LMFDB - Embedded newform 1450.2.b.a.349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1450,2,Mod(349,1450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("1450.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1450 = 2 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1450.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$11.5783082931$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1450.349
Dual form			1450.2.b.a.349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -4.00000 q^{21} -6.00000i q^{22} +2.00000 q^{24} +2.00000 q^{26} +4.00000i q^{27} -2.00000i q^{28} +1.00000 q^{29} -7.00000 q^{31} +1.00000i q^{32} -12.0000i q^{33} +6.00000 q^{34} +1.00000 q^{36} -7.00000i q^{37} -2.00000i q^{38} +4.00000 q^{39} -12.0000 q^{41} -4.00000i q^{42} -8.00000i q^{43} +6.00000 q^{44} +9.00000i q^{47} +2.00000i q^{48} +3.00000 q^{49} +12.0000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -4.00000 q^{54} +2.00000 q^{56} -4.00000i q^{57} +1.00000i q^{58} +9.00000 q^{59} -1.00000 q^{61} -7.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +12.0000 q^{66} -1.00000i q^{67} +6.00000i q^{68} -12.0000 q^{71} +1.00000i q^{72} -2.00000i q^{73} +7.00000 q^{74} +2.00000 q^{76} -12.0000i q^{77} +4.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} -12.0000i q^{82} +4.00000 q^{84} +8.00000 q^{86} +2.00000i q^{87} +6.00000i q^{88} +6.00000 q^{89} +4.00000 q^{91} -14.0000i q^{93} -9.00000 q^{94} -2.00000 q^{96} -4.00000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 12 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{19} - 8 q^{21} + 4 q^{24} + 4 q^{26} + 2 q^{29} - 14 q^{31} + 12 q^{34} + 2 q^{36} + 8 q^{39} - 24 q^{41} + 12 q^{44} + 6 q^{49}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 12 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^19 - 8 * q^21 + 4 * q^24 + 4 * q^26 + 2 * q^29 - 14 * q^31 + 12 * q^34 + 2 * q^36 + 8 * q^39 - 24 * q^41 + 12 * q^44 + 6 * q^49 + 24 * q^51 - 8 * q^54 + 4 * q^56 + 18 * q^59 - 2 * q^61 - 2 * q^64 + 24 * q^66 - 24 * q^71 + 14 * q^74 + 4 * q^76 - 16 * q^79 - 22 * q^81 + 8 * q^84 + 16 * q^86 + 12 * q^89 + 8 * q^91 - 18 * q^94 - 4 * q^96 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	− 6.00000i	− 1.27920i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	2.00000	0.392232
$27$	4.00000i	0.769800i
$28$	− 2.00000i	− 0.377964i
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	1.00000i	0.176777i
$33$	− 12.0000i	− 2.08893i
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	− 2.00000i	− 0.324443i
$39$	4.00000	0.640513
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	− 4.00000i	− 0.617213i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	0	0
$47$	9.00000i	1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$	9.00000i	1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$	2.00000i	0.288675i
$49$	3.00000	0.428571
$50$	0	0
$51$	12.0000	1.68034
$52$	2.00000i	0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	2.00000	0.267261
$57$	− 4.00000i	− 0.529813i
$58$	1.00000i	0.131306i
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	− 7.00000i	− 0.889001i
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	12.0000	1.47710
$67$	− 1.00000i	− 0.122169i	−0.998133	−	0.0610847i	$-0.980544\pi$
$67$	− 1.00000i	− 0.122169i	0.998133	−	0.0610847i	$-0.0194560\pi$
$68$	6.00000i	0.727607i
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000i	0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	2.00000	0.229416
$77$	− 12.0000i	− 1.36753i
$78$	4.00000i	0.452911i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 12.0000i	− 1.32518i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	8.00000	0.862662
$87$	2.00000i	0.214423i
$88$	6.00000i	0.639602i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	− 14.0000i	− 1.45173i
$94$	−9.00000	−0.928279
$95$	0	0
$96$	−2.00000	−0.204124
$97$	− 4.00000i	− 0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$	− 4.00000i	− 0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	3.00000i	0.303046i
$99$	6.00000	0.603023
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	12.0000i	1.18818i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 3.00000i	− 0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$	− 3.00000i	− 0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$	− 4.00000i	− 0.384900i
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	14.0000	1.32882
$112$	2.00000i	0.188982i
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	2.00000i	0.184900i
$118$	9.00000i	0.828517i
$119$	12.0000	1.10004
$120$	0	0
$121$	25.0000	2.27273
$122$	− 1.00000i	− 0.0905357i
$123$	− 24.0000i	− 2.16401i
$124$	7.00000	0.628619
$125$	0	0
$126$	2.00000	0.178174
$127$	− 7.00000i	− 0.621150i	−0.950549	−	0.310575i	$-0.899478\pi$
$127$	− 7.00000i	− 0.621150i	0.950549	−	0.310575i	$-0.100522\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	16.0000	1.40872
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	12.0000i	1.04447i
$133$	− 4.00000i	− 0.346844i
$134$	1.00000	0.0863868
$135$	0	0
$136$	−6.00000	−0.514496
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	−18.0000	−1.51587
$142$	− 12.0000i	− 1.00702i
$143$	12.0000i	1.00349i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	6.00000i	0.494872i
$148$	7.00000i	0.575396i
$149$	−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
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	2.00000i	0.162221i
$153$	6.00000i	0.485071i
$154$	12.0000	0.966988
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 7.00000i	− 0.558661i	−0.960195	−	0.279330i	$-0.909888\pi$
$157$	− 7.00000i	− 0.558661i	0.960195	−	0.279330i	$-0.0901125\pi$
$158$	− 8.00000i	− 0.636446i
$159$	−12.0000	−0.951662
$160$	0	0
$161$	0	0
$162$	− 11.0000i	− 0.864242i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	4.00000i	0.308607i
$169$	9.00000	0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	8.00000i	0.609994i
$173$	− 24.0000i	− 1.82469i	−0.409426	−	0.912343i	$-0.634271\pi$
$173$	− 24.0000i	− 1.82469i	0.409426	−	0.912343i	$-0.365729\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−6.00000	−0.452267
$177$	18.0000i	1.35296i
$178$	6.00000i	0.449719i
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	4.00000i	0.296500i
$183$	− 2.00000i	− 0.147844i
$184$	0	0
$185$	0	0
$186$	14.0000	1.02653
$187$	36.0000i	2.63258i
$188$	− 9.00000i	− 0.656392i
$189$	−8.00000	−0.581914
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	− 2.00000i	− 0.144338i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	6.00000i	0.426401i
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	3.00000i	0.211079i
$203$	2.00000i	0.140372i
$204$	−12.0000	−0.840168
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	12.0000	0.830057
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 24.0000i	− 1.64445i
$214$	3.00000	0.205076
$215$	0	0
$216$	4.00000	0.272166
$217$	− 14.0000i	− 0.950382i
$218$	4.00000i	0.270914i
$219$	4.00000	0.270295
$220$	0	0
$221$	−12.0000	−0.807207
$222$	14.0000i	0.939618i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−18.0000	−1.19734
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	4.00000i	0.264906i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	24.0000	1.57908
$232$	− 1.00000i	− 0.0656532i
$233$	− 15.0000i	− 0.982683i	−0.870967	−	0.491341i	$-0.836507\pi$
$233$	− 15.0000i	− 0.982683i	0.870967	−	0.491341i	$-0.163493\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−9.00000	−0.585850
$237$	− 16.0000i	− 1.03931i
$238$	12.0000i	0.777844i
$239$	−30.0000	−1.94054	−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000	−1.94054	−0.970269	+	0.242028i	$0.922188\pi$
$240$	0	0
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	25.0000i	1.60706i
$243$	− 10.0000i	− 0.641500i
$244$	1.00000	0.0640184
$245$	0	0
$246$	24.0000	1.53018
$247$	4.00000i	0.254514i
$248$	7.00000i	0.444500i
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	2.00000i	0.125988i
$253$	0	0
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	16.0000i	0.996116i
$259$	14.0000	0.869918
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	− 18.0000i	− 1.11204i
$263$	− 27.0000i	− 1.66489i	−0.554107	−	0.832446i	$-0.686940\pi$
$263$	− 27.0000i	− 1.66489i	0.554107	−	0.832446i	$-0.313060\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	4.00000	0.245256
$267$	12.0000i	0.734388i
$268$	1.00000i	0.0610847i
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	− 6.00000i	− 0.363803i
$273$	8.00000i	0.484182i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	− 5.00000i	− 0.299880i
$279$	7.00000	0.419079
$280$	0	0
$281$	−33.0000	−1.96861	−0.984307	−	0.176462i	$-0.943535\pi$
$281$	−33.0000	−1.96861	−0.984307	+	0.176462i	$0.943535\pi$
$282$	− 18.0000i	− 1.07188i
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−12.0000	−0.709575
$287$	− 24.0000i	− 1.41668i
$288$	− 1.00000i	− 0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	8.00000	0.468968
$292$	2.00000i	0.117041i
$293$	− 9.00000i	− 0.525786i	−0.964825	−	0.262893i	$-0.915323\pi$
$293$	− 9.00000i	− 0.525786i	0.964825	−	0.262893i	$-0.0846766\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	−7.00000	−0.406867
$297$	− 24.0000i	− 1.39262i
$298$	− 18.0000i	− 1.04271i
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	20.0000i	1.15087i
$303$	6.00000i	0.344691i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	−6.00000	−0.342997
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	12.0000i	0.683763i
$309$	−8.00000	−0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	− 4.00000i	− 0.226455i
$313$	25.0000i	1.41308i	0.707671	+	0.706542i	$0.249746\pi$
$313$	25.0000i	1.41308i	−0.707671	+	0.706542i	$0.750254\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	8.00000	0.450035
$317$	− 15.0000i	− 0.842484i	−0.906948	−	0.421242i	$-0.861594\pi$
$317$	− 15.0000i	− 0.842484i	0.906948	−	0.421242i	$-0.138406\pi$
$318$	− 12.0000i	− 0.672927i
$319$	−6.00000	−0.335936
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	12.0000i	0.667698i
$324$	11.0000	0.611111
$325$	0	0
$326$	−4.00000	−0.221540
$327$	8.00000i	0.442401i
$328$	12.0000i	0.662589i
$329$	−18.0000	−0.992372
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	7.00000i	0.383598i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	−4.00000	−0.218218
$337$	26.0000i	1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$	26.0000i	1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$	9.00000i	0.489535i
$339$	−36.0000	−1.95525
$340$	0	0
$341$	42.0000	2.27443
$342$	2.00000i	0.108148i
$343$	20.0000i	1.07990i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	24.0000	1.29025
$347$	− 3.00000i	− 0.161048i	−0.996753	−	0.0805242i	$-0.974341\pi$
$347$	− 3.00000i	− 0.161048i	0.996753	−	0.0805242i	$-0.0256594\pi$
$348$	− 2.00000i	− 0.107211i
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	− 6.00000i	− 0.319801i
$353$	33.0000i	1.75641i	0.478282	+	0.878206i	$0.341260\pi$
$353$	33.0000i	1.75641i	−0.478282	+	0.878206i	$0.658740\pi$
$354$	−18.0000	−0.956689
$355$	0	0
$356$	−6.00000	−0.317999
$357$	24.0000i	1.27021i
$358$	− 21.0000i	− 1.10988i
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	2.00000i	0.105118i
$363$	50.0000i	2.62432i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	2.00000	0.104542
$367$	− 19.0000i	− 0.991792i	−0.868382	−	0.495896i	$-0.834840\pi$
$367$	− 19.0000i	− 0.991792i	0.868382	−	0.495896i	$-0.165160\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	−12.0000	−0.623009
$372$	14.0000i	0.725866i
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	−36.0000	−1.86152
$375$	0	0
$376$	9.00000	0.464140
$377$	− 2.00000i	− 0.103005i
$378$	− 8.00000i	− 0.411476i
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	14.0000	0.717242
$382$	15.0000i	0.767467i
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	−10.0000	−0.508987
$387$	8.00000i	0.406663i
$388$	4.00000i	0.203069i
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	0	0
$391$	0	0
$392$	− 3.00000i	− 0.151523i
$393$	− 36.0000i	− 1.81596i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	−6.00000	−0.301511
$397$	− 34.0000i	− 1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	− 8.00000i	− 0.401004i
$399$	8.00000	0.400501
$400$	0	0
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	2.00000i	0.0997509i
$403$	14.0000i	0.697390i
$404$	−3.00000	−0.149256
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	42.0000i	2.08186i
$408$	− 12.0000i	− 0.594089i
$409$	28.0000	1.38451	0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000	1.38451	0.692255	+	0.721653i	$0.256617\pi$
$410$	0	0
$411$	0	0
$412$	− 4.00000i	− 0.197066i
$413$	18.0000i	0.885722i
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	− 10.0000i	− 0.489702i
$418$	12.0000i	0.586939i
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	− 10.0000i	− 0.486792i
$423$	− 9.00000i	− 0.437595i
$424$	6.00000	0.291386
$425$	0	0
$426$	24.0000	1.16280
$427$	− 2.00000i	− 0.0967868i
$428$	3.00000i	0.145010i
$429$	−24.0000	−1.15873
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	4.00000i	0.192450i
$433$	4.00000i	0.192228i	0.995370	+	0.0961139i	$0.0306413\pi$
$433$	4.00000i	0.192228i	−0.995370	+	0.0961139i	$0.969359\pi$
$434$	14.0000	0.672022
$435$	0	0
$436$	−4.00000	−0.191565
$437$	0	0
$438$	4.00000i	0.191127i
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 12.0000i	− 0.570782i
$443$	− 18.0000i	− 0.855206i	−0.903967	−	0.427603i	$-0.859358\pi$
$443$	− 18.0000i	− 0.855206i	0.903967	−	0.427603i	$-0.140642\pi$
$444$	−14.0000	−0.664411
$445$	0	0
$446$	−16.0000	−0.757622
$447$	− 36.0000i	− 1.70274i
$448$	− 2.00000i	− 0.0944911i
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	72.0000	3.39035
$452$	− 18.0000i	− 0.846649i
$453$	40.0000i	1.87936i
$454$	−3.00000	−0.140797
$455$	0	0
$456$	−4.00000	−0.187317
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	− 14.0000i	− 0.654177i
$459$	24.0000	1.12022
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	24.0000i	1.11658i
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	15.0000	0.694862
$467$	42.0000i	1.94353i	0.235954	+	0.971764i	$0.424178\pi$
$467$	42.0000i	1.94353i	−0.235954	+	0.971764i	$0.575822\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	2.00000	0.0923514
$470$	0	0
$471$	14.0000	0.645086
$472$	− 9.00000i	− 0.414259i
$473$	48.0000i	2.20704i
$474$	16.0000	0.734904
$475$	0	0
$476$	−12.0000	−0.550019
$477$	− 6.00000i	− 0.274721i
$478$	− 30.0000i	− 1.37217i
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	− 13.0000i	− 0.592134i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	10.0000	0.453609
$487$	38.0000i	1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$	38.0000i	1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$	1.00000i	0.0452679i
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	24.0000i	1.08200i
$493$	− 6.00000i	− 0.270226i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−7.00000	−0.314309
$497$	− 24.0000i	− 1.07655i
$498$	0	0
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	− 12.0000i	− 0.535586i
$503$	21.0000i	0.936344i	0.883637	+	0.468172i	$0.155087\pi$
$503$	21.0000i	0.936344i	−0.883637	+	0.468172i	$0.844913\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	18.0000i	0.799408i
$508$	7.00000i	0.310575i
$509$	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
$511$	4.00000	0.176950
$512$	1.00000i	0.0441942i
$513$	− 8.00000i	− 0.353209i
$514$	6.00000	0.264649
$515$	0	0
$516$	−16.0000	−0.704361
$517$	− 54.0000i	− 2.37492i
$518$	14.0000i	0.615125i
$519$	48.0000	2.10697
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	− 1.00000i	− 0.0437688i
$523$	− 29.0000i	− 1.26808i	−0.773300	−	0.634041i	$-0.781395\pi$
$523$	− 29.0000i	− 1.26808i	0.773300	−	0.634041i	$-0.218605\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	27.0000	1.17726
$527$	42.0000i	1.82955i
$528$	− 12.0000i	− 0.522233i
$529$	23.0000	1.00000
$530$	0	0
$531$	−9.00000	−0.390567
$532$	4.00000i	0.173422i
$533$	24.0000i	1.03956i
$534$	−12.0000	−0.519291
$535$	0	0
$536$	−1.00000	−0.0431934
$537$	− 42.0000i	− 1.81243i
$538$	− 15.0000i	− 0.646696i
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	− 7.00000i	− 0.300676i
$543$	4.00000i	0.171656i
$544$	6.00000	0.257248
$545$	0	0
$546$	−8.00000	−0.342368
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	1.00000	0.0426790
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	−26.0000	−1.10463
$555$	0	0
$556$	5.00000	0.212047
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	7.00000i	0.296334i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	−72.0000	−3.03984
$562$	− 33.0000i	− 1.39202i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	18.0000	0.757937
$565$	0	0
$566$	20.0000	0.840663
$567$	− 22.0000i	− 0.923913i
$568$	12.0000i	0.503509i
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	29.0000	1.21361	0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000	1.21361	0.606806	+	0.794850i	$0.292450\pi$
$572$	− 12.0000i	− 0.501745i
$573$	30.0000i	1.25327i
$574$	24.0000	1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	14.0000i	0.582828i	0.956597	+	0.291414i	$0.0941257\pi$
$577$	14.0000i	0.582828i	−0.956597	+	0.291414i	$0.905874\pi$
$578$	− 19.0000i	− 0.790296i
$579$	−20.0000	−0.831172
$580$	0	0
$581$	0	0
$582$	8.00000i	0.331611i
$583$	− 36.0000i	− 1.49097i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	9.00000	0.371787
$587$	15.0000i	0.619116i	0.950881	+	0.309558i	$0.100181\pi$
$587$	15.0000i	0.619116i	−0.950881	+	0.309558i	$0.899819\pi$
$588$	− 6.00000i	− 0.247436i
$589$	14.0000	0.576860
$590$	0	0
$591$	−36.0000	−1.48084
$592$	− 7.00000i	− 0.287698i
$593$	33.0000i	1.35515i	0.735455	+	0.677574i	$0.236969\pi$
$593$	33.0000i	1.35515i	−0.735455	+	0.677574i	$0.763031\pi$
$594$	24.0000	0.984732
$595$	0	0
$596$	18.0000	0.737309
$597$	− 16.0000i	− 0.654836i
$598$	0	0
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	16.0000i	0.652111i
$603$	1.00000i	0.0407231i
$604$	−20.0000	−0.813788
$605$	0	0
$606$	−6.00000	−0.243733
$607$	41.0000i	1.66414i	0.554672	+	0.832069i	$0.312844\pi$
$607$	41.0000i	1.66414i	−0.554672	+	0.832069i	$0.687156\pi$
$608$	− 2.00000i	− 0.0811107i
$609$	−4.00000	−0.162088
$610$	0	0
$611$	18.0000	0.728202
$612$	− 6.00000i	− 0.242536i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	−12.0000	−0.483494
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	− 8.00000i	− 0.321807i
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0000i	0.480770i
$624$	4.00000	0.160128
$625$	0	0
$626$	−25.0000	−0.999201
$627$	24.0000i	0.958468i
$628$	7.00000i	0.279330i
$629$	−42.0000	−1.67465
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	8.00000i	0.318223i
$633$	− 20.0000i	− 0.794929i
$634$	15.0000	0.595726
$635$	0	0
$636$	12.0000	0.475831
$637$	− 6.00000i	− 0.237729i
$638$	− 6.00000i	− 0.237542i
$639$	12.0000	0.474713
$640$	0	0
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	6.00000i	0.236801i
$643$	− 23.0000i	− 0.907031i	−0.891248	−	0.453516i	$-0.850170\pi$
$643$	− 23.0000i	− 0.907031i	0.891248	−	0.453516i	$-0.149830\pi$
$644$	0	0
$645$	0	0
$646$	−12.0000	−0.472134
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	11.0000i	0.432121i
$649$	−54.0000	−2.11969
$650$	0	0
$651$	28.0000	1.09741
$652$	− 4.00000i	− 0.156652i
$653$	− 39.0000i	− 1.52619i	−0.646288	−	0.763094i	$-0.723679\pi$
$653$	− 39.0000i	− 1.52619i	0.646288	−	0.763094i	$-0.276321\pi$
$654$	−8.00000	−0.312825
$655$	0	0
$656$	−12.0000	−0.468521
$657$	2.00000i	0.0780274i
$658$	− 18.0000i	− 0.701713i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	8.00000i	0.310929i
$663$	− 24.0000i	− 0.932083i
$664$	0	0
$665$	0	0
$666$	−7.00000	−0.271244
$667$	0	0
$668$	− 12.0000i	− 0.464294i
$669$	−32.0000	−1.23719
$670$	0	0
$671$	6.00000	0.231627
$672$	− 4.00000i	− 0.154303i
$673$	1.00000i	0.0385472i	0.999814	+	0.0192736i	$0.00613535\pi$
$673$	1.00000i	0.0385472i	−0.999814	+	0.0192736i	$0.993865\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	33.0000i	1.26829i	0.773213	+	0.634147i	$0.218648\pi$
$677$	33.0000i	1.26829i	−0.773213	+	0.634147i	$0.781352\pi$
$678$	− 36.0000i	− 1.38257i
$679$	8.00000	0.307012
$680$	0	0
$681$	−6.00000	−0.229920
$682$	42.0000i	1.60826i
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	−20.0000	−0.763604
$687$	− 28.0000i	− 1.06827i
$688$	− 8.00000i	− 0.304997i
$689$	12.0000	0.457164
$690$	0	0
$691$	−1.00000	−0.0380418	−0.0190209	−	0.999819i	$-0.506055\pi$
$691$	−1.00000	−0.0380418	−0.0190209	+	0.999819i	$0.506055\pi$
$692$	24.0000i	0.912343i
$693$	12.0000i	0.455842i
$694$	3.00000	0.113878
$695$	0	0
$696$	2.00000	0.0758098
$697$	72.0000i	2.72719i
$698$	− 26.0000i	− 0.984115i
$699$	30.0000	1.13470
$700$	0	0
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	8.00000i	0.301941i
$703$	14.0000i	0.528020i
$704$	6.00000	0.226134
$705$	0	0
$706$	−33.0000	−1.24197
$707$	6.00000i	0.225653i
$708$	− 18.0000i	− 0.676481i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	− 6.00000i	− 0.224860i
$713$	0	0
$714$	−24.0000	−0.898177
$715$	0	0
$716$	21.0000	0.784807
$717$	− 60.0000i	− 2.24074i
$718$	− 9.00000i	− 0.335877i
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 15.0000i	− 0.558242i
$723$	− 26.0000i	− 0.966950i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−50.0000	−1.85567
$727$	− 52.0000i	− 1.92857i	−0.264861	−	0.964287i	$-0.585326\pi$
$727$	− 52.0000i	− 1.92857i	0.264861	−	0.964287i	$-0.414674\pi$
$728$	− 4.00000i	− 0.148250i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−48.0000	−1.77534
$732$	2.00000i	0.0739221i
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	19.0000	0.701303
$735$	0	0
$736$	0	0
$737$	6.00000i	0.221013i
$738$	12.0000i	0.441726i
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	− 12.0000i	− 0.440534i
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	−14.0000	−0.513265
$745$	0	0
$746$	−4.00000	−0.146450
$747$	0	0
$748$	− 36.0000i	− 1.31629i
$749$	6.00000	0.219235
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	9.00000i	0.328196i
$753$	− 24.0000i	− 0.874609i
$754$	2.00000	0.0728357
$755$	0	0
$756$	8.00000	0.290957
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	− 2.00000i	− 0.0726433i
$759$	0	0
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	14.0000i	0.507166i
$763$	8.00000i	0.289619i
$764$	−15.0000	−0.542681
$765$	0	0
$766$	−6.00000	−0.216789
$767$	− 18.0000i	− 0.649942i
$768$	2.00000i	0.0721688i
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	− 10.0000i	− 0.359908i
$773$	39.0000i	1.40273i	0.712801	+	0.701366i	$0.247426\pi$
$773$	39.0000i	1.40273i	−0.712801	+	0.701366i	$0.752574\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	−4.00000	−0.143592
$777$	28.0000i	1.00449i
$778$	9.00000i	0.322666i
$779$	24.0000	0.859889
$780$	0	0
$781$	72.0000	2.57636
$782$	0	0
$783$	4.00000i	0.142948i
$784$	3.00000	0.107143
$785$	0	0
$786$	36.0000	1.28408
$787$	− 31.0000i	− 1.10503i	−0.833503	−	0.552515i	$-0.813668\pi$
$787$	− 31.0000i	− 1.10503i	0.833503	−	0.552515i	$-0.186332\pi$
$788$	− 18.0000i	− 0.641223i
$789$	54.0000	1.92245
$790$	0	0
$791$	−36.0000	−1.28001
$792$	− 6.00000i	− 0.213201i
$793$	2.00000i	0.0710221i
$794$	34.0000	1.20661
$795$	0	0
$796$	8.00000	0.283552
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	8.00000i	0.283197i
$799$	54.0000	1.91038
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 15.0000i	− 0.529668i
$803$	12.0000i	0.423471i
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	−14.0000	−0.493129
$807$	− 30.0000i	− 1.05605i
$808$	− 3.00000i	− 0.105540i
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	−19.0000	−0.667180	−0.333590	−	0.942718i	$-0.608260\pi$
$811$	−19.0000	−0.667180	−0.333590	+	0.942718i	$0.608260\pi$
$812$	− 2.00000i	− 0.0701862i
$813$	− 14.0000i	− 0.491001i
$814$	−42.0000	−1.47210
$815$	0	0
$816$	12.0000	0.420084
$817$	16.0000i	0.559769i
$818$	28.0000i	0.978997i
$819$	−4.00000	−0.139771
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−18.0000	−0.626300
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−52.0000	−1.80386
$832$	2.00000i	0.0693375i
$833$	− 18.0000i	− 0.623663i
$834$	10.0000	0.346272
$835$	0	0
$836$	−12.0000	−0.415029
$837$	− 28.0000i	− 0.967822i
$838$	9.00000i	0.310900i
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	17.0000i	0.585859i
$843$	− 66.0000i	− 2.27316i
$844$	10.0000	0.344214
$845$	0	0
$846$	9.00000	0.309426
$847$	50.0000i	1.71802i
$848$	6.00000i	0.206041i
$849$	40.0000	1.37280
$850$	0	0
$851$	0	0
$852$	24.0000i	0.822226i
$853$	− 50.0000i	− 1.71197i	−0.517003	−	0.855984i	$-0.672952\pi$
$853$	− 50.0000i	− 1.71197i	0.517003	−	0.855984i	$-0.327048\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	−3.00000	−0.102538
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	− 24.0000i	− 0.819346i
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	48.0000	1.63584
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	−4.00000	−0.135926
$867$	− 38.0000i	− 1.29055i
$868$	14.0000i	0.475191i
$869$	48.0000	1.62829
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	− 4.00000i	− 0.135457i
$873$	4.00000i	0.135379i
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	8.00000i	0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$	8.00000i	0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$	28.0000i	0.944954i
$879$	18.0000	0.607125
$880$	0	0
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	− 3.00000i	− 0.101015i
$883$	− 17.0000i	− 0.572096i	−0.958215	−	0.286048i	$-0.907658\pi$
$883$	− 17.0000i	− 0.572096i	0.958215	−	0.286048i	$-0.0923416\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	18.0000	0.604722
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	− 14.0000i	− 0.469809i
$889$	14.0000	0.469545
$890$	0	0
$891$	66.0000	2.21108
$892$	− 16.0000i	− 0.535720i
$893$	− 18.0000i	− 0.602347i
$894$	36.0000	1.20402
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	− 12.0000i	− 0.400445i
$899$	−7.00000	−0.233463
$900$	0	0
$901$	36.0000	1.19933
$902$	72.0000i	2.39734i
$903$	32.0000i	1.06489i
$904$	18.0000	0.598671
$905$	0	0
$906$	−40.0000	−1.32891
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	− 3.00000i	− 0.0995585i
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	27.0000	0.894550	0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000	0.894550	0.447275	+	0.894397i	$0.352395\pi$
$912$	− 4.00000i	− 0.132453i
$913$	0	0
$914$	10.0000	0.330771
$915$	0	0
$916$	14.0000	0.462573
$917$	− 36.0000i	− 1.18882i
$918$	24.0000i	0.792118i
$919$	46.0000	1.51740	0.758700	−	0.651440i	$-0.225835\pi$
$919$	46.0000	1.51740	0.758700	+	0.651440i	$0.225835\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	− 30.0000i	− 0.987997i
$923$	24.0000i	0.789970i
$924$	−24.0000	−0.789542
$925$	0	0
$926$	−4.00000	−0.131448
$927$	− 4.00000i	− 0.131377i
$928$	1.00000i	0.0328266i
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	15.0000i	0.491341i
$933$	0	0
$934$	−42.0000	−1.37428
$935$	0	0
$936$	2.00000	0.0653720
$937$	11.0000i	0.359354i	0.983726	+	0.179677i	$0.0575053\pi$
$937$	11.0000i	0.359354i	−0.983726	+	0.179677i	$0.942495\pi$
$938$	2.00000i	0.0653023i
$939$	−50.0000	−1.63169
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	14.0000i	0.456145i
$943$	0	0
$944$	9.00000	0.292925
$945$	0	0
$946$	−48.0000	−1.56061
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	16.0000i	0.519656i
$949$	−4.00000	−0.129845
$950$	0	0
$951$	30.0000	0.972817
$952$	− 12.0000i	− 0.388922i
$953$	30.0000i	0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$	30.0000i	0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	30.0000	0.970269
$957$	− 12.0000i	− 0.387905i
$958$	36.0000i	1.16311i
$959$	0	0
$960$	0	0
$961$	18.0000	0.580645
$962$	− 14.0000i	− 0.451378i
$963$	3.00000i	0.0966736i
$964$	13.0000	0.418702
$965$	0	0
$966$	0	0
$967$	− 4.00000i	− 0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$	− 4.00000i	− 0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$	− 25.0000i	− 0.803530i
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	10.0000i	0.320750i
$973$	− 10.0000i	− 0.320585i
$974$	−38.0000	−1.21760
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	− 57.0000i	− 1.82359i	−0.410644	−	0.911796i	$-0.634696\pi$
$977$	− 57.0000i	− 1.82359i	0.410644	−	0.911796i	$-0.365304\pi$
$978$	− 8.00000i	− 0.255812i
$979$	−36.0000	−1.15056
$980$	0	0
$981$	−4.00000	−0.127710
$982$	− 30.0000i	− 0.957338i
$983$	39.0000i	1.24391i	0.783054	+	0.621953i	$0.213661\pi$
$983$	39.0000i	1.24391i	−0.783054	+	0.621953i	$0.786339\pi$
$984$	−24.0000	−0.765092
$985$	0	0
$986$	6.00000	0.191079
$987$	− 36.0000i	− 1.14589i
$988$	− 4.00000i	− 0.127257i
$989$	0	0
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	− 7.00000i	− 0.222250i
$993$	16.0000i	0.507745i
$994$	24.0000	0.761234
$995$	0	0
$996$	0	0
$997$	− 1.00000i	− 0.0316703i	−0.999875	−	0.0158352i	$-0.994959\pi$
$997$	− 1.00000i	− 0.0316703i	0.999875	−	0.0158352i	$-0.00504070\pi$
$998$	13.0000i	0.411508i
$999$	28.0000	0.885881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1450.2.b.a.349.2		2
5.2	odd	4		1450.2.a.d.1.1	✓	1
5.3	odd	4		1450.2.a.e.1.1	yes	1
5.4	even	2	inner	1450.2.b.a.349.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1450.2.a.d.1.1	✓	1	5.2	odd	4
1450.2.a.e.1.1	yes	1	5.3	odd	4
1450.2.b.a.349.1		2	5.4	even	2	inner
1450.2.b.a.349.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1450.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -4.00000 q^{21} -6.00000i q^{22} +2.00000 q^{24} +2.00000 q^{26} +4.00000i q^{27} -2.00000i q^{28} +1.00000 q^{29} -7.00000 q^{31} +1.00000i q^{32} -12.0000i q^{33} +6.00000 q^{34} +1.00000 q^{36} -7.00000i q^{37} -2.00000i q^{38} +4.00000 q^{39} -12.0000 q^{41} -4.00000i q^{42} -8.00000i q^{43} +6.00000 q^{44} +9.00000i q^{47} +2.00000i q^{48} +3.00000 q^{49} +12.0000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -4.00000 q^{54} +2.00000 q^{56} -4.00000i q^{57} +1.00000i q^{58} +9.00000 q^{59} -1.00000 q^{61} -7.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +12.0000 q^{66} -1.00000i q^{67} +6.00000i q^{68} -12.0000 q^{71} +1.00000i q^{72} -2.00000i q^{73} +7.00000 q^{74} +2.00000 q^{76} -12.0000i q^{77} +4.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} -12.0000i q^{82} +4.00000 q^{84} +8.00000 q^{86} +2.00000i q^{87} +6.00000i q^{88} +6.00000 q^{89} +4.00000 q^{91} -14.0000i q^{93} -9.00000 q^{94} -2.00000 q^{96} -4.00000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 12 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{19} - 8 q^{21} + 4 q^{24} + 4 q^{26} + 2 q^{29} - 14 q^{31} + 12 q^{34} + 2 q^{36} + 8 q^{39} - 24 q^{41} + 12 q^{44} + 6 q^{49}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 12 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^19 - 8 * q^21 + 4 * q^24 + 4 * q^26 + 2 * q^29 - 14 * q^31 + 12 * q^34 + 2 * q^36 + 8 * q^39 - 24 * q^41 + 12 * q^44 + 6 * q^49 + 24 * q^51 - 8 * q^54 + 4 * q^56 + 18 * q^59 - 2 * q^61 - 2 * q^64 + 24 * q^66 - 24 * q^71 + 14 * q^74 + 4 * q^76 - 16 * q^79 - 22 * q^81 + 8 * q^84 + 16 * q^86 + 12 * q^89 + 8 * q^91 - 18 * q^94 - 4 * q^96 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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