LMFDB - Embedded newform 200.14.a.l.1.3

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			1.3
Root			$908.118$ 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)

$f(q)$	$=$	$q-1688.24 q^{3} -194262. q^{7} +1.25582e6 q^{9} +O(q^{10})$	$q-1688.24 q^{3} -194262. q^{7} +1.25582e6 q^{9} +868250. q^{11} +2.98364e7 q^{13} -7.65460e7 q^{17} +3.16696e7 q^{19} +3.27961e8 q^{21} +5.57011e8 q^{23} +5.71479e8 q^{27} +5.34821e8 q^{29} +3.08392e9 q^{31} -1.46581e9 q^{33} +2.44396e10 q^{37} -5.03709e10 q^{39} -2.81307e10 q^{41} +1.44237e10 q^{43} -1.43202e11 q^{47} -5.91511e10 q^{49} +1.29228e11 q^{51} +1.30455e11 q^{53} -5.34657e10 q^{57} -3.37155e11 q^{59} +6.55981e11 q^{61} -2.43958e11 q^{63} +3.41908e10 q^{67} -9.40366e11 q^{69} -1.46435e12 q^{71} +3.00965e11 q^{73} -1.68668e11 q^{77} -5.87262e11 q^{79} -2.96697e12 q^{81} +3.05623e12 q^{83} -9.02904e11 q^{87} +2.32598e12 q^{89} -5.79609e12 q^{91} -5.20639e12 q^{93} +1.24467e13 q^{97} +1.09036e12 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1688.24	−1.33704	−0.668521	−	0.743693i	$-0.733072\pi$
$3$	−1688.24	−1.33704	−0.668521	+	0.743693i	$0.733072\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−194262.	−0.624096	−0.312048	−	0.950066i	$-0.601015\pi$
$7$	−194262.	−0.624096	−0.312048	+	0.950066i	$0.601015\pi$
$8$	0	0
$9$	1.25582e6	0.787680
$10$	0	0
$11$	868250.	0.147772	0.0738860	−	0.997267i	$-0.476460\pi$
$11$	868250.	0.147772	0.0738860	+	0.997267i	$0.476460\pi$
$12$	0	0
$13$	2.98364e7	1.71441	0.857205	−	0.514975i	$-0.172199\pi$
$13$	2.98364e7	1.71441	0.857205	+	0.514975i	$0.172199\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.65460e7	−0.769139	−0.384569	−	0.923096i	$-0.625650\pi$
$17$	−7.65460e7	−0.769139	−0.384569	+	0.923096i	$0.625650\pi$
$18$	0	0
$19$	3.16696e7	0.154434	0.0772171	−	0.997014i	$-0.475397\pi$
$19$	3.16696e7	0.154434	0.0772171	+	0.997014i	$0.475397\pi$
$20$	0	0
$21$	3.27961e8	0.834442
$22$	0	0
$23$	5.57011e8	0.784572	0.392286	−	0.919843i	$-0.371684\pi$
$23$	5.57011e8	0.784572	0.392286	+	0.919843i	$0.371684\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.71479e8	0.283881
$28$	0	0
$29$	5.34821e8	0.166963	0.0834817	−	0.996509i	$-0.473396\pi$
$29$	5.34821e8	0.166963	0.0834817	+	0.996509i	$0.473396\pi$
$30$	0	0
$31$	3.08392e9	0.624097	0.312049	−	0.950066i	$-0.398985\pi$
$31$	3.08392e9	0.624097	0.312049	+	0.950066i	$0.398985\pi$
$32$	0	0
$33$	−1.46581e9	−0.197577
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.44396e10	1.56597	0.782985	−	0.622041i	$-0.213696\pi$
$37$	2.44396e10	1.56597	0.782985	+	0.622041i	$0.213696\pi$
$38$	0	0
$39$	−5.03709e10	−2.29224
$40$	0	0
$41$	−2.81307e10	−0.924879	−0.462440	−	0.886651i	$-0.653026\pi$
$41$	−2.81307e10	−0.924879	−0.462440	+	0.886651i	$0.653026\pi$
$42$	0	0
$43$	1.44237e10	0.347961	0.173980	−	0.984749i	$-0.444337\pi$
$43$	1.44237e10	0.347961	0.173980	+	0.984749i	$0.444337\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.43202e11	−1.93782	−0.968908	−	0.247420i	$-0.920417\pi$
$47$	−1.43202e11	−1.93782	−0.968908	+	0.247420i	$0.920417\pi$
$48$	0	0
$49$	−5.91511e10	−0.610504
$50$	0	0
$51$	1.29228e11	1.02837
$52$	0	0
$53$	1.30455e11	0.808476	0.404238	−	0.914654i	$-0.367537\pi$
$53$	1.30455e11	0.808476	0.404238	+	0.914654i	$0.367537\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.34657e10	−0.206485
$58$	0	0
$59$	−3.37155e11	−1.04062	−0.520309	−	0.853978i	$-0.674183\pi$
$59$	−3.37155e11	−1.04062	−0.520309	+	0.853978i	$0.674183\pi$
$60$	0	0
$61$	6.55981e11	1.63022	0.815112	−	0.579303i	$-0.196675\pi$
$61$	6.55981e11	1.63022	0.815112	+	0.579303i	$0.196675\pi$
$62$	0	0
$63$	−2.43958e11	−0.491588
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.41908e10	0.0461768	0.0230884	−	0.999733i	$-0.492650\pi$
$67$	3.41908e10	0.0461768	0.0230884	+	0.999733i	$0.492650\pi$
$68$	0	0
$69$	−9.40366e11	−1.04901
$70$	0	0
$71$	−1.46435e12	−1.35664	−0.678319	−	0.734767i	$-0.737291\pi$
$71$	−1.46435e12	−1.35664	−0.678319	+	0.734767i	$0.737291\pi$
$72$	0	0
$73$	3.00965e11	0.232765	0.116382	−	0.993204i	$-0.462870\pi$
$73$	3.00965e11	0.232765	0.116382	+	0.993204i	$0.462870\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.68668e11	−0.0922240
$78$	0	0
$79$	−5.87262e11	−0.271804	−0.135902	−	0.990722i	$-0.543393\pi$
$79$	−5.87262e11	−0.271804	−0.135902	+	0.990722i	$0.543393\pi$
$80$	0	0
$81$	−2.96697e12	−1.16724
$82$	0	0
$83$	3.05623e12	1.02607	0.513036	−	0.858367i	$-0.328521\pi$
$83$	3.05623e12	1.02607	0.513036	+	0.858367i	$0.328521\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.02904e11	−0.223237
$88$	0	0
$89$	2.32598e12	0.496101	0.248051	−	0.968747i	$-0.420210\pi$
$89$	2.32598e12	0.496101	0.248051	+	0.968747i	$0.420210\pi$
$90$	0	0
$91$	−5.79609e12	−1.06996
$92$	0	0
$93$	−5.20639e12	−0.834444
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.24467e13	1.51718	0.758591	−	0.651567i	$-0.225888\pi$
$97$	1.24467e13	1.51718	0.758591	+	0.651567i	$0.225888\pi$
$98$	0	0
$99$	1.09036e12	0.116397
$100$	0	0
$101$	−1.83448e13	−1.71959	−0.859793	−	0.510643i	$-0.829407\pi$
$101$	−1.83448e13	−1.71959	−0.859793	+	0.510643i	$0.829407\pi$
$102$	0	0
$103$	−8.00813e12	−0.660829	−0.330414	−	0.943836i	$-0.607189\pi$
$103$	−8.00813e12	−0.660829	−0.330414	+	0.943836i	$0.607189\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.17238e13	−1.39940	−0.699700	−	0.714437i	$-0.746683\pi$
$107$	−2.17238e13	−1.39940	−0.699700	+	0.714437i	$0.746683\pi$
$108$	0	0
$109$	−1.62751e13	−0.929506	−0.464753	−	0.885440i	$-0.653857\pi$
$109$	−1.62751e13	−0.929506	−0.464753	+	0.885440i	$0.653857\pi$
$110$	0	0
$111$	−4.12598e13	−2.09377
$112$	0	0
$113$	−7.03486e12	−0.317867	−0.158934	−	0.987289i	$-0.550806\pi$
$113$	−7.03486e12	−0.317867	−0.158934	+	0.987289i	$0.550806\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.74691e13	1.35041
$118$	0	0
$119$	1.48700e13	0.480016
$120$	0	0
$121$	−3.37689e13	−0.978163
$122$	0	0
$123$	4.74912e13	1.23660
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.41395e13	−1.56793	−0.783963	−	0.620808i	$-0.786805\pi$
$127$	−7.41395e13	−1.56793	−0.783963	+	0.620808i	$0.786805\pi$
$128$	0	0
$129$	−2.43505e13	−0.465238
$130$	0	0
$131$	−6.49645e12	−0.112308	−0.0561542	−	0.998422i	$-0.517884\pi$
$131$	−6.49645e12	−0.112308	−0.0561542	+	0.998422i	$0.517884\pi$
$132$	0	0
$133$	−6.15220e12	−0.0963818
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.82802e13	0.882290	0.441145	−	0.897436i	$-0.354573\pi$
$137$	6.82802e13	0.882290	0.441145	+	0.897436i	$0.354573\pi$
$138$	0	0
$139$	4.81592e13	0.566347	0.283174	−	0.959069i	$-0.408613\pi$
$139$	4.81592e13	0.566347	0.283174	+	0.959069i	$0.408613\pi$
$140$	0	0
$141$	2.41759e14	2.59094
$142$	0	0
$143$	2.59055e13	0.253342
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.98611e13	0.816269
$148$	0	0
$149$	−1.10304e14	−0.825814	−0.412907	−	0.910773i	$-0.635486\pi$
$149$	−1.10304e14	−0.825814	−0.412907	+	0.910773i	$0.635486\pi$
$150$	0	0
$151$	2.50615e14	1.72051	0.860254	−	0.509866i	$-0.170305\pi$
$151$	2.50615e14	1.72051	0.860254	+	0.509866i	$0.170305\pi$
$152$	0	0
$153$	−9.61277e13	−0.605835
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.65998e13	−0.301625	−0.150813	−	0.988562i	$-0.548189\pi$
$157$	−5.65998e13	−0.301625	−0.150813	+	0.988562i	$0.548189\pi$
$158$	0	0
$159$	−2.20238e14	−1.08097
$160$	0	0
$161$	−1.08206e14	−0.489648
$162$	0	0
$163$	3.55354e14	1.48403	0.742013	−	0.670386i	$-0.233871\pi$
$163$	3.55354e14	1.48403	0.742013	+	0.670386i	$0.233871\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.42474e14	−1.93518	−0.967592	−	0.252520i	$-0.918741\pi$
$167$	−5.42474e14	−1.93518	−0.967592	+	0.252520i	$0.918741\pi$
$168$	0	0
$169$	5.87336e14	1.93920
$170$	0	0
$171$	3.97711e13	0.121645
$172$	0	0
$173$	4.89252e14	1.38750	0.693750	−	0.720216i	$-0.255957\pi$
$173$	4.89252e14	1.38750	0.693750	+	0.720216i	$0.255957\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.69196e14	1.39135
$178$	0	0
$179$	−1.19260e14	−0.270988	−0.135494	−	0.990778i	$-0.543262\pi$
$179$	−1.19260e14	−0.270988	−0.135494	+	0.990778i	$0.543262\pi$
$180$	0	0
$181$	3.99883e14	0.845321	0.422661	−	0.906288i	$-0.361096\pi$
$181$	3.99883e14	0.845321	0.422661	+	0.906288i	$0.361096\pi$
$182$	0	0
$183$	−1.10745e15	−2.17968
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.64611e13	−0.113657
$188$	0	0
$189$	−1.11017e14	−0.177169
$190$	0	0
$191$	3.07518e14	0.458304	0.229152	−	0.973391i	$-0.426405\pi$
$191$	3.07518e14	0.458304	0.229152	+	0.973391i	$0.426405\pi$
$192$	0	0
$193$	8.63930e14	1.20325	0.601625	−	0.798778i	$-0.294520\pi$
$193$	8.63930e14	1.20325	0.601625	+	0.798778i	$0.294520\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.27017e15	1.54821	0.774106	−	0.633056i	$-0.218200\pi$
$197$	1.27017e15	1.54821	0.774106	+	0.633056i	$0.218200\pi$
$198$	0	0
$199$	1.63551e15	1.86684	0.933420	−	0.358785i	$-0.116809\pi$
$199$	1.63551e15	1.86684	0.933420	+	0.358785i	$0.116809\pi$
$200$	0	0
$201$	−5.77222e13	−0.0617403
$202$	0	0
$203$	−1.03896e14	−0.104201
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.99504e14	0.617992
$208$	0	0
$209$	2.74971e13	0.0228211
$210$	0	0
$211$	−9.79973e14	−0.764501	−0.382251	−	0.924059i	$-0.624851\pi$
$211$	−9.79973e14	−0.764501	−0.382251	+	0.924059i	$0.624851\pi$
$212$	0	0
$213$	2.47216e15	1.81388
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.99090e14	−0.389497
$218$	0	0
$219$	−5.08100e14	−0.311216
$220$	0	0
$221$	−2.28386e15	−1.31862
$222$	0	0
$223$	−1.87884e13	−0.0102307	−0.00511537	−	0.999987i	$-0.501628\pi$
$223$	−1.87884e13	−0.0102307	−0.00511537	+	0.999987i	$0.501628\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.12331e15	−0.544917	−0.272458	−	0.962168i	$-0.587837\pi$
$227$	−1.12331e15	−0.544917	−0.272458	+	0.962168i	$0.587837\pi$
$228$	0	0
$229$	6.29602e14	0.288493	0.144247	−	0.989542i	$-0.453924\pi$
$229$	6.29602e14	0.288493	0.144247	+	0.989542i	$0.453924\pi$
$230$	0	0
$231$	2.84752e14	0.123307
$232$	0	0
$233$	2.58795e15	1.05960	0.529800	−	0.848123i	$-0.322267\pi$
$233$	2.58795e15	1.05960	0.529800	+	0.848123i	$0.322267\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.91437e14	0.363413
$238$	0	0
$239$	−3.86400e15	−1.34107	−0.670534	−	0.741879i	$-0.733935\pi$
$239$	−3.86400e15	−1.34107	−0.670534	+	0.741879i	$0.733935\pi$
$240$	0	0
$241$	−4.09124e14	−0.134507	−0.0672534	−	0.997736i	$-0.521424\pi$
$241$	−4.09124e14	−0.134507	−0.0672534	+	0.997736i	$0.521424\pi$
$242$	0	0
$243$	4.09782e15	1.27677
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.44906e14	0.264764
$248$	0	0
$249$	−5.15963e15	−1.37190
$250$	0	0
$251$	−3.03938e14	−0.0767195	−0.0383598	−	0.999264i	$-0.512213\pi$
$251$	−3.03938e14	−0.0767195	−0.0383598	+	0.999264i	$0.512213\pi$
$252$	0	0
$253$	4.83625e14	0.115938
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.00086e15	0.649651	0.324825	−	0.945774i	$-0.394694\pi$
$257$	3.00086e15	0.649651	0.324825	+	0.945774i	$0.394694\pi$
$258$	0	0
$259$	−4.74770e15	−0.977315
$260$	0	0
$261$	6.71637e14	0.131514
$262$	0	0
$263$	4.14288e15	0.771952	0.385976	−	0.922509i	$-0.373865\pi$
$263$	4.14288e15	0.771952	0.385976	+	0.922509i	$0.373865\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.92680e15	−0.663308
$268$	0	0
$269$	2.99184e15	0.481448	0.240724	−	0.970594i	$-0.422615\pi$
$269$	2.99184e15	0.481448	0.240724	+	0.970594i	$0.422615\pi$
$270$	0	0
$271$	−1.59837e15	−0.245119	−0.122560	−	0.992461i	$-0.539110\pi$
$271$	−1.59837e15	−0.245119	−0.122560	+	0.992461i	$0.539110\pi$
$272$	0	0
$273$	9.78517e15	1.43058
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.03302e15	1.20147	0.600737	−	0.799447i	$-0.294874\pi$
$277$	9.03302e15	1.20147	0.600737	+	0.799447i	$0.294874\pi$
$278$	0	0
$279$	3.87284e15	0.491589
$280$	0	0
$281$	2.83380e15	0.343382	0.171691	−	0.985151i	$-0.445077\pi$
$281$	2.83380e15	0.343382	0.171691	+	0.985151i	$0.445077\pi$
$282$	0	0
$283$	1.23473e16	1.42877	0.714383	−	0.699755i	$-0.246707\pi$
$283$	1.23473e16	1.42877	0.714383	+	0.699755i	$0.246707\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.46473e15	0.577213
$288$	0	0
$289$	−4.04529e15	−0.408426
$290$	0	0
$291$	−2.10129e16	−2.02854
$292$	0	0
$293$	−2.56346e14	−0.0236694	−0.0118347	−	0.999930i	$-0.503767\pi$
$293$	−2.56346e14	−0.0236694	−0.0118347	+	0.999930i	$0.503767\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.96187e14	0.0419496
$298$	0	0
$299$	1.66192e16	1.34508
$300$	0	0
$301$	−2.80197e15	−0.217161
$302$	0	0
$303$	3.09703e16	2.29916
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.53164e15	0.581611	0.290806	−	0.956782i	$-0.406077\pi$
$307$	8.53164e15	0.581611	0.290806	+	0.956782i	$0.406077\pi$
$308$	0	0
$309$	1.35196e16	0.883556
$310$	0	0
$311$	1.66356e16	1.04254	0.521272	−	0.853390i	$-0.325458\pi$
$311$	1.66356e16	1.04254	0.521272	+	0.853390i	$0.325458\pi$
$312$	0	0
$313$	−1.60994e16	−0.967771	−0.483885	−	0.875131i	$-0.660775\pi$
$313$	−1.60994e16	−0.967771	−0.483885	+	0.875131i	$0.660775\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.33475e16	0.738779	0.369390	−	0.929275i	$-0.379567\pi$
$317$	1.33475e16	0.738779	0.369390	+	0.929275i	$0.379567\pi$
$318$	0	0
$319$	4.64358e14	0.0246725
$320$	0	0
$321$	3.66749e16	1.87105
$322$	0	0
$323$	−2.42418e15	−0.118781
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.74762e16	1.24279
$328$	0	0
$329$	2.78187e16	1.20938
$330$	0	0
$331$	2.09834e16	0.876989	0.438494	−	0.898734i	$-0.355512\pi$
$331$	2.09834e16	0.876989	0.438494	+	0.898734i	$0.355512\pi$
$332$	0	0
$333$	3.06917e16	1.23348
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.26719e16	−1.58689	−0.793447	−	0.608639i	$-0.791716\pi$
$337$	−4.26719e16	−1.58689	−0.793447	+	0.608639i	$0.791716\pi$
$338$	0	0
$339$	1.18765e16	0.425001
$340$	0	0
$341$	2.67762e15	0.0922242
$342$	0	0
$343$	3.03127e16	1.00511
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.74501e14	0.0299668	0.0149834	−	0.999888i	$-0.495230\pi$
$347$	9.74501e14	0.0299668	0.0149834	+	0.999888i	$0.495230\pi$
$348$	0	0
$349$	−3.72844e16	−1.10449	−0.552244	−	0.833682i	$-0.686228\pi$
$349$	−3.72844e16	−1.10449	−0.552244	+	0.833682i	$0.686228\pi$
$350$	0	0
$351$	1.70509e16	0.486688
$352$	0	0
$353$	4.47917e16	1.23215	0.616073	−	0.787689i	$-0.288723\pi$
$353$	4.47917e16	1.23215	0.616073	+	0.787689i	$0.288723\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.51041e16	−0.641802
$358$	0	0
$359$	−2.28594e16	−0.563574	−0.281787	−	0.959477i	$-0.590927\pi$
$359$	−2.28594e16	−0.563574	−0.281787	+	0.959477i	$0.590927\pi$
$360$	0	0
$361$	−4.10500e16	−0.976150
$362$	0	0
$363$	5.70098e16	1.30785
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.81667e16	1.66991	0.834953	−	0.550321i	$-0.185495\pi$
$367$	7.81667e16	1.66991	0.834953	+	0.550321i	$0.185495\pi$
$368$	0	0
$369$	−3.53269e16	−0.728509
$370$	0	0
$371$	−2.53425e16	−0.504566
$372$	0	0
$373$	2.45816e16	0.472609	0.236304	−	0.971679i	$-0.424064\pi$
$373$	2.45816e16	0.472609	0.236304	+	0.971679i	$0.424064\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.59571e16	0.286244
$378$	0	0
$379$	−1.01346e17	−1.75652	−0.878259	−	0.478186i	$-0.841295\pi$
$379$	−1.01346e17	−1.75652	−0.878259	+	0.478186i	$0.841295\pi$
$380$	0	0
$381$	1.25165e17	2.09638
$382$	0	0
$383$	−1.12440e17	−1.82023	−0.910117	−	0.414352i	$-0.864008\pi$
$383$	−1.12440e17	−1.82023	−0.910117	+	0.414352i	$0.864008\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.81135e16	0.274082
$388$	0	0
$389$	2.01860e16	0.295378	0.147689	−	0.989034i	$-0.452816\pi$
$389$	2.01860e16	0.295378	0.147689	+	0.989034i	$0.452816\pi$
$390$	0	0
$391$	−4.26370e16	−0.603445
$392$	0	0
$393$	1.09675e16	0.150161
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.72354e16	−0.477329	−0.238665	−	0.971102i	$-0.576710\pi$
$397$	−3.72354e16	−0.477329	−0.238665	+	0.971102i	$0.576710\pi$
$398$	0	0
$399$	1.03864e16	0.128866
$400$	0	0
$401$	−4.16458e16	−0.500188	−0.250094	−	0.968222i	$-0.580461\pi$
$401$	−4.16458e16	−0.500188	−0.250094	+	0.968222i	$0.580461\pi$
$402$	0	0
$403$	9.20131e16	1.06996
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.12197e16	0.231407
$408$	0	0
$409$	1.12701e17	1.19049	0.595247	−	0.803543i	$-0.297054\pi$
$409$	1.12701e17	1.19049	0.595247	+	0.803543i	$0.297054\pi$
$410$	0	0
$411$	−1.15273e17	−1.17966
$412$	0	0
$413$	6.54964e16	0.649445
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8.13041e16	−0.757230
$418$	0	0
$419$	1.24853e17	1.12721	0.563607	−	0.826043i	$-0.309413\pi$
$419$	1.24853e17	1.12721	0.563607	+	0.826043i	$0.309413\pi$
$420$	0	0
$421$	−7.75156e16	−0.678509	−0.339254	−	0.940695i	$-0.610175\pi$
$421$	−7.75156e16	−0.678509	−0.339254	+	0.940695i	$0.610175\pi$
$422$	0	0
$423$	−1.79835e17	−1.52638
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.27432e17	−1.01742
$428$	0	0
$429$	−4.37345e16	−0.338729
$430$	0	0
$431$	−2.38047e17	−1.78879	−0.894397	−	0.447273i	$-0.852395\pi$
$431$	−2.38047e17	−1.78879	−0.894397	+	0.447273i	$0.852395\pi$
$432$	0	0
$433$	−5.48335e16	−0.399829	−0.199915	−	0.979813i	$-0.564067\pi$
$433$	−5.48335e16	−0.399829	−0.199915	+	0.979813i	$0.564067\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.76403e16	0.121165
$438$	0	0
$439$	−2.24372e17	−1.49606	−0.748032	−	0.663663i	$-0.769001\pi$
$439$	−2.24372e17	−1.49606	−0.748032	+	0.663663i	$0.769001\pi$
$440$	0	0
$441$	−7.42830e16	−0.480882
$442$	0	0
$443$	8.14266e16	0.511850	0.255925	−	0.966697i	$-0.417620\pi$
$443$	8.14266e16	0.511850	0.255925	+	0.966697i	$0.417620\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.86220e17	1.10415
$448$	0	0
$449$	2.39661e17	1.38037	0.690187	−	0.723631i	$-0.257528\pi$
$449$	2.39661e17	1.38037	0.690187	+	0.723631i	$0.257528\pi$
$450$	0	0
$451$	−2.44245e16	−0.136671
$452$	0	0
$453$	−4.23097e17	−2.30039
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.64051e17	0.842410	0.421205	−	0.906965i	$-0.361607\pi$
$457$	1.64051e17	0.842410	0.421205	+	0.906965i	$0.361607\pi$
$458$	0	0
$459$	−4.37444e16	−0.218344
$460$	0	0
$461$	−1.39716e17	−0.677938	−0.338969	−	0.940798i	$-0.610078\pi$
$461$	−1.39716e17	−0.677938	−0.338969	+	0.940798i	$0.610078\pi$
$462$	0	0
$463$	−1.99040e17	−0.938997	−0.469498	−	0.882933i	$-0.655565\pi$
$463$	−1.99040e17	−0.938997	−0.469498	+	0.882933i	$0.655565\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.70801e17	−1.20807	−0.604034	−	0.796959i	$-0.706441\pi$
$467$	−2.70801e17	−1.20807	−0.604034	+	0.796959i	$0.706441\pi$
$468$	0	0
$469$	−6.64199e15	−0.0288187
$470$	0	0
$471$	9.55539e16	0.403285
$472$	0	0
$473$	1.25233e16	0.0514189
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.63827e17	0.636820
$478$	0	0
$479$	9.41881e16	0.356299	0.178150	−	0.984003i	$-0.442989\pi$
$479$	9.41881e16	0.356299	0.178150	+	0.984003i	$0.442989\pi$
$480$	0	0
$481$	7.29191e17	2.68471
$482$	0	0
$483$	1.82678e17	0.654680
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.39109e17	−1.15186	−0.575931	−	0.817498i	$-0.695360\pi$
$487$	−3.39109e17	−1.15186	−0.575931	+	0.817498i	$0.695360\pi$
$488$	0	0
$489$	−5.99921e17	−1.98420
$490$	0	0
$491$	−1.02069e17	−0.328750	−0.164375	−	0.986398i	$-0.552561\pi$
$491$	−1.02069e17	−0.328750	−0.164375	+	0.986398i	$0.552561\pi$
$492$	0	0
$493$	−4.09384e16	−0.128418
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.84467e17	0.846673
$498$	0	0
$499$	5.16767e17	1.49845	0.749223	−	0.662317i	$-0.230427\pi$
$499$	5.16767e17	1.49845	0.749223	+	0.662317i	$0.230427\pi$
$500$	0	0
$501$	9.15824e17	2.58742
$502$	0	0
$503$	−6.32558e17	−1.74144	−0.870719	−	0.491780i	$-0.836346\pi$
$503$	−6.32558e17	−1.74144	−0.870719	+	0.491780i	$0.836346\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.91562e17	−2.59279
$508$	0	0
$509$	4.02904e17	1.02692	0.513459	−	0.858114i	$-0.328364\pi$
$509$	4.02904e17	1.02692	0.513459	+	0.858114i	$0.328364\pi$
$510$	0	0
$511$	−5.84661e16	−0.145268
$512$	0	0
$513$	1.80985e16	0.0438409
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.24335e17	−0.286355
$518$	0	0
$519$	−8.25972e17	−1.85514
$520$	0	0
$521$	4.60327e17	1.00837	0.504187	−	0.863595i	$-0.331792\pi$
$521$	4.60327e17	1.00837	0.504187	+	0.863595i	$0.331792\pi$
$522$	0	0
$523$	−1.09610e17	−0.234201	−0.117101	−	0.993120i	$-0.537360\pi$
$523$	−1.09610e17	−0.234201	−0.117101	+	0.993120i	$0.537360\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.36062e17	−0.480017
$528$	0	0
$529$	−1.93775e17	−0.384446
$530$	0	0
$531$	−4.23404e17	−0.819673
$532$	0	0
$533$	−8.39318e17	−1.58562
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.01339e17	0.362322
$538$	0	0
$539$	−5.13580e16	−0.0902155
$540$	0	0
$541$	1.23493e17	0.211767	0.105884	−	0.994379i	$-0.466233\pi$
$541$	1.23493e17	0.211767	0.105884	+	0.994379i	$0.466233\pi$
$542$	0	0
$543$	−6.75096e17	−1.13023
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.31289e17	−0.209561	−0.104780	−	0.994495i	$-0.533414\pi$
$547$	−1.31289e17	−0.209561	−0.104780	+	0.994495i	$0.533414\pi$
$548$	0	0
$549$	8.23792e17	1.28410
$550$	0	0
$551$	1.69375e16	0.0257849
$552$	0	0
$553$	1.14083e17	0.169632
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.79764e17	0.680720	0.340360	−	0.940295i	$-0.389451\pi$
$557$	4.79764e17	0.680720	0.340360	+	0.940295i	$0.389451\pi$
$558$	0	0
$559$	4.30350e17	0.596548
$560$	0	0
$561$	1.12202e17	0.151964
$562$	0	0
$563$	−4.15214e17	−0.549500	−0.274750	−	0.961516i	$-0.588595\pi$
$563$	−4.15214e17	−0.549500	−0.274750	+	0.961516i	$0.588595\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.76370e17	0.728470
$568$	0	0
$569$	1.18396e18	1.46254	0.731272	−	0.682086i	$-0.238927\pi$
$569$	1.18396e18	1.46254	0.731272	+	0.682086i	$0.238927\pi$
$570$	0	0
$571$	7.94940e16	0.0959842	0.0479921	−	0.998848i	$-0.484718\pi$
$571$	7.94940e16	0.0959842	0.0479921	+	0.998848i	$0.484718\pi$
$572$	0	0
$573$	−5.19163e17	−0.612772
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.56265e17	−0.853161	−0.426581	−	0.904450i	$-0.640282\pi$
$577$	−7.56265e17	−0.853161	−0.426581	+	0.904450i	$0.640282\pi$
$578$	0	0
$579$	−1.45852e18	−1.60880
$580$	0	0
$581$	−5.93710e17	−0.640368
$582$	0	0
$583$	1.13267e17	0.119470
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.94283e17	0.599578	0.299789	−	0.954006i	$-0.403084\pi$
$587$	5.94283e17	0.599578	0.299789	+	0.954006i	$0.403084\pi$
$588$	0	0
$589$	9.76664e16	0.0963820
$590$	0	0
$591$	−2.14434e18	−2.07002
$592$	0	0
$593$	4.44896e17	0.420149	0.210074	−	0.977685i	$-0.432629\pi$
$593$	4.44896e17	0.420149	0.210074	+	0.977685i	$0.432629\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.76112e18	−2.49604
$598$	0	0
$599$	7.18429e17	0.635492	0.317746	−	0.948176i	$-0.397074\pi$
$599$	7.18429e17	0.635492	0.317746	+	0.948176i	$0.397074\pi$
$600$	0	0
$601$	4.88540e17	0.422879	0.211440	−	0.977391i	$-0.432185\pi$
$601$	4.88540e17	0.422879	0.211440	+	0.977391i	$0.432185\pi$
$602$	0	0
$603$	4.29374e16	0.0363725
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.27437e18	1.84559	0.922794	−	0.385293i	$-0.125900\pi$
$607$	2.27437e18	1.84559	0.922794	+	0.385293i	$0.125900\pi$
$608$	0	0
$609$	1.75400e17	0.139321
$610$	0	0
$611$	−4.27263e18	−3.32221
$612$	0	0
$613$	−1.08936e18	−0.829234	−0.414617	−	0.909996i	$-0.636084\pi$
$613$	−1.08936e18	−0.829234	−0.414617	+	0.909996i	$0.636084\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.02910e18	0.750941	0.375470	−	0.926834i	$-0.377481\pi$
$617$	1.02910e18	0.750941	0.375470	+	0.926834i	$0.377481\pi$
$618$	0	0
$619$	−1.20966e18	−0.864320	−0.432160	−	0.901797i	$-0.642248\pi$
$619$	−1.20966e18	−0.864320	−0.432160	+	0.901797i	$0.642248\pi$
$620$	0	0
$621$	3.18320e17	0.222725
$622$	0	0
$623$	−4.51850e17	−0.309615
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.64216e16	−0.0305127
$628$	0	0
$629$	−1.87076e18	−1.20445
$630$	0	0
$631$	7.05752e17	0.445104	0.222552	−	0.974921i	$-0.428561\pi$
$631$	7.05752e17	0.445104	0.222552	+	0.974921i	$0.428561\pi$
$632$	0	0
$633$	1.65443e18	1.02217
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.76486e18	−1.04665
$638$	0	0
$639$	−1.83895e18	−1.06860
$640$	0	0
$641$	−9.29161e17	−0.529071	−0.264535	−	0.964376i	$-0.585219\pi$
$641$	−9.29161e17	−0.529071	−0.264535	+	0.964376i	$0.585219\pi$
$642$	0	0
$643$	−1.27076e18	−0.709073	−0.354537	−	0.935042i	$-0.615361\pi$
$643$	−1.27076e18	−0.709073	−0.354537	+	0.935042i	$0.615361\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.39517e17	0.396342	0.198171	−	0.980167i	$-0.436500\pi$
$647$	7.39517e17	0.396342	0.198171	+	0.980167i	$0.436500\pi$
$648$	0	0
$649$	−2.92734e17	−0.153774
$650$	0	0
$651$	1.01140e18	0.520773
$652$	0	0
$653$	−1.61890e18	−0.817117	−0.408558	−	0.912732i	$-0.633968\pi$
$653$	−1.61890e18	−0.817117	−0.408558	+	0.912732i	$0.633968\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.77957e17	0.183344
$658$	0	0
$659$	3.27650e18	1.55831	0.779156	−	0.626830i	$-0.215648\pi$
$659$	3.27650e18	1.55831	0.779156	+	0.626830i	$0.215648\pi$
$660$	0	0
$661$	3.67960e18	1.71590	0.857949	−	0.513735i	$-0.171738\pi$
$661$	3.67960e18	1.71590	0.857949	+	0.513735i	$0.171738\pi$
$662$	0	0
$663$	3.85569e18	1.76305
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.97901e17	0.130995
$668$	0	0
$669$	3.17192e16	0.0136789
$670$	0	0
$671$	5.69556e17	0.240902
$672$	0	0
$673$	4.30029e18	1.78402	0.892010	−	0.452015i	$-0.149295\pi$
$673$	4.30029e18	1.78402	0.892010	+	0.452015i	$0.149295\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.04054e18	−1.21374	−0.606868	−	0.794802i	$-0.707574\pi$
$677$	−3.04054e18	−1.21374	−0.606868	+	0.794802i	$0.707574\pi$
$678$	0	0
$679$	−2.41792e18	−0.946867
$680$	0	0
$681$	1.89641e18	0.728576
$682$	0	0
$683$	3.62530e18	1.36650	0.683250	−	0.730185i	$-0.260566\pi$
$683$	3.62530e18	1.36650	0.683250	+	0.730185i	$0.260566\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.06292e18	−0.385727
$688$	0	0
$689$	3.89230e18	1.38606
$690$	0	0
$691$	−3.33110e17	−0.116407	−0.0582036	−	0.998305i	$-0.518537\pi$
$691$	−3.33110e17	−0.116407	−0.0582036	+	0.998305i	$0.518537\pi$
$692$	0	0
$693$	−2.11816e17	−0.0726430
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.15329e18	0.711360
$698$	0	0
$699$	−4.36907e18	−1.41673
$700$	0	0
$701$	4.85234e18	1.54449	0.772244	−	0.635326i	$-0.219134\pi$
$701$	4.85234e18	1.54449	0.772244	+	0.635326i	$0.219134\pi$
$702$	0	0
$703$	7.73992e17	0.241839
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.56370e18	1.07319
$708$	0	0
$709$	9.22088e17	0.272629	0.136314	−	0.990666i	$-0.456474\pi$
$709$	9.22088e17	0.272629	0.136314	+	0.990666i	$0.456474\pi$
$710$	0	0
$711$	−7.37494e17	−0.214095
$712$	0	0
$713$	1.71778e18	0.489649
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.52334e18	1.79306
$718$	0	0
$719$	3.94997e18	1.06624	0.533122	−	0.846039i	$-0.321019\pi$
$719$	3.94997e18	1.06624	0.533122	+	0.846039i	$0.321019\pi$
$720$	0	0
$721$	1.55568e18	0.412421
$722$	0	0
$723$	6.90698e17	0.179841
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.11213e17	0.0781779	0.0390889	−	0.999236i	$-0.487554\pi$
$727$	3.11213e17	0.0781779	0.0390889	+	0.999236i	$0.487554\pi$
$728$	0	0
$729$	−2.18778e18	−0.539852
$730$	0	0
$731$	−1.10407e18	−0.267630
$732$	0	0
$733$	−1.24989e18	−0.297642	−0.148821	−	0.988864i	$-0.547548\pi$
$733$	−1.24989e18	−0.297642	−0.148821	+	0.988864i	$0.547548\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.96862e16	0.00682364
$738$	0	0
$739$	2.89015e18	0.652727	0.326364	−	0.945244i	$-0.394177\pi$
$739$	2.89015e18	0.652727	0.326364	+	0.945244i	$0.394177\pi$
$740$	0	0
$741$	−1.59522e18	−0.354000
$742$	0	0
$743$	5.88473e18	1.28321	0.641607	−	0.767033i	$-0.278268\pi$
$743$	5.88473e18	1.28321	0.641607	+	0.767033i	$0.278268\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.83806e18	0.808217
$748$	0	0
$749$	4.22012e18	0.873359
$750$	0	0
$751$	−4.06231e18	−0.826253	−0.413126	−	0.910674i	$-0.635563\pi$
$751$	−4.06231e18	−0.826253	−0.413126	+	0.910674i	$0.635563\pi$
$752$	0	0
$753$	5.13119e17	0.102577
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−4.15968e18	−0.803410	−0.401705	−	0.915769i	$-0.631582\pi$
$757$	−4.15968e18	−0.803410	−0.401705	+	0.915769i	$0.631582\pi$
$758$	0	0
$759$	−8.16473e17	−0.155014
$760$	0	0
$761$	5.25629e18	0.981023	0.490511	−	0.871435i	$-0.336810\pi$
$761$	5.25629e18	0.981023	0.490511	+	0.871435i	$0.336810\pi$
$762$	0	0
$763$	3.16164e18	0.580101
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.00595e19	−1.78404
$768$	0	0
$769$	5.16139e18	0.900006	0.450003	−	0.893027i	$-0.351423\pi$
$769$	5.16139e18	0.900006	0.450003	+	0.893027i	$0.351423\pi$
$770$	0	0
$771$	−5.06615e18	−0.868610
$772$	0	0
$773$	−4.53280e18	−0.764187	−0.382094	−	0.924124i	$-0.624797\pi$
$773$	−4.53280e18	−0.764187	−0.382094	+	0.924124i	$0.624797\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.01523e18	1.30671
$778$	0	0
$779$	−8.90886e17	−0.142833
$780$	0	0
$781$	−1.27142e18	−0.200473
$782$	0	0
$783$	3.05639e17	0.0473977
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−5.30317e17	−0.0795609	−0.0397805	−	0.999208i	$-0.512666\pi$
$787$	−5.30317e17	−0.0795609	−0.0397805	+	0.999208i	$0.512666\pi$
$788$	0	0
$789$	−6.99416e18	−1.03213
$790$	0	0
$791$	1.36661e18	0.198380
$792$	0	0
$793$	1.95721e19	2.79487
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.11170e19	1.53641	0.768207	−	0.640201i	$-0.221149\pi$
$797$	1.11170e19	1.53641	0.768207	+	0.640201i	$0.221149\pi$
$798$	0	0
$799$	1.09615e19	1.49045
$800$	0	0
$801$	2.92100e18	0.390769
$802$	0	0
$803$	2.61313e17	0.0343961
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.05094e18	−0.643716
$808$	0	0
$809$	−7.61888e18	−0.955489	−0.477745	−	0.878499i	$-0.658546\pi$
$809$	−7.61888e18	−0.955489	−0.477745	+	0.878499i	$0.658546\pi$
$810$	0	0
$811$	9.64634e18	1.19049	0.595246	−	0.803543i	$-0.297054\pi$
$811$	9.64634e18	1.19049	0.595246	+	0.803543i	$0.297054\pi$
$812$	0	0
$813$	2.69843e18	0.327734
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.56791e17	0.0537371
$818$	0	0
$819$	−7.27883e18	−0.842783
$820$	0	0
$821$	−7.34997e18	−0.837636	−0.418818	−	0.908070i	$-0.637555\pi$
$821$	−7.34997e18	−0.837636	−0.418818	+	0.908070i	$0.637555\pi$
$822$	0	0
$823$	−7.39775e18	−0.829852	−0.414926	−	0.909855i	$-0.636193\pi$
$823$	−7.39775e18	−0.829852	−0.414926	+	0.909855i	$0.636193\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.12975e18	−0.448887	−0.224444	−	0.974487i	$-0.572056\pi$
$827$	−4.12975e18	−0.448887	−0.224444	+	0.974487i	$0.572056\pi$
$828$	0	0
$829$	1.44078e19	1.54168	0.770840	−	0.637028i	$-0.219837\pi$
$829$	1.44078e19	1.54168	0.770840	+	0.637028i	$0.219837\pi$
$830$	0	0
$831$	−1.52499e19	−1.60642
$832$	0	0
$833$	4.52778e18	0.469562
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.76240e18	0.177169
$838$	0	0
$839$	−1.29819e19	−1.28495	−0.642473	−	0.766308i	$-0.722091\pi$
$839$	−1.29819e19	−1.28495	−0.642473	+	0.766308i	$0.722091\pi$
$840$	0	0
$841$	−9.97460e18	−0.972123
$842$	0	0
$843$	−4.78412e18	−0.459116
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.56002e18	0.610468
$848$	0	0
$849$	−2.08452e19	−1.91032
$850$	0	0
$851$	1.36131e19	1.22862
$852$	0	0
$853$	−1.71365e19	−1.52319	−0.761595	−	0.648054i	$-0.775583\pi$
$853$	−1.71365e19	−1.52319	−0.761595	+	0.648054i	$0.775583\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.15098e18	0.530357	0.265179	−	0.964199i	$-0.414569\pi$
$857$	6.15098e18	0.530357	0.265179	+	0.964199i	$0.414569\pi$
$858$	0	0
$859$	−9.20029e18	−0.781351	−0.390675	−	0.920529i	$-0.627758\pi$
$859$	−9.20029e18	−0.781351	−0.390675	+	0.920529i	$0.627758\pi$
$860$	0	0
$861$	−9.22575e18	−0.771758
$862$	0	0
$863$	1.52883e19	1.25977	0.629883	−	0.776690i	$-0.283103\pi$
$863$	1.52883e19	1.25977	0.629883	+	0.776690i	$0.283103\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.82939e18	0.546082
$868$	0	0
$869$	−5.09891e17	−0.0401651
$870$	0	0
$871$	1.02013e18	0.0791659
$872$	0	0
$873$	1.56308e19	1.19505
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.81479e19	−1.34688	−0.673442	−	0.739240i	$-0.735185\pi$
$877$	−1.81479e19	−1.34688	−0.673442	+	0.739240i	$0.735185\pi$
$878$	0	0
$879$	4.32772e17	0.0316469
$880$	0	0
$881$	8.86470e17	0.0638735	0.0319367	−	0.999490i	$-0.489832\pi$
$881$	8.86470e17	0.0638735	0.0319367	+	0.999490i	$0.489832\pi$
$882$	0	0
$883$	−1.59579e19	−1.13300	−0.566502	−	0.824060i	$-0.691704\pi$
$883$	−1.59579e19	−1.13300	−0.566502	+	0.824060i	$0.691704\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.37675e19	−1.63862	−0.819312	−	0.573348i	$-0.805644\pi$
$887$	−2.37675e19	−1.63862	−0.819312	+	0.573348i	$0.805644\pi$
$888$	0	0
$889$	1.44025e19	0.978536
$890$	0	0
$891$	−2.57607e18	−0.172486
$892$	0	0
$893$	−4.53514e18	−0.299265
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.80571e19	−1.79843
$898$	0	0
$899$	1.64935e18	0.104201
$900$	0	0
$901$	−9.98580e18	−0.621830
$902$	0	0
$903$	4.73039e18	0.290353
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.21707e19	1.91873	0.959363	−	0.282175i	$-0.0910558\pi$
$907$	3.21707e19	1.91873	0.959363	+	0.282175i	$0.0910558\pi$
$908$	0	0
$909$	−2.30377e19	−1.35448
$910$	0	0
$911$	2.16631e19	1.25560	0.627801	−	0.778374i	$-0.283955\pi$
$911$	2.16631e19	1.25560	0.627801	+	0.778374i	$0.283955\pi$
$912$	0	0
$913$	2.65357e18	0.151625
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.26201e18	0.0700912
$918$	0	0
$919$	2.12417e19	1.16316	0.581579	−	0.813490i	$-0.302435\pi$
$919$	2.12417e19	1.16316	0.581579	+	0.813490i	$0.302435\pi$
$920$	0	0
$921$	−1.44034e19	−0.777639
$922$	0	0
$923$	−4.36908e19	−2.32584
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.00567e19	−0.520522
$928$	0	0
$929$	1.28318e19	0.654914	0.327457	−	0.944866i	$-0.393808\pi$
$929$	1.28318e19	0.654914	0.327457	+	0.944866i	$0.393808\pi$
$930$	0	0
$931$	−1.87329e18	−0.0942827
$932$	0	0
$933$	−2.80847e19	−1.39393
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.69182e19	0.816668	0.408334	−	0.912833i	$-0.366110\pi$
$937$	1.69182e19	0.816668	0.408334	+	0.912833i	$0.366110\pi$
$938$	0	0
$939$	2.71796e19	1.29395
$940$	0	0
$941$	2.77704e19	1.30391	0.651957	−	0.758256i	$-0.273948\pi$
$941$	2.77704e19	1.30391	0.651957	+	0.758256i	$0.273948\pi$
$942$	0	0
$943$	−1.56691e19	−0.725634
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.03601e18	−0.271941	−0.135971	−	0.990713i	$-0.543415\pi$
$947$	−6.03601e18	−0.271941	−0.135971	+	0.990713i	$0.543415\pi$
$948$	0	0
$949$	8.97971e18	0.399054
$950$	0	0
$951$	−2.25337e19	−0.987778
$952$	0	0
$953$	−4.84037e18	−0.209303	−0.104651	−	0.994509i	$-0.533373\pi$
$953$	−4.84037e18	−0.209303	−0.104651	+	0.994509i	$0.533373\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−7.83946e17	−0.0329882
$958$	0	0
$959$	−1.32643e19	−0.550633
$960$	0	0
$961$	−1.49070e19	−0.610503
$962$	0	0
$963$	−2.72811e19	−1.10228
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.33595e19	0.525435	0.262717	−	0.964873i	$-0.415381\pi$
$967$	1.33595e19	0.525435	0.262717	+	0.964873i	$0.415381\pi$
$968$	0	0
$969$	4.09258e18	0.158816
$970$	0	0
$971$	2.18572e19	0.836892	0.418446	−	0.908242i	$-0.362575\pi$
$971$	2.18572e19	0.836892	0.418446	+	0.908242i	$0.362575\pi$
$972$	0	0
$973$	−9.35552e18	−0.353455
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.78484e19	1.02444	0.512219	−	0.858855i	$-0.328824\pi$
$977$	2.78484e19	1.02444	0.512219	+	0.858855i	$0.328824\pi$
$978$	0	0
$979$	2.01953e18	0.0733099
$980$	0	0
$981$	−2.04386e19	−0.732153
$982$	0	0
$983$	3.02967e19	1.07102	0.535510	−	0.844529i	$-0.320120\pi$
$983$	3.02967e19	1.07102	0.535510	+	0.844529i	$0.320120\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.69646e19	−1.61700
$988$	0	0
$989$	8.03414e18	0.273000
$990$	0	0
$991$	5.09074e19	1.70727	0.853634	−	0.520873i	$-0.174393\pi$
$991$	5.09074e19	1.70727	0.853634	+	0.520873i	$0.174393\pi$
$992$	0	0
$993$	−3.54249e19	−1.17257
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.18017e19	−1.02549	−0.512745	−	0.858541i	$-0.671371\pi$
$997$	−3.18017e19	−1.02549	−0.512745	+	0.858541i	$0.671371\pi$
$998$	0	0
$999$	1.39667e19	0.444548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.14.a.l.1.3		10
5.2	odd	4		40.14.c.a.9.16	yes	20
5.3	odd	4		40.14.c.a.9.5	✓	20
5.4	even	2		200.14.a.k.1.8		10
20.3	even	4		80.14.c.d.49.16		20
20.7	even	4		80.14.c.d.49.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.14.c.a.9.5	✓	20	5.3	odd	4
40.14.c.a.9.16	yes	20	5.2	odd	4
80.14.c.d.49.5		20	20.7	even	4
80.14.c.d.49.16		20	20.3	even	4
200.14.a.k.1.8		10	5.4	even	2
200.14.a.l.1.3		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)

\(f(q)\)	\(=\)	\(q-1688.24 q^{3} -194262. q^{7} +1.25582e6 q^{9} +O(q^{10})\)	\(q-1688.24 q^{3} -194262. q^{7} +1.25582e6 q^{9} +868250. q^{11} +2.98364e7 q^{13} -7.65460e7 q^{17} +3.16696e7 q^{19} +3.27961e8 q^{21} +5.57011e8 q^{23} +5.71479e8 q^{27} +5.34821e8 q^{29} +3.08392e9 q^{31} -1.46581e9 q^{33} +2.44396e10 q^{37} -5.03709e10 q^{39} -2.81307e10 q^{41} +1.44237e10 q^{43} -1.43202e11 q^{47} -5.91511e10 q^{49} +1.29228e11 q^{51} +1.30455e11 q^{53} -5.34657e10 q^{57} -3.37155e11 q^{59} +6.55981e11 q^{61} -2.43958e11 q^{63} +3.41908e10 q^{67} -9.40366e11 q^{69} -1.46435e12 q^{71} +3.00965e11 q^{73} -1.68668e11 q^{77} -5.87262e11 q^{79} -2.96697e12 q^{81} +3.05623e12 q^{83} -9.02904e11 q^{87} +2.32598e12 q^{89} -5.79609e12 q^{91} -5.20639e12 q^{93} +1.24467e13 q^{97} +1.09036e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 1276 q^{3} + 341068 q^{7} + 6557498 q^{9}+O(q^{10}) \) 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

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