LMFDB - Embedded newform 18.12.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,12,Mod(1,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	18.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:	$13.8301772501$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	18.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+32.0000 q^{2} +1024.00 q^{4} +11730.0 q^{5} -50008.0 q^{7} +32768.0 q^{8} +O(q^{10})$

\(q+32.0000 q^{2} +1024.00 q^{4} +11730.0 q^{5} -50008.0 q^{7} +32768.0 q^{8} +375360. q^{10} +531420. q^{11} +1.33257e6 q^{13} -1.60026e6 q^{14} +1.04858e6 q^{16} +5.10968e6 q^{17} +2.90140e6 q^{19} +1.20115e7 q^{20} +1.70054e7 q^{22} -3.05970e7 q^{23} +8.87648e7 q^{25} +4.26421e7 q^{26} -5.12082e7 q^{28} +7.70066e7 q^{29} -2.39418e8 q^{31} +3.35544e7 q^{32} +1.63510e8 q^{34} -5.86594e8 q^{35} -7.85042e8 q^{37} +9.28449e7 q^{38} +3.84369e8 q^{40} -4.11253e8 q^{41} +3.51233e8 q^{43} +5.44174e8 q^{44} -9.79104e8 q^{46} -9.58217e7 q^{47} +5.23473e8 q^{49} +2.84047e9 q^{50} +1.36455e9 q^{52} +1.46586e9 q^{53} +6.23356e9 q^{55} -1.63866e9 q^{56} +2.46421e9 q^{58} -5.62115e9 q^{59} -1.04736e10 q^{61} -7.66139e9 q^{62} +1.07374e9 q^{64} +1.56310e10 q^{65} +4.51531e9 q^{67} +5.23231e9 q^{68} -1.87710e10 q^{70} +8.50958e9 q^{71} +2.01250e9 q^{73} -2.51213e10 q^{74} +2.97104e9 q^{76} -2.65753e10 q^{77} -2.22384e10 q^{79} +1.22998e10 q^{80} -1.31601e10 q^{82} -6.32865e9 q^{83} +5.99365e10 q^{85} +1.12395e10 q^{86} +1.74136e10 q^{88} +5.01237e10 q^{89} -6.66390e10 q^{91} -3.13313e10 q^{92} -3.06629e9 q^{94} +3.40335e10 q^{95} +9.48060e10 q^{97} +1.67511e10 q^{98} +O(q^{100})\)

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 $

$2$	32.0000	0.707107
$2$	32.0000	0.707107
$3$	0	0
$3$	0	0
$4$	1024.00	0.500000
$5$	11730.0	1.67866	0.839330	−	0.543622i	$-0.182947\pi$
$5$	11730.0	1.67866	0.839330	+	0.543622i	$0.182947\pi$
$6$	0	0
$7$	−50008.0	−1.12461	−0.562303	−	0.826931i	$-0.690084\pi$
$7$	−50008.0	−1.12461	−0.562303	+	0.826931i	$0.690084\pi$
$8$	32768.0	0.353553
$9$	0	0
$10$	375360.	1.18699
$11$	531420.	0.994897	0.497449	−	0.867494i	$-0.334270\pi$
$11$	531420.	0.994897	0.497449	+	0.867494i	$0.334270\pi$
$12$	0	0
$13$	1.33257e6	0.995406	0.497703	−	0.867347i	$-0.334177\pi$
$13$	1.33257e6	0.995406	0.497703	+	0.867347i	$0.334177\pi$
$14$	−1.60026e6	−0.795216
$15$	0	0
$16$	1.04858e6	0.250000
$17$	5.10968e6	0.872820	0.436410	−	0.899748i	$-0.356250\pi$
$17$	5.10968e6	0.872820	0.436410	+	0.899748i	$0.356250\pi$
$18$	0	0
$19$	2.90140e6	0.268821	0.134411	−	0.990926i	$-0.457086\pi$
$19$	2.90140e6	0.268821	0.134411	+	0.990926i	$0.457086\pi$
$20$	1.20115e7	0.839330
$21$	0	0
$22$	1.70054e7	0.703498
$23$	−3.05970e7	−0.991233	−0.495616	−	0.868541i	$-0.665058\pi$
$23$	−3.05970e7	−0.991233	−0.495616	+	0.868541i	$0.665058\pi$
$24$	0	0
$25$	8.87648e7	1.81790
$26$	4.26421e7	0.703858
$27$	0	0
$28$	−5.12082e7	−0.562303
$29$	7.70066e7	0.697171	0.348585	−	0.937277i	$-0.386662\pi$
$29$	7.70066e7	0.697171	0.348585	+	0.937277i	$0.386662\pi$
$30$	0	0
$31$	−2.39418e8	−1.50199	−0.750997	−	0.660306i	$-0.770427\pi$
$31$	−2.39418e8	−1.50199	−0.750997	+	0.660306i	$0.770427\pi$
$32$	3.35544e7	0.176777
$33$	0	0
$34$	1.63510e8	0.617177
$35$	−5.86594e8	−1.88783
$36$	0	0
$37$	−7.85042e8	−1.86116	−0.930579	−	0.366091i	$-0.880696\pi$
$37$	−7.85042e8	−1.86116	−0.930579	+	0.366091i	$0.880696\pi$
$38$	9.28449e7	0.190085
$39$	0	0
$40$	3.84369e8	0.593496
$41$	−4.11253e8	−0.554368	−0.277184	−	0.960817i	$-0.589401\pi$
$41$	−4.11253e8	−0.554368	−0.277184	+	0.960817i	$0.589401\pi$
$42$	0	0
$43$	3.51233e8	0.364350	0.182175	−	0.983266i	$-0.441686\pi$
$43$	3.51233e8	0.364350	0.182175	+	0.983266i	$0.441686\pi$
$44$	5.44174e8	0.497449
$45$	0	0
$46$	−9.79104e8	−0.700908
$47$	−9.58217e7	−0.0609432	−0.0304716	−	0.999536i	$-0.509701\pi$
$47$	−9.58217e7	−0.0609432	−0.0304716	+	0.999536i	$0.509701\pi$
$48$	0	0
$49$	5.23473e8	0.264738
$50$	2.84047e9	1.28545
$51$	0	0
$52$	1.36455e9	0.497703
$53$	1.46586e9	0.481476	0.240738	−	0.970590i	$-0.422611\pi$
$53$	1.46586e9	0.481476	0.240738	+	0.970590i	$0.422611\pi$
$54$	0	0
$55$	6.23356e9	1.67009
$56$	−1.63866e9	−0.397608
$57$	0	0
$58$	2.46421e9	0.492974
$59$	−5.62115e9	−1.02362	−0.511811	−	0.859098i	$-0.671025\pi$
$59$	−5.62115e9	−1.02362	−0.511811	+	0.859098i	$0.671025\pi$
$60$	0	0
$61$	−1.04736e10	−1.58775	−0.793874	−	0.608083i	$-0.791939\pi$
$61$	−1.04736e10	−1.58775	−0.793874	+	0.608083i	$0.791939\pi$
$62$	−7.66139e9	−1.06207
$63$	0	0
$64$	1.07374e9	0.125000
$65$	1.56310e10	1.67095
$66$	0	0
$67$	4.51531e9	0.408579	0.204289	−	0.978911i	$-0.434512\pi$
$67$	4.51531e9	0.408579	0.204289	+	0.978911i	$0.434512\pi$
$68$	5.23231e9	0.436410
$69$	0	0
$70$	−1.87710e10	−1.33490
$71$	8.50958e9	0.559741	0.279871	−	0.960038i	$-0.409708\pi$
$71$	8.50958e9	0.559741	0.279871	+	0.960038i	$0.409708\pi$
$72$	0	0
$73$	2.01250e9	0.113621	0.0568106	−	0.998385i	$-0.481907\pi$
$73$	2.01250e9	0.113621	0.0568106	+	0.998385i	$0.481907\pi$
$74$	−2.51213e10	−1.31604
$75$	0	0
$76$	2.97104e9	0.134411
$77$	−2.65753e10	−1.11887
$78$	0	0
$79$	−2.22384e10	−0.813120	−0.406560	−	0.913624i	$-0.633272\pi$
$79$	−2.22384e10	−0.813120	−0.406560	+	0.913624i	$0.633272\pi$
$80$	1.22998e10	0.419665
$81$	0	0
$82$	−1.31601e10	−0.391997
$83$	−6.32865e9	−0.176352	−0.0881762	−	0.996105i	$-0.528104\pi$
$83$	−6.32865e9	−0.176352	−0.0881762	+	0.996105i	$0.528104\pi$
$84$	0	0
$85$	5.99365e10	1.46517
$86$	1.12395e10	0.257635
$87$	0	0
$88$	1.74136e10	0.351749
$89$	5.01237e10	0.951477	0.475738	−	0.879587i	$-0.342181\pi$
$89$	5.01237e10	0.951477	0.475738	+	0.879587i	$0.342181\pi$
$90$	0	0
$91$	−6.66390e10	−1.11944
$92$	−3.13313e10	−0.495616
$93$	0	0
$94$	−3.06629e9	−0.0430934
$95$	3.40335e10	0.451260
$96$	0	0
$97$	9.48060e10	1.12096	0.560481	−	0.828167i	$-0.310616\pi$
$97$	9.48060e10	1.12096	0.560481	+	0.828167i	$0.310616\pi$
$98$	1.67511e10	0.187198
$99$	0	0
$100$	9.08951e10	0.908951
$101$	−1.37952e10	−0.130605	−0.0653026	−	0.997866i	$-0.520801\pi$
$101$	−1.37952e10	−0.130605	−0.0653026	+	0.997866i	$0.520801\pi$
$102$	0	0
$103$	6.51836e10	0.554031	0.277015	−	0.960865i	$-0.410655\pi$
$103$	6.51836e10	0.554031	0.277015	+	0.960865i	$0.410655\pi$
$104$	4.36655e10	0.351929
$105$	0	0
$106$	4.69074e10	0.340455
$107$	−9.33399e10	−0.643363	−0.321682	−	0.946848i	$-0.604248\pi$
$107$	−9.33399e10	−0.643363	−0.321682	+	0.946848i	$0.604248\pi$
$108$	0	0
$109$	−1.51369e11	−0.942307	−0.471154	−	0.882051i	$-0.656162\pi$
$109$	−1.51369e11	−0.942307	−0.471154	+	0.882051i	$0.656162\pi$
$110$	1.99474e11	1.18094
$111$	0	0
$112$	−5.24372e10	−0.281151
$113$	−2.37349e11	−1.21187	−0.605935	−	0.795514i	$-0.707201\pi$
$113$	−2.37349e11	−1.21187	−0.605935	+	0.795514i	$0.707201\pi$
$114$	0	0
$115$	−3.58903e11	−1.66394
$116$	7.88548e10	0.348585
$117$	0	0
$118$	−1.79877e11	−0.723809
$119$	−2.55525e11	−0.981578
$120$	0	0
$121$	−2.90445e9	−0.0101799
$122$	−3.35155e11	−1.12271
$123$	0	0
$124$	−2.45164e11	−0.750997
$125$	4.68457e11	1.37298
$126$	0	0
$127$	5.14414e11	1.38163	0.690816	−	0.723031i	$-0.257251\pi$
$127$	5.14414e11	1.38163	0.690816	+	0.723031i	$0.257251\pi$
$128$	3.43597e10	0.0883883
$129$	0	0
$130$	5.00192e11	1.18154
$131$	−2.98572e11	−0.676171	−0.338086	−	0.941115i	$-0.609779\pi$
$131$	−2.98572e11	−0.676171	−0.338086	+	0.941115i	$0.609779\pi$
$132$	0	0
$133$	−1.45093e11	−0.302318
$134$	1.44490e11	0.288909
$135$	0	0
$136$	1.67434e11	0.308588
$137$	6.16116e11	1.09069	0.545343	−	0.838213i	$-0.316400\pi$
$137$	6.16116e11	1.09069	0.545343	+	0.838213i	$0.316400\pi$
$138$	0	0
$139$	−5.22814e11	−0.854606	−0.427303	−	0.904109i	$-0.640536\pi$
$139$	−5.22814e11	−0.854606	−0.427303	+	0.904109i	$0.640536\pi$
$140$	−6.00672e11	−0.943916
$141$	0	0
$142$	2.72307e11	0.395797
$143$	7.08152e11	0.990327
$144$	0	0
$145$	9.03288e11	1.17031
$146$	6.43999e10	0.0803423
$147$	0	0
$148$	−8.03883e11	−0.930579
$149$	1.27015e12	1.41687	0.708434	−	0.705777i	$-0.249402\pi$
$149$	1.27015e12	1.41687	0.708434	+	0.705777i	$0.249402\pi$
$150$	0	0
$151$	1.25371e12	1.29964	0.649822	−	0.760086i	$-0.274843\pi$
$151$	1.25371e12	1.29964	0.649822	+	0.760086i	$0.274843\pi$
$152$	9.50732e10	0.0950426
$153$	0	0
$154$	−8.50408e11	−0.791158
$155$	−2.80838e12	−2.52134
$156$	0	0
$157$	6.06708e11	0.507611	0.253806	−	0.967255i	$-0.418318\pi$
$157$	6.06708e11	0.507611	0.253806	+	0.967255i	$0.418318\pi$
$158$	−7.11629e11	−0.574963
$159$	0	0
$160$	3.93593e11	0.296748
$161$	1.53009e12	1.11475
$162$	0	0
$163$	−1.58401e12	−1.07827	−0.539133	−	0.842221i	$-0.681248\pi$
$163$	−1.58401e12	−1.07827	−0.539133	+	0.842221i	$0.681248\pi$
$164$	−4.21123e11	−0.277184
$165$	0	0
$166$	−2.02517e11	−0.124700
$167$	−2.95272e12	−1.75907	−0.879533	−	0.475837i	$-0.842145\pi$
$167$	−2.95272e12	−1.75907	−0.879533	+	0.475837i	$0.842145\pi$
$168$	0	0
$169$	−1.64282e10	−0.00916673
$170$	1.91797e12	1.03603
$171$	0	0
$172$	3.59663e11	0.182175
$173$	2.65410e12	1.30216	0.651079	−	0.759010i	$-0.274316\pi$
$173$	2.65410e12	1.30216	0.651079	+	0.759010i	$0.274316\pi$
$174$	0	0
$175$	−4.43895e12	−2.04442
$176$	5.57234e11	0.248724
$177$	0	0
$178$	1.60396e12	0.672796
$179$	4.06450e12	1.65316	0.826580	−	0.562819i	$-0.190283\pi$
$179$	4.06450e12	1.65316	0.826580	+	0.562819i	$0.190283\pi$
$180$	0	0
$181$	−1.68073e12	−0.643082	−0.321541	−	0.946896i	$-0.604201\pi$
$181$	−1.68073e12	−0.643082	−0.321541	+	0.946896i	$0.604201\pi$
$182$	−2.13245e12	−0.791563
$183$	0	0
$184$	−1.00260e12	−0.350454
$185$	−9.20854e12	−3.12425
$186$	0	0
$187$	2.71539e12	0.868366
$188$	−9.81214e10	−0.0304716
$189$	0	0
$190$	1.08907e12	0.319089
$191$	−4.09135e11	−0.116462	−0.0582309	−	0.998303i	$-0.518546\pi$
$191$	−4.09135e11	−0.116462	−0.0582309	+	0.998303i	$0.518546\pi$
$192$	0	0
$193$	4.71906e12	1.26850	0.634249	−	0.773129i	$-0.281309\pi$
$193$	4.71906e12	1.26850	0.634249	+	0.773129i	$0.281309\pi$
$194$	3.03379e12	0.792640
$195$	0	0
$196$	5.36037e11	0.132369
$197$	3.60899e12	0.866605	0.433303	−	0.901249i	$-0.357348\pi$
$197$	3.60899e12	0.866605	0.433303	+	0.901249i	$0.357348\pi$
$198$	0	0
$199$	1.72140e12	0.391012	0.195506	−	0.980703i	$-0.437365\pi$
$199$	1.72140e12	0.391012	0.195506	+	0.980703i	$0.437365\pi$
$200$	2.90864e12	0.642726
$201$	0	0
$202$	−4.41446e11	−0.0923518
$203$	−3.85095e12	−0.784042
$204$	0	0
$205$	−4.82400e12	−0.930595
$206$	2.08588e12	0.391759
$207$	0	0
$208$	1.39730e12	0.248852
$209$	1.54186e12	0.267449
$210$	0	0
$211$	1.70345e12	0.280398	0.140199	−	0.990123i	$-0.455226\pi$
$211$	1.70345e12	0.280398	0.140199	+	0.990123i	$0.455226\pi$
$212$	1.50104e12	0.240738
$213$	0	0
$214$	−2.98688e12	−0.454927
$215$	4.11997e12	0.611621
$216$	0	0
$217$	1.19728e13	1.68915
$218$	−4.84382e12	−0.666312
$219$	0	0
$220$	6.38316e12	0.835047
$221$	6.80898e12	0.868810
$222$	0	0
$223$	−1.46253e13	−1.77594	−0.887969	−	0.459904i	$-0.847884\pi$
$223$	−1.46253e13	−1.77594	−0.887969	+	0.459904i	$0.847884\pi$
$224$	−1.67799e12	−0.198804
$225$	0	0
$226$	−7.59517e12	−0.856921
$227$	−3.36457e12	−0.370499	−0.185249	−	0.982692i	$-0.559309\pi$
$227$	−3.36457e12	−0.370499	−0.185249	+	0.982692i	$0.559309\pi$
$228$	0	0
$229$	1.89920e12	0.199286	0.0996429	−	0.995023i	$-0.468230\pi$
$229$	1.89920e12	0.199286	0.0996429	+	0.995023i	$0.468230\pi$
$230$	−1.14849e13	−1.17659
$231$	0	0
$232$	2.52335e12	0.246487
$233$	−1.51901e12	−0.144912	−0.0724560	−	0.997372i	$-0.523084\pi$
$233$	−1.51901e12	−0.144912	−0.0724560	+	0.997372i	$0.523084\pi$
$234$	0	0
$235$	−1.12399e12	−0.102303
$236$	−5.75606e12	−0.511811
$237$	0	0
$238$	−8.17679e12	−0.694080
$239$	−8.31497e12	−0.689719	−0.344859	−	0.938654i	$-0.612073\pi$
$239$	−8.31497e12	−0.689719	−0.344859	+	0.938654i	$0.612073\pi$
$240$	0	0
$241$	−1.30126e13	−1.03103	−0.515515	−	0.856881i	$-0.672399\pi$
$241$	−1.30126e13	−1.03103	−0.515515	+	0.856881i	$0.672399\pi$
$242$	−9.29425e10	−0.00719830
$243$	0	0
$244$	−1.07250e13	−0.793874
$245$	6.14034e12	0.444405
$246$	0	0
$247$	3.86631e12	0.267586
$248$	−7.84526e12	−0.531035
$249$	0	0
$250$	1.49906e13	0.970844
$251$	1.87138e13	1.18565	0.592825	−	0.805331i	$-0.298013\pi$
$251$	1.87138e13	1.18565	0.592825	+	0.805331i	$0.298013\pi$
$252$	0	0
$253$	−1.62599e13	−0.986175
$254$	1.64613e13	0.976961
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	5.48427e12	0.305131	0.152566	−	0.988293i	$-0.451246\pi$
$257$	5.48427e12	0.305131	0.152566	+	0.988293i	$0.451246\pi$
$258$	0	0
$259$	3.92584e13	2.09307
$260$	1.60061e13	0.835475
$261$	0	0
$262$	−9.55430e12	−0.478125
$263$	−1.95941e13	−0.960218	−0.480109	−	0.877209i	$-0.659403\pi$
$263$	−1.95941e13	−0.960218	−0.480109	+	0.877209i	$0.659403\pi$
$264$	0	0
$265$	1.71945e13	0.808235
$266$	−4.64299e12	−0.213771
$267$	0	0
$268$	4.62367e12	0.204289
$269$	−3.57029e13	−1.54549	−0.772745	−	0.634717i	$-0.781117\pi$
$269$	−3.57029e13	−1.54549	−0.772745	+	0.634717i	$0.781117\pi$
$270$	0	0
$271$	3.60387e13	1.49774	0.748872	−	0.662714i	$-0.230596\pi$
$271$	3.60387e13	1.49774	0.748872	+	0.662714i	$0.230596\pi$
$272$	5.35789e12	0.218205
$273$	0	0
$274$	1.97157e13	0.771231
$275$	4.71714e13	1.80863
$276$	0	0
$277$	−4.07304e13	−1.50065	−0.750326	−	0.661068i	$-0.770103\pi$
$277$	−4.07304e13	−1.50065	−0.750326	+	0.661068i	$0.770103\pi$
$278$	−1.67300e13	−0.604297
$279$	0	0
$280$	−1.92215e13	−0.667449
$281$	−4.08826e12	−0.139205	−0.0696023	−	0.997575i	$-0.522173\pi$
$281$	−4.08826e12	−0.139205	−0.0696023	+	0.997575i	$0.522173\pi$
$282$	0	0
$283$	−1.33555e13	−0.437356	−0.218678	−	0.975797i	$-0.570174\pi$
$283$	−1.33555e13	−0.437356	−0.218678	+	0.975797i	$0.570174\pi$
$284$	8.71381e12	0.279871
$285$	0	0
$286$	2.26609e13	0.700267
$287$	2.05659e13	0.623445
$288$	0	0
$289$	−8.16309e12	−0.238186
$290$	2.89052e13	0.827536
$291$	0	0
$292$	2.06080e12	0.0568106
$293$	−2.19433e13	−0.593651	−0.296825	−	0.954932i	$-0.595928\pi$
$293$	−2.19433e13	−0.593651	−0.296825	+	0.954932i	$0.595928\pi$
$294$	0	0
$295$	−6.59361e13	−1.71831
$296$	−2.57242e13	−0.658019
$297$	0	0
$298$	4.06447e13	1.00188
$299$	−4.07725e13	−0.986679
$300$	0	0
$301$	−1.75645e13	−0.409751
$302$	4.01188e13	0.918987
$303$	0	0
$304$	3.04234e12	0.0672053
$305$	−1.22855e14	−2.66529
$306$	0	0
$307$	2.52177e11	0.00527770	0.00263885	−	0.999997i	$-0.499160\pi$
$307$	2.52177e11	0.00527770	0.00263885	+	0.999997i	$0.499160\pi$
$308$	−2.72131e13	−0.559433
$309$	0	0
$310$	−8.98681e13	−1.78286
$311$	4.95752e13	0.966234	0.483117	−	0.875556i	$-0.339505\pi$
$311$	4.95752e13	0.966234	0.483117	+	0.875556i	$0.339505\pi$
$312$	0	0
$313$	3.24318e13	0.610207	0.305104	−	0.952319i	$-0.401309\pi$
$313$	3.24318e13	0.610207	0.305104	+	0.952319i	$0.401309\pi$
$314$	1.94146e13	0.358935
$315$	0	0
$316$	−2.27721e13	−0.406560
$317$	9.56422e13	1.67812	0.839061	−	0.544037i	$-0.183105\pi$
$317$	9.56422e13	1.67812	0.839061	+	0.544037i	$0.183105\pi$
$318$	0	0
$319$	4.09229e13	0.693613
$320$	1.25950e13	0.209833
$321$	0	0
$322$	4.89630e13	0.788245
$323$	1.48252e13	0.234632
$324$	0	0
$325$	1.18285e14	1.80955
$326$	−5.06883e13	−0.762449
$327$	0	0
$328$	−1.34759e13	−0.195999
$329$	4.79185e12	0.0685371
$330$	0	0
$331$	−1.21579e14	−1.68192	−0.840959	−	0.541099i	$-0.818008\pi$
$331$	−1.21579e14	−1.68192	−0.840959	+	0.541099i	$0.818008\pi$
$332$	−6.48054e12	−0.0881762
$333$	0	0
$334$	−9.44872e13	−1.24385
$335$	5.29646e13	0.685865
$336$	0	0
$337$	−8.38096e13	−1.05034	−0.525169	−	0.850998i	$-0.675998\pi$
$337$	−8.38096e13	−1.05034	−0.525169	+	0.850998i	$0.675998\pi$
$338$	−5.25704e11	−0.00648186
$339$	0	0
$340$	6.13750e13	0.732584
$341$	−1.27232e14	−1.49433
$342$	0	0
$343$	7.27043e13	0.826880
$344$	1.15092e13	0.128817
$345$	0	0
$346$	8.49312e13	0.920765
$347$	7.79457e13	0.831725	0.415863	−	0.909427i	$-0.363480\pi$
$347$	7.79457e13	0.831725	0.415863	+	0.909427i	$0.363480\pi$
$348$	0	0
$349$	1.12179e14	1.15977	0.579884	−	0.814699i	$-0.303098\pi$
$349$	1.12179e14	1.15977	0.579884	+	0.814699i	$0.303098\pi$
$350$	−1.42046e14	−1.44563
$351$	0	0
$352$	1.78315e13	0.175875
$353$	1.59721e14	1.55096	0.775480	−	0.631372i	$-0.217508\pi$
$353$	1.59721e14	1.55096	0.775480	+	0.631372i	$0.217508\pi$
$354$	0	0
$355$	9.98174e13	0.939616
$356$	5.13267e13	0.475738
$357$	0	0
$358$	1.30064e14	1.16896
$359$	2.46378e12	0.0218063	0.0109032	−	0.999941i	$-0.496529\pi$
$359$	2.46378e12	0.0218063	0.0109032	+	0.999941i	$0.496529\pi$
$360$	0	0
$361$	−1.08072e14	−0.927735
$362$	−5.37834e13	−0.454728
$363$	0	0
$364$	−6.82383e13	−0.559720
$365$	2.36066e13	0.190732
$366$	0	0
$367$	1.34080e14	1.05124	0.525620	−	0.850720i	$-0.323833\pi$
$367$	1.34080e14	1.05124	0.525620	+	0.850720i	$0.323833\pi$
$368$	−3.20833e13	−0.247808
$369$	0	0
$370$	−2.94673e14	−2.20918
$371$	−7.33046e13	−0.541470
$372$	0	0
$373$	−1.25684e14	−0.901328	−0.450664	−	0.892694i	$-0.648813\pi$
$373$	−1.25684e14	−0.901328	−0.450664	+	0.892694i	$0.648813\pi$
$374$	8.68923e13	0.614027
$375$	0	0
$376$	−3.13988e12	−0.0215467
$377$	1.02616e14	0.693968
$378$	0	0
$379$	5.62301e13	0.369363	0.184681	−	0.982798i	$-0.440875\pi$
$379$	5.62301e13	0.369363	0.184681	+	0.982798i	$0.440875\pi$
$380$	3.48503e13	0.225630
$381$	0	0
$382$	−1.30923e13	−0.0823509
$383$	−1.07826e14	−0.668543	−0.334272	−	0.942477i	$-0.608490\pi$
$383$	−1.07826e14	−0.668543	−0.334272	+	0.942477i	$0.608490\pi$
$384$	0	0
$385$	−3.11728e14	−1.87820
$386$	1.51010e14	0.896964
$387$	0	0
$388$	9.70813e13	0.560481
$389$	−2.37629e14	−1.35262	−0.676311	−	0.736616i	$-0.736423\pi$
$389$	−2.37629e14	−1.35262	−0.676311	+	0.736616i	$0.736423\pi$
$390$	0	0
$391$	−1.56341e14	−0.865168
$392$	1.71532e13	0.0935990
$393$	0	0
$394$	1.15488e14	0.612782
$395$	−2.60857e14	−1.36495
$396$	0	0
$397$	2.85970e14	1.45537	0.727683	−	0.685914i	$-0.240597\pi$
$397$	2.85970e14	1.45537	0.727683	+	0.685914i	$0.240597\pi$
$398$	5.50848e13	0.276487
$399$	0	0
$400$	9.30766e13	0.454476
$401$	9.31707e13	0.448730	0.224365	−	0.974505i	$-0.427969\pi$
$401$	9.31707e13	0.448730	0.224365	+	0.974505i	$0.427969\pi$
$402$	0	0
$403$	−3.19041e14	−1.49509
$404$	−1.41263e13	−0.0653026
$405$	0	0
$406$	−1.23230e14	−0.554402
$407$	−4.17187e14	−1.85166
$408$	0	0
$409$	2.56950e14	1.11012	0.555061	−	0.831810i	$-0.312695\pi$
$409$	2.56950e14	1.11012	0.555061	+	0.831810i	$0.312695\pi$
$410$	−1.54368e14	−0.658030
$411$	0	0
$412$	6.67481e13	0.277015
$413$	2.81103e14	1.15117
$414$	0	0
$415$	−7.42350e13	−0.296036
$416$	4.47135e13	0.175965
$417$	0	0
$418$	4.93397e13	0.189115
$419$	2.96091e14	1.12008	0.560039	−	0.828466i	$-0.310786\pi$
$419$	2.96091e14	1.12008	0.560039	+	0.828466i	$0.310786\pi$
$420$	0	0
$421$	−3.40485e14	−1.25472	−0.627360	−	0.778730i	$-0.715864\pi$
$421$	−3.40485e14	−1.25472	−0.627360	+	0.778730i	$0.715864\pi$
$422$	5.45103e13	0.198271
$423$	0	0
$424$	4.80332e13	0.170227
$425$	4.53559e14	1.58670
$426$	0	0
$427$	5.23763e14	1.78559
$428$	−9.55800e13	−0.321682
$429$	0	0
$430$	1.31839e14	0.432481
$431$	−1.37789e14	−0.446261	−0.223130	−	0.974789i	$-0.571628\pi$
$431$	−1.37789e14	−0.446261	−0.223130	+	0.974789i	$0.571628\pi$
$432$	0	0
$433$	3.52377e14	1.11256	0.556280	−	0.830995i	$-0.312228\pi$
$433$	3.52377e14	1.11256	0.556280	+	0.830995i	$0.312228\pi$
$434$	3.83131e14	1.19441
$435$	0	0
$436$	−1.55002e14	−0.471154
$437$	−8.87743e13	−0.266464
$438$	0	0
$439$	−4.29185e14	−1.25629	−0.628145	−	0.778097i	$-0.716185\pi$
$439$	−4.29185e14	−1.25629	−0.628145	+	0.778097i	$0.716185\pi$
$440$	2.04261e14	0.590468
$441$	0	0
$442$	2.17887e14	0.614341
$443$	3.61081e14	1.00551	0.502753	−	0.864430i	$-0.332321\pi$
$443$	3.61081e14	1.00551	0.502753	+	0.864430i	$0.332321\pi$
$444$	0	0
$445$	5.87951e14	1.59721
$446$	−4.68009e14	−1.25578
$447$	0	0
$448$	−5.36957e13	−0.140576
$449$	6.27688e12	0.0162326	0.00811632	−	0.999967i	$-0.497416\pi$
$449$	6.27688e12	0.0162326	0.00811632	+	0.999967i	$0.497416\pi$
$450$	0	0
$451$	−2.18548e14	−0.551539
$452$	−2.43045e14	−0.605935
$453$	0	0
$454$	−1.07666e14	−0.261982
$455$	−7.81675e14	−1.87916
$456$	0	0
$457$	4.60907e14	1.08162	0.540809	−	0.841146i	$-0.318118\pi$
$457$	4.60907e14	1.08162	0.540809	+	0.841146i	$0.318118\pi$
$458$	6.07745e13	0.140916
$459$	0	0
$460$	−3.67516e14	−0.831972
$461$	−7.97399e14	−1.78369	−0.891847	−	0.452337i	$-0.850590\pi$
$461$	−7.97399e14	−1.78369	−0.891847	+	0.452337i	$0.850590\pi$
$462$	0	0
$463$	−5.80897e14	−1.26883	−0.634416	−	0.772992i	$-0.718759\pi$
$463$	−5.80897e14	−1.26883	−0.634416	+	0.772992i	$0.718759\pi$
$464$	8.07473e13	0.174293
$465$	0	0
$466$	−4.86085e13	−0.102468
$467$	8.22532e14	1.71360	0.856801	−	0.515647i	$-0.172448\pi$
$467$	8.22532e14	1.71360	0.856801	+	0.515647i	$0.172448\pi$
$468$	0	0
$469$	−2.25801e14	−0.459490
$470$	−3.59676e13	−0.0723392
$471$	0	0
$472$	−1.84194e14	−0.361905
$473$	1.86652e14	0.362491
$474$	0	0
$475$	2.57542e14	0.488691
$476$	−2.61657e14	−0.490789
$477$	0	0
$478$	−2.66079e14	−0.487705
$479$	4.88235e14	0.884675	0.442337	−	0.896849i	$-0.354149\pi$
$479$	4.88235e14	0.884675	0.442337	+	0.896849i	$0.354149\pi$
$480$	0	0
$481$	−1.04612e15	−1.85261
$482$	−4.16404e14	−0.729048
$483$	0	0
$484$	−2.97416e12	−0.00508997
$485$	1.11207e15	1.88172
$486$	0	0
$487$	−2.97281e13	−0.0491766	−0.0245883	−	0.999698i	$-0.507827\pi$
$487$	−2.97281e13	−0.0491766	−0.0245883	+	0.999698i	$0.507827\pi$
$488$	−3.43199e14	−0.561353
$489$	0	0
$490$	1.96491e14	0.314242
$491$	−8.16135e14	−1.29067	−0.645333	−	0.763901i	$-0.723281\pi$
$491$	−8.16135e14	−1.29067	−0.645333	+	0.763901i	$0.723281\pi$
$492$	0	0
$493$	3.93479e14	0.608504
$494$	1.23722e14	0.189212
$495$	0	0
$496$	−2.51048e14	−0.375498
$497$	−4.25547e14	−0.629488
$498$	0	0
$499$	6.09917e14	0.882506	0.441253	−	0.897383i	$-0.354534\pi$
$499$	6.09917e14	0.882506	0.441253	+	0.897383i	$0.354534\pi$
$500$	4.79700e14	0.686491
$501$	0	0
$502$	5.98842e14	0.838381
$503$	−2.40472e14	−0.332998	−0.166499	−	0.986042i	$-0.553246\pi$
$503$	−2.40472e14	−0.332998	−0.166499	+	0.986042i	$0.553246\pi$
$504$	0	0
$505$	−1.61818e14	−0.219242
$506$	−5.20315e14	−0.697331
$507$	0	0
$508$	5.26760e14	0.690816
$509$	1.56720e14	0.203318	0.101659	−	0.994819i	$-0.467585\pi$
$509$	1.56720e14	0.203318	0.101659	+	0.994819i	$0.467585\pi$
$510$	0	0
$511$	−1.00641e14	−0.127779
$512$	3.51844e13	0.0441942
$513$	0	0
$514$	1.75497e14	0.215760
$515$	7.64604e14	0.930030
$516$	0	0
$517$	−5.09216e13	−0.0606323
$518$	1.25627e15	1.48002
$519$	0	0
$520$	5.12197e14	0.590770
$521$	1.00579e14	0.114788	0.0573942	−	0.998352i	$-0.481721\pi$
$521$	1.00579e14	0.114788	0.0573942	+	0.998352i	$0.481721\pi$
$522$	0	0
$523$	1.22443e15	1.36827	0.684137	−	0.729354i	$-0.260179\pi$
$523$	1.22443e15	1.36827	0.684137	+	0.729354i	$0.260179\pi$
$524$	−3.05737e14	−0.338086
$525$	0	0
$526$	−6.27013e14	−0.678977
$527$	−1.22335e15	−1.31097
$528$	0	0
$529$	−1.66333e13	−0.0174572
$530$	5.50224e14	0.571508
$531$	0	0
$532$	−1.48576e14	−0.151159
$533$	−5.48022e14	−0.551821
$534$	0	0
$535$	−1.09488e15	−1.07999
$536$	1.47958e14	0.144454
$537$	0	0
$538$	−1.14249e15	−1.09283
$539$	2.78184e14	0.263387
$540$	0	0
$541$	−3.62345e14	−0.336154	−0.168077	−	0.985774i	$-0.553756\pi$
$541$	−3.62345e14	−0.336154	−0.168077	+	0.985774i	$0.553756\pi$
$542$	1.15324e15	1.05907
$543$	0	0
$544$	1.71452e14	0.154294
$545$	−1.77556e15	−1.58181
$546$	0	0
$547$	1.32630e15	1.15801	0.579006	−	0.815324i	$-0.303441\pi$
$547$	1.32630e15	1.15801	0.579006	+	0.815324i	$0.303441\pi$
$548$	6.30903e14	0.545343
$549$	0	0
$550$	1.50948e15	1.27889
$551$	2.23427e14	0.187414
$552$	0	0
$553$	1.11210e15	0.914440
$554$	−1.30337e15	−1.06112
$555$	0	0
$556$	−5.35361e14	−0.427303
$557$	2.24474e15	1.77404	0.887018	−	0.461735i	$-0.152773\pi$
$557$	2.24474e15	1.77404	0.887018	+	0.461735i	$0.152773\pi$
$558$	0	0
$559$	4.68042e14	0.362677
$560$	−6.15088e14	−0.471958
$561$	0	0
$562$	−1.30824e14	−0.0984326
$563$	1.10178e15	0.820915	0.410457	−	0.911880i	$-0.365369\pi$
$563$	1.10178e15	0.820915	0.410457	+	0.911880i	$0.365369\pi$
$564$	0	0
$565$	−2.78410e15	−2.03432
$566$	−4.27376e14	−0.309257
$567$	0	0
$568$	2.78842e14	0.197898
$569$	−1.69107e15	−1.18863	−0.594313	−	0.804234i	$-0.702576\pi$
$569$	−1.69107e15	−1.18863	−0.594313	+	0.804234i	$0.702576\pi$
$570$	0	0
$571$	−2.36575e15	−1.63106	−0.815529	−	0.578716i	$-0.803554\pi$
$571$	−2.36575e15	−1.63106	−0.815529	+	0.578716i	$0.803554\pi$
$572$	7.25148e14	0.495163
$573$	0	0
$574$	6.58110e14	0.440842
$575$	−2.71594e15	−1.80197
$576$	0	0
$577$	−3.25902e14	−0.212139	−0.106069	−	0.994359i	$-0.533827\pi$
$577$	−3.25902e14	−0.212139	−0.106069	+	0.994359i	$0.533827\pi$
$578$	−2.61219e14	−0.168423
$579$	0	0
$580$	9.24967e14	0.585157
$581$	3.16483e14	0.198327
$582$	0	0
$583$	7.78986e14	0.479019
$584$	6.59455e13	0.0401712
$585$	0	0
$586$	−7.02187e14	−0.419774
$587$	2.76684e15	1.63861	0.819303	−	0.573361i	$-0.194361\pi$
$587$	2.76684e15	1.63861	0.819303	+	0.573361i	$0.194361\pi$
$588$	0	0
$589$	−6.94649e14	−0.403768
$590$	−2.10996e15	−1.21503
$591$	0	0
$592$	−8.23176e14	−0.465289
$593$	−1.95966e15	−1.09744	−0.548719	−	0.836007i	$-0.684884\pi$
$593$	−1.95966e15	−1.09744	−0.548719	+	0.836007i	$0.684884\pi$
$594$	0	0
$595$	−2.99731e15	−1.64774
$596$	1.30063e15	0.708434
$597$	0	0
$598$	−1.30472e15	−0.697688
$599$	3.37393e15	1.78767	0.893837	−	0.448393i	$-0.148003\pi$
$599$	3.37393e15	1.78767	0.893837	+	0.448393i	$0.148003\pi$
$600$	0	0
$601$	3.13585e15	1.63135	0.815673	−	0.578513i	$-0.196367\pi$
$601$	3.13585e15	1.63135	0.815673	+	0.578513i	$0.196367\pi$
$602$	−5.62063e14	−0.289737
$603$	0	0
$604$	1.28380e15	0.649822
$605$	−3.40692e13	−0.0170887
$606$	0	0
$607$	3.67568e15	1.81051	0.905253	−	0.424873i	$-0.139681\pi$
$607$	3.67568e15	1.81051	0.905253	+	0.424873i	$0.139681\pi$
$608$	9.73550e13	0.0475213
$609$	0	0
$610$	−3.93137e15	−1.88464
$611$	−1.27689e14	−0.0606633
$612$	0	0
$613$	−9.78886e14	−0.456772	−0.228386	−	0.973571i	$-0.573345\pi$
$613$	−9.78886e14	−0.456772	−0.228386	+	0.973571i	$0.573345\pi$
$614$	8.06966e12	0.00373189
$615$	0	0
$616$	−8.70818e14	−0.395579
$617$	1.46255e15	0.658481	0.329241	−	0.944246i	$-0.393207\pi$
$617$	1.46255e15	0.658481	0.329241	+	0.944246i	$0.393207\pi$
$618$	0	0
$619$	4.30101e15	1.90227	0.951135	−	0.308776i	$-0.0999194\pi$
$619$	4.30101e15	1.90227	0.951135	+	0.308776i	$0.0999194\pi$
$620$	−2.87578e15	−1.26067
$621$	0	0
$622$	1.58641e15	0.683231
$623$	−2.50659e15	−1.07004
$624$	0	0
$625$	1.16078e15	0.486867
$626$	1.03782e15	0.431482
$627$	0	0
$628$	6.21269e14	0.253806
$629$	−4.01131e15	−1.62446
$630$	0	0
$631$	−5.95346e14	−0.236924	−0.118462	−	0.992959i	$-0.537796\pi$
$631$	−5.95346e14	−0.236924	−0.118462	+	0.992959i	$0.537796\pi$
$632$	−7.28708e14	−0.287481
$633$	0	0
$634$	3.06055e15	1.18661
$635$	6.03408e15	2.31929
$636$	0	0
$637$	6.97563e14	0.263522
$638$	1.30953e15	0.490459
$639$	0	0
$640$	4.03040e14	0.148374
$641$	7.51262e14	0.274203	0.137101	−	0.990557i	$-0.456221\pi$
$641$	7.51262e14	0.274203	0.137101	+	0.990557i	$0.456221\pi$
$642$	0	0
$643$	1.46144e15	0.524351	0.262175	−	0.965020i	$-0.415560\pi$
$643$	1.46144e15	0.524351	0.262175	+	0.965020i	$0.415560\pi$
$644$	1.56682e15	0.557373
$645$	0	0
$646$	4.74408e14	0.165910
$647$	3.57862e15	1.24091	0.620457	−	0.784241i	$-0.286947\pi$
$647$	3.57862e15	1.24091	0.620457	+	0.784241i	$0.286947\pi$
$648$	0	0
$649$	−2.98719e15	−1.01840
$650$	3.78512e15	1.27955
$651$	0	0
$652$	−1.62202e15	−0.539133
$653$	−7.31395e14	−0.241063	−0.120531	−	0.992710i	$-0.538460\pi$
$653$	−7.31395e14	−0.241063	−0.120531	+	0.992710i	$0.538460\pi$
$654$	0	0
$655$	−3.50225e15	−1.13506
$656$	−4.31230e14	−0.138592
$657$	0	0
$658$	1.53339e14	0.0484631
$659$	1.69987e15	0.532778	0.266389	−	0.963866i	$-0.414169\pi$
$659$	1.69987e15	0.532778	0.266389	+	0.963866i	$0.414169\pi$
$660$	0	0
$661$	−1.78604e14	−0.0550534	−0.0275267	−	0.999621i	$-0.508763\pi$
$661$	−1.78604e14	−0.0550534	−0.0275267	+	0.999621i	$0.508763\pi$
$662$	−3.89053e15	−1.18930
$663$	0	0
$664$	−2.07377e14	−0.0623500
$665$	−1.70195e15	−0.507489
$666$	0	0
$667$	−2.35617e15	−0.691059
$668$	−3.02359e15	−0.879533
$669$	0	0
$670$	1.69487e15	0.484980
$671$	−5.56587e15	−1.57965
$672$	0	0
$673$	−9.26676e14	−0.258729	−0.129364	−	0.991597i	$-0.541294\pi$
$673$	−9.26676e14	−0.258729	−0.129364	+	0.991597i	$0.541294\pi$
$674$	−2.68191e15	−0.742702
$675$	0	0
$676$	−1.68225e13	−0.00458336
$677$	1.49664e14	0.0404465	0.0202232	−	0.999795i	$-0.493562\pi$
$677$	1.49664e14	0.0404465	0.0202232	+	0.999795i	$0.493562\pi$
$678$	0	0
$679$	−4.74106e15	−1.26064
$680$	1.96400e15	0.518015
$681$	0	0
$682$	−4.07141e15	−1.05665
$683$	−6.74856e15	−1.73739	−0.868694	−	0.495348i	$-0.835040\pi$
$683$	−6.74856e15	−1.73739	−0.868694	+	0.495348i	$0.835040\pi$
$684$	0	0
$685$	7.22704e15	1.83089
$686$	2.32654e15	0.584692
$687$	0	0
$688$	3.68295e14	0.0910876
$689$	1.95335e15	0.479264
$690$	0	0
$691$	−1.61272e15	−0.389430	−0.194715	−	0.980860i	$-0.562378\pi$
$691$	−1.61272e15	−0.389430	−0.194715	+	0.980860i	$0.562378\pi$
$692$	2.71780e15	0.651079
$693$	0	0
$694$	2.49426e15	0.588119
$695$	−6.13261e15	−1.43459
$696$	0	0
$697$	−2.10137e15	−0.483863
$698$	3.58972e15	0.820080
$699$	0	0
$700$	−4.54548e15	−1.02221
$701$	−4.16261e14	−0.0928789	−0.0464394	−	0.998921i	$-0.514787\pi$
$701$	−4.16261e14	−0.0928789	−0.0464394	+	0.998921i	$0.514787\pi$
$702$	0	0
$703$	−2.27772e15	−0.500319
$704$	5.70608e14	0.124362
$705$	0	0
$706$	5.11106e15	1.09669
$707$	6.89870e14	0.146879
$708$	0	0
$709$	−2.82972e15	−0.593184	−0.296592	−	0.955004i	$-0.595850\pi$
$709$	−2.82972e15	−0.593184	−0.296592	+	0.955004i	$0.595850\pi$
$710$	3.19416e15	0.664408
$711$	0	0
$712$	1.64245e15	0.336398
$713$	7.32548e15	1.48883
$714$	0	0
$715$	8.30663e15	1.66242
$716$	4.16204e15	0.826580
$717$	0	0
$718$	7.88410e13	0.0154194
$719$	−6.81122e15	−1.32195	−0.660977	−	0.750406i	$-0.729858\pi$
$719$	−6.81122e15	−1.32195	−0.660977	+	0.750406i	$0.729858\pi$
$720$	0	0
$721$	−3.25970e15	−0.623066
$722$	−3.45831e15	−0.656008
$723$	0	0
$724$	−1.72107e15	−0.321541
$725$	6.83548e15	1.26739
$726$	0	0
$727$	2.73191e15	0.498916	0.249458	−	0.968386i	$-0.419748\pi$
$727$	2.73191e15	0.498916	0.249458	+	0.968386i	$0.419748\pi$
$728$	−2.18363e15	−0.395782
$729$	0	0
$730$	7.55411e14	0.134868
$731$	1.79469e15	0.318012
$732$	0	0
$733$	−6.58368e15	−1.14920	−0.574602	−	0.818433i	$-0.694843\pi$
$733$	−6.58368e15	−1.14920	−0.574602	+	0.818433i	$0.694843\pi$
$734$	4.29057e15	0.743339
$735$	0	0
$736$	−1.02666e15	−0.175227
$737$	2.39952e15	0.406494
$738$	0	0
$739$	4.37564e15	0.730294	0.365147	−	0.930950i	$-0.381019\pi$
$739$	4.37564e15	0.730294	0.365147	+	0.930950i	$0.381019\pi$
$740$	−9.42954e15	−1.56213
$741$	0	0
$742$	−2.34575e15	−0.382877
$743$	−4.90059e15	−0.793981	−0.396990	−	0.917823i	$-0.629945\pi$
$743$	−4.90059e15	−0.793981	−0.396990	+	0.917823i	$0.629945\pi$
$744$	0	0
$745$	1.48988e16	2.37844
$746$	−4.02190e15	−0.637335
$747$	0	0
$748$	2.78055e15	0.434183
$749$	4.66774e15	0.723530
$750$	0	0
$751$	−7.04936e14	−0.107679	−0.0538394	−	0.998550i	$-0.517146\pi$
$751$	−7.04936e14	−0.107679	−0.0538394	+	0.998550i	$0.517146\pi$
$752$	−1.00476e14	−0.0152358
$753$	0	0
$754$	3.28373e15	0.490709
$755$	1.47060e16	2.18166
$756$	0	0
$757$	5.25516e15	0.768348	0.384174	−	0.923261i	$-0.374486\pi$
$757$	5.25516e15	0.768348	0.384174	+	0.923261i	$0.374486\pi$
$758$	1.79936e15	0.261179
$759$	0	0
$760$	1.11521e15	0.159544
$761$	−3.81336e15	−0.541617	−0.270808	−	0.962633i	$-0.587291\pi$
$761$	−3.81336e15	−0.541617	−0.270808	+	0.962633i	$0.587291\pi$
$762$	0	0
$763$	7.56968e15	1.05972
$764$	−4.18954e14	−0.0582309
$765$	0	0
$766$	−3.45043e15	−0.472732
$767$	−7.49056e15	−1.01892
$768$	0	0
$769$	−9.81241e15	−1.31577	−0.657886	−	0.753117i	$-0.728549\pi$
$769$	−9.81241e15	−1.31577	−0.657886	+	0.753117i	$0.728549\pi$
$770$	−9.97529e15	−1.32809
$771$	0	0
$772$	4.83231e15	0.634249
$773$	−7.09136e15	−0.924149	−0.462075	−	0.886841i	$-0.652895\pi$
$773$	−7.09136e15	−0.924149	−0.462075	+	0.886841i	$0.652895\pi$
$774$	0	0
$775$	−2.12519e16	−2.73048
$776$	3.10660e15	0.396320
$777$	0	0
$778$	−7.60413e15	−0.956448
$779$	−1.19321e15	−0.149026
$780$	0	0
$781$	4.52216e15	0.556885
$782$	−5.00291e15	−0.611766
$783$	0	0
$784$	5.48902e14	0.0661845
$785$	7.11668e15	0.852107
$786$	0	0
$787$	−2.46400e15	−0.290924	−0.145462	−	0.989364i	$-0.546467\pi$
$787$	−2.46400e15	−0.290924	−0.145462	+	0.989364i	$0.546467\pi$
$788$	3.69560e15	0.433303
$789$	0	0
$790$	−8.34741e15	−0.965168
$791$	1.18693e16	1.36288
$792$	0	0
$793$	−1.39567e16	−1.58045
$794$	9.15103e15	1.02910
$795$	0	0
$796$	1.76271e15	0.195506
$797$	−3.04936e15	−0.335883	−0.167941	−	0.985797i	$-0.553712\pi$
$797$	−3.04936e15	−0.335883	−0.167941	+	0.985797i	$0.553712\pi$
$798$	0	0
$799$	−4.89618e14	−0.0531925
$800$	2.97845e15	0.321363
$801$	0	0
$802$	2.98146e15	0.317300
$803$	1.06948e15	0.113041
$804$	0	0
$805$	1.79480e16	1.87128
$806$	−1.02093e16	−1.05719
$807$	0	0
$808$	−4.52041e14	−0.0461759
$809$	3.92265e15	0.397982	0.198991	−	0.980001i	$-0.436234\pi$
$809$	3.92265e15	0.397982	0.198991	+	0.980001i	$0.436234\pi$
$810$	0	0
$811$	1.75012e16	1.75168	0.875838	−	0.482606i	$-0.160310\pi$
$811$	1.75012e16	1.75168	0.875838	+	0.482606i	$0.160310\pi$
$812$	−3.94337e15	−0.392021
$813$	0	0
$814$	−1.33500e16	−1.30932
$815$	−1.85804e16	−1.81004
$816$	0	0
$817$	1.01907e15	0.0979451
$818$	8.22240e15	0.784974
$819$	0	0
$820$	−4.93977e15	−0.465298
$821$	−4.48148e15	−0.419309	−0.209655	−	0.977775i	$-0.567234\pi$
$821$	−4.48148e15	−0.419309	−0.209655	+	0.977775i	$0.567234\pi$
$822$	0	0
$823$	−2.38819e15	−0.220480	−0.110240	−	0.993905i	$-0.535162\pi$
$823$	−2.38819e15	−0.220480	−0.110240	+	0.993905i	$0.535162\pi$
$824$	2.13594e15	0.195879
$825$	0	0
$826$	8.99528e15	0.814000
$827$	−3.42245e15	−0.307650	−0.153825	−	0.988098i	$-0.549159\pi$
$827$	−3.42245e15	−0.307650	−0.153825	+	0.988098i	$0.549159\pi$
$828$	0	0
$829$	7.18881e15	0.637686	0.318843	−	0.947808i	$-0.396706\pi$
$829$	7.18881e15	0.637686	0.318843	+	0.947808i	$0.396706\pi$
$830$	−2.37552e15	−0.209329
$831$	0	0
$832$	1.43083e15	0.124426
$833$	2.67478e15	0.231068
$834$	0	0
$835$	−3.46355e16	−2.95288
$836$	1.57887e15	0.133725
$837$	0	0
$838$	9.47492e15	0.792015
$839$	−1.25101e16	−1.03889	−0.519444	−	0.854504i	$-0.673861\pi$
$839$	−1.25101e16	−1.03889	−0.519444	+	0.854504i	$0.673861\pi$
$840$	0	0
$841$	−6.27049e15	−0.513953
$842$	−1.08955e16	−0.887220
$843$	0	0
$844$	1.74433e15	0.140199
$845$	−1.92703e14	−0.0153878
$846$	0	0
$847$	1.45246e14	0.0114484
$848$	1.53706e15	0.120369
$849$	0	0
$850$	1.45139e16	1.12197
$851$	2.40199e16	1.84484
$852$	0	0
$853$	5.88516e15	0.446210	0.223105	−	0.974794i	$-0.428381\pi$
$853$	5.88516e15	0.446210	0.223105	+	0.974794i	$0.428381\pi$
$854$	1.67604e16	1.26260
$855$	0	0
$856$	−3.05856e15	−0.227463
$857$	−2.57995e15	−0.190641	−0.0953207	−	0.995447i	$-0.530388\pi$
$857$	−2.57995e15	−0.190641	−0.0953207	+	0.995447i	$0.530388\pi$
$858$	0	0
$859$	−3.14874e15	−0.229707	−0.114854	−	0.993382i	$-0.536640\pi$
$859$	−3.14874e15	−0.229707	−0.114854	+	0.993382i	$0.536640\pi$
$860$	4.21885e15	0.305810
$861$	0	0
$862$	−4.40924e15	−0.315554
$863$	2.28539e16	1.62518	0.812589	−	0.582837i	$-0.198058\pi$
$863$	2.28539e16	1.62518	0.812589	+	0.582837i	$0.198058\pi$
$864$	0	0
$865$	3.11326e16	2.18588
$866$	1.12761e16	0.786699
$867$	0	0
$868$	1.22602e16	0.844575
$869$	−1.18179e16	−0.808971
$870$	0	0
$871$	6.01695e15	0.406702
$872$	−4.96007e15	−0.333156
$873$	0	0
$874$	−2.84078e15	−0.188419
$875$	−2.34266e16	−1.54406
$876$	0	0
$877$	−1.83695e16	−1.19564	−0.597820	−	0.801630i	$-0.703966\pi$
$877$	−1.83695e16	−1.19564	−0.597820	+	0.801630i	$0.703966\pi$
$878$	−1.37339e16	−0.888331
$879$	0	0
$880$	6.53636e15	0.417524
$881$	1.17507e16	0.745925	0.372963	−	0.927846i	$-0.378342\pi$
$881$	1.17507e16	0.745925	0.372963	+	0.927846i	$0.378342\pi$
$882$	0	0
$883$	−7.76791e15	−0.486990	−0.243495	−	0.969902i	$-0.578294\pi$
$883$	−7.76791e15	−0.486990	−0.243495	+	0.969902i	$0.578294\pi$
$884$	6.97240e15	0.434405
$885$	0	0
$886$	1.15546e16	0.711000
$887$	−4.62750e15	−0.282987	−0.141493	−	0.989939i	$-0.545190\pi$
$887$	−4.62750e15	−0.282987	−0.141493	+	0.989939i	$0.545190\pi$
$888$	0	0
$889$	−2.57248e16	−1.55379
$890$	1.88144e16	1.12940
$891$	0	0
$892$	−1.49763e16	−0.887969
$893$	−2.78017e14	−0.0163828
$894$	0	0
$895$	4.76765e16	2.77510
$896$	−1.71826e15	−0.0994020
$897$	0	0
$898$	2.00860e14	0.0114782
$899$	−1.84368e16	−1.04715
$900$	0	0
$901$	7.49006e15	0.420241
$902$	−6.99354e15	−0.389997
$903$	0	0
$904$	−7.77745e15	−0.428461
$905$	−1.97150e16	−1.07952
$906$	0	0
$907$	1.92511e16	1.04140	0.520698	−	0.853741i	$-0.325672\pi$
$907$	1.92511e16	1.04140	0.520698	+	0.853741i	$0.325672\pi$
$908$	−3.44532e15	−0.185249
$909$	0	0
$910$	−2.50136e16	−1.32877
$911$	1.80098e16	0.950952	0.475476	−	0.879729i	$-0.342276\pi$
$911$	1.80098e16	0.950952	0.475476	+	0.879729i	$0.342276\pi$
$912$	0	0
$913$	−3.36317e15	−0.175452
$914$	1.47490e16	0.764819
$915$	0	0
$916$	1.94478e15	0.0996429
$917$	1.49310e16	0.760426
$918$	0	0
$919$	−5.42882e15	−0.273193	−0.136597	−	0.990627i	$-0.543616\pi$
$919$	−5.42882e15	−0.273193	−0.136597	+	0.990627i	$0.543616\pi$
$920$	−1.17605e16	−0.588293
$921$	0	0
$922$	−2.55168e16	−1.26126
$923$	1.13396e16	0.557170
$924$	0	0
$925$	−6.96840e16	−3.38340
$926$	−1.85887e16	−0.897199
$927$	0	0
$928$	2.58391e15	0.123244
$929$	−2.76310e16	−1.31012	−0.655059	−	0.755578i	$-0.727356\pi$
$929$	−2.76310e16	−1.31012	−0.655059	+	0.755578i	$0.727356\pi$
$930$	0	0
$931$	1.51881e15	0.0711671
$932$	−1.55547e15	−0.0724560
$933$	0	0
$934$	2.63210e16	1.21170
$935$	3.18515e16	1.45769
$936$	0	0
$937$	−2.46060e16	−1.11294	−0.556472	−	0.830867i	$-0.687845\pi$
$937$	−2.46060e16	−1.11294	−0.556472	+	0.830867i	$0.687845\pi$
$938$	−7.22565e15	−0.324909
$939$	0	0
$940$	−1.15096e15	−0.0511515
$941$	−3.11892e16	−1.37804	−0.689019	−	0.724743i	$-0.741958\pi$
$941$	−3.11892e16	−1.37804	−0.689019	+	0.724743i	$0.741958\pi$
$942$	0	0
$943$	1.25831e16	0.549507
$944$	−5.89421e15	−0.255905
$945$	0	0
$946$	5.97288e15	0.256320
$947$	3.17189e16	1.35330	0.676649	−	0.736305i	$-0.263431\pi$
$947$	3.17189e16	1.35330	0.676649	+	0.736305i	$0.263431\pi$
$948$	0	0
$949$	2.68179e15	0.113099
$950$	8.24136e15	0.345556
$951$	0	0
$952$	−8.37304e15	−0.347040
$953$	−9.40992e15	−0.387771	−0.193885	−	0.981024i	$-0.562109\pi$
$953$	−9.40992e15	−0.387771	−0.193885	+	0.981024i	$0.562109\pi$
$954$	0	0
$955$	−4.79916e15	−0.195500
$956$	−8.51453e15	−0.344859
$957$	0	0
$958$	1.56235e16	0.625559
$959$	−3.08107e16	−1.22659
$960$	0	0
$961$	3.19127e16	1.25599
$962$	−3.34758e16	−1.30999
$963$	0	0
$964$	−1.33249e16	−0.515515
$965$	5.53545e16	2.12938
$966$	0	0
$967$	−1.27731e16	−0.485792	−0.242896	−	0.970052i	$-0.578097\pi$
$967$	−1.27731e16	−0.485792	−0.242896	+	0.970052i	$0.578097\pi$
$968$	−9.51732e13	−0.00359915
$969$	0	0
$970$	3.55864e16	1.33057
$971$	−3.51105e16	−1.30536	−0.652681	−	0.757632i	$-0.726356\pi$
$971$	−3.51105e16	−1.30536	−0.652681	+	0.757632i	$0.726356\pi$
$972$	0	0
$973$	2.61449e16	0.961094
$974$	−9.51300e14	−0.0347731
$975$	0	0
$976$	−1.09824e16	−0.396937
$977$	5.08668e16	1.82816	0.914081	−	0.405532i	$-0.132914\pi$
$977$	5.08668e16	1.82816	0.914081	+	0.405532i	$0.132914\pi$
$978$	0	0
$979$	2.66367e16	0.946621
$980$	6.28771e15	0.222203
$981$	0	0
$982$	−2.61163e16	−0.912639
$983$	4.25676e16	1.47923	0.739614	−	0.673032i	$-0.235008\pi$
$983$	4.25676e16	1.47923	0.739614	+	0.673032i	$0.235008\pi$
$984$	0	0
$985$	4.23334e16	1.45474
$986$	1.25913e16	0.430277
$987$	0	0
$988$	3.95910e15	0.133793
$989$	−1.07467e16	−0.361156
$990$	0	0
$991$	2.24124e16	0.744873	0.372437	−	0.928058i	$-0.378522\pi$
$991$	2.24124e16	0.744873	0.372437	+	0.928058i	$0.378522\pi$
$992$	−8.03355e15	−0.265517
$993$	0	0
$994$	−1.36175e16	−0.445115
$995$	2.01920e16	0.656376
$996$	0	0
$997$	−2.03268e16	−0.653499	−0.326750	−	0.945111i	$-0.605953\pi$
$997$	−2.03268e16	−0.653499	−0.326750	+	0.945111i	$0.605953\pi$
$998$	1.95173e16	0.624026
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.12.a.e.1.1		1
3.2	odd	2		6.12.a.b.1.1	✓	1
4.3	odd	2		144.12.a.o.1.1		1
9.2	odd	6		162.12.c.j.109.1		2
9.4	even	3		162.12.c.a.55.1		2
9.5	odd	6		162.12.c.j.55.1		2
9.7	even	3		162.12.c.a.109.1		2
12.11	even	2		48.12.a.a.1.1		1
15.2	even	4		150.12.c.b.49.1		2
15.8	even	4		150.12.c.b.49.2		2
15.14	odd	2		150.12.a.f.1.1		1
24.5	odd	2		192.12.a.j.1.1		1
24.11	even	2		192.12.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.12.a.b.1.1	✓	1	3.2	odd	2
18.12.a.e.1.1		1	1.1	even	1	trivial
48.12.a.a.1.1		1	12.11	even	2
144.12.a.o.1.1		1	4.3	odd	2
150.12.a.f.1.1		1	15.14	odd	2
150.12.c.b.49.1		2	15.2	even	4
150.12.c.b.49.2		2	15.8	even	4
162.12.c.a.55.1		2	9.4	even	3
162.12.c.a.109.1		2	9.7	even	3
162.12.c.j.55.1		2	9.5	odd	6
162.12.c.j.109.1		2	9.2	odd	6
192.12.a.j.1.1		1	24.5	odd	2
192.12.a.t.1.1		1	24.11	even	2

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:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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