LMFDB - Embedded newform 1225.4.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.31662$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.31662 q^{2} +5.00000 q^{3} +10.6332 q^{4} -21.5831 q^{6} -11.3668 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-4.31662 q^{2} +5.00000 q^{3} +10.6332 q^{4} -21.5831 q^{6} -11.3668 q^{8} -2.00000 q^{9} +19.7335 q^{11} +53.1662 q^{12} +71.3325 q^{13} -36.0000 q^{16} -31.3325 q^{17} +8.63325 q^{18} -136.332 q^{19} -85.1821 q^{22} +100.865 q^{23} -56.8338 q^{24} -307.916 q^{26} -145.000 q^{27} -288.198 q^{29} +208.997 q^{31} +246.332 q^{32} +98.6675 q^{33} +135.251 q^{34} -21.2665 q^{36} -309.931 q^{37} +588.496 q^{38} +356.662 q^{39} +181.662 q^{41} +18.2005 q^{43} +209.831 q^{44} -435.398 q^{46} -147.665 q^{47} -180.000 q^{48} -156.662 q^{51} +758.496 q^{52} +127.995 q^{53} +625.911 q^{54} -681.662 q^{57} +1244.04 q^{58} -322.665 q^{59} -341.003 q^{61} -902.164 q^{62} -775.325 q^{64} -425.911 q^{66} +84.3960 q^{67} -333.166 q^{68} +504.327 q^{69} -315.736 q^{71} +22.7335 q^{72} -1093.32 q^{73} +1337.86 q^{74} -1449.66 q^{76} -1539.58 q^{78} +1233.19 q^{79} -671.000 q^{81} -784.169 q^{82} -643.325 q^{83} -78.5647 q^{86} -1440.99 q^{87} -224.306 q^{88} -1140.32 q^{89} +1072.53 q^{92} +1044.99 q^{93} +637.414 q^{94} +1231.66 q^{96} +1411.99 q^{97} -39.4670 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 10 q^{6} - 36 q^{8} - 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 10 * q^3 + 8 * q^4 - 10 * q^6 - 36 * q^8 - 4 * q^9	$ 2 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 10 q^{6} - 36 q^{8} - 4 q^{9} + 66 q^{11} + 40 q^{12} + 10 q^{13} - 72 q^{16} + 70 q^{17} + 4 q^{18} - 140 q^{19} + 22 q^{22} + 16 q^{23} - 180 q^{24} - 450 q^{26} - 290 q^{27} - 258 q^{29} + 20 q^{31} + 360 q^{32} + 330 q^{33} + 370 q^{34} - 16 q^{36} - 328 q^{37} + 580 q^{38} + 50 q^{39} - 300 q^{41} + 116 q^{43} + 88 q^{44} - 632 q^{46} - 30 q^{47} - 360 q^{48} + 350 q^{51} + 920 q^{52} - 540 q^{53} + 290 q^{54} - 700 q^{57} + 1314 q^{58} - 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} + 110 q^{66} - 468 q^{67} - 600 q^{68} + 80 q^{69} - 1056 q^{71} + 72 q^{72} - 860 q^{73} + 1296 q^{74} - 1440 q^{76} - 2250 q^{78} + 158 q^{79} - 1342 q^{81} - 1900 q^{82} + 40 q^{83} + 148 q^{86} - 1290 q^{87} - 1364 q^{88} + 240 q^{89} + 1296 q^{92} + 100 q^{93} + 910 q^{94} + 1800 q^{96} + 1630 q^{97} - 132 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 10 * q^3 + 8 * q^4 - 10 * q^6 - 36 * q^8 - 4 * q^9 + 66 * q^11 + 40 * q^12 + 10 * q^13 - 72 * q^16 + 70 * q^17 + 4 * q^18 - 140 * q^19 + 22 * q^22 + 16 * q^23 - 180 * q^24 - 450 * q^26 - 290 * q^27 - 258 * q^29 + 20 * q^31 + 360 * q^32 + 330 * q^33 + 370 * q^34 - 16 * q^36 - 328 * q^37 + 580 * q^38 + 50 * q^39 - 300 * q^41 + 116 * q^43 + 88 * q^44 - 632 * q^46 - 30 * q^47 - 360 * q^48 + 350 * q^51 + 920 * q^52 - 540 * q^53 + 290 * q^54 - 700 * q^57 + 1314 * q^58 - 380 * q^59 - 1080 * q^61 - 1340 * q^62 - 224 * q^64 + 110 * q^66 - 468 * q^67 - 600 * q^68 + 80 * q^69 - 1056 * q^71 + 72 * q^72 - 860 * q^73 + 1296 * q^74 - 1440 * q^76 - 2250 * q^78 + 158 * q^79 - 1342 * q^81 - 1900 * q^82 + 40 * q^83 + 148 * q^86 - 1290 * q^87 - 1364 * q^88 + 240 * q^89 + 1296 * q^92 + 100 * q^93 + 910 * q^94 + 1800 * q^96 + 1630 * q^97 - 132 * 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 $

$2$	−4.31662	−1.52616	−0.763079	−	0.646306i	$-0.776313\pi$
$2$	−4.31662	−1.52616	−0.763079	+	0.646306i	$0.776313\pi$
$3$	5.00000	0.962250	0.481125	−	0.876652i	$-0.340228\pi$
$3$	5.00000	0.962250	0.481125	+	0.876652i	$0.340228\pi$
$4$	10.6332	1.32916
$5$	0	0
$5$	0	0
$6$	−21.5831	−1.46855
$7$	0	0
$7$	0	0
$8$	−11.3668	−0.502344
$9$	−2.00000	−0.0740741
$10$	0	0
$11$	19.7335	0.540898	0.270449	−	0.962734i	$-0.412828\pi$
$11$	19.7335	0.540898	0.270449	+	0.962734i	$0.412828\pi$
$12$	53.1662	1.27898
$13$	71.3325	1.52185	0.760926	−	0.648839i	$-0.224745\pi$
$13$	71.3325	1.52185	0.760926	+	0.648839i	$0.224745\pi$
$14$	0	0
$15$	0	0
$16$	−36.0000	−0.562500
$17$	−31.3325	−0.447014	−0.223507	−	0.974702i	$-0.571751\pi$
$17$	−31.3325	−0.447014	−0.223507	+	0.974702i	$0.571751\pi$
$18$	8.63325	0.113049
$19$	−136.332	−1.64615	−0.823074	−	0.567934i	$-0.807743\pi$
$19$	−136.332	−1.64615	−0.823074	+	0.567934i	$0.807743\pi$
$20$	0	0
$21$	0	0
$22$	−85.1821	−0.825495
$23$	100.865	0.914431	0.457215	−	0.889356i	$-0.348847\pi$
$23$	100.865	0.914431	0.457215	+	0.889356i	$0.348847\pi$
$24$	−56.8338	−0.483381
$25$	0	0
$26$	−307.916	−2.32259
$27$	−145.000	−1.03353
$28$	0	0
$29$	−288.198	−1.84541	−0.922707	−	0.385501i	$-0.874029\pi$
$29$	−288.198	−1.84541	−0.922707	+	0.385501i	$0.874029\pi$
$30$	0	0
$31$	208.997	1.21087	0.605436	−	0.795894i	$-0.292999\pi$
$31$	208.997	1.21087	0.605436	+	0.795894i	$0.292999\pi$
$32$	246.332	1.36081
$33$	98.6675	0.520479
$34$	135.251	0.682214
$35$	0	0
$36$	−21.2665	−0.0984560
$37$	−309.931	−1.37709	−0.688546	−	0.725192i	$-0.741751\pi$
$37$	−309.931	−1.37709	−0.688546	+	0.725192i	$0.741751\pi$
$38$	588.496	2.51228
$39$	356.662	1.46440
$40$	0	0
$41$	181.662	0.691973	0.345987	−	0.938239i	$-0.387544\pi$
$41$	181.662	0.691973	0.345987	+	0.938239i	$0.387544\pi$
$42$	0	0
$43$	18.2005	0.0645477	0.0322738	−	0.999479i	$-0.489725\pi$
$43$	18.2005	0.0645477	0.0322738	+	0.999479i	$0.489725\pi$
$44$	209.831	0.718937
$45$	0	0
$46$	−435.398	−1.39557
$47$	−147.665	−0.458280	−0.229140	−	0.973393i	$-0.573591\pi$
$47$	−147.665	−0.458280	−0.229140	+	0.973393i	$0.573591\pi$
$48$	−180.000	−0.541266
$49$	0	0
$50$	0	0
$51$	−156.662	−0.430140
$52$	758.496	2.02278
$53$	127.995	0.331726	0.165863	−	0.986149i	$-0.446959\pi$
$53$	127.995	0.331726	0.165863	+	0.986149i	$0.446959\pi$
$54$	625.911	1.57733
$55$	0	0
$56$	0	0
$57$	−681.662	−1.58401
$58$	1244.04	2.81639
$59$	−322.665	−0.711990	−0.355995	−	0.934488i	$-0.615858\pi$
$59$	−322.665	−0.711990	−0.355995	+	0.934488i	$0.615858\pi$
$60$	0	0
$61$	−341.003	−0.715752	−0.357876	−	0.933769i	$-0.616499\pi$
$61$	−341.003	−0.715752	−0.357876	+	0.933769i	$0.616499\pi$
$62$	−902.164	−1.84798
$63$	0	0
$64$	−775.325	−1.51431
$65$	0	0
$66$	−425.911	−0.794333
$67$	84.3960	0.153890	0.0769449	−	0.997035i	$-0.475483\pi$
$67$	84.3960	0.153890	0.0769449	+	0.997035i	$0.475483\pi$
$68$	−333.166	−0.594152
$69$	504.327	0.879911
$70$	0	0
$71$	−315.736	−0.527760	−0.263880	−	0.964555i	$-0.585002\pi$
$71$	−315.736	−0.527760	−0.263880	+	0.964555i	$0.585002\pi$
$72$	22.7335	0.0372107
$73$	−1093.32	−1.75293	−0.876466	−	0.481464i	$-0.840105\pi$
$73$	−1093.32	−1.75293	−0.876466	+	0.481464i	$0.840105\pi$
$74$	1337.86	2.10166
$75$	0	0
$76$	−1449.66	−2.18799
$77$	0	0
$78$	−1539.58	−2.23491
$79$	1233.19	1.75626	0.878128	−	0.478426i	$-0.158793\pi$
$79$	1233.19	1.75626	0.878128	+	0.478426i	$0.158793\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	−784.169	−1.05606
$83$	−643.325	−0.850772	−0.425386	−	0.905012i	$-0.639862\pi$
$83$	−643.325	−0.850772	−0.425386	+	0.905012i	$0.639862\pi$
$84$	0	0
$85$	0	0
$86$	−78.5647	−0.0985099
$87$	−1440.99	−1.77575
$88$	−224.306	−0.271717
$89$	−1140.32	−1.35813	−0.679064	−	0.734079i	$-0.737614\pi$
$89$	−1140.32	−1.35813	−0.679064	+	0.734079i	$0.737614\pi$
$90$	0	0
$91$	0	0
$92$	1072.53	1.21542
$93$	1044.99	1.16516
$94$	637.414	0.699407
$95$	0	0
$96$	1231.66	1.30944
$97$	1411.99	1.47800	0.739001	−	0.673705i	$-0.235298\pi$
$97$	1411.99	1.47800	0.739001	+	0.673705i	$0.235298\pi$
$98$	0	0
$99$	−39.4670	−0.0400665
$100$	0	0
$101$	545.673	0.537589	0.268794	−	0.963198i	$-0.413375\pi$
$101$	545.673	0.537589	0.268794	+	0.963198i	$0.413375\pi$
$102$	676.253	0.656461
$103$	780.990	0.747119	0.373559	−	0.927606i	$-0.378137\pi$
$103$	780.990	0.747119	0.373559	+	0.927606i	$0.378137\pi$
$104$	−810.819	−0.764493
$105$	0	0
$106$	−552.506	−0.506266
$107$	620.660	0.560761	0.280381	−	0.959889i	$-0.409539\pi$
$107$	620.660	0.560761	0.280381	+	0.959889i	$0.409539\pi$
$108$	−1541.82	−1.37372
$109$	4.18794	0.00368011	0.00184005	−	0.999998i	$-0.499414\pi$
$109$	4.18794	0.00368011	0.00184005	+	0.999998i	$0.499414\pi$
$110$	0	0
$111$	−1549.66	−1.32511
$112$	0	0
$113$	−1413.53	−1.17676	−0.588379	−	0.808585i	$-0.700234\pi$
$113$	−1413.53	−1.17676	−0.588379	+	0.808585i	$0.700234\pi$
$114$	2942.48	2.41744
$115$	0	0
$116$	−3064.48	−2.45284
$117$	−142.665	−0.112730
$118$	1392.82	1.08661
$119$	0	0
$120$	0	0
$121$	−941.589	−0.707430
$122$	1471.98	1.09235
$123$	908.312	0.665852
$124$	2222.32	1.60944
$125$	0	0
$126$	0	0
$127$	2046.26	1.42974	0.714868	−	0.699259i	$-0.246487\pi$
$127$	2046.26	1.42974	0.714868	+	0.699259i	$0.246487\pi$
$128$	1376.13	0.950262
$129$	91.0025	0.0621110
$130$	0	0
$131$	−1140.32	−0.760534	−0.380267	−	0.924877i	$-0.624168\pi$
$131$	−1140.32	−0.760534	−0.380267	+	0.924877i	$0.624168\pi$
$132$	1049.16	0.691798
$133$	0	0
$134$	−364.306	−0.234860
$135$	0	0
$136$	356.149	0.224555
$137$	490.343	0.305787	0.152893	−	0.988243i	$-0.451141\pi$
$137$	490.343	0.305787	0.152893	+	0.988243i	$0.451141\pi$
$138$	−2176.99	−1.34288
$139$	−2800.00	−1.70858	−0.854291	−	0.519795i	$-0.826008\pi$
$139$	−2800.00	−1.70858	−0.854291	+	0.519795i	$0.826008\pi$
$140$	0	0
$141$	−738.325	−0.440980
$142$	1362.91	0.805445
$143$	1407.64	0.823166
$144$	72.0000	0.0416667
$145$	0	0
$146$	4719.47	2.67525
$147$	0	0
$148$	−3295.58	−1.83037
$149$	1166.12	0.641154	0.320577	−	0.947222i	$-0.396123\pi$
$149$	1166.12	0.641154	0.320577	+	0.947222i	$0.396123\pi$
$150$	0	0
$151$	959.581	0.517150	0.258575	−	0.965991i	$-0.416747\pi$
$151$	959.581	0.517150	0.258575	+	0.965991i	$0.416747\pi$
$152$	1549.66	0.826933
$153$	62.6650	0.0331122
$154$	0	0
$155$	0	0
$156$	3792.48	1.94642
$157$	−1020.66	−0.518838	−0.259419	−	0.965765i	$-0.583531\pi$
$157$	−1020.66	−0.518838	−0.259419	+	0.965765i	$0.583531\pi$
$158$	−5323.20	−2.68032
$159$	639.975	0.319203
$160$	0	0
$161$	0	0
$162$	2896.46	1.40473
$163$	1566.02	0.752514	0.376257	−	0.926515i	$-0.377211\pi$
$163$	1566.02	0.752514	0.376257	+	0.926515i	$0.377211\pi$
$164$	1931.66	0.919741
$165$	0	0
$166$	2776.99	1.29841
$167$	−1130.30	−0.523746	−0.261873	−	0.965102i	$-0.584340\pi$
$167$	−1130.30	−0.523746	−0.261873	+	0.965102i	$0.584340\pi$
$168$	0	0
$169$	2891.32	1.31603
$170$	0	0
$171$	272.665	0.121937
$172$	193.530	0.0857940
$173$	2543.34	1.11772	0.558862	−	0.829260i	$-0.311238\pi$
$173$	2543.34	1.11772	0.558862	+	0.829260i	$0.311238\pi$
$174$	6220.21	2.71008
$175$	0	0
$176$	−710.406	−0.304255
$177$	−1613.32	−0.685113
$178$	4922.32	2.07272
$179$	−1210.65	−0.505521	−0.252760	−	0.967529i	$-0.581338\pi$
$179$	−1210.65	−0.505521	−0.252760	+	0.967529i	$0.581338\pi$
$180$	0	0
$181$	3031.32	1.24484	0.622421	−	0.782683i	$-0.286149\pi$
$181$	3031.32	1.24484	0.622421	+	0.782683i	$0.286149\pi$
$182$	0	0
$183$	−1705.01	−0.688733
$184$	−1146.51	−0.459359
$185$	0	0
$186$	−4510.82	−1.77822
$187$	−618.300	−0.241789
$188$	−1570.16	−0.609125
$189$	0	0
$190$	0	0
$191$	−2168.64	−0.821555	−0.410778	−	0.911736i	$-0.634743\pi$
$191$	−2168.64	−0.821555	−0.410778	+	0.911736i	$0.634743\pi$
$192$	−3876.62	−1.45714
$193$	−1490.48	−0.555892	−0.277946	−	0.960597i	$-0.589654\pi$
$193$	−1490.48	−0.555892	−0.277946	+	0.960597i	$0.589654\pi$
$194$	−6095.04	−2.25566
$195$	0	0
$196$	0	0
$197$	−3380.57	−1.22262	−0.611308	−	0.791393i	$-0.709356\pi$
$197$	−3380.57	−1.22262	−0.611308	+	0.791393i	$0.709356\pi$
$198$	170.364	0.0611478
$199$	−4595.33	−1.63696	−0.818478	−	0.574538i	$-0.805182\pi$
$199$	−4595.33	−1.63696	−0.818478	+	0.574538i	$0.805182\pi$
$200$	0	0
$201$	421.980	0.148080
$202$	−2355.46	−0.820445
$203$	0	0
$204$	−1665.83	−0.571723
$205$	0	0
$206$	−3371.24	−1.14022
$207$	−201.731	−0.0677356
$208$	−2567.97	−0.856042
$209$	−2690.32	−0.890398
$210$	0	0
$211$	−4988.66	−1.62765	−0.813824	−	0.581112i	$-0.802618\pi$
$211$	−4988.66	−1.62765	−0.813824	+	0.581112i	$0.802618\pi$
$212$	1361.00	0.440915
$213$	−1578.68	−0.507837
$214$	−2679.16	−0.855810
$215$	0	0
$216$	1648.18	0.519187
$217$	0	0
$218$	−18.0778	−0.00561643
$219$	−5466.62	−1.68676
$220$	0	0
$221$	−2235.03	−0.680290
$222$	6689.29	2.02232
$223$	3792.97	1.13900	0.569498	−	0.821993i	$-0.307138\pi$
$223$	3792.97	1.13900	0.569498	+	0.821993i	$0.307138\pi$
$224$	0	0
$225$	0	0
$226$	6101.68	1.79592
$227$	3910.96	1.14352	0.571762	−	0.820420i	$-0.306260\pi$
$227$	3910.96	1.14352	0.571762	+	0.820420i	$0.306260\pi$
$228$	−7248.29	−2.10539
$229$	−354.327	−0.102247	−0.0511236	−	0.998692i	$-0.516280\pi$
$229$	−354.327	−0.102247	−0.0511236	+	0.998692i	$0.516280\pi$
$230$	0	0
$231$	0	0
$232$	3275.87	0.927033
$233$	−6492.48	−1.82548	−0.912739	−	0.408543i	$-0.866037\pi$
$233$	−6492.48	−1.82548	−0.912739	+	0.408543i	$0.866037\pi$
$234$	615.831	0.172043
$235$	0	0
$236$	−3430.98	−0.946346
$237$	6165.93	1.68996
$238$	0	0
$239$	−342.688	−0.0927474	−0.0463737	−	0.998924i	$-0.514766\pi$
$239$	−342.688	−0.0927474	−0.0463737	+	0.998924i	$0.514766\pi$
$240$	0	0
$241$	−2313.67	−0.618408	−0.309204	−	0.950996i	$-0.600063\pi$
$241$	−2313.67	−0.618408	−0.309204	+	0.950996i	$0.600063\pi$
$242$	4064.49	1.07965
$243$	560.000	0.147835
$244$	−3625.96	−0.951347
$245$	0	0
$246$	−3920.84	−1.01619
$247$	−9724.94	−2.50519
$248$	−2375.62	−0.608275
$249$	−3216.62	−0.818656
$250$	0	0
$251$	−3989.29	−1.00319	−0.501597	−	0.865101i	$-0.667254\pi$
$251$	−3989.29	−1.00319	−0.501597	+	0.865101i	$0.667254\pi$
$252$	0	0
$253$	1990.43	0.494614
$254$	−8832.95	−2.18200
$255$	0	0
$256$	262.376	0.0640566
$257$	2291.32	0.556142	0.278071	−	0.960560i	$-0.410305\pi$
$257$	2291.32	0.556142	0.278071	+	0.960560i	$0.410305\pi$
$258$	−392.824	−0.0947912
$259$	0	0
$260$	0	0
$261$	576.396	0.136697
$262$	4922.32	1.16070
$263$	−6360.47	−1.49127	−0.745634	−	0.666356i	$-0.767853\pi$
$263$	−6360.47	−1.49127	−0.745634	+	0.666356i	$0.767853\pi$
$264$	−1121.53	−0.261460
$265$	0	0
$266$	0	0
$267$	−5701.59	−1.30686
$268$	897.404	0.204543
$269$	991.345	0.224697	0.112348	−	0.993669i	$-0.464163\pi$
$269$	991.345	0.224697	0.112348	+	0.993669i	$0.464163\pi$
$270$	0	0
$271$	−730.977	−0.163851	−0.0819257	−	0.996638i	$-0.526107\pi$
$271$	−730.977	−0.163851	−0.0819257	+	0.996638i	$0.526107\pi$
$272$	1127.97	0.251446
$273$	0	0
$274$	−2116.62	−0.466679
$275$	0	0
$276$	5362.64	1.16954
$277$	3538.63	0.767566	0.383783	−	0.923423i	$-0.374621\pi$
$277$	3538.63	0.767566	0.383783	+	0.923423i	$0.374621\pi$
$278$	12086.5	2.60756
$279$	−417.995	−0.0896943
$280$	0	0
$281$	−4663.20	−0.989975	−0.494988	−	0.868900i	$-0.664827\pi$
$281$	−4663.20	−0.989975	−0.494988	+	0.868900i	$0.664827\pi$
$282$	3187.07	0.673005
$283$	−2104.95	−0.442142	−0.221071	−	0.975258i	$-0.570955\pi$
$283$	−2104.95	−0.442142	−0.221071	+	0.975258i	$0.570955\pi$
$284$	−3357.30	−0.701476
$285$	0	0
$286$	−6076.25	−1.25628
$287$	0	0
$288$	−492.665	−0.100801
$289$	−3931.27	−0.800178
$290$	0	0
$291$	7059.96	1.42221
$292$	−11625.6	−2.32992
$293$	−6594.66	−1.31489	−0.657447	−	0.753501i	$-0.728364\pi$
$293$	−6594.66	−1.31489	−0.657447	+	0.753501i	$0.728364\pi$
$294$	0	0
$295$	0	0
$296$	3522.91	0.691774
$297$	−2861.36	−0.559033
$298$	−5033.69	−0.978503
$299$	7194.99	1.39163
$300$	0	0
$301$	0	0
$302$	−4142.15	−0.789252
$303$	2728.36	0.517295
$304$	4907.97	0.925958
$305$	0	0
$306$	−270.501	−0.0505344
$307$	−2672.97	−0.496920	−0.248460	−	0.968642i	$-0.579924\pi$
$307$	−2672.97	−0.496920	−0.248460	+	0.968642i	$0.579924\pi$
$308$	0	0
$309$	3904.95	0.718915
$310$	0	0
$311$	855.698	0.156020	0.0780099	−	0.996953i	$-0.475143\pi$
$311$	855.698	0.156020	0.0780099	+	0.996953i	$0.475143\pi$
$312$	−4054.09	−0.735634
$313$	−3349.99	−0.604960	−0.302480	−	0.953156i	$-0.597815\pi$
$313$	−3349.99	−0.604960	−0.302480	+	0.953156i	$0.597815\pi$
$314$	4405.81	0.791828
$315$	0	0
$316$	13112.8	2.33434
$317$	−7633.05	−1.35241	−0.676206	−	0.736712i	$-0.736377\pi$
$317$	−7633.05	−1.35241	−0.676206	+	0.736712i	$0.736377\pi$
$318$	−2762.53	−0.487154
$319$	−5687.16	−0.998180
$320$	0	0
$321$	3103.30	0.539593
$322$	0	0
$323$	4271.64	0.735852
$324$	−7134.91	−1.22341
$325$	0	0
$326$	−6759.90	−1.14845
$327$	20.9397	0.00354119
$328$	−2064.91	−0.347609
$329$	0	0
$330$	0	0
$331$	−3321.70	−0.551593	−0.275796	−	0.961216i	$-0.588942\pi$
$331$	−3321.70	−0.551593	−0.275796	+	0.961216i	$0.588942\pi$
$332$	−6840.63	−1.13081
$333$	619.863	0.102007
$334$	4879.10	0.799319
$335$	0	0
$336$	0	0
$337$	2233.98	0.361107	0.180553	−	0.983565i	$-0.442211\pi$
$337$	2233.98	0.361107	0.180553	+	0.983565i	$0.442211\pi$
$338$	−12480.8	−2.00847
$339$	−7067.65	−1.13234
$340$	0	0
$341$	4124.25	0.654958
$342$	−1176.99	−0.186095
$343$	0	0
$344$	−206.881	−0.0324252
$345$	0	0
$346$	−10978.6	−1.70582
$347$	−2528.61	−0.391190	−0.195595	−	0.980685i	$-0.562664\pi$
$347$	−2528.61	−0.391190	−0.195595	+	0.980685i	$0.562664\pi$
$348$	−15322.4	−2.36025
$349$	−1291.00	−0.198011	−0.0990054	−	0.995087i	$-0.531566\pi$
$349$	−1291.00	−0.198011	−0.0990054	+	0.995087i	$0.531566\pi$
$350$	0	0
$351$	−10343.2	−1.57288
$352$	4861.00	0.736058
$353$	−7768.64	−1.17134	−0.585670	−	0.810550i	$-0.699169\pi$
$353$	−7768.64	−1.17134	−0.585670	+	0.810550i	$0.699169\pi$
$354$	6964.12	1.04559
$355$	0	0
$356$	−12125.3	−1.80516
$357$	0	0
$358$	5225.92	0.771504
$359$	−2284.14	−0.335800	−0.167900	−	0.985804i	$-0.553699\pi$
$359$	−2284.14	−0.335800	−0.167900	+	0.985804i	$0.553699\pi$
$360$	0	0
$361$	11727.5	1.70980
$362$	−13085.1	−1.89982
$363$	−4707.94	−0.680725
$364$	0	0
$365$	0	0
$366$	7359.90	1.05112
$367$	10707.0	1.52289	0.761446	−	0.648229i	$-0.224490\pi$
$367$	10707.0	1.52289	0.761446	+	0.648229i	$0.224490\pi$
$368$	−3631.16	−0.514367
$369$	−363.325	−0.0512573
$370$	0	0
$371$	0	0
$372$	11111.6	1.54868
$373$	830.429	0.115276	0.0576381	−	0.998338i	$-0.481643\pi$
$373$	830.429	0.115276	0.0576381	+	0.998338i	$0.481643\pi$
$374$	2668.97	0.369008
$375$	0	0
$376$	1678.47	0.230214
$377$	−20557.9	−2.80845
$378$	0	0
$379$	5253.17	0.711972	0.355986	−	0.934491i	$-0.384145\pi$
$379$	5253.17	0.711972	0.355986	+	0.934491i	$0.384145\pi$
$380$	0	0
$381$	10231.3	1.37576
$382$	9361.19	1.25382
$383$	−11243.9	−1.50010	−0.750048	−	0.661383i	$-0.769970\pi$
$383$	−11243.9	−1.50010	−0.750048	+	0.661383i	$0.769970\pi$
$384$	6880.63	0.914390
$385$	0	0
$386$	6433.84	0.848378
$387$	−36.4010	−0.00478131
$388$	15014.1	1.96449
$389$	−8506.85	−1.10878	−0.554389	−	0.832258i	$-0.687048\pi$
$389$	−8506.85	−1.10878	−0.554389	+	0.832258i	$0.687048\pi$
$390$	0	0
$391$	−3160.37	−0.408764
$392$	0	0
$393$	−5701.59	−0.731824
$394$	14592.6	1.86590
$395$	0	0
$396$	−419.662	−0.0532546
$397$	3123.24	0.394838	0.197419	−	0.980319i	$-0.436744\pi$
$397$	3123.24	0.394838	0.197419	+	0.980319i	$0.436744\pi$
$398$	19836.3	2.49825
$399$	0	0
$400$	0	0
$401$	−11255.3	−1.40166	−0.700828	−	0.713330i	$-0.747186\pi$
$401$	−11255.3	−1.40166	−0.700828	+	0.713330i	$0.747186\pi$
$402$	−1821.53	−0.225994
$403$	14908.3	1.84277
$404$	5802.27	0.714539
$405$	0	0
$406$	0	0
$407$	−6116.03	−0.744866
$408$	1780.74	0.216078
$409$	7919.92	0.957494	0.478747	−	0.877953i	$-0.341091\pi$
$409$	7919.92	0.957494	0.478747	+	0.877953i	$0.341091\pi$
$410$	0	0
$411$	2451.71	0.294243
$412$	8304.46	0.993037
$413$	0	0
$414$	870.797	0.103375
$415$	0	0
$416$	17571.5	2.07095
$417$	−14000.0	−1.64408
$418$	11613.1	1.35889
$419$	5257.28	0.612972	0.306486	−	0.951875i	$-0.400847\pi$
$419$	5257.28	0.612972	0.306486	+	0.951875i	$0.400847\pi$
$420$	0	0
$421$	1457.36	0.168711	0.0843556	−	0.996436i	$-0.473117\pi$
$421$	1457.36	0.168711	0.0843556	+	0.996436i	$0.473117\pi$
$422$	21534.2	2.48405
$423$	295.330	0.0339467
$424$	−1454.89	−0.166640
$425$	0	0
$426$	6814.57	0.775040
$427$	0	0
$428$	6599.63	0.745339
$429$	7038.20	0.792092
$430$	0	0
$431$	−15291.2	−1.70893	−0.854467	−	0.519506i	$-0.826116\pi$
$431$	−15291.2	−1.70893	−0.854467	+	0.519506i	$0.826116\pi$
$432$	5220.00	0.581360
$433$	187.260	0.0207832	0.0103916	−	0.999946i	$-0.496692\pi$
$433$	187.260	0.0207832	0.0103916	+	0.999946i	$0.496692\pi$
$434$	0	0
$435$	0	0
$436$	44.5314	0.00489144
$437$	−13751.2	−1.50529
$438$	23597.4	2.57426
$439$	−3587.92	−0.390073	−0.195037	−	0.980796i	$-0.562483\pi$
$439$	−3587.92	−0.390073	−0.195037	+	0.980796i	$0.562483\pi$
$440$	0	0
$441$	0	0
$442$	9647.76	1.03823
$443$	−4915.65	−0.527200	−0.263600	−	0.964632i	$-0.584910\pi$
$443$	−4915.65	−0.527200	−0.263600	+	0.964632i	$0.584910\pi$
$444$	−16477.9	−1.76128
$445$	0	0
$446$	−16372.8	−1.73829
$447$	5830.58	0.616951
$448$	0	0
$449$	7091.12	0.745324	0.372662	−	0.927967i	$-0.378445\pi$
$449$	7091.12	0.745324	0.372662	+	0.927967i	$0.378445\pi$
$450$	0	0
$451$	3584.84	0.374287
$452$	−15030.4	−1.56410
$453$	4797.91	0.497628
$454$	−16882.2	−1.74520
$455$	0	0
$456$	7748.29	0.795717
$457$	−5051.81	−0.517098	−0.258549	−	0.965998i	$-0.583244\pi$
$457$	−5051.81	−0.517098	−0.258549	+	0.965998i	$0.583244\pi$
$458$	1529.50	0.156045
$459$	4543.21	0.462002
$460$	0	0
$461$	16681.3	1.68531	0.842653	−	0.538456i	$-0.180992\pi$
$461$	16681.3	1.68531	0.842653	+	0.538456i	$0.180992\pi$
$462$	0	0
$463$	−15569.6	−1.56280	−0.781402	−	0.624027i	$-0.785495\pi$
$463$	−15569.6	−1.56280	−0.781402	+	0.624027i	$0.785495\pi$
$464$	10375.1	1.03805
$465$	0	0
$466$	28025.6	2.78597
$467$	3328.35	0.329802	0.164901	−	0.986310i	$-0.447269\pi$
$467$	3328.35	0.329802	0.164901	+	0.986310i	$0.447269\pi$
$468$	−1516.99	−0.149835
$469$	0	0
$470$	0	0
$471$	−5103.30	−0.499252
$472$	3667.65	0.357664
$473$	359.160	0.0349137
$474$	−26616.0	−2.57914
$475$	0	0
$476$	0	0
$477$	−255.990	−0.0245723
$478$	1479.25	0.141547
$479$	14607.5	1.39339	0.696695	−	0.717367i	$-0.254653\pi$
$479$	14607.5	1.39339	0.696695	+	0.717367i	$0.254653\pi$
$480$	0	0
$481$	−22108.2	−2.09573
$482$	9987.23	0.943789
$483$	0	0
$484$	−10012.2	−0.940285
$485$	0	0
$486$	−2417.31	−0.225620
$487$	1879.49	0.174882	0.0874412	−	0.996170i	$-0.472131\pi$
$487$	1879.49	0.174882	0.0874412	+	0.996170i	$0.472131\pi$
$488$	3876.09	0.359554
$489$	7830.08	0.724107
$490$	0	0
$491$	3221.13	0.296064	0.148032	−	0.988983i	$-0.452706\pi$
$491$	3221.13	0.296064	0.148032	+	0.988983i	$0.452706\pi$
$492$	9658.31	0.885021
$493$	9029.96	0.824927
$494$	41978.9	3.82332
$495$	0	0
$496$	−7523.91	−0.681116
$497$	0	0
$498$	13885.0	1.24940
$499$	9713.81	0.871443	0.435721	−	0.900082i	$-0.356493\pi$
$499$	9713.81	0.871443	0.435721	+	0.900082i	$0.356493\pi$
$500$	0	0
$501$	−5651.52	−0.503975
$502$	17220.3	1.53103
$503$	2078.32	0.184230	0.0921152	−	0.995748i	$-0.470637\pi$
$503$	2078.32	0.184230	0.0921152	+	0.995748i	$0.470637\pi$
$504$	0	0
$505$	0	0
$506$	−8591.94	−0.754858
$507$	14456.6	1.26635
$508$	21758.4	1.90034
$509$	18974.5	1.65232	0.826158	−	0.563439i	$-0.190522\pi$
$509$	18974.5	1.65232	0.826158	+	0.563439i	$0.190522\pi$
$510$	0	0
$511$	0	0
$512$	−12141.6	−1.04802
$513$	19768.2	1.70134
$514$	−9890.77	−0.848761
$515$	0	0
$516$	967.652	0.0825553
$517$	−2913.95	−0.247883
$518$	0	0
$519$	12716.7	1.07553
$520$	0	0
$521$	−17523.6	−1.47355	−0.736777	−	0.676136i	$-0.763653\pi$
$521$	−17523.6	−1.47355	−0.736777	+	0.676136i	$0.763653\pi$
$522$	−2488.09	−0.208622
$523$	15218.6	1.27239	0.636197	−	0.771527i	$-0.280507\pi$
$523$	15218.6	1.27239	0.636197	+	0.771527i	$0.280507\pi$
$524$	−12125.3	−1.01087
$525$	0	0
$526$	27455.8	2.27591
$527$	−6548.41	−0.541278
$528$	−3552.03	−0.292769
$529$	−1993.15	−0.163816
$530$	0	0
$531$	645.330	0.0527400
$532$	0	0
$533$	12958.4	1.05308
$534$	24611.6	1.99447
$535$	0	0
$536$	−959.308	−0.0773056
$537$	−6053.25	−0.486438
$538$	−4279.26	−0.342922
$539$	0	0
$540$	0	0
$541$	−15559.3	−1.23650	−0.618249	−	0.785983i	$-0.712158\pi$
$541$	−15559.3	−1.23650	−0.618249	+	0.785983i	$0.712158\pi$
$542$	3155.36	0.250063
$543$	15156.6	1.19785
$544$	−7718.21	−0.608301
$545$	0	0
$546$	0	0
$547$	8690.70	0.679319	0.339660	−	0.940548i	$-0.389688\pi$
$547$	8690.70	0.679319	0.339660	+	0.940548i	$0.389688\pi$
$548$	5213.93	0.406438
$549$	682.005	0.0530187
$550$	0	0
$551$	39290.8	3.03783
$552$	−5732.56	−0.442018
$553$	0	0
$554$	−15274.9	−1.17143
$555$	0	0
$556$	−29773.1	−2.27097
$557$	7376.26	0.561117	0.280559	−	0.959837i	$-0.409480\pi$
$557$	7376.26	0.561117	0.280559	+	0.959837i	$0.409480\pi$
$558$	1804.33	0.136888
$559$	1298.29	0.0982320
$560$	0	0
$561$	−3091.50	−0.232662
$562$	20129.3	1.51086
$563$	12875.4	0.963825	0.481913	−	0.876219i	$-0.339942\pi$
$563$	12875.4	0.963825	0.481913	+	0.876219i	$0.339942\pi$
$564$	−7850.79	−0.586131
$565$	0	0
$566$	9086.28	0.674779
$567$	0	0
$568$	3588.89	0.265117
$569$	−12064.6	−0.888882	−0.444441	−	0.895808i	$-0.646598\pi$
$569$	−12064.6	−0.888882	−0.444441	+	0.895808i	$0.646598\pi$
$570$	0	0
$571$	23745.6	1.74032	0.870158	−	0.492772i	$-0.164016\pi$
$571$	23745.6	1.74032	0.870158	+	0.492772i	$0.164016\pi$
$572$	14967.8	1.09412
$573$	−10843.2	−0.790542
$574$	0	0
$575$	0	0
$576$	1550.65	0.112171
$577$	−9846.08	−0.710394	−0.355197	−	0.934791i	$-0.615586\pi$
$577$	−9846.08	−0.710394	−0.355197	+	0.934791i	$0.615586\pi$
$578$	16969.8	1.22120
$579$	−7452.40	−0.534907
$580$	0	0
$581$	0	0
$582$	−30475.2	−2.17051
$583$	2525.79	0.179430
$584$	12427.6	0.880575
$585$	0	0
$586$	28466.7	2.00674
$587$	10074.7	0.708392	0.354196	−	0.935171i	$-0.384755\pi$
$587$	10074.7	0.708392	0.354196	+	0.935171i	$0.384755\pi$
$588$	0	0
$589$	−28493.1	−1.99328
$590$	0	0
$591$	−16902.8	−1.17646
$592$	11157.5	0.774615
$593$	−7387.25	−0.511565	−0.255782	−	0.966734i	$-0.582333\pi$
$593$	−7387.25	−0.511565	−0.255782	+	0.966734i	$0.582333\pi$
$594$	12351.4	0.853172
$595$	0	0
$596$	12399.6	0.852194
$597$	−22976.6	−1.57516
$598$	−31058.1	−2.12384
$599$	1252.73	0.0854510	0.0427255	−	0.999087i	$-0.486396\pi$
$599$	1252.73	0.0854510	0.0427255	+	0.999087i	$0.486396\pi$
$600$	0	0
$601$	1800.81	0.122224	0.0611120	−	0.998131i	$-0.480535\pi$
$601$	1800.81	0.122224	0.0611120	+	0.998131i	$0.480535\pi$
$602$	0	0
$603$	−168.792	−0.0113992
$604$	10203.5	0.687373
$605$	0	0
$606$	−11777.3	−0.789473
$607$	2497.06	0.166973	0.0834863	−	0.996509i	$-0.473395\pi$
$607$	2497.06	0.166973	0.0834863	+	0.996509i	$0.473395\pi$
$608$	−33583.1	−2.24009
$609$	0	0
$610$	0	0
$611$	−10533.3	−0.697434
$612$	666.332	0.0440113
$613$	19750.8	1.30135	0.650674	−	0.759357i	$-0.274487\pi$
$613$	19750.8	1.30135	0.650674	+	0.759357i	$0.274487\pi$
$614$	11538.2	0.758378
$615$	0	0
$616$	0	0
$617$	−16797.4	−1.09601	−0.548004	−	0.836476i	$-0.684612\pi$
$617$	−16797.4	−1.09601	−0.548004	+	0.836476i	$0.684612\pi$
$618$	−16856.2	−1.09718
$619$	−26547.4	−1.72379	−0.861897	−	0.507084i	$-0.830723\pi$
$619$	−26547.4	−1.72379	−0.861897	+	0.507084i	$0.830723\pi$
$620$	0	0
$621$	−14625.5	−0.945090
$622$	−3693.73	−0.238111
$623$	0	0
$624$	−12839.8	−0.823727
$625$	0	0
$626$	14460.6	0.923264
$627$	−13451.6	−0.856786
$628$	−10852.9	−0.689616
$629$	9710.93	0.615580
$630$	0	0
$631$	5394.86	0.340358	0.170179	−	0.985413i	$-0.445565\pi$
$631$	5394.86	0.340358	0.170179	+	0.985413i	$0.445565\pi$
$632$	−14017.3	−0.882245
$633$	−24943.3	−1.56620
$634$	32949.0	2.06399
$635$	0	0
$636$	6805.01	0.424271
$637$	0	0
$638$	24549.3	1.52338
$639$	631.472	0.0390933
$640$	0	0
$641$	2452.41	0.151114	0.0755572	−	0.997141i	$-0.475926\pi$
$641$	2452.41	0.151114	0.0755572	+	0.997141i	$0.475926\pi$
$642$	−13395.8	−0.823504
$643$	7074.97	0.433919	0.216959	−	0.976181i	$-0.430386\pi$
$643$	7074.97	0.433919	0.216959	+	0.976181i	$0.430386\pi$
$644$	0	0
$645$	0	0
$646$	−18439.1	−1.12303
$647$	3341.37	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$647$	3341.37	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$648$	7627.09	0.462377
$649$	−6367.31	−0.385114
$650$	0	0
$651$	0	0
$652$	16651.8	1.00021
$653$	23061.6	1.38204	0.691019	−	0.722837i	$-0.257162\pi$
$653$	23061.6	1.38204	0.691019	+	0.722837i	$0.257162\pi$
$654$	−90.3888	−0.00540441
$655$	0	0
$656$	−6539.85	−0.389235
$657$	2186.65	0.129847
$658$	0	0
$659$	1742.64	0.103010	0.0515049	−	0.998673i	$-0.483598\pi$
$659$	1742.64	0.103010	0.0515049	+	0.998673i	$0.483598\pi$
$660$	0	0
$661$	12576.5	0.740046	0.370023	−	0.929023i	$-0.379350\pi$
$661$	12576.5	0.740046	0.370023	+	0.929023i	$0.379350\pi$
$662$	14338.5	0.841817
$663$	−11175.1	−0.654609
$664$	7312.51	0.427380
$665$	0	0
$666$	−2675.72	−0.155679
$667$	−29069.2	−1.68750
$668$	−12018.8	−0.696141
$669$	18964.8	1.09600
$670$	0	0
$671$	−6729.17	−0.387149
$672$	0	0
$673$	10680.8	0.611760	0.305880	−	0.952070i	$-0.401049\pi$
$673$	10680.8	0.611760	0.305880	+	0.952070i	$0.401049\pi$
$674$	−9643.27	−0.551105
$675$	0	0
$676$	30744.2	1.74921
$677$	−29559.1	−1.67806	−0.839032	−	0.544082i	$-0.816878\pi$
$677$	−29559.1	−1.67806	−0.839032	+	0.544082i	$0.816878\pi$
$678$	30508.4	1.72812
$679$	0	0
$680$	0	0
$681$	19554.8	1.10036
$682$	−17802.8	−0.999569
$683$	−10250.4	−0.574263	−0.287132	−	0.957891i	$-0.592702\pi$
$683$	−10250.4	−0.574263	−0.287132	+	0.957891i	$0.592702\pi$
$684$	2899.31	0.162073
$685$	0	0
$686$	0	0
$687$	−1771.64	−0.0983875
$688$	−655.218	−0.0363081
$689$	9130.20	0.504837
$690$	0	0
$691$	8874.04	0.488544	0.244272	−	0.969707i	$-0.421451\pi$
$691$	8874.04	0.488544	0.244272	+	0.969707i	$0.421451\pi$
$692$	27043.9	1.48563
$693$	0	0
$694$	10915.1	0.597018
$695$	0	0
$696$	16379.4	0.892038
$697$	−5691.94	−0.309322
$698$	5572.77	0.302196
$699$	−32462.4	−1.75657
$700$	0	0
$701$	−22086.2	−1.18999	−0.594996	−	0.803729i	$-0.702846\pi$
$701$	−22086.2	−1.18999	−0.594996	+	0.803729i	$0.702846\pi$
$702$	44647.8	2.40046
$703$	42253.7	2.26690
$704$	−15299.9	−0.819085
$705$	0	0
$706$	33534.3	1.78765
$707$	0	0
$708$	−17154.9	−0.910622
$709$	−27878.9	−1.47675	−0.738373	−	0.674392i	$-0.764406\pi$
$709$	−27878.9	−1.47675	−0.738373	+	0.674392i	$0.764406\pi$
$710$	0	0
$711$	−2466.37	−0.130093
$712$	12961.7	0.682248
$713$	21080.6	1.10726
$714$	0	0
$715$	0	0
$716$	−12873.1	−0.671916
$717$	−1713.44	−0.0892462
$718$	9859.76	0.512483
$719$	−25863.3	−1.34150	−0.670750	−	0.741684i	$-0.734027\pi$
$719$	−25863.3	−1.34150	−0.670750	+	0.741684i	$0.734027\pi$
$720$	0	0
$721$	0	0
$722$	−50623.4	−2.60943
$723$	−11568.3	−0.595064
$724$	32232.8	1.65459
$725$	0	0
$726$	20322.4	1.03889
$727$	29157.0	1.48744	0.743722	−	0.668489i	$-0.233059\pi$
$727$	29157.0	1.48744	0.743722	+	0.668489i	$0.233059\pi$
$728$	0	0
$729$	20917.0	1.06269
$730$	0	0
$731$	−570.267	−0.0288538
$732$	−18129.8	−0.915434
$733$	11006.1	0.554595	0.277297	−	0.960784i	$-0.410561\pi$
$733$	11006.1	0.554595	0.277297	+	0.960784i	$0.410561\pi$
$734$	−46218.1	−2.32417
$735$	0	0
$736$	24846.4	1.24436
$737$	1665.43	0.0832386
$738$	1568.34	0.0782267
$739$	−37214.4	−1.85244	−0.926221	−	0.376982i	$-0.876962\pi$
$739$	−37214.4	−1.85244	−0.926221	+	0.376982i	$0.876962\pi$
$740$	0	0
$741$	−48624.7	−2.41062
$742$	0	0
$743$	−11214.5	−0.553730	−0.276865	−	0.960909i	$-0.589296\pi$
$743$	−11214.5	−0.553730	−0.276865	+	0.960909i	$0.589296\pi$
$744$	−11878.1	−0.585313
$745$	0	0
$746$	−3584.65	−0.175930
$747$	1286.65	0.0630202
$748$	−6574.54	−0.321375
$749$	0	0
$750$	0	0
$751$	6965.26	0.338437	0.169218	−	0.985579i	$-0.445876\pi$
$751$	6965.26	0.338437	0.169218	+	0.985579i	$0.445876\pi$
$752$	5315.94	0.257782
$753$	−19946.4	−0.965324
$754$	88740.7	4.28613
$755$	0	0
$756$	0	0
$757$	19352.8	0.929180	0.464590	−	0.885526i	$-0.346202\pi$
$757$	19352.8	0.929180	0.464590	+	0.885526i	$0.346202\pi$
$758$	−22676.0	−1.08658
$759$	9952.15	0.475942
$760$	0	0
$761$	−32383.6	−1.54258	−0.771291	−	0.636483i	$-0.780388\pi$
$761$	−32383.6	−1.54258	−0.771291	+	0.636483i	$0.780388\pi$
$762$	−44164.8	−2.09963
$763$	0	0
$764$	−23059.7	−1.09198
$765$	0	0
$766$	48535.7	2.28938
$767$	−23016.5	−1.08354
$768$	1311.88	0.0616385
$769$	25353.9	1.18893	0.594463	−	0.804123i	$-0.297365\pi$
$769$	25353.9	1.18893	0.594463	+	0.804123i	$0.297365\pi$
$770$	0	0
$771$	11456.6	0.535148
$772$	−15848.6	−0.738867
$773$	26117.0	1.21522	0.607610	−	0.794236i	$-0.292128\pi$
$773$	26117.0	1.21522	0.607610	+	0.794236i	$0.292128\pi$
$774$	157.129	0.00729703
$775$	0	0
$776$	−16049.8	−0.742465
$777$	0	0
$778$	36720.9	1.69217
$779$	−24766.5	−1.13909
$780$	0	0
$781$	−6230.58	−0.285464
$782$	13642.1	0.623838
$783$	41788.7	1.90729
$784$	0	0
$785$	0	0
$786$	24611.6	1.11688
$787$	−2273.38	−0.102970	−0.0514849	−	0.998674i	$-0.516395\pi$
$787$	−2273.38	−0.102970	−0.0514849	+	0.998674i	$0.516395\pi$
$788$	−35946.4	−1.62505
$789$	−31802.4	−1.43497
$790$	0	0
$791$	0	0
$792$	448.612	0.0201272
$793$	−24324.6	−1.08927
$794$	−13481.8	−0.602585
$795$	0	0
$796$	−48863.3	−2.17577
$797$	2937.42	0.130551	0.0652753	−	0.997867i	$-0.479207\pi$
$797$	2937.42	0.130551	0.0652753	+	0.997867i	$0.479207\pi$
$798$	0	0
$799$	4626.71	0.204858
$800$	0	0
$801$	2280.63	0.100602
$802$	48585.0	2.13915
$803$	−21575.1	−0.948157
$804$	4487.02	0.196822
$805$	0	0
$806$	−64353.6	−2.81236
$807$	4956.73	0.216214
$808$	−6202.52	−0.270054
$809$	−4317.51	−0.187634	−0.0938169	−	0.995589i	$-0.529907\pi$
$809$	−4317.51	−0.187634	−0.0938169	+	0.995589i	$0.529907\pi$
$810$	0	0
$811$	−1286.12	−0.0556863	−0.0278432	−	0.999612i	$-0.508864\pi$
$811$	−1286.12	−0.0556863	−0.0278432	+	0.999612i	$0.508864\pi$
$812$	0	0
$813$	−3654.89	−0.157666
$814$	26400.6	1.13678
$815$	0	0
$816$	5639.85	0.241954
$817$	−2481.32	−0.106255
$818$	−34187.3	−1.46129
$819$	0	0
$820$	0	0
$821$	26350.2	1.12013	0.560066	−	0.828448i	$-0.310776\pi$
$821$	26350.2	1.12013	0.560066	+	0.828448i	$0.310776\pi$
$822$	−10583.1	−0.449062
$823$	−9820.05	−0.415924	−0.207962	−	0.978137i	$-0.566683\pi$
$823$	−9820.05	−0.415924	−0.207962	+	0.978137i	$0.566683\pi$
$824$	−8877.32	−0.375311
$825$	0	0
$826$	0	0
$827$	30370.7	1.27702	0.638509	−	0.769615i	$-0.279552\pi$
$827$	30370.7	1.27702	0.638509	+	0.769615i	$0.279552\pi$
$828$	−2145.06	−0.0900312
$829$	30817.7	1.29112	0.645562	−	0.763708i	$-0.276623\pi$
$829$	30817.7	1.29112	0.645562	+	0.763708i	$0.276623\pi$
$830$	0	0
$831$	17693.1	0.738590
$832$	−55305.9	−2.30455
$833$	0	0
$834$	60432.7	2.50913
$835$	0	0
$836$	−28606.8	−1.18348
$837$	−30304.6	−1.25147
$838$	−22693.7	−0.935491
$839$	24746.0	1.01827	0.509134	−	0.860688i	$-0.329966\pi$
$839$	24746.0	1.01827	0.509134	+	0.860688i	$0.329966\pi$
$840$	0	0
$841$	58669.1	2.40556
$842$	−6290.88	−0.257480
$843$	−23316.0	−0.952604
$844$	−53045.7	−2.16340
$845$	0	0
$846$	−1274.83	−0.0518079
$847$	0	0
$848$	−4607.82	−0.186596
$849$	−10524.7	−0.425452
$850$	0	0
$851$	−31261.4	−1.25926
$852$	−16786.5	−0.674995
$853$	−11812.1	−0.474135	−0.237067	−	0.971493i	$-0.576186\pi$
$853$	−11812.1	−0.474135	−0.237067	+	0.971493i	$0.576186\pi$
$854$	0	0
$855$	0	0
$856$	−7054.89	−0.281695
$857$	−23440.6	−0.934322	−0.467161	−	0.884172i	$-0.654723\pi$
$857$	−23440.6	−0.934322	−0.467161	+	0.884172i	$0.654723\pi$
$858$	−30381.3	−1.20886
$859$	−8945.58	−0.355319	−0.177660	−	0.984092i	$-0.556853\pi$
$859$	−8945.58	−0.355319	−0.177660	+	0.984092i	$0.556853\pi$
$860$	0	0
$861$	0	0
$862$	66006.3	2.60810
$863$	−19313.5	−0.761806	−0.380903	−	0.924615i	$-0.624387\pi$
$863$	−19313.5	−0.761806	−0.380903	+	0.924615i	$0.624387\pi$
$864$	−35718.2	−1.40643
$865$	0	0
$866$	−808.330	−0.0317184
$867$	−19656.4	−0.769972
$868$	0	0
$869$	24335.1	0.949955
$870$	0	0
$871$	6020.18	0.234197
$872$	−47.6033	−0.00184868
$873$	−2823.98	−0.109482
$874$	59359.0	2.29731
$875$	0	0
$876$	−58128.0	−2.24197
$877$	12154.8	0.468001	0.234001	−	0.972236i	$-0.424818\pi$
$877$	12154.8	0.468001	0.234001	+	0.972236i	$0.424818\pi$
$878$	15487.7	0.595313
$879$	−32973.3	−1.26526
$880$	0	0
$881$	29390.4	1.12394	0.561968	−	0.827159i	$-0.310044\pi$
$881$	29390.4	1.12394	0.561968	+	0.827159i	$0.310044\pi$
$882$	0	0
$883$	−4180.02	−0.159308	−0.0796540	−	0.996823i	$-0.525382\pi$
$883$	−4180.02	−0.159308	−0.0796540	+	0.996823i	$0.525382\pi$
$884$	−23765.6	−0.904211
$885$	0	0
$886$	21219.0	0.804590
$887$	21825.8	0.826198	0.413099	−	0.910686i	$-0.364446\pi$
$887$	21825.8	0.826198	0.413099	+	0.910686i	$0.364446\pi$
$888$	17614.6	0.665660
$889$	0	0
$890$	0	0
$891$	−13241.2	−0.497863
$892$	40331.6	1.51390
$893$	20131.5	0.754397
$894$	−25168.4	−0.941565
$895$	0	0
$896$	0	0
$897$	35974.9	1.33909
$898$	−30609.7	−1.13748
$899$	−60232.7	−2.23456
$900$	0	0
$901$	−4010.40	−0.148286
$902$	−15474.4	−0.571221
$903$	0	0
$904$	16067.2	0.591138
$905$	0	0
$906$	−20710.8	−0.759458
$907$	8356.11	0.305910	0.152955	−	0.988233i	$-0.451121\pi$
$907$	8356.11	0.305910	0.152955	+	0.988233i	$0.451121\pi$
$908$	41586.3	1.51992
$909$	−1091.35	−0.0398214
$910$	0	0
$911$	−4419.80	−0.160740	−0.0803701	−	0.996765i	$-0.525610\pi$
$911$	−4419.80	−0.160740	−0.0803701	+	0.996765i	$0.525610\pi$
$912$	24539.8	0.891004
$913$	−12695.1	−0.460181
$914$	21806.8	0.789173
$915$	0	0
$916$	−3767.65	−0.135903
$917$	0	0
$918$	−19611.3	−0.705088
$919$	−39257.6	−1.40913	−0.704563	−	0.709641i	$-0.748857\pi$
$919$	−39257.6	−1.40913	−0.704563	+	0.709641i	$0.748857\pi$
$920$	0	0
$921$	−13364.8	−0.478162
$922$	−72007.0	−2.57204
$923$	−22522.2	−0.803173
$924$	0	0
$925$	0	0
$926$	67207.9	2.38509
$927$	−1561.98	−0.0553421
$928$	−70992.5	−2.51125
$929$	13399.9	0.473235	0.236618	−	0.971603i	$-0.423961\pi$
$929$	13399.9	0.473235	0.236618	+	0.971603i	$0.423961\pi$
$930$	0	0
$931$	0	0
$932$	−69036.1	−2.42635
$933$	4278.49	0.150130
$934$	−14367.2	−0.503330
$935$	0	0
$936$	1621.64	0.0566291
$937$	27539.8	0.960176	0.480088	−	0.877220i	$-0.340605\pi$
$937$	27539.8	0.960176	0.480088	+	0.877220i	$0.340605\pi$
$938$	0	0
$939$	−16749.9	−0.582123
$940$	0	0
$941$	−14363.8	−0.497605	−0.248802	−	0.968554i	$-0.580037\pi$
$941$	−14363.8	−0.497605	−0.248802	+	0.968554i	$0.580037\pi$
$942$	22029.0	0.761937
$943$	18323.5	0.632762
$944$	11615.9	0.400494
$945$	0	0
$946$	−1550.36	−0.0532838
$947$	6372.12	0.218655	0.109327	−	0.994006i	$-0.465130\pi$
$947$	6372.12	0.218655	0.109327	+	0.994006i	$0.465130\pi$
$948$	65563.8	2.24622
$949$	−77989.6	−2.66770
$950$	0	0
$951$	−38165.3	−1.30136
$952$	0	0
$953$	−958.776	−0.0325895	−0.0162948	−	0.999867i	$-0.505187\pi$
$953$	−958.776	−0.0325895	−0.0162948	+	0.999867i	$0.505187\pi$
$954$	1105.01	0.0375012
$955$	0	0
$956$	−3643.88	−0.123276
$957$	−28435.8	−0.960500
$958$	−63055.1	−2.12653
$959$	0	0
$960$	0	0
$961$	13888.9	0.466213
$962$	95432.7	3.19842
$963$	−1241.32	−0.0415379
$964$	−24601.8	−0.821961
$965$	0	0
$966$	0	0
$967$	40104.5	1.33369	0.666843	−	0.745198i	$-0.267645\pi$
$967$	40104.5	1.33369	0.666843	+	0.745198i	$0.267645\pi$
$968$	10702.8	0.355373
$969$	21358.2	0.708074
$970$	0	0
$971$	−12397.4	−0.409732	−0.204866	−	0.978790i	$-0.565676\pi$
$971$	−12397.4	−0.409732	−0.204866	+	0.978790i	$0.565676\pi$
$972$	5954.62	0.196496
$973$	0	0
$974$	−8113.05	−0.266898
$975$	0	0
$976$	12276.1	0.402611
$977$	−44982.9	−1.47301	−0.736506	−	0.676432i	$-0.763525\pi$
$977$	−44982.9	−1.47301	−0.736506	+	0.676432i	$0.763525\pi$
$978$	−33799.5	−1.10510
$979$	−22502.5	−0.734608
$980$	0	0
$981$	−8.37588	−0.000272601 0
$982$	−13904.4	−0.451841
$983$	−7895.76	−0.256191	−0.128095	−	0.991762i	$-0.540886\pi$
$983$	−7895.76	−0.256191	−0.128095	+	0.991762i	$0.540886\pi$
$984$	−10324.6	−0.334487
$985$	0	0
$986$	−38979.0	−1.25897
$987$	0	0
$988$	−103408.	−3.32979
$989$	1835.80	0.0590244
$990$	0	0
$991$	54534.9	1.74809	0.874046	−	0.485844i	$-0.161488\pi$
$991$	54534.9	1.74809	0.874046	+	0.485844i	$0.161488\pi$
$992$	51482.9	1.64776
$993$	−16608.5	−0.530770
$994$	0	0
$995$	0	0
$996$	−34203.2	−1.08812
$997$	−6028.06	−0.191485	−0.0957425	−	0.995406i	$-0.530523\pi$
$997$	−6028.06	−0.191485	−0.0957425	+	0.995406i	$0.530523\pi$
$998$	−41930.9	−1.32996
$999$	44940.1	1.42326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.q.1.1		2
5.4	even	2		245.4.a.i.1.2	✓	2
7.6	odd	2		1225.4.a.p.1.1		2
15.14	odd	2		2205.4.a.x.1.1		2
35.4	even	6		245.4.e.k.226.1		4
35.9	even	6		245.4.e.k.116.1		4
35.19	odd	6		245.4.e.j.116.1		4
35.24	odd	6		245.4.e.j.226.1		4
35.34	odd	2		245.4.a.j.1.2	yes	2
105.104	even	2		2205.4.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.i.1.2	✓	2	5.4	even	2
245.4.a.j.1.2	yes	2	35.34	odd	2
245.4.e.j.116.1		4	35.19	odd	6
245.4.e.j.226.1		4	35.24	odd	6
245.4.e.k.116.1		4	35.9	even	6
245.4.e.k.226.1		4	35.4	even	6
1225.4.a.p.1.1		2	7.6	odd	2
1225.4.a.q.1.1		2	1.1	even	1	trivial
2205.4.a.w.1.1		2	105.104	even	2
2205.4.a.x.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-4.31662 q^{2} +5.00000 q^{3} +10.6332 q^{4} -21.5831 q^{6} -11.3668 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-4.31662 q^{2} +5.00000 q^{3} +10.6332 q^{4} -21.5831 q^{6} -11.3668 q^{8} -2.00000 q^{9} +19.7335 q^{11} +53.1662 q^{12} +71.3325 q^{13} -36.0000 q^{16} -31.3325 q^{17} +8.63325 q^{18} -136.332 q^{19} -85.1821 q^{22} +100.865 q^{23} -56.8338 q^{24} -307.916 q^{26} -145.000 q^{27} -288.198 q^{29} +208.997 q^{31} +246.332 q^{32} +98.6675 q^{33} +135.251 q^{34} -21.2665 q^{36} -309.931 q^{37} +588.496 q^{38} +356.662 q^{39} +181.662 q^{41} +18.2005 q^{43} +209.831 q^{44} -435.398 q^{46} -147.665 q^{47} -180.000 q^{48} -156.662 q^{51} +758.496 q^{52} +127.995 q^{53} +625.911 q^{54} -681.662 q^{57} +1244.04 q^{58} -322.665 q^{59} -341.003 q^{61} -902.164 q^{62} -775.325 q^{64} -425.911 q^{66} +84.3960 q^{67} -333.166 q^{68} +504.327 q^{69} -315.736 q^{71} +22.7335 q^{72} -1093.32 q^{73} +1337.86 q^{74} -1449.66 q^{76} -1539.58 q^{78} +1233.19 q^{79} -671.000 q^{81} -784.169 q^{82} -643.325 q^{83} -78.5647 q^{86} -1440.99 q^{87} -224.306 q^{88} -1140.32 q^{89} +1072.53 q^{92} +1044.99 q^{93} +637.414 q^{94} +1231.66 q^{96} +1411.99 q^{97} -39.4670 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 10 q^{6} - 36 q^{8} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 10 * q^3 + 8 * q^4 - 10 * q^6 - 36 * q^8 - 4 * q^9	\( 2 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 10 q^{6} - 36 q^{8} - 4 q^{9} + 66 q^{11} + 40 q^{12} + 10 q^{13} - 72 q^{16} + 70 q^{17} + 4 q^{18} - 140 q^{19} + 22 q^{22} + 16 q^{23} - 180 q^{24} - 450 q^{26} - 290 q^{27} - 258 q^{29} + 20 q^{31} + 360 q^{32} + 330 q^{33} + 370 q^{34} - 16 q^{36} - 328 q^{37} + 580 q^{38} + 50 q^{39} - 300 q^{41} + 116 q^{43} + 88 q^{44} - 632 q^{46} - 30 q^{47} - 360 q^{48} + 350 q^{51} + 920 q^{52} - 540 q^{53} + 290 q^{54} - 700 q^{57} + 1314 q^{58} - 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} + 110 q^{66} - 468 q^{67} - 600 q^{68} + 80 q^{69} - 1056 q^{71} + 72 q^{72} - 860 q^{73} + 1296 q^{74} - 1440 q^{76} - 2250 q^{78} + 158 q^{79} - 1342 q^{81} - 1900 q^{82} + 40 q^{83} + 148 q^{86} - 1290 q^{87} - 1364 q^{88} + 240 q^{89} + 1296 q^{92} + 100 q^{93} + 910 q^{94} + 1800 q^{96} + 1630 q^{97} - 132 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 10 * q^3 + 8 * q^4 - 10 * q^6 - 36 * q^8 - 4 * q^9 + 66 * q^11 + 40 * q^12 + 10 * q^13 - 72 * q^16 + 70 * q^17 + 4 * q^18 - 140 * q^19 + 22 * q^22 + 16 * q^23 - 180 * q^24 - 450 * q^26 - 290 * q^27 - 258 * q^29 + 20 * q^31 + 360 * q^32 + 330 * q^33 + 370 * q^34 - 16 * q^36 - 328 * q^37 + 580 * q^38 + 50 * q^39 - 300 * q^41 + 116 * q^43 + 88 * q^44 - 632 * q^46 - 30 * q^47 - 360 * q^48 + 350 * q^51 + 920 * q^52 - 540 * q^53 + 290 * q^54 - 700 * q^57 + 1314 * q^58 - 380 * q^59 - 1080 * q^61 - 1340 * q^62 - 224 * q^64 + 110 * q^66 - 468 * q^67 - 600 * q^68 + 80 * q^69 - 1056 * q^71 + 72 * q^72 - 860 * q^73 + 1296 * q^74 - 1440 * q^76 - 2250 * q^78 + 158 * q^79 - 1342 * q^81 - 1900 * q^82 + 40 * q^83 + 148 * q^86 - 1290 * q^87 - 1364 * q^88 + 240 * q^89 + 1296 * q^92 + 100 * q^93 + 910 * q^94 + 1800 * q^96 + 1630 * q^97 - 132 * q^99

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