LMFDB - Embedded newform 4028.2.a.d.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.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:	$32.1637419342$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 30 x^{17} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 $ x^19 - 4x^18 - 30x^17 + 132x^16 + 332x^15 - 1714x^14 - 1598x^13 + 11179x^12 + 2544x^11 - 38897x^10 + 3416x^9 + 71354x^8 - 10941x^7 - 64854x^6 + 219x^5 + 26515x^4 + 5796x^3 - 2375x^2 - 826x - 49
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			1.14
Root			$-0.755832$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.755832 q^{3} +4.26919 q^{5} -2.09147 q^{7} -2.42872 q^{9} +O(q^{10})$	$q+0.755832 q^{3} +4.26919 q^{5} -2.09147 q^{7} -2.42872 q^{9} -4.58245 q^{11} +1.90593 q^{13} +3.22679 q^{15} -5.40460 q^{17} -1.00000 q^{19} -1.58080 q^{21} -4.46639 q^{23} +13.2260 q^{25} -4.10320 q^{27} +1.04048 q^{29} -6.77185 q^{31} -3.46356 q^{33} -8.92889 q^{35} -1.41651 q^{37} +1.44056 q^{39} +4.93319 q^{41} +1.29411 q^{43} -10.3687 q^{45} +8.54538 q^{47} -2.62575 q^{49} -4.08497 q^{51} +1.00000 q^{53} -19.5633 q^{55} -0.755832 q^{57} -10.5166 q^{59} +1.99109 q^{61} +5.07959 q^{63} +8.13676 q^{65} -9.79064 q^{67} -3.37584 q^{69} +4.71806 q^{71} -6.64663 q^{73} +9.99661 q^{75} +9.58406 q^{77} -5.80655 q^{79} +4.18482 q^{81} -14.1602 q^{83} -23.0733 q^{85} +0.786425 q^{87} -12.5773 q^{89} -3.98619 q^{91} -5.11838 q^{93} -4.26919 q^{95} -9.06337 q^{97} +11.1295 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * 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
$3$	0.755832	0.436380	0.218190	−	0.975906i	$-0.429985\pi$
$3$	0.755832	0.436380	0.218190	+	0.975906i	$0.429985\pi$
$4$	0	0
$5$	4.26919	1.90924	0.954619	−	0.297828i	$-0.0962623\pi$
$5$	4.26919	1.90924	0.954619	+	0.297828i	$0.0962623\pi$
$6$	0	0
$7$	−2.09147	−0.790502	−0.395251	−	0.918573i	$-0.629342\pi$
$7$	−2.09147	−0.790502	−0.395251	+	0.918573i	$0.629342\pi$
$8$	0	0
$9$	−2.42872	−0.809573
$10$	0	0
$11$	−4.58245	−1.38166	−0.690830	−	0.723017i	$-0.742755\pi$
$11$	−4.58245	−1.38166	−0.690830	+	0.723017i	$0.742755\pi$
$12$	0	0
$13$	1.90593	0.528609	0.264305	−	0.964439i	$-0.414858\pi$
$13$	1.90593	0.528609	0.264305	+	0.964439i	$0.414858\pi$
$14$	0	0
$15$	3.22679	0.833153
$16$	0	0
$17$	−5.40460	−1.31081	−0.655404	−	0.755278i	$-0.727502\pi$
$17$	−5.40460	−1.31081	−0.655404	+	0.755278i	$0.727502\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.58080	−0.344959
$22$	0	0
$23$	−4.46639	−0.931307	−0.465654	−	0.884967i	$-0.654181\pi$
$23$	−4.46639	−0.931307	−0.465654	+	0.884967i	$0.654181\pi$
$24$	0	0
$25$	13.2260	2.64519
$26$	0	0
$27$	−4.10320	−0.789661
$28$	0	0
$29$	1.04048	0.193211	0.0966057	−	0.995323i	$-0.469201\pi$
$29$	1.04048	0.193211	0.0966057	+	0.995323i	$0.469201\pi$
$30$	0	0
$31$	−6.77185	−1.21626	−0.608130	−	0.793837i	$-0.708080\pi$
$31$	−6.77185	−1.21626	−0.608130	+	0.793837i	$0.708080\pi$
$32$	0	0
$33$	−3.46356	−0.602928
$34$	0	0
$35$	−8.92889	−1.50926
$36$	0	0
$37$	−1.41651	−0.232873	−0.116437	−	0.993198i	$-0.537147\pi$
$37$	−1.41651	−0.232873	−0.116437	+	0.993198i	$0.537147\pi$
$38$	0	0
$39$	1.44056	0.230674
$40$	0	0
$41$	4.93319	0.770435	0.385217	−	0.922826i	$-0.374126\pi$
$41$	4.93319	0.770435	0.385217	+	0.922826i	$0.374126\pi$
$42$	0	0
$43$	1.29411	0.197350	0.0986749	−	0.995120i	$-0.468540\pi$
$43$	1.29411	0.197350	0.0986749	+	0.995120i	$0.468540\pi$
$44$	0	0
$45$	−10.3687	−1.54567
$46$	0	0
$47$	8.54538	1.24647	0.623236	−	0.782034i	$-0.285818\pi$
$47$	8.54538	1.24647	0.623236	+	0.782034i	$0.285818\pi$
$48$	0	0
$49$	−2.62575	−0.375107
$50$	0	0
$51$	−4.08497	−0.572010
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−19.5633	−2.63792
$56$	0	0
$57$	−0.755832	−0.100112
$58$	0	0
$59$	−10.5166	−1.36915	−0.684575	−	0.728942i	$-0.740012\pi$
$59$	−10.5166	−1.36915	−0.684575	+	0.728942i	$0.740012\pi$
$60$	0	0
$61$	1.99109	0.254932	0.127466	−	0.991843i	$-0.459316\pi$
$61$	1.99109	0.254932	0.127466	+	0.991843i	$0.459316\pi$
$62$	0	0
$63$	5.07959	0.639969
$64$	0	0
$65$	8.13676	1.00924
$66$	0	0
$67$	−9.79064	−1.19612	−0.598059	−	0.801452i	$-0.704061\pi$
$67$	−9.79064	−1.19612	−0.598059	+	0.801452i	$0.704061\pi$
$68$	0	0
$69$	−3.37584	−0.406404
$70$	0	0
$71$	4.71806	0.559931	0.279965	−	0.960010i	$-0.409677\pi$
$71$	4.71806	0.559931	0.279965	+	0.960010i	$0.409677\pi$
$72$	0	0
$73$	−6.64663	−0.777929	−0.388965	−	0.921253i	$-0.627167\pi$
$73$	−6.64663	−0.777929	−0.388965	+	0.921253i	$0.627167\pi$
$74$	0	0
$75$	9.99661	1.15431
$76$	0	0
$77$	9.58406	1.09220
$78$	0	0
$79$	−5.80655	−0.653287	−0.326644	−	0.945148i	$-0.605918\pi$
$79$	−5.80655	−0.653287	−0.326644	+	0.945148i	$0.605918\pi$
$80$	0	0
$81$	4.18482	0.464981
$82$	0	0
$83$	−14.1602	−1.55428	−0.777142	−	0.629325i	$-0.783331\pi$
$83$	−14.1602	−1.55428	−0.777142	+	0.629325i	$0.783331\pi$
$84$	0	0
$85$	−23.0733	−2.50265
$86$	0	0
$87$	0.786425	0.0843136
$88$	0	0
$89$	−12.5773	−1.33319	−0.666595	−	0.745420i	$-0.732249\pi$
$89$	−12.5773	−1.33319	−0.666595	+	0.745420i	$0.732249\pi$
$90$	0	0
$91$	−3.98619	−0.417866
$92$	0	0
$93$	−5.11838	−0.530751
$94$	0	0
$95$	−4.26919	−0.438009
$96$	0	0
$97$	−9.06337	−0.920246	−0.460123	−	0.887855i	$-0.652195\pi$
$97$	−9.06337	−0.920246	−0.460123	+	0.887855i	$0.652195\pi$
$98$	0	0
$99$	11.1295	1.11855
$100$	0	0
$101$	6.13657	0.610612	0.305306	−	0.952254i	$-0.401241\pi$
$101$	6.13657	0.610612	0.305306	+	0.952254i	$0.401241\pi$
$102$	0	0
$103$	4.88502	0.481335	0.240667	−	0.970608i	$-0.422634\pi$
$103$	4.88502	0.481335	0.240667	+	0.970608i	$0.422634\pi$
$104$	0	0
$105$	−6.74874	−0.658609
$106$	0	0
$107$	−1.16312	−0.112443	−0.0562217	−	0.998418i	$-0.517905\pi$
$107$	−1.16312	−0.112443	−0.0562217	+	0.998418i	$0.517905\pi$
$108$	0	0
$109$	−16.1455	−1.54646	−0.773229	−	0.634127i	$-0.781360\pi$
$109$	−16.1455	−1.54646	−0.773229	+	0.634127i	$0.781360\pi$
$110$	0	0
$111$	−1.07065	−0.101621
$112$	0	0
$113$	−1.12181	−0.105531	−0.0527654	−	0.998607i	$-0.516804\pi$
$113$	−1.12181	−0.105531	−0.0527654	+	0.998607i	$0.516804\pi$
$114$	0	0
$115$	−19.0679	−1.77809
$116$	0	0
$117$	−4.62896	−0.427947
$118$	0	0
$119$	11.3036	1.03620
$120$	0	0
$121$	9.99882	0.908984
$122$	0	0
$123$	3.72866	0.336202
$124$	0	0
$125$	35.1182	3.14107
$126$	0	0
$127$	12.4138	1.10154	0.550772	−	0.834656i	$-0.314334\pi$
$127$	12.4138	1.10154	0.550772	+	0.834656i	$0.314334\pi$
$128$	0	0
$129$	0.978129	0.0861194
$130$	0	0
$131$	−3.81267	−0.333115	−0.166557	−	0.986032i	$-0.553265\pi$
$131$	−3.81267	−0.333115	−0.166557	+	0.986032i	$0.553265\pi$
$132$	0	0
$133$	2.09147	0.181354
$134$	0	0
$135$	−17.5173	−1.50765
$136$	0	0
$137$	−8.05715	−0.688369	−0.344184	−	0.938902i	$-0.611845\pi$
$137$	−8.05715	−0.688369	−0.344184	+	0.938902i	$0.611845\pi$
$138$	0	0
$139$	19.6871	1.66984	0.834918	−	0.550375i	$-0.185515\pi$
$139$	19.6871	1.66984	0.834918	+	0.550375i	$0.185515\pi$
$140$	0	0
$141$	6.45887	0.543935
$142$	0	0
$143$	−8.73381	−0.730358
$144$	0	0
$145$	4.44198	0.368887
$146$	0	0
$147$	−1.98462	−0.163689
$148$	0	0
$149$	−12.7977	−1.04843	−0.524216	−	0.851586i	$-0.675641\pi$
$149$	−12.7977	−1.04843	−0.524216	+	0.851586i	$0.675641\pi$
$150$	0	0
$151$	−21.7008	−1.76599	−0.882993	−	0.469387i	$-0.844475\pi$
$151$	−21.7008	−1.76599	−0.882993	+	0.469387i	$0.844475\pi$
$152$	0	0
$153$	13.1263	1.06119
$154$	0	0
$155$	−28.9103	−2.32213
$156$	0	0
$157$	17.1131	1.36577	0.682886	−	0.730525i	$-0.260725\pi$
$157$	17.1131	1.36577	0.682886	+	0.730525i	$0.260725\pi$
$158$	0	0
$159$	0.755832	0.0599414
$160$	0	0
$161$	9.34133	0.736200
$162$	0	0
$163$	−11.6452	−0.912119	−0.456059	−	0.889949i	$-0.650740\pi$
$163$	−11.6452	−0.912119	−0.456059	+	0.889949i	$0.650740\pi$
$164$	0	0
$165$	−14.7866	−1.15113
$166$	0	0
$167$	16.8173	1.30136	0.650679	−	0.759353i	$-0.274484\pi$
$167$	16.8173	1.30136	0.650679	+	0.759353i	$0.274484\pi$
$168$	0	0
$169$	−9.36744	−0.720572
$170$	0	0
$171$	2.42872	0.185729
$172$	0	0
$173$	10.8256	0.823054	0.411527	−	0.911397i	$-0.364996\pi$
$173$	10.8256	0.823054	0.411527	+	0.911397i	$0.364996\pi$
$174$	0	0
$175$	−27.6617	−2.09103
$176$	0	0
$177$	−7.94881	−0.597469
$178$	0	0
$179$	17.8683	1.33554	0.667770	−	0.744368i	$-0.267249\pi$
$179$	17.8683	1.33554	0.667770	+	0.744368i	$0.267249\pi$
$180$	0	0
$181$	18.5908	1.38184	0.690922	−	0.722929i	$-0.257205\pi$
$181$	18.5908	1.38184	0.690922	+	0.722929i	$0.257205\pi$
$182$	0	0
$183$	1.50493	0.111247
$184$	0	0
$185$	−6.04736	−0.444611
$186$	0	0
$187$	24.7663	1.81109
$188$	0	0
$189$	8.58172	0.624229
$190$	0	0
$191$	−2.06510	−0.149426	−0.0747128	−	0.997205i	$-0.523804\pi$
$191$	−2.06510	−0.149426	−0.0747128	+	0.997205i	$0.523804\pi$
$192$	0	0
$193$	13.4781	0.970178	0.485089	−	0.874465i	$-0.338787\pi$
$193$	13.4781	0.970178	0.485089	+	0.874465i	$0.338787\pi$
$194$	0	0
$195$	6.15002	0.440412
$196$	0	0
$197$	12.6889	0.904049	0.452025	−	0.892005i	$-0.350702\pi$
$197$	12.6889	0.904049	0.452025	+	0.892005i	$0.350702\pi$
$198$	0	0
$199$	−0.932888	−0.0661306	−0.0330653	−	0.999453i	$-0.510527\pi$
$199$	−0.932888	−0.0661306	−0.0330653	+	0.999453i	$0.510527\pi$
$200$	0	0
$201$	−7.40008	−0.521962
$202$	0	0
$203$	−2.17612	−0.152734
$204$	0	0
$205$	21.0607	1.47094
$206$	0	0
$207$	10.8476	0.753961
$208$	0	0
$209$	4.58245	0.316974
$210$	0	0
$211$	0.487907	0.0335889	0.0167945	−	0.999859i	$-0.494654\pi$
$211$	0.487907	0.0335889	0.0167945	+	0.999859i	$0.494654\pi$
$212$	0	0
$213$	3.56606	0.244342
$214$	0	0
$215$	5.52479	0.376788
$216$	0	0
$217$	14.1631	0.961456
$218$	0	0
$219$	−5.02374	−0.339473
$220$	0	0
$221$	−10.3008	−0.692905
$222$	0	0
$223$	−16.4800	−1.10358	−0.551791	−	0.833982i	$-0.686056\pi$
$223$	−16.4800	−1.10358	−0.551791	+	0.833982i	$0.686056\pi$
$224$	0	0
$225$	−32.1221	−2.14148
$226$	0	0
$227$	−13.7045	−0.909598	−0.454799	−	0.890594i	$-0.650289\pi$
$227$	−13.7045	−0.909598	−0.454799	+	0.890594i	$0.650289\pi$
$228$	0	0
$229$	13.8828	0.917404	0.458702	−	0.888590i	$-0.348314\pi$
$229$	13.8828	0.917404	0.458702	+	0.888590i	$0.348314\pi$
$230$	0	0
$231$	7.24394	0.476616
$232$	0	0
$233$	−28.3208	−1.85536	−0.927680	−	0.373376i	$-0.878200\pi$
$233$	−28.3208	−1.85536	−0.927680	+	0.373376i	$0.878200\pi$
$234$	0	0
$235$	36.4818	2.37981
$236$	0	0
$237$	−4.38877	−0.285081
$238$	0	0
$239$	20.1363	1.30251	0.651255	−	0.758859i	$-0.274243\pi$
$239$	20.1363	1.30251	0.651255	+	0.758859i	$0.274243\pi$
$240$	0	0
$241$	−6.37249	−0.410488	−0.205244	−	0.978711i	$-0.565799\pi$
$241$	−6.37249	−0.410488	−0.205244	+	0.978711i	$0.565799\pi$
$242$	0	0
$243$	15.4726	0.992569
$244$	0	0
$245$	−11.2098	−0.716168
$246$	0	0
$247$	−1.90593	−0.121271
$248$	0	0
$249$	−10.7027	−0.678258
$250$	0	0
$251$	−6.33554	−0.399896	−0.199948	−	0.979807i	$-0.564077\pi$
$251$	−6.33554	−0.399896	−0.199948	+	0.979807i	$0.564077\pi$
$252$	0	0
$253$	20.4670	1.28675
$254$	0	0
$255$	−17.4395	−1.09210
$256$	0	0
$257$	15.3878	0.959862	0.479931	−	0.877306i	$-0.340662\pi$
$257$	15.3878	0.959862	0.479931	+	0.877306i	$0.340662\pi$
$258$	0	0
$259$	2.96260	0.184087
$260$	0	0
$261$	−2.52702	−0.156419
$262$	0	0
$263$	1.65417	0.102000	0.0510002	−	0.998699i	$-0.483759\pi$
$263$	1.65417	0.102000	0.0510002	+	0.998699i	$0.483759\pi$
$264$	0	0
$265$	4.26919	0.262254
$266$	0	0
$267$	−9.50632	−0.581777
$268$	0	0
$269$	23.2770	1.41922	0.709612	−	0.704593i	$-0.248870\pi$
$269$	23.2770	1.41922	0.709612	+	0.704593i	$0.248870\pi$
$270$	0	0
$271$	8.81860	0.535692	0.267846	−	0.963462i	$-0.413688\pi$
$271$	8.81860	0.535692	0.267846	+	0.963462i	$0.413688\pi$
$272$	0	0
$273$	−3.01289	−0.182349
$274$	0	0
$275$	−60.6073	−3.65476
$276$	0	0
$277$	5.30805	0.318930	0.159465	−	0.987204i	$-0.449023\pi$
$277$	5.30805	0.318930	0.159465	+	0.987204i	$0.449023\pi$
$278$	0	0
$279$	16.4469	0.984651
$280$	0	0
$281$	−5.32370	−0.317585	−0.158793	−	0.987312i	$-0.550760\pi$
$281$	−5.32370	−0.317585	−0.158793	+	0.987312i	$0.550760\pi$
$282$	0	0
$283$	−31.0893	−1.84807	−0.924033	−	0.382313i	$-0.875128\pi$
$283$	−31.0893	−1.84807	−0.924033	+	0.382313i	$0.875128\pi$
$284$	0	0
$285$	−3.22679	−0.191138
$286$	0	0
$287$	−10.3176	−0.609030
$288$	0	0
$289$	12.2097	0.718218
$290$	0	0
$291$	−6.85038	−0.401577
$292$	0	0
$293$	−25.0297	−1.46225	−0.731127	−	0.682242i	$-0.761005\pi$
$293$	−25.0297	−1.46225	−0.731127	+	0.682242i	$0.761005\pi$
$294$	0	0
$295$	−44.8975	−2.61403
$296$	0	0
$297$	18.8027	1.09104
$298$	0	0
$299$	−8.51262	−0.492297
$300$	0	0
$301$	−2.70659	−0.156005
$302$	0	0
$303$	4.63822	0.266459
$304$	0	0
$305$	8.50032	0.486727
$306$	0	0
$307$	26.9466	1.53793	0.768963	−	0.639293i	$-0.220773\pi$
$307$	26.9466	1.53793	0.768963	+	0.639293i	$0.220773\pi$
$308$	0	0
$309$	3.69225	0.210045
$310$	0	0
$311$	21.6551	1.22795	0.613974	−	0.789326i	$-0.289570\pi$
$311$	21.6551	1.22795	0.613974	+	0.789326i	$0.289570\pi$
$312$	0	0
$313$	2.58150	0.145915	0.0729574	−	0.997335i	$-0.476756\pi$
$313$	2.58150	0.145915	0.0729574	+	0.997335i	$0.476756\pi$
$314$	0	0
$315$	21.6857	1.22185
$316$	0	0
$317$	1.40108	0.0786926	0.0393463	−	0.999226i	$-0.487472\pi$
$317$	1.40108	0.0786926	0.0393463	+	0.999226i	$0.487472\pi$
$318$	0	0
$319$	−4.76792	−0.266952
$320$	0	0
$321$	−0.879126	−0.0490680
$322$	0	0
$323$	5.40460	0.300720
$324$	0	0
$325$	25.2077	1.39827
$326$	0	0
$327$	−12.2033	−0.674843
$328$	0	0
$329$	−17.8724	−0.985338
$330$	0	0
$331$	−11.7553	−0.646131	−0.323066	−	0.946377i	$-0.604713\pi$
$331$	−11.7553	−0.646131	−0.323066	+	0.946377i	$0.604713\pi$
$332$	0	0
$333$	3.44031	0.188528
$334$	0	0
$335$	−41.7981	−2.28367
$336$	0	0
$337$	27.5609	1.50134	0.750669	−	0.660679i	$-0.229731\pi$
$337$	27.5609	1.50134	0.750669	+	0.660679i	$0.229731\pi$
$338$	0	0
$339$	−0.847898	−0.0460515
$340$	0	0
$341$	31.0316	1.68046
$342$	0	0
$343$	20.1320	1.08702
$344$	0	0
$345$	−14.4121	−0.775922
$346$	0	0
$347$	−1.43946	−0.0772743	−0.0386372	−	0.999253i	$-0.512302\pi$
$347$	−1.43946	−0.0772743	−0.0386372	+	0.999253i	$0.512302\pi$
$348$	0	0
$349$	−7.49282	−0.401081	−0.200541	−	0.979685i	$-0.564270\pi$
$349$	−7.49282	−0.401081	−0.200541	+	0.979685i	$0.564270\pi$
$350$	0	0
$351$	−7.82040	−0.417422
$352$	0	0
$353$	−13.3430	−0.710178	−0.355089	−	0.934832i	$-0.615549\pi$
$353$	−13.3430	−0.710178	−0.355089	+	0.934832i	$0.615549\pi$
$354$	0	0
$355$	20.1423	1.06904
$356$	0	0
$357$	8.54360	0.452175
$358$	0	0
$359$	23.8047	1.25637	0.628183	−	0.778065i	$-0.283799\pi$
$359$	23.8047	1.25637	0.628183	+	0.778065i	$0.283799\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.55743	0.396662
$364$	0	0
$365$	−28.3757	−1.48525
$366$	0	0
$367$	−8.11128	−0.423405	−0.211703	−	0.977334i	$-0.567901\pi$
$367$	−8.11128	−0.423405	−0.211703	+	0.977334i	$0.567901\pi$
$368$	0	0
$369$	−11.9813	−0.623723
$370$	0	0
$371$	−2.09147	−0.108584
$372$	0	0
$373$	35.9853	1.86325	0.931625	−	0.363421i	$-0.118391\pi$
$373$	35.9853	1.86325	0.931625	+	0.363421i	$0.118391\pi$
$374$	0	0
$375$	26.5435	1.37070
$376$	0	0
$377$	1.98307	0.102133
$378$	0	0
$379$	−33.9515	−1.74397	−0.871986	−	0.489531i	$-0.837168\pi$
$379$	−33.9515	−1.74397	−0.871986	+	0.489531i	$0.837168\pi$
$380$	0	0
$381$	9.38272	0.480691
$382$	0	0
$383$	−29.3988	−1.50221	−0.751105	−	0.660182i	$-0.770479\pi$
$383$	−29.3988	−1.50221	−0.751105	+	0.660182i	$0.770479\pi$
$384$	0	0
$385$	40.9161	2.08528
$386$	0	0
$387$	−3.14303	−0.159769
$388$	0	0
$389$	−4.31425	−0.218741	−0.109370	−	0.994001i	$-0.534883\pi$
$389$	−4.31425	−0.218741	−0.109370	+	0.994001i	$0.534883\pi$
$390$	0	0
$391$	24.1391	1.22076
$392$	0	0
$393$	−2.88174	−0.145365
$394$	0	0
$395$	−24.7892	−1.24728
$396$	0	0
$397$	0.996779	0.0500269	0.0250135	−	0.999687i	$-0.492037\pi$
$397$	0.996779	0.0500269	0.0250135	+	0.999687i	$0.492037\pi$
$398$	0	0
$399$	1.58080	0.0791390
$400$	0	0
$401$	−30.8397	−1.54006	−0.770030	−	0.638007i	$-0.779759\pi$
$401$	−30.8397	−1.54006	−0.770030	+	0.638007i	$0.779759\pi$
$402$	0	0
$403$	−12.9067	−0.642926
$404$	0	0
$405$	17.8658	0.887759
$406$	0	0
$407$	6.49110	0.321752
$408$	0	0
$409$	−29.2447	−1.44606	−0.723029	−	0.690818i	$-0.757251\pi$
$409$	−29.2447	−1.44606	−0.723029	+	0.690818i	$0.757251\pi$
$410$	0	0
$411$	−6.08985	−0.300390
$412$	0	0
$413$	21.9953	1.08232
$414$	0	0
$415$	−60.4526	−2.96750
$416$	0	0
$417$	14.8801	0.728683
$418$	0	0
$419$	−24.5587	−1.19977	−0.599886	−	0.800086i	$-0.704787\pi$
$419$	−24.5587	−1.19977	−0.599886	+	0.800086i	$0.704787\pi$
$420$	0	0
$421$	−13.6879	−0.667108	−0.333554	−	0.942731i	$-0.608248\pi$
$421$	−13.6879	−0.667108	−0.333554	+	0.942731i	$0.608248\pi$
$422$	0	0
$423$	−20.7543	−1.00911
$424$	0	0
$425$	−71.4811	−3.46734
$426$	0	0
$427$	−4.16430	−0.201524
$428$	0	0
$429$	−6.60129	−0.318713
$430$	0	0
$431$	−8.76877	−0.422377	−0.211188	−	0.977445i	$-0.567733\pi$
$431$	−8.76877	−0.422377	−0.211188	+	0.977445i	$0.567733\pi$
$432$	0	0
$433$	−22.3080	−1.07206	−0.536028	−	0.844200i	$-0.680076\pi$
$433$	−22.3080	−1.07206	−0.536028	+	0.844200i	$0.680076\pi$
$434$	0	0
$435$	3.35739	0.160975
$436$	0	0
$437$	4.46639	0.213657
$438$	0	0
$439$	27.9897	1.33587	0.667937	−	0.744218i	$-0.267178\pi$
$439$	27.9897	1.33587	0.667937	+	0.744218i	$0.267178\pi$
$440$	0	0
$441$	6.37720	0.303676
$442$	0	0
$443$	−8.56270	−0.406826	−0.203413	−	0.979093i	$-0.565203\pi$
$443$	−8.56270	−0.406826	−0.203413	+	0.979093i	$0.565203\pi$
$444$	0	0
$445$	−53.6948	−2.54538
$446$	0	0
$447$	−9.67294	−0.457514
$448$	0	0
$449$	−13.3613	−0.630559	−0.315279	−	0.948999i	$-0.602098\pi$
$449$	−13.3613	−0.630559	−0.315279	+	0.948999i	$0.602098\pi$
$450$	0	0
$451$	−22.6061	−1.06448
$452$	0	0
$453$	−16.4022	−0.770640
$454$	0	0
$455$	−17.0178	−0.797807
$456$	0	0
$457$	−30.5153	−1.42745	−0.713723	−	0.700428i	$-0.752992\pi$
$457$	−30.5153	−1.42745	−0.713723	+	0.700428i	$0.752992\pi$
$458$	0	0
$459$	22.1762	1.03509
$460$	0	0
$461$	28.0025	1.30421	0.652103	−	0.758130i	$-0.273887\pi$
$461$	28.0025	1.30421	0.652103	+	0.758130i	$0.273887\pi$
$462$	0	0
$463$	−0.235147	−0.0109282	−0.00546411	−	0.999985i	$-0.501739\pi$
$463$	−0.235147	−0.0109282	−0.00546411	+	0.999985i	$0.501739\pi$
$464$	0	0
$465$	−21.8513	−1.01333
$466$	0	0
$467$	33.7515	1.56183	0.780917	−	0.624635i	$-0.214752\pi$
$467$	33.7515	1.56183	0.780917	+	0.624635i	$0.214752\pi$
$468$	0	0
$469$	20.4769	0.945533
$470$	0	0
$471$	12.9346	0.595995
$472$	0	0
$473$	−5.93019	−0.272670
$474$	0	0
$475$	−13.2260	−0.606849
$476$	0	0
$477$	−2.42872	−0.111203
$478$	0	0
$479$	−26.5955	−1.21518	−0.607591	−	0.794250i	$-0.707864\pi$
$479$	−26.5955	−1.21518	−0.607591	+	0.794250i	$0.707864\pi$
$480$	0	0
$481$	−2.69977	−0.123099
$482$	0	0
$483$	7.06048	0.321263
$484$	0	0
$485$	−38.6932	−1.75697
$486$	0	0
$487$	3.39782	0.153970	0.0769849	−	0.997032i	$-0.475471\pi$
$487$	3.39782	0.153970	0.0769849	+	0.997032i	$0.475471\pi$
$488$	0	0
$489$	−8.80178	−0.398030
$490$	0	0
$491$	27.6717	1.24881	0.624403	−	0.781102i	$-0.285342\pi$
$491$	27.6717	1.24881	0.624403	+	0.781102i	$0.285342\pi$
$492$	0	0
$493$	−5.62335	−0.253263
$494$	0	0
$495$	47.5138	2.13559
$496$	0	0
$497$	−9.86769	−0.442626
$498$	0	0
$499$	15.9106	0.712255	0.356128	−	0.934437i	$-0.384097\pi$
$499$	15.9106	0.712255	0.356128	+	0.934437i	$0.384097\pi$
$500$	0	0
$501$	12.7110	0.567887
$502$	0	0
$503$	−10.8530	−0.483912	−0.241956	−	0.970287i	$-0.577789\pi$
$503$	−10.8530	−0.483912	−0.241956	+	0.970287i	$0.577789\pi$
$504$	0	0
$505$	26.1982	1.16580
$506$	0	0
$507$	−7.08021	−0.314443
$508$	0	0
$509$	−10.9377	−0.484806	−0.242403	−	0.970176i	$-0.577936\pi$
$509$	−10.9377	−0.484806	−0.242403	+	0.970176i	$0.577936\pi$
$510$	0	0
$511$	13.9012	0.614955
$512$	0	0
$513$	4.10320	0.181161
$514$	0	0
$515$	20.8551	0.918983
$516$	0	0
$517$	−39.1587	−1.72220
$518$	0	0
$519$	8.18233	0.359164
$520$	0	0
$521$	−24.1315	−1.05722	−0.528609	−	0.848865i	$-0.677286\pi$
$521$	−24.1315	−1.05722	−0.528609	+	0.848865i	$0.677286\pi$
$522$	0	0
$523$	2.11348	0.0924159	0.0462079	−	0.998932i	$-0.485286\pi$
$523$	2.11348	0.0924159	0.0462079	+	0.998932i	$0.485286\pi$
$524$	0	0
$525$	−20.9076	−0.912483
$526$	0	0
$527$	36.5991	1.59428
$528$	0	0
$529$	−3.05134	−0.132667
$530$	0	0
$531$	25.5420	1.10843
$532$	0	0
$533$	9.40230	0.407259
$534$	0	0
$535$	−4.96559	−0.214681
$536$	0	0
$537$	13.5054	0.582803
$538$	0	0
$539$	12.0323	0.518270
$540$	0	0
$541$	8.28065	0.356013	0.178006	−	0.984029i	$-0.443035\pi$
$541$	8.28065	0.356013	0.178006	+	0.984029i	$0.443035\pi$
$542$	0	0
$543$	14.0515	0.603009
$544$	0	0
$545$	−68.9282	−2.95256
$546$	0	0
$547$	5.35720	0.229057	0.114529	−	0.993420i	$-0.463464\pi$
$547$	5.35720	0.229057	0.114529	+	0.993420i	$0.463464\pi$
$548$	0	0
$549$	−4.83578	−0.206386
$550$	0	0
$551$	−1.04048	−0.0443257
$552$	0	0
$553$	12.1442	0.516425
$554$	0	0
$555$	−4.57079	−0.194019
$556$	0	0
$557$	38.0926	1.61404	0.807018	−	0.590527i	$-0.201080\pi$
$557$	38.0926	1.61404	0.807018	+	0.590527i	$0.201080\pi$
$558$	0	0
$559$	2.46648	0.104321
$560$	0	0
$561$	18.7192	0.790323
$562$	0	0
$563$	−33.2774	−1.40247	−0.701237	−	0.712928i	$-0.747369\pi$
$563$	−33.2774	−1.40247	−0.701237	+	0.712928i	$0.747369\pi$
$564$	0	0
$565$	−4.78921	−0.201483
$566$	0	0
$567$	−8.75244	−0.367568
$568$	0	0
$569$	39.1870	1.64280	0.821402	−	0.570350i	$-0.193192\pi$
$569$	39.1870	1.64280	0.821402	+	0.570350i	$0.193192\pi$
$570$	0	0
$571$	32.2438	1.34936	0.674681	−	0.738110i	$-0.264281\pi$
$571$	32.2438	1.34936	0.674681	+	0.738110i	$0.264281\pi$
$572$	0	0
$573$	−1.56087	−0.0652063
$574$	0	0
$575$	−59.0723	−2.46349
$576$	0	0
$577$	−13.8047	−0.574698	−0.287349	−	0.957826i	$-0.592774\pi$
$577$	−13.8047	−0.574698	−0.287349	+	0.957826i	$0.592774\pi$
$578$	0	0
$579$	10.1872	0.423366
$580$	0	0
$581$	29.6157	1.22866
$582$	0	0
$583$	−4.58245	−0.189786
$584$	0	0
$585$	−19.7619	−0.817054
$586$	0	0
$587$	14.5565	0.600811	0.300406	−	0.953812i	$-0.402878\pi$
$587$	14.5565	0.600811	0.300406	+	0.953812i	$0.402878\pi$
$588$	0	0
$589$	6.77185	0.279029
$590$	0	0
$591$	9.59070	0.394509
$592$	0	0
$593$	18.6298	0.765033	0.382517	−	0.923949i	$-0.375058\pi$
$593$	18.6298	0.765033	0.382517	+	0.923949i	$0.375058\pi$
$594$	0	0
$595$	48.2571	1.97835
$596$	0	0
$597$	−0.705106	−0.0288581
$598$	0	0
$599$	−13.5356	−0.553051	−0.276526	−	0.961007i	$-0.589183\pi$
$599$	−13.5356	−0.553051	−0.276526	+	0.961007i	$0.589183\pi$
$600$	0	0
$601$	−39.8556	−1.62574	−0.812872	−	0.582443i	$-0.802097\pi$
$601$	−39.8556	−1.62574	−0.812872	+	0.582443i	$0.802097\pi$
$602$	0	0
$603$	23.7787	0.968344
$604$	0	0
$605$	42.6868	1.73547
$606$	0	0
$607$	23.8760	0.969097	0.484549	−	0.874764i	$-0.338984\pi$
$607$	23.8760	0.969097	0.484549	+	0.874764i	$0.338984\pi$
$608$	0	0
$609$	−1.64478	−0.0666500
$610$	0	0
$611$	16.2869	0.658896
$612$	0	0
$613$	−35.2215	−1.42258	−0.711291	−	0.702897i	$-0.751889\pi$
$613$	−35.2215	−1.42258	−0.711291	+	0.702897i	$0.751889\pi$
$614$	0	0
$615$	15.9184	0.641890
$616$	0	0
$617$	−14.4938	−0.583496	−0.291748	−	0.956495i	$-0.594237\pi$
$617$	−14.4938	−0.583496	−0.291748	+	0.956495i	$0.594237\pi$
$618$	0	0
$619$	9.77791	0.393007	0.196504	−	0.980503i	$-0.437041\pi$
$619$	9.77791	0.393007	0.196504	+	0.980503i	$0.437041\pi$
$620$	0	0
$621$	18.3265	0.735417
$622$	0	0
$623$	26.3050	1.05389
$624$	0	0
$625$	83.7963	3.35185
$626$	0	0
$627$	3.46356	0.138321
$628$	0	0
$629$	7.65569	0.305252
$630$	0	0
$631$	24.9888	0.994789	0.497394	−	0.867525i	$-0.334290\pi$
$631$	24.9888	0.994789	0.497394	+	0.867525i	$0.334290\pi$
$632$	0	0
$633$	0.368776	0.0146575
$634$	0	0
$635$	52.9967	2.10311
$636$	0	0
$637$	−5.00448	−0.198285
$638$	0	0
$639$	−11.4588	−0.453305
$640$	0	0
$641$	3.11175	0.122907	0.0614533	−	0.998110i	$-0.480426\pi$
$641$	3.11175	0.122907	0.0614533	+	0.998110i	$0.480426\pi$
$642$	0	0
$643$	20.1805	0.795841	0.397921	−	0.917420i	$-0.369732\pi$
$643$	20.1805	0.795841	0.397921	+	0.917420i	$0.369732\pi$
$644$	0	0
$645$	4.17582	0.164423
$646$	0	0
$647$	7.61650	0.299436	0.149718	−	0.988729i	$-0.452163\pi$
$647$	7.61650	0.299436	0.149718	+	0.988729i	$0.452163\pi$
$648$	0	0
$649$	48.1920	1.89170
$650$	0	0
$651$	10.7049	0.419560
$652$	0	0
$653$	19.5573	0.765335	0.382667	−	0.923886i	$-0.375005\pi$
$653$	19.5573	0.765335	0.382667	+	0.923886i	$0.375005\pi$
$654$	0	0
$655$	−16.2770	−0.635996
$656$	0	0
$657$	16.1428	0.629790
$658$	0	0
$659$	−12.4315	−0.484263	−0.242132	−	0.970243i	$-0.577847\pi$
$659$	−12.4315	−0.484263	−0.242132	+	0.970243i	$0.577847\pi$
$660$	0	0
$661$	24.3990	0.949010	0.474505	−	0.880253i	$-0.342627\pi$
$661$	24.3990	0.949010	0.474505	+	0.880253i	$0.342627\pi$
$662$	0	0
$663$	−7.78566	−0.302370
$664$	0	0
$665$	8.92889	0.346247
$666$	0	0
$667$	−4.64717	−0.179939
$668$	0	0
$669$	−12.4561	−0.481581
$670$	0	0
$671$	−9.12404	−0.352230
$672$	0	0
$673$	31.3700	1.20923	0.604613	−	0.796520i	$-0.293328\pi$
$673$	31.3700	1.20923	0.604613	+	0.796520i	$0.293328\pi$
$674$	0	0
$675$	−54.2688	−2.08881
$676$	0	0
$677$	−20.8259	−0.800406	−0.400203	−	0.916427i	$-0.631060\pi$
$677$	−20.8259	−0.800406	−0.400203	+	0.916427i	$0.631060\pi$
$678$	0	0
$679$	18.9558	0.727456
$680$	0	0
$681$	−10.3583	−0.396930
$682$	0	0
$683$	39.2654	1.50245	0.751224	−	0.660047i	$-0.229464\pi$
$683$	39.2654	1.50245	0.751224	+	0.660047i	$0.229464\pi$
$684$	0	0
$685$	−34.3975	−1.31426
$686$	0	0
$687$	10.4931	0.400337
$688$	0	0
$689$	1.90593	0.0726100
$690$	0	0
$691$	−15.1167	−0.575068	−0.287534	−	0.957770i	$-0.592835\pi$
$691$	−15.1167	−0.575068	−0.287534	+	0.957770i	$0.592835\pi$
$692$	0	0
$693$	−23.2770	−0.884219
$694$	0	0
$695$	84.0478	3.18812
$696$	0	0
$697$	−26.6619	−1.00989
$698$	0	0
$699$	−21.4058	−0.809642
$700$	0	0
$701$	−12.1392	−0.458493	−0.229246	−	0.973368i	$-0.573626\pi$
$701$	−12.1392	−0.458493	−0.229246	+	0.973368i	$0.573626\pi$
$702$	0	0
$703$	1.41651	0.0534248
$704$	0	0
$705$	27.5741	1.03850
$706$	0	0
$707$	−12.8345	−0.482690
$708$	0	0
$709$	−38.8422	−1.45875	−0.729374	−	0.684115i	$-0.760189\pi$
$709$	−38.8422	−1.45875	−0.729374	+	0.684115i	$0.760189\pi$
$710$	0	0
$711$	14.1025	0.528884
$712$	0	0
$713$	30.2457	1.13271
$714$	0	0
$715$	−37.2863	−1.39443
$716$	0	0
$717$	15.2197	0.568389
$718$	0	0
$719$	−8.63184	−0.321913	−0.160957	−	0.986961i	$-0.551458\pi$
$719$	−8.63184	−0.321913	−0.160957	+	0.986961i	$0.551458\pi$
$720$	0	0
$721$	−10.2169	−0.380496
$722$	0	0
$723$	−4.81653	−0.179129
$724$	0	0
$725$	13.7613	0.511082
$726$	0	0
$727$	8.11390	0.300928	0.150464	−	0.988615i	$-0.451923\pi$
$727$	8.11390	0.300928	0.150464	+	0.988615i	$0.451923\pi$
$728$	0	0
$729$	−0.859772	−0.0318434
$730$	0	0
$731$	−6.99414	−0.258688
$732$	0	0
$733$	−25.7078	−0.949539	−0.474769	−	0.880110i	$-0.657469\pi$
$733$	−25.7078	−0.949539	−0.474769	+	0.880110i	$0.657469\pi$
$734$	0	0
$735$	−8.47273	−0.312521
$736$	0	0
$737$	44.8651	1.65263
$738$	0	0
$739$	−29.1335	−1.07169	−0.535846	−	0.844316i	$-0.680007\pi$
$739$	−29.1335	−1.07169	−0.535846	+	0.844316i	$0.680007\pi$
$740$	0	0
$741$	−1.44056	−0.0529203
$742$	0	0
$743$	−1.80153	−0.0660918	−0.0330459	−	0.999454i	$-0.510521\pi$
$743$	−1.80153	−0.0660918	−0.0330459	+	0.999454i	$0.510521\pi$
$744$	0	0
$745$	−54.6359	−2.00171
$746$	0	0
$747$	34.3911	1.25831
$748$	0	0
$749$	2.43264	0.0888868
$750$	0	0
$751$	36.4147	1.32879	0.664395	−	0.747381i	$-0.268689\pi$
$751$	36.4147	1.32879	0.664395	+	0.747381i	$0.268689\pi$
$752$	0	0
$753$	−4.78861	−0.174507
$754$	0	0
$755$	−92.6447	−3.37169
$756$	0	0
$757$	28.8853	1.04985	0.524926	−	0.851148i	$-0.324093\pi$
$757$	28.8853	1.04985	0.524926	+	0.851148i	$0.324093\pi$
$758$	0	0
$759$	15.4696	0.561512
$760$	0	0
$761$	8.04178	0.291514	0.145757	−	0.989320i	$-0.453438\pi$
$761$	8.04178	0.291514	0.145757	+	0.989320i	$0.453438\pi$
$762$	0	0
$763$	33.7679	1.22248
$764$	0	0
$765$	56.0384	2.02607
$766$	0	0
$767$	−20.0440	−0.723745
$768$	0	0
$769$	−33.6211	−1.21241	−0.606204	−	0.795309i	$-0.707308\pi$
$769$	−33.6211	−1.21241	−0.606204	+	0.795309i	$0.707308\pi$
$770$	0	0
$771$	11.6306	0.418864
$772$	0	0
$773$	42.3933	1.52478	0.762391	−	0.647117i	$-0.224026\pi$
$773$	42.3933	1.52478	0.762391	+	0.647117i	$0.224026\pi$
$774$	0	0
$775$	−89.5643	−3.21724
$776$	0	0
$777$	2.23923	0.0803318
$778$	0	0
$779$	−4.93319	−0.176750
$780$	0	0
$781$	−21.6203	−0.773634
$782$	0	0
$783$	−4.26928	−0.152572
$784$	0	0
$785$	73.0589	2.60759
$786$	0	0
$787$	−37.4901	−1.33638	−0.668189	−	0.743992i	$-0.732930\pi$
$787$	−37.4901	−1.33638	−0.668189	+	0.743992i	$0.732930\pi$
$788$	0	0
$789$	1.25027	0.0445110
$790$	0	0
$791$	2.34623	0.0834223
$792$	0	0
$793$	3.79486	0.134760
$794$	0	0
$795$	3.22679	0.114442
$796$	0	0
$797$	37.4896	1.32795	0.663975	−	0.747755i	$-0.268868\pi$
$797$	37.4896	1.32795	0.663975	+	0.747755i	$0.268868\pi$
$798$	0	0
$799$	−46.1844	−1.63389
$800$	0	0
$801$	30.5467	1.07931
$802$	0	0
$803$	30.4578	1.07483
$804$	0	0
$805$	39.8799	1.40558
$806$	0	0
$807$	17.5935	0.619321
$808$	0	0
$809$	30.0452	1.05633	0.528167	−	0.849141i	$-0.322880\pi$
$809$	30.0452	1.05633	0.528167	+	0.849141i	$0.322880\pi$
$810$	0	0
$811$	47.0199	1.65109	0.825546	−	0.564335i	$-0.190867\pi$
$811$	47.0199	1.65109	0.825546	+	0.564335i	$0.190867\pi$
$812$	0	0
$813$	6.66538	0.233765
$814$	0	0
$815$	−49.7154	−1.74145
$816$	0	0
$817$	−1.29411	−0.0452751
$818$	0	0
$819$	9.68134	0.338293
$820$	0	0
$821$	51.0087	1.78022	0.890108	−	0.455749i	$-0.150629\pi$
$821$	51.0087	1.78022	0.890108	+	0.455749i	$0.150629\pi$
$822$	0	0
$823$	−25.2441	−0.879953	−0.439977	−	0.898009i	$-0.645013\pi$
$823$	−25.2441	−0.879953	−0.439977	+	0.898009i	$0.645013\pi$
$824$	0	0
$825$	−45.8089	−1.59486
$826$	0	0
$827$	19.3992	0.674578	0.337289	−	0.941401i	$-0.390490\pi$
$827$	19.3992	0.674578	0.337289	+	0.941401i	$0.390490\pi$
$828$	0	0
$829$	44.5506	1.54731	0.773653	−	0.633610i	$-0.218427\pi$
$829$	44.5506	1.54731	0.773653	+	0.633610i	$0.218427\pi$
$830$	0	0
$831$	4.01200	0.139175
$832$	0	0
$833$	14.1911	0.491693
$834$	0	0
$835$	71.7960	2.48460
$836$	0	0
$837$	27.7862	0.960433
$838$	0	0
$839$	−24.0866	−0.831563	−0.415781	−	0.909465i	$-0.636492\pi$
$839$	−24.0866	−0.831563	−0.415781	+	0.909465i	$0.636492\pi$
$840$	0	0
$841$	−27.9174	−0.962669
$842$	0	0
$843$	−4.02382	−0.138588
$844$	0	0
$845$	−39.9914	−1.37574
$846$	0	0
$847$	−20.9122	−0.718553
$848$	0	0
$849$	−23.4983	−0.806459
$850$	0	0
$851$	6.32670	0.216877
$852$	0	0
$853$	21.7912	0.746115	0.373058	−	0.927808i	$-0.378309\pi$
$853$	21.7912	0.746115	0.373058	+	0.927808i	$0.378309\pi$
$854$	0	0
$855$	10.3687	0.354600
$856$	0	0
$857$	−2.54144	−0.0868141	−0.0434070	−	0.999057i	$-0.513821\pi$
$857$	−2.54144	−0.0868141	−0.0434070	+	0.999057i	$0.513821\pi$
$858$	0	0
$859$	−40.7092	−1.38898	−0.694491	−	0.719502i	$-0.744370\pi$
$859$	−40.7092	−1.38898	−0.694491	+	0.719502i	$0.744370\pi$
$860$	0	0
$861$	−7.79839	−0.265768
$862$	0	0
$863$	−14.4803	−0.492916	−0.246458	−	0.969154i	$-0.579267\pi$
$863$	−14.4803	−0.492916	−0.246458	+	0.969154i	$0.579267\pi$
$864$	0	0
$865$	46.2165	1.57141
$866$	0	0
$867$	9.22849	0.313416
$868$	0	0
$869$	26.6082	0.902621
$870$	0	0
$871$	−18.6603	−0.632279
$872$	0	0
$873$	22.0124	0.745006
$874$	0	0
$875$	−73.4487	−2.48302
$876$	0	0
$877$	−30.9769	−1.04601	−0.523007	−	0.852328i	$-0.675190\pi$
$877$	−30.9769	−1.04601	−0.523007	+	0.852328i	$0.675190\pi$
$878$	0	0
$879$	−18.9183	−0.638098
$880$	0	0
$881$	−47.1900	−1.58987	−0.794936	−	0.606693i	$-0.792496\pi$
$881$	−47.1900	−1.58987	−0.794936	+	0.606693i	$0.792496\pi$
$882$	0	0
$883$	6.60510	0.222279	0.111140	−	0.993805i	$-0.464550\pi$
$883$	6.60510	0.222279	0.111140	+	0.993805i	$0.464550\pi$
$884$	0	0
$885$	−33.9350	−1.14071
$886$	0	0
$887$	−4.64869	−0.156088	−0.0780439	−	0.996950i	$-0.524867\pi$
$887$	−4.64869	−0.156088	−0.0780439	+	0.996950i	$0.524867\pi$
$888$	0	0
$889$	−25.9630	−0.870772
$890$	0	0
$891$	−19.1767	−0.642445
$892$	0	0
$893$	−8.54538	−0.285960
$894$	0	0
$895$	76.2831	2.54986
$896$	0	0
$897$	−6.43411	−0.214829
$898$	0	0
$899$	−7.04594	−0.234995
$900$	0	0
$901$	−5.40460	−0.180053
$902$	0	0
$903$	−2.04573	−0.0680776
$904$	0	0
$905$	79.3676	2.63827
$906$	0	0
$907$	−50.9420	−1.69150	−0.845751	−	0.533577i	$-0.820847\pi$
$907$	−50.9420	−1.69150	−0.845751	+	0.533577i	$0.820847\pi$
$908$	0	0
$909$	−14.9040	−0.494335
$910$	0	0
$911$	−11.8612	−0.392979	−0.196490	−	0.980506i	$-0.562954\pi$
$911$	−11.8612	−0.392979	−0.196490	+	0.980506i	$0.562954\pi$
$912$	0	0
$913$	64.8884	2.14749
$914$	0	0
$915$	6.42481	0.212398
$916$	0	0
$917$	7.97410	0.263328
$918$	0	0
$919$	−32.4608	−1.07078	−0.535392	−	0.844604i	$-0.679836\pi$
$919$	−32.4608	−1.07078	−0.535392	+	0.844604i	$0.679836\pi$
$920$	0	0
$921$	20.3671	0.671120
$922$	0	0
$923$	8.99228	0.295984
$924$	0	0
$925$	−18.7348	−0.615995
$926$	0	0
$927$	−11.8643	−0.389676
$928$	0	0
$929$	−54.2865	−1.78108	−0.890542	−	0.454902i	$-0.849674\pi$
$929$	−54.2865	−1.78108	−0.890542	+	0.454902i	$0.849674\pi$
$930$	0	0
$931$	2.62575	0.0860554
$932$	0	0
$933$	16.3676	0.535852
$934$	0	0
$935$	105.732	3.45781
$936$	0	0
$937$	2.35844	0.0770468	0.0385234	−	0.999258i	$-0.487735\pi$
$937$	2.35844	0.0770468	0.0385234	+	0.999258i	$0.487735\pi$
$938$	0	0
$939$	1.95118	0.0636743
$940$	0	0
$941$	−4.54445	−0.148145	−0.0740724	−	0.997253i	$-0.523600\pi$
$941$	−4.54445	−0.148145	−0.0740724	+	0.997253i	$0.523600\pi$
$942$	0	0
$943$	−22.0336	−0.717511
$944$	0	0
$945$	36.6370	1.19180
$946$	0	0
$947$	30.0143	0.975334	0.487667	−	0.873030i	$-0.337848\pi$
$947$	30.0143	0.975334	0.487667	+	0.873030i	$0.337848\pi$
$948$	0	0
$949$	−12.6680	−0.411220
$950$	0	0
$951$	1.05898	0.0343399
$952$	0	0
$953$	1.85207	0.0599944	0.0299972	−	0.999550i	$-0.490450\pi$
$953$	1.85207	0.0599944	0.0299972	+	0.999550i	$0.490450\pi$
$954$	0	0
$955$	−8.81631	−0.285289
$956$	0	0
$957$	−3.60375	−0.116493
$958$	0	0
$959$	16.8513	0.544157
$960$	0	0
$961$	14.8580	0.479289
$962$	0	0
$963$	2.82490	0.0910311
$964$	0	0
$965$	57.5408	1.85230
$966$	0	0
$967$	−5.99575	−0.192810	−0.0964052	−	0.995342i	$-0.530734\pi$
$967$	−5.99575	−0.192810	−0.0964052	+	0.995342i	$0.530734\pi$
$968$	0	0
$969$	4.08497	0.131228
$970$	0	0
$971$	−20.7011	−0.664329	−0.332165	−	0.943221i	$-0.607779\pi$
$971$	−20.7011	−0.664329	−0.332165	+	0.943221i	$0.607779\pi$
$972$	0	0
$973$	−41.1750	−1.32001
$974$	0	0
$975$	19.0528	0.610178
$976$	0	0
$977$	19.9594	0.638556	0.319278	−	0.947661i	$-0.396560\pi$
$977$	19.9594	0.638556	0.319278	+	0.947661i	$0.396560\pi$
$978$	0	0
$979$	57.6348	1.84202
$980$	0	0
$981$	39.2129	1.25197
$982$	0	0
$983$	−52.0436	−1.65993	−0.829966	−	0.557814i	$-0.811640\pi$
$983$	−52.0436	−1.65993	−0.829966	+	0.557814i	$0.811640\pi$
$984$	0	0
$985$	54.1715	1.72605
$986$	0	0
$987$	−13.5085	−0.429982
$988$	0	0
$989$	−5.78000	−0.183793
$990$	0	0
$991$	12.5003	0.397087	0.198543	−	0.980092i	$-0.436379\pi$
$991$	12.5003	0.397087	0.198543	+	0.980092i	$0.436379\pi$
$992$	0	0
$993$	−8.88506	−0.281959
$994$	0	0
$995$	−3.98267	−0.126259
$996$	0	0
$997$	−54.9379	−1.73990	−0.869951	−	0.493139i	$-0.835849\pi$
$997$	−54.9379	−1.73990	−0.869951	+	0.493139i	$0.835849\pi$
$998$	0	0
$999$	5.81224	0.183891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.14	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.14	✓	19	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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q+0.755832 q^{3} +4.26919 q^{5} -2.09147 q^{7} -2.42872 q^{9} +O(q^{10})\)	\(q+0.755832 q^{3} +4.26919 q^{5} -2.09147 q^{7} -2.42872 q^{9} -4.58245 q^{11} +1.90593 q^{13} +3.22679 q^{15} -5.40460 q^{17} -1.00000 q^{19} -1.58080 q^{21} -4.46639 q^{23} +13.2260 q^{25} -4.10320 q^{27} +1.04048 q^{29} -6.77185 q^{31} -3.46356 q^{33} -8.92889 q^{35} -1.41651 q^{37} +1.44056 q^{39} +4.93319 q^{41} +1.29411 q^{43} -10.3687 q^{45} +8.54538 q^{47} -2.62575 q^{49} -4.08497 q^{51} +1.00000 q^{53} -19.5633 q^{55} -0.755832 q^{57} -10.5166 q^{59} +1.99109 q^{61} +5.07959 q^{63} +8.13676 q^{65} -9.79064 q^{67} -3.37584 q^{69} +4.71806 q^{71} -6.64663 q^{73} +9.99661 q^{75} +9.58406 q^{77} -5.80655 q^{79} +4.18482 q^{81} -14.1602 q^{83} -23.0733 q^{85} +0.786425 q^{87} -12.5773 q^{89} -3.98619 q^{91} -5.11838 q^{93} -4.26919 q^{95} -9.06337 q^{97} +11.1295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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