LMFDB - Embedded newform 9016.2.a.bs.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9016,2,Mod(1,9016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9016, 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("9016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9016 = 2^{3} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9016.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:	$71.9931224624$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 28x^{12} + 293x^{10} - 1394x^{8} + 2848x^{6} - 1722x^{4} + 332x^{2} - 8 $ x^14 - 28x^12 + 293x^10 - 1394x^8 + 2848x^6 - 1722x^4 + 332x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.12838$ of defining polynomial
Character	$\chi$	$=$	9016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12838 q^{3} -0.504697 q^{5} +1.53001 q^{9} +O(q^{10})$	$q-2.12838 q^{3} -0.504697 q^{5} +1.53001 q^{9} +2.36044 q^{11} +1.21887 q^{13} +1.07419 q^{15} +3.03702 q^{17} +2.40080 q^{19} -1.00000 q^{23} -4.74528 q^{25} +3.12869 q^{27} -0.856064 q^{29} +4.69317 q^{31} -5.02392 q^{33} -11.9421 q^{37} -2.59422 q^{39} -9.46633 q^{41} -12.8001 q^{43} -0.772194 q^{45} -6.72467 q^{47} -6.46395 q^{51} +9.51381 q^{53} -1.19131 q^{55} -5.10982 q^{57} -6.44317 q^{59} +10.0581 q^{61} -0.615159 q^{65} +6.38042 q^{67} +2.12838 q^{69} -6.50520 q^{71} +14.8834 q^{73} +10.0998 q^{75} +3.50252 q^{79} -11.2491 q^{81} +4.31583 q^{83} -1.53278 q^{85} +1.82203 q^{87} +14.7229 q^{89} -9.98886 q^{93} -1.21168 q^{95} +12.5696 q^{97} +3.61150 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{9}+O(q^{10}) $ 14 * q + 14 * q^9	$ 14 q + 14 q^{9} - 8 q^{11} - 28 q^{15} - 14 q^{23} + 10 q^{25} + 4 q^{29} + 12 q^{37} - 24 q^{39} - 44 q^{43} - 28 q^{51} - 4 q^{53} - 8 q^{57} - 44 q^{65} - 28 q^{67} - 28 q^{71} - 36 q^{79} + 18 q^{81} + 8 q^{85} + 44 q^{93} - 8 q^{95} - 60 q^{99}+O(q^{100}) $ 14 * q + 14 * q^9 - 8 * q^11 - 28 * q^15 - 14 * q^23 + 10 * q^25 + 4 * q^29 + 12 * q^37 - 24 * q^39 - 44 * q^43 - 28 * q^51 - 4 * q^53 - 8 * q^57 - 44 * q^65 - 28 * q^67 - 28 * q^71 - 36 * q^79 + 18 * q^81 + 8 * q^85 + 44 * q^93 - 8 * q^95 - 60 * 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
$3$	−2.12838	−1.22882	−0.614411	−	0.788986i	$-0.710606\pi$
$3$	−2.12838	−1.22882	−0.614411	+	0.788986i	$0.710606\pi$
$4$	0	0
$5$	−0.504697	−0.225708	−0.112854	−	0.993612i	$-0.535999\pi$
$5$	−0.504697	−0.225708	−0.112854	+	0.993612i	$0.535999\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.53001	0.510005
$10$	0	0
$11$	2.36044	0.711699	0.355850	−	0.934543i	$-0.384192\pi$
$11$	2.36044	0.711699	0.355850	+	0.934543i	$0.384192\pi$
$12$	0	0
$13$	1.21887	0.338053	0.169026	−	0.985612i	$-0.445938\pi$
$13$	1.21887	0.338053	0.169026	+	0.985612i	$0.445938\pi$
$14$	0	0
$15$	1.07419	0.277355
$16$	0	0
$17$	3.03702	0.736587	0.368293	−	0.929710i	$-0.379942\pi$
$17$	3.03702	0.736587	0.368293	+	0.929710i	$0.379942\pi$
$18$	0	0
$19$	2.40080	0.550781	0.275390	−	0.961333i	$-0.411193\pi$
$19$	2.40080	0.550781	0.275390	+	0.961333i	$0.411193\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.74528	−0.949056
$26$	0	0
$27$	3.12869	0.602117
$28$	0	0
$29$	−0.856064	−0.158967	−0.0794835	−	0.996836i	$-0.525327\pi$
$29$	−0.856064	−0.158967	−0.0794835	+	0.996836i	$0.525327\pi$
$30$	0	0
$31$	4.69317	0.842918	0.421459	−	0.906847i	$-0.361518\pi$
$31$	4.69317	0.842918	0.421459	+	0.906847i	$0.361518\pi$
$32$	0	0
$33$	−5.02392	−0.874552
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.9421	−1.96327	−0.981637	−	0.190757i	$-0.938906\pi$
$37$	−11.9421	−1.96327	−0.981637	+	0.190757i	$0.938906\pi$
$38$	0	0
$39$	−2.59422	−0.415407
$40$	0	0
$41$	−9.46633	−1.47839	−0.739196	−	0.673490i	$-0.764794\pi$
$41$	−9.46633	−1.47839	−0.739196	+	0.673490i	$0.764794\pi$
$42$	0	0
$43$	−12.8001	−1.95199	−0.975995	−	0.217792i	$-0.930115\pi$
$43$	−12.8001	−1.95199	−0.975995	+	0.217792i	$0.930115\pi$
$44$	0	0
$45$	−0.772194	−0.115112
$46$	0	0
$47$	−6.72467	−0.980894	−0.490447	−	0.871471i	$-0.663167\pi$
$47$	−6.72467	−0.980894	−0.490447	+	0.871471i	$0.663167\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.46395	−0.905134
$52$	0	0
$53$	9.51381	1.30682	0.653411	−	0.757003i	$-0.273337\pi$
$53$	9.51381	1.30682	0.653411	+	0.757003i	$0.273337\pi$
$54$	0	0
$55$	−1.19131	−0.160636
$56$	0	0
$57$	−5.10982	−0.676812
$58$	0	0
$59$	−6.44317	−0.838830	−0.419415	−	0.907795i	$-0.637765\pi$
$59$	−6.44317	−0.838830	−0.419415	+	0.907795i	$0.637765\pi$
$60$	0	0
$61$	10.0581	1.28781	0.643903	−	0.765107i	$-0.277314\pi$
$61$	10.0581	1.28781	0.643903	+	0.765107i	$0.277314\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.615159	−0.0763011
$66$	0	0
$67$	6.38042	0.779492	0.389746	−	0.920922i	$-0.372563\pi$
$67$	6.38042	0.779492	0.389746	+	0.920922i	$0.372563\pi$
$68$	0	0
$69$	2.12838	0.256227
$70$	0	0
$71$	−6.50520	−0.772025	−0.386013	−	0.922493i	$-0.626148\pi$
$71$	−6.50520	−0.772025	−0.386013	+	0.922493i	$0.626148\pi$
$72$	0	0
$73$	14.8834	1.74197	0.870983	−	0.491313i	$-0.163483\pi$
$73$	14.8834	1.74197	0.870983	+	0.491313i	$0.163483\pi$
$74$	0	0
$75$	10.0998	1.16622
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.50252	0.394064	0.197032	−	0.980397i	$-0.436870\pi$
$79$	3.50252	0.394064	0.197032	+	0.980397i	$0.436870\pi$
$80$	0	0
$81$	−11.2491	−1.24990
$82$	0	0
$83$	4.31583	0.473724	0.236862	−	0.971543i	$-0.423881\pi$
$83$	4.31583	0.473724	0.236862	+	0.971543i	$0.423881\pi$
$84$	0	0
$85$	−1.53278	−0.166253
$86$	0	0
$87$	1.82203	0.195342
$88$	0	0
$89$	14.7229	1.56063	0.780314	−	0.625388i	$-0.215059\pi$
$89$	14.7229	1.56063	0.780314	+	0.625388i	$0.215059\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.98886	−1.03580
$94$	0	0
$95$	−1.21168	−0.124315
$96$	0	0
$97$	12.5696	1.27625	0.638125	−	0.769933i	$-0.279710\pi$
$97$	12.5696	1.27625	0.638125	+	0.769933i	$0.279710\pi$
$98$	0	0
$99$	3.61150	0.362970
$100$	0	0
$101$	−3.09143	−0.307609	−0.153805	−	0.988101i	$-0.549153\pi$
$101$	−3.09143	−0.307609	−0.153805	+	0.988101i	$0.549153\pi$
$102$	0	0
$103$	−0.996437	−0.0981819	−0.0490909	−	0.998794i	$-0.515632\pi$
$103$	−0.996437	−0.0981819	−0.0490909	+	0.998794i	$0.515632\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.45859	−0.431028	−0.215514	−	0.976501i	$-0.569143\pi$
$107$	−4.45859	−0.431028	−0.215514	+	0.976501i	$0.569143\pi$
$108$	0	0
$109$	6.00958	0.575613	0.287807	−	0.957689i	$-0.407074\pi$
$109$	6.00958	0.575613	0.287807	+	0.957689i	$0.407074\pi$
$110$	0	0
$111$	25.4174	2.41252
$112$	0	0
$113$	9.22346	0.867670	0.433835	−	0.900992i	$-0.357160\pi$
$113$	9.22346	0.867670	0.433835	+	0.900992i	$0.357160\pi$
$114$	0	0
$115$	0.504697	0.0470633
$116$	0	0
$117$	1.86488	0.172408
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.42833	−0.493484
$122$	0	0
$123$	20.1480	1.81668
$124$	0	0
$125$	4.91842	0.439917
$126$	0	0
$127$	−13.3516	−1.18476	−0.592382	−	0.805658i	$-0.701812\pi$
$127$	−13.3516	−1.18476	−0.592382	+	0.805658i	$0.701812\pi$
$128$	0	0
$129$	27.2434	2.39865
$130$	0	0
$131$	−8.44668	−0.737990	−0.368995	−	0.929431i	$-0.620298\pi$
$131$	−8.44668	−0.737990	−0.368995	+	0.929431i	$0.620298\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.57904	−0.135902
$136$	0	0
$137$	15.6195	1.33447	0.667233	−	0.744849i	$-0.267478\pi$
$137$	15.6195	1.33447	0.667233	+	0.744849i	$0.267478\pi$
$138$	0	0
$139$	−18.9029	−1.60333	−0.801663	−	0.597776i	$-0.796051\pi$
$139$	−18.9029	−1.60333	−0.801663	+	0.597776i	$0.796051\pi$
$140$	0	0
$141$	14.3127	1.20534
$142$	0	0
$143$	2.87706	0.240592
$144$	0	0
$145$	0.432053	0.0358801
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.88580	0.564107	0.282053	−	0.959399i	$-0.408984\pi$
$149$	6.88580	0.564107	0.282053	+	0.959399i	$0.408984\pi$
$150$	0	0
$151$	−4.34777	−0.353816	−0.176908	−	0.984227i	$-0.556610\pi$
$151$	−4.34777	−0.353816	−0.176908	+	0.984227i	$0.556610\pi$
$152$	0	0
$153$	4.64669	0.375663
$154$	0	0
$155$	−2.36863	−0.190253
$156$	0	0
$157$	10.7111	0.854835	0.427418	−	0.904054i	$-0.359423\pi$
$157$	10.7111	0.854835	0.427418	+	0.904054i	$0.359423\pi$
$158$	0	0
$159$	−20.2490	−1.60585
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.33669	0.183024	0.0915119	−	0.995804i	$-0.470830\pi$
$163$	2.33669	0.183024	0.0915119	+	0.995804i	$0.470830\pi$
$164$	0	0
$165$	2.53556	0.197393
$166$	0	0
$167$	−8.21625	−0.635792	−0.317896	−	0.948126i	$-0.602976\pi$
$167$	−8.21625	−0.635792	−0.317896	+	0.948126i	$0.602976\pi$
$168$	0	0
$169$	−11.5144	−0.885720
$170$	0	0
$171$	3.67325	0.280901
$172$	0	0
$173$	−15.0186	−1.14185	−0.570923	−	0.821004i	$-0.693414\pi$
$173$	−15.0186	−1.14185	−0.570923	+	0.821004i	$0.693414\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.7135	1.03077
$178$	0	0
$179$	−5.58888	−0.417733	−0.208866	−	0.977944i	$-0.566977\pi$
$179$	−5.58888	−0.417733	−0.208866	+	0.977944i	$0.566977\pi$
$180$	0	0
$181$	6.37855	0.474114	0.237057	−	0.971496i	$-0.423817\pi$
$181$	6.37855	0.474114	0.237057	+	0.971496i	$0.423817\pi$
$182$	0	0
$183$	−21.4075	−1.58249
$184$	0	0
$185$	6.02716	0.443126
$186$	0	0
$187$	7.16871	0.524228
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.6028	−0.839551	−0.419775	−	0.907628i	$-0.637891\pi$
$191$	−11.6028	−0.839551	−0.419775	+	0.907628i	$0.637891\pi$
$192$	0	0
$193$	5.81634	0.418669	0.209335	−	0.977844i	$-0.432870\pi$
$193$	5.81634	0.418669	0.209335	+	0.977844i	$0.432870\pi$
$194$	0	0
$195$	1.30929	0.0937605
$196$	0	0
$197$	−2.78326	−0.198299	−0.0991494	−	0.995073i	$-0.531612\pi$
$197$	−2.78326	−0.198299	−0.0991494	+	0.995073i	$0.531612\pi$
$198$	0	0
$199$	−9.06182	−0.642376	−0.321188	−	0.947016i	$-0.604082\pi$
$199$	−9.06182	−0.642376	−0.321188	+	0.947016i	$0.604082\pi$
$200$	0	0
$201$	−13.5800	−0.957857
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.77763	0.333684
$206$	0	0
$207$	−1.53001	−0.106343
$208$	0	0
$209$	5.66694	0.391990
$210$	0	0
$211$	−19.6639	−1.35372	−0.676858	−	0.736114i	$-0.736659\pi$
$211$	−19.6639	−1.35372	−0.676858	+	0.736114i	$0.736659\pi$
$212$	0	0
$213$	13.8456	0.948682
$214$	0	0
$215$	6.46016	0.440579
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−31.6775	−2.14057
$220$	0	0
$221$	3.70173	0.249005
$222$	0	0
$223$	−8.85405	−0.592911	−0.296456	−	0.955047i	$-0.595805\pi$
$223$	−8.85405	−0.592911	−0.296456	+	0.955047i	$0.595805\pi$
$224$	0	0
$225$	−7.26034	−0.484023
$226$	0	0
$227$	−14.2242	−0.944096	−0.472048	−	0.881573i	$-0.656485\pi$
$227$	−14.2242	−0.944096	−0.472048	+	0.881573i	$0.656485\pi$
$228$	0	0
$229$	−19.8203	−1.30976	−0.654880	−	0.755733i	$-0.727281\pi$
$229$	−19.8203	−1.30976	−0.654880	+	0.755733i	$0.727281\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.8413	−0.906772	−0.453386	−	0.891314i	$-0.649784\pi$
$233$	−13.8413	−0.906772	−0.453386	+	0.891314i	$0.649784\pi$
$234$	0	0
$235$	3.39392	0.221395
$236$	0	0
$237$	−7.45469	−0.484234
$238$	0	0
$239$	18.7762	1.21453	0.607266	−	0.794499i	$-0.292266\pi$
$239$	18.7762	1.21453	0.607266	+	0.794499i	$0.292266\pi$
$240$	0	0
$241$	−22.4290	−1.44478	−0.722391	−	0.691485i	$-0.756957\pi$
$241$	−22.4290	−1.44478	−0.722391	+	0.691485i	$0.756957\pi$
$242$	0	0
$243$	14.5563	0.933788
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.92625	0.186193
$248$	0	0
$249$	−9.18575	−0.582123
$250$	0	0
$251$	17.5003	1.10461	0.552306	−	0.833642i	$-0.313748\pi$
$251$	17.5003	1.10461	0.552306	+	0.833642i	$0.313748\pi$
$252$	0	0
$253$	−2.36044	−0.148400
$254$	0	0
$255$	3.26234	0.204296
$256$	0	0
$257$	−15.2249	−0.949705	−0.474852	−	0.880065i	$-0.657499\pi$
$257$	−15.2249	−0.949705	−0.474852	+	0.880065i	$0.657499\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.30979	−0.0810739
$262$	0	0
$263$	24.6775	1.52168	0.760841	−	0.648938i	$-0.224787\pi$
$263$	24.6775	1.52168	0.760841	+	0.648938i	$0.224787\pi$
$264$	0	0
$265$	−4.80159	−0.294960
$266$	0	0
$267$	−31.3360	−1.91773
$268$	0	0
$269$	24.5844	1.49894	0.749469	−	0.662039i	$-0.230309\pi$
$269$	24.5844	1.49894	0.749469	+	0.662039i	$0.230309\pi$
$270$	0	0
$271$	11.8878	0.722133	0.361066	−	0.932540i	$-0.382413\pi$
$271$	11.8878	0.722133	0.361066	+	0.932540i	$0.382413\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.2009	−0.675443
$276$	0	0
$277$	22.8343	1.37198	0.685990	−	0.727611i	$-0.259369\pi$
$277$	22.8343	1.37198	0.685990	+	0.727611i	$0.259369\pi$
$278$	0	0
$279$	7.18061	0.429892
$280$	0	0
$281$	−15.9909	−0.953939	−0.476970	−	0.878920i	$-0.658265\pi$
$281$	−15.9909	−0.953939	−0.476970	+	0.878920i	$0.658265\pi$
$282$	0	0
$283$	24.1707	1.43680	0.718400	−	0.695631i	$-0.244875\pi$
$283$	24.1707	1.43680	0.718400	+	0.695631i	$0.244875\pi$
$284$	0	0
$285$	2.57891	0.152761
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−7.77648	−0.457440
$290$	0	0
$291$	−26.7529	−1.56828
$292$	0	0
$293$	20.4562	1.19506	0.597532	−	0.801845i	$-0.296148\pi$
$293$	20.4562	1.19506	0.597532	+	0.801845i	$0.296148\pi$
$294$	0	0
$295$	3.25185	0.189330
$296$	0	0
$297$	7.38509	0.428526
$298$	0	0
$299$	−1.21887	−0.0704889
$300$	0	0
$301$	0	0
$302$	0	0
$303$	6.57975	0.377997
$304$	0	0
$305$	−5.07629	−0.290668
$306$	0	0
$307$	16.9095	0.965077	0.482539	−	0.875875i	$-0.339715\pi$
$307$	16.9095	0.965077	0.482539	+	0.875875i	$0.339715\pi$
$308$	0	0
$309$	2.12080	0.120648
$310$	0	0
$311$	−22.0462	−1.25013	−0.625063	−	0.780575i	$-0.714927\pi$
$311$	−22.0462	−1.25013	−0.625063	+	0.780575i	$0.714927\pi$
$312$	0	0
$313$	−33.8457	−1.91307	−0.956536	−	0.291613i	$-0.905808\pi$
$313$	−33.8457	−1.91307	−0.956536	+	0.291613i	$0.905808\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.7076	1.05072	0.525362	−	0.850879i	$-0.323930\pi$
$317$	18.7076	1.05072	0.525362	+	0.850879i	$0.323930\pi$
$318$	0	0
$319$	−2.02069	−0.113137
$320$	0	0
$321$	9.48959	0.529657
$322$	0	0
$323$	7.29128	0.405698
$324$	0	0
$325$	−5.78386	−0.320831
$326$	0	0
$327$	−12.7907	−0.707326
$328$	0	0
$329$	0	0
$330$	0	0
$331$	30.5042	1.67666	0.838331	−	0.545162i	$-0.183532\pi$
$331$	30.5042	1.67666	0.838331	+	0.545162i	$0.183532\pi$
$332$	0	0
$333$	−18.2716	−1.00128
$334$	0	0
$335$	−3.22018	−0.175937
$336$	0	0
$337$	−3.10760	−0.169282	−0.0846409	−	0.996412i	$-0.526974\pi$
$337$	−3.10760	−0.169282	−0.0846409	+	0.996412i	$0.526974\pi$
$338$	0	0
$339$	−19.6311	−1.06621
$340$	0	0
$341$	11.0779	0.599904
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.07419	−0.0578324
$346$	0	0
$347$	−1.10271	−0.0591966	−0.0295983	−	0.999562i	$-0.509423\pi$
$347$	−1.10271	−0.0591966	−0.0295983	+	0.999562i	$0.509423\pi$
$348$	0	0
$349$	−15.2435	−0.815963	−0.407982	−	0.912990i	$-0.633767\pi$
$349$	−15.2435	−0.815963	−0.407982	+	0.912990i	$0.633767\pi$
$350$	0	0
$351$	3.81346	0.203547
$352$	0	0
$353$	−7.55961	−0.402357	−0.201179	−	0.979555i	$-0.564477\pi$
$353$	−7.55961	−0.402357	−0.201179	+	0.979555i	$0.564477\pi$
$354$	0	0
$355$	3.28316	0.174252
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.226793	−0.0119697	−0.00598484	−	0.999982i	$-0.501905\pi$
$359$	−0.226793	−0.0119697	−0.00598484	+	0.999982i	$0.501905\pi$
$360$	0	0
$361$	−13.2362	−0.696641
$362$	0	0
$363$	11.5536	0.606404
$364$	0	0
$365$	−7.51160	−0.393175
$366$	0	0
$367$	−18.0257	−0.940935	−0.470467	−	0.882417i	$-0.655915\pi$
$367$	−18.0257	−0.940935	−0.470467	+	0.882417i	$0.655915\pi$
$368$	0	0
$369$	−14.4836	−0.753987
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−23.4259	−1.21295	−0.606473	−	0.795104i	$-0.707416\pi$
$373$	−23.4259	−1.21295	−0.606473	+	0.795104i	$0.707416\pi$
$374$	0	0
$375$	−10.4683	−0.540580
$376$	0	0
$377$	−1.04343	−0.0537393
$378$	0	0
$379$	−16.4590	−0.845441	−0.422720	−	0.906260i	$-0.638925\pi$
$379$	−16.4590	−0.845441	−0.422720	+	0.906260i	$0.638925\pi$
$380$	0	0
$381$	28.4173	1.45586
$382$	0	0
$383$	−10.4299	−0.532945	−0.266473	−	0.963842i	$-0.585858\pi$
$383$	−10.4299	−0.532945	−0.266473	+	0.963842i	$0.585858\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−19.5843	−0.995524
$388$	0	0
$389$	−32.0918	−1.62712	−0.813561	−	0.581480i	$-0.802474\pi$
$389$	−32.0918	−1.62712	−0.813561	+	0.581480i	$0.802474\pi$
$390$	0	0
$391$	−3.03702	−0.153589
$392$	0	0
$393$	17.9778	0.906859
$394$	0	0
$395$	−1.76771	−0.0889432
$396$	0	0
$397$	3.08390	0.154777	0.0773883	−	0.997001i	$-0.475342\pi$
$397$	3.08390	0.154777	0.0773883	+	0.997001i	$0.475342\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.5888	−1.07809	−0.539045	−	0.842277i	$-0.681215\pi$
$401$	−21.5888	−1.07809	−0.539045	+	0.842277i	$0.681215\pi$
$402$	0	0
$403$	5.72035	0.284951
$404$	0	0
$405$	5.67739	0.282112
$406$	0	0
$407$	−28.1887	−1.39726
$408$	0	0
$409$	23.6803	1.17092	0.585459	−	0.810702i	$-0.300914\pi$
$409$	23.6803	1.17092	0.585459	+	0.810702i	$0.300914\pi$
$410$	0	0
$411$	−33.2443	−1.63982
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2.17819	−0.106923
$416$	0	0
$417$	40.2327	1.97020
$418$	0	0
$419$	13.2202	0.645848	0.322924	−	0.946425i	$-0.395334\pi$
$419$	13.2202	0.645848	0.322924	+	0.946425i	$0.395334\pi$
$420$	0	0
$421$	27.1735	1.32436	0.662179	−	0.749346i	$-0.269632\pi$
$421$	27.1735	1.32436	0.662179	+	0.749346i	$0.269632\pi$
$422$	0	0
$423$	−10.2888	−0.500261
$424$	0	0
$425$	−14.4115	−0.699062
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6.12349	−0.295645
$430$	0	0
$431$	−14.1737	−0.682724	−0.341362	−	0.939932i	$-0.610888\pi$
$431$	−14.1737	−0.682724	−0.341362	+	0.939932i	$0.610888\pi$
$432$	0	0
$433$	−2.37810	−0.114284	−0.0571421	−	0.998366i	$-0.518199\pi$
$433$	−2.37810	−0.114284	−0.0571421	+	0.998366i	$0.518199\pi$
$434$	0	0
$435$	−0.919575	−0.0440902
$436$	0	0
$437$	−2.40080	−0.114846
$438$	0	0
$439$	−23.3186	−1.11293	−0.556467	−	0.830870i	$-0.687843\pi$
$439$	−23.3186	−1.11293	−0.556467	+	0.830870i	$0.687843\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.4851	−0.925766	−0.462883	−	0.886419i	$-0.653185\pi$
$443$	−19.4851	−0.925766	−0.462883	+	0.886419i	$0.653185\pi$
$444$	0	0
$445$	−7.43063	−0.352245
$446$	0	0
$447$	−14.6556	−0.693187
$448$	0	0
$449$	−21.6922	−1.02372	−0.511860	−	0.859069i	$-0.671043\pi$
$449$	−21.6922	−1.02372	−0.511860	+	0.859069i	$0.671043\pi$
$450$	0	0
$451$	−22.3447	−1.05217
$452$	0	0
$453$	9.25371	0.434777
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.6891	0.687128	0.343564	−	0.939129i	$-0.388366\pi$
$457$	14.6891	0.687128	0.343564	+	0.939129i	$0.388366\pi$
$458$	0	0
$459$	9.50192	0.443512
$460$	0	0
$461$	21.2042	0.987578	0.493789	−	0.869582i	$-0.335612\pi$
$461$	21.2042	0.987578	0.493789	+	0.869582i	$0.335612\pi$
$462$	0	0
$463$	−29.6054	−1.37588	−0.687941	−	0.725767i	$-0.741485\pi$
$463$	−29.6054	−1.37588	−0.687941	+	0.725767i	$0.741485\pi$
$464$	0	0
$465$	5.04135	0.233787
$466$	0	0
$467$	14.7383	0.682005	0.341003	−	0.940062i	$-0.389234\pi$
$467$	14.7383	0.682005	0.341003	+	0.940062i	$0.389234\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−22.7972	−1.05044
$472$	0	0
$473$	−30.2138	−1.38923
$474$	0	0
$475$	−11.3925	−0.522722
$476$	0	0
$477$	14.5563	0.666485
$478$	0	0
$479$	4.27868	0.195498	0.0977490	−	0.995211i	$-0.468836\pi$
$479$	4.27868	0.195498	0.0977490	+	0.995211i	$0.468836\pi$
$480$	0	0
$481$	−14.5559	−0.663690
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.34384	−0.288059
$486$	0	0
$487$	−28.9130	−1.31017	−0.655087	−	0.755554i	$-0.727368\pi$
$487$	−28.9130	−1.31017	−0.655087	+	0.755554i	$0.727368\pi$
$488$	0	0
$489$	−4.97338	−0.224904
$490$	0	0
$491$	−42.2239	−1.90554	−0.952769	−	0.303697i	$-0.901779\pi$
$491$	−42.2239	−1.90554	−0.952769	+	0.303697i	$0.901779\pi$
$492$	0	0
$493$	−2.59989	−0.117093
$494$	0	0
$495$	−1.82272	−0.0819251
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−22.3512	−1.00058	−0.500289	−	0.865858i	$-0.666773\pi$
$499$	−22.3512	−1.00058	−0.500289	+	0.865858i	$0.666773\pi$
$500$	0	0
$501$	17.4873	0.781276
$502$	0	0
$503$	10.5541	0.470585	0.235292	−	0.971925i	$-0.424395\pi$
$503$	10.5541	0.470585	0.235292	+	0.971925i	$0.424395\pi$
$504$	0	0
$505$	1.56024	0.0694297
$506$	0	0
$507$	24.5070	1.08839
$508$	0	0
$509$	−17.6123	−0.780652	−0.390326	−	0.920677i	$-0.627638\pi$
$509$	−17.6123	−0.780652	−0.390326	+	0.920677i	$0.627638\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.51136	0.331635
$514$	0	0
$515$	0.502899	0.0221604
$516$	0	0
$517$	−15.8732	−0.698102
$518$	0	0
$519$	31.9654	1.40313
$520$	0	0
$521$	−2.99407	−0.131173	−0.0655864	−	0.997847i	$-0.520892\pi$
$521$	−2.99407	−0.131173	−0.0655864	+	0.997847i	$0.520892\pi$
$522$	0	0
$523$	−13.0249	−0.569537	−0.284769	−	0.958596i	$-0.591917\pi$
$523$	−13.0249	−0.569537	−0.284769	+	0.958596i	$0.591917\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.2533	0.620882
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.85814	−0.427807
$532$	0	0
$533$	−11.5382	−0.499775
$534$	0	0
$535$	2.25024	0.0972863
$536$	0	0
$537$	11.8953	0.513319
$538$	0	0
$539$	0	0
$540$	0	0
$541$	41.8722	1.80023	0.900113	−	0.435656i	$-0.143483\pi$
$541$	41.8722	1.80023	0.900113	+	0.435656i	$0.143483\pi$
$542$	0	0
$543$	−13.5760	−0.582602
$544$	0	0
$545$	−3.03302	−0.129920
$546$	0	0
$547$	20.3301	0.869253	0.434626	−	0.900611i	$-0.356880\pi$
$547$	20.3301	0.869253	0.434626	+	0.900611i	$0.356880\pi$
$548$	0	0
$549$	15.3890	0.656787
$550$	0	0
$551$	−2.05524	−0.0875560
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−12.8281	−0.544523
$556$	0	0
$557$	−22.1456	−0.938339	−0.469169	−	0.883108i	$-0.655447\pi$
$557$	−22.1456	−0.938339	−0.469169	+	0.883108i	$0.655447\pi$
$558$	0	0
$559$	−15.6016	−0.659876
$560$	0	0
$561$	−15.2578	−0.644183
$562$	0	0
$563$	−17.8553	−0.752513	−0.376256	−	0.926516i	$-0.622789\pi$
$563$	−17.8553	−0.752513	−0.376256	+	0.926516i	$0.622789\pi$
$564$	0	0
$565$	−4.65506	−0.195840
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.3568	−0.769557	−0.384778	−	0.923009i	$-0.625722\pi$
$569$	−18.3568	−0.769557	−0.384778	+	0.923009i	$0.625722\pi$
$570$	0	0
$571$	−1.53024	−0.0640386	−0.0320193	−	0.999487i	$-0.510194\pi$
$571$	−1.53024	−0.0640386	−0.0320193	+	0.999487i	$0.510194\pi$
$572$	0	0
$573$	24.6952	1.03166
$574$	0	0
$575$	4.74528	0.197892
$576$	0	0
$577$	−10.8818	−0.453016	−0.226508	−	0.974009i	$-0.572731\pi$
$577$	−10.8818	−0.453016	−0.226508	+	0.974009i	$0.572731\pi$
$578$	0	0
$579$	−12.3794	−0.514470
$580$	0	0
$581$	0	0
$582$	0	0
$583$	22.4568	0.930064
$584$	0	0
$585$	−0.941202	−0.0389139
$586$	0	0
$587$	−26.3921	−1.08932	−0.544660	−	0.838657i	$-0.683341\pi$
$587$	−26.3921	−1.08932	−0.544660	+	0.838657i	$0.683341\pi$
$588$	0	0
$589$	11.2673	0.464263
$590$	0	0
$591$	5.92384	0.243674
$592$	0	0
$593$	−18.4149	−0.756209	−0.378104	−	0.925763i	$-0.623424\pi$
$593$	−18.4149	−0.756209	−0.378104	+	0.925763i	$0.623424\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.2870	0.789365
$598$	0	0
$599$	−43.2006	−1.76513	−0.882565	−	0.470190i	$-0.844185\pi$
$599$	−43.2006	−1.76513	−0.882565	+	0.470190i	$0.844185\pi$
$600$	0	0
$601$	16.2922	0.664574	0.332287	−	0.943178i	$-0.392180\pi$
$601$	16.2922	0.664574	0.332287	+	0.943178i	$0.392180\pi$
$602$	0	0
$603$	9.76212	0.397544
$604$	0	0
$605$	2.73966	0.111383
$606$	0	0
$607$	2.19119	0.0889375	0.0444688	−	0.999011i	$-0.485840\pi$
$607$	2.19119	0.0889375	0.0444688	+	0.999011i	$0.485840\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.19648	−0.331594
$612$	0	0
$613$	6.55377	0.264704	0.132352	−	0.991203i	$-0.457747\pi$
$613$	6.55377	0.264704	0.132352	+	0.991203i	$0.457747\pi$
$614$	0	0
$615$	−10.1686	−0.410039
$616$	0	0
$617$	18.8705	0.759697	0.379849	−	0.925049i	$-0.375976\pi$
$617$	18.8705	0.759697	0.379849	+	0.925049i	$0.375976\pi$
$618$	0	0
$619$	−3.68844	−0.148251	−0.0741255	−	0.997249i	$-0.523617\pi$
$619$	−3.68844	−0.148251	−0.0741255	+	0.997249i	$0.523617\pi$
$620$	0	0
$621$	−3.12869	−0.125550
$622$	0	0
$623$	0	0
$624$	0	0
$625$	21.2441	0.849764
$626$	0	0
$627$	−12.0614	−0.481686
$628$	0	0
$629$	−36.2686	−1.44612
$630$	0	0
$631$	−19.4847	−0.775674	−0.387837	−	0.921728i	$-0.626778\pi$
$631$	−19.4847	−0.775674	−0.387837	+	0.921728i	$0.626778\pi$
$632$	0	0
$633$	41.8522	1.66348
$634$	0	0
$635$	6.73852	0.267410
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9.95305	−0.393736
$640$	0	0
$641$	21.7076	0.857399	0.428700	−	0.903447i	$-0.358972\pi$
$641$	21.7076	0.857399	0.428700	+	0.903447i	$0.358972\pi$
$642$	0	0
$643$	15.6702	0.617972	0.308986	−	0.951067i	$-0.400010\pi$
$643$	15.6702	0.617972	0.308986	+	0.951067i	$0.400010\pi$
$644$	0	0
$645$	−13.7497	−0.541393
$646$	0	0
$647$	−42.3081	−1.66330	−0.831652	−	0.555298i	$-0.812604\pi$
$647$	−42.3081	−1.66330	−0.831652	+	0.555298i	$0.812604\pi$
$648$	0	0
$649$	−15.2087	−0.596995
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.6785	0.457015	0.228508	−	0.973542i	$-0.426615\pi$
$653$	11.6785	0.457015	0.228508	+	0.973542i	$0.426615\pi$
$654$	0	0
$655$	4.26302	0.166570
$656$	0	0
$657$	22.7718	0.888411
$658$	0	0
$659$	−46.6022	−1.81537	−0.907683	−	0.419657i	$-0.862150\pi$
$659$	−46.6022	−1.81537	−0.907683	+	0.419657i	$0.862150\pi$
$660$	0	0
$661$	44.2362	1.72059	0.860294	−	0.509798i	$-0.170280\pi$
$661$	44.2362	1.72059	0.860294	+	0.509798i	$0.170280\pi$
$662$	0	0
$663$	−7.87870	−0.305983
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.856064	0.0331469
$668$	0	0
$669$	18.8448	0.728583
$670$	0	0
$671$	23.7415	0.916531
$672$	0	0
$673$	−17.6726	−0.681230	−0.340615	−	0.940203i	$-0.610635\pi$
$673$	−17.6726	−0.681230	−0.340615	+	0.940203i	$0.610635\pi$
$674$	0	0
$675$	−14.8465	−0.571443
$676$	0	0
$677$	−46.3393	−1.78096	−0.890482	−	0.455019i	$-0.849632\pi$
$677$	−46.3393	−1.78096	−0.890482	+	0.455019i	$0.849632\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	30.2746	1.16013
$682$	0	0
$683$	−6.18783	−0.236771	−0.118385	−	0.992968i	$-0.537772\pi$
$683$	−6.18783	−0.236771	−0.118385	+	0.992968i	$0.537772\pi$
$684$	0	0
$685$	−7.88314	−0.301199
$686$	0	0
$687$	42.1851	1.60946
$688$	0	0
$689$	11.5961	0.441775
$690$	0	0
$691$	33.7549	1.28410	0.642049	−	0.766664i	$-0.278085\pi$
$691$	33.7549	1.28410	0.642049	+	0.766664i	$0.278085\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.54026	0.361883
$696$	0	0
$697$	−28.7495	−1.08896
$698$	0	0
$699$	29.4595	1.11426
$700$	0	0
$701$	−5.83626	−0.220432	−0.110216	−	0.993908i	$-0.535154\pi$
$701$	−5.83626	−0.220432	−0.110216	+	0.993908i	$0.535154\pi$
$702$	0	0
$703$	−28.6706	−1.08133
$704$	0	0
$705$	−7.22357	−0.272055
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−9.73923	−0.365765	−0.182882	−	0.983135i	$-0.558543\pi$
$709$	−9.73923	−0.365765	−0.182882	+	0.983135i	$0.558543\pi$
$710$	0	0
$711$	5.35890	0.200974
$712$	0	0
$713$	−4.69317	−0.175761
$714$	0	0
$715$	−1.45205	−0.0543034
$716$	0	0
$717$	−39.9630	−1.49244
$718$	0	0
$719$	0.496383	0.0185120	0.00925598	−	0.999957i	$-0.497054\pi$
$719$	0.496383	0.0185120	0.00925598	+	0.999957i	$0.497054\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	47.7376	1.77538
$724$	0	0
$725$	4.06226	0.150869
$726$	0	0
$727$	39.7533	1.47437	0.737184	−	0.675692i	$-0.236155\pi$
$727$	39.7533	1.47437	0.737184	+	0.675692i	$0.236155\pi$
$728$	0	0
$729$	2.76590	0.102441
$730$	0	0
$731$	−38.8741	−1.43781
$732$	0	0
$733$	−9.08958	−0.335731	−0.167866	−	0.985810i	$-0.553687\pi$
$733$	−9.08958	−0.335731	−0.167866	+	0.985810i	$0.553687\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.0606	0.554764
$738$	0	0
$739$	14.6281	0.538102	0.269051	−	0.963126i	$-0.413290\pi$
$739$	14.6281	0.538102	0.269051	+	0.963126i	$0.413290\pi$
$740$	0	0
$741$	−6.22818	−0.228798
$742$	0	0
$743$	−7.08804	−0.260035	−0.130018	−	0.991512i	$-0.541503\pi$
$743$	−7.08804	−0.260035	−0.130018	+	0.991512i	$0.541503\pi$
$744$	0	0
$745$	−3.47525	−0.127323
$746$	0	0
$747$	6.60329	0.241602
$748$	0	0
$749$	0	0
$750$	0	0
$751$	5.91172	0.215722	0.107861	−	0.994166i	$-0.465600\pi$
$751$	5.91172	0.215722	0.107861	+	0.994166i	$0.465600\pi$
$752$	0	0
$753$	−37.2474	−1.35737
$754$	0	0
$755$	2.19431	0.0798590
$756$	0	0
$757$	11.6553	0.423621	0.211810	−	0.977311i	$-0.432064\pi$
$757$	11.6553	0.423621	0.211810	+	0.977311i	$0.432064\pi$
$758$	0	0
$759$	5.02392	0.182357
$760$	0	0
$761$	−5.13146	−0.186015	−0.0930076	−	0.995665i	$-0.529648\pi$
$761$	−5.13146	−0.186015	−0.0930076	+	0.995665i	$0.529648\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.34517	−0.0847899
$766$	0	0
$767$	−7.85337	−0.283569
$768$	0	0
$769$	−28.3422	−1.02205	−0.511023	−	0.859567i	$-0.670733\pi$
$769$	−28.3422	−1.02205	−0.511023	+	0.859567i	$0.670733\pi$
$770$	0	0
$771$	32.4045	1.16702
$772$	0	0
$773$	−2.19531	−0.0789597	−0.0394798	−	0.999220i	$-0.512570\pi$
$773$	−2.19531	−0.0789597	−0.0394798	+	0.999220i	$0.512570\pi$
$774$	0	0
$775$	−22.2704	−0.799976
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−22.7267	−0.814270
$780$	0	0
$781$	−15.3551	−0.549450
$782$	0	0
$783$	−2.67836	−0.0957168
$784$	0	0
$785$	−5.40584	−0.192943
$786$	0	0
$787$	26.4524	0.942928	0.471464	−	0.881885i	$-0.343726\pi$
$787$	26.4524	0.942928	0.471464	+	0.881885i	$0.343726\pi$
$788$	0	0
$789$	−52.5233	−1.86988
$790$	0	0
$791$	0	0
$792$	0	0
$793$	12.2595	0.435347
$794$	0	0
$795$	10.2196	0.362453
$796$	0	0
$797$	−4.86695	−0.172396	−0.0861981	−	0.996278i	$-0.527472\pi$
$797$	−4.86695	−0.172396	−0.0861981	+	0.996278i	$0.527472\pi$
$798$	0	0
$799$	−20.4230	−0.722514
$800$	0	0
$801$	22.5263	0.795927
$802$	0	0
$803$	35.1313	1.23976
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−52.3250	−1.84193
$808$	0	0
$809$	31.1587	1.09548	0.547740	−	0.836648i	$-0.315488\pi$
$809$	31.1587	1.09548	0.547740	+	0.836648i	$0.315488\pi$
$810$	0	0
$811$	30.8318	1.08265	0.541325	−	0.840814i	$-0.317923\pi$
$811$	30.8318	1.08265	0.541325	+	0.840814i	$0.317923\pi$
$812$	0	0
$813$	−25.3018	−0.887373
$814$	0	0
$815$	−1.17932	−0.0413099
$816$	0	0
$817$	−30.7303	−1.07512
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.9359	−0.835370	−0.417685	−	0.908592i	$-0.637158\pi$
$821$	−23.9359	−0.835370	−0.417685	+	0.908592i	$0.637158\pi$
$822$	0	0
$823$	−32.8660	−1.14564	−0.572818	−	0.819683i	$-0.694150\pi$
$823$	−32.8660	−1.14564	−0.572818	+	0.819683i	$0.694150\pi$
$824$	0	0
$825$	23.8399	0.829999
$826$	0	0
$827$	50.6542	1.76142	0.880709	−	0.473657i	$-0.157066\pi$
$827$	50.6542	1.76142	0.880709	+	0.473657i	$0.157066\pi$
$828$	0	0
$829$	−10.1368	−0.352067	−0.176033	−	0.984384i	$-0.556327\pi$
$829$	−10.1368	−0.352067	−0.176033	+	0.984384i	$0.556327\pi$
$830$	0	0
$831$	−48.6001	−1.68592
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4.14672	0.143503
$836$	0	0
$837$	14.6835	0.507536
$838$	0	0
$839$	−37.3963	−1.29106	−0.645531	−	0.763734i	$-0.723364\pi$
$839$	−37.3963	−1.29106	−0.645531	+	0.763734i	$0.723364\pi$
$840$	0	0
$841$	−28.2672	−0.974729
$842$	0	0
$843$	34.0348	1.17222
$844$	0	0
$845$	5.81127	0.199914
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−51.4445	−1.76557
$850$	0	0
$851$	11.9421	0.409371
$852$	0	0
$853$	−32.2713	−1.10495	−0.552475	−	0.833530i	$-0.686316\pi$
$853$	−32.2713	−1.10495	−0.552475	+	0.833530i	$0.686316\pi$
$854$	0	0
$855$	−1.85388	−0.0634014
$856$	0	0
$857$	−28.2546	−0.965157	−0.482579	−	0.875853i	$-0.660300\pi$
$857$	−28.2546	−0.965157	−0.482579	+	0.875853i	$0.660300\pi$
$858$	0	0
$859$	−32.6887	−1.11532	−0.557662	−	0.830068i	$-0.688301\pi$
$859$	−32.6887	−1.11532	−0.557662	+	0.830068i	$0.688301\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.0999	1.29694	0.648468	−	0.761242i	$-0.275410\pi$
$863$	38.0999	1.29694	0.648468	+	0.761242i	$0.275410\pi$
$864$	0	0
$865$	7.57987	0.257723
$866$	0	0
$867$	16.5513	0.562113
$868$	0	0
$869$	8.26747	0.280455
$870$	0	0
$871$	7.77688	0.263509
$872$	0	0
$873$	19.2317	0.650893
$874$	0	0
$875$	0	0
$876$	0	0
$877$	56.1519	1.89611	0.948057	−	0.318100i	$-0.103045\pi$
$877$	56.1519	1.89611	0.948057	+	0.318100i	$0.103045\pi$
$878$	0	0
$879$	−43.5386	−1.46852
$880$	0	0
$881$	15.5452	0.523731	0.261865	−	0.965104i	$-0.415662\pi$
$881$	15.5452	0.523731	0.261865	+	0.965104i	$0.415662\pi$
$882$	0	0
$883$	−33.0469	−1.11212	−0.556058	−	0.831143i	$-0.687687\pi$
$883$	−33.0469	−1.11212	−0.556058	+	0.831143i	$0.687687\pi$
$884$	0	0
$885$	−6.92119	−0.232653
$886$	0	0
$887$	−51.2820	−1.72188	−0.860941	−	0.508704i	$-0.830125\pi$
$887$	−51.2820	−1.72188	−0.860941	+	0.508704i	$0.830125\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−26.5528	−0.889553
$892$	0	0
$893$	−16.1446	−0.540258
$894$	0	0
$895$	2.82070	0.0942854
$896$	0	0
$897$	2.59422	0.0866183
$898$	0	0
$899$	−4.01765	−0.133996
$900$	0	0
$901$	28.8937	0.962588
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.21924	−0.107011
$906$	0	0
$907$	−46.8241	−1.55477	−0.777385	−	0.629025i	$-0.783454\pi$
$907$	−46.8241	−1.55477	−0.777385	+	0.629025i	$0.783454\pi$
$908$	0	0
$909$	−4.72993	−0.156882
$910$	0	0
$911$	39.8447	1.32011	0.660056	−	0.751216i	$-0.270532\pi$
$911$	39.8447	1.32011	0.660056	+	0.751216i	$0.270532\pi$
$912$	0	0
$913$	10.1873	0.337149
$914$	0	0
$915$	10.8043	0.357179
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−42.6479	−1.40682	−0.703412	−	0.710782i	$-0.748341\pi$
$919$	−42.6479	−1.40682	−0.703412	+	0.710782i	$0.748341\pi$
$920$	0	0
$921$	−35.9899	−1.18591
$922$	0	0
$923$	−7.92897	−0.260985
$924$	0	0
$925$	56.6688	1.86326
$926$	0	0
$927$	−1.52456	−0.0500732
$928$	0	0
$929$	10.1838	0.334120	0.167060	−	0.985947i	$-0.446573\pi$
$929$	10.1838	0.334120	0.167060	+	0.985947i	$0.446573\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	46.9227	1.53618
$934$	0	0
$935$	−3.61803	−0.118322
$936$	0	0
$937$	31.1361	1.01717	0.508586	−	0.861011i	$-0.330168\pi$
$937$	31.1361	1.01717	0.508586	+	0.861011i	$0.330168\pi$
$938$	0	0
$939$	72.0366	2.35083
$940$	0	0
$941$	−18.2450	−0.594771	−0.297386	−	0.954757i	$-0.596115\pi$
$941$	−18.2450	−0.594771	−0.297386	+	0.954757i	$0.596115\pi$
$942$	0	0
$943$	9.46633	0.308266
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.9879	−0.909484	−0.454742	−	0.890623i	$-0.650268\pi$
$947$	−27.9879	−0.909484	−0.454742	+	0.890623i	$0.650268\pi$
$948$	0	0
$949$	18.1408	0.588877
$950$	0	0
$951$	−39.8169	−1.29115
$952$	0	0
$953$	−29.2453	−0.947349	−0.473675	−	0.880700i	$-0.657073\pi$
$953$	−29.2453	−0.947349	−0.473675	+	0.880700i	$0.657073\pi$
$954$	0	0
$955$	5.85591	0.189493
$956$	0	0
$957$	4.30079	0.139025
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−8.97417	−0.289489
$962$	0	0
$963$	−6.82170	−0.219826
$964$	0	0
$965$	−2.93549	−0.0944969
$966$	0	0
$967$	26.7646	0.860692	0.430346	−	0.902664i	$-0.358392\pi$
$967$	26.7646	0.860692	0.430346	+	0.902664i	$0.358392\pi$
$968$	0	0
$969$	−15.5186	−0.498530
$970$	0	0
$971$	27.9362	0.896514	0.448257	−	0.893905i	$-0.352045\pi$
$971$	27.9362	0.896514	0.448257	+	0.893905i	$0.352045\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	12.3103	0.394244
$976$	0	0
$977$	13.1154	0.419600	0.209800	−	0.977744i	$-0.432719\pi$
$977$	13.1154	0.419600	0.209800	+	0.977744i	$0.432719\pi$
$978$	0	0
$979$	34.7526	1.11070
$980$	0	0
$981$	9.19474	0.293565
$982$	0	0
$983$	27.9107	0.890212	0.445106	−	0.895478i	$-0.353166\pi$
$983$	27.9107	0.890212	0.445106	+	0.895478i	$0.353166\pi$
$984$	0	0
$985$	1.40470	0.0447575
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.8001	0.407018
$990$	0	0
$991$	−39.9723	−1.26976	−0.634881	−	0.772610i	$-0.718951\pi$
$991$	−39.9723	−1.26976	−0.634881	+	0.772610i	$0.718951\pi$
$992$	0	0
$993$	−64.9246	−2.06032
$994$	0	0
$995$	4.57348	0.144989
$996$	0	0
$997$	−1.37756	−0.0436278	−0.0218139	−	0.999762i	$-0.506944\pi$
$997$	−1.37756	−0.0436278	−0.0218139	+	0.999762i	$0.506944\pi$
$998$	0	0
$999$	−37.3633	−1.18212

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.bs.1.4	✓	14
7.6	odd	2	inner	9016.2.a.bs.1.11	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9016.2.a.bs.1.4	✓	14	1.1	even	1	trivial
9016.2.a.bs.1.11	yes	14	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9016.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12838 q^{3} -0.504697 q^{5} +1.53001 q^{9} +O(q^{10})\)	\(q-2.12838 q^{3} -0.504697 q^{5} +1.53001 q^{9} +2.36044 q^{11} +1.21887 q^{13} +1.07419 q^{15} +3.03702 q^{17} +2.40080 q^{19} -1.00000 q^{23} -4.74528 q^{25} +3.12869 q^{27} -0.856064 q^{29} +4.69317 q^{31} -5.02392 q^{33} -11.9421 q^{37} -2.59422 q^{39} -9.46633 q^{41} -12.8001 q^{43} -0.772194 q^{45} -6.72467 q^{47} -6.46395 q^{51} +9.51381 q^{53} -1.19131 q^{55} -5.10982 q^{57} -6.44317 q^{59} +10.0581 q^{61} -0.615159 q^{65} +6.38042 q^{67} +2.12838 q^{69} -6.50520 q^{71} +14.8834 q^{73} +10.0998 q^{75} +3.50252 q^{79} -11.2491 q^{81} +4.31583 q^{83} -1.53278 q^{85} +1.82203 q^{87} +14.7229 q^{89} -9.98886 q^{93} -1.21168 q^{95} +12.5696 q^{97} +3.61150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{9}+O(q^{10}) \) 14 * q + 14 * q^9	\( 14 q + 14 q^{9} - 8 q^{11} - 28 q^{15} - 14 q^{23} + 10 q^{25} + 4 q^{29} + 12 q^{37} - 24 q^{39} - 44 q^{43} - 28 q^{51} - 4 q^{53} - 8 q^{57} - 44 q^{65} - 28 q^{67} - 28 q^{71} - 36 q^{79} + 18 q^{81} + 8 q^{85} + 44 q^{93} - 8 q^{95} - 60 q^{99}+O(q^{100}) \) 14 * q + 14 * q^9 - 8 * q^11 - 28 * q^15 - 14 * q^23 + 10 * q^25 + 4 * q^29 + 12 * q^37 - 24 * q^39 - 44 * q^43 - 28 * q^51 - 4 * q^53 - 8 * q^57 - 44 * q^65 - 28 * q^67 - 28 * q^71 - 36 * q^79 + 18 * q^81 + 8 * q^85 + 44 * q^93 - 8 * q^95 - 60 * q^99

Introduction

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