LMFDB - Embedded newform 1596.2.a.f.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.64575 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.29150 q^{11} -1.64575 q^{15} +5.64575 q^{17} +1.00000 q^{19} +1.00000 q^{21} -5.29150 q^{23} -2.29150 q^{25} -1.00000 q^{27} -4.35425 q^{29} -6.00000 q^{31} +5.29150 q^{33} -1.64575 q^{35} -5.29150 q^{37} +7.29150 q^{41} +9.29150 q^{43} +1.64575 q^{45} -9.64575 q^{47} +1.00000 q^{49} -5.64575 q^{51} -8.35425 q^{53} -8.70850 q^{55} -1.00000 q^{57} -8.00000 q^{59} -1.29150 q^{61} -1.00000 q^{63} -3.29150 q^{67} +5.29150 q^{69} +6.93725 q^{71} -2.00000 q^{73} +2.29150 q^{75} +5.29150 q^{77} +7.29150 q^{79} +1.00000 q^{81} +0.937254 q^{83} +9.29150 q^{85} +4.35425 q^{87} +7.29150 q^{89} +6.00000 q^{93} +1.64575 q^{95} -16.5830 q^{97} -5.29150 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.64575	0.736002	0.368001	−	0.929825i	$-0.380042\pi$
$5$	1.64575	0.736002	0.368001	+	0.929825i	$0.380042\pi$
$6$	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	$-0.793963\pi$
$11$	−5.29150	−1.59545	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−1.64575	−0.424931
$16$	0	0
$17$	5.64575	1.36930	0.684648	−	0.728874i	$-0.259956\pi$
$17$	5.64575	1.36930	0.684648	+	0.728874i	$0.259956\pi$
$18$	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	$-0.686012\pi$
$23$	−5.29150	−1.10335	−0.551677	+	0.834058i	$0.686012\pi$
$24$	0	0
$25$	−2.29150	−0.458301
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.35425	−0.808564	−0.404282	−	0.914634i	$-0.632479\pi$
$29$	−4.35425	−0.808564	−0.404282	+	0.914634i	$0.632479\pi$
$30$	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
$35$	−1.64575	−0.278183
$36$	0	0
$37$	−5.29150	−0.869918	−0.434959	−	0.900450i	$-0.643237\pi$
$37$	−5.29150	−0.869918	−0.434959	+	0.900450i	$0.643237\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.29150	1.13874	0.569371	−	0.822081i	$-0.307187\pi$
$41$	7.29150	1.13874	0.569371	+	0.822081i	$0.307187\pi$
$42$	0	0
$43$	9.29150	1.41694	0.708470	−	0.705740i	$-0.249386\pi$
$43$	9.29150	1.41694	0.708470	+	0.705740i	$0.249386\pi$
$44$	0	0
$45$	1.64575	0.245334
$46$	0	0
$47$	−9.64575	−1.40698	−0.703489	−	0.710706i	$-0.748375\pi$
$47$	−9.64575	−1.40698	−0.703489	+	0.710706i	$0.748375\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.64575	−0.790563
$52$	0	0
$53$	−8.35425	−1.14754	−0.573772	−	0.819015i	$-0.694521\pi$
$53$	−8.35425	−1.14754	−0.573772	+	0.819015i	$0.694521\pi$
$54$	0	0
$55$	−8.70850	−1.17425
$56$	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
$61$	−1.29150	−0.165360	−0.0826800	−	0.996576i	$-0.526348\pi$
$61$	−1.29150	−0.165360	−0.0826800	+	0.996576i	$0.526348\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.29150	−0.402121	−0.201061	−	0.979579i	$-0.564439\pi$
$67$	−3.29150	−0.402121	−0.201061	+	0.979579i	$0.564439\pi$
$68$	0	0
$69$	5.29150	0.637022
$70$	0	0
$71$	6.93725	0.823301	0.411650	−	0.911342i	$-0.364953\pi$
$71$	6.93725	0.823301	0.411650	+	0.911342i	$0.364953\pi$
$72$	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
$75$	2.29150	0.264600
$76$	0	0
$77$	5.29150	0.603023
$78$	0	0
$79$	7.29150	0.820358	0.410179	−	0.912005i	$-0.365466\pi$
$79$	7.29150	0.820358	0.410179	+	0.912005i	$0.365466\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.937254	0.102877	0.0514385	−	0.998676i	$-0.483619\pi$
$83$	0.937254	0.102877	0.0514385	+	0.998676i	$0.483619\pi$
$84$	0	0
$85$	9.29150	1.00780
$86$	0	0
$87$	4.35425	0.466824
$88$	0	0
$89$	7.29150	0.772898	0.386449	−	0.922311i	$-0.373702\pi$
$89$	7.29150	0.772898	0.386449	+	0.922311i	$0.373702\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.00000	0.622171
$94$	0	0
$95$	1.64575	0.168851
$96$	0	0
$97$	−16.5830	−1.68375	−0.841875	−	0.539673i	$-0.818548\pi$
$97$	−16.5830	−1.68375	−0.841875	+	0.539673i	$0.818548\pi$
$98$	0	0
$99$	−5.29150	−0.531816
$100$	0	0
$101$	−17.6458	−1.75582	−0.877909	−	0.478828i	$-0.841062\pi$
$101$	−17.6458	−1.75582	−0.877909	+	0.478828i	$0.841062\pi$
$102$	0	0
$103$	−6.58301	−0.648643	−0.324321	−	0.945947i	$-0.605136\pi$
$103$	−6.58301	−0.648643	−0.324321	+	0.945947i	$0.605136\pi$
$104$	0	0
$105$	1.64575	0.160609
$106$	0	0
$107$	−8.35425	−0.807636	−0.403818	−	0.914839i	$-0.632317\pi$
$107$	−8.35425	−0.807636	−0.403818	+	0.914839i	$0.632317\pi$
$108$	0	0
$109$	−6.70850	−0.642558	−0.321279	−	0.946985i	$-0.604113\pi$
$109$	−6.70850	−0.642558	−0.321279	+	0.946985i	$0.604113\pi$
$110$	0	0
$111$	5.29150	0.502247
$112$	0	0
$113$	−6.93725	−0.652602	−0.326301	−	0.945266i	$-0.605802\pi$
$113$	−6.93725	−0.652602	−0.326301	+	0.945266i	$0.605802\pi$
$114$	0	0
$115$	−8.70850	−0.812072
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.64575	−0.517545
$120$	0	0
$121$	17.0000	1.54545
$122$	0	0
$123$	−7.29150	−0.657453
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	7.29150	0.647016	0.323508	−	0.946225i	$-0.395138\pi$
$127$	7.29150	0.647016	0.323508	+	0.946225i	$0.395138\pi$
$128$	0	0
$129$	−9.29150	−0.818071
$130$	0	0
$131$	−10.3542	−0.904655	−0.452327	−	0.891852i	$-0.649406\pi$
$131$	−10.3542	−0.904655	−0.452327	+	0.891852i	$0.649406\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−1.64575	−0.141644
$136$	0	0
$137$	−19.1660	−1.63746	−0.818731	−	0.574177i	$-0.805322\pi$
$137$	−19.1660	−1.63746	−0.818731	+	0.574177i	$0.805322\pi$
$138$	0	0
$139$	−10.5830	−0.897639	−0.448819	−	0.893622i	$-0.648155\pi$
$139$	−10.5830	−0.897639	−0.448819	+	0.893622i	$0.648155\pi$
$140$	0	0
$141$	9.64575	0.812319
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−7.16601	−0.595105
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	10.7085	0.877274	0.438637	−	0.898664i	$-0.355461\pi$
$149$	10.7085	0.877274	0.438637	+	0.898664i	$0.355461\pi$
$150$	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
$153$	5.64575	0.456432
$154$	0	0
$155$	−9.87451	−0.793140
$156$	0	0
$157$	9.29150	0.741543	0.370771	−	0.928724i	$-0.379093\pi$
$157$	9.29150	0.741543	0.370771	+	0.928724i	$0.379093\pi$
$158$	0	0
$159$	8.35425	0.662535
$160$	0	0
$161$	5.29150	0.417029
$162$	0	0
$163$	23.8745	1.87000	0.934998	−	0.354653i	$-0.115401\pi$
$163$	23.8745	1.87000	0.934998	+	0.354653i	$0.115401\pi$
$164$	0	0
$165$	8.70850	0.677956
$166$	0	0
$167$	−3.29150	−0.254704	−0.127352	−	0.991858i	$-0.540648\pi$
$167$	−3.29150	−0.254704	−0.127352	+	0.991858i	$0.540648\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−15.2915	−1.16259	−0.581296	−	0.813692i	$-0.697454\pi$
$173$	−15.2915	−1.16259	−0.581296	+	0.813692i	$0.697454\pi$
$174$	0	0
$175$	2.29150	0.173221
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	−15.6458	−1.16942	−0.584709	−	0.811243i	$-0.698791\pi$
$179$	−15.6458	−1.16942	−0.584709	+	0.811243i	$0.698791\pi$
$180$	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
$183$	1.29150	0.0954706
$184$	0	0
$185$	−8.70850	−0.640261
$186$	0	0
$187$	−29.8745	−2.18464
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	1.29150	0.0934499	0.0467249	−	0.998908i	$-0.485122\pi$
$191$	1.29150	0.0934499	0.0467249	+	0.998908i	$0.485122\pi$
$192$	0	0
$193$	23.8745	1.71852	0.859262	−	0.511535i	$-0.170923\pi$
$193$	23.8745	1.71852	0.859262	+	0.511535i	$0.170923\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.5830	1.18149	0.590745	−	0.806858i	$-0.298834\pi$
$197$	16.5830	1.18149	0.590745	+	0.806858i	$0.298834\pi$
$198$	0	0
$199$	17.8745	1.26709	0.633545	−	0.773706i	$-0.281599\pi$
$199$	17.8745	1.26709	0.633545	+	0.773706i	$0.281599\pi$
$200$	0	0
$201$	3.29150	0.232165
$202$	0	0
$203$	4.35425	0.305608
$204$	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	$-0.381314\pi$
$211$	10.5830	0.728564	0.364282	+	0.931289i	$0.381314\pi$
$212$	0	0
$213$	−6.93725	−0.475333
$214$	0	0
$215$	15.2915	1.04287
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.1660	1.28345	0.641725	−	0.766934i	$-0.278219\pi$
$223$	19.1660	1.28345	0.641725	+	0.766934i	$0.278219\pi$
$224$	0	0
$225$	−2.29150	−0.152767
$226$	0	0
$227$	16.7085	1.10898	0.554491	−	0.832190i	$-0.312913\pi$
$227$	16.7085	1.10898	0.554491	+	0.832190i	$0.312913\pi$
$228$	0	0
$229$	7.87451	0.520362	0.260181	−	0.965560i	$-0.416218\pi$
$229$	7.87451	0.520362	0.260181	+	0.965560i	$0.416218\pi$
$230$	0	0
$231$	−5.29150	−0.348155
$232$	0	0
$233$	9.29150	0.608707	0.304353	−	0.952559i	$-0.401560\pi$
$233$	9.29150	0.608707	0.304353	+	0.952559i	$0.401560\pi$
$234$	0	0
$235$	−15.8745	−1.03554
$236$	0	0
$237$	−7.29150	−0.473634
$238$	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
$241$	16.5830	1.06821	0.534103	−	0.845420i	$-0.320650\pi$
$241$	16.5830	1.06821	0.534103	+	0.845420i	$0.320650\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.64575	0.105143
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.937254	−0.0593961
$250$	0	0
$251$	−7.06275	−0.445797	−0.222898	−	0.974842i	$-0.571552\pi$
$251$	−7.06275	−0.445797	−0.222898	+	0.974842i	$0.571552\pi$
$252$	0	0
$253$	28.0000	1.76034
$254$	0	0
$255$	−9.29150	−0.581856
$256$	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
$261$	−4.35425	−0.269521
$262$	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
$265$	−13.7490	−0.844595
$266$	0	0
$267$	−7.29150	−0.446233
$268$	0	0
$269$	−22.5830	−1.37691	−0.688455	−	0.725279i	$-0.741711\pi$
$269$	−22.5830	−1.37691	−0.688455	+	0.725279i	$0.741711\pi$
$270$	0	0
$271$	4.70850	0.286021	0.143010	−	0.989721i	$-0.454322\pi$
$271$	4.70850	0.286021	0.143010	+	0.989721i	$0.454322\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.1255	0.731195
$276$	0	0
$277$	32.4575	1.95018	0.975091	−	0.221803i	$-0.0711942\pi$
$277$	32.4575	1.95018	0.975091	+	0.221803i	$0.0711942\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	1.77124	0.105664	0.0528318	−	0.998603i	$-0.483175\pi$
$281$	1.77124	0.105664	0.0528318	+	0.998603i	$0.483175\pi$
$282$	0	0
$283$	25.8745	1.53808	0.769040	−	0.639201i	$-0.220735\pi$
$283$	25.8745	1.53808	0.769040	+	0.639201i	$0.220735\pi$
$284$	0	0
$285$	−1.64575	−0.0974859
$286$	0	0
$287$	−7.29150	−0.430404
$288$	0	0
$289$	14.8745	0.874971
$290$	0	0
$291$	16.5830	0.972113
$292$	0	0
$293$	−19.2915	−1.12702	−0.563511	−	0.826109i	$-0.690550\pi$
$293$	−19.2915	−1.12702	−0.563511	+	0.826109i	$0.690550\pi$
$294$	0	0
$295$	−13.1660	−0.766555
$296$	0	0
$297$	5.29150	0.307044
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−9.29150	−0.535553
$302$	0	0
$303$	17.6458	1.01372
$304$	0	0
$305$	−2.12549	−0.121705
$306$	0	0
$307$	0.583005	0.0332739	0.0166369	−	0.999862i	$-0.494704\pi$
$307$	0.583005	0.0332739	0.0166369	+	0.999862i	$0.494704\pi$
$308$	0	0
$309$	6.58301	0.374494
$310$	0	0
$311$	−0.937254	−0.0531468	−0.0265734	−	0.999647i	$-0.508460\pi$
$311$	−0.937254	−0.0531468	−0.0265734	+	0.999647i	$0.508460\pi$
$312$	0	0
$313$	14.7085	0.831373	0.415687	−	0.909508i	$-0.363541\pi$
$313$	14.7085	0.831373	0.415687	+	0.909508i	$0.363541\pi$
$314$	0	0
$315$	−1.64575	−0.0927276
$316$	0	0
$317$	1.06275	0.0596898	0.0298449	−	0.999555i	$-0.490499\pi$
$317$	1.06275	0.0596898	0.0298449	+	0.999555i	$0.490499\pi$
$318$	0	0
$319$	23.0405	1.29002
$320$	0	0
$321$	8.35425	0.466289
$322$	0	0
$323$	5.64575	0.314138
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.70850	0.370981
$328$	0	0
$329$	9.64575	0.531787
$330$	0	0
$331$	−30.5830	−1.68099	−0.840497	−	0.541816i	$-0.817737\pi$
$331$	−30.5830	−1.68099	−0.840497	+	0.541816i	$0.817737\pi$
$332$	0	0
$333$	−5.29150	−0.289973
$334$	0	0
$335$	−5.41699	−0.295962
$336$	0	0
$337$	22.7085	1.23701	0.618505	−	0.785781i	$-0.287739\pi$
$337$	22.7085	1.23701	0.618505	+	0.785781i	$0.287739\pi$
$338$	0	0
$339$	6.93725	0.376780
$340$	0	0
$341$	31.7490	1.71931
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	8.70850	0.468850
$346$	0	0
$347$	11.1660	0.599423	0.299711	−	0.954030i	$-0.403110\pi$
$347$	11.1660	0.599423	0.299711	+	0.954030i	$0.403110\pi$
$348$	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
$353$	28.9373	1.54017	0.770087	−	0.637939i	$-0.220213\pi$
$353$	28.9373	1.54017	0.770087	+	0.637939i	$0.220213\pi$
$354$	0	0
$355$	11.4170	0.605951
$356$	0	0
$357$	5.64575	0.298805
$358$	0	0
$359$	22.4575	1.18526	0.592631	−	0.805474i	$-0.298089\pi$
$359$	22.4575	1.18526	0.592631	+	0.805474i	$0.298089\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−17.0000	−0.892269
$364$	0	0
$365$	−3.29150	−0.172285
$366$	0	0
$367$	−13.8745	−0.724243	−0.362122	−	0.932131i	$-0.617948\pi$
$367$	−13.8745	−0.724243	−0.362122	+	0.932131i	$0.617948\pi$
$368$	0	0
$369$	7.29150	0.379580
$370$	0	0
$371$	8.35425	0.433731
$372$	0	0
$373$	−16.5830	−0.858635	−0.429318	−	0.903154i	$-0.641246\pi$
$373$	−16.5830	−0.858635	−0.429318	+	0.903154i	$0.641246\pi$
$374$	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	$-0.587621\pi$
$379$	−10.5830	−0.543612	−0.271806	+	0.962352i	$0.587621\pi$
$380$	0	0
$381$	−7.29150	−0.373555
$382$	0	0
$383$	−1.87451	−0.0957829	−0.0478914	−	0.998853i	$-0.515250\pi$
$383$	−1.87451	−0.0957829	−0.0478914	+	0.998853i	$0.515250\pi$
$384$	0	0
$385$	8.70850	0.443826
$386$	0	0
$387$	9.29150	0.472314
$388$	0	0
$389$	−26.4575	−1.34145	−0.670725	−	0.741707i	$-0.734017\pi$
$389$	−26.4575	−1.34145	−0.670725	+	0.741707i	$0.734017\pi$
$390$	0	0
$391$	−29.8745	−1.51082
$392$	0	0
$393$	10.3542	0.522303
$394$	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
$401$	−10.9373	−0.546180	−0.273090	−	0.961988i	$-0.588046\pi$
$401$	−10.9373	−0.546180	−0.273090	+	0.961988i	$0.588046\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.64575	0.0817780
$406$	0	0
$407$	28.0000	1.38791
$408$	0	0
$409$	−18.5830	−0.918870	−0.459435	−	0.888211i	$-0.651948\pi$
$409$	−18.5830	−0.918870	−0.459435	+	0.888211i	$0.651948\pi$
$410$	0	0
$411$	19.1660	0.945390
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	1.54249	0.0757177
$416$	0	0
$417$	10.5830	0.518252
$418$	0	0
$419$	−16.9373	−0.827439	−0.413720	−	0.910404i	$-0.635771\pi$
$419$	−16.9373	−0.827439	−0.413720	+	0.910404i	$0.635771\pi$
$420$	0	0
$421$	−20.5830	−1.00315	−0.501577	−	0.865113i	$-0.667247\pi$
$421$	−20.5830	−1.00315	−0.501577	+	0.865113i	$0.667247\pi$
$422$	0	0
$423$	−9.64575	−0.468992
$424$	0	0
$425$	−12.9373	−0.627549
$426$	0	0
$427$	1.29150	0.0625002
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.64575	0.175610	0.0878048	−	0.996138i	$-0.472015\pi$
$431$	3.64575	0.175610	0.0878048	+	0.996138i	$0.472015\pi$
$432$	0	0
$433$	−24.5830	−1.18138	−0.590692	−	0.806897i	$-0.701145\pi$
$433$	−24.5830	−1.18138	−0.590692	+	0.806897i	$0.701145\pi$
$434$	0	0
$435$	7.16601	0.343584
$436$	0	0
$437$	−5.29150	−0.253127
$438$	0	0
$439$	11.1660	0.532925	0.266462	−	0.963845i	$-0.414145\pi$
$439$	11.1660	0.532925	0.266462	+	0.963845i	$0.414145\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	31.1660	1.48074	0.740371	−	0.672199i	$-0.234650\pi$
$443$	31.1660	1.48074	0.740371	+	0.672199i	$0.234650\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	0	0
$447$	−10.7085	−0.506495
$448$	0	0
$449$	27.3948	1.29284	0.646419	−	0.762982i	$-0.276266\pi$
$449$	27.3948	1.29284	0.646419	+	0.762982i	$0.276266\pi$
$450$	0	0
$451$	−38.5830	−1.81680
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.2915	1.08953	0.544765	−	0.838589i	$-0.316619\pi$
$457$	23.2915	1.08953	0.544765	+	0.838589i	$0.316619\pi$
$458$	0	0
$459$	−5.64575	−0.263521
$460$	0	0
$461$	35.5203	1.65434	0.827172	−	0.561949i	$-0.189948\pi$
$461$	35.5203	1.65434	0.827172	+	0.561949i	$0.189948\pi$
$462$	0	0
$463$	−25.1660	−1.16956	−0.584782	−	0.811191i	$-0.698820\pi$
$463$	−25.1660	−1.16956	−0.584782	+	0.811191i	$0.698820\pi$
$464$	0	0
$465$	9.87451	0.457919
$466$	0	0
$467$	16.2288	0.750977	0.375489	−	0.926827i	$-0.377475\pi$
$467$	16.2288	0.750977	0.375489	+	0.926827i	$0.377475\pi$
$468$	0	0
$469$	3.29150	0.151987
$470$	0	0
$471$	−9.29150	−0.428130
$472$	0	0
$473$	−49.1660	−2.26066
$474$	0	0
$475$	−2.29150	−0.105141
$476$	0	0
$477$	−8.35425	−0.382515
$478$	0	0
$479$	−11.5203	−0.526374	−0.263187	−	0.964745i	$-0.584774\pi$
$479$	−11.5203	−0.526374	−0.263187	+	0.964745i	$0.584774\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−5.29150	−0.240772
$484$	0	0
$485$	−27.2915	−1.23924
$486$	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
$489$	−23.8745	−1.07964
$490$	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
$493$	−24.5830	−1.10716
$494$	0	0
$495$	−8.70850	−0.391418
$496$	0	0
$497$	−6.93725	−0.311178
$498$	0	0
$499$	13.4170	0.600627	0.300314	−	0.953841i	$-0.402909\pi$
$499$	13.4170	0.600627	0.300314	+	0.953841i	$0.402909\pi$
$500$	0	0
$501$	3.29150	0.147053
$502$	0	0
$503$	24.2288	1.08031	0.540153	−	0.841567i	$-0.318366\pi$
$503$	24.2288	1.08031	0.540153	+	0.841567i	$0.318366\pi$
$504$	0	0
$505$	−29.0405	−1.29229
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	−9.41699	−0.417401	−0.208700	−	0.977980i	$-0.566923\pi$
$509$	−9.41699	−0.417401	−0.208700	+	0.977980i	$0.566923\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−10.8340	−0.477403
$516$	0	0
$517$	51.0405	2.24476
$518$	0	0
$519$	15.2915	0.671222
$520$	0	0
$521$	−14.5830	−0.638893	−0.319447	−	0.947604i	$-0.603497\pi$
$521$	−14.5830	−0.638893	−0.319447	+	0.947604i	$0.603497\pi$
$522$	0	0
$523$	−27.1660	−1.18789	−0.593943	−	0.804507i	$-0.702430\pi$
$523$	−27.1660	−1.18789	−0.593943	+	0.804507i	$0.702430\pi$
$524$	0	0
$525$	−2.29150	−0.100009
$526$	0	0
$527$	−33.8745	−1.47560
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−13.7490	−0.594422
$536$	0	0
$537$	15.6458	0.675164
$538$	0	0
$539$	−5.29150	−0.227921
$540$	0	0
$541$	−23.1660	−0.995985	−0.497992	−	0.867181i	$-0.665929\pi$
$541$	−23.1660	−0.995985	−0.497992	+	0.867181i	$0.665929\pi$
$542$	0	0
$543$	−6.00000	−0.257485
$544$	0	0
$545$	−11.0405	−0.472924
$546$	0	0
$547$	24.4575	1.04573	0.522864	−	0.852416i	$-0.324864\pi$
$547$	24.4575	1.04573	0.522864	+	0.852416i	$0.324864\pi$
$548$	0	0
$549$	−1.29150	−0.0551200
$550$	0	0
$551$	−4.35425	−0.185497
$552$	0	0
$553$	−7.29150	−0.310066
$554$	0	0
$555$	8.70850	0.369655
$556$	0	0
$557$	7.87451	0.333654	0.166827	−	0.985986i	$-0.446648\pi$
$557$	7.87451	0.333654	0.166827	+	0.985986i	$0.446648\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	29.8745	1.26130
$562$	0	0
$563$	−8.70850	−0.367019	−0.183510	−	0.983018i	$-0.558746\pi$
$563$	−8.70850	−0.367019	−0.183510	+	0.983018i	$0.558746\pi$
$564$	0	0
$565$	−11.4170	−0.480317
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	22.9373	0.961580	0.480790	−	0.876836i	$-0.340350\pi$
$569$	22.9373	0.961580	0.480790	+	0.876836i	$0.340350\pi$
$570$	0	0
$571$	−29.1660	−1.22056	−0.610280	−	0.792186i	$-0.708943\pi$
$571$	−29.1660	−1.22056	−0.610280	+	0.792186i	$0.708943\pi$
$572$	0	0
$573$	−1.29150	−0.0539533
$574$	0	0
$575$	12.1255	0.505668
$576$	0	0
$577$	19.8745	0.827387	0.413693	−	0.910416i	$-0.364239\pi$
$577$	19.8745	0.827387	0.413693	+	0.910416i	$0.364239\pi$
$578$	0	0
$579$	−23.8745	−0.992191
$580$	0	0
$581$	−0.937254	−0.0388838
$582$	0	0
$583$	44.2065	1.83085
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.2288	−1.16513	−0.582563	−	0.812786i	$-0.697950\pi$
$587$	−28.2288	−1.16513	−0.582563	+	0.812786i	$0.697950\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	0	0
$591$	−16.5830	−0.682134
$592$	0	0
$593$	0.228757	0.00939391	0.00469695	−	0.999989i	$-0.498505\pi$
$593$	0.228757	0.00939391	0.00469695	+	0.999989i	$0.498505\pi$
$594$	0	0
$595$	−9.29150	−0.380914
$596$	0	0
$597$	−17.8745	−0.731555
$598$	0	0
$599$	−30.2288	−1.23511	−0.617557	−	0.786526i	$-0.711877\pi$
$599$	−30.2288	−1.23511	−0.617557	+	0.786526i	$0.711877\pi$
$600$	0	0
$601$	−36.5830	−1.49225	−0.746126	−	0.665805i	$-0.768088\pi$
$601$	−36.5830	−1.49225	−0.746126	+	0.665805i	$0.768088\pi$
$602$	0	0
$603$	−3.29150	−0.134040
$604$	0	0
$605$	27.9778	1.13746
$606$	0	0
$607$	38.5830	1.56604	0.783018	−	0.621999i	$-0.213679\pi$
$607$	38.5830	1.56604	0.783018	+	0.621999i	$0.213679\pi$
$608$	0	0
$609$	−4.35425	−0.176443
$610$	0	0
$611$	0	0
$612$	0	0
$613$	36.4575	1.47251	0.736253	−	0.676707i	$-0.236594\pi$
$613$	36.4575	1.47251	0.736253	+	0.676707i	$0.236594\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	43.1660	1.73780	0.868899	−	0.494989i	$-0.164828\pi$
$617$	43.1660	1.73780	0.868899	+	0.494989i	$0.164828\pi$
$618$	0	0
$619$	−15.7490	−0.633006	−0.316503	−	0.948591i	$-0.602509\pi$
$619$	−15.7490	−0.633006	−0.316503	+	0.948591i	$0.602509\pi$
$620$	0	0
$621$	5.29150	0.212341
$622$	0	0
$623$	−7.29150	−0.292128
$624$	0	0
$625$	−8.29150	−0.331660
$626$	0	0
$627$	5.29150	0.211322
$628$	0	0
$629$	−29.8745	−1.19117
$630$	0	0
$631$	23.8745	0.950429	0.475215	−	0.879870i	$-0.342370\pi$
$631$	23.8745	0.950429	0.475215	+	0.879870i	$0.342370\pi$
$632$	0	0
$633$	−10.5830	−0.420637
$634$	0	0
$635$	12.0000	0.476205
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.93725	0.274434
$640$	0	0
$641$	1.77124	0.0699599	0.0349800	−	0.999388i	$-0.488863\pi$
$641$	1.77124	0.0699599	0.0349800	+	0.999388i	$0.488863\pi$
$642$	0	0
$643$	8.45751	0.333532	0.166766	−	0.985997i	$-0.446668\pi$
$643$	8.45751	0.333532	0.166766	+	0.985997i	$0.446668\pi$
$644$	0	0
$645$	−15.2915	−0.602102
$646$	0	0
$647$	−14.8118	−0.582310	−0.291155	−	0.956676i	$-0.594040\pi$
$647$	−14.8118	−0.582310	−0.291155	+	0.956676i	$0.594040\pi$
$648$	0	0
$649$	42.3320	1.66168
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	−37.7490	−1.47723	−0.738617	−	0.674126i	$-0.764521\pi$
$653$	−37.7490	−1.47723	−0.738617	+	0.674126i	$0.764521\pi$
$654$	0	0
$655$	−17.0405	−0.665828
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	19.3948	0.755513	0.377756	−	0.925905i	$-0.376696\pi$
$659$	19.3948	0.755513	0.377756	+	0.925905i	$0.376696\pi$
$660$	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
$665$	−1.64575	−0.0638195
$666$	0	0
$667$	23.0405	0.892132
$668$	0	0
$669$	−19.1660	−0.741001
$670$	0	0
$671$	6.83399	0.263823
$672$	0	0
$673$	−43.8745	−1.69124	−0.845619	−	0.533787i	$-0.820768\pi$
$673$	−43.8745	−1.69124	−0.845619	+	0.533787i	$0.820768\pi$
$674$	0	0
$675$	2.29150	0.0882000
$676$	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
$679$	16.5830	0.636397
$680$	0	0
$681$	−16.7085	−0.640271
$682$	0	0
$683$	22.9373	0.877670	0.438835	−	0.898568i	$-0.355391\pi$
$683$	22.9373	0.877670	0.438835	+	0.898568i	$0.355391\pi$
$684$	0	0
$685$	−31.5425	−1.20518
$686$	0	0
$687$	−7.87451	−0.300431
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0.708497	0.0269525	0.0134763	−	0.999909i	$-0.495710\pi$
$691$	0.708497	0.0269525	0.0134763	+	0.999909i	$0.495710\pi$
$692$	0	0
$693$	5.29150	0.201008
$694$	0	0
$695$	−17.4170	−0.660664
$696$	0	0
$697$	41.1660	1.55927
$698$	0	0
$699$	−9.29150	−0.351437
$700$	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
$707$	17.6458	0.663637
$708$	0	0
$709$	17.8745	0.671291	0.335646	−	0.941988i	$-0.391046\pi$
$709$	17.8745	0.671291	0.335646	+	0.941988i	$0.391046\pi$
$710$	0	0
$711$	7.29150	0.273453
$712$	0	0
$713$	31.7490	1.18901
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.00000	0.224074
$718$	0	0
$719$	14.3542	0.535323	0.267662	−	0.963513i	$-0.413749\pi$
$719$	14.3542	0.535323	0.267662	+	0.963513i	$0.413749\pi$
$720$	0	0
$721$	6.58301	0.245164
$722$	0	0
$723$	−16.5830	−0.616729
$724$	0	0
$725$	9.97777	0.370565
$726$	0	0
$727$	−24.7085	−0.916387	−0.458194	−	0.888852i	$-0.651503\pi$
$727$	−24.7085	−0.916387	−0.458194	+	0.888852i	$0.651503\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	52.4575	1.94021
$732$	0	0
$733$	27.4170	1.01267	0.506335	−	0.862337i	$-0.331000\pi$
$733$	27.4170	1.01267	0.506335	+	0.862337i	$0.331000\pi$
$734$	0	0
$735$	−1.64575	−0.0607045
$736$	0	0
$737$	17.4170	0.641563
$738$	0	0
$739$	−42.4575	−1.56182	−0.780912	−	0.624641i	$-0.785246\pi$
$739$	−42.4575	−1.56182	−0.780912	+	0.624641i	$0.785246\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.9373	−1.28172	−0.640862	−	0.767656i	$-0.721423\pi$
$743$	−34.9373	−1.28172	−0.640862	+	0.767656i	$0.721423\pi$
$744$	0	0
$745$	17.6235	0.645676
$746$	0	0
$747$	0.937254	0.0342923
$748$	0	0
$749$	8.35425	0.305258
$750$	0	0
$751$	53.1660	1.94006	0.970028	−	0.242995i	$-0.0781297\pi$
$751$	53.1660	1.94006	0.970028	+	0.242995i	$0.0781297\pi$
$752$	0	0
$753$	7.06275	0.257381
$754$	0	0
$755$	−13.1660	−0.479160
$756$	0	0
$757$	8.58301	0.311955	0.155977	−	0.987761i	$-0.450147\pi$
$757$	8.58301	0.311955	0.155977	+	0.987761i	$0.450147\pi$
$758$	0	0
$759$	−28.0000	−1.01634
$760$	0	0
$761$	15.0627	0.546024	0.273012	−	0.962011i	$-0.411980\pi$
$761$	15.0627	0.546024	0.273012	+	0.962011i	$0.411980\pi$
$762$	0	0
$763$	6.70850	0.242864
$764$	0	0
$765$	9.29150	0.335935
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−9.29150	−0.335060	−0.167530	−	0.985867i	$-0.553579\pi$
$769$	−9.29150	−0.335060	−0.167530	+	0.985867i	$0.553579\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−35.0405	−1.26032	−0.630160	−	0.776465i	$-0.717011\pi$
$773$	−35.0405	−1.26032	−0.630160	+	0.776465i	$0.717011\pi$
$774$	0	0
$775$	13.7490	0.493879
$776$	0	0
$777$	−5.29150	−0.189832
$778$	0	0
$779$	7.29150	0.261245
$780$	0	0
$781$	−36.7085	−1.31353
$782$	0	0
$783$	4.35425	0.155608
$784$	0	0
$785$	15.2915	0.545777
$786$	0	0
$787$	35.1660	1.25353	0.626766	−	0.779207i	$-0.284378\pi$
$787$	35.1660	1.25353	0.626766	+	0.779207i	$0.284378\pi$
$788$	0	0
$789$	18.0000	0.640817
$790$	0	0
$791$	6.93725	0.246660
$792$	0	0
$793$	0	0
$794$	0	0
$795$	13.7490	0.487627
$796$	0	0
$797$	−39.0405	−1.38289	−0.691443	−	0.722431i	$-0.743025\pi$
$797$	−39.0405	−1.38289	−0.691443	+	0.722431i	$0.743025\pi$
$798$	0	0
$799$	−54.4575	−1.92657
$800$	0	0
$801$	7.29150	0.257633
$802$	0	0
$803$	10.5830	0.373466
$804$	0	0
$805$	8.70850	0.306934
$806$	0	0
$807$	22.5830	0.794959
$808$	0	0
$809$	−23.4170	−0.823298	−0.411649	−	0.911343i	$-0.635047\pi$
$809$	−23.4170	−0.823298	−0.411649	+	0.911343i	$0.635047\pi$
$810$	0	0
$811$	−39.7490	−1.39578	−0.697888	−	0.716207i	$-0.745877\pi$
$811$	−39.7490	−1.39578	−0.697888	+	0.716207i	$0.745877\pi$
$812$	0	0
$813$	−4.70850	−0.165134
$814$	0	0
$815$	39.2915	1.37632
$816$	0	0
$817$	9.29150	0.325069
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.70850	−0.0945272	−0.0472636	−	0.998882i	$-0.515050\pi$
$821$	−2.70850	−0.0945272	−0.0472636	+	0.998882i	$0.515050\pi$
$822$	0	0
$823$	12.1255	0.422668	0.211334	−	0.977414i	$-0.432219\pi$
$823$	12.1255	0.422668	0.211334	+	0.977414i	$0.432219\pi$
$824$	0	0
$825$	−12.1255	−0.422155
$826$	0	0
$827$	−13.0627	−0.454236	−0.227118	−	0.973867i	$-0.572930\pi$
$827$	−13.0627	−0.454236	−0.227118	+	0.973867i	$0.572930\pi$
$828$	0	0
$829$	−48.3320	−1.67864	−0.839320	−	0.543637i	$-0.817047\pi$
$829$	−48.3320	−1.67864	−0.839320	+	0.543637i	$0.817047\pi$
$830$	0	0
$831$	−32.4575	−1.12594
$832$	0	0
$833$	5.64575	0.195614
$834$	0	0
$835$	−5.41699	−0.187463
$836$	0	0
$837$	6.00000	0.207390
$838$	0	0
$839$	−42.5830	−1.47013	−0.735064	−	0.677997i	$-0.762848\pi$
$839$	−42.5830	−1.47013	−0.735064	+	0.677997i	$0.762848\pi$
$840$	0	0
$841$	−10.0405	−0.346225
$842$	0	0
$843$	−1.77124	−0.0610049
$844$	0	0
$845$	−21.3948	−0.736002
$846$	0	0
$847$	−17.0000	−0.584127
$848$	0	0
$849$	−25.8745	−0.888011
$850$	0	0
$851$	28.0000	0.959828
$852$	0	0
$853$	7.16601	0.245360	0.122680	−	0.992446i	$-0.460851\pi$
$853$	7.16601	0.245360	0.122680	+	0.992446i	$0.460851\pi$
$854$	0	0
$855$	1.64575	0.0562835
$856$	0	0
$857$	−14.1255	−0.482518	−0.241259	−	0.970461i	$-0.577560\pi$
$857$	−14.1255	−0.482518	−0.241259	+	0.970461i	$0.577560\pi$
$858$	0	0
$859$	−45.1660	−1.54104	−0.770522	−	0.637413i	$-0.780004\pi$
$859$	−45.1660	−1.54104	−0.770522	+	0.637413i	$0.780004\pi$
$860$	0	0
$861$	7.29150	0.248494
$862$	0	0
$863$	−31.3948	−1.06869	−0.534345	−	0.845266i	$-0.679442\pi$
$863$	−31.3948	−1.06869	−0.534345	+	0.845266i	$0.679442\pi$
$864$	0	0
$865$	−25.1660	−0.855670
$866$	0	0
$867$	−14.8745	−0.505165
$868$	0	0
$869$	−38.5830	−1.30884
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−16.5830	−0.561250
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	−16.5830	−0.559968	−0.279984	−	0.960005i	$-0.590329\pi$
$877$	−16.5830	−0.559968	−0.279984	+	0.960005i	$0.590329\pi$
$878$	0	0
$879$	19.2915	0.650686
$880$	0	0
$881$	24.6863	0.831702	0.415851	−	0.909433i	$-0.363484\pi$
$881$	24.6863	0.831702	0.415851	+	0.909433i	$0.363484\pi$
$882$	0	0
$883$	1.16601	0.0392394	0.0196197	−	0.999808i	$-0.493754\pi$
$883$	1.16601	0.0392394	0.0196197	+	0.999808i	$0.493754\pi$
$884$	0	0
$885$	13.1660	0.442571
$886$	0	0
$887$	33.8745	1.13739	0.568697	−	0.822547i	$-0.307448\pi$
$887$	33.8745	1.13739	0.568697	+	0.822547i	$0.307448\pi$
$888$	0	0
$889$	−7.29150	−0.244549
$890$	0	0
$891$	−5.29150	−0.177272
$892$	0	0
$893$	−9.64575	−0.322783
$894$	0	0
$895$	−25.7490	−0.860695
$896$	0	0
$897$	0	0
$898$	0	0
$899$	26.1255	0.871334
$900$	0	0
$901$	−47.1660	−1.57133
$902$	0	0
$903$	9.29150	0.309202
$904$	0	0
$905$	9.87451	0.328240
$906$	0	0
$907$	−58.3320	−1.93688	−0.968441	−	0.249241i	$-0.919819\pi$
$907$	−58.3320	−1.93688	−0.968441	+	0.249241i	$0.919819\pi$
$908$	0	0
$909$	−17.6458	−0.585273
$910$	0	0
$911$	37.0627	1.22794	0.613972	−	0.789328i	$-0.289571\pi$
$911$	37.0627	1.22794	0.613972	+	0.789328i	$0.289571\pi$
$912$	0	0
$913$	−4.95948	−0.164135
$914$	0	0
$915$	2.12549	0.0702666
$916$	0	0
$917$	10.3542	0.341927
$918$	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
$921$	−0.583005	−0.0192107
$922$	0	0
$923$	0	0
$924$	0	0
$925$	12.1255	0.398684
$926$	0	0
$927$	−6.58301	−0.216214
$928$	0	0
$929$	−24.6863	−0.809930	−0.404965	−	0.914332i	$-0.632716\pi$
$929$	−24.6863	−0.809930	−0.404965	+	0.914332i	$0.632716\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0.937254	0.0306843
$934$	0	0
$935$	−49.1660	−1.60790
$936$	0	0
$937$	45.7490	1.49456	0.747278	−	0.664512i	$-0.231361\pi$
$937$	45.7490	1.49456	0.747278	+	0.664512i	$0.231361\pi$
$938$	0	0
$939$	−14.7085	−0.479994
$940$	0	0
$941$	47.7490	1.55657	0.778287	−	0.627909i	$-0.216089\pi$
$941$	47.7490	1.55657	0.778287	+	0.627909i	$0.216089\pi$
$942$	0	0
$943$	−38.5830	−1.25644
$944$	0	0
$945$	1.64575	0.0535363
$946$	0	0
$947$	43.8745	1.42573	0.712865	−	0.701301i	$-0.247397\pi$
$947$	43.8745	1.42573	0.712865	+	0.701301i	$0.247397\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−1.06275	−0.0344619
$952$	0	0
$953$	−5.52026	−0.178819	−0.0894094	−	0.995995i	$-0.528498\pi$
$953$	−5.52026	−0.178819	−0.0894094	+	0.995995i	$0.528498\pi$
$954$	0	0
$955$	2.12549	0.0687793
$956$	0	0
$957$	−23.0405	−0.744794
$958$	0	0
$959$	19.1660	0.618903
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−8.35425	−0.269212
$964$	0	0
$965$	39.2915	1.26484
$966$	0	0
$967$	21.2915	0.684689	0.342344	−	0.939575i	$-0.388779\pi$
$967$	21.2915	0.684689	0.342344	+	0.939575i	$0.388779\pi$
$968$	0	0
$969$	−5.64575	−0.181368
$970$	0	0
$971$	−31.7490	−1.01887	−0.509437	−	0.860508i	$-0.670146\pi$
$971$	−31.7490	−1.01887	−0.509437	+	0.860508i	$0.670146\pi$
$972$	0	0
$973$	10.5830	0.339276
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.8118	1.04974	0.524871	−	0.851182i	$-0.324114\pi$
$977$	32.8118	1.04974	0.524871	+	0.851182i	$0.324114\pi$
$978$	0	0
$979$	−38.5830	−1.23312
$980$	0	0
$981$	−6.70850	−0.214186
$982$	0	0
$983$	55.7490	1.77812	0.889059	−	0.457793i	$-0.151360\pi$
$983$	55.7490	1.77812	0.889059	+	0.457793i	$0.151360\pi$
$984$	0	0
$985$	27.2915	0.869580
$986$	0	0
$987$	−9.64575	−0.307028
$988$	0	0
$989$	−49.1660	−1.56339
$990$	0	0
$991$	54.3320	1.72591	0.862957	−	0.505278i	$-0.168610\pi$
$991$	54.3320	1.72591	0.862957	+	0.505278i	$0.168610\pi$
$992$	0	0
$993$	30.5830	0.970522
$994$	0	0
$995$	29.4170	0.932581
$996$	0	0
$997$	−39.6235	−1.25489	−0.627445	−	0.778661i	$-0.715899\pi$
$997$	−39.6235	−1.25489	−0.627445	+	0.778661i	$0.715899\pi$
$998$	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.1.2	✓	2
3.2	odd	2		4788.2.a.k.1.1		2
4.3	odd	2		6384.2.a.bo.1.2		2

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.64575 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.29150 q^{11} -1.64575 q^{15} +5.64575 q^{17} +1.00000 q^{19} +1.00000 q^{21} -5.29150 q^{23} -2.29150 q^{25} -1.00000 q^{27} -4.35425 q^{29} -6.00000 q^{31} +5.29150 q^{33} -1.64575 q^{35} -5.29150 q^{37} +7.29150 q^{41} +9.29150 q^{43} +1.64575 q^{45} -9.64575 q^{47} +1.00000 q^{49} -5.64575 q^{51} -8.35425 q^{53} -8.70850 q^{55} -1.00000 q^{57} -8.00000 q^{59} -1.29150 q^{61} -1.00000 q^{63} -3.29150 q^{67} +5.29150 q^{69} +6.93725 q^{71} -2.00000 q^{73} +2.29150 q^{75} +5.29150 q^{77} +7.29150 q^{79} +1.00000 q^{81} +0.937254 q^{83} +9.29150 q^{85} +4.35425 q^{87} +7.29150 q^{89} +6.00000 q^{93} +1.64575 q^{95} -16.5830 q^{97} -5.29150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} + 6 q^{25} - 2 q^{27} - 14 q^{29} - 12 q^{31} + 2 q^{35} + 4 q^{41} + 8 q^{43} - 2 q^{45} - 14 q^{47} + 2 q^{49} - 6 q^{51}+ \cdots - 12 q^{97}+O(q^{100}) \) 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

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