LMFDB - Embedded newform 1680.4.a.bd.1.2

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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +50.4721 q^{11} -80.9706 q^{13} +15.0000 q^{15} +76.3870 q^{17} -4.13777 q^{19} -21.0000 q^{21} +204.721 q^{23} +25.0000 q^{25} -27.0000 q^{27} -91.1672 q^{29} -198.079 q^{31} -151.416 q^{33} -35.0000 q^{35} +155.666 q^{37} +242.912 q^{39} -156.885 q^{41} -354.217 q^{43} -45.0000 q^{45} +175.659 q^{47} +49.0000 q^{49} -229.161 q^{51} +200.302 q^{53} -252.361 q^{55} +12.4133 q^{57} +312.498 q^{59} -154.170 q^{61} +63.0000 q^{63} +404.853 q^{65} -734.715 q^{67} -614.164 q^{69} +678.577 q^{71} -60.8003 q^{73} -75.0000 q^{75} +353.305 q^{77} +1286.26 q^{79} +81.0000 q^{81} -116.170 q^{83} -381.935 q^{85} +273.502 q^{87} -916.440 q^{89} -566.794 q^{91} +594.237 q^{93} +20.6888 q^{95} -1416.30 q^{97} +454.249 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 10 * q^5 + 14 * q^7 + 18 * q^9	$ 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}+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

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 $

$2$	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
$11$	50.4721	1.38345	0.691724	−	0.722162i	$-0.256852\pi$
$11$	50.4721	1.38345	0.691724	+	0.722162i	$0.256852\pi$
$12$	0	0
$13$	−80.9706	−1.72748	−0.863738	−	0.503940i	$-0.831883\pi$
$13$	−80.9706	−1.72748	−0.863738	+	0.503940i	$0.831883\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	76.3870	1.08980	0.544899	−	0.838502i	$-0.316568\pi$
$17$	76.3870	1.08980	0.544899	+	0.838502i	$0.316568\pi$
$18$	0	0
$19$	−4.13777	−0.0499615	−0.0249808	−	0.999688i	$-0.507952\pi$
$19$	−4.13777	−0.0499615	−0.0249808	+	0.999688i	$0.507952\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	204.721	1.85597	0.927986	−	0.372615i	$-0.121539\pi$
$23$	204.721	1.85597	0.927986	+	0.372615i	$0.121539\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−91.1672	−0.583770	−0.291885	−	0.956453i	$-0.594282\pi$
$29$	−91.1672	−0.583770	−0.291885	+	0.956453i	$0.594282\pi$
$30$	0	0
$31$	−198.079	−1.14761	−0.573807	−	0.818991i	$-0.694534\pi$
$31$	−198.079	−1.14761	−0.573807	+	0.818991i	$0.694534\pi$
$32$	0	0
$33$	−151.416	−0.798734
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	155.666	0.691656	0.345828	−	0.938298i	$-0.387598\pi$
$37$	155.666	0.691656	0.345828	+	0.938298i	$0.387598\pi$
$38$	0	0
$39$	242.912	0.997359
$40$	0	0
$41$	−156.885	−0.597595	−0.298797	−	0.954317i	$-0.596585\pi$
$41$	−156.885	−0.597595	−0.298797	+	0.954317i	$0.596585\pi$
$42$	0	0
$43$	−354.217	−1.25622	−0.628111	−	0.778124i	$-0.716172\pi$
$43$	−354.217	−1.25622	−0.628111	+	0.778124i	$0.716172\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	175.659	0.545161	0.272580	−	0.962133i	$-0.412123\pi$
$47$	175.659	0.545161	0.272580	+	0.962133i	$0.412123\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−229.161	−0.629195
$52$	0	0
$53$	200.302	0.519124	0.259562	−	0.965726i	$-0.416422\pi$
$53$	200.302	0.519124	0.259562	+	0.965726i	$0.416422\pi$
$54$	0	0
$55$	−252.361	−0.618696
$56$	0	0
$57$	12.4133	0.0288453
$58$	0	0
$59$	312.498	0.689556	0.344778	−	0.938684i	$-0.387954\pi$
$59$	312.498	0.689556	0.344778	+	0.938684i	$0.387954\pi$
$60$	0	0
$61$	−154.170	−0.323598	−0.161799	−	0.986824i	$-0.551730\pi$
$61$	−154.170	−0.323598	−0.161799	+	0.986824i	$0.551730\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	404.853	0.772551
$66$	0	0
$67$	−734.715	−1.33970	−0.669849	−	0.742498i	$-0.733641\pi$
$67$	−734.715	−1.33970	−0.669849	+	0.742498i	$0.733641\pi$
$68$	0	0
$69$	−614.164	−1.07155
$70$	0	0
$71$	678.577	1.13426	0.567129	−	0.823629i	$-0.308054\pi$
$71$	678.577	1.13426	0.567129	+	0.823629i	$0.308054\pi$
$72$	0	0
$73$	−60.8003	−0.0974813	−0.0487407	−	0.998811i	$-0.515521\pi$
$73$	−60.8003	−0.0974813	−0.0487407	+	0.998811i	$0.515521\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	353.305	0.522894
$78$	0	0
$79$	1286.26	1.83184	0.915919	−	0.401363i	$-0.131463\pi$
$79$	1286.26	1.83184	0.915919	+	0.401363i	$0.131463\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−116.170	−0.153631	−0.0768153	−	0.997045i	$-0.524475\pi$
$83$	−116.170	−0.153631	−0.0768153	+	0.997045i	$0.524475\pi$
$84$	0	0
$85$	−381.935	−0.487372
$86$	0	0
$87$	273.502	0.337040
$88$	0	0
$89$	−916.440	−1.09149	−0.545744	−	0.837952i	$-0.683753\pi$
$89$	−916.440	−1.09149	−0.545744	+	0.837952i	$0.683753\pi$
$90$	0	0
$91$	−566.794	−0.652925
$92$	0	0
$93$	594.237	0.662575
$94$	0	0
$95$	20.6888	0.0223435
$96$	0	0
$97$	−1416.30	−1.48251	−0.741256	−	0.671222i	$-0.765770\pi$
$97$	−1416.30	−1.48251	−0.741256	+	0.671222i	$0.765770\pi$
$98$	0	0
$99$	454.249	0.461149
$100$	0	0
$101$	−1379.19	−1.35876	−0.679381	−	0.733785i	$-0.737752\pi$
$101$	−1379.19	−1.35876	−0.679381	+	0.733785i	$0.737752\pi$
$102$	0	0
$103$	1308.43	1.25169	0.625844	−	0.779949i	$-0.284755\pi$
$103$	1308.43	1.25169	0.625844	+	0.779949i	$0.284755\pi$
$104$	0	0
$105$	105.000	0.0975900
$106$	0	0
$107$	−1265.65	−1.14351	−0.571754	−	0.820425i	$-0.693737\pi$
$107$	−1265.65	−1.14351	−0.571754	+	0.820425i	$0.693737\pi$
$108$	0	0
$109$	2069.32	1.81839	0.909196	−	0.416368i	$-0.136697\pi$
$109$	2069.32	1.81839	0.909196	+	0.416368i	$0.136697\pi$
$110$	0	0
$111$	−466.997	−0.399328
$112$	0	0
$113$	−1953.89	−1.62661	−0.813303	−	0.581840i	$-0.802333\pi$
$113$	−1953.89	−1.62661	−0.813303	+	0.581840i	$0.802333\pi$
$114$	0	0
$115$	−1023.61	−0.830016
$116$	0	0
$117$	−728.735	−0.575826
$118$	0	0
$119$	534.709	0.411905
$120$	0	0
$121$	1216.44	0.913927
$122$	0	0
$123$	470.656	0.345022
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−224.251	−0.156685	−0.0783426	−	0.996926i	$-0.524963\pi$
$127$	−224.251	−0.156685	−0.0783426	+	0.996926i	$0.524963\pi$
$128$	0	0
$129$	1062.65	0.725280
$130$	0	0
$131$	490.898	0.327404	0.163702	−	0.986510i	$-0.447656\pi$
$131$	490.898	0.327404	0.163702	+	0.986510i	$0.447656\pi$
$132$	0	0
$133$	−28.9644	−0.0188837
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	1831.12	1.14192	0.570961	−	0.820977i	$-0.306571\pi$
$137$	1831.12	1.14192	0.570961	+	0.820977i	$0.306571\pi$
$138$	0	0
$139$	3050.84	1.86165	0.930823	−	0.365469i	$-0.119091\pi$
$139$	3050.84	1.86165	0.930823	+	0.365469i	$0.119091\pi$
$140$	0	0
$141$	−526.978	−0.314749
$142$	0	0
$143$	−4086.76	−2.38987
$144$	0	0
$145$	455.836	0.261070
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	2246.55	1.23520	0.617599	−	0.786493i	$-0.288105\pi$
$149$	2246.55	1.23520	0.617599	+	0.786493i	$0.288105\pi$
$150$	0	0
$151$	1311.53	0.706826	0.353413	−	0.935467i	$-0.385021\pi$
$151$	1311.53	0.706826	0.353413	+	0.935467i	$0.385021\pi$
$152$	0	0
$153$	687.483	0.363266
$154$	0	0
$155$	990.395	0.513228
$156$	0	0
$157$	1790.94	0.910398	0.455199	−	0.890390i	$-0.349568\pi$
$157$	1790.94	0.910398	0.455199	+	0.890390i	$0.349568\pi$
$158$	0	0
$159$	−600.906	−0.299716
$160$	0	0
$161$	1433.05	0.701491
$162$	0	0
$163$	491.108	0.235991	0.117996	−	0.993014i	$-0.462353\pi$
$163$	491.108	0.235991	0.117996	+	0.993014i	$0.462353\pi$
$164$	0	0
$165$	757.082	0.357205
$166$	0	0
$167$	826.059	0.382769	0.191384	−	0.981515i	$-0.438702\pi$
$167$	826.059	0.382769	0.191384	+	0.981515i	$0.438702\pi$
$168$	0	0
$169$	4359.24	1.98418
$170$	0	0
$171$	−37.2399	−0.0166538
$172$	0	0
$173$	2918.00	1.28238	0.641190	−	0.767382i	$-0.278441\pi$
$173$	2918.00	1.28238	0.641190	+	0.767382i	$0.278441\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	−937.495	−0.398116
$178$	0	0
$179$	−955.745	−0.399082	−0.199541	−	0.979889i	$-0.563945\pi$
$179$	−955.745	−0.399082	−0.199541	+	0.979889i	$0.563945\pi$
$180$	0	0
$181$	206.080	0.0846289	0.0423145	−	0.999104i	$-0.486527\pi$
$181$	206.080	0.0846289	0.0423145	+	0.999104i	$0.486527\pi$
$182$	0	0
$183$	462.511	0.186829
$184$	0	0
$185$	−778.328	−0.309318
$186$	0	0
$187$	3855.41	1.50768
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	2018.63	0.764727	0.382364	−	0.924012i	$-0.375110\pi$
$191$	2018.63	0.764727	0.382364	+	0.924012i	$0.375110\pi$
$192$	0	0
$193$	1031.69	0.384781	0.192390	−	0.981318i	$-0.438376\pi$
$193$	1031.69	0.384781	0.192390	+	0.981318i	$0.438376\pi$
$194$	0	0
$195$	−1214.56	−0.446033
$196$	0	0
$197$	−205.955	−0.0744857	−0.0372429	−	0.999306i	$-0.511858\pi$
$197$	−205.955	−0.0744857	−0.0372429	+	0.999306i	$0.511858\pi$
$198$	0	0
$199$	1831.38	0.652376	0.326188	−	0.945305i	$-0.394236\pi$
$199$	1831.38	0.652376	0.326188	+	0.945305i	$0.394236\pi$
$200$	0	0
$201$	2204.15	0.773475
$202$	0	0
$203$	−638.170	−0.220644
$204$	0	0
$205$	784.427	0.267253
$206$	0	0
$207$	1842.49	0.618657
$208$	0	0
$209$	−208.842	−0.0691191
$210$	0	0
$211$	−1030.19	−0.336119	−0.168060	−	0.985777i	$-0.553750\pi$
$211$	−1030.19	−0.336119	−0.168060	+	0.985777i	$0.553750\pi$
$212$	0	0
$213$	−2035.73	−0.654864
$214$	0	0
$215$	1771.08	0.561800
$216$	0	0
$217$	−1386.55	−0.433757
$218$	0	0
$219$	182.401	0.0562809
$220$	0	0
$221$	−6185.10	−1.88260
$222$	0	0
$223$	−5368.67	−1.61217	−0.806083	−	0.591803i	$-0.798416\pi$
$223$	−5368.67	−1.61217	−0.806083	+	0.591803i	$0.798416\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	932.121	0.272542	0.136271	−	0.990672i	$-0.456488\pi$
$227$	932.121	0.272542	0.136271	+	0.990672i	$0.456488\pi$
$228$	0	0
$229$	3163.05	0.912752	0.456376	−	0.889787i	$-0.349147\pi$
$229$	3163.05	0.912752	0.456376	+	0.889787i	$0.349147\pi$
$230$	0	0
$231$	−1059.91	−0.301893
$232$	0	0
$233$	436.562	0.122747	0.0613737	−	0.998115i	$-0.480452\pi$
$233$	436.562	0.122747	0.0613737	+	0.998115i	$0.480452\pi$
$234$	0	0
$235$	−878.297	−0.243803
$236$	0	0
$237$	−3858.77	−1.05761
$238$	0	0
$239$	1980.82	0.536103	0.268051	−	0.963405i	$-0.413620\pi$
$239$	1980.82	0.536103	0.268051	+	0.963405i	$0.413620\pi$
$240$	0	0
$241$	5303.55	1.41756	0.708780	−	0.705430i	$-0.249246\pi$
$241$	5303.55	1.41756	0.708780	+	0.705430i	$0.249246\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	335.037	0.0863074
$248$	0	0
$249$	348.511	0.0886987
$250$	0	0
$251$	2996.04	0.753420	0.376710	−	0.926331i	$-0.377055\pi$
$251$	2996.04	0.753420	0.376710	+	0.926331i	$0.377055\pi$
$252$	0	0
$253$	10332.7	2.56764
$254$	0	0
$255$	1145.80	0.281385
$256$	0	0
$257$	968.861	0.235159	0.117580	−	0.993063i	$-0.462487\pi$
$257$	968.861	0.235159	0.117580	+	0.993063i	$0.462487\pi$
$258$	0	0
$259$	1089.66	0.261421
$260$	0	0
$261$	−820.505	−0.194590
$262$	0	0
$263$	4830.18	1.13248	0.566239	−	0.824241i	$-0.308398\pi$
$263$	4830.18	1.13248	0.566239	+	0.824241i	$0.308398\pi$
$264$	0	0
$265$	−1001.51	−0.232159
$266$	0	0
$267$	2749.32	0.630171
$268$	0	0
$269$	−4774.97	−1.08229	−0.541143	−	0.840930i	$-0.682008\pi$
$269$	−4774.97	−1.08229	−0.541143	+	0.840930i	$0.682008\pi$
$270$	0	0
$271$	−141.909	−0.0318094	−0.0159047	−	0.999874i	$-0.505063\pi$
$271$	−141.909	−0.0318094	−0.0159047	+	0.999874i	$0.505063\pi$
$272$	0	0
$273$	1700.38	0.376966
$274$	0	0
$275$	1261.80	0.276689
$276$	0	0
$277$	−3621.13	−0.785460	−0.392730	−	0.919654i	$-0.628469\pi$
$277$	−3621.13	−0.785460	−0.392730	+	0.919654i	$0.628469\pi$
$278$	0	0
$279$	−1782.71	−0.382538
$280$	0	0
$281$	5790.87	1.22937	0.614687	−	0.788771i	$-0.289282\pi$
$281$	5790.87	1.22937	0.614687	+	0.788771i	$0.289282\pi$
$282$	0	0
$283$	−526.043	−0.110495	−0.0552474	−	0.998473i	$-0.517595\pi$
$283$	−526.043	−0.110495	−0.0552474	+	0.998473i	$0.517595\pi$
$284$	0	0
$285$	−62.0665	−0.0129000
$286$	0	0
$287$	−1098.20	−0.225870
$288$	0	0
$289$	921.972	0.187660
$290$	0	0
$291$	4248.91	0.855929
$292$	0	0
$293$	1914.88	0.381803	0.190901	−	0.981609i	$-0.438859\pi$
$293$	1914.88	0.381803	0.190901	+	0.981609i	$0.438859\pi$
$294$	0	0
$295$	−1562.49	−0.308379
$296$	0	0
$297$	−1362.75	−0.266245
$298$	0	0
$299$	−16576.4	−3.20615
$300$	0	0
$301$	−2479.52	−0.474807
$302$	0	0
$303$	4137.58	0.784482
$304$	0	0
$305$	770.851	0.144717
$306$	0	0
$307$	6244.17	1.16083	0.580413	−	0.814322i	$-0.302891\pi$
$307$	6244.17	1.16083	0.580413	+	0.814322i	$0.302891\pi$
$308$	0	0
$309$	−3925.30	−0.722662
$310$	0	0
$311$	9658.78	1.76109	0.880545	−	0.473962i	$-0.157177\pi$
$311$	9658.78	1.76109	0.880545	+	0.473962i	$0.157177\pi$
$312$	0	0
$313$	2198.34	0.396988	0.198494	−	0.980102i	$-0.436395\pi$
$313$	2198.34	0.396988	0.198494	+	0.980102i	$0.436395\pi$
$314$	0	0
$315$	−315.000	−0.0563436
$316$	0	0
$317$	3030.78	0.536990	0.268495	−	0.963281i	$-0.413474\pi$
$317$	3030.78	0.536990	0.268495	+	0.963281i	$0.413474\pi$
$318$	0	0
$319$	−4601.40	−0.807615
$320$	0	0
$321$	3796.96	0.660204
$322$	0	0
$323$	−316.072	−0.0544480
$324$	0	0
$325$	−2024.26	−0.345495
$326$	0	0
$327$	−6207.96	−1.04985
$328$	0	0
$329$	1229.62	0.206051
$330$	0	0
$331$	−4753.74	−0.789393	−0.394696	−	0.918812i	$-0.629150\pi$
$331$	−4753.74	−0.789393	−0.394696	+	0.918812i	$0.629150\pi$
$332$	0	0
$333$	1400.99	0.230552
$334$	0	0
$335$	3673.58	0.599131
$336$	0	0
$337$	8824.40	1.42640	0.713199	−	0.700962i	$-0.247246\pi$
$337$	8824.40	1.42640	0.713199	+	0.700962i	$0.247246\pi$
$338$	0	0
$339$	5861.67	0.939122
$340$	0	0
$341$	−9997.47	−1.58766
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	3070.82	0.479210
$346$	0	0
$347$	3413.97	0.528160	0.264080	−	0.964501i	$-0.414932\pi$
$347$	3413.97	0.528160	0.264080	+	0.964501i	$0.414932\pi$
$348$	0	0
$349$	−5676.32	−0.870621	−0.435310	−	0.900280i	$-0.643361\pi$
$349$	−5676.32	−0.870621	−0.435310	+	0.900280i	$0.643361\pi$
$350$	0	0
$351$	2186.21	0.332453
$352$	0	0
$353$	6225.80	0.938713	0.469357	−	0.883009i	$-0.344486\pi$
$353$	6225.80	0.938713	0.469357	+	0.883009i	$0.344486\pi$
$354$	0	0
$355$	−3392.89	−0.507256
$356$	0	0
$357$	−1604.13	−0.237813
$358$	0	0
$359$	4907.73	0.721505	0.360752	−	0.932662i	$-0.382520\pi$
$359$	4907.73	0.721505	0.360752	+	0.932662i	$0.382520\pi$
$360$	0	0
$361$	−6841.88	−0.997504
$362$	0	0
$363$	−3649.31	−0.527656
$364$	0	0
$365$	304.001	0.0435950
$366$	0	0
$367$	3906.48	0.555631	0.277816	−	0.960634i	$-0.410390\pi$
$367$	3906.48	0.555631	0.277816	+	0.960634i	$0.410390\pi$
$368$	0	0
$369$	−1411.97	−0.199198
$370$	0	0
$371$	1402.11	0.196210
$372$	0	0
$373$	−2102.52	−0.291862	−0.145931	−	0.989295i	$-0.546618\pi$
$373$	−2102.52	−0.291862	−0.145931	+	0.989295i	$0.546618\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	7381.86	1.00845
$378$	0	0
$379$	6612.76	0.896239	0.448120	−	0.893974i	$-0.352094\pi$
$379$	6612.76	0.896239	0.448120	+	0.893974i	$0.352094\pi$
$380$	0	0
$381$	672.752	0.0904623
$382$	0	0
$383$	−2.37457	−0.000316801 0	−0.000158401	−	1.00000i	$-0.500050\pi$
$383$	−2.37457	−0.000316801 0	−0.000158401		1.00000i	$0.500050\pi$
$384$	0	0
$385$	−1766.52	−0.233845
$386$	0	0
$387$	−3187.95	−0.418741
$388$	0	0
$389$	7716.98	1.00583	0.502913	−	0.864337i	$-0.332261\pi$
$389$	7716.98	1.00583	0.502913	+	0.864337i	$0.332261\pi$
$390$	0	0
$391$	15638.0	2.02263
$392$	0	0
$393$	−1472.69	−0.189027
$394$	0	0
$395$	−6431.28	−0.819223
$396$	0	0
$397$	6403.95	0.809584	0.404792	−	0.914409i	$-0.367344\pi$
$397$	6403.95	0.809584	0.404792	+	0.914409i	$0.367344\pi$
$398$	0	0
$399$	86.8931	0.0109025
$400$	0	0
$401$	−10969.9	−1.36611	−0.683054	−	0.730368i	$-0.739349\pi$
$401$	−10969.9	−1.36611	−0.683054	+	0.730368i	$0.739349\pi$
$402$	0	0
$403$	16038.6	1.98248
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	7856.78	0.956870
$408$	0	0
$409$	−400.353	−0.0484015	−0.0242007	−	0.999707i	$-0.507704\pi$
$409$	−400.353	−0.0484015	−0.0242007	+	0.999707i	$0.507704\pi$
$410$	0	0
$411$	−5493.37	−0.659289
$412$	0	0
$413$	2187.49	0.260628
$414$	0	0
$415$	580.851	0.0687057
$416$	0	0
$417$	−9152.52	−1.07482
$418$	0	0
$419$	15815.4	1.84399	0.921995	−	0.387201i	$-0.126558\pi$
$419$	15815.4	1.84399	0.921995	+	0.387201i	$0.126558\pi$
$420$	0	0
$421$	1936.53	0.224182	0.112091	−	0.993698i	$-0.464245\pi$
$421$	1936.53	0.224182	0.112091	+	0.993698i	$0.464245\pi$
$422$	0	0
$423$	1580.93	0.181720
$424$	0	0
$425$	1909.67	0.217960
$426$	0	0
$427$	−1079.19	−0.122309
$428$	0	0
$429$	12260.3	1.37979
$430$	0	0
$431$	−2030.91	−0.226973	−0.113487	−	0.993540i	$-0.536202\pi$
$431$	−2030.91	−0.226973	−0.113487	+	0.993540i	$0.536202\pi$
$432$	0	0
$433$	−10784.1	−1.19689	−0.598443	−	0.801165i	$-0.704214\pi$
$433$	−10784.1	−1.19689	−0.598443	+	0.801165i	$0.704214\pi$
$434$	0	0
$435$	−1367.51	−0.150729
$436$	0	0
$437$	−847.089	−0.0927272
$438$	0	0
$439$	6304.19	0.685382	0.342691	−	0.939448i	$-0.388662\pi$
$439$	6304.19	0.685382	0.342691	+	0.939448i	$0.388662\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−15494.8	−1.66181	−0.830905	−	0.556414i	$-0.812177\pi$
$443$	−15494.8	−1.66181	−0.830905	+	0.556414i	$0.812177\pi$
$444$	0	0
$445$	4582.20	0.488128
$446$	0	0
$447$	−6739.65	−0.713142
$448$	0	0
$449$	−242.018	−0.0254377	−0.0127189	−	0.999919i	$-0.504049\pi$
$449$	−242.018	−0.0254377	−0.0127189	+	0.999919i	$0.504049\pi$
$450$	0	0
$451$	−7918.34	−0.826741
$452$	0	0
$453$	−3934.59	−0.408086
$454$	0	0
$455$	2833.97	0.291997
$456$	0	0
$457$	−11670.4	−1.19457	−0.597283	−	0.802030i	$-0.703753\pi$
$457$	−11670.4	−1.19457	−0.597283	+	0.802030i	$0.703753\pi$
$458$	0	0
$459$	−2062.45	−0.209732
$460$	0	0
$461$	12128.1	1.22530	0.612649	−	0.790355i	$-0.290104\pi$
$461$	12128.1	1.22530	0.612649	+	0.790355i	$0.290104\pi$
$462$	0	0
$463$	−5161.64	−0.518103	−0.259051	−	0.965864i	$-0.583410\pi$
$463$	−5161.64	−0.518103	−0.259051	+	0.965864i	$0.583410\pi$
$464$	0	0
$465$	−2971.18	−0.296313
$466$	0	0
$467$	−11680.2	−1.15738	−0.578691	−	0.815547i	$-0.696436\pi$
$467$	−11680.2	−1.15738	−0.578691	+	0.815547i	$0.696436\pi$
$468$	0	0
$469$	−5143.01	−0.506358
$470$	0	0
$471$	−5372.82	−0.525619
$472$	0	0
$473$	−17878.1	−1.73792
$474$	0	0
$475$	−103.444	−0.00999230
$476$	0	0
$477$	1802.72	0.173041
$478$	0	0
$479$	18458.7	1.76075	0.880373	−	0.474282i	$-0.157292\pi$
$479$	18458.7	1.76075	0.880373	+	0.474282i	$0.157292\pi$
$480$	0	0
$481$	−12604.3	−1.19482
$482$	0	0
$483$	−4299.15	−0.405006
$484$	0	0
$485$	7081.51	0.663000
$486$	0	0
$487$	−8630.11	−0.803014	−0.401507	−	0.915856i	$-0.631513\pi$
$487$	−8630.11	−0.803014	−0.401507	+	0.915856i	$0.631513\pi$
$488$	0	0
$489$	−1473.33	−0.136250
$490$	0	0
$491$	−17801.6	−1.63621	−0.818103	−	0.575072i	$-0.804974\pi$
$491$	−17801.6	−1.63621	−0.818103	+	0.575072i	$0.804974\pi$
$492$	0	0
$493$	−6963.99	−0.636191
$494$	0	0
$495$	−2271.25	−0.206232
$496$	0	0
$497$	4750.04	0.428709
$498$	0	0
$499$	−200.167	−0.0179574	−0.00897868	−	0.999960i	$-0.502858\pi$
$499$	−200.167	−0.0179574	−0.00897868	+	0.999960i	$0.502858\pi$
$500$	0	0
$501$	−2478.18	−0.220992
$502$	0	0
$503$	−16400.4	−1.45379	−0.726896	−	0.686747i	$-0.759038\pi$
$503$	−16400.4	−1.45379	−0.726896	+	0.686747i	$0.759038\pi$
$504$	0	0
$505$	6895.97	0.607657
$506$	0	0
$507$	−13077.7	−1.14556
$508$	0	0
$509$	−15006.6	−1.30679	−0.653394	−	0.757018i	$-0.726656\pi$
$509$	−15006.6	−1.30679	−0.653394	+	0.757018i	$0.726656\pi$
$510$	0	0
$511$	−425.602	−0.0368445
$512$	0	0
$513$	111.720	0.00961510
$514$	0	0
$515$	−6542.17	−0.559772
$516$	0	0
$517$	8865.91	0.754201
$518$	0	0
$519$	−8754.01	−0.740382
$520$	0	0
$521$	7113.18	0.598146	0.299073	−	0.954230i	$-0.403323\pi$
$521$	7113.18	0.598146	0.299073	+	0.954230i	$0.403323\pi$
$522$	0	0
$523$	8888.46	0.743146	0.371573	−	0.928404i	$-0.378819\pi$
$523$	8888.46	0.743146	0.371573	+	0.928404i	$0.378819\pi$
$524$	0	0
$525$	−525.000	−0.0436436
$526$	0	0
$527$	−15130.7	−1.25067
$528$	0	0
$529$	29743.8	2.44463
$530$	0	0
$531$	2812.49	0.229852
$532$	0	0
$533$	12703.1	1.03233
$534$	0	0
$535$	6328.27	0.511392
$536$	0	0
$537$	2867.23	0.230410
$538$	0	0
$539$	2473.13	0.197635
$540$	0	0
$541$	−653.827	−0.0519597	−0.0259799	−	0.999662i	$-0.508271\pi$
$541$	−653.827	−0.0519597	−0.0259799	+	0.999662i	$0.508271\pi$
$542$	0	0
$543$	−618.241	−0.0488605
$544$	0	0
$545$	−10346.6	−0.813210
$546$	0	0
$547$	−1138.52	−0.0889940	−0.0444970	−	0.999010i	$-0.514169\pi$
$547$	−1138.52	−0.0889940	−0.0444970	+	0.999010i	$0.514169\pi$
$548$	0	0
$549$	−1387.53	−0.107866
$550$	0	0
$551$	377.229	0.0291660
$552$	0	0
$553$	9003.80	0.692370
$554$	0	0
$555$	2334.98	0.178585
$556$	0	0
$557$	−19804.8	−1.50657	−0.753283	−	0.657696i	$-0.771531\pi$
$557$	−19804.8	−1.50657	−0.753283	+	0.657696i	$0.771531\pi$
$558$	0	0
$559$	28681.1	2.17009
$560$	0	0
$561$	−11566.2	−0.870458
$562$	0	0
$563$	−10276.3	−0.769265	−0.384632	−	0.923070i	$-0.625672\pi$
$563$	−10276.3	−0.769265	−0.384632	+	0.923070i	$0.625672\pi$
$564$	0	0
$565$	9769.45	0.727440
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	4139.03	0.304951	0.152475	−	0.988307i	$-0.451276\pi$
$569$	4139.03	0.304951	0.152475	+	0.988307i	$0.451276\pi$
$570$	0	0
$571$	4486.81	0.328839	0.164420	−	0.986390i	$-0.447425\pi$
$571$	4486.81	0.328839	0.164420	+	0.986390i	$0.447425\pi$
$572$	0	0
$573$	−6055.89	−0.441516
$574$	0	0
$575$	5118.03	0.371194
$576$	0	0
$577$	−1104.77	−0.0797093	−0.0398547	−	0.999205i	$-0.512689\pi$
$577$	−1104.77	−0.0797093	−0.0398547	+	0.999205i	$0.512689\pi$
$578$	0	0
$579$	−3095.07	−0.222153
$580$	0	0
$581$	−813.192	−0.0580669
$582$	0	0
$583$	10109.7	0.718181
$584$	0	0
$585$	3643.68	0.257517
$586$	0	0
$587$	10413.2	0.732199	0.366100	−	0.930576i	$-0.380693\pi$
$587$	10413.2	0.732199	0.366100	+	0.930576i	$0.380693\pi$
$588$	0	0
$589$	819.605	0.0573365
$590$	0	0
$591$	617.865	0.0430044
$592$	0	0
$593$	−3235.16	−0.224034	−0.112017	−	0.993706i	$-0.535731\pi$
$593$	−3235.16	−0.224034	−0.112017	+	0.993706i	$0.535731\pi$
$594$	0	0
$595$	−2673.54	−0.184209
$596$	0	0
$597$	−5494.13	−0.376649
$598$	0	0
$599$	−569.048	−0.0388158	−0.0194079	−	0.999812i	$-0.506178\pi$
$599$	−569.048	−0.0388158	−0.0194079	+	0.999812i	$0.506178\pi$
$600$	0	0
$601$	3760.89	0.255258	0.127629	−	0.991822i	$-0.459263\pi$
$601$	3760.89	0.255258	0.127629	+	0.991822i	$0.459263\pi$
$602$	0	0
$603$	−6612.44	−0.446566
$604$	0	0
$605$	−6082.18	−0.408720
$606$	0	0
$607$	2224.05	0.148717	0.0743585	−	0.997232i	$-0.476309\pi$
$607$	2224.05	0.148717	0.0743585	+	0.997232i	$0.476309\pi$
$608$	0	0
$609$	1914.51	0.127389
$610$	0	0
$611$	−14223.2	−0.941753
$612$	0	0
$613$	5914.50	0.389697	0.194849	−	0.980833i	$-0.437578\pi$
$613$	5914.50	0.389697	0.194849	+	0.980833i	$0.437578\pi$
$614$	0	0
$615$	−2353.28	−0.154298
$616$	0	0
$617$	−18591.2	−1.21306	−0.606528	−	0.795062i	$-0.707438\pi$
$617$	−18591.2	−1.21306	−0.606528	+	0.795062i	$0.707438\pi$
$618$	0	0
$619$	−5125.97	−0.332844	−0.166422	−	0.986055i	$-0.553221\pi$
$619$	−5125.97	−0.332844	−0.166422	+	0.986055i	$0.553221\pi$
$620$	0	0
$621$	−5527.48	−0.357182
$622$	0	0
$623$	−6415.08	−0.412544
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	626.526	0.0399060
$628$	0	0
$629$	11890.8	0.753765
$630$	0	0
$631$	10649.0	0.671839	0.335919	−	0.941891i	$-0.390953\pi$
$631$	10649.0	0.671839	0.335919	+	0.941891i	$0.390953\pi$
$632$	0	0
$633$	3090.57	0.194058
$634$	0	0
$635$	1121.25	0.0700718
$636$	0	0
$637$	−3967.56	−0.246782
$638$	0	0
$639$	6107.20	0.378086
$640$	0	0
$641$	24025.7	1.48044	0.740218	−	0.672367i	$-0.234722\pi$
$641$	24025.7	1.48044	0.740218	+	0.672367i	$0.234722\pi$
$642$	0	0
$643$	−14929.3	−0.915638	−0.457819	−	0.889045i	$-0.651369\pi$
$643$	−14929.3	−0.915638	−0.457819	+	0.889045i	$0.651369\pi$
$644$	0	0
$645$	−5313.25	−0.324355
$646$	0	0
$647$	−14479.1	−0.879801	−0.439901	−	0.898046i	$-0.644986\pi$
$647$	−14479.1	−0.879801	−0.439901	+	0.898046i	$0.644986\pi$
$648$	0	0
$649$	15772.5	0.953965
$650$	0	0
$651$	4159.66	0.250430
$652$	0	0
$653$	898.168	0.0538255	0.0269127	−	0.999638i	$-0.491432\pi$
$653$	898.168	0.0538255	0.0269127	+	0.999638i	$0.491432\pi$
$654$	0	0
$655$	−2454.49	−0.146420
$656$	0	0
$657$	−547.203	−0.0324938
$658$	0	0
$659$	30198.0	1.78505	0.892526	−	0.450997i	$-0.148931\pi$
$659$	30198.0	1.78505	0.892526	+	0.450997i	$0.148931\pi$
$660$	0	0
$661$	−19337.8	−1.13790	−0.568952	−	0.822371i	$-0.692651\pi$
$661$	−19337.8	−1.13790	−0.568952	+	0.822371i	$0.692651\pi$
$662$	0	0
$663$	18555.3	1.08692
$664$	0	0
$665$	144.822	0.00844504
$666$	0	0
$667$	−18663.9	−1.08346
$668$	0	0
$669$	16106.0	0.930784
$670$	0	0
$671$	−7781.30	−0.447681
$672$	0	0
$673$	10132.2	0.580336	0.290168	−	0.956976i	$-0.406289\pi$
$673$	10132.2	0.580336	0.290168	+	0.956976i	$0.406289\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−33177.3	−1.88347	−0.941733	−	0.336361i	$-0.890804\pi$
$677$	−33177.3	−1.88347	−0.941733	+	0.336361i	$0.890804\pi$
$678$	0	0
$679$	−9914.11	−0.560337
$680$	0	0
$681$	−2796.36	−0.157352
$682$	0	0
$683$	11423.6	0.639987	0.319994	−	0.947420i	$-0.396319\pi$
$683$	11423.6	0.639987	0.319994	+	0.947420i	$0.396319\pi$
$684$	0	0
$685$	−9155.61	−0.510683
$686$	0	0
$687$	−9489.15	−0.526978
$688$	0	0
$689$	−16218.6	−0.896775
$690$	0	0
$691$	19737.4	1.08661	0.543304	−	0.839536i	$-0.317173\pi$
$691$	19737.4	1.08661	0.543304	+	0.839536i	$0.317173\pi$
$692$	0	0
$693$	3179.74	0.174298
$694$	0	0
$695$	−15254.2	−0.832554
$696$	0	0
$697$	−11984.0	−0.651258
$698$	0	0
$699$	−1309.69	−0.0708682
$700$	0	0
$701$	11307.8	0.609258	0.304629	−	0.952471i	$-0.401468\pi$
$701$	11307.8	0.609258	0.304629	+	0.952471i	$0.401468\pi$
$702$	0	0
$703$	−644.108	−0.0345562
$704$	0	0
$705$	2634.89	0.140760
$706$	0	0
$707$	−9654.36	−0.513564
$708$	0	0
$709$	30859.2	1.63461	0.817307	−	0.576202i	$-0.195466\pi$
$709$	30859.2	1.63461	0.817307	+	0.576202i	$0.195466\pi$
$710$	0	0
$711$	11576.3	0.610613
$712$	0	0
$713$	−40551.0	−2.12994
$714$	0	0
$715$	20433.8	1.06878
$716$	0	0
$717$	−5942.46	−0.309519
$718$	0	0
$719$	−33152.4	−1.71958	−0.859789	−	0.510650i	$-0.829405\pi$
$719$	−33152.4	−1.71958	−0.859789	+	0.510650i	$0.829405\pi$
$720$	0	0
$721$	9159.03	0.473093
$722$	0	0
$723$	−15910.7	−0.818428
$724$	0	0
$725$	−2279.18	−0.116754
$726$	0	0
$727$	16743.0	0.854146	0.427073	−	0.904217i	$-0.359545\pi$
$727$	16743.0	0.854146	0.427073	+	0.904217i	$0.359545\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−27057.5	−1.36903
$732$	0	0
$733$	−8827.55	−0.444820	−0.222410	−	0.974953i	$-0.571392\pi$
$733$	−8827.55	−0.444820	−0.222410	+	0.974953i	$0.571392\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	−37082.6	−1.85340
$738$	0	0
$739$	−36154.0	−1.79966	−0.899829	−	0.436243i	$-0.856309\pi$
$739$	−36154.0	−1.79966	−0.899829	+	0.436243i	$0.856309\pi$
$740$	0	0
$741$	−1005.11	−0.0498296
$742$	0	0
$743$	1820.69	0.0898987	0.0449494	−	0.998989i	$-0.485687\pi$
$743$	1820.69	0.0898987	0.0449494	+	0.998989i	$0.485687\pi$
$744$	0	0
$745$	−11232.8	−0.552398
$746$	0	0
$747$	−1045.53	−0.0512102
$748$	0	0
$749$	−8859.57	−0.432205
$750$	0	0
$751$	−27764.4	−1.34905	−0.674526	−	0.738251i	$-0.735652\pi$
$751$	−27764.4	−1.34905	−0.674526	+	0.738251i	$0.735652\pi$
$752$	0	0
$753$	−8988.12	−0.434987
$754$	0	0
$755$	−6557.65	−0.316102
$756$	0	0
$757$	−13518.3	−0.649050	−0.324525	−	0.945877i	$-0.605205\pi$
$757$	−13518.3	−0.649050	−0.324525	+	0.945877i	$0.605205\pi$
$758$	0	0
$759$	−30998.2	−1.48243
$760$	0	0
$761$	30695.2	1.46216	0.731078	−	0.682294i	$-0.239017\pi$
$761$	30695.2	1.46216	0.731078	+	0.682294i	$0.239017\pi$
$762$	0	0
$763$	14485.2	0.687288
$764$	0	0
$765$	−3437.41	−0.162457
$766$	0	0
$767$	−25303.2	−1.19119
$768$	0	0
$769$	4536.39	0.212726	0.106363	−	0.994327i	$-0.466079\pi$
$769$	4536.39	0.212726	0.106363	+	0.994327i	$0.466079\pi$
$770$	0	0
$771$	−2906.58	−0.135769
$772$	0	0
$773$	31238.9	1.45354	0.726768	−	0.686883i	$-0.241022\pi$
$773$	31238.9	1.45354	0.726768	+	0.686883i	$0.241022\pi$
$774$	0	0
$775$	−4951.97	−0.229523
$776$	0	0
$777$	−3268.98	−0.150932
$778$	0	0
$779$	649.155	0.0298567
$780$	0	0
$781$	34249.2	1.56919
$782$	0	0
$783$	2461.51	0.112347
$784$	0	0
$785$	−8954.70	−0.407143
$786$	0	0
$787$	−39597.2	−1.79350	−0.896752	−	0.442534i	$-0.854079\pi$
$787$	−39597.2	−1.79350	−0.896752	+	0.442534i	$0.854079\pi$
$788$	0	0
$789$	−14490.5	−0.653836
$790$	0	0
$791$	−13677.2	−0.614799
$792$	0	0
$793$	12483.3	0.559008
$794$	0	0
$795$	3004.53	0.134037
$796$	0	0
$797$	−16567.0	−0.736302	−0.368151	−	0.929766i	$-0.620009\pi$
$797$	−16567.0	−0.736302	−0.368151	+	0.929766i	$0.620009\pi$
$798$	0	0
$799$	13418.1	0.594115
$800$	0	0
$801$	−8247.96	−0.363829
$802$	0	0
$803$	−3068.72	−0.134860
$804$	0	0
$805$	−7165.25	−0.313717
$806$	0	0
$807$	14324.9	0.624859
$808$	0	0
$809$	−12141.6	−0.527657	−0.263828	−	0.964570i	$-0.584985\pi$
$809$	−12141.6	−0.527657	−0.263828	+	0.964570i	$0.584985\pi$
$810$	0	0
$811$	−30295.6	−1.31174	−0.655870	−	0.754873i	$-0.727698\pi$
$811$	−30295.6	−1.31174	−0.655870	+	0.754873i	$0.727698\pi$
$812$	0	0
$813$	425.726	0.0183651
$814$	0	0
$815$	−2455.54	−0.105538
$816$	0	0
$817$	1465.67	0.0627628
$818$	0	0
$819$	−5101.15	−0.217642
$820$	0	0
$821$	−14914.8	−0.634018	−0.317009	−	0.948423i	$-0.602678\pi$
$821$	−14914.8	−0.634018	−0.317009	+	0.948423i	$0.602678\pi$
$822$	0	0
$823$	31077.6	1.31628	0.658138	−	0.752897i	$-0.271344\pi$
$823$	31077.6	1.31628	0.658138	+	0.752897i	$0.271344\pi$
$824$	0	0
$825$	−3785.41	−0.159747
$826$	0	0
$827$	−15527.8	−0.652908	−0.326454	−	0.945213i	$-0.605854\pi$
$827$	−15527.8	−0.652908	−0.326454	+	0.945213i	$0.605854\pi$
$828$	0	0
$829$	−40221.5	−1.68510	−0.842551	−	0.538617i	$-0.818947\pi$
$829$	−40221.5	−1.68510	−0.842551	+	0.538617i	$0.818947\pi$
$830$	0	0
$831$	10863.4	0.453486
$832$	0	0
$833$	3742.96	0.155685
$834$	0	0
$835$	−4130.29	−0.171179
$836$	0	0
$837$	5348.13	0.220858
$838$	0	0
$839$	21153.7	0.870448	0.435224	−	0.900322i	$-0.356669\pi$
$839$	21153.7	0.870448	0.435224	+	0.900322i	$0.356669\pi$
$840$	0	0
$841$	−16077.5	−0.659213
$842$	0	0
$843$	−17372.6	−0.709780
$844$	0	0
$845$	−21796.2	−0.887351
$846$	0	0
$847$	8515.06	0.345432
$848$	0	0
$849$	1578.13	0.0637942
$850$	0	0
$851$	31868.1	1.28369
$852$	0	0
$853$	636.075	0.0255320	0.0127660	−	0.999919i	$-0.495936\pi$
$853$	636.075	0.0255320	0.0127660	+	0.999919i	$0.495936\pi$
$854$	0	0
$855$	186.200	0.00744782
$856$	0	0
$857$	−3941.46	−0.157104	−0.0785518	−	0.996910i	$-0.525030\pi$
$857$	−3941.46	−0.157104	−0.0785518	+	0.996910i	$0.525030\pi$
$858$	0	0
$859$	21781.6	0.865165	0.432583	−	0.901594i	$-0.357602\pi$
$859$	21781.6	0.865165	0.432583	+	0.901594i	$0.357602\pi$
$860$	0	0
$861$	3294.59	0.130406
$862$	0	0
$863$	44697.9	1.76308	0.881538	−	0.472112i	$-0.156508\pi$
$863$	44697.9	1.76308	0.881538	+	0.472112i	$0.156508\pi$
$864$	0	0
$865$	−14590.0	−0.573498
$866$	0	0
$867$	−2765.92	−0.108345
$868$	0	0
$869$	64920.1	2.53425
$870$	0	0
$871$	59490.3	2.31430
$872$	0	0
$873$	−12746.7	−0.494171
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−20171.7	−0.776682	−0.388341	−	0.921516i	$-0.626952\pi$
$877$	−20171.7	−0.776682	−0.388341	+	0.921516i	$0.626952\pi$
$878$	0	0
$879$	−5744.63	−0.220434
$880$	0	0
$881$	11577.6	0.442744	0.221372	−	0.975189i	$-0.428946\pi$
$881$	11577.6	0.442744	0.221372	+	0.975189i	$0.428946\pi$
$882$	0	0
$883$	35388.3	1.34871	0.674355	−	0.738407i	$-0.264422\pi$
$883$	35388.3	1.34871	0.674355	+	0.738407i	$0.264422\pi$
$884$	0	0
$885$	4687.48	0.178043
$886$	0	0
$887$	41705.0	1.57871	0.789356	−	0.613936i	$-0.210415\pi$
$887$	41705.0	1.57871	0.789356	+	0.613936i	$0.210415\pi$
$888$	0	0
$889$	−1569.75	−0.0592215
$890$	0	0
$891$	4088.24	0.153716
$892$	0	0
$893$	−726.838	−0.0272371
$894$	0	0
$895$	4778.72	0.178475
$896$	0	0
$897$	49729.2	1.85107
$898$	0	0
$899$	18058.3	0.669942
$900$	0	0
$901$	15300.5	0.565740
$902$	0	0
$903$	7438.55	0.274130
$904$	0	0
$905$	−1030.40	−0.0378472
$906$	0	0
$907$	28645.1	1.04867	0.524336	−	0.851511i	$-0.324313\pi$
$907$	28645.1	1.04867	0.524336	+	0.851511i	$0.324313\pi$
$908$	0	0
$909$	−12412.8	−0.452921
$910$	0	0
$911$	−13337.8	−0.485074	−0.242537	−	0.970142i	$-0.577980\pi$
$911$	−13337.8	−0.485074	−0.242537	+	0.970142i	$0.577980\pi$
$912$	0	0
$913$	−5863.36	−0.212540
$914$	0	0
$915$	−2312.55	−0.0835527
$916$	0	0
$917$	3436.29	0.123747
$918$	0	0
$919$	28911.0	1.03774	0.518871	−	0.854853i	$-0.326353\pi$
$919$	28911.0	1.03774	0.518871	+	0.854853i	$0.326353\pi$
$920$	0	0
$921$	−18732.5	−0.670203
$922$	0	0
$923$	−54944.8	−1.95940
$924$	0	0
$925$	3891.64	0.138331
$926$	0	0
$927$	11775.9	0.417229
$928$	0	0
$929$	7093.88	0.250530	0.125265	−	0.992123i	$-0.460022\pi$
$929$	7093.88	0.250530	0.125265	+	0.992123i	$0.460022\pi$
$930$	0	0
$931$	−202.751	−0.00713736
$932$	0	0
$933$	−28976.3	−1.01677
$934$	0	0
$935$	−19277.1	−0.674254
$936$	0	0
$937$	−19271.1	−0.671888	−0.335944	−	0.941882i	$-0.609055\pi$
$937$	−19271.1	−0.671888	−0.335944	+	0.941882i	$0.609055\pi$
$938$	0	0
$939$	−6595.01	−0.229201
$940$	0	0
$941$	−18115.2	−0.627563	−0.313782	−	0.949495i	$-0.601596\pi$
$941$	−18115.2	−0.627563	−0.313782	+	0.949495i	$0.601596\pi$
$942$	0	0
$943$	−32117.8	−1.10912
$944$	0	0
$945$	945.000	0.0325300
$946$	0	0
$947$	2475.55	0.0849468	0.0424734	−	0.999098i	$-0.486476\pi$
$947$	2475.55	0.0849468	0.0424734	+	0.999098i	$0.486476\pi$
$948$	0	0
$949$	4923.04	0.168397
$950$	0	0
$951$	−9092.35	−0.310031
$952$	0	0
$953$	12866.6	0.437344	0.218672	−	0.975798i	$-0.429827\pi$
$953$	12866.6	0.437344	0.218672	+	0.975798i	$0.429827\pi$
$954$	0	0
$955$	−10093.2	−0.341997
$956$	0	0
$957$	13804.2	0.466277
$958$	0	0
$959$	12817.9	0.431606
$960$	0	0
$961$	9444.26	0.317017
$962$	0	0
$963$	−11390.9	−0.381169
$964$	0	0
$965$	−5158.45	−0.172079
$966$	0	0
$967$	2142.86	0.0712614	0.0356307	−	0.999365i	$-0.488656\pi$
$967$	2142.86	0.0712614	0.0356307	+	0.999365i	$0.488656\pi$
$968$	0	0
$969$	948.215	0.0314356
$970$	0	0
$971$	−2879.06	−0.0951529	−0.0475765	−	0.998868i	$-0.515150\pi$
$971$	−2879.06	−0.0951529	−0.0475765	+	0.998868i	$0.515150\pi$
$972$	0	0
$973$	21355.9	0.703636
$974$	0	0
$975$	6072.79	0.199472
$976$	0	0
$977$	48741.1	1.59607	0.798037	−	0.602608i	$-0.205872\pi$
$977$	48741.1	1.59607	0.798037	+	0.602608i	$0.205872\pi$
$978$	0	0
$979$	−46254.7	−1.51002
$980$	0	0
$981$	18623.9	0.606131
$982$	0	0
$983$	45756.8	1.48466	0.742328	−	0.670037i	$-0.233722\pi$
$983$	45756.8	1.48466	0.742328	+	0.670037i	$0.233722\pi$
$984$	0	0
$985$	1029.78	0.0333110
$986$	0	0
$987$	−3688.85	−0.118964
$988$	0	0
$989$	−72515.7	−2.33151
$990$	0	0
$991$	51552.1	1.65248	0.826240	−	0.563319i	$-0.190476\pi$
$991$	51552.1	1.65248	0.826240	+	0.563319i	$0.190476\pi$
$992$	0	0
$993$	14261.2	0.455756
$994$	0	0
$995$	−9156.88	−0.291751
$996$	0	0
$997$	25565.3	0.812097	0.406048	−	0.913852i	$-0.366906\pi$
$997$	25565.3	0.812097	0.406048	+	0.913852i	$0.366906\pi$
$998$	0	0
$999$	−4202.97	−0.133109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.4.a.bd.1.2		2
4.3	odd	2		105.4.a.d.1.2	✓	2
12.11	even	2		315.4.a.l.1.1		2
20.3	even	4		525.4.d.k.274.2		4
20.7	even	4		525.4.d.k.274.3		4
20.19	odd	2		525.4.a.o.1.1		2
28.27	even	2		735.4.a.m.1.2		2
60.59	even	2		1575.4.a.n.1.2		2
84.83	odd	2		2205.4.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.d.1.2	✓	2	4.3	odd	2
315.4.a.l.1.1		2	12.11	even	2
525.4.a.o.1.1		2	20.19	odd	2
525.4.d.k.274.2		4	20.3	even	4
525.4.d.k.274.3		4	20.7	even	4
735.4.a.m.1.2		2	28.27	even	2
1575.4.a.n.1.2		2	60.59	even	2
1680.4.a.bd.1.2		2	1.1	even	1	trivial
2205.4.a.be.1.1		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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +50.4721 q^{11} -80.9706 q^{13} +15.0000 q^{15} +76.3870 q^{17} -4.13777 q^{19} -21.0000 q^{21} +204.721 q^{23} +25.0000 q^{25} -27.0000 q^{27} -91.1672 q^{29} -198.079 q^{31} -151.416 q^{33} -35.0000 q^{35} +155.666 q^{37} +242.912 q^{39} -156.885 q^{41} -354.217 q^{43} -45.0000 q^{45} +175.659 q^{47} +49.0000 q^{49} -229.161 q^{51} +200.302 q^{53} -252.361 q^{55} +12.4133 q^{57} +312.498 q^{59} -154.170 q^{61} +63.0000 q^{63} +404.853 q^{65} -734.715 q^{67} -614.164 q^{69} +678.577 q^{71} -60.8003 q^{73} -75.0000 q^{75} +353.305 q^{77} +1286.26 q^{79} +81.0000 q^{81} -116.170 q^{83} -381.935 q^{85} +273.502 q^{87} -916.440 q^{89} -566.794 q^{91} +594.237 q^{93} +20.6888 q^{95} -1416.30 q^{97} +454.249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 10 * q^5 + 14 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9} + 92 q^{11} + 8 q^{13} + 30 q^{15} - 44 q^{17} + 108 q^{19} - 42 q^{21} + 320 q^{23} + 50 q^{25} - 54 q^{27} - 236 q^{29} + 60 q^{31} - 276 q^{33} - 70 q^{35} + 204 q^{37} - 24 q^{39} + 44 q^{41} - 136 q^{43} - 90 q^{45} - 400 q^{47} + 98 q^{49} + 132 q^{51} + 16 q^{53} - 460 q^{55} - 324 q^{57} + 464 q^{59} - 684 q^{61} + 126 q^{63} - 40 q^{65} - 736 q^{67} - 960 q^{69} + 740 q^{71} + 424 q^{73} - 150 q^{75} + 644 q^{77} + 408 q^{79} + 162 q^{81} - 608 q^{83} + 220 q^{85} + 708 q^{87} - 1332 q^{89} + 56 q^{91} - 180 q^{93} - 540 q^{95} - 2448 q^{97} + 828 q^{99}+O(q^{100}) \) 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

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