LMFDB - Embedded newform 5550.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5550,2,Mod(1,5550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5550, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5550.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:	$44.3169731218$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{21} -2.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -2.00000 q^{42} +8.00000 q^{43} -2.00000 q^{44} -6.00000 q^{46} -7.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{51} -1.00000 q^{52} -3.00000 q^{53} +1.00000 q^{54} -2.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +7.00000 q^{59} -14.0000 q^{61} -5.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} -3.00000 q^{67} +1.00000 q^{68} -6.00000 q^{69} -7.00000 q^{71} +1.00000 q^{72} +1.00000 q^{73} +1.00000 q^{74} -4.00000 q^{76} +4.00000 q^{77} -1.00000 q^{78} -13.0000 q^{79} +1.00000 q^{81} -14.0000 q^{83} -2.00000 q^{84} +8.00000 q^{86} +2.00000 q^{87} -2.00000 q^{88} -6.00000 q^{89} +2.00000 q^{91} -6.00000 q^{92} -5.00000 q^{93} -7.00000 q^{94} +1.00000 q^{96} +18.0000 q^{97} -3.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	−2.00000	−0.426401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	1.00000	0.176777
$33$	−2.00000	−0.348155
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−4.00000	−0.648886
$39$	−1.00000	−0.160128
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−2.00000	−0.308607
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	1.00000	0.140028
$52$	−1.00000	−0.138675
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	−4.00000	−0.529813
$58$	2.00000	0.262613
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	−5.00000	−0.635001
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	1.00000	0.121268
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−7.00000	−0.830747	−0.415374	−	0.909651i	$-0.636349\pi$
$71$	−7.00000	−0.830747	−0.415374	+	0.909651i	$0.636349\pi$
$72$	1.00000	0.117851
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	4.00000	0.455842
$78$	−1.00000	−0.113228
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	8.00000	0.862662
$87$	2.00000	0.214423
$88$	−2.00000	−0.213201
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−6.00000	−0.625543
$93$	−5.00000	−0.518476
$94$	−7.00000	−0.721995
$95$	0	0
$96$	1.00000	0.102062
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	−3.00000	−0.303046
$99$	−2.00000	−0.201008
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	1.00000	0.0990148
$103$	−9.00000	−0.886796	−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000	−0.886796	−0.443398	+	0.896325i	$0.646227\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−3.00000	−0.291386
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	1.00000	0.0962250
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	−2.00000	−0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	−1.00000	−0.0924500
$118$	7.00000	0.644402
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−14.0000	−1.26750
$123$	0	0
$124$	−5.00000	−0.449013
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	0	0
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	−2.00000	−0.174078
$133$	8.00000	0.693688
$134$	−3.00000	−0.259161
$135$	0	0
$136$	1.00000	0.0857493
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	−6.00000	−0.510754
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	0	0
$141$	−7.00000	−0.589506
$142$	−7.00000	−0.587427
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	0	0
$146$	1.00000	0.0827606
$147$	−3.00000	−0.247436
$148$	1.00000	0.0821995
$149$	1.00000	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−4.00000	−0.324443
$153$	1.00000	0.0808452
$154$	4.00000	0.322329
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	16.0000	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$158$	−13.0000	−1.03422
$159$	−3.00000	−0.237915
$160$	0	0
$161$	12.0000	0.945732
$162$	1.00000	0.0785674
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	0	0
$165$	0	0
$166$	−14.0000	−1.08661
$167$	10.0000	0.773823	0.386912	−	0.922117i	$-0.373542\pi$
$167$	10.0000	0.773823	0.386912	+	0.922117i	$0.373542\pi$
$168$	−2.00000	−0.154303
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−4.00000	−0.305888
$172$	8.00000	0.609994
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	−2.00000	−0.150756
$177$	7.00000	0.526152
$178$	−6.00000	−0.449719
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	2.00000	0.148250
$183$	−14.0000	−1.03491
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−5.00000	−0.366618
$187$	−2.00000	−0.146254
$188$	−7.00000	−0.510527
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	1.00000	0.0721688
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	18.0000	1.29232
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−2.00000	−0.142134
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−3.00000	−0.211604
$202$	3.00000	0.211079
$203$	−4.00000	−0.280745
$204$	1.00000	0.0700140
$205$	0	0
$206$	−9.00000	−0.627060
$207$	−6.00000	−0.417029
$208$	−1.00000	−0.0693375
$209$	8.00000	0.553372
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	−3.00000	−0.206041
$213$	−7.00000	−0.479632
$214$	10.0000	0.683586
$215$	0	0
$216$	1.00000	0.0680414
$217$	10.0000	0.678844
$218$	5.00000	0.338643
$219$	1.00000	0.0675737
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	1.00000	0.0671156
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	−4.00000	−0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	2.00000	0.131306
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	7.00000	0.455661
$237$	−13.0000	−0.844441
$238$	−2.00000	−0.129641
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	−5.00000	−0.317500
$249$	−14.0000	−0.887214
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−2.00000	−0.125988
$253$	12.0000	0.754434
$254$	−6.00000	−0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	8.00000	0.498058
$259$	−2.00000	−0.124274
$260$	0	0
$261$	2.00000	0.123797
$262$	1.00000	0.0617802
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	8.00000	0.490511
$267$	−6.00000	−0.367194
$268$	−3.00000	−0.183254
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	1.00000	0.0606339
$273$	2.00000	0.121046
$274$	12.0000	0.724947
$275$	0	0
$276$	−6.00000	−0.361158
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	1.00000	0.0599760
$279$	−5.00000	−0.299342
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	−7.00000	−0.416844
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	−7.00000	−0.415374
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	1.00000	0.0589256
$289$	−16.0000	−0.941176
$290$	0	0
$291$	18.0000	1.05518
$292$	1.00000	0.0585206
$293$	1.00000	0.0584206	0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000	0.0584206	0.0292103	+	0.999573i	$0.490701\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	1.00000	0.0581238
$297$	−2.00000	−0.116052
$298$	1.00000	0.0579284
$299$	6.00000	0.346989
$300$	0	0
$301$	−16.0000	−0.922225
$302$	4.00000	0.230174
$303$	3.00000	0.172345
$304$	−4.00000	−0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	9.00000	0.513657	0.256829	−	0.966457i	$-0.417322\pi$
$307$	9.00000	0.513657	0.256829	+	0.966457i	$0.417322\pi$
$308$	4.00000	0.227921
$309$	−9.00000	−0.511992
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	−1.00000	−0.0566139
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	16.0000	0.902932
$315$	0	0
$316$	−13.0000	−0.731307
$317$	33.0000	1.85346	0.926732	−	0.375722i	$-0.122605\pi$
$317$	33.0000	1.85346	0.926732	+	0.375722i	$0.122605\pi$
$318$	−3.00000	−0.168232
$319$	−4.00000	−0.223957
$320$	0	0
$321$	10.0000	0.558146
$322$	12.0000	0.668734
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	−22.0000	−1.21847
$327$	5.00000	0.276501
$328$	0	0
$329$	14.0000	0.771845
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−14.0000	−0.768350
$333$	1.00000	0.0547997
$334$	10.0000	0.547176
$335$	0	0
$336$	−2.00000	−0.109109
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−12.0000	−0.652714
$339$	−6.00000	−0.325875
$340$	0	0
$341$	10.0000	0.541530
$342$	−4.00000	−0.216295
$343$	20.0000	1.07990
$344$	8.00000	0.431331
$345$	0	0
$346$	−13.0000	−0.698884
$347$	21.0000	1.12734	0.563670	−	0.826000i	$-0.309389\pi$
$347$	21.0000	1.12734	0.563670	+	0.826000i	$0.309389\pi$
$348$	2.00000	0.107211
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−2.00000	−0.106600
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	7.00000	0.372046
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−2.00000	−0.105851
$358$	3.00000	0.158555
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	18.0000	0.946059
$363$	−7.00000	−0.367405
$364$	2.00000	0.104828
$365$	0	0
$366$	−14.0000	−0.731792
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	−5.00000	−0.259238
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	−7.00000	−0.360997
$377$	−2.00000	−0.103005
$378$	−2.00000	−0.102869
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	−18.0000	−0.920960
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	6.00000	0.305392
$387$	8.00000	0.406663
$388$	18.0000	0.913812
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	−3.00000	−0.151523
$393$	1.00000	0.0504433
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−2.00000	−0.100504
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	−8.00000	−0.401004
$399$	8.00000	0.400501
$400$	0	0
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	−3.00000	−0.149626
$403$	5.00000	0.249068
$404$	3.00000	0.149256
$405$	0	0
$406$	−4.00000	−0.198517
$407$	−2.00000	−0.0991363
$408$	1.00000	0.0495074
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	−9.00000	−0.443398
$413$	−14.0000	−0.688895
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	1.00000	0.0489702
$418$	8.00000	0.391293
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−35.0000	−1.70580	−0.852898	−	0.522078i	$-0.825157\pi$
$421$	−35.0000	−1.70580	−0.852898	+	0.522078i	$0.825157\pi$
$422$	13.0000	0.632830
$423$	−7.00000	−0.340352
$424$	−3.00000	−0.145693
$425$	0	0
$426$	−7.00000	−0.339151
$427$	28.0000	1.35501
$428$	10.0000	0.483368
$429$	2.00000	0.0965609
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	1.00000	0.0481125
$433$	−9.00000	−0.432512	−0.216256	−	0.976337i	$-0.569385\pi$
$433$	−9.00000	−0.432512	−0.216256	+	0.976337i	$0.569385\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	5.00000	0.239457
$437$	24.0000	1.14808
$438$	1.00000	0.0477818
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−1.00000	−0.0475651
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	1.00000	0.0474579
$445$	0	0
$446$	2.00000	0.0947027
$447$	1.00000	0.0472984
$448$	−2.00000	−0.0944911
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	4.00000	0.187936
$454$	3.00000	0.140797
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	14.0000	0.654177
$459$	1.00000	0.0466760
$460$	0	0
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	4.00000	0.186097
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−22.0000	−1.01913
$467$	−5.00000	−0.231372	−0.115686	−	0.993286i	$-0.536907\pi$
$467$	−5.00000	−0.231372	−0.115686	+	0.993286i	$0.536907\pi$
$468$	−1.00000	−0.0462250
$469$	6.00000	0.277054
$470$	0	0
$471$	16.0000	0.737241
$472$	7.00000	0.322201
$473$	−16.0000	−0.735681
$474$	−13.0000	−0.597110
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	−3.00000	−0.137361
$478$	−20.0000	−0.914779
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	4.00000	0.182195
$483$	12.0000	0.546019
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	33.0000	1.49537	0.747686	−	0.664052i	$-0.231165\pi$
$487$	33.0000	1.49537	0.747686	+	0.664052i	$0.231165\pi$
$488$	−14.0000	−0.633750
$489$	−22.0000	−0.994874
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	4.00000	0.179969
$495$	0	0
$496$	−5.00000	−0.224507
$497$	14.0000	0.627986
$498$	−14.0000	−0.627355
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	10.0000	0.446767
$502$	12.0000	0.535586
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	12.0000	0.533465
$507$	−12.0000	−0.532939
$508$	−6.00000	−0.266207
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	21.0000	0.926270
$515$	0	0
$516$	8.00000	0.352180
$517$	14.0000	0.615719
$518$	−2.00000	−0.0878750
$519$	−13.0000	−0.570637
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	2.00000	0.0875376
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	1.00000	0.0436852
$525$	0	0
$526$	8.00000	0.348817
$527$	−5.00000	−0.217803
$528$	−2.00000	−0.0870388
$529$	13.0000	0.565217
$530$	0	0
$531$	7.00000	0.303774
$532$	8.00000	0.346844
$533$	0	0
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−3.00000	−0.129580
$537$	3.00000	0.129460
$538$	−9.00000	−0.388018
$539$	6.00000	0.258438
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−14.0000	−0.601351
$543$	18.0000	0.772454
$544$	1.00000	0.0428746
$545$	0	0
$546$	2.00000	0.0855921
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	12.0000	0.512615
$549$	−14.0000	−0.597505
$550$	0	0
$551$	−8.00000	−0.340811
$552$	−6.00000	−0.255377
$553$	26.0000	1.10563
$554$	−11.0000	−0.467345
$555$	0	0
$556$	1.00000	0.0424094
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	−5.00000	−0.211667
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	−15.0000	−0.632737
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	−7.00000	−0.294753
$565$	0	0
$566$	2.00000	0.0840663
$567$	−2.00000	−0.0839921
$568$	−7.00000	−0.293713
$569$	−5.00000	−0.209611	−0.104805	−	0.994493i	$-0.533422\pi$
$569$	−5.00000	−0.209611	−0.104805	+	0.994493i	$0.533422\pi$
$570$	0	0
$571$	29.0000	1.21361	0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000	1.21361	0.606806	+	0.794850i	$0.292450\pi$
$572$	2.00000	0.0836242
$573$	−18.0000	−0.751961
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	−16.0000	−0.665512
$579$	6.00000	0.249351
$580$	0	0
$581$	28.0000	1.16164
$582$	18.0000	0.746124
$583$	6.00000	0.248495
$584$	1.00000	0.0413803
$585$	0	0
$586$	1.00000	0.0413096
$587$	−44.0000	−1.81607	−0.908037	−	0.418890i	$-0.862419\pi$
$587$	−44.0000	−1.81607	−0.908037	+	0.418890i	$0.862419\pi$
$588$	−3.00000	−0.123718
$589$	20.0000	0.824086
$590$	0	0
$591$	−6.00000	−0.246807
$592$	1.00000	0.0410997
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	1.00000	0.0409616
$597$	−8.00000	−0.327418
$598$	6.00000	0.245358
$599$	−1.00000	−0.0408589	−0.0204294	−	0.999791i	$-0.506503\pi$
$599$	−1.00000	−0.0408589	−0.0204294	+	0.999791i	$0.506503\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	−16.0000	−0.652111
$603$	−3.00000	−0.122169
$604$	4.00000	0.162758
$605$	0	0
$606$	3.00000	0.121867
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	−4.00000	−0.162221
$609$	−4.00000	−0.162088
$610$	0	0
$611$	7.00000	0.283190
$612$	1.00000	0.0404226
$613$	36.0000	1.45403	0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000	1.45403	0.727013	+	0.686624i	$0.240908\pi$
$614$	9.00000	0.363210
$615$	0	0
$616$	4.00000	0.161165
$617$	28.0000	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$618$	−9.00000	−0.362033
$619$	21.0000	0.844061	0.422031	−	0.906582i	$-0.361317\pi$
$619$	21.0000	0.844061	0.422031	+	0.906582i	$0.361317\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	−4.00000	−0.160385
$623$	12.0000	0.480770
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	14.0000	0.559553
$627$	8.00000	0.319489
$628$	16.0000	0.638470
$629$	1.00000	0.0398726
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−13.0000	−0.517112
$633$	13.0000	0.516704
$634$	33.0000	1.31060
$635$	0	0
$636$	−3.00000	−0.118958
$637$	3.00000	0.118864
$638$	−4.00000	−0.158362
$639$	−7.00000	−0.276916
$640$	0	0
$641$	−20.0000	−0.789953	−0.394976	−	0.918691i	$-0.629247\pi$
$641$	−20.0000	−0.789953	−0.394976	+	0.918691i	$0.629247\pi$
$642$	10.0000	0.394669
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−4.00000	−0.157378
$647$	20.0000	0.786281	0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000	0.786281	0.393141	+	0.919478i	$0.371389\pi$
$648$	1.00000	0.0392837
$649$	−14.0000	−0.549548
$650$	0	0
$651$	10.0000	0.391931
$652$	−22.0000	−0.861586
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	5.00000	0.195515
$655$	0	0
$656$	0	0
$657$	1.00000	0.0390137
$658$	14.0000	0.545777
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	8.00000	0.310929
$663$	−1.00000	−0.0388368
$664$	−14.0000	−0.543305
$665$	0	0
$666$	1.00000	0.0387492
$667$	−12.0000	−0.464642
$668$	10.0000	0.386912
$669$	2.00000	0.0773245
$670$	0	0
$671$	28.0000	1.08093
$672$	−2.00000	−0.0771517
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−12.0000	−0.461538
$677$	−17.0000	−0.653363	−0.326682	−	0.945134i	$-0.605930\pi$
$677$	−17.0000	−0.653363	−0.326682	+	0.945134i	$0.605930\pi$
$678$	−6.00000	−0.230429
$679$	−36.0000	−1.38155
$680$	0	0
$681$	3.00000	0.114960
$682$	10.0000	0.382920
$683$	−51.0000	−1.95146	−0.975730	−	0.218975i	$-0.929729\pi$
$683$	−51.0000	−1.95146	−0.975730	+	0.218975i	$0.929729\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	20.0000	0.763604
$687$	14.0000	0.534133
$688$	8.00000	0.304997
$689$	3.00000	0.114291
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−13.0000	−0.494186
$693$	4.00000	0.151947
$694$	21.0000	0.797149
$695$	0	0
$696$	2.00000	0.0758098
$697$	0	0
$698$	2.00000	0.0757011
$699$	−22.0000	−0.832116
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−1.00000	−0.0377426
$703$	−4.00000	−0.150863
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	9.00000	0.338719
$707$	−6.00000	−0.225653
$708$	7.00000	0.263076
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	−13.0000	−0.487538
$712$	−6.00000	−0.224860
$713$	30.0000	1.12351
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	3.00000	0.112115
$717$	−20.0000	−0.746914
$718$	−15.0000	−0.559795
$719$	1.00000	0.0372937	0.0186469	−	0.999826i	$-0.494064\pi$
$719$	1.00000	0.0372937	0.0186469	+	0.999826i	$0.494064\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	−3.00000	−0.111648
$723$	4.00000	0.148762
$724$	18.0000	0.668965
$725$	0	0
$726$	−7.00000	−0.259794
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	−14.0000	−0.517455
$733$	−16.0000	−0.590973	−0.295487	−	0.955347i	$-0.595482\pi$
$733$	−16.0000	−0.590973	−0.295487	+	0.955347i	$0.595482\pi$
$734$	−2.00000	−0.0738213
$735$	0	0
$736$	−6.00000	−0.221163
$737$	6.00000	0.221013
$738$	0	0
$739$	−23.0000	−0.846069	−0.423034	−	0.906114i	$-0.639035\pi$
$739$	−23.0000	−0.846069	−0.423034	+	0.906114i	$0.639035\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	6.00000	0.220267
$743$	−9.00000	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$744$	−5.00000	−0.183309
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	−14.0000	−0.512233
$748$	−2.00000	−0.0731272
$749$	−20.0000	−0.730784
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	−7.00000	−0.255264
$753$	12.0000	0.437304
$754$	−2.00000	−0.0728357
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	45.0000	1.63555	0.817776	−	0.575536i	$-0.195207\pi$
$757$	45.0000	1.63555	0.817776	+	0.575536i	$0.195207\pi$
$758$	4.00000	0.145287
$759$	12.0000	0.435572
$760$	0	0
$761$	40.0000	1.45000	0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000	1.45000	0.724999	+	0.688749i	$0.241840\pi$
$762$	−6.00000	−0.217357
$763$	−10.0000	−0.362024
$764$	−18.0000	−0.651217
$765$	0	0
$766$	−4.00000	−0.144526
$767$	−7.00000	−0.252755
$768$	1.00000	0.0360844
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	21.0000	0.756297
$772$	6.00000	0.215945
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	18.0000	0.646162
$777$	−2.00000	−0.0717496
$778$	18.0000	0.645331
$779$	0	0
$780$	0	0
$781$	14.0000	0.500959
$782$	−6.00000	−0.214560
$783$	2.00000	0.0714742
$784$	−3.00000	−0.107143
$785$	0	0
$786$	1.00000	0.0356688
$787$	−11.0000	−0.392108	−0.196054	−	0.980593i	$-0.562813\pi$
$787$	−11.0000	−0.392108	−0.196054	+	0.980593i	$0.562813\pi$
$788$	−6.00000	−0.213741
$789$	8.00000	0.284808
$790$	0	0
$791$	12.0000	0.426671
$792$	−2.00000	−0.0710669
$793$	14.0000	0.497155
$794$	12.0000	0.425864
$795$	0	0
$796$	−8.00000	−0.283552
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	8.00000	0.283197
$799$	−7.00000	−0.247642
$800$	0	0
$801$	−6.00000	−0.212000
$802$	−15.0000	−0.529668
$803$	−2.00000	−0.0705785
$804$	−3.00000	−0.105802
$805$	0	0
$806$	5.00000	0.176117
$807$	−9.00000	−0.316815
$808$	3.00000	0.105540
$809$	34.0000	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$810$	0	0
$811$	−13.0000	−0.456492	−0.228246	−	0.973604i	$-0.573299\pi$
$811$	−13.0000	−0.456492	−0.228246	+	0.973604i	$0.573299\pi$
$812$	−4.00000	−0.140372
$813$	−14.0000	−0.491001
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	1.00000	0.0350070
$817$	−32.0000	−1.11954
$818$	38.0000	1.32864
$819$	2.00000	0.0698857
$820$	0	0
$821$	13.0000	0.453703	0.226852	−	0.973929i	$-0.427157\pi$
$821$	13.0000	0.453703	0.226852	+	0.973929i	$0.427157\pi$
$822$	12.0000	0.418548
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	−9.00000	−0.313530
$825$	0	0
$826$	−14.0000	−0.487122
$827$	5.00000	0.173867	0.0869335	−	0.996214i	$-0.472293\pi$
$827$	5.00000	0.173867	0.0869335	+	0.996214i	$0.472293\pi$
$828$	−6.00000	−0.208514
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	0	0
$831$	−11.0000	−0.381586
$832$	−1.00000	−0.0346688
$833$	−3.00000	−0.103944
$834$	1.00000	0.0346272
$835$	0	0
$836$	8.00000	0.276686
$837$	−5.00000	−0.172825
$838$	−30.0000	−1.03633
$839$	−9.00000	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$839$	−9.00000	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−35.0000	−1.20618
$843$	−15.0000	−0.516627
$844$	13.0000	0.447478
$845$	0	0
$846$	−7.00000	−0.240665
$847$	14.0000	0.481046
$848$	−3.00000	−0.103020
$849$	2.00000	0.0686398
$850$	0	0
$851$	−6.00000	−0.205677
$852$	−7.00000	−0.239816
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	28.0000	0.958140
$855$	0	0
$856$	10.0000	0.341793
$857$	−17.0000	−0.580709	−0.290354	−	0.956919i	$-0.593773\pi$
$857$	−17.0000	−0.580709	−0.290354	+	0.956919i	$0.593773\pi$
$858$	2.00000	0.0682789
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	0	0
$862$	20.0000	0.681203
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−9.00000	−0.305832
$867$	−16.0000	−0.543388
$868$	10.0000	0.339422
$869$	26.0000	0.881990
$870$	0	0
$871$	3.00000	0.101651
$872$	5.00000	0.169321
$873$	18.0000	0.609208
$874$	24.0000	0.811812
$875$	0	0
$876$	1.00000	0.0337869
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−8.00000	−0.269987
$879$	1.00000	0.0337292
$880$	0	0
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	−3.00000	−0.101015
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	−1.00000	−0.0336336
$885$	0	0
$886$	0	0
$887$	−19.0000	−0.637958	−0.318979	−	0.947762i	$-0.603340\pi$
$887$	−19.0000	−0.637958	−0.318979	+	0.947762i	$0.603340\pi$
$888$	1.00000	0.0335578
$889$	12.0000	0.402467
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	2.00000	0.0669650
$893$	28.0000	0.936984
$894$	1.00000	0.0334450
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	6.00000	0.200334
$898$	−9.00000	−0.300334
$899$	−10.0000	−0.333519
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	0	0
$903$	−16.0000	−0.532447
$904$	−6.00000	−0.199557
$905$	0	0
$906$	4.00000	0.132891
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	3.00000	0.0995585
$909$	3.00000	0.0995037
$910$	0	0
$911$	−10.0000	−0.331315	−0.165657	−	0.986183i	$-0.552975\pi$
$911$	−10.0000	−0.331315	−0.165657	+	0.986183i	$0.552975\pi$
$912$	−4.00000	−0.132453
$913$	28.0000	0.926665
$914$	−14.0000	−0.463079
$915$	0	0
$916$	14.0000	0.462573
$917$	−2.00000	−0.0660458
$918$	1.00000	0.0330049
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	0	0
$921$	9.00000	0.296560
$922$	−28.0000	−0.922131
$923$	7.00000	0.230408
$924$	4.00000	0.131590
$925$	0	0
$926$	36.0000	1.18303
$927$	−9.00000	−0.295599
$928$	2.00000	0.0656532
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	−22.0000	−0.720634
$933$	−4.00000	−0.130954
$934$	−5.00000	−0.163605
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	−55.0000	−1.79677	−0.898386	−	0.439207i	$-0.855259\pi$
$937$	−55.0000	−1.79677	−0.898386	+	0.439207i	$0.855259\pi$
$938$	6.00000	0.195907
$939$	14.0000	0.456873
$940$	0	0
$941$	−13.0000	−0.423788	−0.211894	−	0.977293i	$-0.567963\pi$
$941$	−13.0000	−0.423788	−0.211894	+	0.977293i	$0.567963\pi$
$942$	16.0000	0.521308
$943$	0	0
$944$	7.00000	0.227831
$945$	0	0
$946$	−16.0000	−0.520205
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	−13.0000	−0.422220
$949$	−1.00000	−0.0324614
$950$	0	0
$951$	33.0000	1.07010
$952$	−2.00000	−0.0648204
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	−3.00000	−0.0971286
$955$	0	0
$956$	−20.0000	−0.646846
$957$	−4.00000	−0.129302
$958$	16.0000	0.516937
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−1.00000	−0.0322413
$963$	10.0000	0.322245
$964$	4.00000	0.128831
$965$	0	0
$966$	12.0000	0.386094
$967$	−61.0000	−1.96163	−0.980814	−	0.194946i	$-0.937547\pi$
$967$	−61.0000	−1.96163	−0.980814	+	0.194946i	$0.937547\pi$
$968$	−7.00000	−0.224989
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	1.00000	0.0320750
$973$	−2.00000	−0.0641171
$974$	33.0000	1.05739
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−7.00000	−0.223950	−0.111975	−	0.993711i	$-0.535718\pi$
$977$	−7.00000	−0.223950	−0.111975	+	0.993711i	$0.535718\pi$
$978$	−22.0000	−0.703482
$979$	12.0000	0.383522
$980$	0	0
$981$	5.00000	0.159638
$982$	−28.0000	−0.893516
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	0	0
$986$	2.00000	0.0636930
$987$	14.0000	0.445625
$988$	4.00000	0.127257
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−11.0000	−0.349427	−0.174713	−	0.984619i	$-0.555900\pi$
$991$	−11.0000	−0.349427	−0.174713	+	0.984619i	$0.555900\pi$
$992$	−5.00000	−0.158750
$993$	8.00000	0.253872
$994$	14.0000	0.444053
$995$	0	0
$996$	−14.0000	−0.443607
$997$	−3.00000	−0.0950110	−0.0475055	−	0.998871i	$-0.515127\pi$
$997$	−3.00000	−0.0950110	−0.0475055	+	0.998871i	$0.515127\pi$
$998$	28.0000	0.886325
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5550.2.a.bj.1.1	yes	1
5.4	even	2		5550.2.a.i.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5550.2.a.i.1.1	✓	1	5.4	even	2
5550.2.a.bj.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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