LMFDB - Embedded newform 4200.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4200,2,Mod(1,4200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4200.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:	$33.5371688489$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	4200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -0.561553 q^{11} +6.68466 q^{13} +5.56155 q^{17} -1.12311 q^{19} +1.00000 q^{21} -4.12311 q^{23} +1.00000 q^{27} +8.12311 q^{29} -6.68466 q^{31} -0.561553 q^{33} -0.561553 q^{37} +6.68466 q^{39} +0.684658 q^{41} +0.123106 q^{43} +5.12311 q^{47} +1.00000 q^{49} +5.56155 q^{51} -0.438447 q^{53} -1.12311 q^{57} -2.68466 q^{59} -2.43845 q^{61} +1.00000 q^{63} -12.8078 q^{67} -4.12311 q^{69} +12.8078 q^{71} -9.36932 q^{73} -0.561553 q^{77} +12.5616 q^{79} +1.00000 q^{81} -2.68466 q^{83} +8.12311 q^{87} +10.0000 q^{89} +6.68466 q^{91} -6.68466 q^{93} +6.00000 q^{97} -0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + 3 q^{11} + q^{13} + 7 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{27} + 8 q^{29} - q^{31} + 3 q^{33} + 3 q^{37} + q^{39} - 11 q^{41} - 8 q^{43} + 2 q^{47} + 2 q^{49} + 7 q^{51} - 5 q^{53} + 6 q^{57} + 7 q^{59} - 9 q^{61} + 2 q^{63} - 5 q^{67} + 5 q^{71} + 6 q^{73} + 3 q^{77} + 21 q^{79} + 2 q^{81} + 7 q^{83} + 8 q^{87} + 20 q^{89} + q^{91} - q^{93} + 12 q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 + 3 * q^11 + q^13 + 7 * q^17 + 6 * q^19 + 2 * q^21 + 2 * q^27 + 8 * q^29 - q^31 + 3 * q^33 + 3 * q^37 + q^39 - 11 * q^41 - 8 * q^43 + 2 * q^47 + 2 * q^49 + 7 * q^51 - 5 * q^53 + 6 * q^57 + 7 * q^59 - 9 * q^61 + 2 * q^63 - 5 * q^67 + 5 * q^71 + 6 * q^73 + 3 * q^77 + 21 * q^79 + 2 * q^81 + 7 * q^83 + 8 * q^87 + 20 * q^89 + q^91 - q^93 + 12 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.561553	−0.169315	−0.0846573	−	0.996410i	$-0.526980\pi$
$11$	−0.561553	−0.169315	−0.0846573	+	0.996410i	$0.526980\pi$
$12$	0	0
$13$	6.68466	1.85399	0.926995	−	0.375073i	$-0.122382\pi$
$13$	6.68466	1.85399	0.926995	+	0.375073i	$0.122382\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.56155	1.34887	0.674437	−	0.738332i	$-0.264386\pi$
$17$	5.56155	1.34887	0.674437	+	0.738332i	$0.264386\pi$
$18$	0	0
$19$	−1.12311	−0.257658	−0.128829	−	0.991667i	$-0.541122\pi$
$19$	−1.12311	−0.257658	−0.128829	+	0.991667i	$0.541122\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−4.12311	−0.859727	−0.429863	−	0.902894i	$-0.641438\pi$
$23$	−4.12311	−0.859727	−0.429863	+	0.902894i	$0.641438\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.12311	1.50842	0.754211	−	0.656632i	$-0.228019\pi$
$29$	8.12311	1.50842	0.754211	+	0.656632i	$0.228019\pi$
$30$	0	0
$31$	−6.68466	−1.20060	−0.600300	−	0.799775i	$-0.704952\pi$
$31$	−6.68466	−1.20060	−0.600300	+	0.799775i	$0.704952\pi$
$32$	0	0
$33$	−0.561553	−0.0977538
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.561553	−0.0923187	−0.0461594	−	0.998934i	$-0.514698\pi$
$37$	−0.561553	−0.0923187	−0.0461594	+	0.998934i	$0.514698\pi$
$38$	0	0
$39$	6.68466	1.07040
$40$	0	0
$41$	0.684658	0.106926	0.0534628	−	0.998570i	$-0.482974\pi$
$41$	0.684658	0.106926	0.0534628	+	0.998570i	$0.482974\pi$
$42$	0	0
$43$	0.123106	0.0187734	0.00938672	−	0.999956i	$-0.497012\pi$
$43$	0.123106	0.0187734	0.00938672	+	0.999956i	$0.497012\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.12311	0.747282	0.373641	−	0.927573i	$-0.378109\pi$
$47$	5.12311	0.747282	0.373641	+	0.927573i	$0.378109\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.56155	0.778773
$52$	0	0
$53$	−0.438447	−0.0602254	−0.0301127	−	0.999547i	$-0.509587\pi$
$53$	−0.438447	−0.0602254	−0.0301127	+	0.999547i	$0.509587\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.12311	−0.148759
$58$	0	0
$59$	−2.68466	−0.349513	−0.174756	−	0.984612i	$-0.555914\pi$
$59$	−2.68466	−0.349513	−0.174756	+	0.984612i	$0.555914\pi$
$60$	0	0
$61$	−2.43845	−0.312211	−0.156106	−	0.987740i	$-0.549894\pi$
$61$	−2.43845	−0.312211	−0.156106	+	0.987740i	$0.549894\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.8078	−1.56472	−0.782359	−	0.622828i	$-0.785984\pi$
$67$	−12.8078	−1.56472	−0.782359	+	0.622828i	$0.785984\pi$
$68$	0	0
$69$	−4.12311	−0.496364
$70$	0	0
$71$	12.8078	1.52000	0.760001	−	0.649922i	$-0.225198\pi$
$71$	12.8078	1.52000	0.760001	+	0.649922i	$0.225198\pi$
$72$	0	0
$73$	−9.36932	−1.09660	−0.548298	−	0.836283i	$-0.684724\pi$
$73$	−9.36932	−1.09660	−0.548298	+	0.836283i	$0.684724\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.561553	−0.0639949
$78$	0	0
$79$	12.5616	1.41329	0.706643	−	0.707571i	$-0.250209\pi$
$79$	12.5616	1.41329	0.706643	+	0.707571i	$0.250209\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.68466	−0.294680	−0.147340	−	0.989086i	$-0.547071\pi$
$83$	−2.68466	−0.294680	−0.147340	+	0.989086i	$0.547071\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.12311	0.870888
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	6.68466	0.700743
$92$	0	0
$93$	−6.68466	−0.693167
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	−0.561553	−0.0564382
$100$	0	0
$101$	−16.4924	−1.64106	−0.820529	−	0.571605i	$-0.806321\pi$
$101$	−16.4924	−1.64106	−0.820529	+	0.571605i	$0.806321\pi$
$102$	0	0
$103$	−18.6847	−1.84105	−0.920527	−	0.390679i	$-0.872240\pi$
$103$	−18.6847	−1.84105	−0.920527	+	0.390679i	$0.872240\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.12311	0.688617	0.344308	−	0.938857i	$-0.388113\pi$
$107$	7.12311	0.688617	0.344308	+	0.938857i	$0.388113\pi$
$108$	0	0
$109$	−3.68466	−0.352926	−0.176463	−	0.984307i	$-0.556466\pi$
$109$	−3.68466	−0.352926	−0.176463	+	0.984307i	$0.556466\pi$
$110$	0	0
$111$	−0.561553	−0.0533002
$112$	0	0
$113$	10.8078	1.01671	0.508354	−	0.861148i	$-0.330254\pi$
$113$	10.8078	1.01671	0.508354	+	0.861148i	$0.330254\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.68466	0.617997
$118$	0	0
$119$	5.56155	0.509827
$120$	0	0
$121$	−10.6847	−0.971333
$122$	0	0
$123$	0.684658	0.0617336
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.80776	0.604091	0.302046	−	0.953293i	$-0.402330\pi$
$127$	6.80776	0.604091	0.302046	+	0.953293i	$0.402330\pi$
$128$	0	0
$129$	0.123106	0.0108388
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	−1.12311	−0.0973856
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.1231	1.80467	0.902334	−	0.431037i	$-0.141852\pi$
$137$	21.1231	1.80467	0.902334	+	0.431037i	$0.141852\pi$
$138$	0	0
$139$	18.2462	1.54762	0.773812	−	0.633416i	$-0.218348\pi$
$139$	18.2462	1.54762	0.773812	+	0.633416i	$0.218348\pi$
$140$	0	0
$141$	5.12311	0.431443
$142$	0	0
$143$	−3.75379	−0.313908
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	7.00000	0.573462	0.286731	−	0.958011i	$-0.407431\pi$
$149$	7.00000	0.573462	0.286731	+	0.958011i	$0.407431\pi$
$150$	0	0
$151$	5.68466	0.462611	0.231305	−	0.972881i	$-0.425700\pi$
$151$	5.68466	0.462611	0.231305	+	0.972881i	$0.425700\pi$
$152$	0	0
$153$	5.56155	0.449625
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.3693	1.06699	0.533494	−	0.845804i	$-0.320879\pi$
$157$	13.3693	1.06699	0.533494	+	0.845804i	$0.320879\pi$
$158$	0	0
$159$	−0.438447	−0.0347711
$160$	0	0
$161$	−4.12311	−0.324946
$162$	0	0
$163$	6.93087	0.542868	0.271434	−	0.962457i	$-0.412502\pi$
$163$	6.93087	0.542868	0.271434	+	0.962457i	$0.412502\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.4924	−1.43099	−0.715493	−	0.698620i	$-0.753798\pi$
$167$	−18.4924	−1.43099	−0.715493	+	0.698620i	$0.753798\pi$
$168$	0	0
$169$	31.6847	2.43728
$170$	0	0
$171$	−1.12311	−0.0858860
$172$	0	0
$173$	9.12311	0.693617	0.346808	−	0.937936i	$-0.387265\pi$
$173$	9.12311	0.693617	0.346808	+	0.937936i	$0.387265\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.68466	−0.201791
$178$	0	0
$179$	−8.87689	−0.663490	−0.331745	−	0.943369i	$-0.607637\pi$
$179$	−8.87689	−0.663490	−0.331745	+	0.943369i	$0.607637\pi$
$180$	0	0
$181$	−16.8769	−1.25445	−0.627225	−	0.778838i	$-0.715809\pi$
$181$	−16.8769	−1.25445	−0.627225	+	0.778838i	$0.715809\pi$
$182$	0	0
$183$	−2.43845	−0.180255
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.12311	−0.228384
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	4.19224	0.303340	0.151670	−	0.988431i	$-0.451535\pi$
$191$	4.19224	0.303340	0.151670	+	0.988431i	$0.451535\pi$
$192$	0	0
$193$	−15.4384	−1.11128	−0.555642	−	0.831422i	$-0.687527\pi$
$193$	−15.4384	−1.11128	−0.555642	+	0.831422i	$0.687527\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.12311	−0.293759	−0.146880	−	0.989154i	$-0.546923\pi$
$197$	−4.12311	−0.293759	−0.146880	+	0.989154i	$0.546923\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−12.8078	−0.903390
$202$	0	0
$203$	8.12311	0.570130
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.12311	−0.286576
$208$	0	0
$209$	0.630683	0.0436253
$210$	0	0
$211$	−2.93087	−0.201769	−0.100885	−	0.994898i	$-0.532167\pi$
$211$	−2.93087	−0.201769	−0.100885	+	0.994898i	$0.532167\pi$
$212$	0	0
$213$	12.8078	0.877574
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6.68466	−0.453784
$218$	0	0
$219$	−9.36932	−0.633120
$220$	0	0
$221$	37.1771	2.50080
$222$	0	0
$223$	−17.5616	−1.17601	−0.588004	−	0.808858i	$-0.700086\pi$
$223$	−17.5616	−1.17601	−0.588004	+	0.808858i	$0.700086\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.19224	0.145504	0.0727519	−	0.997350i	$-0.476822\pi$
$227$	2.19224	0.145504	0.0727519	+	0.997350i	$0.476822\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−0.561553	−0.0369475
$232$	0	0
$233$	−15.6847	−1.02754	−0.513768	−	0.857929i	$-0.671751\pi$
$233$	−15.6847	−1.02754	−0.513768	+	0.857929i	$0.671751\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.5616	0.815961
$238$	0	0
$239$	2.24621	0.145295	0.0726477	−	0.997358i	$-0.476855\pi$
$239$	2.24621	0.145295	0.0726477	+	0.997358i	$0.476855\pi$
$240$	0	0
$241$	14.4924	0.933539	0.466769	−	0.884379i	$-0.345418\pi$
$241$	14.4924	0.933539	0.466769	+	0.884379i	$0.345418\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7.50758	−0.477696
$248$	0	0
$249$	−2.68466	−0.170133
$250$	0	0
$251$	−1.56155	−0.0985643	−0.0492822	−	0.998785i	$-0.515693\pi$
$251$	−1.56155	−0.0985643	−0.0492822	+	0.998785i	$0.515693\pi$
$252$	0	0
$253$	2.31534	0.145564
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.19224	0.136748	0.0683740	−	0.997660i	$-0.478219\pi$
$257$	2.19224	0.136748	0.0683740	+	0.997660i	$0.478219\pi$
$258$	0	0
$259$	−0.561553	−0.0348932
$260$	0	0
$261$	8.12311	0.502808
$262$	0	0
$263$	−29.2462	−1.80340	−0.901699	−	0.432364i	$-0.857680\pi$
$263$	−29.2462	−1.80340	−0.901699	+	0.432364i	$0.857680\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	22.0000	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$270$	0	0
$271$	27.3693	1.66257	0.831284	−	0.555848i	$-0.187606\pi$
$271$	27.3693	1.66257	0.831284	+	0.555848i	$0.187606\pi$
$272$	0	0
$273$	6.68466	0.404574
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.75379	−0.465880	−0.232940	−	0.972491i	$-0.574835\pi$
$277$	−7.75379	−0.465880	−0.232940	+	0.972491i	$0.574835\pi$
$278$	0	0
$279$	−6.68466	−0.400200
$280$	0	0
$281$	−13.9309	−0.831046	−0.415523	−	0.909583i	$-0.636401\pi$
$281$	−13.9309	−0.831046	−0.415523	+	0.909583i	$0.636401\pi$
$282$	0	0
$283$	−19.3693	−1.15139	−0.575693	−	0.817666i	$-0.695268\pi$
$283$	−19.3693	−1.15139	−0.575693	+	0.817666i	$0.695268\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.684658	0.0404141
$288$	0	0
$289$	13.9309	0.819463
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	−25.6155	−1.49648	−0.748238	−	0.663431i	$-0.769100\pi$
$293$	−25.6155	−1.49648	−0.748238	+	0.663431i	$0.769100\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.561553	−0.0325846
$298$	0	0
$299$	−27.5616	−1.59393
$300$	0	0
$301$	0.123106	0.00709569
$302$	0	0
$303$	−16.4924	−0.947465
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.1231	0.863121	0.431561	−	0.902084i	$-0.357963\pi$
$307$	15.1231	0.863121	0.431561	+	0.902084i	$0.357963\pi$
$308$	0	0
$309$	−18.6847	−1.06293
$310$	0	0
$311$	5.36932	0.304466	0.152233	−	0.988345i	$-0.451354\pi$
$311$	5.36932	0.304466	0.152233	+	0.988345i	$0.451354\pi$
$312$	0	0
$313$	−20.2462	−1.14438	−0.572192	−	0.820120i	$-0.693907\pi$
$313$	−20.2462	−1.14438	−0.572192	+	0.820120i	$0.693907\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.87689	0.442410	0.221205	−	0.975227i	$-0.429001\pi$
$317$	7.87689	0.442410	0.221205	+	0.975227i	$0.429001\pi$
$318$	0	0
$319$	−4.56155	−0.255398
$320$	0	0
$321$	7.12311	0.397573
$322$	0	0
$323$	−6.24621	−0.347548
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.68466	−0.203762
$328$	0	0
$329$	5.12311	0.282446
$330$	0	0
$331$	3.87689	0.213093	0.106547	−	0.994308i	$-0.466021\pi$
$331$	3.87689	0.213093	0.106547	+	0.994308i	$0.466021\pi$
$332$	0	0
$333$	−0.561553	−0.0307729
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−21.8078	−1.18794	−0.593972	−	0.804485i	$-0.702441\pi$
$337$	−21.8078	−1.18794	−0.593972	+	0.804485i	$0.702441\pi$
$338$	0	0
$339$	10.8078	0.586997
$340$	0	0
$341$	3.75379	0.203279
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.1771	1.40526	0.702630	−	0.711556i	$-0.252009\pi$
$347$	26.1771	1.40526	0.702630	+	0.711556i	$0.252009\pi$
$348$	0	0
$349$	21.8078	1.16734	0.583671	−	0.811990i	$-0.301616\pi$
$349$	21.8078	1.16734	0.583671	+	0.811990i	$0.301616\pi$
$350$	0	0
$351$	6.68466	0.356801
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.56155	0.294349
$358$	0	0
$359$	−29.7386	−1.56955	−0.784773	−	0.619784i	$-0.787220\pi$
$359$	−29.7386	−1.56955	−0.784773	+	0.619784i	$0.787220\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	−10.6847	−0.560799
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−26.9309	−1.40578	−0.702890	−	0.711299i	$-0.748107\pi$
$367$	−26.9309	−1.40578	−0.702890	+	0.711299i	$0.748107\pi$
$368$	0	0
$369$	0.684658	0.0356419
$370$	0	0
$371$	−0.438447	−0.0227630
$372$	0	0
$373$	32.1771	1.66607	0.833033	−	0.553223i	$-0.186602\pi$
$373$	32.1771	1.66607	0.833033	+	0.553223i	$0.186602\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	54.3002	2.79660
$378$	0	0
$379$	−13.4924	−0.693059	−0.346530	−	0.938039i	$-0.612640\pi$
$379$	−13.4924	−0.693059	−0.346530	+	0.938039i	$0.612640\pi$
$380$	0	0
$381$	6.80776	0.348772
$382$	0	0
$383$	17.6155	0.900111	0.450056	−	0.893000i	$-0.351404\pi$
$383$	17.6155	0.900111	0.450056	+	0.893000i	$0.351404\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.123106	0.00625781
$388$	0	0
$389$	−2.31534	−0.117392	−0.0586962	−	0.998276i	$-0.518694\pi$
$389$	−2.31534	−0.117392	−0.0586962	+	0.998276i	$0.518694\pi$
$390$	0	0
$391$	−22.9309	−1.15966
$392$	0	0
$393$	16.0000	0.807093
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−28.9309	−1.45200	−0.725999	−	0.687695i	$-0.758622\pi$
$397$	−28.9309	−1.45200	−0.725999	+	0.687695i	$0.758622\pi$
$398$	0	0
$399$	−1.12311	−0.0562256
$400$	0	0
$401$	33.3002	1.66293	0.831466	−	0.555576i	$-0.187502\pi$
$401$	33.3002	1.66293	0.831466	+	0.555576i	$0.187502\pi$
$402$	0	0
$403$	−44.6847	−2.22590
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.315342	0.0156309
$408$	0	0
$409$	−6.49242	−0.321030	−0.160515	−	0.987033i	$-0.551315\pi$
$409$	−6.49242	−0.321030	−0.160515	+	0.987033i	$0.551315\pi$
$410$	0	0
$411$	21.1231	1.04193
$412$	0	0
$413$	−2.68466	−0.132103
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.2462	0.893521
$418$	0	0
$419$	11.1771	0.546036	0.273018	−	0.962009i	$-0.411978\pi$
$419$	11.1771	0.546036	0.273018	+	0.962009i	$0.411978\pi$
$420$	0	0
$421$	−5.68466	−0.277053	−0.138527	−	0.990359i	$-0.544237\pi$
$421$	−5.68466	−0.277053	−0.138527	+	0.990359i	$0.544237\pi$
$422$	0	0
$423$	5.12311	0.249094
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.43845	−0.118005
$428$	0	0
$429$	−3.75379	−0.181235
$430$	0	0
$431$	−24.3002	−1.17050	−0.585249	−	0.810853i	$-0.699003\pi$
$431$	−24.3002	−1.17050	−0.585249	+	0.810853i	$0.699003\pi$
$432$	0	0
$433$	33.8617	1.62729	0.813646	−	0.581361i	$-0.197480\pi$
$433$	33.8617	1.62729	0.813646	+	0.581361i	$0.197480\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.63068	0.221516
$438$	0	0
$439$	15.5616	0.742712	0.371356	−	0.928490i	$-0.378893\pi$
$439$	15.5616	0.742712	0.371356	+	0.928490i	$0.378893\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.00000	0.331089
$448$	0	0
$449$	1.43845	0.0678845	0.0339423	−	0.999424i	$-0.489194\pi$
$449$	1.43845	0.0678845	0.0339423	+	0.999424i	$0.489194\pi$
$450$	0	0
$451$	−0.384472	−0.0181041
$452$	0	0
$453$	5.68466	0.267089
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	5.56155	0.259591
$460$	0	0
$461$	−35.3693	−1.64731	−0.823657	−	0.567089i	$-0.808070\pi$
$461$	−35.3693	−1.64731	−0.823657	+	0.567089i	$0.808070\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.19224	−0.101445	−0.0507223	−	0.998713i	$-0.516152\pi$
$467$	−2.19224	−0.101445	−0.0507223	+	0.998713i	$0.516152\pi$
$468$	0	0
$469$	−12.8078	−0.591408
$470$	0	0
$471$	13.3693	0.616026
$472$	0	0
$473$	−0.0691303	−0.00317862
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.438447	−0.0200751
$478$	0	0
$479$	17.6155	0.804874	0.402437	−	0.915448i	$-0.368163\pi$
$479$	17.6155	0.804874	0.402437	+	0.915448i	$0.368163\pi$
$480$	0	0
$481$	−3.75379	−0.171158
$482$	0	0
$483$	−4.12311	−0.187608
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−36.5616	−1.65676	−0.828381	−	0.560165i	$-0.810738\pi$
$487$	−36.5616	−1.65676	−0.828381	+	0.560165i	$0.810738\pi$
$488$	0	0
$489$	6.93087	0.313425
$490$	0	0
$491$	−12.5616	−0.566895	−0.283447	−	0.958988i	$-0.591478\pi$
$491$	−12.5616	−0.566895	−0.283447	+	0.958988i	$0.591478\pi$
$492$	0	0
$493$	45.1771	2.03467
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.8078	0.574507
$498$	0	0
$499$	9.17708	0.410823	0.205411	−	0.978676i	$-0.434147\pi$
$499$	9.17708	0.410823	0.205411	+	0.978676i	$0.434147\pi$
$500$	0	0
$501$	−18.4924	−0.826181
$502$	0	0
$503$	18.4924	0.824536	0.412268	−	0.911063i	$-0.364737\pi$
$503$	18.4924	0.824536	0.412268	+	0.911063i	$0.364737\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	31.6847	1.40717
$508$	0	0
$509$	−0.630683	−0.0279545	−0.0139773	−	0.999902i	$-0.504449\pi$
$509$	−0.630683	−0.0279545	−0.0139773	+	0.999902i	$0.504449\pi$
$510$	0	0
$511$	−9.36932	−0.414474
$512$	0	0
$513$	−1.12311	−0.0495863
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.87689	−0.126526
$518$	0	0
$519$	9.12311	0.400460
$520$	0	0
$521$	12.9309	0.566512	0.283256	−	0.959044i	$-0.408585\pi$
$521$	12.9309	0.566512	0.283256	+	0.959044i	$0.408585\pi$
$522$	0	0
$523$	−22.2462	−0.972759	−0.486379	−	0.873748i	$-0.661683\pi$
$523$	−22.2462	−0.972759	−0.486379	+	0.873748i	$0.661683\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.1771	−1.61946
$528$	0	0
$529$	−6.00000	−0.260870
$530$	0	0
$531$	−2.68466	−0.116504
$532$	0	0
$533$	4.57671	0.198239
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−8.87689	−0.383066
$538$	0	0
$539$	−0.561553	−0.0241878
$540$	0	0
$541$	−20.8078	−0.894596	−0.447298	−	0.894385i	$-0.647614\pi$
$541$	−20.8078	−0.894596	−0.447298	+	0.894385i	$0.647614\pi$
$542$	0	0
$543$	−16.8769	−0.724257
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18.1231	0.774888	0.387444	−	0.921893i	$-0.373358\pi$
$547$	18.1231	0.774888	0.387444	+	0.921893i	$0.373358\pi$
$548$	0	0
$549$	−2.43845	−0.104070
$550$	0	0
$551$	−9.12311	−0.388657
$552$	0	0
$553$	12.5616	0.534172
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.4233	−1.54330	−0.771652	−	0.636045i	$-0.780569\pi$
$557$	−36.4233	−1.54330	−0.771652	+	0.636045i	$0.780569\pi$
$558$	0	0
$559$	0.822919	0.0348058
$560$	0	0
$561$	−3.12311	−0.131858
$562$	0	0
$563$	24.0540	1.01375	0.506877	−	0.862018i	$-0.330800\pi$
$563$	24.0540	1.01375	0.506877	+	0.862018i	$0.330800\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	6.94602	0.291193	0.145596	−	0.989344i	$-0.453490\pi$
$569$	6.94602	0.291193	0.145596	+	0.989344i	$0.453490\pi$
$570$	0	0
$571$	13.7386	0.574944	0.287472	−	0.957789i	$-0.407185\pi$
$571$	13.7386	0.574944	0.287472	+	0.957789i	$0.407185\pi$
$572$	0	0
$573$	4.19224	0.175133
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−30.4924	−1.26942	−0.634708	−	0.772752i	$-0.718880\pi$
$577$	−30.4924	−1.26942	−0.634708	+	0.772752i	$0.718880\pi$
$578$	0	0
$579$	−15.4384	−0.641600
$580$	0	0
$581$	−2.68466	−0.111378
$582$	0	0
$583$	0.246211	0.0101970
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.6847	0.853747	0.426874	−	0.904311i	$-0.359615\pi$
$587$	20.6847	0.853747	0.426874	+	0.904311i	$0.359615\pi$
$588$	0	0
$589$	7.50758	0.309344
$590$	0	0
$591$	−4.12311	−0.169602
$592$	0	0
$593$	−41.3693	−1.69883	−0.849417	−	0.527722i	$-0.823046\pi$
$593$	−41.3693	−1.69883	−0.849417	+	0.527722i	$0.823046\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	−44.6155	−1.82294	−0.911471	−	0.411365i	$-0.865052\pi$
$599$	−44.6155	−1.82294	−0.911471	+	0.411365i	$0.865052\pi$
$600$	0	0
$601$	−30.2462	−1.23377	−0.616884	−	0.787054i	$-0.711605\pi$
$601$	−30.2462	−1.23377	−0.616884	+	0.787054i	$0.711605\pi$
$602$	0	0
$603$	−12.8078	−0.521572
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−19.3693	−0.786176	−0.393088	−	0.919501i	$-0.628593\pi$
$607$	−19.3693	−0.786176	−0.393088	+	0.919501i	$0.628593\pi$
$608$	0	0
$609$	8.12311	0.329165
$610$	0	0
$611$	34.2462	1.38545
$612$	0	0
$613$	−39.7926	−1.60721	−0.803604	−	0.595164i	$-0.797087\pi$
$613$	−39.7926	−1.60721	−0.803604	+	0.595164i	$0.797087\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.4233	0.661177	0.330588	−	0.943775i	$-0.392753\pi$
$617$	16.4233	0.661177	0.330588	+	0.943775i	$0.392753\pi$
$618$	0	0
$619$	−6.00000	−0.241160	−0.120580	−	0.992704i	$-0.538475\pi$
$619$	−6.00000	−0.241160	−0.120580	+	0.992704i	$0.538475\pi$
$620$	0	0
$621$	−4.12311	−0.165455
$622$	0	0
$623$	10.0000	0.400642
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.630683	0.0251871
$628$	0	0
$629$	−3.12311	−0.124526
$630$	0	0
$631$	−39.4384	−1.57002	−0.785010	−	0.619483i	$-0.787342\pi$
$631$	−39.4384	−1.57002	−0.785010	+	0.619483i	$0.787342\pi$
$632$	0	0
$633$	−2.93087	−0.116492
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.68466	0.264856
$638$	0	0
$639$	12.8078	0.506667
$640$	0	0
$641$	15.4384	0.609782	0.304891	−	0.952387i	$-0.401380\pi$
$641$	15.4384	0.609782	0.304891	+	0.952387i	$0.401380\pi$
$642$	0	0
$643$	24.8769	0.981049	0.490524	−	0.871427i	$-0.336805\pi$
$643$	24.8769	0.981049	0.490524	+	0.871427i	$0.336805\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.50758	−0.216525	−0.108263	−	0.994122i	$-0.534529\pi$
$647$	−5.50758	−0.216525	−0.108263	+	0.994122i	$0.534529\pi$
$648$	0	0
$649$	1.50758	0.0591776
$650$	0	0
$651$	−6.68466	−0.261992
$652$	0	0
$653$	32.7386	1.28116	0.640581	−	0.767891i	$-0.278694\pi$
$653$	32.7386	1.28116	0.640581	+	0.767891i	$0.278694\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.36932	−0.365532
$658$	0	0
$659$	−21.3693	−0.832430	−0.416215	−	0.909266i	$-0.636644\pi$
$659$	−21.3693	−0.832430	−0.416215	+	0.909266i	$0.636644\pi$
$660$	0	0
$661$	−20.2462	−0.787486	−0.393743	−	0.919220i	$-0.628820\pi$
$661$	−20.2462	−0.787486	−0.393743	+	0.919220i	$0.628820\pi$
$662$	0	0
$663$	37.1771	1.44384
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−33.4924	−1.29683
$668$	0	0
$669$	−17.5616	−0.678969
$670$	0	0
$671$	1.36932	0.0528619
$672$	0	0
$673$	36.4384	1.40460	0.702299	−	0.711882i	$-0.252157\pi$
$673$	36.4384	1.40460	0.702299	+	0.711882i	$0.252157\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.61553	0.215822	0.107911	−	0.994161i	$-0.465584\pi$
$677$	5.61553	0.215822	0.107911	+	0.994161i	$0.465584\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	2.19224	0.0840067
$682$	0	0
$683$	−22.8078	−0.872715	−0.436357	−	0.899773i	$-0.643732\pi$
$683$	−22.8078	−0.872715	−0.436357	+	0.899773i	$0.643732\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	−2.93087	−0.111657
$690$	0	0
$691$	22.6307	0.860912	0.430456	−	0.902612i	$-0.358353\pi$
$691$	22.6307	0.860912	0.430456	+	0.902612i	$0.358353\pi$
$692$	0	0
$693$	−0.561553	−0.0213316
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.80776	0.144229
$698$	0	0
$699$	−15.6847	−0.593248
$700$	0	0
$701$	43.1771	1.63078	0.815388	−	0.578915i	$-0.196524\pi$
$701$	43.1771	1.63078	0.815388	+	0.578915i	$0.196524\pi$
$702$	0	0
$703$	0.630683	0.0237867
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.4924	−0.620261
$708$	0	0
$709$	−37.1231	−1.39419	−0.697094	−	0.716980i	$-0.745524\pi$
$709$	−37.1231	−1.39419	−0.697094	+	0.716980i	$0.745524\pi$
$710$	0	0
$711$	12.5616	0.471095
$712$	0	0
$713$	27.5616	1.03219
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.24621	0.0838863
$718$	0	0
$719$	−31.1231	−1.16070	−0.580348	−	0.814369i	$-0.697083\pi$
$719$	−31.1231	−1.16070	−0.580348	+	0.814369i	$0.697083\pi$
$720$	0	0
$721$	−18.6847	−0.695853
$722$	0	0
$723$	14.4924	0.538979
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.31534	−0.122959	−0.0614796	−	0.998108i	$-0.519582\pi$
$727$	−3.31534	−0.122959	−0.0614796	+	0.998108i	$0.519582\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.684658	0.0253230
$732$	0	0
$733$	−11.3153	−0.417942	−0.208971	−	0.977922i	$-0.567011\pi$
$733$	−11.3153	−0.417942	−0.208971	+	0.977922i	$0.567011\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.19224	0.264929
$738$	0	0
$739$	−46.1231	−1.69667	−0.848333	−	0.529463i	$-0.822393\pi$
$739$	−46.1231	−1.69667	−0.848333	+	0.529463i	$0.822393\pi$
$740$	0	0
$741$	−7.50758	−0.275798
$742$	0	0
$743$	13.0691	0.479460	0.239730	−	0.970840i	$-0.422941\pi$
$743$	13.0691	0.479460	0.239730	+	0.970840i	$0.422941\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.68466	−0.0982265
$748$	0	0
$749$	7.12311	0.260273
$750$	0	0
$751$	2.24621	0.0819654	0.0409827	−	0.999160i	$-0.486951\pi$
$751$	2.24621	0.0819654	0.0409827	+	0.999160i	$0.486951\pi$
$752$	0	0
$753$	−1.56155	−0.0569061
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−6.31534	−0.229535	−0.114767	−	0.993392i	$-0.536612\pi$
$757$	−6.31534	−0.229535	−0.114767	+	0.993392i	$0.536612\pi$
$758$	0	0
$759$	2.31534	0.0840416
$760$	0	0
$761$	−6.63068	−0.240362	−0.120181	−	0.992752i	$-0.538348\pi$
$761$	−6.63068	−0.240362	−0.120181	+	0.992752i	$0.538348\pi$
$762$	0	0
$763$	−3.68466	−0.133394
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−17.9460	−0.647993
$768$	0	0
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	0	0
$771$	2.19224	0.0789514
$772$	0	0
$773$	47.3693	1.70376	0.851878	−	0.523740i	$-0.175464\pi$
$773$	47.3693	1.70376	0.851878	+	0.523740i	$0.175464\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.561553	−0.0201456
$778$	0	0
$779$	−0.768944	−0.0275503
$780$	0	0
$781$	−7.19224	−0.257358
$782$	0	0
$783$	8.12311	0.290296
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−0.876894	−0.0312579	−0.0156290	−	0.999878i	$-0.504975\pi$
$787$	−0.876894	−0.0312579	−0.0156290	+	0.999878i	$0.504975\pi$
$788$	0	0
$789$	−29.2462	−1.04119
$790$	0	0
$791$	10.8078	0.384280
$792$	0	0
$793$	−16.3002	−0.578837
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.8617	−0.916070	−0.458035	−	0.888934i	$-0.651447\pi$
$797$	−25.8617	−0.916070	−0.458035	+	0.888934i	$0.651447\pi$
$798$	0	0
$799$	28.4924	1.00799
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	5.26137	0.185670
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.0000	0.774437
$808$	0	0
$809$	28.6695	1.00797	0.503983	−	0.863714i	$-0.331867\pi$
$809$	28.6695	1.00797	0.503983	+	0.863714i	$0.331867\pi$
$810$	0	0
$811$	0.246211	0.00864565	0.00432282	−	0.999991i	$-0.498624\pi$
$811$	0.246211	0.00864565	0.00432282	+	0.999991i	$0.498624\pi$
$812$	0	0
$813$	27.3693	0.959884
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.138261	−0.00483713
$818$	0	0
$819$	6.68466	0.233581
$820$	0	0
$821$	11.7538	0.410210	0.205105	−	0.978740i	$-0.434246\pi$
$821$	11.7538	0.410210	0.205105	+	0.978740i	$0.434246\pi$
$822$	0	0
$823$	−6.94602	−0.242123	−0.121062	−	0.992645i	$-0.538630\pi$
$823$	−6.94602	−0.242123	−0.121062	+	0.992645i	$0.538630\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.1922	0.736926	0.368463	−	0.929642i	$-0.379884\pi$
$827$	21.1922	0.736926	0.368463	+	0.929642i	$0.379884\pi$
$828$	0	0
$829$	37.6695	1.30832	0.654158	−	0.756358i	$-0.273023\pi$
$829$	37.6695	1.30832	0.654158	+	0.756358i	$0.273023\pi$
$830$	0	0
$831$	−7.75379	−0.268976
$832$	0	0
$833$	5.56155	0.192696
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.68466	−0.231056
$838$	0	0
$839$	−30.2462	−1.04422	−0.522108	−	0.852880i	$-0.674854\pi$
$839$	−30.2462	−1.04422	−0.522108	+	0.852880i	$0.674854\pi$
$840$	0	0
$841$	36.9848	1.27534
$842$	0	0
$843$	−13.9309	−0.479805
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.6847	−0.367129
$848$	0	0
$849$	−19.3693	−0.664753
$850$	0	0
$851$	2.31534	0.0793689
$852$	0	0
$853$	−26.0540	−0.892071	−0.446036	−	0.895015i	$-0.647165\pi$
$853$	−26.0540	−0.892071	−0.446036	+	0.895015i	$0.647165\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36.8769	1.25969	0.629845	−	0.776721i	$-0.283118\pi$
$857$	36.8769	1.25969	0.629845	+	0.776721i	$0.283118\pi$
$858$	0	0
$859$	54.1080	1.84614	0.923070	−	0.384633i	$-0.125672\pi$
$859$	54.1080	1.84614	0.923070	+	0.384633i	$0.125672\pi$
$860$	0	0
$861$	0.684658	0.0233331
$862$	0	0
$863$	−21.9309	−0.746535	−0.373268	−	0.927724i	$-0.621763\pi$
$863$	−21.9309	−0.746535	−0.373268	+	0.927724i	$0.621763\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.9309	0.473117
$868$	0	0
$869$	−7.05398	−0.239290
$870$	0	0
$871$	−85.6155	−2.90097
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.3693	0.518985	0.259492	−	0.965745i	$-0.416445\pi$
$877$	15.3693	0.518985	0.259492	+	0.965745i	$0.416445\pi$
$878$	0	0
$879$	−25.6155	−0.863990
$880$	0	0
$881$	−40.9309	−1.37900	−0.689498	−	0.724288i	$-0.742169\pi$
$881$	−40.9309	−1.37900	−0.689498	+	0.724288i	$0.742169\pi$
$882$	0	0
$883$	20.8617	0.702053	0.351027	−	0.936365i	$-0.385833\pi$
$883$	20.8617	0.702053	0.351027	+	0.936365i	$0.385833\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.8769	−0.700978	−0.350489	−	0.936567i	$-0.613985\pi$
$887$	−20.8769	−0.700978	−0.350489	+	0.936567i	$0.613985\pi$
$888$	0	0
$889$	6.80776	0.228325
$890$	0	0
$891$	−0.561553	−0.0188127
$892$	0	0
$893$	−5.75379	−0.192543
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−27.5616	−0.920253
$898$	0	0
$899$	−54.3002	−1.81101
$900$	0	0
$901$	−2.43845	−0.0812365
$902$	0	0
$903$	0.123106	0.00409670
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.68466	0.155552	0.0777758	−	0.996971i	$-0.475218\pi$
$907$	4.68466	0.155552	0.0777758	+	0.996971i	$0.475218\pi$
$908$	0	0
$909$	−16.4924	−0.547019
$910$	0	0
$911$	−53.0000	−1.75597	−0.877984	−	0.478690i	$-0.841112\pi$
$911$	−53.0000	−1.75597	−0.877984	+	0.478690i	$0.841112\pi$
$912$	0	0
$913$	1.50758	0.0498935
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.0000	0.528367
$918$	0	0
$919$	23.6847	0.781285	0.390642	−	0.920543i	$-0.372253\pi$
$919$	23.6847	0.781285	0.390642	+	0.920543i	$0.372253\pi$
$920$	0	0
$921$	15.1231	0.498323
$922$	0	0
$923$	85.6155	2.81807
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−18.6847	−0.613685
$928$	0	0
$929$	−15.8078	−0.518636	−0.259318	−	0.965792i	$-0.583498\pi$
$929$	−15.8078	−0.518636	−0.259318	+	0.965792i	$0.583498\pi$
$930$	0	0
$931$	−1.12311	−0.0368083
$932$	0	0
$933$	5.36932	0.175784
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−39.6155	−1.29418	−0.647091	−	0.762412i	$-0.724015\pi$
$937$	−39.6155	−1.29418	−0.647091	+	0.762412i	$0.724015\pi$
$938$	0	0
$939$	−20.2462	−0.660710
$940$	0	0
$941$	−38.1080	−1.24228	−0.621142	−	0.783698i	$-0.713331\pi$
$941$	−38.1080	−1.24228	−0.621142	+	0.783698i	$0.713331\pi$
$942$	0	0
$943$	−2.82292	−0.0919269
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.24621	0.0729921	0.0364960	−	0.999334i	$-0.488380\pi$
$947$	2.24621	0.0729921	0.0364960	+	0.999334i	$0.488380\pi$
$948$	0	0
$949$	−62.6307	−2.03308
$950$	0	0
$951$	7.87689	0.255426
$952$	0	0
$953$	16.1771	0.524027	0.262014	−	0.965064i	$-0.415613\pi$
$953$	16.1771	0.524027	0.262014	+	0.965064i	$0.415613\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.56155	−0.147454
$958$	0	0
$959$	21.1231	0.682101
$960$	0	0
$961$	13.6847	0.441441
$962$	0	0
$963$	7.12311	0.229539
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−34.7386	−1.11712	−0.558560	−	0.829464i	$-0.688646\pi$
$967$	−34.7386	−1.11712	−0.558560	+	0.829464i	$0.688646\pi$
$968$	0	0
$969$	−6.24621	−0.200657
$970$	0	0
$971$	−7.86174	−0.252295	−0.126148	−	0.992011i	$-0.540261\pi$
$971$	−7.86174	−0.252295	−0.126148	+	0.992011i	$0.540261\pi$
$972$	0	0
$973$	18.2462	0.584947
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.8078	−0.473742	−0.236871	−	0.971541i	$-0.576122\pi$
$977$	−14.8078	−0.473742	−0.236871	+	0.971541i	$0.576122\pi$
$978$	0	0
$979$	−5.61553	−0.179473
$980$	0	0
$981$	−3.68466	−0.117642
$982$	0	0
$983$	−11.8617	−0.378331	−0.189165	−	0.981945i	$-0.560578\pi$
$983$	−11.8617	−0.378331	−0.189165	+	0.981945i	$0.560578\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.12311	0.163070
$988$	0	0
$989$	−0.507577	−0.0161400
$990$	0	0
$991$	15.1922	0.482597	0.241299	−	0.970451i	$-0.422427\pi$
$991$	15.1922	0.482597	0.241299	+	0.970451i	$0.422427\pi$
$992$	0	0
$993$	3.87689	0.123030
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−32.2462	−1.02125	−0.510624	−	0.859804i	$-0.670586\pi$
$997$	−32.2462	−1.02125	−0.510624	+	0.859804i	$0.670586\pi$
$998$	0	0
$999$	−0.561553	−0.0177667

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4200.2.a.bm.1.1	yes	2
4.3	odd	2		8400.2.a.cu.1.2		2
5.2	odd	4		4200.2.t.w.1849.1		4
5.3	odd	4		4200.2.t.w.1849.3		4
5.4	even	2		4200.2.a.bi.1.1	✓	2
20.19	odd	2		8400.2.a.dc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4200.2.a.bi.1.1	✓	2	5.4	even	2
4200.2.a.bm.1.1	yes	2	1.1	even	1	trivial
4200.2.t.w.1849.1		4	5.2	odd	4
4200.2.t.w.1849.3		4	5.3	odd	4
8400.2.a.cu.1.2		2	4.3	odd	2
8400.2.a.dc.1.2		2	20.19	odd	2

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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:	\( N \)	\(=\)	\( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4200.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -0.561553 q^{11} +6.68466 q^{13} +5.56155 q^{17} -1.12311 q^{19} +1.00000 q^{21} -4.12311 q^{23} +1.00000 q^{27} +8.12311 q^{29} -6.68466 q^{31} -0.561553 q^{33} -0.561553 q^{37} +6.68466 q^{39} +0.684658 q^{41} +0.123106 q^{43} +5.12311 q^{47} +1.00000 q^{49} +5.56155 q^{51} -0.438447 q^{53} -1.12311 q^{57} -2.68466 q^{59} -2.43845 q^{61} +1.00000 q^{63} -12.8078 q^{67} -4.12311 q^{69} +12.8078 q^{71} -9.36932 q^{73} -0.561553 q^{77} +12.5616 q^{79} +1.00000 q^{81} -2.68466 q^{83} +8.12311 q^{87} +10.0000 q^{89} +6.68466 q^{91} -6.68466 q^{93} +6.00000 q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + 3 q^{11} + q^{13} + 7 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{27} + 8 q^{29} - q^{31} + 3 q^{33} + 3 q^{37} + q^{39} - 11 q^{41} - 8 q^{43} + 2 q^{47} + 2 q^{49} + 7 q^{51} - 5 q^{53} + 6 q^{57} + 7 q^{59} - 9 q^{61} + 2 q^{63} - 5 q^{67} + 5 q^{71} + 6 q^{73} + 3 q^{77} + 21 q^{79} + 2 q^{81} + 7 q^{83} + 8 q^{87} + 20 q^{89} + q^{91} - q^{93} + 12 q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 + 3 * q^11 + q^13 + 7 * q^17 + 6 * q^19 + 2 * q^21 + 2 * q^27 + 8 * q^29 - q^31 + 3 * q^33 + 3 * q^37 + q^39 - 11 * q^41 - 8 * q^43 + 2 * q^47 + 2 * q^49 + 7 * q^51 - 5 * q^53 + 6 * q^57 + 7 * q^59 - 9 * q^61 + 2 * q^63 - 5 * q^67 + 5 * q^71 + 6 * q^73 + 3 * q^77 + 21 * q^79 + 2 * q^81 + 7 * q^83 + 8 * q^87 + 20 * q^89 + q^91 - q^93 + 12 * q^97 + 3 * q^99

Introduction

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