LMFDB - Embedded newform 200.14.a.l.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,14,Mod(1,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("200.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	200.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:	$214.461857904$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 2792237 x^{8} + 94050736 x^{7} + 2615193085270 x^{6} - 30081688120700 x^{5} + \cdots + 14\!\cdots\!75 $ x^10 - 2x^9 - 2792237x^8 + 94050736x^7 + 2615193085270x^6 - 30081688120700x^5 - 919603429545956750x^4 - 94224520282915730000x^3 + 105082623154370946015625x^2 + 25030138583264107436718750*x + 1405679570677464988623046875
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{55}\cdot 3^{5}\cdot 5^{16} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1048.51$ of defining polynomial
Character	$\chi$	$=$	200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2225.01 q^{3} -497533. q^{7} +3.35636e6 q^{9} +O(q^{10})$	$q+2225.01 q^{3} -497533. q^{7} +3.35636e6 q^{9} -5.23136e6 q^{11} -2.26332e7 q^{13} -1.13459e8 q^{17} -8.56912e7 q^{19} -1.10702e9 q^{21} +9.79305e8 q^{23} +3.92056e9 q^{27} -5.34290e8 q^{29} +3.41559e9 q^{31} -1.16398e10 q^{33} +3.08019e10 q^{37} -5.03591e10 q^{39} -7.78417e9 q^{41} +3.72189e10 q^{43} +2.23220e10 q^{47} +1.50650e11 q^{49} -2.52448e11 q^{51} -9.47457e10 q^{53} -1.90664e11 q^{57} -1.49984e11 q^{59} +4.53934e11 q^{61} -1.66990e12 q^{63} +6.15149e11 q^{67} +2.17897e12 q^{69} +1.30848e12 q^{71} +2.03177e12 q^{73} +2.60277e12 q^{77} +2.09382e12 q^{79} +3.37218e12 q^{81} -1.10129e11 q^{83} -1.18880e12 q^{87} +6.25166e12 q^{89} +1.12607e13 q^{91} +7.59974e12 q^{93} -5.81165e12 q^{97} -1.75583e13 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 1276 q^{3} + 341068 q^{7} + 6557498 q^{9}+O(q^{10}) $ 10 * q + 1276 * q^3 + 341068 * q^7 + 6557498 * q^9	$ 10 q + 1276 q^{3} + 341068 q^{7} + 6557498 q^{9} - 4074536 q^{11} - 7720792 q^{13} + 29092704 q^{17} - 82552936 q^{19} - 140909032 q^{21} + 448872532 q^{23} + 6653011048 q^{27} + 4118753908 q^{29} - 161162304 q^{31} + 6635864336 q^{33} + 37799831624 q^{37} + 2651467216 q^{39} + 31205287876 q^{41} + 48886150956 q^{43} - 54068936100 q^{47} + 313063319346 q^{49} - 179707160640 q^{51} - 116117700520 q^{53} - 863281677424 q^{57} - 658612309176 q^{59} + 405225797804 q^{61} - 169179586580 q^{63} - 291232794812 q^{67} + 1989925462312 q^{69} - 956854398704 q^{71} + 262407355312 q^{73} + 3471232133584 q^{77} - 3176139465248 q^{79} + 3191121792770 q^{81} + 5123747754524 q^{83} + 9347096848952 q^{87} + 453451956924 q^{89} + 6935945759120 q^{91} - 806946503040 q^{93} - 11480198579200 q^{97} + 18677113053976 q^{99}+O(q^{100}) $ 10 * q + 1276 * q^3 + 341068 * q^7 + 6557498 * q^9 - 4074536 * q^11 - 7720792 * q^13 + 29092704 * q^17 - 82552936 * q^19 - 140909032 * q^21 + 448872532 * q^23 + 6653011048 * q^27 + 4118753908 * q^29 - 161162304 * q^31 + 6635864336 * q^33 + 37799831624 * q^37 + 2651467216 * q^39 + 31205287876 * q^41 + 48886150956 * q^43 - 54068936100 * q^47 + 313063319346 * q^49 - 179707160640 * q^51 - 116117700520 * q^53 - 863281677424 * q^57 - 658612309176 * q^59 + 405225797804 * q^61 - 169179586580 * q^63 - 291232794812 * q^67 + 1989925462312 * q^69 - 956854398704 * q^71 + 262407355312 * q^73 + 3471232133584 * q^77 - 3176139465248 * q^79 + 3191121792770 * q^81 + 5123747754524 * q^83 + 9347096848952 * q^87 + 453451956924 * q^89 + 6935945759120 * q^91 - 806946503040 * q^93 - 11480198579200 * q^97 + 18677113053976 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2225.01	1.76216	0.881078	−	0.472970i	$-0.156818\pi$
$3$	2225.01	1.76216	0.881078	+	0.472970i	$0.156818\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−497533.	−1.59840	−0.799198	−	0.601067i	$-0.794742\pi$
$7$	−497533.	−1.59840	−0.799198	+	0.601067i	$0.794742\pi$
$8$	0	0
$9$	3.35636e6	2.10520
$10$	0	0
$11$	−5.23136e6	−0.890352	−0.445176	−	0.895443i	$-0.646859\pi$
$11$	−5.23136e6	−0.890352	−0.445176	+	0.895443i	$0.646859\pi$
$12$	0	0
$13$	−2.26332e7	−1.30051	−0.650255	−	0.759716i	$-0.725338\pi$
$13$	−2.26332e7	−1.30051	−0.650255	+	0.759716i	$0.725338\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.13459e8	−1.14004	−0.570021	−	0.821630i	$-0.693065\pi$
$17$	−1.13459e8	−1.14004	−0.570021	+	0.821630i	$0.693065\pi$
$18$	0	0
$19$	−8.56912e7	−0.417867	−0.208933	−	0.977930i	$-0.566999\pi$
$19$	−8.56912e7	−0.417867	−0.208933	+	0.977930i	$0.566999\pi$
$20$	0	0
$21$	−1.10702e9	−2.81663
$22$	0	0
$23$	9.79305e8	1.37939	0.689695	−	0.724100i	$-0.257745\pi$
$23$	9.79305e8	1.37939	0.689695	+	0.724100i	$0.257745\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.92056e9	1.94753
$28$	0	0
$29$	−5.34290e8	−0.166798	−0.0833988	−	0.996516i	$-0.526578\pi$
$29$	−5.34290e8	−0.166798	−0.0833988	+	0.996516i	$0.526578\pi$
$30$	0	0
$31$	3.41559e9	0.691218	0.345609	−	0.938379i	$-0.387672\pi$
$31$	3.41559e9	0.691218	0.345609	+	0.938379i	$0.387672\pi$
$32$	0	0
$33$	−1.16398e10	−1.56894
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.08019e10	1.97363	0.986817	−	0.161841i	$-0.0517433\pi$
$37$	3.08019e10	1.97363	0.986817	+	0.161841i	$0.0517433\pi$
$38$	0	0
$39$	−5.03591e10	−2.29170
$40$	0	0
$41$	−7.78417e9	−0.255928	−0.127964	−	0.991779i	$-0.540844\pi$
$41$	−7.78417e9	−0.255928	−0.127964	+	0.991779i	$0.540844\pi$
$42$	0	0
$43$	3.72189e10	0.897881	0.448941	−	0.893562i	$-0.351801\pi$
$43$	3.72189e10	0.897881	0.448941	+	0.893562i	$0.351801\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.23220e10	0.302063	0.151031	−	0.988529i	$-0.451741\pi$
$47$	2.23220e10	0.302063	0.151031	+	0.988529i	$0.451741\pi$
$48$	0	0
$49$	1.50650e11	1.55487
$50$	0	0
$51$	−2.52448e11	−2.00893
$52$	0	0
$53$	−9.47457e10	−0.587173	−0.293587	−	0.955932i	$-0.594849\pi$
$53$	−9.47457e10	−0.587173	−0.293587	+	0.955932i	$0.594849\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.90664e11	−0.736347
$58$	0	0
$59$	−1.49984e11	−0.462922	−0.231461	−	0.972844i	$-0.574351\pi$
$59$	−1.49984e11	−0.462922	−0.231461	+	0.972844i	$0.574351\pi$
$60$	0	0
$61$	4.53934e11	1.12810	0.564051	−	0.825740i	$-0.309242\pi$
$61$	4.53934e11	1.12810	0.564051	+	0.825740i	$0.309242\pi$
$62$	0	0
$63$	−1.66990e12	−3.36494
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.15149e11	0.830796	0.415398	−	0.909640i	$-0.363642\pi$
$67$	6.15149e11	0.830796	0.415398	+	0.909640i	$0.363642\pi$
$68$	0	0
$69$	2.17897e12	2.43070
$70$	0	0
$71$	1.30848e12	1.21223	0.606117	−	0.795375i	$-0.292726\pi$
$71$	1.30848e12	1.21223	0.606117	+	0.795375i	$0.292726\pi$
$72$	0	0
$73$	2.03177e12	1.57136	0.785680	−	0.618633i	$-0.212313\pi$
$73$	2.03177e12	1.57136	0.785680	+	0.618633i	$0.212313\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.60277e12	1.42314
$78$	0	0
$79$	2.09382e12	0.969089	0.484544	−	0.874767i	$-0.338985\pi$
$79$	2.09382e12	0.969089	0.484544	+	0.874767i	$0.338985\pi$
$80$	0	0
$81$	3.37218e12	1.32665
$82$	0	0
$83$	−1.10129e11	−0.0369738	−0.0184869	−	0.999829i	$-0.505885\pi$
$83$	−1.10129e11	−0.0369738	−0.0184869	+	0.999829i	$0.505885\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.18880e12	−0.293923
$88$	0	0
$89$	6.25166e12	1.33340	0.666699	−	0.745327i	$-0.267707\pi$
$89$	6.25166e12	1.33340	0.666699	+	0.745327i	$0.267707\pi$
$90$	0	0
$91$	1.12607e13	2.07873
$92$	0	0
$93$	7.59974e12	1.21803
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.81165e12	−0.708407	−0.354204	−	0.935168i	$-0.615248\pi$
$97$	−5.81165e12	−0.708407	−0.354204	+	0.935168i	$0.615248\pi$
$98$	0	0
$99$	−1.75583e13	−1.87437
$100$	0	0
$101$	4.36952e12	0.409585	0.204793	−	0.978805i	$-0.434348\pi$
$101$	4.36952e12	0.409585	0.204793	+	0.978805i	$0.434348\pi$
$102$	0	0
$103$	−3.71585e12	−0.306631	−0.153316	−	0.988177i	$-0.548995\pi$
$103$	−3.71585e12	−0.306631	−0.153316	+	0.988177i	$0.548995\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.67384e12	−0.429913	−0.214957	−	0.976624i	$-0.568961\pi$
$107$	−6.67384e12	−0.429913	−0.214957	+	0.976624i	$0.568961\pi$
$108$	0	0
$109$	−2.73131e13	−1.55991	−0.779954	−	0.625837i	$-0.784758\pi$
$109$	−2.73131e13	−1.55991	−0.779954	+	0.625837i	$0.784758\pi$
$110$	0	0
$111$	6.85347e13	3.47785
$112$	0	0
$113$	6.29589e12	0.284477	0.142239	−	0.989832i	$-0.454570\pi$
$113$	6.29589e12	0.284477	0.142239	+	0.989832i	$0.454570\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.59651e13	−2.73783
$118$	0	0
$119$	5.64495e13	1.82224
$120$	0	0
$121$	−7.15561e12	−0.207273
$122$	0	0
$123$	−1.73199e13	−0.450985
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.18558e13	1.94260	0.971298	−	0.237867i	$-0.0764484\pi$
$127$	9.18558e13	1.94260	0.971298	+	0.237867i	$0.0764484\pi$
$128$	0	0
$129$	8.28126e13	1.58221
$130$	0	0
$131$	−3.99245e13	−0.690203	−0.345101	−	0.938565i	$-0.612155\pi$
$131$	−3.99245e13	−0.690203	−0.345101	+	0.938565i	$0.612155\pi$
$132$	0	0
$133$	4.26342e13	0.667917
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.75752e13	1.00240	0.501198	−	0.865333i	$-0.332893\pi$
$137$	7.75752e13	1.00240	0.501198	+	0.865333i	$0.332893\pi$
$138$	0	0
$139$	−5.09202e13	−0.598817	−0.299408	−	0.954125i	$-0.596789\pi$
$139$	−5.09202e13	−0.598817	−0.299408	+	0.954125i	$0.596789\pi$
$140$	0	0
$141$	4.96668e13	0.532282
$142$	0	0
$143$	1.18402e14	1.15791
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.35198e14	2.73993
$148$	0	0
$149$	9.21164e13	0.689646	0.344823	−	0.938668i	$-0.387939\pi$
$149$	9.21164e13	0.689646	0.344823	+	0.938668i	$0.387939\pi$
$150$	0	0
$151$	−1.45811e14	−1.00101	−0.500506	−	0.865733i	$-0.666853\pi$
$151$	−1.45811e14	−1.00101	−0.500506	+	0.865733i	$0.666853\pi$
$152$	0	0
$153$	−3.80809e14	−2.40001
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.97614e14	−1.58601	−0.793005	−	0.609215i	$-0.791485\pi$
$157$	−2.97614e14	−1.58601	−0.793005	+	0.609215i	$0.791485\pi$
$158$	0	0
$159$	−2.10810e14	−1.03469
$160$	0	0
$161$	−4.87237e14	−2.20481
$162$	0	0
$163$	4.60725e13	0.192408	0.0962039	−	0.995362i	$-0.469330\pi$
$163$	4.60725e13	0.192408	0.0962039	+	0.995362i	$0.469330\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.84971e14	−1.01659	−0.508293	−	0.861184i	$-0.669723\pi$
$167$	−2.84971e14	−1.01659	−0.508293	+	0.861184i	$0.669723\pi$
$168$	0	0
$169$	2.09385e14	0.691325
$170$	0	0
$171$	−2.87611e14	−0.879692
$172$	0	0
$173$	1.06602e14	0.302320	0.151160	−	0.988509i	$-0.451699\pi$
$173$	1.06602e14	0.302320	0.151160	+	0.988509i	$0.451699\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.33717e14	−0.815741
$178$	0	0
$179$	−4.21296e14	−0.957288	−0.478644	−	0.878009i	$-0.658871\pi$
$179$	−4.21296e14	−0.957288	−0.478644	+	0.878009i	$0.658871\pi$
$180$	0	0
$181$	−4.44348e14	−0.939318	−0.469659	−	0.882848i	$-0.655623\pi$
$181$	−4.44348e14	−0.939318	−0.469659	+	0.882848i	$0.655623\pi$
$182$	0	0
$183$	1.01001e15	1.98789
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.93544e14	1.01504
$188$	0	0
$189$	−1.95061e15	−3.11292
$190$	0	0
$191$	−2.66867e14	−0.397721	−0.198860	−	0.980028i	$-0.563724\pi$
$191$	−2.66867e14	−0.397721	−0.198860	+	0.980028i	$0.563724\pi$
$192$	0	0
$193$	1.94784e14	0.271288	0.135644	−	0.990758i	$-0.456690\pi$
$193$	1.94784e14	0.271288	0.135644	+	0.990758i	$0.456690\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.16793e14	−0.751812	−0.375906	−	0.926658i	$-0.622668\pi$
$197$	−6.16793e14	−0.751812	−0.375906	+	0.926658i	$0.622668\pi$
$198$	0	0
$199$	9.33142e13	0.106513	0.0532565	−	0.998581i	$-0.483040\pi$
$199$	9.33142e13	0.106513	0.0532565	+	0.998581i	$0.483040\pi$
$200$	0	0
$201$	1.36872e15	1.46399
$202$	0	0
$203$	2.65827e14	0.266609
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.28690e15	2.90389
$208$	0	0
$209$	4.48281e14	0.372049
$210$	0	0
$211$	−1.85043e15	−1.44357	−0.721783	−	0.692119i	$-0.756677\pi$
$211$	−1.85043e15	−1.44357	−0.721783	+	0.692119i	$0.756677\pi$
$212$	0	0
$213$	2.91138e15	2.13615
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.69937e15	−1.10484
$218$	0	0
$219$	4.52071e15	2.76898
$220$	0	0
$221$	2.56793e15	1.48263
$222$	0	0
$223$	2.18557e15	1.19010	0.595051	−	0.803688i	$-0.297132\pi$
$223$	2.18557e15	1.19010	0.595051	+	0.803688i	$0.297132\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.00051e15	−1.45555	−0.727774	−	0.685817i	$-0.759445\pi$
$227$	−3.00051e15	−1.45555	−0.727774	+	0.685817i	$0.759445\pi$
$228$	0	0
$229$	6.55798e14	0.300496	0.150248	−	0.988648i	$-0.451993\pi$
$229$	6.55798e14	0.300496	0.150248	+	0.988648i	$0.451993\pi$
$230$	0	0
$231$	5.79120e15	2.50779
$232$	0	0
$233$	−2.64302e15	−1.08215	−0.541075	−	0.840974i	$-0.681983\pi$
$233$	−2.64302e15	−1.08215	−0.541075	+	0.840974i	$0.681983\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.65878e15	1.70769
$238$	0	0
$239$	2.80852e15	0.974744	0.487372	−	0.873194i	$-0.337956\pi$
$239$	2.80852e15	0.974744	0.487372	+	0.873194i	$0.337956\pi$
$240$	0	0
$241$	−5.62309e14	−0.184869	−0.0924345	−	0.995719i	$-0.529465\pi$
$241$	−5.62309e14	−0.184869	−0.0924345	+	0.995719i	$0.529465\pi$
$242$	0	0
$243$	1.25250e15	0.390244
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.93946e15	0.543440
$248$	0	0
$249$	−2.45038e14	−0.0651536
$250$	0	0
$251$	−1.14714e15	−0.289560	−0.144780	−	0.989464i	$-0.546247\pi$
$251$	−1.14714e15	−0.289560	−0.144780	+	0.989464i	$0.546247\pi$
$252$	0	0
$253$	−5.12309e15	−1.22814
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.16628e15	−0.901951	−0.450976	−	0.892536i	$-0.648924\pi$
$257$	−4.16628e15	−0.901951	−0.450976	+	0.892536i	$0.648924\pi$
$258$	0	0
$259$	−1.53250e16	−3.15465
$260$	0	0
$261$	−1.79327e15	−0.351142
$262$	0	0
$263$	5.11944e15	0.953916	0.476958	−	0.878926i	$-0.341739\pi$
$263$	5.11944e15	0.953916	0.476958	+	0.878926i	$0.341739\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.39100e16	2.34966
$268$	0	0
$269$	1.00289e16	1.61385	0.806924	−	0.590655i	$-0.201130\pi$
$269$	1.00289e16	1.61385	0.806924	+	0.590655i	$0.201130\pi$
$270$	0	0
$271$	3.58519e15	0.549809	0.274904	−	0.961472i	$-0.411354\pi$
$271$	3.58519e15	0.549809	0.274904	+	0.961472i	$0.411354\pi$
$272$	0	0
$273$	2.50553e16	3.66305
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.35242e15	0.977939	0.488969	−	0.872301i	$-0.337373\pi$
$277$	7.35242e15	0.977939	0.488969	+	0.872301i	$0.337373\pi$
$278$	0	0
$279$	1.14640e16	1.45515
$280$	0	0
$281$	1.38116e16	1.67361	0.836803	−	0.547504i	$-0.184422\pi$
$281$	1.38116e16	1.67361	0.836803	+	0.547504i	$0.184422\pi$
$282$	0	0
$283$	1.36548e16	1.58006	0.790029	−	0.613070i	$-0.210066\pi$
$283$	1.36548e16	1.58006	0.790029	+	0.613070i	$0.210066\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.87288e15	0.409074
$288$	0	0
$289$	2.96834e15	0.299694
$290$	0	0
$291$	−1.29310e16	−1.24832
$292$	0	0
$293$	1.58062e15	0.145944	0.0729722	−	0.997334i	$-0.476752\pi$
$293$	1.58062e15	0.145944	0.0729722	+	0.997334i	$0.476752\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.05099e16	−1.73399
$298$	0	0
$299$	−2.21648e16	−1.79391
$300$	0	0
$301$	−1.85176e16	−1.43517
$302$	0	0
$303$	9.72223e15	0.721754
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.87367e15	0.264073	0.132036	−	0.991245i	$-0.457848\pi$
$307$	3.87367e15	0.264073	0.132036	+	0.991245i	$0.457848\pi$
$308$	0	0
$309$	−8.26782e15	−0.540332
$310$	0	0
$311$	2.50260e16	1.56837	0.784185	−	0.620527i	$-0.213081\pi$
$311$	2.50260e16	1.56837	0.784185	+	0.620527i	$0.213081\pi$
$312$	0	0
$313$	−4.11831e15	−0.247560	−0.123780	−	0.992310i	$-0.539502\pi$
$313$	−4.11831e15	−0.247560	−0.123780	+	0.992310i	$0.539502\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.07654e16	1.70285	0.851426	−	0.524475i	$-0.175739\pi$
$317$	3.07654e16	1.70285	0.851426	+	0.524475i	$0.175739\pi$
$318$	0	0
$319$	2.79506e15	0.148509
$320$	0	0
$321$	−1.48494e16	−0.757575
$322$	0	0
$323$	9.72243e15	0.476386
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.07721e16	−2.74880
$328$	0	0
$329$	−1.11059e16	−0.482816
$330$	0	0
$331$	−2.54394e16	−1.06322	−0.531612	−	0.846988i	$-0.678413\pi$
$331$	−2.54394e16	−1.06322	−0.531612	+	0.846988i	$0.678413\pi$
$332$	0	0
$333$	1.03382e17	4.15489
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.84380e16	1.05756	0.528779	−	0.848759i	$-0.322650\pi$
$337$	2.84380e16	1.05756	0.528779	+	0.848759i	$0.322650\pi$
$338$	0	0
$339$	1.40085e16	0.501294
$340$	0	0
$341$	−1.78682e16	−0.615427
$342$	0	0
$343$	−2.67479e16	−0.886905
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.12432e16	−0.960757	−0.480379	−	0.877061i	$-0.659501\pi$
$347$	−3.12432e16	−0.960757	−0.480379	+	0.877061i	$0.659501\pi$
$348$	0	0
$349$	5.43320e15	0.160950	0.0804749	−	0.996757i	$-0.474356\pi$
$349$	5.43320e15	0.160950	0.0804749	+	0.996757i	$0.474356\pi$
$350$	0	0
$351$	−8.87347e16	−2.53278
$352$	0	0
$353$	−1.19480e16	−0.328670	−0.164335	−	0.986405i	$-0.552548\pi$
$353$	−1.19480e16	−0.328670	−0.164335	+	0.986405i	$0.552548\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.25601e17	3.21107
$358$	0	0
$359$	−2.62711e16	−0.647685	−0.323842	−	0.946111i	$-0.604975\pi$
$359$	−2.62711e16	−0.647685	−0.323842	+	0.946111i	$0.604975\pi$
$360$	0	0
$361$	−3.47100e16	−0.825387
$362$	0	0
$363$	−1.59213e16	−0.365247
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.97984e16	0.422962	0.211481	−	0.977382i	$-0.432171\pi$
$367$	1.97984e16	0.422962	0.211481	+	0.977382i	$0.432171\pi$
$368$	0	0
$369$	−2.61265e16	−0.538778
$370$	0	0
$371$	4.71391e16	0.938536
$372$	0	0
$373$	2.48267e16	0.477322	0.238661	−	0.971103i	$-0.423292\pi$
$373$	2.48267e16	0.477322	0.238661	+	0.971103i	$0.423292\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.20927e16	0.216922
$378$	0	0
$379$	1.96724e16	0.340960	0.170480	−	0.985361i	$-0.445468\pi$
$379$	1.96724e16	0.340960	0.170480	+	0.985361i	$0.445468\pi$
$380$	0	0
$381$	2.04380e17	3.42316
$382$	0	0
$383$	7.77108e16	1.25802	0.629012	−	0.777395i	$-0.283460\pi$
$383$	7.77108e16	1.25802	0.629012	+	0.777395i	$0.283460\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.24920e17	1.89022
$388$	0	0
$389$	−3.95198e16	−0.578286	−0.289143	−	0.957286i	$-0.593370\pi$
$389$	−3.95198e16	−0.578286	−0.289143	+	0.957286i	$0.593370\pi$
$390$	0	0
$391$	−1.11111e17	−1.57256
$392$	0	0
$393$	−8.88327e16	−1.21624
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.66074e16	−0.341086	−0.170543	−	0.985350i	$-0.554552\pi$
$397$	−2.66074e16	−0.341086	−0.170543	+	0.985350i	$0.554552\pi$
$398$	0	0
$399$	9.48617e16	1.17697
$400$	0	0
$401$	1.06905e16	0.128398	0.0641991	−	0.997937i	$-0.479551\pi$
$401$	1.06905e16	0.128398	0.0641991	+	0.997937i	$0.479551\pi$
$402$	0	0
$403$	−7.73056e16	−0.898935
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.61136e17	−1.75723
$408$	0	0
$409$	1.16605e17	1.23174	0.615868	−	0.787850i	$-0.288806\pi$
$409$	1.16605e17	1.23174	0.615868	+	0.787850i	$0.288806\pi$
$410$	0	0
$411$	1.72606e17	1.76638
$412$	0	0
$413$	7.46221e16	0.739933
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.13298e17	−1.05521
$418$	0	0
$419$	1.15914e16	0.104651	0.0523255	−	0.998630i	$-0.483337\pi$
$419$	1.15914e16	0.104651	0.0523255	+	0.998630i	$0.483337\pi$
$420$	0	0
$421$	4.16863e16	0.364888	0.182444	−	0.983216i	$-0.441599\pi$
$421$	4.16863e16	0.364888	0.182444	+	0.983216i	$0.441599\pi$
$422$	0	0
$423$	7.49207e16	0.635901
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.25847e17	−1.80316
$428$	0	0
$429$	2.63446e17	2.04042
$430$	0	0
$431$	−4.68226e15	−0.0351847	−0.0175923	−	0.999845i	$-0.505600\pi$
$431$	−4.68226e15	−0.0351847	−0.0175923	+	0.999845i	$0.505600\pi$
$432$	0	0
$433$	−6.02602e16	−0.439399	−0.219699	−	0.975568i	$-0.570508\pi$
$433$	−6.02602e16	−0.439399	−0.219699	+	0.975568i	$0.570508\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.39179e16	−0.576401
$438$	0	0
$439$	−1.16184e17	−0.774686	−0.387343	−	0.921936i	$-0.626607\pi$
$439$	−1.16184e17	−0.774686	−0.387343	+	0.921936i	$0.626607\pi$
$440$	0	0
$441$	5.05636e17	3.27331
$442$	0	0
$443$	2.15208e17	1.35280	0.676400	−	0.736535i	$-0.263539\pi$
$443$	2.15208e17	1.35280	0.676400	+	0.736535i	$0.263539\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.04960e17	1.21526
$448$	0	0
$449$	−9.18109e16	−0.528802	−0.264401	−	0.964413i	$-0.585174\pi$
$449$	−9.18109e16	−0.528802	−0.264401	+	0.964413i	$0.585174\pi$
$450$	0	0
$451$	4.07218e16	0.227866
$452$	0	0
$453$	−3.24431e17	−1.76394
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.39168e16	0.328216	0.164108	−	0.986442i	$-0.447525\pi$
$457$	6.39168e16	0.328216	0.164108	+	0.986442i	$0.447525\pi$
$458$	0	0
$459$	−4.44823e17	−2.22026
$460$	0	0
$461$	1.06709e17	0.517781	0.258891	−	0.965907i	$-0.416643\pi$
$461$	1.06709e17	0.517781	0.258891	+	0.965907i	$0.416643\pi$
$462$	0	0
$463$	1.82731e17	0.862059	0.431029	−	0.902338i	$-0.358151\pi$
$463$	1.82731e17	0.862059	0.431029	+	0.902338i	$0.358151\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.05552e17	1.36309	0.681546	−	0.731775i	$-0.261308\pi$
$467$	3.05552e17	1.36309	0.681546	+	0.731775i	$0.261308\pi$
$468$	0	0
$469$	−3.06057e17	−1.32794
$470$	0	0
$471$	−6.62196e17	−2.79480
$472$	0	0
$473$	−1.94706e17	−0.799431
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.18001e17	−1.23612
$478$	0	0
$479$	3.66538e17	1.38656	0.693280	−	0.720668i	$-0.256165\pi$
$479$	3.66538e17	1.38656	0.693280	+	0.720668i	$0.256165\pi$
$480$	0	0
$481$	−6.97145e17	−2.56673
$482$	0	0
$483$	−1.08411e18	−3.88522
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.56489e17	0.871226	0.435613	−	0.900134i	$-0.356532\pi$
$487$	2.56489e17	0.871226	0.435613	+	0.900134i	$0.356532\pi$
$488$	0	0
$489$	1.02512e17	0.339053
$490$	0	0
$491$	4.07478e17	1.31243	0.656213	−	0.754576i	$-0.272157\pi$
$491$	4.07478e17	1.31243	0.656213	+	0.754576i	$0.272157\pi$
$492$	0	0
$493$	6.06199e16	0.190156
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.51010e17	−1.93763
$498$	0	0
$499$	−1.48328e17	−0.430101	−0.215051	−	0.976603i	$-0.568992\pi$
$499$	−1.48328e17	−0.430101	−0.215051	+	0.976603i	$0.568992\pi$
$500$	0	0
$501$	−6.34065e17	−1.79138
$502$	0	0
$503$	2.00392e17	0.551681	0.275840	−	0.961203i	$-0.411044\pi$
$503$	2.00392e17	0.551681	0.275840	+	0.961203i	$0.411044\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.65885e17	1.21822
$508$	0	0
$509$	−1.65450e17	−0.421697	−0.210849	−	0.977519i	$-0.567623\pi$
$509$	−1.65450e17	−0.421697	−0.210849	+	0.977519i	$0.567623\pi$
$510$	0	0
$511$	−1.01087e18	−2.51166
$512$	0	0
$513$	−3.35958e17	−0.813808
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.16774e17	−0.268942
$518$	0	0
$519$	2.37192e17	0.532735
$520$	0	0
$521$	4.99446e17	1.09407	0.547033	−	0.837111i	$-0.315757\pi$
$521$	4.99446e17	1.09407	0.547033	+	0.837111i	$0.315757\pi$
$522$	0	0
$523$	−7.68859e17	−1.64280	−0.821401	−	0.570350i	$-0.806807\pi$
$523$	−7.68859e17	−1.64280	−0.821401	+	0.570350i	$0.806807\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.87529e17	−0.788017
$528$	0	0
$529$	4.55002e17	0.902717
$530$	0	0
$531$	−5.03402e17	−0.974542
$532$	0	0
$533$	1.76180e17	0.332836
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.37389e17	−1.68689
$538$	0	0
$539$	−7.88104e17	−1.38438
$540$	0	0
$541$	5.68095e17	0.974179	0.487090	−	0.873352i	$-0.338058\pi$
$541$	5.68095e17	0.974179	0.487090	+	0.873352i	$0.338058\pi$
$542$	0	0
$543$	−9.88681e17	−1.65523
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.62589e17	1.21723	0.608616	−	0.793465i	$-0.291725\pi$
$547$	7.62589e17	1.21723	0.608616	+	0.793465i	$0.291725\pi$
$548$	0	0
$549$	1.52357e18	2.37488
$550$	0	0
$551$	4.57839e16	0.0696992
$552$	0	0
$553$	−1.04175e18	−1.54899
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.42611e17	0.769891	0.384946	−	0.922939i	$-0.374220\pi$
$557$	5.42611e17	0.769891	0.384946	+	0.922939i	$0.374220\pi$
$558$	0	0
$559$	−8.42382e17	−1.16770
$560$	0	0
$561$	1.32064e18	1.78866
$562$	0	0
$563$	7.22031e17	0.955545	0.477772	−	0.878484i	$-0.341444\pi$
$563$	7.22031e17	0.955545	0.477772	+	0.878484i	$0.341444\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.67777e18	−2.12052
$568$	0	0
$569$	−3.81421e17	−0.471166	−0.235583	−	0.971854i	$-0.575700\pi$
$569$	−3.81421e17	−0.471166	−0.235583	+	0.971854i	$0.575700\pi$
$570$	0	0
$571$	−9.27717e16	−0.112016	−0.0560081	−	0.998430i	$-0.517837\pi$
$571$	−9.27717e16	−0.112016	−0.0560081	+	0.998430i	$0.517837\pi$
$572$	0	0
$573$	−5.93783e17	−0.700846
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.56031e18	1.76022	0.880110	−	0.474770i	$-0.157469\pi$
$577$	1.56031e18	1.76022	0.880110	+	0.474770i	$0.157469\pi$
$578$	0	0
$579$	4.33397e17	0.478052
$580$	0	0
$581$	5.47928e16	0.0590988
$582$	0	0
$583$	4.95649e17	0.522791
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.86952e18	−1.88618	−0.943089	−	0.332541i	$-0.892094\pi$
$587$	−1.86952e18	−1.88618	−0.943089	+	0.332541i	$0.892094\pi$
$588$	0	0
$589$	−2.92686e17	−0.288837
$590$	0	0
$591$	−1.37237e18	−1.32481
$592$	0	0
$593$	1.38827e18	1.31105	0.655524	−	0.755174i	$-0.272448\pi$
$593$	1.38827e18	1.31105	0.655524	+	0.755174i	$0.272448\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.07625e17	0.187693
$598$	0	0
$599$	−1.62907e18	−1.44101	−0.720504	−	0.693451i	$-0.756089\pi$
$599$	−1.62907e18	−1.44101	−0.720504	+	0.693451i	$0.756089\pi$
$600$	0	0
$601$	1.88662e18	1.63306	0.816528	−	0.577307i	$-0.195896\pi$
$601$	1.88662e18	1.63306	0.816528	+	0.577307i	$0.195896\pi$
$602$	0	0
$603$	2.06466e18	1.74899
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.87475e17	0.639014	0.319507	−	0.947584i	$-0.396483\pi$
$607$	7.87475e17	0.639014	0.319507	+	0.947584i	$0.396483\pi$
$608$	0	0
$609$	5.91468e17	0.469806
$610$	0	0
$611$	−5.05218e17	−0.392835
$612$	0	0
$613$	6.33652e17	0.482345	0.241172	−	0.970482i	$-0.422468\pi$
$613$	6.33652e17	0.482345	0.241172	+	0.970482i	$0.422468\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.41320e16	−0.0686884	−0.0343442	−	0.999410i	$-0.510934\pi$
$617$	−9.41320e16	−0.0686884	−0.0343442	+	0.999410i	$0.510934\pi$
$618$	0	0
$619$	1.05666e18	0.755001	0.377501	−	0.926009i	$-0.376784\pi$
$619$	1.05666e18	0.755001	0.377501	+	0.926009i	$0.376784\pi$
$620$	0	0
$621$	3.83943e18	2.68640
$622$	0	0
$623$	−3.11040e18	−2.13130
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9.97432e17	0.655608
$628$	0	0
$629$	−3.49475e18	−2.25002
$630$	0	0
$631$	2.47390e18	1.56024	0.780118	−	0.625632i	$-0.215159\pi$
$631$	2.47390e18	1.56024	0.780118	+	0.625632i	$0.215159\pi$
$632$	0	0
$633$	−4.11723e18	−2.54379
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.40969e18	−2.02213
$638$	0	0
$639$	4.39172e18	2.55199
$640$	0	0
$641$	1.32920e18	0.756857	0.378429	−	0.925630i	$-0.376465\pi$
$641$	1.32920e18	0.756857	0.378429	+	0.925630i	$0.376465\pi$
$642$	0	0
$643$	−2.24470e17	−0.125253	−0.0626263	−	0.998037i	$-0.519948\pi$
$643$	−2.24470e17	−0.125253	−0.0626263	+	0.998037i	$0.519948\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.62151e18	−1.40499	−0.702495	−	0.711689i	$-0.747931\pi$
$647$	−2.62151e18	−1.40499	−0.702495	+	0.711689i	$0.747931\pi$
$648$	0	0
$649$	7.84622e17	0.412164
$650$	0	0
$651$	−3.78112e18	−1.94690
$652$	0	0
$653$	2.82894e18	1.42787	0.713935	−	0.700212i	$-0.246911\pi$
$653$	2.82894e18	1.42787	0.713935	+	0.700212i	$0.246911\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.81935e18	3.30802
$658$	0	0
$659$	−1.48763e18	−0.707522	−0.353761	−	0.935336i	$-0.615097\pi$
$659$	−1.48763e18	−0.707522	−0.353761	+	0.935336i	$0.615097\pi$
$660$	0	0
$661$	3.17303e18	1.47967	0.739834	−	0.672790i	$-0.234904\pi$
$661$	3.17303e18	1.47967	0.739834	+	0.672790i	$0.234904\pi$
$662$	0	0
$663$	5.71369e18	2.61263
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.23233e17	−0.230079
$668$	0	0
$669$	4.86293e18	2.09714
$670$	0	0
$671$	−2.37469e18	−1.00441
$672$	0	0
$673$	−2.95100e18	−1.22425	−0.612127	−	0.790759i	$-0.709686\pi$
$673$	−2.95100e18	−1.22425	−0.612127	+	0.790759i	$0.709686\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.21889e18	0.885746	0.442873	−	0.896584i	$-0.353959\pi$
$677$	2.21889e18	0.885746	0.442873	+	0.896584i	$0.353959\pi$
$678$	0	0
$679$	2.89149e18	1.13232
$680$	0	0
$681$	−6.67617e18	−2.56490
$682$	0	0
$683$	−3.29072e18	−1.24038	−0.620192	−	0.784450i	$-0.712945\pi$
$683$	−3.29072e18	−1.24038	−0.620192	+	0.784450i	$0.712945\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.45916e18	0.529522
$688$	0	0
$689$	2.14440e18	0.763625
$690$	0	0
$691$	−3.78086e18	−1.32124	−0.660622	−	0.750719i	$-0.729707\pi$
$691$	−3.78086e18	−1.32124	−0.660622	+	0.750719i	$0.729707\pi$
$692$	0	0
$693$	8.73585e18	2.99598
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.83183e17	0.291768
$698$	0	0
$699$	−5.88076e18	−1.90692
$700$	0	0
$701$	−4.69433e18	−1.49419	−0.747097	−	0.664715i	$-0.768553\pi$
$701$	−4.69433e18	−1.49419	−0.747097	+	0.664715i	$0.768553\pi$
$702$	0	0
$703$	−2.63945e18	−0.824716
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.17398e18	−0.654680
$708$	0	0
$709$	−1.81561e18	−0.536810	−0.268405	−	0.963306i	$-0.586497\pi$
$709$	−1.81561e18	−0.536810	−0.268405	+	0.963306i	$0.586497\pi$
$710$	0	0
$711$	7.02762e18	2.04012
$712$	0	0
$713$	3.34491e18	0.953459
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.24899e18	1.71765
$718$	0	0
$719$	−4.62083e18	−1.24733	−0.623666	−	0.781691i	$-0.714358\pi$
$719$	−4.62083e18	−1.24733	−0.623666	+	0.781691i	$0.714358\pi$
$720$	0	0
$721$	1.84876e18	0.490119
$722$	0	0
$723$	−1.25114e18	−0.325768
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.23477e18	0.561383	0.280691	−	0.959798i	$-0.409436\pi$
$727$	2.23477e18	0.561383	0.280691	+	0.959798i	$0.409436\pi$
$728$	0	0
$729$	−2.58951e18	−0.638983
$730$	0	0
$731$	−4.22282e18	−1.02362
$732$	0	0
$733$	8.03894e18	1.91436	0.957178	−	0.289499i	$-0.0934887\pi$
$733$	8.03894e18	1.91436	0.957178	+	0.289499i	$0.0934887\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.21806e18	−0.739701
$738$	0	0
$739$	−3.24131e18	−0.732035	−0.366018	−	0.930608i	$-0.619279\pi$
$739$	−3.24131e18	−0.732035	−0.366018	+	0.930608i	$0.619279\pi$
$740$	0	0
$741$	4.31533e18	0.957626
$742$	0	0
$743$	5.07492e18	1.10663	0.553314	−	0.832973i	$-0.313363\pi$
$743$	5.07492e18	1.10663	0.553314	+	0.832973i	$0.313363\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3.69633e17	−0.0778370
$748$	0	0
$749$	3.32045e18	0.687172
$750$	0	0
$751$	−6.16208e18	−1.25334	−0.626668	−	0.779286i	$-0.715582\pi$
$751$	−6.16208e18	−1.25334	−0.626668	+	0.779286i	$0.715582\pi$
$752$	0	0
$753$	−2.55241e18	−0.510249
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.80501e18	0.541765	0.270883	−	0.962612i	$-0.412684\pi$
$757$	2.80501e18	0.541765	0.270883	+	0.962612i	$0.412684\pi$
$758$	0	0
$759$	−1.13990e19	−2.16418
$760$	0	0
$761$	5.54228e16	0.0103440	0.00517200	−	0.999987i	$-0.498354\pi$
$761$	5.54228e16	0.0103440	0.00517200	+	0.999987i	$0.498354\pi$
$762$	0	0
$763$	1.35892e19	2.49335
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.39462e18	0.602035
$768$	0	0
$769$	−2.53133e18	−0.441395	−0.220697	−	0.975342i	$-0.570833\pi$
$769$	−2.53133e18	−0.441395	−0.220697	+	0.975342i	$0.570833\pi$
$770$	0	0
$771$	−9.27003e18	−1.58938
$772$	0	0
$773$	1.08313e19	1.82606	0.913032	−	0.407889i	$-0.133735\pi$
$773$	1.08313e19	1.82606	0.913032	+	0.407889i	$0.133735\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.40983e19	−5.55899
$778$	0	0
$779$	6.67035e17	0.106944
$780$	0	0
$781$	−6.84511e18	−1.07932
$782$	0	0
$783$	−2.09472e18	−0.324843
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.27038e18	−0.340614	−0.170307	−	0.985391i	$-0.554476\pi$
$787$	−2.27038e18	−0.340614	−0.170307	+	0.985391i	$0.554476\pi$
$788$	0	0
$789$	1.13908e19	1.68095
$790$	0	0
$791$	−3.13241e18	−0.454708
$792$	0	0
$793$	−1.02740e19	−1.46711
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.17724e19	1.62699	0.813494	−	0.581573i	$-0.197563\pi$
$797$	1.17724e19	1.62699	0.813494	+	0.581573i	$0.197563\pi$
$798$	0	0
$799$	−2.53263e18	−0.344364
$800$	0	0
$801$	2.09828e19	2.80707
$802$	0	0
$803$	−1.06289e19	−1.39906
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.23144e19	2.84385
$808$	0	0
$809$	−7.77987e18	−0.975679	−0.487839	−	0.872933i	$-0.662215\pi$
$809$	−7.77987e18	−0.975679	−0.487839	+	0.872933i	$0.662215\pi$
$810$	0	0
$811$	1.31650e19	1.62475	0.812373	−	0.583139i	$-0.198176\pi$
$811$	1.31650e19	1.62475	0.812373	+	0.583139i	$0.198176\pi$
$812$	0	0
$813$	7.97709e18	0.968849
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.18934e18	−0.375195
$818$	0	0
$819$	3.77951e19	4.37613
$820$	0	0
$821$	−3.61367e18	−0.411830	−0.205915	−	0.978570i	$-0.566017\pi$
$821$	−3.61367e18	−0.411830	−0.205915	+	0.978570i	$0.566017\pi$
$822$	0	0
$823$	4.02897e18	0.451955	0.225977	−	0.974133i	$-0.427442\pi$
$823$	4.02897e18	0.451955	0.225977	+	0.974133i	$0.427442\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.11439e18	0.664610	0.332305	−	0.943172i	$-0.392174\pi$
$827$	6.11439e18	0.664610	0.332305	+	0.943172i	$0.392174\pi$
$828$	0	0
$829$	−1.26414e19	−1.35267	−0.676336	−	0.736593i	$-0.736433\pi$
$829$	−1.26414e19	−1.35267	−0.676336	+	0.736593i	$0.736433\pi$
$830$	0	0
$831$	1.63592e19	1.72328
$832$	0	0
$833$	−1.70926e19	−1.77262
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.33910e19	1.34617
$838$	0	0
$839$	−4.38090e17	−0.0433621	−0.0216810	−	0.999765i	$-0.506902\pi$
$839$	−4.38090e17	−0.0433621	−0.0216810	+	0.999765i	$0.506902\pi$
$840$	0	0
$841$	−9.97516e18	−0.972179
$842$	0	0
$843$	3.07310e19	2.94916
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.56015e18	0.331304
$848$	0	0
$849$	3.03821e19	2.78431
$850$	0	0
$851$	3.01645e19	2.72241
$852$	0	0
$853$	−1.60292e19	−1.42476	−0.712381	−	0.701793i	$-0.752383\pi$
$853$	−1.60292e19	−1.42476	−0.712381	+	0.701793i	$0.752383\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.23153e19	1.92410	0.962049	−	0.272878i	$-0.0879755\pi$
$857$	2.23153e19	1.92410	0.962049	+	0.272878i	$0.0879755\pi$
$858$	0	0
$859$	7.29169e18	0.619260	0.309630	−	0.950857i	$-0.399795\pi$
$859$	7.29169e18	0.619260	0.309630	+	0.950857i	$0.399795\pi$
$860$	0	0
$861$	8.61721e18	0.720852
$862$	0	0
$863$	7.94788e18	0.654909	0.327455	−	0.944867i	$-0.393809\pi$
$863$	7.94788e18	0.654909	0.327455	+	0.944867i	$0.393809\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.60460e18	0.528108
$868$	0	0
$869$	−1.09535e19	−0.862831
$870$	0	0
$871$	−1.39228e19	−1.08046
$872$	0	0
$873$	−1.95060e19	−1.49134
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.17789e19	−1.61636	−0.808181	−	0.588935i	$-0.799548\pi$
$877$	−2.17789e19	−1.61636	−0.808181	+	0.588935i	$0.799548\pi$
$878$	0	0
$879$	3.51690e18	0.257177
$880$	0	0
$881$	−1.07275e19	−0.772955	−0.386477	−	0.922299i	$-0.626308\pi$
$881$	−1.07275e19	−0.772955	−0.386477	+	0.922299i	$0.626308\pi$
$882$	0	0
$883$	6.72945e18	0.477787	0.238894	−	0.971046i	$-0.423215\pi$
$883$	6.72945e18	0.477787	0.238894	+	0.971046i	$0.423215\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.28317e19	−1.57411	−0.787053	−	0.616885i	$-0.788394\pi$
$887$	−2.28317e19	−1.57411	−0.787053	+	0.616885i	$0.788394\pi$
$888$	0	0
$889$	−4.57013e19	−3.10504
$890$	0	0
$891$	−1.76411e19	−1.18119
$892$	0	0
$893$	−1.91280e18	−0.126222
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.93169e19	−3.16115
$898$	0	0
$899$	−1.82491e18	−0.115293
$900$	0	0
$901$	1.07497e19	0.669402
$902$	0	0
$903$	−4.12020e19	−2.52900
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.21092e18	−0.251148	−0.125574	−	0.992084i	$-0.540077\pi$
$907$	−4.21092e18	−0.251148	−0.125574	+	0.992084i	$0.540077\pi$
$908$	0	0
$909$	1.46657e19	0.862258
$910$	0	0
$911$	−2.32762e19	−1.34910	−0.674548	−	0.738231i	$-0.735661\pi$
$911$	−2.32762e19	−1.34910	−0.674548	+	0.738231i	$0.735661\pi$
$912$	0	0
$913$	5.76124e17	0.0329197
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.98638e19	1.10322
$918$	0	0
$919$	−1.40968e19	−0.771913	−0.385957	−	0.922517i	$-0.626129\pi$
$919$	−1.40968e19	−0.771913	−0.385957	+	0.922517i	$0.626129\pi$
$920$	0	0
$921$	8.61898e18	0.465338
$922$	0	0
$923$	−2.96150e19	−1.57652
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.24717e19	−0.645519
$928$	0	0
$929$	2.34139e19	1.19501	0.597504	−	0.801866i	$-0.296159\pi$
$929$	2.34139e19	1.19501	0.597504	+	0.801866i	$0.296159\pi$
$930$	0	0
$931$	−1.29094e19	−0.649729
$932$	0	0
$933$	5.56831e19	2.76371
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.36822e18	0.307405	0.153702	−	0.988117i	$-0.450880\pi$
$937$	6.36822e18	0.307405	0.153702	+	0.988117i	$0.450880\pi$
$938$	0	0
$939$	−9.16331e18	−0.436240
$940$	0	0
$941$	−9.88316e17	−0.0464048	−0.0232024	−	0.999731i	$-0.507386\pi$
$941$	−9.88316e17	−0.0464048	−0.0232024	+	0.999731i	$0.507386\pi$
$942$	0	0
$943$	−7.62308e18	−0.353024
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.43099e18	−0.424896	−0.212448	−	0.977172i	$-0.568144\pi$
$947$	−9.43099e18	−0.424896	−0.212448	+	0.977172i	$0.568144\pi$
$948$	0	0
$949$	−4.59854e19	−2.04357
$950$	0	0
$951$	6.84533e19	3.00069
$952$	0	0
$953$	−3.34443e19	−1.44617	−0.723084	−	0.690760i	$-0.757276\pi$
$953$	−3.34443e19	−1.44617	−0.723084	+	0.690760i	$0.757276\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.21905e18	0.261695
$958$	0	0
$959$	−3.85962e19	−1.60223
$960$	0	0
$961$	−1.27513e19	−0.522218
$962$	0	0
$963$	−2.23998e19	−0.905052
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−9.00952e18	−0.354348	−0.177174	−	0.984180i	$-0.556696\pi$
$967$	−9.00952e18	−0.354348	−0.177174	+	0.984180i	$0.556696\pi$
$968$	0	0
$969$	2.16325e19	0.839466
$970$	0	0
$971$	4.49062e17	0.0171942	0.00859709	−	0.999963i	$-0.497263\pi$
$971$	4.49062e17	0.0171942	0.00859709	+	0.999963i	$0.497263\pi$
$972$	0	0
$973$	2.53345e19	0.957146
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.37369e17	−0.0344823	−0.0172411	−	0.999851i	$-0.505488\pi$
$977$	−9.37369e17	−0.0344823	−0.0172411	+	0.999851i	$0.505488\pi$
$978$	0	0
$979$	−3.27046e19	−1.18719
$980$	0	0
$981$	−9.16727e19	−3.28391
$982$	0	0
$983$	−9.36467e18	−0.331051	−0.165525	−	0.986206i	$-0.552932\pi$
$983$	−9.36467e18	−0.331051	−0.165525	+	0.986206i	$0.552932\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.47108e19	−0.850797
$988$	0	0
$989$	3.64487e19	1.23853
$990$	0	0
$991$	−3.11736e19	−1.04546	−0.522731	−	0.852497i	$-0.675087\pi$
$991$	−3.11736e19	−1.04546	−0.522731	+	0.852497i	$0.675087\pi$
$992$	0	0
$993$	−5.66030e19	−1.87357
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.98985e19	1.28658	0.643291	−	0.765622i	$-0.277568\pi$
$997$	3.98985e19	1.28658	0.643291	+	0.765622i	$0.277568\pi$
$998$	0	0
$999$	1.20761e20	3.84371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.14.a.l.1.9		10
5.2	odd	4		40.14.c.a.9.2	✓	20
5.3	odd	4		40.14.c.a.9.19	yes	20
5.4	even	2		200.14.a.k.1.2		10
20.3	even	4		80.14.c.d.49.2		20
20.7	even	4		80.14.c.d.49.19		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.14.c.a.9.2	✓	20	5.2	odd	4
40.14.c.a.9.19	yes	20	5.3	odd	4
80.14.c.d.49.2		20	20.3	even	4
80.14.c.d.49.19		20	20.7	even	4
200.14.a.k.1.2		10	5.4	even	2
200.14.a.l.1.9		10	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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)

\(f(q)\)	\(=\)	\(q+2225.01 q^{3} -497533. q^{7} +3.35636e6 q^{9} +O(q^{10})\)	\(q+2225.01 q^{3} -497533. q^{7} +3.35636e6 q^{9} -5.23136e6 q^{11} -2.26332e7 q^{13} -1.13459e8 q^{17} -8.56912e7 q^{19} -1.10702e9 q^{21} +9.79305e8 q^{23} +3.92056e9 q^{27} -5.34290e8 q^{29} +3.41559e9 q^{31} -1.16398e10 q^{33} +3.08019e10 q^{37} -5.03591e10 q^{39} -7.78417e9 q^{41} +3.72189e10 q^{43} +2.23220e10 q^{47} +1.50650e11 q^{49} -2.52448e11 q^{51} -9.47457e10 q^{53} -1.90664e11 q^{57} -1.49984e11 q^{59} +4.53934e11 q^{61} -1.66990e12 q^{63} +6.15149e11 q^{67} +2.17897e12 q^{69} +1.30848e12 q^{71} +2.03177e12 q^{73} +2.60277e12 q^{77} +2.09382e12 q^{79} +3.37218e12 q^{81} -1.10129e11 q^{83} -1.18880e12 q^{87} +6.25166e12 q^{89} +1.12607e13 q^{91} +7.59974e12 q^{93} -5.81165e12 q^{97} -1.75583e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 1276 q^{3} + 341068 q^{7} + 6557498 q^{9}+O(q^{10}) \) 10 * q + 1276 * q^3 + 341068 * q^7 + 6557498 * q^9	\( 10 q + 1276 q^{3} + 341068 q^{7} + 6557498 q^{9} - 4074536 q^{11} - 7720792 q^{13} + 29092704 q^{17} - 82552936 q^{19} - 140909032 q^{21} + 448872532 q^{23} + 6653011048 q^{27} + 4118753908 q^{29} - 161162304 q^{31} + 6635864336 q^{33} + 37799831624 q^{37} + 2651467216 q^{39} + 31205287876 q^{41} + 48886150956 q^{43} - 54068936100 q^{47} + 313063319346 q^{49} - 179707160640 q^{51} - 116117700520 q^{53} - 863281677424 q^{57} - 658612309176 q^{59} + 405225797804 q^{61} - 169179586580 q^{63} - 291232794812 q^{67} + 1989925462312 q^{69} - 956854398704 q^{71} + 262407355312 q^{73} + 3471232133584 q^{77} - 3176139465248 q^{79} + 3191121792770 q^{81} + 5123747754524 q^{83} + 9347096848952 q^{87} + 453451956924 q^{89} + 6935945759120 q^{91} - 806946503040 q^{93} - 11480198579200 q^{97} + 18677113053976 q^{99}+O(q^{100}) \) 10 * q + 1276 * q^3 + 341068 * q^7 + 6557498 * q^9 - 4074536 * q^11 - 7720792 * q^13 + 29092704 * q^17 - 82552936 * q^19 - 140909032 * q^21 + 448872532 * q^23 + 6653011048 * q^27 + 4118753908 * q^29 - 161162304 * q^31 + 6635864336 * q^33 + 37799831624 * q^37 + 2651467216 * q^39 + 31205287876 * q^41 + 48886150956 * q^43 - 54068936100 * q^47 + 313063319346 * q^49 - 179707160640 * q^51 - 116117700520 * q^53 - 863281677424 * q^57 - 658612309176 * q^59 + 405225797804 * q^61 - 169179586580 * q^63 - 291232794812 * q^67 + 1989925462312 * q^69 - 956854398704 * q^71 + 262407355312 * q^73 + 3471232133584 * q^77 - 3176139465248 * q^79 + 3191121792770 * q^81 + 5123747754524 * q^83 + 9347096848952 * q^87 + 453451956924 * q^89 + 6935945759120 * q^91 - 806946503040 * q^93 - 11480198579200 * q^97 + 18677113053976 * q^99

Introduction

L-functions

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