LMFDB - Embedded newform 4200.2.a.bi.1.2

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			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +3.56155 q^{11} +5.68466 q^{13} -1.43845 q^{17} +7.12311 q^{19} +1.00000 q^{21} -4.12311 q^{23} -1.00000 q^{27} -0.123106 q^{29} +5.68466 q^{31} -3.56155 q^{33} -3.56155 q^{37} -5.68466 q^{39} -11.6847 q^{41} +8.12311 q^{43} +3.12311 q^{47} +1.00000 q^{49} +1.43845 q^{51} +4.56155 q^{53} -7.12311 q^{57} +9.68466 q^{59} -6.56155 q^{61} -1.00000 q^{63} -7.80776 q^{67} +4.12311 q^{69} -7.80776 q^{71} -15.3693 q^{73} -3.56155 q^{77} +8.43845 q^{79} +1.00000 q^{81} -9.68466 q^{83} +0.123106 q^{87} +10.0000 q^{89} -5.68466 q^{91} -5.68466 q^{93} -6.00000 q^{97} +3.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	3.56155	1.07385	0.536924	−	0.843630i	$-0.319586\pi$
$11$	3.56155	1.07385	0.536924	+	0.843630i	$0.319586\pi$
$12$	0	0
$13$	5.68466	1.57664	0.788320	−	0.615265i	$-0.210951\pi$
$13$	5.68466	1.57664	0.788320	+	0.615265i	$0.210951\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.43845	−0.348875	−0.174437	−	0.984668i	$-0.555811\pi$
$17$	−1.43845	−0.348875	−0.174437	+	0.984668i	$0.555811\pi$
$18$	0	0
$19$	7.12311	1.63415	0.817076	−	0.576530i	$-0.195593\pi$
$19$	7.12311	1.63415	0.817076	+	0.576530i	$0.195593\pi$
$20$	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
$29$	−0.123106	−0.0228601	−0.0114301	−	0.999935i	$-0.503638\pi$
$29$	−0.123106	−0.0228601	−0.0114301	+	0.999935i	$0.503638\pi$
$30$	0	0
$31$	5.68466	1.02099	0.510497	−	0.859879i	$-0.329461\pi$
$31$	5.68466	1.02099	0.510497	+	0.859879i	$0.329461\pi$
$32$	0	0
$33$	−3.56155	−0.619987
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.56155	−0.585516	−0.292758	−	0.956187i	$-0.594573\pi$
$37$	−3.56155	−0.585516	−0.292758	+	0.956187i	$0.594573\pi$
$38$	0	0
$39$	−5.68466	−0.910274
$40$	0	0
$41$	−11.6847	−1.82484	−0.912419	−	0.409258i	$-0.865787\pi$
$41$	−11.6847	−1.82484	−0.912419	+	0.409258i	$0.865787\pi$
$42$	0	0
$43$	8.12311	1.23876	0.619381	−	0.785091i	$-0.287384\pi$
$43$	8.12311	1.23876	0.619381	+	0.785091i	$0.287384\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.12311	0.455552	0.227776	−	0.973714i	$-0.426855\pi$
$47$	3.12311	0.455552	0.227776	+	0.973714i	$0.426855\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.43845	0.201423
$52$	0	0
$53$	4.56155	0.626577	0.313289	−	0.949658i	$-0.398569\pi$
$53$	4.56155	0.626577	0.313289	+	0.949658i	$0.398569\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.12311	−0.943478
$58$	0	0
$59$	9.68466	1.26084	0.630418	−	0.776256i	$-0.282884\pi$
$59$	9.68466	1.26084	0.630418	+	0.776256i	$0.282884\pi$
$60$	0	0
$61$	−6.56155	−0.840121	−0.420060	−	0.907496i	$-0.637991\pi$
$61$	−6.56155	−0.840121	−0.420060	+	0.907496i	$0.637991\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.80776	−0.953870	−0.476935	−	0.878939i	$-0.658252\pi$
$67$	−7.80776	−0.953870	−0.476935	+	0.878939i	$0.658252\pi$
$68$	0	0
$69$	4.12311	0.496364
$70$	0	0
$71$	−7.80776	−0.926611	−0.463306	−	0.886199i	$-0.653337\pi$
$71$	−7.80776	−0.926611	−0.463306	+	0.886199i	$0.653337\pi$
$72$	0	0
$73$	−15.3693	−1.79884	−0.899421	−	0.437083i	$-0.856012\pi$
$73$	−15.3693	−1.79884	−0.899421	+	0.437083i	$0.856012\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.56155	−0.405877
$78$	0	0
$79$	8.43845	0.949399	0.474700	−	0.880148i	$-0.342557\pi$
$79$	8.43845	0.949399	0.474700	+	0.880148i	$0.342557\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.68466	−1.06303	−0.531515	−	0.847049i	$-0.678377\pi$
$83$	−9.68466	−1.06303	−0.531515	+	0.847049i	$0.678377\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.123106	0.0131983
$88$	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
$91$	−5.68466	−0.595914
$92$	0	0
$93$	−5.68466	−0.589472
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	3.56155	0.357950
$100$	0	0
$101$	16.4924	1.64106	0.820529	−	0.571605i	$-0.193679\pi$
$101$	16.4924	1.64106	0.820529	+	0.571605i	$0.193679\pi$
$102$	0	0
$103$	6.31534	0.622269	0.311135	−	0.950366i	$-0.399291\pi$
$103$	6.31534	0.622269	0.311135	+	0.950366i	$0.399291\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.12311	0.108575	0.0542874	−	0.998525i	$-0.482711\pi$
$107$	1.12311	0.108575	0.0542874	+	0.998525i	$0.482711\pi$
$108$	0	0
$109$	8.68466	0.831839	0.415920	−	0.909401i	$-0.363460\pi$
$109$	8.68466	0.831839	0.415920	+	0.909401i	$0.363460\pi$
$110$	0	0
$111$	3.56155	0.338048
$112$	0	0
$113$	9.80776	0.922637	0.461318	−	0.887235i	$-0.347377\pi$
$113$	9.80776	0.922637	0.461318	+	0.887235i	$0.347377\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.68466	0.525547
$118$	0	0
$119$	1.43845	0.131862
$120$	0	0
$121$	1.68466	0.153151
$122$	0	0
$123$	11.6847	1.05357
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.8078	1.22524	0.612620	−	0.790377i	$-0.290115\pi$
$127$	13.8078	1.22524	0.612620	+	0.790377i	$0.290115\pi$
$128$	0	0
$129$	−8.12311	−0.715200
$130$	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
$133$	−7.12311	−0.617652
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.8769	−1.10015	−0.550074	−	0.835116i	$-0.685400\pi$
$137$	−12.8769	−1.10015	−0.550074	+	0.835116i	$0.685400\pi$
$138$	0	0
$139$	1.75379	0.148754	0.0743772	−	0.997230i	$-0.476303\pi$
$139$	1.75379	0.148754	0.0743772	+	0.997230i	$0.476303\pi$
$140$	0	0
$141$	−3.12311	−0.263013
$142$	0	0
$143$	20.2462	1.69307
$144$	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
$151$	−6.68466	−0.543990	−0.271995	−	0.962299i	$-0.587683\pi$
$151$	−6.68466	−0.543990	−0.271995	+	0.962299i	$0.587683\pi$
$152$	0	0
$153$	−1.43845	−0.116292
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.3693	0.907370	0.453685	−	0.891162i	$-0.350109\pi$
$157$	11.3693	0.907370	0.453685	+	0.891162i	$0.350109\pi$
$158$	0	0
$159$	−4.56155	−0.361755
$160$	0	0
$161$	4.12311	0.324946
$162$	0	0
$163$	21.9309	1.71776	0.858879	−	0.512178i	$-0.171161\pi$
$163$	21.9309	1.71776	0.858879	+	0.512178i	$0.171161\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.4924	−1.12146	−0.560729	−	0.828000i	$-0.689479\pi$
$167$	−14.4924	−1.12146	−0.560729	+	0.828000i	$0.689479\pi$
$168$	0	0
$169$	19.3153	1.48580
$170$	0	0
$171$	7.12311	0.544718
$172$	0	0
$173$	−0.876894	−0.0666690	−0.0333345	−	0.999444i	$-0.510613\pi$
$173$	−0.876894	−0.0666690	−0.0333345	+	0.999444i	$0.510613\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.68466	−0.727944
$178$	0	0
$179$	−17.1231	−1.27984	−0.639921	−	0.768441i	$-0.721033\pi$
$179$	−17.1231	−1.27984	−0.639921	+	0.768441i	$0.721033\pi$
$180$	0	0
$181$	−25.1231	−1.86739	−0.933693	−	0.358075i	$-0.883433\pi$
$181$	−25.1231	−1.86739	−0.933693	+	0.358075i	$0.883433\pi$
$182$	0	0
$183$	6.56155	0.485044
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.12311	−0.374639
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	24.8078	1.79503	0.897513	−	0.440987i	$-0.145372\pi$
$191$	24.8078	1.79503	0.897513	+	0.440987i	$0.145372\pi$
$192$	0	0
$193$	19.5616	1.40807	0.704036	−	0.710165i	$-0.251379\pi$
$193$	19.5616	1.40807	0.704036	+	0.710165i	$0.251379\pi$
$194$	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
$201$	7.80776	0.550717
$202$	0	0
$203$	0.123106	0.00864032
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.12311	−0.286576
$208$	0	0
$209$	25.3693	1.75483
$210$	0	0
$211$	25.9309	1.78515	0.892577	−	0.450894i	$-0.148895\pi$
$211$	25.9309	1.78515	0.892577	+	0.450894i	$0.148895\pi$
$212$	0	0
$213$	7.80776	0.534979
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.68466	−0.385900
$218$	0	0
$219$	15.3693	1.03856
$220$	0	0
$221$	−8.17708	−0.550050
$222$	0	0
$223$	13.4384	0.899905	0.449952	−	0.893052i	$-0.351441\pi$
$223$	13.4384	0.899905	0.449952	+	0.893052i	$0.351441\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.8078	−1.51380	−0.756902	−	0.653528i	$-0.773288\pi$
$227$	−22.8078	−1.51380	−0.756902	+	0.653528i	$0.773288\pi$
$228$	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
$231$	3.56155	0.234333
$232$	0	0
$233$	3.31534	0.217195	0.108598	−	0.994086i	$-0.465364\pi$
$233$	3.31534	0.217195	0.108598	+	0.994086i	$0.465364\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.43845	−0.548136
$238$	0	0
$239$	−14.2462	−0.921511	−0.460755	−	0.887527i	$-0.652421\pi$
$239$	−14.2462	−0.921511	−0.460755	+	0.887527i	$0.652421\pi$
$240$	0	0
$241$	−18.4924	−1.19120	−0.595601	−	0.803281i	$-0.703086\pi$
$241$	−18.4924	−1.19120	−0.595601	+	0.803281i	$0.703086\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	40.4924	2.57647
$248$	0	0
$249$	9.68466	0.613740
$250$	0	0
$251$	2.56155	0.161684	0.0808419	−	0.996727i	$-0.474239\pi$
$251$	2.56155	0.161684	0.0808419	+	0.996727i	$0.474239\pi$
$252$	0	0
$253$	−14.6847	−0.923217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.8078	−1.42271	−0.711355	−	0.702833i	$-0.751918\pi$
$257$	−22.8078	−1.42271	−0.711355	+	0.702833i	$0.751918\pi$
$258$	0	0
$259$	3.56155	0.221304
$260$	0	0
$261$	−0.123106	−0.00762005
$262$	0	0
$263$	12.7538	0.786432	0.393216	−	0.919446i	$-0.371362\pi$
$263$	12.7538	0.786432	0.393216	+	0.919446i	$0.371362\pi$
$264$	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
$271$	2.63068	0.159803	0.0799013	−	0.996803i	$-0.474539\pi$
$271$	2.63068	0.159803	0.0799013	+	0.996803i	$0.474539\pi$
$272$	0	0
$273$	5.68466	0.344051
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.2462	1.45681	0.728407	−	0.685145i	$-0.240261\pi$
$277$	24.2462	1.45681	0.728407	+	0.685145i	$0.240261\pi$
$278$	0	0
$279$	5.68466	0.340332
$280$	0	0
$281$	14.9309	0.890701	0.445351	−	0.895356i	$-0.353079\pi$
$281$	14.9309	0.890701	0.445351	+	0.895356i	$0.353079\pi$
$282$	0	0
$283$	−5.36932	−0.319173	−0.159586	−	0.987184i	$-0.551016\pi$
$283$	−5.36932	−0.319173	−0.159586	+	0.987184i	$0.551016\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.6847	0.689724
$288$	0	0
$289$	−14.9309	−0.878286
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	−15.6155	−0.912269	−0.456134	−	0.889911i	$-0.650766\pi$
$293$	−15.6155	−0.912269	−0.456134	+	0.889911i	$0.650766\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.56155	−0.206662
$298$	0	0
$299$	−23.4384	−1.35548
$300$	0	0
$301$	−8.12311	−0.468208
$302$	0	0
$303$	−16.4924	−0.947465
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6.87689	−0.392485	−0.196243	−	0.980555i	$-0.562874\pi$
$307$	−6.87689	−0.392485	−0.196243	+	0.980555i	$0.562874\pi$
$308$	0	0
$309$	−6.31534	−0.359267
$310$	0	0
$311$	−19.3693	−1.09833	−0.549167	−	0.835713i	$-0.685055\pi$
$311$	−19.3693	−1.09833	−0.549167	+	0.835713i	$0.685055\pi$
$312$	0	0
$313$	3.75379	0.212177	0.106088	−	0.994357i	$-0.466167\pi$
$313$	3.75379	0.212177	0.106088	+	0.994357i	$0.466167\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.1231	−0.905564	−0.452782	−	0.891621i	$-0.649568\pi$
$317$	−16.1231	−0.905564	−0.452782	+	0.891621i	$0.649568\pi$
$318$	0	0
$319$	−0.438447	−0.0245483
$320$	0	0
$321$	−1.12311	−0.0626856
$322$	0	0
$323$	−10.2462	−0.570114
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.68466	−0.480263
$328$	0	0
$329$	−3.12311	−0.172182
$330$	0	0
$331$	12.1231	0.666346	0.333173	−	0.942866i	$-0.391881\pi$
$331$	12.1231	0.666346	0.333173	+	0.942866i	$0.391881\pi$
$332$	0	0
$333$	−3.56155	−0.195172
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.19224	0.0649452	0.0324726	−	0.999473i	$-0.489662\pi$
$337$	1.19224	0.0649452	0.0324726	+	0.999473i	$0.489662\pi$
$338$	0	0
$339$	−9.80776	−0.532685
$340$	0	0
$341$	20.2462	1.09639
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.1771	1.02948	0.514740	−	0.857346i	$-0.327889\pi$
$347$	19.1771	1.02948	0.514740	+	0.857346i	$0.327889\pi$
$348$	0	0
$349$	1.19224	0.0638189	0.0319095	−	0.999491i	$-0.489841\pi$
$349$	1.19224	0.0638189	0.0319095	+	0.999491i	$0.489841\pi$
$350$	0	0
$351$	−5.68466	−0.303425
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.43845	−0.0761307
$358$	0	0
$359$	19.7386	1.04177	0.520883	−	0.853628i	$-0.325603\pi$
$359$	19.7386	1.04177	0.520883	+	0.853628i	$0.325603\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	0	0
$363$	−1.68466	−0.0884216
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.93087	−0.100791	−0.0503953	−	0.998729i	$-0.516048\pi$
$367$	−1.93087	−0.100791	−0.0503953	+	0.998729i	$0.516048\pi$
$368$	0	0
$369$	−11.6847	−0.608279
$370$	0	0
$371$	−4.56155	−0.236824
$372$	0	0
$373$	13.1771	0.682283	0.341142	−	0.940012i	$-0.389186\pi$
$373$	13.1771	0.682283	0.341142	+	0.940012i	$0.389186\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.699813	−0.0360422
$378$	0	0
$379$	19.4924	1.00126	0.500629	−	0.865662i	$-0.333102\pi$
$379$	19.4924	1.00126	0.500629	+	0.865662i	$0.333102\pi$
$380$	0	0
$381$	−13.8078	−0.707393
$382$	0	0
$383$	23.6155	1.20670	0.603349	−	0.797478i	$-0.293833\pi$
$383$	23.6155	1.20670	0.603349	+	0.797478i	$0.293833\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.12311	0.412921
$388$	0	0
$389$	−14.6847	−0.744542	−0.372271	−	0.928124i	$-0.621421\pi$
$389$	−14.6847	−0.744542	−0.372271	+	0.928124i	$0.621421\pi$
$390$	0	0
$391$	5.93087	0.299937
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.0691303	0.00346955	0.00173478	−	0.999998i	$-0.499448\pi$
$397$	0.0691303	0.00346955	0.00173478	+	0.999998i	$0.499448\pi$
$398$	0	0
$399$	7.12311	0.356601
$400$	0	0
$401$	−20.3002	−1.01374	−0.506871	−	0.862022i	$-0.669198\pi$
$401$	−20.3002	−1.01374	−0.506871	+	0.862022i	$0.669198\pi$
$402$	0	0
$403$	32.3153	1.60974
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.6847	−0.628755
$408$	0	0
$409$	26.4924	1.30997	0.654983	−	0.755644i	$-0.272676\pi$
$409$	26.4924	1.30997	0.654983	+	0.755644i	$0.272676\pi$
$410$	0	0
$411$	12.8769	0.635170
$412$	0	0
$413$	−9.68466	−0.476551
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.75379	−0.0858834
$418$	0	0
$419$	−34.1771	−1.66966	−0.834830	−	0.550508i	$-0.814434\pi$
$419$	−34.1771	−1.66966	−0.834830	+	0.550508i	$0.814434\pi$
$420$	0	0
$421$	6.68466	0.325790	0.162895	−	0.986643i	$-0.447917\pi$
$421$	6.68466	0.325790	0.162895	+	0.986643i	$0.447917\pi$
$422$	0	0
$423$	3.12311	0.151851
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.56155	0.317536
$428$	0	0
$429$	−20.2462	−0.977496
$430$	0	0
$431$	29.3002	1.41134	0.705670	−	0.708540i	$-0.250646\pi$
$431$	29.3002	1.41134	0.705670	+	0.708540i	$0.250646\pi$
$432$	0	0
$433$	23.8617	1.14672	0.573361	−	0.819303i	$-0.305639\pi$
$433$	23.8617	1.14672	0.573361	+	0.819303i	$0.305639\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−29.3693	−1.40492
$438$	0	0
$439$	11.4384	0.545927	0.272964	−	0.962024i	$-0.411996\pi$
$439$	11.4384	0.545927	0.272964	+	0.962024i	$0.411996\pi$
$440$	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
$449$	5.56155	0.262466	0.131233	−	0.991352i	$-0.458106\pi$
$449$	5.56155	0.262466	0.131233	+	0.991352i	$0.458106\pi$
$450$	0	0
$451$	−41.6155	−1.95960
$452$	0	0
$453$	6.68466	0.314073
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	0	0
$459$	1.43845	0.0671410
$460$	0	0
$461$	−10.6307	−0.495120	−0.247560	−	0.968873i	$-0.579629\pi$
$461$	−10.6307	−0.495120	−0.247560	+	0.968873i	$0.579629\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.8078	1.05542	0.527709	−	0.849425i	$-0.323051\pi$
$467$	22.8078	1.05542	0.527709	+	0.849425i	$0.323051\pi$
$468$	0	0
$469$	7.80776	0.360529
$470$	0	0
$471$	−11.3693	−0.523870
$472$	0	0
$473$	28.9309	1.33024
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.56155	0.208859
$478$	0	0
$479$	−23.6155	−1.07902	−0.539511	−	0.841979i	$-0.681391\pi$
$479$	−23.6155	−1.07902	−0.539511	+	0.841979i	$0.681391\pi$
$480$	0	0
$481$	−20.2462	−0.923148
$482$	0	0
$483$	−4.12311	−0.187608
$484$	0	0
$485$	0	0
$486$	0	0
$487$	32.4384	1.46993	0.734963	−	0.678107i	$-0.237199\pi$
$487$	32.4384	1.46993	0.734963	+	0.678107i	$0.237199\pi$
$488$	0	0
$489$	−21.9309	−0.991748
$490$	0	0
$491$	−8.43845	−0.380822	−0.190411	−	0.981704i	$-0.560982\pi$
$491$	−8.43845	−0.380822	−0.190411	+	0.981704i	$0.560982\pi$
$492$	0	0
$493$	0.177081	0.00797532
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.80776	0.350226
$498$	0	0
$499$	−36.1771	−1.61951	−0.809754	−	0.586769i	$-0.800400\pi$
$499$	−36.1771	−1.61951	−0.809754	+	0.586769i	$0.800400\pi$
$500$	0	0
$501$	14.4924	0.647474
$502$	0	0
$503$	14.4924	0.646185	0.323093	−	0.946367i	$-0.395277\pi$
$503$	14.4924	0.646185	0.323093	+	0.946367i	$0.395277\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−19.3153	−0.857824
$508$	0	0
$509$	−25.3693	−1.12448	−0.562238	−	0.826976i	$-0.690059\pi$
$509$	−25.3693	−1.12448	−0.562238	+	0.826976i	$0.690059\pi$
$510$	0	0
$511$	15.3693	0.679899
$512$	0	0
$513$	−7.12311	−0.314493
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.1231	0.489194
$518$	0	0
$519$	0.876894	0.0384914
$520$	0	0
$521$	−15.9309	−0.697944	−0.348972	−	0.937133i	$-0.613469\pi$
$521$	−15.9309	−0.697944	−0.348972	+	0.937133i	$0.613469\pi$
$522$	0	0
$523$	5.75379	0.251596	0.125798	−	0.992056i	$-0.459851\pi$
$523$	5.75379	0.251596	0.125798	+	0.992056i	$0.459851\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.17708	−0.356199
$528$	0	0
$529$	−6.00000	−0.260870
$530$	0	0
$531$	9.68466	0.420278
$532$	0	0
$533$	−66.4233	−2.87711
$534$	0	0
$535$	0	0
$536$	0	0
$537$	17.1231	0.738917
$538$	0	0
$539$	3.56155	0.153407
$540$	0	0
$541$	−0.192236	−0.00826487	−0.00413243	−	0.999991i	$-0.501315\pi$
$541$	−0.192236	−0.00826487	−0.00413243	+	0.999991i	$0.501315\pi$
$542$	0	0
$543$	25.1231	1.07814
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9.87689	−0.422306	−0.211153	−	0.977453i	$-0.567722\pi$
$547$	−9.87689	−0.422306	−0.211153	+	0.977453i	$0.567722\pi$
$548$	0	0
$549$	−6.56155	−0.280040
$550$	0	0
$551$	−0.876894	−0.0373570
$552$	0	0
$553$	−8.43845	−0.358839
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.4233	−1.07722	−0.538610	−	0.842555i	$-0.681050\pi$
$557$	−25.4233	−1.07722	−0.538610	+	0.842555i	$0.681050\pi$
$558$	0	0
$559$	46.1771	1.95308
$560$	0	0
$561$	5.12311	0.216298
$562$	0	0
$563$	13.0540	0.550159	0.275080	−	0.961421i	$-0.411296\pi$
$563$	13.0540	0.550159	0.275080	+	0.961421i	$0.411296\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	44.0540	1.84684	0.923419	−	0.383793i	$-0.125382\pi$
$569$	44.0540	1.84684	0.923419	+	0.383793i	$0.125382\pi$
$570$	0	0
$571$	−35.7386	−1.49562	−0.747808	−	0.663915i	$-0.768893\pi$
$571$	−35.7386	−1.49562	−0.747808	+	0.663915i	$0.768893\pi$
$572$	0	0
$573$	−24.8078	−1.03636
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.49242	−0.103761	−0.0518805	−	0.998653i	$-0.516521\pi$
$577$	−2.49242	−0.103761	−0.0518805	+	0.998653i	$0.516521\pi$
$578$	0	0
$579$	−19.5616	−0.812950
$580$	0	0
$581$	9.68466	0.401787
$582$	0	0
$583$	16.2462	0.672849
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.31534	−0.343211	−0.171605	−	0.985166i	$-0.554895\pi$
$587$	−8.31534	−0.343211	−0.171605	+	0.985166i	$0.554895\pi$
$588$	0	0
$589$	40.4924	1.66846
$590$	0	0
$591$	4.12311	0.169602
$592$	0	0
$593$	16.6307	0.682940	0.341470	−	0.939893i	$-0.389075\pi$
$593$	16.6307	0.682940	0.341470	+	0.939893i	$0.389075\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−3.38447	−0.138286	−0.0691429	−	0.997607i	$-0.522026\pi$
$599$	−3.38447	−0.138286	−0.0691429	+	0.997607i	$0.522026\pi$
$600$	0	0
$601$	−13.7538	−0.561029	−0.280514	−	0.959850i	$-0.590505\pi$
$601$	−13.7538	−0.561029	−0.280514	+	0.959850i	$0.590505\pi$
$602$	0	0
$603$	−7.80776	−0.317957
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−5.36932	−0.217934	−0.108967	−	0.994045i	$-0.534754\pi$
$607$	−5.36932	−0.217934	−0.108967	+	0.994045i	$0.534754\pi$
$608$	0	0
$609$	−0.123106	−0.00498849
$610$	0	0
$611$	17.7538	0.718241
$612$	0	0
$613$	−46.7926	−1.88994	−0.944968	−	0.327163i	$-0.893907\pi$
$613$	−46.7926	−1.88994	−0.944968	+	0.327163i	$0.893907\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.4233	1.82867	0.914336	−	0.404955i	$-0.132713\pi$
$617$	45.4233	1.82867	0.914336	+	0.404955i	$0.132713\pi$
$618$	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
$627$	−25.3693	−1.01315
$628$	0	0
$629$	5.12311	0.204272
$630$	0	0
$631$	−43.5616	−1.73416	−0.867079	−	0.498171i	$-0.834005\pi$
$631$	−43.5616	−1.73416	−0.867079	+	0.498171i	$0.834005\pi$
$632$	0	0
$633$	−25.9309	−1.03066
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.68466	0.225234
$638$	0	0
$639$	−7.80776	−0.308870
$640$	0	0
$641$	19.5616	0.772635	0.386317	−	0.922366i	$-0.373747\pi$
$641$	19.5616	0.772635	0.386317	+	0.922366i	$0.373747\pi$
$642$	0	0
$643$	−33.1231	−1.30625	−0.653124	−	0.757251i	$-0.726542\pi$
$643$	−33.1231	−1.30625	−0.653124	+	0.757251i	$0.726542\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.4924	1.51329	0.756647	−	0.653824i	$-0.226836\pi$
$647$	38.4924	1.51329	0.756647	+	0.653824i	$0.226836\pi$
$648$	0	0
$649$	34.4924	1.35395
$650$	0	0
$651$	5.68466	0.222799
$652$	0	0
$653$	16.7386	0.655033	0.327517	−	0.944845i	$-0.393788\pi$
$653$	16.7386	0.655033	0.327517	+	0.944845i	$0.393788\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−15.3693	−0.599614
$658$	0	0
$659$	3.36932	0.131250	0.0656250	−	0.997844i	$-0.479096\pi$
$659$	3.36932	0.131250	0.0656250	+	0.997844i	$0.479096\pi$
$660$	0	0
$661$	−3.75379	−0.146005	−0.0730027	−	0.997332i	$-0.523258\pi$
$661$	−3.75379	−0.146005	−0.0730027	+	0.997332i	$0.523258\pi$
$662$	0	0
$663$	8.17708	0.317572
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.507577	0.0196535
$668$	0	0
$669$	−13.4384	−0.519560
$670$	0	0
$671$	−23.3693	−0.902162
$672$	0	0
$673$	−40.5616	−1.56353	−0.781766	−	0.623571i	$-0.785681\pi$
$673$	−40.5616	−1.56353	−0.781766	+	0.623571i	$0.785681\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.6155	1.36882	0.684408	−	0.729099i	$-0.260061\pi$
$677$	35.6155	1.36882	0.684408	+	0.729099i	$0.260061\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	22.8078	0.873995
$682$	0	0
$683$	2.19224	0.0838836	0.0419418	−	0.999120i	$-0.486646\pi$
$683$	2.19224	0.0838836	0.0419418	+	0.999120i	$0.486646\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	25.9309	0.987887
$690$	0	0
$691$	47.3693	1.80201	0.901007	−	0.433805i	$-0.142829\pi$
$691$	47.3693	1.80201	0.901007	+	0.433805i	$0.142829\pi$
$692$	0	0
$693$	−3.56155	−0.135292
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.8078	0.636639
$698$	0	0
$699$	−3.31534	−0.125398
$700$	0	0
$701$	−2.17708	−0.0822272	−0.0411136	−	0.999154i	$-0.513091\pi$
$701$	−2.17708	−0.0822272	−0.0411136	+	0.999154i	$0.513091\pi$
$702$	0	0
$703$	−25.3693	−0.956822
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.4924	−0.620261
$708$	0	0
$709$	−28.8769	−1.08449	−0.542247	−	0.840219i	$-0.682426\pi$
$709$	−28.8769	−1.08449	−0.542247	+	0.840219i	$0.682426\pi$
$710$	0	0
$711$	8.43845	0.316466
$712$	0	0
$713$	−23.4384	−0.877777
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.2462	0.532035
$718$	0	0
$719$	−22.8769	−0.853164	−0.426582	−	0.904449i	$-0.640282\pi$
$719$	−22.8769	−0.853164	−0.426582	+	0.904449i	$0.640282\pi$
$720$	0	0
$721$	−6.31534	−0.235196
$722$	0	0
$723$	18.4924	0.687741
$724$	0	0
$725$	0	0
$726$	0	0
$727$	15.6847	0.581712	0.290856	−	0.956767i	$-0.406060\pi$
$727$	15.6847	0.581712	0.290856	+	0.956767i	$0.406060\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.6847	−0.432173
$732$	0	0
$733$	23.6847	0.874813	0.437406	−	0.899264i	$-0.355897\pi$
$733$	23.6847	0.874813	0.437406	+	0.899264i	$0.355897\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27.8078	−1.02431
$738$	0	0
$739$	−37.8769	−1.39332	−0.696662	−	0.717399i	$-0.745332\pi$
$739$	−37.8769	−1.39332	−0.696662	+	0.717399i	$0.745332\pi$
$740$	0	0
$741$	−40.4924	−1.48753
$742$	0	0
$743$	−41.9309	−1.53829	−0.769147	−	0.639072i	$-0.779319\pi$
$743$	−41.9309	−1.53829	−0.769147	+	0.639072i	$0.779319\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.68466	−0.354343
$748$	0	0
$749$	−1.12311	−0.0410374
$750$	0	0
$751$	−14.2462	−0.519852	−0.259926	−	0.965629i	$-0.583698\pi$
$751$	−14.2462	−0.519852	−0.259926	+	0.965629i	$0.583698\pi$
$752$	0	0
$753$	−2.56155	−0.0933482
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.6847	0.679105	0.339553	−	0.940587i	$-0.389724\pi$
$757$	18.6847	0.679105	0.339553	+	0.940587i	$0.389724\pi$
$758$	0	0
$759$	14.6847	0.533019
$760$	0	0
$761$	−31.3693	−1.13714	−0.568568	−	0.822636i	$-0.692503\pi$
$761$	−31.3693	−1.13714	−0.568568	+	0.822636i	$0.692503\pi$
$762$	0	0
$763$	−8.68466	−0.314406
$764$	0	0
$765$	0	0
$766$	0	0
$767$	55.0540	1.98788
$768$	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
$771$	22.8078	0.821402
$772$	0	0
$773$	−22.6307	−0.813969	−0.406985	−	0.913435i	$-0.633420\pi$
$773$	−22.6307	−0.813969	−0.406985	+	0.913435i	$0.633420\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.56155	−0.127770
$778$	0	0
$779$	−83.2311	−2.98206
$780$	0	0
$781$	−27.8078	−0.995040
$782$	0	0
$783$	0.123106	0.00439944
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9.12311	0.325204	0.162602	−	0.986692i	$-0.448011\pi$
$787$	9.12311	0.325204	0.162602	+	0.986692i	$0.448011\pi$
$788$	0	0
$789$	−12.7538	−0.454047
$790$	0	0
$791$	−9.80776	−0.348724
$792$	0	0
$793$	−37.3002	−1.32457
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.8617	−1.12860	−0.564300	−	0.825570i	$-0.690854\pi$
$797$	−31.8617	−1.12860	−0.564300	+	0.825570i	$0.690854\pi$
$798$	0	0
$799$	−4.49242	−0.158930
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	−54.7386	−1.93168
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−22.0000	−0.774437
$808$	0	0
$809$	−49.6695	−1.74629	−0.873143	−	0.487463i	$-0.837922\pi$
$809$	−49.6695	−1.74629	−0.873143	+	0.487463i	$0.837922\pi$
$810$	0	0
$811$	−16.2462	−0.570482	−0.285241	−	0.958456i	$-0.592074\pi$
$811$	−16.2462	−0.570482	−0.285241	+	0.958456i	$0.592074\pi$
$812$	0	0
$813$	−2.63068	−0.0922621
$814$	0	0
$815$	0	0
$816$	0	0
$817$	57.8617	2.02433
$818$	0	0
$819$	−5.68466	−0.198638
$820$	0	0
$821$	28.2462	0.985800	0.492900	−	0.870086i	$-0.335937\pi$
$821$	28.2462	0.985800	0.492900	+	0.870086i	$0.335937\pi$
$822$	0	0
$823$	44.0540	1.53563	0.767813	−	0.640675i	$-0.221345\pi$
$823$	44.0540	1.53563	0.767813	+	0.640675i	$0.221345\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.8078	−1.45380	−0.726899	−	0.686744i	$-0.759039\pi$
$827$	−41.8078	−1.45380	−0.726899	+	0.686744i	$0.759039\pi$
$828$	0	0
$829$	−40.6695	−1.41251	−0.706255	−	0.707957i	$-0.749617\pi$
$829$	−40.6695	−1.41251	−0.706255	+	0.707957i	$0.749617\pi$
$830$	0	0
$831$	−24.2462	−0.841092
$832$	0	0
$833$	−1.43845	−0.0498392
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−5.68466	−0.196491
$838$	0	0
$839$	−13.7538	−0.474834	−0.237417	−	0.971408i	$-0.576301\pi$
$839$	−13.7538	−0.474834	−0.237417	+	0.971408i	$0.576301\pi$
$840$	0	0
$841$	−28.9848	−0.999477
$842$	0	0
$843$	−14.9309	−0.514246
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.68466	−0.0578855
$848$	0	0
$849$	5.36932	0.184274
$850$	0	0
$851$	14.6847	0.503384
$852$	0	0
$853$	−11.0540	−0.378481	−0.189240	−	0.981931i	$-0.560603\pi$
$853$	−11.0540	−0.378481	−0.189240	+	0.981931i	$0.560603\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−45.1231	−1.54138	−0.770688	−	0.637213i	$-0.780087\pi$
$857$	−45.1231	−1.54138	−0.770688	+	0.637213i	$0.780087\pi$
$858$	0	0
$859$	−20.1080	−0.686074	−0.343037	−	0.939322i	$-0.611456\pi$
$859$	−20.1080	−0.686074	−0.343037	+	0.939322i	$0.611456\pi$
$860$	0	0
$861$	−11.6847	−0.398212
$862$	0	0
$863$	−6.93087	−0.235930	−0.117965	−	0.993018i	$-0.537637\pi$
$863$	−6.93087	−0.235930	−0.117965	+	0.993018i	$0.537637\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14.9309	0.507079
$868$	0	0
$869$	30.0540	1.01951
$870$	0	0
$871$	−44.3845	−1.50391
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9.36932	0.316379	0.158190	−	0.987409i	$-0.449434\pi$
$877$	9.36932	0.316379	0.158190	+	0.987409i	$0.449434\pi$
$878$	0	0
$879$	15.6155	0.526699
$880$	0	0
$881$	−12.0691	−0.406619	−0.203310	−	0.979114i	$-0.565170\pi$
$881$	−12.0691	−0.406619	−0.203310	+	0.979114i	$0.565170\pi$
$882$	0	0
$883$	36.8617	1.24050	0.620248	−	0.784406i	$-0.287032\pi$
$883$	36.8617	1.24050	0.620248	+	0.784406i	$0.287032\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29.1231	0.977858	0.488929	−	0.872324i	$-0.337388\pi$
$887$	29.1231	0.977858	0.488929	+	0.872324i	$0.337388\pi$
$888$	0	0
$889$	−13.8078	−0.463098
$890$	0	0
$891$	3.56155	0.119317
$892$	0	0
$893$	22.2462	0.744441
$894$	0	0
$895$	0	0
$896$	0	0
$897$	23.4384	0.782587
$898$	0	0
$899$	−0.699813	−0.0233401
$900$	0	0
$901$	−6.56155	−0.218597
$902$	0	0
$903$	8.12311	0.270320
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.68466	0.255165	0.127582	−	0.991828i	$-0.459278\pi$
$907$	7.68466	0.255165	0.127582	+	0.991828i	$0.459278\pi$
$908$	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
$913$	−34.4924	−1.14153
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16.0000	−0.528367
$918$	0	0
$919$	11.3153	0.373259	0.186629	−	0.982430i	$-0.440244\pi$
$919$	11.3153	0.373259	0.186629	+	0.982430i	$0.440244\pi$
$920$	0	0
$921$	6.87689	0.226601
$922$	0	0
$923$	−44.3845	−1.46093
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.31534	0.207423
$928$	0	0
$929$	4.80776	0.157738	0.0788688	−	0.996885i	$-0.474869\pi$
$929$	4.80776	0.157738	0.0788688	+	0.996885i	$0.474869\pi$
$930$	0	0
$931$	7.12311	0.233450
$932$	0	0
$933$	19.3693	0.634123
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.61553	−0.0527770	−0.0263885	−	0.999652i	$-0.508401\pi$
$937$	−1.61553	−0.0527770	−0.0263885	+	0.999652i	$0.508401\pi$
$938$	0	0
$939$	−3.75379	−0.122500
$940$	0	0
$941$	36.1080	1.17709	0.588543	−	0.808466i	$-0.299702\pi$
$941$	36.1080	1.17709	0.588543	+	0.808466i	$0.299702\pi$
$942$	0	0
$943$	48.1771	1.56886
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.2462	0.462940	0.231470	−	0.972842i	$-0.425647\pi$
$947$	14.2462	0.462940	0.231470	+	0.972842i	$0.425647\pi$
$948$	0	0
$949$	−87.3693	−2.83613
$950$	0	0
$951$	16.1231	0.522828
$952$	0	0
$953$	29.1771	0.945138	0.472569	−	0.881294i	$-0.343327\pi$
$953$	29.1771	0.945138	0.472569	+	0.881294i	$0.343327\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.438447	0.0141730
$958$	0	0
$959$	12.8769	0.415817
$960$	0	0
$961$	1.31534	0.0424304
$962$	0	0
$963$	1.12311	0.0361916
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−14.7386	−0.473963	−0.236981	−	0.971514i	$-0.576158\pi$
$967$	−14.7386	−0.473963	−0.236981	+	0.971514i	$0.576158\pi$
$968$	0	0
$969$	10.2462	0.329156
$970$	0	0
$971$	49.8617	1.60014	0.800070	−	0.599907i	$-0.204796\pi$
$971$	49.8617	1.60014	0.800070	+	0.599907i	$0.204796\pi$
$972$	0	0
$973$	−1.75379	−0.0562239
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.80776	−0.185807	−0.0929034	−	0.995675i	$-0.529615\pi$
$977$	−5.80776	−0.185807	−0.0929034	+	0.995675i	$0.529615\pi$
$978$	0	0
$979$	35.6155	1.13828
$980$	0	0
$981$	8.68466	0.277280
$982$	0	0
$983$	−45.8617	−1.46276	−0.731381	−	0.681969i	$-0.761124\pi$
$983$	−45.8617	−1.46276	−0.731381	+	0.681969i	$0.761124\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.12311	0.0994095
$988$	0	0
$989$	−33.4924	−1.06500
$990$	0	0
$991$	35.8078	1.13747	0.568736	−	0.822520i	$-0.307433\pi$
$991$	35.8078	1.13747	0.568736	+	0.822520i	$0.307433\pi$
$992$	0	0
$993$	−12.1231	−0.384715
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15.7538	0.498927	0.249464	−	0.968384i	$-0.419746\pi$
$997$	15.7538	0.498927	0.249464	+	0.968384i	$0.419746\pi$
$998$	0	0
$999$	3.56155	0.112683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4200.2.a.bi.1.2	✓	2
4.3	odd	2		8400.2.a.dc.1.1		2
5.2	odd	4		4200.2.t.w.1849.4		4
5.3	odd	4		4200.2.t.w.1849.2		4
5.4	even	2		4200.2.a.bm.1.2	yes	2
20.19	odd	2		8400.2.a.cu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4200.2.a.bi.1.2	✓	2	1.1	even	1	trivial
4200.2.a.bm.1.2	yes	2	5.4	even	2
4200.2.t.w.1849.2		4	5.3	odd	4
4200.2.t.w.1849.4		4	5.2	odd	4
8400.2.a.cu.1.1		2	20.19	odd	2
8400.2.a.dc.1.1		2	4.3	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +3.56155 q^{11} +5.68466 q^{13} -1.43845 q^{17} +7.12311 q^{19} +1.00000 q^{21} -4.12311 q^{23} -1.00000 q^{27} -0.123106 q^{29} +5.68466 q^{31} -3.56155 q^{33} -3.56155 q^{37} -5.68466 q^{39} -11.6847 q^{41} +8.12311 q^{43} +3.12311 q^{47} +1.00000 q^{49} +1.43845 q^{51} +4.56155 q^{53} -7.12311 q^{57} +9.68466 q^{59} -6.56155 q^{61} -1.00000 q^{63} -7.80776 q^{67} +4.12311 q^{69} -7.80776 q^{71} -15.3693 q^{73} -3.56155 q^{77} +8.43845 q^{79} +1.00000 q^{81} -9.68466 q^{83} +0.123106 q^{87} +10.0000 q^{89} -5.68466 q^{91} -5.68466 q^{93} -6.00000 q^{97} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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

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