LMFDB - Embedded newform 675.3.d.a.674.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,3,Mod(674,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.674"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	675.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-6,0,10,0,0,0,-6,0,0,0,0,0,0,0,-22,-36,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$18.3924178443$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			674.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	675.674
Dual form			675.3.d.a.674.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{2} +5.00000 q^{4} +5.00000i q^{7} -3.00000 q^{8} -15.0000i q^{11} +10.0000i q^{13} -15.0000i q^{14} -11.0000 q^{16} -18.0000 q^{17} +16.0000 q^{19} +45.0000i q^{22} -12.0000 q^{23} -30.0000i q^{26} +25.0000i q^{28} -30.0000i q^{29} -1.00000 q^{31} +45.0000 q^{32} +54.0000 q^{34} +20.0000i q^{37} -48.0000 q^{38} +60.0000i q^{41} -50.0000i q^{43} -75.0000i q^{44} +36.0000 q^{46} +6.00000 q^{47} +24.0000 q^{49} +50.0000i q^{52} -27.0000 q^{53} -15.0000i q^{56} +90.0000i q^{58} +30.0000i q^{59} -76.0000 q^{61} +3.00000 q^{62} -91.0000 q^{64} -10.0000i q^{67} -90.0000 q^{68} -90.0000i q^{71} -65.0000i q^{73} -60.0000i q^{74} +80.0000 q^{76} +75.0000 q^{77} -14.0000 q^{79} -180.000i q^{82} +3.00000 q^{83} +150.000i q^{86} +45.0000i q^{88} -90.0000i q^{89} -50.0000 q^{91} -60.0000 q^{92} -18.0000 q^{94} -85.0000i q^{97} -72.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{2} + 10 q^{4} - 6 q^{8} - 22 q^{16} - 36 q^{17} + 32 q^{19} - 24 q^{23} - 2 q^{31} + 90 q^{32} + 108 q^{34} - 96 q^{38} + 72 q^{46} + 12 q^{47} + 48 q^{49} - 54 q^{53} - 152 q^{61} + 6 q^{62}+ \cdots - 144 q^{98}+O(q^{100}) $ 2 * q - 6 * q^2 + 10 * q^4 - 6 * q^8 - 22 * q^16 - 36 * q^17 + 32 * q^19 - 24 * q^23 - 2 * q^31 + 90 * q^32 + 108 * q^34 - 96 * q^38 + 72 * q^46 + 12 * q^47 + 48 * q^49 - 54 * q^53 - 152 * q^61 + 6 * q^62 - 182 * q^64 - 180 * q^68 + 160 * q^76 + 150 * q^77 - 28 * q^79 + 6 * q^83 - 100 * q^91 - 120 * q^92 - 36 * q^94 - 144 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−3.00000	−1.50000	−0.750000	−	0.661438i	$-0.769947\pi$
$2$	−3.00000	−1.50000	−0.750000	+	0.661438i	$0.769947\pi$
$3$	0	0
$3$	0	0
$4$	5.00000	1.25000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.00000i	0.714286i	0.934050	+	0.357143i	$0.116249\pi$
$7$	5.00000i	0.714286i	−0.934050	+	0.357143i	$0.883751\pi$
$8$	−3.00000	−0.375000
$9$	0	0
$10$	0	0
$11$	− 15.0000i	− 1.36364i	−0.731522	−	0.681818i	$-0.761190\pi$
$11$	− 15.0000i	− 1.36364i	0.731522	−	0.681818i	$-0.238810\pi$
$12$	0	0
$13$	10.0000i	0.769231i	0.923077	+	0.384615i	$0.125666\pi$
$13$	10.0000i	0.769231i	−0.923077	+	0.384615i	$0.874334\pi$
$14$	− 15.0000i	− 1.07143i
$15$	0	0
$16$	−11.0000	−0.687500
$17$	−18.0000	−1.05882	−0.529412	−	0.848365i	$-0.677587\pi$
$17$	−18.0000	−1.05882	−0.529412	+	0.848365i	$0.677587\pi$
$18$	0	0
$19$	16.0000	0.842105	0.421053	−	0.907036i	$-0.361661\pi$
$19$	16.0000	0.842105	0.421053	+	0.907036i	$0.361661\pi$
$20$	0	0
$21$	0	0
$22$	45.0000i	2.04545i
$23$	−12.0000	−0.521739	−0.260870	−	0.965374i	$-0.584009\pi$
$23$	−12.0000	−0.521739	−0.260870	+	0.965374i	$0.584009\pi$
$24$	0	0
$25$	0	0
$26$	− 30.0000i	− 1.15385i
$27$	0	0
$28$	25.0000i	0.892857i
$29$	− 30.0000i	− 1.03448i	−0.855840	−	0.517241i	$-0.826959\pi$
$29$	− 30.0000i	− 1.03448i	0.855840	−	0.517241i	$-0.173041\pi$
$30$	0	0
$31$	−1.00000	−0.0322581	−0.0161290	−	0.999870i	$-0.505134\pi$
$31$	−1.00000	−0.0322581	−0.0161290	+	0.999870i	$0.505134\pi$
$32$	45.0000	1.40625
$33$	0	0
$34$	54.0000	1.58824
$35$	0	0
$36$	0	0
$37$	20.0000i	0.540541i	0.962784	+	0.270270i	$0.0871131\pi$
$37$	20.0000i	0.540541i	−0.962784	+	0.270270i	$0.912887\pi$
$38$	−48.0000	−1.26316
$39$	0	0
$40$	0	0
$41$	60.0000i	1.46341i	0.681619	+	0.731707i	$0.261276\pi$
$41$	60.0000i	1.46341i	−0.681619	+	0.731707i	$0.738724\pi$
$42$	0	0
$43$	− 50.0000i	− 1.16279i	−0.813621	−	0.581395i	$-0.802507\pi$
$43$	− 50.0000i	− 1.16279i	0.813621	−	0.581395i	$-0.197493\pi$
$44$	− 75.0000i	− 1.70455i
$45$	0	0
$46$	36.0000	0.782609
$47$	6.00000	0.127660	0.0638298	−	0.997961i	$-0.479669\pi$
$47$	6.00000	0.127660	0.0638298	+	0.997961i	$0.479669\pi$
$48$	0	0
$49$	24.0000	0.489796
$50$	0	0
$51$	0	0
$52$	50.0000i	0.961538i
$53$	−27.0000	−0.509434	−0.254717	−	0.967016i	$-0.581982\pi$
$53$	−27.0000	−0.509434	−0.254717	+	0.967016i	$0.581982\pi$
$54$	0	0
$55$	0	0
$56$	− 15.0000i	− 0.267857i
$57$	0	0
$58$	90.0000i	1.55172i
$59$	30.0000i	0.508475i	0.967142	+	0.254237i	$0.0818244\pi$
$59$	30.0000i	0.508475i	−0.967142	+	0.254237i	$0.918176\pi$
$60$	0	0
$61$	−76.0000	−1.24590	−0.622951	−	0.782261i	$-0.714066\pi$
$61$	−76.0000	−1.24590	−0.622951	+	0.782261i	$0.714066\pi$
$62$	3.00000	0.0483871
$63$	0	0
$64$	−91.0000	−1.42188
$65$	0	0
$66$	0	0
$67$	− 10.0000i	− 0.149254i	−0.997212	−	0.0746269i	$-0.976223\pi$
$67$	− 10.0000i	− 0.149254i	0.997212	−	0.0746269i	$-0.0237766\pi$
$68$	−90.0000	−1.32353
$69$	0	0
$70$	0	0
$71$	− 90.0000i	− 1.26761i	−0.773495	−	0.633803i	$-0.781493\pi$
$71$	− 90.0000i	− 1.26761i	0.773495	−	0.633803i	$-0.218507\pi$
$72$	0	0
$73$	− 65.0000i	− 0.890411i	−0.895428	−	0.445205i	$-0.853131\pi$
$73$	− 65.0000i	− 0.890411i	0.895428	−	0.445205i	$-0.146869\pi$
$74$	− 60.0000i	− 0.810811i
$75$	0	0
$76$	80.0000	1.05263
$77$	75.0000	0.974026
$78$	0	0
$79$	−14.0000	−0.177215	−0.0886076	−	0.996067i	$-0.528242\pi$
$79$	−14.0000	−0.177215	−0.0886076	+	0.996067i	$0.528242\pi$
$80$	0	0
$81$	0	0
$82$	− 180.000i	− 2.19512i
$83$	3.00000	0.0361446	0.0180723	−	0.999837i	$-0.494247\pi$
$83$	3.00000	0.0361446	0.0180723	+	0.999837i	$0.494247\pi$
$84$	0	0
$85$	0	0
$86$	150.000i	1.74419i
$87$	0	0
$88$	45.0000i	0.511364i
$89$	− 90.0000i	− 1.01124i	−0.862757	−	0.505618i	$-0.831265\pi$
$89$	− 90.0000i	− 1.01124i	0.862757	−	0.505618i	$-0.168735\pi$
$90$	0	0
$91$	−50.0000	−0.549451
$92$	−60.0000	−0.652174
$93$	0	0
$94$	−18.0000	−0.191489
$95$	0	0
$96$	0	0
$97$	− 85.0000i	− 0.876289i	−0.898905	−	0.438144i	$-0.855636\pi$
$97$	− 85.0000i	− 0.876289i	0.898905	−	0.438144i	$-0.144364\pi$
$98$	−72.0000	−0.734694
$99$	0	0
$100$	0	0
$101$	− 195.000i	− 1.93069i	−0.260971	−	0.965347i	$-0.584043\pi$
$101$	− 195.000i	− 1.93069i	0.260971	−	0.965347i	$-0.415957\pi$
$102$	0	0
$103$	− 170.000i	− 1.65049i	−0.564778	−	0.825243i	$-0.691038\pi$
$103$	− 170.000i	− 1.65049i	0.564778	−	0.825243i	$-0.308962\pi$
$104$	− 30.0000i	− 0.288462i
$105$	0	0
$106$	81.0000	0.764151
$107$	−189.000	−1.76636	−0.883178	−	0.469039i	$-0.844600\pi$
$107$	−189.000	−1.76636	−0.883178	+	0.469039i	$0.844600\pi$
$108$	0	0
$109$	−164.000	−1.50459	−0.752294	−	0.658828i	$-0.771052\pi$
$109$	−164.000	−1.50459	−0.752294	+	0.658828i	$0.771052\pi$
$110$	0	0
$111$	0	0
$112$	− 55.0000i	− 0.491071i
$113$	24.0000	0.212389	0.106195	−	0.994345i	$-0.466133\pi$
$113$	24.0000	0.212389	0.106195	+	0.994345i	$0.466133\pi$
$114$	0	0
$115$	0	0
$116$	− 150.000i	− 1.29310i
$117$	0	0
$118$	− 90.0000i	− 0.762712i
$119$	− 90.0000i	− 0.756303i
$120$	0	0
$121$	−104.000	−0.859504
$122$	228.000	1.86885
$123$	0	0
$124$	−5.00000	−0.0403226
$125$	0	0
$126$	0	0
$127$	− 205.000i	− 1.61417i	−0.590433	−	0.807087i	$-0.701043\pi$
$127$	− 205.000i	− 1.61417i	0.590433	−	0.807087i	$-0.298957\pi$
$128$	93.0000	0.726562
$129$	0	0
$130$	0	0
$131$	15.0000i	0.114504i	0.998360	+	0.0572519i	$0.0182338\pi$
$131$	15.0000i	0.114504i	−0.998360	+	0.0572519i	$0.981766\pi$
$132$	0	0
$133$	80.0000i	0.601504i
$134$	30.0000i	0.223881i
$135$	0	0
$136$	54.0000	0.397059
$137$	−138.000	−1.00730	−0.503650	−	0.863908i	$-0.668010\pi$
$137$	−138.000	−1.00730	−0.503650	+	0.863908i	$0.668010\pi$
$138$	0	0
$139$	28.0000	0.201439	0.100719	−	0.994915i	$-0.467886\pi$
$139$	28.0000	0.201439	0.100719	+	0.994915i	$0.467886\pi$
$140$	0	0
$141$	0	0
$142$	270.000i	1.90141i
$143$	150.000	1.04895
$144$	0	0
$145$	0	0
$146$	195.000i	1.33562i
$147$	0	0
$148$	100.000i	0.675676i
$149$	75.0000i	0.503356i	0.967811	+	0.251678i	$0.0809823\pi$
$149$	75.0000i	0.503356i	−0.967811	+	0.251678i	$0.919018\pi$
$150$	0	0
$151$	77.0000	0.509934	0.254967	−	0.966950i	$-0.417935\pi$
$151$	77.0000	0.509934	0.254967	+	0.966950i	$0.417935\pi$
$152$	−48.0000	−0.315789
$153$	0	0
$154$	−225.000	−1.46104
$155$	0	0
$156$	0	0
$157$	− 100.000i	− 0.636943i	−0.947932	−	0.318471i	$-0.896831\pi$
$157$	− 100.000i	− 0.636943i	0.947932	−	0.318471i	$-0.103169\pi$
$158$	42.0000	0.265823
$159$	0	0
$160$	0	0
$161$	− 60.0000i	− 0.372671i
$162$	0	0
$163$	− 110.000i	− 0.674847i	−0.941353	−	0.337423i	$-0.890445\pi$
$163$	− 110.000i	− 0.674847i	0.941353	−	0.337423i	$-0.109555\pi$
$164$	300.000i	1.82927i
$165$	0	0
$166$	−9.00000	−0.0542169
$167$	−78.0000	−0.467066	−0.233533	−	0.972349i	$-0.575029\pi$
$167$	−78.0000	−0.467066	−0.233533	+	0.972349i	$0.575029\pi$
$168$	0	0
$169$	69.0000	0.408284
$170$	0	0
$171$	0	0
$172$	− 250.000i	− 1.45349i
$173$	−177.000	−1.02312	−0.511561	−	0.859247i	$-0.670932\pi$
$173$	−177.000	−1.02312	−0.511561	+	0.859247i	$0.670932\pi$
$174$	0	0
$175$	0	0
$176$	165.000i	0.937500i
$177$	0	0
$178$	270.000i	1.51685i
$179$	225.000i	1.25698i	0.777816	+	0.628492i	$0.216327\pi$
$179$	225.000i	1.25698i	−0.777816	+	0.628492i	$0.783673\pi$
$180$	0	0
$181$	−16.0000	−0.0883978	−0.0441989	−	0.999023i	$-0.514074\pi$
$181$	−16.0000	−0.0883978	−0.0441989	+	0.999023i	$0.514074\pi$
$182$	150.000	0.824176
$183$	0	0
$184$	36.0000	0.195652
$185$	0	0
$186$	0	0
$187$	270.000i	1.44385i
$188$	30.0000	0.159574
$189$	0	0
$190$	0	0
$191$	30.0000i	0.157068i	0.996911	+	0.0785340i	$0.0250239\pi$
$191$	30.0000i	0.157068i	−0.996911	+	0.0785340i	$0.974976\pi$
$192$	0	0
$193$	− 215.000i	− 1.11399i	−0.830516	−	0.556995i	$-0.811954\pi$
$193$	− 215.000i	− 1.11399i	0.830516	−	0.556995i	$-0.188046\pi$
$194$	255.000i	1.31443i
$195$	0	0
$196$	120.000	0.612245
$197$	207.000	1.05076	0.525381	−	0.850867i	$-0.323923\pi$
$197$	207.000	1.05076	0.525381	+	0.850867i	$0.323923\pi$
$198$	0	0
$199$	223.000	1.12060	0.560302	−	0.828289i	$-0.310685\pi$
$199$	223.000	1.12060	0.560302	+	0.828289i	$0.310685\pi$
$200$	0	0
$201$	0	0
$202$	585.000i	2.89604i
$203$	150.000	0.738916
$204$	0	0
$205$	0	0
$206$	510.000i	2.47573i
$207$	0	0
$208$	− 110.000i	− 0.528846i
$209$	− 240.000i	− 1.14833i
$210$	0	0
$211$	−316.000	−1.49763	−0.748815	−	0.662779i	$-0.769377\pi$
$211$	−316.000	−1.49763	−0.748815	+	0.662779i	$0.769377\pi$
$212$	−135.000	−0.636792
$213$	0	0
$214$	567.000	2.64953
$215$	0	0
$216$	0	0
$217$	− 5.00000i	− 0.0230415i
$218$	492.000	2.25688
$219$	0	0
$220$	0	0
$221$	− 180.000i	− 0.814480i
$222$	0	0
$223$	130.000i	0.582960i	0.956577	+	0.291480i	$0.0941476\pi$
$223$	130.000i	0.582960i	−0.956577	+	0.291480i	$0.905852\pi$
$224$	225.000i	1.00446i
$225$	0	0
$226$	−72.0000	−0.318584
$227$	42.0000	0.185022	0.0925110	−	0.995712i	$-0.470511\pi$
$227$	42.0000	0.185022	0.0925110	+	0.995712i	$0.470511\pi$
$228$	0	0
$229$	226.000	0.986900	0.493450	−	0.869774i	$-0.335736\pi$
$229$	226.000	0.986900	0.493450	+	0.869774i	$0.335736\pi$
$230$	0	0
$231$	0	0
$232$	90.0000i	0.387931i
$233$	234.000	1.00429	0.502146	−	0.864783i	$-0.332544\pi$
$233$	234.000	1.00429	0.502146	+	0.864783i	$0.332544\pi$
$234$	0	0
$235$	0	0
$236$	150.000i	0.635593i
$237$	0	0
$238$	270.000i	1.13445i
$239$	120.000i	0.502092i	0.967975	+	0.251046i	$0.0807746\pi$
$239$	120.000i	0.502092i	−0.967975	+	0.251046i	$0.919225\pi$
$240$	0	0
$241$	14.0000	0.0580913	0.0290456	−	0.999578i	$-0.490753\pi$
$241$	14.0000	0.0580913	0.0290456	+	0.999578i	$0.490753\pi$
$242$	312.000	1.28926
$243$	0	0
$244$	−380.000	−1.55738
$245$	0	0
$246$	0	0
$247$	160.000i	0.647773i
$248$	3.00000	0.0120968
$249$	0	0
$250$	0	0
$251$	90.0000i	0.358566i	0.983798	+	0.179283i	$0.0573777\pi$
$251$	90.0000i	0.358566i	−0.983798	+	0.179283i	$0.942622\pi$
$252$	0	0
$253$	180.000i	0.711462i
$254$	615.000i	2.42126i
$255$	0	0
$256$	85.0000	0.332031
$257$	−438.000	−1.70428	−0.852140	−	0.523314i	$-0.824696\pi$
$257$	−438.000	−1.70428	−0.852140	+	0.523314i	$0.824696\pi$
$258$	0	0
$259$	−100.000	−0.386100
$260$	0	0
$261$	0	0
$262$	− 45.0000i	− 0.171756i
$263$	−276.000	−1.04943	−0.524715	−	0.851278i	$-0.675828\pi$
$263$	−276.000	−1.04943	−0.524715	+	0.851278i	$0.675828\pi$
$264$	0	0
$265$	0	0
$266$	− 240.000i	− 0.902256i
$267$	0	0
$268$	− 50.0000i	− 0.186567i
$269$	− 270.000i	− 1.00372i	−0.864950	−	0.501859i	$-0.832650\pi$
$269$	− 270.000i	− 1.00372i	0.864950	−	0.501859i	$-0.167350\pi$
$270$	0	0
$271$	299.000	1.10332	0.551661	−	0.834069i	$-0.313994\pi$
$271$	299.000	1.10332	0.551661	+	0.834069i	$0.313994\pi$
$272$	198.000	0.727941
$273$	0	0
$274$	414.000	1.51095
$275$	0	0
$276$	0	0
$277$	140.000i	0.505415i	0.967543	+	0.252708i	$0.0813211\pi$
$277$	140.000i	0.505415i	−0.967543	+	0.252708i	$0.918679\pi$
$278$	−84.0000	−0.302158
$279$	0	0
$280$	0	0
$281$	− 150.000i	− 0.533808i	−0.963723	−	0.266904i	$-0.913999\pi$
$281$	− 150.000i	− 0.533808i	0.963723	−	0.266904i	$-0.0860006\pi$
$282$	0	0
$283$	280.000i	0.989399i	0.869064	+	0.494700i	$0.164722\pi$
$283$	280.000i	0.989399i	−0.869064	+	0.494700i	$0.835278\pi$
$284$	− 450.000i	− 1.58451i
$285$	0	0
$286$	−450.000	−1.57343
$287$	−300.000	−1.04530
$288$	0	0
$289$	35.0000	0.121107
$290$	0	0
$291$	0	0
$292$	− 325.000i	− 1.11301i
$293$	258.000	0.880546	0.440273	−	0.897864i	$-0.354882\pi$
$293$	258.000	0.880546	0.440273	+	0.897864i	$0.354882\pi$
$294$	0	0
$295$	0	0
$296$	− 60.0000i	− 0.202703i
$297$	0	0
$298$	− 225.000i	− 0.755034i
$299$	− 120.000i	− 0.401338i
$300$	0	0
$301$	250.000	0.830565
$302$	−231.000	−0.764901
$303$	0	0
$304$	−176.000	−0.578947
$305$	0	0
$306$	0	0
$307$	290.000i	0.944625i	0.881431	+	0.472313i	$0.156581\pi$
$307$	290.000i	0.944625i	−0.881431	+	0.472313i	$0.843419\pi$
$308$	375.000	1.21753
$309$	0	0
$310$	0	0
$311$	− 480.000i	− 1.54341i	−0.635982	−	0.771704i	$-0.719405\pi$
$311$	− 480.000i	− 1.54341i	0.635982	−	0.771704i	$-0.280595\pi$
$312$	0	0
$313$	− 185.000i	− 0.591054i	−0.955334	−	0.295527i	$-0.904505\pi$
$313$	− 185.000i	− 0.591054i	0.955334	−	0.295527i	$-0.0954953\pi$
$314$	300.000i	0.955414i
$315$	0	0
$316$	−70.0000	−0.221519
$317$	−183.000	−0.577287	−0.288644	−	0.957437i	$-0.593204\pi$
$317$	−183.000	−0.577287	−0.288644	+	0.957437i	$0.593204\pi$
$318$	0	0
$319$	−450.000	−1.41066
$320$	0	0
$321$	0	0
$322$	180.000i	0.559006i
$323$	−288.000	−0.891641
$324$	0	0
$325$	0	0
$326$	330.000i	1.01227i
$327$	0	0
$328$	− 180.000i	− 0.548780i
$329$	30.0000i	0.0911854i
$330$	0	0
$331$	−238.000	−0.719033	−0.359517	−	0.933139i	$-0.617058\pi$
$331$	−238.000	−0.719033	−0.359517	+	0.933139i	$0.617058\pi$
$332$	15.0000	0.0451807
$333$	0	0
$334$	234.000	0.700599
$335$	0	0
$336$	0	0
$337$	− 10.0000i	− 0.0296736i	−0.999890	−	0.0148368i	$-0.995277\pi$
$337$	− 10.0000i	− 0.0296736i	0.999890	−	0.0148368i	$-0.00472287\pi$
$338$	−207.000	−0.612426
$339$	0	0
$340$	0	0
$341$	15.0000i	0.0439883i
$342$	0	0
$343$	365.000i	1.06414i
$344$	150.000i	0.436047i
$345$	0	0
$346$	531.000	1.53468
$347$	−69.0000	−0.198847	−0.0994236	−	0.995045i	$-0.531700\pi$
$347$	−69.0000	−0.198847	−0.0994236	+	0.995045i	$0.531700\pi$
$348$	0	0
$349$	256.000	0.733524	0.366762	−	0.930315i	$-0.380466\pi$
$349$	256.000	0.733524	0.366762	+	0.930315i	$0.380466\pi$
$350$	0	0
$351$	0	0
$352$	− 675.000i	− 1.91761i
$353$	−456.000	−1.29178	−0.645892	−	0.763428i	$-0.723515\pi$
$353$	−456.000	−1.29178	−0.645892	+	0.763428i	$0.723515\pi$
$354$	0	0
$355$	0	0
$356$	− 450.000i	− 1.26404i
$357$	0	0
$358$	− 675.000i	− 1.88547i
$359$	450.000i	1.25348i	0.779228	+	0.626741i	$0.215612\pi$
$359$	450.000i	1.25348i	−0.779228	+	0.626741i	$0.784388\pi$
$360$	0	0
$361$	−105.000	−0.290859
$362$	48.0000	0.132597
$363$	0	0
$364$	−250.000	−0.686813
$365$	0	0
$366$	0	0
$367$	− 625.000i	− 1.70300i	−0.524357	−	0.851499i	$-0.675694\pi$
$367$	− 625.000i	− 1.70300i	0.524357	−	0.851499i	$-0.324306\pi$
$368$	132.000	0.358696
$369$	0	0
$370$	0	0
$371$	− 135.000i	− 0.363881i
$372$	0	0
$373$	− 170.000i	− 0.455764i	−0.973689	−	0.227882i	$-0.926820\pi$
$373$	− 170.000i	− 0.455764i	0.973689	−	0.227882i	$-0.0731801\pi$
$374$	− 810.000i	− 2.16578i
$375$	0	0
$376$	−18.0000	−0.0478723
$377$	300.000	0.795756
$378$	0	0
$379$	−704.000	−1.85752	−0.928760	−	0.370682i	$-0.879124\pi$
$379$	−704.000	−1.85752	−0.928760	+	0.370682i	$0.879124\pi$
$380$	0	0
$381$	0	0
$382$	− 90.0000i	− 0.235602i
$383$	618.000	1.61358	0.806789	−	0.590840i	$-0.201204\pi$
$383$	618.000	1.61358	0.806789	+	0.590840i	$0.201204\pi$
$384$	0	0
$385$	0	0
$386$	645.000i	1.67098i
$387$	0	0
$388$	− 425.000i	− 1.09536i
$389$	− 525.000i	− 1.34961i	−0.737994	−	0.674807i	$-0.764227\pi$
$389$	− 525.000i	− 1.34961i	0.737994	−	0.674807i	$-0.235773\pi$
$390$	0	0
$391$	216.000	0.552430
$392$	−72.0000	−0.183673
$393$	0	0
$394$	−621.000	−1.57614
$395$	0	0
$396$	0	0
$397$	− 70.0000i	− 0.176322i	−0.996106	−	0.0881612i	$-0.971901\pi$
$397$	− 70.0000i	− 0.176322i	0.996106	−	0.0881612i	$-0.0280991\pi$
$398$	−669.000	−1.68090
$399$	0	0
$400$	0	0
$401$	− 120.000i	− 0.299252i	−0.988743	−	0.149626i	$-0.952193\pi$
$401$	− 120.000i	− 0.299252i	0.988743	−	0.149626i	$-0.0478069\pi$
$402$	0	0
$403$	− 10.0000i	− 0.0248139i
$404$	− 975.000i	− 2.41337i
$405$	0	0
$406$	−450.000	−1.10837
$407$	300.000	0.737101
$408$	0	0
$409$	−269.000	−0.657702	−0.328851	−	0.944382i	$-0.606661\pi$
$409$	−269.000	−0.657702	−0.328851	+	0.944382i	$0.606661\pi$
$410$	0	0
$411$	0	0
$412$	− 850.000i	− 2.06311i
$413$	−150.000	−0.363196
$414$	0	0
$415$	0	0
$416$	450.000i	1.08173i
$417$	0	0
$418$	720.000i	1.72249i
$419$	210.000i	0.501193i	0.968092	+	0.250597i	$0.0806268\pi$
$419$	210.000i	0.501193i	−0.968092	+	0.250597i	$0.919373\pi$
$420$	0	0
$421$	644.000	1.52969	0.764846	−	0.644214i	$-0.222815\pi$
$421$	644.000	1.52969	0.764846	+	0.644214i	$0.222815\pi$
$422$	948.000	2.24645
$423$	0	0
$424$	81.0000	0.191038
$425$	0	0
$426$	0	0
$427$	− 380.000i	− 0.889930i
$428$	−945.000	−2.20794
$429$	0	0
$430$	0	0
$431$	270.000i	0.626450i	0.949679	+	0.313225i	$0.101409\pi$
$431$	270.000i	0.626450i	−0.949679	+	0.313225i	$0.898591\pi$
$432$	0	0
$433$	565.000i	1.30485i	0.757853	+	0.652425i	$0.226248\pi$
$433$	565.000i	1.30485i	−0.757853	+	0.652425i	$0.773752\pi$
$434$	15.0000i	0.0345622i
$435$	0	0
$436$	−820.000	−1.88073
$437$	−192.000	−0.439359
$438$	0	0
$439$	211.000	0.480638	0.240319	−	0.970694i	$-0.422748\pi$
$439$	211.000	0.480638	0.240319	+	0.970694i	$0.422748\pi$
$440$	0	0
$441$	0	0
$442$	540.000i	1.22172i
$443$	498.000	1.12415	0.562077	−	0.827085i	$-0.310003\pi$
$443$	498.000	1.12415	0.562077	+	0.827085i	$0.310003\pi$
$444$	0	0
$445$	0	0
$446$	− 390.000i	− 0.874439i
$447$	0	0
$448$	− 455.000i	− 1.01562i
$449$	360.000i	0.801782i	0.916126	+	0.400891i	$0.131299\pi$
$449$	360.000i	0.801782i	−0.916126	+	0.400891i	$0.868701\pi$
$450$	0	0
$451$	900.000	1.99557
$452$	120.000	0.265487
$453$	0	0
$454$	−126.000	−0.277533
$455$	0	0
$456$	0	0
$457$	365.000i	0.798687i	0.916801	+	0.399344i	$0.130762\pi$
$457$	365.000i	0.798687i	−0.916801	+	0.399344i	$0.869238\pi$
$458$	−678.000	−1.48035
$459$	0	0
$460$	0	0
$461$	− 105.000i	− 0.227766i	−0.993494	−	0.113883i	$-0.963671\pi$
$461$	− 105.000i	− 0.227766i	0.993494	−	0.113883i	$-0.0363289\pi$
$462$	0	0
$463$	− 215.000i	− 0.464363i	−0.972672	−	0.232181i	$-0.925414\pi$
$463$	− 215.000i	− 0.464363i	0.972672	−	0.232181i	$-0.0745863\pi$
$464$	330.000i	0.711207i
$465$	0	0
$466$	−702.000	−1.50644
$467$	−63.0000	−0.134904	−0.0674518	−	0.997723i	$-0.521487\pi$
$467$	−63.0000	−0.134904	−0.0674518	+	0.997723i	$0.521487\pi$
$468$	0	0
$469$	50.0000	0.106610
$470$	0	0
$471$	0	0
$472$	− 90.0000i	− 0.190678i
$473$	−750.000	−1.58562
$474$	0	0
$475$	0	0
$476$	− 450.000i	− 0.945378i
$477$	0	0
$478$	− 360.000i	− 0.753138i
$479$	− 750.000i	− 1.56576i	−0.622171	−	0.782881i	$-0.713749\pi$
$479$	− 750.000i	− 1.56576i	0.622171	−	0.782881i	$-0.286251\pi$
$480$	0	0
$481$	−200.000	−0.415800
$482$	−42.0000	−0.0871369
$483$	0	0
$484$	−520.000	−1.07438
$485$	0	0
$486$	0	0
$487$	110.000i	0.225873i	0.993602	+	0.112936i	$0.0360256\pi$
$487$	110.000i	0.225873i	−0.993602	+	0.112936i	$0.963974\pi$
$488$	228.000	0.467213
$489$	0	0
$490$	0	0
$491$	645.000i	1.31365i	0.754045	+	0.656823i	$0.228100\pi$
$491$	645.000i	1.31365i	−0.754045	+	0.656823i	$0.771900\pi$
$492$	0	0
$493$	540.000i	1.09533i
$494$	− 480.000i	− 0.971660i
$495$	0	0
$496$	11.0000	0.0221774
$497$	450.000	0.905433
$498$	0	0
$499$	766.000	1.53507	0.767535	−	0.641007i	$-0.221483\pi$
$499$	766.000	1.53507	0.767535	+	0.641007i	$0.221483\pi$
$500$	0	0
$501$	0	0
$502$	− 270.000i	− 0.537849i
$503$	828.000	1.64612	0.823062	−	0.567952i	$-0.192264\pi$
$503$	828.000	1.64612	0.823062	+	0.567952i	$0.192264\pi$
$504$	0	0
$505$	0	0
$506$	− 540.000i	− 1.06719i
$507$	0	0
$508$	− 1025.00i	− 2.01772i
$509$	− 555.000i	− 1.09037i	−0.838315	−	0.545187i	$-0.816459\pi$
$509$	− 555.000i	− 1.09037i	0.838315	−	0.545187i	$-0.183541\pi$
$510$	0	0
$511$	325.000	0.636008
$512$	−627.000	−1.22461
$513$	0	0
$514$	1314.00	2.55642
$515$	0	0
$516$	0	0
$517$	− 90.0000i	− 0.174081i
$518$	300.000	0.579151
$519$	0	0
$520$	0	0
$521$	− 450.000i	− 0.863724i	−0.901940	−	0.431862i	$-0.857857\pi$
$521$	− 450.000i	− 0.863724i	0.901940	−	0.431862i	$-0.142143\pi$
$522$	0	0
$523$	250.000i	0.478011i	0.971018	+	0.239006i	$0.0768215\pi$
$523$	250.000i	0.478011i	−0.971018	+	0.239006i	$0.923179\pi$
$524$	75.0000i	0.143130i
$525$	0	0
$526$	828.000	1.57414
$527$	18.0000	0.0341556
$528$	0	0
$529$	−385.000	−0.727788
$530$	0	0
$531$	0	0
$532$	400.000i	0.751880i
$533$	−600.000	−1.12570
$534$	0	0
$535$	0	0
$536$	30.0000i	0.0559701i
$537$	0	0
$538$	810.000i	1.50558i
$539$	− 360.000i	− 0.667904i
$540$	0	0
$541$	−268.000	−0.495379	−0.247689	−	0.968839i	$-0.579671\pi$
$541$	−268.000	−0.495379	−0.247689	+	0.968839i	$0.579671\pi$
$542$	−897.000	−1.65498
$543$	0	0
$544$	−810.000	−1.48897
$545$	0	0
$546$	0	0
$547$	410.000i	0.749543i	0.927117	+	0.374771i	$0.122279\pi$
$547$	410.000i	0.749543i	−0.927117	+	0.374771i	$0.877721\pi$
$548$	−690.000	−1.25912
$549$	0	0
$550$	0	0
$551$	− 480.000i	− 0.871143i
$552$	0	0
$553$	− 70.0000i	− 0.126582i
$554$	− 420.000i	− 0.758123i
$555$	0	0
$556$	140.000	0.251799
$557$	−639.000	−1.14722	−0.573609	−	0.819130i	$-0.694457\pi$
$557$	−639.000	−1.14722	−0.573609	+	0.819130i	$0.694457\pi$
$558$	0	0
$559$	500.000	0.894454
$560$	0	0
$561$	0	0
$562$	450.000i	0.800712i
$563$	−201.000	−0.357016	−0.178508	−	0.983938i	$-0.557127\pi$
$563$	−201.000	−0.357016	−0.178508	+	0.983938i	$0.557127\pi$
$564$	0	0
$565$	0	0
$566$	− 840.000i	− 1.48410i
$567$	0	0
$568$	270.000i	0.475352i
$569$	240.000i	0.421793i	0.977508	+	0.210896i	$0.0676382\pi$
$569$	240.000i	0.421793i	−0.977508	+	0.210896i	$0.932362\pi$
$570$	0	0
$571$	−946.000	−1.65674	−0.828371	−	0.560179i	$-0.810732\pi$
$571$	−946.000	−1.65674	−0.828371	+	0.560179i	$0.810732\pi$
$572$	750.000	1.31119
$573$	0	0
$574$	900.000	1.56794
$575$	0	0
$576$	0	0
$577$	830.000i	1.43847i	0.694764	+	0.719237i	$0.255509\pi$
$577$	830.000i	1.43847i	−0.694764	+	0.719237i	$0.744491\pi$
$578$	−105.000	−0.181661
$579$	0	0
$580$	0	0
$581$	15.0000i	0.0258176i
$582$	0	0
$583$	405.000i	0.694683i
$584$	195.000i	0.333904i
$585$	0	0
$586$	−774.000	−1.32082
$587$	−453.000	−0.771721	−0.385860	−	0.922557i	$-0.626095\pi$
$587$	−453.000	−0.771721	−0.385860	+	0.922557i	$0.626095\pi$
$588$	0	0
$589$	−16.0000	−0.0271647
$590$	0	0
$591$	0	0
$592$	− 220.000i	− 0.371622i
$593$	−702.000	−1.18381	−0.591906	−	0.806007i	$-0.701624\pi$
$593$	−702.000	−1.18381	−0.591906	+	0.806007i	$0.701624\pi$
$594$	0	0
$595$	0	0
$596$	375.000i	0.629195i
$597$	0	0
$598$	360.000i	0.602007i
$599$	1110.00i	1.85309i	0.376186	+	0.926544i	$0.377235\pi$
$599$	1110.00i	1.85309i	−0.376186	+	0.926544i	$0.622765\pi$
$600$	0	0
$601$	869.000	1.44592	0.722962	−	0.690888i	$-0.242780\pi$
$601$	869.000	1.44592	0.722962	+	0.690888i	$0.242780\pi$
$602$	−750.000	−1.24585
$603$	0	0
$604$	385.000	0.637417
$605$	0	0
$606$	0	0
$607$	530.000i	0.873147i	0.899669	+	0.436573i	$0.143808\pi$
$607$	530.000i	0.873147i	−0.899669	+	0.436573i	$0.856192\pi$
$608$	720.000	1.18421
$609$	0	0
$610$	0	0
$611$	60.0000i	0.0981997i
$612$	0	0
$613$	70.0000i	0.114192i	0.998369	+	0.0570962i	$0.0181842\pi$
$613$	70.0000i	0.114192i	−0.998369	+	0.0570962i	$0.981816\pi$
$614$	− 870.000i	− 1.41694i
$615$	0	0
$616$	−225.000	−0.365260
$617$	552.000	0.894652	0.447326	−	0.894371i	$-0.352376\pi$
$617$	552.000	0.894652	0.447326	+	0.894371i	$0.352376\pi$
$618$	0	0
$619$	−662.000	−1.06947	−0.534733	−	0.845021i	$-0.679588\pi$
$619$	−662.000	−1.06947	−0.534733	+	0.845021i	$0.679588\pi$
$620$	0	0
$621$	0	0
$622$	1440.00i	2.31511i
$623$	450.000	0.722311
$624$	0	0
$625$	0	0
$626$	555.000i	0.886581i
$627$	0	0
$628$	− 500.000i	− 0.796178i
$629$	− 360.000i	− 0.572337i
$630$	0	0
$631$	−331.000	−0.524564	−0.262282	−	0.964991i	$-0.584475\pi$
$631$	−331.000	−0.524564	−0.262282	+	0.964991i	$0.584475\pi$
$632$	42.0000	0.0664557
$633$	0	0
$634$	549.000	0.865931
$635$	0	0
$636$	0	0
$637$	240.000i	0.376766i
$638$	1350.00	2.11599
$639$	0	0
$640$	0	0
$641$	− 60.0000i	− 0.0936037i	−0.998904	−	0.0468019i	$-0.985097\pi$
$641$	− 60.0000i	− 0.0936037i	0.998904	−	0.0468019i	$-0.0149029\pi$
$642$	0	0
$643$	− 440.000i	− 0.684292i	−0.939647	−	0.342146i	$-0.888846\pi$
$643$	− 440.000i	− 0.684292i	0.939647	−	0.342146i	$-0.111154\pi$
$644$	− 300.000i	− 0.465839i
$645$	0	0
$646$	864.000	1.33746
$647$	972.000	1.50232	0.751159	−	0.660121i	$-0.229495\pi$
$647$	972.000	1.50232	0.751159	+	0.660121i	$0.229495\pi$
$648$	0	0
$649$	450.000	0.693374
$650$	0	0
$651$	0	0
$652$	− 550.000i	− 0.843558i
$653$	483.000	0.739663	0.369832	−	0.929099i	$-0.379415\pi$
$653$	483.000	0.739663	0.369832	+	0.929099i	$0.379415\pi$
$654$	0	0
$655$	0	0
$656$	− 660.000i	− 1.00610i
$657$	0	0
$658$	− 90.0000i	− 0.136778i
$659$	825.000i	1.25190i	0.779864	+	0.625948i	$0.215288\pi$
$659$	825.000i	1.25190i	−0.779864	+	0.625948i	$0.784712\pi$
$660$	0	0
$661$	−928.000	−1.40393	−0.701967	−	0.712210i	$-0.747694\pi$
$661$	−928.000	−1.40393	−0.701967	+	0.712210i	$0.747694\pi$
$662$	714.000	1.07855
$663$	0	0
$664$	−9.00000	−0.0135542
$665$	0	0
$666$	0	0
$667$	360.000i	0.539730i
$668$	−390.000	−0.583832
$669$	0	0
$670$	0	0
$671$	1140.00i	1.69896i
$672$	0	0
$673$	985.000i	1.46360i	0.681522	+	0.731798i	$0.261319\pi$
$673$	985.000i	1.46360i	−0.681522	+	0.731798i	$0.738681\pi$
$674$	30.0000i	0.0445104i
$675$	0	0
$676$	345.000	0.510355
$677$	−354.000	−0.522895	−0.261448	−	0.965218i	$-0.584200\pi$
$677$	−354.000	−0.522895	−0.261448	+	0.965218i	$0.584200\pi$
$678$	0	0
$679$	425.000	0.625920
$680$	0	0
$681$	0	0
$682$	− 45.0000i	− 0.0659824i
$683$	198.000	0.289898	0.144949	−	0.989439i	$-0.453698\pi$
$683$	198.000	0.289898	0.144949	+	0.989439i	$0.453698\pi$
$684$	0	0
$685$	0	0
$686$	− 1095.00i	− 1.59621i
$687$	0	0
$688$	550.000i	0.799419i
$689$	− 270.000i	− 0.391872i
$690$	0	0
$691$	−436.000	−0.630970	−0.315485	−	0.948931i	$-0.602167\pi$
$691$	−436.000	−0.630970	−0.315485	+	0.948931i	$0.602167\pi$
$692$	−885.000	−1.27890
$693$	0	0
$694$	207.000	0.298271
$695$	0	0
$696$	0	0
$697$	− 1080.00i	− 1.54950i
$698$	−768.000	−1.10029
$699$	0	0
$700$	0	0
$701$	135.000i	0.192582i	0.995353	+	0.0962910i	$0.0306979\pi$
$701$	135.000i	0.192582i	−0.995353	+	0.0962910i	$0.969302\pi$
$702$	0	0
$703$	320.000i	0.455192i
$704$	1365.00i	1.93892i
$705$	0	0
$706$	1368.00	1.93768
$707$	975.000	1.37907
$708$	0	0
$709$	−32.0000	−0.0451340	−0.0225670	−	0.999745i	$-0.507184\pi$
$709$	−32.0000	−0.0451340	−0.0225670	+	0.999745i	$0.507184\pi$
$710$	0	0
$711$	0	0
$712$	270.000i	0.379213i
$713$	12.0000	0.0168303
$714$	0	0
$715$	0	0
$716$	1125.00i	1.57123i
$717$	0	0
$718$	− 1350.00i	− 1.88022i
$719$	900.000i	1.25174i	0.779928	+	0.625869i	$0.215256\pi$
$719$	900.000i	1.25174i	−0.779928	+	0.625869i	$0.784744\pi$
$720$	0	0
$721$	850.000	1.17892
$722$	315.000	0.436288
$723$	0	0
$724$	−80.0000	−0.110497
$725$	0	0
$726$	0	0
$727$	− 175.000i	− 0.240715i	−0.992731	−	0.120358i	$-0.961596\pi$
$727$	− 175.000i	− 0.240715i	0.992731	−	0.120358i	$-0.0384041\pi$
$728$	150.000	0.206044
$729$	0	0
$730$	0	0
$731$	900.000i	1.23119i
$732$	0	0
$733$	− 1160.00i	− 1.58254i	−0.611469	−	0.791269i	$-0.709421\pi$
$733$	− 1160.00i	− 1.58254i	0.611469	−	0.791269i	$-0.290579\pi$
$734$	1875.00i	2.55450i
$735$	0	0
$736$	−540.000	−0.733696
$737$	−150.000	−0.203528
$738$	0	0
$739$	1006.00	1.36130	0.680650	−	0.732609i	$-0.261698\pi$
$739$	1006.00	1.36130	0.680650	+	0.732609i	$0.261698\pi$
$740$	0	0
$741$	0	0
$742$	405.000i	0.545822i
$743$	114.000	0.153432	0.0767160	−	0.997053i	$-0.475557\pi$
$743$	114.000	0.153432	0.0767160	+	0.997053i	$0.475557\pi$
$744$	0	0
$745$	0	0
$746$	510.000i	0.683646i
$747$	0	0
$748$	1350.00i	1.80481i
$749$	− 945.000i	− 1.26168i
$750$	0	0
$751$	359.000	0.478029	0.239015	−	0.971016i	$-0.423176\pi$
$751$	359.000	0.478029	0.239015	+	0.971016i	$0.423176\pi$
$752$	−66.0000	−0.0877660
$753$	0	0
$754$	−900.000	−1.19363
$755$	0	0
$756$	0	0
$757$	− 430.000i	− 0.568032i	−0.958820	−	0.284016i	$-0.908333\pi$
$757$	− 430.000i	− 0.568032i	0.958820	−	0.284016i	$-0.0916668\pi$
$758$	2112.00	2.78628
$759$	0	0
$760$	0	0
$761$	1320.00i	1.73456i	0.497821	+	0.867280i	$0.334134\pi$
$761$	1320.00i	1.73456i	−0.497821	+	0.867280i	$0.665866\pi$
$762$	0	0
$763$	− 820.000i	− 1.07471i
$764$	150.000i	0.196335i
$765$	0	0
$766$	−1854.00	−2.42037
$767$	−300.000	−0.391134
$768$	0	0
$769$	−1259.00	−1.63719	−0.818596	−	0.574370i	$-0.805247\pi$
$769$	−1259.00	−1.63719	−0.818596	+	0.574370i	$0.805247\pi$
$770$	0	0
$771$	0	0
$772$	− 1075.00i	− 1.39249i
$773$	−522.000	−0.675291	−0.337646	−	0.941273i	$-0.609631\pi$
$773$	−522.000	−0.675291	−0.337646	+	0.941273i	$0.609631\pi$
$774$	0	0
$775$	0	0
$776$	255.000i	0.328608i
$777$	0	0
$778$	1575.00i	2.02442i
$779$	960.000i	1.23235i
$780$	0	0
$781$	−1350.00	−1.72855
$782$	−648.000	−0.828645
$783$	0	0
$784$	−264.000	−0.336735
$785$	0	0
$786$	0	0
$787$	− 460.000i	− 0.584498i	−0.956342	−	0.292249i	$-0.905596\pi$
$787$	− 460.000i	− 0.584498i	0.956342	−	0.292249i	$-0.0944036\pi$
$788$	1035.00	1.31345
$789$	0	0
$790$	0	0
$791$	120.000i	0.151707i
$792$	0	0
$793$	− 760.000i	− 0.958386i
$794$	210.000i	0.264484i
$795$	0	0
$796$	1115.00	1.40075
$797$	237.000	0.297365	0.148683	−	0.988885i	$-0.452497\pi$
$797$	237.000	0.297365	0.148683	+	0.988885i	$0.452497\pi$
$798$	0	0
$799$	−108.000	−0.135169
$800$	0	0
$801$	0	0
$802$	360.000i	0.448878i
$803$	−975.000	−1.21420
$804$	0	0
$805$	0	0
$806$	30.0000i	0.0372208i
$807$	0	0
$808$	585.000i	0.724010i
$809$	− 810.000i	− 1.00124i	−0.865668	−	0.500618i	$-0.833106\pi$
$809$	− 810.000i	− 1.00124i	0.865668	−	0.500618i	$-0.166894\pi$
$810$	0	0
$811$	272.000	0.335388	0.167694	−	0.985839i	$-0.446368\pi$
$811$	272.000	0.335388	0.167694	+	0.985839i	$0.446368\pi$
$812$	750.000	0.923645
$813$	0	0
$814$	−900.000	−1.10565
$815$	0	0
$816$	0	0
$817$	− 800.000i	− 0.979192i
$818$	807.000	0.986553
$819$	0	0
$820$	0	0
$821$	390.000i	0.475030i	0.971384	+	0.237515i	$0.0763330\pi$
$821$	390.000i	0.475030i	−0.971384	+	0.237515i	$0.923667\pi$
$822$	0	0
$823$	− 1205.00i	− 1.46416i	−0.681221	−	0.732078i	$-0.738551\pi$
$823$	− 1205.00i	− 1.46416i	0.681221	−	0.732078i	$-0.261449\pi$
$824$	510.000i	0.618932i
$825$	0	0
$826$	450.000	0.544794
$827$	−18.0000	−0.0217654	−0.0108827	−	0.999941i	$-0.503464\pi$
$827$	−18.0000	−0.0217654	−0.0108827	+	0.999941i	$0.503464\pi$
$828$	0	0
$829$	−1442.00	−1.73945	−0.869723	−	0.493541i	$-0.835702\pi$
$829$	−1442.00	−1.73945	−0.869723	+	0.493541i	$0.835702\pi$
$830$	0	0
$831$	0	0
$832$	− 910.000i	− 1.09375i
$833$	−432.000	−0.518607
$834$	0	0
$835$	0	0
$836$	− 1200.00i	− 1.43541i
$837$	0	0
$838$	− 630.000i	− 0.751790i
$839$	1590.00i	1.89511i	0.319588	+	0.947557i	$0.396456\pi$
$839$	1590.00i	1.89511i	−0.319588	+	0.947557i	$0.603544\pi$
$840$	0	0
$841$	−59.0000	−0.0701546
$842$	−1932.00	−2.29454
$843$	0	0
$844$	−1580.00	−1.87204
$845$	0	0
$846$	0	0
$847$	− 520.000i	− 0.613932i
$848$	297.000	0.350236
$849$	0	0
$850$	0	0
$851$	− 240.000i	− 0.282021i
$852$	0	0
$853$	− 590.000i	− 0.691676i	−0.938294	−	0.345838i	$-0.887595\pi$
$853$	− 590.000i	− 0.691676i	0.938294	−	0.345838i	$-0.112405\pi$
$854$	1140.00i	1.33489i
$855$	0	0
$856$	567.000	0.662383
$857$	1302.00	1.51925	0.759627	−	0.650359i	$-0.225382\pi$
$857$	1302.00	1.51925	0.759627	+	0.650359i	$0.225382\pi$
$858$	0	0
$859$	316.000	0.367870	0.183935	−	0.982938i	$-0.441116\pi$
$859$	316.000	0.367870	0.183935	+	0.982938i	$0.441116\pi$
$860$	0	0
$861$	0	0
$862$	− 810.000i	− 0.939675i
$863$	1188.00	1.37659	0.688297	−	0.725429i	$-0.258359\pi$
$863$	1188.00	1.37659	0.688297	+	0.725429i	$0.258359\pi$
$864$	0	0
$865$	0	0
$866$	− 1695.00i	− 1.95727i
$867$	0	0
$868$	− 25.0000i	− 0.0288018i
$869$	210.000i	0.241657i
$870$	0	0
$871$	100.000	0.114811
$872$	492.000	0.564220
$873$	0	0
$874$	576.000	0.659039
$875$	0	0
$876$	0	0
$877$	− 550.000i	− 0.627138i	−0.949565	−	0.313569i	$-0.898475\pi$
$877$	− 550.000i	− 0.627138i	0.949565	−	0.313569i	$-0.101525\pi$
$878$	−633.000	−0.720957
$879$	0	0
$880$	0	0
$881$	− 90.0000i	− 0.102157i	−0.998695	−	0.0510783i	$-0.983734\pi$
$881$	− 90.0000i	− 0.102157i	0.998695	−	0.0510783i	$-0.0162658\pi$
$882$	0	0
$883$	880.000i	0.996602i	0.867004	+	0.498301i	$0.166043\pi$
$883$	880.000i	0.996602i	−0.867004	+	0.498301i	$0.833957\pi$
$884$	− 900.000i	− 1.01810i
$885$	0	0
$886$	−1494.00	−1.68623
$887$	282.000	0.317926	0.158963	−	0.987285i	$-0.449185\pi$
$887$	282.000	0.317926	0.158963	+	0.987285i	$0.449185\pi$
$888$	0	0
$889$	1025.00	1.15298
$890$	0	0
$891$	0	0
$892$	650.000i	0.728700i
$893$	96.0000	0.107503
$894$	0	0
$895$	0	0
$896$	465.000i	0.518973i
$897$	0	0
$898$	− 1080.00i	− 1.20267i
$899$	30.0000i	0.0333704i
$900$	0	0
$901$	486.000	0.539401
$902$	−2700.00	−2.99335
$903$	0	0
$904$	−72.0000	−0.0796460
$905$	0	0
$906$	0	0
$907$	− 1300.00i	− 1.43330i	−0.697435	−	0.716648i	$-0.745675\pi$
$907$	− 1300.00i	− 1.43330i	0.697435	−	0.716648i	$-0.254325\pi$
$908$	210.000	0.231278
$909$	0	0
$910$	0	0
$911$	210.000i	0.230516i	0.993336	+	0.115258i	$0.0367695\pi$
$911$	210.000i	0.230516i	−0.993336	+	0.115258i	$0.963231\pi$
$912$	0	0
$913$	− 45.0000i	− 0.0492881i
$914$	− 1095.00i	− 1.19803i
$915$	0	0
$916$	1130.00	1.23362
$917$	−75.0000	−0.0817884
$918$	0	0
$919$	−137.000	−0.149075	−0.0745375	−	0.997218i	$-0.523748\pi$
$919$	−137.000	−0.149075	−0.0745375	+	0.997218i	$0.523748\pi$
$920$	0	0
$921$	0	0
$922$	315.000i	0.341649i
$923$	900.000	0.975081
$924$	0	0
$925$	0	0
$926$	645.000i	0.696544i
$927$	0	0
$928$	− 1350.00i	− 1.45474i
$929$	− 660.000i	− 0.710441i	−0.934782	−	0.355221i	$-0.884406\pi$
$929$	− 660.000i	− 0.710441i	0.934782	−	0.355221i	$-0.115594\pi$
$930$	0	0
$931$	384.000	0.412460
$932$	1170.00	1.25536
$933$	0	0
$934$	189.000	0.202355
$935$	0	0
$936$	0	0
$937$	605.000i	0.645678i	0.946454	+	0.322839i	$0.104637\pi$
$937$	605.000i	0.645678i	−0.946454	+	0.322839i	$0.895363\pi$
$938$	−150.000	−0.159915
$939$	0	0
$940$	0	0
$941$	− 1605.00i	− 1.70563i	−0.522211	−	0.852816i	$-0.674893\pi$
$941$	− 1605.00i	− 1.70563i	0.522211	−	0.852816i	$-0.325107\pi$
$942$	0	0
$943$	− 720.000i	− 0.763521i
$944$	− 330.000i	− 0.349576i
$945$	0	0
$946$	2250.00	2.37844
$947$	−543.000	−0.573390	−0.286695	−	0.958022i	$-0.592557\pi$
$947$	−543.000	−0.573390	−0.286695	+	0.958022i	$0.592557\pi$
$948$	0	0
$949$	650.000	0.684932
$950$	0	0
$951$	0	0
$952$	270.000i	0.283613i
$953$	144.000	0.151102	0.0755509	−	0.997142i	$-0.475928\pi$
$953$	144.000	0.151102	0.0755509	+	0.997142i	$0.475928\pi$
$954$	0	0
$955$	0	0
$956$	600.000i	0.627615i
$957$	0	0
$958$	2250.00i	2.34864i
$959$	− 690.000i	− 0.719499i
$960$	0	0
$961$	−960.000	−0.998959
$962$	600.000	0.623701
$963$	0	0
$964$	70.0000	0.0726141
$965$	0	0
$966$	0	0
$967$	845.000i	0.873837i	0.899501	+	0.436918i	$0.143930\pi$
$967$	845.000i	0.873837i	−0.899501	+	0.436918i	$0.856070\pi$
$968$	312.000	0.322314
$969$	0	0
$970$	0	0
$971$	405.000i	0.417096i	0.978012	+	0.208548i	$0.0668737\pi$
$971$	405.000i	0.417096i	−0.978012	+	0.208548i	$0.933126\pi$
$972$	0	0
$973$	140.000i	0.143885i
$974$	− 330.000i	− 0.338809i
$975$	0	0
$976$	836.000	0.856557
$977$	246.000	0.251791	0.125896	−	0.992043i	$-0.459820\pi$
$977$	246.000	0.251791	0.125896	+	0.992043i	$0.459820\pi$
$978$	0	0
$979$	−1350.00	−1.37896
$980$	0	0
$981$	0	0
$982$	− 1935.00i	− 1.97047i
$983$	1038.00	1.05595	0.527976	−	0.849260i	$-0.322951\pi$
$983$	1038.00	1.05595	0.527976	+	0.849260i	$0.322951\pi$
$984$	0	0
$985$	0	0
$986$	− 1620.00i	− 1.64300i
$987$	0	0
$988$	800.000i	0.809717i
$989$	600.000i	0.606673i
$990$	0	0
$991$	−1501.00	−1.51463	−0.757316	−	0.653049i	$-0.773490\pi$
$991$	−1501.00	−1.51463	−0.757316	+	0.653049i	$0.773490\pi$
$992$	−45.0000	−0.0453629
$993$	0	0
$994$	−1350.00	−1.35815
$995$	0	0
$996$	0	0
$997$	770.000i	0.772317i	0.922432	+	0.386158i	$0.126198\pi$
$997$	770.000i	0.772317i	−0.922432	+	0.386158i	$0.873802\pi$
$998$	−2298.00	−2.30261
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.3.d.a.674.2		2
3.2	odd	2		675.3.d.d.674.2		2
5.2	odd	4		675.3.c.h.26.1		2
5.3	odd	4		27.3.b.b.26.2	yes	2
5.4	even	2		675.3.d.d.674.1		2
15.2	even	4		675.3.c.h.26.2		2
15.8	even	4		27.3.b.b.26.1	✓	2
15.14	odd	2	inner	675.3.d.a.674.1		2
20.3	even	4		432.3.e.c.161.1		2
40.3	even	4		1728.3.e.g.1025.2		2
40.13	odd	4		1728.3.e.m.1025.2		2
45.13	odd	12		81.3.d.b.26.1		4
45.23	even	12		81.3.d.b.26.2		4
45.38	even	12		81.3.d.b.53.1		4
45.43	odd	12		81.3.d.b.53.2		4
60.23	odd	4		432.3.e.c.161.2		2
120.53	even	4		1728.3.e.m.1025.1		2
120.83	odd	4		1728.3.e.g.1025.1		2
180.23	odd	12		1296.3.q.j.593.2		4
180.43	even	12		1296.3.q.j.1025.2		4
180.83	odd	12		1296.3.q.j.1025.1		4
180.103	even	12		1296.3.q.j.593.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.b.b.26.1	✓	2	15.8	even	4
27.3.b.b.26.2	yes	2	5.3	odd	4
81.3.d.b.26.1		4	45.13	odd	12
81.3.d.b.26.2		4	45.23	even	12
81.3.d.b.53.1		4	45.38	even	12
81.3.d.b.53.2		4	45.43	odd	12
432.3.e.c.161.1		2	20.3	even	4
432.3.e.c.161.2		2	60.23	odd	4
675.3.c.h.26.1		2	5.2	odd	4
675.3.c.h.26.2		2	15.2	even	4
675.3.d.a.674.1		2	15.14	odd	2	inner
675.3.d.a.674.2		2	1.1	even	1	trivial
675.3.d.d.674.1		2	5.4	even	2
675.3.d.d.674.2		2	3.2	odd	2
1296.3.q.j.593.1		4	180.103	even	12
1296.3.q.j.593.2		4	180.23	odd	12
1296.3.q.j.1025.1		4	180.83	odd	12
1296.3.q.j.1025.2		4	180.43	even	12
1728.3.e.g.1025.1		2	120.83	odd	4
1728.3.e.g.1025.2		2	40.3	even	4
1728.3.e.m.1025.1		2	120.53	even	4
1728.3.e.m.1025.2		2	40.13	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	675.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{2} +5.00000 q^{4} +5.00000i q^{7} -3.00000 q^{8} -15.0000i q^{11} +10.0000i q^{13} -15.0000i q^{14} -11.0000 q^{16} -18.0000 q^{17} +16.0000 q^{19} +45.0000i q^{22} -12.0000 q^{23} -30.0000i q^{26} +25.0000i q^{28} -30.0000i q^{29} -1.00000 q^{31} +45.0000 q^{32} +54.0000 q^{34} +20.0000i q^{37} -48.0000 q^{38} +60.0000i q^{41} -50.0000i q^{43} -75.0000i q^{44} +36.0000 q^{46} +6.00000 q^{47} +24.0000 q^{49} +50.0000i q^{52} -27.0000 q^{53} -15.0000i q^{56} +90.0000i q^{58} +30.0000i q^{59} -76.0000 q^{61} +3.00000 q^{62} -91.0000 q^{64} -10.0000i q^{67} -90.0000 q^{68} -90.0000i q^{71} -65.0000i q^{73} -60.0000i q^{74} +80.0000 q^{76} +75.0000 q^{77} -14.0000 q^{79} -180.000i q^{82} +3.00000 q^{83} +150.000i q^{86} +45.0000i q^{88} -90.0000i q^{89} -50.0000 q^{91} -60.0000 q^{92} -18.0000 q^{94} -85.0000i q^{97} -72.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} + 10 q^{4} - 6 q^{8} - 22 q^{16} - 36 q^{17} + 32 q^{19} - 24 q^{23} - 2 q^{31} + 90 q^{32} + 108 q^{34} - 96 q^{38} + 72 q^{46} + 12 q^{47} + 48 q^{49} - 54 q^{53} - 152 q^{61} + 6 q^{62}+ \cdots - 144 q^{98}+O(q^{100}) \) 2 * q - 6 * q^2 + 10 * q^4 - 6 * q^8 - 22 * q^16 - 36 * q^17 + 32 * q^19 - 24 * q^23 - 2 * q^31 + 90 * q^32 + 108 * q^34 - 96 * q^38 + 72 * q^46 + 12 * q^47 + 48 * q^49 - 54 * q^53 - 152 * q^61 + 6 * q^62 - 182 * q^64 - 180 * q^68 + 160 * q^76 + 150 * q^77 - 28 * q^79 + 6 * q^83 - 100 * q^91 - 120 * q^92 - 36 * q^94 - 144 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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