.139424
$58$	−30.0000	−0.0679171
$59$	730.301	1.61148	0.805738	−	0.592272i	$-0.201769\pi$
$59$	730.301	1.61148	0.805738	+	0.592272i	$0.201769\pi$
$60$	0	0
$61$	−732.901	−1.53833	−0.769167	−	0.639048i	$-0.779329\pi$
$61$	−732.901	−1.53833	−0.769167	+	0.639048i	$0.779329\pi$
$62$	−657.001	−1.34579
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	0	0
$66$	−301.801	−0.562865
$67$	899.301	1.63981	0.819904	−	0.572501i	$-0.194027\pi$
$67$	899.301	1.63981	0.819904	+	0.572501i	$0.194027\pi$
$68$	−333.200	−0.594213
$69$	454.801	0.793501
$70$	0	0
$71$	104.300	0.174340	0.0871700	−	0.996193i	$-0.472218\pi$
$71$	104.300	0.174340	0.0871700	+	0.996193i	$0.472218\pi$
$72$	−72.0000	−0.117851
$73$	−172.800	−0.277050	−0.138525	−	0.990359i	$-0.544236\pi$
$73$	−172.800	−0.277050	−0.138525	+	0.990359i	$0.544236\pi$
$74$	615.001	0.966114
$75$	0	0
$76$	80.0000	0.120745
$77$	352.101	0.521112
$78$	271.801	0.394556
$79$	−799.102	−1.13805	−0.569025	−	0.822320i	$-0.692679\pi$
$79$	−799.102	−1.13805	−0.569025	+	0.822320i	$0.692679\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−178.600	−0.240526
$83$	157.502	0.208290	0.104145	−	0.994562i	$-0.466789\pi$
$83$	157.502	0.208290	0.104145	+	0.994562i	$0.466789\pi$
$84$	84.0000	0.109109
$85$	0	0
$86$	772.401	0.968490
$87$	−45.0000	−0.0554541
$88$	402.401	0.487455
$89$	−962.201	−1.14599	−0.572995	−	0.819559i	$-0.694219\pi$
$89$	−962.201	−1.14599	−0.572995	+	0.819559i	$0.694219\pi$
$90$	0	0
$91$	−317.101	−0.365288
$92$	−606.401	−0.687192
$93$	−985.502	−1.09884
$94$	−306.401	−0.336200
$95$	0	0
$96$	96.0000	0.102062
$97$	−252.002	−0.263783	−0.131891	−	0.991264i	$-0.542105\pi$
$97$	−252.002	−0.263783	−0.131891	+	0.991264i	$0.542105\pi$
$98$	−98.0000	−0.101015
$99$	−452.701	−0.459577
$100$	0	0
$101$	−1049.40	−1.03386	−0.516928	−	0.856029i	$-0.672924\pi$
$101$	−1049.40	−1.03386	−0.516928	+	0.856029i	$0.672924\pi$
$102$	−499.801	−0.485173
$103$	63.7010	0.0609383	0.0304691	−	0.999536i	$-0.490300\pi$
$103$	63.7010	0.0609383	0.0304691	+	0.999536i	$0.490300\pi$
$104$	−362.401	−0.341695
$105$	0	0
$106$	787.801	0.721868
$107$	409.400	0.369889	0.184945	−	0.982749i	$-0.440789\pi$
$107$	409.400	0.369889	0.184945	+	0.982749i	$0.440789\pi$
$108$	−108.000	−0.0962250
$109$	1842.90	1.61943	0.809716	−	0.586822i	$-0.199621\pi$
$109$	1842.90	1.61943	0.809716	+	0.586822i	$0.199621\pi$
$110$	0	0
$111$	922.502	0.788829
$112$	−112.000	−0.0944911
$113$	−1823.50	−1.51806	−0.759029	−	0.651057i	$-0.774326\pi$
$113$	−1823.50	−1.51806	−0.759029	+	0.651057i	$0.774326\pi$
$114$	120.000	0.0985880
$115$	0	0
$116$	60.0000	0.0480247
$117$	407.701	0.322154
$118$	−1460.60	−1.13949
$119$	583.101	0.449183
$120$	0	0
$121$	1199.10	0.900902
$122$	1465.80	1.08777
$123$	−267.900	−0.196388
$124$	1314.00	0.951620
$125$	0	0
$126$	126.000	0.0890871
$127$	1607.90	1.12345	0.561724	−	0.827324i	$-0.310138\pi$
$127$	1607.90	1.12345	0.561724	+	0.827324i	$0.310138\pi$
$128$	−128.000	−0.0883883
$129$	1158.60	0.790769
$130$	0	0
$131$	2628.60	1.75315	0.876573	−	0.481268i	$-0.159824\pi$
$131$	2628.60	1.75315	0.876573	+	0.481268i	$0.159824\pi$
$132$	603.601	0.398006
$133$	−140.000	−0.0912747
$134$	−1798.60	−1.15952
$135$	0	0
$136$	666.401	0.420172
$137$	1519.40	0.947526	0.473763	−	0.880652i	$-0.342895\pi$
$137$	1519.40	0.947526	0.473763	+	0.880652i	$0.342895\pi$
$138$	−909.601	−0.561090
$139$	1473.80	0.899325	0.449662	−	0.893199i	$-0.351544\pi$
$139$	1473.80	0.899325	0.449662	+	0.893199i	$0.351544\pi$
$140$	0	0
$141$	−459.601	−0.274506
$142$	−208.600	−0.123277
$143$	−2278.60	−1.33249
$144$	144.000	0.0833333
$145$	0	0
$146$	345.599	0.195904
$147$	−147.000	−0.0824786
$148$	−1230.00	−0.683146
$149$	268.801	0.147792	0.0738960	−	0.997266i	$-0.476457\pi$
$149$	268.801	0.147792	0.0738960	+	0.997266i	$0.476457\pi$
$150$	0	0
$151$	−870.103	−0.468927	−0.234463	−	0.972125i	$-0.575333\pi$
$151$	−870.103	−0.468927	−0.234463	+	0.972125i	$0.575333\pi$
$152$	−160.000	−0.0853797
$153$	−749.701	−0.396142
$154$	−704.201	−0.368482
$155$	0	0
$156$	−543.601	−0.278993
$157$	2507.80	1.27480	0.637402	−	0.770531i	$-0.280009\pi$
$157$	2507.80	1.27480	0.637402	+	0.770531i	$0.280009\pi$
$158$	1598.20	0.804723
$159$	1181.70	0.589402
$160$	0	0
$161$	1061.20	0.519468
$162$	−162.000	−0.0785674
$163$	2394.50	1.15062	0.575312	−	0.817934i	$-0.304881\pi$
$163$	2394.50	1.15062	0.575312	+	0.817934i	$0.304881\pi$
$164$	357.200	0.170077
$165$	0	0
$166$	−315.003	−0.147283
$167$	3418.60	1.58407	0.792034	−	0.610477i	$-0.209022\pi$
$167$	3418.60	1.58407	0.792034	+	0.610477i	$0.209022\pi$
$168$	−168.000	−0.0771517
$169$	−144.900	−0.0659537
$170$	0	0
$171$	180.000	0.0804967
$172$	−1544.80	−0.684826
$173$	−2301.20	−1.01131	−0.505656	−	0.862735i	$-0.668750\pi$
$173$	−2301.20	−1.01131	−0.505656	+	0.862735i	$0.668750\pi$
$174$	90.0000	0.0392120
$175$	0	0
$176$	−804.802	−0.344683
$177$	−2190.90	−0.930386
$178$	1924.40	0.810338
$179$	−1815.80	−0.758210	−0.379105	−	0.925354i	$-0.623768\pi$
$179$	−1815.80	−0.758210	−0.379105	+	0.925354i	$0.623768\pi$
$180$	0	0
$181$	3684.80	1.51320	0.756600	−	0.653878i	$-0.226859\pi$
$181$	3684.80	1.51320	0.756600	+	0.653878i	$0.226859\pi$
$182$	634.201	0.258297
$183$	2198.70	0.888158
$184$	1212.80	0.485918
$185$	0	0
$186$	1971.00	0.776994
$187$	4190.00	1.63852
$188$	612.802	0.237730
$189$	189.000	0.0727393
$190$	0	0
$191$	1845.50	0.699140	0.349570	−	0.936910i	$-0.386328\pi$
$191$	1845.50	0.699140	0.349570	+	0.936910i	$0.386328\pi$
$192$	−192.000	−0.0721688
$193$	1955.10	0.729176	0.364588	−	0.931169i	$-0.381210\pi$
$193$	1955.10	0.729176	0.364588	+	0.931169i	$0.381210\pi$
$194$	504.004	0.186523
$195$	0	0
$196$	196.000	0.0714286
$197$	4326.60	1.56476	0.782380	−	0.622802i	$-0.214006\pi$
$197$	4326.60	1.56476	0.782380	+	0.622802i	$0.214006\pi$
$198$	905.402	0.324970
$199$	1104.80	0.393553	0.196776	−	0.980448i	$-0.436953\pi$
$199$	1104.80	0.393553	0.196776	+	0.980448i	$0.436953\pi$
$200$	0	0
$201$	−2697.90	−0.946743
$202$	2098.80	0.731046
$203$	−105.000	−0.0363032
$204$	999.601	0.343069
$205$	0	0
$206$	−127.402	−0.0430899
$207$	−1364.40	−0.458128
$208$	724.802	0.241615
$209$	−1006.00	−0.332950
$210$	0	0
$211$	3367.70	1.09878	0.549389	−	0.835567i	$-0.314861\pi$
$211$	3367.70	1.09878	0.549389	+	0.835567i	$0.314861\pi$
$212$	−1575.60	−0.510437
$213$	−312.900	−0.100655
$214$	−818.800	−0.261551
$215$	0	0
$216$	216.000	0.0680414
$217$	−2299.50	−0.719357
$218$	−3685.80	−1.14511
$219$	518.399	0.159955
$220$	0	0
$221$	−3773.50	−1.14857
$222$	−1845.00	−0.557786
$223$	2505.50	0.752379	0.376189	−	0.926543i	$-0.377234\pi$
$223$	2505.50	0.752379	0.376189	+	0.926543i	$0.377234\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	3647.00	1.07343
$227$	−1649.71	−0.482356	−0.241178	−	0.970481i	$-0.577534\pi$
$227$	−1649.71	−0.482356	−0.241178	+	0.970481i	$0.577534\pi$
$228$	−240.000	−0.0697122
$229$	−3494.40	−1.00837	−0.504185	−	0.863596i	$-0.668207\pi$
$229$	−3494.40	−1.00837	−0.504185	+	0.863596i	$0.668207\pi$
$230$	0	0
$231$	−1056.30	−0.300864
$232$	−120.000	−0.0339586
$233$	578.898	0.162768	0.0813839	−	0.996683i	$-0.474066\pi$
$233$	578.898	0.162768	0.0813839	+	0.996683i	$0.474066\pi$
$234$	−815.402	−0.227797
$235$	0	0
$236$	2921.20	0.805738
$237$	2397.31	0.657054
$238$	−1166.20	−0.317620
$239$	6617.61	1.79104	0.895518	−	0.445025i	$-0.146805\pi$
$239$	6617.61	1.79104	0.895518	+	0.445025i	$0.146805\pi$
$240$	0	0
$241$	3500.20	0.935552	0.467776	−	0.883847i	$-0.345055\pi$
$241$	3500.20	0.935552	0.467776	+	0.883847i	$0.345055\pi$
$242$	−2398.20	−0.637034
$243$	−243.000	−0.0641500
$244$	−2931.61	−0.769167
$245$	0	0
$246$	535.801	0.138867
$247$	906.002	0.233391
$248$	−2628.00	−0.672897
$249$	−472.505	−0.120256
$250$	0	0
$251$	−5275.11	−1.32654	−0.663271	−	0.748379i	$-0.730832\pi$
$251$	−5275.11	−1.32654	−0.663271	+	0.748379i	$0.730832\pi$
$252$	−252.000	−0.0629941
$253$	7625.51	1.89491
$254$	−3215.80	−0.794398
$255$	0	0
$256$	256.000	0.0625000
$257$	−2146.10	−0.520896	−0.260448	−	0.965488i	$-0.583870\pi$
$257$	−2146.10	−0.520896	−0.260448	+	0.965488i	$0.583870\pi$
$258$	−2317.20	−0.559158
$259$	2152.50	0.516410
$260$	0	0
$261$	135.000	0.0320164
$262$	−5257.21	−1.23966
$263$	−1337.41	−0.313566	−0.156783	−	0.987633i	$-0.550112\pi$
$263$	−1337.41	−0.313566	−0.156783	+	0.987633i	$0.550112\pi$
$264$	−1207.20	−0.281433
$265$	0	0
$266$	280.000	0.0645410
$267$	2886.60	0.661638
$268$	3597.20	0.819904
$269$	2042.40	0.462927	0.231463	−	0.972844i	$-0.425649\pi$
$269$	2042.40	0.462927	0.231463	+	0.972844i	$0.425649\pi$
$270$	0	0
$271$	4946.81	1.10885	0.554423	−	0.832235i	$-0.312939\pi$
$271$	4946.81	1.10885	0.554423	+	0.832235i	$0.312939\pi$
$272$	−1332.80	−0.297107
$273$	951.302	0.210899
$274$	−3038.80	−0.670002
$275$	0	0
$276$	1819.20	0.396750
$277$	3937.61	0.854109	0.427055	−	0.904226i	$-0.359551\pi$
$277$	3937.61	0.854109	0.427055	+	0.904226i	$0.359551\pi$
$278$	−2947.60	−0.635919
$279$	2956.50	0.634413
$280$	0	0
$281$	3634.31	0.771547	0.385774	−	0.922593i	$-0.373935\pi$
$281$	3634.31	0.771547	0.385774	+	0.922593i	$0.373935\pi$
$282$	919.203	0.194105
$283$	−36.2058	−0.00760498	−0.00380249	−	0.999993i	$-0.501210\pi$
$283$	−36.2058	−0.00760498	−0.00380249	+	0.999993i	$0.501210\pi$
$284$	417.200	0.0871700
$285$	0	0
$286$	4557.20	0.942213
$287$	−625.101	−0.128566
$288$	−288.000	−0.0589256
$289$	2025.91	0.412357
$290$	0	0
$291$	756.006	0.152295
$292$	−691.198	−0.138525
$293$	2013.61	0.401489	0.200744	−	0.979644i	$-0.435664\pi$
$293$	2013.61	0.401489	0.200744	+	0.979644i	$0.435664\pi$
$294$	294.000	0.0583212
$295$	0	0
$296$	2460.00	0.483057
$297$	1358.10	0.265337
$298$	−537.601	−0.104505
$299$	−6867.51	−1.32829
$300$	0	0
$301$	2703.40	0.517680
$302$	1740.21	0.331581
$303$	3148.21	0.596897
$304$	320.000	0.0603726
$305$	0	0
$306$	1499.40	0.280115
$307$	−861.203	−0.160102	−0.0800512	−	0.996791i	$-0.525508\pi$
$307$	−861.203	−0.160102	−0.0800512	+	0.996791i	$0.525508\pi$
$308$	1408.40	0.260556
$309$	−191.103	−0.0351827
$310$	0	0
$311$	2266.40	0.413234	0.206617	−	0.978422i	$-0.433755\pi$
$311$	2266.40	0.413234	0.206617	+	0.978422i	$0.433755\pi$
$312$	1087.20	0.197278
$313$	−1381.80	−0.249534	−0.124767	−	0.992186i	$-0.539818\pi$
$313$	−1381.80	−0.249534	−0.124767	+	0.992186i	$0.539818\pi$
$314$	−5015.60	−0.901423
$315$	0	0
$316$	−3196.41	−0.569025
$317$	8243.82	1.46063	0.730314	−	0.683112i	$-0.239374\pi$
$317$	8243.82	1.46063	0.730314	+	0.683112i	$0.239374\pi$
$318$	−2363.40	−0.416770
$319$	−754.502	−0.132426
$320$	0	0
$321$	−1228.20	−0.213556
$322$	−2122.40	−0.367320
$323$	−1666.00	−0.286993
$324$	324.000	0.0555556
$325$	0	0
$326$	−4789.00	−0.813614
$327$	−5528.71	−0.934979
$328$	−714.401	−0.120263
$329$	−1072.40	−0.179707
$330$	0	0
$331$	−11466.6	−1.90412	−0.952058	−	0.305916i	$-0.901037\pi$
$331$	−11466.6	−1.90412	−0.952058	+	0.305916i	$0.901037\pi$
$332$	630.006	0.104145
$333$	−2767.50	−0.455430
$334$	−6837.20	−1.12010
$335$	0	0
$336$	336.000	0.0545545
$337$	−3305.32	−0.534279	−0.267139	−	0.963658i	$-0.586078\pi$
$337$	−3305.32	−0.534279	−0.267139	+	0.963658i	$0.586078\pi$
$338$	289.801	0.0466363
$339$	5470.50	0.876451
$340$	0	0
$341$	−16523.6	−2.62406
$342$	−360.000	−0.0569198
$343$	−343.000	−0.0539949
$344$	3089.60	0.484245
$345$	0	0
$346$	4602.40	0.715106
$347$	−8561.72	−1.32454	−0.662272	−	0.749263i	$-0.730408\pi$
$347$	−8561.72	−1.32454	−0.662272	+	0.749263i	$0.730408\pi$
$348$	−180.000	−0.0277270
$349$	−553.894	−0.0849549	−0.0424775	−	0.999097i	$-0.513525\pi$
$349$	−553.894	−0.0849549	−0.0424775	+	0.999097i	$0.513525\pi$
$350$	0	0
$351$	−1223.10	−0.185995
$352$	1609.60	0.243728
$353$	778.810	0.117427	0.0587137	−	0.998275i	$-0.481300\pi$
$353$	778.810	0.117427	0.0587137	+	0.998275i	$0.481300\pi$
$354$	4381.81	0.657883
$355$	0	0
$356$	−3848.81	−0.572995
$357$	−1749.30	−0.259336
$358$	3631.61	0.536136
$359$	6319.01	0.928982	0.464491	−	0.885578i	$-0.346237\pi$
$359$	6319.01	0.928982	0.464491	+	0.885578i	$0.346237\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	−7369.61	−1.06999
$363$	−3597.30	−0.520136
$364$	−1268.40	−0.182644
$365$	0	0
$366$	−4397.41	−0.628022
$367$	7227.90	1.02805	0.514024	−	0.857776i	$-0.328154\pi$
$367$	7227.90	1.02805	0.514024	+	0.857776i	$0.328154\pi$
$368$	−2425.60	−0.343596
$369$	803.701	0.113385
$370$	0	0
$371$	2757.30	0.385854
$372$	−3942.01	−0.549418
$373$	7980.10	1.10776	0.553879	−	0.832597i	$-0.313147\pi$
$373$	7980.10	1.10776	0.553879	+	0.832597i	$0.313147\pi$
$374$	−8380.01	−1.15861
$375$	0	0
$376$	−1225.60	−0.168100
$377$	679.502	0.0928279
$378$	−378.000	−0.0514344
$379$	−10341.6	−1.40162	−0.700809	−	0.713349i	$-0.747177\pi$
$379$	−10341.6	−1.40162	−0.700809	+	0.713349i	$0.747177\pi$
$380$	0	0
$381$	−4823.70	−0.648623
$382$	−3691.00	−0.494366
$383$	−6339.60	−0.845792	−0.422896	−	0.906178i	$-0.638987\pi$
$383$	−6339.60	−0.845792	−0.422896	+	0.906178i	$0.638987\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−3910.20	−0.515605
$387$	−3475.80	−0.456550
$388$	−1008.01	−0.131891
$389$	1216.50	0.158557	0.0792786	−	0.996852i	$-0.474738\pi$
$389$	1216.50	0.158557	0.0792786	+	0.996852i	$0.474738\pi$
$390$	0	0
$391$	12628.3	1.63335
$392$	−392.000	−0.0505076
$393$	−7885.81	−1.01218
$394$	−8653.20	−1.10645
$395$	0	0
$396$	−1810.80	−0.229789
$397$	−12690.3	−1.60430	−0.802152	−	0.597119i	$-0.796312\pi$
$397$	−12690.3	−1.60430	−0.802152	+	0.597119i	$0.796312\pi$
$398$	−2209.59	−0.278284
$399$	420.000	0.0526975
$400$	0	0
$401$	9047.13	1.12666	0.563332	−	0.826231i	$-0.309519\pi$
$401$	9047.13	1.12666	0.563332	+	0.826231i	$0.309519\pi$
$402$	5395.81	0.669449
$403$	14881.1	1.83941
$404$	−4197.61	−0.516928
$405$	0	0
$406$	210.000	0.0256703
$407$	15467.3	1.88375
$408$	−1999.20	−0.242586
$409$	5002.20	0.604750	0.302375	−	0.953189i	$-0.402220\pi$
$409$	5002.20	0.604750	0.302375	+	0.953189i	$0.402220\pi$
$410$	0	0
$411$	−4558.20	−0.547054
$412$	254.804	0.0304691
$413$	−5112.11	−0.609081
$414$	2728.80	0.323945
$415$	0	0
$416$	−1449.60	−0.170848
$417$	−4421.40	−0.519225
$418$	2012.00	0.235431
$419$	−7398.91	−0.862674	−0.431337	−	0.902191i	$-0.641958\pi$
$419$	−7398.91	−0.862674	−0.431337	+	0.902191i	$0.641958\pi$
$420$	0	0
$421$	−11235.9	−1.30072	−0.650362	−	0.759625i	$-0.725383\pi$
$421$	−11235.9	−1.30072	−0.650362	+	0.759625i	$0.725383\pi$
$422$	−6735.40	−0.776953
$423$	1378.80	0.158486
$424$	3151.20	0.360934
$425$	0	0
$426$	625.801	0.0711740
$427$	5130.31	0.581436
$428$	1637.60	0.184945
$429$	6835.80	0.769314
$430$	0	0
$431$	−5382.89	−0.601589	−0.300794	−	0.953689i	$-0.597252\pi$
$431$	−5382.89	−0.601589	−0.300794	+	0.953689i	$0.597252\pi$
$432$	−432.000	−0.0481125
$433$	2532.20	0.281038	0.140519	−	0.990078i	$-0.455123\pi$
$433$	2532.20	0.281038	0.140519	+	0.990078i	$0.455123\pi$
$434$	4599.01	0.508662
$435$	0	0
$436$	7371.61	0.809716
$437$	−3032.00	−0.331900
$438$	−1036.80	−0.113105
$439$	−10530.1	−1.14482	−0.572408	−	0.819969i	$-0.693991\pi$
$439$	−10530.1	−1.14482	−0.572408	+	0.819969i	$0.693991\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	7547.01	0.812160
$443$	−2166.81	−0.232389	−0.116194	−	0.993227i	$-0.537070\pi$
$443$	−2166.81	−0.232389	−0.116194	+	0.993227i	$0.537070\pi$
$444$	3690.01	0.394414
$445$	0	0
$446$	−5010.99	−0.532012
$447$	−806.402	−0.0853277
$448$	−448.000	−0.0472456
$449$	7809.69	0.820851	0.410426	−	0.911894i	$-0.365380\pi$
$449$	7809.69	0.820851	0.410426	+	0.911894i	$0.365380\pi$
$450$	0	0
$451$	−4491.80	−0.468982
$452$	−7294.00	−0.759029
$453$	2610.31	0.270735
$454$	3299.41	0.341078
$455$	0	0
$456$	480.000	0.0492940
$457$	−2581.80	−0.264270	−0.132135	−	0.991232i	$-0.542183\pi$
$457$	−2581.80	−0.264270	−0.132135	+	0.991232i	$0.542183\pi$
$458$	6988.81	0.713025
$459$	2249.10	0.228713
$460$	0	0
$461$	−2009.20	−0.202989	−0.101495	−	0.994836i	$-0.532362\pi$
$461$	−2009.20	−0.202989	−0.101495	+	0.994836i	$0.532362\pi$
$462$	2112.60	0.212743
$463$	7694.79	0.772370	0.386185	−	0.922421i	$-0.373793\pi$
$463$	7694.79	0.772370	0.386185	+	0.922421i	$0.373793\pi$
$464$	240.000	0.0240123
$465$	0	0
$466$	−1157.80	−0.115094
$467$	15375.5	1.52354	0.761772	−	0.647845i	$-0.224330\pi$
$467$	15375.5	1.52354	0.761772	+	0.647845i	$0.224330\pi$
$468$	1630.80	0.161077
$469$	−6295.11	−0.619789
$470$	0	0
$471$	−7523.40	−0.736009
$472$	−5842.41	−0.569743
$473$	19425.9	1.88838
$474$	−4794.61	−0.464607
$475$	0	0
$476$	2332.40	0.224591
$477$	−3545.10	−0.340292
$478$	−13235.2	−1.26645
$479$	14048.4	1.34006	0.670030	−	0.742334i	$-0.266281\pi$
$479$	14048.4	1.34006	0.670030	+	0.742334i	$0.266281\pi$
$480$	0	0
$481$	−13929.8	−1.32047
$482$	−7000.41	−0.661535
$483$	−3183.60	−0.299915
$484$	4796.40	0.450451
$485$	0	0
$486$	486.000	0.0453609
$487$	−5605.90	−0.521617	−0.260809	−	0.965391i	$-0.583989\pi$
$487$	−5605.90	−0.521617	−0.260809	+	0.965391i	$0.583989\pi$
$488$	5863.21	0.543883
$489$	−7183.50	−0.664313
$490$	0	0
$491$	−15035.1	−1.38193	−0.690963	−	0.722890i	$-0.742813\pi$
$491$	−15035.1	−1.38193	−0.690963	+	0.722890i	$0.742813\pi$
$492$	−1071.60	−0.0981941
$493$	−1249.50	−0.114148
$494$	−1812.00	−0.165032
$495$	0	0
$496$	5256.01	0.475810
$497$	−730.101	−0.0658944
$498$	945.010	0.0850339
$499$	−11160.9	−1.00126	−0.500632	−	0.865660i	$-0.666899\pi$
$499$	−11160.9	−1.00126	−0.500632	+	0.865660i	$0.666899\pi$
$500$	0	0
$501$	−10255.8	−0.914562
$502$	10550.2	0.938007
$503$	−12143.2	−1.07642	−0.538209	−	0.842811i	$-0.680899\pi$
$503$	−12143.2	−1.07642	−0.538209	+	0.842811i	$0.680899\pi$
$504$	504.000	0.0445435
$505$	0	0
$506$	−15251.0	−1.33990
$507$	434.701	0.0380784
$508$	6431.60	0.561724
$509$	17416.4	1.51664	0.758320	−	0.651883i	$-0.226021\pi$
$509$	17416.4	1.51664	0.758320	+	0.651883i	$0.226021\pi$
$510$	0	0
$511$	1209.60	0.104715
$512$	−512.000	−0.0441942
$513$	−540.000	−0.0464748
$514$	4292.21	0.368329
$515$	0	0
$516$	4634.41	0.395384
$517$	−7706.00	−0.655531
$518$	−4305.01	−0.365157
$519$	6903.60	0.583881
$520$	0	0
$521$	−12235.5	−1.02888	−0.514441	−	0.857526i	$-0.672000\pi$
$521$	−12235.5	−1.02888	−0.514441	+	0.857526i	$0.672000\pi$
$522$	−270.000	−0.0226390
$523$	2844.39	0.237813	0.118907	−	0.992905i	$-0.462061\pi$
$523$	2844.39	0.237813	0.118907	+	0.992905i	$0.462061\pi$
$524$	10514.4	0.876573
$525$	0	0
$526$	2674.81	0.221725
$527$	−27364.1	−2.26186
$528$	2414.41	0.199003
$529$	10815.6	0.888931
$530$	0	0
$531$	6572.71	0.537159
$532$	−560.000	−0.0456374
$533$	4045.30	0.328746
$534$	−5773.21	−0.467849
$535$	0	0
$536$	−7194.41	−0.579760
$537$	5447.41	0.437753
$538$	−4084.80	−0.327339
$539$	−2464.71	−0.196962
$540$	0	0
$541$	671.923	0.0533978	0.0266989	−	0.999644i	$-0.491500\pi$
$541$	671.923	0.0533978	0.0266989	+	0.999644i	$0.491500\pi$
$542$	−9893.62	−0.784072
$543$	−11054.4	−0.873647
$544$	2665.60	0.210086
$545$	0	0
$546$	−1902.60	−0.149128
$547$	24786.6	1.93748	0.968738	−	0.248087i	$-0.0798020\pi$
$547$	24786.6	1.93748	0.968738	+	0.248087i	$0.0798020\pi$
$548$	6077.60	0.473763
$549$	−6596.11	−0.512778
$550$	0	0
$551$	300.000	0.0231950
$552$	−3638.41	−0.280545
$553$	5593.71	0.430143
$554$	−7875.23	−0.603947
$555$	0	0
$556$	5895.20	0.449662
$557$	−4759.90	−0.362089	−0.181044	−	0.983475i	$-0.557948\pi$
$557$	−4759.90	−0.362089	−0.181044	+	0.983475i	$0.557948\pi$
$558$	−5913.01	−0.448598
$559$	−17494.9	−1.32371
$560$	0	0
$561$	−12570.0	−0.946001
$562$	−7268.62	−0.545566
$563$	−2145.30	−0.160593	−0.0802964	−	0.996771i	$-0.525587\pi$
$563$	−2145.30	−0.160593	−0.0802964	+	0.996771i	$0.525587\pi$
$564$	−1838.41	−0.137253
$565$	0	0
$566$	72.4115	0.00537753
$567$	−567.000	−0.0419961
$568$	−834.401	−0.0616385
$569$	−10928.7	−0.805196	−0.402598	−	0.915377i	$-0.631893\pi$
$569$	−10928.7	−0.805196	−0.402598	+	0.915377i	$0.631893\pi$
$570$	0	0
$571$	−4437.59	−0.325232	−0.162616	−	0.986689i	$-0.551993\pi$
$571$	−4437.59	−0.325232	−0.162616	+	0.986689i	$0.551993\pi$
$572$	−9114.40	−0.666245
$573$	−5536.50	−0.403648
$574$	1250.20	0.0909101
$575$	0	0
$576$	576.000	0.0416667
$577$	13851.4	0.999379	0.499690	−	0.866204i	$-0.333447\pi$
$577$	13851.4	0.999379	0.499690	+	0.866204i	$0.333447\pi$
$578$	−4051.82	−0.291580
$579$	−5865.29	−0.420990
$580$	0	0
$581$	−1102.51	−0.0787261
$582$	−1512.01	−0.107689
$583$	19813.2	1.40751
$584$	1382.40	0.0979520
$585$	0	0
$586$	−4027.22	−0.283895
$587$	−15538.5	−1.09258	−0.546289	−	0.837597i	$-0.683960\pi$
$587$	−15538.5	−1.09258	−0.546289	+	0.837597i	$0.683960\pi$
$588$	−588.000	−0.0412393
$589$	6570.01	0.459614
$590$	0	0
$591$	−12979.8	−0.903414
$592$	−4920.01	−0.341573
$593$	−24054.6	−1.66578	−0.832888	−	0.553442i	$-0.813314\pi$
$593$	−24054.6	−1.66578	−0.832888	+	0.553442i	$0.813314\pi$
$594$	−2716.21	−0.187622
$595$	0	0
$596$	1075.20	0.0738960
$597$	−3314.39	−0.227218
$598$	13735.0	0.939241
$599$	2390.03	0.163028	0.0815141	−	0.996672i	$-0.474024\pi$
$599$	2390.03	0.163028	0.0815141	+	0.996672i	$0.474024\pi$
$600$	0	0
$601$	370.379	0.0251383	0.0125691	−	0.999921i	$-0.495999\pi$
$601$	370.379	0.0251383	0.0125691	+	0.999921i	$0.495999\pi$
$602$	−5406.81	−0.366055
$603$	8093.71	0.546603
$604$	−3480.41	−0.234463
$605$	0	0
$606$	−6296.41	−0.422070
$607$	−13637.2	−0.911891	−0.455946	−	0.890008i	$-0.650699\pi$
$607$	−13637.2	−0.911891	−0.455946	+	0.890008i	$0.650699\pi$
$608$	−640.000	−0.0426898
$609$	315.000	0.0209597
$610$	0	0
$611$	6940.00	0.459513
$612$	−2998.80	−0.198071
$613$	27116.5	1.78667	0.893333	−	0.449396i	$-0.148361\pi$
$613$	27116.5	1.78667	0.893333	+	0.449396i	$0.148361\pi$
$614$	1722.41	0.113209
$615$	0	0
$616$	−2816.81	−0.184241
$617$	8155.08	0.532109	0.266055	−	0.963958i	$-0.414280\pi$
$617$	8155.08	0.532109	0.266055	+	0.963958i	$0.414280\pi$
$618$	382.206	0.0248779
$619$	5352.61	0.347560	0.173780	−	0.984785i	$-0.444402\pi$
$619$	5352.61	0.347560	0.173780	+	0.984785i	$0.444402\pi$
$620$	0	0
$621$	4093.21	0.264500
$622$	−4532.81	−0.292201
$623$	6735.41	0.433144
$624$	−2174.41	−0.139497
$625$	0	0
$626$	2763.61	0.176447
$627$	3018.01	0.192229
$628$	10031.2	0.637402
$629$	25614.8	1.62374
$630$	0	0
$631$	2654.52	0.167472	0.0837359	−	0.996488i	$-0.473315\pi$
$631$	2654.52	0.167472	0.0837359	+	0.996488i	$0.473315\pi$
$632$	6392.81	0.402362
$633$	−10103.1	−0.634380
$634$	−16487.6	−1.03282
$635$	0	0
$636$	4726.80	0.294701
$637$	2219.71	0.138066
$638$	1509.00	0.0936395
$639$	938.701	0.0581134
$640$	0	0
$641$	−10059.9	−0.619880	−0.309940	−	0.950756i	$-0.600309\pi$
$641$	−10059.9	−0.619880	−0.309940	+	0.950756i	$0.600309\pi$
$642$	2456.40	0.151007
$643$	24993.0	1.53286	0.766430	−	0.642328i	$-0.222031\pi$
$643$	24993.0	1.53286	0.766430	+	0.642328i	$0.222031\pi$
$644$	4244.81	0.259734
$645$	0	0
$646$	3332.00	0.202935
$647$	27503.6	1.67122	0.835610	−	0.549323i	$-0.185114\pi$
$647$	27503.6	1.67122	0.835610	+	0.549323i	$0.185114\pi$
$648$	−648.000	−0.0392837
$649$	−36734.2	−2.22179
$650$	0	0
$651$	6898.51	0.415321
$652$	9578.00	0.575312
$653$	−19166.0	−1.14858	−0.574292	−	0.818651i	$-0.694722\pi$
$653$	−19166.0	−1.14858	−0.574292	+	0.818651i	$0.694722\pi$
$654$	11057.4	0.661130
$655$	0	0
$656$	1428.80	0.0850386
$657$	−1555.20	−0.0923500
$658$	2144.81	0.127072
$659$	20770.6	1.22778	0.613891	−	0.789391i	$-0.289603\pi$
$659$	20770.6	1.22778	0.613891	+	0.789391i	$0.289603\pi$
$660$	0	0
$661$	−11326.6	−0.666498	−0.333249	−	0.942839i	$-0.608145\pi$
$661$	−11326.6	−0.666498	−0.333249	+	0.942839i	$0.608145\pi$
$662$	22933.2	1.34641
$663$	11320.5	0.663126
$664$	−1260.01	−0.0736415
$665$	0	0
$666$	5535.01	0.322038
$667$	−2274.00	−0.132009
$668$	13674.4	0.792034
$669$	−7516.49	−0.434386
$670$	0	0
$671$	36865.0	2.12095
$672$	−672.000	−0.0385758
$673$	−19478.9	−1.11569	−0.557844	−	0.829946i	$-0.688371\pi$
$673$	−19478.9	−1.11569	−0.557844	+	0.829946i	$0.688371\pi$
$674$	6610.63	0.377792
$675$	0	0
$676$	−579.601	−0.0329769
$677$	−29813.4	−1.69250	−0.846250	−	0.532786i	$-0.821145\pi$
$677$	−29813.4	−1.69250	−0.846250	+	0.532786i	$0.821145\pi$
$678$	−10941.0	−0.619744
$679$	1764.01	0.0997006
$680$	0	0
$681$	4949.12	0.278489
$682$	33047.2	1.85549
$683$	7486.29	0.419407	0.209704	−	0.977765i	$-0.432750\pi$
$683$	7486.29	0.419407	0.209704	+	0.977765i	$0.432750\pi$
$684$	720.000	0.0402484
$685$	0	0
$686$	686.000	0.0381802
$687$	10483.2	0.582182
$688$	−6179.21	−0.342413
$689$	−17843.7	−0.986635
$690$	0	0
$691$	32741.4	1.80252	0.901261	−	0.433277i	$-0.142643\pi$
$691$	32741.4	1.80252	0.901261	+	0.433277i	$0.142643\pi$
$692$	−9204.80	−0.505656
$693$	3168.91	0.173704
$694$	17123.4	0.936595
$695$	0	0
$696$	360.000	0.0196060
$697$	−7438.71	−0.404248
$698$	1107.79	0.0600722
$699$	−1736.69	−0.0939740
$700$	0	0
$701$	19570.5	1.05445	0.527225	−	0.849726i	$-0.323233\pi$
$701$	19570.5	1.05445	0.527225	+	0.849726i	$0.323233\pi$
$702$	2446.21	0.131519
$703$	−6150.01	−0.329946
$704$	−3219.21	−0.172342
$705$	0	0
$706$	−1557.62	−0.0830337
$707$	7345.81	0.390761
$708$	−8763.61	−0.465193
$709$	3115.82	0.165045	0.0825225	−	0.996589i	$-0.473702\pi$
$709$	3115.82	0.165045	0.0825225	+	0.996589i	$0.473702\pi$
$710$	0	0
$711$	−7191.92	−0.379350
$712$	7697.61	0.405169
$713$	−49800.8	−2.61578
$714$	3498.60	0.183378
$715$	0	0
$716$	−7263.22	−0.379105
$717$	−19852.8	−1.03406
$718$	−12638.0	−0.656889
$719$	−18545.4	−0.961930	−0.480965	−	0.876740i	$-0.659714\pi$
$719$	−18545.4	−0.961930	−0.480965	+	0.876740i	$0.659714\pi$
$720$	0	0
$721$	−445.907	−0.0230325
$722$	12918.0	0.665870
$723$	−10500.6	−0.540141
$724$	14739.2	0.756600
$725$	0	0
$726$	7194.60	0.367792
$727$	−21559.3	−1.09985	−0.549925	−	0.835214i	$-0.685344\pi$
$727$	−21559.3	−1.09985	−0.549925	+	0.835214i	$0.685344\pi$
$728$	2536.81	0.129149
$729$	729.000	0.0370370
$730$	0	0
$731$	32170.5	1.62773
$732$	8794.82	0.444079
$733$	13924.7	0.701666	0.350833	−	0.936438i	$-0.385898\pi$
$733$	13924.7	0.701666	0.350833	+	0.936438i	$0.385898\pi$
$734$	−14455.8	−0.726940
$735$	0	0
$736$	4851.21	0.242959
$737$	−45234.9	−2.26086
$738$	−1607.40	−0.0801752
$739$	−30999.8	−1.54309	−0.771547	−	0.636172i	$-0.780517\pi$
$739$	−30999.8	−1.54309	−0.771547	+	0.636172i	$0.780517\pi$
$740$	0	0
$741$	−2718.01	−0.134748
$742$	−5514.60	−0.272840
$743$	29325.9	1.44800	0.724000	−	0.689800i	$-0.242301\pi$
$743$	29325.9	1.44800	0.724000	+	0.689800i	$0.242301\pi$
$744$	7884.01	0.388497
$745$	0	0
$746$	−15960.2	−0.783304
$747$	1417.51	0.0694299
$748$	16760.0	0.819261
$749$	−2865.80	−0.139805
$750$	0	0
$751$	153.578	0.00746223	0.00373111	−	0.999993i	$-0.498812\pi$
$751$	153.578	0.00746223	0.00373111	+	0.999993i	$0.498812\pi$
$752$	2451.21	0.118865
$753$	15825.3	0.765880
$754$	−1359.00	−0.0656392
$755$	0	0
$756$	756.000	0.0363696
$757$	34232.5	1.64360	0.821798	−	0.569779i	$-0.192971\pi$
$757$	34232.5	1.64360	0.821798	+	0.569779i	$0.192971\pi$
$758$	20683.2	0.991093
$759$	−22876.5	−1.09403
$760$	0	0
$761$	10335.8	0.492341	0.246171	−	0.969227i	$-0.420828\pi$
$761$	10335.8	0.492341	0.246171	+	0.969227i	$0.420828\pi$
$762$	9647.40	0.458646
$763$	−12900.3	−0.612088
$764$	7382.00	0.349570
$765$	0	0
$766$	12679.2	0.598066
$767$	33082.7	1.55743
$768$	−768.000	−0.0360844
$769$	2106.00	0.0987573	0.0493787	−	0.998780i	$-0.484276\pi$
$769$	2106.00	0.0987573	0.0493787	+	0.998780i	$0.484276\pi$
$770$	0	0
$771$	6438.31	0.300739
$772$	7820.39	0.364588
$773$	−6386.20	−0.297148	−0.148574	−	0.988901i	$-0.547468\pi$
$773$	−6386.20	−0.297148	−0.148574	+	0.988901i	$0.547468\pi$
$774$	6951.61	0.322830
$775$	0	0
$776$	2016.02	0.0932613
$777$	−6457.51	−0.298149
$778$	−2432.99	−0.112117
$779$	1786.00	0.0821440
$780$	0	0
$781$	−5246.31	−0.240368
$782$	−25256.6	−1.15496
$783$	−405.000	−0.0184847
$784$	784.000	0.0357143
$785$	0	0
$786$	15771.6	0.715719
$787$	28805.2	1.30470	0.652348	−	0.757920i	$-0.273784\pi$
$787$	28805.2	1.30470	0.652348	+	0.757920i	$0.273784\pi$
$788$	17306.4	0.782380
$789$	4012.22	0.181038
$790$	0	0
$791$	12764.5	0.573772
$792$	3621.61	0.162485
$793$	−33200.5	−1.48674
$794$	25380.6	1.13441
$795$	0	0
$796$	4419.19	0.196776
$797$	20319.7	0.903086	0.451543	−	0.892249i	$-0.350874\pi$
$797$	20319.7	0.903086	0.451543	+	0.892249i	$0.350874\pi$
$798$	−840.000	−0.0372628
$799$	−12761.6	−0.565048
$800$	0	0
$801$	−8659.81	−0.381997
$802$	−18094.3	−0.796671
$803$	8691.84	0.381978
$804$	−10791.6	−0.473372
$805$	0	0
$806$	−29762.2	−1.30066
$807$	−6127.20	−0.267271
$808$	8395.22	0.365523
$809$	−7633.11	−0.331725	−0.165863	−	0.986149i	$-0.553041\pi$
$809$	−7633.11	−0.331725	−0.165863	+	0.986149i	$0.553041\pi$
$810$	0	0
$811$	8238.36	0.356705	0.178353	−	0.983967i	$-0.442923\pi$
$811$	8238.36	0.356705	0.178353	+	0.983967i	$0.442923\pi$
$812$	−420.000	−0.0181516
$813$	−14840.4	−0.640192
$814$	−30934.6	−1.33201
$815$	0	0
$816$	3998.41	0.171535
$817$	−7724.01	−0.330757
$818$	−10004.4	−0.427623
$819$	−2853.91	−0.121763
$820$	0	0
$821$	−5571.25	−0.236831	−0.118415	−	0.992964i	$-0.537781\pi$
$821$	−5571.25	−0.236831	−0.118415	+	0.992964i	$0.537781\pi$
$822$	9116.40	0.386826
$823$	−13207.0	−0.559375	−0.279688	−	0.960091i	$-0.590231\pi$
$823$	−13207.0	−0.559375	−0.279688	+	0.960091i	$0.590231\pi$
$824$	−509.608	−0.0215449
$825$	0	0
$826$	10224.2	0.430685
$827$	−12274.6	−0.516116	−0.258058	−	0.966129i	$-0.583083\pi$
$827$	−12274.6	−0.516116	−0.258058	+	0.966129i	$0.583083\pi$
$828$	−5457.61	−0.229064
$829$	12992.3	0.544320	0.272160	−	0.962252i	$-0.412262\pi$
$829$	12992.3	0.544320	0.272160	+	0.962252i	$0.412262\pi$
$830$	0	0
$831$	−11812.8	−0.493120
$832$	2899.21	0.120808
$833$	−4081.71	−0.169775
$834$	8842.80	0.367148
$835$	0	0
$836$	−4024.01	−0.166475
$837$	−8869.51	−0.366279
$838$	14797.8	0.610003
$839$	−46595.5	−1.91735	−0.958674	−	0.284507i	$-0.908170\pi$
$839$	−46595.5	−1.91735	−0.958674	+	0.284507i	$0.908170\pi$
$840$	0	0
$841$	−24164.0	−0.990775
$842$	22471.8	0.919750
$843$	−10902.9	−0.445453
$844$	13470.8	0.549389
$845$	0	0
$846$	−2757.61	−0.112067
$847$	−8393.71	−0.340509
$848$	−6302.41	−0.255219
$849$	108.617	0.00439074
$850$	0	0
$851$	46617.1	1.87781
$852$	−1251.60	−0.0503276
$853$	−26660.1	−1.07014	−0.535068	−	0.844809i	$-0.679714\pi$
$853$	−26660.1	−1.07014	−0.535068	+	0.844809i	$0.679714\pi$
$854$	−10260.6	−0.411137
$855$	0	0
$856$	−3275.20	−0.130776
$857$	9000.97	0.358771	0.179386	−	0.983779i	$-0.442589\pi$
$857$	9000.97	0.358771	0.179386	+	0.983779i	$0.442589\pi$
$858$	−13671.6	−0.543987
$859$	1461.84	0.0580644	0.0290322	−	0.999578i	$-0.490757\pi$
$859$	1461.84	0.0580644	0.0290322	+	0.999578i	$0.490757\pi$
$860$	0	0
$861$	1875.30	0.0742278
$862$	10765.8	0.425387
$863$	36079.0	1.42311	0.711554	−	0.702631i	$-0.247992\pi$
$863$	36079.0	1.42311	0.711554	+	0.702631i	$0.247992\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	−5064.39	−0.198724
$867$	−6077.72	−0.238074
$868$	−9198.01	−0.359679
$869$	40194.9	1.56907
$870$	0	0
$871$	40738.4	1.58481
$872$	−14743.2	−0.572556
$873$	−2268.02	−0.0879276
$874$	6064.01	0.234689
$875$	0	0
$876$	2073.59	0.0799775
$877$	−2218.96	−0.0854377	−0.0427189	−	0.999087i	$-0.513602\pi$
$877$	−2218.96	−0.0854377	−0.0427189	+	0.999087i	$0.513602\pi$
$878$	21060.2	0.809507
$879$	−6040.82	−0.231800
$880$	0	0
$881$	39834.3	1.52333	0.761665	−	0.647972i	$-0.224382\pi$
$881$	39834.3	1.52333	0.761665	+	0.647972i	$0.224382\pi$
$882$	−882.000	−0.0336718
$883$	45397.8	1.73019	0.865096	−	0.501607i	$-0.167258\pi$
$883$	45397.8	1.73019	0.865096	+	0.501607i	$0.167258\pi$
$884$	−15094.0	−0.574284
$885$	0	0
$886$	4333.62	0.164324
$887$	−38834.2	−1.47004	−0.735020	−	0.678046i	$-0.762827\pi$
$887$	−38834.2	−1.47004	−0.735020	+	0.678046i	$0.762827\pi$
$888$	−7380.01	−0.278893
$889$	−11255.3	−0.424624
$890$	0	0
$891$	−4074.31	−0.153192
$892$	10022.0	0.376189
$893$	3064.01	0.114819
$894$	1612.80	0.0603358
$895$	0	0
$896$	896.000	0.0334077
$897$	20602.5	0.766887
$898$	−15619.4	−0.580430
$899$	4927.51	0.182805
$900$	0	0
$901$	32811.9	1.21323
$902$	8983.61	0.331620
$903$	−8110.21	−0.298882
$904$	14588.0	0.536714
$905$	0	0
$906$	−5220.62	−0.191439
$907$	−1549.94	−0.0567418	−0.0283709	−	0.999597i	$-0.509032\pi$
$907$	−1549.94	−0.0567418	−0.0283709	+	0.999597i	$0.509032\pi$
$908$	−6598.83	−0.241178
$909$	−9444.62	−0.344618
$910$	0	0
$911$	−22788.7	−0.828784	−0.414392	−	0.910098i	$-0.636006\pi$
$911$	−22788.7	−0.828784	−0.414392	+	0.910098i	$0.636006\pi$
$912$	−960.000	−0.0348561
$913$	−7922.35	−0.287176
$914$	5163.60	0.186867
$915$	0	0
$916$	−13977.6	−0.504185
$917$	−18400.2	−0.662627
$918$	−4498.21	−0.161724
$919$	46832.8	1.68103	0.840517	−	0.541785i	$-0.182251\pi$
$919$	46832.8	1.68103	0.840517	+	0.541785i	$0.182251\pi$
$920$	0	0
$921$	2583.61	0.0924351
$922$	4018.41	0.143535
$923$	4724.81	0.168493
$924$	−4225.21	−0.150432
$925$	0	0
$926$	−15389.6	−0.546148
$927$	573.309	0.0203128
$928$	−480.000	−0.0169793
$929$	42103.9	1.48696	0.743479	−	0.668759i	$-0.233174\pi$
$929$	42103.9	1.48696	0.743479	+	0.668759i	$0.233174\pi$
$930$	0	0
$931$	980.000	0.0344986
$932$	2315.59	0.0813839
$933$	−6799.21	−0.238581
$934$	−30751.1	−1.07731
$935$	0	0
$936$	−3261.61	−0.113898
$937$	5009.07	0.174642	0.0873208	−	0.996180i	$-0.472169\pi$
$937$	5009.07	0.174642	0.0873208	+	0.996180i	$0.472169\pi$
$938$	12590.2	0.438257
$939$	4145.41	0.144068
$940$	0	0
$941$	−4072.21	−0.141073	−0.0705367	−	0.997509i	$-0.522471\pi$
$941$	−4072.21	−0.141073	−0.0705367	+	0.997509i	$0.522471\pi$
$942$	15046.8	0.520437
$943$	−13537.9	−0.467503
$944$	11684.8	0.402869
$945$	0	0
$946$	−38851.8	−1.33529
$947$	42738.1	1.46653	0.733264	−	0.679945i	$-0.237996\pi$
$947$	42738.1	1.46653	0.733264	+	0.679945i	$0.237996\pi$
$948$	9589.22	0.328527
$949$	−7827.84	−0.267758
$950$	0	0
$951$	−24731.5	−0.843294
$952$	−4664.81	−0.158810
$953$	−15552.9	−0.528654	−0.264327	−	0.964433i	$-0.585150\pi$
$953$	−15552.9	−0.528654	−0.264327	+	0.964433i	$0.585150\pi$
$954$	7090.21	0.240623
$955$	0	0
$956$	26470.4	0.895518
$957$	2263.50	0.0764563
$958$	−28096.8	−0.947565
$959$	−10635.8	−0.358131
$960$	0	0
$961$	78121.6	2.62232
$962$	27859.6	0.933711
$963$	3684.60	0.123296
$964$	14000.8	0.467776
$965$	0	0
$966$	6367.21	0.212072
$967$	23514.0	0.781964	0.390982	−	0.920398i	$-0.372135\pi$
$967$	23514.0	0.781964	0.390982	+	0.920398i	$0.372135\pi$
$968$	−9592.81	−0.318517
$969$	4998.01	0.165696
$970$	0	0
$971$	14763.6	0.487936	0.243968	−	0.969783i	$-0.421551\pi$
$971$	14763.6	0.487936	0.243968	+	0.969783i	$0.421551\pi$
$972$	−972.000	−0.0320750
$973$	−10316.6	−0.339913
$974$	11211.8	0.368839
$975$	0	0
$976$	−11726.4	−0.384584
$977$	38229.9	1.25188	0.625939	−	0.779872i	$-0.284716\pi$
$977$	38229.9	1.25188	0.625939	+	0.779872i	$0.284716\pi$
$978$	14367.0	0.469740
$979$	48398.8	1.58001
$980$	0	0
$981$	16586.1	0.539811
$982$	30070.2	0.977169
$983$	−34284.3	−1.11241	−0.556205	−	0.831045i	$-0.687743\pi$
$983$	−34284.3	−1.11241	−0.556205	+	0.831045i	$0.687743\pi$
$984$	2143.20	0.0694337
$985$	0	0
$986$	2499.00	0.0807145
$987$	3217.21	0.103754
$988$	3624.01	0.116695
$989$	58548.1	1.88243
$990$	0	0
$991$	−38985.3	−1.24966	−0.624828	−	0.780762i	$-0.714831\pi$
$991$	−38985.3	−1.24966	−0.624828	+	0.780762i	$0.714831\pi$
$992$	−10512.0	−0.336448
$993$	34399.9	1.09934
$994$	1460.20	0.0465943
$995$	0	0
$996$	−1890.02	−0.0601281
$997$	−16314.3	−0.518232	−0.259116	−	0.965846i	$-0.583431\pi$
$997$	−16314.3	−0.518232	−0.259116	+	0.965846i	$0.583431\pi$
$998$	22321.8	0.708001
$999$	8302.51	0.262943

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.4.a.y.1.1	✓	2
5.2	odd	4		1050.4.g.u.799.1		4
5.3	odd	4		1050.4.g.u.799.3		4
5.4	even	2		1050.4.a.bi.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.4.a.y.1.1	✓	2	1.1	even	1	trivial
1050.4.a.bi.1.1	yes	2	5.4	even	2
1050.4.g.u.799.1		4	5.2	odd	4
1050.4.g.u.799.3		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -50.3001 q^{11} -12.0000 q^{12} +45.3001 q^{13} +14.0000 q^{14} +16.0000 q^{16} -83.3001 q^{17} -18.0000 q^{18} +20.0000 q^{19} +21.0000 q^{21} +100.600 q^{22} -151.600 q^{23} +24.0000 q^{24} -90.6002 q^{26} -27.0000 q^{27} -28.0000 q^{28} +15.0000 q^{29} +328.501 q^{31} -32.0000 q^{32} +150.900 q^{33} +166.600 q^{34} +36.0000 q^{36} -307.501 q^{37} -40.0000 q^{38} -135.900 q^{39} +89.3001 q^{41} -42.0000 q^{42} -386.200 q^{43} -201.200 q^{44} +303.200 q^{46} +153.200 q^{47} -48.0000 q^{48} +49.0000 q^{49} +249.900 q^{51} +181.200 q^{52} -393.900 q^{53} +54.0000 q^{54} +56.0000 q^{56} -60.0000 q^{57} -30.0000 q^{58} +730.301 q^{59} -732.901 q^{61} -657.001 q^{62} -63.0000 q^{63} +64.0000 q^{64} -301.801 q^{66} +899.301 q^{67} -333.200 q^{68} +454.801 q^{69} +104.300 q^{71} -72.0000 q^{72} -172.800 q^{73} +615.001 q^{74} +80.0000 q^{76} +352.101 q^{77} +271.801 q^{78} -799.102 q^{79} +81.0000 q^{81} -178.600 q^{82} +157.502 q^{83} +84.0000 q^{84} +772.401 q^{86} -45.0000 q^{87} +402.401 q^{88} -962.201 q^{89} -317.101 q^{91} -606.401 q^{92} -985.502 q^{93} -306.401 q^{94} +96.0000 q^{96} -252.002 q^{97} -98.0000 q^{98} -452.701 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 + 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9} - 7 q^{11} - 24 q^{12} - 3 q^{13} + 28 q^{14} + 32 q^{16} - 73 q^{17} - 36 q^{18} + 40 q^{19} + 42 q^{21} + 14 q^{22} - 116 q^{23} + 48 q^{24} + 6 q^{26} - 54 q^{27} - 56 q^{28} + 30 q^{29} + 189 q^{31} - 64 q^{32} + 21 q^{33} + 146 q^{34} + 72 q^{36} - 147 q^{37} - 80 q^{38} + 9 q^{39} + 85 q^{41} - 84 q^{42} - 398 q^{43} - 28 q^{44} + 232 q^{46} - 68 q^{47} - 96 q^{48} + 98 q^{49} + 219 q^{51} - 12 q^{52} - 507 q^{53} + 108 q^{54} + 112 q^{56} - 120 q^{57} - 60 q^{58} + 431 q^{59} - 249 q^{61} - 378 q^{62} - 126 q^{63} + 128 q^{64} - 42 q^{66} + 769 q^{67} - 292 q^{68} + 348 q^{69} + 115 q^{71} - 144 q^{72} - 720 q^{73} + 294 q^{74} + 160 q^{76} + 49 q^{77} - 18 q^{78} - 7 q^{79} + 162 q^{81} - 170 q^{82} - 1089 q^{83} + 168 q^{84} + 796 q^{86} - 90 q^{87} + 56 q^{88} - 614 q^{89} + 21 q^{91} - 464 q^{92} - 567 q^{93} + 136 q^{94} + 192 q^{96} + 1368 q^{97} - 196 q^{98} - 63 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 + 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9 - 7 * q^11 - 24 * q^12 - 3 * q^13 + 28 * q^14 + 32 * q^16 - 73 * q^17 - 36 * q^18 + 40 * q^19 + 42 * q^21 + 14 * q^22 - 116 * q^23 + 48 * q^24 + 6 * q^26 - 54 * q^27 - 56 * q^28 + 30 * q^29 + 189 * q^31 - 64 * q^32 + 21 * q^33 + 146 * q^34 + 72 * q^36 - 147 * q^37 - 80 * q^38 + 9 * q^39 + 85 * q^41 - 84 * q^42 - 398 * q^43 - 28 * q^44 + 232 * q^46 - 68 * q^47 - 96 * q^48 + 98 * q^49 + 219 * q^51 - 12 * q^52 - 507 * q^53 + 108 * q^54 + 112 * q^56 - 120 * q^57 - 60 * q^58 + 431 * q^59 - 249 * q^61 - 378 * q^62 - 126 * q^63 + 128 * q^64 - 42 * q^66 + 769 * q^67 - 292 * q^68 + 348 * q^69 + 115 * q^71 - 144 * q^72 - 720 * q^73 + 294 * q^74 + 160 * q^76 + 49 * q^77 - 18 * q^78 - 7 * q^79 + 162 * q^81 - 170 * q^82 - 1089 * q^83 + 168 * q^84 + 796 * q^86 - 90 * q^87 + 56 * q^88 - 614 * q^89 + 21 * q^91 - 464 * q^92 - 567 * q^93 + 136 * q^94 + 192 * q^96 + 1368 * q^97 - 196 * q^98 - 63 * q^99

Introduction

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