LMFDB - Embedded newform 1500.2.a.c.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:	$0$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.23607 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.23607 q^{7} +1.00000 q^{9} +3.23607 q^{11} -2.61803 q^{17} -0.854102 q^{19} +1.23607 q^{21} +1.85410 q^{23} -1.00000 q^{27} -0.472136 q^{29} +5.38197 q^{31} -3.23607 q^{33} -2.00000 q^{37} +4.47214 q^{41} +7.70820 q^{43} +2.09017 q^{47} -5.47214 q^{49} +2.61803 q^{51} -6.09017 q^{53} +0.854102 q^{57} +6.76393 q^{59} +14.5623 q^{61} -1.23607 q^{63} +2.29180 q^{67} -1.85410 q^{69} +14.4721 q^{71} +12.4721 q^{73} -4.00000 q^{77} +7.14590 q^{79} +1.00000 q^{81} -9.32624 q^{83} +0.472136 q^{87} -7.23607 q^{89} -5.38197 q^{93} +12.0000 q^{97} +3.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} + 2 q^{11} - 3 q^{17} + 5 q^{19} - 2 q^{21} - 3 q^{23} - 2 q^{27} + 8 q^{29} + 13 q^{31} - 2 q^{33} - 4 q^{37} + 2 q^{43} - 7 q^{47} - 2 q^{49} + 3 q^{51} - q^{53} - 5 q^{57} + 18 q^{59} + 9 q^{61} + 2 q^{63} + 18 q^{67} + 3 q^{69} + 20 q^{71} + 16 q^{73} - 8 q^{77} + 21 q^{79} + 2 q^{81} - 3 q^{83} - 8 q^{87} - 10 q^{89} - 13 q^{93} + 24 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 + 2 * q^11 - 3 * q^17 + 5 * q^19 - 2 * q^21 - 3 * q^23 - 2 * q^27 + 8 * q^29 + 13 * q^31 - 2 * q^33 - 4 * q^37 + 2 * q^43 - 7 * q^47 - 2 * q^49 + 3 * q^51 - q^53 - 5 * q^57 + 18 * q^59 + 9 * q^61 + 2 * q^63 + 18 * q^67 + 3 * q^69 + 20 * q^71 + 16 * q^73 - 8 * q^77 + 21 * q^79 + 2 * q^81 - 3 * q^83 - 8 * q^87 - 10 * q^89 - 13 * q^93 + 24 * 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
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.23607	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	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
$19$	−0.854102	−0.195944	−0.0979722	−	0.995189i	$-0.531236\pi$
$19$	−0.854102	−0.195944	−0.0979722	+	0.995189i	$0.531236\pi$
$20$	0	0
$21$	1.23607	0.269732
$22$	0	0
$23$	1.85410	0.386607	0.193303	−	0.981139i	$-0.438080\pi$
$23$	1.85410	0.386607	0.193303	+	0.981139i	$0.438080\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	0	0
$31$	5.38197	0.966630	0.483315	−	0.875447i	$-0.339433\pi$
$31$	5.38197	0.966630	0.483315	+	0.875447i	$0.339433\pi$
$32$	0	0
$33$	−3.23607	−0.563327
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.47214	0.698430	0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214	0.698430	0.349215	+	0.937043i	$0.386448\pi$
$42$	0	0
$43$	7.70820	1.17549	0.587745	−	0.809046i	$-0.300016\pi$
$43$	7.70820	1.17549	0.587745	+	0.809046i	$0.300016\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.09017	0.304883	0.152441	−	0.988313i	$-0.451286\pi$
$47$	2.09017	0.304883	0.152441	+	0.988313i	$0.451286\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	2.61803	0.366598
$52$	0	0
$53$	−6.09017	−0.836549	−0.418275	−	0.908321i	$-0.637365\pi$
$53$	−6.09017	−0.836549	−0.418275	+	0.908321i	$0.637365\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.854102	0.113129
$58$	0	0
$59$	6.76393	0.880589	0.440294	−	0.897853i	$-0.354874\pi$
$59$	6.76393	0.880589	0.440294	+	0.897853i	$0.354874\pi$
$60$	0	0
$61$	14.5623	1.86451	0.932256	−	0.361799i	$-0.117837\pi$
$61$	14.5623	1.86451	0.932256	+	0.361799i	$0.117837\pi$
$62$	0	0
$63$	−1.23607	−0.155730
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.29180	0.279987	0.139994	−	0.990152i	$-0.455292\pi$
$67$	2.29180	0.279987	0.139994	+	0.990152i	$0.455292\pi$
$68$	0	0
$69$	−1.85410	−0.223208
$70$	0	0
$71$	14.4721	1.71753	0.858763	−	0.512373i	$-0.171233\pi$
$71$	14.4721	1.71753	0.858763	+	0.512373i	$0.171233\pi$
$72$	0	0
$73$	12.4721	1.45975	0.729877	−	0.683579i	$-0.239578\pi$
$73$	12.4721	1.45975	0.729877	+	0.683579i	$0.239578\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	7.14590	0.803976	0.401988	−	0.915645i	$-0.368319\pi$
$79$	7.14590	0.803976	0.401988	+	0.915645i	$0.368319\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.32624	−1.02369	−0.511844	−	0.859079i	$-0.671037\pi$
$83$	−9.32624	−1.02369	−0.511844	+	0.859079i	$0.671037\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.472136	0.0506183
$88$	0	0
$89$	−7.23607	−0.767022	−0.383511	−	0.923536i	$-0.625285\pi$
$89$	−7.23607	−0.767022	−0.383511	+	0.923536i	$0.625285\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.38197	−0.558084
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	3.23607	0.325237
$100$	0	0
$101$	5.23607	0.521008	0.260504	−	0.965473i	$-0.416111\pi$
$101$	5.23607	0.521008	0.260504	+	0.965473i	$0.416111\pi$
$102$	0	0
$103$	15.7082	1.54778	0.773888	−	0.633323i	$-0.218309\pi$
$103$	15.7082	1.54778	0.773888	+	0.633323i	$0.218309\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.6180	1.60653	0.803263	−	0.595625i	$-0.203095\pi$
$107$	16.6180	1.60653	0.803263	+	0.595625i	$0.203095\pi$
$108$	0	0
$109$	−0.618034	−0.0591969	−0.0295985	−	0.999562i	$-0.509423\pi$
$109$	−0.618034	−0.0591969	−0.0295985	+	0.999562i	$0.509423\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−17.3262	−1.62992	−0.814958	−	0.579520i	$-0.803240\pi$
$113$	−17.3262	−1.62992	−0.814958	+	0.579520i	$0.803240\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.23607	0.296650
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	−4.47214	−0.403239
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.4164	−1.54546	−0.772728	−	0.634737i	$-0.781108\pi$
$127$	−17.4164	−1.54546	−0.772728	+	0.634737i	$0.781108\pi$
$128$	0	0
$129$	−7.70820	−0.678670
$130$	0	0
$131$	−6.94427	−0.606724	−0.303362	−	0.952875i	$-0.598109\pi$
$131$	−6.94427	−0.606724	−0.303362	+	0.952875i	$0.598109\pi$
$132$	0	0
$133$	1.05573	0.0915432
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.52786	−0.643149	−0.321574	−	0.946884i	$-0.604212\pi$
$137$	−7.52786	−0.643149	−0.321574	+	0.946884i	$0.604212\pi$
$138$	0	0
$139$	11.8541	1.00545	0.502726	−	0.864446i	$-0.332331\pi$
$139$	11.8541	1.00545	0.502726	+	0.864446i	$0.332331\pi$
$140$	0	0
$141$	−2.09017	−0.176024
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.47214	0.451334
$148$	0	0
$149$	−17.1246	−1.40290	−0.701451	−	0.712717i	$-0.747464\pi$
$149$	−17.1246	−1.40290	−0.701451	+	0.712717i	$0.747464\pi$
$150$	0	0
$151$	0.854102	0.0695058	0.0347529	−	0.999396i	$-0.488936\pi$
$151$	0.854102	0.0695058	0.0347529	+	0.999396i	$0.488936\pi$
$152$	0	0
$153$	−2.61803	−0.211656
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.2361	1.53521	0.767603	−	0.640926i	$-0.221449\pi$
$157$	19.2361	1.53521	0.767603	+	0.640926i	$0.221449\pi$
$158$	0	0
$159$	6.09017	0.482982
$160$	0	0
$161$	−2.29180	−0.180619
$162$	0	0
$163$	7.23607	0.566773	0.283386	−	0.959006i	$-0.408542\pi$
$163$	7.23607	0.566773	0.283386	+	0.959006i	$0.408542\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.14590	−0.0886723	−0.0443361	−	0.999017i	$-0.514117\pi$
$167$	−1.14590	−0.0886723	−0.0443361	+	0.999017i	$0.514117\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−0.854102	−0.0653148
$172$	0	0
$173$	11.8885	0.903869	0.451935	−	0.892051i	$-0.350734\pi$
$173$	11.8885	0.903869	0.451935	+	0.892051i	$0.350734\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.76393	−0.508408
$178$	0	0
$179$	11.8885	0.888591	0.444296	−	0.895880i	$-0.353454\pi$
$179$	11.8885	0.888591	0.444296	+	0.895880i	$0.353454\pi$
$180$	0	0
$181$	2.56231	0.190455	0.0952273	−	0.995456i	$-0.469642\pi$
$181$	2.56231	0.190455	0.0952273	+	0.995456i	$0.469642\pi$
$182$	0	0
$183$	−14.5623	−1.07648
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−8.47214	−0.619544
$188$	0	0
$189$	1.23607	0.0899107
$190$	0	0
$191$	−4.94427	−0.357755	−0.178877	−	0.983871i	$-0.557247\pi$
$191$	−4.94427	−0.357755	−0.178877	+	0.983871i	$0.557247\pi$
$192$	0	0
$193$	19.1246	1.37662	0.688310	−	0.725417i	$-0.258353\pi$
$193$	19.1246	1.37662	0.688310	+	0.725417i	$0.258353\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.03444	0.429936	0.214968	−	0.976621i	$-0.431035\pi$
$197$	6.03444	0.429936	0.214968	+	0.976621i	$0.431035\pi$
$198$	0	0
$199$	12.7984	0.907253	0.453626	−	0.891192i	$-0.350130\pi$
$199$	12.7984	0.907253	0.453626	+	0.891192i	$0.350130\pi$
$200$	0	0
$201$	−2.29180	−0.161651
$202$	0	0
$203$	0.583592	0.0409601
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.85410	0.128869
$208$	0	0
$209$	−2.76393	−0.191185
$210$	0	0
$211$	−7.56231	−0.520611	−0.260305	−	0.965526i	$-0.583823\pi$
$211$	−7.56231	−0.520611	−0.260305	+	0.965526i	$0.583823\pi$
$212$	0	0
$213$	−14.4721	−0.991614
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6.65248	−0.451599
$218$	0	0
$219$	−12.4721	−0.842789
$220$	0	0
$221$	0	0
$222$	0	0
$223$	26.8328	1.79686	0.898429	−	0.439119i	$-0.144709\pi$
$223$	26.8328	1.79686	0.898429	+	0.439119i	$0.144709\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.0902	−1.53255	−0.766274	−	0.642514i	$-0.777892\pi$
$227$	−23.0902	−1.53255	−0.766274	+	0.642514i	$0.777892\pi$
$228$	0	0
$229$	−9.85410	−0.651177	−0.325589	−	0.945512i	$-0.605562\pi$
$229$	−9.85410	−0.651177	−0.325589	+	0.945512i	$0.605562\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	−13.4164	−0.878938	−0.439469	−	0.898258i	$-0.644833\pi$
$233$	−13.4164	−0.878938	−0.439469	+	0.898258i	$0.644833\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−7.14590	−0.464176
$238$	0	0
$239$	5.23607	0.338693	0.169347	−	0.985557i	$-0.445834\pi$
$239$	5.23607	0.338693	0.169347	+	0.985557i	$0.445834\pi$
$240$	0	0
$241$	−21.2705	−1.37015	−0.685077	−	0.728471i	$-0.740231\pi$
$241$	−21.2705	−1.37015	−0.685077	+	0.728471i	$0.740231\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	9.32624	0.591026
$250$	0	0
$251$	−8.18034	−0.516338	−0.258169	−	0.966100i	$-0.583119\pi$
$251$	−8.18034	−0.516338	−0.258169	+	0.966100i	$0.583119\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.6180	0.724713	0.362357	−	0.932040i	$-0.381972\pi$
$257$	11.6180	0.724713	0.362357	+	0.932040i	$0.381972\pi$
$258$	0	0
$259$	2.47214	0.153611
$260$	0	0
$261$	−0.472136	−0.0292245
$262$	0	0
$263$	−15.3820	−0.948493	−0.474246	−	0.880392i	$-0.657279\pi$
$263$	−15.3820	−0.948493	−0.474246	+	0.880392i	$0.657279\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.23607	0.442840
$268$	0	0
$269$	11.1246	0.678280	0.339140	−	0.940736i	$-0.389864\pi$
$269$	11.1246	0.678280	0.339140	+	0.940736i	$0.389864\pi$
$270$	0	0
$271$	−15.2705	−0.927617	−0.463809	−	0.885935i	$-0.653518\pi$
$271$	−15.2705	−0.927617	−0.463809	+	0.885935i	$0.653518\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−11.4164	−0.685945	−0.342973	−	0.939345i	$-0.611434\pi$
$277$	−11.4164	−0.685945	−0.342973	+	0.939345i	$0.611434\pi$
$278$	0	0
$279$	5.38197	0.322210
$280$	0	0
$281$	−31.2361	−1.86339	−0.931694	−	0.363245i	$-0.881669\pi$
$281$	−31.2361	−1.86339	−0.931694	+	0.363245i	$0.881669\pi$
$282$	0	0
$283$	−18.1803	−1.08071	−0.540355	−	0.841437i	$-0.681710\pi$
$283$	−18.1803	−1.08071	−0.540355	+	0.841437i	$0.681710\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.52786	−0.326299
$288$	0	0
$289$	−10.1459	−0.596818
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	10.0344	0.586218	0.293109	−	0.956079i	$-0.405310\pi$
$293$	10.0344	0.586218	0.293109	+	0.956079i	$0.405310\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.23607	−0.187776
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−9.52786	−0.549177
$302$	0	0
$303$	−5.23607	−0.300804
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.81966	0.103853	0.0519267	−	0.998651i	$-0.483464\pi$
$307$	1.81966	0.103853	0.0519267	+	0.998651i	$0.483464\pi$
$308$	0	0
$309$	−15.7082	−0.893609
$310$	0	0
$311$	−27.7082	−1.57119	−0.785594	−	0.618742i	$-0.787643\pi$
$311$	−27.7082	−1.57119	−0.785594	+	0.618742i	$0.787643\pi$
$312$	0	0
$313$	2.29180	0.129540	0.0647700	−	0.997900i	$-0.479369\pi$
$313$	2.29180	0.129540	0.0647700	+	0.997900i	$0.479369\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.8328	−1.17009	−0.585044	−	0.811002i	$-0.698923\pi$
$317$	−20.8328	−1.17009	−0.585044	+	0.811002i	$0.698923\pi$
$318$	0	0
$319$	−1.52786	−0.0855440
$320$	0	0
$321$	−16.6180	−0.927528
$322$	0	0
$323$	2.23607	0.124418
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.618034	0.0341774
$328$	0	0
$329$	−2.58359	−0.142438
$330$	0	0
$331$	−5.03444	−0.276718	−0.138359	−	0.990382i	$-0.544183\pi$
$331$	−5.03444	−0.276718	−0.138359	+	0.990382i	$0.544183\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.18034	0.227718	0.113859	−	0.993497i	$-0.463679\pi$
$337$	4.18034	0.227718	0.113859	+	0.993497i	$0.463679\pi$
$338$	0	0
$339$	17.3262	0.941032
$340$	0	0
$341$	17.4164	0.943151
$342$	0	0
$343$	15.4164	0.832408
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.5066	1.58400	0.791998	−	0.610524i	$-0.209041\pi$
$347$	29.5066	1.58400	0.791998	+	0.610524i	$0.209041\pi$
$348$	0	0
$349$	1.85410	0.0992478	0.0496239	−	0.998768i	$-0.484198\pi$
$349$	1.85410	0.0992478	0.0496239	+	0.998768i	$0.484198\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.1459	−0.859359	−0.429680	−	0.902981i	$-0.641374\pi$
$353$	−16.1459	−0.859359	−0.429680	+	0.902981i	$0.641374\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.23607	−0.171271
$358$	0	0
$359$	−1.52786	−0.0806376	−0.0403188	−	0.999187i	$-0.512837\pi$
$359$	−1.52786	−0.0806376	−0.0403188	+	0.999187i	$0.512837\pi$
$360$	0	0
$361$	−18.2705	−0.961606
$362$	0	0
$363$	0.527864	0.0277057
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7.41641	−0.387133	−0.193567	−	0.981087i	$-0.562006\pi$
$367$	−7.41641	−0.387133	−0.193567	+	0.981087i	$0.562006\pi$
$368$	0	0
$369$	4.47214	0.232810
$370$	0	0
$371$	7.52786	0.390827
$372$	0	0
$373$	30.4721	1.57779	0.788894	−	0.614530i	$-0.210654\pi$
$373$	30.4721	1.57779	0.788894	+	0.614530i	$0.210654\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	17.4164	0.892270
$382$	0	0
$383$	−30.2148	−1.54390	−0.771952	−	0.635681i	$-0.780719\pi$
$383$	−30.2148	−1.54390	−0.771952	+	0.635681i	$0.780719\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.70820	0.391830
$388$	0	0
$389$	−24.6525	−1.24993	−0.624965	−	0.780653i	$-0.714887\pi$
$389$	−24.6525	−1.24993	−0.624965	+	0.780653i	$0.714887\pi$
$390$	0	0
$391$	−4.85410	−0.245482
$392$	0	0
$393$	6.94427	0.350292
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.7082	0.989126	0.494563	−	0.869142i	$-0.335328\pi$
$397$	19.7082	0.989126	0.494563	+	0.869142i	$0.335328\pi$
$398$	0	0
$399$	−1.05573	−0.0528525
$400$	0	0
$401$	−26.3607	−1.31639	−0.658195	−	0.752848i	$-0.728680\pi$
$401$	−26.3607	−1.31639	−0.658195	+	0.752848i	$0.728680\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.47214	−0.320812
$408$	0	0
$409$	−1.43769	−0.0710894	−0.0355447	−	0.999368i	$-0.511317\pi$
$409$	−1.43769	−0.0710894	−0.0355447	+	0.999368i	$0.511317\pi$
$410$	0	0
$411$	7.52786	0.371322
$412$	0	0
$413$	−8.36068	−0.411402
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−11.8541	−0.580498
$418$	0	0
$419$	28.1803	1.37670	0.688350	−	0.725379i	$-0.258335\pi$
$419$	28.1803	1.37670	0.688350	+	0.725379i	$0.258335\pi$
$420$	0	0
$421$	−4.67376	−0.227785	−0.113893	−	0.993493i	$-0.536332\pi$
$421$	−4.67376	−0.227785	−0.113893	+	0.993493i	$0.536332\pi$
$422$	0	0
$423$	2.09017	0.101628
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−18.0000	−0.871081
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.4721	−0.793435	−0.396717	−	0.917941i	$-0.629851\pi$
$431$	−16.4721	−0.793435	−0.396717	+	0.917941i	$0.629851\pi$
$432$	0	0
$433$	0.583592	0.0280456	0.0140228	−	0.999902i	$-0.495536\pi$
$433$	0.583592	0.0280456	0.0140228	+	0.999902i	$0.495536\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.58359	−0.0757535
$438$	0	0
$439$	11.4164	0.544875	0.272438	−	0.962173i	$-0.412170\pi$
$439$	11.4164	0.544875	0.272438	+	0.962173i	$0.412170\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	0	0
$443$	3.79837	0.180466	0.0902331	−	0.995921i	$-0.471239\pi$
$443$	3.79837	0.180466	0.0902331	+	0.995921i	$0.471239\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17.1246	0.809966
$448$	0	0
$449$	−28.1803	−1.32991	−0.664956	−	0.746882i	$-0.731550\pi$
$449$	−28.1803	−1.32991	−0.664956	+	0.746882i	$0.731550\pi$
$450$	0	0
$451$	14.4721	0.681466
$452$	0	0
$453$	−0.854102	−0.0401292
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.1246	1.64306	0.821530	−	0.570165i	$-0.193121\pi$
$457$	35.1246	1.64306	0.821530	+	0.570165i	$0.193121\pi$
$458$	0	0
$459$	2.61803	0.122199
$460$	0	0
$461$	2.18034	0.101549	0.0507743	−	0.998710i	$-0.483831\pi$
$461$	2.18034	0.101549	0.0507743	+	0.998710i	$0.483831\pi$
$462$	0	0
$463$	−13.0557	−0.606751	−0.303376	−	0.952871i	$-0.598114\pi$
$463$	−13.0557	−0.606751	−0.303376	+	0.952871i	$0.598114\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.7426	1.70025	0.850123	−	0.526583i	$-0.176527\pi$
$467$	36.7426	1.70025	0.850123	+	0.526583i	$0.176527\pi$
$468$	0	0
$469$	−2.83282	−0.130807
$470$	0	0
$471$	−19.2361	−0.886351
$472$	0	0
$473$	24.9443	1.14694
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.09017	−0.278850
$478$	0	0
$479$	29.1246	1.33074	0.665369	−	0.746515i	$-0.268274\pi$
$479$	29.1246	1.33074	0.665369	+	0.746515i	$0.268274\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	2.29180	0.104280
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−15.2361	−0.690412	−0.345206	−	0.938527i	$-0.612191\pi$
$487$	−15.2361	−0.690412	−0.345206	+	0.938527i	$0.612191\pi$
$488$	0	0
$489$	−7.23607	−0.327226
$490$	0	0
$491$	20.8328	0.940172	0.470086	−	0.882621i	$-0.344223\pi$
$491$	20.8328	0.940172	0.470086	+	0.882621i	$0.344223\pi$
$492$	0	0
$493$	1.23607	0.0556697
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.8885	−0.802411
$498$	0	0
$499$	31.6869	1.41850	0.709251	−	0.704956i	$-0.249033\pi$
$499$	31.6869	1.41850	0.709251	+	0.704956i	$0.249033\pi$
$500$	0	0
$501$	1.14590	0.0511949
$502$	0	0
$503$	15.0557	0.671302	0.335651	−	0.941986i	$-0.391044\pi$
$503$	15.0557	0.671302	0.335651	+	0.941986i	$0.391044\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	−5.88854	−0.261005	−0.130503	−	0.991448i	$-0.541659\pi$
$509$	−5.88854	−0.261005	−0.130503	+	0.991448i	$0.541659\pi$
$510$	0	0
$511$	−15.4164	−0.681982
$512$	0	0
$513$	0.854102	0.0377095
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.76393	0.297477
$518$	0	0
$519$	−11.8885	−0.521849
$520$	0	0
$521$	−5.88854	−0.257982	−0.128991	−	0.991646i	$-0.541174\pi$
$521$	−5.88854	−0.257982	−0.128991	+	0.991646i	$0.541174\pi$
$522$	0	0
$523$	−4.11146	−0.179781	−0.0898907	−	0.995952i	$-0.528652\pi$
$523$	−4.11146	−0.179781	−0.0898907	+	0.995952i	$0.528652\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.0902	−0.613777
$528$	0	0
$529$	−19.5623	−0.850535
$530$	0	0
$531$	6.76393	0.293530
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11.8885	−0.513029
$538$	0	0
$539$	−17.7082	−0.762746
$540$	0	0
$541$	17.2705	0.742517	0.371259	−	0.928530i	$-0.378926\pi$
$541$	17.2705	0.742517	0.371259	+	0.928530i	$0.378926\pi$
$542$	0	0
$543$	−2.56231	−0.109959
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.83282	0.377664	0.188832	−	0.982009i	$-0.439530\pi$
$547$	8.83282	0.377664	0.188832	+	0.982009i	$0.439530\pi$
$548$	0	0
$549$	14.5623	0.621504
$550$	0	0
$551$	0.403252	0.0171791
$552$	0	0
$553$	−8.83282	−0.375610
$554$	0	0
$555$	0	0
$556$	0	0
$557$	34.3607	1.45591	0.727954	−	0.685626i	$-0.240471\pi$
$557$	34.3607	1.45591	0.727954	+	0.685626i	$0.240471\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.47214	0.357694
$562$	0	0
$563$	8.61803	0.363207	0.181603	−	0.983372i	$-0.441871\pi$
$563$	8.61803	0.363207	0.181603	+	0.983372i	$0.441871\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.23607	−0.0519100
$568$	0	0
$569$	37.3050	1.56390	0.781952	−	0.623338i	$-0.214224\pi$
$569$	37.3050	1.56390	0.781952	+	0.623338i	$0.214224\pi$
$570$	0	0
$571$	−2.11146	−0.0883617	−0.0441808	−	0.999024i	$-0.514068\pi$
$571$	−2.11146	−0.0883617	−0.0441808	+	0.999024i	$0.514068\pi$
$572$	0	0
$573$	4.94427	0.206550
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.1246	0.712907	0.356453	−	0.934313i	$-0.383986\pi$
$577$	17.1246	0.712907	0.356453	+	0.934313i	$0.383986\pi$
$578$	0	0
$579$	−19.1246	−0.794792
$580$	0	0
$581$	11.5279	0.478256
$582$	0	0
$583$	−19.7082	−0.816230
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.0344	−0.661812	−0.330906	−	0.943664i	$-0.607354\pi$
$587$	−16.0344	−0.661812	−0.330906	+	0.943664i	$0.607354\pi$
$588$	0	0
$589$	−4.59675	−0.189406
$590$	0	0
$591$	−6.03444	−0.248224
$592$	0	0
$593$	17.5066	0.718909	0.359454	−	0.933163i	$-0.382963\pi$
$593$	17.5066	0.718909	0.359454	+	0.933163i	$0.382963\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12.7984	−0.523803
$598$	0	0
$599$	−20.8328	−0.851206	−0.425603	−	0.904910i	$-0.639938\pi$
$599$	−20.8328	−0.851206	−0.425603	+	0.904910i	$0.639938\pi$
$600$	0	0
$601$	−6.14590	−0.250696	−0.125348	−	0.992113i	$-0.540005\pi$
$601$	−6.14590	−0.250696	−0.125348	+	0.992113i	$0.540005\pi$
$602$	0	0
$603$	2.29180	0.0933292
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−32.2492	−1.30896	−0.654478	−	0.756081i	$-0.727111\pi$
$607$	−32.2492	−1.30896	−0.654478	+	0.756081i	$0.727111\pi$
$608$	0	0
$609$	−0.583592	−0.0236483
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−41.5967	−1.68008	−0.840038	−	0.542527i	$-0.817468\pi$
$613$	−41.5967	−1.68008	−0.840038	+	0.542527i	$0.817468\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.4508	−0.863579	−0.431789	−	0.901974i	$-0.642118\pi$
$617$	−21.4508	−0.863579	−0.431789	+	0.901974i	$0.642118\pi$
$618$	0	0
$619$	−0.798374	−0.0320894	−0.0160447	−	0.999871i	$-0.505107\pi$
$619$	−0.798374	−0.0320894	−0.0160447	+	0.999871i	$0.505107\pi$
$620$	0	0
$621$	−1.85410	−0.0744025
$622$	0	0
$623$	8.94427	0.358345
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.76393	0.110381
$628$	0	0
$629$	5.23607	0.208776
$630$	0	0
$631$	33.3050	1.32585	0.662925	−	0.748686i	$-0.269315\pi$
$631$	33.3050	1.32585	0.662925	+	0.748686i	$0.269315\pi$
$632$	0	0
$633$	7.56231	0.300575
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	14.4721	0.572509
$640$	0	0
$641$	29.7771	1.17612	0.588062	−	0.808816i	$-0.299891\pi$
$641$	29.7771	1.17612	0.588062	+	0.808816i	$0.299891\pi$
$642$	0	0
$643$	−26.6525	−1.05107	−0.525536	−	0.850772i	$-0.676135\pi$
$643$	−26.6525	−1.05107	−0.525536	+	0.850772i	$0.676135\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.9443	1.45243	0.726215	−	0.687468i	$-0.241278\pi$
$647$	36.9443	1.45243	0.726215	+	0.687468i	$0.241278\pi$
$648$	0	0
$649$	21.8885	0.859201
$650$	0	0
$651$	6.65248	0.260731
$652$	0	0
$653$	13.6869	0.535610	0.267805	−	0.963473i	$-0.413702\pi$
$653$	13.6869	0.535610	0.267805	+	0.963473i	$0.413702\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.4721	0.486584
$658$	0	0
$659$	−26.0689	−1.01550	−0.507750	−	0.861505i	$-0.669523\pi$
$659$	−26.0689	−1.01550	−0.507750	+	0.861505i	$0.669523\pi$
$660$	0	0
$661$	42.7984	1.66466	0.832332	−	0.554278i	$-0.187005\pi$
$661$	42.7984	1.66466	0.832332	+	0.554278i	$0.187005\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.875388	−0.0338952
$668$	0	0
$669$	−26.8328	−1.03742
$670$	0	0
$671$	47.1246	1.81923
$672$	0	0
$673$	−27.8885	−1.07502	−0.537512	−	0.843256i	$-0.680636\pi$
$673$	−27.8885	−1.07502	−0.537512	+	0.843256i	$0.680636\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−39.9787	−1.53651	−0.768253	−	0.640146i	$-0.778874\pi$
$677$	−39.9787	−1.53651	−0.768253	+	0.640146i	$0.778874\pi$
$678$	0	0
$679$	−14.8328	−0.569231
$680$	0	0
$681$	23.0902	0.884817
$682$	0	0
$683$	3.67376	0.140573	0.0702863	−	0.997527i	$-0.477609\pi$
$683$	3.67376	0.140573	0.0702863	+	0.997527i	$0.477609\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.85410	0.375957
$688$	0	0
$689$	0	0
$690$	0	0
$691$	15.5623	0.592018	0.296009	−	0.955185i	$-0.404344\pi$
$691$	15.5623	0.592018	0.296009	+	0.955185i	$0.404344\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−11.7082	−0.443480
$698$	0	0
$699$	13.4164	0.507455
$700$	0	0
$701$	33.8885	1.27995	0.639976	−	0.768395i	$-0.278944\pi$
$701$	33.8885	1.27995	0.639976	+	0.768395i	$0.278944\pi$
$702$	0	0
$703$	1.70820	0.0644261
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.47214	−0.243410
$708$	0	0
$709$	10.3820	0.389903	0.194951	−	0.980813i	$-0.437545\pi$
$709$	10.3820	0.389903	0.194951	+	0.980813i	$0.437545\pi$
$710$	0	0
$711$	7.14590	0.267992
$712$	0	0
$713$	9.97871	0.373706
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.23607	−0.195545
$718$	0	0
$719$	25.2361	0.941147	0.470573	−	0.882361i	$-0.344047\pi$
$719$	25.2361	0.941147	0.470573	+	0.882361i	$0.344047\pi$
$720$	0	0
$721$	−19.4164	−0.723105
$722$	0	0
$723$	21.2705	0.791059
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−34.8328	−1.29188	−0.645939	−	0.763389i	$-0.723534\pi$
$727$	−34.8328	−1.29188	−0.645939	+	0.763389i	$0.723534\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−20.1803	−0.746397
$732$	0	0
$733$	12.1803	0.449891	0.224946	−	0.974371i	$-0.427780\pi$
$733$	12.1803	0.449891	0.224946	+	0.974371i	$0.427780\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.41641	0.273187
$738$	0	0
$739$	7.56231	0.278184	0.139092	−	0.990279i	$-0.455582\pi$
$739$	7.56231	0.278184	0.139092	+	0.990279i	$0.455582\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.4164	−0.712319	−0.356159	−	0.934425i	$-0.615914\pi$
$743$	−19.4164	−0.712319	−0.356159	+	0.934425i	$0.615914\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.32624	−0.341229
$748$	0	0
$749$	−20.5410	−0.750553
$750$	0	0
$751$	20.0902	0.733101	0.366550	−	0.930398i	$-0.380539\pi$
$751$	20.0902	0.733101	0.366550	+	0.930398i	$0.380539\pi$
$752$	0	0
$753$	8.18034	0.298108
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.4853	1.50781	0.753904	−	0.656984i	$-0.228168\pi$
$757$	41.4853	1.50781	0.753904	+	0.656984i	$0.228168\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−22.4721	−0.814614	−0.407307	−	0.913291i	$-0.633532\pi$
$761$	−22.4721	−0.814614	−0.407307	+	0.913291i	$0.633532\pi$
$762$	0	0
$763$	0.763932	0.0276562
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−8.85410	−0.319287	−0.159644	−	0.987175i	$-0.551034\pi$
$769$	−8.85410	−0.319287	−0.159644	+	0.987175i	$0.551034\pi$
$770$	0	0
$771$	−11.6180	−0.418413
$772$	0	0
$773$	−12.9230	−0.464807	−0.232404	−	0.972619i	$-0.574659\pi$
$773$	−12.9230	−0.464807	−0.232404	+	0.972619i	$0.574659\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.47214	−0.0886874
$778$	0	0
$779$	−3.81966	−0.136854
$780$	0	0
$781$	46.8328	1.67581
$782$	0	0
$783$	0.472136	0.0168728
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−26.2918	−0.937201	−0.468601	−	0.883410i	$-0.655242\pi$
$787$	−26.2918	−0.937201	−0.468601	+	0.883410i	$0.655242\pi$
$788$	0	0
$789$	15.3820	0.547612
$790$	0	0
$791$	21.4164	0.761480
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.7426	−0.770164	−0.385082	−	0.922882i	$-0.625827\pi$
$797$	−21.7426	−0.770164	−0.385082	+	0.922882i	$0.625827\pi$
$798$	0	0
$799$	−5.47214	−0.193590
$800$	0	0
$801$	−7.23607	−0.255674
$802$	0	0
$803$	40.3607	1.42430
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11.1246	−0.391605
$808$	0	0
$809$	22.5836	0.793997	0.396998	−	0.917819i	$-0.370052\pi$
$809$	22.5836	0.793997	0.396998	+	0.917819i	$0.370052\pi$
$810$	0	0
$811$	22.5623	0.792270	0.396135	−	0.918192i	$-0.370351\pi$
$811$	22.5623	0.792270	0.396135	+	0.918192i	$0.370351\pi$
$812$	0	0
$813$	15.2705	0.535560
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.58359	−0.230331
$818$	0	0
$819$	0	0
$820$	0	0
$821$	50.4721	1.76149	0.880745	−	0.473591i	$-0.157043\pi$
$821$	50.4721	1.76149	0.880745	+	0.473591i	$0.157043\pi$
$822$	0	0
$823$	−19.2361	−0.670527	−0.335264	−	0.942124i	$-0.608825\pi$
$823$	−19.2361	−0.670527	−0.335264	+	0.942124i	$0.608825\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.8541	0.794715	0.397357	−	0.917664i	$-0.369927\pi$
$827$	22.8541	0.794715	0.397357	+	0.917664i	$0.369927\pi$
$828$	0	0
$829$	31.1459	1.08174	0.540871	−	0.841106i	$-0.318095\pi$
$829$	31.1459	1.08174	0.540871	+	0.841106i	$0.318095\pi$
$830$	0	0
$831$	11.4164	0.396031
$832$	0	0
$833$	14.3262	0.496375
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−5.38197	−0.186028
$838$	0	0
$839$	−49.0132	−1.69212	−0.846061	−	0.533086i	$-0.821032\pi$
$839$	−49.0132	−1.69212	−0.846061	+	0.533086i	$0.821032\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	31.2361	1.07583
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.652476	0.0224193
$848$	0	0
$849$	18.1803	0.623948
$850$	0	0
$851$	−3.70820	−0.127116
$852$	0	0
$853$	−39.7082	−1.35958	−0.679792	−	0.733405i	$-0.737930\pi$
$853$	−39.7082	−1.35958	−0.679792	+	0.733405i	$0.737930\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−46.0902	−1.57441	−0.787205	−	0.616691i	$-0.788473\pi$
$857$	−46.0902	−1.57441	−0.787205	+	0.616691i	$0.788473\pi$
$858$	0	0
$859$	−26.8328	−0.915524	−0.457762	−	0.889075i	$-0.651349\pi$
$859$	−26.8328	−0.915524	−0.457762	+	0.889075i	$0.651349\pi$
$860$	0	0
$861$	5.52786	0.188389
$862$	0	0
$863$	26.4721	0.901122	0.450561	−	0.892746i	$-0.351224\pi$
$863$	26.4721	0.901122	0.450561	+	0.892746i	$0.351224\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10.1459	0.344573
$868$	0	0
$869$	23.1246	0.784449
$870$	0	0
$871$	0	0
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.5410	1.09883	0.549416	−	0.835549i	$-0.314850\pi$
$877$	32.5410	1.09883	0.549416	+	0.835549i	$0.314850\pi$
$878$	0	0
$879$	−10.0344	−0.338453
$880$	0	0
$881$	−43.4164	−1.46274	−0.731368	−	0.681983i	$-0.761118\pi$
$881$	−43.4164	−1.46274	−0.731368	+	0.681983i	$0.761118\pi$
$882$	0	0
$883$	−45.8885	−1.54427	−0.772136	−	0.635457i	$-0.780812\pi$
$883$	−45.8885	−1.54427	−0.772136	+	0.635457i	$0.780812\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.0902	1.14464	0.572318	−	0.820032i	$-0.306044\pi$
$887$	34.0902	1.14464	0.572318	+	0.820032i	$0.306044\pi$
$888$	0	0
$889$	21.5279	0.722021
$890$	0	0
$891$	3.23607	0.108412
$892$	0	0
$893$	−1.78522	−0.0597401
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.54102	−0.0847478
$900$	0	0
$901$	15.9443	0.531181
$902$	0	0
$903$	9.52786	0.317067
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−40.2918	−1.33787	−0.668934	−	0.743322i	$-0.733249\pi$
$907$	−40.2918	−1.33787	−0.668934	+	0.743322i	$0.733249\pi$
$908$	0	0
$909$	5.23607	0.173669
$910$	0	0
$911$	2.06888	0.0685452	0.0342726	−	0.999413i	$-0.489089\pi$
$911$	2.06888	0.0685452	0.0342726	+	0.999413i	$0.489089\pi$
$912$	0	0
$913$	−30.1803	−0.998823
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.58359	0.283455
$918$	0	0
$919$	−45.7426	−1.50891	−0.754455	−	0.656351i	$-0.772099\pi$
$919$	−45.7426	−1.50891	−0.754455	+	0.656351i	$0.772099\pi$
$920$	0	0
$921$	−1.81966	−0.0599598
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15.7082	0.515925
$928$	0	0
$929$	35.4853	1.16423	0.582117	−	0.813105i	$-0.302224\pi$
$929$	35.4853	1.16423	0.582117	+	0.813105i	$0.302224\pi$
$930$	0	0
$931$	4.67376	0.153176
$932$	0	0
$933$	27.7082	0.907126
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41.4164	1.35302	0.676508	−	0.736436i	$-0.263493\pi$
$937$	41.4164	1.35302	0.676508	+	0.736436i	$0.263493\pi$
$938$	0	0
$939$	−2.29180	−0.0747899
$940$	0	0
$941$	51.7082	1.68564	0.842820	−	0.538196i	$-0.180894\pi$
$941$	51.7082	1.68564	0.842820	+	0.538196i	$0.180894\pi$
$942$	0	0
$943$	8.29180	0.270018
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.2148	−0.981848	−0.490924	−	0.871202i	$-0.663341\pi$
$947$	−30.2148	−0.981848	−0.490924	+	0.871202i	$0.663341\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	20.8328	0.675550
$952$	0	0
$953$	−29.7426	−0.963459	−0.481729	−	0.876320i	$-0.659991\pi$
$953$	−29.7426	−0.963459	−0.481729	+	0.876320i	$0.659991\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.52786	0.0493888
$958$	0	0
$959$	9.30495	0.300473
$960$	0	0
$961$	−2.03444	−0.0656272
$962$	0	0
$963$	16.6180	0.535509
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.05573	−0.0339499	−0.0169750	−	0.999856i	$-0.505404\pi$
$967$	−1.05573	−0.0339499	−0.0169750	+	0.999856i	$0.505404\pi$
$968$	0	0
$969$	−2.23607	−0.0718329
$970$	0	0
$971$	36.5410	1.17266	0.586329	−	0.810073i	$-0.300573\pi$
$971$	36.5410	1.17266	0.586329	+	0.810073i	$0.300573\pi$
$972$	0	0
$973$	−14.6525	−0.469737
$974$	0	0
$975$	0	0
$976$	0	0
$977$	51.2705	1.64029	0.820144	−	0.572156i	$-0.193893\pi$
$977$	51.2705	1.64029	0.820144	+	0.572156i	$0.193893\pi$
$978$	0	0
$979$	−23.4164	−0.748392
$980$	0	0
$981$	−0.618034	−0.0197323
$982$	0	0
$983$	−10.4934	−0.334688	−0.167344	−	0.985899i	$-0.553519\pi$
$983$	−10.4934	−0.334688	−0.167344	+	0.985899i	$0.553519\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.58359	0.0822366
$988$	0	0
$989$	14.2918	0.454453
$990$	0	0
$991$	−14.5623	−0.462587	−0.231293	−	0.972884i	$-0.574296\pi$
$991$	−14.5623	−0.462587	−0.231293	+	0.972884i	$0.574296\pi$
$992$	0	0
$993$	5.03444	0.159763
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1500.2.a.c.1.1	✓	2
3.2	odd	2		4500.2.a.h.1.1		2
4.3	odd	2		6000.2.a.s.1.2		2
5.2	odd	4		1500.2.d.b.1249.3		4
5.3	odd	4		1500.2.d.b.1249.2		4
5.4	even	2		1500.2.a.g.1.2	yes	2
15.2	even	4		4500.2.d.a.4249.2		4
15.8	even	4		4500.2.d.a.4249.3		4
15.14	odd	2		4500.2.a.d.1.2		2
20.3	even	4		6000.2.f.f.1249.3		4
20.7	even	4		6000.2.f.f.1249.2		4
20.19	odd	2		6000.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1500.2.a.c.1.1	✓	2	1.1	even	1	trivial
1500.2.a.g.1.2	yes	2	5.4	even	2
1500.2.d.b.1249.2		4	5.3	odd	4
1500.2.d.b.1249.3		4	5.2	odd	4
4500.2.a.d.1.2		2	15.14	odd	2
4500.2.a.h.1.1		2	3.2	odd	2
4500.2.d.a.4249.2		4	15.2	even	4
4500.2.d.a.4249.3		4	15.8	even	4
6000.2.a.i.1.1		2	20.19	odd	2
6000.2.a.s.1.2		2	4.3	odd	2
6000.2.f.f.1249.2		4	20.7	even	4
6000.2.f.f.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.23607 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.23607 q^{7} +1.00000 q^{9} +3.23607 q^{11} -2.61803 q^{17} -0.854102 q^{19} +1.23607 q^{21} +1.85410 q^{23} -1.00000 q^{27} -0.472136 q^{29} +5.38197 q^{31} -3.23607 q^{33} -2.00000 q^{37} +4.47214 q^{41} +7.70820 q^{43} +2.09017 q^{47} -5.47214 q^{49} +2.61803 q^{51} -6.09017 q^{53} +0.854102 q^{57} +6.76393 q^{59} +14.5623 q^{61} -1.23607 q^{63} +2.29180 q^{67} -1.85410 q^{69} +14.4721 q^{71} +12.4721 q^{73} -4.00000 q^{77} +7.14590 q^{79} +1.00000 q^{81} -9.32624 q^{83} +0.472136 q^{87} -7.23607 q^{89} -5.38197 q^{93} +12.0000 q^{97} +3.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} + 2 q^{11} - 3 q^{17} + 5 q^{19} - 2 q^{21} - 3 q^{23} - 2 q^{27} + 8 q^{29} + 13 q^{31} - 2 q^{33} - 4 q^{37} + 2 q^{43} - 7 q^{47} - 2 q^{49} + 3 q^{51} - q^{53} - 5 q^{57} + 18 q^{59} + 9 q^{61} + 2 q^{63} + 18 q^{67} + 3 q^{69} + 20 q^{71} + 16 q^{73} - 8 q^{77} + 21 q^{79} + 2 q^{81} - 3 q^{83} - 8 q^{87} - 10 q^{89} - 13 q^{93} + 24 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 + 2 * q^11 - 3 * q^17 + 5 * q^19 - 2 * q^21 - 3 * q^23 - 2 * q^27 + 8 * q^29 + 13 * q^31 - 2 * q^33 - 4 * q^37 + 2 * q^43 - 7 * q^47 - 2 * q^49 + 3 * q^51 - q^53 - 5 * q^57 + 18 * q^59 + 9 * q^61 + 2 * q^63 + 18 * q^67 + 3 * q^69 + 20 * q^71 + 16 * q^73 - 8 * q^77 + 21 * q^79 + 2 * q^81 - 3 * q^83 - 8 * q^87 - 10 * q^89 - 13 * q^93 + 24 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more