LMFDB - Embedded newform 9800.2.a.cj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.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:	$78.2533939809$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.43449.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 2x + 1 $ x^4 - x^3 - 7x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1400)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.534166$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.465834 q^{3} -2.78300 q^{9} +O(q^{10})$	$q-0.465834 q^{3} -2.78300 q^{9} +1.37675 q^{11} -5.12091 q^{13} -0.337912 q^{17} +5.99299 q^{19} +6.54790 q^{23} +2.69392 q^{27} -7.99299 q^{29} -7.24883 q^{31} -0.641339 q^{33} +6.05258 q^{37} +2.38550 q^{39} +6.68457 q^{41} -5.02949 q^{43} +6.64634 q^{47} +0.157411 q^{51} +4.87208 q^{53} -2.79174 q^{57} +0.602497 q^{59} -13.0821 q^{61} +12.1052 q^{67} -3.05024 q^{69} -1.39251 q^{71} -1.68518 q^{73} +7.61623 q^{79} +7.09408 q^{81} -7.49532 q^{83} +3.72341 q^{87} -12.3905 q^{89} +3.37675 q^{93} +0.691577 q^{97} -3.83151 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + 5 q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 + 5 * q^9	$ 4 q - 3 q^{3} + 5 q^{9} + 4 q^{13} + 7 q^{17} - 10 q^{19} - 12 q^{27} + 2 q^{29} - 14 q^{31} - 2 q^{33} + 2 q^{37} - 10 q^{39} - 4 q^{41} - 15 q^{43} + 15 q^{47} + 5 q^{51} + 10 q^{53} + 19 q^{57} - q^{59} - 25 q^{61} + 4 q^{67} - 16 q^{69} - 20 q^{71} + 2 q^{73} + 2 q^{79} + 24 q^{81} - 26 q^{83} - 13 q^{87} - 19 q^{89} + 8 q^{93} + 6 q^{97} - 51 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + 5 * q^9 + 4 * q^13 + 7 * q^17 - 10 * q^19 - 12 * q^27 + 2 * q^29 - 14 * q^31 - 2 * q^33 + 2 * q^37 - 10 * q^39 - 4 * q^41 - 15 * q^43 + 15 * q^47 + 5 * q^51 + 10 * q^53 + 19 * q^57 - q^59 - 25 * q^61 + 4 * q^67 - 16 * q^69 - 20 * q^71 + 2 * q^73 + 2 * q^79 + 24 * q^81 - 26 * q^83 - 13 * q^87 - 19 * q^89 + 8 * q^93 + 6 * q^97 - 51 * 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
$3$	−0.465834	−0.268950	−0.134475	−	0.990917i	$-0.542935\pi$
$3$	−0.465834	−0.268950	−0.134475	+	0.990917i	$0.542935\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.78300	−0.927666
$10$	0	0
$11$	1.37675	0.415107	0.207554	−	0.978224i	$-0.433450\pi$
$11$	1.37675	0.415107	0.207554	+	0.978224i	$0.433450\pi$
$12$	0	0
$13$	−5.12091	−1.42029	−0.710143	−	0.704058i	$-0.751370\pi$
$13$	−5.12091	−1.42029	−0.710143	+	0.704058i	$0.751370\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.337912	−0.0819558	−0.0409779	−	0.999160i	$-0.513047\pi$
$17$	−0.337912	−0.0819558	−0.0409779	+	0.999160i	$0.513047\pi$
$18$	0	0
$19$	5.99299	1.37489	0.687443	−	0.726238i	$-0.258733\pi$
$19$	5.99299	1.37489	0.687443	+	0.726238i	$0.258733\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.54790	1.36533	0.682666	−	0.730730i	$-0.260820\pi$
$23$	6.54790	1.36533	0.682666	+	0.730730i	$0.260820\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.69392	0.518445
$28$	0	0
$29$	−7.99299	−1.48426	−0.742130	−	0.670256i	$-0.766184\pi$
$29$	−7.99299	−1.48426	−0.742130	+	0.670256i	$0.766184\pi$
$30$	0	0
$31$	−7.24883	−1.30193	−0.650964	−	0.759108i	$-0.725635\pi$
$31$	−7.24883	−1.30193	−0.650964	+	0.759108i	$0.725635\pi$
$32$	0	0
$33$	−0.641339	−0.111643
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.05258	0.995038	0.497519	−	0.867453i	$-0.334244\pi$
$37$	6.05258	0.995038	0.497519	+	0.867453i	$0.334244\pi$
$38$	0	0
$39$	2.38550	0.381985
$40$	0	0
$41$	6.68457	1.04395	0.521977	−	0.852960i	$-0.325195\pi$
$41$	6.68457	1.04395	0.521977	+	0.852960i	$0.325195\pi$
$42$	0	0
$43$	−5.02949	−0.766990	−0.383495	−	0.923543i	$-0.625280\pi$
$43$	−5.02949	−0.766990	−0.383495	+	0.923543i	$0.625280\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.64634	0.969468	0.484734	−	0.874662i	$-0.338916\pi$
$47$	6.64634	0.969468	0.484734	+	0.874662i	$0.338916\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.157411	0.0220420
$52$	0	0
$53$	4.87208	0.669231	0.334616	−	0.942355i	$-0.391393\pi$
$53$	4.87208	0.669231	0.334616	+	0.942355i	$0.391393\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.79174	−0.369775
$58$	0	0
$59$	0.602497	0.0784385	0.0392192	−	0.999231i	$-0.487513\pi$
$59$	0.602497	0.0784385	0.0392192	+	0.999231i	$0.487513\pi$
$60$	0	0
$61$	−13.0821	−1.67499	−0.837494	−	0.546447i	$-0.815980\pi$
$61$	−13.0821	−1.67499	−0.837494	+	0.546447i	$0.815980\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.1052	1.47888	0.739440	−	0.673222i	$-0.235090\pi$
$67$	12.1052	1.47888	0.739440	+	0.673222i	$0.235090\pi$
$68$	0	0
$69$	−3.05024	−0.367205
$70$	0	0
$71$	−1.39251	−0.165260	−0.0826301	−	0.996580i	$-0.526332\pi$
$71$	−1.39251	−0.165260	−0.0826301	+	0.996580i	$0.526332\pi$
$72$	0	0
$73$	−1.68518	−0.197235	−0.0986176	−	0.995125i	$-0.531442\pi$
$73$	−1.68518	−0.197235	−0.0986176	+	0.995125i	$0.531442\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.61623	0.856893	0.428447	−	0.903567i	$-0.359061\pi$
$79$	7.61623	0.856893	0.428447	+	0.903567i	$0.359061\pi$
$80$	0	0
$81$	7.09408	0.788231
$82$	0	0
$83$	−7.49532	−0.822719	−0.411359	−	0.911473i	$-0.634946\pi$
$83$	−7.49532	−0.822719	−0.411359	+	0.911473i	$0.634946\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.72341	0.399191
$88$	0	0
$89$	−12.3905	−1.31339	−0.656695	−	0.754156i	$-0.728046\pi$
$89$	−12.3905	−1.31339	−0.656695	+	0.754156i	$0.728046\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.37675	0.350153
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.691577	0.0702190	0.0351095	−	0.999383i	$-0.488822\pi$
$97$	0.691577	0.0702190	0.0351095	+	0.999383i	$0.488822\pi$
$98$	0	0
$99$	−3.83151	−0.385081
$100$	0	0
$101$	4.01374	0.399382	0.199691	−	0.979859i	$-0.436006\pi$
$101$	4.01374	0.399382	0.199691	+	0.979859i	$0.436006\pi$
$102$	0	0
$103$	6.79674	0.669702	0.334851	−	0.942271i	$-0.391314\pi$
$103$	6.79674	0.669702	0.334851	+	0.942271i	$0.391314\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.78300	−0.945758	−0.472879	−	0.881127i	$-0.656785\pi$
$107$	−9.78300	−0.945758	−0.472879	+	0.881127i	$0.656785\pi$
$108$	0	0
$109$	4.58175	0.438852	0.219426	−	0.975629i	$-0.429582\pi$
$109$	4.58175	0.438852	0.219426	+	0.975629i	$0.429582\pi$
$110$	0	0
$111$	−2.81950	−0.267615
$112$	0	0
$113$	−17.6688	−1.66214	−0.831071	−	0.556166i	$-0.812272\pi$
$113$	−17.6688	−1.66214	−0.831071	+	0.556166i	$0.812272\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	14.2515	1.31755
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.10455	−0.827686
$122$	0	0
$123$	−3.11390	−0.280771
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.8513	1.31784	0.658921	−	0.752212i	$-0.271013\pi$
$127$	14.8513	1.31784	0.658921	+	0.752212i	$0.271013\pi$
$128$	0	0
$129$	2.34291	0.206282
$130$	0	0
$131$	−1.05459	−0.0921403	−0.0460702	−	0.998938i	$-0.514670\pi$
$131$	−1.05459	−0.0921403	−0.0460702	+	0.998938i	$0.514670\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.2506	−1.38838	−0.694190	−	0.719792i	$-0.744237\pi$
$137$	−16.2506	−1.38838	−0.694190	+	0.719792i	$0.744237\pi$
$138$	0	0
$139$	−4.32591	−0.366918	−0.183459	−	0.983027i	$-0.558730\pi$
$139$	−4.32591	−0.366918	−0.183459	+	0.983027i	$0.558730\pi$
$140$	0	0
$141$	−3.09609	−0.260738
$142$	0	0
$143$	−7.05024	−0.589570
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.67817	0.629020	0.314510	−	0.949254i	$-0.398160\pi$
$149$	7.67817	0.629020	0.314510	+	0.949254i	$0.398160\pi$
$150$	0	0
$151$	2.93868	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$151$	2.93868	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$152$	0	0
$153$	0.940410	0.0760276
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.2783	1.21934	0.609671	−	0.792654i	$-0.291301\pi$
$157$	15.2783	1.21934	0.609671	+	0.792654i	$0.291301\pi$
$158$	0	0
$159$	−2.26958	−0.179989
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.18223	−0.484230	−0.242115	−	0.970248i	$-0.577841\pi$
$163$	−6.18223	−0.484230	−0.242115	+	0.970248i	$0.577841\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.1162	−1.40188	−0.700938	−	0.713222i	$-0.747235\pi$
$167$	−18.1162	−1.40188	−0.700938	+	0.713222i	$0.747235\pi$
$168$	0	0
$169$	13.2237	1.01721
$170$	0	0
$171$	−16.6785	−1.27544
$172$	0	0
$173$	−20.5704	−1.56394	−0.781969	−	0.623318i	$-0.785784\pi$
$173$	−20.5704	−1.56394	−0.781969	+	0.623318i	$0.785784\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.280664	−0.0210960
$178$	0	0
$179$	−9.25117	−0.691465	−0.345733	−	0.938333i	$-0.612370\pi$
$179$	−9.25117	−0.691465	−0.345733	+	0.938333i	$0.612370\pi$
$180$	0	0
$181$	−14.2763	−1.06115	−0.530575	−	0.847638i	$-0.678024\pi$
$181$	−14.2763	−1.06115	−0.530575	+	0.847638i	$0.678024\pi$
$182$	0	0
$183$	6.09408	0.450487
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.465222	−0.0340204
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.4836	1.69922	0.849608	−	0.527414i	$-0.176838\pi$
$191$	23.4836	1.69922	0.849608	+	0.527414i	$0.176838\pi$
$192$	0	0
$193$	13.9772	1.00610	0.503052	−	0.864256i	$-0.332211\pi$
$193$	13.9772	1.00610	0.503052	+	0.864256i	$0.332211\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.69017	0.619149	0.309575	−	0.950875i	$-0.399813\pi$
$197$	8.69017	0.619149	0.309575	+	0.950875i	$0.399813\pi$
$198$	0	0
$199$	−8.56865	−0.607416	−0.303708	−	0.952765i	$-0.598225\pi$
$199$	−8.56865	−0.607416	−0.303708	+	0.952765i	$0.598225\pi$
$200$	0	0
$201$	−5.63900	−0.397744
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−18.2228	−1.26657
$208$	0	0
$209$	8.25087	0.570725
$210$	0	0
$211$	−23.1457	−1.59342	−0.796709	−	0.604363i	$-0.793428\pi$
$211$	−23.1457	−1.59342	−0.796709	+	0.604363i	$0.793428\pi$
$212$	0	0
$213$	0.648677	0.0444466
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.785013	0.0530463
$220$	0	0
$221$	1.73042	0.116401
$222$	0	0
$223$	2.69158	0.180241	0.0901207	−	0.995931i	$-0.471275\pi$
$223$	2.69158	0.180241	0.0901207	+	0.995931i	$0.471275\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.0231	−0.731628	−0.365814	−	0.930688i	$-0.619209\pi$
$227$	−11.0231	−0.731628	−0.365814	+	0.930688i	$0.619209\pi$
$228$	0	0
$229$	−2.97752	−0.196760	−0.0983801	−	0.995149i	$-0.531366\pi$
$229$	−2.97752	−0.196760	−0.0983801	+	0.995149i	$0.531366\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.22574	0.0803011	0.0401505	−	0.999194i	$-0.487216\pi$
$233$	1.22574	0.0803011	0.0401505	+	0.999194i	$0.487216\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.54790	−0.230461
$238$	0	0
$239$	−17.2381	−1.11504	−0.557519	−	0.830164i	$-0.688247\pi$
$239$	−17.2381	−1.11504	−0.557519	+	0.830164i	$0.688247\pi$
$240$	0	0
$241$	4.14400	0.266939	0.133469	−	0.991053i	$-0.457388\pi$
$241$	4.14400	0.266939	0.133469	+	0.991053i	$0.457388\pi$
$242$	0	0
$243$	−11.3864	−0.730439
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−30.6896	−1.95273
$248$	0	0
$249$	3.49158	0.221270
$250$	0	0
$251$	−18.6481	−1.17706	−0.588528	−	0.808477i	$-0.700292\pi$
$251$	−18.6481	−1.17706	−0.588528	+	0.808477i	$0.700292\pi$
$252$	0	0
$253$	9.01486	0.566759
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.46084	0.590151	0.295075	−	0.955474i	$-0.404655\pi$
$257$	9.46084	0.590151	0.295075	+	0.955474i	$0.404655\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	22.2445	1.37690
$262$	0	0
$263$	2.73776	0.168817	0.0844087	−	0.996431i	$-0.473100\pi$
$263$	2.73776	0.168817	0.0844087	+	0.996431i	$0.473100\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.77192	0.353235
$268$	0	0
$269$	−18.2801	−1.11455	−0.557277	−	0.830327i	$-0.688154\pi$
$269$	−18.2801	−1.11455	−0.557277	+	0.830327i	$0.688154\pi$
$270$	0	0
$271$	−31.4173	−1.90847	−0.954233	−	0.299063i	$-0.903326\pi$
$271$	−31.4173	−1.90847	−0.954233	+	0.299063i	$0.903326\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.1785	0.611566	0.305783	−	0.952101i	$-0.401082\pi$
$277$	10.1785	0.611566	0.305783	+	0.952101i	$0.401082\pi$
$278$	0	0
$279$	20.1735	1.20776
$280$	0	0
$281$	−26.4261	−1.57645	−0.788224	−	0.615389i	$-0.788999\pi$
$281$	−26.4261	−1.57645	−0.788224	+	0.615389i	$0.788999\pi$
$282$	0	0
$283$	−8.99500	−0.534697	−0.267349	−	0.963600i	$-0.586148\pi$
$283$	−8.99500	−0.534697	−0.267349	+	0.963600i	$0.586148\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.8858	−0.993283
$290$	0	0
$291$	−0.322160	−0.0188854
$292$	0	0
$293$	34.1092	1.99268	0.996341	−	0.0854640i	$-0.0272372\pi$
$293$	34.1092	1.99268	0.996341	+	0.0854640i	$0.0272372\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.70887	0.215210
$298$	0	0
$299$	−33.5312	−1.93916
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.86974	−0.107414
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.5815	0.946354	0.473177	−	0.880967i	$-0.343107\pi$
$307$	16.5815	0.946354	0.473177	+	0.880967i	$0.343107\pi$
$308$	0	0
$309$	−3.16615	−0.180116
$310$	0	0
$311$	−30.0161	−1.70206	−0.851028	−	0.525121i	$-0.824020\pi$
$311$	−30.0161	−1.70206	−0.851028	+	0.525121i	$0.824020\pi$
$312$	0	0
$313$	−9.60250	−0.542765	−0.271383	−	0.962472i	$-0.587481\pi$
$313$	−9.60250	−0.542765	−0.271383	+	0.962472i	$0.587481\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.7602	1.39067	0.695337	−	0.718684i	$-0.255255\pi$
$317$	24.7602	1.39067	0.695337	+	0.718684i	$0.255255\pi$
$318$	0	0
$319$	−11.0044	−0.616127
$320$	0	0
$321$	4.55726	0.254361
$322$	0	0
$323$	−2.02511	−0.112680
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.13434	−0.118029
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−19.9652	−1.09739	−0.548694	−	0.836023i	$-0.684875\pi$
$331$	−19.9652	−1.09739	−0.548694	+	0.836023i	$0.684875\pi$
$332$	0	0
$333$	−16.8443	−0.923063
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.9421	1.30421	0.652106	−	0.758128i	$-0.273886\pi$
$337$	23.9421	1.30421	0.652106	+	0.758128i	$0.273886\pi$
$338$	0	0
$339$	8.23074	0.447032
$340$	0	0
$341$	−9.97986	−0.540440
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.15914	0.169592	0.0847958	−	0.996398i	$-0.472976\pi$
$347$	3.15914	0.169592	0.0847958	+	0.996398i	$0.472976\pi$
$348$	0	0
$349$	−34.6022	−1.85221	−0.926107	−	0.377261i	$-0.876866\pi$
$349$	−34.6022	−1.85221	−0.926107	+	0.377261i	$0.876866\pi$
$350$	0	0
$351$	−13.7953	−0.736340
$352$	0	0
$353$	−10.7973	−0.574685	−0.287342	−	0.957828i	$-0.592772\pi$
$353$	−10.7973	−0.574685	−0.287342	+	0.957828i	$0.592772\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.62123	−0.191121	−0.0955606	−	0.995424i	$-0.530464\pi$
$359$	−3.62123	−0.191121	−0.0955606	+	0.995424i	$0.530464\pi$
$360$	0	0
$361$	16.9159	0.890311
$362$	0	0
$363$	4.24121	0.222606
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−9.73042	−0.507924	−0.253962	−	0.967214i	$-0.581734\pi$
$367$	−9.73042	−0.507924	−0.253962	+	0.967214i	$0.581734\pi$
$368$	0	0
$369$	−18.6031	−0.968441
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5.40688	−0.279958	−0.139979	−	0.990154i	$-0.544703\pi$
$373$	−5.40688	−0.279958	−0.139979	+	0.990154i	$0.544703\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	40.9314	2.10807
$378$	0	0
$379$	−9.67144	−0.496789	−0.248394	−	0.968659i	$-0.579903\pi$
$379$	−9.67144	−0.496789	−0.248394	+	0.968659i	$0.579903\pi$
$380$	0	0
$381$	−6.91826	−0.354433
$382$	0	0
$383$	−32.4471	−1.65797	−0.828986	−	0.559270i	$-0.811082\pi$
$383$	−32.4471	−1.65797	−0.828986	+	0.559270i	$0.811082\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.9971	0.711511
$388$	0	0
$389$	5.34431	0.270967	0.135484	−	0.990780i	$-0.456741\pi$
$389$	5.34431	0.270967	0.135484	+	0.990780i	$0.456741\pi$
$390$	0	0
$391$	−2.21262	−0.111897
$392$	0	0
$393$	0.491266	0.0247811
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.1326	0.558729	0.279365	−	0.960185i	$-0.409876\pi$
$397$	11.1326	0.558729	0.279365	+	0.960185i	$0.409876\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.1960	1.80754	0.903770	−	0.428018i	$-0.140788\pi$
$401$	36.1960	1.80754	0.903770	+	0.428018i	$0.140788\pi$
$402$	0	0
$403$	37.1206	1.84911
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.33292	0.413047
$408$	0	0
$409$	−34.5660	−1.70918	−0.854589	−	0.519305i	$-0.826191\pi$
$409$	−34.5660	−1.70918	−0.854589	+	0.519305i	$0.826191\pi$
$410$	0	0
$411$	7.57007	0.373404
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.01515	0.0986826
$418$	0	0
$419$	−2.60984	−0.127499	−0.0637494	−	0.997966i	$-0.520306\pi$
$419$	−2.60984	−0.127499	−0.0637494	+	0.997966i	$0.520306\pi$
$420$	0	0
$421$	−27.9541	−1.36240	−0.681201	−	0.732097i	$-0.738542\pi$
$421$	−27.9541	−1.36240	−0.681201	+	0.732097i	$0.738542\pi$
$422$	0	0
$423$	−18.4967	−0.899342
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3.28424	0.158565
$430$	0	0
$431$	38.1866	1.83938	0.919692	−	0.392640i	$-0.128438\pi$
$431$	38.1866	1.83938	0.919692	+	0.392640i	$0.128438\pi$
$432$	0	0
$433$	−13.7257	−0.659617	−0.329809	−	0.944048i	$-0.606984\pi$
$433$	−13.7257	−0.659617	−0.329809	+	0.944048i	$0.606984\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	39.2415	1.87718
$438$	0	0
$439$	16.9866	0.810726	0.405363	−	0.914156i	$-0.367145\pi$
$439$	16.9866	0.810726	0.405363	+	0.914156i	$0.367145\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.54291	0.120817	0.0604086	−	0.998174i	$-0.480760\pi$
$443$	2.54291	0.120817	0.0604086	+	0.998174i	$0.480760\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.57675	−0.169175
$448$	0	0
$449$	34.5205	1.62912	0.814561	−	0.580078i	$-0.196978\pi$
$449$	34.5205	1.62912	0.814561	+	0.580078i	$0.196978\pi$
$450$	0	0
$451$	9.20301	0.433353
$452$	0	0
$453$	−1.36894	−0.0643183
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.2144	−0.898811	−0.449405	−	0.893328i	$-0.648364\pi$
$457$	−19.2144	−0.898811	−0.449405	+	0.893328i	$0.648364\pi$
$458$	0	0
$459$	−0.910308	−0.0424896
$460$	0	0
$461$	5.48457	0.255442	0.127721	−	0.991810i	$-0.459234\pi$
$461$	5.48457	0.255442	0.127721	+	0.991810i	$0.459234\pi$
$462$	0	0
$463$	−2.55056	−0.118534	−0.0592672	−	0.998242i	$-0.518876\pi$
$463$	−2.55056	−0.118534	−0.0592672	+	0.998242i	$0.518876\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.6977	1.00405	0.502025	−	0.864853i	$-0.332589\pi$
$467$	21.6977	1.00405	0.502025	+	0.864853i	$0.332589\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−7.11717	−0.327942
$472$	0	0
$473$	−6.92437	−0.318383
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−13.5590	−0.620823
$478$	0	0
$479$	38.0674	1.73935	0.869673	−	0.493629i	$-0.164330\pi$
$479$	38.0674	1.73935	0.869673	+	0.493629i	$0.164330\pi$
$480$	0	0
$481$	−30.9947	−1.41324
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0.533554	0.0241776	0.0120888	−	0.999927i	$-0.496152\pi$
$487$	0.533554	0.0241776	0.0120888	+	0.999927i	$0.496152\pi$
$488$	0	0
$489$	2.87990	0.130233
$490$	0	0
$491$	22.7255	1.02559	0.512793	−	0.858512i	$-0.328611\pi$
$491$	22.7255	1.02559	0.512793	+	0.858512i	$0.328611\pi$
$492$	0	0
$493$	2.70093	0.121644
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	14.0657	0.629667	0.314834	−	0.949147i	$-0.398051\pi$
$499$	14.0657	0.629667	0.314834	+	0.949147i	$0.398051\pi$
$500$	0	0
$501$	8.43917	0.377034
$502$	0	0
$503$	−15.2757	−0.681108	−0.340554	−	0.940225i	$-0.610615\pi$
$503$	−15.2757	−0.681108	−0.340554	+	0.940225i	$0.610615\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.16007	−0.273578
$508$	0	0
$509$	21.4743	0.951831	0.475916	−	0.879491i	$-0.342117\pi$
$509$	21.4743	0.951831	0.475916	+	0.879491i	$0.342117\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	16.1446	0.712803
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.15037	0.402433
$518$	0	0
$519$	9.58239	0.420620
$520$	0	0
$521$	−38.3029	−1.67808	−0.839039	−	0.544071i	$-0.816882\pi$
$521$	−38.3029	−1.67808	−0.839039	+	0.544071i	$0.816882\pi$
$522$	0	0
$523$	−36.7777	−1.60818	−0.804089	−	0.594509i	$-0.797346\pi$
$523$	−36.7777	−1.60818	−0.804089	+	0.594509i	$0.797346\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.44947	0.106701
$528$	0	0
$529$	19.8750	0.864132
$530$	0	0
$531$	−1.67675	−0.0727647
$532$	0	0
$533$	−34.2311	−1.48271
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.30951	0.185969
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16.6337	0.715139	0.357570	−	0.933886i	$-0.383605\pi$
$541$	16.6337	0.715139	0.357570	+	0.933886i	$0.383605\pi$
$542$	0	0
$543$	6.65039	0.285396
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.0493	0.942761	0.471380	−	0.881930i	$-0.343756\pi$
$547$	22.0493	0.942761	0.471380	+	0.881930i	$0.343756\pi$
$548$	0	0
$549$	36.4074	1.55383
$550$	0	0
$551$	−47.9019	−2.04069
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.6691	−1.38424	−0.692118	−	0.721784i	$-0.743322\pi$
$557$	−32.6691	−1.38424	−0.692118	+	0.721784i	$0.743322\pi$
$558$	0	0
$559$	25.7556	1.08934
$560$	0	0
$561$	0.216717	0.00914978
$562$	0	0
$563$	−35.4086	−1.49229	−0.746147	−	0.665781i	$-0.768098\pi$
$563$	−35.4086	−1.49229	−0.746147	+	0.665781i	$0.768098\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.9822	−0.544244	−0.272122	−	0.962263i	$-0.587725\pi$
$569$	−12.9822	−0.544244	−0.272122	+	0.962263i	$0.587725\pi$
$570$	0	0
$571$	5.00500	0.209453	0.104726	−	0.994501i	$-0.466603\pi$
$571$	5.00500	0.209453	0.104726	+	0.994501i	$0.466603\pi$
$572$	0	0
$573$	−10.9395	−0.457004
$574$	0	0
$575$	0	0
$576$	0	0
$577$	44.8222	1.86597	0.932986	−	0.359912i	$-0.117193\pi$
$577$	44.8222	1.86597	0.932986	+	0.359912i	$0.117193\pi$
$578$	0	0
$579$	−6.51108	−0.270591
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.70766	0.277803
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.1092	−0.953819	−0.476910	−	0.878952i	$-0.658243\pi$
$587$	−23.1092	−0.953819	−0.476910	+	0.878952i	$0.658243\pi$
$588$	0	0
$589$	−43.4422	−1.79000
$590$	0	0
$591$	−4.04818	−0.166520
$592$	0	0
$593$	40.8245	1.67646	0.838231	−	0.545315i	$-0.183590\pi$
$593$	40.8245	1.67646	0.838231	+	0.545315i	$0.183590\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.99157	0.163364
$598$	0	0
$599$	38.9787	1.59263	0.796313	−	0.604885i	$-0.206781\pi$
$599$	38.9787	1.59263	0.796313	+	0.604885i	$0.206781\pi$
$600$	0	0
$601$	8.31217	0.339060	0.169530	−	0.985525i	$-0.445775\pi$
$601$	8.31217	0.339060	0.169530	+	0.985525i	$0.445775\pi$
$602$	0	0
$603$	−33.6886	−1.37191
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.20061	0.0487313	0.0243656	−	0.999703i	$-0.492243\pi$
$607$	1.20061	0.0487313	0.0243656	+	0.999703i	$0.492243\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−34.0353	−1.37692
$612$	0	0
$613$	−18.5147	−0.747800	−0.373900	−	0.927469i	$-0.621980\pi$
$613$	−18.5147	−0.747800	−0.373900	+	0.927469i	$0.621980\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.81137	−0.113181	−0.0565907	−	0.998397i	$-0.518023\pi$
$617$	−2.81137	−0.113181	−0.0565907	+	0.998397i	$0.518023\pi$
$618$	0	0
$619$	−44.9659	−1.80733	−0.903665	−	0.428239i	$-0.859134\pi$
$619$	−44.9659	−1.80733	−0.903665	+	0.428239i	$0.859134\pi$
$620$	0	0
$621$	17.6395	0.707849
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.84354	−0.153496
$628$	0	0
$629$	−2.04524	−0.0815491
$630$	0	0
$631$	−9.37270	−0.373121	−0.186561	−	0.982443i	$-0.559734\pi$
$631$	−9.37270	−0.373121	−0.186561	+	0.982443i	$0.559734\pi$
$632$	0	0
$633$	10.7821	0.428549
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.87534	0.153306
$640$	0	0
$641$	−44.8236	−1.77043	−0.885213	−	0.465186i	$-0.845987\pi$
$641$	−44.8236	−1.77043	−0.885213	+	0.465186i	$0.845987\pi$
$642$	0	0
$643$	−18.1709	−0.716589	−0.358294	−	0.933609i	$-0.616642\pi$
$643$	−18.1709	−0.716589	−0.358294	+	0.933609i	$0.616642\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.1092	−0.790575	−0.395288	−	0.918557i	$-0.629355\pi$
$647$	−20.1092	−0.790575	−0.395288	+	0.918557i	$0.629355\pi$
$648$	0	0
$649$	0.829491	0.0325604
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.50641	0.293748	0.146874	−	0.989155i	$-0.453079\pi$
$653$	7.50641	0.293748	0.146874	+	0.989155i	$0.453079\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.68985	0.182968
$658$	0	0
$659$	−41.5476	−1.61847	−0.809233	−	0.587488i	$-0.800117\pi$
$659$	−41.5476	−1.61847	−0.809233	+	0.587488i	$0.800117\pi$
$660$	0	0
$661$	−1.26397	−0.0491629	−0.0245814	−	0.999698i	$-0.507825\pi$
$661$	−1.26397	−0.0491629	−0.0245814	+	0.999698i	$0.507825\pi$
$662$	0	0
$663$	−0.806089	−0.0313059
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−52.3373	−2.02651
$668$	0	0
$669$	−1.25383	−0.0484758
$670$	0	0
$671$	−18.0108	−0.695299
$672$	0	0
$673$	−24.0432	−0.926798	−0.463399	−	0.886150i	$-0.653370\pi$
$673$	−24.0432	−0.926798	−0.463399	+	0.886150i	$0.653370\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.4860	0.864206	0.432103	−	0.901824i	$-0.357772\pi$
$677$	22.4860	0.864206	0.432103	+	0.901824i	$0.357772\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.13493	0.196771
$682$	0	0
$683$	−14.9410	−0.571703	−0.285852	−	0.958274i	$-0.592276\pi$
$683$	−14.9410	−0.571703	−0.285852	+	0.958274i	$0.592276\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.38703	0.0529186
$688$	0	0
$689$	−24.9495	−0.950499
$690$	0	0
$691$	−37.4317	−1.42397	−0.711985	−	0.702195i	$-0.752203\pi$
$691$	−37.4317	−1.42397	−0.711985	+	0.702195i	$0.752203\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2.25880	−0.0855581
$698$	0	0
$699$	−0.570993	−0.0215969
$700$	0	0
$701$	−44.8376	−1.69349	−0.846747	−	0.531996i	$-0.821442\pi$
$701$	−44.8376	−1.69349	−0.846747	+	0.531996i	$0.821442\pi$
$702$	0	0
$703$	36.2730	1.36806
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	19.2918	0.724518	0.362259	−	0.932078i	$-0.382006\pi$
$709$	19.2918	0.724518	0.362259	+	0.932078i	$0.382006\pi$
$710$	0	0
$711$	−21.1960	−0.794911
$712$	0	0
$713$	−47.4647	−1.77757
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.03009	0.299889
$718$	0	0
$719$	4.40016	0.164098	0.0820491	−	0.996628i	$-0.473854\pi$
$719$	4.40016	0.164098	0.0820491	+	0.996628i	$0.473854\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.93042	−0.0717930
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−43.4378	−1.61102	−0.805509	−	0.592583i	$-0.798108\pi$
$727$	−43.4378	−1.61102	−0.805509	+	0.592583i	$0.798108\pi$
$728$	0	0
$729$	−15.9780	−0.591779
$730$	0	0
$731$	1.69953	0.0628593
$732$	0	0
$733$	−10.2883	−0.380007	−0.190003	−	0.981783i	$-0.560850\pi$
$733$	−10.2883	−0.380007	−0.190003	+	0.981783i	$0.560850\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.6658	0.613894
$738$	0	0
$739$	29.4535	1.08347	0.541733	−	0.840551i	$-0.317768\pi$
$739$	29.4535	1.08347	0.541733	+	0.840551i	$0.317768\pi$
$740$	0	0
$741$	14.2962	0.525186
$742$	0	0
$743$	17.9240	0.657569	0.328785	−	0.944405i	$-0.393361\pi$
$743$	17.9240	0.657569	0.328785	+	0.944405i	$0.393361\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	20.8595	0.763208
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−23.2181	−0.847241	−0.423621	−	0.905840i	$-0.639241\pi$
$751$	−23.2181	−0.847241	−0.423621	+	0.905840i	$0.639241\pi$
$752$	0	0
$753$	8.68691	0.316569
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−40.8014	−1.48295	−0.741477	−	0.670978i	$-0.765874\pi$
$757$	−40.8014	−1.48295	−0.741477	+	0.670978i	$0.765874\pi$
$758$	0	0
$759$	−4.19943	−0.152430
$760$	0	0
$761$	−4.42933	−0.160563	−0.0802816	−	0.996772i	$-0.525582\pi$
$761$	−4.42933	−0.160563	−0.0802816	+	0.996772i	$0.525582\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.08533	−0.111405
$768$	0	0
$769$	−3.66676	−0.132227	−0.0661133	−	0.997812i	$-0.521060\pi$
$769$	−3.66676	−0.132227	−0.0661133	+	0.997812i	$0.521060\pi$
$770$	0	0
$771$	−4.40718	−0.158721
$772$	0	0
$773$	6.97051	0.250712	0.125356	−	0.992112i	$-0.459993\pi$
$773$	6.97051	0.250712	0.125356	+	0.992112i	$0.459993\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40.0605	1.43532
$780$	0	0
$781$	−1.91714	−0.0686007
$782$	0	0
$783$	−21.5325	−0.769507
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−26.3537	−0.939407	−0.469703	−	0.882824i	$-0.655639\pi$
$787$	−26.3537	−0.939407	−0.469703	+	0.882824i	$0.655639\pi$
$788$	0	0
$789$	−1.27534	−0.0454033
$790$	0	0
$791$	0	0
$792$	0	0
$793$	66.9921	2.37896
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.31451	−0.117406	−0.0587030	−	0.998275i	$-0.518696\pi$
$797$	−3.31451	−0.117406	−0.0587030	+	0.998275i	$0.518696\pi$
$798$	0	0
$799$	−2.24588	−0.0794535
$800$	0	0
$801$	34.4827	1.21839
$802$	0	0
$803$	−2.32008	−0.0818737
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.51547	0.299759
$808$	0	0
$809$	−32.9871	−1.15976	−0.579882	−	0.814700i	$-0.696901\pi$
$809$	−32.9871	−1.15976	−0.579882	+	0.814700i	$0.696901\pi$
$810$	0	0
$811$	30.1575	1.05897	0.529486	−	0.848319i	$-0.322385\pi$
$811$	30.1575	1.05897	0.529486	+	0.848319i	$0.322385\pi$
$812$	0	0
$813$	14.6353	0.513281
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−30.1417	−1.05452
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.84903	−0.204133	−0.102066	−	0.994778i	$-0.532545\pi$
$821$	−5.84903	−0.204133	−0.102066	+	0.994778i	$0.532545\pi$
$822$	0	0
$823$	−3.77660	−0.131644	−0.0658220	−	0.997831i	$-0.520967\pi$
$823$	−3.77660	−0.131644	−0.0658220	+	0.997831i	$0.520967\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19.7854	−0.688005	−0.344002	−	0.938969i	$-0.611783\pi$
$827$	−19.7854	−0.688005	−0.344002	+	0.938969i	$0.611783\pi$
$828$	0	0
$829$	−42.3891	−1.47223	−0.736117	−	0.676854i	$-0.763343\pi$
$829$	−42.3891	−1.47223	−0.736117	+	0.676854i	$0.763343\pi$
$830$	0	0
$831$	−4.74149	−0.164480
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−19.5278	−0.674978
$838$	0	0
$839$	−33.0442	−1.14081	−0.570406	−	0.821363i	$-0.693214\pi$
$839$	−33.0442	−1.14081	−0.570406	+	0.821363i	$0.693214\pi$
$840$	0	0
$841$	34.8879	1.20303
$842$	0	0
$843$	12.3102	0.423985
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	4.19018	0.143807
$850$	0	0
$851$	39.6317	1.35856
$852$	0	0
$853$	39.3487	1.34727	0.673637	−	0.739062i	$-0.264731\pi$
$853$	39.3487	1.34727	0.673637	+	0.739062i	$0.264731\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.1615	1.30357	0.651786	−	0.758403i	$-0.274020\pi$
$857$	38.1615	1.30357	0.651786	+	0.758403i	$0.274020\pi$
$858$	0	0
$859$	1.57240	0.0536495	0.0268247	−	0.999640i	$-0.491460\pi$
$859$	1.57240	0.0536495	0.0268247	+	0.999640i	$0.491460\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.2398	−1.09746	−0.548728	−	0.836001i	$-0.684888\pi$
$863$	−32.2398	−1.09746	−0.548728	+	0.836001i	$0.684888\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.86599	0.267143
$868$	0	0
$869$	10.4857	0.355703
$870$	0	0
$871$	−61.9894	−2.10043
$872$	0	0
$873$	−1.92466	−0.0651398
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.7844	−0.769376	−0.384688	−	0.923047i	$-0.625691\pi$
$877$	−22.7844	−0.769376	−0.384688	+	0.923047i	$0.625691\pi$
$878$	0	0
$879$	−15.8892	−0.535931
$880$	0	0
$881$	−10.2816	−0.346396	−0.173198	−	0.984887i	$-0.555410\pi$
$881$	−10.2816	−0.346396	−0.173198	+	0.984887i	$0.555410\pi$
$882$	0	0
$883$	−36.8161	−1.23896	−0.619480	−	0.785013i	$-0.712656\pi$
$883$	−36.8161	−1.23896	−0.619480	+	0.785013i	$0.712656\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.6598	0.727264	0.363632	−	0.931543i	$-0.381536\pi$
$887$	21.6598	0.727264	0.363632	+	0.931543i	$0.381536\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.76680	0.327200
$892$	0	0
$893$	39.8314	1.33291
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15.6200	0.521536
$898$	0	0
$899$	57.9398	1.93240
$900$	0	0
$901$	−1.64634	−0.0548474
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	17.8841	0.593831	0.296916	−	0.954904i	$-0.404042\pi$
$907$	17.8841	0.593831	0.296916	+	0.954904i	$0.404042\pi$
$908$	0	0
$909$	−11.1702	−0.370493
$910$	0	0
$911$	−13.5178	−0.447865	−0.223932	−	0.974605i	$-0.571889\pi$
$911$	−13.5178	−0.447865	−0.223932	+	0.974605i	$0.571889\pi$
$912$	0	0
$913$	−10.3192	−0.341516
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.04198	0.0343716	0.0171858	−	0.999852i	$-0.494529\pi$
$919$	1.04198	0.0343716	0.0171858	+	0.999852i	$0.494529\pi$
$920$	0	0
$921$	−7.72421	−0.254522
$922$	0	0
$923$	7.13090	0.234717
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−18.9153	−0.621260
$928$	0	0
$929$	−6.33263	−0.207767	−0.103883	−	0.994589i	$-0.533127\pi$
$929$	−6.33263	−0.207767	−0.103883	+	0.994589i	$0.533127\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	13.9825	0.457767
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.2670	0.694761	0.347381	−	0.937724i	$-0.387071\pi$
$937$	21.2670	0.694761	0.347381	+	0.937724i	$0.387071\pi$
$938$	0	0
$939$	4.47317	0.145976
$940$	0	0
$941$	16.1128	0.525262	0.262631	−	0.964896i	$-0.415410\pi$
$941$	16.1128	0.525262	0.262631	+	0.964896i	$0.415410\pi$
$942$	0	0
$943$	43.7699	1.42534
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.1828	−0.525872	−0.262936	−	0.964813i	$-0.584691\pi$
$947$	−16.1828	−0.525872	−0.262936	+	0.964813i	$0.584691\pi$
$948$	0	0
$949$	8.62964	0.280130
$950$	0	0
$951$	−11.5342	−0.374021
$952$	0	0
$953$	−12.4950	−0.404753	−0.202377	−	0.979308i	$-0.564866\pi$
$953$	−12.4950	−0.404753	−0.202377	+	0.979308i	$0.564866\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.12622	0.165707
$958$	0	0
$959$	0	0
$960$	0	0
$961$	21.5456	0.695019
$962$	0	0
$963$	27.2261	0.877348
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28.6562	0.921521	0.460761	−	0.887524i	$-0.347577\pi$
$967$	28.6562	0.921521	0.460761	+	0.887524i	$0.347577\pi$
$968$	0	0
$969$	0.943363	0.0303052
$970$	0	0
$971$	15.8741	0.509424	0.254712	−	0.967017i	$-0.418019\pi$
$971$	15.8741	0.509424	0.254712	+	0.967017i	$0.418019\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.3292	−0.810352	−0.405176	−	0.914239i	$-0.632790\pi$
$977$	−25.3292	−0.810352	−0.405176	+	0.914239i	$0.632790\pi$
$978$	0	0
$979$	−17.0587	−0.545197
$980$	0	0
$981$	−12.7510	−0.407108
$982$	0	0
$983$	−9.93666	−0.316930	−0.158465	−	0.987365i	$-0.550655\pi$
$983$	−9.93666	−0.316930	−0.158465	+	0.987365i	$0.550655\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32.9326	−1.04720
$990$	0	0
$991$	30.3738	0.964856	0.482428	−	0.875936i	$-0.339755\pi$
$991$	30.3738	0.964856	0.482428	+	0.875936i	$0.339755\pi$
$992$	0	0
$993$	9.30049	0.295142
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−0.676437	−0.0214230	−0.0107115	−	0.999943i	$-0.503410\pi$
$997$	−0.676437	−0.0214230	−0.0107115	+	0.999943i	$0.503410\pi$
$998$	0	0
$999$	16.3052	0.515872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.cj.1.3		4
5.4	even	2		9800.2.a.ct.1.2		4
7.2	even	3		1400.2.q.m.1201.2	yes	8
7.4	even	3		1400.2.q.m.401.2	yes	8
7.6	odd	2		9800.2.a.cu.1.2		4
35.2	odd	12		1400.2.bh.j.249.4		16
35.4	even	6		1400.2.q.l.401.3	✓	8
35.9	even	6		1400.2.q.l.1201.3	yes	8
35.18	odd	12		1400.2.bh.j.849.4		16
35.23	odd	12		1400.2.bh.j.249.5		16
35.32	odd	12		1400.2.bh.j.849.5		16
35.34	odd	2		9800.2.a.ck.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.q.l.401.3	✓	8	35.4	even	6
1400.2.q.l.1201.3	yes	8	35.9	even	6
1400.2.q.m.401.2	yes	8	7.4	even	3
1400.2.q.m.1201.2	yes	8	7.2	even	3
1400.2.bh.j.249.4		16	35.2	odd	12
1400.2.bh.j.249.5		16	35.23	odd	12
1400.2.bh.j.849.4		16	35.18	odd	12
1400.2.bh.j.849.5		16	35.32	odd	12
9800.2.a.cj.1.3		4	1.1	even	1	trivial
9800.2.a.ck.1.3		4	35.34	odd	2
9800.2.a.ct.1.2		4	5.4	even	2
9800.2.a.cu.1.2		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q-0.465834 q^{3} -2.78300 q^{9} +O(q^{10})\)	\(q-0.465834 q^{3} -2.78300 q^{9} +1.37675 q^{11} -5.12091 q^{13} -0.337912 q^{17} +5.99299 q^{19} +6.54790 q^{23} +2.69392 q^{27} -7.99299 q^{29} -7.24883 q^{31} -0.641339 q^{33} +6.05258 q^{37} +2.38550 q^{39} +6.68457 q^{41} -5.02949 q^{43} +6.64634 q^{47} +0.157411 q^{51} +4.87208 q^{53} -2.79174 q^{57} +0.602497 q^{59} -13.0821 q^{61} +12.1052 q^{67} -3.05024 q^{69} -1.39251 q^{71} -1.68518 q^{73} +7.61623 q^{79} +7.09408 q^{81} -7.49532 q^{83} +3.72341 q^{87} -12.3905 q^{89} +3.37675 q^{93} +0.691577 q^{97} -3.83151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + 5 q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 + 5 * q^9	\( 4 q - 3 q^{3} + 5 q^{9} + 4 q^{13} + 7 q^{17} - 10 q^{19} - 12 q^{27} + 2 q^{29} - 14 q^{31} - 2 q^{33} + 2 q^{37} - 10 q^{39} - 4 q^{41} - 15 q^{43} + 15 q^{47} + 5 q^{51} + 10 q^{53} + 19 q^{57} - q^{59} - 25 q^{61} + 4 q^{67} - 16 q^{69} - 20 q^{71} + 2 q^{73} + 2 q^{79} + 24 q^{81} - 26 q^{83} - 13 q^{87} - 19 q^{89} + 8 q^{93} + 6 q^{97} - 51 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + 5 * q^9 + 4 * q^13 + 7 * q^17 - 10 * q^19 - 12 * q^27 + 2 * q^29 - 14 * q^31 - 2 * q^33 + 2 * q^37 - 10 * q^39 - 4 * q^41 - 15 * q^43 + 15 * q^47 + 5 * q^51 + 10 * q^53 + 19 * q^57 - q^59 - 25 * q^61 + 4 * q^67 - 16 * q^69 - 20 * q^71 + 2 * q^73 + 2 * q^79 + 24 * q^81 - 26 * q^83 - 13 * q^87 - 19 * q^89 + 8 * q^93 + 6 * q^97 - 51 * q^99

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