LMFDB - Embedded newform 2112.2.h.d.1759.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(1759,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2112.1759");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.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:	$16.8644049069$
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} - 2 x^{15} + 5 x^{14} + 20 x^{13} - 49 x^{12} + 56 x^{11} + 174 x^{10} - 572 x^{9} + 120 x^{8} + \cdots + 433 $ x^16 - 2x^15 + 5x^14 + 20x^13 - 49x^12 + 56x^11 + 174x^10 - 572x^9 + 120x^8 + 1148x^7 - 2520x^6 - 296x^5 + 4721x^4 - 1622x^3 - 1429x^2 + 788*x + 433
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1759.6
Root			$0.626493 - 0.442020i$ of defining polynomial
Character	$\chi$	$=$	2112.1759
Dual form			2112.2.h.d.1759.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.66719i q^{5} +3.05522 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.66719i q^{5} +3.05522 q^{7} +1.00000 q^{9} +(-3.27743 + 0.508401i) q^{11} -6.95805 q^{13} -1.66719i q^{15} +1.66625i q^{17} -4.44420i q^{19} +3.05522 q^{21} -7.13129i q^{23} +2.22047 q^{25} +1.00000 q^{27} +6.68078 q^{29} -3.22047i q^{31} +(-3.27743 + 0.508401i) q^{33} -5.09364i q^{35} -8.01896i q^{37} -6.95805 q^{39} -11.2330i q^{41} +7.77669i q^{43} -1.66719i q^{45} -8.22205i q^{47} +2.33438 q^{49} +1.66625i q^{51} -7.44252i q^{53} +(0.847602 + 5.46410i) q^{55} -4.44420i q^{57} -11.0421 q^{59} +11.0349 q^{61} +3.05522 q^{63} +11.6004i q^{65} -3.88924 q^{67} -7.13129i q^{69} -0.887662i q^{71} +16.6940i q^{73} +2.22047 q^{75} +(-10.0133 + 1.55328i) q^{77} +5.08883 q^{79} +1.00000 q^{81} +13.9161i q^{83} +2.77795 q^{85} +6.68078 q^{87} -4.92820 q^{89} -21.2584 q^{91} -3.22047i q^{93} -7.40933 q^{95} +12.4441 q^{97} +(-3.27743 + 0.508401i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 16 * q^9	$ 16 q + 16 q^{3} + 16 q^{9} - 8 q^{11} - 16 q^{25} + 16 q^{27} - 8 q^{33} - 80 q^{59} + 64 q^{67} - 16 q^{75} + 16 q^{81} + 32 q^{89} - 16 q^{97} - 8 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 16 * q^9 - 8 * q^11 - 16 * q^25 + 16 * q^27 - 8 * q^33 - 80 * q^59 + 64 * q^67 - 16 * q^75 + 16 * q^81 + 32 * q^89 - 16 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$		−	1.66719i		−	0.745591i	−0.927914	−	0.372796i	$-0.878399\pi$
$5$		−	1.66719i		−	0.745591i	0.927914	−	0.372796i	$-0.121601\pi$
$6$		0			0
$7$	3.05522			1.15477			0.577383	−	0.816474i	$-0.304074\pi$
$7$	3.05522			1.15477			0.577383	+	0.816474i	$0.304074\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	−3.27743	+	0.508401i	−0.988181	+	0.153289i
$11$	−3.27743	+	0.508401i	−0.988181	+	0.153289i
$12$		0			0
$13$	−6.95805			−1.92981			−0.964907	−	0.262590i	$-0.915423\pi$
$13$	−6.95805			−1.92981			−0.964907	+	0.262590i	$0.915423\pi$
$14$		0			0
$15$		−	1.66719i		−	0.430467i
$16$		0			0
$17$			1.66625i			0.404124i	0.979373	+	0.202062i	$0.0647643\pi$
$17$			1.66625i			0.404124i	−0.979373	+	0.202062i	$0.935236\pi$
$18$		0			0
$19$		−	4.44420i		−	1.01957i	−0.860302	−	0.509785i	$-0.829725\pi$
$19$		−	4.44420i		−	1.01957i	0.860302	−	0.509785i	$-0.170275\pi$
$20$		0			0
$21$	3.05522			0.666704
$22$		0			0
$23$		−	7.13129i		−	1.48698i	−0.668748	−	0.743489i	$-0.733170\pi$
$23$		−	7.13129i		−	1.48698i	0.668748	−	0.743489i	$-0.266830\pi$
$24$		0			0
$25$	2.22047			0.444094
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	6.68078			1.24059			0.620295	−	0.784369i	$-0.287013\pi$
$29$	6.68078			1.24059			0.620295	+	0.784369i	$0.287013\pi$
$30$		0			0
$31$		−	3.22047i		−	0.578413i	−0.957267	−	0.289207i	$-0.906609\pi$
$31$		−	3.22047i		−	0.578413i	0.957267	−	0.289207i	$-0.0933915\pi$
$32$		0			0
$33$	−3.27743	+	0.508401i	−0.570527	+	0.0885012i
$34$		0			0
$35$		−	5.09364i		−	0.860983i
$36$		0			0
$37$		−	8.01896i		−	1.31831i	−0.752008	−	0.659154i	$-0.770914\pi$
$37$		−	8.01896i		−	1.31831i	0.752008	−	0.659154i	$-0.229086\pi$
$38$		0			0
$39$	−6.95805			−1.11418
$40$		0			0
$41$		−	11.2330i		−	1.75431i	−0.480210	−	0.877154i	$-0.659440\pi$
$41$		−	11.2330i		−	1.75431i	0.480210	−	0.877154i	$-0.340560\pi$
$42$		0			0
$43$			7.77669i			1.18593i	0.805227	+	0.592967i	$0.202044\pi$
$43$			7.77669i			1.18593i	−0.805227	+	0.592967i	$0.797956\pi$
$44$		0			0
$45$		−	1.66719i		−	0.248530i
$46$		0			0
$47$		−	8.22205i		−	1.19931i	−0.800259	−	0.599654i	$-0.795305\pi$
$47$		−	8.22205i		−	1.19931i	0.800259	−	0.599654i	$-0.204695\pi$
$48$		0			0
$49$	2.33438			0.333484
$50$		0			0
$51$			1.66625i			0.233321i
$52$		0			0
$53$		−	7.44252i		−	1.02231i	−0.859489	−	0.511154i	$-0.829218\pi$
$53$		−	7.44252i		−	1.02231i	0.859489	−	0.511154i	$-0.170782\pi$
$54$		0			0
$55$	0.847602	+	5.46410i	0.114291	+	0.736779i
$56$		0			0
$57$		−	4.44420i		−	0.588649i
$58$		0			0
$59$	−11.0421			−1.43756			−0.718781	−	0.695237i	$-0.755299\pi$
$59$	−11.0421			−1.43756			−0.718781	+	0.695237i	$0.755299\pi$
$60$		0			0
$61$	11.0349			1.41287			0.706436	−	0.707777i	$-0.250302\pi$
$61$	11.0349			1.41287			0.706436	+	0.707777i	$0.250302\pi$
$62$		0			0
$63$	3.05522			0.384922
$64$		0			0
$65$			11.6004i			1.43885i
$66$		0			0
$67$	−3.88924			−0.475146			−0.237573	−	0.971370i	$-0.576352\pi$
$67$	−3.88924			−0.475146			−0.237573	+	0.971370i	$0.576352\pi$
$68$		0			0
$69$		−	7.13129i		−	0.858507i
$70$		0			0
$71$		−	0.887662i		−	0.105346i	−0.998612	−	0.0526730i	$-0.983226\pi$
$71$		−	0.887662i		−	0.105346i	0.998612	−	0.0526730i	$-0.0167741\pi$
$72$		0			0
$73$			16.6940i			1.95389i	0.213490	+	0.976945i	$0.431517\pi$
$73$			16.6940i			1.95389i	−0.213490	+	0.976945i	$0.568483\pi$
$74$		0			0
$75$	2.22047			0.256398
$76$		0			0
$77$	−10.0133	+	1.55328i	−1.14112	+	0.177012i
$78$		0			0
$79$	5.08883			0.572538			0.286269	−	0.958149i	$-0.407585\pi$
$79$	5.08883			0.572538			0.286269	+	0.958149i	$0.407585\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			13.9161i			1.52749i	0.645518	+	0.763745i	$0.276641\pi$
$83$			13.9161i			1.52749i	−0.645518	+	0.763745i	$0.723359\pi$
$84$		0			0
$85$	2.77795			0.301311
$86$		0			0
$87$	6.68078			0.716254
$88$		0			0
$89$	−4.92820			−0.522388			−0.261194	−	0.965286i	$-0.584116\pi$
$89$	−4.92820			−0.522388			−0.261194	+	0.965286i	$0.584116\pi$
$90$		0			0
$91$	−21.2584			−2.22848
$92$		0			0
$93$		−	3.22047i		−	0.333947i
$94$		0			0
$95$	−7.40933			−0.760182
$96$		0			0
$97$	12.4441			1.26351			0.631753	−	0.775170i	$-0.282336\pi$
$97$	12.4441			1.26351			0.631753	+	0.775170i	$0.282336\pi$
$98$		0			0
$99$	−3.27743	+	0.508401i	−0.329394	+	0.0510962i
$100$		0			0
$101$	0.570332			0.0567502			0.0283751	−	0.999597i	$-0.490967\pi$
$101$	0.570332			0.0567502			0.0283751	+	0.999597i	$0.490967\pi$
$102$		0			0
$103$			12.8174i			1.26294i	0.775400	+	0.631470i	$0.217548\pi$
$103$			12.8174i			1.26294i	−0.775400	+	0.631470i	$0.782452\pi$
$104$		0			0
$105$		−	5.09364i		−	0.497089i
$106$		0			0
$107$		−	15.6772i		−	1.51558i	−0.652501	−	0.757788i	$-0.726280\pi$
$107$		−	15.6772i		−	1.51558i	0.652501	−	0.757788i	$-0.273720\pi$
$108$		0			0
$109$	1.59195			0.152481			0.0762407	−	0.997089i	$-0.475708\pi$
$109$	1.59195			0.152481			0.0762407	+	0.997089i	$0.475708\pi$
$110$		0			0
$111$		−	8.01896i		−	0.761126i
$112$		0			0
$113$	2.00000			0.188144			0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000			0.188144			0.0940721	+	0.995565i	$0.470012\pi$
$114$		0			0
$115$	−11.8892			−1.10868
$116$		0			0
$117$	−6.95805			−0.643272
$118$		0			0
$119$			5.09075i			0.466669i
$120$		0			0
$121$	10.4831	−	3.33249i	0.953005	−	0.302954i
$122$		0			0
$123$		−	11.2330i		−	1.01285i
$124$		0			0
$125$		−	12.0379i		−	1.07670i
$126$		0			0
$127$	−14.3832			−1.27630			−0.638150	−	0.769912i	$-0.720300\pi$
$127$	−14.3832			−1.27630			−0.638150	+	0.769912i	$0.720300\pi$
$128$		0			0
$129$			7.77669i			0.684699i
$130$		0			0
$131$		−	2.77795i		−	0.242711i	−0.992609	−	0.121355i	$-0.961276\pi$
$131$		−	2.77795i		−	0.242711i	0.992609	−	0.121355i	$-0.0387241\pi$
$132$		0			0
$133$		−	13.5780i		−	1.17736i
$134$		0			0
$135$		−	1.66719i		−	0.143489i
$136$		0			0
$137$	0.440939			0.0376720			0.0188360	−	0.999823i	$-0.494004\pi$
$137$	0.440939			0.0376720			0.0188360	+	0.999823i	$0.494004\pi$
$138$		0			0
$139$		−	4.44420i		−	0.376952i	−0.982078	−	0.188476i	$-0.939645\pi$
$139$		−	4.44420i		−	0.376952i	0.982078	−	0.188476i	$-0.0603548\pi$
$140$		0			0
$141$		−	8.22205i		−	0.692421i
$142$		0			0
$143$	22.8045	−	3.53748i	1.90701	−	0.295819i
$144$		0			0
$145$		−	11.1381i		−	0.924972i
$146$		0			0
$147$	2.33438			0.192537
$148$		0			0
$149$	4.98557			0.408434			0.204217	−	0.978926i	$-0.434535\pi$
$149$	4.98557			0.408434			0.204217	+	0.978926i	$0.434535\pi$
$150$		0			0
$151$	−16.4168			−1.33598			−0.667989	−	0.744171i	$-0.732845\pi$
$151$	−16.4168			−1.33598			−0.667989	+	0.744171i	$0.732845\pi$
$152$		0			0
$153$			1.66625i			0.134708i
$154$		0			0
$155$	−5.36914			−0.431260
$156$		0			0
$157$			12.6224i			1.00738i	0.863884	+	0.503690i	$0.168025\pi$
$157$			12.6224i			1.00738i	−0.863884	+	0.503690i	$0.831975\pi$
$158$		0			0
$159$		−	7.44252i		−	0.590230i
$160$		0			0
$161$		−	21.7877i		−	1.71711i
$162$		0			0
$163$	2.44094			0.191189			0.0955946	−	0.995420i	$-0.469525\pi$
$163$	2.44094			0.191189			0.0955946	+	0.995420i	$0.469525\pi$
$164$		0			0
$165$	0.847602	+	5.46410i	0.0659857	+	0.425380i
$166$		0			0
$167$	−4.81156			−0.372329			−0.186165	−	0.982519i	$-0.559606\pi$
$167$	−4.81156			−0.372329			−0.186165	+	0.982519i	$0.559606\pi$
$168$		0			0
$169$	35.4144			2.72419
$170$		0			0
$171$		−	4.44420i		−	0.339856i
$172$		0			0
$173$	5.54011			0.421207			0.210603	−	0.977572i	$-0.432457\pi$
$173$	5.54011			0.421207			0.210603	+	0.977572i	$0.432457\pi$
$174$		0			0
$175$	6.78403			0.512824
$176$		0			0
$177$	−11.0421			−0.829976
$178$		0			0
$179$	−5.10971			−0.381918			−0.190959	−	0.981598i	$-0.561160\pi$
$179$	−5.10971			−0.381918			−0.190959	+	0.981598i	$0.561160\pi$
$180$		0			0
$181$			11.5938i			0.861762i	0.902409	+	0.430881i	$0.141797\pi$
$181$			11.5938i			0.861762i	−0.902409	+	0.430881i	$0.858203\pi$
$182$		0			0
$183$	11.0349			0.815722
$184$		0			0
$185$	−13.3691			−0.982919
$186$		0			0
$187$	−0.847121	−	5.46100i	−0.0619476	−	0.399348i
$188$		0			0
$189$	3.05522			0.222235
$190$		0			0
$191$		−	8.40040i		−	0.607831i	−0.952699	−	0.303916i	$-0.901706\pi$
$191$		−	8.40040i		−	0.607831i	0.952699	−	0.303916i	$-0.0982941\pi$
$192$		0			0
$193$		−	7.80565i		−	0.561863i	−0.959728	−	0.280931i	$-0.909357\pi$
$193$		−	7.80565i		−	0.561863i	0.959728	−	0.280931i	$-0.0906434\pi$
$194$		0			0
$195$			11.6004i			0.830722i
$196$		0			0
$197$	−12.7912			−0.911337			−0.455668	−	0.890150i	$-0.650600\pi$
$197$	−12.7912			−0.911337			−0.455668	+	0.890150i	$0.650600\pi$
$198$		0			0
$199$			8.73636i			0.619304i	0.950850	+	0.309652i	$0.100213\pi$
$199$			8.73636i			0.619304i	−0.950850	+	0.309652i	$0.899787\pi$
$200$		0			0
$201$	−3.88924			−0.274326
$202$		0			0
$203$	20.4113			1.43259
$204$		0			0
$205$	−18.7276			−1.30800
$206$		0			0
$207$		−	7.13129i		−	0.495659i
$208$		0			0
$209$	2.25943	+	14.5655i	0.156288	+	1.00752i
$210$		0			0
$211$		−	6.47780i		−	0.445950i	−0.974824	−	0.222975i	$-0.928423\pi$
$211$		−	6.47780i		−	0.445950i	0.974824	−	0.222975i	$-0.0715769\pi$
$212$		0			0
$213$		−	0.887662i		−	0.0608216i
$214$		0			0
$215$	12.9652			0.884222
$216$		0			0
$217$		−	9.83925i		−	0.667932i
$218$		0			0
$219$			16.6940i			1.12808i
$220$		0			0
$221$		−	11.5938i		−	0.779885i
$222$		0			0
$223$		−	14.5549i		−	0.974665i	−0.873216	−	0.487333i	$-0.837970\pi$
$223$		−	14.5549i		−	0.974665i	0.873216	−	0.487333i	$-0.162030\pi$
$224$		0			0
$225$	2.22047			0.148031
$226$		0			0
$227$			9.90520i			0.657431i	0.944429	+	0.328716i	$0.106616\pi$
$227$			9.90520i			0.657431i	−0.944429	+	0.328716i	$0.893384\pi$
$228$		0			0
$229$		−	0.197307i		−	0.0130384i	−0.999979	−	0.00651921i	$-0.997925\pi$
$229$		−	0.197307i		−	0.0130384i	0.999979	−	0.00651921i	$-0.00207514\pi$
$230$		0			0
$231$	−10.0133	+	1.55328i	−0.658825	+	0.102198i
$232$		0			0
$233$			28.8200i			1.88806i	0.329855	+	0.944032i	$0.393000\pi$
$233$			28.8200i			1.88806i	−0.329855	+	0.944032i	$0.607000\pi$
$234$		0			0
$235$	−13.7077			−0.894194
$236$		0			0
$237$	5.08883			0.330555
$238$		0			0
$239$	−16.3556			−1.05796			−0.528979	−	0.848635i	$-0.677425\pi$
$239$	−16.3556			−1.05796			−0.528979	+	0.848635i	$0.677425\pi$
$240$		0			0
$241$			14.6604i			0.944362i	0.881502	+	0.472181i	$0.156533\pi$
$241$			14.6604i			0.944362i	−0.881502	+	0.472181i	$0.843467\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	3.89187i		−	0.248642i
$246$		0			0
$247$			30.9229i			1.96758i
$248$		0			0
$249$			13.9161i			0.881897i
$250$		0			0
$251$	26.4789			1.67133			0.835665	−	0.549239i	$-0.185082\pi$
$251$	26.4789			1.67133			0.835665	+	0.549239i	$0.185082\pi$
$252$		0			0
$253$	3.62555	+	23.3723i	0.227937	+	1.46940i
$254$		0			0
$255$	2.77795			0.173962
$256$		0			0
$257$	23.8564			1.48812			0.744061	−	0.668112i	$-0.232897\pi$
$257$	23.8564			1.48812			0.744061	+	0.668112i	$0.232897\pi$
$258$		0			0
$259$		−	24.4997i		−	1.52234i
$260$		0			0
$261$	6.68078			0.413530
$262$		0			0
$263$	0.744350			0.0458986			0.0229493	−	0.999737i	$-0.492694\pi$
$263$	0.744350			0.0458986			0.0229493	+	0.999737i	$0.492694\pi$
$264$		0			0
$265$	−12.4081			−0.762224
$266$		0			0
$267$	−4.92820			−0.301601
$268$		0			0
$269$			5.74828i			0.350479i	0.984526	+	0.175239i	$0.0560699\pi$
$269$			5.74828i			0.350479i	−0.984526	+	0.175239i	$0.943930\pi$
$270$		0			0
$271$	17.5259			1.06462			0.532310	−	0.846550i	$-0.321324\pi$
$271$	17.5259			1.06462			0.532310	+	0.846550i	$0.321324\pi$
$272$		0			0
$273$	−21.2584			−1.28662
$274$		0			0
$275$	−7.27743	+	1.12889i	−0.438845	+	0.0680745i
$276$		0			0
$277$	8.59534			0.516444			0.258222	−	0.966086i	$-0.416863\pi$
$277$	8.59534			0.516444			0.258222	+	0.966086i	$0.416863\pi$
$278$		0			0
$279$		−	3.22047i		−	0.192804i
$280$		0			0
$281$			11.8535i			0.707123i	0.935411	+	0.353561i	$0.115029\pi$
$281$			11.8535i			0.707123i	−0.935411	+	0.353561i	$0.884971\pi$
$282$		0			0
$283$			23.5778i			1.40155i	0.713380	+	0.700777i	$0.247163\pi$
$283$			23.5778i			1.40155i	−0.713380	+	0.700777i	$0.752837\pi$
$284$		0			0
$285$	−7.40933			−0.438891
$286$		0			0
$287$		−	34.3195i		−	2.02581i
$288$		0			0
$289$	14.2236			0.836684
$290$		0			0
$291$	12.4441			0.729486
$292$		0			0
$293$	21.1514			1.23568			0.617839	−	0.786304i	$-0.288008\pi$
$293$	21.1514			1.23568			0.617839	+	0.786304i	$0.288008\pi$
$294$		0			0
$295$			18.4093i			1.07183i
$296$		0			0
$297$	−3.27743	+	0.508401i	−0.190176	+	0.0295004i
$298$		0			0
$299$			49.6199i			2.86959i
$300$		0			0
$301$			23.7595i			1.36948i
$302$		0			0
$303$	0.570332			0.0327647
$304$		0			0
$305$		−	18.3973i		−	1.05343i
$306$		0			0
$307$		−	14.4417i		−	0.824230i	−0.911132	−	0.412115i	$-0.864790\pi$
$307$		−	14.4417i		−	0.824230i	0.911132	−	0.412115i	$-0.135210\pi$
$308$		0			0
$309$			12.8174i			0.729159i
$310$		0			0
$311$			16.4194i			0.931056i	0.885033	+	0.465528i	$0.154136\pi$
$311$			16.4194i			0.931056i	−0.885033	+	0.465528i	$0.845864\pi$
$312$		0			0
$313$	−6.24910			−0.353220			−0.176610	−	0.984281i	$-0.556513\pi$
$313$	−6.24910			−0.353220			−0.176610	+	0.984281i	$0.556513\pi$
$314$		0			0
$315$		−	5.09364i		−	0.286994i
$316$		0			0
$317$		−	16.5143i		−	0.927536i	−0.885957	−	0.463768i	$-0.846497\pi$
$317$		−	16.5143i		−	0.927536i	0.885957	−	0.463768i	$-0.153503\pi$
$318$		0			0
$319$	−21.8958	+	3.39651i	−1.22593	+	0.190168i
$320$		0			0
$321$		−	15.6772i		−	0.875018i
$322$		0			0
$323$	7.40513			0.412032
$324$		0			0
$325$	−15.4501			−0.857019
$326$		0			0
$327$	1.59195			0.0880351
$328$		0			0
$329$		−	25.1202i		−	1.38492i
$330$		0			0
$331$	14.9282			0.820528			0.410264	−	0.911967i	$-0.365437\pi$
$331$	14.9282			0.820528			0.410264	+	0.911967i	$0.365437\pi$
$332$		0			0
$333$		−	8.01896i		−	0.439436i
$334$		0			0
$335$			6.48411i			0.354265i
$336$		0			0
$337$		−	13.1717i		−	0.717511i	−0.933432	−	0.358755i	$-0.883201\pi$
$337$		−	13.1717i		−	0.717511i	0.933432	−	0.358755i	$-0.116799\pi$
$338$		0			0
$339$	2.00000			0.108625
$340$		0			0
$341$	1.63729	+	10.5549i	0.0886642	+	0.571577i
$342$		0			0
$343$	−14.2545			−0.769670
$344$		0			0
$345$	−11.8892			−0.640095
$346$		0			0
$347$		−	26.1370i		−	1.40311i	−0.712617	−	0.701553i	$-0.752490\pi$
$347$		−	26.1370i		−	1.40311i	0.712617	−	0.701553i	$-0.247510\pi$
$348$		0			0
$349$	2.14649			0.114899			0.0574495	−	0.998348i	$-0.481703\pi$
$349$	2.14649			0.114899			0.0574495	+	0.998348i	$0.481703\pi$
$350$		0			0
$351$	−6.95805			−0.371393
$352$		0			0
$353$	−32.1221			−1.70969			−0.854845	−	0.518884i	$-0.826348\pi$
$353$	−32.1221			−1.70969			−0.854845	+	0.518884i	$0.826348\pi$
$354$		0			0
$355$	−1.47990			−0.0785451
$356$		0			0
$357$			5.09075i			0.269431i
$358$		0			0
$359$	−17.5870			−0.928206			−0.464103	−	0.885781i	$-0.653623\pi$
$359$	−17.5870			−0.928206			−0.464103	+	0.885781i	$0.653623\pi$
$360$		0			0
$361$	−0.750904			−0.0395213
$362$		0			0
$363$	10.4831	−	3.33249i	0.550218	−	0.174910i
$364$		0			0
$365$	27.8322			1.45680
$366$		0			0
$367$			23.3395i			1.21831i	0.793051	+	0.609155i	$0.208491\pi$
$367$			23.3395i			1.21831i	−0.793051	+	0.609155i	$0.791509\pi$
$368$		0			0
$369$		−	11.2330i		−	0.584769i
$370$		0			0
$371$		−	22.7385i		−	1.18053i
$372$		0			0
$373$	6.21370			0.321733			0.160867	−	0.986976i	$-0.448571\pi$
$373$	6.21370			0.321733			0.160867	+	0.986976i	$0.448571\pi$
$374$		0			0
$375$		−	12.0379i		−	0.621635i
$376$		0			0
$377$	−46.4852			−2.39411
$378$		0			0
$379$	−7.74057			−0.397606			−0.198803	−	0.980039i	$-0.563705\pi$
$379$	−7.74057			−0.397606			−0.198803	+	0.980039i	$0.563705\pi$
$380$		0			0
$381$	−14.3832			−0.736872
$382$		0			0
$383$		−	24.9877i		−	1.27681i	−0.769700	−	0.638406i	$-0.779594\pi$
$383$		−	24.9877i		−	1.27681i	0.769700	−	0.638406i	$-0.220406\pi$
$384$		0			0
$385$	2.58961	+	16.6940i	0.131979	+	0.850807i
$386$		0			0
$387$			7.77669i			0.395311i
$388$		0			0
$389$		−	4.85483i		−	0.246150i	−0.992397	−	0.123075i	$-0.960724\pi$
$389$		−	4.85483i		−	0.246150i	0.992397	−	0.123075i	$-0.0392755\pi$
$390$		0			0
$391$	11.8825			0.600923
$392$		0			0
$393$		−	2.77795i		−	0.140129i
$394$		0			0
$395$		−	8.48405i		−	0.426879i
$396$		0			0
$397$			24.0569i			1.20738i	0.797219	+	0.603690i	$0.206303\pi$
$397$			24.0569i			1.20738i	−0.797219	+	0.603690i	$0.793697\pi$
$398$		0			0
$399$		−	13.5780i		−	0.679751i
$400$		0			0
$401$	4.08424			0.203957			0.101979	−	0.994787i	$-0.467483\pi$
$401$	4.08424			0.203957			0.101979	+	0.994787i	$0.467483\pi$
$402$		0			0
$403$			22.4082i			1.11623i
$404$		0			0
$405$		−	1.66719i		−	0.0828435i
$406$		0			0
$407$	4.07684	+	26.2815i	0.202082	+	1.30273i
$408$		0			0
$409$			10.2452i			0.506593i	0.967389	+	0.253296i	$0.0815148\pi$
$409$			10.2452i			0.506593i	−0.967389	+	0.253296i	$0.918485\pi$
$410$		0			0
$411$	0.440939			0.0217499
$412$		0			0
$413$	−33.7361			−1.66005
$414$		0			0
$415$	23.2008			1.13888
$416$		0			0
$417$		−	4.44420i		−	0.217633i
$418$		0			0
$419$	18.1815			0.888225			0.444112	−	0.895971i	$-0.353519\pi$
$419$	18.1815			0.888225			0.444112	+	0.895971i	$0.353519\pi$
$420$		0			0
$421$		−	12.9250i		−	0.629928i	−0.949104	−	0.314964i	$-0.898007\pi$
$421$		−	12.9250i		−	0.629928i	0.949104	−	0.314964i	$-0.101993\pi$
$422$		0			0
$423$		−	8.22205i		−	0.399770i
$424$		0			0
$425$			3.69985i			0.179469i
$426$		0			0
$427$	33.7140			1.63154
$428$		0			0
$429$	22.8045	−	3.53748i	1.10101	−	0.170791i
$430$		0			0
$431$	−0.554539			−0.0267112			−0.0133556	−	0.999911i	$-0.504251\pi$
$431$	−0.554539			−0.0267112			−0.0133556	+	0.999911i	$0.504251\pi$
$432$		0			0
$433$	−5.36722			−0.257932			−0.128966	−	0.991649i	$-0.541166\pi$
$433$	−5.36722			−0.257932			−0.128966	+	0.991649i	$0.541166\pi$
$434$		0			0
$435$		−	11.1381i		−	0.534033i
$436$		0			0
$437$	−31.6929			−1.51608
$438$		0			0
$439$	7.68660			0.366862			0.183431	−	0.983033i	$-0.441280\pi$
$439$	7.68660			0.366862			0.183431	+	0.983033i	$0.441280\pi$
$440$		0			0
$441$	2.33438			0.111161
$442$		0			0
$443$	27.0800			1.28661			0.643306	−	0.765610i	$-0.277563\pi$
$443$	27.0800			1.28661			0.643306	+	0.765610i	$0.277563\pi$
$444$		0			0
$445$			8.21626i			0.389488i
$446$		0			0
$447$	4.98557			0.235810
$448$		0			0
$449$	−27.8943			−1.31641			−0.658207	−	0.752837i	$-0.728685\pi$
$449$	−27.8943			−1.31641			−0.658207	+	0.752837i	$0.728685\pi$
$450$		0			0
$451$	5.71089	+	36.8155i	0.268915	+	1.73357i
$452$		0			0
$453$	−16.4168			−0.771327
$454$		0			0
$455$			35.4418i			1.66154i
$456$		0			0
$457$		−	15.5534i		−	0.727557i	−0.931486	−	0.363778i	$-0.881487\pi$
$457$		−	15.5534i		−	0.727557i	0.931486	−	0.363778i	$-0.118513\pi$
$458$		0			0
$459$			1.66625i			0.0777737i
$460$		0			0
$461$	−10.4192			−0.485271			−0.242636	−	0.970117i	$-0.578012\pi$
$461$	−10.4192			−0.485271			−0.242636	+	0.970117i	$0.578012\pi$
$462$		0			0
$463$		−	8.45443i		−	0.392911i	−0.980513	−	0.196455i	$-0.937057\pi$
$463$		−	8.45443i		−	0.392911i	0.980513	−	0.196455i	$-0.0629431\pi$
$464$		0			0
$465$	−5.36914			−0.248988
$466$		0			0
$467$	15.0800			0.697821			0.348910	−	0.937156i	$-0.386552\pi$
$467$	15.0800			0.697821			0.348910	+	0.937156i	$0.386552\pi$
$468$		0			0
$469$	−11.8825			−0.548682
$470$		0			0
$471$			12.6224i			0.581612i
$472$		0			0
$473$	−3.95367	−	25.4875i	−0.181790	−	1.17192i
$474$		0			0
$475$		−	9.86821i		−	0.452784i
$476$		0			0
$477$		−	7.44252i		−	0.340769i
$478$		0			0
$479$	21.7217			0.992492			0.496246	−	0.868182i	$-0.334711\pi$
$479$	21.7217			0.992492			0.496246	+	0.868182i	$0.334711\pi$
$480$		0			0
$481$			55.7963i			2.54409i
$482$		0			0
$483$		−	21.7877i		−	0.991374i
$484$		0			0
$485$		−	20.7467i		−	0.942059i
$486$		0			0
$487$			13.2236i			0.599220i	0.954062	+	0.299610i	$0.0968565\pi$
$487$			13.2236i			0.599220i	−0.954062	+	0.299610i	$0.903144\pi$
$488$		0			0
$489$	2.44094			0.110383
$490$		0			0
$491$		−	15.4295i		−	0.696324i	−0.937434	−	0.348162i	$-0.886806\pi$
$491$		−	15.4295i		−	0.696324i	0.937434	−	0.348162i	$-0.113194\pi$
$492$		0			0
$493$			11.1318i			0.501352i
$494$		0			0
$495$	0.847602	+	5.46410i	0.0380969	+	0.245593i
$496$		0			0
$497$		−	2.71200i		−	0.121650i
$498$		0			0
$499$	−29.5970			−1.32494			−0.662471	−	0.749087i	$-0.730492\pi$
$499$	−29.5970			−1.32494			−0.662471	+	0.749087i	$0.730492\pi$
$500$		0			0
$501$	−4.81156			−0.214964
$502$		0			0
$503$	−4.62174			−0.206073			−0.103037	−	0.994678i	$-0.532856\pi$
$503$	−4.62174			−0.206073			−0.103037	+	0.994678i	$0.532856\pi$
$504$		0			0
$505$		−	0.950853i		−	0.0423124i
$506$		0			0
$507$	35.4144			1.57281
$508$		0			0
$509$			20.8548i			0.924374i	0.886782	+	0.462187i	$0.152935\pi$
$509$			20.8548i			0.924374i	−0.886782	+	0.462187i	$0.847065\pi$
$510$		0			0
$511$			51.0040i			2.25629i
$512$		0			0
$513$		−	4.44420i		−	0.196216i
$514$		0			0
$515$	21.3691			0.941637
$516$		0			0
$517$	4.18009	+	26.9472i	0.183840	+	1.18513i
$518$		0			0
$519$	5.54011			0.243184
$520$		0			0
$521$	19.8564			0.869925			0.434962	−	0.900449i	$-0.356762\pi$
$521$	19.8564			0.869925			0.434962	+	0.900449i	$0.356762\pi$
$522$		0			0
$523$			13.3905i			0.585526i	0.956185	+	0.292763i	$0.0945747\pi$
$523$			13.3905i			0.585526i	−0.956185	+	0.292763i	$0.905425\pi$
$524$		0			0
$525$	6.78403			0.296079
$526$		0			0
$527$	5.36609			0.233751
$528$		0			0
$529$	−27.8554			−1.21110
$530$		0			0
$531$	−11.0421			−0.479187
$532$		0			0
$533$			78.1601i			3.38549i
$534$		0			0
$535$	−26.1370			−1.13000
$536$		0			0
$537$	−5.10971			−0.220500
$538$		0			0
$539$	−7.65078	+	1.18680i	−0.329542	+	0.0511192i
$540$		0			0
$541$	−3.96395			−0.170424			−0.0852118	−	0.996363i	$-0.527157\pi$
$541$	−3.96395			−0.170424			−0.0852118	+	0.996363i	$0.527157\pi$
$542$		0			0
$543$			11.5938i			0.497538i
$544$		0			0
$545$		−	2.65409i		−	0.113689i
$546$		0			0
$547$			5.74309i			0.245557i	0.992434	+	0.122778i	$0.0391804\pi$
$547$			5.74309i			0.245557i	−0.992434	+	0.122778i	$0.960820\pi$
$548$		0			0
$549$	11.0349			0.470958
$550$		0			0
$551$		−	29.6907i		−	1.26487i
$552$		0			0
$553$	15.5475			0.661147
$554$		0			0
$555$	−13.3691			−0.567488
$556$		0			0
$557$	24.1095			1.02155			0.510777	−	0.859713i	$-0.329358\pi$
$557$	24.1095			1.02155			0.510777	+	0.859713i	$0.329358\pi$
$558$		0			0
$559$		−	54.1106i		−	2.28863i
$560$		0			0
$561$	−0.847121	−	5.46100i	−0.0357655	−	0.230564i
$562$		0			0
$563$		−	14.3468i		−	0.604644i	−0.953206	−	0.302322i	$-0.902238\pi$
$563$		−	14.3468i		−	0.604644i	0.953206	−	0.302322i	$-0.0977618\pi$
$564$		0			0
$565$		−	3.33438i		−	0.140279i
$566$		0			0
$567$	3.05522			0.128307
$568$		0			0
$569$			5.18855i			0.217515i	0.994068	+	0.108758i	$0.0346872\pi$
$569$			5.18855i			0.217515i	−0.994068	+	0.108758i	$0.965313\pi$
$570$		0			0
$571$			5.74309i			0.240341i	0.992753	+	0.120170i	$0.0383441\pi$
$571$			5.74309i			0.240341i	−0.992753	+	0.120170i	$0.961656\pi$
$572$		0			0
$573$		−	8.40040i		−	0.350932i
$574$		0			0
$575$		−	15.8348i		−	0.660358i
$576$		0			0
$577$	18.8638			0.785309			0.392654	−	0.919686i	$-0.371557\pi$
$577$	18.8638			0.785309			0.392654	+	0.919686i	$0.371557\pi$
$578$		0			0
$579$		−	7.80565i		−	0.324392i
$580$		0			0
$581$			42.5168i			1.76389i
$582$		0			0
$583$	3.78378	+	24.3923i	0.156708	+	1.01023i
$584$		0			0
$585$			11.6004i			0.479618i
$586$		0			0
$587$	−21.2552			−0.877297			−0.438649	−	0.898659i	$-0.644543\pi$
$587$	−21.2552			−0.877297			−0.438649	+	0.898659i	$0.644543\pi$
$588$		0			0
$589$	−14.3124			−0.589732
$590$		0			0
$591$	−12.7912			−0.526161
$592$		0			0
$593$		−	14.8696i		−	0.610620i	−0.952253	−	0.305310i	$-0.901240\pi$
$593$		−	14.8696i		−	0.610620i	0.952253	−	0.305310i	$-0.0987601\pi$
$594$		0			0
$595$	8.48726			0.347944
$596$		0			0
$597$			8.73636i			0.357555i
$598$		0			0
$599$			25.2155i			1.03028i	0.857106	+	0.515139i	$0.172260\pi$
$599$			25.2155i			1.03028i	−0.857106	+	0.515139i	$0.827740\pi$
$600$		0			0
$601$		−	17.4288i		−	0.710934i	−0.934689	−	0.355467i	$-0.884322\pi$
$601$		−	17.4288i		−	0.710934i	0.934689	−	0.355467i	$-0.115678\pi$
$602$		0			0
$603$	−3.88924			−0.158382
$604$		0			0
$605$	−5.55591	−	17.4773i	−0.225880	−	0.710552i
$606$		0			0
$607$	−3.64135			−0.147798			−0.0738989	−	0.997266i	$-0.523544\pi$
$607$	−3.64135			−0.147798			−0.0738989	+	0.997266i	$0.523544\pi$
$608$		0			0
$609$	20.4113			0.827106
$610$		0			0
$611$			57.2094i			2.31444i
$612$		0			0
$613$	15.1021			0.609968			0.304984	−	0.952357i	$-0.401349\pi$
$613$	15.1021			0.609968			0.304984	+	0.952357i	$0.401349\pi$
$614$		0			0
$615$	−18.7276			−0.755172
$616$		0			0
$617$	38.1415			1.53552			0.767759	−	0.640738i	$-0.221372\pi$
$617$	38.1415			1.53552			0.767759	+	0.640738i	$0.221372\pi$
$618$		0			0
$619$	15.1741			0.609900			0.304950	−	0.952368i	$-0.401360\pi$
$619$	15.1741			0.609900			0.304950	+	0.952368i	$0.401360\pi$
$620$		0			0
$621$		−	7.13129i		−	0.286169i
$622$		0			0
$623$	−15.0568			−0.603236
$624$		0			0
$625$	−8.96717			−0.358687
$626$		0			0
$627$	2.25943	+	14.5655i	0.0902331	+	0.581692i
$628$		0			0
$629$	13.3616			0.532760
$630$		0			0
$631$			33.3426i			1.32735i	0.748022	+	0.663674i	$0.231004\pi$
$631$			33.3426i			1.32735i	−0.748022	+	0.663674i	$0.768996\pi$
$632$		0			0
$633$		−	6.47780i		−	0.257469i
$634$		0			0
$635$			23.9795i			0.951598i
$636$		0			0
$637$	−16.2428			−0.643562
$638$		0			0
$639$		−	0.887662i		−	0.0351154i
$640$		0			0
$641$	−0.921895			−0.0364126			−0.0182063	−	0.999834i	$-0.505796\pi$
$641$	−0.921895			−0.0364126			−0.0182063	+	0.999834i	$0.505796\pi$
$642$		0			0
$643$	8.14867			0.321352			0.160676	−	0.987007i	$-0.448633\pi$
$643$	8.14867			0.321352			0.160676	+	0.987007i	$0.448633\pi$
$644$		0			0
$645$	12.9652			0.510506
$646$		0			0
$647$		−	10.1406i		−	0.398667i	−0.979932	−	0.199334i	$-0.936122\pi$
$647$		−	10.1406i		−	0.398667i	0.979932	−	0.199334i	$-0.0638778\pi$
$648$		0			0
$649$	36.1897	−	5.61382i	1.42057	−	0.220362i
$650$		0			0
$651$		−	9.83925i		−	0.385631i
$652$		0			0
$653$			27.8487i			1.08980i	0.838500	+	0.544902i	$0.183433\pi$
$653$			27.8487i			1.08980i	−0.838500	+	0.544902i	$0.816567\pi$
$654$		0			0
$655$	−4.63138			−0.180963
$656$		0			0
$657$			16.6940i			0.651297i
$658$		0			0
$659$		−	17.6529i		−	0.687661i	−0.939032	−	0.343830i	$-0.888275\pi$
$659$		−	17.6529i		−	0.687661i	0.939032	−	0.343830i	$-0.111725\pi$
$660$		0			0
$661$		−	6.76604i		−	0.263168i	−0.991305	−	0.131584i	$-0.957994\pi$
$661$		−	6.76604i		−	0.263168i	0.991305	−	0.131584i	$-0.0420064\pi$
$662$		0			0
$663$		−	11.5938i		−	0.450267i
$664$		0			0
$665$	−22.6372			−0.877832
$666$		0			0
$667$		−	47.6426i		−	1.84473i
$668$		0			0
$669$		−	14.5549i		−	0.562723i
$670$		0			0
$671$	−36.1660	+	5.61014i	−1.39617	+	0.216577i
$672$		0			0
$673$		−	19.8104i		−	0.763635i	−0.924238	−	0.381817i	$-0.875298\pi$
$673$		−	19.8104i		−	0.763635i	0.924238	−	0.381817i	$-0.124702\pi$
$674$		0			0
$675$	2.22047			0.0854659
$676$		0			0
$677$	16.5297			0.635287			0.317643	−	0.948210i	$-0.397109\pi$
$677$	16.5297			0.635287			0.317643	+	0.948210i	$0.397109\pi$
$678$		0			0
$679$	38.0195			1.45905
$680$		0			0
$681$			9.90520i			0.379568i
$682$		0			0
$683$	6.76604			0.258895			0.129448	−	0.991586i	$-0.458680\pi$
$683$	6.76604			0.258895			0.129448	+	0.991586i	$0.458680\pi$
$684$		0			0
$685$		−	0.735130i		−	0.0280879i
$686$		0			0
$687$		−	0.197307i		−	0.00752773i
$688$		0			0
$689$			51.7854i			1.97287i
$690$		0			0
$691$	16.5980			0.631419			0.315709	−	0.948856i	$-0.397758\pi$
$691$	16.5980			0.631419			0.315709	+	0.948856i	$0.397758\pi$
$692$		0			0
$693$	−10.0133	+	1.55328i	−0.380373	+	0.0590041i
$694$		0			0
$695$	−7.40933			−0.281052
$696$		0			0
$697$	18.7170			0.708958
$698$		0			0
$699$			28.8200i			1.09007i
$700$		0			0
$701$	2.94233			0.111130			0.0555652	−	0.998455i	$-0.482304\pi$
$701$	2.94233			0.111130			0.0555652	+	0.998455i	$0.482304\pi$
$702$		0			0
$703$	−35.6378			−1.34411
$704$		0			0
$705$	−13.7077			−0.516263
$706$		0			0
$707$	1.74249			0.0655331
$708$		0			0
$709$		−	35.4292i		−	1.33057i	−0.746589	−	0.665285i	$-0.768310\pi$
$709$		−	35.4292i		−	1.33057i	0.746589	−	0.665285i	$-0.231690\pi$
$710$		0			0
$711$	5.08883			0.190846
$712$		0			0
$713$	−22.9661			−0.860088
$714$		0			0
$715$	−5.89765	−	38.0195i	−0.220560	−	1.42185i
$716$		0			0
$717$	−16.3556			−0.610813
$718$		0			0
$719$		−	11.8815i		−	0.443106i	−0.975148	−	0.221553i	$-0.928887\pi$
$719$		−	11.8815i		−	0.443106i	0.975148	−	0.221553i	$-0.0711126\pi$
$720$		0			0
$721$			39.1601i			1.45840i
$722$		0			0
$723$			14.6604i			0.545228i
$724$		0			0
$725$	14.8345			0.550938
$726$		0			0
$727$		−	16.2761i		−	0.603646i	−0.953364	−	0.301823i	$-0.902405\pi$
$727$		−	16.2761i		−	0.603646i	0.953364	−	0.301823i	$-0.0975952\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−12.9579			−0.479265
$732$		0			0
$733$	41.6450			1.53819			0.769097	−	0.639132i	$-0.220706\pi$
$733$	41.6450			1.53819			0.769097	+	0.639132i	$0.220706\pi$
$734$		0			0
$735$		−	3.89187i		−	0.143554i
$736$		0			0
$737$	12.7467	−	1.97729i	0.469531	−	0.0728345i
$738$		0			0
$739$		−	9.26539i		−	0.340833i	−0.985372	−	0.170416i	$-0.945489\pi$
$739$		−	9.26539i		−	0.340833i	0.985372	−	0.170416i	$-0.0545113\pi$
$740$		0			0
$741$			30.9229i			1.13598i
$742$		0			0
$743$	−29.9977			−1.10051			−0.550254	−	0.834997i	$-0.685469\pi$
$743$	−29.9977			−1.10051			−0.550254	+	0.834997i	$0.685469\pi$
$744$		0			0
$745$		−	8.31191i		−	0.304525i
$746$		0			0
$747$			13.9161i			0.509163i
$748$		0			0
$749$		−	47.8975i		−	1.75014i
$750$		0			0
$751$		−	9.66141i		−	0.352550i	−0.984341	−	0.176275i	$-0.943595\pi$
$751$		−	9.66141i		−	0.352550i	0.984341	−	0.176275i	$-0.0564048\pi$
$752$		0			0
$753$	26.4789			0.964943
$754$		0			0
$755$			27.3699i			0.996094i
$756$		0			0
$757$		−	6.71194i		−	0.243950i	−0.992533	−	0.121975i	$-0.961077\pi$
$757$		−	6.71194i		−	0.243950i	0.992533	−	0.121975i	$-0.0389227\pi$
$758$		0			0
$759$	3.62555	+	23.3723i	0.131599	+	0.848361i
$760$		0			0
$761$		−	5.73345i		−	0.207837i	−0.994586	−	0.103919i	$-0.966862\pi$
$761$		−	5.73345i		−	0.207837i	0.994586	−	0.103919i	$-0.0331382\pi$
$762$		0			0
$763$	4.86377			0.176080
$764$		0			0
$765$	2.77795			0.100437
$766$		0			0
$767$	76.8316			2.77423
$768$		0			0
$769$		−	47.3138i		−	1.70618i	−0.521764	−	0.853090i	$-0.674726\pi$
$769$		−	47.3138i		−	1.70618i	0.521764	−	0.853090i	$-0.325274\pi$
$770$		0			0
$771$	23.8564			0.859167
$772$		0			0
$773$			11.6047i			0.417391i	0.977981	+	0.208696i	$0.0669218\pi$
$773$			11.6047i			0.417391i	−0.977981	+	0.208696i	$0.933078\pi$
$774$		0			0
$775$		−	7.15095i		−	0.256870i
$776$		0			0
$777$		−	24.4997i		−	0.878922i
$778$		0			0
$779$	−49.9219			−1.78864
$780$		0			0
$781$	0.451288	+	2.90925i	0.0161484	+	0.104101i
$782$		0			0
$783$	6.68078			0.238751
$784$		0			0
$785$	21.0440			0.751094
$786$		0			0
$787$			21.2965i			0.759137i	0.925164	+	0.379569i	$0.123928\pi$
$787$			21.2965i			0.759137i	−0.925164	+	0.379569i	$0.876072\pi$
$788$		0			0
$789$	0.744350			0.0264996
$790$		0			0
$791$	6.11045			0.217262
$792$		0			0
$793$	−76.7813			−2.72658
$794$		0			0
$795$	−12.4081			−0.440070
$796$		0			0
$797$		−	7.40460i		−	0.262285i	−0.991364	−	0.131142i	$-0.958136\pi$
$797$		−	7.40460i		−	0.262285i	0.991364	−	0.131142i	$-0.0418645\pi$
$798$		0			0
$799$	13.7000			0.484670
$800$		0			0
$801$	−4.92820			−0.174129
$802$		0			0
$803$	−8.48726	−	54.7135i	−0.299509	−	1.93080i
$804$		0			0
$805$	−36.3243			−1.28026
$806$		0			0
$807$			5.74828i			0.202349i
$808$		0			0
$809$			31.5883i			1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$809$			31.5883i			1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$810$		0			0
$811$		−	7.09989i		−	0.249311i	−0.992200	−	0.124655i	$-0.960217\pi$
$811$		−	7.09989i		−	0.249311i	0.992200	−	0.124655i	$-0.0397826\pi$
$812$		0			0
$813$	17.5259			0.614659
$814$		0			0
$815$		−	4.06952i		−	0.142549i
$816$		0			0
$817$	34.5612			1.20914
$818$		0			0
$819$	−21.2584			−0.742828
$820$		0			0
$821$	−53.5211			−1.86790			−0.933950	−	0.357405i	$-0.883662\pi$
$821$	−53.5211			−1.86790			−0.933950	+	0.357405i	$0.883662\pi$
$822$		0			0
$823$		−	40.4555i		−	1.41019i	−0.709113	−	0.705095i	$-0.750904\pi$
$823$		−	40.4555i		−	1.41019i	0.709113	−	0.705095i	$-0.249096\pi$
$824$		0			0
$825$	−7.27743	+	1.12889i	−0.253367	+	0.0393028i
$826$		0			0
$827$			31.3449i			1.08997i	0.838447	+	0.544984i	$0.183464\pi$
$827$			31.3449i			1.08997i	−0.838447	+	0.544984i	$0.816536\pi$
$828$		0			0
$829$			42.8977i			1.48990i	0.667121	+	0.744949i	$0.267526\pi$
$829$			42.8977i			1.48990i	−0.667121	+	0.744949i	$0.732474\pi$
$830$		0			0
$831$	8.59534			0.298169
$832$		0			0
$833$			3.88966i			0.134769i
$834$		0			0
$835$			8.02179i			0.277605i
$836$		0			0
$837$		−	3.22047i		−	0.111316i
$838$		0			0
$839$			15.5375i			0.536413i	0.963361	+	0.268207i	$0.0864310\pi$
$839$			15.5375i			0.536413i	−0.963361	+	0.268207i	$0.913569\pi$
$840$		0			0
$841$	15.6328			0.539061
$842$		0			0
$843$			11.8535i			0.408257i
$844$		0			0
$845$		−	59.0426i		−	2.03113i
$846$		0			0
$847$	32.0281	−	10.1815i	1.10050	−	0.349841i
$848$		0			0
$849$			23.5778i			0.809188i
$850$		0			0
$851$	−57.1855			−1.96029
$852$		0			0
$853$	18.7730			0.642775			0.321387	−	0.946948i	$-0.395851\pi$
$853$	18.7730			0.642775			0.321387	+	0.946948i	$0.395851\pi$
$854$		0			0
$855$	−7.40933			−0.253394
$856$		0			0
$857$		−	11.9098i		−	0.406833i	−0.979092	−	0.203416i	$-0.934796\pi$
$857$		−	11.9098i		−	0.406833i	0.979092	−	0.203416i	$-0.0652045\pi$
$858$		0			0
$859$	−14.6751			−0.500707			−0.250354	−	0.968154i	$-0.580547\pi$
$859$	−14.6751			−0.500707			−0.250354	+	0.968154i	$0.580547\pi$
$860$		0			0
$861$		−	34.3195i		−	1.16960i
$862$		0			0
$863$			3.35282i			0.114131i	0.998370	+	0.0570656i	$0.0181744\pi$
$863$			3.35282i			0.114131i	−0.998370	+	0.0570656i	$0.981826\pi$
$864$		0			0
$865$		−	9.23643i		−	0.314048i
$866$		0			0
$867$	14.2236			0.483060
$868$		0			0
$869$	−16.6783	+	2.58716i	−0.565771	+	0.0877635i
$870$		0			0
$871$	27.0615			0.916944
$872$		0			0
$873$	12.4441			0.421169
$874$		0			0
$875$		−	36.7785i		−	1.24334i
$876$		0			0
$877$	−32.2928			−1.09045			−0.545225	−	0.838290i	$-0.683556\pi$
$877$	−32.2928			−1.09045			−0.545225	+	0.838290i	$0.683556\pi$
$878$		0			0
$879$	21.1514			0.713419
$880$		0			0
$881$	−40.4195			−1.36177			−0.680884	−	0.732392i	$-0.738404\pi$
$881$	−40.4195			−1.36177			−0.680884	+	0.732392i	$0.738404\pi$
$882$		0			0
$883$	24.4801			0.823820			0.411910	−	0.911224i	$-0.364862\pi$
$883$	24.4801			0.823820			0.411910	+	0.911224i	$0.364862\pi$
$884$		0			0
$885$			18.4093i			0.618823i
$886$		0			0
$887$	25.5509			0.857914			0.428957	−	0.903325i	$-0.358881\pi$
$887$	25.5509			0.857914			0.428957	+	0.903325i	$0.358881\pi$
$888$		0			0
$889$	−43.9438			−1.47383
$890$		0			0
$891$	−3.27743	+	0.508401i	−0.109798	+	0.0170321i
$892$		0			0
$893$	−36.5404			−1.22278
$894$		0			0
$895$			8.51887i			0.284754i
$896$		0			0
$897$			49.6199i			1.65676i
$898$		0			0
$899$		−	21.5152i		−	0.717573i
$900$		0			0
$901$	12.4011			0.413139
$902$		0			0
$903$			23.7595i			0.790667i
$904$		0			0
$905$	19.3291			0.642522
$906$		0			0
$907$	−29.1611			−0.968279			−0.484139	−	0.874991i	$-0.660867\pi$
$907$	−29.1611			−0.968279			−0.484139	+	0.874991i	$0.660867\pi$
$908$		0			0
$909$	0.570332			0.0189167
$910$		0			0
$911$			10.1757i			0.337137i	0.985690	+	0.168568i	$0.0539144\pi$
$911$			10.1757i			0.337137i	−0.985690	+	0.168568i	$0.946086\pi$
$912$		0			0
$913$	−7.07495	−	45.6090i	−0.234147	−	1.50944i
$914$		0			0
$915$		−	18.3973i		−	0.608195i
$916$		0			0
$917$		−	8.48726i		−	0.280274i
$918$		0			0
$919$	−9.72021			−0.320640			−0.160320	−	0.987065i	$-0.551253\pi$
$919$	−9.72021			−0.320640			−0.160320	+	0.987065i	$0.551253\pi$
$920$		0			0
$921$		−	14.4417i		−	0.475869i
$922$		0			0
$923$			6.17639i			0.203298i
$924$		0			0
$925$		−	17.8058i		−	0.585453i
$926$		0			0
$927$			12.8174i			0.420980i
$928$		0			0
$929$	41.3670			1.35721			0.678604	−	0.734504i	$-0.262585\pi$
$929$	41.3670			1.35721			0.678604	+	0.734504i	$0.262585\pi$
$930$		0			0
$931$		−	10.3745i		−	0.340010i
$932$		0			0
$933$			16.4194i			0.537546i
$934$		0			0
$935$	−9.10454	+	1.41231i	−0.297750	+	0.0461876i
$936$		0			0
$937$			2.71040i			0.0885449i	0.999019	+	0.0442725i	$0.0140970\pi$
$937$			2.71040i			0.0885449i	−0.999019	+	0.0442725i	$0.985903\pi$
$938$		0			0
$939$	−6.24910			−0.203932
$940$		0			0
$941$	44.3619			1.44616			0.723078	−	0.690767i	$-0.242727\pi$
$941$	44.3619			1.44616			0.723078	+	0.690767i	$0.242727\pi$
$942$		0			0
$943$	−80.1062			−2.60862
$944$		0			0
$945$		−	5.09364i		−	0.165696i
$946$		0			0
$947$	−37.0504			−1.20397			−0.601987	−	0.798506i	$-0.705624\pi$
$947$	−37.0504			−1.20397			−0.601987	+	0.798506i	$0.705624\pi$
$948$		0			0
$949$		−	116.158i		−	3.77065i
$950$		0			0
$951$		−	16.5143i		−	0.535513i
$952$		0			0
$953$		−	11.9098i		−	0.385798i	−0.981219	−	0.192899i	$-0.938211\pi$
$953$		−	11.9098i		−	0.385798i	0.981219	−	0.192899i	$-0.0617889\pi$
$954$		0			0
$955$	−14.0051			−0.453194
$956$		0			0
$957$	−21.8958	+	3.39651i	−0.707789	+	0.109794i
$958$		0			0
$959$	1.34717			0.0435023
$960$		0			0
$961$	20.6286			0.665438
$962$		0			0
$963$		−	15.6772i		−	0.505192i
$964$		0			0
$965$	−13.0135			−0.418920
$966$		0			0
$967$	−53.2956			−1.71387			−0.856935	−	0.515424i	$-0.827634\pi$
$967$	−53.2956			−1.71387			−0.856935	+	0.515424i	$0.827634\pi$
$968$		0			0
$969$	7.40513			0.237887
$970$		0			0
$971$	40.7086			1.30640			0.653201	−	0.757185i	$-0.273426\pi$
$971$	40.7086			1.30640			0.653201	+	0.757185i	$0.273426\pi$
$972$		0			0
$973$		−	13.5780i		−	0.435291i
$974$		0			0
$975$	−15.4501			−0.494800
$976$		0			0
$977$	−33.9027			−1.08464			−0.542322	−	0.840171i	$-0.682455\pi$
$977$	−33.9027			−1.08464			−0.542322	+	0.840171i	$0.682455\pi$
$978$		0			0
$979$	16.1518	−	2.50550i	0.516215	−	0.0800762i
$980$		0			0
$981$	1.59195			0.0508271
$982$		0			0
$983$			26.3684i			0.841022i	0.907288	+	0.420511i	$0.138149\pi$
$983$			26.3684i			0.841022i	−0.907288	+	0.420511i	$0.861851\pi$
$984$		0			0
$985$			21.3254i			0.679485i
$986$		0			0
$987$		−	25.1202i		−	0.799584i
$988$		0			0
$989$	55.4579			1.76346
$990$		0			0
$991$		−	26.0893i		−	0.828754i	−0.910105	−	0.414377i	$-0.863999\pi$
$991$		−	26.0893i		−	0.828754i	0.910105	−	0.414377i	$-0.136001\pi$
$992$		0			0
$993$	14.9282			0.473732
$994$		0			0
$995$	14.5652			0.461748
$996$		0			0
$997$	36.5498			1.15754			0.578772	−	0.815490i	$-0.303532\pi$
$997$	36.5498			1.15754			0.578772	+	0.815490i	$0.303532\pi$
$998$		0			0
$999$		−	8.01896i		−	0.253709i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.h.d.1759.6	yes	16
4.3	odd	2		2112.2.h.c.1759.5	✓	16
8.3	odd	2	inner	2112.2.h.d.1759.11	yes	16
8.5	even	2		2112.2.h.c.1759.12	yes	16
11.10	odd	2	inner	2112.2.h.d.1759.5	yes	16
44.43	even	2		2112.2.h.c.1759.6	yes	16
88.21	odd	2		2112.2.h.c.1759.11	yes	16
88.43	even	2	inner	2112.2.h.d.1759.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.h.c.1759.5	✓	16	4.3	odd	2
2112.2.h.c.1759.6	yes	16	44.43	even	2
2112.2.h.c.1759.11	yes	16	88.21	odd	2
2112.2.h.c.1759.12	yes	16	8.5	even	2
2112.2.h.d.1759.5	yes	16	11.10	odd	2	inner
2112.2.h.d.1759.6	yes	16	1.1	even	1	trivial
2112.2.h.d.1759.11	yes	16	8.3	odd	2	inner
2112.2.h.d.1759.12	yes	16	88.43	even	2	inner

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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.66719i q^{5} +3.05522 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.66719i q^{5} +3.05522 q^{7} +1.00000 q^{9} +(-3.27743 + 0.508401i) q^{11} -6.95805 q^{13} -1.66719i q^{15} +1.66625i q^{17} -4.44420i q^{19} +3.05522 q^{21} -7.13129i q^{23} +2.22047 q^{25} +1.00000 q^{27} +6.68078 q^{29} -3.22047i q^{31} +(-3.27743 + 0.508401i) q^{33} -5.09364i q^{35} -8.01896i q^{37} -6.95805 q^{39} -11.2330i q^{41} +7.77669i q^{43} -1.66719i q^{45} -8.22205i q^{47} +2.33438 q^{49} +1.66625i q^{51} -7.44252i q^{53} +(0.847602 + 5.46410i) q^{55} -4.44420i q^{57} -11.0421 q^{59} +11.0349 q^{61} +3.05522 q^{63} +11.6004i q^{65} -3.88924 q^{67} -7.13129i q^{69} -0.887662i q^{71} +16.6940i q^{73} +2.22047 q^{75} +(-10.0133 + 1.55328i) q^{77} +5.08883 q^{79} +1.00000 q^{81} +13.9161i q^{83} +2.77795 q^{85} +6.68078 q^{87} -4.92820 q^{89} -21.2584 q^{91} -3.22047i q^{93} -7.40933 q^{95} +12.4441 q^{97} +(-3.27743 + 0.508401i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 16 * q^9	\( 16 q + 16 q^{3} + 16 q^{9} - 8 q^{11} - 16 q^{25} + 16 q^{27} - 8 q^{33} - 80 q^{59} + 64 q^{67} - 16 q^{75} + 16 q^{81} + 32 q^{89} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 16 * q^9 - 8 * q^11 - 16 * q^25 + 16 * q^27 - 8 * q^33 - 80 * q^59 + 64 * q^67 - 16 * q^75 + 16 * q^81 + 32 * q^89 - 16 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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