LMFDB - Embedded newform 1680.4.a.y.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1.1
Root			$3.70156$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -37.4031 q^{11} +29.0156 q^{13} +15.0000 q^{15} +58.4187 q^{17} +54.5969 q^{19} +21.0000 q^{21} -161.675 q^{23} +25.0000 q^{25} -27.0000 q^{27} +137.581 q^{29} -154.659 q^{31} +112.209 q^{33} +35.0000 q^{35} -350.125 q^{37} -87.0469 q^{39} +353.769 q^{41} +518.156 q^{43} -45.0000 q^{45} +542.219 q^{47} +49.0000 q^{49} -175.256 q^{51} +305.309 q^{53} +187.016 q^{55} -163.791 q^{57} -14.6813 q^{59} -171.069 q^{61} -63.0000 q^{63} -145.078 q^{65} -551.956 q^{67} +485.025 q^{69} +120.334 q^{71} +284.659 q^{73} -75.0000 q^{75} +261.822 q^{77} -941.612 q^{79} +81.0000 q^{81} -377.150 q^{83} -292.094 q^{85} -412.744 q^{87} -677.725 q^{89} -203.109 q^{91} +463.978 q^{93} -272.984 q^{95} -1225.03 q^{97} -336.628 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 62 q^{11} - 6 q^{13} + 30 q^{15} + 40 q^{17} + 122 q^{19} + 42 q^{21} - 16 q^{23} + 50 q^{25} - 54 q^{27} + 352 q^{29} - 66 q^{31} + 186 q^{33} + 70 q^{35} - 188 q^{37} + 18 q^{39} + 16 q^{41} + 396 q^{43} - 90 q^{45} + 188 q^{47} + 98 q^{49} - 120 q^{51} + 982 q^{53} + 310 q^{55} - 366 q^{57} - 516 q^{59} - 880 q^{61} - 126 q^{63} + 30 q^{65} + 356 q^{67} + 48 q^{69} - 310 q^{71} + 326 q^{73} - 150 q^{75} + 434 q^{77} - 1832 q^{79} + 162 q^{81} + 680 q^{83} - 200 q^{85} - 1056 q^{87} + 796 q^{89} + 42 q^{91} + 198 q^{93} - 610 q^{95} - 670 q^{97} - 558 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 10 * q^5 - 14 * q^7 + 18 * q^9 - 62 * q^11 - 6 * q^13 + 30 * q^15 + 40 * q^17 + 122 * q^19 + 42 * q^21 - 16 * q^23 + 50 * q^25 - 54 * q^27 + 352 * q^29 - 66 * q^31 + 186 * q^33 + 70 * q^35 - 188 * q^37 + 18 * q^39 + 16 * q^41 + 396 * q^43 - 90 * q^45 + 188 * q^47 + 98 * q^49 - 120 * q^51 + 982 * q^53 + 310 * q^55 - 366 * q^57 - 516 * q^59 - 880 * q^61 - 126 * q^63 + 30 * q^65 + 356 * q^67 + 48 * q^69 - 310 * q^71 + 326 * q^73 - 150 * q^75 + 434 * q^77 - 1832 * q^79 + 162 * q^81 + 680 * q^83 - 200 * q^85 - 1056 * q^87 + 796 * q^89 + 42 * q^91 + 198 * q^93 - 610 * q^95 - 670 * q^97 - 558 * 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
$11$	−37.4031	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$11$	−37.4031	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$12$	0	0
$13$	29.0156	0.619037	0.309519	−	0.950893i	$-0.399832\pi$
$13$	29.0156	0.619037	0.309519	+	0.950893i	$0.399832\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	58.4187	0.833449	0.416724	−	0.909033i	$-0.363178\pi$
$17$	58.4187	0.833449	0.416724	+	0.909033i	$0.363178\pi$
$18$	0	0
$19$	54.5969	0.659231	0.329615	−	0.944115i	$-0.393081\pi$
$19$	54.5969	0.659231	0.329615	+	0.944115i	$0.393081\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	−161.675	−1.46572	−0.732860	−	0.680379i	$-0.761815\pi$
$23$	−161.675	−1.46572	−0.732860	+	0.680379i	$0.761815\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	137.581	0.880972	0.440486	−	0.897759i	$-0.354806\pi$
$29$	137.581	0.880972	0.440486	+	0.897759i	$0.354806\pi$
$30$	0	0
$31$	−154.659	−0.896053	−0.448026	−	0.894020i	$-0.647873\pi$
$31$	−154.659	−0.896053	−0.448026	+	0.894020i	$0.647873\pi$
$32$	0	0
$33$	112.209	0.591913
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	−350.125	−1.55568	−0.777840	−	0.628462i	$-0.783685\pi$
$37$	−350.125	−1.55568	−0.777840	+	0.628462i	$0.783685\pi$
$38$	0	0
$39$	−87.0469	−0.357401
$40$	0	0
$41$	353.769	1.34755	0.673773	−	0.738938i	$-0.264673\pi$
$41$	353.769	1.34755	0.673773	+	0.738938i	$0.264673\pi$
$42$	0	0
$43$	518.156	1.83763	0.918815	−	0.394689i	$-0.129148\pi$
$43$	518.156	1.83763	0.918815	+	0.394689i	$0.129148\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	542.219	1.68278	0.841391	−	0.540427i	$-0.181737\pi$
$47$	542.219	1.68278	0.841391	+	0.540427i	$0.181737\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−175.256	−0.481192
$52$	0	0
$53$	305.309	0.791273	0.395637	−	0.918407i	$-0.370524\pi$
$53$	305.309	0.791273	0.395637	+	0.918407i	$0.370524\pi$
$54$	0	0
$55$	187.016	0.458494
$56$	0	0
$57$	−163.791	−0.380607
$58$	0	0
$59$	−14.6813	−0.0323956	−0.0161978	−	0.999869i	$-0.505156\pi$
$59$	−14.6813	−0.0323956	−0.0161978	+	0.999869i	$0.505156\pi$
$60$	0	0
$61$	−171.069	−0.359067	−0.179534	−	0.983752i	$-0.557459\pi$
$61$	−171.069	−0.359067	−0.179534	+	0.983752i	$0.557459\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−145.078	−0.276842
$66$	0	0
$67$	−551.956	−1.00645	−0.503225	−	0.864155i	$-0.667853\pi$
$67$	−551.956	−1.00645	−0.503225	+	0.864155i	$0.667853\pi$
$68$	0	0
$69$	485.025	0.846234
$70$	0	0
$71$	120.334	0.201142	0.100571	−	0.994930i	$-0.467933\pi$
$71$	120.334	0.201142	0.100571	+	0.994930i	$0.467933\pi$
$72$	0	0
$73$	284.659	0.456395	0.228198	−	0.973615i	$-0.426717\pi$
$73$	284.659	0.456395	0.228198	+	0.973615i	$0.426717\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	261.822	0.387498
$78$	0	0
$79$	−941.612	−1.34101	−0.670504	−	0.741906i	$-0.733922\pi$
$79$	−941.612	−1.34101	−0.670504	+	0.741906i	$0.733922\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−377.150	−0.498766	−0.249383	−	0.968405i	$-0.580228\pi$
$83$	−377.150	−0.498766	−0.249383	+	0.968405i	$0.580228\pi$
$84$	0	0
$85$	−292.094	−0.372730
$86$	0	0
$87$	−412.744	−0.508630
$88$	0	0
$89$	−677.725	−0.807176	−0.403588	−	0.914941i	$-0.632237\pi$
$89$	−677.725	−0.807176	−0.403588	+	0.914941i	$0.632237\pi$
$90$	0	0
$91$	−203.109	−0.233974
$92$	0	0
$93$	463.978	0.517336
$94$	0	0
$95$	−272.984	−0.294817
$96$	0	0
$97$	−1225.03	−1.28230	−0.641151	−	0.767414i	$-0.721543\pi$
$97$	−1225.03	−1.28230	−0.641151	+	0.767414i	$0.721543\pi$
$98$	0	0
$99$	−336.628	−0.341741
$100$	0	0
$101$	−338.144	−0.333134	−0.166567	−	0.986030i	$-0.553268\pi$
$101$	−338.144	−0.333134	−0.166567	+	0.986030i	$0.553268\pi$
$102$	0	0
$103$	566.700	0.542122	0.271061	−	0.962562i	$-0.412625\pi$
$103$	566.700	0.542122	0.271061	+	0.962562i	$0.412625\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	562.531	0.508242	0.254121	−	0.967172i	$-0.418214\pi$
$107$	562.531	0.508242	0.254121	+	0.967172i	$0.418214\pi$
$108$	0	0
$109$	1830.79	1.60879	0.804396	−	0.594094i	$-0.202489\pi$
$109$	1830.79	1.60879	0.804396	+	0.594094i	$0.202489\pi$
$110$	0	0
$111$	1050.37	0.898173
$112$	0	0
$113$	−31.8032	−0.0264761	−0.0132380	−	0.999912i	$-0.504214\pi$
$113$	−31.8032	−0.0264761	−0.0132380	+	0.999912i	$0.504214\pi$
$114$	0	0
$115$	808.375	0.655490
$116$	0	0
$117$	261.141	0.206346
$118$	0	0
$119$	−408.931	−0.315014
$120$	0	0
$121$	67.9937	0.0510847
$122$	0	0
$123$	−1061.31	−0.778006
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2220.81	−1.55169	−0.775847	−	0.630921i	$-0.782677\pi$
$127$	−2220.81	−1.55169	−0.775847	+	0.630921i	$0.782677\pi$
$128$	0	0
$129$	−1554.47	−1.06096
$130$	0	0
$131$	−646.512	−0.431191	−0.215596	−	0.976483i	$-0.569169\pi$
$131$	−646.512	−0.431191	−0.215596	+	0.976483i	$0.569169\pi$
$132$	0	0
$133$	−382.178	−0.249166
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−896.009	−0.558768	−0.279384	−	0.960179i	$-0.590130\pi$
$137$	−896.009	−0.558768	−0.279384	+	0.960179i	$0.590130\pi$
$138$	0	0
$139$	2313.61	1.41178	0.705891	−	0.708320i	$-0.250547\pi$
$139$	2313.61	1.41178	0.705891	+	0.708320i	$0.250547\pi$
$140$	0	0
$141$	−1626.66	−0.971554
$142$	0	0
$143$	−1085.27	−0.634652
$144$	0	0
$145$	−687.906	−0.393983
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	819.337	0.450488	0.225244	−	0.974302i	$-0.427682\pi$
$149$	819.337	0.450488	0.225244	+	0.974302i	$0.427682\pi$
$150$	0	0
$151$	−534.744	−0.288191	−0.144095	−	0.989564i	$-0.546027\pi$
$151$	−534.744	−0.288191	−0.144095	+	0.989564i	$0.546027\pi$
$152$	0	0
$153$	525.769	0.277816
$154$	0	0
$155$	773.297	0.400727
$156$	0	0
$157$	1564.76	0.795423	0.397711	−	0.917511i	$-0.369805\pi$
$157$	1564.76	0.795423	0.397711	+	0.917511i	$0.369805\pi$
$158$	0	0
$159$	−915.928	−0.456842
$160$	0	0
$161$	1131.72	0.553990
$162$	0	0
$163$	1114.31	0.535455	0.267728	−	0.963495i	$-0.413727\pi$
$163$	1114.31	0.535455	0.267728	+	0.963495i	$0.413727\pi$
$164$	0	0
$165$	−561.047	−0.264712
$166$	0	0
$167$	1774.47	0.822231	0.411115	−	0.911583i	$-0.365139\pi$
$167$	1774.47	0.822231	0.411115	+	0.911583i	$0.365139\pi$
$168$	0	0
$169$	−1355.09	−0.616793
$170$	0	0
$171$	491.372	0.219744
$172$	0	0
$173$	−4215.88	−1.85276	−0.926380	−	0.376590i	$-0.877097\pi$
$173$	−4215.88	−1.85276	−0.926380	+	0.376590i	$0.877097\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	44.0438	0.0187036
$178$	0	0
$179$	2430.70	1.01497	0.507483	−	0.861662i	$-0.330576\pi$
$179$	2430.70	1.01497	0.507483	+	0.861662i	$0.330576\pi$
$180$	0	0
$181$	−2700.91	−1.10916	−0.554578	−	0.832132i	$-0.687120\pi$
$181$	−2700.91	−1.10916	−0.554578	+	0.832132i	$0.687120\pi$
$182$	0	0
$183$	513.206	0.207308
$184$	0	0
$185$	1750.62	0.695722
$186$	0	0
$187$	−2185.04	−0.854472
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−3611.10	−1.36801	−0.684005	−	0.729478i	$-0.739763\pi$
$191$	−3611.10	−1.36801	−0.684005	+	0.729478i	$0.739763\pi$
$192$	0	0
$193$	−4468.33	−1.66651	−0.833257	−	0.552886i	$-0.813526\pi$
$193$	−4468.33	−1.66651	−0.833257	+	0.552886i	$0.813526\pi$
$194$	0	0
$195$	435.234	0.159835
$196$	0	0
$197$	−434.422	−0.157113	−0.0785566	−	0.996910i	$-0.525031\pi$
$197$	−434.422	−0.157113	−0.0785566	+	0.996910i	$0.525031\pi$
$198$	0	0
$199$	468.915	0.167038	0.0835189	−	0.996506i	$-0.473384\pi$
$199$	468.915	0.167038	0.0835189	+	0.996506i	$0.473384\pi$
$200$	0	0
$201$	1655.87	0.581074
$202$	0	0
$203$	−963.069	−0.332976
$204$	0	0
$205$	−1768.84	−0.602641
$206$	0	0
$207$	−1455.07	−0.488573
$208$	0	0
$209$	−2042.09	−0.675859
$210$	0	0
$211$	−3735.51	−1.21878	−0.609392	−	0.792869i	$-0.708586\pi$
$211$	−3735.51	−1.21878	−0.609392	+	0.792869i	$0.708586\pi$
$212$	0	0
$213$	−361.003	−0.116129
$214$	0	0
$215$	−2590.78	−0.821813
$216$	0	0
$217$	1082.62	0.338676
$218$	0	0
$219$	−853.978	−0.263500
$220$	0	0
$221$	1695.06	0.515936
$222$	0	0
$223$	−842.806	−0.253087	−0.126544	−	0.991961i	$-0.540388\pi$
$223$	−842.806	−0.253087	−0.126544	+	0.991961i	$0.540388\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−992.150	−0.290094	−0.145047	−	0.989425i	$-0.546333\pi$
$227$	−992.150	−0.290094	−0.145047	+	0.989425i	$0.546333\pi$
$228$	0	0
$229$	−6411.39	−1.85012	−0.925059	−	0.379825i	$-0.875984\pi$
$229$	−6411.39	−1.85012	−0.925059	+	0.379825i	$0.875984\pi$
$230$	0	0
$231$	−785.466	−0.223722
$232$	0	0
$233$	−2274.35	−0.639476	−0.319738	−	0.947506i	$-0.603595\pi$
$233$	−2274.35	−0.639476	−0.319738	+	0.947506i	$0.603595\pi$
$234$	0	0
$235$	−2711.09	−0.752563
$236$	0	0
$237$	2824.84	0.774232
$238$	0	0
$239$	−2863.12	−0.774893	−0.387447	−	0.921892i	$-0.626643\pi$
$239$	−2863.12	−0.774893	−0.387447	+	0.921892i	$0.626643\pi$
$240$	0	0
$241$	−5364.23	−1.43378	−0.716889	−	0.697187i	$-0.754435\pi$
$241$	−5364.23	−1.43378	−0.716889	+	0.697187i	$0.754435\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	1584.16	0.408088
$248$	0	0
$249$	1131.45	0.287963
$250$	0	0
$251$	−5569.81	−1.40065	−0.700325	−	0.713824i	$-0.746961\pi$
$251$	−5569.81	−1.40065	−0.700325	+	0.713824i	$0.746961\pi$
$252$	0	0
$253$	6047.15	1.50269
$254$	0	0
$255$	876.281	0.215196
$256$	0	0
$257$	2095.36	0.508580	0.254290	−	0.967128i	$-0.418158\pi$
$257$	2095.36	0.508580	0.254290	+	0.967128i	$0.418158\pi$
$258$	0	0
$259$	2450.87	0.587992
$260$	0	0
$261$	1238.23	0.293657
$262$	0	0
$263$	7465.88	1.75044	0.875220	−	0.483724i	$-0.160716\pi$
$263$	7465.88	1.75044	0.875220	+	0.483724i	$0.160716\pi$
$264$	0	0
$265$	−1526.55	−0.353868
$266$	0	0
$267$	2033.17	0.466023
$268$	0	0
$269$	−6521.38	−1.47812	−0.739062	−	0.673637i	$-0.764731\pi$
$269$	−6521.38	−1.47812	−0.739062	+	0.673637i	$0.764731\pi$
$270$	0	0
$271$	−2409.70	−0.540144	−0.270072	−	0.962840i	$-0.587048\pi$
$271$	−2409.70	−0.540144	−0.270072	+	0.962840i	$0.587048\pi$
$272$	0	0
$273$	609.328	0.135085
$274$	0	0
$275$	−935.078	−0.205045
$276$	0	0
$277$	−2219.83	−0.481503	−0.240752	−	0.970587i	$-0.577394\pi$
$277$	−2219.83	−0.481503	−0.240752	+	0.970587i	$0.577394\pi$
$278$	0	0
$279$	−1391.93	−0.298684
$280$	0	0
$281$	5838.56	1.23950	0.619749	−	0.784800i	$-0.287234\pi$
$281$	5838.56	1.23950	0.619749	+	0.784800i	$0.287234\pi$
$282$	0	0
$283$	3645.04	0.765636	0.382818	−	0.923824i	$-0.374954\pi$
$283$	3645.04	0.765636	0.382818	+	0.923824i	$0.374954\pi$
$284$	0	0
$285$	818.953	0.170213
$286$	0	0
$287$	−2476.38	−0.509325
$288$	0	0
$289$	−1500.25	−0.305363
$290$	0	0
$291$	3675.10	0.740338
$292$	0	0
$293$	3777.91	0.753268	0.376634	−	0.926362i	$-0.377081\pi$
$293$	3777.91	0.753268	0.376634	+	0.926362i	$0.377081\pi$
$294$	0	0
$295$	73.4064	0.0144877
$296$	0	0
$297$	1009.88	0.197304
$298$	0	0
$299$	−4691.10	−0.907336
$300$	0	0
$301$	−3627.09	−0.694559
$302$	0	0
$303$	1014.43	0.192335
$304$	0	0
$305$	855.344	0.160580
$306$	0	0
$307$	4799.64	0.892281	0.446140	−	0.894963i	$-0.352798\pi$
$307$	4799.64	0.892281	0.446140	+	0.894963i	$0.352798\pi$
$308$	0	0
$309$	−1700.10	−0.312994
$310$	0	0
$311$	−580.113	−0.105772	−0.0528861	−	0.998601i	$-0.516842\pi$
$311$	−580.113	−0.105772	−0.0528861	+	0.998601i	$0.516842\pi$
$312$	0	0
$313$	−6114.78	−1.10424	−0.552121	−	0.833764i	$-0.686182\pi$
$313$	−6114.78	−1.10424	−0.552121	+	0.833764i	$0.686182\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	−4300.63	−0.761979	−0.380989	−	0.924579i	$-0.624417\pi$
$317$	−4300.63	−0.761979	−0.380989	+	0.924579i	$0.624417\pi$
$318$	0	0
$319$	−5145.97	−0.903194
$320$	0	0
$321$	−1687.59	−0.293434
$322$	0	0
$323$	3189.48	0.549435
$324$	0	0
$325$	725.391	0.123807
$326$	0	0
$327$	−5492.38	−0.928836
$328$	0	0
$329$	−3795.53	−0.636032
$330$	0	0
$331$	6687.54	1.11051	0.555257	−	0.831679i	$-0.312620\pi$
$331$	6687.54	1.11051	0.555257	+	0.831679i	$0.312620\pi$
$332$	0	0
$333$	−3151.12	−0.518560
$334$	0	0
$335$	2759.78	0.450098
$336$	0	0
$337$	5869.28	0.948723	0.474362	−	0.880330i	$-0.342679\pi$
$337$	5869.28	0.948723	0.474362	+	0.880330i	$0.342679\pi$
$338$	0	0
$339$	95.4097	0.0152860
$340$	0	0
$341$	5784.74	0.918655
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−2425.12	−0.378447
$346$	0	0
$347$	−1937.22	−0.299699	−0.149850	−	0.988709i	$-0.547879\pi$
$347$	−1937.22	−0.299699	−0.149850	+	0.988709i	$0.547879\pi$
$348$	0	0
$349$	−9748.82	−1.49525	−0.747625	−	0.664121i	$-0.768806\pi$
$349$	−9748.82	−1.49525	−0.747625	+	0.664121i	$0.768806\pi$
$350$	0	0
$351$	−783.422	−0.119134
$352$	0	0
$353$	−4576.61	−0.690052	−0.345026	−	0.938593i	$-0.612130\pi$
$353$	−4576.61	−0.690052	−0.345026	+	0.938593i	$0.612130\pi$
$354$	0	0
$355$	−601.672	−0.0899533
$356$	0	0
$357$	1226.79	0.181873
$358$	0	0
$359$	−10849.9	−1.59509	−0.797546	−	0.603258i	$-0.793869\pi$
$359$	−10849.9	−1.59509	−0.797546	+	0.603258i	$0.793869\pi$
$360$	0	0
$361$	−3878.18	−0.565415
$362$	0	0
$363$	−203.981	−0.0294938
$364$	0	0
$365$	−1423.30	−0.204106
$366$	0	0
$367$	−11467.7	−1.63108	−0.815541	−	0.578699i	$-0.803561\pi$
$367$	−11467.7	−1.63108	−0.815541	+	0.578699i	$0.803561\pi$
$368$	0	0
$369$	3183.92	0.449182
$370$	0	0
$371$	−2137.17	−0.299073
$372$	0	0
$373$	539.982	0.0749576	0.0374788	−	0.999297i	$-0.488067\pi$
$373$	539.982	0.0749576	0.0374788	+	0.999297i	$0.488067\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	3992.01	0.545355
$378$	0	0
$379$	−8577.57	−1.16253	−0.581267	−	0.813713i	$-0.697443\pi$
$379$	−8577.57	−1.16253	−0.581267	+	0.813713i	$0.697443\pi$
$380$	0	0
$381$	6662.44	0.895871
$382$	0	0
$383$	−8627.96	−1.15109	−0.575546	−	0.817770i	$-0.695210\pi$
$383$	−8627.96	−1.15109	−0.575546	+	0.817770i	$0.695210\pi$
$384$	0	0
$385$	−1309.11	−0.173295
$386$	0	0
$387$	4663.41	0.612543
$388$	0	0
$389$	9234.06	1.20356	0.601781	−	0.798661i	$-0.294458\pi$
$389$	9234.06	1.20356	0.601781	+	0.798661i	$0.294458\pi$
$390$	0	0
$391$	−9444.85	−1.22160
$392$	0	0
$393$	1939.54	0.248948
$394$	0	0
$395$	4708.06	0.599717
$396$	0	0
$397$	11618.0	1.46874	0.734372	−	0.678747i	$-0.237477\pi$
$397$	11618.0	1.46874	0.734372	+	0.678747i	$0.237477\pi$
$398$	0	0
$399$	1146.53	0.143856
$400$	0	0
$401$	11157.1	1.38942	0.694711	−	0.719289i	$-0.255532\pi$
$401$	11157.1	1.38942	0.694711	+	0.719289i	$0.255532\pi$
$402$	0	0
$403$	−4487.54	−0.554690
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	13095.8	1.59492
$408$	0	0
$409$	−7428.08	−0.898031	−0.449015	−	0.893524i	$-0.648225\pi$
$409$	−7428.08	−0.898031	−0.449015	+	0.893524i	$0.648225\pi$
$410$	0	0
$411$	2688.03	0.322605
$412$	0	0
$413$	102.769	0.0122444
$414$	0	0
$415$	1885.75	0.223055
$416$	0	0
$417$	−6940.83	−0.815093
$418$	0	0
$419$	−9644.74	−1.12453	−0.562263	−	0.826959i	$-0.690069\pi$
$419$	−9644.74	−1.12453	−0.562263	+	0.826959i	$0.690069\pi$
$420$	0	0
$421$	9918.18	1.14818	0.574088	−	0.818793i	$-0.305357\pi$
$421$	9918.18	1.14818	0.574088	+	0.818793i	$0.305357\pi$
$422$	0	0
$423$	4879.97	0.560927
$424$	0	0
$425$	1460.47	0.166690
$426$	0	0
$427$	1197.48	0.135715
$428$	0	0
$429$	3255.82	0.366417
$430$	0	0
$431$	16324.1	1.82437	0.912185	−	0.409779i	$-0.134394\pi$
$431$	16324.1	1.82437	0.912185	+	0.409779i	$0.134394\pi$
$432$	0	0
$433$	−5168.75	−0.573659	−0.286829	−	0.957982i	$-0.592601\pi$
$433$	−5168.75	−0.573659	−0.286829	+	0.957982i	$0.592601\pi$
$434$	0	0
$435$	2063.72	0.227466
$436$	0	0
$437$	−8826.95	−0.966248
$438$	0	0
$439$	−18339.0	−1.99378	−0.996892	−	0.0787782i	$-0.974898\pi$
$439$	−18339.0	−1.99378	−0.996892	+	0.0787782i	$0.974898\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	1613.28	0.173023	0.0865113	−	0.996251i	$-0.472428\pi$
$443$	1613.28	0.173023	0.0865113	+	0.996251i	$0.472428\pi$
$444$	0	0
$445$	3388.62	0.360980
$446$	0	0
$447$	−2458.01	−0.260089
$448$	0	0
$449$	−886.750	−0.0932034	−0.0466017	−	0.998914i	$-0.514839\pi$
$449$	−886.750	−0.0932034	−0.0466017	+	0.998914i	$0.514839\pi$
$450$	0	0
$451$	−13232.1	−1.38154
$452$	0	0
$453$	1604.23	0.166387
$454$	0	0
$455$	1015.55	0.104636
$456$	0	0
$457$	7391.22	0.756557	0.378279	−	0.925692i	$-0.376516\pi$
$457$	7391.22	0.756557	0.378279	+	0.925692i	$0.376516\pi$
$458$	0	0
$459$	−1577.31	−0.160397
$460$	0	0
$461$	−7133.35	−0.720679	−0.360340	−	0.932821i	$-0.617339\pi$
$461$	−7133.35	−0.720679	−0.360340	+	0.932821i	$0.617339\pi$
$462$	0	0
$463$	−14461.8	−1.45162	−0.725808	−	0.687897i	$-0.758534\pi$
$463$	−14461.8	−1.45162	−0.725808	+	0.687897i	$0.758534\pi$
$464$	0	0
$465$	−2319.89	−0.231360
$466$	0	0
$467$	16393.5	1.62441	0.812206	−	0.583370i	$-0.198266\pi$
$467$	16393.5	1.62441	0.812206	+	0.583370i	$0.198266\pi$
$468$	0	0
$469$	3863.69	0.380403
$470$	0	0
$471$	−4694.28	−0.459238
$472$	0	0
$473$	−19380.7	−1.88398
$474$	0	0
$475$	1364.92	0.131846
$476$	0	0
$477$	2747.78	0.263758
$478$	0	0
$479$	12991.4	1.23923	0.619617	−	0.784904i	$-0.287288\pi$
$479$	12991.4	1.23923	0.619617	+	0.784904i	$0.287288\pi$
$480$	0	0
$481$	−10159.1	−0.963025
$482$	0	0
$483$	−3395.17	−0.319846
$484$	0	0
$485$	6125.17	0.573463
$486$	0	0
$487$	12863.5	1.19692	0.598461	−	0.801152i	$-0.295779\pi$
$487$	12863.5	1.19692	0.598461	+	0.801152i	$0.295779\pi$
$488$	0	0
$489$	−3342.92	−0.309145
$490$	0	0
$491$	4898.10	0.450200	0.225100	−	0.974336i	$-0.427729\pi$
$491$	4898.10	0.450200	0.225100	+	0.974336i	$0.427729\pi$
$492$	0	0
$493$	8037.32	0.734245
$494$	0	0
$495$	1683.14	0.152831
$496$	0	0
$497$	−842.340	−0.0760244
$498$	0	0
$499$	−10308.0	−0.924746	−0.462373	−	0.886686i	$-0.653002\pi$
$499$	−10308.0	−0.924746	−0.462373	+	0.886686i	$0.653002\pi$
$500$	0	0
$501$	−5323.41	−0.474715
$502$	0	0
$503$	15119.6	1.34026	0.670130	−	0.742244i	$-0.266238\pi$
$503$	15119.6	1.34026	0.670130	+	0.742244i	$0.266238\pi$
$504$	0	0
$505$	1690.72	0.148982
$506$	0	0
$507$	4065.28	0.356105
$508$	0	0
$509$	14183.8	1.23514	0.617571	−	0.786515i	$-0.288117\pi$
$509$	14183.8	1.23514	0.617571	+	0.786515i	$0.288117\pi$
$510$	0	0
$511$	−1992.62	−0.172501
$512$	0	0
$513$	−1474.12	−0.126869
$514$	0	0
$515$	−2833.50	−0.242444
$516$	0	0
$517$	−20280.7	−1.72523
$518$	0	0
$519$	12647.6	1.06969
$520$	0	0
$521$	−7464.08	−0.627653	−0.313827	−	0.949480i	$-0.601611\pi$
$521$	−7464.08	−0.627653	−0.313827	+	0.949480i	$0.601611\pi$
$522$	0	0
$523$	16642.9	1.39148	0.695739	−	0.718295i	$-0.255077\pi$
$523$	16642.9	1.39148	0.695739	+	0.718295i	$0.255077\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	−9035.01	−0.746814
$528$	0	0
$529$	13971.8	1.14834
$530$	0	0
$531$	−132.132	−0.0107985
$532$	0	0
$533$	10264.8	0.834181
$534$	0	0
$535$	−2812.66	−0.227293
$536$	0	0
$537$	−7292.09	−0.585991
$538$	0	0
$539$	−1832.75	−0.146461
$540$	0	0
$541$	67.8755	0.00539408	0.00269704	−	0.999996i	$-0.499142\pi$
$541$	67.8755	0.00539408	0.00269704	+	0.999996i	$0.499142\pi$
$542$	0	0
$543$	8102.74	0.640372
$544$	0	0
$545$	−9153.97	−0.719473
$546$	0	0
$547$	9212.91	0.720138	0.360069	−	0.932926i	$-0.382753\pi$
$547$	9212.91	0.720138	0.360069	+	0.932926i	$0.382753\pi$
$548$	0	0
$549$	−1539.62	−0.119689
$550$	0	0
$551$	7511.51	0.580764
$552$	0	0
$553$	6591.29	0.506854
$554$	0	0
$555$	−5251.87	−0.401675
$556$	0	0
$557$	16699.6	1.27035	0.635173	−	0.772370i	$-0.280929\pi$
$557$	16699.6	1.27035	0.635173	+	0.772370i	$0.280929\pi$
$558$	0	0
$559$	15034.6	1.13756
$560$	0	0
$561$	6555.13	0.493329
$562$	0	0
$563$	−14772.8	−1.10586	−0.552931	−	0.833227i	$-0.686491\pi$
$563$	−14772.8	−1.10586	−0.552931	+	0.833227i	$0.686491\pi$
$564$	0	0
$565$	159.016	0.0118405
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	5663.76	0.417289	0.208644	−	0.977992i	$-0.433095\pi$
$569$	5663.76	0.417289	0.208644	+	0.977992i	$0.433095\pi$
$570$	0	0
$571$	−5579.58	−0.408929	−0.204464	−	0.978874i	$-0.565545\pi$
$571$	−5579.58	−0.408929	−0.204464	+	0.978874i	$0.565545\pi$
$572$	0	0
$573$	10833.3	0.789821
$574$	0	0
$575$	−4041.87	−0.293144
$576$	0	0
$577$	−2301.23	−0.166034	−0.0830170	−	0.996548i	$-0.526456\pi$
$577$	−2301.23	−0.166034	−0.0830170	+	0.996548i	$0.526456\pi$
$578$	0	0
$579$	13405.0	0.962162
$580$	0	0
$581$	2640.05	0.188516
$582$	0	0
$583$	−11419.5	−0.811232
$584$	0	0
$585$	−1305.70	−0.0922806
$586$	0	0
$587$	16470.4	1.15810	0.579052	−	0.815291i	$-0.303423\pi$
$587$	16470.4	1.15810	0.579052	+	0.815291i	$0.303423\pi$
$588$	0	0
$589$	−8443.92	−0.590706
$590$	0	0
$591$	1303.27	0.0907093
$592$	0	0
$593$	−13570.0	−0.939715	−0.469858	−	0.882742i	$-0.655695\pi$
$593$	−13570.0	−0.939715	−0.469858	+	0.882742i	$0.655695\pi$
$594$	0	0
$595$	2044.66	0.140879
$596$	0	0
$597$	−1406.75	−0.0964393
$598$	0	0
$599$	−27814.1	−1.89725	−0.948625	−	0.316403i	$-0.897525\pi$
$599$	−27814.1	−1.89725	−0.948625	+	0.316403i	$0.897525\pi$
$600$	0	0
$601$	20646.1	1.40128	0.700641	−	0.713514i	$-0.252898\pi$
$601$	20646.1	1.40128	0.700641	+	0.713514i	$0.252898\pi$
$602$	0	0
$603$	−4967.61	−0.335483
$604$	0	0
$605$	−339.969	−0.0228458
$606$	0	0
$607$	−3315.28	−0.221686	−0.110843	−	0.993838i	$-0.535355\pi$
$607$	−3315.28	−0.221686	−0.110843	+	0.993838i	$0.535355\pi$
$608$	0	0
$609$	2889.21	0.192244
$610$	0	0
$611$	15732.8	1.04170
$612$	0	0
$613$	−11113.9	−0.732278	−0.366139	−	0.930560i	$-0.619320\pi$
$613$	−11113.9	−0.732278	−0.366139	+	0.930560i	$0.619320\pi$
$614$	0	0
$615$	5306.53	0.347935
$616$	0	0
$617$	7871.34	0.513595	0.256797	−	0.966465i	$-0.417333\pi$
$617$	7871.34	0.513595	0.256797	+	0.966465i	$0.417333\pi$
$618$	0	0
$619$	19107.1	1.24068	0.620339	−	0.784334i	$-0.286995\pi$
$619$	19107.1	1.24068	0.620339	+	0.784334i	$0.286995\pi$
$620$	0	0
$621$	4365.22	0.282078
$622$	0	0
$623$	4744.07	0.305084
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	6126.28	0.390208
$628$	0	0
$629$	−20453.9	−1.29658
$630$	0	0
$631$	25769.9	1.62580	0.812902	−	0.582401i	$-0.197887\pi$
$631$	25769.9	1.62580	0.812902	+	0.582401i	$0.197887\pi$
$632$	0	0
$633$	11206.5	0.703665
$634$	0	0
$635$	11104.1	0.693939
$636$	0	0
$637$	1421.77	0.0884339
$638$	0	0
$639$	1083.01	0.0670472
$640$	0	0
$641$	−1954.61	−0.120440	−0.0602202	−	0.998185i	$-0.519180\pi$
$641$	−1954.61	−0.120440	−0.0602202	+	0.998185i	$0.519180\pi$
$642$	0	0
$643$	19396.5	1.18961	0.594807	−	0.803868i	$-0.297228\pi$
$643$	19396.5	1.18961	0.594807	+	0.803868i	$0.297228\pi$
$644$	0	0
$645$	7772.34	0.474474
$646$	0	0
$647$	−31264.3	−1.89973	−0.949865	−	0.312661i	$-0.898780\pi$
$647$	−31264.3	−1.89973	−0.949865	+	0.312661i	$0.898780\pi$
$648$	0	0
$649$	549.126	0.0332127
$650$	0	0
$651$	−3247.85	−0.195535
$652$	0	0
$653$	6442.75	0.386102	0.193051	−	0.981189i	$-0.438162\pi$
$653$	6442.75	0.386102	0.193051	+	0.981189i	$0.438162\pi$
$654$	0	0
$655$	3232.56	0.192835
$656$	0	0
$657$	2561.93	0.152132
$658$	0	0
$659$	3584.70	0.211897	0.105949	−	0.994372i	$-0.466212\pi$
$659$	3584.70	0.211897	0.105949	+	0.994372i	$0.466212\pi$
$660$	0	0
$661$	6294.43	0.370386	0.185193	−	0.982702i	$-0.440709\pi$
$661$	6294.43	0.370386	0.185193	+	0.982702i	$0.440709\pi$
$662$	0	0
$663$	−5085.17	−0.297876
$664$	0	0
$665$	1910.89	0.111430
$666$	0	0
$667$	−22243.4	−1.29126
$668$	0	0
$669$	2528.42	0.146120
$670$	0	0
$671$	6398.51	0.368125
$672$	0	0
$673$	−10233.9	−0.586162	−0.293081	−	0.956088i	$-0.594681\pi$
$673$	−10233.9	−0.586162	−0.293081	+	0.956088i	$0.594681\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−7100.75	−0.403108	−0.201554	−	0.979477i	$-0.564599\pi$
$677$	−7100.75	−0.403108	−0.201554	+	0.979477i	$0.564599\pi$
$678$	0	0
$679$	8575.24	0.484665
$680$	0	0
$681$	2976.45	0.167486
$682$	0	0
$683$	35274.6	1.97620	0.988100	−	0.153813i	$-0.0491554\pi$
$683$	35274.6	1.97620	0.988100	+	0.153813i	$0.0491554\pi$
$684$	0	0
$685$	4480.05	0.249889
$686$	0	0
$687$	19234.2	1.06817
$688$	0	0
$689$	8858.74	0.489828
$690$	0	0
$691$	−4945.12	−0.272245	−0.136122	−	0.990692i	$-0.543464\pi$
$691$	−4945.12	−0.272245	−0.136122	+	0.990692i	$0.543464\pi$
$692$	0	0
$693$	2356.40	0.129166
$694$	0	0
$695$	−11568.0	−0.631368
$696$	0	0
$697$	20666.7	1.12311
$698$	0	0
$699$	6823.06	0.369201
$700$	0	0
$701$	−15300.4	−0.824379	−0.412190	−	0.911098i	$-0.635236\pi$
$701$	−15300.4	−0.824379	−0.412190	+	0.911098i	$0.635236\pi$
$702$	0	0
$703$	−19115.7	−1.02555
$704$	0	0
$705$	8133.28	0.434492
$706$	0	0
$707$	2367.01	0.125913
$708$	0	0
$709$	−28297.4	−1.49892	−0.749458	−	0.662052i	$-0.769686\pi$
$709$	−28297.4	−1.49892	−0.749458	+	0.662052i	$0.769686\pi$
$710$	0	0
$711$	−8474.51	−0.447003
$712$	0	0
$713$	25004.5	1.31336
$714$	0	0
$715$	5426.37	0.283825
$716$	0	0
$717$	8589.35	0.447385
$718$	0	0
$719$	−8548.96	−0.443425	−0.221712	−	0.975112i	$-0.571165\pi$
$719$	−8548.96	−0.443425	−0.221712	+	0.975112i	$0.571165\pi$
$720$	0	0
$721$	−3966.90	−0.204903
$722$	0	0
$723$	16092.7	0.827792
$724$	0	0
$725$	3439.53	0.176194
$726$	0	0
$727$	−14345.3	−0.731827	−0.365913	−	0.930649i	$-0.619243\pi$
$727$	−14345.3	−0.731827	−0.365913	+	0.930649i	$0.619243\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	30270.0	1.53157
$732$	0	0
$733$	−22624.0	−1.14002	−0.570012	−	0.821637i	$-0.693061\pi$
$733$	−22624.0	−1.14002	−0.570012	+	0.821637i	$0.693061\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	20644.9	1.03184
$738$	0	0
$739$	−14837.3	−0.738566	−0.369283	−	0.929317i	$-0.620397\pi$
$739$	−14837.3	−0.738566	−0.369283	+	0.929317i	$0.620397\pi$
$740$	0	0
$741$	−4752.49	−0.235610
$742$	0	0
$743$	−13073.1	−0.645497	−0.322749	−	0.946485i	$-0.604607\pi$
$743$	−13073.1	−0.645497	−0.322749	+	0.946485i	$0.604607\pi$
$744$	0	0
$745$	−4096.69	−0.201464
$746$	0	0
$747$	−3394.35	−0.166255
$748$	0	0
$749$	−3937.72	−0.192098
$750$	0	0
$751$	−16213.3	−0.787791	−0.393895	−	0.919155i	$-0.628873\pi$
$751$	−16213.3	−0.787791	−0.393895	+	0.919155i	$0.628873\pi$
$752$	0	0
$753$	16709.4	0.808665
$754$	0	0
$755$	2673.72	0.128883
$756$	0	0
$757$	19903.9	0.955642	0.477821	−	0.878457i	$-0.341427\pi$
$757$	19903.9	0.955642	0.477821	+	0.878457i	$0.341427\pi$
$758$	0	0
$759$	−18141.4	−0.867580
$760$	0	0
$761$	−30125.5	−1.43502	−0.717509	−	0.696549i	$-0.754718\pi$
$761$	−30125.5	−1.43502	−0.717509	+	0.696549i	$0.754718\pi$
$762$	0	0
$763$	−12815.6	−0.608066
$764$	0	0
$765$	−2628.84	−0.124243
$766$	0	0
$767$	−425.986	−0.0200541
$768$	0	0
$769$	−36049.1	−1.69046	−0.845230	−	0.534402i	$-0.820537\pi$
$769$	−36049.1	−1.69046	−0.845230	+	0.534402i	$0.820537\pi$
$770$	0	0
$771$	−6286.09	−0.293629
$772$	0	0
$773$	−9644.77	−0.448769	−0.224384	−	0.974501i	$-0.572037\pi$
$773$	−9644.77	−0.448769	−0.224384	+	0.974501i	$0.572037\pi$
$774$	0	0
$775$	−3866.48	−0.179211
$776$	0	0
$777$	−7352.62	−0.339477
$778$	0	0
$779$	19314.7	0.888344
$780$	0	0
$781$	−4500.88	−0.206215
$782$	0	0
$783$	−3714.69	−0.169543
$784$	0	0
$785$	−7823.80	−0.355724
$786$	0	0
$787$	25218.6	1.14224	0.571122	−	0.820865i	$-0.306508\pi$
$787$	25218.6	1.14224	0.571122	+	0.820865i	$0.306508\pi$
$788$	0	0
$789$	−22397.6	−1.01062
$790$	0	0
$791$	222.623	0.0100070
$792$	0	0
$793$	−4963.67	−0.222276
$794$	0	0
$795$	4579.64	0.204306
$796$	0	0
$797$	32042.3	1.42409	0.712044	−	0.702135i	$-0.247770\pi$
$797$	32042.3	1.42409	0.712044	+	0.702135i	$0.247770\pi$
$798$	0	0
$799$	31675.7	1.40251
$800$	0	0
$801$	−6099.52	−0.269059
$802$	0	0
$803$	−10647.1	−0.467908
$804$	0	0
$805$	−5658.62	−0.247752
$806$	0	0
$807$	19564.1	0.853396
$808$	0	0
$809$	3427.90	0.148972	0.0744860	−	0.997222i	$-0.476268\pi$
$809$	3427.90	0.148972	0.0744860	+	0.997222i	$0.476268\pi$
$810$	0	0
$811$	−23094.4	−0.999943	−0.499972	−	0.866042i	$-0.666656\pi$
$811$	−23094.4	−0.999943	−0.499972	+	0.866042i	$0.666656\pi$
$812$	0	0
$813$	7229.11	0.311852
$814$	0	0
$815$	−5571.53	−0.239463
$816$	0	0
$817$	28289.7	1.21142
$818$	0	0
$819$	−1827.98	−0.0779914
$820$	0	0
$821$	474.741	0.0201810	0.0100905	−	0.999949i	$-0.496788\pi$
$821$	474.741	0.0201810	0.0100905	+	0.999949i	$0.496788\pi$
$822$	0	0
$823$	−24159.8	−1.02328	−0.511638	−	0.859201i	$-0.670961\pi$
$823$	−24159.8	−1.02328	−0.511638	+	0.859201i	$0.670961\pi$
$824$	0	0
$825$	2805.23	0.118383
$826$	0	0
$827$	7566.35	0.318147	0.159074	−	0.987267i	$-0.449149\pi$
$827$	7566.35	0.318147	0.159074	+	0.987267i	$0.449149\pi$
$828$	0	0
$829$	−10580.9	−0.443295	−0.221648	−	0.975127i	$-0.571143\pi$
$829$	−10580.9	−0.443295	−0.221648	+	0.975127i	$0.571143\pi$
$830$	0	0
$831$	6659.48	0.277996
$832$	0	0
$833$	2862.52	0.119064
$834$	0	0
$835$	−8872.34	−0.367713
$836$	0	0
$837$	4175.80	0.172445
$838$	0	0
$839$	−15315.6	−0.630218	−0.315109	−	0.949055i	$-0.602041\pi$
$839$	−15315.6	−0.630218	−0.315109	+	0.949055i	$0.602041\pi$
$840$	0	0
$841$	−5460.40	−0.223888
$842$	0	0
$843$	−17515.7	−0.715625
$844$	0	0
$845$	6775.47	0.275838
$846$	0	0
$847$	−475.956	−0.0193082
$848$	0	0
$849$	−10935.1	−0.442040
$850$	0	0
$851$	56606.4	2.28019
$852$	0	0
$853$	18598.2	0.746528	0.373264	−	0.927725i	$-0.378239\pi$
$853$	18598.2	0.746528	0.373264	+	0.927725i	$0.378239\pi$
$854$	0	0
$855$	−2456.86	−0.0982723
$856$	0	0
$857$	41775.3	1.66513	0.832566	−	0.553926i	$-0.186871\pi$
$857$	41775.3	1.66513	0.832566	+	0.553926i	$0.186871\pi$
$858$	0	0
$859$	−32414.7	−1.28752	−0.643758	−	0.765229i	$-0.722626\pi$
$859$	−32414.7	−1.28752	−0.643758	+	0.765229i	$0.722626\pi$
$860$	0	0
$861$	7429.14	0.294059
$862$	0	0
$863$	31299.7	1.23459	0.617297	−	0.786730i	$-0.288228\pi$
$863$	31299.7	1.23459	0.617297	+	0.786730i	$0.288228\pi$
$864$	0	0
$865$	21079.4	0.828580
$866$	0	0
$867$	4500.75	0.176302
$868$	0	0
$869$	35219.2	1.37483
$870$	0	0
$871$	−16015.4	−0.623030
$872$	0	0
$873$	−11025.3	−0.427434
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−19973.0	−0.769031	−0.384515	−	0.923119i	$-0.625631\pi$
$877$	−19973.0	−0.769031	−0.384515	+	0.923119i	$0.625631\pi$
$878$	0	0
$879$	−11333.7	−0.434900
$880$	0	0
$881$	17367.9	0.664176	0.332088	−	0.943248i	$-0.392247\pi$
$881$	17367.9	0.664176	0.332088	+	0.943248i	$0.392247\pi$
$882$	0	0
$883$	14364.2	0.547446	0.273723	−	0.961809i	$-0.411745\pi$
$883$	14364.2	0.547446	0.273723	+	0.961809i	$0.411745\pi$
$884$	0	0
$885$	−220.219	−0.00836450
$886$	0	0
$887$	33738.1	1.27713	0.638564	−	0.769568i	$-0.279529\pi$
$887$	33738.1	1.27713	0.638564	+	0.769568i	$0.279529\pi$
$888$	0	0
$889$	15545.7	0.586485
$890$	0	0
$891$	−3029.65	−0.113914
$892$	0	0
$893$	29603.4	1.10934
$894$	0	0
$895$	−12153.5	−0.453906
$896$	0	0
$897$	14073.3	0.523850
$898$	0	0
$899$	−21278.2	−0.789398
$900$	0	0
$901$	17835.8	0.659485
$902$	0	0
$903$	10881.3	0.401004
$904$	0	0
$905$	13504.6	0.496030
$906$	0	0
$907$	32998.6	1.20805	0.604024	−	0.796966i	$-0.293563\pi$
$907$	32998.6	1.20805	0.604024	+	0.796966i	$0.293563\pi$
$908$	0	0
$909$	−3043.29	−0.111045
$910$	0	0
$911$	−33446.3	−1.21638	−0.608192	−	0.793790i	$-0.708105\pi$
$911$	−33446.3	−1.21638	−0.608192	+	0.793790i	$0.708105\pi$
$912$	0	0
$913$	14106.6	0.511347
$914$	0	0
$915$	−2566.03	−0.0927108
$916$	0	0
$917$	4525.59	0.162975
$918$	0	0
$919$	−41708.7	−1.49711	−0.748554	−	0.663074i	$-0.769252\pi$
$919$	−41708.7	−1.49711	−0.748554	+	0.663074i	$0.769252\pi$
$920$	0	0
$921$	−14398.9	−0.515158
$922$	0	0
$923$	3491.58	0.124514
$924$	0	0
$925$	−8753.12	−0.311136
$926$	0	0
$927$	5100.30	0.180707
$928$	0	0
$929$	49024.6	1.73137	0.865686	−	0.500587i	$-0.166883\pi$
$929$	49024.6	1.73137	0.865686	+	0.500587i	$0.166883\pi$
$930$	0	0
$931$	2675.25	0.0941758
$932$	0	0
$933$	1740.34	0.0610677
$934$	0	0
$935$	10925.2	0.382131
$936$	0	0
$937$	−5447.58	−0.189930	−0.0949651	−	0.995481i	$-0.530274\pi$
$937$	−5447.58	−0.189930	−0.0949651	+	0.995481i	$0.530274\pi$
$938$	0	0
$939$	18344.4	0.637535
$940$	0	0
$941$	4125.75	0.142928	0.0714642	−	0.997443i	$-0.477233\pi$
$941$	4125.75	0.142928	0.0714642	+	0.997443i	$0.477233\pi$
$942$	0	0
$943$	−57195.5	−1.97513
$944$	0	0
$945$	−945.000	−0.0325300
$946$	0	0
$947$	−17332.3	−0.594747	−0.297374	−	0.954761i	$-0.596111\pi$
$947$	−17332.3	−0.594747	−0.297374	+	0.954761i	$0.596111\pi$
$948$	0	0
$949$	8259.57	0.282526
$950$	0	0
$951$	12901.9	0.439929
$952$	0	0
$953$	56839.4	1.93201	0.966007	−	0.258517i	$-0.0832337\pi$
$953$	56839.4	1.93201	0.966007	+	0.258517i	$0.0832337\pi$
$954$	0	0
$955$	18055.5	0.611792
$956$	0	0
$957$	15437.9	0.521459
$958$	0	0
$959$	6272.07	0.211195
$960$	0	0
$961$	−5871.48	−0.197089
$962$	0	0
$963$	5062.78	0.169414
$964$	0	0
$965$	22341.6	0.745287
$966$	0	0
$967$	13284.2	0.441769	0.220884	−	0.975300i	$-0.429106\pi$
$967$	13284.2	0.441769	0.220884	+	0.975300i	$0.429106\pi$
$968$	0	0
$969$	−9568.44	−0.317216
$970$	0	0
$971$	−12153.0	−0.401658	−0.200829	−	0.979626i	$-0.564364\pi$
$971$	−12153.0	−0.401658	−0.200829	+	0.979626i	$0.564364\pi$
$972$	0	0
$973$	−16195.3	−0.533604
$974$	0	0
$975$	−2176.17	−0.0714803
$976$	0	0
$977$	−37999.3	−1.24433	−0.622163	−	0.782888i	$-0.713746\pi$
$977$	−37999.3	−1.24433	−0.622163	+	0.782888i	$0.713746\pi$
$978$	0	0
$979$	25349.0	0.827537
$980$	0	0
$981$	16477.1	0.536264
$982$	0	0
$983$	−22375.0	−0.725993	−0.362996	−	0.931791i	$-0.618246\pi$
$983$	−22375.0	−0.725993	−0.362996	+	0.931791i	$0.618246\pi$
$984$	0	0
$985$	2172.11	0.0702631
$986$	0	0
$987$	11386.6	0.367213
$988$	0	0
$989$	−83772.9	−2.69345
$990$	0	0
$991$	−18985.3	−0.608564	−0.304282	−	0.952582i	$-0.598417\pi$
$991$	−18985.3	−0.608564	−0.304282	+	0.952582i	$0.598417\pi$
$992$	0	0
$993$	−20062.6	−0.641156
$994$	0	0
$995$	−2344.58	−0.0747016
$996$	0	0
$997$	−56476.5	−1.79401	−0.897006	−	0.442019i	$-0.854262\pi$
$997$	−56476.5	−1.79401	−0.897006	+	0.442019i	$0.854262\pi$
$998$	0	0
$999$	9453.37	0.299391

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.4.a.y.1.1		2
4.3	odd	2		105.4.a.g.1.1	✓	2
12.11	even	2		315.4.a.g.1.2		2
20.3	even	4		525.4.d.j.274.3		4
20.7	even	4		525.4.d.j.274.2		4
20.19	odd	2		525.4.a.i.1.2		2
28.27	even	2		735.4.a.q.1.1		2
60.59	even	2		1575.4.a.y.1.1		2
84.83	odd	2		2205.4.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.g.1.1	✓	2	4.3	odd	2
315.4.a.g.1.2		2	12.11	even	2
525.4.a.i.1.2		2	20.19	odd	2
525.4.d.j.274.2		4	20.7	even	4
525.4.d.j.274.3		4	20.3	even	4
735.4.a.q.1.1		2	28.27	even	2
1575.4.a.y.1.1		2	60.59	even	2
1680.4.a.y.1.1		2	1.1	even	1	trivial
2205.4.a.v.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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -37.4031 q^{11} +29.0156 q^{13} +15.0000 q^{15} +58.4187 q^{17} +54.5969 q^{19} +21.0000 q^{21} -161.675 q^{23} +25.0000 q^{25} -27.0000 q^{27} +137.581 q^{29} -154.659 q^{31} +112.209 q^{33} +35.0000 q^{35} -350.125 q^{37} -87.0469 q^{39} +353.769 q^{41} +518.156 q^{43} -45.0000 q^{45} +542.219 q^{47} +49.0000 q^{49} -175.256 q^{51} +305.309 q^{53} +187.016 q^{55} -163.791 q^{57} -14.6813 q^{59} -171.069 q^{61} -63.0000 q^{63} -145.078 q^{65} -551.956 q^{67} +485.025 q^{69} +120.334 q^{71} +284.659 q^{73} -75.0000 q^{75} +261.822 q^{77} -941.612 q^{79} +81.0000 q^{81} -377.150 q^{83} -292.094 q^{85} -412.744 q^{87} -677.725 q^{89} -203.109 q^{91} +463.978 q^{93} -272.984 q^{95} -1225.03 q^{97} -336.628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 10 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 10 * q^5 - 14 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 10 q^{5} - 14 q^{7} + 18 q^{9} - 62 q^{11} - 6 q^{13} + 30 q^{15} + 40 q^{17} + 122 q^{19} + 42 q^{21} - 16 q^{23} + 50 q^{25} - 54 q^{27} + 352 q^{29} - 66 q^{31} + 186 q^{33} + 70 q^{35} - 188 q^{37} + 18 q^{39} + 16 q^{41} + 396 q^{43} - 90 q^{45} + 188 q^{47} + 98 q^{49} - 120 q^{51} + 982 q^{53} + 310 q^{55} - 366 q^{57} - 516 q^{59} - 880 q^{61} - 126 q^{63} + 30 q^{65} + 356 q^{67} + 48 q^{69} - 310 q^{71} + 326 q^{73} - 150 q^{75} + 434 q^{77} - 1832 q^{79} + 162 q^{81} + 680 q^{83} - 200 q^{85} - 1056 q^{87} + 796 q^{89} + 42 q^{91} + 198 q^{93} - 610 q^{95} - 670 q^{97} - 558 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 10 * q^5 - 14 * q^7 + 18 * q^9 - 62 * q^11 - 6 * q^13 + 30 * q^15 + 40 * q^17 + 122 * q^19 + 42 * q^21 - 16 * q^23 + 50 * q^25 - 54 * q^27 + 352 * q^29 - 66 * q^31 + 186 * q^33 + 70 * q^35 - 188 * q^37 + 18 * q^39 + 16 * q^41 + 396 * q^43 - 90 * q^45 + 188 * q^47 + 98 * q^49 - 120 * q^51 + 982 * q^53 + 310 * q^55 - 366 * q^57 - 516 * q^59 - 880 * q^61 - 126 * q^63 + 30 * q^65 + 356 * q^67 + 48 * q^69 - 310 * q^71 + 326 * q^73 - 150 * q^75 + 434 * q^77 - 1832 * q^79 + 162 * q^81 + 680 * q^83 - 200 * q^85 - 1056 * q^87 + 796 * q^89 + 42 * q^91 + 198 * q^93 - 610 * q^95 - 670 * q^97 - 558 * q^99

Introduction

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