LMFDB - Embedded newform 825.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.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:	$6.58765816676$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} +1.46410 q^{13} +3.46410 q^{14} -5.00000 q^{16} -1.73205 q^{18} -1.46410 q^{19} +2.00000 q^{21} +1.73205 q^{22} +6.92820 q^{23} -1.73205 q^{24} -2.53590 q^{26} -1.00000 q^{27} -2.00000 q^{28} +3.46410 q^{29} +2.92820 q^{31} +5.19615 q^{32} +1.00000 q^{33} +1.00000 q^{36} -8.92820 q^{37} +2.53590 q^{38} -1.46410 q^{39} -3.46410 q^{41} -3.46410 q^{42} -8.92820 q^{43} -1.00000 q^{44} -12.0000 q^{46} -6.92820 q^{47} +5.00000 q^{48} -3.00000 q^{49} +1.46410 q^{52} +12.9282 q^{53} +1.73205 q^{54} -3.46410 q^{56} +1.46410 q^{57} -6.00000 q^{58} +6.92820 q^{59} +2.00000 q^{61} -5.07180 q^{62} -2.00000 q^{63} +1.00000 q^{64} -1.73205 q^{66} -8.00000 q^{67} -6.92820 q^{69} -13.8564 q^{71} +1.73205 q^{72} -12.3923 q^{73} +15.4641 q^{74} -1.46410 q^{76} +2.00000 q^{77} +2.53590 q^{78} -13.4641 q^{79} +1.00000 q^{81} +6.00000 q^{82} -15.4641 q^{83} +2.00000 q^{84} +15.4641 q^{86} -3.46410 q^{87} -1.73205 q^{88} -12.9282 q^{89} -2.92820 q^{91} +6.92820 q^{92} -2.92820 q^{93} +12.0000 q^{94} -5.19615 q^{96} +10.0000 q^{97} +5.19615 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^4 - 4 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{4} - 4 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{12} - 4 q^{13} - 10 q^{16} + 4 q^{19} + 4 q^{21} - 12 q^{26} - 2 q^{27} - 4 q^{28} - 8 q^{31} + 2 q^{33} + 2 q^{36} - 4 q^{37} + 12 q^{38} + 4 q^{39} - 4 q^{43} - 2 q^{44} - 24 q^{46} + 10 q^{48} - 6 q^{49} - 4 q^{52} + 12 q^{53} - 4 q^{57} - 12 q^{58} + 4 q^{61} - 24 q^{62} - 4 q^{63} + 2 q^{64} - 16 q^{67} - 4 q^{73} + 24 q^{74} + 4 q^{76} + 4 q^{77} + 12 q^{78} - 20 q^{79} + 2 q^{81} + 12 q^{82} - 24 q^{83} + 4 q^{84} + 24 q^{86} - 12 q^{89} + 8 q^{91} + 8 q^{93} + 24 q^{94} + 20 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 - 4 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^12 - 4 * q^13 - 10 * q^16 + 4 * q^19 + 4 * q^21 - 12 * q^26 - 2 * q^27 - 4 * q^28 - 8 * q^31 + 2 * q^33 + 2 * q^36 - 4 * q^37 + 12 * q^38 + 4 * q^39 - 4 * q^43 - 2 * q^44 - 24 * q^46 + 10 * q^48 - 6 * q^49 - 4 * q^52 + 12 * q^53 - 4 * q^57 - 12 * q^58 + 4 * q^61 - 24 * q^62 - 4 * q^63 + 2 * q^64 - 16 * q^67 - 4 * q^73 + 24 * q^74 + 4 * q^76 + 4 * q^77 + 12 * q^78 - 20 * q^79 + 2 * q^81 + 12 * q^82 - 24 * q^83 + 4 * q^84 + 24 * q^86 - 12 * q^89 + 8 * q^91 + 8 * q^93 + 24 * q^94 + 20 * q^97 - 2 * q^99

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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.73205	0.707107
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.73205	0.612372
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	3.46410	0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.73205	−0.408248
$19$	−1.46410	−0.335888	−0.167944	−	0.985797i	$-0.553713\pi$
$19$	−1.46410	−0.335888	−0.167944	+	0.985797i	$0.553713\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	1.73205	0.369274
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	−1.73205	−0.353553
$25$	0	0
$26$	−2.53590	−0.497331
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	2.92820	0.525921	0.262960	−	0.964807i	$-0.415301\pi$
$31$	2.92820	0.525921	0.262960	+	0.964807i	$0.415301\pi$
$32$	5.19615	0.918559
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.92820	−1.46779	−0.733894	−	0.679264i	$-0.762299\pi$
$37$	−8.92820	−1.46779	−0.733894	+	0.679264i	$0.762299\pi$
$38$	2.53590	0.411377
$39$	−1.46410	−0.234444
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	−3.46410	−0.534522
$43$	−8.92820	−1.36154	−0.680769	−	0.732498i	$-0.738354\pi$
$43$	−8.92820	−1.36154	−0.680769	+	0.732498i	$0.738354\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−12.0000	−1.76930
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	5.00000	0.721688
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	1.46410	0.203034
$53$	12.9282	1.77583	0.887913	−	0.460012i	$-0.152155\pi$
$53$	12.9282	1.77583	0.887913	+	0.460012i	$0.152155\pi$
$54$	1.73205	0.235702
$55$	0	0
$56$	−3.46410	−0.462910
$57$	1.46410	0.193925
$58$	−6.00000	−0.787839
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−5.07180	−0.644119
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.73205	−0.213201
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−6.92820	−0.834058
$70$	0	0
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	1.73205	0.204124
$73$	−12.3923	−1.45041	−0.725205	−	0.688533i	$-0.758255\pi$
$73$	−12.3923	−1.45041	−0.725205	+	0.688533i	$0.758255\pi$
$74$	15.4641	1.79767
$75$	0	0
$76$	−1.46410	−0.167944
$77$	2.00000	0.227921
$78$	2.53590	0.287134
$79$	−13.4641	−1.51483	−0.757415	−	0.652934i	$-0.773538\pi$
$79$	−13.4641	−1.51483	−0.757415	+	0.652934i	$0.773538\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−15.4641	−1.69741	−0.848703	−	0.528870i	$-0.822616\pi$
$83$	−15.4641	−1.69741	−0.848703	+	0.528870i	$0.822616\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	15.4641	1.66754
$87$	−3.46410	−0.371391
$88$	−1.73205	−0.184637
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	−2.92820	−0.306959
$92$	6.92820	0.722315
$93$	−2.92820	−0.303641
$94$	12.0000	1.23771
$95$	0	0
$96$	−5.19615	−0.530330
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	5.19615	0.524891
$99$	−1.00000	−0.100504
$100$	0	0
$101$	10.3923	1.03407	0.517036	−	0.855963i	$-0.327035\pi$
$101$	10.3923	1.03407	0.517036	+	0.855963i	$0.327035\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	2.53590	0.248665
$105$	0	0
$106$	−22.3923	−2.17493
$107$	−15.4641	−1.49497	−0.747486	−	0.664278i	$-0.768739\pi$
$107$	−15.4641	−1.49497	−0.747486	+	0.664278i	$0.768739\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	8.92820	0.847428
$112$	10.0000	0.944911
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	−2.53590	−0.237509
$115$	0	0
$116$	3.46410	0.321634
$117$	1.46410	0.135356
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.46410	−0.313625
$123$	3.46410	0.312348
$124$	2.92820	0.262960
$125$	0	0
$126$	3.46410	0.308607
$127$	4.92820	0.437307	0.218654	−	0.975803i	$-0.429834\pi$
$127$	4.92820	0.437307	0.218654	+	0.975803i	$0.429834\pi$
$128$	−12.1244	−1.07165
$129$	8.92820	0.786084
$130$	0	0
$131$	5.07180	0.443125	0.221562	−	0.975146i	$-0.428884\pi$
$131$	5.07180	0.443125	0.221562	+	0.975146i	$0.428884\pi$
$132$	1.00000	0.0870388
$133$	2.92820	0.253907
$134$	13.8564	1.19701
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	12.0000	1.02151
$139$	−8.39230	−0.711826	−0.355913	−	0.934519i	$-0.615830\pi$
$139$	−8.39230	−0.711826	−0.355913	+	0.934519i	$0.615830\pi$
$140$	0	0
$141$	6.92820	0.583460
$142$	24.0000	2.01404
$143$	−1.46410	−0.122434
$144$	−5.00000	−0.416667
$145$	0	0
$146$	21.4641	1.77638
$147$	3.00000	0.247436
$148$	−8.92820	−0.733894
$149$	−8.53590	−0.699288	−0.349644	−	0.936883i	$-0.613697\pi$
$149$	−8.53590	−0.699288	−0.349644	+	0.936883i	$0.613697\pi$
$150$	0	0
$151$	0.392305	0.0319253	0.0159627	−	0.999873i	$-0.494919\pi$
$151$	0.392305	0.0319253	0.0159627	+	0.999873i	$0.494919\pi$
$152$	−2.53590	−0.205689
$153$	0	0
$154$	−3.46410	−0.279145
$155$	0	0
$156$	−1.46410	−0.117222
$157$	16.9282	1.35102	0.675509	−	0.737352i	$-0.263924\pi$
$157$	16.9282	1.35102	0.675509	+	0.737352i	$0.263924\pi$
$158$	23.3205	1.85528
$159$	−12.9282	−1.02527
$160$	0	0
$161$	−13.8564	−1.09204
$162$	−1.73205	−0.136083
$163$	17.8564	1.39862	0.699311	−	0.714818i	$-0.253490\pi$
$163$	17.8564	1.39862	0.699311	+	0.714818i	$0.253490\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	26.7846	2.07889
$167$	−10.3923	−0.804181	−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923	−0.804181	−0.402090	+	0.915600i	$0.631716\pi$
$168$	3.46410	0.267261
$169$	−10.8564	−0.835108
$170$	0	0
$171$	−1.46410	−0.111963
$172$	−8.92820	−0.680769
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	5.00000	0.376889
$177$	−6.92820	−0.520756
$178$	22.3923	1.67837
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	−11.8564	−0.881280	−0.440640	−	0.897684i	$-0.645248\pi$
$181$	−11.8564	−0.881280	−0.440640	+	0.897684i	$0.645248\pi$
$182$	5.07180	0.375947
$183$	−2.00000	−0.147844
$184$	12.0000	0.884652
$185$	0	0
$186$	5.07180	0.371882
$187$	0	0
$188$	−6.92820	−0.505291
$189$	2.00000	0.145479
$190$	0	0
$191$	−5.07180	−0.366982	−0.183491	−	0.983021i	$-0.558740\pi$
$191$	−5.07180	−0.366982	−0.183491	+	0.983021i	$0.558740\pi$
$192$	−1.00000	−0.0721688
$193$	−3.60770	−0.259688	−0.129844	−	0.991534i	$-0.541448\pi$
$193$	−3.60770	−0.259688	−0.129844	+	0.991534i	$0.541448\pi$
$194$	−17.3205	−1.24354
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	1.73205	0.123091
$199$	16.7846	1.18983	0.594915	−	0.803789i	$-0.297186\pi$
$199$	16.7846	1.18983	0.594915	+	0.803789i	$0.297186\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	−18.0000	−1.26648
$203$	−6.92820	−0.486265
$204$	0	0
$205$	0	0
$206$	13.8564	0.965422
$207$	6.92820	0.481543
$208$	−7.32051	−0.507586
$209$	1.46410	0.101274
$210$	0	0
$211$	12.3923	0.853121	0.426561	−	0.904459i	$-0.359725\pi$
$211$	12.3923	0.853121	0.426561	+	0.904459i	$0.359725\pi$
$212$	12.9282	0.887913
$213$	13.8564	0.949425
$214$	26.7846	1.83096
$215$	0	0
$216$	−1.73205	−0.117851
$217$	−5.85641	−0.397559
$218$	17.3205	1.17309
$219$	12.3923	0.837394
$220$	0	0
$221$	0	0
$222$	−15.4641	−1.03788
$223$	17.8564	1.19575	0.597877	−	0.801588i	$-0.296011\pi$
$223$	17.8564	1.19575	0.597877	+	0.801588i	$0.296011\pi$
$224$	−10.3923	−0.694365
$225$	0	0
$226$	1.60770	0.106942
$227$	−8.53590	−0.566547	−0.283274	−	0.959039i	$-0.591420\pi$
$227$	−8.53590	−0.566547	−0.283274	+	0.959039i	$0.591420\pi$
$228$	1.46410	0.0969625
$229$	3.85641	0.254839	0.127419	−	0.991849i	$-0.459331\pi$
$229$	3.85641	0.254839	0.127419	+	0.991849i	$0.459331\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	6.00000	0.393919
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	−2.53590	−0.165777
$235$	0	0
$236$	6.92820	0.450988
$237$	13.4641	0.874587
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	27.8564	1.79439	0.897194	−	0.441636i	$-0.145602\pi$
$241$	27.8564	1.79439	0.897194	+	0.441636i	$0.145602\pi$
$242$	−1.73205	−0.111340
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−2.14359	−0.136394
$248$	5.07180	0.322059
$249$	15.4641	0.979998
$250$	0	0
$251$	25.8564	1.63204	0.816021	−	0.578022i	$-0.196175\pi$
$251$	25.8564	1.63204	0.816021	+	0.578022i	$0.196175\pi$
$252$	−2.00000	−0.125988
$253$	−6.92820	−0.435572
$254$	−8.53590	−0.535590
$255$	0	0
$256$	19.0000	1.18750
$257$	−7.85641	−0.490069	−0.245035	−	0.969514i	$-0.578799\pi$
$257$	−7.85641	−0.490069	−0.245035	+	0.969514i	$0.578799\pi$
$258$	−15.4641	−0.962753
$259$	17.8564	1.10954
$260$	0	0
$261$	3.46410	0.214423
$262$	−8.78461	−0.542715
$263$	−27.4641	−1.69351	−0.846755	−	0.531984i	$-0.821447\pi$
$263$	−27.4641	−1.69351	−0.846755	+	0.531984i	$0.821447\pi$
$264$	1.73205	0.106600
$265$	0	0
$266$	−5.07180	−0.310972
$267$	12.9282	0.791193
$268$	−8.00000	−0.488678
$269$	−7.85641	−0.479014	−0.239507	−	0.970895i	$-0.576986\pi$
$269$	−7.85641	−0.479014	−0.239507	+	0.970895i	$0.576986\pi$
$270$	0	0
$271$	−32.3923	−1.96769	−0.983846	−	0.179016i	$-0.942709\pi$
$271$	−32.3923	−1.96769	−0.983846	+	0.179016i	$0.942709\pi$
$272$	0	0
$273$	2.92820	0.177223
$274$	−31.1769	−1.88347
$275$	0	0
$276$	−6.92820	−0.417029
$277$	−22.5359	−1.35405	−0.677025	−	0.735960i	$-0.736731\pi$
$277$	−22.5359	−1.35405	−0.677025	+	0.735960i	$0.736731\pi$
$278$	14.5359	0.871805
$279$	2.92820	0.175307
$280$	0	0
$281$	−3.46410	−0.206651	−0.103325	−	0.994648i	$-0.532948\pi$
$281$	−3.46410	−0.206651	−0.103325	+	0.994648i	$0.532948\pi$
$282$	−12.0000	−0.714590
$283$	−8.92820	−0.530727	−0.265363	−	0.964148i	$-0.585492\pi$
$283$	−8.92820	−0.530727	−0.265363	+	0.964148i	$0.585492\pi$
$284$	−13.8564	−0.822226
$285$	0	0
$286$	2.53590	0.149951
$287$	6.92820	0.408959
$288$	5.19615	0.306186
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−10.0000	−0.586210
$292$	−12.3923	−0.725205
$293$	−13.8564	−0.809500	−0.404750	−	0.914427i	$-0.632641\pi$
$293$	−13.8564	−0.809500	−0.404750	+	0.914427i	$0.632641\pi$
$294$	−5.19615	−0.303046
$295$	0	0
$296$	−15.4641	−0.898833
$297$	1.00000	0.0580259
$298$	14.7846	0.856449
$299$	10.1436	0.586619
$300$	0	0
$301$	17.8564	1.02923
$302$	−0.679492	−0.0391004
$303$	−10.3923	−0.597022
$304$	7.32051	0.419860
$305$	0	0
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	2.00000	0.113961
$309$	8.00000	0.455104
$310$	0	0
$311$	18.9282	1.07332	0.536660	−	0.843799i	$-0.319686\pi$
$311$	18.9282	1.07332	0.536660	+	0.843799i	$0.319686\pi$
$312$	−2.53590	−0.143567
$313$	−7.07180	−0.399722	−0.199861	−	0.979824i	$-0.564049\pi$
$313$	−7.07180	−0.399722	−0.199861	+	0.979824i	$0.564049\pi$
$314$	−29.3205	−1.65465
$315$	0	0
$316$	−13.4641	−0.757415
$317$	11.0718	0.621854	0.310927	−	0.950434i	$-0.399361\pi$
$317$	11.0718	0.621854	0.310927	+	0.950434i	$0.399361\pi$
$318$	22.3923	1.25570
$319$	−3.46410	−0.193952
$320$	0	0
$321$	15.4641	0.863122
$322$	24.0000	1.33747
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	−30.9282	−1.71295
$327$	10.0000	0.553001
$328$	−6.00000	−0.331295
$329$	13.8564	0.763928
$330$	0	0
$331$	−17.8564	−0.981477	−0.490738	−	0.871307i	$-0.663273\pi$
$331$	−17.8564	−0.981477	−0.490738	+	0.871307i	$0.663273\pi$
$332$	−15.4641	−0.848703
$333$	−8.92820	−0.489263
$334$	18.0000	0.984916
$335$	0	0
$336$	−10.0000	−0.545545
$337$	29.1769	1.58937	0.794684	−	0.607023i	$-0.207637\pi$
$337$	29.1769	1.58937	0.794684	+	0.607023i	$0.207637\pi$
$338$	18.8038	1.02279
$339$	0.928203	0.0504131
$340$	0	0
$341$	−2.92820	−0.158571
$342$	2.53590	0.137126
$343$	20.0000	1.07990
$344$	−15.4641	−0.833768
$345$	0	0
$346$	−20.7846	−1.11739
$347$	−1.60770	−0.0863056	−0.0431528	−	0.999068i	$-0.513740\pi$
$347$	−1.60770	−0.0863056	−0.0431528	+	0.999068i	$0.513740\pi$
$348$	−3.46410	−0.185695
$349$	−35.8564	−1.91935	−0.959675	−	0.281113i	$-0.909296\pi$
$349$	−35.8564	−1.91935	−0.959675	+	0.281113i	$0.909296\pi$
$350$	0	0
$351$	−1.46410	−0.0781480
$352$	−5.19615	−0.276956
$353$	0.928203	0.0494033	0.0247016	−	0.999695i	$-0.492136\pi$
$353$	0.928203	0.0494033	0.0247016	+	0.999695i	$0.492136\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−12.9282	−0.685193
$357$	0	0
$358$	12.0000	0.634220
$359$	20.7846	1.09697	0.548485	−	0.836160i	$-0.315205\pi$
$359$	20.7846	1.09697	0.548485	+	0.836160i	$0.315205\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	20.5359	1.07934
$363$	−1.00000	−0.0524864
$364$	−2.92820	−0.153480
$365$	0	0
$366$	3.46410	0.181071
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	−34.6410	−1.80579
$369$	−3.46410	−0.180334
$370$	0	0
$371$	−25.8564	−1.34240
$372$	−2.92820	−0.151820
$373$	−0.392305	−0.0203128	−0.0101564	−	0.999948i	$-0.503233\pi$
$373$	−0.392305	−0.0203128	−0.0101564	+	0.999948i	$0.503233\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	5.07180	0.261211
$378$	−3.46410	−0.178174
$379$	9.85641	0.506290	0.253145	−	0.967428i	$-0.418535\pi$
$379$	9.85641	0.506290	0.253145	+	0.967428i	$0.418535\pi$
$380$	0	0
$381$	−4.92820	−0.252479
$382$	8.78461	0.449460
$383$	−13.8564	−0.708029	−0.354015	−	0.935240i	$-0.615184\pi$
$383$	−13.8564	−0.708029	−0.354015	+	0.935240i	$0.615184\pi$
$384$	12.1244	0.618718
$385$	0	0
$386$	6.24871	0.318051
$387$	−8.92820	−0.453846
$388$	10.0000	0.507673
$389$	24.9282	1.26391	0.631955	−	0.775005i	$-0.282253\pi$
$389$	24.9282	1.26391	0.631955	+	0.775005i	$0.282253\pi$
$390$	0	0
$391$	0	0
$392$	−5.19615	−0.262445
$393$	−5.07180	−0.255838
$394$	−20.7846	−1.04711
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−29.0718	−1.45724
$399$	−2.92820	−0.146594
$400$	0	0
$401$	19.8564	0.991582	0.495791	−	0.868442i	$-0.334878\pi$
$401$	19.8564	0.991582	0.495791	+	0.868442i	$0.334878\pi$
$402$	−13.8564	−0.691095
$403$	4.28719	0.213560
$404$	10.3923	0.517036
$405$	0	0
$406$	12.0000	0.595550
$407$	8.92820	0.442555
$408$	0	0
$409$	34.7846	1.71999	0.859994	−	0.510304i	$-0.170467\pi$
$409$	34.7846	1.71999	0.859994	+	0.510304i	$0.170467\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	−8.00000	−0.394132
$413$	−13.8564	−0.681829
$414$	−12.0000	−0.589768
$415$	0	0
$416$	7.60770	0.372998
$417$	8.39230	0.410973
$418$	−2.53590	−0.124035
$419$	17.0718	0.834012	0.417006	−	0.908904i	$-0.363079\pi$
$419$	17.0718	0.834012	0.417006	+	0.908904i	$0.363079\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−21.4641	−1.04486
$423$	−6.92820	−0.336861
$424$	22.3923	1.08747
$425$	0	0
$426$	−24.0000	−1.16280
$427$	−4.00000	−0.193574
$428$	−15.4641	−0.747486
$429$	1.46410	0.0706875
$430$	0	0
$431$	−32.7846	−1.57918	−0.789590	−	0.613635i	$-0.789706\pi$
$431$	−32.7846	−1.57918	−0.789590	+	0.613635i	$0.789706\pi$
$432$	5.00000	0.240563
$433$	−27.8564	−1.33869	−0.669347	−	0.742950i	$-0.733426\pi$
$433$	−27.8564	−1.33869	−0.669347	+	0.742950i	$0.733426\pi$
$434$	10.1436	0.486908
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−10.1436	−0.485234
$438$	−21.4641	−1.02559
$439$	−29.1769	−1.39254	−0.696269	−	0.717781i	$-0.745158\pi$
$439$	−29.1769	−1.39254	−0.696269	+	0.717781i	$0.745158\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	8.92820	0.423714
$445$	0	0
$446$	−30.9282	−1.46449
$447$	8.53590	0.403734
$448$	−2.00000	−0.0944911
$449$	14.7846	0.697729	0.348864	−	0.937173i	$-0.386567\pi$
$449$	14.7846	0.697729	0.348864	+	0.937173i	$0.386567\pi$
$450$	0	0
$451$	3.46410	0.163118
$452$	−0.928203	−0.0436590
$453$	−0.392305	−0.0184321
$454$	14.7846	0.693876
$455$	0	0
$456$	2.53590	0.118754
$457$	8.39230	0.392575	0.196288	−	0.980546i	$-0.437111\pi$
$457$	8.39230	0.392575	0.196288	+	0.980546i	$0.437111\pi$
$458$	−6.67949	−0.312112
$459$	0	0
$460$	0	0
$461$	−12.2487	−0.570479	−0.285240	−	0.958456i	$-0.592073\pi$
$461$	−12.2487	−0.570479	−0.285240	+	0.958456i	$0.592073\pi$
$462$	3.46410	0.161165
$463$	28.0000	1.30127	0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000	1.30127	0.650635	+	0.759390i	$0.274503\pi$
$464$	−17.3205	−0.804084
$465$	0	0
$466$	20.7846	0.962828
$467$	−18.9282	−0.875893	−0.437946	−	0.899001i	$-0.644294\pi$
$467$	−18.9282	−0.875893	−0.437946	+	0.899001i	$0.644294\pi$
$468$	1.46410	0.0676781
$469$	16.0000	0.738811
$470$	0	0
$471$	−16.9282	−0.780010
$472$	12.0000	0.552345
$473$	8.92820	0.410519
$474$	−23.3205	−1.07115
$475$	0	0
$476$	0	0
$477$	12.9282	0.591942
$478$	20.7846	0.950666
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−13.0718	−0.596023
$482$	−48.2487	−2.19767
$483$	13.8564	0.630488
$484$	1.00000	0.0454545
$485$	0	0
$486$	1.73205	0.0785674
$487$	−23.7128	−1.07453	−0.537265	−	0.843413i	$-0.680543\pi$
$487$	−23.7128	−1.07453	−0.537265	+	0.843413i	$0.680543\pi$
$488$	3.46410	0.156813
$489$	−17.8564	−0.807495
$490$	0	0
$491$	−17.0718	−0.770439	−0.385220	−	0.922825i	$-0.625874\pi$
$491$	−17.0718	−0.770439	−0.385220	+	0.922825i	$0.625874\pi$
$492$	3.46410	0.156174
$493$	0	0
$494$	3.71281	0.167047
$495$	0	0
$496$	−14.6410	−0.657401
$497$	27.7128	1.24309
$498$	−26.7846	−1.20025
$499$	−12.7846	−0.572318	−0.286159	−	0.958182i	$-0.592379\pi$
$499$	−12.7846	−0.572318	−0.286159	+	0.958182i	$0.592379\pi$
$500$	0	0
$501$	10.3923	0.464294
$502$	−44.7846	−1.99883
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	−3.46410	−0.154303
$505$	0	0
$506$	12.0000	0.533465
$507$	10.8564	0.482150
$508$	4.92820	0.218654
$509$	−7.85641	−0.348229	−0.174115	−	0.984725i	$-0.555706\pi$
$509$	−7.85641	−0.348229	−0.174115	+	0.984725i	$0.555706\pi$
$510$	0	0
$511$	24.7846	1.09641
$512$	−8.66025	−0.382733
$513$	1.46410	0.0646417
$514$	13.6077	0.600210
$515$	0	0
$516$	8.92820	0.393042
$517$	6.92820	0.304702
$518$	−30.9282	−1.35891
$519$	−12.0000	−0.526742
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−6.00000	−0.262613
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	5.07180	0.221562
$525$	0	0
$526$	47.5692	2.07412
$527$	0	0
$528$	−5.00000	−0.217597
$529$	25.0000	1.08696
$530$	0	0
$531$	6.92820	0.300658
$532$	2.92820	0.126954
$533$	−5.07180	−0.219684
$534$	−22.3923	−0.969010
$535$	0	0
$536$	−13.8564	−0.598506
$537$	6.92820	0.298974
$538$	13.6077	0.586669
$539$	3.00000	0.129219
$540$	0	0
$541$	0.143594	0.00617357	0.00308678	−	0.999995i	$-0.499017\pi$
$541$	0.143594	0.00617357	0.00308678	+	0.999995i	$0.499017\pi$
$542$	56.1051	2.40992
$543$	11.8564	0.508807
$544$	0	0
$545$	0	0
$546$	−5.07180	−0.217053
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	18.0000	0.768922
$549$	2.00000	0.0853579
$550$	0	0
$551$	−5.07180	−0.216066
$552$	−12.0000	−0.510754
$553$	26.9282	1.14510
$554$	39.0333	1.65837
$555$	0	0
$556$	−8.39230	−0.355913
$557$	−44.7846	−1.89758	−0.948792	−	0.315900i	$-0.897694\pi$
$557$	−44.7846	−1.89758	−0.948792	+	0.315900i	$0.897694\pi$
$558$	−5.07180	−0.214706
$559$	−13.0718	−0.552878
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	10.3923	0.437983	0.218992	−	0.975727i	$-0.429723\pi$
$563$	10.3923	0.437983	0.218992	+	0.975727i	$0.429723\pi$
$564$	6.92820	0.291730
$565$	0	0
$566$	15.4641	0.650005
$567$	−2.00000	−0.0839921
$568$	−24.0000	−1.00702
$569$	−29.3205	−1.22918	−0.614590	−	0.788847i	$-0.710678\pi$
$569$	−29.3205	−1.22918	−0.614590	+	0.788847i	$0.710678\pi$
$570$	0	0
$571$	24.3923	1.02079	0.510393	−	0.859941i	$-0.329500\pi$
$571$	24.3923	1.02079	0.510393	+	0.859941i	$0.329500\pi$
$572$	−1.46410	−0.0612172
$573$	5.07180	0.211877
$574$	−12.0000	−0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	−22.7846	−0.948536	−0.474268	−	0.880381i	$-0.657287\pi$
$577$	−22.7846	−0.948536	−0.474268	+	0.880381i	$0.657287\pi$
$578$	29.4449	1.22474
$579$	3.60770	0.149931
$580$	0	0
$581$	30.9282	1.28312
$582$	17.3205	0.717958
$583$	−12.9282	−0.535431
$584$	−21.4641	−0.888191
$585$	0	0
$586$	24.0000	0.991431
$587$	5.07180	0.209335	0.104668	−	0.994507i	$-0.466622\pi$
$587$	5.07180	0.209335	0.104668	+	0.994507i	$0.466622\pi$
$588$	3.00000	0.123718
$589$	−4.28719	−0.176650
$590$	0	0
$591$	−12.0000	−0.493614
$592$	44.6410	1.83473
$593$	32.7846	1.34630	0.673151	−	0.739505i	$-0.264940\pi$
$593$	32.7846	1.34630	0.673151	+	0.739505i	$0.264940\pi$
$594$	−1.73205	−0.0710669
$595$	0	0
$596$	−8.53590	−0.349644
$597$	−16.7846	−0.686948
$598$	−17.5692	−0.718459
$599$	−10.1436	−0.414456	−0.207228	−	0.978293i	$-0.566444\pi$
$599$	−10.1436	−0.414456	−0.207228	+	0.978293i	$0.566444\pi$
$600$	0	0
$601$	36.6410	1.49462	0.747309	−	0.664477i	$-0.231345\pi$
$601$	36.6410	1.49462	0.747309	+	0.664477i	$0.231345\pi$
$602$	−30.9282	−1.26054
$603$	−8.00000	−0.325785
$604$	0.392305	0.0159627
$605$	0	0
$606$	18.0000	0.731200
$607$	−22.7846	−0.924799	−0.462399	−	0.886672i	$-0.653011\pi$
$607$	−22.7846	−0.924799	−0.462399	+	0.886672i	$0.653011\pi$
$608$	−7.60770	−0.308533
$609$	6.92820	0.280745
$610$	0	0
$611$	−10.1436	−0.410366
$612$	0	0
$613$	−0.392305	−0.0158450	−0.00792252	−	0.999969i	$-0.502522\pi$
$613$	−0.392305	−0.0158450	−0.00792252	+	0.999969i	$0.502522\pi$
$614$	24.2487	0.978598
$615$	0	0
$616$	3.46410	0.139573
$617$	23.0718	0.928836	0.464418	−	0.885616i	$-0.346264\pi$
$617$	23.0718	0.928836	0.464418	+	0.885616i	$0.346264\pi$
$618$	−13.8564	−0.557386
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−6.92820	−0.278019
$622$	−32.7846	−1.31454
$623$	25.8564	1.03592
$624$	7.32051	0.293055
$625$	0	0
$626$	12.2487	0.489557
$627$	−1.46410	−0.0584706
$628$	16.9282	0.675509
$629$	0	0
$630$	0	0
$631$	−34.9282	−1.39047	−0.695235	−	0.718783i	$-0.744700\pi$
$631$	−34.9282	−1.39047	−0.695235	+	0.718783i	$0.744700\pi$
$632$	−23.3205	−0.927640
$633$	−12.3923	−0.492550
$634$	−19.1769	−0.761613
$635$	0	0
$636$	−12.9282	−0.512637
$637$	−4.39230	−0.174029
$638$	6.00000	0.237542
$639$	−13.8564	−0.548151
$640$	0	0
$641$	0.928203	0.0366618	0.0183309	−	0.999832i	$-0.494165\pi$
$641$	0.928203	0.0366618	0.0183309	+	0.999832i	$0.494165\pi$
$642$	−26.7846	−1.05710
$643$	45.5692	1.79707	0.898537	−	0.438897i	$-0.144631\pi$
$643$	45.5692	1.79707	0.898537	+	0.438897i	$0.144631\pi$
$644$	−13.8564	−0.546019
$645$	0	0
$646$	0	0
$647$	27.7128	1.08950	0.544752	−	0.838597i	$-0.316624\pi$
$647$	27.7128	1.08950	0.544752	+	0.838597i	$0.316624\pi$
$648$	1.73205	0.0680414
$649$	−6.92820	−0.271956
$650$	0	0
$651$	5.85641	0.229531
$652$	17.8564	0.699311
$653$	7.85641	0.307445	0.153722	−	0.988114i	$-0.450874\pi$
$653$	7.85641	0.307445	0.153722	+	0.988114i	$0.450874\pi$
$654$	−17.3205	−0.677285
$655$	0	0
$656$	17.3205	0.676252
$657$	−12.3923	−0.483470
$658$	−24.0000	−0.935617
$659$	39.7128	1.54699	0.773496	−	0.633801i	$-0.218506\pi$
$659$	39.7128	1.54699	0.773496	+	0.633801i	$0.218506\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	30.9282	1.20206
$663$	0	0
$664$	−26.7846	−1.03944
$665$	0	0
$666$	15.4641	0.599222
$667$	24.0000	0.929284
$668$	−10.3923	−0.402090
$669$	−17.8564	−0.690369
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	10.3923	0.400892
$673$	−31.3205	−1.20732	−0.603658	−	0.797243i	$-0.706291\pi$
$673$	−31.3205	−1.20732	−0.603658	+	0.797243i	$0.706291\pi$
$674$	−50.5359	−1.94657
$675$	0	0
$676$	−10.8564	−0.417554
$677$	32.7846	1.26001	0.630007	−	0.776589i	$-0.283052\pi$
$677$	32.7846	1.26001	0.630007	+	0.776589i	$0.283052\pi$
$678$	−1.60770	−0.0617432
$679$	−20.0000	−0.767530
$680$	0	0
$681$	8.53590	0.327096
$682$	5.07180	0.194209
$683$	−8.78461	−0.336134	−0.168067	−	0.985776i	$-0.553752\pi$
$683$	−8.78461	−0.336134	−0.168067	+	0.985776i	$0.553752\pi$
$684$	−1.46410	−0.0559813
$685$	0	0
$686$	−34.6410	−1.32260
$687$	−3.85641	−0.147131
$688$	44.6410	1.70192
$689$	18.9282	0.721107
$690$	0	0
$691$	−7.71281	−0.293409	−0.146705	−	0.989180i	$-0.546867\pi$
$691$	−7.71281	−0.293409	−0.146705	+	0.989180i	$0.546867\pi$
$692$	12.0000	0.456172
$693$	2.00000	0.0759737
$694$	2.78461	0.105702
$695$	0	0
$696$	−6.00000	−0.227429
$697$	0	0
$698$	62.1051	2.35071
$699$	12.0000	0.453882
$700$	0	0
$701$	32.5359	1.22886	0.614432	−	0.788970i	$-0.289385\pi$
$701$	32.5359	1.22886	0.614432	+	0.788970i	$0.289385\pi$
$702$	2.53590	0.0957113
$703$	13.0718	0.493012
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−1.60770	−0.0605064
$707$	−20.7846	−0.781686
$708$	−6.92820	−0.260378
$709$	15.8564	0.595500	0.297750	−	0.954644i	$-0.403764\pi$
$709$	15.8564	0.595500	0.297750	+	0.954644i	$0.403764\pi$
$710$	0	0
$711$	−13.4641	−0.504943
$712$	−22.3923	−0.839187
$713$	20.2872	0.759761
$714$	0	0
$715$	0	0
$716$	−6.92820	−0.258919
$717$	12.0000	0.448148
$718$	−36.0000	−1.34351
$719$	−18.9282	−0.705903	−0.352951	−	0.935642i	$-0.614822\pi$
$719$	−18.9282	−0.705903	−0.352951	+	0.935642i	$0.614822\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	29.1962	1.08657
$723$	−27.8564	−1.03599
$724$	−11.8564	−0.440640
$725$	0	0
$726$	1.73205	0.0642824
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−5.07180	−0.187973
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−2.00000	−0.0739221
$733$	49.9615	1.84537	0.922686	−	0.385553i	$-0.125989\pi$
$733$	49.9615	1.84537	0.922686	+	0.385553i	$0.125989\pi$
$734$	34.6410	1.27862
$735$	0	0
$736$	36.0000	1.32698
$737$	8.00000	0.294684
$738$	6.00000	0.220863
$739$	10.5359	0.387569	0.193785	−	0.981044i	$-0.437924\pi$
$739$	10.5359	0.387569	0.193785	+	0.981044i	$0.437924\pi$
$740$	0	0
$741$	2.14359	0.0787469
$742$	44.7846	1.64409
$743$	−46.3923	−1.70197	−0.850984	−	0.525191i	$-0.823994\pi$
$743$	−46.3923	−1.70197	−0.850984	+	0.525191i	$0.823994\pi$
$744$	−5.07180	−0.185941
$745$	0	0
$746$	0.679492	0.0248780
$747$	−15.4641	−0.565802
$748$	0	0
$749$	30.9282	1.13009
$750$	0	0
$751$	13.0718	0.476997	0.238498	−	0.971143i	$-0.423345\pi$
$751$	13.0718	0.476997	0.238498	+	0.971143i	$0.423345\pi$
$752$	34.6410	1.26323
$753$	−25.8564	−0.942260
$754$	−8.78461	−0.319917
$755$	0	0
$756$	2.00000	0.0727393
$757$	6.78461	0.246591	0.123295	−	0.992370i	$-0.460654\pi$
$757$	6.78461	0.246591	0.123295	+	0.992370i	$0.460654\pi$
$758$	−17.0718	−0.620076
$759$	6.92820	0.251478
$760$	0	0
$761$	−39.4641	−1.43057	−0.715286	−	0.698832i	$-0.753704\pi$
$761$	−39.4641	−1.43057	−0.715286	+	0.698832i	$0.753704\pi$
$762$	8.53590	0.309223
$763$	20.0000	0.724049
$764$	−5.07180	−0.183491
$765$	0	0
$766$	24.0000	0.867155
$767$	10.1436	0.366264
$768$	−19.0000	−0.685603
$769$	−46.4974	−1.67674	−0.838370	−	0.545102i	$-0.816491\pi$
$769$	−46.4974	−1.67674	−0.838370	+	0.545102i	$0.816491\pi$
$770$	0	0
$771$	7.85641	0.282942
$772$	−3.60770	−0.129844
$773$	31.8564	1.14580	0.572898	−	0.819627i	$-0.305819\pi$
$773$	31.8564	1.14580	0.572898	+	0.819627i	$0.305819\pi$
$774$	15.4641	0.555846
$775$	0	0
$776$	17.3205	0.621770
$777$	−17.8564	−0.640595
$778$	−43.1769	−1.54797
$779$	5.07180	0.181716
$780$	0	0
$781$	13.8564	0.495821
$782$	0	0
$783$	−3.46410	−0.123797
$784$	15.0000	0.535714
$785$	0	0
$786$	8.78461	0.313337
$787$	18.7846	0.669599	0.334800	−	0.942289i	$-0.391331\pi$
$787$	18.7846	0.669599	0.334800	+	0.942289i	$0.391331\pi$
$788$	12.0000	0.427482
$789$	27.4641	0.977748
$790$	0	0
$791$	1.85641	0.0660062
$792$	−1.73205	−0.0615457
$793$	2.92820	0.103984
$794$	3.46410	0.122936
$795$	0	0
$796$	16.7846	0.594915
$797$	−16.6410	−0.589455	−0.294728	−	0.955581i	$-0.595229\pi$
$797$	−16.6410	−0.589455	−0.294728	+	0.955581i	$0.595229\pi$
$798$	5.07180	0.179540
$799$	0	0
$800$	0	0
$801$	−12.9282	−0.456796
$802$	−34.3923	−1.21443
$803$	12.3923	0.437315
$804$	8.00000	0.282138
$805$	0	0
$806$	−7.42563	−0.261557
$807$	7.85641	0.276559
$808$	18.0000	0.633238
$809$	−8.53590	−0.300106	−0.150053	−	0.988678i	$-0.547944\pi$
$809$	−8.53590	−0.300106	−0.150053	+	0.988678i	$0.547944\pi$
$810$	0	0
$811$	−8.39230	−0.294694	−0.147347	−	0.989085i	$-0.547073\pi$
$811$	−8.39230	−0.294694	−0.147347	+	0.989085i	$0.547073\pi$
$812$	−6.92820	−0.243132
$813$	32.3923	1.13605
$814$	−15.4641	−0.542016
$815$	0	0
$816$	0	0
$817$	13.0718	0.457324
$818$	−60.2487	−2.10655
$819$	−2.92820	−0.102320
$820$	0	0
$821$	27.4641	0.958504	0.479252	−	0.877677i	$-0.340908\pi$
$821$	27.4641	0.958504	0.479252	+	0.877677i	$0.340908\pi$
$822$	31.1769	1.08742
$823$	−49.5692	−1.72787	−0.863937	−	0.503600i	$-0.832009\pi$
$823$	−49.5692	−1.72787	−0.863937	+	0.503600i	$0.832009\pi$
$824$	−13.8564	−0.482711
$825$	0	0
$826$	24.0000	0.835067
$827$	−1.60770	−0.0559050	−0.0279525	−	0.999609i	$-0.508899\pi$
$827$	−1.60770	−0.0559050	−0.0279525	+	0.999609i	$0.508899\pi$
$828$	6.92820	0.240772
$829$	−25.7128	−0.893043	−0.446521	−	0.894773i	$-0.647337\pi$
$829$	−25.7128	−0.893043	−0.446521	+	0.894773i	$0.647337\pi$
$830$	0	0
$831$	22.5359	0.781762
$832$	1.46410	0.0507586
$833$	0	0
$834$	−14.5359	−0.503337
$835$	0	0
$836$	1.46410	0.0506370
$837$	−2.92820	−0.101214
$838$	−29.5692	−1.02145
$839$	15.2154	0.525294	0.262647	−	0.964892i	$-0.415405\pi$
$839$	15.2154	0.525294	0.262647	+	0.964892i	$0.415405\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	−3.46410	−0.119381
$843$	3.46410	0.119310
$844$	12.3923	0.426561
$845$	0	0
$846$	12.0000	0.412568
$847$	−2.00000	−0.0687208
$848$	−64.6410	−2.21978
$849$	8.92820	0.306415
$850$	0	0
$851$	−61.8564	−2.12041
$852$	13.8564	0.474713
$853$	−24.3923	−0.835177	−0.417588	−	0.908636i	$-0.637125\pi$
$853$	−24.3923	−0.835177	−0.417588	+	0.908636i	$0.637125\pi$
$854$	6.92820	0.237078
$855$	0	0
$856$	−26.7846	−0.915479
$857$	10.1436	0.346499	0.173249	−	0.984878i	$-0.444573\pi$
$857$	10.1436	0.346499	0.173249	+	0.984878i	$0.444573\pi$
$858$	−2.53590	−0.0865741
$859$	47.7128	1.62794	0.813970	−	0.580907i	$-0.197302\pi$
$859$	47.7128	1.62794	0.813970	+	0.580907i	$0.197302\pi$
$860$	0	0
$861$	−6.92820	−0.236113
$862$	56.7846	1.93409
$863$	10.1436	0.345292	0.172646	−	0.984984i	$-0.444768\pi$
$863$	10.1436	0.345292	0.172646	+	0.984984i	$0.444768\pi$
$864$	−5.19615	−0.176777
$865$	0	0
$866$	48.2487	1.63956
$867$	17.0000	0.577350
$868$	−5.85641	−0.198779
$869$	13.4641	0.456738
$870$	0	0
$871$	−11.7128	−0.396874
$872$	−17.3205	−0.586546
$873$	10.0000	0.338449
$874$	17.5692	0.594288
$875$	0	0
$876$	12.3923	0.418697
$877$	−14.2487	−0.481145	−0.240572	−	0.970631i	$-0.577335\pi$
$877$	−14.2487	−0.481145	−0.240572	+	0.970631i	$0.577335\pi$
$878$	50.5359	1.70550
$879$	13.8564	0.467365
$880$	0	0
$881$	12.9282	0.435562	0.217781	−	0.975998i	$-0.430118\pi$
$881$	12.9282	0.435562	0.217781	+	0.975998i	$0.430118\pi$
$882$	5.19615	0.174964
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	0	0
$886$	20.7846	0.698273
$887$	36.2487	1.21711	0.608556	−	0.793511i	$-0.291749\pi$
$887$	36.2487	1.21711	0.608556	+	0.793511i	$0.291749\pi$
$888$	15.4641	0.518941
$889$	−9.85641	−0.330573
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	17.8564	0.597877
$893$	10.1436	0.339442
$894$	−14.7846	−0.494471
$895$	0	0
$896$	24.2487	0.810093
$897$	−10.1436	−0.338685
$898$	−25.6077	−0.854540
$899$	10.1436	0.338308
$900$	0	0
$901$	0	0
$902$	−6.00000	−0.199778
$903$	−17.8564	−0.594224
$904$	−1.60770	−0.0534711
$905$	0	0
$906$	0.679492	0.0225746
$907$	−45.8564	−1.52264	−0.761318	−	0.648378i	$-0.775448\pi$
$907$	−45.8564	−1.52264	−0.761318	+	0.648378i	$0.775448\pi$
$908$	−8.53590	−0.283274
$909$	10.3923	0.344691
$910$	0	0
$911$	5.07180	0.168036	0.0840181	−	0.996464i	$-0.473225\pi$
$911$	5.07180	0.168036	0.0840181	+	0.996464i	$0.473225\pi$
$912$	−7.32051	−0.242406
$913$	15.4641	0.511787
$914$	−14.5359	−0.480805
$915$	0	0
$916$	3.85641	0.127419
$917$	−10.1436	−0.334971
$918$	0	0
$919$	−11.6077	−0.382903	−0.191451	−	0.981502i	$-0.561319\pi$
$919$	−11.6077	−0.382903	−0.191451	+	0.981502i	$0.561319\pi$
$920$	0	0
$921$	14.0000	0.461316
$922$	21.2154	0.698692
$923$	−20.2872	−0.667761
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	−48.4974	−1.59372
$927$	−8.00000	−0.262754
$928$	18.0000	0.590879
$929$	−38.7846	−1.27248	−0.636241	−	0.771490i	$-0.719512\pi$
$929$	−38.7846	−1.27248	−0.636241	+	0.771490i	$0.719512\pi$
$930$	0	0
$931$	4.39230	0.143952
$932$	−12.0000	−0.393073
$933$	−18.9282	−0.619682
$934$	32.7846	1.07275
$935$	0	0
$936$	2.53590	0.0828884
$937$	−0.392305	−0.0128160	−0.00640802	−	0.999979i	$-0.502040\pi$
$937$	−0.392305	−0.0128160	−0.00640802	+	0.999979i	$0.502040\pi$
$938$	−27.7128	−0.904855
$939$	7.07180	0.230779
$940$	0	0
$941$	20.5359	0.669451	0.334726	−	0.942316i	$-0.391356\pi$
$941$	20.5359	0.669451	0.334726	+	0.942316i	$0.391356\pi$
$942$	29.3205	0.955314
$943$	−24.0000	−0.781548
$944$	−34.6410	−1.12747
$945$	0	0
$946$	−15.4641	−0.502781
$947$	5.07180	0.164811	0.0824056	−	0.996599i	$-0.473740\pi$
$947$	5.07180	0.164811	0.0824056	+	0.996599i	$0.473740\pi$
$948$	13.4641	0.437294
$949$	−18.1436	−0.588966
$950$	0	0
$951$	−11.0718	−0.359028
$952$	0	0
$953$	44.7846	1.45072	0.725358	−	0.688372i	$-0.241674\pi$
$953$	44.7846	1.45072	0.725358	+	0.688372i	$0.241674\pi$
$954$	−22.3923	−0.724978
$955$	0	0
$956$	−12.0000	−0.388108
$957$	3.46410	0.111979
$958$	−20.7846	−0.671520
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−22.4256	−0.723407
$962$	22.6410	0.729976
$963$	−15.4641	−0.498324
$964$	27.8564	0.897194
$965$	0	0
$966$	−24.0000	−0.772187
$967$	18.7846	0.604072	0.302036	−	0.953296i	$-0.402334\pi$
$967$	18.7846	0.604072	0.302036	+	0.953296i	$0.402334\pi$
$968$	1.73205	0.0556702
$969$	0	0
$970$	0	0
$971$	−1.85641	−0.0595749	−0.0297875	−	0.999556i	$-0.509483\pi$
$971$	−1.85641	−0.0595749	−0.0297875	+	0.999556i	$0.509483\pi$
$972$	−1.00000	−0.0320750
$973$	16.7846	0.538090
$974$	41.0718	1.31603
$975$	0	0
$976$	−10.0000	−0.320092
$977$	35.5692	1.13796	0.568980	−	0.822351i	$-0.307338\pi$
$977$	35.5692	1.13796	0.568980	+	0.822351i	$0.307338\pi$
$978$	30.9282	0.988975
$979$	12.9282	0.413187
$980$	0	0
$981$	−10.0000	−0.319275
$982$	29.5692	0.943592
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	0	0
$987$	−13.8564	−0.441054
$988$	−2.14359	−0.0681968
$989$	−61.8564	−1.96692
$990$	0	0
$991$	−48.7846	−1.54969	−0.774847	−	0.632149i	$-0.782173\pi$
$991$	−48.7846	−1.54969	−0.774847	+	0.632149i	$0.782173\pi$
$992$	15.2154	0.483089
$993$	17.8564	0.566656
$994$	−48.0000	−1.52247
$995$	0	0
$996$	15.4641	0.489999
$997$	−48.3923	−1.53260	−0.766300	−	0.642483i	$-0.777904\pi$
$997$	−48.3923	−1.53260	−0.766300	+	0.642483i	$0.777904\pi$
$998$	22.1436	0.700943
$999$	8.92820	0.282476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.a.e.1.1		2
3.2	odd	2		2475.2.a.r.1.2		2
5.2	odd	4		825.2.c.c.199.2		4
5.3	odd	4		825.2.c.c.199.3		4
5.4	even	2		165.2.a.b.1.2	✓	2
11.10	odd	2		9075.2.a.bh.1.2		2
15.2	even	4		2475.2.c.n.199.3		4
15.8	even	4		2475.2.c.n.199.2		4
15.14	odd	2		495.2.a.c.1.1		2
20.19	odd	2		2640.2.a.x.1.1		2
35.34	odd	2		8085.2.a.bd.1.2		2
55.54	odd	2		1815.2.a.i.1.1		2
60.59	even	2		7920.2.a.bz.1.1		2
165.164	even	2		5445.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.b.1.2	✓	2	5.4	even	2
495.2.a.c.1.1		2	15.14	odd	2
825.2.a.e.1.1		2	1.1	even	1	trivial
825.2.c.c.199.2		4	5.2	odd	4
825.2.c.c.199.3		4	5.3	odd	4
1815.2.a.i.1.1		2	55.54	odd	2
2475.2.a.r.1.2		2	3.2	odd	2
2475.2.c.n.199.2		4	15.8	even	4
2475.2.c.n.199.3		4	15.2	even	4
2640.2.a.x.1.1		2	20.19	odd	2
5445.2.a.s.1.2		2	165.164	even	2
7920.2.a.bz.1.1		2	60.59	even	2
8085.2.a.bd.1.2		2	35.34	odd	2
9075.2.a.bh.1.2		2	11.10	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} +1.46410 q^{13} +3.46410 q^{14} -5.00000 q^{16} -1.73205 q^{18} -1.46410 q^{19} +2.00000 q^{21} +1.73205 q^{22} +6.92820 q^{23} -1.73205 q^{24} -2.53590 q^{26} -1.00000 q^{27} -2.00000 q^{28} +3.46410 q^{29} +2.92820 q^{31} +5.19615 q^{32} +1.00000 q^{33} +1.00000 q^{36} -8.92820 q^{37} +2.53590 q^{38} -1.46410 q^{39} -3.46410 q^{41} -3.46410 q^{42} -8.92820 q^{43} -1.00000 q^{44} -12.0000 q^{46} -6.92820 q^{47} +5.00000 q^{48} -3.00000 q^{49} +1.46410 q^{52} +12.9282 q^{53} +1.73205 q^{54} -3.46410 q^{56} +1.46410 q^{57} -6.00000 q^{58} +6.92820 q^{59} +2.00000 q^{61} -5.07180 q^{62} -2.00000 q^{63} +1.00000 q^{64} -1.73205 q^{66} -8.00000 q^{67} -6.92820 q^{69} -13.8564 q^{71} +1.73205 q^{72} -12.3923 q^{73} +15.4641 q^{74} -1.46410 q^{76} +2.00000 q^{77} +2.53590 q^{78} -13.4641 q^{79} +1.00000 q^{81} +6.00000 q^{82} -15.4641 q^{83} +2.00000 q^{84} +15.4641 q^{86} -3.46410 q^{87} -1.73205 q^{88} -12.9282 q^{89} -2.92820 q^{91} +6.92820 q^{92} -2.92820 q^{93} +12.0000 q^{94} -5.19615 q^{96} +10.0000 q^{97} +5.19615 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^4 - 4 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{4} - 4 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{12} - 4 q^{13} - 10 q^{16} + 4 q^{19} + 4 q^{21} - 12 q^{26} - 2 q^{27} - 4 q^{28} - 8 q^{31} + 2 q^{33} + 2 q^{36} - 4 q^{37} + 12 q^{38} + 4 q^{39} - 4 q^{43} - 2 q^{44} - 24 q^{46} + 10 q^{48} - 6 q^{49} - 4 q^{52} + 12 q^{53} - 4 q^{57} - 12 q^{58} + 4 q^{61} - 24 q^{62} - 4 q^{63} + 2 q^{64} - 16 q^{67} - 4 q^{73} + 24 q^{74} + 4 q^{76} + 4 q^{77} + 12 q^{78} - 20 q^{79} + 2 q^{81} + 12 q^{82} - 24 q^{83} + 4 q^{84} + 24 q^{86} - 12 q^{89} + 8 q^{91} + 8 q^{93} + 24 q^{94} + 20 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 - 4 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^12 - 4 * q^13 - 10 * q^16 + 4 * q^19 + 4 * q^21 - 12 * q^26 - 2 * q^27 - 4 * q^28 - 8 * q^31 + 2 * q^33 + 2 * q^36 - 4 * q^37 + 12 * q^38 + 4 * q^39 - 4 * q^43 - 2 * q^44 - 24 * q^46 + 10 * q^48 - 6 * q^49 - 4 * q^52 + 12 * q^53 - 4 * q^57 - 12 * q^58 + 4 * q^61 - 24 * q^62 - 4 * q^63 + 2 * q^64 - 16 * q^67 - 4 * q^73 + 24 * q^74 + 4 * q^76 + 4 * q^77 + 12 * q^78 - 20 * q^79 + 2 * q^81 + 12 * q^82 - 24 * q^83 + 4 * q^84 + 24 * q^86 - 12 * q^89 + 8 * q^91 + 8 * q^93 + 24 * q^94 + 20 * q^97 - 2 * q^99

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