LMFDB - Embedded newform 1596.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1596,2,Mod(1,1596)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1596, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1596.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1596 = 2^{2} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1596.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:	$12.7441241626$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
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			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	1596.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.64575 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.29150 q^{11} +3.64575 q^{15} +0.354249 q^{17} +1.00000 q^{19} +1.00000 q^{21} +5.29150 q^{23} +8.29150 q^{25} -1.00000 q^{27} -9.64575 q^{29} -6.00000 q^{31} -5.29150 q^{33} +3.64575 q^{35} +5.29150 q^{37} -3.29150 q^{41} -1.29150 q^{43} -3.64575 q^{45} -4.35425 q^{47} +1.00000 q^{49} -0.354249 q^{51} -13.6458 q^{53} -19.2915 q^{55} -1.00000 q^{57} -8.00000 q^{59} +9.29150 q^{61} -1.00000 q^{63} +7.29150 q^{67} -5.29150 q^{69} -8.93725 q^{71} -2.00000 q^{73} -8.29150 q^{75} -5.29150 q^{77} -3.29150 q^{79} +1.00000 q^{81} -14.9373 q^{83} -1.29150 q^{85} +9.64575 q^{87} -3.29150 q^{89} +6.00000 q^{93} -3.64575 q^{95} +4.58301 q^{97} +5.29150 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} + 6 q^{25} - 2 q^{27} - 14 q^{29} - 12 q^{31} + 2 q^{35} + 4 q^{41} + 8 q^{43} - 2 q^{45} - 14 q^{47} + 2 q^{49} - 6 q^{51}+ \cdots - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 + 6 * q^25 - 2 * q^27 - 14 * q^29 - 12 * q^31 + 2 * q^35 + 4 * q^41 + 8 * q^43 - 2 * q^45 - 14 * q^47 + 2 * q^49 - 6 * q^51 - 22 * q^53 - 28 * q^55 - 2 * q^57 - 16 * q^59 + 8 * q^61 - 2 * q^63 + 4 * q^67 - 2 * q^71 - 4 * q^73 - 6 * q^75 + 4 * q^79 + 2 * q^81 - 14 * q^83 + 8 * q^85 + 14 * q^87 + 4 * q^89 + 12 * q^93 - 2 * q^95 - 12 * q^97

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.64575	−1.63043	−0.815215	−	0.579159i	$-0.803381\pi$
$5$	−3.64575	−1.63043	−0.815215	+	0.579159i	$0.803381\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.29150	1.59545	0.797724	−	0.603023i	$-0.206037\pi$
$11$	5.29150	1.59545	0.797724	+	0.603023i	$0.206037\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	3.64575	0.941329
$16$	0	0
$17$	0.354249	0.0859179	0.0429590	−	0.999077i	$-0.486322\pi$
$17$	0.354249	0.0859179	0.0429590	+	0.999077i	$0.486322\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	5.29150	1.10335	0.551677	−	0.834058i	$-0.313988\pi$
$23$	5.29150	1.10335	0.551677	+	0.834058i	$0.313988\pi$
$24$	0	0
$25$	8.29150	1.65830
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.64575	−1.79117	−0.895586	−	0.444889i	$-0.853243\pi$
$29$	−9.64575	−1.79117	−0.895586	+	0.444889i	$0.853243\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	−5.29150	−0.921132
$34$	0	0
$35$	3.64575	0.616244
$36$	0	0
$37$	5.29150	0.869918	0.434959	−	0.900450i	$-0.356763\pi$
$37$	5.29150	0.869918	0.434959	+	0.900450i	$0.356763\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.29150	−0.514046	−0.257023	−	0.966405i	$-0.582742\pi$
$41$	−3.29150	−0.514046	−0.257023	+	0.966405i	$0.582742\pi$
$42$	0	0
$43$	−1.29150	−0.196952	−0.0984762	−	0.995139i	$-0.531397\pi$
$43$	−1.29150	−0.196952	−0.0984762	+	0.995139i	$0.531397\pi$
$44$	0	0
$45$	−3.64575	−0.543477
$46$	0	0
$47$	−4.35425	−0.635132	−0.317566	−	0.948236i	$-0.602866\pi$
$47$	−4.35425	−0.635132	−0.317566	+	0.948236i	$0.602866\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.354249	−0.0496047
$52$	0	0
$53$	−13.6458	−1.87439	−0.937194	−	0.348808i	$-0.886586\pi$
$53$	−13.6458	−1.87439	−0.937194	+	0.348808i	$0.886586\pi$
$54$	0	0
$55$	−19.2915	−2.60127
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	9.29150	1.18966	0.594828	−	0.803853i	$-0.297220\pi$
$61$	9.29150	1.18966	0.594828	+	0.803853i	$0.297220\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.29150	0.890799	0.445399	−	0.895332i	$-0.353062\pi$
$67$	7.29150	0.890799	0.445399	+	0.895332i	$0.353062\pi$
$68$	0	0
$69$	−5.29150	−0.637022
$70$	0	0
$71$	−8.93725	−1.06066	−0.530328	−	0.847792i	$-0.677931\pi$
$71$	−8.93725	−1.06066	−0.530328	+	0.847792i	$0.677931\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−8.29150	−0.957420
$76$	0	0
$77$	−5.29150	−0.603023
$78$	0	0
$79$	−3.29150	−0.370323	−0.185161	−	0.982708i	$-0.559281\pi$
$79$	−3.29150	−0.370323	−0.185161	+	0.982708i	$0.559281\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.9373	−1.63958	−0.819788	−	0.572667i	$-0.805909\pi$
$83$	−14.9373	−1.63958	−0.819788	+	0.572667i	$0.805909\pi$
$84$	0	0
$85$	−1.29150	−0.140083
$86$	0	0
$87$	9.64575	1.03413
$88$	0	0
$89$	−3.29150	−0.348899	−0.174449	−	0.984666i	$-0.555815\pi$
$89$	−3.29150	−0.348899	−0.174449	+	0.984666i	$0.555815\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.00000	0.622171
$94$	0	0
$95$	−3.64575	−0.374046
$96$	0	0
$97$	4.58301	0.465334	0.232667	−	0.972556i	$-0.425255\pi$
$97$	4.58301	0.465334	0.232667	+	0.972556i	$0.425255\pi$
$98$	0	0
$99$	5.29150	0.531816
$100$	0	0
$101$	−12.3542	−1.22929	−0.614647	−	0.788802i	$-0.710701\pi$
$101$	−12.3542	−1.22929	−0.614647	+	0.788802i	$0.710701\pi$
$102$	0	0
$103$	14.5830	1.43691	0.718453	−	0.695575i	$-0.244850\pi$
$103$	14.5830	1.43691	0.718453	+	0.695575i	$0.244850\pi$
$104$	0	0
$105$	−3.64575	−0.355789
$106$	0	0
$107$	−13.6458	−1.31918	−0.659592	−	0.751624i	$-0.729271\pi$
$107$	−13.6458	−1.31918	−0.659592	+	0.751624i	$0.729271\pi$
$108$	0	0
$109$	−17.2915	−1.65623	−0.828113	−	0.560562i	$-0.810585\pi$
$109$	−17.2915	−1.65623	−0.828113	+	0.560562i	$0.810585\pi$
$110$	0	0
$111$	−5.29150	−0.502247
$112$	0	0
$113$	8.93725	0.840746	0.420373	−	0.907351i	$-0.361899\pi$
$113$	8.93725	0.840746	0.420373	+	0.907351i	$0.361899\pi$
$114$	0	0
$115$	−19.2915	−1.79894
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.354249	−0.0324739
$120$	0	0
$121$	17.0000	1.54545
$122$	0	0
$123$	3.29150	0.296785
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−3.29150	−0.292074	−0.146037	−	0.989279i	$-0.546652\pi$
$127$	−3.29150	−0.292074	−0.146037	+	0.989279i	$0.546652\pi$
$128$	0	0
$129$	1.29150	0.113710
$130$	0	0
$131$	−15.6458	−1.36698	−0.683488	−	0.729962i	$-0.739538\pi$
$131$	−15.6458	−1.36698	−0.683488	+	0.729962i	$0.739538\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	3.64575	0.313776
$136$	0	0
$137$	23.1660	1.97921	0.989603	−	0.143826i	$-0.0459407\pi$
$137$	23.1660	1.97921	0.989603	+	0.143826i	$0.0459407\pi$
$138$	0	0
$139$	10.5830	0.897639	0.448819	−	0.893622i	$-0.351845\pi$
$139$	10.5830	0.897639	0.448819	+	0.893622i	$0.351845\pi$
$140$	0	0
$141$	4.35425	0.366694
$142$	0	0
$143$	0	0
$144$	0	0
$145$	35.1660	2.92038
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	21.2915	1.74427	0.872134	−	0.489267i	$-0.162736\pi$
$149$	21.2915	1.74427	0.872134	+	0.489267i	$0.162736\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0.354249	0.0286393
$154$	0	0
$155$	21.8745	1.75700
$156$	0	0
$157$	−1.29150	−0.103073	−0.0515366	−	0.998671i	$-0.516412\pi$
$157$	−1.29150	−0.103073	−0.0515366	+	0.998671i	$0.516412\pi$
$158$	0	0
$159$	13.6458	1.08218
$160$	0	0
$161$	−5.29150	−0.417029
$162$	0	0
$163$	−7.87451	−0.616779	−0.308390	−	0.951260i	$-0.599790\pi$
$163$	−7.87451	−0.616779	−0.308390	+	0.951260i	$0.599790\pi$
$164$	0	0
$165$	19.2915	1.50184
$166$	0	0
$167$	7.29150	0.564233	0.282117	−	0.959380i	$-0.408963\pi$
$167$	7.29150	0.564233	0.282117	+	0.959380i	$0.408963\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−4.70850	−0.357980	−0.178990	−	0.983851i	$-0.557283\pi$
$173$	−4.70850	−0.357980	−0.178990	+	0.983851i	$0.557283\pi$
$174$	0	0
$175$	−8.29150	−0.626779
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	−10.3542	−0.773913	−0.386956	−	0.922098i	$-0.626474\pi$
$179$	−10.3542	−0.773913	−0.386956	+	0.922098i	$0.626474\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	−9.29150	−0.686848
$184$	0	0
$185$	−19.2915	−1.41834
$186$	0	0
$187$	1.87451	0.137078
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−9.29150	−0.672310	−0.336155	−	0.941807i	$-0.609127\pi$
$191$	−9.29150	−0.672310	−0.336155	+	0.941807i	$0.609127\pi$
$192$	0	0
$193$	−7.87451	−0.566819	−0.283410	−	0.958999i	$-0.591466\pi$
$193$	−7.87451	−0.566819	−0.283410	+	0.958999i	$0.591466\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.58301	−0.326526	−0.163263	−	0.986583i	$-0.552202\pi$
$197$	−4.58301	−0.326526	−0.163263	+	0.986583i	$0.552202\pi$
$198$	0	0
$199$	−13.8745	−0.983538	−0.491769	−	0.870726i	$-0.663650\pi$
$199$	−13.8745	−0.983538	−0.491769	+	0.870726i	$0.663650\pi$
$200$	0	0
$201$	−7.29150	−0.514303
$202$	0	0
$203$	9.64575	0.676999
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	5.29150	0.367785
$208$	0	0
$209$	5.29150	0.366021
$210$	0	0
$211$	−10.5830	−0.728564	−0.364282	−	0.931289i	$-0.618686\pi$
$211$	−10.5830	−0.728564	−0.364282	+	0.931289i	$0.618686\pi$
$212$	0	0
$213$	8.93725	0.612371
$214$	0	0
$215$	4.70850	0.321117
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−23.1660	−1.55131	−0.775655	−	0.631157i	$-0.782581\pi$
$223$	−23.1660	−1.55131	−0.775655	+	0.631157i	$0.782581\pi$
$224$	0	0
$225$	8.29150	0.552767
$226$	0	0
$227$	27.2915	1.81140	0.905700	−	0.423919i	$-0.139346\pi$
$227$	27.2915	1.81140	0.905700	+	0.423919i	$0.139346\pi$
$228$	0	0
$229$	−23.8745	−1.57767	−0.788836	−	0.614604i	$-0.789316\pi$
$229$	−23.8745	−1.57767	−0.788836	+	0.614604i	$0.789316\pi$
$230$	0	0
$231$	5.29150	0.348155
$232$	0	0
$233$	−1.29150	−0.0846091	−0.0423046	−	0.999105i	$-0.513470\pi$
$233$	−1.29150	−0.0846091	−0.0423046	+	0.999105i	$0.513470\pi$
$234$	0	0
$235$	15.8745	1.03554
$236$	0	0
$237$	3.29150	0.213806
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−4.58301	−0.295217	−0.147609	−	0.989046i	$-0.547158\pi$
$241$	−4.58301	−0.295217	−0.147609	+	0.989046i	$0.547158\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.64575	−0.232919
$246$	0	0
$247$	0	0
$248$	0	0
$249$	14.9373	0.946610
$250$	0	0
$251$	−22.9373	−1.44779	−0.723893	−	0.689912i	$-0.757649\pi$
$251$	−22.9373	−1.44779	−0.723893	+	0.689912i	$0.757649\pi$
$252$	0	0
$253$	28.0000	1.76034
$254$	0	0
$255$	1.29150	0.0808770
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−5.29150	−0.328798
$260$	0	0
$261$	−9.64575	−0.597057
$262$	0	0
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	49.7490	3.05606
$266$	0	0
$267$	3.29150	0.201437
$268$	0	0
$269$	−1.41699	−0.0863957	−0.0431978	−	0.999067i	$-0.513755\pi$
$269$	−1.41699	−0.0863957	−0.0431978	+	0.999067i	$0.513755\pi$
$270$	0	0
$271$	15.2915	0.928893	0.464446	−	0.885601i	$-0.346253\pi$
$271$	15.2915	0.928893	0.464446	+	0.885601i	$0.346253\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	43.8745	2.64573
$276$	0	0
$277$	−20.4575	−1.22917	−0.614586	−	0.788850i	$-0.710677\pi$
$277$	−20.4575	−1.22917	−0.614586	+	0.788850i	$0.710677\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	28.2288	1.68399	0.841993	−	0.539488i	$-0.181382\pi$
$281$	28.2288	1.68399	0.841993	+	0.539488i	$0.181382\pi$
$282$	0	0
$283$	−5.87451	−0.349203	−0.174602	−	0.984639i	$-0.555864\pi$
$283$	−5.87451	−0.349203	−0.174602	+	0.984639i	$0.555864\pi$
$284$	0	0
$285$	3.64575	0.215956
$286$	0	0
$287$	3.29150	0.194291
$288$	0	0
$289$	−16.8745	−0.992618
$290$	0	0
$291$	−4.58301	−0.268661
$292$	0	0
$293$	−8.70850	−0.508756	−0.254378	−	0.967105i	$-0.581871\pi$
$293$	−8.70850	−0.508756	−0.254378	+	0.967105i	$0.581871\pi$
$294$	0	0
$295$	29.1660	1.69811
$296$	0	0
$297$	−5.29150	−0.307044
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.29150	0.0744410
$302$	0	0
$303$	12.3542	0.709733
$304$	0	0
$305$	−33.8745	−1.93965
$306$	0	0
$307$	−20.5830	−1.17473	−0.587367	−	0.809321i	$-0.699835\pi$
$307$	−20.5830	−1.17473	−0.587367	+	0.809321i	$0.699835\pi$
$308$	0	0
$309$	−14.5830	−0.829598
$310$	0	0
$311$	14.9373	0.847014	0.423507	−	0.905893i	$-0.360799\pi$
$311$	14.9373	0.847014	0.423507	+	0.905893i	$0.360799\pi$
$312$	0	0
$313$	25.2915	1.42956	0.714780	−	0.699349i	$-0.246527\pi$
$313$	25.2915	1.42956	0.714780	+	0.699349i	$0.246527\pi$
$314$	0	0
$315$	3.64575	0.205415
$316$	0	0
$317$	16.9373	0.951291	0.475645	−	0.879637i	$-0.342215\pi$
$317$	16.9373	0.951291	0.475645	+	0.879637i	$0.342215\pi$
$318$	0	0
$319$	−51.0405	−2.85772
$320$	0	0
$321$	13.6458	0.761632
$322$	0	0
$323$	0.354249	0.0197109
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.2915	0.956222
$328$	0	0
$329$	4.35425	0.240058
$330$	0	0
$331$	−9.41699	−0.517605	−0.258802	−	0.965930i	$-0.583328\pi$
$331$	−9.41699	−0.517605	−0.258802	+	0.965930i	$0.583328\pi$
$332$	0	0
$333$	5.29150	0.289973
$334$	0	0
$335$	−26.5830	−1.45238
$336$	0	0
$337$	33.2915	1.81350	0.906752	−	0.421665i	$-0.138554\pi$
$337$	33.2915	1.81350	0.906752	+	0.421665i	$0.138554\pi$
$338$	0	0
$339$	−8.93725	−0.485405
$340$	0	0
$341$	−31.7490	−1.71931
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	19.2915	1.03862
$346$	0	0
$347$	−31.1660	−1.67308	−0.836540	−	0.547907i	$-0.815425\pi$
$347$	−31.1660	−1.67308	−0.836540	+	0.547907i	$0.815425\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.0627	0.695260	0.347630	−	0.937632i	$-0.386987\pi$
$353$	13.0627	0.695260	0.347630	+	0.937632i	$0.386987\pi$
$354$	0	0
$355$	32.5830	1.72933
$356$	0	0
$357$	0.354249	0.0187488
$358$	0	0
$359$	−30.4575	−1.60749	−0.803743	−	0.594977i	$-0.797161\pi$
$359$	−30.4575	−1.60749	−0.803743	+	0.594977i	$0.797161\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−17.0000	−0.892269
$364$	0	0
$365$	7.29150	0.381655
$366$	0	0
$367$	17.8745	0.933042	0.466521	−	0.884510i	$-0.345507\pi$
$367$	17.8745	0.933042	0.466521	+	0.884510i	$0.345507\pi$
$368$	0	0
$369$	−3.29150	−0.171349
$370$	0	0
$371$	13.6458	0.708452
$372$	0	0
$373$	4.58301	0.237299	0.118650	−	0.992936i	$-0.462144\pi$
$373$	4.58301	0.237299	0.118650	+	0.992936i	$0.462144\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	0	0
$378$	0	0
$379$	10.5830	0.543612	0.271806	−	0.962352i	$-0.412379\pi$
$379$	10.5830	0.543612	0.271806	+	0.962352i	$0.412379\pi$
$380$	0	0
$381$	3.29150	0.168629
$382$	0	0
$383$	29.8745	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$383$	29.8745	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$384$	0	0
$385$	19.2915	0.983186
$386$	0	0
$387$	−1.29150	−0.0656508
$388$	0	0
$389$	26.4575	1.34145	0.670725	−	0.741707i	$-0.265983\pi$
$389$	26.4575	1.34145	0.670725	+	0.741707i	$0.265983\pi$
$390$	0	0
$391$	1.87451	0.0947979
$392$	0	0
$393$	15.6458	0.789224
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	4.93725	0.246555	0.123277	−	0.992372i	$-0.460660\pi$
$401$	4.93725	0.246555	0.123277	+	0.992372i	$0.460660\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−3.64575	−0.181159
$406$	0	0
$407$	28.0000	1.38791
$408$	0	0
$409$	2.58301	0.127721	0.0638607	−	0.997959i	$-0.479659\pi$
$409$	2.58301	0.127721	0.0638607	+	0.997959i	$0.479659\pi$
$410$	0	0
$411$	−23.1660	−1.14270
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	54.4575	2.67321
$416$	0	0
$417$	−10.5830	−0.518252
$418$	0	0
$419$	−1.06275	−0.0519185	−0.0259593	−	0.999663i	$-0.508264\pi$
$419$	−1.06275	−0.0519185	−0.0259593	+	0.999663i	$0.508264\pi$
$420$	0	0
$421$	0.583005	0.0284139	0.0142070	−	0.999899i	$-0.495478\pi$
$421$	0.583005	0.0284139	0.0142070	+	0.999899i	$0.495478\pi$
$422$	0	0
$423$	−4.35425	−0.211711
$424$	0	0
$425$	2.93725	0.142478
$426$	0	0
$427$	−9.29150	−0.449647
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.64575	−0.0792731	−0.0396365	−	0.999214i	$-0.512620\pi$
$431$	−1.64575	−0.0792731	−0.0396365	+	0.999214i	$0.512620\pi$
$432$	0	0
$433$	−3.41699	−0.164210	−0.0821051	−	0.996624i	$-0.526164\pi$
$433$	−3.41699	−0.164210	−0.0821051	+	0.996624i	$0.526164\pi$
$434$	0	0
$435$	−35.1660	−1.68608
$436$	0	0
$437$	5.29150	0.253127
$438$	0	0
$439$	−31.1660	−1.48747	−0.743736	−	0.668473i	$-0.766948\pi$
$439$	−31.1660	−1.48747	−0.743736	+	0.668473i	$0.766948\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−11.1660	−0.530513	−0.265257	−	0.964178i	$-0.585457\pi$
$443$	−11.1660	−0.530513	−0.265257	+	0.964178i	$0.585457\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	0	0
$447$	−21.2915	−1.00705
$448$	0	0
$449$	−41.3948	−1.95354	−0.976770	−	0.214291i	$-0.931256\pi$
$449$	−41.3948	−1.95354	−0.976770	+	0.214291i	$0.931256\pi$
$450$	0	0
$451$	−17.4170	−0.820134
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.7085	0.594478	0.297239	−	0.954803i	$-0.403934\pi$
$457$	12.7085	0.594478	0.297239	+	0.954803i	$0.403934\pi$
$458$	0	0
$459$	−0.354249	−0.0165349
$460$	0	0
$461$	−1.52026	−0.0708055	−0.0354028	−	0.999373i	$-0.511271\pi$
$461$	−1.52026	−0.0708055	−0.0354028	+	0.999373i	$0.511271\pi$
$462$	0	0
$463$	17.1660	0.797772	0.398886	−	0.917000i	$-0.369397\pi$
$463$	17.1660	0.797772	0.398886	+	0.917000i	$0.369397\pi$
$464$	0	0
$465$	−21.8745	−1.01441
$466$	0	0
$467$	−10.2288	−0.473330	−0.236665	−	0.971591i	$-0.576054\pi$
$467$	−10.2288	−0.473330	−0.236665	+	0.971591i	$0.576054\pi$
$468$	0	0
$469$	−7.29150	−0.336690
$470$	0	0
$471$	1.29150	0.0595093
$472$	0	0
$473$	−6.83399	−0.314227
$474$	0	0
$475$	8.29150	0.380440
$476$	0	0
$477$	−13.6458	−0.624796
$478$	0	0
$479$	25.5203	1.16605	0.583025	−	0.812454i	$-0.301869\pi$
$479$	25.5203	1.16605	0.583025	+	0.812454i	$0.301869\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	5.29150	0.240772
$484$	0	0
$485$	−16.7085	−0.758694
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	0	0
$489$	7.87451	0.356098
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	−3.41699	−0.153894
$494$	0	0
$495$	−19.2915	−0.867089
$496$	0	0
$497$	8.93725	0.400891
$498$	0	0
$499$	34.5830	1.54815	0.774074	−	0.633095i	$-0.218216\pi$
$499$	34.5830	1.54815	0.774074	+	0.633095i	$0.218216\pi$
$500$	0	0
$501$	−7.29150	−0.325760
$502$	0	0
$503$	−2.22876	−0.0993753	−0.0496877	−	0.998765i	$-0.515823\pi$
$503$	−2.22876	−0.0993753	−0.0496877	+	0.998765i	$0.515823\pi$
$504$	0	0
$505$	45.0405	2.00428
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	−30.5830	−1.35557	−0.677784	−	0.735261i	$-0.737060\pi$
$509$	−30.5830	−1.35557	−0.677784	+	0.735261i	$0.737060\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−53.1660	−2.34277
$516$	0	0
$517$	−23.0405	−1.01332
$518$	0	0
$519$	4.70850	0.206680
$520$	0	0
$521$	6.58301	0.288407	0.144203	−	0.989548i	$-0.453938\pi$
$521$	6.58301	0.288407	0.144203	+	0.989548i	$0.453938\pi$
$522$	0	0
$523$	15.1660	0.663163	0.331582	−	0.943427i	$-0.392418\pi$
$523$	15.1660	0.663163	0.331582	+	0.943427i	$0.392418\pi$
$524$	0	0
$525$	8.29150	0.361871
$526$	0	0
$527$	−2.12549	−0.0925879
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	0	0
$534$	0	0
$535$	49.7490	2.15084
$536$	0	0
$537$	10.3542	0.446819
$538$	0	0
$539$	5.29150	0.227921
$540$	0	0
$541$	19.1660	0.824011	0.412006	−	0.911181i	$-0.364828\pi$
$541$	19.1660	0.824011	0.412006	+	0.911181i	$0.364828\pi$
$542$	0	0
$543$	−6.00000	−0.257485
$544$	0	0
$545$	63.0405	2.70036
$546$	0	0
$547$	−28.4575	−1.21676	−0.608378	−	0.793648i	$-0.708179\pi$
$547$	−28.4575	−1.21676	−0.608378	+	0.793648i	$0.708179\pi$
$548$	0	0
$549$	9.29150	0.396552
$550$	0	0
$551$	−9.64575	−0.410923
$552$	0	0
$553$	3.29150	0.139969
$554$	0	0
$555$	19.2915	0.818879
$556$	0	0
$557$	−23.8745	−1.01160	−0.505798	−	0.862652i	$-0.668802\pi$
$557$	−23.8745	−1.01160	−0.505798	+	0.862652i	$0.668802\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−1.87451	−0.0791418
$562$	0	0
$563$	−19.2915	−0.813040	−0.406520	−	0.913642i	$-0.633258\pi$
$563$	−19.2915	−0.813040	−0.406520	+	0.913642i	$0.633258\pi$
$564$	0	0
$565$	−32.5830	−1.37078
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	7.06275	0.296086	0.148043	−	0.988981i	$-0.452703\pi$
$569$	7.06275	0.296086	0.148043	+	0.988981i	$0.452703\pi$
$570$	0	0
$571$	13.1660	0.550980	0.275490	−	0.961304i	$-0.411160\pi$
$571$	13.1660	0.550980	0.275490	+	0.961304i	$0.411160\pi$
$572$	0	0
$573$	9.29150	0.388158
$574$	0	0
$575$	43.8745	1.82969
$576$	0	0
$577$	−11.8745	−0.494342	−0.247171	−	0.968972i	$-0.579501\pi$
$577$	−11.8745	−0.494342	−0.247171	+	0.968972i	$0.579501\pi$
$578$	0	0
$579$	7.87451	0.327253
$580$	0	0
$581$	14.9373	0.619702
$582$	0	0
$583$	−72.2065	−2.99049
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.77124	−0.0731070	−0.0365535	−	0.999332i	$-0.511638\pi$
$587$	−1.77124	−0.0731070	−0.0365535	+	0.999332i	$0.511638\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	0	0
$591$	4.58301	0.188520
$592$	0	0
$593$	−26.2288	−1.07709	−0.538543	−	0.842598i	$-0.681025\pi$
$593$	−26.2288	−1.07709	−0.538543	+	0.842598i	$0.681025\pi$
$594$	0	0
$595$	1.29150	0.0529464
$596$	0	0
$597$	13.8745	0.567846
$598$	0	0
$599$	−3.77124	−0.154089	−0.0770444	−	0.997028i	$-0.524548\pi$
$599$	−3.77124	−0.154089	−0.0770444	+	0.997028i	$0.524548\pi$
$600$	0	0
$601$	−15.4170	−0.628872	−0.314436	−	0.949279i	$-0.601815\pi$
$601$	−15.4170	−0.628872	−0.314436	+	0.949279i	$0.601815\pi$
$602$	0	0
$603$	7.29150	0.296933
$604$	0	0
$605$	−61.9778	−2.51975
$606$	0	0
$607$	17.4170	0.706934	0.353467	−	0.935447i	$-0.385003\pi$
$607$	17.4170	0.706934	0.353467	+	0.935447i	$0.385003\pi$
$608$	0	0
$609$	−9.64575	−0.390866
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−16.4575	−0.664713	−0.332356	−	0.943154i	$-0.607844\pi$
$613$	−16.4575	−0.664713	−0.332356	+	0.943154i	$0.607844\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	0.833990	0.0335752	0.0167876	−	0.999859i	$-0.494656\pi$
$617$	0.833990	0.0335752	0.0167876	+	0.999859i	$0.494656\pi$
$618$	0	0
$619$	47.7490	1.91919	0.959597	−	0.281376i	$-0.0907909\pi$
$619$	47.7490	1.91919	0.959597	+	0.281376i	$0.0907909\pi$
$620$	0	0
$621$	−5.29150	−0.212341
$622$	0	0
$623$	3.29150	0.131871
$624$	0	0
$625$	2.29150	0.0916601
$626$	0	0
$627$	−5.29150	−0.211322
$628$	0	0
$629$	1.87451	0.0747415
$630$	0	0
$631$	−7.87451	−0.313479	−0.156740	−	0.987640i	$-0.550098\pi$
$631$	−7.87451	−0.313479	−0.156740	+	0.987640i	$0.550098\pi$
$632$	0	0
$633$	10.5830	0.420637
$634$	0	0
$635$	12.0000	0.476205
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.93725	−0.353552
$640$	0	0
$641$	28.2288	1.11497	0.557484	−	0.830187i	$-0.311767\pi$
$641$	28.2288	1.11497	0.557484	+	0.830187i	$0.311767\pi$
$642$	0	0
$643$	−44.4575	−1.75323	−0.876617	−	0.481190i	$-0.840205\pi$
$643$	−44.4575	−1.75323	−0.876617	+	0.481190i	$0.840205\pi$
$644$	0	0
$645$	−4.70850	−0.185397
$646$	0	0
$647$	32.8118	1.28996	0.644982	−	0.764198i	$-0.276865\pi$
$647$	32.8118	1.28996	0.644982	+	0.764198i	$0.276865\pi$
$648$	0	0
$649$	−42.3320	−1.66168
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	25.7490	1.00764	0.503818	−	0.863810i	$-0.331928\pi$
$653$	25.7490	1.00764	0.503818	+	0.863810i	$0.331928\pi$
$654$	0	0
$655$	57.0405	2.22876
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	−49.3948	−1.92415	−0.962073	−	0.272790i	$-0.912053\pi$
$659$	−49.3948	−1.92415	−0.962073	+	0.272790i	$0.912053\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.64575	0.141376
$666$	0	0
$667$	−51.0405	−1.97630
$668$	0	0
$669$	23.1660	0.895650
$670$	0	0
$671$	49.1660	1.89803
$672$	0	0
$673$	−12.1255	−0.467403	−0.233702	−	0.972308i	$-0.575084\pi$
$673$	−12.1255	−0.467403	−0.233702	+	0.972308i	$0.575084\pi$
$674$	0	0
$675$	−8.29150	−0.319140
$676$	0	0
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	0	0
$679$	−4.58301	−0.175880
$680$	0	0
$681$	−27.2915	−1.04581
$682$	0	0
$683$	7.06275	0.270248	0.135124	−	0.990829i	$-0.456857\pi$
$683$	7.06275	0.270248	0.135124	+	0.990829i	$0.456857\pi$
$684$	0	0
$685$	−84.4575	−3.22696
$686$	0	0
$687$	23.8745	0.910869
$688$	0	0
$689$	0	0
$690$	0	0
$691$	11.2915	0.429549	0.214775	−	0.976664i	$-0.431098\pi$
$691$	11.2915	0.429549	0.214775	+	0.976664i	$0.431098\pi$
$692$	0	0
$693$	−5.29150	−0.201008
$694$	0	0
$695$	−38.5830	−1.46354
$696$	0	0
$697$	−1.16601	−0.0441658
$698$	0	0
$699$	1.29150	0.0488491
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	5.29150	0.199573
$704$	0	0
$705$	−15.8745	−0.597869
$706$	0	0
$707$	12.3542	0.464629
$708$	0	0
$709$	−13.8745	−0.521068	−0.260534	−	0.965465i	$-0.583899\pi$
$709$	−13.8745	−0.521068	−0.260534	+	0.965465i	$0.583899\pi$
$710$	0	0
$711$	−3.29150	−0.123441
$712$	0	0
$713$	−31.7490	−1.18901
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.00000	0.224074
$718$	0	0
$719$	19.6458	0.732663	0.366331	−	0.930484i	$-0.380614\pi$
$719$	19.6458	0.732663	0.366331	+	0.930484i	$0.380614\pi$
$720$	0	0
$721$	−14.5830	−0.543099
$722$	0	0
$723$	4.58301	0.170444
$724$	0	0
$725$	−79.9778	−2.97030
$726$	0	0
$727$	−35.2915	−1.30889	−0.654445	−	0.756110i	$-0.727098\pi$
$727$	−35.2915	−1.30889	−0.654445	+	0.756110i	$0.727098\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.457513	−0.0169217
$732$	0	0
$733$	48.5830	1.79445	0.897227	−	0.441569i	$-0.145578\pi$
$733$	48.5830	1.79445	0.897227	+	0.441569i	$0.145578\pi$
$734$	0	0
$735$	3.64575	0.134476
$736$	0	0
$737$	38.5830	1.42122
$738$	0	0
$739$	10.4575	0.384686	0.192343	−	0.981328i	$-0.438391\pi$
$739$	10.4575	0.384686	0.192343	+	0.981328i	$0.438391\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.0627	−0.699344	−0.349672	−	0.936872i	$-0.613707\pi$
$743$	−19.0627	−0.699344	−0.349672	+	0.936872i	$0.613707\pi$
$744$	0	0
$745$	−77.6235	−2.84391
$746$	0	0
$747$	−14.9373	−0.546526
$748$	0	0
$749$	13.6458	0.498605
$750$	0	0
$751$	10.8340	0.395338	0.197669	−	0.980269i	$-0.436663\pi$
$751$	10.8340	0.395338	0.197669	+	0.980269i	$0.436663\pi$
$752$	0	0
$753$	22.9373	0.835880
$754$	0	0
$755$	29.1660	1.06146
$756$	0	0
$757$	−12.5830	−0.457337	−0.228669	−	0.973504i	$-0.573437\pi$
$757$	−12.5830	−0.457337	−0.228669	+	0.973504i	$0.573437\pi$
$758$	0	0
$759$	−28.0000	−1.01634
$760$	0	0
$761$	30.9373	1.12147	0.560737	−	0.827994i	$-0.310518\pi$
$761$	30.9373	1.12147	0.560737	+	0.827994i	$0.310518\pi$
$762$	0	0
$763$	17.2915	0.625994
$764$	0	0
$765$	−1.29150	−0.0466944
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1.29150	0.0465728	0.0232864	−	0.999729i	$-0.492587\pi$
$769$	1.29150	0.0465728	0.0232864	+	0.999729i	$0.492587\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39.0405	1.40419	0.702095	−	0.712083i	$-0.252248\pi$
$773$	39.0405	1.40419	0.702095	+	0.712083i	$0.252248\pi$
$774$	0	0
$775$	−49.7490	−1.78704
$776$	0	0
$777$	5.29150	0.189832
$778$	0	0
$779$	−3.29150	−0.117930
$780$	0	0
$781$	−47.2915	−1.69222
$782$	0	0
$783$	9.64575	0.344711
$784$	0	0
$785$	4.70850	0.168053
$786$	0	0
$787$	−7.16601	−0.255441	−0.127720	−	0.991810i	$-0.540766\pi$
$787$	−7.16601	−0.255441	−0.127720	+	0.991810i	$0.540766\pi$
$788$	0	0
$789$	18.0000	0.640817
$790$	0	0
$791$	−8.93725	−0.317772
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−49.7490	−1.76442
$796$	0	0
$797$	35.0405	1.24120	0.620599	−	0.784128i	$-0.286889\pi$
$797$	35.0405	1.24120	0.620599	+	0.784128i	$0.286889\pi$
$798$	0	0
$799$	−1.54249	−0.0545693
$800$	0	0
$801$	−3.29150	−0.116300
$802$	0	0
$803$	−10.5830	−0.373466
$804$	0	0
$805$	19.2915	0.679936
$806$	0	0
$807$	1.41699	0.0498806
$808$	0	0
$809$	−44.5830	−1.56745	−0.783727	−	0.621105i	$-0.786684\pi$
$809$	−44.5830	−1.56745	−0.783727	+	0.621105i	$0.786684\pi$
$810$	0	0
$811$	23.7490	0.833941	0.416970	−	0.908920i	$-0.363092\pi$
$811$	23.7490	0.833941	0.416970	+	0.908920i	$0.363092\pi$
$812$	0	0
$813$	−15.2915	−0.536296
$814$	0	0
$815$	28.7085	1.00561
$816$	0	0
$817$	−1.29150	−0.0451840
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.2915	−0.463877	−0.231938	−	0.972730i	$-0.574507\pi$
$821$	−13.2915	−0.463877	−0.231938	+	0.972730i	$0.574507\pi$
$822$	0	0
$823$	43.8745	1.52937	0.764685	−	0.644405i	$-0.222895\pi$
$823$	43.8745	1.52937	0.764685	+	0.644405i	$0.222895\pi$
$824$	0	0
$825$	−43.8745	−1.52751
$826$	0	0
$827$	−28.9373	−1.00625	−0.503123	−	0.864215i	$-0.667816\pi$
$827$	−28.9373	−1.00625	−0.503123	+	0.864215i	$0.667816\pi$
$828$	0	0
$829$	36.3320	1.26186	0.630932	−	0.775838i	$-0.282673\pi$
$829$	36.3320	1.26186	0.630932	+	0.775838i	$0.282673\pi$
$830$	0	0
$831$	20.4575	0.709663
$832$	0	0
$833$	0.354249	0.0122740
$834$	0	0
$835$	−26.5830	−0.919943
$836$	0	0
$837$	6.00000	0.207390
$838$	0	0
$839$	−21.4170	−0.739397	−0.369698	−	0.929152i	$-0.620539\pi$
$839$	−21.4170	−0.739397	−0.369698	+	0.929152i	$0.620539\pi$
$840$	0	0
$841$	64.0405	2.20829
$842$	0	0
$843$	−28.2288	−0.972250
$844$	0	0
$845$	47.3948	1.63043
$846$	0	0
$847$	−17.0000	−0.584127
$848$	0	0
$849$	5.87451	0.201613
$850$	0	0
$851$	28.0000	0.959828
$852$	0	0
$853$	−35.1660	−1.20406	−0.602031	−	0.798473i	$-0.705641\pi$
$853$	−35.1660	−1.20406	−0.602031	+	0.798473i	$0.705641\pi$
$854$	0	0
$855$	−3.64575	−0.124682
$856$	0	0
$857$	−45.8745	−1.56704	−0.783522	−	0.621364i	$-0.786579\pi$
$857$	−45.8745	−1.56704	−0.783522	+	0.621364i	$0.786579\pi$
$858$	0	0
$859$	−2.83399	−0.0966945	−0.0483472	−	0.998831i	$-0.515395\pi$
$859$	−2.83399	−0.0966945	−0.0483472	+	0.998831i	$0.515395\pi$
$860$	0	0
$861$	−3.29150	−0.112174
$862$	0	0
$863$	37.3948	1.27293	0.636466	−	0.771304i	$-0.280395\pi$
$863$	37.3948	1.27293	0.636466	+	0.771304i	$0.280395\pi$
$864$	0	0
$865$	17.1660	0.583662
$866$	0	0
$867$	16.8745	0.573088
$868$	0	0
$869$	−17.4170	−0.590831
$870$	0	0
$871$	0	0
$872$	0	0
$873$	4.58301	0.155111
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	4.58301	0.154757	0.0773785	−	0.997002i	$-0.475345\pi$
$877$	4.58301	0.154757	0.0773785	+	0.997002i	$0.475345\pi$
$878$	0	0
$879$	8.70850	0.293730
$880$	0	0
$881$	−54.6863	−1.84243	−0.921214	−	0.389057i	$-0.872801\pi$
$881$	−54.6863	−1.84243	−0.921214	+	0.389057i	$0.872801\pi$
$882$	0	0
$883$	−41.1660	−1.38535	−0.692673	−	0.721252i	$-0.743567\pi$
$883$	−41.1660	−1.38535	−0.692673	+	0.721252i	$0.743567\pi$
$884$	0	0
$885$	−29.1660	−0.980405
$886$	0	0
$887$	2.12549	0.0713670	0.0356835	−	0.999363i	$-0.488639\pi$
$887$	2.12549	0.0713670	0.0356835	+	0.999363i	$0.488639\pi$
$888$	0	0
$889$	3.29150	0.110393
$890$	0	0
$891$	5.29150	0.177272
$892$	0	0
$893$	−4.35425	−0.145709
$894$	0	0
$895$	37.7490	1.26181
$896$	0	0
$897$	0	0
$898$	0	0
$899$	57.8745	1.93022
$900$	0	0
$901$	−4.83399	−0.161044
$902$	0	0
$903$	−1.29150	−0.0429785
$904$	0	0
$905$	−21.8745	−0.727133
$906$	0	0
$907$	26.3320	0.874340	0.437170	−	0.899379i	$-0.355981\pi$
$907$	26.3320	0.874340	0.437170	+	0.899379i	$0.355981\pi$
$908$	0	0
$909$	−12.3542	−0.409765
$910$	0	0
$911$	52.9373	1.75389	0.876945	−	0.480591i	$-0.159578\pi$
$911$	52.9373	1.75389	0.876945	+	0.480591i	$0.159578\pi$
$912$	0	0
$913$	−79.0405	−2.61586
$914$	0	0
$915$	33.8745	1.11986
$916$	0	0
$917$	15.6458	0.516668
$918$	0	0
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	20.5830	0.678233
$922$	0	0
$923$	0	0
$924$	0	0
$925$	43.8745	1.44258
$926$	0	0
$927$	14.5830	0.478969
$928$	0	0
$929$	54.6863	1.79420	0.897099	−	0.441829i	$-0.145670\pi$
$929$	54.6863	1.79420	0.897099	+	0.441829i	$0.145670\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−14.9373	−0.489024
$934$	0	0
$935$	−6.83399	−0.223495
$936$	0	0
$937$	−17.7490	−0.579835	−0.289918	−	0.957052i	$-0.593628\pi$
$937$	−17.7490	−0.579835	−0.289918	+	0.957052i	$0.593628\pi$
$938$	0	0
$939$	−25.2915	−0.825357
$940$	0	0
$941$	−15.7490	−0.513403	−0.256702	−	0.966491i	$-0.582636\pi$
$941$	−15.7490	−0.513403	−0.256702	+	0.966491i	$0.582636\pi$
$942$	0	0
$943$	−17.4170	−0.567175
$944$	0	0
$945$	−3.64575	−0.118596
$946$	0	0
$947$	12.1255	0.394026	0.197013	−	0.980401i	$-0.436876\pi$
$947$	12.1255	0.394026	0.197013	+	0.980401i	$0.436876\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−16.9373	−0.549228
$952$	0	0
$953$	31.5203	1.02104	0.510521	−	0.859865i	$-0.329453\pi$
$953$	31.5203	1.02104	0.510521	+	0.859865i	$0.329453\pi$
$954$	0	0
$955$	33.8745	1.09615
$956$	0	0
$957$	51.0405	1.64991
$958$	0	0
$959$	−23.1660	−0.748069
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−13.6458	−0.439728
$964$	0	0
$965$	28.7085	0.924159
$966$	0	0
$967$	10.7085	0.344362	0.172181	−	0.985065i	$-0.444919\pi$
$967$	10.7085	0.344362	0.172181	+	0.985065i	$0.444919\pi$
$968$	0	0
$969$	−0.354249	−0.0113801
$970$	0	0
$971$	31.7490	1.01887	0.509437	−	0.860508i	$-0.329854\pi$
$971$	31.7490	1.01887	0.509437	+	0.860508i	$0.329854\pi$
$972$	0	0
$973$	−10.5830	−0.339276
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.8118	−0.473870	−0.236935	−	0.971525i	$-0.576143\pi$
$977$	−14.8118	−0.473870	−0.236935	+	0.971525i	$0.576143\pi$
$978$	0	0
$979$	−17.4170	−0.556650
$980$	0	0
$981$	−17.2915	−0.552075
$982$	0	0
$983$	−7.74902	−0.247155	−0.123578	−	0.992335i	$-0.539437\pi$
$983$	−7.74902	−0.247155	−0.123578	+	0.992335i	$0.539437\pi$
$984$	0	0
$985$	16.7085	0.532377
$986$	0	0
$987$	−4.35425	−0.138597
$988$	0	0
$989$	−6.83399	−0.217308
$990$	0	0
$991$	−30.3320	−0.963528	−0.481764	−	0.876301i	$-0.660004\pi$
$991$	−30.3320	−0.963528	−0.481764	+	0.876301i	$0.660004\pi$
$992$	0	0
$993$	9.41699	0.298839
$994$	0	0
$995$	50.5830	1.60359
$996$	0	0
$997$	55.6235	1.76161	0.880807	−	0.473475i	$-0.157001\pi$
$997$	55.6235	1.76161	0.880807	+	0.473475i	$0.157001\pi$
$998$	0	0
$999$	−5.29150	−0.167416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1596.2.a.f.1.1	✓	2
3.2	odd	2		4788.2.a.k.1.2		2
4.3	odd	2		6384.2.a.bo.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.f.1.1	✓	2	1.1	even	1	trivial
4788.2.a.k.1.2		2	3.2	odd	2
6384.2.a.bo.1.1		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 1596 = 2^{2} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1596.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.64575 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.29150 q^{11} +3.64575 q^{15} +0.354249 q^{17} +1.00000 q^{19} +1.00000 q^{21} +5.29150 q^{23} +8.29150 q^{25} -1.00000 q^{27} -9.64575 q^{29} -6.00000 q^{31} -5.29150 q^{33} +3.64575 q^{35} +5.29150 q^{37} -3.29150 q^{41} -1.29150 q^{43} -3.64575 q^{45} -4.35425 q^{47} +1.00000 q^{49} -0.354249 q^{51} -13.6458 q^{53} -19.2915 q^{55} -1.00000 q^{57} -8.00000 q^{59} +9.29150 q^{61} -1.00000 q^{63} +7.29150 q^{67} -5.29150 q^{69} -8.93725 q^{71} -2.00000 q^{73} -8.29150 q^{75} -5.29150 q^{77} -3.29150 q^{79} +1.00000 q^{81} -14.9373 q^{83} -1.29150 q^{85} +9.64575 q^{87} -3.29150 q^{89} +6.00000 q^{93} -3.64575 q^{95} +4.58301 q^{97} +5.29150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} + 6 q^{25} - 2 q^{27} - 14 q^{29} - 12 q^{31} + 2 q^{35} + 4 q^{41} + 8 q^{43} - 2 q^{45} - 14 q^{47} + 2 q^{49} - 6 q^{51}+ \cdots - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 + 6 * q^25 - 2 * q^27 - 14 * q^29 - 12 * q^31 + 2 * q^35 + 4 * q^41 + 8 * q^43 - 2 * q^45 - 14 * q^47 + 2 * q^49 - 6 * q^51 - 22 * q^53 - 28 * q^55 - 2 * q^57 - 16 * q^59 + 8 * q^61 - 2 * q^63 + 4 * q^67 - 2 * q^71 - 4 * q^73 - 6 * q^75 + 4 * q^79 + 2 * q^81 - 14 * q^83 + 8 * q^85 + 14 * q^87 + 4 * q^89 + 12 * q^93 - 2 * q^95 - 12 * q^97

Introduction

L-functions

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