LMFDB - Embedded newform 1680.4.a.y.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:	$1$
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			1.2
Root			$-2.70156$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -24.5969 q^{11} -35.0156 q^{13} +15.0000 q^{15} -18.4187 q^{17} +67.4031 q^{19} +21.0000 q^{21} +145.675 q^{23} +25.0000 q^{25} -27.0000 q^{27} +214.419 q^{29} +88.6594 q^{31} +73.7906 q^{33} +35.0000 q^{35} +162.125 q^{37} +105.047 q^{39} -337.769 q^{41} -122.156 q^{43} -45.0000 q^{45} -354.219 q^{47} +49.0000 q^{49} +55.2562 q^{51} +676.691 q^{53} +122.984 q^{55} -202.209 q^{57} -501.319 q^{59} -708.931 q^{61} -63.0000 q^{63} +175.078 q^{65} +907.956 q^{67} -437.025 q^{69} -430.334 q^{71} +41.3406 q^{73} -75.0000 q^{75} +172.178 q^{77} -890.388 q^{79} +81.0000 q^{81} +1057.15 q^{83} +92.0937 q^{85} -643.256 q^{87} +1473.72 q^{89} +245.109 q^{91} -265.978 q^{93} -337.016 q^{95} +555.034 q^{97} -221.372 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	−24.5969	−0.674203	−0.337102	−	0.941468i	$-0.609447\pi$
$11$	−24.5969	−0.674203	−0.337102	+	0.941468i	$0.609447\pi$
$12$	0	0
$13$	−35.0156	−0.747045	−0.373523	−	0.927621i	$-0.621850\pi$
$13$	−35.0156	−0.747045	−0.373523	+	0.927621i	$0.621850\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−18.4187	−0.262777	−0.131388	−	0.991331i	$-0.541943\pi$
$17$	−18.4187	−0.262777	−0.131388	+	0.991331i	$0.541943\pi$
$18$	0	0
$19$	67.4031	0.813860	0.406930	−	0.913459i	$-0.366599\pi$
$19$	67.4031	0.813860	0.406930	+	0.913459i	$0.366599\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	145.675	1.32067	0.660333	−	0.750973i	$-0.270415\pi$
$23$	145.675	1.32067	0.660333	+	0.750973i	$0.270415\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	214.419	1.37298	0.686492	−	0.727137i	$-0.259149\pi$
$29$	214.419	1.37298	0.686492	+	0.727137i	$0.259149\pi$
$30$	0	0
$31$	88.6594	0.513667	0.256834	−	0.966456i	$-0.417321\pi$
$31$	88.6594	0.513667	0.256834	+	0.966456i	$0.417321\pi$
$32$	0	0
$33$	73.7906	0.389251
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	162.125	0.720356	0.360178	−	0.932884i	$-0.382716\pi$
$37$	162.125	0.720356	0.360178	+	0.932884i	$0.382716\pi$
$38$	0	0
$39$	105.047	0.431307
$40$	0	0
$41$	−337.769	−1.28660	−0.643300	−	0.765614i	$-0.722435\pi$
$41$	−337.769	−1.28660	−0.643300	+	0.765614i	$0.722435\pi$
$42$	0	0
$43$	−122.156	−0.433224	−0.216612	−	0.976258i	$-0.569501\pi$
$43$	−122.156	−0.433224	−0.216612	+	0.976258i	$0.569501\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−354.219	−1.09932	−0.549661	−	0.835388i	$-0.685243\pi$
$47$	−354.219	−1.09932	−0.549661	+	0.835388i	$0.685243\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	55.2562	0.151714
$52$	0	0
$53$	676.691	1.75378	0.876892	−	0.480687i	$-0.159613\pi$
$53$	676.691	1.75378	0.876892	+	0.480687i	$0.159613\pi$
$54$	0	0
$55$	122.984	0.301513
$56$	0	0
$57$	−202.209	−0.469882
$58$	0	0
$59$	−501.319	−1.10621	−0.553103	−	0.833113i	$-0.686556\pi$
$59$	−501.319	−1.10621	−0.553103	+	0.833113i	$0.686556\pi$
$60$	0	0
$61$	−708.931	−1.48802	−0.744011	−	0.668167i	$-0.767079\pi$
$61$	−708.931	−1.48802	−0.744011	+	0.668167i	$0.767079\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	175.078	0.334089
$66$	0	0
$67$	907.956	1.65559	0.827795	−	0.561031i	$-0.189595\pi$
$67$	907.956	1.65559	0.827795	+	0.561031i	$0.189595\pi$
$68$	0	0
$69$	−437.025	−0.762487
$70$	0	0
$71$	−430.334	−0.719314	−0.359657	−	0.933085i	$-0.617106\pi$
$71$	−430.334	−0.719314	−0.359657	+	0.933085i	$0.617106\pi$
$72$	0	0
$73$	41.3406	0.0662816	0.0331408	−	0.999451i	$-0.489449\pi$
$73$	41.3406	0.0662816	0.0331408	+	0.999451i	$0.489449\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	172.178	0.254825
$78$	0	0
$79$	−890.388	−1.26806	−0.634028	−	0.773310i	$-0.718600\pi$
$79$	−890.388	−1.26806	−0.634028	+	0.773310i	$0.718600\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1057.15	1.39804	0.699020	−	0.715102i	$-0.253620\pi$
$83$	1057.15	1.39804	0.699020	+	0.715102i	$0.253620\pi$
$84$	0	0
$85$	92.0937	0.117517
$86$	0	0
$87$	−643.256	−0.792693
$88$	0	0
$89$	1473.72	1.75522	0.877610	−	0.479376i	$-0.159137\pi$
$89$	1473.72	1.75522	0.877610	+	0.479376i	$0.159137\pi$
$90$	0	0
$91$	245.109	0.282356
$92$	0	0
$93$	−265.978	−0.296566
$94$	0	0
$95$	−337.016	−0.363969
$96$	0	0
$97$	555.034	0.580981	0.290491	−	0.956878i	$-0.406181\pi$
$97$	555.034	0.580981	0.290491	+	0.956878i	$0.406181\pi$
$98$	0	0
$99$	−221.372	−0.224734
$100$	0	0
$101$	1890.14	1.86214	0.931071	−	0.364838i	$-0.118876\pi$
$101$	1890.14	1.86214	0.931071	+	0.364838i	$0.118876\pi$
$102$	0	0
$103$	−662.700	−0.633959	−0.316979	−	0.948432i	$-0.602669\pi$
$103$	−662.700	−0.633959	−0.316979	+	0.948432i	$0.602669\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	−1614.53	−1.45872	−0.729358	−	0.684132i	$-0.760181\pi$
$107$	−1614.53	−1.45872	−0.729358	+	0.684132i	$0.760181\pi$
$108$	0	0
$109$	217.206	0.190868	0.0954339	−	0.995436i	$-0.469576\pi$
$109$	217.206	0.190868	0.0954339	+	0.995436i	$0.469576\pi$
$110$	0	0
$111$	−486.375	−0.415898
$112$	0	0
$113$	−1658.20	−1.38044	−0.690221	−	0.723598i	$-0.742487\pi$
$113$	−1658.20	−1.38044	−0.690221	+	0.723598i	$0.742487\pi$
$114$	0	0
$115$	−728.375	−0.590620
$116$	0	0
$117$	−315.141	−0.249015
$118$	0	0
$119$	128.931	0.0993202
$120$	0	0
$121$	−725.994	−0.545450
$122$	0	0
$123$	1013.31	0.742819
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1108.81	0.774734	0.387367	−	0.921926i	$-0.373385\pi$
$127$	1108.81	0.774734	0.387367	+	0.921926i	$0.373385\pi$
$128$	0	0
$129$	366.469	0.250122
$130$	0	0
$131$	−185.488	−0.123711	−0.0618554	−	0.998085i	$-0.519702\pi$
$131$	−185.488	−0.123711	−0.0618554	+	0.998085i	$0.519702\pi$
$132$	0	0
$133$	−471.822	−0.307610
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−37.9907	−0.0236917	−0.0118458	−	0.999930i	$-0.503771\pi$
$137$	−37.9907	−0.0236917	−0.0118458	+	0.999930i	$0.503771\pi$
$138$	0	0
$139$	−183.609	−0.112040	−0.0560199	−	0.998430i	$-0.517841\pi$
$139$	−183.609	−0.112040	−0.0560199	+	0.998430i	$0.517841\pi$
$140$	0	0
$141$	1062.66	0.634694
$142$	0	0
$143$	861.275	0.503660
$144$	0	0
$145$	−1072.09	−0.614018
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−1383.34	−0.760587	−0.380293	−	0.924866i	$-0.624177\pi$
$149$	−1383.34	−0.760587	−0.380293	+	0.924866i	$0.624177\pi$
$150$	0	0
$151$	−765.256	−0.412422	−0.206211	−	0.978508i	$-0.566113\pi$
$151$	−765.256	−0.412422	−0.206211	+	0.978508i	$0.566113\pi$
$152$	0	0
$153$	−165.769	−0.0875922
$154$	0	0
$155$	−443.297	−0.229719
$156$	0	0
$157$	−2366.76	−1.20311	−0.601554	−	0.798832i	$-0.705452\pi$
$157$	−2366.76	−1.20311	−0.601554	+	0.798832i	$0.705452\pi$
$158$	0	0
$159$	−2030.07	−1.01255
$160$	0	0
$161$	−1019.72	−0.499165
$162$	0	0
$163$	3137.69	1.50775	0.753875	−	0.657018i	$-0.228183\pi$
$163$	3137.69	1.50775	0.753875	+	0.657018i	$0.228183\pi$
$164$	0	0
$165$	−368.953	−0.174079
$166$	0	0
$167$	−146.469	−0.0678688	−0.0339344	−	0.999424i	$-0.510804\pi$
$167$	−146.469	−0.0678688	−0.0339344	+	0.999424i	$0.510804\pi$
$168$	0	0
$169$	−970.906	−0.441924
$170$	0	0
$171$	606.628	0.271287
$172$	0	0
$173$	−1424.12	−0.625860	−0.312930	−	0.949776i	$-0.601311\pi$
$173$	−1424.12	−0.625860	−0.312930	+	0.949776i	$0.601311\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	1503.96	0.638668
$178$	0	0
$179$	−1244.70	−0.519737	−0.259869	−	0.965644i	$-0.583679\pi$
$179$	−1244.70	−0.519737	−0.259869	+	0.965644i	$0.583679\pi$
$180$	0	0
$181$	−3879.09	−1.59299	−0.796493	−	0.604648i	$-0.793314\pi$
$181$	−3879.09	−1.59299	−0.796493	+	0.604648i	$0.793314\pi$
$182$	0	0
$183$	2126.79	0.859110
$184$	0	0
$185$	−810.625	−0.322153
$186$	0	0
$187$	453.044	0.177165
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−1574.90	−0.596628	−0.298314	−	0.954468i	$-0.596424\pi$
$191$	−1574.90	−0.596628	−0.298314	+	0.954468i	$0.596424\pi$
$192$	0	0
$193$	−4775.67	−1.78114	−0.890572	−	0.454843i	$-0.849695\pi$
$193$	−4775.67	−1.78114	−0.890572	+	0.454843i	$0.849695\pi$
$194$	0	0
$195$	−525.234	−0.192886
$196$	0	0
$197$	−2803.58	−1.01394	−0.506971	−	0.861963i	$-0.669235\pi$
$197$	−2803.58	−1.01394	−0.506971	+	0.861963i	$0.669235\pi$
$198$	0	0
$199$	−4102.92	−1.46155	−0.730774	−	0.682620i	$-0.760841\pi$
$199$	−4102.92	−1.46155	−0.730774	+	0.682620i	$0.760841\pi$
$200$	0	0
$201$	−2723.87	−0.955855
$202$	0	0
$203$	−1500.93	−0.518940
$204$	0	0
$205$	1688.84	0.575385
$206$	0	0
$207$	1311.07	0.440222
$208$	0	0
$209$	−1657.91	−0.548707
$210$	0	0
$211$	823.512	0.268687	0.134343	−	0.990935i	$-0.457107\pi$
$211$	823.512	0.268687	0.134343	+	0.990935i	$0.457107\pi$
$212$	0	0
$213$	1291.00	0.415296
$214$	0	0
$215$	610.781	0.193744
$216$	0	0
$217$	−620.616	−0.194148
$218$	0	0
$219$	−124.022	−0.0382677
$220$	0	0
$221$	644.944	0.196306
$222$	0	0
$223$	−817.194	−0.245396	−0.122698	−	0.992444i	$-0.539155\pi$
$223$	−817.194	−0.245396	−0.122698	+	0.992444i	$0.539155\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−3655.85	−1.06893	−0.534465	−	0.845190i	$-0.679487\pi$
$227$	−3655.85	−1.06893	−0.534465	+	0.845190i	$0.679487\pi$
$228$	0	0
$229$	939.393	0.271078	0.135539	−	0.990772i	$-0.456723\pi$
$229$	939.393	0.271078	0.135539	+	0.990772i	$0.456723\pi$
$230$	0	0
$231$	−516.534	−0.147123
$232$	0	0
$233$	−7.64701	−0.00215010	−0.00107505	−	0.999999i	$-0.500342\pi$
$233$	−7.64701	−0.00215010	−0.00107505	+	0.999999i	$0.500342\pi$
$234$	0	0
$235$	1771.09	0.491631
$236$	0	0
$237$	2671.16	0.732112
$238$	0	0
$239$	889.115	0.240636	0.120318	−	0.992735i	$-0.461609\pi$
$239$	889.115	0.240636	0.120318	+	0.992735i	$0.461609\pi$
$240$	0	0
$241$	2140.23	0.572051	0.286026	−	0.958222i	$-0.407666\pi$
$241$	2140.23	0.572051	0.286026	+	0.958222i	$0.407666\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−2360.16	−0.607990
$248$	0	0
$249$	−3171.45	−0.807158
$250$	0	0
$251$	6749.81	1.69739	0.848693	−	0.528886i	$-0.177390\pi$
$251$	6749.81	1.69739	0.848693	+	0.528886i	$0.177390\pi$
$252$	0	0
$253$	−3583.15	−0.890398
$254$	0	0
$255$	−276.281	−0.0678486
$256$	0	0
$257$	3068.64	0.744811	0.372405	−	0.928070i	$-0.378533\pi$
$257$	3068.64	0.744811	0.372405	+	0.928070i	$0.378533\pi$
$258$	0	0
$259$	−1134.87	−0.272269
$260$	0	0
$261$	1929.77	0.457662
$262$	0	0
$263$	4674.12	1.09589	0.547944	−	0.836515i	$-0.315411\pi$
$263$	4674.12	1.09589	0.547944	+	0.836515i	$0.315411\pi$
$264$	0	0
$265$	−3383.45	−0.784316
$266$	0	0
$267$	−4421.17	−1.01338
$268$	0	0
$269$	2417.38	0.547919	0.273960	−	0.961741i	$-0.411667\pi$
$269$	2417.38	0.547919	0.273960	+	0.961741i	$0.411667\pi$
$270$	0	0
$271$	−7724.30	−1.73143	−0.865715	−	0.500537i	$-0.833136\pi$
$271$	−7724.30	−1.73143	−0.865715	+	0.500537i	$0.833136\pi$
$272$	0	0
$273$	−735.328	−0.163019
$274$	0	0
$275$	−614.922	−0.134841
$276$	0	0
$277$	−4576.17	−0.992620	−0.496310	−	0.868145i	$-0.665312\pi$
$277$	−4576.17	−0.992620	−0.496310	+	0.868145i	$0.665312\pi$
$278$	0	0
$279$	797.934	0.171222
$280$	0	0
$281$	−1358.56	−0.288415	−0.144208	−	0.989547i	$-0.546063\pi$
$281$	−1358.56	−0.288415	−0.144208	+	0.989547i	$0.546063\pi$
$282$	0	0
$283$	−3885.04	−0.816048	−0.408024	−	0.912971i	$-0.633782\pi$
$283$	−3885.04	−0.816048	−0.408024	+	0.912971i	$0.633782\pi$
$284$	0	0
$285$	1011.05	0.210138
$286$	0	0
$287$	2364.38	0.486289
$288$	0	0
$289$	−4573.75	−0.930948
$290$	0	0
$291$	−1665.10	−0.335430
$292$	0	0
$293$	−4033.91	−0.804312	−0.402156	−	0.915571i	$-0.631739\pi$
$293$	−4033.91	−0.804312	−0.402156	+	0.915571i	$0.631739\pi$
$294$	0	0
$295$	2506.59	0.494710
$296$	0	0
$297$	664.116	0.129750
$298$	0	0
$299$	−5100.90	−0.986598
$300$	0	0
$301$	855.093	0.163743
$302$	0	0
$303$	−5670.43	−1.07511
$304$	0	0
$305$	3544.66	0.665464
$306$	0	0
$307$	4620.36	0.858950	0.429475	−	0.903079i	$-0.358699\pi$
$307$	4620.36	0.858950	0.429475	+	0.903079i	$0.358699\pi$
$308$	0	0
$309$	1988.10	0.366016
$310$	0	0
$311$	−6675.89	−1.21722	−0.608609	−	0.793470i	$-0.708272\pi$
$311$	−6675.89	−1.21722	−0.608609	+	0.793470i	$0.708272\pi$
$312$	0	0
$313$	2836.78	0.512283	0.256141	−	0.966639i	$-0.417549\pi$
$313$	2836.78	0.512283	0.256141	+	0.966639i	$0.417549\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	4010.63	0.710597	0.355299	−	0.934753i	$-0.384379\pi$
$317$	4010.63	0.710597	0.355299	+	0.934753i	$0.384379\pi$
$318$	0	0
$319$	−5274.03	−0.925671
$320$	0	0
$321$	4843.59	0.842190
$322$	0	0
$323$	−1241.48	−0.213863
$324$	0	0
$325$	−875.391	−0.149409
$326$	0	0
$327$	−651.619	−0.110198
$328$	0	0
$329$	2479.53	0.415504
$330$	0	0
$331$	−11087.5	−1.84117	−0.920583	−	0.390546i	$-0.872286\pi$
$331$	−11087.5	−1.84117	−0.920583	+	0.390546i	$0.872286\pi$
$332$	0	0
$333$	1459.12	0.240119
$334$	0	0
$335$	−4539.78	−0.740402
$336$	0	0
$337$	12118.7	1.95890	0.979450	−	0.201689i	$-0.0646431\pi$
$337$	12118.7	1.95890	0.979450	+	0.201689i	$0.0646431\pi$
$338$	0	0
$339$	4974.59	0.796999
$340$	0	0
$341$	−2180.74	−0.346316
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	2185.12	0.340995
$346$	0	0
$347$	6361.22	0.984116	0.492058	−	0.870562i	$-0.336245\pi$
$347$	6361.22	0.984116	0.492058	+	0.870562i	$0.336245\pi$
$348$	0	0
$349$	−3115.18	−0.477799	−0.238899	−	0.971044i	$-0.576787\pi$
$349$	−3115.18	−0.477799	−0.238899	+	0.971044i	$0.576787\pi$
$350$	0	0
$351$	945.422	0.143769
$352$	0	0
$353$	−11927.4	−1.79839	−0.899194	−	0.437550i	$-0.855846\pi$
$353$	−11927.4	−1.79839	−0.899194	+	0.437550i	$0.855846\pi$
$354$	0	0
$355$	2151.67	0.321687
$356$	0	0
$357$	−386.794	−0.0573426
$358$	0	0
$359$	6143.95	0.903245	0.451623	−	0.892209i	$-0.350845\pi$
$359$	6143.95	0.903245	0.451623	+	0.892209i	$0.350845\pi$
$360$	0	0
$361$	−2315.82	−0.337632
$362$	0	0
$363$	2177.98	0.314916
$364$	0	0
$365$	−206.703	−0.0296420
$366$	0	0
$367$	1927.67	0.274178	0.137089	−	0.990559i	$-0.456225\pi$
$367$	1927.67	0.274178	0.137089	+	0.990559i	$0.456225\pi$
$368$	0	0
$369$	−3039.92	−0.428867
$370$	0	0
$371$	−4736.83	−0.662868
$372$	0	0
$373$	10452.0	1.45090	0.725449	−	0.688276i	$-0.241632\pi$
$373$	10452.0	1.45090	0.725449	+	0.688276i	$0.241632\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−7508.01	−1.02568
$378$	0	0
$379$	−7066.43	−0.957726	−0.478863	−	0.877890i	$-0.658951\pi$
$379$	−7066.43	−0.957726	−0.478863	+	0.877890i	$0.658951\pi$
$380$	0	0
$381$	−3326.44	−0.447293
$382$	0	0
$383$	−7168.04	−0.956318	−0.478159	−	0.878273i	$-0.658696\pi$
$383$	−7168.04	−0.956318	−0.478159	+	0.878273i	$0.658696\pi$
$384$	0	0
$385$	−860.891	−0.113961
$386$	0	0
$387$	−1099.41	−0.144408
$388$	0	0
$389$	−7414.06	−0.966344	−0.483172	−	0.875525i	$-0.660515\pi$
$389$	−7414.06	−0.966344	−0.483172	+	0.875525i	$0.660515\pi$
$390$	0	0
$391$	−2683.15	−0.347040
$392$	0	0
$393$	556.463	0.0714245
$394$	0	0
$395$	4451.94	0.567092
$396$	0	0
$397$	−8936.01	−1.12969	−0.564843	−	0.825198i	$-0.691063\pi$
$397$	−8936.01	−1.12969	−0.564843	+	0.825198i	$0.691063\pi$
$398$	0	0
$399$	1415.47	0.177599
$400$	0	0
$401$	1782.91	0.222031	0.111015	−	0.993819i	$-0.464590\pi$
$401$	1782.91	0.222031	0.111015	+	0.993819i	$0.464590\pi$
$402$	0	0
$403$	−3104.46	−0.383733
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−3987.77	−0.485667
$408$	0	0
$409$	−8759.92	−1.05905	−0.529524	−	0.848295i	$-0.677629\pi$
$409$	−8759.92	−1.05905	−0.529524	+	0.848295i	$0.677629\pi$
$410$	0	0
$411$	113.972	0.0136784
$412$	0	0
$413$	3509.23	0.418106
$414$	0	0
$415$	−5285.75	−0.625222
$416$	0	0
$417$	550.828	0.0646862
$418$	0	0
$419$	3212.74	0.374588	0.187294	−	0.982304i	$-0.440028\pi$
$419$	3212.74	0.374588	0.187294	+	0.982304i	$0.440028\pi$
$420$	0	0
$421$	15757.8	1.82420	0.912101	−	0.409965i	$-0.134459\pi$
$421$	15757.8	1.82420	0.912101	+	0.409965i	$0.134459\pi$
$422$	0	0
$423$	−3187.97	−0.366440
$424$	0	0
$425$	−460.469	−0.0525553
$426$	0	0
$427$	4962.52	0.562419
$428$	0	0
$429$	−2583.82	−0.290788
$430$	0	0
$431$	405.917	0.0453650	0.0226825	−	0.999743i	$-0.492779\pi$
$431$	405.917	0.0453650	0.0226825	+	0.999743i	$0.492779\pi$
$432$	0	0
$433$	−7845.25	−0.870713	−0.435357	−	0.900258i	$-0.643378\pi$
$433$	−7845.25	−0.870713	−0.435357	+	0.900258i	$0.643378\pi$
$434$	0	0
$435$	3216.28	0.354503
$436$	0	0
$437$	9818.95	1.07484
$438$	0	0
$439$	−423.029	−0.0459911	−0.0229955	−	0.999736i	$-0.507320\pi$
$439$	−423.029	−0.0459911	−0.0229955	+	0.999736i	$0.507320\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	16058.7	1.72229	0.861143	−	0.508362i	$-0.169749\pi$
$443$	16058.7	1.72229	0.861143	+	0.508362i	$0.169749\pi$
$444$	0	0
$445$	−7368.62	−0.784958
$446$	0	0
$447$	4150.01	0.439125
$448$	0	0
$449$	2186.75	0.229842	0.114921	−	0.993375i	$-0.463338\pi$
$449$	2186.75	0.229842	0.114921	+	0.993375i	$0.463338\pi$
$450$	0	0
$451$	8308.05	0.867430
$452$	0	0
$453$	2295.77	0.238112
$454$	0	0
$455$	−1225.55	−0.126274
$456$	0	0
$457$	−5799.22	−0.593602	−0.296801	−	0.954939i	$-0.595920\pi$
$457$	−5799.22	−0.593602	−0.296801	+	0.954939i	$0.595920\pi$
$458$	0	0
$459$	497.306	0.0505714
$460$	0	0
$461$	9873.35	0.997500	0.498750	−	0.866746i	$-0.333793\pi$
$461$	9873.35	0.997500	0.498750	+	0.866746i	$0.333793\pi$
$462$	0	0
$463$	6181.84	0.620506	0.310253	−	0.950654i	$-0.399586\pi$
$463$	6181.84	0.620506	0.310253	+	0.950654i	$0.399586\pi$
$464$	0	0
$465$	1329.89	0.132628
$466$	0	0
$467$	−6145.50	−0.608950	−0.304475	−	0.952520i	$-0.598481\pi$
$467$	−6145.50	−0.608950	−0.304475	+	0.952520i	$0.598481\pi$
$468$	0	0
$469$	−6355.69	−0.625754
$470$	0	0
$471$	7100.28	0.694615
$472$	0	0
$473$	3004.66	0.292081
$474$	0	0
$475$	1685.08	0.162772
$476$	0	0
$477$	6090.22	0.584595
$478$	0	0
$479$	−10879.4	−1.03777	−0.518887	−	0.854843i	$-0.673653\pi$
$479$	−10879.4	−1.03777	−0.518887	+	0.854843i	$0.673653\pi$
$480$	0	0
$481$	−5676.91	−0.538139
$482$	0	0
$483$	3059.17	0.288193
$484$	0	0
$485$	−2775.17	−0.259823
$486$	0	0
$487$	−8087.51	−0.752526	−0.376263	−	0.926513i	$-0.622791\pi$
$487$	−8087.51	−0.752526	−0.376263	+	0.926513i	$0.622791\pi$
$488$	0	0
$489$	−9413.08	−0.870499
$490$	0	0
$491$	6959.90	0.639707	0.319853	−	0.947467i	$-0.396366\pi$
$491$	6959.90	0.639707	0.319853	+	0.947467i	$0.396366\pi$
$492$	0	0
$493$	−3949.32	−0.360788
$494$	0	0
$495$	1106.86	0.100504
$496$	0	0
$497$	3012.34	0.271875
$498$	0	0
$499$	−18632.0	−1.67151	−0.835756	−	0.549101i	$-0.814970\pi$
$499$	−18632.0	−1.67151	−0.835756	+	0.549101i	$0.814970\pi$
$500$	0	0
$501$	439.406	0.0391840
$502$	0	0
$503$	−4627.62	−0.410209	−0.205105	−	0.978740i	$-0.565753\pi$
$503$	−4627.62	−0.410209	−0.205105	+	0.978740i	$0.565753\pi$
$504$	0	0
$505$	−9450.72	−0.832775
$506$	0	0
$507$	2912.72	0.255145
$508$	0	0
$509$	−11351.8	−0.988528	−0.494264	−	0.869312i	$-0.664562\pi$
$509$	−11351.8	−0.988528	−0.494264	+	0.869312i	$0.664562\pi$
$510$	0	0
$511$	−289.384	−0.0250521
$512$	0	0
$513$	−1819.88	−0.156627
$514$	0	0
$515$	3313.50	0.283515
$516$	0	0
$517$	8712.67	0.741166
$518$	0	0
$519$	4272.36	0.361340
$520$	0	0
$521$	19096.1	1.60579	0.802893	−	0.596123i	$-0.203293\pi$
$521$	19096.1	1.60579	0.802893	+	0.596123i	$0.203293\pi$
$522$	0	0
$523$	3145.11	0.262956	0.131478	−	0.991319i	$-0.458028\pi$
$523$	3145.11	0.262956	0.131478	+	0.991319i	$0.458028\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	−1632.99	−0.134980
$528$	0	0
$529$	9054.20	0.744160
$530$	0	0
$531$	−4511.87	−0.368735
$532$	0	0
$533$	11827.2	0.961148
$534$	0	0
$535$	8072.66	0.652358
$536$	0	0
$537$	3734.09	0.300071
$538$	0	0
$539$	−1205.25	−0.0963148
$540$	0	0
$541$	8776.12	0.697440	0.348720	−	0.937227i	$-0.386616\pi$
$541$	8776.12	0.697440	0.348720	+	0.937227i	$0.386616\pi$
$542$	0	0
$543$	11637.3	0.919710
$544$	0	0
$545$	−1086.03	−0.0853587
$546$	0	0
$547$	13695.1	1.07049	0.535247	−	0.844696i	$-0.320219\pi$
$547$	13695.1	1.07049	0.535247	+	0.844696i	$0.320219\pi$
$548$	0	0
$549$	−6380.38	−0.496007
$550$	0	0
$551$	14452.5	1.11742
$552$	0	0
$553$	6232.71	0.479280
$554$	0	0
$555$	2431.87	0.185995
$556$	0	0
$557$	7850.44	0.597188	0.298594	−	0.954380i	$-0.403482\pi$
$557$	7850.44	0.597188	0.298594	+	0.954380i	$0.403482\pi$
$558$	0	0
$559$	4277.38	0.323638
$560$	0	0
$561$	−1359.13	−0.102286
$562$	0	0
$563$	4948.81	0.370457	0.185229	−	0.982695i	$-0.440697\pi$
$563$	4948.81	0.370457	0.185229	+	0.982695i	$0.440697\pi$
$564$	0	0
$565$	8290.98	0.617353
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−8115.76	−0.597945	−0.298972	−	0.954262i	$-0.596644\pi$
$569$	−8115.76	−0.597945	−0.298972	+	0.954262i	$0.596644\pi$
$570$	0	0
$571$	−5656.42	−0.414560	−0.207280	−	0.978282i	$-0.566461\pi$
$571$	−5656.42	−0.414560	−0.207280	+	0.978282i	$0.566461\pi$
$572$	0	0
$573$	4724.71	0.344463
$574$	0	0
$575$	3641.87	0.264133
$576$	0	0
$577$	−9536.77	−0.688078	−0.344039	−	0.938955i	$-0.611795\pi$
$577$	−9536.77	−0.688078	−0.344039	+	0.938955i	$0.611795\pi$
$578$	0	0
$579$	14327.0	1.02834
$580$	0	0
$581$	−7400.05	−0.528409
$582$	0	0
$583$	−16644.5	−1.18241
$584$	0	0
$585$	1575.70	0.111363
$586$	0	0
$587$	13089.6	0.920383	0.460191	−	0.887820i	$-0.347781\pi$
$587$	13089.6	0.920383	0.460191	+	0.887820i	$0.347781\pi$
$588$	0	0
$589$	5975.92	0.418053
$590$	0	0
$591$	8410.73	0.585400
$592$	0	0
$593$	4281.96	0.296524	0.148262	−	0.988948i	$-0.452632\pi$
$593$	4281.96	0.296524	0.148262	+	0.988948i	$0.452632\pi$
$594$	0	0
$595$	−644.656	−0.0444173
$596$	0	0
$597$	12308.7	0.843825
$598$	0	0
$599$	−3699.92	−0.252378	−0.126189	−	0.992006i	$-0.540275\pi$
$599$	−3699.92	−0.252378	−0.126189	+	0.992006i	$0.540275\pi$
$600$	0	0
$601$	−17286.1	−1.17323	−0.586616	−	0.809865i	$-0.699540\pi$
$601$	−17286.1	−1.17323	−0.586616	+	0.809865i	$0.699540\pi$
$602$	0	0
$603$	8171.61	0.551863
$604$	0	0
$605$	3629.97	0.243933
$606$	0	0
$607$	−14456.7	−0.966689	−0.483344	−	0.875430i	$-0.660578\pi$
$607$	−14456.7	−0.966689	−0.483344	+	0.875430i	$0.660578\pi$
$608$	0	0
$609$	4502.79	0.299610
$610$	0	0
$611$	12403.2	0.821243
$612$	0	0
$613$	17981.9	1.18480	0.592400	−	0.805644i	$-0.298181\pi$
$613$	17981.9	1.18480	0.592400	+	0.805644i	$0.298181\pi$
$614$	0	0
$615$	−5066.53	−0.332199
$616$	0	0
$617$	19614.7	1.27983	0.639916	−	0.768445i	$-0.278969\pi$
$617$	19614.7	1.27983	0.639916	+	0.768445i	$0.278969\pi$
$618$	0	0
$619$	10462.9	0.679385	0.339692	−	0.940537i	$-0.389677\pi$
$619$	10462.9	0.679385	0.339692	+	0.940537i	$0.389677\pi$
$620$	0	0
$621$	−3933.22	−0.254162
$622$	0	0
$623$	−10316.1	−0.663411
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4973.72	0.316796
$628$	0	0
$629$	−2986.14	−0.189293
$630$	0	0
$631$	−24481.9	−1.54454	−0.772272	−	0.635292i	$-0.780880\pi$
$631$	−24481.9	−1.54454	−0.772272	+	0.635292i	$0.780880\pi$
$632$	0	0
$633$	−2470.54	−0.155126
$634$	0	0
$635$	−5544.06	−0.346471
$636$	0	0
$637$	−1715.77	−0.106721
$638$	0	0
$639$	−3873.01	−0.239771
$640$	0	0
$641$	−1109.39	−0.0683595	−0.0341797	−	0.999416i	$-0.510882\pi$
$641$	−1109.39	−0.0683595	−0.0341797	+	0.999416i	$0.510882\pi$
$642$	0	0
$643$	−30112.5	−1.84684	−0.923422	−	0.383787i	$-0.874620\pi$
$643$	−30112.5	−1.84684	−0.923422	+	0.383787i	$0.874620\pi$
$644$	0	0
$645$	−1832.34	−0.111858
$646$	0	0
$647$	4260.27	0.258869	0.129435	−	0.991588i	$-0.458684\pi$
$647$	4260.27	0.258869	0.129435	+	0.991588i	$0.458684\pi$
$648$	0	0
$649$	12330.9	0.745808
$650$	0	0
$651$	1861.85	0.112091
$652$	0	0
$653$	−10576.8	−0.633844	−0.316922	−	0.948452i	$-0.602649\pi$
$653$	−10576.8	−0.633844	−0.316922	+	0.948452i	$0.602649\pi$
$654$	0	0
$655$	927.438	0.0553252
$656$	0	0
$657$	372.066	0.0220939
$658$	0	0
$659$	−3394.70	−0.200666	−0.100333	−	0.994954i	$-0.531991\pi$
$659$	−3394.70	−0.200666	−0.100333	+	0.994954i	$0.531991\pi$
$660$	0	0
$661$	−33174.4	−1.95210	−0.976048	−	0.217554i	$-0.930192\pi$
$661$	−33174.4	−1.95210	−0.976048	+	0.217554i	$0.930192\pi$
$662$	0	0
$663$	−1934.83	−0.113337
$664$	0	0
$665$	2359.11	0.137567
$666$	0	0
$667$	31235.4	1.81326
$668$	0	0
$669$	2451.58	0.141680
$670$	0	0
$671$	17437.5	1.00323
$672$	0	0
$673$	753.881	0.0431797	0.0215899	−	0.999767i	$-0.493127\pi$
$673$	753.881	0.0431797	0.0215899	+	0.999767i	$0.493127\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	15668.8	0.889511	0.444756	−	0.895652i	$-0.353291\pi$
$677$	15668.8	0.889511	0.444756	+	0.895652i	$0.353291\pi$
$678$	0	0
$679$	−3885.24	−0.219590
$680$	0	0
$681$	10967.5	0.617147
$682$	0	0
$683$	11557.4	0.647485	0.323742	−	0.946145i	$-0.395059\pi$
$683$	11557.4	0.647485	0.323742	+	0.946145i	$0.395059\pi$
$684$	0	0
$685$	189.953	0.0105952
$686$	0	0
$687$	−2818.18	−0.156507
$688$	0	0
$689$	−23694.7	−1.31016
$690$	0	0
$691$	18503.1	1.01866	0.509328	−	0.860572i	$-0.329894\pi$
$691$	18503.1	1.01866	0.509328	+	0.860572i	$0.329894\pi$
$692$	0	0
$693$	1549.60	0.0849416
$694$	0	0
$695$	918.046	0.0501057
$696$	0	0
$697$	6221.28	0.338088
$698$	0	0
$699$	22.9410	0.00124136
$700$	0	0
$701$	22580.4	1.21662	0.608311	−	0.793699i	$-0.291847\pi$
$701$	22580.4	1.21662	0.608311	+	0.793699i	$0.291847\pi$
$702$	0	0
$703$	10927.7	0.586269
$704$	0	0
$705$	−5313.28	−0.283844
$706$	0	0
$707$	−13231.0	−0.703823
$708$	0	0
$709$	−27426.6	−1.45279	−0.726394	−	0.687278i	$-0.758805\pi$
$709$	−27426.6	−1.45279	−0.726394	+	0.687278i	$0.758805\pi$
$710$	0	0
$711$	−8013.49	−0.422685
$712$	0	0
$713$	12915.5	0.678383
$714$	0	0
$715$	−4306.37	−0.225244
$716$	0	0
$717$	−2667.35	−0.138931
$718$	0	0
$719$	−19383.0	−1.00538	−0.502688	−	0.864468i	$-0.667656\pi$
$719$	−19383.0	−1.00538	−0.502688	+	0.864468i	$0.667656\pi$
$720$	0	0
$721$	4638.90	0.239614
$722$	0	0
$723$	−6420.69	−0.330274
$724$	0	0
$725$	5360.47	0.274597
$726$	0	0
$727$	12317.3	0.628368	0.314184	−	0.949362i	$-0.398269\pi$
$727$	12317.3	0.628368	0.314184	+	0.949362i	$0.398269\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	2249.96	0.113841
$732$	0	0
$733$	1234.02	0.0621822	0.0310911	−	0.999517i	$-0.490102\pi$
$733$	1234.02	0.0621822	0.0310911	+	0.999517i	$0.490102\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	−22332.9	−1.11620
$738$	0	0
$739$	15257.3	0.759473	0.379736	−	0.925095i	$-0.376015\pi$
$739$	15257.3	0.759473	0.379736	+	0.925095i	$0.376015\pi$
$740$	0	0
$741$	7080.49	0.351023
$742$	0	0
$743$	35565.1	1.75606	0.878032	−	0.478602i	$-0.158856\pi$
$743$	35565.1	1.75606	0.878032	+	0.478602i	$0.158856\pi$
$744$	0	0
$745$	6916.69	0.340145
$746$	0	0
$747$	9514.35	0.466013
$748$	0	0
$749$	11301.7	0.551343
$750$	0	0
$751$	−14266.7	−0.693209	−0.346605	−	0.938011i	$-0.612665\pi$
$751$	−14266.7	−0.693209	−0.346605	+	0.938011i	$0.612665\pi$
$752$	0	0
$753$	−20249.4	−0.979986
$754$	0	0
$755$	3826.28	0.184441
$756$	0	0
$757$	−15927.9	−0.764744	−0.382372	−	0.924009i	$-0.624893\pi$
$757$	−15927.9	−0.764744	−0.382372	+	0.924009i	$0.624893\pi$
$758$	0	0
$759$	10749.4	0.514071
$760$	0	0
$761$	−2566.48	−0.122253	−0.0611266	−	0.998130i	$-0.519469\pi$
$761$	−2566.48	−0.122253	−0.0611266	+	0.998130i	$0.519469\pi$
$762$	0	0
$763$	−1520.44	−0.0721413
$764$	0	0
$765$	828.844	0.0391724
$766$	0	0
$767$	17554.0	0.826386
$768$	0	0
$769$	14433.1	0.676816	0.338408	−	0.940999i	$-0.390112\pi$
$769$	14433.1	0.676816	0.338408	+	0.940999i	$0.390112\pi$
$770$	0	0
$771$	−9205.91	−0.430017
$772$	0	0
$773$	−29443.2	−1.36999	−0.684993	−	0.728550i	$-0.740195\pi$
$773$	−29443.2	−1.36999	−0.684993	+	0.728550i	$0.740195\pi$
$774$	0	0
$775$	2216.48	0.102733
$776$	0	0
$777$	3404.62	0.157195
$778$	0	0
$779$	−22766.7	−1.04711
$780$	0	0
$781$	10584.9	0.484964
$782$	0	0
$783$	−5789.31	−0.264231
$784$	0	0
$785$	11833.8	0.538046
$786$	0	0
$787$	−26390.6	−1.19533	−0.597664	−	0.801747i	$-0.703904\pi$
$787$	−26390.6	−1.19533	−0.597664	+	0.801747i	$0.703904\pi$
$788$	0	0
$789$	−14022.4	−0.632711
$790$	0	0
$791$	11607.4	0.521758
$792$	0	0
$793$	24823.7	1.11162
$794$	0	0
$795$	10150.4	0.452825
$796$	0	0
$797$	−3738.33	−0.166146	−0.0830730	−	0.996543i	$-0.526473\pi$
$797$	−3738.33	−0.166146	−0.0830730	+	0.996543i	$0.526473\pi$
$798$	0	0
$799$	6524.26	0.288876
$800$	0	0
$801$	13263.5	0.585073
$802$	0	0
$803$	−1016.85	−0.0446873
$804$	0	0
$805$	5098.62	0.223233
$806$	0	0
$807$	−7252.14	−0.316341
$808$	0	0
$809$	43204.1	1.87760	0.938798	−	0.344468i	$-0.111941\pi$
$809$	43204.1	1.87760	0.938798	+	0.344468i	$0.111941\pi$
$810$	0	0
$811$	30192.4	1.30727	0.653637	−	0.756809i	$-0.273242\pi$
$811$	30192.4	1.30727	0.653637	+	0.756809i	$0.273242\pi$
$812$	0	0
$813$	23172.9	0.999642
$814$	0	0
$815$	−15688.5	−0.674286
$816$	0	0
$817$	−8233.71	−0.352584
$818$	0	0
$819$	2205.98	0.0941188
$820$	0	0
$821$	−40274.7	−1.71206	−0.856028	−	0.516929i	$-0.827075\pi$
$821$	−40274.7	−1.71206	−0.856028	+	0.516929i	$0.827075\pi$
$822$	0	0
$823$	−25184.2	−1.06667	−0.533334	−	0.845905i	$-0.679061\pi$
$823$	−25184.2	−1.06667	−0.533334	+	0.845905i	$0.679061\pi$
$824$	0	0
$825$	1844.77	0.0778503
$826$	0	0
$827$	38941.7	1.63741	0.818703	−	0.574218i	$-0.194694\pi$
$827$	38941.7	1.63741	0.818703	+	0.574218i	$0.194694\pi$
$828$	0	0
$829$	−8327.05	−0.348867	−0.174433	−	0.984669i	$-0.555809\pi$
$829$	−8327.05	−0.348867	−0.174433	+	0.984669i	$0.555809\pi$
$830$	0	0
$831$	13728.5	0.573089
$832$	0	0
$833$	−902.519	−0.0375395
$834$	0	0
$835$	732.343	0.0303518
$836$	0	0
$837$	−2393.80	−0.0988553
$838$	0	0
$839$	−8784.41	−0.361468	−0.180734	−	0.983532i	$-0.557847\pi$
$839$	−8784.41	−0.361468	−0.180734	+	0.983532i	$0.557847\pi$
$840$	0	0
$841$	21586.4	0.885087
$842$	0	0
$843$	4075.67	0.166517
$844$	0	0
$845$	4854.53	0.197634
$846$	0	0
$847$	5081.96	0.206161
$848$	0	0
$849$	11655.1	0.471145
$850$	0	0
$851$	23617.6	0.951350
$852$	0	0
$853$	−9076.15	−0.364316	−0.182158	−	0.983269i	$-0.558308\pi$
$853$	−9076.15	−0.364316	−0.182158	+	0.983269i	$0.558308\pi$
$854$	0	0
$855$	−3033.14	−0.121323
$856$	0	0
$857$	36396.7	1.45074	0.725372	−	0.688357i	$-0.241668\pi$
$857$	36396.7	1.45074	0.725372	+	0.688357i	$0.241668\pi$
$858$	0	0
$859$	−8915.27	−0.354115	−0.177058	−	0.984200i	$-0.556658\pi$
$859$	−8915.27	−0.354115	−0.177058	+	0.984200i	$0.556658\pi$
$860$	0	0
$861$	−7093.14	−0.280759
$862$	0	0
$863$	6148.26	0.242514	0.121257	−	0.992621i	$-0.461308\pi$
$863$	6148.26	0.242514	0.121257	+	0.992621i	$0.461308\pi$
$864$	0	0
$865$	7120.59	0.279893
$866$	0	0
$867$	13721.2	0.537483
$868$	0	0
$869$	21900.8	0.854928
$870$	0	0
$871$	−31792.6	−1.23680
$872$	0	0
$873$	4995.31	0.193660
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−14287.0	−0.550101	−0.275050	−	0.961430i	$-0.588694\pi$
$877$	−14287.0	−0.550101	−0.275050	+	0.961430i	$0.588694\pi$
$878$	0	0
$879$	12101.7	0.464370
$880$	0	0
$881$	−13315.9	−0.509221	−0.254610	−	0.967044i	$-0.581947\pi$
$881$	−13315.9	−0.509221	−0.254610	+	0.967044i	$0.581947\pi$
$882$	0	0
$883$	5271.78	0.200917	0.100458	−	0.994941i	$-0.467969\pi$
$883$	5271.78	0.200917	0.100458	+	0.994941i	$0.467969\pi$
$884$	0	0
$885$	−7519.78	−0.285621
$886$	0	0
$887$	−2606.07	−0.0986507	−0.0493253	−	0.998783i	$-0.515707\pi$
$887$	−2606.07	−0.0986507	−0.0493253	+	0.998783i	$0.515707\pi$
$888$	0	0
$889$	−7761.69	−0.292822
$890$	0	0
$891$	−1992.35	−0.0749115
$892$	0	0
$893$	−23875.4	−0.894694
$894$	0	0
$895$	6223.48	0.232434
$896$	0	0
$897$	15302.7	0.569612
$898$	0	0
$899$	19010.2	0.705258
$900$	0	0
$901$	−12463.8	−0.460854
$902$	0	0
$903$	−2565.28	−0.0945373
$904$	0	0
$905$	19395.4	0.712405
$906$	0	0
$907$	−18610.6	−0.681317	−0.340659	−	0.940187i	$-0.610650\pi$
$907$	−18610.6	−0.681317	−0.340659	+	0.940187i	$0.610650\pi$
$908$	0	0
$909$	17011.3	0.620714
$910$	0	0
$911$	−41091.7	−1.49443	−0.747216	−	0.664581i	$-0.768610\pi$
$911$	−41091.7	−1.49443	−0.747216	+	0.664581i	$0.768610\pi$
$912$	0	0
$913$	−26002.6	−0.942563
$914$	0	0
$915$	−10634.0	−0.384206
$916$	0	0
$917$	1298.41	0.0467583
$918$	0	0
$919$	−38891.3	−1.39598	−0.697990	−	0.716107i	$-0.745922\pi$
$919$	−38891.3	−1.39598	−0.697990	+	0.716107i	$0.745922\pi$
$920$	0	0
$921$	−13861.1	−0.495915
$922$	0	0
$923$	15068.4	0.537360
$924$	0	0
$925$	4053.12	0.144071
$926$	0	0
$927$	−5964.30	−0.211320
$928$	0	0
$929$	18699.4	0.660396	0.330198	−	0.943912i	$-0.392885\pi$
$929$	18699.4	0.660396	0.330198	+	0.943912i	$0.392885\pi$
$930$	0	0
$931$	3302.75	0.116266
$932$	0	0
$933$	20027.7	0.702761
$934$	0	0
$935$	−2265.22	−0.0792305
$936$	0	0
$937$	21509.6	0.749933	0.374967	−	0.927038i	$-0.377654\pi$
$937$	21509.6	0.749933	0.374967	+	0.927038i	$0.377654\pi$
$938$	0	0
$939$	−8510.35	−0.295767
$940$	0	0
$941$	−11241.7	−0.389448	−0.194724	−	0.980858i	$-0.562381\pi$
$941$	−11241.7	−0.389448	−0.194724	+	0.980858i	$0.562381\pi$
$942$	0	0
$943$	−49204.5	−1.69917
$944$	0	0
$945$	−945.000	−0.0325300
$946$	0	0
$947$	36556.3	1.25441	0.627203	−	0.778856i	$-0.284200\pi$
$947$	36556.3	1.25441	0.627203	+	0.778856i	$0.284200\pi$
$948$	0	0
$949$	−1447.57	−0.0495153
$950$	0	0
$951$	−12031.9	−0.410263
$952$	0	0
$953$	−36633.4	−1.24520	−0.622598	−	0.782542i	$-0.713923\pi$
$953$	−36633.4	−1.24520	−0.622598	+	0.782542i	$0.713923\pi$
$954$	0	0
$955$	7874.52	0.266820
$956$	0	0
$957$	15822.1	0.534436
$958$	0	0
$959$	265.935	0.00895462
$960$	0	0
$961$	−21930.5	−0.736146
$962$	0	0
$963$	−14530.8	−0.486239
$964$	0	0
$965$	23878.4	0.796551
$966$	0	0
$967$	35515.8	1.18109	0.590544	−	0.807006i	$-0.298913\pi$
$967$	35515.8	1.18109	0.590544	+	0.807006i	$0.298913\pi$
$968$	0	0
$969$	3724.44	0.123474
$970$	0	0
$971$	39661.0	1.31080	0.655398	−	0.755283i	$-0.272501\pi$
$971$	39661.0	1.31080	0.655398	+	0.755283i	$0.272501\pi$
$972$	0	0
$973$	1285.26	0.0423471
$974$	0	0
$975$	2626.17	0.0862613
$976$	0	0
$977$	50325.3	1.64795	0.823977	−	0.566624i	$-0.191751\pi$
$977$	50325.3	1.64795	0.823977	+	0.566624i	$0.191751\pi$
$978$	0	0
$979$	−36249.0	−1.18337
$980$	0	0
$981$	1954.86	0.0636226
$982$	0	0
$983$	−51189.0	−1.66091	−0.830456	−	0.557084i	$-0.811920\pi$
$983$	−51189.0	−1.66091	−0.830456	+	0.557084i	$0.811920\pi$
$984$	0	0
$985$	14017.9	0.453449
$986$	0	0
$987$	−7438.59	−0.239892
$988$	0	0
$989$	−17795.1	−0.572145
$990$	0	0
$991$	55137.3	1.76740	0.883700	−	0.468054i	$-0.155045\pi$
$991$	55137.3	1.76740	0.883700	+	0.468054i	$0.155045\pi$
$992$	0	0
$993$	33262.6	1.06300
$994$	0	0
$995$	20514.6	0.653624
$996$	0	0
$997$	41606.5	1.32166	0.660828	−	0.750537i	$-0.270205\pi$
$997$	41606.5	1.32166	0.660828	+	0.750537i	$0.270205\pi$
$998$	0	0
$999$	−4377.37	−0.138633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.4.a.y.1.2		2
4.3	odd	2		105.4.a.g.1.2	✓	2
12.11	even	2		315.4.a.g.1.1		2
20.3	even	4		525.4.d.j.274.1		4
20.7	even	4		525.4.d.j.274.4		4
20.19	odd	2		525.4.a.i.1.1		2
28.27	even	2		735.4.a.q.1.2		2
60.59	even	2		1575.4.a.y.1.2		2
84.83	odd	2		2205.4.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.g.1.2	✓	2	4.3	odd	2
315.4.a.g.1.1		2	12.11	even	2
525.4.a.i.1.1		2	20.19	odd	2
525.4.d.j.274.1		4	20.3	even	4
525.4.d.j.274.4		4	20.7	even	4
735.4.a.q.1.2		2	28.27	even	2
1575.4.a.y.1.2		2	60.59	even	2
1680.4.a.y.1.2		2	1.1	even	1	trivial
2205.4.a.v.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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -24.5969 q^{11} -35.0156 q^{13} +15.0000 q^{15} -18.4187 q^{17} +67.4031 q^{19} +21.0000 q^{21} +145.675 q^{23} +25.0000 q^{25} -27.0000 q^{27} +214.419 q^{29} +88.6594 q^{31} +73.7906 q^{33} +35.0000 q^{35} +162.125 q^{37} +105.047 q^{39} -337.769 q^{41} -122.156 q^{43} -45.0000 q^{45} -354.219 q^{47} +49.0000 q^{49} +55.2562 q^{51} +676.691 q^{53} +122.984 q^{55} -202.209 q^{57} -501.319 q^{59} -708.931 q^{61} -63.0000 q^{63} +175.078 q^{65} +907.956 q^{67} -437.025 q^{69} -430.334 q^{71} +41.3406 q^{73} -75.0000 q^{75} +172.178 q^{77} -890.388 q^{79} +81.0000 q^{81} +1057.15 q^{83} +92.0937 q^{85} -643.256 q^{87} +1473.72 q^{89} +245.109 q^{91} -265.978 q^{93} -337.016 q^{95} +555.034 q^{97} -221.372 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 10 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) \) 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

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