LMFDB - Embedded newform 9996.2.a.z.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9996,2,Mod(1,9996)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9996, 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("9996.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9996 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9996.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:	$79.8184618605$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7053.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 3x + 3 $ x^4 - 2x^3 - 4x^2 + 3*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1428)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.36361$ of defining polynomial
Character	$\chi$	$=$	9996.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.95025 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.95025 q^{5} +1.00000 q^{9} +2.42308 q^{11} +0.636391 q^{13} -3.95025 q^{15} +1.00000 q^{17} +8.55989 q^{19} +3.92726 q^{23} +10.6045 q^{25} -1.00000 q^{27} +6.00973 q^{29} +3.30926 q^{31} -2.42308 q^{33} +11.6737 q^{37} -0.636391 q^{39} -2.99624 q^{41} -2.47146 q^{43} +3.95025 q^{45} +6.09083 q^{47} -1.00000 q^{51} -11.7221 q^{53} +9.57179 q^{55} -8.55989 q^{57} +8.94053 q^{59} -11.5815 q^{61} +2.51391 q^{65} +1.43122 q^{67} -3.92726 q^{69} -5.75021 q^{71} -8.71833 q^{73} -10.6045 q^{75} -12.8008 q^{79} +1.00000 q^{81} -4.17705 q^{83} +3.95025 q^{85} -6.00973 q^{87} -12.6410 q^{89} -3.30926 q^{93} +33.8138 q^{95} -9.25439 q^{97} +2.42308 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^9	$ 4 q - 4 q^{3} - 2 q^{5} + 4 q^{9} + q^{11} + 10 q^{13} + 2 q^{15} + 4 q^{17} + 9 q^{19} - 5 q^{23} + 18 q^{25} - 4 q^{27} + 5 q^{29} + 24 q^{31} - q^{33} + 2 q^{37} - 10 q^{39} - 14 q^{43} - 2 q^{45} + 2 q^{47} - 4 q^{51} - 15 q^{53} + 30 q^{55} - 9 q^{57} + 37 q^{59} - 19 q^{61} - 21 q^{65} + 2 q^{67} + 5 q^{69} - 11 q^{71} + 9 q^{73} - 18 q^{75} - 9 q^{79} + 4 q^{81} - 8 q^{83} - 2 q^{85} - 5 q^{87} - 22 q^{89} - 24 q^{93} + 33 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^9 + q^11 + 10 * q^13 + 2 * q^15 + 4 * q^17 + 9 * q^19 - 5 * q^23 + 18 * q^25 - 4 * q^27 + 5 * q^29 + 24 * q^31 - q^33 + 2 * q^37 - 10 * q^39 - 14 * q^43 - 2 * q^45 + 2 * q^47 - 4 * q^51 - 15 * q^53 + 30 * q^55 - 9 * q^57 + 37 * q^59 - 19 * q^61 - 21 * q^65 + 2 * q^67 + 5 * q^69 - 11 * q^71 + 9 * q^73 - 18 * q^75 - 9 * q^79 + 4 * q^81 - 8 * q^83 - 2 * q^85 - 5 * q^87 - 22 * q^89 - 24 * q^93 + 33 * q^95 - 13 * q^97 + 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.95025	1.76661	0.883304	−	0.468801i	$-0.155314\pi$
$5$	3.95025	1.76661	0.883304	+	0.468801i	$0.155314\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.42308	0.730587	0.365293	−	0.930892i	$-0.380969\pi$
$11$	2.42308	0.730587	0.365293	+	0.930892i	$0.380969\pi$
$12$	0	0
$13$	0.636391	0.176503	0.0882516	−	0.996098i	$-0.471872\pi$
$13$	0.636391	0.176503	0.0882516	+	0.996098i	$0.471872\pi$
$14$	0	0
$15$	−3.95025	−1.01995
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	8.55989	1.96377	0.981887	−	0.189467i	$-0.0606761\pi$
$19$	8.55989	1.96377	0.981887	+	0.189467i	$0.0606761\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.92726	0.818891	0.409445	−	0.912335i	$-0.365722\pi$
$23$	3.92726	0.818891	0.409445	+	0.912335i	$0.365722\pi$
$24$	0	0
$25$	10.6045	2.12090
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.00973	1.11598	0.557989	−	0.829848i	$-0.311573\pi$
$29$	6.00973	1.11598	0.557989	+	0.829848i	$0.311573\pi$
$30$	0	0
$31$	3.30926	0.594361	0.297181	−	0.954821i	$-0.403954\pi$
$31$	3.30926	0.594361	0.297181	+	0.954821i	$0.403954\pi$
$32$	0	0
$33$	−2.42308	−0.421804
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.6737	1.91915	0.959573	−	0.281459i	$-0.0908185\pi$
$37$	11.6737	1.91915	0.959573	+	0.281459i	$0.0908185\pi$
$38$	0	0
$39$	−0.636391	−0.101904
$40$	0	0
$41$	−2.99624	−0.467934	−0.233967	−	0.972245i	$-0.575171\pi$
$41$	−2.99624	−0.467934	−0.233967	+	0.972245i	$0.575171\pi$
$42$	0	0
$43$	−2.47146	−0.376894	−0.188447	−	0.982083i	$-0.560345\pi$
$43$	−2.47146	−0.376894	−0.188447	+	0.982083i	$0.560345\pi$
$44$	0	0
$45$	3.95025	0.588869
$46$	0	0
$47$	6.09083	0.888438	0.444219	−	0.895918i	$-0.353481\pi$
$47$	6.09083	0.888438	0.444219	+	0.895918i	$0.353481\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−11.7221	−1.61015	−0.805076	−	0.593171i	$-0.797876\pi$
$53$	−11.7221	−1.61015	−0.805076	+	0.593171i	$0.797876\pi$
$54$	0	0
$55$	9.57179	1.29066
$56$	0	0
$57$	−8.55989	−1.13379
$58$	0	0
$59$	8.94053	1.16396	0.581979	−	0.813204i	$-0.302279\pi$
$59$	8.94053	1.16396	0.581979	+	0.813204i	$0.302279\pi$
$60$	0	0
$61$	−11.5815	−1.48286	−0.741431	−	0.671029i	$-0.765852\pi$
$61$	−11.5815	−1.48286	−0.741431	+	0.671029i	$0.765852\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.51391	0.311812
$66$	0	0
$67$	1.43122	0.174851	0.0874256	−	0.996171i	$-0.472136\pi$
$67$	1.43122	0.174851	0.0874256	+	0.996171i	$0.472136\pi$
$68$	0	0
$69$	−3.92726	−0.472787
$70$	0	0
$71$	−5.75021	−0.682424	−0.341212	−	0.939986i	$-0.610837\pi$
$71$	−5.75021	−0.682424	−0.341212	+	0.939986i	$0.610837\pi$
$72$	0	0
$73$	−8.71833	−1.02040	−0.510202	−	0.860055i	$-0.670429\pi$
$73$	−8.71833	−1.02040	−0.510202	+	0.860055i	$0.670429\pi$
$74$	0	0
$75$	−10.6045	−1.22450
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.8008	−1.44020	−0.720101	−	0.693869i	$-0.755905\pi$
$79$	−12.8008	−1.44020	−0.720101	+	0.693869i	$0.755905\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.17705	−0.458491	−0.229246	−	0.973369i	$-0.573626\pi$
$83$	−4.17705	−0.458491	−0.229246	+	0.973369i	$0.573626\pi$
$84$	0	0
$85$	3.95025	0.428465
$86$	0	0
$87$	−6.00973	−0.644310
$88$	0	0
$89$	−12.6410	−1.33994	−0.669971	−	0.742387i	$-0.733694\pi$
$89$	−12.6410	−1.33994	−0.669971	+	0.742387i	$0.733694\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.30926	−0.343155
$94$	0	0
$95$	33.8138	3.46922
$96$	0	0
$97$	−9.25439	−0.939641	−0.469820	−	0.882762i	$-0.655681\pi$
$97$	−9.25439	−0.939641	−0.469820	+	0.882762i	$0.655681\pi$
$98$	0	0
$99$	2.42308	0.243529
$100$	0	0
$101$	−3.15008	−0.313444	−0.156722	−	0.987643i	$-0.550093\pi$
$101$	−3.15008	−0.313444	−0.156722	+	0.987643i	$0.550093\pi$
$102$	0	0
$103$	10.3926	1.02401	0.512005	−	0.858982i	$-0.328903\pi$
$103$	10.3926	1.02401	0.512005	+	0.858982i	$0.328903\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.79129	0.849886	0.424943	−	0.905220i	$-0.360294\pi$
$107$	8.79129	0.849886	0.424943	+	0.905220i	$0.360294\pi$
$108$	0	0
$109$	15.1319	1.44937	0.724687	−	0.689078i	$-0.241984\pi$
$109$	15.1319	1.44937	0.724687	+	0.689078i	$0.241984\pi$
$110$	0	0
$111$	−11.6737	−1.10802
$112$	0	0
$113$	−12.3412	−1.16097	−0.580483	−	0.814273i	$-0.697136\pi$
$113$	−12.3412	−1.16097	−0.580483	+	0.814273i	$0.697136\pi$
$114$	0	0
$115$	15.5137	1.44666
$116$	0	0
$117$	0.636391	0.0588344
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.12867	−0.466243
$122$	0	0
$123$	2.99624	0.270162
$124$	0	0
$125$	22.1392	1.98019
$126$	0	0
$127$	2.38523	0.211655	0.105828	−	0.994384i	$-0.466251\pi$
$127$	2.38523	0.211655	0.105828	+	0.994384i	$0.466251\pi$
$128$	0	0
$129$	2.47146	0.217600
$130$	0	0
$131$	8.77917	0.767040	0.383520	−	0.923533i	$-0.374712\pi$
$131$	8.77917	0.767040	0.383520	+	0.923533i	$0.374712\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.95025	−0.339984
$136$	0	0
$137$	−13.8627	−1.18437	−0.592184	−	0.805803i	$-0.701734\pi$
$137$	−13.8627	−1.18437	−0.592184	+	0.805803i	$0.701734\pi$
$138$	0	0
$139$	5.85430	0.496555	0.248278	−	0.968689i	$-0.420135\pi$
$139$	5.85430	0.496555	0.248278	+	0.968689i	$0.420135\pi$
$140$	0	0
$141$	−6.09083	−0.512940
$142$	0	0
$143$	1.54203	0.128951
$144$	0	0
$145$	23.7400	1.97150
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.1457	−0.995014	−0.497507	−	0.867460i	$-0.665751\pi$
$149$	−12.1457	−0.995014	−0.497507	+	0.867460i	$0.665751\pi$
$150$	0	0
$151$	9.63285	0.783911	0.391955	−	0.919984i	$-0.371799\pi$
$151$	9.63285	0.783911	0.391955	+	0.919984i	$0.371799\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	13.0724	1.05000
$156$	0	0
$157$	11.0373	0.880874	0.440437	−	0.897784i	$-0.354824\pi$
$157$	11.0373	0.880874	0.440437	+	0.897784i	$0.354824\pi$
$158$	0	0
$159$	11.7221	0.929622
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.4929	−1.29182	−0.645912	−	0.763412i	$-0.723523\pi$
$163$	−16.4929	−1.29182	−0.645912	+	0.763412i	$0.723523\pi$
$164$	0	0
$165$	−9.57179	−0.745163
$166$	0	0
$167$	19.3222	1.49520	0.747599	−	0.664150i	$-0.231206\pi$
$167$	19.3222	1.49520	0.747599	+	0.664150i	$0.231206\pi$
$168$	0	0
$169$	−12.5950	−0.968847
$170$	0	0
$171$	8.55989	0.654591
$172$	0	0
$173$	−9.74645	−0.741009	−0.370504	−	0.928831i	$-0.620815\pi$
$173$	−9.74645	−0.741009	−0.370504	+	0.928831i	$0.620815\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.94053	−0.672011
$178$	0	0
$179$	−11.4372	−0.854855	−0.427428	−	0.904050i	$-0.640580\pi$
$179$	−11.4372	−0.854855	−0.427428	+	0.904050i	$0.640580\pi$
$180$	0	0
$181$	−15.0819	−1.12103	−0.560516	−	0.828144i	$-0.689397\pi$
$181$	−15.0819	−1.12103	−0.560516	+	0.828144i	$0.689397\pi$
$182$	0	0
$183$	11.5815	0.856130
$184$	0	0
$185$	46.1141	3.39038
$186$	0	0
$187$	2.42308	0.177193
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.366002	−0.0264830	−0.0132415	−	0.999912i	$-0.504215\pi$
$191$	−0.366002	−0.0264830	−0.0132415	+	0.999912i	$0.504215\pi$
$192$	0	0
$193$	−17.8238	−1.28298	−0.641492	−	0.767130i	$-0.721684\pi$
$193$	−17.8238	−1.28298	−0.641492	+	0.767130i	$0.721684\pi$
$194$	0	0
$195$	−2.51391	−0.180025
$196$	0	0
$197$	27.5919	1.96584	0.982919	−	0.184037i	$-0.0589166\pi$
$197$	27.5919	1.96584	0.982919	+	0.184037i	$0.0589166\pi$
$198$	0	0
$199$	−9.24687	−0.655493	−0.327747	−	0.944766i	$-0.606289\pi$
$199$	−9.24687	−0.655493	−0.327747	+	0.944766i	$0.606289\pi$
$200$	0	0
$201$	−1.43122	−0.100950
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−11.8359	−0.826656
$206$	0	0
$207$	3.92726	0.272964
$208$	0	0
$209$	20.7413	1.43471
$210$	0	0
$211$	−19.0130	−1.30891	−0.654453	−	0.756103i	$-0.727101\pi$
$211$	−19.0130	−1.30891	−0.654453	+	0.756103i	$0.727101\pi$
$212$	0	0
$213$	5.75021	0.393998
$214$	0	0
$215$	−9.76290	−0.665824
$216$	0	0
$217$	0	0
$218$	0	0
$219$	8.71833	0.589130
$220$	0	0
$221$	0.636391	0.0428083
$222$	0	0
$223$	−28.9027	−1.93547	−0.967734	−	0.251972i	$-0.918921\pi$
$223$	−28.9027	−1.93547	−0.967734	+	0.251972i	$0.918921\pi$
$224$	0	0
$225$	10.6045	0.706967
$226$	0	0
$227$	−5.35034	−0.355115	−0.177557	−	0.984110i	$-0.556820\pi$
$227$	−5.35034	−0.355115	−0.177557	+	0.984110i	$0.556820\pi$
$228$	0	0
$229$	−20.6880	−1.36710	−0.683552	−	0.729902i	$-0.739566\pi$
$229$	−20.6880	−1.36710	−0.683552	+	0.729902i	$0.739566\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.44170	−0.618546	−0.309273	−	0.950973i	$-0.600086\pi$
$233$	−9.44170	−0.618546	−0.309273	+	0.950973i	$0.600086\pi$
$234$	0	0
$235$	24.0603	1.56952
$236$	0	0
$237$	12.8008	0.831501
$238$	0	0
$239$	−5.75481	−0.372248	−0.186124	−	0.982526i	$-0.559593\pi$
$239$	−5.75481	−0.372248	−0.186124	+	0.982526i	$0.559593\pi$
$240$	0	0
$241$	9.55613	0.615565	0.307782	−	0.951457i	$-0.400413\pi$
$241$	9.55613	0.615565	0.307782	+	0.951457i	$0.400413\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.44744	0.346612
$248$	0	0
$249$	4.17705	0.264710
$250$	0	0
$251$	21.8232	1.37747	0.688733	−	0.725015i	$-0.258167\pi$
$251$	21.8232	1.37747	0.688733	+	0.725015i	$0.258167\pi$
$252$	0	0
$253$	9.51608	0.598271
$254$	0	0
$255$	−3.95025	−0.247375
$256$	0	0
$257$	6.51904	0.406646	0.203323	−	0.979112i	$-0.434826\pi$
$257$	6.51904	0.406646	0.203323	+	0.979112i	$0.434826\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00973	0.371993
$262$	0	0
$263$	−21.7789	−1.34294	−0.671471	−	0.741031i	$-0.734337\pi$
$263$	−21.7789	−1.34294	−0.671471	+	0.741031i	$0.734337\pi$
$264$	0	0
$265$	−46.3052	−2.84451
$266$	0	0
$267$	12.6410	0.773616
$268$	0	0
$269$	23.8375	1.45340	0.726699	−	0.686956i	$-0.241054\pi$
$269$	23.8375	1.45340	0.726699	+	0.686956i	$0.241054\pi$
$270$	0	0
$271$	18.9862	1.15333	0.576665	−	0.816981i	$-0.304354\pi$
$271$	18.9862	1.15333	0.576665	+	0.816981i	$0.304354\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25.6956	1.54950
$276$	0	0
$277$	−6.78293	−0.407547	−0.203773	−	0.979018i	$-0.565321\pi$
$277$	−6.78293	−0.407547	−0.203773	+	0.979018i	$0.565321\pi$
$278$	0	0
$279$	3.30926	0.198120
$280$	0	0
$281$	3.20442	0.191160	0.0955799	−	0.995422i	$-0.469529\pi$
$281$	3.20442	0.191160	0.0955799	+	0.995422i	$0.469529\pi$
$282$	0	0
$283$	9.18355	0.545905	0.272953	−	0.962027i	$-0.412000\pi$
$283$	9.18355	0.545905	0.272953	+	0.962027i	$0.412000\pi$
$284$	0	0
$285$	−33.8138	−2.00295
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	9.25439	0.542502
$292$	0	0
$293$	−2.63502	−0.153940	−0.0769699	−	0.997033i	$-0.524525\pi$
$293$	−2.63502	−0.153940	−0.0769699	+	0.997033i	$0.524525\pi$
$294$	0	0
$295$	35.3174	2.05626
$296$	0	0
$297$	−2.42308	−0.140601
$298$	0	0
$299$	2.49928	0.144537
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3.15008	0.180967
$304$	0	0
$305$	−45.7499	−2.61963
$306$	0	0
$307$	−6.69074	−0.381860	−0.190930	−	0.981604i	$-0.561150\pi$
$307$	−6.69074	−0.381860	−0.190930	+	0.981604i	$0.561150\pi$
$308$	0	0
$309$	−10.3926	−0.591213
$310$	0	0
$311$	−1.84564	−0.104656	−0.0523282	−	0.998630i	$-0.516664\pi$
$311$	−1.84564	−0.104656	−0.0523282	+	0.998630i	$0.516664\pi$
$312$	0	0
$313$	14.1674	0.800789	0.400395	−	0.916343i	$-0.368873\pi$
$313$	14.1674	0.800789	0.400395	+	0.916343i	$0.368873\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.99562	−0.505245	−0.252622	−	0.967565i	$-0.581293\pi$
$317$	−8.99562	−0.505245	−0.252622	+	0.967565i	$0.581293\pi$
$318$	0	0
$319$	14.5621	0.815319
$320$	0	0
$321$	−8.79129	−0.490682
$322$	0	0
$323$	8.55989	0.476285
$324$	0	0
$325$	6.74862	0.374346
$326$	0	0
$327$	−15.1319	−0.836796
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6.12867	−0.336862	−0.168431	−	0.985713i	$-0.553870\pi$
$331$	−6.12867	−0.336862	−0.168431	+	0.985713i	$0.553870\pi$
$332$	0	0
$333$	11.6737	0.639715
$334$	0	0
$335$	5.65368	0.308893
$336$	0	0
$337$	−10.6061	−0.577751	−0.288876	−	0.957367i	$-0.593281\pi$
$337$	−10.6061	−0.577751	−0.288876	+	0.957367i	$0.593281\pi$
$338$	0	0
$339$	12.3412	0.670284
$340$	0	0
$341$	8.01862	0.434232
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−15.5137	−0.835229
$346$	0	0
$347$	−30.4406	−1.63414	−0.817070	−	0.576539i	$-0.804403\pi$
$347$	−30.4406	−1.63414	−0.817070	+	0.576539i	$0.804403\pi$
$348$	0	0
$349$	2.53743	0.135825	0.0679127	−	0.997691i	$-0.478366\pi$
$349$	2.53743	0.135825	0.0679127	+	0.997691i	$0.478366\pi$
$350$	0	0
$351$	−0.636391	−0.0339681
$352$	0	0
$353$	20.0782	1.06865	0.534327	−	0.845278i	$-0.320565\pi$
$353$	20.0782	1.06865	0.534327	+	0.845278i	$0.320565\pi$
$354$	0	0
$355$	−22.7148	−1.20558
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.4245	−1.23630	−0.618150	−	0.786060i	$-0.712118\pi$
$359$	−23.4245	−1.23630	−0.618150	+	0.786060i	$0.712118\pi$
$360$	0	0
$361$	54.2718	2.85641
$362$	0	0
$363$	5.12867	0.269186
$364$	0	0
$365$	−34.4396	−1.80265
$366$	0	0
$367$	−23.5463	−1.22911	−0.614554	−	0.788875i	$-0.710664\pi$
$367$	−23.5463	−1.22911	−0.614554	+	0.788875i	$0.710664\pi$
$368$	0	0
$369$	−2.99624	−0.155978
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−36.6815	−1.89930	−0.949649	−	0.313315i	$-0.898560\pi$
$373$	−36.6815	−1.89930	−0.949649	+	0.313315i	$0.898560\pi$
$374$	0	0
$375$	−22.1392	−1.14327
$376$	0	0
$377$	3.82454	0.196974
$378$	0	0
$379$	9.70183	0.498350	0.249175	−	0.968459i	$-0.419841\pi$
$379$	9.70183	0.498350	0.249175	+	0.968459i	$0.419841\pi$
$380$	0	0
$381$	−2.38523	−0.122199
$382$	0	0
$383$	12.7575	0.651880	0.325940	−	0.945390i	$-0.394319\pi$
$383$	12.7575	0.651880	0.325940	+	0.945390i	$0.394319\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.47146	−0.125631
$388$	0	0
$389$	19.3666	0.981924	0.490962	−	0.871181i	$-0.336645\pi$
$389$	19.3666	0.981924	0.490962	+	0.871181i	$0.336645\pi$
$390$	0	0
$391$	3.92726	0.198610
$392$	0	0
$393$	−8.77917	−0.442850
$394$	0	0
$395$	−50.5664	−2.54427
$396$	0	0
$397$	−0.977008	−0.0490346	−0.0245173	−	0.999699i	$-0.507805\pi$
$397$	−0.977008	−0.0490346	−0.0245173	+	0.999699i	$0.507805\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.1655	−1.10689	−0.553445	−	0.832886i	$-0.686687\pi$
$401$	−22.1655	−1.10689	−0.553445	+	0.832886i	$0.686687\pi$
$402$	0	0
$403$	2.10599	0.104907
$404$	0	0
$405$	3.95025	0.196290
$406$	0	0
$407$	28.2864	1.40210
$408$	0	0
$409$	11.3130	0.559393	0.279697	−	0.960088i	$-0.409766\pi$
$409$	11.3130	0.559393	0.279697	+	0.960088i	$0.409766\pi$
$410$	0	0
$411$	13.8627	0.683795
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16.5004	−0.809974
$416$	0	0
$417$	−5.85430	−0.286686
$418$	0	0
$419$	19.2619	0.941006	0.470503	−	0.882398i	$-0.344072\pi$
$419$	19.2619	0.941006	0.470503	+	0.882398i	$0.344072\pi$
$420$	0	0
$421$	26.5226	1.29263	0.646316	−	0.763070i	$-0.276309\pi$
$421$	26.5226	1.29263	0.646316	+	0.763070i	$0.276309\pi$
$422$	0	0
$423$	6.09083	0.296146
$424$	0	0
$425$	10.6045	0.514394
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.54203	−0.0744498
$430$	0	0
$431$	−18.5199	−0.892071	−0.446035	−	0.895015i	$-0.647164\pi$
$431$	−18.5199	−0.892071	−0.446035	+	0.895015i	$0.647164\pi$
$432$	0	0
$433$	1.66097	0.0798214	0.0399107	−	0.999203i	$-0.487293\pi$
$433$	1.66097	0.0798214	0.0399107	+	0.999203i	$0.487293\pi$
$434$	0	0
$435$	−23.7400	−1.13824
$436$	0	0
$437$	33.6169	1.60812
$438$	0	0
$439$	−11.0687	−0.528279	−0.264139	−	0.964484i	$-0.585088\pi$
$439$	−11.0687	−0.528279	−0.264139	+	0.964484i	$0.585088\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.11081	−0.290333	−0.145167	−	0.989407i	$-0.546372\pi$
$443$	−6.11081	−0.290333	−0.145167	+	0.989407i	$0.546372\pi$
$444$	0	0
$445$	−49.9351	−2.36715
$446$	0	0
$447$	12.1457	0.574472
$448$	0	0
$449$	5.73151	0.270487	0.135243	−	0.990812i	$-0.456818\pi$
$449$	5.73151	0.270487	0.135243	+	0.990812i	$0.456818\pi$
$450$	0	0
$451$	−7.26013	−0.341866
$452$	0	0
$453$	−9.63285	−0.452591
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.10630	0.332419	0.166209	−	0.986091i	$-0.446847\pi$
$457$	7.10630	0.332419	0.166209	+	0.986091i	$0.446847\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	19.5426	0.910188	0.455094	−	0.890443i	$-0.349606\pi$
$461$	19.5426	0.910188	0.455094	+	0.890443i	$0.349606\pi$
$462$	0	0
$463$	−21.9206	−1.01874	−0.509368	−	0.860549i	$-0.670121\pi$
$463$	−21.9206	−1.01874	−0.509368	+	0.860549i	$0.670121\pi$
$464$	0	0
$465$	−13.0724	−0.606220
$466$	0	0
$467$	−1.92275	−0.0889742	−0.0444871	−	0.999010i	$-0.514165\pi$
$467$	−1.92275	−0.0889742	−0.0444871	+	0.999010i	$0.514165\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−11.0373	−0.508573
$472$	0	0
$473$	−5.98855	−0.275354
$474$	0	0
$475$	90.7735	4.16497
$476$	0	0
$477$	−11.7221	−0.536718
$478$	0	0
$479$	11.2539	0.514202	0.257101	−	0.966385i	$-0.417233\pi$
$479$	11.2539	0.514202	0.257101	+	0.966385i	$0.417233\pi$
$480$	0	0
$481$	7.42905	0.338735
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−36.5572	−1.65998
$486$	0	0
$487$	−8.24015	−0.373397	−0.186698	−	0.982417i	$-0.559779\pi$
$487$	−8.24015	−0.373397	−0.186698	+	0.982417i	$0.559779\pi$
$488$	0	0
$489$	16.4929	0.745835
$490$	0	0
$491$	24.9659	1.12670	0.563349	−	0.826219i	$-0.309513\pi$
$491$	24.9659	1.12670	0.563349	+	0.826219i	$0.309513\pi$
$492$	0	0
$493$	6.00973	0.270665
$494$	0	0
$495$	9.57179	0.430220
$496$	0	0
$497$	0	0
$498$	0	0
$499$	30.1639	1.35032	0.675160	−	0.737671i	$-0.264074\pi$
$499$	30.1639	1.35032	0.675160	+	0.737671i	$0.264074\pi$
$500$	0	0
$501$	−19.3222	−0.863253
$502$	0	0
$503$	−7.07141	−0.315298	−0.157649	−	0.987495i	$-0.550392\pi$
$503$	−7.07141	−0.315298	−0.157649	+	0.987495i	$0.550392\pi$
$504$	0	0
$505$	−12.4436	−0.553733
$506$	0	0
$507$	12.5950	0.559364
$508$	0	0
$509$	−18.8797	−0.836827	−0.418414	−	0.908257i	$-0.637414\pi$
$509$	−18.8797	−0.836827	−0.418414	+	0.908257i	$0.637414\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8.55989	−0.377928
$514$	0	0
$515$	41.0533	1.80902
$516$	0	0
$517$	14.7586	0.649081
$518$	0	0
$519$	9.74645	0.427822
$520$	0	0
$521$	−25.1157	−1.10034	−0.550170	−	0.835053i	$-0.685437\pi$
$521$	−25.1157	−1.10034	−0.550170	+	0.835053i	$0.685437\pi$
$522$	0	0
$523$	−3.66814	−0.160396	−0.0801982	−	0.996779i	$-0.525555\pi$
$523$	−3.66814	−0.160396	−0.0801982	+	0.996779i	$0.525555\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.30926	0.144154
$528$	0	0
$529$	−7.57661	−0.329418
$530$	0	0
$531$	8.94053	0.387986
$532$	0	0
$533$	−1.90678	−0.0825918
$534$	0	0
$535$	34.7278	1.50142
$536$	0	0
$537$	11.4372	0.493551
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.99925	−0.257928	−0.128964	−	0.991649i	$-0.541165\pi$
$541$	−5.99925	−0.257928	−0.128964	+	0.991649i	$0.541165\pi$
$542$	0	0
$543$	15.0819	0.647228
$544$	0	0
$545$	59.7749	2.56047
$546$	0	0
$547$	29.4844	1.26066	0.630331	−	0.776326i	$-0.282919\pi$
$547$	29.4844	1.26066	0.630331	+	0.776326i	$0.282919\pi$
$548$	0	0
$549$	−11.5815	−0.494287
$550$	0	0
$551$	51.4426	2.19153
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−46.1141	−1.95744
$556$	0	0
$557$	−5.90617	−0.250252	−0.125126	−	0.992141i	$-0.539934\pi$
$557$	−5.90617	−0.250252	−0.125126	+	0.992141i	$0.539934\pi$
$558$	0	0
$559$	−1.57282	−0.0665230
$560$	0	0
$561$	−2.42308	−0.102303
$562$	0	0
$563$	15.0051	0.632391	0.316195	−	0.948694i	$-0.397594\pi$
$563$	15.0051	0.632391	0.316195	+	0.948694i	$0.397594\pi$
$564$	0	0
$565$	−48.7510	−2.05097
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.1308	−1.47276	−0.736379	−	0.676569i	$-0.763466\pi$
$569$	−35.1308	−1.47276	−0.736379	+	0.676569i	$0.763466\pi$
$570$	0	0
$571$	30.0463	1.25740	0.628699	−	0.777649i	$-0.283588\pi$
$571$	30.0463	1.25740	0.628699	+	0.777649i	$0.283588\pi$
$572$	0	0
$573$	0.366002	0.0152900
$574$	0	0
$575$	41.6467	1.73679
$576$	0	0
$577$	8.91024	0.370938	0.185469	−	0.982650i	$-0.440620\pi$
$577$	8.91024	0.370938	0.185469	+	0.982650i	$0.440620\pi$
$578$	0	0
$579$	17.8238	0.740731
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−28.4036	−1.17636
$584$	0	0
$585$	2.51391	0.103937
$586$	0	0
$587$	−27.5879	−1.13868	−0.569338	−	0.822104i	$-0.692800\pi$
$587$	−27.5879	−1.13868	−0.569338	+	0.822104i	$0.692800\pi$
$588$	0	0
$589$	28.3269	1.16719
$590$	0	0
$591$	−27.5919	−1.13498
$592$	0	0
$593$	−38.9712	−1.60035	−0.800177	−	0.599764i	$-0.795261\pi$
$593$	−38.9712	−1.60035	−0.800177	+	0.599764i	$0.795261\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.24687	0.378449
$598$	0	0
$599$	−10.3143	−0.421429	−0.210715	−	0.977548i	$-0.567579\pi$
$599$	−10.3143	−0.421429	−0.210715	+	0.977548i	$0.567579\pi$
$600$	0	0
$601$	17.5483	0.715809	0.357904	−	0.933758i	$-0.383491\pi$
$601$	17.5483	0.715809	0.357904	+	0.933758i	$0.383491\pi$
$602$	0	0
$603$	1.43122	0.0582837
$604$	0	0
$605$	−20.2596	−0.823669
$606$	0	0
$607$	1.01964	0.0413860	0.0206930	−	0.999786i	$-0.493413\pi$
$607$	1.01964	0.0413860	0.0206930	+	0.999786i	$0.493413\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.87615	0.156812
$612$	0	0
$613$	27.9575	1.12919	0.564596	−	0.825368i	$-0.309032\pi$
$613$	27.9575	1.12919	0.564596	+	0.825368i	$0.309032\pi$
$614$	0	0
$615$	11.8359	0.477270
$616$	0	0
$617$	−43.2182	−1.73990	−0.869950	−	0.493140i	$-0.835849\pi$
$617$	−43.2182	−1.73990	−0.869950	+	0.493140i	$0.835849\pi$
$618$	0	0
$619$	−2.21635	−0.0890828	−0.0445414	−	0.999008i	$-0.514183\pi$
$619$	−2.21635	−0.0890828	−0.0445414	+	0.999008i	$0.514183\pi$
$620$	0	0
$621$	−3.92726	−0.157596
$622$	0	0
$623$	0	0
$624$	0	0
$625$	34.4331	1.37732
$626$	0	0
$627$	−20.7413	−0.828329
$628$	0	0
$629$	11.6737	0.465461
$630$	0	0
$631$	−3.63366	−0.144654	−0.0723268	−	0.997381i	$-0.523042\pi$
$631$	−3.63366	−0.144654	−0.0723268	+	0.997381i	$0.523042\pi$
$632$	0	0
$633$	19.0130	0.755697
$634$	0	0
$635$	9.42228	0.373912
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5.75021	−0.227475
$640$	0	0
$641$	−47.8248	−1.88897	−0.944484	−	0.328557i	$-0.893438\pi$
$641$	−47.8248	−1.88897	−0.944484	+	0.328557i	$0.893438\pi$
$642$	0	0
$643$	38.8354	1.53152	0.765760	−	0.643127i	$-0.222363\pi$
$643$	38.8354	1.53152	0.765760	+	0.643127i	$0.222363\pi$
$644$	0	0
$645$	9.76290	0.384414
$646$	0	0
$647$	16.1352	0.634341	0.317170	−	0.948369i	$-0.397267\pi$
$647$	16.1352	0.634341	0.317170	+	0.948369i	$0.397267\pi$
$648$	0	0
$649$	21.6636	0.850372
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.580268	−0.0227076	−0.0113538	−	0.999936i	$-0.503614\pi$
$653$	−0.580268	−0.0227076	−0.0113538	+	0.999936i	$0.503614\pi$
$654$	0	0
$655$	34.6800	1.35506
$656$	0	0
$657$	−8.71833	−0.340134
$658$	0	0
$659$	1.78833	0.0696635	0.0348318	−	0.999393i	$-0.488910\pi$
$659$	1.78833	0.0696635	0.0348318	+	0.999393i	$0.488910\pi$
$660$	0	0
$661$	4.36246	0.169680	0.0848401	−	0.996395i	$-0.472962\pi$
$661$	4.36246	0.169680	0.0848401	+	0.996395i	$0.472962\pi$
$662$	0	0
$663$	−0.636391	−0.0247154
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23.6018	0.913864
$668$	0	0
$669$	28.9027	1.11744
$670$	0	0
$671$	−28.0630	−1.08336
$672$	0	0
$673$	−28.2352	−1.08839	−0.544194	−	0.838959i	$-0.683165\pi$
$673$	−28.2352	−1.08839	−0.544194	+	0.838959i	$0.683165\pi$
$674$	0	0
$675$	−10.6045	−0.408168
$676$	0	0
$677$	36.2191	1.39201	0.696006	−	0.718036i	$-0.254959\pi$
$677$	36.2191	1.39201	0.696006	+	0.718036i	$0.254959\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.35034	0.205026
$682$	0	0
$683$	−16.0056	−0.612438	−0.306219	−	0.951961i	$-0.599064\pi$
$683$	−16.0056	−0.612438	−0.306219	+	0.951961i	$0.599064\pi$
$684$	0	0
$685$	−54.7610	−2.09231
$686$	0	0
$687$	20.6880	0.789298
$688$	0	0
$689$	−7.45984	−0.284197
$690$	0	0
$691$	5.76108	0.219162	0.109581	−	0.993978i	$-0.465049\pi$
$691$	5.76108	0.219162	0.109581	+	0.993978i	$0.465049\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.1260	0.877218
$696$	0	0
$697$	−2.99624	−0.113491
$698$	0	0
$699$	9.44170	0.357118
$700$	0	0
$701$	−23.1863	−0.875733	−0.437866	−	0.899040i	$-0.644266\pi$
$701$	−23.1863	−0.875733	−0.437866	+	0.899040i	$0.644266\pi$
$702$	0	0
$703$	99.9257	3.76877
$704$	0	0
$705$	−24.0603	−0.906164
$706$	0	0
$707$	0	0
$708$	0	0
$709$	49.7935	1.87003	0.935017	−	0.354603i	$-0.115384\pi$
$709$	49.7935	1.87003	0.935017	+	0.354603i	$0.115384\pi$
$710$	0	0
$711$	−12.8008	−0.480067
$712$	0	0
$713$	12.9963	0.486717
$714$	0	0
$715$	6.09140	0.227806
$716$	0	0
$717$	5.75481	0.214917
$718$	0	0
$719$	52.4581	1.95636	0.978178	−	0.207767i	$-0.0666195\pi$
$719$	52.4581	1.95636	0.978178	+	0.207767i	$0.0666195\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9.55613	−0.355396
$724$	0	0
$725$	63.7302	2.36688
$726$	0	0
$727$	−24.8364	−0.921132	−0.460566	−	0.887625i	$-0.652354\pi$
$727$	−24.8364	−0.921132	−0.460566	+	0.887625i	$0.652354\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.47146	−0.0914103
$732$	0	0
$733$	21.1721	0.782011	0.391006	−	0.920388i	$-0.372127\pi$
$733$	21.1721	0.782011	0.391006	+	0.920388i	$0.372127\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.46796	0.127744
$738$	0	0
$739$	9.56661	0.351914	0.175957	−	0.984398i	$-0.443698\pi$
$739$	9.56661	0.351914	0.175957	+	0.984398i	$0.443698\pi$
$740$	0	0
$741$	−5.44744	−0.200117
$742$	0	0
$743$	−44.3767	−1.62802	−0.814011	−	0.580850i	$-0.802720\pi$
$743$	−44.3767	−1.62802	−0.814011	+	0.580850i	$0.802720\pi$
$744$	0	0
$745$	−47.9786	−1.75780
$746$	0	0
$747$	−4.17705	−0.152830
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.82808	−0.176179	−0.0880895	−	0.996113i	$-0.528076\pi$
$751$	−4.82808	−0.176179	−0.0880895	+	0.996113i	$0.528076\pi$
$752$	0	0
$753$	−21.8232	−0.795281
$754$	0	0
$755$	38.0522	1.38486
$756$	0	0
$757$	9.42171	0.342438	0.171219	−	0.985233i	$-0.445229\pi$
$757$	9.42171	0.342438	0.171219	+	0.985233i	$0.445229\pi$
$758$	0	0
$759$	−9.51608	−0.345412
$760$	0	0
$761$	44.5148	1.61366	0.806831	−	0.590783i	$-0.201181\pi$
$761$	44.5148	1.61366	0.806831	+	0.590783i	$0.201181\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.95025	0.142822
$766$	0	0
$767$	5.68967	0.205442
$768$	0	0
$769$	−37.8126	−1.36356	−0.681779	−	0.731558i	$-0.738793\pi$
$769$	−37.8126	−1.36356	−0.681779	+	0.731558i	$0.738793\pi$
$770$	0	0
$771$	−6.51904	−0.234777
$772$	0	0
$773$	−25.9808	−0.934465	−0.467232	−	0.884135i	$-0.654749\pi$
$773$	−25.9808	−0.934465	−0.467232	+	0.884135i	$0.654749\pi$
$774$	0	0
$775$	35.0931	1.26058
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−25.6475	−0.918917
$780$	0	0
$781$	−13.9332	−0.498570
$782$	0	0
$783$	−6.00973	−0.214770
$784$	0	0
$785$	43.6002	1.55616
$786$	0	0
$787$	33.4015	1.19063	0.595317	−	0.803491i	$-0.297027\pi$
$787$	33.4015	1.19063	0.595317	+	0.803491i	$0.297027\pi$
$788$	0	0
$789$	21.7789	0.775348
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−7.37038	−0.261730
$794$	0	0
$795$	46.3052	1.64228
$796$	0	0
$797$	8.76454	0.310456	0.155228	−	0.987879i	$-0.450389\pi$
$797$	8.76454	0.310456	0.155228	+	0.987879i	$0.450389\pi$
$798$	0	0
$799$	6.09083	0.215478
$800$	0	0
$801$	−12.6410	−0.446647
$802$	0	0
$803$	−21.1252	−0.745493
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−23.8375	−0.839120
$808$	0	0
$809$	48.6656	1.71099	0.855495	−	0.517811i	$-0.173253\pi$
$809$	48.6656	1.71099	0.855495	+	0.517811i	$0.173253\pi$
$810$	0	0
$811$	39.0617	1.37164	0.685821	−	0.727771i	$-0.259443\pi$
$811$	39.0617	1.37164	0.685821	+	0.727771i	$0.259443\pi$
$812$	0	0
$813$	−18.9862	−0.665875
$814$	0	0
$815$	−65.1511	−2.28215
$816$	0	0
$817$	−21.1554	−0.740135
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.45120	0.155348	0.0776740	−	0.996979i	$-0.475251\pi$
$821$	4.45120	0.155348	0.0776740	+	0.996979i	$0.475251\pi$
$822$	0	0
$823$	51.0098	1.77809	0.889045	−	0.457819i	$-0.151369\pi$
$823$	51.0098	1.77809	0.889045	+	0.457819i	$0.151369\pi$
$824$	0	0
$825$	−25.6956	−0.894606
$826$	0	0
$827$	−27.0947	−0.942175	−0.471088	−	0.882086i	$-0.656138\pi$
$827$	−27.0947	−0.942175	−0.471088	+	0.882086i	$0.656138\pi$
$828$	0	0
$829$	17.5812	0.610621	0.305310	−	0.952253i	$-0.401240\pi$
$829$	17.5812	0.610621	0.305310	+	0.952253i	$0.401240\pi$
$830$	0	0
$831$	6.78293	0.235297
$832$	0	0
$833$	0	0
$834$	0	0
$835$	76.3277	2.64143
$836$	0	0
$837$	−3.30926	−0.114385
$838$	0	0
$839$	−8.86969	−0.306216	−0.153108	−	0.988209i	$-0.548928\pi$
$839$	−8.86969	−0.306216	−0.153108	+	0.988209i	$0.548928\pi$
$840$	0	0
$841$	7.11683	0.245408
$842$	0	0
$843$	−3.20442	−0.110366
$844$	0	0
$845$	−49.7535	−1.71157
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9.18355	−0.315179
$850$	0	0
$851$	45.8457	1.57157
$852$	0	0
$853$	−15.9303	−0.545442	−0.272721	−	0.962093i	$-0.587924\pi$
$853$	−15.9303	−0.545442	−0.272721	+	0.962093i	$0.587924\pi$
$854$	0	0
$855$	33.8138	1.15641
$856$	0	0
$857$	−4.24341	−0.144952	−0.0724761	−	0.997370i	$-0.523090\pi$
$857$	−4.24341	−0.144952	−0.0724761	+	0.997370i	$0.523090\pi$
$858$	0	0
$859$	−12.3129	−0.420112	−0.210056	−	0.977689i	$-0.567365\pi$
$859$	−12.3129	−0.420112	−0.210056	+	0.977689i	$0.567365\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.72426	−0.0927348	−0.0463674	−	0.998924i	$-0.514765\pi$
$863$	−2.72426	−0.0927348	−0.0463674	+	0.998924i	$0.514765\pi$
$864$	0	0
$865$	−38.5010	−1.30907
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−31.0174	−1.05219
$870$	0	0
$871$	0.910815	0.0308618
$872$	0	0
$873$	−9.25439	−0.313214
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27.5720	0.931041	0.465520	−	0.885037i	$-0.345867\pi$
$877$	27.5720	0.931041	0.465520	+	0.885037i	$0.345867\pi$
$878$	0	0
$879$	2.63502	0.0888772
$880$	0	0
$881$	−4.78814	−0.161317	−0.0806583	−	0.996742i	$-0.525702\pi$
$881$	−4.78814	−0.161317	−0.0806583	+	0.996742i	$0.525702\pi$
$882$	0	0
$883$	−37.5632	−1.26410	−0.632051	−	0.774927i	$-0.717786\pi$
$883$	−37.5632	−1.26410	−0.632051	+	0.774927i	$0.717786\pi$
$884$	0	0
$885$	−35.3174	−1.18718
$886$	0	0
$887$	−46.7709	−1.57041	−0.785206	−	0.619234i	$-0.787443\pi$
$887$	−46.7709	−1.57041	−0.785206	+	0.619234i	$0.787443\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.42308	0.0811763
$892$	0	0
$893$	52.1368	1.74469
$894$	0	0
$895$	−45.1798	−1.51019
$896$	0	0
$897$	−2.49928	−0.0834484
$898$	0	0
$899$	19.8878	0.663294
$900$	0	0
$901$	−11.7221	−0.390519
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−59.5775	−1.98042
$906$	0	0
$907$	48.1139	1.59759	0.798797	−	0.601600i	$-0.205470\pi$
$907$	48.1139	1.59759	0.798797	+	0.601600i	$0.205470\pi$
$908$	0	0
$909$	−3.15008	−0.104481
$910$	0	0
$911$	11.4794	0.380330	0.190165	−	0.981752i	$-0.439098\pi$
$911$	11.4794	0.380330	0.190165	+	0.981752i	$0.439098\pi$
$912$	0	0
$913$	−10.1213	−0.334967
$914$	0	0
$915$	45.7499	1.51245
$916$	0	0
$917$	0	0
$918$	0	0
$919$	15.3870	0.507570	0.253785	−	0.967261i	$-0.418324\pi$
$919$	15.3870	0.507570	0.253785	+	0.967261i	$0.418324\pi$
$920$	0	0
$921$	6.69074	0.220467
$922$	0	0
$923$	−3.65938	−0.120450
$924$	0	0
$925$	123.794	4.07032
$926$	0	0
$927$	10.3926	0.341337
$928$	0	0
$929$	−34.8651	−1.14389	−0.571943	−	0.820293i	$-0.693810\pi$
$929$	−34.8651	−1.14389	−0.571943	+	0.820293i	$0.693810\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.84564	0.0604234
$934$	0	0
$935$	9.57179	0.313031
$936$	0	0
$937$	11.1099	0.362944	0.181472	−	0.983396i	$-0.441914\pi$
$937$	11.1099	0.362944	0.181472	+	0.983396i	$0.441914\pi$
$938$	0	0
$939$	−14.1674	−0.462336
$940$	0	0
$941$	−43.2075	−1.40852	−0.704262	−	0.709941i	$-0.748722\pi$
$941$	−43.2075	−1.40852	−0.704262	+	0.709941i	$0.748722\pi$
$942$	0	0
$943$	−11.7670	−0.383187
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.1522	1.69472	0.847359	−	0.531020i	$-0.178191\pi$
$947$	52.1522	1.69472	0.847359	+	0.531020i	$0.178191\pi$
$948$	0	0
$949$	−5.54827	−0.180104
$950$	0	0
$951$	8.99562	0.291703
$952$	0	0
$953$	50.5711	1.63816	0.819079	−	0.573681i	$-0.194485\pi$
$953$	50.5711	1.63816	0.819079	+	0.573681i	$0.194485\pi$
$954$	0	0
$955$	−1.44580	−0.0467850
$956$	0	0
$957$	−14.5621	−0.470725
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.0488	−0.646735
$962$	0	0
$963$	8.79129	0.283295
$964$	0	0
$965$	−70.4085	−2.26653
$966$	0	0
$967$	−45.7355	−1.47075	−0.735377	−	0.677658i	$-0.762995\pi$
$967$	−45.7355	−1.47075	−0.735377	+	0.677658i	$0.762995\pi$
$968$	0	0
$969$	−8.55989	−0.274983
$970$	0	0
$971$	2.19107	0.0703148	0.0351574	−	0.999382i	$-0.488807\pi$
$971$	2.19107	0.0703148	0.0351574	+	0.999382i	$0.488807\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−6.74862	−0.216129
$976$	0	0
$977$	−1.86832	−0.0597728	−0.0298864	−	0.999553i	$-0.509515\pi$
$977$	−1.86832	−0.0597728	−0.0298864	+	0.999553i	$0.509515\pi$
$978$	0	0
$979$	−30.6302	−0.978944
$980$	0	0
$981$	15.1319	0.483125
$982$	0	0
$983$	28.6797	0.914740	0.457370	−	0.889277i	$-0.348792\pi$
$983$	28.6797	0.914740	0.457370	+	0.889277i	$0.348792\pi$
$984$	0	0
$985$	108.995	3.47287
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.70607	−0.308635
$990$	0	0
$991$	−8.33005	−0.264613	−0.132306	−	0.991209i	$-0.542238\pi$
$991$	−8.33005	−0.264613	−0.132306	+	0.991209i	$0.542238\pi$
$992$	0	0
$993$	6.12867	0.194488
$994$	0	0
$995$	−36.5275	−1.15800
$996$	0	0
$997$	−7.82340	−0.247769	−0.123885	−	0.992297i	$-0.539535\pi$
$997$	−7.82340	−0.247769	−0.123885	+	0.992297i	$0.539535\pi$
$998$	0	0
$999$	−11.6737	−0.369340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9996.2.a.z.1.4		4
7.3	odd	6		1428.2.q.e.205.4	✓	8
7.5	odd	6		1428.2.q.e.613.4	yes	8
7.6	odd	2		9996.2.a.bd.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1428.2.q.e.205.4	✓	8	7.3	odd	6
1428.2.q.e.613.4	yes	8	7.5	odd	6
9996.2.a.z.1.4		4	1.1	even	1	trivial
9996.2.a.bd.1.1		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:	\( N \)	\(=\)	\( 9996 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9996.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.95025 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.95025 q^{5} +1.00000 q^{9} +2.42308 q^{11} +0.636391 q^{13} -3.95025 q^{15} +1.00000 q^{17} +8.55989 q^{19} +3.92726 q^{23} +10.6045 q^{25} -1.00000 q^{27} +6.00973 q^{29} +3.30926 q^{31} -2.42308 q^{33} +11.6737 q^{37} -0.636391 q^{39} -2.99624 q^{41} -2.47146 q^{43} +3.95025 q^{45} +6.09083 q^{47} -1.00000 q^{51} -11.7221 q^{53} +9.57179 q^{55} -8.55989 q^{57} +8.94053 q^{59} -11.5815 q^{61} +2.51391 q^{65} +1.43122 q^{67} -3.92726 q^{69} -5.75021 q^{71} -8.71833 q^{73} -10.6045 q^{75} -12.8008 q^{79} +1.00000 q^{81} -4.17705 q^{83} +3.95025 q^{85} -6.00973 q^{87} -12.6410 q^{89} -3.30926 q^{93} +33.8138 q^{95} -9.25439 q^{97} +2.42308 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^9	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{9} + q^{11} + 10 q^{13} + 2 q^{15} + 4 q^{17} + 9 q^{19} - 5 q^{23} + 18 q^{25} - 4 q^{27} + 5 q^{29} + 24 q^{31} - q^{33} + 2 q^{37} - 10 q^{39} - 14 q^{43} - 2 q^{45} + 2 q^{47} - 4 q^{51} - 15 q^{53} + 30 q^{55} - 9 q^{57} + 37 q^{59} - 19 q^{61} - 21 q^{65} + 2 q^{67} + 5 q^{69} - 11 q^{71} + 9 q^{73} - 18 q^{75} - 9 q^{79} + 4 q^{81} - 8 q^{83} - 2 q^{85} - 5 q^{87} - 22 q^{89} - 24 q^{93} + 33 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^9 + q^11 + 10 * q^13 + 2 * q^15 + 4 * q^17 + 9 * q^19 - 5 * q^23 + 18 * q^25 - 4 * q^27 + 5 * q^29 + 24 * q^31 - q^33 + 2 * q^37 - 10 * q^39 - 14 * q^43 - 2 * q^45 + 2 * q^47 - 4 * q^51 - 15 * q^53 + 30 * q^55 - 9 * q^57 + 37 * q^59 - 19 * q^61 - 21 * q^65 + 2 * q^67 + 5 * q^69 - 11 * q^71 + 9 * q^73 - 18 * q^75 - 9 * q^79 + 4 * q^81 - 8 * q^83 - 2 * q^85 - 5 * q^87 - 22 * q^89 - 24 * q^93 + 33 * q^95 - 13 * q^97 + q^99

Introduction

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