LMFDB - Embedded newform 840.2.t.c.169.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [840,2,Mod(169,840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("840.169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	840.t (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.70743376979$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.1
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	840.169
Dual form			840.2.t.c.169.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +4.47214i q^{13} +2.23607i q^{15} +2.47214i q^{17} -2.00000 q^{19} -1.00000 q^{21} +4.00000i q^{23} +5.00000 q^{25} +1.00000i q^{27} -0.472136 q^{29} +8.47214 q^{31} +2.00000i q^{33} +2.23607i q^{35} +6.47214i q^{37} +4.47214 q^{39} -12.4721 q^{41} +6.47214i q^{43} +2.23607 q^{45} +2.47214i q^{47} -1.00000 q^{49} +2.47214 q^{51} +2.00000i q^{53} +4.47214 q^{55} +2.00000i q^{57} -12.4721 q^{61} +1.00000i q^{63} -10.0000i q^{65} -10.4721i q^{67} +4.00000 q^{69} +3.52786 q^{71} +16.4721i q^{73} -5.00000i q^{75} +2.00000i q^{77} +8.94427 q^{79} +1.00000 q^{81} +12.9443i q^{83} -5.52786i q^{85} +0.472136i q^{87} -9.41641 q^{89} +4.47214 q^{91} -8.47214i q^{93} +4.47214 q^{95} -12.4721i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 8 q^{11} - 8 q^{19} - 4 q^{21} + 20 q^{25} + 16 q^{29} + 16 q^{31} - 32 q^{41} - 4 q^{49} - 8 q^{51} - 32 q^{61} + 16 q^{69} + 32 q^{71} + 4 q^{81} + 16 q^{89} + 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 8 * q^11 - 8 * q^19 - 4 * q^21 + 20 * q^25 + 16 * q^29 + 16 * q^31 - 32 * q^41 - 4 * q^49 - 8 * q^51 - 32 * q^61 + 16 * q^69 + 32 * q^71 + 4 * q^81 + 16 * q^89 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	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
$13$	4.47214i	1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$	4.47214i	1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	2.23607i	0.577350i
$16$	0	0
$17$	2.47214i	0.599581i	0.954005	+	0.299791i	$0.0969168\pi$
$17$	2.47214i	0.599581i	−0.954005	+	0.299791i	$0.903083\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	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
$33$	2.00000i	0.348155i
$34$	0	0
$35$	2.23607i	0.377964i
$36$	0	0
$37$	6.47214i	1.06401i	0.846740	+	0.532006i	$0.178562\pi$
$37$	6.47214i	1.06401i	−0.846740	+	0.532006i	$0.821438\pi$
$38$	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
$43$	6.47214i	0.986991i	0.869748	+	0.493496i	$0.164281\pi$
$43$	6.47214i	0.986991i	−0.869748	+	0.493496i	$0.835719\pi$
$44$	0	0
$45$	2.23607	0.333333
$46$	0	0
$47$	2.47214i	0.360598i	0.983612	+	0.180299i	$0.0577065\pi$
$47$	2.47214i	0.360598i	−0.983612	+	0.180299i	$0.942293\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	2.47214	0.346168
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	4.47214	0.603023
$56$	0	0
$57$	2.00000i	0.264906i
$58$	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
$63$	1.00000i	0.125988i
$64$	0	0
$65$	− 10.0000i	− 1.24035i
$66$	0	0
$67$	− 10.4721i	− 1.27938i	−0.768635	−	0.639688i	$-0.779064\pi$
$67$	− 10.4721i	− 1.27938i	0.768635	−	0.639688i	$-0.220936\pi$
$68$	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
$73$	16.4721i	1.92792i	0.266051	+	0.963959i	$0.414281\pi$
$73$	16.4721i	1.92792i	−0.266051	+	0.963959i	$0.585719\pi$
$74$	0	0
$75$	− 5.00000i	− 0.577350i
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.9443i	1.42082i	0.703789	+	0.710409i	$0.251490\pi$
$83$	12.9443i	1.42082i	−0.703789	+	0.710409i	$0.748510\pi$
$84$	0	0
$85$	− 5.52786i	− 0.599581i
$86$	0	0
$87$	0.472136i	0.0506183i
$88$	0	0
$89$	−9.41641	−0.998137	−0.499069	−	0.866562i	$-0.666324\pi$
$89$	−9.41641	−0.998137	−0.499069	+	0.866562i	$0.666324\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	0	0
$93$	− 8.47214i	− 0.878520i
$94$	0	0
$95$	4.47214	0.458831
$96$	0	0
$97$	− 12.4721i	− 1.26635i	−0.774007	−	0.633177i	$-0.781751\pi$
$97$	− 12.4721i	− 1.26635i	0.774007	−	0.633177i	$-0.218249\pi$
$98$	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
$103$	− 4.94427i	− 0.487174i	−0.969879	−	0.243587i	$-0.921676\pi$
$103$	− 4.94427i	− 0.487174i	0.969879	−	0.243587i	$-0.0783241\pi$
$104$	0	0
$105$	2.23607	0.218218
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	0	0
$111$	6.47214	0.614308
$112$	0	0
$113$	− 2.94427i	− 0.276974i	−0.990364	−	0.138487i	$-0.955776\pi$
$113$	− 2.94427i	− 0.276974i	0.990364	−	0.138487i	$-0.0442239\pi$
$114$	0	0
$115$	− 8.94427i	− 0.834058i
$116$	0	0
$117$	− 4.47214i	− 0.413449i
$118$	0	0
$119$	2.47214	0.226620
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	12.4721i	1.12457i
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	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
$133$	2.00000i	0.173422i
$134$	0	0
$135$	− 2.23607i	− 0.192450i
$136$	0	0
$137$	− 15.8885i	− 1.35745i	−0.734393	−	0.678725i	$-0.762533\pi$
$137$	− 15.8885i	− 1.35745i	0.734393	−	0.678725i	$-0.237467\pi$
$138$	0	0
$139$	−14.9443	−1.26756	−0.633778	−	0.773515i	$-0.718497\pi$
$139$	−14.9443	−1.26756	−0.633778	+	0.773515i	$0.718497\pi$
$140$	0	0
$141$	2.47214	0.208191
$142$	0	0
$143$	− 8.94427i	− 0.747958i
$144$	0	0
$145$	1.05573	0.0876734
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	3.52786	0.289014	0.144507	−	0.989504i	$-0.453840\pi$
$149$	3.52786	0.289014	0.144507	+	0.989504i	$0.453840\pi$
$150$	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
$153$	− 2.47214i	− 0.199860i
$154$	0	0
$155$	−18.9443	−1.52164
$156$	0	0
$157$	17.4164i	1.38998i	0.719019	+	0.694990i	$0.244591\pi$
$157$	17.4164i	1.38998i	−0.719019	+	0.694990i	$0.755409\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	− 3.41641i	− 0.267594i	−0.991009	−	0.133797i	$-0.957283\pi$
$163$	− 3.41641i	− 0.267594i	0.991009	−	0.133797i	$-0.0427170\pi$
$164$	0	0
$165$	− 4.47214i	− 0.348155i
$166$	0	0
$167$	− 1.52786i	− 0.118230i	−0.998251	−	0.0591148i	$-0.981172\pi$
$167$	− 1.52786i	− 0.118230i	0.998251	−	0.0591148i	$-0.0188278\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	− 5.00000i	− 0.377964i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.05573	0.377883	0.188941	−	0.981988i	$-0.439494\pi$
$179$	5.05573	0.377883	0.188941	+	0.981988i	$0.439494\pi$
$180$	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
$183$	12.4721i	0.921967i
$184$	0	0
$185$	− 14.4721i	− 1.06401i
$186$	0	0
$187$	− 4.94427i	− 0.361561i
$188$	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
$193$	− 4.94427i	− 0.355896i	−0.984040	−	0.177948i	$-0.943054\pi$
$193$	− 4.94427i	− 0.355896i	0.984040	−	0.177948i	$-0.0569460\pi$
$194$	0	0
$195$	−10.0000	−0.716115
$196$	0	0
$197$	− 15.8885i	− 1.13201i	−0.824401	−	0.566006i	$-0.808488\pi$
$197$	− 15.8885i	− 1.13201i	0.824401	−	0.566006i	$-0.191512\pi$
$198$	0	0
$199$	−11.5279	−0.817189	−0.408594	−	0.912716i	$-0.633981\pi$
$199$	−11.5279	−0.817189	−0.408594	+	0.912716i	$0.633981\pi$
$200$	0	0
$201$	−10.4721	−0.738648
$202$	0	0
$203$	0.472136i	0.0331374i
$204$	0	0
$205$	27.8885	1.94782
$206$	0	0
$207$	− 4.00000i	− 0.278019i
$208$	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
$213$	− 3.52786i	− 0.241725i
$214$	0	0
$215$	− 14.4721i	− 0.986991i
$216$	0	0
$217$	− 8.47214i	− 0.575126i
$218$	0	0
$219$	16.4721	1.11308
$220$	0	0
$221$	−11.0557	−0.743689
$222$	0	0
$223$	3.05573i	0.204627i	0.994752	+	0.102313i	$0.0326244\pi$
$223$	3.05573i	0.204627i	−0.994752	+	0.102313i	$0.967376\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	− 25.8885i	− 1.71828i	−0.511738	−	0.859142i	$-0.670998\pi$
$227$	− 25.8885i	− 1.71828i	0.511738	−	0.859142i	$-0.329002\pi$
$228$	0	0
$229$	−18.3607	−1.21331	−0.606654	−	0.794966i	$-0.707489\pi$
$229$	−18.3607	−1.21331	−0.606654	+	0.794966i	$0.707489\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	14.9443i	0.979032i	0.871994	+	0.489516i	$0.162826\pi$
$233$	14.9443i	0.979032i	−0.871994	+	0.489516i	$0.837174\pi$
$234$	0	0
$235$	− 5.52786i	− 0.360598i
$236$	0	0
$237$	− 8.94427i	− 0.580993i
$238$	0	0
$239$	−24.4721	−1.58297	−0.791485	−	0.611188i	$-0.790692\pi$
$239$	−24.4721	−1.58297	−0.791485	+	0.611188i	$0.790692\pi$
$240$	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
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	2.23607	0.142857
$246$	0	0
$247$	− 8.94427i	− 0.569110i
$248$	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
$253$	− 8.00000i	− 0.502956i
$254$	0	0
$255$	−5.52786	−0.346168
$256$	0	0
$257$	3.41641i	0.213110i	0.994307	+	0.106555i	$0.0339820\pi$
$257$	3.41641i	0.213110i	−0.994307	+	0.106555i	$0.966018\pi$
$258$	0	0
$259$	6.47214	0.402159
$260$	0	0
$261$	0.472136	0.0292245
$262$	0	0
$263$	4.00000i	0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$	4.00000i	0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	0	0
$265$	− 4.47214i	− 0.274721i
$266$	0	0
$267$	9.41641i	0.576275i
$268$	0	0
$269$	14.3607	0.875586	0.437793	−	0.899076i	$-0.355760\pi$
$269$	14.3607	0.875586	0.437793	+	0.899076i	$0.355760\pi$
$270$	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
$273$	− 4.47214i	− 0.270666i
$274$	0	0
$275$	−10.0000	−0.603023
$276$	0	0
$277$	3.41641i	0.205272i	0.994719	+	0.102636i	$0.0327277\pi$
$277$	3.41641i	0.205272i	−0.994719	+	0.102636i	$0.967272\pi$
$278$	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
$283$	− 0.944272i	− 0.0561311i	−0.999606	−	0.0280656i	$-0.991065\pi$
$283$	− 0.944272i	− 0.0561311i	0.999606	−	0.0280656i	$-0.00893472\pi$
$284$	0	0
$285$	− 4.47214i	− 0.264906i
$286$	0	0
$287$	12.4721i	0.736207i
$288$	0	0
$289$	10.8885	0.640503
$290$	0	0
$291$	−12.4721	−0.731130
$292$	0	0
$293$	7.05573i	0.412200i	0.978531	+	0.206100i	$0.0660772\pi$
$293$	7.05573i	0.412200i	−0.978531	+	0.206100i	$0.933923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.00000i	− 0.116052i
$298$	0	0
$299$	−17.8885	−1.03452
$300$	0	0
$301$	6.47214	0.373048
$302$	0	0
$303$	4.47214i	0.256917i
$304$	0	0
$305$	27.8885	1.59689
$306$	0	0
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	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
$313$	1.41641i	0.0800601i	0.999198	+	0.0400301i	$0.0127454\pi$
$313$	1.41641i	0.0800601i	−0.999198	+	0.0400301i	$0.987255\pi$
$314$	0	0
$315$	− 2.23607i	− 0.125988i
$316$	0	0
$317$	10.9443i	0.614692i	0.951598	+	0.307346i	$0.0994408\pi$
$317$	10.9443i	0.614692i	−0.951598	+	0.307346i	$0.900559\pi$
$318$	0	0
$319$	0.944272	0.0528691
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 4.94427i	− 0.275107i
$324$	0	0
$325$	22.3607i	1.24035i
$326$	0	0
$327$	− 2.94427i	− 0.162819i
$328$	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
$333$	− 6.47214i	− 0.354671i
$334$	0	0
$335$	23.4164i	1.27938i
$336$	0	0
$337$	28.9443i	1.57669i	0.615230	+	0.788347i	$0.289063\pi$
$337$	28.9443i	1.57669i	−0.615230	+	0.788347i	$0.710937\pi$
$338$	0	0
$339$	−2.94427	−0.159911
$340$	0	0
$341$	−16.9443	−0.917584
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	−8.94427	−0.481543
$346$	0	0
$347$	22.8328i	1.22573i	0.790188	+	0.612865i	$0.209983\pi$
$347$	22.8328i	1.22573i	−0.790188	+	0.612865i	$0.790017\pi$
$348$	0	0
$349$	1.41641	0.0758186	0.0379093	−	0.999281i	$-0.487930\pi$
$349$	1.41641	0.0758186	0.0379093	+	0.999281i	$0.487930\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	0	0
$353$	26.4721i	1.40897i	0.709719	+	0.704485i	$0.248822\pi$
$353$	26.4721i	1.40897i	−0.709719	+	0.704485i	$0.751178\pi$
$354$	0	0
$355$	−7.88854	−0.418680
$356$	0	0
$357$	− 2.47214i	− 0.130839i
$358$	0	0
$359$	13.4164	0.708091	0.354045	−	0.935228i	$-0.384806\pi$
$359$	13.4164	0.708091	0.354045	+	0.935228i	$0.384806\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	7.00000i	0.367405i
$364$	0	0
$365$	− 36.8328i	− 1.92792i
$366$	0	0
$367$	− 9.88854i	− 0.516178i	−0.966121	−	0.258089i	$-0.916907\pi$
$367$	− 9.88854i	− 0.516178i	0.966121	−	0.258089i	$-0.0830927\pi$
$368$	0	0
$369$	12.4721	0.649273
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	28.3607i	1.46846i	0.678901	+	0.734230i	$0.262457\pi$
$373$	28.3607i	1.46846i	−0.678901	+	0.734230i	$0.737543\pi$
$374$	0	0
$375$	11.1803i	0.577350i
$376$	0	0
$377$	− 2.11146i	− 0.108746i
$378$	0	0
$379$	−29.8885	−1.53527	−0.767636	−	0.640886i	$-0.778567\pi$
$379$	−29.8885	−1.53527	−0.767636	+	0.640886i	$0.778567\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	24.3607i	1.24477i	0.782710	+	0.622386i	$0.213837\pi$
$383$	24.3607i	1.24477i	−0.782710	+	0.622386i	$0.786163\pi$
$384$	0	0
$385$	− 4.47214i	− 0.227921i
$386$	0	0
$387$	− 6.47214i	− 0.328997i
$388$	0	0
$389$	29.4164	1.49147	0.745736	−	0.666242i	$-0.232098\pi$
$389$	29.4164	1.49147	0.745736	+	0.666242i	$0.232098\pi$
$390$	0	0
$391$	−9.88854	−0.500085
$392$	0	0
$393$	21.8885i	1.10413i
$394$	0	0
$395$	−20.0000	−1.00631
$396$	0	0
$397$	16.4721i	0.826713i	0.910569	+	0.413356i	$0.135644\pi$
$397$	16.4721i	0.826713i	−0.910569	+	0.413356i	$0.864356\pi$
$398$	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
$403$	37.8885i	1.88736i
$404$	0	0
$405$	−2.23607	−0.111111
$406$	0	0
$407$	− 12.9443i	− 0.641624i
$408$	0	0
$409$	−27.8885	−1.37900	−0.689500	−	0.724286i	$-0.742170\pi$
$409$	−27.8885	−1.37900	−0.689500	+	0.724286i	$0.742170\pi$
$410$	0	0
$411$	−15.8885	−0.783724
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 28.9443i	− 1.42082i
$416$	0	0
$417$	14.9443i	0.731824i
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	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
$423$	− 2.47214i	− 0.120199i
$424$	0	0
$425$	12.3607i	0.599581i
$426$	0	0
$427$	12.4721i	0.603569i
$428$	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
$433$	− 5.41641i	− 0.260296i	−0.991495	−	0.130148i	$-0.958455\pi$
$433$	− 5.41641i	− 0.260296i	0.991495	−	0.130148i	$-0.0415452\pi$
$434$	0	0
$435$	− 1.05573i	− 0.0506183i
$436$	0	0
$437$	− 8.00000i	− 0.382692i
$438$	0	0
$439$	−40.4721	−1.93163	−0.965815	−	0.259233i	$-0.916530\pi$
$439$	−40.4721	−1.93163	−0.965815	+	0.259233i	$0.916530\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	− 19.0557i	− 0.905365i	−0.891672	−	0.452682i	$-0.850467\pi$
$443$	− 19.0557i	− 0.905365i	0.891672	−	0.452682i	$-0.149533\pi$
$444$	0	0
$445$	21.0557	0.998137
$446$	0	0
$447$	− 3.52786i	− 0.166862i
$448$	0	0
$449$	26.9443	1.27158	0.635789	−	0.771863i	$-0.280675\pi$
$449$	26.9443	1.27158	0.635789	+	0.771863i	$0.280675\pi$
$450$	0	0
$451$	24.9443	1.17458
$452$	0	0
$453$	− 17.8885i	− 0.840477i
$454$	0	0
$455$	−10.0000	−0.468807
$456$	0	0
$457$	− 8.94427i	− 0.418395i	−0.977873	−	0.209198i	$-0.932915\pi$
$457$	− 8.94427i	− 0.418395i	0.977873	−	0.209198i	$-0.0670852\pi$
$458$	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
$463$	18.8328i	0.875235i	0.899161	+	0.437618i	$0.144178\pi$
$463$	18.8328i	0.875235i	−0.899161	+	0.437618i	$0.855822\pi$
$464$	0	0
$465$	18.9443i	0.878520i
$466$	0	0
$467$	36.9443i	1.70958i	0.518976	+	0.854789i	$0.326313\pi$
$467$	36.9443i	1.70958i	−0.518976	+	0.854789i	$0.673687\pi$
$468$	0	0
$469$	−10.4721	−0.483558
$470$	0	0
$471$	17.4164	0.802506
$472$	0	0
$473$	− 12.9443i	− 0.595178i
$474$	0	0
$475$	−10.0000	−0.458831
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−28.9443	−1.31975
$482$	0	0
$483$	− 4.00000i	− 0.182006i
$484$	0	0
$485$	27.8885i	1.26635i
$486$	0	0
$487$	21.8885i	0.991865i	0.868361	+	0.495932i	$0.165174\pi$
$487$	21.8885i	0.991865i	−0.868361	+	0.495932i	$0.834826\pi$
$488$	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
$493$	− 1.16718i	− 0.0525673i
$494$	0	0
$495$	−4.47214	−0.201008
$496$	0	0
$497$	− 3.52786i	− 0.158246i
$498$	0	0
$499$	−29.8885	−1.33799	−0.668997	−	0.743265i	$-0.733276\pi$
$499$	−29.8885	−1.33799	−0.668997	+	0.743265i	$0.733276\pi$
$500$	0	0
$501$	−1.52786	−0.0682599
$502$	0	0
$503$	− 12.5836i	− 0.561075i	−0.959843	−	0.280537i	$-0.909487\pi$
$503$	− 12.5836i	− 0.561075i	0.959843	−	0.280537i	$-0.0905126\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	7.00000i	0.310881i
$508$	0	0
$509$	37.4164	1.65845	0.829227	−	0.558913i	$-0.188781\pi$
$509$	37.4164	1.65845	0.829227	+	0.558913i	$0.188781\pi$
$510$	0	0
$511$	16.4721	0.728684
$512$	0	0
$513$	− 2.00000i	− 0.0883022i
$514$	0	0
$515$	11.0557i	0.487174i
$516$	0	0
$517$	− 4.94427i	− 0.217449i
$518$	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
$523$	− 24.9443i	− 1.09074i	−0.838196	−	0.545368i	$-0.816390\pi$
$523$	− 24.9443i	− 1.09074i	0.838196	−	0.545368i	$-0.183610\pi$
$524$	0	0
$525$	−5.00000	−0.218218
$526$	0	0
$527$	20.9443i	0.912347i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 55.7771i	− 2.41597i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 5.05573i	− 0.218171i
$538$	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
$543$	− 7.52786i	− 0.323052i
$544$	0	0
$545$	−6.58359	−0.282010
$546$	0	0
$547$	25.3050i	1.08196i	0.841035	+	0.540981i	$0.181947\pi$
$547$	25.3050i	1.08196i	−0.841035	+	0.540981i	$0.818053\pi$
$548$	0	0
$549$	12.4721	0.532298
$550$	0	0
$551$	0.944272	0.0402273
$552$	0	0
$553$	− 8.94427i	− 0.380349i
$554$	0	0
$555$	−14.4721	−0.614308
$556$	0	0
$557$	− 22.9443i	− 0.972180i	−0.873909	−	0.486090i	$-0.838423\pi$
$557$	− 22.9443i	− 0.972180i	0.873909	−	0.486090i	$-0.161577\pi$
$558$	0	0
$559$	−28.9443	−1.22421
$560$	0	0
$561$	−4.94427	−0.208747
$562$	0	0
$563$	− 18.8328i	− 0.793709i	−0.917882	−	0.396854i	$-0.870102\pi$
$563$	− 18.8328i	− 0.793709i	0.917882	−	0.396854i	$-0.129898\pi$
$564$	0	0
$565$	6.58359i	0.276974i
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	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
$573$	− 9.41641i	− 0.393376i
$574$	0	0
$575$	20.0000i	0.834058i
$576$	0	0
$577$	− 15.5279i	− 0.646433i	−0.946325	−	0.323217i	$-0.895236\pi$
$577$	− 15.5279i	− 0.646433i	0.946325	−	0.323217i	$-0.104764\pi$
$578$	0	0
$579$	−4.94427	−0.205477
$580$	0	0
$581$	12.9443	0.537019
$582$	0	0
$583$	− 4.00000i	− 0.165663i
$584$	0	0
$585$	10.0000i	0.413449i
$586$	0	0
$587$	32.9443i	1.35976i	0.733325	+	0.679878i	$0.237967\pi$
$587$	32.9443i	1.35976i	−0.733325	+	0.679878i	$0.762033\pi$
$588$	0	0
$589$	−16.9443	−0.698177
$590$	0	0
$591$	−15.8885	−0.653567
$592$	0	0
$593$	41.3050i	1.69619i	0.529843	+	0.848096i	$0.322251\pi$
$593$	41.3050i	1.69619i	−0.529843	+	0.848096i	$0.677749\pi$
$594$	0	0
$595$	−5.52786	−0.226620
$596$	0	0
$597$	11.5279i	0.471804i
$598$	0	0
$599$	26.3607	1.07707	0.538534	−	0.842604i	$-0.318978\pi$
$599$	26.3607	1.07707	0.538534	+	0.842604i	$0.318978\pi$
$600$	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
$603$	10.4721i	0.426458i
$604$	0	0
$605$	15.6525	0.636364
$606$	0	0
$607$	− 12.9443i	− 0.525392i	−0.964879	−	0.262696i	$-0.915388\pi$
$607$	− 12.9443i	− 0.525392i	0.964879	−	0.262696i	$-0.0846116\pi$
$608$	0	0
$609$	0.472136	0.0191319
$610$	0	0
$611$	−11.0557	−0.447267
$612$	0	0
$613$	19.4164i	0.784221i	0.919918	+	0.392111i	$0.128255\pi$
$613$	19.4164i	0.784221i	−0.919918	+	0.392111i	$0.871745\pi$
$614$	0	0
$615$	− 27.8885i	− 1.12457i
$616$	0	0
$617$	12.1115i	0.487589i	0.969827	+	0.243794i	$0.0783922\pi$
$617$	12.1115i	0.487589i	−0.969827	+	0.243794i	$0.921608\pi$
$618$	0	0
$619$	23.8885	0.960162	0.480081	−	0.877224i	$-0.340607\pi$
$619$	23.8885	0.960162	0.480081	+	0.877224i	$0.340607\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	9.41641i	0.377260i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	− 4.00000i	− 0.159745i
$628$	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
$633$	9.88854i	0.393034i
$634$	0	0
$635$	8.94427i	0.354943i
$636$	0	0
$637$	− 4.47214i	− 0.177192i
$638$	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
$643$	− 24.9443i	− 0.983706i	−0.870678	−	0.491853i	$-0.836320\pi$
$643$	− 24.9443i	− 0.983706i	0.870678	−	0.491853i	$-0.163680\pi$
$644$	0	0
$645$	−14.4721	−0.569840
$646$	0	0
$647$	− 42.2492i	− 1.66099i	−0.557027	−	0.830494i	$-0.688058\pi$
$647$	− 42.2492i	− 1.66099i	0.557027	−	0.830494i	$-0.311942\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−8.47214	−0.332049
$652$	0	0
$653$	18.9443i	0.741347i	0.928763	+	0.370673i	$0.120873\pi$
$653$	18.9443i	0.741347i	−0.928763	+	0.370673i	$0.879127\pi$
$654$	0	0
$655$	48.9443	1.91241
$656$	0	0
$657$	− 16.4721i	− 0.642639i
$658$	0	0
$659$	47.8885	1.86547	0.932736	−	0.360559i	$-0.117414\pi$
$659$	47.8885	1.86547	0.932736	+	0.360559i	$0.117414\pi$
$660$	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
$663$	11.0557i	0.429369i
$664$	0	0
$665$	− 4.47214i	− 0.173422i
$666$	0	0
$667$	− 1.88854i	− 0.0731247i
$668$	0	0
$669$	3.05573	0.118141
$670$	0	0
$671$	24.9443	0.962963
$672$	0	0
$673$	− 13.8885i	− 0.535364i	−0.963507	−	0.267682i	$-0.913742\pi$
$673$	− 13.8885i	− 0.535364i	0.963507	−	0.267682i	$-0.0862577\pi$
$674$	0	0
$675$	5.00000i	0.192450i
$676$	0	0
$677$	− 37.8885i	− 1.45618i	−0.685484	−	0.728088i	$-0.740409\pi$
$677$	− 37.8885i	− 1.45618i	0.685484	−	0.728088i	$-0.259591\pi$
$678$	0	0
$679$	−12.4721	−0.478637
$680$	0	0
$681$	−25.8885	−0.992051
$682$	0	0
$683$	− 49.8885i	− 1.90893i	−0.298320	−	0.954466i	$-0.596426\pi$
$683$	− 49.8885i	− 1.90893i	0.298320	−	0.954466i	$-0.403574\pi$
$684$	0	0
$685$	35.5279i	1.35745i
$686$	0	0
$687$	18.3607i	0.700504i
$688$	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
$693$	− 2.00000i	− 0.0759737i
$694$	0	0
$695$	33.4164	1.26756
$696$	0	0
$697$	− 30.8328i	− 1.16788i
$698$	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
$703$	− 12.9443i	− 0.488202i
$704$	0	0
$705$	−5.52786	−0.208191
$706$	0	0
$707$	4.47214i	0.168192i
$708$	0	0
$709$	−0.111456	−0.00418582	−0.00209291	−	0.999998i	$-0.500666\pi$
$709$	−0.111456	−0.00418582	−0.00209291	+	0.999998i	$0.500666\pi$
$710$	0	0
$711$	−8.94427	−0.335436
$712$	0	0
$713$	33.8885i	1.26914i
$714$	0	0
$715$	20.0000i	0.747958i
$716$	0	0
$717$	24.4721i	0.913929i
$718$	0	0
$719$	8.94427	0.333565	0.166783	−	0.985994i	$-0.446662\pi$
$719$	8.94427	0.333565	0.166783	+	0.985994i	$0.446662\pi$
$720$	0	0
$721$	−4.94427	−0.184134
$722$	0	0
$723$	− 23.8885i	− 0.888425i
$724$	0	0
$725$	−2.36068	−0.0876734
$726$	0	0
$727$	36.9443i	1.37019i	0.728455	+	0.685094i	$0.240239\pi$
$727$	36.9443i	1.37019i	−0.728455	+	0.685094i	$0.759761\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	− 50.3607i	− 1.86011i	−0.367415	−	0.930057i	$-0.619757\pi$
$733$	− 50.3607i	− 1.86011i	0.367415	−	0.930057i	$-0.380243\pi$
$734$	0	0
$735$	− 2.23607i	− 0.0824786i
$736$	0	0
$737$	20.9443i	0.771492i
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−8.94427	−0.328576
$742$	0	0
$743$	− 50.8328i	− 1.86488i	−0.361331	−	0.932438i	$-0.617678\pi$
$743$	− 50.8328i	− 1.86488i	0.361331	−	0.932438i	$-0.382322\pi$
$744$	0	0
$745$	−7.88854	−0.289014
$746$	0	0
$747$	− 12.9443i	− 0.473606i
$748$	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
$753$	− 24.0000i	− 0.874609i
$754$	0	0
$755$	−40.0000	−1.45575
$756$	0	0
$757$	21.5279i	0.782444i	0.920296	+	0.391222i	$0.127947\pi$
$757$	21.5279i	0.782444i	−0.920296	+	0.391222i	$0.872053\pi$
$758$	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
$763$	− 2.94427i	− 0.106590i
$764$	0	0
$765$	5.52786i	0.199860i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−18.9443	−0.683148	−0.341574	−	0.939855i	$-0.610960\pi$
$769$	−18.9443	−0.683148	−0.341574	+	0.939855i	$0.610960\pi$
$770$	0	0
$771$	3.41641	0.123039
$772$	0	0
$773$	0.944272i	0.0339631i	0.999856	+	0.0169815i	$0.00540566\pi$
$773$	0.944272i	0.0339631i	−0.999856	+	0.0169815i	$0.994594\pi$
$774$	0	0
$775$	42.3607	1.52164
$776$	0	0
$777$	− 6.47214i	− 0.232187i
$778$	0	0
$779$	24.9443	0.893721
$780$	0	0
$781$	−7.05573	−0.252474
$782$	0	0
$783$	− 0.472136i	− 0.0168728i
$784$	0	0
$785$	− 38.9443i	− 1.38998i
$786$	0	0
$787$	− 36.0000i	− 1.28326i	−0.767014	−	0.641631i	$-0.778258\pi$
$787$	− 36.0000i	− 1.28326i	0.767014	−	0.641631i	$-0.221742\pi$
$788$	0	0
$789$	4.00000	0.142404
$790$	0	0
$791$	−2.94427	−0.104686
$792$	0	0
$793$	− 55.7771i	− 1.98070i
$794$	0	0
$795$	−4.47214	−0.158610
$796$	0	0
$797$	− 28.0000i	− 0.991811i	−0.868377	−	0.495905i	$-0.834836\pi$
$797$	− 28.0000i	− 0.991811i	0.868377	−	0.495905i	$-0.165164\pi$
$798$	0	0
$799$	−6.11146	−0.216208
$800$	0	0
$801$	9.41641	0.332712
$802$	0	0
$803$	− 32.9443i	− 1.16258i
$804$	0	0
$805$	−8.94427	−0.315244
$806$	0	0
$807$	− 14.3607i	− 0.505520i
$808$	0	0
$809$	−28.8328	−1.01371	−0.506854	−	0.862032i	$-0.669192\pi$
$809$	−28.8328	−1.01371	−0.506854	+	0.862032i	$0.669192\pi$
$810$	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
$813$	− 15.5279i	− 0.544586i
$814$	0	0
$815$	7.63932i	0.267594i
$816$	0	0
$817$	− 12.9443i	− 0.452863i
$818$	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
$823$	32.9443i	1.14837i	0.818727	+	0.574183i	$0.194680\pi$
$823$	32.9443i	1.14837i	−0.818727	+	0.574183i	$0.805320\pi$
$824$	0	0
$825$	10.0000i	0.348155i
$826$	0	0
$827$	− 56.7214i	− 1.97239i	−0.165573	−	0.986197i	$-0.552947\pi$
$827$	− 56.7214i	− 1.97239i	0.165573	−	0.986197i	$-0.447053\pi$
$828$	0	0
$829$	−12.4721	−0.433175	−0.216588	−	0.976263i	$-0.569493\pi$
$829$	−12.4721	−0.433175	−0.216588	+	0.976263i	$0.569493\pi$
$830$	0	0
$831$	3.41641	0.118514
$832$	0	0
$833$	− 2.47214i	− 0.0856544i
$834$	0	0
$835$	3.41641i	0.118230i
$836$	0	0
$837$	8.47214i	0.292840i
$838$	0	0
$839$	−15.0557	−0.519781	−0.259891	−	0.965638i	$-0.583687\pi$
$839$	−15.0557	−0.519781	−0.259891	+	0.965638i	$0.583687\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	14.9443i	0.514708i
$844$	0	0
$845$	15.6525	0.538462
$846$	0	0
$847$	7.00000i	0.240523i
$848$	0	0
$849$	−0.944272	−0.0324073
$850$	0	0
$851$	−25.8885	−0.887448
$852$	0	0
$853$	− 40.4721i	− 1.38574i	−0.721063	−	0.692870i	$-0.756346\pi$
$853$	− 40.4721i	− 1.38574i	0.721063	−	0.692870i	$-0.243654\pi$
$854$	0	0
$855$	−4.47214	−0.152944
$856$	0	0
$857$	43.4164i	1.48308i	0.670911	+	0.741538i	$0.265903\pi$
$857$	43.4164i	1.48308i	−0.670911	+	0.741538i	$0.734097\pi$
$858$	0	0
$859$	38.9443	1.32876	0.664381	−	0.747394i	$-0.268695\pi$
$859$	38.9443	1.32876	0.664381	+	0.747394i	$0.268695\pi$
$860$	0	0
$861$	12.4721	0.425049
$862$	0	0
$863$	− 34.8328i	− 1.18572i	−0.805305	−	0.592861i	$-0.797998\pi$
$863$	− 34.8328i	− 1.18572i	0.805305	−	0.592861i	$-0.202002\pi$
$864$	0	0
$865$	− 26.8328i	− 0.912343i
$866$	0	0
$867$	− 10.8885i	− 0.369794i
$868$	0	0
$869$	−17.8885	−0.606827
$870$	0	0
$871$	46.8328	1.58687
$872$	0	0
$873$	12.4721i	0.422118i
$874$	0	0
$875$	11.1803i	0.377964i
$876$	0	0
$877$	13.3050i	0.449276i	0.974442	+	0.224638i	$0.0721200\pi$
$877$	13.3050i	0.449276i	−0.974442	+	0.224638i	$0.927880\pi$
$878$	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
$883$	28.3607i	0.954413i	0.878791	+	0.477206i	$0.158351\pi$
$883$	28.3607i	0.954413i	−0.878791	+	0.477206i	$0.841649\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.4164i	1.05486i	0.849599	+	0.527430i	$0.176844\pi$
$887$	31.4164i	1.05486i	−0.849599	+	0.527430i	$0.823156\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	− 4.94427i	− 0.165454i
$894$	0	0
$895$	−11.3050	−0.377883
$896$	0	0
$897$	17.8885i	0.597281i
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−4.94427	−0.164718
$902$	0	0
$903$	− 6.47214i	− 0.215379i
$904$	0	0
$905$	−16.8328	−0.559542
$906$	0	0
$907$	− 28.3607i	− 0.941701i	−0.882213	−	0.470850i	$-0.843947\pi$
$907$	− 28.3607i	− 0.941701i	0.882213	−	0.470850i	$-0.156053\pi$
$908$	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
$913$	− 25.8885i	− 0.856786i
$914$	0	0
$915$	− 27.8885i	− 0.921967i
$916$	0	0
$917$	21.8885i	0.722823i
$918$	0	0
$919$	−52.7214	−1.73912	−0.869559	−	0.493830i	$-0.835597\pi$
$919$	−52.7214	−1.73912	−0.869559	+	0.493830i	$0.835597\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	15.7771i	0.519309i
$924$	0	0
$925$	32.3607i	1.06401i
$926$	0	0
$927$	4.94427i	0.162391i
$928$	0	0
$929$	43.3050	1.42079	0.710395	−	0.703804i	$-0.248516\pi$
$929$	43.3050	1.42079	0.710395	+	0.703804i	$0.248516\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	− 16.0000i	− 0.523816i
$934$	0	0
$935$	11.0557i	0.361561i
$936$	0	0
$937$	− 47.3050i	− 1.54539i	−0.634780	−	0.772693i	$-0.718909\pi$
$937$	− 47.3050i	− 1.54539i	0.634780	−	0.772693i	$-0.281091\pi$
$938$	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
$943$	− 49.8885i	− 1.62459i
$944$	0	0
$945$	−2.23607	−0.0727393
$946$	0	0
$947$	16.0000i	0.519930i	0.965618	+	0.259965i	$0.0837111\pi$
$947$	16.0000i	0.519930i	−0.965618	+	0.259965i	$0.916289\pi$
$948$	0	0
$949$	−73.6656	−2.39129
$950$	0	0
$951$	10.9443	0.354892
$952$	0	0
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	0	0
$955$	−21.0557	−0.681347
$956$	0	0
$957$	− 0.944272i	− 0.0305240i
$958$	0	0
$959$	−15.8885	−0.513068
$960$	0	0
$961$	40.7771	1.31539
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.0557i	0.355896i
$966$	0	0
$967$	9.88854i	0.317994i	0.987279	+	0.158997i	$0.0508260\pi$
$967$	9.88854i	0.317994i	−0.987279	+	0.158997i	$0.949174\pi$
$968$	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
$973$	14.9443i	0.479091i
$974$	0	0
$975$	22.3607	0.716115
$976$	0	0
$977$	− 0.832816i	− 0.0266441i	−0.999911	−	0.0133221i	$-0.995759\pi$
$977$	− 0.832816i	− 0.0266441i	0.999911	−	0.0133221i	$-0.00424067\pi$
$978$	0	0
$979$	18.8328	0.601899
$980$	0	0
$981$	−2.94427	−0.0940034
$982$	0	0
$983$	− 34.4721i	− 1.09949i	−0.835332	−	0.549745i	$-0.814725\pi$
$983$	− 34.4721i	− 1.09949i	0.835332	−	0.549745i	$-0.185275\pi$
$984$	0	0
$985$	35.5279i	1.13201i
$986$	0	0
$987$	− 2.47214i	− 0.0786890i
$988$	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
$993$	1.88854i	0.0599311i
$994$	0	0
$995$	25.7771	0.817189
$996$	0	0
$997$	50.3607i	1.59494i	0.603359	+	0.797469i	$0.293828\pi$
$997$	50.3607i	1.59494i	−0.603359	+	0.797469i	$0.706172\pi$
$998$	0	0
$999$	−6.47214	−0.204769

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.c.169.1	✓	4	1.1	even	1	trivial
840.2.t.c.169.3	yes	4	5.4	even	2	inner
1680.2.t.h.1009.1		4	20.19	odd	2
1680.2.t.h.1009.3		4	4.3	odd	2
2520.2.t.f.1009.3		4	3.2	odd	2
2520.2.t.f.1009.4		4	15.14	odd	2
4200.2.a.bj.1.2		2	5.2	odd	4
4200.2.a.bk.1.1		2	5.3	odd	4
5040.2.t.u.1009.3		4	60.59	even	2
5040.2.t.u.1009.4		4	12.11	even	2
8400.2.a.cz.1.1		2	20.3	even	4
8400.2.a.db.1.2		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.t (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +4.47214i q^{13} +2.23607i q^{15} +2.47214i q^{17} -2.00000 q^{19} -1.00000 q^{21} +4.00000i q^{23} +5.00000 q^{25} +1.00000i q^{27} -0.472136 q^{29} +8.47214 q^{31} +2.00000i q^{33} +2.23607i q^{35} +6.47214i q^{37} +4.47214 q^{39} -12.4721 q^{41} +6.47214i q^{43} +2.23607 q^{45} +2.47214i q^{47} -1.00000 q^{49} +2.47214 q^{51} +2.00000i q^{53} +4.47214 q^{55} +2.00000i q^{57} -12.4721 q^{61} +1.00000i q^{63} -10.0000i q^{65} -10.4721i q^{67} +4.00000 q^{69} +3.52786 q^{71} +16.4721i q^{73} -5.00000i q^{75} +2.00000i q^{77} +8.94427 q^{79} +1.00000 q^{81} +12.9443i q^{83} -5.52786i q^{85} +0.472136i q^{87} -9.41641 q^{89} +4.47214 q^{91} -8.47214i q^{93} +4.47214 q^{95} -12.4721i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 8 q^{11} - 8 q^{19} - 4 q^{21} + 20 q^{25} + 16 q^{29} + 16 q^{31} - 32 q^{41} - 4 q^{49} - 8 q^{51} - 32 q^{61} + 16 q^{69} + 32 q^{71} + 4 q^{81} + 16 q^{89} + 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 8 * q^11 - 8 * q^19 - 4 * q^21 + 20 * q^25 + 16 * q^29 + 16 * q^31 - 32 * q^41 - 4 * q^49 - 8 * q^51 - 32 * q^61 + 16 * q^69 + 32 * q^71 + 4 * q^81 + 16 * q^89 + 8 * q^99

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