LMFDB - Embedded newform 1380.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.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:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{15}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 15 $ x^2 - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.87298$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -2.87298 q^{11} +4.87298 q^{13} +1.00000 q^{15} -3.87298 q^{17} +4.87298 q^{19} +3.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.87298 q^{29} +3.00000 q^{31} -2.87298 q^{33} +3.00000 q^{35} +1.00000 q^{37} +4.87298 q^{39} +1.87298 q^{41} -11.7460 q^{43} +1.00000 q^{45} +0.872983 q^{47} +2.00000 q^{49} -3.87298 q^{51} +3.87298 q^{53} -2.87298 q^{55} +4.87298 q^{57} -1.87298 q^{59} +1.12702 q^{61} +3.00000 q^{63} +4.87298 q^{65} -4.74597 q^{67} -1.00000 q^{69} +9.61895 q^{71} +4.87298 q^{73} +1.00000 q^{75} -8.61895 q^{77} +4.00000 q^{79} +1.00000 q^{81} -7.87298 q^{83} -3.87298 q^{85} +1.87298 q^{87} -13.7460 q^{89} +14.6190 q^{91} +3.00000 q^{93} +4.87298 q^{95} +8.00000 q^{97} -2.87298 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{19} + 6 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 2 q^{33} + 6 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} - 6 q^{47} + 4 q^{49} + 2 q^{55} + 2 q^{57} + 4 q^{59} + 10 q^{61} + 6 q^{63} + 2 q^{65} + 6 q^{67} - 2 q^{69} - 4 q^{71} + 2 q^{73} + 2 q^{75} + 6 q^{77} + 8 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{87} - 12 q^{89} + 6 q^{91} + 6 q^{93} + 2 q^{95} + 16 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 + 2 * q^19 + 6 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 6 * q^31 + 2 * q^33 + 6 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 - 6 * q^47 + 4 * q^49 + 2 * q^55 + 2 * q^57 + 4 * q^59 + 10 * q^61 + 6 * q^63 + 2 * q^65 + 6 * q^67 - 2 * q^69 - 4 * q^71 + 2 * q^73 + 2 * q^75 + 6 * q^77 + 8 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^87 - 12 * q^89 + 6 * q^91 + 6 * q^93 + 2 * q^95 + 16 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.87298	−0.866237	−0.433119	−	0.901337i	$-0.642587\pi$
$11$	−2.87298	−0.866237	−0.433119	+	0.901337i	$0.642587\pi$
$12$	0	0
$13$	4.87298	1.35152	0.675761	−	0.737121i	$-0.263815\pi$
$13$	4.87298	1.35152	0.675761	+	0.737121i	$0.263815\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.87298	−0.939336	−0.469668	−	0.882843i	$-0.655626\pi$
$17$	−3.87298	−0.939336	−0.469668	+	0.882843i	$0.655626\pi$
$18$	0	0
$19$	4.87298	1.11794	0.558970	−	0.829188i	$-0.311197\pi$
$19$	4.87298	1.11794	0.558970	+	0.829188i	$0.311197\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.87298	0.347804	0.173902	−	0.984763i	$-0.444362\pi$
$29$	1.87298	0.347804	0.173902	+	0.984763i	$0.444362\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	−2.87298	−0.500122
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	0	0
$39$	4.87298	0.780302
$40$	0	0
$41$	1.87298	0.292511	0.146255	−	0.989247i	$-0.453278\pi$
$41$	1.87298	0.292511	0.146255	+	0.989247i	$0.453278\pi$
$42$	0	0
$43$	−11.7460	−1.79124	−0.895622	−	0.444817i	$-0.853269\pi$
$43$	−11.7460	−1.79124	−0.895622	+	0.444817i	$0.853269\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0.872983	0.127338	0.0636689	−	0.997971i	$-0.479720\pi$
$47$	0.872983	0.127338	0.0636689	+	0.997971i	$0.479720\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−3.87298	−0.542326
$52$	0	0
$53$	3.87298	0.531995	0.265998	−	0.963974i	$-0.414299\pi$
$53$	3.87298	0.531995	0.265998	+	0.963974i	$0.414299\pi$
$54$	0	0
$55$	−2.87298	−0.387393
$56$	0	0
$57$	4.87298	0.645442
$58$	0	0
$59$	−1.87298	−0.243842	−0.121921	−	0.992540i	$-0.538905\pi$
$59$	−1.87298	−0.243842	−0.121921	+	0.992540i	$0.538905\pi$
$60$	0	0
$61$	1.12702	0.144300	0.0721498	−	0.997394i	$-0.477014\pi$
$61$	1.12702	0.144300	0.0721498	+	0.997394i	$0.477014\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	4.87298	0.604419
$66$	0	0
$67$	−4.74597	−0.579812	−0.289906	−	0.957055i	$-0.593624\pi$
$67$	−4.74597	−0.579812	−0.289906	+	0.957055i	$0.593624\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	9.61895	1.14156	0.570780	−	0.821103i	$-0.306641\pi$
$71$	9.61895	1.14156	0.570780	+	0.821103i	$0.306641\pi$
$72$	0	0
$73$	4.87298	0.570340	0.285170	−	0.958477i	$-0.407950\pi$
$73$	4.87298	0.570340	0.285170	+	0.958477i	$0.407950\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−8.61895	−0.982221
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.87298	−0.864172	−0.432086	−	0.901832i	$-0.642222\pi$
$83$	−7.87298	−0.864172	−0.432086	+	0.901832i	$0.642222\pi$
$84$	0	0
$85$	−3.87298	−0.420084
$86$	0	0
$87$	1.87298	0.200805
$88$	0	0
$89$	−13.7460	−1.45707	−0.728535	−	0.685009i	$-0.759798\pi$
$89$	−13.7460	−1.45707	−0.728535	+	0.685009i	$0.759798\pi$
$90$	0	0
$91$	14.6190	1.53248
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	4.87298	0.499958
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	−2.87298	−0.288746
$100$	0	0
$101$	3.87298	0.385376	0.192688	−	0.981260i	$-0.438279\pi$
$101$	3.87298	0.385376	0.192688	+	0.981260i	$0.438279\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	17.6190	1.70329	0.851644	−	0.524121i	$-0.175606\pi$
$107$	17.6190	1.70329	0.851644	+	0.524121i	$0.175606\pi$
$108$	0	0
$109$	−6.61895	−0.633980	−0.316990	−	0.948429i	$-0.602672\pi$
$109$	−6.61895	−0.633980	−0.316990	+	0.948429i	$0.602672\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	3.87298	0.364340	0.182170	−	0.983267i	$-0.441688\pi$
$113$	3.87298	0.364340	0.182170	+	0.983267i	$0.441688\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	4.87298	0.450507
$118$	0	0
$119$	−11.6190	−1.06511
$120$	0	0
$121$	−2.74597	−0.249633
$122$	0	0
$123$	1.87298	0.168881
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.6190	1.47469	0.737347	−	0.675515i	$-0.236078\pi$
$127$	16.6190	1.47469	0.737347	+	0.675515i	$0.236078\pi$
$128$	0	0
$129$	−11.7460	−1.03417
$130$	0	0
$131$	−19.4919	−1.70302	−0.851509	−	0.524340i	$-0.824312\pi$
$131$	−19.4919	−1.70302	−0.851509	+	0.524340i	$0.824312\pi$
$132$	0	0
$133$	14.6190	1.26762
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	10.4919	0.889914	0.444957	−	0.895552i	$-0.353219\pi$
$139$	10.4919	0.889914	0.444957	+	0.895552i	$0.353219\pi$
$140$	0	0
$141$	0.872983	0.0735185
$142$	0	0
$143$	−14.0000	−1.17074
$144$	0	0
$145$	1.87298	0.155543
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	−3.12702	−0.256175	−0.128088	−	0.991763i	$-0.540884\pi$
$149$	−3.12702	−0.256175	−0.128088	+	0.991763i	$0.540884\pi$
$150$	0	0
$151$	−11.7460	−0.955873	−0.477937	−	0.878394i	$-0.658615\pi$
$151$	−11.7460	−0.955873	−0.477937	+	0.878394i	$0.658615\pi$
$152$	0	0
$153$	−3.87298	−0.313112
$154$	0	0
$155$	3.00000	0.240966
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	0	0
$159$	3.87298	0.307148
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	0	0
$165$	−2.87298	−0.223661
$166$	0	0
$167$	−16.8730	−1.30567	−0.652835	−	0.757500i	$-0.726421\pi$
$167$	−16.8730	−1.30567	−0.652835	+	0.757500i	$0.726421\pi$
$168$	0	0
$169$	10.7460	0.826613
$170$	0	0
$171$	4.87298	0.372646
$172$	0	0
$173$	−7.49193	−0.569601	−0.284801	−	0.958587i	$-0.591927\pi$
$173$	−7.49193	−0.569601	−0.284801	+	0.958587i	$0.591927\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	0	0
$177$	−1.87298	−0.140782
$178$	0	0
$179$	−0.254033	−0.0189873	−0.00949367	−	0.999955i	$-0.503022\pi$
$179$	−0.254033	−0.0189873	−0.00949367	+	0.999955i	$0.503022\pi$
$180$	0	0
$181$	−6.25403	−0.464859	−0.232429	−	0.972613i	$-0.574667\pi$
$181$	−6.25403	−0.464859	−0.232429	+	0.972613i	$0.574667\pi$
$182$	0	0
$183$	1.12702	0.0833115
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	11.1270	0.813688
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	−11.1270	−0.805123	−0.402561	−	0.915393i	$-0.631880\pi$
$191$	−11.1270	−0.805123	−0.402561	+	0.915393i	$0.631880\pi$
$192$	0	0
$193$	−5.74597	−0.413604	−0.206802	−	0.978383i	$-0.566306\pi$
$193$	−5.74597	−0.413604	−0.206802	+	0.978383i	$0.566306\pi$
$194$	0	0
$195$	4.87298	0.348962
$196$	0	0
$197$	19.7460	1.40684	0.703421	−	0.710774i	$-0.251655\pi$
$197$	19.7460	1.40684	0.703421	+	0.710774i	$0.251655\pi$
$198$	0	0
$199$	−27.2379	−1.93084	−0.965422	−	0.260693i	$-0.916049\pi$
$199$	−27.2379	−1.93084	−0.965422	+	0.260693i	$0.916049\pi$
$200$	0	0
$201$	−4.74597	−0.334755
$202$	0	0
$203$	5.61895	0.394373
$204$	0	0
$205$	1.87298	0.130815
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−14.0000	−0.968400
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	0	0
$213$	9.61895	0.659080
$214$	0	0
$215$	−11.7460	−0.801068
$216$	0	0
$217$	9.00000	0.610960
$218$	0	0
$219$	4.87298	0.329286
$220$	0	0
$221$	−18.8730	−1.26953
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−23.4919	−1.55921	−0.779607	−	0.626269i	$-0.784581\pi$
$227$	−23.4919	−1.55921	−0.779607	+	0.626269i	$0.784581\pi$
$228$	0	0
$229$	9.74597	0.644032	0.322016	−	0.946734i	$-0.395640\pi$
$229$	9.74597	0.644032	0.322016	+	0.946734i	$0.395640\pi$
$230$	0	0
$231$	−8.61895	−0.567085
$232$	0	0
$233$	−15.4919	−1.01491	−0.507455	−	0.861678i	$-0.669414\pi$
$233$	−15.4919	−1.01491	−0.507455	+	0.861678i	$0.669414\pi$
$234$	0	0
$235$	0.872983	0.0569472
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	−10.1270	−0.655062	−0.327531	−	0.944840i	$-0.606217\pi$
$239$	−10.1270	−0.655062	−0.327531	+	0.944840i	$0.606217\pi$
$240$	0	0
$241$	−12.8730	−0.829222	−0.414611	−	0.909999i	$-0.636082\pi$
$241$	−12.8730	−0.829222	−0.414611	+	0.909999i	$0.636082\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	23.7460	1.51092
$248$	0	0
$249$	−7.87298	−0.498930
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	2.87298	0.180623
$254$	0	0
$255$	−3.87298	−0.242536
$256$	0	0
$257$	6.61895	0.412879	0.206439	−	0.978459i	$-0.433812\pi$
$257$	6.61895	0.412879	0.206439	+	0.978459i	$0.433812\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	1.87298	0.115935
$262$	0	0
$263$	−14.1270	−0.871109	−0.435555	−	0.900162i	$-0.643448\pi$
$263$	−14.1270	−0.871109	−0.435555	+	0.900162i	$0.643448\pi$
$264$	0	0
$265$	3.87298	0.237915
$266$	0	0
$267$	−13.7460	−0.841239
$268$	0	0
$269$	15.3649	0.936816	0.468408	−	0.883512i	$-0.344828\pi$
$269$	15.3649	0.936816	0.468408	+	0.883512i	$0.344828\pi$
$270$	0	0
$271$	−22.7460	−1.38172	−0.690860	−	0.722989i	$-0.742768\pi$
$271$	−22.7460	−1.38172	−0.690860	+	0.722989i	$0.742768\pi$
$272$	0	0
$273$	14.6190	0.884779
$274$	0	0
$275$	−2.87298	−0.173247
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	3.00000	0.179605
$280$	0	0
$281$	22.3649	1.33418	0.667090	−	0.744978i	$-0.267540\pi$
$281$	22.3649	1.33418	0.667090	+	0.744978i	$0.267540\pi$
$282$	0	0
$283$	−24.2379	−1.44079	−0.720397	−	0.693562i	$-0.756040\pi$
$283$	−24.2379	−1.44079	−0.720397	+	0.693562i	$0.756040\pi$
$284$	0	0
$285$	4.87298	0.288651
$286$	0	0
$287$	5.61895	0.331676
$288$	0	0
$289$	−2.00000	−0.117647
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	9.61895	0.561945	0.280973	−	0.959716i	$-0.409343\pi$
$293$	9.61895	0.561945	0.280973	+	0.959716i	$0.409343\pi$
$294$	0	0
$295$	−1.87298	−0.109049
$296$	0	0
$297$	−2.87298	−0.166707
$298$	0	0
$299$	−4.87298	−0.281812
$300$	0	0
$301$	−35.2379	−2.03108
$302$	0	0
$303$	3.87298	0.222497
$304$	0	0
$305$	1.12702	0.0645328
$306$	0	0
$307$	16.8730	0.962992	0.481496	−	0.876448i	$-0.340094\pi$
$307$	16.8730	0.962992	0.481496	+	0.876448i	$0.340094\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	7.00000	0.395663	0.197832	−	0.980236i	$-0.436610\pi$
$313$	7.00000	0.395663	0.197832	+	0.980236i	$0.436610\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−16.3649	−0.919145	−0.459573	−	0.888140i	$-0.651997\pi$
$317$	−16.3649	−0.919145	−0.459573	+	0.888140i	$0.651997\pi$
$318$	0	0
$319$	−5.38105	−0.301281
$320$	0	0
$321$	17.6190	0.983394
$322$	0	0
$323$	−18.8730	−1.05012
$324$	0	0
$325$	4.87298	0.270304
$326$	0	0
$327$	−6.61895	−0.366029
$328$	0	0
$329$	2.61895	0.144387
$330$	0	0
$331$	24.2379	1.33224	0.666118	−	0.745847i	$-0.267955\pi$
$331$	24.2379	1.33224	0.666118	+	0.745847i	$0.267955\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	−4.74597	−0.259300
$336$	0	0
$337$	29.4919	1.60653	0.803264	−	0.595623i	$-0.203095\pi$
$337$	29.4919	1.60653	0.803264	+	0.595623i	$0.203095\pi$
$338$	0	0
$339$	3.87298	0.210352
$340$	0	0
$341$	−8.61895	−0.466742
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−10.2540	−0.550465	−0.275233	−	0.961378i	$-0.588755\pi$
$347$	−10.2540	−0.550465	−0.275233	+	0.961378i	$0.588755\pi$
$348$	0	0
$349$	0.745967	0.0399307	0.0199653	−	0.999801i	$-0.493644\pi$
$349$	0.745967	0.0399307	0.0199653	+	0.999801i	$0.493644\pi$
$350$	0	0
$351$	4.87298	0.260101
$352$	0	0
$353$	−8.87298	−0.472261	−0.236131	−	0.971721i	$-0.575879\pi$
$353$	−8.87298	−0.472261	−0.236131	+	0.971721i	$0.575879\pi$
$354$	0	0
$355$	9.61895	0.510521
$356$	0	0
$357$	−11.6190	−0.614940
$358$	0	0
$359$	−28.8730	−1.52386	−0.761929	−	0.647661i	$-0.775747\pi$
$359$	−28.8730	−1.52386	−0.761929	+	0.647661i	$0.775747\pi$
$360$	0	0
$361$	4.74597	0.249788
$362$	0	0
$363$	−2.74597	−0.144126
$364$	0	0
$365$	4.87298	0.255064
$366$	0	0
$367$	−35.9839	−1.87834	−0.939171	−	0.343449i	$-0.888405\pi$
$367$	−35.9839	−1.87834	−0.939171	+	0.343449i	$0.888405\pi$
$368$	0	0
$369$	1.87298	0.0975036
$370$	0	0
$371$	11.6190	0.603226
$372$	0	0
$373$	−13.7460	−0.711739	−0.355870	−	0.934536i	$-0.615815\pi$
$373$	−13.7460	−0.711739	−0.355870	+	0.934536i	$0.615815\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	9.12702	0.470065
$378$	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
$381$	16.6190	0.851415
$382$	0	0
$383$	−2.38105	−0.121666	−0.0608330	−	0.998148i	$-0.519376\pi$
$383$	−2.38105	−0.121666	−0.0608330	+	0.998148i	$0.519376\pi$
$384$	0	0
$385$	−8.61895	−0.439262
$386$	0	0
$387$	−11.7460	−0.597081
$388$	0	0
$389$	−11.7460	−0.595544	−0.297772	−	0.954637i	$-0.596244\pi$
$389$	−11.7460	−0.595544	−0.297772	+	0.954637i	$0.596244\pi$
$390$	0	0
$391$	3.87298	0.195865
$392$	0	0
$393$	−19.4919	−0.983238
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	−11.4919	−0.576764	−0.288382	−	0.957516i	$-0.593117\pi$
$397$	−11.4919	−0.576764	−0.288382	+	0.957516i	$0.593117\pi$
$398$	0	0
$399$	14.6190	0.731863
$400$	0	0
$401$	−19.2379	−0.960695	−0.480347	−	0.877078i	$-0.659489\pi$
$401$	−19.2379	−0.960695	−0.480347	+	0.877078i	$0.659489\pi$
$402$	0	0
$403$	14.6190	0.728222
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−2.87298	−0.142408
$408$	0	0
$409$	−24.2379	−1.19849	−0.599244	−	0.800567i	$-0.704532\pi$
$409$	−24.2379	−1.19849	−0.599244	+	0.800567i	$0.704532\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	−5.61895	−0.276490
$414$	0	0
$415$	−7.87298	−0.386470
$416$	0	0
$417$	10.4919	0.513792
$418$	0	0
$419$	−20.6190	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$419$	−20.6190	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$420$	0	0
$421$	−31.1270	−1.51704	−0.758519	−	0.651651i	$-0.774077\pi$
$421$	−31.1270	−1.51704	−0.758519	+	0.651651i	$0.774077\pi$
$422$	0	0
$423$	0.872983	0.0424459
$424$	0	0
$425$	−3.87298	−0.187867
$426$	0	0
$427$	3.38105	0.163620
$428$	0	0
$429$	−14.0000	−0.675926
$430$	0	0
$431$	−9.74597	−0.469447	−0.234723	−	0.972062i	$-0.575418\pi$
$431$	−9.74597	−0.469447	−0.234723	+	0.972062i	$0.575418\pi$
$432$	0	0
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	1.87298	0.0898027
$436$	0	0
$437$	−4.87298	−0.233106
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−1.38105	−0.0656157	−0.0328078	−	0.999462i	$-0.510445\pi$
$443$	−1.38105	−0.0656157	−0.0328078	+	0.999462i	$0.510445\pi$
$444$	0	0
$445$	−13.7460	−0.651621
$446$	0	0
$447$	−3.12702	−0.147903
$448$	0	0
$449$	−14.1270	−0.666695	−0.333348	−	0.942804i	$-0.608178\pi$
$449$	−14.1270	−0.666695	−0.333348	+	0.942804i	$0.608178\pi$
$450$	0	0
$451$	−5.38105	−0.253384
$452$	0	0
$453$	−11.7460	−0.551874
$454$	0	0
$455$	14.6190	0.685347
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	−3.87298	−0.180775
$460$	0	0
$461$	16.2540	0.757026	0.378513	−	0.925596i	$-0.376436\pi$
$461$	16.2540	0.757026	0.378513	+	0.925596i	$0.376436\pi$
$462$	0	0
$463$	27.1270	1.26070	0.630350	−	0.776311i	$-0.282912\pi$
$463$	27.1270	1.26070	0.630350	+	0.776311i	$0.282912\pi$
$464$	0	0
$465$	3.00000	0.139122
$466$	0	0
$467$	17.6190	0.815308	0.407654	−	0.913137i	$-0.366347\pi$
$467$	17.6190	0.815308	0.407654	+	0.913137i	$0.366347\pi$
$468$	0	0
$469$	−14.2379	−0.657445
$470$	0	0
$471$	17.0000	0.783319
$472$	0	0
$473$	33.7460	1.55164
$474$	0	0
$475$	4.87298	0.223588
$476$	0	0
$477$	3.87298	0.177332
$478$	0	0
$479$	−27.1270	−1.23947	−0.619733	−	0.784813i	$-0.712759\pi$
$479$	−27.1270	−1.23947	−0.619733	+	0.784813i	$0.712759\pi$
$480$	0	0
$481$	4.87298	0.222189
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	8.00000	0.363261
$486$	0	0
$487$	28.1109	1.27383	0.636913	−	0.770936i	$-0.280211\pi$
$487$	28.1109	1.27383	0.636913	+	0.770936i	$0.280211\pi$
$488$	0	0
$489$	−22.0000	−0.994874
$490$	0	0
$491$	35.8730	1.61893	0.809463	−	0.587172i	$-0.199759\pi$
$491$	35.8730	1.61893	0.809463	+	0.587172i	$0.199759\pi$
$492$	0	0
$493$	−7.25403	−0.326705
$494$	0	0
$495$	−2.87298	−0.129131
$496$	0	0
$497$	28.8569	1.29441
$498$	0	0
$499$	18.7460	0.839185	0.419592	−	0.907713i	$-0.362173\pi$
$499$	18.7460	0.839185	0.419592	+	0.907713i	$0.362173\pi$
$500$	0	0
$501$	−16.8730	−0.753829
$502$	0	0
$503$	17.1109	0.762937	0.381468	−	0.924382i	$-0.375419\pi$
$503$	17.1109	0.762937	0.381468	+	0.924382i	$0.375419\pi$
$504$	0	0
$505$	3.87298	0.172345
$506$	0	0
$507$	10.7460	0.477245
$508$	0	0
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	14.6190	0.646704
$512$	0	0
$513$	4.87298	0.215147
$514$	0	0
$515$	−6.00000	−0.264392
$516$	0	0
$517$	−2.50807	−0.110305
$518$	0	0
$519$	−7.49193	−0.328859
$520$	0	0
$521$	33.1270	1.45132	0.725660	−	0.688053i	$-0.241534\pi$
$521$	33.1270	1.45132	0.725660	+	0.688053i	$0.241534\pi$
$522$	0	0
$523$	42.9839	1.87955	0.939777	−	0.341789i	$-0.111033\pi$
$523$	42.9839	1.87955	0.939777	+	0.341789i	$0.111033\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	−11.6190	−0.506129
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.87298	−0.0812806
$532$	0	0
$533$	9.12702	0.395335
$534$	0	0
$535$	17.6190	0.761734
$536$	0	0
$537$	−0.254033	−0.0109623
$538$	0	0
$539$	−5.74597	−0.247496
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	0	0
$543$	−6.25403	−0.268386
$544$	0	0
$545$	−6.61895	−0.283525
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	0	0
$549$	1.12702	0.0480999
$550$	0	0
$551$	9.12702	0.388824
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	1.00000	0.0424476
$556$	0	0
$557$	35.8730	1.51999	0.759994	−	0.649931i	$-0.225202\pi$
$557$	35.8730	1.51999	0.759994	+	0.649931i	$0.225202\pi$
$558$	0	0
$559$	−57.2379	−2.42091
$560$	0	0
$561$	11.1270	0.469783
$562$	0	0
$563$	−21.1109	−0.889718	−0.444859	−	0.895601i	$-0.646746\pi$
$563$	−21.1109	−0.889718	−0.444859	+	0.895601i	$0.646746\pi$
$564$	0	0
$565$	3.87298	0.162938
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	19.7460	0.827794	0.413897	−	0.910324i	$-0.364167\pi$
$569$	19.7460	0.827794	0.413897	+	0.910324i	$0.364167\pi$
$570$	0	0
$571$	−2.36492	−0.0989687	−0.0494843	−	0.998775i	$-0.515758\pi$
$571$	−2.36492	−0.0989687	−0.0494843	+	0.998775i	$0.515758\pi$
$572$	0	0
$573$	−11.1270	−0.464838
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	0	0
$579$	−5.74597	−0.238794
$580$	0	0
$581$	−23.6190	−0.979879
$582$	0	0
$583$	−11.1270	−0.460834
$584$	0	0
$585$	4.87298	0.201473
$586$	0	0
$587$	−17.4919	−0.721969	−0.360985	−	0.932572i	$-0.617559\pi$
$587$	−17.4919	−0.721969	−0.360985	+	0.932572i	$0.617559\pi$
$588$	0	0
$589$	14.6190	0.602363
$590$	0	0
$591$	19.7460	0.812241
$592$	0	0
$593$	21.3810	0.878014	0.439007	−	0.898484i	$-0.355330\pi$
$593$	21.3810	0.878014	0.439007	+	0.898484i	$0.355330\pi$
$594$	0	0
$595$	−11.6190	−0.476331
$596$	0	0
$597$	−27.2379	−1.11477
$598$	0	0
$599$	6.25403	0.255533	0.127766	−	0.991804i	$-0.459219\pi$
$599$	6.25403	0.255533	0.127766	+	0.991804i	$0.459219\pi$
$600$	0	0
$601$	2.74597	0.112010	0.0560052	−	0.998430i	$-0.482164\pi$
$601$	2.74597	0.112010	0.0560052	+	0.998430i	$0.482164\pi$
$602$	0	0
$603$	−4.74597	−0.193271
$604$	0	0
$605$	−2.74597	−0.111639
$606$	0	0
$607$	38.8730	1.57781	0.788903	−	0.614518i	$-0.210649\pi$
$607$	38.8730	1.57781	0.788903	+	0.614518i	$0.210649\pi$
$608$	0	0
$609$	5.61895	0.227691
$610$	0	0
$611$	4.25403	0.172100
$612$	0	0
$613$	−12.9839	−0.524413	−0.262207	−	0.965012i	$-0.584450\pi$
$613$	−12.9839	−0.524413	−0.262207	+	0.965012i	$0.584450\pi$
$614$	0	0
$615$	1.87298	0.0755260
$616$	0	0
$617$	5.61895	0.226210	0.113105	−	0.993583i	$-0.463920\pi$
$617$	5.61895	0.226210	0.113105	+	0.993583i	$0.463920\pi$
$618$	0	0
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−41.2379	−1.65216
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−14.0000	−0.559106
$628$	0	0
$629$	−3.87298	−0.154426
$630$	0	0
$631$	−6.36492	−0.253383	−0.126692	−	0.991942i	$-0.540436\pi$
$631$	−6.36492	−0.253383	−0.126692	+	0.991942i	$0.540436\pi$
$632$	0	0
$633$	1.00000	0.0397464
$634$	0	0
$635$	16.6190	0.659503
$636$	0	0
$637$	9.74597	0.386149
$638$	0	0
$639$	9.61895	0.380520
$640$	0	0
$641$	44.1109	1.74228	0.871138	−	0.491039i	$-0.163383\pi$
$641$	44.1109	1.74228	0.871138	+	0.491039i	$0.163383\pi$
$642$	0	0
$643$	8.49193	0.334889	0.167445	−	0.985881i	$-0.446448\pi$
$643$	8.49193	0.334889	0.167445	+	0.985881i	$0.446448\pi$
$644$	0	0
$645$	−11.7460	−0.462497
$646$	0	0
$647$	−41.8569	−1.64556	−0.822781	−	0.568358i	$-0.807579\pi$
$647$	−41.8569	−1.64556	−0.822781	+	0.568358i	$0.807579\pi$
$648$	0	0
$649$	5.38105	0.211225
$650$	0	0
$651$	9.00000	0.352738
$652$	0	0
$653$	44.1109	1.72619	0.863096	−	0.505040i	$-0.168522\pi$
$653$	44.1109	1.72619	0.863096	+	0.505040i	$0.168522\pi$
$654$	0	0
$655$	−19.4919	−0.761613
$656$	0	0
$657$	4.87298	0.190113
$658$	0	0
$659$	21.1270	0.822992	0.411496	−	0.911412i	$-0.365006\pi$
$659$	21.1270	0.822992	0.411496	+	0.911412i	$0.365006\pi$
$660$	0	0
$661$	3.23790	0.125940	0.0629699	−	0.998015i	$-0.479943\pi$
$661$	3.23790	0.125940	0.0629699	+	0.998015i	$0.479943\pi$
$662$	0	0
$663$	−18.8730	−0.732966
$664$	0	0
$665$	14.6190	0.566899
$666$	0	0
$667$	−1.87298	−0.0725222
$668$	0	0
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	−3.23790	−0.124998
$672$	0	0
$673$	26.8730	1.03588	0.517939	−	0.855418i	$-0.326700\pi$
$673$	26.8730	1.03588	0.517939	+	0.855418i	$0.326700\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	47.1109	1.81062	0.905309	−	0.424753i	$-0.139639\pi$
$677$	47.1109	1.81062	0.905309	+	0.424753i	$0.139639\pi$
$678$	0	0
$679$	24.0000	0.921035
$680$	0	0
$681$	−23.4919	−0.900213
$682$	0	0
$683$	24.8730	0.951738	0.475869	−	0.879516i	$-0.342134\pi$
$683$	24.8730	0.951738	0.475869	+	0.879516i	$0.342134\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	0	0
$687$	9.74597	0.371832
$688$	0	0
$689$	18.8730	0.719003
$690$	0	0
$691$	−30.7298	−1.16902	−0.584509	−	0.811387i	$-0.698713\pi$
$691$	−30.7298	−1.16902	−0.584509	+	0.811387i	$0.698713\pi$
$692$	0	0
$693$	−8.61895	−0.327407
$694$	0	0
$695$	10.4919	0.397982
$696$	0	0
$697$	−7.25403	−0.274766
$698$	0	0
$699$	−15.4919	−0.585959
$700$	0	0
$701$	−39.8569	−1.50537	−0.752686	−	0.658379i	$-0.771242\pi$
$701$	−39.8569	−1.50537	−0.752686	+	0.658379i	$0.771242\pi$
$702$	0	0
$703$	4.87298	0.183788
$704$	0	0
$705$	0.872983	0.0328785
$706$	0	0
$707$	11.6190	0.436976
$708$	0	0
$709$	16.6190	0.624138	0.312069	−	0.950059i	$-0.398978\pi$
$709$	16.6190	0.624138	0.312069	+	0.950059i	$0.398978\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−3.00000	−0.112351
$714$	0	0
$715$	−14.0000	−0.523570
$716$	0	0
$717$	−10.1270	−0.378200
$718$	0	0
$719$	−0.635083	−0.0236846	−0.0118423	−	0.999930i	$-0.503770\pi$
$719$	−0.635083	−0.0236846	−0.0118423	+	0.999930i	$0.503770\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	−12.8730	−0.478751
$724$	0	0
$725$	1.87298	0.0695609
$726$	0	0
$727$	26.7460	0.991953	0.495976	−	0.868336i	$-0.334810\pi$
$727$	26.7460	0.991953	0.495976	+	0.868336i	$0.334810\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	45.4919	1.68258
$732$	0	0
$733$	7.25403	0.267934	0.133967	−	0.990986i	$-0.457228\pi$
$733$	7.25403	0.267934	0.133967	+	0.990986i	$0.457228\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	13.6351	0.502255
$738$	0	0
$739$	28.7460	1.05744	0.528719	−	0.848797i	$-0.322673\pi$
$739$	28.7460	1.05744	0.528719	+	0.848797i	$0.322673\pi$
$740$	0	0
$741$	23.7460	0.872330
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−3.12702	−0.114565
$746$	0	0
$747$	−7.87298	−0.288057
$748$	0	0
$749$	52.8569	1.93135
$750$	0	0
$751$	39.8569	1.45440	0.727199	−	0.686427i	$-0.240822\pi$
$751$	39.8569	1.45440	0.727199	+	0.686427i	$0.240822\pi$
$752$	0	0
$753$	2.00000	0.0728841
$754$	0	0
$755$	−11.7460	−0.427479
$756$	0	0
$757$	10.7460	0.390569	0.195284	−	0.980747i	$-0.437437\pi$
$757$	10.7460	0.390569	0.195284	+	0.980747i	$0.437437\pi$
$758$	0	0
$759$	2.87298	0.104283
$760$	0	0
$761$	54.8569	1.98856	0.994280	−	0.106808i	$-0.0340631\pi$
$761$	54.8569	1.98856	0.994280	+	0.106808i	$0.0340631\pi$
$762$	0	0
$763$	−19.8569	−0.718866
$764$	0	0
$765$	−3.87298	−0.140028
$766$	0	0
$767$	−9.12702	−0.329557
$768$	0	0
$769$	21.1270	0.761860	0.380930	−	0.924604i	$-0.375604\pi$
$769$	21.1270	0.761860	0.380930	+	0.924604i	$0.375604\pi$
$770$	0	0
$771$	6.61895	0.238376
$772$	0	0
$773$	19.7460	0.710213	0.355107	−	0.934826i	$-0.384445\pi$
$773$	19.7460	0.710213	0.355107	+	0.934826i	$0.384445\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	0	0
$777$	3.00000	0.107624
$778$	0	0
$779$	9.12702	0.327009
$780$	0	0
$781$	−27.6351	−0.988861
$782$	0	0
$783$	1.87298	0.0669350
$784$	0	0
$785$	17.0000	0.606756
$786$	0	0
$787$	25.0000	0.891154	0.445577	−	0.895244i	$-0.352999\pi$
$787$	25.0000	0.891154	0.445577	+	0.895244i	$0.352999\pi$
$788$	0	0
$789$	−14.1270	−0.502935
$790$	0	0
$791$	11.6190	0.413122
$792$	0	0
$793$	5.49193	0.195024
$794$	0	0
$795$	3.87298	0.137361
$796$	0	0
$797$	4.12702	0.146186	0.0730932	−	0.997325i	$-0.476713\pi$
$797$	4.12702	0.146186	0.0730932	+	0.997325i	$0.476713\pi$
$798$	0	0
$799$	−3.38105	−0.119613
$800$	0	0
$801$	−13.7460	−0.485690
$802$	0	0
$803$	−14.0000	−0.494049
$804$	0	0
$805$	−3.00000	−0.105736
$806$	0	0
$807$	15.3649	0.540871
$808$	0	0
$809$	−44.8569	−1.57708	−0.788541	−	0.614982i	$-0.789163\pi$
$809$	−44.8569	−1.57708	−0.788541	+	0.614982i	$0.789163\pi$
$810$	0	0
$811$	−2.49193	−0.0875036	−0.0437518	−	0.999042i	$-0.513931\pi$
$811$	−2.49193	−0.0875036	−0.0437518	+	0.999042i	$0.513931\pi$
$812$	0	0
$813$	−22.7460	−0.797736
$814$	0	0
$815$	−22.0000	−0.770626
$816$	0	0
$817$	−57.2379	−2.00250
$818$	0	0
$819$	14.6190	0.510827
$820$	0	0
$821$	27.4919	0.959475	0.479738	−	0.877412i	$-0.340732\pi$
$821$	27.4919	0.959475	0.479738	+	0.877412i	$0.340732\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	−2.87298	−0.100024
$826$	0	0
$827$	39.1109	1.36002	0.680009	−	0.733203i	$-0.261976\pi$
$827$	39.1109	1.36002	0.680009	+	0.733203i	$0.261976\pi$
$828$	0	0
$829$	34.4919	1.19795	0.598977	−	0.800766i	$-0.295574\pi$
$829$	34.4919	1.19795	0.598977	+	0.800766i	$0.295574\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	−7.74597	−0.268382
$834$	0	0
$835$	−16.8730	−0.583914
$836$	0	0
$837$	3.00000	0.103695
$838$	0	0
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	−25.4919	−0.879032
$842$	0	0
$843$	22.3649	0.770289
$844$	0	0
$845$	10.7460	0.369672
$846$	0	0
$847$	−8.23790	−0.283058
$848$	0	0
$849$	−24.2379	−0.831843
$850$	0	0
$851$	−1.00000	−0.0342796
$852$	0	0
$853$	−28.2540	−0.967400	−0.483700	−	0.875234i	$-0.660707\pi$
$853$	−28.2540	−0.967400	−0.483700	+	0.875234i	$0.660707\pi$
$854$	0	0
$855$	4.87298	0.166653
$856$	0	0
$857$	−45.4919	−1.55397	−0.776987	−	0.629516i	$-0.783253\pi$
$857$	−45.4919	−1.55397	−0.776987	+	0.629516i	$0.783253\pi$
$858$	0	0
$859$	29.0000	0.989467	0.494734	−	0.869045i	$-0.335266\pi$
$859$	29.0000	0.989467	0.494734	+	0.869045i	$0.335266\pi$
$860$	0	0
$861$	5.61895	0.191493
$862$	0	0
$863$	23.7460	0.808322	0.404161	−	0.914688i	$-0.367564\pi$
$863$	23.7460	0.808322	0.404161	+	0.914688i	$0.367564\pi$
$864$	0	0
$865$	−7.49193	−0.254733
$866$	0	0
$867$	−2.00000	−0.0679236
$868$	0	0
$869$	−11.4919	−0.389837
$870$	0	0
$871$	−23.1270	−0.783629
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	−22.7298	−0.767532	−0.383766	−	0.923430i	$-0.625373\pi$
$877$	−22.7298	−0.767532	−0.383766	+	0.923430i	$0.625373\pi$
$878$	0	0
$879$	9.61895	0.324439
$880$	0	0
$881$	−50.1109	−1.68828	−0.844139	−	0.536124i	$-0.819888\pi$
$881$	−50.1109	−1.68828	−0.844139	+	0.536124i	$0.819888\pi$
$882$	0	0
$883$	−45.3488	−1.52611	−0.763054	−	0.646335i	$-0.776301\pi$
$883$	−45.3488	−1.52611	−0.763054	+	0.646335i	$0.776301\pi$
$884$	0	0
$885$	−1.87298	−0.0629596
$886$	0	0
$887$	4.25403	0.142836	0.0714182	−	0.997446i	$-0.477248\pi$
$887$	4.25403	0.142836	0.0714182	+	0.997446i	$0.477248\pi$
$888$	0	0
$889$	49.8569	1.67215
$890$	0	0
$891$	−2.87298	−0.0962486
$892$	0	0
$893$	4.25403	0.142356
$894$	0	0
$895$	−0.254033	−0.00849140
$896$	0	0
$897$	−4.87298	−0.162704
$898$	0	0
$899$	5.61895	0.187402
$900$	0	0
$901$	−15.0000	−0.499722
$902$	0	0
$903$	−35.2379	−1.17264
$904$	0	0
$905$	−6.25403	−0.207891
$906$	0	0
$907$	15.2540	0.506502	0.253251	−	0.967401i	$-0.418500\pi$
$907$	15.2540	0.506502	0.253251	+	0.967401i	$0.418500\pi$
$908$	0	0
$909$	3.87298	0.128459
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	22.6190	0.748578
$914$	0	0
$915$	1.12702	0.0372580
$916$	0	0
$917$	−58.4758	−1.93104
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	16.8730	0.555984
$922$	0	0
$923$	46.8730	1.54284
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−6.00000	−0.197066
$928$	0	0
$929$	−7.36492	−0.241635	−0.120818	−	0.992675i	$-0.538552\pi$
$929$	−7.36492	−0.241635	−0.120818	+	0.992675i	$0.538552\pi$
$930$	0	0
$931$	9.74597	0.319411
$932$	0	0
$933$	−4.00000	−0.130954
$934$	0	0
$935$	11.1270	0.363892
$936$	0	0
$937$	−38.4758	−1.25695	−0.628475	−	0.777830i	$-0.716320\pi$
$937$	−38.4758	−1.25695	−0.628475	+	0.777830i	$0.716320\pi$
$938$	0	0
$939$	7.00000	0.228436
$940$	0	0
$941$	38.1109	1.24238	0.621190	−	0.783660i	$-0.286650\pi$
$941$	38.1109	1.24238	0.621190	+	0.783660i	$0.286650\pi$
$942$	0	0
$943$	−1.87298	−0.0609927
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	21.2379	0.690139	0.345070	−	0.938577i	$-0.387855\pi$
$947$	21.2379	0.690139	0.345070	+	0.938577i	$0.387855\pi$
$948$	0	0
$949$	23.7460	0.770827
$950$	0	0
$951$	−16.3649	−0.530669
$952$	0	0
$953$	−52.9839	−1.71632	−0.858158	−	0.513386i	$-0.828391\pi$
$953$	−52.9839	−1.71632	−0.858158	+	0.513386i	$0.828391\pi$
$954$	0	0
$955$	−11.1270	−0.360062
$956$	0	0
$957$	−5.38105	−0.173945
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	17.6190	0.567763
$964$	0	0
$965$	−5.74597	−0.184969
$966$	0	0
$967$	44.6190	1.43485	0.717424	−	0.696636i	$-0.245321\pi$
$967$	44.6190	1.43485	0.717424	+	0.696636i	$0.245321\pi$
$968$	0	0
$969$	−18.8730	−0.606288
$970$	0	0
$971$	38.2540	1.22763	0.613815	−	0.789450i	$-0.289634\pi$
$971$	38.2540	1.22763	0.613815	+	0.789450i	$0.289634\pi$
$972$	0	0
$973$	31.4758	1.00907
$974$	0	0
$975$	4.87298	0.156060
$976$	0	0
$977$	−44.1270	−1.41175	−0.705874	−	0.708337i	$-0.749446\pi$
$977$	−44.1270	−1.41175	−0.705874	+	0.708337i	$0.749446\pi$
$978$	0	0
$979$	39.4919	1.26217
$980$	0	0
$981$	−6.61895	−0.211327
$982$	0	0
$983$	−31.3649	−1.00039	−0.500193	−	0.865914i	$-0.666738\pi$
$983$	−31.3649	−1.00039	−0.500193	+	0.865914i	$0.666738\pi$
$984$	0	0
$985$	19.7460	0.629159
$986$	0	0
$987$	2.61895	0.0833621
$988$	0	0
$989$	11.7460	0.373500
$990$	0	0
$991$	−42.4919	−1.34980	−0.674900	−	0.737909i	$-0.735813\pi$
$991$	−42.4919	−1.34980	−0.674900	+	0.737909i	$0.735813\pi$
$992$	0	0
$993$	24.2379	0.769167
$994$	0	0
$995$	−27.2379	−0.863499
$996$	0	0
$997$	−26.2540	−0.831474	−0.415737	−	0.909485i	$-0.636476\pi$
$997$	−26.2540	−0.831474	−0.415737	+	0.909485i	$0.636476\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.a.i.1.1	✓	2
3.2	odd	2		4140.2.a.p.1.2		2
4.3	odd	2		5520.2.a.bj.1.2		2
5.2	odd	4		6900.2.f.o.6349.1		4
5.3	odd	4		6900.2.f.o.6349.3		4
5.4	even	2		6900.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.i.1.1	✓	2	1.1	even	1	trivial
4140.2.a.p.1.2		2	3.2	odd	2
5520.2.a.bj.1.2		2	4.3	odd	2
6900.2.a.j.1.1		2	5.4	even	2
6900.2.f.o.6349.1		4	5.2	odd	4
6900.2.f.o.6349.3		4	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -2.87298 q^{11} +4.87298 q^{13} +1.00000 q^{15} -3.87298 q^{17} +4.87298 q^{19} +3.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.87298 q^{29} +3.00000 q^{31} -2.87298 q^{33} +3.00000 q^{35} +1.00000 q^{37} +4.87298 q^{39} +1.87298 q^{41} -11.7460 q^{43} +1.00000 q^{45} +0.872983 q^{47} +2.00000 q^{49} -3.87298 q^{51} +3.87298 q^{53} -2.87298 q^{55} +4.87298 q^{57} -1.87298 q^{59} +1.12702 q^{61} +3.00000 q^{63} +4.87298 q^{65} -4.74597 q^{67} -1.00000 q^{69} +9.61895 q^{71} +4.87298 q^{73} +1.00000 q^{75} -8.61895 q^{77} +4.00000 q^{79} +1.00000 q^{81} -7.87298 q^{83} -3.87298 q^{85} +1.87298 q^{87} -13.7460 q^{89} +14.6190 q^{91} +3.00000 q^{93} +4.87298 q^{95} +8.00000 q^{97} -2.87298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{19} + 6 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 2 q^{33} + 6 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} - 6 q^{47} + 4 q^{49} + 2 q^{55} + 2 q^{57} + 4 q^{59} + 10 q^{61} + 6 q^{63} + 2 q^{65} + 6 q^{67} - 2 q^{69} - 4 q^{71} + 2 q^{73} + 2 q^{75} + 6 q^{77} + 8 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{87} - 12 q^{89} + 6 q^{91} + 6 q^{93} + 2 q^{95} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 + 2 * q^19 + 6 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 6 * q^31 + 2 * q^33 + 6 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 - 6 * q^47 + 4 * q^49 + 2 * q^55 + 2 * q^57 + 4 * q^59 + 10 * q^61 + 6 * q^63 + 2 * q^65 + 6 * q^67 - 2 * q^69 - 4 * q^71 + 2 * q^73 + 2 * q^75 + 6 * q^77 + 8 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^87 - 12 * q^89 + 6 * q^91 + 6 * q^93 + 2 * q^95 + 16 * q^97 + 2 * q^99

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