LMFDB - Embedded newform 1470.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,2,Mod(1,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, 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("1470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1470.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:	$11.7380090971$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.41421 q^{11} +1.00000 q^{12} +1.00000 q^{15} +1.00000 q^{16} -1.41421 q^{17} +1.00000 q^{18} +2.82843 q^{19} +1.00000 q^{20} +3.41421 q^{22} -0.828427 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -0.242641 q^{29} +1.00000 q^{30} -9.07107 q^{31} +1.00000 q^{32} +3.41421 q^{33} -1.41421 q^{34} +1.00000 q^{36} +1.41421 q^{37} +2.82843 q^{38} +1.00000 q^{40} -3.17157 q^{41} +7.41421 q^{43} +3.41421 q^{44} +1.00000 q^{45} -0.828427 q^{46} +5.07107 q^{47} +1.00000 q^{48} +1.00000 q^{50} -1.41421 q^{51} +13.3137 q^{53} +1.00000 q^{54} +3.41421 q^{55} +2.82843 q^{57} -0.242641 q^{58} -14.4853 q^{59} +1.00000 q^{60} -0.343146 q^{61} -9.07107 q^{62} +1.00000 q^{64} +3.41421 q^{66} -11.8995 q^{67} -1.41421 q^{68} -0.828427 q^{69} +5.17157 q^{71} +1.00000 q^{72} -3.65685 q^{73} +1.41421 q^{74} +1.00000 q^{75} +2.82843 q^{76} +11.3137 q^{79} +1.00000 q^{80} +1.00000 q^{81} -3.17157 q^{82} -10.8284 q^{83} -1.41421 q^{85} +7.41421 q^{86} -0.242641 q^{87} +3.41421 q^{88} +10.4853 q^{89} +1.00000 q^{90} -0.828427 q^{92} -9.07107 q^{93} +5.07107 q^{94} +2.82843 q^{95} +1.00000 q^{96} +3.17157 q^{97} +3.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 4 q^{11} + 2 q^{12} + 2 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{20} + 4 q^{22} + 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 4 * q^11 + 2 * q^12 + 2 * q^15 + 2 * q^16 + 2 * q^18 + 2 * q^20 + 4 * q^22 + 4 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^27 + 8 * q^29 + 2 * q^30 - 4 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^36 + 2 * q^40 - 12 * q^41 + 12 * q^43 + 4 * q^44 + 2 * q^45 + 4 * q^46 - 4 * q^47 + 2 * q^48 + 2 * q^50 + 4 * q^53 + 2 * q^54 + 4 * q^55 + 8 * q^58 - 12 * q^59 + 2 * q^60 - 12 * q^61 - 4 * q^62 + 2 * q^64 + 4 * q^66 - 4 * q^67 + 4 * q^69 + 16 * q^71 + 2 * q^72 + 4 * q^73 + 2 * q^75 + 2 * q^80 + 2 * q^81 - 12 * q^82 - 16 * q^83 + 12 * q^86 + 8 * q^87 + 4 * q^88 + 4 * q^89 + 2 * q^90 + 4 * q^92 - 4 * q^93 - 4 * q^94 + 2 * q^96 + 12 * q^97 + 4 * q^99

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	3.41421	1.02942	0.514712	−	0.857363i	$-0.327899\pi$
$11$	3.41421	1.02942	0.514712	+	0.857363i	$0.327899\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	1.00000	0.235702
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	3.41421	0.727913
$23$	−0.828427	−0.172739	−0.0863695	−	0.996263i	$-0.527527\pi$
$23$	−0.828427	−0.172739	−0.0863695	+	0.996263i	$0.527527\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.242641	−0.0450572	−0.0225286	−	0.999746i	$-0.507172\pi$
$29$	−0.242641	−0.0450572	−0.0225286	+	0.999746i	$0.507172\pi$
$30$	1.00000	0.182574
$31$	−9.07107	−1.62921	−0.814606	−	0.580015i	$-0.803047\pi$
$31$	−9.07107	−1.62921	−0.814606	+	0.580015i	$0.803047\pi$
$32$	1.00000	0.176777
$33$	3.41421	0.594338
$34$	−1.41421	−0.242536
$35$	0	0
$36$	1.00000	0.166667
$37$	1.41421	0.232495	0.116248	−	0.993220i	$-0.462913\pi$
$37$	1.41421	0.232495	0.116248	+	0.993220i	$0.462913\pi$
$38$	2.82843	0.458831
$39$	0	0
$40$	1.00000	0.158114
$41$	−3.17157	−0.495316	−0.247658	−	0.968847i	$-0.579661\pi$
$41$	−3.17157	−0.495316	−0.247658	+	0.968847i	$0.579661\pi$
$42$	0	0
$43$	7.41421	1.13066	0.565328	−	0.824866i	$-0.308749\pi$
$43$	7.41421	1.13066	0.565328	+	0.824866i	$0.308749\pi$
$44$	3.41421	0.514712
$45$	1.00000	0.149071
$46$	−0.828427	−0.122145
$47$	5.07107	0.739691	0.369846	−	0.929093i	$-0.379411\pi$
$47$	5.07107	0.739691	0.369846	+	0.929093i	$0.379411\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	−1.41421	−0.198030
$52$	0	0
$53$	13.3137	1.82878	0.914389	−	0.404836i	$-0.132671\pi$
$53$	13.3137	1.82878	0.914389	+	0.404836i	$0.132671\pi$
$54$	1.00000	0.136083
$55$	3.41421	0.460372
$56$	0	0
$57$	2.82843	0.374634
$58$	−0.242641	−0.0318603
$59$	−14.4853	−1.88582	−0.942912	−	0.333043i	$-0.891924\pi$
$59$	−14.4853	−1.88582	−0.942912	+	0.333043i	$0.891924\pi$
$60$	1.00000	0.129099
$61$	−0.343146	−0.0439353	−0.0219677	−	0.999759i	$-0.506993\pi$
$61$	−0.343146	−0.0439353	−0.0219677	+	0.999759i	$0.506993\pi$
$62$	−9.07107	−1.15203
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	3.41421	0.420261
$67$	−11.8995	−1.45375	−0.726877	−	0.686767i	$-0.759029\pi$
$67$	−11.8995	−1.45375	−0.726877	+	0.686767i	$0.759029\pi$
$68$	−1.41421	−0.171499
$69$	−0.828427	−0.0997309
$70$	0	0
$71$	5.17157	0.613753	0.306876	−	0.951749i	$-0.400716\pi$
$71$	5.17157	0.613753	0.306876	+	0.951749i	$0.400716\pi$
$72$	1.00000	0.117851
$73$	−3.65685	−0.428002	−0.214001	−	0.976833i	$-0.568650\pi$
$73$	−3.65685	−0.428002	−0.214001	+	0.976833i	$0.568650\pi$
$74$	1.41421	0.164399
$75$	1.00000	0.115470
$76$	2.82843	0.324443
$77$	0	0
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−3.17157	−0.350242
$83$	−10.8284	−1.18857	−0.594287	−	0.804253i	$-0.702566\pi$
$83$	−10.8284	−1.18857	−0.594287	+	0.804253i	$0.702566\pi$
$84$	0	0
$85$	−1.41421	−0.153393
$86$	7.41421	0.799495
$87$	−0.242641	−0.0260138
$88$	3.41421	0.363956
$89$	10.4853	1.11144	0.555719	−	0.831370i	$-0.312443\pi$
$89$	10.4853	1.11144	0.555719	+	0.831370i	$0.312443\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−0.828427	−0.0863695
$93$	−9.07107	−0.940626
$94$	5.07107	0.523041
$95$	2.82843	0.290191
$96$	1.00000	0.102062
$97$	3.17157	0.322024	0.161012	−	0.986952i	$-0.448524\pi$
$97$	3.17157	0.322024	0.161012	+	0.986952i	$0.448524\pi$
$98$	0	0
$99$	3.41421	0.343141
$100$	1.00000	0.100000
$101$	−17.6569	−1.75692	−0.878461	−	0.477813i	$-0.841429\pi$
$101$	−17.6569	−1.75692	−0.878461	+	0.477813i	$0.841429\pi$
$102$	−1.41421	−0.140028
$103$	18.1421	1.78760	0.893799	−	0.448468i	$-0.148030\pi$
$103$	18.1421	1.78760	0.893799	+	0.448468i	$0.148030\pi$
$104$	0	0
$105$	0	0
$106$	13.3137	1.29314
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	−17.3137	−1.65835	−0.829176	−	0.558987i	$-0.811190\pi$
$109$	−17.3137	−1.65835	−0.829176	+	0.558987i	$0.811190\pi$
$110$	3.41421	0.325532
$111$	1.41421	0.134231
$112$	0	0
$113$	−13.3137	−1.25245	−0.626224	−	0.779643i	$-0.715401\pi$
$113$	−13.3137	−1.25245	−0.626224	+	0.779643i	$0.715401\pi$
$114$	2.82843	0.264906
$115$	−0.828427	−0.0772512
$116$	−0.242641	−0.0225286
$117$	0	0
$118$	−14.4853	−1.33348
$119$	0	0
$120$	1.00000	0.0912871
$121$	0.656854	0.0597140
$122$	−0.343146	−0.0310670
$123$	−3.17157	−0.285971
$124$	−9.07107	−0.814606
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.1421	1.25491	0.627456	−	0.778652i	$-0.284096\pi$
$127$	14.1421	1.25491	0.627456	+	0.778652i	$0.284096\pi$
$128$	1.00000	0.0883883
$129$	7.41421	0.652785
$130$	0	0
$131$	−11.1716	−0.976065	−0.488032	−	0.872825i	$-0.662285\pi$
$131$	−11.1716	−0.976065	−0.488032	+	0.872825i	$0.662285\pi$
$132$	3.41421	0.297169
$133$	0	0
$134$	−11.8995	−1.02796
$135$	1.00000	0.0860663
$136$	−1.41421	−0.121268
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	−0.828427	−0.0705204
$139$	6.34315	0.538019	0.269009	−	0.963138i	$-0.413304\pi$
$139$	6.34315	0.538019	0.269009	+	0.963138i	$0.413304\pi$
$140$	0	0
$141$	5.07107	0.427061
$142$	5.17157	0.433989
$143$	0	0
$144$	1.00000	0.0833333
$145$	−0.242641	−0.0201502
$146$	−3.65685	−0.302643
$147$	0	0
$148$	1.41421	0.116248
$149$	5.89949	0.483305	0.241653	−	0.970363i	$-0.422311\pi$
$149$	5.89949	0.483305	0.241653	+	0.970363i	$0.422311\pi$
$150$	1.00000	0.0816497
$151$	−21.7990	−1.77398	−0.886988	−	0.461792i	$-0.847207\pi$
$151$	−21.7990	−1.77398	−0.886988	+	0.461792i	$0.847207\pi$
$152$	2.82843	0.229416
$153$	−1.41421	−0.114332
$154$	0	0
$155$	−9.07107	−0.728606
$156$	0	0
$157$	11.6569	0.930318	0.465159	−	0.885227i	$-0.345997\pi$
$157$	11.6569	0.930318	0.465159	+	0.885227i	$0.345997\pi$
$158$	11.3137	0.900070
$159$	13.3137	1.05585
$160$	1.00000	0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−10.7279	−0.840276	−0.420138	−	0.907460i	$-0.638018\pi$
$163$	−10.7279	−0.840276	−0.420138	+	0.907460i	$0.638018\pi$
$164$	−3.17157	−0.247658
$165$	3.41421	0.265796
$166$	−10.8284	−0.840449
$167$	−13.0711	−1.01147	−0.505735	−	0.862689i	$-0.668779\pi$
$167$	−13.0711	−1.01147	−0.505735	+	0.862689i	$0.668779\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−1.41421	−0.108465
$171$	2.82843	0.216295
$172$	7.41421	0.565328
$173$	1.51472	0.115162	0.0575810	−	0.998341i	$-0.481661\pi$
$173$	1.51472	0.115162	0.0575810	+	0.998341i	$0.481661\pi$
$174$	−0.242641	−0.0183945
$175$	0	0
$176$	3.41421	0.257356
$177$	−14.4853	−1.08878
$178$	10.4853	0.785905
$179$	3.41421	0.255190	0.127595	−	0.991826i	$-0.459274\pi$
$179$	3.41421	0.255190	0.127595	+	0.991826i	$0.459274\pi$
$180$	1.00000	0.0745356
$181$	−3.17157	−0.235741	−0.117871	−	0.993029i	$-0.537607\pi$
$181$	−3.17157	−0.235741	−0.117871	+	0.993029i	$0.537607\pi$
$182$	0	0
$183$	−0.343146	−0.0253661
$184$	−0.828427	−0.0610725
$185$	1.41421	0.103975
$186$	−9.07107	−0.665123
$187$	−4.82843	−0.353090
$188$	5.07107	0.369846
$189$	0	0
$190$	2.82843	0.205196
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	1.00000	0.0721688
$193$	17.7990	1.28120	0.640600	−	0.767875i	$-0.278686\pi$
$193$	17.7990	1.28120	0.640600	+	0.767875i	$0.278686\pi$
$194$	3.17157	0.227706
$195$	0	0
$196$	0	0
$197$	−15.1716	−1.08093	−0.540465	−	0.841367i	$-0.681752\pi$
$197$	−15.1716	−1.08093	−0.540465	+	0.841367i	$0.681752\pi$
$198$	3.41421	0.242638
$199$	17.0711	1.21014	0.605068	−	0.796174i	$-0.293146\pi$
$199$	17.0711	1.21014	0.605068	+	0.796174i	$0.293146\pi$
$200$	1.00000	0.0707107
$201$	−11.8995	−0.839326
$202$	−17.6569	−1.24233
$203$	0	0
$204$	−1.41421	−0.0990148
$205$	−3.17157	−0.221512
$206$	18.1421	1.26402
$207$	−0.828427	−0.0575797
$208$	0	0
$209$	9.65685	0.667979
$210$	0	0
$211$	9.65685	0.664805	0.332403	−	0.943138i	$-0.392141\pi$
$211$	9.65685	0.664805	0.332403	+	0.943138i	$0.392141\pi$
$212$	13.3137	0.914389
$213$	5.17157	0.354350
$214$	−4.00000	−0.273434
$215$	7.41421	0.505645
$216$	1.00000	0.0680414
$217$	0	0
$218$	−17.3137	−1.17263
$219$	−3.65685	−0.247107
$220$	3.41421	0.230186
$221$	0	0
$222$	1.41421	0.0949158
$223$	6.34315	0.424768	0.212384	−	0.977186i	$-0.431877\pi$
$223$	6.34315	0.424768	0.212384	+	0.977186i	$0.431877\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−13.3137	−0.885615
$227$	−2.34315	−0.155520	−0.0777600	−	0.996972i	$-0.524777\pi$
$227$	−2.34315	−0.155520	−0.0777600	+	0.996972i	$0.524777\pi$
$228$	2.82843	0.187317
$229$	−13.7990	−0.911863	−0.455931	−	0.890015i	$-0.650694\pi$
$229$	−13.7990	−0.911863	−0.455931	+	0.890015i	$0.650694\pi$
$230$	−0.828427	−0.0546249
$231$	0	0
$232$	−0.242641	−0.0159301
$233$	−26.9706	−1.76690	−0.883450	−	0.468525i	$-0.844786\pi$
$233$	−26.9706	−1.76690	−0.883450	+	0.468525i	$0.844786\pi$
$234$	0	0
$235$	5.07107	0.330800
$236$	−14.4853	−0.942912
$237$	11.3137	0.734904
$238$	0	0
$239$	0.686292	0.0443925	0.0221963	−	0.999754i	$-0.492934\pi$
$239$	0.686292	0.0443925	0.0221963	+	0.999754i	$0.492934\pi$
$240$	1.00000	0.0645497
$241$	−7.75736	−0.499695	−0.249848	−	0.968285i	$-0.580381\pi$
$241$	−7.75736	−0.499695	−0.249848	+	0.968285i	$0.580381\pi$
$242$	0.656854	0.0422242
$243$	1.00000	0.0641500
$244$	−0.343146	−0.0219677
$245$	0	0
$246$	−3.17157	−0.202212
$247$	0	0
$248$	−9.07107	−0.576013
$249$	−10.8284	−0.686224
$250$	1.00000	0.0632456
$251$	−6.34315	−0.400376	−0.200188	−	0.979758i	$-0.564155\pi$
$251$	−6.34315	−0.400376	−0.200188	+	0.979758i	$0.564155\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	14.1421	0.887357
$255$	−1.41421	−0.0885615
$256$	1.00000	0.0625000
$257$	−17.8995	−1.11654	−0.558270	−	0.829659i	$-0.688535\pi$
$257$	−17.8995	−1.11654	−0.558270	+	0.829659i	$0.688535\pi$
$258$	7.41421	0.461589
$259$	0	0
$260$	0	0
$261$	−0.242641	−0.0150191
$262$	−11.1716	−0.690182
$263$	17.7990	1.09753	0.548766	−	0.835976i	$-0.315098\pi$
$263$	17.7990	1.09753	0.548766	+	0.835976i	$0.315098\pi$
$264$	3.41421	0.210130
$265$	13.3137	0.817855
$266$	0	0
$267$	10.4853	0.641689
$268$	−11.8995	−0.726877
$269$	−3.65685	−0.222962	−0.111481	−	0.993767i	$-0.535559\pi$
$269$	−3.65685	−0.222962	−0.111481	+	0.993767i	$0.535559\pi$
$270$	1.00000	0.0608581
$271$	−5.75736	−0.349735	−0.174867	−	0.984592i	$-0.555950\pi$
$271$	−5.75736	−0.349735	−0.174867	+	0.984592i	$0.555950\pi$
$272$	−1.41421	−0.0857493
$273$	0	0
$274$	−16.0000	−0.966595
$275$	3.41421	0.205885
$276$	−0.828427	−0.0498655
$277$	2.10051	0.126207	0.0631036	−	0.998007i	$-0.479900\pi$
$277$	2.10051	0.126207	0.0631036	+	0.998007i	$0.479900\pi$
$278$	6.34315	0.380437
$279$	−9.07107	−0.543071
$280$	0	0
$281$	−30.4853	−1.81860	−0.909300	−	0.416142i	$-0.863382\pi$
$281$	−30.4853	−1.81860	−0.909300	+	0.416142i	$0.863382\pi$
$282$	5.07107	0.301978
$283$	13.5147	0.803367	0.401683	−	0.915779i	$-0.368425\pi$
$283$	13.5147	0.803367	0.401683	+	0.915779i	$0.368425\pi$
$284$	5.17157	0.306876
$285$	2.82843	0.167542
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	−15.0000	−0.882353
$290$	−0.242641	−0.0142484
$291$	3.17157	0.185921
$292$	−3.65685	−0.214001
$293$	16.6274	0.971384	0.485692	−	0.874130i	$-0.338568\pi$
$293$	16.6274	0.971384	0.485692	+	0.874130i	$0.338568\pi$
$294$	0	0
$295$	−14.4853	−0.843366
$296$	1.41421	0.0821995
$297$	3.41421	0.198113
$298$	5.89949	0.341749
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	−21.7990	−1.25439
$303$	−17.6569	−1.01436
$304$	2.82843	0.162221
$305$	−0.343146	−0.0196485
$306$	−1.41421	−0.0808452
$307$	15.3137	0.874000	0.437000	−	0.899462i	$-0.356041\pi$
$307$	15.3137	0.874000	0.437000	+	0.899462i	$0.356041\pi$
$308$	0	0
$309$	18.1421	1.03207
$310$	−9.07107	−0.515202
$311$	−34.8284	−1.97494	−0.987469	−	0.157810i	$-0.949557\pi$
$311$	−34.8284	−1.97494	−0.987469	+	0.157810i	$0.949557\pi$
$312$	0	0
$313$	23.1716	1.30973	0.654867	−	0.755744i	$-0.272724\pi$
$313$	23.1716	1.30973	0.654867	+	0.755744i	$0.272724\pi$
$314$	11.6569	0.657834
$315$	0	0
$316$	11.3137	0.636446
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	13.3137	0.746596
$319$	−0.828427	−0.0463830
$320$	1.00000	0.0559017
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	−10.7279	−0.594165
$327$	−17.3137	−0.957450
$328$	−3.17157	−0.175121
$329$	0	0
$330$	3.41421	0.187946
$331$	26.6274	1.46358	0.731788	−	0.681533i	$-0.238686\pi$
$331$	26.6274	1.46358	0.731788	+	0.681533i	$0.238686\pi$
$332$	−10.8284	−0.594287
$333$	1.41421	0.0774984
$334$	−13.0711	−0.715217
$335$	−11.8995	−0.650139
$336$	0	0
$337$	−8.82843	−0.480915	−0.240458	−	0.970660i	$-0.577297\pi$
$337$	−8.82843	−0.480915	−0.240458	+	0.970660i	$0.577297\pi$
$338$	−13.0000	−0.707107
$339$	−13.3137	−0.723101
$340$	−1.41421	−0.0766965
$341$	−30.9706	−1.67715
$342$	2.82843	0.152944
$343$	0	0
$344$	7.41421	0.399748
$345$	−0.828427	−0.0446010
$346$	1.51472	0.0814318
$347$	−9.65685	−0.518407	−0.259204	−	0.965823i	$-0.583460\pi$
$347$	−9.65685	−0.518407	−0.259204	+	0.965823i	$0.583460\pi$
$348$	−0.242641	−0.0130069
$349$	33.7990	1.80922	0.904609	−	0.426242i	$-0.140163\pi$
$349$	33.7990	1.80922	0.904609	+	0.426242i	$0.140163\pi$
$350$	0	0
$351$	0	0
$352$	3.41421	0.181978
$353$	24.7279	1.31613	0.658067	−	0.752959i	$-0.271374\pi$
$353$	24.7279	1.31613	0.658067	+	0.752959i	$0.271374\pi$
$354$	−14.4853	−0.769884
$355$	5.17157	0.274479
$356$	10.4853	0.555719
$357$	0	0
$358$	3.41421	0.180447
$359$	13.1716	0.695169	0.347585	−	0.937649i	$-0.387002\pi$
$359$	13.1716	0.695169	0.347585	+	0.937649i	$0.387002\pi$
$360$	1.00000	0.0527046
$361$	−11.0000	−0.578947
$362$	−3.17157	−0.166694
$363$	0.656854	0.0344759
$364$	0	0
$365$	−3.65685	−0.191408
$366$	−0.343146	−0.0179365
$367$	−0.485281	−0.0253315	−0.0126657	−	0.999920i	$-0.504032\pi$
$367$	−0.485281	−0.0253315	−0.0126657	+	0.999920i	$0.504032\pi$
$368$	−0.828427	−0.0431847
$369$	−3.17157	−0.165105
$370$	1.41421	0.0735215
$371$	0	0
$372$	−9.07107	−0.470313
$373$	−3.07107	−0.159014	−0.0795069	−	0.996834i	$-0.525335\pi$
$373$	−3.07107	−0.159014	−0.0795069	+	0.996834i	$0.525335\pi$
$374$	−4.82843	−0.249672
$375$	1.00000	0.0516398
$376$	5.07107	0.261520
$377$	0	0
$378$	0	0
$379$	−7.51472	−0.386005	−0.193003	−	0.981198i	$-0.561823\pi$
$379$	−7.51472	−0.386005	−0.193003	+	0.981198i	$0.561823\pi$
$380$	2.82843	0.145095
$381$	14.1421	0.724524
$382$	11.3137	0.578860
$383$	−3.89949	−0.199255	−0.0996274	−	0.995025i	$-0.531765\pi$
$383$	−3.89949	−0.199255	−0.0996274	+	0.995025i	$0.531765\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	17.7990	0.905945
$387$	7.41421	0.376886
$388$	3.17157	0.161012
$389$	12.9289	0.655523	0.327761	−	0.944761i	$-0.393706\pi$
$389$	12.9289	0.655523	0.327761	+	0.944761i	$0.393706\pi$
$390$	0	0
$391$	1.17157	0.0592490
$392$	0	0
$393$	−11.1716	−0.563531
$394$	−15.1716	−0.764333
$395$	11.3137	0.569254
$396$	3.41421	0.171571
$397$	−24.6274	−1.23601	−0.618007	−	0.786172i	$-0.712060\pi$
$397$	−24.6274	−1.23601	−0.618007	+	0.786172i	$0.712060\pi$
$398$	17.0711	0.855695
$399$	0	0
$400$	1.00000	0.0500000
$401$	−2.68629	−0.134147	−0.0670735	−	0.997748i	$-0.521366\pi$
$401$	−2.68629	−0.134147	−0.0670735	+	0.997748i	$0.521366\pi$
$402$	−11.8995	−0.593493
$403$	0	0
$404$	−17.6569	−0.878461
$405$	1.00000	0.0496904
$406$	0	0
$407$	4.82843	0.239336
$408$	−1.41421	−0.0700140
$409$	18.3848	0.909069	0.454534	−	0.890729i	$-0.349806\pi$
$409$	18.3848	0.909069	0.454534	+	0.890729i	$0.349806\pi$
$410$	−3.17157	−0.156633
$411$	−16.0000	−0.789222
$412$	18.1421	0.893799
$413$	0	0
$414$	−0.828427	−0.0407150
$415$	−10.8284	−0.531547
$416$	0	0
$417$	6.34315	0.310625
$418$	9.65685	0.472332
$419$	−6.34315	−0.309883	−0.154941	−	0.987924i	$-0.549519\pi$
$419$	−6.34315	−0.309883	−0.154941	+	0.987924i	$0.549519\pi$
$420$	0	0
$421$	−28.6274	−1.39521	−0.697607	−	0.716480i	$-0.745752\pi$
$421$	−28.6274	−1.39521	−0.697607	+	0.716480i	$0.745752\pi$
$422$	9.65685	0.470088
$423$	5.07107	0.246564
$424$	13.3137	0.646571
$425$	−1.41421	−0.0685994
$426$	5.17157	0.250564
$427$	0	0
$428$	−4.00000	−0.193347
$429$	0	0
$430$	7.41421	0.357545
$431$	−14.1421	−0.681203	−0.340601	−	0.940208i	$-0.610631\pi$
$431$	−14.1421	−0.681203	−0.340601	+	0.940208i	$0.610631\pi$
$432$	1.00000	0.0481125
$433$	3.17157	0.152416	0.0762080	−	0.997092i	$-0.475719\pi$
$433$	3.17157	0.152416	0.0762080	+	0.997092i	$0.475719\pi$
$434$	0	0
$435$	−0.242641	−0.0116337
$436$	−17.3137	−0.829176
$437$	−2.34315	−0.112088
$438$	−3.65685	−0.174731
$439$	2.44365	0.116629	0.0583145	−	0.998298i	$-0.481427\pi$
$439$	2.44365	0.116629	0.0583145	+	0.998298i	$0.481427\pi$
$440$	3.41421	0.162766
$441$	0	0
$442$	0	0
$443$	2.34315	0.111326	0.0556631	−	0.998450i	$-0.482273\pi$
$443$	2.34315	0.111326	0.0556631	+	0.998450i	$0.482273\pi$
$444$	1.41421	0.0671156
$445$	10.4853	0.497050
$446$	6.34315	0.300357
$447$	5.89949	0.279037
$448$	0	0
$449$	37.1127	1.75146	0.875728	−	0.482804i	$-0.160382\pi$
$449$	37.1127	1.75146	0.875728	+	0.482804i	$0.160382\pi$
$450$	1.00000	0.0471405
$451$	−10.8284	−0.509891
$452$	−13.3137	−0.626224
$453$	−21.7990	−1.02421
$454$	−2.34315	−0.109969
$455$	0	0
$456$	2.82843	0.132453
$457$	29.7990	1.39394	0.696969	−	0.717101i	$-0.254532\pi$
$457$	29.7990	1.39394	0.696969	+	0.717101i	$0.254532\pi$
$458$	−13.7990	−0.644784
$459$	−1.41421	−0.0660098
$460$	−0.828427	−0.0386256
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	22.3431	1.03837	0.519187	−	0.854661i	$-0.326235\pi$
$463$	22.3431	1.03837	0.519187	+	0.854661i	$0.326235\pi$
$464$	−0.242641	−0.0112643
$465$	−9.07107	−0.420661
$466$	−26.9706	−1.24939
$467$	−3.51472	−0.162642	−0.0813209	−	0.996688i	$-0.525914\pi$
$467$	−3.51472	−0.162642	−0.0813209	+	0.996688i	$0.525914\pi$
$468$	0	0
$469$	0	0
$470$	5.07107	0.233911
$471$	11.6569	0.537119
$472$	−14.4853	−0.666739
$473$	25.3137	1.16393
$474$	11.3137	0.519656
$475$	2.82843	0.129777
$476$	0	0
$477$	13.3137	0.609593
$478$	0.686292	0.0313902
$479$	−0.485281	−0.0221731	−0.0110865	−	0.999939i	$-0.503529\pi$
$479$	−0.485281	−0.0221731	−0.0110865	+	0.999939i	$0.503529\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−7.75736	−0.353338
$483$	0	0
$484$	0.656854	0.0298570
$485$	3.17157	0.144014
$486$	1.00000	0.0453609
$487$	7.51472	0.340524	0.170262	−	0.985399i	$-0.445539\pi$
$487$	7.51472	0.340524	0.170262	+	0.985399i	$0.445539\pi$
$488$	−0.343146	−0.0155335
$489$	−10.7279	−0.485133
$490$	0	0
$491$	−7.89949	−0.356499	−0.178250	−	0.983985i	$-0.557043\pi$
$491$	−7.89949	−0.356499	−0.178250	+	0.983985i	$0.557043\pi$
$492$	−3.17157	−0.142986
$493$	0.343146	0.0154545
$494$	0	0
$495$	3.41421	0.153457
$496$	−9.07107	−0.407303
$497$	0	0
$498$	−10.8284	−0.485233
$499$	−23.7990	−1.06539	−0.532695	−	0.846308i	$-0.678821\pi$
$499$	−23.7990	−1.06539	−0.532695	+	0.846308i	$0.678821\pi$
$500$	1.00000	0.0447214
$501$	−13.0711	−0.583972
$502$	−6.34315	−0.283108
$503$	4.10051	0.182832	0.0914162	−	0.995813i	$-0.470861\pi$
$503$	4.10051	0.182832	0.0914162	+	0.995813i	$0.470861\pi$
$504$	0	0
$505$	−17.6569	−0.785720
$506$	−2.82843	−0.125739
$507$	−13.0000	−0.577350
$508$	14.1421	0.627456
$509$	−29.6569	−1.31452	−0.657258	−	0.753665i	$-0.728284\pi$
$509$	−29.6569	−1.31452	−0.657258	+	0.753665i	$0.728284\pi$
$510$	−1.41421	−0.0626224
$511$	0	0
$512$	1.00000	0.0441942
$513$	2.82843	0.124878
$514$	−17.8995	−0.789513
$515$	18.1421	0.799438
$516$	7.41421	0.326393
$517$	17.3137	0.761456
$518$	0	0
$519$	1.51472	0.0664888
$520$	0	0
$521$	16.1421	0.707200	0.353600	−	0.935397i	$-0.384957\pi$
$521$	16.1421	0.707200	0.353600	+	0.935397i	$0.384957\pi$
$522$	−0.242641	−0.0106201
$523$	41.1127	1.79773	0.898866	−	0.438223i	$-0.144392\pi$
$523$	41.1127	1.79773	0.898866	+	0.438223i	$0.144392\pi$
$524$	−11.1716	−0.488032
$525$	0	0
$526$	17.7990	0.776073
$527$	12.8284	0.558815
$528$	3.41421	0.148585
$529$	−22.3137	−0.970161
$530$	13.3137	0.578311
$531$	−14.4853	−0.628608
$532$	0	0
$533$	0	0
$534$	10.4853	0.453743
$535$	−4.00000	−0.172935
$536$	−11.8995	−0.513980
$537$	3.41421	0.147334
$538$	−3.65685	−0.157658
$539$	0	0
$540$	1.00000	0.0430331
$541$	14.4853	0.622771	0.311385	−	0.950284i	$-0.399207\pi$
$541$	14.4853	0.622771	0.311385	+	0.950284i	$0.399207\pi$
$542$	−5.75736	−0.247300
$543$	−3.17157	−0.136105
$544$	−1.41421	−0.0606339
$545$	−17.3137	−0.741638
$546$	0	0
$547$	24.5858	1.05121	0.525606	−	0.850728i	$-0.323839\pi$
$547$	24.5858	1.05121	0.525606	+	0.850728i	$0.323839\pi$
$548$	−16.0000	−0.683486
$549$	−0.343146	−0.0146451
$550$	3.41421	0.145583
$551$	−0.686292	−0.0292370
$552$	−0.828427	−0.0352602
$553$	0	0
$554$	2.10051	0.0892419
$555$	1.41421	0.0600300
$556$	6.34315	0.269009
$557$	−32.6274	−1.38247	−0.691234	−	0.722631i	$-0.742933\pi$
$557$	−32.6274	−1.38247	−0.691234	+	0.722631i	$0.742933\pi$
$558$	−9.07107	−0.384009
$559$	0	0
$560$	0	0
$561$	−4.82843	−0.203856
$562$	−30.4853	−1.28594
$563$	2.14214	0.0902803	0.0451401	−	0.998981i	$-0.485627\pi$
$563$	2.14214	0.0902803	0.0451401	+	0.998981i	$0.485627\pi$
$564$	5.07107	0.213530
$565$	−13.3137	−0.560112
$566$	13.5147	0.568066
$567$	0	0
$568$	5.17157	0.216994
$569$	1.02944	0.0431563	0.0215781	−	0.999767i	$-0.493131\pi$
$569$	1.02944	0.0431563	0.0215781	+	0.999767i	$0.493131\pi$
$570$	2.82843	0.118470
$571$	27.3137	1.14304	0.571522	−	0.820587i	$-0.306353\pi$
$571$	27.3137	1.14304	0.571522	+	0.820587i	$0.306353\pi$
$572$	0	0
$573$	11.3137	0.472637
$574$	0	0
$575$	−0.828427	−0.0345478
$576$	1.00000	0.0416667
$577$	16.1421	0.672006	0.336003	−	0.941861i	$-0.390925\pi$
$577$	16.1421	0.672006	0.336003	+	0.941861i	$0.390925\pi$
$578$	−15.0000	−0.623918
$579$	17.7990	0.739701
$580$	−0.242641	−0.0100751
$581$	0	0
$582$	3.17157	0.131466
$583$	45.4558	1.88259
$584$	−3.65685	−0.151322
$585$	0	0
$586$	16.6274	0.686872
$587$	−19.7990	−0.817192	−0.408596	−	0.912715i	$-0.633981\pi$
$587$	−19.7990	−0.817192	−0.408596	+	0.912715i	$0.633981\pi$
$588$	0	0
$589$	−25.6569	−1.05717
$590$	−14.4853	−0.596350
$591$	−15.1716	−0.624075
$592$	1.41421	0.0581238
$593$	−5.21320	−0.214081	−0.107040	−	0.994255i	$-0.534137\pi$
$593$	−5.21320	−0.214081	−0.107040	+	0.994255i	$0.534137\pi$
$594$	3.41421	0.140087
$595$	0	0
$596$	5.89949	0.241653
$597$	17.0711	0.698672
$598$	0	0
$599$	27.3137	1.11601	0.558004	−	0.829838i	$-0.311567\pi$
$599$	27.3137	1.11601	0.558004	+	0.829838i	$0.311567\pi$
$600$	1.00000	0.0408248
$601$	−34.1838	−1.39438	−0.697192	−	0.716884i	$-0.745568\pi$
$601$	−34.1838	−1.39438	−0.697192	+	0.716884i	$0.745568\pi$
$602$	0	0
$603$	−11.8995	−0.484585
$604$	−21.7990	−0.886988
$605$	0.656854	0.0267049
$606$	−17.6569	−0.717261
$607$	28.9706	1.17588	0.587939	−	0.808905i	$-0.299939\pi$
$607$	28.9706	1.17588	0.587939	+	0.808905i	$0.299939\pi$
$608$	2.82843	0.114708
$609$	0	0
$610$	−0.343146	−0.0138936
$611$	0	0
$612$	−1.41421	−0.0571662
$613$	37.2132	1.50303	0.751514	−	0.659718i	$-0.229324\pi$
$613$	37.2132	1.50303	0.751514	+	0.659718i	$0.229324\pi$
$614$	15.3137	0.618011
$615$	−3.17157	−0.127890
$616$	0	0
$617$	12.2843	0.494546	0.247273	−	0.968946i	$-0.420466\pi$
$617$	12.2843	0.494546	0.247273	+	0.968946i	$0.420466\pi$
$618$	18.1421	0.729784
$619$	30.1421	1.21151	0.605757	−	0.795649i	$-0.292870\pi$
$619$	30.1421	1.21151	0.605757	+	0.795649i	$0.292870\pi$
$620$	−9.07107	−0.364303
$621$	−0.828427	−0.0332436
$622$	−34.8284	−1.39649
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	23.1716	0.926122
$627$	9.65685	0.385658
$628$	11.6569	0.465159
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	19.1716	0.763208	0.381604	−	0.924326i	$-0.375372\pi$
$631$	19.1716	0.763208	0.381604	+	0.924326i	$0.375372\pi$
$632$	11.3137	0.450035
$633$	9.65685	0.383825
$634$	−18.0000	−0.714871
$635$	14.1421	0.561214
$636$	13.3137	0.527923
$637$	0	0
$638$	−0.828427	−0.0327977
$639$	5.17157	0.204584
$640$	1.00000	0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−4.00000	−0.157867
$643$	−36.4264	−1.43652	−0.718259	−	0.695776i	$-0.755061\pi$
$643$	−36.4264	−1.43652	−0.718259	+	0.695776i	$0.755061\pi$
$644$	0	0
$645$	7.41421	0.291934
$646$	−4.00000	−0.157378
$647$	18.9289	0.744173	0.372087	−	0.928198i	$-0.378642\pi$
$647$	18.9289	0.744173	0.372087	+	0.928198i	$0.378642\pi$
$648$	1.00000	0.0392837
$649$	−49.4558	−1.94131
$650$	0	0
$651$	0	0
$652$	−10.7279	−0.420138
$653$	27.1716	1.06331	0.531653	−	0.846962i	$-0.321571\pi$
$653$	27.1716	1.06331	0.531653	+	0.846962i	$0.321571\pi$
$654$	−17.3137	−0.677020
$655$	−11.1716	−0.436509
$656$	−3.17157	−0.123829
$657$	−3.65685	−0.142667
$658$	0	0
$659$	39.8995	1.55426	0.777132	−	0.629338i	$-0.216674\pi$
$659$	39.8995	1.55426	0.777132	+	0.629338i	$0.216674\pi$
$660$	3.41421	0.132898
$661$	−16.3431	−0.635675	−0.317837	−	0.948145i	$-0.602957\pi$
$661$	−16.3431	−0.635675	−0.317837	+	0.948145i	$0.602957\pi$
$662$	26.6274	1.03490
$663$	0	0
$664$	−10.8284	−0.420224
$665$	0	0
$666$	1.41421	0.0547997
$667$	0.201010	0.00778314
$668$	−13.0711	−0.505735
$669$	6.34315	0.245240
$670$	−11.8995	−0.459718
$671$	−1.17157	−0.0452281
$672$	0	0
$673$	−37.5980	−1.44930	−0.724648	−	0.689119i	$-0.757998\pi$
$673$	−37.5980	−1.44930	−0.724648	+	0.689119i	$0.757998\pi$
$674$	−8.82843	−0.340058
$675$	1.00000	0.0384900
$676$	−13.0000	−0.500000
$677$	16.8284	0.646769	0.323384	−	0.946268i	$-0.395179\pi$
$677$	16.8284	0.646769	0.323384	+	0.946268i	$0.395179\pi$
$678$	−13.3137	−0.511310
$679$	0	0
$680$	−1.41421	−0.0542326
$681$	−2.34315	−0.0897895
$682$	−30.9706	−1.18592
$683$	−7.79899	−0.298420	−0.149210	−	0.988806i	$-0.547673\pi$
$683$	−7.79899	−0.298420	−0.149210	+	0.988806i	$0.547673\pi$
$684$	2.82843	0.108148
$685$	−16.0000	−0.611329
$686$	0	0
$687$	−13.7990	−0.526464
$688$	7.41421	0.282664
$689$	0	0
$690$	−0.828427	−0.0315377
$691$	44.2843	1.68465	0.842327	−	0.538968i	$-0.181185\pi$
$691$	44.2843	1.68465	0.842327	+	0.538968i	$0.181185\pi$
$692$	1.51472	0.0575810
$693$	0	0
$694$	−9.65685	−0.366569
$695$	6.34315	0.240609
$696$	−0.242641	−0.00919727
$697$	4.48528	0.169892
$698$	33.7990	1.27931
$699$	−26.9706	−1.02012
$700$	0	0
$701$	19.5563	0.738633	0.369317	−	0.929304i	$-0.379592\pi$
$701$	19.5563	0.738633	0.369317	+	0.929304i	$0.379592\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	3.41421	0.128678
$705$	5.07107	0.190987
$706$	24.7279	0.930648
$707$	0	0
$708$	−14.4853	−0.544390
$709$	17.1127	0.642681	0.321340	−	0.946964i	$-0.395867\pi$
$709$	17.1127	0.642681	0.321340	+	0.946964i	$0.395867\pi$
$710$	5.17157	0.194086
$711$	11.3137	0.424297
$712$	10.4853	0.392953
$713$	7.51472	0.281428
$714$	0	0
$715$	0	0
$716$	3.41421	0.127595
$717$	0.686292	0.0256300
$718$	13.1716	0.491559
$719$	44.2843	1.65152	0.825762	−	0.564018i	$-0.190745\pi$
$719$	44.2843	1.65152	0.825762	+	0.564018i	$0.190745\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−11.0000	−0.409378
$723$	−7.75736	−0.288499
$724$	−3.17157	−0.117871
$725$	−0.242641	−0.00901145
$726$	0.656854	0.0243781
$727$	34.3431	1.27372	0.636858	−	0.770981i	$-0.280234\pi$
$727$	34.3431	1.27372	0.636858	+	0.770981i	$0.280234\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−3.65685	−0.135346
$731$	−10.4853	−0.387812
$732$	−0.343146	−0.0126830
$733$	51.9411	1.91849	0.959245	−	0.282577i	$-0.0911893\pi$
$733$	51.9411	1.91849	0.959245	+	0.282577i	$0.0911893\pi$
$734$	−0.485281	−0.0179121
$735$	0	0
$736$	−0.828427	−0.0305362
$737$	−40.6274	−1.49653
$738$	−3.17157	−0.116747
$739$	19.1127	0.703072	0.351536	−	0.936174i	$-0.385659\pi$
$739$	19.1127	0.703072	0.351536	+	0.936174i	$0.385659\pi$
$740$	1.41421	0.0519875
$741$	0	0
$742$	0	0
$743$	−5.65685	−0.207530	−0.103765	−	0.994602i	$-0.533089\pi$
$743$	−5.65685	−0.207530	−0.103765	+	0.994602i	$0.533089\pi$
$744$	−9.07107	−0.332561
$745$	5.89949	0.216141
$746$	−3.07107	−0.112440
$747$	−10.8284	−0.396191
$748$	−4.82843	−0.176545
$749$	0	0
$750$	1.00000	0.0365148
$751$	42.4853	1.55031	0.775155	−	0.631771i	$-0.217672\pi$
$751$	42.4853	1.55031	0.775155	+	0.631771i	$0.217672\pi$
$752$	5.07107	0.184923
$753$	−6.34315	−0.231157
$754$	0	0
$755$	−21.7990	−0.793346
$756$	0	0
$757$	32.2426	1.17188	0.585939	−	0.810355i	$-0.300726\pi$
$757$	32.2426	1.17188	0.585939	+	0.810355i	$0.300726\pi$
$758$	−7.51472	−0.272947
$759$	−2.82843	−0.102665
$760$	2.82843	0.102598
$761$	14.9706	0.542682	0.271341	−	0.962483i	$-0.412533\pi$
$761$	14.9706	0.542682	0.271341	+	0.962483i	$0.412533\pi$
$762$	14.1421	0.512316
$763$	0	0
$764$	11.3137	0.409316
$765$	−1.41421	−0.0511310
$766$	−3.89949	−0.140894
$767$	0	0
$768$	1.00000	0.0360844
$769$	−33.4142	−1.20495	−0.602474	−	0.798139i	$-0.705818\pi$
$769$	−33.4142	−1.20495	−0.602474	+	0.798139i	$0.705818\pi$
$770$	0	0
$771$	−17.8995	−0.644635
$772$	17.7990	0.640600
$773$	39.6569	1.42636	0.713179	−	0.700982i	$-0.247255\pi$
$773$	39.6569	1.42636	0.713179	+	0.700982i	$0.247255\pi$
$774$	7.41421	0.266498
$775$	−9.07107	−0.325842
$776$	3.17157	0.113853
$777$	0	0
$778$	12.9289	0.463525
$779$	−8.97056	−0.321404
$780$	0	0
$781$	17.6569	0.631812
$782$	1.17157	0.0418954
$783$	−0.242641	−0.00867127
$784$	0	0
$785$	11.6569	0.416051
$786$	−11.1716	−0.398477
$787$	−4.97056	−0.177181	−0.0885907	−	0.996068i	$-0.528236\pi$
$787$	−4.97056	−0.177181	−0.0885907	+	0.996068i	$0.528236\pi$
$788$	−15.1716	−0.540465
$789$	17.7990	0.633661
$790$	11.3137	0.402524
$791$	0	0
$792$	3.41421	0.121319
$793$	0	0
$794$	−24.6274	−0.873994
$795$	13.3137	0.472189
$796$	17.0711	0.605068
$797$	9.51472	0.337029	0.168514	−	0.985699i	$-0.446103\pi$
$797$	9.51472	0.337029	0.168514	+	0.985699i	$0.446103\pi$
$798$	0	0
$799$	−7.17157	−0.253712
$800$	1.00000	0.0353553
$801$	10.4853	0.370479
$802$	−2.68629	−0.0948563
$803$	−12.4853	−0.440596
$804$	−11.8995	−0.419663
$805$	0	0
$806$	0	0
$807$	−3.65685	−0.128727
$808$	−17.6569	−0.621166
$809$	−37.1127	−1.30481	−0.652406	−	0.757869i	$-0.726240\pi$
$809$	−37.1127	−1.30481	−0.652406	+	0.757869i	$0.726240\pi$
$810$	1.00000	0.0351364
$811$	−36.9706	−1.29821	−0.649106	−	0.760698i	$-0.724857\pi$
$811$	−36.9706	−1.29821	−0.649106	+	0.760698i	$0.724857\pi$
$812$	0	0
$813$	−5.75736	−0.201919
$814$	4.82843	0.169236
$815$	−10.7279	−0.375783
$816$	−1.41421	−0.0495074
$817$	20.9706	0.733667
$818$	18.3848	0.642809
$819$	0	0
$820$	−3.17157	−0.110756
$821$	−35.5563	−1.24093	−0.620463	−	0.784236i	$-0.713055\pi$
$821$	−35.5563	−1.24093	−0.620463	+	0.784236i	$0.713055\pi$
$822$	−16.0000	−0.558064
$823$	−40.2843	−1.40422	−0.702111	−	0.712068i	$-0.747759\pi$
$823$	−40.2843	−1.40422	−0.702111	+	0.712068i	$0.747759\pi$
$824$	18.1421	0.632011
$825$	3.41421	0.118868
$826$	0	0
$827$	51.1127	1.77736	0.888681	−	0.458525i	$-0.151622\pi$
$827$	51.1127	1.77736	0.888681	+	0.458525i	$0.151622\pi$
$828$	−0.828427	−0.0287898
$829$	−22.6863	−0.787927	−0.393964	−	0.919126i	$-0.628896\pi$
$829$	−22.6863	−0.787927	−0.393964	+	0.919126i	$0.628896\pi$
$830$	−10.8284	−0.375860
$831$	2.10051	0.0728657
$832$	0	0
$833$	0	0
$834$	6.34315	0.219645
$835$	−13.0711	−0.452343
$836$	9.65685	0.333989
$837$	−9.07107	−0.313542
$838$	−6.34315	−0.219120
$839$	10.8284	0.373839	0.186919	−	0.982375i	$-0.440150\pi$
$839$	10.8284	0.373839	0.186919	+	0.982375i	$0.440150\pi$
$840$	0	0
$841$	−28.9411	−0.997970
$842$	−28.6274	−0.986566
$843$	−30.4853	−1.04997
$844$	9.65685	0.332403
$845$	−13.0000	−0.447214
$846$	5.07107	0.174347
$847$	0	0
$848$	13.3137	0.457195
$849$	13.5147	0.463824
$850$	−1.41421	−0.0485071
$851$	−1.17157	−0.0401610
$852$	5.17157	0.177175
$853$	29.3137	1.00368	0.501841	−	0.864960i	$-0.332656\pi$
$853$	29.3137	1.00368	0.501841	+	0.864960i	$0.332656\pi$
$854$	0	0
$855$	2.82843	0.0967302
$856$	−4.00000	−0.136717
$857$	52.7279	1.80115	0.900576	−	0.434699i	$-0.143145\pi$
$857$	52.7279	1.80115	0.900576	+	0.434699i	$0.143145\pi$
$858$	0	0
$859$	−16.9706	−0.579028	−0.289514	−	0.957174i	$-0.593494\pi$
$859$	−16.9706	−0.579028	−0.289514	+	0.957174i	$0.593494\pi$
$860$	7.41421	0.252823
$861$	0	0
$862$	−14.1421	−0.481683
$863$	32.9706	1.12233	0.561166	−	0.827704i	$-0.310353\pi$
$863$	32.9706	1.12233	0.561166	+	0.827704i	$0.310353\pi$
$864$	1.00000	0.0340207
$865$	1.51472	0.0515020
$866$	3.17157	0.107774
$867$	−15.0000	−0.509427
$868$	0	0
$869$	38.6274	1.31035
$870$	−0.242641	−0.00822629
$871$	0	0
$872$	−17.3137	−0.586316
$873$	3.17157	0.107341
$874$	−2.34315	−0.0792581
$875$	0	0
$876$	−3.65685	−0.123554
$877$	18.5858	0.627597	0.313799	−	0.949490i	$-0.398398\pi$
$877$	18.5858	0.627597	0.313799	+	0.949490i	$0.398398\pi$
$878$	2.44365	0.0824692
$879$	16.6274	0.560829
$880$	3.41421	0.115093
$881$	20.3431	0.685378	0.342689	−	0.939449i	$-0.388662\pi$
$881$	20.3431	0.685378	0.342689	+	0.939449i	$0.388662\pi$
$882$	0	0
$883$	38.0416	1.28020	0.640101	−	0.768290i	$-0.278892\pi$
$883$	38.0416	1.28020	0.640101	+	0.768290i	$0.278892\pi$
$884$	0	0
$885$	−14.4853	−0.486917
$886$	2.34315	0.0787195
$887$	35.6985	1.19864	0.599319	−	0.800510i	$-0.295438\pi$
$887$	35.6985	1.19864	0.599319	+	0.800510i	$0.295438\pi$
$888$	1.41421	0.0474579
$889$	0	0
$890$	10.4853	0.351467
$891$	3.41421	0.114380
$892$	6.34315	0.212384
$893$	14.3431	0.479975
$894$	5.89949	0.197309
$895$	3.41421	0.114125
$896$	0	0
$897$	0	0
$898$	37.1127	1.23847
$899$	2.20101	0.0734078
$900$	1.00000	0.0333333
$901$	−18.8284	−0.627266
$902$	−10.8284	−0.360547
$903$	0	0
$904$	−13.3137	−0.442807
$905$	−3.17157	−0.105427
$906$	−21.7990	−0.724223
$907$	0.786797	0.0261252	0.0130626	−	0.999915i	$-0.495842\pi$
$907$	0.786797	0.0261252	0.0130626	+	0.999915i	$0.495842\pi$
$908$	−2.34315	−0.0777600
$909$	−17.6569	−0.585641
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	2.82843	0.0936586
$913$	−36.9706	−1.22355
$914$	29.7990	0.985663
$915$	−0.343146	−0.0113440
$916$	−13.7990	−0.455931
$917$	0	0
$918$	−1.41421	−0.0466760
$919$	−46.7696	−1.54279	−0.771393	−	0.636360i	$-0.780439\pi$
$919$	−46.7696	−1.54279	−0.771393	+	0.636360i	$0.780439\pi$
$920$	−0.828427	−0.0273124
$921$	15.3137	0.504604
$922$	−14.0000	−0.461065
$923$	0	0
$924$	0	0
$925$	1.41421	0.0464991
$926$	22.3431	0.734241
$927$	18.1421	0.595866
$928$	−0.242641	−0.00796507
$929$	−0.343146	−0.0112582	−0.00562912	−	0.999984i	$-0.501792\pi$
$929$	−0.343146	−0.0112582	−0.00562912	+	0.999984i	$0.501792\pi$
$930$	−9.07107	−0.297452
$931$	0	0
$932$	−26.9706	−0.883450
$933$	−34.8284	−1.14023
$934$	−3.51472	−0.115005
$935$	−4.82843	−0.157906
$936$	0	0
$937$	−50.9706	−1.66514	−0.832568	−	0.553923i	$-0.813130\pi$
$937$	−50.9706	−1.66514	−0.832568	+	0.553923i	$0.813130\pi$
$938$	0	0
$939$	23.1716	0.756176
$940$	5.07107	0.165400
$941$	−20.2843	−0.661248	−0.330624	−	0.943763i	$-0.607259\pi$
$941$	−20.2843	−0.661248	−0.330624	+	0.943763i	$0.607259\pi$
$942$	11.6569	0.379801
$943$	2.62742	0.0855605
$944$	−14.4853	−0.471456
$945$	0	0
$946$	25.3137	0.823020
$947$	19.5147	0.634143	0.317072	−	0.948402i	$-0.397300\pi$
$947$	19.5147	0.634143	0.317072	+	0.948402i	$0.397300\pi$
$948$	11.3137	0.367452
$949$	0	0
$950$	2.82843	0.0917663
$951$	−18.0000	−0.583690
$952$	0	0
$953$	55.5980	1.80100	0.900498	−	0.434861i	$-0.143202\pi$
$953$	55.5980	1.80100	0.900498	+	0.434861i	$0.143202\pi$
$954$	13.3137	0.431047
$955$	11.3137	0.366103
$956$	0.686292	0.0221963
$957$	−0.828427	−0.0267792
$958$	−0.485281	−0.0156787
$959$	0	0
$960$	1.00000	0.0322749
$961$	51.2843	1.65433
$962$	0	0
$963$	−4.00000	−0.128898
$964$	−7.75736	−0.249848
$965$	17.7990	0.572970
$966$	0	0
$967$	12.2843	0.395036	0.197518	−	0.980299i	$-0.436712\pi$
$967$	12.2843	0.395036	0.197518	+	0.980299i	$0.436712\pi$
$968$	0.656854	0.0211121
$969$	−4.00000	−0.128499
$970$	3.17157	0.101833
$971$	11.0294	0.353951	0.176976	−	0.984215i	$-0.443369\pi$
$971$	11.0294	0.353951	0.176976	+	0.984215i	$0.443369\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	7.51472	0.240787
$975$	0	0
$976$	−0.343146	−0.0109838
$977$	−17.3137	−0.553915	−0.276957	−	0.960882i	$-0.589326\pi$
$977$	−17.3137	−0.553915	−0.276957	+	0.960882i	$0.589326\pi$
$978$	−10.7279	−0.343041
$979$	35.7990	1.14414
$980$	0	0
$981$	−17.3137	−0.552784
$982$	−7.89949	−0.252083
$983$	36.8701	1.17597	0.587986	−	0.808871i	$-0.299921\pi$
$983$	36.8701	1.17597	0.587986	+	0.808871i	$0.299921\pi$
$984$	−3.17157	−0.101106
$985$	−15.1716	−0.483407
$986$	0.343146	0.0109280
$987$	0	0
$988$	0	0
$989$	−6.14214	−0.195309
$990$	3.41421	0.108511
$991$	15.1716	0.481941	0.240970	−	0.970532i	$-0.422534\pi$
$991$	15.1716	0.481941	0.240970	+	0.970532i	$0.422534\pi$
$992$	−9.07107	−0.288007
$993$	26.6274	0.844996
$994$	0	0
$995$	17.0711	0.541189
$996$	−10.8284	−0.343112
$997$	−25.3137	−0.801693	−0.400847	−	0.916145i	$-0.631284\pi$
$997$	−25.3137	−0.801693	−0.400847	+	0.916145i	$0.631284\pi$
$998$	−23.7990	−0.753344
$999$	1.41421	0.0447437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.2.a.v.1.2	yes	2
3.2	odd	2		4410.2.a.bn.1.1		2
5.4	even	2		7350.2.a.dd.1.2		2
7.2	even	3		1470.2.i.u.361.1		4
7.3	odd	6		1470.2.i.v.961.1		4
7.4	even	3		1470.2.i.u.961.1		4
7.5	odd	6		1470.2.i.v.361.1		4
7.6	odd	2		1470.2.a.u.1.2	✓	2
21.20	even	2		4410.2.a.br.1.1		2
35.34	odd	2		7350.2.a.df.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.a.u.1.2	✓	2	7.6	odd	2
1470.2.a.v.1.2	yes	2	1.1	even	1	trivial
1470.2.i.u.361.1		4	7.2	even	3
1470.2.i.u.961.1		4	7.4	even	3
1470.2.i.v.361.1		4	7.5	odd	6
1470.2.i.v.961.1		4	7.3	odd	6
4410.2.a.bn.1.1		2	3.2	odd	2
4410.2.a.br.1.1		2	21.20	even	2
7350.2.a.dd.1.2		2	5.4	even	2
7350.2.a.df.1.2		2	35.34	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:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1470.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.41421 q^{11} +1.00000 q^{12} +1.00000 q^{15} +1.00000 q^{16} -1.41421 q^{17} +1.00000 q^{18} +2.82843 q^{19} +1.00000 q^{20} +3.41421 q^{22} -0.828427 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -0.242641 q^{29} +1.00000 q^{30} -9.07107 q^{31} +1.00000 q^{32} +3.41421 q^{33} -1.41421 q^{34} +1.00000 q^{36} +1.41421 q^{37} +2.82843 q^{38} +1.00000 q^{40} -3.17157 q^{41} +7.41421 q^{43} +3.41421 q^{44} +1.00000 q^{45} -0.828427 q^{46} +5.07107 q^{47} +1.00000 q^{48} +1.00000 q^{50} -1.41421 q^{51} +13.3137 q^{53} +1.00000 q^{54} +3.41421 q^{55} +2.82843 q^{57} -0.242641 q^{58} -14.4853 q^{59} +1.00000 q^{60} -0.343146 q^{61} -9.07107 q^{62} +1.00000 q^{64} +3.41421 q^{66} -11.8995 q^{67} -1.41421 q^{68} -0.828427 q^{69} +5.17157 q^{71} +1.00000 q^{72} -3.65685 q^{73} +1.41421 q^{74} +1.00000 q^{75} +2.82843 q^{76} +11.3137 q^{79} +1.00000 q^{80} +1.00000 q^{81} -3.17157 q^{82} -10.8284 q^{83} -1.41421 q^{85} +7.41421 q^{86} -0.242641 q^{87} +3.41421 q^{88} +10.4853 q^{89} +1.00000 q^{90} -0.828427 q^{92} -9.07107 q^{93} +5.07107 q^{94} +2.82843 q^{95} +1.00000 q^{96} +3.17157 q^{97} +3.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 4 q^{11} + 2 q^{12} + 2 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{20} + 4 q^{22} + 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 4 * q^11 + 2 * q^12 + 2 * q^15 + 2 * q^16 + 2 * q^18 + 2 * q^20 + 4 * q^22 + 4 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^27 + 8 * q^29 + 2 * q^30 - 4 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^36 + 2 * q^40 - 12 * q^41 + 12 * q^43 + 4 * q^44 + 2 * q^45 + 4 * q^46 - 4 * q^47 + 2 * q^48 + 2 * q^50 + 4 * q^53 + 2 * q^54 + 4 * q^55 + 8 * q^58 - 12 * q^59 + 2 * q^60 - 12 * q^61 - 4 * q^62 + 2 * q^64 + 4 * q^66 - 4 * q^67 + 4 * q^69 + 16 * q^71 + 2 * q^72 + 4 * q^73 + 2 * q^75 + 2 * q^80 + 2 * q^81 - 12 * q^82 - 16 * q^83 + 12 * q^86 + 8 * q^87 + 4 * q^88 + 4 * q^89 + 2 * q^90 + 4 * q^92 - 4 * q^93 - 4 * q^94 + 2 * q^96 + 12 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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