LMFDB - Embedded newform 1680.4.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,4,Mod(1,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1680.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1680.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:	$99.1232088096$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +41.5279 q^{11} +88.9706 q^{13} +15.0000 q^{15} -120.387 q^{17} +112.138 q^{19} -21.0000 q^{21} +115.279 q^{23} +25.0000 q^{25} -27.0000 q^{27} -144.833 q^{29} +258.079 q^{31} -124.584 q^{33} -35.0000 q^{35} +48.3344 q^{37} -266.912 q^{39} +200.885 q^{41} +218.217 q^{43} -45.0000 q^{45} -575.659 q^{47} +49.0000 q^{49} +361.161 q^{51} -184.302 q^{53} -207.639 q^{55} -336.413 q^{57} +151.502 q^{59} -529.830 q^{61} +63.0000 q^{63} -444.853 q^{65} -1.28485 q^{67} -345.836 q^{69} +61.4226 q^{71} +484.800 q^{73} -75.0000 q^{75} +290.695 q^{77} -878.257 q^{79} +81.0000 q^{81} -491.830 q^{83} +601.935 q^{85} +434.498 q^{87} -415.560 q^{89} +622.794 q^{91} -774.237 q^{93} -560.689 q^{95} -1031.70 q^{97} +373.751 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9} + 92 q^{11} + 8 q^{13} + 30 q^{15} - 44 q^{17} + 108 q^{19} - 42 q^{21} + 320 q^{23} + 50 q^{25} - 54 q^{27} - 236 q^{29} + 60 q^{31} - 276 q^{33} - 70 q^{35}+ \cdots + 828 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 10 * q^5 + 14 * q^7 + 18 * q^9 + 92 * q^11 + 8 * q^13 + 30 * q^15 - 44 * q^17 + 108 * q^19 - 42 * q^21 + 320 * q^23 + 50 * q^25 - 54 * q^27 - 236 * q^29 + 60 * q^31 - 276 * q^33 - 70 * q^35 + 204 * q^37 - 24 * q^39 + 44 * q^41 - 136 * q^43 - 90 * q^45 - 400 * q^47 + 98 * q^49 + 132 * q^51 + 16 * q^53 - 460 * q^55 - 324 * q^57 + 464 * q^59 - 684 * q^61 + 126 * q^63 - 40 * q^65 - 736 * q^67 - 960 * q^69 + 740 * q^71 + 424 * q^73 - 150 * q^75 + 644 * q^77 + 408 * q^79 + 162 * q^81 - 608 * q^83 + 220 * q^85 + 708 * q^87 - 1332 * q^89 + 56 * q^91 - 180 * q^93 - 540 * q^95 - 2448 * q^97 + 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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	41.5279	1.13828	0.569142	−	0.822239i	$-0.307275\pi$
$11$	41.5279	1.13828	0.569142	+	0.822239i	$0.307275\pi$
$12$	0	0
$13$	88.9706	1.89815	0.949077	−	0.315044i	$-0.102019\pi$
$13$	88.9706	1.89815	0.949077	+	0.315044i	$0.102019\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−120.387	−1.71754	−0.858769	−	0.512364i	$-0.828770\pi$
$17$	−120.387	−1.71754	−0.858769	+	0.512364i	$0.828770\pi$
$18$	0	0
$19$	112.138	1.35401	0.677004	−	0.735979i	$-0.263278\pi$
$19$	112.138	1.35401	0.677004	+	0.735979i	$0.263278\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	115.279	1.04510	0.522549	−	0.852609i	$-0.324981\pi$
$23$	115.279	1.04510	0.522549	+	0.852609i	$0.324981\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−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	$-0.231197\pi$
$31$	258.079	1.49524	0.747618	+	0.664128i	$0.231197\pi$
$32$	0	0
$33$	−124.584	−0.657188
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	48.3344	0.214760	0.107380	−	0.994218i	$-0.465754\pi$
$37$	48.3344	0.214760	0.107380	+	0.994218i	$0.465754\pi$
$38$	0	0
$39$	−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	$0.375029\pi$
$42$	0	0
$43$	218.217	0.773901	0.386950	−	0.922101i	$-0.373528\pi$
$43$	218.217	0.773901	0.386950	+	0.922101i	$0.373528\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−575.659	−1.78657	−0.893283	−	0.449496i	$-0.851604\pi$
$47$	−575.659	−1.78657	−0.893283	+	0.449496i	$0.851604\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	361.161	0.991621
$52$	0	0
$53$	−184.302	−0.477657	−0.238828	−	0.971062i	$-0.576763\pi$
$53$	−184.302	−0.477657	−0.238828	+	0.971062i	$0.576763\pi$
$54$	0	0
$55$	−207.639	−0.509056
$56$	0	0
$57$	−336.413	−0.781737
$58$	0	0
$59$	151.502	0.334302	0.167151	−	0.985931i	$-0.446543\pi$
$59$	151.502	0.334302	0.167151	+	0.985931i	$0.446543\pi$
$60$	0	0
$61$	−529.830	−1.11209	−0.556047	−	0.831151i	$-0.687683\pi$
$61$	−529.830	−1.11209	−0.556047	+	0.831151i	$0.687683\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−444.853	−0.848880
$66$	0	0
$67$	−1.28485	−0.00234283	−0.00117142	−	0.999999i	$-0.500373\pi$
$67$	−1.28485	−0.00234283	−0.00117142	+	0.999999i	$0.500373\pi$
$68$	0	0
$69$	−345.836	−0.603388
$70$	0	0
$71$	61.4226	0.102669	0.0513347	−	0.998682i	$-0.483652\pi$
$71$	61.4226	0.102669	0.0513347	+	0.998682i	$0.483652\pi$
$72$	0	0
$73$	484.800	0.777282	0.388641	−	0.921389i	$-0.372945\pi$
$73$	484.800	0.777282	0.388641	+	0.921389i	$0.372945\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	290.695	0.430231
$78$	0	0
$79$	−878.257	−1.25078	−0.625390	−	0.780312i	$-0.715060\pi$
$79$	−878.257	−1.25078	−0.625390	+	0.780312i	$0.715060\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−491.830	−0.650426	−0.325213	−	0.945641i	$-0.605436\pi$
$83$	−491.830	−0.650426	−0.325213	+	0.945641i	$0.605436\pi$
$84$	0	0
$85$	601.935	0.768106
$86$	0	0
$87$	434.498	0.535438
$88$	0	0
$89$	−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
$92$	0	0
$93$	−774.237	−0.863275
$94$	0	0
$95$	−560.689	−0.605531
$96$	0	0
$97$	−1031.70	−1.07993	−0.539964	−	0.841688i	$-0.681562\pi$
$97$	−1031.70	−1.07993	−0.539964	+	0.841688i	$0.681562\pi$
$98$	0	0
$99$	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	$0.247391\pi$
$102$	0	0
$103$	163.567	0.156473	0.0782364	−	0.996935i	$-0.475071\pi$
$103$	163.567	0.156473	0.0782364	+	0.996935i	$0.475071\pi$
$104$	0	0
$105$	105.000	0.0975900
$106$	0	0
$107$	129.653	0.117141	0.0585703	−	0.998283i	$-0.481346\pi$
$107$	129.653	0.117141	0.0585703	+	0.998283i	$0.481346\pi$
$108$	0	0
$109$	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
$112$	0	0
$113$	809.890	0.674230	0.337115	−	0.941463i	$-0.390549\pi$
$113$	809.890	0.674230	0.337115	+	0.941463i	$0.390549\pi$
$114$	0	0
$115$	−576.393	−0.467382
$116$	0	0
$117$	800.735	0.632718
$118$	0	0
$119$	−842.709	−0.649168
$120$	0	0
$121$	393.563	0.295690
$122$	0	0
$123$	−602.656	−0.441786
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2584.25	1.80563	0.902816	−	0.430028i	$-0.141496\pi$
$127$	2584.25	1.80563	0.902816	+	0.430028i	$0.141496\pi$
$128$	0	0
$129$	−654.650	−0.446812
$130$	0	0
$131$	1421.10	0.947804	0.473902	−	0.880578i	$-0.342845\pi$
$131$	1421.10	0.947804	0.473902	+	0.880578i	$0.342845\pi$
$132$	0	0
$133$	784.964	0.511767
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	104.878	0.0654037	0.0327019	−	0.999465i	$-0.489589\pi$
$137$	104.878	0.0654037	0.0327019	+	0.999465i	$0.489589\pi$
$138$	0	0
$139$	913.160	0.557217	0.278609	−	0.960405i	$-0.410127\pi$
$139$	913.160	0.557217	0.278609	+	0.960405i	$0.410127\pi$
$140$	0	0
$141$	1726.98	1.03147
$142$	0	0
$143$	3694.76	2.16064
$144$	0	0
$145$	724.164	0.414749
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	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	$-0.623829\pi$
$151$	−1407.53	−0.758564	−0.379282	+	0.925281i	$0.623829\pi$
$152$	0	0
$153$	−1083.48	−0.572512
$154$	0	0
$155$	−1290.39	−0.668690
$156$	0	0
$157$	−1598.94	−0.812798	−0.406399	−	0.913696i	$-0.633216\pi$
$157$	−1598.94	−0.812798	−0.406399	+	0.913696i	$0.633216\pi$
$158$	0	0
$159$	552.906	0.275775
$160$	0	0
$161$	806.950	0.395010
$162$	0	0
$163$	204.892	0.0984562	0.0492281	−	0.998788i	$-0.484324\pi$
$163$	204.892	0.0984562	0.0492281	+	0.998788i	$0.484324\pi$
$164$	0	0
$165$	622.918	0.293904
$166$	0	0
$167$	1165.94	0.540259	0.270129	−	0.962824i	$-0.412933\pi$
$167$	1165.94	0.540259	0.270129	+	0.962824i	$0.412933\pi$
$168$	0	0
$169$	5718.76	2.60299
$170$	0	0
$171$	1009.24	0.451336
$172$	0	0
$173$	−2538.00	−1.11538	−0.557690	−	0.830049i	$-0.688312\pi$
$173$	−2538.00	−1.11538	−0.557690	+	0.830049i	$0.688312\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	−454.505	−0.193009
$178$	0	0
$179$	−392.255	−0.163791	−0.0818954	−	0.996641i	$-0.526097\pi$
$179$	−392.255	−0.163791	−0.0818954	+	0.996641i	$0.526097\pi$
$180$	0	0
$181$	−2978.08	−1.22298	−0.611489	−	0.791253i	$-0.709429\pi$
$181$	−2978.08	−1.22298	−0.611489	+	0.791253i	$0.709429\pi$
$182$	0	0
$183$	1589.49	0.642068
$184$	0	0
$185$	−241.672	−0.0960436
$186$	0	0
$187$	−4999.41	−1.95504
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	1097.37	0.415722	0.207861	−	0.978158i	$-0.433350\pi$
$191$	1097.37	0.415722	0.207861	+	0.978158i	$0.433350\pi$
$192$	0	0
$193$	3500.31	1.30548	0.652740	−	0.757582i	$-0.273619\pi$
$193$	3500.31	1.30548	0.652740	+	0.757582i	$0.273619\pi$
$194$	0	0
$195$	1334.56	0.490101
$196$	0	0
$197$	1573.96	0.569237	0.284618	−	0.958641i	$-0.408133\pi$
$197$	1573.96	0.569237	0.284618	+	0.958641i	$0.408133\pi$
$198$	0	0
$199$	3396.62	1.20995	0.604976	−	0.796244i	$-0.293183\pi$
$199$	3396.62	1.20995	0.604976	+	0.796244i	$0.293183\pi$
$200$	0	0
$201$	3.85456	0.00135263
$202$	0	0
$203$	−1013.83	−0.350527
$204$	0	0
$205$	−1004.43	−0.342206
$206$	0	0
$207$	1037.51	0.348366
$208$	0	0
$209$	4656.84	1.54125
$210$	0	0
$211$	−3337.81	−1.08903	−0.544513	−	0.838753i	$-0.683285\pi$
$211$	−3337.81	−1.08903	−0.544513	+	0.838753i	$0.683285\pi$
$212$	0	0
$213$	−184.268	−0.0592762
$214$	0	0
$215$	−1091.08	−0.346099
$216$	0	0
$217$	1806.55	0.565146
$218$	0	0
$219$	−1454.40	−0.448764
$220$	0	0
$221$	−10710.9	−3.26015
$222$	0	0
$223$	−127.328	−0.0382356	−0.0191178	−	0.999817i	$-0.506086\pi$
$223$	−127.328	−0.0382356	−0.0191178	+	0.999817i	$0.506086\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−3844.12	−1.12398	−0.561990	−	0.827144i	$-0.689964\pi$
$227$	−3844.12	−1.12398	−0.561990	+	0.827144i	$0.689964\pi$
$228$	0	0
$229$	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
$232$	0	0
$233$	3987.44	1.12114	0.560570	−	0.828107i	$-0.310582\pi$
$233$	3987.44	1.12114	0.560570	+	0.828107i	$0.310582\pi$
$234$	0	0
$235$	2878.30	0.798976
$236$	0	0
$237$	2634.77	0.722138
$238$	0	0
$239$	3367.18	0.911317	0.455659	−	0.890155i	$-0.349404\pi$
$239$	3367.18	0.911317	0.455659	+	0.890155i	$0.349404\pi$
$240$	0	0
$241$	−939.551	−0.251128	−0.125564	−	0.992086i	$-0.540074\pi$
$241$	−939.551	−0.251128	−0.125564	+	0.992086i	$0.540074\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	9976.96	2.57012
$248$	0	0
$249$	1475.49	0.375523
$250$	0	0
$251$	1403.96	0.353056	0.176528	−	0.984296i	$-0.443513\pi$
$251$	1403.96	0.353056	0.176528	+	0.984296i	$0.443513\pi$
$252$	0	0
$253$	4787.28	1.18962
$254$	0	0
$255$	−1805.80	−0.443466
$256$	0	0
$257$	−1964.86	−0.476905	−0.238453	−	0.971154i	$-0.576640\pi$
$257$	−1964.86	−0.476905	−0.238453	+	0.971154i	$0.576640\pi$
$258$	0	0
$259$	338.341	0.0811717
$260$	0	0
$261$	−1303.50	−0.309135
$262$	0	0
$263$	393.821	0.0923347	0.0461673	−	0.998934i	$-0.485299\pi$
$263$	393.821	0.0923347	0.0461673	+	0.998934i	$0.485299\pi$
$264$	0	0
$265$	921.509	0.213615
$266$	0	0
$267$	1246.68	0.285751
$268$	0	0
$269$	−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	$-0.475363\pi$
$271$	689.909	0.154646	0.0773228	+	0.997006i	$0.475363\pi$
$272$	0	0
$273$	−1868.38	−0.414211
$274$	0	0
$275$	1038.20	0.227657
$276$	0	0
$277$	6289.13	1.36418	0.682088	−	0.731270i	$-0.261072\pi$
$277$	6289.13	1.36418	0.682088	+	0.731270i	$0.261072\pi$
$278$	0	0
$279$	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	$0.566534\pi$
$282$	0	0
$283$	−5033.96	−1.05738	−0.528688	−	0.848816i	$-0.677316\pi$
$283$	−5033.96	−1.05738	−0.528688	+	0.848816i	$0.677316\pi$
$284$	0	0
$285$	1682.07	0.349604
$286$	0	0
$287$	1406.20	0.289217
$288$	0	0
$289$	9580.03	1.94993
$290$	0	0
$291$	3095.09	0.623497
$292$	0	0
$293$	6369.12	1.26993	0.634963	−	0.772543i	$-0.281015\pi$
$293$	6369.12	1.26993	0.634963	+	0.772543i	$0.281015\pi$
$294$	0	0
$295$	−757.508	−0.149504
$296$	0	0
$297$	−1121.25	−0.219063
$298$	0	0
$299$	10256.4	1.98376
$300$	0	0
$301$	1527.52	0.292507
$302$	0	0
$303$	−4341.58	−0.823160
$304$	0	0
$305$	2649.15	0.497344
$306$	0	0
$307$	6619.83	1.23066	0.615332	−	0.788268i	$-0.289022\pi$
$307$	6619.83	1.23066	0.615332	+	0.788268i	$0.289022\pi$
$308$	0	0
$309$	−490.700	−0.0903396
$310$	0	0
$311$	9909.22	1.80675	0.903377	−	0.428848i	$-0.141080\pi$
$311$	9909.22	1.80675	0.903377	+	0.428848i	$0.141080\pi$
$312$	0	0
$313$	−422.336	−0.0762678	−0.0381339	−	0.999273i	$-0.512141\pi$
$313$	−422.336	−0.0762678	−0.0381339	+	0.999273i	$0.512141\pi$
$314$	0	0
$315$	−315.000	−0.0563436
$316$	0	0
$317$	−4902.78	−0.868668	−0.434334	−	0.900752i	$-0.643016\pi$
$317$	−4902.78	−0.868668	−0.434334	+	0.900752i	$0.643016\pi$
$318$	0	0
$319$	−6014.60	−1.05565
$320$	0	0
$321$	−388.960	−0.0676312
$322$	0	0
$323$	−13499.9	−2.32556
$324$	0	0
$325$	2224.26	0.379631
$326$	0	0
$327$	−1700.04	−0.287500
$328$	0	0
$329$	−4029.62	−0.675258
$330$	0	0
$331$	5281.74	0.877071	0.438535	−	0.898714i	$-0.355497\pi$
$331$	5281.74	0.877071	0.438535	+	0.898714i	$0.355497\pi$
$332$	0	0
$333$	435.009	0.0715867
$334$	0	0
$335$	6.42426	0.00104775
$336$	0	0
$337$	4459.60	0.720860	0.360430	−	0.932786i	$-0.382630\pi$
$337$	4459.60	0.720860	0.360430	+	0.932786i	$0.382630\pi$
$338$	0	0
$339$	−2429.67	−0.389267
$340$	0	0
$341$	10717.5	1.70200
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	1729.18	0.269843
$346$	0	0
$347$	−5261.97	−0.814056	−0.407028	−	0.913416i	$-0.633435\pi$
$347$	−5261.97	−0.814056	−0.407028	+	0.913416i	$0.633435\pi$
$348$	0	0
$349$	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
$352$	0	0
$353$	−8925.80	−1.34581	−0.672907	−	0.739727i	$-0.734955\pi$
$353$	−8925.80	−1.34581	−0.672907	+	0.739727i	$0.734955\pi$
$354$	0	0
$355$	−307.113	−0.0459151
$356$	0	0
$357$	2528.13	0.374797
$358$	0	0
$359$	3056.27	0.449314	0.224657	−	0.974438i	$-0.427874\pi$
$359$	3056.27	0.449314	0.224657	+	0.974438i	$0.427874\pi$
$360$	0	0
$361$	5715.88	0.833340
$362$	0	0
$363$	−1180.69	−0.170717
$364$	0	0
$365$	−2424.00	−0.347611
$366$	0	0
$367$	1813.52	0.257943	0.128971	−	0.991648i	$-0.458832\pi$
$367$	1813.52	0.257943	0.128971	+	0.991648i	$0.458832\pi$
$368$	0	0
$369$	1807.97	0.255065
$370$	0	0
$371$	−1290.11	−0.180537
$372$	0	0
$373$	−4517.48	−0.627094	−0.313547	−	0.949573i	$-0.601517\pi$
$373$	−4517.48	−0.627094	−0.313547	+	0.949573i	$0.601517\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−12885.9	−1.76036
$378$	0	0
$379$	4931.24	0.668340	0.334170	−	0.942513i	$-0.391544\pi$
$379$	4931.24	0.668340	0.334170	+	0.942513i	$0.391544\pi$
$380$	0	0
$381$	−7752.75	−1.04248
$382$	0	0
$383$	1482.37	0.197770	0.0988849	−	0.995099i	$-0.468472\pi$
$383$	1482.37	0.197770	0.0988849	+	0.995099i	$0.468472\pi$
$384$	0	0
$385$	−1453.48	−0.192405
$386$	0	0
$387$	1963.95	0.257967
$388$	0	0
$389$	−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
$392$	0	0
$393$	−4263.31	−0.547215
$394$	0	0
$395$	4391.28	0.559366
$396$	0	0
$397$	−13675.9	−1.72891	−0.864453	−	0.502713i	$-0.832335\pi$
$397$	−13675.9	−1.72891	−0.864453	+	0.502713i	$0.832335\pi$
$398$	0	0
$399$	−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	$0.158498\pi$
$402$	0	0
$403$	22961.4	2.83819
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	2007.22	0.244458
$408$	0	0
$409$	−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
$412$	0	0
$413$	1060.51	0.126354
$414$	0	0
$415$	2459.15	0.290879
$416$	0	0
$417$	−2739.48	−0.321709
$418$	0	0
$419$	9840.61	1.14736	0.573682	−	0.819078i	$-0.305515\pi$
$419$	9840.61	1.14736	0.573682	+	0.819078i	$0.305515\pi$
$420$	0	0
$421$	−12660.5	−1.46564	−0.732822	−	0.680420i	$-0.761797\pi$
$421$	−12660.5	−1.46564	−0.732822	+	0.680420i	$0.761797\pi$
$422$	0	0
$423$	−5180.93	−0.595522
$424$	0	0
$425$	−3009.67	−0.343507
$426$	0	0
$427$	−3708.81	−0.420332
$428$	0	0
$429$	−11084.3	−1.24744
$430$	0	0
$431$	4578.91	0.511736	0.255868	−	0.966712i	$-0.417639\pi$
$431$	4578.91	0.511736	0.255868	+	0.966712i	$0.417639\pi$
$432$	0	0
$433$	−3279.88	−0.364020	−0.182010	−	0.983297i	$-0.558260\pi$
$433$	−3279.88	−0.364020	−0.182010	+	0.983297i	$0.558260\pi$
$434$	0	0
$435$	−2172.49	−0.239455
$436$	0	0
$437$	12927.1	1.41507
$438$	0	0
$439$	427.807	0.0465105	0.0232552	−	0.999730i	$-0.492597\pi$
$439$	427.807	0.0465105	0.0232552	+	0.999730i	$0.492597\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−15441.2	−1.65605	−0.828027	−	0.560688i	$-0.810537\pi$
$443$	−15441.2	−1.65605	−0.828027	+	0.560688i	$0.810537\pi$
$444$	0	0
$445$	2077.80	0.221342
$446$	0	0
$447$	−5344.35	−0.565501
$448$	0	0
$449$	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
$452$	0	0
$453$	4222.59	0.437957
$454$	0	0
$455$	−3113.97	−0.320847
$456$	0	0
$457$	13570.4	1.38905	0.694524	−	0.719469i	$-0.255615\pi$
$457$	13570.4	1.38905	0.694524	+	0.719469i	$0.255615\pi$
$458$	0	0
$459$	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	$0.479857\pi$
$462$	0	0
$463$	−7934.36	−0.796417	−0.398209	−	0.917295i	$-0.630368\pi$
$463$	−7934.36	−0.796417	−0.398209	+	0.917295i	$0.630368\pi$
$464$	0	0
$465$	3871.18	0.386069
$466$	0	0
$467$	−7583.76	−0.751466	−0.375733	−	0.926728i	$-0.622609\pi$
$467$	−7583.76	−0.751466	−0.375733	+	0.926728i	$0.622609\pi$
$468$	0	0
$469$	−8.99396	−0.000885507 0
$470$	0	0
$471$	4796.82	0.469269
$472$	0	0
$473$	9062.07	0.880919
$474$	0	0
$475$	2803.44	0.270802
$476$	0	0
$477$	−1658.72	−0.159219
$478$	0	0
$479$	5829.34	0.556053	0.278027	−	0.960573i	$-0.410320\pi$
$479$	5829.34	0.556053	0.278027	+	0.960573i	$0.410320\pi$
$480$	0	0
$481$	4300.34	0.407648
$482$	0	0
$483$	−2420.85	−0.228059
$484$	0	0
$485$	5158.49	0.482959
$486$	0	0
$487$	19902.1	1.85185	0.925925	−	0.377708i	$-0.123288\pi$
$487$	19902.1	1.85185	0.925925	+	0.377708i	$0.123288\pi$
$488$	0	0
$489$	−614.675	−0.0568437
$490$	0	0
$491$	16821.6	1.54613	0.773065	−	0.634327i	$-0.218723\pi$
$491$	16821.6	1.54613	0.773065	+	0.634327i	$0.218723\pi$
$492$	0	0
$493$	17436.0	1.59285
$494$	0	0
$495$	−1868.75	−0.169685
$496$	0	0
$497$	429.958	0.0388054
$498$	0	0
$499$	−6031.83	−0.541126	−0.270563	−	0.962702i	$-0.587210\pi$
$499$	−6031.83	−0.541126	−0.270563	+	0.962702i	$0.587210\pi$
$500$	0	0
$501$	−3497.82	−0.311919
$502$	0	0
$503$	17176.4	1.52258	0.761290	−	0.648412i	$-0.224566\pi$
$503$	17176.4	1.52258	0.761290	+	0.648412i	$0.224566\pi$
$504$	0	0
$505$	−7235.97	−0.637617
$506$	0	0
$507$	−17156.3	−1.50284
$508$	0	0
$509$	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
$512$	0	0
$513$	−3027.72	−0.260579
$514$	0	0
$515$	−817.833	−0.0699768
$516$	0	0
$517$	−23905.9	−2.03362
$518$	0	0
$519$	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	$0.617150\pi$
$522$	0	0
$523$	−18248.5	−1.52572	−0.762858	−	0.646566i	$-0.776204\pi$
$523$	−18248.5	−1.52572	−0.762858	+	0.646566i	$0.776204\pi$
$524$	0	0
$525$	−525.000	−0.0436436
$526$	0	0
$527$	−31069.3	−2.56813
$528$	0	0
$529$	1122.16	0.0922302
$530$	0	0
$531$	1363.51	0.111434
$532$	0	0
$533$	17872.9	1.45246
$534$	0	0
$535$	−648.266	−0.0523869
$536$	0	0
$537$	1176.77	0.0945646
$538$	0	0
$539$	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	$0.573169\pi$
$542$	0	0
$543$	8934.24	0.706087
$544$	0	0
$545$	−2833.41	−0.222697
$546$	0	0
$547$	8002.52	0.625527	0.312763	−	0.949831i	$-0.398745\pi$
$547$	8002.52	0.625527	0.312763	+	0.949831i	$0.398745\pi$
$548$	0	0
$549$	−4768.47	−0.370698
$550$	0	0
$551$	−16241.2	−1.25572
$552$	0	0
$553$	−6147.80	−0.472750
$554$	0	0
$555$	725.016	0.0554508
$556$	0	0
$557$	1276.82	0.0971289	0.0485644	−	0.998820i	$-0.484535\pi$
$557$	1276.82	0.0971289	0.0485644	+	0.998820i	$0.484535\pi$
$558$	0	0
$559$	19414.9	1.46898
$560$	0	0
$561$	14998.2	1.12875
$562$	0	0
$563$	−11027.7	−0.825507	−0.412753	−	0.910843i	$-0.635433\pi$
$563$	−11027.7	−0.825507	−0.412753	+	0.910843i	$0.635433\pi$
$564$	0	0
$565$	−4049.45	−0.301525
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−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	$-0.542101\pi$
$571$	−3598.81	−0.263758	−0.131879	+	0.991266i	$0.542101\pi$
$572$	0	0
$573$	−3292.11	−0.240017
$574$	0	0
$575$	2881.97	0.209020
$576$	0	0
$577$	−3439.23	−0.248140	−0.124070	−	0.992273i	$-0.539595\pi$
$577$	−3439.23	−0.248140	−0.124070	+	0.992273i	$0.539595\pi$
$578$	0	0
$579$	−10500.9	−0.753720
$580$	0	0
$581$	−3442.81	−0.245838
$582$	0	0
$583$	−7653.66	−0.543709
$584$	0	0
$585$	−4003.68	−0.282960
$586$	0	0
$587$	−21285.2	−1.49665	−0.748327	−	0.663330i	$-0.769143\pi$
$587$	−21285.2	−1.49665	−0.748327	+	0.663330i	$0.769143\pi$
$588$	0	0
$589$	28940.4	2.02456
$590$	0	0
$591$	−4721.87	−0.328649
$592$	0	0
$593$	−14200.8	−0.983404	−0.491702	−	0.870764i	$-0.663625\pi$
$593$	−14200.8	−0.983404	−0.491702	+	0.870764i	$0.663625\pi$
$594$	0	0
$595$	4213.54	0.290317
$596$	0	0
$597$	−10189.9	−0.698566
$598$	0	0
$599$	8885.05	0.606065	0.303033	−	0.952980i	$-0.402001\pi$
$599$	8885.05	0.606065	0.303033	+	0.952980i	$0.402001\pi$
$600$	0	0
$601$	−2052.89	−0.139333	−0.0696664	−	0.997570i	$-0.522193\pi$
$601$	−2052.89	−0.139333	−0.0696664	+	0.997570i	$0.522193\pi$
$602$	0	0
$603$	−11.5637	−0.000780943 0
$604$	0	0
$605$	−1967.82	−0.132237
$606$	0	0
$607$	−10280.0	−0.687404	−0.343702	−	0.939079i	$-0.611681\pi$
$607$	−10280.0	−0.687404	−0.343702	+	0.939079i	$0.611681\pi$
$608$	0	0
$609$	3041.49	0.202377
$610$	0	0
$611$	−51216.8	−3.39118
$612$	0	0
$613$	23409.5	1.54242	0.771208	−	0.636584i	$-0.219653\pi$
$613$	23409.5	1.54242	0.771208	+	0.636584i	$0.219653\pi$
$614$	0	0
$615$	3013.28	0.197573
$616$	0	0
$617$	−6632.75	−0.432779	−0.216389	−	0.976307i	$-0.569428\pi$
$617$	−6632.75	−0.432779	−0.216389	+	0.976307i	$0.569428\pi$
$618$	0	0
$619$	−10734.0	−0.696990	−0.348495	−	0.937311i	$-0.613307\pi$
$619$	−10734.0	−0.696990	−0.348495	+	0.937311i	$0.613307\pi$
$620$	0	0
$621$	−3112.52	−0.201129
$622$	0	0
$623$	−2908.92	−0.187068
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−13970.5	−0.889839
$628$	0	0
$629$	−5818.83	−0.368858
$630$	0	0
$631$	17071.0	1.07700	0.538499	−	0.842626i	$-0.318992\pi$
$631$	17071.0	1.07700	0.538499	+	0.842626i	$0.318992\pi$
$632$	0	0
$633$	10013.4	0.628749
$634$	0	0
$635$	−12921.3	−0.807503
$636$	0	0
$637$	4359.56	0.271165
$638$	0	0
$639$	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	$0.703794\pi$
$642$	0	0
$643$	25409.3	1.55839	0.779196	−	0.626780i	$-0.215628\pi$
$643$	25409.3	1.55839	0.779196	+	0.626780i	$0.215628\pi$
$644$	0	0
$645$	3273.25	0.199820
$646$	0	0
$647$	6039.08	0.366956	0.183478	−	0.983024i	$-0.441264\pi$
$647$	6039.08	0.366956	0.183478	+	0.983024i	$0.441264\pi$
$648$	0	0
$649$	6291.54	0.380531
$650$	0	0
$651$	−5419.66	−0.326287
$652$	0	0
$653$	−30666.2	−1.83776	−0.918882	−	0.394532i	$-0.870907\pi$
$653$	−30666.2	−1.83776	−0.918882	+	0.394532i	$0.870907\pi$
$654$	0	0
$655$	−7105.51	−0.423871
$656$	0	0
$657$	4363.20	0.259094
$658$	0	0
$659$	2765.96	0.163500	0.0817500	−	0.996653i	$-0.473949\pi$
$659$	2765.96	0.163500	0.0817500	+	0.996653i	$0.473949\pi$
$660$	0	0
$661$	27261.8	1.60418	0.802089	−	0.597204i	$-0.203722\pi$
$661$	27261.8	1.60418	0.802089	+	0.597204i	$0.203722\pi$
$662$	0	0
$663$	32132.7	1.88225
$664$	0	0
$665$	−3924.82	−0.228869
$666$	0	0
$667$	−16696.1	−0.969230
$668$	0	0
$669$	381.985	0.0220753
$670$	0	0
$671$	−22002.7	−1.26588
$672$	0	0
$673$	−1048.17	−0.0600356	−0.0300178	−	0.999549i	$-0.509556\pi$
$673$	−1048.17	−0.0600356	−0.0300178	+	0.999549i	$0.509556\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−34554.7	−1.96166	−0.980831	−	0.194860i	$-0.937575\pi$
$677$	−34554.7	−1.96166	−0.980831	+	0.194860i	$0.937575\pi$
$678$	0	0
$679$	−7221.89	−0.408175
$680$	0	0
$681$	11532.4	0.648930
$682$	0	0
$683$	−14711.6	−0.824192	−0.412096	−	0.911140i	$-0.635203\pi$
$683$	−14711.6	−0.824192	−0.412096	+	0.911140i	$0.635203\pi$
$684$	0	0
$685$	−524.389	−0.0292494
$686$	0	0
$687$	−7610.85	−0.422667
$688$	0	0
$689$	−16397.4	−0.906666
$690$	0	0
$691$	24522.6	1.35005	0.675024	−	0.737796i	$-0.264133\pi$
$691$	24522.6	1.35005	0.675024	+	0.737796i	$0.264133\pi$
$692$	0	0
$693$	2616.26	0.143410
$694$	0	0
$695$	−4565.80	−0.249195
$696$	0	0
$697$	−24184.0	−1.31425
$698$	0	0
$699$	−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	$0.319773\pi$
$702$	0	0
$703$	5420.11	0.290787
$704$	0	0
$705$	−8634.89	−0.461289
$706$	0	0
$707$	10130.4	0.538885
$708$	0	0
$709$	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
$712$	0	0
$713$	29751.0	1.56267
$714$	0	0
$715$	−18473.8	−0.966267
$716$	0	0
$717$	−10101.5	−0.526149
$718$	0	0
$719$	−13063.6	−0.677593	−0.338797	−	0.940860i	$-0.610020\pi$
$719$	−13063.6	−0.677593	−0.338797	+	0.940860i	$0.610020\pi$
$720$	0	0
$721$	1144.97	0.0591411
$722$	0	0
$723$	2818.65	0.144989
$724$	0	0
$725$	−3620.82	−0.185481
$726$	0	0
$727$	12897.0	0.657940	0.328970	−	0.944340i	$-0.393298\pi$
$727$	12897.0	0.657940	0.328970	+	0.944340i	$0.393298\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−26270.5	−1.32920
$732$	0	0
$733$	11699.6	0.589540	0.294770	−	0.955568i	$-0.404757\pi$
$733$	11699.6	0.589540	0.294770	+	0.955568i	$0.404757\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	−53.3571	−0.00266681
$738$	0	0
$739$	−14974.0	−0.745368	−0.372684	−	0.927958i	$-0.621562\pi$
$739$	−14974.0	−0.745368	−0.372684	+	0.927958i	$0.621562\pi$
$740$	0	0
$741$	−29930.9	−1.48386
$742$	0	0
$743$	−18500.7	−0.913492	−0.456746	−	0.889597i	$-0.650985\pi$
$743$	−18500.7	−0.913492	−0.456746	+	0.889597i	$0.650985\pi$
$744$	0	0
$745$	−8907.24	−0.438035
$746$	0	0
$747$	−4426.47	−0.216809
$748$	0	0
$749$	907.572	0.0442750
$750$	0	0
$751$	26348.4	1.28025	0.640125	−	0.768271i	$-0.278883\pi$
$751$	26348.4	1.28025	0.640125	+	0.768271i	$0.278883\pi$
$752$	0	0
$753$	−4211.88	−0.203837
$754$	0	0
$755$	7037.65	0.339240
$756$	0	0
$757$	−28061.7	−1.34732	−0.673659	−	0.739042i	$-0.735278\pi$
$757$	−28061.7	−1.34732	−0.673659	+	0.739042i	$0.735278\pi$
$758$	0	0
$759$	−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	$0.527168\pi$
$762$	0	0
$763$	3966.77	0.188213
$764$	0	0
$765$	5417.41	0.256035
$766$	0	0
$767$	13479.2	0.634557
$768$	0	0
$769$	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
$772$	0	0
$773$	10005.1	0.465537	0.232769	−	0.972532i	$-0.425222\pi$
$773$	10005.1	0.465537	0.232769	+	0.972532i	$0.425222\pi$
$774$	0	0
$775$	6451.97	0.299047
$776$	0	0
$777$	−1015.02	−0.0468645
$778$	0	0
$779$	22526.8	1.03608
$780$	0	0
$781$	2550.75	0.116867
$782$	0	0
$783$	3910.49	0.178479
$784$	0	0
$785$	7994.70	0.363494
$786$	0	0
$787$	−17826.8	−0.807443	−0.403721	−	0.914882i	$-0.632284\pi$
$787$	−17826.8	−0.807443	−0.403721	+	0.914882i	$0.632284\pi$
$788$	0	0
$789$	−1181.46	−0.0533094
$790$	0	0
$791$	5669.23	0.254835
$792$	0	0
$793$	−47139.3	−2.11093
$794$	0	0
$795$	−2764.53	−0.123330
$796$	0	0
$797$	36723.0	1.63211	0.816057	−	0.577971i	$-0.196155\pi$
$797$	36723.0	1.63211	0.816057	+	0.577971i	$0.196155\pi$
$798$	0	0
$799$	69301.9	3.06849
$800$	0	0
$801$	−3740.04	−0.164979
$802$	0	0
$803$	20132.7	0.884767
$804$	0	0
$805$	−4034.75	−0.176654
$806$	0	0
$807$	5631.08	0.245630
$808$	0	0
$809$	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	$-0.552139\pi$
$811$	−7532.41	−0.326139	−0.163070	+	0.986615i	$0.552139\pi$
$812$	0	0
$813$	−2069.73	−0.0892847
$814$	0	0
$815$	−1024.46	−0.0440309
$816$	0	0
$817$	24470.3	1.04787
$818$	0	0
$819$	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	$0.544044\pi$
$822$	0	0
$823$	−7901.57	−0.334668	−0.167334	−	0.985900i	$-0.553516\pi$
$823$	−7901.57	−0.334668	−0.167334	+	0.985900i	$0.553516\pi$
$824$	0	0
$825$	−3114.59	−0.131438
$826$	0	0
$827$	37815.8	1.59007	0.795033	−	0.606566i	$-0.207453\pi$
$827$	37815.8	1.59007	0.795033	+	0.606566i	$0.207453\pi$
$828$	0	0
$829$	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
$832$	0	0
$833$	−5898.96	−0.245362
$834$	0	0
$835$	−5829.71	−0.241611
$836$	0	0
$837$	−6968.13	−0.287758
$838$	0	0
$839$	15590.3	0.641523	0.320762	−	0.947160i	$-0.396061\pi$
$839$	15590.3	0.641523	0.320762	+	0.947160i	$0.396061\pi$
$840$	0	0
$841$	−3412.46	−0.139918
$842$	0	0
$843$	5864.61	0.239606
$844$	0	0
$845$	−28593.8	−1.16409
$846$	0	0
$847$	2754.94	0.111760
$848$	0	0
$849$	15101.9	0.610477
$850$	0	0
$851$	5571.92	0.224445
$852$	0	0
$853$	−17476.1	−0.701488	−0.350744	−	0.936471i	$-0.614071\pi$
$853$	−17476.1	−0.701488	−0.350744	+	0.936471i	$0.614071\pi$
$854$	0	0
$855$	−5046.20	−0.201844
$856$	0	0
$857$	−5694.54	−0.226980	−0.113490	−	0.993539i	$-0.536203\pi$
$857$	−5694.54	−0.226980	−0.113490	+	0.993539i	$0.536203\pi$
$858$	0	0
$859$	−27313.6	−1.08490	−0.542448	−	0.840089i	$-0.682503\pi$
$859$	−27313.6	−1.08490	−0.542448	+	0.840089i	$0.682503\pi$
$860$	0	0
$861$	−4218.59	−0.166979
$862$	0	0
$863$	9046.07	0.356815	0.178408	−	0.983957i	$-0.442905\pi$
$863$	9046.07	0.356815	0.178408	+	0.983957i	$0.442905\pi$
$864$	0	0
$865$	12690.0	0.498813
$866$	0	0
$867$	−28740.1	−1.12580
$868$	0	0
$869$	−36472.1	−1.42374
$870$	0	0
$871$	−114.314	−0.00444705
$872$	0	0
$873$	−9285.28	−0.359976
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−2104.29	−0.0810224	−0.0405112	−	0.999179i	$-0.512899\pi$
$877$	−2104.29	−0.0810224	−0.0405112	+	0.999179i	$0.512899\pi$
$878$	0	0
$879$	−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	$0.642167\pi$
$882$	0	0
$883$	2419.71	0.0922193	0.0461096	−	0.998936i	$-0.485318\pi$
$883$	2419.71	0.0922193	0.0461096	+	0.998936i	$0.485318\pi$
$884$	0	0
$885$	2272.52	0.0863164
$886$	0	0
$887$	−13177.0	−0.498806	−0.249403	−	0.968400i	$-0.580234\pi$
$887$	−13177.0	−0.498806	−0.249403	+	0.968400i	$0.580234\pi$
$888$	0	0
$889$	18089.8	0.682464
$890$	0	0
$891$	3363.76	0.126476
$892$	0	0
$893$	−64553.2	−2.41902
$894$	0	0
$895$	1961.28	0.0732495
$896$	0	0
$897$	−30769.2	−1.14532
$898$	0	0
$899$	−37378.3	−1.38669
$900$	0	0
$901$	22187.5	0.820393
$902$	0	0
$903$	−4582.55	−0.168879
$904$	0	0
$905$	14890.4	0.546932
$906$	0	0
$907$	−9189.14	−0.336406	−0.168203	−	0.985752i	$-0.553796\pi$
$907$	−9189.14	−0.336406	−0.168203	+	0.985752i	$0.553796\pi$
$908$	0	0
$909$	13024.8	0.475252
$910$	0	0
$911$	17045.8	0.619928	0.309964	−	0.950748i	$-0.399683\pi$
$911$	17045.8	0.619928	0.309964	+	0.950748i	$0.399683\pi$
$912$	0	0
$913$	−20424.6	−0.740369
$914$	0	0
$915$	−7947.45	−0.287141
$916$	0	0
$917$	9947.71	0.358236
$918$	0	0
$919$	30825.0	1.10645	0.553223	−	0.833033i	$-0.313398\pi$
$919$	30825.0	1.10645	0.553223	+	0.833033i	$0.313398\pi$
$920$	0	0
$921$	−19859.5	−0.710524
$922$	0	0
$923$	5464.81	0.194882
$924$	0	0
$925$	1208.36	0.0429520
$926$	0	0
$927$	1472.10	0.0521576
$928$	0	0
$929$	−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
$932$	0	0
$933$	−29727.7	−1.04313
$934$	0	0
$935$	24997.1	0.874323
$936$	0	0
$937$	−13680.9	−0.476986	−0.238493	−	0.971144i	$-0.576653\pi$
$937$	−13680.9	−0.476986	−0.238493	+	0.971144i	$0.576653\pi$
$938$	0	0
$939$	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	$0.788492\pi$
$942$	0	0
$943$	23157.8	0.799705
$944$	0	0
$945$	945.000	0.0325300
$946$	0	0
$947$	7788.45	0.267255	0.133628	−	0.991032i	$-0.457337\pi$
$947$	7788.45	0.267255	0.133628	+	0.991032i	$0.457337\pi$
$948$	0	0
$949$	43133.0	1.47540
$950$	0	0
$951$	14708.4	0.501526
$952$	0	0
$953$	6149.43	0.209024	0.104512	−	0.994524i	$-0.466672\pi$
$953$	6149.43	0.209024	0.104512	+	0.994524i	$0.466672\pi$
$954$	0	0
$955$	−5486.85	−0.185917
$956$	0	0
$957$	18043.8	0.609481
$958$	0	0
$959$	734.144	0.0247203
$960$	0	0
$961$	36813.7	1.23573
$962$	0	0
$963$	1166.88	0.0390469
$964$	0	0
$965$	−17501.5	−0.583829
$966$	0	0
$967$	−23902.9	−0.794896	−0.397448	−	0.917625i	$-0.630104\pi$
$967$	−23902.9	−0.794896	−0.397448	+	0.917625i	$0.630104\pi$
$968$	0	0
$969$	40499.8	1.34266
$970$	0	0
$971$	8015.06	0.264898	0.132449	−	0.991190i	$-0.457716\pi$
$971$	8015.06	0.264898	0.132449	+	0.991190i	$0.457716\pi$
$972$	0	0
$973$	6392.12	0.210608
$974$	0	0
$975$	−6672.79	−0.219180
$976$	0	0
$977$	−34861.1	−1.14156	−0.570780	−	0.821103i	$-0.693359\pi$
$977$	−34861.1	−1.14156	−0.570780	+	0.821103i	$0.693359\pi$
$978$	0	0
$979$	−17257.3	−0.563378
$980$	0	0
$981$	5100.13	0.165988
$982$	0	0
$983$	−6620.83	−0.214824	−0.107412	−	0.994215i	$-0.534256\pi$
$983$	−6620.83	−0.214824	−0.107412	+	0.994215i	$0.534256\pi$
$984$	0	0
$985$	−7869.78	−0.254570
$986$	0	0
$987$	12088.8	0.389860
$988$	0	0
$989$	25155.7	0.808802
$990$	0	0
$991$	−10360.1	−0.332089	−0.166045	−	0.986118i	$-0.553100\pi$
$991$	−10360.1	−0.332089	−0.166045	+	0.986118i	$0.553100\pi$
$992$	0	0
$993$	−15845.2	−0.506377
$994$	0	0
$995$	−16983.1	−0.541107
$996$	0	0
$997$	−40309.3	−1.28045	−0.640225	−	0.768188i	$-0.721159\pi$
$997$	−40309.3	−1.28045	−0.640225	+	0.768188i	$0.721159\pi$
$998$	0	0
$999$	−1305.03	−0.0413306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.4.a.bd.1.1		2
4.3	odd	2		105.4.a.d.1.1	✓	2
12.11	even	2		315.4.a.l.1.2		2
20.3	even	4		525.4.d.k.274.4		4
20.7	even	4		525.4.d.k.274.1		4
20.19	odd	2		525.4.a.o.1.2		2
28.27	even	2		735.4.a.m.1.1		2
60.59	even	2		1575.4.a.n.1.1		2
84.83	odd	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	4.3	odd	2
315.4.a.l.1.2		2	12.11	even	2
525.4.a.o.1.2		2	20.19	odd	2
525.4.d.k.274.1		4	20.7	even	4
525.4.d.k.274.4		4	20.3	even	4
735.4.a.m.1.1		2	28.27	even	2
1575.4.a.n.1.1		2	60.59	even	2
1680.4.a.bd.1.1		2	1.1	even	1	trivial
2205.4.a.be.1.2		2	84.83	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +41.5279 q^{11} +88.9706 q^{13} +15.0000 q^{15} -120.387 q^{17} +112.138 q^{19} -21.0000 q^{21} +115.279 q^{23} +25.0000 q^{25} -27.0000 q^{27} -144.833 q^{29} +258.079 q^{31} -124.584 q^{33} -35.0000 q^{35} +48.3344 q^{37} -266.912 q^{39} +200.885 q^{41} +218.217 q^{43} -45.0000 q^{45} -575.659 q^{47} +49.0000 q^{49} +361.161 q^{51} -184.302 q^{53} -207.639 q^{55} -336.413 q^{57} +151.502 q^{59} -529.830 q^{61} +63.0000 q^{63} -444.853 q^{65} -1.28485 q^{67} -345.836 q^{69} +61.4226 q^{71} +484.800 q^{73} -75.0000 q^{75} +290.695 q^{77} -878.257 q^{79} +81.0000 q^{81} -491.830 q^{83} +601.935 q^{85} +434.498 q^{87} -415.560 q^{89} +622.794 q^{91} -774.237 q^{93} -560.689 q^{95} -1031.70 q^{97} +373.751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9} + 92 q^{11} + 8 q^{13} + 30 q^{15} - 44 q^{17} + 108 q^{19} - 42 q^{21} + 320 q^{23} + 50 q^{25} - 54 q^{27} - 236 q^{29} + 60 q^{31} - 276 q^{33} - 70 q^{35}+ \cdots + 828 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 10 * q^5 + 14 * q^7 + 18 * q^9 + 92 * q^11 + 8 * q^13 + 30 * q^15 - 44 * q^17 + 108 * q^19 - 42 * q^21 + 320 * q^23 + 50 * q^25 - 54 * q^27 - 236 * q^29 + 60 * q^31 - 276 * q^33 - 70 * q^35 + 204 * q^37 - 24 * q^39 + 44 * q^41 - 136 * q^43 - 90 * q^45 - 400 * q^47 + 98 * q^49 + 132 * q^51 + 16 * q^53 - 460 * q^55 - 324 * q^57 + 464 * q^59 - 684 * q^61 + 126 * q^63 - 40 * q^65 - 736 * q^67 - 960 * q^69 + 740 * q^71 + 424 * q^73 - 150 * q^75 + 644 * q^77 + 408 * q^79 + 162 * q^81 - 608 * q^83 + 220 * q^85 + 708 * q^87 - 1332 * q^89 + 56 * q^91 - 180 * q^93 - 540 * q^95 - 2448 * q^97 + 828 * q^99

Introduction

L-functions

Modular forms

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