LMFDB - Embedded newform 153.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,4,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("153.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	153.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:	$9.02729223088$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -7.00000 q^{4} +10.0000 q^{5} -8.00000 q^{7} +15.0000 q^{8} +O(q^{10})$

\(q-1.00000 q^{2} -7.00000 q^{4} +10.0000 q^{5} -8.00000 q^{7} +15.0000 q^{8} -10.0000 q^{10} -12.0000 q^{11} -26.0000 q^{13} +8.00000 q^{14} +41.0000 q^{16} -17.0000 q^{17} -148.000 q^{19} -70.0000 q^{20} +12.0000 q^{22} -152.000 q^{23} -25.0000 q^{25} +26.0000 q^{26} +56.0000 q^{28} +66.0000 q^{29} -32.0000 q^{31} -161.000 q^{32} +17.0000 q^{34} -80.0000 q^{35} -266.000 q^{37} +148.000 q^{38} +150.000 q^{40} +6.00000 q^{41} -92.0000 q^{43} +84.0000 q^{44} +152.000 q^{46} +288.000 q^{47} -279.000 q^{49} +25.0000 q^{50} +182.000 q^{52} +546.000 q^{53} -120.000 q^{55} -120.000 q^{56} -66.0000 q^{58} -420.000 q^{59} +350.000 q^{61} +32.0000 q^{62} -167.000 q^{64} -260.000 q^{65} +940.000 q^{67} +119.000 q^{68} +80.0000 q^{70} -424.000 q^{71} +378.000 q^{73} +266.000 q^{74} +1036.00 q^{76} +96.0000 q^{77} +288.000 q^{79} +410.000 q^{80} -6.00000 q^{82} -748.000 q^{83} -170.000 q^{85} +92.0000 q^{86} -180.000 q^{88} +1558.00 q^{89} +208.000 q^{91} +1064.00 q^{92} -288.000 q^{94} -1480.00 q^{95} +530.000 q^{97} +279.000 q^{98} +O(q^{100})\)

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 $

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	10.0000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	10.0000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−8.00000	−0.431959	−0.215980	−	0.976398i	$-0.569295\pi$
$7$	−8.00000	−0.431959	−0.215980	+	0.976398i	$0.569295\pi$
$8$	15.0000	0.662913
$9$	0	0
$10$	−10.0000	−0.316228
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	−26.0000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−26.0000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	8.00000	0.152721
$15$	0	0
$16$	41.0000	0.640625
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−148.000	−1.78703	−0.893514	−	0.449036i	$-0.851768\pi$
$19$	−148.000	−1.78703	−0.893514	+	0.449036i	$0.851768\pi$
$20$	−70.0000	−0.782624
$21$	0	0
$22$	12.0000	0.116291
$23$	−152.000	−1.37801	−0.689004	−	0.724757i	$-0.741952\pi$
$23$	−152.000	−1.37801	−0.689004	+	0.724757i	$0.741952\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	26.0000	0.196116
$27$	0	0
$28$	56.0000	0.377964
$29$	66.0000	0.422617	0.211308	−	0.977419i	$-0.432228\pi$
$29$	66.0000	0.422617	0.211308	+	0.977419i	$0.432228\pi$
$30$	0	0
$31$	−32.0000	−0.185399	−0.0926995	−	0.995694i	$-0.529550\pi$
$31$	−32.0000	−0.185399	−0.0926995	+	0.995694i	$0.529550\pi$
$32$	−161.000	−0.889408
$33$	0	0
$34$	17.0000	0.0857493
$35$	−80.0000	−0.386356
$36$	0	0
$37$	−266.000	−1.18190	−0.590948	−	0.806710i	$-0.701246\pi$
$37$	−266.000	−1.18190	−0.590948	+	0.806710i	$0.701246\pi$
$38$	148.000	0.631810
$39$	0	0
$40$	150.000	0.592927
$41$	6.00000	0.0228547	0.0114273	−	0.999935i	$-0.496362\pi$
$41$	6.00000	0.0228547	0.0114273	+	0.999935i	$0.496362\pi$
$42$	0	0
$43$	−92.0000	−0.326276	−0.163138	−	0.986603i	$-0.552162\pi$
$43$	−92.0000	−0.326276	−0.163138	+	0.986603i	$0.552162\pi$
$44$	84.0000	0.287806
$45$	0	0
$46$	152.000	0.487200
$47$	288.000	0.893811	0.446906	−	0.894581i	$-0.352526\pi$
$47$	288.000	0.893811	0.446906	+	0.894581i	$0.352526\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	25.0000	0.0707107
$51$	0	0
$52$	182.000	0.485363
$53$	546.000	1.41507	0.707536	−	0.706677i	$-0.249807\pi$
$53$	546.000	1.41507	0.707536	+	0.706677i	$0.249807\pi$
$54$	0	0
$55$	−120.000	−0.294196
$56$	−120.000	−0.286351
$57$	0	0
$58$	−66.0000	−0.149418
$59$	−420.000	−0.926769	−0.463384	−	0.886157i	$-0.653365\pi$
$59$	−420.000	−0.926769	−0.463384	+	0.886157i	$0.653365\pi$
$60$	0	0
$61$	350.000	0.734638	0.367319	−	0.930095i	$-0.380276\pi$
$61$	350.000	0.734638	0.367319	+	0.930095i	$0.380276\pi$
$62$	32.0000	0.0655485
$63$	0	0
$64$	−167.000	−0.326172
$65$	−260.000	−0.496139
$66$	0	0
$67$	940.000	1.71402	0.857010	−	0.515301i	$-0.172320\pi$
$67$	940.000	1.71402	0.857010	+	0.515301i	$0.172320\pi$
$68$	119.000	0.212219
$69$	0	0
$70$	80.0000	0.136598
$71$	−424.000	−0.708726	−0.354363	−	0.935108i	$-0.615302\pi$
$71$	−424.000	−0.708726	−0.354363	+	0.935108i	$0.615302\pi$
$72$	0	0
$73$	378.000	0.606049	0.303024	−	0.952983i	$-0.402004\pi$
$73$	378.000	0.606049	0.303024	+	0.952983i	$0.402004\pi$
$74$	266.000	0.417863
$75$	0	0
$76$	1036.00	1.56365
$77$	96.0000	0.142081
$78$	0	0
$79$	288.000	0.410159	0.205079	−	0.978745i	$-0.434255\pi$
$79$	288.000	0.410159	0.205079	+	0.978745i	$0.434255\pi$
$80$	410.000	0.572992
$81$	0	0
$82$	−6.00000	−0.00808036
$83$	−748.000	−0.989201	−0.494600	−	0.869121i	$-0.664686\pi$
$83$	−748.000	−0.989201	−0.494600	+	0.869121i	$0.664686\pi$
$84$	0	0
$85$	−170.000	−0.216930
$86$	92.0000	0.115356
$87$	0	0
$88$	−180.000	−0.218046
$89$	1558.00	1.85559	0.927796	−	0.373088i	$-0.121701\pi$
$89$	1558.00	1.85559	0.927796	+	0.373088i	$0.121701\pi$
$90$	0	0
$91$	208.000	0.239608
$92$	1064.00	1.20576
$93$	0	0
$94$	−288.000	−0.316010
$95$	−1480.00	−1.59837
$96$	0	0
$97$	530.000	0.554777	0.277388	−	0.960758i	$-0.410531\pi$
$97$	530.000	0.554777	0.277388	+	0.960758i	$0.410531\pi$
$98$	279.000	0.287584
$99$	0	0
$100$	175.000	0.175000
$101$	1010.00	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	1010.00	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−1624.00	−1.55357	−0.776784	−	0.629767i	$-0.783150\pi$
$103$	−1624.00	−1.55357	−0.776784	+	0.629767i	$0.783150\pi$
$104$	−390.000	−0.367718
$105$	0	0
$106$	−546.000	−0.500304
$107$	−796.000	−0.719180	−0.359590	−	0.933110i	$-0.617083\pi$
$107$	−796.000	−0.719180	−0.359590	+	0.933110i	$0.617083\pi$
$108$	0	0
$109$	−1698.00	−1.49210	−0.746050	−	0.665890i	$-0.768052\pi$
$109$	−1698.00	−1.49210	−0.746050	+	0.665890i	$0.768052\pi$
$110$	120.000	0.104014
$111$	0	0
$112$	−328.000	−0.276724
$113$	−1730.00	−1.44022	−0.720109	−	0.693861i	$-0.755908\pi$
$113$	−1730.00	−1.44022	−0.720109	+	0.693861i	$0.755908\pi$
$114$	0	0
$115$	−1520.00	−1.23253
$116$	−462.000	−0.369790
$117$	0	0
$118$	420.000	0.327662
$119$	136.000	0.104766
$120$	0	0
$121$	−1187.00	−0.891811
$122$	−350.000	−0.259734
$123$	0	0
$124$	224.000	0.162224
$125$	−1500.00	−1.07331
$126$	0	0
$127$	848.000	0.592503	0.296251	−	0.955110i	$-0.404263\pi$
$127$	848.000	0.592503	0.296251	+	0.955110i	$0.404263\pi$
$128$	1455.00	1.00473
$129$	0	0
$130$	260.000	0.175412
$131$	1372.00	0.915055	0.457527	−	0.889195i	$-0.348735\pi$
$131$	1372.00	0.915055	0.457527	+	0.889195i	$0.348735\pi$
$132$	0	0
$133$	1184.00	0.771923
$134$	−940.000	−0.605997
$135$	0	0
$136$	−255.000	−0.160780
$137$	−666.000	−0.415330	−0.207665	−	0.978200i	$-0.566586\pi$
$137$	−666.000	−0.415330	−0.207665	+	0.978200i	$0.566586\pi$
$138$	0	0
$139$	−2756.00	−1.68173	−0.840866	−	0.541243i	$-0.817954\pi$
$139$	−2756.00	−1.68173	−0.840866	+	0.541243i	$0.817954\pi$
$140$	560.000	0.338062
$141$	0	0
$142$	424.000	0.250572
$143$	312.000	0.182453
$144$	0	0
$145$	660.000	0.378000
$146$	−378.000	−0.214271
$147$	0	0
$148$	1862.00	1.03416
$149$	626.000	0.344187	0.172094	−	0.985081i	$-0.444947\pi$
$149$	626.000	0.344187	0.172094	+	0.985081i	$0.444947\pi$
$150$	0	0
$151$	2936.00	1.58231	0.791153	−	0.611618i	$-0.209481\pi$
$151$	2936.00	1.58231	0.791153	+	0.611618i	$0.209481\pi$
$152$	−2220.00	−1.18464
$153$	0	0
$154$	−96.0000	−0.0502331
$155$	−320.000	−0.165826
$156$	0	0
$157$	−602.000	−0.306018	−0.153009	−	0.988225i	$-0.548896\pi$
$157$	−602.000	−0.306018	−0.153009	+	0.988225i	$0.548896\pi$
$158$	−288.000	−0.145013
$159$	0	0
$160$	−1610.00	−0.795510
$161$	1216.00	0.595244
$162$	0	0
$163$	36.0000	0.0172990	0.00864950	−	0.999963i	$-0.497247\pi$
$163$	36.0000	0.0172990	0.00864950	+	0.999963i	$0.497247\pi$
$164$	−42.0000	−0.0199979
$165$	0	0
$166$	748.000	0.349735
$167$	−2008.00	−0.930441	−0.465221	−	0.885195i	$-0.654025\pi$
$167$	−2008.00	−0.930441	−0.465221	+	0.885195i	$0.654025\pi$
$168$	0	0
$169$	−1521.00	−0.692308
$170$	170.000	0.0766965
$171$	0	0
$172$	644.000	0.285492
$173$	4114.00	1.80799	0.903993	−	0.427547i	$-0.140622\pi$
$173$	4114.00	1.80799	0.903993	+	0.427547i	$0.140622\pi$
$174$	0	0
$175$	200.000	0.0863919
$176$	−492.000	−0.210715
$177$	0	0
$178$	−1558.00	−0.656051
$179$	804.000	0.335719	0.167860	−	0.985811i	$-0.446314\pi$
$179$	804.000	0.335719	0.167860	+	0.985811i	$0.446314\pi$
$180$	0	0
$181$	−3194.00	−1.31165	−0.655824	−	0.754914i	$-0.727679\pi$
$181$	−3194.00	−1.31165	−0.655824	+	0.754914i	$0.727679\pi$
$182$	−208.000	−0.0847142
$183$	0	0
$184$	−2280.00	−0.913499
$185$	−2660.00	−1.05712
$186$	0	0
$187$	204.000	0.0797752
$188$	−2016.00	−0.782085
$189$	0	0
$190$	1480.00	0.565108
$191$	4880.00	1.84871	0.924357	−	0.381528i	$-0.124602\pi$
$191$	4880.00	1.84871	0.924357	+	0.381528i	$0.124602\pi$
$192$	0	0
$193$	3698.00	1.37921	0.689606	−	0.724185i	$-0.257784\pi$
$193$	3698.00	1.37921	0.689606	+	0.724185i	$0.257784\pi$
$194$	−530.000	−0.196143
$195$	0	0
$196$	1953.00	0.711735
$197$	1914.00	0.692218	0.346109	−	0.938194i	$-0.387503\pi$
$197$	1914.00	0.692218	0.346109	+	0.938194i	$0.387503\pi$
$198$	0	0
$199$	520.000	0.185235	0.0926176	−	0.995702i	$-0.470477\pi$
$199$	520.000	0.185235	0.0926176	+	0.995702i	$0.470477\pi$
$200$	−375.000	−0.132583
$201$	0	0
$202$	−1010.00	−0.351799
$203$	−528.000	−0.182553
$204$	0	0
$205$	60.0000	0.0204419
$206$	1624.00	0.549269
$207$	0	0
$208$	−1066.00	−0.355355
$209$	1776.00	0.587792
$210$	0	0
$211$	2564.00	0.836555	0.418277	−	0.908319i	$-0.362634\pi$
$211$	2564.00	0.836555	0.418277	+	0.908319i	$0.362634\pi$
$212$	−3822.00	−1.23819
$213$	0	0
$214$	796.000	0.254268
$215$	−920.000	−0.291830
$216$	0	0
$217$	256.000	0.0800848
$218$	1698.00	0.527537
$219$	0	0
$220$	840.000	0.257422
$221$	442.000	0.134535
$222$	0	0
$223$	−3936.00	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$223$	−3936.00	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$224$	1288.00	0.384188
$225$	0	0
$226$	1730.00	0.509194
$227$	1596.00	0.466653	0.233327	−	0.972398i	$-0.425039\pi$
$227$	1596.00	0.466653	0.233327	+	0.972398i	$0.425039\pi$
$228$	0	0
$229$	1278.00	0.368789	0.184394	−	0.982852i	$-0.440968\pi$
$229$	1278.00	0.368789	0.184394	+	0.982852i	$0.440968\pi$
$230$	1520.00	0.435764
$231$	0	0
$232$	990.000	0.280158
$233$	3414.00	0.959908	0.479954	−	0.877294i	$-0.340653\pi$
$233$	3414.00	0.959908	0.479954	+	0.877294i	$0.340653\pi$
$234$	0	0
$235$	2880.00	0.799449
$236$	2940.00	0.810922
$237$	0	0
$238$	−136.000	−0.0370402
$239$	−3904.00	−1.05661	−0.528303	−	0.849056i	$-0.677171\pi$
$239$	−3904.00	−1.05661	−0.528303	+	0.849056i	$0.677171\pi$
$240$	0	0
$241$	4866.00	1.30061	0.650304	−	0.759674i	$-0.274641\pi$
$241$	4866.00	1.30061	0.650304	+	0.759674i	$0.274641\pi$
$242$	1187.00	0.315303
$243$	0	0
$244$	−2450.00	−0.642808
$245$	−2790.00	−0.727537
$246$	0	0
$247$	3848.00	0.991265
$248$	−480.000	−0.122903
$249$	0	0
$250$	1500.00	0.379473
$251$	−6276.00	−1.57824	−0.789119	−	0.614241i	$-0.789462\pi$
$251$	−6276.00	−1.57824	−0.789119	+	0.614241i	$0.789462\pi$
$252$	0	0
$253$	1824.00	0.453257
$254$	−848.000	−0.209481
$255$	0	0
$256$	−119.000	−0.0290527
$257$	−5826.00	−1.41407	−0.707035	−	0.707179i	$-0.749968\pi$
$257$	−5826.00	−1.41407	−0.707035	+	0.707179i	$0.749968\pi$
$258$	0	0
$259$	2128.00	0.510531
$260$	1820.00	0.434122
$261$	0	0
$262$	−1372.00	−0.323521
$263$	−3448.00	−0.808414	−0.404207	−	0.914668i	$-0.632452\pi$
$263$	−3448.00	−0.808414	−0.404207	+	0.914668i	$0.632452\pi$
$264$	0	0
$265$	5460.00	1.26568
$266$	−1184.00	−0.272916
$267$	0	0
$268$	−6580.00	−1.49977
$269$	4114.00	0.932472	0.466236	−	0.884660i	$-0.345610\pi$
$269$	4114.00	0.932472	0.466236	+	0.884660i	$0.345610\pi$
$270$	0	0
$271$	−3856.00	−0.864337	−0.432168	−	0.901793i	$-0.642251\pi$
$271$	−3856.00	−0.864337	−0.432168	+	0.901793i	$0.642251\pi$
$272$	−697.000	−0.155374
$273$	0	0
$274$	666.000	0.146841
$275$	300.000	0.0657843
$276$	0	0
$277$	278.000	0.0603011	0.0301505	−	0.999545i	$-0.490401\pi$
$277$	278.000	0.0603011	0.0301505	+	0.999545i	$0.490401\pi$
$278$	2756.00	0.594582
$279$	0	0
$280$	−1200.00	−0.256120
$281$	3222.00	0.684016	0.342008	−	0.939697i	$-0.388893\pi$
$281$	3222.00	0.684016	0.342008	+	0.939697i	$0.388893\pi$
$282$	0	0
$283$	−7684.00	−1.61402	−0.807008	−	0.590541i	$-0.798914\pi$
$283$	−7684.00	−1.61402	−0.807008	+	0.590541i	$0.798914\pi$
$284$	2968.00	0.620135
$285$	0	0
$286$	−312.000	−0.0645068
$287$	−48.0000	−0.00987230
$288$	0	0
$289$	289.000	0.0588235
$290$	−660.000	−0.133643
$291$	0	0
$292$	−2646.00	−0.530293
$293$	−3678.00	−0.733348	−0.366674	−	0.930349i	$-0.619504\pi$
$293$	−3678.00	−0.733348	−0.366674	+	0.930349i	$0.619504\pi$
$294$	0	0
$295$	−4200.00	−0.828927
$296$	−3990.00	−0.783493
$297$	0	0
$298$	−626.000	−0.121689
$299$	3952.00	0.764381
$300$	0	0
$301$	736.000	0.140938
$302$	−2936.00	−0.559430
$303$	0	0
$304$	−6068.00	−1.14481
$305$	3500.00	0.657080
$306$	0	0
$307$	−4884.00	−0.907963	−0.453981	−	0.891011i	$-0.649997\pi$
$307$	−4884.00	−0.907963	−0.453981	+	0.891011i	$0.649997\pi$
$308$	−672.000	−0.124321
$309$	0	0
$310$	320.000	0.0586283
$311$	9400.00	1.71391	0.856954	−	0.515394i	$-0.172354\pi$
$311$	9400.00	1.71391	0.856954	+	0.515394i	$0.172354\pi$
$312$	0	0
$313$	5690.00	1.02753	0.513766	−	0.857930i	$-0.328250\pi$
$313$	5690.00	1.02753	0.513766	+	0.857930i	$0.328250\pi$
$314$	602.000	0.108194
$315$	0	0
$316$	−2016.00	−0.358889
$317$	−6478.00	−1.14776	−0.573881	−	0.818939i	$-0.694563\pi$
$317$	−6478.00	−1.14776	−0.573881	+	0.818939i	$0.694563\pi$
$318$	0	0
$319$	−792.000	−0.139008
$320$	−1670.00	−0.291737
$321$	0	0
$322$	−1216.00	−0.210450
$323$	2516.00	0.433418
$324$	0	0
$325$	650.000	0.110940
$326$	−36.0000	−0.00611612
$327$	0	0
$328$	90.0000	0.0151507
$329$	−2304.00	−0.386090
$330$	0	0
$331$	−6140.00	−1.01959	−0.509796	−	0.860295i	$-0.670279\pi$
$331$	−6140.00	−1.01959	−0.509796	+	0.860295i	$0.670279\pi$
$332$	5236.00	0.865551
$333$	0	0
$334$	2008.00	0.328961
$335$	9400.00	1.53307
$336$	0	0
$337$	1042.00	0.168431	0.0842157	−	0.996448i	$-0.473162\pi$
$337$	1042.00	0.168431	0.0842157	+	0.996448i	$0.473162\pi$
$338$	1521.00	0.244768
$339$	0	0
$340$	1190.00	0.189814
$341$	384.000	0.0609817
$342$	0	0
$343$	4976.00	0.783320
$344$	−1380.00	−0.216292
$345$	0	0
$346$	−4114.00	−0.639220
$347$	−10828.0	−1.67515	−0.837576	−	0.546321i	$-0.816028\pi$
$347$	−10828.0	−1.67515	−0.837576	+	0.546321i	$0.816028\pi$
$348$	0	0
$349$	1014.00	0.155525	0.0777624	−	0.996972i	$-0.475222\pi$
$349$	1014.00	0.155525	0.0777624	+	0.996972i	$0.475222\pi$
$350$	−200.000	−0.0305441
$351$	0	0
$352$	1932.00	0.292545
$353$	−7842.00	−1.18240	−0.591200	−	0.806525i	$-0.701346\pi$
$353$	−7842.00	−1.18240	−0.591200	+	0.806525i	$0.701346\pi$
$354$	0	0
$355$	−4240.00	−0.633904
$356$	−10906.0	−1.62364
$357$	0	0
$358$	−804.000	−0.118695
$359$	−744.000	−0.109378	−0.0546892	−	0.998503i	$-0.517417\pi$
$359$	−744.000	−0.109378	−0.0546892	+	0.998503i	$0.517417\pi$
$360$	0	0
$361$	15045.0	2.19347
$362$	3194.00	0.463737
$363$	0	0
$364$	−1456.00	−0.209657
$365$	3780.00	0.542066
$366$	0	0
$367$	−6944.00	−0.987667	−0.493834	−	0.869556i	$-0.664405\pi$
$367$	−6944.00	−0.987667	−0.493834	+	0.869556i	$0.664405\pi$
$368$	−6232.00	−0.882787
$369$	0	0
$370$	2660.00	0.373748
$371$	−4368.00	−0.611254
$372$	0	0
$373$	−3522.00	−0.488907	−0.244453	−	0.969661i	$-0.578609\pi$
$373$	−3522.00	−0.488907	−0.244453	+	0.969661i	$0.578609\pi$
$374$	−204.000	−0.0282048
$375$	0	0
$376$	4320.00	0.592519
$377$	−1716.00	−0.234426
$378$	0	0
$379$	−1124.00	−0.152338	−0.0761689	−	0.997095i	$-0.524269\pi$
$379$	−1124.00	−0.152338	−0.0761689	+	0.997095i	$0.524269\pi$
$380$	10360.0	1.39857
$381$	0	0
$382$	−4880.00	−0.653619
$383$	−9120.00	−1.21674	−0.608368	−	0.793655i	$-0.708176\pi$
$383$	−9120.00	−1.21674	−0.608368	+	0.793655i	$0.708176\pi$
$384$	0	0
$385$	960.000	0.127081
$386$	−3698.00	−0.487625
$387$	0	0
$388$	−3710.00	−0.485430
$389$	−4926.00	−0.642052	−0.321026	−	0.947070i	$-0.604028\pi$
$389$	−4926.00	−0.642052	−0.321026	+	0.947070i	$0.604028\pi$
$390$	0	0
$391$	2584.00	0.334216
$392$	−4185.00	−0.539220
$393$	0	0
$394$	−1914.00	−0.244736
$395$	2880.00	0.366857
$396$	0	0
$397$	942.000	0.119087	0.0595436	−	0.998226i	$-0.481035\pi$
$397$	942.000	0.119087	0.0595436	+	0.998226i	$0.481035\pi$
$398$	−520.000	−0.0654906
$399$	0	0
$400$	−1025.00	−0.128125
$401$	11182.0	1.39252	0.696262	−	0.717787i	$-0.254845\pi$
$401$	11182.0	1.39252	0.696262	+	0.717787i	$0.254845\pi$
$402$	0	0
$403$	832.000	0.102841
$404$	−7070.00	−0.870658
$405$	0	0
$406$	528.000	0.0645424
$407$	3192.00	0.388751
$408$	0	0
$409$	−1654.00	−0.199963	−0.0999817	−	0.994989i	$-0.531878\pi$
$409$	−1654.00	−0.199963	−0.0999817	+	0.994989i	$0.531878\pi$
$410$	−60.0000	−0.00722729
$411$	0	0
$412$	11368.0	1.35937
$413$	3360.00	0.400326
$414$	0	0
$415$	−7480.00	−0.884768
$416$	4186.00	0.493355
$417$	0	0
$418$	−1776.00	−0.207816
$419$	8444.00	0.984526	0.492263	−	0.870447i	$-0.336170\pi$
$419$	8444.00	0.984526	0.492263	+	0.870447i	$0.336170\pi$
$420$	0	0
$421$	7726.00	0.894400	0.447200	−	0.894434i	$-0.352421\pi$
$421$	7726.00	0.894400	0.447200	+	0.894434i	$0.352421\pi$
$422$	−2564.00	−0.295767
$423$	0	0
$424$	8190.00	0.938070
$425$	425.000	0.0485071
$426$	0	0
$427$	−2800.00	−0.317334
$428$	5572.00	0.629282
$429$	0	0
$430$	920.000	0.103178
$431$	15648.0	1.74881	0.874406	−	0.485196i	$-0.161252\pi$
$431$	15648.0	1.74881	0.874406	+	0.485196i	$0.161252\pi$
$432$	0	0
$433$	1298.00	0.144060	0.0720299	−	0.997402i	$-0.477052\pi$
$433$	1298.00	0.144060	0.0720299	+	0.997402i	$0.477052\pi$
$434$	−256.000	−0.0283143
$435$	0	0
$436$	11886.0	1.30559
$437$	22496.0	2.46254
$438$	0	0
$439$	−5288.00	−0.574903	−0.287452	−	0.957795i	$-0.592808\pi$
$439$	−5288.00	−0.574903	−0.287452	+	0.957795i	$0.592808\pi$
$440$	−1800.00	−0.195026
$441$	0	0
$442$	−442.000	−0.0475651
$443$	−13892.0	−1.48991	−0.744954	−	0.667116i	$-0.767528\pi$
$443$	−13892.0	−1.48991	−0.744954	+	0.667116i	$0.767528\pi$
$444$	0	0
$445$	15580.0	1.65969
$446$	3936.00	0.417881
$447$	0	0
$448$	1336.00	0.140893
$449$	−16322.0	−1.71555	−0.857776	−	0.514024i	$-0.828154\pi$
$449$	−16322.0	−1.71555	−0.857776	+	0.514024i	$0.828154\pi$
$450$	0	0
$451$	−72.0000	−0.00751740
$452$	12110.0	1.26019
$453$	0	0
$454$	−1596.00	−0.164987
$455$	2080.00	0.214312
$456$	0	0
$457$	−10278.0	−1.05204	−0.526022	−	0.850471i	$-0.676317\pi$
$457$	−10278.0	−1.05204	−0.526022	+	0.850471i	$0.676317\pi$
$458$	−1278.00	−0.130387
$459$	0	0
$460$	10640.0	1.07846
$461$	1306.00	0.131945	0.0659723	−	0.997821i	$-0.478985\pi$
$461$	1306.00	0.131945	0.0659723	+	0.997821i	$0.478985\pi$
$462$	0	0
$463$	320.000	0.0321202	0.0160601	−	0.999871i	$-0.494888\pi$
$463$	320.000	0.0321202	0.0160601	+	0.999871i	$0.494888\pi$
$464$	2706.00	0.270739
$465$	0	0
$466$	−3414.00	−0.339379
$467$	8676.00	0.859695	0.429847	−	0.902902i	$-0.358567\pi$
$467$	8676.00	0.859695	0.429847	+	0.902902i	$0.358567\pi$
$468$	0	0
$469$	−7520.00	−0.740387
$470$	−2880.00	−0.282648
$471$	0	0
$472$	−6300.00	−0.614367
$473$	1104.00	0.107319
$474$	0	0
$475$	3700.00	0.357406
$476$	−952.000	−0.0916698
$477$	0	0
$478$	3904.00	0.373567
$479$	−8976.00	−0.856209	−0.428104	−	0.903729i	$-0.640818\pi$
$479$	−8976.00	−0.856209	−0.428104	+	0.903729i	$0.640818\pi$
$480$	0	0
$481$	6916.00	0.655598
$482$	−4866.00	−0.459834
$483$	0	0
$484$	8309.00	0.780334
$485$	5300.00	0.496207
$486$	0	0
$487$	−19096.0	−1.77684	−0.888421	−	0.459029i	$-0.848197\pi$
$487$	−19096.0	−1.77684	−0.888421	+	0.459029i	$0.848197\pi$
$488$	5250.00	0.487001
$489$	0	0
$490$	2790.00	0.257223
$491$	−964.000	−0.0886043	−0.0443021	−	0.999018i	$-0.514106\pi$
$491$	−964.000	−0.0886043	−0.0443021	+	0.999018i	$0.514106\pi$
$492$	0	0
$493$	−1122.00	−0.102500
$494$	−3848.00	−0.350465
$495$	0	0
$496$	−1312.00	−0.118771
$497$	3392.00	0.306141
$498$	0	0
$499$	2756.00	0.247245	0.123623	−	0.992329i	$-0.460549\pi$
$499$	2756.00	0.247245	0.123623	+	0.992329i	$0.460549\pi$
$500$	10500.0	0.939149
$501$	0	0
$502$	6276.00	0.557991
$503$	−17208.0	−1.52538	−0.762691	−	0.646763i	$-0.776122\pi$
$503$	−17208.0	−1.52538	−0.762691	+	0.646763i	$0.776122\pi$
$504$	0	0
$505$	10100.0	0.889988
$506$	−1824.00	−0.160250
$507$	0	0
$508$	−5936.00	−0.518440
$509$	15802.0	1.37605	0.688027	−	0.725685i	$-0.258477\pi$
$509$	15802.0	1.37605	0.688027	+	0.725685i	$0.258477\pi$
$510$	0	0
$511$	−3024.00	−0.261788
$512$	−11521.0	−0.994455
$513$	0	0
$514$	5826.00	0.499949
$515$	−16240.0	−1.38955
$516$	0	0
$517$	−3456.00	−0.293994
$518$	−2128.00	−0.180500
$519$	0	0
$520$	−3900.00	−0.328897
$521$	−922.000	−0.0775308	−0.0387654	−	0.999248i	$-0.512343\pi$
$521$	−922.000	−0.0775308	−0.0387654	+	0.999248i	$0.512343\pi$
$522$	0	0
$523$	6372.00	0.532750	0.266375	−	0.963870i	$-0.414174\pi$
$523$	6372.00	0.532750	0.266375	+	0.963870i	$0.414174\pi$
$524$	−9604.00	−0.800673
$525$	0	0
$526$	3448.00	0.285817
$527$	544.000	0.0449659
$528$	0	0
$529$	10937.0	0.898907
$530$	−5460.00	−0.447485
$531$	0	0
$532$	−8288.00	−0.675433
$533$	−156.000	−0.0126775
$534$	0	0
$535$	−7960.00	−0.643254
$536$	14100.0	1.13624
$537$	0	0
$538$	−4114.00	−0.329679
$539$	3348.00	0.267548
$540$	0	0
$541$	−10050.0	−0.798675	−0.399338	−	0.916804i	$-0.630760\pi$
$541$	−10050.0	−0.798675	−0.399338	+	0.916804i	$0.630760\pi$
$542$	3856.00	0.305589
$543$	0	0
$544$	2737.00	0.215713
$545$	−16980.0	−1.33457
$546$	0	0
$547$	−15356.0	−1.20032	−0.600160	−	0.799880i	$-0.704896\pi$
$547$	−15356.0	−1.20032	−0.600160	+	0.799880i	$0.704896\pi$
$548$	4662.00	0.363414
$549$	0	0
$550$	−300.000	−0.0232583
$551$	−9768.00	−0.755228
$552$	0	0
$553$	−2304.00	−0.177172
$554$	−278.000	−0.0213197
$555$	0	0
$556$	19292.0	1.47152
$557$	−5382.00	−0.409412	−0.204706	−	0.978823i	$-0.565624\pi$
$557$	−5382.00	−0.409412	−0.204706	+	0.978823i	$0.565624\pi$
$558$	0	0
$559$	2392.00	0.180985
$560$	−3280.00	−0.247509
$561$	0	0
$562$	−3222.00	−0.241836
$563$	4340.00	0.324883	0.162442	−	0.986718i	$-0.448063\pi$
$563$	4340.00	0.324883	0.162442	+	0.986718i	$0.448063\pi$
$564$	0	0
$565$	−17300.0	−1.28817
$566$	7684.00	0.570641
$567$	0	0
$568$	−6360.00	−0.469823
$569$	−13034.0	−0.960305	−0.480153	−	0.877185i	$-0.659419\pi$
$569$	−13034.0	−0.960305	−0.480153	+	0.877185i	$0.659419\pi$
$570$	0	0
$571$	22924.0	1.68010	0.840052	−	0.542506i	$-0.182524\pi$
$571$	22924.0	1.68010	0.840052	+	0.542506i	$0.182524\pi$
$572$	−2184.00	−0.159646
$573$	0	0
$574$	48.0000	0.00349039
$575$	3800.00	0.275602
$576$	0	0
$577$	−18494.0	−1.33434	−0.667171	−	0.744905i	$-0.732495\pi$
$577$	−18494.0	−1.33434	−0.667171	+	0.744905i	$0.732495\pi$
$578$	−289.000	−0.0207973
$579$	0	0
$580$	−4620.00	−0.330750
$581$	5984.00	0.427295
$582$	0	0
$583$	−6552.00	−0.465448
$584$	5670.00	0.401757
$585$	0	0
$586$	3678.00	0.259278
$587$	−2180.00	−0.153285	−0.0766424	−	0.997059i	$-0.524420\pi$
$587$	−2180.00	−0.153285	−0.0766424	+	0.997059i	$0.524420\pi$
$588$	0	0
$589$	4736.00	0.331313
$590$	4200.00	0.293070
$591$	0	0
$592$	−10906.0	−0.757152
$593$	−15186.0	−1.05163	−0.525813	−	0.850600i	$-0.676239\pi$
$593$	−15186.0	−1.05163	−0.525813	+	0.850600i	$0.676239\pi$
$594$	0	0
$595$	1360.00	0.0937051
$596$	−4382.00	−0.301164
$597$	0	0
$598$	−3952.00	−0.270250
$599$	−840.000	−0.0572979	−0.0286490	−	0.999590i	$-0.509120\pi$
$599$	−840.000	−0.0572979	−0.0286490	+	0.999590i	$0.509120\pi$
$600$	0	0
$601$	8266.00	0.561027	0.280513	−	0.959850i	$-0.409495\pi$
$601$	8266.00	0.561027	0.280513	+	0.959850i	$0.409495\pi$
$602$	−736.000	−0.0498291
$603$	0	0
$604$	−20552.0	−1.38452
$605$	−11870.0	−0.797660
$606$	0	0
$607$	−15712.0	−1.05063	−0.525313	−	0.850909i	$-0.676052\pi$
$607$	−15712.0	−1.05063	−0.525313	+	0.850909i	$0.676052\pi$
$608$	23828.0	1.58940
$609$	0	0
$610$	−3500.00	−0.232313
$611$	−7488.00	−0.495797
$612$	0	0
$613$	−24098.0	−1.58778	−0.793890	−	0.608061i	$-0.791947\pi$
$613$	−24098.0	−1.58778	−0.793890	+	0.608061i	$0.791947\pi$
$614$	4884.00	0.321013
$615$	0	0
$616$	1440.00	0.0941871
$617$	12342.0	0.805300	0.402650	−	0.915354i	$-0.368089\pi$
$617$	12342.0	0.805300	0.402650	+	0.915354i	$0.368089\pi$
$618$	0	0
$619$	22140.0	1.43761	0.718806	−	0.695211i	$-0.244689\pi$
$619$	22140.0	1.43761	0.718806	+	0.695211i	$0.244689\pi$
$620$	2240.00	0.145098
$621$	0	0
$622$	−9400.00	−0.605958
$623$	−12464.0	−0.801540
$624$	0	0
$625$	−11875.0	−0.760000
$626$	−5690.00	−0.363288
$627$	0	0
$628$	4214.00	0.267766
$629$	4522.00	0.286652
$630$	0	0
$631$	−8792.00	−0.554681	−0.277341	−	0.960772i	$-0.589453\pi$
$631$	−8792.00	−0.554681	−0.277341	+	0.960772i	$0.589453\pi$
$632$	4320.00	0.271899
$633$	0	0
$634$	6478.00	0.405795
$635$	8480.00	0.529950
$636$	0	0
$637$	7254.00	0.451199
$638$	792.000	0.0491467
$639$	0	0
$640$	14550.0	0.898655
$641$	−10050.0	−0.619269	−0.309634	−	0.950856i	$-0.600207\pi$
$641$	−10050.0	−0.619269	−0.309634	+	0.950856i	$0.600207\pi$
$642$	0	0
$643$	5588.00	0.342720	0.171360	−	0.985208i	$-0.445184\pi$
$643$	5588.00	0.342720	0.171360	+	0.985208i	$0.445184\pi$
$644$	−8512.00	−0.520838
$645$	0	0
$646$	−2516.00	−0.153236
$647$	9464.00	0.575067	0.287533	−	0.957771i	$-0.407165\pi$
$647$	9464.00	0.575067	0.287533	+	0.957771i	$0.407165\pi$
$648$	0	0
$649$	5040.00	0.304834
$650$	−650.000	−0.0392232
$651$	0	0
$652$	−252.000	−0.0151366
$653$	15474.0	0.927327	0.463663	−	0.886011i	$-0.346535\pi$
$653$	15474.0	0.927327	0.463663	+	0.886011i	$0.346535\pi$
$654$	0	0
$655$	13720.0	0.818450
$656$	246.000	0.0146413
$657$	0	0
$658$	2304.00	0.136503
$659$	7780.00	0.459887	0.229944	−	0.973204i	$-0.426146\pi$
$659$	7780.00	0.459887	0.229944	+	0.973204i	$0.426146\pi$
$660$	0	0
$661$	−3490.00	−0.205363	−0.102682	−	0.994714i	$-0.532742\pi$
$661$	−3490.00	−0.205363	−0.102682	+	0.994714i	$0.532742\pi$
$662$	6140.00	0.360480
$663$	0	0
$664$	−11220.0	−0.655754
$665$	11840.0	0.690429
$666$	0	0
$667$	−10032.0	−0.582370
$668$	14056.0	0.814136
$669$	0	0
$670$	−9400.00	−0.542020
$671$	−4200.00	−0.241638
$672$	0	0
$673$	15794.0	0.904627	0.452313	−	0.891859i	$-0.350599\pi$
$673$	15794.0	0.904627	0.452313	+	0.891859i	$0.350599\pi$
$674$	−1042.00	−0.0595495
$675$	0	0
$676$	10647.0	0.605769
$677$	−10374.0	−0.588929	−0.294465	−	0.955662i	$-0.595141\pi$
$677$	−10374.0	−0.588929	−0.294465	+	0.955662i	$0.595141\pi$
$678$	0	0
$679$	−4240.00	−0.239641
$680$	−2550.00	−0.143806
$681$	0	0
$682$	−384.000	−0.0215603
$683$	10292.0	0.576592	0.288296	−	0.957541i	$-0.406911\pi$
$683$	10292.0	0.576592	0.288296	+	0.957541i	$0.406911\pi$
$684$	0	0
$685$	−6660.00	−0.371483
$686$	−4976.00	−0.276945
$687$	0	0
$688$	−3772.00	−0.209021
$689$	−14196.0	−0.784941
$690$	0	0
$691$	−4316.00	−0.237610	−0.118805	−	0.992918i	$-0.537906\pi$
$691$	−4316.00	−0.237610	−0.118805	+	0.992918i	$0.537906\pi$
$692$	−28798.0	−1.58199
$693$	0	0
$694$	10828.0	0.592255
$695$	−27560.0	−1.50419
$696$	0	0
$697$	−102.000	−0.00554308
$698$	−1014.00	−0.0549863
$699$	0	0
$700$	−1400.00	−0.0755929
$701$	30858.0	1.66261	0.831306	−	0.555815i	$-0.187594\pi$
$701$	30858.0	1.66261	0.831306	+	0.555815i	$0.187594\pi$
$702$	0	0
$703$	39368.0	2.11208
$704$	2004.00	0.107285
$705$	0	0
$706$	7842.00	0.418042
$707$	−8080.00	−0.429816
$708$	0	0
$709$	−11386.0	−0.603117	−0.301559	−	0.953448i	$-0.597507\pi$
$709$	−11386.0	−0.603117	−0.301559	+	0.953448i	$0.597507\pi$
$710$	4240.00	0.224119
$711$	0	0
$712$	23370.0	1.23010
$713$	4864.00	0.255481
$714$	0	0
$715$	3120.00	0.163191
$716$	−5628.00	−0.293755
$717$	0	0
$718$	744.000	0.0386711
$719$	−13088.0	−0.678860	−0.339430	−	0.940631i	$-0.610234\pi$
$719$	−13088.0	−0.678860	−0.339430	+	0.940631i	$0.610234\pi$
$720$	0	0
$721$	12992.0	0.671078
$722$	−15045.0	−0.775508
$723$	0	0
$724$	22358.0	1.14769
$725$	−1650.00	−0.0845234
$726$	0	0
$727$	−7352.00	−0.375063	−0.187531	−	0.982259i	$-0.560049\pi$
$727$	−7352.00	−0.375063	−0.187531	+	0.982259i	$0.560049\pi$
$728$	3120.00	0.158839
$729$	0	0
$730$	−3780.00	−0.191649
$731$	1564.00	0.0791336
$732$	0	0
$733$	10390.0	0.523552	0.261776	−	0.965129i	$-0.415692\pi$
$733$	10390.0	0.523552	0.261776	+	0.965129i	$0.415692\pi$
$734$	6944.00	0.349193
$735$	0	0
$736$	24472.0	1.22561
$737$	−11280.0	−0.563778
$738$	0	0
$739$	10012.0	0.498373	0.249186	−	0.968456i	$-0.419837\pi$
$739$	10012.0	0.498373	0.249186	+	0.968456i	$0.419837\pi$
$740$	18620.0	0.924979
$741$	0	0
$742$	4368.00	0.216111
$743$	15336.0	0.757232	0.378616	−	0.925554i	$-0.376400\pi$
$743$	15336.0	0.757232	0.378616	+	0.925554i	$0.376400\pi$
$744$	0	0
$745$	6260.00	0.307851
$746$	3522.00	0.172855
$747$	0	0
$748$	−1428.00	−0.0698033
$749$	6368.00	0.310656
$750$	0	0
$751$	−7648.00	−0.371610	−0.185805	−	0.982587i	$-0.559489\pi$
$751$	−7648.00	−0.371610	−0.185805	+	0.982587i	$0.559489\pi$
$752$	11808.0	0.572598
$753$	0	0
$754$	1716.00	0.0828820
$755$	29360.0	1.41526
$756$	0	0
$757$	−14546.0	−0.698393	−0.349196	−	0.937050i	$-0.613545\pi$
$757$	−14546.0	−0.698393	−0.349196	+	0.937050i	$0.613545\pi$
$758$	1124.00	0.0538595
$759$	0	0
$760$	−22200.0	−1.05958
$761$	−2026.00	−0.0965078	−0.0482539	−	0.998835i	$-0.515366\pi$
$761$	−2026.00	−0.0965078	−0.0482539	+	0.998835i	$0.515366\pi$
$762$	0	0
$763$	13584.0	0.644527
$764$	−34160.0	−1.61762
$765$	0	0
$766$	9120.00	0.430181
$767$	10920.0	0.514079
$768$	0	0
$769$	−32798.0	−1.53801	−0.769003	−	0.639246i	$-0.779247\pi$
$769$	−32798.0	−1.53801	−0.769003	+	0.639246i	$0.779247\pi$
$770$	−960.000	−0.0449299
$771$	0	0
$772$	−25886.0	−1.20681
$773$	14530.0	0.676077	0.338039	−	0.941132i	$-0.390237\pi$
$773$	14530.0	0.676077	0.338039	+	0.941132i	$0.390237\pi$
$774$	0	0
$775$	800.000	0.0370798
$776$	7950.00	0.367769
$777$	0	0
$778$	4926.00	0.227000
$779$	−888.000	−0.0408420
$780$	0	0
$781$	5088.00	0.233115
$782$	−2584.00	−0.118163
$783$	0	0
$784$	−11439.0	−0.521091
$785$	−6020.00	−0.273711
$786$	0	0
$787$	33364.0	1.51118	0.755590	−	0.655045i	$-0.227350\pi$
$787$	33364.0	1.51118	0.755590	+	0.655045i	$0.227350\pi$
$788$	−13398.0	−0.605690
$789$	0	0
$790$	−2880.00	−0.129704
$791$	13840.0	0.622116
$792$	0	0
$793$	−9100.00	−0.407504
$794$	−942.000	−0.0421037
$795$	0	0
$796$	−3640.00	−0.162081
$797$	26698.0	1.18656	0.593282	−	0.804995i	$-0.297832\pi$
$797$	26698.0	1.18656	0.593282	+	0.804995i	$0.297832\pi$
$798$	0	0
$799$	−4896.00	−0.216781
$800$	4025.00	0.177882
$801$	0	0
$802$	−11182.0	−0.492332
$803$	−4536.00	−0.199342
$804$	0	0
$805$	12160.0	0.532402
$806$	−832.000	−0.0363597
$807$	0	0
$808$	15150.0	0.659623
$809$	26342.0	1.14479	0.572395	−	0.819978i	$-0.306014\pi$
$809$	26342.0	1.14479	0.572395	+	0.819978i	$0.306014\pi$
$810$	0	0
$811$	26092.0	1.12973	0.564867	−	0.825182i	$-0.308928\pi$
$811$	26092.0	1.12973	0.564867	+	0.825182i	$0.308928\pi$
$812$	3696.00	0.159734
$813$	0	0
$814$	−3192.00	−0.137444
$815$	360.000	0.0154727
$816$	0	0
$817$	13616.0	0.583064
$818$	1654.00	0.0706977
$819$	0	0
$820$	−420.000	−0.0178866
$821$	−39702.0	−1.68771	−0.843855	−	0.536572i	$-0.819719\pi$
$821$	−39702.0	−1.68771	−0.843855	+	0.536572i	$0.819719\pi$
$822$	0	0
$823$	−2344.00	−0.0992791	−0.0496396	−	0.998767i	$-0.515807\pi$
$823$	−2344.00	−0.0992791	−0.0496396	+	0.998767i	$0.515807\pi$
$824$	−24360.0	−1.02988
$825$	0	0
$826$	−3360.00	−0.141537
$827$	−5916.00	−0.248754	−0.124377	−	0.992235i	$-0.539693\pi$
$827$	−5916.00	−0.248754	−0.124377	+	0.992235i	$0.539693\pi$
$828$	0	0
$829$	−26554.0	−1.11250	−0.556248	−	0.831017i	$-0.687759\pi$
$829$	−26554.0	−1.11250	−0.556248	+	0.831017i	$0.687759\pi$
$830$	7480.00	0.312813
$831$	0	0
$832$	4342.00	0.180928
$833$	4743.00	0.197281
$834$	0	0
$835$	−20080.0	−0.832212
$836$	−12432.0	−0.514318
$837$	0	0
$838$	−8444.00	−0.348083
$839$	−20248.0	−0.833181	−0.416590	−	0.909094i	$-0.636775\pi$
$839$	−20248.0	−0.833181	−0.416590	+	0.909094i	$0.636775\pi$
$840$	0	0
$841$	−20033.0	−0.821395
$842$	−7726.00	−0.316218
$843$	0	0
$844$	−17948.0	−0.731985
$845$	−15210.0	−0.619219
$846$	0	0
$847$	9496.00	0.385226
$848$	22386.0	0.906531
$849$	0	0
$850$	−425.000	−0.0171499
$851$	40432.0	1.62866
$852$	0	0
$853$	14198.0	0.569907	0.284953	−	0.958541i	$-0.408022\pi$
$853$	14198.0	0.569907	0.284953	+	0.958541i	$0.408022\pi$
$854$	2800.00	0.112194
$855$	0	0
$856$	−11940.0	−0.476753
$857$	28534.0	1.13734	0.568671	−	0.822565i	$-0.307458\pi$
$857$	28534.0	1.13734	0.568671	+	0.822565i	$0.307458\pi$
$858$	0	0
$859$	1684.00	0.0668886	0.0334443	−	0.999441i	$-0.489352\pi$
$859$	1684.00	0.0668886	0.0334443	+	0.999441i	$0.489352\pi$
$860$	6440.00	0.255351
$861$	0	0
$862$	−15648.0	−0.618298
$863$	−37760.0	−1.48942	−0.744708	−	0.667391i	$-0.767411\pi$
$863$	−37760.0	−1.48942	−0.744708	+	0.667391i	$0.767411\pi$
$864$	0	0
$865$	41140.0	1.61711
$866$	−1298.00	−0.0509328
$867$	0	0
$868$	−1792.00	−0.0700742
$869$	−3456.00	−0.134910
$870$	0	0
$871$	−24440.0	−0.950767
$872$	−25470.0	−0.989132
$873$	0	0
$874$	−22496.0	−0.870639
$875$	12000.0	0.463627
$876$	0	0
$877$	−18882.0	−0.727024	−0.363512	−	0.931590i	$-0.618422\pi$
$877$	−18882.0	−0.727024	−0.363512	+	0.931590i	$0.618422\pi$
$878$	5288.00	0.203259
$879$	0	0
$880$	−4920.00	−0.188470
$881$	46142.0	1.76454	0.882272	−	0.470740i	$-0.156013\pi$
$881$	46142.0	1.76454	0.882272	+	0.470740i	$0.156013\pi$
$882$	0	0
$883$	60.0000	0.00228671	0.00114335	−	0.999999i	$-0.499636\pi$
$883$	60.0000	0.00228671	0.00114335	+	0.999999i	$0.499636\pi$
$884$	−3094.00	−0.117718
$885$	0	0
$886$	13892.0	0.526762
$887$	24360.0	0.922129	0.461065	−	0.887367i	$-0.347468\pi$
$887$	24360.0	0.922129	0.461065	+	0.887367i	$0.347468\pi$
$888$	0	0
$889$	−6784.00	−0.255937
$890$	−15580.0	−0.586790
$891$	0	0
$892$	27552.0	1.03420
$893$	−42624.0	−1.59727
$894$	0	0
$895$	8040.00	0.300277
$896$	−11640.0	−0.434001
$897$	0	0
$898$	16322.0	0.606539
$899$	−2112.00	−0.0783528
$900$	0	0
$901$	−9282.00	−0.343206
$902$	72.0000	0.00265780
$903$	0	0
$904$	−25950.0	−0.954739
$905$	−31940.0	−1.17317
$906$	0	0
$907$	−13396.0	−0.490416	−0.245208	−	0.969471i	$-0.578856\pi$
$907$	−13396.0	−0.490416	−0.245208	+	0.969471i	$0.578856\pi$
$908$	−11172.0	−0.408321
$909$	0	0
$910$	−2080.00	−0.0757707
$911$	−40816.0	−1.48441	−0.742203	−	0.670175i	$-0.766219\pi$
$911$	−40816.0	−1.48441	−0.742203	+	0.670175i	$0.766219\pi$
$912$	0	0
$913$	8976.00	0.325369
$914$	10278.0	0.371954
$915$	0	0
$916$	−8946.00	−0.322690
$917$	−10976.0	−0.395267
$918$	0	0
$919$	20232.0	0.726216	0.363108	−	0.931747i	$-0.381716\pi$
$919$	20232.0	0.726216	0.363108	+	0.931747i	$0.381716\pi$
$920$	−22800.0	−0.817058
$921$	0	0
$922$	−1306.00	−0.0466495
$923$	11024.0	0.393130
$924$	0	0
$925$	6650.00	0.236379
$926$	−320.000	−0.0113562
$927$	0	0
$928$	−10626.0	−0.375879
$929$	18750.0	0.662183	0.331091	−	0.943599i	$-0.392583\pi$
$929$	18750.0	0.662183	0.331091	+	0.943599i	$0.392583\pi$
$930$	0	0
$931$	41292.0	1.45359
$932$	−23898.0	−0.839920
$933$	0	0
$934$	−8676.00	−0.303948
$935$	2040.00	0.0713531
$936$	0	0
$937$	−2246.00	−0.0783070	−0.0391535	−	0.999233i	$-0.512466\pi$
$937$	−2246.00	−0.0783070	−0.0391535	+	0.999233i	$0.512466\pi$
$938$	7520.00	0.261766
$939$	0	0
$940$	−20160.0	−0.699518
$941$	−25182.0	−0.872380	−0.436190	−	0.899855i	$-0.643672\pi$
$941$	−25182.0	−0.872380	−0.436190	+	0.899855i	$0.643672\pi$
$942$	0	0
$943$	−912.000	−0.0314940
$944$	−17220.0	−0.593711
$945$	0	0
$946$	−1104.00	−0.0379431
$947$	−1588.00	−0.0544911	−0.0272455	−	0.999629i	$-0.508674\pi$
$947$	−1588.00	−0.0544911	−0.0272455	+	0.999629i	$0.508674\pi$
$948$	0	0
$949$	−9828.00	−0.336175
$950$	−3700.00	−0.126362
$951$	0	0
$952$	2040.00	0.0694504
$953$	12566.0	0.427128	0.213564	−	0.976929i	$-0.431493\pi$
$953$	12566.0	0.427128	0.213564	+	0.976929i	$0.431493\pi$
$954$	0	0
$955$	48800.0	1.65354
$956$	27328.0	0.924530
$957$	0	0
$958$	8976.00	0.302715
$959$	5328.00	0.179406
$960$	0	0
$961$	−28767.0	−0.965627
$962$	−6916.00	−0.231789
$963$	0	0
$964$	−34062.0	−1.13803
$965$	36980.0	1.23360
$966$	0	0
$967$	−36072.0	−1.19958	−0.599792	−	0.800156i	$-0.704750\pi$
$967$	−36072.0	−1.19958	−0.599792	+	0.800156i	$0.704750\pi$
$968$	−17805.0	−0.591193
$969$	0	0
$970$	−5300.00	−0.175436
$971$	50220.0	1.65977	0.829885	−	0.557935i	$-0.188406\pi$
$971$	50220.0	1.65977	0.829885	+	0.557935i	$0.188406\pi$
$972$	0	0
$973$	22048.0	0.726440
$974$	19096.0	0.628209
$975$	0	0
$976$	14350.0	0.470627
$977$	−43506.0	−1.42465	−0.712324	−	0.701851i	$-0.752357\pi$
$977$	−43506.0	−1.42465	−0.712324	+	0.701851i	$0.752357\pi$
$978$	0	0
$979$	−18696.0	−0.610344
$980$	19530.0	0.636595
$981$	0	0
$982$	964.000	0.0313264
$983$	456.000	0.0147957	0.00739783	−	0.999973i	$-0.497645\pi$
$983$	456.000	0.0147957	0.00739783	+	0.999973i	$0.497645\pi$
$984$	0	0
$985$	19140.0	0.619138
$986$	1122.00	0.0362391
$987$	0	0
$988$	−26936.0	−0.867357
$989$	13984.0	0.449611
$990$	0	0
$991$	−11664.0	−0.373884	−0.186942	−	0.982371i	$-0.559858\pi$
$991$	−11664.0	−0.373884	−0.186942	+	0.982371i	$0.559858\pi$
$992$	5152.00	0.164895
$993$	0	0
$994$	−3392.00	−0.108237
$995$	5200.00	0.165679
$996$	0	0
$997$	50710.0	1.61083	0.805417	−	0.592708i	$-0.201941\pi$
$997$	50710.0	1.61083	0.805417	+	0.592708i	$0.201941\pi$
$998$	−2756.00	−0.0874145
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.a.1.1		1
3.2	odd	2		51.4.a.c.1.1	✓	1
4.3	odd	2		2448.4.a.n.1.1		1
12.11	even	2		816.4.a.f.1.1		1
15.14	odd	2		1275.4.a.d.1.1		1
21.20	even	2		2499.4.a.i.1.1		1
51.50	odd	2		867.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.c.1.1	✓	1	3.2	odd	2
153.4.a.a.1.1		1	1.1	even	1	trivial
816.4.a.f.1.1		1	12.11	even	2
867.4.a.e.1.1		1	51.50	odd	2
1275.4.a.d.1.1		1	15.14	odd	2
2448.4.a.n.1.1		1	4.3	odd	2
2499.4.a.i.1.1		1	21.20	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:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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