LMFDB - Embedded newform 4200.2.a.bk.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 840)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -4.47214 q^{13} +2.47214 q^{17} +2.00000 q^{19} -1.00000 q^{21} -4.00000 q^{23} +1.00000 q^{27} +0.472136 q^{29} +8.47214 q^{31} -2.00000 q^{33} +6.47214 q^{37} -4.47214 q^{39} -12.4721 q^{41} -6.47214 q^{43} +2.47214 q^{47} +1.00000 q^{49} +2.47214 q^{51} -2.00000 q^{53} +2.00000 q^{57} -12.4721 q^{61} -1.00000 q^{63} -10.4721 q^{67} -4.00000 q^{69} +3.52786 q^{71} -16.4721 q^{73} +2.00000 q^{77} -8.94427 q^{79} +1.00000 q^{81} -12.9443 q^{83} +0.472136 q^{87} +9.41641 q^{89} +4.47214 q^{91} +8.47214 q^{93} -12.4721 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{11} - 4 q^{17} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{27} - 8 q^{29} + 8 q^{31} - 4 q^{33} + 4 q^{37} - 16 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 4 q^{57} - 16 q^{61} - 2 q^{63} - 12 q^{67} - 8 q^{69} + 16 q^{71} - 24 q^{73} + 4 q^{77} + 2 q^{81} - 8 q^{83} - 8 q^{87} - 8 q^{89} + 8 q^{93} - 16 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 4 * q^17 + 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^27 - 8 * q^29 + 8 * q^31 - 4 * q^33 + 4 * q^37 - 16 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 + 4 * q^57 - 16 * q^61 - 2 * q^63 - 12 * q^67 - 8 * q^69 + 16 * q^71 - 24 * q^73 + 4 * q^77 + 2 * q^81 - 8 * q^83 - 8 * q^87 - 8 * q^89 + 8 * q^93 - 16 * q^97 - 4 * 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
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	8.47214	1.52164	0.760820	−	0.648963i	$-0.224797\pi$
$31$	8.47214	1.52164	0.760820	+	0.648963i	$0.224797\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.47214	1.06401	0.532006	−	0.846740i	$-0.321438\pi$
$37$	6.47214	1.06401	0.532006	+	0.846740i	$0.321438\pi$
$38$	0	0
$39$	−4.47214	−0.716115
$40$	0	0
$41$	−12.4721	−1.94782	−0.973910	−	0.226934i	$-0.927130\pi$
$41$	−12.4721	−1.94782	−0.973910	+	0.226934i	$0.927130\pi$
$42$	0	0
$43$	−6.47214	−0.986991	−0.493496	−	0.869748i	$-0.664281\pi$
$43$	−6.47214	−0.986991	−0.493496	+	0.869748i	$0.664281\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.47214	0.360598	0.180299	−	0.983612i	$-0.442293\pi$
$47$	2.47214	0.360598	0.180299	+	0.983612i	$0.442293\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.47214	0.346168
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−12.4721	−1.59689	−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721	−1.59689	−0.798447	+	0.602066i	$0.794345\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.4721	−1.27938	−0.639688	−	0.768635i	$-0.720936\pi$
$67$	−10.4721	−1.27938	−0.639688	+	0.768635i	$0.720936\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	3.52786	0.418680	0.209340	−	0.977843i	$-0.432868\pi$
$71$	3.52786	0.418680	0.209340	+	0.977843i	$0.432868\pi$
$72$	0	0
$73$	−16.4721	−1.92792	−0.963959	−	0.266051i	$-0.914281\pi$
$73$	−16.4721	−1.92792	−0.963959	+	0.266051i	$0.914281\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.9443	−1.42082	−0.710409	−	0.703789i	$-0.751490\pi$
$83$	−12.9443	−1.42082	−0.710409	+	0.703789i	$0.751490\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.472136	0.0506183
$88$	0	0
$89$	9.41641	0.998137	0.499069	−	0.866562i	$-0.333676\pi$
$89$	9.41641	0.998137	0.499069	+	0.866562i	$0.333676\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	0	0
$93$	8.47214	0.878520
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.4721	−1.26635	−0.633177	−	0.774007i	$-0.718249\pi$
$97$	−12.4721	−1.26635	−0.633177	+	0.774007i	$0.718249\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−4.47214	−0.444994	−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214	−0.444994	−0.222497	+	0.974933i	$0.571421\pi$
$102$	0	0
$103$	4.94427	0.487174	0.243587	−	0.969879i	$-0.421676\pi$
$103$	4.94427	0.487174	0.243587	+	0.969879i	$0.421676\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−2.94427	−0.282010	−0.141005	−	0.990009i	$-0.545033\pi$
$109$	−2.94427	−0.282010	−0.141005	+	0.990009i	$0.545033\pi$
$110$	0	0
$111$	6.47214	0.614308
$112$	0	0
$113$	2.94427	0.276974	0.138487	−	0.990364i	$-0.455776\pi$
$113$	2.94427	0.276974	0.138487	+	0.990364i	$0.455776\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.47214	−0.413449
$118$	0	0
$119$	−2.47214	−0.226620
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−12.4721	−1.12457
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	−6.47214	−0.569840
$130$	0	0
$131$	−21.8885	−1.91241	−0.956205	−	0.292696i	$-0.905448\pi$
$131$	−21.8885	−1.91241	−0.956205	+	0.292696i	$0.905448\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.8885	−1.35745	−0.678725	−	0.734393i	$-0.737467\pi$
$137$	−15.8885	−1.35745	−0.678725	+	0.734393i	$0.737467\pi$
$138$	0	0
$139$	14.9443	1.26756	0.633778	−	0.773515i	$-0.281503\pi$
$139$	14.9443	1.26756	0.633778	+	0.773515i	$0.281503\pi$
$140$	0	0
$141$	2.47214	0.208191
$142$	0	0
$143$	8.94427	0.747958
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−3.52786	−0.289014	−0.144507	−	0.989504i	$-0.546160\pi$
$149$	−3.52786	−0.289014	−0.144507	+	0.989504i	$0.546160\pi$
$150$	0	0
$151$	17.8885	1.45575	0.727875	−	0.685710i	$-0.240508\pi$
$151$	17.8885	1.45575	0.727875	+	0.685710i	$0.240508\pi$
$152$	0	0
$153$	2.47214	0.199860
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.4164	1.38998	0.694990	−	0.719019i	$-0.255409\pi$
$157$	17.4164	1.38998	0.694990	+	0.719019i	$0.255409\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	3.41641	0.267594	0.133797	−	0.991009i	$-0.457283\pi$
$163$	3.41641	0.267594	0.133797	+	0.991009i	$0.457283\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.52786	−0.118230	−0.0591148	−	0.998251i	$-0.518828\pi$
$167$	−1.52786	−0.118230	−0.0591148	+	0.998251i	$0.518828\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.05573	−0.377883	−0.188941	−	0.981988i	$-0.560506\pi$
$179$	−5.05573	−0.377883	−0.188941	+	0.981988i	$0.560506\pi$
$180$	0	0
$181$	7.52786	0.559542	0.279771	−	0.960067i	$-0.409742\pi$
$181$	7.52786	0.559542	0.279771	+	0.960067i	$0.409742\pi$
$182$	0	0
$183$	−12.4721	−0.921967
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.94427	−0.361561
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	9.41641	0.681347	0.340674	−	0.940182i	$-0.389345\pi$
$191$	9.41641	0.681347	0.340674	+	0.940182i	$0.389345\pi$
$192$	0	0
$193$	4.94427	0.355896	0.177948	−	0.984040i	$-0.443054\pi$
$193$	4.94427	0.355896	0.177948	+	0.984040i	$0.443054\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.8885	−1.13201	−0.566006	−	0.824401i	$-0.691512\pi$
$197$	−15.8885	−1.13201	−0.566006	+	0.824401i	$0.691512\pi$
$198$	0	0
$199$	11.5279	0.817189	0.408594	−	0.912716i	$-0.366019\pi$
$199$	11.5279	0.817189	0.408594	+	0.912716i	$0.366019\pi$
$200$	0	0
$201$	−10.4721	−0.738648
$202$	0	0
$203$	−0.472136	−0.0331374
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−9.88854	−0.680755	−0.340378	−	0.940289i	$-0.610555\pi$
$211$	−9.88854	−0.680755	−0.340378	+	0.940289i	$0.610555\pi$
$212$	0	0
$213$	3.52786	0.241725
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.47214	−0.575126
$218$	0	0
$219$	−16.4721	−1.11308
$220$	0	0
$221$	−11.0557	−0.743689
$222$	0	0
$223$	−3.05573	−0.204627	−0.102313	−	0.994752i	$-0.532624\pi$
$223$	−3.05573	−0.204627	−0.102313	+	0.994752i	$0.532624\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.8885	−1.71828	−0.859142	−	0.511738i	$-0.829002\pi$
$227$	−25.8885	−1.71828	−0.859142	+	0.511738i	$0.829002\pi$
$228$	0	0
$229$	18.3607	1.21331	0.606654	−	0.794966i	$-0.292511\pi$
$229$	18.3607	1.21331	0.606654	+	0.794966i	$0.292511\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−14.9443	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$233$	−14.9443	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.94427	−0.580993
$238$	0	0
$239$	24.4721	1.58297	0.791485	−	0.611188i	$-0.209308\pi$
$239$	24.4721	1.58297	0.791485	+	0.611188i	$0.209308\pi$
$240$	0	0
$241$	23.8885	1.53880	0.769398	−	0.638769i	$-0.220556\pi$
$241$	23.8885	1.53880	0.769398	+	0.638769i	$0.220556\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.94427	−0.569110
$248$	0	0
$249$	−12.9443	−0.820310
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.41641	0.213110	0.106555	−	0.994307i	$-0.466018\pi$
$257$	3.41641	0.213110	0.106555	+	0.994307i	$0.466018\pi$
$258$	0	0
$259$	−6.47214	−0.402159
$260$	0	0
$261$	0.472136	0.0292245
$262$	0	0
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.41641	0.576275
$268$	0	0
$269$	−14.3607	−0.875586	−0.437793	−	0.899076i	$-0.644240\pi$
$269$	−14.3607	−0.875586	−0.437793	+	0.899076i	$0.644240\pi$
$270$	0	0
$271$	15.5279	0.943251	0.471625	−	0.881799i	$-0.343668\pi$
$271$	15.5279	0.943251	0.471625	+	0.881799i	$0.343668\pi$
$272$	0	0
$273$	4.47214	0.270666
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.41641	0.205272	0.102636	−	0.994719i	$-0.467272\pi$
$277$	3.41641	0.205272	0.102636	+	0.994719i	$0.467272\pi$
$278$	0	0
$279$	8.47214	0.507214
$280$	0	0
$281$	−14.9443	−0.891501	−0.445750	−	0.895157i	$-0.647063\pi$
$281$	−14.9443	−0.891501	−0.445750	+	0.895157i	$0.647063\pi$
$282$	0	0
$283$	0.944272	0.0561311	0.0280656	−	0.999606i	$-0.491065\pi$
$283$	0.944272	0.0561311	0.0280656	+	0.999606i	$0.491065\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.4721	0.736207
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	−12.4721	−0.731130
$292$	0	0
$293$	−7.05573	−0.412200	−0.206100	−	0.978531i	$-0.566077\pi$
$293$	−7.05573	−0.412200	−0.206100	+	0.978531i	$0.566077\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	17.8885	1.03452
$300$	0	0
$301$	6.47214	0.373048
$302$	0	0
$303$	−4.47214	−0.256917
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	4.94427	0.281270
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−1.41641	−0.0800601	−0.0400301	−	0.999198i	$-0.512745\pi$
$313$	−1.41641	−0.0800601	−0.0400301	+	0.999198i	$0.512745\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.9443	0.614692	0.307346	−	0.951598i	$-0.400559\pi$
$317$	10.9443	0.614692	0.307346	+	0.951598i	$0.400559\pi$
$318$	0	0
$319$	−0.944272	−0.0528691
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.94427	0.275107
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.94427	−0.162819
$328$	0	0
$329$	−2.47214	−0.136293
$330$	0	0
$331$	−1.88854	−0.103804	−0.0519019	−	0.998652i	$-0.516528\pi$
$331$	−1.88854	−0.103804	−0.0519019	+	0.998652i	$0.516528\pi$
$332$	0	0
$333$	6.47214	0.354671
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.9443	1.57669	0.788347	−	0.615230i	$-0.210937\pi$
$337$	28.9443	1.57669	0.788347	+	0.615230i	$0.210937\pi$
$338$	0	0
$339$	2.94427	0.159911
$340$	0	0
$341$	−16.9443	−0.917584
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.8328	1.22573	0.612865	−	0.790188i	$-0.290017\pi$
$347$	22.8328	1.22573	0.612865	+	0.790188i	$0.290017\pi$
$348$	0	0
$349$	−1.41641	−0.0758186	−0.0379093	−	0.999281i	$-0.512070\pi$
$349$	−1.41641	−0.0758186	−0.0379093	+	0.999281i	$0.512070\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	0	0
$353$	−26.4721	−1.40897	−0.704485	−	0.709719i	$-0.748822\pi$
$353$	−26.4721	−1.40897	−0.704485	+	0.709719i	$0.748822\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.47214	−0.130839
$358$	0	0
$359$	−13.4164	−0.708091	−0.354045	−	0.935228i	$-0.615194\pi$
$359$	−13.4164	−0.708091	−0.354045	+	0.935228i	$0.615194\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−9.88854	−0.516178	−0.258089	−	0.966121i	$-0.583093\pi$
$367$	−9.88854	−0.516178	−0.258089	+	0.966121i	$0.583093\pi$
$368$	0	0
$369$	−12.4721	−0.649273
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	−28.3607	−1.46846	−0.734230	−	0.678901i	$-0.762457\pi$
$373$	−28.3607	−1.46846	−0.734230	+	0.678901i	$0.762457\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.11146	−0.108746
$378$	0	0
$379$	29.8885	1.53527	0.767636	−	0.640886i	$-0.221433\pi$
$379$	29.8885	1.53527	0.767636	+	0.640886i	$0.221433\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−24.3607	−1.24477	−0.622386	−	0.782710i	$-0.713837\pi$
$383$	−24.3607	−1.24477	−0.622386	+	0.782710i	$0.713837\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.47214	−0.328997
$388$	0	0
$389$	−29.4164	−1.49147	−0.745736	−	0.666242i	$-0.767902\pi$
$389$	−29.4164	−1.49147	−0.745736	+	0.666242i	$0.767902\pi$
$390$	0	0
$391$	−9.88854	−0.500085
$392$	0	0
$393$	−21.8885	−1.10413
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.4721	0.826713	0.413356	−	0.910569i	$-0.364356\pi$
$397$	16.4721	0.826713	0.413356	+	0.910569i	$0.364356\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	19.8885	0.993186	0.496593	−	0.867983i	$-0.334584\pi$
$401$	19.8885	0.993186	0.496593	+	0.867983i	$0.334584\pi$
$402$	0	0
$403$	−37.8885	−1.88736
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.9443	−0.641624
$408$	0	0
$409$	27.8885	1.37900	0.689500	−	0.724286i	$-0.257830\pi$
$409$	27.8885	1.37900	0.689500	+	0.724286i	$0.257830\pi$
$410$	0	0
$411$	−15.8885	−0.783724
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.9443	0.731824
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	6.94427	0.338443	0.169222	−	0.985578i	$-0.445875\pi$
$421$	6.94427	0.338443	0.169222	+	0.985578i	$0.445875\pi$
$422$	0	0
$423$	2.47214	0.120199
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.4721	0.603569
$428$	0	0
$429$	8.94427	0.431834
$430$	0	0
$431$	5.41641	0.260899	0.130450	−	0.991455i	$-0.458358\pi$
$431$	5.41641	0.260899	0.130450	+	0.991455i	$0.458358\pi$
$432$	0	0
$433$	5.41641	0.260296	0.130148	−	0.991495i	$-0.458455\pi$
$433$	5.41641	0.260296	0.130148	+	0.991495i	$0.458455\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	40.4721	1.93163	0.965815	−	0.259233i	$-0.0834697\pi$
$439$	40.4721	1.93163	0.965815	+	0.259233i	$0.0834697\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	19.0557	0.905365	0.452682	−	0.891672i	$-0.350467\pi$
$443$	19.0557	0.905365	0.452682	+	0.891672i	$0.350467\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.52786	−0.166862
$448$	0	0
$449$	−26.9443	−1.27158	−0.635789	−	0.771863i	$-0.719325\pi$
$449$	−26.9443	−1.27158	−0.635789	+	0.771863i	$0.719325\pi$
$450$	0	0
$451$	24.9443	1.17458
$452$	0	0
$453$	17.8885	0.840477
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.94427	−0.418395	−0.209198	−	0.977873i	$-0.567085\pi$
$457$	−8.94427	−0.418395	−0.209198	+	0.977873i	$0.567085\pi$
$458$	0	0
$459$	2.47214	0.115389
$460$	0	0
$461$	−3.52786	−0.164309	−0.0821545	−	0.996620i	$-0.526180\pi$
$461$	−3.52786	−0.164309	−0.0821545	+	0.996620i	$0.526180\pi$
$462$	0	0
$463$	−18.8328	−0.875235	−0.437618	−	0.899161i	$-0.644178\pi$
$463$	−18.8328	−0.875235	−0.437618	+	0.899161i	$0.644178\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.9443	1.70958	0.854789	−	0.518976i	$-0.173687\pi$
$467$	36.9443	1.70958	0.854789	+	0.518976i	$0.173687\pi$
$468$	0	0
$469$	10.4721	0.483558
$470$	0	0
$471$	17.4164	0.802506
$472$	0	0
$473$	12.9443	0.595178
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−28.9443	−1.31975
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	0	0
$486$	0	0
$487$	21.8885	0.991865	0.495932	−	0.868361i	$-0.334826\pi$
$487$	21.8885	0.991865	0.495932	+	0.868361i	$0.334826\pi$
$488$	0	0
$489$	3.41641	0.154495
$490$	0	0
$491$	18.9443	0.854943	0.427472	−	0.904029i	$-0.359404\pi$
$491$	18.9443	0.854943	0.427472	+	0.904029i	$0.359404\pi$
$492$	0	0
$493$	1.16718	0.0525673
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.52786	−0.158246
$498$	0	0
$499$	29.8885	1.33799	0.668997	−	0.743265i	$-0.266724\pi$
$499$	29.8885	1.33799	0.668997	+	0.743265i	$0.266724\pi$
$500$	0	0
$501$	−1.52786	−0.0682599
$502$	0	0
$503$	12.5836	0.561075	0.280537	−	0.959843i	$-0.409487\pi$
$503$	12.5836	0.561075	0.280537	+	0.959843i	$0.409487\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.00000	0.310881
$508$	0	0
$509$	−37.4164	−1.65845	−0.829227	−	0.558913i	$-0.811219\pi$
$509$	−37.4164	−1.65845	−0.829227	+	0.558913i	$0.811219\pi$
$510$	0	0
$511$	16.4721	0.728684
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.94427	−0.217449
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	17.4164	0.763027	0.381513	−	0.924363i	$-0.375403\pi$
$521$	17.4164	0.763027	0.381513	+	0.924363i	$0.375403\pi$
$522$	0	0
$523$	24.9443	1.09074	0.545368	−	0.838196i	$-0.316390\pi$
$523$	24.9443	1.09074	0.545368	+	0.838196i	$0.316390\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.9443	0.912347
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	55.7771	2.41597
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.05573	−0.218171
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−5.05573	−0.217363	−0.108681	−	0.994077i	$-0.534663\pi$
$541$	−5.05573	−0.217363	−0.108681	+	0.994077i	$0.534663\pi$
$542$	0	0
$543$	7.52786	0.323052
$544$	0	0
$545$	0	0
$546$	0	0
$547$	25.3050	1.08196	0.540981	−	0.841035i	$-0.318053\pi$
$547$	25.3050	1.08196	0.540981	+	0.841035i	$0.318053\pi$
$548$	0	0
$549$	−12.4721	−0.532298
$550$	0	0
$551$	0.944272	0.0402273
$552$	0	0
$553$	8.94427	0.380349
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.9443	−0.972180	−0.486090	−	0.873909i	$-0.661577\pi$
$557$	−22.9443	−0.972180	−0.486090	+	0.873909i	$0.661577\pi$
$558$	0	0
$559$	28.9443	1.22421
$560$	0	0
$561$	−4.94427	−0.208747
$562$	0	0
$563$	18.8328	0.793709	0.396854	−	0.917882i	$-0.370102\pi$
$563$	18.8328	0.793709	0.396854	+	0.917882i	$0.370102\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−45.8885	−1.92038	−0.960188	−	0.279355i	$-0.909879\pi$
$571$	−45.8885	−1.92038	−0.960188	+	0.279355i	$0.909879\pi$
$572$	0	0
$573$	9.41641	0.393376
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−15.5279	−0.646433	−0.323217	−	0.946325i	$-0.604764\pi$
$577$	−15.5279	−0.646433	−0.323217	+	0.946325i	$0.604764\pi$
$578$	0	0
$579$	4.94427	0.205477
$580$	0	0
$581$	12.9443	0.537019
$582$	0	0
$583$	4.00000	0.165663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.9443	1.35976	0.679878	−	0.733325i	$-0.262033\pi$
$587$	32.9443	1.35976	0.679878	+	0.733325i	$0.262033\pi$
$588$	0	0
$589$	16.9443	0.698177
$590$	0	0
$591$	−15.8885	−0.653567
$592$	0	0
$593$	−41.3050	−1.69619	−0.848096	−	0.529843i	$-0.822251\pi$
$593$	−41.3050	−1.69619	−0.848096	+	0.529843i	$0.822251\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.5279	0.471804
$598$	0	0
$599$	−26.3607	−1.07707	−0.538534	−	0.842604i	$-0.681022\pi$
$599$	−26.3607	−1.07707	−0.538534	+	0.842604i	$0.681022\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	−10.4721	−0.426458
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.9443	−0.525392	−0.262696	−	0.964879i	$-0.584612\pi$
$607$	−12.9443	−0.525392	−0.262696	+	0.964879i	$0.584612\pi$
$608$	0	0
$609$	−0.472136	−0.0191319
$610$	0	0
$611$	−11.0557	−0.447267
$612$	0	0
$613$	−19.4164	−0.784221	−0.392111	−	0.919918i	$-0.628255\pi$
$613$	−19.4164	−0.784221	−0.392111	+	0.919918i	$0.628255\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.1115	0.487589	0.243794	−	0.969827i	$-0.421608\pi$
$617$	12.1115	0.487589	0.243794	+	0.969827i	$0.421608\pi$
$618$	0	0
$619$	−23.8885	−0.960162	−0.480081	−	0.877224i	$-0.659393\pi$
$619$	−23.8885	−0.960162	−0.480081	+	0.877224i	$0.659393\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−9.41641	−0.377260
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	16.0000	0.637962
$630$	0	0
$631$	40.9443	1.62997	0.814983	−	0.579485i	$-0.196746\pi$
$631$	40.9443	1.62997	0.814983	+	0.579485i	$0.196746\pi$
$632$	0	0
$633$	−9.88854	−0.393034
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.47214	−0.177192
$638$	0	0
$639$	3.52786	0.139560
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	24.9443	0.983706	0.491853	−	0.870678i	$-0.336320\pi$
$643$	24.9443	0.983706	0.491853	+	0.870678i	$0.336320\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−42.2492	−1.66099	−0.830494	−	0.557027i	$-0.811942\pi$
$647$	−42.2492	−1.66099	−0.830494	+	0.557027i	$0.811942\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−8.47214	−0.332049
$652$	0	0
$653$	−18.9443	−0.741347	−0.370673	−	0.928763i	$-0.620873\pi$
$653$	−18.9443	−0.741347	−0.370673	+	0.928763i	$0.620873\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−16.4721	−0.642639
$658$	0	0
$659$	−47.8885	−1.86547	−0.932736	−	0.360559i	$-0.882586\pi$
$659$	−47.8885	−1.86547	−0.932736	+	0.360559i	$0.882586\pi$
$660$	0	0
$661$	−32.2492	−1.25435	−0.627175	−	0.778879i	$-0.715789\pi$
$661$	−32.2492	−1.25435	−0.627175	+	0.778879i	$0.715789\pi$
$662$	0	0
$663$	−11.0557	−0.429369
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.88854	−0.0731247
$668$	0	0
$669$	−3.05573	−0.118141
$670$	0	0
$671$	24.9443	0.962963
$672$	0	0
$673$	13.8885	0.535364	0.267682	−	0.963507i	$-0.413742\pi$
$673$	13.8885	0.535364	0.267682	+	0.963507i	$0.413742\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−37.8885	−1.45618	−0.728088	−	0.685484i	$-0.759591\pi$
$677$	−37.8885	−1.45618	−0.728088	+	0.685484i	$0.759591\pi$
$678$	0	0
$679$	12.4721	0.478637
$680$	0	0
$681$	−25.8885	−0.992051
$682$	0	0
$683$	49.8885	1.90893	0.954466	−	0.298320i	$-0.0964261\pi$
$683$	49.8885	1.90893	0.954466	+	0.298320i	$0.0964261\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	18.3607	0.700504
$688$	0	0
$689$	8.94427	0.340750
$690$	0	0
$691$	4.83282	0.183849	0.0919245	−	0.995766i	$-0.470698\pi$
$691$	4.83282	0.183849	0.0919245	+	0.995766i	$0.470698\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−30.8328	−1.16788
$698$	0	0
$699$	−14.9443	−0.565244
$700$	0	0
$701$	8.47214	0.319988	0.159994	−	0.987118i	$-0.448852\pi$
$701$	8.47214	0.319988	0.159994	+	0.987118i	$0.448852\pi$
$702$	0	0
$703$	12.9443	0.488202
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.47214	0.168192
$708$	0	0
$709$	0.111456	0.00418582	0.00209291	−	0.999998i	$-0.499334\pi$
$709$	0.111456	0.00418582	0.00209291	+	0.999998i	$0.499334\pi$
$710$	0	0
$711$	−8.94427	−0.335436
$712$	0	0
$713$	−33.8885	−1.26914
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.4721	0.913929
$718$	0	0
$719$	−8.94427	−0.333565	−0.166783	−	0.985994i	$-0.553338\pi$
$719$	−8.94427	−0.333565	−0.166783	+	0.985994i	$0.553338\pi$
$720$	0	0
$721$	−4.94427	−0.184134
$722$	0	0
$723$	23.8885	0.888425
$724$	0	0
$725$	0	0
$726$	0	0
$727$	36.9443	1.37019	0.685094	−	0.728455i	$-0.259761\pi$
$727$	36.9443	1.37019	0.685094	+	0.728455i	$0.259761\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	50.3607	1.86011	0.930057	−	0.367415i	$-0.119757\pi$
$733$	50.3607	1.86011	0.930057	+	0.367415i	$0.119757\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20.9443	0.771492
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−8.94427	−0.328576
$742$	0	0
$743$	50.8328	1.86488	0.932438	−	0.361331i	$-0.117678\pi$
$743$	50.8328	1.86488	0.932438	+	0.361331i	$0.117678\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12.9443	−0.473606
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26.8328	−0.979143	−0.489572	−	0.871963i	$-0.662847\pi$
$751$	−26.8328	−0.979143	−0.489572	+	0.871963i	$0.662847\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21.5279	0.782444	0.391222	−	0.920296i	$-0.372053\pi$
$757$	21.5279	0.782444	0.391222	+	0.920296i	$0.372053\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	−15.5279	−0.562885	−0.281442	−	0.959578i	$-0.590813\pi$
$761$	−15.5279	−0.562885	−0.281442	+	0.959578i	$0.590813\pi$
$762$	0	0
$763$	2.94427	0.106590
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	18.9443	0.683148	0.341574	−	0.939855i	$-0.389040\pi$
$769$	18.9443	0.683148	0.341574	+	0.939855i	$0.389040\pi$
$770$	0	0
$771$	3.41641	0.123039
$772$	0	0
$773$	−0.944272	−0.0339631	−0.0169815	−	0.999856i	$-0.505406\pi$
$773$	−0.944272	−0.0339631	−0.0169815	+	0.999856i	$0.505406\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6.47214	−0.232187
$778$	0	0
$779$	−24.9443	−0.893721
$780$	0	0
$781$	−7.05573	−0.252474
$782$	0	0
$783$	0.472136	0.0168728
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	−2.94427	−0.104686
$792$	0	0
$793$	55.7771	1.98070
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.0000	−0.991811	−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000	−0.991811	−0.495905	+	0.868377i	$0.665164\pi$
$798$	0	0
$799$	6.11146	0.216208
$800$	0	0
$801$	9.41641	0.332712
$802$	0	0
$803$	32.9443	1.16258
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.3607	−0.505520
$808$	0	0
$809$	28.8328	1.01371	0.506854	−	0.862032i	$-0.330808\pi$
$809$	28.8328	1.01371	0.506854	+	0.862032i	$0.330808\pi$
$810$	0	0
$811$	30.9443	1.08660	0.543300	−	0.839539i	$-0.317175\pi$
$811$	30.9443	1.08660	0.543300	+	0.839539i	$0.317175\pi$
$812$	0	0
$813$	15.5279	0.544586
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−12.9443	−0.452863
$818$	0	0
$819$	4.47214	0.156269
$820$	0	0
$821$	−43.3050	−1.51135	−0.755677	−	0.654945i	$-0.772692\pi$
$821$	−43.3050	−1.51135	−0.755677	+	0.654945i	$0.772692\pi$
$822$	0	0
$823$	−32.9443	−1.14837	−0.574183	−	0.818727i	$-0.694680\pi$
$823$	−32.9443	−1.14837	−0.574183	+	0.818727i	$0.694680\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−56.7214	−1.97239	−0.986197	−	0.165573i	$-0.947053\pi$
$827$	−56.7214	−1.97239	−0.986197	+	0.165573i	$0.947053\pi$
$828$	0	0
$829$	12.4721	0.433175	0.216588	−	0.976263i	$-0.430507\pi$
$829$	12.4721	0.433175	0.216588	+	0.976263i	$0.430507\pi$
$830$	0	0
$831$	3.41641	0.118514
$832$	0	0
$833$	2.47214	0.0856544
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.47214	0.292840
$838$	0	0
$839$	15.0557	0.519781	0.259891	−	0.965638i	$-0.416313\pi$
$839$	15.0557	0.519781	0.259891	+	0.965638i	$0.416313\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	−14.9443	−0.514708
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	0.944272	0.0324073
$850$	0	0
$851$	−25.8885	−0.887448
$852$	0	0
$853$	40.4721	1.38574	0.692870	−	0.721063i	$-0.256346\pi$
$853$	40.4721	1.38574	0.692870	+	0.721063i	$0.256346\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.4164	1.48308	0.741538	−	0.670911i	$-0.234097\pi$
$857$	43.4164	1.48308	0.741538	+	0.670911i	$0.234097\pi$
$858$	0	0
$859$	−38.9443	−1.32876	−0.664381	−	0.747394i	$-0.731305\pi$
$859$	−38.9443	−1.32876	−0.664381	+	0.747394i	$0.731305\pi$
$860$	0	0
$861$	12.4721	0.425049
$862$	0	0
$863$	34.8328	1.18572	0.592861	−	0.805305i	$-0.297998\pi$
$863$	34.8328	1.18572	0.592861	+	0.805305i	$0.297998\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−10.8885	−0.369794
$868$	0	0
$869$	17.8885	0.606827
$870$	0	0
$871$	46.8328	1.58687
$872$	0	0
$873$	−12.4721	−0.422118
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.3050	0.449276	0.224638	−	0.974442i	$-0.427880\pi$
$877$	13.3050	0.449276	0.224638	+	0.974442i	$0.427880\pi$
$878$	0	0
$879$	−7.05573	−0.237984
$880$	0	0
$881$	36.4721	1.22878	0.614389	−	0.789003i	$-0.289403\pi$
$881$	36.4721	1.22878	0.614389	+	0.789003i	$0.289403\pi$
$882$	0	0
$883$	−28.3607	−0.954413	−0.477206	−	0.878791i	$-0.658351\pi$
$883$	−28.3607	−0.954413	−0.477206	+	0.878791i	$0.658351\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.4164	1.05486	0.527430	−	0.849599i	$-0.323156\pi$
$887$	31.4164	1.05486	0.527430	+	0.849599i	$0.323156\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	4.94427	0.165454
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17.8885	0.597281
$898$	0	0
$899$	4.00000	0.133407
$900$	0	0
$901$	−4.94427	−0.164718
$902$	0	0
$903$	6.47214	0.215379
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−28.3607	−0.941701	−0.470850	−	0.882213i	$-0.656053\pi$
$907$	−28.3607	−0.941701	−0.470850	+	0.882213i	$0.656053\pi$
$908$	0	0
$909$	−4.47214	−0.148331
$910$	0	0
$911$	1.41641	0.0469277	0.0234638	−	0.999725i	$-0.492531\pi$
$911$	1.41641	0.0469277	0.0234638	+	0.999725i	$0.492531\pi$
$912$	0	0
$913$	25.8885	0.856786
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.8885	0.722823
$918$	0	0
$919$	52.7214	1.73912	0.869559	−	0.493830i	$-0.164403\pi$
$919$	52.7214	1.73912	0.869559	+	0.493830i	$0.164403\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	−15.7771	−0.519309
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.94427	0.162391
$928$	0	0
$929$	−43.3050	−1.42079	−0.710395	−	0.703804i	$-0.751484\pi$
$929$	−43.3050	−1.42079	−0.710395	+	0.703804i	$0.751484\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	16.0000	0.523816
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−47.3050	−1.54539	−0.772693	−	0.634780i	$-0.781091\pi$
$937$	−47.3050	−1.54539	−0.772693	+	0.634780i	$0.781091\pi$
$938$	0	0
$939$	−1.41641	−0.0462227
$940$	0	0
$941$	44.4721	1.44975	0.724875	−	0.688880i	$-0.241897\pi$
$941$	44.4721	1.44975	0.724875	+	0.688880i	$0.241897\pi$
$942$	0	0
$943$	49.8885	1.62459
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	0	0
$949$	73.6656	2.39129
$950$	0	0
$951$	10.9443	0.354892
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.944272	−0.0305240
$958$	0	0
$959$	15.8885	0.513068
$960$	0	0
$961$	40.7771	1.31539
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	9.88854	0.317994	0.158997	−	0.987279i	$-0.449174\pi$
$967$	9.88854	0.317994	0.158997	+	0.987279i	$0.449174\pi$
$968$	0	0
$969$	4.94427	0.158833
$970$	0	0
$971$	1.88854	0.0606063	0.0303031	−	0.999541i	$-0.490353\pi$
$971$	1.88854	0.0606063	0.0303031	+	0.999541i	$0.490353\pi$
$972$	0	0
$973$	−14.9443	−0.479091
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−0.832816	−0.0266441	−0.0133221	−	0.999911i	$-0.504241\pi$
$977$	−0.832816	−0.0266441	−0.0133221	+	0.999911i	$0.504241\pi$
$978$	0	0
$979$	−18.8328	−0.601899
$980$	0	0
$981$	−2.94427	−0.0940034
$982$	0	0
$983$	34.4721	1.09949	0.549745	−	0.835332i	$-0.314725\pi$
$983$	34.4721	1.09949	0.549745	+	0.835332i	$0.314725\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.47214	−0.0786890
$988$	0	0
$989$	25.8885	0.823208
$990$	0	0
$991$	−39.0557	−1.24065	−0.620323	−	0.784346i	$-0.712998\pi$
$991$	−39.0557	−1.24065	−0.620323	+	0.784346i	$0.712998\pi$
$992$	0	0
$993$	−1.88854	−0.0599311
$994$	0	0
$995$	0	0
$996$	0	0
$997$	50.3607	1.59494	0.797469	−	0.603359i	$-0.206172\pi$
$997$	50.3607	1.59494	0.797469	+	0.603359i	$0.206172\pi$
$998$	0	0
$999$	6.47214	0.204769

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4200.2.a.bk.1.1		2
4.3	odd	2		8400.2.a.cz.1.1		2
5.2	odd	4		840.2.t.c.169.1	✓	4
5.3	odd	4		840.2.t.c.169.3	yes	4
5.4	even	2		4200.2.a.bj.1.2		2
15.2	even	4		2520.2.t.f.1009.3		4
15.8	even	4		2520.2.t.f.1009.4		4
20.3	even	4		1680.2.t.h.1009.1		4
20.7	even	4		1680.2.t.h.1009.3		4
20.19	odd	2		8400.2.a.db.1.2		2
60.23	odd	4		5040.2.t.u.1009.3		4
60.47	odd	4		5040.2.t.u.1009.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.c.169.1	✓	4	5.2	odd	4
840.2.t.c.169.3	yes	4	5.3	odd	4
1680.2.t.h.1009.1		4	20.3	even	4
1680.2.t.h.1009.3		4	20.7	even	4
2520.2.t.f.1009.3		4	15.2	even	4
2520.2.t.f.1009.4		4	15.8	even	4
4200.2.a.bj.1.2		2	5.4	even	2
4200.2.a.bk.1.1		2	1.1	even	1	trivial
5040.2.t.u.1009.3		4	60.23	odd	4
5040.2.t.u.1009.4		4	60.47	odd	4
8400.2.a.cz.1.1		2	4.3	odd	2
8400.2.a.db.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -4.47214 q^{13} +2.47214 q^{17} +2.00000 q^{19} -1.00000 q^{21} -4.00000 q^{23} +1.00000 q^{27} +0.472136 q^{29} +8.47214 q^{31} -2.00000 q^{33} +6.47214 q^{37} -4.47214 q^{39} -12.4721 q^{41} -6.47214 q^{43} +2.47214 q^{47} +1.00000 q^{49} +2.47214 q^{51} -2.00000 q^{53} +2.00000 q^{57} -12.4721 q^{61} -1.00000 q^{63} -10.4721 q^{67} -4.00000 q^{69} +3.52786 q^{71} -16.4721 q^{73} +2.00000 q^{77} -8.94427 q^{79} +1.00000 q^{81} -12.9443 q^{83} +0.472136 q^{87} +9.41641 q^{89} +4.47214 q^{91} +8.47214 q^{93} -12.4721 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 4 q^{17} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{27} - 8 q^{29} + 8 q^{31} - 4 q^{33} + 4 q^{37} - 16 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 4 q^{57} - 16 q^{61} - 2 q^{63} - 12 q^{67} - 8 q^{69} + 16 q^{71} - 24 q^{73} + 4 q^{77} + 2 q^{81} - 8 q^{83} - 8 q^{87} - 8 q^{89} + 8 q^{93} - 16 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 4 * q^17 + 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^27 - 8 * q^29 + 8 * q^31 - 4 * q^33 + 4 * q^37 - 16 * q^41 - 4 * q^43 - 4 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 + 4 * q^57 - 16 * q^61 - 2 * q^63 - 12 * q^67 - 8 * q^69 + 16 * q^71 - 24 * q^73 + 4 * q^77 + 2 * q^81 - 8 * q^83 - 8 * q^87 - 8 * q^89 + 8 * q^93 - 16 * q^97 - 4 * q^99

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