LMFDB - Embedded newform 1450.2.b.f.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:	$0$
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:	no (minimal twist has level 58)
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.f.349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} +3.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -6.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} +3.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -6.00000 q^{9} -1.00000 q^{11} +3.00000i q^{12} +3.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} -6.00000i q^{18} +8.00000 q^{19} +6.00000 q^{21} -1.00000i q^{22} -3.00000 q^{24} -3.00000 q^{26} +9.00000i q^{27} -2.00000i q^{28} +1.00000 q^{29} +3.00000 q^{31} +1.00000i q^{32} +3.00000i q^{33} -4.00000 q^{34} +6.00000 q^{36} +8.00000i q^{37} +8.00000i q^{38} +9.00000 q^{39} -2.00000 q^{41} +6.00000i q^{42} +7.00000i q^{43} +1.00000 q^{44} -11.0000i q^{47} -3.00000i q^{48} +3.00000 q^{49} +12.0000 q^{51} -3.00000i q^{52} +1.00000i q^{53} -9.00000 q^{54} +2.00000 q^{56} -24.0000i q^{57} +1.00000i q^{58} +4.00000 q^{59} +4.00000 q^{61} +3.00000i q^{62} -12.0000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +4.00000i q^{67} -4.00000i q^{68} -2.00000 q^{71} +6.00000i q^{72} -12.0000i q^{73} -8.00000 q^{74} -8.00000 q^{76} -2.00000i q^{77} +9.00000i q^{78} +7.00000 q^{79} +9.00000 q^{81} -2.00000i q^{82} -6.00000 q^{84} -7.00000 q^{86} -3.00000i q^{87} +1.00000i q^{88} +6.00000 q^{89} -6.00000 q^{91} -9.00000i q^{93} +11.0000 q^{94} +3.00000 q^{96} +6.00000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 6 * q^6 - 12 * q^9	$ 2 q - 2 q^{4} + 6 q^{6} - 12 q^{9} - 2 q^{11} - 4 q^{14} + 2 q^{16} + 16 q^{19} + 12 q^{21} - 6 q^{24} - 6 q^{26} + 2 q^{29} + 6 q^{31} - 8 q^{34} + 12 q^{36} + 18 q^{39} - 4 q^{41} + 2 q^{44} + 6 q^{49} + 24 q^{51} - 18 q^{54} + 4 q^{56} + 8 q^{59} + 8 q^{61} - 2 q^{64} - 6 q^{66} - 4 q^{71} - 16 q^{74} - 16 q^{76} + 14 q^{79} + 18 q^{81} - 12 q^{84} - 14 q^{86} + 12 q^{89} - 12 q^{91} + 22 q^{94} + 6 q^{96} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^6 - 12 * q^9 - 2 * q^11 - 4 * q^14 + 2 * q^16 + 16 * q^19 + 12 * q^21 - 6 * q^24 - 6 * q^26 + 2 * q^29 + 6 * q^31 - 8 * q^34 + 12 * q^36 + 18 * q^39 - 4 * q^41 + 2 * q^44 + 6 * q^49 + 24 * q^51 - 18 * q^54 + 4 * q^56 + 8 * q^59 + 8 * q^61 - 2 * q^64 - 6 * q^66 - 4 * q^71 - 16 * q^74 - 16 * q^76 + 14 * q^79 + 18 * q^81 - 12 * q^84 - 14 * q^86 + 12 * q^89 - 12 * q^91 + 22 * q^94 + 6 * q^96 + 12 * q^99

Display 100 coefficients 10 coefficients

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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	3.00000	1.22474
$7$	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
$9$	−6.00000	−2.00000
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	3.00000i	0.866025i
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	− 6.00000i	− 1.41421i
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	− 1.00000i	− 0.213201i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	−3.00000	−0.588348
$27$	9.00000i	1.73205i
$28$	− 2.00000i	− 0.377964i
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	1.00000i	0.176777i
$33$	3.00000i	0.522233i
$34$	−4.00000	−0.685994
$35$	0	0
$36$	6.00000	1.00000
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	8.00000i	1.29777i
$39$	9.00000	1.44115
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	6.00000i	0.925820i
$43$	7.00000i	1.06749i	0.845645	+	0.533745i	$0.179216\pi$
$43$	7.00000i	1.06749i	−0.845645	+	0.533745i	$0.820784\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	0	0
$47$	− 11.0000i	− 1.60451i	−0.596978	−	0.802257i	$-0.703632\pi$
$47$	− 11.0000i	− 1.60451i	0.596978	−	0.802257i	$-0.296368\pi$
$48$	− 3.00000i	− 0.433013i
$49$	3.00000	0.428571
$50$	0	0
$51$	12.0000	1.68034
$52$	− 3.00000i	− 0.416025i
$53$	1.00000i	0.137361i	0.997639	+	0.0686803i	$0.0218788\pi$
$53$	1.00000i	0.137361i	−0.997639	+	0.0686803i	$0.978121\pi$
$54$	−9.00000	−1.22474
$55$	0	0
$56$	2.00000	0.267261
$57$	− 24.0000i	− 3.17888i
$58$	1.00000i	0.131306i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	3.00000i	0.381000i
$63$	− 12.0000i	− 1.51186i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 4.00000i	− 0.485071i
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	6.00000i	0.707107i
$73$	− 12.0000i	− 1.40449i	−0.711934	−	0.702247i	$-0.752180\pi$
$73$	− 12.0000i	− 1.40449i	0.711934	−	0.702247i	$-0.247820\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	−8.00000	−0.917663
$77$	− 2.00000i	− 0.227921i
$78$	9.00000i	1.01905i
$79$	7.00000	0.787562	0.393781	−	0.919204i	$-0.371167\pi$
$79$	7.00000	0.787562	0.393781	+	0.919204i	$0.371167\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	− 2.00000i	− 0.220863i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−6.00000	−0.654654
$85$	0	0
$86$	−7.00000	−0.754829
$87$	− 3.00000i	− 0.321634i
$88$	1.00000i	0.106600i
$89$	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
$91$	−6.00000	−0.628971
$92$	0	0
$93$	− 9.00000i	− 0.933257i
$94$	11.0000	1.13456
$95$	0	0
$96$	3.00000	0.306186
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	3.00000i	0.303046i
$99$	6.00000	0.603023
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	12.0000i	1.18818i
$103$	− 6.00000i	− 0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$	− 6.00000i	− 0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	− 9.00000i	− 0.866025i
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	24.0000	2.27798
$112$	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	$0.678617\pi$
$114$	24.0000	2.24781
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	− 18.0000i	− 1.66410i
$118$	4.00000i	0.368230i
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−10.0000	−0.909091
$122$	4.00000i	0.362143i
$123$	6.00000i	0.541002i
$124$	−3.00000	−0.269408
$125$	0	0
$126$	12.0000	1.06904
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	21.0000	1.84895
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	− 3.00000i	− 0.261116i
$133$	16.0000i	1.38738i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	4.00000	0.342997
$137$	20.0000i	1.70872i	0.519685	+	0.854358i	$0.326049\pi$
$137$	20.0000i	1.70872i	−0.519685	+	0.854358i	$0.673951\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−33.0000	−2.77910
$142$	− 2.00000i	− 0.167836i
$143$	− 3.00000i	− 0.250873i
$144$	−6.00000	−0.500000
$145$	0	0
$146$	12.0000	0.993127
$147$	− 9.00000i	− 0.742307i
$148$	− 8.00000i	− 0.657596i
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	− 8.00000i	− 0.648886i
$153$	− 24.0000i	− 1.94029i
$154$	2.00000	0.161165
$155$	0	0
$156$	−9.00000	−0.720577
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	7.00000i	0.556890i
$159$	3.00000	0.237915
$160$	0	0
$161$	0	0
$162$	9.00000i	0.707107i
$163$	19.0000i	1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$	19.0000i	1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	0	0
$167$	22.0000i	1.70241i	0.524832	+	0.851206i	$0.324128\pi$
$167$	22.0000i	1.70241i	−0.524832	+	0.851206i	$0.675872\pi$
$168$	− 6.00000i	− 0.462910i
$169$	4.00000	0.307692
$170$	0	0
$171$	−48.0000	−3.67065
$172$	− 7.00000i	− 0.533745i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	− 12.0000i	− 0.901975i
$178$	6.00000i	0.449719i
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	0	0
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	− 6.00000i	− 0.444750i
$183$	− 12.0000i	− 0.887066i
$184$	0	0
$185$	0	0
$186$	9.00000	0.659912
$187$	− 4.00000i	− 0.292509i
$188$	11.0000i	0.802257i
$189$	−18.0000	−1.30931
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	3.00000i	0.216506i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	6.00000i	0.426401i
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	8.00000i	0.562878i
$203$	2.00000i	0.140372i
$204$	−12.0000	−0.840168
$205$	0	0
$206$	6.00000	0.418040
$207$	0	0
$208$	3.00000i	0.208013i
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	− 1.00000i	− 0.0686803i
$213$	6.00000i	0.411113i
$214$	−2.00000	−0.136717
$215$	0	0
$216$	9.00000	0.612372
$217$	6.00000i	0.407307i
$218$	− 1.00000i	− 0.0677285i
$219$	−36.0000	−2.43265
$220$	0	0
$221$	−12.0000	−0.807207
$222$	24.0000i	1.61077i
$223$	26.0000i	1.74109i	0.492090	+	0.870544i	$0.336233\pi$
$223$	26.0000i	1.74109i	−0.492090	+	0.870544i	$0.663767\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−18.0000	−1.19734
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	24.0000i	1.58944i
$229$	−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
$231$	−6.00000	−0.394771
$232$	− 1.00000i	− 0.0656532i
$233$	− 25.0000i	− 1.63780i	−0.573933	−	0.818902i	$-0.694583\pi$
$233$	− 25.0000i	− 1.63780i	0.573933	−	0.818902i	$-0.305417\pi$
$234$	18.0000	1.17670
$235$	0	0
$236$	−4.00000	−0.260378
$237$	− 21.0000i	− 1.36410i
$238$	− 8.00000i	− 0.518563i
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	− 10.0000i	− 0.642824i
$243$	0	0
$244$	−4.00000	−0.256074
$245$	0	0
$246$	−6.00000	−0.382546
$247$	24.0000i	1.52708i
$248$	− 3.00000i	− 0.190500i
$249$	0	0
$250$	0	0
$251$	−7.00000	−0.441836	−0.220918	−	0.975292i	$-0.570905\pi$
$251$	−7.00000	−0.441836	−0.220918	+	0.975292i	$0.570905\pi$
$252$	12.0000i	0.755929i
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 21.0000i	− 1.30994i	−0.755653	−	0.654972i	$-0.772680\pi$
$257$	− 21.0000i	− 1.30994i	0.755653	−	0.654972i	$-0.227320\pi$
$258$	21.0000i	1.30740i
$259$	−16.0000	−0.994192
$260$	0	0
$261$	−6.00000	−0.371391
$262$	12.0000i	0.741362i
$263$	− 17.0000i	− 1.04826i	−0.851637	−	0.524132i	$-0.824390\pi$
$263$	− 17.0000i	− 1.04826i	0.851637	−	0.524132i	$-0.175610\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	−16.0000	−0.981023
$267$	− 18.0000i	− 1.10158i
$268$	− 4.00000i	− 0.244339i
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	4.00000i	0.242536i
$273$	18.0000i	1.08941i
$274$	−20.0000	−1.20824
$275$	0	0
$276$	0	0
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	0	0
$279$	−18.0000	−1.07763
$280$	0	0
$281$	−13.0000	−0.775515	−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000	−0.775515	−0.387757	+	0.921761i	$0.626750\pi$
$282$	− 33.0000i	− 1.96512i
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	3.00000	0.177394
$287$	− 4.00000i	− 0.236113i
$288$	− 6.00000i	− 0.353553i
$289$	1.00000	0.0588235
$290$	0	0
$291$	18.0000	1.05518
$292$	12.0000i	0.702247i
$293$	− 34.0000i	− 1.98630i	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	− 34.0000i	− 1.98630i	0.116841	−	0.993151i	$-0.462723\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	8.00000	0.464991
$297$	− 9.00000i	− 0.522233i
$298$	− 3.00000i	− 0.173785i
$299$	0	0
$300$	0	0
$301$	−14.0000	−0.806947
$302$	10.0000i	0.575435i
$303$	− 24.0000i	− 1.37876i
$304$	8.00000	0.458831
$305$	0	0
$306$	24.0000	1.37199
$307$	29.0000i	1.65512i	0.561379	+	0.827559i	$0.310271\pi$
$307$	29.0000i	1.65512i	−0.561379	+	0.827559i	$0.689729\pi$
$308$	2.00000i	0.113961i
$309$	−18.0000	−1.02398
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	− 9.00000i	− 0.509525i
$313$	25.0000i	1.41308i	0.707671	+	0.706542i	$0.249746\pi$
$313$	25.0000i	1.41308i	−0.707671	+	0.706542i	$0.750254\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	−7.00000	−0.393781
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	3.00000i	0.168232i
$319$	−1.00000	−0.0559893
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	32.0000i	1.78053i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−19.0000	−1.05231
$327$	3.00000i	0.165900i
$328$	2.00000i	0.110432i
$329$	22.0000	1.21290
$330$	0	0
$331$	3.00000	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$331$	3.00000	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$332$	0	0
$333$	− 48.0000i	− 2.63038i
$334$	−22.0000	−1.20379
$335$	0	0
$336$	6.00000	0.327327
$337$	− 4.00000i	− 0.217894i	−0.994048	−	0.108947i	$-0.965252\pi$
$337$	− 4.00000i	− 0.217894i	0.994048	−	0.108947i	$-0.0347479\pi$
$338$	4.00000i	0.217571i
$339$	54.0000	2.93288
$340$	0	0
$341$	−3.00000	−0.162459
$342$	− 48.0000i	− 2.59554i
$343$	20.0000i	1.07990i
$344$	7.00000	0.377415
$345$	0	0
$346$	14.0000	0.752645
$347$	− 18.0000i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$	− 18.0000i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$	3.00000i	0.160817i
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	−27.0000	−1.44115
$352$	− 1.00000i	− 0.0533002i
$353$	− 2.00000i	− 0.106449i	−0.998583	−	0.0532246i	$-0.983050\pi$
$353$	− 2.00000i	− 0.106449i	0.998583	−	0.0532246i	$-0.0169499\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−6.00000	−0.317999
$357$	24.0000i	1.27021i
$358$	14.0000i	0.739923i
$359$	−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
$361$	45.0000	2.36842
$362$	− 13.0000i	− 0.683265i
$363$	30.0000i	1.57459i
$364$	6.00000	0.314485
$365$	0	0
$366$	12.0000	0.627250
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	−2.00000	−0.103835
$372$	9.00000i	0.466628i
$373$	− 1.00000i	− 0.0517780i	−0.999665	−	0.0258890i	$-0.991758\pi$
$373$	− 1.00000i	− 0.0517780i	0.999665	−	0.0258890i	$-0.00824165\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−11.0000	−0.567282
$377$	3.00000i	0.154508i
$378$	− 18.0000i	− 0.925820i
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	24.0000	1.22956
$382$	0	0
$383$	− 34.0000i	− 1.73732i	−0.495410	−	0.868659i	$-0.664982\pi$
$383$	− 34.0000i	− 1.73732i	0.495410	−	0.868659i	$-0.335018\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−10.0000	−0.508987
$387$	− 42.0000i	− 2.13498i
$388$	− 6.00000i	− 0.304604i
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	0	0
$392$	− 3.00000i	− 0.151523i
$393$	− 36.0000i	− 1.81596i
$394$	2.00000	0.100759
$395$	0	0
$396$	−6.00000	−0.301511
$397$	− 19.0000i	− 0.953583i	−0.879017	−	0.476791i	$-0.841800\pi$
$397$	− 19.0000i	− 0.953583i	0.879017	−	0.476791i	$-0.158200\pi$
$398$	2.00000i	0.100251i
$399$	48.0000	2.40301
$400$	0	0
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	12.0000i	0.598506i
$403$	9.00000i	0.448322i
$404$	−8.00000	−0.398015
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	− 8.00000i	− 0.396545i
$408$	− 12.0000i	− 0.594089i
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	60.0000	2.95958
$412$	6.00000i	0.295599i
$413$	8.00000i	0.393654i
$414$	0	0
$415$	0	0
$416$	−3.00000	−0.147087
$417$	0	0
$418$	− 8.00000i	− 0.391293i
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	− 25.0000i	− 1.21698i
$423$	66.0000i	3.20903i
$424$	1.00000	0.0485643
$425$	0	0
$426$	−6.00000	−0.290701
$427$	8.00000i	0.387147i
$428$	− 2.00000i	− 0.0966736i
$429$	−9.00000	−0.434524
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	9.00000i	0.433013i
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	1.00000	0.0478913
$437$	0	0
$438$	− 36.0000i	− 1.72015i
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	− 12.0000i	− 0.570782i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	−24.0000	−1.13899
$445$	0	0
$446$	−26.0000	−1.23114
$447$	9.00000i	0.425685i
$448$	− 2.00000i	− 0.0944911i
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	− 18.0000i	− 0.846649i
$453$	− 30.0000i	− 1.40952i
$454$	−18.0000	−0.844782
$455$	0	0
$456$	−24.0000	−1.12390
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	− 14.0000i	− 0.654177i
$459$	−36.0000	−1.68034
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	− 6.00000i	− 0.279145i
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	25.0000	1.15810
$467$	− 23.0000i	− 1.06431i	−0.846646	−	0.532157i	$-0.821382\pi$
$467$	− 23.0000i	− 1.06431i	0.846646	−	0.532157i	$-0.178618\pi$
$468$	18.0000i	0.832050i
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−66.0000	−3.04112
$472$	− 4.00000i	− 0.184115i
$473$	− 7.00000i	− 0.321860i
$474$	21.0000	0.964562
$475$	0	0
$476$	8.00000	0.366679
$477$	− 6.00000i	− 0.274721i
$478$	− 20.0000i	− 0.914779i
$479$	11.0000	0.502603	0.251301	−	0.967909i	$-0.419141\pi$
$479$	11.0000	0.502603	0.251301	+	0.967909i	$0.419141\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	17.0000i	0.774329i
$483$	0	0
$484$	10.0000	0.454545
$485$	0	0
$486$	0	0
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	− 4.00000i	− 0.181071i
$489$	57.0000	2.57763
$490$	0	0
$491$	5.00000	0.225647	0.112823	−	0.993615i	$-0.464011\pi$
$491$	5.00000	0.225647	0.112823	+	0.993615i	$0.464011\pi$
$492$	− 6.00000i	− 0.270501i
$493$	4.00000i	0.180151i
$494$	−24.0000	−1.07981
$495$	0	0
$496$	3.00000	0.134704
$497$	− 4.00000i	− 0.179425i
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	66.0000	2.94866
$502$	− 7.00000i	− 0.312425i
$503$	− 19.0000i	− 0.847168i	−0.905857	−	0.423584i	$-0.860772\pi$
$503$	− 19.0000i	− 0.847168i	0.905857	−	0.423584i	$-0.139228\pi$
$504$	−12.0000	−0.534522
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	− 8.00000i	− 0.354943i
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	1.00000i	0.0441942i
$513$	72.0000i	3.17888i
$514$	21.0000	0.926270
$515$	0	0
$516$	−21.0000	−0.924473
$517$	11.0000i	0.483779i
$518$	− 16.0000i	− 0.703000i
$519$	−42.0000	−1.84360
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	− 6.00000i	− 0.262613i
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	17.0000	0.741235
$527$	12.0000i	0.522728i
$528$	3.00000i	0.130558i
$529$	23.0000	1.00000
$530$	0	0
$531$	−24.0000	−1.04151
$532$	− 16.0000i	− 0.693688i
$533$	− 6.00000i	− 0.259889i
$534$	18.0000	0.778936
$535$	0	0
$536$	4.00000	0.172774
$537$	− 42.0000i	− 1.81243i
$538$	− 20.0000i	− 0.862261i
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	13.0000i	0.558398i
$543$	39.0000i	1.67365i
$544$	−4.00000	−0.171499
$545$	0	0
$546$	−18.0000	−0.770329
$547$	2.00000i	0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$547$	2.00000i	0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$548$	− 20.0000i	− 0.854358i
$549$	−24.0000	−1.02430
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	14.0000i	0.595341i
$554$	14.0000	0.594803
$555$	0	0
$556$	0	0
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	− 18.0000i	− 0.762001i
$559$	−21.0000	−0.888205
$560$	0	0
$561$	−12.0000	−0.506640
$562$	− 13.0000i	− 0.548372i
$563$	− 9.00000i	− 0.379305i	−0.981851	−	0.189652i	$-0.939264\pi$
$563$	− 9.00000i	− 0.379305i	0.981851	−	0.189652i	$-0.0607361\pi$
$564$	33.0000	1.38955
$565$	0	0
$566$	0	0
$567$	18.0000i	0.755929i
$568$	2.00000i	0.0839181i
$569$	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
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	3.00000i	0.125436i
$573$	0	0
$574$	4.00000	0.166957
$575$	0	0
$576$	6.00000	0.250000
$577$	44.0000i	1.83174i	0.401470	+	0.915872i	$0.368499\pi$
$577$	44.0000i	1.83174i	−0.401470	+	0.915872i	$0.631501\pi$
$578$	1.00000i	0.0415945i
$579$	30.0000	1.24676
$580$	0	0
$581$	0	0
$582$	18.0000i	0.746124i
$583$	− 1.00000i	− 0.0414158i
$584$	−12.0000	−0.496564
$585$	0	0
$586$	34.0000	1.40453
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	9.00000i	0.371154i
$589$	24.0000	0.988903
$590$	0	0
$591$	−6.00000	−0.246807
$592$	8.00000i	0.328798i
$593$	− 17.0000i	− 0.698106i	−0.937103	−	0.349053i	$-0.886503\pi$
$593$	− 17.0000i	− 0.698106i	0.937103	−	0.349053i	$-0.113497\pi$
$594$	9.00000	0.369274
$595$	0	0
$596$	3.00000	0.122885
$597$	− 6.00000i	− 0.245564i
$598$	0	0
$599$	−13.0000	−0.531166	−0.265583	−	0.964088i	$-0.585564\pi$
$599$	−13.0000	−0.531166	−0.265583	+	0.964088i	$0.585564\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	− 14.0000i	− 0.570597i
$603$	− 24.0000i	− 0.977356i
$604$	−10.0000	−0.406894
$605$	0	0
$606$	24.0000	0.974933
$607$	− 29.0000i	− 1.17707i	−0.808470	−	0.588537i	$-0.799704\pi$
$607$	− 29.0000i	− 1.17707i	0.808470	−	0.588537i	$-0.200296\pi$
$608$	8.00000i	0.324443i
$609$	6.00000	0.243132
$610$	0	0
$611$	33.0000	1.33504
$612$	24.0000i	0.970143i
$613$	− 27.0000i	− 1.09052i	−0.838267	−	0.545260i	$-0.816431\pi$
$613$	− 27.0000i	− 1.09052i	0.838267	−	0.545260i	$-0.183569\pi$
$614$	−29.0000	−1.17034
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	− 18.0000i	− 0.724066i
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0000i	0.480770i
$624$	9.00000	0.360288
$625$	0	0
$626$	−25.0000	−0.999201
$627$	24.0000i	0.958468i
$628$	22.0000i	0.877896i
$629$	−32.0000	−1.27592
$630$	0	0
$631$	22.0000	0.875806	0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000	0.875806	0.437903	+	0.899022i	$0.355721\pi$
$632$	− 7.00000i	− 0.278445i
$633$	75.0000i	2.98098i
$634$	0	0
$635$	0	0
$636$	−3.00000	−0.118958
$637$	9.00000i	0.356593i
$638$	− 1.00000i	− 0.0395904i
$639$	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
$643$	2.00000i	0.0788723i	0.999222	+	0.0394362i	$0.0125562\pi$
$643$	2.00000i	0.0788723i	−0.999222	+	0.0394362i	$0.987444\pi$
$644$	0	0
$645$	0	0
$646$	−32.0000	−1.25902
$647$	20.0000i	0.786281i	0.919478	+	0.393141i	$0.128611\pi$
$647$	20.0000i	0.786281i	−0.919478	+	0.393141i	$0.871389\pi$
$648$	− 9.00000i	− 0.353553i
$649$	−4.00000	−0.157014
$650$	0	0
$651$	18.0000	0.705476
$652$	− 19.0000i	− 0.744097i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	−3.00000	−0.117309
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	72.0000i	2.80899i
$658$	22.0000i	0.857649i
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	0	0
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	3.00000i	0.116598i
$663$	36.0000i	1.39812i
$664$	0	0
$665$	0	0
$666$	48.0000	1.85996
$667$	0	0
$668$	− 22.0000i	− 0.851206i
$669$	78.0000	3.01565
$670$	0	0
$671$	−4.00000	−0.154418
$672$	6.00000i	0.231455i
$673$	1.00000i	0.0385472i	0.999814	+	0.0192736i	$0.00613535\pi$
$673$	1.00000i	0.0385472i	−0.999814	+	0.0192736i	$0.993865\pi$
$674$	4.00000	0.154074
$675$	0	0
$676$	−4.00000	−0.153846
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	54.0000i	2.07386i
$679$	−12.0000	−0.460518
$680$	0	0
$681$	54.0000	2.06928
$682$	− 3.00000i	− 0.114876i
$683$	− 16.0000i	− 0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$	− 16.0000i	− 0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$	48.0000	1.83533
$685$	0	0
$686$	−20.0000	−0.763604
$687$	42.0000i	1.60240i
$688$	7.00000i	0.266872i
$689$	−3.00000	−0.114291
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	14.0000i	0.532200i
$693$	12.0000i	0.455842i
$694$	18.0000	0.683271
$695$	0	0
$696$	−3.00000	−0.113715
$697$	− 8.00000i	− 0.303022i
$698$	19.0000i	0.719161i
$699$	−75.0000	−2.83676
$700$	0	0
$701$	31.0000	1.17085	0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000	1.17085	0.585427	+	0.810725i	$0.300927\pi$
$702$	− 27.0000i	− 1.01905i
$703$	64.0000i	2.41381i
$704$	1.00000	0.0376889
$705$	0	0
$706$	2.00000	0.0752710
$707$	16.0000i	0.601742i
$708$	12.0000i	0.450988i
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	−42.0000	−1.57512
$712$	− 6.00000i	− 0.224860i
$713$	0	0
$714$	−24.0000	−0.898177
$715$	0	0
$716$	−14.0000	−0.523205
$717$	60.0000i	2.24074i
$718$	− 9.00000i	− 0.335877i
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	45.0000i	1.67473i
$723$	− 51.0000i	− 1.89671i
$724$	13.0000	0.483141
$725$	0	0
$726$	−30.0000	−1.11340
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	6.00000i	0.222375i
$729$	27.0000	1.00000
$730$	0	0
$731$	−28.0000	−1.03562
$732$	12.0000i	0.443533i
$733$	− 48.0000i	− 1.77292i	−0.462805	−	0.886460i	$-0.653157\pi$
$733$	− 48.0000i	− 1.77292i	0.462805	−	0.886460i	$-0.346843\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	0	0
$737$	− 4.00000i	− 0.147342i
$738$	12.0000i	0.441726i
$739$	−1.00000	−0.0367856	−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000	−0.0367856	−0.0183928	+	0.999831i	$0.505855\pi$
$740$	0	0
$741$	72.0000	2.64499
$742$	− 2.00000i	− 0.0734223i
$743$	− 28.0000i	− 1.02722i	−0.858024	−	0.513610i	$-0.828308\pi$
$743$	− 28.0000i	− 1.02722i	0.858024	−	0.513610i	$-0.171692\pi$
$744$	−9.00000	−0.329956
$745$	0	0
$746$	1.00000	0.0366126
$747$	0	0
$748$	4.00000i	0.146254i
$749$	−4.00000	−0.146157
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	− 11.0000i	− 0.401129i
$753$	21.0000i	0.765283i
$754$	−3.00000	−0.109254
$755$	0	0
$756$	18.0000	0.654654
$757$	28.0000i	1.01768i	0.860862	+	0.508839i	$0.169925\pi$
$757$	28.0000i	1.01768i	−0.860862	+	0.508839i	$0.830075\pi$
$758$	28.0000i	1.01701i
$759$	0	0
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	24.0000i	0.869428i
$763$	− 2.00000i	− 0.0724049i
$764$	0	0
$765$	0	0
$766$	34.0000	1.22847
$767$	12.0000i	0.433295i
$768$	− 3.00000i	− 0.108253i
$769$	−48.0000	−1.73092	−0.865462	−	0.500974i	$-0.832975\pi$
$769$	−48.0000	−1.73092	−0.865462	+	0.500974i	$0.832975\pi$
$770$	0	0
$771$	−63.0000	−2.26889
$772$	− 10.0000i	− 0.359908i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	42.0000	1.50966
$775$	0	0
$776$	6.00000	0.215387
$777$	48.0000i	1.72199i
$778$	− 16.0000i	− 0.573628i
$779$	−16.0000	−0.573259
$780$	0	0
$781$	2.00000	0.0715656
$782$	0	0
$783$	9.00000i	0.321634i
$784$	3.00000	0.107143
$785$	0	0
$786$	36.0000	1.28408
$787$	− 46.0000i	− 1.63972i	−0.572562	−	0.819861i	$-0.694050\pi$
$787$	− 46.0000i	− 1.63972i	0.572562	−	0.819861i	$-0.305950\pi$
$788$	2.00000i	0.0712470i
$789$	−51.0000	−1.81565
$790$	0	0
$791$	−36.0000	−1.28001
$792$	− 6.00000i	− 0.213201i
$793$	12.0000i	0.426132i
$794$	19.0000	0.674285
$795$	0	0
$796$	−2.00000	−0.0708881
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	48.0000i	1.69918i
$799$	44.0000	1.55661
$800$	0	0
$801$	−36.0000	−1.27200
$802$	− 5.00000i	− 0.176556i
$803$	12.0000i	0.423471i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−9.00000	−0.317011
$807$	60.0000i	2.11210i
$808$	− 8.00000i	− 0.281439i
$809$	−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
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	− 2.00000i	− 0.0701862i
$813$	− 39.0000i	− 1.36779i
$814$	8.00000	0.280400
$815$	0	0
$816$	12.0000	0.420084
$817$	56.0000i	1.95919i
$818$	− 22.0000i	− 0.769212i
$819$	36.0000	1.25794
$820$	0	0
$821$	−37.0000	−1.29131	−0.645654	−	0.763630i	$-0.723415\pi$
$821$	−37.0000	−1.29131	−0.645654	+	0.763630i	$0.723415\pi$
$822$	60.0000i	2.09274i
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	−8.00000	−0.278356
$827$	− 23.0000i	− 0.799788i	−0.916561	−	0.399894i	$-0.869047\pi$
$827$	− 23.0000i	− 0.799788i	0.916561	−	0.399894i	$-0.130953\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	−42.0000	−1.45696
$832$	− 3.00000i	− 0.104006i
$833$	12.0000i	0.415775i
$834$	0	0
$835$	0	0
$836$	8.00000	0.276686
$837$	27.0000i	0.933257i
$838$	14.0000i	0.483622i
$839$	45.0000	1.55357	0.776786	−	0.629764i	$-0.216849\pi$
$839$	45.0000	1.55357	0.776786	+	0.629764i	$0.216849\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 28.0000i	− 0.964944i
$843$	39.0000i	1.34323i
$844$	25.0000	0.860535
$845$	0	0
$846$	−66.0000	−2.26913
$847$	− 20.0000i	− 0.687208i
$848$	1.00000i	0.0343401i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	− 6.00000i	− 0.205557i
$853$	30.0000i	1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$	30.0000i	1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	2.00000	0.0683586
$857$	11.0000i	0.375753i	0.982193	+	0.187876i	$0.0601604\pi$
$857$	11.0000i	0.375753i	−0.982193	+	0.187876i	$0.939840\pi$
$858$	− 9.00000i	− 0.307255i
$859$	51.0000	1.74010	0.870049	−	0.492966i	$-0.164087\pi$
$859$	51.0000	1.74010	0.870049	+	0.492966i	$0.164087\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	− 20.0000i	− 0.681203i
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	−9.00000	−0.306186
$865$	0	0
$866$	16.0000	0.543702
$867$	− 3.00000i	− 0.101885i
$868$	− 6.00000i	− 0.203653i
$869$	−7.00000	−0.237459
$870$	0	0
$871$	−12.0000	−0.406604
$872$	1.00000i	0.0338643i
$873$	− 36.0000i	− 1.21842i
$874$	0	0
$875$	0	0
$876$	36.0000	1.21633
$877$	23.0000i	0.776655i	0.921521	+	0.388327i	$0.126947\pi$
$877$	23.0000i	0.776655i	−0.921521	+	0.388327i	$0.873053\pi$
$878$	8.00000i	0.269987i
$879$	−102.000	−3.44037
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	− 18.0000i	− 0.606092i
$883$	− 42.0000i	− 1.41341i	−0.707507	−	0.706706i	$-0.750180\pi$
$883$	− 42.0000i	− 1.41341i	0.707507	−	0.706706i	$-0.249820\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−12.0000	−0.403148
$887$	1.00000i	0.0335767i	0.999859	+	0.0167884i	$0.00534415\pi$
$887$	1.00000i	0.0335767i	−0.999859	+	0.0167884i	$0.994656\pi$
$888$	− 24.0000i	− 0.805387i
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−9.00000	−0.301511
$892$	− 26.0000i	− 0.870544i
$893$	− 88.0000i	− 2.94481i
$894$	−9.00000	−0.301005
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	− 22.0000i	− 0.734150i
$899$	3.00000	0.100056
$900$	0	0
$901$	−4.00000	−0.133259
$902$	2.00000i	0.0665927i
$903$	42.0000i	1.39767i
$904$	18.0000	0.598671
$905$	0	0
$906$	30.0000	0.996683
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	− 18.0000i	− 0.597351i
$909$	−48.0000	−1.59206
$910$	0	0
$911$	−3.00000	−0.0993944	−0.0496972	−	0.998764i	$-0.515826\pi$
$911$	−3.00000	−0.0993944	−0.0496972	+	0.998764i	$0.515826\pi$
$912$	− 24.0000i	− 0.794719i
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	14.0000	0.462573
$917$	24.0000i	0.792550i
$918$	− 36.0000i	− 1.18818i
$919$	6.00000	0.197922	0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000	0.197922	0.0989609	+	0.995091i	$0.468448\pi$
$920$	0	0
$921$	87.0000	2.86675
$922$	− 30.0000i	− 0.987997i
$923$	− 6.00000i	− 0.197492i
$924$	6.00000	0.197386
$925$	0	0
$926$	16.0000	0.525793
$927$	36.0000i	1.18240i
$928$	1.00000i	0.0328266i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	25.0000i	0.818902i
$933$	0	0
$934$	23.0000	0.752583
$935$	0	0
$936$	−18.0000	−0.588348
$937$	− 14.0000i	− 0.457360i	−0.973502	−	0.228680i	$-0.926559\pi$
$937$	− 14.0000i	− 0.457360i	0.973502	−	0.228680i	$-0.0734410\pi$
$938$	− 8.00000i	− 0.261209i
$939$	75.0000	2.44753
$940$	0	0
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	− 66.0000i	− 2.15040i
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	7.00000	0.227590
$947$	− 51.0000i	− 1.65728i	−0.559784	−	0.828639i	$-0.689116\pi$
$947$	− 51.0000i	− 1.65728i	0.559784	−	0.828639i	$-0.310884\pi$
$948$	21.0000i	0.682048i
$949$	36.0000	1.16861
$950$	0	0
$951$	0	0
$952$	8.00000i	0.259281i
$953$	15.0000i	0.485898i	0.970039	+	0.242949i	$0.0781147\pi$
$953$	15.0000i	0.485898i	−0.970039	+	0.242949i	$0.921885\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	20.0000	0.646846
$957$	3.00000i	0.0969762i
$958$	11.0000i	0.355394i
$959$	−40.0000	−1.29167
$960$	0	0
$961$	−22.0000	−0.709677
$962$	− 24.0000i	− 0.773791i
$963$	− 12.0000i	− 0.386695i
$964$	−17.0000	−0.547533
$965$	0	0
$966$	0	0
$967$	− 59.0000i	− 1.89731i	−0.316310	−	0.948656i	$-0.602444\pi$
$967$	− 59.0000i	− 1.89731i	0.316310	−	0.948656i	$-0.397556\pi$
$968$	10.0000i	0.321412i
$969$	96.0000	3.08396
$970$	0	0
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	0	0
$973$	0	0
$974$	2.00000	0.0640841
$975$	0	0
$976$	4.00000	0.128037
$977$	− 37.0000i	− 1.18373i	−0.806035	−	0.591867i	$-0.798391\pi$
$977$	− 37.0000i	− 1.18373i	0.806035	−	0.591867i	$-0.201609\pi$
$978$	57.0000i	1.82266i
$979$	−6.00000	−0.191761
$980$	0	0
$981$	6.00000	0.191565
$982$	5.00000i	0.159556i
$983$	39.0000i	1.24391i	0.783054	+	0.621953i	$0.213661\pi$
$983$	39.0000i	1.24391i	−0.783054	+	0.621953i	$0.786339\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	−4.00000	−0.127386
$987$	− 66.0000i	− 2.10080i
$988$	− 24.0000i	− 0.763542i
$989$	0	0
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	3.00000i	0.0952501i
$993$	− 9.00000i	− 0.285606i
$994$	4.00000	0.126872
$995$	0	0
$996$	0	0
$997$	− 16.0000i	− 0.506725i	−0.967371	−	0.253363i	$-0.918463\pi$
$997$	− 16.0000i	− 0.506725i	0.967371	−	0.253363i	$-0.0815366\pi$
$998$	8.00000i	0.253236i
$999$	−72.0000	−2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1450.2.b.f.349.2		2
5.2	odd	4		58.2.a.a.1.1	✓	1
5.3	odd	4		1450.2.a.i.1.1		1
5.4	even	2	inner	1450.2.b.f.349.1		2
15.2	even	4		522.2.a.k.1.1		1
20.7	even	4		464.2.a.f.1.1		1
35.27	even	4		2842.2.a.d.1.1		1
40.27	even	4		1856.2.a.b.1.1		1
40.37	odd	4		1856.2.a.p.1.1		1
55.32	even	4		7018.2.a.c.1.1		1
60.47	odd	4		4176.2.a.bh.1.1		1
65.12	odd	4		9802.2.a.d.1.1		1
145.12	even	4		1682.2.b.e.1681.1		2
145.17	even	4		1682.2.b.e.1681.2		2
145.57	odd	4		1682.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.a.a.1.1	✓	1	5.2	odd	4
464.2.a.f.1.1		1	20.7	even	4
522.2.a.k.1.1		1	15.2	even	4
1450.2.a.i.1.1		1	5.3	odd	4
1450.2.b.f.349.1		2	5.4	even	2	inner
1450.2.b.f.349.2		2	1.1	even	1	trivial
1682.2.a.j.1.1		1	145.57	odd	4
1682.2.b.e.1681.1		2	145.12	even	4
1682.2.b.e.1681.2		2	145.17	even	4
1856.2.a.b.1.1		1	40.27	even	4
1856.2.a.p.1.1		1	40.37	odd	4
2842.2.a.d.1.1		1	35.27	even	4
4176.2.a.bh.1.1		1	60.47	odd	4
7018.2.a.c.1.1		1	55.32	even	4
9802.2.a.d.1.1		1	65.12	odd	4

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} +3.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -6.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} +3.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -6.00000 q^{9} -1.00000 q^{11} +3.00000i q^{12} +3.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} -6.00000i q^{18} +8.00000 q^{19} +6.00000 q^{21} -1.00000i q^{22} -3.00000 q^{24} -3.00000 q^{26} +9.00000i q^{27} -2.00000i q^{28} +1.00000 q^{29} +3.00000 q^{31} +1.00000i q^{32} +3.00000i q^{33} -4.00000 q^{34} +6.00000 q^{36} +8.00000i q^{37} +8.00000i q^{38} +9.00000 q^{39} -2.00000 q^{41} +6.00000i q^{42} +7.00000i q^{43} +1.00000 q^{44} -11.0000i q^{47} -3.00000i q^{48} +3.00000 q^{49} +12.0000 q^{51} -3.00000i q^{52} +1.00000i q^{53} -9.00000 q^{54} +2.00000 q^{56} -24.0000i q^{57} +1.00000i q^{58} +4.00000 q^{59} +4.00000 q^{61} +3.00000i q^{62} -12.0000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +4.00000i q^{67} -4.00000i q^{68} -2.00000 q^{71} +6.00000i q^{72} -12.0000i q^{73} -8.00000 q^{74} -8.00000 q^{76} -2.00000i q^{77} +9.00000i q^{78} +7.00000 q^{79} +9.00000 q^{81} -2.00000i q^{82} -6.00000 q^{84} -7.00000 q^{86} -3.00000i q^{87} +1.00000i q^{88} +6.00000 q^{89} -6.00000 q^{91} -9.00000i q^{93} +11.0000 q^{94} +3.00000 q^{96} +6.00000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 6 * q^6 - 12 * q^9	\( 2 q - 2 q^{4} + 6 q^{6} - 12 q^{9} - 2 q^{11} - 4 q^{14} + 2 q^{16} + 16 q^{19} + 12 q^{21} - 6 q^{24} - 6 q^{26} + 2 q^{29} + 6 q^{31} - 8 q^{34} + 12 q^{36} + 18 q^{39} - 4 q^{41} + 2 q^{44} + 6 q^{49} + 24 q^{51} - 18 q^{54} + 4 q^{56} + 8 q^{59} + 8 q^{61} - 2 q^{64} - 6 q^{66} - 4 q^{71} - 16 q^{74} - 16 q^{76} + 14 q^{79} + 18 q^{81} - 12 q^{84} - 14 q^{86} + 12 q^{89} - 12 q^{91} + 22 q^{94} + 6 q^{96} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^6 - 12 * q^9 - 2 * q^11 - 4 * q^14 + 2 * q^16 + 16 * q^19 + 12 * q^21 - 6 * q^24 - 6 * q^26 + 2 * q^29 + 6 * q^31 - 8 * q^34 + 12 * q^36 + 18 * q^39 - 4 * q^41 + 2 * q^44 + 6 * q^49 + 24 * q^51 - 18 * q^54 + 4 * q^56 + 8 * q^59 + 8 * q^61 - 2 * q^64 - 6 * q^66 - 4 * q^71 - 16 * q^74 - 16 * q^76 + 14 * q^79 + 18 * q^81 - 12 * q^84 - 14 * q^86 + 12 * q^89 - 12 * q^91 + 22 * q^94 + 6 * q^96 + 12 * q^99

Introduction

L-functions

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Database

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