LMFDB - Embedded newform 4028.2.a.e.1.4

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} - 3 x^{18} - 35 x^{17} + 103 x^{16} + 501 x^{15} - 1437 x^{14} - 3775 x^{13} + 10450 x^{12} + \cdots - 4 $ x^19 - 3x^18 - 35x^17 + 103x^16 + 501x^15 - 1437x^14 - 3775x^13 + 10450x^12 + 16076x^11 - 42255x^10 - 38701x^9 + 93907x^8 + 49522x^7 - 106284x^6 - 26703x^5 + 50522x^4 - 446x^3 - 6265x^2 + 719x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.37182$ 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)

$f(q)$	$=$	$q-2.37182 q^{3} +1.99718 q^{5} +3.57468 q^{7} +2.62553 q^{9} +O(q^{10})$	$q-2.37182 q^{3} +1.99718 q^{5} +3.57468 q^{7} +2.62553 q^{9} -4.27104 q^{11} +1.29162 q^{13} -4.73696 q^{15} -6.52056 q^{17} +1.00000 q^{19} -8.47851 q^{21} +3.63134 q^{23} -1.01126 q^{25} +0.888175 q^{27} -0.456081 q^{29} +1.90460 q^{31} +10.1301 q^{33} +7.13930 q^{35} +5.50495 q^{37} -3.06350 q^{39} -3.36509 q^{41} +6.83432 q^{43} +5.24366 q^{45} +6.24765 q^{47} +5.77836 q^{49} +15.4656 q^{51} +1.00000 q^{53} -8.53006 q^{55} -2.37182 q^{57} +7.27476 q^{59} +5.03294 q^{61} +9.38544 q^{63} +2.57961 q^{65} -9.61867 q^{67} -8.61287 q^{69} -2.70160 q^{71} +1.18849 q^{73} +2.39853 q^{75} -15.2676 q^{77} +15.5737 q^{79} -9.98318 q^{81} -15.5143 q^{83} -13.0227 q^{85} +1.08174 q^{87} +8.27024 q^{89} +4.61714 q^{91} -4.51737 q^{93} +1.99718 q^{95} -18.1377 q^{97} -11.2138 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * 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
$3$	−2.37182	−1.36937	−0.684685	−	0.728839i	$-0.740060\pi$
$3$	−2.37182	−1.36937	−0.684685	+	0.728839i	$0.740060\pi$
$4$	0	0
$5$	1.99718	0.893167	0.446584	−	0.894742i	$-0.352640\pi$
$5$	1.99718	0.893167	0.446584	+	0.894742i	$0.352640\pi$
$6$	0	0
$7$	3.57468	1.35110	0.675552	−	0.737313i	$-0.263906\pi$
$7$	3.57468	1.35110	0.675552	+	0.737313i	$0.263906\pi$
$8$	0	0
$9$	2.62553	0.875177
$10$	0	0
$11$	−4.27104	−1.28777	−0.643884	−	0.765123i	$-0.722678\pi$
$11$	−4.27104	−1.28777	−0.643884	+	0.765123i	$0.722678\pi$
$12$	0	0
$13$	1.29162	0.358232	0.179116	−	0.983828i	$-0.442676\pi$
$13$	1.29162	0.358232	0.179116	+	0.983828i	$0.442676\pi$
$14$	0	0
$15$	−4.73696	−1.22308
$16$	0	0
$17$	−6.52056	−1.58147	−0.790734	−	0.612160i	$-0.790301\pi$
$17$	−6.52056	−1.58147	−0.790734	+	0.612160i	$0.790301\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−8.47851	−1.85016
$22$	0	0
$23$	3.63134	0.757186	0.378593	−	0.925563i	$-0.376408\pi$
$23$	3.63134	0.757186	0.378593	+	0.925563i	$0.376408\pi$
$24$	0	0
$25$	−1.01126	−0.202252
$26$	0	0
$27$	0.888175	0.170929
$28$	0	0
$29$	−0.456081	−0.0846921	−0.0423461	−	0.999103i	$-0.513483\pi$
$29$	−0.456081	−0.0846921	−0.0423461	+	0.999103i	$0.513483\pi$
$30$	0	0
$31$	1.90460	0.342076	0.171038	−	0.985264i	$-0.445288\pi$
$31$	1.90460	0.342076	0.171038	+	0.985264i	$0.445288\pi$
$32$	0	0
$33$	10.1301	1.76343
$34$	0	0
$35$	7.13930	1.20676
$36$	0	0
$37$	5.50495	0.905008	0.452504	−	0.891762i	$-0.350531\pi$
$37$	5.50495	0.905008	0.452504	+	0.891762i	$0.350531\pi$
$38$	0	0
$39$	−3.06350	−0.490552
$40$	0	0
$41$	−3.36509	−0.525539	−0.262770	−	0.964859i	$-0.584636\pi$
$41$	−3.36509	−0.525539	−0.262770	+	0.964859i	$0.584636\pi$
$42$	0	0
$43$	6.83432	1.04222	0.521112	−	0.853488i	$-0.325517\pi$
$43$	6.83432	1.04222	0.521112	+	0.853488i	$0.325517\pi$
$44$	0	0
$45$	5.24366	0.781679
$46$	0	0
$47$	6.24765	0.911313	0.455656	−	0.890156i	$-0.349405\pi$
$47$	6.24765	0.911313	0.455656	+	0.890156i	$0.349405\pi$
$48$	0	0
$49$	5.77836	0.825480
$50$	0	0
$51$	15.4656	2.16562
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−8.53006	−1.15019
$56$	0	0
$57$	−2.37182	−0.314155
$58$	0	0
$59$	7.27476	0.947093	0.473546	−	0.880769i	$-0.342974\pi$
$59$	7.27476	0.947093	0.473546	+	0.880769i	$0.342974\pi$
$60$	0	0
$61$	5.03294	0.644402	0.322201	−	0.946671i	$-0.395577\pi$
$61$	5.03294	0.644402	0.322201	+	0.946671i	$0.395577\pi$
$62$	0	0
$63$	9.38544	1.18245
$64$	0	0
$65$	2.57961	0.319961
$66$	0	0
$67$	−9.61867	−1.17511	−0.587554	−	0.809185i	$-0.699909\pi$
$67$	−9.61867	−1.17511	−0.587554	+	0.809185i	$0.699909\pi$
$68$	0	0
$69$	−8.61287	−1.03687
$70$	0	0
$71$	−2.70160	−0.320621	−0.160310	−	0.987067i	$-0.551250\pi$
$71$	−2.70160	−0.320621	−0.160310	+	0.987067i	$0.551250\pi$
$72$	0	0
$73$	1.18849	0.139103	0.0695513	−	0.997578i	$-0.477843\pi$
$73$	1.18849	0.139103	0.0695513	+	0.997578i	$0.477843\pi$
$74$	0	0
$75$	2.39853	0.276958
$76$	0	0
$77$	−15.2676	−1.73991
$78$	0	0
$79$	15.5737	1.75218	0.876088	−	0.482151i	$-0.160145\pi$
$79$	15.5737	1.75218	0.876088	+	0.482151i	$0.160145\pi$
$80$	0	0
$81$	−9.98318	−1.10924
$82$	0	0
$83$	−15.5143	−1.70292	−0.851460	−	0.524419i	$-0.824282\pi$
$83$	−15.5143	−1.70292	−0.851460	+	0.524419i	$0.824282\pi$
$84$	0	0
$85$	−13.0227	−1.41252
$86$	0	0
$87$	1.08174	0.115975
$88$	0	0
$89$	8.27024	0.876644	0.438322	−	0.898818i	$-0.355573\pi$
$89$	8.27024	0.876644	0.438322	+	0.898818i	$0.355573\pi$
$90$	0	0
$91$	4.61714	0.484008
$92$	0	0
$93$	−4.51737	−0.468429
$94$	0	0
$95$	1.99718	0.204907
$96$	0	0
$97$	−18.1377	−1.84160	−0.920800	−	0.390036i	$-0.872463\pi$
$97$	−18.1377	−1.84160	−0.920800	+	0.390036i	$0.872463\pi$
$98$	0	0
$99$	−11.2138	−1.12702
$100$	0	0
$101$	17.5754	1.74881	0.874407	−	0.485194i	$-0.161251\pi$
$101$	17.5754	1.74881	0.874407	+	0.485194i	$0.161251\pi$
$102$	0	0
$103$	9.89831	0.975309	0.487655	−	0.873037i	$-0.337853\pi$
$103$	9.89831	0.975309	0.487655	+	0.873037i	$0.337853\pi$
$104$	0	0
$105$	−16.9331	−1.65250
$106$	0	0
$107$	−6.83121	−0.660398	−0.330199	−	0.943911i	$-0.607116\pi$
$107$	−6.83121	−0.660398	−0.330199	+	0.943911i	$0.607116\pi$
$108$	0	0
$109$	13.9252	1.33379	0.666894	−	0.745152i	$-0.267623\pi$
$109$	13.9252	1.33379	0.666894	+	0.745152i	$0.267623\pi$
$110$	0	0
$111$	−13.0568	−1.23929
$112$	0	0
$113$	−7.20979	−0.678240	−0.339120	−	0.940743i	$-0.610129\pi$
$113$	−7.20979	−0.678240	−0.339120	+	0.940743i	$0.610129\pi$
$114$	0	0
$115$	7.25244	0.676294
$116$	0	0
$117$	3.39120	0.313516
$118$	0	0
$119$	−23.3089	−2.13673
$120$	0	0
$121$	7.24182	0.658347
$122$	0	0
$123$	7.98140	0.719658
$124$	0	0
$125$	−12.0056	−1.07381
$126$	0	0
$127$	15.3077	1.35834	0.679168	−	0.733983i	$-0.262341\pi$
$127$	15.3077	1.35834	0.679168	+	0.733983i	$0.262341\pi$
$128$	0	0
$129$	−16.2098	−1.42719
$130$	0	0
$131$	21.0113	1.83576	0.917882	−	0.396854i	$-0.129898\pi$
$131$	21.0113	1.83576	0.917882	+	0.396854i	$0.129898\pi$
$132$	0	0
$133$	3.57468	0.309964
$134$	0	0
$135$	1.77385	0.152669
$136$	0	0
$137$	−18.4593	−1.57709	−0.788544	−	0.614979i	$-0.789165\pi$
$137$	−18.4593	−1.57709	−0.788544	+	0.614979i	$0.789165\pi$
$138$	0	0
$139$	1.59092	0.134940	0.0674701	−	0.997721i	$-0.478507\pi$
$139$	1.59092	0.134940	0.0674701	+	0.997721i	$0.478507\pi$
$140$	0	0
$141$	−14.8183	−1.24793
$142$	0	0
$143$	−5.51658	−0.461320
$144$	0	0
$145$	−0.910878	−0.0756443
$146$	0	0
$147$	−13.7052	−1.13039
$148$	0	0
$149$	−8.41803	−0.689632	−0.344816	−	0.938670i	$-0.612059\pi$
$149$	−8.41803	−0.689632	−0.344816	+	0.938670i	$0.612059\pi$
$150$	0	0
$151$	3.48748	0.283807	0.141903	−	0.989880i	$-0.454678\pi$
$151$	3.48748	0.283807	0.141903	+	0.989880i	$0.454678\pi$
$152$	0	0
$153$	−17.1199	−1.38406
$154$	0	0
$155$	3.80383	0.305531
$156$	0	0
$157$	1.12708	0.0899505	0.0449752	−	0.998988i	$-0.485679\pi$
$157$	1.12708	0.0899505	0.0449752	+	0.998988i	$0.485679\pi$
$158$	0	0
$159$	−2.37182	−0.188098
$160$	0	0
$161$	12.9809	1.02304
$162$	0	0
$163$	22.8140	1.78693	0.893466	−	0.449130i	$-0.148266\pi$
$163$	22.8140	1.78693	0.893466	+	0.449130i	$0.148266\pi$
$164$	0	0
$165$	20.2318	1.57504
$166$	0	0
$167$	22.3137	1.72669	0.863344	−	0.504616i	$-0.168366\pi$
$167$	22.3137	1.72669	0.863344	+	0.504616i	$0.168366\pi$
$168$	0	0
$169$	−11.3317	−0.871670
$170$	0	0
$171$	2.62553	0.200779
$172$	0	0
$173$	4.25484	0.323489	0.161745	−	0.986833i	$-0.448288\pi$
$173$	4.25484	0.323489	0.161745	+	0.986833i	$0.448288\pi$
$174$	0	0
$175$	−3.61493	−0.273263
$176$	0	0
$177$	−17.2544	−1.29692
$178$	0	0
$179$	0.252022	0.0188370	0.00941851	−	0.999956i	$-0.497002\pi$
$179$	0.252022	0.0188370	0.00941851	+	0.999956i	$0.497002\pi$
$180$	0	0
$181$	3.54152	0.263239	0.131620	−	0.991300i	$-0.457982\pi$
$181$	3.54152	0.263239	0.131620	+	0.991300i	$0.457982\pi$
$182$	0	0
$183$	−11.9372	−0.882426
$184$	0	0
$185$	10.9944	0.808324
$186$	0	0
$187$	27.8496	2.03656
$188$	0	0
$189$	3.17494	0.230943
$190$	0	0
$191$	−16.0125	−1.15862	−0.579310	−	0.815107i	$-0.696678\pi$
$191$	−16.0125	−1.15862	−0.579310	+	0.815107i	$0.696678\pi$
$192$	0	0
$193$	18.2944	1.31686	0.658430	−	0.752642i	$-0.271221\pi$
$193$	18.2944	1.31686	0.658430	+	0.752642i	$0.271221\pi$
$194$	0	0
$195$	−6.11837	−0.438145
$196$	0	0
$197$	0.838573	0.0597458	0.0298729	−	0.999554i	$-0.490490\pi$
$197$	0.838573	0.0597458	0.0298729	+	0.999554i	$0.490490\pi$
$198$	0	0
$199$	11.7546	0.833265	0.416632	−	0.909075i	$-0.363210\pi$
$199$	11.7546	0.833265	0.416632	+	0.909075i	$0.363210\pi$
$200$	0	0
$201$	22.8138	1.60916
$202$	0	0
$203$	−1.63035	−0.114428
$204$	0	0
$205$	−6.72071	−0.469395
$206$	0	0
$207$	9.53418	0.662671
$208$	0	0
$209$	−4.27104	−0.295434
$210$	0	0
$211$	16.4828	1.13472	0.567361	−	0.823469i	$-0.307964\pi$
$211$	16.4828	1.13472	0.567361	+	0.823469i	$0.307964\pi$
$212$	0	0
$213$	6.40771	0.439049
$214$	0	0
$215$	13.6494	0.930880
$216$	0	0
$217$	6.80834	0.462180
$218$	0	0
$219$	−2.81889	−0.190483
$220$	0	0
$221$	−8.42210	−0.566532
$222$	0	0
$223$	−8.08756	−0.541583	−0.270792	−	0.962638i	$-0.587285\pi$
$223$	−8.08756	−0.541583	−0.270792	+	0.962638i	$0.587285\pi$
$224$	0	0
$225$	−2.65509	−0.177006
$226$	0	0
$227$	13.3452	0.885750	0.442875	−	0.896583i	$-0.353959\pi$
$227$	13.3452	0.885750	0.442875	+	0.896583i	$0.353959\pi$
$228$	0	0
$229$	−12.1395	−0.802203	−0.401102	−	0.916034i	$-0.631373\pi$
$229$	−12.1395	−0.802203	−0.401102	+	0.916034i	$0.631373\pi$
$230$	0	0
$231$	36.2121	2.38258
$232$	0	0
$233$	7.29556	0.477948	0.238974	−	0.971026i	$-0.423189\pi$
$233$	7.29556	0.477948	0.238974	+	0.971026i	$0.423189\pi$
$234$	0	0
$235$	12.4777	0.813955
$236$	0	0
$237$	−36.9380	−2.39938
$238$	0	0
$239$	3.29752	0.213299	0.106650	−	0.994297i	$-0.465988\pi$
$239$	3.29752	0.213299	0.106650	+	0.994297i	$0.465988\pi$
$240$	0	0
$241$	16.1122	1.03788	0.518938	−	0.854812i	$-0.326327\pi$
$241$	16.1122	1.03788	0.518938	+	0.854812i	$0.326327\pi$
$242$	0	0
$243$	21.0138	1.34803
$244$	0	0
$245$	11.5404	0.737292
$246$	0	0
$247$	1.29162	0.0821840
$248$	0	0
$249$	36.7972	2.33193
$250$	0	0
$251$	3.30585	0.208663	0.104332	−	0.994543i	$-0.466730\pi$
$251$	3.30585	0.208663	0.104332	+	0.994543i	$0.466730\pi$
$252$	0	0
$253$	−15.5096	−0.975080
$254$	0	0
$255$	30.8876	1.93426
$256$	0	0
$257$	−11.1980	−0.698512	−0.349256	−	0.937027i	$-0.613566\pi$
$257$	−11.1980	−0.698512	−0.349256	+	0.937027i	$0.613566\pi$
$258$	0	0
$259$	19.6785	1.22276
$260$	0	0
$261$	−1.19745	−0.0741206
$262$	0	0
$263$	18.4695	1.13888	0.569439	−	0.822033i	$-0.307160\pi$
$263$	18.4695	1.13888	0.569439	+	0.822033i	$0.307160\pi$
$264$	0	0
$265$	1.99718	0.122686
$266$	0	0
$267$	−19.6155	−1.20045
$268$	0	0
$269$	−17.2298	−1.05052	−0.525260	−	0.850942i	$-0.676032\pi$
$269$	−17.2298	−1.05052	−0.525260	+	0.850942i	$0.676032\pi$
$270$	0	0
$271$	−29.6573	−1.80155	−0.900776	−	0.434284i	$-0.857001\pi$
$271$	−29.6573	−1.80155	−0.900776	+	0.434284i	$0.857001\pi$
$272$	0	0
$273$	−10.9510	−0.662787
$274$	0	0
$275$	4.31913	0.260454
$276$	0	0
$277$	−13.3382	−0.801413	−0.400706	−	0.916207i	$-0.631235\pi$
$277$	−13.3382	−0.801413	−0.400706	+	0.916207i	$0.631235\pi$
$278$	0	0
$279$	5.00058	0.299377
$280$	0	0
$281$	17.2915	1.03153	0.515763	−	0.856732i	$-0.327509\pi$
$281$	17.2915	1.03153	0.515763	+	0.856732i	$0.327509\pi$
$282$	0	0
$283$	−19.0724	−1.13374	−0.566868	−	0.823809i	$-0.691845\pi$
$283$	−19.0724	−1.13374	−0.566868	+	0.823809i	$0.691845\pi$
$284$	0	0
$285$	−4.73696	−0.280593
$286$	0	0
$287$	−12.0291	−0.710058
$288$	0	0
$289$	25.5177	1.50104
$290$	0	0
$291$	43.0192	2.52183
$292$	0	0
$293$	27.8612	1.62767	0.813834	−	0.581097i	$-0.197376\pi$
$293$	27.8612	1.62767	0.813834	+	0.581097i	$0.197376\pi$
$294$	0	0
$295$	14.5290	0.845912
$296$	0	0
$297$	−3.79343	−0.220117
$298$	0	0
$299$	4.69032	0.271248
$300$	0	0
$301$	24.4305	1.40815
$302$	0	0
$303$	−41.6856	−2.39477
$304$	0	0
$305$	10.0517	0.575559
$306$	0	0
$307$	13.7483	0.784655	0.392328	−	0.919826i	$-0.371670\pi$
$307$	13.7483	0.784655	0.392328	+	0.919826i	$0.371670\pi$
$308$	0	0
$309$	−23.4770	−1.33556
$310$	0	0
$311$	−21.1813	−1.20108	−0.600541	−	0.799594i	$-0.705048\pi$
$311$	−21.1813	−1.20108	−0.600541	+	0.799594i	$0.705048\pi$
$312$	0	0
$313$	−6.90953	−0.390550	−0.195275	−	0.980749i	$-0.562560\pi$
$313$	−6.90953	−0.390550	−0.195275	+	0.980749i	$0.562560\pi$
$314$	0	0
$315$	18.7444	1.05613
$316$	0	0
$317$	30.4288	1.70905	0.854525	−	0.519410i	$-0.173848\pi$
$317$	30.4288	1.70905	0.854525	+	0.519410i	$0.173848\pi$
$318$	0	0
$319$	1.94794	0.109064
$320$	0	0
$321$	16.2024	0.904330
$322$	0	0
$323$	−6.52056	−0.362814
$324$	0	0
$325$	−1.30617	−0.0724531
$326$	0	0
$327$	−33.0280	−1.82645
$328$	0	0
$329$	22.3334	1.23128
$330$	0	0
$331$	3.22230	0.177113	0.0885567	−	0.996071i	$-0.471775\pi$
$331$	3.22230	0.177113	0.0885567	+	0.996071i	$0.471775\pi$
$332$	0	0
$333$	14.4534	0.792042
$334$	0	0
$335$	−19.2102	−1.04957
$336$	0	0
$337$	16.3816	0.892363	0.446181	−	0.894943i	$-0.352784\pi$
$337$	16.3816	0.892363	0.446181	+	0.894943i	$0.352784\pi$
$338$	0	0
$339$	17.1003	0.928763
$340$	0	0
$341$	−8.13463	−0.440515
$342$	0	0
$343$	−4.36697	−0.235794
$344$	0	0
$345$	−17.2015	−0.926097
$346$	0	0
$347$	18.6334	1.00029	0.500146	−	0.865941i	$-0.333280\pi$
$347$	18.6334	1.00029	0.500146	+	0.865941i	$0.333280\pi$
$348$	0	0
$349$	30.0788	1.61008	0.805042	−	0.593218i	$-0.202143\pi$
$349$	30.0788	1.61008	0.805042	+	0.593218i	$0.202143\pi$
$350$	0	0
$351$	1.14719	0.0612323
$352$	0	0
$353$	−5.78759	−0.308042	−0.154021	−	0.988068i	$-0.549222\pi$
$353$	−5.78759	−0.308042	−0.154021	+	0.988068i	$0.549222\pi$
$354$	0	0
$355$	−5.39559	−0.286368
$356$	0	0
$357$	55.2846	2.92597
$358$	0	0
$359$	−5.88858	−0.310787	−0.155394	−	0.987853i	$-0.549665\pi$
$359$	−5.88858	−0.310787	−0.155394	+	0.987853i	$0.549665\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−17.1763	−0.901521
$364$	0	0
$365$	2.37364	0.124242
$366$	0	0
$367$	20.7238	1.08177	0.540887	−	0.841095i	$-0.318089\pi$
$367$	20.7238	1.08177	0.540887	+	0.841095i	$0.318089\pi$
$368$	0	0
$369$	−8.83516	−0.459940
$370$	0	0
$371$	3.57468	0.185588
$372$	0	0
$373$	9.19668	0.476186	0.238093	−	0.971242i	$-0.423478\pi$
$373$	9.19668	0.476186	0.238093	+	0.971242i	$0.423478\pi$
$374$	0	0
$375$	28.4751	1.47045
$376$	0	0
$377$	−0.589085	−0.0303394
$378$	0	0
$379$	15.4914	0.795740	0.397870	−	0.917442i	$-0.369750\pi$
$379$	15.4914	0.795740	0.397870	+	0.917442i	$0.369750\pi$
$380$	0	0
$381$	−36.3070	−1.86007
$382$	0	0
$383$	−11.0620	−0.565241	−0.282621	−	0.959232i	$-0.591204\pi$
$383$	−11.0620	−0.565241	−0.282621	+	0.959232i	$0.591204\pi$
$384$	0	0
$385$	−30.4923	−1.55403
$386$	0	0
$387$	17.9437	0.912130
$388$	0	0
$389$	−8.49290	−0.430607	−0.215304	−	0.976547i	$-0.569074\pi$
$389$	−8.49290	−0.430607	−0.215304	+	0.976547i	$0.569074\pi$
$390$	0	0
$391$	−23.6783	−1.19746
$392$	0	0
$393$	−49.8350	−2.51384
$394$	0	0
$395$	31.1035	1.56499
$396$	0	0
$397$	−2.22037	−0.111437	−0.0557186	−	0.998447i	$-0.517745\pi$
$397$	−2.22037	−0.111437	−0.0557186	+	0.998447i	$0.517745\pi$
$398$	0	0
$399$	−8.47851	−0.424456
$400$	0	0
$401$	20.8986	1.04363	0.521813	−	0.853060i	$-0.325256\pi$
$401$	20.8986	1.04363	0.521813	+	0.853060i	$0.325256\pi$
$402$	0	0
$403$	2.46002	0.122543
$404$	0	0
$405$	−19.9382	−0.990739
$406$	0	0
$407$	−23.5119	−1.16544
$408$	0	0
$409$	−28.3149	−1.40008	−0.700042	−	0.714102i	$-0.746835\pi$
$409$	−28.3149	−1.40008	−0.700042	+	0.714102i	$0.746835\pi$
$410$	0	0
$411$	43.7822	2.15962
$412$	0	0
$413$	26.0049	1.27962
$414$	0	0
$415$	−30.9850	−1.52099
$416$	0	0
$417$	−3.77338	−0.184783
$418$	0	0
$419$	7.74372	0.378306	0.189153	−	0.981948i	$-0.439426\pi$
$419$	7.74372	0.378306	0.189153	+	0.981948i	$0.439426\pi$
$420$	0	0
$421$	20.2771	0.988244	0.494122	−	0.869393i	$-0.335490\pi$
$421$	20.2771	0.988244	0.494122	+	0.869393i	$0.335490\pi$
$422$	0	0
$423$	16.4034	0.797560
$424$	0	0
$425$	6.59397	0.319855
$426$	0	0
$427$	17.9912	0.870654
$428$	0	0
$429$	13.0843	0.631718
$430$	0	0
$431$	−17.0662	−0.822048	−0.411024	−	0.911625i	$-0.634829\pi$
$431$	−17.0662	−0.822048	−0.411024	+	0.911625i	$0.634829\pi$
$432$	0	0
$433$	10.2595	0.493039	0.246519	−	0.969138i	$-0.420713\pi$
$433$	10.2595	0.493039	0.246519	+	0.969138i	$0.420713\pi$
$434$	0	0
$435$	2.16044	0.103585
$436$	0	0
$437$	3.63134	0.173710
$438$	0	0
$439$	−30.3107	−1.44665	−0.723325	−	0.690508i	$-0.757387\pi$
$439$	−30.3107	−1.44665	−0.723325	+	0.690508i	$0.757387\pi$
$440$	0	0
$441$	15.1713	0.722441
$442$	0	0
$443$	12.3451	0.586533	0.293266	−	0.956031i	$-0.405258\pi$
$443$	12.3451	0.586533	0.293266	+	0.956031i	$0.405258\pi$
$444$	0	0
$445$	16.5172	0.782990
$446$	0	0
$447$	19.9660	0.944362
$448$	0	0
$449$	−29.0269	−1.36986	−0.684932	−	0.728607i	$-0.740168\pi$
$449$	−29.0269	−1.36986	−0.684932	+	0.728607i	$0.740168\pi$
$450$	0	0
$451$	14.3725	0.676773
$452$	0	0
$453$	−8.27167	−0.388637
$454$	0	0
$455$	9.22128	0.432300
$456$	0	0
$457$	22.0152	1.02983	0.514913	−	0.857242i	$-0.327824\pi$
$457$	22.0152	1.02983	0.514913	+	0.857242i	$0.327824\pi$
$458$	0	0
$459$	−5.79140	−0.270319
$460$	0	0
$461$	38.4352	1.79011	0.895053	−	0.445959i	$-0.147137\pi$
$461$	38.4352	1.79011	0.895053	+	0.445959i	$0.147137\pi$
$462$	0	0
$463$	1.90589	0.0885744	0.0442872	−	0.999019i	$-0.485898\pi$
$463$	1.90589	0.0885744	0.0442872	+	0.999019i	$0.485898\pi$
$464$	0	0
$465$	−9.02201	−0.418386
$466$	0	0
$467$	−30.7981	−1.42517	−0.712584	−	0.701587i	$-0.752475\pi$
$467$	−30.7981	−1.42517	−0.712584	+	0.701587i	$0.752475\pi$
$468$	0	0
$469$	−34.3837	−1.58769
$470$	0	0
$471$	−2.67322	−0.123176
$472$	0	0
$473$	−29.1897	−1.34214
$474$	0	0
$475$	−1.01126	−0.0463998
$476$	0	0
$477$	2.62553	0.120215
$478$	0	0
$479$	−6.12520	−0.279868	−0.139934	−	0.990161i	$-0.544689\pi$
$479$	−6.12520	−0.279868	−0.139934	+	0.990161i	$0.544689\pi$
$480$	0	0
$481$	7.11032	0.324203
$482$	0	0
$483$	−30.7883	−1.40092
$484$	0	0
$485$	−36.2242	−1.64486
$486$	0	0
$487$	−32.3204	−1.46458	−0.732288	−	0.680995i	$-0.761548\pi$
$487$	−32.3204	−1.46458	−0.732288	+	0.680995i	$0.761548\pi$
$488$	0	0
$489$	−54.1108	−2.44697
$490$	0	0
$491$	−8.88521	−0.400984	−0.200492	−	0.979695i	$-0.564254\pi$
$491$	−8.88521	−0.400984	−0.200492	+	0.979695i	$0.564254\pi$
$492$	0	0
$493$	2.97390	0.133938
$494$	0	0
$495$	−22.3959	−1.00662
$496$	0	0
$497$	−9.65736	−0.433192
$498$	0	0
$499$	−1.45164	−0.0649843	−0.0324921	−	0.999472i	$-0.510344\pi$
$499$	−1.45164	−0.0649843	−0.0324921	+	0.999472i	$0.510344\pi$
$500$	0	0
$501$	−52.9241	−2.36448
$502$	0	0
$503$	25.3476	1.13020	0.565098	−	0.825024i	$-0.308838\pi$
$503$	25.3476	1.13020	0.565098	+	0.825024i	$0.308838\pi$
$504$	0	0
$505$	35.1012	1.56198
$506$	0	0
$507$	26.8768	1.19364
$508$	0	0
$509$	−25.0544	−1.11052	−0.555258	−	0.831678i	$-0.687381\pi$
$509$	−25.0544	−1.11052	−0.555258	+	0.831678i	$0.687381\pi$
$510$	0	0
$511$	4.24849	0.187942
$512$	0	0
$513$	0.888175	0.0392139
$514$	0	0
$515$	19.7687	0.871115
$516$	0	0
$517$	−26.6840	−1.17356
$518$	0	0
$519$	−10.0917	−0.442977
$520$	0	0
$521$	12.9321	0.566567	0.283284	−	0.959036i	$-0.408576\pi$
$521$	12.9321	0.566567	0.283284	+	0.959036i	$0.408576\pi$
$522$	0	0
$523$	−0.151235	−0.00661305	−0.00330653	−	0.999995i	$-0.501053\pi$
$523$	−0.151235	−0.00661305	−0.00330653	+	0.999995i	$0.501053\pi$
$524$	0	0
$525$	8.57397	0.374199
$526$	0	0
$527$	−12.4190	−0.540982
$528$	0	0
$529$	−9.81340	−0.426670
$530$	0	0
$531$	19.1001	0.828873
$532$	0	0
$533$	−4.34643	−0.188265
$534$	0	0
$535$	−13.6432	−0.589846
$536$	0	0
$537$	−0.597751	−0.0257949
$538$	0	0
$539$	−24.6796	−1.06303
$540$	0	0
$541$	7.72300	0.332038	0.166019	−	0.986123i	$-0.446909\pi$
$541$	7.72300	0.332038	0.166019	+	0.986123i	$0.446909\pi$
$542$	0	0
$543$	−8.39986	−0.360472
$544$	0	0
$545$	27.8111	1.19130
$546$	0	0
$547$	16.8117	0.718818	0.359409	−	0.933180i	$-0.382978\pi$
$547$	16.8117	0.718818	0.359409	+	0.933180i	$0.382978\pi$
$548$	0	0
$549$	13.2141	0.563966
$550$	0	0
$551$	−0.456081	−0.0194297
$552$	0	0
$553$	55.6710	2.36737
$554$	0	0
$555$	−26.0767	−1.10690
$556$	0	0
$557$	10.3607	0.438999	0.219499	−	0.975613i	$-0.429558\pi$
$557$	10.3607	0.438999	0.219499	+	0.975613i	$0.429558\pi$
$558$	0	0
$559$	8.82736	0.373358
$560$	0	0
$561$	−66.0542	−2.78881
$562$	0	0
$563$	32.5701	1.37267	0.686333	−	0.727288i	$-0.259219\pi$
$563$	32.5701	1.37267	0.686333	+	0.727288i	$0.259219\pi$
$564$	0	0
$565$	−14.3993	−0.605782
$566$	0	0
$567$	−35.6867	−1.49870
$568$	0	0
$569$	−12.3015	−0.515705	−0.257853	−	0.966184i	$-0.583015\pi$
$569$	−12.3015	−0.515705	−0.257853	+	0.966184i	$0.583015\pi$
$570$	0	0
$571$	−19.5606	−0.818587	−0.409294	−	0.912403i	$-0.634225\pi$
$571$	−19.5606	−0.818587	−0.409294	+	0.912403i	$0.634225\pi$
$572$	0	0
$573$	37.9787	1.58658
$574$	0	0
$575$	−3.67222	−0.153142
$576$	0	0
$577$	38.0387	1.58357	0.791787	−	0.610798i	$-0.209151\pi$
$577$	38.0387	1.58357	0.791787	+	0.610798i	$0.209151\pi$
$578$	0	0
$579$	−43.3910	−1.80327
$580$	0	0
$581$	−55.4589	−2.30082
$582$	0	0
$583$	−4.27104	−0.176889
$584$	0	0
$585$	6.77284	0.280022
$586$	0	0
$587$	20.0460	0.827385	0.413692	−	0.910417i	$-0.364239\pi$
$587$	20.0460	0.827385	0.413692	+	0.910417i	$0.364239\pi$
$588$	0	0
$589$	1.90460	0.0784776
$590$	0	0
$591$	−1.98894	−0.0818142
$592$	0	0
$593$	7.38007	0.303063	0.151531	−	0.988452i	$-0.451580\pi$
$593$	7.38007	0.303063	0.151531	+	0.988452i	$0.451580\pi$
$594$	0	0
$595$	−46.5522	−1.90845
$596$	0	0
$597$	−27.8799	−1.14105
$598$	0	0
$599$	38.6665	1.57987	0.789935	−	0.613190i	$-0.210114\pi$
$599$	38.6665	1.57987	0.789935	+	0.613190i	$0.210114\pi$
$600$	0	0
$601$	−45.5027	−1.85609	−0.928047	−	0.372463i	$-0.878513\pi$
$601$	−45.5027	−1.85609	−0.928047	+	0.372463i	$0.878513\pi$
$602$	0	0
$603$	−25.2541	−1.02843
$604$	0	0
$605$	14.4632	0.588014
$606$	0	0
$607$	23.3944	0.949548	0.474774	−	0.880108i	$-0.342530\pi$
$607$	23.3944	0.949548	0.474774	+	0.880108i	$0.342530\pi$
$608$	0	0
$609$	3.86689	0.156694
$610$	0	0
$611$	8.06960	0.326461
$612$	0	0
$613$	6.77428	0.273611	0.136805	−	0.990598i	$-0.456317\pi$
$613$	6.77428	0.273611	0.136805	+	0.990598i	$0.456317\pi$
$614$	0	0
$615$	15.9403	0.642775
$616$	0	0
$617$	−28.3870	−1.14282	−0.571408	−	0.820666i	$-0.693603\pi$
$617$	−28.3870	−1.14282	−0.571408	+	0.820666i	$0.693603\pi$
$618$	0	0
$619$	5.09177	0.204656	0.102328	−	0.994751i	$-0.467371\pi$
$619$	5.09177	0.204656	0.102328	+	0.994751i	$0.467371\pi$
$620$	0	0
$621$	3.22526	0.129425
$622$	0	0
$623$	29.5635	1.18444
$624$	0	0
$625$	−18.9211	−0.756842
$626$	0	0
$627$	10.1301	0.404559
$628$	0	0
$629$	−35.8953	−1.43124
$630$	0	0
$631$	−35.6283	−1.41834	−0.709171	−	0.705037i	$-0.750931\pi$
$631$	−35.6283	−1.41834	−0.709171	+	0.705037i	$0.750931\pi$
$632$	0	0
$633$	−39.0942	−1.55386
$634$	0	0
$635$	30.5722	1.21322
$636$	0	0
$637$	7.46346	0.295713
$638$	0	0
$639$	−7.09313	−0.280600
$640$	0	0
$641$	−21.0710	−0.832255	−0.416128	−	0.909306i	$-0.636613\pi$
$641$	−21.0710	−0.832255	−0.416128	+	0.909306i	$0.636613\pi$
$642$	0	0
$643$	−23.6581	−0.932983	−0.466492	−	0.884526i	$-0.654482\pi$
$643$	−23.6581	−0.932983	−0.466492	+	0.884526i	$0.654482\pi$
$644$	0	0
$645$	−32.3739	−1.27472
$646$	0	0
$647$	27.2666	1.07196	0.535980	−	0.844231i	$-0.319942\pi$
$647$	27.2666	1.07196	0.535980	+	0.844231i	$0.319942\pi$
$648$	0	0
$649$	−31.0708	−1.21964
$650$	0	0
$651$	−16.1482	−0.632896
$652$	0	0
$653$	34.9981	1.36958	0.684791	−	0.728739i	$-0.259893\pi$
$653$	34.9981	1.36958	0.684791	+	0.728739i	$0.259893\pi$
$654$	0	0
$655$	41.9634	1.63964
$656$	0	0
$657$	3.12043	0.121739
$658$	0	0
$659$	28.7943	1.12167	0.560834	−	0.827929i	$-0.310481\pi$
$659$	28.7943	1.12167	0.560834	+	0.827929i	$0.310481\pi$
$660$	0	0
$661$	33.0098	1.28393	0.641967	−	0.766732i	$-0.278119\pi$
$661$	33.0098	1.28393	0.641967	+	0.766732i	$0.278119\pi$
$662$	0	0
$663$	19.9757	0.775792
$664$	0	0
$665$	7.13930	0.276850
$666$	0	0
$667$	−1.65618	−0.0641277
$668$	0	0
$669$	19.1822	0.741628
$670$	0	0
$671$	−21.4959	−0.829841
$672$	0	0
$673$	−32.8395	−1.26587	−0.632936	−	0.774204i	$-0.718150\pi$
$673$	−32.8395	−1.26587	−0.632936	+	0.774204i	$0.718150\pi$
$674$	0	0
$675$	−0.898175	−0.0345708
$676$	0	0
$677$	−27.8622	−1.07083	−0.535415	−	0.844589i	$-0.679845\pi$
$677$	−27.8622	−1.07083	−0.535415	+	0.844589i	$0.679845\pi$
$678$	0	0
$679$	−64.8364	−2.48819
$680$	0	0
$681$	−31.6523	−1.21292
$682$	0	0
$683$	−13.8619	−0.530410	−0.265205	−	0.964192i	$-0.585440\pi$
$683$	−13.8619	−0.530410	−0.265205	+	0.964192i	$0.585440\pi$
$684$	0	0
$685$	−36.8667	−1.40860
$686$	0	0
$687$	28.7928	1.09851
$688$	0	0
$689$	1.29162	0.0492069
$690$	0	0
$691$	23.1570	0.880934	0.440467	−	0.897769i	$-0.354813\pi$
$691$	23.1570	0.880934	0.440467	+	0.897769i	$0.354813\pi$
$692$	0	0
$693$	−40.0856	−1.52273
$694$	0	0
$695$	3.17736	0.120524
$696$	0	0
$697$	21.9423	0.831123
$698$	0	0
$699$	−17.3037	−0.654488
$700$	0	0
$701$	−32.0764	−1.21151	−0.605754	−	0.795652i	$-0.707128\pi$
$701$	−32.0764	−1.21151	−0.605754	+	0.795652i	$0.707128\pi$
$702$	0	0
$703$	5.50495	0.207623
$704$	0	0
$705$	−29.5948	−1.11461
$706$	0	0
$707$	62.8263	2.36283
$708$	0	0
$709$	−14.6117	−0.548755	−0.274377	−	0.961622i	$-0.588472\pi$
$709$	−14.6117	−0.548755	−0.274377	+	0.961622i	$0.588472\pi$
$710$	0	0
$711$	40.8892	1.53346
$712$	0	0
$713$	6.91624	0.259015
$714$	0	0
$715$	−11.0176	−0.412036
$716$	0	0
$717$	−7.82113	−0.292086
$718$	0	0
$719$	−14.1204	−0.526601	−0.263300	−	0.964714i	$-0.584811\pi$
$719$	−14.1204	−0.526601	−0.263300	+	0.964714i	$0.584811\pi$
$720$	0	0
$721$	35.3833	1.31774
$722$	0	0
$723$	−38.2152	−1.42124
$724$	0	0
$725$	0.461216	0.0171291
$726$	0	0
$727$	3.96290	0.146976	0.0734879	−	0.997296i	$-0.476587\pi$
$727$	3.96290	0.146976	0.0734879	+	0.997296i	$0.476587\pi$
$728$	0	0
$729$	−19.8914	−0.736717
$730$	0	0
$731$	−44.5636	−1.64824
$732$	0	0
$733$	−45.2769	−1.67234	−0.836171	−	0.548469i	$-0.815211\pi$
$733$	−45.2769	−1.67234	−0.836171	+	0.548469i	$0.815211\pi$
$734$	0	0
$735$	−27.3719	−1.00963
$736$	0	0
$737$	41.0818	1.51327
$738$	0	0
$739$	−14.2976	−0.525947	−0.262973	−	0.964803i	$-0.584703\pi$
$739$	−14.2976	−0.525947	−0.262973	+	0.964803i	$0.584703\pi$
$740$	0	0
$741$	−3.06350	−0.112540
$742$	0	0
$743$	−32.3378	−1.18636	−0.593180	−	0.805070i	$-0.702128\pi$
$743$	−32.3378	−1.18636	−0.593180	+	0.805070i	$0.702128\pi$
$744$	0	0
$745$	−16.8123	−0.615957
$746$	0	0
$747$	−40.7334	−1.49036
$748$	0	0
$749$	−24.4194	−0.892266
$750$	0	0
$751$	−4.30819	−0.157208	−0.0786040	−	0.996906i	$-0.525046\pi$
$751$	−4.30819	−0.157208	−0.0786040	+	0.996906i	$0.525046\pi$
$752$	0	0
$753$	−7.84087	−0.285737
$754$	0	0
$755$	6.96513	0.253487
$756$	0	0
$757$	8.42280	0.306132	0.153066	−	0.988216i	$-0.451085\pi$
$757$	8.42280	0.306132	0.153066	+	0.988216i	$0.451085\pi$
$758$	0	0
$759$	36.7860	1.33525
$760$	0	0
$761$	−10.7562	−0.389911	−0.194956	−	0.980812i	$-0.562456\pi$
$761$	−10.7562	−0.389911	−0.194956	+	0.980812i	$0.562456\pi$
$762$	0	0
$763$	49.7781	1.80209
$764$	0	0
$765$	−34.1916	−1.23620
$766$	0	0
$767$	9.39624	0.339279
$768$	0	0
$769$	−31.4767	−1.13508	−0.567539	−	0.823346i	$-0.692104\pi$
$769$	−31.4767	−1.13508	−0.567539	+	0.823346i	$0.692104\pi$
$770$	0	0
$771$	26.5596	0.956522
$772$	0	0
$773$	−44.4806	−1.59986	−0.799928	−	0.600095i	$-0.795129\pi$
$773$	−44.4806	−1.59986	−0.799928	+	0.600095i	$0.795129\pi$
$774$	0	0
$775$	−1.92604	−0.0691855
$776$	0	0
$777$	−46.6737	−1.67441
$778$	0	0
$779$	−3.36509	−0.120567
$780$	0	0
$781$	11.5387	0.412885
$782$	0	0
$783$	−0.405080	−0.0144764
$784$	0	0
$785$	2.25098	0.0803408
$786$	0	0
$787$	−30.8036	−1.09803	−0.549014	−	0.835813i	$-0.684997\pi$
$787$	−30.8036	−1.09803	−0.549014	+	0.835813i	$0.684997\pi$
$788$	0	0
$789$	−43.8064	−1.55955
$790$	0	0
$791$	−25.7727	−0.916373
$792$	0	0
$793$	6.50067	0.230845
$794$	0	0
$795$	−4.73696	−0.168003
$796$	0	0
$797$	−27.4187	−0.971222	−0.485611	−	0.874175i	$-0.661403\pi$
$797$	−27.4187	−0.971222	−0.485611	+	0.874175i	$0.661403\pi$
$798$	0	0
$799$	−40.7381	−1.44121
$800$	0	0
$801$	21.7138	0.767218
$802$	0	0
$803$	−5.07611	−0.179132
$804$	0	0
$805$	25.9252	0.913743
$806$	0	0
$807$	40.8660	1.43855
$808$	0	0
$809$	−30.6302	−1.07690	−0.538451	−	0.842657i	$-0.680990\pi$
$809$	−30.6302	−1.07690	−0.538451	+	0.842657i	$0.680990\pi$
$810$	0	0
$811$	39.7799	1.39686	0.698430	−	0.715678i	$-0.253882\pi$
$811$	39.7799	1.39686	0.698430	+	0.715678i	$0.253882\pi$
$812$	0	0
$813$	70.3417	2.46699
$814$	0	0
$815$	45.5638	1.59603
$816$	0	0
$817$	6.83432	0.239102
$818$	0	0
$819$	12.1225	0.423593
$820$	0	0
$821$	−1.48233	−0.0517336	−0.0258668	−	0.999665i	$-0.508235\pi$
$821$	−1.48233	−0.0517336	−0.0258668	+	0.999665i	$0.508235\pi$
$822$	0	0
$823$	−30.4168	−1.06026	−0.530132	−	0.847915i	$-0.677858\pi$
$823$	−30.4168	−1.06026	−0.530132	+	0.847915i	$0.677858\pi$
$824$	0	0
$825$	−10.2442	−0.356657
$826$	0	0
$827$	46.8398	1.62878	0.814390	−	0.580318i	$-0.197072\pi$
$827$	46.8398	1.62878	0.814390	+	0.580318i	$0.197072\pi$
$828$	0	0
$829$	28.3308	0.983971	0.491985	−	0.870603i	$-0.336271\pi$
$829$	28.3308	0.983971	0.491985	+	0.870603i	$0.336271\pi$
$830$	0	0
$831$	31.6357	1.09743
$832$	0	0
$833$	−37.6781	−1.30547
$834$	0	0
$835$	44.5646	1.54222
$836$	0	0
$837$	1.69162	0.0584709
$838$	0	0
$839$	34.5364	1.19233	0.596164	−	0.802863i	$-0.296691\pi$
$839$	34.5364	1.19233	0.596164	+	0.802863i	$0.296691\pi$
$840$	0	0
$841$	−28.7920	−0.992827
$842$	0	0
$843$	−41.0124	−1.41254
$844$	0	0
$845$	−22.6315	−0.778547
$846$	0	0
$847$	25.8872	0.889495
$848$	0	0
$849$	45.2363	1.55250
$850$	0	0
$851$	19.9903	0.685259
$852$	0	0
$853$	50.8057	1.73956	0.869778	−	0.493444i	$-0.164262\pi$
$853$	50.8057	1.73956	0.869778	+	0.493444i	$0.164262\pi$
$854$	0	0
$855$	5.24366	0.179330
$856$	0	0
$857$	31.0966	1.06224	0.531120	−	0.847297i	$-0.321771\pi$
$857$	31.0966	1.06224	0.531120	+	0.847297i	$0.321771\pi$
$858$	0	0
$859$	9.40978	0.321058	0.160529	−	0.987031i	$-0.448680\pi$
$859$	9.40978	0.321058	0.160529	+	0.987031i	$0.448680\pi$
$860$	0	0
$861$	28.5310	0.972333
$862$	0	0
$863$	10.5378	0.358712	0.179356	−	0.983784i	$-0.442599\pi$
$863$	10.5378	0.358712	0.179356	+	0.983784i	$0.442599\pi$
$864$	0	0
$865$	8.49769	0.288930
$866$	0	0
$867$	−60.5233	−2.05548
$868$	0	0
$869$	−66.5159	−2.25640
$870$	0	0
$871$	−12.4237	−0.420961
$872$	0	0
$873$	−47.6209	−1.61172
$874$	0	0
$875$	−42.9162	−1.45083
$876$	0	0
$877$	−14.3029	−0.482975	−0.241488	−	0.970404i	$-0.577635\pi$
$877$	−14.3029	−0.482975	−0.241488	+	0.970404i	$0.577635\pi$
$878$	0	0
$879$	−66.0818	−2.22888
$880$	0	0
$881$	50.2344	1.69244	0.846220	−	0.532833i	$-0.178873\pi$
$881$	50.2344	1.69244	0.846220	+	0.532833i	$0.178873\pi$
$882$	0	0
$883$	−46.5772	−1.56745	−0.783724	−	0.621109i	$-0.786682\pi$
$883$	−46.5772	−1.56745	−0.783724	+	0.621109i	$0.786682\pi$
$884$	0	0
$885$	−34.4602	−1.15837
$886$	0	0
$887$	−6.07349	−0.203928	−0.101964	−	0.994788i	$-0.532513\pi$
$887$	−6.07349	−0.203928	−0.101964	+	0.994788i	$0.532513\pi$
$888$	0	0
$889$	54.7200	1.83525
$890$	0	0
$891$	42.6386	1.42845
$892$	0	0
$893$	6.24765	0.209070
$894$	0	0
$895$	0.503335	0.0168246
$896$	0	0
$897$	−11.1246	−0.371439
$898$	0	0
$899$	−0.868652	−0.0289712
$900$	0	0
$901$	−6.52056	−0.217231
$902$	0	0
$903$	−57.9448	−1.92828
$904$	0	0
$905$	7.07307	0.235117
$906$	0	0
$907$	52.8315	1.75424	0.877120	−	0.480272i	$-0.159462\pi$
$907$	52.8315	1.75424	0.877120	+	0.480272i	$0.159462\pi$
$908$	0	0
$909$	46.1446	1.53052
$910$	0	0
$911$	−13.0083	−0.430984	−0.215492	−	0.976506i	$-0.569136\pi$
$911$	−13.0083	−0.430984	−0.215492	+	0.976506i	$0.569136\pi$
$912$	0	0
$913$	66.2624	2.19297
$914$	0	0
$915$	−23.8408	−0.788154
$916$	0	0
$917$	75.1087	2.48031
$918$	0	0
$919$	−0.0937423	−0.00309227	−0.00154614	−	0.999999i	$-0.500492\pi$
$919$	−0.0937423	−0.00309227	−0.00154614	+	0.999999i	$0.500492\pi$
$920$	0	0
$921$	−32.6084	−1.07448
$922$	0	0
$923$	−3.48945	−0.114857
$924$	0	0
$925$	−5.56693	−0.183040
$926$	0	0
$927$	25.9883	0.853568
$928$	0	0
$929$	−3.04521	−0.0999101	−0.0499551	−	0.998751i	$-0.515908\pi$
$929$	−3.04521	−0.0999101	−0.0499551	+	0.998751i	$0.515908\pi$
$930$	0	0
$931$	5.77836	0.189378
$932$	0	0
$933$	50.2383	1.64473
$934$	0	0
$935$	55.6207	1.81899
$936$	0	0
$937$	7.98292	0.260791	0.130395	−	0.991462i	$-0.458375\pi$
$937$	7.98292	0.260791	0.130395	+	0.991462i	$0.458375\pi$
$938$	0	0
$939$	16.3882	0.534807
$940$	0	0
$941$	−38.8426	−1.26623	−0.633116	−	0.774057i	$-0.718224\pi$
$941$	−38.8426	−1.26623	−0.633116	+	0.774057i	$0.718224\pi$
$942$	0	0
$943$	−12.2198	−0.397931
$944$	0	0
$945$	6.34095	0.206271
$946$	0	0
$947$	37.2708	1.21114	0.605570	−	0.795792i	$-0.292945\pi$
$947$	37.2708	1.21114	0.605570	+	0.795792i	$0.292945\pi$
$948$	0	0
$949$	1.53509	0.0498310
$950$	0	0
$951$	−72.1716	−2.34032
$952$	0	0
$953$	−34.9771	−1.13302	−0.566509	−	0.824055i	$-0.691706\pi$
$953$	−34.9771	−1.13302	−0.566509	+	0.824055i	$0.691706\pi$
$954$	0	0
$955$	−31.9798	−1.03484
$956$	0	0
$957$	−4.62017	−0.149349
$958$	0	0
$959$	−65.9863	−2.13081
$960$	0	0
$961$	−27.3725	−0.882984
$962$	0	0
$963$	−17.9356	−0.577965
$964$	0	0
$965$	36.5373	1.17618
$966$	0	0
$967$	34.4641	1.10829	0.554146	−	0.832420i	$-0.313045\pi$
$967$	34.4641	1.10829	0.554146	+	0.832420i	$0.313045\pi$
$968$	0	0
$969$	15.4656	0.496826
$970$	0	0
$971$	31.2400	1.00254	0.501270	−	0.865291i	$-0.332866\pi$
$971$	31.2400	1.00254	0.501270	+	0.865291i	$0.332866\pi$
$972$	0	0
$973$	5.68704	0.182318
$974$	0	0
$975$	3.09799	0.0992151
$976$	0	0
$977$	40.4002	1.29252	0.646258	−	0.763119i	$-0.276333\pi$
$977$	40.4002	1.29252	0.646258	+	0.763119i	$0.276333\pi$
$978$	0	0
$979$	−35.3226	−1.12891
$980$	0	0
$981$	36.5609	1.16730
$982$	0	0
$983$	−14.3269	−0.456957	−0.228479	−	0.973549i	$-0.573375\pi$
$983$	−14.3269	−0.456957	−0.228479	+	0.973549i	$0.573375\pi$
$984$	0	0
$985$	1.67478	0.0533630
$986$	0	0
$987$	−52.9707	−1.68608
$988$	0	0
$989$	24.8177	0.789157
$990$	0	0
$991$	−0.725495	−0.0230461	−0.0115230	−	0.999934i	$-0.503668\pi$
$991$	−0.725495	−0.0230461	−0.0115230	+	0.999934i	$0.503668\pi$
$992$	0	0
$993$	−7.64270	−0.242534
$994$	0	0
$995$	23.4762	0.744245
$996$	0	0
$997$	−60.8688	−1.92773	−0.963866	−	0.266386i	$-0.914170\pi$
$997$	−60.8688	−1.92773	−0.963866	+	0.266386i	$0.914170\pi$
$998$	0	0
$999$	4.88936	0.154692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.e.1.4	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.e.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-2.37182 q^{3} +1.99718 q^{5} +3.57468 q^{7} +2.62553 q^{9} +O(q^{10})\)	\(q-2.37182 q^{3} +1.99718 q^{5} +3.57468 q^{7} +2.62553 q^{9} -4.27104 q^{11} +1.29162 q^{13} -4.73696 q^{15} -6.52056 q^{17} +1.00000 q^{19} -8.47851 q^{21} +3.63134 q^{23} -1.01126 q^{25} +0.888175 q^{27} -0.456081 q^{29} +1.90460 q^{31} +10.1301 q^{33} +7.13930 q^{35} +5.50495 q^{37} -3.06350 q^{39} -3.36509 q^{41} +6.83432 q^{43} +5.24366 q^{45} +6.24765 q^{47} +5.77836 q^{49} +15.4656 q^{51} +1.00000 q^{53} -8.53006 q^{55} -2.37182 q^{57} +7.27476 q^{59} +5.03294 q^{61} +9.38544 q^{63} +2.57961 q^{65} -9.61867 q^{67} -8.61287 q^{69} -2.70160 q^{71} +1.18849 q^{73} +2.39853 q^{75} -15.2676 q^{77} +15.5737 q^{79} -9.98318 q^{81} -15.5143 q^{83} -13.0227 q^{85} +1.08174 q^{87} +8.27024 q^{89} +4.61714 q^{91} -4.51737 q^{93} +1.99718 q^{95} -18.1377 q^{97} -11.2138 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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