LMFDB - Embedded newform 5550.2.a.t.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:	no (minimal twist has level 1110)
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)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +3.00000 q^{21} +5.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} +3.00000 q^{28} -1.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} -5.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} +6.00000 q^{38} +2.00000 q^{39} -7.00000 q^{41} -3.00000 q^{42} -3.00000 q^{43} -5.00000 q^{44} -4.00000 q^{46} +1.00000 q^{48} +2.00000 q^{49} -3.00000 q^{51} +2.00000 q^{52} -5.00000 q^{53} -1.00000 q^{54} -3.00000 q^{56} -6.00000 q^{57} +1.00000 q^{58} +6.00000 q^{59} +5.00000 q^{61} +3.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +5.00000 q^{66} +4.00000 q^{67} -3.00000 q^{68} +4.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} -1.00000 q^{74} -6.00000 q^{76} -15.0000 q^{77} -2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} +7.00000 q^{82} -6.00000 q^{83} +3.00000 q^{84} +3.00000 q^{86} -1.00000 q^{87} +5.00000 q^{88} -18.0000 q^{89} +6.00000 q^{91} +4.00000 q^{92} -3.00000 q^{93} -1.00000 q^{96} +13.0000 q^{97} -2.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−1.00000	−0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	5.00000	1.06600
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	3.00000	0.566947
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−1.00000	−0.176777
$33$	−5.00000	−0.870388
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	6.00000	0.973329
$39$	2.00000	0.320256
$40$	0	0
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	−3.00000	−0.462910
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	−4.00000	−0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	−3.00000	−0.420084
$52$	2.00000	0.277350
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−3.00000	−0.400892
$57$	−6.00000	−0.794719
$58$	1.00000	0.131306
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	3.00000	0.381000
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	5.00000	0.615457
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−3.00000	−0.363803
$69$	4.00000	0.481543
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−6.00000	−0.688247
$77$	−15.0000	−1.70941
$78$	−2.00000	−0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.00000	0.773021
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	3.00000	0.323498
$87$	−1.00000	−0.107211
$88$	5.00000	0.533002
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	4.00000	0.417029
$93$	−3.00000	−0.311086
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	−2.00000	−0.202031
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	3.00000	0.297044
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	5.00000	0.485643
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	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
$112$	3.00000	0.283473
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	6.00000	0.561951
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	2.00000	0.184900
$118$	−6.00000	−0.552345
$119$	−9.00000	−0.825029
$120$	0	0
$121$	14.0000	1.27273
$122$	−5.00000	−0.452679
$123$	−7.00000	−0.631169
$124$	−3.00000	−0.269408
$125$	0	0
$126$	−3.00000	−0.267261
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	−3.00000	−0.264135
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	−5.00000	−0.435194
$133$	−18.0000	−1.56080
$134$	−4.00000	−0.345547
$135$	0	0
$136$	3.00000	0.257248
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	−4.00000	−0.340503
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	−10.0000	−0.836242
$144$	1.00000	0.0833333
$145$	0	0
$146$	0	0
$147$	2.00000	0.164957
$148$	1.00000	0.0821995
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	6.00000	0.486664
$153$	−3.00000	−0.242536
$154$	15.0000	1.20873
$155$	0	0
$156$	2.00000	0.160128
$157$	−11.0000	−0.877896	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000	−0.877896	−0.438948	+	0.898513i	$0.644649\pi$
$158$	4.00000	0.318223
$159$	−5.00000	−0.396526
$160$	0	0
$161$	12.0000	0.945732
$162$	−1.00000	−0.0785674
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	−7.00000	−0.546608
$165$	0	0
$166$	6.00000	0.465690
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	−3.00000	−0.231455
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	−3.00000	−0.228748
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	−5.00000	−0.376889
$177$	6.00000	0.450988
$178$	18.0000	1.34916
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	−6.00000	−0.444750
$183$	5.00000	0.369611
$184$	−4.00000	−0.294884
$185$	0	0
$186$	3.00000	0.219971
$187$	15.0000	1.09691
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	1.00000	0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−13.0000	−0.933346
$195$	0	0
$196$	2.00000	0.142857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	5.00000	0.355335
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	12.0000	0.844317
$203$	−3.00000	−0.210559
$204$	−3.00000	−0.210042
$205$	0	0
$206$	16.0000	1.11477
$207$	4.00000	0.278019
$208$	2.00000	0.138675
$209$	30.0000	2.07514
$210$	0	0
$211$	19.0000	1.30801	0.654007	−	0.756489i	$-0.273087\pi$
$211$	19.0000	1.30801	0.654007	+	0.756489i	$0.273087\pi$
$212$	−5.00000	−0.343401
$213$	−12.0000	−0.822226
$214$	2.00000	0.136717
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−9.00000	−0.610960
$218$	−5.00000	−0.338643
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	−1.00000	−0.0671156
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	9.00000	0.598671
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	−6.00000	−0.397360
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	1.00000	0.0656532
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	−4.00000	−0.259828
$238$	9.00000	0.583383
$239$	29.0000	1.87585	0.937927	−	0.346833i	$-0.112743\pi$
$239$	29.0000	1.87585	0.937927	+	0.346833i	$0.112743\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−14.0000	−0.899954
$243$	1.00000	0.0641500
$244$	5.00000	0.320092
$245$	0	0
$246$	7.00000	0.446304
$247$	−12.0000	−0.763542
$248$	3.00000	0.190500
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	3.00000	0.188982
$253$	−20.0000	−1.25739
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	3.00000	0.186772
$259$	3.00000	0.186411
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−10.0000	−0.617802
$263$	−5.00000	−0.308313	−0.154157	−	0.988046i	$-0.549266\pi$
$263$	−5.00000	−0.308313	−0.154157	+	0.988046i	$0.549266\pi$
$264$	5.00000	0.307729
$265$	0	0
$266$	18.0000	1.10365
$267$	−18.0000	−1.10158
$268$	4.00000	0.244339
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−3.00000	−0.181902
$273$	6.00000	0.363137
$274$	−10.0000	−0.604122
$275$	0	0
$276$	4.00000	0.240772
$277$	16.0000	0.961347	0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000	0.961347	0.480673	+	0.876900i	$0.340392\pi$
$278$	−5.00000	−0.299880
$279$	−3.00000	−0.179605
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	10.0000	0.591312
$287$	−21.0000	−1.23959
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	13.0000	0.762073
$292$	0	0
$293$	−7.00000	−0.408944	−0.204472	−	0.978872i	$-0.565548\pi$
$293$	−7.00000	−0.408944	−0.204472	+	0.978872i	$0.565548\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	−5.00000	−0.290129
$298$	18.0000	1.04271
$299$	8.00000	0.462652
$300$	0	0
$301$	−9.00000	−0.518751
$302$	−10.0000	−0.575435
$303$	−12.0000	−0.689382
$304$	−6.00000	−0.344124
$305$	0	0
$306$	3.00000	0.171499
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	−15.0000	−0.854704
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−31.0000	−1.75785	−0.878924	−	0.476961i	$-0.841738\pi$
$311$	−31.0000	−1.75785	−0.878924	+	0.476961i	$0.841738\pi$
$312$	−2.00000	−0.113228
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	11.0000	0.620766
$315$	0	0
$316$	−4.00000	−0.225018
$317$	27.0000	1.51647	0.758236	−	0.651981i	$-0.226062\pi$
$317$	27.0000	1.51647	0.758236	+	0.651981i	$0.226062\pi$
$318$	5.00000	0.280386
$319$	5.00000	0.279946
$320$	0	0
$321$	−2.00000	−0.111629
$322$	−12.0000	−0.668734
$323$	18.0000	1.00155
$324$	1.00000	0.0555556
$325$	0	0
$326$	11.0000	0.609234
$327$	5.00000	0.276501
$328$	7.00000	0.386510
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−6.00000	−0.329293
$333$	1.00000	0.0547997
$334$	−24.0000	−1.31322
$335$	0	0
$336$	3.00000	0.163663
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	9.00000	0.489535
$339$	−9.00000	−0.488813
$340$	0	0
$341$	15.0000	0.812296
$342$	6.00000	0.324443
$343$	−15.0000	−0.809924
$344$	3.00000	0.161749
$345$	0	0
$346$	13.0000	0.698884
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−1.00000	−0.0536056
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	5.00000	0.266501
$353$	37.0000	1.96931	0.984656	−	0.174509i	$-0.0558337\pi$
$353$	37.0000	1.96931	0.984656	+	0.174509i	$0.0558337\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	−18.0000	−0.953998
$357$	−9.00000	−0.476331
$358$	−12.0000	−0.634220
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−10.0000	−0.525588
$363$	14.0000	0.734809
$364$	6.00000	0.314485
$365$	0	0
$366$	−5.00000	−0.261354
$367$	3.00000	0.156599	0.0782994	−	0.996930i	$-0.475051\pi$
$367$	3.00000	0.156599	0.0782994	+	0.996930i	$0.475051\pi$
$368$	4.00000	0.208514
$369$	−7.00000	−0.364405
$370$	0	0
$371$	−15.0000	−0.778761
$372$	−3.00000	−0.155543
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	−15.0000	−0.775632
$375$	0	0
$376$	0	0
$377$	−2.00000	−0.103005
$378$	−3.00000	−0.154303
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	3.00000	0.153493
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	18.0000	0.916176
$387$	−3.00000	−0.152499
$388$	13.0000	0.659975
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	−2.00000	−0.101015
$393$	10.0000	0.504433
$394$	−2.00000	−0.100759
$395$	0	0
$396$	−5.00000	−0.251259
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	4.00000	0.200502
$399$	−18.0000	−0.901127
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	−4.00000	−0.199502
$403$	−6.00000	−0.298881
$404$	−12.0000	−0.597022
$405$	0	0
$406$	3.00000	0.148888
$407$	−5.00000	−0.247841
$408$	3.00000	0.148522
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	−16.0000	−0.788263
$413$	18.0000	0.885722
$414$	−4.00000	−0.196589
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	5.00000	0.244851
$418$	−30.0000	−1.46735
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−19.0000	−0.924906
$423$	0	0
$424$	5.00000	0.242821
$425$	0	0
$426$	12.0000	0.581402
$427$	15.0000	0.725901
$428$	−2.00000	−0.0966736
$429$	−10.0000	−0.482805
$430$	0	0
$431$	−7.00000	−0.337178	−0.168589	−	0.985686i	$-0.553921\pi$
$431$	−7.00000	−0.337178	−0.168589	+	0.985686i	$0.553921\pi$
$432$	1.00000	0.0481125
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	9.00000	0.432014
$435$	0	0
$436$	5.00000	0.239457
$437$	−24.0000	−1.14808
$438$	0	0
$439$	−29.0000	−1.38409	−0.692047	−	0.721852i	$-0.743291\pi$
$439$	−29.0000	−1.38409	−0.692047	+	0.721852i	$0.743291\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	6.00000	0.285391
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	1.00000	0.0474579
$445$	0	0
$446$	7.00000	0.331460
$447$	−18.0000	−0.851371
$448$	3.00000	0.141737
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	35.0000	1.64809
$452$	−9.00000	−0.423324
$453$	10.0000	0.469841
$454$	−21.0000	−0.985579
$455$	0	0
$456$	6.00000	0.280976
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	−12.0000	−0.560723
$459$	−3.00000	−0.140028
$460$	0	0
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	15.0000	0.697863
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	0	0
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	2.00000	0.0924500
$469$	12.0000	0.554109
$470$	0	0
$471$	−11.0000	−0.506853
$472$	−6.00000	−0.276172
$473$	15.0000	0.689701
$474$	4.00000	0.183726
$475$	0	0
$476$	−9.00000	−0.412514
$477$	−5.00000	−0.228934
$478$	−29.0000	−1.32643
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	10.0000	0.455488
$483$	12.0000	0.546019
$484$	14.0000	0.636364
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	6.00000	0.271886	0.135943	−	0.990717i	$-0.456594\pi$
$487$	6.00000	0.271886	0.135943	+	0.990717i	$0.456594\pi$
$488$	−5.00000	−0.226339
$489$	−11.0000	−0.497437
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	−7.00000	−0.315584
$493$	3.00000	0.135113
$494$	12.0000	0.539906
$495$	0	0
$496$	−3.00000	−0.134704
$497$	−36.0000	−1.61482
$498$	6.00000	0.268866
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	24.0000	1.07117
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	20.0000	0.889108
$507$	−9.00000	−0.399704
$508$	−8.00000	−0.354943
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−6.00000	−0.264906
$514$	26.0000	1.14681
$515$	0	0
$516$	−3.00000	−0.132068
$517$	0	0
$518$	−3.00000	−0.131812
$519$	−13.0000	−0.570637
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	1.00000	0.0437688
$523$	40.0000	1.74908	0.874539	−	0.484955i	$-0.161164\pi$
$523$	40.0000	1.74908	0.874539	+	0.484955i	$0.161164\pi$
$524$	10.0000	0.436852
$525$	0	0
$526$	5.00000	0.218010
$527$	9.00000	0.392046
$528$	−5.00000	−0.217597
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.00000	0.260378
$532$	−18.0000	−0.780399
$533$	−14.0000	−0.606407
$534$	18.0000	0.778936
$535$	0	0
$536$	−4.00000	−0.172774
$537$	12.0000	0.517838
$538$	16.0000	0.689809
$539$	−10.0000	−0.430730
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	−8.00000	−0.343629
$543$	10.0000	0.429141
$544$	3.00000	0.128624
$545$	0	0
$546$	−6.00000	−0.256776
$547$	−19.0000	−0.812381	−0.406191	−	0.913788i	$-0.633143\pi$
$547$	−19.0000	−0.812381	−0.406191	+	0.913788i	$0.633143\pi$
$548$	10.0000	0.427179
$549$	5.00000	0.213395
$550$	0	0
$551$	6.00000	0.255609
$552$	−4.00000	−0.170251
$553$	−12.0000	−0.510292
$554$	−16.0000	−0.679775
$555$	0	0
$556$	5.00000	0.212047
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	3.00000	0.127000
$559$	−6.00000	−0.253773
$560$	0	0
$561$	15.0000	0.633300
$562$	−2.00000	−0.0843649
$563$	17.0000	0.716465	0.358232	−	0.933632i	$-0.383380\pi$
$563$	17.0000	0.716465	0.358232	+	0.933632i	$0.383380\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	3.00000	0.125988
$568$	12.0000	0.503509
$569$	46.0000	1.92842	0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000	1.92842	0.964210	+	0.265139i	$0.0854179\pi$
$570$	0	0
$571$	−21.0000	−0.878823	−0.439411	−	0.898286i	$-0.644813\pi$
$571$	−21.0000	−0.878823	−0.439411	+	0.898286i	$0.644813\pi$
$572$	−10.0000	−0.418121
$573$	−3.00000	−0.125327
$574$	21.0000	0.876523
$575$	0	0
$576$	1.00000	0.0416667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	8.00000	0.332756
$579$	−18.0000	−0.748054
$580$	0	0
$581$	−18.0000	−0.746766
$582$	−13.0000	−0.538867
$583$	25.0000	1.03539
$584$	0	0
$585$	0	0
$586$	7.00000	0.289167
$587$	−1.00000	−0.0412744	−0.0206372	−	0.999787i	$-0.506569\pi$
$587$	−1.00000	−0.0412744	−0.0206372	+	0.999787i	$0.506569\pi$
$588$	2.00000	0.0824786
$589$	18.0000	0.741677
$590$	0	0
$591$	2.00000	0.0822690
$592$	1.00000	0.0410997
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	5.00000	0.205152
$595$	0	0
$596$	−18.0000	−0.737309
$597$	−4.00000	−0.163709
$598$	−8.00000	−0.327144
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	−31.0000	−1.26452	−0.632258	−	0.774758i	$-0.717872\pi$
$601$	−31.0000	−1.26452	−0.632258	+	0.774758i	$0.717872\pi$
$602$	9.00000	0.366813
$603$	4.00000	0.162893
$604$	10.0000	0.406894
$605$	0	0
$606$	12.0000	0.487467
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	6.00000	0.243332
$609$	−3.00000	−0.121566
$610$	0	0
$611$	0	0
$612$	−3.00000	−0.121268
$613$	43.0000	1.73675	0.868377	−	0.495905i	$-0.165164\pi$
$613$	43.0000	1.73675	0.868377	+	0.495905i	$0.165164\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	15.0000	0.604367
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	16.0000	0.643614
$619$	−27.0000	−1.08522	−0.542611	−	0.839984i	$-0.682564\pi$
$619$	−27.0000	−1.08522	−0.542611	+	0.839984i	$0.682564\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	31.0000	1.24299
$623$	−54.0000	−2.16346
$624$	2.00000	0.0800641
$625$	0	0
$626$	−30.0000	−1.19904
$627$	30.0000	1.19808
$628$	−11.0000	−0.438948
$629$	−3.00000	−0.119618
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	4.00000	0.159111
$633$	19.0000	0.755182
$634$	−27.0000	−1.07231
$635$	0	0
$636$	−5.00000	−0.198263
$637$	4.00000	0.158486
$638$	−5.00000	−0.197952
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−23.0000	−0.908445	−0.454223	−	0.890888i	$-0.650083\pi$
$641$	−23.0000	−0.908445	−0.454223	+	0.890888i	$0.650083\pi$
$642$	2.00000	0.0789337
$643$	−31.0000	−1.22252	−0.611260	−	0.791430i	$-0.709337\pi$
$643$	−31.0000	−1.22252	−0.611260	+	0.791430i	$0.709337\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−18.0000	−0.708201
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	−1.00000	−0.0392837
$649$	−30.0000	−1.17760
$650$	0	0
$651$	−9.00000	−0.352738
$652$	−11.0000	−0.430793
$653$	−16.0000	−0.626128	−0.313064	−	0.949732i	$-0.601356\pi$
$653$	−16.0000	−0.626128	−0.313064	+	0.949732i	$0.601356\pi$
$654$	−5.00000	−0.195515
$655$	0	0
$656$	−7.00000	−0.273304
$657$	0	0
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−29.0000	−1.12797	−0.563985	−	0.825785i	$-0.690732\pi$
$661$	−29.0000	−1.12797	−0.563985	+	0.825785i	$0.690732\pi$
$662$	20.0000	0.777322
$663$	−6.00000	−0.233021
$664$	6.00000	0.232845
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	−4.00000	−0.154881
$668$	24.0000	0.928588
$669$	−7.00000	−0.270636
$670$	0	0
$671$	−25.0000	−0.965114
$672$	−3.00000	−0.115728
$673$	−48.0000	−1.85026	−0.925132	−	0.379646i	$-0.876046\pi$
$673$	−48.0000	−1.85026	−0.925132	+	0.379646i	$0.876046\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	9.00000	0.345643
$679$	39.0000	1.49668
$680$	0	0
$681$	21.0000	0.804722
$682$	−15.0000	−0.574380
$683$	29.0000	1.10965	0.554827	−	0.831966i	$-0.312784\pi$
$683$	29.0000	1.10965	0.554827	+	0.831966i	$0.312784\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	15.0000	0.572703
$687$	12.0000	0.457829
$688$	−3.00000	−0.114374
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−33.0000	−1.25538	−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000	−1.25538	−0.627690	+	0.778464i	$0.715999\pi$
$692$	−13.0000	−0.494186
$693$	−15.0000	−0.569803
$694$	−20.0000	−0.759190
$695$	0	0
$696$	1.00000	0.0379049
$697$	21.0000	0.795432
$698$	20.0000	0.757011
$699$	0	0
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	−2.00000	−0.0754851
$703$	−6.00000	−0.226294
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−37.0000	−1.39251
$707$	−36.0000	−1.35392
$708$	6.00000	0.225494
$709$	49.0000	1.84023	0.920117	−	0.391644i	$-0.128094\pi$
$709$	49.0000	1.84023	0.920117	+	0.391644i	$0.128094\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	18.0000	0.674579
$713$	−12.0000	−0.449404
$714$	9.00000	0.336817
$715$	0	0
$716$	12.0000	0.448461
$717$	29.0000	1.08302
$718$	30.0000	1.11959
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	−17.0000	−0.632674
$723$	−10.0000	−0.371904
$724$	10.0000	0.371647
$725$	0	0
$726$	−14.0000	−0.519589
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.00000	0.332877
$732$	5.00000	0.184805
$733$	51.0000	1.88373	0.941864	−	0.335994i	$-0.109072\pi$
$733$	51.0000	1.88373	0.941864	+	0.335994i	$0.109072\pi$
$734$	−3.00000	−0.110732
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−20.0000	−0.736709
$738$	7.00000	0.257674
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	15.0000	0.550667
$743$	3.00000	0.110059	0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000	0.110059	0.0550297	+	0.998485i	$0.482475\pi$
$744$	3.00000	0.109985
$745$	0	0
$746$	26.0000	0.951928
$747$	−6.00000	−0.219529
$748$	15.0000	0.548454
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	−24.0000	−0.874609
$754$	2.00000	0.0728357
$755$	0	0
$756$	3.00000	0.109109
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	−32.0000	−1.16229
$759$	−20.0000	−0.725954
$760$	0	0
$761$	25.0000	0.906249	0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000	0.906249	0.453125	+	0.891447i	$0.350309\pi$
$762$	8.00000	0.289809
$763$	15.0000	0.543036
$764$	−3.00000	−0.108536
$765$	0	0
$766$	−20.0000	−0.722629
$767$	12.0000	0.433295
$768$	1.00000	0.0360844
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	−26.0000	−0.936367
$772$	−18.0000	−0.647834
$773$	17.0000	0.611448	0.305724	−	0.952120i	$-0.401102\pi$
$773$	17.0000	0.611448	0.305724	+	0.952120i	$0.401102\pi$
$774$	3.00000	0.107833
$775$	0	0
$776$	−13.0000	−0.466673
$777$	3.00000	0.107624
$778$	3.00000	0.107555
$779$	42.0000	1.50481
$780$	0	0
$781$	60.0000	2.14697
$782$	12.0000	0.429119
$783$	−1.00000	−0.0357371
$784$	2.00000	0.0714286
$785$	0	0
$786$	−10.0000	−0.356688
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	2.00000	0.0712470
$789$	−5.00000	−0.178005
$790$	0	0
$791$	−27.0000	−0.960009
$792$	5.00000	0.177667
$793$	10.0000	0.355110
$794$	18.0000	0.638796
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−40.0000	−1.41687	−0.708436	−	0.705775i	$-0.750599\pi$
$797$	−40.0000	−1.41687	−0.708436	+	0.705775i	$0.750599\pi$
$798$	18.0000	0.637193
$799$	0	0
$800$	0	0
$801$	−18.0000	−0.635999
$802$	2.00000	0.0706225
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	6.00000	0.211341
$807$	−16.0000	−0.563227
$808$	12.0000	0.422159
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−3.00000	−0.105279
$813$	8.00000	0.280572
$814$	5.00000	0.175250
$815$	0	0
$816$	−3.00000	−0.105021
$817$	18.0000	0.629740
$818$	−6.00000	−0.209785
$819$	6.00000	0.209657
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	−10.0000	−0.348790
$823$	−56.0000	−1.95204	−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000	−1.95204	−0.976019	+	0.217687i	$0.930149\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	−18.0000	−0.626300
$827$	−23.0000	−0.799788	−0.399894	−	0.916561i	$-0.630953\pi$
$827$	−23.0000	−0.799788	−0.399894	+	0.916561i	$0.630953\pi$
$828$	4.00000	0.139010
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	16.0000	0.555034
$832$	2.00000	0.0693375
$833$	−6.00000	−0.207888
$834$	−5.00000	−0.173136
$835$	0	0
$836$	30.0000	1.03757
$837$	−3.00000	−0.103695
$838$	12.0000	0.414533
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−2.00000	−0.0689246
$843$	2.00000	0.0688837
$844$	19.0000	0.654007
$845$	0	0
$846$	0	0
$847$	42.0000	1.44314
$848$	−5.00000	−0.171701
$849$	−4.00000	−0.137280
$850$	0	0
$851$	4.00000	0.137118
$852$	−12.0000	−0.411113
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	−15.0000	−0.513289
$855$	0	0
$856$	2.00000	0.0683586
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	10.0000	0.341394
$859$	−26.0000	−0.887109	−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000	−0.887109	−0.443554	+	0.896248i	$0.646283\pi$
$860$	0	0
$861$	−21.0000	−0.715678
$862$	7.00000	0.238421
$863$	−27.0000	−0.919091	−0.459545	−	0.888154i	$-0.651988\pi$
$863$	−27.0000	−0.919091	−0.459545	+	0.888154i	$0.651988\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−8.00000	−0.271851
$867$	−8.00000	−0.271694
$868$	−9.00000	−0.305480
$869$	20.0000	0.678454
$870$	0	0
$871$	8.00000	0.271070
$872$	−5.00000	−0.169321
$873$	13.0000	0.439983
$874$	24.0000	0.811812
$875$	0	0
$876$	0	0
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	29.0000	0.978703
$879$	−7.00000	−0.236104
$880$	0	0
$881$	−39.0000	−1.31394	−0.656972	−	0.753915i	$-0.728163\pi$
$881$	−39.0000	−1.31394	−0.656972	+	0.753915i	$0.728163\pi$
$882$	−2.00000	−0.0673435
$883$	3.00000	0.100958	0.0504790	−	0.998725i	$-0.483925\pi$
$883$	3.00000	0.100958	0.0504790	+	0.998725i	$0.483925\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	14.0000	0.470339
$887$	7.00000	0.235037	0.117518	−	0.993071i	$-0.462506\pi$
$887$	7.00000	0.235037	0.117518	+	0.993071i	$0.462506\pi$
$888$	−1.00000	−0.0335578
$889$	−24.0000	−0.804934
$890$	0	0
$891$	−5.00000	−0.167506
$892$	−7.00000	−0.234377
$893$	0	0
$894$	18.0000	0.602010
$895$	0	0
$896$	−3.00000	−0.100223
$897$	8.00000	0.267112
$898$	36.0000	1.20134
$899$	3.00000	0.100056
$900$	0	0
$901$	15.0000	0.499722
$902$	−35.0000	−1.16537
$903$	−9.00000	−0.299501
$904$	9.00000	0.299336
$905$	0	0
$906$	−10.0000	−0.332228
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	21.0000	0.696909
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−6.00000	−0.198680
$913$	30.0000	0.992855
$914$	−5.00000	−0.165385
$915$	0	0
$916$	12.0000	0.396491
$917$	30.0000	0.990687
$918$	3.00000	0.0990148
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	−21.0000	−0.691598
$923$	−24.0000	−0.789970
$924$	−15.0000	−0.493464
$925$	0	0
$926$	34.0000	1.11731
$927$	−16.0000	−0.525509
$928$	1.00000	0.0328266
$929$	−1.00000	−0.0328089	−0.0164045	−	0.999865i	$-0.505222\pi$
$929$	−1.00000	−0.0328089	−0.0164045	+	0.999865i	$0.505222\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	0	0
$933$	−31.0000	−1.01489
$934$	−15.0000	−0.490815
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	40.0000	1.30674	0.653372	−	0.757037i	$-0.273354\pi$
$937$	40.0000	1.30674	0.653372	+	0.757037i	$0.273354\pi$
$938$	−12.0000	−0.391814
$939$	30.0000	0.979013
$940$	0	0
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	11.0000	0.358399
$943$	−28.0000	−0.911805
$944$	6.00000	0.195283
$945$	0	0
$946$	−15.0000	−0.487692
$947$	29.0000	0.942373	0.471187	−	0.882034i	$-0.343826\pi$
$947$	29.0000	0.942373	0.471187	+	0.882034i	$0.343826\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	0	0
$951$	27.0000	0.875535
$952$	9.00000	0.291692
$953$	40.0000	1.29573	0.647864	−	0.761756i	$-0.275663\pi$
$953$	40.0000	1.29573	0.647864	+	0.761756i	$0.275663\pi$
$954$	5.00000	0.161881
$955$	0	0
$956$	29.0000	0.937927
$957$	5.00000	0.161627
$958$	16.0000	0.516937
$959$	30.0000	0.968751
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−2.00000	−0.0644826
$963$	−2.00000	−0.0644491
$964$	−10.0000	−0.322078
$965$	0	0
$966$	−12.0000	−0.386094
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−14.0000	−0.449977
$969$	18.0000	0.578243
$970$	0	0
$971$	−35.0000	−1.12320	−0.561602	−	0.827408i	$-0.689815\pi$
$971$	−35.0000	−1.12320	−0.561602	+	0.827408i	$0.689815\pi$
$972$	1.00000	0.0320750
$973$	15.0000	0.480878
$974$	−6.00000	−0.192252
$975$	0	0
$976$	5.00000	0.160046
$977$	25.0000	0.799821	0.399910	−	0.916554i	$-0.369041\pi$
$977$	25.0000	0.799821	0.399910	+	0.916554i	$0.369041\pi$
$978$	11.0000	0.351741
$979$	90.0000	2.87641
$980$	0	0
$981$	5.00000	0.159638
$982$	−4.00000	−0.127645
$983$	29.0000	0.924956	0.462478	−	0.886631i	$-0.346960\pi$
$983$	29.0000	0.924956	0.462478	+	0.886631i	$0.346960\pi$
$984$	7.00000	0.223152
$985$	0	0
$986$	−3.00000	−0.0955395
$987$	0	0
$988$	−12.0000	−0.381771
$989$	−12.0000	−0.381578
$990$	0	0
$991$	7.00000	0.222362	0.111181	−	0.993800i	$-0.464537\pi$
$991$	7.00000	0.222362	0.111181	+	0.993800i	$0.464537\pi$
$992$	3.00000	0.0952501
$993$	−20.0000	−0.634681
$994$	36.0000	1.14185
$995$	0	0
$996$	−6.00000	−0.190117
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5550.2.a.t.1.1		1
5.4	even	2		1110.2.a.j.1.1	✓	1
15.14	odd	2		3330.2.a.b.1.1		1
20.19	odd	2		8880.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.j.1.1	✓	1	5.4	even	2
3330.2.a.b.1.1		1	15.14	odd	2
5550.2.a.t.1.1		1	1.1	even	1	trivial
8880.2.a.bc.1.1		1	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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