LMFDB - Embedded newform 4028.2.a.f.1.10

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			1.10
Root			$0.154590$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.154590 q^{3} +0.882499 q^{5} -1.05782 q^{7} -2.97610 q^{9} +O(q^{10})$	$q+0.154590 q^{3} +0.882499 q^{5} -1.05782 q^{7} -2.97610 q^{9} -4.14996 q^{11} -3.65371 q^{13} +0.136426 q^{15} +4.60286 q^{17} -1.00000 q^{19} -0.163528 q^{21} +5.28840 q^{23} -4.22119 q^{25} -0.923845 q^{27} +4.39547 q^{29} -3.48449 q^{31} -0.641542 q^{33} -0.933523 q^{35} -3.30116 q^{37} -0.564826 q^{39} +7.84277 q^{41} +7.44429 q^{43} -2.62641 q^{45} +9.33965 q^{47} -5.88102 q^{49} +0.711556 q^{51} -1.00000 q^{53} -3.66234 q^{55} -0.154590 q^{57} +7.01306 q^{59} +4.58270 q^{61} +3.14817 q^{63} -3.22439 q^{65} +0.201310 q^{67} +0.817534 q^{69} +5.78213 q^{71} +8.99842 q^{73} -0.652554 q^{75} +4.38990 q^{77} -1.88324 q^{79} +8.78549 q^{81} +14.4439 q^{83} +4.06202 q^{85} +0.679495 q^{87} +13.6353 q^{89} +3.86495 q^{91} -0.538668 q^{93} -0.882499 q^{95} -17.6435 q^{97} +12.3507 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.154590	0.0892526	0.0446263	−	0.999004i	$-0.485790\pi$
$3$	0.154590	0.0892526	0.0446263	+	0.999004i	$0.485790\pi$
$4$	0	0
$5$	0.882499	0.394666	0.197333	−	0.980337i	$-0.436772\pi$
$5$	0.882499	0.394666	0.197333	+	0.980337i	$0.436772\pi$
$6$	0	0
$7$	−1.05782	−0.399818	−0.199909	−	0.979815i	$-0.564065\pi$
$7$	−1.05782	−0.399818	−0.199909	+	0.979815i	$0.564065\pi$
$8$	0	0
$9$	−2.97610	−0.992034
$10$	0	0
$11$	−4.14996	−1.25126	−0.625630	−	0.780120i	$-0.715158\pi$
$11$	−4.14996	−1.25126	−0.625630	+	0.780120i	$0.715158\pi$
$12$	0	0
$13$	−3.65371	−1.01336	−0.506678	−	0.862136i	$-0.669127\pi$
$13$	−3.65371	−1.01336	−0.506678	+	0.862136i	$0.669127\pi$
$14$	0	0
$15$	0.136426	0.0352249
$16$	0	0
$17$	4.60286	1.11636	0.558179	−	0.829721i	$-0.311500\pi$
$17$	4.60286	1.11636	0.558179	+	0.829721i	$0.311500\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.163528	−0.0356847
$22$	0	0
$23$	5.28840	1.10271	0.551354	−	0.834271i	$-0.314111\pi$
$23$	5.28840	1.10271	0.551354	+	0.834271i	$0.314111\pi$
$24$	0	0
$25$	−4.22119	−0.844239
$26$	0	0
$27$	−0.923845	−0.177794
$28$	0	0
$29$	4.39547	0.816217	0.408109	−	0.912933i	$-0.366188\pi$
$29$	4.39547	0.816217	0.408109	+	0.912933i	$0.366188\pi$
$30$	0	0
$31$	−3.48449	−0.625833	−0.312917	−	0.949781i	$-0.601306\pi$
$31$	−3.48449	−0.625833	−0.312917	+	0.949781i	$0.601306\pi$
$32$	0	0
$33$	−0.641542	−0.111678
$34$	0	0
$35$	−0.933523	−0.157794
$36$	0	0
$37$	−3.30116	−0.542707	−0.271354	−	0.962480i	$-0.587471\pi$
$37$	−3.30116	−0.542707	−0.271354	+	0.962480i	$0.587471\pi$
$38$	0	0
$39$	−0.564826	−0.0904446
$40$	0	0
$41$	7.84277	1.22483	0.612417	−	0.790535i	$-0.290197\pi$
$41$	7.84277	1.22483	0.612417	+	0.790535i	$0.290197\pi$
$42$	0	0
$43$	7.44429	1.13524	0.567622	−	0.823289i	$-0.307864\pi$
$43$	7.44429	1.13524	0.567622	+	0.823289i	$0.307864\pi$
$44$	0	0
$45$	−2.62641	−0.391522
$46$	0	0
$47$	9.33965	1.36233	0.681164	−	0.732131i	$-0.261474\pi$
$47$	9.33965	1.36233	0.681164	+	0.732131i	$0.261474\pi$
$48$	0	0
$49$	−5.88102	−0.840146
$50$	0	0
$51$	0.711556	0.0996378
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	−3.66234	−0.493829
$56$	0	0
$57$	−0.154590	−0.0204759
$58$	0	0
$59$	7.01306	0.913023	0.456512	−	0.889717i	$-0.349099\pi$
$59$	7.01306	0.913023	0.456512	+	0.889717i	$0.349099\pi$
$60$	0	0
$61$	4.58270	0.586754	0.293377	−	0.955997i	$-0.405221\pi$
$61$	4.58270	0.586754	0.293377	+	0.955997i	$0.405221\pi$
$62$	0	0
$63$	3.14817	0.396633
$64$	0	0
$65$	−3.22439	−0.399937
$66$	0	0
$67$	0.201310	0.0245940	0.0122970	−	0.999924i	$-0.496086\pi$
$67$	0.201310	0.0245940	0.0122970	+	0.999924i	$0.496086\pi$
$68$	0	0
$69$	0.817534	0.0984195
$70$	0	0
$71$	5.78213	0.686212	0.343106	−	0.939297i	$-0.388521\pi$
$71$	5.78213	0.686212	0.343106	+	0.939297i	$0.388521\pi$
$72$	0	0
$73$	8.99842	1.05318	0.526592	−	0.850118i	$-0.323469\pi$
$73$	8.99842	1.05318	0.526592	+	0.850118i	$0.323469\pi$
$74$	0	0
$75$	−0.652554	−0.0753505
$76$	0	0
$77$	4.38990	0.500276
$78$	0	0
$79$	−1.88324	−0.211881	−0.105940	−	0.994372i	$-0.533785\pi$
$79$	−1.88324	−0.211881	−0.105940	+	0.994372i	$0.533785\pi$
$80$	0	0
$81$	8.78549	0.976165
$82$	0	0
$83$	14.4439	1.58542	0.792712	−	0.609596i	$-0.208668\pi$
$83$	14.4439	1.58542	0.792712	+	0.609596i	$0.208668\pi$
$84$	0	0
$85$	4.06202	0.440588
$86$	0	0
$87$	0.679495	0.0728495
$88$	0	0
$89$	13.6353	1.44534	0.722671	−	0.691192i	$-0.242914\pi$
$89$	13.6353	1.44534	0.722671	+	0.691192i	$0.242914\pi$
$90$	0	0
$91$	3.86495	0.405157
$92$	0	0
$93$	−0.538668	−0.0558572
$94$	0	0
$95$	−0.882499	−0.0905425
$96$	0	0
$97$	−17.6435	−1.79143	−0.895714	−	0.444631i	$-0.853335\pi$
$97$	−17.6435	−1.79143	−0.895714	+	0.444631i	$0.853335\pi$
$98$	0	0
$99$	12.3507	1.24129
$100$	0	0
$101$	−2.24045	−0.222933	−0.111467	−	0.993768i	$-0.535555\pi$
$101$	−2.24045	−0.222933	−0.111467	+	0.993768i	$0.535555\pi$
$102$	0	0
$103$	−2.91887	−0.287605	−0.143802	−	0.989606i	$-0.545933\pi$
$103$	−2.91887	−0.287605	−0.143802	+	0.989606i	$0.545933\pi$
$104$	0	0
$105$	−0.144313	−0.0140835
$106$	0	0
$107$	2.62923	0.254177	0.127088	−	0.991891i	$-0.459437\pi$
$107$	2.62923	0.254177	0.127088	+	0.991891i	$0.459437\pi$
$108$	0	0
$109$	11.2638	1.07888	0.539439	−	0.842025i	$-0.318636\pi$
$109$	11.2638	1.07888	0.539439	+	0.842025i	$0.318636\pi$
$110$	0	0
$111$	−0.510326	−0.0484380
$112$	0	0
$113$	−12.0627	−1.13476	−0.567381	−	0.823455i	$-0.692043\pi$
$113$	−12.0627	−1.13476	−0.567381	+	0.823455i	$0.692043\pi$
$114$	0	0
$115$	4.66701	0.435201
$116$	0	0
$117$	10.8738	1.00528
$118$	0	0
$119$	−4.86899	−0.446339
$120$	0	0
$121$	6.22217	0.565652
$122$	0	0
$123$	1.21241	0.109320
$124$	0	0
$125$	−8.13770	−0.727858
$126$	0	0
$127$	15.2890	1.35668	0.678338	−	0.734750i	$-0.262701\pi$
$127$	15.2890	1.35668	0.678338	+	0.734750i	$0.262701\pi$
$128$	0	0
$129$	1.15081	0.101323
$130$	0	0
$131$	0.320120	0.0279690	0.0139845	−	0.999902i	$-0.495548\pi$
$131$	0.320120	0.0279690	0.0139845	+	0.999902i	$0.495548\pi$
$132$	0	0
$133$	1.05782	0.0917244
$134$	0	0
$135$	−0.815293	−0.0701693
$136$	0	0
$137$	−8.81396	−0.753027	−0.376514	−	0.926411i	$-0.622877\pi$
$137$	−8.81396	−0.753027	−0.376514	+	0.926411i	$0.622877\pi$
$138$	0	0
$139$	10.5742	0.896889	0.448444	−	0.893811i	$-0.351978\pi$
$139$	10.5742	0.896889	0.448444	+	0.893811i	$0.351978\pi$
$140$	0	0
$141$	1.44382	0.121591
$142$	0	0
$143$	15.1627	1.26797
$144$	0	0
$145$	3.87900	0.322133
$146$	0	0
$147$	−0.909147	−0.0749852
$148$	0	0
$149$	18.2785	1.49743	0.748717	−	0.662890i	$-0.230670\pi$
$149$	18.2785	1.49743	0.748717	+	0.662890i	$0.230670\pi$
$150$	0	0
$151$	3.12381	0.254212	0.127106	−	0.991889i	$-0.459431\pi$
$151$	3.12381	0.254212	0.127106	+	0.991889i	$0.459431\pi$
$152$	0	0
$153$	−13.6986	−1.10746
$154$	0	0
$155$	−3.07506	−0.246995
$156$	0	0
$157$	11.1637	0.890960	0.445480	−	0.895292i	$-0.353033\pi$
$157$	11.1637	0.890960	0.445480	+	0.895292i	$0.353033\pi$
$158$	0	0
$159$	−0.154590	−0.0122598
$160$	0	0
$161$	−5.59417	−0.440882
$162$	0	0
$163$	−13.6408	−1.06843	−0.534215	−	0.845348i	$-0.679393\pi$
$163$	−13.6408	−1.06843	−0.534215	+	0.845348i	$0.679393\pi$
$164$	0	0
$165$	−0.566161	−0.0440755
$166$	0	0
$167$	−19.1473	−1.48167	−0.740833	−	0.671689i	$-0.765569\pi$
$167$	−19.1473	−1.48167	−0.740833	+	0.671689i	$0.765569\pi$
$168$	0	0
$169$	0.349563	0.0268895
$170$	0	0
$171$	2.97610	0.227588
$172$	0	0
$173$	−16.9445	−1.28826	−0.644132	−	0.764915i	$-0.722781\pi$
$173$	−16.9445	−1.28826	−0.644132	+	0.764915i	$0.722781\pi$
$174$	0	0
$175$	4.46525	0.337542
$176$	0	0
$177$	1.08415	0.0814897
$178$	0	0
$179$	10.4418	0.780456	0.390228	−	0.920718i	$-0.372396\pi$
$179$	10.4418	0.780456	0.390228	+	0.920718i	$0.372396\pi$
$180$	0	0
$181$	1.15443	0.0858084	0.0429042	−	0.999079i	$-0.486339\pi$
$181$	1.15443	0.0858084	0.0429042	+	0.999079i	$0.486339\pi$
$182$	0	0
$183$	0.708439	0.0523693
$184$	0	0
$185$	−2.91327	−0.214188
$186$	0	0
$187$	−19.1017	−1.39685
$188$	0	0
$189$	0.977260	0.0710852
$190$	0	0
$191$	−17.6860	−1.27971	−0.639857	−	0.768494i	$-0.721006\pi$
$191$	−17.6860	−1.27971	−0.639857	+	0.768494i	$0.721006\pi$
$192$	0	0
$193$	−1.60671	−0.115653	−0.0578267	−	0.998327i	$-0.518417\pi$
$193$	−1.60671	−0.115653	−0.0578267	+	0.998327i	$0.518417\pi$
$194$	0	0
$195$	−0.498459	−0.0356954
$196$	0	0
$197$	−3.70590	−0.264034	−0.132017	−	0.991247i	$-0.542145\pi$
$197$	−3.70590	−0.264034	−0.132017	+	0.991247i	$0.542145\pi$
$198$	0	0
$199$	−12.0438	−0.853763	−0.426882	−	0.904308i	$-0.640388\pi$
$199$	−12.0438	−0.853763	−0.426882	+	0.904308i	$0.640388\pi$
$200$	0	0
$201$	0.0311205	0.00219507
$202$	0	0
$203$	−4.64960	−0.326338
$204$	0	0
$205$	6.92124	0.483400
$206$	0	0
$207$	−15.7388	−1.09392
$208$	0	0
$209$	4.14996	0.287059
$210$	0	0
$211$	1.59223	0.109614	0.0548068	−	0.998497i	$-0.482546\pi$
$211$	1.59223	0.109614	0.0548068	+	0.998497i	$0.482546\pi$
$212$	0	0
$213$	0.893859	0.0612462
$214$	0	0
$215$	6.56958	0.448042
$216$	0	0
$217$	3.68596	0.250219
$218$	0	0
$219$	1.39106	0.0939995
$220$	0	0
$221$	−16.8175	−1.13127
$222$	0	0
$223$	−9.67033	−0.647573	−0.323787	−	0.946130i	$-0.604956\pi$
$223$	−9.67033	−0.647573	−0.323787	+	0.946130i	$0.604956\pi$
$224$	0	0
$225$	12.5627	0.837514
$226$	0	0
$227$	29.3787	1.94993	0.974967	−	0.222349i	$-0.0713724\pi$
$227$	29.3787	1.94993	0.974967	+	0.222349i	$0.0713724\pi$
$228$	0	0
$229$	−24.0905	−1.59194	−0.795972	−	0.605334i	$-0.793040\pi$
$229$	−24.0905	−1.59194	−0.795972	+	0.605334i	$0.793040\pi$
$230$	0	0
$231$	0.678635	0.0446509
$232$	0	0
$233$	−15.4250	−1.01053	−0.505263	−	0.862965i	$-0.668605\pi$
$233$	−15.4250	−1.01053	−0.505263	+	0.862965i	$0.668605\pi$
$234$	0	0
$235$	8.24224	0.537664
$236$	0	0
$237$	−0.291129	−0.0189109
$238$	0	0
$239$	−5.04278	−0.326190	−0.163095	−	0.986610i	$-0.552148\pi$
$239$	−5.04278	−0.326190	−0.163095	+	0.986610i	$0.552148\pi$
$240$	0	0
$241$	30.4060	1.95862	0.979312	−	0.202355i	$-0.0648596\pi$
$241$	30.4060	1.95862	0.979312	+	0.202355i	$0.0648596\pi$
$242$	0	0
$243$	4.12968	0.264919
$244$	0	0
$245$	−5.19000	−0.331577
$246$	0	0
$247$	3.65371	0.232480
$248$	0	0
$249$	2.23288	0.141503
$250$	0	0
$251$	9.87473	0.623287	0.311644	−	0.950199i	$-0.399120\pi$
$251$	9.87473	0.623287	0.311644	+	0.950199i	$0.399120\pi$
$252$	0	0
$253$	−21.9467	−1.37977
$254$	0	0
$255$	0.627948	0.0393236
$256$	0	0
$257$	−0.0572198	−0.00356927	−0.00178464	−	0.999998i	$-0.500568\pi$
$257$	−0.0572198	−0.00356927	−0.00178464	+	0.999998i	$0.500568\pi$
$258$	0	0
$259$	3.49203	0.216984
$260$	0	0
$261$	−13.0814	−0.809715
$262$	0	0
$263$	13.9753	0.861755	0.430877	−	0.902410i	$-0.358204\pi$
$263$	13.9753	0.861755	0.430877	+	0.902410i	$0.358204\pi$
$264$	0	0
$265$	−0.882499	−0.0542115
$266$	0	0
$267$	2.10789	0.129000
$268$	0	0
$269$	6.26865	0.382206	0.191103	−	0.981570i	$-0.438793\pi$
$269$	6.26865	0.382206	0.191103	+	0.981570i	$0.438793\pi$
$270$	0	0
$271$	21.2508	1.29089	0.645447	−	0.763805i	$-0.276671\pi$
$271$	21.2508	1.29089	0.645447	+	0.763805i	$0.276671\pi$
$272$	0	0
$273$	0.597483	0.0361613
$274$	0	0
$275$	17.5178	1.05636
$276$	0	0
$277$	4.82577	0.289952	0.144976	−	0.989435i	$-0.453689\pi$
$277$	4.82577	0.289952	0.144976	+	0.989435i	$0.453689\pi$
$278$	0	0
$279$	10.3702	0.620848
$280$	0	0
$281$	−3.23117	−0.192755	−0.0963777	−	0.995345i	$-0.530726\pi$
$281$	−3.23117	−0.192755	−0.0963777	+	0.995345i	$0.530726\pi$
$282$	0	0
$283$	8.44782	0.502171	0.251085	−	0.967965i	$-0.419212\pi$
$283$	8.44782	0.502171	0.251085	+	0.967965i	$0.419212\pi$
$284$	0	0
$285$	−0.136426	−0.00808115
$286$	0	0
$287$	−8.29622	−0.489710
$288$	0	0
$289$	4.18633	0.246255
$290$	0	0
$291$	−2.72751	−0.159890
$292$	0	0
$293$	0.666509	0.0389378	0.0194689	−	0.999810i	$-0.493802\pi$
$293$	0.666509	0.0389378	0.0194689	+	0.999810i	$0.493802\pi$
$294$	0	0
$295$	6.18902	0.360339
$296$	0	0
$297$	3.83392	0.222467
$298$	0	0
$299$	−19.3223	−1.11744
$300$	0	0
$301$	−7.87470	−0.453890
$302$	0	0
$303$	−0.346351	−0.0198974
$304$	0	0
$305$	4.04423	0.231572
$306$	0	0
$307$	4.34701	0.248097	0.124049	−	0.992276i	$-0.460412\pi$
$307$	4.34701	0.248097	0.124049	+	0.992276i	$0.460412\pi$
$308$	0	0
$309$	−0.451228	−0.0256695
$310$	0	0
$311$	22.8486	1.29563	0.647813	−	0.761799i	$-0.275684\pi$
$311$	22.8486	1.29563	0.647813	+	0.761799i	$0.275684\pi$
$312$	0	0
$313$	23.2249	1.31275	0.656375	−	0.754435i	$-0.272089\pi$
$313$	23.2249	1.31275	0.656375	+	0.754435i	$0.272089\pi$
$314$	0	0
$315$	2.77826	0.156537
$316$	0	0
$317$	−9.67485	−0.543394	−0.271697	−	0.962383i	$-0.587585\pi$
$317$	−9.67485	−0.543394	−0.271697	+	0.962383i	$0.587585\pi$
$318$	0	0
$319$	−18.2410	−1.02130
$320$	0	0
$321$	0.406452	0.0226859
$322$	0	0
$323$	−4.60286	−0.256110
$324$	0	0
$325$	15.4230	0.855514
$326$	0	0
$327$	1.74127	0.0962926
$328$	0	0
$329$	−9.87965	−0.544683
$330$	0	0
$331$	10.6238	0.583939	0.291970	−	0.956428i	$-0.405689\pi$
$331$	10.6238	0.583939	0.291970	+	0.956428i	$0.405689\pi$
$332$	0	0
$333$	9.82459	0.538384
$334$	0	0
$335$	0.177656	0.00970639
$336$	0	0
$337$	5.52503	0.300968	0.150484	−	0.988612i	$-0.451917\pi$
$337$	5.52503	0.300968	0.150484	+	0.988612i	$0.451917\pi$
$338$	0	0
$339$	−1.86477	−0.101280
$340$	0	0
$341$	14.4605	0.783080
$342$	0	0
$343$	13.6258	0.735723
$344$	0	0
$345$	0.721473	0.0388428
$346$	0	0
$347$	17.7937	0.955218	0.477609	−	0.878572i	$-0.341504\pi$
$347$	17.7937	0.955218	0.477609	+	0.878572i	$0.341504\pi$
$348$	0	0
$349$	12.2858	0.657644	0.328822	−	0.944392i	$-0.393348\pi$
$349$	12.2858	0.657644	0.328822	+	0.944392i	$0.393348\pi$
$350$	0	0
$351$	3.37546	0.180169
$352$	0	0
$353$	−36.0413	−1.91829	−0.959143	−	0.282921i	$-0.908696\pi$
$353$	−36.0413	−1.91829	−0.959143	+	0.282921i	$0.908696\pi$
$354$	0	0
$355$	5.10272	0.270825
$356$	0	0
$357$	−0.752697	−0.0398369
$358$	0	0
$359$	−26.3256	−1.38941	−0.694707	−	0.719293i	$-0.744466\pi$
$359$	−26.3256	−1.38941	−0.694707	+	0.719293i	$0.744466\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.961885	0.0504859
$364$	0	0
$365$	7.94110	0.415656
$366$	0	0
$367$	14.9085	0.778218	0.389109	−	0.921192i	$-0.372783\pi$
$367$	14.9085	0.778218	0.389109	+	0.921192i	$0.372783\pi$
$368$	0	0
$369$	−23.3409	−1.21508
$370$	0	0
$371$	1.05782	0.0549192
$372$	0	0
$373$	−8.49133	−0.439665	−0.219832	−	0.975538i	$-0.570551\pi$
$373$	−8.49133	−0.439665	−0.219832	+	0.975538i	$0.570551\pi$
$374$	0	0
$375$	−1.25801	−0.0649632
$376$	0	0
$377$	−16.0597	−0.827118
$378$	0	0
$379$	1.13802	0.0584564	0.0292282	−	0.999573i	$-0.490695\pi$
$379$	1.13802	0.0584564	0.0292282	+	0.999573i	$0.490695\pi$
$380$	0	0
$381$	2.36352	0.121087
$382$	0	0
$383$	38.9063	1.98802	0.994011	−	0.109284i	$-0.0348559\pi$
$383$	38.9063	1.98802	0.994011	+	0.109284i	$0.0348559\pi$
$384$	0	0
$385$	3.87408	0.197442
$386$	0	0
$387$	−22.1550	−1.12620
$388$	0	0
$389$	34.1323	1.73057	0.865287	−	0.501276i	$-0.167136\pi$
$389$	34.1323	1.73057	0.865287	+	0.501276i	$0.167136\pi$
$390$	0	0
$391$	24.3418	1.23102
$392$	0	0
$393$	0.0494874	0.00249631
$394$	0	0
$395$	−1.66195	−0.0836220
$396$	0	0
$397$	10.4522	0.524580	0.262290	−	0.964989i	$-0.415522\pi$
$397$	10.4522	0.524580	0.262290	+	0.964989i	$0.415522\pi$
$398$	0	0
$399$	0.163528	0.00818664
$400$	0	0
$401$	−15.5046	−0.774261	−0.387130	−	0.922025i	$-0.626534\pi$
$401$	−15.5046	−0.774261	−0.387130	+	0.922025i	$0.626534\pi$
$402$	0	0
$403$	12.7313	0.634192
$404$	0	0
$405$	7.75319	0.385259
$406$	0	0
$407$	13.6997	0.679068
$408$	0	0
$409$	−9.29074	−0.459398	−0.229699	−	0.973262i	$-0.573774\pi$
$409$	−9.29074	−0.459398	−0.229699	+	0.973262i	$0.573774\pi$
$410$	0	0
$411$	−1.36255	−0.0672096
$412$	0	0
$413$	−7.41854	−0.365043
$414$	0	0
$415$	12.7467	0.625713
$416$	0	0
$417$	1.63466	0.0800496
$418$	0	0
$419$	−10.4673	−0.511362	−0.255681	−	0.966761i	$-0.582300\pi$
$419$	−10.4673	−0.511362	−0.255681	+	0.966761i	$0.582300\pi$
$420$	0	0
$421$	−13.0029	−0.633724	−0.316862	−	0.948472i	$-0.602629\pi$
$421$	−13.0029	−0.633724	−0.316862	+	0.948472i	$0.602629\pi$
$422$	0	0
$423$	−27.7958	−1.35148
$424$	0	0
$425$	−19.4296	−0.942473
$426$	0	0
$427$	−4.84766	−0.234595
$428$	0	0
$429$	2.34401	0.113170
$430$	0	0
$431$	−31.0682	−1.49650	−0.748251	−	0.663416i	$-0.769106\pi$
$431$	−31.0682	−1.49650	−0.748251	+	0.663416i	$0.769106\pi$
$432$	0	0
$433$	−10.1465	−0.487608	−0.243804	−	0.969824i	$-0.578395\pi$
$433$	−10.1465	−0.487608	−0.243804	+	0.969824i	$0.578395\pi$
$434$	0	0
$435$	0.599654	0.0287512
$436$	0	0
$437$	−5.28840	−0.252979
$438$	0	0
$439$	8.64887	0.412788	0.206394	−	0.978469i	$-0.433827\pi$
$439$	8.64887	0.412788	0.206394	+	0.978469i	$0.433827\pi$
$440$	0	0
$441$	17.5025	0.833453
$442$	0	0
$443$	−16.5973	−0.788562	−0.394281	−	0.918990i	$-0.629006\pi$
$443$	−16.5973	−0.788562	−0.394281	+	0.918990i	$0.629006\pi$
$444$	0	0
$445$	12.0332	0.570427
$446$	0	0
$447$	2.82567	0.133650
$448$	0	0
$449$	35.4384	1.67244	0.836220	−	0.548394i	$-0.184760\pi$
$449$	35.4384	1.67244	0.836220	+	0.548394i	$0.184760\pi$
$450$	0	0
$451$	−32.5472	−1.53259
$452$	0	0
$453$	0.482910	0.0226891
$454$	0	0
$455$	3.41082	0.159902
$456$	0	0
$457$	25.0651	1.17249	0.586247	−	0.810132i	$-0.300605\pi$
$457$	25.0651	1.17249	0.586247	+	0.810132i	$0.300605\pi$
$458$	0	0
$459$	−4.25233	−0.198482
$460$	0	0
$461$	9.32373	0.434249	0.217125	−	0.976144i	$-0.430332\pi$
$461$	9.32373	0.434249	0.217125	+	0.976144i	$0.430332\pi$
$462$	0	0
$463$	14.6505	0.680866	0.340433	−	0.940269i	$-0.389426\pi$
$463$	14.6505	0.680866	0.340433	+	0.940269i	$0.389426\pi$
$464$	0	0
$465$	−0.475374	−0.0220449
$466$	0	0
$467$	12.7161	0.588430	0.294215	−	0.955739i	$-0.404942\pi$
$467$	12.7161	0.588430	0.294215	+	0.955739i	$0.404942\pi$
$468$	0	0
$469$	−0.212949	−0.00983309
$470$	0	0
$471$	1.72580	0.0795205
$472$	0	0
$473$	−30.8935	−1.42049
$474$	0	0
$475$	4.22119	0.193682
$476$	0	0
$477$	2.97610	0.136266
$478$	0	0
$479$	−8.46030	−0.386561	−0.193280	−	0.981144i	$-0.561913\pi$
$479$	−8.46030	−0.386561	−0.193280	+	0.981144i	$0.561913\pi$
$480$	0	0
$481$	12.0615	0.549956
$482$	0	0
$483$	−0.864802	−0.0393499
$484$	0	0
$485$	−15.5704	−0.707015
$486$	0	0
$487$	12.9226	0.585578	0.292789	−	0.956177i	$-0.405417\pi$
$487$	12.9226	0.585578	0.292789	+	0.956177i	$0.405417\pi$
$488$	0	0
$489$	−2.10873	−0.0953602
$490$	0	0
$491$	41.8078	1.88676	0.943379	−	0.331717i	$-0.107628\pi$
$491$	41.8078	1.88676	0.943379	+	0.331717i	$0.107628\pi$
$492$	0	0
$493$	20.2317	0.911191
$494$	0	0
$495$	10.8995	0.489896
$496$	0	0
$497$	−6.11644	−0.274360
$498$	0	0
$499$	−39.0643	−1.74876	−0.874379	−	0.485244i	$-0.838731\pi$
$499$	−39.0643	−1.74876	−0.874379	+	0.485244i	$0.838731\pi$
$500$	0	0
$501$	−2.95999	−0.132242
$502$	0	0
$503$	−1.99819	−0.0890950	−0.0445475	−	0.999007i	$-0.514185\pi$
$503$	−1.99819	−0.0890950	−0.0445475	+	0.999007i	$0.514185\pi$
$504$	0	0
$505$	−1.97720	−0.0879841
$506$	0	0
$507$	0.0540389	0.00239995
$508$	0	0
$509$	20.2998	0.899772	0.449886	−	0.893086i	$-0.351465\pi$
$509$	20.2998	0.899772	0.449886	+	0.893086i	$0.351465\pi$
$510$	0	0
$511$	−9.51868	−0.421082
$512$	0	0
$513$	0.923845	0.0407888
$514$	0	0
$515$	−2.57590	−0.113508
$516$	0	0
$517$	−38.7592	−1.70463
$518$	0	0
$519$	−2.61944	−0.114981
$520$	0	0
$521$	−6.71724	−0.294288	−0.147144	−	0.989115i	$-0.547008\pi$
$521$	−6.71724	−0.294288	−0.147144	+	0.989115i	$0.547008\pi$
$522$	0	0
$523$	20.8090	0.909915	0.454958	−	0.890513i	$-0.349654\pi$
$523$	20.8090	0.909915	0.454958	+	0.890513i	$0.349654\pi$
$524$	0	0
$525$	0.690284	0.0301264
$526$	0	0
$527$	−16.0386	−0.698654
$528$	0	0
$529$	4.96721	0.215966
$530$	0	0
$531$	−20.8716	−0.905750
$532$	0	0
$533$	−28.6552	−1.24119
$534$	0	0
$535$	2.32029	0.100315
$536$	0	0
$537$	1.61420	0.0696577
$538$	0	0
$539$	24.4060	1.05124
$540$	0	0
$541$	6.81251	0.292893	0.146446	−	0.989219i	$-0.453216\pi$
$541$	6.81251	0.292893	0.146446	+	0.989219i	$0.453216\pi$
$542$	0	0
$543$	0.178464	0.00765862
$544$	0	0
$545$	9.94031	0.425796
$546$	0	0
$547$	−28.3831	−1.21357	−0.606787	−	0.794865i	$-0.707542\pi$
$547$	−28.3831	−1.21357	−0.606787	+	0.794865i	$0.707542\pi$
$548$	0	0
$549$	−13.6386	−0.582080
$550$	0	0
$551$	−4.39547	−0.187253
$552$	0	0
$553$	1.99212	0.0847136
$554$	0	0
$555$	−0.450363	−0.0191168
$556$	0	0
$557$	−40.9675	−1.73585	−0.867924	−	0.496697i	$-0.834546\pi$
$557$	−40.9675	−1.73585	−0.867924	+	0.496697i	$0.834546\pi$
$558$	0	0
$559$	−27.1993	−1.15041
$560$	0	0
$561$	−2.95293	−0.124673
$562$	0	0
$563$	3.03616	0.127959	0.0639795	−	0.997951i	$-0.479621\pi$
$563$	3.03616	0.127959	0.0639795	+	0.997951i	$0.479621\pi$
$564$	0	0
$565$	−10.6453	−0.447852
$566$	0	0
$567$	−9.29345	−0.390288
$568$	0	0
$569$	−28.0050	−1.17403	−0.587015	−	0.809576i	$-0.699697\pi$
$569$	−28.0050	−1.17403	−0.587015	+	0.809576i	$0.699697\pi$
$570$	0	0
$571$	−28.8630	−1.20788	−0.603939	−	0.797031i	$-0.706403\pi$
$571$	−28.8630	−1.20788	−0.603939	+	0.797031i	$0.706403\pi$
$572$	0	0
$573$	−2.73408	−0.114218
$574$	0	0
$575$	−22.3234	−0.930949
$576$	0	0
$577$	−12.8038	−0.533029	−0.266514	−	0.963831i	$-0.585872\pi$
$577$	−12.8038	−0.533029	−0.266514	+	0.963831i	$0.585872\pi$
$578$	0	0
$579$	−0.248381	−0.0103224
$580$	0	0
$581$	−15.2790	−0.633881
$582$	0	0
$583$	4.14996	0.171874
$584$	0	0
$585$	9.59612	0.396751
$586$	0	0
$587$	29.6017	1.22179	0.610896	−	0.791711i	$-0.290809\pi$
$587$	29.6017	1.22179	0.610896	+	0.791711i	$0.290809\pi$
$588$	0	0
$589$	3.48449	0.143576
$590$	0	0
$591$	−0.572895	−0.0235657
$592$	0	0
$593$	−3.08102	−0.126522	−0.0632611	−	0.997997i	$-0.520150\pi$
$593$	−3.08102	−0.126522	−0.0632611	+	0.997997i	$0.520150\pi$
$594$	0	0
$595$	−4.29688	−0.176155
$596$	0	0
$597$	−1.86185	−0.0762006
$598$	0	0
$599$	38.2681	1.56359	0.781796	−	0.623535i	$-0.214304\pi$
$599$	38.2681	1.56359	0.781796	+	0.623535i	$0.214304\pi$
$600$	0	0
$601$	−25.4513	−1.03818	−0.519090	−	0.854720i	$-0.673729\pi$
$601$	−25.4513	−1.03818	−0.519090	+	0.854720i	$0.673729\pi$
$602$	0	0
$603$	−0.599120	−0.0243980
$604$	0	0
$605$	5.49106	0.223243
$606$	0	0
$607$	4.89483	0.198675	0.0993376	−	0.995054i	$-0.468328\pi$
$607$	4.89483	0.198675	0.0993376	+	0.995054i	$0.468328\pi$
$608$	0	0
$609$	−0.718782	−0.0291265
$610$	0	0
$611$	−34.1243	−1.38052
$612$	0	0
$613$	−8.59698	−0.347229	−0.173614	−	0.984814i	$-0.555545\pi$
$613$	−8.59698	−0.347229	−0.173614	+	0.984814i	$0.555545\pi$
$614$	0	0
$615$	1.06995	0.0431447
$616$	0	0
$617$	−0.154132	−0.00620513	−0.00310256	−	0.999995i	$-0.500988\pi$
$617$	−0.154132	−0.00620513	−0.00310256	+	0.999995i	$0.500988\pi$
$618$	0	0
$619$	25.8329	1.03831	0.519156	−	0.854680i	$-0.326246\pi$
$619$	25.8329	1.03831	0.519156	+	0.854680i	$0.326246\pi$
$620$	0	0
$621$	−4.88567	−0.196055
$622$	0	0
$623$	−14.4237	−0.577873
$624$	0	0
$625$	13.9245	0.556978
$626$	0	0
$627$	0.641542	0.0256207
$628$	0	0
$629$	−15.1948	−0.605856
$630$	0	0
$631$	19.3290	0.769476	0.384738	−	0.923026i	$-0.374292\pi$
$631$	19.3290	0.769476	0.384738	+	0.923026i	$0.374292\pi$
$632$	0	0
$633$	0.246143	0.00978329
$634$	0	0
$635$	13.4925	0.535433
$636$	0	0
$637$	21.4875	0.851367
$638$	0	0
$639$	−17.2082	−0.680746
$640$	0	0
$641$	11.5763	0.457236	0.228618	−	0.973516i	$-0.426579\pi$
$641$	11.5763	0.457236	0.228618	+	0.973516i	$0.426579\pi$
$642$	0	0
$643$	−11.1849	−0.441090	−0.220545	−	0.975377i	$-0.570784\pi$
$643$	−11.1849	−0.441090	−0.220545	+	0.975377i	$0.570784\pi$
$644$	0	0
$645$	1.01559	0.0399889
$646$	0	0
$647$	27.5334	1.08245	0.541225	−	0.840878i	$-0.317961\pi$
$647$	27.5334	1.08245	0.541225	+	0.840878i	$0.317961\pi$
$648$	0	0
$649$	−29.1039	−1.14243
$650$	0	0
$651$	0.569812	0.0223327
$652$	0	0
$653$	24.9130	0.974922	0.487461	−	0.873145i	$-0.337923\pi$
$653$	24.9130	0.974922	0.487461	+	0.873145i	$0.337923\pi$
$654$	0	0
$655$	0.282506	0.0110384
$656$	0	0
$657$	−26.7802	−1.04480
$658$	0	0
$659$	16.5228	0.643638	0.321819	−	0.946801i	$-0.395706\pi$
$659$	16.5228	0.643638	0.321819	+	0.946801i	$0.395706\pi$
$660$	0	0
$661$	−29.3858	−1.14298	−0.571488	−	0.820611i	$-0.693634\pi$
$661$	−29.3858	−1.14298	−0.571488	+	0.820611i	$0.693634\pi$
$662$	0	0
$663$	−2.59982	−0.100969
$664$	0	0
$665$	0.933523	0.0362005
$666$	0	0
$667$	23.2450	0.900050
$668$	0	0
$669$	−1.49494	−0.0577976
$670$	0	0
$671$	−19.0180	−0.734182
$672$	0	0
$673$	−37.6317	−1.45059	−0.725297	−	0.688436i	$-0.758298\pi$
$673$	−37.6317	−1.45059	−0.725297	+	0.688436i	$0.758298\pi$
$674$	0	0
$675$	3.89973	0.150101
$676$	0	0
$677$	19.9016	0.764880	0.382440	−	0.923980i	$-0.375084\pi$
$677$	19.9016	0.764880	0.382440	+	0.923980i	$0.375084\pi$
$678$	0	0
$679$	18.6636	0.716244
$680$	0	0
$681$	4.54166	0.174037
$682$	0	0
$683$	18.1526	0.694591	0.347296	−	0.937756i	$-0.387100\pi$
$683$	18.1526	0.694591	0.347296	+	0.937756i	$0.387100\pi$
$684$	0	0
$685$	−7.77831	−0.297194
$686$	0	0
$687$	−3.72415	−0.142085
$688$	0	0
$689$	3.65371	0.139195
$690$	0	0
$691$	−4.98566	−0.189664	−0.0948318	−	0.995493i	$-0.530231\pi$
$691$	−4.98566	−0.189664	−0.0948318	+	0.995493i	$0.530231\pi$
$692$	0	0
$693$	−13.0648	−0.496290
$694$	0	0
$695$	9.33169	0.353971
$696$	0	0
$697$	36.0992	1.36735
$698$	0	0
$699$	−2.38455	−0.0901921
$700$	0	0
$701$	−10.4014	−0.392854	−0.196427	−	0.980518i	$-0.562934\pi$
$701$	−10.4014	−0.392854	−0.196427	+	0.980518i	$0.562934\pi$
$702$	0	0
$703$	3.30116	0.124506
$704$	0	0
$705$	1.27417	0.0479879
$706$	0	0
$707$	2.36999	0.0891327
$708$	0	0
$709$	19.8302	0.744737	0.372369	−	0.928085i	$-0.378546\pi$
$709$	19.8302	0.744737	0.372369	+	0.928085i	$0.378546\pi$
$710$	0	0
$711$	5.60470	0.210193
$712$	0	0
$713$	−18.4274	−0.690111
$714$	0	0
$715$	13.3811	0.500425
$716$	0	0
$717$	−0.779563	−0.0291133
$718$	0	0
$719$	−29.7744	−1.11040	−0.555199	−	0.831717i	$-0.687358\pi$
$719$	−29.7744	−1.11040	−0.555199	+	0.831717i	$0.687358\pi$
$720$	0	0
$721$	3.08763	0.114989
$722$	0	0
$723$	4.70047	0.174812
$724$	0	0
$725$	−18.5541	−0.689083
$726$	0	0
$727$	−15.8852	−0.589149	−0.294575	−	0.955628i	$-0.595178\pi$
$727$	−15.8852	−0.589149	−0.294575	+	0.955628i	$0.595178\pi$
$728$	0	0
$729$	−25.7181	−0.952521
$730$	0	0
$731$	34.2650	1.26734
$732$	0	0
$733$	−41.9793	−1.55054	−0.775270	−	0.631630i	$-0.782386\pi$
$733$	−41.9793	−1.55054	−0.775270	+	0.631630i	$0.782386\pi$
$734$	0	0
$735$	−0.802322	−0.0295941
$736$	0	0
$737$	−0.835429	−0.0307734
$738$	0	0
$739$	12.8416	0.472387	0.236193	−	0.971706i	$-0.424100\pi$
$739$	12.8416	0.472387	0.236193	+	0.971706i	$0.424100\pi$
$740$	0	0
$741$	0.564826	0.0207494
$742$	0	0
$743$	47.8455	1.75528	0.877641	−	0.479318i	$-0.159116\pi$
$743$	47.8455	1.75528	0.877641	+	0.479318i	$0.159116\pi$
$744$	0	0
$745$	16.1308	0.590985
$746$	0	0
$747$	−42.9865	−1.57280
$748$	0	0
$749$	−2.78124	−0.101624
$750$	0	0
$751$	−14.4369	−0.526811	−0.263405	−	0.964685i	$-0.584846\pi$
$751$	−14.4369	−0.526811	−0.263405	+	0.964685i	$0.584846\pi$
$752$	0	0
$753$	1.52653	0.0556300
$754$	0	0
$755$	2.75676	0.100329
$756$	0	0
$757$	48.1742	1.75092	0.875460	−	0.483290i	$-0.160558\pi$
$757$	48.1742	1.75092	0.875460	+	0.483290i	$0.160558\pi$
$758$	0	0
$759$	−3.39273	−0.123148
$760$	0	0
$761$	11.9605	0.433569	0.216784	−	0.976220i	$-0.430443\pi$
$761$	11.9605	0.433569	0.216784	+	0.976220i	$0.430443\pi$
$762$	0	0
$763$	−11.9151	−0.431354
$764$	0	0
$765$	−12.0890	−0.437078
$766$	0	0
$767$	−25.6237	−0.925217
$768$	0	0
$769$	13.7524	0.495924	0.247962	−	0.968770i	$-0.420239\pi$
$769$	13.7524	0.495924	0.247962	+	0.968770i	$0.420239\pi$
$770$	0	0
$771$	−0.00884560	−0.000318567 0
$772$	0	0
$773$	16.3427	0.587807	0.293903	−	0.955835i	$-0.405046\pi$
$773$	16.3427	0.587807	0.293903	+	0.955835i	$0.405046\pi$
$774$	0	0
$775$	14.7087	0.528353
$776$	0	0
$777$	0.539832	0.0193664
$778$	0	0
$779$	−7.84277	−0.280996
$780$	0	0
$781$	−23.9956	−0.858630
$782$	0	0
$783$	−4.06073	−0.145119
$784$	0	0
$785$	9.85196	0.351631
$786$	0	0
$787$	−49.1190	−1.75090	−0.875452	−	0.483306i	$-0.839436\pi$
$787$	−49.1190	−1.75090	−0.875452	+	0.483306i	$0.839436\pi$
$788$	0	0
$789$	2.16044	0.0769138
$790$	0	0
$791$	12.7601	0.453698
$792$	0	0
$793$	−16.7438	−0.594591
$794$	0	0
$795$	−0.136426	−0.00483852
$796$	0	0
$797$	−34.9980	−1.23969	−0.619846	−	0.784724i	$-0.712805\pi$
$797$	−34.9980	−1.23969	−0.619846	+	0.784724i	$0.712805\pi$
$798$	0	0
$799$	42.9891	1.52085
$800$	0	0
$801$	−40.5801	−1.43383
$802$	0	0
$803$	−37.3431	−1.31781
$804$	0	0
$805$	−4.93685	−0.174001
$806$	0	0
$807$	0.969070	0.0341129
$808$	0	0
$809$	−13.0801	−0.459872	−0.229936	−	0.973206i	$-0.573852\pi$
$809$	−13.0801	−0.459872	−0.229936	+	0.973206i	$0.573852\pi$
$810$	0	0
$811$	34.4857	1.21096	0.605478	−	0.795862i	$-0.292982\pi$
$811$	34.4857	1.21096	0.605478	+	0.795862i	$0.292982\pi$
$812$	0	0
$813$	3.28516	0.115216
$814$	0	0
$815$	−12.0380	−0.421673
$816$	0	0
$817$	−7.44429	−0.260443
$818$	0	0
$819$	−11.5025	−0.401930
$820$	0	0
$821$	−40.2699	−1.40543	−0.702715	−	0.711471i	$-0.748029\pi$
$821$	−40.2699	−1.40543	−0.702715	+	0.711471i	$0.748029\pi$
$822$	0	0
$823$	−4.29380	−0.149672	−0.0748361	−	0.997196i	$-0.523843\pi$
$823$	−4.29380	−0.149672	−0.0748361	+	0.997196i	$0.523843\pi$
$824$	0	0
$825$	2.70807	0.0942831
$826$	0	0
$827$	−20.1341	−0.700130	−0.350065	−	0.936725i	$-0.613841\pi$
$827$	−20.1341	−0.700130	−0.350065	+	0.936725i	$0.613841\pi$
$828$	0	0
$829$	−16.9690	−0.589359	−0.294679	−	0.955596i	$-0.595213\pi$
$829$	−16.9690	−0.589359	−0.294679	+	0.955596i	$0.595213\pi$
$830$	0	0
$831$	0.746016	0.0258790
$832$	0	0
$833$	−27.0695	−0.937904
$834$	0	0
$835$	−16.8975	−0.584763
$836$	0	0
$837$	3.21913	0.111269
$838$	0	0
$839$	−30.6461	−1.05802	−0.529011	−	0.848615i	$-0.677437\pi$
$839$	−30.6461	−1.05802	−0.529011	+	0.848615i	$0.677437\pi$
$840$	0	0
$841$	−9.67988	−0.333789
$842$	0	0
$843$	−0.499507	−0.0172039
$844$	0	0
$845$	0.308489	0.0106123
$846$	0	0
$847$	−6.58192	−0.226157
$848$	0	0
$849$	1.30595	0.0448200
$850$	0	0
$851$	−17.4579	−0.598448
$852$	0	0
$853$	29.6005	1.01350	0.506750	−	0.862093i	$-0.330847\pi$
$853$	29.6005	1.01350	0.506750	+	0.862093i	$0.330847\pi$
$854$	0	0
$855$	2.62641	0.0898213
$856$	0	0
$857$	28.4478	0.971758	0.485879	−	0.874026i	$-0.338500\pi$
$857$	28.4478	0.971758	0.485879	+	0.874026i	$0.338500\pi$
$858$	0	0
$859$	42.8543	1.46217	0.731086	−	0.682286i	$-0.239014\pi$
$859$	42.8543	1.46217	0.731086	+	0.682286i	$0.239014\pi$
$860$	0	0
$861$	−1.28251	−0.0437079
$862$	0	0
$863$	42.4521	1.44509	0.722543	−	0.691325i	$-0.242973\pi$
$863$	42.4521	1.44509	0.722543	+	0.691325i	$0.242973\pi$
$864$	0	0
$865$	−14.9535	−0.508433
$866$	0	0
$867$	0.647165	0.0219789
$868$	0	0
$869$	7.81535	0.265118
$870$	0	0
$871$	−0.735528	−0.0249224
$872$	0	0
$873$	52.5089	1.77716
$874$	0	0
$875$	8.60820	0.291010
$876$	0	0
$877$	46.7975	1.58024	0.790120	−	0.612952i	$-0.210018\pi$
$877$	46.7975	1.58024	0.790120	+	0.612952i	$0.210018\pi$
$878$	0	0
$879$	0.103036	0.00347530
$880$	0	0
$881$	−2.39194	−0.0805866	−0.0402933	−	0.999188i	$-0.512829\pi$
$881$	−2.39194	−0.0805866	−0.0402933	+	0.999188i	$0.512829\pi$
$882$	0	0
$883$	−36.4647	−1.22713	−0.613567	−	0.789643i	$-0.710266\pi$
$883$	−36.4647	−1.22713	−0.613567	+	0.789643i	$0.710266\pi$
$884$	0	0
$885$	0.956761	0.0321612
$886$	0	0
$887$	−21.3845	−0.718020	−0.359010	−	0.933334i	$-0.616886\pi$
$887$	−21.3845	−0.718020	−0.359010	+	0.933334i	$0.616886\pi$
$888$	0	0
$889$	−16.1729	−0.542423
$890$	0	0
$891$	−36.4594	−1.22144
$892$	0	0
$893$	−9.33965	−0.312540
$894$	0	0
$895$	9.21487	0.308019
$896$	0	0
$897$	−2.98703	−0.0997340
$898$	0	0
$899$	−15.3160	−0.510816
$900$	0	0
$901$	−4.60286	−0.153344
$902$	0	0
$903$	−1.21735	−0.0405109
$904$	0	0
$905$	1.01879	0.0338656
$906$	0	0
$907$	−12.3003	−0.408425	−0.204213	−	0.978927i	$-0.565463\pi$
$907$	−12.3003	−0.408425	−0.204213	+	0.978927i	$0.565463\pi$
$908$	0	0
$909$	6.66781	0.221157
$910$	0	0
$911$	−56.6327	−1.87632	−0.938162	−	0.346196i	$-0.887473\pi$
$911$	−56.6327	−1.87632	−0.938162	+	0.346196i	$0.887473\pi$
$912$	0	0
$913$	−59.9416	−1.98378
$914$	0	0
$915$	0.625197	0.0206684
$916$	0	0
$917$	−0.338629	−0.0111825
$918$	0	0
$919$	42.4766	1.40117	0.700586	−	0.713568i	$-0.252922\pi$
$919$	42.4766	1.40117	0.700586	+	0.713568i	$0.252922\pi$
$920$	0	0
$921$	0.672005	0.0221433
$922$	0	0
$923$	−21.1262	−0.695377
$924$	0	0
$925$	13.9348	0.458175
$926$	0	0
$927$	8.68685	0.285314
$928$	0	0
$929$	48.6268	1.59539	0.797697	−	0.603058i	$-0.206051\pi$
$929$	48.6268	1.59539	0.797697	+	0.603058i	$0.206051\pi$
$930$	0	0
$931$	5.88102	0.192743
$932$	0	0
$933$	3.53217	0.115638
$934$	0	0
$935$	−16.8572	−0.551290
$936$	0	0
$937$	−1.96785	−0.0642868	−0.0321434	−	0.999483i	$-0.510233\pi$
$937$	−1.96785	−0.0642868	−0.0321434	+	0.999483i	$0.510233\pi$
$938$	0	0
$939$	3.59034	0.117166
$940$	0	0
$941$	−39.1798	−1.27722	−0.638612	−	0.769529i	$-0.720491\pi$
$941$	−39.1798	−1.27722	−0.638612	+	0.769529i	$0.720491\pi$
$942$	0	0
$943$	41.4757	1.35064
$944$	0	0
$945$	0.862431	0.0280549
$946$	0	0
$947$	−35.2257	−1.14468	−0.572341	−	0.820016i	$-0.693965\pi$
$947$	−35.2257	−1.14468	−0.572341	+	0.820016i	$0.693965\pi$
$948$	0	0
$949$	−32.8776	−1.06725
$950$	0	0
$951$	−1.49563	−0.0484993
$952$	0	0
$953$	20.6427	0.668684	0.334342	−	0.942452i	$-0.391486\pi$
$953$	20.6427	0.668684	0.334342	+	0.942452i	$0.391486\pi$
$954$	0	0
$955$	−15.6079	−0.505059
$956$	0	0
$957$	−2.81988	−0.0911537
$958$	0	0
$959$	9.32356	0.301074
$960$	0	0
$961$	−18.8583	−0.608333
$962$	0	0
$963$	−7.82485	−0.252152
$964$	0	0
$965$	−1.41792	−0.0456444
$966$	0	0
$967$	9.16439	0.294707	0.147353	−	0.989084i	$-0.452925\pi$
$967$	9.16439	0.294707	0.147353	+	0.989084i	$0.452925\pi$
$968$	0	0
$969$	−0.711556	−0.0228585
$970$	0	0
$971$	6.22320	0.199712	0.0998560	−	0.995002i	$-0.468162\pi$
$971$	6.22320	0.199712	0.0998560	+	0.995002i	$0.468162\pi$
$972$	0	0
$973$	−11.1855	−0.358592
$974$	0	0
$975$	2.38424	0.0763568
$976$	0	0
$977$	−29.7230	−0.950924	−0.475462	−	0.879736i	$-0.657719\pi$
$977$	−29.7230	−0.950924	−0.475462	+	0.879736i	$0.657719\pi$
$978$	0	0
$979$	−56.5861	−1.80850
$980$	0	0
$981$	−33.5223	−1.07028
$982$	0	0
$983$	33.8094	1.07835	0.539176	−	0.842193i	$-0.318736\pi$
$983$	33.8094	1.07835	0.539176	+	0.842193i	$0.318736\pi$
$984$	0	0
$985$	−3.27045	−0.104205
$986$	0	0
$987$	−1.52730	−0.0486143
$988$	0	0
$989$	39.3684	1.25184
$990$	0	0
$991$	29.4702	0.936153	0.468076	−	0.883688i	$-0.344947\pi$
$991$	29.4702	0.936153	0.468076	+	0.883688i	$0.344947\pi$
$992$	0	0
$993$	1.64234	0.0521181
$994$	0	0
$995$	−10.6287	−0.336951
$996$	0	0
$997$	25.7233	0.814666	0.407333	−	0.913280i	$-0.366459\pi$
$997$	25.7233	0.814666	0.407333	+	0.913280i	$0.366459\pi$
$998$	0	0
$999$	3.04976	0.0964902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.f.1.10	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.f.1.10	✓	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)

\(f(q)\)	\(=\)	\(q+0.154590 q^{3} +0.882499 q^{5} -1.05782 q^{7} -2.97610 q^{9} +O(q^{10})\)	\(q+0.154590 q^{3} +0.882499 q^{5} -1.05782 q^{7} -2.97610 q^{9} -4.14996 q^{11} -3.65371 q^{13} +0.136426 q^{15} +4.60286 q^{17} -1.00000 q^{19} -0.163528 q^{21} +5.28840 q^{23} -4.22119 q^{25} -0.923845 q^{27} +4.39547 q^{29} -3.48449 q^{31} -0.641542 q^{33} -0.933523 q^{35} -3.30116 q^{37} -0.564826 q^{39} +7.84277 q^{41} +7.44429 q^{43} -2.62641 q^{45} +9.33965 q^{47} -5.88102 q^{49} +0.711556 q^{51} -1.00000 q^{53} -3.66234 q^{55} -0.154590 q^{57} +7.01306 q^{59} +4.58270 q^{61} +3.14817 q^{63} -3.22439 q^{65} +0.201310 q^{67} +0.817534 q^{69} +5.78213 q^{71} +8.99842 q^{73} -0.652554 q^{75} +4.38990 q^{77} -1.88324 q^{79} +8.78549 q^{81} +14.4439 q^{83} +4.06202 q^{85} +0.679495 q^{87} +13.6353 q^{89} +3.86495 q^{91} -0.538668 q^{93} -0.882499 q^{95} -17.6435 q^{97} +12.3507 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) 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

Properties

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