LMFDB - Embedded newform 1692.2.h.a.845.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1692,2,Mod(845,1692)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1692, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1692.845");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1692 = 2^{2} \cdot 3^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1692.h (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$13.5106880220$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26 x^{14} - 16 x^{13} + 269 x^{12} + 288 x^{11} + 850 x^{10} + 2032 x^{9} + 6628 x^{8} + \cdots + 253609 $ x^16 - 26x^14 - 16x^13 + 269x^12 + 288x^11 + 850x^10 + 2032x^9 + 6628x^8 + 7312x^7 + 53790x^6 + 18944x^5 + 201613x^4 + 27344x^3 + 355010x^2 + 13984x + 253609
Coefficient ring:	$\Z[a_1, \ldots, a_{47}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			845.15
Root			$1.69301 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	1692.845
Dual form			1692.2.h.a.845.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.93643 q^{5} +1.10235 q^{7} +O(q^{10})$	$q+3.93643 q^{5} +1.10235 q^{7} -2.28367 q^{11} -1.45355i q^{13} -5.57708i q^{17} -0.359263i q^{19} +2.79175 q^{23} +10.4955 q^{25} +5.48399 q^{29} -6.66125i q^{31} +4.33931 q^{35} +1.60831 q^{37} +3.20032 q^{41} -5.04242i q^{43} +(-5.99207 + 3.33094i) q^{47} -5.78483 q^{49} +9.41052i q^{53} -8.98953 q^{55} +8.55024i q^{59} +4.71065 q^{61} -5.72182i q^{65} -5.05047i q^{67} +14.8429i q^{71} -0.412535i q^{73} -2.51740 q^{77} +0.897653 q^{79} +8.40550i q^{83} -21.9538i q^{85} -8.18099i q^{89} -1.60232i q^{91} -1.41421i q^{95} +11.7002 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{25} + 8 q^{37} + 24 q^{49} + 8 q^{55} + 40 q^{61} + 32 q^{79}+O(q^{100}) $ 16 * q + 16 * q^25 + 8 * q^37 + 24 * q^49 + 8 * q^55 + 40 * q^61 + 32 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1692\mathbb{Z}\right)^\times$.

$n$	$757$	$847$	$1505$
$\chi(n)$	$-1$	$1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	3.93643			1.76043			0.880213	−	0.474580i	$-0.157400\pi$
$5$	3.93643			1.76043			0.880213	+	0.474580i	$0.157400\pi$
$6$		0			0
$7$	1.10235			0.416648			0.208324	−	0.978060i	$-0.433199\pi$
$7$	1.10235			0.416648			0.208324	+	0.978060i	$0.433199\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.28367			−0.688554			−0.344277	−	0.938868i	$-0.611876\pi$
$11$	−2.28367			−0.688554			−0.344277	+	0.938868i	$0.611876\pi$
$12$		0			0
$13$		−	1.45355i		−	0.403144i	−0.979474	−	0.201572i	$-0.935395\pi$
$13$		−	1.45355i		−	0.403144i	0.979474	−	0.201572i	$-0.0646049\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.57708i		−	1.35264i	−0.736608	−	0.676320i	$-0.763574\pi$
$17$		−	5.57708i		−	1.35264i	0.736608	−	0.676320i	$-0.236426\pi$
$18$		0			0
$19$		−	0.359263i		−	0.0824206i	−0.999150	−	0.0412103i	$-0.986879\pi$
$19$		−	0.359263i		−	0.0824206i	0.999150	−	0.0412103i	$-0.0131214\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.79175			0.582120			0.291060	−	0.956705i	$-0.405992\pi$
$23$	2.79175			0.582120			0.291060	+	0.956705i	$0.405992\pi$
$24$		0			0
$25$	10.4955			2.09910
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.48399			1.01835			0.509176	−	0.860662i	$-0.329950\pi$
$29$	5.48399			1.01835			0.509176	+	0.860662i	$0.329950\pi$
$30$		0			0
$31$		−	6.66125i		−	1.19639i	−0.801349	−	0.598197i	$-0.795884\pi$
$31$		−	6.66125i		−	1.19639i	0.801349	−	0.598197i	$-0.204116\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.33931			0.733477
$36$		0			0
$37$	1.60831			0.264404			0.132202	−	0.991223i	$-0.457795\pi$
$37$	1.60831			0.264404			0.132202	+	0.991223i	$0.457795\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.20032			0.499806			0.249903	−	0.968271i	$-0.419601\pi$
$41$	3.20032			0.499806			0.249903	+	0.968271i	$0.419601\pi$
$42$		0			0
$43$		−	5.04242i		−	0.768962i	−0.923133	−	0.384481i	$-0.874380\pi$
$43$		−	5.04242i		−	0.768962i	0.923133	−	0.384481i	$-0.125620\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.99207	+	3.33094i	−0.874033	+	0.485867i
$47$	−5.99207	+	3.33094i	−0.874033	+	0.485867i
$48$		0			0
$49$	−5.78483			−0.826405
$50$		0			0
$51$		0			0
$52$		0			0
$53$			9.41052i			1.29263i	0.763069	+	0.646317i	$0.223692\pi$
$53$			9.41052i			1.29263i	−0.763069	+	0.646317i	$0.776308\pi$
$54$		0			0
$55$	−8.98953			−1.21215
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.55024i			1.11315i	0.830798	+	0.556574i	$0.187884\pi$
$59$			8.55024i			1.11315i	−0.830798	+	0.556574i	$0.812116\pi$
$60$		0			0
$61$	4.71065			0.603137			0.301569	−	0.953444i	$-0.402490\pi$
$61$	4.71065			0.603137			0.301569	+	0.953444i	$0.402490\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	5.72182i		−	0.709704i
$66$		0			0
$67$		−	5.05047i		−	0.617013i	−0.951222	−	0.308506i	$-0.900171\pi$
$67$		−	5.05047i		−	0.617013i	0.951222	−	0.308506i	$-0.0998291\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			14.8429i			1.76152i	0.473560	+	0.880762i	$0.342969\pi$
$71$			14.8429i			1.76152i	−0.473560	+	0.880762i	$0.657031\pi$
$72$		0			0
$73$		−	0.412535i		−	0.0482836i	−0.999709	−	0.0241418i	$-0.992315\pi$
$73$		−	0.412535i		−	0.0482836i	0.999709	−	0.0241418i	$-0.00768532\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.51740			−0.286884
$78$		0			0
$79$	0.897653			0.100994			0.0504969	−	0.998724i	$-0.483919\pi$
$79$	0.897653			0.100994			0.0504969	+	0.998724i	$0.483919\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			8.40550i			0.922624i	0.887238	+	0.461312i	$0.152621\pi$
$83$			8.40550i			0.922624i	−0.887238	+	0.461312i	$0.847379\pi$
$84$		0			0
$85$		−	21.9538i		−	2.38122i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	8.18099i		−	0.867183i	−0.901110	−	0.433591i	$-0.857246\pi$
$89$		−	8.18099i		−	0.867183i	0.901110	−	0.433591i	$-0.142754\pi$
$90$		0			0
$91$		−	1.60232i		−	0.167969i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	1.41421i		−	0.145095i
$96$		0			0
$97$	11.7002			1.18797			0.593987	−	0.804475i	$-0.297553\pi$
$97$	11.7002			1.18797			0.593987	+	0.804475i	$0.297553\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			13.2839i			1.32180i	0.750475	+	0.660899i	$0.229825\pi$
$101$			13.2839i			1.32180i	−0.750475	+	0.660899i	$0.770175\pi$
$102$		0			0
$103$	−8.78483			−0.865595			−0.432798	−	0.901491i	$-0.642474\pi$
$103$	−8.78483			−0.865595			−0.432798	+	0.901491i	$0.642474\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.31953			0.127563			0.0637816	−	0.997964i	$-0.479684\pi$
$107$	1.31953			0.127563			0.0637816	+	0.997964i	$0.479684\pi$
$108$		0			0
$109$		−	16.5356i		−	1.58382i	−0.610636	−	0.791911i	$-0.709086\pi$
$109$		−	16.5356i		−	1.58382i	0.610636	−	0.791911i	$-0.290914\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.613268			0.0576914			0.0288457	−	0.999584i	$-0.490817\pi$
$113$	0.613268			0.0576914			0.0288457	+	0.999584i	$0.490817\pi$
$114$		0			0
$115$	10.9895			1.02478
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	6.14787i		−	0.563575i
$120$		0			0
$121$	−5.78483			−0.525894
$122$		0			0
$123$		0			0
$124$		0			0
$125$	21.6326			1.93488
$126$		0			0
$127$		−	6.09994i		−	0.541282i	−0.962680	−	0.270641i	$-0.912764\pi$
$127$		−	6.09994i		−	0.541282i	0.962680	−	0.270641i	$-0.0872357\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	10.4555i		−	0.913499i	−0.889595	−	0.456750i	$-0.849014\pi$
$131$		−	10.4555i		−	0.913499i	0.889595	−	0.456750i	$-0.150986\pi$
$132$		0			0
$133$		−	0.396032i		−	0.0343404i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.7875			0.921634			0.460817	−	0.887495i	$-0.347556\pi$
$137$	10.7875			0.921634			0.460817	+	0.887495i	$0.347556\pi$
$138$		0			0
$139$			20.7188i			1.75734i	0.477427	+	0.878671i	$0.341569\pi$
$139$			20.7188i			1.75734i	−0.477427	+	0.878671i	$0.658431\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3.31945i			0.277586i
$144$		0			0
$145$	21.5874			1.79273
$146$		0			0
$147$		0			0
$148$		0			0
$149$			4.53212i			0.371286i	0.982617	+	0.185643i	$0.0594368\pi$
$149$			4.53212i			0.371286i	−0.982617	+	0.185643i	$0.940563\pi$
$150$		0			0
$151$			17.3826i			1.41458i	0.706925	+	0.707289i	$0.250082\pi$
$151$			17.3826i			1.41458i	−0.706925	+	0.707289i	$0.749918\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	26.2215i		−	2.10616i
$156$		0			0
$157$	−14.8767			−1.18729			−0.593645	−	0.804727i	$-0.702312\pi$
$157$	−14.8767			−1.18729			−0.593645	+	0.804727i	$0.702312\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.07748			0.242539
$162$		0			0
$163$		−	4.63793i		−	0.363271i	−0.983366	−	0.181635i	$-0.941861\pi$
$163$		−	4.63793i		−	0.363271i	0.983366	−	0.181635i	$-0.0581391\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−18.3037			−1.41639			−0.708193	−	0.706019i	$-0.750489\pi$
$167$	−18.3037			−1.41639			−0.708193	+	0.706019i	$0.750489\pi$
$168$		0			0
$169$	10.8872			0.837475
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	6.55707i		−	0.498525i	−0.968436	−	0.249262i	$-0.919812\pi$
$173$		−	6.55707i		−	0.498525i	0.968436	−	0.249262i	$-0.0801882\pi$
$174$		0			0
$175$	11.5697			0.874584
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.2320			1.13849			0.569245	−	0.822168i	$-0.307236\pi$
$179$	15.2320			1.13849			0.569245	+	0.822168i	$0.307236\pi$
$180$		0			0
$181$		−	0.412535i		−	0.0306635i	−0.999882	−	0.0153318i	$-0.995120\pi$
$181$		−	0.412535i		−	0.0306635i	0.999882	−	0.0153318i	$-0.00488044\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.33099			0.465463
$186$		0			0
$187$			12.7362i			0.931365i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	8.28579i		−	0.599538i	−0.954012	−	0.299769i	$-0.903090\pi$
$191$		−	8.28579i		−	0.599538i	0.954012	−	0.299769i	$-0.0969097\pi$
$192$		0			0
$193$		−	10.9686i		−	0.789540i	−0.918780	−	0.394770i	$-0.870824\pi$
$193$		−	10.9686i		−	0.789540i	0.918780	−	0.394770i	$-0.129176\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			13.2189i			0.941811i	0.882184	+	0.470905i	$0.156073\pi$
$197$			13.2189i			0.941811i	−0.882184	+	0.470905i	$0.843927\pi$
$198$		0			0
$199$			11.8977i			0.843403i	0.906735	+	0.421701i	$0.138567\pi$
$199$			11.8977i			0.843403i	−0.906735	+	0.421701i	$0.861433\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.04526			0.424294
$204$		0			0
$205$	12.5978			0.879871
$206$		0			0
$207$		0			0
$208$		0			0
$209$			0.820440i			0.0567510i
$210$		0			0
$211$		−	19.8937i		−	1.36954i	−0.728760	−	0.684769i	$-0.759903\pi$
$211$		−	19.8937i		−	1.36954i	0.728760	−	0.684769i	$-0.240097\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	19.8491i		−	1.35370i
$216$		0			0
$217$		−	7.34300i		−	0.498475i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.10659			−0.545308
$222$		0			0
$223$			6.33029i			0.423908i	0.977280	+	0.211954i	$0.0679827\pi$
$223$			6.33029i			0.423908i	−0.977280	+	0.211954i	$0.932017\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−23.6882			−1.57224			−0.786122	−	0.618072i	$-0.787914\pi$
$227$	−23.6882			−1.57224			−0.786122	+	0.618072i	$0.787914\pi$
$228$		0			0
$229$			8.78851i			0.580761i	0.956911	+	0.290381i	$0.0937819\pi$
$229$			8.78851i			0.580761i	−0.956911	+	0.290381i	$0.906218\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.15046			−0.402930			−0.201465	−	0.979496i	$-0.564570\pi$
$233$	−6.15046			−0.402930			−0.201465	+	0.979496i	$0.564570\pi$
$234$		0			0
$235$	−23.5874	+	13.1120i	−1.53867	+	0.855332i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	14.1672i		−	0.916397i	−0.888850	−	0.458199i	$-0.848495\pi$
$239$		−	14.1672i		−	0.916397i	0.888850	−	0.458199i	$-0.151505\pi$
$240$		0			0
$241$	−18.2980			−1.17868			−0.589339	−	0.807886i	$-0.700612\pi$
$241$	−18.2980			−1.17868			−0.589339	+	0.807886i	$0.700612\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−22.7716			−1.45482
$246$		0			0
$247$	−0.522208			−0.0332273
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	6.55707i		−	0.413879i	−0.978354	−	0.206939i	$-0.933650\pi$
$251$		−	6.55707i		−	0.413879i	0.978354	−	0.206939i	$-0.0663503\pi$
$252$		0			0
$253$	−6.37544			−0.400821
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.3407			0.769792			0.384896	−	0.922960i	$-0.374237\pi$
$257$	12.3407			0.769792			0.384896	+	0.922960i	$0.374237\pi$
$258$		0			0
$259$	1.77291			0.110163
$260$		0			0
$261$		0			0
$262$		0			0
$263$			24.6478i			1.51985i	0.650013	+	0.759923i	$0.274763\pi$
$263$			24.6478i			1.51985i	−0.650013	+	0.759923i	$0.725237\pi$
$264$		0			0
$265$			37.0439i			2.27559i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			28.8484i			1.75892i	0.475974	+	0.879460i	$0.342096\pi$
$269$			28.8484i			1.75892i	−0.475974	+	0.879460i	$0.657904\pi$
$270$		0			0
$271$	0.989525			0.0601094			0.0300547	−	0.999548i	$-0.490432\pi$
$271$	0.989525			0.0601094			0.0300547	+	0.999548i	$0.490432\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−23.9683			−1.44534
$276$		0			0
$277$	−21.4731			−1.29019			−0.645096	−	0.764101i	$-0.723183\pi$
$277$	−21.4731			−1.29019			−0.645096	+	0.764101i	$0.723183\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.88099			−0.589451			−0.294725	−	0.955582i	$-0.595228\pi$
$281$	−9.88099			−0.589451			−0.294725	+	0.955582i	$0.595228\pi$
$282$		0			0
$283$	6.78483			0.403316			0.201658	−	0.979456i	$-0.435367\pi$
$283$	6.78483			0.403316			0.201658	+	0.979456i	$0.435367\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.52786			0.208243
$288$		0			0
$289$	−14.1038			−0.829635
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.66208			−0.213941			−0.106971	−	0.994262i	$-0.534115\pi$
$293$	−3.66208			−0.213941			−0.106971	+	0.994262i	$0.534115\pi$
$294$		0			0
$295$			33.6574i			1.95961i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	4.05796i		−	0.234678i
$300$		0			0
$301$		−	5.55850i		−	0.320386i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	18.5432			1.06178
$306$		0			0
$307$	−10.9330			−0.623982			−0.311991	−	0.950085i	$-0.600996\pi$
$307$	−10.9330			−0.623982			−0.311991	+	0.950085i	$0.600996\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.28731			0.243111			0.121556	−	0.992585i	$-0.461212\pi$
$311$	4.28731			0.243111			0.121556	+	0.992585i	$0.461212\pi$
$312$		0			0
$313$		−	29.7680i		−	1.68259i	−0.540577	−	0.841295i	$-0.681794\pi$
$313$		−	29.7680i		−	1.68259i	0.540577	−	0.841295i	$-0.318206\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.5116			−0.646556			−0.323278	−	0.946304i	$-0.604785\pi$
$317$	−11.5116			−0.646556			−0.323278	+	0.946304i	$0.604785\pi$
$318$		0			0
$319$	−12.5237			−0.701190
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.00364			−0.111485
$324$		0			0
$325$		−	15.2558i		−	0.846237i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.60534	+	3.67185i	−0.364164	+	0.202435i
$330$		0			0
$331$	−4.69441			−0.258028			−0.129014	−	0.991643i	$-0.541181\pi$
$331$	−4.69441			−0.258028			−0.129014	+	0.991643i	$0.541181\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	19.8808i		−	1.08620i
$336$		0			0
$337$	2.58014			0.140549			0.0702745	−	0.997528i	$-0.477612\pi$
$337$	2.58014			0.140549			0.0702745	+	0.997528i	$0.477612\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			15.2121i			0.823782i
$342$		0			0
$343$	−14.0933			−0.760968
$344$		0			0
$345$		0			0
$346$		0			0
$347$			14.7381i			0.791180i	0.918427	+	0.395590i	$0.129460\pi$
$347$			14.7381i			0.791180i	−0.918427	+	0.395590i	$0.870540\pi$
$348$		0			0
$349$		−	3.52714i		−	0.188803i	−0.995534	−	0.0944016i	$-0.969906\pi$
$349$		−	3.52714i		−	0.188803i	0.995534	−	0.0944016i	$-0.0300938\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	0.306338i		−	0.0163047i	−0.999967	−	0.00815236i	$-0.997405\pi$
$353$		−	0.306338i		−	0.0163047i	0.999967	−	0.00815236i	$-0.00259500\pi$
$354$		0			0
$355$			58.4279i			3.10103i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−25.6165			−1.35199			−0.675994	−	0.736907i	$-0.736286\pi$
$359$	−25.6165			−1.35199			−0.675994	+	0.736907i	$0.736286\pi$
$360$		0			0
$361$	18.8709			0.993207
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	1.62392i		−	0.0849997i
$366$		0			0
$367$		−	4.22540i		−	0.220564i	−0.993900	−	0.110282i	$-0.964825\pi$
$367$		−	4.22540i		−	0.220564i	0.993900	−	0.110282i	$-0.0351754\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			10.3737i			0.538573i
$372$		0			0
$373$		−	25.9015i		−	1.34113i	−0.741851	−	0.670564i	$-0.766052\pi$
$373$		−	25.9015i		−	1.34113i	0.741851	−	0.670564i	$-0.233948\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	7.97128i		−	0.410542i
$378$		0			0
$379$	−14.1143			−0.725001			−0.362501	−	0.931984i	$-0.618077\pi$
$379$	−14.1143			−0.725001			−0.362501	+	0.931984i	$0.618077\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	24.6876i		−	1.26148i	−0.775995	−	0.630739i	$-0.782752\pi$
$383$		−	24.6876i		−	1.26148i	0.775995	−	0.630739i	$-0.217248\pi$
$384$		0			0
$385$	−9.90957			−0.505039
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−29.3306			−1.48712			−0.743561	−	0.668669i	$-0.766865\pi$
$389$	−29.3306			−1.48712			−0.743561	+	0.668669i	$0.766865\pi$
$390$		0			0
$391$		−	15.5698i		−	0.787399i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.53355			0.177792
$396$		0			0
$397$	14.6141			0.733460			0.366730	−	0.930327i	$-0.380477\pi$
$397$	14.6141			0.733460			0.366730	+	0.930327i	$0.380477\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			16.4668i			0.822311i	0.911565	+	0.411156i	$0.134875\pi$
$401$			16.4668i			0.822311i	−0.911565	+	0.411156i	$0.865125\pi$
$402$		0			0
$403$	−9.68249			−0.482319
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.67285			−0.182056
$408$		0			0
$409$			33.4837i			1.65566i	0.560977	+	0.827832i	$0.310426\pi$
$409$			33.4837i			1.65566i	−0.560977	+	0.827832i	$0.689574\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			9.42533i			0.463790i
$414$		0			0
$415$			33.0877i			1.62421i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−28.8157			−1.40774			−0.703868	−	0.710330i	$-0.748546\pi$
$419$	−28.8157			−1.40774			−0.703868	+	0.710330i	$0.748546\pi$
$420$		0			0
$421$			16.9903i			0.828059i	0.910263	+	0.414029i	$0.135879\pi$
$421$			16.9903i			0.828059i	−0.910263	+	0.414029i	$0.864121\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	58.5341i		−	2.83932i
$426$		0			0
$427$	5.19277			0.251296
$428$		0			0
$429$		0			0
$430$		0			0
$431$			4.63692i			0.223353i	0.993745	+	0.111676i	$0.0356220\pi$
$431$			4.63692i			0.223353i	−0.993745	+	0.111676i	$0.964378\pi$
$432$		0			0
$433$			31.7914i			1.52779i	0.645338	+	0.763897i	$0.276717\pi$
$433$			31.7914i			1.52779i	−0.645338	+	0.763897i	$0.723283\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.00297i		−	0.0479787i
$438$		0			0
$439$	24.1498			1.15261			0.576304	−	0.817236i	$-0.304495\pi$
$439$	24.1498			1.15261			0.576304	+	0.817236i	$0.304495\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.55903			0.311629			0.155814	−	0.987786i	$-0.450200\pi$
$443$	6.55903			0.311629			0.155814	+	0.987786i	$0.450200\pi$
$444$		0			0
$445$		−	32.2039i		−	1.52661i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−23.5944			−1.11349			−0.556745	−	0.830684i	$-0.687950\pi$
$449$	−23.5944			−1.11349			−0.556745	+	0.830684i	$0.687950\pi$
$450$		0			0
$451$	−7.30849			−0.344143
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	6.30743i		−	0.295697i
$456$		0			0
$457$	−19.0058			−0.889053			−0.444526	−	0.895766i	$-0.646628\pi$
$457$	−19.0058			−0.889053			−0.444526	+	0.895766i	$0.646628\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.5152			0.629468			0.314734	−	0.949180i	$-0.398085\pi$
$461$	13.5152			0.629468			0.314734	+	0.949180i	$0.398085\pi$
$462$		0			0
$463$			32.1667i			1.49491i	0.664310	+	0.747457i	$0.268725\pi$
$463$			32.1667i			1.49491i	−0.664310	+	0.747457i	$0.731275\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.1963			1.11967			0.559836	−	0.828604i	$-0.310864\pi$
$467$	24.1963			1.11967			0.559836	+	0.828604i	$0.310864\pi$
$468$		0			0
$469$		−	5.56737i		−	0.257077i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			11.5152i			0.529472i
$474$		0			0
$475$		−	3.77064i		−	0.173009i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	18.8860i		−	0.862924i	−0.902131	−	0.431462i	$-0.857998\pi$
$479$		−	18.8860i		−	0.862924i	0.902131	−	0.431462i	$-0.142002\pi$
$480$		0			0
$481$		−	2.33776i		−	0.106593i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	46.0569			2.09134
$486$		0			0
$487$	−32.5606			−1.47546			−0.737732	−	0.675094i	$-0.764103\pi$
$487$	−32.5606			−1.47546			−0.737732	+	0.675094i	$0.764103\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	27.2245i		−	1.22862i	−0.789063	−	0.614312i	$-0.789434\pi$
$491$		−	27.2245i		−	1.22862i	0.789063	−	0.614312i	$-0.210566\pi$
$492$		0			0
$493$		−	30.5847i		−	1.37746i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			16.3620i			0.733935i
$498$		0			0
$499$		−	19.4812i		−	0.872096i	−0.899923	−	0.436048i	$-0.856378\pi$
$499$		−	19.4812i		−	0.872096i	0.899923	−	0.436048i	$-0.143622\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	43.2828			1.92989			0.964943	−	0.262459i	$-0.0845333\pi$
$503$	43.2828			1.92989			0.964943	+	0.262459i	$0.0845333\pi$
$504$		0			0
$505$			52.2912i			2.32693i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.0882			0.580124			0.290062	−	0.957008i	$-0.406324\pi$
$509$	13.0882			0.580124			0.290062	+	0.957008i	$0.406324\pi$
$510$		0			0
$511$		−	0.454757i		−	0.0201173i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−34.5809			−1.52382
$516$		0			0
$517$	13.6839	−	7.60677i	0.601819	−	0.334545i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			7.33757i			0.321465i	0.986998	+	0.160732i	$0.0513856\pi$
$521$			7.33757i			0.321465i	−0.986998	+	0.160732i	$0.948614\pi$
$522$		0			0
$523$	44.6749			1.95350			0.976749	−	0.214388i	$-0.0687758\pi$
$523$	44.6749			1.95350			0.976749	+	0.214388i	$0.0687758\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−37.1503			−1.61829
$528$		0			0
$529$	−15.2061			−0.661136
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	4.65184i		−	0.201493i
$534$		0			0
$535$	5.19422			0.224566
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13.2107			0.569024
$540$		0			0
$541$	15.7982			0.679218			0.339609	−	0.940567i	$-0.389705\pi$
$541$	15.7982			0.679218			0.339609	+	0.940567i	$0.389705\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	65.0912i		−	2.78820i
$546$		0			0
$547$			32.8691i			1.40538i	0.711495	+	0.702692i	$0.248019\pi$
$547$			32.8691i			1.40538i	−0.711495	+	0.702692i	$0.751981\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.97020i		−	0.0839331i
$552$		0			0
$553$	0.989525			0.0420789
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.9366			0.505771			0.252886	−	0.967496i	$-0.418620\pi$
$557$	11.9366			0.505771			0.252886	+	0.967496i	$0.418620\pi$
$558$		0			0
$559$	−7.32944			−0.310002
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−41.3446			−1.74247			−0.871234	−	0.490869i	$-0.836680\pi$
$563$	−41.3446			−1.74247			−0.871234	+	0.490869i	$0.836680\pi$
$564$		0			0
$565$	2.41409			0.101561
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.36154			−0.392456			−0.196228	−	0.980558i	$-0.562869\pi$
$569$	−9.36154			−0.392456			−0.196228	+	0.980558i	$0.562869\pi$
$570$		0			0
$571$	32.7088			1.36882			0.684411	−	0.729096i	$-0.260059\pi$
$571$	32.7088			1.36882			0.684411	+	0.729096i	$0.260059\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	29.3008			1.22193
$576$		0			0
$577$			41.8923i			1.74400i	0.489507	+	0.871999i	$0.337177\pi$
$577$			41.8923i			1.74400i	−0.489507	+	0.871999i	$0.662823\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			9.26578i			0.384409i
$582$		0			0
$583$		−	21.4906i		−	0.890048i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	39.3763			1.62523			0.812616	−	0.582799i	$-0.198043\pi$
$587$	39.3763			1.62523			0.812616	+	0.582799i	$0.198043\pi$
$588$		0			0
$589$	−2.39314			−0.0986076
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.6136			0.559042			0.279521	−	0.960140i	$-0.409824\pi$
$593$	13.6136			0.559042			0.279521	+	0.960140i	$0.409824\pi$
$594$		0			0
$595$		−	24.2007i		−	0.992131i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−30.0249			−1.22678			−0.613392	−	0.789778i	$-0.710196\pi$
$599$	−30.0249			−1.22678			−0.613392	+	0.789778i	$0.710196\pi$
$600$		0			0
$601$	−37.0651			−1.51192			−0.755959	−	0.654618i	$-0.772829\pi$
$601$	−37.0651			−1.51192			−0.755959	+	0.654618i	$0.772829\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.7716			−0.925797
$606$		0			0
$607$			19.8937i			0.807460i	0.914878	+	0.403730i	$0.132287\pi$
$607$			19.8937i			0.807460i	−0.914878	+	0.403730i	$0.867713\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.84170	+	8.70980i	0.195874	+	0.352361i
$612$		0			0
$613$	41.5146			1.67676			0.838380	−	0.545086i	$-0.183503\pi$
$613$	41.5146			1.67676			0.838380	+	0.545086i	$0.183503\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	10.3655i		−	0.417299i	−0.977990	−	0.208649i	$-0.933093\pi$
$617$		−	10.3655i		−	0.417299i	0.977990	−	0.208649i	$-0.0669068\pi$
$618$		0			0
$619$	−28.9121			−1.16208			−0.581038	−	0.813877i	$-0.697353\pi$
$619$	−28.9121			−1.16208			−0.581038	+	0.813877i	$0.697353\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	9.01828i		−	0.361310i
$624$		0			0
$625$	32.6778			1.30711
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	8.96965i		−	0.357643i
$630$		0			0
$631$		−	35.0493i		−	1.39529i	−0.716444	−	0.697645i	$-0.754231\pi$
$631$		−	35.0493i		−	1.39529i	0.716444	−	0.697645i	$-0.245769\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	24.0120i		−	0.952887i
$636$		0			0
$637$			8.40857i			0.333160i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−28.0397			−1.10750			−0.553752	−	0.832682i	$-0.686804\pi$
$641$	−28.0397			−1.10750			−0.553752	+	0.832682i	$0.686804\pi$
$642$		0			0
$643$	−34.8572			−1.37463			−0.687317	−	0.726358i	$-0.741212\pi$
$643$	−34.8572			−1.37463			−0.687317	+	0.726358i	$0.741212\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			32.2625i			1.26837i	0.773182	+	0.634184i	$0.218664\pi$
$647$			32.2625i			1.26837i	−0.773182	+	0.634184i	$0.781336\pi$
$648$		0			0
$649$		−	19.5260i		−	0.766462i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			35.5103i			1.38962i	0.719191	+	0.694812i	$0.244513\pi$
$653$			35.5103i			1.38962i	−0.719191	+	0.694812i	$0.755487\pi$
$654$		0			0
$655$		−	41.1573i		−	1.60815i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			8.28589i			0.322773i	0.986891	+	0.161386i	$0.0515965\pi$
$659$			8.28589i			0.322773i	−0.986891	+	0.161386i	$0.948404\pi$
$660$		0			0
$661$	−25.5534			−0.993913			−0.496956	−	0.867776i	$-0.665549\pi$
$661$	−25.5534			−0.993913			−0.496956	+	0.867776i	$0.665549\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1.55895i		−	0.0604536i
$666$		0			0
$667$	15.3099			0.592803
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.7576			−0.415292
$672$		0			0
$673$		−	35.5983i		−	1.37222i	−0.727500	−	0.686108i	$-0.759318\pi$
$673$		−	35.5983i		−	1.37222i	0.727500	−	0.686108i	$-0.240682\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.5133			0.711524			0.355762	−	0.934577i	$-0.384221\pi$
$677$	18.5133			0.711524			0.355762	+	0.934577i	$0.384221\pi$
$678$		0			0
$679$	12.8977			0.494967
$680$		0			0
$681$		0			0
$682$		0			0
$683$			30.3045i			1.15957i	0.814770	+	0.579785i	$0.196863\pi$
$683$			30.3045i			1.15957i	−0.814770	+	0.579785i	$0.803137\pi$
$684$		0			0
$685$	42.4641			1.62247
$686$		0			0
$687$		0			0
$688$		0			0
$689$	13.6787			0.521117
$690$		0			0
$691$		−	16.0030i		−	0.608783i	−0.952547	−	0.304392i	$-0.901547\pi$
$691$		−	16.0030i		−	0.608783i	0.952547	−	0.304392i	$-0.0984532\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			81.5580i			3.09367i
$696$		0			0
$697$		−	17.8484i		−	0.676057i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	31.7183			1.19798			0.598992	−	0.800755i	$-0.295568\pi$
$701$	31.7183			1.19798			0.598992	+	0.800755i	$0.295568\pi$
$702$		0			0
$703$		−	0.577805i		−	0.0217923i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			14.6435i			0.550724i
$708$		0			0
$709$	−48.7267			−1.82997			−0.914985	−	0.403489i	$-0.867797\pi$
$709$	−48.7267			−1.82997			−0.914985	+	0.403489i	$0.867797\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	18.5965i		−	0.696445i
$714$		0			0
$715$			13.0668i			0.488669i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			10.7470i			0.400796i	0.979715	+	0.200398i	$0.0642234\pi$
$719$			10.7470i			0.400796i	−0.979715	+	0.200398i	$0.935777\pi$
$720$		0			0
$721$	−9.68393			−0.360648
$722$		0			0
$723$		0			0
$724$		0			0
$725$	57.5572			2.13762
$726$		0			0
$727$		−	30.6721i		−	1.13757i	−0.822488	−	0.568783i	$-0.807415\pi$
$727$		−	30.6721i		−	1.13757i	0.822488	−	0.568783i	$-0.192585\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−28.1220			−1.04013
$732$		0			0
$733$	37.6274			1.38980			0.694901	−	0.719106i	$-0.255448\pi$
$733$	37.6274			1.38980			0.694901	+	0.719106i	$0.255448\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			11.5336i			0.424846i
$738$		0			0
$739$	−47.3455			−1.74163			−0.870816	−	0.491610i	$-0.836409\pi$
$739$	−47.3455			−1.74163			−0.870816	+	0.491610i	$0.836409\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7.72585			0.283434			0.141717	−	0.989907i	$-0.454738\pi$
$743$	7.72585			0.283434			0.141717	+	0.989907i	$0.454738\pi$
$744$		0			0
$745$			17.8404i			0.653621i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.45457			0.0531490
$750$		0			0
$751$			24.1478i			0.881166i	0.897712	+	0.440583i	$0.145228\pi$
$751$			24.1478i			0.881166i	−0.897712	+	0.440583i	$0.854772\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			68.4255i			2.49026i
$756$		0			0
$757$		−	23.7949i		−	0.864841i	−0.901672	−	0.432420i	$-0.857660\pi$
$757$		−	23.7949i		−	0.864841i	0.901672	−	0.432420i	$-0.142340\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			27.9150i			1.01192i	0.862557	+	0.505959i	$0.168861\pi$
$761$			27.9150i			1.01192i	−0.862557	+	0.505959i	$0.831139\pi$
$762$		0			0
$763$		−	18.2280i		−	0.659896i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.4282			0.448758
$768$		0			0
$769$	14.7462			0.531761			0.265881	−	0.964006i	$-0.414337\pi$
$769$	14.7462			0.531761			0.265881	+	0.964006i	$0.414337\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	20.5626i		−	0.739586i	−0.929114	−	0.369793i	$-0.879429\pi$
$773$		−	20.5626i		−	0.739586i	0.929114	−	0.369793i	$-0.120571\pi$
$774$		0			0
$775$		−	69.9130i		−	2.51135i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1.14976i		−	0.0411943i
$780$		0			0
$781$		−	33.8963i		−	1.21290i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−58.5611			−2.09014
$786$		0			0
$787$		−	4.18317i		−	0.149114i	−0.997217	−	0.0745570i	$-0.976246\pi$
$787$		−	4.18317i		−	0.149114i	0.997217	−	0.0745570i	$-0.0237543\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.676034			0.0240370
$792$		0			0
$793$		−	6.84719i		−	0.243151i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.96759			−0.246805			−0.123402	−	0.992357i	$-0.539381\pi$
$797$	−6.96759			−0.246805			−0.123402	+	0.992357i	$0.539381\pi$
$798$		0			0
$799$	18.5769	+	33.4182i	0.657203	+	1.18225i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			0.942096i			0.0332459i
$804$		0			0
$805$	12.1143			0.426972
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.03624			0.106749			0.0533743	−	0.998575i	$-0.483002\pi$
$809$	3.03624			0.106749			0.0533743	+	0.998575i	$0.483002\pi$
$810$		0			0
$811$	10.9005			0.382770			0.191385	−	0.981515i	$-0.438702\pi$
$811$	10.9005			0.382770			0.191385	+	0.981515i	$0.438702\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	18.2569i		−	0.639511i
$816$		0			0
$817$	−1.81155			−0.0633783
$818$		0			0
$819$		0			0
$820$		0			0
$821$	44.3465			1.54770			0.773852	−	0.633367i	$-0.218328\pi$
$821$	44.3465			1.54770			0.773852	+	0.633367i	$0.218328\pi$
$822$		0			0
$823$	−28.7075			−1.00068			−0.500341	−	0.865828i	$-0.666792\pi$
$823$	−28.7075			−1.00068			−0.500341	+	0.865828i	$0.666792\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	42.9277i		−	1.49274i	−0.665529	−	0.746372i	$-0.731794\pi$
$827$		−	42.9277i		−	1.49274i	0.665529	−	0.746372i	$-0.268206\pi$
$828$		0			0
$829$			47.1955i			1.63917i	0.572960	+	0.819583i	$0.305795\pi$
$829$			47.1955i			1.63917i	−0.572960	+	0.819583i	$0.694205\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			32.2625i			1.11783i
$834$		0			0
$835$	−72.0514			−2.49344
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−27.5132			−0.949861			−0.474931	−	0.880023i	$-0.657527\pi$
$839$	−27.5132			−0.949861			−0.474931	+	0.880023i	$0.657527\pi$
$840$		0			0
$841$	1.07418			0.0370406
$842$		0			0
$843$		0			0
$844$		0			0
$845$	42.8566			1.47431
$846$		0			0
$847$	−6.37689			−0.219113
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.48999			0.153915
$852$		0			0
$853$	35.6051			1.21909			0.609547	−	0.792750i	$-0.291351\pi$
$853$	35.6051			1.21909			0.609547	+	0.792750i	$0.291351\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.8811			−0.508328			−0.254164	−	0.967161i	$-0.581800\pi$
$857$	−14.8811			−0.508328			−0.254164	+	0.967161i	$0.581800\pi$
$858$		0			0
$859$		−	22.5849i		−	0.770585i	−0.922794	−	0.385293i	$-0.874101\pi$
$859$		−	22.5849i		−	0.770585i	0.922794	−	0.385293i	$-0.125899\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	3.28573i		−	0.111848i	−0.998435	−	0.0559238i	$-0.982190\pi$
$863$		−	3.28573i		−	0.111848i	0.998435	−	0.0559238i	$-0.0178104\pi$
$864$		0			0
$865$		−	25.8115i		−	0.877616i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.04995			−0.0695397
$870$		0			0
$871$	−7.34113			−0.248745
$872$		0			0
$873$		0			0
$874$		0			0
$875$	23.8466			0.806163
$876$		0			0
$877$		−	50.0193i		−	1.68903i	−0.535530	−	0.844516i	$-0.679888\pi$
$877$		−	50.0193i		−	1.68903i	0.535530	−	0.844516i	$-0.320112\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−44.8489			−1.51100			−0.755499	−	0.655150i	$-0.772605\pi$
$881$	−44.8489			−1.51100			−0.755499	+	0.655150i	$0.772605\pi$
$882$		0			0
$883$	27.3679			0.921002			0.460501	−	0.887659i	$-0.347670\pi$
$883$	27.3679			0.921002			0.460501	+	0.887659i	$0.347670\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	46.1863			1.55078			0.775392	−	0.631480i	$-0.217552\pi$
$887$	46.1863			1.55078			0.775392	+	0.631480i	$0.217552\pi$
$888$		0			0
$889$		−	6.72425i		−	0.225524i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.19668	+	2.15273i	0.0400454	+	0.0720383i
$894$		0			0
$895$	59.9595			2.00423
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	36.5302i		−	1.21835i
$900$		0			0
$901$	52.4832			1.74847
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1.62392i		−	0.0539808i
$906$		0			0
$907$	25.7403			0.854692			0.427346	−	0.904088i	$-0.359449\pi$
$907$	25.7403			0.854692			0.427346	+	0.904088i	$0.359449\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	9.80490i		−	0.324851i	−0.986721	−	0.162425i	$-0.948068\pi$
$911$		−	9.80490i		−	0.324851i	0.986721	−	0.162425i	$-0.0519317\pi$
$912$		0			0
$913$		−	19.1954i		−	0.635276i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	11.5256i		−	0.380608i
$918$		0			0
$919$			22.4533i			0.740668i	0.928899	+	0.370334i	$0.120757\pi$
$919$			22.4533i			0.740668i	−0.928899	+	0.370334i	$0.879243\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	21.5749			0.710147
$924$		0			0
$925$	16.8800			0.555010
$926$		0			0
$927$		0			0
$928$		0			0
$929$			12.0523i			0.395424i	0.980260	+	0.197712i	$0.0633511\pi$
$929$			12.0523i			0.395424i	−0.980260	+	0.197712i	$0.936649\pi$
$930$		0			0
$931$			2.07828i			0.0681127i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			50.1353i			1.63960i
$936$		0			0
$937$		−	40.6125i		−	1.32675i	−0.748287	−	0.663376i	$-0.769123\pi$
$937$		−	40.6125i		−	1.32675i	0.748287	−	0.663376i	$-0.230877\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	35.3007i		−	1.15077i	−0.817883	−	0.575385i	$-0.804852\pi$
$941$		−	35.3007i		−	1.15077i	0.817883	−	0.575385i	$-0.195148\pi$
$942$		0			0
$943$	8.93449			0.290947
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	11.3639i		−	0.369276i	−0.982807	−	0.184638i	$-0.940889\pi$
$947$		−	11.3639i		−	0.369276i	0.982807	−	0.184638i	$-0.0591112\pi$
$948$		0			0
$949$	−0.599643			−0.0194652
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.57592			−0.0834421			−0.0417211	−	0.999129i	$-0.513284\pi$
$953$	−2.57592			−0.0834421			−0.0417211	+	0.999129i	$0.513284\pi$
$954$		0			0
$955$		−	32.6164i		−	1.05544i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11.8915			0.383997
$960$		0			0
$961$	−13.3722			−0.431361
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	43.1773i		−	1.38993i
$966$		0			0
$967$	−13.8886			−0.446628			−0.223314	−	0.974747i	$-0.571688\pi$
$967$	−13.8886			−0.446628			−0.223314	+	0.974747i	$0.571688\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	59.8857			1.92182			0.960912	−	0.276852i	$-0.0892913\pi$
$971$	59.8857			1.92182			0.960912	+	0.276852i	$0.0892913\pi$
$972$		0			0
$973$			22.8393i			0.732193i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			17.3148i			0.553949i	0.960877	+	0.276975i	$0.0893318\pi$
$977$			17.3148i			0.553949i	−0.960877	+	0.276975i	$0.910668\pi$
$978$		0			0
$979$			18.6827i			0.597102i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	5.42511			0.173034			0.0865171	−	0.996250i	$-0.472426\pi$
$983$	5.42511			0.173034			0.0865171	+	0.996250i	$0.472426\pi$
$984$		0			0
$985$			52.0354i			1.65799i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	14.0772i		−	0.447628i
$990$		0			0
$991$	33.4228			1.06171			0.530854	−	0.847463i	$-0.321871\pi$
$991$	33.4228			1.06171			0.530854	+	0.847463i	$0.321871\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			46.8343i			1.48475i
$996$		0			0
$997$		−	3.51104i		−	0.111196i	−0.998453	−	0.0555980i	$-0.982293\pi$
$997$		−	3.51104i		−	0.111196i	0.998453	−	0.0555980i	$-0.0177065\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1692.2.h.a.845.15	yes	16
3.2	odd	2	inner	1692.2.h.a.845.1	✓	16
4.3	odd	2		6768.2.o.d.5921.15		16
12.11	even	2		6768.2.o.d.5921.1		16
47.46	odd	2	inner	1692.2.h.a.845.2	yes	16
141.140	even	2	inner	1692.2.h.a.845.16	yes	16
188.187	even	2		6768.2.o.d.5921.2		16
564.563	odd	2		6768.2.o.d.5921.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1692.2.h.a.845.1	✓	16	3.2	odd	2	inner
1692.2.h.a.845.2	yes	16	47.46	odd	2	inner
1692.2.h.a.845.15	yes	16	1.1	even	1	trivial
1692.2.h.a.845.16	yes	16	141.140	even	2	inner
6768.2.o.d.5921.1		16	12.11	even	2
6768.2.o.d.5921.2		16	188.187	even	2
6768.2.o.d.5921.15		16	4.3	odd	2
6768.2.o.d.5921.16		16	564.563	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:	\( N \)	\(=\)	\( 1692 = 2^{2} \cdot 3^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1692.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.93643 q^{5} +1.10235 q^{7} +O(q^{10})\)	\(q+3.93643 q^{5} +1.10235 q^{7} -2.28367 q^{11} -1.45355i q^{13} -5.57708i q^{17} -0.359263i q^{19} +2.79175 q^{23} +10.4955 q^{25} +5.48399 q^{29} -6.66125i q^{31} +4.33931 q^{35} +1.60831 q^{37} +3.20032 q^{41} -5.04242i q^{43} +(-5.99207 + 3.33094i) q^{47} -5.78483 q^{49} +9.41052i q^{53} -8.98953 q^{55} +8.55024i q^{59} +4.71065 q^{61} -5.72182i q^{65} -5.05047i q^{67} +14.8429i q^{71} -0.412535i q^{73} -2.51740 q^{77} +0.897653 q^{79} +8.40550i q^{83} -21.9538i q^{85} -8.18099i q^{89} -1.60232i q^{91} -1.41421i q^{95} +11.7002 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{25} + 8 q^{37} + 24 q^{49} + 8 q^{55} + 40 q^{61} + 32 q^{79}+O(q^{100}) \) 16 * q + 16 * q^25 + 8 * q^37 + 24 * q^49 + 8 * q^55 + 40 * q^61 + 32 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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