LMFDB - Embedded newform 4028.2.a.f.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} - 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			1.4
Root			$-1.93943$ 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)

$f(q)$	$=$	$q-1.93943 q^{3} +0.842053 q^{5} -2.29879 q^{7} +0.761401 q^{9} +O(q^{10})$	$q-1.93943 q^{3} +0.842053 q^{5} -2.29879 q^{7} +0.761401 q^{9} -4.68475 q^{11} -6.07426 q^{13} -1.63311 q^{15} -5.58884 q^{17} -1.00000 q^{19} +4.45834 q^{21} -7.82359 q^{23} -4.29095 q^{25} +4.34161 q^{27} -8.09724 q^{29} +9.03212 q^{31} +9.08576 q^{33} -1.93570 q^{35} +2.03569 q^{37} +11.7806 q^{39} -4.95946 q^{41} -0.970721 q^{43} +0.641140 q^{45} +7.88034 q^{47} -1.71558 q^{49} +10.8392 q^{51} -1.00000 q^{53} -3.94481 q^{55} +1.93943 q^{57} +0.226530 q^{59} +5.53150 q^{61} -1.75030 q^{63} -5.11485 q^{65} +11.3747 q^{67} +15.1733 q^{69} -6.09899 q^{71} +2.94500 q^{73} +8.32200 q^{75} +10.7692 q^{77} -9.44518 q^{79} -10.7045 q^{81} -2.94000 q^{83} -4.70610 q^{85} +15.7041 q^{87} +4.88041 q^{89} +13.9634 q^{91} -17.5172 q^{93} -0.842053 q^{95} +0.524856 q^{97} -3.56697 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.93943	−1.11973	−0.559866	−	0.828583i	$-0.689147\pi$
$3$	−1.93943	−1.11973	−0.559866	+	0.828583i	$0.689147\pi$
$4$	0	0
$5$	0.842053	0.376578	0.188289	−	0.982114i	$-0.439706\pi$
$5$	0.842053	0.376578	0.188289	+	0.982114i	$0.439706\pi$
$6$	0	0
$7$	−2.29879	−0.868859	−0.434430	−	0.900706i	$-0.643050\pi$
$7$	−2.29879	−0.868859	−0.434430	+	0.900706i	$0.643050\pi$
$8$	0	0
$9$	0.761401	0.253800
$10$	0	0
$11$	−4.68475	−1.41250	−0.706252	−	0.707960i	$-0.749616\pi$
$11$	−4.68475	−1.41250	−0.706252	+	0.707960i	$0.749616\pi$
$12$	0	0
$13$	−6.07426	−1.68470	−0.842348	−	0.538934i	$-0.818827\pi$
$13$	−6.07426	−1.68470	−0.842348	+	0.538934i	$0.818827\pi$
$14$	0	0
$15$	−1.63311	−0.421666
$16$	0	0
$17$	−5.58884	−1.35549	−0.677746	−	0.735296i	$-0.737043\pi$
$17$	−5.58884	−1.35549	−0.677746	+	0.735296i	$0.737043\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	4.45834	0.972890
$22$	0	0
$23$	−7.82359	−1.63133	−0.815666	−	0.578523i	$-0.803629\pi$
$23$	−7.82359	−1.63133	−0.815666	+	0.578523i	$0.803629\pi$
$24$	0	0
$25$	−4.29095	−0.858189
$26$	0	0
$27$	4.34161	0.835544
$28$	0	0
$29$	−8.09724	−1.50362	−0.751810	−	0.659380i	$-0.770819\pi$
$29$	−8.09724	−1.50362	−0.751810	+	0.659380i	$0.770819\pi$
$30$	0	0
$31$	9.03212	1.62222	0.811109	−	0.584895i	$-0.198864\pi$
$31$	9.03212	1.62222	0.811109	+	0.584895i	$0.198864\pi$
$32$	0	0
$33$	9.08576	1.58163
$34$	0	0
$35$	−1.93570	−0.327193
$36$	0	0
$37$	2.03569	0.334666	0.167333	−	0.985900i	$-0.446484\pi$
$37$	2.03569	0.334666	0.167333	+	0.985900i	$0.446484\pi$
$38$	0	0
$39$	11.7806	1.88641
$40$	0	0
$41$	−4.95946	−0.774538	−0.387269	−	0.921967i	$-0.626582\pi$
$41$	−4.95946	−0.774538	−0.387269	+	0.921967i	$0.626582\pi$
$42$	0	0
$43$	−0.970721	−0.148033	−0.0740167	−	0.997257i	$-0.523582\pi$
$43$	−0.970721	−0.148033	−0.0740167	+	0.997257i	$0.523582\pi$
$44$	0	0
$45$	0.641140	0.0955756
$46$	0	0
$47$	7.88034	1.14947	0.574733	−	0.818341i	$-0.305106\pi$
$47$	7.88034	1.14947	0.574733	+	0.818341i	$0.305106\pi$
$48$	0	0
$49$	−1.71558	−0.245083
$50$	0	0
$51$	10.8392	1.51779
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	−3.94481	−0.531918
$56$	0	0
$57$	1.93943	0.256884
$58$	0	0
$59$	0.226530	0.0294918	0.0147459	−	0.999891i	$-0.495306\pi$
$59$	0.226530	0.0294918	0.0147459	+	0.999891i	$0.495306\pi$
$60$	0	0
$61$	5.53150	0.708236	0.354118	−	0.935201i	$-0.384781\pi$
$61$	5.53150	0.708236	0.354118	+	0.935201i	$0.384781\pi$
$62$	0	0
$63$	−1.75030	−0.220517
$64$	0	0
$65$	−5.11485	−0.634419
$66$	0	0
$67$	11.3747	1.38964	0.694820	−	0.719184i	$-0.255484\pi$
$67$	11.3747	1.38964	0.694820	+	0.719184i	$0.255484\pi$
$68$	0	0
$69$	15.1733	1.82665
$70$	0	0
$71$	−6.09899	−0.723817	−0.361908	−	0.932214i	$-0.617875\pi$
$71$	−6.09899	−0.723817	−0.361908	+	0.932214i	$0.617875\pi$
$72$	0	0
$73$	2.94500	0.344686	0.172343	−	0.985037i	$-0.444866\pi$
$73$	2.94500	0.344686	0.172343	+	0.985037i	$0.444866\pi$
$74$	0	0
$75$	8.32200	0.960942
$76$	0	0
$77$	10.7692	1.22727
$78$	0	0
$79$	−9.44518	−1.06267	−0.531333	−	0.847163i	$-0.678309\pi$
$79$	−9.44518	−1.06267	−0.531333	+	0.847163i	$0.678309\pi$
$80$	0	0
$81$	−10.7045	−1.18939
$82$	0	0
$83$	−2.94000	−0.322707	−0.161354	−	0.986897i	$-0.551586\pi$
$83$	−2.94000	−0.322707	−0.161354	+	0.986897i	$0.551586\pi$
$84$	0	0
$85$	−4.70610	−0.510448
$86$	0	0
$87$	15.7041	1.68365
$88$	0	0
$89$	4.88041	0.517322	0.258661	−	0.965968i	$-0.416719\pi$
$89$	4.88041	0.517322	0.258661	+	0.965968i	$0.416719\pi$
$90$	0	0
$91$	13.9634	1.46376
$92$	0	0
$93$	−17.5172	−1.81645
$94$	0	0
$95$	−0.842053	−0.0863929
$96$	0	0
$97$	0.524856	0.0532911	0.0266455	−	0.999645i	$-0.491517\pi$
$97$	0.524856	0.0532911	0.0266455	+	0.999645i	$0.491517\pi$
$98$	0	0
$99$	−3.56697	−0.358494
$100$	0	0
$101$	−10.9444	−1.08901	−0.544504	−	0.838758i	$-0.683282\pi$
$101$	−10.9444	−1.08901	−0.544504	+	0.838758i	$0.683282\pi$
$102$	0	0
$103$	1.00784	0.0993057	0.0496528	−	0.998767i	$-0.484189\pi$
$103$	1.00784	0.0993057	0.0496528	+	0.998767i	$0.484189\pi$
$104$	0	0
$105$	3.75416	0.366369
$106$	0	0
$107$	13.4722	1.30241	0.651206	−	0.758901i	$-0.274264\pi$
$107$	13.4722	1.30241	0.651206	+	0.758901i	$0.274264\pi$
$108$	0	0
$109$	−20.1459	−1.92963	−0.964815	−	0.262931i	$-0.915311\pi$
$109$	−20.1459	−1.92963	−0.964815	+	0.262931i	$0.915311\pi$
$110$	0	0
$111$	−3.94809	−0.374736
$112$	0	0
$113$	−6.52351	−0.613680	−0.306840	−	0.951761i	$-0.599272\pi$
$113$	−6.52351	−0.613680	−0.306840	+	0.951761i	$0.599272\pi$
$114$	0	0
$115$	−6.58788	−0.614323
$116$	0	0
$117$	−4.62495	−0.427576
$118$	0	0
$119$	12.8475	1.17773
$120$	0	0
$121$	10.9469	0.995170
$122$	0	0
$123$	9.61855	0.867275
$124$	0	0
$125$	−7.82347	−0.699753
$126$	0	0
$127$	21.1119	1.87338	0.936691	−	0.350158i	$-0.113872\pi$
$127$	21.1119	1.87338	0.936691	+	0.350158i	$0.113872\pi$
$128$	0	0
$129$	1.88265	0.165758
$130$	0	0
$131$	−8.17981	−0.714674	−0.357337	−	0.933976i	$-0.616315\pi$
$131$	−8.17981	−0.714674	−0.357337	+	0.933976i	$0.616315\pi$
$132$	0	0
$133$	2.29879	0.199330
$134$	0	0
$135$	3.65587	0.314647
$136$	0	0
$137$	−17.8489	−1.52493	−0.762467	−	0.647027i	$-0.776012\pi$
$137$	−17.8489	−1.52493	−0.762467	+	0.647027i	$0.776012\pi$
$138$	0	0
$139$	−21.4810	−1.82199	−0.910997	−	0.412413i	$-0.864686\pi$
$139$	−21.4810	−1.82199	−0.910997	+	0.412413i	$0.864686\pi$
$140$	0	0
$141$	−15.2834	−1.28709
$142$	0	0
$143$	28.4564	2.37964
$144$	0	0
$145$	−6.81831	−0.566230
$146$	0	0
$147$	3.32726	0.274428
$148$	0	0
$149$	−5.18112	−0.424454	−0.212227	−	0.977220i	$-0.568072\pi$
$149$	−5.18112	−0.424454	−0.212227	+	0.977220i	$0.568072\pi$
$150$	0	0
$151$	−17.9051	−1.45709	−0.728546	−	0.684996i	$-0.759804\pi$
$151$	−17.9051	−1.45709	−0.728546	+	0.684996i	$0.759804\pi$
$152$	0	0
$153$	−4.25535	−0.344024
$154$	0	0
$155$	7.60553	0.610891
$156$	0	0
$157$	5.09844	0.406900	0.203450	−	0.979085i	$-0.434785\pi$
$157$	5.09844	0.406900	0.203450	+	0.979085i	$0.434785\pi$
$158$	0	0
$159$	1.93943	0.153807
$160$	0	0
$161$	17.9848	1.41740
$162$	0	0
$163$	22.9001	1.79367	0.896835	−	0.442365i	$-0.145860\pi$
$163$	22.9001	1.79367	0.896835	+	0.442365i	$0.145860\pi$
$164$	0	0
$165$	7.65069	0.595606
$166$	0	0
$167$	−6.14217	−0.475296	−0.237648	−	0.971351i	$-0.576376\pi$
$167$	−6.14217	−0.475296	−0.237648	+	0.971351i	$0.576376\pi$
$168$	0	0
$169$	23.8966	1.83820
$170$	0	0
$171$	−0.761401	−0.0582258
$172$	0	0
$173$	4.07177	0.309571	0.154785	−	0.987948i	$-0.450531\pi$
$173$	4.07177	0.309571	0.154785	+	0.987948i	$0.450531\pi$
$174$	0	0
$175$	9.86397	0.745646
$176$	0	0
$177$	−0.439341	−0.0330229
$178$	0	0
$179$	−17.6200	−1.31698	−0.658492	−	0.752588i	$-0.728805\pi$
$179$	−17.6200	−1.31698	−0.658492	+	0.752588i	$0.728805\pi$
$180$	0	0
$181$	19.5638	1.45417	0.727083	−	0.686549i	$-0.240875\pi$
$181$	19.5638	1.45417	0.727083	+	0.686549i	$0.240875\pi$
$182$	0	0
$183$	−10.7280	−0.793035
$184$	0	0
$185$	1.71416	0.126028
$186$	0	0
$187$	26.1823	1.91464
$188$	0	0
$189$	−9.98044	−0.725970
$190$	0	0
$191$	10.6951	0.773870	0.386935	−	0.922107i	$-0.373534\pi$
$191$	10.6951	0.773870	0.386935	+	0.922107i	$0.373534\pi$
$192$	0	0
$193$	−19.8368	−1.42788	−0.713941	−	0.700206i	$-0.753091\pi$
$193$	−19.8368	−1.42788	−0.713941	+	0.700206i	$0.753091\pi$
$194$	0	0
$195$	9.91991	0.710379
$196$	0	0
$197$	−3.53154	−0.251612	−0.125806	−	0.992055i	$-0.540152\pi$
$197$	−3.53154	−0.251612	−0.125806	+	0.992055i	$0.540152\pi$
$198$	0	0
$199$	−16.9879	−1.20424	−0.602120	−	0.798406i	$-0.705677\pi$
$199$	−16.9879	−1.20424	−0.602120	+	0.798406i	$0.705677\pi$
$200$	0	0
$201$	−22.0605	−1.55602
$202$	0	0
$203$	18.6138	1.30643
$204$	0	0
$205$	−4.17613	−0.291674
$206$	0	0
$207$	−5.95689	−0.414033
$208$	0	0
$209$	4.68475	0.324051
$210$	0	0
$211$	−23.0278	−1.58530	−0.792650	−	0.609677i	$-0.791299\pi$
$211$	−23.0278	−1.58530	−0.792650	+	0.609677i	$0.791299\pi$
$212$	0	0
$213$	11.8286	0.810481
$214$	0	0
$215$	−0.817399	−0.0557461
$216$	0	0
$217$	−20.7629	−1.40948
$218$	0	0
$219$	−5.71163	−0.385956
$220$	0	0
$221$	33.9480	2.28359
$222$	0	0
$223$	6.10474	0.408804	0.204402	−	0.978887i	$-0.434475\pi$
$223$	6.10474	0.408804	0.204402	+	0.978887i	$0.434475\pi$
$224$	0	0
$225$	−3.26713	−0.217809
$226$	0	0
$227$	24.6034	1.63298	0.816492	−	0.577357i	$-0.195916\pi$
$227$	24.6034	1.63298	0.816492	+	0.577357i	$0.195916\pi$
$228$	0	0
$229$	−9.29086	−0.613957	−0.306979	−	0.951716i	$-0.599318\pi$
$229$	−9.29086	−0.613957	−0.306979	+	0.951716i	$0.599318\pi$
$230$	0	0
$231$	−20.8862	−1.37421
$232$	0	0
$233$	17.4056	1.14028	0.570139	−	0.821548i	$-0.306889\pi$
$233$	17.4056	1.14028	0.570139	+	0.821548i	$0.306889\pi$
$234$	0	0
$235$	6.63567	0.432863
$236$	0	0
$237$	18.3183	1.18990
$238$	0	0
$239$	−5.32819	−0.344652	−0.172326	−	0.985040i	$-0.555128\pi$
$239$	−5.32819	−0.344652	−0.172326	+	0.985040i	$0.555128\pi$
$240$	0	0
$241$	−3.06349	−0.197337	−0.0986684	−	0.995120i	$-0.531458\pi$
$241$	−3.06349	−0.197337	−0.0986684	+	0.995120i	$0.531458\pi$
$242$	0	0
$243$	7.73577	0.496250
$244$	0	0
$245$	−1.44461	−0.0922930
$246$	0	0
$247$	6.07426	0.386496
$248$	0	0
$249$	5.70194	0.361346
$250$	0	0
$251$	18.5931	1.17358	0.586792	−	0.809738i	$-0.300391\pi$
$251$	18.5931	1.17358	0.586792	+	0.809738i	$0.300391\pi$
$252$	0	0
$253$	36.6516	2.30426
$254$	0	0
$255$	9.12717	0.571565
$256$	0	0
$257$	5.28579	0.329719	0.164859	−	0.986317i	$-0.447283\pi$
$257$	5.28579	0.329719	0.164859	+	0.986317i	$0.447283\pi$
$258$	0	0
$259$	−4.67962	−0.290778
$260$	0	0
$261$	−6.16525	−0.381619
$262$	0	0
$263$	−30.8381	−1.90156	−0.950780	−	0.309866i	$-0.899716\pi$
$263$	−30.8381	−1.90156	−0.950780	+	0.309866i	$0.899716\pi$
$264$	0	0
$265$	−0.842053	−0.0517269
$266$	0	0
$267$	−9.46523	−0.579262
$268$	0	0
$269$	17.7622	1.08298	0.541492	−	0.840706i	$-0.317860\pi$
$269$	17.7622	1.08298	0.541492	+	0.840706i	$0.317860\pi$
$270$	0	0
$271$	−13.8330	−0.840294	−0.420147	−	0.907456i	$-0.638022\pi$
$271$	−13.8330	−0.840294	−0.420147	+	0.907456i	$0.638022\pi$
$272$	0	0
$273$	−27.0811	−1.63902
$274$	0	0
$275$	20.1020	1.21220
$276$	0	0
$277$	−5.33671	−0.320652	−0.160326	−	0.987064i	$-0.551255\pi$
$277$	−5.33671	−0.320652	−0.160326	+	0.987064i	$0.551255\pi$
$278$	0	0
$279$	6.87707	0.411719
$280$	0	0
$281$	−16.3319	−0.974281	−0.487140	−	0.873324i	$-0.661960\pi$
$281$	−16.3319	−0.974281	−0.487140	+	0.873324i	$0.661960\pi$
$282$	0	0
$283$	15.0677	0.895684	0.447842	−	0.894113i	$-0.352193\pi$
$283$	15.0677	0.895684	0.447842	+	0.894113i	$0.352193\pi$
$284$	0	0
$285$	1.63311	0.0967369
$286$	0	0
$287$	11.4007	0.672965
$288$	0	0
$289$	14.2351	0.837360
$290$	0	0
$291$	−1.01792	−0.0596717
$292$	0	0
$293$	−3.36337	−0.196490	−0.0982452	−	0.995162i	$-0.531323\pi$
$293$	−3.36337	−0.196490	−0.0982452	+	0.995162i	$0.531323\pi$
$294$	0	0
$295$	0.190751	0.0111059
$296$	0	0
$297$	−20.3394	−1.18021
$298$	0	0
$299$	47.5225	2.74830
$300$	0	0
$301$	2.23148	0.128620
$302$	0	0
$303$	21.2259	1.21940
$304$	0	0
$305$	4.65782	0.266706
$306$	0	0
$307$	−26.3108	−1.50164	−0.750819	−	0.660508i	$-0.770341\pi$
$307$	−26.3108	−1.50164	−0.750819	+	0.660508i	$0.770341\pi$
$308$	0	0
$309$	−1.95464	−0.111196
$310$	0	0
$311$	−9.30266	−0.527506	−0.263753	−	0.964590i	$-0.584960\pi$
$311$	−9.30266	−0.527506	−0.263753	+	0.964590i	$0.584960\pi$
$312$	0	0
$313$	−6.67207	−0.377128	−0.188564	−	0.982061i	$-0.560383\pi$
$313$	−6.67207	−0.377128	−0.188564	+	0.982061i	$0.560383\pi$
$314$	0	0
$315$	−1.47384	−0.0830417
$316$	0	0
$317$	7.25633	0.407556	0.203778	−	0.979017i	$-0.434678\pi$
$317$	7.25633	0.407556	0.203778	+	0.979017i	$0.434678\pi$
$318$	0	0
$319$	37.9335	2.12387
$320$	0	0
$321$	−26.1285	−1.45835
$322$	0	0
$323$	5.58884	0.310971
$324$	0	0
$325$	26.0643	1.44579
$326$	0	0
$327$	39.0717	2.16067
$328$	0	0
$329$	−18.1152	−0.998724
$330$	0	0
$331$	29.1966	1.60479	0.802396	−	0.596792i	$-0.203558\pi$
$331$	29.1966	1.60479	0.802396	+	0.596792i	$0.203558\pi$
$332$	0	0
$333$	1.54998	0.0849384
$334$	0	0
$335$	9.57810	0.523307
$336$	0	0
$337$	3.56052	0.193954	0.0969771	−	0.995287i	$-0.469083\pi$
$337$	3.56052	0.193954	0.0969771	+	0.995287i	$0.469083\pi$
$338$	0	0
$339$	12.6519	0.687158
$340$	0	0
$341$	−42.3132	−2.29139
$342$	0	0
$343$	20.0353	1.08180
$344$	0	0
$345$	12.7768	0.687878
$346$	0	0
$347$	8.26260	0.443559	0.221780	−	0.975097i	$-0.428813\pi$
$347$	8.26260	0.443559	0.221780	+	0.975097i	$0.428813\pi$
$348$	0	0
$349$	25.3532	1.35713	0.678563	−	0.734543i	$-0.262603\pi$
$349$	25.3532	1.35713	0.678563	+	0.734543i	$0.262603\pi$
$350$	0	0
$351$	−26.3721	−1.40764
$352$	0	0
$353$	17.7452	0.944481	0.472241	−	0.881470i	$-0.343445\pi$
$353$	17.7452	0.944481	0.472241	+	0.881470i	$0.343445\pi$
$354$	0	0
$355$	−5.13568	−0.272573
$356$	0	0
$357$	−24.9169	−1.31874
$358$	0	0
$359$	14.1165	0.745041	0.372521	−	0.928024i	$-0.378494\pi$
$359$	14.1165	0.745041	0.372521	+	0.928024i	$0.378494\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−21.2307	−1.11432
$364$	0	0
$365$	2.47985	0.129801
$366$	0	0
$367$	−13.2679	−0.692580	−0.346290	−	0.938127i	$-0.612559\pi$
$367$	−13.2679	−0.692580	−0.346290	+	0.938127i	$0.612559\pi$
$368$	0	0
$369$	−3.77614	−0.196578
$370$	0	0
$371$	2.29879	0.119347
$372$	0	0
$373$	29.2428	1.51414	0.757068	−	0.653336i	$-0.226631\pi$
$373$	29.2428	1.51414	0.757068	+	0.653336i	$0.226631\pi$
$374$	0	0
$375$	15.1731	0.783536
$376$	0	0
$377$	49.1847	2.53314
$378$	0	0
$379$	−35.5319	−1.82515	−0.912577	−	0.408906i	$-0.865911\pi$
$379$	−35.5319	−1.82515	−0.912577	+	0.408906i	$0.865911\pi$
$380$	0	0
$381$	−40.9452	−2.09769
$382$	0	0
$383$	−26.1465	−1.33602	−0.668012	−	0.744150i	$-0.732855\pi$
$383$	−26.1465	−1.33602	−0.668012	+	0.744150i	$0.732855\pi$
$384$	0	0
$385$	9.06827	0.462162
$386$	0	0
$387$	−0.739108	−0.0375709
$388$	0	0
$389$	−1.76059	−0.0892654	−0.0446327	−	0.999003i	$-0.514212\pi$
$389$	−1.76059	−0.0892654	−0.0446327	+	0.999003i	$0.514212\pi$
$390$	0	0
$391$	43.7248	2.21126
$392$	0	0
$393$	15.8642	0.800243
$394$	0	0
$395$	−7.95335	−0.400176
$396$	0	0
$397$	−21.9181	−1.10004	−0.550020	−	0.835152i	$-0.685380\pi$
$397$	−21.9181	−1.10004	−0.550020	+	0.835152i	$0.685380\pi$
$398$	0	0
$399$	−4.45834	−0.223196
$400$	0	0
$401$	−17.5680	−0.877305	−0.438652	−	0.898657i	$-0.644544\pi$
$401$	−17.5680	−0.877305	−0.438652	+	0.898657i	$0.644544\pi$
$402$	0	0
$403$	−54.8634	−2.73294
$404$	0	0
$405$	−9.01374	−0.447896
$406$	0	0
$407$	−9.53672	−0.472717
$408$	0	0
$409$	−16.5362	−0.817664	−0.408832	−	0.912610i	$-0.634064\pi$
$409$	−16.5362	−0.817664	−0.408832	+	0.912610i	$0.634064\pi$
$410$	0	0
$411$	34.6167	1.70752
$412$	0	0
$413$	−0.520745	−0.0256242
$414$	0	0
$415$	−2.47564	−0.121524
$416$	0	0
$417$	41.6610	2.04015
$418$	0	0
$419$	−1.91098	−0.0933575	−0.0466787	−	0.998910i	$-0.514864\pi$
$419$	−1.91098	−0.0933575	−0.0466787	+	0.998910i	$0.514864\pi$
$420$	0	0
$421$	11.9812	0.583927	0.291964	−	0.956429i	$-0.405691\pi$
$421$	11.9812	0.583927	0.291964	+	0.956429i	$0.405691\pi$
$422$	0	0
$423$	6.00010	0.291735
$424$	0	0
$425$	23.9814	1.16327
$426$	0	0
$427$	−12.7157	−0.615358
$428$	0	0
$429$	−55.1892	−2.66456
$430$	0	0
$431$	37.9721	1.82905	0.914525	−	0.404529i	$-0.132565\pi$
$431$	37.9721	1.82905	0.914525	+	0.404529i	$0.132565\pi$
$432$	0	0
$433$	−25.0993	−1.20619	−0.603097	−	0.797668i	$-0.706067\pi$
$433$	−25.0993	−1.20619	−0.603097	+	0.797668i	$0.706067\pi$
$434$	0	0
$435$	13.2237	0.634026
$436$	0	0
$437$	7.82359	0.374253
$438$	0	0
$439$	−12.1078	−0.577871	−0.288936	−	0.957349i	$-0.593301\pi$
$439$	−12.1078	−0.577871	−0.288936	+	0.957349i	$0.593301\pi$
$440$	0	0
$441$	−1.30625	−0.0622023
$442$	0	0
$443$	0.169276	0.00804253	0.00402126	−	0.999992i	$-0.498720\pi$
$443$	0.169276	0.00804253	0.00402126	+	0.999992i	$0.498720\pi$
$444$	0	0
$445$	4.10956	0.194812
$446$	0	0
$447$	10.0484	0.475275
$448$	0	0
$449$	−19.8046	−0.934635	−0.467317	−	0.884090i	$-0.654779\pi$
$449$	−19.8046	−0.934635	−0.467317	+	0.884090i	$0.654779\pi$
$450$	0	0
$451$	23.2338	1.09404
$452$	0	0
$453$	34.7257	1.63155
$454$	0	0
$455$	11.7579	0.551221
$456$	0	0
$457$	−16.1336	−0.754699	−0.377350	−	0.926071i	$-0.623164\pi$
$457$	−16.1336	−0.754699	−0.377350	+	0.926071i	$0.623164\pi$
$458$	0	0
$459$	−24.2646	−1.13257
$460$	0	0
$461$	−14.8193	−0.690205	−0.345102	−	0.938565i	$-0.612156\pi$
$461$	−14.8193	−0.690205	−0.345102	+	0.938565i	$0.612156\pi$
$462$	0	0
$463$	15.7489	0.731913	0.365956	−	0.930632i	$-0.380742\pi$
$463$	15.7489	0.731913	0.365956	+	0.930632i	$0.380742\pi$
$464$	0	0
$465$	−14.7504	−0.684034
$466$	0	0
$467$	−3.92158	−0.181469	−0.0907346	−	0.995875i	$-0.528922\pi$
$467$	−3.92158	−0.181469	−0.0907346	+	0.995875i	$0.528922\pi$
$468$	0	0
$469$	−26.1480	−1.20740
$470$	0	0
$471$	−9.88809	−0.455619
$472$	0	0
$473$	4.54758	0.209098
$474$	0	0
$475$	4.29095	0.196882
$476$	0	0
$477$	−0.761401	−0.0348622
$478$	0	0
$479$	6.08447	0.278007	0.139003	−	0.990292i	$-0.455610\pi$
$479$	6.08447	0.278007	0.139003	+	0.990292i	$0.455610\pi$
$480$	0	0
$481$	−12.3653	−0.563811
$482$	0	0
$483$	−34.8802	−1.58711
$484$	0	0
$485$	0.441957	0.0200682
$486$	0	0
$487$	−3.51717	−0.159378	−0.0796890	−	0.996820i	$-0.525393\pi$
$487$	−3.51717	−0.159378	−0.0796890	+	0.996820i	$0.525393\pi$
$488$	0	0
$489$	−44.4131	−2.00843
$490$	0	0
$491$	7.52119	0.339426	0.169713	−	0.985493i	$-0.445716\pi$
$491$	7.52119	0.339426	0.169713	+	0.985493i	$0.445716\pi$
$492$	0	0
$493$	45.2542	2.03815
$494$	0	0
$495$	−3.00358	−0.135001
$496$	0	0
$497$	14.0203	0.628895
$498$	0	0
$499$	−28.8057	−1.28952	−0.644761	−	0.764384i	$-0.723043\pi$
$499$	−28.8057	−1.28952	−0.644761	+	0.764384i	$0.723043\pi$
$500$	0	0
$501$	11.9123	0.532204
$502$	0	0
$503$	−41.2210	−1.83795	−0.918977	−	0.394310i	$-0.870984\pi$
$503$	−41.2210	−1.83795	−0.918977	+	0.394310i	$0.870984\pi$
$504$	0	0
$505$	−9.21576	−0.410096
$506$	0	0
$507$	−46.3459	−2.05829
$508$	0	0
$509$	−36.7692	−1.62977	−0.814884	−	0.579624i	$-0.803199\pi$
$509$	−36.7692	−1.62977	−0.814884	+	0.579624i	$0.803199\pi$
$510$	0	0
$511$	−6.76992	−0.299484
$512$	0	0
$513$	−4.34161	−0.191687
$514$	0	0
$515$	0.848657	0.0373963
$516$	0	0
$517$	−36.9174	−1.62363
$518$	0	0
$519$	−7.89692	−0.346636
$520$	0	0
$521$	11.8434	0.518867	0.259433	−	0.965761i	$-0.416464\pi$
$521$	11.8434	0.518867	0.259433	+	0.965761i	$0.416464\pi$
$522$	0	0
$523$	14.8712	0.650274	0.325137	−	0.945667i	$-0.394590\pi$
$523$	14.8712	0.650274	0.325137	+	0.945667i	$0.394590\pi$
$524$	0	0
$525$	−19.1305	−0.834924
$526$	0	0
$527$	−50.4791	−2.19890
$528$	0	0
$529$	38.2086	1.66124
$530$	0	0
$531$	0.172481	0.00748502
$532$	0	0
$533$	30.1251	1.30486
$534$	0	0
$535$	11.3444	0.490459
$536$	0	0
$537$	34.1729	1.47467
$538$	0	0
$539$	8.03708	0.346182
$540$	0	0
$541$	27.9415	1.20130	0.600649	−	0.799513i	$-0.294909\pi$
$541$	27.9415	1.20130	0.600649	+	0.799513i	$0.294909\pi$
$542$	0	0
$543$	−37.9427	−1.62828
$544$	0	0
$545$	−16.9639	−0.726656
$546$	0	0
$547$	24.4376	1.04488	0.522438	−	0.852677i	$-0.325022\pi$
$547$	24.4376	1.04488	0.522438	+	0.852677i	$0.325022\pi$
$548$	0	0
$549$	4.21169	0.179751
$550$	0	0
$551$	8.09724	0.344954
$552$	0	0
$553$	21.7124	0.923307
$554$	0	0
$555$	−3.32451	−0.141117
$556$	0	0
$557$	−6.22718	−0.263854	−0.131927	−	0.991259i	$-0.542116\pi$
$557$	−6.22718	−0.263854	−0.131927	+	0.991259i	$0.542116\pi$
$558$	0	0
$559$	5.89641	0.249391
$560$	0	0
$561$	−50.7788	−2.14388
$562$	0	0
$563$	−14.2762	−0.601670	−0.300835	−	0.953676i	$-0.597265\pi$
$563$	−14.2762	−0.601670	−0.300835	+	0.953676i	$0.597265\pi$
$564$	0	0
$565$	−5.49314	−0.231098
$566$	0	0
$567$	24.6073	1.03341
$568$	0	0
$569$	2.33506	0.0978908	0.0489454	−	0.998801i	$-0.484414\pi$
$569$	2.33506	0.0978908	0.0489454	+	0.998801i	$0.484414\pi$
$570$	0	0
$571$	−5.28236	−0.221060	−0.110530	−	0.993873i	$-0.535255\pi$
$571$	−5.28236	−0.221060	−0.110530	+	0.993873i	$0.535255\pi$
$572$	0	0
$573$	−20.7424	−0.866527
$574$	0	0
$575$	33.5706	1.39999
$576$	0	0
$577$	19.5401	0.813464	0.406732	−	0.913548i	$-0.366668\pi$
$577$	19.5401	0.813464	0.406732	+	0.913548i	$0.366668\pi$
$578$	0	0
$579$	38.4721	1.59885
$580$	0	0
$581$	6.75844	0.280387
$582$	0	0
$583$	4.68475	0.194022
$584$	0	0
$585$	−3.89445	−0.161016
$586$	0	0
$587$	3.59759	0.148488	0.0742442	−	0.997240i	$-0.476346\pi$
$587$	3.59759	0.148488	0.0742442	+	0.997240i	$0.476346\pi$
$588$	0	0
$589$	−9.03212	−0.372162
$590$	0	0
$591$	6.84919	0.281738
$592$	0	0
$593$	−0.0121920	−0.000500667 0	−0.000250334	−	1.00000i	$-0.500080\pi$
$593$	−0.0121920	−0.000500667 0	−0.000250334		1.00000i	$0.500080\pi$
$594$	0	0
$595$	10.8183	0.443508
$596$	0	0
$597$	32.9469	1.34843
$598$	0	0
$599$	−11.0176	−0.450166	−0.225083	−	0.974340i	$-0.572265\pi$
$599$	−11.0176	−0.450166	−0.225083	+	0.974340i	$0.572265\pi$
$600$	0	0
$601$	36.2922	1.48039	0.740195	−	0.672392i	$-0.234733\pi$
$601$	36.2922	1.48039	0.740195	+	0.672392i	$0.234733\pi$
$602$	0	0
$603$	8.66070	0.352691
$604$	0	0
$605$	9.21785	0.374759
$606$	0	0
$607$	−45.9044	−1.86320	−0.931601	−	0.363482i	$-0.881588\pi$
$607$	−45.9044	−1.86320	−0.931601	+	0.363482i	$0.881588\pi$
$608$	0	0
$609$	−36.1003	−1.46286
$610$	0	0
$611$	−47.8672	−1.93650
$612$	0	0
$613$	25.9232	1.04703	0.523514	−	0.852017i	$-0.324621\pi$
$613$	25.9232	1.04703	0.523514	+	0.852017i	$0.324621\pi$
$614$	0	0
$615$	8.09933	0.326597
$616$	0	0
$617$	23.5286	0.947226	0.473613	−	0.880733i	$-0.342950\pi$
$617$	23.5286	0.947226	0.473613	+	0.880733i	$0.342950\pi$
$618$	0	0
$619$	−7.13510	−0.286784	−0.143392	−	0.989666i	$-0.545801\pi$
$619$	−7.13510	−0.286784	−0.143392	+	0.989666i	$0.545801\pi$
$620$	0	0
$621$	−33.9670	−1.36305
$622$	0	0
$623$	−11.2190	−0.449480
$624$	0	0
$625$	14.8669	0.594678
$626$	0	0
$627$	−9.08576	−0.362850
$628$	0	0
$629$	−11.3772	−0.453637
$630$	0	0
$631$	17.8606	0.711019	0.355510	−	0.934673i	$-0.384307\pi$
$631$	17.8606	0.711019	0.355510	+	0.934673i	$0.384307\pi$
$632$	0	0
$633$	44.6609	1.77511
$634$	0	0
$635$	17.7774	0.705474
$636$	0	0
$637$	10.4209	0.412891
$638$	0	0
$639$	−4.64378	−0.183705
$640$	0	0
$641$	−12.7942	−0.505340	−0.252670	−	0.967553i	$-0.581309\pi$
$641$	−12.7942	−0.505340	−0.252670	+	0.967553i	$0.581309\pi$
$642$	0	0
$643$	17.5265	0.691177	0.345589	−	0.938386i	$-0.387679\pi$
$643$	17.5265	0.691177	0.345589	+	0.938386i	$0.387679\pi$
$644$	0	0
$645$	1.58529	0.0624207
$646$	0	0
$647$	5.51443	0.216795	0.108397	−	0.994108i	$-0.465428\pi$
$647$	5.51443	0.216795	0.108397	+	0.994108i	$0.465428\pi$
$648$	0	0
$649$	−1.06124	−0.0416572
$650$	0	0
$651$	40.2683	1.57824
$652$	0	0
$653$	−14.3573	−0.561843	−0.280921	−	0.959731i	$-0.590640\pi$
$653$	−14.3573	−0.561843	−0.280921	+	0.959731i	$0.590640\pi$
$654$	0	0
$655$	−6.88784	−0.269130
$656$	0	0
$657$	2.24233	0.0874815
$658$	0	0
$659$	−23.1967	−0.903617	−0.451808	−	0.892115i	$-0.649221\pi$
$659$	−23.1967	−0.903617	−0.451808	+	0.892115i	$0.649221\pi$
$660$	0	0
$661$	−36.6655	−1.42612	−0.713061	−	0.701102i	$-0.752692\pi$
$661$	−36.6655	−1.42612	−0.713061	+	0.701102i	$0.752692\pi$
$662$	0	0
$663$	−65.8400	−2.55701
$664$	0	0
$665$	1.93570	0.0750632
$666$	0	0
$667$	63.3495	2.45290
$668$	0	0
$669$	−11.8397	−0.457751
$670$	0	0
$671$	−25.9137	−1.00039
$672$	0	0
$673$	−48.2672	−1.86057	−0.930283	−	0.366843i	$-0.880439\pi$
$673$	−48.2672	−1.86057	−0.930283	+	0.366843i	$0.880439\pi$
$674$	0	0
$675$	−18.6296	−0.717055
$676$	0	0
$677$	−43.1037	−1.65661	−0.828305	−	0.560278i	$-0.810694\pi$
$677$	−43.1037	−1.65661	−0.828305	+	0.560278i	$0.810694\pi$
$678$	0	0
$679$	−1.20653	−0.0463024
$680$	0	0
$681$	−47.7166	−1.82850
$682$	0	0
$683$	−16.1259	−0.617040	−0.308520	−	0.951218i	$-0.599834\pi$
$683$	−16.1259	−0.617040	−0.308520	+	0.951218i	$0.599834\pi$
$684$	0	0
$685$	−15.0297	−0.574256
$686$	0	0
$687$	18.0190	0.687468
$688$	0	0
$689$	6.07426	0.231411
$690$	0	0
$691$	10.2937	0.391592	0.195796	−	0.980645i	$-0.437271\pi$
$691$	10.2937	0.391592	0.195796	+	0.980645i	$0.437271\pi$
$692$	0	0
$693$	8.19970	0.311481
$694$	0	0
$695$	−18.0881	−0.686122
$696$	0	0
$697$	27.7176	1.04988
$698$	0	0
$699$	−33.7570	−1.27681
$700$	0	0
$701$	14.1832	0.535693	0.267846	−	0.963462i	$-0.413688\pi$
$701$	14.1832	0.535693	0.267846	+	0.963462i	$0.413688\pi$
$702$	0	0
$703$	−2.03569	−0.0767777
$704$	0	0
$705$	−12.8694	−0.484691
$706$	0	0
$707$	25.1588	0.946194
$708$	0	0
$709$	33.8191	1.27010	0.635051	−	0.772470i	$-0.280979\pi$
$709$	33.8191	1.27010	0.635051	+	0.772470i	$0.280979\pi$
$710$	0	0
$711$	−7.19157	−0.269705
$712$	0	0
$713$	−70.6637	−2.64637
$714$	0	0
$715$	23.9618	0.896120
$716$	0	0
$717$	10.3337	0.385918
$718$	0	0
$719$	26.7465	0.997477	0.498739	−	0.866752i	$-0.333797\pi$
$719$	26.7465	0.997477	0.498739	+	0.866752i	$0.333797\pi$
$720$	0	0
$721$	−2.31681	−0.0862826
$722$	0	0
$723$	5.94144	0.220964
$724$	0	0
$725$	34.7448	1.29039
$726$	0	0
$727$	17.1344	0.635479	0.317740	−	0.948178i	$-0.397076\pi$
$727$	17.1344	0.635479	0.317740	+	0.948178i	$0.397076\pi$
$728$	0	0
$729$	17.1104	0.633719
$730$	0	0
$731$	5.42520	0.200658
$732$	0	0
$733$	−44.0707	−1.62779	−0.813894	−	0.581013i	$-0.802656\pi$
$733$	−44.0707	−1.62779	−0.813894	+	0.581013i	$0.802656\pi$
$734$	0	0
$735$	2.80173	0.103343
$736$	0	0
$737$	−53.2876	−1.96287
$738$	0	0
$739$	2.21495	0.0814784	0.0407392	−	0.999170i	$-0.487029\pi$
$739$	2.21495	0.0814784	0.0407392	+	0.999170i	$0.487029\pi$
$740$	0	0
$741$	−11.7806	−0.432772
$742$	0	0
$743$	−35.9900	−1.32034	−0.660172	−	0.751114i	$-0.729517\pi$
$743$	−35.9900	−1.32034	−0.660172	+	0.751114i	$0.729517\pi$
$744$	0	0
$745$	−4.36278	−0.159840
$746$	0	0
$747$	−2.23852	−0.0819032
$748$	0	0
$749$	−30.9698	−1.13161
$750$	0	0
$751$	−2.59632	−0.0947409	−0.0473705	−	0.998877i	$-0.515084\pi$
$751$	−2.59632	−0.0947409	−0.0473705	+	0.998877i	$0.515084\pi$
$752$	0	0
$753$	−36.0600	−1.31410
$754$	0	0
$755$	−15.0770	−0.548709
$756$	0	0
$757$	−34.2776	−1.24584	−0.622920	−	0.782286i	$-0.714054\pi$
$757$	−34.2776	−1.24584	−0.622920	+	0.782286i	$0.714054\pi$
$758$	0	0
$759$	−71.0833	−2.58016
$760$	0	0
$761$	10.7794	0.390752	0.195376	−	0.980728i	$-0.437407\pi$
$761$	10.7794	0.390752	0.195376	+	0.980728i	$0.437407\pi$
$762$	0	0
$763$	46.3112	1.67658
$764$	0	0
$765$	−3.58323	−0.129552
$766$	0	0
$767$	−1.37600	−0.0496846
$768$	0	0
$769$	−16.7688	−0.604697	−0.302349	−	0.953197i	$-0.597771\pi$
$769$	−16.7688	−0.604697	−0.302349	+	0.953197i	$0.597771\pi$
$770$	0	0
$771$	−10.2514	−0.369196
$772$	0	0
$773$	3.66231	0.131724	0.0658620	−	0.997829i	$-0.479020\pi$
$773$	3.66231	0.131724	0.0658620	+	0.997829i	$0.479020\pi$
$774$	0	0
$775$	−38.7564	−1.39217
$776$	0	0
$777$	9.07582	0.325593
$778$	0	0
$779$	4.95946	0.177691
$780$	0	0
$781$	28.5722	1.02239
$782$	0	0
$783$	−35.1551	−1.25634
$784$	0	0
$785$	4.29316	0.153229
$786$	0	0
$787$	15.3897	0.548584	0.274292	−	0.961646i	$-0.411557\pi$
$787$	15.3897	0.548584	0.274292	+	0.961646i	$0.411557\pi$
$788$	0	0
$789$	59.8085	2.12924
$790$	0	0
$791$	14.9962	0.533202
$792$	0	0
$793$	−33.5998	−1.19316
$794$	0	0
$795$	1.63311	0.0579203
$796$	0	0
$797$	−4.02018	−0.142402	−0.0712011	−	0.997462i	$-0.522683\pi$
$797$	−4.02018	−0.142402	−0.0712011	+	0.997462i	$0.522683\pi$
$798$	0	0
$799$	−44.0419	−1.55809
$800$	0	0
$801$	3.71595	0.131297
$802$	0	0
$803$	−13.7966	−0.486871
$804$	0	0
$805$	15.1441	0.533761
$806$	0	0
$807$	−34.4487	−1.21265
$808$	0	0
$809$	−13.4510	−0.472911	−0.236455	−	0.971642i	$-0.575986\pi$
$809$	−13.4510	−0.472911	−0.236455	+	0.971642i	$0.575986\pi$
$810$	0	0
$811$	−0.560997	−0.0196993	−0.00984963	−	0.999951i	$-0.503135\pi$
$811$	−0.560997	−0.0196993	−0.00984963	+	0.999951i	$0.503135\pi$
$812$	0	0
$813$	26.8282	0.940905
$814$	0	0
$815$	19.2831	0.675456
$816$	0	0
$817$	0.970721	0.0339612
$818$	0	0
$819$	10.6318	0.371504
$820$	0	0
$821$	−55.3381	−1.93131	−0.965656	−	0.259824i	$-0.916336\pi$
$821$	−55.3381	−1.93131	−0.965656	+	0.259824i	$0.916336\pi$
$822$	0	0
$823$	33.6085	1.17152	0.585760	−	0.810485i	$-0.300796\pi$
$823$	33.6085	1.17152	0.585760	+	0.810485i	$0.300796\pi$
$824$	0	0
$825$	−38.9865	−1.35734
$826$	0	0
$827$	−5.71742	−0.198814	−0.0994071	−	0.995047i	$-0.531695\pi$
$827$	−5.71742	−0.198814	−0.0994071	+	0.995047i	$0.531695\pi$
$828$	0	0
$829$	20.1062	0.698318	0.349159	−	0.937064i	$-0.386467\pi$
$829$	20.1062	0.698318	0.349159	+	0.937064i	$0.386467\pi$
$830$	0	0
$831$	10.3502	0.359044
$832$	0	0
$833$	9.58812	0.332209
$834$	0	0
$835$	−5.17204	−0.178986
$836$	0	0
$837$	39.2140	1.35543
$838$	0	0
$839$	−28.7904	−0.993956	−0.496978	−	0.867763i	$-0.665557\pi$
$839$	−28.7904	−0.993956	−0.496978	+	0.867763i	$0.665557\pi$
$840$	0	0
$841$	36.5653	1.26087
$842$	0	0
$843$	31.6747	1.09093
$844$	0	0
$845$	20.1222	0.692226
$846$	0	0
$847$	−25.1645	−0.864663
$848$	0	0
$849$	−29.2229	−1.00293
$850$	0	0
$851$	−15.9264	−0.545951
$852$	0	0
$853$	22.2263	0.761013	0.380506	−	0.924778i	$-0.375750\pi$
$853$	22.2263	0.761013	0.380506	+	0.924778i	$0.375750\pi$
$854$	0	0
$855$	−0.641140	−0.0219265
$856$	0	0
$857$	−18.1475	−0.619908	−0.309954	−	0.950752i	$-0.600314\pi$
$857$	−18.1475	−0.619908	−0.309954	+	0.950752i	$0.600314\pi$
$858$	0	0
$859$	−33.0275	−1.12688	−0.563441	−	0.826156i	$-0.690523\pi$
$859$	−33.0275	−1.12688	−0.563441	+	0.826156i	$0.690523\pi$
$860$	0	0
$861$	−22.1110	−0.753540
$862$	0	0
$863$	23.7750	0.809311	0.404655	−	0.914469i	$-0.367391\pi$
$863$	23.7750	0.809311	0.404655	+	0.914469i	$0.367391\pi$
$864$	0	0
$865$	3.42864	0.116577
$866$	0	0
$867$	−27.6081	−0.937619
$868$	0	0
$869$	44.2483	1.50102
$870$	0	0
$871$	−69.0928	−2.34112
$872$	0	0
$873$	0.399626	0.0135253
$874$	0	0
$875$	17.9845	0.607987
$876$	0	0
$877$	1.80351	0.0609001	0.0304501	−	0.999536i	$-0.490306\pi$
$877$	1.80351	0.0609001	0.0304501	+	0.999536i	$0.490306\pi$
$878$	0	0
$879$	6.52304	0.220017
$880$	0	0
$881$	−26.1628	−0.881447	−0.440724	−	0.897643i	$-0.645278\pi$
$881$	−26.1628	−0.881447	−0.440724	+	0.897643i	$0.645278\pi$
$882$	0	0
$883$	29.5327	0.993854	0.496927	−	0.867792i	$-0.334462\pi$
$883$	29.5327	0.993854	0.496927	+	0.867792i	$0.334462\pi$
$884$	0	0
$885$	−0.369948	−0.0124357
$886$	0	0
$887$	−12.4641	−0.418502	−0.209251	−	0.977862i	$-0.567103\pi$
$887$	−12.4641	−0.418502	−0.209251	+	0.977862i	$0.567103\pi$
$888$	0	0
$889$	−48.5318	−1.62770
$890$	0	0
$891$	50.1478	1.68001
$892$	0	0
$893$	−7.88034	−0.263705
$894$	0	0
$895$	−14.8370	−0.495947
$896$	0	0
$897$	−92.1668	−3.07736
$898$	0	0
$899$	−73.1353	−2.43920
$900$	0	0
$901$	5.58884	0.186191
$902$	0	0
$903$	−4.32780	−0.144020
$904$	0	0
$905$	16.4738	0.547607
$906$	0	0
$907$	−11.0018	−0.365309	−0.182654	−	0.983177i	$-0.558469\pi$
$907$	−11.0018	−0.365309	−0.182654	+	0.983177i	$0.558469\pi$
$908$	0	0
$909$	−8.33307	−0.276390
$910$	0	0
$911$	−4.97095	−0.164695	−0.0823474	−	0.996604i	$-0.526242\pi$
$911$	−4.97095	−0.164695	−0.0823474	+	0.996604i	$0.526242\pi$
$912$	0	0
$913$	13.7732	0.455826
$914$	0	0
$915$	−9.03353	−0.298639
$916$	0	0
$917$	18.8036	0.620951
$918$	0	0
$919$	−52.3942	−1.72833	−0.864163	−	0.503211i	$-0.832152\pi$
$919$	−52.3942	−1.72833	−0.864163	+	0.503211i	$0.832152\pi$
$920$	0	0
$921$	51.0281	1.68143
$922$	0	0
$923$	37.0468	1.21941
$924$	0	0
$925$	−8.73505	−0.287207
$926$	0	0
$927$	0.767372	0.0252038
$928$	0	0
$929$	−4.85913	−0.159423	−0.0797114	−	0.996818i	$-0.525400\pi$
$929$	−4.85913	−0.159423	−0.0797114	+	0.996818i	$0.525400\pi$
$930$	0	0
$931$	1.71558	0.0562260
$932$	0	0
$933$	18.0419	0.590665
$934$	0	0
$935$	22.0469	0.721011
$936$	0	0
$937$	24.8552	0.811983	0.405992	−	0.913877i	$-0.366926\pi$
$937$	24.8552	0.811983	0.405992	+	0.913877i	$0.366926\pi$
$938$	0	0
$939$	12.9400	0.422282
$940$	0	0
$941$	6.50019	0.211900	0.105950	−	0.994371i	$-0.466212\pi$
$941$	6.50019	0.211900	0.105950	+	0.994371i	$0.466212\pi$
$942$	0	0
$943$	38.8008	1.26353
$944$	0	0
$945$	−8.40406	−0.273384
$946$	0	0
$947$	−19.7535	−0.641903	−0.320951	−	0.947096i	$-0.604003\pi$
$947$	−19.7535	−0.641903	−0.320951	+	0.947096i	$0.604003\pi$
$948$	0	0
$949$	−17.8887	−0.580691
$950$	0	0
$951$	−14.0732	−0.456354
$952$	0	0
$953$	−54.4838	−1.76490	−0.882452	−	0.470403i	$-0.844109\pi$
$953$	−54.4838	−1.76490	−0.882452	+	0.470403i	$0.844109\pi$
$954$	0	0
$955$	9.00584	0.291422
$956$	0	0
$957$	−73.5696	−2.37817
$958$	0	0
$959$	41.0308	1.32495
$960$	0	0
$961$	50.5793	1.63159
$962$	0	0
$963$	10.2578	0.330552
$964$	0	0
$965$	−16.7036	−0.537708
$966$	0	0
$967$	19.9435	0.641339	0.320669	−	0.947191i	$-0.396092\pi$
$967$	19.9435	0.641339	0.320669	+	0.947191i	$0.396092\pi$
$968$	0	0
$969$	−10.8392	−0.348205
$970$	0	0
$971$	−8.01490	−0.257210	−0.128605	−	0.991696i	$-0.541050\pi$
$971$	−8.01490	−0.257210	−0.128605	+	0.991696i	$0.541050\pi$
$972$	0	0
$973$	49.3802	1.58306
$974$	0	0
$975$	−50.5500	−1.61890
$976$	0	0
$977$	−1.00918	−0.0322865	−0.0161432	−	0.999870i	$-0.505139\pi$
$977$	−1.00918	−0.0322865	−0.0161432	+	0.999870i	$0.505139\pi$
$978$	0	0
$979$	−22.8635	−0.730720
$980$	0	0
$981$	−15.3391	−0.489741
$982$	0	0
$983$	−44.9820	−1.43470	−0.717351	−	0.696712i	$-0.754646\pi$
$983$	−44.9820	−1.43470	−0.717351	+	0.696712i	$0.754646\pi$
$984$	0	0
$985$	−2.97375	−0.0947514
$986$	0	0
$987$	35.1332	1.11830
$988$	0	0
$989$	7.59452	0.241492
$990$	0	0
$991$	26.6235	0.845723	0.422861	−	0.906194i	$-0.361026\pi$
$991$	26.6235	0.845723	0.422861	+	0.906194i	$0.361026\pi$
$992$	0	0
$993$	−56.6249	−1.79694
$994$	0	0
$995$	−14.3047	−0.453490
$996$	0	0
$997$	46.2496	1.46474	0.732370	−	0.680907i	$-0.238414\pi$
$997$	46.2496	1.46474	0.732370	+	0.680907i	$0.238414\pi$
$998$	0	0
$999$	8.83820	0.279628

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.f.1.4	✓	19	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.93943 q^{3} +0.842053 q^{5} -2.29879 q^{7} +0.761401 q^{9} +O(q^{10})\)	\(q-1.93943 q^{3} +0.842053 q^{5} -2.29879 q^{7} +0.761401 q^{9} -4.68475 q^{11} -6.07426 q^{13} -1.63311 q^{15} -5.58884 q^{17} -1.00000 q^{19} +4.45834 q^{21} -7.82359 q^{23} -4.29095 q^{25} +4.34161 q^{27} -8.09724 q^{29} +9.03212 q^{31} +9.08576 q^{33} -1.93570 q^{35} +2.03569 q^{37} +11.7806 q^{39} -4.95946 q^{41} -0.970721 q^{43} +0.641140 q^{45} +7.88034 q^{47} -1.71558 q^{49} +10.8392 q^{51} -1.00000 q^{53} -3.94481 q^{55} +1.93943 q^{57} +0.226530 q^{59} +5.53150 q^{61} -1.75030 q^{63} -5.11485 q^{65} +11.3747 q^{67} +15.1733 q^{69} -6.09899 q^{71} +2.94500 q^{73} +8.32200 q^{75} +10.7692 q^{77} -9.44518 q^{79} -10.7045 q^{81} -2.94000 q^{83} -4.70610 q^{85} +15.7041 q^{87} +4.88041 q^{89} +13.9634 q^{91} -17.5172 q^{93} -0.842053 q^{95} +0.524856 q^{97} -3.56697 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) 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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