LMFDB - Embedded newform 1638.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,4,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1638.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1638.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:	$96.6451285894$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} -16.0000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} -16.0000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +32.0000 q^{10} +15.0000 q^{11} -13.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} +44.0000 q^{17} -138.000 q^{19} -64.0000 q^{20} -30.0000 q^{22} -111.000 q^{23} +131.000 q^{25} +26.0000 q^{26} +28.0000 q^{28} +12.0000 q^{29} +215.000 q^{31} -32.0000 q^{32} -88.0000 q^{34} -112.000 q^{35} +55.0000 q^{37} +276.000 q^{38} +128.000 q^{40} +133.000 q^{41} -180.000 q^{43} +60.0000 q^{44} +222.000 q^{46} -471.000 q^{47} +49.0000 q^{49} -262.000 q^{50} -52.0000 q^{52} +260.000 q^{53} -240.000 q^{55} -56.0000 q^{56} -24.0000 q^{58} -110.000 q^{59} -271.000 q^{61} -430.000 q^{62} +64.0000 q^{64} +208.000 q^{65} -799.000 q^{67} +176.000 q^{68} +224.000 q^{70} -912.000 q^{71} +747.000 q^{73} -110.000 q^{74} -552.000 q^{76} +105.000 q^{77} -883.000 q^{79} -256.000 q^{80} -266.000 q^{82} +924.000 q^{83} -704.000 q^{85} +360.000 q^{86} -120.000 q^{88} -142.000 q^{89} -91.0000 q^{91} -444.000 q^{92} +942.000 q^{94} +2208.00 q^{95} -1407.00 q^{97} -98.0000 q^{98} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−16.0000	−1.43108	−0.715542	−	0.698570i	$-0.753820\pi$
$5$	−16.0000	−1.43108	−0.715542	+	0.698570i	$0.753820\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	32.0000	1.01193
$11$	15.0000	0.411152	0.205576	−	0.978641i	$-0.434093\pi$
$11$	15.0000	0.411152	0.205576	+	0.978641i	$0.434093\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	44.0000	0.627739	0.313870	−	0.949466i	$-0.398375\pi$
$17$	44.0000	0.627739	0.313870	+	0.949466i	$0.398375\pi$
$18$	0	0
$19$	−138.000	−1.66628	−0.833141	−	0.553060i	$-0.813460\pi$
$19$	−138.000	−1.66628	−0.833141	+	0.553060i	$0.813460\pi$
$20$	−64.0000	−0.715542
$21$	0	0
$22$	−30.0000	−0.290728
$23$	−111.000	−1.00631	−0.503154	−	0.864197i	$-0.667827\pi$
$23$	−111.000	−1.00631	−0.503154	+	0.864197i	$0.667827\pi$
$24$	0	0
$25$	131.000	1.04800
$26$	26.0000	0.196116
$27$	0	0
$28$	28.0000	0.188982
$29$	12.0000	0.0768395	0.0384197	−	0.999262i	$-0.487768\pi$
$29$	12.0000	0.0768395	0.0384197	+	0.999262i	$0.487768\pi$
$30$	0	0
$31$	215.000	1.24565	0.622825	−	0.782361i	$-0.285985\pi$
$31$	215.000	1.24565	0.622825	+	0.782361i	$0.285985\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−88.0000	−0.443879
$35$	−112.000	−0.540899
$36$	0	0
$37$	55.0000	0.244377	0.122188	−	0.992507i	$-0.461009\pi$
$37$	55.0000	0.244377	0.122188	+	0.992507i	$0.461009\pi$
$38$	276.000	1.17824
$39$	0	0
$40$	128.000	0.505964
$41$	133.000	0.506612	0.253306	−	0.967386i	$-0.418482\pi$
$41$	133.000	0.506612	0.253306	+	0.967386i	$0.418482\pi$
$42$	0	0
$43$	−180.000	−0.638366	−0.319183	−	0.947693i	$-0.603408\pi$
$43$	−180.000	−0.638366	−0.319183	+	0.947693i	$0.603408\pi$
$44$	60.0000	0.205576
$45$	0	0
$46$	222.000	0.711568
$47$	−471.000	−1.46175	−0.730877	−	0.682510i	$-0.760889\pi$
$47$	−471.000	−1.46175	−0.730877	+	0.682510i	$0.760889\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−262.000	−0.741048
$51$	0	0
$52$	−52.0000	−0.138675
$53$	260.000	0.673844	0.336922	−	0.941533i	$-0.390614\pi$
$53$	260.000	0.673844	0.336922	+	0.941533i	$0.390614\pi$
$54$	0	0
$55$	−240.000	−0.588393
$56$	−56.0000	−0.133631
$57$	0	0
$58$	−24.0000	−0.0543337
$59$	−110.000	−0.242725	−0.121363	−	0.992608i	$-0.538726\pi$
$59$	−110.000	−0.242725	−0.121363	+	0.992608i	$0.538726\pi$
$60$	0	0
$61$	−271.000	−0.568820	−0.284410	−	0.958703i	$-0.591798\pi$
$61$	−271.000	−0.568820	−0.284410	+	0.958703i	$0.591798\pi$
$62$	−430.000	−0.880807
$63$	0	0
$64$	64.0000	0.125000
$65$	208.000	0.396911
$66$	0	0
$67$	−799.000	−1.45692	−0.728458	−	0.685090i	$-0.759763\pi$
$67$	−799.000	−1.45692	−0.728458	+	0.685090i	$0.759763\pi$
$68$	176.000	0.313870
$69$	0	0
$70$	224.000	0.382473
$71$	−912.000	−1.52443	−0.762215	−	0.647324i	$-0.775888\pi$
$71$	−912.000	−1.52443	−0.762215	+	0.647324i	$0.775888\pi$
$72$	0	0
$73$	747.000	1.19767	0.598834	−	0.800873i	$-0.295631\pi$
$73$	747.000	1.19767	0.598834	+	0.800873i	$0.295631\pi$
$74$	−110.000	−0.172801
$75$	0	0
$76$	−552.000	−0.833141
$77$	105.000	0.155401
$78$	0	0
$79$	−883.000	−1.25753	−0.628767	−	0.777593i	$-0.716440\pi$
$79$	−883.000	−1.25753	−0.628767	+	0.777593i	$0.716440\pi$
$80$	−256.000	−0.357771
$81$	0	0
$82$	−266.000	−0.358229
$83$	924.000	1.22195	0.610977	−	0.791648i	$-0.290777\pi$
$83$	924.000	1.22195	0.610977	+	0.791648i	$0.290777\pi$
$84$	0	0
$85$	−704.000	−0.898347
$86$	360.000	0.451393
$87$	0	0
$88$	−120.000	−0.145364
$89$	−142.000	−0.169123	−0.0845616	−	0.996418i	$-0.526949\pi$
$89$	−142.000	−0.169123	−0.0845616	+	0.996418i	$0.526949\pi$
$90$	0	0
$91$	−91.0000	−0.104828
$92$	−444.000	−0.503154
$93$	0	0
$94$	942.000	1.03362
$95$	2208.00	2.38459
$96$	0	0
$97$	−1407.00	−1.47278	−0.736388	−	0.676560i	$-0.763470\pi$
$97$	−1407.00	−1.47278	−0.736388	+	0.676560i	$0.763470\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	524.000	0.524000
$101$	−19.0000	−0.0187185	−0.00935926	−	0.999956i	$-0.502979\pi$
$101$	−19.0000	−0.0187185	−0.00935926	+	0.999956i	$0.502979\pi$
$102$	0	0
$103$	338.000	0.323341	0.161671	−	0.986845i	$-0.448312\pi$
$103$	338.000	0.323341	0.161671	+	0.986845i	$0.448312\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	−520.000	−0.476480
$107$	400.000	0.361397	0.180698	−	0.983539i	$-0.442164\pi$
$107$	400.000	0.361397	0.180698	+	0.983539i	$0.442164\pi$
$108$	0	0
$109$	790.000	0.694204	0.347102	−	0.937827i	$-0.387166\pi$
$109$	790.000	0.694204	0.347102	+	0.937827i	$0.387166\pi$
$110$	480.000	0.416056
$111$	0	0
$112$	112.000	0.0944911
$113$	−513.000	−0.427071	−0.213535	−	0.976935i	$-0.568498\pi$
$113$	−513.000	−0.427071	−0.213535	+	0.976935i	$0.568498\pi$
$114$	0	0
$115$	1776.00	1.44011
$116$	48.0000	0.0384197
$117$	0	0
$118$	220.000	0.171633
$119$	308.000	0.237263
$120$	0	0
$121$	−1106.00	−0.830954
$122$	542.000	0.402216
$123$	0	0
$124$	860.000	0.622825
$125$	−96.0000	−0.0686920
$126$	0	0
$127$	2291.00	1.60074	0.800368	−	0.599510i	$-0.204638\pi$
$127$	2291.00	1.60074	0.800368	+	0.599510i	$0.204638\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−416.000	−0.280659
$131$	−1584.00	−1.05645	−0.528224	−	0.849105i	$-0.677142\pi$
$131$	−1584.00	−1.05645	−0.528224	+	0.849105i	$0.677142\pi$
$132$	0	0
$133$	−966.000	−0.629796
$134$	1598.00	1.03020
$135$	0	0
$136$	−352.000	−0.221939
$137$	−90.0000	−0.0561257	−0.0280628	−	0.999606i	$-0.508934\pi$
$137$	−90.0000	−0.0561257	−0.0280628	+	0.999606i	$0.508934\pi$
$138$	0	0
$139$	3220.00	1.96487	0.982435	−	0.186607i	$-0.0597491\pi$
$139$	3220.00	1.96487	0.982435	+	0.186607i	$0.0597491\pi$
$140$	−448.000	−0.270449
$141$	0	0
$142$	1824.00	1.07793
$143$	−195.000	−0.114033
$144$	0	0
$145$	−192.000	−0.109964
$146$	−1494.00	−0.846879
$147$	0	0
$148$	220.000	0.122188
$149$	2409.00	1.32452	0.662258	−	0.749276i	$-0.269598\pi$
$149$	2409.00	1.32452	0.662258	+	0.749276i	$0.269598\pi$
$150$	0	0
$151$	−1100.00	−0.592826	−0.296413	−	0.955060i	$-0.595790\pi$
$151$	−1100.00	−0.592826	−0.296413	+	0.955060i	$0.595790\pi$
$152$	1104.00	0.589120
$153$	0	0
$154$	−210.000	−0.109885
$155$	−3440.00	−1.78263
$156$	0	0
$157$	3421.00	1.73902	0.869508	−	0.493919i	$-0.164436\pi$
$157$	3421.00	1.73902	0.869508	+	0.493919i	$0.164436\pi$
$158$	1766.00	0.889211
$159$	0	0
$160$	512.000	0.252982
$161$	−777.000	−0.380349
$162$	0	0
$163$	−1156.00	−0.555490	−0.277745	−	0.960655i	$-0.589587\pi$
$163$	−1156.00	−0.555490	−0.277745	+	0.960655i	$0.589587\pi$
$164$	532.000	0.253306
$165$	0	0
$166$	−1848.00	−0.864052
$167$	−3080.00	−1.42717	−0.713585	−	0.700568i	$-0.752930\pi$
$167$	−3080.00	−1.42717	−0.713585	+	0.700568i	$0.752930\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	1408.00	0.635227
$171$	0	0
$172$	−720.000	−0.319183
$173$	1622.00	0.712823	0.356411	−	0.934329i	$-0.384000\pi$
$173$	1622.00	0.712823	0.356411	+	0.934329i	$0.384000\pi$
$174$	0	0
$175$	917.000	0.396107
$176$	240.000	0.102788
$177$	0	0
$178$	284.000	0.119588
$179$	−1770.00	−0.739084	−0.369542	−	0.929214i	$-0.620485\pi$
$179$	−1770.00	−0.739084	−0.369542	+	0.929214i	$0.620485\pi$
$180$	0	0
$181$	4375.00	1.79664	0.898318	−	0.439345i	$-0.144790\pi$
$181$	4375.00	1.79664	0.898318	+	0.439345i	$0.144790\pi$
$182$	182.000	0.0741249
$183$	0	0
$184$	888.000	0.355784
$185$	−880.000	−0.349724
$186$	0	0
$187$	660.000	0.258096
$188$	−1884.00	−0.730877
$189$	0	0
$190$	−4416.00	−1.68616
$191$	−2232.00	−0.845559	−0.422780	−	0.906232i	$-0.638946\pi$
$191$	−2232.00	−0.845559	−0.422780	+	0.906232i	$0.638946\pi$
$192$	0	0
$193$	−1268.00	−0.472915	−0.236458	−	0.971642i	$-0.575986\pi$
$193$	−1268.00	−0.472915	−0.236458	+	0.971642i	$0.575986\pi$
$194$	2814.00	1.04141
$195$	0	0
$196$	196.000	0.0714286
$197$	1405.00	0.508133	0.254066	−	0.967187i	$-0.418232\pi$
$197$	1405.00	0.508133	0.254066	+	0.967187i	$0.418232\pi$
$198$	0	0
$199$	3708.00	1.32087	0.660435	−	0.750883i	$-0.270372\pi$
$199$	3708.00	1.32087	0.660435	+	0.750883i	$0.270372\pi$
$200$	−1048.00	−0.370524
$201$	0	0
$202$	38.0000	0.0132360
$203$	84.0000	0.0290426
$204$	0	0
$205$	−2128.00	−0.725005
$206$	−676.000	−0.228637
$207$	0	0
$208$	−208.000	−0.0693375
$209$	−2070.00	−0.685095
$210$	0	0
$211$	4622.00	1.50802	0.754009	−	0.656865i	$-0.228118\pi$
$211$	4622.00	1.50802	0.754009	+	0.656865i	$0.228118\pi$
$212$	1040.00	0.336922
$213$	0	0
$214$	−800.000	−0.255546
$215$	2880.00	0.913555
$216$	0	0
$217$	1505.00	0.470811
$218$	−1580.00	−0.490877
$219$	0	0
$220$	−960.000	−0.294196
$221$	−572.000	−0.174104
$222$	0	0
$223$	3101.00	0.931203	0.465602	−	0.884994i	$-0.345838\pi$
$223$	3101.00	0.931203	0.465602	+	0.884994i	$0.345838\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	1026.00	0.301985
$227$	6484.00	1.89585	0.947926	−	0.318492i	$-0.103176\pi$
$227$	6484.00	1.89585	0.947926	+	0.318492i	$0.103176\pi$
$228$	0	0
$229$	−4688.00	−1.35280	−0.676401	−	0.736533i	$-0.736461\pi$
$229$	−4688.00	−1.35280	−0.676401	+	0.736533i	$0.736461\pi$
$230$	−3552.00	−1.01831
$231$	0	0
$232$	−96.0000	−0.0271668
$233$	3389.00	0.952879	0.476439	−	0.879207i	$-0.341927\pi$
$233$	3389.00	0.952879	0.476439	+	0.879207i	$0.341927\pi$
$234$	0	0
$235$	7536.00	2.09189
$236$	−440.000	−0.121363
$237$	0	0
$238$	−616.000	−0.167770
$239$	26.0000	0.00703682	0.00351841	−	0.999994i	$-0.498880\pi$
$239$	26.0000	0.00703682	0.00351841	+	0.999994i	$0.498880\pi$
$240$	0	0
$241$	−3390.00	−0.906096	−0.453048	−	0.891486i	$-0.649663\pi$
$241$	−3390.00	−0.906096	−0.453048	+	0.891486i	$0.649663\pi$
$242$	2212.00	0.587573
$243$	0	0
$244$	−1084.00	−0.284410
$245$	−784.000	−0.204441
$246$	0	0
$247$	1794.00	0.462144
$248$	−1720.00	−0.440404
$249$	0	0
$250$	192.000	0.0485726
$251$	1435.00	0.360862	0.180431	−	0.983588i	$-0.442251\pi$
$251$	1435.00	0.360862	0.180431	+	0.983588i	$0.442251\pi$
$252$	0	0
$253$	−1665.00	−0.413746
$254$	−4582.00	−1.13189
$255$	0	0
$256$	256.000	0.0625000
$257$	2756.00	0.668928	0.334464	−	0.942408i	$-0.391445\pi$
$257$	2756.00	0.668928	0.334464	+	0.942408i	$0.391445\pi$
$258$	0	0
$259$	385.000	0.0923658
$260$	832.000	0.198456
$261$	0	0
$262$	3168.00	0.747022
$263$	2696.00	0.632101	0.316050	−	0.948742i	$-0.397643\pi$
$263$	2696.00	0.632101	0.316050	+	0.948742i	$0.397643\pi$
$264$	0	0
$265$	−4160.00	−0.964327
$266$	1932.00	0.445333
$267$	0	0
$268$	−3196.00	−0.728458
$269$	8031.00	1.82029	0.910146	−	0.414287i	$-0.135969\pi$
$269$	8031.00	1.82029	0.910146	+	0.414287i	$0.135969\pi$
$270$	0	0
$271$	849.000	0.190307	0.0951533	−	0.995463i	$-0.469666\pi$
$271$	849.000	0.190307	0.0951533	+	0.995463i	$0.469666\pi$
$272$	704.000	0.156935
$273$	0	0
$274$	180.000	0.0396869
$275$	1965.00	0.430887
$276$	0	0
$277$	−3634.00	−0.788252	−0.394126	−	0.919056i	$-0.628953\pi$
$277$	−3634.00	−0.788252	−0.394126	+	0.919056i	$0.628953\pi$
$278$	−6440.00	−1.38937
$279$	0	0
$280$	896.000	0.191237
$281$	2280.00	0.484033	0.242017	−	0.970272i	$-0.422191\pi$
$281$	2280.00	0.484033	0.242017	+	0.970272i	$0.422191\pi$
$282$	0	0
$283$	2693.00	0.565662	0.282831	−	0.959170i	$-0.408727\pi$
$283$	2693.00	0.565662	0.282831	+	0.959170i	$0.408727\pi$
$284$	−3648.00	−0.762215
$285$	0	0
$286$	390.000	0.0806335
$287$	931.000	0.191482
$288$	0	0
$289$	−2977.00	−0.605943
$290$	384.000	0.0777561
$291$	0	0
$292$	2988.00	0.598834
$293$	−7938.00	−1.58274	−0.791370	−	0.611337i	$-0.790632\pi$
$293$	−7938.00	−1.58274	−0.791370	+	0.611337i	$0.790632\pi$
$294$	0	0
$295$	1760.00	0.347360
$296$	−440.000	−0.0864003
$297$	0	0
$298$	−4818.00	−0.936575
$299$	1443.00	0.279100
$300$	0	0
$301$	−1260.00	−0.241280
$302$	2200.00	0.419191
$303$	0	0
$304$	−2208.00	−0.416571
$305$	4336.00	0.814028
$306$	0	0
$307$	1680.00	0.312321	0.156161	−	0.987732i	$-0.450088\pi$
$307$	1680.00	0.312321	0.156161	+	0.987732i	$0.450088\pi$
$308$	420.000	0.0777004
$309$	0	0
$310$	6880.00	1.26051
$311$	−8342.00	−1.52100	−0.760501	−	0.649337i	$-0.775046\pi$
$311$	−8342.00	−1.52100	−0.760501	+	0.649337i	$0.775046\pi$
$312$	0	0
$313$	−2294.00	−0.414264	−0.207132	−	0.978313i	$-0.566413\pi$
$313$	−2294.00	−0.414264	−0.207132	+	0.978313i	$0.566413\pi$
$314$	−6842.00	−1.22967
$315$	0	0
$316$	−3532.00	−0.628767
$317$	4509.00	0.798898	0.399449	−	0.916755i	$-0.369201\pi$
$317$	4509.00	0.798898	0.399449	+	0.916755i	$0.369201\pi$
$318$	0	0
$319$	180.000	0.0315927
$320$	−1024.00	−0.178885
$321$	0	0
$322$	1554.00	0.268947
$323$	−6072.00	−1.04599
$324$	0	0
$325$	−1703.00	−0.290663
$326$	2312.00	0.392791
$327$	0	0
$328$	−1064.00	−0.179115
$329$	−3297.00	−0.552491
$330$	0	0
$331$	10121.0	1.68067	0.840333	−	0.542070i	$-0.182359\pi$
$331$	10121.0	1.68067	0.840333	+	0.542070i	$0.182359\pi$
$332$	3696.00	0.610977
$333$	0	0
$334$	6160.00	1.00916
$335$	12784.0	2.08497
$336$	0	0
$337$	−1055.00	−0.170533	−0.0852663	−	0.996358i	$-0.527174\pi$
$337$	−1055.00	−0.170533	−0.0852663	+	0.996358i	$0.527174\pi$
$338$	−338.000	−0.0543928
$339$	0	0
$340$	−2816.00	−0.449174
$341$	3225.00	0.512151
$342$	0	0
$343$	343.000	0.0539949
$344$	1440.00	0.225697
$345$	0	0
$346$	−3244.00	−0.504042
$347$	6280.00	0.971551	0.485775	−	0.874084i	$-0.338537\pi$
$347$	6280.00	0.971551	0.485775	+	0.874084i	$0.338537\pi$
$348$	0	0
$349$	10514.0	1.61261	0.806306	−	0.591499i	$-0.201464\pi$
$349$	10514.0	1.61261	0.806306	+	0.591499i	$0.201464\pi$
$350$	−1834.00	−0.280090
$351$	0	0
$352$	−480.000	−0.0726821
$353$	−2007.00	−0.302611	−0.151306	−	0.988487i	$-0.548348\pi$
$353$	−2007.00	−0.302611	−0.151306	+	0.988487i	$0.548348\pi$
$354$	0	0
$355$	14592.0	2.18159
$356$	−568.000	−0.0845616
$357$	0	0
$358$	3540.00	0.522611
$359$	5972.00	0.877967	0.438983	−	0.898495i	$-0.355339\pi$
$359$	5972.00	0.877967	0.438983	+	0.898495i	$0.355339\pi$
$360$	0	0
$361$	12185.0	1.77650
$362$	−8750.00	−1.27041
$363$	0	0
$364$	−364.000	−0.0524142
$365$	−11952.0	−1.71396
$366$	0	0
$367$	−72.0000	−0.0102408	−0.00512039	−	0.999987i	$-0.501630\pi$
$367$	−72.0000	−0.0102408	−0.00512039	+	0.999987i	$0.501630\pi$
$368$	−1776.00	−0.251577
$369$	0	0
$370$	1760.00	0.247292
$371$	1820.00	0.254689
$372$	0	0
$373$	9988.00	1.38649	0.693243	−	0.720704i	$-0.256181\pi$
$373$	9988.00	1.38649	0.693243	+	0.720704i	$0.256181\pi$
$374$	−1320.00	−0.182502
$375$	0	0
$376$	3768.00	0.516808
$377$	−156.000	−0.0213114
$378$	0	0
$379$	7296.00	0.988840	0.494420	−	0.869223i	$-0.335380\pi$
$379$	7296.00	0.988840	0.494420	+	0.869223i	$0.335380\pi$
$380$	8832.00	1.19229
$381$	0	0
$382$	4464.00	0.597901
$383$	7103.00	0.947641	0.473820	−	0.880622i	$-0.342875\pi$
$383$	7103.00	0.947641	0.473820	+	0.880622i	$0.342875\pi$
$384$	0	0
$385$	−1680.00	−0.222392
$386$	2536.00	0.334402
$387$	0	0
$388$	−5628.00	−0.736388
$389$	5048.00	0.657953	0.328977	−	0.944338i	$-0.393296\pi$
$389$	5048.00	0.657953	0.328977	+	0.944338i	$0.393296\pi$
$390$	0	0
$391$	−4884.00	−0.631699
$392$	−392.000	−0.0505076
$393$	0	0
$394$	−2810.00	−0.359304
$395$	14128.0	1.79964
$396$	0	0
$397$	3208.00	0.405554	0.202777	−	0.979225i	$-0.435003\pi$
$397$	3208.00	0.405554	0.202777	+	0.979225i	$0.435003\pi$
$398$	−7416.00	−0.933996
$399$	0	0
$400$	2096.00	0.262000
$401$	−9372.00	−1.16712	−0.583560	−	0.812070i	$-0.698341\pi$
$401$	−9372.00	−1.16712	−0.583560	+	0.812070i	$0.698341\pi$
$402$	0	0
$403$	−2795.00	−0.345481
$404$	−76.0000	−0.00935926
$405$	0	0
$406$	−168.000	−0.0205362
$407$	825.000	0.100476
$408$	0	0
$409$	13018.0	1.57384	0.786918	−	0.617058i	$-0.211676\pi$
$409$	13018.0	1.57384	0.786918	+	0.617058i	$0.211676\pi$
$410$	4256.00	0.512656
$411$	0	0
$412$	1352.00	0.161671
$413$	−770.000	−0.0917415
$414$	0	0
$415$	−14784.0	−1.74872
$416$	416.000	0.0490290
$417$	0	0
$418$	4140.00	0.484435
$419$	3591.00	0.418692	0.209346	−	0.977842i	$-0.432867\pi$
$419$	3591.00	0.418692	0.209346	+	0.977842i	$0.432867\pi$
$420$	0	0
$421$	−13851.0	−1.60346	−0.801730	−	0.597687i	$-0.796087\pi$
$421$	−13851.0	−1.60346	−0.801730	+	0.597687i	$0.796087\pi$
$422$	−9244.00	−1.06633
$423$	0	0
$424$	−2080.00	−0.238240
$425$	5764.00	0.657871
$426$	0	0
$427$	−1897.00	−0.214994
$428$	1600.00	0.180698
$429$	0	0
$430$	−5760.00	−0.645981
$431$	−5942.00	−0.664074	−0.332037	−	0.943266i	$-0.607736\pi$
$431$	−5942.00	−0.664074	−0.332037	+	0.943266i	$0.607736\pi$
$432$	0	0
$433$	5698.00	0.632398	0.316199	−	0.948693i	$-0.397593\pi$
$433$	5698.00	0.632398	0.316199	+	0.948693i	$0.397593\pi$
$434$	−3010.00	−0.332914
$435$	0	0
$436$	3160.00	0.347102
$437$	15318.0	1.67679
$438$	0	0
$439$	15682.0	1.70492	0.852461	−	0.522790i	$-0.175109\pi$
$439$	15682.0	1.70492	0.852461	+	0.522790i	$0.175109\pi$
$440$	1920.00	0.208028
$441$	0	0
$442$	1144.00	0.123110
$443$	−5844.00	−0.626765	−0.313382	−	0.949627i	$-0.601462\pi$
$443$	−5844.00	−0.626765	−0.313382	+	0.949627i	$0.601462\pi$
$444$	0	0
$445$	2272.00	0.242030
$446$	−6202.00	−0.658460
$447$	0	0
$448$	448.000	0.0472456
$449$	−3194.00	−0.335711	−0.167855	−	0.985812i	$-0.553684\pi$
$449$	−3194.00	−0.335711	−0.167855	+	0.985812i	$0.553684\pi$
$450$	0	0
$451$	1995.00	0.208295
$452$	−2052.00	−0.213535
$453$	0	0
$454$	−12968.0	−1.34057
$455$	1456.00	0.150018
$456$	0	0
$457$	−13994.0	−1.43241	−0.716205	−	0.697890i	$-0.754123\pi$
$457$	−13994.0	−1.43241	−0.716205	+	0.697890i	$0.754123\pi$
$458$	9376.00	0.956576
$459$	0	0
$460$	7104.00	0.720056
$461$	10052.0	1.01555	0.507775	−	0.861490i	$-0.330468\pi$
$461$	10052.0	1.01555	0.507775	+	0.861490i	$0.330468\pi$
$462$	0	0
$463$	−5276.00	−0.529582	−0.264791	−	0.964306i	$-0.585303\pi$
$463$	−5276.00	−0.529582	−0.264791	+	0.964306i	$0.585303\pi$
$464$	192.000	0.0192099
$465$	0	0
$466$	−6778.00	−0.673787
$467$	9924.00	0.983358	0.491679	−	0.870777i	$-0.336383\pi$
$467$	9924.00	0.983358	0.491679	+	0.870777i	$0.336383\pi$
$468$	0	0
$469$	−5593.00	−0.550663
$470$	−15072.0	−1.47919
$471$	0	0
$472$	880.000	0.0858163
$473$	−2700.00	−0.262465
$474$	0	0
$475$	−18078.0	−1.74626
$476$	1232.00	0.118632
$477$	0	0
$478$	−52.0000	−0.00497578
$479$	11328.0	1.08056	0.540281	−	0.841484i	$-0.318318\pi$
$479$	11328.0	1.08056	0.540281	+	0.841484i	$0.318318\pi$
$480$	0	0
$481$	−715.000	−0.0677779
$482$	6780.00	0.640707
$483$	0	0
$484$	−4424.00	−0.415477
$485$	22512.0	2.10766
$486$	0	0
$487$	−7470.00	−0.695068	−0.347534	−	0.937667i	$-0.612981\pi$
$487$	−7470.00	−0.695068	−0.347534	+	0.937667i	$0.612981\pi$
$488$	2168.00	0.201108
$489$	0	0
$490$	1568.00	0.144561
$491$	572.000	0.0525743	0.0262872	−	0.999654i	$-0.491632\pi$
$491$	572.000	0.0525743	0.0262872	+	0.999654i	$0.491632\pi$
$492$	0	0
$493$	528.000	0.0482351
$494$	−3588.00	−0.326785
$495$	0	0
$496$	3440.00	0.311412
$497$	−6384.00	−0.576180
$498$	0	0
$499$	−12125.0	−1.08775	−0.543877	−	0.839165i	$-0.683044\pi$
$499$	−12125.0	−1.08775	−0.543877	+	0.839165i	$0.683044\pi$
$500$	−384.000	−0.0343460
$501$	0	0
$502$	−2870.00	−0.255168
$503$	−7336.00	−0.650290	−0.325145	−	0.945664i	$-0.605413\pi$
$503$	−7336.00	−0.650290	−0.325145	+	0.945664i	$0.605413\pi$
$504$	0	0
$505$	304.000	0.0267878
$506$	3330.00	0.292562
$507$	0	0
$508$	9164.00	0.800368
$509$	20970.0	1.82609	0.913044	−	0.407861i	$-0.133725\pi$
$509$	20970.0	1.82609	0.913044	+	0.407861i	$0.133725\pi$
$510$	0	0
$511$	5229.00	0.452676
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−5512.00	−0.473004
$515$	−5408.00	−0.462728
$516$	0	0
$517$	−7065.00	−0.601003
$518$	−770.000	−0.0653125
$519$	0	0
$520$	−1664.00	−0.140329
$521$	−15202.0	−1.27833	−0.639167	−	0.769068i	$-0.720721\pi$
$521$	−15202.0	−1.27833	−0.639167	+	0.769068i	$0.720721\pi$
$522$	0	0
$523$	12357.0	1.03314	0.516572	−	0.856244i	$-0.327208\pi$
$523$	12357.0	1.03314	0.516572	+	0.856244i	$0.327208\pi$
$524$	−6336.00	−0.528224
$525$	0	0
$526$	−5392.00	−0.446963
$527$	9460.00	0.781943
$528$	0	0
$529$	154.000	0.0126572
$530$	8320.00	0.681882
$531$	0	0
$532$	−3864.00	−0.314898
$533$	−1729.00	−0.140509
$534$	0	0
$535$	−6400.00	−0.517189
$536$	6392.00	0.515098
$537$	0	0
$538$	−16062.0	−1.28714
$539$	735.000	0.0587360
$540$	0	0
$541$	10618.0	0.843814	0.421907	−	0.906639i	$-0.361361\pi$
$541$	10618.0	0.843814	0.421907	+	0.906639i	$0.361361\pi$
$542$	−1698.00	−0.134567
$543$	0	0
$544$	−1408.00	−0.110970
$545$	−12640.0	−0.993464
$546$	0	0
$547$	−3232.00	−0.252633	−0.126317	−	0.991990i	$-0.540316\pi$
$547$	−3232.00	−0.252633	−0.126317	+	0.991990i	$0.540316\pi$
$548$	−360.000	−0.0280628
$549$	0	0
$550$	−3930.00	−0.304683
$551$	−1656.00	−0.128036
$552$	0	0
$553$	−6181.00	−0.475304
$554$	7268.00	0.557379
$555$	0	0
$556$	12880.0	0.982435
$557$	−21671.0	−1.64853	−0.824264	−	0.566206i	$-0.808411\pi$
$557$	−21671.0	−1.64853	−0.824264	+	0.566206i	$0.808411\pi$
$558$	0	0
$559$	2340.00	0.177051
$560$	−1792.00	−0.135225
$561$	0	0
$562$	−4560.00	−0.342263
$563$	−14397.0	−1.07773	−0.538864	−	0.842393i	$-0.681146\pi$
$563$	−14397.0	−1.07773	−0.538864	+	0.842393i	$0.681146\pi$
$564$	0	0
$565$	8208.00	0.611174
$566$	−5386.00	−0.399983
$567$	0	0
$568$	7296.00	0.538967
$569$	14715.0	1.08416	0.542078	−	0.840328i	$-0.317638\pi$
$569$	14715.0	1.08416	0.542078	+	0.840328i	$0.317638\pi$
$570$	0	0
$571$	3002.00	0.220017	0.110009	−	0.993931i	$-0.464912\pi$
$571$	3002.00	0.220017	0.110009	+	0.993931i	$0.464912\pi$
$572$	−780.000	−0.0570165
$573$	0	0
$574$	−1862.00	−0.135398
$575$	−14541.0	−1.05461
$576$	0	0
$577$	13074.0	0.943289	0.471644	−	0.881789i	$-0.343661\pi$
$577$	13074.0	0.943289	0.471644	+	0.881789i	$0.343661\pi$
$578$	5954.00	0.428467
$579$	0	0
$580$	−768.000	−0.0549818
$581$	6468.00	0.461855
$582$	0	0
$583$	3900.00	0.277052
$584$	−5976.00	−0.423439
$585$	0	0
$586$	15876.0	1.11917
$587$	9282.00	0.652656	0.326328	−	0.945257i	$-0.394189\pi$
$587$	9282.00	0.652656	0.326328	+	0.945257i	$0.394189\pi$
$588$	0	0
$589$	−29670.0	−2.07560
$590$	−3520.00	−0.245621
$591$	0	0
$592$	880.000	0.0610942
$593$	−18146.0	−1.25661	−0.628303	−	0.777969i	$-0.716250\pi$
$593$	−18146.0	−1.25661	−0.628303	+	0.777969i	$0.716250\pi$
$594$	0	0
$595$	−4928.00	−0.339543
$596$	9636.00	0.662258
$597$	0	0
$598$	−2886.00	−0.197353
$599$	11859.0	0.808924	0.404462	−	0.914555i	$-0.367459\pi$
$599$	11859.0	0.808924	0.404462	+	0.914555i	$0.367459\pi$
$600$	0	0
$601$	5866.00	0.398135	0.199067	−	0.979986i	$-0.436209\pi$
$601$	5866.00	0.398135	0.199067	+	0.979986i	$0.436209\pi$
$602$	2520.00	0.170611
$603$	0	0
$604$	−4400.00	−0.296413
$605$	17696.0	1.18916
$606$	0	0
$607$	−26486.0	−1.77106	−0.885530	−	0.464582i	$-0.846205\pi$
$607$	−26486.0	−1.77106	−0.885530	+	0.464582i	$0.846205\pi$
$608$	4416.00	0.294560
$609$	0	0
$610$	−8672.00	−0.575605
$611$	6123.00	0.405417
$612$	0	0
$613$	2225.00	0.146602	0.0733009	−	0.997310i	$-0.476647\pi$
$613$	2225.00	0.146602	0.0733009	+	0.997310i	$0.476647\pi$
$614$	−3360.00	−0.220845
$615$	0	0
$616$	−840.000	−0.0549425
$617$	14138.0	0.922487	0.461243	−	0.887274i	$-0.347403\pi$
$617$	14138.0	0.922487	0.461243	+	0.887274i	$0.347403\pi$
$618$	0	0
$619$	−17110.0	−1.11100	−0.555500	−	0.831517i	$-0.687473\pi$
$619$	−17110.0	−1.11100	−0.555500	+	0.831517i	$0.687473\pi$
$620$	−13760.0	−0.891314
$621$	0	0
$622$	16684.0	1.07551
$623$	−994.000	−0.0639226
$624$	0	0
$625$	−14839.0	−0.949696
$626$	4588.00	0.292929
$627$	0	0
$628$	13684.0	0.869508
$629$	2420.00	0.153405
$630$	0	0
$631$	−10750.0	−0.678210	−0.339105	−	0.940748i	$-0.610124\pi$
$631$	−10750.0	−0.678210	−0.339105	+	0.940748i	$0.610124\pi$
$632$	7064.00	0.444606
$633$	0	0
$634$	−9018.00	−0.564906
$635$	−36656.0	−2.29079
$636$	0	0
$637$	−637.000	−0.0396214
$638$	−360.000	−0.0223394
$639$	0	0
$640$	2048.00	0.126491
$641$	8289.00	0.510758	0.255379	−	0.966841i	$-0.417800\pi$
$641$	8289.00	0.510758	0.255379	+	0.966841i	$0.417800\pi$
$642$	0	0
$643$	−1316.00	−0.0807122	−0.0403561	−	0.999185i	$-0.512849\pi$
$643$	−1316.00	−0.0807122	−0.0403561	+	0.999185i	$0.512849\pi$
$644$	−3108.00	−0.190174
$645$	0	0
$646$	12144.0	0.739627
$647$	−22356.0	−1.35843	−0.679216	−	0.733939i	$-0.737680\pi$
$647$	−22356.0	−1.35843	−0.679216	+	0.733939i	$0.737680\pi$
$648$	0	0
$649$	−1650.00	−0.0997969
$650$	3406.00	0.205530
$651$	0	0
$652$	−4624.00	−0.277745
$653$	22884.0	1.37139	0.685697	−	0.727887i	$-0.259498\pi$
$653$	22884.0	1.37139	0.685697	+	0.727887i	$0.259498\pi$
$654$	0	0
$655$	25344.0	1.51187
$656$	2128.00	0.126653
$657$	0	0
$658$	6594.00	0.390670
$659$	−15318.0	−0.905470	−0.452735	−	0.891645i	$-0.649552\pi$
$659$	−15318.0	−0.905470	−0.452735	+	0.891645i	$0.649552\pi$
$660$	0	0
$661$	−19420.0	−1.14274	−0.571369	−	0.820693i	$-0.693588\pi$
$661$	−19420.0	−1.14274	−0.571369	+	0.820693i	$0.693588\pi$
$662$	−20242.0	−1.18841
$663$	0	0
$664$	−7392.00	−0.432026
$665$	15456.0	0.901290
$666$	0	0
$667$	−1332.00	−0.0773242
$668$	−12320.0	−0.713585
$669$	0	0
$670$	−25568.0	−1.47430
$671$	−4065.00	−0.233871
$672$	0	0
$673$	695.000	0.0398073	0.0199036	−	0.999802i	$-0.493664\pi$
$673$	695.000	0.0398073	0.0199036	+	0.999802i	$0.493664\pi$
$674$	2110.00	0.120585
$675$	0	0
$676$	676.000	0.0384615
$677$	4625.00	0.262560	0.131280	−	0.991345i	$-0.458091\pi$
$677$	4625.00	0.262560	0.131280	+	0.991345i	$0.458091\pi$
$678$	0	0
$679$	−9849.00	−0.556657
$680$	5632.00	0.317614
$681$	0	0
$682$	−6450.00	−0.362146
$683$	−7153.00	−0.400735	−0.200367	−	0.979721i	$-0.564214\pi$
$683$	−7153.00	−0.400735	−0.200367	+	0.979721i	$0.564214\pi$
$684$	0	0
$685$	1440.00	0.0803205
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−2880.00	−0.159592
$689$	−3380.00	−0.186891
$690$	0	0
$691$	−15748.0	−0.866979	−0.433489	−	0.901159i	$-0.642718\pi$
$691$	−15748.0	−0.866979	−0.433489	+	0.901159i	$0.642718\pi$
$692$	6488.00	0.356411
$693$	0	0
$694$	−12560.0	−0.686990
$695$	−51520.0	−2.81189
$696$	0	0
$697$	5852.00	0.318021
$698$	−21028.0	−1.14029
$699$	0	0
$700$	3668.00	0.198053
$701$	−16620.0	−0.895476	−0.447738	−	0.894165i	$-0.647770\pi$
$701$	−16620.0	−0.895476	−0.447738	+	0.894165i	$0.647770\pi$
$702$	0	0
$703$	−7590.00	−0.407201
$704$	960.000	0.0513940
$705$	0	0
$706$	4014.00	0.213979
$707$	−133.000	−0.00707494
$708$	0	0
$709$	−19475.0	−1.03159	−0.515796	−	0.856711i	$-0.672504\pi$
$709$	−19475.0	−1.03159	−0.515796	+	0.856711i	$0.672504\pi$
$710$	−29184.0	−1.54261
$711$	0	0
$712$	1136.00	0.0597941
$713$	−23865.0	−1.25351
$714$	0	0
$715$	3120.00	0.163191
$716$	−7080.00	−0.369542
$717$	0	0
$718$	−11944.0	−0.620816
$719$	−6386.00	−0.331235	−0.165617	−	0.986190i	$-0.552962\pi$
$719$	−6386.00	−0.331235	−0.165617	+	0.986190i	$0.552962\pi$
$720$	0	0
$721$	2366.00	0.122211
$722$	−24370.0	−1.25617
$723$	0	0
$724$	17500.0	0.898318
$725$	1572.00	0.0805277
$726$	0	0
$727$	24402.0	1.24487	0.622435	−	0.782672i	$-0.286144\pi$
$727$	24402.0	1.24487	0.622435	+	0.782672i	$0.286144\pi$
$728$	728.000	0.0370625
$729$	0	0
$730$	23904.0	1.21195
$731$	−7920.00	−0.400727
$732$	0	0
$733$	31852.0	1.60502	0.802511	−	0.596638i	$-0.203497\pi$
$733$	31852.0	1.60502	0.802511	+	0.596638i	$0.203497\pi$
$734$	144.000	0.00724133
$735$	0	0
$736$	3552.00	0.177892
$737$	−11985.0	−0.599014
$738$	0	0
$739$	−38256.0	−1.90429	−0.952145	−	0.305648i	$-0.901127\pi$
$739$	−38256.0	−1.90429	−0.952145	+	0.305648i	$0.901127\pi$
$740$	−3520.00	−0.174862
$741$	0	0
$742$	−3640.00	−0.180092
$743$	−31012.0	−1.53125	−0.765626	−	0.643286i	$-0.777571\pi$
$743$	−31012.0	−1.53125	−0.765626	+	0.643286i	$0.777571\pi$
$744$	0	0
$745$	−38544.0	−1.89549
$746$	−19976.0	−0.980393
$747$	0	0
$748$	2640.00	0.129048
$749$	2800.00	0.136595
$750$	0	0
$751$	−8093.00	−0.393233	−0.196616	−	0.980480i	$-0.562995\pi$
$751$	−8093.00	−0.393233	−0.196616	+	0.980480i	$0.562995\pi$
$752$	−7536.00	−0.365438
$753$	0	0
$754$	312.000	0.0150695
$755$	17600.0	0.848384
$756$	0	0
$757$	23998.0	1.15221	0.576104	−	0.817376i	$-0.304572\pi$
$757$	23998.0	1.15221	0.576104	+	0.817376i	$0.304572\pi$
$758$	−14592.0	−0.699215
$759$	0	0
$760$	−17664.0	−0.843080
$761$	−21499.0	−1.02410	−0.512049	−	0.858956i	$-0.671113\pi$
$761$	−21499.0	−1.02410	−0.512049	+	0.858956i	$0.671113\pi$
$762$	0	0
$763$	5530.00	0.262385
$764$	−8928.00	−0.422780
$765$	0	0
$766$	−14206.0	−0.670083
$767$	1430.00	0.0673198
$768$	0	0
$769$	7931.00	0.371911	0.185955	−	0.982558i	$-0.440462\pi$
$769$	7931.00	0.371911	0.185955	+	0.982558i	$0.440462\pi$
$770$	3360.00	0.157255
$771$	0	0
$772$	−5072.00	−0.236458
$773$	14310.0	0.665841	0.332920	−	0.942955i	$-0.391966\pi$
$773$	14310.0	0.665841	0.332920	+	0.942955i	$0.391966\pi$
$774$	0	0
$775$	28165.0	1.30544
$776$	11256.0	0.520705
$777$	0	0
$778$	−10096.0	−0.465243
$779$	−18354.0	−0.844160
$780$	0	0
$781$	−13680.0	−0.626772
$782$	9768.00	0.446679
$783$	0	0
$784$	784.000	0.0357143
$785$	−54736.0	−2.48868
$786$	0	0
$787$	−38936.0	−1.76356	−0.881778	−	0.471665i	$-0.843653\pi$
$787$	−38936.0	−1.76356	−0.881778	+	0.471665i	$0.843653\pi$
$788$	5620.00	0.254066
$789$	0	0
$790$	−28256.0	−1.27254
$791$	−3591.00	−0.161418
$792$	0	0
$793$	3523.00	0.157762
$794$	−6416.00	−0.286770
$795$	0	0
$796$	14832.0	0.660435
$797$	−16975.0	−0.754436	−0.377218	−	0.926125i	$-0.623119\pi$
$797$	−16975.0	−0.754436	−0.377218	+	0.926125i	$0.623119\pi$
$798$	0	0
$799$	−20724.0	−0.917600
$800$	−4192.00	−0.185262
$801$	0	0
$802$	18744.0	0.825279
$803$	11205.0	0.492423
$804$	0	0
$805$	12432.0	0.544311
$806$	5590.00	0.244292
$807$	0	0
$808$	152.000	0.00661800
$809$	35162.0	1.52810	0.764048	−	0.645159i	$-0.223209\pi$
$809$	35162.0	1.52810	0.764048	+	0.645159i	$0.223209\pi$
$810$	0	0
$811$	−31164.0	−1.34934	−0.674671	−	0.738119i	$-0.735714\pi$
$811$	−31164.0	−1.34934	−0.674671	+	0.738119i	$0.735714\pi$
$812$	336.000	0.0145213
$813$	0	0
$814$	−1650.00	−0.0710473
$815$	18496.0	0.794953
$816$	0	0
$817$	24840.0	1.06370
$818$	−26036.0	−1.11287
$819$	0	0
$820$	−8512.00	−0.362502
$821$	17634.0	0.749611	0.374806	−	0.927103i	$-0.377709\pi$
$821$	17634.0	0.749611	0.374806	+	0.927103i	$0.377709\pi$
$822$	0	0
$823$	40235.0	1.70414	0.852068	−	0.523431i	$-0.175348\pi$
$823$	40235.0	1.70414	0.852068	+	0.523431i	$0.175348\pi$
$824$	−2704.00	−0.114318
$825$	0	0
$826$	1540.00	0.0648710
$827$	36636.0	1.54046	0.770229	−	0.637768i	$-0.220142\pi$
$827$	36636.0	1.54046	0.770229	+	0.637768i	$0.220142\pi$
$828$	0	0
$829$	−36430.0	−1.52626	−0.763128	−	0.646247i	$-0.776337\pi$
$829$	−36430.0	−1.52626	−0.763128	+	0.646247i	$0.776337\pi$
$830$	29568.0	1.23653
$831$	0	0
$832$	−832.000	−0.0346688
$833$	2156.00	0.0896770
$834$	0	0
$835$	49280.0	2.04240
$836$	−8280.00	−0.342548
$837$	0	0
$838$	−7182.00	−0.296060
$839$	40999.0	1.68706	0.843530	−	0.537083i	$-0.180474\pi$
$839$	40999.0	1.68706	0.843530	+	0.537083i	$0.180474\pi$
$840$	0	0
$841$	−24245.0	−0.994096
$842$	27702.0	1.13382
$843$	0	0
$844$	18488.0	0.754009
$845$	−2704.00	−0.110083
$846$	0	0
$847$	−7742.00	−0.314071
$848$	4160.00	0.168461
$849$	0	0
$850$	−11528.0	−0.465185
$851$	−6105.00	−0.245919
$852$	0	0
$853$	22960.0	0.921612	0.460806	−	0.887501i	$-0.347560\pi$
$853$	22960.0	0.921612	0.460806	+	0.887501i	$0.347560\pi$
$854$	3794.00	0.152023
$855$	0	0
$856$	−3200.00	−0.127773
$857$	7646.00	0.304764	0.152382	−	0.988322i	$-0.451306\pi$
$857$	7646.00	0.304764	0.152382	+	0.988322i	$0.451306\pi$
$858$	0	0
$859$	−28901.0	−1.14795	−0.573975	−	0.818873i	$-0.694599\pi$
$859$	−28901.0	−1.14795	−0.573975	+	0.818873i	$0.694599\pi$
$860$	11520.0	0.456778
$861$	0	0
$862$	11884.0	0.469572
$863$	6056.00	0.238874	0.119437	−	0.992842i	$-0.461891\pi$
$863$	6056.00	0.238874	0.119437	+	0.992842i	$0.461891\pi$
$864$	0	0
$865$	−25952.0	−1.02011
$866$	−11396.0	−0.447173
$867$	0	0
$868$	6020.00	0.235406
$869$	−13245.0	−0.517038
$870$	0	0
$871$	10387.0	0.404076
$872$	−6320.00	−0.245438
$873$	0	0
$874$	−30636.0	−1.18567
$875$	−672.000	−0.0259631
$876$	0	0
$877$	−34203.0	−1.31694	−0.658468	−	0.752609i	$-0.728795\pi$
$877$	−34203.0	−1.31694	−0.658468	+	0.752609i	$0.728795\pi$
$878$	−31364.0	−1.20556
$879$	0	0
$880$	−3840.00	−0.147098
$881$	−21350.0	−0.816458	−0.408229	−	0.912879i	$-0.633854\pi$
$881$	−21350.0	−0.816458	−0.408229	+	0.912879i	$0.633854\pi$
$882$	0	0
$883$	11482.0	0.437599	0.218800	−	0.975770i	$-0.429786\pi$
$883$	11482.0	0.437599	0.218800	+	0.975770i	$0.429786\pi$
$884$	−2288.00	−0.0870518
$885$	0	0
$886$	11688.0	0.443190
$887$	23910.0	0.905095	0.452547	−	0.891740i	$-0.350515\pi$
$887$	23910.0	0.905095	0.452547	+	0.891740i	$0.350515\pi$
$888$	0	0
$889$	16037.0	0.605021
$890$	−4544.00	−0.171141
$891$	0	0
$892$	12404.0	0.465602
$893$	64998.0	2.43569
$894$	0	0
$895$	28320.0	1.05769
$896$	−896.000	−0.0334077
$897$	0	0
$898$	6388.00	0.237383
$899$	2580.00	0.0957150
$900$	0	0
$901$	11440.0	0.422999
$902$	−3990.00	−0.147287
$903$	0	0
$904$	4104.00	0.150992
$905$	−70000.0	−2.57114
$906$	0	0
$907$	52282.0	1.91400	0.956999	−	0.290093i	$-0.0936862\pi$
$907$	52282.0	1.91400	0.956999	+	0.290093i	$0.0936862\pi$
$908$	25936.0	0.947926
$909$	0	0
$910$	−2912.00	−0.106079
$911$	34340.0	1.24889	0.624443	−	0.781070i	$-0.285326\pi$
$911$	34340.0	1.24889	0.624443	+	0.781070i	$0.285326\pi$
$912$	0	0
$913$	13860.0	0.502409
$914$	27988.0	1.01287
$915$	0	0
$916$	−18752.0	−0.676401
$917$	−11088.0	−0.399300
$918$	0	0
$919$	−18803.0	−0.674922	−0.337461	−	0.941339i	$-0.609568\pi$
$919$	−18803.0	−0.674922	−0.337461	+	0.941339i	$0.609568\pi$
$920$	−14208.0	−0.509156
$921$	0	0
$922$	−20104.0	−0.718102
$923$	11856.0	0.422801
$924$	0	0
$925$	7205.00	0.256107
$926$	10552.0	0.374471
$927$	0	0
$928$	−384.000	−0.0135834
$929$	40241.0	1.42117	0.710584	−	0.703613i	$-0.248431\pi$
$929$	40241.0	1.42117	0.710584	+	0.703613i	$0.248431\pi$
$930$	0	0
$931$	−6762.00	−0.238040
$932$	13556.0	0.476439
$933$	0	0
$934$	−19848.0	−0.695339
$935$	−10560.0	−0.369357
$936$	0	0
$937$	7616.00	0.265532	0.132766	−	0.991147i	$-0.457614\pi$
$937$	7616.00	0.265532	0.132766	+	0.991147i	$0.457614\pi$
$938$	11186.0	0.389377
$939$	0	0
$940$	30144.0	1.04595
$941$	−52638.0	−1.82354	−0.911769	−	0.410703i	$-0.865283\pi$
$941$	−52638.0	−1.82354	−0.911769	+	0.410703i	$0.865283\pi$
$942$	0	0
$943$	−14763.0	−0.509809
$944$	−1760.00	−0.0606813
$945$	0	0
$946$	5400.00	0.185591
$947$	−24576.0	−0.843308	−0.421654	−	0.906757i	$-0.638550\pi$
$947$	−24576.0	−0.843308	−0.421654	+	0.906757i	$0.638550\pi$
$948$	0	0
$949$	−9711.00	−0.332173
$950$	36156.0	1.23480
$951$	0	0
$952$	−2464.00	−0.0838852
$953$	52218.0	1.77493	0.887464	−	0.460876i	$-0.152465\pi$
$953$	52218.0	1.77493	0.887464	+	0.460876i	$0.152465\pi$
$954$	0	0
$955$	35712.0	1.21007
$956$	104.000	0.00351841
$957$	0	0
$958$	−22656.0	−0.764073
$959$	−630.000	−0.0212135
$960$	0	0
$961$	16434.0	0.551643
$962$	1430.00	0.0479262
$963$	0	0
$964$	−13560.0	−0.453048
$965$	20288.0	0.676781
$966$	0	0
$967$	7086.00	0.235647	0.117823	−	0.993035i	$-0.462408\pi$
$967$	7086.00	0.235647	0.117823	+	0.993035i	$0.462408\pi$
$968$	8848.00	0.293787
$969$	0	0
$970$	−45024.0	−1.49034
$971$	24281.0	0.802486	0.401243	−	0.915972i	$-0.368578\pi$
$971$	24281.0	0.802486	0.401243	+	0.915972i	$0.368578\pi$
$972$	0	0
$973$	22540.0	0.742651
$974$	14940.0	0.491487
$975$	0	0
$976$	−4336.00	−0.142205
$977$	12874.0	0.421572	0.210786	−	0.977532i	$-0.432398\pi$
$977$	12874.0	0.421572	0.210786	+	0.977532i	$0.432398\pi$
$978$	0	0
$979$	−2130.00	−0.0695353
$980$	−3136.00	−0.102220
$981$	0	0
$982$	−1144.00	−0.0371757
$983$	−18072.0	−0.586376	−0.293188	−	0.956055i	$-0.594716\pi$
$983$	−18072.0	−0.586376	−0.293188	+	0.956055i	$0.594716\pi$
$984$	0	0
$985$	−22480.0	−0.727180
$986$	−1056.00	−0.0341074
$987$	0	0
$988$	7176.00	0.231072
$989$	19980.0	0.642393
$990$	0	0
$991$	−31095.0	−0.996736	−0.498368	−	0.866966i	$-0.666067\pi$
$991$	−31095.0	−0.996736	−0.498368	+	0.866966i	$0.666067\pi$
$992$	−6880.00	−0.220202
$993$	0	0
$994$	12768.0	0.407421
$995$	−59328.0	−1.89028
$996$	0	0
$997$	−11419.0	−0.362732	−0.181366	−	0.983416i	$-0.558052\pi$
$997$	−11419.0	−0.362732	−0.181366	+	0.983416i	$0.558052\pi$
$998$	24250.0	0.769159
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.4.a.a.1.1		1
3.2	odd	2		182.4.a.d.1.1	✓	1
12.11	even	2		1456.4.a.c.1.1		1
21.20	even	2		1274.4.a.c.1.1		1
39.38	odd	2		2366.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.4.a.d.1.1	✓	1	3.2	odd	2
1274.4.a.c.1.1		1	21.20	even	2
1456.4.a.c.1.1		1	12.11	even	2
1638.4.a.a.1.1		1	1.1	even	1	trivial
2366.4.a.e.1.1		1	39.38	odd	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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