LMFDB - Embedded newform 2100.3.j.a.601.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,3,Mod(601,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2100.601");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2100.j (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:	$57.2208555157$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			601.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2100.601
Dual form			2100.3.j.a.601.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -7.00000 q^{7} -3.00000 q^{9} +3.00000 q^{11} +13.8564i q^{13} -20.7846i q^{17} +20.7846i q^{19} +12.1244i q^{21} +3.00000 q^{23} +5.19615i q^{27} +3.00000 q^{29} +6.92820i q^{31} -5.19615i q^{33} +55.0000 q^{37} +24.0000 q^{39} -20.7846i q^{41} +7.00000 q^{43} -62.3538i q^{47} +49.0000 q^{49} -36.0000 q^{51} -102.000 q^{53} +36.0000 q^{57} +41.5692i q^{59} -20.7846i q^{61} +21.0000 q^{63} -49.0000 q^{67} -5.19615i q^{69} +123.000 q^{71} -27.7128i q^{73} -21.0000 q^{77} +47.0000 q^{79} +9.00000 q^{81} -103.923i q^{83} -5.19615i q^{87} -96.9948i q^{91} +12.0000 q^{93} -145.492i q^{97} -9.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{7} - 6 q^{9} + 6 q^{11} + 6 q^{23} + 6 q^{29} + 110 q^{37} + 48 q^{39} + 14 q^{43} + 98 q^{49} - 72 q^{51} - 204 q^{53} + 72 q^{57} + 42 q^{63} - 98 q^{67} + 246 q^{71} - 42 q^{77} + 94 q^{79}+ \cdots - 18 q^{99}+O(q^{100}) $ 2 * q - 14 * q^7 - 6 * q^9 + 6 * q^11 + 6 * q^23 + 6 * q^29 + 110 * q^37 + 48 * q^39 + 14 * q^43 + 98 * q^49 - 72 * q^51 - 204 * q^53 + 72 * q^57 + 42 * q^63 - 98 * q^67 + 246 * q^71 - 42 * q^77 + 94 * q^79 + 18 * q^81 + 24 * q^93 - 18 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−1.00000
$7$	−7.00000	−1.00000
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	3.00000	0.272727	0.136364	−	0.990659i	$-0.456458\pi$
$11$	3.00000	0.272727	0.136364	+	0.990659i	$0.456458\pi$
$12$	0	0
$13$	13.8564i	1.06588i	0.846154	+	0.532939i	$0.178912\pi$
$13$	13.8564i	1.06588i	−0.846154	+	0.532939i	$0.821088\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 20.7846i	− 1.22262i	−0.791390	−	0.611312i	$-0.790642\pi$
$17$	− 20.7846i	− 1.22262i	0.791390	−	0.611312i	$-0.209358\pi$
$18$	0	0
$19$	20.7846i	1.09393i	0.837157	+	0.546963i	$0.184216\pi$
$19$	20.7846i	1.09393i	−0.837157	+	0.546963i	$0.815784\pi$
$20$	0	0
$21$	12.1244i	0.577350i
$22$	0	0
$23$	3.00000	0.130435	0.0652174	−	0.997871i	$-0.479226\pi$
$23$	3.00000	0.130435	0.0652174	+	0.997871i	$0.479226\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	3.00000	0.103448	0.0517241	−	0.998661i	$-0.483528\pi$
$29$	3.00000	0.103448	0.0517241	+	0.998661i	$0.483528\pi$
$30$	0	0
$31$	6.92820i	0.223490i	0.993737	+	0.111745i	$0.0356441\pi$
$31$	6.92820i	0.223490i	−0.993737	+	0.111745i	$0.964356\pi$
$32$	0	0
$33$	− 5.19615i	− 0.157459i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	55.0000	1.48649	0.743243	−	0.669021i	$-0.233287\pi$
$37$	55.0000	1.48649	0.743243	+	0.669021i	$0.233287\pi$
$38$	0	0
$39$	24.0000	0.615385
$40$	0	0
$41$	− 20.7846i	− 0.506942i	−0.967343	−	0.253471i	$-0.918428\pi$
$41$	− 20.7846i	− 0.506942i	0.967343	−	0.253471i	$-0.0815722\pi$
$42$	0	0
$43$	7.00000	0.162791	0.0813953	−	0.996682i	$-0.474062\pi$
$43$	7.00000	0.162791	0.0813953	+	0.996682i	$0.474062\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 62.3538i	− 1.32668i	−0.748319	−	0.663339i	$-0.769139\pi$
$47$	− 62.3538i	− 1.32668i	0.748319	−	0.663339i	$-0.230861\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	0	0
$51$	−36.0000	−0.705882
$52$	0	0
$53$	−102.000	−1.92453	−0.962264	−	0.272117i	$-0.912276\pi$
$53$	−102.000	−1.92453	−0.962264	+	0.272117i	$0.912276\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	36.0000	0.631579
$58$	0	0
$59$	41.5692i	0.704563i	0.935894	+	0.352282i	$0.114594\pi$
$59$	41.5692i	0.704563i	−0.935894	+	0.352282i	$0.885406\pi$
$60$	0	0
$61$	− 20.7846i	− 0.340731i	−0.985381	−	0.170366i	$-0.945505\pi$
$61$	− 20.7846i	− 0.340731i	0.985381	−	0.170366i	$-0.0544949\pi$
$62$	0	0
$63$	21.0000	0.333333
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−49.0000	−0.731343	−0.365672	−	0.930744i	$-0.619161\pi$
$67$	−49.0000	−0.731343	−0.365672	+	0.930744i	$0.619161\pi$
$68$	0	0
$69$	− 5.19615i	− 0.0753066i
$70$	0	0
$71$	123.000	1.73239	0.866197	−	0.499702i	$-0.166557\pi$
$71$	123.000	1.73239	0.866197	+	0.499702i	$0.166557\pi$
$72$	0	0
$73$	− 27.7128i	− 0.379628i	−0.981820	−	0.189814i	$-0.939212\pi$
$73$	− 27.7128i	− 0.379628i	0.981820	−	0.189814i	$-0.0607884\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−21.0000	−0.272727
$78$	0	0
$79$	47.0000	0.594937	0.297468	−	0.954732i	$-0.403858\pi$
$79$	47.0000	0.594937	0.297468	+	0.954732i	$0.403858\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 103.923i	− 1.25208i	−0.779789	−	0.626042i	$-0.784674\pi$
$83$	− 103.923i	− 1.25208i	0.779789	−	0.626042i	$-0.215326\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.19615i	− 0.0597259i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	− 96.9948i	− 1.06588i
$92$	0	0
$93$	12.0000	0.129032
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 145.492i	− 1.49992i	−0.661483	−	0.749960i	$-0.730073\pi$
$97$	− 145.492i	− 1.49992i	0.661483	−	0.749960i	$-0.269927\pi$
$98$	0	0
$99$	−9.00000	−0.0909091
$100$	0	0
$101$	− 124.708i	− 1.23473i	−0.786677	−	0.617365i	$-0.788200\pi$
$101$	− 124.708i	− 1.23473i	0.786677	−	0.617365i	$-0.211800\pi$
$102$	0	0
$103$	− 48.4974i	− 0.470849i	−0.971893	−	0.235424i	$-0.924352\pi$
$103$	− 48.4974i	− 0.470849i	0.971893	−	0.235424i	$-0.0756480\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.0560748	−0.0280374	−	0.999607i	$-0.508926\pi$
$107$	−6.00000	−0.0560748	−0.0280374	+	0.999607i	$0.508926\pi$
$108$	0	0
$109$	−89.0000	−0.816514	−0.408257	−	0.912867i	$-0.633863\pi$
$109$	−89.0000	−0.816514	−0.408257	+	0.912867i	$0.633863\pi$
$110$	0	0
$111$	− 95.2628i	− 0.858223i
$112$	0	0
$113$	99.0000	0.876106	0.438053	−	0.898949i	$-0.355668\pi$
$113$	99.0000	0.876106	0.438053	+	0.898949i	$0.355668\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 41.5692i	− 0.355292i
$118$	0	0
$119$	145.492i	1.22262i
$120$	0	0
$121$	−112.000	−0.925620
$122$	0	0
$123$	−36.0000	−0.292683
$124$	0	0
$125$	0	0
$126$	0	0
$127$	119.000	0.937008	0.468504	−	0.883461i	$-0.344793\pi$
$127$	119.000	0.937008	0.468504	+	0.883461i	$0.344793\pi$
$128$	0	0
$129$	− 12.1244i	− 0.0939873i
$130$	0	0
$131$	− 20.7846i	− 0.158661i	−0.996848	−	0.0793306i	$-0.974722\pi$
$131$	− 20.7846i	− 0.158661i	0.996848	−	0.0793306i	$-0.0252783\pi$
$132$	0	0
$133$	− 145.492i	− 1.09393i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−78.0000	−0.569343	−0.284672	−	0.958625i	$-0.591885\pi$
$137$	−78.0000	−0.569343	−0.284672	+	0.958625i	$0.591885\pi$
$138$	0	0
$139$	96.9948i	0.697805i	0.937159	+	0.348902i	$0.113446\pi$
$139$	96.9948i	0.697805i	−0.937159	+	0.348902i	$0.886554\pi$
$140$	0	0
$141$	−108.000	−0.765957
$142$	0	0
$143$	41.5692i	0.290694i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 84.8705i	− 0.577350i
$148$	0	0
$149$	27.0000	0.181208	0.0906040	−	0.995887i	$-0.471120\pi$
$149$	27.0000	0.181208	0.0906040	+	0.995887i	$0.471120\pi$
$150$	0	0
$151$	31.0000	0.205298	0.102649	−	0.994718i	$-0.467268\pi$
$151$	31.0000	0.205298	0.102649	+	0.994718i	$0.467268\pi$
$152$	0	0
$153$	62.3538i	0.407541i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 297.913i	− 1.89753i	−0.315977	−	0.948767i	$-0.602332\pi$
$157$	− 297.913i	− 1.89753i	0.315977	−	0.948767i	$-0.397668\pi$
$158$	0	0
$159$	176.669i	1.11113i
$160$	0	0
$161$	−21.0000	−0.130435
$162$	0	0
$163$	26.0000	0.159509	0.0797546	−	0.996815i	$-0.474586\pi$
$163$	26.0000	0.159509	0.0797546	+	0.996815i	$0.474586\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 166.277i	− 0.995670i	−0.867272	−	0.497835i	$-0.834129\pi$
$167$	− 166.277i	− 0.995670i	0.867272	−	0.497835i	$-0.165871\pi$
$168$	0	0
$169$	−23.0000	−0.136095
$170$	0	0
$171$	− 62.3538i	− 0.364642i
$172$	0	0
$173$	20.7846i	0.120142i	0.998194	+	0.0600711i	$0.0191328\pi$
$173$	20.7846i	0.120142i	−0.998194	+	0.0600711i	$0.980867\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	72.0000	0.406780
$178$	0	0
$179$	−54.0000	−0.301676	−0.150838	−	0.988558i	$-0.548197\pi$
$179$	−54.0000	−0.301676	−0.150838	+	0.988558i	$0.548197\pi$
$180$	0	0
$181$	− 193.990i	− 1.07177i	−0.844292	−	0.535883i	$-0.819979\pi$
$181$	− 193.990i	− 1.07177i	0.844292	−	0.535883i	$-0.180021\pi$
$182$	0	0
$183$	−36.0000	−0.196721
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 62.3538i	− 0.333443i
$188$	0	0
$189$	− 36.3731i	− 0.192450i
$190$	0	0
$191$	210.000	1.09948	0.549738	−	0.835337i	$-0.314727\pi$
$191$	210.000	1.09948	0.549738	+	0.835337i	$0.314727\pi$
$192$	0	0
$193$	−161.000	−0.834197	−0.417098	−	0.908861i	$-0.636953\pi$
$193$	−161.000	−0.834197	−0.417098	+	0.908861i	$0.636953\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−213.000	−1.08122	−0.540609	−	0.841274i	$-0.681806\pi$
$197$	−213.000	−1.08122	−0.540609	+	0.841274i	$0.681806\pi$
$198$	0	0
$199$	159.349i	0.800747i	0.916352	+	0.400374i	$0.131120\pi$
$199$	159.349i	0.800747i	−0.916352	+	0.400374i	$0.868880\pi$
$200$	0	0
$201$	84.8705i	0.422241i
$202$	0	0
$203$	−21.0000	−0.103448
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−9.00000	−0.0434783
$208$	0	0
$209$	62.3538i	0.298344i
$210$	0	0
$211$	−230.000	−1.09005	−0.545024	−	0.838421i	$-0.683479\pi$
$211$	−230.000	−1.09005	−0.545024	+	0.838421i	$0.683479\pi$
$212$	0	0
$213$	− 213.042i	− 1.00020i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 48.4974i	− 0.223490i
$218$	0	0
$219$	−48.0000	−0.219178
$220$	0	0
$221$	288.000	1.30317
$222$	0	0
$223$	96.9948i	0.434954i	0.976065	+	0.217477i	$0.0697828\pi$
$223$	96.9948i	0.434954i	−0.976065	+	0.217477i	$0.930217\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 228.631i	− 1.00718i	−0.863942	−	0.503592i	$-0.832012\pi$
$227$	− 228.631i	− 1.00718i	0.863942	−	0.503592i	$-0.167988\pi$
$228$	0	0
$229$	304.841i	1.33118i	0.746316	+	0.665592i	$0.231821\pi$
$229$	304.841i	1.33118i	−0.746316	+	0.665592i	$0.768179\pi$
$230$	0	0
$231$	36.3731i	0.157459i
$232$	0	0
$233$	339.000	1.45494	0.727468	−	0.686142i	$-0.240697\pi$
$233$	339.000	1.45494	0.727468	+	0.686142i	$0.240697\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 81.4064i	− 0.343487i
$238$	0	0
$239$	258.000	1.07950	0.539749	−	0.841826i	$-0.318519\pi$
$239$	258.000	1.07950	0.539749	+	0.841826i	$0.318519\pi$
$240$	0	0
$241$	173.205i	0.718693i	0.933204	+	0.359347i	$0.117000\pi$
$241$	173.205i	0.718693i	−0.933204	+	0.359347i	$0.883000\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−288.000	−1.16599
$248$	0	0
$249$	−180.000	−0.722892
$250$	0	0
$251$	20.7846i	0.0828072i	0.999143	+	0.0414036i	$0.0131829\pi$
$251$	20.7846i	0.0828072i	−0.999143	+	0.0414036i	$0.986817\pi$
$252$	0	0
$253$	9.00000	0.0355731
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 478.046i	− 1.86010i	−0.367431	−	0.930051i	$-0.619763\pi$
$257$	− 478.046i	− 1.86010i	0.367431	−	0.930051i	$-0.380237\pi$
$258$	0	0
$259$	−385.000	−1.48649
$260$	0	0
$261$	−9.00000	−0.0344828
$262$	0	0
$263$	435.000	1.65399	0.826996	−	0.562208i	$-0.190048\pi$
$263$	435.000	1.65399	0.826996	+	0.562208i	$0.190048\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 145.492i	− 0.540863i	−0.962739	−	0.270432i	$-0.912834\pi$
$269$	− 145.492i	− 0.540863i	0.962739	−	0.270432i	$-0.0871664\pi$
$270$	0	0
$271$	− 346.410i	− 1.27827i	−0.769096	−	0.639133i	$-0.779293\pi$
$271$	− 346.410i	− 1.27827i	0.769096	−	0.639133i	$-0.220707\pi$
$272$	0	0
$273$	−168.000	−0.615385
$274$	0	0
$275$	0	0
$276$	0	0
$277$	362.000	1.30686	0.653430	−	0.756987i	$-0.273330\pi$
$277$	362.000	1.30686	0.653430	+	0.756987i	$0.273330\pi$
$278$	0	0
$279$	− 20.7846i	− 0.0744968i
$280$	0	0
$281$	−309.000	−1.09964	−0.549822	−	0.835282i	$-0.685304\pi$
$281$	−309.000	−1.09964	−0.549822	+	0.835282i	$0.685304\pi$
$282$	0	0
$283$	318.697i	1.12614i	0.826410	+	0.563070i	$0.190380\pi$
$283$	318.697i	1.12614i	−0.826410	+	0.563070i	$0.809620\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	145.492i	0.506942i
$288$	0	0
$289$	−143.000	−0.494810
$290$	0	0
$291$	−252.000	−0.865979
$292$	0	0
$293$	− 498.831i	− 1.70249i	−0.524766	−	0.851247i	$-0.675847\pi$
$293$	− 498.831i	− 1.70249i	0.524766	−	0.851247i	$-0.324153\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	15.5885i	0.0524864i
$298$	0	0
$299$	41.5692i	0.139027i
$300$	0	0
$301$	−49.0000	−0.162791
$302$	0	0
$303$	−216.000	−0.712871
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.92820i	0.0225674i	0.999936	+	0.0112837i	$0.00359180\pi$
$307$	6.92820i	0.0225674i	−0.999936	+	0.0112837i	$0.996408\pi$
$308$	0	0
$309$	−84.0000	−0.271845
$310$	0	0
$311$	− 561.184i	− 1.80445i	−0.431264	−	0.902226i	$-0.641932\pi$
$311$	− 561.184i	− 1.80445i	0.431264	−	0.902226i	$-0.358068\pi$
$312$	0	0
$313$	− 547.328i	− 1.74865i	−0.485339	−	0.874326i	$-0.661304\pi$
$313$	− 547.328i	− 1.74865i	0.485339	−	0.874326i	$-0.338696\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−165.000	−0.520505	−0.260252	−	0.965541i	$-0.583806\pi$
$317$	−165.000	−0.520505	−0.260252	+	0.965541i	$0.583806\pi$
$318$	0	0
$319$	9.00000	0.0282132
$320$	0	0
$321$	10.3923i	0.0323748i
$322$	0	0
$323$	432.000	1.33746
$324$	0	0
$325$	0	0
$326$	0	0
$327$	154.153i	0.471414i
$328$	0	0
$329$	436.477i	1.32668i
$330$	0	0
$331$	−65.0000	−0.196375	−0.0981873	−	0.995168i	$-0.531304\pi$
$331$	−65.0000	−0.196375	−0.0981873	+	0.995168i	$0.531304\pi$
$332$	0	0
$333$	−165.000	−0.495495
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−14.0000	−0.0415430	−0.0207715	−	0.999784i	$-0.506612\pi$
$337$	−14.0000	−0.0415430	−0.0207715	+	0.999784i	$0.506612\pi$
$338$	0	0
$339$	− 171.473i	− 0.505820i
$340$	0	0
$341$	20.7846i	0.0609519i
$342$	0	0
$343$	−343.000	−1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−309.000	−0.890490	−0.445245	−	0.895409i	$-0.646883\pi$
$347$	−309.000	−0.890490	−0.445245	+	0.895409i	$0.646883\pi$
$348$	0	0
$349$	− 297.913i	− 0.853618i	−0.904342	−	0.426809i	$-0.859638\pi$
$349$	− 297.913i	− 0.853618i	0.904342	−	0.426809i	$-0.140362\pi$
$350$	0	0
$351$	−72.0000	−0.205128
$352$	0	0
$353$	145.492i	0.412159i	0.978535	+	0.206080i	$0.0660706\pi$
$353$	145.492i	0.412159i	−0.978535	+	0.206080i	$0.933929\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	252.000	0.705882
$358$	0	0
$359$	243.000	0.676880	0.338440	−	0.940988i	$-0.390101\pi$
$359$	243.000	0.676880	0.338440	+	0.940988i	$0.390101\pi$
$360$	0	0
$361$	−71.0000	−0.196676
$362$	0	0
$363$	193.990i	0.534407i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 339.482i	− 0.925019i	−0.886614	−	0.462509i	$-0.846949\pi$
$367$	− 339.482i	− 0.925019i	0.886614	−	0.462509i	$-0.153051\pi$
$368$	0	0
$369$	62.3538i	0.168981i
$370$	0	0
$371$	714.000	1.92453
$372$	0	0
$373$	−289.000	−0.774799	−0.387399	−	0.921912i	$-0.626627\pi$
$373$	−289.000	−0.774799	−0.387399	+	0.921912i	$0.626627\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	41.5692i	0.110263i
$378$	0	0
$379$	271.000	0.715040	0.357520	−	0.933906i	$-0.383622\pi$
$379$	271.000	0.715040	0.357520	+	0.933906i	$0.383622\pi$
$380$	0	0
$381$	− 206.114i	− 0.540982i
$382$	0	0
$383$	124.708i	0.325607i	0.986658	+	0.162804i	$0.0520537\pi$
$383$	124.708i	0.325607i	−0.986658	+	0.162804i	$0.947946\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−21.0000	−0.0542636
$388$	0	0
$389$	339.000	0.871465	0.435733	−	0.900076i	$-0.356489\pi$
$389$	339.000	0.871465	0.435733	+	0.900076i	$0.356489\pi$
$390$	0	0
$391$	− 62.3538i	− 0.159473i
$392$	0	0
$393$	−36.0000	−0.0916031
$394$	0	0
$395$	0	0
$396$	0	0
$397$	512.687i	1.29140i	0.763590	+	0.645702i	$0.223435\pi$
$397$	512.687i	1.29140i	−0.763590	+	0.645702i	$0.776565\pi$
$398$	0	0
$399$	−252.000	−0.631579
$400$	0	0
$401$	387.000	0.965087	0.482544	−	0.875872i	$-0.339713\pi$
$401$	387.000	0.965087	0.482544	+	0.875872i	$0.339713\pi$
$402$	0	0
$403$	−96.0000	−0.238213
$404$	0	0
$405$	0	0
$406$	0	0
$407$	165.000	0.405405
$408$	0	0
$409$	− 533.472i	− 1.30433i	−0.758076	−	0.652166i	$-0.773861\pi$
$409$	− 533.472i	− 1.30433i	0.758076	−	0.652166i	$-0.226139\pi$
$410$	0	0
$411$	135.100i	0.328710i
$412$	0	0
$413$	− 290.985i	− 0.704563i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	168.000	0.402878
$418$	0	0
$419$	394.908i	0.942500i	0.882000	+	0.471250i	$0.156197\pi$
$419$	394.908i	0.942500i	−0.882000	+	0.471250i	$0.843803\pi$
$420$	0	0
$421$	−169.000	−0.401425	−0.200713	−	0.979650i	$-0.564326\pi$
$421$	−169.000	−0.401425	−0.200713	+	0.979650i	$0.564326\pi$
$422$	0	0
$423$	187.061i	0.442226i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	145.492i	0.340731i
$428$	0	0
$429$	72.0000	0.167832
$430$	0	0
$431$	450.000	1.04408	0.522042	−	0.852920i	$-0.325171\pi$
$431$	450.000	1.04408	0.522042	+	0.852920i	$0.325171\pi$
$432$	0	0
$433$	96.9948i	0.224007i	0.993708	+	0.112003i	$0.0357267\pi$
$433$	96.9948i	0.224007i	−0.993708	+	0.112003i	$0.964273\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	62.3538i	0.142686i
$438$	0	0
$439$	263.272i	0.599708i	0.953985	+	0.299854i	$0.0969379\pi$
$439$	263.272i	0.599708i	−0.953985	+	0.299854i	$0.903062\pi$
$440$	0	0
$441$	−147.000	−0.333333
$442$	0	0
$443$	−54.0000	−0.121896	−0.0609481	−	0.998141i	$-0.519412\pi$
$443$	−54.0000	−0.121896	−0.0609481	+	0.998141i	$0.519412\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 46.7654i	− 0.104621i
$448$	0	0
$449$	387.000	0.861915	0.430958	−	0.902372i	$-0.358176\pi$
$449$	387.000	0.861915	0.430958	+	0.902372i	$0.358176\pi$
$450$	0	0
$451$	− 62.3538i	− 0.138257i
$452$	0	0
$453$	− 53.6936i	− 0.118529i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	607.000	1.32823	0.664114	−	0.747632i	$-0.268809\pi$
$457$	607.000	1.32823	0.664114	+	0.747632i	$0.268809\pi$
$458$	0	0
$459$	108.000	0.235294
$460$	0	0
$461$	249.415i	0.541031i	0.962716	+	0.270516i	$0.0871941\pi$
$461$	249.415i	0.541031i	−0.962716	+	0.270516i	$0.912806\pi$
$462$	0	0
$463$	338.000	0.730022	0.365011	−	0.931003i	$-0.381065\pi$
$463$	338.000	0.730022	0.365011	+	0.931003i	$0.381065\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	789.815i	1.69125i	0.533775	+	0.845627i	$0.320773\pi$
$467$	789.815i	1.69125i	−0.533775	+	0.845627i	$0.679227\pi$
$468$	0	0
$469$	343.000	0.731343
$470$	0	0
$471$	−516.000	−1.09554
$472$	0	0
$473$	21.0000	0.0443975
$474$	0	0
$475$	0	0
$476$	0	0
$477$	306.000	0.641509
$478$	0	0
$479$	− 748.246i	− 1.56210i	−0.624468	−	0.781050i	$-0.714684\pi$
$479$	− 748.246i	− 1.56210i	0.624468	−	0.781050i	$-0.285316\pi$
$480$	0	0
$481$	762.102i	1.58441i
$482$	0	0
$483$	36.3731i	0.0753066i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.0000	−0.0349076	−0.0174538	−	0.999848i	$-0.505556\pi$
$487$	−17.0000	−0.0349076	−0.0174538	+	0.999848i	$0.505556\pi$
$488$	0	0
$489$	− 45.0333i	− 0.0920927i
$490$	0	0
$491$	579.000	1.17923	0.589613	−	0.807686i	$-0.299280\pi$
$491$	579.000	1.17923	0.589613	+	0.807686i	$0.299280\pi$
$492$	0	0
$493$	− 62.3538i	− 0.126478i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−861.000	−1.73239
$498$	0	0
$499$	58.0000	0.116232	0.0581162	−	0.998310i	$-0.481491\pi$
$499$	58.0000	0.116232	0.0581162	+	0.998310i	$0.481491\pi$
$500$	0	0
$501$	−288.000	−0.574850
$502$	0	0
$503$	956.092i	1.90078i	0.311062	+	0.950390i	$0.399315\pi$
$503$	956.092i	1.90078i	−0.311062	+	0.950390i	$0.600685\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	39.8372i	0.0785743i
$508$	0	0
$509$	− 332.554i	− 0.653347i	−0.945137	−	0.326674i	$-0.894072\pi$
$509$	− 332.554i	− 0.653347i	0.945137	−	0.326674i	$-0.105928\pi$
$510$	0	0
$511$	193.990i	0.379628i
$512$	0	0
$513$	−108.000	−0.210526
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 187.061i	− 0.361821i
$518$	0	0
$519$	36.0000	0.0693642
$520$	0	0
$521$	706.677i	1.35639i	0.734884	+	0.678193i	$0.237237\pi$
$521$	706.677i	1.35639i	−0.734884	+	0.678193i	$0.762763\pi$
$522$	0	0
$523$	332.554i	0.635858i	0.948115	+	0.317929i	$0.102987\pi$
$523$	332.554i	0.635858i	−0.948115	+	0.317929i	$0.897013\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	144.000	0.273245
$528$	0	0
$529$	−520.000	−0.982987
$530$	0	0
$531$	− 124.708i	− 0.234854i
$532$	0	0
$533$	288.000	0.540338
$534$	0	0
$535$	0	0
$536$	0	0
$537$	93.5307i	0.174173i
$538$	0	0
$539$	147.000	0.272727
$540$	0	0
$541$	−457.000	−0.844732	−0.422366	−	0.906425i	$-0.638800\pi$
$541$	−457.000	−0.844732	−0.422366	+	0.906425i	$0.638800\pi$
$542$	0	0
$543$	−336.000	−0.618785
$544$	0	0
$545$	0	0
$546$	0	0
$547$	551.000	1.00731	0.503656	−	0.863904i	$-0.331988\pi$
$547$	551.000	1.00731	0.503656	+	0.863904i	$0.331988\pi$
$548$	0	0
$549$	62.3538i	0.113577i
$550$	0	0
$551$	62.3538i	0.113165i
$552$	0	0
$553$	−329.000	−0.594937
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−333.000	−0.597846	−0.298923	−	0.954277i	$-0.596627\pi$
$557$	−333.000	−0.597846	−0.298923	+	0.954277i	$0.596627\pi$
$558$	0	0
$559$	96.9948i	0.173515i
$560$	0	0
$561$	−108.000	−0.192513
$562$	0	0
$563$	706.677i	1.25520i	0.778537	+	0.627599i	$0.215962\pi$
$563$	706.677i	1.25520i	−0.778537	+	0.627599i	$0.784038\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−63.0000	−0.111111
$568$	0	0
$569$	−477.000	−0.838313	−0.419156	−	0.907914i	$-0.637674\pi$
$569$	−477.000	−0.838313	−0.419156	+	0.907914i	$0.637674\pi$
$570$	0	0
$571$	−649.000	−1.13660	−0.568301	−	0.822821i	$-0.692399\pi$
$571$	−649.000	−1.13660	−0.568301	+	0.822821i	$0.692399\pi$
$572$	0	0
$573$	− 363.731i	− 0.634783i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	228.631i	0.396240i	0.980178	+	0.198120i	$0.0634836\pi$
$577$	228.631i	0.396240i	−0.980178	+	0.198120i	$0.936516\pi$
$578$	0	0
$579$	278.860i	0.481624i
$580$	0	0
$581$	727.461i	1.25208i
$582$	0	0
$583$	−306.000	−0.524871
$584$	0	0
$585$	0	0
$586$	0	0
$587$	665.108i	1.13306i	0.824040	+	0.566531i	$0.191715\pi$
$587$	665.108i	1.13306i	−0.824040	+	0.566531i	$0.808285\pi$
$588$	0	0
$589$	−144.000	−0.244482
$590$	0	0
$591$	368.927i	0.624242i
$592$	0	0
$593$	− 540.400i	− 0.911298i	−0.890159	−	0.455649i	$-0.849407\pi$
$593$	− 540.400i	− 0.911298i	0.890159	−	0.455649i	$-0.150593\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	276.000	0.462312
$598$	0	0
$599$	243.000	0.405676	0.202838	−	0.979212i	$-0.434983\pi$
$599$	243.000	0.405676	0.202838	+	0.979212i	$0.434983\pi$
$600$	0	0
$601$	803.672i	1.33722i	0.743611	+	0.668612i	$0.233111\pi$
$601$	803.672i	1.33722i	−0.743611	+	0.668612i	$0.766889\pi$
$602$	0	0
$603$	147.000	0.243781
$604$	0	0
$605$	0	0
$606$	0	0
$607$	526.543i	0.867452i	0.901045	+	0.433726i	$0.142801\pi$
$607$	526.543i	0.867452i	−0.901045	+	0.433726i	$0.857199\pi$
$608$	0	0
$609$	36.3731i	0.0597259i
$610$	0	0
$611$	864.000	1.41408
$612$	0	0
$613$	−361.000	−0.588907	−0.294454	−	0.955666i	$-0.595138\pi$
$613$	−361.000	−0.588907	−0.294454	+	0.955666i	$0.595138\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−717.000	−1.16207	−0.581037	−	0.813877i	$-0.697353\pi$
$617$	−717.000	−1.16207	−0.581037	+	0.813877i	$0.697353\pi$
$618$	0	0
$619$	− 1039.23i	− 1.67889i	−0.543448	−	0.839443i	$-0.682881\pi$
$619$	− 1039.23i	− 1.67889i	0.543448	−	0.839443i	$-0.317119\pi$
$620$	0	0
$621$	15.5885i	0.0251022i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	108.000	0.172249
$628$	0	0
$629$	− 1143.15i	− 1.81741i
$630$	0	0
$631$	−145.000	−0.229794	−0.114897	−	0.993377i	$-0.536654\pi$
$631$	−145.000	−0.229794	−0.114897	+	0.993377i	$0.536654\pi$
$632$	0	0
$633$	398.372i	0.629339i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	678.964i	1.06588i
$638$	0	0
$639$	−369.000	−0.577465
$640$	0	0
$641$	1035.00	1.61466	0.807332	−	0.590097i	$-0.200911\pi$
$641$	1035.00	1.61466	0.807332	+	0.590097i	$0.200911\pi$
$642$	0	0
$643$	124.708i	0.193947i	0.995287	+	0.0969733i	$0.0309161\pi$
$643$	124.708i	0.193947i	−0.995287	+	0.0969733i	$0.969084\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 249.415i	− 0.385495i	−0.981248	−	0.192748i	$-0.938260\pi$
$647$	− 249.415i	− 0.385495i	0.981248	−	0.192748i	$-0.0617398\pi$
$648$	0	0
$649$	124.708i	0.192154i
$650$	0	0
$651$	−84.0000	−0.129032
$652$	0	0
$653$	858.000	1.31394	0.656968	−	0.753919i	$-0.271839\pi$
$653$	858.000	1.31394	0.656968	+	0.753919i	$0.271839\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	83.1384i	0.126543i
$658$	0	0
$659$	282.000	0.427921	0.213961	−	0.976842i	$-0.431364\pi$
$659$	282.000	0.427921	0.213961	+	0.976842i	$0.431364\pi$
$660$	0	0
$661$	− 949.164i	− 1.43595i	−0.696068	−	0.717976i	$-0.745069\pi$
$661$	− 949.164i	− 1.43595i	0.696068	−	0.717976i	$-0.254931\pi$
$662$	0	0
$663$	− 498.831i	− 0.752384i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.00000	0.0134933
$668$	0	0
$669$	168.000	0.251121
$670$	0	0
$671$	− 62.3538i	− 0.0929267i
$672$	0	0
$673$	466.000	0.692422	0.346211	−	0.938157i	$-0.387468\pi$
$673$	466.000	0.692422	0.346211	+	0.938157i	$0.387468\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	706.677i	1.04384i	0.852996	+	0.521918i	$0.174783\pi$
$677$	706.677i	1.04384i	−0.852996	+	0.521918i	$0.825217\pi$
$678$	0	0
$679$	1018.45i	1.49992i
$680$	0	0
$681$	−396.000	−0.581498
$682$	0	0
$683$	−813.000	−1.19034	−0.595168	−	0.803601i	$-0.702915\pi$
$683$	−813.000	−1.19034	−0.595168	+	0.803601i	$0.702915\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	528.000	0.768559
$688$	0	0
$689$	− 1413.35i	− 2.05131i
$690$	0	0
$691$	− 20.7846i	− 0.0300790i	−0.999887	−	0.0150395i	$-0.995213\pi$
$691$	− 20.7846i	− 0.0300790i	0.999887	−	0.0150395i	$-0.00478741\pi$
$692$	0	0
$693$	63.0000	0.0909091
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−432.000	−0.619799
$698$	0	0
$699$	− 587.165i	− 0.840007i
$700$	0	0
$701$	1146.00	1.63481	0.817404	−	0.576065i	$-0.195413\pi$
$701$	1146.00	1.63481	0.817404	+	0.576065i	$0.195413\pi$
$702$	0	0
$703$	1143.15i	1.62611i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	872.954i	1.23473i
$708$	0	0
$709$	778.000	1.09732	0.548660	−	0.836046i	$-0.315138\pi$
$709$	778.000	1.09732	0.548660	+	0.836046i	$0.315138\pi$
$710$	0	0
$711$	−141.000	−0.198312
$712$	0	0
$713$	20.7846i	0.0291509i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 446.869i	− 0.623248i
$718$	0	0
$719$	1184.72i	1.64774i	0.566781	+	0.823868i	$0.308189\pi$
$719$	1184.72i	1.64774i	−0.566781	+	0.823868i	$0.691811\pi$
$720$	0	0
$721$	339.482i	0.470849i
$722$	0	0
$723$	300.000	0.414938
$724$	0	0
$725$	0	0
$726$	0	0
$727$	796.743i	1.09593i	0.836500	+	0.547967i	$0.184598\pi$
$727$	796.743i	1.09593i	−0.836500	+	0.547967i	$0.815402\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 145.492i	− 0.199032i
$732$	0	0
$733$	− 166.277i	− 0.226844i	−0.993547	−	0.113422i	$-0.963819\pi$
$733$	− 166.277i	− 0.226844i	0.993547	−	0.113422i	$-0.0361813\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−147.000	−0.199457
$738$	0	0
$739$	−97.0000	−0.131258	−0.0656292	−	0.997844i	$-0.520905\pi$
$739$	−97.0000	−0.131258	−0.0656292	+	0.997844i	$0.520905\pi$
$740$	0	0
$741$	498.831i	0.673186i
$742$	0	0
$743$	210.000	0.282638	0.141319	−	0.989964i	$-0.454866\pi$
$743$	210.000	0.282638	0.141319	+	0.989964i	$0.454866\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	311.769i	0.417362i
$748$	0	0
$749$	42.0000	0.0560748
$750$	0	0
$751$	−1166.00	−1.55260	−0.776298	−	0.630366i	$-0.782905\pi$
$751$	−1166.00	−1.55260	−0.776298	+	0.630366i	$0.782905\pi$
$752$	0	0
$753$	36.0000	0.0478088
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1087.00	1.43593	0.717966	−	0.696079i	$-0.245073\pi$
$757$	1087.00	1.43593	0.717966	+	0.696079i	$0.245073\pi$
$758$	0	0
$759$	− 15.5885i	− 0.0205382i
$760$	0	0
$761$	415.692i	0.546245i	0.961979	+	0.273122i	$0.0880564\pi$
$761$	415.692i	0.546245i	−0.961979	+	0.273122i	$0.911944\pi$
$762$	0	0
$763$	623.000	0.816514
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−576.000	−0.750978
$768$	0	0
$769$	− 1198.58i	− 1.55862i	−0.626638	−	0.779310i	$-0.715570\pi$
$769$	− 1198.58i	− 1.55862i	0.626638	−	0.779310i	$-0.284430\pi$
$770$	0	0
$771$	−828.000	−1.07393
$772$	0	0
$773$	145.492i	0.188218i	0.995562	+	0.0941088i	$0.0300002\pi$
$773$	145.492i	0.188218i	−0.995562	+	0.0941088i	$0.970000\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	666.840i	0.858223i
$778$	0	0
$779$	432.000	0.554557
$780$	0	0
$781$	369.000	0.472471
$782$	0	0
$783$	15.5885i	0.0199086i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1080.80i	1.37332i	0.726981	+	0.686658i	$0.240923\pi$
$787$	1080.80i	1.37332i	−0.726981	+	0.686658i	$0.759077\pi$
$788$	0	0
$789$	− 753.442i	− 0.954933i
$790$	0	0
$791$	−693.000	−0.876106
$792$	0	0
$793$	288.000	0.363178
$794$	0	0
$795$	0	0
$796$	0	0
$797$	436.477i	0.547650i	0.961780	+	0.273825i	$0.0882889\pi$
$797$	436.477i	0.547650i	−0.961780	+	0.273825i	$0.911711\pi$
$798$	0	0
$799$	−1296.00	−1.62203
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 83.1384i	− 0.103535i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−252.000	−0.312268
$808$	0	0
$809$	−1245.00	−1.53894	−0.769468	−	0.638685i	$-0.779479\pi$
$809$	−1245.00	−1.53894	−0.769468	+	0.638685i	$0.779479\pi$
$810$	0	0
$811$	13.8564i	0.0170856i	0.999964	+	0.00854279i	$0.00271929\pi$
$811$	13.8564i	0.0170856i	−0.999964	+	0.00854279i	$0.997281\pi$
$812$	0	0
$813$	−600.000	−0.738007
$814$	0	0
$815$	0	0
$816$	0	0
$817$	145.492i	0.178081i
$818$	0	0
$819$	290.985i	0.355292i
$820$	0	0
$821$	474.000	0.577345	0.288672	−	0.957428i	$-0.406786\pi$
$821$	474.000	0.577345	0.288672	+	0.957428i	$0.406786\pi$
$822$	0	0
$823$	−625.000	−0.759417	−0.379708	−	0.925106i	$-0.623976\pi$
$823$	−625.000	−0.759417	−0.379708	+	0.925106i	$0.623976\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1221.00	−1.47642	−0.738210	−	0.674571i	$-0.764329\pi$
$827$	−1221.00	−1.47642	−0.738210	+	0.674571i	$0.764329\pi$
$828$	0	0
$829$	− 1323.29i	− 1.59624i	−0.602495	−	0.798122i	$-0.705827\pi$
$829$	− 1323.29i	− 1.59624i	0.602495	−	0.798122i	$-0.294173\pi$
$830$	0	0
$831$	− 627.002i	− 0.754516i
$832$	0	0
$833$	− 1018.45i	− 1.22262i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−36.0000	−0.0430108
$838$	0	0
$839$	415.692i	0.495461i	0.968829	+	0.247731i	$0.0796848\pi$
$839$	415.692i	0.495461i	−0.968829	+	0.247731i	$0.920315\pi$
$840$	0	0
$841$	−832.000	−0.989298
$842$	0	0
$843$	535.204i	0.634880i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	784.000	0.925620
$848$	0	0
$849$	552.000	0.650177
$850$	0	0
$851$	165.000	0.193890
$852$	0	0
$853$	96.9948i	0.113710i	0.998382	+	0.0568551i	$0.0181073\pi$
$853$	96.9948i	0.113710i	−0.998382	+	0.0568551i	$0.981893\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1018.45i	− 1.18838i	−0.804323	−	0.594192i	$-0.797472\pi$
$857$	− 1018.45i	− 1.18838i	0.804323	−	0.594192i	$-0.202528\pi$
$858$	0	0
$859$	27.7128i	0.0322617i	0.999870	+	0.0161309i	$0.00513483\pi$
$859$	27.7128i	0.0322617i	−0.999870	+	0.0161309i	$0.994865\pi$
$860$	0	0
$861$	252.000	0.292683
$862$	0	0
$863$	867.000	1.00463	0.502317	−	0.864683i	$-0.332481\pi$
$863$	867.000	1.00463	0.502317	+	0.864683i	$0.332481\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	247.683i	0.285679i
$868$	0	0
$869$	141.000	0.162255
$870$	0	0
$871$	− 678.964i	− 0.779522i
$872$	0	0
$873$	436.477i	0.499973i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1414.00	−1.61231	−0.806157	−	0.591701i	$-0.798457\pi$
$877$	−1414.00	−1.61231	−0.806157	+	0.591701i	$0.798457\pi$
$878$	0	0
$879$	−864.000	−0.982935
$880$	0	0
$881$	− 1018.45i	− 1.15601i	−0.816033	−	0.578006i	$-0.803831\pi$
$881$	− 1018.45i	− 1.15601i	0.816033	−	0.578006i	$-0.196169\pi$
$882$	0	0
$883$	−1145.00	−1.29672	−0.648358	−	0.761336i	$-0.724544\pi$
$883$	−1145.00	−1.29672	−0.648358	+	0.761336i	$0.724544\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1351.00i	1.52311i	0.648100	+	0.761556i	$0.275564\pi$
$887$	1351.00i	1.52311i	−0.648100	+	0.761556i	$0.724436\pi$
$888$	0	0
$889$	−833.000	−0.937008
$890$	0	0
$891$	27.0000	0.0303030
$892$	0	0
$893$	1296.00	1.45129
$894$	0	0
$895$	0	0
$896$	0	0
$897$	72.0000	0.0802676
$898$	0	0
$899$	20.7846i	0.0231197i
$900$	0	0
$901$	2120.03i	2.35297i
$902$	0	0
$903$	84.8705i	0.0939873i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	106.000	0.116869	0.0584344	−	0.998291i	$-0.481389\pi$
$907$	106.000	0.116869	0.0584344	+	0.998291i	$0.481389\pi$
$908$	0	0
$909$	374.123i	0.411576i
$910$	0	0
$911$	−741.000	−0.813392	−0.406696	−	0.913564i	$-0.633319\pi$
$911$	−741.000	−0.813392	−0.406696	+	0.913564i	$0.633319\pi$
$912$	0	0
$913$	− 311.769i	− 0.341478i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	145.492i	0.158661i
$918$	0	0
$919$	−953.000	−1.03700	−0.518498	−	0.855079i	$-0.673509\pi$
$919$	−953.000	−1.03700	−0.518498	+	0.855079i	$0.673509\pi$
$920$	0	0
$921$	12.0000	0.0130293
$922$	0	0
$923$	1704.34i	1.84652i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	145.492i	0.156950i
$928$	0	0
$929$	1122.37i	1.20815i	0.796928	+	0.604074i	$0.206457\pi$
$929$	1122.37i	1.20815i	−0.796928	+	0.604074i	$0.793543\pi$
$930$	0	0
$931$	1018.45i	1.09393i
$932$	0	0
$933$	−972.000	−1.04180
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 685.892i	− 0.732009i	−0.930613	−	0.366004i	$-0.880726\pi$
$937$	− 685.892i	− 0.732009i	0.930613	−	0.366004i	$-0.119274\pi$
$938$	0	0
$939$	−948.000	−1.00958
$940$	0	0
$941$	893.738i	0.949775i	0.880047	+	0.474887i	$0.157511\pi$
$941$	893.738i	0.949775i	−0.880047	+	0.474887i	$0.842489\pi$
$942$	0	0
$943$	− 62.3538i	− 0.0661228i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	810.000	0.855333	0.427666	−	0.903937i	$-0.359336\pi$
$947$	810.000	0.855333	0.427666	+	0.903937i	$0.359336\pi$
$948$	0	0
$949$	384.000	0.404636
$950$	0	0
$951$	285.788i	0.300514i
$952$	0	0
$953$	939.000	0.985310	0.492655	−	0.870225i	$-0.336027\pi$
$953$	939.000	0.985310	0.492655	+	0.870225i	$0.336027\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 15.5885i	− 0.0162889i
$958$	0	0
$959$	546.000	0.569343
$960$	0	0
$961$	913.000	0.950052
$962$	0	0
$963$	18.0000	0.0186916
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1246.00	−1.28852	−0.644261	−	0.764806i	$-0.722835\pi$
$967$	−1246.00	−1.28852	−0.644261	+	0.764806i	$0.722835\pi$
$968$	0	0
$969$	− 748.246i	− 0.772184i
$970$	0	0
$971$	436.477i	0.449513i	0.974415	+	0.224756i	$0.0721586\pi$
$971$	436.477i	0.449513i	−0.974415	+	0.224756i	$0.927841\pi$
$972$	0	0
$973$	− 678.964i	− 0.697805i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	627.000	0.641760	0.320880	−	0.947120i	$-0.396021\pi$
$977$	627.000	0.641760	0.320880	+	0.947120i	$0.396021\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	267.000	0.272171
$982$	0	0
$983$	436.477i	0.444025i	0.975044	+	0.222013i	$0.0712626\pi$
$983$	436.477i	0.444025i	−0.975044	+	0.222013i	$0.928737\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	756.000	0.765957
$988$	0	0
$989$	21.0000	0.0212336
$990$	0	0
$991$	583.000	0.588295	0.294147	−	0.955760i	$-0.404964\pi$
$991$	583.000	0.588295	0.294147	+	0.955760i	$0.404964\pi$
$992$	0	0
$993$	112.583i	0.113377i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1309.43i	− 1.31337i	−0.754165	−	0.656685i	$-0.771958\pi$
$997$	− 1309.43i	− 1.31337i	0.754165	−	0.656685i	$-0.228042\pi$
$998$	0	0
$999$	285.788i	0.286074i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.3.j.a.601.1	✓	2
5.2	odd	4		2100.3.p.a.349.1		4
5.3	odd	4		2100.3.p.a.349.4		4
5.4	even	2		2100.3.j.b.601.2	yes	2
7.6	odd	2	inner	2100.3.j.a.601.2	yes	2
35.13	even	4		2100.3.p.a.349.2		4
35.27	even	4		2100.3.p.a.349.3		4
35.34	odd	2		2100.3.j.b.601.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.3.j.a.601.1	✓	2	1.1	even	1	trivial
2100.3.j.a.601.2	yes	2	7.6	odd	2	inner
2100.3.j.b.601.1	yes	2	35.34	odd	2
2100.3.j.b.601.2	yes	2	5.4	even	2
2100.3.p.a.349.1		4	5.2	odd	4
2100.3.p.a.349.2		4	35.13	even	4
2100.3.p.a.349.3		4	35.27	even	4
2100.3.p.a.349.4		4	5.3	odd	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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2100.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -7.00000 q^{7} -3.00000 q^{9} +3.00000 q^{11} +13.8564i q^{13} -20.7846i q^{17} +20.7846i q^{19} +12.1244i q^{21} +3.00000 q^{23} +5.19615i q^{27} +3.00000 q^{29} +6.92820i q^{31} -5.19615i q^{33} +55.0000 q^{37} +24.0000 q^{39} -20.7846i q^{41} +7.00000 q^{43} -62.3538i q^{47} +49.0000 q^{49} -36.0000 q^{51} -102.000 q^{53} +36.0000 q^{57} +41.5692i q^{59} -20.7846i q^{61} +21.0000 q^{63} -49.0000 q^{67} -5.19615i q^{69} +123.000 q^{71} -27.7128i q^{73} -21.0000 q^{77} +47.0000 q^{79} +9.00000 q^{81} -103.923i q^{83} -5.19615i q^{87} -96.9948i q^{91} +12.0000 q^{93} -145.492i q^{97} -9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{7} - 6 q^{9} + 6 q^{11} + 6 q^{23} + 6 q^{29} + 110 q^{37} + 48 q^{39} + 14 q^{43} + 98 q^{49} - 72 q^{51} - 204 q^{53} + 72 q^{57} + 42 q^{63} - 98 q^{67} + 246 q^{71} - 42 q^{77} + 94 q^{79}+ \cdots - 18 q^{99}+O(q^{100}) \) 2 * q - 14 * q^7 - 6 * q^9 + 6 * q^11 + 6 * q^23 + 6 * q^29 + 110 * q^37 + 48 * q^39 + 14 * q^43 + 98 * q^49 - 72 * q^51 - 204 * q^53 + 72 * q^57 + 42 * q^63 - 98 * q^67 + 246 * q^71 - 42 * q^77 + 94 * q^79 + 18 * q^81 + 24 * q^93 - 18 * q^99

Introduction

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