LMFDB - Embedded newform 1500.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1500,2,Mod(1,1500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1500 = 2^{2} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1500.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:	$11.9775603032$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.61803 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.61803 q^{7} +1.00000 q^{9} +1.00000 q^{11} +2.23607 q^{13} -2.61803 q^{17} +2.23607 q^{19} +2.61803 q^{21} -1.76393 q^{23} -1.00000 q^{27} +6.23607 q^{29} -9.09017 q^{31} -1.00000 q^{33} -8.70820 q^{37} -2.23607 q^{39} +1.38197 q^{41} +4.09017 q^{43} -12.7082 q^{47} -0.145898 q^{49} +2.61803 q^{51} +0.0901699 q^{53} -2.23607 q^{57} -9.09017 q^{59} -6.61803 q^{61} -2.61803 q^{63} -1.52786 q^{67} +1.76393 q^{69} +0.854102 q^{71} +3.32624 q^{73} -2.61803 q^{77} -0.291796 q^{79} +1.00000 q^{81} -9.85410 q^{83} -6.23607 q^{87} -3.94427 q^{89} -5.85410 q^{91} +9.09017 q^{93} -12.7984 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 3 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 3 q^{7} + 2 q^{9} + 2 q^{11} - 3 q^{17} + 3 q^{21} - 8 q^{23} - 2 q^{27} + 8 q^{29} - 7 q^{31} - 2 q^{33} - 4 q^{37} + 5 q^{41} - 3 q^{43} - 12 q^{47} - 7 q^{49} + 3 q^{51} - 11 q^{53} - 7 q^{59} - 11 q^{61} - 3 q^{63} - 12 q^{67} + 8 q^{69} - 5 q^{71} - 9 q^{73} - 3 q^{77} - 14 q^{79} + 2 q^{81} - 13 q^{83} - 8 q^{87} + 10 q^{89} - 5 q^{91} + 7 q^{93} - q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 3 * q^7 + 2 * q^9 + 2 * q^11 - 3 * q^17 + 3 * q^21 - 8 * q^23 - 2 * q^27 + 8 * q^29 - 7 * q^31 - 2 * q^33 - 4 * q^37 + 5 * q^41 - 3 * q^43 - 12 * q^47 - 7 * q^49 + 3 * q^51 - 11 * q^53 - 7 * q^59 - 11 * q^61 - 3 * q^63 - 12 * q^67 + 8 * q^69 - 5 * q^71 - 9 * q^73 - 3 * q^77 - 14 * q^79 + 2 * q^81 - 13 * q^83 - 8 * q^87 + 10 * q^89 - 5 * q^91 + 7 * q^93 - q^97 + 2 * 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
$7$	−2.61803	−0.989524	−0.494762	−	0.869029i	$-0.664745\pi$
$7$	−2.61803	−0.989524	−0.494762	+	0.869029i	$0.664745\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	2.23607	0.620174	0.310087	−	0.950708i	$-0.399642\pi$
$13$	2.23607	0.620174	0.310087	+	0.950708i	$0.399642\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.61803	−0.634967	−0.317483	−	0.948264i	$-0.602838\pi$
$17$	−2.61803	−0.634967	−0.317483	+	0.948264i	$0.602838\pi$
$18$	0	0
$19$	2.23607	0.512989	0.256495	−	0.966546i	$-0.417432\pi$
$19$	2.23607	0.512989	0.256495	+	0.966546i	$0.417432\pi$
$20$	0	0
$21$	2.61803	0.571302
$22$	0	0
$23$	−1.76393	−0.367805	−0.183903	−	0.982944i	$-0.558873\pi$
$23$	−1.76393	−0.367805	−0.183903	+	0.982944i	$0.558873\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.23607	1.15801	0.579004	−	0.815324i	$-0.303441\pi$
$29$	6.23607	1.15801	0.579004	+	0.815324i	$0.303441\pi$
$30$	0	0
$31$	−9.09017	−1.63264	−0.816321	−	0.577598i	$-0.803990\pi$
$31$	−9.09017	−1.63264	−0.816321	+	0.577598i	$0.803990\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.70820	−1.43162	−0.715810	−	0.698295i	$-0.753942\pi$
$37$	−8.70820	−1.43162	−0.715810	+	0.698295i	$0.753942\pi$
$38$	0	0
$39$	−2.23607	−0.358057
$40$	0	0
$41$	1.38197	0.215827	0.107913	−	0.994160i	$-0.465583\pi$
$41$	1.38197	0.215827	0.107913	+	0.994160i	$0.465583\pi$
$42$	0	0
$43$	4.09017	0.623745	0.311873	−	0.950124i	$-0.399044\pi$
$43$	4.09017	0.623745	0.311873	+	0.950124i	$0.399044\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.7082	−1.85368	−0.926841	−	0.375454i	$-0.877487\pi$
$47$	−12.7082	−1.85368	−0.926841	+	0.375454i	$0.877487\pi$
$48$	0	0
$49$	−0.145898	−0.0208426
$50$	0	0
$51$	2.61803	0.366598
$52$	0	0
$53$	0.0901699	0.0123858	0.00619290	−	0.999981i	$-0.498029\pi$
$53$	0.0901699	0.0123858	0.00619290	+	0.999981i	$0.498029\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.23607	−0.296174
$58$	0	0
$59$	−9.09017	−1.18344	−0.591720	−	0.806144i	$-0.701551\pi$
$59$	−9.09017	−1.18344	−0.591720	+	0.806144i	$0.701551\pi$
$60$	0	0
$61$	−6.61803	−0.847352	−0.423676	−	0.905814i	$-0.639261\pi$
$61$	−6.61803	−0.847352	−0.423676	+	0.905814i	$0.639261\pi$
$62$	0	0
$63$	−2.61803	−0.329841
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.52786	−0.186658	−0.0933292	−	0.995635i	$-0.529751\pi$
$67$	−1.52786	−0.186658	−0.0933292	+	0.995635i	$0.529751\pi$
$68$	0	0
$69$	1.76393	0.212352
$70$	0	0
$71$	0.854102	0.101363	0.0506816	−	0.998715i	$-0.483861\pi$
$71$	0.854102	0.101363	0.0506816	+	0.998715i	$0.483861\pi$
$72$	0	0
$73$	3.32624	0.389307	0.194653	−	0.980872i	$-0.437642\pi$
$73$	3.32624	0.389307	0.194653	+	0.980872i	$0.437642\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.61803	−0.298353
$78$	0	0
$79$	−0.291796	−0.0328296	−0.0164148	−	0.999865i	$-0.505225\pi$
$79$	−0.291796	−0.0328296	−0.0164148	+	0.999865i	$0.505225\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.85410	−1.08163	−0.540814	−	0.841142i	$-0.681884\pi$
$83$	−9.85410	−1.08163	−0.540814	+	0.841142i	$0.681884\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.23607	−0.668577
$88$	0	0
$89$	−3.94427	−0.418092	−0.209046	−	0.977906i	$-0.567036\pi$
$89$	−3.94427	−0.418092	−0.209046	+	0.977906i	$0.567036\pi$
$90$	0	0
$91$	−5.85410	−0.613677
$92$	0	0
$93$	9.09017	0.942607
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.7984	−1.29948	−0.649739	−	0.760157i	$-0.725122\pi$
$97$	−12.7984	−1.29948	−0.649739	+	0.760157i	$0.725122\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	16.9443	1.68602	0.843009	−	0.537899i	$-0.180782\pi$
$101$	16.9443	1.68602	0.843009	+	0.537899i	$0.180782\pi$
$102$	0	0
$103$	−16.8541	−1.66068	−0.830342	−	0.557254i	$-0.811855\pi$
$103$	−16.8541	−1.66068	−0.830342	+	0.557254i	$0.811855\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.8541	−1.24265	−0.621326	−	0.783552i	$-0.713406\pi$
$107$	−12.8541	−1.24265	−0.621326	+	0.783552i	$0.713406\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	8.70820	0.826546
$112$	0	0
$113$	19.7082	1.85399	0.926996	−	0.375071i	$-0.122382\pi$
$113$	19.7082	1.85399	0.926996	+	0.375071i	$0.122382\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.23607	0.206725
$118$	0	0
$119$	6.85410	0.628314
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−1.38197	−0.124608
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.94427	0.882411	0.441206	−	0.897406i	$-0.354551\pi$
$127$	9.94427	0.882411	0.441206	+	0.897406i	$0.354551\pi$
$128$	0	0
$129$	−4.09017	−0.360119
$130$	0	0
$131$	9.56231	0.835463	0.417731	−	0.908571i	$-0.362825\pi$
$131$	9.56231	0.835463	0.417731	+	0.908571i	$0.362825\pi$
$132$	0	0
$133$	−5.85410	−0.507615
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.52786	0.728585	0.364292	−	0.931285i	$-0.381311\pi$
$137$	8.52786	0.728585	0.364292	+	0.931285i	$0.381311\pi$
$138$	0	0
$139$	−4.52786	−0.384048	−0.192024	−	0.981390i	$-0.561505\pi$
$139$	−4.52786	−0.384048	−0.192024	+	0.981390i	$0.561505\pi$
$140$	0	0
$141$	12.7082	1.07022
$142$	0	0
$143$	2.23607	0.186989
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.145898	0.0120335
$148$	0	0
$149$	−1.47214	−0.120602	−0.0603010	−	0.998180i	$-0.519206\pi$
$149$	−1.47214	−0.120602	−0.0603010	+	0.998180i	$0.519206\pi$
$150$	0	0
$151$	−20.6525	−1.68067	−0.840337	−	0.542064i	$-0.817643\pi$
$151$	−20.6525	−1.68067	−0.840337	+	0.542064i	$0.817643\pi$
$152$	0	0
$153$	−2.61803	−0.211656
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.0902	−0.885092	−0.442546	−	0.896746i	$-0.645925\pi$
$157$	−11.0902	−0.885092	−0.442546	+	0.896746i	$0.645925\pi$
$158$	0	0
$159$	−0.0901699	−0.00715094
$160$	0	0
$161$	4.61803	0.363952
$162$	0	0
$163$	−8.41641	−0.659224	−0.329612	−	0.944116i	$-0.606918\pi$
$163$	−8.41641	−0.659224	−0.329612	+	0.944116i	$0.606918\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.8885	1.22949	0.614746	−	0.788725i	$-0.289258\pi$
$167$	15.8885	1.22949	0.614746	+	0.788725i	$0.289258\pi$
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	2.23607	0.170996
$172$	0	0
$173$	8.79837	0.668928	0.334464	−	0.942409i	$-0.391445\pi$
$173$	8.79837	0.668928	0.334464	+	0.942409i	$0.391445\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.09017	0.683259
$178$	0	0
$179$	−5.47214	−0.409007	−0.204503	−	0.978866i	$-0.565558\pi$
$179$	−5.47214	−0.409007	−0.204503	+	0.978866i	$0.565558\pi$
$180$	0	0
$181$	−18.9443	−1.40812	−0.704058	−	0.710142i	$-0.748631\pi$
$181$	−18.9443	−1.40812	−0.704058	+	0.710142i	$0.748631\pi$
$182$	0	0
$183$	6.61803	0.489219
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.61803	−0.191450
$188$	0	0
$189$	2.61803	0.190434
$190$	0	0
$191$	−11.6525	−0.843144	−0.421572	−	0.906795i	$-0.638521\pi$
$191$	−11.6525	−0.843144	−0.421572	+	0.906795i	$0.638521\pi$
$192$	0	0
$193$	23.4721	1.68956	0.844781	−	0.535113i	$-0.179731\pi$
$193$	23.4721	1.68956	0.844781	+	0.535113i	$0.179731\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.3607	−1.66438	−0.832190	−	0.554491i	$-0.812913\pi$
$197$	−23.3607	−1.66438	−0.832190	+	0.554491i	$0.812913\pi$
$198$	0	0
$199$	25.3607	1.79777	0.898885	−	0.438184i	$-0.144378\pi$
$199$	25.3607	1.79777	0.898885	+	0.438184i	$0.144378\pi$
$200$	0	0
$201$	1.52786	0.107767
$202$	0	0
$203$	−16.3262	−1.14588
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.76393	−0.122602
$208$	0	0
$209$	2.23607	0.154672
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	−0.854102	−0.0585221
$214$	0	0
$215$	0	0
$216$	0	0
$217$	23.7984	1.61554
$218$	0	0
$219$	−3.32624	−0.224766
$220$	0	0
$221$	−5.85410	−0.393790
$222$	0	0
$223$	−9.47214	−0.634301	−0.317151	−	0.948375i	$-0.602726\pi$
$223$	−9.47214	−0.634301	−0.317151	+	0.948375i	$0.602726\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.1246	1.66758	0.833790	−	0.552082i	$-0.186167\pi$
$227$	25.1246	1.66758	0.833790	+	0.552082i	$0.186167\pi$
$228$	0	0
$229$	−6.76393	−0.446973	−0.223487	−	0.974707i	$-0.571744\pi$
$229$	−6.76393	−0.446973	−0.223487	+	0.974707i	$0.571744\pi$
$230$	0	0
$231$	2.61803	0.172254
$232$	0	0
$233$	−11.5066	−0.753821	−0.376910	−	0.926250i	$-0.623014\pi$
$233$	−11.5066	−0.753821	−0.376910	+	0.926250i	$0.623014\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.291796	0.0189542
$238$	0	0
$239$	12.1459	0.785653	0.392826	−	0.919613i	$-0.371497\pi$
$239$	12.1459	0.785653	0.392826	+	0.919613i	$0.371497\pi$
$240$	0	0
$241$	−4.76393	−0.306872	−0.153436	−	0.988159i	$-0.549034\pi$
$241$	−4.76393	−0.306872	−0.153436	+	0.988159i	$0.549034\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.00000	0.318142
$248$	0	0
$249$	9.85410	0.624478
$250$	0	0
$251$	12.2705	0.774508	0.387254	−	0.921973i	$-0.373424\pi$
$251$	12.2705	0.774508	0.387254	+	0.921973i	$0.373424\pi$
$252$	0	0
$253$	−1.76393	−0.110897
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.4164	1.64781	0.823905	−	0.566728i	$-0.191791\pi$
$257$	26.4164	1.64781	0.823905	+	0.566728i	$0.191791\pi$
$258$	0	0
$259$	22.7984	1.41662
$260$	0	0
$261$	6.23607	0.386003
$262$	0	0
$263$	−3.27051	−0.201668	−0.100834	−	0.994903i	$-0.532151\pi$
$263$	−3.27051	−0.201668	−0.100834	+	0.994903i	$0.532151\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.94427	0.241386
$268$	0	0
$269$	−5.38197	−0.328144	−0.164072	−	0.986448i	$-0.552463\pi$
$269$	−5.38197	−0.328144	−0.164072	+	0.986448i	$0.552463\pi$
$270$	0	0
$271$	−17.5066	−1.06345	−0.531724	−	0.846917i	$-0.678456\pi$
$271$	−17.5066	−1.06345	−0.531724	+	0.846917i	$0.678456\pi$
$272$	0	0
$273$	5.85410	0.354306
$274$	0	0
$275$	0	0
$276$	0	0
$277$	21.7984	1.30974	0.654869	−	0.755743i	$-0.272724\pi$
$277$	21.7984	1.30974	0.654869	+	0.755743i	$0.272724\pi$
$278$	0	0
$279$	−9.09017	−0.544214
$280$	0	0
$281$	19.6180	1.17031	0.585157	−	0.810920i	$-0.301033\pi$
$281$	19.6180	1.17031	0.585157	+	0.810920i	$0.301033\pi$
$282$	0	0
$283$	−10.4164	−0.619191	−0.309596	−	0.950868i	$-0.600194\pi$
$283$	−10.4164	−0.619191	−0.309596	+	0.950868i	$0.600194\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.61803	−0.213566
$288$	0	0
$289$	−10.1459	−0.596818
$290$	0	0
$291$	12.7984	0.750254
$292$	0	0
$293$	22.4721	1.31284	0.656418	−	0.754397i	$-0.272071\pi$
$293$	22.4721	1.31284	0.656418	+	0.754397i	$0.272071\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−3.94427	−0.228103
$300$	0	0
$301$	−10.7082	−0.617211
$302$	0	0
$303$	−16.9443	−0.973423
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.236068	0.0134731	0.00673656	−	0.999977i	$-0.497856\pi$
$307$	0.236068	0.0134731	0.00673656	+	0.999977i	$0.497856\pi$
$308$	0	0
$309$	16.8541	0.958796
$310$	0	0
$311$	22.0902	1.25262	0.626309	−	0.779575i	$-0.284565\pi$
$311$	22.0902	1.25262	0.626309	+	0.779575i	$0.284565\pi$
$312$	0	0
$313$	32.6180	1.84368	0.921840	−	0.387570i	$-0.126686\pi$
$313$	32.6180	1.84368	0.921840	+	0.387570i	$0.126686\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.1803	0.964944	0.482472	−	0.875911i	$-0.339739\pi$
$317$	17.1803	0.964944	0.482472	+	0.875911i	$0.339739\pi$
$318$	0	0
$319$	6.23607	0.349153
$320$	0	0
$321$	12.8541	0.717446
$322$	0	0
$323$	−5.85410	−0.325731
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.00000	0.387101
$328$	0	0
$329$	33.2705	1.83426
$330$	0	0
$331$	1.14590	0.0629843	0.0314921	−	0.999504i	$-0.489974\pi$
$331$	1.14590	0.0629843	0.0314921	+	0.999504i	$0.489974\pi$
$332$	0	0
$333$	−8.70820	−0.477207
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.0000	1.52526	0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000	1.52526	0.762629	+	0.646837i	$0.223908\pi$
$338$	0	0
$339$	−19.7082	−1.07040
$340$	0	0
$341$	−9.09017	−0.492260
$342$	0	0
$343$	18.7082	1.01015
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.14590	0.383612	0.191806	−	0.981433i	$-0.438566\pi$
$347$	7.14590	0.383612	0.191806	+	0.981433i	$0.438566\pi$
$348$	0	0
$349$	−22.0902	−1.18246	−0.591230	−	0.806503i	$-0.701357\pi$
$349$	−22.0902	−1.18246	−0.591230	+	0.806503i	$0.701357\pi$
$350$	0	0
$351$	−2.23607	−0.119352
$352$	0	0
$353$	−21.2705	−1.13212	−0.566058	−	0.824366i	$-0.691532\pi$
$353$	−21.2705	−1.13212	−0.566058	+	0.824366i	$0.691532\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.85410	−0.362758
$358$	0	0
$359$	−29.4164	−1.55254	−0.776269	−	0.630401i	$-0.782890\pi$
$359$	−29.4164	−1.55254	−0.776269	+	0.630401i	$0.782890\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	23.6869	1.23645	0.618224	−	0.786002i	$-0.287853\pi$
$367$	23.6869	1.23645	0.618224	+	0.786002i	$0.287853\pi$
$368$	0	0
$369$	1.38197	0.0719423
$370$	0	0
$371$	−0.236068	−0.0122560
$372$	0	0
$373$	13.0344	0.674898	0.337449	−	0.941344i	$-0.390436\pi$
$373$	13.0344	0.674898	0.337449	+	0.941344i	$0.390436\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.9443	0.718167
$378$	0	0
$379$	−16.7984	−0.862875	−0.431437	−	0.902143i	$-0.641993\pi$
$379$	−16.7984	−0.862875	−0.431437	+	0.902143i	$0.641993\pi$
$380$	0	0
$381$	−9.94427	−0.509460
$382$	0	0
$383$	−21.2705	−1.08687	−0.543436	−	0.839451i	$-0.682877\pi$
$383$	−21.2705	−1.08687	−0.543436	+	0.839451i	$0.682877\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.09017	0.207915
$388$	0	0
$389$	26.8541	1.36156	0.680779	−	0.732489i	$-0.261642\pi$
$389$	26.8541	1.36156	0.680779	+	0.732489i	$0.261642\pi$
$390$	0	0
$391$	4.61803	0.233544
$392$	0	0
$393$	−9.56231	−0.482355
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−24.3607	−1.22263	−0.611314	−	0.791388i	$-0.709359\pi$
$397$	−24.3607	−1.22263	−0.611314	+	0.791388i	$0.709359\pi$
$398$	0	0
$399$	5.85410	0.293072
$400$	0	0
$401$	−2.81966	−0.140807	−0.0704036	−	0.997519i	$-0.522429\pi$
$401$	−2.81966	−0.140807	−0.0704036	+	0.997519i	$0.522429\pi$
$402$	0	0
$403$	−20.3262	−1.01252
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.70820	−0.431650
$408$	0	0
$409$	7.70820	0.381146	0.190573	−	0.981673i	$-0.438965\pi$
$409$	7.70820	0.381146	0.190573	+	0.981673i	$0.438965\pi$
$410$	0	0
$411$	−8.52786	−0.420649
$412$	0	0
$413$	23.7984	1.17104
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.52786	0.221730
$418$	0	0
$419$	−21.6180	−1.05611	−0.528055	−	0.849210i	$-0.677079\pi$
$419$	−21.6180	−1.05611	−0.528055	+	0.849210i	$0.677079\pi$
$420$	0	0
$421$	13.4164	0.653876	0.326938	−	0.945046i	$-0.393983\pi$
$421$	13.4164	0.653876	0.326938	+	0.945046i	$0.393983\pi$
$422$	0	0
$423$	−12.7082	−0.617894
$424$	0	0
$425$	0	0
$426$	0	0
$427$	17.3262	0.838475
$428$	0	0
$429$	−2.23607	−0.107958
$430$	0	0
$431$	24.3050	1.17073	0.585364	−	0.810770i	$-0.300952\pi$
$431$	24.3050	1.17073	0.585364	+	0.810770i	$0.300952\pi$
$432$	0	0
$433$	−3.88854	−0.186871	−0.0934357	−	0.995625i	$-0.529785\pi$
$433$	−3.88854	−0.186871	−0.0934357	+	0.995625i	$0.529785\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.94427	−0.188680
$438$	0	0
$439$	12.2705	0.585639	0.292820	−	0.956168i	$-0.405406\pi$
$439$	12.2705	0.585639	0.292820	+	0.956168i	$0.405406\pi$
$440$	0	0
$441$	−0.145898	−0.00694753
$442$	0	0
$443$	26.0344	1.23693	0.618467	−	0.785811i	$-0.287754\pi$
$443$	26.0344	1.23693	0.618467	+	0.785811i	$0.287754\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.47214	0.0696296
$448$	0	0
$449$	−35.2148	−1.66189	−0.830944	−	0.556356i	$-0.812199\pi$
$449$	−35.2148	−1.66189	−0.830944	+	0.556356i	$0.812199\pi$
$450$	0	0
$451$	1.38197	0.0650742
$452$	0	0
$453$	20.6525	0.970338
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.7984	0.692239	0.346119	−	0.938190i	$-0.387499\pi$
$457$	14.7984	0.692239	0.346119	+	0.938190i	$0.387499\pi$
$458$	0	0
$459$	2.61803	0.122199
$460$	0	0
$461$	−34.7771	−1.61973	−0.809865	−	0.586616i	$-0.800460\pi$
$461$	−34.7771	−1.61973	−0.809865	+	0.586616i	$0.800460\pi$
$462$	0	0
$463$	−6.79837	−0.315947	−0.157974	−	0.987443i	$-0.550496\pi$
$463$	−6.79837	−0.315947	−0.157974	+	0.987443i	$0.550496\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.9787	0.646858	0.323429	−	0.946252i	$-0.395164\pi$
$467$	13.9787	0.646858	0.323429	+	0.946252i	$0.395164\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	11.0902	0.511008
$472$	0	0
$473$	4.09017	0.188066
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.0901699	0.00412860
$478$	0	0
$479$	21.2361	0.970301	0.485150	−	0.874431i	$-0.338765\pi$
$479$	21.2361	0.970301	0.485150	+	0.874431i	$0.338765\pi$
$480$	0	0
$481$	−19.4721	−0.887853
$482$	0	0
$483$	−4.61803	−0.210128
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−21.4164	−0.970470	−0.485235	−	0.874384i	$-0.661266\pi$
$487$	−21.4164	−0.970470	−0.485235	+	0.874384i	$0.661266\pi$
$488$	0	0
$489$	8.41641	0.380603
$490$	0	0
$491$	−25.7984	−1.16426	−0.582132	−	0.813094i	$-0.697781\pi$
$491$	−25.7984	−1.16426	−0.582132	+	0.813094i	$0.697781\pi$
$492$	0	0
$493$	−16.3262	−0.735297
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.23607	−0.100301
$498$	0	0
$499$	0.0557281	0.00249473	0.00124737	−	0.999999i	$-0.499603\pi$
$499$	0.0557281	0.00249473	0.00124737	+	0.999999i	$0.499603\pi$
$500$	0	0
$501$	−15.8885	−0.709848
$502$	0	0
$503$	−7.90983	−0.352682	−0.176341	−	0.984329i	$-0.556426\pi$
$503$	−7.90983	−0.352682	−0.176341	+	0.984329i	$0.556426\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.00000	0.355292
$508$	0	0
$509$	−30.7639	−1.36359	−0.681794	−	0.731545i	$-0.738800\pi$
$509$	−30.7639	−1.36359	−0.681794	+	0.731545i	$0.738800\pi$
$510$	0	0
$511$	−8.70820	−0.385228
$512$	0	0
$513$	−2.23607	−0.0987248
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.7082	−0.558906
$518$	0	0
$519$	−8.79837	−0.386206
$520$	0	0
$521$	0.0901699	0.00395042	0.00197521	−	0.999998i	$-0.499371\pi$
$521$	0.0901699	0.00395042	0.00197521	+	0.999998i	$0.499371\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23.7984	1.03667
$528$	0	0
$529$	−19.8885	−0.864719
$530$	0	0
$531$	−9.09017	−0.394480
$532$	0	0
$533$	3.09017	0.133850
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.47214	0.236140
$538$	0	0
$539$	−0.145898	−0.00628427
$540$	0	0
$541$	20.6869	0.889400	0.444700	−	0.895680i	$-0.353310\pi$
$541$	20.6869	0.889400	0.444700	+	0.895680i	$0.353310\pi$
$542$	0	0
$543$	18.9443	0.812977
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−27.2705	−1.16600	−0.583001	−	0.812471i	$-0.698122\pi$
$547$	−27.2705	−1.16600	−0.583001	+	0.812471i	$0.698122\pi$
$548$	0	0
$549$	−6.61803	−0.282451
$550$	0	0
$551$	13.9443	0.594046
$552$	0	0
$553$	0.763932	0.0324857
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.0344	−1.06074	−0.530372	−	0.847765i	$-0.677948\pi$
$557$	−25.0344	−1.06074	−0.530372	+	0.847765i	$0.677948\pi$
$558$	0	0
$559$	9.14590	0.386830
$560$	0	0
$561$	2.61803	0.110533
$562$	0	0
$563$	−33.2148	−1.39984	−0.699918	−	0.714223i	$-0.746780\pi$
$563$	−33.2148	−1.39984	−0.699918	+	0.714223i	$0.746780\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.61803	−0.109947
$568$	0	0
$569$	44.0132	1.84513	0.922564	−	0.385845i	$-0.126090\pi$
$569$	44.0132	1.84513	0.922564	+	0.385845i	$0.126090\pi$
$570$	0	0
$571$	23.6180	0.988384	0.494192	−	0.869353i	$-0.335464\pi$
$571$	23.6180	0.988384	0.494192	+	0.869353i	$0.335464\pi$
$572$	0	0
$573$	11.6525	0.486789
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.65248	0.318577	0.159288	−	0.987232i	$-0.449080\pi$
$577$	7.65248	0.318577	0.159288	+	0.987232i	$0.449080\pi$
$578$	0	0
$579$	−23.4721	−0.975469
$580$	0	0
$581$	25.7984	1.07030
$582$	0	0
$583$	0.0901699	0.00373446
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.8885	−1.10981	−0.554904	−	0.831914i	$-0.687245\pi$
$587$	−26.8885	−1.10981	−0.554904	+	0.831914i	$0.687245\pi$
$588$	0	0
$589$	−20.3262	−0.837528
$590$	0	0
$591$	23.3607	0.960930
$592$	0	0
$593$	3.76393	0.154566	0.0772831	−	0.997009i	$-0.475375\pi$
$593$	3.76393	0.154566	0.0772831	+	0.997009i	$0.475375\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−25.3607	−1.03794
$598$	0	0
$599$	−36.6869	−1.49899	−0.749493	−	0.662012i	$-0.769703\pi$
$599$	−36.6869	−1.49899	−0.749493	+	0.662012i	$0.769703\pi$
$600$	0	0
$601$	−11.4721	−0.467958	−0.233979	−	0.972242i	$-0.575175\pi$
$601$	−11.4721	−0.467958	−0.233979	+	0.972242i	$0.575175\pi$
$602$	0	0
$603$	−1.52786	−0.0622194
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.9443	−0.850102	−0.425051	−	0.905169i	$-0.639744\pi$
$607$	−20.9443	−0.850102	−0.425051	+	0.905169i	$0.639744\pi$
$608$	0	0
$609$	16.3262	0.661573
$610$	0	0
$611$	−28.4164	−1.14960
$612$	0	0
$613$	5.36068	0.216516	0.108258	−	0.994123i	$-0.465473\pi$
$613$	5.36068	0.216516	0.108258	+	0.994123i	$0.465473\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.0557281	0.00224353	0.00112176	−	0.999999i	$-0.499643\pi$
$617$	0.0557281	0.00224353	0.00112176	+	0.999999i	$0.499643\pi$
$618$	0	0
$619$	6.11146	0.245640	0.122820	−	0.992429i	$-0.460806\pi$
$619$	6.11146	0.245640	0.122820	+	0.992429i	$0.460806\pi$
$620$	0	0
$621$	1.76393	0.0707842
$622$	0	0
$623$	10.3262	0.413712
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.23607	−0.0893000
$628$	0	0
$629$	22.7984	0.909031
$630$	0	0
$631$	27.0000	1.07485	0.537427	−	0.843311i	$-0.319397\pi$
$631$	27.0000	1.07485	0.537427	+	0.843311i	$0.319397\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.326238	−0.0129260
$638$	0	0
$639$	0.854102	0.0337878
$640$	0	0
$641$	−10.0689	−0.397697	−0.198848	−	0.980030i	$-0.563720\pi$
$641$	−10.0689	−0.397697	−0.198848	+	0.980030i	$0.563720\pi$
$642$	0	0
$643$	11.8885	0.468838	0.234419	−	0.972136i	$-0.424681\pi$
$643$	11.8885	0.468838	0.234419	+	0.972136i	$0.424681\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.61803	0.260182	0.130091	−	0.991502i	$-0.458473\pi$
$647$	6.61803	0.260182	0.130091	+	0.991502i	$0.458473\pi$
$648$	0	0
$649$	−9.09017	−0.356820
$650$	0	0
$651$	−23.7984	−0.932732
$652$	0	0
$653$	−7.81966	−0.306007	−0.153003	−	0.988226i	$-0.548895\pi$
$653$	−7.81966	−0.306007	−0.153003	+	0.988226i	$0.548895\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.32624	0.129769
$658$	0	0
$659$	25.3607	0.987912	0.493956	−	0.869487i	$-0.335550\pi$
$659$	25.3607	0.987912	0.493956	+	0.869487i	$0.335550\pi$
$660$	0	0
$661$	34.5066	1.34215	0.671075	−	0.741389i	$-0.265833\pi$
$661$	34.5066	1.34215	0.671075	+	0.741389i	$0.265833\pi$
$662$	0	0
$663$	5.85410	0.227354
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.0000	−0.425922
$668$	0	0
$669$	9.47214	0.366214
$670$	0	0
$671$	−6.61803	−0.255486
$672$	0	0
$673$	−5.52786	−0.213083	−0.106542	−	0.994308i	$-0.533978\pi$
$673$	−5.52786	−0.213083	−0.106542	+	0.994308i	$0.533978\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−35.7082	−1.37238	−0.686189	−	0.727423i	$-0.740718\pi$
$677$	−35.7082	−1.37238	−0.686189	+	0.727423i	$0.740718\pi$
$678$	0	0
$679$	33.5066	1.28586
$680$	0	0
$681$	−25.1246	−0.962777
$682$	0	0
$683$	35.8328	1.37110	0.685552	−	0.728023i	$-0.259561\pi$
$683$	35.8328	1.37110	0.685552	+	0.728023i	$0.259561\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.76393	0.258060
$688$	0	0
$689$	0.201626	0.00768134
$690$	0	0
$691$	36.4164	1.38535	0.692673	−	0.721252i	$-0.256433\pi$
$691$	36.4164	1.38535	0.692673	+	0.721252i	$0.256433\pi$
$692$	0	0
$693$	−2.61803	−0.0994509
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3.61803	−0.137043
$698$	0	0
$699$	11.5066	0.435219
$700$	0	0
$701$	24.5410	0.926902	0.463451	−	0.886123i	$-0.346611\pi$
$701$	24.5410	0.926902	0.463451	+	0.886123i	$0.346611\pi$
$702$	0	0
$703$	−19.4721	−0.734406
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−44.3607	−1.66836
$708$	0	0
$709$	7.29180	0.273849	0.136925	−	0.990581i	$-0.456278\pi$
$709$	7.29180	0.273849	0.136925	+	0.990581i	$0.456278\pi$
$710$	0	0
$711$	−0.291796	−0.0109432
$712$	0	0
$713$	16.0344	0.600495
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12.1459	−0.453597
$718$	0	0
$719$	−20.8197	−0.776442	−0.388221	−	0.921566i	$-0.626910\pi$
$719$	−20.8197	−0.776442	−0.388221	+	0.921566i	$0.626910\pi$
$720$	0	0
$721$	44.1246	1.64329
$722$	0	0
$723$	4.76393	0.177173
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.8328	−0.735558	−0.367779	−	0.929913i	$-0.619882\pi$
$727$	−19.8328	−0.735558	−0.367779	+	0.929913i	$0.619882\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.7082	−0.396057
$732$	0	0
$733$	−52.9443	−1.95554	−0.977771	−	0.209677i	$-0.932759\pi$
$733$	−52.9443	−1.95554	−0.977771	+	0.209677i	$0.932759\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.52786	−0.0562796
$738$	0	0
$739$	18.6180	0.684875	0.342438	−	0.939541i	$-0.388747\pi$
$739$	18.6180	0.684875	0.342438	+	0.939541i	$0.388747\pi$
$740$	0	0
$741$	−5.00000	−0.183680
$742$	0	0
$743$	18.0689	0.662883	0.331442	−	0.943476i	$-0.392465\pi$
$743$	18.0689	0.662883	0.331442	+	0.943476i	$0.392465\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.85410	−0.360543
$748$	0	0
$749$	33.6525	1.22963
$750$	0	0
$751$	−11.7426	−0.428495	−0.214248	−	0.976779i	$-0.568730\pi$
$751$	−11.7426	−0.428495	−0.214248	+	0.976779i	$0.568730\pi$
$752$	0	0
$753$	−12.2705	−0.447162
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−19.9443	−0.724887	−0.362443	−	0.932006i	$-0.618057\pi$
$757$	−19.9443	−0.724887	−0.362443	+	0.932006i	$0.618057\pi$
$758$	0	0
$759$	1.76393	0.0640267
$760$	0	0
$761$	−37.0689	−1.34375	−0.671873	−	0.740666i	$-0.734510\pi$
$761$	−37.0689	−1.34375	−0.671873	+	0.740666i	$0.734510\pi$
$762$	0	0
$763$	18.3262	0.663454
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20.3262	−0.733938
$768$	0	0
$769$	−29.3050	−1.05676	−0.528382	−	0.849007i	$-0.677201\pi$
$769$	−29.3050	−1.05676	−0.528382	+	0.849007i	$0.677201\pi$
$770$	0	0
$771$	−26.4164	−0.951364
$772$	0	0
$773$	−21.2918	−0.765813	−0.382906	−	0.923787i	$-0.625077\pi$
$773$	−21.2918	−0.765813	−0.382906	+	0.923787i	$0.625077\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−22.7984	−0.817887
$778$	0	0
$779$	3.09017	0.110717
$780$	0	0
$781$	0.854102	0.0305622
$782$	0	0
$783$	−6.23607	−0.222859
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−27.1459	−0.967647	−0.483823	−	0.875166i	$-0.660752\pi$
$787$	−27.1459	−0.967647	−0.483823	+	0.875166i	$0.660752\pi$
$788$	0	0
$789$	3.27051	0.116433
$790$	0	0
$791$	−51.5967	−1.83457
$792$	0	0
$793$	−14.7984	−0.525506
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36.7426	−1.30149	−0.650746	−	0.759296i	$-0.725544\pi$
$797$	−36.7426	−1.30149	−0.650746	+	0.759296i	$0.725544\pi$
$798$	0	0
$799$	33.2705	1.17703
$800$	0	0
$801$	−3.94427	−0.139364
$802$	0	0
$803$	3.32624	0.117380
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.38197	0.189454
$808$	0	0
$809$	−48.7984	−1.71566	−0.857830	−	0.513934i	$-0.828188\pi$
$809$	−48.7984	−1.71566	−0.857830	+	0.513934i	$0.828188\pi$
$810$	0	0
$811$	5.32624	0.187030	0.0935148	−	0.995618i	$-0.470190\pi$
$811$	5.32624	0.187030	0.0935148	+	0.995618i	$0.470190\pi$
$812$	0	0
$813$	17.5066	0.613982
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.14590	0.319974
$818$	0	0
$819$	−5.85410	−0.204559
$820$	0	0
$821$	−27.5410	−0.961188	−0.480594	−	0.876943i	$-0.659579\pi$
$821$	−27.5410	−0.961188	−0.480594	+	0.876943i	$0.659579\pi$
$822$	0	0
$823$	35.8885	1.25100	0.625498	−	0.780226i	$-0.284896\pi$
$823$	35.8885	1.25100	0.625498	+	0.780226i	$0.284896\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.7295	0.442648	0.221324	−	0.975200i	$-0.428962\pi$
$827$	12.7295	0.442648	0.221324	+	0.975200i	$0.428962\pi$
$828$	0	0
$829$	−41.0132	−1.42445	−0.712223	−	0.701953i	$-0.752311\pi$
$829$	−41.0132	−1.42445	−0.712223	+	0.701953i	$0.752311\pi$
$830$	0	0
$831$	−21.7984	−0.756177
$832$	0	0
$833$	0.381966	0.0132343
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.09017	0.314202
$838$	0	0
$839$	44.1246	1.52335	0.761675	−	0.647959i	$-0.224377\pi$
$839$	44.1246	1.52335	0.761675	+	0.647959i	$0.224377\pi$
$840$	0	0
$841$	9.88854	0.340984
$842$	0	0
$843$	−19.6180	−0.675681
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.1803	0.899567
$848$	0	0
$849$	10.4164	0.357490
$850$	0	0
$851$	15.3607	0.526557
$852$	0	0
$853$	23.8328	0.816020	0.408010	−	0.912977i	$-0.366223\pi$
$853$	23.8328	0.816020	0.408010	+	0.912977i	$0.366223\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.4508	0.766906	0.383453	−	0.923560i	$-0.374735\pi$
$857$	22.4508	0.766906	0.383453	+	0.923560i	$0.374735\pi$
$858$	0	0
$859$	48.4164	1.65195	0.825973	−	0.563709i	$-0.190626\pi$
$859$	48.4164	1.65195	0.825973	+	0.563709i	$0.190626\pi$
$860$	0	0
$861$	3.61803	0.123302
$862$	0	0
$863$	−51.2148	−1.74337	−0.871686	−	0.490065i	$-0.836973\pi$
$863$	−51.2148	−1.74337	−0.871686	+	0.490065i	$0.836973\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10.1459	0.344573
$868$	0	0
$869$	−0.291796	−0.00989850
$870$	0	0
$871$	−3.41641	−0.115761
$872$	0	0
$873$	−12.7984	−0.433159
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−48.2361	−1.62882	−0.814408	−	0.580292i	$-0.802938\pi$
$877$	−48.2361	−1.62882	−0.814408	+	0.580292i	$0.802938\pi$
$878$	0	0
$879$	−22.4721	−0.757966
$880$	0	0
$881$	25.1246	0.846470	0.423235	−	0.906020i	$-0.360895\pi$
$881$	25.1246	0.846470	0.423235	+	0.906020i	$0.360895\pi$
$882$	0	0
$883$	42.7771	1.43956	0.719782	−	0.694200i	$-0.244242\pi$
$883$	42.7771	1.43956	0.719782	+	0.694200i	$0.244242\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27.9443	−0.938277	−0.469138	−	0.883125i	$-0.655435\pi$
$887$	−27.9443	−0.938277	−0.469138	+	0.883125i	$0.655435\pi$
$888$	0	0
$889$	−26.0344	−0.873167
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−28.4164	−0.950919
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.94427	0.131695
$898$	0	0
$899$	−56.6869	−1.89061
$900$	0	0
$901$	−0.236068	−0.00786456
$902$	0	0
$903$	10.7082	0.356347
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.34752	0.0779483	0.0389741	−	0.999240i	$-0.487591\pi$
$907$	2.34752	0.0779483	0.0389741	+	0.999240i	$0.487591\pi$
$908$	0	0
$909$	16.9443	0.562006
$910$	0	0
$911$	49.3820	1.63610	0.818049	−	0.575149i	$-0.195056\pi$
$911$	49.3820	1.63610	0.818049	+	0.575149i	$0.195056\pi$
$912$	0	0
$913$	−9.85410	−0.326123
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−25.0344	−0.826710
$918$	0	0
$919$	1.61803	0.0533740	0.0266870	−	0.999644i	$-0.491504\pi$
$919$	1.61803	0.0533740	0.0266870	+	0.999644i	$0.491504\pi$
$920$	0	0
$921$	−0.236068	−0.00777870
$922$	0	0
$923$	1.90983	0.0628628
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.8541	−0.553561
$928$	0	0
$929$	59.6312	1.95644	0.978218	−	0.207581i	$-0.0665589\pi$
$929$	59.6312	1.95644	0.978218	+	0.207581i	$0.0665589\pi$
$930$	0	0
$931$	−0.326238	−0.0106920
$932$	0	0
$933$	−22.0902	−0.723200
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.1803	0.789937	0.394969	−	0.918695i	$-0.370755\pi$
$937$	24.1803	0.789937	0.394969	+	0.918695i	$0.370755\pi$
$938$	0	0
$939$	−32.6180	−1.06445
$940$	0	0
$941$	−39.9230	−1.30145	−0.650726	−	0.759313i	$-0.725535\pi$
$941$	−39.9230	−1.30145	−0.650726	+	0.759313i	$0.725535\pi$
$942$	0	0
$943$	−2.43769	−0.0793822
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.8885	0.678786	0.339393	−	0.940645i	$-0.389778\pi$
$947$	20.8885	0.678786	0.339393	+	0.940645i	$0.389778\pi$
$948$	0	0
$949$	7.43769	0.241438
$950$	0	0
$951$	−17.1803	−0.557111
$952$	0	0
$953$	14.2016	0.460036	0.230018	−	0.973186i	$-0.426122\pi$
$953$	14.2016	0.460036	0.230018	+	0.973186i	$0.426122\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.23607	−0.201583
$958$	0	0
$959$	−22.3262	−0.720952
$960$	0	0
$961$	51.6312	1.66552
$962$	0	0
$963$	−12.8541	−0.414218
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−36.3050	−1.16749	−0.583744	−	0.811938i	$-0.698413\pi$
$967$	−36.3050	−1.16749	−0.583744	+	0.811938i	$0.698413\pi$
$968$	0	0
$969$	5.85410	0.188061
$970$	0	0
$971$	−54.7639	−1.75746	−0.878729	−	0.477321i	$-0.841608\pi$
$971$	−54.7639	−1.75746	−0.878729	+	0.477321i	$0.841608\pi$
$972$	0	0
$973$	11.8541	0.380025
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.85410	−0.283268	−0.141634	−	0.989919i	$-0.545236\pi$
$977$	−8.85410	−0.283268	−0.141634	+	0.989919i	$0.545236\pi$
$978$	0	0
$979$	−3.94427	−0.126059
$980$	0	0
$981$	−7.00000	−0.223493
$982$	0	0
$983$	38.4508	1.22639	0.613196	−	0.789931i	$-0.289884\pi$
$983$	38.4508	1.22639	0.613196	+	0.789931i	$0.289884\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−33.2705	−1.05901
$988$	0	0
$989$	−7.21478	−0.229417
$990$	0	0
$991$	41.2148	1.30923	0.654615	−	0.755962i	$-0.272831\pi$
$991$	41.2148	1.30923	0.654615	+	0.755962i	$0.272831\pi$
$992$	0	0
$993$	−1.14590	−0.0363640
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−57.1803	−1.81092	−0.905460	−	0.424432i	$-0.860474\pi$
$997$	−57.1803	−1.81092	−0.905460	+	0.424432i	$0.860474\pi$
$998$	0	0
$999$	8.70820	0.275515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1500.2.a.a.1.1	✓	2
3.2	odd	2		4500.2.a.c.1.1		2
4.3	odd	2		6000.2.a.z.1.2		2
5.2	odd	4		1500.2.d.d.1249.3		4
5.3	odd	4		1500.2.d.d.1249.2		4
5.4	even	2		1500.2.a.h.1.2	yes	2
15.2	even	4		4500.2.d.c.4249.1		4
15.8	even	4		4500.2.d.c.4249.4		4
15.14	odd	2		4500.2.a.l.1.2		2
20.3	even	4		6000.2.f.e.1249.3		4
20.7	even	4		6000.2.f.e.1249.2		4
20.19	odd	2		6000.2.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1500.2.a.a.1.1	✓	2	1.1	even	1	trivial
1500.2.a.h.1.2	yes	2	5.4	even	2
1500.2.d.d.1249.2		4	5.3	odd	4
1500.2.d.d.1249.3		4	5.2	odd	4
4500.2.a.c.1.1		2	3.2	odd	2
4500.2.a.l.1.2		2	15.14	odd	2
4500.2.d.c.4249.1		4	15.2	even	4
4500.2.d.c.4249.4		4	15.8	even	4
6000.2.a.c.1.1		2	20.19	odd	2
6000.2.a.z.1.2		2	4.3	odd	2
6000.2.f.e.1249.2		4	20.7	even	4
6000.2.f.e.1249.3		4	20.3	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:	\( N \)	\(=\)	\( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1500.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.61803 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.61803 q^{7} +1.00000 q^{9} +1.00000 q^{11} +2.23607 q^{13} -2.61803 q^{17} +2.23607 q^{19} +2.61803 q^{21} -1.76393 q^{23} -1.00000 q^{27} +6.23607 q^{29} -9.09017 q^{31} -1.00000 q^{33} -8.70820 q^{37} -2.23607 q^{39} +1.38197 q^{41} +4.09017 q^{43} -12.7082 q^{47} -0.145898 q^{49} +2.61803 q^{51} +0.0901699 q^{53} -2.23607 q^{57} -9.09017 q^{59} -6.61803 q^{61} -2.61803 q^{63} -1.52786 q^{67} +1.76393 q^{69} +0.854102 q^{71} +3.32624 q^{73} -2.61803 q^{77} -0.291796 q^{79} +1.00000 q^{81} -9.85410 q^{83} -6.23607 q^{87} -3.94427 q^{89} -5.85410 q^{91} +9.09017 q^{93} -12.7984 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 3 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 3 q^{7} + 2 q^{9} + 2 q^{11} - 3 q^{17} + 3 q^{21} - 8 q^{23} - 2 q^{27} + 8 q^{29} - 7 q^{31} - 2 q^{33} - 4 q^{37} + 5 q^{41} - 3 q^{43} - 12 q^{47} - 7 q^{49} + 3 q^{51} - 11 q^{53} - 7 q^{59} - 11 q^{61} - 3 q^{63} - 12 q^{67} + 8 q^{69} - 5 q^{71} - 9 q^{73} - 3 q^{77} - 14 q^{79} + 2 q^{81} - 13 q^{83} - 8 q^{87} + 10 q^{89} - 5 q^{91} + 7 q^{93} - q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 3 * q^7 + 2 * q^9 + 2 * q^11 - 3 * q^17 + 3 * q^21 - 8 * q^23 - 2 * q^27 + 8 * q^29 - 7 * q^31 - 2 * q^33 - 4 * q^37 + 5 * q^41 - 3 * q^43 - 12 * q^47 - 7 * q^49 + 3 * q^51 - 11 * q^53 - 7 * q^59 - 11 * q^61 - 3 * q^63 - 12 * q^67 + 8 * q^69 - 5 * q^71 - 9 * q^73 - 3 * q^77 - 14 * q^79 + 2 * q^81 - 13 * q^83 - 8 * q^87 + 10 * q^89 - 5 * q^91 + 7 * q^93 - q^97 + 2 * q^99

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