LMFDB - Embedded newform 9016.2.a.bs.1.14

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			1.14
Root			$3.03220$ 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)

$f(q)$	$=$	$q+3.03220 q^{3} -3.31370 q^{5} +6.19425 q^{9} +O(q^{10})$	$q+3.03220 q^{3} -3.31370 q^{5} +6.19425 q^{9} -3.36153 q^{11} +1.69571 q^{13} -10.0478 q^{15} -3.08231 q^{17} -0.856882 q^{19} -1.00000 q^{23} +5.98060 q^{25} +9.68560 q^{27} +2.19791 q^{29} +7.39935 q^{31} -10.1928 q^{33} +3.81703 q^{37} +5.14174 q^{39} +2.47141 q^{41} -8.16910 q^{43} -20.5259 q^{45} -9.13217 q^{47} -9.34619 q^{51} -0.517176 q^{53} +11.1391 q^{55} -2.59824 q^{57} -13.0473 q^{59} +7.12135 q^{61} -5.61907 q^{65} +3.21484 q^{67} -3.03220 q^{69} -8.90909 q^{71} +6.79985 q^{73} +18.1344 q^{75} -13.3477 q^{79} +10.7860 q^{81} -10.3999 q^{83} +10.2139 q^{85} +6.66451 q^{87} -5.12244 q^{89} +22.4363 q^{93} +2.83945 q^{95} +2.17377 q^{97} -20.8221 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.03220	1.75064	0.875321	−	0.483542i	$-0.160650\pi$
$3$	3.03220	1.75064	0.875321	+	0.483542i	$0.160650\pi$
$4$	0	0
$5$	−3.31370	−1.48193	−0.740966	−	0.671543i	$-0.765632\pi$
$5$	−3.31370	−1.48193	−0.740966	+	0.671543i	$0.765632\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.19425	2.06475
$10$	0	0
$11$	−3.36153	−1.01354	−0.506769	−	0.862082i	$-0.669160\pi$
$11$	−3.36153	−1.01354	−0.506769	+	0.862082i	$0.669160\pi$
$12$	0	0
$13$	1.69571	0.470306	0.235153	−	0.971958i	$-0.424441\pi$
$13$	1.69571	0.470306	0.235153	+	0.971958i	$0.424441\pi$
$14$	0	0
$15$	−10.0478	−2.59433
$16$	0	0
$17$	−3.08231	−0.747570	−0.373785	−	0.927515i	$-0.621940\pi$
$17$	−3.08231	−0.747570	−0.373785	+	0.927515i	$0.621940\pi$
$18$	0	0
$19$	−0.856882	−0.196582	−0.0982911	−	0.995158i	$-0.531338\pi$
$19$	−0.856882	−0.196582	−0.0982911	+	0.995158i	$0.531338\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.98060	1.19612
$26$	0	0
$27$	9.68560	1.86399
$28$	0	0
$29$	2.19791	0.408142	0.204071	−	0.978956i	$-0.434583\pi$
$29$	2.19791	0.408142	0.204071	+	0.978956i	$0.434583\pi$
$30$	0	0
$31$	7.39935	1.32896	0.664481	−	0.747305i	$-0.268653\pi$
$31$	7.39935	1.32896	0.664481	+	0.747305i	$0.268653\pi$
$32$	0	0
$33$	−10.1928	−1.77434
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.81703	0.627516	0.313758	−	0.949503i	$-0.398412\pi$
$37$	3.81703	0.627516	0.313758	+	0.949503i	$0.398412\pi$
$38$	0	0
$39$	5.14174	0.823337
$40$	0	0
$41$	2.47141	0.385969	0.192985	−	0.981202i	$-0.438183\pi$
$41$	2.47141	0.385969	0.192985	+	0.981202i	$0.438183\pi$
$42$	0	0
$43$	−8.16910	−1.24578	−0.622888	−	0.782311i	$-0.714041\pi$
$43$	−8.16910	−1.24578	−0.622888	+	0.782311i	$0.714041\pi$
$44$	0	0
$45$	−20.5259	−3.05982
$46$	0	0
$47$	−9.13217	−1.33206	−0.666032	−	0.745923i	$-0.732008\pi$
$47$	−9.13217	−1.33206	−0.666032	+	0.745923i	$0.732008\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−9.34619	−1.30873
$52$	0	0
$53$	−0.517176	−0.0710396	−0.0355198	−	0.999369i	$-0.511309\pi$
$53$	−0.517176	−0.0710396	−0.0355198	+	0.999369i	$0.511309\pi$
$54$	0	0
$55$	11.1391	1.50199
$56$	0	0
$57$	−2.59824	−0.344145
$58$	0	0
$59$	−13.0473	−1.69861	−0.849304	−	0.527904i	$-0.822978\pi$
$59$	−13.0473	−1.69861	−0.849304	+	0.527904i	$0.822978\pi$
$60$	0	0
$61$	7.12135	0.911796	0.455898	−	0.890032i	$-0.349318\pi$
$61$	7.12135	0.911796	0.455898	+	0.890032i	$0.349318\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.61907	−0.696960
$66$	0	0
$67$	3.21484	0.392755	0.196377	−	0.980528i	$-0.437082\pi$
$67$	3.21484	0.392755	0.196377	+	0.980528i	$0.437082\pi$
$68$	0	0
$69$	−3.03220	−0.365034
$70$	0	0
$71$	−8.90909	−1.05731	−0.528657	−	0.848835i	$-0.677304\pi$
$71$	−8.90909	−1.05731	−0.528657	+	0.848835i	$0.677304\pi$
$72$	0	0
$73$	6.79985	0.795863	0.397931	−	0.917415i	$-0.369728\pi$
$73$	6.79985	0.795863	0.397931	+	0.917415i	$0.369728\pi$
$74$	0	0
$75$	18.1344	2.09398
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.3477	−1.50173	−0.750864	−	0.660457i	$-0.770363\pi$
$79$	−13.3477	−1.50173	−0.750864	+	0.660457i	$0.770363\pi$
$80$	0	0
$81$	10.7860	1.19844
$82$	0	0
$83$	−10.3999	−1.14154	−0.570770	−	0.821110i	$-0.693355\pi$
$83$	−10.3999	−1.14154	−0.570770	+	0.821110i	$0.693355\pi$
$84$	0	0
$85$	10.2139	1.10785
$86$	0	0
$87$	6.66451	0.714510
$88$	0	0
$89$	−5.12244	−0.542978	−0.271489	−	0.962442i	$-0.587516\pi$
$89$	−5.12244	−0.542978	−0.271489	+	0.962442i	$0.587516\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	22.4363	2.32654
$94$	0	0
$95$	2.83945	0.291321
$96$	0	0
$97$	2.17377	0.220713	0.110357	−	0.993892i	$-0.464801\pi$
$97$	2.17377	0.220713	0.110357	+	0.993892i	$0.464801\pi$
$98$	0	0
$99$	−20.8221	−2.09270
$100$	0	0
$101$	6.63910	0.660615	0.330307	−	0.943873i	$-0.392848\pi$
$101$	6.63910	0.660615	0.330307	+	0.943873i	$0.392848\pi$
$102$	0	0
$103$	−1.27527	−0.125657	−0.0628283	−	0.998024i	$-0.520012\pi$
$103$	−1.27527	−0.125657	−0.0628283	+	0.998024i	$0.520012\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.83395	−0.370642	−0.185321	−	0.982678i	$-0.559332\pi$
$107$	−3.83395	−0.370642	−0.185321	+	0.982678i	$0.559332\pi$
$108$	0	0
$109$	5.74547	0.550316	0.275158	−	0.961399i	$-0.411270\pi$
$109$	5.74547	0.550316	0.275158	+	0.961399i	$0.411270\pi$
$110$	0	0
$111$	11.5740	1.09856
$112$	0	0
$113$	−20.1438	−1.89497	−0.947486	−	0.319799i	$-0.896385\pi$
$113$	−20.1438	−1.89497	−0.947486	+	0.319799i	$0.896385\pi$
$114$	0	0
$115$	3.31370	0.309004
$116$	0	0
$117$	10.5037	0.971063
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0.299852	0.0272593
$122$	0	0
$123$	7.49381	0.675694
$124$	0	0
$125$	−3.24941	−0.290636
$126$	0	0
$127$	−17.7224	−1.57261	−0.786303	−	0.617841i	$-0.788007\pi$
$127$	−17.7224	−1.57261	−0.786303	+	0.617841i	$0.788007\pi$
$128$	0	0
$129$	−24.7704	−2.18091
$130$	0	0
$131$	−4.16548	−0.363940	−0.181970	−	0.983304i	$-0.558247\pi$
$131$	−4.16548	−0.363940	−0.181970	+	0.983304i	$0.558247\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−32.0952	−2.76231
$136$	0	0
$137$	−10.8593	−0.927773	−0.463886	−	0.885895i	$-0.653545\pi$
$137$	−10.8593	−0.927773	−0.463886	+	0.885895i	$0.653545\pi$
$138$	0	0
$139$	−14.6339	−1.24123	−0.620617	−	0.784114i	$-0.713118\pi$
$139$	−14.6339	−1.24123	−0.620617	+	0.784114i	$0.713118\pi$
$140$	0	0
$141$	−27.6906	−2.33197
$142$	0	0
$143$	−5.70017	−0.476673
$144$	0	0
$145$	−7.28321	−0.604838
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.7494	−1.37216	−0.686082	−	0.727525i	$-0.740671\pi$
$149$	−16.7494	−1.37216	−0.686082	+	0.727525i	$0.740671\pi$
$150$	0	0
$151$	21.9064	1.78272	0.891361	−	0.453295i	$-0.149751\pi$
$151$	21.9064	1.78272	0.891361	+	0.453295i	$0.149751\pi$
$152$	0	0
$153$	−19.0926	−1.54355
$154$	0	0
$155$	−24.5192	−1.96943
$156$	0	0
$157$	20.1730	1.60998	0.804989	−	0.593290i	$-0.202171\pi$
$157$	20.1730	1.60998	0.804989	+	0.593290i	$0.202171\pi$
$158$	0	0
$159$	−1.56818	−0.124365
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−13.5789	−1.06358	−0.531790	−	0.846876i	$-0.678480\pi$
$163$	−13.5789	−1.06358	−0.531790	+	0.846876i	$0.678480\pi$
$164$	0	0
$165$	33.7759	2.62945
$166$	0	0
$167$	−10.5071	−0.813065	−0.406532	−	0.913636i	$-0.633262\pi$
$167$	−10.5071	−0.813065	−0.406532	+	0.913636i	$0.633262\pi$
$168$	0	0
$169$	−10.1246	−0.778813
$170$	0	0
$171$	−5.30774	−0.405893
$172$	0	0
$173$	−12.4274	−0.944836	−0.472418	−	0.881375i	$-0.656619\pi$
$173$	−12.4274	−0.944836	−0.472418	+	0.881375i	$0.656619\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−39.5619	−2.97365
$178$	0	0
$179$	22.6975	1.69649	0.848246	−	0.529603i	$-0.177659\pi$
$179$	22.6975	1.69649	0.848246	+	0.529603i	$0.177659\pi$
$180$	0	0
$181$	−23.3413	−1.73495	−0.867474	−	0.497483i	$-0.834258\pi$
$181$	−23.3413	−1.73495	−0.867474	+	0.497483i	$0.834258\pi$
$182$	0	0
$183$	21.5934	1.59623
$184$	0	0
$185$	−12.6485	−0.929935
$186$	0	0
$187$	10.3613	0.757691
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.56800	0.619959	0.309979	−	0.950743i	$-0.399678\pi$
$191$	8.56800	0.619959	0.309979	+	0.950743i	$0.399678\pi$
$192$	0	0
$193$	−19.7843	−1.42411	−0.712053	−	0.702126i	$-0.752235\pi$
$193$	−19.7843	−1.42411	−0.712053	+	0.702126i	$0.752235\pi$
$194$	0	0
$195$	−17.0382	−1.22013
$196$	0	0
$197$	19.2727	1.37312	0.686560	−	0.727073i	$-0.259120\pi$
$197$	19.2727	1.37312	0.686560	+	0.727073i	$0.259120\pi$
$198$	0	0
$199$	15.9555	1.13106	0.565528	−	0.824729i	$-0.308673\pi$
$199$	15.9555	1.13106	0.565528	+	0.824729i	$0.308673\pi$
$200$	0	0
$201$	9.74803	0.687573
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−8.18951	−0.571980
$206$	0	0
$207$	−6.19425	−0.430530
$208$	0	0
$209$	2.88043	0.199244
$210$	0	0
$211$	−24.4446	−1.68283	−0.841417	−	0.540387i	$-0.818278\pi$
$211$	−24.4446	−1.68283	−0.841417	+	0.540387i	$0.818278\pi$
$212$	0	0
$213$	−27.0142	−1.85098
$214$	0	0
$215$	27.0699	1.84615
$216$	0	0
$217$	0	0
$218$	0	0
$219$	20.6185	1.39327
$220$	0	0
$221$	−5.22671	−0.351586
$222$	0	0
$223$	−20.7277	−1.38803	−0.694013	−	0.719962i	$-0.744159\pi$
$223$	−20.7277	−1.38803	−0.694013	+	0.719962i	$0.744159\pi$
$224$	0	0
$225$	37.0453	2.46969
$226$	0	0
$227$	4.58825	0.304533	0.152266	−	0.988339i	$-0.451343\pi$
$227$	4.58825	0.304533	0.152266	+	0.988339i	$0.451343\pi$
$228$	0	0
$229$	−22.4756	−1.48523	−0.742616	−	0.669717i	$-0.766415\pi$
$229$	−22.4756	−1.48523	−0.742616	+	0.669717i	$0.766415\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.1672	1.32120	0.660600	−	0.750738i	$-0.270302\pi$
$233$	20.1672	1.32120	0.660600	+	0.750738i	$0.270302\pi$
$234$	0	0
$235$	30.2612	1.97403
$236$	0	0
$237$	−40.4728	−2.62899
$238$	0	0
$239$	12.7046	0.821795	0.410898	−	0.911682i	$-0.365215\pi$
$239$	12.7046	0.821795	0.410898	+	0.911682i	$0.365215\pi$
$240$	0	0
$241$	−11.7846	−0.759110	−0.379555	−	0.925169i	$-0.623923\pi$
$241$	−11.7846	−0.759110	−0.379555	+	0.925169i	$0.623923\pi$
$242$	0	0
$243$	3.64839	0.234044
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.45302	−0.0924537
$248$	0	0
$249$	−31.5347	−1.99843
$250$	0	0
$251$	1.38462	0.0873964	0.0436982	−	0.999045i	$-0.486086\pi$
$251$	1.38462	0.0873964	0.0436982	+	0.999045i	$0.486086\pi$
$252$	0	0
$253$	3.36153	0.211337
$254$	0	0
$255$	30.9705	1.93945
$256$	0	0
$257$	8.32256	0.519147	0.259574	−	0.965723i	$-0.416418\pi$
$257$	8.32256	0.519147	0.259574	+	0.965723i	$0.416418\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	13.6144	0.842710
$262$	0	0
$263$	−13.7539	−0.848101	−0.424050	−	0.905639i	$-0.639392\pi$
$263$	−13.7539	−0.848101	−0.424050	+	0.905639i	$0.639392\pi$
$264$	0	0
$265$	1.71376	0.105276
$266$	0	0
$267$	−15.5323	−0.950560
$268$	0	0
$269$	10.7651	0.656360	0.328180	−	0.944615i	$-0.393565\pi$
$269$	10.7651	0.656360	0.328180	+	0.944615i	$0.393565\pi$
$270$	0	0
$271$	−5.05260	−0.306923	−0.153462	−	0.988155i	$-0.549042\pi$
$271$	−5.05260	−0.306923	−0.153462	+	0.988155i	$0.549042\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.1039	−1.21231
$276$	0	0
$277$	13.1007	0.787145	0.393572	−	0.919294i	$-0.371239\pi$
$277$	13.1007	0.787145	0.393572	+	0.919294i	$0.371239\pi$
$278$	0	0
$279$	45.8334	2.74397
$280$	0	0
$281$	−1.66787	−0.0994970	−0.0497485	−	0.998762i	$-0.515842\pi$
$281$	−1.66787	−0.0994970	−0.0497485	+	0.998762i	$0.515842\pi$
$282$	0	0
$283$	5.56543	0.330831	0.165415	−	0.986224i	$-0.447104\pi$
$283$	5.56543	0.330831	0.165415	+	0.986224i	$0.447104\pi$
$284$	0	0
$285$	8.60979	0.510000
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−7.49935	−0.441138
$290$	0	0
$291$	6.59131	0.386390
$292$	0	0
$293$	−8.38592	−0.489910	−0.244955	−	0.969534i	$-0.578773\pi$
$293$	−8.38592	−0.489910	−0.244955	+	0.969534i	$0.578773\pi$
$294$	0	0
$295$	43.2347	2.51722
$296$	0	0
$297$	−32.5584	−1.88923
$298$	0	0
$299$	−1.69571	−0.0980655
$300$	0	0
$301$	0	0
$302$	0	0
$303$	20.1311	1.15650
$304$	0	0
$305$	−23.5980	−1.35122
$306$	0	0
$307$	25.7897	1.47190	0.735949	−	0.677037i	$-0.236736\pi$
$307$	25.7897	1.47190	0.735949	+	0.677037i	$0.236736\pi$
$308$	0	0
$309$	−3.86689	−0.219980
$310$	0	0
$311$	32.5915	1.84810	0.924049	−	0.382275i	$-0.124859\pi$
$311$	32.5915	1.84810	0.924049	+	0.382275i	$0.124859\pi$
$312$	0	0
$313$	26.5049	1.49815	0.749073	−	0.662488i	$-0.230499\pi$
$313$	26.5049	1.49815	0.749073	+	0.662488i	$0.230499\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.28841	−0.184695	−0.0923477	−	0.995727i	$-0.529437\pi$
$317$	−3.28841	−0.184695	−0.0923477	+	0.995727i	$0.529437\pi$
$318$	0	0
$319$	−7.38833	−0.413667
$320$	0	0
$321$	−11.6253	−0.648862
$322$	0	0
$323$	2.64118	0.146959
$324$	0	0
$325$	10.1414	0.562542
$326$	0	0
$327$	17.4214	0.963407
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.49104	−0.136920	−0.0684601	−	0.997654i	$-0.521809\pi$
$331$	−2.49104	−0.136920	−0.0684601	+	0.997654i	$0.521809\pi$
$332$	0	0
$333$	23.6436	1.29566
$334$	0	0
$335$	−10.6530	−0.582036
$336$	0	0
$337$	29.3210	1.59722	0.798609	−	0.601850i	$-0.205569\pi$
$337$	29.3210	1.59722	0.798609	+	0.601850i	$0.205569\pi$
$338$	0	0
$339$	−61.0801	−3.31742
$340$	0	0
$341$	−24.8731	−1.34695
$342$	0	0
$343$	0	0
$344$	0	0
$345$	10.0478	0.540956
$346$	0	0
$347$	−12.0836	−0.648681	−0.324341	−	0.945940i	$-0.605142\pi$
$347$	−12.0836	−0.648681	−0.324341	+	0.945940i	$0.605142\pi$
$348$	0	0
$349$	8.76755	0.469316	0.234658	−	0.972078i	$-0.424603\pi$
$349$	8.76755	0.469316	0.234658	+	0.972078i	$0.424603\pi$
$350$	0	0
$351$	16.4240	0.876647
$352$	0	0
$353$	−9.49088	−0.505149	−0.252574	−	0.967578i	$-0.581277\pi$
$353$	−9.49088	−0.505149	−0.252574	+	0.967578i	$0.581277\pi$
$354$	0	0
$355$	29.5220	1.56687
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.3556	−0.599325	−0.299663	−	0.954045i	$-0.596874\pi$
$359$	−11.3556	−0.599325	−0.299663	+	0.954045i	$0.596874\pi$
$360$	0	0
$361$	−18.2658	−0.961355
$362$	0	0
$363$	0.909212	0.0477213
$364$	0	0
$365$	−22.5327	−1.17941
$366$	0	0
$367$	10.0297	0.523549	0.261774	−	0.965129i	$-0.415692\pi$
$367$	10.0297	0.523549	0.261774	+	0.965129i	$0.415692\pi$
$368$	0	0
$369$	15.3085	0.796930
$370$	0	0
$371$	0	0
$372$	0	0
$373$	18.7997	0.973410	0.486705	−	0.873566i	$-0.338199\pi$
$373$	18.7997	0.973410	0.486705	+	0.873566i	$0.338199\pi$
$374$	0	0
$375$	−9.85287	−0.508800
$376$	0	0
$377$	3.72702	0.191951
$378$	0	0
$379$	19.9476	1.02464	0.512320	−	0.858795i	$-0.328786\pi$
$379$	19.9476	1.02464	0.512320	+	0.858795i	$0.328786\pi$
$380$	0	0
$381$	−53.7378	−2.75307
$382$	0	0
$383$	−23.1096	−1.18084	−0.590422	−	0.807095i	$-0.701039\pi$
$383$	−23.1096	−1.18084	−0.590422	+	0.807095i	$0.701039\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−50.6014	−2.57222
$388$	0	0
$389$	17.4139	0.882918	0.441459	−	0.897281i	$-0.354461\pi$
$389$	17.4139	0.882918	0.441459	+	0.897281i	$0.354461\pi$
$390$	0	0
$391$	3.08231	0.155879
$392$	0	0
$393$	−12.6306	−0.637129
$394$	0	0
$395$	44.2301	2.22546
$396$	0	0
$397$	9.42740	0.473148	0.236574	−	0.971614i	$-0.423975\pi$
$397$	9.42740	0.473148	0.236574	+	0.971614i	$0.423975\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.2310	−1.50967	−0.754833	−	0.655917i	$-0.772282\pi$
$401$	−30.2310	−1.50967	−0.754833	+	0.655917i	$0.772282\pi$
$402$	0	0
$403$	12.5472	0.625018
$404$	0	0
$405$	−35.7414	−1.77600
$406$	0	0
$407$	−12.8310	−0.636011
$408$	0	0
$409$	−15.6354	−0.773122	−0.386561	−	0.922264i	$-0.626337\pi$
$409$	−15.6354	−0.773122	−0.386561	+	0.922264i	$0.626337\pi$
$410$	0	0
$411$	−32.9276	−1.62420
$412$	0	0
$413$	0	0
$414$	0	0
$415$	34.4622	1.69168
$416$	0	0
$417$	−44.3730	−2.17296
$418$	0	0
$419$	21.6891	1.05958	0.529792	−	0.848128i	$-0.322270\pi$
$419$	21.6891	1.05958	0.529792	+	0.848128i	$0.322270\pi$
$420$	0	0
$421$	−20.9122	−1.01920	−0.509599	−	0.860412i	$-0.670206\pi$
$421$	−20.9122	−1.01920	−0.509599	+	0.860412i	$0.670206\pi$
$422$	0	0
$423$	−56.5669	−2.75038
$424$	0	0
$425$	−18.4341	−0.894184
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−17.2841	−0.834483
$430$	0	0
$431$	−32.9942	−1.58928	−0.794638	−	0.607083i	$-0.792340\pi$
$431$	−32.9942	−1.58928	−0.794638	+	0.607083i	$0.792340\pi$
$432$	0	0
$433$	21.4609	1.03135	0.515673	−	0.856785i	$-0.327542\pi$
$433$	21.4609	1.03135	0.515673	+	0.856785i	$0.327542\pi$
$434$	0	0
$435$	−22.0842	−1.05885
$436$	0	0
$437$	0.856882	0.0409902
$438$	0	0
$439$	−23.4348	−1.11848	−0.559242	−	0.829004i	$-0.688908\pi$
$439$	−23.4348	−1.11848	−0.559242	+	0.829004i	$0.688908\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.2958	1.77198	0.885988	−	0.463707i	$-0.153481\pi$
$443$	37.2958	1.77198	0.885988	+	0.463707i	$0.153481\pi$
$444$	0	0
$445$	16.9742	0.804656
$446$	0	0
$447$	−50.7875	−2.40217
$448$	0	0
$449$	−13.9420	−0.657963	−0.328982	−	0.944336i	$-0.606705\pi$
$449$	−13.9420	−0.657963	−0.328982	+	0.944336i	$0.606705\pi$
$450$	0	0
$451$	−8.30770	−0.391194
$452$	0	0
$453$	66.4248	3.12091
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.3628	−1.32676	−0.663379	−	0.748284i	$-0.730878\pi$
$457$	−28.3628	−1.32676	−0.663379	+	0.748284i	$0.730878\pi$
$458$	0	0
$459$	−29.8540	−1.39347
$460$	0	0
$461$	36.1314	1.68281	0.841404	−	0.540406i	$-0.181729\pi$
$461$	36.1314	1.68281	0.841404	+	0.540406i	$0.181729\pi$
$462$	0	0
$463$	1.14633	0.0532744	0.0266372	−	0.999645i	$-0.491520\pi$
$463$	1.14633	0.0532744	0.0266372	+	0.999645i	$0.491520\pi$
$464$	0	0
$465$	−74.3472	−3.44777
$466$	0	0
$467$	−17.6741	−0.817858	−0.408929	−	0.912566i	$-0.634098\pi$
$467$	−17.6741	−0.817858	−0.408929	+	0.912566i	$0.634098\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	61.1685	2.81850
$472$	0	0
$473$	27.4606	1.26264
$474$	0	0
$475$	−5.12467	−0.235136
$476$	0	0
$477$	−3.20351	−0.146679
$478$	0	0
$479$	39.1021	1.78662	0.893310	−	0.449441i	$-0.148377\pi$
$479$	39.1021	1.78662	0.893310	+	0.449441i	$0.148377\pi$
$480$	0	0
$481$	6.47257	0.295124
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.20322	−0.327081
$486$	0	0
$487$	−25.5233	−1.15657	−0.578285	−	0.815835i	$-0.696278\pi$
$487$	−25.5233	−1.15657	−0.578285	+	0.815835i	$0.696278\pi$
$488$	0	0
$489$	−41.1739	−1.86195
$490$	0	0
$491$	−27.5183	−1.24188	−0.620942	−	0.783857i	$-0.713250\pi$
$491$	−27.5183	−1.24188	−0.620942	+	0.783857i	$0.713250\pi$
$492$	0	0
$493$	−6.77464	−0.305115
$494$	0	0
$495$	68.9982	3.10124
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−21.6867	−0.970831	−0.485415	−	0.874284i	$-0.661332\pi$
$499$	−21.6867	−0.970831	−0.485415	+	0.874284i	$0.661332\pi$
$500$	0	0
$501$	−31.8597	−1.42339
$502$	0	0
$503$	36.1453	1.61164	0.805820	−	0.592161i	$-0.201725\pi$
$503$	36.1453	1.61164	0.805820	+	0.592161i	$0.201725\pi$
$504$	0	0
$505$	−22.0000	−0.978985
$506$	0	0
$507$	−30.6997	−1.36342
$508$	0	0
$509$	21.3661	0.947037	0.473518	−	0.880784i	$-0.342984\pi$
$509$	21.3661	0.947037	0.473518	+	0.880784i	$0.342984\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8.29942	−0.366428
$514$	0	0
$515$	4.22588	0.186214
$516$	0	0
$517$	30.6980	1.35010
$518$	0	0
$519$	−37.6823	−1.65407
$520$	0	0
$521$	−6.94994	−0.304482	−0.152241	−	0.988343i	$-0.548649\pi$
$521$	−6.94994	−0.304482	−0.152241	+	0.988343i	$0.548649\pi$
$522$	0	0
$523$	15.4069	0.673695	0.336847	−	0.941559i	$-0.390639\pi$
$523$	15.4069	0.673695	0.336847	+	0.941559i	$0.390639\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−22.8071	−0.993493
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−80.8179	−3.50720
$532$	0	0
$533$	4.19079	0.181523
$534$	0	0
$535$	12.7046	0.549266
$536$	0	0
$537$	68.8234	2.96995
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.03913	−0.302636	−0.151318	−	0.988485i	$-0.548352\pi$
$541$	−7.03913	−0.302636	−0.151318	+	0.988485i	$0.548352\pi$
$542$	0	0
$543$	−70.7756	−3.03727
$544$	0	0
$545$	−19.0388	−0.815531
$546$	0	0
$547$	−7.62171	−0.325881	−0.162940	−	0.986636i	$-0.552098\pi$
$547$	−7.62171	−0.325881	−0.162940	+	0.986636i	$0.552098\pi$
$548$	0	0
$549$	44.1114	1.88263
$550$	0	0
$551$	−1.88335	−0.0802334
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−38.3527	−1.62798
$556$	0	0
$557$	−19.9960	−0.847259	−0.423630	−	0.905835i	$-0.639244\pi$
$557$	−19.9960	−0.847259	−0.423630	+	0.905835i	$0.639244\pi$
$558$	0	0
$559$	−13.8524	−0.585895
$560$	0	0
$561$	31.4175	1.32645
$562$	0	0
$563$	12.0850	0.509322	0.254661	−	0.967030i	$-0.418036\pi$
$563$	12.0850	0.509322	0.254661	+	0.967030i	$0.418036\pi$
$564$	0	0
$565$	66.7505	2.80822
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.8154	0.621095	0.310548	−	0.950558i	$-0.399488\pi$
$569$	14.8154	0.621095	0.310548	+	0.950558i	$0.399488\pi$
$570$	0	0
$571$	−30.6103	−1.28100	−0.640500	−	0.767958i	$-0.721273\pi$
$571$	−30.6103	−1.28100	−0.640500	+	0.767958i	$0.721273\pi$
$572$	0	0
$573$	25.9799	1.08533
$574$	0	0
$575$	−5.98060	−0.249408
$576$	0	0
$577$	−39.3782	−1.63934	−0.819668	−	0.572839i	$-0.805842\pi$
$577$	−39.3782	−1.63934	−0.819668	+	0.572839i	$0.805842\pi$
$578$	0	0
$579$	−59.9900	−2.49310
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.73850	0.0720013
$584$	0	0
$585$	−34.8059	−1.43905
$586$	0	0
$587$	−20.4546	−0.844253	−0.422126	−	0.906537i	$-0.638716\pi$
$587$	−20.4546	−0.844253	−0.422126	+	0.906537i	$0.638716\pi$
$588$	0	0
$589$	−6.34037	−0.261250
$590$	0	0
$591$	58.4386	2.40384
$592$	0	0
$593$	−19.0538	−0.782446	−0.391223	−	0.920296i	$-0.627948\pi$
$593$	−19.0538	−0.782446	−0.391223	+	0.920296i	$0.627948\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	48.3803	1.98007
$598$	0	0
$599$	8.37261	0.342096	0.171048	−	0.985263i	$-0.445285\pi$
$599$	8.37261	0.342096	0.171048	+	0.985263i	$0.445285\pi$
$600$	0	0
$601$	31.9348	1.30265	0.651325	−	0.758799i	$-0.274214\pi$
$601$	31.9348	1.30265	0.651325	+	0.758799i	$0.274214\pi$
$602$	0	0
$603$	19.9135	0.810940
$604$	0	0
$605$	−0.993620	−0.0403964
$606$	0	0
$607$	−14.3295	−0.581617	−0.290809	−	0.956781i	$-0.593924\pi$
$607$	−14.3295	−0.581617	−0.290809	+	0.956781i	$0.593924\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.4855	−0.626477
$612$	0	0
$613$	10.4701	0.422883	0.211441	−	0.977391i	$-0.432184\pi$
$613$	10.4701	0.422883	0.211441	+	0.977391i	$0.432184\pi$
$614$	0	0
$615$	−24.8322	−1.00133
$616$	0	0
$617$	28.3758	1.14237	0.571183	−	0.820823i	$-0.306485\pi$
$617$	28.3758	1.14237	0.571183	+	0.820823i	$0.306485\pi$
$618$	0	0
$619$	13.0106	0.522938	0.261469	−	0.965212i	$-0.415793\pi$
$619$	13.0106	0.522938	0.261469	+	0.965212i	$0.415793\pi$
$620$	0	0
$621$	−9.68560	−0.388670
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.1354	−0.765417
$626$	0	0
$627$	8.73405	0.348804
$628$	0	0
$629$	−11.7653	−0.469112
$630$	0	0
$631$	−46.6293	−1.85628	−0.928141	−	0.372229i	$-0.878594\pi$
$631$	−46.6293	−1.85628	−0.928141	+	0.372229i	$0.878594\pi$
$632$	0	0
$633$	−74.1209	−2.94604
$634$	0	0
$635$	58.7266	2.33049
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−55.1851	−2.18309
$640$	0	0
$641$	−21.6422	−0.854817	−0.427409	−	0.904059i	$-0.640573\pi$
$641$	−21.6422	−0.854817	−0.427409	+	0.904059i	$0.640573\pi$
$642$	0	0
$643$	46.5704	1.83656	0.918279	−	0.395934i	$-0.129579\pi$
$643$	46.5704	1.83656	0.918279	+	0.395934i	$0.129579\pi$
$644$	0	0
$645$	82.0815	3.23196
$646$	0	0
$647$	−0.624776	−0.0245625	−0.0122812	−	0.999925i	$-0.503909\pi$
$647$	−0.624776	−0.0245625	−0.0122812	+	0.999925i	$0.503909\pi$
$648$	0	0
$649$	43.8587	1.72160
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.3860	1.07170	0.535849	−	0.844314i	$-0.319991\pi$
$653$	27.3860	1.07170	0.535849	+	0.844314i	$0.319991\pi$
$654$	0	0
$655$	13.8032	0.539334
$656$	0	0
$657$	42.1200	1.64326
$658$	0	0
$659$	−3.99773	−0.155729	−0.0778647	−	0.996964i	$-0.524810\pi$
$659$	−3.99773	−0.155729	−0.0778647	+	0.996964i	$0.524810\pi$
$660$	0	0
$661$	32.0617	1.24706	0.623528	−	0.781801i	$-0.285699\pi$
$661$	32.0617	1.24706	0.623528	+	0.781801i	$0.285699\pi$
$662$	0	0
$663$	−15.8484	−0.615502
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.19791	−0.0851034
$668$	0	0
$669$	−62.8504	−2.42994
$670$	0	0
$671$	−23.9386	−0.924140
$672$	0	0
$673$	32.5090	1.25313	0.626564	−	0.779370i	$-0.284461\pi$
$673$	32.5090	1.25313	0.626564	+	0.779370i	$0.284461\pi$
$674$	0	0
$675$	57.9257	2.22956
$676$	0	0
$677$	3.22665	0.124010	0.0620052	−	0.998076i	$-0.480250\pi$
$677$	3.22665	0.124010	0.0620052	+	0.998076i	$0.480250\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	13.9125	0.533128
$682$	0	0
$683$	16.4426	0.629158	0.314579	−	0.949231i	$-0.398137\pi$
$683$	16.4426	0.629158	0.314579	+	0.949231i	$0.398137\pi$
$684$	0	0
$685$	35.9845	1.37490
$686$	0	0
$687$	−68.1507	−2.60011
$688$	0	0
$689$	−0.876980	−0.0334103
$690$	0	0
$691$	15.2338	0.579521	0.289760	−	0.957099i	$-0.406424\pi$
$691$	15.2338	0.579521	0.289760	+	0.957099i	$0.406424\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	48.4924	1.83942
$696$	0	0
$697$	−7.61765	−0.288539
$698$	0	0
$699$	61.1511	2.31295
$700$	0	0
$701$	−40.9400	−1.54628	−0.773142	−	0.634233i	$-0.781316\pi$
$701$	−40.9400	−1.54628	−0.773142	+	0.634233i	$0.781316\pi$
$702$	0	0
$703$	−3.27074	−0.123358
$704$	0	0
$705$	91.7582	3.45581
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−0.143769	−0.00539935	−0.00269967	−	0.999996i	$-0.500859\pi$
$709$	−0.143769	−0.00539935	−0.00269967	+	0.999996i	$0.500859\pi$
$710$	0	0
$711$	−82.6787	−3.10069
$712$	0	0
$713$	−7.39935	−0.277108
$714$	0	0
$715$	18.8887	0.706396
$716$	0	0
$717$	38.5230	1.43867
$718$	0	0
$719$	16.2125	0.604622	0.302311	−	0.953209i	$-0.402242\pi$
$719$	16.2125	0.604622	0.302311	+	0.953209i	$0.402242\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−35.7332	−1.32893
$724$	0	0
$725$	13.1448	0.488186
$726$	0	0
$727$	7.23984	0.268511	0.134255	−	0.990947i	$-0.457136\pi$
$727$	7.23984	0.268511	0.134255	+	0.990947i	$0.457136\pi$
$728$	0	0
$729$	−21.2952	−0.788712
$730$	0	0
$731$	25.1797	0.931306
$732$	0	0
$733$	25.2628	0.933105	0.466552	−	0.884494i	$-0.345496\pi$
$733$	25.2628	0.933105	0.466552	+	0.884494i	$0.345496\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.8068	−0.398072
$738$	0	0
$739$	37.8587	1.39265	0.696327	−	0.717725i	$-0.254816\pi$
$739$	37.8587	1.39265	0.696327	+	0.717725i	$0.254816\pi$
$740$	0	0
$741$	−4.40586	−0.161853
$742$	0	0
$743$	30.5929	1.12234	0.561172	−	0.827699i	$-0.310351\pi$
$743$	30.5929	1.12234	0.561172	+	0.827699i	$0.310351\pi$
$744$	0	0
$745$	55.5024	2.03345
$746$	0	0
$747$	−64.4197	−2.35699
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−43.5943	−1.59078	−0.795389	−	0.606099i	$-0.792734\pi$
$751$	−43.5943	−1.59078	−0.795389	+	0.606099i	$0.792734\pi$
$752$	0	0
$753$	4.19844	0.153000
$754$	0	0
$755$	−72.5914	−2.64187
$756$	0	0
$757$	−0.377943	−0.0137366	−0.00686828	−	0.999976i	$-0.502186\pi$
$757$	−0.377943	−0.0137366	−0.00686828	+	0.999976i	$0.502186\pi$
$758$	0	0
$759$	10.1928	0.369976
$760$	0	0
$761$	−36.8897	−1.33725	−0.668626	−	0.743599i	$-0.733117\pi$
$761$	−36.8897	−1.33725	−0.668626	+	0.743599i	$0.733117\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	63.2671	2.28743
$766$	0	0
$767$	−22.1244	−0.798864
$768$	0	0
$769$	53.9555	1.94568	0.972842	−	0.231469i	$-0.0743533\pi$
$769$	53.9555	1.94568	0.972842	+	0.231469i	$0.0743533\pi$
$770$	0	0
$771$	25.2357	0.908841
$772$	0	0
$773$	−19.3492	−0.695942	−0.347971	−	0.937505i	$-0.613129\pi$
$773$	−19.3492	−0.695942	−0.347971	+	0.937505i	$0.613129\pi$
$774$	0	0
$775$	44.2525	1.58960
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.11771	−0.0758747
$780$	0	0
$781$	29.9481	1.07163
$782$	0	0
$783$	21.2881	0.760774
$784$	0	0
$785$	−66.8471	−2.38588
$786$	0	0
$787$	−42.6957	−1.52194	−0.760969	−	0.648788i	$-0.775276\pi$
$787$	−42.6957	−1.52194	−0.760969	+	0.648788i	$0.775276\pi$
$788$	0	0
$789$	−41.7045	−1.48472
$790$	0	0
$791$	0	0
$792$	0	0
$793$	12.0758	0.428823
$794$	0	0
$795$	5.19648	0.184300
$796$	0	0
$797$	−1.63677	−0.0579775	−0.0289887	−	0.999580i	$-0.509229\pi$
$797$	−1.63677	−0.0579775	−0.0289887	+	0.999580i	$0.509229\pi$
$798$	0	0
$799$	28.1482	0.995811
$800$	0	0
$801$	−31.7297	−1.12111
$802$	0	0
$803$	−22.8579	−0.806637
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32.6420	1.14905
$808$	0	0
$809$	35.7134	1.25562	0.627808	−	0.778369i	$-0.283952\pi$
$809$	35.7134	1.25562	0.627808	+	0.778369i	$0.283952\pi$
$810$	0	0
$811$	15.2326	0.534889	0.267445	−	0.963573i	$-0.413821\pi$
$811$	15.2326	0.534889	0.267445	+	0.963573i	$0.413821\pi$
$812$	0	0
$813$	−15.3205	−0.537313
$814$	0	0
$815$	44.9963	1.57615
$816$	0	0
$817$	6.99996	0.244898
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−51.3054	−1.79057	−0.895286	−	0.445493i	$-0.853029\pi$
$821$	−51.3054	−1.79057	−0.895286	+	0.445493i	$0.853029\pi$
$822$	0	0
$823$	−2.95146	−0.102881	−0.0514407	−	0.998676i	$-0.516381\pi$
$823$	−2.95146	−0.102881	−0.0514407	+	0.998676i	$0.516381\pi$
$824$	0	0
$825$	−60.9592	−2.12233
$826$	0	0
$827$	−12.2083	−0.424526	−0.212263	−	0.977213i	$-0.568083\pi$
$827$	−12.2083	−0.424526	−0.212263	+	0.977213i	$0.568083\pi$
$828$	0	0
$829$	23.2835	0.808670	0.404335	−	0.914611i	$-0.367503\pi$
$829$	23.2835	0.808670	0.404335	+	0.914611i	$0.367503\pi$
$830$	0	0
$831$	39.7240	1.37801
$832$	0	0
$833$	0	0
$834$	0	0
$835$	34.8174	1.20491
$836$	0	0
$837$	71.6671	2.47718
$838$	0	0
$839$	2.71680	0.0937944	0.0468972	−	0.998900i	$-0.485067\pi$
$839$	2.71680	0.0937944	0.0468972	+	0.998900i	$0.485067\pi$
$840$	0	0
$841$	−24.1692	−0.833420
$842$	0	0
$843$	−5.05733	−0.174184
$844$	0	0
$845$	33.5498	1.15415
$846$	0	0
$847$	0	0
$848$	0	0
$849$	16.8755	0.579166
$850$	0	0
$851$	−3.81703	−0.130846
$852$	0	0
$853$	−10.1446	−0.347345	−0.173673	−	0.984803i	$-0.555563\pi$
$853$	−10.1446	−0.347345	−0.173673	+	0.984803i	$0.555563\pi$
$854$	0	0
$855$	17.5883	0.601506
$856$	0	0
$857$	14.4948	0.495133	0.247567	−	0.968871i	$-0.420369\pi$
$857$	14.4948	0.495133	0.247567	+	0.968871i	$0.420369\pi$
$858$	0	0
$859$	−12.8908	−0.439827	−0.219913	−	0.975519i	$-0.570577\pi$
$859$	−12.8908	−0.439827	−0.219913	+	0.975519i	$0.570577\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.53916	0.188555	0.0942777	−	0.995546i	$-0.469946\pi$
$863$	5.53916	0.188555	0.0942777	+	0.995546i	$0.469946\pi$
$864$	0	0
$865$	41.1806	1.40018
$866$	0	0
$867$	−22.7396	−0.772276
$868$	0	0
$869$	44.8685	1.52206
$870$	0	0
$871$	5.45143	0.184715
$872$	0	0
$873$	13.4649	0.455717
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5.37921	−0.181643	−0.0908214	−	0.995867i	$-0.528949\pi$
$877$	−5.37921	−0.181643	−0.0908214	+	0.995867i	$0.528949\pi$
$878$	0	0
$879$	−25.4278	−0.857658
$880$	0	0
$881$	27.7496	0.934908	0.467454	−	0.884017i	$-0.345171\pi$
$881$	27.7496	0.934908	0.467454	+	0.884017i	$0.345171\pi$
$882$	0	0
$883$	−12.7300	−0.428400	−0.214200	−	0.976790i	$-0.568714\pi$
$883$	−12.7300	−0.428400	−0.214200	+	0.976790i	$0.568714\pi$
$884$	0	0
$885$	131.096	4.40675
$886$	0	0
$887$	24.1984	0.812502	0.406251	−	0.913761i	$-0.366836\pi$
$887$	24.1984	0.812502	0.406251	+	0.913761i	$0.366836\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−36.2573	−1.21466
$892$	0	0
$893$	7.82519	0.261860
$894$	0	0
$895$	−75.2127	−2.51408
$896$	0	0
$897$	−5.14174	−0.171678
$898$	0	0
$899$	16.2631	0.542405
$900$	0	0
$901$	1.59410	0.0531071
$902$	0	0
$903$	0	0
$904$	0	0
$905$	77.3462	2.57107
$906$	0	0
$907$	44.4539	1.47607	0.738034	−	0.674764i	$-0.235754\pi$
$907$	44.4539	1.47607	0.738034	+	0.674764i	$0.235754\pi$
$908$	0	0
$909$	41.1242	1.36400
$910$	0	0
$911$	−11.9597	−0.396244	−0.198122	−	0.980177i	$-0.563484\pi$
$911$	−11.9597	−0.396244	−0.198122	+	0.980177i	$0.563484\pi$
$912$	0	0
$913$	34.9596	1.15699
$914$	0	0
$915$	−71.5540	−2.36550
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−2.25216	−0.0742920	−0.0371460	−	0.999310i	$-0.511827\pi$
$919$	−2.25216	−0.0742920	−0.0371460	+	0.999310i	$0.511827\pi$
$920$	0	0
$921$	78.1997	2.57677
$922$	0	0
$923$	−15.1072	−0.497261
$924$	0	0
$925$	22.8281	0.750584
$926$	0	0
$927$	−7.89937	−0.259449
$928$	0	0
$929$	−27.8606	−0.914077	−0.457038	−	0.889447i	$-0.651090\pi$
$929$	−27.8606	−0.914077	−0.457038	+	0.889447i	$0.651090\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	98.8241	3.23536
$934$	0	0
$935$	−34.3341	−1.12285
$936$	0	0
$937$	−33.0116	−1.07844	−0.539221	−	0.842164i	$-0.681281\pi$
$937$	−33.0116	−1.07844	−0.539221	+	0.842164i	$0.681281\pi$
$938$	0	0
$939$	80.3682	2.62272
$940$	0	0
$941$	11.1957	0.364968	0.182484	−	0.983209i	$-0.441586\pi$
$941$	11.1957	0.364968	0.182484	+	0.983209i	$0.441586\pi$
$942$	0	0
$943$	−2.47141	−0.0804802
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.1039	1.20572	0.602858	−	0.797849i	$-0.294029\pi$
$947$	37.1039	1.20572	0.602858	+	0.797849i	$0.294029\pi$
$948$	0	0
$949$	11.5306	0.374299
$950$	0	0
$951$	−9.97112	−0.323336
$952$	0	0
$953$	−28.1582	−0.912134	−0.456067	−	0.889945i	$-0.650742\pi$
$953$	−28.1582	−0.912134	−0.456067	+	0.889945i	$0.650742\pi$
$954$	0	0
$955$	−28.3918	−0.918736
$956$	0	0
$957$	−22.4029	−0.724183
$958$	0	0
$959$	0	0
$960$	0	0
$961$	23.7503	0.766140
$962$	0	0
$963$	−23.7484	−0.765283
$964$	0	0
$965$	65.5593	2.11043
$966$	0	0
$967$	45.6189	1.46700	0.733502	−	0.679687i	$-0.237885\pi$
$967$	45.6189	1.46700	0.733502	+	0.679687i	$0.237885\pi$
$968$	0	0
$969$	8.00859	0.257273
$970$	0	0
$971$	42.5541	1.36562	0.682812	−	0.730594i	$-0.260757\pi$
$971$	42.5541	1.36562	0.682812	+	0.730594i	$0.260757\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	30.7507	0.984810
$976$	0	0
$977$	25.5760	0.818250	0.409125	−	0.912478i	$-0.365834\pi$
$977$	25.5760	0.818250	0.409125	+	0.912478i	$0.365834\pi$
$978$	0	0
$979$	17.2192	0.550329
$980$	0	0
$981$	35.5889	1.13627
$982$	0	0
$983$	−49.7589	−1.58706	−0.793532	−	0.608529i	$-0.791760\pi$
$983$	−49.7589	−1.58706	−0.793532	+	0.608529i	$0.791760\pi$
$984$	0	0
$985$	−63.8638	−2.03487
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.16910	0.259762
$990$	0	0
$991$	−29.6447	−0.941693	−0.470847	−	0.882215i	$-0.656052\pi$
$991$	−29.6447	−0.941693	−0.470847	+	0.882215i	$0.656052\pi$
$992$	0	0
$993$	−7.55334	−0.239698
$994$	0	0
$995$	−52.8717	−1.67615
$996$	0	0
$997$	5.11076	0.161859	0.0809297	−	0.996720i	$-0.474211\pi$
$997$	5.11076	0.161859	0.0809297	+	0.996720i	$0.474211\pi$
$998$	0	0
$999$	36.9702	1.16969

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.bs.1.14	yes	14
7.6	odd	2	inner	9016.2.a.bs.1.1	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9016.2.a.bs.1.1	✓	14	7.6	odd	2	inner
9016.2.a.bs.1.14	yes	14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.03220 q^{3} -3.31370 q^{5} +6.19425 q^{9} +O(q^{10})\)	\(q+3.03220 q^{3} -3.31370 q^{5} +6.19425 q^{9} -3.36153 q^{11} +1.69571 q^{13} -10.0478 q^{15} -3.08231 q^{17} -0.856882 q^{19} -1.00000 q^{23} +5.98060 q^{25} +9.68560 q^{27} +2.19791 q^{29} +7.39935 q^{31} -10.1928 q^{33} +3.81703 q^{37} +5.14174 q^{39} +2.47141 q^{41} -8.16910 q^{43} -20.5259 q^{45} -9.13217 q^{47} -9.34619 q^{51} -0.517176 q^{53} +11.1391 q^{55} -2.59824 q^{57} -13.0473 q^{59} +7.12135 q^{61} -5.61907 q^{65} +3.21484 q^{67} -3.03220 q^{69} -8.90909 q^{71} +6.79985 q^{73} +18.1344 q^{75} -13.3477 q^{79} +10.7860 q^{81} -10.3999 q^{83} +10.2139 q^{85} +6.66451 q^{87} -5.12244 q^{89} +22.4363 q^{93} +2.83945 q^{95} +2.17377 q^{97} -20.8221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{9}+O(q^{10}) \) 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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