LMFDB - Embedded newform 2016.3.g.b.1135.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,3,Mod(1135,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2016.1135");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$54.9320212950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.292213762624.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 $ x^8 - x^7 - 2x^6 - 2x^5 + 24x^4 - 8x^3 - 32x^2 - 64x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1135.5
Root			$1.85837 + 0.739226i$ of defining polynomial
Character	$\chi$	$=$	2016.1135
Dual form			2016.3.g.b.1135.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46547i q^{5} -2.64575i q^{7} +O(q^{10})$	$q+3.46547i q^{5} -2.64575i q^{7} -2.92866 q^{11} -19.1586i q^{13} +14.3897 q^{17} -8.09744 q^{19} +16.7598i q^{23} +12.9905 q^{25} +27.1649i q^{29} -44.8923i q^{31} +9.16878 q^{35} +39.5687i q^{37} -45.8766 q^{41} -61.0334 q^{43} -46.2793i q^{47} -7.00000 q^{49} -9.69424i q^{53} -10.1492i q^{55} -114.554 q^{59} -7.48032i q^{61} +66.3935 q^{65} +12.0590 q^{67} -129.187i q^{71} -18.2854 q^{73} +7.74851i q^{77} +42.6168i q^{79} -109.670 q^{83} +49.8673i q^{85} +80.9162 q^{89} -50.6889 q^{91} -28.0614i q^{95} +162.086 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 32 q^{11} + 80 q^{17} - 56 q^{19} - 16 q^{25} + 56 q^{35} - 128 q^{41} - 56 q^{49} + 104 q^{59} + 72 q^{65} - 304 q^{67} - 112 q^{73} + 72 q^{83} + 512 q^{89} + 56 q^{91} + 64 q^{97}+O(q^{100}) $ 8 * q - 32 * q^11 + 80 * q^17 - 56 * q^19 - 16 * q^25 + 56 * q^35 - 128 * q^41 - 56 * q^49 + 104 * q^59 + 72 * q^65 - 304 * q^67 - 112 * q^73 + 72 * q^83 + 512 * q^89 + 56 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.46547i	0.693094i	0.938033	+	0.346547i	$0.112646\pi$
$5$	3.46547i	0.693094i	−0.938033	+	0.346547i	$0.887354\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.92866	−0.266242	−0.133121	−	0.991100i	$-0.542500\pi$
$11$	−2.92866	−0.266242	−0.133121	+	0.991100i	$0.542500\pi$
$12$	0	0
$13$	− 19.1586i	− 1.47374i	−0.676036	−	0.736869i	$-0.736304\pi$
$13$	− 19.1586i	− 1.47374i	0.676036	−	0.736869i	$-0.263696\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.3897	0.846456	0.423228	−	0.906023i	$-0.360897\pi$
$17$	14.3897	0.846456	0.423228	+	0.906023i	$0.360897\pi$
$18$	0	0
$19$	−8.09744	−0.426181	−0.213090	−	0.977032i	$-0.568353\pi$
$19$	−8.09744	−0.426181	−0.213090	+	0.977032i	$0.568353\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	16.7598i	0.728687i	0.931265	+	0.364344i	$0.118707\pi$
$23$	16.7598i	0.728687i	−0.931265	+	0.364344i	$0.881293\pi$
$24$	0	0
$25$	12.9905	0.519620
$26$	0	0
$27$	0	0
$28$	0	0
$29$	27.1649i	0.936720i	0.883538	+	0.468360i	$0.155155\pi$
$29$	27.1649i	0.936720i	−0.883538	+	0.468360i	$0.844845\pi$
$30$	0	0
$31$	− 44.8923i	− 1.44814i	−0.689728	−	0.724069i	$-0.742270\pi$
$31$	− 44.8923i	− 1.44814i	0.689728	−	0.724069i	$-0.257730\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.16878	0.261965
$36$	0	0
$37$	39.5687i	1.06943i	0.845034	+	0.534713i	$0.179580\pi$
$37$	39.5687i	1.06943i	−0.845034	+	0.534713i	$0.820420\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−45.8766	−1.11894	−0.559471	−	0.828850i	$-0.688996\pi$
$41$	−45.8766	−1.11894	−0.559471	+	0.828850i	$0.688996\pi$
$42$	0	0
$43$	−61.0334	−1.41938	−0.709690	−	0.704514i	$-0.751165\pi$
$43$	−61.0334	−1.41938	−0.709690	+	0.704514i	$0.751165\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 46.2793i	− 0.984666i	−0.870407	−	0.492333i	$-0.836144\pi$
$47$	− 46.2793i	− 0.984666i	0.870407	−	0.492333i	$-0.163856\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.69424i	− 0.182910i	−0.995809	−	0.0914551i	$-0.970848\pi$
$53$	− 9.69424i	− 0.182910i	0.995809	−	0.0914551i	$-0.0291518\pi$
$54$	0	0
$55$	− 10.1492i	− 0.184531i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−114.554	−1.94159	−0.970796	−	0.239907i	$-0.922883\pi$
$59$	−114.554	−1.94159	−0.970796	+	0.239907i	$0.922883\pi$
$60$	0	0
$61$	− 7.48032i	− 0.122628i	−0.998119	−	0.0613141i	$-0.980471\pi$
$61$	− 7.48032i	− 0.122628i	0.998119	−	0.0613141i	$-0.0195291\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	66.3935	1.02144
$66$	0	0
$67$	12.0590	0.179985	0.0899925	−	0.995942i	$-0.471316\pi$
$67$	12.0590	0.179985	0.0899925	+	0.995942i	$0.471316\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 129.187i	− 1.81953i	−0.415124	−	0.909765i	$-0.636262\pi$
$71$	− 129.187i	− 1.81953i	0.415124	−	0.909765i	$-0.363738\pi$
$72$	0	0
$73$	−18.2854	−0.250484	−0.125242	−	0.992126i	$-0.539971\pi$
$73$	−18.2854	−0.250484	−0.125242	+	0.992126i	$0.539971\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.74851i	0.100630i
$78$	0	0
$79$	42.6168i	0.539454i	0.962937	+	0.269727i	$0.0869334\pi$
$79$	42.6168i	0.539454i	−0.962937	+	0.269727i	$0.913067\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−109.670	−1.32133	−0.660663	−	0.750683i	$-0.729725\pi$
$83$	−109.670	−1.32133	−0.660663	+	0.750683i	$0.729725\pi$
$84$	0	0
$85$	49.8673i	0.586674i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	80.9162	0.909170	0.454585	−	0.890703i	$-0.349788\pi$
$89$	80.9162	0.909170	0.454585	+	0.890703i	$0.349788\pi$
$90$	0	0
$91$	−50.6889	−0.557020
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 28.0614i	− 0.295384i
$96$	0	0
$97$	162.086	1.67099	0.835495	−	0.549498i	$-0.185181\pi$
$97$	162.086	1.67099	0.835495	+	0.549498i	$0.185181\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 106.827i	− 1.05769i	−0.848717	−	0.528847i	$-0.822625\pi$
$101$	− 106.827i	− 1.05769i	0.848717	−	0.528847i	$-0.177375\pi$
$102$	0	0
$103$	− 126.626i	− 1.22938i	−0.788768	−	0.614691i	$-0.789281\pi$
$103$	− 126.626i	− 1.22938i	0.788768	−	0.614691i	$-0.210719\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	87.0191	0.813263	0.406632	−	0.913592i	$-0.366703\pi$
$107$	87.0191	0.813263	0.406632	+	0.913592i	$0.366703\pi$
$108$	0	0
$109$	− 189.921i	− 1.74240i	−0.490930	−	0.871199i	$-0.663343\pi$
$109$	− 189.921i	− 1.74240i	0.490930	−	0.871199i	$-0.336657\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	40.1848	0.355617	0.177809	−	0.984065i	$-0.443099\pi$
$113$	40.1848	0.355617	0.177809	+	0.984065i	$0.443099\pi$
$114$	0	0
$115$	−58.0806	−0.505049
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 38.0717i	− 0.319930i
$120$	0	0
$121$	−112.423	−0.929115
$122$	0	0
$123$	0	0
$124$	0	0
$125$	131.655i	1.05324i
$126$	0	0
$127$	− 153.657i	− 1.20989i	−0.796266	−	0.604947i	$-0.793194\pi$
$127$	− 153.657i	− 1.20989i	0.796266	−	0.604947i	$-0.206806\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	61.8649	0.472251	0.236126	−	0.971723i	$-0.424122\pi$
$131$	61.8649	0.472251	0.236126	+	0.971723i	$0.424122\pi$
$132$	0	0
$133$	21.4238i	0.161081i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−105.943	−0.773307	−0.386653	−	0.922225i	$-0.626369\pi$
$137$	−105.943	−0.773307	−0.386653	+	0.922225i	$0.626369\pi$
$138$	0	0
$139$	185.384	1.33370	0.666848	−	0.745194i	$-0.267643\pi$
$139$	185.384	1.33370	0.666848	+	0.745194i	$0.267643\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	56.1090i	0.392371i
$144$	0	0
$145$	−94.1392	−0.649236
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 47.4096i	− 0.318185i	−0.987264	−	0.159093i	$-0.949143\pi$
$149$	− 47.4096i	− 0.318185i	0.987264	−	0.159093i	$-0.0508568\pi$
$150$	0	0
$151$	− 114.576i	− 0.758780i	−0.925237	−	0.379390i	$-0.876134\pi$
$151$	− 114.576i	− 0.758780i	0.925237	−	0.379390i	$-0.123866\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	155.573	1.00370
$156$	0	0
$157$	− 294.095i	− 1.87322i	−0.350378	−	0.936608i	$-0.613947\pi$
$157$	− 294.095i	− 1.87322i	0.350378	−	0.936608i	$-0.386053\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	44.3423	0.275418
$162$	0	0
$163$	−171.021	−1.04921	−0.524603	−	0.851347i	$-0.675786\pi$
$163$	−171.021	−1.04921	−0.524603	+	0.851347i	$0.675786\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	120.657i	0.722499i	0.932469	+	0.361249i	$0.117650\pi$
$167$	120.657i	0.722499i	−0.932469	+	0.361249i	$0.882350\pi$
$168$	0	0
$169$	−198.052	−1.17190
$170$	0	0
$171$	0	0
$172$	0	0
$173$	108.339i	0.626236i	0.949714	+	0.313118i	$0.101374\pi$
$173$	108.339i	0.626236i	−0.949714	+	0.313118i	$0.898626\pi$
$174$	0	0
$175$	− 34.3696i	− 0.196398i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−161.438	−0.901886	−0.450943	−	0.892553i	$-0.648912\pi$
$179$	−161.438	−0.901886	−0.450943	+	0.892553i	$0.648912\pi$
$180$	0	0
$181$	− 7.14696i	− 0.0394860i	−0.999805	−	0.0197430i	$-0.993715\pi$
$181$	− 7.14696i	− 0.0394860i	0.999805	−	0.0197430i	$-0.00628479\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−137.124	−0.741212
$186$	0	0
$187$	−42.1427	−0.225362
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 73.2983i	− 0.383761i	−0.981418	−	0.191880i	$-0.938541\pi$
$191$	− 73.2983i	− 0.383761i	0.981418	−	0.191880i	$-0.0614586\pi$
$192$	0	0
$193$	−85.0705	−0.440780	−0.220390	−	0.975412i	$-0.570733\pi$
$193$	−85.0705	−0.440780	−0.220390	+	0.975412i	$0.570733\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 140.460i	− 0.712996i	−0.934296	−	0.356498i	$-0.883971\pi$
$197$	− 140.460i	− 0.712996i	0.934296	−	0.356498i	$-0.116029\pi$
$198$	0	0
$199$	143.082i	0.719006i	0.933144	+	0.359503i	$0.117054\pi$
$199$	143.082i	0.719006i	−0.933144	+	0.359503i	$0.882946\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	71.8715	0.354047
$204$	0	0
$205$	− 158.984i	− 0.775532i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	23.7146	0.113467
$210$	0	0
$211$	111.955	0.530591	0.265295	−	0.964167i	$-0.414531\pi$
$211$	111.955	0.530591	0.265295	+	0.964167i	$0.414531\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 211.509i	− 0.983765i
$216$	0	0
$217$	−118.774	−0.547345
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 275.687i	− 1.24745i
$222$	0	0
$223$	− 311.438i	− 1.39658i	−0.715814	−	0.698291i	$-0.753944\pi$
$223$	− 311.438i	− 1.39658i	0.715814	−	0.698291i	$-0.246056\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	74.3581	0.327569	0.163784	−	0.986496i	$-0.447630\pi$
$227$	74.3581	0.327569	0.163784	+	0.986496i	$0.447630\pi$
$228$	0	0
$229$	− 78.2710i	− 0.341795i	−0.985289	−	0.170897i	$-0.945333\pi$
$229$	− 78.2710i	− 0.341795i	0.985289	−	0.170897i	$-0.0546667\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−93.0573	−0.399388	−0.199694	−	0.979858i	$-0.563995\pi$
$233$	−93.0573	−0.399388	−0.199694	+	0.979858i	$0.563995\pi$
$234$	0	0
$235$	160.380	0.682466
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 291.605i	− 1.22011i	−0.792361	−	0.610053i	$-0.791148\pi$
$239$	− 291.605i	− 1.22011i	0.792361	−	0.610053i	$-0.208852\pi$
$240$	0	0
$241$	223.748	0.928413	0.464207	−	0.885727i	$-0.346340\pi$
$241$	223.748	0.928413	0.464207	+	0.885727i	$0.346340\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 24.2583i	− 0.0990135i
$246$	0	0
$247$	155.135i	0.628079i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−310.605	−1.23747	−0.618734	−	0.785600i	$-0.712354\pi$
$251$	−310.605	−1.23747	−0.618734	+	0.785600i	$0.712354\pi$
$252$	0	0
$253$	− 49.0838i	− 0.194007i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−175.472	−0.682769	−0.341385	−	0.939924i	$-0.610896\pi$
$257$	−175.472	−0.682769	−0.341385	+	0.939924i	$0.610896\pi$
$258$	0	0
$259$	104.689	0.404205
$260$	0	0
$261$	0	0
$262$	0	0
$263$	312.127i	1.18680i	0.804910	+	0.593398i	$0.202214\pi$
$263$	312.127i	1.18680i	−0.804910	+	0.593398i	$0.797786\pi$
$264$	0	0
$265$	33.5951	0.126774
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 142.817i	− 0.530918i	−0.964122	−	0.265459i	$-0.914477\pi$
$269$	− 142.817i	− 0.530918i	0.964122	−	0.265459i	$-0.0855234\pi$
$270$	0	0
$271$	266.117i	0.981982i	0.871165	+	0.490991i	$0.163365\pi$
$271$	266.117i	0.981982i	−0.871165	+	0.490991i	$0.836635\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−38.0448	−0.138345
$276$	0	0
$277$	366.740i	1.32397i	0.749516	+	0.661986i	$0.230286\pi$
$277$	366.740i	1.32397i	−0.749516	+	0.661986i	$0.769714\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−147.977	−0.526607	−0.263303	−	0.964713i	$-0.584812\pi$
$281$	−147.977	−0.526607	−0.263303	+	0.964713i	$0.584812\pi$
$282$	0	0
$283$	−327.739	−1.15809	−0.579043	−	0.815297i	$-0.696574\pi$
$283$	−327.739	−1.15809	−0.579043	+	0.815297i	$0.696574\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	121.378i	0.422920i
$288$	0	0
$289$	−81.9352	−0.283513
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 259.881i	− 0.886966i	−0.896283	−	0.443483i	$-0.853743\pi$
$293$	− 259.881i	− 0.886966i	0.896283	−	0.443483i	$-0.146257\pi$
$294$	0	0
$295$	− 396.983i	− 1.34571i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	321.094	1.07389
$300$	0	0
$301$	161.479i	0.536475i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	25.9228	0.0849929
$306$	0	0
$307$	−290.462	−0.946131	−0.473065	−	0.881027i	$-0.656853\pi$
$307$	−290.462	−0.946131	−0.473065	+	0.881027i	$0.656853\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 74.9081i	− 0.240862i	−0.992722	−	0.120431i	$-0.961572\pi$
$311$	− 74.9081i	− 0.240862i	0.992722	−	0.120431i	$-0.0384276\pi$
$312$	0	0
$313$	284.507	0.908969	0.454485	−	0.890755i	$-0.349823\pi$
$313$	284.507	0.908969	0.454485	+	0.890755i	$0.349823\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.2631i	− 0.0386850i	−0.999813	−	0.0193425i	$-0.993843\pi$
$317$	− 12.2631i	− 0.0386850i	0.999813	−	0.0193425i	$-0.00615729\pi$
$318$	0	0
$319$	− 79.5567i	− 0.249394i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−116.520	−0.360743
$324$	0	0
$325$	− 248.880i	− 0.765784i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−122.443	−0.372169
$330$	0	0
$331$	−194.466	−0.587510	−0.293755	−	0.955881i	$-0.594905\pi$
$331$	−194.466	−0.587510	−0.293755	+	0.955881i	$0.594905\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	41.7901i	0.124747i
$336$	0	0
$337$	0.596077	0.00176877	0.000884387	−	1.00000i	$-0.499718\pi$
$337$	0.596077	0.00176877	0.000884387		1.00000i	$0.499718\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	131.474i	0.385555i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−204.867	−0.590394	−0.295197	−	0.955436i	$-0.595385\pi$
$347$	−204.867	−0.590394	−0.295197	+	0.955436i	$0.595385\pi$
$348$	0	0
$349$	− 128.396i	− 0.367898i	−0.982936	−	0.183949i	$-0.941112\pi$
$349$	− 128.396i	− 0.367898i	0.982936	−	0.183949i	$-0.0588882\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	190.841	0.540627	0.270314	−	0.962772i	$-0.412873\pi$
$353$	190.841	0.540627	0.270314	+	0.962772i	$0.412873\pi$
$354$	0	0
$355$	447.692	1.26111
$356$	0	0
$357$	0	0
$358$	0	0
$359$	215.704i	0.600847i	0.953806	+	0.300424i	$0.0971281\pi$
$359$	215.704i	0.600847i	−0.953806	+	0.300424i	$0.902872\pi$
$360$	0	0
$361$	−295.432	−0.818370
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 63.3674i	− 0.173609i
$366$	0	0
$367$	454.789i	1.23921i	0.784915	+	0.619604i	$0.212707\pi$
$367$	454.789i	1.23921i	−0.784915	+	0.619604i	$0.787293\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−25.6485	−0.0691335
$372$	0	0
$373$	− 360.748i	− 0.967153i	−0.875302	−	0.483576i	$-0.839338\pi$
$373$	− 360.748i	− 0.967153i	0.875302	−	0.483576i	$-0.160662\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	520.441	1.38048
$378$	0	0
$379$	268.427	0.708250	0.354125	−	0.935198i	$-0.384779\pi$
$379$	268.427	0.708250	0.354125	+	0.935198i	$0.384779\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	581.532i	1.51836i	0.650881	+	0.759180i	$0.274400\pi$
$383$	581.532i	1.51836i	−0.650881	+	0.759180i	$0.725600\pi$
$384$	0	0
$385$	−26.8522	−0.0697461
$386$	0	0
$387$	0	0
$388$	0	0
$389$	512.278i	1.31691i	0.752620	+	0.658455i	$0.228790\pi$
$389$	512.278i	1.31691i	−0.752620	+	0.658455i	$0.771210\pi$
$390$	0	0
$391$	241.169i	0.616801i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−147.687	−0.373892
$396$	0	0
$397$	− 81.3250i	− 0.204849i	−0.994741	−	0.102424i	$-0.967340\pi$
$397$	− 81.3250i	− 0.204849i	0.994741	−	0.102424i	$-0.0326600\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−527.441	−1.31531	−0.657657	−	0.753318i	$-0.728452\pi$
$401$	−527.441	−1.31531	−0.657657	+	0.753318i	$0.728452\pi$
$402$	0	0
$403$	−860.073	−2.13418
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 115.883i	− 0.284726i
$408$	0	0
$409$	−58.6727	−0.143454	−0.0717270	−	0.997424i	$-0.522851\pi$
$409$	−58.6727	−0.143454	−0.0717270	+	0.997424i	$0.522851\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	303.081i	0.733853i
$414$	0	0
$415$	− 380.058i	− 0.915803i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	760.704	1.81552	0.907761	−	0.419487i	$-0.137790\pi$
$419$	760.704	1.81552	0.907761	+	0.419487i	$0.137790\pi$
$420$	0	0
$421$	− 46.2918i	− 0.109957i	−0.998488	−	0.0549784i	$-0.982491\pi$
$421$	− 46.2918i	− 0.109957i	0.998488	−	0.0549784i	$-0.0175090\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	186.930	0.439835
$426$	0	0
$427$	−19.7911	−0.0463491
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 336.176i	− 0.779991i	−0.920817	−	0.389996i	$-0.872477\pi$
$431$	− 336.176i	− 0.779991i	0.920817	−	0.389996i	$-0.127523\pi$
$432$	0	0
$433$	−372.694	−0.860725	−0.430363	−	0.902656i	$-0.641614\pi$
$433$	−372.694	−0.860725	−0.430363	+	0.902656i	$0.641614\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 135.711i	− 0.310553i
$438$	0	0
$439$	− 397.478i	− 0.905418i	−0.891658	−	0.452709i	$-0.850458\pi$
$439$	− 397.478i	− 0.905418i	0.891658	−	0.452709i	$-0.149542\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	273.530	0.617450	0.308725	−	0.951151i	$-0.400098\pi$
$443$	273.530	0.617450	0.308725	+	0.951151i	$0.400098\pi$
$444$	0	0
$445$	280.413i	0.630141i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−428.702	−0.954792	−0.477396	−	0.878688i	$-0.658419\pi$
$449$	−428.702	−0.954792	−0.477396	+	0.878688i	$0.658419\pi$
$450$	0	0
$451$	134.357	0.297909
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 175.661i	− 0.386068i
$456$	0	0
$457$	10.0500	0.0219913	0.0109956	−	0.999940i	$-0.496500\pi$
$457$	10.0500	0.0219913	0.0109956	+	0.999940i	$0.496500\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 825.802i	− 1.79133i	−0.444732	−	0.895664i	$-0.646701\pi$
$461$	− 825.802i	− 1.79133i	0.444732	−	0.895664i	$-0.353299\pi$
$462$	0	0
$463$	− 114.707i	− 0.247748i	−0.992298	−	0.123874i	$-0.960468\pi$
$463$	− 114.707i	− 0.247748i	0.992298	−	0.123874i	$-0.0395318\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−201.727	−0.431964	−0.215982	−	0.976397i	$-0.569295\pi$
$467$	−201.727	−0.431964	−0.215982	+	0.976397i	$0.569295\pi$
$468$	0	0
$469$	− 31.9051i	− 0.0680280i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	178.746	0.377899
$474$	0	0
$475$	−105.190	−0.221452
$476$	0	0
$477$	0	0
$478$	0	0
$479$	597.538i	1.24747i	0.781636	+	0.623735i	$0.214385\pi$
$479$	597.538i	1.24747i	−0.781636	+	0.623735i	$0.785615\pi$
$480$	0	0
$481$	758.081	1.57605
$482$	0	0
$483$	0	0
$484$	0	0
$485$	561.705i	1.15815i
$486$	0	0
$487$	345.125i	0.708675i	0.935118	+	0.354337i	$0.115294\pi$
$487$	345.125i	0.708675i	−0.935118	+	0.354337i	$0.884706\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	373.498	0.760689	0.380344	−	0.924845i	$-0.375805\pi$
$491$	373.498	0.760689	0.380344	+	0.924845i	$0.375805\pi$
$492$	0	0
$493$	390.896i	0.792892i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−341.796	−0.687718
$498$	0	0
$499$	850.317	1.70404	0.852021	−	0.523508i	$-0.175377\pi$
$499$	850.317	1.70404	0.852021	+	0.523508i	$0.175377\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	459.256i	0.913033i	0.889715	+	0.456517i	$0.150903\pi$
$503$	459.256i	0.913033i	−0.889715	+	0.456517i	$0.849097\pi$
$504$	0	0
$505$	370.206	0.733082
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 673.910i	− 1.32399i	−0.749509	−	0.661994i	$-0.769710\pi$
$509$	− 673.910i	− 1.32399i	0.749509	−	0.661994i	$-0.230290\pi$
$510$	0	0
$511$	48.3785i	0.0946742i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	438.820	0.852078
$516$	0	0
$517$	135.536i	0.262159i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	486.473	0.933729	0.466864	−	0.884329i	$-0.345384\pi$
$521$	486.473	0.933729	0.466864	+	0.884329i	$0.345384\pi$
$522$	0	0
$523$	680.087	1.30036	0.650179	−	0.759781i	$-0.274694\pi$
$523$	680.087	1.30036	0.650179	+	0.759781i	$0.274694\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 645.988i	− 1.22578i
$528$	0	0
$529$	248.109	0.469015
$530$	0	0
$531$	0	0
$532$	0	0
$533$	878.931i	1.64903i
$534$	0	0
$535$	301.562i	0.563668i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.5006	0.0380346
$540$	0	0
$541$	794.999i	1.46950i	0.678339	+	0.734749i	$0.262700\pi$
$541$	794.999i	1.46950i	−0.678339	+	0.734749i	$0.737300\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	658.167	1.20765
$546$	0	0
$547$	736.752	1.34690	0.673448	−	0.739235i	$-0.264813\pi$
$547$	736.752	1.34690	0.673448	+	0.739235i	$0.264813\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 219.966i	− 0.399212i
$552$	0	0
$553$	112.754	0.203894
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 415.758i	− 0.746423i	−0.927746	−	0.373212i	$-0.878257\pi$
$557$	− 415.758i	− 0.746423i	0.927746	−	0.373212i	$-0.121743\pi$
$558$	0	0
$559$	1169.31i	2.09179i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−96.4996	−0.171402	−0.0857012	−	0.996321i	$-0.527313\pi$
$563$	−96.4996	−0.171402	−0.0857012	+	0.996321i	$0.527313\pi$
$564$	0	0
$565$	139.259i	0.246476i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−347.953	−0.611517	−0.305759	−	0.952109i	$-0.598910\pi$
$569$	−347.953	−0.611517	−0.305759	+	0.952109i	$0.598910\pi$
$570$	0	0
$571$	13.7251	0.0240370	0.0120185	−	0.999928i	$-0.496174\pi$
$571$	13.7251	0.0240370	0.0120185	+	0.999928i	$0.496174\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	217.718i	0.378641i
$576$	0	0
$577$	−827.320	−1.43383	−0.716915	−	0.697161i	$-0.754446\pi$
$577$	−827.320	−1.43383	−0.716915	+	0.697161i	$0.754446\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	290.160i	0.499414i
$582$	0	0
$583$	28.3911i	0.0486983i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−675.987	−1.15160	−0.575798	−	0.817592i	$-0.695308\pi$
$587$	−675.987	−1.15160	−0.575798	+	0.817592i	$0.695308\pi$
$588$	0	0
$589$	363.512i	0.617169i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	414.116	0.698341	0.349171	−	0.937059i	$-0.386463\pi$
$593$	414.116	0.698341	0.349171	+	0.937059i	$0.386463\pi$
$594$	0	0
$595$	131.936	0.221742
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 723.303i	− 1.20752i	−0.797167	−	0.603759i	$-0.793669\pi$
$599$	− 723.303i	− 1.20752i	0.797167	−	0.603759i	$-0.206331\pi$
$600$	0	0
$601$	68.7503	0.114393	0.0571966	−	0.998363i	$-0.481784\pi$
$601$	68.7503	0.114393	0.0571966	+	0.998363i	$0.481784\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 389.599i	− 0.643965i
$606$	0	0
$607$	141.263i	0.232724i	0.993207	+	0.116362i	$0.0371232\pi$
$607$	141.263i	0.232724i	−0.993207	+	0.116362i	$0.962877\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−886.646	−1.45114
$612$	0	0
$613$	− 96.7370i	− 0.157809i	−0.996882	−	0.0789046i	$-0.974858\pi$
$613$	− 96.7370i	− 0.157809i	0.996882	−	0.0789046i	$-0.0251422\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−580.418	−0.940709	−0.470355	−	0.882478i	$-0.655874\pi$
$617$	−580.418	−0.940709	−0.470355	+	0.882478i	$0.655874\pi$
$618$	0	0
$619$	−157.945	−0.255161	−0.127581	−	0.991828i	$-0.540721\pi$
$619$	−157.945	−0.255161	−0.127581	+	0.991828i	$0.540721\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 214.084i	− 0.343634i
$624$	0	0
$625$	−131.484	−0.210375
$626$	0	0
$627$	0	0
$628$	0	0
$629$	569.384i	0.905221i
$630$	0	0
$631$	771.793i	1.22313i	0.791195	+	0.611564i	$0.209459\pi$
$631$	771.793i	1.22313i	−0.791195	+	0.611564i	$0.790541\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	532.493	0.838571
$636$	0	0
$637$	134.110i	0.210534i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	586.903	0.915605	0.457802	−	0.889054i	$-0.348637\pi$
$641$	586.903	0.915605	0.457802	+	0.889054i	$0.348637\pi$
$642$	0	0
$643$	−865.328	−1.34577	−0.672883	−	0.739749i	$-0.734944\pi$
$643$	−865.328	−1.34577	−0.672883	+	0.739749i	$0.734944\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	132.883i	0.205383i	0.994713	+	0.102691i	$0.0327454\pi$
$647$	132.883i	0.205383i	−0.994713	+	0.102691i	$0.967255\pi$
$648$	0	0
$649$	335.489	0.516933
$650$	0	0
$651$	0	0
$652$	0	0
$653$	364.309i	0.557900i	0.960306	+	0.278950i	$0.0899865\pi$
$653$	364.309i	0.557900i	−0.960306	+	0.278950i	$0.910014\pi$
$654$	0	0
$655$	214.391i	0.327315i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.8972	−0.0286756	−0.0143378	−	0.999897i	$-0.504564\pi$
$659$	−18.8972	−0.0286756	−0.0143378	+	0.999897i	$0.504564\pi$
$660$	0	0
$661$	339.106i	0.513019i	0.966542	+	0.256510i	$0.0825725\pi$
$661$	339.106i	0.513019i	−0.966542	+	0.256510i	$0.917427\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−74.2436	−0.111644
$666$	0	0
$667$	−455.278	−0.682576
$668$	0	0
$669$	0	0
$670$	0	0
$671$	21.9073i	0.0326487i
$672$	0	0
$673$	674.869	1.00278	0.501389	−	0.865222i	$-0.332823\pi$
$673$	674.869	1.00278	0.501389	+	0.865222i	$0.332823\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	988.747i	1.46048i	0.683189	+	0.730242i	$0.260593\pi$
$677$	988.747i	1.46048i	−0.683189	+	0.730242i	$0.739407\pi$
$678$	0	0
$679$	− 428.839i	− 0.631575i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−518.125	−0.758602	−0.379301	−	0.925273i	$-0.623835\pi$
$683$	−518.125	−0.758602	−0.379301	+	0.925273i	$0.623835\pi$
$684$	0	0
$685$	− 367.143i	− 0.535975i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−185.728	−0.269562
$690$	0	0
$691$	617.021	0.892940	0.446470	−	0.894799i	$-0.352681\pi$
$691$	617.021	0.892940	0.446470	+	0.894799i	$0.352681\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	642.442i	0.924377i
$696$	0	0
$697$	−660.153	−0.947135
$698$	0	0
$699$	0	0
$700$	0	0
$701$	97.6954i	0.139366i	0.997569	+	0.0696829i	$0.0221987\pi$
$701$	97.6954i	0.139366i	−0.997569	+	0.0696829i	$0.977801\pi$
$702$	0	0
$703$	− 320.405i	− 0.455768i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−282.638	−0.399771
$708$	0	0
$709$	1249.74i	1.76269i	0.472476	+	0.881343i	$0.343360\pi$
$709$	1249.74i	1.76269i	−0.472476	+	0.881343i	$0.656640\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	752.386	1.05524
$714$	0	0
$715$	−194.444	−0.271950
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 424.744i	− 0.590743i	−0.955382	−	0.295372i	$-0.904557\pi$
$719$	− 424.744i	− 0.590743i	0.955382	−	0.295372i	$-0.0954435\pi$
$720$	0	0
$721$	−335.022	−0.464663
$722$	0	0
$723$	0	0
$724$	0	0
$725$	352.886i	0.486739i
$726$	0	0
$727$	79.1445i	0.108865i	0.998517	+	0.0544323i	$0.0173349\pi$
$727$	79.1445i	0.108865i	−0.998517	+	0.0544323i	$0.982665\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−878.255	−1.20144
$732$	0	0
$733$	− 663.766i	− 0.905548i	−0.891625	−	0.452774i	$-0.850434\pi$
$733$	− 663.766i	− 0.905548i	0.891625	−	0.452774i	$-0.149566\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35.3167	−0.0479196
$738$	0	0
$739$	−832.112	−1.12600	−0.562998	−	0.826458i	$-0.690352\pi$
$739$	−832.112	−1.12600	−0.562998	+	0.826458i	$0.690352\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	283.217i	0.381180i	0.981670	+	0.190590i	$0.0610401\pi$
$743$	283.217i	0.381180i	−0.981670	+	0.190590i	$0.938960\pi$
$744$	0	0
$745$	164.297	0.220532
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 230.231i	− 0.307385i
$750$	0	0
$751$	374.981i	0.499309i	0.968335	+	0.249654i	$0.0803170\pi$
$751$	374.981i	0.499309i	−0.968335	+	0.249654i	$0.919683\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	397.059	0.525906
$756$	0	0
$757$	− 63.0951i	− 0.0833488i	−0.999131	−	0.0416744i	$-0.986731\pi$
$757$	− 63.0951i	− 0.0833488i	0.999131	−	0.0416744i	$-0.0132692\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−467.505	−0.614330	−0.307165	−	0.951656i	$-0.599380\pi$
$761$	−467.505	−0.614330	−0.307165	+	0.951656i	$0.599380\pi$
$762$	0	0
$763$	−502.485	−0.658564
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2194.69i	2.86140i
$768$	0	0
$769$	900.573	1.17110	0.585548	−	0.810638i	$-0.300879\pi$
$769$	900.573	1.17110	0.585548	+	0.810638i	$0.300879\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1222.76i	1.58184i	0.611920	+	0.790919i	$0.290397\pi$
$773$	1222.76i	1.58184i	−0.611920	+	0.790919i	$0.709603\pi$
$774$	0	0
$775$	− 583.173i	− 0.752482i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	371.483	0.476872
$780$	0	0
$781$	378.344i	0.484435i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1019.18	1.29832
$786$	0	0
$787$	−862.942	−1.09650	−0.548248	−	0.836316i	$-0.684705\pi$
$787$	−862.942	−1.09650	−0.548248	+	0.836316i	$0.684705\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 106.319i	− 0.134411i
$792$	0	0
$793$	−143.312	−0.180722
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1078.34i	− 1.35300i	−0.736442	−	0.676500i	$-0.763496\pi$
$797$	− 1078.34i	− 1.35300i	0.736442	−	0.676500i	$-0.236504\pi$
$798$	0	0
$799$	− 665.947i	− 0.833476i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	53.5516	0.0666894
$804$	0	0
$805$	153.667i	0.190891i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−333.388	−0.412099	−0.206050	−	0.978542i	$-0.566061\pi$
$809$	−333.388	−0.412099	−0.206050	+	0.978542i	$0.566061\pi$
$810$	0	0
$811$	1246.04	1.53642	0.768211	−	0.640197i	$-0.221147\pi$
$811$	1246.04	1.53642	0.768211	+	0.640197i	$0.221147\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 592.667i	− 0.727199i
$816$	0	0
$817$	494.214	0.604913
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1458.68i	1.77671i	0.459162	+	0.888353i	$0.348150\pi$
$821$	1458.68i	1.77671i	−0.459162	+	0.888353i	$0.651850\pi$
$822$	0	0
$823$	464.047i	0.563848i	0.959437	+	0.281924i	$0.0909725\pi$
$823$	464.047i	0.563848i	−0.959437	+	0.281924i	$0.909027\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1077.41	1.30279	0.651394	−	0.758739i	$-0.274184\pi$
$827$	1077.41	1.30279	0.651394	+	0.758739i	$0.274184\pi$
$828$	0	0
$829$	35.9354i	0.0433479i	0.999765	+	0.0216740i	$0.00689958\pi$
$829$	35.9354i	0.0433479i	−0.999765	+	0.0216740i	$0.993100\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−100.728	−0.120922
$834$	0	0
$835$	−418.134	−0.500760
$836$	0	0
$837$	0	0
$838$	0	0
$839$	734.676i	0.875656i	0.899059	+	0.437828i	$0.144252\pi$
$839$	734.676i	0.875656i	−0.899059	+	0.437828i	$0.855748\pi$
$840$	0	0
$841$	103.069	0.122555
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 686.342i	− 0.812239i
$846$	0	0
$847$	297.443i	0.351173i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−663.164	−0.779276
$852$	0	0
$853$	402.566i	0.471942i	0.971760	+	0.235971i	$0.0758270\pi$
$853$	402.566i	0.471942i	−0.971760	+	0.235971i	$0.924173\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−552.003	−0.644110	−0.322055	−	0.946721i	$-0.604374\pi$
$857$	−552.003	−0.644110	−0.322055	+	0.946721i	$0.604374\pi$
$858$	0	0
$859$	330.117	0.384303	0.192152	−	0.981365i	$-0.438453\pi$
$859$	330.117	0.384303	0.192152	+	0.981365i	$0.438453\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1116.67i	− 1.29394i	−0.762516	−	0.646969i	$-0.776036\pi$
$863$	− 1116.67i	− 1.29394i	0.762516	−	0.646969i	$-0.223964\pi$
$864$	0	0
$865$	−375.445	−0.434041
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 124.810i	− 0.143625i
$870$	0	0
$871$	− 231.033i	− 0.265251i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	348.326	0.398087
$876$	0	0
$877$	0.747441i	0 0.000852270i	1.00000		0.000426135i	$0.000135643\pi$
$877$	0.747441i	0 0.000852270i	−1.00000		0.000426135i	$0.999864\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	247.826	0.281301	0.140650	−	0.990059i	$-0.455081\pi$
$881$	247.826	0.281301	0.140650	+	0.990059i	$0.455081\pi$
$882$	0	0
$883$	1613.74	1.82757	0.913784	−	0.406200i	$-0.133146\pi$
$883$	1613.74	1.82757	0.913784	+	0.406200i	$0.133146\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1110.59i	− 1.25207i	−0.779795	−	0.626034i	$-0.784677\pi$
$887$	− 1110.59i	− 1.25207i	0.779795	−	0.626034i	$-0.215323\pi$
$888$	0	0
$889$	−406.537	−0.457297
$890$	0	0
$891$	0	0
$892$	0	0
$893$	374.744i	0.419646i
$894$	0	0
$895$	− 559.458i	− 0.625092i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1219.49	1.35650
$900$	0	0
$901$	− 139.498i	− 0.154825i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.7676	0.0273675
$906$	0	0
$907$	518.009	0.571123	0.285562	−	0.958360i	$-0.407820\pi$
$907$	518.009	0.571123	0.285562	+	0.958360i	$0.407820\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1065.61i	− 1.16972i	−0.811135	−	0.584860i	$-0.801150\pi$
$911$	− 1065.61i	− 1.16972i	0.811135	−	0.584860i	$-0.198850\pi$
$912$	0	0
$913$	321.186	0.351792
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 163.679i	− 0.178494i
$918$	0	0
$919$	1183.66i	1.28799i	0.765030	+	0.643994i	$0.222724\pi$
$919$	1183.66i	1.28799i	−0.765030	+	0.643994i	$0.777276\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2475.03	−2.68151
$924$	0	0
$925$	514.018i	0.555695i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	379.019	0.407986	0.203993	−	0.978972i	$-0.434608\pi$
$929$	379.019	0.407986	0.203993	+	0.978972i	$0.434608\pi$
$930$	0	0
$931$	56.6821	0.0608830
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 146.044i	− 0.156197i
$936$	0	0
$937$	1316.09	1.40458	0.702289	−	0.711892i	$-0.252161\pi$
$937$	1316.09	1.40458	0.702289	+	0.711892i	$0.252161\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	497.031i	0.528194i	0.964496	+	0.264097i	$0.0850739\pi$
$941$	497.031i	0.528194i	−0.964496	+	0.264097i	$0.914926\pi$
$942$	0	0
$943$	− 768.883i	− 0.815359i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	808.487	0.853735	0.426867	−	0.904314i	$-0.359617\pi$
$947$	808.487	0.853735	0.426867	+	0.904314i	$0.359617\pi$
$948$	0	0
$949$	350.322i	0.369148i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1345.58	1.41194	0.705969	−	0.708243i	$-0.250512\pi$
$953$	1345.58	1.41194	0.705969	+	0.708243i	$0.250512\pi$
$954$	0	0
$955$	254.013	0.265982
$956$	0	0
$957$	0	0
$958$	0	0
$959$	280.299i	0.292283i
$960$	0	0
$961$	−1054.32	−1.09710
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 294.809i	− 0.305502i
$966$	0	0
$967$	140.279i	0.145066i	0.997366	+	0.0725330i	$0.0231083\pi$
$967$	140.279i	0.145066i	−0.997366	+	0.0725330i	$0.976892\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1438.67	1.48164	0.740820	−	0.671704i	$-0.234437\pi$
$971$	1438.67	1.48164	0.740820	+	0.671704i	$0.234437\pi$
$972$	0	0
$973$	− 490.479i	− 0.504090i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−902.190	−0.923428	−0.461714	−	0.887029i	$-0.652765\pi$
$977$	−902.190	−0.923428	−0.461714	+	0.887029i	$0.652765\pi$
$978$	0	0
$979$	−236.976	−0.242059
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 503.193i	− 0.511896i	−0.966691	−	0.255948i	$-0.917612\pi$
$983$	− 503.193i	− 0.511896i	0.966691	−	0.255948i	$-0.0823875\pi$
$984$	0	0
$985$	486.761	0.494173
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1022.91i	− 1.03428i
$990$	0	0
$991$	906.322i	0.914553i	0.889325	+	0.457277i	$0.151175\pi$
$991$	906.322i	0.914553i	−0.889325	+	0.457277i	$0.848825\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−495.847	−0.498339
$996$	0	0
$997$	608.625i	0.610457i	0.952279	+	0.305228i	$0.0987328\pi$
$997$	608.625i	0.610457i	−0.952279	+	0.305228i	$0.901267\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.g.b.1135.5		8
3.2	odd	2		224.3.g.b.15.3		8
4.3	odd	2		504.3.g.b.379.1		8
8.3	odd	2	inner	2016.3.g.b.1135.4		8
8.5	even	2		504.3.g.b.379.2		8
12.11	even	2		56.3.g.b.43.8	yes	8
21.20	even	2		1568.3.g.m.687.6		8
24.5	odd	2		56.3.g.b.43.7	✓	8
24.11	even	2		224.3.g.b.15.4		8
48.5	odd	4		1792.3.d.j.1023.10		16
48.11	even	4		1792.3.d.j.1023.8		16
48.29	odd	4		1792.3.d.j.1023.7		16
48.35	even	4		1792.3.d.j.1023.9		16
84.11	even	6		392.3.k.o.275.2		16
84.23	even	6		392.3.k.o.67.4		16
84.47	odd	6		392.3.k.n.67.4		16
84.59	odd	6		392.3.k.n.275.2		16
84.83	odd	2		392.3.g.m.99.8		8
168.5	even	6		392.3.k.n.67.2		16
168.53	odd	6		392.3.k.o.275.4		16
168.83	odd	2		1568.3.g.m.687.5		8
168.101	even	6		392.3.k.n.275.4		16
168.125	even	2		392.3.g.m.99.7		8
168.149	odd	6		392.3.k.o.67.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.b.43.7	✓	8	24.5	odd	2
56.3.g.b.43.8	yes	8	12.11	even	2
224.3.g.b.15.3		8	3.2	odd	2
224.3.g.b.15.4		8	24.11	even	2
392.3.g.m.99.7		8	168.125	even	2
392.3.g.m.99.8		8	84.83	odd	2
392.3.k.n.67.2		16	168.5	even	6
392.3.k.n.67.4		16	84.47	odd	6
392.3.k.n.275.2		16	84.59	odd	6
392.3.k.n.275.4		16	168.101	even	6
392.3.k.o.67.2		16	168.149	odd	6
392.3.k.o.67.4		16	84.23	even	6
392.3.k.o.275.2		16	84.11	even	6
392.3.k.o.275.4		16	168.53	odd	6
504.3.g.b.379.1		8	4.3	odd	2
504.3.g.b.379.2		8	8.5	even	2
1568.3.g.m.687.5		8	168.83	odd	2
1568.3.g.m.687.6		8	21.20	even	2
1792.3.d.j.1023.7		16	48.29	odd	4
1792.3.d.j.1023.8		16	48.11	even	4
1792.3.d.j.1023.9		16	48.35	even	4
1792.3.d.j.1023.10		16	48.5	odd	4
2016.3.g.b.1135.4		8	8.3	odd	2	inner
2016.3.g.b.1135.5		8	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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2016.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.46547i q^{5} -2.64575i q^{7} +O(q^{10})\)	\(q+3.46547i q^{5} -2.64575i q^{7} -2.92866 q^{11} -19.1586i q^{13} +14.3897 q^{17} -8.09744 q^{19} +16.7598i q^{23} +12.9905 q^{25} +27.1649i q^{29} -44.8923i q^{31} +9.16878 q^{35} +39.5687i q^{37} -45.8766 q^{41} -61.0334 q^{43} -46.2793i q^{47} -7.00000 q^{49} -9.69424i q^{53} -10.1492i q^{55} -114.554 q^{59} -7.48032i q^{61} +66.3935 q^{65} +12.0590 q^{67} -129.187i q^{71} -18.2854 q^{73} +7.74851i q^{77} +42.6168i q^{79} -109.670 q^{83} +49.8673i q^{85} +80.9162 q^{89} -50.6889 q^{91} -28.0614i q^{95} +162.086 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 32 q^{11} + 80 q^{17} - 56 q^{19} - 16 q^{25} + 56 q^{35} - 128 q^{41} - 56 q^{49} + 104 q^{59} + 72 q^{65} - 304 q^{67} - 112 q^{73} + 72 q^{83} + 512 q^{89} + 56 q^{91} + 64 q^{97}+O(q^{100}) \) 8 * q - 32 * q^11 + 80 * q^17 - 56 * q^19 - 16 * q^25 + 56 * q^35 - 128 * q^41 - 56 * q^49 + 104 * q^59 + 72 * q^65 - 304 * q^67 - 112 * q^73 + 72 * q^83 + 512 * q^89 + 56 * q^91 + 64 * q^97

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