LMFDB - Embedded newform 525.4.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, 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("525.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.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:	$30.9760027530$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.23607 q^{2} -3.00000 q^{3} +9.94427 q^{4} -12.7082 q^{6} +7.00000 q^{7} +8.23607 q^{8} +9.00000 q^{9} -41.5279 q^{11} -29.8328 q^{12} -88.9706 q^{13} +29.6525 q^{14} -44.6656 q^{16} +120.387 q^{17} +38.1246 q^{18} -112.138 q^{19} -21.0000 q^{21} -175.915 q^{22} +115.279 q^{23} -24.7082 q^{24} -376.885 q^{26} -27.0000 q^{27} +69.6099 q^{28} -144.833 q^{29} -258.079 q^{31} -255.095 q^{32} +124.584 q^{33} +509.967 q^{34} +89.4984 q^{36} -48.3344 q^{37} -475.023 q^{38} +266.912 q^{39} +200.885 q^{41} -88.9574 q^{42} +218.217 q^{43} -412.964 q^{44} +488.328 q^{46} -575.659 q^{47} +133.997 q^{48} +49.0000 q^{49} -361.161 q^{51} -884.748 q^{52} +184.302 q^{53} -114.374 q^{54} +57.6525 q^{56} +336.413 q^{57} -613.522 q^{58} -151.502 q^{59} -529.830 q^{61} -1093.24 q^{62} +63.0000 q^{63} -723.276 q^{64} +527.745 q^{66} -1.28485 q^{67} +1197.16 q^{68} -345.836 q^{69} -61.4226 q^{71} +74.1246 q^{72} -484.800 q^{73} -204.748 q^{74} -1115.13 q^{76} -290.695 q^{77} +1130.66 q^{78} +878.257 q^{79} +81.0000 q^{81} +850.964 q^{82} -491.830 q^{83} -208.830 q^{84} +924.381 q^{86} +434.498 q^{87} -342.026 q^{88} -415.560 q^{89} -622.794 q^{91} +1146.36 q^{92} +774.237 q^{93} -2438.53 q^{94} +765.286 q^{96} +1031.70 q^{97} +207.567 q^{98} -373.751 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 6 q^{3} + 2 q^{4} - 12 q^{6} + 14 q^{7} + 12 q^{8} + 18 q^{9} - 92 q^{11} - 6 q^{12} - 8 q^{13} + 28 q^{14} + 18 q^{16} + 44 q^{17} + 36 q^{18} - 108 q^{19} - 42 q^{21} - 164 q^{22} + 320 q^{23}+ \cdots - 828 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - 6 * q^3 + 2 * q^4 - 12 * q^6 + 14 * q^7 + 12 * q^8 + 18 * q^9 - 92 * q^11 - 6 * q^12 - 8 * q^13 + 28 * q^14 + 18 * q^16 + 44 * q^17 + 36 * q^18 - 108 * q^19 - 42 * q^21 - 164 * q^22 + 320 * q^23 - 36 * q^24 - 396 * q^26 - 54 * q^27 + 14 * q^28 - 236 * q^29 - 60 * q^31 - 300 * q^32 + 276 * q^33 + 528 * q^34 + 18 * q^36 - 204 * q^37 - 476 * q^38 + 24 * q^39 + 44 * q^41 - 84 * q^42 - 136 * q^43 - 12 * q^44 + 440 * q^46 - 400 * q^47 - 54 * q^48 + 98 * q^49 - 132 * q^51 - 1528 * q^52 - 16 * q^53 - 108 * q^54 + 84 * q^56 + 324 * q^57 - 592 * q^58 - 464 * q^59 - 684 * q^61 - 1140 * q^62 + 126 * q^63 - 1214 * q^64 + 492 * q^66 - 736 * q^67 + 1804 * q^68 - 960 * q^69 - 740 * q^71 + 108 * q^72 - 424 * q^73 - 168 * q^74 - 1148 * q^76 - 644 * q^77 + 1188 * q^78 - 408 * q^79 + 162 * q^81 + 888 * q^82 - 608 * q^83 - 42 * q^84 + 1008 * q^86 + 708 * q^87 - 532 * q^88 - 1332 * q^89 - 56 * q^91 - 480 * q^92 + 180 * q^93 - 2480 * q^94 + 900 * q^96 + 2448 * q^97 + 196 * q^98 - 828 * q^99

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 $

$2$	4.23607	1.49768	0.748838	−	0.662753i	$-0.230612\pi$
$2$	4.23607	1.49768	0.748838	+	0.662753i	$0.230612\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	9.94427	1.24303
$5$	0	0
$5$	0	0
$6$	−12.7082	−0.864684
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.23607	0.363986
$9$	9.00000	0.333333
$10$	0	0
$11$	−41.5279	−1.13828	−0.569142	−	0.822239i	$-0.692725\pi$
$11$	−41.5279	−1.13828	−0.569142	+	0.822239i	$0.692725\pi$
$12$	−29.8328	−0.717666
$13$	−88.9706	−1.89815	−0.949077	−	0.315044i	$-0.897981\pi$
$13$	−88.9706	−1.89815	−0.949077	+	0.315044i	$0.897981\pi$
$14$	29.6525	0.566068
$15$	0	0
$16$	−44.6656	−0.697900
$17$	120.387	1.71754	0.858769	−	0.512364i	$-0.171230\pi$
$17$	120.387	1.71754	0.858769	+	0.512364i	$0.171230\pi$
$18$	38.1246	0.499225
$19$	−112.138	−1.35401	−0.677004	−	0.735979i	$-0.736722\pi$
$19$	−112.138	−1.35401	−0.677004	+	0.735979i	$0.736722\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	−175.915	−1.70478
$23$	115.279	1.04510	0.522549	−	0.852609i	$-0.324981\pi$
$23$	115.279	1.04510	0.522549	+	0.852609i	$0.324981\pi$
$24$	−24.7082	−0.210148
$25$	0	0
$26$	−376.885	−2.84282
$27$	−27.0000	−0.192450
$28$	69.6099	0.469823
$29$	−144.833	−0.927406	−0.463703	−	0.885991i	$-0.653480\pi$
$29$	−144.833	−0.927406	−0.463703	+	0.885991i	$0.653480\pi$
$30$	0	0
$31$	−258.079	−1.49524	−0.747618	−	0.664128i	$-0.768803\pi$
$31$	−258.079	−1.49524	−0.747618	+	0.664128i	$0.768803\pi$
$32$	−255.095	−1.40922
$33$	124.584	0.657188
$34$	509.967	2.57231
$35$	0	0
$36$	89.4984	0.414345
$37$	−48.3344	−0.214760	−0.107380	−	0.994218i	$-0.534246\pi$
$37$	−48.3344	−0.214760	−0.107380	+	0.994218i	$0.534246\pi$
$38$	−475.023	−2.02787
$39$	266.912	1.09590
$40$	0	0
$41$	200.885	0.765196	0.382598	−	0.923915i	$-0.375029\pi$
$41$	200.885	0.765196	0.382598	+	0.923915i	$0.375029\pi$
$42$	−88.9574	−0.326820
$43$	218.217	0.773901	0.386950	−	0.922101i	$-0.373528\pi$
$43$	218.217	0.773901	0.386950	+	0.922101i	$0.373528\pi$
$44$	−412.964	−1.41493
$45$	0	0
$46$	488.328	1.56522
$47$	−575.659	−1.78657	−0.893283	−	0.449496i	$-0.851604\pi$
$47$	−575.659	−1.78657	−0.893283	+	0.449496i	$0.851604\pi$
$48$	133.997	0.402933
$49$	49.0000	0.142857
$50$	0	0
$51$	−361.161	−0.991621
$52$	−884.748	−2.35947
$53$	184.302	0.477657	0.238828	−	0.971062i	$-0.423237\pi$
$53$	184.302	0.477657	0.238828	+	0.971062i	$0.423237\pi$
$54$	−114.374	−0.288228
$55$	0	0
$56$	57.6525	0.137574
$57$	336.413	0.781737
$58$	−613.522	−1.38895
$59$	−151.502	−0.334302	−0.167151	−	0.985931i	$-0.553457\pi$
$59$	−151.502	−0.334302	−0.167151	+	0.985931i	$0.553457\pi$
$60$	0	0
$61$	−529.830	−1.11209	−0.556047	−	0.831151i	$-0.687683\pi$
$61$	−529.830	−1.11209	−0.556047	+	0.831151i	$0.687683\pi$
$62$	−1093.24	−2.23938
$63$	63.0000	0.125988
$64$	−723.276	−1.41265
$65$	0	0
$66$	527.745	0.984256
$67$	−1.28485	−0.00234283	−0.00117142	−	0.999999i	$-0.500373\pi$
$67$	−1.28485	−0.00234283	−0.00117142	+	0.999999i	$0.500373\pi$
$68$	1197.16	2.13496
$69$	−345.836	−0.603388
$70$	0	0
$71$	−61.4226	−0.102669	−0.0513347	−	0.998682i	$-0.516348\pi$
$71$	−61.4226	−0.102669	−0.0513347	+	0.998682i	$0.516348\pi$
$72$	74.1246	0.121329
$73$	−484.800	−0.777282	−0.388641	−	0.921389i	$-0.627055\pi$
$73$	−484.800	−0.777282	−0.388641	+	0.921389i	$0.627055\pi$
$74$	−204.748	−0.321641
$75$	0	0
$76$	−1115.13	−1.68308
$77$	−290.695	−0.430231
$78$	1130.66	1.64130
$79$	878.257	1.25078	0.625390	−	0.780312i	$-0.284940\pi$
$79$	878.257	1.25078	0.625390	+	0.780312i	$0.284940\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	850.964	1.14602
$83$	−491.830	−0.650426	−0.325213	−	0.945641i	$-0.605436\pi$
$83$	−491.830	−0.650426	−0.325213	+	0.945641i	$0.605436\pi$
$84$	−208.830	−0.271252
$85$	0	0
$86$	924.381	1.15905
$87$	434.498	0.535438
$88$	−342.026	−0.414320
$89$	−415.560	−0.494936	−0.247468	−	0.968896i	$-0.579599\pi$
$89$	−415.560	−0.494936	−0.247468	+	0.968896i	$0.579599\pi$
$90$	0	0
$91$	−622.794	−0.717435
$92$	1146.36	1.29909
$93$	774.237	0.863275
$94$	−2438.53	−2.67570
$95$	0	0
$96$	765.286	0.813611
$97$	1031.70	1.07993	0.539964	−	0.841688i	$-0.318438\pi$
$97$	1031.70	1.07993	0.539964	+	0.841688i	$0.318438\pi$
$98$	207.567	0.213954
$99$	−373.751	−0.379428
$100$	0	0
$101$	1447.19	1.42576	0.712878	−	0.701288i	$-0.247391\pi$
$101$	1447.19	1.42576	0.712878	+	0.701288i	$0.247391\pi$
$102$	−1529.90	−1.48513
$103$	163.567	0.156473	0.0782364	−	0.996935i	$-0.475071\pi$
$103$	163.567	0.156473	0.0782364	+	0.996935i	$0.475071\pi$
$104$	−732.768	−0.690902
$105$	0	0
$106$	780.715	0.715375
$107$	129.653	0.117141	0.0585703	−	0.998283i	$-0.481346\pi$
$107$	129.653	0.117141	0.0585703	+	0.998283i	$0.481346\pi$
$108$	−268.495	−0.239222
$109$	566.681	0.497965	0.248983	−	0.968508i	$-0.419904\pi$
$109$	566.681	0.497965	0.248983	+	0.968508i	$0.419904\pi$
$110$	0	0
$111$	145.003	0.123992
$112$	−312.659	−0.263782
$113$	−809.890	−0.674230	−0.337115	−	0.941463i	$-0.609451\pi$
$113$	−809.890	−0.674230	−0.337115	+	0.941463i	$0.609451\pi$
$114$	1425.07	1.17079
$115$	0	0
$116$	−1440.26	−1.15280
$117$	−800.735	−0.632718
$118$	−641.771	−0.500676
$119$	842.709	0.649168
$120$	0	0
$121$	393.563	0.295690
$122$	−2244.39	−1.66556
$123$	−602.656	−0.441786
$124$	−2566.41	−1.85863
$125$	0	0
$126$	266.872	0.188689
$127$	2584.25	1.80563	0.902816	−	0.430028i	$-0.141496\pi$
$127$	2584.25	1.80563	0.902816	+	0.430028i	$0.141496\pi$
$128$	−1023.08	−0.706473
$129$	−654.650	−0.446812
$130$	0	0
$131$	−1421.10	−0.947804	−0.473902	−	0.880578i	$-0.657155\pi$
$131$	−1421.10	−0.947804	−0.473902	+	0.880578i	$0.657155\pi$
$132$	1238.89	0.816908
$133$	−784.964	−0.511767
$134$	−5.44272	−0.00350880
$135$	0	0
$136$	991.515	0.625160
$137$	−104.878	−0.0654037	−0.0327019	−	0.999465i	$-0.510411\pi$
$137$	−104.878	−0.0654037	−0.0327019	+	0.999465i	$0.510411\pi$
$138$	−1464.98	−0.903679
$139$	−913.160	−0.557217	−0.278609	−	0.960405i	$-0.589873\pi$
$139$	−913.160	−0.557217	−0.278609	+	0.960405i	$0.589873\pi$
$140$	0	0
$141$	1726.98	1.03147
$142$	−260.190	−0.153765
$143$	3694.76	2.16064
$144$	−401.991	−0.232633
$145$	0	0
$146$	−2053.65	−1.16412
$147$	−147.000	−0.0824786
$148$	−480.650	−0.266954
$149$	1781.45	0.979476	0.489738	−	0.871870i	$-0.337092\pi$
$149$	1781.45	0.979476	0.489738	+	0.871870i	$0.337092\pi$
$150$	0	0
$151$	1407.53	0.758564	0.379282	−	0.925281i	$-0.376171\pi$
$151$	1407.53	0.758564	0.379282	+	0.925281i	$0.376171\pi$
$152$	−923.574	−0.492841
$153$	1083.48	0.572512
$154$	−1231.40	−0.644346
$155$	0	0
$156$	2654.24	1.36224
$157$	1598.94	0.812798	0.406399	−	0.913696i	$-0.366784\pi$
$157$	1598.94	0.812798	0.406399	+	0.913696i	$0.366784\pi$
$158$	3720.36	1.87326
$159$	−552.906	−0.275775
$160$	0	0
$161$	806.950	0.395010
$162$	343.122	0.166408
$163$	204.892	0.0984562	0.0492281	−	0.998788i	$-0.484324\pi$
$163$	204.892	0.0984562	0.0492281	+	0.998788i	$0.484324\pi$
$164$	1997.66	0.951165
$165$	0	0
$166$	−2083.42	−0.974127
$167$	1165.94	0.540259	0.270129	−	0.962824i	$-0.412933\pi$
$167$	1165.94	0.540259	0.270129	+	0.962824i	$0.412933\pi$
$168$	−172.957	−0.0794283
$169$	5718.76	2.60299
$170$	0	0
$171$	−1009.24	−0.451336
$172$	2170.01	0.961985
$173$	2538.00	1.11538	0.557690	−	0.830049i	$-0.311688\pi$
$173$	2538.00	1.11538	0.557690	+	0.830049i	$0.311688\pi$
$174$	1840.56	0.801913
$175$	0	0
$176$	1854.87	0.794409
$177$	454.505	0.193009
$178$	−1760.34	−0.741254
$179$	392.255	0.163791	0.0818954	−	0.996641i	$-0.473903\pi$
$179$	392.255	0.163791	0.0818954	+	0.996641i	$0.473903\pi$
$180$	0	0
$181$	−2978.08	−1.22298	−0.611489	−	0.791253i	$-0.709429\pi$
$181$	−2978.08	−1.22298	−0.611489	+	0.791253i	$0.709429\pi$
$182$	−2638.20	−1.07448
$183$	1589.49	0.642068
$184$	949.443	0.380401
$185$	0	0
$186$	3279.72	1.29291
$187$	−4999.41	−1.95504
$188$	−5724.51	−2.22076
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−1097.37	−0.415722	−0.207861	−	0.978158i	$-0.566650\pi$
$191$	−1097.37	−0.415722	−0.207861	+	0.978158i	$0.566650\pi$
$192$	2169.83	0.815592
$193$	−3500.31	−1.30548	−0.652740	−	0.757582i	$-0.726381\pi$
$193$	−3500.31	−1.30548	−0.652740	+	0.757582i	$0.726381\pi$
$194$	4370.34	1.61738
$195$	0	0
$196$	487.269	0.177576
$197$	−1573.96	−0.569237	−0.284618	−	0.958641i	$-0.591867\pi$
$197$	−1573.96	−0.569237	−0.284618	+	0.958641i	$0.591867\pi$
$198$	−1583.23	−0.568260
$199$	−3396.62	−1.20995	−0.604976	−	0.796244i	$-0.706817\pi$
$199$	−3396.62	−1.20995	−0.604976	+	0.796244i	$0.706817\pi$
$200$	0	0
$201$	3.85456	0.00135263
$202$	6130.42	2.13532
$203$	−1013.83	−0.350527
$204$	−3591.48	−1.23262
$205$	0	0
$206$	692.879	0.234346
$207$	1037.51	0.348366
$208$	3973.93	1.32472
$209$	4656.84	1.54125
$210$	0	0
$211$	3337.81	1.08903	0.544513	−	0.838753i	$-0.316715\pi$
$211$	3337.81	1.08903	0.544513	+	0.838753i	$0.316715\pi$
$212$	1832.75	0.593744
$213$	184.268	0.0592762
$214$	549.220	0.175439
$215$	0	0
$216$	−222.374	−0.0700492
$217$	−1806.55	−0.565146
$218$	2400.50	0.745791
$219$	1454.40	0.448764
$220$	0	0
$221$	−10710.9	−3.26015
$222$	614.243	0.185700
$223$	−127.328	−0.0382356	−0.0191178	−	0.999817i	$-0.506086\pi$
$223$	−127.328	−0.0382356	−0.0191178	+	0.999817i	$0.506086\pi$
$224$	−1785.67	−0.532633
$225$	0	0
$226$	−3430.75	−1.00978
$227$	−3844.12	−1.12398	−0.561990	−	0.827144i	$-0.689964\pi$
$227$	−3844.12	−1.12398	−0.561990	+	0.827144i	$0.689964\pi$
$228$	3345.39	0.971726
$229$	2536.95	0.732080	0.366040	−	0.930599i	$-0.380713\pi$
$229$	2536.95	0.732080	0.366040	+	0.930599i	$0.380713\pi$
$230$	0	0
$231$	872.085	0.248394
$232$	−1192.85	−0.337563
$233$	−3987.44	−1.12114	−0.560570	−	0.828107i	$-0.689418\pi$
$233$	−3987.44	−1.12114	−0.560570	+	0.828107i	$0.689418\pi$
$234$	−3391.97	−0.947607
$235$	0	0
$236$	−1506.57	−0.415549
$237$	−2634.77	−0.722138
$238$	3569.77	0.972244
$239$	−3367.18	−0.911317	−0.455659	−	0.890155i	$-0.650596\pi$
$239$	−3367.18	−0.911317	−0.455659	+	0.890155i	$0.650596\pi$
$240$	0	0
$241$	−939.551	−0.251128	−0.125564	−	0.992086i	$-0.540074\pi$
$241$	−939.551	−0.251128	−0.125564	+	0.992086i	$0.540074\pi$
$242$	1667.16	0.442848
$243$	−243.000	−0.0641500
$244$	−5268.77	−1.38237
$245$	0	0
$246$	−2552.89	−0.661653
$247$	9976.96	2.57012
$248$	−2125.56	−0.544246
$249$	1475.49	0.375523
$250$	0	0
$251$	−1403.96	−0.353056	−0.176528	−	0.984296i	$-0.556487\pi$
$251$	−1403.96	−0.353056	−0.176528	+	0.984296i	$0.556487\pi$
$252$	626.489	0.156608
$253$	−4787.28	−1.18962
$254$	10947.1	2.70425
$255$	0	0
$256$	1452.36	0.354579
$257$	1964.86	0.476905	0.238453	−	0.971154i	$-0.423360\pi$
$257$	1964.86	0.476905	0.238453	+	0.971154i	$0.423360\pi$
$258$	−2773.14	−0.669179
$259$	−338.341	−0.0811717
$260$	0	0
$261$	−1303.50	−0.309135
$262$	−6019.89	−1.41950
$263$	393.821	0.0923347	0.0461673	−	0.998934i	$-0.485299\pi$
$263$	393.821	0.0923347	0.0461673	+	0.998934i	$0.485299\pi$
$264$	1026.08	0.239208
$265$	0	0
$266$	−3325.16	−0.766462
$267$	1246.68	0.285751
$268$	−12.7769	−0.00291222
$269$	−1877.03	−0.425444	−0.212722	−	0.977113i	$-0.568233\pi$
$269$	−1877.03	−0.425444	−0.212722	+	0.977113i	$0.568233\pi$
$270$	0	0
$271$	−689.909	−0.154646	−0.0773228	−	0.997006i	$-0.524637\pi$
$271$	−689.909	−0.154646	−0.0773228	+	0.997006i	$0.524637\pi$
$272$	−5377.16	−1.19867
$273$	1868.38	0.414211
$274$	−444.269	−0.0979536
$275$	0	0
$276$	−3439.09	−0.750031
$277$	−6289.13	−1.36418	−0.682088	−	0.731270i	$-0.738928\pi$
$277$	−6289.13	−1.36418	−0.682088	+	0.731270i	$0.738928\pi$
$278$	−3868.21	−0.834531
$279$	−2322.71	−0.498412
$280$	0	0
$281$	−1954.87	−0.415010	−0.207505	−	0.978234i	$-0.566534\pi$
$281$	−1954.87	−0.415010	−0.207505	+	0.978234i	$0.566534\pi$
$282$	7315.60	1.54481
$283$	−5033.96	−1.05738	−0.528688	−	0.848816i	$-0.677316\pi$
$283$	−5033.96	−1.05738	−0.528688	+	0.848816i	$0.677316\pi$
$284$	−610.803	−0.127621
$285$	0	0
$286$	15651.2	3.23594
$287$	1406.20	0.289217
$288$	−2295.86	−0.469738
$289$	9580.03	1.94993
$290$	0	0
$291$	−3095.09	−0.623497
$292$	−4820.99	−0.966188
$293$	−6369.12	−1.26993	−0.634963	−	0.772543i	$-0.718985\pi$
$293$	−6369.12	−1.26993	−0.634963	+	0.772543i	$0.718985\pi$
$294$	−622.702	−0.123526
$295$	0	0
$296$	−398.085	−0.0781697
$297$	1121.25	0.219063
$298$	7546.34	1.46694
$299$	−10256.4	−1.98376
$300$	0	0
$301$	1527.52	0.292507
$302$	5962.39	1.13608
$303$	−4341.58	−0.823160
$304$	5008.70	0.944963
$305$	0	0
$306$	4589.71	0.857438
$307$	6619.83	1.23066	0.615332	−	0.788268i	$-0.289022\pi$
$307$	6619.83	1.23066	0.615332	+	0.788268i	$0.289022\pi$
$308$	−2890.75	−0.534792
$309$	−490.700	−0.0903396
$310$	0	0
$311$	−9909.22	−1.80675	−0.903377	−	0.428848i	$-0.858920\pi$
$311$	−9909.22	−1.80675	−0.903377	+	0.428848i	$0.858920\pi$
$312$	2198.30	0.398892
$313$	422.336	0.0762678	0.0381339	−	0.999273i	$-0.487859\pi$
$313$	422.336	0.0762678	0.0381339	+	0.999273i	$0.487859\pi$
$314$	6773.22	1.21731
$315$	0	0
$316$	8733.63	1.55476
$317$	4902.78	0.868668	0.434334	−	0.900752i	$-0.356984\pi$
$317$	4902.78	0.868668	0.434334	+	0.900752i	$0.356984\pi$
$318$	−2342.15	−0.413022
$319$	6014.60	1.05565
$320$	0	0
$321$	−388.960	−0.0676312
$322$	3418.30	0.591597
$323$	−13499.9	−2.32556
$324$	805.486	0.138115
$325$	0	0
$326$	867.935	0.147455
$327$	−1700.04	−0.287500
$328$	1654.51	0.278521
$329$	−4029.62	−0.675258
$330$	0	0
$331$	−5281.74	−0.877071	−0.438535	−	0.898714i	$-0.644503\pi$
$331$	−5281.74	−0.877071	−0.438535	+	0.898714i	$0.644503\pi$
$332$	−4890.89	−0.808501
$333$	−435.009	−0.0715867
$334$	4939.01	0.809133
$335$	0	0
$336$	937.978	0.152294
$337$	−4459.60	−0.720860	−0.360430	−	0.932786i	$-0.617370\pi$
$337$	−4459.60	−0.720860	−0.360430	+	0.932786i	$0.617370\pi$
$338$	24225.1	3.89843
$339$	2429.67	0.389267
$340$	0	0
$341$	10717.5	1.70200
$342$	−4275.21	−0.675956
$343$	343.000	0.0539949
$344$	1797.25	0.281689
$345$	0	0
$346$	10751.2	1.67048
$347$	−5261.97	−0.814056	−0.407028	−	0.913416i	$-0.633435\pi$
$347$	−5261.97	−0.814056	−0.407028	+	0.913416i	$0.633435\pi$
$348$	4320.77	0.665568
$349$	960.325	0.147292	0.0736461	−	0.997284i	$-0.476536\pi$
$349$	960.325	0.147292	0.0736461	+	0.997284i	$0.476536\pi$
$350$	0	0
$351$	2402.21	0.365300
$352$	10593.6	1.60409
$353$	8925.80	1.34581	0.672907	−	0.739727i	$-0.265045\pi$
$353$	8925.80	1.34581	0.672907	+	0.739727i	$0.265045\pi$
$354$	1925.31	0.289066
$355$	0	0
$356$	−4132.45	−0.615222
$357$	−2528.13	−0.374797
$358$	1661.62	0.245306
$359$	−3056.27	−0.449314	−0.224657	−	0.974438i	$-0.572126\pi$
$359$	−3056.27	−0.449314	−0.224657	+	0.974438i	$0.572126\pi$
$360$	0	0
$361$	5715.88	0.833340
$362$	−12615.4	−1.83162
$363$	−1180.69	−0.170717
$364$	−6193.23	−0.891796
$365$	0	0
$366$	6733.18	0.961610
$367$	1813.52	0.257943	0.128971	−	0.991648i	$-0.458832\pi$
$367$	1813.52	0.257943	0.128971	+	0.991648i	$0.458832\pi$
$368$	−5148.99	−0.729375
$369$	1807.97	0.255065
$370$	0	0
$371$	1290.11	0.180537
$372$	7699.22	1.07308
$373$	4517.48	0.627094	0.313547	−	0.949573i	$-0.398483\pi$
$373$	4517.48	0.627094	0.313547	+	0.949573i	$0.398483\pi$
$374$	−21177.9	−2.92802
$375$	0	0
$376$	−4741.17	−0.650285
$377$	12885.9	1.76036
$378$	−800.617	−0.108940
$379$	−4931.24	−0.668340	−0.334170	−	0.942513i	$-0.608456\pi$
$379$	−4931.24	−0.668340	−0.334170	+	0.942513i	$0.608456\pi$
$380$	0	0
$381$	−7752.75	−1.04248
$382$	−4648.53	−0.622617
$383$	1482.37	0.197770	0.0988849	−	0.995099i	$-0.468472\pi$
$383$	1482.37	0.197770	0.0988849	+	0.995099i	$0.468472\pi$
$384$	3069.25	0.407883
$385$	0	0
$386$	−14827.5	−1.95519
$387$	1963.95	0.257967
$388$	10259.5	1.34239
$389$	−5448.98	−0.710217	−0.355109	−	0.934825i	$-0.615556\pi$
$389$	−5448.98	−0.710217	−0.355109	+	0.934825i	$0.615556\pi$
$390$	0	0
$391$	13878.0	1.79500
$392$	403.567	0.0519980
$393$	4263.31	0.547215
$394$	−6667.38	−0.852532
$395$	0	0
$396$	−3716.68	−0.471642
$397$	13675.9	1.72891	0.864453	−	0.502713i	$-0.167665\pi$
$397$	13675.9	1.72891	0.864453	+	0.502713i	$0.167665\pi$
$398$	−14388.3	−1.81212
$399$	2354.89	0.295469
$400$	0	0
$401$	14109.9	1.75714	0.878570	−	0.477613i	$-0.158498\pi$
$401$	14109.9	1.75714	0.878570	+	0.477613i	$0.158498\pi$
$402$	16.3282	0.00202581
$403$	22961.4	2.83819
$404$	14391.3	1.77226
$405$	0	0
$406$	−4294.65	−0.524975
$407$	2007.22	0.244458
$408$	−2974.55	−0.360936
$409$	−13995.6	−1.69203	−0.846015	−	0.533159i	$-0.821005\pi$
$409$	−13995.6	−1.69203	−0.846015	+	0.533159i	$0.821005\pi$
$410$	0	0
$411$	314.633	0.0377609
$412$	1626.55	0.194501
$413$	−1060.51	−0.126354
$414$	4394.95	0.521740
$415$	0	0
$416$	22696.0	2.67491
$417$	2739.48	0.321709
$418$	19726.7	2.30829
$419$	−9840.61	−1.14736	−0.573682	−	0.819078i	$-0.694485\pi$
$419$	−9840.61	−1.14736	−0.573682	+	0.819078i	$0.694485\pi$
$420$	0	0
$421$	−12660.5	−1.46564	−0.732822	−	0.680420i	$-0.761797\pi$
$421$	−12660.5	−1.46564	−0.732822	+	0.680420i	$0.761797\pi$
$422$	14139.2	1.63101
$423$	−5180.93	−0.595522
$424$	1517.92	0.173860
$425$	0	0
$426$	780.571	0.0887765
$427$	−3708.81	−0.420332
$428$	1289.31	0.145610
$429$	−11084.3	−1.24744
$430$	0	0
$431$	−4578.91	−0.511736	−0.255868	−	0.966712i	$-0.582361\pi$
$431$	−4578.91	−0.511736	−0.255868	+	0.966712i	$0.582361\pi$
$432$	1205.97	0.134311
$433$	3279.88	0.364020	0.182010	−	0.983297i	$-0.441740\pi$
$433$	3279.88	0.364020	0.182010	+	0.983297i	$0.441740\pi$
$434$	−7652.68	−0.846406
$435$	0	0
$436$	5635.23	0.618988
$437$	−12927.1	−1.41507
$438$	6160.94	0.672103
$439$	−427.807	−0.0465105	−0.0232552	−	0.999730i	$-0.507403\pi$
$439$	−427.807	−0.0465105	−0.0232552	+	0.999730i	$0.507403\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	−45372.1	−4.88265
$443$	−15441.2	−1.65605	−0.828027	−	0.560688i	$-0.810537\pi$
$443$	−15441.2	−1.65605	−0.828027	+	0.560688i	$0.810537\pi$
$444$	1441.95	0.154126
$445$	0	0
$446$	−539.371	−0.0572645
$447$	−5344.35	−0.565501
$448$	−5062.93	−0.533931
$449$	9382.02	0.986113	0.493057	−	0.869997i	$-0.335880\pi$
$449$	9382.02	0.986113	0.493057	+	0.869997i	$0.335880\pi$
$450$	0	0
$451$	−8342.34	−0.871010
$452$	−8053.77	−0.838091
$453$	−4222.59	−0.437957
$454$	−16284.0	−1.68336
$455$	0	0
$456$	2770.72	0.284542
$457$	−13570.4	−1.38905	−0.694524	−	0.719469i	$-0.744385\pi$
$457$	−13570.4	−1.38905	−0.694524	+	0.719469i	$0.744385\pi$
$458$	10746.7	1.09642
$459$	−3250.45	−0.330540
$460$	0	0
$461$	1251.88	0.126477	0.0632386	−	0.997998i	$-0.479857\pi$
$461$	1251.88	0.126477	0.0632386	+	0.997998i	$0.479857\pi$
$462$	3694.21	0.372014
$463$	−7934.36	−0.796417	−0.398209	−	0.917295i	$-0.630368\pi$
$463$	−7934.36	−0.796417	−0.398209	+	0.917295i	$0.630368\pi$
$464$	6469.05	0.647237
$465$	0	0
$466$	−16891.1	−1.67911
$467$	−7583.76	−0.751466	−0.375733	−	0.926728i	$-0.622609\pi$
$467$	−7583.76	−0.751466	−0.375733	+	0.926728i	$0.622609\pi$
$468$	−7962.73	−0.786490
$469$	−8.99396	−0.000885507 0
$470$	0	0
$471$	−4796.82	−0.469269
$472$	−1247.78	−0.121681
$473$	−9062.07	−0.880919
$474$	−11161.1	−1.08153
$475$	0	0
$476$	8380.13	0.806938
$477$	1658.72	0.159219
$478$	−14263.6	−1.36486
$479$	−5829.34	−0.556053	−0.278027	−	0.960573i	$-0.589680\pi$
$479$	−5829.34	−0.556053	−0.278027	+	0.960573i	$0.589680\pi$
$480$	0	0
$481$	4300.34	0.407648
$482$	−3980.00	−0.376108
$483$	−2420.85	−0.228059
$484$	3913.70	0.367553
$485$	0	0
$486$	−1029.36	−0.0960760
$487$	19902.1	1.85185	0.925925	−	0.377708i	$-0.123288\pi$
$487$	19902.1	1.85185	0.925925	+	0.377708i	$0.123288\pi$
$488$	−4363.71	−0.404787
$489$	−614.675	−0.0568437
$490$	0	0
$491$	−16821.6	−1.54613	−0.773065	−	0.634327i	$-0.781277\pi$
$491$	−16821.6	−1.54613	−0.773065	+	0.634327i	$0.781277\pi$
$492$	−5992.98	−0.549155
$493$	−17436.0	−1.59285
$494$	42263.1	3.84920
$495$	0	0
$496$	11527.3	1.04353
$497$	−429.958	−0.0388054
$498$	6250.27	0.562412
$499$	6031.83	0.541126	0.270563	−	0.962702i	$-0.412790\pi$
$499$	6031.83	0.541126	0.270563	+	0.962702i	$0.412790\pi$
$500$	0	0
$501$	−3497.82	−0.311919
$502$	−5947.27	−0.528764
$503$	17176.4	1.52258	0.761290	−	0.648412i	$-0.224566\pi$
$503$	17176.4	1.52258	0.761290	+	0.648412i	$0.224566\pi$
$504$	518.872	0.0458580
$505$	0	0
$506$	−20279.2	−1.78166
$507$	−17156.3	−1.50284
$508$	25698.5	2.24446
$509$	4706.59	0.409854	0.204927	−	0.978777i	$-0.434304\pi$
$509$	4706.59	0.409854	0.204927	+	0.978777i	$0.434304\pi$
$510$	0	0
$511$	−3393.60	−0.293785
$512$	14336.9	1.23752
$513$	3027.72	0.260579
$514$	8323.28	0.714250
$515$	0	0
$516$	−6510.02	−0.555402
$517$	23905.9	2.03362
$518$	−1433.23	−0.121569
$519$	−7614.01	−0.643965
$520$	0	0
$521$	−8557.18	−0.719572	−0.359786	−	0.933035i	$-0.617150\pi$
$521$	−8557.18	−0.719572	−0.359786	+	0.933035i	$0.617150\pi$
$522$	−5521.69	−0.462985
$523$	−18248.5	−1.52572	−0.762858	−	0.646566i	$-0.776204\pi$
$523$	−18248.5	−1.52572	−0.762858	+	0.646566i	$0.776204\pi$
$524$	−14131.8	−1.17815
$525$	0	0
$526$	1668.25	0.138287
$527$	−31069.3	−2.56813
$528$	−5564.60	−0.458652
$529$	1122.16	0.0922302
$530$	0	0
$531$	−1363.51	−0.111434
$532$	−7805.90	−0.636144
$533$	−17872.9	−1.45246
$534$	5281.03	0.427963
$535$	0	0
$536$	−10.5821	−0.000852758 0
$537$	−1176.77	−0.0945646
$538$	−7951.22	−0.637177
$539$	−2034.87	−0.162612
$540$	0	0
$541$	−5734.17	−0.455696	−0.227848	−	0.973697i	$-0.573169\pi$
$541$	−5734.17	−0.455696	−0.227848	+	0.973697i	$0.573169\pi$
$542$	−2922.50	−0.231609
$543$	8934.24	0.706087
$544$	−30710.1	−2.42038
$545$	0	0
$546$	7914.59	0.620354
$547$	8002.52	0.625527	0.312763	−	0.949831i	$-0.398745\pi$
$547$	8002.52	0.625527	0.312763	+	0.949831i	$0.398745\pi$
$548$	−1042.93	−0.0812991
$549$	−4768.47	−0.370698
$550$	0	0
$551$	16241.2	1.25572
$552$	−2848.33	−0.219625
$553$	6147.80	0.472750
$554$	−26641.2	−2.04310
$555$	0	0
$556$	−9080.71	−0.692640
$557$	−1276.82	−0.0971289	−0.0485644	−	0.998820i	$-0.515465\pi$
$557$	−1276.82	−0.0971289	−0.0485644	+	0.998820i	$0.515465\pi$
$558$	−9839.16	−0.746460
$559$	−19414.9	−1.46898
$560$	0	0
$561$	14998.2	1.12875
$562$	−8280.96	−0.621550
$563$	−11027.7	−0.825507	−0.412753	−	0.910843i	$-0.635433\pi$
$563$	−11027.7	−0.825507	−0.412753	+	0.910843i	$0.635433\pi$
$564$	17173.5	1.28216
$565$	0	0
$566$	−21324.2	−1.58361
$567$	567.000	0.0419961
$568$	−505.881	−0.0373702
$569$	−4519.03	−0.332948	−0.166474	−	0.986046i	$-0.553238\pi$
$569$	−4519.03	−0.332948	−0.166474	+	0.986046i	$0.553238\pi$
$570$	0	0
$571$	3598.81	0.263758	0.131879	−	0.991266i	$-0.457899\pi$
$571$	3598.81	0.263758	0.131879	+	0.991266i	$0.457899\pi$
$572$	36741.7	2.68575
$573$	3292.11	0.240017
$574$	5956.75	0.433153
$575$	0	0
$576$	−6509.48	−0.470883
$577$	3439.23	0.248140	0.124070	−	0.992273i	$-0.460405\pi$
$577$	3439.23	0.248140	0.124070	+	0.992273i	$0.460405\pi$
$578$	40581.6	2.92037
$579$	10500.9	0.753720
$580$	0	0
$581$	−3442.81	−0.245838
$582$	−13111.0	−0.933797
$583$	−7653.66	−0.543709
$584$	−3992.85	−0.282920
$585$	0	0
$586$	−26980.0	−1.90194
$587$	−21285.2	−1.49665	−0.748327	−	0.663330i	$-0.769143\pi$
$587$	−21285.2	−1.49665	−0.748327	+	0.663330i	$0.769143\pi$
$588$	−1461.81	−0.102524
$589$	28940.4	2.02456
$590$	0	0
$591$	4721.87	0.328649
$592$	2158.89	0.149881
$593$	14200.8	0.983404	0.491702	−	0.870764i	$-0.336375\pi$
$593$	14200.8	0.983404	0.491702	+	0.870764i	$0.336375\pi$
$594$	4749.70	0.328085
$595$	0	0
$596$	17715.2	1.21752
$597$	10189.9	0.698566
$598$	−43446.8	−2.97103
$599$	−8885.05	−0.606065	−0.303033	−	0.952980i	$-0.597999\pi$
$599$	−8885.05	−0.606065	−0.303033	+	0.952980i	$0.597999\pi$
$600$	0	0
$601$	−2052.89	−0.139333	−0.0696664	−	0.997570i	$-0.522193\pi$
$601$	−2052.89	−0.139333	−0.0696664	+	0.997570i	$0.522193\pi$
$602$	6470.67	0.438081
$603$	−11.5637	−0.000780943 0
$604$	13996.9	0.942920
$605$	0	0
$606$	−18391.2	−1.23283
$607$	−10280.0	−0.687404	−0.343702	−	0.939079i	$-0.611681\pi$
$607$	−10280.0	−0.687404	−0.343702	+	0.939079i	$0.611681\pi$
$608$	28605.8	1.90809
$609$	3041.49	0.202377
$610$	0	0
$611$	51216.8	3.39118
$612$	10774.4	0.711652
$613$	−23409.5	−1.54242	−0.771208	−	0.636584i	$-0.780347\pi$
$613$	−23409.5	−1.54242	−0.771208	+	0.636584i	$0.780347\pi$
$614$	28042.0	1.84314
$615$	0	0
$616$	−2394.18	−0.156598
$617$	6632.75	0.432779	0.216389	−	0.976307i	$-0.430572\pi$
$617$	6632.75	0.432779	0.216389	+	0.976307i	$0.430572\pi$
$618$	−2078.64	−0.135299
$619$	10734.0	0.696990	0.348495	−	0.937311i	$-0.386693\pi$
$619$	10734.0	0.696990	0.348495	+	0.937311i	$0.386693\pi$
$620$	0	0
$621$	−3112.52	−0.201129
$622$	−41976.1	−2.70593
$623$	−2908.92	−0.187068
$624$	−11921.8	−0.764829
$625$	0	0
$626$	1789.04	0.114225
$627$	−13970.5	−0.889839
$628$	15900.3	1.01034
$629$	−5818.83	−0.368858
$630$	0	0
$631$	−17071.0	−1.07700	−0.538499	−	0.842626i	$-0.681008\pi$
$631$	−17071.0	−1.07700	−0.538499	+	0.842626i	$0.681008\pi$
$632$	7233.38	0.455267
$633$	−10013.4	−0.628749
$634$	20768.5	1.30098
$635$	0	0
$636$	−5498.24	−0.342798
$637$	−4359.56	−0.271165
$638$	25478.2	1.58102
$639$	−552.804	−0.0342231
$640$	0	0
$641$	−19389.7	−1.19477	−0.597386	−	0.801954i	$-0.703794\pi$
$641$	−19389.7	−1.19477	−0.597386	+	0.801954i	$0.703794\pi$
$642$	−1647.66	−0.101290
$643$	25409.3	1.55839	0.779196	−	0.626780i	$-0.215628\pi$
$643$	25409.3	1.55839	0.779196	+	0.626780i	$0.215628\pi$
$644$	8024.54	0.491011
$645$	0	0
$646$	−57186.6	−3.48294
$647$	6039.08	0.366956	0.183478	−	0.983024i	$-0.441264\pi$
$647$	6039.08	0.366956	0.183478	+	0.983024i	$0.441264\pi$
$648$	667.122	0.0404429
$649$	6291.54	0.380531
$650$	0	0
$651$	5419.66	0.326287
$652$	2037.50	0.122384
$653$	30666.2	1.83776	0.918882	−	0.394532i	$-0.129093\pi$
$653$	30666.2	1.83776	0.918882	+	0.394532i	$0.129093\pi$
$654$	−7201.50	−0.430582
$655$	0	0
$656$	−8972.67	−0.534031
$657$	−4363.20	−0.259094
$658$	−17069.7	−1.01132
$659$	−2765.96	−0.163500	−0.0817500	−	0.996653i	$-0.526051\pi$
$659$	−2765.96	−0.163500	−0.0817500	+	0.996653i	$0.526051\pi$
$660$	0	0
$661$	27261.8	1.60418	0.802089	−	0.597204i	$-0.203722\pi$
$661$	27261.8	1.60418	0.802089	+	0.597204i	$0.203722\pi$
$662$	−22373.8	−1.31357
$663$	32132.7	1.88225
$664$	−4050.74	−0.236746
$665$	0	0
$666$	−1842.73	−0.107214
$667$	−16696.1	−0.969230
$668$	11594.4	0.671560
$669$	381.985	0.0220753
$670$	0	0
$671$	22002.7	1.26588
$672$	5357.00	0.307516
$673$	1048.17	0.0600356	0.0300178	−	0.999549i	$-0.490444\pi$
$673$	1048.17	0.0600356	0.0300178	+	0.999549i	$0.490444\pi$
$674$	−18891.2	−1.07961
$675$	0	0
$676$	56869.0	3.23560
$677$	34554.7	1.96166	0.980831	−	0.194860i	$-0.0624252\pi$
$677$	34554.7	1.96166	0.980831	+	0.194860i	$0.0624252\pi$
$678$	10292.2	0.582996
$679$	7221.89	0.408175
$680$	0	0
$681$	11532.4	0.648930
$682$	45399.9	2.54905
$683$	−14711.6	−0.824192	−0.412096	−	0.911140i	$-0.635203\pi$
$683$	−14711.6	−0.824192	−0.412096	+	0.911140i	$0.635203\pi$
$684$	−10036.2	−0.561026
$685$	0	0
$686$	1452.97	0.0808669
$687$	−7610.85	−0.422667
$688$	−9746.79	−0.540106
$689$	−16397.4	−0.906666
$690$	0	0
$691$	−24522.6	−1.35005	−0.675024	−	0.737796i	$-0.735867\pi$
$691$	−24522.6	−1.35005	−0.675024	+	0.737796i	$0.735867\pi$
$692$	25238.6	1.38646
$693$	−2616.26	−0.143410
$694$	−22290.1	−1.21919
$695$	0	0
$696$	3578.56	0.194892
$697$	24184.0	1.31425
$698$	4068.00	0.220596
$699$	11962.3	0.647291
$700$	0	0
$701$	19912.2	1.07286	0.536429	−	0.843946i	$-0.319773\pi$
$701$	19912.2	1.07286	0.536429	+	0.843946i	$0.319773\pi$
$702$	10175.9	0.547101
$703$	5420.11	0.290787
$704$	30036.1	1.60799
$705$	0	0
$706$	37810.3	2.01559
$707$	10130.4	0.538885
$708$	4519.72	0.239917
$709$	6208.79	0.328880	0.164440	−	0.986387i	$-0.447418\pi$
$709$	6208.79	0.328880	0.164440	+	0.986387i	$0.447418\pi$
$710$	0	0
$711$	7904.31	0.416927
$712$	−3422.58	−0.180150
$713$	−29751.0	−1.56267
$714$	−10709.3	−0.561325
$715$	0	0
$716$	3900.69	0.203597
$717$	10101.5	0.526149
$718$	−12946.6	−0.672927
$719$	13063.6	0.677593	0.338797	−	0.940860i	$-0.389980\pi$
$719$	13063.6	0.677593	0.338797	+	0.940860i	$0.389980\pi$
$720$	0	0
$721$	1144.97	0.0591411
$722$	24212.9	1.24807
$723$	2818.65	0.144989
$724$	−29614.8	−1.52020
$725$	0	0
$726$	−5001.49	−0.255678
$727$	12897.0	0.657940	0.328970	−	0.944340i	$-0.393298\pi$
$727$	12897.0	0.657940	0.328970	+	0.944340i	$0.393298\pi$
$728$	−5129.37	−0.261136
$729$	729.000	0.0370370
$730$	0	0
$731$	26270.5	1.32920
$732$	15806.3	0.798112
$733$	−11699.6	−0.589540	−0.294770	−	0.955568i	$-0.595243\pi$
$733$	−11699.6	−0.589540	−0.294770	+	0.955568i	$0.595243\pi$
$734$	7682.19	0.386315
$735$	0	0
$736$	−29407.0	−1.47277
$737$	53.3571	0.00266681
$738$	7658.68	0.382005
$739$	14974.0	0.745368	0.372684	−	0.927958i	$-0.378438\pi$
$739$	14974.0	0.745368	0.372684	+	0.927958i	$0.378438\pi$
$740$	0	0
$741$	−29930.9	−1.48386
$742$	5465.01	0.270386
$743$	−18500.7	−0.913492	−0.456746	−	0.889597i	$-0.650985\pi$
$743$	−18500.7	−0.913492	−0.456746	+	0.889597i	$0.650985\pi$
$744$	6376.67	0.314220
$745$	0	0
$746$	19136.3	0.939184
$747$	−4426.47	−0.216809
$748$	−49715.5	−2.43019
$749$	907.572	0.0442750
$750$	0	0
$751$	−26348.4	−1.28025	−0.640125	−	0.768271i	$-0.721117\pi$
$751$	−26348.4	−1.28025	−0.640125	+	0.768271i	$0.721117\pi$
$752$	25712.2	1.24684
$753$	4211.88	0.203837
$754$	54585.4	2.63645
$755$	0	0
$756$	−1879.47	−0.0904174
$757$	28061.7	1.34732	0.673659	−	0.739042i	$-0.264722\pi$
$757$	28061.7	1.34732	0.673659	+	0.739042i	$0.264722\pi$
$758$	−20889.1	−1.00096
$759$	14361.8	0.686826
$760$	0	0
$761$	−3579.22	−0.170495	−0.0852476	−	0.996360i	$-0.527168\pi$
$761$	−3579.22	−0.170495	−0.0852476	+	0.996360i	$0.527168\pi$
$762$	−32841.2	−1.56130
$763$	3966.77	0.188213
$764$	−10912.5	−0.516757
$765$	0	0
$766$	6279.44	0.296195
$767$	13479.2	0.634557
$768$	−4357.07	−0.204716
$769$	4339.61	0.203499	0.101749	−	0.994810i	$-0.467556\pi$
$769$	4339.61	0.203499	0.101749	+	0.994810i	$0.467556\pi$
$770$	0	0
$771$	−5894.58	−0.275341
$772$	−34808.0	−1.62276
$773$	−10005.1	−0.465537	−0.232769	−	0.972532i	$-0.574778\pi$
$773$	−10005.1	−0.465537	−0.232769	+	0.972532i	$0.574778\pi$
$774$	8319.43	0.386351
$775$	0	0
$776$	8497.14	0.393079
$777$	1015.02	0.0468645
$778$	−23082.3	−1.06368
$779$	−22526.8	−1.03608
$780$	0	0
$781$	2550.75	0.116867
$782$	58788.4	2.68832
$783$	3910.49	0.178479
$784$	−2188.62	−0.0997001
$785$	0	0
$786$	18059.7	0.819550
$787$	−17826.8	−0.807443	−0.403721	−	0.914882i	$-0.632284\pi$
$787$	−17826.8	−0.807443	−0.403721	+	0.914882i	$0.632284\pi$
$788$	−15651.8	−0.707581
$789$	−1181.46	−0.0533094
$790$	0	0
$791$	−5669.23	−0.254835
$792$	−3078.24	−0.138107
$793$	47139.3	2.11093
$794$	57932.2	2.58934
$795$	0	0
$796$	−33777.0	−1.50401
$797$	−36723.0	−1.63211	−0.816057	−	0.577971i	$-0.803845\pi$
$797$	−36723.0	−1.63211	−0.816057	+	0.577971i	$0.803845\pi$
$798$	9975.49	0.442517
$799$	−69301.9	−3.06849
$800$	0	0
$801$	−3740.04	−0.164979
$802$	59770.4	2.63163
$803$	20132.7	0.884767
$804$	38.3307	0.00168137
$805$	0	0
$806$	97266.2	4.25069
$807$	5631.08	0.245630
$808$	11919.2	0.518955
$809$	5657.55	0.245870	0.122935	−	0.992415i	$-0.460769\pi$
$809$	5657.55	0.245870	0.122935	+	0.992415i	$0.460769\pi$
$810$	0	0
$811$	7532.41	0.326139	0.163070	−	0.986615i	$-0.447861\pi$
$811$	7532.41	0.326139	0.163070	+	0.986615i	$0.447861\pi$
$812$	−10081.8	−0.435716
$813$	2069.73	0.0892847
$814$	8502.73	0.366119
$815$	0	0
$816$	16131.5	0.692053
$817$	−24470.3	−1.04787
$818$	−59286.5	−2.53411
$819$	−5605.15	−0.239145
$820$	0	0
$821$	−6489.25	−0.275854	−0.137927	−	0.990442i	$-0.544044\pi$
$821$	−6489.25	−0.275854	−0.137927	+	0.990442i	$0.544044\pi$
$822$	1332.81	0.0565535
$823$	−7901.57	−0.334668	−0.167334	−	0.985900i	$-0.553516\pi$
$823$	−7901.57	−0.334668	−0.167334	+	0.985900i	$0.553516\pi$
$824$	1347.15	0.0569539
$825$	0	0
$826$	−4492.40	−0.189238
$827$	37815.8	1.59007	0.795033	−	0.606566i	$-0.207453\pi$
$827$	37815.8	1.59007	0.795033	+	0.606566i	$0.207453\pi$
$828$	10317.3	0.433031
$829$	26073.5	1.09236	0.546182	−	0.837667i	$-0.316081\pi$
$829$	26073.5	1.09236	0.546182	+	0.837667i	$0.316081\pi$
$830$	0	0
$831$	18867.4	0.787608
$832$	64350.2	2.68142
$833$	5898.96	0.245362
$834$	11604.6	0.481817
$835$	0	0
$836$	46308.9	1.91582
$837$	6968.13	0.287758
$838$	−41685.5	−1.71838
$839$	−15590.3	−0.641523	−0.320762	−	0.947160i	$-0.603939\pi$
$839$	−15590.3	−0.641523	−0.320762	+	0.947160i	$0.603939\pi$
$840$	0	0
$841$	−3412.46	−0.139918
$842$	−53630.8	−2.19506
$843$	5864.61	0.239606
$844$	33192.1	1.35370
$845$	0	0
$846$	−21946.8	−0.891899
$847$	2754.94	0.111760
$848$	−8231.96	−0.333357
$849$	15101.9	0.610477
$850$	0	0
$851$	−5571.92	−0.224445
$852$	1832.41	0.0736823
$853$	17476.1	0.701488	0.350744	−	0.936471i	$-0.385929\pi$
$853$	17476.1	0.701488	0.350744	+	0.936471i	$0.385929\pi$
$854$	−15710.8	−0.629521
$855$	0	0
$856$	1067.83	0.0426376
$857$	5694.54	0.226980	0.113490	−	0.993539i	$-0.463797\pi$
$857$	5694.54	0.226980	0.113490	+	0.993539i	$0.463797\pi$
$858$	−46953.7	−1.86827
$859$	27313.6	1.08490	0.542448	−	0.840089i	$-0.317497\pi$
$859$	27313.6	1.08490	0.542448	+	0.840089i	$0.317497\pi$
$860$	0	0
$861$	−4218.59	−0.166979
$862$	−19396.6	−0.766415
$863$	9046.07	0.356815	0.178408	−	0.983957i	$-0.442905\pi$
$863$	9046.07	0.356815	0.178408	+	0.983957i	$0.442905\pi$
$864$	6887.57	0.271204
$865$	0	0
$866$	13893.8	0.545185
$867$	−28740.1	−1.12580
$868$	−17964.8	−0.702496
$869$	−36472.1	−1.42374
$870$	0	0
$871$	114.314	0.00444705
$872$	4667.22	0.181252
$873$	9285.28	0.359976
$874$	−54760.0	−2.11932
$875$	0	0
$876$	14463.0	0.557829
$877$	2104.29	0.0810224	0.0405112	−	0.999179i	$-0.487101\pi$
$877$	2104.29	0.0810224	0.0405112	+	0.999179i	$0.487101\pi$
$878$	−1812.22	−0.0696576
$879$	19107.4	0.733192
$880$	0	0
$881$	−22589.6	−0.863861	−0.431931	−	0.901907i	$-0.642167\pi$
$881$	−22589.6	−0.863861	−0.431931	+	0.901907i	$0.642167\pi$
$882$	1868.11	0.0713179
$883$	2419.71	0.0922193	0.0461096	−	0.998936i	$-0.485318\pi$
$883$	2419.71	0.0922193	0.0461096	+	0.998936i	$0.485318\pi$
$884$	−106512.	−4.05248
$885$	0	0
$886$	−65409.8	−2.48023
$887$	−13177.0	−0.498806	−0.249403	−	0.968400i	$-0.580234\pi$
$887$	−13177.0	−0.498806	−0.249403	+	0.968400i	$0.580234\pi$
$888$	1194.26	0.0451313
$889$	18089.8	0.682464
$890$	0	0
$891$	−3363.76	−0.126476
$892$	−1266.19	−0.0475281
$893$	64553.2	2.41902
$894$	−22639.0	−0.846937
$895$	0	0
$896$	−7161.58	−0.267022
$897$	30769.2	1.14532
$898$	39742.9	1.47688
$899$	37378.3	1.38669
$900$	0	0
$901$	22187.5	0.820393
$902$	−35338.7	−1.30449
$903$	−4582.55	−0.168879
$904$	−6670.31	−0.245411
$905$	0	0
$906$	−17887.2	−0.655918
$907$	−9189.14	−0.336406	−0.168203	−	0.985752i	$-0.553796\pi$
$907$	−9189.14	−0.336406	−0.168203	+	0.985752i	$0.553796\pi$
$908$	−38227.0	−1.39714
$909$	13024.8	0.475252
$910$	0	0
$911$	−17045.8	−0.619928	−0.309964	−	0.950748i	$-0.600317\pi$
$911$	−17045.8	−0.619928	−0.309964	+	0.950748i	$0.600317\pi$
$912$	−15026.1	−0.545575
$913$	20424.6	0.740369
$914$	−57485.0	−2.08034
$915$	0	0
$916$	25228.1	0.910001
$917$	−9947.71	−0.358236
$918$	−13769.1	−0.495042
$919$	−30825.0	−1.10645	−0.553223	−	0.833033i	$-0.686602\pi$
$919$	−30825.0	−1.10645	−0.553223	+	0.833033i	$0.686602\pi$
$920$	0	0
$921$	−19859.5	−0.710524
$922$	5303.06	0.189422
$923$	5464.81	0.194882
$924$	8672.25	0.308762
$925$	0	0
$926$	−33610.5	−1.19277
$927$	1472.10	0.0521576
$928$	36946.2	1.30691
$929$	−5785.88	−0.204336	−0.102168	−	0.994767i	$-0.532578\pi$
$929$	−5785.88	−0.204336	−0.102168	+	0.994767i	$0.532578\pi$
$930$	0	0
$931$	−5494.75	−0.193430
$932$	−39652.2	−1.39362
$933$	29727.7	1.04313
$934$	−32125.3	−1.12545
$935$	0	0
$936$	−6594.91	−0.230301
$937$	13680.9	0.476986	0.238493	−	0.971144i	$-0.423347\pi$
$937$	13680.9	0.476986	0.238493	+	0.971144i	$0.423347\pi$
$938$	−38.0990	−0.00132620
$939$	−1267.01	−0.0440333
$940$	0	0
$941$	−45448.8	−1.57448	−0.787242	−	0.616644i	$-0.788492\pi$
$941$	−45448.8	−1.57448	−0.787242	+	0.616644i	$0.788492\pi$
$942$	−20319.6	−0.702813
$943$	23157.8	0.799705
$944$	6766.91	0.233310
$945$	0	0
$946$	−38387.6	−1.31933
$947$	7788.45	0.267255	0.133628	−	0.991032i	$-0.457337\pi$
$947$	7788.45	0.267255	0.133628	+	0.991032i	$0.457337\pi$
$948$	−26200.9	−0.897642
$949$	43133.0	1.47540
$950$	0	0
$951$	−14708.4	−0.501526
$952$	6940.61	0.236288
$953$	−6149.43	−0.209024	−0.104512	−	0.994524i	$-0.533328\pi$
$953$	−6149.43	−0.209024	−0.104512	+	0.994524i	$0.533328\pi$
$954$	7026.44	0.238458
$955$	0	0
$956$	−33484.2	−1.13280
$957$	−18043.8	−0.609481
$958$	−24693.5	−0.832788
$959$	−734.144	−0.0247203
$960$	0	0
$961$	36813.7	1.23573
$962$	18216.5	0.610524
$963$	1166.88	0.0390469
$964$	−9343.15	−0.312160
$965$	0	0
$966$	−10254.9	−0.341559
$967$	−23902.9	−0.794896	−0.397448	−	0.917625i	$-0.630104\pi$
$967$	−23902.9	−0.794896	−0.397448	+	0.917625i	$0.630104\pi$
$968$	3241.42	0.107627
$969$	40499.8	1.34266
$970$	0	0
$971$	−8015.06	−0.264898	−0.132449	−	0.991190i	$-0.542284\pi$
$971$	−8015.06	−0.264898	−0.132449	+	0.991190i	$0.542284\pi$
$972$	−2416.46	−0.0797407
$973$	−6392.12	−0.210608
$974$	84306.7	2.77347
$975$	0	0
$976$	23665.2	0.776131
$977$	34861.1	1.14156	0.570780	−	0.821103i	$-0.306641\pi$
$977$	34861.1	1.14156	0.570780	+	0.821103i	$0.306641\pi$
$978$	−2603.80	−0.0851334
$979$	17257.3	0.563378
$980$	0	0
$981$	5100.13	0.165988
$982$	−71257.6	−2.31560
$983$	−6620.83	−0.214824	−0.107412	−	0.994215i	$-0.534256\pi$
$983$	−6620.83	−0.214824	−0.107412	+	0.994215i	$0.534256\pi$
$984$	−4963.52	−0.160804
$985$	0	0
$986$	−73860.0	−2.38558
$987$	12088.8	0.389860
$988$	99213.6	3.19474
$989$	25155.7	0.808802
$990$	0	0
$991$	10360.1	0.332089	0.166045	−	0.986118i	$-0.446900\pi$
$991$	10360.1	0.332089	0.166045	+	0.986118i	$0.446900\pi$
$992$	65834.7	2.10711
$993$	15845.2	0.506377
$994$	−1821.33	−0.0581179
$995$	0	0
$996$	14672.7	0.466788
$997$	40309.3	1.28045	0.640225	−	0.768188i	$-0.278841\pi$
$997$	40309.3	1.28045	0.640225	+	0.768188i	$0.278841\pi$
$998$	25551.3	0.810432
$999$	1305.03	0.0413306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.a.o.1.2		2
3.2	odd	2		1575.4.a.n.1.1		2
5.2	odd	4		525.4.d.k.274.4		4
5.3	odd	4		525.4.d.k.274.1		4
5.4	even	2		105.4.a.d.1.1	✓	2
15.14	odd	2		315.4.a.l.1.2		2
20.19	odd	2		1680.4.a.bd.1.1		2
35.34	odd	2		735.4.a.m.1.1		2
105.104	even	2		2205.4.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.d.1.1	✓	2	5.4	even	2
315.4.a.l.1.2		2	15.14	odd	2
525.4.a.o.1.2		2	1.1	even	1	trivial
525.4.d.k.274.1		4	5.3	odd	4
525.4.d.k.274.4		4	5.2	odd	4
735.4.a.m.1.1		2	35.34	odd	2
1575.4.a.n.1.1		2	3.2	odd	2
1680.4.a.bd.1.1		2	20.19	odd	2
2205.4.a.be.1.2		2	105.104	even	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q+4.23607 q^{2} -3.00000 q^{3} +9.94427 q^{4} -12.7082 q^{6} +7.00000 q^{7} +8.23607 q^{8} +9.00000 q^{9} -41.5279 q^{11} -29.8328 q^{12} -88.9706 q^{13} +29.6525 q^{14} -44.6656 q^{16} +120.387 q^{17} +38.1246 q^{18} -112.138 q^{19} -21.0000 q^{21} -175.915 q^{22} +115.279 q^{23} -24.7082 q^{24} -376.885 q^{26} -27.0000 q^{27} +69.6099 q^{28} -144.833 q^{29} -258.079 q^{31} -255.095 q^{32} +124.584 q^{33} +509.967 q^{34} +89.4984 q^{36} -48.3344 q^{37} -475.023 q^{38} +266.912 q^{39} +200.885 q^{41} -88.9574 q^{42} +218.217 q^{43} -412.964 q^{44} +488.328 q^{46} -575.659 q^{47} +133.997 q^{48} +49.0000 q^{49} -361.161 q^{51} -884.748 q^{52} +184.302 q^{53} -114.374 q^{54} +57.6525 q^{56} +336.413 q^{57} -613.522 q^{58} -151.502 q^{59} -529.830 q^{61} -1093.24 q^{62} +63.0000 q^{63} -723.276 q^{64} +527.745 q^{66} -1.28485 q^{67} +1197.16 q^{68} -345.836 q^{69} -61.4226 q^{71} +74.1246 q^{72} -484.800 q^{73} -204.748 q^{74} -1115.13 q^{76} -290.695 q^{77} +1130.66 q^{78} +878.257 q^{79} +81.0000 q^{81} +850.964 q^{82} -491.830 q^{83} -208.830 q^{84} +924.381 q^{86} +434.498 q^{87} -342.026 q^{88} -415.560 q^{89} -622.794 q^{91} +1146.36 q^{92} +774.237 q^{93} -2438.53 q^{94} +765.286 q^{96} +1031.70 q^{97} +207.567 q^{98} -373.751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 6 q^{3} + 2 q^{4} - 12 q^{6} + 14 q^{7} + 12 q^{8} + 18 q^{9} - 92 q^{11} - 6 q^{12} - 8 q^{13} + 28 q^{14} + 18 q^{16} + 44 q^{17} + 36 q^{18} - 108 q^{19} - 42 q^{21} - 164 q^{22} + 320 q^{23}+ \cdots - 828 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 6 * q^3 + 2 * q^4 - 12 * q^6 + 14 * q^7 + 12 * q^8 + 18 * q^9 - 92 * q^11 - 6 * q^12 - 8 * q^13 + 28 * q^14 + 18 * q^16 + 44 * q^17 + 36 * q^18 - 108 * q^19 - 42 * q^21 - 164 * q^22 + 320 * q^23 - 36 * q^24 - 396 * q^26 - 54 * q^27 + 14 * q^28 - 236 * q^29 - 60 * q^31 - 300 * q^32 + 276 * q^33 + 528 * q^34 + 18 * q^36 - 204 * q^37 - 476 * q^38 + 24 * q^39 + 44 * q^41 - 84 * q^42 - 136 * q^43 - 12 * q^44 + 440 * q^46 - 400 * q^47 - 54 * q^48 + 98 * q^49 - 132 * q^51 - 1528 * q^52 - 16 * q^53 - 108 * q^54 + 84 * q^56 + 324 * q^57 - 592 * q^58 - 464 * q^59 - 684 * q^61 - 1140 * q^62 + 126 * q^63 - 1214 * q^64 + 492 * q^66 - 736 * q^67 + 1804 * q^68 - 960 * q^69 - 740 * q^71 + 108 * q^72 - 424 * q^73 - 168 * q^74 - 1148 * q^76 - 644 * q^77 + 1188 * q^78 - 408 * q^79 + 162 * q^81 + 888 * q^82 - 608 * q^83 - 42 * q^84 + 1008 * q^86 + 708 * q^87 - 532 * q^88 - 1332 * q^89 - 56 * q^91 - 480 * q^92 + 180 * q^93 - 2480 * q^94 + 900 * q^96 + 2448 * q^97 + 196 * q^98 - 828 * q^99

Introduction

L-functions

Modular forms

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