LMFDB - Embedded newform 8024.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.642074 q^{3} +1.47283 q^{5} +4.58774 q^{7} -2.58774 q^{9} +O(q^{10})$	$q-0.642074 q^{3} +1.47283 q^{5} +4.58774 q^{7} -2.58774 q^{9} +4.75698 q^{11} +6.11491 q^{13} -0.945668 q^{15} +1.00000 q^{17} -6.81756 q^{19} -2.94567 q^{21} +5.39905 q^{23} -2.83076 q^{25} +3.58774 q^{27} +6.47283 q^{29} +3.00000 q^{31} -3.05433 q^{33} +6.75698 q^{35} +1.41226 q^{37} -3.92622 q^{39} +0.696406 q^{41} -6.62887 q^{43} -3.81131 q^{45} -1.83076 q^{47} +14.0474 q^{49} -0.642074 q^{51} +2.35793 q^{53} +7.00624 q^{55} +4.37737 q^{57} -1.00000 q^{59} +3.87189 q^{61} -11.8719 q^{63} +9.00624 q^{65} +7.94567 q^{67} -3.46659 q^{69} +8.88509 q^{71} +3.18244 q^{73} +1.81756 q^{75} +21.8238 q^{77} -11.7026 q^{79} +5.45963 q^{81} -9.45963 q^{83} +1.47283 q^{85} -4.15604 q^{87} -1.45339 q^{89} +28.0536 q^{91} -1.92622 q^{93} -10.0411 q^{95} +0.243019 q^{97} -12.3098 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q - q^3 - q^5 + 2 * q^7 + 4 * q^9	$ 3 q - q^{3} - q^{5} + 2 q^{7} + 4 q^{9} + 7 q^{11} + 12 q^{13} + 8 q^{15} + 3 q^{17} + 4 q^{19} + 2 q^{21} + 8 q^{23} - 4 q^{25} - q^{27} + 14 q^{29} + 9 q^{31} - 20 q^{33} + 13 q^{35} + 16 q^{37} - 9 q^{39} + 12 q^{41} + q^{43} - 15 q^{45} - q^{47} + 5 q^{49} - q^{51} + 8 q^{53} - 7 q^{55} + 6 q^{57} - 3 q^{59} - 2 q^{61} - 22 q^{63} - q^{65} + 13 q^{67} - 33 q^{69} + 33 q^{71} + 34 q^{73} - 19 q^{75} + 13 q^{77} - 17 q^{79} - 9 q^{81} - 3 q^{83} - q^{85} + 3 q^{87} - 7 q^{89} + 19 q^{91} - 3 q^{93} - 21 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) $ 3 * q - q^3 - q^5 + 2 * q^7 + 4 * q^9 + 7 * q^11 + 12 * q^13 + 8 * q^15 + 3 * q^17 + 4 * q^19 + 2 * q^21 + 8 * q^23 - 4 * q^25 - q^27 + 14 * q^29 + 9 * q^31 - 20 * q^33 + 13 * q^35 + 16 * q^37 - 9 * q^39 + 12 * q^41 + q^43 - 15 * q^45 - q^47 + 5 * q^49 - q^51 + 8 * q^53 - 7 * q^55 + 6 * q^57 - 3 * q^59 - 2 * q^61 - 22 * q^63 - q^65 + 13 * q^67 - 33 * q^69 + 33 * q^71 + 34 * q^73 - 19 * q^75 + 13 * q^77 - 17 * q^79 - 9 * q^81 - 3 * q^83 - q^85 + 3 * q^87 - 7 * q^89 + 19 * q^91 - 3 * q^93 - 21 * q^95 + 8 * q^97 + 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
$3$	−0.642074	−0.370701	−0.185351	−	0.982672i	$-0.559342\pi$
$3$	−0.642074	−0.370701	−0.185351	+	0.982672i	$0.559342\pi$
$4$	0	0
$5$	1.47283	0.658671	0.329336	−	0.944213i	$-0.393175\pi$
$5$	1.47283	0.658671	0.329336	+	0.944213i	$0.393175\pi$
$6$	0	0
$7$	4.58774	1.73400	0.867002	−	0.498305i	$-0.166044\pi$
$7$	4.58774	1.73400	0.867002	+	0.498305i	$0.166044\pi$
$8$	0	0
$9$	−2.58774	−0.862580
$10$	0	0
$11$	4.75698	1.43428	0.717142	−	0.696927i	$-0.245450\pi$
$11$	4.75698	1.43428	0.717142	+	0.696927i	$0.245450\pi$
$12$	0	0
$13$	6.11491	1.69597	0.847985	−	0.530020i	$-0.177816\pi$
$13$	6.11491	1.69597	0.847985	+	0.530020i	$0.177816\pi$
$14$	0	0
$15$	−0.945668	−0.244170
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−6.81756	−1.56405	−0.782027	−	0.623244i	$-0.785814\pi$
$19$	−6.81756	−1.56405	−0.782027	+	0.623244i	$0.785814\pi$
$20$	0	0
$21$	−2.94567	−0.642797
$22$	0	0
$23$	5.39905	1.12578	0.562890	−	0.826532i	$-0.309689\pi$
$23$	5.39905	1.12578	0.562890	+	0.826532i	$0.309689\pi$
$24$	0	0
$25$	−2.83076	−0.566152
$26$	0	0
$27$	3.58774	0.690461
$28$	0	0
$29$	6.47283	1.20198	0.600988	−	0.799258i	$-0.294774\pi$
$29$	6.47283	1.20198	0.600988	+	0.799258i	$0.294774\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	−3.05433	−0.531691
$34$	0	0
$35$	6.75698	1.14214
$36$	0	0
$37$	1.41226	0.232174	0.116087	−	0.993239i	$-0.462965\pi$
$37$	1.41226	0.232174	0.116087	+	0.993239i	$0.462965\pi$
$38$	0	0
$39$	−3.92622	−0.628699
$40$	0	0
$41$	0.696406	0.108760	0.0543802	−	0.998520i	$-0.482682\pi$
$41$	0.696406	0.108760	0.0543802	+	0.998520i	$0.482682\pi$
$42$	0	0
$43$	−6.62887	−1.01089	−0.505447	−	0.862858i	$-0.668672\pi$
$43$	−6.62887	−1.01089	−0.505447	+	0.862858i	$0.668672\pi$
$44$	0	0
$45$	−3.81131	−0.568157
$46$	0	0
$47$	−1.83076	−0.267044	−0.133522	−	0.991046i	$-0.542629\pi$
$47$	−1.83076	−0.267044	−0.133522	+	0.991046i	$0.542629\pi$
$48$	0	0
$49$	14.0474	2.00677
$50$	0	0
$51$	−0.642074	−0.0899083
$52$	0	0
$53$	2.35793	0.323886	0.161943	−	0.986800i	$-0.448224\pi$
$53$	2.35793	0.323886	0.161943	+	0.986800i	$0.448224\pi$
$54$	0	0
$55$	7.00624	0.944722
$56$	0	0
$57$	4.37737	0.579797
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	3.87189	0.495745	0.247872	−	0.968793i	$-0.420269\pi$
$61$	3.87189	0.495745	0.247872	+	0.968793i	$0.420269\pi$
$62$	0	0
$63$	−11.8719	−1.49572
$64$	0	0
$65$	9.00624	1.11709
$66$	0	0
$67$	7.94567	0.970718	0.485359	−	0.874315i	$-0.338689\pi$
$67$	7.94567	0.970718	0.485359	+	0.874315i	$0.338689\pi$
$68$	0	0
$69$	−3.46659	−0.417328
$70$	0	0
$71$	8.88509	1.05447	0.527233	−	0.849721i	$-0.323229\pi$
$71$	8.88509	1.05447	0.527233	+	0.849721i	$0.323229\pi$
$72$	0	0
$73$	3.18244	0.372477	0.186238	−	0.982505i	$-0.440370\pi$
$73$	3.18244	0.372477	0.186238	+	0.982505i	$0.440370\pi$
$74$	0	0
$75$	1.81756	0.209873
$76$	0	0
$77$	21.8238	2.48705
$78$	0	0
$79$	−11.7026	−1.31665	−0.658325	−	0.752733i	$-0.728735\pi$
$79$	−11.7026	−1.31665	−0.658325	+	0.752733i	$0.728735\pi$
$80$	0	0
$81$	5.45963	0.606626
$82$	0	0
$83$	−9.45963	−1.03833	−0.519165	−	0.854674i	$-0.673757\pi$
$83$	−9.45963	−1.03833	−0.519165	+	0.854674i	$0.673757\pi$
$84$	0	0
$85$	1.47283	0.159751
$86$	0	0
$87$	−4.15604	−0.445574
$88$	0	0
$89$	−1.45339	−0.154059	−0.0770294	−	0.997029i	$-0.524544\pi$
$89$	−1.45339	−0.154059	−0.0770294	+	0.997029i	$0.524544\pi$
$90$	0	0
$91$	28.0536	2.94082
$92$	0	0
$93$	−1.92622	−0.199740
$94$	0	0
$95$	−10.0411	−1.03020
$96$	0	0
$97$	0.243019	0.0246748	0.0123374	−	0.999924i	$-0.496073\pi$
$97$	0.243019	0.0246748	0.0123374	+	0.999924i	$0.496073\pi$
$98$	0	0
$99$	−12.3098	−1.23719
$100$	0	0
$101$	−11.6072	−1.15496	−0.577479	−	0.816405i	$-0.695964\pi$
$101$	−11.6072	−1.15496	−0.577479	+	0.816405i	$0.695964\pi$
$102$	0	0
$103$	3.54037	0.348843	0.174422	−	0.984671i	$-0.444194\pi$
$103$	3.54037	0.348843	0.174422	+	0.984671i	$0.444194\pi$
$104$	0	0
$105$	−4.33848	−0.423392
$106$	0	0
$107$	9.62887	0.930858	0.465429	−	0.885085i	$-0.345900\pi$
$107$	9.62887	0.930858	0.465429	+	0.885085i	$0.345900\pi$
$108$	0	0
$109$	−17.5140	−1.67753	−0.838767	−	0.544491i	$-0.816723\pi$
$109$	−17.5140	−1.67753	−0.838767	+	0.544491i	$0.816723\pi$
$110$	0	0
$111$	−0.906774	−0.0860672
$112$	0	0
$113$	−6.21037	−0.584222	−0.292111	−	0.956384i	$-0.594358\pi$
$113$	−6.21037	−0.584222	−0.292111	+	0.956384i	$0.594358\pi$
$114$	0	0
$115$	7.95191	0.741520
$116$	0	0
$117$	−15.8238	−1.46291
$118$	0	0
$119$	4.58774	0.420558
$120$	0	0
$121$	11.6289	1.05717
$122$	0	0
$123$	−0.447144	−0.0403176
$124$	0	0
$125$	−11.5334	−1.03158
$126$	0	0
$127$	5.71585	0.507200	0.253600	−	0.967309i	$-0.418385\pi$
$127$	5.71585	0.507200	0.253600	+	0.967309i	$0.418385\pi$
$128$	0	0
$129$	4.25622	0.374740
$130$	0	0
$131$	9.51396	0.831239	0.415619	−	0.909539i	$-0.363565\pi$
$131$	9.51396	0.831239	0.415619	+	0.909539i	$0.363565\pi$
$132$	0	0
$133$	−31.2772	−2.71208
$134$	0	0
$135$	5.28415	0.454787
$136$	0	0
$137$	2.66152	0.227389	0.113695	−	0.993516i	$-0.463731\pi$
$137$	2.66152	0.227389	0.113695	+	0.993516i	$0.463731\pi$
$138$	0	0
$139$	−8.32304	−0.705951	−0.352976	−	0.935633i	$-0.614830\pi$
$139$	−8.32304	−0.705951	−0.352976	+	0.935633i	$0.614830\pi$
$140$	0	0
$141$	1.17548	0.0989935
$142$	0	0
$143$	29.0885	2.43250
$144$	0	0
$145$	9.53341	0.791707
$146$	0	0
$147$	−9.01945	−0.743911
$148$	0	0
$149$	−5.94567	−0.487088	−0.243544	−	0.969890i	$-0.578310\pi$
$149$	−5.94567	−0.487088	−0.243544	+	0.969890i	$0.578310\pi$
$150$	0	0
$151$	4.54661	0.369998	0.184999	−	0.982739i	$-0.440772\pi$
$151$	4.54661	0.369998	0.184999	+	0.982739i	$0.440772\pi$
$152$	0	0
$153$	−2.58774	−0.209206
$154$	0	0
$155$	4.41850	0.354903
$156$	0	0
$157$	−0.114908	−0.00917062	−0.00458531	−	0.999989i	$-0.501460\pi$
$157$	−0.114908	−0.00917062	−0.00458531	+	0.999989i	$0.501460\pi$
$158$	0	0
$159$	−1.51396	−0.120065
$160$	0	0
$161$	24.7695	1.95211
$162$	0	0
$163$	−7.16228	−0.560993	−0.280496	−	0.959855i	$-0.590499\pi$
$163$	−7.16228	−0.560993	−0.280496	+	0.959855i	$0.590499\pi$
$164$	0	0
$165$	−4.49852	−0.350210
$166$	0	0
$167$	−4.75698	−0.368106	−0.184053	−	0.982916i	$-0.558922\pi$
$167$	−4.75698	−0.368106	−0.184053	+	0.982916i	$0.558922\pi$
$168$	0	0
$169$	24.3921	1.87631
$170$	0	0
$171$	17.6421	1.34912
$172$	0	0
$173$	10.7089	0.814182	0.407091	−	0.913388i	$-0.366543\pi$
$173$	10.7089	0.814182	0.407091	+	0.913388i	$0.366543\pi$
$174$	0	0
$175$	−12.9868	−0.981710
$176$	0	0
$177$	0.642074	0.0482612
$178$	0	0
$179$	−17.8976	−1.33773	−0.668864	−	0.743385i	$-0.733219\pi$
$179$	−17.8976	−1.33773	−0.668864	+	0.743385i	$0.733219\pi$
$180$	0	0
$181$	−9.69641	−0.720728	−0.360364	−	0.932812i	$-0.617348\pi$
$181$	−9.69641	−0.720728	−0.360364	+	0.932812i	$0.617348\pi$
$182$	0	0
$183$	−2.48604	−0.183773
$184$	0	0
$185$	2.08002	0.152926
$186$	0	0
$187$	4.75698	0.347865
$188$	0	0
$189$	16.4596	1.19726
$190$	0	0
$191$	−2.71585	−0.196512	−0.0982561	−	0.995161i	$-0.531326\pi$
$191$	−2.71585	−0.196512	−0.0982561	+	0.995161i	$0.531326\pi$
$192$	0	0
$193$	−25.9930	−1.87102	−0.935510	−	0.353300i	$-0.885059\pi$
$193$	−25.9930	−1.87102	−0.935510	+	0.353300i	$0.885059\pi$
$194$	0	0
$195$	−5.78267	−0.414106
$196$	0	0
$197$	−8.57454	−0.610911	−0.305455	−	0.952206i	$-0.598809\pi$
$197$	−8.57454	−0.610911	−0.305455	+	0.952206i	$0.598809\pi$
$198$	0	0
$199$	−26.2034	−1.85751	−0.928755	−	0.370694i	$-0.879120\pi$
$199$	−26.2034	−1.85751	−0.928755	+	0.370694i	$0.879120\pi$
$200$	0	0
$201$	−5.10170	−0.359846
$202$	0	0
$203$	29.6957	2.08423
$204$	0	0
$205$	1.02569	0.0716373
$206$	0	0
$207$	−13.9714	−0.971077
$208$	0	0
$209$	−32.4310	−2.24330
$210$	0	0
$211$	−9.30359	−0.640486	−0.320243	−	0.947335i	$-0.603765\pi$
$211$	−9.30359	−0.640486	−0.320243	+	0.947335i	$0.603765\pi$
$212$	0	0
$213$	−5.70488	−0.390892
$214$	0	0
$215$	−9.76322	−0.665846
$216$	0	0
$217$	13.7632	0.934309
$218$	0	0
$219$	−2.04336	−0.138078
$220$	0	0
$221$	6.11491	0.411333
$222$	0	0
$223$	11.4185	0.764639	0.382320	−	0.924030i	$-0.375125\pi$
$223$	11.4185	0.764639	0.382320	+	0.924030i	$0.375125\pi$
$224$	0	0
$225$	7.32528	0.488352
$226$	0	0
$227$	28.8587	1.91542	0.957709	−	0.287738i	$-0.0929033\pi$
$227$	28.8587	1.91542	0.957709	+	0.287738i	$0.0929033\pi$
$228$	0	0
$229$	0.351683	0.0232399	0.0116199	−	0.999932i	$-0.496301\pi$
$229$	0.351683	0.0232399	0.0116199	+	0.999932i	$0.496301\pi$
$230$	0	0
$231$	−14.0125	−0.921954
$232$	0	0
$233$	7.55286	0.494804	0.247402	−	0.968913i	$-0.420423\pi$
$233$	7.55286	0.494804	0.247402	+	0.968913i	$0.420423\pi$
$234$	0	0
$235$	−2.69641	−0.175894
$236$	0	0
$237$	7.51396	0.488084
$238$	0	0
$239$	−10.7959	−0.698327	−0.349164	−	0.937062i	$-0.613534\pi$
$239$	−10.7959	−0.698327	−0.349164	+	0.937062i	$0.613534\pi$
$240$	0	0
$241$	−21.7151	−1.39879	−0.699397	−	0.714733i	$-0.746548\pi$
$241$	−21.7151	−1.39879	−0.699397	+	0.714733i	$0.746548\pi$
$242$	0	0
$243$	−14.2687	−0.915338
$244$	0	0
$245$	20.6894	1.32180
$246$	0	0
$247$	−41.6887	−2.65259
$248$	0	0
$249$	6.07378	0.384910
$250$	0	0
$251$	13.2166	0.834225	0.417113	−	0.908855i	$-0.363042\pi$
$251$	13.2166	0.834225	0.417113	+	0.908855i	$0.363042\pi$
$252$	0	0
$253$	25.6832	1.61469
$254$	0	0
$255$	−0.945668	−0.0592200
$256$	0	0
$257$	−23.0823	−1.43983	−0.719916	−	0.694061i	$-0.755820\pi$
$257$	−23.0823	−1.43983	−0.719916	+	0.694061i	$0.755820\pi$
$258$	0	0
$259$	6.47908	0.402590
$260$	0	0
$261$	−16.7500	−1.03680
$262$	0	0
$263$	13.9993	0.863233	0.431616	−	0.902057i	$-0.357943\pi$
$263$	13.9993	0.863233	0.431616	+	0.902057i	$0.357943\pi$
$264$	0	0
$265$	3.47283	0.213334
$266$	0	0
$267$	0.933181	0.0571098
$268$	0	0
$269$	−0.520206	−0.0317175	−0.0158587	−	0.999874i	$-0.505048\pi$
$269$	−0.520206	−0.0317175	−0.0158587	+	0.999874i	$0.505048\pi$
$270$	0	0
$271$	25.3859	1.54208	0.771040	−	0.636786i	$-0.219737\pi$
$271$	25.3859	1.54208	0.771040	+	0.636786i	$0.219737\pi$
$272$	0	0
$273$	−18.0125	−1.09017
$274$	0	0
$275$	−13.4659	−0.812023
$276$	0	0
$277$	15.7570	0.946745	0.473373	−	0.880862i	$-0.343036\pi$
$277$	15.7570	0.946745	0.473373	+	0.880862i	$0.343036\pi$
$278$	0	0
$279$	−7.76322	−0.464772
$280$	0	0
$281$	−28.4853	−1.69929	−0.849646	−	0.527354i	$-0.823184\pi$
$281$	−28.4853	−1.69929	−0.849646	+	0.527354i	$0.823184\pi$
$282$	0	0
$283$	31.3572	1.86399	0.931996	−	0.362468i	$-0.118066\pi$
$283$	31.3572	1.86399	0.931996	+	0.362468i	$0.118066\pi$
$284$	0	0
$285$	6.44714	0.381896
$286$	0	0
$287$	3.19493	0.188591
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−0.156036	−0.00914699
$292$	0	0
$293$	−22.3532	−1.30589	−0.652944	−	0.757406i	$-0.726466\pi$
$293$	−22.3532	−1.30589	−0.652944	+	0.757406i	$0.726466\pi$
$294$	0	0
$295$	−1.47283	−0.0857517
$296$	0	0
$297$	17.0668	0.990317
$298$	0	0
$299$	33.0147	1.90929
$300$	0	0
$301$	−30.4115	−1.75289
$302$	0	0
$303$	7.45267	0.428145
$304$	0	0
$305$	5.70265	0.326533
$306$	0	0
$307$	5.75074	0.328212	0.164106	−	0.986443i	$-0.447526\pi$
$307$	5.75074	0.328212	0.164106	+	0.986443i	$0.447526\pi$
$308$	0	0
$309$	−2.27318	−0.129317
$310$	0	0
$311$	−14.7911	−0.838729	−0.419365	−	0.907818i	$-0.637747\pi$
$311$	−14.7911	−0.838729	−0.419365	+	0.907818i	$0.637747\pi$
$312$	0	0
$313$	21.2904	1.20340	0.601702	−	0.798721i	$-0.294489\pi$
$313$	21.2904	1.20340	0.601702	+	0.798721i	$0.294489\pi$
$314$	0	0
$315$	−17.4853	−0.985186
$316$	0	0
$317$	−7.69417	−0.432148	−0.216074	−	0.976377i	$-0.569325\pi$
$317$	−7.69417	−0.432148	−0.216074	+	0.976377i	$0.569325\pi$
$318$	0	0
$319$	30.7911	1.72397
$320$	0	0
$321$	−6.18244	−0.345070
$322$	0	0
$323$	−6.81756	−0.379339
$324$	0	0
$325$	−17.3098	−0.960177
$326$	0	0
$327$	11.2453	0.621864
$328$	0	0
$329$	−8.39905	−0.463055
$330$	0	0
$331$	−6.75002	−0.371015	−0.185507	−	0.982643i	$-0.559393\pi$
$331$	−6.75002	−0.371015	−0.185507	+	0.982643i	$0.559393\pi$
$332$	0	0
$333$	−3.65456	−0.200269
$334$	0	0
$335$	11.7026	0.639384
$336$	0	0
$337$	4.46587	0.243272	0.121636	−	0.992575i	$-0.461186\pi$
$337$	4.46587	0.243272	0.121636	+	0.992575i	$0.461186\pi$
$338$	0	0
$339$	3.98751	0.216572
$340$	0	0
$341$	14.2709	0.772815
$342$	0	0
$343$	32.3315	1.74574
$344$	0	0
$345$	−5.10571	−0.274882
$346$	0	0
$347$	21.7563	1.16794	0.583969	−	0.811776i	$-0.301499\pi$
$347$	21.7563	1.16794	0.583969	+	0.811776i	$0.301499\pi$
$348$	0	0
$349$	−2.66848	−0.142841	−0.0714203	−	0.997446i	$-0.522753\pi$
$349$	−2.66848	−0.142841	−0.0714203	+	0.997446i	$0.522753\pi$
$350$	0	0
$351$	21.9387	1.17100
$352$	0	0
$353$	−1.58150	−0.0841747	−0.0420873	−	0.999114i	$-0.513401\pi$
$353$	−1.58150	−0.0841747	−0.0420873	+	0.999114i	$0.513401\pi$
$354$	0	0
$355$	13.0863	0.694547
$356$	0	0
$357$	−2.94567	−0.155901
$358$	0	0
$359$	−1.81756	−0.0959270	−0.0479635	−	0.998849i	$-0.515273\pi$
$359$	−1.81756	−0.0959270	−0.0479635	+	0.998849i	$0.515273\pi$
$360$	0	0
$361$	27.4791	1.44627
$362$	0	0
$363$	−7.46659	−0.391894
$364$	0	0
$365$	4.68721	0.245340
$366$	0	0
$367$	−25.4813	−1.33011	−0.665057	−	0.746793i	$-0.731593\pi$
$367$	−25.4813	−1.33011	−0.665057	+	0.746793i	$0.731593\pi$
$368$	0	0
$369$	−1.80212	−0.0938145
$370$	0	0
$371$	10.8176	0.561620
$372$	0	0
$373$	11.8983	0.616070	0.308035	−	0.951375i	$-0.400329\pi$
$373$	11.8983	0.616070	0.308035	+	0.951375i	$0.400329\pi$
$374$	0	0
$375$	7.40530	0.382408
$376$	0	0
$377$	39.5808	2.03851
$378$	0	0
$379$	−30.9170	−1.58810	−0.794050	−	0.607852i	$-0.792031\pi$
$379$	−30.9170	−1.58810	−0.794050	+	0.607852i	$0.792031\pi$
$380$	0	0
$381$	−3.67000	−0.188020
$382$	0	0
$383$	8.49852	0.434254	0.217127	−	0.976143i	$-0.430331\pi$
$383$	8.49852	0.434254	0.217127	+	0.976143i	$0.430331\pi$
$384$	0	0
$385$	32.1428	1.63815
$386$	0	0
$387$	17.1538	0.871977
$388$	0	0
$389$	−29.0404	−1.47241	−0.736204	−	0.676760i	$-0.763383\pi$
$389$	−29.0404	−1.47241	−0.736204	+	0.676760i	$0.763383\pi$
$390$	0	0
$391$	5.39905	0.273042
$392$	0	0
$393$	−6.10866	−0.308141
$394$	0	0
$395$	−17.2361	−0.867240
$396$	0	0
$397$	21.3253	1.07028	0.535142	−	0.844762i	$-0.320258\pi$
$397$	21.3253	1.07028	0.535142	+	0.844762i	$0.320258\pi$
$398$	0	0
$399$	20.0823	1.00537
$400$	0	0
$401$	7.94567	0.396788	0.198394	−	0.980122i	$-0.436427\pi$
$401$	7.94567	0.396788	0.198394	+	0.980122i	$0.436427\pi$
$402$	0	0
$403$	18.3447	0.913816
$404$	0	0
$405$	8.04113	0.399567
$406$	0	0
$407$	6.71809	0.333003
$408$	0	0
$409$	14.4721	0.715600	0.357800	−	0.933798i	$-0.383527\pi$
$409$	14.4721	0.715600	0.357800	+	0.933798i	$0.383527\pi$
$410$	0	0
$411$	−1.70889	−0.0842934
$412$	0	0
$413$	−4.58774	−0.225748
$414$	0	0
$415$	−13.9325	−0.683918
$416$	0	0
$417$	5.34401	0.261697
$418$	0	0
$419$	−0.0264075	−0.00129009	−0.000645045	−	1.00000i	$-0.500205\pi$
$419$	−0.0264075	−0.00129009	−0.000645045		1.00000i	$0.500205\pi$
$420$	0	0
$421$	−6.75475	−0.329206	−0.164603	−	0.986360i	$-0.552634\pi$
$421$	−6.75475	−0.329206	−0.164603	+	0.986360i	$0.552634\pi$
$422$	0	0
$423$	4.73753	0.230347
$424$	0	0
$425$	−2.83076	−0.137312
$426$	0	0
$427$	17.7632	0.859623
$428$	0	0
$429$	−18.6770	−0.901732
$430$	0	0
$431$	2.81756	0.135717	0.0678585	−	0.997695i	$-0.478383\pi$
$431$	2.81756	0.135717	0.0678585	+	0.997695i	$0.478383\pi$
$432$	0	0
$433$	28.3378	1.36183	0.680913	−	0.732364i	$-0.261583\pi$
$433$	28.3378	1.36183	0.680913	+	0.732364i	$0.261583\pi$
$434$	0	0
$435$	−6.12115	−0.293487
$436$	0	0
$437$	−36.8084	−1.76078
$438$	0	0
$439$	36.3098	1.73297	0.866487	−	0.499200i	$-0.166373\pi$
$439$	36.3098	1.73297	0.866487	+	0.499200i	$0.166373\pi$
$440$	0	0
$441$	−36.3510	−1.73100
$442$	0	0
$443$	14.2229	0.675748	0.337874	−	0.941191i	$-0.390292\pi$
$443$	14.2229	0.675748	0.337874	+	0.941191i	$0.390292\pi$
$444$	0	0
$445$	−2.14060	−0.101474
$446$	0	0
$447$	3.81756	0.180564
$448$	0	0
$449$	−24.8998	−1.17509	−0.587547	−	0.809190i	$-0.699906\pi$
$449$	−24.8998	−1.17509	−0.587547	+	0.809190i	$0.699906\pi$
$450$	0	0
$451$	3.31279	0.155993
$452$	0	0
$453$	−2.91926	−0.137159
$454$	0	0
$455$	41.3183	1.93703
$456$	0	0
$457$	6.45267	0.301843	0.150922	−	0.988546i	$-0.451776\pi$
$457$	6.45267	0.301843	0.150922	+	0.988546i	$0.451776\pi$
$458$	0	0
$459$	3.58774	0.167461
$460$	0	0
$461$	−19.5070	−0.908532	−0.454266	−	0.890866i	$-0.650098\pi$
$461$	−19.5070	−0.908532	−0.454266	+	0.890866i	$0.650098\pi$
$462$	0	0
$463$	33.1685	1.54147	0.770736	−	0.637155i	$-0.219889\pi$
$463$	33.1685	1.54147	0.770736	+	0.637155i	$0.219889\pi$
$464$	0	0
$465$	−2.83700	−0.131563
$466$	0	0
$467$	23.3851	1.08213	0.541067	−	0.840979i	$-0.318020\pi$
$467$	23.3851	1.08213	0.541067	+	0.840979i	$0.318020\pi$
$468$	0	0
$469$	36.4527	1.68323
$470$	0	0
$471$	0.0737791	0.00339956
$472$	0	0
$473$	−31.5334	−1.44991
$474$	0	0
$475$	19.2989	0.885493
$476$	0	0
$477$	−6.10170	−0.279378
$478$	0	0
$479$	16.9526	0.774585	0.387293	−	0.921957i	$-0.373410\pi$
$479$	16.9526	0.774585	0.387293	+	0.921957i	$0.373410\pi$
$480$	0	0
$481$	8.63583	0.393760
$482$	0	0
$483$	−15.9038	−0.723649
$484$	0	0
$485$	0.357926	0.0162526
$486$	0	0
$487$	22.0753	1.00033	0.500164	−	0.865931i	$-0.333273\pi$
$487$	22.0753	1.00033	0.500164	+	0.865931i	$0.333273\pi$
$488$	0	0
$489$	4.59871	0.207961
$490$	0	0
$491$	−18.7547	−0.846390	−0.423195	−	0.906039i	$-0.639091\pi$
$491$	−18.7547	−0.846390	−0.423195	+	0.906039i	$0.639091\pi$
$492$	0	0
$493$	6.47283	0.291522
$494$	0	0
$495$	−18.1303	−0.814898
$496$	0	0
$497$	40.7625	1.82845
$498$	0	0
$499$	27.7478	1.24216	0.621081	−	0.783747i	$-0.286694\pi$
$499$	27.7478	1.24216	0.621081	+	0.783747i	$0.286694\pi$
$500$	0	0
$501$	3.05433	0.136457
$502$	0	0
$503$	17.5397	0.782054	0.391027	−	0.920379i	$-0.372120\pi$
$503$	17.5397	0.782054	0.391027	+	0.920379i	$0.372120\pi$
$504$	0	0
$505$	−17.0955	−0.760738
$506$	0	0
$507$	−15.6615	−0.695553
$508$	0	0
$509$	−32.1336	−1.42430	−0.712149	−	0.702028i	$-0.752278\pi$
$509$	−32.1336	−1.42430	−0.712149	+	0.702028i	$0.752278\pi$
$510$	0	0
$511$	14.6002	0.645876
$512$	0	0
$513$	−24.4596	−1.07992
$514$	0	0
$515$	5.21438	0.229773
$516$	0	0
$517$	−8.70889	−0.383017
$518$	0	0
$519$	−6.87590	−0.301818
$520$	0	0
$521$	−12.8106	−0.561242	−0.280621	−	0.959819i	$-0.590540\pi$
$521$	−12.8106	−0.561242	−0.280621	+	0.959819i	$0.590540\pi$
$522$	0	0
$523$	8.41850	0.368115	0.184058	−	0.982915i	$-0.441077\pi$
$523$	8.41850	0.368115	0.184058	+	0.982915i	$0.441077\pi$
$524$	0	0
$525$	8.33848	0.363921
$526$	0	0
$527$	3.00000	0.130682
$528$	0	0
$529$	6.14979	0.267382
$530$	0	0
$531$	2.58774	0.112298
$532$	0	0
$533$	4.25846	0.184454
$534$	0	0
$535$	14.1817	0.613129
$536$	0	0
$537$	11.4916	0.495898
$538$	0	0
$539$	66.8231	2.87827
$540$	0	0
$541$	−24.5202	−1.05421	−0.527103	−	0.849801i	$-0.676722\pi$
$541$	−24.5202	−1.05421	−0.527103	+	0.849801i	$0.676722\pi$
$542$	0	0
$543$	6.22581	0.267175
$544$	0	0
$545$	−25.7952	−1.10494
$546$	0	0
$547$	−33.9233	−1.45045	−0.725227	−	0.688510i	$-0.758265\pi$
$547$	−33.9233	−1.45045	−0.725227	+	0.688510i	$0.758265\pi$
$548$	0	0
$549$	−10.0194	−0.427620
$550$	0	0
$551$	−44.1289	−1.87995
$552$	0	0
$553$	−53.6887	−2.28308
$554$	0	0
$555$	−1.33553	−0.0566900
$556$	0	0
$557$	7.02569	0.297688	0.148844	−	0.988861i	$-0.452445\pi$
$557$	7.02569	0.297688	0.148844	+	0.988861i	$0.452445\pi$
$558$	0	0
$559$	−40.5349	−1.71444
$560$	0	0
$561$	−3.05433	−0.128954
$562$	0	0
$563$	−12.8524	−0.541666	−0.270833	−	0.962626i	$-0.587299\pi$
$563$	−12.8524	−0.541666	−0.270833	+	0.962626i	$0.587299\pi$
$564$	0	0
$565$	−9.14684	−0.384810
$566$	0	0
$567$	25.0474	1.05189
$568$	0	0
$569$	32.1964	1.34975	0.674873	−	0.737934i	$-0.264199\pi$
$569$	32.1964	1.34975	0.674873	+	0.737934i	$0.264199\pi$
$570$	0	0
$571$	36.8906	1.54382	0.771912	−	0.635729i	$-0.219300\pi$
$571$	36.8906	1.54382	0.771912	+	0.635729i	$0.219300\pi$
$572$	0	0
$573$	1.74378	0.0728473
$574$	0	0
$575$	−15.2834	−0.637363
$576$	0	0
$577$	−17.1670	−0.714672	−0.357336	−	0.933976i	$-0.616315\pi$
$577$	−17.1670	−0.714672	−0.357336	+	0.933976i	$0.616315\pi$
$578$	0	0
$579$	16.6894	0.693590
$580$	0	0
$581$	−43.3983	−1.80047
$582$	0	0
$583$	11.2166	0.464545
$584$	0	0
$585$	−23.3058	−0.963577
$586$	0	0
$587$	−3.36489	−0.138884	−0.0694419	−	0.997586i	$-0.522122\pi$
$587$	−3.36489	−0.138884	−0.0694419	+	0.997586i	$0.522122\pi$
$588$	0	0
$589$	−20.4527	−0.842738
$590$	0	0
$591$	5.50548	0.226465
$592$	0	0
$593$	−5.12811	−0.210586	−0.105293	−	0.994441i	$-0.533578\pi$
$593$	−5.12811	−0.210586	−0.105293	+	0.994441i	$0.533578\pi$
$594$	0	0
$595$	6.75698	0.277009
$596$	0	0
$597$	16.8245	0.688582
$598$	0	0
$599$	−9.91926	−0.405290	−0.202645	−	0.979252i	$-0.564954\pi$
$599$	−9.91926	−0.405290	−0.202645	+	0.979252i	$0.564954\pi$
$600$	0	0
$601$	−37.4574	−1.52792	−0.763960	−	0.645264i	$-0.776747\pi$
$601$	−37.4574	−1.52792	−0.763960	+	0.645264i	$0.776747\pi$
$602$	0	0
$603$	−20.5613	−0.837322
$604$	0	0
$605$	17.1274	0.696328
$606$	0	0
$607$	−33.9255	−1.37699	−0.688497	−	0.725239i	$-0.741729\pi$
$607$	−33.9255	−1.37699	−0.688497	+	0.725239i	$0.741729\pi$
$608$	0	0
$609$	−19.0668	−0.772626
$610$	0	0
$611$	−11.1949	−0.452898
$612$	0	0
$613$	42.9185	1.73346	0.866732	−	0.498774i	$-0.166216\pi$
$613$	42.9185	1.73346	0.866732	+	0.498774i	$0.166216\pi$
$614$	0	0
$615$	−0.658569	−0.0265560
$616$	0	0
$617$	−42.2034	−1.69905	−0.849523	−	0.527552i	$-0.823110\pi$
$617$	−42.2034	−1.69905	−0.849523	+	0.527552i	$0.823110\pi$
$618$	0	0
$619$	7.14132	0.287034	0.143517	−	0.989648i	$-0.454159\pi$
$619$	7.14132	0.287034	0.143517	+	0.989648i	$0.454159\pi$
$620$	0	0
$621$	19.3704	0.777308
$622$	0	0
$623$	−6.66776	−0.267138
$624$	0	0
$625$	−2.83299	−0.113320
$626$	0	0
$627$	20.8231	0.831594
$628$	0	0
$629$	1.41226	0.0563104
$630$	0	0
$631$	−15.1498	−0.603104	−0.301552	−	0.953450i	$-0.597505\pi$
$631$	−15.1498	−0.603104	−0.301552	+	0.953450i	$0.597505\pi$
$632$	0	0
$633$	5.97359	0.237429
$634$	0	0
$635$	8.41850	0.334078
$636$	0	0
$637$	85.8984	3.40342
$638$	0	0
$639$	−22.9923	−0.909562
$640$	0	0
$641$	22.8781	0.903632	0.451816	−	0.892111i	$-0.350776\pi$
$641$	22.8781	0.903632	0.451816	+	0.892111i	$0.350776\pi$
$642$	0	0
$643$	9.71585	0.383156	0.191578	−	0.981477i	$-0.438640\pi$
$643$	9.71585	0.383156	0.191578	+	0.981477i	$0.438640\pi$
$644$	0	0
$645$	6.26871	0.246830
$646$	0	0
$647$	−24.7974	−0.974886	−0.487443	−	0.873155i	$-0.662070\pi$
$647$	−24.7974	−0.974886	−0.487443	+	0.873155i	$0.662070\pi$
$648$	0	0
$649$	−4.75698	−0.186728
$650$	0	0
$651$	−8.83700	−0.346349
$652$	0	0
$653$	−0.991522	−0.0388013	−0.0194006	−	0.999812i	$-0.506176\pi$
$653$	−0.991522	−0.0388013	−0.0194006	+	0.999812i	$0.506176\pi$
$654$	0	0
$655$	14.0125	0.547513
$656$	0	0
$657$	−8.23534	−0.321291
$658$	0	0
$659$	20.7757	0.809307	0.404653	−	0.914470i	$-0.367392\pi$
$659$	20.7757	0.809307	0.404653	+	0.914470i	$0.367392\pi$
$660$	0	0
$661$	9.29111	0.361382	0.180691	−	0.983540i	$-0.442167\pi$
$661$	9.29111	0.361382	0.180691	+	0.983540i	$0.442167\pi$
$662$	0	0
$663$	−3.92622	−0.152482
$664$	0	0
$665$	−46.0661	−1.78637
$666$	0	0
$667$	34.9472	1.35316
$668$	0	0
$669$	−7.33152	−0.283453
$670$	0	0
$671$	18.4185	0.711038
$672$	0	0
$673$	−25.7328	−0.991927	−0.495963	−	0.868343i	$-0.665185\pi$
$673$	−25.7328	−0.991927	−0.495963	+	0.868343i	$0.665185\pi$
$674$	0	0
$675$	−10.1560	−0.390906
$676$	0	0
$677$	34.2750	1.31729	0.658647	−	0.752452i	$-0.271129\pi$
$677$	34.2750	1.31729	0.658647	+	0.752452i	$0.271129\pi$
$678$	0	0
$679$	1.11491	0.0427862
$680$	0	0
$681$	−18.5294	−0.710048
$682$	0	0
$683$	37.3425	1.42887	0.714435	−	0.699702i	$-0.246684\pi$
$683$	37.3425	1.42887	0.714435	+	0.699702i	$0.246684\pi$
$684$	0	0
$685$	3.91998	0.149775
$686$	0	0
$687$	−0.225807	−0.00861506
$688$	0	0
$689$	14.4185	0.549301
$690$	0	0
$691$	20.0364	0.762221	0.381110	−	0.924530i	$-0.375542\pi$
$691$	20.0364	0.762221	0.381110	+	0.924530i	$0.375542\pi$
$692$	0	0
$693$	−56.4744	−2.14528
$694$	0	0
$695$	−12.2585	−0.464990
$696$	0	0
$697$	0.696406	0.0263783
$698$	0	0
$699$	−4.84949	−0.183425
$700$	0	0
$701$	24.8844	0.939870	0.469935	−	0.882701i	$-0.344277\pi$
$701$	24.8844	0.939870	0.469935	+	0.882701i	$0.344277\pi$
$702$	0	0
$703$	−9.62815	−0.363133
$704$	0	0
$705$	1.73129	0.0652042
$706$	0	0
$707$	−53.2508	−2.00270
$708$	0	0
$709$	−39.4938	−1.48322	−0.741610	−	0.670831i	$-0.765938\pi$
$709$	−39.4938	−1.48322	−0.741610	+	0.670831i	$0.765938\pi$
$710$	0	0
$711$	30.2834	1.13572
$712$	0	0
$713$	16.1972	0.606589
$714$	0	0
$715$	42.8425	1.60222
$716$	0	0
$717$	6.93175	0.258871
$718$	0	0
$719$	−36.2423	−1.35161	−0.675805	−	0.737081i	$-0.736204\pi$
$719$	−36.2423	−1.35161	−0.675805	+	0.737081i	$0.736204\pi$
$720$	0	0
$721$	16.2423	0.604895
$722$	0	0
$723$	13.9427	0.518535
$724$	0	0
$725$	−18.3230	−0.680501
$726$	0	0
$727$	27.2027	1.00889	0.504446	−	0.863443i	$-0.331697\pi$
$727$	27.2027	1.00889	0.504446	+	0.863443i	$0.331697\pi$
$728$	0	0
$729$	−7.21733	−0.267308
$730$	0	0
$731$	−6.62887	−0.245178
$732$	0	0
$733$	−30.6219	−1.13105	−0.565523	−	0.824733i	$-0.691326\pi$
$733$	−30.6219	−1.13105	−0.565523	+	0.824733i	$0.691326\pi$
$734$	0	0
$735$	−13.2841	−0.489993
$736$	0	0
$737$	37.7974	1.39228
$738$	0	0
$739$	16.3161	0.600197	0.300098	−	0.953908i	$-0.402980\pi$
$739$	16.3161	0.600197	0.300098	+	0.953908i	$0.402980\pi$
$740$	0	0
$741$	26.7672	0.983319
$742$	0	0
$743$	−13.1685	−0.483106	−0.241553	−	0.970388i	$-0.577657\pi$
$743$	−13.1685	−0.483106	−0.241553	+	0.970388i	$0.577657\pi$
$744$	0	0
$745$	−8.75698	−0.320831
$746$	0	0
$747$	24.4791	0.895643
$748$	0	0
$749$	44.1748	1.61411
$750$	0	0
$751$	42.5629	1.55314	0.776570	−	0.630031i	$-0.216958\pi$
$751$	42.5629	1.55314	0.776570	+	0.630031i	$0.216958\pi$
$752$	0	0
$753$	−8.48604	−0.309248
$754$	0	0
$755$	6.69641	0.243707
$756$	0	0
$757$	24.8106	0.901756	0.450878	−	0.892585i	$-0.351111\pi$
$757$	24.8106	0.901756	0.450878	+	0.892585i	$0.351111\pi$
$758$	0	0
$759$	−16.4905	−0.598567
$760$	0	0
$761$	8.65456	0.313728	0.156864	−	0.987620i	$-0.449862\pi$
$761$	8.65456	0.313728	0.156864	+	0.987620i	$0.449862\pi$
$762$	0	0
$763$	−80.3495	−2.90885
$764$	0	0
$765$	−3.81131	−0.137798
$766$	0	0
$767$	−6.11491	−0.220797
$768$	0	0
$769$	40.3268	1.45422	0.727111	−	0.686520i	$-0.240863\pi$
$769$	40.3268	1.45422	0.727111	+	0.686520i	$0.240863\pi$
$770$	0	0
$771$	14.8205	0.533748
$772$	0	0
$773$	36.4200	1.30994	0.654969	−	0.755656i	$-0.272682\pi$
$773$	36.4200	1.30994	0.654969	+	0.755656i	$0.272682\pi$
$774$	0	0
$775$	−8.49228	−0.305052
$776$	0	0
$777$	−4.16004	−0.149241
$778$	0	0
$779$	−4.74779	−0.170107
$780$	0	0
$781$	42.2662	1.51240
$782$	0	0
$783$	23.2229	0.829917
$784$	0	0
$785$	−0.169240	−0.00604042
$786$	0	0
$787$	41.4923	1.47904	0.739520	−	0.673135i	$-0.235053\pi$
$787$	41.4923	1.47904	0.739520	+	0.673135i	$0.235053\pi$
$788$	0	0
$789$	−8.98857	−0.320002
$790$	0	0
$791$	−28.4916	−1.01304
$792$	0	0
$793$	23.6762	0.840768
$794$	0	0
$795$	−2.22982	−0.0790834
$796$	0	0
$797$	22.7299	0.805133	0.402566	−	0.915391i	$-0.368118\pi$
$797$	22.7299	0.805133	0.402566	+	0.915391i	$0.368118\pi$
$798$	0	0
$799$	−1.83076	−0.0647676
$800$	0	0
$801$	3.76099	0.132888
$802$	0	0
$803$	15.1388	0.534237
$804$	0	0
$805$	36.4813	1.28580
$806$	0	0
$807$	0.334010	0.0117577
$808$	0	0
$809$	−16.2687	−0.571977	−0.285989	−	0.958233i	$-0.592322\pi$
$809$	−16.2687	−0.571977	−0.285989	+	0.958233i	$0.592322\pi$
$810$	0	0
$811$	−40.0078	−1.40486	−0.702431	−	0.711752i	$-0.747902\pi$
$811$	−40.0078	−1.40486	−0.702431	+	0.711752i	$0.747902\pi$
$812$	0	0
$813$	−16.2996	−0.571651
$814$	0	0
$815$	−10.5488	−0.369510
$816$	0	0
$817$	45.1927	1.58109
$818$	0	0
$819$	−72.5955	−2.53669
$820$	0	0
$821$	1.00624	0.0351181	0.0175591	−	0.999846i	$-0.494410\pi$
$821$	1.00624	0.0351181	0.0175591	+	0.999846i	$0.494410\pi$
$822$	0	0
$823$	17.0085	0.592878	0.296439	−	0.955052i	$-0.404201\pi$
$823$	17.0085	0.592878	0.296439	+	0.955052i	$0.404201\pi$
$824$	0	0
$825$	8.64608	0.301018
$826$	0	0
$827$	9.03889	0.314313	0.157157	−	0.987574i	$-0.449767\pi$
$827$	9.03889	0.314313	0.157157	+	0.987574i	$0.449767\pi$
$828$	0	0
$829$	56.5544	1.96421	0.982107	−	0.188322i	$-0.0603049\pi$
$829$	56.5544	1.96421	0.982107	+	0.188322i	$0.0603049\pi$
$830$	0	0
$831$	−10.1171	−0.350960
$832$	0	0
$833$	14.0474	0.486713
$834$	0	0
$835$	−7.00624	−0.242461
$836$	0	0
$837$	10.7632	0.372031
$838$	0	0
$839$	−34.1296	−1.17829	−0.589143	−	0.808029i	$-0.700534\pi$
$839$	−34.1296	−1.17829	−0.589143	+	0.808029i	$0.700534\pi$
$840$	0	0
$841$	12.8976	0.444744
$842$	0	0
$843$	18.2897	0.629930
$844$	0	0
$845$	35.9255	1.23587
$846$	0	0
$847$	53.3502	1.83314
$848$	0	0
$849$	−20.1336	−0.690985
$850$	0	0
$851$	7.62486	0.261377
$852$	0	0
$853$	−38.0319	−1.30219	−0.651094	−	0.758997i	$-0.725690\pi$
$853$	−38.0319	−1.30219	−0.651094	+	0.758997i	$0.725690\pi$
$854$	0	0
$855$	25.9838	0.888629
$856$	0	0
$857$	27.7167	0.946783	0.473391	−	0.880852i	$-0.343030\pi$
$857$	27.7167	0.946783	0.473391	+	0.880852i	$0.343030\pi$
$858$	0	0
$859$	−30.2577	−1.03238	−0.516190	−	0.856474i	$-0.672650\pi$
$859$	−30.2577	−1.03238	−0.516190	+	0.856474i	$0.672650\pi$
$860$	0	0
$861$	−2.05138	−0.0699109
$862$	0	0
$863$	−46.4046	−1.57963	−0.789815	−	0.613345i	$-0.789824\pi$
$863$	−46.4046	−1.57963	−0.789815	+	0.613345i	$0.789824\pi$
$864$	0	0
$865$	15.7724	0.536278
$866$	0	0
$867$	−0.642074	−0.0218060
$868$	0	0
$869$	−55.6693	−1.88845
$870$	0	0
$871$	48.5870	1.64631
$872$	0	0
$873$	−0.628870	−0.0212840
$874$	0	0
$875$	−52.9123	−1.78876
$876$	0	0
$877$	37.8253	1.27727	0.638635	−	0.769510i	$-0.279499\pi$
$877$	37.8253	1.27727	0.638635	+	0.769510i	$0.279499\pi$
$878$	0	0
$879$	14.3524	0.484094
$880$	0	0
$881$	−43.0103	−1.44905	−0.724526	−	0.689247i	$-0.757941\pi$
$881$	−43.0103	−1.44905	−0.724526	+	0.689247i	$0.757941\pi$
$882$	0	0
$883$	−11.8936	−0.400251	−0.200125	−	0.979770i	$-0.564135\pi$
$883$	−11.8936	−0.400251	−0.200125	+	0.979770i	$0.564135\pi$
$884$	0	0
$885$	0.945668	0.0317883
$886$	0	0
$887$	−8.41778	−0.282642	−0.141321	−	0.989964i	$-0.545135\pi$
$887$	−8.41778	−0.282642	−0.141321	+	0.989964i	$0.545135\pi$
$888$	0	0
$889$	26.2229	0.879486
$890$	0	0
$891$	25.9714	0.870073
$892$	0	0
$893$	12.4813	0.417671
$894$	0	0
$895$	−26.3602	−0.881123
$896$	0	0
$897$	−21.1979	−0.707777
$898$	0	0
$899$	19.4185	0.647643
$900$	0	0
$901$	2.35793	0.0785539
$902$	0	0
$903$	19.5264	0.649800
$904$	0	0
$905$	−14.2812	−0.474723
$906$	0	0
$907$	53.3418	1.77118	0.885592	−	0.464464i	$-0.153753\pi$
$907$	53.3418	1.77118	0.885592	+	0.464464i	$0.153753\pi$
$908$	0	0
$909$	30.0364	0.996245
$910$	0	0
$911$	10.6942	0.354314	0.177157	−	0.984183i	$-0.443310\pi$
$911$	10.6942	0.354314	0.177157	+	0.984183i	$0.443310\pi$
$912$	0	0
$913$	−44.9993	−1.48926
$914$	0	0
$915$	−3.66152	−0.121046
$916$	0	0
$917$	43.6476	1.44137
$918$	0	0
$919$	−19.8044	−0.653285	−0.326643	−	0.945148i	$-0.605917\pi$
$919$	−19.8044	−0.653285	−0.326643	+	0.945148i	$0.605917\pi$
$920$	0	0
$921$	−3.69240	−0.121669
$922$	0	0
$923$	54.3315	1.78834
$924$	0	0
$925$	−3.99777	−0.131446
$926$	0	0
$927$	−9.16156	−0.300905
$928$	0	0
$929$	27.4178	0.899548	0.449774	−	0.893142i	$-0.351504\pi$
$929$	27.4178	0.899548	0.449774	+	0.893142i	$0.351504\pi$
$930$	0	0
$931$	−95.7688	−3.13869
$932$	0	0
$933$	9.49701	0.310918
$934$	0	0
$935$	7.00624	0.229129
$936$	0	0
$937$	−36.2034	−1.18271	−0.591357	−	0.806410i	$-0.701408\pi$
$937$	−36.2034	−1.18271	−0.591357	+	0.806410i	$0.701408\pi$
$938$	0	0
$939$	−13.6700	−0.446104
$940$	0	0
$941$	−7.94639	−0.259045	−0.129522	−	0.991576i	$-0.541344\pi$
$941$	−7.94639	−0.259045	−0.129522	+	0.991576i	$0.541344\pi$
$942$	0	0
$943$	3.75993	0.122440
$944$	0	0
$945$	24.2423	0.788602
$946$	0	0
$947$	32.2508	1.04801	0.524005	−	0.851715i	$-0.324437\pi$
$947$	32.2508	1.04801	0.524005	+	0.851715i	$0.324437\pi$
$948$	0	0
$949$	19.4603	0.631710
$950$	0	0
$951$	4.94022	0.160198
$952$	0	0
$953$	−24.2361	−0.785083	−0.392541	−	0.919734i	$-0.628404\pi$
$953$	−24.2361	−0.785083	−0.392541	+	0.919734i	$0.628404\pi$
$954$	0	0
$955$	−4.00000	−0.129437
$956$	0	0
$957$	−19.7702	−0.639079
$958$	0	0
$959$	12.2104	0.394293
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−24.9170	−0.802940
$964$	0	0
$965$	−38.2834	−1.23239
$966$	0	0
$967$	−15.8991	−0.511280	−0.255640	−	0.966772i	$-0.582286\pi$
$967$	−15.8991	−0.511280	−0.255640	+	0.966772i	$0.582286\pi$
$968$	0	0
$969$	4.37737	0.140621
$970$	0	0
$971$	52.7500	1.69283	0.846414	−	0.532525i	$-0.178757\pi$
$971$	52.7500	1.69283	0.846414	+	0.532525i	$0.178757\pi$
$972$	0	0
$973$	−38.1840	−1.22412
$974$	0	0
$975$	11.1142	0.355939
$976$	0	0
$977$	−39.9519	−1.27817	−0.639087	−	0.769134i	$-0.720688\pi$
$977$	−39.9519	−1.27817	−0.639087	+	0.769134i	$0.720688\pi$
$978$	0	0
$979$	−6.91373	−0.220964
$980$	0	0
$981$	45.3216	1.44701
$982$	0	0
$983$	−25.9519	−0.827737	−0.413869	−	0.910337i	$-0.635823\pi$
$983$	−25.9519	−0.827737	−0.413869	+	0.910337i	$0.635823\pi$
$984$	0	0
$985$	−12.6289	−0.402389
$986$	0	0
$987$	5.39281	0.171655
$988$	0	0
$989$	−35.7896	−1.13804
$990$	0	0
$991$	21.3859	0.679344	0.339672	−	0.940544i	$-0.389684\pi$
$991$	21.3859	0.679344	0.339672	+	0.940544i	$0.389684\pi$
$992$	0	0
$993$	4.33401	0.137536
$994$	0	0
$995$	−38.5933	−1.22349
$996$	0	0
$997$	−19.7261	−0.624732	−0.312366	−	0.949962i	$-0.601122\pi$
$997$	−19.7261	−0.624732	−0.312366	+	0.949962i	$0.601122\pi$
$998$	0	0
$999$	5.06682	0.160307

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.s.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.s.1.2	✓	3	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.642074 q^{3} +1.47283 q^{5} +4.58774 q^{7} -2.58774 q^{9} +O(q^{10})\)	\(q-0.642074 q^{3} +1.47283 q^{5} +4.58774 q^{7} -2.58774 q^{9} +4.75698 q^{11} +6.11491 q^{13} -0.945668 q^{15} +1.00000 q^{17} -6.81756 q^{19} -2.94567 q^{21} +5.39905 q^{23} -2.83076 q^{25} +3.58774 q^{27} +6.47283 q^{29} +3.00000 q^{31} -3.05433 q^{33} +6.75698 q^{35} +1.41226 q^{37} -3.92622 q^{39} +0.696406 q^{41} -6.62887 q^{43} -3.81131 q^{45} -1.83076 q^{47} +14.0474 q^{49} -0.642074 q^{51} +2.35793 q^{53} +7.00624 q^{55} +4.37737 q^{57} -1.00000 q^{59} +3.87189 q^{61} -11.8719 q^{63} +9.00624 q^{65} +7.94567 q^{67} -3.46659 q^{69} +8.88509 q^{71} +3.18244 q^{73} +1.81756 q^{75} +21.8238 q^{77} -11.7026 q^{79} +5.45963 q^{81} -9.45963 q^{83} +1.47283 q^{85} -4.15604 q^{87} -1.45339 q^{89} +28.0536 q^{91} -1.92622 q^{93} -10.0411 q^{95} +0.243019 q^{97} -12.3098 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q - q^3 - q^5 + 2 * q^7 + 4 * q^9	\( 3 q - q^{3} - q^{5} + 2 q^{7} + 4 q^{9} + 7 q^{11} + 12 q^{13} + 8 q^{15} + 3 q^{17} + 4 q^{19} + 2 q^{21} + 8 q^{23} - 4 q^{25} - q^{27} + 14 q^{29} + 9 q^{31} - 20 q^{33} + 13 q^{35} + 16 q^{37} - 9 q^{39} + 12 q^{41} + q^{43} - 15 q^{45} - q^{47} + 5 q^{49} - q^{51} + 8 q^{53} - 7 q^{55} + 6 q^{57} - 3 q^{59} - 2 q^{61} - 22 q^{63} - q^{65} + 13 q^{67} - 33 q^{69} + 33 q^{71} + 34 q^{73} - 19 q^{75} + 13 q^{77} - 17 q^{79} - 9 q^{81} - 3 q^{83} - q^{85} + 3 q^{87} - 7 q^{89} + 19 q^{91} - 3 q^{93} - 21 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) \) 3 * q - q^3 - q^5 + 2 * q^7 + 4 * q^9 + 7 * q^11 + 12 * q^13 + 8 * q^15 + 3 * q^17 + 4 * q^19 + 2 * q^21 + 8 * q^23 - 4 * q^25 - q^27 + 14 * q^29 + 9 * q^31 - 20 * q^33 + 13 * q^35 + 16 * q^37 - 9 * q^39 + 12 * q^41 + q^43 - 15 * q^45 - q^47 + 5 * q^49 - q^51 + 8 * q^53 - 7 * q^55 + 6 * q^57 - 3 * q^59 - 2 * q^61 - 22 * q^63 - q^65 + 13 * q^67 - 33 * q^69 + 33 * q^71 + 34 * q^73 - 19 * q^75 + 13 * q^77 - 17 * q^79 - 9 * q^81 - 3 * q^83 - q^85 + 3 * q^87 - 7 * q^89 + 19 * q^91 - 3 * q^93 - 21 * q^95 + 8 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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