LMFDB - Embedded newform 693.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} -4.00000 q^{4} -1.00000 q^{5} -7.00000 q^{7} -24.0000 q^{8} -2.00000 q^{10} +11.0000 q^{11} +7.00000 q^{13} -14.0000 q^{14} -16.0000 q^{16} +14.0000 q^{17} -45.0000 q^{19} +4.00000 q^{20} +22.0000 q^{22} +88.0000 q^{23} -124.000 q^{25} +14.0000 q^{26} +28.0000 q^{28} +69.0000 q^{29} +22.0000 q^{31} +160.000 q^{32} +28.0000 q^{34} +7.00000 q^{35} +57.0000 q^{37} -90.0000 q^{38} +24.0000 q^{40} +380.000 q^{41} +48.0000 q^{43} -44.0000 q^{44} +176.000 q^{46} +385.000 q^{47} +49.0000 q^{49} -248.000 q^{50} -28.0000 q^{52} +672.000 q^{53} -11.0000 q^{55} +168.000 q^{56} +138.000 q^{58} +469.000 q^{59} -342.000 q^{61} +44.0000 q^{62} +448.000 q^{64} -7.00000 q^{65} -139.000 q^{67} -56.0000 q^{68} +14.0000 q^{70} -132.000 q^{71} +145.000 q^{73} +114.000 q^{74} +180.000 q^{76} -77.0000 q^{77} +1244.00 q^{79} +16.0000 q^{80} +760.000 q^{82} -522.000 q^{83} -14.0000 q^{85} +96.0000 q^{86} -264.000 q^{88} -822.000 q^{89} -49.0000 q^{91} -352.000 q^{92} +770.000 q^{94} +45.0000 q^{95} +272.000 q^{97} +98.0000 q^{98} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	2.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	−1.00000	−0.0894427	−0.0447214	−	0.998999i	$-0.514240\pi$
$5$	−1.00000	−0.0894427	−0.0447214	+	0.998999i	$0.514240\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−24.0000	−1.06066
$9$	0	0
$10$	−2.00000	−0.0632456
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	7.00000	0.149342	0.0746712	−	0.997208i	$-0.476209\pi$
$13$	7.00000	0.149342	0.0746712	+	0.997208i	$0.476209\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	−16.0000	−0.250000
$17$	14.0000	0.199735	0.0998676	−	0.995001i	$-0.468158\pi$
$17$	14.0000	0.199735	0.0998676	+	0.995001i	$0.468158\pi$
$18$	0	0
$19$	−45.0000	−0.543353	−0.271677	−	0.962389i	$-0.587578\pi$
$19$	−45.0000	−0.543353	−0.271677	+	0.962389i	$0.587578\pi$
$20$	4.00000	0.0447214
$21$	0	0
$22$	22.0000	0.213201
$23$	88.0000	0.797794	0.398897	−	0.916996i	$-0.369393\pi$
$23$	88.0000	0.797794	0.398897	+	0.916996i	$0.369393\pi$
$24$	0	0
$25$	−124.000	−0.992000
$26$	14.0000	0.105601
$27$	0	0
$28$	28.0000	0.188982
$29$	69.0000	0.441827	0.220913	−	0.975293i	$-0.429096\pi$
$29$	69.0000	0.441827	0.220913	+	0.975293i	$0.429096\pi$
$30$	0	0
$31$	22.0000	0.127462	0.0637309	−	0.997967i	$-0.479700\pi$
$31$	22.0000	0.127462	0.0637309	+	0.997967i	$0.479700\pi$
$32$	160.000	0.883883
$33$	0	0
$34$	28.0000	0.141234
$35$	7.00000	0.0338062
$36$	0	0
$37$	57.0000	0.253263	0.126632	−	0.991950i	$-0.459583\pi$
$37$	57.0000	0.253263	0.126632	+	0.991950i	$0.459583\pi$
$38$	−90.0000	−0.384209
$39$	0	0
$40$	24.0000	0.0948683
$41$	380.000	1.44746	0.723732	−	0.690081i	$-0.242425\pi$
$41$	380.000	1.44746	0.723732	+	0.690081i	$0.242425\pi$
$42$	0	0
$43$	48.0000	0.170231	0.0851155	−	0.996371i	$-0.472874\pi$
$43$	48.0000	0.170231	0.0851155	+	0.996371i	$0.472874\pi$
$44$	−44.0000	−0.150756
$45$	0	0
$46$	176.000	0.564126
$47$	385.000	1.19485	0.597426	−	0.801924i	$-0.296190\pi$
$47$	385.000	1.19485	0.597426	+	0.801924i	$0.296190\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−248.000	−0.701450
$51$	0	0
$52$	−28.0000	−0.0746712
$53$	672.000	1.74163	0.870814	−	0.491612i	$-0.163592\pi$
$53$	672.000	1.74163	0.870814	+	0.491612i	$0.163592\pi$
$54$	0	0
$55$	−11.0000	−0.0269680
$56$	168.000	0.400892
$57$	0	0
$58$	138.000	0.312419
$59$	469.000	1.03489	0.517446	−	0.855716i	$-0.326883\pi$
$59$	469.000	1.03489	0.517446	+	0.855716i	$0.326883\pi$
$60$	0	0
$61$	−342.000	−0.717846	−0.358923	−	0.933367i	$-0.616856\pi$
$61$	−342.000	−0.717846	−0.358923	+	0.933367i	$0.616856\pi$
$62$	44.0000	0.0901291
$63$	0	0
$64$	448.000	0.875000
$65$	−7.00000	−0.0133576
$66$	0	0
$67$	−139.000	−0.253456	−0.126728	−	0.991938i	$-0.540448\pi$
$67$	−139.000	−0.253456	−0.126728	+	0.991938i	$0.540448\pi$
$68$	−56.0000	−0.0998676
$69$	0	0
$70$	14.0000	0.0239046
$71$	−132.000	−0.220641	−0.110321	−	0.993896i	$-0.535188\pi$
$71$	−132.000	−0.220641	−0.110321	+	0.993896i	$0.535188\pi$
$72$	0	0
$73$	145.000	0.232479	0.116239	−	0.993221i	$-0.462916\pi$
$73$	145.000	0.232479	0.116239	+	0.993221i	$0.462916\pi$
$74$	114.000	0.179084
$75$	0	0
$76$	180.000	0.271677
$77$	−77.0000	−0.113961
$78$	0	0
$79$	1244.00	1.77166	0.885829	−	0.464012i	$-0.153591\pi$
$79$	1244.00	1.77166	0.885829	+	0.464012i	$0.153591\pi$
$80$	16.0000	0.0223607
$81$	0	0
$82$	760.000	1.02351
$83$	−522.000	−0.690325	−0.345162	−	0.938543i	$-0.612176\pi$
$83$	−522.000	−0.690325	−0.345162	+	0.938543i	$0.612176\pi$
$84$	0	0
$85$	−14.0000	−0.0178649
$86$	96.0000	0.120371
$87$	0	0
$88$	−264.000	−0.319801
$89$	−822.000	−0.979009	−0.489505	−	0.872001i	$-0.662822\pi$
$89$	−822.000	−0.979009	−0.489505	+	0.872001i	$0.662822\pi$
$90$	0	0
$91$	−49.0000	−0.0564461
$92$	−352.000	−0.398897
$93$	0	0
$94$	770.000	0.844888
$95$	45.0000	0.0485990
$96$	0	0
$97$	272.000	0.284716	0.142358	−	0.989815i	$-0.454532\pi$
$97$	272.000	0.284716	0.142358	+	0.989815i	$0.454532\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	496.000	0.496000
$101$	874.000	0.861052	0.430526	−	0.902578i	$-0.358328\pi$
$101$	874.000	0.861052	0.430526	+	0.902578i	$0.358328\pi$
$102$	0	0
$103$	−826.000	−0.790177	−0.395088	−	0.918643i	$-0.629286\pi$
$103$	−826.000	−0.790177	−0.395088	+	0.918643i	$0.629286\pi$
$104$	−168.000	−0.158401
$105$	0	0
$106$	1344.00	1.23152
$107$	−219.000	−0.197865	−0.0989324	−	0.995094i	$-0.531543\pi$
$107$	−219.000	−0.197865	−0.0989324	+	0.995094i	$0.531543\pi$
$108$	0	0
$109$	1426.00	1.25308	0.626541	−	0.779388i	$-0.284470\pi$
$109$	1426.00	1.25308	0.626541	+	0.779388i	$0.284470\pi$
$110$	−22.0000	−0.0190693
$111$	0	0
$112$	112.000	0.0944911
$113$	−882.000	−0.734262	−0.367131	−	0.930169i	$-0.619660\pi$
$113$	−882.000	−0.734262	−0.367131	+	0.930169i	$0.619660\pi$
$114$	0	0
$115$	−88.0000	−0.0713569
$116$	−276.000	−0.220913
$117$	0	0
$118$	938.000	0.731779
$119$	−98.0000	−0.0754928
$120$	0	0
$121$	121.000	0.0909091
$122$	−684.000	−0.507594
$123$	0	0
$124$	−88.0000	−0.0637309
$125$	249.000	0.178170
$126$	0	0
$127$	1826.00	1.27584	0.637918	−	0.770104i	$-0.279796\pi$
$127$	1826.00	1.27584	0.637918	+	0.770104i	$0.279796\pi$
$128$	−384.000	−0.265165
$129$	0	0
$130$	−14.0000	−0.00944524
$131$	−84.0000	−0.0560238	−0.0280119	−	0.999608i	$-0.508918\pi$
$131$	−84.0000	−0.0560238	−0.0280119	+	0.999608i	$0.508918\pi$
$132$	0	0
$133$	315.000	0.205368
$134$	−278.000	−0.179220
$135$	0	0
$136$	−336.000	−0.211851
$137$	1834.00	1.14372	0.571858	−	0.820352i	$-0.306223\pi$
$137$	1834.00	1.14372	0.571858	+	0.820352i	$0.306223\pi$
$138$	0	0
$139$	2416.00	1.47426	0.737131	−	0.675750i	$-0.236180\pi$
$139$	2416.00	1.47426	0.737131	+	0.675750i	$0.236180\pi$
$140$	−28.0000	−0.0169031
$141$	0	0
$142$	−264.000	−0.156017
$143$	77.0000	0.0450284
$144$	0	0
$145$	−69.0000	−0.0395182
$146$	290.000	0.164387
$147$	0	0
$148$	−228.000	−0.126632
$149$	−1895.00	−1.04191	−0.520955	−	0.853584i	$-0.674424\pi$
$149$	−1895.00	−1.04191	−0.520955	+	0.853584i	$0.674424\pi$
$150$	0	0
$151$	−3478.00	−1.87441	−0.937204	−	0.348782i	$-0.886596\pi$
$151$	−3478.00	−1.87441	−0.937204	+	0.348782i	$0.886596\pi$
$152$	1080.00	0.576313
$153$	0	0
$154$	−154.000	−0.0805823
$155$	−22.0000	−0.0114005
$156$	0	0
$157$	−952.000	−0.483935	−0.241968	−	0.970284i	$-0.577793\pi$
$157$	−952.000	−0.483935	−0.241968	+	0.970284i	$0.577793\pi$
$158$	2488.00	1.25275
$159$	0	0
$160$	−160.000	−0.0790569
$161$	−616.000	−0.301538
$162$	0	0
$163$	−2483.00	−1.19315	−0.596575	−	0.802557i	$-0.703472\pi$
$163$	−2483.00	−1.19315	−0.596575	+	0.802557i	$0.703472\pi$
$164$	−1520.00	−0.723732
$165$	0	0
$166$	−1044.00	−0.488133
$167$	−468.000	−0.216856	−0.108428	−	0.994104i	$-0.534582\pi$
$167$	−468.000	−0.216856	−0.108428	+	0.994104i	$0.534582\pi$
$168$	0	0
$169$	−2148.00	−0.977697
$170$	−28.0000	−0.0126324
$171$	0	0
$172$	−192.000	−0.0851155
$173$	2148.00	0.943985	0.471993	−	0.881603i	$-0.343535\pi$
$173$	2148.00	0.943985	0.471993	+	0.881603i	$0.343535\pi$
$174$	0	0
$175$	868.000	0.374941
$176$	−176.000	−0.0753778
$177$	0	0
$178$	−1644.00	−0.692264
$179$	1464.00	0.611310	0.305655	−	0.952142i	$-0.401125\pi$
$179$	1464.00	0.611310	0.305655	+	0.952142i	$0.401125\pi$
$180$	0	0
$181$	1432.00	0.588065	0.294032	−	0.955795i	$-0.405003\pi$
$181$	1432.00	0.588065	0.294032	+	0.955795i	$0.405003\pi$
$182$	−98.0000	−0.0399134
$183$	0	0
$184$	−2112.00	−0.846189
$185$	−57.0000	−0.0226526
$186$	0	0
$187$	154.000	0.0602224
$188$	−1540.00	−0.597426
$189$	0	0
$190$	90.0000	0.0343647
$191$	1330.00	0.503850	0.251925	−	0.967747i	$-0.418936\pi$
$191$	1330.00	0.503850	0.251925	+	0.967747i	$0.418936\pi$
$192$	0	0
$193$	−2540.00	−0.947322	−0.473661	−	0.880707i	$-0.657068\pi$
$193$	−2540.00	−0.947322	−0.473661	+	0.880707i	$0.657068\pi$
$194$	544.000	0.201324
$195$	0	0
$196$	−196.000	−0.0714286
$197$	−1606.00	−0.580826	−0.290413	−	0.956901i	$-0.593793\pi$
$197$	−1606.00	−0.580826	−0.290413	+	0.956901i	$0.593793\pi$
$198$	0	0
$199$	288.000	0.102592	0.0512959	−	0.998683i	$-0.483665\pi$
$199$	288.000	0.102592	0.0512959	+	0.998683i	$0.483665\pi$
$200$	2976.00	1.05217
$201$	0	0
$202$	1748.00	0.608856
$203$	−483.000	−0.166995
$204$	0	0
$205$	−380.000	−0.129465
$206$	−1652.00	−0.558739
$207$	0	0
$208$	−112.000	−0.0373356
$209$	−495.000	−0.163827
$210$	0	0
$211$	1482.00	0.483531	0.241766	−	0.970335i	$-0.422273\pi$
$211$	1482.00	0.483531	0.241766	+	0.970335i	$0.422273\pi$
$212$	−2688.00	−0.870814
$213$	0	0
$214$	−438.000	−0.139912
$215$	−48.0000	−0.0152259
$216$	0	0
$217$	−154.000	−0.0481760
$218$	2852.00	0.886063
$219$	0	0
$220$	44.0000	0.0134840
$221$	98.0000	0.0298289
$222$	0	0
$223$	3502.00	1.05162	0.525810	−	0.850602i	$-0.323762\pi$
$223$	3502.00	1.05162	0.525810	+	0.850602i	$0.323762\pi$
$224$	−1120.00	−0.334077
$225$	0	0
$226$	−1764.00	−0.519201
$227$	−1830.00	−0.535072	−0.267536	−	0.963548i	$-0.586209\pi$
$227$	−1830.00	−0.535072	−0.267536	+	0.963548i	$0.586209\pi$
$228$	0	0
$229$	−2268.00	−0.654470	−0.327235	−	0.944943i	$-0.606117\pi$
$229$	−2268.00	−0.654470	−0.327235	+	0.944943i	$0.606117\pi$
$230$	−176.000	−0.0504569
$231$	0	0
$232$	−1656.00	−0.468628
$233$	−2430.00	−0.683239	−0.341619	−	0.939838i	$-0.610975\pi$
$233$	−2430.00	−0.683239	−0.341619	+	0.939838i	$0.610975\pi$
$234$	0	0
$235$	−385.000	−0.106871
$236$	−1876.00	−0.517446
$237$	0	0
$238$	−196.000	−0.0533815
$239$	2151.00	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	2151.00	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	2803.00	0.749200	0.374600	−	0.927187i	$-0.377780\pi$
$241$	2803.00	0.749200	0.374600	+	0.927187i	$0.377780\pi$
$242$	242.000	0.0642824
$243$	0	0
$244$	1368.00	0.358923
$245$	−49.0000	−0.0127775
$246$	0	0
$247$	−315.000	−0.0811456
$248$	−528.000	−0.135194
$249$	0	0
$250$	498.000	0.125985
$251$	5339.00	1.34261	0.671304	−	0.741182i	$-0.265734\pi$
$251$	5339.00	1.34261	0.671304	+	0.741182i	$0.265734\pi$
$252$	0	0
$253$	968.000	0.240544
$254$	3652.00	0.902153
$255$	0	0
$256$	−4352.00	−1.06250
$257$	4439.00	1.07742	0.538711	−	0.842491i	$-0.318912\pi$
$257$	4439.00	1.07742	0.538711	+	0.842491i	$0.318912\pi$
$258$	0	0
$259$	−399.000	−0.0957245
$260$	28.0000	0.00667879
$261$	0	0
$262$	−168.000	−0.0396148
$263$	−1271.00	−0.297997	−0.148999	−	0.988837i	$-0.547605\pi$
$263$	−1271.00	−0.297997	−0.148999	+	0.988837i	$0.547605\pi$
$264$	0	0
$265$	−672.000	−0.155776
$266$	630.000	0.145217
$267$	0	0
$268$	556.000	0.126728
$269$	−3682.00	−0.834556	−0.417278	−	0.908779i	$-0.637016\pi$
$269$	−3682.00	−0.834556	−0.417278	+	0.908779i	$0.637016\pi$
$270$	0	0
$271$	4127.00	0.925083	0.462541	−	0.886598i	$-0.346938\pi$
$271$	4127.00	0.925083	0.462541	+	0.886598i	$0.346938\pi$
$272$	−224.000	−0.0499338
$273$	0	0
$274$	3668.00	0.808730
$275$	−1364.00	−0.299099
$276$	0	0
$277$	−44.0000	−0.00954406	−0.00477203	−	0.999989i	$-0.501519\pi$
$277$	−44.0000	−0.00954406	−0.00477203	+	0.999989i	$0.501519\pi$
$278$	4832.00	1.04246
$279$	0	0
$280$	−168.000	−0.0358569
$281$	561.000	0.119098	0.0595489	−	0.998225i	$-0.481034\pi$
$281$	561.000	0.119098	0.0595489	+	0.998225i	$0.481034\pi$
$282$	0	0
$283$	−3845.00	−0.807638	−0.403819	−	0.914839i	$-0.632317\pi$
$283$	−3845.00	−0.807638	−0.403819	+	0.914839i	$0.632317\pi$
$284$	528.000	0.110321
$285$	0	0
$286$	154.000	0.0318399
$287$	−2660.00	−0.547090
$288$	0	0
$289$	−4717.00	−0.960106
$290$	−138.000	−0.0279436
$291$	0	0
$292$	−580.000	−0.116239
$293$	−912.000	−0.181842	−0.0909208	−	0.995858i	$-0.528981\pi$
$293$	−912.000	−0.181842	−0.0909208	+	0.995858i	$0.528981\pi$
$294$	0	0
$295$	−469.000	−0.0925635
$296$	−1368.00	−0.268626
$297$	0	0
$298$	−3790.00	−0.736741
$299$	616.000	0.119144
$300$	0	0
$301$	−336.000	−0.0643413
$302$	−6956.00	−1.32541
$303$	0	0
$304$	720.000	0.135838
$305$	342.000	0.0642061
$306$	0	0
$307$	6860.00	1.27531	0.637656	−	0.770321i	$-0.279904\pi$
$307$	6860.00	1.27531	0.637656	+	0.770321i	$0.279904\pi$
$308$	308.000	0.0569803
$309$	0	0
$310$	−44.0000	−0.00806139
$311$	−5472.00	−0.997713	−0.498856	−	0.866685i	$-0.666246\pi$
$311$	−5472.00	−0.997713	−0.498856	+	0.866685i	$0.666246\pi$
$312$	0	0
$313$	6422.00	1.15972	0.579861	−	0.814716i	$-0.303107\pi$
$313$	6422.00	1.15972	0.579861	+	0.814716i	$0.303107\pi$
$314$	−1904.00	−0.342194
$315$	0	0
$316$	−4976.00	−0.885829
$317$	6476.00	1.14741	0.573704	−	0.819063i	$-0.305506\pi$
$317$	6476.00	1.14741	0.573704	+	0.819063i	$0.305506\pi$
$318$	0	0
$319$	759.000	0.133216
$320$	−448.000	−0.0782624
$321$	0	0
$322$	−1232.00	−0.213219
$323$	−630.000	−0.108527
$324$	0	0
$325$	−868.000	−0.148148
$326$	−4966.00	−0.843685
$327$	0	0
$328$	−9120.00	−1.53527
$329$	−2695.00	−0.451611
$330$	0	0
$331$	−11840.0	−1.96612	−0.983059	−	0.183288i	$-0.941326\pi$
$331$	−11840.0	−1.96612	−0.983059	+	0.183288i	$0.941326\pi$
$332$	2088.00	0.345162
$333$	0	0
$334$	−936.000	−0.153340
$335$	139.000	0.0226698
$336$	0	0
$337$	1640.00	0.265093	0.132547	−	0.991177i	$-0.457685\pi$
$337$	1640.00	0.265093	0.132547	+	0.991177i	$0.457685\pi$
$338$	−4296.00	−0.691336
$339$	0	0
$340$	56.0000	0.00893243
$341$	242.000	0.0384312
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−1152.00	−0.180557
$345$	0	0
$346$	4296.00	0.667498
$347$	−2032.00	−0.314362	−0.157181	−	0.987570i	$-0.550241\pi$
$347$	−2032.00	−0.314362	−0.157181	+	0.987570i	$0.550241\pi$
$348$	0	0
$349$	1755.00	0.269178	0.134589	−	0.990902i	$-0.457029\pi$
$349$	1755.00	0.269178	0.134589	+	0.990902i	$0.457029\pi$
$350$	1736.00	0.265123
$351$	0	0
$352$	1760.00	0.266501
$353$	1101.00	0.166007	0.0830033	−	0.996549i	$-0.473549\pi$
$353$	1101.00	0.166007	0.0830033	+	0.996549i	$0.473549\pi$
$354$	0	0
$355$	132.000	0.0197347
$356$	3288.00	0.489505
$357$	0	0
$358$	2928.00	0.432261
$359$	−5020.00	−0.738010	−0.369005	−	0.929427i	$-0.620301\pi$
$359$	−5020.00	−0.738010	−0.369005	+	0.929427i	$0.620301\pi$
$360$	0	0
$361$	−4834.00	−0.704767
$362$	2864.00	0.415825
$363$	0	0
$364$	196.000	0.0282231
$365$	−145.000	−0.0207936
$366$	0	0
$367$	11638.0	1.65531	0.827655	−	0.561237i	$-0.189675\pi$
$367$	11638.0	1.65531	0.827655	+	0.561237i	$0.189675\pi$
$368$	−1408.00	−0.199449
$369$	0	0
$370$	−114.000	−0.0160178
$371$	−4704.00	−0.658274
$372$	0	0
$373$	−8834.00	−1.22629	−0.613146	−	0.789969i	$-0.710096\pi$
$373$	−8834.00	−1.22629	−0.613146	+	0.789969i	$0.710096\pi$
$374$	308.000	0.0425837
$375$	0	0
$376$	−9240.00	−1.26733
$377$	483.000	0.0659835
$378$	0	0
$379$	7507.00	1.01744	0.508719	−	0.860933i	$-0.330119\pi$
$379$	7507.00	1.01744	0.508719	+	0.860933i	$0.330119\pi$
$380$	−180.000	−0.0242995
$381$	0	0
$382$	2660.00	0.356276
$383$	−3608.00	−0.481358	−0.240679	−	0.970605i	$-0.577370\pi$
$383$	−3608.00	−0.481358	−0.240679	+	0.970605i	$0.577370\pi$
$384$	0	0
$385$	77.0000	0.0101929
$386$	−5080.00	−0.669858
$387$	0	0
$388$	−1088.00	−0.142358
$389$	7896.00	1.02916	0.514580	−	0.857442i	$-0.327948\pi$
$389$	7896.00	1.02916	0.514580	+	0.857442i	$0.327948\pi$
$390$	0	0
$391$	1232.00	0.159348
$392$	−1176.00	−0.151523
$393$	0	0
$394$	−3212.00	−0.410706
$395$	−1244.00	−0.158462
$396$	0	0
$397$	6934.00	0.876593	0.438297	−	0.898830i	$-0.355582\pi$
$397$	6934.00	0.876593	0.438297	+	0.898830i	$0.355582\pi$
$398$	576.000	0.0725434
$399$	0	0
$400$	1984.00	0.248000
$401$	700.000	0.0871729	0.0435864	−	0.999050i	$-0.486122\pi$
$401$	700.000	0.0871729	0.0435864	+	0.999050i	$0.486122\pi$
$402$	0	0
$403$	154.000	0.0190355
$404$	−3496.00	−0.430526
$405$	0	0
$406$	−966.000	−0.118083
$407$	627.000	0.0763618
$408$	0	0
$409$	−11550.0	−1.39636	−0.698179	−	0.715923i	$-0.746006\pi$
$409$	−11550.0	−1.39636	−0.698179	+	0.715923i	$0.746006\pi$
$410$	−760.000	−0.0915457
$411$	0	0
$412$	3304.00	0.395088
$413$	−3283.00	−0.391152
$414$	0	0
$415$	522.000	0.0617445
$416$	1120.00	0.132001
$417$	0	0
$418$	−990.000	−0.115843
$419$	4555.00	0.531089	0.265545	−	0.964099i	$-0.414448\pi$
$419$	4555.00	0.531089	0.265545	+	0.964099i	$0.414448\pi$
$420$	0	0
$421$	−161.000	−0.0186381	−0.00931907	−	0.999957i	$-0.502966\pi$
$421$	−161.000	−0.0186381	−0.00931907	+	0.999957i	$0.502966\pi$
$422$	2964.00	0.341908
$423$	0	0
$424$	−16128.0	−1.84728
$425$	−1736.00	−0.198137
$426$	0	0
$427$	2394.00	0.271320
$428$	876.000	0.0989324
$429$	0	0
$430$	−96.0000	−0.0107664
$431$	3249.00	0.363106	0.181553	−	0.983381i	$-0.441888\pi$
$431$	3249.00	0.363106	0.181553	+	0.983381i	$0.441888\pi$
$432$	0	0
$433$	1792.00	0.198887	0.0994434	−	0.995043i	$-0.468294\pi$
$433$	1792.00	0.198887	0.0994434	+	0.995043i	$0.468294\pi$
$434$	−308.000	−0.0340656
$435$	0	0
$436$	−5704.00	−0.626541
$437$	−3960.00	−0.433484
$438$	0	0
$439$	9319.00	1.01315	0.506574	−	0.862197i	$-0.330912\pi$
$439$	9319.00	1.01315	0.506574	+	0.862197i	$0.330912\pi$
$440$	264.000	0.0286039
$441$	0	0
$442$	196.000	0.0210922
$443$	−5608.00	−0.601454	−0.300727	−	0.953710i	$-0.597229\pi$
$443$	−5608.00	−0.601454	−0.300727	+	0.953710i	$0.597229\pi$
$444$	0	0
$445$	822.000	0.0875653
$446$	7004.00	0.743608
$447$	0	0
$448$	−3136.00	−0.330719
$449$	4384.00	0.460788	0.230394	−	0.973097i	$-0.425999\pi$
$449$	4384.00	0.460788	0.230394	+	0.973097i	$0.425999\pi$
$450$	0	0
$451$	4180.00	0.436427
$452$	3528.00	0.367131
$453$	0	0
$454$	−3660.00	−0.378353
$455$	49.0000	0.00504869
$456$	0	0
$457$	324.000	0.0331643	0.0165821	−	0.999863i	$-0.494721\pi$
$457$	324.000	0.0331643	0.0165821	+	0.999863i	$0.494721\pi$
$458$	−4536.00	−0.462780
$459$	0	0
$460$	352.000	0.0356784
$461$	18360.0	1.85490	0.927452	−	0.373943i	$-0.121994\pi$
$461$	18360.0	1.85490	0.927452	+	0.373943i	$0.121994\pi$
$462$	0	0
$463$	−1667.00	−0.167326	−0.0836631	−	0.996494i	$-0.526662\pi$
$463$	−1667.00	−0.167326	−0.0836631	+	0.996494i	$0.526662\pi$
$464$	−1104.00	−0.110457
$465$	0	0
$466$	−4860.00	−0.483123
$467$	−10839.0	−1.07402	−0.537012	−	0.843575i	$-0.680447\pi$
$467$	−10839.0	−1.07402	−0.537012	+	0.843575i	$0.680447\pi$
$468$	0	0
$469$	973.000	0.0957974
$470$	−770.000	−0.0755690
$471$	0	0
$472$	−11256.0	−1.09767
$473$	528.000	0.0513266
$474$	0	0
$475$	5580.00	0.539006
$476$	392.000	0.0377464
$477$	0	0
$478$	4302.00	0.411650
$479$	910.000	0.0868037	0.0434018	−	0.999058i	$-0.486180\pi$
$479$	910.000	0.0868037	0.0434018	+	0.999058i	$0.486180\pi$
$480$	0	0
$481$	399.000	0.0378229
$482$	5606.00	0.529764
$483$	0	0
$484$	−484.000	−0.0454545
$485$	−272.000	−0.0254657
$486$	0	0
$487$	12032.0	1.11955	0.559776	−	0.828644i	$-0.310887\pi$
$487$	12032.0	1.11955	0.559776	+	0.828644i	$0.310887\pi$
$488$	8208.00	0.761391
$489$	0	0
$490$	−98.0000	−0.00903508
$491$	−4243.00	−0.389988	−0.194994	−	0.980804i	$-0.562469\pi$
$491$	−4243.00	−0.389988	−0.194994	+	0.980804i	$0.562469\pi$
$492$	0	0
$493$	966.000	0.0882484
$494$	−630.000	−0.0573786
$495$	0	0
$496$	−352.000	−0.0318655
$497$	924.000	0.0833945
$498$	0	0
$499$	−5869.00	−0.526518	−0.263259	−	0.964725i	$-0.584797\pi$
$499$	−5869.00	−0.526518	−0.263259	+	0.964725i	$0.584797\pi$
$500$	−996.000	−0.0890849
$501$	0	0
$502$	10678.0	0.949367
$503$	−148.000	−0.0131193	−0.00655964	−	0.999978i	$-0.502088\pi$
$503$	−148.000	−0.0131193	−0.00655964	+	0.999978i	$0.502088\pi$
$504$	0	0
$505$	−874.000	−0.0770148
$506$	1936.00	0.170090
$507$	0	0
$508$	−7304.00	−0.637918
$509$	−12114.0	−1.05490	−0.527450	−	0.849586i	$-0.676852\pi$
$509$	−12114.0	−1.05490	−0.527450	+	0.849586i	$0.676852\pi$
$510$	0	0
$511$	−1015.00	−0.0878688
$512$	−5632.00	−0.486136
$513$	0	0
$514$	8878.00	0.761852
$515$	826.000	0.0706756
$516$	0	0
$517$	4235.00	0.360261
$518$	−798.000	−0.0676875
$519$	0	0
$520$	168.000	0.0141679
$521$	14943.0	1.25655	0.628277	−	0.777990i	$-0.283760\pi$
$521$	14943.0	1.25655	0.628277	+	0.777990i	$0.283760\pi$
$522$	0	0
$523$	16589.0	1.38697	0.693486	−	0.720470i	$-0.256074\pi$
$523$	16589.0	1.38697	0.693486	+	0.720470i	$0.256074\pi$
$524$	336.000	0.0280119
$525$	0	0
$526$	−2542.00	−0.210716
$527$	308.000	0.0254586
$528$	0	0
$529$	−4423.00	−0.363524
$530$	−1344.00	−0.110150
$531$	0	0
$532$	−1260.00	−0.102684
$533$	2660.00	0.216168
$534$	0	0
$535$	219.000	0.0176976
$536$	3336.00	0.268831
$537$	0	0
$538$	−7364.00	−0.590120
$539$	539.000	0.0430730
$540$	0	0
$541$	3232.00	0.256848	0.128424	−	0.991719i	$-0.459008\pi$
$541$	3232.00	0.256848	0.128424	+	0.991719i	$0.459008\pi$
$542$	8254.00	0.654132
$543$	0	0
$544$	2240.00	0.176543
$545$	−1426.00	−0.112079
$546$	0	0
$547$	−16468.0	−1.28724	−0.643621	−	0.765345i	$-0.722568\pi$
$547$	−16468.0	−1.28724	−0.643621	+	0.765345i	$0.722568\pi$
$548$	−7336.00	−0.571858
$549$	0	0
$550$	−2728.00	−0.211495
$551$	−3105.00	−0.240068
$552$	0	0
$553$	−8708.00	−0.669624
$554$	−88.0000	−0.00674867
$555$	0	0
$556$	−9664.00	−0.737131
$557$	−14143.0	−1.07587	−0.537934	−	0.842987i	$-0.680795\pi$
$557$	−14143.0	−1.07587	−0.537934	+	0.842987i	$0.680795\pi$
$558$	0	0
$559$	336.000	0.0254227
$560$	−112.000	−0.00845154
$561$	0	0
$562$	1122.00	0.0842148
$563$	22916.0	1.71544	0.857721	−	0.514115i	$-0.171879\pi$
$563$	22916.0	1.71544	0.857721	+	0.514115i	$0.171879\pi$
$564$	0	0
$565$	882.000	0.0656744
$566$	−7690.00	−0.571086
$567$	0	0
$568$	3168.00	0.234025
$569$	294.000	0.0216610	0.0108305	−	0.999941i	$-0.496552\pi$
$569$	294.000	0.0216610	0.0108305	+	0.999941i	$0.496552\pi$
$570$	0	0
$571$	13292.0	0.974173	0.487087	−	0.873354i	$-0.338060\pi$
$571$	13292.0	0.974173	0.487087	+	0.873354i	$0.338060\pi$
$572$	−308.000	−0.0225142
$573$	0	0
$574$	−5320.00	−0.386851
$575$	−10912.0	−0.791412
$576$	0	0
$577$	9630.00	0.694804	0.347402	−	0.937716i	$-0.387064\pi$
$577$	9630.00	0.694804	0.347402	+	0.937716i	$0.387064\pi$
$578$	−9434.00	−0.678897
$579$	0	0
$580$	276.000	0.0197591
$581$	3654.00	0.260918
$582$	0	0
$583$	7392.00	0.525121
$584$	−3480.00	−0.246581
$585$	0	0
$586$	−1824.00	−0.128581
$587$	17015.0	1.19640	0.598198	−	0.801348i	$-0.295884\pi$
$587$	17015.0	1.19640	0.598198	+	0.801348i	$0.295884\pi$
$588$	0	0
$589$	−990.000	−0.0692568
$590$	−938.000	−0.0654523
$591$	0	0
$592$	−912.000	−0.0633158
$593$	12028.0	0.832936	0.416468	−	0.909150i	$-0.363268\pi$
$593$	12028.0	0.832936	0.416468	+	0.909150i	$0.363268\pi$
$594$	0	0
$595$	98.0000	0.00675228
$596$	7580.00	0.520955
$597$	0	0
$598$	1232.00	0.0842479
$599$	−1252.00	−0.0854012	−0.0427006	−	0.999088i	$-0.513596\pi$
$599$	−1252.00	−0.0854012	−0.0427006	+	0.999088i	$0.513596\pi$
$600$	0	0
$601$	−19699.0	−1.33700	−0.668502	−	0.743711i	$-0.733064\pi$
$601$	−19699.0	−1.33700	−0.668502	+	0.743711i	$0.733064\pi$
$602$	−672.000	−0.0454961
$603$	0	0
$604$	13912.0	0.937204
$605$	−121.000	−0.00813116
$606$	0	0
$607$	29855.0	1.99634	0.998169	−	0.0604881i	$-0.0192657\pi$
$607$	29855.0	1.99634	0.998169	+	0.0604881i	$0.0192657\pi$
$608$	−7200.00	−0.480261
$609$	0	0
$610$	684.000	0.0454006
$611$	2695.00	0.178442
$612$	0	0
$613$	−3178.00	−0.209393	−0.104697	−	0.994504i	$-0.533387\pi$
$613$	−3178.00	−0.209393	−0.104697	+	0.994504i	$0.533387\pi$
$614$	13720.0	0.901782
$615$	0	0
$616$	1848.00	0.120873
$617$	14394.0	0.939191	0.469595	−	0.882882i	$-0.344400\pi$
$617$	14394.0	0.939191	0.469595	+	0.882882i	$0.344400\pi$
$618$	0	0
$619$	3878.00	0.251809	0.125905	−	0.992042i	$-0.459817\pi$
$619$	3878.00	0.251809	0.125905	+	0.992042i	$0.459817\pi$
$620$	88.0000	0.00570027
$621$	0	0
$622$	−10944.0	−0.705489
$623$	5754.00	0.370031
$624$	0	0
$625$	15251.0	0.976064
$626$	12844.0	0.820047
$627$	0	0
$628$	3808.00	0.241968
$629$	798.000	0.0505856
$630$	0	0
$631$	18100.0	1.14192	0.570958	−	0.820979i	$-0.306572\pi$
$631$	18100.0	1.14192	0.570958	+	0.820979i	$0.306572\pi$
$632$	−29856.0	−1.87913
$633$	0	0
$634$	12952.0	0.811340
$635$	−1826.00	−0.114114
$636$	0	0
$637$	343.000	0.0213346
$638$	1518.00	0.0941978
$639$	0	0
$640$	384.000	0.0237171
$641$	7470.00	0.460292	0.230146	−	0.973156i	$-0.426080\pi$
$641$	7470.00	0.460292	0.230146	+	0.973156i	$0.426080\pi$
$642$	0	0
$643$	25400.0	1.55782	0.778910	−	0.627136i	$-0.215773\pi$
$643$	25400.0	1.55782	0.778910	+	0.627136i	$0.215773\pi$
$644$	2464.00	0.150769
$645$	0	0
$646$	−1260.00	−0.0767400
$647$	−25061.0	−1.52280	−0.761398	−	0.648284i	$-0.775487\pi$
$647$	−25061.0	−1.52280	−0.761398	+	0.648284i	$0.775487\pi$
$648$	0	0
$649$	5159.00	0.312032
$650$	−1736.00	−0.104756
$651$	0	0
$652$	9932.00	0.596575
$653$	1832.00	0.109788	0.0548941	−	0.998492i	$-0.482518\pi$
$653$	1832.00	0.109788	0.0548941	+	0.998492i	$0.482518\pi$
$654$	0	0
$655$	84.0000	0.00501092
$656$	−6080.00	−0.361866
$657$	0	0
$658$	−5390.00	−0.319338
$659$	27809.0	1.64383	0.821916	−	0.569609i	$-0.192905\pi$
$659$	27809.0	1.64383	0.821916	+	0.569609i	$0.192905\pi$
$660$	0	0
$661$	−29612.0	−1.74247	−0.871235	−	0.490865i	$-0.836681\pi$
$661$	−29612.0	−1.74247	−0.871235	+	0.490865i	$0.836681\pi$
$662$	−23680.0	−1.39026
$663$	0	0
$664$	12528.0	0.732200
$665$	−315.000	−0.0183687
$666$	0	0
$667$	6072.00	0.352487
$668$	1872.00	0.108428
$669$	0	0
$670$	278.000	0.0160300
$671$	−3762.00	−0.216439
$672$	0	0
$673$	−3892.00	−0.222921	−0.111460	−	0.993769i	$-0.535553\pi$
$673$	−3892.00	−0.222921	−0.111460	+	0.993769i	$0.535553\pi$
$674$	3280.00	0.187449
$675$	0	0
$676$	8592.00	0.488848
$677$	−10518.0	−0.597104	−0.298552	−	0.954393i	$-0.596504\pi$
$677$	−10518.0	−0.597104	−0.298552	+	0.954393i	$0.596504\pi$
$678$	0	0
$679$	−1904.00	−0.107612
$680$	336.000	0.0189485
$681$	0	0
$682$	484.000	0.0271750
$683$	−12902.0	−0.722813	−0.361407	−	0.932408i	$-0.617703\pi$
$683$	−12902.0	−0.722813	−0.361407	+	0.932408i	$0.617703\pi$
$684$	0	0
$685$	−1834.00	−0.102297
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−768.000	−0.0425577
$689$	4704.00	0.260099
$690$	0	0
$691$	30386.0	1.67285	0.836424	−	0.548083i	$-0.184642\pi$
$691$	30386.0	1.67285	0.836424	+	0.548083i	$0.184642\pi$
$692$	−8592.00	−0.471993
$693$	0	0
$694$	−4064.00	−0.222287
$695$	−2416.00	−0.131862
$696$	0	0
$697$	5320.00	0.289110
$698$	3510.00	0.190337
$699$	0	0
$700$	−3472.00	−0.187470
$701$	26842.0	1.44623	0.723116	−	0.690727i	$-0.242709\pi$
$701$	26842.0	1.44623	0.723116	+	0.690727i	$0.242709\pi$
$702$	0	0
$703$	−2565.00	−0.137611
$704$	4928.00	0.263822
$705$	0	0
$706$	2202.00	0.117384
$707$	−6118.00	−0.325447
$708$	0	0
$709$	−5701.00	−0.301982	−0.150991	−	0.988535i	$-0.548247\pi$
$709$	−5701.00	−0.301982	−0.150991	+	0.988535i	$0.548247\pi$
$710$	264.000	0.0139546
$711$	0	0
$712$	19728.0	1.03840
$713$	1936.00	0.101688
$714$	0	0
$715$	−77.0000	−0.00402746
$716$	−5856.00	−0.305655
$717$	0	0
$718$	−10040.0	−0.521852
$719$	23919.0	1.24065	0.620326	−	0.784344i	$-0.287000\pi$
$719$	23919.0	1.24065	0.620326	+	0.784344i	$0.287000\pi$
$720$	0	0
$721$	5782.00	0.298659
$722$	−9668.00	−0.498346
$723$	0	0
$724$	−5728.00	−0.294032
$725$	−8556.00	−0.438292
$726$	0	0
$727$	19188.0	0.978877	0.489438	−	0.872038i	$-0.337202\pi$
$727$	19188.0	0.978877	0.489438	+	0.872038i	$0.337202\pi$
$728$	1176.00	0.0598701
$729$	0	0
$730$	−290.000	−0.0147033
$731$	672.000	0.0340011
$732$	0	0
$733$	−5670.00	−0.285711	−0.142856	−	0.989744i	$-0.545628\pi$
$733$	−5670.00	−0.285711	−0.142856	+	0.989744i	$0.545628\pi$
$734$	23276.0	1.17048
$735$	0	0
$736$	14080.0	0.705157
$737$	−1529.00	−0.0764199
$738$	0	0
$739$	17276.0	0.859957	0.429978	−	0.902839i	$-0.358521\pi$
$739$	17276.0	0.859957	0.429978	+	0.902839i	$0.358521\pi$
$740$	228.000	0.0113263
$741$	0	0
$742$	−9408.00	−0.465470
$743$	−15367.0	−0.758763	−0.379381	−	0.925240i	$-0.623863\pi$
$743$	−15367.0	−0.758763	−0.379381	+	0.925240i	$0.623863\pi$
$744$	0	0
$745$	1895.00	0.0931912
$746$	−17668.0	−0.867120
$747$	0	0
$748$	−616.000	−0.0301112
$749$	1533.00	0.0747858
$750$	0	0
$751$	−38123.0	−1.85237	−0.926184	−	0.377072i	$-0.876931\pi$
$751$	−38123.0	−1.85237	−0.926184	+	0.377072i	$0.876931\pi$
$752$	−6160.00	−0.298713
$753$	0	0
$754$	966.000	0.0466574
$755$	3478.00	0.167652
$756$	0	0
$757$	23417.0	1.12431	0.562157	−	0.827031i	$-0.309972\pi$
$757$	23417.0	1.12431	0.562157	+	0.827031i	$0.309972\pi$
$758$	15014.0	0.719437
$759$	0	0
$760$	−1080.00	−0.0515470
$761$	16332.0	0.777969	0.388985	−	0.921244i	$-0.372826\pi$
$761$	16332.0	0.777969	0.388985	+	0.921244i	$0.372826\pi$
$762$	0	0
$763$	−9982.00	−0.473621
$764$	−5320.00	−0.251925
$765$	0	0
$766$	−7216.00	−0.340372
$767$	3283.00	0.154553
$768$	0	0
$769$	19523.0	0.915497	0.457749	−	0.889082i	$-0.348656\pi$
$769$	19523.0	0.915497	0.457749	+	0.889082i	$0.348656\pi$
$770$	154.000	0.00720750
$771$	0	0
$772$	10160.0	0.473661
$773$	−21525.0	−1.00155	−0.500776	−	0.865577i	$-0.666952\pi$
$773$	−21525.0	−1.00155	−0.500776	+	0.865577i	$0.666952\pi$
$774$	0	0
$775$	−2728.00	−0.126442
$776$	−6528.00	−0.301987
$777$	0	0
$778$	15792.0	0.727726
$779$	−17100.0	−0.786484
$780$	0	0
$781$	−1452.00	−0.0665258
$782$	2464.00	0.112676
$783$	0	0
$784$	−784.000	−0.0357143
$785$	952.000	0.0432845
$786$	0	0
$787$	−3707.00	−0.167904	−0.0839519	−	0.996470i	$-0.526754\pi$
$787$	−3707.00	−0.167904	−0.0839519	+	0.996470i	$0.526754\pi$
$788$	6424.00	0.290413
$789$	0	0
$790$	−2488.00	−0.112049
$791$	6174.00	0.277525
$792$	0	0
$793$	−2394.00	−0.107205
$794$	13868.0	0.619845
$795$	0	0
$796$	−1152.00	−0.0512959
$797$	7621.00	0.338707	0.169354	−	0.985555i	$-0.445832\pi$
$797$	7621.00	0.338707	0.169354	+	0.985555i	$0.445832\pi$
$798$	0	0
$799$	5390.00	0.238654
$800$	−19840.0	−0.876812
$801$	0	0
$802$	1400.00	0.0616405
$803$	1595.00	0.0700951
$804$	0	0
$805$	616.000	0.0269704
$806$	308.000	0.0134601
$807$	0	0
$808$	−20976.0	−0.913284
$809$	−23451.0	−1.01915	−0.509576	−	0.860426i	$-0.670198\pi$
$809$	−23451.0	−1.01915	−0.509576	+	0.860426i	$0.670198\pi$
$810$	0	0
$811$	−4499.00	−0.194798	−0.0973990	−	0.995245i	$-0.531052\pi$
$811$	−4499.00	−0.194798	−0.0973990	+	0.995245i	$0.531052\pi$
$812$	1932.00	0.0834974
$813$	0	0
$814$	1254.00	0.0539959
$815$	2483.00	0.106719
$816$	0	0
$817$	−2160.00	−0.0924955
$818$	−23100.0	−0.987375
$819$	0	0
$820$	1520.00	0.0647326
$821$	30215.0	1.28442	0.642211	−	0.766528i	$-0.278017\pi$
$821$	30215.0	1.28442	0.642211	+	0.766528i	$0.278017\pi$
$822$	0	0
$823$	−19947.0	−0.844847	−0.422423	−	0.906399i	$-0.638820\pi$
$823$	−19947.0	−0.844847	−0.422423	+	0.906399i	$0.638820\pi$
$824$	19824.0	0.838109
$825$	0	0
$826$	−6566.00	−0.276586
$827$	20341.0	0.855291	0.427646	−	0.903946i	$-0.359343\pi$
$827$	20341.0	0.855291	0.427646	+	0.903946i	$0.359343\pi$
$828$	0	0
$829$	−14024.0	−0.587544	−0.293772	−	0.955876i	$-0.594911\pi$
$829$	−14024.0	−0.587544	−0.293772	+	0.955876i	$0.594911\pi$
$830$	1044.00	0.0436600
$831$	0	0
$832$	3136.00	0.130675
$833$	686.000	0.0285336
$834$	0	0
$835$	468.000	0.0193962
$836$	1980.00	0.0819136
$837$	0	0
$838$	9110.00	0.375537
$839$	37193.0	1.53045	0.765223	−	0.643765i	$-0.222628\pi$
$839$	37193.0	1.53045	0.765223	+	0.643765i	$0.222628\pi$
$840$	0	0
$841$	−19628.0	−0.804789
$842$	−322.000	−0.0131792
$843$	0	0
$844$	−5928.00	−0.241766
$845$	2148.00	0.0874479
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−10752.0	−0.435407
$849$	0	0
$850$	−3472.00	−0.140104
$851$	5016.00	0.202052
$852$	0	0
$853$	−37042.0	−1.48686	−0.743431	−	0.668812i	$-0.766803\pi$
$853$	−37042.0	−1.48686	−0.743431	+	0.668812i	$0.766803\pi$
$854$	4788.00	0.191852
$855$	0	0
$856$	5256.00	0.209867
$857$	41962.0	1.67257	0.836286	−	0.548293i	$-0.184722\pi$
$857$	41962.0	1.67257	0.836286	+	0.548293i	$0.184722\pi$
$858$	0	0
$859$	−9372.00	−0.372257	−0.186128	−	0.982525i	$-0.559594\pi$
$859$	−9372.00	−0.372257	−0.186128	+	0.982525i	$0.559594\pi$
$860$	192.000	0.00761296
$861$	0	0
$862$	6498.00	0.256755
$863$	−18096.0	−0.713783	−0.356892	−	0.934146i	$-0.616163\pi$
$863$	−18096.0	−0.713783	−0.356892	+	0.934146i	$0.616163\pi$
$864$	0	0
$865$	−2148.00	−0.0844326
$866$	3584.00	0.140634
$867$	0	0
$868$	616.000	0.0240880
$869$	13684.0	0.534175
$870$	0	0
$871$	−973.000	−0.0378517
$872$	−34224.0	−1.32910
$873$	0	0
$874$	−7920.00	−0.306519
$875$	−1743.00	−0.0673419
$876$	0	0
$877$	−28848.0	−1.11075	−0.555375	−	0.831600i	$-0.687425\pi$
$877$	−28848.0	−1.11075	−0.555375	+	0.831600i	$0.687425\pi$
$878$	18638.0	0.716403
$879$	0	0
$880$	176.000	0.00674200
$881$	−26273.0	−1.00472	−0.502361	−	0.864658i	$-0.667535\pi$
$881$	−26273.0	−1.00472	−0.502361	+	0.864658i	$0.667535\pi$
$882$	0	0
$883$	−33821.0	−1.28898	−0.644489	−	0.764614i	$-0.722930\pi$
$883$	−33821.0	−1.28898	−0.644489	+	0.764614i	$0.722930\pi$
$884$	−392.000	−0.0149145
$885$	0	0
$886$	−11216.0	−0.425292
$887$	14134.0	0.535032	0.267516	−	0.963553i	$-0.413797\pi$
$887$	14134.0	0.535032	0.267516	+	0.963553i	$0.413797\pi$
$888$	0	0
$889$	−12782.0	−0.482221
$890$	1644.00	0.0619180
$891$	0	0
$892$	−14008.0	−0.525810
$893$	−17325.0	−0.649226
$894$	0	0
$895$	−1464.00	−0.0546772
$896$	2688.00	0.100223
$897$	0	0
$898$	8768.00	0.325826
$899$	1518.00	0.0563161
$900$	0	0
$901$	9408.00	0.347865
$902$	8360.00	0.308600
$903$	0	0
$904$	21168.0	0.778802
$905$	−1432.00	−0.0525981
$906$	0	0
$907$	−35748.0	−1.30870	−0.654351	−	0.756191i	$-0.727058\pi$
$907$	−35748.0	−1.30870	−0.654351	+	0.756191i	$0.727058\pi$
$908$	7320.00	0.267536
$909$	0	0
$910$	98.0000	0.00356997
$911$	9522.00	0.346299	0.173149	−	0.984896i	$-0.444606\pi$
$911$	9522.00	0.346299	0.173149	+	0.984896i	$0.444606\pi$
$912$	0	0
$913$	−5742.00	−0.208141
$914$	648.000	0.0234507
$915$	0	0
$916$	9072.00	0.327235
$917$	588.000	0.0211750
$918$	0	0
$919$	−33260.0	−1.19385	−0.596924	−	0.802298i	$-0.703611\pi$
$919$	−33260.0	−1.19385	−0.596924	+	0.802298i	$0.703611\pi$
$920$	2112.00	0.0756854
$921$	0	0
$922$	36720.0	1.31161
$923$	−924.000	−0.0329511
$924$	0	0
$925$	−7068.00	−0.251237
$926$	−3334.00	−0.118318
$927$	0	0
$928$	11040.0	0.390523
$929$	55101.0	1.94597	0.972984	−	0.230871i	$-0.0741574\pi$
$929$	55101.0	1.94597	0.972984	+	0.230871i	$0.0741574\pi$
$930$	0	0
$931$	−2205.00	−0.0776219
$932$	9720.00	0.341619
$933$	0	0
$934$	−21678.0	−0.759449
$935$	−154.000	−0.00538646
$936$	0	0
$937$	−19474.0	−0.678962	−0.339481	−	0.940613i	$-0.610252\pi$
$937$	−19474.0	−0.678962	−0.339481	+	0.940613i	$0.610252\pi$
$938$	1946.00	0.0677390
$939$	0	0
$940$	1540.00	0.0534354
$941$	−52104.0	−1.80504	−0.902520	−	0.430649i	$-0.858285\pi$
$941$	−52104.0	−1.80504	−0.902520	+	0.430649i	$0.858285\pi$
$942$	0	0
$943$	33440.0	1.15478
$944$	−7504.00	−0.258723
$945$	0	0
$946$	1056.00	0.0362934
$947$	−16830.0	−0.577510	−0.288755	−	0.957403i	$-0.593241\pi$
$947$	−16830.0	−0.577510	−0.288755	+	0.957403i	$0.593241\pi$
$948$	0	0
$949$	1015.00	0.0347190
$950$	11160.0	0.381135
$951$	0	0
$952$	2352.00	0.0800722
$953$	13997.0	0.475768	0.237884	−	0.971294i	$-0.423546\pi$
$953$	13997.0	0.475768	0.237884	+	0.971294i	$0.423546\pi$
$954$	0	0
$955$	−1330.00	−0.0450657
$956$	−8604.00	−0.291081
$957$	0	0
$958$	1820.00	0.0613795
$959$	−12838.0	−0.432284
$960$	0	0
$961$	−29307.0	−0.983753
$962$	798.000	0.0267449
$963$	0	0
$964$	−11212.0	−0.374600
$965$	2540.00	0.0847311
$966$	0	0
$967$	9574.00	0.318386	0.159193	−	0.987247i	$-0.449111\pi$
$967$	9574.00	0.318386	0.159193	+	0.987247i	$0.449111\pi$
$968$	−2904.00	−0.0964237
$969$	0	0
$970$	−544.000	−0.0180070
$971$	−32585.0	−1.07693	−0.538467	−	0.842647i	$-0.680996\pi$
$971$	−32585.0	−1.07693	−0.538467	+	0.842647i	$0.680996\pi$
$972$	0	0
$973$	−16912.0	−0.557219
$974$	24064.0	0.791643
$975$	0	0
$976$	5472.00	0.179462
$977$	−42036.0	−1.37651	−0.688255	−	0.725469i	$-0.741623\pi$
$977$	−42036.0	−1.37651	−0.688255	+	0.725469i	$0.741623\pi$
$978$	0	0
$979$	−9042.00	−0.295182
$980$	196.000	0.00638877
$981$	0	0
$982$	−8486.00	−0.275763
$983$	−42336.0	−1.37366	−0.686830	−	0.726818i	$-0.740999\pi$
$983$	−42336.0	−1.37366	−0.686830	+	0.726818i	$0.740999\pi$
$984$	0	0
$985$	1606.00	0.0519507
$986$	1932.00	0.0624010
$987$	0	0
$988$	1260.00	0.0405728
$989$	4224.00	0.135809
$990$	0	0
$991$	−4691.00	−0.150368	−0.0751839	−	0.997170i	$-0.523954\pi$
$991$	−4691.00	−0.150368	−0.0751839	+	0.997170i	$0.523954\pi$
$992$	3520.00	0.112661
$993$	0	0
$994$	1848.00	0.0589688
$995$	−288.000	−0.00917609
$996$	0	0
$997$	18518.0	0.588236	0.294118	−	0.955769i	$-0.404974\pi$
$997$	18518.0	0.588236	0.294118	+	0.955769i	$0.404974\pi$
$998$	−11738.0	−0.372305
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.e.1.1		1
3.2	odd	2		231.4.a.b.1.1	✓	1
21.20	even	2		1617.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.b.1.1	✓	1	3.2	odd	2
693.4.a.e.1.1		1	1.1	even	1	trivial
1617.4.a.c.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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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