LMFDB - Embedded newform 1792.3.d.j.1023.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,3,Mod(1023,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1792.1023");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1792.d (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:	$48.8284633734$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 5x^{14} + 48x^{12} + 180x^{10} + 1056x^{8} + 2880x^{6} + 12288x^{4} + 20480x^{2} + 65536 $ x^16 + 5x^14 + 48x^12 + 180x^10 + 1056x^8 + 2880x^6 + 12288x^4 + 20480*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{38} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1023.10
Root			$-0.739226 + 1.85837i$ of defining polynomial
Character	$\chi$	$=$	1792.1023
Dual form			1792.3.d.j.1023.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0974366i q^{3} +3.46547 q^{5} +2.64575i q^{7} +8.99051 q^{9} +O(q^{10})$	$q+0.0974366i q^{3} +3.46547 q^{5} +2.64575i q^{7} +8.99051 q^{9} +2.92866i q^{11} -19.1586 q^{13} +0.337664i q^{15} -14.3897 q^{17} +8.09744i q^{19} -0.257793 q^{21} +16.7598i q^{23} -12.9905 q^{25} +1.75293i q^{27} -27.1649 q^{29} -44.8923i q^{31} -0.285359 q^{33} +9.16878i q^{35} -39.5687 q^{37} -1.86675i q^{39} -45.8766 q^{41} -61.0334i q^{43} +31.1563 q^{45} +46.2793i q^{47} -7.00000 q^{49} -1.40209i q^{51} -9.69424 q^{53} +10.1492i q^{55} -0.788986 q^{57} +114.554i q^{59} -7.48032 q^{61} +23.7866i q^{63} -66.3935 q^{65} -12.0590i q^{67} -1.63302 q^{69} -129.187i q^{71} +18.2854 q^{73} -1.26575i q^{75} -7.74851 q^{77} +42.6168i q^{79} +80.7438 q^{81} -109.670i q^{83} -49.8673 q^{85} -2.64685i q^{87} +80.9162 q^{89} -50.6889i q^{91} +4.37415 q^{93} +28.0614i q^{95} +162.086 q^{97} +26.3301i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 96 q^{9}+O(q^{10}) $ 16 * q - 96 * q^9	$ 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) $ 16 * q - 96 * q^9 - 160 * q^17 + 32 * q^25 + 64 * q^33 - 256 * q^41 - 112 * q^49 - 112 * q^57 - 144 * q^65 + 224 * q^73 + 96 * q^81 + 1024 * q^89 + 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0.0974366i	0.0324789i	0.999868	+	0.0162394i	$0.00516940\pi$
$3$	0.0974366i	0.0324789i	−0.999868	+	0.0162394i	$0.994831\pi$
$4$	0	0
$5$	3.46547	0.693094	0.346547	−	0.938033i	$-0.387354\pi$
$5$	3.46547	0.693094	0.346547	+	0.938033i	$0.387354\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	8.99051	0.998945
$10$	0	0
$11$	2.92866i	0.266242i	0.991100	+	0.133121i	$0.0424999\pi$
$11$	2.92866i	0.266242i	−0.991100	+	0.133121i	$0.957500\pi$
$12$	0	0
$13$	−19.1586	−1.47374	−0.736869	−	0.676036i	$-0.763696\pi$
$13$	−19.1586	−1.47374	−0.736869	+	0.676036i	$0.763696\pi$
$14$	0	0
$15$	0.337664i	0.0225109i
$16$	0	0
$17$	−14.3897	−0.846456	−0.423228	−	0.906023i	$-0.639103\pi$
$17$	−14.3897	−0.846456	−0.423228	+	0.906023i	$0.639103\pi$
$18$	0	0
$19$	8.09744i	0.426181i	0.977032	+	0.213090i	$0.0683529\pi$
$19$	8.09744i	0.426181i	−0.977032	+	0.213090i	$0.931647\pi$
$20$	0	0
$21$	−0.257793	−0.0122759
$22$	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
$27$	1.75293i	0.0649234i
$28$	0	0
$29$	−27.1649	−0.936720	−0.468360	−	0.883538i	$-0.655155\pi$
$29$	−27.1649	−0.936720	−0.468360	+	0.883538i	$0.655155\pi$
$30$	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
$33$	−0.285359	−0.00864723
$34$	0	0
$35$	9.16878i	0.261965i
$36$	0	0
$37$	−39.5687	−1.06943	−0.534713	−	0.845034i	$-0.679580\pi$
$37$	−39.5687	−1.06943	−0.534713	+	0.845034i	$0.679580\pi$
$38$	0	0
$39$	− 1.86675i	− 0.0478653i
$40$	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
$43$	− 61.0334i	− 1.41938i	−0.704514	−	0.709690i	$-0.748835\pi$
$43$	− 61.0334i	− 1.41938i	0.704514	−	0.709690i	$-0.251165\pi$
$44$	0	0
$45$	31.1563	0.692363
$46$	0	0
$47$	46.2793i	0.984666i	0.870407	+	0.492333i	$0.163856\pi$
$47$	46.2793i	0.984666i	−0.870407	+	0.492333i	$0.836144\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 1.40209i	− 0.0274919i
$52$	0	0
$53$	−9.69424	−0.182910	−0.0914551	−	0.995809i	$-0.529152\pi$
$53$	−9.69424	−0.182910	−0.0914551	+	0.995809i	$0.529152\pi$
$54$	0	0
$55$	10.1492i	0.184531i
$56$	0	0
$57$	−0.788986	−0.0138419
$58$	0	0
$59$	114.554i	1.94159i	0.239907	+	0.970796i	$0.422883\pi$
$59$	114.554i	1.94159i	−0.239907	+	0.970796i	$0.577117\pi$
$60$	0	0
$61$	−7.48032	−0.122628	−0.0613141	−	0.998119i	$-0.519529\pi$
$61$	−7.48032	−0.122628	−0.0613141	+	0.998119i	$0.519529\pi$
$62$	0	0
$63$	23.7866i	0.377566i
$64$	0	0
$65$	−66.3935	−1.02144
$66$	0	0
$67$	− 12.0590i	− 0.179985i	−0.995942	−	0.0899925i	$-0.971316\pi$
$67$	− 12.0590i	− 0.179985i	0.995942	−	0.0899925i	$-0.0286843\pi$
$68$	0	0
$69$	−1.63302	−0.0236669
$70$	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	$-0.460029\pi$
$73$	18.2854	0.250484	0.125242	+	0.992126i	$0.460029\pi$
$74$	0	0
$75$	− 1.26575i	− 0.0168767i
$76$	0	0
$77$	−7.74851	−0.100630
$78$	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
$81$	80.7438	0.996836
$82$	0	0
$83$	− 109.670i	− 1.32133i	−0.750683	−	0.660663i	$-0.770275\pi$
$83$	− 109.670i	− 1.32133i	0.750683	−	0.660663i	$-0.229725\pi$
$84$	0	0
$85$	−49.8673	−0.586674
$86$	0	0
$87$	− 2.64685i	− 0.0304236i
$88$	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
$91$	− 50.6889i	− 0.557020i
$92$	0	0
$93$	4.37415	0.0470339
$94$	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
$99$	26.3301i	0.265961i
$100$	0	0
$101$	−106.827	−1.05769	−0.528847	−	0.848717i	$-0.677375\pi$
$101$	−106.827	−1.05769	−0.528847	+	0.848717i	$0.677375\pi$
$102$	0	0
$103$	126.626i	1.22938i	0.788768	+	0.614691i	$0.210719\pi$
$103$	126.626i	1.22938i	−0.788768	+	0.614691i	$0.789281\pi$
$104$	0	0
$105$	−0.893374	−0.00850832
$106$	0	0
$107$	− 87.0191i	− 0.813263i	−0.913592	−	0.406632i	$-0.866703\pi$
$107$	− 87.0191i	− 0.813263i	0.913592	−	0.406632i	$-0.133297\pi$
$108$	0	0
$109$	−189.921	−1.74240	−0.871199	−	0.490930i	$-0.836657\pi$
$109$	−189.921	−1.74240	−0.871199	+	0.490930i	$0.836657\pi$
$110$	0	0
$111$	− 3.85544i	− 0.0347337i
$112$	0	0
$113$	−40.1848	−0.355617	−0.177809	−	0.984065i	$-0.556901\pi$
$113$	−40.1848	−0.355617	−0.177809	+	0.984065i	$0.556901\pi$
$114$	0	0
$115$	58.0806i	0.505049i
$116$	0	0
$117$	−172.245	−1.47218
$118$	0	0
$119$	− 38.0717i	− 0.319930i
$120$	0	0
$121$	112.423	0.929115
$122$	0	0
$123$	− 4.47006i	− 0.0363420i
$124$	0	0
$125$	−131.655	−1.05324
$126$	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
$129$	5.94688	0.0460999
$130$	0	0
$131$	61.8649i	0.472251i	0.971723	+	0.236126i	$0.0758777\pi$
$131$	61.8649i	0.472251i	−0.971723	+	0.236126i	$0.924122\pi$
$132$	0	0
$133$	−21.4238	−0.161081
$134$	0	0
$135$	6.07474i	0.0449981i
$136$	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
$139$	185.384i	1.33370i	0.745194	+	0.666848i	$0.232357\pi$
$139$	185.384i	1.33370i	−0.745194	+	0.666848i	$0.767643\pi$
$140$	0	0
$141$	−4.50930	−0.0319808
$142$	0	0
$143$	− 56.1090i	− 0.392371i
$144$	0	0
$145$	−94.1392	−0.649236
$146$	0	0
$147$	− 0.682056i	− 0.00463984i
$148$	0	0
$149$	−47.4096	−0.318185	−0.159093	−	0.987264i	$-0.550857\pi$
$149$	−47.4096	−0.318185	−0.159093	+	0.987264i	$0.550857\pi$
$150$	0	0
$151$	114.576i	0.758780i	0.925237	+	0.379390i	$0.123866\pi$
$151$	114.576i	0.758780i	−0.925237	+	0.379390i	$0.876134\pi$
$152$	0	0
$153$	−129.371	−0.845563
$154$	0	0
$155$	− 155.573i	− 1.00370i
$156$	0	0
$157$	−294.095	−1.87322	−0.936608	−	0.350378i	$-0.886053\pi$
$157$	−294.095	−1.87322	−0.936608	+	0.350378i	$0.886053\pi$
$158$	0	0
$159$	− 0.944573i	− 0.00594071i
$160$	0	0
$161$	−44.3423	−0.275418
$162$	0	0
$163$	171.021i	1.04921i	0.851347	+	0.524603i	$0.175786\pi$
$163$	171.021i	1.04921i	−0.851347	+	0.524603i	$0.824214\pi$
$164$	0	0
$165$	−0.988902	−0.00599335
$166$	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
$171$	72.8001i	0.425731i
$172$	0	0
$173$	−108.339	−0.626236	−0.313118	−	0.949714i	$-0.601374\pi$
$173$	−108.339	−0.626236	−0.313118	+	0.949714i	$0.601374\pi$
$174$	0	0
$175$	− 34.3696i	− 0.196398i
$176$	0	0
$177$	−11.1617	−0.0630607
$178$	0	0
$179$	− 161.438i	− 0.901886i	−0.892553	−	0.450943i	$-0.851088\pi$
$179$	− 161.438i	− 0.901886i	0.892553	−	0.450943i	$-0.148912\pi$
$180$	0	0
$181$	7.14696	0.0394860	0.0197430	−	0.999805i	$-0.493715\pi$
$181$	7.14696	0.0394860	0.0197430	+	0.999805i	$0.493715\pi$
$182$	0	0
$183$	− 0.728856i	− 0.00398282i
$184$	0	0
$185$	−137.124	−0.741212
$186$	0	0
$187$	− 42.1427i	− 0.225362i
$188$	0	0
$189$	−4.63782	−0.0245388
$190$	0	0
$191$	73.2983i	0.383761i	0.981418	+	0.191880i	$0.0614586\pi$
$191$	73.2983i	0.383761i	−0.981418	+	0.191880i	$0.938541\pi$
$192$	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
$195$	− 6.46916i	− 0.0331752i
$196$	0	0
$197$	−140.460	−0.712996	−0.356498	−	0.934296i	$-0.616029\pi$
$197$	−140.460	−0.712996	−0.356498	+	0.934296i	$0.616029\pi$
$198$	0	0
$199$	− 143.082i	− 0.719006i	−0.933144	−	0.359503i	$-0.882946\pi$
$199$	− 143.082i	− 0.719006i	0.933144	−	0.359503i	$-0.117054\pi$
$200$	0	0
$201$	1.17499	0.00584571
$202$	0	0
$203$	− 71.8715i	− 0.354047i
$204$	0	0
$205$	−158.984	−0.775532
$206$	0	0
$207$	150.679i	0.727919i
$208$	0	0
$209$	−23.7146	−0.113467
$210$	0	0
$211$	− 111.955i	− 0.530591i	−0.964167	−	0.265295i	$-0.914531\pi$
$211$	− 111.955i	− 0.530591i	0.964167	−	0.265295i	$-0.0854694\pi$
$212$	0	0
$213$	12.5875	0.0590962
$214$	0	0
$215$	− 211.509i	− 0.983765i
$216$	0	0
$217$	118.774	0.547345
$218$	0	0
$219$	1.78166i	0.00813545i
$220$	0	0
$221$	275.687	1.24745
$222$	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
$225$	−116.791	−0.519072
$226$	0	0
$227$	74.3581i	0.327569i	0.986496	+	0.163784i	$0.0523701\pi$
$227$	74.3581i	0.327569i	−0.986496	+	0.163784i	$0.947630\pi$
$228$	0	0
$229$	78.2710	0.341795	0.170897	−	0.985289i	$-0.445333\pi$
$229$	78.2710	0.341795	0.170897	+	0.985289i	$0.445333\pi$
$230$	0	0
$231$	− 0.754988i	− 0.00326835i
$232$	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
$235$	160.380i	0.682466i
$236$	0	0
$237$	−4.15244	−0.0175208
$238$	0	0
$239$	291.605i	1.22011i	0.792361	+	0.610053i	$0.208852\pi$
$239$	291.605i	1.22011i	−0.792361	+	0.610053i	$0.791148\pi$
$240$	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
$243$	23.6438i	0.0972996i
$244$	0	0
$245$	−24.2583	−0.0990135
$246$	0	0
$247$	− 155.135i	− 0.628079i
$248$	0	0
$249$	10.6859	0.0429151
$250$	0	0
$251$	310.605i	1.23747i	0.785600	+	0.618734i	$0.212354\pi$
$251$	310.605i	1.23747i	−0.785600	+	0.618734i	$0.787646\pi$
$252$	0	0
$253$	−49.0838	−0.194007
$254$	0	0
$255$	− 4.85889i	− 0.0190545i
$256$	0	0
$257$	175.472	0.682769	0.341385	−	0.939924i	$-0.389104\pi$
$257$	175.472	0.682769	0.341385	+	0.939924i	$0.389104\pi$
$258$	0	0
$259$	− 104.689i	− 0.404205i
$260$	0	0
$261$	−244.226	−0.935732
$262$	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
$267$	7.88419i	0.0295288i
$268$	0	0
$269$	142.817	0.530918	0.265459	−	0.964122i	$-0.414477\pi$
$269$	142.817	0.530918	0.265459	+	0.964122i	$0.414477\pi$
$270$	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
$273$	4.93895	0.0180914
$274$	0	0
$275$	− 38.0448i	− 0.138345i
$276$	0	0
$277$	−366.740	−1.32397	−0.661986	−	0.749516i	$-0.730286\pi$
$277$	−366.740	−1.32397	−0.661986	+	0.749516i	$0.730286\pi$
$278$	0	0
$279$	− 403.604i	− 1.44661i
$280$	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
$283$	− 327.739i	− 1.15809i	−0.815297	−	0.579043i	$-0.803426\pi$
$283$	− 327.739i	− 1.15809i	0.815297	−	0.579043i	$-0.196574\pi$
$284$	0	0
$285$	−2.73421	−0.00959372
$286$	0	0
$287$	− 121.378i	− 0.422920i
$288$	0	0
$289$	−81.9352	−0.283513
$290$	0	0
$291$	15.7931i	0.0542718i
$292$	0	0
$293$	−259.881	−0.886966	−0.443483	−	0.896283i	$-0.646257\pi$
$293$	−259.881	−0.886966	−0.443483	+	0.896283i	$0.646257\pi$
$294$	0	0
$295$	396.983i	1.34571i
$296$	0	0
$297$	−5.13375	−0.0172853
$298$	0	0
$299$	− 321.094i	− 1.07389i
$300$	0	0
$301$	161.479	0.536475
$302$	0	0
$303$	− 10.4089i	− 0.0343527i
$304$	0	0
$305$	−25.9228	−0.0849929
$306$	0	0
$307$	290.462i	0.946131i	0.881027	+	0.473065i	$0.156853\pi$
$307$	290.462i	0.946131i	−0.881027	+	0.473065i	$0.843147\pi$
$308$	0	0
$309$	−12.3380	−0.0399289
$310$	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	$-0.650177\pi$
$313$	−284.507	−0.908969	−0.454485	+	0.890755i	$0.650177\pi$
$314$	0	0
$315$	82.4319i	0.261689i
$316$	0	0
$317$	12.2631	0.0386850	0.0193425	−	0.999813i	$-0.493843\pi$
$317$	12.2631	0.0386850	0.0193425	+	0.999813i	$0.493843\pi$
$318$	0	0
$319$	− 79.5567i	− 0.249394i
$320$	0	0
$321$	8.47885	0.0264139
$322$	0	0
$323$	− 116.520i	− 0.360743i
$324$	0	0
$325$	248.880	0.765784
$326$	0	0
$327$	− 18.5053i	− 0.0565911i
$328$	0	0
$329$	−122.443	−0.372169
$330$	0	0
$331$	− 194.466i	− 0.587510i	−0.955881	−	0.293755i	$-0.905095\pi$
$331$	− 194.466i	− 0.587510i	0.955881	−	0.293755i	$-0.0949050\pi$
$332$	0	0
$333$	−355.743	−1.06830
$334$	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
$339$	− 3.91547i	− 0.0115500i
$340$	0	0
$341$	131.474	0.385555
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−5.65918	−0.0164034
$346$	0	0
$347$	204.867i	0.590394i	0.955436	+	0.295197i	$0.0953852\pi$
$347$	204.867i	0.590394i	−0.955436	+	0.295197i	$0.904615\pi$
$348$	0	0
$349$	−128.396	−0.367898	−0.183949	−	0.982936i	$-0.558888\pi$
$349$	−128.396	−0.367898	−0.183949	+	0.982936i	$0.558888\pi$
$350$	0	0
$351$	− 33.5837i	− 0.0956801i
$352$	0	0
$353$	−190.841	−0.540627	−0.270314	−	0.962772i	$-0.587127\pi$
$353$	−190.841	−0.540627	−0.270314	+	0.962772i	$0.587127\pi$
$354$	0	0
$355$	− 447.692i	− 1.26111i
$356$	0	0
$357$	3.70957	0.0103910
$358$	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
$363$	10.9541i	0.0301766i
$364$	0	0
$365$	63.3674	0.173609
$366$	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
$369$	−412.454	−1.11776
$370$	0	0
$371$	− 25.6485i	− 0.0691335i
$372$	0	0
$373$	360.748	0.967153	0.483576	−	0.875302i	$-0.339338\pi$
$373$	360.748	0.967153	0.483576	+	0.875302i	$0.339338\pi$
$374$	0	0
$375$	− 12.8280i	− 0.0342080i
$376$	0	0
$377$	520.441	1.38048
$378$	0	0
$379$	268.427i	0.708250i	0.935198	+	0.354125i	$0.115221\pi$
$379$	268.427i	0.708250i	−0.935198	+	0.354125i	$0.884779\pi$
$380$	0	0
$381$	14.9718	0.0392960
$382$	0	0
$383$	− 581.532i	− 1.51836i	−0.650881	−	0.759180i	$-0.725600\pi$
$383$	− 581.532i	− 1.51836i	0.650881	−	0.759180i	$-0.274400\pi$
$384$	0	0
$385$	−26.8522	−0.0697461
$386$	0	0
$387$	− 548.721i	− 1.41788i
$388$	0	0
$389$	512.278	1.31691	0.658455	−	0.752620i	$-0.271210\pi$
$389$	512.278	1.31691	0.658455	+	0.752620i	$0.271210\pi$
$390$	0	0
$391$	− 241.169i	− 0.616801i
$392$	0	0
$393$	−6.02790	−0.0153382
$394$	0	0
$395$	147.687i	0.373892i
$396$	0	0
$397$	−81.3250	−0.204849	−0.102424	−	0.994741i	$-0.532660\pi$
$397$	−81.3250	−0.204849	−0.102424	+	0.994741i	$0.532660\pi$
$398$	0	0
$399$	− 2.08746i	− 0.00523173i
$400$	0	0
$401$	527.441	1.31531	0.657657	−	0.753318i	$-0.271548\pi$
$401$	527.441	1.31531	0.657657	+	0.753318i	$0.271548\pi$
$402$	0	0
$403$	860.073i	2.13418i
$404$	0	0
$405$	279.815	0.690902
$406$	0	0
$407$	− 115.883i	− 0.284726i
$408$	0	0
$409$	58.6727	0.143454	0.0717270	−	0.997424i	$-0.477149\pi$
$409$	58.6727	0.143454	0.0717270	+	0.997424i	$0.477149\pi$
$410$	0	0
$411$	− 10.3227i	− 0.0251161i
$412$	0	0
$413$	−303.081	−0.733853
$414$	0	0
$415$	− 380.058i	− 0.915803i
$416$	0	0
$417$	−18.0632	−0.0433169
$418$	0	0
$419$	760.704i	1.81552i	0.419487	+	0.907761i	$0.362210\pi$
$419$	760.704i	1.81552i	−0.419487	+	0.907761i	$0.637790\pi$
$420$	0	0
$421$	46.2918	0.109957	0.0549784	−	0.998488i	$-0.482491\pi$
$421$	46.2918	0.109957	0.0549784	+	0.998488i	$0.482491\pi$
$422$	0	0
$423$	416.074i	0.983627i
$424$	0	0
$425$	186.930	0.439835
$426$	0	0
$427$	− 19.7911i	− 0.0463491i
$428$	0	0
$429$	5.46707	0.0127437
$430$	0	0
$431$	336.176i	0.779991i	0.920817	+	0.389996i	$0.127523\pi$
$431$	336.176i	0.779991i	−0.920817	+	0.389996i	$0.872477\pi$
$432$	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
$435$	− 9.17260i	− 0.0210864i
$436$	0	0
$437$	−135.711	−0.310553
$438$	0	0
$439$	397.478i	0.905418i	0.891658	+	0.452709i	$0.149542\pi$
$439$	397.478i	0.905418i	−0.891658	+	0.452709i	$0.850458\pi$
$440$	0	0
$441$	−62.9335	−0.142706
$442$	0	0
$443$	− 273.530i	− 0.617450i	−0.951151	−	0.308725i	$-0.900098\pi$
$443$	− 273.530i	− 0.617450i	0.951151	−	0.308725i	$-0.0999022\pi$
$444$	0	0
$445$	280.413	0.630141
$446$	0	0
$447$	− 4.61943i	− 0.0103343i
$448$	0	0
$449$	428.702	0.954792	0.477396	−	0.878688i	$-0.341581\pi$
$449$	428.702	0.954792	0.477396	+	0.878688i	$0.341581\pi$
$450$	0	0
$451$	− 134.357i	− 0.297909i
$452$	0	0
$453$	−11.1639	−0.0246443
$454$	0	0
$455$	− 175.661i	− 0.386068i
$456$	0	0
$457$	−10.0500	−0.0219913	−0.0109956	−	0.999940i	$-0.503500\pi$
$457$	−10.0500	−0.0219913	−0.0109956	+	0.999940i	$0.503500\pi$
$458$	0	0
$459$	− 25.2243i	− 0.0549548i
$460$	0	0
$461$	825.802	1.79133	0.895664	−	0.444732i	$-0.146701\pi$
$461$	825.802	1.79133	0.895664	+	0.444732i	$0.146701\pi$
$462$	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
$465$	15.1585	0.0325989
$466$	0	0
$467$	− 201.727i	− 0.431964i	−0.976397	−	0.215982i	$-0.930705\pi$
$467$	− 201.727i	− 0.431964i	0.976397	−	0.215982i	$-0.0692953\pi$
$468$	0	0
$469$	31.9051	0.0680280
$470$	0	0
$471$	− 28.6556i	− 0.0608399i
$472$	0	0
$473$	178.746	0.377899
$474$	0	0
$475$	− 105.190i	− 0.221452i
$476$	0	0
$477$	−87.1561	−0.182717
$478$	0	0
$479$	− 597.538i	− 1.24747i	−0.781636	−	0.623735i	$-0.785615\pi$
$479$	− 597.538i	− 1.24747i	0.781636	−	0.623735i	$-0.214385\pi$
$480$	0	0
$481$	758.081	1.57605
$482$	0	0
$483$	− 4.32056i	− 0.00894526i
$484$	0	0
$485$	561.705	1.15815
$486$	0	0
$487$	− 345.125i	− 0.708675i	−0.935118	−	0.354337i	$-0.884706\pi$
$487$	− 345.125i	− 0.708675i	0.935118	−	0.354337i	$-0.115294\pi$
$488$	0	0
$489$	−16.6637	−0.0340770
$490$	0	0
$491$	− 373.498i	− 0.760689i	−0.924845	−	0.380344i	$-0.875805\pi$
$491$	− 373.498i	− 0.760689i	0.924845	−	0.380344i	$-0.124195\pi$
$492$	0	0
$493$	390.896	0.792892
$494$	0	0
$495$	91.2464i	0.184336i
$496$	0	0
$497$	341.796	0.687718
$498$	0	0
$499$	− 850.317i	− 1.70404i	−0.523508	−	0.852021i	$-0.675377\pi$
$499$	− 850.317i	− 1.70404i	0.523508	−	0.852021i	$-0.324623\pi$
$500$	0	0
$501$	−11.7564	−0.0234659
$502$	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
$507$	19.2975i	0.0380620i
$508$	0	0
$509$	673.910	1.32399	0.661994	−	0.749509i	$-0.269710\pi$
$509$	673.910	1.32399	0.661994	+	0.749509i	$0.269710\pi$
$510$	0	0
$511$	48.3785i	0.0946742i
$512$	0	0
$513$	−14.1943	−0.0276691
$514$	0	0
$515$	438.820i	0.852078i
$516$	0	0
$517$	−135.536	−0.262159
$518$	0	0
$519$	− 10.5562i	− 0.0203394i
$520$	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
$523$	680.087i	1.30036i	0.759781	+	0.650179i	$0.225306\pi$
$523$	680.087i	1.30036i	−0.759781	+	0.650179i	$0.774694\pi$
$524$	0	0
$525$	3.34886	0.00637878
$526$	0	0
$527$	645.988i	1.22578i
$528$	0	0
$529$	248.109	0.469015
$530$	0	0
$531$	1029.90i	1.93954i
$532$	0	0
$533$	878.931	1.64903
$534$	0	0
$535$	− 301.562i	− 0.563668i
$536$	0	0
$537$	15.7299	0.0292922
$538$	0	0
$539$	− 20.5006i	− 0.0380346i
$540$	0	0
$541$	794.999	1.46950	0.734749	−	0.678339i	$-0.237300\pi$
$541$	794.999	1.46950	0.734749	+	0.678339i	$0.237300\pi$
$542$	0	0
$543$	0.696375i	0.00128246i
$544$	0	0
$545$	−658.167	−1.20765
$546$	0	0
$547$	− 736.752i	− 1.34690i	−0.739235	−	0.673448i	$-0.764813\pi$
$547$	− 736.752i	− 1.34690i	0.739235	−	0.673448i	$-0.235187\pi$
$548$	0	0
$549$	−67.2518	−0.122499
$550$	0	0
$551$	− 219.966i	− 0.399212i
$552$	0	0
$553$	−112.754	−0.203894
$554$	0	0
$555$	− 13.3609i	− 0.0240737i
$556$	0	0
$557$	415.758	0.746423	0.373212	−	0.927746i	$-0.378257\pi$
$557$	415.758	0.746423	0.373212	+	0.927746i	$0.378257\pi$
$558$	0	0
$559$	1169.31i	2.09179i
$560$	0	0
$561$	4.10624	0.00731950
$562$	0	0
$563$	− 96.4996i	− 0.171402i	−0.996321	−	0.0857012i	$-0.972687\pi$
$563$	− 96.4996i	− 0.171402i	0.996321	−	0.0857012i	$-0.0273130\pi$
$564$	0	0
$565$	−139.259	−0.246476
$566$	0	0
$567$	213.628i	0.376769i
$568$	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
$571$	13.7251i	0.0240370i	0.999928	+	0.0120185i	$0.00382569\pi$
$571$	13.7251i	0.0240370i	−0.999928	+	0.0120185i	$0.996174\pi$
$572$	0	0
$573$	−7.14193	−0.0124641
$574$	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
$579$	− 8.28898i	− 0.0143160i
$580$	0	0
$581$	290.160	0.499414
$582$	0	0
$583$	− 28.3911i	− 0.0486983i
$584$	0	0
$585$	−596.912	−1.02036
$586$	0	0
$587$	675.987i	1.15160i	0.817592	+	0.575798i	$0.195308\pi$
$587$	675.987i	1.15160i	−0.817592	+	0.575798i	$0.804692\pi$
$588$	0	0
$589$	363.512	0.617169
$590$	0	0
$591$	− 13.6860i	− 0.0231573i
$592$	0	0
$593$	−414.116	−0.698341	−0.349171	−	0.937059i	$-0.613537\pi$
$593$	−414.116	−0.698341	−0.349171	+	0.937059i	$0.613537\pi$
$594$	0	0
$595$	− 131.936i	− 0.221742i
$596$	0	0
$597$	13.9414	0.0233525
$598$	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	$-0.518216\pi$
$601$	−68.7503	−0.114393	−0.0571966	+	0.998363i	$0.518216\pi$
$602$	0	0
$603$	− 108.416i	− 0.179795i
$604$	0	0
$605$	389.599	0.643965
$606$	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
$609$	7.00292	0.0114990
$610$	0	0
$611$	− 886.646i	− 1.45114i
$612$	0	0
$613$	96.7370	0.157809	0.0789046	−	0.996882i	$-0.474858\pi$
$613$	96.7370	0.157809	0.0789046	+	0.996882i	$0.474858\pi$
$614$	0	0
$615$	− 15.4909i	− 0.0251884i
$616$	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
$619$	− 157.945i	− 0.255161i	−0.991828	−	0.127581i	$-0.959279\pi$
$619$	− 157.945i	− 0.255161i	0.991828	−	0.127581i	$-0.0407212\pi$
$620$	0	0
$621$	−29.3788	−0.0473089
$622$	0	0
$623$	214.084i	0.343634i
$624$	0	0
$625$	−131.484	−0.210375
$626$	0	0
$627$	− 2.31067i	− 0.00368528i
$628$	0	0
$629$	569.384	0.905221
$630$	0	0
$631$	− 771.793i	− 1.22313i	−0.791195	−	0.611564i	$-0.790541\pi$
$631$	− 771.793i	− 1.22313i	0.791195	−	0.611564i	$-0.209459\pi$
$632$	0	0
$633$	10.9085	0.0172330
$634$	0	0
$635$	− 532.493i	− 0.838571i
$636$	0	0
$637$	134.110	0.210534
$638$	0	0
$639$	− 1161.45i	− 1.81761i
$640$	0	0
$641$	−586.903	−0.915605	−0.457802	−	0.889054i	$-0.651363\pi$
$641$	−586.903	−0.915605	−0.457802	+	0.889054i	$0.651363\pi$
$642$	0	0
$643$	865.328i	1.34577i	0.739749	+	0.672883i	$0.234944\pi$
$643$	865.328i	1.34577i	−0.739749	+	0.672883i	$0.765056\pi$
$644$	0	0
$645$	20.6087	0.0319515
$646$	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
$651$	11.5729i	0.0177771i
$652$	0	0
$653$	−364.309	−0.557900	−0.278950	−	0.960306i	$-0.589986\pi$
$653$	−364.309	−0.557900	−0.278950	+	0.960306i	$0.589986\pi$
$654$	0	0
$655$	214.391i	0.327315i
$656$	0	0
$657$	164.395	0.250220
$658$	0	0
$659$	− 18.8972i	− 0.0286756i	−0.999897	−	0.0143378i	$-0.995436\pi$
$659$	− 18.8972i	− 0.0286756i	0.999897	−	0.0143378i	$-0.00456402\pi$
$660$	0	0
$661$	−339.106	−0.513019	−0.256510	−	0.966542i	$-0.582573\pi$
$661$	−339.106	−0.513019	−0.256510	+	0.966542i	$0.582573\pi$
$662$	0	0
$663$	26.8620i	0.0405159i
$664$	0	0
$665$	−74.2436	−0.111644
$666$	0	0
$667$	− 455.278i	− 0.682576i
$668$	0	0
$669$	30.3454	0.0453594
$670$	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
$675$	− 22.7715i	− 0.0337355i
$676$	0	0
$677$	988.747	1.46048	0.730242	−	0.683189i	$-0.239407\pi$
$677$	988.747	1.46048	0.730242	+	0.683189i	$0.239407\pi$
$678$	0	0
$679$	428.839i	0.631575i
$680$	0	0
$681$	−7.24520	−0.0106391
$682$	0	0
$683$	518.125i	0.758602i	0.925273	+	0.379301i	$0.123835\pi$
$683$	518.125i	0.758602i	−0.925273	+	0.379301i	$0.876165\pi$
$684$	0	0
$685$	−367.143	−0.535975
$686$	0	0
$687$	7.62646i	0.0111011i
$688$	0	0
$689$	185.728	0.269562
$690$	0	0
$691$	− 617.021i	− 0.892940i	−0.894799	−	0.446470i	$-0.852681\pi$
$691$	− 617.021i	− 0.892940i	0.894799	−	0.446470i	$-0.147319\pi$
$692$	0	0
$693$	−69.6630	−0.100524
$694$	0	0
$695$	642.442i	0.924377i
$696$	0	0
$697$	660.153	0.947135
$698$	0	0
$699$	− 9.06718i	− 0.0129717i
$700$	0	0
$701$	−97.6954	−0.139366	−0.0696829	−	0.997569i	$-0.522199\pi$
$701$	−97.6954	−0.139366	−0.0696829	+	0.997569i	$0.522199\pi$
$702$	0	0
$703$	− 320.405i	− 0.455768i
$704$	0	0
$705$	−15.6268	−0.0221657
$706$	0	0
$707$	− 282.638i	− 0.399771i
$708$	0	0
$709$	−1249.74	−1.76269	−0.881343	−	0.472476i	$-0.843360\pi$
$709$	−1249.74	−1.76269	−0.881343	+	0.472476i	$0.843360\pi$
$710$	0	0
$711$	383.147i	0.538885i
$712$	0	0
$713$	752.386	1.05524
$714$	0	0
$715$	− 194.444i	− 0.271950i
$716$	0	0
$717$	−28.4130	−0.0396277
$718$	0	0
$719$	424.744i	0.590743i	0.955382	+	0.295372i	$0.0954435\pi$
$719$	424.744i	0.590743i	−0.955382	+	0.295372i	$0.904557\pi$
$720$	0	0
$721$	−335.022	−0.464663
$722$	0	0
$723$	21.8012i	0.0301538i
$724$	0	0
$725$	352.886	0.486739
$726$	0	0
$727$	− 79.1445i	− 0.108865i	−0.998517	−	0.0544323i	$-0.982665\pi$
$727$	− 79.1445i	− 0.108865i	0.998517	−	0.0544323i	$-0.0173349\pi$
$728$	0	0
$729$	724.390	0.993676
$730$	0	0
$731$	878.255i	1.20144i
$732$	0	0
$733$	−663.766	−0.905548	−0.452774	−	0.891625i	$-0.649566\pi$
$733$	−663.766	−0.905548	−0.452774	+	0.891625i	$0.649566\pi$
$734$	0	0
$735$	− 2.36365i	− 0.00321584i
$736$	0	0
$737$	35.3167	0.0479196
$738$	0	0
$739$	832.112i	1.12600i	0.826458	+	0.562998i	$0.190352\pi$
$739$	832.112i	1.12600i	−0.826458	+	0.562998i	$0.809648\pi$
$740$	0	0
$741$	15.1159	0.0203993
$742$	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
$747$	− 985.989i	− 1.31993i
$748$	0	0
$749$	230.231	0.307385
$750$	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
$753$	−30.2642	−0.0401916
$754$	0	0
$755$	397.059i	0.525906i
$756$	0	0
$757$	63.0951	0.0833488	0.0416744	−	0.999131i	$-0.486731\pi$
$757$	63.0951	0.0833488	0.0416744	+	0.999131i	$0.486731\pi$
$758$	0	0
$759$	− 4.78255i	− 0.00630113i
$760$	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
$763$	− 502.485i	− 0.658564i
$764$	0	0
$765$	−448.332	−0.586055
$766$	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
$771$	17.0974i	0.0221756i
$772$	0	0
$773$	1222.76	1.58184	0.790919	−	0.611920i	$-0.209603\pi$
$773$	1222.76	1.58184	0.790919	+	0.611920i	$0.209603\pi$
$774$	0	0
$775$	583.173i	0.752482i
$776$	0	0
$777$	10.2005	0.0131281
$778$	0	0
$779$	− 371.483i	− 0.476872i
$780$	0	0
$781$	378.344	0.484435
$782$	0	0
$783$	− 47.6182i	− 0.0608151i
$784$	0	0
$785$	−1019.18	−1.29832
$786$	0	0
$787$	862.942i	1.09650i	0.836316	+	0.548248i	$0.184705\pi$
$787$	862.942i	1.09650i	−0.836316	+	0.548248i	$0.815295\pi$
$788$	0	0
$789$	−30.4126	−0.0385457
$790$	0	0
$791$	− 106.319i	− 0.134411i
$792$	0	0
$793$	143.312	0.180722
$794$	0	0
$795$	− 3.27339i	− 0.00411747i
$796$	0	0
$797$	1078.34	1.35300	0.676500	−	0.736442i	$-0.263496\pi$
$797$	1078.34	1.35300	0.676500	+	0.736442i	$0.263496\pi$
$798$	0	0
$799$	− 665.947i	− 0.833476i
$800$	0	0
$801$	727.477	0.908211
$802$	0	0
$803$	53.5516i	0.0666894i
$804$	0	0
$805$	−153.667	−0.190891
$806$	0	0
$807$	13.9156i	0.0172436i
$808$	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
$811$	1246.04i	1.53642i	0.640197	+	0.768211i	$0.278853\pi$
$811$	1246.04i	1.53642i	−0.640197	+	0.768211i	$0.721147\pi$
$812$	0	0
$813$	−25.9295	−0.0318936
$814$	0	0
$815$	592.667i	0.727199i
$816$	0	0
$817$	494.214	0.604913
$818$	0	0
$819$	− 455.719i	− 0.556433i
$820$	0	0
$821$	1458.68	1.77671	0.888353	−	0.459162i	$-0.151850\pi$
$821$	1458.68	1.77671	0.888353	+	0.459162i	$0.151850\pi$
$822$	0	0
$823$	− 464.047i	− 0.563848i	−0.959437	−	0.281924i	$-0.909027\pi$
$823$	− 464.047i	− 0.563848i	0.959437	−	0.281924i	$-0.0909725\pi$
$824$	0	0
$825$	3.70695	0.00449328
$826$	0	0
$827$	− 1077.41i	− 1.30279i	−0.758739	−	0.651394i	$-0.774184\pi$
$827$	− 1077.41i	− 1.30279i	0.758739	−	0.651394i	$-0.225816\pi$
$828$	0	0
$829$	35.9354	0.0433479	0.0216740	−	0.999765i	$-0.493100\pi$
$829$	35.9354	0.0433479	0.0216740	+	0.999765i	$0.493100\pi$
$830$	0	0
$831$	− 35.7339i	− 0.0430011i
$832$	0	0
$833$	100.728	0.120922
$834$	0	0
$835$	418.134i	0.500760i
$836$	0	0
$837$	78.6932	0.0940181
$838$	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
$843$	− 14.4183i	− 0.0171036i
$844$	0	0
$845$	686.342	0.812239
$846$	0	0
$847$	297.443i	0.351173i
$848$	0	0
$849$	31.9337	0.0376133
$850$	0	0
$851$	− 663.164i	− 0.779276i
$852$	0	0
$853$	−402.566	−0.471942	−0.235971	−	0.971760i	$-0.575827\pi$
$853$	−402.566	−0.471942	−0.235971	+	0.971760i	$0.575827\pi$
$854$	0	0
$855$	252.287i	0.295072i
$856$	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
$859$	330.117i	0.384303i	0.981365	+	0.192152i	$0.0615466\pi$
$859$	330.117i	0.384303i	−0.981365	+	0.192152i	$0.938453\pi$
$860$	0	0
$861$	11.8267	0.0137360
$862$	0	0
$863$	1116.67i	1.29394i	0.762516	+	0.646969i	$0.223964\pi$
$863$	1116.67i	1.29394i	−0.762516	+	0.646969i	$0.776036\pi$
$864$	0	0
$865$	−375.445	−0.434041
$866$	0	0
$867$	− 7.98348i	− 0.00920817i
$868$	0	0
$869$	−124.810	−0.143625
$870$	0	0
$871$	231.033i	0.265251i
$872$	0	0
$873$	1457.24	1.66923
$874$	0	0
$875$	− 348.326i	− 0.398087i
$876$	0	0
$877$	0.747441	0.000852270 0	0.000426135	−	1.00000i	$-0.499864\pi$
$877$	0.747441	0.000852270 0	0.000426135		1.00000i	$0.499864\pi$
$878$	0	0
$879$	− 25.3219i	− 0.0288076i
$880$	0	0
$881$	−247.826	−0.281301	−0.140650	−	0.990059i	$-0.544919\pi$
$881$	−247.826	−0.281301	−0.140650	+	0.990059i	$0.544919\pi$
$882$	0	0
$883$	− 1613.74i	− 1.82757i	−0.406200	−	0.913784i	$-0.633146\pi$
$883$	− 1613.74i	− 1.82757i	0.406200	−	0.913784i	$-0.366854\pi$
$884$	0	0
$885$	−38.6807	−0.0437070
$886$	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
$891$	236.471i	0.265400i
$892$	0	0
$893$	−374.744	−0.419646
$894$	0	0
$895$	− 559.458i	− 0.625092i
$896$	0	0
$897$	31.2863	0.0348788
$898$	0	0
$899$	1219.49i	1.35650i
$900$	0	0
$901$	139.498	0.154825
$902$	0	0
$903$	15.7340i	0.0174241i
$904$	0	0
$905$	24.7676	0.0273675
$906$	0	0
$907$	518.009i	0.571123i	0.958360	+	0.285562i	$0.0921802\pi$
$907$	518.009i	0.571123i	−0.958360	+	0.285562i	$0.907820\pi$
$908$	0	0
$909$	−960.430	−1.05658
$910$	0	0
$911$	1065.61i	1.16972i	0.811135	+	0.584860i	$0.198850\pi$
$911$	1065.61i	1.16972i	−0.811135	+	0.584860i	$0.801150\pi$
$912$	0	0
$913$	321.186	0.351792
$914$	0	0
$915$	− 2.52583i	− 0.00276047i
$916$	0	0
$917$	−163.679	−0.178494
$918$	0	0
$919$	− 1183.66i	− 1.28799i	−0.765030	−	0.643994i	$-0.777276\pi$
$919$	− 1183.66i	− 1.28799i	0.765030	−	0.643994i	$-0.222724\pi$
$920$	0	0
$921$	−28.3016	−0.0307292
$922$	0	0
$923$	2475.03i	2.68151i
$924$	0	0
$925$	514.018	0.555695
$926$	0	0
$927$	1138.43i	1.22809i
$928$	0	0
$929$	−379.019	−0.407986	−0.203993	−	0.978972i	$-0.565392\pi$
$929$	−379.019	−0.407986	−0.203993	+	0.978972i	$0.565392\pi$
$930$	0	0
$931$	− 56.6821i	− 0.0608830i
$932$	0	0
$933$	7.29878	0.00782292
$934$	0	0
$935$	− 146.044i	− 0.156197i
$936$	0	0
$937$	−1316.09	−1.40458	−0.702289	−	0.711892i	$-0.747839\pi$
$937$	−1316.09	−1.40458	−0.702289	+	0.711892i	$0.747839\pi$
$938$	0	0
$939$	− 27.7214i	− 0.0295223i
$940$	0	0
$941$	−497.031	−0.528194	−0.264097	−	0.964496i	$-0.585074\pi$
$941$	−497.031	−0.528194	−0.264097	+	0.964496i	$0.585074\pi$
$942$	0	0
$943$	− 768.883i	− 0.815359i
$944$	0	0
$945$	−16.0723	−0.0170077
$946$	0	0
$947$	808.487i	0.853735i	0.904314	+	0.426867i	$0.140383\pi$
$947$	808.487i	0.853735i	−0.904314	+	0.426867i	$0.859617\pi$
$948$	0	0
$949$	−350.322	−0.369148
$950$	0	0
$951$	1.19488i	0.00125644i
$952$	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
$955$	254.013i	0.265982i
$956$	0	0
$957$	7.75173	0.00810004
$958$	0	0
$959$	− 280.299i	− 0.292283i
$960$	0	0
$961$	−1054.32	−1.09710
$962$	0	0
$963$	− 782.346i	− 0.812405i
$964$	0	0
$965$	−294.809	−0.305502
$966$	0	0
$967$	− 140.279i	− 0.145066i	−0.997366	−	0.0725330i	$-0.976892\pi$
$967$	− 140.279i	− 0.145066i	0.997366	−	0.0725330i	$-0.0231083\pi$
$968$	0	0
$969$	11.3533	0.0117165
$970$	0	0
$971$	− 1438.67i	− 1.48164i	−0.671704	−	0.740820i	$-0.734437\pi$
$971$	− 1438.67i	− 1.48164i	0.671704	−	0.740820i	$-0.265563\pi$
$972$	0	0
$973$	−490.479	−0.504090
$974$	0	0
$975$	24.2500i	0.0248718i
$976$	0	0
$977$	902.190	0.923428	0.461714	−	0.887029i	$-0.347235\pi$
$977$	902.190	0.923428	0.461714	+	0.887029i	$0.347235\pi$
$978$	0	0
$979$	236.976i	0.242059i
$980$	0	0
$981$	−1707.49	−1.74056
$982$	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
$987$	− 11.9305i	− 0.0120876i
$988$	0	0
$989$	1022.91	1.03428
$990$	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
$993$	18.9481	0.0190816
$994$	0	0
$995$	− 495.847i	− 0.498339i
$996$	0	0
$997$	−608.625	−0.610457	−0.305228	−	0.952279i	$-0.598733\pi$
$997$	−608.625	−0.610457	−0.305228	+	0.952279i	$0.598733\pi$
$998$	0	0
$999$	− 69.3613i	− 0.0694308i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.d.j.1023.10		16
4.3	odd	2	inner	1792.3.d.j.1023.8		16
8.3	odd	2	inner	1792.3.d.j.1023.9		16
8.5	even	2	inner	1792.3.d.j.1023.7		16
16.3	odd	4		56.3.g.b.43.8	yes	8
16.5	even	4		56.3.g.b.43.7	✓	8
16.11	odd	4		224.3.g.b.15.4		8
16.13	even	4		224.3.g.b.15.3		8
48.5	odd	4		504.3.g.b.379.2		8
48.11	even	4		2016.3.g.b.1135.4		8
48.29	odd	4		2016.3.g.b.1135.5		8
48.35	even	4		504.3.g.b.379.1		8
112.3	even	12		392.3.k.n.275.2		16
112.5	odd	12		392.3.k.n.67.2		16
112.13	odd	4		1568.3.g.m.687.6		8
112.19	even	12		392.3.k.n.67.4		16
112.27	even	4		1568.3.g.m.687.5		8
112.37	even	12		392.3.k.o.67.2		16
112.51	odd	12		392.3.k.o.67.4		16
112.53	even	12		392.3.k.o.275.4		16
112.67	odd	12		392.3.k.o.275.2		16
112.69	odd	4		392.3.g.m.99.7		8
112.83	even	4		392.3.g.m.99.8		8
112.101	odd	12		392.3.k.n.275.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.b.43.7	✓	8	16.5	even	4
56.3.g.b.43.8	yes	8	16.3	odd	4
224.3.g.b.15.3		8	16.13	even	4
224.3.g.b.15.4		8	16.11	odd	4
392.3.g.m.99.7		8	112.69	odd	4
392.3.g.m.99.8		8	112.83	even	4
392.3.k.n.67.2		16	112.5	odd	12
392.3.k.n.67.4		16	112.19	even	12
392.3.k.n.275.2		16	112.3	even	12
392.3.k.n.275.4		16	112.101	odd	12
392.3.k.o.67.2		16	112.37	even	12
392.3.k.o.67.4		16	112.51	odd	12
392.3.k.o.275.2		16	112.67	odd	12
392.3.k.o.275.4		16	112.53	even	12
504.3.g.b.379.1		8	48.35	even	4
504.3.g.b.379.2		8	48.5	odd	4
1568.3.g.m.687.5		8	112.27	even	4
1568.3.g.m.687.6		8	112.13	odd	4
1792.3.d.j.1023.7		16	8.5	even	2	inner
1792.3.d.j.1023.8		16	4.3	odd	2	inner
1792.3.d.j.1023.9		16	8.3	odd	2	inner
1792.3.d.j.1023.10		16	1.1	even	1	trivial
2016.3.g.b.1135.4		8	48.11	even	4
2016.3.g.b.1135.5		8	48.29	odd	4

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1792.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.0974366i q^{3} +3.46547 q^{5} +2.64575i q^{7} +8.99051 q^{9} +O(q^{10})\)	\(q+0.0974366i q^{3} +3.46547 q^{5} +2.64575i q^{7} +8.99051 q^{9} +2.92866i q^{11} -19.1586 q^{13} +0.337664i q^{15} -14.3897 q^{17} +8.09744i q^{19} -0.257793 q^{21} +16.7598i q^{23} -12.9905 q^{25} +1.75293i q^{27} -27.1649 q^{29} -44.8923i q^{31} -0.285359 q^{33} +9.16878i q^{35} -39.5687 q^{37} -1.86675i q^{39} -45.8766 q^{41} -61.0334i q^{43} +31.1563 q^{45} +46.2793i q^{47} -7.00000 q^{49} -1.40209i q^{51} -9.69424 q^{53} +10.1492i q^{55} -0.788986 q^{57} +114.554i q^{59} -7.48032 q^{61} +23.7866i q^{63} -66.3935 q^{65} -12.0590i q^{67} -1.63302 q^{69} -129.187i q^{71} +18.2854 q^{73} -1.26575i q^{75} -7.74851 q^{77} +42.6168i q^{79} +80.7438 q^{81} -109.670i q^{83} -49.8673 q^{85} -2.64685i q^{87} +80.9162 q^{89} -50.6889i q^{91} +4.37415 q^{93} +28.0614i q^{95} +162.086 q^{97} +26.3301i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 96 q^{9}+O(q^{10}) \) 16 * q - 96 * q^9	\( 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) \) 16 * q - 96 * q^9 - 160 * q^17 + 32 * q^25 + 64 * q^33 - 256 * q^41 - 112 * q^49 - 112 * q^57 - 144 * q^65 + 224 * q^73 + 96 * q^81 + 1024 * q^89 + 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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