LMFDB - Embedded newform 4028.2.a.f.1.18

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:	$0$
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} + 124 x^{16} + 364 x^{15} - 1554 x^{14} - 2310 x^{13} + 10113 x^{12} + \cdots + 139 $ x^19 - 4x^18 - 30x^17 + 124x^16 + 364x^15 - 1554x^14 - 2310x^13 + 10113x^12 + 8368x^11 - 36567x^10 - 18074x^9 + 72868x^8 + 23819x^7 - 73816x^6 - 17877x^5 + 30973x^4 + 5074x^3 - 4299x^2 - 454x + 139
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			1.18
Root			$3.13171$ 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)

$f(q)$	$=$	$q+3.13171 q^{3} +3.22705 q^{5} -4.22839 q^{7} +6.80761 q^{9} +O(q^{10})$	$q+3.13171 q^{3} +3.22705 q^{5} -4.22839 q^{7} +6.80761 q^{9} +0.360842 q^{11} +0.769242 q^{13} +10.1062 q^{15} -0.871414 q^{17} -1.00000 q^{19} -13.2421 q^{21} +2.99494 q^{23} +5.41386 q^{25} +11.9243 q^{27} -0.0640563 q^{29} +10.6052 q^{31} +1.13005 q^{33} -13.6452 q^{35} +0.722667 q^{37} +2.40904 q^{39} +1.46209 q^{41} +0.946959 q^{43} +21.9685 q^{45} +3.57846 q^{47} +10.8793 q^{49} -2.72902 q^{51} -1.00000 q^{53} +1.16446 q^{55} -3.13171 q^{57} -9.43009 q^{59} +3.77899 q^{61} -28.7852 q^{63} +2.48238 q^{65} +13.1340 q^{67} +9.37929 q^{69} +12.3901 q^{71} -6.74607 q^{73} +16.9547 q^{75} -1.52578 q^{77} -3.97473 q^{79} +16.9208 q^{81} +9.49368 q^{83} -2.81210 q^{85} -0.200606 q^{87} -7.31280 q^{89} -3.25265 q^{91} +33.2124 q^{93} -3.22705 q^{95} -18.3040 q^{97} +2.45647 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9	$ 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) $ 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9 + q^11 - q^13 + 8 * q^15 + 3 * q^17 - 19 * q^19 + 8 * q^21 + 10 * q^23 + 21 * q^25 + 28 * q^27 + 2 * q^29 + 25 * q^31 + q^33 + 20 * q^35 + 19 * q^37 + 37 * q^39 - 9 * q^41 + 35 * q^43 + 37 * q^45 + 23 * q^47 + 30 * q^49 + 34 * q^51 - 19 * q^53 + 40 * q^55 - 4 * q^57 + 16 * q^59 + 21 * q^61 + 3 * q^63 - 10 * q^65 + 67 * q^67 + 23 * q^69 + 18 * q^71 - 20 * q^73 + 33 * q^75 + 37 * q^77 + 2 * q^79 + 23 * q^81 + 38 * q^83 + 8 * q^85 + 18 * q^87 - q^89 - 9 * q^91 + 14 * q^93 - 4 * q^95 - 21 * q^97 + 33 * 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
$3$	3.13171	1.80809	0.904047	−	0.427433i	$-0.140582\pi$
$3$	3.13171	1.80809	0.904047	+	0.427433i	$0.140582\pi$
$4$	0	0
$5$	3.22705	1.44318	0.721591	−	0.692320i	$-0.243411\pi$
$5$	3.22705	1.44318	0.721591	+	0.692320i	$0.243411\pi$
$6$	0	0
$7$	−4.22839	−1.59818	−0.799090	−	0.601211i	$-0.794685\pi$
$7$	−4.22839	−1.59818	−0.799090	+	0.601211i	$0.794685\pi$
$8$	0	0
$9$	6.80761	2.26920
$10$	0	0
$11$	0.360842	0.108798	0.0543990	−	0.998519i	$-0.482676\pi$
$11$	0.360842	0.108798	0.0543990	+	0.998519i	$0.482676\pi$
$12$	0	0
$13$	0.769242	0.213349	0.106675	−	0.994294i	$-0.465980\pi$
$13$	0.769242	0.213349	0.106675	+	0.994294i	$0.465980\pi$
$14$	0	0
$15$	10.1062	2.60941
$16$	0	0
$17$	−0.871414	−0.211349	−0.105674	−	0.994401i	$-0.533700\pi$
$17$	−0.871414	−0.211349	−0.105674	+	0.994401i	$0.533700\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−13.2421	−2.88966
$22$	0	0
$23$	2.99494	0.624488	0.312244	−	0.950002i	$-0.398919\pi$
$23$	2.99494	0.624488	0.312244	+	0.950002i	$0.398919\pi$
$24$	0	0
$25$	5.41386	1.08277
$26$	0	0
$27$	11.9243	2.29484
$28$	0	0
$29$	−0.0640563	−0.0118950	−0.00594748	−	0.999982i	$-0.501893\pi$
$29$	−0.0640563	−0.0118950	−0.00594748	+	0.999982i	$0.501893\pi$
$30$	0	0
$31$	10.6052	1.90475	0.952373	−	0.304935i	$-0.0986349\pi$
$31$	10.6052	1.90475	0.952373	+	0.304935i	$0.0986349\pi$
$32$	0	0
$33$	1.13005	0.196717
$34$	0	0
$35$	−13.6452	−2.30646
$36$	0	0
$37$	0.722667	0.118806	0.0594029	−	0.998234i	$-0.481080\pi$
$37$	0.722667	0.118806	0.0594029	+	0.998234i	$0.481080\pi$
$38$	0	0
$39$	2.40904	0.385756
$40$	0	0
$41$	1.46209	0.228340	0.114170	−	0.993461i	$-0.463579\pi$
$41$	1.46209	0.228340	0.114170	+	0.993461i	$0.463579\pi$
$42$	0	0
$43$	0.946959	0.144410	0.0722049	−	0.997390i	$-0.476996\pi$
$43$	0.946959	0.144410	0.0722049	+	0.997390i	$0.476996\pi$
$44$	0	0
$45$	21.9685	3.27487
$46$	0	0
$47$	3.57846	0.521972	0.260986	−	0.965343i	$-0.415952\pi$
$47$	3.57846	0.521972	0.260986	+	0.965343i	$0.415952\pi$
$48$	0	0
$49$	10.8793	1.55418
$50$	0	0
$51$	−2.72902	−0.382139
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	1.16446	0.157015
$56$	0	0
$57$	−3.13171	−0.414805
$58$	0	0
$59$	−9.43009	−1.22769	−0.613846	−	0.789425i	$-0.710379\pi$
$59$	−9.43009	−1.22769	−0.613846	+	0.789425i	$0.710379\pi$
$60$	0	0
$61$	3.77899	0.483851	0.241925	−	0.970295i	$-0.422221\pi$
$61$	3.77899	0.483851	0.241925	+	0.970295i	$0.422221\pi$
$62$	0	0
$63$	−28.7852	−3.62660
$64$	0	0
$65$	2.48238	0.307902
$66$	0	0
$67$	13.1340	1.60457	0.802284	−	0.596943i	$-0.203618\pi$
$67$	13.1340	1.60457	0.802284	+	0.596943i	$0.203618\pi$
$68$	0	0
$69$	9.37929	1.12913
$70$	0	0
$71$	12.3901	1.47043	0.735215	−	0.677834i	$-0.237081\pi$
$71$	12.3901	1.47043	0.735215	+	0.677834i	$0.237081\pi$
$72$	0	0
$73$	−6.74607	−0.789567	−0.394784	−	0.918774i	$-0.629180\pi$
$73$	−6.74607	−0.789567	−0.394784	+	0.918774i	$0.629180\pi$
$74$	0	0
$75$	16.9547	1.95775
$76$	0	0
$77$	−1.52578	−0.173879
$78$	0	0
$79$	−3.97473	−0.447192	−0.223596	−	0.974682i	$-0.571780\pi$
$79$	−3.97473	−0.447192	−0.223596	+	0.974682i	$0.571780\pi$
$80$	0	0
$81$	16.9208	1.88009
$82$	0	0
$83$	9.49368	1.04207	0.521033	−	0.853536i	$-0.325547\pi$
$83$	9.49368	1.04207	0.521033	+	0.853536i	$0.325547\pi$
$84$	0	0
$85$	−2.81210	−0.305015
$86$	0	0
$87$	−0.200606	−0.0215072
$88$	0	0
$89$	−7.31280	−0.775155	−0.387577	−	0.921837i	$-0.626688\pi$
$89$	−7.31280	−0.775155	−0.387577	+	0.921837i	$0.626688\pi$
$90$	0	0
$91$	−3.25265	−0.340971
$92$	0	0
$93$	33.2124	3.44396
$94$	0	0
$95$	−3.22705	−0.331089
$96$	0	0
$97$	−18.3040	−1.85849	−0.929243	−	0.369470i	$-0.879539\pi$
$97$	−18.3040	−1.85849	−0.929243	+	0.369470i	$0.879539\pi$
$98$	0	0
$99$	2.45647	0.246885
$100$	0	0
$101$	−12.6422	−1.25795	−0.628974	−	0.777427i	$-0.716525\pi$
$101$	−12.6422	−1.25795	−0.628974	+	0.777427i	$0.716525\pi$
$102$	0	0
$103$	0.580850	0.0572329	0.0286164	−	0.999590i	$-0.490890\pi$
$103$	0.580850	0.0572329	0.0286164	+	0.999590i	$0.490890\pi$
$104$	0	0
$105$	−42.7329	−4.17030
$106$	0	0
$107$	9.51425	0.919777	0.459889	−	0.887977i	$-0.347889\pi$
$107$	9.51425	0.919777	0.459889	+	0.887977i	$0.347889\pi$
$108$	0	0
$109$	−14.3204	−1.37165	−0.685823	−	0.727768i	$-0.740558\pi$
$109$	−14.3204	−1.37165	−0.685823	+	0.727768i	$0.740558\pi$
$110$	0	0
$111$	2.26319	0.214812
$112$	0	0
$113$	−0.118840	−0.0111796	−0.00558978	−	0.999984i	$-0.501779\pi$
$113$	−0.118840	−0.0111796	−0.00558978	+	0.999984i	$0.501779\pi$
$114$	0	0
$115$	9.66483	0.901250
$116$	0	0
$117$	5.23670	0.484133
$118$	0	0
$119$	3.68468	0.337774
$120$	0	0
$121$	−10.8698	−0.988163
$122$	0	0
$123$	4.57885	0.412861
$124$	0	0
$125$	1.33556	0.119456
$126$	0	0
$127$	−7.35577	−0.652719	−0.326359	−	0.945246i	$-0.605822\pi$
$127$	−7.35577	−0.652719	−0.326359	+	0.945246i	$0.605822\pi$
$128$	0	0
$129$	2.96560	0.261107
$130$	0	0
$131$	13.7008	1.19704	0.598521	−	0.801107i	$-0.295755\pi$
$131$	13.7008	1.19704	0.598521	+	0.801107i	$0.295755\pi$
$132$	0	0
$133$	4.22839	0.366648
$134$	0	0
$135$	38.4805	3.31187
$136$	0	0
$137$	−4.16172	−0.355560	−0.177780	−	0.984070i	$-0.556892\pi$
$137$	−4.16172	−0.355560	−0.177780	+	0.984070i	$0.556892\pi$
$138$	0	0
$139$	−17.5247	−1.48642	−0.743211	−	0.669057i	$-0.766698\pi$
$139$	−17.5247	−1.48642	−0.743211	+	0.669057i	$0.766698\pi$
$140$	0	0
$141$	11.2067	0.943775
$142$	0	0
$143$	0.277575	0.0232120
$144$	0	0
$145$	−0.206713	−0.0171666
$146$	0	0
$147$	34.0707	2.81010
$148$	0	0
$149$	7.59371	0.622101	0.311050	−	0.950393i	$-0.399319\pi$
$149$	7.59371	0.622101	0.311050	+	0.950393i	$0.399319\pi$
$150$	0	0
$151$	19.8465	1.61508	0.807541	−	0.589811i	$-0.200798\pi$
$151$	19.8465	1.61508	0.807541	+	0.589811i	$0.200798\pi$
$152$	0	0
$153$	−5.93225	−0.479594
$154$	0	0
$155$	34.2235	2.74889
$156$	0	0
$157$	−5.23036	−0.417428	−0.208714	−	0.977977i	$-0.566928\pi$
$157$	−5.23036	−0.417428	−0.208714	+	0.977977i	$0.566928\pi$
$158$	0	0
$159$	−3.13171	−0.248361
$160$	0	0
$161$	−12.6638	−0.998045
$162$	0	0
$163$	−2.85778	−0.223839	−0.111919	−	0.993717i	$-0.535700\pi$
$163$	−2.85778	−0.223839	−0.111919	+	0.993717i	$0.535700\pi$
$164$	0	0
$165$	3.64674	0.283898
$166$	0	0
$167$	2.00931	0.155485	0.0777427	−	0.996973i	$-0.475229\pi$
$167$	2.00931	0.155485	0.0777427	+	0.996973i	$0.475229\pi$
$168$	0	0
$169$	−12.4083	−0.954482
$170$	0	0
$171$	−6.80761	−0.520591
$172$	0	0
$173$	10.5880	0.804994	0.402497	−	0.915421i	$-0.368142\pi$
$173$	10.5880	0.804994	0.402497	+	0.915421i	$0.368142\pi$
$174$	0	0
$175$	−22.8919	−1.73047
$176$	0	0
$177$	−29.5323	−2.21978
$178$	0	0
$179$	8.51654	0.636556	0.318278	−	0.947997i	$-0.396895\pi$
$179$	8.51654	0.636556	0.318278	+	0.947997i	$0.396895\pi$
$180$	0	0
$181$	−3.51992	−0.261634	−0.130817	−	0.991407i	$-0.541760\pi$
$181$	−3.51992	−0.261634	−0.130817	+	0.991407i	$0.541760\pi$
$182$	0	0
$183$	11.8347	0.874847
$184$	0	0
$185$	2.33209	0.171458
$186$	0	0
$187$	−0.314443	−0.0229943
$188$	0	0
$189$	−50.4208	−3.66757
$190$	0	0
$191$	−6.20537	−0.449004	−0.224502	−	0.974474i	$-0.572076\pi$
$191$	−6.20537	−0.449004	−0.224502	+	0.974474i	$0.572076\pi$
$192$	0	0
$193$	−15.8906	−1.14383	−0.571916	−	0.820312i	$-0.693800\pi$
$193$	−15.8906	−1.14383	−0.571916	+	0.820312i	$0.693800\pi$
$194$	0	0
$195$	7.77411	0.556715
$196$	0	0
$197$	6.81356	0.485446	0.242723	−	0.970096i	$-0.421959\pi$
$197$	6.81356	0.485446	0.242723	+	0.970096i	$0.421959\pi$
$198$	0	0
$199$	11.6939	0.828959	0.414479	−	0.910059i	$-0.363964\pi$
$199$	11.6939	0.828959	0.414479	+	0.910059i	$0.363964\pi$
$200$	0	0
$201$	41.1317	2.90121
$202$	0	0
$203$	0.270855	0.0190103
$204$	0	0
$205$	4.71824	0.329536
$206$	0	0
$207$	20.3884	1.41709
$208$	0	0
$209$	−0.360842	−0.0249600
$210$	0	0
$211$	−14.5546	−1.00198	−0.500992	−	0.865452i	$-0.667031\pi$
$211$	−14.5546	−1.00198	−0.500992	+	0.865452i	$0.667031\pi$
$212$	0	0
$213$	38.8021	2.65868
$214$	0	0
$215$	3.05589	0.208410
$216$	0	0
$217$	−44.8428	−3.04413
$218$	0	0
$219$	−21.1267	−1.42761
$220$	0	0
$221$	−0.670328	−0.0450912
$222$	0	0
$223$	5.44021	0.364303	0.182152	−	0.983270i	$-0.441694\pi$
$223$	5.44021	0.364303	0.182152	+	0.983270i	$0.441694\pi$
$224$	0	0
$225$	36.8555	2.45703
$226$	0	0
$227$	−8.64850	−0.574021	−0.287011	−	0.957927i	$-0.592662\pi$
$227$	−8.64850	−0.574021	−0.287011	+	0.957927i	$0.592662\pi$
$228$	0	0
$229$	−17.3986	−1.14973	−0.574866	−	0.818247i	$-0.694946\pi$
$229$	−17.3986	−1.14973	−0.574866	+	0.818247i	$0.694946\pi$
$230$	0	0
$231$	−4.77830	−0.314389
$232$	0	0
$233$	−14.5518	−0.953317	−0.476658	−	0.879089i	$-0.658152\pi$
$233$	−14.5518	−0.953317	−0.476658	+	0.879089i	$0.658152\pi$
$234$	0	0
$235$	11.5479	0.753301
$236$	0	0
$237$	−12.4477	−0.808566
$238$	0	0
$239$	−19.1284	−1.23731	−0.618657	−	0.785661i	$-0.712323\pi$
$239$	−19.1284	−1.23731	−0.618657	+	0.785661i	$0.712323\pi$
$240$	0	0
$241$	8.91777	0.574444	0.287222	−	0.957864i	$-0.407268\pi$
$241$	8.91777	0.574444	0.287222	+	0.957864i	$0.407268\pi$
$242$	0	0
$243$	17.2179	1.10453
$244$	0	0
$245$	35.1079	2.24296
$246$	0	0
$247$	−0.769242	−0.0489457
$248$	0	0
$249$	29.7315	1.88416
$250$	0	0
$251$	12.6873	0.800815	0.400407	−	0.916337i	$-0.368869\pi$
$251$	12.6873	0.800815	0.400407	+	0.916337i	$0.368869\pi$
$252$	0	0
$253$	1.08070	0.0679431
$254$	0	0
$255$	−8.80668	−0.551496
$256$	0	0
$257$	−20.5871	−1.28419	−0.642095	−	0.766625i	$-0.721934\pi$
$257$	−20.5871	−1.28419	−0.642095	+	0.766625i	$0.721934\pi$
$258$	0	0
$259$	−3.05572	−0.189873
$260$	0	0
$261$	−0.436071	−0.0269921
$262$	0	0
$263$	0.708235	0.0436716	0.0218358	−	0.999762i	$-0.493049\pi$
$263$	0.708235	0.0436716	0.0218358	+	0.999762i	$0.493049\pi$
$264$	0	0
$265$	−3.22705	−0.198236
$266$	0	0
$267$	−22.9016	−1.40155
$268$	0	0
$269$	−11.1401	−0.679224	−0.339612	−	0.940566i	$-0.610296\pi$
$269$	−11.1401	−0.679224	−0.339612	+	0.940566i	$0.610296\pi$
$270$	0	0
$271$	−15.8985	−0.965762	−0.482881	−	0.875686i	$-0.660410\pi$
$271$	−15.8985	−0.965762	−0.482881	+	0.875686i	$0.660410\pi$
$272$	0	0
$273$	−10.1864	−0.616507
$274$	0	0
$275$	1.95355	0.117804
$276$	0	0
$277$	11.9763	0.719587	0.359794	−	0.933032i	$-0.382847\pi$
$277$	11.9763	0.719587	0.359794	+	0.933032i	$0.382847\pi$
$278$	0	0
$279$	72.1960	4.32226
$280$	0	0
$281$	14.3249	0.854550	0.427275	−	0.904122i	$-0.359474\pi$
$281$	14.3249	0.854550	0.427275	+	0.904122i	$0.359474\pi$
$282$	0	0
$283$	−8.34304	−0.495942	−0.247971	−	0.968767i	$-0.579764\pi$
$283$	−8.34304	−0.495942	−0.247971	+	0.968767i	$0.579764\pi$
$284$	0	0
$285$	−10.1062	−0.598639
$286$	0	0
$287$	−6.18229	−0.364929
$288$	0	0
$289$	−16.2406	−0.955332
$290$	0	0
$291$	−57.3227	−3.36032
$292$	0	0
$293$	−9.80348	−0.572725	−0.286363	−	0.958121i	$-0.592446\pi$
$293$	−9.80348	−0.572725	−0.286363	+	0.958121i	$0.592446\pi$
$294$	0	0
$295$	−30.4314	−1.77178
$296$	0	0
$297$	4.30281	0.249674
$298$	0	0
$299$	2.30383	0.133234
$300$	0	0
$301$	−4.00411	−0.230793
$302$	0	0
$303$	−39.5918	−2.27449
$304$	0	0
$305$	12.1950	0.698284
$306$	0	0
$307$	−5.63978	−0.321879	−0.160940	−	0.986964i	$-0.551452\pi$
$307$	−5.63978	−0.321879	−0.160940	+	0.986964i	$0.551452\pi$
$308$	0	0
$309$	1.81905	0.103482
$310$	0	0
$311$	−24.8535	−1.40931	−0.704657	−	0.709548i	$-0.748899\pi$
$311$	−24.8535	−1.40931	−0.704657	+	0.709548i	$0.748899\pi$
$312$	0	0
$313$	4.82402	0.272670	0.136335	−	0.990663i	$-0.456468\pi$
$313$	4.82402	0.272670	0.136335	+	0.990663i	$0.456468\pi$
$314$	0	0
$315$	−92.8914	−5.23384
$316$	0	0
$317$	−24.6935	−1.38693	−0.693463	−	0.720493i	$-0.743916\pi$
$317$	−24.6935	−1.38693	−0.693463	+	0.720493i	$0.743916\pi$
$318$	0	0
$319$	−0.0231142	−0.00129415
$320$	0	0
$321$	29.7959	1.66304
$322$	0	0
$323$	0.871414	0.0484868
$324$	0	0
$325$	4.16457	0.231009
$326$	0	0
$327$	−44.8474	−2.48007
$328$	0	0
$329$	−15.1311	−0.834206
$330$	0	0
$331$	−35.2442	−1.93720	−0.968599	−	0.248626i	$-0.920021\pi$
$331$	−35.2442	−1.93720	−0.968599	+	0.248626i	$0.920021\pi$
$332$	0	0
$333$	4.91964	0.269595
$334$	0	0
$335$	42.3839	2.31568
$336$	0	0
$337$	−35.0418	−1.90885	−0.954424	−	0.298453i	$-0.903529\pi$
$337$	−35.0418	−1.90885	−0.954424	+	0.298453i	$0.903529\pi$
$338$	0	0
$339$	−0.372174	−0.0202137
$340$	0	0
$341$	3.82680	0.207233
$342$	0	0
$343$	−16.4030	−0.885678
$344$	0	0
$345$	30.2675	1.62955
$346$	0	0
$347$	−26.6248	−1.42929	−0.714647	−	0.699486i	$-0.753412\pi$
$347$	−26.6248	−1.42929	−0.714647	+	0.699486i	$0.753412\pi$
$348$	0	0
$349$	−18.7292	−1.00255	−0.501275	−	0.865288i	$-0.667136\pi$
$349$	−18.7292	−1.00255	−0.501275	+	0.865288i	$0.667136\pi$
$350$	0	0
$351$	9.17271	0.489603
$352$	0	0
$353$	−7.46048	−0.397081	−0.198541	−	0.980093i	$-0.563620\pi$
$353$	−7.46048	−0.397081	−0.198541	+	0.980093i	$0.563620\pi$
$354$	0	0
$355$	39.9834	2.12210
$356$	0	0
$357$	11.5393	0.610727
$358$	0	0
$359$	27.0233	1.42623	0.713117	−	0.701045i	$-0.247283\pi$
$359$	27.0233	1.42623	0.713117	+	0.701045i	$0.247283\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−34.0411	−1.78669
$364$	0	0
$365$	−21.7699	−1.13949
$366$	0	0
$367$	12.7237	0.664174	0.332087	−	0.943249i	$-0.392247\pi$
$367$	12.7237	0.664174	0.332087	+	0.943249i	$0.392247\pi$
$368$	0	0
$369$	9.95335	0.518151
$370$	0	0
$371$	4.22839	0.219527
$372$	0	0
$373$	28.9162	1.49723	0.748613	−	0.663008i	$-0.230720\pi$
$373$	28.9162	1.49723	0.748613	+	0.663008i	$0.230720\pi$
$374$	0	0
$375$	4.18258	0.215987
$376$	0	0
$377$	−0.0492748	−0.00253778
$378$	0	0
$379$	5.62047	0.288704	0.144352	−	0.989526i	$-0.453890\pi$
$379$	5.62047	0.288704	0.144352	+	0.989526i	$0.453890\pi$
$380$	0	0
$381$	−23.0361	−1.18018
$382$	0	0
$383$	−6.85863	−0.350459	−0.175230	−	0.984528i	$-0.556067\pi$
$383$	−6.85863	−0.350459	−0.175230	+	0.984528i	$0.556067\pi$
$384$	0	0
$385$	−4.92377	−0.250939
$386$	0	0
$387$	6.44653	0.327696
$388$	0	0
$389$	25.2609	1.28078	0.640388	−	0.768052i	$-0.278774\pi$
$389$	25.2609	1.28078	0.640388	+	0.768052i	$0.278774\pi$
$390$	0	0
$391$	−2.60983	−0.131985
$392$	0	0
$393$	42.9069	2.16437
$394$	0	0
$395$	−12.8267	−0.645380
$396$	0	0
$397$	24.0714	1.20811	0.604055	−	0.796942i	$-0.293551\pi$
$397$	24.0714	1.20811	0.604055	+	0.796942i	$0.293551\pi$
$398$	0	0
$399$	13.2421	0.662933
$400$	0	0
$401$	−21.1878	−1.05807	−0.529035	−	0.848600i	$-0.677446\pi$
$401$	−21.1878	−1.05807	−0.529035	+	0.848600i	$0.677446\pi$
$402$	0	0
$403$	8.15795	0.406376
$404$	0	0
$405$	54.6042	2.71330
$406$	0	0
$407$	0.260769	0.0129258
$408$	0	0
$409$	−5.14858	−0.254581	−0.127290	−	0.991865i	$-0.540628\pi$
$409$	−5.14858	−0.254581	−0.127290	+	0.991865i	$0.540628\pi$
$410$	0	0
$411$	−13.0333	−0.642886
$412$	0	0
$413$	39.8741	1.96207
$414$	0	0
$415$	30.6366	1.50389
$416$	0	0
$417$	−54.8822	−2.68759
$418$	0	0
$419$	−14.0355	−0.685680	−0.342840	−	0.939394i	$-0.611389\pi$
$419$	−14.0355	−0.685680	−0.342840	+	0.939394i	$0.611389\pi$
$420$	0	0
$421$	32.7713	1.59718	0.798588	−	0.601878i	$-0.205580\pi$
$421$	32.7713	1.59718	0.798588	+	0.601878i	$0.205580\pi$
$422$	0	0
$423$	24.3608	1.18446
$424$	0	0
$425$	−4.71772	−0.228843
$426$	0	0
$427$	−15.9790	−0.773280
$428$	0	0
$429$	0.869284	0.0419694
$430$	0	0
$431$	24.8404	1.19652	0.598259	−	0.801303i	$-0.295859\pi$
$431$	24.8404	1.19652	0.598259	+	0.801303i	$0.295859\pi$
$432$	0	0
$433$	39.9264	1.91874	0.959370	−	0.282153i	$-0.0910484\pi$
$433$	39.9264	1.91874	0.959370	+	0.282153i	$0.0910484\pi$
$434$	0	0
$435$	−0.647365	−0.0310388
$436$	0	0
$437$	−2.99494	−0.143267
$438$	0	0
$439$	−11.8010	−0.563233	−0.281617	−	0.959527i	$-0.590871\pi$
$439$	−11.8010	−0.563233	−0.281617	+	0.959527i	$0.590871\pi$
$440$	0	0
$441$	74.0618	3.52675
$442$	0	0
$443$	−1.31388	−0.0624241	−0.0312121	−	0.999513i	$-0.509937\pi$
$443$	−1.31388	−0.0624241	−0.0312121	+	0.999513i	$0.509937\pi$
$444$	0	0
$445$	−23.5988	−1.11869
$446$	0	0
$447$	23.7813	1.12482
$448$	0	0
$449$	28.0980	1.32603	0.663014	−	0.748607i	$-0.269277\pi$
$449$	28.0980	1.32603	0.663014	+	0.748607i	$0.269277\pi$
$450$	0	0
$451$	0.527584	0.0248430
$452$	0	0
$453$	62.1534	2.92022
$454$	0	0
$455$	−10.4965	−0.492082
$456$	0	0
$457$	28.5839	1.33710	0.668548	−	0.743669i	$-0.266916\pi$
$457$	28.5839	1.33710	0.668548	+	0.743669i	$0.266916\pi$
$458$	0	0
$459$	−10.3910	−0.485012
$460$	0	0
$461$	25.3851	1.18230	0.591152	−	0.806560i	$-0.298673\pi$
$461$	25.3851	1.18230	0.591152	+	0.806560i	$0.298673\pi$
$462$	0	0
$463$	41.5765	1.93222	0.966111	−	0.258126i	$-0.0831050\pi$
$463$	41.5765	1.93222	0.966111	+	0.258126i	$0.0831050\pi$
$464$	0	0
$465$	107.178	4.97026
$466$	0	0
$467$	−21.0326	−0.973273	−0.486637	−	0.873604i	$-0.661776\pi$
$467$	−21.0326	−0.973273	−0.486637	+	0.873604i	$0.661776\pi$
$468$	0	0
$469$	−55.5354	−2.56439
$470$	0	0
$471$	−16.3800	−0.754749
$472$	0	0
$473$	0.341703	0.0157115
$474$	0	0
$475$	−5.41386	−0.248405
$476$	0	0
$477$	−6.80761	−0.311699
$478$	0	0
$479$	−27.8659	−1.27323	−0.636614	−	0.771183i	$-0.719665\pi$
$479$	−27.8659	−1.27323	−0.636614	+	0.771183i	$0.719665\pi$
$480$	0	0
$481$	0.555906	0.0253471
$482$	0	0
$483$	−39.6593	−1.80456
$484$	0	0
$485$	−59.0678	−2.68213
$486$	0	0
$487$	15.9462	0.722590	0.361295	−	0.932451i	$-0.382335\pi$
$487$	15.9462	0.722590	0.361295	+	0.932451i	$0.382335\pi$
$488$	0	0
$489$	−8.94974	−0.404721
$490$	0	0
$491$	34.1127	1.53948	0.769742	−	0.638355i	$-0.220385\pi$
$491$	34.1127	1.53948	0.769742	+	0.638355i	$0.220385\pi$
$492$	0	0
$493$	0.0558196	0.00251399
$494$	0	0
$495$	7.92717	0.356300
$496$	0	0
$497$	−52.3900	−2.35001
$498$	0	0
$499$	−10.2032	−0.456756	−0.228378	−	0.973573i	$-0.573342\pi$
$499$	−10.2032	−0.456756	−0.228378	+	0.973573i	$0.573342\pi$
$500$	0	0
$501$	6.29259	0.281132
$502$	0	0
$503$	4.90289	0.218609	0.109305	−	0.994008i	$-0.465138\pi$
$503$	4.90289	0.218609	0.109305	+	0.994008i	$0.465138\pi$
$504$	0	0
$505$	−40.7971	−1.81545
$506$	0	0
$507$	−38.8591	−1.72579
$508$	0	0
$509$	14.2904	0.633409	0.316704	−	0.948524i	$-0.397424\pi$
$509$	14.2904	0.633409	0.316704	+	0.948524i	$0.397424\pi$
$510$	0	0
$511$	28.5250	1.26187
$512$	0	0
$513$	−11.9243	−0.526473
$514$	0	0
$515$	1.87443	0.0825974
$516$	0	0
$517$	1.29126	0.0567896
$518$	0	0
$519$	33.1587	1.45550
$520$	0	0
$521$	−8.73978	−0.382897	−0.191448	−	0.981503i	$-0.561318\pi$
$521$	−8.73978	−0.382897	−0.191448	+	0.981503i	$0.561318\pi$
$522$	0	0
$523$	9.16263	0.400654	0.200327	−	0.979729i	$-0.435800\pi$
$523$	9.16263	0.400654	0.200327	+	0.979729i	$0.435800\pi$
$524$	0	0
$525$	−71.6908	−3.12884
$526$	0	0
$527$	−9.24150	−0.402566
$528$	0	0
$529$	−14.0303	−0.610014
$530$	0	0
$531$	−64.1964	−2.78589
$532$	0	0
$533$	1.12470	0.0487162
$534$	0	0
$535$	30.7030	1.32741
$536$	0	0
$537$	26.6713	1.15095
$538$	0	0
$539$	3.92569	0.169092
$540$	0	0
$541$	−43.1657	−1.85584	−0.927919	−	0.372781i	$-0.878404\pi$
$541$	−43.1657	−1.85584	−0.927919	+	0.372781i	$0.878404\pi$
$542$	0	0
$543$	−11.0234	−0.473058
$544$	0	0
$545$	−46.2127	−1.97953
$546$	0	0
$547$	17.1047	0.731344	0.365672	−	0.930744i	$-0.380839\pi$
$547$	17.1047	0.731344	0.365672	+	0.930744i	$0.380839\pi$
$548$	0	0
$549$	25.7259	1.09796
$550$	0	0
$551$	0.0640563	0.00272889
$552$	0	0
$553$	16.8067	0.714694
$554$	0	0
$555$	7.30342	0.310013
$556$	0	0
$557$	28.3186	1.19990	0.599948	−	0.800039i	$-0.295188\pi$
$557$	28.3186	1.19990	0.599948	+	0.800039i	$0.295188\pi$
$558$	0	0
$559$	0.728440	0.0308097
$560$	0	0
$561$	−0.984744	−0.0415759
$562$	0	0
$563$	15.2138	0.641185	0.320593	−	0.947217i	$-0.396118\pi$
$563$	15.2138	0.641185	0.320593	+	0.947217i	$0.396118\pi$
$564$	0	0
$565$	−0.383504	−0.0161341
$566$	0	0
$567$	−71.5475	−3.00471
$568$	0	0
$569$	21.4804	0.900505	0.450252	−	0.892901i	$-0.351334\pi$
$569$	21.4804	0.900505	0.450252	+	0.892901i	$0.351334\pi$
$570$	0	0
$571$	−18.2461	−0.763575	−0.381788	−	0.924250i	$-0.624691\pi$
$571$	−18.2461	−0.763575	−0.381788	+	0.924250i	$0.624691\pi$
$572$	0	0
$573$	−19.4334	−0.811842
$574$	0	0
$575$	16.2142	0.676179
$576$	0	0
$577$	−8.89770	−0.370416	−0.185208	−	0.982699i	$-0.559296\pi$
$577$	−8.89770	−0.370416	−0.185208	+	0.982699i	$0.559296\pi$
$578$	0	0
$579$	−49.7648	−2.06816
$580$	0	0
$581$	−40.1430	−1.66541
$582$	0	0
$583$	−0.360842	−0.0149446
$584$	0	0
$585$	16.8991	0.698692
$586$	0	0
$587$	−19.1363	−0.789841	−0.394920	−	0.918715i	$-0.629228\pi$
$587$	−19.1363	−0.789841	−0.394920	+	0.918715i	$0.629228\pi$
$588$	0	0
$589$	−10.6052	−0.436979
$590$	0	0
$591$	21.3381	0.877732
$592$	0	0
$593$	19.4971	0.800650	0.400325	−	0.916373i	$-0.368897\pi$
$593$	19.4971	0.800650	0.400325	+	0.916373i	$0.368897\pi$
$594$	0	0
$595$	11.8906	0.487469
$596$	0	0
$597$	36.6219	1.49884
$598$	0	0
$599$	−41.7147	−1.70442	−0.852208	−	0.523203i	$-0.824737\pi$
$599$	−41.7147	−1.70442	−0.852208	+	0.523203i	$0.824737\pi$
$600$	0	0
$601$	−42.1059	−1.71753	−0.858767	−	0.512367i	$-0.828769\pi$
$601$	−42.1059	−1.71753	−0.858767	+	0.512367i	$0.828769\pi$
$602$	0	0
$603$	89.4109	3.64109
$604$	0	0
$605$	−35.0774	−1.42610
$606$	0	0
$607$	14.8627	0.603260	0.301630	−	0.953425i	$-0.402469\pi$
$607$	14.8627	0.603260	0.301630	+	0.953425i	$0.402469\pi$
$608$	0	0
$609$	0.848239	0.0343724
$610$	0	0
$611$	2.75270	0.111362
$612$	0	0
$613$	35.8195	1.44674	0.723369	−	0.690462i	$-0.242593\pi$
$613$	35.8195	1.44674	0.723369	+	0.690462i	$0.242593\pi$
$614$	0	0
$615$	14.7762	0.595833
$616$	0	0
$617$	36.5574	1.47175	0.735873	−	0.677119i	$-0.236772\pi$
$617$	36.5574	1.47175	0.735873	+	0.677119i	$0.236772\pi$
$618$	0	0
$619$	12.2603	0.492784	0.246392	−	0.969170i	$-0.420755\pi$
$619$	12.2603	0.492784	0.246392	+	0.969170i	$0.420755\pi$
$620$	0	0
$621$	35.7127	1.43310
$622$	0	0
$623$	30.9213	1.23884
$624$	0	0
$625$	−22.7594	−0.910376
$626$	0	0
$627$	−1.13005	−0.0451300
$628$	0	0
$629$	−0.629743	−0.0251095
$630$	0	0
$631$	−14.4075	−0.573554	−0.286777	−	0.957997i	$-0.592584\pi$
$631$	−14.4075	−0.573554	−0.286777	+	0.957997i	$0.592584\pi$
$632$	0	0
$633$	−45.5809	−1.81168
$634$	0	0
$635$	−23.7374	−0.941992
$636$	0	0
$637$	8.36878	0.331583
$638$	0	0
$639$	84.3467	3.33671
$640$	0	0
$641$	43.2404	1.70789	0.853946	−	0.520362i	$-0.174203\pi$
$641$	43.2404	1.70789	0.853946	+	0.520362i	$0.174203\pi$
$642$	0	0
$643$	−24.2316	−0.955599	−0.477800	−	0.878469i	$-0.658565\pi$
$643$	−24.2316	−0.955599	−0.477800	+	0.878469i	$0.658565\pi$
$644$	0	0
$645$	9.57015	0.376824
$646$	0	0
$647$	10.6256	0.417736	0.208868	−	0.977944i	$-0.433022\pi$
$647$	10.6256	0.417736	0.208868	+	0.977944i	$0.433022\pi$
$648$	0	0
$649$	−3.40277	−0.133571
$650$	0	0
$651$	−140.435	−5.50407
$652$	0	0
$653$	23.8274	0.932437	0.466218	−	0.884670i	$-0.345616\pi$
$653$	23.8274	0.932437	0.466218	+	0.884670i	$0.345616\pi$
$654$	0	0
$655$	44.2131	1.72755
$656$	0	0
$657$	−45.9246	−1.79169
$658$	0	0
$659$	46.5363	1.81280	0.906399	−	0.422424i	$-0.138821\pi$
$659$	46.5363	1.81280	0.906399	+	0.422424i	$0.138821\pi$
$660$	0	0
$661$	27.8430	1.08297	0.541484	−	0.840711i	$-0.317863\pi$
$661$	27.8430	1.08297	0.541484	+	0.840711i	$0.317863\pi$
$662$	0	0
$663$	−2.09927	−0.0815291
$664$	0	0
$665$	13.6452	0.529139
$666$	0	0
$667$	−0.191845	−0.00742826
$668$	0	0
$669$	17.0372	0.658694
$670$	0	0
$671$	1.36362	0.0526420
$672$	0	0
$673$	23.4679	0.904622	0.452311	−	0.891860i	$-0.350600\pi$
$673$	23.4679	0.904622	0.452311	+	0.891860i	$0.350600\pi$
$674$	0	0
$675$	64.5568	2.48479
$676$	0	0
$677$	−23.6461	−0.908795	−0.454398	−	0.890799i	$-0.650145\pi$
$677$	−23.6461	−0.908795	−0.454398	+	0.890799i	$0.650145\pi$
$678$	0	0
$679$	77.3962	2.97019
$680$	0	0
$681$	−27.0846	−1.03788
$682$	0	0
$683$	−40.9696	−1.56766	−0.783829	−	0.620977i	$-0.786736\pi$
$683$	−40.9696	−1.56766	−0.783829	+	0.620977i	$0.786736\pi$
$684$	0	0
$685$	−13.4301	−0.513138
$686$	0	0
$687$	−54.4874	−2.07883
$688$	0	0
$689$	−0.769242	−0.0293058
$690$	0	0
$691$	21.9362	0.834492	0.417246	−	0.908794i	$-0.362995\pi$
$691$	21.9362	0.834492	0.417246	+	0.908794i	$0.362995\pi$
$692$	0	0
$693$	−10.3869	−0.394567
$694$	0	0
$695$	−56.5530	−2.14518
$696$	0	0
$697$	−1.27409	−0.0482595
$698$	0	0
$699$	−45.5719	−1.72369
$700$	0	0
$701$	−34.9975	−1.32184	−0.660919	−	0.750457i	$-0.729833\pi$
$701$	−34.9975	−1.32184	−0.660919	+	0.750457i	$0.729833\pi$
$702$	0	0
$703$	−0.722667	−0.0272559
$704$	0	0
$705$	36.1646	1.36204
$706$	0	0
$707$	53.4562	2.01043
$708$	0	0
$709$	33.1216	1.24391	0.621955	−	0.783053i	$-0.286339\pi$
$709$	33.1216	1.24391	0.621955	+	0.783053i	$0.286339\pi$
$710$	0	0
$711$	−27.0585	−1.01477
$712$	0	0
$713$	31.7619	1.18949
$714$	0	0
$715$	0.895749	0.0334991
$716$	0	0
$717$	−59.9047	−2.23718
$718$	0	0
$719$	−45.8060	−1.70828	−0.854139	−	0.520045i	$-0.825915\pi$
$719$	−45.8060	−1.70828	−0.854139	+	0.520045i	$0.825915\pi$
$720$	0	0
$721$	−2.45606	−0.0914684
$722$	0	0
$723$	27.9279	1.03865
$724$	0	0
$725$	−0.346792	−0.0128795
$726$	0	0
$727$	−11.4064	−0.423040	−0.211520	−	0.977374i	$-0.567841\pi$
$727$	−11.4064	−0.423040	−0.211520	+	0.977374i	$0.567841\pi$
$728$	0	0
$729$	3.15922	0.117008
$730$	0	0
$731$	−0.825193	−0.0305209
$732$	0	0
$733$	−14.9016	−0.550404	−0.275202	−	0.961386i	$-0.588745\pi$
$733$	−14.9016	−0.550404	−0.275202	+	0.961386i	$0.588745\pi$
$734$	0	0
$735$	109.948	4.05549
$736$	0	0
$737$	4.73928	0.174574
$738$	0	0
$739$	3.12370	0.114907	0.0574536	−	0.998348i	$-0.481702\pi$
$739$	3.12370	0.114907	0.0574536	+	0.998348i	$0.481702\pi$
$740$	0	0
$741$	−2.40904	−0.0884984
$742$	0	0
$743$	−8.62910	−0.316571	−0.158285	−	0.987393i	$-0.550597\pi$
$743$	−8.62910	−0.316571	−0.158285	+	0.987393i	$0.550597\pi$
$744$	0	0
$745$	24.5053	0.897805
$746$	0	0
$747$	64.6293	2.36466
$748$	0	0
$749$	−40.2299	−1.46997
$750$	0	0
$751$	−15.2777	−0.557490	−0.278745	−	0.960365i	$-0.589918\pi$
$751$	−15.2777	−0.557490	−0.278745	+	0.960365i	$0.589918\pi$
$752$	0	0
$753$	39.7329	1.44795
$754$	0	0
$755$	64.0456	2.33086
$756$	0	0
$757$	14.5188	0.527696	0.263848	−	0.964564i	$-0.415008\pi$
$757$	14.5188	0.527696	0.263848	+	0.964564i	$0.415008\pi$
$758$	0	0
$759$	3.38444	0.122848
$760$	0	0
$761$	35.9031	1.30148	0.650742	−	0.759299i	$-0.274458\pi$
$761$	35.9031	1.30148	0.650742	+	0.759299i	$0.274458\pi$
$762$	0	0
$763$	60.5522	2.19214
$764$	0	0
$765$	−19.1437	−0.692141
$766$	0	0
$767$	−7.25402	−0.261927
$768$	0	0
$769$	6.05103	0.218206	0.109103	−	0.994030i	$-0.465202\pi$
$769$	6.05103	0.218206	0.109103	+	0.994030i	$0.465202\pi$
$770$	0	0
$771$	−64.4729	−2.32194
$772$	0	0
$773$	33.1814	1.19345	0.596726	−	0.802445i	$-0.296468\pi$
$773$	33.1814	1.19345	0.596726	+	0.802445i	$0.296468\pi$
$774$	0	0
$775$	57.4150	2.06241
$776$	0	0
$777$	−9.56962	−0.343308
$778$	0	0
$779$	−1.46209	−0.0523848
$780$	0	0
$781$	4.47086	0.159980
$782$	0	0
$783$	−0.763830	−0.0272970
$784$	0	0
$785$	−16.8786	−0.602424
$786$	0	0
$787$	−30.7600	−1.09648	−0.548238	−	0.836322i	$-0.684701\pi$
$787$	−30.7600	−1.09648	−0.548238	+	0.836322i	$0.684701\pi$
$788$	0	0
$789$	2.21799	0.0789624
$790$	0	0
$791$	0.502503	0.0178669
$792$	0	0
$793$	2.90696	0.103229
$794$	0	0
$795$	−10.1062	−0.358430
$796$	0	0
$797$	−38.0718	−1.34857	−0.674286	−	0.738470i	$-0.735549\pi$
$797$	−38.0718	−1.34857	−0.674286	+	0.738470i	$0.735549\pi$
$798$	0	0
$799$	−3.11832	−0.110318
$800$	0	0
$801$	−49.7827	−1.75898
$802$	0	0
$803$	−2.43426	−0.0859033
$804$	0	0
$805$	−40.8666	−1.44036
$806$	0	0
$807$	−34.8876	−1.22810
$808$	0	0
$809$	24.6594	0.866978	0.433489	−	0.901159i	$-0.357282\pi$
$809$	24.6594	0.866978	0.433489	+	0.901159i	$0.357282\pi$
$810$	0	0
$811$	−31.2710	−1.09807	−0.549036	−	0.835799i	$-0.685005\pi$
$811$	−31.2710	−1.09807	−0.549036	+	0.835799i	$0.685005\pi$
$812$	0	0
$813$	−49.7894	−1.74619
$814$	0	0
$815$	−9.22221	−0.323040
$816$	0	0
$817$	−0.946959	−0.0331299
$818$	0	0
$819$	−22.1428	−0.773732
$820$	0	0
$821$	−10.9884	−0.383497	−0.191748	−	0.981444i	$-0.561416\pi$
$821$	−10.9884	−0.383497	−0.191748	+	0.981444i	$0.561416\pi$
$822$	0	0
$823$	27.7397	0.966945	0.483473	−	0.875360i	$-0.339375\pi$
$823$	27.7397	0.966945	0.483473	+	0.875360i	$0.339375\pi$
$824$	0	0
$825$	6.11795	0.213000
$826$	0	0
$827$	−14.3020	−0.497330	−0.248665	−	0.968590i	$-0.579992\pi$
$827$	−14.3020	−0.497330	−0.248665	+	0.968590i	$0.579992\pi$
$828$	0	0
$829$	−33.8878	−1.17697	−0.588485	−	0.808508i	$-0.700276\pi$
$829$	−33.8878	−1.17697	−0.588485	+	0.808508i	$0.700276\pi$
$830$	0	0
$831$	37.5064	1.30108
$832$	0	0
$833$	−9.48034	−0.328474
$834$	0	0
$835$	6.48416	0.224394
$836$	0	0
$837$	126.460	4.37109
$838$	0	0
$839$	−17.8896	−0.617619	−0.308809	−	0.951124i	$-0.599931\pi$
$839$	−17.8896	−0.617619	−0.308809	+	0.951124i	$0.599931\pi$
$840$	0	0
$841$	−28.9959	−0.999859
$842$	0	0
$843$	44.8614	1.54511
$844$	0	0
$845$	−40.0421	−1.37749
$846$	0	0
$847$	45.9617	1.57926
$848$	0	0
$849$	−26.1280	−0.896710
$850$	0	0
$851$	2.16435	0.0741928
$852$	0	0
$853$	−47.7153	−1.63374	−0.816870	−	0.576822i	$-0.804293\pi$
$853$	−47.7153	−1.63374	−0.816870	+	0.576822i	$0.804293\pi$
$854$	0	0
$855$	−21.9685	−0.751308
$856$	0	0
$857$	43.2058	1.47588	0.737940	−	0.674866i	$-0.235799\pi$
$857$	43.2058	1.47588	0.737940	+	0.674866i	$0.235799\pi$
$858$	0	0
$859$	−52.4451	−1.78940	−0.894701	−	0.446665i	$-0.852612\pi$
$859$	−52.4451	−1.78940	−0.894701	+	0.446665i	$0.852612\pi$
$860$	0	0
$861$	−19.3611	−0.659826
$862$	0	0
$863$	−19.4690	−0.662733	−0.331367	−	0.943502i	$-0.607510\pi$
$863$	−19.4690	−0.662733	−0.331367	+	0.943502i	$0.607510\pi$
$864$	0	0
$865$	34.1681	1.16175
$866$	0	0
$867$	−50.8610	−1.72733
$868$	0	0
$869$	−1.43425	−0.0486537
$870$	0	0
$871$	10.1032	0.342333
$872$	0	0
$873$	−124.606	−4.21728
$874$	0	0
$875$	−5.64725	−0.190912
$876$	0	0
$877$	16.2266	0.547932	0.273966	−	0.961739i	$-0.411664\pi$
$877$	16.2266	0.547932	0.273966	+	0.961739i	$0.411664\pi$
$878$	0	0
$879$	−30.7017	−1.03554
$880$	0	0
$881$	−15.6336	−0.526710	−0.263355	−	0.964699i	$-0.584829\pi$
$881$	−15.6336	−0.526710	−0.263355	+	0.964699i	$0.584829\pi$
$882$	0	0
$883$	−6.73490	−0.226647	−0.113324	−	0.993558i	$-0.536150\pi$
$883$	−6.73490	−0.226647	−0.113324	+	0.993558i	$0.536150\pi$
$884$	0	0
$885$	−95.3023	−3.20355
$886$	0	0
$887$	18.3074	0.614701	0.307350	−	0.951596i	$-0.400558\pi$
$887$	18.3074	0.614701	0.307350	+	0.951596i	$0.400558\pi$
$888$	0	0
$889$	31.1030	1.04316
$890$	0	0
$891$	6.10573	0.204550
$892$	0	0
$893$	−3.57846	−0.119749
$894$	0	0
$895$	27.4833	0.918665
$896$	0	0
$897$	7.21494	0.240900
$898$	0	0
$899$	−0.679328	−0.0226569
$900$	0	0
$901$	0.871414	0.0290310
$902$	0	0
$903$	−12.5397	−0.417295
$904$	0	0
$905$	−11.3590	−0.377585
$906$	0	0
$907$	26.1716	0.869013	0.434507	−	0.900669i	$-0.356923\pi$
$907$	26.1716	0.869013	0.434507	+	0.900669i	$0.356923\pi$
$908$	0	0
$909$	−86.0633	−2.85454
$910$	0	0
$911$	8.41152	0.278686	0.139343	−	0.990244i	$-0.455501\pi$
$911$	8.41152	0.278686	0.139343	+	0.990244i	$0.455501\pi$
$912$	0	0
$913$	3.42572	0.113375
$914$	0	0
$915$	38.1912	1.26256
$916$	0	0
$917$	−57.9322	−1.91309
$918$	0	0
$919$	−12.1184	−0.399748	−0.199874	−	0.979822i	$-0.564053\pi$
$919$	−12.1184	−0.399748	−0.199874	+	0.979822i	$0.564053\pi$
$920$	0	0
$921$	−17.6622	−0.581988
$922$	0	0
$923$	9.53095	0.313715
$924$	0	0
$925$	3.91242	0.128640
$926$	0	0
$927$	3.95420	0.129873
$928$	0	0
$929$	−50.2497	−1.64864	−0.824320	−	0.566124i	$-0.808442\pi$
$929$	−50.2497	−1.64864	−0.824320	+	0.566124i	$0.808442\pi$
$930$	0	0
$931$	−10.8793	−0.356553
$932$	0	0
$933$	−77.8341	−2.54817
$934$	0	0
$935$	−1.01472	−0.0331850
$936$	0	0
$937$	−36.3462	−1.18738	−0.593689	−	0.804695i	$-0.702329\pi$
$937$	−36.3462	−1.18738	−0.593689	+	0.804695i	$0.702329\pi$
$938$	0	0
$939$	15.1074	0.493012
$940$	0	0
$941$	22.1795	0.723030	0.361515	−	0.932366i	$-0.382260\pi$
$941$	22.1795	0.723030	0.361515	+	0.932366i	$0.382260\pi$
$942$	0	0
$943$	4.37888	0.142596
$944$	0	0
$945$	−162.710	−5.29297
$946$	0	0
$947$	45.9683	1.49377	0.746885	−	0.664953i	$-0.231548\pi$
$947$	45.9683	1.49377	0.746885	+	0.664953i	$0.231548\pi$
$948$	0	0
$949$	−5.18936	−0.168454
$950$	0	0
$951$	−77.3329	−2.50769
$952$	0	0
$953$	−15.1162	−0.489663	−0.244832	−	0.969566i	$-0.578733\pi$
$953$	−15.1162	−0.489663	−0.244832	+	0.969566i	$0.578733\pi$
$954$	0	0
$955$	−20.0250	−0.647995
$956$	0	0
$957$	−0.0723871	−0.00233994
$958$	0	0
$959$	17.5974	0.568249
$960$	0	0
$961$	81.4698	2.62806
$962$	0	0
$963$	64.7693	2.08716
$964$	0	0
$965$	−51.2798	−1.65076
$966$	0	0
$967$	23.7091	0.762433	0.381217	−	0.924486i	$-0.375505\pi$
$967$	23.7091	0.762433	0.381217	+	0.924486i	$0.375505\pi$
$968$	0	0
$969$	2.72902	0.0876687
$970$	0	0
$971$	13.9339	0.447161	0.223581	−	0.974685i	$-0.428225\pi$
$971$	13.9339	0.447161	0.223581	+	0.974685i	$0.428225\pi$
$972$	0	0
$973$	74.1010	2.37557
$974$	0	0
$975$	13.0422	0.417686
$976$	0	0
$977$	−6.72120	−0.215030	−0.107515	−	0.994203i	$-0.534289\pi$
$977$	−6.72120	−0.215030	−0.107515	+	0.994203i	$0.534289\pi$
$978$	0	0
$979$	−2.63876	−0.0843353
$980$	0	0
$981$	−97.4878	−3.11255
$982$	0	0
$983$	2.85133	0.0909433	0.0454716	−	0.998966i	$-0.485521\pi$
$983$	2.85133	0.0909433	0.0454716	+	0.998966i	$0.485521\pi$
$984$	0	0
$985$	21.9877	0.700587
$986$	0	0
$987$	−47.3863	−1.50832
$988$	0	0
$989$	2.83609	0.0901823
$990$	0	0
$991$	14.0776	0.447189	0.223595	−	0.974682i	$-0.428221\pi$
$991$	14.0776	0.447189	0.223595	+	0.974682i	$0.428221\pi$
$992$	0	0
$993$	−110.375	−3.50264
$994$	0	0
$995$	37.7368	1.19634
$996$	0	0
$997$	38.1112	1.20699	0.603496	−	0.797366i	$-0.293774\pi$
$997$	38.1112	1.20699	0.603496	+	0.797366i	$0.293774\pi$
$998$	0	0
$999$	8.61734	0.272640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.f.1.18	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.f.1.18	✓	19	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q+3.13171 q^{3} +3.22705 q^{5} -4.22839 q^{7} +6.80761 q^{9} +O(q^{10})\)	\(q+3.13171 q^{3} +3.22705 q^{5} -4.22839 q^{7} +6.80761 q^{9} +0.360842 q^{11} +0.769242 q^{13} +10.1062 q^{15} -0.871414 q^{17} -1.00000 q^{19} -13.2421 q^{21} +2.99494 q^{23} +5.41386 q^{25} +11.9243 q^{27} -0.0640563 q^{29} +10.6052 q^{31} +1.13005 q^{33} -13.6452 q^{35} +0.722667 q^{37} +2.40904 q^{39} +1.46209 q^{41} +0.946959 q^{43} +21.9685 q^{45} +3.57846 q^{47} +10.8793 q^{49} -2.72902 q^{51} -1.00000 q^{53} +1.16446 q^{55} -3.13171 q^{57} -9.43009 q^{59} +3.77899 q^{61} -28.7852 q^{63} +2.48238 q^{65} +13.1340 q^{67} +9.37929 q^{69} +12.3901 q^{71} -6.74607 q^{73} +16.9547 q^{75} -1.52578 q^{77} -3.97473 q^{79} +16.9208 q^{81} +9.49368 q^{83} -2.81210 q^{85} -0.200606 q^{87} -7.31280 q^{89} -3.25265 q^{91} +33.2124 q^{93} -3.22705 q^{95} -18.3040 q^{97} +2.45647 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) \) 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9 + q^11 - q^13 + 8 * q^15 + 3 * q^17 - 19 * q^19 + 8 * q^21 + 10 * q^23 + 21 * q^25 + 28 * q^27 + 2 * q^29 + 25 * q^31 + q^33 + 20 * q^35 + 19 * q^37 + 37 * q^39 - 9 * q^41 + 35 * q^43 + 37 * q^45 + 23 * q^47 + 30 * q^49 + 34 * q^51 - 19 * q^53 + 40 * q^55 - 4 * q^57 + 16 * q^59 + 21 * q^61 + 3 * q^63 - 10 * q^65 + 67 * q^67 + 23 * q^69 + 18 * q^71 - 20 * q^73 + 33 * q^75 + 37 * q^77 + 2 * q^79 + 23 * q^81 + 38 * q^83 + 8 * q^85 + 18 * q^87 - q^89 - 9 * q^91 + 14 * q^93 - 4 * q^95 - 21 * q^97 + 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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