LMFDB - Embedded newform 4028.2.a.e.1.2

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			1.2
Root			$-2.82248$ 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)

$f(q)$	$=$	$q-2.82248 q^{3} +0.0936572 q^{5} -0.650141 q^{7} +4.96640 q^{9} +O(q^{10})$	$q-2.82248 q^{3} +0.0936572 q^{5} -0.650141 q^{7} +4.96640 q^{9} +5.87938 q^{11} +5.73453 q^{13} -0.264346 q^{15} +2.88721 q^{17} +1.00000 q^{19} +1.83501 q^{21} +7.57364 q^{23} -4.99123 q^{25} -5.55011 q^{27} +0.702692 q^{29} +10.9328 q^{31} -16.5944 q^{33} -0.0608903 q^{35} +0.728585 q^{37} -16.1856 q^{39} +1.10054 q^{41} -1.59121 q^{43} +0.465139 q^{45} +10.1029 q^{47} -6.57732 q^{49} -8.14910 q^{51} +1.00000 q^{53} +0.550646 q^{55} -2.82248 q^{57} +6.09513 q^{59} -12.4313 q^{61} -3.22886 q^{63} +0.537080 q^{65} +3.05415 q^{67} -21.3765 q^{69} -10.2572 q^{71} -15.7286 q^{73} +14.0876 q^{75} -3.82242 q^{77} -2.39139 q^{79} +0.765900 q^{81} +3.94838 q^{83} +0.270408 q^{85} -1.98334 q^{87} +9.35346 q^{89} -3.72825 q^{91} -30.8577 q^{93} +0.0936572 q^{95} +17.1878 q^{97} +29.1993 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.82248	−1.62956	−0.814780	−	0.579770i	$-0.803142\pi$
$3$	−2.82248	−1.62956	−0.814780	+	0.579770i	$0.803142\pi$
$4$	0	0
$5$	0.0936572	0.0418848	0.0209424	−	0.999781i	$-0.493333\pi$
$5$	0.0936572	0.0418848	0.0209424	+	0.999781i	$0.493333\pi$
$6$	0	0
$7$	−0.650141	−0.245730	−0.122865	−	0.992423i	$-0.539208\pi$
$7$	−0.650141	−0.245730	−0.122865	+	0.992423i	$0.539208\pi$
$8$	0	0
$9$	4.96640	1.65547
$10$	0	0
$11$	5.87938	1.77270	0.886350	−	0.463016i	$-0.153233\pi$
$11$	5.87938	1.77270	0.886350	+	0.463016i	$0.153233\pi$
$12$	0	0
$13$	5.73453	1.59047	0.795236	−	0.606300i	$-0.207347\pi$
$13$	5.73453	1.59047	0.795236	+	0.606300i	$0.207347\pi$
$14$	0	0
$15$	−0.264346	−0.0682537
$16$	0	0
$17$	2.88721	0.700252	0.350126	−	0.936703i	$-0.386139\pi$
$17$	2.88721	0.700252	0.350126	+	0.936703i	$0.386139\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.83501	0.400432
$22$	0	0
$23$	7.57364	1.57921	0.789607	−	0.613613i	$-0.210284\pi$
$23$	7.57364	1.57921	0.789607	+	0.613613i	$0.210284\pi$
$24$	0	0
$25$	−4.99123	−0.998246
$26$	0	0
$27$	−5.55011	−1.06812
$28$	0	0
$29$	0.702692	0.130487	0.0652433	−	0.997869i	$-0.479218\pi$
$29$	0.702692	0.130487	0.0652433	+	0.997869i	$0.479218\pi$
$30$	0	0
$31$	10.9328	1.96359	0.981796	−	0.189939i	$-0.0608290\pi$
$31$	10.9328	1.96359	0.981796	+	0.189939i	$0.0608290\pi$
$32$	0	0
$33$	−16.5944	−2.88872
$34$	0	0
$35$	−0.0608903	−0.0102923
$36$	0	0
$37$	0.728585	0.119779	0.0598893	−	0.998205i	$-0.480925\pi$
$37$	0.728585	0.119779	0.0598893	+	0.998205i	$0.480925\pi$
$38$	0	0
$39$	−16.1856	−2.59177
$40$	0	0
$41$	1.10054	0.171875	0.0859377	−	0.996301i	$-0.472611\pi$
$41$	1.10054	0.171875	0.0859377	+	0.996301i	$0.472611\pi$
$42$	0	0
$43$	−1.59121	−0.242658	−0.121329	−	0.992612i	$-0.538716\pi$
$43$	−1.59121	−0.242658	−0.121329	+	0.992612i	$0.538716\pi$
$44$	0	0
$45$	0.465139	0.0693388
$46$	0	0
$47$	10.1029	1.47366	0.736830	−	0.676078i	$-0.236322\pi$
$47$	10.1029	1.47366	0.736830	+	0.676078i	$0.236322\pi$
$48$	0	0
$49$	−6.57732	−0.939617
$50$	0	0
$51$	−8.14910	−1.14110
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	0.550646	0.0742491
$56$	0	0
$57$	−2.82248	−0.373847
$58$	0	0
$59$	6.09513	0.793518	0.396759	−	0.917923i	$-0.370135\pi$
$59$	6.09513	0.793518	0.396759	+	0.917923i	$0.370135\pi$
$60$	0	0
$61$	−12.4313	−1.59166	−0.795830	−	0.605520i	$-0.792965\pi$
$61$	−12.4313	−1.59166	−0.795830	+	0.605520i	$0.792965\pi$
$62$	0	0
$63$	−3.22886	−0.406798
$64$	0	0
$65$	0.537080	0.0666165
$66$	0	0
$67$	3.05415	0.373124	0.186562	−	0.982443i	$-0.440265\pi$
$67$	3.05415	0.373124	0.186562	+	0.982443i	$0.440265\pi$
$68$	0	0
$69$	−21.3765	−2.57342
$70$	0	0
$71$	−10.2572	−1.21730	−0.608652	−	0.793437i	$-0.708289\pi$
$71$	−10.2572	−1.21730	−0.608652	+	0.793437i	$0.708289\pi$
$72$	0	0
$73$	−15.7286	−1.84090	−0.920450	−	0.390861i	$-0.872177\pi$
$73$	−15.7286	−1.84090	−0.920450	+	0.390861i	$0.872177\pi$
$74$	0	0
$75$	14.0876	1.62670
$76$	0	0
$77$	−3.82242	−0.435606
$78$	0	0
$79$	−2.39139	−0.269052	−0.134526	−	0.990910i	$-0.542951\pi$
$79$	−2.39139	−0.269052	−0.134526	+	0.990910i	$0.542951\pi$
$80$	0	0
$81$	0.765900	0.0851000
$82$	0	0
$83$	3.94838	0.433391	0.216695	−	0.976239i	$-0.430472\pi$
$83$	3.94838	0.433391	0.216695	+	0.976239i	$0.430472\pi$
$84$	0	0
$85$	0.270408	0.0293299
$86$	0	0
$87$	−1.98334	−0.212636
$88$	0	0
$89$	9.35346	0.991465	0.495732	−	0.868475i	$-0.334900\pi$
$89$	9.35346	0.991465	0.495732	+	0.868475i	$0.334900\pi$
$90$	0	0
$91$	−3.72825	−0.390827
$92$	0	0
$93$	−30.8577	−3.19979
$94$	0	0
$95$	0.0936572	0.00960902
$96$	0	0
$97$	17.1878	1.74516	0.872581	−	0.488470i	$-0.162445\pi$
$97$	17.1878	1.74516	0.872581	+	0.488470i	$0.162445\pi$
$98$	0	0
$99$	29.1993	2.93464
$100$	0	0
$101$	−8.00447	−0.796475	−0.398237	−	0.917282i	$-0.630378\pi$
$101$	−8.00447	−0.796475	−0.398237	+	0.917282i	$0.630378\pi$
$102$	0	0
$103$	2.16047	0.212878	0.106439	−	0.994319i	$-0.466055\pi$
$103$	2.16047	0.212878	0.106439	+	0.994319i	$0.466055\pi$
$104$	0	0
$105$	0.171862	0.0167720
$106$	0	0
$107$	−6.80400	−0.657768	−0.328884	−	0.944370i	$-0.606672\pi$
$107$	−6.80400	−0.657768	−0.328884	+	0.944370i	$0.606672\pi$
$108$	0	0
$109$	2.84583	0.272581	0.136291	−	0.990669i	$-0.456482\pi$
$109$	2.84583	0.272581	0.136291	+	0.990669i	$0.456482\pi$
$110$	0	0
$111$	−2.05642	−0.195186
$112$	0	0
$113$	−12.4376	−1.17003	−0.585014	−	0.811023i	$-0.698911\pi$
$113$	−12.4376	−1.17003	−0.585014	+	0.811023i	$0.698911\pi$
$114$	0	0
$115$	0.709326	0.0661450
$116$	0	0
$117$	28.4799	2.63297
$118$	0	0
$119$	−1.87709	−0.172073
$120$	0	0
$121$	23.5671	2.14246
$122$	0	0
$123$	−3.10625	−0.280081
$124$	0	0
$125$	−0.935750	−0.0836961
$126$	0	0
$127$	9.21396	0.817607	0.408803	−	0.912623i	$-0.365946\pi$
$127$	9.21396	0.817607	0.408803	+	0.912623i	$0.365946\pi$
$128$	0	0
$129$	4.49116	0.395425
$130$	0	0
$131$	−5.87354	−0.513173	−0.256587	−	0.966521i	$-0.582598\pi$
$131$	−5.87354	−0.513173	−0.256587	+	0.966521i	$0.582598\pi$
$132$	0	0
$133$	−0.650141	−0.0563743
$134$	0	0
$135$	−0.519808	−0.0447380
$136$	0	0
$137$	18.7030	1.59790	0.798952	−	0.601394i	$-0.205388\pi$
$137$	18.7030	1.59790	0.798952	+	0.601394i	$0.205388\pi$
$138$	0	0
$139$	−2.04591	−0.173532	−0.0867659	−	0.996229i	$-0.527653\pi$
$139$	−2.04591	−0.173532	−0.0867659	+	0.996229i	$0.527653\pi$
$140$	0	0
$141$	−28.5153	−2.40142
$142$	0	0
$143$	33.7155	2.81943
$144$	0	0
$145$	0.0658122	0.00546540
$146$	0	0
$147$	18.5643	1.53116
$148$	0	0
$149$	−3.21316	−0.263232	−0.131616	−	0.991301i	$-0.542017\pi$
$149$	−3.21316	−0.263232	−0.131616	+	0.991301i	$0.542017\pi$
$150$	0	0
$151$	−2.97817	−0.242360	−0.121180	−	0.992631i	$-0.538668\pi$
$151$	−2.97817	−0.242360	−0.121180	+	0.992631i	$0.538668\pi$
$152$	0	0
$153$	14.3390	1.15924
$154$	0	0
$155$	1.02394	0.0822446
$156$	0	0
$157$	−24.3306	−1.94180	−0.970898	−	0.239494i	$-0.923018\pi$
$157$	−24.3306	−1.94180	−0.970898	+	0.239494i	$0.923018\pi$
$158$	0	0
$159$	−2.82248	−0.223837
$160$	0	0
$161$	−4.92393	−0.388060
$162$	0	0
$163$	20.8029	1.62941	0.814704	−	0.579877i	$-0.196899\pi$
$163$	20.8029	1.62941	0.814704	+	0.579877i	$0.196899\pi$
$164$	0	0
$165$	−1.55419	−0.120993
$166$	0	0
$167$	−10.3801	−0.803235	−0.401618	−	0.915807i	$-0.631552\pi$
$167$	−10.3801	−0.803235	−0.401618	+	0.915807i	$0.631552\pi$
$168$	0	0
$169$	19.8848	1.52960
$170$	0	0
$171$	4.96640	0.379790
$172$	0	0
$173$	−1.15325	−0.0876801	−0.0438401	−	0.999039i	$-0.513959\pi$
$173$	−1.15325	−0.0876801	−0.0438401	+	0.999039i	$0.513959\pi$
$174$	0	0
$175$	3.24500	0.245299
$176$	0	0
$177$	−17.2034	−1.29309
$178$	0	0
$179$	5.74649	0.429513	0.214757	−	0.976668i	$-0.431104\pi$
$179$	5.74649	0.429513	0.214757	+	0.976668i	$0.431104\pi$
$180$	0	0
$181$	1.96507	0.146062	0.0730312	−	0.997330i	$-0.476733\pi$
$181$	1.96507	0.146062	0.0730312	+	0.997330i	$0.476733\pi$
$182$	0	0
$183$	35.0870	2.59371
$184$	0	0
$185$	0.0682372	0.00501690
$186$	0	0
$187$	16.9750	1.24134
$188$	0	0
$189$	3.60835	0.262469
$190$	0	0
$191$	−15.6026	−1.12896	−0.564482	−	0.825445i	$-0.690924\pi$
$191$	−15.6026	−1.12896	−0.564482	+	0.825445i	$0.690924\pi$
$192$	0	0
$193$	−3.19675	−0.230107	−0.115054	−	0.993359i	$-0.536704\pi$
$193$	−3.19675	−0.230107	−0.115054	+	0.993359i	$0.536704\pi$
$194$	0	0
$195$	−1.51590	−0.108556
$196$	0	0
$197$	0.410184	0.0292244	0.0146122	−	0.999893i	$-0.495349\pi$
$197$	0.410184	0.0292244	0.0146122	+	0.999893i	$0.495349\pi$
$198$	0	0
$199$	−19.9737	−1.41590	−0.707948	−	0.706265i	$-0.750379\pi$
$199$	−19.9737	−1.41590	−0.707948	+	0.706265i	$0.750379\pi$
$200$	0	0
$201$	−8.62029	−0.608028
$202$	0	0
$203$	−0.456849	−0.0320645
$204$	0	0
$205$	0.103073	0.00719897
$206$	0	0
$207$	37.6137	2.61433
$208$	0	0
$209$	5.87938	0.406685
$210$	0	0
$211$	18.6289	1.28246	0.641232	−	0.767347i	$-0.278424\pi$
$211$	18.6289	1.28246	0.641232	+	0.767347i	$0.278424\pi$
$212$	0	0
$213$	28.9507	1.98367
$214$	0	0
$215$	−0.149028	−0.0101637
$216$	0	0
$217$	−7.10787	−0.482513
$218$	0	0
$219$	44.3938	2.99986
$220$	0	0
$221$	16.5568	1.11373
$222$	0	0
$223$	−0.809021	−0.0541760	−0.0270880	−	0.999633i	$-0.508623\pi$
$223$	−0.809021	−0.0541760	−0.0270880	+	0.999633i	$0.508623\pi$
$224$	0	0
$225$	−24.7884	−1.65256
$226$	0	0
$227$	9.54991	0.633850	0.316925	−	0.948451i	$-0.397350\pi$
$227$	9.54991	0.633850	0.316925	+	0.948451i	$0.397350\pi$
$228$	0	0
$229$	4.88646	0.322907	0.161453	−	0.986880i	$-0.448382\pi$
$229$	4.88646	0.322907	0.161453	+	0.986880i	$0.448382\pi$
$230$	0	0
$231$	10.7887	0.709845
$232$	0	0
$233$	−3.70817	−0.242930	−0.121465	−	0.992596i	$-0.538759\pi$
$233$	−3.70817	−0.242930	−0.121465	+	0.992596i	$0.538759\pi$
$234$	0	0
$235$	0.946210	0.0617239
$236$	0	0
$237$	6.74965	0.438437
$238$	0	0
$239$	−13.6680	−0.884107	−0.442053	−	0.896989i	$-0.645750\pi$
$239$	−13.6680	−0.884107	−0.442053	+	0.896989i	$0.645750\pi$
$240$	0	0
$241$	−2.71389	−0.174817	−0.0874084	−	0.996173i	$-0.527859\pi$
$241$	−2.71389	−0.174817	−0.0874084	+	0.996173i	$0.527859\pi$
$242$	0	0
$243$	14.4886	0.929444
$244$	0	0
$245$	−0.616013	−0.0393556
$246$	0	0
$247$	5.73453	0.364879
$248$	0	0
$249$	−11.1442	−0.706236
$250$	0	0
$251$	8.55691	0.540108	0.270054	−	0.962845i	$-0.412959\pi$
$251$	8.55691	0.540108	0.270054	+	0.962845i	$0.412959\pi$
$252$	0	0
$253$	44.5283	2.79947
$254$	0	0
$255$	−0.763222	−0.0477948
$256$	0	0
$257$	7.61768	0.475178	0.237589	−	0.971366i	$-0.423643\pi$
$257$	7.61768	0.475178	0.237589	+	0.971366i	$0.423643\pi$
$258$	0	0
$259$	−0.473683	−0.0294332
$260$	0	0
$261$	3.48985	0.216016
$262$	0	0
$263$	9.21661	0.568321	0.284160	−	0.958777i	$-0.408285\pi$
$263$	9.21661	0.568321	0.284160	+	0.958777i	$0.408285\pi$
$264$	0	0
$265$	0.0936572	0.00575332
$266$	0	0
$267$	−26.4000	−1.61565
$268$	0	0
$269$	22.9227	1.39762	0.698812	−	0.715305i	$-0.253712\pi$
$269$	22.9227	1.39762	0.698812	+	0.715305i	$0.253712\pi$
$270$	0	0
$271$	−22.6512	−1.37596	−0.687980	−	0.725729i	$-0.741503\pi$
$271$	−22.6512	−1.37596	−0.687980	+	0.725729i	$0.741503\pi$
$272$	0	0
$273$	10.5229	0.636875
$274$	0	0
$275$	−29.3453	−1.76959
$276$	0	0
$277$	23.8375	1.43225	0.716127	−	0.697970i	$-0.245913\pi$
$277$	23.8375	1.43225	0.716127	+	0.697970i	$0.245913\pi$
$278$	0	0
$279$	54.2967	3.25066
$280$	0	0
$281$	−2.40913	−0.143717	−0.0718583	−	0.997415i	$-0.522893\pi$
$281$	−2.40913	−0.143717	−0.0718583	+	0.997415i	$0.522893\pi$
$282$	0	0
$283$	−13.4124	−0.797287	−0.398643	−	0.917106i	$-0.630519\pi$
$283$	−13.4124	−0.797287	−0.398643	+	0.917106i	$0.630519\pi$
$284$	0	0
$285$	−0.264346	−0.0156585
$286$	0	0
$287$	−0.715506	−0.0422350
$288$	0	0
$289$	−8.66401	−0.509648
$290$	0	0
$291$	−48.5124	−2.84385
$292$	0	0
$293$	−17.5822	−1.02716	−0.513582	−	0.858041i	$-0.671682\pi$
$293$	−17.5822	−1.02716	−0.513582	+	0.858041i	$0.671682\pi$
$294$	0	0
$295$	0.570853	0.0332363
$296$	0	0
$297$	−32.6312	−1.89346
$298$	0	0
$299$	43.4313	2.51169
$300$	0	0
$301$	1.03451	0.0596282
$302$	0	0
$303$	22.5925	1.29790
$304$	0	0
$305$	−1.16428	−0.0666663
$306$	0	0
$307$	−28.8122	−1.64440	−0.822200	−	0.569199i	$-0.807253\pi$
$307$	−28.8122	−1.64440	−0.822200	+	0.569199i	$0.807253\pi$
$308$	0	0
$309$	−6.09789	−0.346897
$310$	0	0
$311$	−32.1786	−1.82468	−0.912341	−	0.409431i	$-0.865727\pi$
$311$	−32.1786	−1.82468	−0.912341	+	0.409431i	$0.865727\pi$
$312$	0	0
$313$	6.89605	0.389788	0.194894	−	0.980824i	$-0.437564\pi$
$313$	6.89605	0.389788	0.194894	+	0.980824i	$0.437564\pi$
$314$	0	0
$315$	−0.302406	−0.0170386
$316$	0	0
$317$	23.3002	1.30867	0.654335	−	0.756204i	$-0.272948\pi$
$317$	23.3002	1.30867	0.654335	+	0.756204i	$0.272948\pi$
$318$	0	0
$319$	4.13139	0.231314
$320$	0	0
$321$	19.2042	1.07187
$322$	0	0
$323$	2.88721	0.160649
$324$	0	0
$325$	−28.6223	−1.58768
$326$	0	0
$327$	−8.03231	−0.444188
$328$	0	0
$329$	−6.56831	−0.362123
$330$	0	0
$331$	7.97717	0.438465	0.219232	−	0.975673i	$-0.429645\pi$
$331$	7.97717	0.438465	0.219232	+	0.975673i	$0.429645\pi$
$332$	0	0
$333$	3.61844	0.198289
$334$	0	0
$335$	0.286043	0.0156282
$336$	0	0
$337$	−14.3441	−0.781371	−0.390685	−	0.920524i	$-0.627762\pi$
$337$	−14.3441	−0.781371	−0.390685	+	0.920524i	$0.627762\pi$
$338$	0	0
$339$	35.1048	1.90663
$340$	0	0
$341$	64.2782	3.48086
$342$	0	0
$343$	8.82716	0.476622
$344$	0	0
$345$	−2.00206	−0.107787
$346$	0	0
$347$	25.2126	1.35348	0.676741	−	0.736221i	$-0.263392\pi$
$347$	25.2126	1.35348	0.676741	+	0.736221i	$0.263392\pi$
$348$	0	0
$349$	−20.9422	−1.12101	−0.560504	−	0.828152i	$-0.689393\pi$
$349$	−20.9422	−1.12101	−0.560504	+	0.828152i	$0.689393\pi$
$350$	0	0
$351$	−31.8273	−1.69881
$352$	0	0
$353$	−15.8173	−0.841871	−0.420935	−	0.907091i	$-0.638298\pi$
$353$	−15.8173	−0.841871	−0.420935	+	0.907091i	$0.638298\pi$
$354$	0	0
$355$	−0.960659	−0.0509865
$356$	0	0
$357$	5.29806	0.280403
$358$	0	0
$359$	−22.6865	−1.19735	−0.598675	−	0.800992i	$-0.704306\pi$
$359$	−22.6865	−1.19735	−0.598675	+	0.800992i	$0.704306\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−66.5177	−3.49127
$364$	0	0
$365$	−1.47310	−0.0771056
$366$	0	0
$367$	7.21968	0.376864	0.188432	−	0.982086i	$-0.439659\pi$
$367$	7.21968	0.376864	0.188432	+	0.982086i	$0.439659\pi$
$368$	0	0
$369$	5.46572	0.284534
$370$	0	0
$371$	−0.650141	−0.0337536
$372$	0	0
$373$	−8.13340	−0.421131	−0.210566	−	0.977580i	$-0.567531\pi$
$373$	−8.13340	−0.421131	−0.210566	+	0.977580i	$0.567531\pi$
$374$	0	0
$375$	2.64114	0.136388
$376$	0	0
$377$	4.02961	0.207535
$378$	0	0
$379$	17.4431	0.895993	0.447996	−	0.894035i	$-0.352138\pi$
$379$	17.4431	0.895993	0.447996	+	0.894035i	$0.352138\pi$
$380$	0	0
$381$	−26.0062	−1.33234
$382$	0	0
$383$	−14.1510	−0.723082	−0.361541	−	0.932356i	$-0.617749\pi$
$383$	−14.1510	−0.723082	−0.361541	+	0.932356i	$0.617749\pi$
$384$	0	0
$385$	−0.357997	−0.0182452
$386$	0	0
$387$	−7.90259	−0.401711
$388$	0	0
$389$	22.0758	1.11929	0.559644	−	0.828733i	$-0.310938\pi$
$389$	22.0758	1.11929	0.559644	+	0.828733i	$0.310938\pi$
$390$	0	0
$391$	21.8667	1.10585
$392$	0	0
$393$	16.5779	0.836247
$394$	0	0
$395$	−0.223971	−0.0112692
$396$	0	0
$397$	15.5585	0.780857	0.390429	−	0.920633i	$-0.372327\pi$
$397$	15.5585	0.780857	0.390429	+	0.920633i	$0.372327\pi$
$398$	0	0
$399$	1.83501	0.0918654
$400$	0	0
$401$	−9.31707	−0.465272	−0.232636	−	0.972564i	$-0.574735\pi$
$401$	−9.31707	−0.465272	−0.232636	+	0.972564i	$0.574735\pi$
$402$	0	0
$403$	62.6945	3.12304
$404$	0	0
$405$	0.0717320	0.00356439
$406$	0	0
$407$	4.28363	0.212332
$408$	0	0
$409$	−37.1572	−1.83730	−0.918652	−	0.395067i	$-0.870721\pi$
$409$	−37.1572	−1.83730	−0.918652	+	0.395067i	$0.870721\pi$
$410$	0	0
$411$	−52.7888	−2.60388
$412$	0	0
$413$	−3.96269	−0.194991
$414$	0	0
$415$	0.369794	0.0181525
$416$	0	0
$417$	5.77454	0.282781
$418$	0	0
$419$	−5.22322	−0.255171	−0.127585	−	0.991828i	$-0.540723\pi$
$419$	−5.22322	−0.255171	−0.127585	+	0.991828i	$0.540723\pi$
$420$	0	0
$421$	28.5676	1.39230	0.696150	−	0.717896i	$-0.254895\pi$
$421$	28.5676	1.39230	0.696150	+	0.717896i	$0.254895\pi$
$422$	0	0
$423$	50.1750	2.43959
$424$	0	0
$425$	−14.4107	−0.699023
$426$	0	0
$427$	8.08207	0.391119
$428$	0	0
$429$	−95.1612	−4.59443
$430$	0	0
$431$	23.2488	1.11985	0.559927	−	0.828542i	$-0.310829\pi$
$431$	23.2488	1.11985	0.559927	+	0.828542i	$0.310829\pi$
$432$	0	0
$433$	12.5833	0.604714	0.302357	−	0.953195i	$-0.402226\pi$
$433$	12.5833	0.604714	0.302357	+	0.953195i	$0.402226\pi$
$434$	0	0
$435$	−0.185754	−0.00890620
$436$	0	0
$437$	7.57364	0.362297
$438$	0	0
$439$	−14.1191	−0.673869	−0.336935	−	0.941528i	$-0.609390\pi$
$439$	−14.1191	−0.673869	−0.336935	+	0.941528i	$0.609390\pi$
$440$	0	0
$441$	−32.6656	−1.55550
$442$	0	0
$443$	−5.42255	−0.257633	−0.128817	−	0.991668i	$-0.541118\pi$
$443$	−5.42255	−0.257633	−0.128817	+	0.991668i	$0.541118\pi$
$444$	0	0
$445$	0.876019	0.0415273
$446$	0	0
$447$	9.06907	0.428952
$448$	0	0
$449$	−18.3659	−0.866743	−0.433371	−	0.901215i	$-0.642676\pi$
$449$	−18.3659	−0.866743	−0.433371	+	0.901215i	$0.642676\pi$
$450$	0	0
$451$	6.47049	0.304684
$452$	0	0
$453$	8.40582	0.394940
$454$	0	0
$455$	−0.349177	−0.0163697
$456$	0	0
$457$	4.70723	0.220195	0.110097	−	0.993921i	$-0.464884\pi$
$457$	4.70723	0.220195	0.110097	+	0.993921i	$0.464884\pi$
$458$	0	0
$459$	−16.0244	−0.747953
$460$	0	0
$461$	−13.3922	−0.623736	−0.311868	−	0.950125i	$-0.600955\pi$
$461$	−13.3922	−0.623736	−0.311868	+	0.950125i	$0.600955\pi$
$462$	0	0
$463$	14.5228	0.674931	0.337465	−	0.941338i	$-0.390430\pi$
$463$	14.5228	0.674931	0.337465	+	0.941338i	$0.390430\pi$
$464$	0	0
$465$	−2.89004	−0.134022
$466$	0	0
$467$	1.52068	0.0703687	0.0351843	−	0.999381i	$-0.488798\pi$
$467$	1.52068	0.0703687	0.0351843	+	0.999381i	$0.488798\pi$
$468$	0	0
$469$	−1.98563	−0.0916878
$470$	0	0
$471$	68.6727	3.16427
$472$	0	0
$473$	−9.35534	−0.430159
$474$	0	0
$475$	−4.99123	−0.229013
$476$	0	0
$477$	4.96640	0.227396
$478$	0	0
$479$	−24.9306	−1.13911	−0.569554	−	0.821954i	$-0.692884\pi$
$479$	−24.9306	−1.13911	−0.569554	+	0.821954i	$0.692884\pi$
$480$	0	0
$481$	4.17809	0.190505
$482$	0	0
$483$	13.8977	0.632367
$484$	0	0
$485$	1.60977	0.0730957
$486$	0	0
$487$	18.9077	0.856790	0.428395	−	0.903592i	$-0.359079\pi$
$487$	18.9077	0.856790	0.428395	+	0.903592i	$0.359079\pi$
$488$	0	0
$489$	−58.7158	−2.65522
$490$	0	0
$491$	−27.8607	−1.25734	−0.628668	−	0.777674i	$-0.716399\pi$
$491$	−27.8607	−1.25734	−0.628668	+	0.777674i	$0.716399\pi$
$492$	0	0
$493$	2.02882	0.0913735
$494$	0	0
$495$	2.73473	0.122917
$496$	0	0
$497$	6.66861	0.299128
$498$	0	0
$499$	−18.5544	−0.830609	−0.415305	−	0.909682i	$-0.636325\pi$
$499$	−18.5544	−0.830609	−0.415305	+	0.909682i	$0.636325\pi$
$500$	0	0
$501$	29.2976	1.30892
$502$	0	0
$503$	7.77031	0.346461	0.173230	−	0.984881i	$-0.444579\pi$
$503$	7.77031	0.346461	0.173230	+	0.984881i	$0.444579\pi$
$504$	0	0
$505$	−0.749676	−0.0333602
$506$	0	0
$507$	−56.1244	−2.49257
$508$	0	0
$509$	32.3489	1.43384	0.716921	−	0.697155i	$-0.245551\pi$
$509$	32.3489	1.43384	0.716921	+	0.697155i	$0.245551\pi$
$510$	0	0
$511$	10.2258	0.452364
$512$	0	0
$513$	−5.55011	−0.245044
$514$	0	0
$515$	0.202344	0.00891633
$516$	0	0
$517$	59.3988	2.61236
$518$	0	0
$519$	3.25503	0.142880
$520$	0	0
$521$	−8.94847	−0.392039	−0.196020	−	0.980600i	$-0.562802\pi$
$521$	−8.94847	−0.392039	−0.196020	+	0.980600i	$0.562802\pi$
$522$	0	0
$523$	7.20749	0.315162	0.157581	−	0.987506i	$-0.449631\pi$
$523$	7.20749	0.315162	0.157581	+	0.987506i	$0.449631\pi$
$524$	0	0
$525$	−9.15895	−0.399729
$526$	0	0
$527$	31.5654	1.37501
$528$	0	0
$529$	34.3601	1.49392
$530$	0	0
$531$	30.2708	1.31364
$532$	0	0
$533$	6.31108	0.273363
$534$	0	0
$535$	−0.637244	−0.0275504
$536$	0	0
$537$	−16.2194	−0.699917
$538$	0	0
$539$	−38.6705	−1.66566
$540$	0	0
$541$	−41.4971	−1.78410	−0.892050	−	0.451936i	$-0.850733\pi$
$541$	−41.4971	−1.78410	−0.892050	+	0.451936i	$0.850733\pi$
$542$	0	0
$543$	−5.54637	−0.238018
$544$	0	0
$545$	0.266533	0.0114170
$546$	0	0
$547$	25.9801	1.11083	0.555415	−	0.831573i	$-0.312559\pi$
$547$	25.9801	1.11083	0.555415	+	0.831573i	$0.312559\pi$
$548$	0	0
$549$	−61.7386	−2.63494
$550$	0	0
$551$	0.702692	0.0299357
$552$	0	0
$553$	1.55474	0.0661142
$554$	0	0
$555$	−0.192598	−0.00817534
$556$	0	0
$557$	9.90023	0.419486	0.209743	−	0.977757i	$-0.432737\pi$
$557$	9.90023	0.419486	0.209743	+	0.977757i	$0.432737\pi$
$558$	0	0
$559$	−9.12485	−0.385940
$560$	0	0
$561$	−47.9116	−2.02283
$562$	0	0
$563$	−33.4501	−1.40975	−0.704876	−	0.709330i	$-0.748998\pi$
$563$	−33.4501	−1.40975	−0.704876	+	0.709330i	$0.748998\pi$
$564$	0	0
$565$	−1.16487	−0.0490064
$566$	0	0
$567$	−0.497942	−0.0209116
$568$	0	0
$569$	−30.2043	−1.26623	−0.633115	−	0.774058i	$-0.718224\pi$
$569$	−30.2043	−1.26623	−0.633115	+	0.774058i	$0.718224\pi$
$570$	0	0
$571$	24.8112	1.03832	0.519158	−	0.854678i	$-0.326246\pi$
$571$	24.8112	1.03832	0.519158	+	0.854678i	$0.326246\pi$
$572$	0	0
$573$	44.0380	1.83972
$574$	0	0
$575$	−37.8018	−1.57644
$576$	0	0
$577$	−4.44965	−0.185241	−0.0926206	−	0.995701i	$-0.529524\pi$
$577$	−4.44965	−0.185241	−0.0926206	+	0.995701i	$0.529524\pi$
$578$	0	0
$579$	9.02277	0.374973
$580$	0	0
$581$	−2.56700	−0.106497
$582$	0	0
$583$	5.87938	0.243499
$584$	0	0
$585$	2.66735	0.110281
$586$	0	0
$587$	8.36650	0.345323	0.172661	−	0.984981i	$-0.444763\pi$
$587$	8.36650	0.345323	0.172661	+	0.984981i	$0.444763\pi$
$588$	0	0
$589$	10.9328	0.450479
$590$	0	0
$591$	−1.15774	−0.0476229
$592$	0	0
$593$	−5.02494	−0.206349	−0.103175	−	0.994663i	$-0.532900\pi$
$593$	−5.02494	−0.206349	−0.103175	+	0.994663i	$0.532900\pi$
$594$	0	0
$595$	−0.175803	−0.00720723
$596$	0	0
$597$	56.3753	2.30729
$598$	0	0
$599$	3.48609	0.142438	0.0712188	−	0.997461i	$-0.477311\pi$
$599$	3.48609	0.142438	0.0712188	+	0.997461i	$0.477311\pi$
$600$	0	0
$601$	−41.6082	−1.69723	−0.848616	−	0.529009i	$-0.822564\pi$
$601$	−41.6082	−1.69723	−0.848616	+	0.529009i	$0.822564\pi$
$602$	0	0
$603$	15.1681	0.617694
$604$	0	0
$605$	2.20723	0.0897366
$606$	0	0
$607$	15.1203	0.613714	0.306857	−	0.951756i	$-0.400723\pi$
$607$	15.1203	0.613714	0.306857	+	0.951756i	$0.400723\pi$
$608$	0	0
$609$	1.28945	0.0522510
$610$	0	0
$611$	57.9354	2.34382
$612$	0	0
$613$	−27.6346	−1.11615	−0.558076	−	0.829790i	$-0.688460\pi$
$613$	−27.6346	−1.11615	−0.558076	+	0.829790i	$0.688460\pi$
$614$	0	0
$615$	−0.290923	−0.0117311
$616$	0	0
$617$	−35.9340	−1.44665	−0.723324	−	0.690509i	$-0.757387\pi$
$617$	−35.9340	−1.44665	−0.723324	+	0.690509i	$0.757387\pi$
$618$	0	0
$619$	14.3209	0.575606	0.287803	−	0.957690i	$-0.407075\pi$
$619$	14.3209	0.575606	0.287803	+	0.957690i	$0.407075\pi$
$620$	0	0
$621$	−42.0346	−1.68679
$622$	0	0
$623$	−6.08106	−0.243633
$624$	0	0
$625$	24.8685	0.994740
$626$	0	0
$627$	−16.5944	−0.662718
$628$	0	0
$629$	2.10358	0.0838752
$630$	0	0
$631$	29.4412	1.17204	0.586018	−	0.810298i	$-0.300695\pi$
$631$	29.4412	1.17204	0.586018	+	0.810298i	$0.300695\pi$
$632$	0	0
$633$	−52.5796	−2.08985
$634$	0	0
$635$	0.862953	0.0342453
$636$	0	0
$637$	−37.7178	−1.49443
$638$	0	0
$639$	−50.9412	−2.01520
$640$	0	0
$641$	40.8975	1.61535	0.807676	−	0.589626i	$-0.200725\pi$
$641$	40.8975	1.61535	0.807676	+	0.589626i	$0.200725\pi$
$642$	0	0
$643$	44.7641	1.76532	0.882661	−	0.470010i	$-0.155750\pi$
$643$	44.7641	1.76532	0.882661	+	0.470010i	$0.155750\pi$
$644$	0	0
$645$	0.420630	0.0165623
$646$	0	0
$647$	36.1874	1.42267	0.711336	−	0.702852i	$-0.248090\pi$
$647$	36.1874	1.42267	0.711336	+	0.702852i	$0.248090\pi$
$648$	0	0
$649$	35.8356	1.40667
$650$	0	0
$651$	20.0618	0.786285
$652$	0	0
$653$	−7.71033	−0.301728	−0.150864	−	0.988554i	$-0.548206\pi$
$653$	−7.71033	−0.301728	−0.150864	+	0.988554i	$0.548206\pi$
$654$	0	0
$655$	−0.550099	−0.0214941
$656$	0	0
$657$	−78.1147	−3.04754
$658$	0	0
$659$	1.01541	0.0395547	0.0197774	−	0.999804i	$-0.493704\pi$
$659$	1.01541	0.0395547	0.0197774	+	0.999804i	$0.493704\pi$
$660$	0	0
$661$	−5.65764	−0.220057	−0.110028	−	0.993928i	$-0.535094\pi$
$661$	−5.65764	−0.220057	−0.110028	+	0.993928i	$0.535094\pi$
$662$	0	0
$663$	−46.7312	−1.81489
$664$	0	0
$665$	−0.0608903	−0.00236123
$666$	0	0
$667$	5.32194	0.206066
$668$	0	0
$669$	2.28344	0.0882831
$670$	0	0
$671$	−73.0881	−2.82154
$672$	0	0
$673$	15.4574	0.595839	0.297919	−	0.954591i	$-0.403707\pi$
$673$	15.4574	0.595839	0.297919	+	0.954591i	$0.403707\pi$
$674$	0	0
$675$	27.7019	1.06625
$676$	0	0
$677$	−42.6664	−1.63980	−0.819902	−	0.572504i	$-0.805972\pi$
$677$	−42.6664	−1.63980	−0.819902	+	0.572504i	$0.805972\pi$
$678$	0	0
$679$	−11.1745	−0.428839
$680$	0	0
$681$	−26.9544	−1.03290
$682$	0	0
$683$	−36.9492	−1.41382	−0.706912	−	0.707302i	$-0.749912\pi$
$683$	−36.9492	−1.41382	−0.706912	+	0.707302i	$0.749912\pi$
$684$	0	0
$685$	1.75167	0.0669279
$686$	0	0
$687$	−13.7919	−0.526196
$688$	0	0
$689$	5.73453	0.218468
$690$	0	0
$691$	−1.25085	−0.0475845	−0.0237922	−	0.999717i	$-0.507574\pi$
$691$	−1.25085	−0.0475845	−0.0237922	+	0.999717i	$0.507574\pi$
$692$	0	0
$693$	−18.9837	−0.721130
$694$	0	0
$695$	−0.191614	−0.00726834
$696$	0	0
$697$	3.17749	0.120356
$698$	0	0
$699$	10.4662	0.395869
$700$	0	0
$701$	39.8051	1.50342	0.751710	−	0.659494i	$-0.229229\pi$
$701$	39.8051	1.50342	0.751710	+	0.659494i	$0.229229\pi$
$702$	0	0
$703$	0.728585	0.0274791
$704$	0	0
$705$	−2.67066	−0.100583
$706$	0	0
$707$	5.20403	0.195718
$708$	0	0
$709$	24.7694	0.930236	0.465118	−	0.885249i	$-0.346012\pi$
$709$	24.7694	0.930236	0.465118	+	0.885249i	$0.346012\pi$
$710$	0	0
$711$	−11.8766	−0.445407
$712$	0	0
$713$	82.8013	3.10093
$714$	0	0
$715$	3.15769	0.118091
$716$	0	0
$717$	38.5775	1.44070
$718$	0	0
$719$	−14.1998	−0.529563	−0.264782	−	0.964308i	$-0.585300\pi$
$719$	−14.1998	−0.529563	−0.264782	+	0.964308i	$0.585300\pi$
$720$	0	0
$721$	−1.40461	−0.0523104
$722$	0	0
$723$	7.65989	0.284875
$724$	0	0
$725$	−3.50730	−0.130258
$726$	0	0
$727$	−31.2833	−1.16023	−0.580116	−	0.814534i	$-0.696993\pi$
$727$	−31.2833	−1.16023	−0.580116	+	0.814534i	$0.696993\pi$
$728$	0	0
$729$	−43.1915	−1.59969
$730$	0	0
$731$	−4.59417	−0.169921
$732$	0	0
$733$	−29.1695	−1.07740	−0.538701	−	0.842497i	$-0.681085\pi$
$733$	−29.1695	−1.07740	−0.538701	+	0.842497i	$0.681085\pi$
$734$	0	0
$735$	1.73868	0.0641324
$736$	0	0
$737$	17.9565	0.661437
$738$	0	0
$739$	12.4382	0.457547	0.228774	−	0.973480i	$-0.426528\pi$
$739$	12.4382	0.457547	0.228774	+	0.973480i	$0.426528\pi$
$740$	0	0
$741$	−16.1856	−0.594592
$742$	0	0
$743$	−33.3290	−1.22272	−0.611361	−	0.791352i	$-0.709378\pi$
$743$	−33.3290	−1.22272	−0.611361	+	0.791352i	$0.709378\pi$
$744$	0	0
$745$	−0.300935	−0.0110254
$746$	0	0
$747$	19.6092	0.717463
$748$	0	0
$749$	4.42356	0.161633
$750$	0	0
$751$	4.73182	0.172666	0.0863332	−	0.996266i	$-0.472485\pi$
$751$	4.73182	0.172666	0.0863332	+	0.996266i	$0.472485\pi$
$752$	0	0
$753$	−24.1517	−0.880138
$754$	0	0
$755$	−0.278927	−0.0101512
$756$	0	0
$757$	36.0822	1.31143	0.655715	−	0.755009i	$-0.272367\pi$
$757$	36.0822	1.31143	0.655715	+	0.755009i	$0.272367\pi$
$758$	0	0
$759$	−125.680	−4.56191
$760$	0	0
$761$	−20.8977	−0.757542	−0.378771	−	0.925490i	$-0.623653\pi$
$761$	−20.8977	−0.757542	−0.378771	+	0.925490i	$0.623653\pi$
$762$	0	0
$763$	−1.85019	−0.0669814
$764$	0	0
$765$	1.34295	0.0485546
$766$	0	0
$767$	34.9527	1.26207
$768$	0	0
$769$	−40.4964	−1.46034	−0.730168	−	0.683267i	$-0.760558\pi$
$769$	−40.4964	−1.46034	−0.730168	+	0.683267i	$0.760558\pi$
$770$	0	0
$771$	−21.5008	−0.774331
$772$	0	0
$773$	−17.8443	−0.641814	−0.320907	−	0.947111i	$-0.603988\pi$
$773$	−17.8443	−0.641814	−0.320907	+	0.947111i	$0.603988\pi$
$774$	0	0
$775$	−54.5682	−1.96015
$776$	0	0
$777$	1.33696	0.0479632
$778$	0	0
$779$	1.10054	0.0394309
$780$	0	0
$781$	−60.3059	−2.15791
$782$	0	0
$783$	−3.90002	−0.139375
$784$	0	0
$785$	−2.27874	−0.0813317
$786$	0	0
$787$	−14.0369	−0.500360	−0.250180	−	0.968199i	$-0.580490\pi$
$787$	−14.0369	−0.500360	−0.250180	+	0.968199i	$0.580490\pi$
$788$	0	0
$789$	−26.0137	−0.926113
$790$	0	0
$791$	8.08617	0.287511
$792$	0	0
$793$	−71.2874	−2.53149
$794$	0	0
$795$	−0.264346	−0.00937537
$796$	0	0
$797$	−1.52654	−0.0540730	−0.0270365	−	0.999634i	$-0.508607\pi$
$797$	−1.52654	−0.0540730	−0.0270365	+	0.999634i	$0.508607\pi$
$798$	0	0
$799$	29.1692	1.03193
$800$	0	0
$801$	46.4530	1.64134
$802$	0	0
$803$	−92.4747	−3.26336
$804$	0	0
$805$	−0.461162	−0.0162538
$806$	0	0
$807$	−64.6990	−2.27751
$808$	0	0
$809$	13.9799	0.491505	0.245753	−	0.969333i	$-0.420965\pi$
$809$	13.9799	0.491505	0.245753	+	0.969333i	$0.420965\pi$
$810$	0	0
$811$	−37.6662	−1.32264	−0.661320	−	0.750104i	$-0.730003\pi$
$811$	−37.6662	−1.32264	−0.661320	+	0.750104i	$0.730003\pi$
$812$	0	0
$813$	63.9325	2.24221
$814$	0	0
$815$	1.94834	0.0682474
$816$	0	0
$817$	−1.59121	−0.0556695
$818$	0	0
$819$	−18.5160	−0.647000
$820$	0	0
$821$	−25.8786	−0.903169	−0.451585	−	0.892228i	$-0.649141\pi$
$821$	−25.8786	−0.903169	−0.451585	+	0.892228i	$0.649141\pi$
$822$	0	0
$823$	−18.8354	−0.656560	−0.328280	−	0.944580i	$-0.606469\pi$
$823$	−18.8354	−0.656560	−0.328280	+	0.944580i	$0.606469\pi$
$824$	0	0
$825$	82.8266	2.88365
$826$	0	0
$827$	−33.4326	−1.16256	−0.581282	−	0.813702i	$-0.697449\pi$
$827$	−33.4326	−1.16256	−0.581282	+	0.813702i	$0.697449\pi$
$828$	0	0
$829$	−12.6391	−0.438974	−0.219487	−	0.975615i	$-0.570438\pi$
$829$	−12.6391	−0.438974	−0.219487	+	0.975615i	$0.570438\pi$
$830$	0	0
$831$	−67.2808	−2.33394
$832$	0	0
$833$	−18.9901	−0.657968
$834$	0	0
$835$	−0.972170	−0.0336433
$836$	0	0
$837$	−60.6784	−2.09735
$838$	0	0
$839$	−35.9688	−1.24178	−0.620890	−	0.783898i	$-0.713228\pi$
$839$	−35.9688	−1.24178	−0.620890	+	0.783898i	$0.713228\pi$
$840$	0	0
$841$	−28.5062	−0.982973
$842$	0	0
$843$	6.79972	0.234195
$844$	0	0
$845$	1.86235	0.0640669
$846$	0	0
$847$	−15.3219	−0.526468
$848$	0	0
$849$	37.8563	1.29923
$850$	0	0
$851$	5.51804	0.189156
$852$	0	0
$853$	−33.2648	−1.13896	−0.569482	−	0.822004i	$-0.692856\pi$
$853$	−33.2648	−1.13896	−0.569482	+	0.822004i	$0.692856\pi$
$854$	0	0
$855$	0.465139	0.0159074
$856$	0	0
$857$	1.39360	0.0476046	0.0238023	−	0.999717i	$-0.492423\pi$
$857$	1.39360	0.0476046	0.0238023	+	0.999717i	$0.492423\pi$
$858$	0	0
$859$	43.4395	1.48214	0.741068	−	0.671430i	$-0.234320\pi$
$859$	43.4395	1.48214	0.741068	+	0.671430i	$0.234320\pi$
$860$	0	0
$861$	2.01950	0.0688244
$862$	0	0
$863$	47.0463	1.60148	0.800738	−	0.599015i	$-0.204441\pi$
$863$	47.0463	1.60148	0.800738	+	0.599015i	$0.204441\pi$
$864$	0	0
$865$	−0.108010	−0.00367246
$866$	0	0
$867$	24.4540	0.830501
$868$	0	0
$869$	−14.0599	−0.476949
$870$	0	0
$871$	17.5141	0.593443
$872$	0	0
$873$	85.3617	2.88905
$874$	0	0
$875$	0.608369	0.0205666
$876$	0	0
$877$	−13.8415	−0.467394	−0.233697	−	0.972309i	$-0.575082\pi$
$877$	−13.8415	−0.467394	−0.233697	+	0.972309i	$0.575082\pi$
$878$	0	0
$879$	49.6255	1.67382
$880$	0	0
$881$	31.9384	1.07603	0.538017	−	0.842934i	$-0.319174\pi$
$881$	31.9384	1.07603	0.538017	+	0.842934i	$0.319174\pi$
$882$	0	0
$883$	18.8732	0.635135	0.317568	−	0.948236i	$-0.397134\pi$
$883$	18.8732	0.635135	0.317568	+	0.948236i	$0.397134\pi$
$884$	0	0
$885$	−1.61122	−0.0541606
$886$	0	0
$887$	16.4260	0.551531	0.275765	−	0.961225i	$-0.411069\pi$
$887$	16.4260	0.551531	0.275765	+	0.961225i	$0.411069\pi$
$888$	0	0
$889$	−5.99037	−0.200910
$890$	0	0
$891$	4.50302	0.150857
$892$	0	0
$893$	10.1029	0.338081
$894$	0	0
$895$	0.538201	0.0179901
$896$	0	0
$897$	−122.584	−4.09296
$898$	0	0
$899$	7.68241	0.256223
$900$	0	0
$901$	2.88721	0.0961870
$902$	0	0
$903$	−2.91989	−0.0971678
$904$	0	0
$905$	0.184043	0.00611779
$906$	0	0
$907$	54.5293	1.81062	0.905308	−	0.424755i	$-0.139640\pi$
$907$	54.5293	1.81062	0.905308	+	0.424755i	$0.139640\pi$
$908$	0	0
$909$	−39.7534	−1.31854
$910$	0	0
$911$	7.19145	0.238263	0.119132	−	0.992878i	$-0.461989\pi$
$911$	7.19145	0.238263	0.119132	+	0.992878i	$0.461989\pi$
$912$	0	0
$913$	23.2140	0.768272
$914$	0	0
$915$	3.28615	0.108637
$916$	0	0
$917$	3.81862	0.126102
$918$	0	0
$919$	11.6037	0.382770	0.191385	−	0.981515i	$-0.438702\pi$
$919$	11.6037	0.382770	0.191385	+	0.981515i	$0.438702\pi$
$920$	0	0
$921$	81.3219	2.67965
$922$	0	0
$923$	−58.8201	−1.93609
$924$	0	0
$925$	−3.63653	−0.119569
$926$	0	0
$927$	10.7298	0.352411
$928$	0	0
$929$	−6.56249	−0.215308	−0.107654	−	0.994188i	$-0.534334\pi$
$929$	−6.56249	−0.215308	−0.107654	+	0.994188i	$0.534334\pi$
$930$	0	0
$931$	−6.57732	−0.215563
$932$	0	0
$933$	90.8235	2.97343
$934$	0	0
$935$	1.58983	0.0519931
$936$	0	0
$937$	54.4758	1.77965	0.889823	−	0.456306i	$-0.150828\pi$
$937$	54.4758	1.77965	0.889823	+	0.456306i	$0.150828\pi$
$938$	0	0
$939$	−19.4640	−0.635183
$940$	0	0
$941$	1.16727	0.0380519	0.0190259	−	0.999819i	$-0.493943\pi$
$941$	1.16727	0.0380519	0.0190259	+	0.999819i	$0.493943\pi$
$942$	0	0
$943$	8.33510	0.271428
$944$	0	0
$945$	0.337948	0.0109935
$946$	0	0
$947$	15.2843	0.496673	0.248336	−	0.968674i	$-0.420116\pi$
$947$	15.2843	0.496673	0.248336	+	0.968674i	$0.420116\pi$
$948$	0	0
$949$	−90.1964	−2.92790
$950$	0	0
$951$	−65.7644	−2.13256
$952$	0	0
$953$	−26.0212	−0.842909	−0.421455	−	0.906849i	$-0.638480\pi$
$953$	−26.0212	−0.842909	−0.421455	+	0.906849i	$0.638480\pi$
$954$	0	0
$955$	−1.46130	−0.0472864
$956$	0	0
$957$	−11.6608	−0.376940
$958$	0	0
$959$	−12.1596	−0.392653
$960$	0	0
$961$	88.5265	2.85569
$962$	0	0
$963$	−33.7914	−1.08891
$964$	0	0
$965$	−0.299399	−0.00963799
$966$	0	0
$967$	0.140967	0.00453319	0.00226659	−	0.999997i	$-0.499279\pi$
$967$	0.140967	0.00453319	0.00226659	+	0.999997i	$0.499279\pi$
$968$	0	0
$969$	−8.14910	−0.261787
$970$	0	0
$971$	−28.5405	−0.915907	−0.457954	−	0.888976i	$-0.651417\pi$
$971$	−28.5405	−0.915907	−0.457954	+	0.888976i	$0.651417\pi$
$972$	0	0
$973$	1.33013	0.0426420
$974$	0	0
$975$	80.7860	2.58722
$976$	0	0
$977$	−41.7495	−1.33569	−0.667843	−	0.744302i	$-0.732782\pi$
$977$	−41.7495	−1.33569	−0.667843	+	0.744302i	$0.732782\pi$
$978$	0	0
$979$	54.9925	1.75757
$980$	0	0
$981$	14.1335	0.451249
$982$	0	0
$983$	−25.1070	−0.800790	−0.400395	−	0.916343i	$-0.631127\pi$
$983$	−25.1070	−0.800790	−0.400395	+	0.916343i	$0.631127\pi$
$984$	0	0
$985$	0.0384167	0.00122406
$986$	0	0
$987$	18.5389	0.590101
$988$	0	0
$989$	−12.0513	−0.383208
$990$	0	0
$991$	32.0246	1.01729	0.508647	−	0.860975i	$-0.330146\pi$
$991$	32.0246	1.01729	0.508647	+	0.860975i	$0.330146\pi$
$992$	0	0
$993$	−22.5154	−0.714505
$994$	0	0
$995$	−1.87068	−0.0593045
$996$	0	0
$997$	20.2936	0.642704	0.321352	−	0.946960i	$-0.395863\pi$
$997$	20.2936	0.642704	0.321352	+	0.946960i	$0.395863\pi$
$998$	0	0
$999$	−4.04373	−0.127938

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.e.1.2	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.e.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-2.82248 q^{3} +0.0936572 q^{5} -0.650141 q^{7} +4.96640 q^{9} +O(q^{10})\)	\(q-2.82248 q^{3} +0.0936572 q^{5} -0.650141 q^{7} +4.96640 q^{9} +5.87938 q^{11} +5.73453 q^{13} -0.264346 q^{15} +2.88721 q^{17} +1.00000 q^{19} +1.83501 q^{21} +7.57364 q^{23} -4.99123 q^{25} -5.55011 q^{27} +0.702692 q^{29} +10.9328 q^{31} -16.5944 q^{33} -0.0608903 q^{35} +0.728585 q^{37} -16.1856 q^{39} +1.10054 q^{41} -1.59121 q^{43} +0.465139 q^{45} +10.1029 q^{47} -6.57732 q^{49} -8.14910 q^{51} +1.00000 q^{53} +0.550646 q^{55} -2.82248 q^{57} +6.09513 q^{59} -12.4313 q^{61} -3.22886 q^{63} +0.537080 q^{65} +3.05415 q^{67} -21.3765 q^{69} -10.2572 q^{71} -15.7286 q^{73} +14.0876 q^{75} -3.82242 q^{77} -2.39139 q^{79} +0.765900 q^{81} +3.94838 q^{83} +0.270408 q^{85} -1.98334 q^{87} +9.35346 q^{89} -3.72825 q^{91} -30.8577 q^{93} +0.0936572 q^{95} +17.1878 q^{97} +29.1993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) \) 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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